Proceedings of 2019 Chinese Intelligent Systems Conference: Volume I [1st ed. 2020] 978-981-32-9681-7, 978-981-32-9682-4

Table of contents :
Front Matter ....Pages i-xi
Non-invasive Fetal Electrocardiography Denoising Using Deep Convolutional Encoder-Decoder Networks (Wei Zhong, Xuemei Guo, Guoli Wang)....Pages 1-10
The Construction of Multi-wavelets with Matrix Dilation via the Lifting Scheme with Several Steps (Xiaohui Zhou, Guiding Gu)....Pages 11-23
Research on Regression Model Based on Uncertain System (Sizhe Wang, Zhigang Wang)....Pages 24-29
Online Optimization of Heating Furnace Temperature Set Value Based on Fast Predictive Genetic Algorithm (Zhengguang Xu, Hao Tian, Pengfei Xu, Shuai Liu)....Pages 30-38
Denoising of Carbon Steel Corrosion Monitoring Signal Based on NLM-VMD with MFDFA Technique (Zhuolin Li, Dongmei Fu, Ying Li, Zibo Pei, Qiong Yao, Jinbin Zhao)....Pages 39-47
High Angle of Attack Sliding Mode Control for Aircraft with Thrust Vector Based on ESO (Junjie Liu, Zengqiang Chen, Mingwei Sun, Qinglin Sun)....Pages 48-57
Research and Application of Behavior Detection Based on Smart Phone Sensor (Yongsheng Xie, Linbing Wei)....Pages 58-63
Adaptive NN Control of Discrete-Time Nonlinear Strict-Feedback System Using Disturbance Observer (Bei Wu, Mou Chen, Ronggang Zhu)....Pages 64-72
Fault Detection of High Throughput Screening System (Sijing Zhang, Shuang Fang, Na Lu, Yingying Wu, Yaoyao Li, Xin Zhao)....Pages 73-84
Grey Wolf Optimizer Based Active Disturbance Rejection Controller for Reusable Launch Vehicle (Xia Wu, Shibin Luo, Yuxin Liao, Xiaodong Li, Jun Li)....Pages 85-100
Consensus of Discrete-Time Heterogenous Multi-agent Systems with Nonconvex Velocity and Input Constraints (Lipo Mo, Ju Cheng, Xianbing Cao)....Pages 101-110
The Improved Intelligent Optimal Algorithm Based on the Artificial Bee Colony Algorithm and the Differential Evolution Algorithm (Jingyi Li, Ju Cheng, Lipo Mo)....Pages 111-121
Semantic Analysis Using Convolutional Neural Networks for Assisted Driving (Fen Zhao, Penghua Li, Yinguo Li, Yuanyuan Li)....Pages 122-130
Target Tracking Based on SAGBA Optimized Particle Filter (Siyao Lv, Yihong Zhang, Wuneng Zhou, Longlong Li)....Pages 131-139
\(L_2\) Leader-Following Consensus of Second-Order Nonlinear Multi-agent Systems with Disturbances Under Directed Topology via Event-Triggered Control Scheme (Yuanhong Ren, Wuneng Zhou, Zhiwei Li, Yuqing Sun)....Pages 140-149
Event-Triggered Nonsingular Fast Terminal Sliding Mode Control for Nonlinear Dynamical Systems (Jianlin Feng, Fei Hao)....Pages 150-158
Fault Detection for T-S Fuzzy Networked Control Systems with Signal to Noise Ratio Constrained Channels (Fumin Guo, Xuemei Ren)....Pages 159-166
Modified EM Algorithms for Parameter Estimation in Finite Mixture Models (Weigang Wang, Shengjie Yang, Jinlei Cao, Ruijiao He, Gengxin Xu)....Pages 167-175
The Ridge Iterative Regression and the Data-Augmentation Lasso (Gengxin Xu, Weigang Wang)....Pages 176-184
Adaptive Fault-Tolerant Cooperative Output Regulation for Linear Multi-Agent Systems (Jie Zhang, Dawei Ding, Cuijuan An)....Pages 185-194
Graphic Approach for the Disturbance Decoupling of Boolean Networks (Yifeng Li, Jiandong Zhu)....Pages 195-207
Neural Network Augmented Adaptive Backstepping Control of Spacecraft Proximity Operations (Liang Sun, Bing Zhu)....Pages 208-217
Finite Time Anti-disturbance Control Based on Disturbance Observer for Systems with Disturbance (Hanxu Zhao, Xinjiang Wei)....Pages 218-226
Anti-disturbance Control Based on Disturbance Observer for Dynamic Positioning System of Ships (Yongli Wei, Xinjiang Wei)....Pages 227-235
Research on Video Target Tracking Algorithm Based on Particle Filter and CNN (Longlong Li, Yihong Zhang, Wuneng Zhou, Siyao Lv)....Pages 236-244
Fault Diagnosis and Fault Tolerant Control of High Power Variable Frequency Speed Control System Based on Data Driven (Yuedou Pan, Yongliang Li)....Pages 245-253
Leader-Following Consensus of Discrete-Time Multi-agent Systems with a Smart Leader (Shuang Liang, Fuyong Wang, Zhongxin Liu, Zengqiang Chen)....Pages 254-265
A Group Control Method of Central Air-Conditioning System Based on Reinforcement Learning with the Aim of Energy Saving (Yuedou Pan, Jiaxing Zhao)....Pages 266-275
Comparative Studies on Activity Recognition of Elderly People Living Alone (Zimin Xu, Guoli Wang, Xuemei Guo)....Pages 276-291
The Improved Regularized Extreme Learning Machine for the Estimation of Gas Flow Temperature of Blast Furnace (Xiaoyang Wu, Sen Zhang, Xiaoli Su, Yixin Yin)....Pages 292-300
Research on Soft PLC Programming System Based on IEC61131-3 Standard (Kun Zhao, Yonghui Zhang)....Pages 301-309
Periodic Dynamic Event-Triggered Bipartite Consensus for Multi-agent Systems Associated with Signed Graphs (Junjian Li, Xia Chen, Yanbing Tian)....Pages 310-321
Random Multi-scale Gaussian Kernels Based Relevance Vector Machine (Yinhe Gu, Xuemei Dong, Jian Shi, Xudong Kong)....Pages 322-329
Research on Image Segmentation Based on CBCT Roots (Zhengguang Xu, Shijie Che, Zhaohui Zhou)....Pages 330-339
Efficient and Fast Expression Recognition with Deep Learning CNN-ELM (Yiping Zou, Xuemei Ren)....Pages 340-348
On Collective Behavior of a Group of Autonomous Agents on a Sphere (Yi Guo, Jinxing Zhang, Jiandong Zhu)....Pages 349-356
A Novel Single-Input Rule Module Connected Fuzzy Logic System and Its Applications to Medical Diagnosis (Qiye Zhang, Chunwei Wen)....Pages 357-366
Adaptive Finite-Time Neural Consensus Tracking for Second-Order Nonlinear Multiagent Systems with Unknown Control Directions (Guoqing Liu, Lin Zhao, Jinpeng Yu)....Pages 367-374
Adaptive Finite-Time Bipartite Output Consensus Tracking of Second-Order Nonlinear Multi-agent Systems with Input Saturation (Xiao Chen, Lin Zhao, Ruiping Xu)....Pages 375-383
Neural Network Based Adaptive Backstepping Control of Uncertain Flexible Joint Robot Systems (Dongdong Wang, Lin Zhao, Jinpeng Yu)....Pages 384-392
RSSI Localization Algorithm Based on Electromagnetic Spectrum Detection (Mingming Ma, Yonghui Zhang, Zhenjia Chen, Chao He)....Pages 393-400
Distributed \(H_{\infty }\) Consensus Control with Nonconvex Input and Velocity Constraints (Jiahui Shi, Hongqiu Zhu)....Pages 401-412
Correction-Based Diffusion LMS Algorithms for Secure Distributed Estimation Under Attacks (Huining Chang, Wenling Li, Junping Du)....Pages 413-421
Personal Credit Scoring via Logistic Regression with Elastic Net Penalty (Juntao Li, Mingming Chang, Pengjie Tian, Liuyuan Chen, Xiaoxia Mu)....Pages 422-428
Application of Wireless Sensor Networks in Offshore Electromagnetic Spectrum Monitoring Scenarios (Chao He, Yonghui Zhang, Zhenjia Chen, Xia Guo)....Pages 429-435
Modified ELM-RBF with Finite Perception for Multi-label Classification (Peng Chen, Qing Li, Zitong Zhou, Ziyi Lu, Hao Zhou, Jiarui Cui)....Pages 436-445
An Iterative Learning Scheme-Based Fault Estimator Design for Nonlinear Systems with Quantised Measurements (Xiaoyu Liu, Shanbi Wei, Yi Chai)....Pages 446-456
Smooth Trajectory Generation for Linear Paths with Optimal Polygonal Approximation (Kai Zhao, Shurong Li, Zhongjian Kang)....Pages 457-465
Single Image Defogging Method Combined with Multi-exposure Fusion Approach Based on Parameter Dynamic Selection (Yuanyuan Li, Hexu Hu, Hongyan Wei, Xinhua Huang, Penghua Li, Zhiqin Zhu)....Pages 466-470
Finite-Time Consensus for Second-Order Leader-Following Multi-agent Systems with Disturbances Based on the Event-Triggered Scheme (Yan Cui, Xiaoshan Wang)....Pages 471-486
An Entropy-Based Inertia Weight Krill Herd Algorithm (Chen Zhao, Zhongxin Liu, Zengqiang Chen, Yao Ning)....Pages 487-498
State Estimation for One-Sided Lipschitz System with Markovian Jump Parameters (Zhenkun Zhu, Jun Huang, Ming Yang)....Pages 499-507
Model Reference Adaptive Control with Unknown Gain Sign (Heqing Liu, Tianping Zhang, Ziwen Wu, Yu Hua)....Pages 508-514
Adaptive Neural Network Control of Uncertain Systems with Full State Constraints and Unknown Gain Sign (Heqing Liu, Tianping Zhang, Meizhen Xia, Ziwen Wu)....Pages 515-525
Smooth Globally Convergent Velocity Observer Design for Uncertain Robotic Manipulators (Yuan Liu, Shangzheng Liu)....Pages 526-534
Optimal Sensor Placement for TDOA-Based Source Localization with Distance-Dependent Noises (Yueqian Liang, Changqing Liu)....Pages 535-544
Qualitative Path Reasoning with Incomplete Information in VAR-Space (Xiaodong Wang, Yan Zhang, Nan Xiao, Ming Li)....Pages 545-552
A Novel Competitive Particle Swarm Optimization Algorithm Based on Levy Flight (Yao Ning, Zhongxin Liu, Zengqiang Chen, Chen Zhao)....Pages 553-565
Research on Establishment Method of Steel Composition Model Based on High Dimension and Small Sample Data (Xin Wei, Dongmei Fu, Mindong Chen, Qiong Yao)....Pages 566-573
A Note on Actuator Fault Detection for One-Sided Lipschitz Systems (Ming Yang, Jun Huang, Fei Sun)....Pages 574-581
Aperiodic Control Strategy for Multi-agent Systems with Time-Varying Delays (Hongguang Zhang)....Pages 582-590
Robust Convergence of High-Order Adaptive Iterative Learning Control Against Iteration-Varying Uncertainties (Zirong Guo, Deyuan Meng, Jingyao Zhang)....Pages 591-598
Distributed Robust Control of Signed Networks Subject to External Disturbances (Mingjun Du, Baoli Ma, Deyuan Meng, Hua Yang, Hong Jiang)....Pages 599-608
Obstacle Avoidance Based on 2D-Lidar in Unknown Environment (Kai Yan, Baoli Ma)....Pages 609-618
Fault Tolerant Consensus Control in a Linear Leader-Follower Multi-agent System (Xingxia Wang, Zhongxin Liu, Zengqiang Chen, Fuyong Wang)....Pages 619-628
A Two Phase Method for Skyline Computation (Haipeng Du, Lizhen Shao, Yang You, Zhuolin Li, Dongmei Fu)....Pages 629-637
Synchronization Analysis of Delayed Neural Networks with Stochastic Missing Data (Nan Xiao, Guilai Zhang, Yuan Ma)....Pages 638-648
A Deep Learning Based Method for Low Dose Lung CT Noise Reduction (Yinjin Ma, Peng Feng, Peng He, Zourong Long, Biao Wei)....Pages 649-657
Multi-agent Deep Reinforcement Learning for Pursuit-Evasion Game Scalability (Lin Xu, Bin Hu, Zhihong Guan, Xinming Cheng, Tao Li, Jiangwen Xiao)....Pages 658-669
Event-Triggered Adaptive Path-Following Control for Micro Helicopter (Mingrun Bai, Ming Zhu, Tian Chen, Zewei Zheng)....Pages 670-680
Robust SLAM Algorithm in Dynamic Environment Using Optical Flow (Yiying Ma, Yingmin Jia)....Pages 681-689
Velocity Tracking Control Based on Throttle-Pedal-Moving Data Mapping for the Autonomous Vehicle (Mingxing Li, Yingmin Jia)....Pages 690-698
Consensus Tracking Control for Switched Multiple Non-holonomic Mobile Robots (Lixia Liu, Xiaohua Wang, Lan Xiang, Zhonghua Miao, Jin Zhou)....Pages 699-707
Consensus for a Class of Nonlinear Multi-Agent Systems with Switching Graphs and Arbitrarily Bounded Communication Delays (Yali Liao, Peng Xiang, Mengmeng Duan, Hongqiu Zhu)....Pages 708-715
Cooperative Control of High-Speed Train Systems With Velocity Constraints and Parameter Uncertainties (Yi Huang, Shuai Su, Hialiang Hou, Yonggang Li)....Pages 716-723
Safe-circumnavigation of Multiple Non-stationary Targets with Bearing-only Measurements (Shida Cao, Rui Li, Li Xiao, Yingjing Shi)....Pages 724-734
Dissipative Formation Control for Fuzzy Multi-Agent Systems Under Switching Topologies (Jiafeng Yu, Wen Xing, Jian Wang, Qinsheng Li)....Pages 735-741
One-Shot Chinese Character Recognition Based on Deep Siamese Networks (Huichao Li, Xuemei Ren, Yongfeng Lv)....Pages 742-750
Fuzzy-Model-Based Consensus for Multi-Agent Systems Under Directed Topology (Jiafeng Yu, Jian Wang, Wen Xing, Chunsong Han, Qinsheng Li)....Pages 751-757
Fuzzy Control and Non-contact Free Loop for an Intermittent Web Transport System (Yimin Zhou, Yi Zhang)....Pages 758-767
Back Matter ....Pages 769-771

Citation preview

Lecture Notes in Electrical Engineering 592

Yingmin Jia Junping Du Weicun Zhang Editors

Proceedings of 2019 Chinese Intelligent Systems Conference Volume I

Lecture Notes in Electrical Engineering Volume 592

Series Editors Leopoldo Angrisani, Department of Electrical and Information Technologies Engineering, University of Napoli Federico II, Naples, Italy Marco Arteaga, Departament de Control y Robótica, Universidad Nacional Autónoma de México, Coyoacán, Mexico Bijaya Ketan Panigrahi, Electrical Engineering, Indian Institute of Technology Delhi, New Delhi, Delhi, India Samarjit Chakraborty, Fakultät für Elektrotechnik und Informationstechnik, TU München, Munich, Germany Jiming Chen, Zhejiang University, Hangzhou, Zhejiang, China Shanben Chen, Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai, China Tan Kay Chen, Department of Electrical and Computer Engineering, National University of Singapore, Singapore, Singapore Rüdiger Dillmann, Humanoids and Intelligent Systems Lab, Karlsruhe Institute for Technology, Karlsruhe, Baden-Württemberg, Germany Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, China Gianluigi Ferrari, Università di Parma, Parma, Italy Manuel Ferre, Centre for Automation and Robotics CAR (UPM-CSIC), Universidad Politécnica de Madrid, Madrid, Spain Sandra Hirche, Department of Electrical Engineering and Information Science, Technische Universität München, Munich, Germany Faryar Jabbari, Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, USA Limin Jia, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Alaa Khamis, German University in Egypt El Tagamoa El Khames, New Cairo City, Egypt Torsten Kroeger, Stanford University, Stanford, CA, USA Qilian Liang, Department of Electrical Engineering, University of Texas at Arlington, Arlington, TX, USA Ferran Martin, Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain Tan Cher Ming, College of Engineering, Nanyang Technological University, Singapore, Singapore Wolfgang Minker, Institute of Information Technology, University of Ulm, Ulm, Germany Pradeep Misra, Department of Electrical Engineering, Wright State University, Dayton, OH, USA Sebastian Möller, Quality and Usability Lab, TU Berlin, Berlin, Germany Subhas Mukhopadhyay, School of Engineering & Advanced Technology, Massey University, Palmerston North, Manawatu-Wanganui, New Zealand Cun-Zheng Ning, Electrical Engineering, Arizona State University, Tempe, AZ, USA Toyoaki Nishida, Graduate School of Informatics, Kyoto University, Kyoto, Japan Federica Pascucci, Dipartimento di Ingegneria, Università degli Studi “Roma Tre”, Rome, Italy Yong Qin, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China Gan Woon Seng, School of Electrical & Electronic Engineering, Nanyang Technological University, Singapore, Singapore Joachim Speidel, Institute of Telecommunications, Universität Stuttgart, Stuttgart, Baden-Württemberg, Germany Germano Veiga, Campus da FEUP, INESC Porto, Porto, Portugal Haitao Wu, Academy of Opto-electronics, Chinese Academy of Sciences, Beijing, China Junjie James Zhang, Charlotte, NC, USA

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Yingmin Jia Junping Du Weicun Zhang •

•

Editors

Proceedings of 2019 Chinese Intelligent Systems Conference Volume I

123

Editors Yingmin Jia Beihang University Beijing, China

Junping Du Beijing University of Posts and Telecommunications Beijing, China

Weicun Zhang University of Science and Technology Beijing Beijing, China

ISSN 1876-1100 ISSN 1876-1119 (electronic) Lecture Notes in Electrical Engineering ISBN 978-981-32-9681-7 ISBN 978-981-32-9682-4 (eBook) https://doi.org/10.1007/978-981-32-9682-4 © Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Contents

Non-invasive Fetal Electrocardiography Denoising Using Deep Convolutional Encoder-Decoder Networks . . . . . . . . . . . . . . . . . . . . . . . Wei Zhong, Xuemei Guo, and Guoli Wang

1

The Construction of Multi-wavelets with Matrix Dilation via the Lifting Scheme with Several Steps . . . . . . . . . . . . . . . . . . . . . . . Xiaohui Zhou and Guiding Gu

11

Research on Regression Model Based on Uncertain System . . . . . . . . . . Sizhe Wang and Zhigang Wang

24

Online Optimization of Heating Furnace Temperature Set Value Based on Fast Predictive Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . Zhengguang Xu, Hao Tian, Pengfei Xu, and Shuai Liu

30

Denoising of Carbon Steel Corrosion Monitoring Signal Based on NLM-VMD with MFDFA Technique . . . . . . . . . . . . . . . . . . . . . . . . Zhuolin Li, Dongmei Fu, Ying Li, Zibo Pei, Qiong Yao, and Jinbin Zhao

39

High Angle of Attack Sliding Mode Control for Aircraft with Thrust Vector Based on ESO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Junjie Liu, Zengqiang Chen, Mingwei Sun, and Qinglin Sun

48

Research and Application of Behavior Detection Based on Smart Phone Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yongsheng Xie and Linbing Wei

58

Adaptive NN Control of Discrete-Time Nonlinear Strict-Feedback System Using Disturbance Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bei Wu, Mou Chen, and Ronggang Zhu

64

Fault Detection of High Throughput Screening System . . . . . . . . . . . . . Sijing Zhang, Shuang Fang, Na Lu, Yingying Wu, Yaoyao Li, and Xin Zhao

73

v

vi

Contents

Grey Wolf Optimizer Based Active Disturbance Rejection Controller for Reusable Launch Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xia Wu, Shibin Luo, Yuxin Liao, Xiaodong Li, and Jun Li

85

Consensus of Discrete-Time Heterogenous Multi-agent Systems with Nonconvex Velocity and Input Constraints . . . . . . . . . . . . . . . . . . 101 Lipo Mo, Ju Cheng, and Xianbing Cao The Improved Intelligent Optimal Algorithm Based on the Artificial Bee Colony Algorithm and the Differential Evolution Algorithm . . . . . . 111 Jingyi Li, Ju Cheng, and Lipo Mo Semantic Analysis Using Convolutional Neural Networks for Assisted Driving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Fen Zhao, Penghua Li, Yinguo Li, and Yuanyuan Li Target Tracking Based on SAGBA Optimized Particle Filter . . . . . . . . 131 Siyao Lv, Yihong Zhang, Wuneng Zhou, and Longlong Li L2 Leader-Following Consensus of Second-Order Nonlinear Multi-agent Systems with Disturbances Under Directed Topology via Event-Triggered Control Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Yuanhong Ren, Wuneng Zhou, Zhiwei Li, and Yuqing Sun Event-Triggered Nonsingular Fast Terminal Sliding Mode Control for Nonlinear Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Jianlin Feng and Fei Hao Fault Detection for T-S Fuzzy Networked Control Systems with Signal to Noise Ratio Constrained Channels . . . . . . . . . . . . . . . . . 159 Fumin Guo and Xuemei Ren Modified EM Algorithms for Parameter Estimation in Finite Mixture Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Weigang Wang, Shengjie Yang, Jinlei Cao, Ruijiao He, and Gengxin Xu The Ridge Iterative Regression and the Data-Augmentation Lasso . . . . 176 Gengxin Xu and Weigang Wang Adaptive Fault-Tolerant Cooperative Output Regulation for Linear Multi-Agent Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Jie Zhang, Dawei Ding, and Cuijuan An Graphic Approach for the Disturbance Decoupling of Boolean Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Yifeng Li and Jiandong Zhu Neural Network Augmented Adaptive Backstepping Control of Spacecraft Proximity Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 Liang Sun and Bing Zhu

Contents

vii

Finite Time Anti-disturbance Control Based on Disturbance Observer for Systems with Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . 218 Hanxu Zhao and Xinjiang Wei Anti-disturbance Control Based on Disturbance Observer for Dynamic Positioning System of Ships . . . . . . . . . . . . . . . . . . . . . . . . 227 Yongli Wei and Xinjiang Wei Research on Video Target Tracking Algorithm Based on Particle Filter and CNN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Longlong Li, Yihong Zhang, Wuneng Zhou, and Siyao Lv Fault Diagnosis and Fault Tolerant Control of High Power Variable Frequency Speed Control System Based on Data Driven . . . . . . . . . . . . 245 Yuedou Pan and Yongliang Li Leader-Following Consensus of Discrete-Time Multi-agent Systems with a Smart Leader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 Shuang Liang, Fuyong Wang, Zhongxin Liu, and Zengqiang Chen A Group Control Method of Central Air-Conditioning System Based on Reinforcement Learning with the Aim of Energy Saving . . . . 266 Yuedou Pan and Jiaxing Zhao Comparative Studies on Activity Recognition of Elderly People Living Alone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Zimin Xu, Guoli Wang, and Xuemei Guo The Improved Regularized Extreme Learning Machine for the Estimation of Gas Flow Temperature of Blast Furnace . . . . . . . 292 Xiaoyang Wu, Sen Zhang, Xiaoli Su, and Yixin Yin Research on Soft PLC Programming System Based on IEC61131-3 Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Kun Zhao and Yonghui Zhang Periodic Dynamic Event-Triggered Bipartite Consensus for Multi-agent Systems Associated with Signed Graphs . . . . . . . . . . . . 310 Junjian Li, Xia Chen, and Yanbing Tian Random Multi-scale Gaussian Kernels Based Relevance Vector Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 Yinhe Gu, Xuemei Dong, Jian Shi, and Xudong Kong Research on Image Segmentation Based on CBCT Roots . . . . . . . . . . . 330 Zhengguang Xu, Shijie Che, and Zhaohui Zhou Efficient and Fast Expression Recognition with Deep Learning CNN-ELM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 Yiping Zou and Xuemei Ren

viii

Contents

On Collective Behavior of a Group of Autonomous Agents on a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 Yi Guo, Jinxing Zhang, and Jiandong Zhu A Novel Single-Input Rule Module Connected Fuzzy Logic System and Its Applications to Medical Diagnosis . . . . . . . . . . . . . . . . . . . . . . . 357 Qiye Zhang and Chunwei Wen Adaptive Finite-Time Neural Consensus Tracking for Second-Order Nonlinear Multiagent Systems with Unknown Control Directions . . . . . 367 Guoqing Liu, Lin Zhao, and Jinpeng Yu Adaptive Finite-Time Bipartite Output Consensus Tracking of Second-Order Nonlinear Multi-agent Systems with Input Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 Xiao Chen, Lin Zhao, and Ruiping Xu Neural Network Based Adaptive Backstepping Control of Uncertain Flexible Joint Robot Systems . . . . . . . . . . . . . . . . . . . . . . . 384 Dongdong Wang, Lin Zhao, and Jinpeng Yu RSSI Localization Algorithm Based on Electromagnetic Spectrum Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 Mingming Ma, Yonghui Zhang, Zhenjia Chen, and Chao He Distributed H1 Consensus Control with Nonconvex Input and Velocity Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 Jiahui Shi and Hongqiu Zhu Correction-Based Diffusion LMS Algorithms for Secure Distributed Estimation Under Attacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 Huining Chang, Wenling Li, and Junping Du Personal Credit Scoring via Logistic Regression with Elastic Net Penalty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 Juntao Li, Mingming Chang, Pengjie Tian, Liuyuan Chen, and Xiaoxia Mu Application of Wireless Sensor Networks in Offshore Electromagnetic Spectrum Monitoring Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 Chao He, Yonghui Zhang, Zhenjia Chen, and Xia Guo Modified ELM-RBF with Finite Perception for Multi-label Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 Peng Chen, Qing Li, Zitong Zhou, Ziyi Lu, Hao Zhou, and Jiarui Cui An Iterative Learning Scheme-Based Fault Estimator Design for Nonlinear Systems with Quantised Measurements . . . . . . . . . . . . . . 446 Xiaoyu Liu, Shanbi Wei, and Yi Chai

Contents

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Smooth Trajectory Generation for Linear Paths with Optimal Polygonal Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 Kai Zhao, Shurong Li, and Zhongjian Kang Single Image Defogging Method Combined with Multi-exposure Fusion Approach Based on Parameter Dynamic Selection . . . . . . . . . . . 466 Yuanyuan Li, Hexu Hu, Hongyan Wei, Xinhua Huang, Penghua Li, and Zhiqin Zhu Finite-Time Consensus for Second-Order Leader-Following Multi-agent Systems with Disturbances Based on the Event-Triggered Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 Yan Cui and Xiaoshan Wang An Entropy-Based Inertia Weight Krill Herd Algorithm . . . . . . . . . . . . 487 Chen Zhao, Zhongxin Liu, Zengqiang Chen, and Yao Ning State Estimation for One-Sided Lipschitz System with Markovian Jump Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 Zhenkun Zhu, Jun Huang, and Ming Yang Model Reference Adaptive Control with Unknown Gain Sign . . . . . . . . 508 Heqing Liu, Tianping Zhang, Ziwen Wu, and Yu Hua Adaptive Neural Network Control of Uncertain Systems with Full State Constraints and Unknown Gain Sign . . . . . . . . . . . . . . . . . . . . . . 515 Heqing Liu, Tianping Zhang, Meizhen Xia, and Ziwen Wu Smooth Globally Convergent Velocity Observer Design for Uncertain Robotic Manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 Yuan Liu and Shangzheng Liu Optimal Sensor Placement for TDOA-Based Source Localization with Distance-Dependent Noises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 Yueqian Liang and Changqing Liu Qualitative Path Reasoning with Incomplete Information in VAR-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 Xiaodong Wang, Yan Zhang, Nan Xiao, and Ming Li A Novel Competitive Particle Swarm Optimization Algorithm Based on Levy Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 Yao Ning, Zhongxin Liu, Zengqiang Chen, and Chen Zhao Research on Establishment Method of Steel Composition Model Based on High Dimension and Small Sample Data . . . . . . . . . . . . . . . . 566 Xin Wei, Dongmei Fu, Mindong Chen, and Qiong Yao

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A Note on Actuator Fault Detection for One-Sided Lipschitz Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 Ming Yang, Jun Huang, and Fei Sun Aperiodic Control Strategy for Multi-agent Systems with Time-Varying Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 Hongguang Zhang Robust Convergence of High-Order Adaptive Iterative Learning Control Against Iteration-Varying Uncertainties . . . . . . . . . . . . . . . . . . 591 Zirong Guo, Deyuan Meng, and Jingyao Zhang Distributed Robust Control of Signed Networks Subject to External Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 Mingjun Du, Baoli Ma, Deyuan Meng, Hua Yang, and Hong Jiang Obstacle Avoidance Based on 2D-Lidar in Unknown Environment . . . . 609 Kai Yan and Baoli Ma Fault Tolerant Consensus Control in a Linear Leader-Follower Multi-agent System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 Xingxia Wang, Zhongxin Liu, Zengqiang Chen, and Fuyong Wang A Two Phase Method for Skyline Computation . . . . . . . . . . . . . . . . . . . 629 Haipeng Du, Lizhen Shao, Yang You, Zhuolin Li, and Dongmei Fu Synchronization Analysis of Delayed Neural Networks with Stochastic Missing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 Nan Xiao, Guilai Zhang, and Yuan Ma A Deep Learning Based Method for Low Dose Lung CT Noise Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 Yinjin Ma, Peng Feng, Peng He, Zourong Long, and Biao Wei Multi-agent Deep Reinforcement Learning for Pursuit-Evasion Game Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658 Lin Xu, Bin Hu, Zhihong Guan, Xinming Cheng, Tao Li, and Jiangwen Xiao Event-Triggered Adaptive Path-Following Control for Micro Helicopter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 Mingrun Bai, Ming Zhu, Tian Chen, and Zewei Zheng Robust SLAM Algorithm in Dynamic Environment Using Optical Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 Yiying Ma and Yingmin Jia Velocity Tracking Control Based on Throttle-Pedal-Moving Data Mapping for the Autonomous Vehicle . . . . . . . . . . . . . . . . . . . . . . 690 Mingxing Li and Yingmin Jia

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Consensus Tracking Control for Switched Multiple Non-holonomic Mobile Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 Lixia Liu, Xiaohua Wang, Lan Xiang, Zhonghua Miao, and Jin Zhou Consensus for a Class of Nonlinear Multi-Agent Systems with Switching Graphs and Arbitrarily Bounded Communication Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 Yali Liao, Peng Xiang, Mengmeng Duan, and Hongqiu Zhu Cooperative Control of High-Speed Train Systems With Velocity Constraints and Parameter Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . 716 Yi Huang, Shuai Su, Hailiang Hou, and Yonggang Li Safe-circumnavigation of Multiple Non-stationary Targets with Bearing-only Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 Shida Cao, Rui Li, Li Xiao, and Yingjing Shi Dissipative Formation Control for Fuzzy Multi-Agent Systems Under Switching Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 Jiafeng Yu, Wen Xing, Jian Wang, and Qinsheng Li One-Shot Chinese Character Recognition Based on Deep Siamese Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 Huichao Li, Xuemei Ren, and Yongfeng Lv Fuzzy-Model-Based Consensus for Multi-Agent Systems Under Directed Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751 Jiafeng Yu, Jian Wang, Wen Xing, Chunsong Han, and Qinsheng Li Fuzzy Control and Non-contact Free Loop for an Intermittent Web Transport System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758 Yimin Zhou and Yi Zhang Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769

Non-invasive Fetal Electrocardiography Denoising Using Deep Convolutional Encoder-Decoder Networks Wei Zhong1,2 , Xuemei Guo1,2 , and Guoli Wang1,2(B) 1

School of Data and Computer Science, Sun Yat-sen University, Guangzhou, China [email protected] 2 Key Laboratory of Machine Intelligence and Advanced Computing, Ministry of Education, Guangzhou, China

Abstract. Non-invasive fetal electrocardiography (NI-FECG) plays an important role in fetal monitoring and fetal health assessment. However, NI-FECG is inevitably contaminated by a variety of unwanted noise. Therefore, NI-FECG denoising is a challenging task. There is a need for effective techniques that can preserve most components of the morphology of fetal electrocardiography (FECG) and meanwhile eliminating the noise. In this paper, we propose a deep convolutional encoding-decoding framework for NI-FECG denoising. The network contains multiple convolution and deconvolution layers. End-to-end mappings from corrupted signals to the clean ones are learned in the network. Experimental results on two different datasets show that our network achieves better performance than two other state-of-the-art methods namely band-pass Butterworth filter and wavelet soft-threshold denoising method. The exceeding performance shows a promising method for noise cancellation of NIFECG signals. Keywords: Fetal ECG Fetal monitoring

1

· Denoising · Convolutional networks ·

Introduction

As one of the leading causes of birth defect-related death, congenital heart defect is the subject of considerable research [1,3,4]. The signals of non-invasive fetal electrocardiography (NI-FECG) are recorded from the maternal abdomen. NIFECG signals can not only provide an accurate estimation of the fetal heart rate (FHR), but also have the potential to provide morphological information related to the pathological condition of the fetal heart. In this regard, the NIFECG has been attracting considerable attentions. However, despite the significant advances in the field of adult ECG signals processing, the analysis of NI-FECG signals is still in its infancy. In practice, NI-FECG signals obtained non-invasively from the maternal abdomen are inevitably contaminated by a variety of unwanted noise such as c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 1–10, 2020. https://doi.org/10.1007/978-981-32-9682-4_1

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power-line interference and electrode contact noise. Thus, in essence the abdominal ECG signals can be decomposed into three categories: maternal electrocardiography (MECG), fetal electrocardiography (FECG) and noise [5]. In order to get the clear parameters of the morphology in NI-FECG signals, which plays an important role in recognising the heart activity of fetus and mother, preliminary processing step such as eliminating noise is normally used. However, the NI-FECG denoising is known to be a challenging task. One of the reasons is that NI-FECG signal is a non-stationary biosignal and hard work is needed to denoise. Effective techniques are needed to remove out noise from the NI-FECG signals to get the clean parameter for primary diagnosis. In order to reduce the noise of NI-FECG signals, numerous denoising systems such as digital filters and wavelet transform thresholding methods have been used. In the work of [2], band-pass Butterworth filter (BPB) is used to preprocess NI-FECG signals. In the work of [6], a low-pass (order 459) and a high-pass (order 2234) filter are used to implement a band-pass FIR filter between 2 and 46 Hz. In order to preserve at the most the NI-FECG morphology, zero-phase digital filtering is exploited. The zero-phase digital filtering minimizes start-up and ending transients in the NI-FECG signals. In the work of [7], a denoising method that applies soft-threshold in the wavelet domain has been proposed. It has been proved powerful for noise reduction. Soft-threshold denoising method works well for both one-dimensional and two-dimensional signals. In the work of [17], the performance of wavelet-based thresholding methods with several different wavelet bases is investigated. The result indicates that the combination of coif 5 basis and rigrsure method works well for reducing the noise of ECG signals. In the work of [13], a notch filter [8] is used to eliminate the power-line interference. And then wavelet soft-threshold denoising method (WST) is used to remove the high frequency noise in the NIFECG signals. The wavelet transform is an efficient technique for processing such a non-stationary biosignal. Despite the progress that digital filters and wavelet transform thresholding methods achieved in the field of adult ECG signal processing, some questions remain to be answered. For visualisation purposes, an example of abdominal mixtures and corresponding clean signal is showed in Fig. 1. We can note that the morphology of NI-FECG is covered by the noise. There is a need for effective techniques that can preserve most components of the morphology of NI-FECG and meanwhile eliminating the noise. Recently, with large annotated datasets, deep neural networks (DNNs) based machine learning models have consistently been able to approach and often exceed human performance [9]. It has already proven to be effective in many fields such as healthcare applications [14]. Inspired by the recent superior performance of DNNs, we propose a deep learning framework for NI-FECG denoising. In this contribution, an end-to-end denoising convolutional encoder-decoder networks (DnCED-Net) for NI-FECG noise reduction is introduced for the first time (to the best of our knowledge). Experimental results demonstrate the advantages of our network over two other state-of-the-art ECG denoising methods, namely

NI-FECG Denoising Using DnCED-Net

3

Fig. 1. An example of a 4 s long abdominal mixtures (top) and corresponding clean signal (bottom).

BPB and WST used in this study. It shows the large capacity and fitting ability of deep learning approach for NI-FECG denoising.

2 2.1

Methods Dataset

Data used in this study is collected from two different databases. – The Fetal ECG Synthetic Database (FECGSYNDB) [2]. The FECGSYNDB is a large database, which provides simulated maternal-fetal ECG mixtures. It is a robust resource that enables reproducible research in the field of NIFECG processing. For this database, a total of 10 different pregnancies with 5 noise levels (0, 3, 6, 9 and 12 dB) and 7 physiological events are simulated. The 7 physiological events [2] are described in Table 1. Each event contains five repetitions. The FECGSYNDB is comprised of 1750 five-minute realistic simulations of abdominal mixtures. Each simulation consisted of 34 channels data (32 abdominal and 2 maternal ECG channels) and the reference clean data is available. All the NI-FECG signals are sampled at 250 Hz. – The DaISy database [16]: this database contains eight 10-s real ECG channels (5 abdominal and 3 thoracic). The signals are sampled at frequency 250 Hz, without reference annotations. As in [2], for providing a clearer presentation of the results in this NI-FECG denoising task, a subset of the database containing Case 0, Case 1, Case 2, Case 3 and Case 4 are used for further analysis.

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W. Zhong et al. Table 1. Simulated scenarios of pregnancy’s pathophysiological events. Case

Description

Baseline Abdominal mixture (no noise or events) Case 0

Baseline (no events) + noise

Case 1

Fetal movement + noise

Case 2

MHR (maternal heart rate)/FHR acceleration/decelerations + noise

Case 3

Uterine contraction + noise

Case 4

Ectopic beats (for both fetus and mother) + noise

Case 5

Additional NI-FECG (twin pregnancy) + noise

NI-FECG signals of pregnancies 01-08 are selected to form the training set. NI-FECG signals of pregnancies 09 are selected to form the validation set. The signals of pregnancies 10 and the DaISy database are selected to form the test set. In order to learn the basic features of NI-FECG morphology, 7 channels (1, 6, 11, 16, 21, 26, 31) of abdominal mixtures with 5 noise levels are selected. Only the repetition of number three is used in our study. After collection, the training set, validation set and test set include totalizing 116.7 h, 14.6 h and 14.6 h of data, respectively. Notably, the NI-FECG signals for the training, validation and test sets are mutually exclusive. 2.2

Problem Formulation

The NI-FECG denoising task is a sequence-to-sequence task which takes as input a noisy observation of NI-FECG signals Y = [y1 , y2 , ..., yk ] ∈ Rk×1 . The noisy observation can be represented as: Y = ΦX + V

(1)

where Φ ∈ Rk×k is the sensing matrix. X = [x1 , x2 , ..., xk ] ∈ Rk×1 in this study is the unknown clean source matrix and V ∈ Rk×1 is an unknown noise matrix of NI-FECG signals. ˆ = [ˆ ˆ2 , ..., x ˆk ] ∈ The proposed network outputs a sequence of data X x1 , x ˆ is a predicted clean signal of input Y . And the denoising network Rk×1 . The X aims to learn a mapping function f (Y ) = X to estimate the latent clean signal. ˆ corresponds to a segment of the noisy obserEach output of the network (X) ˆ covers the full sequence of NI-FECG sigvation. Together all the output data (X) nals. In specific, we adopt the averaged mean squared error between the desired clean signal and estimated ones M 1  2 f (Yi ; Θ) − Xi F (Θ) = M i=1

(2)

as the loss function. Here Θ represents the trainable parameters of denoising network and {(Yi , Xi )}M i=1 is a collection of M noisy-clean training sample pairs.

NI-FECG Denoising Using DnCED-Net

2.3

5

Model Architecture

The architecture of proposed network contains a chain of convolutional layers, symmetric deconvolutional layers and a fully connected layer, as in Fig. 2.

Fig. 2. The architecture of the proposed DnCED-Net. Overall, the network contains multiple convolutional layers followed by symmetric deconvolutional layers.

Given the DnCED-Net with depth 2N + 1, there are three types of layers in the deep architecture. (i) Conv + Tanh: for layers 1 ∼ N , 64 filters of length 3 are used in the convolutional layers. Tanh activation function is used for nonlinearity. (ii) Deconv + Tanh: for layers (N +1) ∼ (2N ), 64 feature maps are generated by 64 filters of length 3 in the deconvolutional layers. (iii) Fully connected: for the last layer, a fully connected layer is used to reconstruct the NI-FECG signals. In practice, we prefer using neither pooling nor unpooling in these low-level NI-FECG denoising tasks as usually pooling discards useful signal details that are essential to reconstruct the NI-FECG signals. The network contains three convolutional layers followed by symmetric deconvolutional layers (N = 3). We use Adam as the optimizer and learning rate is set at 10−4 for all the convolutional and deconvolutional layers. To sum up, our DnCED-Net model has two main features: the end-to-end mapping formulation is adopted to learn f (Y ), and activation function is used to incorporate with convolutional and deconvolutional operations. In this study, we are particularly interested in investigating the performance of the proposed deep learning approach for NI-FECG denoising in the presence of such nonstationarities. 2.4

Training

In this section, the validation set is used to train the DnCED-Net. The best model is saved during the optimization process. In order to learn all the possible patterns presented in the NI-FECG signals, the frame size is set to 4 s, with a 2-s overlap to generate the training set. We use 4 s long abdominal mixtures as the input and the corresponding clean signal is used as the ground truth. An example of an input and ground truth in this study is showed in Fig. 1.

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Testing

Unlike the training set, non-overlapping scheme is used to generate the test set. The frame size is also set to 4 s (1, 000 samples). A robust normalization strategy is used to normalize the training, validation and test data before they are fed into the network. We compare our approach with two other state-of-the-art denoising methods on the test set, namely BPB [2] and WST [13]. In our study, experiments with BPB and WST are performed on the MATLAB platform. The band-pass Butterworth filter is cascaded by a fifth-order low-pass filter with cutoff at 100 Hz and fifth-order high-pass filter with cutoff at 3 Hz. Meanwhile zero-phase filtering based on [11] is exploited in the experiment. For the WST, rigrsure is used as the soft thresholding technique and coif 5 basis is used as the wavelet base. In the work of [10], the performance of rigrsure method is better than universal thresholding, minimax and heursure methods.

3 3.1

Results Evaluation Metrics

For the FECGSYNDB, the evaluation metrics typically used in signal denoising are Signal to Noise Ratio (SNR) and Percentage Root mean-square Difference (PRD) [15]. Equations (3) and (4) show the definitions of the 2 metrics   Z2  (3) SN R = 20log10    Z − Zˆ  2

 

2

 K

ˆ

 i=1 Z(i) − Z(i) × 100% P RD =  K 2 i=1 Z (i)

(4)

where Z ∈ RK×1 , Zˆ ∈ RK×1 are the clean signal without noise and the output of denoising methods, respectively. For the DaISy database, since the clean NI-FECG is unavailable, the accurate quantitative measure results on the SNR and PRD can not be obtained. Instead, we employed the signal quality index (SQI) to quantify the quality of the filtered NI-FECG. Concretely, the kSQI, which is the fourth moment (kurtosis) of the signal, is adopted in this study [12]. The formula is shown as: kSQI =

E{X − μ}4 σ4

(5)

where μ corresponds to the mean of X, σ corresponds to the standard deviation of X, E{X − μ} corresponds to the expected value of (X − μ). It is expected that good signal is highly non-Gaussian since it is not very random.

NI-FECG Denoising Using DnCED-Net

3.2

7

Results on FECGSYNDB

In this section, the test set is used to compare the performance of DnCED-Net with other methods. The results of proposed approach and two other stateof-the-art methods for NI-FECG denoising are summarized in Table 2. Table 2 includes the results of SNR and PRD by using different methods. As indicated in Table 2, the DnCED-Net structure achieves the optimal performance. NI-FECG signal is considered as a non-stationary biosignal which is hard to denoise. As a digital filter, BPB can be applied to signal with stationary characteristics in many cases. However, we can note that the BPB achieves worse performance compared with the two other methods in both SNR and PRD in this study, possibly because of its relatively limited capability in processing such a non-stationary signal like NI-FECG. Table 2. Performance comparison among DnCED-Net and two other methods, using the test set by varying the noise levels of input. The best results are highlighted in bold. Noise levels SNR (dB) PRD (%) BPB WST DnCED-Net BPB WST DnCED-Net

3.3

0 dB

0.99

3.09

4.53

0.79

0.49

0.35

3 dB

1.33

3.39

4.89

0.73

0.45

0.32

6 dB

2.79

4.48

5.52

0.52

0.35

0.28

9 dB

3.85

5.05

5.92

0.41

0.31

0.25

12 dB

3.73

5.14

5.93

0.42

0.30

0.25

Total

2.39

4.46

5.45

0.55

0.35

0.28

Results on DaISy

The results of different methods on kSQI are demonstrated in Table 3. Table 3. Denoising performance on kSQI. The best results are highlighted in bold. Methods

Channel 1 Channel 2 Channel 3 Channel 4 Channel 5 Total

BPB

12.57

WST

12.89

DnCED-Net 15.12

16.17

18.30

18.34

18.38

17.03

16.33

18.16

19.11

17.87

17.07

17.97

20.75

24.75

19.59

19.07

Here, we can note that the DnCED-Net outperforms the other two methods in this NI-FECG denoising task.

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Discussions

In this study, we use the validation set to determine the structure of DnCEDNet. We analyze the correlation between denoising performance and structure of convolutional encoding-decoding framework (numbers of layers). Performance of the structures that involve 4, 6, 10 layers are tested. We note that increasing the number of layers from 4 to 6 and 10 does enhance the performance of the convolutional encoding-decoding framework, suggesting that using a deeper structure can lead to better denoising performance. And it is interesting to note that, when the structure goes deeper, the improvement of performance slows down. We further investigate the effectiveness of activation functions, namely sigmoid (Sigm), rectified linear unit (Relu) and hyperbolic tangent (Tanh) on this NI-FECG denoising task. The Sigm, Relu and Tanh activation functions 1 , fr (xi ) = max(0, xi ) and ft (xi ) = are represented by fs (xi ) = 1+exp(−x i) exp(xi )−exp(−xi ) exp(xi )+exp(−xi )

respectively with xi ∈ R. The Tanh activation function outperforms the other two activation functions in this NI-FECG denoising task. The wavelet transform method can decompose a NI-FECG signal in the time-frequency scale plane. In the wavelet transform, the NI-FECG signal is transformed by predefined wavelets namely coif 5 basis in this study. Some coefficients of wavelet transform subsignals of the measured abdominal mixtures are removed or smoothed out by using thresholding technique in wavelet domain. Then the noise content of the abdominal mixtures can be eliminated effectively even under the nonstationary environment. In Table 2, we can note that good denoising performance is achieved by the WST. The evaluation on the proposed DnCED-Net is also shown in Table 2. The DnCED-Net achieves even better denoising performance, which exceeds the WST on SNR by 1.44 dB, 1.50 dB, 1.04 dB, 0.87 dB and 0.79 dB on noise levels of 0, 3, 6, 9 and 12 dB, respectively. The exceeding performance can also be seen on PRD in Table 2. It shows the relatively better capability of DnCED-Net in analysing transient, nonstationary or time-varying signal like NI-FECG. This good property is possibly attributed to the powerful ability of the DnCED-Net model in learning the end-to-end mapping function to estimate the latent clean signal. In the proposed DnCED-Net, the abstraction of NI-FECG signals is captured by the convolutional layers, which act as the feature extractor in the network. Meanwhile, the noise or undesired signals are eliminated step by step in the convolutional operations. After the convolutional operations, primary signal content preserved in convolution is passed to the deconvolution layers. Deconvolution is usually considered as a learnable up-sampling operation which is used to compensate the details. With a relatively better denoising ability, the proposed DnCED-Net achieves better performance with higher SNR and lower PRD. These results show a promising method for noise cancellation of NI-FECG signals.

NI-FECG Denoising Using DnCED-Net

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9

Conclusion

The major interest of this paper is to shed some light to the NI-FECG signals noise reduction using deep learning approach. A deep convolutional encodingdecoding framework is proposed for NI-FECG denoising, where end-to-end learning of mapping function is adopted to separate the latent clean signal from noisy observation. Convolutional layers and deconvolutional layers are integrated to build the networks. The primary signal content is extracted in the convolutional operations. Details of useful signals are recovered after the deconvolutional operations. The presented method is compared with the BPB and WST used in this study. Meanwhile, the best results are obtained by the proposed method. Acknowledgments. This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 61772574, 61375080 and U1811462 and in part by the Key Program of the National Social Science Fund of China with Grant No. 18ZDA308.

References 1. Adkins RM, Krushkal J, Somes G, Fain J, Morrison J, Klauser C, Magann EF (2009) Extensive parent-of-origin genetic effects on fetal growth. BMC Bioinform 10(7):A13. https://doi.org/10.1186/1471-2105-10-S7-A13 2. Andreotti F, Behar J, Zaunseder S, Oster J, Clifford GD (2016) An open-source framework for stress-testing non-invasive foetal ECG extraction algorithms. Physiol Meas 37(5):627 3. Behar J, Andreotti F, Zaunseder S, Oster J, Clifford GD (2016) A practical guide to non-invasive foetal electrocardiogram extraction and analysis. Physiol Meas 37:R1– R35 4. Clifford GD, Azuaje F, McSharry P (2006) Advanced Methods and Tools for ECG Data Analysis. Artech House Inc., Norwood 5. Clifford GD, Silva I, Behar J, Moody GB (2014) Non-invasive fetal ECG analysis. Physiol Meas 35(8):1521 6. Dess´ı A, Pani D, Raffo L (2014) An advanced algorithm for fetal heart rate estimation from non-invasive low electrode density recordings. Physiol Meas 35(8):1621 7. Donoho D (1995) De-noising by soft-thresholding. IEEE Trans Inf Theory 41:613– 627 8. Hamilton PS (1996) A comparison of adaptive and nonadaptive filters for reduction of power line interference in the ECG. IEEE Trans Biomed Eng 43(1):105–109 9. He K, Zhang X, Ren S, Sun J (2015) Delving deep into rectifiers: surpassing humanlevel performance on imagenet classification. In: 2015 IEEE international conference on computer vision (ICCV), pp 1026–1034 10. Joy J, Peter S, John N (2013) Denoising using soft thresholding. Int J Adv Res Electr Electron Instrum Eng 2(3):1027–1032

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W. Zhong et al.

11. Kligfield P, Gettes LS, Bailey JJ, Childers R, Deal BJ, Hancock EW, van Herpen G, Kors JA, Macfarlane P, Mirvis DM, Pahlm O, Rautaharju P, Wagner GS (2007) Recommendations for the standardization and interpretation of the electrocardiogram: part i: the electrocardiogram and its technology a scientific statement from the american heart association electrocardiography and arrhythmias committee, council on clinical cardiology; the american college of cardiology foundation; and the heart rhythm society endorsed by the international society for computerized electrocardiology. J Am Coll Cardiol 49(10):1109–1127 12. Li Q, Mark RG, Clifford GD (2008) Robust heart rate estimation from multiple asynchronous noisy sources using signal quality indices and a Kalman filter. Physiol Meas 29:15–32. https://doi.org/10.1088/0967-3334/29/1/002 13. Liu C, Li P, Di Maria C, Zhao L, Zhang H, Chen Z (2014) A multi-step method with signal quality assessment and fine-tuning procedure to locate maternal and fetal QRS complexes from abdominal ECG recordings. Physiol Meas 35:1665–1683 14. Mao X, Shen C, Yang YB (2016) Image restoration using very deep convolutional encoder-decoder networks with symmetric skip connections. Adv Neural Inf Process Syst 29:2802–2810 15. Mikhled A, Daqrouq K (2008) ECG signal denoising by wavelet transform thresholding. Am J Appl Sci 5:276–281 16. Moor BD, Gersem PD, Schutter BD, Favoreel W (1997) DAISY: a database for identification of systems. J A 30(3):4–5 17. Palanisamy K, Murugappan M, Yaacob S (2012) ECG signal denoising using wavelet thresholding techniques in human stress assessment. Int J Electr Eng Inform 4:306

The Construction of Multi-wavelets with Matrix Dilation via the Lifting Scheme with Several Steps Xiaohui Zhou1,2(B) and Guiding Gu1 1

School of Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China [email protected] 2 Zhejiang University of Finance and Economics Dongfang College, Jiaxing 314400, China

Abstract. In this paper, a multidimensional bi-orthogonal multiwavelets with matrix dilation is constructed via the lifting scheme with several steps. We generalize the lifting scheme in the case of multidimensional bi-orthogonal multi-wavelets with matrix dilation. The lifing and dual Lifing scheme are discussed in the case of |det(M )| ≥ 2, where M denotes a dilation matrix. The general interpolating multiscaling function is also presented and discussed. Finally, an example is given. Keywords: Lifting scheme · Multidimensional bi-orthogonal multi-wavelets

1

· Matrix dilation

Introduction

Lifting is a simple method for constructing bi-orthogonal wavelet bases. In recent years, the lifting scheme has been generalized to the multi-band and multidimensional cases. And it has been used to design M -band interpolating scaling filters and their duals. Moreover, the cases of the multi-wavelets with multidimensional and multi-band have also been discussed. In [1], the lifting scheme has been presented to construct the biorthogonal wavelets. In [2–4], the case of the biorthogonal multi-wavelets has been discussed and the corresponding lifting scheme is also given. In [5], Shui and Bao have presented the M-band bi-orthogonal interpolating wavelets by using lifting scheme. The theory of the interpolating scaling vectors and multi-wavelets with matrix dilation has been established symmetrically by Koch in [6]. Based on Hermite interpolation filter banks, the lifting scheme is used for constructing arbitrary dimensional biorthogonal multi-wavelet in [7]. In this paper, multidimensional bi-orthogonal multi-wavelets with matrix dilation is discussed by using the lifting scheme with several steps. We generalize the lifting scheme to the case of multidimensional bi-orthogonal multi-wavelets with matrix dilation, including the lifting and dual lifting scheme. The general interpolating multi-scaling function is also presented and discussed. c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 11–23, 2020. https://doi.org/10.1007/978-981-32-9682-4_2

12

2

X. Zhou and G. Gu

Preliminary T

Assume that Φ(x) = [ϕ1 (x), ϕ2 (x), ...ϕr (x)] , x ∈ Rn is a scaling function vector with multiplicity r and dimension n, satisfying the following matrix refinable equation:  Φ(x) = Pk Φ(M x − k), Pk ∈ Rr×r (1) k∈Z n

where M ∈ Z n×n is a scaling dilation matrix and all eigenvalues of M are larger than one in modulus, and P := (Pk )k∈Z n . Then Φ(x) is called (P, M )—refinable. The notation m is used to denote m = |det(M )|. For all h, l > 0, the notation h×l r×r denotes the matrix with h × l sequence space. Then P ∈ (Z n ) . (Z n ) ⎛ ⎞ (1,1) (1,r) pk · · · pk ⎜ . . .. ⎟ ⎟ . . Pk = ⎜ (2) . . . ⎠ ⎝ (r,1) (r,r) pk · · · pk Taking the Fourier transform on the component of (1) yields T ˆ ˆ Φ(ω) = P (e−iM ω )Φ(ω), ω ∈ Rn

where P (z) =

1 m

k∈Z n n

(3)

Pk z k , z = e−iω = (e−iω1 , e−iω2 , · · · , e−iωn ) ∈ T n , ωl ∈

R, l = 1, · · · , n, T = {z ∈ C n : |zl | = 1} denotes the n-dimensional torus. Then z k = e−i . For all y ∈ Rn , we define the notation zy := e−i(ω+2πy) . In addi β T T tion, we define z M = e−iM ω , so that z M = z M β and zyM := e−iM (ω+2πy) . ˆ ˆ From the transformed refinement equation (3), we have Φ(0) = P (1)Φ(0), where n ˆ 1 = [1, 1, · · · , 1] ∈ C . So Φ(0) is an eigenvector of the eigenvalue 1 of P (1). On ∞ −jT ˆ ˆ P (e−iM ω )Φ(0). However, if the the other hand, iterating (3) yields Φ(ω) = infinite product matrix

∞ j=1

j=1

P (e−iM

−jT

ω

) converges, then the scaling vector Φ(x)

is completely determined by its symbol or mask. The following Lemma 1, stated in [6], provides us with a sufficient condition for the existence of a compactly supported solution of the refinement equation (1), see also [6]. Lemma 1 [6]. For a mask P := (Pk )k∈Z n , let P (1) have the eigenvalues λ1 = 1, |λ2 | < 1, · · · , |λr | < 1, then the following statements hold: 1. The infinite matrix product

∞ j=1

P (e−iM

−jT

ω

) converges uniformly on compact

sets. 2. Any eigenvector v of the eigenvalue 1 of P (1) defines a compactly supported ∞ −jT ˆ P (e−iM ω )v. distributional solution Φ(x) of (1) via Φ(ω) = j=1

3. If Φ is a nontrivial compactly supported distributional solution of (1), then ˆ Φ(0) is the eigenvector of the eigenvalue 1 of P (1).

Multi-wavelets via the Lifting Scheme with Several Steps

13

For a scaling matrix M , let Γ = {ρ0 , ρ1 , · · · , ρm−1 } denote a complete set of the corresponding coset. representatives of Z n /M Z n and for ρ ∈ Γ , [ρ] denote

[ρ] = (M Z n + ρ). So Z n decomposes into the disjoint union Z n = ρ∈Γ

3

ρ∈Γ

Lifting for Multi-wavelets with Matrix Dilation T

Assume that Φ(x) = [ϕ1 (x), ϕ2 (x), ...ϕr (x)] , x ∈ Rn is a scaling function vector with multiplicity r and dimension n. And it satisfies the following matrix refinable equation: √  Pk Φ(M x − k), Pk ∈ Rr×r (4) Φ(x) = m k∈Z n

where M ∈ Z n×n is a scaling dilation matrix and m = |det(M )|. Taking the Fourier transform on the component of (4) yields √ −T m ˆ ˆ −T ω), ω ∈ Rn Φ(ω) = P (e−iM ω )Φ(M (5) m where P (ω) = Pk e−iω,k . According to the multi-resolution analysis of k∈Z n

L (R ) (see in [6]), there exists m − 1 multi-wavelets Ψ j (x) corresponding to the multi-scaling function. Each Ψ j (x) can be represented as √  j Qk Φ(M x − k) (6) Ψ j (x) = m 2

n

k∈Z n

where Qjk are r × r real matrices and 0 < j < m. Taking the Fourier transform on the component of (6) yields √ m j −iM −T ω ˆ −T Ψˆ j (ω) = Q (e )Φ(M ω) (7) m j iω,k where Qj (ω) = Qk e is the symbol of Ψ j (x). Assume that only a finite k∈Z n

number of Pk and Qjk is nonzero. In [6], we know there are dual multi-scaling ˜ function Φ(x) and dual m − 1 multi-wavelets Ψ˜ j (x), respectively, satisfying the ˜ j . There are matrix similar matrix Eqs. (4) and (6) with coefficients P˜k and Q k ˜ j (ω). The i-th sub-symbol Pρ (ω) of P (ω) is defined by symbols P˜ (ω) and Q i  Pρi (ω) := PM k+ρi eiω,k , 0 ≤ l < m (8) k∈Z n

such that we have the decomposition: P (ω) =

m−1 



l=0 k∈Z n

PM k+ρi e−iω,M k+ρi  =

m−1 

Pρi (M T ω)e−iω,ρi  .

l=0

A complete set of representatives of Z n /M T Z n can be denoted by Γ˜ = {˜ ρ0 , ρ˜1 , · · · , ρ˜m−1 }. The sets Γ and Γ˜ are connected by the following lemma on character sums.

14

X. Zhou and G. Gu

Lemma 2 [6]. For ρi , ρj ∈ Γ and ω ∈ Rn , it holds that 

e−iω+2πM

−T

ρ,ρ ˜ i  iω+2πM −T ρ,ρ ˜ j

e

= mδi,j

(9)

ρ∈ ˜ Γ˜

  ˜ Ψ˜ j to be bi-orthogonal is that the symThe necessary condition for Φ, Ψ j , Φ, bols P (ω) and P˜ (ω) satisfy ρ∈ ˜ Γ˜

∗ P (ω + 2πM −T ρ˜)P˜ (ω + 2πM −T ρ˜) = mIr ,



˜ j (ω + 2πM −T ρ˜)∗ = Or P (ω + 2πM −T ρ˜)Q

ρ∈ ˜ Γ˜

˜ ∗ P (ω + 2πM −T ρ˜)Qj (ω + 2πM −T ρ˜) = Or , ˜ ˜ Γ ρ∈ ˜ i (ω + 2πM −T ρ˜)Qj (ω + 2πM −T ρ˜)∗ = mδl,j Ir Q

(10)

ρ∈ ˜ Γ˜

where 0 < j, l < m. The similar modulation matrix M (ω) can be written as ⎞ P (ω + 2πM −T ρ˜0 ) P (ω + 2πM −T ρ˜1 ) · · · P (ω + 2πM −T ρ˜m−1 ) 1 −T 1 −T 1 −T ⎜ Q (ω + 2πM ρ˜0 ) Q (ω + 2πM ρ˜1 ) · · · Q (ω + 2πM ρ˜m−1 ) ⎟ ⎟ ⎜ M (ω) = ⎜ ⎟ . . . . ⎠ ⎝ . . Qm−1 (ω + 2πM −T ρ˜0 ) Qm−1 (ω + 2πM −T ρ˜1 ) · · · Qm−1 (ω + 2πM −T ρ˜m−1 ) ⎛

(11) ˜ (ω) can be defined. The formula Similarly, the dual modulation matrix M (10) is equivalent to the following matrix equation ˜ (ω)∗ = mImr M (ω)M

(12)

The poly-phase matrix H(ω) can be written as ⎛ ⎞ PM k+ρ0 PM k+ρ1 · · · PM k+ρ Pρo (ω) Pρ1 (ω) · · · Pρm−1 (ω) m−1 ⎜ 1 1 1 1 ⎟ ⎜ Q1 (ω) ⎜ QM k+ρ Q1 Q (ω) · · · Q (ω) ⎟ ⎜ M k+ρ1 · · · QM k+ρm−1 ρo ρ1 ρm−1 ⎜ 0 ⎟ ⎜  ⎜ ⎟ ⎜ H(ω) = ⎜ ⎜ . . . . ⎟= . . . . ⎟ k∈Z n ⎜ ⎜ ⎜ . . . . ⎠ ⎝ ⎝ m−1 m−1 m−1 m−1 m−1 m−1 (ω) · · · Qρ (ω) Qρo (ω) Qρ QM k+ρ QM k+ρ · · · QM k+ρ 1 m−1 0 1 m−1 ⎛

⎞ ⎟ ⎟ ⎟ ⎟ −iω,k ⎟e ⎟ ⎟ ⎠

(13) Then ⎛ T

H(M ω) =

⎜ ⎜ ⎜ 1 M (ω)⎜ ⎜ m ⎜ ⎝



i ω+2πM −T ρ ˜0 ,ρ0

i ω+2πM −T ρ ˜1 ,ρ0

e

e

e

e

. . .



i ω+2πM −T ρ ˜m−1 ,ρ0

e

e



i ω+2πM −T ρ ˜0 ,ρ1

i ω+2πM −T ρ ˜1 ,ρ1



i ω+2πM −T ρ ˜m−1 ,ρ1

e e



i ω+2πM −T ρ ˜0 ,ρm−1

i ω+2πM −T ρ ˜1 ,ρm−1

. . .

e

i ω+2πM −T ρ ˜m−1 ,ρm−1

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

⎠

(14) ˜ The dual poly-phase matrix H(ω) can be defined similarly. So the perfect reconstruction property can be given by ∗

˜ = Imr H(ω)H(ω)

(15)

Multi-wavelets via the Lifting Scheme with Several Steps

15

  ˜ Ψ˜ j be bi-orthogonal, and P (ω), Qj (ω), P˜ (ω), Q ˜ j (ω) Theorem 1. Let Φ, Ψ j , Φ, ˜ are their corresponding matrix symbol. Then H(ω) and H(ω) defined in (14) satisfy the formula (15). Theorem 2. (Lifing and Dual Lifing) Assume that we have two compactly supported multi-wavelet families which satisfy the conditions of bi-orthogonality and sharea same multiscaling function Φ(x) with     dilation matrix M . Denote the fam˜ j (ω) ilies Φ, ΨAj , Φ˜A , Ψ˜Aj and Φ, ΨBj , Φ˜B , Ψ˜Bj . Let {P (ω) , QjA (ω), P˜A (ω), Q A   ˜ j (ω) be the matrix symbols for these functions. and P (ω), QjB (ω), P˜B (ω), Q B Then ˜ 1B (ω), Q ˜ 2B (ω), · · · , Q ˜ m−1 (ω) (ω); P˜B (ω), Q Q1B (ω), Q2B (ω), · · · , QiB (ω), · · · , Qm−1 B B can be generated by matrix (17), where Sij (ω) and Ti (ω) are of finite degree, the determinant of Ti (ω) is a monomial, 0 < i, j < m and m = det(M ). 2

Proof. To prove Theorem 2, we must see that there exists (m − 1) finite-degree matrices Sij (ω) and m − 1 uni-modular matrices Ti (ω) that satisfy HB (ω) = τm−1 (ω)Ωm−1 (ω)τm−2 (ω)Ωm−2 (ω) · · · τ1 (ω)Ω1 (ω)HA (ω) where

⎛

Ir

Or

Or

⎞

⎛

Ir

Or

Or

(16) ⎞

⎟ ⎟ ⎜ ⎜ .. .. ⎟ ⎟ ⎜ ⎜ . . ⎟ ⎟ ⎜ ⎜ i i ⎟ ⎟ ⎜ O S T (ω) O (ω) I S (ω) , Ω (ω) = τi (ω) = ⎜ i r⎟ r i m−1 ⎟ ⎜ r ⎜ 1 ⎟ ⎟ ⎜ ⎜ . . .. .. ⎠ ⎠ ⎝ ⎝ Or Or Ir Or Ir Or Matrix Ti (ω) is the component matrix of τi (ω) in the i + 1-th row and i + 1-th column; the matrix in the i + 1-th row of Ωi (ω) are matrix Sji (ω) except that matrix in the i + 1-th row and i + 1-th column is unit matrix Ir ; ⎛ ⎞ Pρo (ω) Pρ1 (ω) · · · Pρm−1 (ω) ⎜ Q1A,ρo (ω) Q1A,ρ1 (ω) · · · Q1A,ρm−1 (ω) ⎟ ⎜ ⎟ HA (ω) = ⎜ ⎟, .. .. ⎝ ⎠ . . m−1 m−1 m−1 QA,ρo (ω) QA,ρ1 (ω) · · · QA,ρm−1 (ω) ⎛ ⎞ Pρo (ω) Pρ1 (ω) · · · Pρm−1 (ω) 1 1 1 ⎜ QB,ρo (ω) QB,ρ1 (ω) · · · QB,ρm−1 (ω) ⎟ ⎜ ⎟ HB (ω) = ⎜ ⎟ , and 0 < i, j < m. .. .. ⎝ ⎠ . . m−1 m−1 (ω) Q (ω) · · · Q (ω) Qm−1 B,ρo B,ρ1 B,ρm−1   ˜ j (ω) Since both {P (ω) , QjA (ω), P˜A (ω), Q and P (ω), QjB (ω), P˜B (ω), A  ˜ j (ω) satisfy the perfect reconstruction conditions, that is, HA (ω)H ˜ A (ω)∗ = Q B

16

X. Zhou and G. Gu ∗

˜ B (ω) = Imr . The first row of HA (ω) and HB (ω) are the same, Imr , HB (ω)H the rest m − 1 rows are different. It is difficult to change from HA (ω) lifting to HB (ω) via one step. But it is easy to realize that through m − 1 steps. In other words, each step only lifts one multi-wavelet. So let H1,B (ω) be ⎛

⎞ Pρo (ω) Pρ1 (ω) Pρm−1 (ω) ⎜ Q1B,ρo (ω) Q1B,ρ1 (ω) Q1B,ρm−1 (ω) ⎟ ⎜ 2 ⎟ 2 2 ⎟ H1,B (ω) = ⎜ ⎜ QA,ρo (ω) QA,ρ1 (ω) QA,ρm−1 (ω) ⎟ ⎝ ⎠ m−1 m−1 (ω) Q (ω) Q (ω) Qm−1 A,ρo A,ρ1 A,ρm−1 We have ⎛

Ir Or Or m−1 m−1 ⎜ m−1  ∗ 1 ⎜  Q1 ˜ρ (ω)∗  Q1 ˜1 ˜ m−1 (ω)∗ (ω) P (ω) Q (ω) Q (ω) Q ⎜ B,ρl B,ρl A,ρl B,ρl A,ρl l ⎜ l=1 l=1 l=1 ˜ A (ω)∗ = ⎜ H1,B (ω)H O O ⎜ r r ⎜ ⎜ . . .. ⎜ . . ⎝ . . . Or Ir Or

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

˜ A (ω)∗ must be Because H1,B (ω) and HA (ω) are both uni-modular, H1,B (ω)H ∗ ˜ A (ω) ) is equal to the determinant of matrix uni-modular. Since det(H1,B (ω)H m−1 m−1 1 ˜ 1 (ω)∗ . So T1 (ω) = Q1 (ω)Q ˜ 1 (ω)∗ is also uni-modular. QB,ρl (ω)Q A,ρl B,ρl A,ρl l=1

l=1

Let

S11 (ω) = (T1 (ω))−1

m−1 

Q1B,ρl (ω)P˜A,ρl (ω)∗ , Sj1 (ω) = (T1 (ω))−1

l=1

m−1  l=1

˜ j+1 (ω)∗ , Q1B,ρl (ω)Q A,ρl

where j = 2, · · · , m − 1. Then we have the conclusion of the first step ⎞⎛ ⎞ ⎛ Ir Or Or Ir Or Or Or 1 ⎟ ⎜ S11 (ω) Ir S21 (ω) ⎜ Or T1 (ω) Sm−1 (ω) ⎟ ⎟⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ . O I O O r r r r .. H1,B (ω) = ⎜ ⎟⎜ ⎟ HA (ω) ⎟⎜ ⎟ ⎜ . . ⎝ ⎠ ⎠ ⎝ Or Or . Ir Or Or Or Or Or Or Or Ir Ir ˜ 1,B (ω) as Now we construct H ⎞⎛ ∗ Ir Or Or Ir −S11 (ω) −1 ⎟ ⎜ Or ⎜ Or T1 (ω) Ir ⎟⎜ ⎜ ∗ 1 ⎜ ⎟ ⎜ O −S . ˜ 1,B (ω) = ⎜ 2 (ω) .. H ⎟⎜ r ⎟⎜ ⎜ ⎝ Or Or Ir Or ⎠ ⎝ ∗ 1 Or −Sm−1 (ω) Or Or Or Ir ⎛

⎞ Or Or ⎟ ⎟ Or ⎟ ˜ A (ω) ⎟H ⎟ .. ⎠ . Or Ir Or Or Ir

Multi-wavelets via the Lifting Scheme with Several Steps

17

Similar to the first step, let H1,B (ω) be ⎛

Pρo (ω)

Pρ1 (ω)

Pρm−1 (ω)

⎞

⎜ 1 ⎟ ⎜ QB,ρo (ω) Q1B,ρ1 (ω) Q1B,ρm−1 (ω) ⎟ ⎜ ⎟ ⎟ H2,B (ω) = ⎜ ⎜ Q2B,ρo (ω) Q2B,ρ1 (ω) Q2B,ρm−1 (ω) ⎟ ⎜ ⎟ ⎝ ⎠ m−1 m−1 m−1 QA,ρo (ω) QA,ρ1 (ω) QA,ρm−1 (ω) We have T2 (ω) =

m−1 l=1

∗

˜ 2 (ω) is also uni-modular. Let Q2B,ρl (ω)Q A,ρl

S12 (ω) = (T2 (ω))

−1

m−1 

−1

m−1 

−1

m−1 

∗ Q2B,ρl (ω)P˜A,ρl (ω) ,

l=1

S22 (ω) = (T2 (ω))

˜ 1B,ρ (ω)∗ , Q2B,ρl (ω)Q l

l=1

Sj2 (ω) = (T1 (ω))

l=1

˜ j+1 (ω)∗ , j = 3, · · · , m − 1. Q2B,ρl (ω)Q A,ρl

Then we have the conclusion of the second step ⎛ ⎞ ⎛ ⎞ I Or Or Or Ir Or Or Or ⎜ r ⎟ ⎟ ⎜ ⎟ ⎜ Or Ir Or Or ⎜ ⎟ ⎜ Or Ir Or ⎟ Or ⎟ ⎜ ⎟ ⎜ 2 ⎜ ⎟ ⎜ S12 (ω) S22 (ω) Ir Sm−1 (ω) ⎟ H2,B (ω) = ⎜ Or Or T2 (ω) ⎟ H1,B (ω) ⎟⎜ O r ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ .. .. ⎟ ⎝ ⎠⎜ . . ⎝ ⎠ Or Or Or Or Ir Or Or Or Ir ˜ 2,B (ω) as So construct H ⎛

Ir Or

Or

Or

⎞⎛

Ir ⎜ ⎟⎜ ⎜ Or Or ⎟ O O r r ⎟⎜O ⎜ r ⎜ ⎟⎜ ⎜ −1 ˜ ⎜ ⎟ H2,B (ω) = ⎜ Or Or T2 (ω) Or ⎟ ⎜ Or ⎜ ⎟⎜ ⎜ ⎟⎜ .. ⎝ ⎠⎝ . Or Or Or Or Ir

Or −S12 (ω)

∗

Ir

∗

Or

⎞

⎟ Or ⎟ ⎟ ⎟ ˜ H (ω) Or Ir Or ⎟ ⎟ 1,B ⎟ .. ⎠ . ∗ 2 Or −Sm−1 (ω) Ir −S22 (ω)

18

X. Zhou and G. Gu

Continue to do there m − 1 times, then we have ⎛ ⎜ ⎜ ⎜ HB (ω) = ⎜ ⎜ ⎝

Ir

Or

Ir ..

.

⎞ ⎛

Ir

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟·⎜ ⎟ ⎜ ⎠ ⎝

Or

Ir ..

.

Ir m−1 m−1 m−1 O T (ω) S (ω) S (ω) S r m−1 1 2 m−1 (ω) ⎛ ⎞⎛ ⎞ Ir Ir Or ··· Or Ir Or · · · Or Or ⎜ Or Ir ⎜ ⎟ Ir Or Or Or ⎟ ⎜ ⎟ ⎜ Or ⎟ ⎜ ⎜ ⎟ ⎟ .. .. ⎜ ⎟⎜ ⎟··· . . ⎜ ⎟ ⎜ m−2 ⎟ m−2 m−2 ⎝ Or ⎝ ⎠ Tm−2 (ω) S1 (ω) S2 (ω) Ir Sm−1 (ω) ⎠ ··· Or Ir ⎞ ⎛ Or Or ··· Ir Or ⎛ ⎞ Ir Or Ir Or Or Or Or 1 ⎜ Or T1 (ω) ⎟ ⎜ S11 (ω) Ir S21 (ω) Sm−1 (ω) ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎜ ⎟ ⎟ .. Or ⎜ ⎟ ⎜ Or Or Ir ⎟ HA (ω) . ⎜ ⎟⎜ ⎟ . . ⎝O O ⎠ . I O ⎠⎝ r

Ir

r

r

Or Or

r

Or

Or Ir

Or

Or

r

Or

Ir

r

Or

⎞⎛

Or

r

r

Or Ir

⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Ir

∗⎞ Ir Or Or −S1m−1 (ω) ∗ ⎟⎜ ⎜ Ir Ir −S2m−1 (ω) ⎟ ⎟⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ .. .. ˜ B (ω) = ⎜ H ⎟⎜ ⎟ . . ⎟⎜ ⎟ ⎜ ∗⎠ m−1 ⎠⎝ ⎝ Ir Ir −Sm−1 (ω) −1 Or Tm−1 (ω) Or Or Or Ir⎞ ⎛ ⎞⎛ ∗ Ir Or · · · Or Or Ir Or · · · −S1m−2 (ω) Or ∗ ⎜ Or Ir ⎜ Or Or ⎟ −S2m−2 (ω) Or ⎟ ⎜ ⎟ ⎜ Or Ir ⎟ ⎜ ⎜ ⎟ ⎟ . .. .. ⎜ ⎟⎜ ⎟··· . ⎜ ⎜ ⎟ ⎟ −1 ⎝ Or ⎝ ⎠ ⎠ O I O O Tm−2 (ω) r r r r ∗ m−2 ··· Or Ir⎛ Or Or · · · −Sm−1 (ω) ⎞Ir ⎛ Or ⎞ ∗ Ir Or Or Ir −S11 (ω) Or Or ⎜ Or T1 (ω)−1 ⎟ ⎜ Or I O Or ⎟ r r ⎜ ⎟⎜ ⎟ ∗ 1 ⎜ ⎜ ⎟ Ir Or ⎟ .. ˜ (ω) ⎜ ⎟ ⎜ Or −S2 (ω) ⎟H . ⎜ ⎟⎜ ⎟ A . . ⎝O ⎠ . O I O ⎠⎝

⎛

⎞

∗

1 Or −Sm−1 (ω) Or

(17)

Ir

These conclusions imply the Theorem 2. We can see that Theorem 2 gives the lifting scheme in the case of multi-wavelets with matrix dilation. It is a generalization of the lifting scheme for multi-wavelet in [5]. In the case of |det(M )| = 2, the conclusion of Theorem 2 will be easy and obvious. So we have the following theorem.

Multi-wavelets via the Lifting Scheme with Several Steps

19

Theorem 3. Assume that we have two compactly supported multi-wavelet families which satisfy the conditions of bi-orthogonality and share a multi-scaling function and share a same multi-scaling function Φ(x) with matrix  dilation   ˜ ˜ M ,where |det(M )| = 2. Denote the families Φ, ΨA , ΦA , ΨA and Φ, ΨB , Φ˜B ,     ˜ A (ω) and P (ω), QB (ω), P˜B (ω), Q ˜ B (ω) be Ψ˜B . Let {P (ω) , QA (ω), P˜A (ω), Q the matrix symbols for these functions. Then + S(M T ω)PA (ω)) QB (ω) = T (M T ω)(Q

A (ω) ∗ T ˜ A (ω) ˜ ˜ PB (ω) = PA (ω) − S(M ω) Q

 −1 ˜ B (ω) = T (M T ω) ˜ A (ω) Q Q

(18)

where S(ω) and T (ω) are of finite degree and the determinant of T (ω) is a monomial. The conclusion of Theorem 3 can also be expressed by The poly-phase matrix H(ω), which can be written as    Ir Or Ir Or HA (ω) HB (ω) = Or T (ω) S(ω) Ir   ∗ Ir Or −S(ω) I r ˜ A (ω) ˜

H HB (ω) = ∗ −1 Or Ir Or T (ω) 

   Pρo (ω) Pρ1 (ω) Pρo (ω) Pρ1 (ω) , HB (ω) = the where HA (ω) = QA,ρo (ω) QA,ρ1 (ω) QB,ρo (ω) QB,ρ1 (ω) complete set of representatives of Z n /M Z n is Γ = {ρ0 , ρ1 }. According to Theorem 3, we can write the multi-wavelets ΨB as Ψ B (ω) = =

√

2 2 −T ω) = −T ω)  QB (M −T ω)Φ T (ω) QA (M −T ω) + S(ω)PA (M −T ω) Φ(M B (M 2 2

√ √ −T ω) + S(ω) 2 P (M −T ω)Φ(M −T ω)   T (ω) 22 QA (M −T ω)Φ(M A 2 √



 = T (ω) Ψ A (ω) + S(ω)Φ(ω)

This means that ΨB (x) =

 k1 ∈Z n

Tk1 ΨA (x − k1 ) +



Tk1 Sk2 Φ(x − k1 − k2 )

k1 ,k2 ∈Z n

Similarly we obtain  Φ B (ω) = = = =

√ √   2  2 −T ω) = −T ω) − (S(ω))∗ Q −T ω) Φ −T ω)   PB (M −T ω)Φ P B (M A (M A (M B (M 2 2 √ √   2  −T ω) − (S(ω))∗ 2 Q −T ω)Φ −T ω)   PA (M −T ω)Φ B (M A (M B (M 2 2 √ √   2  −T ω) − (S(ω))∗ T (ω) 2 Q −T ω)Φ −T ω)   PA (M −T ω)Φ B (M B (M B (M 2 2 √   2  −T ω) − (S(ω))∗ T (ω)Ψ  PA (M −T ω)Φ B (M B (ω) 2

20

X. Zhou and G. Gu

So, Φ B (x) =

√

2

  PA,k ΦB (M x − k) −

k∈Z n

−1 A (ω) = Q Let T (M T ω)

k1 ,k2

∈Z n

k1 ,k2 ∈Z n

S−k2 Tk1 Ψ B (x − k1 − k2 ).

−iω,k1 +k2   Q Tk−1 . A,k2 e 1

We have  Ψ B (ω) =

That is

√ 2 2

QB (M

−T

 −T ω)Φ ω) = B (M

Ψ B (x) =

√ 2

2 k ,k ∈Z n 1 2



√ 2

k1 ,k2 ∈Z n



−T ω,k +k  −1  −i M −T 1 2 Φ QA,k2 e ω) B (M 1

Tk

  Q Tk−1 A,k2 ΦB (M x − k1 − k2 ) 1

We can summarize these conclusions in the following theorem. Theorem 4. (Lifting scheme) Take a initial compactly supported bi-orthogonal multi-wavelet families Φ, ΨA , Φ˜A , Ψ˜A . Then a new bi-orthogonal multi-wavelet   families Φ, ΨA , Φ˜A , Ψ˜A can be found as ΨB (x) = Tk1 ΨA (x − k1 ) + Tk1 Sk2 Φ(x − k1 − k2 ) n k1 ,k2 ∈Z n 1 ∈Z √ k  P Φ 2 S−k2 Tk1 Ψ B (x) = A,k ΦB (M x − k) − B (x − k1 − k2 ) k∈Z n k1 ,k2 ∈Z n √   Ψ Q 2 Tk−1 B (x) = A,k2 ΦB (M x − k1 − k2 ) 1

(19)

k1 ,k2 ∈Z n

where the matrix coefficient Sk can be freely chosen, but the determinant of T (ω) is a monomial. In general, there are numerous ways to factorize a polyphase matrix of any compactly supported biorthogonal multiwavelet transform into a finite sequence of simple lifting steps. Next, we will explain how to factor a polyphase matrix of a compactly supported biorthogonal multiwavelet with matrix dilation into a simple lifting step. According ⎛ to the definition (13) of a poly-phase ⎞ matrix, we Pρo (ω) Pρ1 (ω) · · · Pρm−1 (ω) ⎜ Q1ρo (ω) Q1ρ1 (ω) · · · Q1ρm−1 (ω) ⎟ ⎜ ⎟ can rewrite H(ω) as: H(ω) = ⎜ ⎟. .. .. ⎝ ⎠ . . m−1 m−1 m−1 Qρo (ω) Qρ1 (ω) · · · Qρm−1 (ω) Because H(ω) is uni-modular, we can factor any unimodular matrix H(ω) into H(ω) = L(ω)U (ω), where ⎛

L1,1 (ω) Or ⎜ L2,1 (ω) L2,2 (ω) ⎜ L(ω) = ⎜ .. ⎝ . Lm,1 (ω) Lm,2 (ω)

Or Or Lm,m (ω)

⎞

⎛

U1,1 (ω) U1,2 (ω) U1,m (ω) ⎜ Or U2,2 (ω) ⎟ U2,m (ω) ⎜ ⎟ ⎟ , U (ω) = ⎜ . .. ⎝ ⎠ Or

Or

Um,m (ω)

⎞ ⎟ ⎟ ⎟ ⎠

Multi-wavelets via the Lifting Scheme with Several Steps

21

Lower triangular matrices Li,i (ω) and upper triangular matrices Ui,i (ω) are inverse. Since ⎞ ⎛ ⎞⎛ Ir Or Or L1,1 (ω) Or Or ⎜ Or L2,2 (ω) ⎟ ⎜ L2,2 (ω)−1 L2,1 (ω) Ir Or Or ⎟ ⎟ ⎜ ⎟⎜ L(ω) = ⎜ ⎜ ⎟··· ⎟ . . .. .. ⎠ ⎝ ⎠⎝ Or Or Lm,m (ω) Or O Ir ⎛ ⎞r Ir Or Or ⎜ Or Ir Or ⎟ ⎜ ⎟ ⎜ ⎟ .. ⎝ ⎠ . −1 −1 Lm,m (ω) Lm,1 (ω) Lm,m (ω) Lm,2 (ω) Ir ⎛

⎞ ⎛ ⎞ −1 −1 Ir Or Um,m (ω) U1,m (ω) Ir U2,2 (ω) U1,2 (ω) Or −1 ⎜ Or Ir ⎜ Ir Or ⎟ Um,m (ω) U2,m (ω) ⎟ ⎜ ⎟ ⎜ Or ⎟ U (ω) = ⎜ · · · ⎜ ⎟ ⎟ .. .. ⎝ ⎠ ⎝ ⎠ . . Or ⎞ Or Ir Ir Or Or ⎛ U1,1 (ω) Or Or ⎜ Or U2,2 (ω) ⎟ Or ⎜ ⎟ ⎜ ⎟ .. ⎝ ⎠ . Or Um,m (ω) Or We have H(ω) = L(ω)U (ω) = A1 (ω)A2 (ω) · · · Am (ω)Bm (ω) · · · B2 (ω)B1 (ω), where ⎛ ⎜ ⎜ ⎜ A1 (ω) = ⎜ ⎜ ⎝

L1,1 (ω) Or Or L2,2 (ω)

Or Or .

..

⎞

⎛

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ , B1 (ω) = ⎜ ⎟ ⎜ ⎠ ⎝

U1,1 (ω) Or Or U2,2 (ω)

Or Or .

..

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Lm,m (ω) Or Um,m (ω) Or ⎛ ⎛ ⎞ ⎞ Ir Or Ir Ui,i (ω)−1 U1,i (ω) Or ⎜ ⎜ ⎟ ⎟ . . . ⎜ ⎜ ⎟ ⎟ .. .. . ⎜ ⎜ ⎟ ⎟ . ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ −1 Ai (ω) = ⎜ Li,i (ω) Li,1 (ω) · · · Ir Or ⎟ , Bi (ω) = ⎜ Or ⎟ , Ir ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ .. .. ⎜ ⎜ ⎟ ⎟ ⎝ ⎝ ⎠ ⎠ . . Or

Or

Or

Ir i = 2, · · · , m.

Or

(20)

Ir

So we have the following theorem. Theorem 5. A polyphase matrix H(ω) corresponding to compactly supported bi-orthogonal multi-wavelets with dilation matrix can be expressed in the form H(ω) = A1 (ω)A2 (ω) · · · Am (ω)Bm (ω) · · · B2 (ω)B1 (ω) where Ai (ω), Bi (ω) are defined in the formula (20).

(21)

22

4

X. Zhou and G. Gu

General Interpolating Scaling Function Vector

Definition 1. Assume that an n−dimension scaling function vector Φ(x) = T [ϕ1 (x), ϕ2 (x), · · · , ϕr (x)] is called to be general interpolating, if Φ(x) satisfies the following equations: ϕl (M −1 N −1 ρj + k) = δ0,k δρl ,ρj+1 l = 1, 2, · · · , r, j = 0, 1, · · · , r − 1, ρj ∈ Γ or   Φ(M −1 N −1 ρ0 + k), Φ(M −1 N −1 ρ1 + k), · · · , Φ(M −1 N −1 ρr−1 + k) = δ0,k Ir (22) It is reasonable to define this general interpolating scaling function vector. In a simple case, if φ(x) ∈ L2 (Rn ) is an n−dimension uni-scaling function, then the scaling function vector defined by Φ(x) = [φ(N M x − ρ0 ), φ(N M x − ρ1 ), · · · , φ(N M x − ρr−1 )]

T

satisfies the interpolating condition (22). For an advantage of the interpolating scaling function vector, it is easy to construct the mask. Next we give the condition which an interpolating mask satisfies. Let T

T

T

e1 = [1, 0, · · · , 0] , e2 = [0, 1, · · · , 0] , · · · , er = [0, 0, · · · , 1] . In one hand, Choose the matrix N ∈ Z n×n such that N M N −1 ∈ Z n×n . So we have Φ(M −1 N −1 ρj + k) = δ0,k ej+1 . In the other hand, there is Φ(M −1 N −1 ρj + k) =



Pl Φ(N −1 ρj + M k − l).

l∈Z d

For N −1 ρj , there exists β ∈ Z n such that N −1 ρj = β + M −1 N −1 ρj  , where ρj  ∈ Γ . In fact, since N M N −1 ∈ Z n×n , then N M N −1 ρj ∈ Z n . So there exists β ∈ Z n , such that N M N −1 ρj = N M β + ρj  .  Φ(M −1 N −1 ρj + k) = Pl Φ(β + M −1 N −1 ρj  + M k − l) = Pβ+M k ej  +1 . l∈Z d

So we have Theorem 6. Let ρj ∈ Z n /(N M ) Z n , where matrix N ∈ Z n×n satisfies N M N −1 ∈ Z n×n . Then there exists β ∈ Z n such that the mask of interpolating scaling function vector satisfies Pβ+M k ej  +1 = δ0,k ej+1 where N −1 ρj = β + M −1 N −1 ρj  .

(23)

Multi-wavelets via the Lifting Scheme with Several Steps

23

Note: the chosen matrix N ∈ Z n×n which satisfies N M N −1 ∈ Z n×n is in existence such as N = M , aIn , a ∈ Z and so on.   1 1 Example 1. Let N = M = , n = 2. The multi-scaling function with 1 −1 matrix dilation M satisfies the following matrix equations:  Pk Φ(M x − k), x ∈ R2 Φ(x) = k∈Z 2

Construct P (ω) as: ⎛

⎛ ⎛ ⎞ ⎞ ⎞ 0000 1000 0010 ⎜ ⎜ ⎟ ⎟ 1 ⎜0 0 0 1⎟ ⎟ eiω2 + 1 ⎜ 0 1 0 0 ⎟ + 1 ⎜ 0 0 0 0 ⎟ e−iω1 , P (ω) = ⎜ ⎝ ⎝ ⎝ ⎠ ⎠ 2 0100 2 0001 2 0 0 0 0⎠ 0000 0010 1000

where ω = (ω1 , ω2 ) ∈ R2 . It is easy to see that P (ω) satisfy the⎞condition (23). The eigenvalues of the ⎛ 1010 ⎜0 1 0 1⎟ T ⎟ matrix symbol P ((1, 1) ) = 12 ⎜ ⎝ 0 1 0 1 ⎠ , ω1 = ω2 = 0, satisfy the condition of 1010 Lemma 1 the multiscaling function Φ(x) generated by P (ω) is general interpolat ing. And its dual multiscaling function Φ(x) is itself, that is, Φ(x) is orthogonal. It is also easy to check that P (ω) satisfies the condition of orthogonality. Acknowledgments. This work is supported by the Planning of philosophy and Social Sciences in Zhejiang Province of China (19NDQN340YB) and National Natural Science Foundation of China (11771100).

References 1. Sweldens W (1996) The lifting scheme: a custom-design construction of biorthogonal wavelets. Appl Comput Harmon Anal 3:186–200 2. Goh SS, Jiang Q, Xia T (2000) Construction of biorthogonal multiwavelets using the lifting scheme. Appl Comput Harmon Anal 9:336–352 3. Averbuch AZ, Zheludev VA (2002) Lifting scheme for biorthogonal multiwavelets originated from Hermite splines. IEEE Trans Signal Process 50:487–500 4. Davis GM, Strela V, Turcajova R (1999) Multiwavelet construction via lifting scheme. In: He TX (ed) Wavelet analysis and multi-resolution methods. Lecture notes in pure and applied mathematics 5. Shui P, Bao Z (2004) M-band biorthogonal interpolating wavelets via lifting scheme. IEEE Trans Signal Process 52(9):2500–2512 6. Koch K (2006) Interpolating scaling vectors and multiwavelets in Rd . Fachbereich Mathematik und Informatikder Philipps-UniversitatMarburg 7. Gao X, Xiao F, Li B (2009) Construction of arbitrary dimensional biorthogonal multiwavelet using lifting scheme. IEEE Trans Image Process 18(5):942–955

Research on Regression Model Based on Uncertain System Sizhe Wang1 and Zhigang Wang2(B) 1

School of Automation, Central South University, Changsha 410083, China [email protected] 2 School of Science, Hainan University, Haikou 570228, China [email protected]

Abstract. The emphasis in this paper is mainly on regression model based on uncertain system. Expert’s experimental data and Delphi method for estimating empirical uncertainty distributions are introduced. By collecting and interpreting expert’s experimental data, we establish an uncertain regression model and get the optimal solution of the parameter estimation. The numerical example shows that the uncertain regression model is superior to the classical regression model. Keywords: Uncertainty distribution · Delphi method · Uncertain regression model · Classical regression model

1

Introduction

Fuzzy theory is a branch of mathematics concerned with analysis of fuzzy phenomena. However, a lot of surveys showed that some information represented usually by human language “like about 1000 km”, “roughly 60 kg”, and “small size” are neither randomness nor fuzziness. In order to model this type of imprecise quantities, uncertainty theory was founded by Liu in 2007 and refined by Liu in 2010. Uncertain statistics is a methodology for collecting and interpreting expert’s experimental data. Liu [1] suggested an empirical uncertainty distribution and proposed a principle of least squares as the method for estimating the unknown parameters of an uncertainty distribution based on the expert’s experimental data. Wang and Gao [2] proposed a method of moments for estimating the unknown parameters. Regression analysis is one of the most frequently used statistical tools, which is concerned with investigating the relationship between random variable and one or more non-random variables. Because of lack of observed data, Ha, Wang and Gao [3] built uncertain linear regression model for collecting and interpreting expert’s experimental data and forecast the value of China’s GDP by using this model. The emphasis in this paper is mainly on linear regression model based on uncertain system. Expert’s experimental data and Delphi method for estimating empirical uncertainty distributions are introduced. An uncertain linear regression model and the optimal solution of the model parameter estimation are c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 24–29, 2020. https://doi.org/10.1007/978-981-32-9682-4_3

Hamiltonian Mechanics

25

given. The numerical example shows that the uncertain linear regression model is superior to n the classical linear regression model.

2

System Description

Let Γ be a nonempty set, and L be a σ-algebra over Γ . Each element Λ ∈ L is called an event. A number M{Λ} indicates the level that Λ will occur. Liu [1] proposed that number M{Λ} has certain mathematical properties. (1) (Normality Axiom) M(Γ ) = 1 for the universal set Γ . (2) (Duality Axiom) M{Λ} + M{Λc } = 1 for any event Λ. (3) (Subadditivity Axiom) For every countable event Λ1 , Λ2 , · · · , we have M{

∞ 

Λn } ≤

n=1

∞ 

M{Λn }.

n=1

(4) (Product Axiom) Let (Γk , Lk , Mk ) be uncertainty spaces for k = 1, 2, · · · . The product uncertain measure M is an uncertain measure satisfying M{

∞ 

k=1

Λk } =

∞ 

Mk {Λk }.

k=1

Where Λk are arbitrarily chosen events from Lk for k = 1, 2, · · · , respectively. The expected value operator of an uncertain variable was defined by Liu [1] as





+∞

M{ξ ≥ r}dr −

E[ξ] = 0

0

−∞

M{ξ ≤ r}dr

provided that at least one of the two integrals is finite. The variance of an uncertain variable ξ with finite expected value e is defined by E[(ξ − e)2 ]. For any x ∈ R the function Φ(x) = M{ξ ≤ x} is called uncertainty distribution of ξ. 2.1

Delphi Method and Uncertainty Distribution

In this section, our method is aimed at estimating the uncertainty distribution Φ(x) of ξ with Delphi method. We first invite m experts to choose some possible values x1 , x2 , · · · , xn that the uncertain variable ξ may take (for all experts, their possible values and the number of values can be different). Without loss of generality, we assume x1 < x2 < · · · < xn . Then the procedure can be summarized as follows. Step 1. The m experts provide their experimental data (xij , αij ) independently, where xij denote the jth possible value provided by the ith expert and αij denote the ith expert’s belief degree that ξ is less then xij , i = 1, 2, · · · , m, j = 1, 2, · · · , ni , respectively.

26

S. Wang and Z. Wang

Step 2. Use the ith expert’s experimental data (xi1 , αi1 ), (xi2 , α12 ), · · · , (xini , αini ) to generate the uncertainty distributions Φ of the ith domain experts, respectively. Step 3. Calculate the number of the possible values of ξ presented by all experts denoted by n, where the same values from different experts are treated as one. Then the possible values of ξ are x1 ≤ x2 ≤ · · · ≤ xn , and compute m

αˆj =

m

1  1  Φi (xj ), j = 1, 2, · · · , n, dj = (Φi (xj )− αˆj )2 , j = 1, 2, · · · , n m i=1 m i=1

Step 4. If dj are less than a given level ε > 0 for all j, then go to Step 5. Otherwise, the ith domain experts receive the summary (for example, the αˆj obtained in the previous round and the reasons of other experts), and then provide a set of revised expert’s experimental data (xi1 , αi1 ), (xi2 , α12 ), · · · , (xini , αini ) for i = 1, 2, · · · , m. Go to Step 2. Step 5. Use the integrated data (x1 , αˆ1 ), (x2 , αˆ2 ), · · · , (xn , αˆn ) to generate the uncertainty distribution Φ(x). 2.2

Uncertain Linear Regression Model and Parameter Estimation

Let Y denote the uncertainty dependent variable and x1 , x2 , · · · , xp denote the explanatory variables, where p is the number of explanatory variables. The functional relationship between Y and x1 , x2 , · · · , xp can be expressed by the uncertain regression model Y = f (x1 , x2 , · · · , xp ) + ε

(1)

where ε is an uncertainty variable which denotes the uncertainty error term and f (x1 , x2 , · · · , xp ) denotes the functional relation. Regularly, ε can be assumed to have an uncertainty distribution with mean zero and variance σ 2 . In particular, the uncertain linear regression model is Y = β0 + β1 x1 + β2 x2 + · · · + βp xp + ε

(2)

where β0 , β1 , · · · , βp denote the unknown regression parameters, ε is a normal uncertain variable with expected value 0 and variance σ 2 . In order to estimate the parameters β0 , β1 , · · · , βp in the model. We present the following empirical uncertainty distribution based on the expert’s experimental data. ⎧ if y < yi1 ⎨ 0, (αi,j+1 −αij )(y−yij ) , if yij ≤ y ≤ yi,j+1 , 1 ≤ j < n Φi (y) = αij + (3) yi,j+1 −yij ⎩ 1, y > yin Where Φi (y) is the empirical uncertainty distribution of Yi whose expected values is yˆi =

n i −1 αi,j+1 − αi,j−1 αi1 + αi2 αi,ni −1 + αini yi1 + yij + (1 − )yini 2 2 2 j=2

(4)

Hamiltonian Mechanics

27

We can use the empirical uncertainty distribution instead of theoretical distribution when dealing with practical problems. Each yi can be written as yˆi = β0 + β1 x1 + β2 x2 + · · · + βp xp + ei .

(5)

where ei = yˆi − (β0 + β1 x1 + β2 x2 + · · · + βp xp ) is an approximation of the uncertain error term εi . We can approximate the sum of the squared difference between Yi and the linear deterministic part β0 + β1 x1 + β2 x2 + · · · + βp xp as follow m  (yˆi − β0 + β1 x1 + β2 x2 + · · · + βp xp )2 .

(6)

i=1

By minimizing (6), we get the least squares estimates of β0 , β1 , · · · , βp , denoted by βˆ0 , βˆ1 , · · · , βˆp . Then we write the fitted uncertain linear regression equation as Yˆ = βˆ0 + βˆ1 x1 + βˆ2 x2 + · · · + βˆp xp We introduce the matrix notation for ⎛ ⎛ ⎞ 1 x11 x12 yˆ1 ⎜ 1 x21 x22 ⎜ yˆ2 ⎟ ⎜ ⎜ ⎟ yˆ = ⎜ . ⎟ , X = ⎜ . . .. ⎝ .. .. ⎝ .. ⎠ . yˆm

(7)

convenience. Let ⎛ˆ ⎞ ⎛ ⎞ β0 e1 ... x1p ⎜ βˆ1 ⎟ ⎜ e2 ... x2p ⎟ ⎜ ⎟ ⎜ ⎟ .. .. ⎟ , βˆ = ⎜ .. ⎟ , e = ⎜ .. ⎝ . ⎠ ⎝ . . . ⎠

1 xm1 xm2 ... xmp

βˆp

⎞ ⎟ ⎟ ⎟ ⎠

em

Then the least squares estimators are given by βˆ = (X T X)−1 X T yˆ.

(8)

Where X T is the transpose of X. 2.3

Numerical Method

In this section, we introduce one example to establish classical linear regression equation and uncertain linear regression equation, respectively. Example. A survey of television advertising and newspaper advertising costs on the company’s weekly income as follows. Table 1. Survey data of the weekly income and advertising costs Variables

Survey data

Weekly income

96 90 95 92 95 95 94 94

Costs of advertisement on TV

1.5 2.0 1.5 2.5 3.3 2.3 4.2 2.0

Costs of advertisement in newspapers 5.0 2.0 4.0 2.5 3.0 3.5 2.5 3.0

28

S. Wang and Z. Wang

By fitting a classic simple regression model to the data in Table 1, we get the model. y = 1.2985x1 + 2.3372x2 + 83.2116.

(9)

Where x1 , x2 denote the costs of advertisement on TV and the costs of advertisement in newspapers, respectively, y denotes the company’s weekly income. Now, we design the uncertain linear regression model. Due to lack of survey data, we invite experts who don’t know the true data of the weekly income and advertising costs. By the method introduced in Sect. 2.1, we get experimental data Table 2 by combining a number of expert opinion. Table 2. Survey data of the weekly income and advertising costs Dependent variable

(x1 , x2 )

Y1 : (90, 0.15), (91, 0.3), (92, 0.5), (93, 0.6), (94, 0.8), (95, 0.95)

(1.5, 3.0)

Y2 : (90, 0.2), (91, 0.3), (92, 0.45), (93, 0.7), (94, 0.85), (95, 1.0)

(2.0, 2.5)

Y3 : (93, 0.2), (94, 0.3), (95, 0.45), (96, 0.6), (97, 0.85), (98, 0.95)

(2.5, 3.5)

Y4 : (95, 0.2), (96, 0.35), (97, 0.65), (98, 0.75), (99, 0.8), (100, 0.95)

(4.0, 3.5)

Y5 : (95, 0.25), (96, 0.35), (97, 0.6), (98, 0.75), (99, 0.85), (101, 0.95) (5.0, 3.0) Y6 : (86, 0.15), (87, 0.3), (88, 0.5), (89, 0.6), (90, 0.8), (91, 0.95)

(1.5, 1.5)

By (4), (7) and (8), we use the empirical uncertainty distribution instead of theoretical distribution, and get the following calculation results. ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ 1 1.5 3.0 92.25 yˆ1 ⎜ 1 2.0 2.5 ⎟ ⎜ yˆ2 ⎟ ⎜ 92.1 ⎟ ⎛ˆ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ β0 ⎜ 1 2.5 3.5 ⎟ ⎜ yˆ3 ⎟ ⎜ 95.225 ⎟ ⎟ , βˆ = ⎝ βˆ1 ⎠ ⎟,X = ⎜ ⎟=⎜ yˆ = ⎜ ⎜ 1 4.0 3.5 ⎟ ⎜ yˆ4 ⎟ ⎜ 96.875 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ βˆ2 ⎝ 1 5.0 3.0 ⎠ ⎝ yˆ5 ⎠ ⎝ 96.95 ⎠ 1 1.5 1.5 88.25 yˆ6 That is y = 1.2746x1 + 2.7276x2 + 82.375.

(10)

Where x1 , x2 denotes the costs of advertisement on TV and the costs of advertisement in newspapers, respectively, y denotes the company’s weekly income. By comparing the adjusted R-squared, F-statistic, the log likelihood, Akaike’s information criterion, Schwarz’s criterion, Durbin-Watson’s statistics and correlation probability, respectively. We get the model (10) is superior to the model (9), apparently. Thought the model (10), we can design the maximum of the company’s weekly income by determining the appropriate the costs of advertisement on TV and the costs of advertisement in newspapers.

Hamiltonian Mechanics

3

29

Conclusions

The main contribution of the present paper is to establish uncertainty linear regression model under the uncertainty system, and get the optimal solution of the model parameter estimation. One example shows that uncertain linear regression model is better than simple linear regression model. Acknowledgments. This work was supported by the Hainan Natural Science Foundation (No. 118MS002), the Funding National Natural Science Foundation of China (No. 11601108).

References 1. Liu B (2015) Uncertainty theory, 5th edn. http://orsc.edu.cn/liu/s 2. Wang X, Gao Z, Guo H (2012) Delphi method for estimating uncertainty distributions. Inf: Int Interdiscip J 15(2):449–460 3. Ha M, Wang X, Gao Z Uncertain linear regression model and its application. http:// orsc.edu.cn/online/130331 4. Wang X, Gao Z, Guo H (2012) Uncertain hypothesis testing for two experts empirical data. Math Comput Model 55:1478–1482 5. Wang Z, Tian F (2012) Strong convergence of independent fuzzy variables sequence. Int J Nonlinear Sci 14(4):392–397 6. Wang Z, Fu Y, Cai B (2017) Consistency test for the empirical uncertainty distribution. Fuzzy Syst Math 31(3):175–182 7. Wang Z, Tian F (2015) The distribution of values of the infinite order random Dirichlet series on the right half-plane. Acta Math Sci 35A(2):245–255 8. Wang Z (2018) Stochastic process. University of Science and Technology of China Press, Beijing

Online Optimization of Heating Furnace Temperature Set Value Based on Fast Predictive Genetic Algorithm Zhengguang Xu, Hao Tian, Pengfei Xu(B) , and Shuai Liu University of Science and Technology Beijing, Beijing 100083, People’s Republic of China [email protected] http://www.ustb.edu.cn/

Abstract. The optimization of furnace temperature set value is the basis of furnace optimization control. Existing method of computing furnace temperature set value consists of offline furnace temperature calculation and online dynamic compensation. In this paper, it is proposed a set value model that computed furnace temperature online. This online model regarded the deviation between the desired and the realized final slab temperature profiles as restrictions. The furnace temperature measured by thermocouples was used to compute slabs’ temperature distribution. The slab temperature distribution prediction model was introduced so that each slab can be tracked, and its temperature distribution would be computed online. Then the improved genetic algorithm is applied to online optimization in that the calculation speed of traditional genetic algorithm is slow. According to this algorithm, the best individual in the next generation can be predicted by using the best individual of each generation. The predicted individuals and the individuals generated by crossover and mutation form the next generation together. It is shown by simulation that, compared with conventional genetic algorithm and neural network prediction genetic algorithm, the convergence rate of fast predicted genetic algorithm show 17 times higher than those of them. The simulation results also indicate that the proposed method of computing furnace temperature presented better performance in terms of slabs discharging temperature. Keywords: Heating furnace · Genetic algorithm · Prediction Temperature optimization · Slab temperature distribution

1

·

Introduction

Heating furnace is the main energy-consuming equipment in iron and steel industry, accounting for about 50% of the whole production line energy consumption [1]. Heating furnace is a complicated system with large inertia, pure lag, nonlinear, time-varying and distributive parameters [2]. Its main function is to heat c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 30–38, 2020. https://doi.org/10.1007/978-981-32-9682-4_4

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slabs until their temperature meets the rolling requirements. Therefore, the temperature control target is not only to heat slabs with qualified discharging temperature but also to consume energy as less as possible. Because of all elements mentioned above, designing the optimal heating furnace temperature control system is a practical method to reduce energy consumption. The traditional method usually optimizes furnace temperature offline with different production rhythms, then compensates the set value online according to practical conditions. Yang and Wu et al. [3,4] proposed an online dynamic optimization control strategy based on the target steel temperature of each zone, and established an online optimization control system with steel temperature negative feedback in industrial field. Nevertheless, it is still based on offline calculation of the ideal slab temperature and the corresponding furnace temperature. In the production process, the furnace conditions are very complex, and the rolling rhythm will change with the production requirements. So, it is difficult for offline furnace temperature optimization to contain each rolling rhythm or adapt to changes in the furnace conditions quickly. To solve this problem, furnace temperature set value optimization online is proposed in this paper. Genetic algorithm has the characteristics of straightforward process and strong robustness, so it was applied to solve this optimization problem. However, the convergence rate of traditional genetic algorithm is slow, and it is easy to fall into “precocious”. As for these shortcomings, Schmidt and Lipson [5] have done a lot of research on evolutionary algorithms, and proposed that the primary mechanism by which fitness approximations improve performance is by providing accurate rankings of individuals, rather that accurate fitness values as originally intended. This method can find a more accurate solution than the method of sorting fitness when solving complex optimization problems. The strategy of setting furnace temperature online is composed of slab temperature prediction model and online optimization of furnace temperature. Firstly, the temperature distribution of all slabs in the furnace at the current moment is computed in real time and predicts their discharging temperature distribution which is based on current furnace temperature. Then the genetic algorithm is used to optimize the furnace temperature in a fixed cycle. Aiming at the existing problems of traditional genetic algorithm, a fast-predictive genetic algorithm is put forward to meet the requirements of online optimization for computational speed.

2

Slab Temperature Distribution Prediction Model

In the part of furnace temperature optimization, the slab temperature distribution will be used to determine what kind of reheating schedule can meet the slab temperature requirements. For this purpose, it is necessary to know slabs temperature distribution. However, because of high furnace temperature and complex conditions, the slab temperature cannot be measured in the real plant. As mechanism method is widely used to compute slab temperature distribution [6], the slab temperature prediction distribution model is established here

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by this method. The so-called mechanism modeling is that establishing thermal differential equation based on heat transfer theory, then using finite difference to discrete differential equation. A series of difference equations for estimating temperature distribution of grid nodes can be obtained. 2.1

Model Establishment

The slab temperature distribution prediction model consists of differential thermal conductivity and boundary conditions. Because of the assumption mentioned above, it is assumed that the slab temperature along the furnace width direction is same. Every slab should be tracked until it is discharged. The temperature in the slab width direction, namely the furnace length direction, can also be assumed to be the same. Therefore, for each slab, one-dimensional heat conduction model along the thickness of the slab can be considered only. The slab thickness direction is x, and the temperature filed is governed by the following the internal differential Eq. [7]: ∂ 2 T (x, t) dT (x, t) =λ 0 ≤ t ≤ tf 0 ≤ x ≤ d (1) dt ∂x2 Where, ρ is the slab’s density, kg/m3 , c is the slab specific heat capacity, J/(kgK) λ is the slab thermal conductivity coefficient, W/(mK), t is time variable, tf is the total time when slabs stay in furnace, d is the thickness of slabs. The initial condition of the model is the charging temperature of slabs. The heat flux density of the slab’s surface forms the second boundary condition of the model. ρc

2.2

The Discretization of the Model

For discretizing the model, a generally uniform grid along the slab thickness direction is introduced, whose size is 5 mm. The model is discretized by means of finite difference method [8]. The initial condition for our model can be considered as t=0

T = T0

(2)

The surface temperature and the internal temperature of the slab can be determined by following difference equation: Δt λΔt q+ [T (2, k) − T (1, k)] ρcΔz ρcΔz 2

(3)

Δtλ T (p − 1, k) − 2T (p, k) + T (p + 1, k) ρc Δz 2

(4)

T (1, k + 1) = T (1, k) +

T (p, k + 1) = T (p, k) +

Where, T (p, k + 1) is the temperature at the moment (k + 1) and the number of layer is p in the thickness direction of the slab. T (1, k + 1) is the surface temperature of the slab at the moment (k + 1), Δt is a discrete time step. According to (2)–(4), the temperature distribution of each slab in the furnace can be obtained.

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3 3.1

33

Optimizing Furnace Temperature Online Objective Function and Constrains

The amount of energy consumption and slab heating quality are determined by whether the temperature set value is reasonable so that finding a reasonable temperature set value curve is crucial [9]. The objective function of this paper is mainly to consider two aspects of production targets, the minimum loss of energy [10] and process indicators. The objective function is showed as follows: J=

N   i=1

L

Tsi (l)dl +

0

N 

|Tsi (L) − Tci (L)| +

i=1

s.t

N 

|Tsi (L) − T ∗ |

(5)

i=1

|Tsi (L) − Tci (L)| ≤ f1 (l)

(6)

|Tsi (L) − T ∗ | ≤ f2 (l)

(7)

Tg min (lj ) ≤ Tg (lj ) ≤ Tg max (lj )

j = 1, 2, 3

(8)

Where, Tsi (l) is the surface temperature of slabs in furnace, Tsi (L) and Tci (L) are the discharging surface temperature and the discharging central temperature of the slab, the subscript i means the label of slabs. T ∗ is the slab’s target temperature; f1 (l) represents the maximum value of the difference between the slab discharging surface temperature and center temperature, f2 (l) is the maximum value of the difference between slab discharging surface temperature and its target temperature, both f2 (l) and f2 (l) are function of l; Tg (lj ) represents the furnace temperature; Tg min (lj ) and Tg max (lj ) are respectively represents the maximum and minimum furnace temperature; Subscript j means different furnace zones. In the objective function, the first term is the sum of the individual slabs in the furnace along the length direction of the furnace. The second term is the sum of all slabs’ difference between discharging surface temperature and discharging central temperature. The third term is the sum of all slabs’ difference between discharging surface temperature and its target temperature. The first term reflects energy consumption, and the latter two terms reflect the process indicators on the slab heating quality. Genetic algorithm was used to solve this problem. 3.2

Fast Predictive Genetic Algorithm (FPGA)

In order to search this best furnace temperature curve quickly, the historical information in the optimization process is fully utilized, which means all best

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individuals in different generations will be saved. Moreover, the optimal value generated by each generation is used to predict the optimal value of the next generation. Every individual in the offspring consists of three parameters, each of which has the potential to increase and decrease over the next generation. Therefore, there is a possibility of eight combinations of the three temperature values, and eight individuals can be predicted. Then the eight individuals can be put into the next generation and form the next generation together with the individuals generated by cross and mutation. The predicted values of the three furnace temperature points are as follows: ˆ2 (α2 , Δ2 ), x ˆ3 (α3 , Δ3 )] x ˆ(α, Δ) = [ˆ x1 (α1 , Δ1 ), x

x ˆi (αi , Δi ) = xi + (−1)α i · Δi · ri

T

i = 1, 2, 3

(9)

(10)

Where, αi ∈ {0, 1}, ri is a random number, and ri ∈ [0, 1], Δi is the difference between the parameters represented by the best individual in current generation and in previous generation, xi is the parameters represented by the best individual in current generation, x ˆ(α, Δ) is a vector of predicted temperature of preheating zone, heating zone and soaking zone. When variable i takes different values, 8 predicted individuals are obtained from formula (9) and (10), and the 8 individuals are used to replace the same number of individuals in the next generation. The specific steps of this algorithm for optimizing the furnace temperature by using the fast-predictive genetic algorithm (FPGA) flow chart is shown in Fig. 1. 3.3

Furnace Temperature Online Setting Strategy

In the second part of this paper, the slab temperature prediction model is established. The third part introduces the objectives and constraints of optimization and the Fast-Predictive Genetic Algorithm (FPGA) in detail. Furnace temperature online setting strategy consists of the slab temperature prediction model and the furnace temperature optimization. The slab temperature prediction model computes all slabs temperature distribution in furnace and forecasts the slab discharging temperature at each moment. These two parts are computed in parallel and the latest calculation results of the slab temperature distribution prediction model are used to optimize the furnace temperature. Once production rhythm changes, the slab movement speed changes correspondingly. Then the slab temperature prediction model and furnace temperature optimization are based on the latest slab velocity. The slab velocity and position in the furnace depends on the speed of the walking beam:  t v(τ )dτ 0 ≤ t ≤ tf (11) Sj (t) = 0

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35

Where, Sj is the position of the slab in the furnace at the t moment, the subscript j represents different slabs; v(t) is the average speed of a slab in the furnace; is the distance of the walking-beam from the entrance; For tracking and computing all slabs temperature distribution, a tracking algorithm is designed. The specific algorithm steps are as follows: The initial temperature of each slab is T = T0 . The flow of slabs temperature tracking algorithm flow chart is shown in Figs. 2 and 3 show the Furnace temperature online setting strategy. Start

Start

Start Determining parameters

Initialization Initialization

Initialization of population

Termination condition satisfied

Y

Output the Optimal solution

Fitting the actual furnace temperature

t

N

Predicting all slabs discharging temperature at current furnace temperature

Calculating the heat flux density of all slabs surface

selecting

Crossing

whether the slab reaches the discharging time

Mutating

Start prediction

Y

prediction and record

N

Fig. 1. fast-predictive genetic algorithm (FPGA)

4

t 1

Calculating the location of all billets

Calculation fitness

N

N

t

nt1 Y

Calculating all slabs temperature

Algorithm 2

t

nt 2

N

Y

Algorithm 1

Y Recording the discharging temperature; updating slabs location

Fig. 2. Flow of slabs temperature tracking algorithm

Fig. 3. Flow chart of algorithm 3, where n is a natural number

Simulation Experiment and Analysis

A vector composed of furnace temperature is used to represent a furnace temperature combination. The ranges of them are 900 ≤ x1 ≤ 1100, 1250 ≤ x2 ≤ 1350 and 1230 ≤ x3 ≤ 1330 respectively. Considering the range of parameters and accuracy, the code method is 8-bit binary. The algorithm can be terminated after 500 generations, then the best individual in the latest population is selected as the final solution. In our experiment, the conventional genetic algorithm, neural network prediction genetic algorithm and the fast prediction genetic algorithm proposed in this paper were simulated. The target temperature of slab was 1100 ◦ C. Three algorithm results are showed in Figs. 4, 5, and 6 respectively. The conventional genetic algorithm converges slowly and easily falls into “precocious”. As shown in Fig. 4, when the population evolves to near 20 and 160 generations, the objective function value does not change for a long time. When the evolutionary algebra is near 300 generations, a new optimal individual is generated to reduce the objective function value.

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Figure 5 shows the result of neural network-genetic algorithm and the optimal solution emerged near 190 generations, after that the optimal solution did not change until the program was executed to 320 generations. Moreover, there is a little frustration at the end of the program. Therefore, this method’s performance has been slightly improved, but it is not satisfying. Meanwhile the neural network has a complex internal structure hence the prediction process also needs time. Overall, this approach still cannot meet the needs of the actual production.

Fig. 4. Basic genetic algorithm optimization results

Fig. 5. Neural network predictive GA simulation results

Fig. 6. Fast predictive genetic algorithm (FPGA)

Fig. 7. All Slabs discharging surface temperature Simulation Results

The fast-predictive genetic algorithm (FPGA) can find the global optimal value within 20 generations, as shown in Fig. 6, and its computational rate is 17 times as fast as that of the conventional genetic algorithm and the neural network - genetic algorithm. Additionally, compared with the neural network prediction, the prediction method proposed in this paper is simple. In the practical field, the furnace temperature set value can be obtained in 20 min by setting reasonable termination condition, which satisfies the real-time requirement of industrial field.

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Figure 7 shows the discharging surface temperatures of 50 consecutive slabs. Each slab temperature is around the target value (1100 ◦ C) without exceeding the error range of the process requirements (±20 ◦ C). The difference between discharging surface temperature of slab and its target value is small, which not only indicated stable operation of the furnace, but the temperature set value did not occur great fluctuations as well. Meanwhile, it can be seen from the experimental results listed in Table 1 that the three zones’ temperature and objective function values obtained by the proposed algorithm were lower than other two algorithms. The global optimal value appears early in the algorithm calculation, and did not fall into “precocious” in the whole calculation process. Table 1. Comparison of optimization results

5

of three parameters/◦ C

Generation that Objective Best value generates the global function x (Pre1 optimal value value heating zone)

x2 (Heating zone)

x3 (Soaking zone)

Conventional genetic algorithm (GA)

310

773.2193

900.8

1349.6

1326.1

Neural network-genetic algorithm

320

773.0650

900.6

1350.0

1324.1

Fast predictive genetic algorithm (FPGA)

18

772.3645

901.6

1350.0

1326.1

Conclusions

The strategy of setting the furnace temperature online was proposed. The slab temperature distribution prediction model was established by means of mechanistic analysis to predict the temperature distribution of each slab in real time. When the furnace temperature is optimizing, the integral value of all slabs’ temperature curve along the furnace length was used to represent the energy consumption. As for optimization algorithm, the traditional genetic algorithm (GA) has the problem of slow convergence and easy to fall “premature”, which cannot meet the requirements of online optimization. For solving this problem, a fast-predictive genetic algorithm (FPGA) was presented. This method used the historical information of the best individual in each population to predict the best possible individuals in the next generation. The predicted individuals are put into the next generation, then form a new generation with the individuals generated by the cross and mutation together. The simulation shows that the gene quality of the population is improved, and the convergence rate is

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greatly increased. Consequently, the calculation time can meet the requirements of online production. Meanwhile, the optimized temperature set value ensured the heating quality of slab. Finally, the proposed algorithm is compared with the traditional genetic algorithm and neural network-genetic algorithm. It turns out that the proposed genetic algorithm is superior to the other two algorithms in terms of convergence speed and accuracy. The slab discharging temperature meets the requirements of rolling with less fluctuation during continuous production, which proves the effectiveness of the on-line temperature setting strategy proposed in this paper. At present, the simulation in laboratory has been completed. As for applying this method to the practical plant, there are other elements we need to consider such as slab oxidation loss and revising the precision of slab temperature distribution prediction model. The coefficients of slab model in this paper was not revised and it must adapt to the furnace we research. Henceforth, the next step of our work is to find a method to identify coefficients of slab model, and develop application program via C++.

References 1. Liu SZ, Zhang JT (2016) Application status and trends of steel scrap in China. Iron Steel 51(6):1–9 2. Chen YW, Chai TY (2007) Research and application on temperature control of industry heating furnace. J Iron Steel Res 22(9):53–57 3. Wu XF (2007) Control and optimization of walking-beam researching furnace. Shanghai Jiaotong University, Shanghai 4. Yang YJ, Jiang ZY, Zhang XX (2012) On-line dynamic optimization control of heating furnace based on target billet temperature. Metall. Ind Autom 36(1):19– 24 5. Schmidt MD, Lipson H (2008) Coevolution of Fitness Predictors. IEEE Trans Evol Comput 12(6):736–749 6. Hsieh C-T, Huang M-J (2008) Numerical modeling of a walking-beam-type slab reheating furnace. Numer Heat Transf 53:966–981 7. He L, Chen K, Ke HL, Zhang W, Peng YH (2014) Two dimension temperature model for billet reheating furnace. Chin J Iron Steel 26(10):21–25 8. Ghader S, Nordstr¨ om J (2015) High-order compact finite difference schemes for the vorticity-divergence representation of the spherical shallow water equation. Int J Numer Methods Fluids 78:709–738 9. Ke HL, Dong B, Ye B (2014) Research and application of slab heating curve in reheating furnace. Metall Ind Autom 28(3):50–55 10. Guo T, Chu M, Liu Z (2014) Numerical simulation on blast furnace operation with hot burden charging. Chin J Iron Steel 21(8):729–736

Denoising of Carbon Steel Corrosion Monitoring Signal Based on NLM-VMD with MFDFA Technique Zhuolin Li1 , Dongmei Fu1(B) , Ying Li1 , Zibo Pei2 , Qiong Yao3 , and Jinbin Zhao4 1

2

School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China fdm [email protected] Corrosion and Protection Center, University of Science and Technology Beijing, Beijing 100083, China 3 Key Laboratory of Space Launching Site Reliability Technology, Xichang Satellite Launch Center, Haikou 571126, China 4 Research Institute, Nanjing Iron and Steel Co., Ltd., Nanjing 210035, China

Abstract. In order to recover effective information of steel atmospheric corrosion from corrosion monitoring signal, an adaptive denoising algorithm based on NLM-VMD with MFDFA technique has been processed in this paper. To eliminate the influence of outliers in the signal, NLM was used for signal preprocessing. The preprocessed signal was decomposed by VMD into several IMFs and then every IMF was analyzed by MFDFA to quantify the noise containing condition. Reconstructing the IMFs which contained none noise could obtain removed noise signal. To evaluate the performance of proposed algorithm, an experiment about artificial analog signal was carried out. The experiment result was compared with the other denoising algorithms based on mode decomposition and the proposed algorithm performed better than others. Using the proposed algorithm in corrosion monitoring signal, the quasi-periodic pulses were excellent preserved. Furthermore, this method provides the basis for research the characteristics of corrosion monitoring signal. Keywords: Corrosion monitoring signal · Variational mode decomposition · Multifractal detrended fluctuation analysis

1 Introduction Real-time and on-line corrosion monitoring has great significance [1]. Acquired corrosion monitoring signal can be used to obtain important corrosion information [2]. Electrical resistance probe technology is an important method for real-time monitoring of changes in corrosion conditions of metallic materials and it has the advantages of mature technology, strong practicability and low cost [3]. However, in the monitoring progress, the electrical resistance probe monitoring signal (hereinafter, corrosion monitoring signal) is susceptible to the fluctuations in electronic circuits and power supplies. It could cause corrosion information that is emerged in the noise and could c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 39–47, 2020. https://doi.org/10.1007/978-981-32-9682-4_5

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change the real corrosion characteristics. Therefore, corrosion monitoring signal needs to be denoised. The denoising method based on Fourier transform has a good effect on smooth signal processing, but it has limited signal processing capability for nonlinear and non-stationary [4]. The wavelet transform algorithm has a small amount of computation, fast speed and high retention of real signals, but its denoising effect depends on the selection of the threshold function and the traditional threshold function is easy to cause excessive denoised results or signal distortion [5]. The denoising method based on mode decomposition has adaptability and advantage in dealing with non-stationary and nonlinear signals compared with short-time Fourier transform and wavelet transform. Empirical mode decomposition (EMD), ensemble empirical mode decomposition (EEMD), complementary ensemble empirical mode decomposition (CEEMD) and modified ensemble empirical mode decomposition (MEEMD) have been found to be suitable for analysing nonstationary signals, but these methods may have a less mathematical theory, have modal aliasing and endpoint effects, introduce new noise interference and consume a lot of calculating time [6–9]. In response to the shortcomings of the EMD method, Dragomiretskiy et al. [10] has proposed a variational mode decomposition (VMD) method. This method has a rigorous mathematical derivation and mathematical foundation, which can effectively overcome the problems of EMD. Considering the features of corrosion monitoring signal and the disadvantages of each denoising method, an adaptive signal denoising algorithm based on VMD is proposed. This algorithm provides an effective method for recovering real signal from acquired corrosion monitoring signal.

2 Materials and Method 2.1

Variational Mode Decomposition (VMD)

VMD is a modern signal decompose technique. The progress of VMD [10, 11] is a special iteration solving variational mode and can non-cursively decompose a multicomponent signal into series of intrinsic mode functions (IMFs) which have limited bandwidth. In order to establish a variational model of signal decomposition, the VMD defines the IMF as the amplitude modulation-frequency modulation (AM-FM) and calculation principle are expressed as follows uk = Ak (t)cos(ϕk (t)),

(1)

ωk = dϕk (t)/dt ≥ 0,

(2)

K

j

∑ ||∂t [(δ (t) + π t ) ∗ uk (t)e− jωkt ]||22 , {uk },{ωk } min

(3)

k=1

K

∑ uk (t) = f (t),

(4)

k=1

where AK (t) is the instantaneous amplitude of uk (t) and ϕk (t) is the instantaneous phase of uk (t), ϕk (t) is the reduction function and ωk (t) is the instantaneous frequency. On the basis of this definition, VMD considers that the input signal f (t) is composed of IMF

Carbon Steel Corrosion Monitoring Signal Denoising

41

components with different center frequencies and limited bandwidths and the minimum sum of the estimated bandwidth of each IMF is almost equal to the input signal f (t). K is the number of IMF component, {uk } = {u1 , u2 , ..., uK } are IMF components and {ωk } = {ω1 , ω2 , ..., ωK } are center frequencies of uk (t). δ (t) is Dirichlet function, ∗ symbol is convolution symbol. 2.2 Non-local Mean (NLM) The processing result of VMD will be not ideal for denoising low signal to noise ratio (SNR) signal and the outliers could have a bad effect on the decomposition result. Therefore, it is important to remove the background noise and outlier interference in signal using non-local mean (NLM) method [13]. Supposing u(t) is the real corrosion monitoring signal, n(t) is the background noise interference and f (t) is the actual acquired corrosion monitoring signal. The signal model can be expressed as f (t) = u(t) + n(t). If N is the search area centered on sample point i, then the set of all points in this search area is N(i). The weighting operation is obtained after weighting all points in the search area and the weighted mean K(i) is calculated as: K(i) =

1 Z(i)

∑

w(i, j)x(i),

(5)

j∈N(i)

where Z(i) is the normalization factor Z(i) =

∑

w(i, j),

(6)

j∈N(i)

the weight w(i, j) can be obtained by: w(i, j) = e

− ∑δ ∈Δ ( f (i+δ )− f ( j+δ ))2 h2

,

(7)

the parameter h is the bandwidth parameter and determines the attenuation speed of the weight w(i, j) which could control the degree of signal filtering. 2.3 Multifractal Detrended Fluctuation Analysis (MFDFA) MFDFA can analyze the noise content of IMF by using the Hurst exponent [14, 15]. Consider one IMF(i), where (i = 1, 2, ..., N). N is the length of IMF. Construct the profile IMF(i) IMF(i) =

i

∑ (IMFk − < IMF >)

i = 1, 2, ..., N,

(8)

k=1

where < IMF >= N1 ∑Nk=1 IMFk . Cut the profile IMF(i) into Ns = [N/s] nonoverlapping segments of equal length s. Consider that the signal length N need not be a multiple of the time scale s, a short part at the end of the profile will remain in most cases. In order to retain the part of the signal, the same procedure is repeated starting from the other

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end of the signal. Thus 2Ns segments are obtained together. The detrending variance of series is calculated by the substraction of the polynomial fits from the profile. ⎧ 2 f (v, s) = 1s ∑si=1 {IMF[(v − 1)s + i] − IMFv (i)}2 , ⎪ ⎪ ⎨ v = 1, 2, ..., Ns , (9) f 2 (v, s) = 1s ∑si=1 {IMF[N − (v − Ns )s + i] − IMFv (i)}2 , ⎪ ⎪ ⎩ v = Ns + 1, Ns + 2, ..., 2Ns average all segments to obtain the q order fluctuation function  F(q, s) = { 2N1 s ∑2v=1 Ns [ f 2 (v, s)]q/2 }1/q , q = 0, 1

2Ns

F(q, s) = e{ 4Ns ∑v=1 In f

2 (v,s)

}, q = 0,

(10)

the scaling exponent h(q) is called generalized Hurst exponent. Where H = h(q = 2), H is called Hurst exponent, if H < 0.5, it indicates short-range correlations of IMF and contains more noise; if 0.5 ≤ H < 1, it indicates long-range correlations of IMF and contains no noise [16]. 2.4

Denoising Effect Evaluation Index

The SNR and root mean square error (RMSE) of the output signal are used as the index evaluation algorithm of denoising performance [17]: SNR = 10lg  RMSE =

N u2 (t) ∑t=1 , N [u(t) − fˆ(t)]2 ∑t=1

(11)

1 N ∑ [u(t) − fˆ(t)]2 , N t=1

(12)

where u(t) is the real signal, fˆ(t) is the denoising signal and N is the signal length

3 Experiment and Result 3.1

Artificial Analog Signal Simulation

The artificial analog signal s(t) (N = 2048) is composed of a periodic vibration signal p(t) with a frequency of 5 Hz and 20 Hz and a long-term trend d(t), which has a certain similarity with the corrosion monitoring signal: s(t) = d(t) + p(t) = 15t 2 + 2sin(10π t) + 4cos(40π t),t ∈ [0, 1),

(13)

adding white noise to s(t) respectively generates a noisy signal f (t) with an input SNR of −5, 0, 5 dB. The artificial analog signals are processed in the article using NLM, EMD, NLM-EMD, EEMD, NLM-EEMD, CEEMD, NLM-CEEMD, MEEMD, NLMMEEMD, VMD and NLM-VMD, respectively. When SNR = 5 dB, f (t) is denoised after determining the optimal IMF number for each above-mentioned algorithm. The

Carbon Steel Corrosion Monitoring Signal Denoising

43

Table 1. Artificial analog signal denoising result −5 dB 0dB SNR RMSE SNR NLM-MFDFA

17.990 0.669

5 dB RMSE SNR

23.094 0.372

RMSE

27.375 0.227

EMD-MFDFA

19.069 0.589

21.931 0.425

15.885 0.852

NLM-EMD-MFDFA

17.665 0.694

25.777 0.273

30.063 0.167

EEMD-MFDFA

22.024 0.420

26.624 0.247

27.592 0.221

NLM-EEMD-MFDFA

23.383 0.359

27.118 0.234

29.588 0.176

CEEMD-MFDFA

22.587 0.394

27.010 0.237

32.607 0.124

NLM-CEEMD-MFDFA 22.764 0.386

26.869 0.241

32.217 0.130

MEEMD-MFDFA

21.985 0.422

24.676 0.309

28.875 0.191

NLM-MEEMD-MFDFA 20.983 0.474

25.760 0.273

30.801 0.153

VMD-MFDFA

22.074 0.418

25.935 0.267

30.261 0.163

NLM-VMD-MFDFA

24.664 0.310

28.593 0.197

33.347 0.114

decomposition result of the NLM-VMD of f (t) is shown in Fig. 1. Calculating the Hurst exponent value of each IMF, it can be found that the denoised signal can be obtained by using IMF11 and IMF12. The denoising effect fˆ(t) of the artificial analog signal through the NLM-VMD-MFDFA algorithm is shown in Fig. 2. The processing result of the algorithm is closer to the original signal and has better visual effect. The denoising effect of the artificial analog signal is evaluated by SNR and RMSE, which are shown in Table 1. Compared with other algorithms, the denoising effect of NLM-VMD-MFDFA has greater the value of SNR and smaller the value of RMSE, which indicates that the signal processed by this algorithm accurately moves out more noise components. Based on the above analysis, it can be further explained that the NLM-VMD-MFDFA algorithm has better denoising effect. 3.2 Corrosion Monitoring Signal Simulation The corrosion monitoring signal is processed using the NLM-VMD-MFDFA algorithm. Exposing the electrical resistant probe to a real atmospheric environment for an outdoor test for 3 months (the fourth quarter of 2017) in Qingdao, China. The ratio of the total resistance of sensor and the resistance of the corrosion part of sensor is collected every hour and the sensor is powered by a constant current source. The actual corrosion monitoring signal and its details are shown in Fig. 3. The corrosion monitoring signal is denoised by the NLM-VMD-MFDFA algorithm. Using NLM-VMD, the corrosion monitoring signal is decomposed into 10 IMFs, as shown in Fig. 4. The Hurst exponent of each IMF is then calculated and the denoising results of the corrosion monitoring signal can be obtained by constructing IMF7, IMF8,

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Fig. 1. Artificial analog signal VMD decomposition result

Fig. 2. Artificial analog signal denoising

IMF9 and IMF10. The denoised signal and its details are shown in Fig. 5. By analyzing the denoising results, it can be conclude that the period of quasi-periodic fluctuations in the corrosion monitoring data is almost closed to 24 h, which may be affected by some periodically environmental variables.

Carbon Steel Corrosion Monitoring Signal Denoising

Fig. 3. Corrosion monitoring signal (resistance ratio) and details

Fig. 4. VMD decomposition result

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Fig. 5. Denoised corrosion monitoring signal (resistance ratio) results and details

4 Conclusions For the corrosion monitoring signal with quasi-periodic fluctuation noise and long-term trend, the NLM-VMD-MFDFA algorithm can remove the noise well and preserve the real signal better. Compared with other methods of the similiar principle, the denoising effect of NLM-VMD-MFDFA is better. Acknowledgments. This work was supported by National Key Research and Development Program of China (Grant No. 2017YFB0702104) and Other Projects of the Ministry of Science and Technology of China (Grant No. 2012FY113000).

References 1. Li Z, Fu D, Li Y, Wang G, Meng J, Zhang D, Yang Z, Ding G, Zhao J (2019) Application of an electrical resistance sensor-based automated corrosion monitor in the study of atmospheric corrosion. Materials 12:1065. https://doi.org/10.3390/ma12071065 2. Shi Y, Fu D, Zhou X, Yang T, Zhi Y, Pei Z, Zhang D, Shao L (2018) Data mining to online galvanic current of zinc/copper Internet atmospheric corrosion monitor. Corros Sci 133:443– 450. https://doi.org/10.1016/j.corsci.2018.02.005 3. Legat A (2007) Monitoring of steel corrosion in concrete by electrode arrays and electrical resistance probes. Electrochim Acta 52:7590–7598. https://doi.org/10.1016/j.electacta.2007. 06.060 4. Offelli C, Petri D (2002) Weighting effect on the discrete time Fourier transform of noisy signals. IEEE Trans Instrum Meas 40:972–981. https://doi.org/10.1109/19.119777 5. Chang G, Yu B, Vetterli M (2000) Adaptive wavelet thresholding for image denoising and compression. IEEE Trans Image Process 9:1532–1546. https://doi.org/10.1109/83.862633

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6. Blanco-Velasco M, Weng B, Barner K (2008) ECG signal denoising and baseline wander correction based on the empirical mode decomposition. Comput Biol Med 38:1–13. https:// doi.org/10.1016/j.compbiomed.2007.06.003 7. Chang K, Liu S (2011) Gaussian noise filtering from ECG by Wiener filter and ensemble empirical mode decomposition. J Signal Process Sys 64:249–264. https://doi.org/10.1007/ s11265-009-0447-z 8. Li C, Zhan L, Shen L (2015) Friction signal denoising using complete ensemble EMD with adaptive noise and mutual information. Entropy 17:5965–5979. https://doi.org/10.3390/ e17095965 9. Yu Y, Li W, Sheng D, Chen J (2015) A novel sensor fault diagnosis method based on modified ensemble empirical mode decomposition and probabilistic neural network. Measurement 68:328–336. https://doi.org/10.1016/j.measurement.2015.03.003 10. Dragomiretskiy K, Zosso D (2014) Variational mode decomposition. IEEE Trans Signal Process 62:531–544. https://doi.org/10.1109/TSP.2013.2288675 11. Liu Y, Wang J, Li Y, Zhao H, Chen S (2017) Feature extraction method based on VMD and MFDFA for fault diagnosis of reciprocating compressor valve. J Vibroeng 19:6007–6020. https://doi.org/10.21595/jve.2017.18726 12. Viswanath A, Jose K, Krishnan N, Kumar S, Soman K (2015) Spike detection of disturbed power signal using VMD. Procedia Comput Sci 46:1087–1094. https://doi.org/10.1016/j. procs.2015.01.021 13. Tracy B, Miller E (2012) Nonlocal means denoising of ECG signals. IEEE Trans Bio-Med Eng 59:2383–2386. https://doi.org/10.1109/TBME.2012.2208964 14. Miriyala S, Koppireddi P, Chanamallu S (2015) Robust detection of ionospheric scintillations using MF-DFA technique. Earth, Planets Space 67:98. https://doi.org/10.1186/s40623-0150268-1 15. Liu Y, Yang G, Li M, Yin H (2016) Variational mode decomposition denoising combined the detrended fluctuation analysis. Signal Process 125:349–364. https://doi.org/10.1016/j.sigpro. 2016.02.011 16. Thlen E (2012) Introduction to multifractal detrended fluctuation analysis in matlab. Front Physiol 3:141. https://doi.org/10.3389/fphys.2012.00141 17. Xiong P, Wang H, Liu M, Zhou S, Hou Z, Liu X (2016) ECG signal enhancement based on improved denoising auto-encoder. Eng Appl Artif Intell 52:194–202. https://doi.org/10. 1016/j.engappai.2016.02.015

High Angle of Attack Sliding Mode Control for Aircraft with Thrust Vector Based on ESO Junjie Liu1,2 , Zengqiang Chen1,2(B) , Mingwei Sun1,2 , and Qinglin Sun1,2 1

College of Artificial Intelligence, Nankai University, Tianjin 300350, China [email protected] 2 Tianjin Key Laboratory of Intelligent Robotics, Nankai University, Tianjin 300350, China

Abstract. This paper proposes a decoupling control strategy for aircraft with thrust vector at high angle of attack. To eliminate strong couplings among different channels, the three-channels controllers are designed independently, respectively. The channels coupling, aerodynamic uncertainties, unmodeled dynamics, and external disturbance are regarded as generalized disturbance and extended into a new state, which is estimated and compensated in real time by extended state observer (ESO). The sliding mode control (SMC) method is employed to achieve the expected control performance. The numerical simulations demonstrate the effectiveness of the proposed control strategy. Keywords: High angle of attack · Thrust vector · Sliding mode control (SMC) · Decoupling control · Extended state observer (ESO)

1

Introduction

With the development of fighter aircrafts, maneuverability and agility become more important. The maneuverability can increase attack opportunities and survival chances, which has become a prominent feature of the new generation fighter aircraft [1,2]. However, at the state of high angle of attack, fighter aircrafts become highly nonlinear and unstable. In addition, it also produce strong coupling and uncertain dynamics [3,4]. Such control difficulties give challenges to flight controller design. Many control methods have been used in high angle of attack, such as the traditional feedback linearization method nonlinear dynamic inversion (NDI) [5,6], sliding mode control (SMC) method [7], fixed H∞ control [8] and so on. However, most of these methods rely on the model, resulting in sensitivity to model uncertainties. Besides, in order to eliminate the coupling among different channels, the traditional control strategy usually needs to design special dynamic decoupling controller based on multivariable control philosophy. These disadvantages are not conducive to practical flight. Active disturbance rejection control (ADRC) is a novel control method, which is not model based c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 48–57, 2020. https://doi.org/10.1007/978-981-32-9682-4_6

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[9]. To simplify parameters adjustment, ADRC is simplified into linear ADRC (LADRC) [10]. Sliding mode control (SMC) is a special kind of nonlinear control and it can eliminate system uncertainties. In order to realize high angle of attack decoupling control, a novel decoupling scheme based on sliding mode control and linear active disturbance rejection control is presented.

2

Model Description

In this paper, We choose a benchmark nonlinear mathematical model [11]. The states at high angle of attack (AOA) may have singularity, thus a mathematical model based on the track coordinate system is derived. In addition, the thrust vector technology needs to be adopted to overcome the poor control effectiveness of aerodynamic actuators. Therefore, a nonlinear mathematical model on the track coordinate system is presented first. 1 V˙ = [Y sinβ − mgsinγ − D + Fx cosαcosβ + Fy sinβ + Fz cosαsinβ] m

(1)

1 [mgcosμcosγ − L + Fz cosα − Fx sinα] mV cosβ (2) 1 1 ˙ [mgsinμcosγ + Y cosβ] − [Fx cosαsinβ β = psinα − rcosα + mV mV (3) − Fy cosβ + Fz sinαsinβ]

α˙ = q − [rsinα + pcosα]tanβ +

1 Fy Fx [−Y cosβsinμ + Lcosμ] − cosβ + [cosαsinβsinμ mV mV mV Fz + sinαcosμ] + [sinαsinμ − cosαcosμ] sinα Fx Fy 1 [Y cosβcosμ + Lsinμ] + [sinαsinμ + χ˙ = mV cosγ mV cosγ mV cosγ Fz [sinαsinβcosμ + cosαsinμ] cosβcosμ − cosαsinβcosμ] − mV cosγ γ˙ =

(4)

(5)

L 1 1 [pcosα + rsinα] + [sinμtanγ + tanβ] + [(Y + Fy )tanγ cosβ mV mV 1 [(Fx sinα − Fz cosα)(tanβ + tanγ (6) cosβcosμ − mgtanβcosγcosμ] + mV 1 [sinβtanγcosμ(Fx cosα + Fz sinα)] sinμ)] − mV

μ˙ =

p˙ =

2 (lT + la )Iz + (na + nT )Ixz + (Ix − Iy + Iz )Ixz pq + (Iz (Iy − Iz ) − Ixz )qr 2 Ix Iz − Ixz (7) ma + mT − (Ix − Iz )pr − (p2 − r2 )Ixz (8) q˙ = Iy

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r˙ =

Ixz la + Ixz lT + Ix na + Ix nT Ix2 − Ix Iy + Iz2 + pq 2 2 Ix Iz − Ixz Ix Iz − Ixz Ix Ixz − Iy Ixz + Iz Ixz qr − 2 Ix Iz − Ixz x˙ E = V cos(γ)cos(χ)

(9)

(10)

y˙ E = V cos(γ)sin(χ)

(11)

z˙E = − V sin(γ)

(12)

We select a fighter aircraft with double-nozzles as controlled plant. The double-nozzles are fixed symmetrically and can rotate in up-down-left-right directions. Through different deflection combinations, the two nozzles can produce the expected torques along three-axes. Then the total deflection angles will become δz = 0.5(δzr + δzl ), δx = 0.5(δzr − δzl ), δy = 0.5(δyr + δyl ).

(13)

In the body coordinate, we can obtain the engine thrust components along the three-axes as ⎡ ⎤ ⎡ ⎤ Fxj cosδzj cosδyj ⎣ Fyj ⎦ = ζf j Fj ⎣ ⎦, j = l, r. sinδyj (14) Fzj −sinδzj cosδyj ζf l , ζf r represent the every engine thrust coefficient respectively. Assuming that the two engines are exactly the same, so ζf r = ζf l . Besides, the thrust generated by each engine has equal value, that is Tr = Tl . The defletion angles δyr = δyl = δy are also considered to be reasonable. Thus, the thrust along three-axes can be obtained as ⎡ ⎤ ⎤ ⎡ Fx cosδx cosδy cosδz ⎣ Fy ⎦ =ζf F ⎣ ⎦ sinδy (15) Fz −cosδx cosδy sinδz F is the total thrust and Fx , Fy , Fz represent the thrust component along threeaxes respectively. Thrust loss coefficient is denoted by ζf . To facilitate controller design, in the case of small deflection angles, we can re-express Eq. (15) approximatively as ⎡ ⎤ ⎤ ⎡ Fx 1 ⎣ Fy ⎦ = ζf F ⎣ δy ⎦. (16) Fz −δz Let iT (i = x, y, z) denote the thrust action point position coordinates, then according to the relationship between force and moment, we can obtain the torque as ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ xT Fx lT ⎣ mT ⎦ = ⎣ yT ⎦ ⊗ ⎣ Fy ⎦. (17) nT zT Fz

High Angle of Attack

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Control Scheme

From the aircraft model and thrust vector model in Sect. 2, we propose a three-channel decoupling control strategy. Independent single-inupt-singleoutput (SISO) controllers are designed in different channels, respectively. The channels coupling is treated as generalized total disturbance and can then be estimated by ESO. The sliding mode control (SMC) method is untilized to eliminate feedback error. The control scheme is showed as Fig. 1. αd , βd , pd represent the desired value respectively. Tc represents the thrust command. δi (i = x, y, z) are the thrust vector deflection angles. The throttle is set to maximum during maneuver. Taking into account the lack of aerodynamic control at high AOA, their deflection angles are set at trim point.

Fig. 1. SMC control structure based on LESO

3.1

Angle of Attack Channel Controller Design

Firstly, we re-express Eq. (2) as α˙ = f1 + q f1 = − [rsinα + pcosα]tanβ +

(18)

1 [mgcosμcosγ − L + Fz cosα − Fx sinα] mV cosβ (19)

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We differentiate Eq. (18) and then combine Eqs. (8) and (17), the second-order differentiator can be obtained as α ¨ = f˙1 + q˙ = f˙1 +

ζf F zT + ma − (Ix pr − Iz pr) − Ixz (p2 − r2 ) ζf F xT +( − b01 )δz Iy Iy

+ b0α δz = F1 + b0α δz (20) Obviously, δz firstly appears in the second-order differential of the angle of attack ˙ x3 = Fα , then the state-space form of Eq. (20) α. Define y = α, x1 = y, x2 = y, can be redescribed as ⎧ x˙ 1 = x2 ⎪ ⎪ ⎨ x˙ 2 = x3 + b0α δz (21) x˙ 3 = Hα ⎪ ⎪ ⎩ y = x1 where Hα denotes the total disturbance which includes strong coupling among channels and aerodynamic uncertainties. Then, we can establish linear extended state observer(LESO) as ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎛ ⎡ ⎤⎞ ⎡ z˙1α 010 z1α 0 z1α β01 ⎣ z˙2α ⎦ = ⎣ 0 0 1 ⎦⎣ z2α ⎦ + ⎣ b0α ⎦δz + ⎣ β02 ⎦ ⎝y − [1, 0, 0] ⎣ z2α ⎦⎠ (22) z˙3α z3α 0 β03 z3α 000 where ziα (i = 1, 2, 3) is the estimation value of xi (i = 1, 2, 3), respectively. Besides, pole placement is untilized to simplify the tuning process. That is, the observer gain parameter can be replaced by β01 = 3ωo , β02 = 3ωo2 , β03 = ωo3 .

(23)

where the observer bandwidth ωo is a tunable parameter. When the control law is chose as −z3α + u0α δz = (24) b01 where u0α denotes virtual control law. Then Eq. (21) can be approximately expressed as y¨ = α ¨ = u0α . (25) For the dynamic system Eq. (25), we can design sliding mode control law. The reference signal and its differential are produced by a fastest tracking differentiator (TD) [12] as  v˙ 1α = v2α (26) v˙ 2α = f han (v1α − αd , v2α , ε, η) ˙ the sliding mode Define the tracking error as Eα = v1α − α, E˙ α = v2α − α, surface can be expressed as s(t) = cα Eα + E˙ α

(27)

High Angle of Attack

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where cα > 0 satisfies Hurwitz. Then ¨α = cα (v2α − α) ˙ + (v˙ 2α − α ¨) s(t) ˙ = cα E˙ α + E = cα (v2α − z2α ) + (α¨d − α ¨ ) = cα (v2α − z2α ) + (α¨d − u0α )

(28)

Exponential approach law is used as s(t) ˙ = −εα sgn(s) − kα s

εα > 0, kα > 0

(29)

Combining Eqs. (28) and (29), the sliding mode control law can be obtained as u0α = cα (v2α − z2α ) + α ¨ d + εα sgn(s) + kα s.

(30)

In order to attenuate the chattering problem, saturated function is used instead of sign function. The convergence proof of ESO can be seen in [13]. The stability of the controller can be proved as follows. Choose Lyapunov function as Vα =

1 2 s 2

(31)

Then, its differential can be obtained as V˙ α = ss˙ = s(cα E˙ α − u0α + α ¨ d ) = s(−εα sgn(s) − kα sgn(s)) = − εα |s| − kα |s| ≤ 0

(32)

When V˙ α ≡ 0 , s ≡ 0, according to LaSalle invariance principle, the system is asymptotically stable, and the convergence speed depends on εα . 3.2

Sideslip Angle Channel Controller Design

From the definition of coordinate system and angular variable, α channel and β channel have symmetry. Thus, the control law for the sideslip angle can be given as −z3β + u0β δy = (33) b02 where z3β can be obtained by ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎛ ⎡ ⎤⎞ ⎡ z˙1β 010 z1β 0 z1β 3ωo ⎣ z˙2β ⎦ = ⎣ 0 0 1 ⎦⎣ z2β ⎦ + ⎣ b02 ⎦δy + ⎣ 3ωo2 ⎦ ⎝β − [1, 0, 0] ⎣ z2β ⎦⎠ (34) z˙3β z3β 0 ωo3 z3β 000 the corresponding sliding mode control law can be designed as u0β = cβ (v2β − z2β ) + β¨d + εβ sgn(s) + kβ s. v1β and v2β are produced by TD as  v˙ 1β = v2β v˙ 2β = f han (v1β − βd , v2β , ε, η)

(35)

(36)

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Roll Angular Rate Channel Controller Design

Taking into account the inaccuracy of the measurement for the bank angle μ, we choose the roll angular rate p as controlled variable and regulate it to track an appropriate value so that the bank angle can change properly. For convenience, we can rewrite Eq. (7) as p˙ = F3 + b03 δx (37) F3 =

Iz (L + F δy xT ) + Ixz na + Ixz nT (Ix Ixz − Iy Ixz + Iz Ixz ) + pq 2 2 Ix Iz − Ixz Ix Iz − Ixz 2 (Iy Iz − Iz2 ) − Ixz Iz F xT + qr + ( − b03 )δx 2 2 Ix Iz − Ixz Ix Iz − Ixz

So we can define

⎧ ⎨y = p x1 = p ⎩ x2 = F3

(38)

(39)

where x2 is a new state which is extended. Thus, the state-space form of p channel can be represented as ⎧ ⎨ x˙ 1 = x2 + b0p δx x˙ 2 = H3 (40) ⎩ y = x1 where H3 represents the total disturbance. The second order ESO for Eq. (40) can be constructed as            2ωo 0 1 z1p b z1p z˙1p = + 03 δx + p − [1, 0] (41) z˙2p 0 ωo2 z2p 0 0 z2p Similarly, the control law for roll angular rate p can be obtained as δx =

u0p − z2p . b03

(42)

Define Ep = v1p − p, v1p is produced by the desired value pd and TD, that is  v˙ 1p = v2p (43) v˙ 2p = f han (v1p − pd , v2p , ε, η) The sliding mode surface can be designed as s(t) = cp Ep

(44)

Then, the sliding mode control law can be obtained as u0p = (v2p + εp sgn(s) + kp s)/cp .

(45)

High Angle of Attack

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55

Simulation Results

In simulations, the initial height is chosen as h0 = −zE0 = 1200 m. The intial AOA is α0 = 10◦ , and the initial flight speed is V = 90 m/s. The related control parameters are adjusted as :ωo = 10, b01 = 2.7, b02 = −0.9, b03 = −0.4, cα = 6, εα = 0.01, kα = 10, cβ = 3, εβ = 0.001, kβ = 1, cp = 1, εp = 0.3, kp = 20, ε = 2, η = 0.01. To verify the performance of the proposed controllers, we choose a Herbst-type maneuver as a benchmark. The simulation results are displayed from Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13. At first, for α and p channel, we let an appropriate curve go through a designed fastest tracking differentiator, and use its smooth output as final reference signal shown in Figs. 2 and 4 respectively. The angle of attack begins to increase at t = 1.5 s and reaches 62.5◦ after 2 s and then keep a certain period of time, while the roll angular rate changes as shown in Fig. 4 to cause the aircraft to finish rolling along the velocity vector. When the rolling is completed, the maneuver ends after the aircraft has been swooped for a while. The velocity changes as Fig. 5. When the angle of attack α and roll angualr rate p change as expected curve, the bank angle μ can change as Fig. 7. The angle which insicates flight direction changes 180◦ as Fig. 8. The deflections of thrust vector are displayed in Figs. 10, 11 and 12. Figure 13 shows that the vertical overload is also moderate. 1

70

d

60

TD

40

/(°)

/(°)

TD

0.5

d

50

30 20

0

-0.5

10 0

0

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10

-1

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t/s

Fig. 2. AOA

Fig. 3. Sideslip angle

30 pd

20

p

V/(m/s)

p/(°/s)

10 0 -10 -20 -30

V

100

pTD

80

60

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Fig. 4. Roll angular rate

0

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Fig. 5. Flight speed

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20

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-40 -60

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Fig. 6. Pitch angular rate

Fig. 7. Roll angle along the velocity vector

200

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30 150

r/(°/s)

/(°)

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10 0

50 -10 0

0

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Fig. 9. Yaw angular rate 20

10

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x

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Fig. 8. Velocity heading angle

-10

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t/s

-10

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t/s

10

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t/s

Fig. 10. Vector nozzles roll deflections

Fig. 11. Vector nozzles yaw deflections

20

7 6 5

nz/(g)

z

/(°)

10

0

4 3 2

-10

1 -20

0

5

10

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t/s

Fig. 12. Vector nozzles pitch deflections

0

0

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10

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t/s

Fig. 13. Vertical overload

High Angle of Attack

5

57

Conclusions

This paper presented a high AOA control strategy for fighter aircraft based on sliding mode control and extended state observer. Taking into account the lack of aerodynamic control, thrust vector technique was untilized instead. Different from traditional multivariate control design, three SISO controllers were designed in the horizontal, vertical and heading channels. The total disturbance including channels coupling and model uncertainties was estimated by the extended state observer and then compensated by sliding mode control law. Numerical simulation results was conducted to verify the effectiveness of the proposed strategy.

References 1. Wu DW, Chen M, Gong HJ (2017) Robust control of post-stall pitching maneuver based on finite-time observer. ISA Trans 70(4):53–63. https://doi.org/10.1016/j. isatra.2017.06.015 2. Yang JL, Zhu JH (2016) A hybrid NDI control method for the high-alpha supermaneuver flight control. In: Proceedings of American Control Conference, Boston, MA, USA, pp 6747–6753. IEEE. https://doi.org/10.1109/ACC.2016.7526734 3. Mukherjee BK, Thomas P, Sinha M (2016) Automatic recovery of a combat aircraft from a completed cobra and herbst maneuver: a sliding mode control based scheme. In: Proceedings of Indian Control Conference, Hyderabad, India, pp 44–46. IEEE. https://doi.org/10.1109/INDIANCC.2016.7441137 4. Sinha M, Kuttieri R, Ghosh A, Misra A (2013) High angle of attack parameter estimation of cascaded fins using neural network. J Guidance Control Dyn 50(1):272–291. https://doi.org/10.2514/1.C031912 5. Snell S, Enns D, Garrard W (1992) Nonlinear inversion flight control for a supermaneuverable aircraft. J Guidance Control Dyn 15(4):976–984. https://doi.org/ 10.2514/3.20932 6. Adams R, Buffington J, Banda S (1994) Design of nonlinear control laws for highangle-of-attack flight. J Guidance Control Dyn 17(4):737–746. https://doi.org/10. 2514/3.21262 7. Seshagiri S, Promtun E (2008) Sliding mode control of F-16 longitudinal dynamics. In: Proceedings of American Control Conference, Seattle, Washington, USA, pp 1520-1532. IEEE. https://doi.org/10.1109/ACC.2008.4586748 8. Chiang R, Safonov M, Haiges K (1993) A fixed H∞ controller for a supermaneuverable fighter performing the herbst maneuver. Automatica 29(1):111–127. https:// doi.org/10.1016/0005-1098(93)90176-T 9. Han JQ (2009) From PID to active disturbance rejection control. IEEE Trans Ind Electron 56(3):900–906. https://doi.org/10.1109/TIE.2008.2011621 10. Gao ZQ (2014) On the centrality of disturbance rejection in automatic control. ISA Trans 53(4):850–857. https://doi.org/10.1016/j.isatra.2013.09.012 11. Sonneveldt L (2010) Nonlinear F-16 model description. Technical report, Delft University of Technology, The Netherlands 12. Han JQ (2008) Active disturbance rejection control technique-the technique for estimating and compensating the uncertainties. National Defense Industry Press, Beijing 13. Zheng Q, Dong LL, Lee D, Gao ZQ (2009) Active disturbance rejection control for MEMS gyroscopes. IEEE Trans Control Syst Technol 17(6):1432–1438. https:// doi.org/10.1016/0005-1098(93)90176-T

Research and Application of Behavior Detection Based on Smart Phone Sensor Yongsheng Xie1(B) and Linbing Wei2

2

1 College of Mathematics and Computer Science, Guangxi Science and Technology Normal University, Laibin 546199, Guangxi, China [email protected] Finance Department, Guangxi Science and Technology Normal University, Laibin 546199, Guangxi, China http://www.gxstnu.edu.cn

Abstract. The advent of intelligence era has made peoples lives colorful. The humanization and intellectualization of smart phone have become a breakthrough to the future era. Smart-phone sensors can be used to detect and recognize human behavior, thus realizing human-computer interaction, virtual reality and so on, which shows the important value of behavior detection. In order to further understand the principle and application value of behavior detection of smart phone sensor, this paper introduces common smart phone sensors, and studies the principle and application of behavior recognition, in order to provide reference for future exploration.

Keywords: Smart phone

1

· Sensor · User behavior detection

Introduction

With the continuous development of science and technology, smart phones have become a necessary tool in life. In the development of smart terminals and mobile internet, great changes have been brought to people’s lives. It is obvious that the demand for location services relying on smart phone sensors is growing rapidly. More and more scene applications need the support of location information [1]. In the location information detection, behavior recognition detection plays an important role in providing rich support for location information by identifying human behavior characteristics [2].

2

Introduction of Smart Phone Sensors

With the gradual improvement of smart phone research and design, the types and functions of sensors in smart phones are also gradually improved. Compared with traditional mobile phones, the main difference between smart phones is the c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 58–63, 2020. https://doi.org/10.1007/978-981-32-9682-4_7

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intelligence of the operating system, the humanization and the diversification of device functions. In the process of realizing these advantages on smart phones, sensors play a key role [3]. There are many kinds of sensors in smart phones. Here are some of the most common ones. 2.1

Acceleration Sensor

When the user holds the smart phone in moving, the acceleration sensor captures the acceleration data in the direction of motion. By analyzing the acceleration data, the user’s moving direction and moving speed can be discerned, which becomes an important parameter in behavior recognition. In the acceleration sensor, an inductive gravity module is included. As the direction of movement of the user changes, the gravity module contacts the piezoelectric elements in different directions to form different forces, and the direction of motion and acceleration can be judged according to the output electrical signal. 2.2

Direction Sensor

Direction sensor plays an important role in smart phones, and the smart navigation of maps involves the sensor. It can recognize the orientation state of the smart phone itself, and output the parameters information of the mobile phone in different states, such as vertical, horizontal and pitch. The data acquired by the direction sensor in the smart phone can be represented by three-axis coordinate values, and the three-axis coordinate value is fed back to the specific state through the angle data. Using the three-axis coordinate value parameter, the posture of the mobile phone can be determined, and the most important one is the azimuth angle, which represents the angle of rotation around the Z-axis. In the behavior recognition, according to the change of the angle, it can be determined how much the user has turned, so as to identify the user’s direction. 2.3

Magnetic Sensor

The sensor is also called electronic compass, which is also called digital compass. It can identify the changes of magnetic field around users. When users approach magnetic materials or large-scale electrical equipment, according to the changes of magnetic field around the environment, it can judge the user’s action trend in a specific scene. These commonly used smart phone sensors are representative and play an important role in the behavior detection based on smart phone sensors.

3

The Principle of Behavior Detection Based on Smart Phone Sensor

Human behavior detection, in a broad sense, is to use sensors of the smart phone to obtain the data formed by the user’s motion process in the real life environment. After analyzing and processing the data, the approximate change of the

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user behavior is obtained according to different algorithms, and then according to the preliminary judgment result, combined with the environment information, the basic behavior of the user can be grasped, thereby more intelligent and humanized functional services can be provided for users. 3.1

Motion Parameter Analysis

Motion behavior parameters are the most direct parameters smart phone sensors can detect. When human body moves in outdoor or indoor environments, such as stationary, squatting, walking, running and so on, sensors in smart phone can get the real information of the movement process. Combining with the characteristics of the user’s surrounding environment, the user’s motion state can be grasped. In this process, smart phone sensors mainly acquire the following parameters [4]. (1) Acceleration parameters: The average acceleration is the average value of the increment of velocity in a period of time, while the instantaneous acceleration is the limit value of acceleration when time tends to infinite. Smart phones use acceleration sensors to obtain the magnitude and direction of acceleration, which can be used to judge the user’s movement tendency. (2) Direction angle parameter: When acquiring the direction data, the smart phone mainly uses the direction sensor to obtain the horizontal rotation angle of the current direction of the mobile phone body and the inclination angle of the body, and perceives the movement direction of the user according to the change of the rotation angle. (3) Magnetic field intensity: The measurement of magnetic field intensity can judge the change of users’ behavior in some special scenes. For example, when users take the elevator, according to the data read by the magnetic sensor, the change of magnetic field intensity outside and inside the elevator can be judged, so as to judge whether they enter or leave the elevator. With the elevator moving, the magnetic field data presents a larger disorder state, which indicates that the user is in the elevator environment, while when the user is outside the elevator, the total magnetic field intensity around the elevator will not be excessively violent, so the change of the data can be used as a judgment of the behavior of the user in the special environment. 3.2

Behavior Recognition Process

In recent years, behavior recognition based on smart phone sensors has become a research hotspot. Human behavior recognition has been widely used in the field of video surveillance, human interaction and virtual reality. Through parameter acquisition of sensors, users’ specific action can be obtained according to the coordination of different sensors, and then combined with the detection of the surrounding environment, users behavior in the environment can be judged more accurately. By using the identification of the smart phone sensor, the motion change, direction, moving speed, angle, etc. generated by the user’s motion can be obtained, and then the data is processed by different methods, and the interference is removed by means of denoising, segmentation, etc. so as to get true

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behavioral data [5]. In this process, if the raw data is preprocessed, the obtained behavioral reflection will not be obvious, so the step of extracting the characteristic data from the original data cannot be reduced. The main extracted feature data includes frequency domain features and time domain features. Multiple feature data can be extracted from a piece of raw data. The multi-dimensional feature matrix is built to analyze the feature values. Finally, the classification model database is established according to different classification and recognition algorithms. By comparing the parameters of different behaviors in the database, users behavior can be determined.

4

Application Research of Behavior Detection Based on Smart-Phone Sensor

The behavior recognition based on smart phone sensor is to obtain the user’s behavior. By data collection and training, the behavior recognition database can be established as the standard of behavior recognition. Then the newly collected data can be compared with it to make a judgment on the newly collected unknown data and identify the user’s behavior when the new data is collected [6]. This function can be widely used in many scenes. In this paper, the smart phone sensor behavior recognition during the process of getting off the bus is studied, and the idea is proposed that mobile phone can be used in paying by distance when taking bus. The specific application research is as follows. 4.1

Scene Analysis of Getting On and Off Buses

The behavior of getting on and off a bus is relatively simple, but it has obvious characteristics. Compared with standing, walking and running, the process of getting on and off a bus can be viewed step by step, that is, getting on a bus = waiting at rest + walking to the door position + getting on the door position + moving with the bus; getting off = standing at the door + bus parking + getting off at the door + walking away. Among these actions, the most obvious change of acceleration data is the action of boarding and getting off at the door position. Through smart phone sampling, the action of boarding and getting off can be identified. Starting and ending signs are recorded according to the time points of the two actions, combined with the change of vehicle positioning during driving, can be used as a judgment of the distance of taking a bus, thus paying for it. 4.2

Establishment of Getting On and Off Bus Characteristic Database

The experimental data are collected, and the data collected by mobile phone sensors are acquired by preset program, and the data collected in the process of up-and-down bus behavior are collected. In order to avoid the data difference caused by the mobile phone’s non-stationary, the mobile phone can be placed

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randomly in the data acquisition process. The concept of space coordinate system in space mathematics is introduced into the scheme design, so that the coordinate system of the acceleration sensor can be transformed into the absolute coordinate system of the earth, so as to ensure the accuracy of the data acquisition results. The 10 times acquisition data of 10 people are aggregated, and the behavior data is established in combination with the characteristics of the behavior of getting on and off bus, and the variation law of sampling time, X-axis acceleration value, Y-axis acceleration value, and Z-axis acceleration value is formed, thereby forming the characteristic database of getting on and off the bus. 4.3

Experiment Analysis

Ten users were selected as experimental objects to obtain walking data, stationary data, running data, upstairs data, downstairs data, boarding data and disembarkation data of the experimental objects, and then compared with the characteristics database of getting on and off buses. Then, an attempt was made to combine FFT components to calculate the classification accuracy of different combinations of features by using FFT component features, so as to ensure the accuracy of feature classification. By identifying the behavior of 10 users, the experimental data are obtained as shown in Table 1. Table 1. The accuracy of feature classification with FFT components Behavior

Accuracy

Stationary

99.42%

Walking

95.53%

Running

96.34%

Going upstairs

75.34%

Going downstairs

76.14%

Getting on the bus 92.87% Getting off the bus 91.56%

Through analysis, it is found that the classification accuracy of stationary, walking and running, especially stationary are high, which is not easy to confuse with other behaviors. Walking upstairs and downstairs is easy to be confused with walking, and the recognition rate is slightly lower than walking, stationary and running. Normally up and down the building has a certain similarity with the getting on and off the vehicle, the data collection results are relatively similar, and it is easy to be confused with each other. Overall, the recognition rate of upper and lower difference is relatively high, partly because of the lack of diversity under experimental conditions. Although behavior similarity is easy to confuse, the recognition rate of getting on and off buses in feature classification is relatively high with the support of FFT component, which is 92.87% and

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91.56% respectively. This shows that using smart phone sensor to identify upper and lower bus behavior has a certain degree of feasibility.

5

Conclusions

Behavior detection based on smart phone sensors is of great significance in improving the interactive services of smart phones. This study analyses the principle of behavior detection based on smart phone sensors, recognizes several common smart phone sensors and their characteristics, and puts forward a concept for the practical application of behavior recognition. It is believed that using smart phone sensor behavior recognition to develop public transportation. The mobile payment mode of vehicle distance difference has certain feasibility. Acknowledgments. This work was supported by the Science and Technology Foundation of Guangxi Province under Grant No.AD16450003 & the basic ability Promotion Project of Young and Middle-aged Teachers in Guangxi Colleges and Universities in 2018 under grant No.2018KY0703.

References 1. Kuang X, He J, Hu Z (2018) Comparison of deep feature learning methods for human behavior recognition. Appl Res Comput 35(9):2815–2822 2. Sun B, Lv W, Li W (2013) Behavior recognition based on smartphone sensor and SC-HMM algorithm. J Jilin Univ (Sci Ed) 51(6):1128–1132 3. Zhu X, Qiu H (2016) Research on human behavior recognition based on smartphone sensor data. Comput Eng Appl 52(23):1–5, 49 4. Li J, Huang H, Deng N (2016) Indoor user behavior recognition based on mobile phone sensor. Sci Technol 5:143 5. Chen B, Yu Q, Chen T (2018) Research on deep learning model based on sensor human body behavior recognition. J Zhejiang Univ Technol 46(4):375–381 6. Guo Y, Kong J, Liu C (2018) Research on daily behavior recognition method based on multi-sensor change of smartphone. J Commun 39(z2):164–172

Adaptive NN Control of Discrete-Time Nonlinear Strict-Feedback System Using Disturbance Observer Bei Wu1,2 , Mou Chen1(B) , and Ronggang Zhu3 1

3

College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, Jiangsu, China [email protected] 2 Nanhang Jincheng College, Nanjing 211156, Jiangsu, China Luoyang Institute of Electro-Optical Equipment of Avic, Luoyang 471000, China http://cae.nuaa.edu.cn/2018/0906/c5454a132086/page.htm

Abstract. A disturbance-observer-based adaptive neural network (ANN) control scheme is discussed for the discrete-time nonlinear system with strict-feedback form in this paper. To monitor the external disturbance, a disturbance observer (DO) is proposed. Then, an ANN controller is developed with the outputs of the DO and the radial basis function neural network (RBFNN). The bounded stability of the whole closed-loop system is guaranteed by choosing the appropriate parameters. At last, the validity of the proposed control scheme is verified by a numerical simulation.

Keywords: Adaptive neural network control Strict-feedback nonlinear system

1

· Disturbance observer ·

Introduction

In practical industrial engineering, most of the systems are nonlinear systems. By the method of linearization, a nonlinear system can be transformed into a linear system. However, this method can not accurately describe the actual system. With the rapid development of industrial technology, the requirement for control accuracy has been more and more higher. As a result, plenty of scholars at home and abroad have proposed a lot of different control methods, such as NN control, adaptive control, sliding mode control, to handle the control problems in nonlinear system [1–6]. Adopting the idea of the one-step ahead predictor, discrete-time nonlinear strict-feedback (NSF) system was transformed into a predictor which has maximum n-step ahead to avoid the causality contradiction problem in [1]. In [2], the NSF system subject to both matched and mismatched uncertainties was further discussed. In [3], a fuzzy tracking adaptive controller was proposed for switched systems with arbitrary switching. With the method of Euler approximation, the continuous-time hypersonic flight vehicle model was c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 64–72, 2020. https://doi.org/10.1007/978-981-32-9682-4_8

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transformed into the discrete-time one, and the NN control technology was used to solve the stability problem in [4]. It is well known that DO has a good ability of disturbance rejection. For the continuous-time nonlinear systems, a Nussbaum DO was studied in [5] and a nonlinear DO was proposed in [6]. Nevertheless, to the author’s knowledge, DO technology was seldom discussed for discrete-time NSF systems. Via the motivation discussed above, a DO-based ANN controller was studied for the discrete-time NSF system in the presence of disturbance in this paper. The paper is arranged as follows: the NSF system, lemmas as well as assumptions are described in Sect. 2. In Sect. 3, the design process about the ANN controller and DO is introduced in detail. Section 4 provides stability analysis. In Sect. 5, the validity of the designed control scheme is demonstrated by a numerical simulation. At last, Sect. 6 concludes the work of this paper.

2

Problem Formulation and Preliminaries

Consider the discrete-time NSF system as follows [1,3] ⎧ ⎨ xi (ϑ + 1) = xi+1 (ϑ), 1 ≤ i ≤ n − 1 xn (ϑ + 1) = θ(x(ϑ)) + τ (x(ϑ))u(ϑ) + ω(ϑ) , ⎩ y(ϑ) = x1 (ϑ)

(1)

where ϑ represents the discrete-time sampling point, the state vector x(ϑ) = T  x1 (ϑ), x2 (ϑ), · · · , xn (ϑ) ∈ Rn , the control input u(ϑ) ∈ R, the external disturbance ω(ϑ) ∈ R, the system output y(ϑ) ∈ R, the smooth function θ(x(ϑ)) is unknown while τ (x(ϑ)) is known. For the convenience of writing and analysis, in the following sections, θ(x(ϑ)) and τ (x(ϑ)) are replaced by θ(ϑ) and τ (ϑ), respectively. In order to design the DO and ANN controller in the next sections, the following assumptions and lemmas need to be given. Lemma 1 [7]. For a nonlinear function ς(z(ϑ)) which is in the presence of unknown terms and defined on a compact set Ωz(ϑ) ⊂ Rq , RBFNN can be applied to approximate it, ς(z(ϑ)) = Ξ ∗ T Φ(z(ϑ)) + ε(z(ϑ)),

(2)

where the base function Φ(z(ϑ)) is a column vector which is constituted by T ˆ [Φ1 (z(ϑ)), Φ2 (z(ϑ)), · · · , Φr (z(ϑ))] , r is the NN node number. Ξ(ϑ) is the esti   mation of Ξ ∗ , Ξ ∗ := arg min sup |ς(z(ϑ)) − Ξˆ T (ϑ)Φ(z(ϑ)) . The comr ˆ Ξ∈R

z(ϑ)∈Ωz(ϑ)

ponents Φi (z(ϑ)), i = 1, 2, · · · , r are usually selected as Gaussian functions, and the approximation error ε(z(ϑ)) has the upper bound ε¯ > 0.

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Assumption 1 [1]: Suppose that the known function τ (ϑ) > 0, and there exist known constants τ¯ > τ > 0 to make sure that τ ≤ τ (ϑ) ≤ τ¯. Assumption 2 [3]: The reference signal yd (ϑ) is assumed to be known and bounded. Furthermore, we assume that yd (ϑ + j) with j = 1, 2, · · · , n are all bounded. Assumption 3 [8]: Assume that the external disturbance ω(ϑ) and Δω(ϑ) = ω(ϑ) − ω(ϑ − 1) are all bounded. In this paper, we aim to design a DO-based ANN controller to make sure that the output signal can track the reference output signal effectively. At the same time, the designed ANN controller can ensure that all signals in the closed-loop system are all bounded and the tracking signals are convergent.

3

Design of ANN Controller via Disturbance Observer

Before we propose the control program of the DO and the ANN controller, the desired state vector xd (ϑ) and the error σ(ϑ) between x(ϑ) and xd (ϑ) should be defined as follows: xd (ϑ) = [yd (ϑ), yd (ϑ + 1), · · · , yd (ϑ + n − 1)]T , σ(ϑ) = x(ϑ) − xd (ϑ) with σ(ϑ) = [σ1 (ϑ), σ2 (ϑ), · · · , σn (ϑ)]T . Then, considering (1), one has

σi (ϑ + 1) = σi+1 (ϑ), 1 ≤ i ≤ n − 1 . (3) σn (ϑ + 1) = θ(ϑ) + τ (ϑ)u(ϑ) + ω(ϑ) − yd (ϑ + n) From the last equation above, the desired controller u∗ (k) is given by u∗ (ϑ) = ς(ϑ) −

1 ω(ϑ), τ (ϑ)

(4)

1 where ς(ϑ) = − τ (ϑ) (θ(ϑ) − yd (ϑ + n)). Owing to the unknown function θ(ϑ), ς(ϑ) can be approximated by RBFNN . Then, one obtain

u∗ (ϑ) = Ξ ∗T Φ(z(ϑ)) + ε(z(ϑ)) −

1 ω(ϑ), τ (ϑ)

(5)

where z(ϑ) = [xT (ϑ), yd (ϑ + n)]T ∈ Rn+1 . Because of the unknown Ξ ∗ , ε(z(ϑ)) and ω(ϑ), one has u(ϑ) = Ξˆ T (ϑ)Φ(z(ϑ)) −

1 ω ˆ (ϑ), τ (ϑ)

(6)

ˆ where Ξ(ϑ) and ω ˆ (ϑ) are the estimations of Ξ(ϑ) and ω(ϑ), respectively. The weight value update law is given by ˆ + 1) = (I − κΥ ) Ξ(ϑ) ˆ Ξ(ϑ − Υ Φ(z(ϑ))σn (ϑ + 1),

(7)

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67

where I is an identity matrix with appropriate dimension, Υ = Υ T > 0 and κ > 0 are to be designed. The disturbance estimation (ϑ) ˜ is defined as (ϑ) ˜ = (ϑ)−(ϑ). ˆ Invoking (3) – (6) result in  σn (ϑ + 1) = τ (ϑ) Ξ˜ T (ϑ)Φ(z(ϑ)) − ε(z(ϑ)) − ω ˜ (ϑ), (8) ˜ ˆ where Ξ(ϑ) = Ξ(ϑ) − Ξ ∗. In order to design the DO, overwrite the dynamic Eq. (8) as σn (ϑ + 1) = p−1 (p (θ(ϑ) + τ (ϑ)u(ϑ))) + ω(ϑ) − yd (ϑ + n)

 = −p−1 Ξp∗T Φp (z0 (ϑ)) + εp (z0 (ϑ)) + ω(ϑ) − yd (ϑ + n),

(9)

where z0 (ϑ) = [xT (ϑ), u(ϑ)]T ∈ Rn+1 and p > 0 is a parameter to be designed. An auxiliary variable is designed as α(ϑ) = σn (ϑ) − μ(ϑ),

(10)

and the dynamic equation of μ(k) is chosen as μ(ϑ + 1) = −p−1 ΞˆpT (ϑ)Φp (z0 (ϑ)) + cα(ϑ) − yd (ϑ + n),

(11)

where c > 0 is a design parameter. Considering (10) and (11), yields α(ϑ + 1) = −cα(ϑ) + p−1 Ξ˜pT (ϑ)Φp (z0 (ϑ)) + εp0 (ϑ), where εp0 (ϑ) = −p−1 εp (z0 (ϑ)) + ω(ϑ). Based on the auxiliary variable α(ϑ), the DO is proposed as

ω ˆ (ϑ) = p(α(ϑ) − ι(ϑ)) . ω (ϑ) − cα(ϑ) ι(ϑ + 1) = (1 − p−1 )ˆ

(12)

(13)

Then, considering (12) and (13), the dynamic equation of the disturbance estimation error is given by ω (ϑ) + εp1 (ϑ), ω ˜ (ϑ + 1) = Ξ˜pT (ϑ)Φp (z0 (ϑ)) + (1 − p)˜

(14)

where εp1 (ϑ) = −εp (z0 (ϑ)) − Δω(ϑ + 1) is bounded with the upper value ε¯p1 . Besides, we define Ξˆp (ϑ + 1) = Ξˆp (ϑ) − Υp p−1 Φp (z0 (ϑ))α(ϑ + 1) − Υp κp Ξˆp (ϑ), where Υp = Υp T > 0 and κp > 0 are to be designed.

(15)

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The DO and the ANN controller are proposed in this section. On the basis of the proposed DO (13) and the controller (6), the stability analysis of the closed-loop system will be carried out below.

4

Stability Analysis

With the aid of the ANN controller (6) and the DO (13), the bounded stability of the closed-loop system and the effectiveness of output tracking are shown by Theorem 1. Theorem 1. Considering the discrete-time NSF system (1) subject to external bounded disturbance, the DO is designed as (13), the controller is chosen as (6) and the NN weight value update laws are given by (7) and (15). Choosing the appropriate parameters to make sure that χ > 0, ν > 0, νp > 0, να1 > 0, να > 0, νβ1 > 0 and νβ > 0, then, the bounded stability of the closed-loop system and the convergence of tracking signals are guaranteed, where χ = 1 − ρ − a − τ¯ρr − ρr, ν = 1−¯ τ ρκ−κρ, νp = 1−2ρp κp , να1 = 1−2ρp −2p−2 ρp rp , να = 1− ρ1p c2 , νβ1 =  2 2 κp − 3rp , νβ = 1 − 3(1 − p)2 − τ¯ , π = 1 ε¯2p0 + κp Ξp∗  + κΞ ∗  + 3¯ ε2p1 + τ¯ ε¯2 , ρ

ρp

ρ

ΦT (z(ϑ)) Φ (z(ϑ)) ≤ r, ρ = λmax (Υ ), ρp = λmax (Υp ), ΦT p (z0 (ϑ)) Φp (z0 (ϑ)) ≤ rp , |εp0 (ϑ)| ≤ ε¯p0 and a > 0. Proof. The Lyapunov function V (ϑ) is selected as V (ϑ) =

n 1 ˜ ˜pT (ϑ)Υp−1 Ξ ˜p (ϑ) + α2 (ϑ) + ω ˜ T (ϑ)Υ −1 Ξ(ϑ) +Ξ σi (ϑ) + Ξ ˜ 2 (ϑ). (16) τ¯ i=1

Then, the first difference of V (ϑ) along the Eqs. (7), (8), (14) and (15) is written as ΔV (ϑ) =

 1 2 σn (ϑ + 1) − σ12 (ϑ) + Σ1 + Σ2 , τ¯

(17)

where ˆ ˆ Σ1 = −2κΞ˜ T (ϑ)Ξ(ϑ) − 2Ξ˜ T (ϑ)Φ(z(ϑ))σn (ϑ + 1) + κ2 Ξˆ T (ϑ)Υ Ξ(ϑ) T ˆ +2κ(Φ(z(ϑ))σn (ϑ + 1)) Υ Ξ(ϑ) + ΦT (z(ϑ))Υ Φ(z(ϑ))σ 2 (ϑ + 1), n

−2

T

Σ2 = α (ϑ + 1) + p {Φp (z0 (ϑ))α(ϑ + 1)} Υp {Φp (z0 (ϑ))α(ϑ + 1)} T −2Ξ˜ T (ϑ)p−1 Φp (z0 (ϑ))α(ϑ + 1) + 2κp p−1 {Φp (z0 (ϑ))α(ϑ + 1)} Υp Ξˆp (ϑ) 2

p

ω (ϑ)εp1 (ϑ) +κ2p ΞˆpT (ϑ)Υp Ξˆp (ϑ) − 2κp Ξ˜pT (ϑ)Ξˆp (ϑ) − α2 (ϑ) + 2(1 − p)˜ 

˜ 2 (ϑ) − 2(1 − p)˜ ω (ϑ)Ξ˜pT (ϑ)Φp (z0 (ϑ)) + −2p + p2 ω  2 (18) −2Ξ˜pT (ϑ)Φp (z0 (ϑ)) εp1 (ϑ) + Ξ˜pT (ϑ)Φp (z0 (ϑ)) + ε2p1 (ϑ).

Hamiltonian Mechanics

Substituting Ξ˜ T (ϑ)S(z(ϑ)) =

1 τ (ϑ)

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(σn (ϑ + 1) + ω ˜ (ϑ)) + ε(z(ϑ)) into Σ1 , one

has 2 2 ˜ (ϑ)σn (ϑ + 1) − 2ε(z(ϑ))σn (ϑ + 1) Σ1 ≤ − σn2 (ϑ + 1) − ω τ¯ τ¯ T ˆ ˆ +2κ(Φ(z(ϑ))σn (ϑ + 1)) Υ Ξ(ϑ) − 2κΞ˜ T (ϑ)Ξ(ϑ) T 2 2 ˆT ˆ +Φ (z(ϑ))Υ Φ(z(ϑ))σn (ϑ + 1) + κ Ξ (ϑ)Υ Ξ(ϑ).

(19)

Considering the following facts that − τ2¯ ω ˜ (ϑ)σn (ϑ + 1) ≤ τ¯1a ω ˜ 2 (ϑ) + τ1¯ aσn2 (ϑ + 1), τ¯ 2 −2ε(z(ϑ))σn (ϑ + 1) ≤ ρ ε¯ + τ1¯ ρσn2 (ϑ + 1),      ˆ 2  ˜ 2 2 ˆ = Ξ(ϑ) 2Ξ˜ T (ϑ)Ξ(ϑ)  + Ξ(ϑ) − Ξ ∗  , ΦT (z(ϑ))Υ Φ(z(ϑ))σn2 (ϑ + 1) ≤ ρrσn2 (ϑ + 1),    ˆ 2 1 T ˆ 2κ(Φ(z(ϑ))σn (ϑ + 1)) Υ Ξ(ϑ) ≤ τ¯ρκ2 Ξ(ϑ)  + τ¯ ρrσn2 (ϑ + 1), 2  ˆ  ˆ ≤ κ2 ρΞ(ϑ) κ2 Ξˆ T (ϑ)Υ Ξ(ϑ)  ,

(20)

where a > 0 is a designed constant and ρ = λmax (Υ ). Substituting (20) into (19), yields   1  ˜ 2 2 Σ1 ≤ − {2 − ρ − a − τ¯ρr − ρr} σn2 (ϑ + 1) + κΞ ∗  − κΞ(ϑ)  τ¯ 2  1 2 τ¯ ˆ  ω ˜ (ϑ) + ε¯2 . −κ {1 − τ¯ρκ − κρ} Ξ(ϑ)  + τ¯a ρ

(21)

Substituting Ξ˜pT (ϑ)Φp (z0 (ϑ)) = pα(ϑ + 1) + pcα(ϑ) − pεp0 (ϑ) into Σ2 , we have Σ2 = −α2 (ϑ + 1) + 2εp0 (ϑ)α(ϑ + 1) − 2cα(ϑ)α(ϑ + 1) − α2 (ϑ) T +κ2p ΞˆpT (ϑ)Υp Ξˆp (ϑ) + 2κp p−1 {Φp (z0 (ϑ))α(ϑ + 1)} Υp Ξˆp (ϑ) +p−2 {Φp (z0 (ϑ))α(ϑ + 1)} Υp {Φp (z0 (ϑ))α(ϑ + 1)}  2

˜ (ϑ) + 2(1 − p)˜ −2κp Ξ˜ T (ϑ)Ξˆp (ϑ) + −2p + p2 ω ω (ϑ)εp1 (ϑ) T

p

+2(1 − p)˜ ω (ϑ)Ξ˜pT (ϑ)Φp (z0 (ϑ)) + 2Ξ˜pT (ϑ)Φp (z0 (ϑ)) εp1 (ϑ) 2  + Ξ˜pT (ϑ)Φp (z0 (ϑ)) + ε2p1 (ϑ).

(22)

Similar to the process analysis in (20), one has 

 1 2 Σ2 ≤ − 1 − 2ρp − 2p ρp rp α (ϑ + 1) − 1 − c α2 (ϑ) ρp   2 2   ˜   − (κp − 3rp ) Ξp (ϑ) − κp − 2ρp κ2p Ξˆp (ϑ)   2 1 2 ˜ 2 (ϑ) + ε¯2p0 + κp Ξp∗  + 3¯ − 1 − 3(1 − p) ω ε2p1 . ρp

−2



2

(23)

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Considering (17), (21) and (23), we obtain     2  1 1   ˆ 2   ˜ 2 ΔV (ϑ) ≤ − χσn2 (ϑ + 1) − σ12 (ϑ) − κν Ξ(ϑ)  − κp νp Ξˆp (ϑ) − κΞ(ϑ)  τ¯ τ¯ 2    −νβ1 Ξ˜p (ϑ) − να1 α2 (ϑ + 1) − να α2 (ϑ) − νβ ω ˜ 2 (ϑ) + π, (24) where χ = 1 − ρ − a − τ¯ρr − ρr, ν = 1 − τ¯ρκ − κρ, νp = 1 − 2ρp κp , να1 = 1 − 2ρp − 2p−2 ρp rp , να = 1 − ρ1p c2 , νβ1 = κp − 3rp , νβ = 1 − 3(1 − p)2 − τ¯1a and  2 2 π = 1 ε¯2p0 + κp Ξp∗  + κΞ ∗  + 3¯ ε2p1 + τ¯ ε¯2 . ρp

ρ

Choosing the appropriate design parameters to make sure that χ > 0, ν > 0, νp > 0, να1 > 0, να > 0, ν β1 > 0 and νβ > 0, then  we have ΔV (ϑ) ≤ 0 once √ |σ1 (ϑ)| ≥ τ¯π, |σn (ϑ)| ≥ τ¯π/χ and |ω(ϑ)| ≥ π/νβ . Through the above analysis, it can be concluded that the closed-loop system is bounded stable. Meanwhile, the tracking error is convergent. This concludes the proof. So far, the design of the controller and DO as well as the stability analysis of the discrete-time NSF system (1) have been completed. Next, simulation will be applied to verify the effectiveness and the correctness of the proposed DO-based ANN control strategy.

5

Simulation Example

For the sake of verifying the correctness and the effectiveness of the designed DO-based NN control strategy, the following nonlinear discrete-time NSF system [1,9] is considered: x1 (ϑ + 1) = x2 (ϑ) x22 (ϑ) + 2.5u(ϑ) + ω(ϑ) x2 (ϑ + 1) = 1 + x21 (ϑ) + x22 (ϑ) y(ϑ) = x1 (ϑ)

(25) (26) (27)

The reference signal yd (ϑ) = 0.18 sin(ϑπ − 0.7π), the disturbance is given by ω(ϑ) = 0.11 sin(ϑπ + π3 ) cos(x1 (ϑ)). Given the initial values as μ (0) = −0.2, T ϑ (0) = 0.15, x (0) = [−0.3, −0.2] . Besides, choosing κ = 2, κp = 20, Υ = 0.01I, Υp = 0.02I, p = 0.8 and c = 0.1. Under the effects of the ANN controller (6) and DO (13), simulation results can be achieved as follows. From Fig. 1, we can see that the disturbance estimation ω ˆ (ϑ) can approximate the disturbance ω(ϑ) quickly and effectively. The control output y(ϑ) in Fig. 2 can effectively track the reference signal yd (ϑ).

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Conclusion

In this paper, the problem of DO-based ANN control scheme has been studied for the discrete-time NSF system in the presence of disturbance. To enhance the disturbance rejection ability, a DO has been proposed. The ANN controller has been studied via the outputs of the developed DO and RBFNN. Choosing the suitable parameters, the bounded stability of the closed-loop system and the convergence of tracking signals were guaranteed. At last, simulation results were provided to verify the validity of the proposed ANN control scheme.

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Acknowledgments. This research is supported by Jiangsu Natural Science Foundation of China (No. BK20171417).

References 1. Ge SS, Li GY, Lee TH (2003) Adaptive NN control for a class of strict-feedback discrete-time nonlinear systems. Automatica 39:807–819 2. Ge SS, Yang C, Dai SL, Jiao Z, Lee TH (2009) Robust adaptive control of a class of nonlinear strict-feedback discrete-time systems with exact output tracking. Automatica 45(11):2537–2545 3. Wang H, Wang Z, Liu YJ, Tong S (2017) Fuzzy tracking adaptive control of discretetime switched nonlinear systems. Fuzzy Sets Syst 316:35–48 4. Xu B, Shi Z, Yang C, Wang S (2013) Neural control of hypersonic flight vehicle model via time-scale decomposition with throttle setting constraint. Nonlinear Dyn 73:1849–1861 5. Chen M, Shao SY, Jiang B (2017) Adaptive neural control of uncertain nonlinear systems using disturbance observer. IEEE Trans Cybern 47:3110–3123 6. Chen M, Tao G, Jiang B (2017) Dynamic surface control using neural networks for a class of uncertain nonlinear systems with input saturation. IEEE Trans Neural Netw Learn Syst 26:2086–2097 7. Li YX, Yang GH (2017) Event-based adaptive NN tracking control of nonlinear discrete-time systems. IEEE Trans Neural Netw Learn Syst 29:4359–4369 8. Wu B, Chen M, Zhang L. Disturbance-observer-based sliding mode control for TS fuzzy discrete-time systems with application to circuit system. Fuzzy Sets Syst. https://doi.org/10.1016/j.fss.2018.10.022 9. Liu YJ, Wen GX, Tong SC (2010) Direct adaptive NN control for a class of discretetime nonlinear strict-feedback systems. Neurocomputing 73:2498–2505

Fault Detection of High Throughput Screening System Sijing Zhang(B) , Shuang Fang, Na Lu, Yingying Wu, Yaoyao Li, and Xin Zhao School of Electrical Engineering, Shanghai Institute of Technology, Shanghai 201418, China [email protected]

Abstract. For a new class of discrete event systems–high throughput screening systems (HTS) can be modeled by dioid Max in [[γ, δ]] in both time-domain and event-domain at the same time. The purpose of this paper is to propose a method of fault detection of high throughput screening systems and these systems can be graphically represented by Timed Event Graph (TEG). First, We introduce the problem of fault detection in HTS systems. Then we propose a conception of fault detection method, which compares the real observed outputs with the expected outputs. Finally, two examples of high throughput screening system are used to illustrate the feasibility of this detection method. Keywords: High throughput screening system Fault detection · Fault diagnosis

1

· Dioid Max in [[γ, δ]] ·

Introduction

Since the 1990s, with the development of high-speed computer technology and high automation technology, especially the extensive application of robots, high throughput screening (HTS) systems – a new class of discrete event systems have emerged. HTS system combines advanced technologies with automated instrumentation into a high-program, high automated system that creates a new program for screening compounds or biological materials. In addition, HTS can rapidly analyze compounds and biological substances through a series of activities including liquid handing, micro-plate operation, labeling, incubation and others at the rate of thousands or even millions of samples a day [1–3]. High throughput screening systems are usually considered as new kinds of timed discrete event systems, they are compared with the traditional discrete event systems such as computer network systems and automated production lines, HTS systems present a series of new features. Events that mark the beginning or ending of these above activities in HTS systems take place without the This work is supported by Shanghai Pujiang Program (14PJ1407900), Open Research Fund of Jiangnan University Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education) (APCLI1402), Shanghai Institute of Technology Foundation for Development of Science and Technology (KJFZ2019-2). c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 73–84, 2020. https://doi.org/10.1007/978-981-32-9682-4_9

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need competing with other events. On the contrary, they take the occurrence of others as a prerequisite for their own occurrence. Besides, there is no buffer area between resources and even one user occupies two resources at the same time when resources are transferred. Moreover, a fault will happen easily during the running process of system, therefore, the issue of fault detection and diagnosis need to be addressed urgently. This paper addresses the problem of fault detection and diagnosis of HTS systems. The primary problem consists in determining whether the flow of observations results from a normal behavior of the system and an abnormal behavior. The methods of fault diagnosis, modeling and localization of timed discrete event systems often involve Finite State Automaton (FSA) [4], Timed Automaton (TA) [5], Durational Graph (DG) [6]. Max-plus Algebra [7,8], and Fault Monitoring [9,10]. These approaches may have been extended to HTS systems and are also used to fault detection of the HTS systems. Specifically, this FSA may have a large number of states especially when there are both short and long delay times between occurrence instances of the successor events. About TA, the fault can be detected by comparing the sojourn times of the system model with the measured ones. Then about DG, it can solve the problem of fault detection in some complex systems such as the nondeterministic and concurrent systems, and it can also predict the timed-event trajectories of systems. In both [7] and [8] based on Max-plus algebra, they have been used for solving synchronization problems. In [9], it proposes the problem of detecting time shift failures in a TDES modeled as a (max, +)-linear system, and introduces the definition of an indicator that relies on the (max, +) algebra framework. In [10], it proposes a localization method of time shift failures. In this paper, we introduce a new way to solve the problem of fault detection and diagnosis in high throughput screening systems by using dioid Max in [[γ, δ]] algebraic techniques to model [11]. In this way, the HTS system is controlled and analyzed conveniently from time-domain and event-domain at the same time. A suitable method for fault detection is to compare two different kinds of outputs to describe the fault information of the system. First, due to the relationship between output and input, obtaining the transfer matrix H of the system by a series of computation so that we get the outputs of the system. Then introducing the comparison using the residuation theory to show the fault information by comparing the expected output with the real observed output of the HTS system. Finally, using two examples to prove the correctness of this method mentioned above. The paper is organized as follows. Section 2 introduces the mathematical background such as dioid, dioid Max in [[γ, δ]] and residuation theory. Section 3 first introduces the model method, then propose the comparison method to show the fault information of HTS system. Section 4 takes two examples to illustrate the correctness of this proposed method of fault detection. Section 5 summarizes the full text and looks forward to the next direction of research.

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Mathematical Background

This section recalls the mathematical background that will be used throughout the paper. If readers are interested in them, you can refer to the literatures [12,13]. 2.1

Dioid

A dioid or idempotent semiring D is an algebraic structure containing two binary operations which are denoted ⊕ and ⊗. ⊕, also called addition of the dioid D, is associative, commutative, idempotent (i.e., ∀a ∈ D, a ⊕ a = a). The zero element of the dioid is denoted by ε, which is the neutral element of addition (∀a ∈ D, a ⊕ ε = ε ⊕ a = a). The other operation, ⊗, also called multiplication of the dioid, is associative, distributes over the addition. The identity or unit element of the dioid is denoted by e, which is the neutral element of multiplication (∀a ∈ D, a ⊗ e = e ⊗ a = a). Furthermore, the neutral element of addition, ε, is absorbing for multiplication, i.e., ∀a ∈ D, a ⊗ ε = ε ⊗ a = ε. As in usual algebra, ⊗ will be omitted when no confusion is possible (i.e., a ⊗ b = ab). A dioid D is said to be complete if it is closed for infinite ⊕ and ⊗ distributes over infinite ⊕ too. Moreover, due to the idempotency, an order relation can be associated with D by the following equivalences: ∀a, b ∈ D, ab⇔a⊕b=b⇔a=a∧b

(1)

Because of the lattice properties of a complete D, a ⊕ b is the least upper bound of D whereas a ∧ b is its greatest lower bound, which is also associative, commutative and idempotent. In this paper, only complete D is considered. As in standard algebra, addition and multiplication can be extended to matrices. For matrices A, B ∈ Dn×p and C ∈ Dp×m , these operations are defined by (A ⊕ B)ij = Aij ⊕ Bij , (A ∧ B)ij = Aij ∧ Bij (A ⊗ C)ij =

p  (Aik ⊗ Ckj )

(2) (3)

k=1

In the algebra, the unit matrix is written as I, when i = j, Iij = e, otherwise, Iij = ε. Correspondingly, the zero matrix is denoted by N , that is, Nij = ε. For high throughput screening systems, which are often targeted at getting the task done early, the following theorem in D is useful for finding the earliest possible time of the event. Theorem 1. The implicit equation x = ax⊕b defined over a complete D admits x = a∗ b, with a∗ = ⊕(i≥0) ai , a0 = e, as the least solution. Some useful properties of the star operator (∗) in D are the following: (a ⊕ b)∗ = a∗ (ba∗ )∗ = (a∗ b)∗ a∗

(4)

(ab)∗ a = a(ba)∗

(5)

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2.2

Dioid Max in [[γ, δ]]

In order to describe high throughput screening system from time domain and event domain conveniently, a specific dioid structure as dioid Max in [[γ, δ]] is proposed. It is defined as the quotient dioid of B[[γ, δ]], the set of formal power series in two variables (γ, δ) with Boolean coefficients B = {ε, e} and with exponents in Z. The traditional addition and multiplication of power series are all given to the set as two operations respectively. The general form of the power series can be written as: S = eγ k δ t ⊕ eγ k+i δ t−j ⊕ · · · ⊕ εγ k+p δ t−q = γ k δ t ⊕ γ k+i δ t−j (i, j, p, q ∈ N0 ) Physically speaking, the information of the monomial γ k δ t ∈ Max in [[γ, δ]] can be interpreted as: the k-th event occurs at the latest at time t or at time t at least k events have occurred. Considering HTS systems, the number of events is non-decreasing over time, similarly, the occurrence time of events is non-decreasing as batches increase. Therefore, the elements meet these characteristics can be selected out through the definition of dioid Max in [[γ, δ]] from B[[γ, δ]]: def

=== B[[γ, δ]]/R(γ,δ) : dioid Max in [[γ, δ]] =

∀S1 , S2 ∈ B[[γ, δ]], S1 ≡ S2 ⇔ S1 R(γ,δ) S2 ⇔ (γ ∗ (δ −1 )∗ )S1 = (γ ∗ (δ −1 )∗ )S2 Where, R(γ,δ) represents one equivalence relation. In dioid Max in [[γ, δ]], there are the following arithmetic rules: γ k δ t ⊕ γ l δ t = γ min(k,l) δ t , γ k δ t ⊕ γ k δ τ = γ k δ max(t,τ )

(6)

γ k δ t ⊗ γ l δ τ = γ k+l δ t+τ , γ k δ t ∧ γ l δ τ = γ max(k,l) δ min(t,τ )

(7)

From a physical point of view, each monomial in the formal power series B[[γ, δ]] represents a point (k, t) in the time-event-domain, whereas for dioid Max in [[γ, δ]], the monomial includes not only the point of (k, t), but also all of the points under the equivalence classes in the time-event-domain.

Fig. 1. γ k δ t in dioid Max in [[γ, δ]]

As shown in Fig. 1, all points are represented by γ k δ t . They are belonging to the southeast cone with coordinates (k, t) described by the two rays that are

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positive in the horizontal axis and negative in the vertical axis. Moreover, the +∞ −∞ δ neutral elements for addition and multiplication of Max in [[γ, δ]] are ε = γ 0 0 +∞ −∞ and e = γ δ . The dioid is complete with top element = γ δ . 2.3

Residuation Theory

Residuation is a general notion in lattice theory which allows for the definition of “pseudo-inverse” of some isotone maps. In particular, the residution theory provides optimal solutions to inequalities such as f (x)  b, where f is an orderpreserving mapping defined over ordered sets. This theory can then be applied over idempotent semiring as follows. Definition 1. (Istone mapping) A mapping f defined on the ordered set D is isotone if a  b ⇒ f (a)  f (b). Definition 2. Let f : D → C be an isotone mapping, where D and C are complete idempotent semirings. Mapping f is said to be residuated if ∀b ∈ C, the greatest element of subset {x ∈ D|f (x)  b}, denoted f  (b), exists and belongs to this subset. Mapping f  is called the residual of f . Theorem 2. Mappings x → a ⊗ x and x → x ⊗ a defined over D are both residuated. Their residuals are usually denoted respectively x → a \ ◦ x and x → x ◦/ a. The greatest solution of a ⊗ x  b is x = a \ ◦ b, and the greatest solution of x ⊗ a  b is x = b /◦ a. ax k t \◦ γ l δ τ = γ l−k δ τ −t and γ k δ t / ◦ γ l δ τ = γ k−l δ t−τ . In M in [[γ, δ]], there are γ δ  m n ki ti lj τj Let A = i=1 γ δ and B = j=1 γ δ be two polynomials composed of m and n monomials respectively, then the following rules hold: (A \◦ B) =

m  n 

γ lj −ki δ τj −ti ,

(8)

γ ki −lj δ ti −τj ,

(9)

i=1 j=1

(B \◦ A) =

n  m  j=1 i=1

This property can be also applied for matrices. Let A ∈ Dm×n , B ∈ Dm×p , C ∈ ◦ A, and the greatest Dp×n , the greatest solution of B ⊗ X  A is X = B \ solution of X ⊗ C  A is X = A /◦ C, that: (B \◦ A) ∈ Dp×n : (B \◦ A)ij =

m 

Bki \ ◦ Akj ,

(10)

Aik / ◦ Cjk .

(11)

k=1

(A /◦ C) ∈ Dm×p : (A /◦ C)ij =

n  k=1

Theorem 3. Let D be a complete dioid and A ∈ Dn×m be a matrix. Then, ◦ A)∗ , (A \◦ A) ∈ Dm×m , (A \◦ A) = (A \

(12)

◦ A)∗ . (A /◦ A) ∈ Dn×n , (A /◦ A) = (A /

(13)

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3

Fault Detection and Diagnosis of High Throughput Screening System

3.1

Dioid Max in [[γ, δ]] Models for HTS Systems

In general, based on dioid Max in [[γ, δ]], a high throughput screening system with p-inputs and q-outputs can be established the following form of system model: X(γδ) = A(γδ)X(γδ) ⊕ B(γδ)U (γδ),

(14)

Y (γδ) = C(γδ)X(γδ). n×n ,B Max in [[γ, δ]]

n×p Max in [[γ, δ]]

(15) q×n Max , in [[γ, δ]]

∈ and C ∈ X, U Where A ∈ and Y are the system matrix, input matrix and output matrix of HTS systems respectively. Then, according to Theorem 1, the system model can be written on dioid Max in [[γ, δ]] as follows: X(γδ) = A∗ (γδ)B(γδ)U (γδ),

(16)

Y (γδ) = C(γδ)A∗ (γδ)B(γδ)U (γδ).

(17)

From Eq. (17), we can get the system transfer matrix H: H(γδ) = C(γδ)A∗ (γδ)B(γδ).

(18)

Taking a HTS system modeled with dioid Max in [[γ, δ]] in Li [14] as an example that two activities are performed in sequence on two resources as R1 and R2. Assuming that the two activities are respectively Acti.1 and Acti.2, 8 time units are required for Acti.1, and 4 time units for Acti.2. xi is the earliest occurrence time of the start event (or end event) of corresponding activity. x1 , x3 will be affected by the inputs of u1 , u2 , and the output after x4 is as the system output y. R1 spends 2 time units to carry out the next batch of activities after finishing current batch of activities, and R2 needs 1 time unit to start next batch. The line from x2 to x1 and from x4 to x3 represent the required time between two batches. And the “/” represents the difference between two batches. The above information can be described in Fig. 2(a). Moreover, this HTS system can also be described in Fig. 2(b) by a Timed Event Graph (TEG). For this HTS system, R1 is a double-capacity resource, which means it can deal with Acti.1 belonging to two batches, so for x1 , it will be affected by the former x2 and u1 of the same batch, x1 = γ 2 δ 2 x2 ⊕ γ 0 δ 0 u1 . In which, the index of γ represents the two “/” as the difference of two batches, and the index of δ represents the difference of two time units. If the index of γ is 0, which means they are in the same batch, similarly, we can get x3 = γ 0 δ 0 x2 ⊕ γ 1 δ 1 x4 ⊕ γ 0 δ 0 u2 . Therefore, considering the system in Fig. 2(b), the matrices A, B and C of this HTS system are given in Max in [[γ, δ]]: ⎤ ⎡ 0 0 ⎤ γ δ ε ε ε γ 2 δ2 ε ⎢ ε ⎢ γ 0 δ8 ε  ε ε ⎥ ε ⎥ 0 0 ⎥ ⎢ ⎥ A(γδ) = ⎢ ⎣ ε γ 0 δ 0 ε γ 1 δ 1 ⎦ , B(γδ) = ⎣ ε γ 0 δ 0 ⎦ , C(γδ) = ε ε ε γ δ . ε ε γ 0 δ4 ε ε ε ⎡

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Fig. 2. An example of HTS system with single output and its TEG q×p The transfer matrix (which is actually a matrix H ∈ Max with in [[γ, δ]] q = 1) of the HTS system is computed:   H(γδ) = C(γδ)A∗ (γδ)B(γδ) = γ 0 δ 12 (γ 1 δ 5 )∗ γ 0 δ 4 (γ 1 δ 5 )∗ .

That is, the system produces the same batch of output after 12 and 4 time units when the occurrence of u1 and u2 respectively. The next batch will enter the system after 5 time units. So, if the following input is given:  0 20  γ δ ⊕ γ 6 δ +∞ U (γδ) = γ 0 δ 20 ⊕ γ 6 δ +∞ the corresponding output is: Y (γδ) = H(γδ)U (γδ) = γ 0 δ 32 ⊕ γ 1 δ 37 ⊕ γ 2 δ 42 ⊕ γ 3 δ 47 ⊕ γ 4 δ 52 ⊕ γ 5 δ 57 ⊕ γ 6 δ +∞ . These processes of computation above can be carried out by programming in the ScicosLab software. 3.2

The Method of Fault Detection and Diagnosis

As presented above in Subsect. 3.1, the HTS systems can be fully described by the transfer function H. The relationship between the input u and the output y is actually the functional model of the HTS system. A system is said to be faulted as soon as the function of the system does not properly operate, that is as soon as the real output y does not match the expected output y˜. Supposing that both y and y˜ are known, now it’s important to find a way to compare y with y˜ and determine the presence of fault between y and y˜. For the detection of fault, we propose here to perform this comparison by using the residuation theory relying on the use of the left quotient \ ◦. Assuming that the fault mentioned in this paper is time shift fault, that means a fault only affect the delay or advance of the time required for the process. Here, an example is used for illustrating the method of using left quotient \ ◦. [[γ, δ]] be the polynomials, such that: For instance, Let A, B ∈ Max in A = γ 0 δ 4 ⊕ γ 1 δ 7 ⊕ γ 2 δ 15 ⊕ γ 3 δ 18 ⊕ γ 4 δ +∞ , B = γ 0 δ 4 ⊕ γ 1 δ 7 ⊕ γ 2 δ 16 ⊕ γ 3 δ 20 ⊕ γ 4 δ +∞ .

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Then, according to Eqs. (8) and (9), the computations of A \ ◦ B and B \ ◦ A provide mathematically these results: C = B \◦ A = γ 0 δ −2 ⊕ γ 1 δ 2 ⊕ γ 2 δ 11 ⊕ γ 3 δ 14 ⊕ γ 4 δ +∞ , D = A \◦ B = γ 0 δ 0 ⊕ γ 1 δ 3 ⊕ γ 2 δ 12 ⊕ γ 3 δ 16 ⊕ γ 4 δ +∞ . According to the result, we define a fault function F to describe whether the difference between A and B exists or not. Supposing F is defined as follows: true, i.e. B \ ◦A=A\ ◦ B, F = (C == D) = f alse, otherwise. Where, C == D is used for checking if the result of B \ ◦ A is equal to the result of A \◦ B. After another calculation, we can get F = (C == D) = f alse. This means that there is difference between A and B. Now, back to the problem of comparing the expected output y˜ and the real output y of the HTS system, then refer to the above fault function, the fault p function for HTS system is determined as follows. Let u ∈ Max in [[γ, δ]] and ax q y ∈ Min [[γ, δ]] be the observable inputs and outputs of HTS system (the system is a single output system when q = 1): true, i.e. yi \ ◦ y˜i = y˜i \ ◦ yi , F = (C == D) = f alse, otherwise. where, C = yi \◦ y˜i and D = y˜i \◦ yi .

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For describing intuitively the fault of HTS system, the fault function mentioned in the Subsect. 3.2 can be applied to the HTS systems that are composed of a set of inputs u and one single or multi outputs y. Figure 2 above illustrates a system with two inputs and one output. And only the inputs and the output of system are observable. Considering the output of system is fully described by its transfer matrix H. By the observation of u, it is therefore possible to compute the expected output of the system, that is: y˜ = Hu.

(19)

Backing to the example in Fig. 2, the expected output of the HTS system can be obtained: y˜ = Hu = γ 0 δ 32 ⊕ γ 1 δ 37 ⊕ γ 2 δ 42 ⊕ γ 3 δ 47 ⊕ γ 4 δ 52 ⊕ γ 5 δ 57 ⊕ γ 6 δ +∞ . For illustrating the result clearly and directly of the system with fault, we assume that there is a fault generating a delay of 3 time units between transitions x3 and x4 , obtaining therefore the result of the real observed output y: y = γ 0 δ 35 ⊕ γ 1 δ 43 ⊕ γ 2 δ 51 ⊕ γ 3 δ 59 ⊕ γ 4 δ 67 ⊕ γ 5 δ 75 ⊕ γ 6 δ +∞ .

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According to the outputs of system, the difference between the real output y and the expected output y˜ can be showed clearly in Fig. 3. And then, the results of left quotient and fault function by computing as follows: C = y \◦ y˜ = γ 0 δ −18 ⊕ γ 1 δ −10 ⊕ γ 2 δ −2 ⊕ γ 3 δ 6 ⊕ γ 4 δ 14 ⊕ γ 5 δ 22 ⊕ γ 6 δ +∞ , D = y˜ \◦ y = γ 0 δ 3 ⊕ γ 1 δ 11 ⊕ γ 2 δ 19 ⊕ γ 3 δ 27 ⊕ γ 4 δ 35 ⊕ γ 5 δ 43 ⊕ γ 6 δ +∞ . F = (C == D) = f alse. In conclusion, there is a fault can be detected in this system due to the result F = f alse.

Fig. 3. The real output and the expected output of the system

The above example is a single output system, now the other example with multi-outputs system (see Fig. 4(a)) is applied for illustrating the correctness of the fault function, for describing the system clearly and specifically, the Timed Event Graph (TEG) of this system is described as Fig. 4(b). According to TEG, we can get the matrices of the HTS system as follows: ⎡ 0 0 ⎤ ⎤ ⎡ γ δ ε γ 1 δ2 ε ε ε ε ⎢ ε ⎢ γ 0 δ3 ε ε ε ε ⎥ ε ⎥ ⎢ ⎥ ⎥ ⎢ 0 5 ⎢ ⎥ ⎢ ε ⎥ , B(γδ) = ⎢ ε ε ⎥ A(γδ) = ⎢ ε γ δ ε ε ⎥, ⎣ ε γ 0 δ0 ⎦ ⎣ ε γ 0 δ4 ε ε γ 2 δ3 ⎦ ε ε ε γ 0 δ6 ε ε ε   ε ε γ 0 δ0 ε ε C(γδ) = . ε ε ε ε γ 0 δ0 The input matrix of system is: 

γ 0 δ 25 ⊕ γ 8 δ +∞ U (γδ) = γ 0 δ 25 ⊕ γ 8 δ +∞



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Fig. 4. A HTS system with multi-outputs and its TEG

And all outputs are observable. Now, assuming that there is a fault leading to a delay of 2 time units between transitions x2 and x3 , then we can obtain the real output and the expected output of this system by computing with transfer matrix H, that: y1 = γ 0 δ 35 ⊕ γ 1 δ 40 ⊕ γ 2 δ 45 ⊕ γ 3 δ 50 ⊕ γ 4 δ 55 ⊕ γ 5 δ 60 ⊕ γ 6 δ 65 ⊕ γ 7 δ 70 ⊕ γ 8 δ +∞ , y˜1 = γ 0 δ 33 ⊕ γ 1 δ 38 ⊕ γ 2 δ 43 ⊕ γ 3 δ 48 ⊕ γ 4 δ 53 ⊕ γ 5 δ 58 ⊕ γ 6 δ 63 ⊕ γ 7 δ 68 ⊕ γ 8 δ +∞ , y2 = y˜2 = γ 0 δ 38 ⊕ γ 1 δ 43 ⊕ γ 2 δ 48 ⊕ γ 3 δ 53 ⊕ γ 4 δ 58 ⊕ γ 5 δ 63 ⊕ γ 6 δ 68 ⊕ γ 7 δ 73 ⊕ γ 8 δ +∞ .

According to the outputs of system, the result can be depicted clearly in Fig. 5.

Fig. 5. The real output and the expected output of system

y1 :

Afterwards, obtaining the results of y \◦ y˜ and y˜ \ ◦ y by computing for output C1 = y1 \◦ y˜1 = γ 0 δ −2 ⊕ γ 1 δ 3 ⊕ γ 2 δ 8 ⊕ γ 3 δ 13 ⊕ γ 4 δ 18 ⊕ γ 5 δ 23 ⊕ γ 6 δ 28 ⊕ γ 7 δ 33 ⊕ γ 8 δ +∞ ,

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D1 = y˜1 \◦ y1 = γ 0 δ 2 ⊕ γ 1 δ 7 ⊕ γ 2 δ 12 ⊕ γ 3 δ 17 ⊕ γ 4 δ 22 ⊕ γ 5 δ 27 ⊕ γ 6 δ 32 ⊕ γ 7 δ 37 ⊕ γ 8 δ +∞ . F1 = (C1 == D1 ) = f alse. Thus there is a time shift fault detected from this output because F1 = f alse. About output y2 , we obtain: C2 = y2 \◦ y˜2 = γ 0 δ 0 ⊕ γ 1 δ 5 ⊕ γ 2 δ 10 ⊕ γ 3 δ 15 ⊕ γ 4 δ 20 ⊕ γ 5 δ 25 ⊕ γ 6 δ 30 ⊕ γ 7 δ 35 ⊕ γ 8 δ +∞ , D2 = y˜2 \◦ y2 = γ 0 δ 0 ⊕ γ 1 δ 5 ⊕ γ 2 δ 10 ⊕ γ 3 δ 15 ⊕ γ 4 δ 20 ⊕ γ 5 δ 25 ⊕ γ 6 δ 30 ⊕ γ 7 δ 35 ⊕ γ 8 δ +∞ . F2 = (C2 == D2 ) = true. From above, there is no fault detected from this output y2 , because F2 = true. Finally, the result of fault function is F = F1 ∨ F2 = f alse ∨ true = f alse, therefore, we can draw a conclusion, there is at least one fault of the HTS system.

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In this paper, we first introduce an issue of fault detection and diagnosis of high throughput screening systems that modeled by dioid Max in [[γ, δ]] from timedomain and event-domain. Then a method is proposed to make sure if the fault exists or not by applying residuation operation to formally compare the expected outputs and the real outputs of the system. The method defines a kind of fault function to describe the fault information of outputs. Finally, two examples (single output system and multi-outputs system) are used to prove the correctness of this method. This proposed method has been fully implemented with the minmaxgd C++ library [15]. This paper is a basis towards a complete fault diagnostic frame of high throughput screening systems, next steps include fault identification, fault localization, and fault tolerance, even the uncertain HTS systems which the time required between places is not a definite time but a possible duration of time. This can be very helpful in that systems where processes times are not exactly known. In that case, the bounds of system behaviors can be obtained, a slowest with all the maximal times and a fastest with all the minimal times. Whatever the direction of next research, the sensor placement matters because the faults of system can be compensated by synchronization phenomena so that they might totally be silent from global point of view. Therefore, how to add local sensors is also an interesting study.

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References 1. Mayer E, Raisch J (2004) Time-optimal scheduling for high throughput screening processes using cyclic discrete event models. Math Comput Simul 66(2–3):181–191 2. Geyer F (2004) Analyse und Optimierung zyklischer ereignis-diskreter Systeme mit Reihenfolgealternativen. Otto-von-Guericke-Universitst Magdeburg, Germany 3. Li D (2008) A hierarchical control structure for a class of timed discrete event systems. Fachgebiet Regelungssysteme, Berlin 4. Hashtrudi-Zad S, Kwong RH, Wonham WM (2005) Fault diagnosis in discrete event system: incorporation timing information. IEEE Trans Autom Control 50(7):1010–1015 5. Supavatanakul P, Lunze J (2006) Diagnosis of timed automata: theory and application to the DAMADICS actuator benchmark. Control Eng Prac 14:609–619 6. Baniardalani S, Askari J (2013) Fault diagnosis of timed discrete event systems using dioid algebra. Int J Control Autom Syst 11(6):1095–1105 7. Supavatanakul P, Schullerus G (2004) A hierarchical heterogeneous approach to diagnosis of discrete event systems. Technical Report, Ruhr-Universitat Bochum 8. Schullerus G, Supavatanakul P, Krebs V, Lunze J (2006) Modelling and hierarchical diagnosis of timed discrete-event systems. Math Comput Model Dyn Syst 12(6):519–542 9. Sahugude A, Corronc EL, Pencol Y (2017) Design of indicators for the detection of time shift failures in (max, +)-linear systems. Int Fed Autom Control 50(1):6813– 6818 10. Corronc EL, Sahugude A, Pencol Y, Paya C (2018) Localization of time shift failures in (max, +)-linear systems. Int Fed Autom Control 51(7):186–191 11. Li D (2017) Dioid-based modeling of high throughput screening systems. Control Theory Appl 34(5):619–626 12. Baccelli F, Cohen G, Olsder GJ, Quadrat JP (1992) Synchronization and linearity, an algebra for discrete event systems. Wiley, New York 13. Blyth TS, Janowitz MF (1972) Residuation theory. Pergamon Press, Oxford 14. Li D (2017) Dioid-based optimal control for high throughput screening systems. Control Dec 32(9):1681–1688 15. Hardouin L, Cottenceau B, Lhommeau M (2013) MinMaxgd, a toolbox to handle periodic series in semiring max in [γ, δ]. University of Angers, France

Grey Wolf Optimizer Based Active Disturbance Rejection Controller for Reusable Launch Vehicle Xia Wu1 , Shibin Luo1,2 , Yuxin Liao2(B) , Xiaodong Li2 , and Jun Li2 1

School of Automation, Central South University, Changsha 410083, Hunan, China 2 School of Aeronautics and Astronautics, Central South University, Changsha 410083, Hunan, China [email protected]

Abstract. This paper refers to finding a new solution to the attitude tracking control problem for reusable launch vehicle in reentry phase with multiple disturbances. We address the problem by a new scheme, which integrating a nonlinear active disturbance rejection controller with a meta-heuristic algorithm. Firstly, the attitude model with strict-feedback form is constructed to facilitate the controller design. Then, an attitude tracking controller based on the nonlinear active disturbance rejection control is built. A linear tracking differentiator is utilized to arrange a transient profile and extract derivatives of the attitude reference input. A linear extended state observer with model-assisted term is chosen to estimate the multiple disturbances. A feedback control law with the model-assisted information and observer estimations is deduced to achieve rapid response. Futhermore, a novel meta-heuristic algorithm named grey wolf optimizer is developed to search for the optimal controller parameters by minimizing a criterion function. Finally, the main simulation results are given to validate the superiority of the proposed approach. Keywords: Reusable launch vehicle · Attitude control · Nonlinear active disturbance rejection control · Grey wolf optimizer

1 Introduction The reusable launch vehicle (RLV) is highly integrated with many aerospace technologies [1]. Comparing with other existing space vehicles, the RLV can be re-used and shows many merits in launch cost, rapidity, reusability, and maintainability, etc. Due to the foregoing advantages, the RLV is of high value in both civil and military and reveals broad prospects for development. However, the controller design for RLV in the reentry phase is considerable difficult due to the fact that the dynamic of the vehicle is strong coupling and the vehicle is influenced by multiple disturbances. In addition, various stringent constraints, including heat flux, structured load and dynamic pressure, reduce the vehicle path into a narrow scope, which increase the difficulties of the controller design [2]. c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 85–100, 2020. https://doi.org/10.1007/978-981-32-9682-4_10

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Researchers have proposed numerous solutions for the control of RLV. Reference [3] designed a novel fuzzy tracking controller with H∞ and decay rate η control methods to solve the robust tracking control problem for RLV. Reference [4] examined the applicability of H∞ and μ -synthesis control for automatic flight control systems of RLV to decrease the sensitivity of system performance to structured plant uncertainty. Reference [5] proposed a flight control strategy based on a novel incremental nonlinear dynamic inversion for attitude control strategy of RLV reentry flight to eliminate the sensitivity to model mismatch and ensure better tracking performance. Reference [6] used a robust adaptive back-stepping control strategy to acquire desired control performance for the reentry process of RLV with high nonlinearity, strong coupling, fast time-variation and great uncertainties. Reference [7] proposed a finite-time adaptive fault tolerant control method for the reentry vehicle to deal with actuator faults, control delay, input saturation, time-varying parameter uncertainties and external disturbances in the attitude control system. Effectiveness of these methods have been demonstrated through simulations. However, these aforementioned algorithms are usually mathematically complex, which may impose restrictions on their practical applications. Active disturbance rejection control (ADRC), as an efficient method in controlling systems with unknown uncertainties and disturbances, achieves a reasonable trade-off between simplicity and performance [8, 9]. The nonlinear tracking differentiator (NTD), as an indispensable part of the ADRC controller, is utilized to generate a smooth approximation of attitude reference and its derivatives. The nonlinear extended state observer (NLESO), as the essence of the ADRC, estimates the unknown uncertainties and disturbances, which can be named as the total disturbance, in real-time. By performing the feedback linearization with the estimated total disturbance, the primal nonlinear system can be simplified into a linear form. Then, a nonlinear state error feedback (NLSEF) control law is designed for the linearized system to achieve desired performance. The original form of ADRC is a general nonlinear controller with over ten parameters. In order to reduce the complicated parameters tuning work, a parameterized linear ADRC (LADRC), which contains the linear tracking differentiator (LTD), linear extended state observer (LESO) and the linear state error feedback (LSEF) control law, was proposed [10, 11]. Both forms of ADRC controllers have been widely used to a variety of different fields and achieved considerable benefits for its simplified design and superior performance [12–17]. The parameter tuning process for ADRC controllers can be counted as an optimization problem. Numerous meta-heuristic algorithms have been presented for controller parameters optimization [18–21]. Among these optimizers, a novel algorithm called grey wolf optimizer (GWO), presented by Mirjalili et al. [22], is enlightened by survival mode and preying style of wolves. The GWO is a population based meta-heuristic algorithm which mimics the living and predatory behaviors of grey wolves in a group. It has been verified that the GWO is dominant to other population based meta-heuristic algorithms, such as particle swarm optimization and differential evolution [23].

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This paper presents a methodology that realizes desired attitude tracking performance for RLV in reentry phase. In the proposed method, a model-assisted ADRC is devoted to control the attitude model of RLV in strict-feedback form. The controller gains tuning stage employs a population based meta-heuristic algorithm, namely GWO. The main research works of this paper mainly include the following parts: i. A model-assisted nonlinear ADRC (M-ADRC), which is established based on the complete framework of ADRC, is presented in this paper for the rotational dynamics and kinematics of RLV in reentry phase. ii. In M-ADRC, LTD is devoted to provide the filtered version of the input and its derivatives. The model-assisted LESO (M-LESO) is used to estimate the total disturbance. The NLSEF with the non-smooth continuous functions f han is used to realize faster convergence of the attitude tracking. iii. The grey wolf optimizer is applied to optimize the parameters of the M-LESO and the NLSEF by minimizing a criterion function to achieve desired response. iv. Comparative studies are performed respectively through numerical simulations to evaluate the performance of the proposed algorithm. The rest of the paper is organized as follows. The mathematical model of the attitude system is described in the upcoming section. Then, the attitude controller design and the grey wolf optimizer are developed in Sect. 3. Comparative simulations and analyses are shown in Sect. 4. The concluding remarks are given in the last section.

2 System Description and Mathematical Modeling Assuming RLV is rigid and plane symmetric, then, its 3-DoF rotational dynamics and kinematics equations can be described as the following form [24] p˙ = (c1 r + c2 p)q + c3 L¯ + c4 N q˙ = c5 pr − c6 (p2 − r2 ) + c7 M r˙ = (c8 p − c2 r)q + c4 L¯ + c9 N

(1)

φ˙ = p + (q sin φ + r cos φ ) tan θ θ˙ = q cos φ − r sin φ ψ˙ = q sin φ sec θ + r cos φ sec θ

(2)

where p, q and r are the roll, pitch, and yaw rates, respectively; c1 · · · c9 are given in ¯ M¯ and N¯ are the roll, pitch, and yaw aerodynamic moments shown in Eq. (4); Eq. (3); L, φ , θ and ψ are the roll, pitch, and yaw angles, respectively. c1 = c5 =

2 (Jyy −Jzz )Jzz −Jxz (J −J +J )J , c2 = xxJ Jyy −Jzz2 xz c3 = J JJzz−J 2 , c4 = J JJxz−J 2 2 Jxx Jzz −Jxz xx zz xx zz xx zz xz xz xz 2 Jxx (Jxx −Jyy )+Jxz Jzz −Jxx Jxz Jxx 1 , c = , c = , c = , c = 7 8 9 6 2 2 Jyy Jyy Jyy Jxx Jzz −Jxz Jxx Jzz −Jxz

(3)

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where Jxx , Jyy and Jzz are the moments of inertia produced by rotating around x, y, z axis; Jxy = Jyx = Jyz = Jzy = 0 are reasonable because RLV is plane symmetric; Jxz , Jzx are inertia products and are equal to each other. ¯ l , M = qS ¯ n L¯ = qS ¯ bC ¯ cC ¯ m , N = qS ¯ bC

(4)

where q¯ = 0.5ρ V 2 refers to the dynamic pressure, V refers to the flight velocity, ρ refers to the air density; S refers to the wing reference area, b¯ is the wing span, c¯ is the wing mean geometric chord; Cl ,Cm and Cn are the roll, pitch, and yaw moment coefficients described as follows β

pb rb Cl = Cl β +Clδa δa +Clδr δr +Clp 2V +Clr 2V β q q c ¯ 0 α δ Cm = Cm +Cm α +Cm β +Cme δe +Cm 2V β pb¯ rb¯ Cn = Cn β +Cnδa δa +Cnδr δr +Cnp 2V +Cnr 2V ¯

¯

(5)

where α , β are the attack angle and sideslip angle, respectively; δa , δe and δr denote × refers to the the aileron, elevator, and rudder deflections of the vehicle, respectively; C× corresponding aerodynamic derivatives. To facilitate the design process of the controller, the model transformation based on rotational dynamics and kinematics Eqs. (1) and (2) is used to develop a second-order strict-feedback control system. The mathematical form of the proce Taking the derivative of Eq. (2), we have [φ¨ , θ¨ , ψ¨ ]T = L(φ , θ )[ f p , fq , fr ]T + g (φ , θ , ψ ) + L(φ , θ )Gu [δa , δe , δr ]T

(6)

where f p = (c1 r + c2 p)q + c3 Mx + c4 Mz , fq = c5 pr− c6 (p2 − r2 ) + c7 My  β pb¯ rb¯ fr = (c8 p − c2 r)q + c4 Mx + c9 Mz , Mx = qS ¯ b¯ Cl β +Clp 2V +Clr 2V     β β qc¯ pb¯ rb¯ , Mz = qS ¯ c¯ Cm0 +Cmα α +Cm β +Cmq 2V ¯ b¯ Cn β +Cnp 2V +Cnr 2V My = qS ⎤ ⎡ δa ⎤ ⎡ ¯ ¯ δr c3 0 c4 bC 0 bC l l Gu = qS ¯ ⎣ 0 c7 0 ⎦ ⎣ 0 cC ¯ mδe 0 ⎦ δ ¯ nδr ¯ n a 0 bC c4 0 c9 bC ⎤ ⎤ ⎡ ⎡ θ˙ ψ˙ sec θ + φ˙ θ˙ tan θ 1 sin φ tan θ cos φ tan θ ⎦ , L(φ , θ ) = ⎣ 0 cos φ − sin φ ⎦ −φ˙ ψ˙ cos θ g (φ , θ , ψ ) = ⎣ 0 sin φ sec θ cos φ sec θ φ˙ θ˙ sec θ + θ˙ ψ˙ tan θ

(7)

In order to acquire a more practical dynamic model of RLV accurately, the model uncertainties and external disturbances should be taken into account. Therefore, a more accurate model is described as follows [φ¨ , θ¨ , ψ¨ ]T = L(φ , θ )[ f p , fq , fr ]T + [Δ f1 , Δ f2 , Δ f3 ]T + g (φ , θ , ψ ) + L(φ , θ ) (Gu + Δ Gu ) [δa , δe , δr ]T + [d1 , d2 , d3 ]T

(8)

where di (i = 1, 2, 3) are the external disturbances, Δ f(x) and Δ Gu are shown as following equations T  Δ f(x) = [Δ f1 , Δ f2 , Δ f3 ]T = L(φ , θ ) c3 Δ Mx + c4 Δ Mz , c7 Δ My , c4 Δ Mx + c9 Δ Mz

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⎤⎡ ⎤ c3 0 c4 b¯ Δ Clδa 0 b¯ Δ Clδr Δ Gu = qS ¯ ⎣ 0 c7 0 ⎦ ⎣ 0 c¯Δ Cmδe 0 ⎦ c4 0 c9 b¯ Δ Cnδa 0 b¯ Δ Cnδr

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where Δ Mx , Δ My and Δ Mz are shown as below   β pb¯ rb¯ Δ Mx = qS ¯ b¯ Δ Cl β + Δ Clp 2V + Δ Clr 2V   β qc¯ Δ My = qS ¯ c¯ Δ Cm0 + Δ Cmα α + Δ Cm β + Δ Cmq 2V   β pb¯ rb¯ Δ Mz = qS ¯ b¯ Δ Cn β + Δ Cnp 2V + Δ Cnr 2V

(10)

(11)

× where Δ C× refers to the corresponding aerodynamic derivatives perturbation of RLV. Equation (8) can be rewritten as the following form

x¨ = f(x) + Δ f(x) + [g(x) + Δ g(x)] u + d (12) y=x

where x = [ϕ , θ , ψ ]T is the system state vector, y is the output vector; u = [δa , δe , δr ]T is the control input vector, d = [d1 , d2 , d3 ] denotes the external disturbances; f(x), g(x) and Δ g(x) are shown as following form ⎧ ⎨ f(x) = L(φ , θ )[ f p , fq , fr ]T + g (φ , θ , ψ ) g(x) = L(φ , θ )Gu (13) ⎩ Δ g(x) = L(φ , θ )Δ Gu = [Δ g1 (x), Δ g2 (x), Δ g3 (x)]T where Δ f(x) and Δ g(x) represent parameters uncertainties of the attitude loop.

3 Attitude Controller Design and Parameters Optimization In this section, the M-ADRC controller is designed for the rotational dynamics and kinematics model of RLV, and the grey wolf optimizer is utilized to optimize the controller parameters so that desired performance can be obtained. 3.1 Controller Design for the Attitude Loop In this paper, a M-ADRC controller is designed for the attitude model of RLV described as the Eq. (12). Figure 1 shows the structure of the M-ADRC controller for RLV. In the system, the known part of the model is regarded as model-assisted part, which is devoted to LESO to reduce computational burden and integrated into the control law to achieve desired output. LTD is used to produce a smooth approximation of the attitude reference. Internal uncertainties and external disturbances are deemed as the total disturbance, estimated by the M-LESO in the corresponding loop. The modelassisted control law (MCL), which including the model-assisted term and the virtual

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Fig. 1. Control structure of the proposed scheme for RLV

control input NLSEF, is used to generate the actual control input u to satisfy the control demands. The GWO based controller for RLV in reentry phase is designed as below. Taking the uncertain dynamics and external disturbances as the total disturbance and extended as a new state of the system, the system (12) can be rewritten as the following form ⎧ x˙ = x2 ⎪ ⎪ ⎨ 1 x˙ 2 = f(x1 , x2 ) + g(x1 , x2 )u + x3 (14) x˙ = h¯ ⎪ ⎪ ⎩ 3 y = x1 where x3 = Δ f(x1 , x2 ) + Δ g(x1 , x2 )u + d is the total disturbance; h¯ is the derivative of the x3 , h¯ is bounded. For the system (14), u is taken as the actual control input to make x1 track its reference command xr = [ϕr , θr , ψr ]T . On purpose to achieve this aim, LTD is designed to generate a smooth approximation x1 ∗ and its derivative x˙ ∗1 . Drawing on system (14), LTD is constructed as

∗ x1 (k + 1) = x1 ∗ (k) + h˙x∗1 (k) (15) x˙ ∗1 (k + 1) = x˙ ∗1 (k) + h(−2R˙x∗1 (k) − R2 (x1 ∗ (k) − xr (k))) where h is the sampling step, R is the tuning parameter to determine the tracking speed. To obtain the estimation of the total disturbance, a M-LESO is designed for system (14) accordingly as follows ⎧ z˙ = z2 − l1 e ⎪ ⎪ ⎨ 1 z˙ 2 = z3 − l2 e + f + gu (16) z˙ = −l3 e ⎪ ⎪ ⎩ 3 e = z1 − x1 T  where l = [l1 , l2 , l3 ]T , li = liϕ liθ l iψ (i = 1, 2, 3) are gain matrixes which are designed to make s3 + l1 j s2 + l2 j s + l3 j = (s + wo j )3 ( j = ϕ , θ , ψ ) is Hurwitz, wo j is the bandwidth of the M-LESO; zi (i = 1, 2, 3) are the estimations of xi .

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Remark 1: Comparing with the original LESO, by applying the model-assisted term f, the bandwidth of M-LESO can be reduced, which can ease the burden of observers. Next, the NLSEF is designed to acquire the desired tracking performance. Integrated state error e1 and its derivative e2 with the existing function f han, we have ⎧ ⎨ e1 = x∗1 − z1 e = x˙ ∗1 − z2 (17) ⎩ 2 u0 = −fhan(e1 , e2 , r, h) where the function f han is described as ⎧ d0 = Rh ⎪ ⎪ ⎪ ⎪ d = hd0 ⎪ ⎪ ⎪ ⎪ y = ⎪ x1 + hx2 ⎪ ⎪ ⎨ a1 = d02 + 8R |y|

a 0 ⎪ x2 + 1 −d ⎪ 2 sign (y), |y| > d ⎪ a = ⎪ y ⎪ + x ⎪ 2 h , |y| ⎪

≤ d ⎪ ⎪ −R · sign (a), |a| > d0 ⎪ ⎪ ⎩ f han (x1 , x2 , R, h) = − Ra d0 , |a| ≤ d0

(18)

Remark 2: Comparing to linear feedback, the function f han in nonlinear feedback shows better control efficiency and convergence owing to its nonlinear and non-smooth continuous structure [17]. The feedback control law with model-assisted information and observer estimations for the rotational dynamics and kinematics of RLV is given by u = g−1 (u0 − z3 − f)

(19)

3.2 Parameters Optimization Based on GWO Comparing to LADRC, the utilization of the function f han make the M-ADRC controller possess more parameters to be tuned although better control is achieved. Finding a set of parameters to achieve the optimum solution in the sense of some indices and avoiding tedious parameter tuning processes is our ultimate goal. Therefore, the optimizer called GWO need to be resorted. Cost Function. The controller parameters to be optimized are placed on observers bandwidth and f han of NLSEF, which are listed as follows C = [wo , r, h] = [woφ , woθ , woψ , roφ , rθ , rψ , hφ , hθ , hψ ]

(20)

where wo =[woφ , woθ , woψ ]T are the observer bandwidth of the roll, pitch and yaw loops; roφ , rθ , rψ , hφ , hθ , hψ are the parameters in NLSEF of the roll, pitch and yaw loops. The purpose of this research is to present a model-assisted control law u to acquire desired dynamic performance, that is the actual output x can track its reference xr

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Fig. 2. Hierarchy of grey wolf (dominance decreases from top down)

rapidly and stably even if the model uncertainties and external disturbances exist. Therefore, the cost function we choose can be described as follows J=

T

|φr − φ | + |θr − θ | + |ψr − ψ | dt

(21)

0

Description of the GWO Method. The Grey Wolf Optimizer, which mimics the survival mode and preying style of wolf pack, is developed by Mirjalili et al. [22]. Each pack has its hierarchical organization and each member in pack has its specialized role. Figure 2 shows the strict social hierarchy of the grey wolves. During the hunting process, the wolf pack is lead by the leader(s) which is fund of knowledge and professional management abilities. Figure 2 shows that in GWO, three group members-known as alpha, beta, and delta-lead the wolf pack in preying. Alphas are the highest rank leaders and the deltas are the lowest. All the leaders are assumed to have better understanding on the position of preys so that other members can update their positions based on the leadership positions. In all of hunting processes, wolves in the pack should follow their leaders [23]. The GWO algorithm begins with a given number of wolves and randomly generated locations. The fittest solution is the alpha. The second and third best solutions are beta and delta respectively. The rest solutions are omega. Leaders have distinctive capability in searching for the potential position of their food and encircling it. The mathematical form of the process of the grey wolves encircle the preys can be developed as bel   (22) D = CX p (t) − X(t) X(t + 1) = X p (t) − A · D

(23)

where t is the current iteration, X p (t) is the position vector of the preys, X is the position vector of a wolf, A, C are calculated as below A = 2a · r1 − a

(24)

C = 2 · r2

(25)

a = 2 − 2 · t/tmax

(26)

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where a is decreased from 2 to 0 linearly based on iterations, tmax is the maximum iterations number of the algorithm, r1 , r2 are random numbers in [0, 1]. Equations (22) to (26) show that as the iteration number increases, wolves get nearer and nearer to the preys, that is, they encircle the preys. In GWO, it is defaulted that positions of leader wolves are reserved and other wolves are compelled to update their positions based on positions of the three types of group leaders according following equations ⎧ ⎨ Dα = |C1 · Xα (t) − X(t)| D = C2 · Xβ (t) − X(t) ⎩ β Dδ = |C3 · Xδ (t) − X(t)|

(27)

⎧ ⎨ X1 = Xα (t) − A1 · Dα X2 = Xβ (t) − A2 · Dβ ⎩ X3 = Xδ (t) − A3 · Dδ

(28)

X1 + X2 + X3 3

(29)

X(t + 1) =

where Ak is calculated as Eq. (24); Ck is calculated as Eq. (25). |A| and |C| are parameters need to be tuned in the GWO. According to Eqs. (24) and (25), when |A| > 1, it forces all of the wolves to keep away from the current preys and search for a better prey. Component |C| provides random weights for preys in order to stochastically emphasize (C > 1) or deemphasize (C < 1) the impact on preys in changing position of wolves. A, C help the GWO algorithm to possess a more random behavior during optimization, especially in the final iteration to avoid local optimum. The flow chart of the GWO algorithm is presented in Fig. 3, and the detailed steps of parameters optimization can be summarized as follows. Step 1: Form an initial grey wolf population (refering to the controller parameters to be tuned in this research), including the population size n, maximum number of iterations tmax , space dimension, upper bound and lower bound of the search agents, and the positions of the search agents Xi (i = 1, · · ·, n). Step 2: Form the initial values of a, A and C. Step 3: Calculate and sort the fitness of each search agent based on the cost function as described in Eq. (20). Step 4: Setting Xα as the position of the best solution, Xβ as the position of the second best solution, Xδ as the position of the third best solution. Step 5: Update the position of all current search agents by Eq. (29). Step 6: Update the value of a, A and C by Eqs. (24) to (26). Step 7: Update the value of fitness of each search agent. If the new best solutions are better than the old best solutions, update the value of Xα , Xβ and Xδ . Step 8: Output the best solution Xα when iteration reaches tmax . Otherwise, go to step 5.

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Fig. 3. Optimization flow chart of GWO

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4 Simulations and Analyses Illustrative numerical simulations are conducted in this section to demonstrate the superiority of the GWO algorithm based M-ADRC method. 4.1 Performance of GWO Scheme To verify the parameters optimization performance of the GWO scheme, another optimizer called dragonfly algorithm (DA) is used to tune the controller parameters. DA optimizer has five parameters to be tuned [25]. Taking the overall fairness into consideration, the population size, maximum number of iterations and search space of the GWO method and DA optimizer are all the same. The only parameter in TD is R = 10. Table 1 shows the initial conditions of RLV and two optimizers. The parameters uncertainties and external disturbances shown in Eq. (30) are added to verify the disturbance rejection ability of the designed M-ADRC controller. Table 2 shows the optimization results of the controller parameters based on the two optimizers. ⎤ ⎡ 0.45 + 0.45 sin(2π t/4) × × (30) Δ C× = 20% ·C× , d = ⎣ 0.90 + 0.90 sin(2π t/4) ⎦ 0.45 + 0.45 sin(2π t/4) The references of attitude angles are shown as the following sine function. ⎤ ⎡ ⎤ ⎡ ϕr 3 sin(2π t/4) xr = ⎣ θr ⎦ = ⎣ 25 + 6 sin(2π t/4) ⎦ ψr 4 sin(2π t/4)

(31)

Table 1. Initial conditions of RLV and two optimizers Initial conditions of RLV

h = 50 km,V = 15Mach, α = 25◦ , θ = 25◦ , β = 0◦ , δa ∈ [−20◦ , 20◦ ], δe ∈ [−20◦ ,20◦ ], δr ∈ [−20◦ , 20◦ ], x(0) = [0◦ ; 0◦ ; 0◦ ], p = q = r = 0◦ /s

Initial conditions of GWO n = 30, woφ = woθ = woψ = [5, 60], rφ = rθ = rψ = [10, 50] tmax = 80,hφ = hψ = [0.1, 0.6], hθ = [0.02, 0.1] Initial conditions of DA

n = 30, woφ = woθ = woψ = [5, 60], rφ = rθ = rψ = [10, 50] tmax = 80, hφ = hψ = [0.1, 0.6], hθ = [0.02, 0.1]

Figure 4 shows the fitness value of the optimized process. Both of the two optimizers can acquire the optimal solutions. Comparing to the DA optimizer, the GWO algorithm can achieve the optimal solutions with a faster speed and less tune parameters. This superiority is attributed to the A, C, which helps the grey wolf population avoid local optimal in the optimized direction.

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DA GWO

fitness

20

15

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50

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Fig. 4. Fitness of the two optimizers

Figures 5, 6, 7, 8, 9 and 10 show the comparative results of the tracking performance in different controller parameters obtained from the two optimizers. It is worth noting that uncertainties and external disturbances have been added. Results show that both optimizers based M-ADRC controller have superior tracking performance. However, the tracking error of the controller based on DA algorithm is bigger than that of the GWO algorithm. That is, the GWO algorithm based controller has better tracking performance with less tuned parameters. Results in Table 2 verify the ability of GWO algorithm in obtaining optimal solutions. It shows that the M-ADRC method can realize better tracking performance with smaller error by employing the GWO algorithm. DA GWO φr

4

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Fig. 6. Tracking performance of pitch angle

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Fig. 8. Tracking error of roll angle

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Fig. 9. Tracking error of pitch angle

Fig. 10. Tracking error of yaw angle

Table 2. Optimization results

ωoφ

ωoθ

ωoψ

GWO 30.872 59.76 DA

rφ

rθ

rψ

hφ

hθ

hψ

Fitness

29.812 12.416 48.977 10.789 0.102 0.020 0.102 11.853

20.507 11.282 60

21.479 47.077 50

0.10

0.022 0.10

11.904

4.2 Performance of M-ADRC Scheme In order to verify the performance of the proposed M-ADRC scheme, the complete LADRC method is used to realize the comparative simulations. The initial conditions and controller parameters are shown in Table 3. Table 3. Initial conditions and parameters of the two ADRC controllers Initial conditions of RLV h = 50 km,V = 15Mach, α = 25◦ , θ = 16◦ ,δa ∈ [−20◦ , 20◦ ] δe ∈ [−20◦ ,20◦ ], δr ∈ [−20◦ , 20◦ ], β = 0, x(0) = [0; 0; 0] M-ADRC

R = 10, woφ = 14.736, woθ = 59.76,woψ = 14.570, rφ = 27.645 rθ = 19.639,rψ = 22.434, hφ = 0.03, hθ = 0.02, hψ = 0.03

LADRC

R = 10, woφ = woθ = woψ = 50, wcφ = wcθ = wcψ = 20

In Table 3, wc = [wcφ ; wcθ ; wcψ ] are the parameters of LSEF in the LADRC. It should be noted that wo = (2 ∼ 5)wc is a empirical formula in the LADRC controller. In addition, it is important to mention that parameters of M-ADRC in Table 3 are obtained by GWO method. Figures 11, 12, 13, 14, 15 and 16 show the tracking performance of the proposed M-ADRC and the LADRC controllers. Results show that comparing with the LADRC controller, the M-ADRC controller can track the reference with faster response speed and smaller steady-state error. In order to verify the superiority of the proposed MADRC method, the values of the cost function in Eq. (20) of both controllers are calculated. The sum of tracking error integral of the M-ADRC controller is 0.596 deg, which is smaller than 3.666 deg of the LADRC controller. Therefore, the M-ADRC controller is more advantageous in dynamic performance than the LADRC controller on account of the use of nonlinear and non-smooth function f han in NLSEF.

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Fig. 14. Tracking error of roll angle 0.6

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Fig. 16. Tracking error of yaw angle

Simulations show that the M-ADRC controller can achieve superior dynamic performance and rapid response although multiple disturbances exist. These merits are owed to the utilization of the model-assisted term and the nonlinear function f han. The smaller value of cost function is attributed to the employment of the GWO method.

5 Conclusions This paper proposes a grey wolf optimizer based M-ADRC controller for the attitude tracking control of RLV in reentry phase. By applying LTD to the attitude system, the smooth approximation of the attitude reference and its derivatives are generated. The M-LESO is used to estimate the system states and the total disturbance. To obtain high tracking efficiency, the NLSEF is conducted by integrating the tracking error and its

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derivative with the non-smooth function f han. Combining the estimated total disturbance and the model-assisted term, the model-assisted control law is generated. A population based meta-heuristic algorithm called GWO is applied to search for the optimal controller parameters by minimizing a criterion function so that desired performance can be achieved. Comparative simulations show the superior optimization performance of the GWO method and better dynamic performance of the proposed M-ADRC controller. In the future, other population based meta-heuristic algorithms together with the modified M-ADRC scheme will be the new research direction. Acknowledgments. This work was supported by the National Natural Science Foundation of China (Grant No. 11272349), and the Key Laboratory of Spacecraft Design Optimization and Dynamic Simulation Technologies (Beihang University), Ministry of Education, China (Grant No. 2019KF006).

References 1. Li MM, Hu J (2018) An approach and landing guidance design for reusable launch vehicle based on adaptive predictor corrector technique. Aerosp Sci Technol 75:13–23 2. Tian BL, Fan WR, Su R et al (2015) Real-time trajectory and attitude coordination control for reusable launch vehicle in reentry phase. IEEE Trans Industr Electron 62(3):1639–1649 3. Wang YH, Wu QX, Jiang CS et al (2009) Reentry attitude tracking control based on fuzzy feedforward for reusable launch vehicle. Int J Control Autom Syst 7(4):503–511 4. Jose S, George R, Safeena MK (2012) Application of H∞ and μ synthesis techniques for reusable launch vehicle control. In: 2012 IEEE aerospace conference. IEEE Press, Bozeman, pp 1–9 5. Prathap H, Brinda V, Ushakumari S (2013) Robust flight control of a typical RLV during re-entry phase. In: 2013 IEEE international conference on control applications. IEEE Press, Hyderabad, pp 718–721 6. Wang Z, Wu Z, Du YJ (2016) Robust adaptive backstepping control for reentry reusable launch vehicles. Acta Astronaut 126:258–264 7. Gao MZ, Yao JY (2018) Finite-time H-infinity adaptive attitude fault-tolerant control for reentry vehicle involving control delay. Aerosp Sci Technol 79:246–254 8. Zhao ZL, Guo BZ (2015) On convergence of nonlinear active disturbance rejection control for SISO nonlinear systems. J Dyn Control Syst 22(2):385–412 9. Li J, Qi XX, Xia YQ et al (2015) On the absolute stability of nonlinear ADRC for SISO systems. In: 34th Chinese control conference. IEEE Press, Hangzhou, pp 1571–1576 10. Pawar SN, Chile RH, Patre BM (2017) Modified reduced order observer based linear active disturbance rejection control for TITO systems. ISA Trans 71:480–494 11. Tan W, Fu CF (2016) Linear active disturbance-rejection control: analysis and tuning via IMC. IEEE Industrial Electronics Society. 63(4):2350–2359 12. Balajiwale S, Arya H, Joshi A (2016) Study of performance of ADRC for longitudinal control of MAV. Int Fed Autom Control 49(1):585–590 13. Miao JM, Wang SP, Zhao ZP et al (2017) Spatial curvilinear path following control of underactuated AUV with multiple uncertainties. ISA Trans 67:107–130 14. Tao J, Sun QL, Tan PL et al (2016) Active disturbance rejection control (ADRC)-based autonomous homing control of powered parafoils. Nonlinear Dyn 86(3):1461–1476 15. Tian JY, Zhang SF, Zhang YH et al (2018) Active disturbance rejection control based robust output feedback autopilot design for airbreathing hypersonic vehicles. ISA Trans 74:45–59

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16. Sun L, Li DH, Gao ZQ et al (2016) Combined feedforward and model-assisted active disturbance rejection control for non-minimum phase system. ISA Trans 64:24–33 17. Yu Y, Wang HL, Li N et al (2018) Finite-time model-assisted active disturbance rejection control with a novel parameters optimizer for hypersonic reentry vehicle subject to multiple disturbances. Aerosp Sci Technol 79:588–600 18. Zhang TJ (2018) Unmanned aerial vehicle formation inspired by bird flocking and foraging behavior. Int J Autom Comput 15(4):402–416 19. Esteveza J, Lopez-Guedea JM, Granaa M (2016) Particle swarm optimization quadrotor control for cooperative aerial transportation of deformable linear objects. Cybern Syst 47(1– 2):4–16 20. Rajasekhar A, Das S, Suganthan PN, (2012) Design of fractional order controller for a servohydraulic positioning system with micro artificial CEE colony algorithm. In: 2012 IEEE congress on evolutionary computation. IEEE Press, Brisbane 21. Abdelaziz AY, Ali ES (2015) Cuckoo search algorithm based load frequency controller design for nonlinear interconnected power system. Int J Electr Power Energy Syst 73:632– 643 22. Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61 23. Komaki GM, Kayvanfar V (2015) Grey wolf optimizer algorithm for the two-stage assembly flow shop scheduling problem with release time. J Comput Sci 8:109–120 24. Mao Q, Dou LQ, Zong Q et al (2017) Attitude control design for reusable launch vehicles using adaptive fuzzy control with compensation controller. J Aerosp Eng 1–14 25. Mirjalili S (2015) Dragonfly algorithm: a new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems. Neural Comput Appl 27(4):1053–1073

Consensus of Discrete-Time Heterogenous Multi-agent Systems with Nonconvex Velocity and Input Constraints Lipo Mo(B) , Ju Cheng, and Xianbing Cao School of Mathematics and Statistics, Beijing Technology and Business University, Beijing 100048, China [email protected]

Abstract. This paper mainly focuses on the nonconvex constrained consensus problem of heterogeneous multi-agent systems under the case of coexistence of velocity and input constraints. By utilizing contraction operator, a novel distributed control law is proposed. Under some mild conditions, it is shown that all agents can reach an agreement on their position states while the velocities of second-order agents and inputs of all agents can stay in their corresponding nonconvex constrained sets. At last, the correctness of theoretical results is verified by simulations. Keywords: Discrete-time · Heterogeneous multi-agent systems · Consensus · Nonconvex constraints

1 Introduction Consensus of multi-agent systems gains rapid development over the past decades. Many results have been reported, such as [1–9] and the references therein. However, most of these works were done under the assumption that there are no constraints on states and inputs of agents. In some situations, the velocity or input of agent is always constrained to remain in a constrained set, such as the dead zone of inputs and velocities of physical vehicles [10]. Recently, the constrained consensus problems have been solved by projection operator when the constrained sets are convex sets [11–14]. In reality, the velocity or input of each agent might be constrained to stay in a nonconvex set, such as quadrotors. By introducing a contraction operator, the nonconvex velocity constrained consensus problem was first investigated in [15, 16] and it was proven that all agents can reach an agreement while their velocities stay in corresponding nonconvex constrained sets. Besides, the nonconvex input constrained consensus problem was also solved by the Metzler matrix theory [17]. Later, these results were extended to the case of coexistence of nonconvex velocity and control input constraints [18] and so on [22]. Most of existing results about nonconvex constrained consensus problem were built based on homogeneous multi-agent systems, i.e. all agents have the same dynamics. Yet, in some practical situations, different agents might be have different dynamics, c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 101–110, 2020. https://doi.org/10.1007/978-981-32-9682-4_11

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i.e., the multi-agent system is a heterogeneous system. The consensus problem of heterogenous multi-agent systems without any constraints was first solved in [19, 20] when the communication is fixed. Then these results were extended to the case of switching communication topologies [21] and the case of existence of communication noises [23]. However, there are few works on constrained consensus problem for heterogeneous multi-agent systems. In [24, 25], the nonconvex constrained consensus problem was studied and it was proven that all agents can reach an agreement on position states while the velocities of second-order agents remain in their corresponding nonconvex sets provided the union graph has a directed. But the nonconvex constrained consensus problem of heterogeneous multi-agent systems is still an open problem when the nonconvex input constraints and velocity constraints coexist. In this paper, we mainly study the consensus problem of heterogeneous multi-agent system with nonconvex velocity and input constraints. A new distributed control law with constraint operator is designed first. Then, by coordination transformation and stochastic matrix theory, we prove that all agents can reach an agreement on their position states, while velocities of second order agents and inputs of all agents are constrained to stay in their corresponding nonconvex constrained sets. Compared with the works in [24, 25], where velocity constrained consensus problems of multi-agent systems with Markovian and arbitrary switching topologies were solved by nonconvex velocity constrained algorithm, this paper considers the case of coexistence of velocity and input constraints. In addition, due to the coexistence of velocity and input constraints, the closed-loop system here is a heterogeneous system composed with secondorder and third-order subsystems in nature, which leads to many challenges to stability analysis.

2 Preliminaries Let G = (VG , EG ) represents a directed graph [26], where VG = {1, · · · , n} is the set of nodes and EG is the set of edges. ai j > 0 if and only if ( j, i) ∈ EG . Denote Ni = { j ∈ V : ( j, i) ∈ E } is the neighbor set of node i. The Laplacian of graph G is denoted as n

L, where [L]ii = ∑ ai j and [L]i j = −ai j for all i = j. The union of finite graphs is a j=1

new graph whose node set and edge set are the unions of the node sets and edge sets of these finite graphs. A series of edges (vik−1 , vik ), (vik−2 , vik−1 ), · · · , (vi1 , vi2 ) is said to be a directed path from node vik to node vi1 . A directed graph has a directed spanning tree if there exists one node such that there exists at least a directed path from the this node to any other node.

3 System Description Consider a heterogeneous multi-agent system, which is composed of m second-order agents and n first-order agents. Suppose that the dynamics of the ith second-order agent has the following form ri (k + 1) = ri (k) + vi (k)T vi (k + 1) = SVi [vi (k) + ui (k)T ],

(1)

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where i ∈ Is = {1, 2, · · · , m}, ri (k), vi (k), ui (k) ∈ Rr are the position state, velocity state and control input of the ith agent, T > 0 is the sample time, and Vi ⊆ Rr , i ∈ Is is nonempty bounded closed sets, which satisfy that sup SVi (x) = ρ i > 0, inf SVi (x) = x∈V / i

x∈Vi

ρ i > 0 and 0 ∈ Vi for all i ∈ {1, 2, · · · , m}, where SVi (·) is a constraint operator with SVi (x) =

x x

sup {β :

0≤β ≤x

αβ x x

∈ Vi , 0 < α < 1} when x = 0 and SVi (0) = 0. Suppose that

the dynamics of the ith first-order agent has the following form ri (k + 1) = ri (k) + ui (k)T,

(2)

where i ∈ I f = {m + 1, m + 2, · · · , m + n}, ri (k), ui (k) ∈ Rr are the position state and control input of the ith agent. It is assumed that ri (k) = ri (0), ui (k) = 0, and vi (k) = 0 for all k ≤ 0 and i ∈ I f ∪ Is . Assumption 1. Let Ui ⊆ Rr , i ∈ Is ∪ I f is nonempty bounded closed sets, which satisfied that sup SUi (x) = ρ i > 0, inf SUi (x) = ρ i > 0 and 0 ∈ Ui for all i, where SUi (·) x∈Ui

x∈U / i

is a constraint operator with SUi (x) = and SUi (0) = 0.

x x

sup {β :

0≤β ≤x

αβ x x

∈ Ui , 0 < α < 1} when x = 0

The purpose of this paper is to design a distributed control law for each agent to assure them reach agreement in position state while their control inputs and velocity states stay in corresponding nonconvex constrained set, i.e., vi (k) ∈ Vi , i ∈ Is , ui (k) ∈ Ui , i ∈ Is ∪ I f for any k.

4 Main Results In this section, we investigate the nonconvex constrained consensus problem of heterogenous multi-agent systems with nonuniform time-delay and switching topologies. To solve this problem, we use the following control law  SUi [ui (k) − pi1 (k)vi (k)T − pi2 (k)ui (k)T + πi (k)], i ∈ Is ui (k + 1) = (3) SUi [ui (k) − pi2 (k)ui (k)T + πi (k)], i ∈ If , where pi1 (k) > 0, i ∈ Is , pi2 (k) > 0, i ∈ Is ∪ I f are feedback gains, πi (k) = ∑ j∈Ni (k) ai j (k)[x j (k − τi j (k)) − xi (k)], 0 ≤ τi j (k) ≤ M are nonuniform time-delay. Suppose that ai j (t) ≥ μc if ai j (t) > 0 where μc > 0 is a positive scalar. Let SUi [ui (k) − pi1 (k)vi (k)T − pi2 (k)ui (k)T + πi (k)] , αi (k) = ui (k) − pi1 (k)vi (k)T − pi2 (k)ui (k)T + πi (k) when ui (k) − pi1 (k)vi (k)T − pi2 (k)ui (k)T + pii (k) = 0, otherwise, αi (k) = 1, for all i ∈ SVi [vi (k)+ui (k)T ] vi (k)+ui (k)T  when vi (t) + ui (t)T = 0, otherwise βi (k) = 1 for all i ∈ Is ; SUi [ui (k)−pi2 (k)ui (k)T +πi (k)] when ui (k) − pi2 (k)ui (k)T + pii (k) = 0, otherwise, ui (k)−pi2 (k)ui (k)T +πi (k)

I f ; βi (k) =

αi (k) =

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αi (k) = 1 for all i ∈ I f . Then the closed-loop system can be rewritten as ri (k + 1) = ri (k) + vi (k)T vi (k + 1) = βi (k)vi (k) + βi (k)ui (k)T, ui (k + 1) = ui (k) − bi (k)ui (k)T − αi (k)pi1 (k)vi (k)T + αi (k)πi (k), i ∈ Is

(4)

ri (k + 1) = ri (k) + ui (k)T ui (k + 1) = ui (k) − bi (k)ui (k)T + αi (k)πi (k), i ∈ I f ,

(5)

and

where bi (k) = 1−[1−pi2T(k)T ]αi (k) , i ∈ I f ∪ Is . For i ∈ Is , define xi (k) = ri (k), yi (k) = ri (k) + c1 vi (k), zi (k) = ri (k) + (c1 + c2 )vi (k) + c1 c2 ui (k); for i ∈ I f , define xi (k) = ri (k), yi (k) = ri (k) + c3 ui (k), where c1 , c2 , c3 are positive scalars. For i ∈ Is , let Ai11 = 1 − cT1 , Ai12 = T T T T T T c1 , Ai21 (k) = 1 − c1 − βi (k) + c1 βi (k), Ai22 (k) = c1 + βi (k) − ( c1 + c2 )βi (k) c1 c2 2 Ai23 (k) = cT2 βi (k), Ai31 (k) = 1 − cT1 − (1 + cc21 )βi (k) + c1c+c 2 βi (k)T + c2 − c1 bi (k)T + c2 αi (k)pi1 (k)T , Ai32 (k) = c2 c1 )

c2 c1 )

T c1

+ (1 +

c2 c1 )βi (k)

1

− (c1 + c2 )( c11c2 + c1 +c2 c1 c2

1 )β (k)T c21 i

− (1 +

+ bi (k)T (1 + − c2 αi (k)pi1 (k)T, Ai33 (k) = βi (k)T + (1 − bi (k)T ). For T T i ∈ I f , let Bi11 = 1 − c3 , Bi12 = c3 , Bi21 (k) = bi (k)T − cT3 , Bi22 (k) = cT3 − bi (k)T + 1. Define ξ (k) = [xs (k)T , ys (k)T , zs (k)T , x f (k)T , y f (k)T ]T , where xs (k) = [x1 (k)T , · · · , xm (k)T ]T , ys (k) = [y1 (k)T , · · · , ym (k)T ]T , zs (k) = [z1 (k)T , · · · , zm (k)T ]T , x f (k) = [xm+1 (k)T , · · · , xm+n (k)T ]T , y f (k) = [ym+1 (k)T , · · · , ym+n (k)T ]T . Denote ⎡ ⎤ A11 A12 0 0 0 ⎢ A21 (k) A22 (k) A23 (k) 0 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ Θ (k) = ⎢ A31 (k) A32 (k) A33 (k) 0 ⎥, ⎣ 0 B12 ⎦ 0 0 B11 0 0 0 B21 (k) B22 (k) where A1i = diag{A11i , A21i , · · · , Am1i }, i ∈ {1, 2}; A2i (k) = diag{A12i (k), A22i (k), · · · , Am2i (k)}, A3i (k) = diag{A13i (k), A23i (k), · · · , Am3i (k)}, i ∈ {1, 2, 3}; B1i = diag{B(m+1)1i , B(m+2)1i , · · · , B(m+n)1i }, B2i (k) = diag{B(m+1)2i (k), B(m+2)2i (k), · · · , B(m+n)2i (k)}, i ∈ {1, 2}. Let ⎡ ⎤ 0 00 0 0 ⎢ 0 00 0 0⎥ ⎢ ⎥ ⎢ 0 0⎥ E0 (k) = ⎢ B(k)L0s (k) 0 0 ⎥, ⎣ 0 00 0 0⎦ 0 0 0 C(k)L0 f (k) 0 ⎡ ⎤ 0 00 0 0 ⎢ 0 00 0 0⎥ ⎢ ⎥ ⎢ Ξl (k) = ⎢ B(k)Φlss (k) 0 0 B(k)Φls f (k) 0 ⎥ ⎥, ⎣ 0 00 0 0⎦ C(k)Φl f s (k) 0 0 C(k)Φl f f (k) 0

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l = 0, 1, · · · , M; B(k) = diag{c1 c2 α1 (k)T, · · · , c1 c2 αm (k)}, C(k) = diag{c3 αm+1 (k)T, · · · , c3 αm+n (k)T }, L0 (k) = diag{L0s (k), L0 f (k)} ∈ R(n+m)×(n+m) is a diagonal matrix and its diagonal elements are the corresponding ones of Laplacian L(k); Φl (k) is square matrix with dimension being (n + m) and [Φl (k)]i j = ai j (k) if τi j (k) = l, otherwise [Φl (k)]i j = 0. Partition the matrix Φl (k) into

Φlss (k) Φls f (k) Φl (k) = , Φl f s (k) Φl f f (k) where Φlss (k) ∈ Rm×m , Φl f f (k) ∈ Rn×n , Φls f (k), Φl f s (k)T ∈ Rm×n . Then system (4) and (5) can be changed into the following form M

ξ (k + 1) = [(Θ (k) − E0 (k)) ⊗ Ir ]ξ (k) + ∑ [Ξl (k) ⊗ Ir ]ξ (k − l),

(6)

l=1

⎡

⎤ Θ (k) − E0 (k) + Ξ0 (k) Ξ1 (k) · · · ΞM−1 (k) ΞM (k) ⎢ I 0 ··· 0 0 ⎥ ⎢ ⎥ ⎢ 0 I ··· 0 0 ⎥ Ψ (k) = ⎢ ⎥ ⎢ .. .. . . .. .. ⎥ ⎣ . . . . . ⎦

Let

0

0

···

I

0

and φ (k) = [ξ (k)T , ξ (k − 1)T , · · · , ξ (k − M)T ]T , then system (6) can be rewritten as

φ (k + 1) = [Ψ (k) ⊗ Ir ]φ (k).

(7)

To analyze the stability of system (7), we make the following assumption. Assumption 2. 1 1 c2 )), i ∈ Is ; T

1 T

> pi1 (k) >

> pi1 (k) > 0;

1 1 1 μi [ c 1 ( T

− c11 ) + c1 dimax ], T1 > pi2 (k) >

> pi2 (k) >

1 T

− σi ( T1 − ( c11 +

+ c3 dimax , i ∈ I f ; c1 > T, c2 > T, c3 > T , ρi ρ c1 c2 < 1, where μi = (n+m)dimax T max j,s {φ j (0)−φs (0)+3ρ } , σi = (1+Ti )ρ , i ∈ Is ; dimax = i i ρ maxk {[L(k)]ii }, i ∈ Is ∪ I f ; δi = (n+m)dimax T max j,s {i φ j (0)−φs (0)+2ρ } , i ∈ I f . i 1 T

1 c3

Remark 1. Assumption 2 is easy to be satisfied when T is small enough. For example, we can take μ = mini {μi }, dmax = maxi {dimax } and c1 = 4T dμmax , c2 < c11 , T <  3μ 2 c3 1 1 min{ 16d , , c1 , c2 , c3 }, then it is easy to verify that Assumption 2 c1 + c2 , 1+c2 d max holds.

3 max

Assumption 3. Suppose there is an infinite time sequence 0 = k0 < k1 < · · · such that 0 < ks+1 − ks ≤ η for any s and the union graph of G (ks ), G (ks + 1), · · · , G (ks+1 − 1) has a directed spanning tree, where η > 0 is a positive scalar. Lemma 1. Under Assumptions 1 and 2, Ψ (k) is a stochastic matrix for any k, and for i ∈ Is , μi ≤ αi (k) ≤ 1, σi ≤ βi (k) ≤ 1; for i ∈ I f , δi ≤ αi (k) ≤ 1.

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Proof. Since vi (k) + ui (k)T  ≤ vi (k) + ui (k) ≤ (1 + T )ρ i and SVi [vi (k) + ui (k)] ≥ ρ i , we have σi ≤ βi (k) ≤ 1. When i ∈ Is , it is clear that Ai11 > 0 and Ai12 > 0. Besides, according to Assumption 2, we have Ai21 (k) = 1 − cT1 − βi (k) + cT1 βi (k) = 1 − cT1 − (1 − cT1 )βi (k) = (1 − cT1 )(1 − βi (k)) > 0, Ai22 (k) = cT1 + βi (k) − ( cT1 + cT2 )βi (k) = cT1 (1 − βi (k)) + (1 − cT2 )βi (k) > 0 Ai23 (k) = cT2 βi (k) > 0 Ai32 (k) = cT1 + (1 + cc21 )βi (k) − (c1 + c2 )( c11c2 + c12 )βi (k)T 1

−(1 + cc21 ) + bi (k)T (1 + cc21 ) − c2 αi (k)pi1 (k)T = cT1 + (1 + cc21 )βi (k)(1 − ( c11 + c12 )T ) −(1 + cc21 )(1 − bi (k)T ) − c2 αi (k)pi1 (k)T ≤ cT1 − c2 T > 0, 2 Ai33 (k) = cc11+c c2 βi (k)T + (1 − bi (k)T ) > 0.

When i ∈ I f , it is clear that Bi11 > 0, Bi12 > 0. Besides, Bi21 − c3 αi (k)T [L(k)]ii = bi (k)T − cT3 −c3 αi (k)T [L(k)]ii ≥ bi (k)T − cT3 −c3 dimax T ≥ (pi2 (k)− c13 −c3 dimax )T > 0, Bi22 (k) = cT3 − bi (k)T + 1 > 0. Clearly, μi ≤ αi (0) ≤ 1, i ∈ Is . Hence, c2 c2 c2 2 Ai31 (0) − c1 c2 αi (0)T [L(0)]ii = 1 − cT − (1 + c−1 )βi (0) + c1c+c 2 βi (k)T + c − c bi (k)T 1

1

1

1

+c2 αi (k)pi1 (k)T − c1 c2 αi (0)T [L(0)]ii ≥ c2 μi pi1 (0)T − cc2 (1 − cT ) − c1 c2 dimax T > 0. 1

1

Hence, Ψ (0) is a nonnegative matrix and it is clear that Ψ (0)1 = 0, which suggests that Ψ (0) is a stochastic matrix and φ (1) = [Ψ (0) ⊗ Ir ]φ (0) is the convex combination of φ j (0), thus, φ j (1) ≤ maxs {φs (0)} and αi (1) ≥ μi . So Ai31 (1) − c1 c2 αi (1)T dimax ≥ c2 μi pi1 (1)T − cc21 (1 − cT1 ) − c1 c2 dimax T > 0. Thus, Ψ (1) is also a stochastic matrix. By induction, we can conclude that Ψ (k) is a stochastic matrix and αi (k) ≥ μi for any k ≥ 0. Lemma 2. Under Assumptions 1, 2 and 3. For any i ∈ {1, 2, · · · , (3m + 2n)(M + 1)} and s ≥ 0, there is a constant 1 ≥ ρi (s) ≥ 0 such that lim [Γ (k, s)] ji = ρi (s) for any j (3m+2n)(M+1)

and ∑i=1

k→∞

ρi (s) = 1, where Γ (k, s) = Ψ (k − 1)Ψ (k − 2) · · · Ψ (s), s < k.

Proof. Let G(k, s) represent the directed graph generated by Γ (k, s). It follows from Assumption 3 that graph G(kr , kr−1 ) has a directed spanning tree. By imitating the proof of Lemma 4 in [15], we can prove this lemma. Theorem 1. Under Assumptions 1, 2 and 3. The control law (3) can drive all agents reach agreement in positions states while the velocity states of second-order agents and control inputs remain in their corresponding nonconvex constrained sets.

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Proof. For all s,t, i ∈ {1, 2, · · · , (3m + 2n)(M + 1)}, it follows from Lemma 2 that lim |[Γ (k, 0)]si − [Γ (k, 0)]ti | = 0. Hence,

t→∞

lim [φs (k) − φt (k)]

t→∞

(3m+n)(M+1)

[Γ (k, 0)]si φi (0) − ∑i=1

(3m+n)(M+1)

([Γ (k, 0)]si − [Γ (k, 0)]ti )φi (0)]

= lim [∑i=1 k→∞

= lim [∑i=1 k→∞

(3m+n)(M+1)

[Γ (k, 0)]ti φi (0)]

= 0.

It follows from the definition of φi (k) that limt→∞ [ri (k) − r j (k)] = 0 for any i, j ∈ {1, 2, · · · , m + n}. In addition, it is easy to see that the inputs remain in their corresponding nonconvex constrained sets by the form of control law (3).

5 Simulations In this section, we will give an example to illustrate the effectiveness of the distributed algorithm proposed in this paper. Figure 1 shows the communication switching topologies, where agent 1 to 3 are second order agents and agent 4 and 5 are first 1]T | 1]T | ≤ 1} ∪ {x|x ≤ 1, |x[0 ≥ 1}, order agents. Assume Ui = Vi = {x|x ≤ 0.8, |x[0 |x[1 0]T | |x[1 0]T | and τ12 (k) = 3; τ13 (k) = 2; τ14 (k) = 1; τ15 (k) = 3; τ 21(k) = 2; τ23 (k) = 1; τ24 (k) = 3; τ25 (k) = 3; τ31 (k) = 2; τ32 (k) = 3; τ34 (k) = 1; τ35 (k) = 3; τ41 (k) = 3; τ42 (k) = 1; τ43 (k) = 1; τ45 (k) = 2; τ51 (k) = 3; τ52 (k) = 2; τ53 (k) = 2; τ54 (k) = 3. The sample time is T = 0.05. Besides, pi1 (k) = 12, pi2 (k) = 8 for i = 1, 2, 3 and all k, pi1 (k) = 6, pi2 (k) = 3 for i = 4, 5 and all k. Suppose the initial values are r1 (s) = [0.25, 0.15]T , r2 (s) = [0.2, 0.35]T , r3 (k) = [0.15, 0.25]T , r4 (s) = [−0.25, −0.1]T , r5 (s) = [−0.2, −0.25]T and the initial input ui (s) = 0 for all s ≤ 0 and i. It is easy to verify that the conditions of Theorem 1 hold. Figure 2 shows all agents’ position states. Figure 3 shows control inputs in the phase plane. From these simulation results, we can see that the all agents reach an agreement on their position state while the control inputs are constrained to stay in the corresponding nonconvex sets.

Fig. 1. Communication topologies.

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Fig. 2. Trajectories of position states of all agents.

Fig. 3. Trajectories of inputs of all agents.

6 Conclusions This paper is devoted to the nonconvex constrained consensus problem of heterogeneous multi-agent systems with nonconvex velocity and input constraints. By introducing two contraction operators, a new distributed control law is designed. Based on a model transformation, the closed-loop nonlinear system is changed into a time varying linear system. Then the nonconvex constrained consensus analysis is completed with the help of stochastic matrix theory. Finally, the simulation results are given to demonstrate the effectiveness of control law proposed in this paper.

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Acknowledgments. This work was supported by the National Natural Science Foundation of China (No. 61973329), the Beijing Educational Committee Foundation (No. KM201910011007, PXM2019 014213 000007) and the Beijing Natural Science Foundation (No. Z180005).

References 1. Moreau L (2005) Stability of multi-agent systems with time-dependent communication links. IEEE Trans Autom Control 50(2):169–182 2. Jia Y (2003) Alternative proofs for improved LMI representations for the analysis and the design of continuous-time systems with polytopic type uncertainty: a predictive approach. IEEE Trans Autom Control 48(8):1413–1416 3. Ren W, Beard R (2005) Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Trans Autom Control 50(5):655–661 4. Lin P, Jia Y (2009) Consensus of second-order discrete-time multi-agent systems with nonuniform time-delays and dynamically changing topologies. Automatica 45(9):2154– 2158 5. Lin P, Ren W, Song Y (2016) Distributed multi-agent optimization subject to nonidentical constraints and communication delays. Automatica 65:120–131 6. Zhao L, Yu J, Lin C (2017) Distributed adaptive fixed-time consensus tracking for secondorder multi-agent systems using modified terminal sliding mode. Appl Math Comput 312:23–35 7. Zhang B, Jia Y (2017) Task-space synchronization of networked mechanical systems with uncertain parameters and communication delays. IEEE Trans Cybern 47(8):2288–2298 8. Mo L, Guo S (2019) Consensus of linear multi-agent systems with persistent disturbances via distributed output feedback. J Syst Sci Complex 32(3):835–845 9. Lin P, Ren W, Wang H, Al-Saggaf UM (2019) Multiagent rendezvous with shortest distance to convex regions with empty intersection: algorithms and experiments. IEEE Trans Cybern 49(3):1026–1034 10. Jia Y (2000) Robust control with decoupling performance for steering and traction of 4WS vehicles under velocity-varying motion. IEEE Trans Control Syst Technol 8(3):554–569 11. Nedi´c A, Ozdaglar A, Parrilo PA (2010) Constrained consensus and optimization in multiagent networks. IEEE Trans Autom Control 55(4):922–938 12. Lin P, Ren W (2014) Constrained consensus in unbalanced networks with communication delays. IEEE Trans Autom Control 59(3):775–781 13. Lin P, Ren W, Farrell JA (2017) Distributed continuous-time optimization: nonuniform gradient gains, finite-time convergence, and convex constraint set. IEEE Trans Autom Control 62(5):2239–2253 14. Lin P, Ren W (2017) Distributed H∞ constrained consensus problem. Syst Control Lett 104:45–48 15. Lin P, Ren W, Gao H (2017) Distributed velocity-constrained consensus of discrete-time multi-agent systems with nonconvex constraints, switching topologies, and delays. IEEE Trans Autom Control 62(11):5788–5794 16. Lin P, Ren W, Yang C et al (2019) Distributed optimization with nonconvex velocity constraints, nonuniform position constraints and nonuniform stepsizes. IEEE Trans Autom Control 64(6):2575–2582 17. Mo L, Lin P (2018) Distributed consensus of second-order multiagent systems with nonconvex input constraints. Int J Robust Nonlinear Control 28(11):3657–3664 18. Lin P, Ren W, Yang C et al (2018) Distributed consensus of second-order multiagent systems with nonconvex velocity and control input constraints. IEEE Trans Autom Control 63(4):1171–1176

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19. Zheng Y, Zhu Y, Wang L (2011) Consensus of heterogeneous multi-agent systems. IET Control Theory Appl 5(16):1881–1888 20. Feng Y, Xu S, Lewis F, Zhang B (2015) Consensus of heterogeneous first- and second-order multi-agent systems with directed communication topologies. Int J Robust Nonlinear Control 25(3):362–375 21. Mo L, Niu G, Pan T (2015) Consensus of heterogeneous multi-agent systems with switching jointly-connected interconnection. Phys A 427:132–140 22. Lin P, Ren W, Yang C, Gui W (2019) Distributed optimization with nonconvex velocity constraints, nonuniform position constraints, and nonuniform stepsizes. IEEE Trans Autom Control 64(6):2575–2582 23. Guo S, Mo L, Yu Y (2018) Mean-square consensus of heterogeneous multi-agent systems with communication noises. J Franklin Inst 355:3717–3736 24. Mo L, Guo S, Yu Y (2018) Mean-square consensus of heterogeneous multi-agent systems with nonconvex constraints, Markovian switching topologies and delays. Neurocomputing 291:167–174 25. Huang H, Mo L, Cao X (2019) Nonconvex constrained consensus of discrete-time heterogeneous multi-agent systems with arbitrarily switching topologies. IEEE Access 7:38157– 38161 26. Godsil C, Royle G (2001) Algebraic Graph Theory. Springer, New York

The Improved Intelligent Optimal Algorithm Based on the Artificial Bee Colony Algorithm and the Differential Evolution Algorithm Jingyi Li1 , Ju Cheng2(B) , and Lipo Mo2 1

School of Mathematics and Systems Science, Beihang University, Beijing 100083, China 2 School of Mathematics and Statistics, Beijing Technology and Business University, Beijing 100048, China [email protected]

Abstract. This paper proposes an improved ABC algorithm to avoid the phenomenon of premature convergence of the basic artificial bee colony (ABC) algorithm. By combining the mutation operators of differential evolution (DE) algorithm, the new search equations of employed bees and onlookers are designed. Besides, a selective probability is leaded into the former one and a development coefficient is added in the latter one. In order to demonstrate the availability of the proposed ABC algorithm, simulations are conducted on a set of benchmark functions. The simulation results show that the improved ABC algorithm performs better than the basic ABC algorithm, DE algorithm and particle swarm optimization (PSO) algorithm. Keywords: Artificial bee colony algorithm · Local convergence · Differential evolution algorithm

1 Introduction The artificial bee colony [1] (ABC) algorithm is a new stochastic optimization algorithm based on swarm intelligence, proposed by Karaboga in 2005. It simulates the foraging behavior of honeybee colonies. In the ABC model, the colony consists of three types of bees: employed bees, onlookers and scouts. They execute different instructions according to their respectively division of work and realize information exchange, thus gaining the optimal solution of the problem. At present, the research and application of ABC algorithm are still at early stage with the lack of theoretical proof. Nevertheless, the ABC algorithm still arouse the attention from scholars due to its simple calculation procedure, fewer parameters, simple structure and easy to implement. Nowadays, the ABC algorithm has been successfully applied in many fields: the optimization of function parameter, chaotic system control [2], digital filter design [3], image processing and analysis [4] and so on. However, there are still some drawbacks with standard ABC algorithm, such as the slow convergence rate, easily getting stuck at local optima, failing to find the optimum c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 111–121, 2020. https://doi.org/10.1007/978-981-32-9682-4_12

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solution for some complex problems, strong searching ability but weak capability of development, which severely limit the practical applications of ABC algorithm. For the drawbacks of the standard ABC algorithm, scholars have proposed many improvement to optimize the performance of ABC algorithm. Based on differential evolution (DE) algorithm, Gao and Liu [5] proposed two modified algorithms with different search equations called “ABC/best/1” and “ABC/best/2” respectively, where the two equations have strong development ability but lack of the ability of exploration. Then, they proposed a modified search equation to improve the ABC algorithm in literature [6], which is similar to the crossover of genetic algorithms and called the crossover artificial bee colony (CABC) algorithm. Nseef et al. [7] divided the population into several subpopulations and run ABC algorithm on sub-populations independently and simultaneously and the adaptive multi-population ABC algorithm proposed there has higher efficiency compared to the single-population one. The combination of ABC algorithm and other algorithms or techniques has been particularly popular by the researchers. Mustafa [8] combines ABC algorithm with particle swarm optimization algorithm (PSO) to design a hybrid algorithm. Chen [9] introduced the simulated annealing operator in the searching process of employed bees which can improve the algorithm’s development capability. However, the above algorithms can hardly get the optimal solution precisely when solving practical optimal problems, in other words, it is difficult to balance the searching ability and the capability of development simultaneously. To deal with this difficulty, the paper presents a new modified ABC algorithm called WPABC for improvement. All the data and experiments herein are implemented by MATLAB programming.

2 An Improved Artificial Bee Colony Algorithm The standard ABC algorithm has strong exploration ability but weak capability of development. As is known to all, the ability of neither exploration nor development can be omitted. However, these two kinds of ability are incompatible in practical situations. Therefore, how to balance the capability of exploration and development is the key point for intelligent algorithm to obtain high performance. Global searching algorithm and local searching algorithm are analogous to exploration and development. The employed bees emphasize global searching and onlookers pay attention to the local one. Hence, the target of this article is to strengthen exploration ability of the employed bees, and to enhance the capability of development of the onlookers. For the disadvantages of the standard ABC algorithm, the article makes the following improvements and proposes the WPABC algorithm based on the DE algorithm. The pseudocode of WPABC is shown in Algorithm 1. (1) The strong exploration ability but weak development ability leads to poor convergence performance of the standard ABC algorithm. As a result, it is easily to trap into local optima. In order to enhance the exploitation ability of the algorithm, we raise the percentage of onlookers in the population. In the standard ABC algorithm, the percentage of onlookers is 0.5. When we lift the proportion of onlookers to 1, the development ability improves. However, if the proportion of onlookers is too high, the development capability will reduce on the contrary. We use 5 test functions with 30-dimensional:

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Sphere, Schwefel2.22, Schwefel2.21, Rastrigin and Ackley to examine how the proportion of onlookers affects the algorithm. Calculate the standard deviation (Std) and the mean best (Mean). The results are shown in Table 1. As we can see, the experimental result is comparatively better when the proportion of onlookers is 0.7. Hence, we suppose the number of the employed bees is CP = 0.3 ∗ NP and the number of onlookers is GP = 0.7 ∗ NP. Algorithm 1. The pseudo-code of WPABC algorithm. 1: Step 1) Initialization: Step 1.1) Generate CP individuals randomly in the search space to form initial population. Step 1.2) Calculate the objective function value of these individuals Step 1.3) iter = 1 2: Step 2) The stage of gather honey bee: for i = 1, 2, . . . ,CP Step 2.1) generate a new candidate solution V1 by using equation (8), and calculate the adaptive value f itVi Step 2.2) if f itVi > f itXi X1 = V1 trial1 = 1 elseif rand(0, 1) < P Step 2.2.1) generate a new candidate solution V1 by using equation (7), and calculate the adaptive value f itVi Step 2.2.2) if f itVi > f itXi X1 = V1 trial1 = 1 else trial1 = trial1 + 1 else trial1 = trial1 + 1 3: Step 3) Using the equation (2) to calculate the value of pv , let t = 0, i = 1 4: Step 4) The stage of observation bee: While t ≤ GP Step 4.1) if rand(0, 1) < p1 Step 4.1.1) generate a new candidate solution V1 by using equation (10), and calculate the adaptive value f itVi Step 4.1.2) if f itVi > f itXi X1 = V1 trial1 = 1 else trial1 = trial1 + 1 Step 4.1.3) let t = t + 1 Step 4.2) Let i = i + 1 if i = CP + 1 i = 1 end 5: Step 5) The stage of scout bee: if maxtrial1 > limit, generate a solution randomly by using equation (3) to replace X1 6: Step 6) iter = iter + 1 7: Step 7) if iter ≥ maxCycle, end the operation, output the optimal solution and optimal individual. else go to Step 2)

(2) In the standard ABC algorithm, the employed bees and onlookers use the same search equation. As a consequence, the algorithm is inefficient and has poor experimental results. For the trait that the employed bees emphasize on exploration and onlookers pay more attention to development, we improve the search equations respectively. We also learn from DE algorithm to enhance the exploration ability of search equations in the stage of employed bees. The vectors in formula (1) are random vectors with the feature of randomness. Therefore, formula (1) has extremely strong ability of exploration. Due to the guidance of the best individual, formula (2) has improved its development ability and simultaneously possesses the ability of exploration. But compared to formula (1), the exploration capability of formula (2) has decreased. Inspired by DE mutation operators, “DE/rand/1” and “DE/current-to-best/1”, we propose two new search equations of the employed bees in the article. The equations are as follows. WPABC/rand/1: (1) Vi, j = Xr1 , j + ϕi, j (Xr2 , j − Xr3 , j ). WPABC/current-to-best/1: Vi, j = Xi, j + ϕi, j (Xbest, j − Xi, j ) + ϕi, j (Xr1 , j − Xr2 , j ),

(2)

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Table 1. The influence of the proportion of onlookers on the performance of the algorithm WPABC. GP = 0.5NP GP = 0.6NP GP = 0.7NP GP = 0.8NP GP = 0.9NP

Function sphere

Mean 5.61E−16

4.28E−16

3.21E−16

3.40E−16

1.69E−07

Std

8.14E−17

7.56E−17

1.09E−16

6.17E−07

8.32E−17

schwefel2.22 Mean 1.33E−15

1.25E−15

1.00E−15

3.33E−06

6.60E−03

Std

1.20E−16

1.24E−16

1.82E−05

3.37E−02

1.72E−16

schwefel2.21 Mean 1.94E+00

1.81E−01

8.34E−01

2.10E+00

3.65E+00

Std

5.87E−02

2.35E−01

3.53E−01

6.03E−01

rastrigin ackley

1.91E+00

Mean 2.95E−01

4.04E−09

0.00E+00

0.00E+00

3.22E−14

Std

2.21E−08

0.00E+00

0.00E+00

2.86E−14

5.88E−01

Mean 1.38E+00

5.72E−11

2.55E−14

3.05E−14

3.06E−14

Std

1.51E−10

3.48E−15

2.35E−15

2.87E−15

1.39E+00

where ϕi, j is a random number in [−1, 1]. r1 = r2 = i are the randomly selected integers in the range of {1, 2, · · · , NP}. Formula (4) has stronger exploration ability and formula (5) has stronger capability of development. In order to balance the ability of exploration and development, we introduce the selection probability P. That is, in each iteration, we select one of the two search equations with a probability of P. The selection probability P plays an important role in balancing the ability of exploration and development. When P = 0, only formula (5) works. The exploration ability increases by the increment of P. And when P = 1, only formula (4) works. We use 5 test functions with 30-dimensional: Sphere, Schwefel2.22, Schwefel2.21 to examine how the proportion of onlookers affects the algorithm. We set the maximum number of iterations 1000 and each function is optimized 30 times independently. Calculate the standard deviation and the mean best. The results are shown in Table 2. As we can see, the experimental result is comparatively better when the proportion of onlookers is 0.7. Hence, we set P = 0.7 in this paper. Table 2. The influence of the selected probability P on the performance of the algorithm WPABC. P = 0.0

Function sphere

P = 0.1

P = 0.3

P = 0.5

P = 0.7

P = 0.9

P=1

Mean 1.84E−11 3.80E−16 3.53E−16 3.40E−16 3.21E−16 3.30E−16 3.43E−16 Std

1.01E−10 9.18E−17 7.74E−17 7.71E−17 7.56E−17 5.84E−17 7.58E−17

schwefel2.22 Mean 1.02E−15 9.51E−06 9.46E−16 1.04E−15 1.00E−15 1.00E−15 1.02E−15 Std

1.50E−16 5.21E−05 1.35E−16 1.28E−16 1.24E−16 1.38E−16 1.50E−16

schwefel2.21 Mean 1.44E+00 Std

1.36E+00

1.10E+00

9.60E−01 8.34E−01 7.20E−01 6.42E−01

3.91E−01 3.74E−01 2.94E−01 2.19E−01 2.35E−01 1.69E−01 1.57E−01

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(3) In order to improve the development ability in the stage of onlookers, we also use the mutation operator of DE algorithm. Due to the new candidate solution generated near Xbest in formula (3), the development ability is stronger than formula (1). To further improve the capability of development, GAO and other scholars increase the guidance of Xbest based on formula (3). Combined with formula (2), we propose the following search equation. ABC/best/double (3) Vi, j = Xbest, j + ϕi, j (Xbest, j − Xk, j ), where k,j are randomly selected subscripts and satisfy that: k ∈ {1.2, · · · ,CP}, j ∈ {1, 2, · · · , D}, k = i, k is the random number in [−1, 1], while Xbest is dominant and the candidates generate near the Xbest . Although the algorithm has strong development capability, it easily plunges into local optima. For the disadvantages of the ABC/best/double algorithm, we consider adding a development coefficient (1 − W ) to make Xbest has little impact on the algorithm. However, as the number of iterations increases, Xbest gradually approaches to the global optima. The impact of Xbest gradually increases and finally holds the dominant position. The modified search equation of this article is shown as below. Vi, j = (1 −W ) ∗ Xbest, j + ϕi, j (Xbest, j − Xk, j ) +W ∗ Xk, j , W=

1 1

(ln(iter + 2)) a

(4)

.

(5)

In order to keep balance of search equations, we add a new item W ∗ Xk, j , where W is the monotone decreasing function in (0, 1). iter is the number of iterations with its value greater than zero. Parameter a plays an important role in the change of coefficient W . To maximize the performance of the algorithm, we use 3 test functions with 30-dimensional: Sphere, Schwefel2.22, Schwefel2.21 to examine how the parameter a affects the algorithm. We select a few integers as the value of a and the result is shown in Table 3. As we can see, the experimental result is comparatively better when a = 3. As the value of a increases after a = 4, the result is getting worse and worse on the contrary. Hence, we set a = 3. Table 3. The influence of the parameter a on the performance of the algorithm WPABC. Function sphere

a=2

a=3

a=4

a=5

a = 10

Mean 2.75E−12 3.21E−16 3.32E−16 3.31E−16 3.55E−16 Std 1.51E−11 7.56E−17 7.18E−17 7.16E−17 8.46E−17

schwefel2.22 Mean 1.01E−15 1.00E−15 1.03E−15 9.86E−16 1.03E−15 Std 1.31E−16 1.24E−16 1.28E−16 1.65E−16 1.38E−16 schwefel2.21 Mean 8.72E−01 8.26E−01 8.00E−01 8.94E−01 8.47E−01 Std 2.51E−01 2.12E−01 2.06E−01 2.22E−01 2.04E−01

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3 Simulation Experiment 3.1

Test Functions and Parameter Settings

To verify the effectiveness of the proposed WPABC algorithm, we select 8 test functions with 30-dimensional or 60-dimensional, 1 test function with 100-dimensional or 200dimensional,1 test function with 10-dimensional. Table 4 is a brief introduction of the test functions used in this article. Appendix A give us their detailed expressions. Simulation experiments of the WPABC algorithm were performed with all the test functions by using MATLAB software. Suppose that the population size is 100, limit = 0.4 ∗ NP/2 ∗ D, the maximum number of iteration is 1000. Each algorithm runs independently 30 times on each test function. The optimal value (Best), Median, Worst, Mean, and Standard deviation (Std) are calculated. Among them, Best, Median and Worst reflect the quality of solutions. Mean reflects the accuracy of the given iteration times and the convergence speed of the algorithm, while Std reflects the stability. 3.2

Selection of Parameter Limit

Parameter limit has important influence on the exploration and development ability of balanced algorithm. Similar to the method in literature [10–13]. We select 5 test functions with 30-dimensional to test the effect of parameter limit on the algorithm. The maximum number of iterations is 1000. The results are shown in Table 5. The function f4 and f6 get better results when limit = 0.4 ∗ NP/2 ∗ D. For the other 3 functions, the performance of the algorithm is influenced less by the parameter limit. Therefore, in this article, the value limit is 0.4 ∗ NP/2 ∗ D. Table 4. The dimension of the test function, the search space, and the optimal value. Function Name

Dimension Search space

Optimal value

f1

Sphere

30, 60

[−100, 100]D

f2

Levy

30, 60

[−10, 10]D

0 0

0

f3

SumSquare

30, 60

[−10, 10]D

f4

Schwefel 2.22 30, 60

[−10, 10]D

0

f5

Schwefel 2.21 30, 60

[−100, 100]D

0

f6

Rastrigin

30, 60

[−5.12, 5.12]D 0

f7

Ackley

30, 60

[−32, 32]D

0 0

f8

Alpine

30, 60

[−10, 10]D

f9

Masters

10

[−5, 5]D

0 −78.33236 0

f10

Himmelblau

100, 200

[−5, 5]D

f11

Elliptic

30, 60

[−100, 100]D

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Table 5. The influence of parameter limit on the performance of the WPABC algorithm. Function

0.2*NP/2*D 0.4*NP/2*D 0.6*NP/2*D 0.8*NP/2*D NP/2*D

f1 Mean 2.77E−16 Std 7.87E−17

3.65E−16 7.65E−17

4.39E−16 7.81E−17

4.78E−16 1.47E−16

4.50E−16 7.51E−17

f4 Mean 8.11E−16 Std 1.40E−16

1.02E−15 1.17E−16

1.05E−15 1.07E−16

1.24E−15 1.16E−16

1.27E−15 1.43E−16

f5 Mean 8.30E−01 Std 2.00E−01

8.52E−01 2.68E−01

8.43E−01 2.41E−01

8.11E−01 2.25E−01

8.04E−01 1.99E−01

f6 Mean 0.00E+00 Std 0.00E+00

0.00E+00 0.00E+00

1.68E−09 9.18E−09

2.27E−14 1.25E−13

3.32E−02 1.82E−01

f7 Mean 2.35E−14

2.72E−14

2.61E−14

2.98E−14

3.25E−14

3.91E−15

4.10E−15

2.23E−15

4.10E−15

Std

3.55E−15

3.3 Experimental Results To further demonstrate the advantages of the new algorithm WPABC, we compare it with standard ABC algorithm, particle swarm algorithm and differential evolution algorithm. Statistical results are shown in Tables 6 and 7. As seen form the tables, the simulation results of WPABC algorithm for most functions are obviously better than the standard ABC algorithm, differential evolution algorithm and particle swarm optimization algorithm, except for the functions f5 , f9 and f10 . For the function f9 , WPABC algorithm has obtained the suboptimal result. For the function f9 , although there’s no obvious difference for the four algorithms on average for the function f9 , the algorithm WPABC gets a smaller standard deviation. 3.4 Performance Analysis In order to illustrate the optimization effect of WPABC algorithm more intuitively and get a better perspective, we compare it with the standard ABC algorithm. Figure 1 are the convergence curves for the related six test functions plotted by the two algorithms ABC and WPABC, respectively. It is not hard to see from the figures that the WPABC algorithm is greatly improved compared with the standard ABC algorithm, both the calculation accuracy and the convergence speed of the algorithm.

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3.50E−13 1.25E−11 2.63E−12 3.68E−12 3.20E−12

WPABC 2.44E−16 4.81E−16 3.06E−16 3.26E−16 6.68E−17 PSO

3.19E−03 2.17E−04 9.24E−04 1.14E−03 8.11E−04

DE

6.96E−02 3.36E−01 1.59E−01 1.80E−01 7.40E−02

60 ABC

1.74E−06 4.83E−05 7.62E−06 1.13E−05 1.05E−05

WPABC 1.14E−15 1.61E−15 1.37E−15 1.32E−15 1.22E−16 PSO

1.04E−01 3.42E−02 6.28E−02 6.65E−02 1.65E−02

DE

6.97E+01

f2 30 ABC

1.68E+02

1.23E+02

1.21E+02

2.42E+01

1.73E−14 3.26E−12 3.78E−13 7.04E−13 7.93E−13

WPABC 7.92E−17 2.91E−16 2.18E−16 2.10E−16 5.04E−17 PSO

6.50E−01 1.94E−04 1.11E−01 1.21E−01 1.70E−01

DE

1.10E−03 7.20E−03 2.75E−03 3.10E−03 1.53E−03

60 ABC

1.66E−06 7.37E−05 6.70E−06 9.72E−06 1.32E−05

WPABC 9.68E−16 1.63E−15 1.40E−15 1.36E−15 1.32E−16 PSO

1.85E+01

2.13E−01 3.70E+00

5.93E+00

5.49E+00

DE

1.86E+00

4.05E+00

2.68E+00

7.17E−01

f3 30 ABC

2.47E+00

1.02E−13 1.46E−12 5.24E−13 6.15E−13 3.97E−13

WPABC 2.02E−16 4.55E−16 3.11E−16 3.22E−16 7.48E−17 PSO

6.34E−02 3.16E−03 1.70E−02 1.99E−02 1.40E−02

DE

9.10E−03 9.73E−01 1.55E−02 6.48E−02 2.14E−01

60 ABC

4.35E−07 3.99E−05 5.30E−06 6.40E−06 6.88E−06

WPABC 9.92E−16 1.64E−15 1.41E−15 1.38E−15 1.50E−16 PSO

7.17E+00

5.92E−01 1.99E+00

2.25E+00

1.18E+00

DE

1.29E+01

3.70E+01

2.24E+01

6.57E+00

f4 30 ABC

2.00E+01

2.18E−07 7.34E−07 4.88E−07 4.78E−07 1.34E−07

WPABC 7.59E−16 1.19E−15 9.45E−16 9.78E−16 1.25E−16 PSO

1.87E+00

DE

3.84E−01 1.82E+00

60 ABC

1.36E−01 5.73E−01 6.89E−01 4.60E−01 7.61E−01 8.82E−01 3.78E−01

2.00E−03 7.90E−03 3.70E−03 4.10E−03 1.70E−03

WPABC 8.73E−15 5.49E−14 1.91E−14 2.44E−14 1.23E−14 PSO

1.10E+01

3.07E+00

4.65E+00

5.15E+00

1.80E+00

DE

1.49E+01

5.00E+01

1.83E+01

1.97E+01

7.64E+00

8.87E+00

2.45E+01

1.89E+01

1.89E+01

4.03E+00

f5 30 ABC

WPABC 6.28E−01 1.26E+00

8.75E−01 9.18E−01 1.99E−01

PSO

6.13E−01 9.67E−02 2.10E−01 2.65E−01 1.46E−01

DE

3.84E−01 1.82E+00

7.61E−01 8.82E−01 3.78E−01

5.43E+01

6.64E+01

6.07E+01

6.03E+01

3.45E+00

WPABC 2.37E+01

3.40E+01

2.87E+01

2.87E+01

2.23E+00

PSO

3.92E+00

1.32E+00

2.28E+00

2.40E+00

7.67E−01

DE

1.49E+01

5.00E+01

1.83E+01

1.97E+01

7.64E+00

60 ABC

The Improved Intelligent Optimal Algorithm Table 7. Performance comparison of the four algorithms: ABC, WPABC, PSO and DE. f6

30

60

f7

30

60

f8

30

60

f9

10

ABC WPABC PSO DE ABC WPABC PSO DE

5.73E−10 0.00E+00 5.59E+01 1.91E+02 3.05E+00 0.00E+00 1.27E+02 1.67E+02

9.97E−01 0.00E+00 1.60E+01 2.26E+02 1.64E+01 0.00E+00 4.13E+01 5.73E+02

1.12E−07 0.00E+00 3.25E+01 2.14E+02 1.15E+01 0.00E+00 7.14E+01 5.21E+02

7.82E−02 0.00E+00 3.36E+01 2.13E+02 1.12E+01 0.00E+00 7.36E+01 5.07E+02

2.58E−01 0.00E+00 1.03E+01 9.07E+00 3.29E+00 0.00E+00 2.06E+01 8.30E+01

ABC WPABC PSO DE ABC WPABC PSO DE

2.46E−06 1.87E−14 4.81E+00 2.08E+01 1.94E−02 2.28E−13 5.56E+00 2.11E+01

2.75E−05 2.93E−14 1.65E+00 2.10E+01 4.24E−01 5.44E−13 3.32E+00 2.12E+01

5.87E−06 2.40E−14 2.70E+00 2.09E+01 7.53E−02 3.79E−13 4.24E+00 2.12E+01

6.86E−06 2.53E−14 2.85E+00 2.09E+01 1.15E−01 3.73E−13 4.28E+00 2.12E+01

5.01E−06 3.82E−15 7.00E−01 5.33E−02 1.09E−01 8.00E−14 6.15E−01 3.12E−02

ABC WPABC PSO DE ABC WPABC PSO DE

3.44E−05 7.40E−16 1.76E−01 3.77E−01 2.16E−02 1.31E−14 8.17E−01 1.72E+01

6.41E−04 2.94E−15 1.07E−02 9.28E+00 2.43E−01 5.34E−14 1.40E−01 3.49E+01

1.05E−04 1.20E−15 3.23E−02 4.10E+00 1.18E−01 2.64E−14 2.22E−01 2.52E+01

1.71E−04 1.44E−15 4.26E−02 4.37E+00 1.19E−01 2.82E−14 3.07E−01 2.59E+01

1.62E−04 5.56E−16 3.32E−02 2.37E+00 6.36E−02 1.07E−14 1.79E−01 5.45E+00

ABC WPABC PSO DE

1.19E−11 0.00E+00 3.24E+00 0.00E+00

7.91E−01 1.02E−06 1.05E+00 0.00E+00

2.69E−07 1.78E−15 2.36E+00 0.00E+00

1.23E−01 3.40E−08 2.20E+00 0.00E+00

2.83E−01 1.86E−07 4.88E−01 0.00E+00

−7.70E+01 −7.83E+01 −6.23E+01 −4.31E+01 −7.14E+01 −7.83E+01 −6.11E+01 −3.70E+01

−7.50E+01 −7.83E+01 −6.78E+01 −3.88E+01 −6.89E+01 −7.82E+01 −6.58E+01 −3.38E+01

−7.60E+01 −7.83E+01 −6.55E+01 −4.00E+01 −7.02E+01 −7.83E+01 −6.33E+01 −3.51E+01

−7.60E+01 −7.83E+01 −6.55E+01 −4.04E+01 −7.02E+01 −7.83E+01 −6.33E+01 −3.52E+01

5.32E−01 1.67E−13 1.35E+00 1.26E+00 6.55E−01 2.58E−02 1.14E+00 9.77E−01

f10 100 ABC WPABC PSO DE 200 ABC WPABC PSO DE

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Fig. 1. Convergence performance comparison of the 6 functions.

4 Conclusions We proposed an improved artificial bee colony algorithm named WPABC in brief, by simulation experiment, we compared the WPABC algorithm with other algorithms, from which we could see that the stability and convergence speed of this algorithm has improved significantly. Our future work will focus on how to design a more reasonable intelligent algorithm to show better performance on more test functions and apply it to real optimal problems.

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Acknowledgments. This work was supported by the Beijing Educational Committee Foundation (No. KM201910011007, PXM2019 014213 000007) and the Beijing Natural Science Foundation (No. Z180005).

References 1. Karaboga D (2005) An idea based on honey bee swarm for numerical optimization. Turkey: Erciyes University, Technical Report-TR06 2. Hu W, Yu YG, Wang S (2015) Parameters estimation of uncertain fractional-order chaotic systems via a modified artificial bee colony algorithm. Entropy 17:692–709 3. Karaboga N (2009) A new design method based on the artificial bee colony algorithm for digital IIR filters. J Franklin Inst 346(4):328–348 4. Horng MH (2011) Multilevel thresholding selection based on the artificial bee colony algorithm for image segmentation. Expert Syst Appl 38(11):13785–13791 5. Cao WF, Liu SY, Huang LL (2012) A global best artificial bee colony algorithm for global optimization. J Comput Appl Math 236(11):2741–2753 6. Cao WF, Liu SY, Huang LL (2013) A novel artificial bee colony algorithm based on modified search equation and orthogonal learning. IEEE Trans Cybern 43:1011–1024 7. Nseef K, Abdullah S, Turky A et al (2016) An adaptive multi-population artificial bee colony algorithm for dynamic optimisation problems. Knowl-Based Syst 104:14–23 8. Mustafa SK, Mesut G (2013) A recombination-based hybridization of particle swarm optimization and artificial bee colony algorithm for continuous optimization problems. Appl Soft Comput 4(13):2188–2203 9. Chen SM, Sarosh A, Dong YF (2012) Simulated annealing based artificial bee colony algorithm for global numerical optimization. Appl Math Comput 219:3575–3589 10. Storn R, Price K (2010) Diferential evolution-a simple and efcient heuristic for global optimization over continuous spaces. J Global Optim 23:689–694 11. Gao WF, Liu SY (2012) A modified artificial bee colony algorihm. Comput Oper Res 39(3):687–697 12. Gao WF (2013) Artificial bee colony algorithm and its application. Xidian University, Xi’an 13. Shan H, Yasuda T, Ohkura K (2015) A self adaptive hybrid enhanced artificial bee colony algorithm for continuous optimization problems. BioSystems 132–133:43–53

Semantic Analysis Using Convolutional Neural Networks for Assisted Driving Fen Zhao1 , Penghua Li2(B) , Yinguo Li1,2 , and Yuanyuan Li2 1

School of Computer Science and Technology, Chongqing University of Posts and Telecommunications, Chongqing 400065, China [email protected] 2 School of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065, China {liph,liyg,liyy}@cqupt.edu.cn

Abstract. As self-driving vehicles have grown in sophistication and ability, they have been deployed on the road in regular private vehicles. This paper presents a new semi-supervised multi-granularity convolutional neural networks (SSMGCNNs) framework for semantic analysis in designing human-vehicle interaction, which is further composed of two parts: a two-view-embedding (TVE) module, and the multi-granularity convolutional neural networks (MGCNNs) module. SSMGCNNs learn embeddings of text regions from the unlabeled user instruction dataset and then integrate the learned tv-embedding into MGCNNs, so that the learned tv-embedded regions are used as an additional input to MGCNNs convolution layer to solve the problem of data annotation. MGCNNs can fully extract feature information hidden in text through multiple convolution kernels of different sizes in the same convolution layer. The experimental results demonstrate the effectiveness of our approach in comparison with baseline systems. . . . Keywords: Semi-supervised multi-granularity convolutional neural networks (SSMGCNNs) · Semantic analysis · Human-vehicle interaction · TV-embedding

1

Introduction

With the progress of the affordable sensing and computing technologies, a wide variety of intelligent technologies have become commercialization recently. The most widely used area of the intelligent application is the area of self-driven cars, which focuses on increasing autonomy in the transportation systems [1]. To reach human-level reasoning of sensing, mapping, and driving policies, which are called the three components of autonomous driving, a great deal of work has been made [2]. At present, in-vehicle intelligent technologies are becoming ubiquitous. Drivers use these intelligent applications because they perceive them as providing valuable services. It is common that drivers operate a car and, meanwhile, c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 122–130, 2020. https://doi.org/10.1007/978-981-32-9682-4_13

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interact with other devices. These in-vehicle intelligent technologies not only entertain the drivers, but assist drivers in performing operation task in a driving vehicle [3]. Cars take over driving operation, but rely on human drivers monitoring work. At present, all assisted car systems require human intervention in particular situations. At this stage, co-driving is our goal. For advanced driver assistance systems (ADAS), deep neural networks (DNNs)-based methods have made breakthroughs in both academia and industry, which advances the development of the area of self-driven cars [4,8]. In this paper, we study an advanced convolutional neural networks (CNNs)-based model in order to realize the intelligent interaction between human drivers and vehicles. With the help of hierarchical concept system, the data-driven CNNs model can understand complicated behaviors in the real world, learn feature from a large amount of data [6]. In addition, since the CNNs model has the ability of learning nonlinear feature, the model can be utilized to simulate the complicated nonlinear change behavior in a driving vehicle [7]. Therefore, the self-driven cars based on the CNNs model have received extensive attention in recent years.

2 2.1

Problem Description and Challenges Problem Description

In recent years, the application requirement for natural language processing (NLP) has increased dramatically. Practical NLP technology will provide convenient, natural and effective humanized services for human-vehicle voice information interaction. However, for large-scale learning tasks in NLP, the training data requires a large number of manual annotations, which increases the labor cost and time expenditure of research and development and limits the NLP general adaptation. 2.2

Working Mechanism of the Intelligent Voice System

The intelligent voice system is an intelligent operation which support the wake words setup and the human-vehicle dialogue. The working mechanism of the human-vehicle voice information interaction system mainly includes the following steps. Firstly, the intelligent system preprocesses the user instructions, which converts the acquired voice user data into text data. Secondly, the model classifies the acquired user instruction information, and then the functions that the user may need in the current context are predicted. Then, the system executes the vehicle function and interacts with the user. Finally, when the user interact with the system, the system will further collect the users feedback information and then learn their behavior patterns in order to improve its ability to predict and make decision.

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Challenges

In order to provide to users a more consistent intelligent experience from driving control to in-vehicle functions, some problems should be considered in the implementation of the DNNs-based intelligent interactions system, namely, how the Human Machine Interaction (HMI) system solves the problem of the large-scale data annotation, and how the system accurately predict the functions that the user may need in the current environment. We conclude the challenges as the following aspects: For large-scale learning tasks in NLP, training data need to be manually annotated, which increases the labor and time costs. It is important to note that while the CNNs-based HMI system has achieved higher intelligent experience than other studies on smart cockpit, significant challenges remain on internal structure of the model such as using only a fixed convolution kernel in the convolution operation to extract text features.

3

Methods

The overview of our framework is shown in Fig. 1. Firstly, we use one-hot-vector (OHV) model to learn vector representations of the user instructions [5]. Then, a TVE learning model is trained to predict the context from each text region and assign a label (e.g., positive/negative) for each small text region. Finally, the learned label data is integrated into the MGCNNs to implement the effective semantic analysis. The framework of SSMGCNNs Data preprocessing

One-hot vector model preprocesses text data, which represents words as vectors.

Tag data acquisition

Pre-train through the T V E learning model to learn embeddings of text regions from the unlabeled dataset to solve the problem of data annotation.

Semantic Analysis

The learned tv-embedded regions are used as an additional input to MGCNNs. MGCNNs can fully extract feature information hidden in text through multiple convolution kernels in the same convolution layer to realize the effective text semantic analysis .

Set up the experiment scene to verify the function and performance

Fig. 1. The overview of the proposed SSMGCNNs model.

3.1

Preliminary: One-Hot - Vector Model for Text Data Preprocessing

Now, we discuss the application of OHV model on text data for the semantic analysis. Suppose that a text data T = (d1 , d2 , · · ·, dn ) with V vocabularys is given. CNNs requires vector representation of text data as input. A straightforward representation way would be to treat the text data T as an image of

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|T | ∗ 1 pixels with |V | channels, treat each word di as a pixel. In this sense, each word (namely, each pixel) is represent as a |V | − dim ensional one-hot vector. As an example, suppose that vocabulary V = {“close”, “don’t”, “musci”, “open”, “the”}, we arrange the words in alphabetical order (as shown above), and that the user instruction T = “open the music”. Then, we have the user instruction T vector: v = [00010 |00001 |00100 ] . Next, we introduce bow-one-hot vector to discuss the application of OHV model on text classification task. For example, the user instruction region vector would be converted to: r0 (v) = [00011] , r1 (v) = [00101]. We call the representation a bow-one-hot vector way. The way loses word order only within small region, but fewer parameters are learned. In this paper, we use the bow-one-hot vector representation to solve the problem of high-dimensional data. 3.2

Semi-supervised Model for Text Categorization

Semi-supervised model directly learns an embedding of text regions from unlabeled data with the TVE model, and then integrates the learned embedding into supervised CNNs. The learned embedding regions (the output of TVE model) are used as an additional input to the supervised model. The first step of the semi-supervised learning model is to learn an region embedding, as illustrated in Fig. 2. The TVE model computers the embedding representation of the text region via: P (X2 |X1 ) = g1 (f1 (X1 ) , X2 ) , (X1 , X2 ) ∈ χ1 ∗ χ2

(1)

where X1 is the input vector, and X2 is the output vector, functions f1 is embodied by the convolution layer, which is a tv-embedding of χ1 w.r.t. χ2 , and g1 is embodied by the top layer. By definition, a view (X1 ) of a tv-embedding, which tv stands for two-view, preserves everything required to predict another view (X2 ). the Forbidden city in Beijing ?

How can I in Beijing ?

How can I get to the ?

Output X2

Top Layer g1

Convolution Layer f1

How can

get to th

Forbidden cit

in Beiji

?

Input X1

Fig. 2. Two-view-embedding (TVE) learning by training to predict the context (region size = 5, strid = 3)

A task on unlabeled text data is created to predict the adjacent text region (namely, context) from each text region of size p. Instead of the ultimate task of classifying the entire text sentence, a sub-task that assigns a label (e.g., positive/negative) to each text region is consider in the TVE model. The TVE

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model is a neural networks equipped with input layer, convolution layer, top layer, and output layer. A convolution layer consists of computation units, each of which responds to a small region of input data (e.g., a small text region of a sentence), and these small text regions collectively cover the entire sentence. Given a sentence s, for the κ − th text region, the TVE models convolution layer computes:   (2) Tκ (s) = σ (T ) W (T ) • rκ(T ) (s) + b(T ) where σ is the activation function. In the paper, we uses the rectified linear unit (T ) (ReLU) activation function, which is widely used at the present. rκ (s) ∈ Rq is the input region vector for the κ − th text region. The input region vector (T ) rκ (s) is a bow-one-hot vector. The top layer uses Tκ (s) as feature vector for prediction task. W (T ) and b(T ) , which are learned through training, are shared by all the computation units in the same layer. The second step of the semisupervised learning model is the integration of tv-embedding region into supervised CNNs, so that the tv-embeddeds are used as an additional label input to the CNNs convolution layer. The tv-embedding, which is obtained from the unlabeled data, is used, by replacing σ (W · rκ (s) + b) with: (3) σ (W · rκ (s) + V · Tκ (s) + b) where Tκ (s) is defined by (2), and Tκ (s) is the output of the TVE model applied to the κ − th text region. We update the weight matrix W , V , bias vector b, and the top-layer parameters, so that we minimize the designated loss function on the labeled training data. 3.3

The Construction of the Multi Granularity Convolution Kernel Model

In this paper, MGCNNs is proposed to extract more feature information hidden in text. Experimental results show that MGCNNs has higher accuracy than single-granularity CNNs. As shown in Fig. 3, A, B, C three kinds of convolution kernels (namely, region size) are showed. The convolution kernel of each size can extract a kind of text expression, namely, the output of the hidden layer is h1 , h2 , h3 . Then, the output of the hidden layer is connected to h = (h1 h2 h3 ) to show the entire text data. In the MGCNNs model, firstly, each word in a sentence is represented as a word vector, that is, the sentence is represented as: S = ς1 , ς2 , · · ·, ςn

(4)

where ςi is the m-dimensional vector representation of the i − th word in the sentence, n is the number of words in a sentence. The convolutional layer contains many kinds of region size of different dimension to extract feature information in different perspective. Multiple feature maps are generated after the convolution operation, and each feature map consists of many neurons. The neurons γi is defined as:

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Region size S2 = 3

h1

Region size S1 = 2

h2

h3

1

How can

1

I get to

2

the gas station

classifier

near here 3

?

Region size S3 = 4

Word vector input

Multi-granularity the global the fully convolution kernel max-pooling connected layer operation convolution

Fig. 3. The structure chart of multi-granularity convolutional neural networks

γi = t (pooling (σ (ωi S + b)))

(5)

where the activation function t, σ stand for tanh and sigmoid, respectively. ωi is the weight parameters of the kernel, b is the bias vector. indicates the operation of pooling, and we apply the global max-pooling operation to select the most discriminative feature information from each feature map and deal with the variable lengths of input. Then, these features are passed to the fully connected layer with ReLU activation function. Finally, the output of fully connected layer is treated as input to the SoftMax layer, and the classification probability formula to the SoftMax layer is: (6) P = σ (Ws γs + bs ) where σ is the activation function sigmoid, γs is the output of neurons after convolution operation and pooling operation, Ws is the weight parameter of SoftMax layer, and bs is the bias parameter of SoftMax layer. The output is the predicted value of target variable.

4

Evaluation

In order to evaluate the model, we use the vehicle operation command dataset to conduct experiments. This dataset is a typical single label text classification dataset. The dataset has 13 categories, which involves opening/closing the windshield wipers, opening/closing headlamps, opening/closing the window, opening/closing the air conditioning and other auto operation command. 4.1

Experimental Results

The evaluation metrics training loss rate is reported. The experimental results demonstrate the effectiveness of our approach in comparison with baseline methods on the user instruction dataset, as shown in Table 1. To be more specific, the

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first row is support vector machine (SVM) model based on optimization theory [9], and the second row is CNNs model [5]. The first thing to note is that on the user instruction datasets, the best-performing SSMGCNNs outperforms the baseline methods, which demonstrates the effectiveness of our method. The best performance 5.04 on the user instruction dataset was obtained by SSMGCNNs, equipped with the TVE model and four region size in the same convolutional layer. Table 1. The training loss rate on user instruction dataset Methods

Training loss rate (%)

Linear SVM

9.35

CNNs

7.59

SSMGCNNs (our) 5.04

4.2

Model Analysis

We also conduct experiments to compare the results using different internal structural. As shown in Fig. 4, the horizontal and vertical coordinates respectively represent the number of training iterations and the training loss rate of the model. We find that the training loss rate of the model will decrease as the number of hidden layers increases. The reason is that the more hidden layers can fully extract feature information for use in the classifier in the top layer. Moreover, we compare to the influence of the number of neurons in the hidden layer on the model performance, as shown in Fig. 5. The results indicate that as the number of neurons in the hidden layer increases, the training loss rate of the model decreases. Besides, we find that the number of convolution kernels (namely, region size) in the same convolution layer has an impact on the model performance, as illustrated in Fig. 6. This indicates that feature extraction is carried out simultaneously on the same convolutional layer with multiple convolution kernels of different dimensions, thus text features of different dimensions are obtained to fully extract the feature hidden in the text. 5.5 5.4

Layers-1 Layers-3 Layers-5 Layers-7 Layers-9

5.2

Training Loss rate (%)

5 4.8 4.6 4.4 4.2 4 3.8 3.6 3.4 3.2 0.03 2

4

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Fig. 4. The change of training loss with the increase of the number of hidden layers

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5.4 Neurons-100 Neurons-300 Neurons-500 Neurons-700 Neurons-900

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5 4.8 4.6 4.4 4.2 4 3.8 3.6 3.4 3.2 2

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Fig. 5. The change of training loss with the increase of neuron number 5.6 5.4

Convolution kernel-1 Convolution kernel-2 Convolution kernel-3 Convolution kernel-4 Convolution kernel-5

Training Loss rate (%)

5.1 4.8 4.5 4.2 3.9 3.6 3.3 3 2.7 2.4 2

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Fig. 6. The change of training loss with the increase of the number of convolution kernels

5

Conclusions

This paper presents a method for human-vehicle voice interaction using SSMGCNNs. SSMGCNNs mainly include two modules: the TVE model and MGCNNs model. SSMGCNNs learn embeddings of text regions from the unlabeled user instruction dataset and then integrate the learned tv-embedding into MGCNNs. The system constructs the automobile semantic analysis, completes the user intention reasoning, and realizes functions such as human-vehicle voice interaction, intelligent navigation, intelligent entertainment, and autonomous control of vehicle. Experimental results on the user instruction dataset demonstrate that our method achieves better or comparable performance comparing with baselines. Acknowledgments. This work was supported by the Common Key Technology Innovation Special of Key Industries of Chongqing science and Technology Commission under Grant No. cstc2017zdcy-zdyfX0067; the Artificial Intelligence Technology Innovation Significant Theme Special Project of Chongqing science and Technology Commission under Grant No. cstc2017rgzn-zdyfX0014 and No. cstc2017rgzn-zdyfX0035.

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References 1. Sotelo M-A, Van Lint JWC, Nunes UJ, Vlacic L (2012) Introduction to the special issue on emergent cooperative technologies in intelligent transportation systems. IEEE Trans Intell Transp Syst 13(1):1–5 2. Benderius O, Berger C, Lundgren VM (2018) The best rated human machine interface design for autonomous vehicles in the 2016 grand cooperative driving challenges. IEEE Trans Intell Transp Syst 19(4):1302–1307 3. Park J, Abdel-Aty M, Wu Y, Mattei I (2018) Enhancing in-vehicle driving assistance information under connected vehicle environment. IEEE Trans Intell Transp Syst PP(99):1–10 4. Huang K, Hu B, Chen L, Knoll A (2017) ADAS on COTS with OpenCL: a case study with lane detection. IEEE Trans Comput PP(99):1 5. Johnson R, Van Lint JWC, Nunes UJ, Vlacic L (2014) Effective use of word order for text categorization with convolutional neural networks. Eprint Arxiv 6. Johnson R, Zhang T (2015) Semi-supervised convolutional neural networks for text categorization via region embedding. In: Advances in neural information processing systems, vol 28, pp 919–927 7. Moeskops P, Viergever MA, Mendrik A, de Vries L (2016) Automatic segmentation of MR brain images with a convolutional neural network. IEEE Trans Med Imaging 35:1252–1261 8. Huang K, Hu B, Chen L, Knoll A (2012) ADAS on COTS with OpenCL: a case study with lane detection. IEEE Trans Comput PP(99):1 9. Leopold E (2002) Text categorization with support vector machines. How to represent texts in input space? Mach Learn 46(1):423–444

Target Tracking Based on SAGBA Optimized Particle Filter Siyao Lv, Yihong Zhang(B) , Wuneng Zhou(B) , and Longlong Li Donghua University College of Information Science and Technology, Shanghai 201620, China [email protected], [email protected]

Abstract. In the field of target tracking, particle filter technology has the advantage of dealing with nonlinear and non-Gaussian problems, but standard particle filter will cause particle depletion when using resampling method to solve the degradation phenomenon, and the filter precision is unstable. In response to this problem, particle filter is improved by using a bat algorithm that combines simulated annealing Gaussian perturbation. In this paper, the bat individual is characterized by particle. The bat group search for the optimal bat in the image area by adjusting the frequency, loudness and pulse emissivity, and this can dynamically control the mutual conversion between the global search and the local search, thereby improving the overall quality of the particles and the rationality of the distribution. The fused simulated annealing Gaussian perturbation strategy can enhance the ability of the algorithm to jump out of local optimum. In order to verify the optimization performance of the proposed algorithm, the performance of this paper and the standard particle filter algorithm are compared. Experimental results show that the filter performance of this algorithm is better than the standard particle filter algorithm. . . . Keywords: Particle filter · Particle depletion · Bat algorithm Simulated annealing · Gaussian perturbation · Target tracking

1

·

Introduction

Visual target tracking has a wide range of applications and covers many areas. Tasks such as video surveillance [1], human-computer interaction [2,3], automatic vehicle control [4] and human behavior analysis [5] all benefit from the visual system. The rapid development of image processing technology and hardware devices has greatly improved the integrated military capabilities of UAVs, enabling multiple tasks to be accomplished through the vision system. Target recognition and tracking technology have also become a core technology for UAVs in visual perception and understanding of the battlefield environment. It not only improves the reconnaissance capability of the UAVs, but also makes the UAVs take a big step to the unmanned combat aircraft. When the UAVs can c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 131–139, 2020. https://doi.org/10.1007/978-981-32-9682-4_14

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not communicate with the ground station, they can independently understand the battlefield environment, independently plan, decision-making and control the launch of the weapon system, and ultimately achieve comprehensive automatic control and operations. Therefore, combined with advanced digital image processing technology and computer vision theory, it explores the autonomous detection and recognition of targets in complex environments and long-term stable tracking become the most important technology of UAVs vision system, and it is also the center of our research. There are two types of methods for visual object tracking: (1) generation type and (2) discriminant type. The discriminant method [6–9] usually requires a large training set because they aim to solve the tracking problem by separating the target object from the background or nontarget object by using the classifier. On the other hand, the generation method [10–12] aims to find the target object by comparing the image area with a specific appearance model. Particle filter is widely used for visual target tracking [13–15]. Particle filter realizes recursive Bayesian filter by non-parametric Monte Carlo simulation, and the accuracy approximates the optimal estimation. It is suitable for dealing with state estimation problems of nonlinear and non-Gaussian time-varying systems. However, the traditional particle filter method (PF) has the problem of weight degradation. Bat Algorithm (BA) [16] was proposed by Professor Yang of Cambridge University in 2010 to achieve intelligent optimization by simulating the predation behavior of bats. The search is optimization mechanism, but the bat algorithm has stronger randomness. [16] has proved that the comprehensive optimization ability of bat algorithm is better than particle swarm algorithm, ant colony algorithm and other mainstream group intelligent optimization algorithms. In [17], we can find that combining bat algorithm with PF further improve the particle filter performance. However, the bat algorithm has a slower convergence rate in the later stage, the convergence accuracy is not high, and it is easy to fall into the local optimum. In this paper, the simulated annealing Gaussian perturbation is added to the bat optimization algorithm to improve the search ability of the algorithm, and it is used in the target tracking based on particle filter. The simulated annealing Gaussian perturbed bat optimization search can be performed before performing re-sampling, which could avoid re-sampling the particle depletion itself.

2 2.1

Proposed Algorithm Particle Filter

In the standard particle filter, the target state vector is xt at time t, and it is considered to find the target state by observing the state vector as zt . The observer is z1:t from the first frame to the t-th frame. In the particle filter, the posterior distribution can be defined using the approximate model of the Chapman-Komogorov equation, using the particle set {xt1 , xt2 , . . . , xtN } and the

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t weight set {w1t , w2t , . . . , wN } [18]. i wti = wt−1

p(zt | xit )p(xit | xit−1 ) q(xit | xit−1 , zt )

(1)

where q is the importance N density function. The previous posterior distribution is p(xt−1 | z1:t−1 ) ≈ i=1 wti δ(xt − xit ), where δ(·) is a Dirac function, and the N constraint condition of the function is that i=1 wti = 1, and 1 ≤ i ≤ N at the previous moment. 2.2

Bat Algorithm

The bat-heuristic-based metaheuristic optimization algorithm, the bat algorithm is proposed by Yang based on the echolocation of microbats. However, the bat algorithm has a slower convergence rate in the later stage, the convergence accuracy is not high, and it is easy to fall into the local optimum. In this paper, the simulated annealing algorithm is fused to the bat algorithm, and a bat algorithm optimized particle filter (SAGBA-PF) based on simulated annealing and Gaussian perturbation is proposed. Gaussian perturbation mutation operation is used to further adjust the set of group to be optimized to achieve the global search ability of the algorithm and the improvement of local search ability. 2.3

The Target Gray Level Description and Objective Function Design of the Algorithm

The gray-scale distribution description of the target is a relatively robust target description method, which can weaken the influence of deformation and rotation on tracking performance to some extent. In this paper, the gray scale distribution is used to describe the target and the system observation model is established. Let the center of the video image tracking area be X = (x, y), the area size be h = (hx , hy ), and the pixel position in the tracking area be Xi = (xi , yi ). Therefore, the kernel probability density estimation is performed to perform the gray scale distribution description of the target. Assuming that the target gray distribution is discretized into B bin intervals, defining a gray level quantization function b(Xi ) : R2 → {1, . . . , B} indicates that the pixel gray level at the position Xi is quantized and assigned to the corresponding gray level bin, where the quantization level of the gray level is represented by B. The grayscale distribution of this defined target is:    M  X−Xi  k   i=1 h  δ(b(Xi ) − u)   (2) pux =  M  X−Xi  i=1 k  h  where δ(·) is the Kronecker Delta function; the total number of pixels in the tracking area is recorded as M , k(·) is Gaussian kernel function.

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In order to ensure the accuracy of the particle filter, it is necessary to introduce the latest observations in the adaptive optimization behavior of the bat. Therefore, the objective function formula of the design is as follows:   I=e 2.4

1 − 2R (znew −zpred (i)) g

(3)

SAGBA-PF Algorithm Design

SAGBA-PF Global Search Optimization Process. For each bat (set to i), its position xg−1 and velocity vig−1 are defined in a d-dimensional search space i and subsequently updated during the iteration. The new solution xgi and speed vig in iteration g are calculated by using a global search strategy, the formula is as follows: fi = fmin + (fmax − fmin )β

(4)

vig+1 = vig + (xgi − x∗ )fi

(5)

= xgi + vig+1 xg+1 i

(6)

where β ∈ [0, 1] is a random vector obtained from a uniform distribution. x∗ is the globally best solution currently found, which is found after compared all solutions between all bats in the current iteration. The values of the frequencies fmin and fmax depend on the domain size of the problem of interest. SAGBA-PF Local Search Optimal Process. For the local search part, once the solution is selected from the current best solution, a new solution for locally generating each bat is generated using the random walk model. The local search method set in this paper is as follows: If rand > ri xnew = xold + εAg

(7)

xnew = xg+1 i

(8)

If rand < ri

where ε ∈ [−1, 1] is a random number and xold is the solution for the current optimal solution set. Ag = is the average loudness of all bats at iteration g. If rand < Agi and I(xgi ) > I(xg+1 ), the current position of the bat is xgi , i otherwise the current position of the bat is xnew . Using the Metropolis criterion equation to choose whether to  accept the new  solution of Eq. (7): If I(xnew ) >

I(xg+1 ) i

or e

g+1 I(xnew )>I(x ) i T

> rand

xg+1 = xnew i

(9)

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SAGBA-PF Global and Local Search Optimization Adjustment Process. By updating the loudness Ai and the pulse rate ri to reflect that if a target is found, the bat reduces the loudness and increases the pulse rate, which is calculated by: Ag+1 = ωAgi i

(10)

rig+1 = ri0 [1 − e−γt ]

(11)

where ri0 is the initial pulse rate, ω is the pulse frequency increase coefficient, and γ is the pulse amplitude attenuation coefficient. For any 0 < ω and γ < 1, we have the following: Agi → 0,

rig = ri0 ,

when

g→∞

(12)

The simulated annealing algorithm is integrated into the bat optimization algorithm, and then the Gaussian perturbation operation is performed on the bat individual, and the further search process can retain the superior bat individuals. As the iterative process advances, the temperature gradually decreases, and the probability of receiving a poor individual solution gradually decreases, thereby improving the convergence of the algorithm. In summary, SAGBA combines the respective advantages of SAG and BA, so that the SAGBA algorithm has stronger global and local search capabilities at the same time, which overcomes the shortcomings of slow convergence in the late BA, low convergence accuracy, and easy to fall into the local best.

3

Algorithm Steps

In this paper, the simulated annealing algorithm is integrated into the bat algorithm, and a Gaussian Disturbed Bat Optimization Algorithm (SAGBA) based on simulated annealing algorithm is proposed. The algorithm flow is shown in Fig. 1. The algorithm process makes full use of the effective information of the entire bat population, helping the bat individuals to jump out of the local optimum. It avoids the iterative process when the state value does not change much, making the algorithm more likely to terminate the optimization, more because the algorithm reaches the set precision stop threshold, thus reducing the algorithm until the set maximum iteration number is reached. The probability of termination will be increased, thereby improving the computational efficiency of the algorithm. In terms of the number of effective particle samples, this method can increase the diversity of the particles and thus the quality of the particle samples.

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Fig. 1. Algorithm step flow.

4

Simulation Experiment and Result Analysis

In this paper, the univariate non-static growth model is selected. We compare particle swarm optimization particle filter (PSO-PF), bat algorithm optimization particle filter (BA-PF) and firefly algorithm optimization particle filter (FA-PF). The process model and observation model of the experimental test object are as follows: Process model: x(t) = 0.5x(t − 1) +

25x(t − 1) + 8 cos [1.2(t − 1)] + net (t) 1 + [x(t − 1)]2

(13)

Observation model: z(t) =

x(t2 ) + wet (t) 20

(14)

In the above equation, net (t) and wet (t) are Gaussian noises with a mean value of zero. In this paper, the system noise variances Q = 1 and Q = 10 are assumed, the observed noise variance is R = 10, the filter time step is 50, the initial oscillation pulse is A0 = 0.5, the initial pulse rate is r0 = 0.5, fmax = 2, fmin = 0, using standard particle filter, particle swarm optimization particle filter and the proposed algorithm optimize the particle filter state estimation and tracking the nonlinear system. In the simulated annealing algorithm part, the initial temperature set by the algorithm is T W 0 = 1000 ◦ C, the equilibrium temperature is T W min = 50 ◦ C,

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the maximum number of iterations is 20, and the annealing method is as shown in Eq. (15), and the degradation coefficient is as = 0.99: T = T ∗ as

(15)

The root mean square error formula used in the algorithm is: 

RM SE

2 T 1 = (xt − x ¯t )2 T 1

(16)

T −1

1. When the number of bats is N = 20 and Q = 10, the simulation results shown in Fig. 2. 2. When the number of bats is N = 50 and Q = 10, the simulation results shown in Fig. 3. 3. When the number of bats is N = 100 and Q = 10, the simulation results shown in Fig. 4. 4. When the number of bats is N = 20 and Q = 1, the simulation results shown in Fig. 5. 5. When the number of bats is N = 50 and Q = 1, the simulation results shown in Fig. 6. 6. When the number of bats is N = 100 and Q = 1, the simulation results shown in Fig. 7.

are are are are are are

It can be seen from Figs. 2, 3, 4, 5, 6 and 7 that compared with the standard particle filter and particle swarm optimize particle filter, the bat algorithm optimized by the simulated annealing algorithm and the Gaussian perturbation algorithm has better state value prediction accuracy because of the fusion simulated annealing algorithm for bat algorithm. A Gaussian perturbation operation is performed on a portion of the bat individual, and further search behavior is performed to store the relative superiority of the bat individual. With the evolution process, the temperature of the simulated annealing algorithm is gradually reduced, and the probability of accepting the poor solution is gradually reduced, thereby improving the performance of the algorithm, so that the algorithm can greatly improve the convergence speed, robustness and optimization ability of the algorithm. It can be seen from Table 1 that under the influence of high noise, the accuracy of the particles number N = 20 in the algorithm is still higher than that in the standard particle filter when the number of particles is N = 100. This shows that the simulated annealing Gaussian perturbed bat algorithm used in this paper optimizes particle filter in a nonlinear, high-noise environment that is better than standard particle filter.

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Fig. 2. Absolute error at N = 20, Q = 10

Fig. 3. Absolute error at N = 50, Q = 10

Fig. 4. Absolute error at N = 100, Q = 10

Fig. 5. Absolute error at N = 20, Q = 1

Fig. 6. Absolute error at N = 50, Q = 1

Fig. 7. Absolute error at N = 100, Q = 1

Table 1. Comparison of root mean square error of experimental result

5

Parameter

RM SE PF PSO-PF BA-PF FA-PF SAGBA-PF

N = 20, Q = 10

5.0419 3.3945

4.4790

3.9520 0.9961

N = 50, Q = 10

4.9143 4.2612

4.2550

4.0822 0.9331

N = 100, Q = 10 4.4381 4.0134

3.8184

3.7837 0.3564

N = 20, Q = 1

1.7933

1.6843 0.6697

1.7985 1.2024

N = 50, Q = 1

1.5430 1.0812

1.4481

1.4438 0.3957

N = 100, Q = 1

1.0837 0.7403

0.8792

0.8217 0.3035

Conclusions

In this paper, a target tracking method based on simulated annealing Gaussian perturbed bat algorithm for intelligent particle filter optimization is proposed. Introducing the bat optimization algorithm combined with simulated annealing Gaussian perturbation to intelligently optimize the particle filter re-sampling process. At the same time, it has global search and local search capabilities, which improves the tracking performance of the tracking algorithm. The experimental results of the test video image also show that the proposed method has better robustness, accuracy, convergence speed and optimization ability in the complex background. In the following work, the target tracking problem of other signal source images will be deeply studied, and the template update and target occlusion in the tracking process will be further optimized.

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Acknowledgments. This work is partially supported by the National Natural Science Foundation of China (No. 61573095).

References 1. Benfold B, Reid I (2011) Stable multi-target tracking in real-time surveillance video. In: Proceedings of the IEEE Conference Computer Vision Pattern Recognition (CVPR), Colorado Springs, CO, USA, pp 3457–3464 2. Bradski GR (1998) Real time face and object tracking as a component of a perceptual user interface. In: Proceedings of the 4th IEEE Workshop Applications Computer Vision (WACV), Princeton, NJ, USA, pp 214–219 3. Bellotto N, Hu H (2009) Multisensor-based human detection and tracking for mobile service robots. IEEE Trans Syst Man Cybern B Cybern 39(1):167–181 4. Vu A, Ramanandan A, Chen A, Farrell JA, Barth M (2012) Real-time computer vision/DGPS-aided inertial navigation system for lane-level vehicle navigation. IEEE Trans Intell Transp Syst 13(2):899–913 5. Cristani M, Raghavendra R, Del Bue A, Murino V (2013) Human behavior analysis in video surveillance: a social signal processing perspective. Neurocomputing 100:86–97 6. Yang F, Lu H, Yang M-H (2014) Robust superpixel tracking. IEEE Trans Image Process 23(4):1639–1651 7. Avidan S (2004) Support vector tracking. IEEE Trans Pattern Anal Mach Intell 26(8):1064–1072 8. Kalal Z, Mikolajczyk K, Matas J (2012) Tracking-learning-detection. IEEE Trans Pattern Anal Mach Intell 34(7):1409–1422 9. Du D, Zhang L, Lu H, Mei X, Li X (2016) Discriminative hash tracking with group sparsity. IEEE Trans Cybern 46(8):1914–1925 10. Wang D, Lu H, Yang M-H (2013) Least soft-threshold squares tracking. In: Proceedings of the IEEE Conference on Computer Vision Pattern Recognition (CVPR), Portland, OR, USA, pp 2371–2378 11. Wang D, Lu H, Yang M-H (2013) Online object tracking with sparse prototypes. IEEE Trans Image Process 22(1):314–325 12. Mei X, Ling H (2009) Robust visual tracking using L1 minimization. In: Proceedings of the IEEE 12th International Conference on Computer Vision, pp 1436–1443 13. Ross DA, Lim J, Lin R-S, Yang M-H (2008) Incremental learning for robust visual tracking. Int J Comput Vis 77(1–3):125–141 14. Khan ZH, Gu IY-H (2013) Nonlinear dynamic model for visual object tracking on Grassmann manifolds with partial occlusion handling. IEEE Trans Cybern 43(6):2005–2019 15. Zhang T, Ghanem B, Liu S, Xu C, Ahuja N (2016) Robust visual tracking via exclusive context modeling. IEEE Trans Cybern 46(1):51–63 16. Yang XS (2010) A new metaheuristic bat-inspired algorithm. In: Nature inspired cooperative strategies for optimization, pp 65–74. Springer, Verlag 17. Chen Z-M, Tian M-C, Wu P-L, Bo Y-M, Gu F-F, Yue C (2017) Intelligent particle filter based on bat algorithm. Acta Phys Sin 66(5) (2017) 18. Maggio E, Smerladi F, Cavallaro A (2007) Adaptive multifeature tracking in a particle filtering framework. IEEE Trans Circ Syst Video Technol 17(10):1348– 1359

L2 Leader-Following Consensus of Second-Order Nonlinear Multi-agent Systems with Disturbances Under Directed Topology via Event-Triggered Control Scheme Yuanhong Ren1 , Wuneng Zhou1(B) , Zhiwei Li1,2 , and Yuqing Sun1 1

College of Information Science and Technology, Donghua University, Shanghai 201620, China [email protected] 2 School of Electronic and Electrical Engineering, Shanghai University of Engineering Science, Shanghai 201620, China

Abstract. This paper addresses the L2 leader-following consensus problem for second-order multi-agent systems with nonlinear dynamics and external disturbances under directed topology. The event-triggered control (ETC) scheme with a new ETC protocol and an event-triggered condition has been introduced. In the control protocol, two parameters which are constrained by the predefined performance index are designed to adjust the convergence rate of the system. Based on the directed graph theory, we construct a Lyapunov function, and then prove theoretically that the system can achieve leader-following consensus without disturbance, and a predefined L2 gain performance index can be guaranteed under the zero initial condition when disturbances exist. A simulation example is provided to verify the theoretical results. Keywords: L2 gain performance · Leader-following consensus · Multi-agent systems · Event-triggered control

1 Introduction The research on consensus of multi-agent systems has become a hot spot in recent years owing to its wide applications, including unmanned air vehicle (UAV) formation control, attitude consistency control of satellite, power sharing of microgrids, and estimation over sensor networks (see [1–5] and references therein). Many researchers have studied consensus problems of first-order/second-order/high-order multi-agent systems with linear/nonlinear dynamics, time-delay and other constraints under undirected/directed topology (see [6, 7]), and these studies can be classed into two categories: one is leaderless consensus (see [8, 9]), and the other is leader-following consensus (see [10, 11]). The main objective of consensus problems is to design a distributed control protocol such that the states of all agents can reach a predefined agreement. Most control protocols are time-triggered which means that the measurement is sampled in periodic c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 140–149, 2020. https://doi.org/10.1007/978-981-32-9682-4_15

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manner and the control input updates continuously, but this is unrealistic in some practical situations when communication channels and bandwidth between the agents are limited. Therefore, an event-triggered control (ETC) scheme has been adopted, which has been proved to be able to effectively reduce the frequency of control updates while ensuring the stability performance of the system (see [12–16]). In [17], two distributed event-based consensus control protocols are proposed for linear multi-agent systems with fixed and switching undirected topologies. In [18], the authors designed centralized and decentralized average-consensus protocols for multi-agent systems under weighted interaction topology digraph which is strongly connected and balanced. [19] analyzed the leader-following consensus of second-order multi-agent systems with fixed and jointly-connected interaction directed topology by an event-triggered distributed sampling scheme. It is noticed that most existing studies address the consensus problem of multi-agent systems without external disturbances. However, in some practical situations, systems are often subjected to external disturbances which always lead to instability and performance degradation. One method to deal with disturbances is to use H∞ control (see [20– 22]). Different from H∞ control, L2 leader-following control can attenuate the external disturbance signal and also limit the tracking error within range of the desired value. [23] studied finite-time L2 leader-following consensus of networked Euler-Lagrange systems with external disturbances and proposed a distributed finite-time L2 control protocol by using back-stepping design. [24] investigated the consensus and its L2 gain performance for multi-agent systems under fixed directed topology where each agent can only communicate with its neighbours intermittently. Usually, information flows among agents are directional, and considering the existences of nonlinearities and disturbances, this paper studies the leader-following consensus problem for second-order systems with nonlinear dynamics and disturbances under directed topology. The main contributions of this paper are listed as follows: (1) L2 leader-following consensus problem is addressed for the second-order multi-agent systems. Unlike H∞ control, L2 leader-following control can attenuate the external disturbance signal and limit the tracking error within range of the desired value. (2) ETC scheme was introduced to reduce the number of transmitted data and the frequency of controller updates. Notation: Let R+ be the set of positive real numbers. Rn denotes n-dimensional real vector space. I n represents the n-dimensional identity matrix. Rm×n and 0m×n represent the m × n dimensional real matrix space and the m × n dimensional matrix space with all elements equal to zero, respectively. For matrix A, A > 0(A < 0) means that A is a positive (negative) definite matrix, and AT denotes the transpose of A. λmax (·) and λmin (·) denote the maximal and minimum eigenvalue of a real symmetric matrix, respectively. Denote  ·  the Euclidean norm for vectors in Rn . L2n [0, +∞) denotes the n-dimensional square integrable function space.

2 Problems Statements We focus on the nonlinear second-order multi-agent systems with N followers (labeled as 1 to N) and one leader (labeled as N+1) under fixed directed topology. The dynamics of the followers are expressed as follows:

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x˙i (t) = vi (t) v˙i (t) = f (t, xi (t), vi (t)) + ui (t) + ωi (t), i = 1, 2, · · · , N,

(1)

where xi (t), vi (t) and ui (t) ∈ Rn denote the position state, the velocity state and the control input of agent i, respectively. The function f: R × Rn × Rn → Rn is the nonlinear term of the systems, and ωi (t) ∈ L2n [0, +∞) describes the external disturbance. Suppose that there are no control input and disturbance for the leader, then the dynamic of the leader is described as  x˙N+1 (t) = vN+1 (t) (2) v˙N+1 (t) = f (t, xN+1 (t), vN+1 (t)). The followers can communicate with their neighbors through directed information flows, which means that the topology is a directed graph. The weighted directed graph G = {V , ε } with a set of nodes V = {υ1 , υ2 , · · · , υN , υN+1 } and edges ε ⊆ V × V is used to model the topology among all agents. We use Ni = {v j ∈ V |(υi , υ j ) ∈ ε } to represent the neighborhood set of agent i, and |Ni | represents the cardinal number of Ni . An edge(υi , υ j ) in G denotes the information flow from node υi to node υ j . A = [ai j ] ∈ R(N+1)×(N+1) is the weighted adjacency matrix of G with ai j > 0 if (υ j , υi ) ∈ ε , and ai j = 0 otherwise. The Laplacian matrix L = [li j ] ∈ R(N+1)×(N+1) of G is defined as lii = ∑N+1 j=1 ai j and li j = −ai j . Then, L can be written in block matrix form as   L f Ll L= , 01×N 0 where L f ∈ RN×N and Ll ∈ RN×1 . Assumption 1. For any given state vector xi (t) and vi (t), there exists two nonnegative constants lx and lv , such that  f (t, xi (t), vi (t)) − f (t, x j (t), v j (t)) ≤ lx xi (t) − x j (t) + lv vi (t) − v j (t). Assumption 2. For each follower, there exists a directed path from the leader to it. Definition 1. [15] For systems (1) and (2), the leader-following consensus can be achieved, if lim xi (t) − xN+1 (t) = 0, lim vi (t) − vN+1 (t) = 0, i = 1, 2, · · · , N.

t→∞

t→∞

Definition 2. [24] Systems (1) and (2) can reach a L2 leader-following consensus with an induced L2 gain less than a positive scalar ρ if the following two conditions are satisfied: (i) when ω (t) ≡ 0, the systems (1) and (2) can reach leader-following consensus. (ii) when ω (t) = 0, the L2 gain performance  ∞ t

s(t)2 dt ≤ ρ 2

 ∞ t

ω (t)2 dt

is guaranteed under the zero initial condition. where ω (t) is the stack vector of ωi (t).

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Lemma 1. [25] Under Assumption 2, L f is a nonsingular M-matrix. Furthermore, there exists a positive vector ζ = (ζ1 , ζ2 , · · · , ζN )T , such that L f ζ = 1N . Denote Θ = diag{1/ζ1 , 1/ζ2 , · · · , 1/ζN } and Ξ = (Θ L f + LTf Θ ), then Θ and Ξ are positive definite. We define s(t) = (c1 x(t), ˆ c2 v(t)) ˆ T as the L2 performance variable with c1 and c2 are positive scalars, and x(t), ˆ v(t) ˆ are defined as  xˆi (t) = xi (t) − xN+1 (t), (3) vˆi (t) = vi (t) − vN+1 (t). i = 1, 2, · · · , N.

3 Main Results In this section, we discuss the L2 leader-following consensus for systems (1) and (2). In order to save network transmission resources, the ETC scheme will be introduced. Under this scheme, the system control input of agent i updates only at the triggering instant of itself. Let {tki , k = 0, 1, · · · } be the triggering instant sequence of agent i, then we design an event-triggered controller as ui (t) = α

N+1

N+1

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i ), ∑ ai j (x j (tki ) − xi (tki )) + β ∑ ai j (v j (tki ) − vi (tki )), t ∈ [tki , tk+1

(4)

where α > 0 and β > 0 are parameters to be designed. First, the state measurement errors of the followers are given as eix (t) = xi (tki ) − xi (t), eiv (t) = vi (tki ) − vi (t), and for the leader, we define eN+1,x (t) = xN+1 (tki ) − xN+1 (t), eN+1,v (t) = vN+1 (tki ) − vN+1 (t). Next, the event-triggering function corresponding to agent i which is used to decide the next triggering instant is designed as  (5) φi (t) = α exi (t) + β evi (t) + α exN+1 (t) + β evN+1 (t) − h gxi (t) + gvi (t) with exi (t) = − ∑Nj=1 li j e jx (t), evi (t) = − ∑Nj=1 li j e jv (t), exN+1 (t) = −li,(N+1) (xN+1 (tki ) − xN+1 (t)), evN+1 (t) = −li,(N+1) (vN+1 (tki ) − vN+1 (t)), gxi (t) = ∑N+1 j=1 ai j (α (x j (t) − 2. a ( β (v (t) − v (t))) xi (t)))2 , and gvi (t) = ∑N+1 ij j i j=1 i is defined iteratively by Then the event triggering instant tk+1 i tk+1 = in f {t|t > tki , φi (t) > 0}.

(6)

Once φi (t) > 0 is satisfied, the current state values of agent i and its neighbors will be sampled and transmitted to the controller. By applying the control protocol (4), the derivative of (3) can be derived as x(t) ˆ˙ =v(t), ˆ ˆ + ex (t)) v(t) ˆ˙ =F(t, x(t), v(t)) − 1N ⊗ f (t, xN+1 (t), vN+1 (t)) − α ((L f ⊗ In )(x(t)

(7)

− α (Ll ⊗ eN+1,x ) − β ((L f ⊗ In )(ev (t) + v(t)) ˆ − β (Ll ⊗ eN+1,v ) + ω (t). where F(t, x(t), v(t)), u(t), ex (t), ev (t), x(t) ˆ and v(t) ˆ are the stack vectors of f (t, xi (t), vi (t)), ui (t), eix (t), eiv (t), xˆi (t), and vˆi (t) for i = 1, 2, · · · , N.

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Theorem 1. Under Assumptions 1 and 2, consider the multi-agent systems (1) and (2) with the controller (4). Suppose the event-triggered condition are given by (5) and (6). For a given desired L2 performance index ρ , the L2 leader-following consensus can be achieved, if the positive scalars α , β , and h satisfy:

α < min{β 2 λmin (Ξ )ζmin , ρ 2 ζmin /η − β }, h ≤ min{

(8)

(α 2 ηλmin (Ξ ) − c21 )ζmin ζmax − (α + 3α l + β l)ηζmin , 2μη (α 2 + αβ )ζmax (β 2 ηλmin (Ξ ) − c22 )ζmin ζmax − (2α + 3β l + α l + β )ηζmin }, 2μη (β 2 + αβ )ζmax )

(9)

where ζmin = min{ζ1 , ζ2 , · · · , ζN }, ζmax = max{ζ1 , ζ2 , · · · , ζN }, l = max(lx , lv ), N¯ = max |Ni |, b = max ai j , d = max ai(N+1) , and μ = 2b2 (N + N¯ − 1) + d 2 .

1≤i≤N

1≤i, j≤N

1≤i≤N

Proof. Choose the following Lyapunov function candidate as   1 αβ Ξ˜ α Θ˜ T V (t) = ξ (t) ⊗ In ξ (t), α Θ˜ β Θ˜ 2 where ξ (t) = (xˆT (t), vˆT (t))T , Ξ˜ = ηΞ , Θ˜ = ηΘ , η is a positive number and Ξ , Θ are as defined in Lemma 1. In order to guarantee the validity of the Lyapunov function V (t), by using Schur’s complement Lemma, one needs (i) β (Θ˜ ⊗ In ) > 0, and (ii) (αβ Ξ˜ − α Θ˜ (β Θ˜ )−1 (α Θ˜ )) ⊗ In > 0. Then it is easy to verify that V (t) > 0 when α < β 2 λmin (Ξ )ζmin . In order to get the results about the L2 gain performance, we define the following function 1 Vˆ (t) = V˙ (t) + (s(t)2 − ρ 2 ω (t)2 ). 2 Calculating the derivative of V (t) along the solution of (7), we have ρ 1 Vˆ (t) = αβ xˆT (t)(Ξ˜ ⊗ In )v(t) ˆ + α vˆT (t)(Θ˜ ⊗ In )v(t) ˆ + sT (t)s(t) − ω T (t)ω (t) 2 2 ˆ + β v(t)) ˆ T (Θ˜ ⊗ In )(F(t, x(t), v(t)) − 1N ⊗ f (t, xN+1 (t), vN+1 (t))) + (α x(t) ˆ + β v(t)) ˆ T (Θ˜ ⊗ In )(α (L f ⊗ In )x(t) ˆ + β (L f ⊗ In )v(t)) ˆ − (α x(t) 2

(10)

ˆ + β v(t)) ˆ T (Θ˜ ⊗ In )Φ (t) + (α x(t) ˆ + β v(t)) ˆ T (Θ˜ ⊗ In )ω (t), + (α x(t)

where Φ (t) = (Φ1 (t), Φ2 (t), · · · , ΦN (t))T and Φi (t) = α exi (t) + β evi (t) + α exN+1 (t) + β evN+1 (t), for i = 1, 2, · · · N. According to the event-triggering condition (5) and (6), we get φi (t) ≤ 0 when t ∈ i ). Based on the Cauchy-inequality, the norm inequality x − y2 ≤ 2x2 + [tki , tk+1 2y2 and the definition of xˆi (t), one has

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Φi (t)2 ≤ α h2

N+1

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ai j (xˆ j (t) − xˆi (t))2 + β h2

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j=1

≤ 2h max 2

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∑ ai j (vˆ j (t) − vˆi (t))2

j=1

(a2i j )

∑ (α xˆi (t)

2

+ α xˆ j (t)2 ) + β vˆi (t)2 + β vˆ j (t)2 )

j∈Ni

+ h2 a2i,N+1 (α xˆi (t)2 + β vˆi (t)2 ). 2 + h2 [2b2 (N + N ¯− Therefore, we get Φ (t)2 ≤ h2 [2b2 (N + N¯ − 1) + d 2 ]α x(t) ˆ 2 2 1) + d ]β v(t) ˆ , and thus

ˆ + β v(t)), ˆ Φ (t) ≤ hμ (α x(t)

(11)

where b, d and μ are as defined in Theorem 1. By using Assumption 1, and based on the inequality xˆi (t)vˆi (t) ≤ 12 (xˆi (t)2 + vˆi (t)2 ), one has (α xˆT (t) + β vˆT (t))(Θ˜ ⊗ In )(F(t, x(t), v(t)) − 1N ⊗ f (t, xN+1 (t), vN+1 (t)) ≤

3α l + β l T 3β l + α l T xˆ (t)(Θ˜ ⊗ In )x(t) vˆ (t)(Θ˜ ⊗ In )v(t), ˆ + ˆ 2 2

(12)

where l = max(lx , lv ). It can be simply calculated that − (α xˆT (t) + β vˆT (t))(Θ˜ ⊗ In )(α ((L f ⊗ In )x(t) ˆ + β ((L f ⊗ In )v(t)) ˆ 1 1 2 T ˆ − αβ xˆT (t)(Ξ˜ ⊗ In )v(t) ˆ − β 2 vˆT (t)(Ξ˜ ⊗ In )v(t). ˆ = − α xˆ (t)(Ξ˜ ⊗ In )x(t) 2 2

(13)

Furthermore, (α xˆT (t) + β vˆT (t))(Θ˜ ⊗ In )ω (t) N η 1 η 1 1 1 ( xˆi (t)2 + ωi (t)2 ) + β ∑ ( vˆi (t)2 + ωi (t)2 ) ζ 2 2 ζ 2 2 i i i=1 i=1 N

≤α ∑ =

(14)

α T β (α + β ) T xˆ (t)(Θ˜ ⊗ In )x(t) ˆ + vˆT (t)(Θ˜ ⊗ In )v(t) ˆ + ω (t)(Θ˜ ⊗ In )ω (t). 2 2 2

In the same way, it follows from (11) that hμη (α 2 + αβ ) hμη (β 2 + αβ ) 2 2 (α xˆT (t) + β vˆT (t))(Θ˜ ⊗ In )Φ (t) ≤ x(t) ˆ + v(t) ˆ . ζmin ζmin (15) Substituting (12), (13), (14) and (15) into (10), we get 1 η (α + 3α l + β l) 2hμη (α 2 + αβ ) Vˆ (t) ≤ − xˆT (t)[(α 2 ηλmin (Ξ ) − − (c21 + )]x(t) ˆ 2 ζmax ζmin 1 η (2α + 3β l + α l + β ) − vˆT (t)[(β 2 ηλmin (Ξ ) − − (c22 (16) 2 ζmax 2hμη (β 2 + αβ ) 1 + )]v(t) ˆ − ω T (t)(ρ 2 INn − (α + β )(Θ˜ ⊗ In ))ω (t). ζmin 2

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If h satisfies (9) in Theorem 1 and α ≤ ρ 2 ζmin /η − β , then we obtain that Vˆ (t) ≤ 0. Next, we will analyze the L2 gain performance of the systems according to Definition 2. When ω (t) ≡ 0, one has Vˆ (t) = V˙ (t) + 12 (s(t)2 ) ≤ 0. Thus we get V˙ (t) ≤ 0, which means that x1 (t) = x2 (t) = · · · = xN+1 (t) and v1 (t) = v2 (t) = · · · = vN+1 (t), i.e., the leader-following consensus can be reached. In the case of ω (t) = 0, based on the result Vˆ (t) ≤ 0, one gets V˙ (t) ≤ 12 (ρ 2 ω (t)2 − s(t)2 ). Integrating the above inequality from t to ∞, and notice that the initial zero initial condition V (t) = 0, one has  ∞ t

s(t)2 dt ≤ ρ 2

 ∞ t

ω (t)2 dt.

Thus, the second condition in Definition 2 is satisfied. This completes the proof. When disturbances are not taken into account, the control gains in (4) can be smaller, and we can select the proper values based on the following corollary. Corollary 1. Under Assumptions 1 and 2, consider the systems (1) and (2) without disturbances. Suppose that the event-triggered condition are given by (5) and (6), then the leader-following consensus can be reached, if the positive scalars α , β , and h satisfy: h ≤ min{

α 2 ηλmin (Ξ )ζmin ζmax − (3α l + β l)ηζmin , 2μη (α 2 + αβ )ζmax (β 2 ηλmin (Ξ ))ζmin ζmax − (2α + 3β l + α l)ηζmin }, 2μη (β 2 + αβ )ζmax )

and α < β 2 λmin (Ξ ), where all the parameters are the same as in Theorem 1. The proof process is similar to that of Theorem 1, so we omit it here.

4 Simulation Example In this section, we borrow the consensus problem model of spacecraft formation flying from [26] to illustrate our results, and take the same nonlinear function as in [26] as follow: ⎞ ⎞ ⎛ ⎛ 0 0 0 0 2γ 0 f (t, xi (t), vi (t)) = ⎝ 0 3γ 2 0 ⎠ xi (t) + ⎝ −2γ 0 0 ⎠ vi (t), 0 0 0 0 0 −γ 2 where xi (t) and vi (t) are the position vector and the velocity vector of the satellite i, and γ denotes the angular rate of the virtual satellite (for more information about the consensus problem model, please refer to the literature [26]). Set γ = 0.1. Then, Assumption 1 holds when we choose l as 0.2. Consider four followers and one leader in the multi-agent system, and the associated Laplacian matrix is given by ⎛ ⎞ 5.4 −1.8 0 −1.6 −2 ⎜ −1.6 3.5 0 −1.9 0 ⎟ ⎜ ⎟ ⎟ L=⎜ ⎜ −1.8 0 4.6 −1.5 −1.3 ⎟ . ⎝ −1.3 0 −1.7 5 −2 ⎠ 0 0 0 0 0

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Fig. 1. Simulation results without disturbances: (a) event-triggered instants; (b) control inputs of four followers; (c) position trajectories of all agents; (d) velocity trajectories of all agents.

If there is no disturbance, according to Corollary 1, the systems (1) and (2) can reach a leading-following consensus when we choose α = 1, β = 1.4 and h = 0.05. Figure 1 depicts the simulation results in this case, and we can see that the position and velocity states data of the four followers and the leader are eventually consistent. Moreover, it can be noticed that the frequency of control updates is effectively reduced by introducing the ETC mechanism from (a). Then, we consider the system (1) with disturbances, and we set ω1 (t) = [cos(2π t), 0.5 sin(2π t), 0.5sin(4π t)]T , ω2 (t) = [0.8cos(π t), sin(π t), 0.5sin(2π t)]T ,ω3 (t) = [cos(0.6π t), 0.4sin(0.6π t), 0.5sin(1.2π t)]T and ω4 (t) = [0.2cos(1.5π t), 0.3sin(1.5π t), 0.5sin(3π t)]T . We choose α = 5.9, β = 6.2 and h = 0.02 when we set c1 = 1, c2 = 1, ρ = 0.5 and η = 0.013, then according to Theorem 1, the L2 leaderfollowing consensus can be achieved. Figure 2 depicts the simulation results in this case. (a) and (b) show the position and velocity trajectories respectively. (c) shows the event-triggered instants in the first two seconds and (d) illustrates the time evolutions ˆ = 0 and of s(t)2 and ρ 2 ω (t)2 with ρ = 0.5 under the zero initial condition x(t) v(t) ˆ = 0. 2

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Fig. 2. Simulation results with disturbances: (a) position trajectories of all agents; (b) velocity trajectories of all agents; (c) event-triggered instants; (d) L2 performance with ρ = 0.5.

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5 Conclusions In this paper, we have investigated L2 leader-following consensus of second-order multi-agent systems with nonlinear dynamics and external disturbances under directed topology. By introducing a new node-based event-triggered controller and the corresponding event triggering condition, the frequency of controller updates is effectively reduced which can be observed from the simulation results. Furthermore, the L2 leaderfollowing consensus can be achieved for a given L2 gain performance index. Finite-time and fixed-time leader-following consensus problems under directed switching topology with time-delay are more challenging, and we will focus on this in the future research. Acknowledgments. This work was partially supported by the National Natural Science Foundation of China (No. 61573095 and No. 61705127).

References 1. Kolaric P, Chen C, Dalal A, Lewis FL (2018) Consensus controller for multi-UAV navigation. Control Theory Technol Control Theory Technol 16(2):110–121 2. Ren W (2007) Formation keeping and attitude alignment for multiple spacecraft through local interactions. J Guid Control Dyn 30(2):633–638 3. Zhao L, Jia Y (2014) Decentralized adaptive attitude synchronization control for spacecraft formation using nonsingular fast terminal sliding mode. Nonlinear Dyn 78(4):2779–2794 4. Schiffer J, Seel T, Raisch J, Sezi T (2016) Voltage stability and reactive power sharing in inverter-based microgrids with consensus-based distributed voltage control. IEEE Trans Control Syst Technol 24(1):96–109 5. Chen Q, Yin C, Zhou J, Wang Y, Wang X, Chen C (2018) Hybrid consensus-based cubature Kalman filtering for distributed state estimation in sensor networks. IEEE Sens J 18(11):4561–4569 6. Lin P, Jia Y (2010) Consensus of a class of second-order multi-agent systems with time-delay and jointly-connected topologies. IEEE Trans Autom Control 55(3):778–784 7. Seo JH, Shim H, Back J (2009) Consensus of high-order linear systems using dynamic output feedback compensator: low gain approach. Automatica 45(11):2659–2664 8. Liu W, Zhou S, Qi Y, Wu X (2016) Leaderless consensus of multi-agent systems with Lipschitz nonlinear dynamics and switching topologies. Neurocomputing 173:1322–1329 9. Feng Y, Zheng WX (2018) Group consensus control for discrete-time heterogeneous firstand second-order multi-agent systems. IET Control Theory Appl 12(6):753–760 10. Ni W, Cheng D (2010) Leader-following consensus of multi-agent systems under fixed and switching topologies. Syst Control Lett 59(3–4):209–217 11. Wen G, Zhao Y, Duan Z, Yu W, Chen G (2016) Containment of higher-order multi-leader multi-agent systems: a dynamic output approach. IEEE Trans Autom Control 61(4):1135– 1140 12. Guo G, Ding L, Han QL (2014) A distributed event-triggered transmission strategy for sampled-data consensus of multi-agent systems. Automatica 50(5):1489–1496 13. Hu W, Liu L, Feng G (2017) Output consensus of heterogeneous linear multi-agent systems by distributed event-triggered/self-triggered strategy. IEEE Trans Cybern 47(8):1914–1924 14. Liu L, Zhou W, Li X, Sun Y (2019) Dynamic event-triggered approach for cluster synchronization of complex dynamical networks with switching via pinning control. Neurocomputing 340:32–41

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15. Xie T, Liao X, Li H (2016) Leader-following consensus in second-order multi-agent systems with input time delay: an event-triggered sampling approach. Neurocomputing 177:130–135 16. Ma L, Wang Z, Lam HK (2017) Event-triggered mean-square consensus control for timevarying stochastic multi-agent system with sensor saturations. IEEE Trans Autom Control 62(7):3524–3531 17. Zhang Z, Hao F, Zhang L, Wang L (2014) Consensus of linear multi-agent systems via eventtriggered control. Int J Control 87(6):1243–1251 18. Liu Z, Chen Z, Yuan Z (2012) Event-triggered average-consensus of multi-agent systems with weighted and direct topology. J Syst Sci Complex 25(5):845–855 19. Li H, Liao X, Huang T, Zhu W (2015) Event-triggering sampling based leader-following consensus in second-order multi-agent systems. IEEE Trans Autom Control 60(7):1998– 2003 20. Li Z, Duan Z, Chen G (2011) On H∞ and H2 performance regions of multi-agent systems. Automatica 47(4):797–803 21. Zhang H, Yang R, Yan H, Yang F (2016) H∞ consensus of event-based multi-agent systems with switching topology. Inf Sci 370–371:623–635 22. Saboori I, Khorasani K (2014) H∞ consensus achievement of multi-agent systems with directed and switching topology networks. IEEE Trans Autom Control 59(11):3104–3109 23. He W, Xu C, Han QL, Qian F, Lang Z (2017) Finite-time L2 leader-follower consensus of networked Euler-Lagrange systems with external disturbances. IEEE Trans Syst Man Cybern: Syst 48(11):1–9 24. Wen G, Duan Z, Li Z, Chen G (2012) Consensus and its L2 -gain performance of multi-agent systems with intermittent information transmissions. Int J Control 85(4):384–396 25. Qu Z (2009) Cooperative control of dynamical systems: applications to autonomous vehicles. Springer, Heidelberg 26. Li Z, Duan Z, Chen G, Huang L (2010) Consensus of multiagent systems and synchronization of complex networks: a unified viewpoint. IEEE Trans Circuits Syst I Regul Pap 57(1):213–224

Event-Triggered Nonsingular Fast Terminal Sliding Mode Control for Nonlinear Dynamical Systems Jianlin Feng and Fei Hao(B) The Seventh Research Division, School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China [email protected], [email protected]

Abstract. This paper proposes an event-triggered control strategy based on nonsingular fast terminal sliding mode for nonlinear dynamical systems with disturbances. It can not only to guarantee the required control performance and reduce the system resource utilization, but also to accelerate the rate of convergence compared with the terminal sliding mode. It is proved that with the sliding mode control the closed-loop system does not exhibit accumulation of triggering instants. Finally, a numerical example is given to illustrate the efficiency and feasibility of the proposed results. Keywords: Event-triggered · Fast terminal sliding mode control · Nonlinear systems

1 Introduction For sample-data systems, almost all the controllers are implemented in time-triggered manner. The most commonly used technique is the sampling and holding strategy where the control is applied at sampling instant and held constant till the next sampling instant. However, this mechanism tends to waste transmission resources since the sampling period is usually conservative for guaranteeing the desired system performance in the worst scenarios. Unlike the traditional digital implementation, the event-triggered strategy can reduce the resources of transmission and computation effectively. In eventtriggered control, evolution of state trajectories of the system with respect to some threshold determines the sampling instants that the control tasks are executed. Over the last decade, significant contributions have been made in event-triggered control. In [1], a new method based on event-triggered state-feedback control was proposed, in which the control input generator simulated continuous feedback between two consecutive event times. In [2], an event-triggered control for nonlinear system was analysed which guaranteed asymptotic stability of the system. However, the disturbance was not taken into account in this analysis. In [3], the authors studied robust stability of linear system with event-triggering technique and sliding mode control (SMC). More recently, event-triggered SMC method has been developed for a broad range of various systems, such as a class of nonlinear systems [4], highly non-linear Euler-Lagrange c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 150–158, 2020. https://doi.org/10.1007/978-981-32-9682-4_16

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systems [5] and decentralized systems with a single input [6]. However, in these studies, all the sliding surface were designed as traditional linear sliding surfaces. In [7], the authors proposed self-triggered nonsingular sliding mode terminal control which allows the system states to achieve finite-time convergence. Motivates by [7], this paper investigates event-triggered nonsingular fast terminal sliding mode control for a faster convergence in comparison with the terminal sliding mode control [8]. The rest of paper is organized as follows. Section 2 introduces the sampled-data control system model based on nonsingular fast terminal SMC. In Sect. 3, the eventtriggered nonsingular fast terminal SMC method and the stability analysis of the closedloop system are presented. Finally, the simulation results are given in Sect. 4 to show the effectiveness of the proposed control method.

2 System Description The dynamics of a class of second-order nonlinear control system are described as following  x˙1 (t) = x2 (t) (1) x˙2 (t) = f (x(t)) + bu(t) + d(t), where x1 ∈ R and x2 ∈ R represents the states of the system, and x(t) = (x1 (t), x2 (t))T . u(t) ∈ R is the control input and d(t) ∈ R is the external disturbance. The known function f (x(t)) ∈ R is an intrinsic nonlinear smooth dynamics and b ∈ R is a known nonzero coefficient. Assumption 1. The external disturbance d(t) is unknown but assumed to be bounded for all time, i.e., supt≥0 |d(t)| ≤ dmax . In this paper, we will design the event-triggered control scheme based on the emulation-based approach [1, 2]. That is, we first design a control strategy in the case of unlimited communication resources, and then transform the strategy into an eventtriggered form. Hence, consider the nonsingular fast terminal sliding mode control which is proved to be an efficient control method for nonlinear control systems. The first step is to construct the nonsingular fast terminal sliding surface, which is shown as follows p g q

h

s(t) = x1 (t) + α [x1 (t)] + β [x2 (t)] ,

(2) g h

p q

p q

where α > 0, β > 0. The odd integers g, h, p and q satisfy that > and 1 < < 2. Based on (2), taking the time derivative along the state trajectory of the system (1), one has g αg βp q −1 h −1 [x1 (t)] x2 + [x2 (t)] × ( f (x) + b · u(t) + d(t)). h q p

s(t) ˙ = x2 (t) +

(3)

Hence, we design the equivalent control variable as follows ueq (t) = −b−1 ( f (x) + η (x2 (t))(1 + φ (x1 (t))), where η (x2 (t)) =

p 2− q

q β p [x2 (t)]

g

, φ (x1 (t)) =

h αg h [x1 (t)]

−1

.

(4)

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The discontinuous switching control is designed that un (t) = −b−1 Ksign(s(t)),

(5)

where K > 0 is the control gain. Combining (4) and (5), the control input can be designed as u(t) = −b−1 [ f (x) + η (x2 (t))(1 + φ (x1 (t)) + Ksign(s(t)))].

(6)

When the communication resources are limited, the feedback control input signals are utilized based on a digital sampling platform. The sampling control signal is obtained at time sequences {ti }∞ i=0 which is decided by some designed event-triggering conditions. Hence, at each sampling instant ti , only the system state x(ti ) could be used to calculate the control input u(ti ). Until the next sampling instant ti+1 , the current control signal u(ti ) will be updated to u(ti+1 ) and input to the controlled plant. The control input implemented in the interval [ti ,ti+1 ) is shown as u(ti ) = −b−1 [ f (x(ti )) + η (x2 (ti ))(1 + φ (x1 (ti )) + Ksign(s(ti )))].

(7)

Hence, we can rewritten the second-order nonlinear sampled-data control system as  x˙1 (t) = x2 (t) (8) x˙2 (t) = f (x(t)) + b · u(ti ) + d(t). For such a sampled-data control system, the objective of this work is to design an event-triggered nonsingular fast terminal sliding mode control strategy to reduce the system resource utilization while ensuring the performance of the system and speeding up the system stabilization.

3 Event-Triggered Nonsingular Fast Terminal Sliding Mode Control This section will provide the main results of this paper. We introduce the design of the event-triggered nonsingular fast terminal sliding mode control method. First, the event condition in terms of the system states is firstly constructed and stability analysis is performed. Then, we calculate the lower bound of the execution interval and prove that there is no Zeno phenomenon [10]. Define an error term ei (t) : t ∈ [ti ,ti+1 ) as ei (t) = x(t) − x(ti ),

(9)

it is generated by the discrete-time control of the system. Due to the discrete-time control implementation, an ideal sliding mode is not available. However, it remains in the vicinity of the sliding manifold. A formal definition is presented below. Definition 1. The system (1) is said to be in practical sliding mode if given any ε > 0 there exists a t1 ≥ 0 such that the system trajectory remains bounded in the ε neighbourhood of the sliding manifold S for all t ≥ t1 . In addition, the system is said to be in ideal sliding mode if the trajectory remains on S (i.e., ε = 0) for t ≥ t1 (see [9]).

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The band size of the practical sliding mode is called practical sliding mode band. It can be seen that the event-triggered implementation of SMC results in the practical sliding mode in the system. Thus, the steady-state performances of the system are significantly improved in the presence of disturbances. Next, we give the sufficient condition for the existence of practical sliding mode in the system. Theorem 1. Consider the system sampled-data control system (8) with the controller (7). Assume that f (x(t)) is Lipschitz continuous on compact sets with the L1 for all t ∈ [ti ,ti+1 ) and θ > 0 be given such that L1 ei (t)∞ < θ

(10)

for all t ≥ 0. Then, practical sliding mode occurs in the system within the region given by 1 + α L4 + β L5 Ω = {x ∈ n : s = Cx < θ} (11) L1 if the switching gain in (7) is selected as  r1  r2 θ ∗ ∗ θ + L3 + dmax + κ , K ≥ θ + (L2 + L2 ) L1 L1

(12)

where L2 , L3 are the Lipschitz constant of η (x2 (t)), φ (x1 (t)) with the order r1 , r2 and g h

p q

L4 , L5 are the Lipschitz constant of [x1 (t)] , [x2 (t)] respectively. Proof. Based on the assumption, the nonlinear function f (x(t)) is Lipschitz continuous on compact sets, and it holds that | f (x(t)) − f (x(ti ))| ≤ L1 x(t) − x(ti )∞ ,

(13)

where L1 > 0. Consider the Lyapunov function candidate V = 12 s2 (t). Taking the derivative yields that V˙ = s(t)s(t) ˙ = s(t){x2 (t) + φ (x1 (t))x2 (t) + ρ (x2 (t))[ f (x(t) − f (x(ti )) − η (x2 (ti ))(1 + φ (x1 (ti )) − Ksign(s(ti ))+d(t))]} (14) ≤ |s(t)| ρ (x2 (t))(L1 |ei (t)| + η (x2 (t))(1 + φ (x1 (t)) − η (x2 (ti ))(1 + φ (x1 (ti ))) + |s(t)| ρ (x2 (t))dmax − ρ (x2 (t))Ks(t)sign(s(ti )), g h −1

p q −1

where φ (x1 (t)) = αhg [x1 (t)] > 0 and ρ (x2 (t)) = βqp [x2 (t)] > 0. Consider that the function η (x2 (t)) and φ (x1 (t)) are uniformly H¨older continuous on compact sets with the order r1 and r2 , one has |η (x2 (t)) − η (x2 (ti ))| ≤ L2 x2 (t) − x2 (ti )r∞1 , |φ (x1 (t)) − φ (x1 (ti ))| ≤ L3 x1 (t) − x1 (ti )r∞2 ,

(15)

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where L2 > 0, L3 > 0. Due to the characteristic of φ (x1 (t)) and η (x2 (t)), their norms are bounded. In the above formula, some of the items can be processed as follows, η (x2 (t))φ (x1 (t)) − η (x2 (ti ))φ (x1 (ti )) = η (x2 (t))φ (x1 (t)) − η (x2 (ti ))φ (x1 (t)) + η (x2 (ti ))φ (x1 (t)) − η (x2 (ti ))φ (x1 (ti )) ≤ L2 x2 (t) − x2 (ti )r∞1 φ (x1 (t)) + L3 x1 (t) − x1 (ti )r∞2 η (x2 (ti )) r1 r2       ≤ L2∗ e1i (t) + L3∗ e2i (t) . ∞

(16)

∞

       Note that ei (t)∞ = max e1i (t)∞ , e2i (t)∞ , one has e2i (t)∞ ≤ ei (t)∞ . When the signs of s(t) and s(ti ) are the same, the above relation can be reduced using (10) as V˙ ≤ |s(t)| ρ (x2 (t)) (L1 ei (t)∞ + L2 ei (t)r∞1 + L2∗ ei (t)r∞1 + L3∗ ei (t)r∞2 − K + dmax )   L∗ L1 r2 ei (t)r∞2 L1 r1 ei (t)r∞1 = |s(t)| ρ (x2 (t)) L1 ei (t)∞ + (L2 + L2∗ ) + 3 − K + dmax r r 1 2 L1 L1   r1  r2  θ θ ∗ ∗ + L3 − K + dmax ≤ |s(t)| ρ (x2 (t)) θ + (L2 + L2 ) L1 L1

(17)

≤ −κ |s(t)| ,

where κ > 0. From this, we see that the system trajectories are attracted toward the sliding manifold as long as sign(s(t)) = sign(s(ti )). In this case, the control system is in the reaching mode. The sliding trajectories are ultimately bounded within this region since V˙ < 0 outside the region with sign(s(t)) = sign(s(ti )). p q

g h

Note that [x1 (t)] and [x2 (t)] are Lipschitz continuous on compact sets, one has  g g   [x1 (t)] h − [x1 (ti )] h  ≤ L4 x1 (t) − x1 (ti ) , ∞    (18) p p   [x2 (t)] q − [x2 (ti )] q  ≤ L5 x2 (t) − x2 (ti ) . ∞   Then, the maximum deviation of the sliding trajectory can be deduced as   p p g g  q q  h h  |s(t) − s(ti )| = x1 (t) + α [x1 (t)] + β [x2 (t)] − x1 (ti ) − α [x1 (ti )] − β [x2 (ti )]              ≤ e1i (t) + α L4 e1i (t) + β L5 e2i (t) ∞

∞

≤ (1 + α L4 + β L5 ) ei (t)∞ ≤

∞

(19)

(1 + α L4 + β L5 ) θ . L1

The maximum value of the band of the sliding mode can be obtained for the case s(ti ) = 0 and is given in (11). Thus, the proof is completed. The stability of the system can be shown by considering the sliding mode dynamics when the system is in practical sliding mode. The system dynamics can then be given as

Event-Triggered

x˙1 (t) = x2 (t), q  g  p 1 h . x2 (t) = s − x1 (t) − α [x1 (t)] β

155

(20)

Consider the Lyapunov function V1 = 12 [x1 (t)]2 . Taking the derivatives of V1 along the system trajectory of (20), we obtain 

q g p 1 ˙ h V1 = x1 (t) . s(t) − x1 (t) − α [x1 (t)] β

(21)

When the system is in the sliding mode, the sliding motion will reach equilibrium g if it always holds that |s(t)| < |x1 (t)| + α |x1 (t)| h . Owing to the sliding trajectory stays in the band in (11), in this case the region that

 g   (22) Ψ = x |s(t)| < |x1 (t)| + α |x1 (t)| h is larger than the region Ω in (11), then the system will converge. g Defining ζ (|x1 (t)|) = |x1 (t)| + α |x1 (t)| h , the derivative of the ζ (|x1 (t)|) along |x1 (t)| can be deduced as g αg |x1 (t)| h −1 > 0. ζ˙ (|x1 (t)|) = 1 + h

(23)

It can be seen that ζ (|x1 (t)|) is monotonically increasing, so there must be a region β L5 )θ of |x1 (t)| such that ζ (|x1 (t)|) ≤ (1+α L4L+ . Therefore the system trajectories remain 1 ultimately bounded. It is seen that the condition (10) is essential for the existence of a practical sliding mode in the system. So, the triggering rule, that ensures stability of the event-triggered control system, is given as ti+1 = inf {t > ti : L1 ei (t)∞ ≥ θ } .

(24)

The event-triggering mechanism proposed in (24) generates the time sequence {ti }∞ i=0 which is Zeno-free as shown in the following theorem. Theorem 2. Consider the control system (8) with control law (7). Let {ti }∞ i=0 be a sequence of triggering instants under the triggering condition (24). Then, the interevent time satisfies   θ 1 ln +1 , (25) Ti ≥ L1 ρ (ti ) + dmax where L1 is the Lipschitz constant of f (x) and ρ (ti ) is defined as a real-valued function     x2 (ti )  . (26) ρ (ti ) =  −η (x2 (ti )) (1 + φ (x1 (ti ))) − Ksign (s (ti )) 

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Proof. Consider the set Γ1 = {t ∈ [ti ,ti+1 ) : ei (t)∞ = 0}. Then, for all time t ∈ [ti ,ti+1 ) \Γ1 , we have 

   d x2 (t) − x2 (ti )  + dmax ei (t)∞ ≤   f (x(t)) − f (x(ti )) ∞ dt 

   x2 (ti ) (27)  +  −η (x2 (ti )) (1 + φ (x1 (ti ))) − Ksigns (ti )  ∞ ≤ L1 ei (t)∞ + ρ (ti ) + dmax where ρ (ti ) is defined by (26). The solution to the differential inequality above can be given with the initial condition ei (ti )∞ = 0 as ei (t)∞ ≤

ρ (ti ) + dmax  L1 (t−ti ) e −1 L1

(28)

for t ∈ [ti ,ti+1 ). Hence, from the event-triggering mechanism (24). The inter-event time Ti must be larger than the time which it takes for error ei (t)∞ to grow from 0 to Lθ1 . One has  θ ρ (ti ) + dmax  L1 Ti e ≤ −1 (29) L1 L1 which leads to (25). Therefore, the proof is completed.

4 Simulation We consider a numerical example to illustrate the efficiency and feasibility of proposed method. Consider the following nonlinear control system x˙1 (t) = x2 (t), x˙2 (t) = −0.1x2 (t)2 + u(t) + d(t),

(30)

where the disturbance is d(t) = 0.5 cos(0.1t) and initial state is x(0) = [3, 3]T . 35

19

The sliding manifold is designed as s(t) = x1 (t) + 2[x1 (t)] 33 + [x2 (t)] 17 with α = 2, β = 1, p = 19, q = 17, g = 35 and h = 33. The parameters are then chosen as follows: consider the system state on compact set [5, 5]. From this we can derive the value of L1 = 1 for f (x (t)) = −0.1x2 (t)2 to satisfy the Lipschitz condition d f (x(t)) ≤ 1 and dt 35

L4 = 1.2 for x1 (t) 33 with its derivative always smaller than 1.2 as well. Similarly it can be derived L5 = 1.4. g 2 33 Concerning φ (x1 (t)) = αhg [x2 (t)] h −1 = 70 33 [x2 (t)] , one can deduce L3 = 2.2 with r=

2 33

since its H¨older condition can be verified that lim

ε →0

|φ (ε )−φ (0)| |ε −0|r

< 2.1. Similarly it

∗ can be derived L2 = 1 with r = 15 17 . Therefore, it can be derived that L2 = φ (x1 (t)) L2 ≤ ∗ 2.3 and L3 = η (x2 (t)) L3 ≤ 8.1. The performance of the event-triggered nonsingular fast terminal SMC strategy is shown in Figs. 1 and 2. The state trajectories converge towards the origin is shown in

Event-Triggered

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State Trajectories

4 2 0 -2 -4

0

1

2

3

4

5

6

7

8

9

10

7

8

9

10

(a) Time (sec)

Sliding surface

15 10 5 0 -5

0

1

2

3

4

5

6

(b) Time (sec)

Fig. 1. System states and sliding surface of the control system

Fig. 1(a). The convergence of sliding trajectories towards the practical sliding mode band is shown in Fig. 1(b). The time evolution of the inter event time is depicted in Fig. 2. It could be seen that the lower bound of the event interval is strictly positive. Hence, the proposed event-triggered SMC are Zeno-free.

Inter-execution time

10 -1

10 -2

10 -3

0

1

2

3

4

5

6

7

8

(c) Time (sec)

Fig. 2. Inter-execution time of the control system

9

10

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5 Conclusions This paper have proposed an event-triggered nonsingular fast terminal sliding mode control method for nonlinear dynamical systems in presence of disturbances. The advantage of the proposed method is that the triggering rule achieves robust performance and a faster convergence rate in comparison with the terminal sliding mode. It is proved that with the sliding mode control the sequence of triggering instants generated by the event-triggering condition does not exhibit accumulation of triggering instants. Finally, a numerical example is given to illustrate the efficiency and feasibility of the proposed method.

References 1. Lunze J, Lehmann D (2010) A state-feedback approach to event-based control. Automatica 46(1):211–215 2. Tabuada P (2007) Event-triggered real-time scheduling of stabilizing control tasks. IEEE Trans Autom Control 52(9):1680–1685 3. Behera AK, Bandyopadhyay B (2017) Robust sliding mode control: an event-triggering approach. IEEE Trans Circ Syst II Express Briefs 64(2):146–150 4. Behera AK, Bandyopadhyay B (2016) Event-triggered sliding mode control for a class of nonlinear systems. Int J Control 1–23 5. Kumari K, Behera AK, Bandyopadhyay B (2018) Event-triggered sliding mode-based tracking control for uncertain Euler-Lagrange systems. IET Control Theory Appl 12(9):1228– 1235 6. Behera AK, Bandyopadhyay B (2015) Decentralized event-triggered sliding mode control. In: 10th Asian control conference (ASCC), Kota Kinabalu, Malaysia. Asian Control Association, pp 1–5 7. Wu P, Zhang WL, Han S (2018) Self-triggered nonsingular terminal sliding mode control. In: 2018 annual American control conference (ACC). IEEE, pp 6513–6520 8. Yu SH, Yu XH, Shirinzadeh B (2005) Continuous finite-time control for robotic manipulators with terminal sliding mode. Automatica 41(11):1957–1964 9. Behera AK, Bandyopadhyay B, Yu XH (2018) Periodic event-triggered sliding mode control. Automatica 96:61–72 10. Borgers DP, Heemel WPMH (2014) Event-separation properties of event-triggered control systems. IEEE Trans Autom Control 59(10):2644–2656

Fault Detection for T-S Fuzzy Networked Control Systems with Signal to Noise Ratio Constrained Channels Fumin Guo1(B) and Xuemei Ren2 1

National Computer Network Emergency Response Technical Team/Coordination Center of China, Beijing 100029, China ggjj [email protected] 2 School of Automation, Beijing Institute of Technology, Beijing 100081, China

Abstract. This paper investigates fault detection for T-S fuzzy networked control systems with signal to noise ratio constrained channels. In order to overcome the unconstrained power distribution phenomenon, a parallel multi-input multi-output additive Gaussian white noise channel is used. For the purpose of detecting the fault, a H− /H∞ robust fault detection filter is designed; then, a dynamic threshold based on the stochastic characteristics of channel noise is constructed, and a constant α is introduced to reduce the conservatism of threshold; finally, a numerical example is given to illustrate the effectiveness of the proposed FD method.

Keywords: Fault detection Threshold

1

· T-S fuzzy · Signal to noise ratio ·

Introduction

Networked control systems (NCSs) are typical feedback systems wherein the distributed system components are connected via communication networks, so the communication channels in NCS are naturally subject to some constraints, such as signal to noise ratio (SNR) constraints. The work in [1] addressed disturbance rejection issues of discrete-time systems with SNR constrained channels, and admissible SNR expressions for stabilization were derived. In addition, for the unstable feedback systems with SNR constraints and model uncertainties, the authors in [2] investigated the influences of model uncertainties on the minimal SNR required for stabilizing the plant. Fault detection (FD) has been one of the hottest research topics since 1970s, which is because of the growing demands of higher performance and reliability of industrial systems. Now, FD has been widely used in engineering areas, such as wireless sensor networks [3]. At the same time, many FD approaches have been proposed, for example, observer-based approaches [4,5]. c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 159–166, 2020. https://doi.org/10.1007/978-981-32-9682-4_17

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Note that FD for NCS with SNR constrained channels is still in its infancy, and furthermore the channels considered in [1,2] are subject to a total power constraint, which may cause an unconstrained power distribution phenomenon. In order to overcome the phenomenon, a multi-input multi-output (MIMO) channel which consists of multiple independent single-input single-output (SISO) channels is considered. Then, for the purpose of detecting the fault, a robust fault detection filter is designed based on H− /H∞ , a dynamic threshold based on the stochastic characteristics of channel noise is constructed, and a constant α is introduced to reduce the conservatism of threshold.

2

Preliminaries and Problem Formulation

Consider the T-S fuzzy NCS over a MIMO additive white Gaussian noise (AWGN) channel, and plant rule i: if ξ1 (k) is M1i ,..., and ξg (k) is Mgi . Then, the final fuzzy system is x (k + 1) =

r 

μi (ξ (k)) (Ai x (k) + Bi u (k)) + Bd d (k) + Bf f (k)

i=1

(1)

y (k) = Cx (k) + Df f (k) where x (k) ∈ Rn is the system state, u (k) ∈ RP is system input, d (k) ∈ Rnd is the unknown disturbance, f (k) ∈ Rnf is the fault, and y (k) ∈ Rm is the output; Ai , Bi , Bd , Bf , C, D are the known matrices with appropriate dimension; ξ (k) = [ξ1 (k) · · · ξg (k)] are the premise variables assumed measureable, M1i , · · i (ξ(k)) ·, M1g are the fuzzy sets and r is the number of rules; μi (ξ (k)) =  , r i (ξ (k)) =

g 

Mji (ξj (k)) and μi (ξ (k)) satisfies

j=1

r 

i=1

i (ξ(k))

μi (ξ (k)) = 1, μi (ξ) ≥ 1.

i=1

In order to estimate the state and sensor fault, an augmented system is constructed by the descriptor technique: Ex (k + 1) =

r 

  μi (ξ (k)) Ai x (k) + B i u (k)

i=1

+B d d (k) + Df xs (k) + B f f (k) y (k) = Cx (k) = C ∗ x (k) + xs (k) where

(2)



 x (k) xs (k) = Df f (k) , x (k) = , xs (k)       I 0 Ai 0 Bd , E= , Ai = , Bi = 0 0 0 0 −I     0 Bf , C ∗ = [C 0] , C = [C I] . Df = , Bf = 0 I

Assumption 1. System output is transmitted to the FD filter through a MIMO AWGN channel which comprised of m independent SISO channels, i.e., w (k) =

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T

y (k) + n (k), where y (k) = [y1 (k) y2 (k) · · · ym (k)] , w (k) = [w1 (k) w2 (k) T T · · · wm (k)] is the channel output, and n (k) = [n1 (k) n2 (k) · · · nm (k)] is the channel noise. In addition, each noise nj (k) is a zero mean Gaussian white one with the variance 0 < σn2 j < ∞ (j = 1, 2, · · ·, m), and each SISO channel is subject to a SNR constraint with the definition: Sj =

Pyj , j = 1, 2, · · ·, m σn2 j

(3)

where Sj and Pyj are the SNR and the channel input power of the jth SISO channel, respectively. Our objective is to design a FD system including a fault detection observer and a threshold to investigate FD for system (1) with a MIMO AWGN channel subjects to SNR constraints shown in (3).

3

Fault Detection Observer Design

In this paper, we will design the following observer Ez (k + 1) =

r 

  μi (ξ (k)) Fi z (k) + B i u (k)

i=1

(4)

 (k) = z (k) + Lw (k) x  (k) y (k) = C ∗ x

 (k) ∈ Rn+m is where z (k) ∈ Rn+m is an auxiliary state of the observer, and x the state estimation of x (k); E, Fi , L are the designed gain matrices. We are interested in estimating the state and sensor fault, i.e., d (k) = 0,  (k) − Ly (k), then we have Bf = 0 and n (k) = 0. Let z (k) = x 

  (k + 1) = E + ELC x (k + 1) − E x 

r    (k) μi (ξ (k)) Fi LC ∗ + Ai x (k) − Fi x i=1   + Fi L + Df xs (k)

(5)

If the following conditions are fulfilled E = E + ELC, Fi = Ai + Fi LC ∗ , Fi L = −Df

(6)

 (k) we can obtain the dynamic of estimation error e (k) = x (k) − x Ee (k + 1) =

r 

μi (ξ (k)) Fi e (k)

(7)

i=1

A solution of constraints (5) is       Ai 0 0 I + QC Q Fi = ,L = ,E = −C −I I RC R

(8)

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Thus, we can obtain e (k + 1) =

r 

μi (ξ (k)) Si e (k)

(9)

i=1

where Si = E −1 Fi . Consider (2) and (4) with the condition (8), we have r  μi (ξ (k)) Si e (k) +T d (k) + Tf f (k) e (k + 1) =  i=1 r (k) = W C ∗ e (k) + Df f (k) + I d (k)

(10)

where r (k) is the residual signal, W is the residual matrix,   Bd −1 T = [Td Si L − L] , Td = E B d = , −CBd Tf = E −1 B f =



 Bf , I = [0 I 0] . −CBf

The H− /H∞ fault detection observer is designed such that r (k) is as sensitive as possible to f (k) and as robust as possible to d (k), so we have: (1) the H− 2 2 fault sensitive r (k)2 > β 2 f (k)2 , i.e., for the system (10) with d (k) = 0, we have r  ef (k + 1) = μi (ξ (k)) Si e (k) + Tf f (k) (11) i=1 r (k) = W (C ∗ e (k) + Df f (k)) 2

2

(2) the H∞ robustness r (k)2 < γ 2 d (k)2 , i.e., for the system (10) with f (k) = 0, we have r  μi (ξ (k)) Si e (k) + T d (k) ed (k + 1) =  i=1  r (k) = W C ∗ e (k) + Id (k)

(12)

Then, the H− performance can be guaranteed if there exist matrices Xi > 0, non-singular matrix G, matrix W and β > 0 such that the following LMI holds: [Υmn ]3×3 > 0

(13)

where Υ11 = Xi + C ∗T W T W C ∗ , Υ21 = DfT W T W C ∗ , Υ22 = DfT W T W Df − β 2 I, Υ31 = GSi , Υ32 = GTf , Υ33 = G + GT − Xj . Similarly, the H∞ performance can be calculated by [Γmn ]4×4 > 0

(14)

where Γ11 = G + GT − Pj , Γ13 = GSi , Γ14 = GT , Γ22 = γI, Γ23 = W C ∗ , Γ24 = W I, Γ33 = Pi , Γ44 = γI, Γmn = 0 otherwise.

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163

Threshold Computation

In this subsection, we will discuss the threshold computation. The threshold constructed here is a dynamic one which includes stochastic characteristics of the channel noise and an upper bound on the residual evaluation function of disturbance, and a constant α is introduced to reduce the conservatism of threshold. As in most works, the threshold Jth (Jth > 0) is adopted according to the following FD logic ref (k)ρ < Jth , f ault − f ree (15) ref (k)ρ ≥ Jth , alarm f or f ault and the residual evaluation function ref (k)ρ is

 k 1  rT (i) r (i) ref (k)ρ =  ρ

(16)

i=k−ρ+1

where ρ is the length of evaluation window. Recall the overall systems (1), e (k) and (10), we have r (k)ρ = rd (k) + rf (k) + rn (k)ρ

(17)

where rd (k) = r (k) |f =0,n=0 , rf (k) = r (k) |d=0,n=0 , rn (k) = r (k) |d=0,f =0 Besides, the fault-free residual evaluation function is rd (k) + rn (k)ρ ≤ rd (k)ρ + rn (k)ρ ≤ α (Jth,d + Jth,n )

(18)

where α ∈ (0, 1] is used to reduce the conservatism of threshold and Jth,d = sup rd (k)ρ , Jth,n = sup rn (k)ρ n

d

Thus, the threshold Jth can be chosen as Jth = α (Jth,d + J th,n )

(19)

and Jth,d can be evaluated by Jth,d = γrd ed,ρ where d (k)ρ ≤ ed,ρ and γrd = sup d

rd (k)2 d(k)2

(20)

can be evaluated by LMI.

Since n (k) is a stochastic process, it is reasonable to set Jth,n based on stochastic characteristics of n (k) [6]       1 (21) Jth,n = sup E n (k)ρ + μ sup σ n (k)ρ α n n

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where E (·) is mathematical expectation, σ (·) is standard deviation, and μ ≥ 1 is a constant. Then, we have (D (·) is variance)     k  2 1 T n (i) n (i) E n (k)ρ = E ρ =

1 ρ

k  i=k−ρ+1

  2 D n (k)ρ = D =

2 ρ2

k  i=k−ρ+1

i=k−ρ+1



 σn2 1 (i) + σn2 2 (i) + · · · + σn2 m (i)

 1 ρ



k 

T

n (i) n (i)

i=k−ρ+1

 4  σn1 (i) + σn4 2 (i) + · · · + σn4 m (i)

On the other hand, the FD system cannot obtain y (k), but y (k) can be estimated by the following inequality 2

2

2 2 u (k)ρ + γjd d (i)ρ yj2 (k) ≤ γju

(22)

where u (k) is assumed to be know online and γju can be calculated by min γju s.t.

⎡

⎤ G + GT − Pj 0 GAi GBi ⎢ ∗ γju I C 0 ⎥ ⎢ ⎥>0 ⎣ ∗ ∗ Pi 0 ⎦ ∗ ∗ ∗ γju I

(23)

min γjd s.t.

⎡

⎤ G + GT − Pj 0 GAi GBd ⎢ ∗ γjd I C 0 ⎥ ⎢ ⎥>0 ⎣ 0 ⎦ ∗ ∗ Pi ∗ ∗ ∗ γjd I

(24)

2

2 2 2 Let Pj (k) = γju u (k)ρ + γjd ed,ρ and according to (3), we have



P1 (k) P2 (k) Pm (k) + Jth,n = α1 S1 + S2 + · · · + Sm

 

2

2

2   P1 (k) P2 (k) Pm (k) μ 2 + + · · · + ρ S1 S2 Sm

(25)

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165

Simulation Results

Consider the T-S model: r = 2,     −0.0027 0.0202 0.0187 , B1 = , A1 = −0.0475 0.9600 −0.0006     0.9045 0.0630 0.0460 , B2 = Bd = A2 = −0.0187 0.9634 −0.0372     −6.6721 −11.2986 −0.0259 0.5361 C= , Bf = , 0 −5.2624 0.0018 −0.9509   0.0559 44.0481 Df = . 9.2414 3.4683 Suppose the sampling time h = 0.02s, ρ = 20, SN R = 35dB, u (k) = 0.1 sin k, and the fault is 0, 0 ≤ k < 50 f (k) = 0.1, 50 ≤ k < 100 Figure 1 shows the fault estimation error.

Fig. 1. The fault estimation error.

From Fig. 1, it can be seen that the fault is estimated in a satisfactory accuracy. The simulation result is shown in Fig. 2. From Fig. 2, we can see that the residual evaluation function value is less than the threshold when there is no fault and exceeds the threshold after the occurrence of fault. After computation, we have ref (50.10)ρ = 0.2303 > Jth (50.10) = 0.2286, that is, the fault can be detected in five time steps after it occurs.

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Fig. 2. Residual evaluation function value and threshold Jth under SN R = 35dB.

6

Conclusion

In this paper, the FD problem of linear T-S fuzzy NCS over the SNR constrained channels has been investigated. In order to detect the fault, a H− /H∞ fault detection observer has been designed, a dynamic threshold has been constructed, and the constant α has been used to reduce the conservatism of threshold. A numerical example has been given to illustrate the feasibility of the proposed method. Acknowledgments. This work was supported by the National Natural Science Foundation of China under Grants 61433003.

References 1. Freudenberg JS, Middleton RH, Solo V (2010) Stabilization and disturbance attenuation over a Gaussian communication channel. IEEE Trans Autom Control 55(3):795–799 2. Rojas AJ, Freudenberg JS (2015) Stabilisation over signal-to-noise ratio constrained channels: robust analysis for the discrete-time case. In: 2015 IEEE conference on control and applications, pp 21–23 3. Liu KB, Ma Q, Gong W, Miao X, Liu YH (2014) Self-diagnosis for detecting system failures in large-scale wireless sensor networks. IEEE Trans Wirel Commun 13(10):5535–5545 4. Li LL, Ding SX, Peng KX, Qiu JB, Yang Y (2019) Fuzzy fault detection filter design for nonlinear distributed parameter systems. IEEE Access 7:11105–1113 5. Yang Y, Ding SX, Li LL (2016) Parameterization of nonlinear observer-based fault detection systems. IEEE Trans Autom Control 61(11):3687–3692 6. Ding SX (2008) Model-based fault diagnosis techniques-design schemes, algorithms and tools. Springer, Berlin

Modified EM Algorithms for Parameter Estimation in Finite Mixture Models Weigang Wang1 , Shengjie Yang1 , Jinlei Cao1 , Ruijiao He1 , and Gengxin Xu2(B) 1

School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China 2 School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China [email protected]

Abstract. There’s no doubt that finite mixture models boom as a result of the proposition and development of the expectation-maximization (EM) algorithm. In this article, we propose three modified EM algorithms for parameter estimation in finite mixture models. The new methods replace the mixing proportions with other estimates while calculating the conditional expectations for the hidden labels given observations and the current estimates for parameters in the E step. We also discuss the convergence properties of the new procedures. Numerical studies show that the new acceleration methods perform much better than the classical EM algorithm in convergence rate and estimation accuracy. Keywords: EM algorithm · Finite mixture model · Acceleration method

1 Introduction Finite mixture models form a flexible tool to deal with complex phenomena. Since almost every probability density function can be approximated by a finite mixed distribution, it has attracted great attention both in theory and practice. However, the parameter estimation in finite mixture models is more complex to handle than that in a common single model. Generally speaking, the method of moments had been used for statistical inference in finite mixture models before the advent of fast computer. [1] and [2] discussed the problem of parameter estimation by the method of moments in mixtures of normal distributions. The method of moments requires solving nonlinear equations, and the number of equations increases with the number of components of the model, which brings difficulties in numerical calculation. The likelihood approach has a lot of advantages compared with the method of moments. With the development of computer, the method of maximum likelihood is becoming more and more popular in the context of finite mixed models. [3] discussed the maximum likelihood estimation for mixtures of distributions from the exponential family. The expectation-maximization (EM) algorithm of [4], which greatly simplifies c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 167–175, 2020. https://doi.org/10.1007/978-981-32-9682-4_18

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the computation of maximum likelihood procedure, has become one of the most popular numerical approaches for statistical analysis [5]. The EM algorithm in finite mixture models has been studied in [6–8], to mention a few. Although the EM algorithm brings convenience to the processing of massive data, it still has some notable drawbacks. Over the past decades, there has been much work applying and generalizing the EM algorithm to a variety of problems. The ECM algorithm of [9] optimizes the M step by the computationally simpler conditional maximization (CM) steps. Further, the ECME algorithm [10] enhances the convergence of EM and ECM. Several acceleration methods have been suggested, such as the quasiNewton EM [11] and the BFGS approach [12]. In addition, [13] discussed the problem to choose initial values for the EM algorithm and some generalizations. These work is surely the impressive improvement of EM algorithm, since they either modify the M step to simplify the calculation or choose sensible initial values to accelerate the convergence. Different with the above work, in this article, we will focus on the E step of the EM algorithm, that is to calculate the conditional expectations for the hidden labels (which define assignment of one observation to specific components in the mixture) given observations and the current estimates for parameters. Note that the mixing proportions are assigned to be the same value for all observations. Our goal is to replace them with other “sensible values” using the sample information. Hence, such doubt use of the sample information (in modified E step and classical M step) are expected to accelerate the convergence, especially in the mixtures with many components. The rest of this article is organized as follows. We introduce the new methods and discuss the convergence properties in Sect. 2. We present the simulation results and a real-data example in Sect. 3. Finally, we present some conclusions and future work in Sect. 4.

2 New Methods for Parameter Estimation in Finite Mixture Models Let F = { f (x; θ ); θ ∈ Θ } be a family of probability density functions with respect to a σ -finite measure ν , and let θ ∈ Θ ⊆ R m , m ≥ 1. The probability density function of a finite mixture distribution is defined by g(x; G) =

K

∑ πk f (x; θk ),

(1)

k=1

where G is called the mixing distribution and ∑Kk=1 πk = 1 and πk ≥ 0 for k = 1, 2, . . . , K. Suppose {x1 , x2 , · · · , xn } are some observations from the finite mixture model (1). By maximizing the log-likelihood function of the parameters, ln (G), we can get the estimates for the mixing proportions (or probabilities) and component parameters. As the classical EM algorithm, we introduce a latent variable zik for the ith observation representing its component membership. Then the complete log-likelihood function is n

lnc (G) = ∑

K

∑ zik [log πk + log{ f (xi ; θk )}].

i=1 k=1

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The E step is to evaluate the conditional expectation of lnc (G) with respect to zik given the observations and assuming the current estimate for G(m) is correct. The conditional expectation is given by n

Q(G; G(m) ) = ∑

K

∑ p(zik ; xi , G(m) )[log πk + log{ f (xi ; θk )}],

(2)

i=1 k=1

where (m)

p(zik ; xi , G

(m)

)=

πk

(m)

f (xi ; θk )

(m)

∑Kj=1 π j

(m)

f (xi ; θ j )

.

(3)

In the M step, we find new estimate G(m+1) by maximizing Q(G; G(m) ). Given some initial values for the mixing proportions and component parameters, the EM algorithm repeats the two steps till the convergence criterion is satisfied. Note that in (3), the mixing proportions πk s are assigned to be the result of the previous iteration and the same value for all observations. Now, we replace these mixing proportions with other values. The first choice is to calculate them separately for each observation, i.e., (m) f (xi ; θk ) πik∗ = . (4) (m) ∑Kj=1 f (xi ; θ j ) Accordingly, we can substitute p(zik ; xi , G(m) ) in (2) using p(z ˜ ik ; xi , G(m) ) =

(m)

πik∗ f (xi ; θk )

(m)

∑Kj=1 πi∗j f (xi ; θ j )

.

The second method is inspired by the sight of the sampling process for finite mixture models. We assign each observation to be the most likely component membership according to the probability density function, and then let the value of πk in (3) be the relative frequency of the observations in the kth component. The last method is a modification of the first method, that is πk∗ = ∑ni=1 πik∗ /n, where πik∗ is defined in (4). Again, (m) substituting πk with πk∗ in (3), we obtain another value for the conditional expectations of zik given the data and the current estimate G(m) . Comparing with the classical EM algorithm, the proposed modifications use the sample information twice in each iteration. Such frequent updating for the parameters accords with the idea of the method of maximum likelihood, that is to find values for the parameters which make the samples the most probable given the mixture model. Therefore, the new methods can reduce the number of iterations for convergence. Our proposed new methods are summarized in the pseudo codes. We next discuss the convergence properties of the new methods. As we elaborated above, the three new methods just replace the mixing proportions with other estimates while calculating the conditional expectations for the hidden labels. Hence, our new methods are monotonic as EM algorithm, that is, with the iterating of the procedures, the (m + 1)th guess G(m+1) will never be less likely than the mth guess G(m) . Further, as the classical EM algorithm, our new methods converge to a local maximum but not an optimal one. The simulation studies and a real-data example show that our new methods require significantly fewer iterations to converge.

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Algorithm 1. New method 1 Input: dataset {x1 , x2 , · · · , xn }, the number of components K; Output: estimate for parameter G; 1: initial G(0) ; 2: repeat (m)

f (xi ;θk )

3:

compute πik∗ =

4:

compute p(z ˜ ik ; xi , G(m) ) =

5:

compute G(m+1) = arg maxG Q(G; G(m) ), where

∑Kj=1

(m)

f (xi ;θ j )

Q(G; G(m) ) =

;

(m) πik∗ f (xi ;θk ) (m) K ∗ ∑ j=1 πi j f (xi ;θ j )

n

;

K

˜ ik ; xi , G(m) )[log πk + log{ f (xi ; θk )}]; ∑ ∑ p(z

(5)

i=1 k=1

6: until convergence.

Algorithm 2. New method 2 Input: dataset {x1 , x2 , · · · , xn }, the number of components K; Output: estimate for parameter G; 1: initial G(0) ; 2: repeat 3: compute πk∗ = card{k|k ∈ A}/n, where (m)

A = {ki |ki = arg max f (xi ; θ j j∈{1,2,··· ,K}

4:

compute p(z ˜ ik ; xi , G(m) ) =

(m) πk∗ f (xi ;θk ) K ∗ f (x ;θ (m) ) π ∑ j=1 j i j

); i ∈ {1, 2, · · · , n}};

;

5: compute G(m+1) = arg maxG Q(G; G(m) ), where Q(G; G(m) ) is defined as in (5); 6: until convergence.

Algorithm 3. New method 3 Input: dataset {x1 , x2 , · · · , xn }, the number of components K; Output: estimate for parameter G; 1: initial G(0) ; 2: repeat 3:

compute πk∗ = ∑ni=1 πik∗ /n, where πik∗ =

4:

compute p(z ˜ ik ; xi , G(m) ) =

(m)

f (xi ;θk ) ∑Kj=1

;

(m)

πk∗ f (xi ;θk )

(m)

;

∑Kj=1 π ∗j f (xi ;θ j ) G(m+1) = arg maxG Q(G; G(m) ), where

5: compute 6: until convergence.

(m)

f (xi ;θ j )

Q(G; G(m) ) is defined as in (5);

3 Numerical Studies Now we compare the performance of the classical EM algorithm and the new methods by Monte Carlo simulation examples and a real-data example. More specifically, we

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carry out the four algorithms to examine their computational efficiency. Particularly, in order to assess the estimates of parameters with the true values in the simulation studies, we calculate the MSE of parameters. All algorithms are implemented by R programming language. 3.1 Simulation Studies The simulated data was generated from the normal mixture models. We report the results out of 500 replications with samples sizes n = 100, 400. The parameter values are given in Table 1, which come from [14]. We set all component standard deviations σk = 1 and consider two cases: σ known and unknown in each model. Table 1. Parameter values for the normal mixture models. Model (π1 , μ1 ) (π2 , μ2 )

( π3 , μ 3 ) ( π4 , μ 4 )

( π5 , μ 5 )

( π6 , μ 6 )

( π7 , μ 7 )

1 2 3 4 5 6 7 8 9 10

(1/4, 6) (1/4, 3) (1/4, 3) (1/7, 6) (1/7, 3) (1/7, 3) (1/7, 3)

(1/7, 12) (1/7, 6) (1/7, 6) (1/7, 9)

(1/7, 15) (1/7, 7.5) (1/7, 9.5) (1/7, 10.5)

(1/7, 18) (1/7, 9) (1/7, 12.5) (1/7, 12)

(1/3, 0) (1/2, 0) (1/2, 0) (1/4, 0) (1/4, 0) (1/4, 0) (1/7, 0) (1/7, 0) (1/7, 0) (1/7, 0)

(2/3, 3) (1/2, 3) (1/2, 1.8) (1/4, 3) (1/4, 1.5) (1/4, 1.5) (1/7, 3) (1/7, 1.5) (1/7, 1.5) (1/7, 1.5)

(1/4, 9) (1/4, 4.5) (1/4, 6) (1/7, 9) (1/7, 4.5) (1/7, 4.5) (1/7, 4.5)

(0)

As in [15], the initial values of mixing coefficients were πk = 1/K, and those of component means were chosen as the 100(k − 1/2)/K% sample quantiles, for k = 1, · · · , K. For the σ -unknown case, the initial values of component standard deviations were setting as the sample standard deviation based on the observations trimmed at the 25% and 75%sample quantiles  as in [16]. In our numerical studies, the algorithm was  (new) (old)  θ θ − terminated if   < 10−5 , where θ includes the mixing proportion π and the parameters in each component (e.g., the component mean), and · is the l2 norm. We summarize the simulation results in terms of mean number of iterations for all methods based on 500 datasets in Table 2. The results show the overall outperformance of the three new methods over the classical EM algorithm (both in convergence rate and stability). Specifically, the new methods decrease to about 2%–33% iterations to converge in the mixture models with number of modes less than that of components. As a concrete illustration, we plot the iteration paths of the four methods based on one dataset generated from the normal mixture model 10 with unknown σ in Fig. 1. Compared with the classical EM algorithm, the paths of our new methods are sharper and require significantly fewer iterations. Further, we observe that new method 1, 2 and 3 converge till the 75th, 1058th and 697th iteration respectively, while the EM algorithm does till the 2738th one.

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Table 2. Algorithm convergence and estimation accuracy of parameters in simulation studies for normal mixture models.

CEM

New1

New2

New3

0.3

pi1 pi2 0.2

pi

pi3 pi4 pi5

0.1

pi6 pi7

0

1000

2000

3000

0

1000

2000

3000

0

1000

2000

3000

0

1000

2000

3000

Steps

(a) mixing proportions CEM

New1

New2

New3

12

mu1 mu2

8

mu

mu3 mu4 mu5

4

mu6 mu7 0 0

1000

2000

3000

0

1000

2000

3000

0

1000

2000

3000

0

1000

2000

3000

Steps

(b) component means CEM

New1

New2

New3

2.5

sd1

sd

2.0

sd2 sd3

1.5

sd4 sd5

1.0

sd6 sd7

0.5

0

1000

2000

3000

0

1000

2000

3000

0

1000

2000

3000

0

1000

2000

3000

Steps

(c) component standard deviations

Fig. 1. Iteration paths for the normal mixture model 10 with unknown σ .

Table 2 also shows that the estimates for the mixing proportions πk s and component parameters θk s. Under the first model with known σ , the classical EM algorithm performs best. However, the estimates by the new methods seem not far from the true

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values. Under the second and third models with known σ , all of the methods do well, but EM suffers from too much variability. Under the mixtures of four components with known σ , models 4–6, the new methods out-perform the EM algorithm, especially in the latter two models whose numbers of modes are less than four. Similarly, under models 7–10, the new methods perform much better than the EM algorithm. Under almost all the model with unknown σ , the new methods perform much better than the classical EM algorithm. Again, as a concrete illustration, we plot the result for the normal mixture model 10 with unknown σ in the box-plot (Fig. 2). It is worth mentioning that under the estimates of component standard deviations by new methods 2 and 3 are closer to the true values, while EM seems to underestimate. Overall, the new methods are consistently better than or comparable with the classical EM algorithm, particularly in the complex case with number of modes less than that of components. 0.5

2.0 1

0.4

0 −1 −2

0.1

0.5

1.5 1.0 0.5

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The simulation results indicate that compared with the classical EM algorithm, the three new modified algorithms require significantly fewer iterations to converge, especially in the model with many components and even with close components. 3.2 Example We now demonstrate the use of the modified EM algorithms for parameter estimation in real applications. This dataset comes from a study in astronomy [18], which is made up of the velocities of 82 galaxies from 6 well-separated conic sections of an unfilled survey of the Corona Borealis region (see the histogram in Fig. 3). Multimodality of the velocities may imply the existence of voids and superclusters in the far universe [19]. Therefore, we model these data using a normal mixtures with a common variance. [20] investigated the galaxy data and deemed that the number of components is between 5 and 7. Using a new penalized-likelihood approach, [16] concluded that be 7. Their work offer an optional range for the number of components in the informationbased methods. We implemented the classical EM algorithm and the new procedures to

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do parameter estimation for the normal mixtures with K = 5, 6, . . . , 11, and then selected the value of K by AIC and BIC. The initial settings and convergence criterion were the same as those in the simulation studies. The experimental result reports that the three new procedures require fewer iterations to converge than the classical EM algorithm in almost all models (except K = 5 for New2 and K = 7 for New3). Moreover, using different methods, both AIC and BIC lead to a choice of eight components, with fitted density function in Fig. 3.

4 Conclusions In this article, we have developed three modified EM algorithms for parameter estimation in univariate finite mixture models. Numerical studies show that the new methods perform much better than the classical EM algorithm in convergence rate and estimation accuracy. Our new methods replace the mixing proportions with other values while calculating the conditional expectations for the hidden labels given observations and the current estimates for parameters in the E step. More generalizations combining such a modification and the other existing EM variants interest us. In addition, as an important and challenging problem in the application of finite mixture models, order selection (i.e. determining the number of components of model), has been studied in recent years. As in the previous analysis for real-world data, our new methods can be used in the penalized-likelihood criteria, AIC and BIC. We also want to investigate the problems of generalizing our new algorithm in this context. Acknowledgments. This work was supported in part by the Natural Science Foundation of Zhejiang Province under Grant LY19A010004, the FirstClass Discipline of Zhejiang-A (Zhejiang Gongshang University-Statistics) and in part by the National Natural Science Foundation of China under Grant 61572536.

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References 1. Clifford Cohen A (1967) Estimation in mixtures of two normal distributions. Technometrics 9(1):15–28 2. Day NE (1969) Estimating the components of a mixture of normal distributions. Biometrika 56(3):463–474 3. Hasselblad V (1969) Estimation of finite mixtures of distributions from the exponential family. J Am Stat Assoc 64(328):1459–1471 4. Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc Ser B (Stat Methodol) 1–38 5. B¨ohning D (1995) A review of reliable maximum likelihood algorithms for semiparametric mixture models. J Stat Plan Inference 47(1–2):5–28 6. Chen J, Li P, Fu Y (2012) Inference on the order of a normal mixture. J Am Stat Assoc 107(499):1096–1105 7. Chen J, Li P (2016) Testing the order of a normal mixture in mean. Commun Math Stat 4(1):21–38 8. Chen J (2017) On finite mixture models. Stat Theory Related Fields 1(1):15–27 9. Meng X, Rubin DB (1993) Maximum likelihood estimation via the ECM algorithm: a general framework. Biometrika 80(2):267–278 10. Liu C, Rubin DB (1994) The ECME algorithm: a simple extension of EM and ECM with faster monotone convergence. Biometrika 81(4):633–648 11. Lange K (1995) A Quasi-Newton acceleration of the EM algorithm. Statistica Sinica 1–18 12. Jamshidian M, Jennrich RI (1997) Acceleration of the EM algorithm by using Quasi-Newton methods. J R Stat Soc: Ser B (Stat Methodol) 59(3):569–587 13. Biernacki C, Celeux G, Govaert G (2003) Choosing starting values for the EM algorithm for getting the highest likelihood in multivariate Gaussian mixture models. Comput Stat Data Anal 41(3):561–575 14. Ishwaran H, James LF, Sun J (2001) Bayesian model selection in finite mixtures by marginal density decompositions. J Am Stat Assoc 96(456):1316–1332 15. Chen J, Khalili A (2008) Order selection in finite mixture models with a nonsmooth penalty. J Am Stat Assoc 103(484):1674–1683 16. Xu C, Chen J (2015) A thresholding algorithm for order selection in finite mixture models. Commun Stat Simul Comput 44(2):433–453 17. Woo M, Sriram TN (2007) Robust estimation of mixture complexity for count data. Comput Stat Data Anal 51(9):4379–4392 18. Postman M, Huchra JP, Geller MJ (1986) Probes of large-scale structure in the Corona Borealis region. Astron J 92:1238–1247 19. Roeder K (1990) Density estimation with confidence sets exemplified by superclusters and voids in the galaxies. J Am Stat Assoc 85(411):617–624 20. Richardson S, Green PJ (1997) On Bayesian analysis of mixtures with an unknown number of components (with discussion). J R Stat Soc: Seri B (Stat Methodol) 59(4):731–792

The Ridge Iterative Regression and the Data-Augmentation Lasso Gengxin Xu1 and Weigang Wang2(B) 1

School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China 2 School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China wwgys [email protected]

Abstract. We propose the ridge iterative regression (RIR) and the dataaugmentation lasso (DAL) to improve the ridge regression and the lasso respectively. We prove that by updating the coefficient itself, the solution of the RIR converges to that of the ordinary least squares when the design matrix X has full column rank. The simulations and real-world data demonstrate that the DAL often outperforms the lasso in sparsity and estimation accuracy for the coefficient and attains smaller prediction error. Keywords: Data-augmentation · Variable selection · Ridge regression · Lasso

1 Introduction and Motivation Consider the usual linear regression model: given an n-by-p regressor matrix X with n observations and p features, we can predict the centered n-by-1 response y by  y = X β , where β is the estimate of the p-dimensional parameter vector β . By minimizing the residual squared error yy − X β 22 , where ·2 is the Euclidian norm, the ordinary least squares (OLS) is the most commonly used method to estimate β because of many good properties (e.g. unbiasedness). However, it is impracticable in the situation that the regressors in X are linearly dependent. This ill-posed problem can be solved by the ridge regression, which estimates the vector of coefficients as follows     2  y  ridge X  √ β β − = arg min  . (1) 0 λ I 2 β Contrasting with the OLS, the ridge regression can be viewed as applying the √ data augmentation technique to X and y . Furthermore, we can clearly see that it is λ I in (1) that helps to decorrelate the columns of the regressor matrix, so the ridge regression naturally solves the singularity problem. There have been many interesting works on the ridge regression, such as [1–4]. However, none of these papers has investigated the problem on the data augmentation technique. c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 176–184, 2020. https://doi.org/10.1007/978-981-32-9682-4_19

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Motivated by the data augmentation technique, we obtain the modified ridge regression estimate as       2 modi f ied ridge X  √ y   √ − β = arg min  β (2)  , λI   λ β aug β 2

in which β aug can be 0 or any other estimates (such as the solution of the OLS or the ridge regression). Our idea is this: on one hand, the new model can solve the singularity problem, on the other hand, the augmentation to y should be “the likely data”. Hence, next section we will propose a new method to achieve the both. However, no matter what β aug is, the modified ridge regression cannot do variable selection. In recent two decades, variable shrinkage methods have received much attention in the fields of modeling and predicting study because they lead to well interpretation and prediction performance of the models. Tibshirani proposed the lasso, for least absolute shrinkage and selection operator, which estimates the parameters and performs variable selection simultaneously [5]. Since then, lots of works have arisen on generalizing the lasso, such as the adaptive lasso [6], the relax lasso [7], the elastic net [8], SCAD [9], MCP [10] and l1/2 regularzation [11]. Furthermore, Li et al. investigated the problem of variable selection for fixed effects varying coefficient models [12]. All these works produce a model with minimal prediction error and select the true underlying variables. In this article, we are interested in whether the modified ridge regression with some  β aug has better performance and, if so, how. We attempt to give an answer. Furthermore, taking feature selection into consideration, we propose a new variable shrinkage method, in which the data augmentation technique and l1 penalty are applied at the same time. Simulation studies and real data example show that the new method performs better than the existing methods in prediction accuracy and sparsity. The rest of this article is organized as follows. In Sect. 2, we introduce the ridge iterative regression, define a new variable shrinkage method and then discuss the choice of tuning parameter and the computations. The performance of the new methods is compared with existing methods through simulations in Sect. 3 and a real prostate cancer data example given in Sect. 4.

2 The Ridge Iterative Regression and the Data-Augmentation Lasso 2.1 The Ridge Iterative Regression In Sect. 1, we obtain the modified ridge regression estimate (2). An advisable choice of β aug is obtained by updating itself through an iterative procedure called the ridge iterative regression (RIR). We define the RIR as follows: for any fixed non-negative λ (0) (i) and an initial value β , we let the ith iterative solution β be       2 y (i) X    √ β = arg min  √  (i−1) − β  , where i = 1, 2, ... λI   λβ β 2

Next, we show its convergence.

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Theorem 1 (convergence). Suppose the design matrix X has full column rank, then (i) the iteration sequence {β } produced by (3) converges to the closed-form solution to OLS, as i → ∞.

Proof. Denote the closed-form solution to OLS by β

OLS

X T X )−1 X T y , the = (X  augment   = √X by  y (i) . Substitute the value X and λI

 and the ith iterative augment y (i) X by X   y (i)  y = √  (i−1) into the closed-form solution to OLS, we can easily obtain λβ (i) (i−1) X T X + λ I )−1 X T y + λ (X X T X + λ I )−1 β β = (X . ridge X T X + λ I )−1 X T y and A = λ (X X T X + λ I )−1 , we have Let β = (X i−1

(i) ridge (i−1) ridge (0) β = β + A β = (II + ∑ A k )β + A i β . k=1

Since X has full column rank, the square matrix X T X can be decomposed as X T X = PT , where the eigenvalue λ j > 0 ( j = 1, 2, ..., p), and P is a p × p P diag(λ1 , λ2 , ..., λ p )P matrix of orthogonal eigenvectors. Hence, we have OLS PT X T y , β = P diag(1/λ1 , , 1/λ2 , ..., 1/λ p )P ridge PT X T y , β = P diag(1/(λ1 + λ ), 1/(λ2 + λ ), ..., 1/(λ p + λ ))P

(4)

and PT . A = P diag(λ /(λ1 + λ ), λ /(λ2 + λ ), ..., λ /(λ p + λ ))P Denote λ /(λ j + λ ) by a j in A , we get PT . A = P diag(a1 , a2 , ..., a p )P Since 0 < a j < 1 for all j, we have i−1

i−1

i−1

k=1

k=1

k=1

(i) ridge (0) PT β Pdiag(ai1 , ai2 , ..., aip )P PT β β = P (II +diag( ∑ ak1 , ∑ ak2 , ..., ∑ akp ))P +P

→ P diag(1−a1 , 1−a2 , ..., 1−a p )−1 P T β

ridge

, as i → ∞.

(5)

Substitute a j = λ /(λ j + λ ) and (4) into the right term of (5), we obtain (i) PT X T y , as i → ∞. β → P diag(1/λ1 , 1/λ2 , ..., 1/λ p )P

Note that the above convergence works in the condition that X has full column (i) rank. However, the iteration sequence {β } produced by (3) is not limited by such constraint. The convergence problem of the RIR in the case that regressors are linearly dependent (especially for high-dimensional data) interests us, though we have not yet (0) rigorously proven it. Moreover, notice that the initial value β can be any value (such as 0 or β = [xxT y /(xxT x j )] p×1 ). We will demonstrate their performance by simulaindv

tions in Sect. 3.

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2.2 The Data-Augmentation Lasso Now applying the data augmentation technique to the penalized least squares problem, that is      2  X   √ y √ − β  + λ1 p(β ),  λ2 I   λ2 β aug 2

where λ1 and λ2 are two tuning parameters, β aug is some vector and p(β ) is some penalty function. Here, by taking the l1 penalty, we obtain the data-augmentation lasso (DAL), which minimizes the criterion      2  y X    √ β  + λ1 β 1 , (6) − L(λ1 , λ2 , β aug , β ) = arg min  √  λ2 I   λ2 β aug 2

for any fixed non-negative λ1 and λ2 , and some β aug . She proved the DAL gives the elastic net estimate when taking β aug = X T y [13]. We conjecture that whenever the RIR improves on the ridge, the DAL with an advisable β will improve the lasso. aug

Here, we replace β aug by a more accurate estimate, i.e. the above RIR. Surprisingly, such fusion not only can do variable selection, but also often shows better performance in learning coefficient and prediction accuracy than the lasso and the elastic net. 2.3 Computations We now discuss the computational issues. Friedman et al. proposed the co-ordinate descent algorithm for many popular models including the lasso, ridge regression and the elastic net [14]. Here, for each fixed λ2 , we regard the DAL as the lasso, so the co-ordinate descent algorithm is applicative. Hence, the parameters (λ1 , λ2 ) are chosen by cross-validation on a two-dimensional surface.

3 A Simulation Study In this section, we demonstrate the performance of the RIR and the DAL on some simulation datasets. Each simulated data was generated from the true model y = X β + σ ε , ε ∼ N(00, I ). As the adaptive lasso paper [6], in order to show the performance in the models with different signal-to-noise ratios (SNR) [15], we assigned two values to σ in all examples. We implemented the co-ordinate descent algorithm by the glmnet R language package [14] to compute the lasso,  and the RIR. Specifically, the algorithm for  the elastic net  (new) (old)  −β RIR was terminated if β  < 10−5 , where ·∞ is the maximum norm. ∞ In addition, we considered other generalizations and variants of the lasso, including the adaptive lasso [6], the relaxed lasso [7], SCAD [9] and MCP [10], which were computed by the parcor [16], relaxo [7] and ncvreg [17] R language package. For all models,

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the five-fold cross-validation was used to estimate the tuning parameters. As the elastic net paper [8], we use “·/·” to denote the number of observations in training data and test data. Five scenarios are as follows: Example 1a (Many uneven and independent effects). We simulated 50 data sets consisting of 20/100 observations. We let β = (3, 1.5, 0, 0.5, 2, 3.5, 0, −2.5)T and x i ’s be independent. We set σ = 3, 6 such that the corresponding SNR was about 3.77 and 0.94. Example 1b (Many uneven and correlated effects). This model is the same as Example 1a, except that the pairwise correlation between x i and x j was set to be corr(i, j) = 0.5|i− j| . We also let σ = 3, 6 and the corresponding SNR was 4.99 and 1.25. Example 2a (A few uneven and independent effects). This example is the same as Example 1a, but with β = (3, 1.5, 0, 0, 2, 0, 0, 0)T . We also set σ = 3, 6 and the corresponding SNR was 1.69 and 0.42. Example 2b (A few uneven and correlated effects). We used the same model as in Example 2a except corr(i, j) = 0.5|i− j| for all i and j; σ = 3, 6 and the corresponding SNR was 2.36 and 0.59. Example 3 (Many equal and independent effects). Same as Example 1a, but with β = (2, 2, 2, 2, 2, 0, 2, 2)T . We let σ = 3, 6 and the corresponding SNR was 3.11 and 0.78. Example 4 (A few equal and correlated effects). This example is the same as Example 2b, except β = (2, 0, 0, 2, 0, 0, 2, 0)T . We let σ = 3, 6 and the corresponding SNR was 1.57 and 0.39. Example 5 (High-dimensional model). This example examines the setting with dimension larger than sample size. We simulated 50 data sets consisting of 100/100 observations and 200 variables. The first 15 variables and the remaining 185 ones were independent; the pairwise correlation between the x i and x j was set to be corr(i, j) = 0.5|i− j| , i, j = 1, ..., 15; the pairwise correlation between the x i and x j was set to be corr(i, j) = 0.5|i− j| , i, j = 16, ..., 200. The coefficient vector was β = (2.5, ..., 2.5, 1.5, ..., 1.5, 0.5, ..., 0.5, 0, ..., 0)T , where each of the first 3 blocks contains 5 repeats and the last block contains 185 ones. We let σ = 1.5, 3 and the corresponding SNR was 50.83 and 12.71. Table 1 presents the performance in variable selection for all examples. In Example 1, we see that the elastic net, DAL.2 and DAL.3 perform well in selecting the nonzero coefficients especially in the case with a high level of SNR, while lasso, adaptive lasso, relax lasso, SCAD and MCP do over-shrinkage. In Example 2, we can see that the DAL.1 outperforms the lasso, which confirms our original idea, improving lasso. Surprisingly, the performance of DAL.1 is comparable with the others. In particular, let’s study two opposite datasets, Examples 3 and 4. The DAL is more applicable for the model with many equal and independent effects than for that with a few equal and correlated effects. Table 2 summarizes the simulation results for all examples. Firstly, DAL.1, DAL.2 and DAL.3 often out-perform the other methods in prediction accuracy, especially in the case with a low level of SNR. We conjecture that it is the data augmentation that makes the new methods better to fit the data with significant noise. Secondly, even though the estimate of RIR is not spare at all, it offers at least a competitive alternative to the usual

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Table 1. Mean number of selected variables for all simulation examples based on 50 replicates Example 1a =3 =6 Example 1b =3 =6 Example 2a =3 =6 Example 2b =3 =6 Example 3 =3 =6 Example 4 =3 =6 Example 5 = 1.5 =3

Ridge

RIR.1

RIR.2

Lasso

8.0(2.0) 8.0(2.0)

8.0(2.0) 8.0(2.0)

8.0(2.0) 8.0(2.0)

8.0(2.0) 8.0(2.0)

8.0(2.0) 8.0(2.0)

8.0(5.0) 8.0(5.0)

ENet

DAL.1

DAL.2

DAL.3

AdapL

RelaL

SCAD

MCP

4.5(0.4) 6.0(1.0) 5.2(0.6) 6.1(0.9) 2.3(0.3) 5.2(1.1) 4.4(0.8) 6.1(1.4)

6.1(0.9) 6.2(1.4)

5.1(0.5) 5.1(0.5) 5.5(0.8) 5.0(0.6) 3.3(0.4) 3.7(0.5) 3.9(0.7) 3.6(0.5)

8.0(2.0) 8.0(2.0)

4.5(0.4) 6.5(1.3) 4.2(0.3) 6.1(1.0) 2.5(0.3) 5.5(1.2) 3.3(0.5) 6.2(1.4)

6.1(1.0) 6.2(1.4)

5.6(0.7) 5.7(0.8) 5.7(0.7) 5.2(0.7) 3.6(0.5) 3.7(0.5) 4.0(0.6) 3.4(0.6)

8.0(5.0) 8.0(5.0)

8.0(5.0) 8.0(5.0)

2.8(0.9) 4.2(1.9) 4.3(1.9) 6.0(3.2) 0.9(0.3) 3.9(2.2) 2.2(0.9) 4.7(2.6)

6.0(3.2) 4.7(2.6)

2.9(0.9) 3.9(1.5) 4.2(1.7) 3.4(1.2) 1.4(0.5) 1.8(0.8) 2.5(1.2) 2.2(1.0)

8.0(5.0) 8.0(5.0)

8.0(5.0) 8.0(5.0)

8.0(5.0) 8.0(5.0)

2.2(0.5) 4.3(1.9) 3.4(1.1) 5.3(2.7) 1.0(0.2) 3.8(2.0) 2.0(0.7) 5.6(3.3)

5.4(2.7) 5.6(3.3)

3.5(1.3) 3.4(1.1) 4.2(1.8) 3.6(1.4) 1.9(0.6) 3.0(1.2) 2.5(1.0) 2.1(0.9)

8.0(1.0) 8.0(1.0)

8.0(1.0) 8.0(1.0)

8.0(1.0) 8.0(1.0)

4.9(0.2) 7.0(0.7) 5.8(0.4) 6.7(0.5) 2.0(0.2) 5.5(0.7) 3.5(0.4) 5.6(0.7)

6.7(0.5) 5.6(0.7)

5.8(0.4) 5.5(0.3) 5.8(0.5) 5.5(0.4) 2.9(0.3) 3.7(0.4) 3.2(0.4) 3.1(0.3)

8.0(5.0) 8.0(5.0)

8.0(5.0) 8.0(5.0)

8.0(5.0) 8.0(5.0)

2.9(0.9) 5.3(2.8) 3.1(1.1) 6.0(3.4) 1.2(0.6) 4.7(2.9) 1.8(1.0) 5.9(3.6)

6.0(3.4) 5.9(3.6)

3.4(1.3) 3.9(1.6) 4.3(1.9) 4.0(1.7) 1.8(0.8) 2.5(1.2) 3.3(1.8) 2.7(1.4)

200.0(185.0) 200.0(185.0) 200.0(185.0) 16.2(2.5) 16.3(2.3) 0.0(0.0) 123.0(108.4) 128.8(114.3) 13.1(0.4) 16.1(2.1) 16.9(3.2) 13.9(1.4) 200.0(185.0) 200.0(185.0) 200.0(185.0) 17.3(4.6) 18.5(5.8) 0.0(0.0) 140.4(125.9) 143.9(129.5) 15.4(3.7) 14.2(2.4) 27.1(14.2) 24.3(11.6)

(NOTE: The number before parenthese gives the mean number of selected non-zero components, and that in parenthese presents the mean number of zero components incorrectly selected into the final model. RIR.1 and RIR.2 initialize with 0 and indv respectively. DAL.1 replaces the aug by indv . DAL.2 and DAL.3 replace aug by the estimate learned by RIR.1 and RIR.2 separately. “ENet”, “AdapL” and “RelaL” are the elastic net, the adaptive lasso and the relaxed lasso respectively.)

Table 2. Prediction MSE for all simulation examples based on 50 replicates Ridge Example 1a =3 =6 Example 1b =3 =6 Example 2a =3 =6 Example 2b =3 =6 Example 3 =3 =6 Example 4 =3 =6 Example 5 = 1.5 =3

RIR.1

RIR.2

Lasso

ENet

DAL.1

DAL.2

DAL.3

AdapL

RelaL

SCAD

MCP

20.8(7.4) 13.8(6.3) 13.8(6.3) 21.6(9.3) 19.0(7.9) 17.6(7.2) 14.6(7.7) 14.6(7.7) 15.2(7.5) 14.4(9.5) 15.6(5.4) 16.1(6.1) 54.9(12.5) 54.2(19.7) 54.2(19.7) 62.5(14.7)57.2(12.1)52.1(13.3) 51.8(19.0) 51.8(19.0) 56.5(16.4)53.2(19.7)55.3(30.5)57.7(30.2) 18.9(6.8) 14.1(5.5) 14.1(5.5) 20.5(10.8)18.1(10.5)20.2(14.5) 16.7(8.9) 16.7(8.9) 15.1(6.1) 14.5(5.5) 15.2(5.6) 15.2(6.5) 57.6(15.8) 52.0(14.4) 52.0(14.5) 59.3(15.0)55.3(14.6)55.4(14.5) 49.5(13.7) 49.5(13.7) 51.9(13.4)54.0(12.7)56.2(13.5)56.3(13.8) 17.9(4.8) 14.3(5.5) 14.3(5.5) 14.4(4.9) 15.1(5.2) 13.3(4.0) 48.9(9.1) 63.6(19.0) 63.6(19.2) 49.0(9.1) 49.1(9.0) 48.1(8.7)

13.0(4.4) 13.0(4.4) 14.3(4.4) 13.3(5.2) 13.9(4.9) 14.3(4.2) 53.7(16.8) 53.7(16.9) 49.5(12.2)51.1(14.1)51.6(16.0)52.3(16.9)

16.4(7.2) 14.3(5.1) 14.3(5.1) 18.5(8.5) 16.6(6.9) 15.2(6.5) 54.6(9.4) 58.3(18.0) 58.6(18.2) 51.7(9.5) 47.8(10.3)47.2(9.0)

14.0(6.1) 14.0(6.1) 13.4(4.6) 12.9(5.1) 14.1(5.9) 13.6(6.0) 52.6(14.1) 52.6(13.9) 46.9(8.8) 46.0(14.7)49.2(15.1)47.6(15.9)

19.5(9.1) 15.4(5.2) 15.4(5.2) 21.3(9.1) 19.8(8.5) 18.7(8.1) 16.2(7.5) 16.2(7.5) 18.9(7.8) 19.4(7.8) 19.2(8.1) 18.7(9.2) 56.1(9.5) 55.6(15.8) 55.6(15.8) 56.4(9.8) 55.1(9.2) 55.1(10.0) 52.0(11.8) 52.0(11.8) 59.9(10.3)60.9(31.1)58.8(12.4)58.9(12.5) 15.9(4.5) 16.8(4.1) 16.8(4.1) 14.6(4.7) 15.2(4.6) 15.0(4.6) 45.6(8.5) 54.1(21.1) 54.2(21.1) 47.3(8.1) 46.0(8.6) 46.2(8.7)

15.3(3.7) 15.3(3.7) 15.5(4.4) 14.3(4.6) 15.7(4.5) 15.6(4.6) 51.4(19.9) 51.4(19.9) 47.1(14.7)45.9(10.0)46.8(16.7)49.5(16.3)

87.9(12.0) 16.8(3.5) 19.8(4.8) 4.3(4.4) 4.6(3.7) 117.9(13.2) 13.3(3.2) 15.7(3.5) 3.2(2.5) 2.9(0.6) 3.3(0.7) 3.0(0.8) 85.3(14.3) 30.8(6.5) 35.9(8.5) 13.6(4.0) 13.7(3.8) 118.1(16.3) 24.1(6.0) 28.7(8.0) 12.6(2.6) 12.2(2.8) 12.9(3.1) 13.0(2.9)

(NOTE: The number before parenthese gives the median of prediction MSE and that in parenthese presents the corresponding standard deviations.)

estimates. Finally, the results show that whether the effects in the model are many or few, uneven or equal, independent or correlated, our new methods RIR and DAL still perform well in both estimating the coefficients and prediction accuracy. At last, we examine the performance of our new methods in a high-dimensional model. The case with σ = 1.5 is the same as the Example 1 in [18]. It is clear that the

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average values of non-zero coefficients produced by the lasso, the elastic net and the oracle methods are close to the true number of non-0s (15), while the DAL.1 carries out over-shrinkage, and DAL.2 and DAL.3 do under-shrinkage as shown in Table 1. However, from Table 2, we see that DAL.2 and DAL.3 are comparable in prediction accuracy. The above examples demonstrate that our new methods RIR and DAL are desirable. They often perform much better than ridge regression and lasso respectively. Further, we see that replacing the β aug by an estimate learned by RIR, DAL offers a natural way to combine the virtues of ridge and lasso. And it is interesting to notice that such fusion performs particularly well to fit the model with relatively more effects and lower level of SNR (as shown in Examples 1, 2 and 3).

4 Example - Prostate Cancer Data The data in this example come from a study of prostate cancer [19]. It is a data frame with 97 observations. The 8 regression features are log(cancer volume) (lcavol), log(prostate weight) (lweight), age, log(benign prostatic hyperplasia amount) (lbph), seminal vesicle invasion (svi), log(capsular penetration) (lcp), Gleason score (gleason) and percentage Gleason scores 4 or 5 (pgg45). The response is log(prostate specific antigen) (lpsa). We divided the dataset into two groups: a training dataset with 67 observations and a test dataset with the remaining 30 observations, and standardized the predictors before applying the algorithms. Since the test error may depend on the partition of the training and testing sets, we implemented 200 bootstrap replications. Five-fold cross-validation on training dataset was carried out to selecting the tuning parameters. We then computed the prediction mean-squared error to compare the performance of different methods. Table 3. Most frequent models selected in the prostate cancer example.

Ridge RIR Lasso Model Proportion Model Proportion Model Proportion 0.260 12345678 1.000 12345678 1.000 125 1258 0.090 12458 0.080 1245 0.060 12 0.055

Enet Model Proportion 125 0.160 12345678 0.150 1258 0.080 15 0.045 1245 0.040

DAL Model Proportion 12345678 0.515 1234578 0.080 1234567 0.060 1234568 0.060 1245678 0.025

SCAD Model Proportion 125 0.095 12345678 0.080 1245 0.060 123457 0.060 12345 0.055

AdapL Model Proportion 125 0.175 123458 0.070 1 0.055 1235 0.050 1234568 0.050

(NOTE: RIR initializes with

indv

RelaL Model Proportion 125 0.185 1234578 0.095 1245 0.055 1234568 0.055 1256 0.045

and DAL replaces the

aug

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Table 3 shows the five most frequent models (non-zero coefficients) selected by the ridge regression, the lasso and some of its generalizations, and the new methods RIR and DAL. We can see that the models selected by DAL tend to contain more features than lasso and its generalizations. Clearly, the RIR and DAL out-perform the others as shown in Fig. 1.

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5 Conclusion In this article we have proposed the ridge iterative regression and the data-augmentation lasso to improve the ridge regression and the lasso respectively. The simulations and real data demonstrate the competitive performance of the DAL in prediction accuracy and sparsity. As a future study, we are eager to solve the convergence problem of the RIR in the case that regressors are linearly dependent. In fact, the result for Example 5 shows us that the DAL is comparable in prediction accuracy, even though it carries out undershrinkage, which gives us a direction to modify this method. And it is necessary to learn in theory why the DAL has such well performance. Finally, motivated by the variants of the lasso, we also want to investigate the problems of generalizing the DAL. Acknowledgments. This work was supported in part by the Natural Science Foundation of Zhejiang Province under Grant LY19A010004, the First Class Discipline of Zhejiang - A (Zhejiang Gongshang University - Statistics) and in part by the National Natural Science Foundation of China under Grant 61572536.

References 1. Hoerl AE, Kennard RW (1970) Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12(1):55–67 2. McDonald GC, Galarneau DI (1975) A Monte Carlo evaluation of some ridge-type estimators. J Am Stat Assoc 70(350):407–416 3. Gibbons DG (1981) A simulation study of some ridge estimators. J Am Stat Assoc 76(373):131–139 4. Arashi M, Tabatabaey S, Bashtian MH (2014) Shrinkage ridge estimators in linear regression. Commun Stat Simul Comput 43(4):871–904

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5. Tibshirani R (1996) Regression shrinkage and selection via the lasso. J R Stat Soc: Ser B (Stat Methodol) 58(1):267–288 6. Zou H (2006) The adaptive lasso and its oracle properties. J Am Stat Assoc 101(476):1418– 1429 7. Meinshausen N (2007) Relaxed lasso. Comput Stat Data Anal 52(1):374–393 8. Zou H, Hastie T (2005) Regularization and variable selection via the elastic net. J R Stat Soc: Ser B (Stat Methodol) 67(5):768–768 9. Fan J, Li R (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. J Am Stat Assoc 96(456):1348–1360 10. Zhang CH (2010) Nearly unbiased variable selection under minimax concave penalty. Ann Stat 38(2):894–942 11. Zhang H, Xu ZB, Wang Y, Chang XY, Liang Y (2014) A sharp nonasymptotic bound and phase diagram of regularization. Acta Mathematica Sinica, English Series 30(7):1242–1258 12. Li GR, Lian H, Lai P, Peng H (2015) Variable selection for fixed effects varying coefficient models. Acta Mathematica Sinica, English Series 31(1):91–110 13. She Y (2010) Sparse regression with exact clustering. Electron J Stat 4:1055–1096 14. Friedman J, Hastie T, Tibshirani R (2010) Regularization paths for generalized linear models via coordinate descent. J Stat Softw 33(1):1–22 15. Czanner G, Sarma SV, Eden UT, Brown EN (2008) A signal-to-noise ratio estimator for generalized linear model systems. In: Proceedings of the world congress on engineering, vol 2 16. Krmer N, Schfer J, Boulesteix AL (2009) Regularized estimation of large-scale gene association networks using graphical Gaussian models. BMC Bioinform 10(1):1–24 17. Breheny P, Huang J (2011) Coordinate descent algorithms for nonconvex penalized regression, with applications to biological feature selection. Ann Appl Stat 5(1):232–253 18. Huang J, Ma S, Zhang CH (2006) Adaptive lasso for sparse high-dimensional regression. Statistica Sinica 18(4):1603–1618 19. Stamey T, Kabalin J, McNeal JE, Johnstone IM, Freiha F, Redwine EA, Yang N (1989) Prostate specific antigen in the diagnosis and treatment of adenocarcinoma of the prostate: Ii. radical prostatectomy treated patients. J Urol 141(5):1076–1083

Adaptive Fault-Tolerant Cooperative Output Regulation for Linear Multi-Agent Systems Jie Zhang1,2 , Dawei Ding1,2 , and Cuijuan An1(B) 1

School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China [email protected] 2 Key Laboratory of Knowledge Automation for Industrial Processes, Ministry of Education, Beijing 100083, China

Abstract. This paper investigates the problem of fault-tolerant cooperative output regulation of linear multi-agent systems (MASs) under a directed topology. A state and fault estimation observer is firstly designed to compensate the fault signals. Then an adaptive fault-tolerant control strategy is proposed such that all followers will track the leader regardless of faults. Simulation results demonstrate the effectiveness of the proposed control law. Keywords: Fault-tolerant · Cooperative output regulation · Multi-agent systems

1 Introduction Over the past decades, the consensus problem of MASs [1–3] has been widely studied due to its great application prospects in practice. Recently, the output regulation theory [4] has been applied to solve the output consensus problem, which can also be called the cooperative output regulation problem (CORP). The aim of CORP is to find a distributed control law such that all the followers can track the leader (exosystem). The faults such as process faults, actuator faults and sensor faults in a dynamic system can result in performance degradation or even instability. Thus the developed observer techniques [5, 6] play an important role in the fault-tolerant control (FTC), which can compensate for the faults by designing FTC law. With the growth of network size and complexity of MASs, safety and reliability have become the urgent demand in industrial control field. Many fault-tolerant protocols, e.g. [7–10], have been applied to the consensus problem with faults. For example, Wang et al. [7] developed the tracking control problem of linear MASs subject to actuator faults and mismatched parameter uncertainties. Ye et al. [8] presented a consensus problem of uncertain multi-agent systems with time-varying actuator fault. It should be noted that the output regulation framework provides a new method for solving the output consensus problem [11–13]. For example, Al et al. [11] proposed a fault-tolerant control law to make two subsystems track a common signal. Qin et al. [12] designed an FTC law for lineaite-time observer. However, the communication topology considered in [12, 13] is undirected. By contrast, this paper aims at designing a fault-tolerant controller based on the output regulation theory to solve the CORP c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 185–194, 2020. https://doi.org/10.1007/978-981-32-9682-4_20

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under a directed topology. In detail, the fault-tolerant CORP of linear multi-agent system with process faults is solved. For each follower, a state and fault estimation observer is designed by utilizing the output estimation error information of neighboring followers. Then the estimations are embedded into a new distributed adaptive FTC law. It is shown that under the proposed FTC law, the CORP can be solved regardless of the process faults. Notations: Through this paper, let R p and Rn×m be the sets of real p-vectors and real n× m matrices, respectively. The n-dimensional real identity matrix can be described by In . 1n and 0n are the n-vector with all elements being 1 and 0, respectively. A = AT > 0 denotes a positive definite symmetric matrix. Denote col(x1 , · · · , xn ) = (x1T , · · · , xnT )T . diag(a1 , · · · , an ) and block(A1 , · · · , An ) are the diagonal matrix and the block diagonal matrix, respectively. The symbol ⊗ denotes the Kronecker product. The term λ (A) represents the spectrum of a square matrix A.

2 Preliminaries and Problem Statement 2.1

Preliminaries

A graph G (V , E , A ) is used to model the communication topology between agents. The node set V = {0, 1, · · · , N} is composed by a leader labeled by node 0 and N followers labeled by nodes 1, · · · , N. The corresponding edge set E ⊆ {(i, j)|i, j ∈ V , i = j} describes the communication links among them. An edge (i, j) ∈ E means that agent i can receive information from agent j directly. Then we say agent j is the neighbor of agent i. For a graph G , if there exists a directed sequence (i1 , i2 ), (i2 , i3 ), · · · , (ik−1 , ik ), then we can say that G contains a spanning tree with agent ik as its root node. A = [ai j ] is the adjacency matrix with ai j = 1 if (i, j) ∈ E , and ai j = 0 otherwise. Further, the Laplacian matrix of graph G is defined as L = [li j ] ∈ Rn×n , where lii = ∑nj=1 ai j and li j = −ai j if i = j. By removing the nodes associated with the leader and the edges between them, we can obtain the induced subgraph G with the corresponding Laplacian matrix L1 . 2.2

Problem Formulation

Consider a linear multi-agent system composed by N followers and one leader (exosystem), which is described as follows:  x˙i (t) = Axi (t) + Bui (t) + E fi (t) , i = 1, · · · , N (1) yi (t) = Cxi (t)  v(t) ˙ = Sv(t) , (2) yr (t) = Cr v(t) where xi ∈ Rd , ui ∈ Rq , yi ∈ R p are the state, the control input and the measured output of follower i, respectively. fi , i = 1, · · · , N with slowly varying rate (i.e. f˙i (t)  0) denote the process faults existing in each follower. In particular, fi , i = 1, · · · , N represent the

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actuator faults when E = B. v ∈ Rh and yr ∈ R p are the state and the output of the leader, respectively. A, B,C, E, S and Cr are constant matrices with appropriate dimensions. Define the tracking errors as ei = yi − yr , i = 1, · · · , N. Then we address the faulttolerant CORP in this paper: Definition 1. Consider a linear multi-agent system (MAS) (1) and (2) with a directed topology determined by G¯. When some faults occur in followers i, = 1, · · · , N, we can find a distributed FTC law ui , i = 1, · · · , N such that all followers will track the leader as time goes to infinity, i.e., (3) lim yi (t) − yr (t) = 0. t→∞

Next, the following standard assumptions for the solvability of the fault-tolerant CORP are needed: Assumption 1. Matrix S has no eigenvalues with negative real parts. Assumption 2. The pair (A, B) is stabilizable and rank(B, E) = rank(B). Assumption 3. The pair (A,C) are observable. Assumption 4. For all σ , the following equation holds   A − σ In B rank = d + p. C D

(4)

Assumption 5. The communication topologies G contains a spanning tree with the node 0 as the root. Remark 1. Assumptions 1–5 are quite standard for the output regulation problem [4]. Assumption 1 is made for convenience and without loss of generality. Assumption 2 guarantees that the system can be locally stabilized by a state feedback control. rank(B, E) = rank(B) means that we can find a matix B∗ such that (I − BB∗ )E = 0, and the detailed proof is given in [6]. Assumption 3 is used to design the state observer for each follower. Assumption 4 is called the transmission zeros condition, which is used to guarantee the existence of the regulator equations. Assumption 5 is a necessary condition to solve CORP.

3 Main Results In this section, an FTC scheme based on the output regulation framework will be given. 3.1 Observer Design Inspired by the adaptive fault estimation algorithm [5, 14], the following observer for state and fault of each follower are designed simultaneously: ⎧ ⎨ x˙ˆi (t) = Axˆi (t) + Bui (t) + E fˆi (t) − H e¯i (t) yˆi (t) = Cxˆi (t) , i = 1, · · · , N (5) ⎩ ˙ˆ fi (t) = −Γ F (e¯i (t) + e˙¯i (t))

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where xˆi (t) ∈ Rd and fˆi ∈ Rr are the state estimation and the fault estimation, respectively. H ∈ Rd×p , Γ ∈ Rr×r and F ∈ Rr×p are observer gain matrices to be designed. It is noted that Γ = Γ T > 0. The measurement error can be defined as follows: e¯i =

N

∑ ai j ((yˆi − yi ) − (yˆ j − y j )) , i = 1, · · · , N.

(6)

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where yˆ0 = y0 . Further, denote the error vectors of state observer and fault observer as: exi = xˆi − xi , e f i = fˆi − fi . The derivatives of these error vectors can be obtained respectively:  N e˙xi = x˙ˆi − x˙i = Aexi + Ee f i − HC ∑ j=0 ai j (exi − ex j ) ,  N e˙ f i = f˙ˆi − f˙i = −Γ FC ∑ j=0 ai j (exi − ex j + e˙xi − e˙x j ) − f˙i .

(7)

(8) (9)

Let ex = col(ex1 , · · · , exn ), e f = col(e f 1 , · · · , e f n ) and f = col( f1 , · · · , fn ). The derivatives of these error vectors can be written as:

 e˙x = IN ⊗ A − L¯1 ⊗ HC ex + (IN ⊗ E) e f , (10) e˙ f = −L¯1 ⊗ (Γ FC) (ex + e˙x ) f˙, where L¯1 = L1 + Λ with Λ = diag(a10 , · · · , aN0 ). If Assumption 5 holds, the matrices L¯1 and L¯1−1 are positive definite and nonsingular [15]. Then we have the following theorem for the observer design: Theorem 1. Under Assumption 3, if there exists a symmetric positive matrix P ∈ Rd×d and a matrix Y = PH satisfying the following conditions: E T P1 = FC   Σ11 Σ12 0.

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Theorem 2. Given the multi-agent system governed by (1) and (2) satisfying Assumptions 1–5 and conditions (11)–(12). Then for any initial values xi (0), i = 1, · · · , N, the fault-tolerant CORP is solved if the following condition matrix equation  XS = AX + BU (18) 0 = CX −Cr has a unique solution (X,U) with K2 = U − K1 X. Proof. Let x = col(x1 , · · · , xn ), z = col(z1 , · · · , zn ), xc = col(x, z), χ = col(xc , ζ ), and cˆ = diag(cˆ1 , · · · , cˆN ). Under the adaptive control law (17), the closed-loop system can be written as the following block matrix form:      Π11 Π12 xc Bc1 ˙χ = Ac χ + Bc = + , (19) 0 Π14 ζ Bc2 where 

Π11 =

     In ⊗ (A + BK1 ) I ⊗ BK1 E 0 In ⊗ BK2 , Bc1 = , , Π12 = N ¯ 0 0 0 In ⊗ S − cˆL1 ⊗ Ψ Λ 1N ⊗ Ψ v

   0 IN ⊗ A − L¯1 ⊗ HC IN ⊗ E

 Π14 = . , Bc2 = − L¯1 ⊗ (Γ FC) e˙x 0 −L¯1 ⊗ (Γ FC)

(20)



(21)

It should be pointed out that A + BK1 can be designed to be Hurwitz by the pole placement technique under Assumption 2. The stability of the closed-loop system (19) is trivially equivalent to the stability of the following system: 

z˙ = IN ⊗ S − cˆL¯1 ⊗ Ψ z + Λ 1N ⊗ Ψ v. (22) Let δ = z − 1N ⊗ v, then we have 

δ˙ = IN ⊗ S − cˆL¯1 ⊗ Ψ δ .

(23)

We can further construct the following Lyapunov function candidate  1

1 V2 = δ T L¯1−1 ⊗ IN δ + 2 2α

∑i=1 c˜2i , N

(24)

where c˜i = cˆi − c and c is positive constant to be determined later. Then the time derivative of (24) can be calculated as

 1 N V˙2 (t) = δ T L¯1−1 ⊗ IN δ˙ + ∑i=1 c˜i c˙˜i α



 1 N = δ T L¯1−1 ⊗ IN IN ⊗ S − cˆL¯1 ⊗ Ψ δ + ∑i=1 c˜i c˙˜i α

 = δ T L¯1−1 ⊗ S − cIN ⊗ Ψ  δ − δ T (cI ˜ N ⊗ Ψ ) δ + ∑Ni=1 c˜i δiT (t)Ψ δi (t) −1 T ¯ = δ L ⊗ S − cIN ⊗ Ψ δ .

(25)

1

Since the matrix L¯1−1 is a positive definite matrix, there exists a unitary matrix M satisfies M T L¯1−1 M = diag(λ1 , λ2 , · · · , λN ) with 0 < λ1 ≤ λ2 ≤ λN , where λi , i =

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Fig. 1. Topology of the multi-vehicle system.

1, · · · , N denote the eigenvalues of L¯1−1 . Then define a matrix r = (M ⊗ IN ) δ , and (25) can be changed to the following form:

 V˙2 (t) = rT M T L¯1−1 M ⊗ S − cM T M ⊗ Ψ r (26) = rT (diag(λ1 , λ2 , · · · , λN ) ⊗ S − cIN ⊗ Ψ ) r. If there exists a sufficiently large constant c satisfies λi S − cΨ < 0, then we can obtain that V˙2 (t) < 0 for any δ = 0. Thus the closed-loop system (19) is asymptotically stable. When time goes to infinity, δi (t) = zi (t)−v(t) → 0. So adaptive parameters cˆi , i = 1, · · · , N are bounded. Moreover, limt→∞ c˙ˆi (t) = 0 implies that the adaptive parameters cˆi , i = 1, · · · , N will approach to some constants as time goes to infinity. Let x˜i = xi − Xv. Some compact vectors can be defined as e = col(e1 , · · · , en ), x˜c = col(x, ˜ δ ), χ˜ = col(x˜c , δ ). Then the system (19) can be rewritten as follows:      ˜ ˙χ˜ = Ac χ˜ + B˜ c = Π11 Π12 x˜c + Bc1 (1N ⊗ v) , 0 Π14 ζ 0 (27) e = (In ⊗C) x˜c + (In ⊗ (CX −Cr )) v. where

  In ⊗ (−XS + AX + BK1 X + BK2 ) ˜ Bc1 = . 0

(28)

If conditions (11) and (12) hold, we have limt→∞ ζ (t) = 0 for i, i = 1, · · · , N. It should be noticed that the matrix Eq. (18) is called regulator equation. If the regulator Eq. (1) has a unique solution (X,U) with K2 = U − K1 X, we can further deduce that limt→∞ x˜c = 0, and thus limt→∞ e = 0. Thus the proof is completed.

4 Simulation Results Consider a multi-robot system consisting of four robots [17]. Figure 1 shows the communication topology. The dynamics of follower robots are given by ⎧ ⎨ x˙i1 = xi2 x˙i2 = −ς xi2 + bui , i ∈ 1, 2, 3, (29) ⎩ yi = xi1

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where xi1 ∈ R1 and xi2 ∈ R1 represent the position and velocity of robot i, respectively. yi is the position output. ς and b are the damping and mass coefficient of vehicle i, respectively. Let ς = 0.5 and b = 1. fi ∈ R1 represents the actuator faults occurring in  T the followers with E = B = 0 1 .

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The labeled by node 0 is governed by v = Sv, where v = col(v1 , v2 ) and  leader robot  0 0.5 S= . The object of our CORP is to control all follower-outputs yi1 = xi1 , i ∈ −0.5 0 1, 2, 3 to track the leader-output yr = v1 in the presence of the faults. And the faults of each follower are given as follows:    0, 0 ≤ t < 10 0, 0 ≤ t < 20 0, 0 ≤ t < 30 f1 = , f2 = , f3 = . 1, t ≥ 10 0.5, t ≥ 20 2, t ≥ 30 For each follower, the observer gains in (5) can be solved by Theorem 1:   9.5126 H= , F=1.1365, Γ = 2.5. 1.8938 The feedback gains K1 and K2 in the controller (17) can be designed as follows:       K1 = −10 −8 , K2 = 9.75 4.25 , B∗ = 1 1 . Simulation results are shown in Figs. 2, 3 and 4, respectively. Figure 2(a) and (b) give fault estimation results. Figure 3(a) shows the position trajectory of three followers and one leader, and the correspond position tracking errors are given in Fig. 3(b). It is observed that with the time goes to infinity, all the followers will track the leader and the tracking errors converge to 0 asymptotically in the presence of the faults fi , i = 1, 2, 3. Figure 4 shows the adaptive parameters cˆi (t), i = 1, 2, 3, which can reach to some constants as time goes to infinity. The simulation results show the effectiveness of the proposed fault-tolerant containment control law.

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Acknowledgements. This work was supported by National Natural Science Foundation of China under Grant Nos. 61873028, 61803026 and Fundamental Research Funds for the Central Universities under Grant Nos. FRF-TP-18-001B1 and FRF-TP-18-032A1.

References 1. Nedic A, Ozdaglar A, Parrilo PA (2010) Constrained consensus and optimization in multiagent networks. IEEE Trans Autom Control 55(4):922–938 2. Zheng Y, Ma J, Wang L (2018) Consensus of hybrid multi-agent systems. IEEE Trans Neural Netw Learn Syst 29(4):1359–1365 3. Nowzari C, Garcia E, Cort´es J (2019) Event-triggered communication and control of networked systems for multi-agent consensus. Automatica 105:1–27 4. Huang J (2004) Nonlinear output regulation: theory and applications. SIAM, Philadelphia 5. Wang H, Daley S (1996) Actuator fault diagnosis: an adaptive observer-based technique. IEEE Trans Autom Control 4(7):1073–1078 6. Jiang B, Staroswiecki M, Cocquempot V (2006) Fault accommodation for nonlinear dynamic systems. IEEE Trans Autom Control 51(9):1578–1583 7. Wang X, Yang GH (2015) Cooperative adaptive fault-tolerant tracking control for a class of multi-agent systems with actuator failures and mismatched parameter uncertainties. IET Control Theory Appl 9(8):1274–1284 8. Ye D, Zhao X, Cao B (2016) Distributed adaptive fault-tolerant consensus tracking of multiagent systems against time-varying actuator faults. IET Control Theory Appl 10(5):554–563 9. Zuo Z, Zhang J, Wang Y (2015) Adaptive fault-tolerant tracking control for linear and Lipschitz nonlinear multi-agent systems. IEEE Trans Industr Electron 62(6):3923–3931 10. Jin X, Wang S, Qin J, Zheng W, Kang Y (2018) Adaptive fault-tolerant consensus for a class of uncertain nonlinear second-order multi-agent systems with circuit implementation. IEEE Trans Circuits Syst I Regul Pap 65(7):2243–2255 11. Al-Bayati AH, Wang H (2015) A passive fault tolerant control for collaborative subsystems based on a new signal output regulation scheme. Glob J Control Eng Technol 1:1–13 12. Qin L, He X, Zhou DH (2017) Fault-tolerant cooperative output regulation for multi-vehicle systems with sensor faults. Int J Control 90(10):2227–2248 13. Deng C, Yang GH (2019) Distributed adaptive fault-tolerant control approach to cooperative output regulation for linear multi-agent systems. Automatica 103:62–68 14. Zhang K, Jiang B, Cocquempot V (2015) Adaptive technique-based distributed fault estimation observer design for multi-agent systems with directed graphs. IET Control Theory Appl 9(18):2619–2625 15. Su Y, Huang J (2012) Cooperative output regulation of linear multi-agent systems. IEEE Trans Autom Control 57(4):1062–1066 16. Li H, Yang Y (2016) Cooperative fault estimation for linear multi-agent systems with undirected graphs. J Eng 79:253–257 17. Cai H, Lewis FL, Hu G, Huang J (2017) The adaptive distributed observer approach to the cooperative output regulation of linear multi-agent systems. Automatica 75:299–305

Graphic Approach for the Disturbance Decoupling of Boolean Networks Yifeng Li and Jiandong Zhu(B) School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, China [email protected]

Abstract. In this paper, the disturbance decoupling problem (DDP) of Boolean networks (BNs) is investigated by graphic approach. Firstly, by referring to the graphic structure of BNs, a necessary and sufficient graphic condition for the disturbance decoupling is proposed. Secondly, an algorithm is designed to search a concolorous perfect equal vertex partition (C-PEVP). By a C-PEVP, we can construct a logical coordinate transformation which makes the DDP solvable for BNs. Finally, an illustrative example is provided to validate the theoretical results. Keywords: Boolean networks · Disturbance decoupling Graphic condition · Semi-tensor product of matrices

1

·

Introduction

BNs are a kind of dynamic systems composed of logical variables and logical functions. They were first introduced by Kauffman [1]. BNs have been proved to be quite useful in modeling and quantitative description of cellular regulator [2– 4]. In the past several decades, with the development of systems biology, the issue of Boolean dynamic models has become a hotspot, which provides an effective tool in simulation and control of genetic regulation networks. Lots of theoretical results on the dynamical behavior of BNs have been published [5–8]. BNs with inputs and outputs are called Boolean control networks (BCNs) [9]. The semi-tensor product (STP) of matrices proposed by Cheng and his collaborators [10], which is a powerful technology to convert logical dynamic systems into discrete-time algebraic systems. Based on the STP, a new theoretical framework for the control theory of BCNs was established in [11]. With the help of this theoretical framework, many fundamental results on traditional linear control system theory have been generalized to BCNs, including controllability [12,13], optimal control [14–17], state observers design [18], stability and stabilization [19,20], system decomposition [21–25], output regulation [26], pinning control [27], l1 -gain problem [28] and so on. In many gene regulatory networks, there are many external disturbances effecting the behavior of the systems. So how to reject disturbance is one of the essential issues in systems biology [29]. Designing controllers such that the c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 195–207, 2020. https://doi.org/10.1007/978-981-32-9682-4_21

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outputs are unaffected by the external disturbances is called the disturbance decoupling problem (DDP). DDP has been widely investigated for linear systems, nonlinear systems, switched systems. But for BCNs, there are still many open problems for DDP. The DDP of BCNs was first investigated in [30], in which the basic idea for solving DDP is to find a coordinate transformation to decompose the control system and subsequently to design controllers such that the outputs are only involved in an undisturbed subsystem. This idea has been successfully applied to solve the DDP of mix-valued logical networks [32,33], switched BCNs [34] and singular BNs [35]. In [36], a state-feedback controller is designed for disturbance decoupling of BCNs by referring to the system decomposition. It is worth mentioning that [37] proposes a new viewpoint to study DDP. A new definition called original disturbance decoupling problem (ODDP) is proposed in [37], which is a generalization of the definition in [30]. To the best of our knowledge, for the disturbance decoupling of BNs, some algebraic conditions have been proposed, but few literatures study the graphic condition. In this paper, we reconsider the DDP of BNs, but we study it from the perspective of graph theory and give a graphic condition for the disturbance decoupling of BNs. Compared with the existing algebraic conditions, the graphic condition can intuitively and clearly reveal the essence of the disturbance decoupling. The main contributions of this paper lie in the following points. A necessary and sufficient graphic condition, i.e., the existence of a C-PEVP, for the solvability of the DDP is derived. For realizing the disturbance decoupling of BNs, an algorithm is designed to compute a logical coordinate transformation. The rest of this paper is organized as follows: Sect. 2 gives some preliminaries. Section 3 is the problem statement. Section 4 presents the main results. Section 5 gives an algorithm and an example. Section 6 is a brief conclusion.

2

Preliminaries

Throughout the whole paper, we use the following notations. Let Nm×n represent the set of m × n nonnegative matrices and Coli (A) be the i-th column of matrix A. Each Coli (Ik ), denoted by δki , is called a k −dimensional logical vector. Let Δk = {δki |i = 1, 2, . . . , k}, where Ik is the k×k identity matrix. A matrix is called a logical matrix if its every column is a logical vector. The set composed of all the m × r logical matrices is denoted by Lm×r . In the following, for convenience, i1 i2 ir δm · · · δm ] is written as δm [i1 i2 · · · ir ]. the logical matrix L = [δm In [10], a generalized matrix product, called left semi-tensor product, is proposed. Assume A ∈ Rm×n , B ∈ Rp×q . Let α = lcm(n, p) be the least common multiple of n and p. The left semi-tensor product of A and B is defined as A  B = (A ⊗ I αn )(B ⊗ I αp ), where ⊗ is the Kronecker product. If n = p, the left semi-tensor product is reduced to the traditional matrix product. For convenience, write AB for A  B. T Letting x ∈ Δm and y ∈ Δn , we have 1T m x = 1, 1n y = 1. So T T x = Im x ⊗ 1T n y = (Im ⊗ 1n )(x ⊗ y) = (Im ⊗ 1n )xy, T T y = 1T m x ⊗ In y = (1m ⊗ In )(x ⊗ y) = (1m ⊗ In )xy.

(1) (2)

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197

In [10], the elements in D are identified with vectors as 1 ∼ δ21 and 0 ∼ δ22 . We denote Δ = {δ21 , δ22 }. Consider the BN with disturbances described by x1 (t + 1) = f1 (x1 (t), . . . , xn (t), ξ1 (t), . . . , ξq (t)), .. . xn (t + 1) = fn (x1 (t), . . . , xn (t), ξ1 (t), . . . , ξq (t)), yj (t) = hj (x1 (t), . . . , xn (t)), j = 1, 2 · · · , p,

(3)

where xi , i = 1, 2, . . . , n are state variables, ξj , j = 1, 2, . . . , q are disturbances, and yl , l = 1, 2, . . . , p are outputs. Let x(t) = ni=1 xi , ξ(t) = qj=1 ξj and y(t) = pl=1 yl , where all the logical variables take values in Δ2 . Then the system (3) is converted into the following algebraic form x(t + 1) = Lξ(t)x(t), y(t) = Hx(t),

(4)

where L ∈ L2n ×2n+q and H ∈ L2p ×2n . The conversion process between the logical form (3) and the algebraic form (4) can be found in [10]. Let i ∼ δ2i n for any i ∈ V . Consider the dynamic equations of state variables 2q in (4) and let L = [L1 L2 · · · L2q ], where Li ∈ L2n ×2n . Set B = i=1 Li , then B ∈ N2n ×2n . Since B is a non-negative matrix, it can be regard as an adjacency matrix of a weighted directed graph G with vertex set V = {1, 2, · · · , 2n }, where {δ21n , δ22n · · · , δ2nn } ∼ {1, 2 · · · , n}. And G has a directed edge (q, p), i.e. state variable x moves from δ2qn to δ2pn , if and only if bpq = 0. We call G the state transition diagram of the dynamic equations of state variables in (4). Considering the dynamic equations of outputs in (4), we have that each state corresponds to an output. Thus the state transition is accompanied by the output transition. Let one color represent one output. In summary, we call G the state-colored transition diagram of BN (4). See Fig. 1 in Example 1 for details. We say that p is an out-neighbor of q if bpq = 0. Let Cq = {p | bpq = 0}. Then Cq is the set of all the out-neighbors of q. Let S be a subset of V and set N (S) = {p | ∃ q ∈ S, bpq = 0} = ∪ Cq . q∈S

(5)

We call N (S) the out-neighborhood of S. Let Sl , l = 1, 2, · · · , μ be some subsets of V . S = {Sl }μl=1 is called a vertex μ

partition of G, if ∪ Sl = V and Si ∩ Sj = ∅ for any i = j. A vertex partition l=1

S = {Sl }μl=1 of G is called an equal vertex partition if |Sl | = |V |/μ for every l = 1, 2, · · · , μ. |V | denotes the number of vertices in V . Definition 1 [38]. Let R and S be two partitions of a set V . Assume that for every R ∈ R, there exists an S ∈ S such that R ⊂ S. Then the partition R is called a refinement of S, and the partition S is called a coarser of R denoted by R  S.

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Definition 2. An equal vertex partition S = {Sl }μl=1 of V is called a concolorous perfect equal vertex partition (C-PEVP), where μ is some positive integer, if 1. for any l ∈ {1, 2, · · · , μ}, all the vertices in Sl have the same color, 2. S = {Sl }μl=1 is perfect for G, that is, for any l ∈ {1, 2, · · · , μ}, there exists an αl such that N (Sl ) ⊂ Sαl . Definition 3 [31]. A logical matrix Q ∈ Lm1 ×m1 m2 is called a regular matrix, if Q1m1 m2 = m2 1m1 . Remark 1 [36]. Since Q1m1 m2 = m2 1m1 , we have that every row of Q has exactly m2 1’s. Therefore there exists a permutation matrix T ∈ Lm1 m2 ×m1 m2 such that (1Tm2 ⊗ Im1 )T = Q.

3

Problem Statement

Consider the BCN with disturbances described by x1 (t + 1) = f1 (x1 (t), . . . , xn (t), u1 (t), . . . , um (t), ξ1 (t), . . . , ξq (t)), .. . xn (t + 1) = fn (x1 (t), . . . , xn (t), u1 (t), . . . , um (t), ξ1 (t), . . . , ξq (t)), yl (t) = hl (x1 (t), . . . , xn (t)), l = 1, 2 · · · , p,

(6)

where xi ’s are state variables, ξj ’s are disturbances, uk ’s are controls and yl ’s are p outputs. Let x(t) = ni=1 xi , ξ(t) = qj=1 ξj , u(t) = m k=1 uk and y(t) = l=1 yl , where all the logical variables take values in Δ2 . Then the system (6) is converted into the following algebraic form x(t + 1) = Ru(t)ξ(t)x(t), y(t) = Hx(t),

(7)

where R ∈ L2n ×2m+n+q and H ∈ L2p ×2n . Consider the logical mapping g : Dn → Dn defined by zi = gi (x1 , x2 , · · · , xn ), i = 1, 2, · · · , n.

(8)

If g : Dn → Dn is a bijection, it is called a logical coordinate transformation [31]. Let z = T x be the algebraic form of the logical coordinate transformation (8), where T ∈ L2n ×2n is the structure matrix of g. It is obvious that g is a logical coordinate transformation if and only if T is a nonsingular logical matrix, i.e., a permutation matrix [31]. In [30], the disturbance decoupling of BCNs is described as follows:

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199

Definition 4 [30]. Consider the BCN (7). The DDP is solvable if there exists a logical coordinate transformation z = T x and a feedback control u(t) = Kx(t) such that under z coordinate frame the closed-loop system becomes z [1] (t + 1) = G1 ξ(t)z(t), z [2] (t + 1) = G2 z [2] (t), y(t) = Ez [2] (t),

(9)

where z [1] = si=1 zi , z [2] = nl=s+1 zl and y(t) = pj=1 yj , G1 ∈ L2s ×2n+q , G2 ∈ L2n−s ×2n−s and E ∈ L2p ×2n−s . For BNs, the Definition 4 can be rewritten as follows: Definition 5. Consider the BN (3). The DDP is solvable if there exists a logical coordinate transformation z = T x such that under z coordinate frame the system becomes (9). In the next section, we study the graphic condition for the disturbance decoupling of BNs.

4

Main Results

Consider the Boolean network (4) with L = [L1 L2 · · · L2q ], where each Lj ∈ 2q L2n ×2n . Set B = i=1 Li , we denote state-colored transition diagram of the system (4) by G. In the following theorem, we propose a graphic condition for the solvability of the disturbance decoupling. Theorem 1. There exists a logical coordinate transformation z = T x such that (4) becomes (9), i.e., the DDP is solvable, if and only if the state-colored trann−s sition diagram G of BN (4) has a C-PEVP S = {Sl }2l=1 with |Sl | = 2s . Proof. (Necessity): From (9), we can see that, letting zs+1 zs+2 . . . zn = δ2l n−s , the set Pl = {z1 z2 · · · zs δ2l n−s | zj ∈ {δ21 , δ22 }, 1 ≤ j ≤ s} has 2s states, i.e., the family n−s

P = {Pl }2l=1

(10) 2n 2n

forms an equal vertex partition of {δ21n , δ22n , · · · , δ }. It follows from the outputs equation of (9) that for any δ2i s δ2l n−s ∈ Pl , y = Ez [2] = Ezs+1 zs+2 . . . zn = Eδ2l n−s = Coll (E),

(11)

i.e., the states in Pl have the same outputs. Let one color represent one output, which implies that all the vertices in Pl have the same color. Furthermore, from (9), we have (12) z [2] (t + 1) = G2 z [2] (t).

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l Thus, z [2] (t + 1) = G2 δ2l n−s = δ2αn−s , which means that the state z(t) moves from Pl to Pαl , i.e., for any given l, there exists an αl such that N (Pl ) ⊂ Pαl . Thus, let x = T T z and Sl = {T T z1 z2 · · · zs δ2l n−s | zj ∈ {δ21 , δ22 }, 1 ≤ j ≤ s}. Then we n−s get a equal vertex partition S = {Sl }2l=1 (|Sl | = 2n−s ) of G is a C-PEVP. (sufficiency): For G, mapping all the vertices in Sl to a single vertex denoted by l (l ∼ δ2l n−s ). If there exists an edge from Sl1 to Sl2 , then construct an edge from vertex l1 to vertex l2 . We denote the resulting diagram by G ∗ . Considering the BN (3), construct a logical matrix Q as follows:

Qδ2qn = δ2l n−s , ∀ δ2qn ∈ Sl , l = 1, 2, · · · , 2n−s .

(13)

By (13), every row of Q has exactly 2s ones. Thus, we can rearrange the columns of 1T2s ⊗ I2n−s to get Q, which means that there exists a permutation matrix T such that (14) Q = (1T2s ⊗ I2n−s )T. Let the logical coordinate transformation be z = z [1] z [2] = T x, where z [1] ∈ Δ2s , z [2] ∈ Δ2n−s , then z [2] = (1T2s ⊗ I2n−s )z [1] z [2] = (1T2s ⊗ I2n−s )T x = Qx. So for any x(t) = δ2qn ∈ Sl , we have z [2] (t) = Qδ2qn = δ2l n−s . And S = {Sl }2l=1 is perfect. Thus, by the construction process of G ∗ , we have that the state-colored transition diagram of z [2] is exactly G ∗ . Since the vertex in Sl has the same color, i.e., the vertex in Sl has the same output, we can determine a logical matrix E ∈ L2p ×2n−s such that n−s

y(t) = Ez [2] (t),

(15)

where Coll (E) is the output of vertex in Sl . Since for any l, there exists an αl such that N (Sl ) ⊂ Sαl . So for any x(t) ∈ Sl , l . x(t + 1) ∈ Sαl . And z [2] (t) = Qx(t) = δ2l n−s , z [2] (t + 1) = Qx(t + 1) = δ2αn−s Thus, there exists a logical matrix G2 ∈ L2n−s ×2n−s such that z [2] (t + 1) = G2 z [2] (t),

(16)

l δ2αn−s .

Thus, by (15) and (16), we can get the form (9) under where G2 δ2l n−s = the logical coordinate transformation z = T x.   Remark 2. From Theorem 1, given a C-PEVP, we can construct a logical coordinate transformation which turns (3) into (9). In order to display the procedure, we consider the following example. Example 1. Consider the BN described by x1 (t + 1) = x1 (t) → ξ(t), x2 (t + 1) = [x1 (t) → ξ(t)] ↔ {[(x1 (t) ∧ x2 (t)) ∨ (¬x1 (t) ∧ ¬x2 (t))] ∨ x3 (t)}, x3 (t + 1) = ¬[x1 (t) ∧ x2 (t)] ∧ [x1 (t) ∨ x2 (t)], ¯ x3 (t))], y(t) = x1 (t) ∧ (x2 (t) ↔ x3 (t)) ∨ [¬x1 (t) ∧ (x2 (t)∨ (17)

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where x1 , x2 , x3 , ξ, y ∈ Δ2 . Let x = x1 x2 x3 . Then the algebraic form of the system (17) is (4) with L = [L1 L2 ] and H = δ2 [1, 2, 2, 1, 2, 1, 1, 2], where L1 = δ8 [2, 2, 1, 3, 1, 3, 2, 2], L2 = δ8 [8, 8, 7, 5, 1, 3, 2, 2]. Let B = L1 + L2 . Considering B and H, we get the state-colored transition diagram G of the system (17) shown in Fig. 1. It follows from the state-colored transition diagram G that the system (17) has a C-PEVP shown in Fig. 1. Construct a graph G ∗ following the sufficiency proof of Theorem 1, shown in Fig. 2. By (13), we get Q = δ4 [3 4 2 1 2 1 3 4]. By (14), we have δ4 [3 4 2 1 2 1 3 4] = (1T2 ⊗ I22 )T. Thus, we get the permutation matrix T = δ8 [3 4 2 1 6 5 7 8]. Under the logical coordinate transformation z = T x, we can get the form (9). From Fig. 2, we directly get G2 = δ4 [2 3 4 4] and E = δ2 [1 2 1 2]. By z = T x and (2), we get G1 = δ16 [1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1]. Using the model construction method given in [11], we get the form (9) z [1] (t + 1) = δ16 [1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1]ξ(t)z(t), z [2] (t + 1) = δ4 [2 3 4 4]z [2] (t), y(t) = δ2 [1 2 1 2]z [2] (t), whose logical form is z1 (t) = ξ(t) ∨ (¬ξ(t) ∧ ¬z1 (t)), z2 (t) = z2 (t) ∧ z3 (t), z3 (t) = z2 (t) ∧ ¬z3 (t), y(t) = z3 (t).   From Theorem 1, for BCN (7), we can directly obtain a graphic condition for the solvability of DDP. Theorem 2. Consider BCN (7). The DDP is solvable if there exists a statefeedback control u(t) = Kx(t) such that the state-colored transition diagram of n−s the closed-loop system has a C-PEVP S = {Sl }2l=1 with |Sl | = 2s . In the following, some results are presented to search a C-PEVP. Lemma 1 [36]. Assume that M ∈ Nm×mn satisfying 1Tm M = k1Tmn . If M (1n ⊗ Im ) = knG, where G is a logical matrix, then M = kG(1Tn ⊗ Im ). Considering the BN (3), from the Theorem 1 in [36], we can directly get the following theorem. Theorem 3. There exists a logical coordinate transformation z = T x such that (3) becomes (9), i.e., the DDP is solvable, if and only if there exist a regular

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Fig. 1. The state-colored diagram G

Fig. 2. The restructuring diagram G ∗ of G

matrix Q ∈ L2n−s ×2n such that 1 2q+s

QBQT ∈ L2n−s ×2n−s ,

1 HQT ∈ L2p ×2n−s , 2s where B = L(12q ⊗ I2n ) =

2q

i=1

(18)

Li .

  From Theorem 3, if the DDP is solvable, then (18) holds. Since Q ∈ L2n−s ×2n is a regular matrix, by Remark 1, there exists a permutation matrix T ∈ L2n ×2n such that (1T2s ⊗ I2n−s )T = Q. Set z = T x, then z [2] = (1T2s ⊗ I2n−s )z(t) = Qx. 1 QBQT = G ∈ L2n−s ×2n−s and 21s HQT = F ∈ L2p ×2n−s . Since Set 2q+s QBQT = QL(12q ⊗ I2n )T T (12s ⊗ I2n−s ) = 2s (2q G) with QL(12q ⊗ I2n )T T being non-negative and satisfying 1T2n−s QL(12q ⊗ I2n )T T = 2q 1T2n . By Lemma 1, we have QL(12q ⊗ I2n )T T = 2q G(1T2s ⊗ I2n−s ).

(19)

Graphic Approach for the Disturbance Decoupling of Boolean Networks

It follows from QL(12q ⊗ I2n )T T = that QLi T T = G(1T2s ⊗ I2n−s ). Similarly, we obtain that

2q

i=1

203

QLi T T and G(1T2s ⊗ I2n−s ) ∈ L2n−s ×2n

HT T = F (1T2s ⊗ I2n−s ).

(20)

By the (19) and (20), we have 21q QB = GQ ∈ L2n−s ×2n and H = F Q ∈ L2p ×2n . Then, 1 1 HB = q F QB = F GQ ∈ L2p ×2n . 2q 2 Repeating this process, we have 1 HB s = F Gs Q ∈ L2p ×2n , (2q )s

(21)

where (21q )s HB s ∈ L2p ×2n . So there exists a r∗ such that (2q1)r HB r ∈ { (21q )i HB i | 0 ≤ i ≤ r∗ } when r > r∗ . Let 1 (22) Rks = {i | Coli ( q s HB s ) = δ2kp }, for k = 1, 2, · · · , 2p , s ≥ 0. (2 ) Here, s = 0 means that Rk = {q| Coli (H) = δ2kp }, k = 1, 2, · · · , 2p . It follows from (22) that for any s, the family p

Rs = {Rks }2k=1 \{∅}

(23)

is a vertex partition of V = {1, 2, · · · , 2n }. And for any q1 , q2 ∈ V belong to the same Rks if and only if Colq1 ( (21q )s HB s ) = Colq2 ( (21q )s HB s ) = δ2kp . n−s

Proposition 1. Consider the BN (3) with algebraic form (4). Let S = {Sl }2l=1 be an equal vertex partition satisfying 1 and 2 in Definition 2 and Rs be defined in (23). Given fixed s, then for any Sl ∈ S, there exists Rβs l ∈ Rs such that Sl ⊂ Rβs l ,

(24)

that is, S  Rs . Proof. Assume that the BN (4) can be turned into (9) under the logical coordinate transformation z = T x. Set Q = (1T2s ⊗ I2n−s )T . It follows from the sufficiency proof of Theorem 1 that Qδ2rn = δ2l n−s for any δ2rn ∈ Sl . Then, for any δ2rn ∈ Sl , it follows from (21) that Colr (

1 HB s ) = F Gs Qδ2rn = F Gs δ2l n−s := δ2βpl . (2q )s

Thus, by (22) and (25), we have r ∈ Rβs l . Therefore (24) holds.

(25)  

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Define a set of logical matrices as H0 := H = δ2p [h01 h02 · · · h02n ], 1 Hi := q i HB i = δ2p [hi1 hi2 · · · hi2n ], i = 1, 2, · · · , r∗ . (2 ) ⎡

Let

h01 h02 ⎢ h11 h12 ⎢ H=⎢ . .. ⎣ .. . hr ∗ 1 h r ∗ 2

⎤ · · · h02n · · · h12n ⎥ ⎥ . ⎥. .. . .. ⎦ · · · h r ∗ 2n

(26)

Construct a vertex partition C = {Cl }ηl=1 of V as follows: any a and b belong to the same Cl if and only if Cola (H) = Colb (H). By the construction of H, we have the following result. n−s

Proposition 2. Consider the BN (3) with algebraic form (4). Let S = {Sl }2l=1 be an equal vertex partition satisfying 1 and 2 in Definition 2 and C = {Cl }ηl=1 is a vertex partition of V . Then for any Sl ∈ S, there exists Cβl ∈ C such that Sl ⊂ Cβl , that is, S  C. Proof. For any q1 , q2 ∈ Sl , it follows from (22), (23) and (24) that Colq1 (Hi ) = Colq2 (Hi ), i = 0, 1, · · · , r∗ .

(27)

By (26), we have hiq1 = hiq2 , i = 0, 1, 2, · · · , r∗ . Thus, Colq1 (H) = Colq2 (H). By the construction of C, there exists a βl such that q1 , q2 ∈ Cβl . Thus, Sl ⊂ Cβl .  

5

Algorithm and Example

For searching a C-PEVP, by Proposition 2, we only need to check equal vertex partition satisfying S  C. Based on the above discussion, we give an algorithm to find a C-PEVP. And then construct a logical coordinate transformation z = T x to realize the disturbance decoupling. Algorithm 1 Consider the BN (3). Step 0. Compute the algebraic form (4) of BN (3). Step 1. Compute the matrix H showed in (26) and get the partition C. n−s Step 2. Search for an equal vertex partition S = {S}2l=1 (|Sl=1 | = 2s ) satisfying S  C. If s = n, stop; otherwise, go to the next step. n−s Step 3. Check whether the equal vertex partition S = {S}2l=1 (|Sl=1 | = 2s ) satisfies 1 and 2 in the Definition 2. If not, repeat Step 2; otherwise, go to the next step.

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Step 4. Construct a logical matrix Q, where Colq (Q) = δ2l n−s for all q ∈ Sl and l = 1, 2, · · · , 2n−s . Step 5. Compute a permutation matrix T satisfying Q = (1T2s ⊗ I2n−s )T . Let z = T x. We reconsider the BN in Example 1 to illustrate the Algorithm 1. Example 2. By the L and H shown in Example 1, we have 1 HB = δ2 [2 2 1 2 1 2 2 2], 2 1 H3 = HB 3 = δ2 [2 2 2 2 2 2 2 2]. 8

H0 = H = δ2 [1 2 2 1 2 1 1 2], H2 =

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1 HB 2 = δ2 [2 2 2 1 2 1 2 2], 4

It follows that

⎡

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From H, we obtain a vertex partition of {1, 2, 3, 4, 5, 6, 7, 8} as C = {{1, 7}, {2, 8}, {3, 5}, {4, 6}}.

(29) n−s

By Proposition 2, we only need to search the equal vertex partition S = {Sl }2l=1 (|Sl | = 2n−s ) satisfying S  C. From (29) and Proposition 2, we know that n − s ≥ 2, and obviously n − s < 3. Thus n − s = 2. Let S1 = {4, 6}, S2 = {3, 5}, S3 = {1, 7}, S4 = {2, 8}. It is easy to check that S = {Sl }4l=1 satisfies 1 and 2 in Definition 2, i.e., S = {Sl }4l=1 is a C-PEVP. The equal vertex partition obtained here just the one shown in Fig. 1. The rest steps of this example have been shown in Example 1.

6

Conclusions

In this paper, we have studied the DDP of BNs using graphic approach. A necessary and sufficient graphic condition has been obtained for disturbance decoupling. For realizing the disturbance decoupling of BNs, an effective algorithm has been designed to construct a logical coordinate transformation.

References 1. Kauffman S (1969) Metabolic stability and epigenesis in randomly constructed genetic. J Theor Biol 22(3):437–467 2. Huang S, Ingber D (2000) Shape-dependent control of cell growth, differentferentiation, and apoptosis: switching between attractors in cell regulatory networks. Exp Cell Res 261(1):91–103

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3. Huang S (2002) Regulation of cellular states in mammalian cells from a genomewide view. In: Julio C-V, Hofestadt R (eds) Gene regulation and metabolism: postgenomic computational approaches. MIT Press, Cambridge, pp 181–220 4. Farrow C, Heidel J, Maloney J, Rogers J (2004) Scalar equations for synchronous Boolean networks with biological applications. IEEE Trans Neural Netw 15(2):348– 354 5. Albert R, Othmer H (2003) The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster. J Theor Biol 75(6):1–18 6. Chaves M, Albert R, Sontag E (2005) Robustness and fragility of Boolean models for genetic regulatory networks. J Theor Biol 235(3):431–449 7. Klmat S, Saez-Rodriguez J, Lindquist JA (2006) A methodology for the structural and functional analysis of signaling and regulatory networks. BMC Bioinform 7(1):56 8. Ching WK, Zhang SQ, Ng MK (2007) An approximation method for solving the steady-state probability distuibution of probabilistic Boolean networks. Bioinformatics 23(12):1511–1518 9. Datta A, Choudhary A, Bittner M et al (2004) External control in Markovian genetic regulatory networks: the imperfect information case. Bioinformatics 20(6):924–930 10. Cheng D, Qi H (2010) A linear representation of dynamics of Boolean networks. IEEE Trans Autom Control 55(10):2251–2258 11. Cheng D, Qi H, Li Z (2011) Analysis and control of Boolean control networks: a semi-tensor product approach. Springer, London 12. Cheng D, Qi H (2008) Controllability and observability of Boolean control networks. Automatica 45(7):1659–1667 13. Liu Y, Chen H, Lu J (2015) Controllability of probabilistic Boolean control networks based on transition probability matrices. Automatica 52(10):340–345 14. Laschov D, Margaliot M (2011) A maximum principle for singke-input Boolean control networks. IEEE Trans Autom Control 56(4):913–917 15. Zhao Y, Li Z, Cheng D (2011) Optimal control of logical control networks. IEEE Trans Autom Control 56(8):1766–1776 16. Li F, Sun J (2012) Controllability and optimal control of a temporal Boolean network. Neural Netw 34(8):10–17 17. Wu Y, Shen T (2015) An algebraic expression of finite horizon optimal control algorithm for stochastic logical dynamical system. Syst Control Lett 82:108–114 18. Fornasini E, Valcher M (2013) Observability, reconstructibility and state observers of Boolean control networks. IEEE Trans Autom Control 26(6):871–885 19. Li R, Yang M, Chu T (2013) State feedback stabilization for Boolean control networks. IEEE Trans Autom Control 58(7):1853–1857 20. Li H, Wang Y (2017) Further results on feedback stabilization control design of Boolean control networks. Automatica 83:303–308 21. Cheng D, Xu X (2013) Bi-decomposition of multi-valued logical functions and its applications. Automatica 49(7):1979–1985 22. Zhao Y, Filippone M (2013) Aggregation algorithm towards large-scale Boolean network analysis. IEEE Trans Autom Control 58(8):1976–1985 23. Zou Y, Zhu J (2014) System decomposition with respect to inputs for Boolean control networks. Automatica 50:1304–1309 24. Zou Y, Zhu J (2017) Graph theory methods for decomposition w.r.t. outputs of Boolean control networks. J Syst Sci Complex 30:519–534

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25. Zou Y, Zhu J (2015) Kalman decomposition for Boolean control networks. Automatica 54:65–71 26. Li H, Xie L, Wang Y (2017) Output regulation of Boolean control networks. IEEE Trans Autom Control 62(6):2993–2998 27. Lu J, Zhong J, Huang C, Cao J (2016) On pinning controllability of Boolean control networks. IEEE Trans Autom Control 61(6):1658–1663 28. Meng M, Lam J, Feng J et al (2016) l1 gain analysis and model reduction problem for Boolean control networks. Inf Sci 348:68–83 29. Chen BS, Wang YC (2016) On the attenuation and amplification of molecular noise in genetic regulatory networks. BMC Bioinform. 7(52):1–14 30. Cheng D (2011) Disturbance decoupling of Boolean control networks. IEEE Trans Autom Control 56(1):2–10 31. Cheng D, Qi H (2010) State-space analysis of Boolean control networks. IEEE Trans Neural Netw 21(4):584–594 32. Liu Z, Wang Y (2012) Disturbance decoupling of mix-valued logical networks via the semi-tensor product method. Automatica 48(8):1839–1844 33. Zhang L, Feng J, Feng X (2014) Further results on disturbance decoupling of mixvalued logical networks. IEEE Trans Autom Control 59(6):1630–1634 34. Li H, Wang Y, Xie L (2014) Disturbance decoupling control design for switched Boolean control networks. Syst Control Lett 72:1–6 35. Meng M, Feng J (2014) Topological structure and the disturbance decoupling problem of singular Boolean networks. IET Control Theory Appl. 8(13):1247–1255 36. Zou Y, Zhu J, Liu Y (2017) State-feedback controller design for disturbance decoupling of Boolean control networks. IET Control Theory Appl. 11(18):3233–3239 37. Li Y, Zhu J (2019) On disturbance decoupling problem of Boolean control network. Asian J Control (in press) 38. Boruvka O (1974) Foundations of the theory of groupoids and groups. VEB Deutscher Verlag der Wissenschaften, Berlin

Neural Network Augmented Adaptive Backstepping Control of Spacecraft Proximity Operations Liang Sun1(B) and Bing Zhu2 1

2

School of Automation and Electrical Engineering and Institute of Artificial Intelligence, University of Science and Technology Beijing, Beijing 100083, China [email protected] The Seventh Research Division, School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China

Abstract. This paper investigates the six degrees-of-freedom relative motion control of spacecraft proximity operations under multiple uncertainties. A radialbasis-function neural network (RBFNN) adaptive backstepping controller is developed. The relative dynamics controller is developed by backstepping technique, and model uncertainties are compensated by RBFNNs. Each RBFNN utilizes a smoothly second-order piecewise function to combine adaptive control and RBFNN. The RBFNN holds the response performance in neural active region, and adaptive control dominates response performance outside the active region. It is proved that the proposed control method can guarantee the relative pose is ultimately uniformly bounded. Simulation demonstrates the performance of the control method. Keywords: Spacecraft control · Proximity operations · Adaptive backstepping control · Neural networks · Model uncertainties

1 Introduction In manu future space programmes, including rendezvous and docking, repair and refuel, hover and capture, and debris remove in orbit [1], closed-range proximity autonomous control is a key technical basis. However, in fulfilling these missions, the practical models of spacecraft relative pose motion are generally uncertain or unknown. These uncertain dynamics always result from the unknown spacecraft parameters and external disturbances, thus these complicated effects cannot negligible such that the relative pose motion control for space proximity missions is a challenging work. This study focuses on the development of an alternative approach to nonlinear feedback control, aimed at improving the control performance of the proximity operations. Since the high dimension and multiple variables in the model of proximity operations and typical cascade structure of this second-order nonlinear system, the backstepping This work was supported by the National Natural Science Foundation of China (No. 61903025) and the Fundamental Research Funds for the Central Universities (No. FRF-GF-18-0028B). c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 208–217, 2020. https://doi.org/10.1007/978-981-32-9682-4_22

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method is employed as a basic control framework. Backstepping provides a systematic, recursive control design methodology that removes the restrictions of matching and growth conditions and remains the useful nonlinear damping terms in systems [2]. Adaptive backstepping technique provides a ways of using adaptive strategy to uncertain parametric systems, such as the spacecraft autonomous proximity systems with uncertain parameters and unknown disturbances. However, if the nonlinear systems have nonlinearly parameterized uncertainty or unknown nonlinearity, then the aforementioned control approaches will be failure [3]. Obviously, with the development of adaptive nonlinear control, neural network control is an alternative method for the highly uncertain and complex systems in view of the advantage that it has an excellent online approximation ability in compact sets [4–6]. In particular, except for stability analysis and simulations, statistical analysis was also developed for neural networkbased robot manipulators controllers in [7] and [8]. In view of the various parametric uncertainties and unknown external disturbances in the relative motion model of spacecraft proximity operations, an augmented RBFNN adaptive backstepping controller is developed for spacecraft proximity operations. The main contributions are as follows. (1) The six degrees-of-freedom relative motion model is deduced with simultaneously considering uncertain mass and inertia matrix of two spacecraft, unknown thrust misalignment of the chaser, and external time-varying disturbances of two spacecraft. (2) RBFNN augmented adaptive backstepping controller is designed by utilizing the RBFNNs to actively compensate the unknown dynamics, where the performance of conventional RBFNN is improved with adaptive technique via smoothly second-order piecewise function to expend the stability region. (3) Ultimately uniformly bounded convergence of relative pose is proved via Lyapunov method, which is better than the asymptotic convergence of relative position and attitude in [10] to avoid the undesired control chattering, and the proposed control algorithm can achieve the attitude synchronization and maintain the constant relative position with respect to the uncontrolled target in spite of the external unknown disturbances and random noises.

2 Preliminaries S(x) ∈ R3×3 is a skew-symmetric matrix derived from any vector x = [x1 , x2 , x3 ]T ∈ R3 and it is denoted by S(x) = [0, −x3 , x2 ; x3 , 0, −x1 ; −x2 , x1 , 0]. It satisfies xT S(x) = 0, S(x) = x, and S(x)y = −S(y)x, y T S(x)y = 0 for any y ∈ R3 . x1 and x denote vector 1-norm and 2-norm of x, respectively; A and AF represents the induced matrix 2-norm and Frobenius-norm of A, respectively. tr{A} denotes the trace of the matrix A. In and On are n × n unit and zero matrices, respectively. Given two constants 0 < c1 < c2 , the defined piecewise function h(x) : Rm → R is a smoothly nth-order nonlinear function [3] ⎧ 0, x ≤ c1 ⎪ ⎪   ⎨ 2 − c2  x π π 1 sinn c1 < x < c2 h(x) = 1 − cosn ⎪ 2 2 c22 − c21 ⎪ ⎩ 1, x ≥ c2

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This piecewise function defines a transient progress between the neural driving region and the adaptive compensation region, which is the key in this paper. Parameters c1 and c2 determine the region of this transient progress.

3 Problem Description The relative attitude between two spacecraft is defined by [9, 10] σe =

σt (σ T σ − 1) + σ(1 − σtT σt ) − 2S(σt )σ 1 + σtT σt σ T σ + 2σtT σ

(1)

The relative position, linear velocity, and angular velocity are described by [10] re = r − Rr pt , ve = v − Rv pt , ωe = ω − Rωt , r pt = rt + pt , v pt = vt + S(ωt )pt

(2)

where R is the rotation matrix from target to chaser, the chaser’s desired position pt ∈ R3 is a constant vector defined in the target’s body-fixed frame. With denoting the system state e1 = [reT , σeT ]T , e2 = [veT , ωeT ]T , then the coupled relative dynamics are [10] e˙ 1 = C1 e1 +C2 e2 , M e˙ 2 = g + n + d + (I6 + A)u

(3)

where C1 = diag{−S(ω), O3 }, C2 = diag{I3 , G(σe )}, M = diag{mI3 , J}, A =  O3 O3 , g = −[mg1T , g2T ]T , n = [nT1 , nT2 ]T , d = [dT1 , dT2 ]T , u = [f T , τ T ]T , g1 = S(ρ) O3 μg (r − re − Rpt ) S(ω)ve + S2 (ω − ωe )Rpt + μ r − T , g2 = S(ω)Jω + JS(ω)ωe , ωt = R (r − re ) − pt 3 m RT (ω − ωe ), n1 = −mRS(pt )Jt−1 S(ωt )Jt ωt , d1 = d − Rdt + mRS(pt )Jt−1 wt , n2 = mt JRJt−1 S(ωt )Jt ωt , d2 = w − JRJt−1 wt . Assumption 1. The chaser’s mass m = m0 + mΔ is a positive constant and its inertial matrix J = J0 + JΔ is a symmetric positive definite matrix, where m0 and J0 are known parts; mΔ and JΔ are uncertain parts. Thrust misalignment vector ρ is an unknown constant vector. Meanwhile, the target’s mass mt and symmetric positive definite inertial matrix Jt are constant and unknown. Time-varying disturbances d, w, dt , wt are unknown but bounded. Assumption 2. The chaser can directly obtain its own states {r, v, σ, ω} and relative states {re , ve , σe , ωe } with the sensors installed on the chaser’s body [11]. Meanwhile, the tumbling target’s states {rt , vt , σt , ωt } cannot be obtained by the chaser. Consider the model (3) under Assumptions 1–2. The aim of driving the chaser to a desired position pt and controlling the chaser’s attitude to coincide with the target’s attitude can be stated as designing control inputs u such that limt→∞ e1 (t) < κ , where κ is a known positive constant.

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4 Controller Design and Stability Analysis The cascaded structure of (3) suggests that the goal of the close proximity operations can be achieved via backstepping technique in two steps. Setp 1: The first step in the backstepping approach involves control of the first subequation of (3). Set the backstepping variables z1 = e1 and z2 = e2 − α, where α ∈ R6 is a virtual control α = −K1C2T z1 , and K1 = diag{K11 , K12 } is a feedback gain matrix, K1i ∈ R3×3 and K1i > 0(i = 1, 2). Define the Lyapunov function V1 = 12 z1T z1 . Based on the fact z1TC1 z1 = 0, the time derivative of V1 is V˙1 = −z1TC2 K1C2T z1 + z1TC2 z2 . Note that C2 K1C2T is a symmetric positive semidefinite matrix. Setp 2: Consider the model (3), a Lyapunov function candidate by augmenting V1 with a quadratic function is chosen as follows V2 = V1 + 12 z2T Mz2 . The time derivative of V2 is given by V˙2 = −z1TC2 K1C2T z1 + z1TC2 z2 + z2T M z˙2 . Differentiating z2 with respect ˙ Since α ˙ = −K1C˙2T z1 − K1C2T z˙1 to time yields M z˙2 = g + n + d + (I6 + A)u − M α. ˙ ˙ ˙ e ) can be found in where z˙1 can be easily derived, C2 = diag{O3 , G(σe )}, and G(σ T T T T ˙ ˙ + δ + (I6 + A)u], where [10]. Thus, V2 = −z1 C2 K1C2 z1 + z1 C2 z2 + z2 [g0 − M0 α g0 = −[m0 g1 , S(ω)J0 ω + J0 S(ω)ωe ], gΔ = −[mΔ g1 , S(ω)JΔ ω + JΔ S(ω)ωe ], M0 = ˙ + n + d. diag{m0 I3 , J0 }, MΔ = diag{mΔ I3 , JΔ }, and δ = gΔ − MΔ α Consider the following desired control law u = (I6 + A)−1 (M0 α ˙ − g0 − δ −C2T z1 − K2 z2 )

(4)

where K2 = K2T > 0 is a 6 × 6 feedback gain matrix. Then V˙2 can be written as V˙2 = −z1TC2 K1C2T z1 − z2T K2 z2 . However, the desired control law (4) is not feasible because the matrix A and the term δ are all unknown. Then, we employ RBFNNs with l nodes to approximate δ (5) δ = V∗T φ(s) + ,   ≤ ε¯ where V∗ ∈ Rl×6 is the weight matrix; φ(s) ∈ Rl is the radial basis function vector; s = [eT1 , eT2 , ω T , r T ]T ∈ R18 is the input vector of the RBFNNs; ∈ R6 is the reconstruction error, ε¯ is an unknown positive constant. In addition, define a smoothly piecewise function as (6) h¯ (s) = max{h(i)}, i = s1 , · · · , s18 where h(i) is shown in Definition 1 with n = 2, and si is the ith element of s. ˆ = [εˆ¯ , δˆ¯ ]T , where δ¯ is the constant bound Define λ = [ε¯ , δ¯ ]T and its estimation λ of δ such that δ ≤ δ¯ . Then, the desired control law (4) is modified as an augmented RBFNN adaptive controller: ˆ −1 [M0 α ˙ − g0 + (1 − h¯ (s))βn + h¯ (s)βr −C2T z1 − K2 z2 ] u = (I6 + A) ˙ˆ = γ (ξ − γ λ) ˆ ρ˙ˆ = γ1 [−ST (f )z22 − γ0 ρ], ˆ V˙ˆ = γ2 [(1 − h¯ (s))φ(s)z2T − γ0Vˆ ], λ 3 0      z2 z2 O3 O3 where Aˆ = , βn = −Vˆ T φ(s) − εˆ¯ tanh , βr = −δˆ¯ tanh ; S(ρ) ˆ O3 η η

(7) (8)

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  T ξ = (1 − h¯ (s))z2T tanh zη2 , h¯ (s)z2T tanh zη2 ; ρˆ is the estimate of thrust misalignment vector; Vˆ is the estimation of V∗ ; z22 denotes the last three element of the z2 , and η > 0, γi > 0(i = 0, 1, 2, 3). Consider the augmented Lyapunov function candidate V3 = V2 +

1 ˜T˜ 1 T 1 λ λ ρ˜ ρ˜ + tr{V˜ TV˜ } + 2γ1 2γ2 2γ3

(9)

˜ =λ ˆ − λ are estimate errors. Note that (I6 + A)u = where ρ˜ = ρˆ − ρ, V˜ = Vˆ −V∗ , and λ T T ˜ ˆ ˜ ˜ with A˜ = Aˆ − A. Define δ˜¯ = δˆ¯ − δ¯ and ε˜¯ = εˆ¯ − ε¯ , (I6 + A)u − Au and z2 Au = z22 S(ρ)f 1 T T and consider σe G(σe ) = 4 (1 + σe σe )σeT , 0 ≤ h¯ (s) ≤ 1, z2T ≤ z2 1 ε¯ , z2T δ ≤ z2 1 δ¯ and xT y = tr{yxT } for any x, y ∈ Rn . Then V˙3 satisfies    z2 V˙3 ≤ − z1T Kz z1 − z2T K2 z2 + z2 1 − z2T tanh [(1 − h¯ (s))ε¯ + h¯ (s)δ¯ ] η ˙ˆ T ˜ Tλ ˜ T ξ + 1 ρ˜ T ρ˙ˆ + 1 tr{V˜ TV˙ˆ } + 1 λ S(ρ)f ˜ − (1 − h¯ (s))z2TV˜ T φ(s) − λ − z22 γ1 γ2 γ3 1 where Kz = diag{K11 , 16 K12 }. Substituting (8) into V˙3 give rise to

1 ˜ 2 ) + b ≤ −aV3 + b V˙3 ≤ −z1T Kz z1 − z2T K2 z2 − γ0 (ρ ˜ 2 + V˜ 2F + λ 2 where a = min{2λm (Kz ), 2λm (K2 )/λM (M), γ0 /γi (i = 1, 2, 3)}, b = 6μη [(1 − h¯ (s))ε¯ + h¯ (s)δ¯ ] + γ20 (ρ2 + V∗ 2F + λ2 ). Theorem 1. Consider the dynamics of spacecraft proximity operations (3) under Assumptions 1–2, the state feedback control law (7) and adaptive law (8) can ensure all signals in overall system are ultimately uniformly bounded and limt→∞ e1 (t) < κ . Proof. From the previous analysis, the closed-loop stability analysis with state feedback control law and adaptive law is made for spacecraft proximity operations under Assumptions 1–2. By solving V˙3 ≤ −aV3 + b, we have V3 (t) ≤ ba + e−at V3 (0). Hence, V3 (t) is bounded. Then based on the definition of V3 (t), the tracking errors z1 , z2 and the ˜ are bounded. Thus, all system signals are ultimately parameter estimate errors ρ, ˜ V˜ , λ uniformly bounded. Furthermore, from the definition of V3 (t), following inequality is also holds: 12 e1 (t)2 = 12 z1 (t)2 ≤ V3 (t) ≤ ba + e−at V3 (0). Thus, tracking errors e1 (t)  eventually converge to a compact set {e1 (∞) ∈ R6 : e1 (∞) ≤ 2b a }. Thus with larger K1 , K2 , γi (1, 2, 3) and smaller η , γ , the ultimate sets can be tuned sufficiently small such 0  that limt→∞ e1 (t) ≤

2b a

≤ κ.

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5 Simulation Results An example of the closed-range proximity mission in orbit is used, where the target has lower dynamic motion conditions, such that the subsequent docking mission can be completed safely. The initial simulation values are [10] r(0) = [1, 1, 1]T × 7.078 × 108 (m), v(0) = [2, 3, −2]T × 104 (m/s), σ(0) = [0, 0, 0]T , ω(0) = = [0.5, −0.6, 0.7]T , ωe (0) = [0.02, −0.02, 0.02]T (rad/s), [0, 0, 0]T (rad/s), √ σe (0) √ re (0) = [50/ 2, 0, −50/ 2]T (m), ve (0) = [0.5, −0.5, 0.5]T (m/s). The desired position for chaser is pt = [0, 5, 0]T (m). Parameters of two spacecraft and external disturbances are [10] ρ = [0.03, 0.02, 0.025]T (m), m = 58.2 (kg), mt = 1425.6 (kg), ⎡ ⎤ ⎡ ⎤ 598.3 −22.5 −51.5 3336.3 −135.4 −154.2 J = ⎣ −22.5 424.4 −27 ⎦ (kgm2 ), Jt = ⎣ −135.4 3184.5 −148.5 ⎦ (kgm2 ). −51.5 −27 263.6 −154.2 −148.5 2423.7 ⎤ 0.3 − 0.2 sin(ωot) − 0.4 cos(ωot) d = dt = ⎣ 0.3 − 0.3 sin(ωot) + 0.2 cos(ωot) ⎦ + ζ¯1 (N), 0.2 + 0.4 sin(ωot) + 0.2 cos(ωot) ⎤ ⎡ 0.06 − 0.04 sin(ωot) + 0.05 cos(ωot) w = wt = ⎣ 0.07 + 0.05 sin(ωot) − 0.04 cos(ωot) ⎦ + ζ¯2 (Nm), 0.04 − 0.03 sin(ωot) + 0.03 cos(ωot) ⎡

 where ωo = μg /r3 and μg = 3.986 × 1014 (m3 /s2 ); ζ¯1 = 0.5 × randn(3, 1), ζ¯2 = 0.25 × randn(3, 1), and randn(3, 1) denotes the random white noise vector. We choose the controller gains K1 = diag{0.05I3 , 0.4I3 }, K2 = diag{5I3 , 250I3 }, γ0 = 0.005, γi = 0.05(i = 1, 2, 3), η = 0.02, and all initial estimations for unknown parameters are set as zero. The number of nodes in RBFNN is l = 200, and the initial weight matrix is set as zero matrix. The input vector of RBFNN is s = [re , σe , ve , ωe , ω, r]T ∈ R18 . The transient response transition between the neural drive region and the adaptive compensation region is set as (re , σe , ve , ωe , ω, r) ∈ ([10, 100], [0.06, 3], [1.5, 10], [0.03, 3], [0.03, 3], [500, 3000]). By applying neural networks as function approximators, stability and convergence of the overall systems can be ensured when the initial online weights are tuned close to the ideal weights. All simulation results of the closed-loop system based on the proposed control method are shown in Figs. 1, 2, 3, 4 and 5. The simulation result implies that the proposed RBFNN augmented adaptive backstepping control approach for the closed-range proximity missions canensure the stability of the controlled system such that the subsequent docking operations can be safely achieved.

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Fig. 5. Estimations for Vˆ F and ρ. ˆ

6 Conclusions A nonlinear control law has been designed for spacecraft proximity operations in the presence of unknown target information and external disturbances. The unknown dynamic couplings and external disturbances have been considered in the nonlinear relative motion mechanical model. The control approach is developed by the nonlinear state feedback strategy with adaptive RBFNNs. The RBFNNs are applied to online compensate the lumped uncertainties. It is proven that all signals in the controlled relative motion closed-loop system are uniformly ultimately bounded. Moreover, all simulation results for a closed-range proximity scenario implies that our control approach is effective with robustness to model uncertainties. The proposed control scheme can drive the chaser spacecraft to the desired position and attitude precisely. From a practical viewpoint, the chaser’s actuator outputs are always limited such that many excellent control methods failure, thus future work will spread the proposed approach to solve the saturation problem for closed-range proximity operations with model uncertainties.

References 1. Goodman JL (2006) History of space shuttle rendezvous and proximity operations. J Spacecr Rockets 43(5):944–959. https://doi.org/10.2514/1.19653 2. Li F, Wu L, Shi P, Lim C-C (2015) State estimation and sliding mode control for semiMarkovian jump systems with mismatched uncertainties. Automatica 51:385–393. https:// doi.org/10.1016/j.automatica.2014.10.065

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3. Wu J, Chen W, Zhao D, Li J (2013) Globally stable direct adaptive backstepping NN control for uncertain nonlinear strict-feedback systems. Neurocomputing 122:134–147. https://doi. org/10.1016/j.neucom.2013.05.042 4. Ren X, Lewis FL, Zhang J (2009) Neural network compensation control for mechanical systems with disturbances. Automatica 45:1221–1226. https://doi.org/10.1016/j.automatica. 2008.12.009 5. Zhu B (2014) Nonlinear adaptive neural network control for a model-scaled unmanned helicopter. Nonlinear Dyn 78:1695–1708. https://doi.org/10.1007/s11071-014-1552-5 6. Sharma R, Kumar V, Gaur P, Mittal AP (2016) An adaptive PID like controller using mix locally recurrent neural network for robotic manipulator with variable payload. ISA Trans 62:258–267. https://doi.org/10.1016/j.isatra.2016.01.016 7. Singh HP, Sukavanam N (2013) Stability analysis of robust adaptive hybrid position/force controller for robot manipulators using neural network with uncertainties. Neural Comput Appl 22(7/8):1745–1755. https://doi.org/10.1007/s00521-012-0966-6 8. Panwar V, Kumar N, Sukavanam N, Borm JH (2012) Adaptive neural controller for cooperative multiple robot manipulator system manipulating a single rigid object. Appl Soft Comput 12:216–227. https://doi.org/10.1016/j.asoc.2011.08.051 9. Shuster MD (1993) A survey of attitude representations. J Astronaut Sci 41(4):439–517 10. Sun L, Huo W (2015) 6-DOF integrated adaptive backstepping control for spacecraft proximity operations. IEEE Trans Aerosp Electron Syst 51(3):2433–2443. https://doi.org/10.1109/ TAES.2015.140339 11. Segal S, Carmi A, Gurfil P (2014) Stereovision-based estimation of relative dynamics between noncooperative satellites: theory and experiments. IEEE Trans Control Syst Technol 22(2):568–584. https://doi.org/10.1109/TCST.2013.2255288

Finite Time Anti-disturbance Control Based on Disturbance Observer for Systems with Disturbance Hanxu Zhao1 and Xinjiang Wei2(B) 1

2

School of Mathematics and Statistics Science, Ludong University, Yantai 264000, China zhao [email protected] School of Information and Electrical Engineering, Ludong University, Yantai 264000, China [email protected]

Abstract. This paper studies the finite time anti-disturbance control (FTADC) scheme for the systems with the harmonic disturbance by Implicit Lyapunov function approach. In order to estimate the harmonic disturbance, a disturbance observer is constructed. Then the FTADC scheme is proposed which guarantees the state of composite system can reach globally finite-time stable. Finally, simulation results show the effechtiveness of the proposed method. Keywords: Disturbance observer · Globally finite time stable Finite time anti-disturbance control

1

·

Introduction

In practical application, in order to improve work efficiency or ensure the normal operation of the machine, the control input needs strict time response constraints. Most of the existing control methods are convergent in infinite time, which may have a great negative impact on the control system in practical application, making the finite time control become a research hot topic. Therefore, the system of finite time stability analysis has attracted extensive attention of scholars [1–6]. The finite time stability analysis is consider for a class of nonlinear systems via the Implicit Lyapunov function approach in [2]. A new concept finite-time input-to-state stability (FTISS) is put forward to analyze the finitetime stability for the nonlinear systems in [3]. The finite-time stabilization for the discrete-time linear systems with disturbances is considered with a sufficient condition for finite-time stabilization via state feedback in [4]. The finite time stabilization problem is also combined with other classical control strategies [7–9]. A continuous finite-time control problem is considered for rigid robotic manipulators by using a new form of terminal sliding modes [7]. The method based on Lyapunov function and homogeneity is given to research global adaptive c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 218–226, 2020. https://doi.org/10.1007/978-981-32-9682-4_23

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finite-time control problems for two special classes of nonlinear control systems in [8]. On the other hand, disturbances exist widely in practical engineering which may affects the performance of the system, as a key anti-disturbance control strategy, disturbance observer-based control (DOBC) had received much attention [10–12]. The advantage of DOBC is that it can reject the disturbance of inner loop and attenuate the outer loop ones respectively, which improves the control accuracy of the system. Recently, DOBC had been extended to a class of systems with multiple disturbances and composite hierarchical anti-disturbance control (CHADC) was proposed [13–16]. However, finite time control via DOBC method has not received enough attention. Therefore, the research of composite finite time control with DOBC is necessary. Aiming at proposing a finite time anti-disturbance control for a class of systems with the harmonic disturbance, the main contributions of this paper are as follows: (1) Extending the current DOBC work to finite time control which can ensures the state converges in finite time. (2) A disturbance observer is designed to estimate the harmonic disturbance, then the FTADC scheme is proposed to guarantee the composite system is globally finite-time stable. The main content of this paper consists of the following parts. In Sect. 2, the problem formulation is presented. In Sect. 3, the main results are displayed, which include a disturbance observer (DO) and a FTADC scheme. In Sect. 4, simulation examples are given to account for the availability of the strategy. In Sect. 5, there is a summary of the paper. Notations. R is the set of real numbers and R+ = {x ∈ R : x > 0}; diag{λi }ni=1 is the diagonal matrix with the elements λi ; ∗ represents the corresponding elements in the symmetric matrix; dV dt |(.) is the time derivative of a function V along the solution of a differential equation numbered as (.);  ·  is the Euclidean norm in Rn ; for a symmetric matrix P = P T the minimal and maximal eigenvalues are denoted by λmin (P ) and λmax (P ); int(Ω) is the interior of the set Ω ⊆ Rn .

2

System Description

The following system with disturbance is described as: x(t) ˙ = Ax(t) + B[u(t) + d(t)],

(1)

where x(t) ∈ Rn×m , u(t) ∈ Rn×m are the system state and the control input, respectively. A ∈ Rn×n and B ∈ Rn×n are the coefficient matrices. d(t) ∈ Rn×m represents the harmonic disturbance, which can be formulated by the following exogenous system ω(t) ˙ = W ω(t), (2) d(t) = Gω(t),

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where W, G are known constant matrices. Assumption 1. (A, B) is controllable and the (W, BG) is observable. The following definition and lemma are introduced by [2]. Definition 1. Consider the system of the form: x(t) ˙ = f (t, x),

x(0) = x0 ,

(3)

where x(t) ∈ Rn is the state vector, f : R+ → Rn is a nonlinear vector field, which can be discontinuous with respect to the state variable. In this case, the solutions of the system (3) are understood in the sense of Filippov. The origin of system (3) is said to be globally finite-time stable if: 1. Finite-time attractivity: there exists a locally bounded function T : Rn \{0} → R+ , such that for all x(0) ∈ Rn \{0}, any solution x(t, x0 ) of the system (3) is defined at least on [0, T (x0 )) and lim x(t, x0 ) = 0. t→T (x0 )

2. Lyapunov stability: ∃δ ∈ κ such that x(t, x0  ≤ δ(x0 ) for all x0 ∈ Rn , t ∈ R+ . where the function T is called settling-time function of system (3). Lemma 1. If there exists a continuous function Q : R+ × Rn → R that satisfies the conditions: (C1) Q is continuously differentiable in R+ × Rn \{0}; (C2) for any x ∈ Rn \{0}, there exist V ∈ R+ such that Q(V, x) = 0; (C3) let Ω = {(V, x) ∈ R+ × Rn : Q(V, x) = 0} and lim V = 0+ , lim+ x = 0,

x→0 (V,x)∈Ω

V →0 (V,x)∈Ω

lim

x→+∞ (V,x)∈Ω

V = +∞;

(C4) the inequality ∂Q(V,x) < 0 holds for all V ∈ R+ and x ∈ Rn \{0}; ∂V (C5) there exist c ∈ R+ and μ ∈ (0, 1] such that sup t∈R+ f (t,x)∈K[f ](t,x)

∂Q(V, x) ∂Q(V, x) f (t, x) ≤ cV 1−μ , (V, x) ∈ Ω; ∂x ∂V

then the origin of system (3) is globally finite-time stable with the following V0µ settling time estimate:T (x0 ) ≤ cμ , where V0 ∈ R+ : Q(V0 , x0 ) = 0.

3

Main Results

In this section, it is supposed that system state x(t) is available. A disturbance observer (DO) is designed to estimate the harmonic disturbance, then the finite time anti-disturbance control (FTADC) scheme is proposed.

Finite Time Anti-disturbance Control

3.1

221

Disturbance Observer (DO)

The disturbance observer is constructed as  v(t) ˙ = (W − LBG)ˆ ω (t) − L[Ax(t) + Bu(t)], ˆ = Gˆ d(t) ω (t), ω ˆ (t) = v(t) + Lx(t),

(4)

where ω ˆ (t) is the estimation of ω(t), and v(t) is the state of the observer. L is the gain of observer to be designed. The estimation error is denoted as eω (t) = ω(t) − ω ˆ (t). Based on (2) and (4), yields e˙ ω (t) = (W − LBG)eω (t).

(5)

Since (W, BG) is observable, the pole of error dynamics (5) can be placed at the left-hand side complex plane according to [17,18]. By adjusting L, the performance requirement of DO can be satisfied. In light of Polyakov, Efimov and Perruquetti (2015) [2], the finite time antidisturbance controller based on disturbance estimation is constructed as  ˆ + uf (t), u(t) = −d(t) (6) uf (t) = V 1−μ KDr (V − )x(t), where Dr (λ) = diag{λri } and V is the Lyapuonv function determined by the Implicit Lyapunov function; r = (r1 , · · · , rn )T , ri = 1 + (n − i)μ, 0 < μ ≤ 1, λ ∈ R+ . Substituting (6) into (1), we have x(t) ˙ = (A + V 1−μ BKDr (V − ))x(t) + BGeω (t). Combining (5) with (7), results in      A + V 1−μ BKDr (V − ) BG x(t) x(t) = . 0 W − LBG eω (t) eω (t)

(7)

(8)

Then the composite system can be depicted by ¯x(t), x ¯(t) = A¯ where

3.2

(9)



   x(t) BG A + V 1−μ BKDr (V − ) ¯ x ¯(t) = . , A= eω (t) 0 W − LBG

(10)

Finite Time Anti-disturbance Control (FTADC)

In this section, we aim to design finite time anti-disturbance control scheme so that the state x ¯(t) of the composite system (9) is globally finite-time stable. The following result can be obtained.

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Theorem 1. For given system (1) with disturbance (2), if Hμ P + P Hμ > 0 and matrices X1 > 0, X2 > 0, R, L satisfying   M1 V −μ BGX2 Θ= < 0, (11) ∗ M2 where M1 = V −μ (AX1 + V 1−μ BR1 + X1 AT + V 1−μ R1T B T ), M2 = V −μ ((W − LBG)X2 + X2 (W T − GT B T LT )). Then, by designing DO (4) with gain L and controller (6) with gain K = R1 Dr−1 (V −1 )X1−1 , the composite system (9) is globally finite-time stable and Vµ T (x0 ) < μ0 . Proof. Consider the following Implicit Lyapunov function Q(V, x ¯) = x ¯T (t)Dr (V −1 )P Dr (V −1 )¯ x(t) − 1.

(12)

Letting 

  −1  P1 0 X1 0 P = = > 0. 0 P2 0 X2−1

(13)

It can be verified that the following inequalities x2 x2 λmin (P )¯ λmax (P )¯ ≤ Q(V, x ¯) + 1 ≤ . 2+2(n−1)μ 2 2+2(n−1)μ max{V ,V } min{V , V 2}

(14)

holds for all V ∈ R+ and x ∈ R+ . The function (12) satisfies conditions (C1) − (C3) of Lemma 2.1. The condition (C4) of Lemma 2.1 also holds, since ∂Q = −V −1 x ¯T (t)Dr (V −1 )(Hμ P + P Hμ )Dr (V −1 )¯ x(t) < 0, ∂V

(15)

where Hμ P + P Hμ > 0, Hμ = diag{ri }ni=1 . Taking into account that ¯ then we can obtain ¯ r (V −1 ) = V −μ A, Dr (V −1 )AD ∂Q ¯ ¯x(t)) (A¯ x(t)) = 2¯ xT (t)Dr (V −1 )P Dr (V −1 )(A¯ ∂x ¯ = V −μ x ¯T (t)Dr (V −1 )(P A¯ + A¯T P )Dr (V −1 )¯ x(t) −1 T −1 = (Dr (V )¯ x(t)) Θ0 (Dr (V )¯ x(t)),

(16)

where Θ0 = V −μ (P A¯ + A¯T P ),

(17)

if Θ0 < 0, it can be obtained that ∂Q ¯ ∂Q (A¯ x(t)) < V 1−μ . ∂x ¯ ∂V

(18)

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Therefore, the condition (C5) of Lemma 2.1 holds. Then, by designing DO (4) with gain L and DOBFTC (6) with gain K = R1 Dr−1 (V −1 )X1−1 , based on Lemma 2.1, the composite system (9) is globally finite-time stable and T (x0 ) < V0µ μ . In the next, we will proof Θ0 < 0. (1): Θ0 < 0 ⇔ Θ1 < 0. Substituting (10) into (19), we can obtain Θ0 < 0 is equivalent to Θ1 < 0, where ⎡ −μ ⎤ V (P1 (A + V 1−μ BKDr (V −1 ) −μ V P1 BG ⎢ + (AT + V 1−μ Dr (V −1 )K T B T )P1 ) ⎥ ⎥ Θ1 = ⎢ −μ ⎣ ⎦ V (P2 (W − LBG) ∗ +(W T − GT B T LT )P2 ) (2): Θ1 < 0 ⇔ Θ2 < 0. Θ1 < 0 is equivalent to Θ2 < 0 by pre-multiplied and post-multiplied diag{X1 , X2 } simultaneously, where   N1 V −μ BGX2 Θ2 = ∗ N2 with N1 = V −μ [(A + V 1−μ BKDr (V −1 ))X1 + X1 (AT + V 1−μ Dr (V −1 )K T B T )], N2 = V −μ ((W − LBG)X2 + X2 (W T − GT B T LT )). (3): Θ2 < 0 ⇔ Θ < 0. Letting K = R1 Dr−1 (V −1 )X1−1 , Θ2 < 0 is equivalent to Θ < 0. It can be knew from above that Θ < 0 ⇔ Θ2 < 0 ⇔ Θ1 < 0, which guarantees Θ0 < 0. Hence, the composite system (9) is globally finite-time stable. The proof is completed.

4

Simulation Example

Consider the system (1) with coefficient matrices as follows     0.2 − 0.97 −2.9 0.92 A= ,B = . −0.98 0.72 1.25 − 1.52 The disturbance d(t) in (2) is given by    0 1.3 0.2 W = ,G = −3.7 0 3.2

 0.39 . − 2.23

Placing pole to λ = [−15, −16], according to pole placement theorem, we can obtain   −12.3267 − 9.7833 L= . −9.9410 − 4.2914

H. Zhao and X. Wei 4

DOBC

FTADC

State−x1

2 0 −2 −4 −6 0

10 15 Time(sec) (a) Comparison of system state x1 between DOBC and FTADC.

5

4

DOBC

FTADC

State−x2

2 0 −2 −4 0

Disturbance estimation error

10 15 Time(sec) (b) Comparison of system state x2 between DOBC and FTADC.

5

6

e (t) ω1

e (t) ω2

4 2 0 −2 −4 0

10 Time(sec) (c) Curves of the disturbance estimation error.

5

10

u1(t)

15

u2(t)

5 Control

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0 −5

−10 0

5

Time(sec)

10

(d) Curves of the controller.

Fig. 1. Responses of composite system.

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The initial value of the state is given to be x(0) = [5, −5]T . Based on Theorem 1, it can be solved that     0.4657 − 0.0000 0.2239 − 0.0371 X1 = , R1 = , −0.0000 0.4657 −0.0371 0.5927     0.0343 0.0018 0.3679 − 0.0637 , K= . X2 = 0.0018 0.0366 −0.0609 1.0181 Figures 1(a) and (b) show the comparison of system responses between DOBC and FTADC. Figure 1(c) shows the disturbance estimation error, which illustrates estimation error of DO is satisfying. Figure 1(d) gives the curve of the controller. The desired system robustness performance can be achieved with the proposed FTADC scheme.

5

Conclusions

In this paper, finite time anti-disturbance control scheme is proposed for the systems with the harmonic disturbances. A disturbance observer(DO) is designed to estimate the harmonic disturbance, then the FTADC scheme is proposed so that the desired dynamic performance of composite system can be achieved. One of the further studies is to extend the range of the disturbance for the finite time control via DOBC. Acknowledgments. This work was supported by the National Natural Science Foundation of China 61973149.

References 1. Dong S, Li S (2009) Stabilization of the attitude of a rigid spacecraft with external disturbances using finite-time control techniques. Aerosp Sci Technol 13(4):256–265 2. Andrey P, Denis E, Wilfrid P (2015) Finite-time and fixed-time stabilization: Implicit Lyapunov function approach. Automatica 51:332–340 3. Hong Y, Jiang Z, Feng G (2008) Finite-time input-to-state stability and applications to finite-time control. IFAC Proc Vol 41(2):2466–2471 4. Amato F, Ariola M (2005) Finite-time control of discrete-time linear system. IEEE Transact Autom Control 50(5):724–729 5. Dong S, Li S, Li Q (2009) Stability analysis for a second-order continuous finitetime control system subject to a disturbance. J Control Theory Appl 7(3):271–276 6. Feng J, Wu Z, Sun J (2005) Finite-time control of linear singular systems with parametric uncertainties and disturbances. Acta Autom Sinica 31(4):634 7. Yu S, Yu X, Shirinzadeh B, Man Z (2005) Continuous finite-time control for robotic manipulators with terminal sliding mode. Automatica 41(11):1957–1964 8. Hong Y, Wang HO, Bushnell LG (2001) Adaptive finite-time control of nonlinear system. IEEE Transact Autom Control 9. Xiang WM, Xiao J (2011) H∞ finite-time control for switched nonlinear discretetime systems with norm-bounded disturbance. J Franklin Inst 348(2):331–352

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10. Guo L, Chen WH (2010) Disturbance attenuation and rejection for systems with nonlinearity via DOBC approach. Int J Robust Nonlinear Control 15(3):109–125 11. Mallon N, Wouw NVD, Putra D (2016) Friction compensation in a controlled onelink robot using a reduced-order observer. IEEE Transact Control Syst Technol 14(2):374–383 12. Yang J, Chen WH, Li SH (2017) Disturbance/uncertainty estimation and attenuation techniques in PMSM drives-a survey. IEEE Transact Ind Electron 64(4):3273– 3285 13. Guo L, Cao S (2014) Anti-Disturbance Control for Systems with Multiple Disturbance. CRC Press, New York 14. Guo L, Cao SY (2014) Anti-disturbance control theory for systems with multiple disturbance: a survey. ISA Transact 53(4):846–849 15. Wei X, Wu Z, Hamid RK (2016) Disturbance observer-based disturbance attenuation control for a class of stochastic systems. Automatica 63(162):21–25 16. Wei XJ, Sun SX (2018) Elegant anti-disturbance control for discrete-time stochastic systems with nonlinearity and multiple disturbances. Int J Control 91(3):706– 714 17. Willems JL, Willems JC (1976) Feedback stabilizability for stochastic systems with state and control dependent noise. Automatica 12:277–283 18. Zhang WH, Chen B (2004) On stabilizability and exact observability of stochastic systems with their applications. Automatica 40(1):87–94

Anti-disturbance Control Based on Disturbance Observer for Dynamic Positioning System of Ships Yongli Wei1 and Xinjiang Wei2(B) 1

2

School of Mathematics and Statistics Science, Ludong University, Yantai 264000, China [email protected] School of Information and Electrical Engineering, Ludong University, Yantai 264000, China [email protected]

Abstract. In this paper, the anti-disturbance control problem is discussed for dynamics positioning (DP) system of ships with stochastic disturbances, which described by exogenous systems. Considering the environmental disturbances, the state space mathematical model of ship is established. A disturbance observer is constructed to estimate the disturbance. Based on observer, a disturbance observer-based on antidisturbance control (DOBADC) scheme for DP system of ships is proposed to make the composite system globally asymptotically stable in probability. Finally, the effectiveness of the proposed control strategy is verified by a simulation research on a supply ship. Keywords: Dynamic positioning system · DOBADC Disturbance observer · Exogenous systems

1

·

Introduction

Dynamic positioning (DP) system of ships technology refers to the thrust generated by the ships own propulsion device, which makes the ship maintain a certain attitude or navigate along a fixed trajectory [1]. It is widely used in salvage boats and offshore drilling platforms on account of its advantages such as not limited by water depth, strong mobility and high positioning accuracy [2]. Ships in the marine environment will be affected by wind, wave, current and other uncertain marine environmental disturbance [3]. It is an important issue in the research of DP ship control to design a controller with strong antidisturbance ability and robustness. At present, the main control methods for the DP system of ships include adaptive control, PID control, backstepping control, neural network control, fuzzy control and other control methods [4–6]. Due to the unpredictability of environmental disturbances, adaptive control method can not effectively solve the control problem of DP system for actual ship motion. c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 227–235, 2020. https://doi.org/10.1007/978-981-32-9682-4_24

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Traditional PID control is difficult to the requirements of high control accuracy and stability of ships [4]. Backstepping control, neural network control, fuzzy control and so on have higher requirements for the control object model and a large amount of calculation, with poor practicability [5,6]. Compared with the above control methods, the disturbance observer-based control strategy (DOBC) can deal with a large class of anti-disturbance problems, and can analyze the stability of the disturbance [7,8]. The structure of DOBC is simple, which greatly simplifies the design process of the controller [9]. The basic idea is to use disturbance observer to estimate the disturbance online, and compensate the disturbance by combining feed-forward compensator and feedback control law [10]. The control gain is solved by linear matrix inequality (LMI) method to achieve the stabilization of the DP system of ships. The DOBC method enables the ship maintain a certain attitude in the marine environment, meets the stability requirements and improves the robustness of the DP system of ship. It avoids the problem that the above method requires a high level of control object model and a large amount of calculation [3]. The purpose of this paper is to present a new disturbance observer-based on anti-disturbance control (DOBADC) scheme for DP system of ships with stochastic disturbance. The main contribution of this paper is summarized as follows: (1) The current DOBC works are extended to a class of dynamics positioning (DP) system of ships with stochastic disturbances, which can be described by exogenous systems. (2) A new disturbance observer (DO) is constructed to estimate the disturbance. Based on the DO, the anti-disturbance control scheme for DP system of ships is designed to attenuate and reduce the stochastic disturbances, such that the composite system is globally asymptotically stable in globally.

2

Mathematical Modeling of Ships

The establishment of dynamic positioning ship mathematical model is divided into two parts: kinematics model of DP ship and dynamics model of DP ship. 2.1

Kinematics Model of Dynamic Positioning Ship

Two right-hand coordinate frames are defined in the Fig. 1. CXY Z is a bodyfixed frame. The center of gravity of the ship is origin C. CX is pointed forward from the tail, CY is pointed to the starboard, CZ is from top to bottom. OX0 Y0 Z0 is the earth-fixed frame. Origin O is any point on the earths surface. The OX0 axis points north, the OY0 axis points east, and the OZ0 axis points to the center of the earth. The vectors (x, y) and yaw angle ψ in the earth-fixed frame are defined as η = [x, y, ψ]T , and the surge velocity u, sway velocity v and yaw angular velocity r in body-fixed frame are defined as υ = [u, v, r]T . Through coordinate transformation, we have: η˙ = R(ψ)υ,

(1)

Anti-disturbance Control

where R(ψ) is the rotation matrix as follows: ⎡ ⎤ cosψ − sinψ 0 cosψ 0⎦. R(ψ) = ⎣ sinψ 0 0 1

229

(2)

Fig. 1. Earth-fixed and body-fixed coordinate frames.

2.2

Dynamics Model of Dynamic Positioning Ship

Dynamics model of dynamic positioning ship is expressed as M v(t) ˙ = −Dv(t) + τ (t) + D0 (t),

(3)

where τ = [τ1 , τ2 , τ3 ] is the surge force τ1 , sway τ2 and sway moment τ3 , respectively. D0 (t) is a vector composed of forces and moments generated by wind, waves and currents. M is a mass matrix, which is expressed as follows: ⎤ ⎡ 0 0 m − Xu˙ mxG − Yr˙ ⎦ , 0 m − Yv˙ (4) M =⎣ 0 mxG − Nv˙ IZ − Nr˙ where m is the mass of the ship, IZ is inertia around the Z axis, and xG is the center of gravity of ship. The addition mass Xu˙ < 0, Yv˙ < 0, Nr˙ < 0, whereas the cross-terms Yr˙ and Nv˙ can have both signs [3]. A damping matrix D as follows: ⎤ ⎡ 0 0 −Xu − Yv − Yr ⎦ . (5) D=⎣ 0 0 − Nv − Nr 2.3

State Space Model of Dynamic Positioning Ship

When the heading angle of the ship in small enough, there is R(ψ) ∼ = I,

(6)

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take U = τ , state space model of dynamic positioning ship is expressed as ˙ X(t) = AX(t) + BU (t) + ED0 (t),     0 0 I X = [η T , υ T ], A = , B = E = , M −1 0 − M −1 D

(7)

where A ∈ IRn×n is the system matrix, X(t) ∈ IRn is the state vector, B ∈ IRn×m is the input matrix, U (t) ∈ IRm is the vector of control inputs, E ∈ IRn×m is the disturbance coefficient matrix, D0 (t) ∈ IRm is the system disturbance. Disturbance D0 (t) can be formulated by the following stochastic exogenous system  D0 (t) = V ω(t), (8) ω(t) ˙ = W ω(t) + B1 X(t)ξ(t), where W ∈ IRr×r , B1 ∈ IRr×ς , and V ∈ IRm×r are properly known matrices. ξ(t) is band-limited white noises, which is considered as independently distributed random vectors, satisfy mathematical expectation E{ξ(t)} = 0 and variance E|ξ(t)|2 = σ, where σ is constant. Notes and Comments. The disturbance D0 (t) represents a class of signals with known frequency, unknown amplitudes and unknown phases in many instances. The stochastic exogenous system (8) can describe D0 (t) when the state is coupled with white noise. It can not only express disturbance with partially-known information, but also describe a stochastic disturbance. Where the waves are mature waves with known frequencies. Assumption 1. (A, B) is controllable and the (W, EV ) is observable. Lemma 1. (see [11]): For system (6), if there exist a C 2 function V (X), and class κ∞ functions α1 and α2 and class κ functions α3 , such that for all X(t) ∈ Rn , t ≥ 0 α1 ( X(t) ) ≤ V (X, t) ≤ α2 ( X(t) ), 1 ∂2V ∂V f (X, t) + T r{g(X, t)T LV (X, t) = g(X, t)} ∂X 2 ∂X 2 ≤ −α3 ( X(t) ).

(9)

Then, the equilibrium X = 0 is globally asymptotically stable in probability.

3 3.1

Main Results Disturbance Observer (DO)

The DO is structured to  dq(t) = (W − LEV )ˆ ω (t)dt − L[AX(t) + BU (t)]dt, ˆ D0 (t) = V ω ˆ (t), ω ˆ (t) = q(t) + LX(t),

(10)

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where D0 (t) is the disturbance estimation, L is the observer gain, and q(t) is the auxiliary vector as the state of DO. Based on (7), (8) and (10), it is shown that the error dynamics satisfies deω (t) = (W − LEV )eω (t)dt + B1 X(t)dW1 (t),

(11)

The poles of error dynamics (5) can be placed at the left-half side complex plane [12,13]. The controller is designed as ˆ 0 (t) + KX(t), U (t) = −D

(12)

where K is the controller gain. Substituting (12) into (7), one have dX(t) = (A + BK)X(t)dt + EV eω (t)dt.

(13)

Combining (13) with (11), the composite system can be described as ¯ ¯ ¯ X(t)dW ¯ d(X(t)) = A¯X(t)dt +B 1 (t), where

 ¯ X(t) =

3.2

  X(t) A + BK ¯ ,A = eω (t) 0

EV W − LEV



(14) 

¯= ,B

 0 0 . B1 0

(15)

Disturbance Observer-Based Disturbance Attenuation Control (DOBADC)

Theorem 1. Consider DP system of ships (7) with disturbance (8) under Assumption 1, if there exist matrices Q1 = P1 −1 > 0, Q2 = P2 −1 > 0, R1 and R2 satisfying the following LMI ⎤ ⎡ 0 Q1 B1 T Λ1 EV Q2 ⎢ ∗ Λ2 0 0 ⎥ ⎥ < 0, (16) Υ =⎢ ⎣ ∗ 0 ⎦ ∗ − Q1 ∗ ∗ ∗ − Q2 where

Λ1 = AQ1 + QT1 AT + BR1 + R1T B T , Λ2 = W Q2 + QT2 W T − LEV Q2 − QT2 V T E T LT .

Then, by designing DO (11) with gain L and DOBC law (13) with gain K = R1 Q−1 1 , the composite system (15) is globally asymptotically stable in probability. Proof. Consider the following Lyapunov function ¯ ¯ T (t)P X(t). ¯ V (X(t), t) = X

(17)

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Selecting



  −1  P1 0 0 Q1 P = = > 0. 0 P2 0 Q−1 2

(18)

Differentiating (17) along with (15), (see [14]) yields ¯ ¯ ¯T ¯T ¯T ¯ ¯ T LV (X(t), t) = ∂∂ν ¯ [AX (t)]dt + T r{X (t)B P B X (t)} X ¯ ¯ +X ¯ T (t)(B ¯ T P B) ¯ X(t) ¯ ¯ T (t)Υ1 X(t), ¯ T (t)(P A¯ + A¯T P )X(t) =X ≤X where



P A¯ + A¯T P Υ1 = ∗

(19)

 ¯T B . − P −1

Then, following inequality (19) can be obtained ¯ ¯ ¯ T (t)Υ1 X(t) LV (X(t), t) ≤ X

(20)

Next, we will prove that Υ < 0 ⇔ Υ1 < 0. (1) To prove Υ1 < 0 ⇔ Υ2 < 0, yields ⎡ Ξ1 P1 EV ⎢ ∗ Ξ2 Υ2 = ⎢ ⎣ ∗ ∗ ∗ ∗ where

0 0 − P1 ∗

⎤ B1T 0 ⎥ ⎥, 0 ⎦ − P2

(21)

Ξ1 = P1 A + AT P1 + P1 BK + K T B T P1 , Ξ2 = P2 W + W T P2 − P2 LEV − V T E T LT P2 .

(2) To prove Υ2 < 0 ⇔ Υ3 < 0. Υ2 < 0 is pre-multiplied and post-multiplied simultaneously by diag{Q1 , Q2 , I, I}, then it is equivalent to Υ3 < 0, yields ⎡ ⎤ Λ1 EV Q2 0 Q1 B1T ⎢ ∗ Λ2 0 0 ⎥ ⎥ (22) Υ3 = ⎢ ⎣ ∗ 0 ⎦ ∗ − Q1 ∗ ∗ ∗ − Q2 where

Λ1 = AQ1 + Q1 AT + BKQ1 + Q1 K T B T , Λ2 = W Q2 + Q2 W T − LEV Q2 − QT2 V T E T LT .

(3) To prove Υ3 < 0 ⇔ Υ < 0. It can be shown that Υ3 < 0 ⇔ Υ < 0 based on K = R1 Q−1 1 in (22). Form (1)–(3), it is obviously that Υ < 0 ⇔ Υ3 < 0 ⇔ Υ2 < 0 ⇔ Υ1 < 0. Therefore Υ < 0 ⇔ Υ1 < 0. Following (17), (20) and (23), there exists a constant α > 0, such that ¯ ¯ ¯ T (t)Υ1 X(t) ≤ −α( X(t) ) LV (X(t), t) ≤ X

(23)

Based on Lemma 1, the composite system (15) satisfies globally asymptotically stable in probability. The proof is completed.

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233

Simulation Example

To prove the effectiveness of the proposed method, a supply ship is simulated. According to [4–6], the length of the ship is 76.2 m, the mass is 4.591 × 106 kg. The dynamic parameters of the DP ship system (3) are as follows: ⎤ ⎡ 0 0 5.3122 × 106 ⎦, 0 0 8.2831 × 106 M =⎣ 6 0 0 3.7454 × 10 ⎡ ⎤ 6 5.0242 × 10 0 0 − 4.3933 × 106 ⎦ . 0 2.7229 × 105 D=⎣ 6 0 − 4.3933 × 10 4.1894 × 108 The initial state is X(0) = [9, 8, 0.5, 30, 20, 20]T , we take ξ(t) as band-limited white noise and the parameters of the disturbance (8) are given by: ⎤ ⎡ 5.0050 × 103 8.000 × 103 6.2300 × 103 − 9.0000 × 103 5.0800 × 104 ⎦ , V = ⎣ 5.0000 × 103 6 3 −9.4620 × 10 6.0800 × 10 − 5.5000 × 103 ⎡

0 3.0000 × 10−1 0 0 W =⎣ 0 3.0000 × 10−1 ⎡

2.1000 × 10−3 ⎢ 2.3000 × 10−3 ⎢ ⎢ 9.0000 × 10−3 B1 = ⎢ ⎢ 2.0000 × 10−3 ⎢ ⎣ 2.0000 × 10−3 1.0000 × 10−3 By placing the poles J1 at ⎡ 0 0.0001 0.0001 L = ⎣0 0 − 0.0001

1.0010 × 10−3 − 3.0000 × 10−3 2.0000 × 10−3 1.0010 × 10−3 2.1000 × 10−3 1.0000 × 10−3

⎤ 0 − 3.0000 × 10−1 ⎦ , 0

⎤T 1.0001 × 10−3 − 4.0000 × 10−3 ⎥ ⎥ ⎥ 0 ⎥ −3 ⎥ . 1.5001 × 10 ⎥ 1.8000 × 10−2 ⎦ 1.0000 × 10−3

[−0.15 − 0.16 − 0.14] in (12), we can obtain ⎤ 0 0.0251 − 0.0488 − 158.5289 0 11.0323 − 51.6379 − 6.4216 ⎦ . 0 33.5301 14.6040 16.9545

Based on Theorem 1, it can be solved that ⎡ −6.6000 × 106 1.0000 × 104 ⎢ 1.0000 × 104 − 1.0300 × 107 ⎢ ⎢ −1.0000 × 104 − 1.0000 × 104 K=⎢ ⎢ −4.3000 × 106 − 1.0000 × 104 ⎢ ⎣ −1.0000 × 104 − 6.5000 × 106 −1.0000 × 104 − 4.4000 × 106

⎤T − 1.0000 × 104 1.1000 × 106 ⎥ ⎥ − 1.0598 × 109 ⎥ ⎥ . − 1.0000 × 104 ⎥ ⎥ − 3.0000 × 105 ⎦ − 2.0506 × 109

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1.5

x 10

X (t) 1

X (t) 2

X (t) 3

X (t) 4

X (t) 5

X (t) 6

1 0.5

State

0 −0.5 −1 −1.5 −2 −2.5 0

5

10

15

Time(Sec)

(a) The response of the system states without control. 30 25

X1(t)

X2(t)

X3(t)

X4(t)

X5(t)

X6(t)

20 15

State

10 5 0 −5 −10 −15 −20 0

5

10

15

Time(Sec)

(b) The response of the system states under DOBADC. 0.5

e (t)

Disturbance estimation error

w1

e (t) w2

e (t) w3

0

−0.5

−1

−1.5

−2 0

5

10

15

Time(Sec)

(c) The response of the disturbance estimation error.

Fig. 2. Responses of composite system.

Comparison of the states between the case without control and the case under DOBADC can be seen in Fig. 2(a), (b). Figure 2(c) shows the response of the estimation errors of disturbance under DO. Figure 2(b) demonstrates that the states of composite system (17) are globally asymptotically stable in probability.

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Conclusions

Aiming at the anti-disturbance control of DP system of ships with stochastic environment disturbances, a DO is designed to estimate the disturbances. Then, the anti-disturbance controller for DP system of ships is designed by combining the disturbance observer, and ensure state in composite system are global asymptotic stability in probability Acknowledgments. This work was supported by the National Natural Science Foundation of China 61973149.

References 1. Xin H, Jialu D, Shi J (2015) Adaptive fuzzy controller design for dynamic positioning system of vessels. Appl Ocean Res 53(4):46–53 2. Sørensen JA (2011) A survey of dynamic positioning control systems. Ann Rev Control 35(1):123–136 3. Fossen TI, Strand JP (1999) Passive nonlinear observer design for ships using Lyapunov methods: full-scale experiments with a supply vessel. Automatica 35(1):3–16 4. Du J, Hu X, Krstic M, Sun Y (2016) Robust dynamic positioning of ships with disturbances under input saturation. Automatica 73(3):207–214 5. Du J, Hu X, Krstic M, Sun Y (2018) Dynamic positioning of ships with unknown parameters and disturbances. Control Eng Pract 76(3):22–30 6. Hu X, Du J, Zhu G, Sun Y (2018) Robust adaptive NN control of dynamically positioned vessels under input constraints. Neurocomputing 318(27):201–212 7. Guo L, Cao SY (2014a) Anti-disturbance control for systems with multiple disturbance. CRC Press, New York 8. Guo L, Cao SY (2014b) Anti-Disturbance control theory for systems with multiple disturbance: a survey. ISA Trans 53(4):846–849 9. Wei XJ, Wu ZJ, Hamid RK (2016) Disturbance observer-based disturbance attenuation control for a class of stochastic systems. Automatica 63(162):21–25 10. Wei XJ, Sun SX (2018) Elegant anti-disturbance control for discrete-time stochastic systems with nonlinearity and multiple disturbances. Int J Control 91(3):706– 714 11. Deng H, Krstic M, Williams R (2001) Stabilization of stochastic nonlinear system driven by noise of unknown covariance. IEEE Trans Autom Control 46(8):1237– 1253 12. Willems JL, Willems JC (1976) Feedback stabilizability for stochastic systems with state and control dependent noise. Automatica 12:277–283 13. Zhang WH, Chen B (2004) On stabilizability and exact observability of stochastic systems with their applications. Automatica 40(1):87–94 14. Mao X, Yuan C (2006) Stochastic differential equations with Markovian switching. Imperial College Press, London

Research on Video Target Tracking Algorithm Based on Particle Filter and CNN Longlong Li, Yihong Zhang(B) , Wuneng Zhou(B) , and Siyao Lv Donghua University College of Information Science and Technology, Shanghai 201620, China [email protected], [email protected]

Abstract. Target recognition and tracking technology has become a core technology for UAVs to visually perceive and understand the battlefield environment. Therefore, this paper proposes an algorithm to study UAVs tracking video targets. In the framework proposed, according to the motion model, the particle filter is used to predict the target position at each frame of the image sequence, the input of the CNN are those particles that round the position predicted, and the adaptive correlation filter is learned on the output of each layer of the CNN to encode the target appearance, and then through a correlation filter maintains the long-term memory of the target’s appearance. Finally, the output of the CNN and the correlation filter is used to determine the particle weights, and the target position of the current sequence of the image sequence is calculated based on the particles and their weight. By using the Visual Tracker Benchmark v1.0 to test and evaluate the algorithm, we can find that the algorithm has good tracking performance. Keywords: UAVs · Video target tracking · Particle filter · CNN · Correlation filter · Visual Tracker Benchmark v1.0

1 Introduction UAVs that appeared in the early 20th century combine the capabilities of reconnaissance surveillance target capture real-time strikes and damage assessment of battlefield targets relying on high sensitivity short working hours and the long-term stay in the air have quickly become the forefront of weapons and equipment in the world. UAVs have broad prospects in the civilian sector due to their low cost strong mobility and survivability. For example, in agriculture pesticides are planted forests are patrolled and inspected and disasters are detected during disasters. It can be seen that in the future war, the status of UAVs in war will become more and more important and the technical level of UAVs will also be related to the victory or defeat of war. The rapid development of image processing technology and imaging hardware devices has greatly improved the integrated military capabilities of UAVs, enabling multiple tasks to be accomplished through the vision system. Target recognition and tracking technology have also become a core technology for UAVs to visually perceive c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 236–244, 2020. https://doi.org/10.1007/978-981-32-9682-4_25

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and understand the battlefield environment. It not only improves the reconnaissance capability of the UAVs, but also makes the UAVs take a big step to the unmanned combat aircraft. When the UAVs can not communicate with the ground station, They can independently understand the battlefield environment make independent planning decision-making and independent control of the launch of the weapon system, and ultimately achieve comprehensive automatic control and operations. Therefore, combined with advanced digital image processing technology and computer vision theory, it explores the autonomous detection and recognition of targets in complex environments and long-term stable tracking become the most important technology of UAVs vision system, and it is also the center of our research. Video-based moving target detection and tracking has a certain research foundation in the development of science and technology and engineering applications [1]. The goal of target tracking is to estimate the unknown state of the target in subsequent frames of the image sequence to achieve more advanced tasks. Despite significant progress in recent years [2–8], target tracking is still challenging because of the large number of appearance changes that cause negative factors including lighting changes occlusions background clutter sudden movements and target removal views. In recent years, researchers have proposed various algorithms to detect and track specific targets in image sequences, such as region-based tracking, feature-based tracking, template-based tracking, and target-based prior knowledge tracking. In the Bayesian filter framework, the target tracking process can be viewed as an adaptive filtering process described by the state space model. Target tracking algorithms based on Bayesian theory include Wiener filter (WF) Kalman filter (KF) and particle filter (PF). Wiener filter is commonly used to solve linear Gaussian stationary signals. It is difficult to find the optimal Bayesian filter directly by calculation, but when the motion of the target conforms to a linear model and the state of the target and the noise conform to the Gaussian distribution, the Kalman filter proves to be the optimal Bayesian filter method. However, these conditions are ideal. The moving targets in the real world are usually nonlinear and non-Gaussian, so it is difficult to directly apply Kalman filtering to achieve target tracking. Many target tracking based on deep learning methods usually make tracking tasks into detection problems in each frame. In [9], there are only two convolutional layers in the proposed CNN tracker. The CNN input uses a strategy that uses three different image hints for per image patch. The CNN’s output strategy is to connect to two fully connected layers to estimate the target location. [10] proposed a CNN tracker called MDNet, in which all domains share three convolutional layers, each of which is assigned a specific layer. In this paper, we use UAVs for video capture to facilitate target tracking of video. Then we use particle filter as the motion model, CNN as the feature extractor and use correlation filter calculate the weights of those particles, what’s more, we use correlation filter maintains the long-term memory of the target’s appearance [11].

2 Algorithm Description In the framework proposed, according to the motion model, the particle filter is used to predict the target position at each frame of the image sequence, the particles around the

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predicted position are used as the input of the CNN, and the adaptive correlation filter is learned on the output of each layer of the CNN to encode the target appearance, and then through a correlation filter maintains the long-term memory of the target’s appearance. Finally, the output of the CNN and the correlation filter is used to determine the particle weights, and the target position of the current sequence of the image sequence is calculated based on the particles and their weight. 2.1

CNN Features

We use the convolution feature map from CNN to encode the target appearance. As features propagate to deeper levels, semantic separation between objects of different categories is enhanced, while spatial resolution is gradually reduced. For UAVs video target tracking, we pay more attention to the precise location of the target, rather than semantic classification, so we removed the fully connected layers in the algorithm because they show a small spatial resolution of 1 × 1 pixel. And only the layered features in the convolutional layer are utilized. When using pooling operations, the spatial resolution of the target to be tracked will gradually decrease as the CNN depth increases. The way we alleviate this problem is to adjust each feature map to a fixed larger size by using bilinear interpolation. The feature vector of the i−th position is: xi = ∑ αik hk

(1)

where the interpolation weight αik are respectively determined by the position of i and k adjacent feature vectors, x denote the upsampled feature map and h denote the feature map. 2.2

Correlation Filter

The correlation filter tracking algorithm has a basic assumption: the more similar the two signals are, the higher the correlation value is, and the target tracking based on the correlation filter is to find the item that has the greatest response to the target to be tracked in the image. In the algorithm of this paper, the output of each convolutional layer is used as a multi-channel feature [8]. Each shifted sample xi, j , (i, j) ∈ {0, 1, . . . , M−1}×{0, 1, . . . , N−1}, where M and N respectively represent the width and height, with Gaussian function label yi j = exp(−(((i − M)/2)2 + (( j − N)/2)2 )/(2σ 2 )), where σ is the kernel width. We set a score y = {yi j |(i, j) ∈ {0, 1, . . . , M − 1} × {0, 1, . . . , N − 1}}. When we get the maximum score, y( M , N ) = 1, indicating that at the target center of zero shift, when the score 2 2 yi j decays rapidly from one to zero, it indicates the position (i, j) gradually away from the target center, we learn the strategy of the correlation filter w with the same size of x to solve the following minimization problem: w∗ = argmin ∑ w · xi j − yi j 2 + λ w22 w

i, j

(2)

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where λ is the regularization parameter (λ ≥ 0). The learned filter for d−th channel of the feature vector is: Wd =

Y  X¯ d i ¯i ∑D d=1 X  X + λ

(3)

where D represents the number of feature channels, d ∈ {1, . . . , D}, Y is the Fourier transformation form of y. For the image patch in the next frame image, the feature vector on the l−th convolution layer is denoted by z, and M × N × D is the size of the feature vector. We obtain the correlation response map of the correlation convolution layer by the following inverse FFT transform formula: D

f (z) = F −1 ( ∑ W d  Z d )

(4)

d=1

2.3 Semantic-to-Space Transform Estimation Given a set of correlation response maps { fl }, we infer the target transition for each ˆ n) ˆ = layer. Let fl (m, n) be the response value of the l−th layer position (m, n), (m, argmax fl (m, n) denotes the position of the maximum value of fl . We locate the target m,n

in the (l − 1) layer by argmax fl−1 (m, n) + μl fl (m, n)

(5)

m,n

2.4 Particle Filter The particle filter (PF) realizes recursive Bayesian filter by non-parametric Monte Carlo simulation, and the accuracy approximates the optimal estimation [12]. The previous posterior distribution at time t: P(x(t)|Yt ) ≈

num

∑ ωi (t)δ (x(t) − xi (t))

(6)

i=1

where xi represents the particles, num represents the total number of particles, and the normalized particle weights is ωi . The target state x(t): x(t) = [u(t), v(t), u(t), ˙ v(t)] ˙ T

(7)

where u(t) and v(t) represent the position of the target on the horizontal and vertical axes in the image, and u(t) ˙ and v(t) ˙ represent the corresponding velocities. In the proposed algorithm, the predicted position x(t) ˆ is calculated by inputting the previous target state x(t − 1) to the motion model. The motion model: x(t) ˆ = Bx(t − 1) + R(t)

(8)

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where R(t) represents the process noise and B represents the process matrix [13]: ⎡ ⎤ 1010 ⎢0 1 0 1⎥ ⎥ B=⎢ ⎣0 0 1 0⎦ 0001

(9)

Then generate an initial particle set xˆi by adding Gaussian noise to the predicted target position. Then after semantic-to-space transform estimation to generate new particle xi . The weight wi of the particle xi is calculated by the sum of all the elements corresponding to the feature map of the particular image patch. Then we use the normalized weight ωi (t) to update the target posterior [12]. Finally, estimate the target position of the current frame based on the following formula [12] x(t) ≈

num

∑ ωi (t)xi (t)

(10)

i=1

2.5

Region Proposals

Using the EdgeBox [14] in this paper, there is a better scale estimation, and it is better to re-detect the target object when the tracking fails: scale proposal BS and Bd with detection recommendations on the entire image. We define b as the candidate bounding box (u, v, w, h), where (u, v) is the center point coordinate and (w, h) is the width and height of the candidate target bounding box. To calculate the confidence score of each candidate region b, we use a correlation filter to preserve the long-term memory of the target’s appearance. Re-detect. Given the estimated target position (ut , vt ) in the t-th frame, we select the patch z centered on the position, and the height and width of this area patch are the same as the estimated target area patch of the previous frame. We can use the threshold T0 to determine if the tracking has failed. If the confidence score g(z) is lower than the threshold T0 , it is judged that the tracking fails and the re-detection target is started. A set of region proposals Bd is generated in the algorithm framework of this paper, and there are big steps in the entire picture to recover the target object. At this time, it is necessary to consider the motion constraint in the case of a drastic change between two consecutive frames, and it is not possible to simply select a candidate region with the greatest confidence as the restored region. The distance D between the current frame and the center position of the previous frame: 1 (uti , vti ) − (ut−1 , vt−1 )2 ) (11) 2σ 2 where σ is the diagonal length of the selected target frame at initialization. Re-detection result can be obtained by minimizing the following issues: D(bti , bt−1 ) = exp(−

argmin g(bti ) + α D(bti , bt−1 )

(12)

g(bti ) > 1.5T0

(13)

i

s.t.

where α acts to balance patch confidence and motion smoothness.

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Scale Estimation. The multi-scale region proposal BS we generated can be used to make scale estimates. When the maximum confidence score {g(b) | b ∈ BS } is greater than g(z), the moving average is used to update the target width wt and height ht : (wt , ht ) = β (wt∗ , ht∗ ) + (1 − β )(wt−1 , ht−1 )

(14)

where wt∗ and ht∗ represent the width and height of the corresponding box, respectively. The effect of the weight factor β is that the size estimate of the target can be smoothly changed. 2.6 Model Update The algorithm framework in this paper can get a better approximation using the moving average update: d Atd = (1 − η )At−1 + η Y  X¯td

(15)

D

d + η ∑ Xti  X¯ti Btd = (1 − η )Bt−1

(16)

i=1

Wtd =

Atd Bdi + λ

(17)

where t represents the video frame number and η represents the learning rate. On the other hand, if the tracking result z has a high confidence score, only the long-term correlation filter needs to be updated.

3 Algorithm Details In the algorithm framework of this paper (see in Algorithm 1), the feature extractor uses VGGNet-19 [15] trained in the dataset ImageNet [16]. The depth feature selects the output of the conv3-4, conv4-4, and conv5-4 convolutional layers. The correlation filter is trained by sharing the parameters on each convolutional layer. In Eq. (2), the λ size is 10−4 and the kernel width is 0.1. The η in Eqs. (14) and (15) is 0.01. The α in the Eq. (12) is 0.1. The β in the Eq. (13) is 0.6. The threshold T0 is 0.2. The number of particles in the particle filter section is set to 50.

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4 Simulation We implemented in the MatConvNet toolbox [17], We tested in the Visual Tracker Benchmark v1.0 [18], including OTB2013 with 50 sequences and OTB2015 with 100 sequences. The Benchmark v1.0 benchmarks trackers for one pass assessment (OPE), spatial robustness assessment (SRE), and time robustness assessment (TRE). We compare our algorithm with some state-of-the-art trackers. The results are in Fig. 1.

Fig. 1. Precision plots of OPE, TRE and SRE and success plots of OPE, TRE and SRE

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Algorithm 1. Proposed tracking algorithm Input: Previous position of the target Output: Estimated object position repeat: Predict x(t) ˆ using Eq.(8) ˆ Additive Gaussian noise to generate an initial particle set (xˆi ) around x(t) Give (xˆi ) to the CNN and then after semantic-to-space transform estimation to get the new xi Get the weight wi of the particle xi by using the sum of all the elements correlation map of the particular image patch Calculate the normalized weight ωi (t) Estimate the target position of the current frame by using Eq.(10) Select new patch z centered the estimated position and calculate the confidence g(z) if g(z) < T0 then continue the target re-detection and update estimated position using Eq.(12) Estimated the new scale using Eq.(13) around the update estimated position and calculate the confidence g(z) if g(z) > T0 then Update g until End of video sequences from UAVs

5 Conclusions The algorithm used in this paper to track the target in the video is to predict the target by particle filter, and then input the particles generated by the particle filter into the CNN to obtain the corresponding three convolution layers, and an adaptive correlation filter is learned on the output of each layer of the CNN to encode the appearance of the object, and then the long-term memory of the target appearance is maintained by the correlation filter. The outputs of the CNN and correlation filters is used to determine the particle weights and to calculate the target position of the current sequence of image sequences based on the particles and their weights. When the estimated target position is obtained, the target position is scored with confidence to determine whether the target tracked is failed, so as to determine whether to re-detect the target and update the model. By testing and evaluating the algorithm using Visual Tracker Benchmark v1.0 with some state-of-the-art methods, we found that our algorithm has good tracking performance. Acknowledgments. This work is partially supported by the National Natural Science Foundation of China (No.61573095).

References 1. Son BJ, Kim YJ, Cho S (2013) Method and terminal for detecting and tracking moving object using real-time camera motion estimation. US, US8611595, pp 33–34 2. Comaniciu D, Ramesh V, Meer P (2003) Kernel-based object tracking. IEEE Trans Pattern Anal Mach Intell 25(5):564–575

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3. Avidan S (2004) Support vector tracking. IEEE Trans Pattern Anal Mach Intell 26(8):1064– 1072 4. Ross DA, Lim J, Lin R-S, Yang M-H (2008) Incremental learning for robust visual tracking. Int. J Comput Vis 77(1–3):125–141 5. Mei X, Ling H (2009) Robust visual tracking using L1 minimization. In: Proceedings of IEEE international conference on computer vision 6. Babenko B, Yang M-H, Belongie S (2011) Robust object tracking with online multiple instance learning. IEEE Trans Pattern Anal Mach Intell 33(8):1619–1632 7. Kalal Z, Mikolajczyk K, Matas J (2012) Tracking-learning-detection. IEEE Trans Pattern Anal Mach Intell 34(7):1409–1422 8. Henriques JF, Caseiro R, Martins P, Batista J (2015) High-speed tracking with kernelized correlation filters. IEEE Trans Pattern Anal Mach Intell 37(3):583–596 9. Li H, Li Y, Porikli F (2016) DeepTrack: learning discriminative feature representations online for robust visual tracking. IEEE Trans Image Process 25(4):1834–1848 10. Nam H, Han B (June 2016) Learning multi-domain convolutional neural networks for visual tracking. In: The IEEE conference on computer vision and pattern recognition (CVPR) 11. Ma C, Huang J-B, Yang X, Yang M-H (2018) Robust visual tracking via hierarchical convolutional features. IEEE Trans Pattern Anal and Mach Intell. https://doi.org/10.1109/TPAMI. 2018.2865311 12. Candy JV (2009) Bayesian signal processing: classical modern and particle filtering methods. Wiley Interscience, New York 13. Mozhdehi RJ, Medeiros H (2017) Deep convolutional particle filter for visual tracking. In: 2017 IEEE International conference on image processing (ICIP) 14. Zitnick CL, Dollar P (2014) Edge boxes: locating object proposals from edges. In: Proceedings of European conference on computer vision 15. Simonyan K, Zisserman A (2015) Very deep convolutional networks for large-scale image recognition. In: Proceedings of international conference on learning representation 16. Deng J, Dong W, Socher R, Li L, Li K, Li F (2009) ImageNet: a largescale hierarchical image database. In: Proceedings of IEEE conference on computer vision and pattern recognition 17. Vedaldi A, Lenc K (2014) MatConvNet C Convolutional Neural Networks for MATLAB. arXiv/1412.4564 18. Wu Y, Lim J, Yang M-H (2013) Online object tracking: a benchmark. In: IEEE conference on computer vision and pattern recognition (CVPR)

Fault Diagnosis and Fault Tolerant Control of High Power Variable Frequency Speed Control System Based on Data Driven Yuedou Pan(B) and Yongliang Li School of Automation and Electrical Engineering, University of Science and Technology Beijng, Beijing 100083, China [email protected]

Abstract. Compared with AC-DC-AC converter, the character of Cycloconverter is no intermediate DC link, small size, light weight and high efficiency of primary power conversion. It is widely used in the main drive speed control system of low-speed, large-capacity hot-rolling and cold-rolling mills. Its safety are very important. Once the Cycloconverter fails, it will result in huge economic losses. At present, there are many researches on fault diagnosis of AC-DC-AC inverter, but the fault diagnosis of Cycloconverter is still in the low stage. Therefore, it is necessary and practical to establish a fault diagnosis system and faulttolerant control method for Cycloconverter. In this paper, a fault diagnosis method based on BP neural network and a fault-tolerant control method of Cycloconverter are designed. The effectiveness of this method is proved by simulation. This provides new methods and ideas for safety issues in industrial production processes. Keywords: Cycloconverter Fault-tolerant control

1

· Fault diagnosis · BP neural network ·

Introduction

At present, high-power synchronous motor drag is used in rolling mill main drive, mine winch, ship propulsion, cement ball mill, etc. [1,2]. Motor-driven mode generally adopts Cycloconverter control. The Cycloconverter uses the power supply voltage to commutate without special conversion circuit. The character of Cycloconverter is no DC link, one conversion and high efficiency. And it is easy to achieve power feedback from the load side to the power side [3]. Failure of the Cycloconverter will cause the motor to run out of control. In severe cases, it will pose a threat to the operator’s personal safety during the production process, causing huge economic losses in industrial production. However, at present, there are many researches on AC-DC-AC converter fault diagnosis and fault-tolerant control. They are mainly based on mathematical analytical model [4], expert system method [5], fault tree model [6], fuzzy theory [7], support Vector machine c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 245–253, 2020. https://doi.org/10.1007/978-981-32-9682-4_26

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[8] and other methods. And fault diagnosis technology and fault-tolerant control method for Cycloconverter are still in the low-level stage. In this paper, the three-phase Cycloconverter is taken as the research object, and the corresponding fault signal is extracted from the output current waveform. The BP neural network is used to realize the accurate positioning of the AC inverter fault, and then the fault-tolerant control is implemented through the designed redundant topology.

2

Fault Analysis of the Cycloconverter

Common faults of the Cycloconverter include: (1) overvoltage; (2) overcurrent; (3) undervoltage; (4) overheating; (5) fuse blown. These faults are mainly caused by faults in electrical components. In terms of reliability, the reliability of computer hardware and software has reached a high level, and the failure of analog components has become the main cause of system failure. According to relevant statistics, eighty percent of control system failures are caused by component failures [9]. Therefore, this paper will research on the fault diagnosis of the thyristor of the Cycloconverter. And because the internal information of the Cycloconverter in production is not easy to obtain, it is chosen to use the input and output voltage or current as the detection object. And the internal fault of the Cycloconverter is diagnosed according to the input and output changes caused by the fault. The thesis selects six-pluse three-phase Cycloconverter as the object, and the main circuit is shown in Fig. 1. The entire Cycloconverter synchronous motor speed control system is modeled and simulated by MATLAB. The simulation model is shown in Fig. 2, where the A phase is expanded as shown in Fig. 3. The no-load starting speed waveform is shown in Fig. 4.

Fig. 1. Three-phase Cycloconverter main circuit

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Fig. 2. Cycloconverter speed control system

Fig. 3. A phase control block diagram

Fig. 4. Speed waveform

The object of this paper is a six-pulse Cycloconverter. There are 36 thyristors in total. The traditional method is to test each thyristor, which takes a long time and has poor effect. In actual production, the more common one is the damage of a thyristor, so this paper first considers the damage of a thyristor, including the thyristor short circuit and open circuit two faults. The Cycloconverter has 36 thyristors, and each 12 thyristors is responsible for one phase of the stator voltage. Every 12 thyristors are identical in structure, so only the 12 thyristors of phase A are simulated for faults. The other two are equally available. When

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the system is in normal operation, short-circuit or open a thyristor at a certain moment to simulate the actual fault. The current waveform changes slowly when the thyristor fails, and the feature is easy to collect, so the current is selected as the diagnostic information. The three-phase current waveform of the previous section of the simulation system during normal operation is shown in Fig. 5. Control AP1 thyristor open circuit in 2.1 s, and the fault current waveform is shown in Fig. 6.

Fig. 5. Normal operation of three-phase current waveform

Fig. 6. AP1 open circuit of three-phase current waveform

It can be seen from the image that after the thyristor AP1 is disconnected, the current signal is distorted, and the different fault waveforms have significant differences. Therefore, the signal characteristics of the corresponding fault can be extracted from the output current waveform.

3

BP Neural Network Model

The BP neural network structure is shown in Fig. 7, and includes an input layer, an hidden layer, and an output layer [10]. Fault location is accomplished by using BP neural network, including pre-collection training samples, mid-term neural network construction, final training neural network, and test neural network. 3.1

Collect Training Samples

Considering the type of fault location can be output by the neural network, this paper uses binary to encode the fault. The highest bit characterizes the fault as

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Fig. 7. BP neural network structure

short circuit or open circuit (0 means short circuit, 1 means open circuit). The second highest bit is characterized by a P-bridge or an N-bridge (when 0, the P-bridge is opened; when it is 1, the N-bridge is opened). The last three digits characterize the thyristor, so a total of 5 bits are used to encode the fault. The fault code is shown in Table 1. Table 1. Fault type code Fault typeAP1 short circuit Code 00001 Fault typeAN1 short circuit Code 01001 Fault typeAP1 open circuit Code 10001 Fault typeAN1 open circuit Code 11001

AP2 short circuit 00010 AN2 short circuit 01010 AP2 open circuit 10010 AN2 open circuit 11010

AP3 short circuit 00011 AN3 short circuit 01011 AP3 open circuit 10011 AN3 open circuit 11011

AP4 short circuit 00100 AN4 short circuit 01100 AP4 open circuit 10100 AN4 open circuit 11100

AP5 short circuit 00101 AN5 short circuit 01101 AP5 open circuit 10101 AN5 open circuit 11101

AP6 short circuit 00110 AN6 short circuit 01110 AP6 open circuit 10110 AN6 open circuit 11110

The characteristic information of the fault waveform is obtained by performing an FFT operation on the output current waveform. Through the statistical study of 24 sets of fault waveforms, the fundamental waveform and the 2-9th harmonic can be taken as input values to fully characterize the fault waveform. Normalize the vector (with the effective value of the output current as the reference value) to form a training sample. The FFT analysis result of the three-phase current waveform output when the AP1 and AN4 thyristors are faulty is a training sample. Table 2 shows the collected data samples. 3.2

Neural Network Construction

This thesis uses MATLAB to build a three-layer BP neural network. According to the fault code of the previous subsection, the output layer has 5 output nodes.

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Table 2. Output FFT analysis results of three-phase current waveforms when AP1 and AN4 thyristors fail (normalized) Harmonic number AP1 short circuit AP4 short circuit A phase B Phase C phase A phase B Phase C phase 1

0.864

0.891

0.956

0.805

0.961

0.875

2

0.185

0.082

0.065

0.191

0.079

0.060

3

0.319

0.207

0.175

0.465

0.113

0.096

4

0.123

0.094

0.068

0.175

0.060

0.018

5

0.293

0.447

0.365

0.231

0.126

0.155

6

0.856

0.036

0.035

0.266

0.013

0.020

7

0.213

0.172

0.115

0. 387

0.237

0.069

8

0.461

0.024

0.022

0.222

0.010

0.015

9

0.061

0.164

0.083

0.102

0.178

0.207

1

0.899

0.942

0.899

0.889

0.945

0.886

2

0.283

0.139

0.156

0.073

0.093

0.080

3

0.154

0.102

0.039

0.146

0.151

0.115

4

0.216

0.072

0.160

0.485

0.113

0.081

5

0.465

0.197

0.292

0.515

0.325

0.363

6

0.693

0.342

0.391

0.219

0.035

0.065

7

0.547

0.343

0.203

0.294

0.167

0.178

8

0.266

0.207

0.060

0.129

0.017

0.019

9

0.0185

0.133

0.109

0.093

0.096

0.075

According to A.J. Maren et al. [4], the optimal number of neural units in the hidden layer should be equal to the geometric mean of the number of input and output neural units. That is, the number of hidden layer units J, the number of input layer units √ N, and the number of output layer units M satisfy the following formula J = M + N . So the hidden layer node is selected as 7. According to many experiments and related experience, the first layer learning rate is 1.1, and the second layer learning rate is 1.3. Train the neural network through the information in Table 2. After training more than 500 times, the squared error of the output of the neural network can finally reach 0.03, which meets the requirements. After the training is finished, input the data of Table 2 to the network to get the diagnosis result. The results are shown in Table 3. 3.3

Results and Test

Fault Diagnosis flow diagram shown in Fig. 8. The FFT analysis result of the three-phase current waveform outputted when the thyristor AP5 is open and the AN6 is short-circuited is selected as a test sample and input to the trained BP neural network. Get the following results (Table 4):

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Table 3. This is the example table taken out of The TEXbook, p. 246 Fault type AP1 short circuit AN4 short circuit AP1 open circuit AN4 open circuit

Expected output Actual output 1 2 3 00001 0.062 0.072 0.055 01100 0.068 0.956 0.975 10001 0.964 0.025 0.042 11100 1.000 0.973 0.979

Sum of squared errors 4 0.048 0.025 0.066 0.015

5 0.944 0.047 0.985 0.062

0.01749 0.01002 0.00827 0.00524

Fig. 8. Fault diagnosis flow diagram Table 4. Test results Fault type

Expected output Actual output Sum of squared errors 1 2 3 4 5 AP5 open circuit 10101 0.945 0.047 0.933 0.063 0.956 0.01562 AN6 short circuit 01110 0.075 0.929 0.915 0.923 0.069 0.02858

The experimental results show that the BP neural network Cycloconverter fault diagnosis system built in this paper meets the requirements of the test sample diagnostic error, and the fault diagnosis result is consistent with the expected output, which realizes the accurate diagnosis of the fault location type. The proof method is feasible for fault diagnosis of the Cycloconverter.

4

Fault Tolerant Control

A fast fuse is usually added to the thyristor structure, so the short circuit fault is equivalent to the open fault of the thyristor. This thesis uses redundant structure for fault tolerant control. The redundancy of the switch tube is mostly used in a highly reliable drive system. After a switch tube fails, the redundant switch tube is put into use. Figure 9 shows a redundant bridge arm topology. In the

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absence of a thyristor fault, the redundant bridge arm does not function. After a thyristor failure, the corresponding single-pole double-throw switch is switched to the redundant phase to achieve the purpose of fault-tolerant control.

Fig. 9. Fault tolerant control structure diagram

After AP1 is opened in 2.1 s, after diagnosing the fault location and type, the single-pole double-throw switch is switched to the redundant phase to achieve fault-tolerant control. The current waveform is shown in the following Fig. 10.

Fig. 10. Current waveforms of fault diagnosis and tolerant control

5

Conclusions

In this paper, a fault diagnosis method based on BP neural network is designed for the case of IGBT open circuit and short circuit fault in high power Cycloconverter. By analyzing the faulty output current waveform, the method can determine the location of the thyristor and the type of fault that the fault occurred. This provides a new method for troubleshooting the thyristor of the Cycloconverter, which saves time. From the simulation results, this method is feasible for fault diagnosis of the Cycloconverter. This paper also designed a redundant thyristor structure for fault-tolerant control of the Cycloconverter, and the effectiveness of the method is proved by simulation. In summary, the data-driven

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high-power variable-frequency drive system fault diagnosis and fault-tolerant control method designed in this paper has the advantages of high diagnostic accuracy, fast diagnosis speed, stable and timely fault-tolerant control. It provides methods and ideas for safety and security issues in industrial production.

References 1. Stemmle H (1994) High-power industrial drives. Proc IEEE 82(8):1266–1286 2. Bhooplapur P (1997) AC drives in metal industries. Power Electron Drive Syst 2:823–828 3. Ma X (1999) High-power Cycloconverter control and vector control technology. Mechanical Industry Press, Beijing, pp 1–11 4. Jing HB, Nian XH, Fan W (2009) Circuit topology and mathematics model of cascaded inverter. High Power Convert Technol (3):10–15 5. Wang X, Sun HN (2015) Fault diagnosis of cascaded inverter based on PSO and neural network. Comput Simul 32(7):421–425 6. Wang XY, Chen T (2011) A fault diagnosis method for inverter based on BAM neural network and fault tree. Electron Opt Control 18(5):85–89 7. Xu CH, Niu JG (2011) Fault diagnosis system of frequency converter used in coal mining machine based on fuzzy neural network. Coal Mine Mach 32(5):236–238 8. Li H, Wang FZ, Wang R (2017) Fault diagnosis for cascaded converter with characteristic entropy of wavelet packet. Meas Control Technol (6):20–23 9. Chiba A, Fukao T, Ichikawa O et al (2005) Magnetic bearings and bearingless drives. Newnes, Tokyo 10. Karayiannis NB, Venetsanopoulos AN (1996) Efficient learning alogrithms for neural networks. IEEE Trans Neural Netw 5(3):498–501

Leader-Following Consensus of Discrete-Time Multi-agent Systems with a Smart Leader Shuang Liang1,2 , Fuyong Wang1,2 , Zhongxin Liu1,2(B) , and Zengqiang Chen1,2 1

College of Artificial Intelligence, Nankai University, Tianjin 300350, China [email protected] 2 Tianjin Key Laboratory of Intelligent Robotics, Nankai University, Tianjin 300350, China

Abstract. In this paper, the consensus problem of discrete-time general linear multi-agent systems with a smart leader is studied. Unlike the previous works, an objective function is designed to decide whether the leader can receive information from the followers, with the purpose of effectively reducing the controller’s energy consumption. In order to track the desired target state, a new distributed control protocol is proposed for the multi-agent systems. By utilizing the Lyapunov function technology, some new sufficient conditions are established, which can ensure the leader-following consensus for discrete-time multi-agent systems with fixed topology and switching topology. In addition, the corresponding gain matrices are also obtained. Finally, simulation results are provided to demonstrate the theoretical results. Keywords: Discrete-time multi-agent systems · Leader-following consensus · Smart leader · Feedback information Energy consumption

1

·

Introduction

Over the past decades, coordination control in multi-agents systems has been widely concerned by researchers, because of its unique applications in different fields, such as mobile robot, formation control and so on (see e.g. [1,2]). The consensus plays a very important role in coordination control of multi-agents systems, whose aim is to find a suitable control algorithm such that the agents eventually reach a common state. There have been many interesting results in the study of the consensus in multi-agent systems, see [3–5] and reference therein. One of the most basic and important types of consensus is leader-following consensus, in which the agents achieve consensus by tracking the leader’s trajectory. Even in leaderless situations [6,7], virtual leaders are often used as a tracking target. It is worth noticing that the dynamics of agents in multi-agent systems plays an important role. Generally, intelligent agents have first-order dynamics, second-order dynamics and even higher-order dynamics. Nevertheless, it is c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 254–265, 2020. https://doi.org/10.1007/978-981-32-9682-4_27

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also very meaningful to study the consensus of multi-agent systems with general linear dynamics [8,9]. For many practical systems, due to limitations of measuring equipment or implementation costs, states of agents cannot be completely measured but only output its neighboring information. Design an observer to evaluate those unmeasurable variables is very necessary. In [10], distributed fault detection and isolation problem for first-order multi-agent systems with uncertainty was investigated, in which a sliding-mode observer was designed. Based on sampled position data, under the assumption of directed topology, the consensus of second-order multi-agent systems was discussed in [11]. Furthermore, there also have been many researches on the consensus of general linear multi-agent systems. In [12], in light of the relative output information, authors addressed the consensus problem of continuous-time and discrete-time general linear multiagent systems and presents two reduced-order protocol algorithms. In [13], the consensus problem of linear multi-agent systems was further studied which had been considered by [12] and a new type distributed reduced-order state-observer was provided. Notably, most of the researches aforementioned concern with the leaderless consensus or common leader-following consensus. In common leader-following configuration, the leader’s decision or behavior is not affected by followers. Different from the traditional leader, a smart leader who can obtain feedback from neighboring followers was proposed in [5,14]. On the basis of the above work, this paper is devoted to considering the leader-following consensus problem with a smart leader and extends their works to discrete-time general linear multi-agent systems. In practice, less energy consumption is expected. When certain conditions are met, the application of smart leader can lead to less energy consumption in the controller. Compared with [5,14], an objective function is designed to automatically select a way with less energy consumption. The main advantages of this paper can be listed as the following three points: (1) A novel leader-following relationship is introduced for general linear multi-agent systems. Different from the previous works, the leader can selectively obtain feedback from neighboring followers based on the designed objective function, which can effectively reduce the energy consumption of the controller. (2) In order to track the desired target state, a new distributed consensus protocol is proposed. Based on matrix transformation, the consensus problem is cast into a robust control problem. (3) By applying Lyapunov function and linear matrix inequality (LMI) technology, new sufficient conditions are established, which can ensure the consensus for discrete-time multi-agent systems with fixed topology and switching topology. The corresponding gain matrices are also obtained. The remainder of this article is arranged as follows. Section 2 gives some preliminaries. In Sect. 3, the main results for discrete multi-agent systems are obtained. In Sect. 4, simulation examples are provided to demonstrate the theoretical results. Section 5 gives some conclusions. Throughout this paper, N + means the set of all non-negative integers and N denotes the set of natural number. Rn and Rn×m represent the n-dimensional real space and the set of all n × m real matrices, respectively. AT denote the

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matrix transpose. The symmetric matrix P > 0(< 0) stands for that the matrix P is positive (negative) definite. For a square matrix X, sym(X) is marked as sym(X) = X + X T . In means the n × n identity matrix. The asterisk ∗ represents the symmetric term in a matrix. ⊗ represents the Kronecker product. Some properties of Kronecker product are useful: (A ⊗ B)T = AT ⊗ B T , (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD), A ⊗ B + A ⊗ C = A ⊗ (B + C).

2 2.1

Problem Statement Preliminaries

¯ suppose that the multi-agent system In the interconnection topology graph G, includes N followers and one leader. The leader is labeled as v0 . Let G = {V, ε, A} describe the topology of the followers, where V = {ν1 , ν2 , · · · , νN } is the nodes set, ε ⊆ V ×V is the edges set and A = (aij )N ×N represents a weighted adjacency matrix, in which the elements aij > 0 when (νj , νi ) ∈ ε, otherwise, aij = 0 and aii = 0 for all i ∈ V . In graph G, node νi denotes the i-th agent. The neighbor set of agent i is represented by Ni = {j ∈ V : (νj , νi ) ∈ ε}. The Laplacian matrix L N is represented by lii = j=1,j=i aij and lij = −aij , i = j. Denote a column vector ¯ = [b1 , b2 , · · · , bN ]T and a matrix H = L + D with D = diag{b1 , b2 , · · · , bN }, D if bi > 0, i ∈ {1, 2, · · · , N }, agent i and the leader v0 are connected, otherwise bi = 0. Furthermore, when the communication topology graph is variable, define a switching signal σ : {1, 2, · · · , k, · · · } → ϕ = {1, 2, · · · , z}, where z represents the number of the switching topology. Because the communication topology is time-varying, the adjacent elements aij (k), the neighbor set Ni (k), Laplacian matrix Lσ (σ ∈ ϕ) and Dσ (σ ∈ ϕ) are time-varying. The dynamics of the i-th agent in the discrete-time multi-agent system has the following form:  xi (k + 1) = Axi (k) + Bui (k), (1) yi (k) = Cxi (k), i = 0, 1, 2, · · · , N, where xi (k) ∈ Rn and ui (k) ∈ Rm represent respectively agent i the state vector and the control input, yi (k) ∈ Rp is the measured output of agent i. x0 (k) ∈ Rn , u0 (k) ∈ Rm and y0 (k) ∈ Rp represent the state vector, the control input and the measured output of leader, respectively. A ∈ Rn×n , B ∈ Rn×m , C ∈ Rp×n are constant matrices. Here, define a desired trajectory that only known by the leader:  xd (k + 1) = Axd (k), (2) yd (k) = Cxd (k). Then the leader-following multi-agent system reaches tracking consensus, if lim xi (k) − xd (k) = 0, ∀i = 0, 1, · · · , N.

k→∞

The following assumptions and lemmas will be used in this paper.

(3)

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Assumption 1. The matrix pair (A, B) is stabilizable and (A, C) is detectable. Assumption 2. Assume that the communication topology is connected. Followers are connected by undirected edges, while the smart leader and followers are connected by directed edges.   S11 S12 T T Lemma 1. For a given matrix S = , S22 = S22 , then with S11 = S11 ∗ S22 S < 0 if and only if −1 T T −1 S11 < 0, S22 − S12 S11 S12 < 0 or S22 < 0, S11 − S12 S22 S12 < 0. Lemma 2 ([15]). If (A, C) is detectable and Q is symmetric positive definite matrix, then there exists a unique positive definite solution P satisfying the following Riccati equation AP AT − P − AP C T (I + CP C T )−1 CP AT + Q = 0.

(4)

Lemma 3 ([9]). For a real scalar ρ > 0, real matrices Λ, Ui , Vi and Wi (i = 1, 2, · · · , N ), if the following condition holds   ··· UN + ρVN Λ U1 + ρV1 < 0, ∗ diag{−ρW1 − ρW1T , · · · , −ρWN − ρWNT } then we have Λ +

N i=1

sym(Ui Wi−1 ViT ) < 0.

Lemma 4 ([16]). For a switching system, if there exists a common Lyapunov function for each subsystem, the stability of switching system can be ensured under arbitrary switching. 2.2

Objective Function Design

In traditional leader-following configuration, the leader can affect the followers, while there is no feedback between followers and the leader. In practice, some followers may fail due to external reasons. Therefore, the leader needs to obtain feedback from the followers to adjust the tracking errors between them. The leader who can receive feedback from the followers is called smart leader. In the protocol of leader, the smart leader can be used or not. So we design an objective function to decide whether the leader kneed to receive information from the followers. For the agent i, let Ei (k) = s=0 u2i (s) denote the energy consumption at time k, i = 1, 2, 3, 4, 5. Then, the leader’s energy consumption, namely the objective function, can be defined as ⎧ k

⎪ ⎪ ⎪ J1 (k) = ⎪ u201 (s), ⎪ ⎨ s=0

k ⎪

⎪ ⎪ ⎪ J (k) = u202 (s), ⎪ ⎩ 2 s=0

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where u01 and u02 indicate that the leader does not use or use feedback information in the control protocol, respectively. Our aim is to ensure that  there is less 0, J1 ≤ J2 , energy consumption, so we introduce a symbolic function η(J) = 1, J1 > J2 .

3

Main Results

In this section, a new distributed consensus protocol is adopted and some sufficient conditions of leader-following consensus are presented. Due to the limitation of measuring equipment or implementation costs, the agents’ states are not generally measurable. To deal with this problem, we design the following state-observer for system (1): xi (k) + Bui (k) + G(yi (k) − yˆi (k)), x ˆi (k + 1) = Aˆ

i = 0, 1, 2, · · · , N,

(5)

where x ˆi (k) and yˆi (k) are respectively used to estimate agent i state and measured output and G ∈ Rn×p is the gain matrix. The output control protocol for follower i is given as follows:

aij (ˆ xj (k) − x ˆi (k)) + bi (ˆ x0 (k) − x ˆi (k))], i = 1, 2, · · · , N, (6) ui (k) = K[ j∈Ni

where K ∈ Rm×p is the consensus control gain matrix. For the smart leader labelled as i = 0, the control law can be given as

xd (k) − x ˆ0 (k))] + K[ a0j (ˆ xj (k) − x ˆ0 (k))]η(J), (7) u0 (k) = γK[(ˆ j∈N0

where γ is given positive scalar. ˆi (k), and x ˜i (k) = x ˆi (k) − xd (k), i = 0, 1, · · · , N. Denote x ¯i (k) = xi (k) − x ˜i (k) are described as follows x ¯i (k + 1) = Then, the error dynamics of x ¯i (k) and x ˜i (k + 1) = A˜ xi (k) + Bui (k) + GC x ¯i (k). Let x ¯(k) = (A − GC)¯ xi (k) and x ¯T1 (k), · · · , x ¯TN (k)]T and x ˜(k) = [˜ xT0 (k), x ˜T1 (k), · · · , x ˜TN (k)]T . Now we con[¯ xT0 (k), x sider the following two cases. If η(J) = 0, the error dynamics can be reformulated in a matrix form  x(k), x ¯(k + 1) = (IN +1 ⊗ (A − GC))¯ (8) x ˜(k + 1) = (IN +1 ⊗ A − M1 ⊗ BK)˜ x(k) + (IN +1 ⊗ GC)¯ x(k), 

 γ 0 ¯ L+D . −D If η(J) = 1, the leader can obtain feedback from neighboring followers to adjust its own state. Hence, an interconnected relationship is established between ˆ the leader and followers and the corresponding graph is denoted by graph G. ˆ ˆ Obviously, graph G is connected. The Laplacian matrix of the graph G is repreˆ The error dynamics can be reformulated as sented by L. where M1 =



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x(k), x ¯(k + 1) = (IN +1 ⊗ (A − GC))¯ x ˜(k + 1) = (IN +1 ⊗ A − M2 ⊗ BK)˜ x(k) + (IN +1 ⊗ GC)¯ x(k),

(9)

ˆ + γΓ , Γ = diag{1, 0, · · · , 0} ∈ R(N +1)×(N +1) . where M2 = L The dynamics of (8)–(9) can be described in a compact form  η(J) = 0, ξ(k + 1) = F1 ξ(k), ξ(k + 1) = F2 ξ(k), η(J) = 1,

(10)

 T T ¯ (k) x ˜T (k) , where ξ(k) = x  IN +1 ⊗ (A − GC) 0 Fi = , i = 1, 2. IN +1 ⊗ A − Mi ⊗ BK IN +1 ⊗ GC Theorem 1. Suppose Assumptions 1–2 hold. The leader-following multi-agent system (1) can reach tracking consensus via the control laws (6) and (7), if there exist P˜ > 0, an invertible matrix X and any matrix Y satisfying the following LMIs and construct the gain matrices G = AP C T (I + CP C T )−1 where P is the solution of Riccati equation (4) and K = X −1 Y : ⎡ ⎤ ˜ 11 Π 0 0 IN +1 ⊗ C T GT P˜ ⎢ ∗ −IN +1 ⊗ P˜ IN +1 ⊗ AT P˜ − M T ⊗ Y T B T ⎥ −M1T ⊗ Y T 1 ⎢ ⎥ < 0, ⎣ ∗ ˜ ˜ ∗ −IN +1 ⊗ P IN +1 ⊗ ρ(P B − BX) ⎦ ∗ ∗ ∗ −ρX − ρX T (11) ⎡ ⎤ ˜ 11 Π 0 0 IN +1 ⊗ C T GT P˜ ⎢ ∗ −IN +1 ⊗ P˜ IN +1 ⊗ AT P˜ − M T ⊗ Y T B T ⎥ −M2T ⊗ Y T 2 ⎢ ⎥ < 0, ⎣ ∗ ∗ −IN +1 ⊗ P˜ IN +1 ⊗ ρ(P˜ B − BX) ⎦ ∗

∗

∗

−ρX − ρX T

(12) ˜ 11 = IN +1 ⊗ ((A − GC)T P¯ (A − GC) − P¯ ). where Π Proof. Choose the following Lyapunov functional candidate for the system (10): V (k) = ξ T (k)T ξ(k),   IN +1 ⊗ P¯ 0 where T = , P¯ and P˜ are symmetric positive definite ∗ IN +1 ⊗ P˜ matrices. For η(J) = 0, the difference of V (k) is given by ΔV (k) = ξ T (k + 1)T ξ(k + 1) − ξ T (k)T ξ(k) = ξ T (k)(F1T T F1 − T )ξ(k) T

= ξ (k)Ω1 ξ(k),  where Ω1 =

 Π11 (IN +1 ⊗ (GC)T )(IN +1 ⊗ P˜ )(IN +1 ⊗ A − M1 ⊗ BK) , ∗ Π22

(13)

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Π11 = IN +1 ⊗ ((A − GC)T P¯ (A − GC) − P¯ + (GC)T P˜ (GC)), Π22 = (IN +1 ⊗ A − M1 ⊗ BK)T (IN +1 ⊗ P˜ )(IN +1 ⊗ A − M1 ⊗ BK) − IN +1 ⊗ P˜ . Applying Lemma 1, we can obtain that Ω1 < 0 if and only if ⎤ ⎡ ˜ 11 Π 0 IN +1 ⊗ C T GT P˜ ¯1 = ⎣ ∗ −IN +1 ⊗ P˜ IN +1 ⊗ AT P˜ − M T ⊗ K T B T P˜ ⎦ < 0. Ω (14) 1 ∗ ∗ −IN +1 ⊗ P˜ Since Q is symmetric positive definite and (A, C) is detectable, then by using the gain matrix G = AP C T (I + CP C T )−1 and Lemma 2, we have (A − GC)P (A − GC)T − P = AP AT − 2AP C T GT + GCP C T GT − P = AP AT − P − AP C T (I + CP C T )−1 CP AT + AP C T (I + CP C T )−1 [CP C T (I + CP C T )−1 − I]CP AT

(15)

= AP AT − P − AP C T (I + CP C T )−1 CP AT − GGT ≤ AP AT − P − AP C T (I + CP C T )−1 CP AT = −Q < 0.   −P A − GC So from (15), we can get < 0. ∗ −P −1 According Lemma 1, one obtain ˜ 11 = IN +1 ⊗ ((A − GC)T P¯ (A − GC) − P¯ ) < 0, Π where P¯ = P −1 . Using the gain matrix K = X −1 Y , it can be established that M1T ⊗ K T B T P˜ = M1T ⊗ Y T B T + M1T ⊗ [Y T (X −1 )T (B T P˜ − X T B T )]. Furthermore, (14) can be rearranged as ⎤ Π11 0 IN +1 ⊗ C T GT P˜ ¯1 = ⎣ ∗ −IN +1 ⊗ P˜ IN +1 ⊗ AT P˜ − M T ⊗ Y T B T ⎦ + sym(U1 W −1 V T ) < 0, Ω 1 1 1 ∗ ∗ −IN +1 ⊗ P˜ (16) T  T where U1 = 0 −M1 ⊗ Y 0 , W1 = IN +1 ⊗ X ,  T V1 = 0 0 IN +1 ⊗ (B T P˜ − X T B T ) . If (11) holds, then by Lemma 3, (16) can be satisfied, which implies Ω1 < 0. For η(J) = 1, the proof is similar to η(J) = 0, which is omitted. According to (12), the consensus of leader-following multi-agent system can be guaranteed when the leader receives information from its neighboring followers. Therefore, the leader-following multi-agent system can reach tracking consensus. The proof of this theorem is completed. ⎡

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It should be noted that the actual communication topology may be changed. As a result, the consensus problem with a smart leader will be discussed for discrete-time multi-agent system under switching topology. Theorem 2. Suppose that Assumptions 1–2 hold and the communication topology is time-invariant in [τr , τr+1 ), r ∈ N, where τr is a positive integer, 0 = τ0 < τ1 < · · · , and switches at time τr , r = 0, 1, · · · , for each graph ¯ σ . Consider the leader-following multi-agent system (1) via the control laws G (6) and (7). The tracking consensus with a smart leader can be guaranteed for multi-agent system under switching topology, if there exist P˜ > 0, an invertible matrix X and any matrix Y satisfying the following LMIs and construct the gain matrices G = AP C T (I + CP C T )−1 where P is the solution of Riccati equation (4) and K = X −1 Y : ⎡

⎤ ˆ 11 Π 0 0 IN +1 ⊗ C T GT P˜ T ⎢ ∗ −IN +1 ⊗ P˜ IN +1 ⊗ AT P˜ − M T ⊗ Y T B T ⎥ −M1σ ⊗YT 1σ ⎢ ⎥ < 0, ⎣ ∗ ˜ ˜ ∗ −IN +1 ⊗ P IN +1 ⊗ ρ(P B − BX) ⎦ ∗ ∗ ∗ −ρX − ρX T (17) ⎡ ⎤ ˆ 11 Π 0 0 IN +1 ⊗ C T GT P˜ T ⎢ ∗ −IN +1 ⊗ P˜ IN +1 ⊗ AT P˜ − M T ⊗ Y T B T ⎥ −M2σ ⊗YT 2σ ⎢ ⎥ < 0, ⎣ ∗ ∗ −IN +1 ⊗ P˜ IN +1 ⊗ ρ(P˜ B − BX) ⎦ ∗ ∗ ∗ −ρX − ρX T (18)   γ 0 T ¯ ˆ ¯ where Π11 = IN +1 ⊗ ((A − GC) P (A − GC) − P ), M1σ = ¯ σ Lσ + Dσ , −D ˆ σ + γΓ. M2σ = L The system (10) under the switching topology can be reformulated as  ξ(k + 1) = F1σ ξ(k), η(J) = 0, (19) ξ(k + 1) = F2σ ξ(k), η(J) = 1, 

 IN +1 ⊗ (A − GC) 0 , i = 1, 2. IN +1 ⊗ GC IN +1 ⊗ A − Miσ ⊗ BK The process of proof is almost the same as Theorem 1, thus omitted.

where Fiσ =

Remark 1. Based on the state observer, some sufficient conditions, under which the multi-agent systems can reach tracking consensus, are established in Theorems 1 and 2. The consensus problems with a smart leader for continuous-time multi-agent systems were investigated in [5,14]. This paper extends their works to discrete-time general linear multi-agent systems, which is challenging. Compared with [12,13], the smart leader is adopted, which can effectively reduce the energy consumption of the controller and have a broader range of applications.

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Numerical Examples

In this part, the simulation results will be presented to illustrate the validity of the main results. Assume that there is a smart leader and five followers in system (1). The ¯ 1 and switches randomly to another network topology starts with the graph G ¯ ¯ ¯ ¯1 − G ¯ 3 , we get within {G1 , G2 , G3 } as shown in Fig. 1. By the topologies of G ⎡

3 ⎢ −3 ⎢ L1 = ⎢ ⎢ 0 ⎣ 0 0

−3 6 −3 0 0

0 −3 6 −3 0

0 0 −3 6 −3

⎤ ⎡ 0 3 ⎢ 0 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎥ , L2 = ⎢ −3 ⎦ ⎣ 0 −3 3 0

0 6 −3 −3 0

−3 −3 6 0 0

0 −3 0 6 −3

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ , D2 = diag{1, 0, 1, 0, 0}, −3 ⎦ 3

⎡

⎤ 3 0 0 0 −3 ⎢ 0 3 0 −3 0 ⎥ ⎢ ⎥ ⎥ L3 = ⎢ ⎢ 0 0 6 −3 −3 ⎥ , D1 = D3 = diag{1, 0, 0, 0, 1}. ⎣ 0 −3 −3 6 0 ⎦ −3 0 −3 0 6 ˆ1 − G ˆ 3 are The Laplacian matrices corresponding to topologies G ⎤ ⎤ ⎡ ⎡ 2 −1 0 0 0 −1 2 −1 0 −1 0 0 ⎢ −1 4 −3 0 0 0 ⎥ ⎢ −1 4 0 −3 0 0 ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ 0 −3 6 −3 0 0 ⎥ ⎢ 0 0 6 −3 −3 0 ⎥ ˆ ˆ ⎥ ⎥ ⎢ ⎢ L1 = ⎢ ⎥ , L2 = ⎢ −1 −3 −3 7 0 0 ⎥ , ⎥ ⎢ 0 0 −3 6 −3 0 ⎥ ⎢ ⎣ 0 0 0 −3 6 −3 ⎦ ⎣ 0 0 −3 0 6 −3 ⎦ −1 0 0 0 −3 4 0 0 0 0 −3 3 ⎤ 2 −1 0 0 0 −1 ⎢ −1 4 0 0 0 −3 ⎥ ⎥ ⎢ ⎥ ⎢ ˆ 3 = ⎢ 0 0 3 0 −3 0 ⎥ . L ⎢ 0 0 0 6 −3 −3 ⎥ ⎥ ⎢ ⎣ 0 0 −3 −3 6 0 ⎦ −1 −3 0 −3 0 7 ⎡

Given the following parameters: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0.2 0.5 0.1 0.1 0.5 0.1 0.1 0 0 1 0.5 0 A = ⎣ 0.5 0.1 0 ⎦ , B = ⎣ 0.5 0.1 0 ⎦ , C = ⎣ 0 0.1 0 ⎦ , Q = ⎣ 0.5 1 0 ⎦ , 0 0 0.1 0 0 0.1 0 0 0.1 0 0 1 γ = 2.5, ρ = 1.2. By solving Riccati equation (4), we can obtain positive definite solution and gain matrix as follows ⎡ ⎤ ⎡ ⎤ 1.6373 0.9951 0.0102 0.0804 0.0933 0.0102 P = ⎣ 0.9951 1.5116 0.0005 ⎦ , G = ⎣ 0.0897 0.0630 0.0005 ⎦ . 0.0102 0.0005 1.0100 0.0001 0.0000 0.0100

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Then, the solutions of the LMIs (17)–(18) is ⎡ ⎤ ⎡ ⎤ 7551.23 −302.99 −23.02 4432.18 274.13 63.41 P˜ = ⎣ −302.99 7788.04 197.72 ⎦ , X = ⎣ 274.13 4402.76 95.62 ⎦ , −23.02 197.72 6962.43 63.41 95.62 5093.65 ⎡ ⎤ 286.1794 95.6802 8.4084 Y = ⎣ 95.6802 276.3263 30.3864 ⎦ . 8.4084 30.3864 57.0279 Applying the formula K = X −1 Y , the controller gain matrix can be obtained as ⎡ ⎤ 0.0635 0.0177 0.0013 K = ⎣ 0.0178 0.0616 0.0066 ⎦ . 0.0005 0.0046 0.0111

Fig. 1. Three possible graphs of the switching topology. Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Leader

2

-2

10

z i (k)

z i (k)

0

-4 -6

Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Leader

5

0

-8 -10 15

-5 5 10 5

-5

0 -10

0 -5

5 -5

5 0

y i (k)

0

10

y i (k)

x i (k)

Fig. 2. The errors trajectories x ¯i , i = 0, 1, · · · , 5.

of

-5 -15

-10

x i (k)

Fig. 3. The errors trajectories of x ˜i , i = 0, 1, · · · , 5.

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70 85

60 50

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40 75 2

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20 The energy comsumption of the common leader The energy comsumption of the smart leader

10 0 0

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Fig. 4. The curves of energy consumption for two kinds of multi-agent systems.

Simulated pictures are shown in Figs. 2, 3 and 4. Furthermore, The state ˜i (k) (i = 0, 1, · · · , 5) for discrete multierrors of x ¯i (k) (i = 0, 1, · · · , 5) and x agent systems are depicted in Figs. 2 and 3, respectively. It is easy to observe that the errors between the smart leader and followers approaches zero. From Fig. 4, we can see that energy consumption of the controller with a smart leader is lower than energy consumption with traditional leader. Thus, it is of both theoretical value and practical significance to introduce a smart leader in the discrete-time multi-agent systems.

5

Conclusions

This paper investigates the consensus problem with a smart leader for discretetime multi-agent systems. Different from the traditional leader, according to the designed objective function, the leader can selectively obtain information from neighboring followers, which can effectively reduce the energy consumption of the controller. Based on reference model, a new distributed control law is provided for the multi-agent systems. By matrix transformation, the leaderfollowing consensus control problem is cast into a robust control problem. In light of the Lyapunov function technology, some new sufficient conditions are established to guarantee the consensus of multi-agent systems under fixed topology and switching topology. Moreover, the corresponding gain matrices are also obtained. In the future work, the consensus of the discrete-time heterogeneous multi-agent systems via a smart leader will be discussed. Acknowledgments. This work was supported by the National Natural Science Foundation of China (Grant No. 61573200, 61573199).

References 1. Huang JS, Wen CY, Wang W, Jiang ZP (2014) Adaptive output feedback tracking control of a nonholonomic mobile robot. Automatica 50(3):821–831. https://doi. org/10.1016/j.automatica.2013.12.036

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2. Lin F, Fardad M, Jovanovic MR (2012) Optimal control of vehicular formations with nearest neighbor interactions. IEEE Trans Autom Control 57(9):2203–2218. https://doi.org/10.1109/TAC.2011.2181790 3. Olfati-Saber R, Murray RM (2004) Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Autom Control 49(9):1520–1533. https://doi.org/10.1109/tac.2004.834113 4. Yu WW, Chen GR, Cao M, Kurths J (2009) Second-order consensus for multiagent systems with directed topologies and nonlinear dynamics. IEEE Trans Syst Man Cybern-Part B: Cybern 40(3):881–891. https://doi.org/10.1109/TSMCB. 2009.2031624 5. Wang FY, Liu ZX, Chen ZQ (2018) A novel leader-following consensus of multiagent systems with smart leader. Int J Control Autom Syst 16(4):1483–1492. https://doi.org/10.1007/s12555-017-0266-0 6. Song Q, Cao JD, Yu WW (2010) Second-order leader-following consensus of nonlinear multi-agent systems via pinning control. Syst Control Lett 59(9):553–562. https://doi.org/10.1016/j.sysconle.2010.06.016 7. Cao YC, Ren W (2012) Distributed coordinated tracking with reduced interaction via a variable structure approach. IEEE Trans Autom Control 57(1):33–48. https://doi.org/10.1109/tac.2011.2146830 8. Wu J, Li HQ, Chen X (2017) Leader-following consensus of nonlinear discrete-time multi-agent systems with limited communication channel capacity. J Franklin Inst 354(10):4179–4195. https://doi.org/10.1016/j.jfranklin.2017.03.005 9. Zhou JP, Sang CY, Li X, Fang MY, Wang Z (2018) H∞ consensus for nonlinear stochastic multi-agent systems with time delay. Appl Math Comput 325:41–58. https://doi.org/10.1016/j.amc.2017.12.020 10. Quan Y, Chen W, Wu ZH, Peng L (2018) Distributed fault detection and isolation for leader-follower multi-agent systems with disturbances using observer techniques. Nonlinear Dyn 93(2):863–871. https://doi.org/10.1007/s11071-018-4232z 11. Huang N, Duan ZS, Chen GR (2016) Some necessary and sufficient conditions for consensus of second-order multi-agent systems with sampled position data. Automatica 63:148–155. https://doi.org/10.1016/j.automatica.2015.10.020 12. Li ZK, Liu XD, Lin P, Ren W (2011) Consensus of linear multi-agent systems with reduced-order observer-based protocols. Syst Control Lett 60(7):510–516. https:// doi.org/10.1016/j.sysconle.2011.04.008 13. Gao LX, Xu BB, Li JW, Zhang H (2015) Distributed reduced-order observerbased approach to consensus problems for linear multi-agent systems. IET Control Theory Appl 9(5):784–792. https://doi.org/10.1049/iet-cta.2013.1104 14. Liu HG, Liu ZX, Chen ZQ (2017) Leader-following consensus of second-order multiagent systems with a smart leader. In: 36th Chinese control conference, Nanchang, pp 8090–8095. https://doi.org/10.23919/ChiCC.2017.8028637 15. Wonham WM (1985) Linear multivariable control. Springer, Heidelberg. https:// doi.org/10.1007/978-1-4612-1082-5 4 16. Semsar-Kazerooni E, Khorasani K (2011) Switching control of a modified leaderfollower team of agents under the leader and network topological changes. IET Control Theory Appl 5(12):1369–1377. https://doi.org/10.3182/20080706-5-kr-1001. 00262

A Group Control Method of Central Air-Conditioning System Based on Reinforcement Learning with the Aim of Energy Saving Yuedou Pan(B) and Jiaxing Zhao School of Automation and Electrical Engineering, University of Science and Technology of Beijing, Beijing 100083, China [email protected], [email protected]

Abstract. This paper proposes a group control method for central air conditioning system based on reinforcement learning to save energy. This method does not depend on the mathematical model of the system, and can automatically improve the control strategy through continuous interaction between the controller and the system. Find the optimal working point of each device under different external conditions by continuous online learning. Even in the case that system parameters changes, the devices can still be guaranteed to work at its optimal performance point. Simulation experiments verify the effectiveness of the proposed method. Keywords: Central air conditioning · Chiller group · Group optimal control · Reinforcement learning

1 Introduction As all knows, almost all large public buildings are equipped with a central air conditioning system to adjust the temperature and humidity of the room. In recent years, many scholars have studied on how to reduce energy consumption while ensuring internal cooling demand. According to the current research status, the study on central airconditioning energy-saving control is mainly divided into two aspects. The first aspect is how to accurately predict the load in the cooling zone, so that the cooling power of the air conditioning and the cooling demand of the internal area are dynamically balanced [1, 2]. The second aspect is the study on group control methods for various components of the central air conditioning system, including chillers, pumps and cooling fans. Taking chillers as an example, there are different distribution methods that can meet a certain demand, but due to the different working characteristics of each chiller, the difference distribution methods will lead to differences in energy consumption. If the allocation of cooling power can be reasonably controlled so that each chiller works at its optimum performance point, the energy consumption will be greatly reduced. This is also the main research of this paper. Li et al. proposed a model-based optimization method for group control of chilled water pumps in [3]. The optimal working point of c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 266–275, 2020. https://doi.org/10.1007/978-981-32-9682-4_28

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each chilled water pump is obtained in real time by the model. However, this method requires high precision for the model. When the model deviates or the working characteristics of the pump change, it will be difficult to get the best working point. Vignail et al. proposed an approximate dynamic programming approach to find the optimal operating point for chillers in [4]. But its in a dilemma, that is when the discrete interval is too large, the approximation error will be increased and when the discrete interval is too small, the state space will become larger and the convergence speed of the algorithm will be slower. This paper proposes an algorithm that does not rely on mathematical model of system and can accurately control chillers to work at their optimal performance point.

2 System Description Although the method proposed in this paper does not depend on the mathematical model of the system, in order to verify the effectiveness of the proposed method, the chiller group energy consumption model, the cooling zone load model and the simplified chiller model are established. In addition, since the group control of the water pumps and the cooling fans has the same form as the chillers, this paper only analyzes the group control strategy of the chillers. 2.1 Chillers Energy Consumption Model Assume that the main grid uses a local packet switching network to supply power to the chillers, and the chillers supply is matched to the required cooling load. n

Pg = ∑ Pc,i

(1)

i=1

Wherein, Pg is power consumed in the grid and Pc,i is power required in the i-th chiller, i = 1, ... , n. The total cooling power Qc that the chiller needs to provide is distributed to each chiller according to (2). (2) Qc,i = αi Qc Wherein, Qc,i is the cooling power required for each chiller and αi i = 1, 2, ... , n is the parameter that determine the i − th chiller’s operating point. It satisfies ∑ni=1 αi = 1 and Its value range is [0, 1]. Note that a single chiller has a maximum cooling power limit. Therefore, the cooling power provided by the i-th chiller must meet 0 ≤ Qc,i ≤ Qmax c,i . In addition, when the i −th chiller is in the OFF state, Pc,i = 0. Conversely, when the i −th chiller is in an ON state, the electric power Pc,i required to generate the cooling power Qc,i is calculated according to a nonlinear static Gordon-Ng model [5, 6].

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Pc,i =

θi,1 Ta Tcwr + θi,2 (Ta − Tcwr ) + θi,4 Ta Qc,i − Qc,i Tcwr − θi,3 Qc,i

(3)

Wherein, θi,k , i = 1, 2, ... , n; k = 1, 2, 3, 4 are the model parameters to be determined, Ta is the outdoor temperature and Tcwr is return water temperature of the chiller. Model (3) shows that when Qc,i is 0, Pc,i is not zero, because a certain amount of power is needed to ensure that the chiller is in ON, so it needs to be slightly modified. The standby state of each chiller is simulated by the operating point parameter αi . When αi = 0, the corresponding i − th chiller is turned off, and the modified model is as follows:  θi,1 Ta Tcwr + θi,2 (Ta − Tcwr ) + θi,4 Ta αi Qc − αi Qc αi = 0 Tcwr −θ i,3α Qc (4) Pc,i = αi = 0 0 The efficiency of chillers can be described using the coefficient of performance COP. COP =

Qc n ∑i=1 Pc,i

(5)

The main work of this paper is to control the operating point parameter vector V = (α1 , α2 , ... , αn ) of chillers in real time so that the chiller works in optimal state. 2.2

Cooling Load Model

The cooling load in the cooling zone mainly comes from two parts. The first part is the heat gain from the internal area, and another part is the gain from the outdoor heat. The internal heat gain is described by the model (6) [7]. Qi = [β1 Ti2 + β2 Ti + β3 ]n + Q+ i

(6)

Wherein, βk , k = 1, 2, 3 are the model parameters to be determined and n is the number of people in the cooling area at a certain moment. The first item represents the heat gain due to the human body inside the area. The second term represents other factors that affect the internal heat gain of the area, such as lights, electronics, and daylight entering from windows, which is calculated as follows: Q+ i = κn + χ + ηI

(7)

Where, κ , χ and η are parameters to be determined. I is the light intensity of daylight. From (7), the internal heat gain generated by electric lights, electronic equipment, etc. consists of two parts. The first part, κ n, represents the heat gain of lamps and electronic devices due to changes in the number of people in the room. The second part χ is the heat gain inherent in the area, which does not change with the number of people in the room. The gain of daylight from windows on internal heat is characterized by η I.

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In the cooling area, the number of people at a certain moment is a random variable, and the process of its change is a random process.The following assumptions are made for the convenience of description: Assumption 1. The cooling zone is opened at a certain time ts every day, and people start to enter or exit the cooling zone after ts . It is closed at a certain time te , and people cannot enter and exit the cooling zone after te . Assumption 2. During [ts ,te ], the number of people entering and leaving the cooling zone in a certain period of time obeys the Poisson distribution, and the corresponding arrival strength and departure strength are λin and λout respectively. Based on the above assumptions, the following model is available.  nin (k + 1) = nin (k) + P(λin (k)) nout (k + 1) = nout (k) + P(λout (k))

(8)

Wherein, nin (k) is the number of people entering the cooling zone at time kτ and nout (k) is the number of people leaving the cooling zone at time kτ . N τ = te − ts , k = 0, 1, 2, ... , N − 1. P(λin (k)) and P(λout (k)) are random numbers obtained from Poisson distribution obeying the strength λin (k) and λout (k) respectively. In summary, n is obtained by (9). n(k) = max(nin (k) − nout (k), 0)

(9)

2.3 Central Air Conditioning System Dynamic Model Since the study focus of this paper is the group control of the central air conditioning system, there is no in-depth research on its dynamic control. For the integrity of the system, a simplified dynamic model of the central air conditioning system is employed [8], in which we do not account for energy accumulation properties of the wall and the time lag caused by pipe. ⎧ Xi kcw (Ti − Tcwr ) + Qi + koi (Ta − Ti ) ˙ ⎪ ⎨ Ti = − Ci + Xi kcw (Ti − Tcwr ) T˙cwr = cw ω (Tcws − TcwrC)cwr ⎪ ⎩˙ Tcws = cw ω (Tcwr − Tcws ) − Qc

(10)

Ccws

Wherein, Ti is internal area temperature, Tcws is chilled supplying water temperature, Xi ∈ [0, 1] is the fraction of available cooling power that is actually provided to the area, kcw and koi are the heat transfer coefficients from chiller returning water and outside to internal area respectively. Ci , Ccwr and Ccws are the thermal capacity of the internal zone, chiller returning water and chiller supplying water respectively, ω is chiller water flow and cw is water heat capacity.

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3 Background of Reinforcement Learning The goal of reinforcement learning is to find a state-to-action mapping π (s) to maximize future expected total discounted rewards or called expected returns. It is described as a value function of the state or the state action pair. The return and value functions are defined as follows [9]: . Gt = Rt+1 + γ Rt+2 + γ 2 Rt+3 + ... =

∞

∑ γ k Rt+k+1

(11)

k=0

where Gt is return, Rt is instant rewards at time t, γ is discount factor, representing the level of trust in future rewards. ∞

. qπ (s, a) = Eπ [Gt |St = s, At = a] = Eπ [ ∑ γ k Rt+k+1 |St = s, At = a]

(12)

k=0

Wherein, qπ (s, a) is value function of state action pair (s, a). E[·] is mathematical expectation. The following recursion formula can be obtained by combining (11) and (12). qπ (s, a) = Eπ [Rt+1 + γ qπ (st+1 , at+1 )|St = s, At = a]

(13)

The core of the reinforcement learning algorithm is to estimate the value function of the state action pair, then select the action according to the value function to maximize the expectation of return. Q-learning is a model-free reinforcement learning algorithm based on value function iteration which can be used to handle optimization problems and multi-agent reinforcement learning problems [10]. The one-step Q-learning algorithm is as follows: Q(St , At ) ← Q(St , At ) + α [Rt+1 + γ maxa Q(St+1 , a) − Q(St , At )]

(14)

where α is learning rate. At each time step, the agent observes the current state St , then selects the action At that can maximize Q(St , ·). In order to balance exploration and exploitation, ε -greedy is usually adopted, which means that selecting action At with a large probability (1 − ε ) or a random action from other actions with a small probability ε . Then observe the environment to get the reward Rt+1 and the next state St+1 and calculate maxa Q(St+1 , a) to complete the update of Q(St , At ). The policy improvement theorem [9] ensures that as the number of iterations increases, the Q value will approach the optimal value Q∗ .

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4 Application of RL in Chillers Croup Control Through Sect. 2 analysis, the group control of chillers is a static optimization problem. According to (4), let S = (Ta , Tcwr , Qc ), A = V , R = −Pg . Since this is a static optimization problem, reinforcement learning is aimed at dynamic optimization problems. There are some differences from the general reinforcement learning problems described in Sect. 3. (1) State St to state St+1 is independent of each other, which means selecting any action At does not affect the next time state St+1 in the current state St . (2) According to (1), because the states are independent of each other, performing the action At at the state St only affects the immediate reward. It will not affect the rewards in the future. That is, in the static optimization problem, instead of maximizing long-term returns, it maximizes immediate reward. Based on the above analysis, in order to apply the idea of reinforcement learning to the group control of chillers, some modifications need to be made to the framework of reinforcement learning in Sect. 3. (1) The return changes to an immediate reward after performing action a under state s. Gt = E[Ras ]

(15)

(2) The value function is denoted by qs (a). It represents the value function of action a under state s, which is defined as follows: qs (a) = Gt

(16)

substituting (15) and (16) into (14) and replacing its corresponding part, the value function update algorithm for the chiller group control problem can be obtained as follows: (17) QSt (At ) ← QSt (At ) + [RAStt − QSt (At )] From (17), The value function in the static reinforcement learning algorithm is QSt (At ) and target of the value function becomes RAStt . If the state and action space are discrete and small enough, we can use the tabular average method to estimate QSt (At ) online. Then an action is generated based on QSt (a). But in this problem, because the state and action are continuous, if we discrete it, we will only get suboptimal results at discrete points. In order to approach the optimal point, it must be ensured that the state and action discrete interval is small enough, which inevitably causes the state and action space to be so large, and it is difficult to estimate the discrete points in all state and action spaces in a limited time. In order to solve above problem, this paper introduces BP neural network to estimate the value function QSt (At ). The weight of the BP neural St (At |μ ), The input of network is μ , the output of the network is the estimated value Q t the network is X = [St , At ] and Its estimated target is R(S )(At ). The output of the BP network is approximated to the true value by the gradient descent method. The neural network weight update algorithm is as follows: St (At |μ ))2 ∂ 12 (RAStt − Q μ = μ −α ∂μ

(18)

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The generalization ability of the BP neural network ensures that even in the case of sparse sample sets, it is still possible to have a good ability to estimate data that does not appear in the sample set. Since the BP neural network is equivalent to a nonlinear function of input to output, it is written as follows for convenience: s (a|μ ) = f (X) = f (s, a) Q

(19)

The goal of reinforcement learning is to obtain an action a that maximizes the value s (a|μ ) in a certain state s. At this point, the state s and the weight μ are all function Q known, so it is equivalent to solving an optimization problem with a known model as follows: s.t. a ∈ g(s) (20) maxa f (s, a) where a ∈ g(s) defines the range of actions allowed in state s and its definition is described in detail in Sect. 2.1. In addition, in order to balance exploration and exploitation, the ε -greedy algorithm is used to generate actions as follows:  St (a) argmaxa Q with probability 1 − ε At ← (21) a random action with probability ε where ε defined as follows: 1 m At St . ε =β ∑ (RSt − Q (At |μ ))2 2m t=1

(22)

where β is a parameter to be determined. (22) means that when the BP network output is close enough to the true value, ε decreases, and the reinforcement learning process is more inclined to make decisions by using existing knowledge. When the characteristics of the chillers change and the BP network output deviates significantly from the true value, ε increases, and the reinforcement learning process is more inclined to explore to discover new knowledge. This design ensures that when the chillers characteristic change, the chillers can still work at the optimal performance point.

5 Numerical Simulation In order to verify the effectiveness of the proposed method, a numerical simulation experiment was designed. The simulation system parameters are on Table 1. The simulation duration is set to 16 days. To verify that the algorithm still enables the chiller to operate at the optimum performance point when the chiller parameters change. At 12 o’clock on the 8th day, the parameter of the chiller 1 was increased by 1.5 times. In the simulation process, the approximation of the value function uses a BP neural network with a structure of 4-12-1. Note that according to Sect. 2.1, if there are n chillers, the input to the network should be n-1 dimensions. To facilitate the discussion of the algorithm, the number of chillers is set to 2, and the output of the network is 1 dimension. In addition, the learning rate in Eq. (18) is set to 0.02, and the exploratory strength

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Table 1. List of simulation system parameters. Model type Central air conditioning system dynamic model

Chillers energy consumption model

Cooling load model

Parameter Value Ci 6092 kJ◦ C−1

Parameter Value koi 0.4625 kW◦ C−1

Ccws Ccwr θ1,1

1.31·103 kJ◦ C−1 kcw 1.31·103 kJ◦ C−1 cw 0.0056 kWK−1 θ1,2

5.29 kW◦ C−1 4.2 kJ(kg·◦ C−1 ) 10.11 kW

θ1,3 θ2,1 θ2,3 Qmax c,1 β1 β3 χ

7 KkW−1 0.0109 kWK−1 3.807 KkW−1 30 kW −0.2199 W◦ C−2 84.9168 W 3 kW

0.9327 20.22 kW 0.9325 30 kW 5.0597 W◦ C−1 0.1 kW per person 0.00001 kW(Lx)−1

θ1,4 θ2,2 θ2,4 Qmax c,2 β2 κ η

Fig. 1. The image on the left is the intensity Lx of daylight on a certain day. The red curve in the middle image is the required cooling power Qc of the area, and the blue curve is the cooling load of the internal area. The red curve in the image on the right is the outside temperature Ta of a certain day, the blue curve is the internal temperature Ti , the purple curve is the chillers returning water temperature Tcwr , and the light blue curve is the chillers supplying water temperature Tcws .

parameter in Eq. (22) is set to 0.5. Figures 1 and 2 show the main numerical simulation results. Through the images on the left side of the top line and the middle line in Fig. 2, the RL algorithm can control the chillers to work at their optimum performance point, so that the energy consumption of the chillers approach the optimal value. According to the images on the bottom line and right size of the top line of Fig. 2, when the parameters of the chiller 1 change, although the action of the RL algorithm deviates from the optimal value in a short period of time, the RL algorithm can still ensure that the chillers are controlled to operate at the optimal performance point after learning and exploring. The image on the left side of the bottom line indicates that the optimal action ratio of the RL algorithm returns to 80% after 8 days of parameters change.

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Fig. 2. The two images in the top line are the chiller operating points on days 7 and 8, respectively, where the red curve is the optimal action value obtained by the analytical method, and the blue curve is the action value obtained by the RL algorithm. The image on the left of the middle line is the energy consumption of the chiller on the 7th day, where the red curve is the optimal energy consumption value obtained by the analytical method, and the blue curve is the energy consumption value obtained by the RL algorithm. The red curve on the right side of the middle line is the energy consumption value predicted by the BP neural network on the 7th day, and the blue curve is the actual energy consumption value (the negative number of the reward in the RL algorithm). The image on the left side of the bottom line is the ratio of the optimal action to the total action during the learning process, and the image on the right is the global error of the neural network during the learning process.

6 Conclusions This paper is devoted to solving the group control optimization problem of central air conditioning systems. A model-free reinforcement learning algorithm is adopted to acquire the knowledge of the system through continuous system interaction and exploration, and gradually approach the optimal solution. In addition, it can also ensure that the system still work at optimal performance point after short learning process, although

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the parameters change. The numerical simulation results verify the effectiveness of the proposed method. Acknowledgments. This work was supported by the National Natural Science Foundation of China (Grant No. 61873025).

References 1. Chang H (2016) Research on the application of fuzzy logic based on weight analysis table in central air conditioning system. School of Control Science and Engineering, Zhejiang University 2. Zhang J (2011) Research of water system control for energy saving in central airconditioning system. School of Control Science and Engineering, University of Science and Technology of China 3. Li H, Shi J, Yang CD, Zhang, Y et al (2018) A group control method of central airconditioning chilled water pumps with the aim of energy saving. In: 2018 International conference on power system technology, pp 4517–4524 4. Vignali RM, Borghesan F, Piroddi L et al (2017) Energy management of a building cooling system with thermal storage: an approximate dynamic programming solution. IEEE Trans Autom Eng 14(2):619–633 5. Gordon JM, Ng KC (1997) Optimizing chiller operation based on finite-time thermodynamics: universal modeling and experimental conformation. Int J Refrig 20(3):191–200 6. Ioli D, Falsone A, Schuler S et al (June 2016) A composiable framework for energy management of a smart grid: a scalable stochastic gybrid model for cooling of a district network. In: IEEE 12th International conference control automation, pp 389-394 7. Ioli D, Falsone A, Prandini M (June 2015) Optimal energy management of a building cooling system with thermal storage: a convex formulation. In: 10th IFAC symposium on advanced control of chemical processes, pp 1150–1155 8. Vaghefi VV, Jafari MA, Zhu J et al (2016) A hybrid physics-based and data driven approach to optimal control of building cooling/heating systems. IEEE Trans Autom Sci Eng 13(2):600–610 9. Sutton RS, Barto AG (2018) Reinforcement learning: an introduction, 2nd edn. The MIT Press, Cambridge 10. Watkins CJCH, Dayan P (1992) Q-learning. Mach Learn 3(8):279–292

Comparative Studies on Activity Recognition of Elderly People Living Alone Zimin Xu1,2 , Guoli Wang1,2 , and Xuemei Guo1,2(B)

2

1 School of Data and Computer Science, Sun Yat-sen University, Guangzhou 510006, China [email protected] Key Laboratory of Machine Intelligence and Advanced Computing, Ministry of Education, Guangzhou, China

Abstract. The social phenomenon of empty nesters is becoming more and more ubiquitous, and it is difficult to keep a watchful eye on their conditions. The advancement of computer technology and the spread of related applications have promoted the study of pervasive computing and smart cities. Ambient assisted living (AAL) systems have also been born in response to trends and demands. Regarding to medical surveillance, the AAL systems can care for the elderly people living alone full time, provide health advice, and initiate an early warning model when an emergency is detected. Main intention of this paper is to make comparative studies on activity recognition of elderly people living alone utilizing 6 classic classification algorithms, namely decision tree (DT), k-nearest neighbor (KNN), support vector machine (SVM), naive Bayes (NB), linear discriminant analysis (LDA), and ensemble learning (EL). And we adopt these models to recognize 10 activities of daily living (ADLs), namely Meal Preparation, Relax, Eating, Work, Sleeping, Wash Dishes, Bed to Toilet, Enter Home, Leave Home and Housekeeping. In this work, we employ the Aruba annotated open dataset that obtained in a smart house where a voluntary single elderly woman has lived for 220 days. After structuring and cleaning data, we slice data according to entire activity and then extract features from these activity units. Results show that most classification algorithms perform well except for NB based on the activity features we extract. In the future, we can boost performance through improving algorithms, automatically extracting features, and changing the way of reproducing activity representation. Keywords: Activity recognition · Classification algorithm · Wireless sensor network · Data preprocessing · Feature extraction

1

Introduction

As the population ages, more and more elderly people need support and waiting upon. And in order to prevent the population from growing too fast, China c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 276–291, 2020. https://doi.org/10.1007/978-981-32-9682-4_29

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implemented family planning policy in the past. The only-child generation has become the pillars of the country and the backbone of the society nowadays. For this reason, empty-nest elderly becomes a common social phenomenon. Whether the children live with their parents or not, it is difficult to constantly pay attention to the situation of the elderly because they have to go out to work. In this context, the research on smart home has become extremely urgent for the reason that the highly intelligent home environment can bring great convenience and security to the life of empty nesters. European Union countries have researched the Ambient Assisted Living (AAL) project since 2007 which aims to enhance the ability of living independently, make life easier and reduce care cost at the same time by means of utilizing intelligent environments and peripheral auxiliary technology [1]. With the development of pervasive computing and the Internet of Things, embedded sensors can be connected to a scalable technology platform to provide corresponding services for monitoring environmental timely response [2], context-aware computing [3], and real-time analysis of the status and behaviors of interested targets. One of the important research directions of AAL is the recognition of human activities of daily living (ADLs) which has become an elegant research field for its remarkable contributions in ubiquitous computing. Modeling the behaviors of the interested target and building a personalized database will help the devices within the environmental system to better understand the behaviors of the interested target and recommend convenient services. In the latest ten years, indoor sensing technology has developed rapidly and a variety of personal information collection programs have appeared frequently. Activity recognition (AR) has become a research hotspot in the computer field, especially in computer vision and pervasive computing. In the study of AR, there are two major methods to recognize ADLs: the first one is the visionbased activity recognition [4,5] and the second one is the sensor-based activity recognition [6]. In this work, we utilize the open dataset offered by Center for Advanced Studies in Adaptive Systems (CASAS), Washington State University (WSU) [7,8], and we make comparative studies on activity recognition of elderly people living alone adopting different classic classification algorithms. The rest of this paper is organized as follows. The next section discusses the related works in the literature followed by Sect. 3 which describes the smart home environment, dataset, data preprocessing, classification algorithms and evaluation. In Sect. 4, experimental results and the corresponding discussions are shown. Finally, the conclusions are summarized in Sect. 5.

2

Related Works

In terms of indoor environment, the recognition systems have to satisfy the following requirements: collecting information all-day around, multi-dimensional recording behavior information of interested targets, non-invasive devices and respecting personal privacy. The camera can capture the target’s motion postures, clothing information and facial expressions in real time, however, it

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depends on the light intensity, therefore the information capture ability is limited when the visibility is low. Furthermore, image information is extremely easy to invade personal privacy, so it is better to choose wireless sensor networks (WSN) as indoor activity recognition systems. WSN can be roughly divided into wearable sensor and non-wearable sensor. Wearable sensors, such as the three-axis accelerometers and gyroscopes, are mostly used in the field of mobile computing. However, the advantages of non-wearable sensors are self-evident from the perspective of user comfort and intelligence, and they can be used in a wide range of smart environment applications and smart home design. Consequently, we employ the non-wearable wireless sensor network here. The physical sensor based AR system is based on the perception of the external environment. In terms of activity recognition based on wearable sensors, researchers usually collect relevant physical and physiological information and then recognize activities based on the collected data [9]. And in the case of the non-wearable sensors based AR systems, i.e. WSN, more and more researchers are joining the AR field based on positioning data or sensor data streams considering user comfort and privacy issues. Tapia et al. have used environmental state change perception sensors to collect behavioral information and recognizes activities through naive Bayes (NB) [10]. Cook et al. have attempted to observe activities at different time intervals [11] and Maurer et al. have attempted to optimize the recognition effect from the sampling frequency of the feature set [12]. With the rapid development of machine learning and deep learning techniques, researchers have combined them with activity recognition to achieve good results. For example, Johnson et al. have used simple Nearest Neighbor (NN) for activity classification [13]. Hidden Markov model (HMM) [14], latent Dirichlet allocation [15], support vector machine (SVM) [6] and other machine learning algorithms are usually adopted in AR systems. Deep belief network (DBN) is one of the robust deep learning techniques using restricted Boltzmann machines (RBMs) during training [16]. Ha et al. have executed a CNN activity recognition model using open datasets collected by tri-axial accelerometers and gyroscopes [17]. Yala et al. have proposed three feature extraction approaches combined with SVM or KNN classification algorithms [6]. Matsui et al. have presented a CNN model for outdoor activity recognition employing an accelerometer, a magnetometer, and a gyroscope [18]. Gowda et al. have provided a CNN model that recognizes human activities from the videos [4]. Wang et al. present a Recurrent CNN (RNN) model that detects daily activities from egocentric videos [19]. Gochoo, et al. have proposed a DCNN model for indoor travel patterns of elderly people living alone and the proposed DCNN classifier can be used to infer dementia through travel pattern matching [20,21]. Subasi et al. have used efficiencies of ensemble classifiers to recognize human activity data taken from body sensors and ensemble classifiers achieve better performance by using a weighted combination of several classifiers [22]. Though deep learning is more efficient than typical neural networks, it consists of two major disadvantages: it has overfitting problem, and it is often much time-consuming.

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Methods

In this section, we will elaborate the smart home environment, dataset, the process of data preprocessing, classification algorithms and evaluation. The flow of activity recognition system is presented in Fig. 1. In general, the data collected is unstructures, so it is not conducive to in-depth exploration and analysis. Therefore, the first step is to structure data. And then we preprocess data and extract features. Finally, various classification algorithms are utilized to recognize activities and we evaluate models with a variety of performance measurements. The following subsections elaborate on these aspects.

Fig. 1. The flow diagram of activity recognition system.

Fig. 2. The layout of Aruba testbed and the locations of sensors.

3.1

Smart Home Environment

The layout of Aruba testbed and the locations of sensors are shown in Fig. 2, which is one of the testbeds of the CASAS project. Clearly, this smart home

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system consists of a living room, a dining room, a kitchen, an office, two bedrooms, two bathrooms, a garage and a backyard. There are 31 motion sensors, 4 door closure sensors and 4 temperature sensors equipped here, however, temperature sensors are not relevant to this study, hence only door closure sensors and motion sensors are taken into account and presented in Fig. 2. Among them, motion area sensors belong to motion sensors, and the only difference is that they have a wider perception range. According to the CASAS project, a single voluntary elderly woman lived in Aruba testbed, and her children and grandchildren visited her regularly during the experiment. In addition to these, it is not able to get more information about her, therefor, we regard her as a healthy person. 3.2

Dataset

There are 1698431 continual events recorded from 31 motion sensors and 4 door closure sensors for 220 days from Nov. 4, 2010 to Jun. 11, 2011. Figure 3 shows the samples of raw dataset where each recording comprises the information of date, time, sensor ID and sensor status. In particular, a corresponding annotation will be added immediately following the sensor status when an activity begins or ends, such as ‘Bed to Toilet begin’, ‘Bed to Toilet end’ and so on. There are 11 activities annotated within the dataset, ‘Meal Preparation’, ‘Relax’, ‘Eating’, ‘Work’, ‘Sleeping’, ‘Wash Dishes’, ‘Bed to Toilet’, ‘Enter Home’, ‘Leave Home’, ‘Housekeeping’ and ‘Respirate’. Respirate is ignored here because the number of this activity appears in the dataset is only 6 which is too small to recognize correctly. Motion sensor and door closure sensor are both trigger-type sensors whose returned values are ‘ON/OFF’ and ‘OPEN/CLOSE’, respectively. And the former is placed throughout the house and the latter is placed on the door panel for detecting the open-close status.

Fig. 3. Samples of the raw dataset.

Table 1 shows some statistical information on each activity. It is quite clear that all activities except for Housekeeping are regional, that is to say, only par-

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Table 1. Statistical information on each activity

Activity

Number of Average Average Average activities number of number of duration events sensors

Meal Preparation 1596 2907 Relax 255 Eating 171 Work 397 Sleeping 64 Wash Dishes 156 Bed to Toilet 427 Enter Home 427 Leave Home 33 Housekeeping

182.8 121.5 71.3 95.4 82.0 162.8 8.5 4.7 4.5 320.9

8.2 4.4 4.0 4.3 3.7 7.4 2.9 2.7 2.7 16.7

470.6 s 2006.5 s 609.0 s 1024.9 s 14501.7 s 434.8 s 163.7 s 6.7 s 6.4 s 1219.4 s

tial sensors in a specific area will be triggered successively when she performs an activity. Meal Preparation and Relax have highest numbers of samples and we can infer that this resident does not regularly do housekeeping because the number of Housekeeping is only 33 and Wash Dishes has the same characteristics. The activity of Enter Home and Leave Home are a pair of short-term activities that include small numbers of sensors and events. The similar features can be found in Bed to Toilet whose average number of events and sensors is only 8.5 and 2.9, respectively. 3.3

Data Preprocessing

Raw data streams are usually not available for feature extraction and need to be preprocessed. The process of data preprocessing consists of multi-data fusion, removing noise, eliminating the false, elimination of redundancy, numeralization and data slicing, as shown in Fig. 4. Every activity can be regarded as several divided atomicity level subactivities and the adjacent sub-activities are correlative with each other, in other words, activity is context aware. The information of different kinds of activities are collected by means of environment embedded sensor network system. In general, the raw data is unstructured, so we have to perform data cleaning and data exploration on the raw data returned by the sensors, and then structure the data. Issues that often need to be settled include noise, outliers, data loss, data duplication, and data errors, where we can take the methods of denoising, de-pseudo, de-duplicating, deleting rows or categories. The structuring process generally refers to codification, which involves digitizing the binary properties of the trigger status and encoding the sensor types. Concretely speaking, the form of date

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Fig. 4. The process of data preprocessing.

Fig. 5. Samples of the converted data.

is converted to ‘yymmdd’, and the time is transformed to the relative timestamp relative to the zero of the day. After that, 31 motion sensors are sequentially represented as 101 to 131, and 4 door closure sensors are denoted as 201 to 204, where their status ‘ON’ or ‘OPEN’ indicated by 1 while 0 for ‘OFF’ or ‘CLOSE’. Finally, expressing the activity of ‘Meal Preparation’, ‘Relax’, ‘Eating’, ‘Work’, ‘Sleeping’, ‘Wash Dishes’, ‘Bed to Toilet’, ‘Enter Home’, ‘Leave Home’, ‘Housekeeping’ as 1 to 10 and the annotations of ‘begin’ and ‘end’ are converted to 1 and 3, respectively. Particularly, we add the annotation ‘2’ in activity status to indicate that the activity is in progress and ‘0’ in activity meaning no activity. In short, we can convert the sensor record to the form of Eq. 1. Si = {di , ti , sii , ssi , ai , asi }

(1)

Where, di and ti mean the the date and timestamp, respectively. sii , ssi , ai and asi indicate the sensor ID, sensor status, activity and activity status, respectively. After a series of transformations, we can get the data form shown in Fig. 5. Then, we slice the data stream according to the full activity to get a series of activity units. In this paper, we propose a method of feature extraction about an entire activity that comprises with five parts: (1) The complete trigger counter of each sensor (motion sensor and door closure sensor included) in this complete activity;

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(2) The trigger duration of each sensor in this entire activity; (3) The duration of this activity; (4) The timestamp when this activity begins; (5) The sensors involved in the events of this activity. That is to say, the feature vector is shown as Eq. 2. Fi = {{ni1 , · · · , ni35 }, {ti1 , · · · , ti35 }, Ti , tsi , {Ii1 , · · · , Ii35 }}

(2)

Where 1–35 represent 31 motion sensors and 4 door closure sensors, and ni1 , · · · , ni35 , ti1 , · · · , ti35 , Ti and tsi correspond to (1)–(4), respectively. The value of Ii1 , · · · , Ii35 is either 1 or 0, and 1 means the corresponding sensor is included in activity events, otherwise it is not involved in this activity. Because the locations of sensors correspond to certain places, and the triggered sequences reflect the time series, the feature vector shows the temporal and spatial information of the activity to a certain extent. 3.4

Classification Algorithms

After extracting features, this paper adopts classic classification algorithms to recognize these 10 activities of daily living. (I) Decision Tree (DT) The decision tree is a tree structure that represents a mapping between object attributes and object values. The decision process comprising of starting from the root node, testing the corresponding feature attributes and selecting the output branch according to its value until the leaf node is reached. Finally, the category stored by the leaf node is used as the decision result. (II) k-Nearest Neighbor (KNN) The core of KNN is that if a sample and most of the k nearest neighbors belong to the same class in the feature space, then this sample also belongs to this class and has the characteristics of the samples of this category. (III) Support Vector Machine (SVM) The basic idea of classification learning is to find a partition hyperplane in the sample space based on the training set, and separate the samples of different categories. (IV) Naive Bayes (NB) The basic principle of naive Bayes is to learn the joint probability distribution of input/output based on the conditional independence hypothesis for a given training dataset; then based on this model, the Bayesian theorem is used to find the output that maximizes the posterior probability for a given input.

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(V) Linear Discriminant Analysis (LDA) The idea of LDA is to project the training samples onto a straight line so that the projection points of the same kind of sample are as close as possible and the projection points of the different samples are as far as possible. A new sample is projected onto the same line and which category this sample belongs to depends on the position of the projected point. (VI) Ensemble Learning (EL) Ensemble learning usually achieves more outstanding performance by combining several learners than single learner. 3.5

Evaluation

The algorithm model first trains the training set and then predicts the test set. And reasonable evaluation of model performance is beneficial for researchers to select algorithm models according to mission requirements, and then optimize models in time. Evaluation method used here is 10-fold cross-validation and the next is a brief description of the performance measures adopted in this paper. This work mainly employs classic models to recognize activities of daily living and the Accuracy (acc), Confusion Matrix and Receiver Operating Characteristic Curve (ROC) are adopted usually as performance measures in the classification task. In the prediction task, given a sample set D = {(x1 , y1 ), (x2 , y2 ), · · · , (xl , yl )}, where yi is the true label of sample xi . To evaluate the performance of the learner f , it is necessary to compare the predicted results of the learner f (x) with the real label y. Here accuracy is defined as Eq. 3. ⎧ ni m    i   ⎪ ni i ⎪ ⎨ acc = l f xj = yj i=1 j=1 (3) m  ⎪ ⎪ ni ⎩l = i=1

Where m is the total number of activity classes, ni is the number of samples in activity category i and l is the total number of samples of all activity categories. ni l indicates the proportion of the activity category i, therefore acc in Eq. 3 denotes the weighted accuracy which can better reflect the recognition effect of the classification model. Furthermore, the confusion matrix can also be used for analysis of unbalanced data sets. It is not a direct performance measure, but the basis of other indicators. For the binary classification, the confusion matrix is shown as Fig. 6. Precision (P ) and Recall (R) are defined as P =

TP TP + FP

(4)

R=

TP TP + FN

(5)

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Fig. 6. The confusion matrix of classification result.

Precision and recall are a pair of contradictory measures. In general, when precision is high, recall tends to be low; while recall is high, precision tends to be low. Therefore, there is a harmonic mean measure F1-Score (F 1) used to describe model classification performance. 2 × TP 2×P ×R = (6) P +R l + TP − TN The ROC curve is a compositional method to reveal the relationship between true positive rate (T P R) and false positive rate (F P R), and it also often used to evaluate the effect of a classifier. T P R and F P R are defined as F1 =

TPR =

TP TP + FN

(7)

FP (8) TN + FP From the definition of R and T P R, we can find that the Recall and True Positive Rate are the same variable. T P R = 1 and F P R = 0 in ideal conditions. The ROC curve is shown in Fig. 7 and the top left corner represents the ideal situation. FPR =

Fig. 7. The ROC curve.

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Results and Discussion

According to the flow diagram of activity recognition system (Fig. 1), after cleaning the Aruba dataset, the activity units are segmented and extracted, the feature vector of each complete activity unit is constructed, and then the feature matrix is trained in different classification models. We do some comparative studies on activity recognition of elderly people living alone using different classic classification algorithms. The algorithms adopted here are: DT, KNN, SVM, NB, LDA and EL. This paper classifies and recognizes 10 kinds of indoor daily life activities whose activity labels are: 1: Meal Preparation; 2: Relax; 3: Eating; 4: Work; 5: Sleeping; 6: Wash Dishes; 7: Bed to Toilet; 8: Enter Home; 9: Leave Home and 10: Housekeeping, respectively. It is easy to see from Table 1 that the numbers of Meal Preparation and Relax are more than the other 8 activities, the down-sampling technique is used in this case to prevent a large sample size gap. Without loss of generality, 500 activity samples of Meal Preparation and 1000 activity samples of Relax are randomly chosen from their activity subdatasets. The experimental results are shown in Figs. 8, 9 and 10. Where Fig. 8(a)(b) present the confusion matrix of decision tree classification algorithm and Fig. 8(c)(d) demonstrate the ROC curve when Meal Preparation and Relax is the positive class, respectively. There are more results of decision tree and the other 5 classification algorithms not presented here because of space cause, but we can infer the other confusion matrixes and ROC curves from the results in Figs. 9 and 10. When computing the true positive rate of a certain activity in Fig. 9, all the other 9 activities are considered as negative class and the interested activity is the positive class. Particularly, the weighted recall of a specific classification algorithm is equivalent to the accuracy of this algorithm, that is to say, either the weighted recall or the accuracy of a given algorithm means the ratio of correctly classified samples to the total samples. It is obvious that the true positive rate of all activities except for Wash Dishes, Enter Home and Leave Home, are high in Figs. 8(a)(b) and 9, and the same phenomenon also appears in the KNN, SVM, LDA and EL algorithm. Investigating the real cause, it is necessary to consider the relevance of these three activities to other activities. Consequently, we turn to explore the similarity measures of these 10 activities. The usual method is to calculate the distance or similarity between samples where Euclidean distance, Manhattan distance, Minkowski distance, cosine similarity and so on are often adopted. Without loss of generality, here shows the results of the Euclidean distance and cosine similarity that are shown in Tables 2 and 3, respectively, both of which are symmetric matrixes. For two N -dimensional vectors x = (x1 , x2 , · · · , xN ) and y = (y1 , y2 , · · · , yN ), their Euclidean distance and cosine similarity are defined as

N T 2 DEu = (x − y)(x − y) = (xi − yi ) (9) i=1

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Fig. 8. The experimental results of decision tree.

cos(x, y) =

x·y =  |x| |y| N

N

i=1

2 i=1 xi ·

xi yi  N

(10)

2 i=1 yi

According to their definitions, the smaller the distance or the greater the similarity, the smaller the gap between the vectors, that is, the more similar the two types of activities are. Here the vector representing one kind of activity used to calculate Euclidean distance and cosine similarity is the averages of all feature vectors of a given type of activity, and these 10 average feature vectors compose a vector matrix that need to be normalized to [0, 1] before computing Euclidean distance and cosine similarity. We can see that the majority of the sample of Wash Dishes is divided into Meal Preparation, Enter Home confuses with Leave Home. The phenomena coincide with Tables 2 and 3 where the Euclidean distance between Meal Preparation and Wash Dishes, Enter Home and Leave Home are smaller than the other activity pairs, and greater for cosine similarity. The fundamental cause of these phenomena is that Meal Preparation (Enter Home) shares the same area and sensors with Wash Dishes (Leave Home). For Meal Preparation and Wash Dishes, their primary areas are both the kitchen, either the trigger sensors or the events or the duration, are extremely similar. Then for Enter Home and Leave Home, besides

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Fig. 9. The recall/true positive rate of each activity in different classification algorithms.

Fig. 10. The weighted precision, recall and F1 of different classification algorithms.

the same characteristics as above, they are short-term activities that makes it harder to distinguish. The other 7 activities have their own unique characteristics so that it is relatively easy to identify. Finally, we can see form Fig. 10, the classification algorithms of decision tree, k-nearest neighbor, support vector machine, linear discriminant analysis and ensemble learning have good performance in various performance measures. However, naive Bayes has poor performance. This could caused by that the feature vector we extract for training can reflect the temporal and spatial information to a certain extent, but they cannot reflect the sequential order of time and space. And as for NB itself, given the output categories, it assumes that the attributes are independent of each other, however, this assumption is often not true here on account of the correlation among 5 parts of the extracted feature. Furthermore, the number of attributes

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Table 2. The Euclidean distance of activities of daily living

1

2

3

4

5

6

7

8

9

10

1

0

4.683 3.777 5.176 6.305 1.747 4.767 4.700 4.656 6.517

2

4.683 0

3

3.777 3.024 0

4

5.176 4.188 3.725 0

5

6.305 5.707 5.589 6.000 0

6

1.747 4.383 3.392 4.757 6.190 0

7

4.767 3.582 2.955 3.789 5.407 4.293 0

8

4.700 3.649 3.112 3.819 5.816 4.320 3.153 0

9

4.656 3.585 3.005 3.775 5.770 4.268 3.002 0.872 0

3.024 4.188 5.707 4.383 3.582 3.649 3.585 6.478 3.725 5.589 3.392 2.955 3.112 3.005 6.634 6.000 4.757 3.789 3.819 3.775 7.466 6.190 5.407 5.816 5.770 7.044 4.293 4.320 4.268 6.858 3.153 3.002 7.220 0.872 7.424 7.394

10 6.517 6.478 6.634 7.466 7.044 6.858 7.220 7.424 7.394 0 Table 3. The cosine similarity of activities of daily living

1

2

3

4

5

6

7

8

9

10

1

1

0.250 0.510 0.116 0.181 0.918 0.008 0.161 0.149 0.483

2

0.250 1

3

0.510 0.390 1

4

0.116 0.133 0.146 1

5

0.181 0.177 0.123 0.109 1

6

0.918 0.218 0.492 0.126 0.137 1

7

0.008 0.010 0.008 0.005 0.166 0.009 1

8

0.161 0.157 0.166 0.158 0.058 0.138 0.000 1

9

0.149 0.137 0.154 0.133 0.047 0.123 0.000 0.939 1

0.390 0.133 0.177 0.218 0.010 0.157 0.137 0.484 0.146 0.123 0.492 0.008 0.166 0.154 0.465 0.109 0.126 0.005 0.158 0.133 0.197 0.137 0.166 0.058 0.047 0.426 0.009 0.138 0.123 0.393 0.000 0.000 0.212 0.939 0.154 0.152

10 0.483 0.484 0.465 0.197 0.426 0.393 0.212 0.154 0.152 1

is relatively large. All in all, the classification effect of naive Bayes is not good for a variety of reasons. Conversely, the other 5 algorithms are not based on the premise that attributes are independent of each other, so the final classification performance is much better than NB.

5

Conclusions

Sensor-based activity recognition is an important and popular research field due to its widespread applications in a variety of areas such as healthcare, assisted living, home monitoring, personal fitness assistants, and terrorist detection. In this paper, we carry out comparative studies on activity recognition of elderly people

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living alone in a smart home adopting various classic classification algorithms. We employ an open dataset collected by 31 motion sensors and 4 door closure sensors for 220 days for training the classifiers, and 10-fold cross-validation is used for the evaluation of the classifiers. We perform the following operations to preprocess data: structuring data, cleaning data, slicing and obtaining time series activity units, and then features of each complete activity are extracted. Results show that the classification models of the decision tree, k-nearest neighbor, support vector machine, linear discriminant analysis and ensemble learning perform well, whose precision, recall and F1-score have reached about 0.9. However, naive Bayes has a poor performance because of the attribute correlation of the extracted activity feature. Although some classifiers can identify different activities at a high rate, it is not enough yet. There are three factors affecting the final result: The limitations of the algorithm itself, the high overlap between some activities and the insufficiency of activity feature. In the future, we can improve performance from the following perspectives: proposing a method to reproduce the motion trail and represent the sequence of space-time, and adopting deep learning technique to automatically extract features. Acknowledgments. This work was supported in part by the National Natural Science Foundation of P. R. China under Grant Nos. 61772574, 61375080 and U1811462 and in part by the Key Program of the National Social Science Fund of China with Grant No. 18ZDA308.

References 1. Fatima I, Halder S, Saleem MA, Batool R (2015) Smart CDSS: integration of social media and interaction engine (SMIE) in healthcare for chronic disease patients. Multimed Tools Appl 74(14):5109–5129 2. Poppe R (2010) A survey on vision-based human action recognition. Image Vis Comput 28(6):976–990 3. Stikic M, Schiele B (2009) Activity recognition from sparsely labeled data using multi-instance learning. In: International symposium on location and context awareness, pp 156–173 4. Gowda SN (2017) Human activity recognition using combinatorial deep belief networks. In: IEEE conference on computer vision & pattern recognition workshops, pp 1589–1594 5. Papakostas M, Giannakopoulos T, Makedon F, Karkaletsis V (2017) Short-term recognition of human activities using convolutional neural networks. In: 2016 12th international conference on signal-image technology & internet-based systems (SITIS), pp 302–307 6. Yala N, Fergani B, Fleury A (2017) Towards improving feature extraction and classification for activity recognition on streaming data. J Ambient Intell Humaniz Comput 8(2):177–189 7. Cook DJ, Crandall AS, Thomas BL, Krishnan NC (2013) CASAS: a smart home in a box. Computer 46(7):62–69 8. Tan TH, Gochoo M, Jean FR, Huang SC, Kuo SY (2017) Front-door event classification algorithm for elderly people living alone in smart house using wireless binary sensors. IEEE Access 5:10734–10743

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9. Lara OD (2012) Centinela: a human activity recognition system based on acceleration and vital sign data. Pervasive Mob Comput 8(5):717–729 10. Tapia EM, Intille SS, Larson K (2004) Activity recognition in the home using simple and ubiquitous sensors. In: International conference on pervasive computing. Springer, Heidelberg, pp 158–175 11. Cook DJ, Schmitter-Edgecombe M (2009) Assessing the quality of activities in a smart environment. Methods Inf Med 48(5):480–485 12. Maurer U, Rowe A, Smailagic A, Siewiorek D (2006) Location and activity recognition using eWatch: a wearable sensor platform. Lecture Notes in Computer Science, vol 3864, pp 86–102 13. Johnson N, Hogg D (1996) Learning the distribution of object trajectories for event recognition. Image Vis Comput 14(8):609–615 14. Van Kasteren T, Krose B (2007) Bayesian Activity Recognition in Residence for Elders. In: IET International Conference on Intelligent Environments, pp 209–212 15. Abidine MB, Fergani B (2016) Comparing HMM, LDA, SVM and smote-SVM algorithms in classifying human activities. In: Proceedings of the Mediterranean conference on information & communication technologies 2015. Springer, Heidelberg 16. Hinton GE, Osindero S, Teh Y-W (2006) A fast learning algorithm for deep belief nets. Neural Comput 18(7):1527–1554 17. Ha S, Yun J, Choi S (2015) Multi-modal convolutional neural networks for activity recognition. In: 2015 IEEE international conference on systems, man, and cybernetics (SMC), pp 3017–3022 (2015) 18. Matsui S, Inoue N, Akagi Y, Nagino G, Shinoda K (2017) User adaptation of convolutional neural network for human activity recognition. In: 25th European signal processing conference, pp 783–787 19. Wang M, Luo C, Ni B, Yuan J, Wang J, Yan S (2017) First-person daily activity recognition with manipulated object proposals and non-linear feature fusion. IEEE Trans Circuits Syst Video Technol 8215(C):1–21 20. Gochoo M, Tan TH, Velusamy V, Liu SH (2017) Device-free non-privacy invasive classification of elderly travel patterns in a smart house Using PIR sensors and DCNN. IEEE Sens J 17(1):390–400 21. Gochoo M, Tan TH, Liu SH, Jean FR (2018) Unobtrusive activity recognition of elderly people living alone using anonymous binary sensors and DCNN. IEEE J Biomed Health Inform 23(2):693–702 22. Subasi A, Dammas DH, Alghamdi RD, Makawi RA, Albiety EA, Brahimi T, Sarirete A (2018) Sensor based human activity recognition using adaboost ensemble classifier. Proc Comput Sci 140:104–111

The Improved Regularized Extreme Learning Machine for the Estimation of Gas Flow Temperature of Blast Furnace Xiaoyang Wu, Sen Zhang(B) , Xiaoli Su, and Yixin Yin University of Science and Technology Beijing, Beijing 100083, China [email protected]

Abstract. Gas flow temperature of blast furnace is one of significant indicators of judging the anterograde state of blast furnace. Aiming at the defects of traditional gas flow temperature prediction model, this paper proposes a prediction model for gas flow temperature based on improved regularized extreme learning machine algorithm (RELM). Firstly, influencing factors are chosen through analyzing the distribution of blast furnace gas flow. Considering the blast furnace production data contain high frequency noise, the Fourier transform method is used for spectrum analysis, and the appropriate digital filter is selected according to the spectrum analysis result to eliminate the high frequency noise. Secondly, this paper establishes the prediction model of gas flow temperature. Regularized extreme learning machine and finite memory recursive least squares are combined to overcome Data saturation problem of ordinary extreme learning machine. And the improved algorithm is named as RFM-RELM. Finally, actual production data are used to train and test this model. Experimental result indicate that the model can quickly and accurately predict gas flow temperature ,which provides effective help and support for blast furnace operators to analyze the conditions of blast furnace. Keywords: Blast furnace · Modeling · Regularized extreme learning machine · Gas flow · Prediction

1 Introduction Iron and steel industry is the pillar industries in the national economy of China. Blast furnace is the core equipment in the blast furnace production process, which is related to the industrial production and energy consumption of the iron and steel industry [1]. Maintaining the ante-grade state of blast furnace is the significant goal pursued by the entire steel industry. In the blast furnace production process, the distribution of blast furnace gas flow directly reflects the utilization of energy. Thus, it is essential to accurately predict the temperature of blast furnace gas flow so as to maintain the ante-grade state of blast furnace [2]. However, the temperature of blast furnace gas flow is difficult to be directly observe. Therefore, the prediction of blast furnace gas flow temperature c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 292–300, 2020. https://doi.org/10.1007/978-981-32-9682-4_30

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has always been a hot research direction in academia and industry [4]. With the development of detection technology, advanced technologies such as infrared imaging and crossing temperature measurement are more mature in industrial field, making gas flow temperature prediction possible. In this paper, a improved RELM algorithm named RFM-RELM is proposed for establish the prediction model of blast furnace gas flow temperature based on finite memory recursive least squares method. Firstly, the model input variables are selected from many influencing factors of blast furnace by mutual information method, so as to realize the data dimensionality reduction of the model. Secondly, the Fourier transform method is used to perform spectrum analysis on the actual data of blast furnace. Then according to the analysis result, this paper selects the appropriate digital filter to remove high frequency noise. Finally, using the denoised data, establishing the prediction model of blast furnace gas flow temperature combined with RELM. Considering that the ordinary RELM algorithm will gradually lose the learning ability as the amount of data increases, this paper adopts the finite memory recursive least squares (RFMLS) method to optimize the RELM algorithm to improve the prediction accuracy of the model. Experimental result indicate that the model can accurately predict gas flow temperature, which provides effective help and support for blast furnace operators to analyze the conditions of blast furnace. The rest of this paper is as follows: Sect. 2 uses mutual information method to analysis the influence factors of blast furnace gas flow. Section 3 is data preparation. Section 4 briefly introduces RFM-RELM and related algorithms, and establishes the prediction model of blast furnace gas flow temperature. Simulation experiment and results are shown in Sect. 5. Conclusions of this paper are in Sect. 6.

2 Gas Flow Temperature Factor Analysis Based on Mutual Information In order to protect the effective information of blast furnace, this paper selects the temperature measurement points in the crossing temperature device and the internal condition information of the blast furnace to analyze the influencing factors of blast furnace gas flow temperature. In this paper, the temperature measurement point T 25 is selected as the output variables. According to the above analysis, this paper selects air volume (FL), wind temperature (FW), wind pressure (FY), top pressure (DY), oxygen enrichment (O2), wind speed (FS), differential pressure (YC) and others 16 temperature points as influencing factors. And this paper use the mutual information method to calculate the degree of mutual information between influencing factors and the output variable T 25. The mutual information method is used to measure the degree of interdependence between two variables [3]. Given two random variables X and Y , Their edge probability density distribution and joint probability distribution are respectively p(x), p(y), and p(x, y), the mutual information I(X : Y ) is, I (X,Y ) = ∑ ∑ p (x, y) log x

y

p (x, y) p (x) p (y)

(1)

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When X and Y are completely unrelated or related independently, Mutual information is the smallest, the result is I(X : Y ) = 0. The real production data which come from the blast furnace are used in this paper. There are 1000 groups, according to the formula (1), the correlation coefficient between each influence factor and T 25 is calculated. The result is shown in the Fig. 1. In order to reduce the complexity of the model, data dimensionality reduction is necessary. This paper selects the influence factor with mutual information greater than 0.8 as the model input variables. Therefore, input variables of the blast furnace gas flow temperature prediction model are: FS, YC, T 23, O2, T 26, T 36, T 21, T 24, T 39, T 33 and T 30. 0.9

Mutual information

0.8

0.7

0.6

0.5

0.4

FL DY FW FY T27 T28 T29 T37 T31 T22 T32 T38 FS YC T23 O2 T26 T36 T21 T24 T39 T33 T30

Influential factor

Fig. 1. The mutual information between the influence factor and T 25

3 Data Preprocessing Based on Fourier Transform Fourier transform can perform spectrum analysis on data, that is transform data from time domain to frequency domain [4]. Then select the appropriate digital filter to strip the high frequency noise in the data. The definition of the Fourier transform is that any one of the periodic signals can be decomposed into the sum of an infinite number of sinusoidal signals of different frequencies. Fourier transform in the form of trigonometric functions is as follows, ∞

x (t) = a0 + ∑ (an cos nω0t + bn sin nω0t)

(2)

n=1

Where, x(t) is the periodic signals. an and bn is Fourier coefficient. According to the conclusion of Sect. 2, this paper selects 12 production parameters FS, YC, T 23, O2, T 26, T 36, T 21, T 24, T 39, T 33, T 30 and T 25. The 12 parameters are separately analyzed by spectrum, and according to the analysis result this paper selects the appropriate digital filter for filtering. This paper takes T 25 as an example to demonstrate the denoising effect. The results of the spectrum analysis are shown in Fig. 2.

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time (minute)

(a) The data of T 25 before data preprocessing

(b) Spectrum analysis of T 25

Fig. 2. The results of the spectrum analysis

4 Blast Furnace Gas Flow Temperature Prediction Model Based on RFM-RELM This section will introduce RFM-RELM and its related algorithms. 4.1 The Theory of Traditional RELM Algorithm Extreme learning machine (ELM), proposed by Huang et al., is a novel machine learning algorithm for single hidden layer feedforward neural networks (SLFNs) [5]. The main feature of ELM is to randomly generate weight matrix and offset vector during model training. Unlike the ordinary ELM algorithm, the main feature of RELM is to use regularization parameters while minimizing training error and the norm of output weight [6]. In the sense of the minimum 2 norm, it can be expressed as, min C e22 + β 22 , s.t : Y − H β = e β

(3)

According to the Lagrange multiplier method, L (β , e, λ ) = C e22 + β 22 + λ T (Y − H β − e)

(4)

Where, e = [e1 , e2 , · · · , eN ]T is N training error parameters, λ is the column vector of the Lagrangian multiplier. According to the Karush-Kuhn-Tucker (KKT) theorem, the optimal condition is, ⎧ ∂L T ⎪ ⎨ ∂ β = 0 ⇒ 2β − H λ = 0 ∂L (5) = 0 ⇒ 2Ce − λ = 0 ⎪ ⎩ ∂∂ Le = 0 ⇒ Y − H β − e = 0 ∂λ This can be obtained βˆ .   I −1 T † T ˆ β =H Y = H H+ H Y C

(6)

Where, H † is the Moore-Penrose generalized inverse of the output matrix H of the hidden layer [7].

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The execution steps of the RELM algorithm are similar to those of the ELM algorithm, except that the regularization parameter C need to be adjusted, and the appropriate regularization parameter C is sought to achieve the purpose of simultaneously minimizing the training error and the norm of the output weight [8]. 4.2

The Introduction for RFM-RELM Algorithm

In order to cope with the changes of the crossing temperature measurement data at different times, the gas flow prediction model needs to read new data in real time for model training. However, traditional extreme learning machine will gradually lose the ability to learn as the training sample increases [9]. To this end, this paper proposes a regularized extreme learning machine based on finite memory recursive least squares algorithm (RFM-RELM). In the model training process, the algorithm removes a historical sample every time a new sample is read. The weight matrix β of the hidden layer to the output layer only depends on the latest sample of the limited number, thus avoiding the problem of “data saturation”. The RFM-RELM algorithm is derived as follows: First, let the model limit the memory length to be Q, then the formula (5) (6) is changed to ⎡

g(w1 xk + b1 ) ⎢ g(w1 xk+1 + b1 ) ⎢ H(k,k+Q−1) = ⎢ .. ⎣ .

Where,

g(w2 xk + b2 ) g(w2 xk+1 + b2 ) .. . g(w1 xk+Q−1 + b1 ) g(w2 xk+Q−1 + b2 )

⎤ · · · g(wL xk + bL ) · · · g(wL xk+1 + bL ) ⎥ ⎥ ⎥ .. .. ⎦ . . · · · g(wL xk+Q−1 + bL ) Q×L

(7)

β = [β1 , β2 , . . . , βL ]T ,Y = [yk , yk+1 , . . . , yk+Q−1 ]T

(8)

  h (k) = g(w1 xk + b1 ) g(w2 xk + b2 ) · · · g(wL xk + bL )

(9)

Add a new data to the model (xk+Q , yk+Q ), Then the current model contains a total of Q + 1 training samples from k to k + Q. According to formula (11), the implicit layer output matrix of RFM-RELM can be derived as,   I −1 T T β (k, k + Q) = H(k,k+Q) H(k,k+Q) + H(k,k+Q)Y(k,k+Q) C

(10)

Assume,

I C According to finite memory recursive least squares method can be derived as, T H(k,k+Q) + P(k,k+Q) = H(k,k+Q)

⎧ ⎪ ⎨ P−1

⎪ ⎩ β (k, k + Q) = β





hT (k+Q)h(k+Q)P−1 (k,k+Q−1) I+h(k+Q)P−1 hT (k+Q) (k,k+Q−1) −1 (k, k + Q − 1) + P(k,k+Q) hT (k + Q) (y(k + Q) − h (k + Q) β

−1 (k,k+Q) = P(k,k+Q−1)

(11)

I−

(12) (k, k + Q − 1))

The above is the recursive formula of the weight matrix β (k, k + Q) when adding a new training sample (xk+Q , yk+Q ), Next, this paper derive the recursive formula of the

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weight matrix β (k + 1, k + Q) when the old training sample (xk , yk ) is removed.   I −1 T T β (k + 1, k + Q) = H(k+1,k+Q) H(k+1,k+Q) + H(k+1,k+Q)Y(k+1,k+Q) C

(13)

Assume,

I (14) C Similarly, according to finite memory recursive least squares method can be derived as,   ⎧ −1 hT (k)h(k)P(k,k+Q) ⎨ P−1 −1 = P(k,k+Q) I + −1 (k+1,k+Q) I−h(k)P(k,k+Q) hT (k) (15) ⎩ −1 β (k + 1, k + Q) = β (k, k + Q) − P(k+1,k+Q) hT (k) (y(k) − h (k) β (k, k + Q)) T H(k+1,k+Q) + P(k+1,k+Q) = H(k+1,k+Q)

The above is the recursive formula of the weight matrix β (k + 1, k + Q) when the old training sample (xk , yk ) is removed. According to the above derivation process, the RFM-ERLM algorithm proposed in this paper can be organized as follows. Algorithm 1. RFM-RELM n STEP 1. Given a set of training data {(xi , yi )}N i=1 ⊂ R × R, Determining the excitation function g (·) of a hidden layer node, Number of hidden layer nodes L, Regularization parameter C and Limited memory length Q. STEP 2. The input weight ωi and the offset vector bi are randomly assigned, where i = 1, 2, · · · , L STEP 3. Train samples from k to k +Q−1 total Q into the model, Calculate the corresponding hidden layer output matrix H(k,k+Q−1) ,where k = 1, 2 · · · (N−Q), And preliminarily estimate the weight matrix β (k, k + Q − 1) of the corresponding hidden layer to the output layer according to formula  (15).  STEP 4. Import a new training sample xk+Q , yk+Q into the model. At this time, the model has a total of Q + 1 training samples from k to k + Q, and estimating the weight matrix β (k, k + Q) of the corresponding hidden layer to the output layer according to formula (17). STEP 5. Then export an old training sample (xk , yk ) to the model, At this time, the model has a total of Q training samples from k + 1 to k + Q, and estimating the weight matrix β (k + 1, k + Q) of the corresponding hidden layer to the output layer according to formula (20). STEP 6. Repeat STEP 4. and STEP 5. until k = N−Q, each training sample has participated in the training of the model. At this time, the model trains a total of N − Q hidden layer to output layer weight matrix βi . According to formula (4), the corresponding output value Yˆi can be estimated, where, i = 1, 2, · · · , N − Q. STEP 7. Calculate the root mean square error (RMSE) of the estimated output value Yˆi and the true value, and select the weight matrix βi corresponding to the estimated output value Yˆi of the smallest RMSE as the model optimal solution. STEP 8. Model training completed.

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5 Example Validation and Analysis The real production data which come from the blast furnace are used in this paper. There are 1000 data pairs, 900 data pairs are used as the training data and the rest 100 data pairs are the testing data. In order to indicate that the blast furnace gas flow temperature prediction model based on REM-RELM has better prediction accuracy and generalization performance, this paper also uses ELM, RELM and WOS-ELM algorithm as comparison algorithms. In this paper, training time, testing time, root mean square error (RMSE) and standard deviation (SD) are used as evaluation criteria. RMSE reflects the difference between the predicted value and the true value, which is expressed as any positive value. SD reflects the distribution of prediction error. The mathematical expressions of RMSE and SD are as follows,     2  n 2  n  Xi −Xˆi −e¯ ∑ (Xi −Xˆi )  Xi   (16) SD = ∑ RMSE = i=1 n n i=1

Where, n indicates the number of data. Xi and Xˆi represent the true and predicted values corresponding to the ith sample. e¯ indicates the mean of the associated errors. In order to establish a blast furnace gas flow temperature prediction model based on the RFM-RELM algorithm, it is necessary to determine the size of the regularization parameter C, the size of the hidden layer node L, and limited memory length Q. After many experiments, the coefficients of the model are determined to be regularization parameter C = 215 , hidden layer nodes L = 20 and defined memory Q = 50. The results is shown in the Fig. 3. 800

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(a) The prediction results of the blast furnace gas flow temperature based on RFM-RELM

-100

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(b) The relative prediction errors of the blast furnace gas flow temperature based on RFM-RELM

Fig. 3. The results of blast furnace gas flow temperature prediction model based on RFM-RELM

According to Fig. 3, it can be seen that RFM-RELM algorithm can well predict the temperature of the blast furnace gas flow. Figure 4 shows a comparison of prediction results based on ELM, RELM, WOS-ELM, and RFM-RELM. It can be known that RFM-RELM algorithm has better prediction results. The evaluation results of the

Hamiltonian Mechanics 800

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Fig. 4. The prediction results of blast furnace gas flow based on all algorithms

prediction models based on all algorithms are given in Table 1. According to Table 1, This shows that the prediction model based on RFM-RELM algorithm has better performance than other algorithms. Although RFM-RELM uses more training time and testing time, it has better performance than other algorithms. Moreover, it does not add too much computational complexity. Based on the above analysis, the prediction model based on RFM-RELM can precisely predict the temperature of blast furnace gas flow, and it also satisfy the production requirements. Table 1. Comparison of prediction results based on different algorithms Algorithm

L

ELM

20

RELM

20

WOS-ELM

10

RFM-RELM 20

C

Training time (s) Testing time (s) RMSE SD 0.0075

225 215

0.0009

0.1294 0.0461

0.0063

0.000625

0.0432 0.0154

0.0284

0.0103

0.059

5.316

0.0116

0.0384 0.0098

0.0161

6 Conclusions Aiming at the defects of traditional gas flow temperature prediction model, this paper proposes a prediction model for gas flow temperature based on RFM-RELM algorithm. Firstly, influencing factors are chosen through analyzing the distribution of blast furnace gas flow. Considering the blast furnace production data contain high frequency noise, the Fourier transform method is used for spectrum analysis, and the appropriate digital filter is selected according to the analysis result to eliminate the high frequency noise. Secondly, the prediction model of gas flow temperature is established. Regularized extreme learning machine and finite memory recursive least squares are combined to overcome Data saturation problem of ordinary extreme learning machine and prediction accuracy is improved. Finally, actual production data are used to train and test this model. Experimental result indicate that the model can quickly and accurately predict gas flow temperature, which provides effective help and support for blast furnace operators to analyze the conditions of blast furnace.

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Acknowledgments. This work was supported by National Natural Science Foundation (NNSF) of China under Grant 61673056, 61673055 and the Beijing Natural Science Foundation under Grant 4182039.

References 1. Zhou P, Song H, Wang H et al (2017) Data-driven nonlinear subspace modeling for prediction and control of molten iron quality indices in blast furnace ironmaking. IEEE Trans Control Syst Technol 25(5):1761–1774 2. Yang Y, Yin Y, Wunsch D et al (2017) Development of blast furnace burden distribution process modeling and control. In: ISIJ International: ISIJINT-2017-002 3. Fan XL, Feng HH, Yuan M (2013) PCA based on mutual information for feature selection. Control and Decision 28(6):915–919 4. Zhang K, Wu M, An J et al (2017) Relation model of burden operation and state variables of blast furnace based on low frequency feature extraction. IFAC-PapersOnLine 50(1):13796– 13801 5. Zhang H, Yin Y, Zhang S (2016) An improved ELM algorithm for the measurement of hot metal temperature in blast furnace. Neurocomputing 174:232–237 6. Bartlett PL (1998) The sample complexity of pattern classification with neural networks: the size of the weights is more important than the size of the network[J]. IEEE Trans Inf Theory 44(2):525–536 7. Zhang K, Luo M (2015) Outlier-robust extreme learning machine for regression problems. Neurocomputing 151:1519–1527 8. Lin Z, Chen M, Ma Y (2010) The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices. arXiv preprint arXiv:1009.5055 9. Mirza B, Lin Z, Toh KA (2013) Weighted online sequential extreme learning machine for class imbalance learning. Neural Process Lett 38(3):465–486

Research on Soft PLC Programming System Based on IEC61131-3 Standard Kun Zhao1 and Yonghui Zhang2(B) 1

Hainan University, Haikou 570228, Hainan, China School of Information and Communication Engineering, No. 58 Renmin Avenue, Meilan District, Haikou, Hainan, China [email protected] 2

Abstract. Aiming at the problems of traditional Programmable Logic Controller, PLC, such as software and hardware architecture closure and mutual incompatibility, according to the actual needs of the field of contemporary industrial control systems, This paper presents a research on the design of soft PLC technology based on IEC61131-3 Standard, and focuses on the programming system of soft PLC. The element design of ladder diagram and the storage method of three-level bi-directional linked list structure of ladder diagram are introduced. The transformation from trapezoidal diagram to instruction table is realized by the method of binary tree directed graph. The Flex tool and the Bison tool are used to realize the conversion of the instruction list to the target code. Finally, the communication between the programming system and the running system can be realized through Ethernet and serial port. Keywords: Soft PLC · IEC61131-3 · Ladder diagram Binary tree directed graph · Object code conversion

1

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Introduction

Traditional hardware-based PLC is widely used in numerical control systems. Although it has the advantages of strong anti-jamming ability, high reliability, easy maintenance and so on, the PLC programming languages of various manufacturers are different. Poor product compatibility and closed hardware and software architecture bring a lot of inconvenience to programmers and can not meet the requirements of open development in the field of industrial control [1]. However, with the rapid development of computer and electronic technology and the formulation and implementation of PLC programming language IEC 61131-3 Standard, a new soft PLC technology is proposed, which solves the limitations of traditional PLC. This paper mainly studies the programming part of soft PLC, which can be developed based on Visusl Studio 2015 platform, and puts forward the modular design and package. This system can use IEC 61131-3 standard of five programming languages programming and several programming languages can be c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 301–309, 2020. https://doi.org/10.1007/978-981-32-9682-4_31

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converted to each other, the programming system and running system of soft PLC can communicate with each other through Ethernet and serial port. This provides users with more normative choices and meets the current development and requirements in the field of industrial control systems.

2

Modular Design of Soft PLC Programming System

The soft PLC system structure is mainly composed of two parts: the programming system and the running system. The programming system can also be called the development system. This paper mainly explores and studies the programming system. The programming system can be divided into modular design, in which the main module is composed of editing module, compilation module, engineering management module, debugging module, simulation module and communication module. The editing module can be edited in five programming languages of IEC 61131-3 standard to make it standardized and efficient. The compilation module mainly converts the IEC 61131-3 compiled language into an instruction list, and converts the instruction list as an intermediate language into object code that the machine can directly recognize; The function of debugging module mainly includes the functions of epicycle validity, differential monitoring and online editing; In the absence of real equipment, the simulation module mainly simulates the operation of real equipment; The communication module is a communication bridge between the soft PLC programming system and the operating system, and the communication between the programming system and the running system can be performed through the Ethernet and the serial port. The overall structure of the soft PLC system is shown in Fig. 1.

Fig. 1. The overall structure of the soft PLC system.

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Design of the Editing Module of the Programming System

In the traditional PLC programming language, each manufacturer’s programming language is different, currently, this greatly increases the programming burden on users, currently, IEC 61131-3 is the world’s only programming language standard for industrial control systems, the IEC61131-3 standard is developed for the standardization of PLC programming methods. The purpose of this standard is to simplify programming methods, reduce the burden of repeated learning by users, and eliminate the irregularities between PLCs, poor compatibility and other problems, at present, more and more manufacturers produce PLCs that comply with the IEC61131-3 standard, IEC61131-3 has thus become the de facto software standard for various industrial control products. The programming system studied in this paper supports five programming languages defined in the standard: Ladder Diagram (LD), Function Block Diagram (FBD), Instruction List (IL), Structured Text (ST), and Sequential Function Chart (SFC). In line with the majority of users, the ladder language is the most widely used and the simplest graphical language in the PLC programming language [2]. This paper mainly uses ladder diagram as the first programming language to carry out research. The editing interface of this programming system can use the Document/View structure of Microsoft MFC to realize the editing interface of ladder diagram, MVC is used as the design mode to design the editing interface. Editing interface mainly consisting of title bar, toolbar, status bar, and menu bar [3]. Before writing a program, you need to open a file to create a new project, and choose which programming language, and then you need to create configuration and resources, in order to call functions and loop circulating operation, etc, so you can create a blank template for the project. This will improve the efficiency of the work, then the ladder programming in the editing area and select the corresponding primitives for drawing, and finally connect all the components by dragging their ends to form a line. A simple Ladder diagram is shown in Fig. 2.

Fig. 2. Ladder diagram.

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Element Design of Ladder Diagram

The ladder diagram is composed of three parts: the contact, the coil, and the instruction represented by the square. The contact represents the logic input condition, the coil usually represents the result of the logic output, and the block is used to represent the timer, counter or mathematical operation [2,4]. Therefore, the primitive design of the ladder diagram is mainly divided according to these three parts. Although the types of primitives of the ladder diagram are various, they contain some common features. By extracting these common features, and using the encapsulation and inheritance of the C++ language, The characteristics of polymorphism are to encapsulate the base class of a element. By inheriting and extending, the instruction elements with various characteristics can be derived, and the elements of the specific ladder diagram can be defined by changing the structure and basic properties of the element [5]. When the user is programming the ladder diagram, simply select the corresponding element in the toolbar with the mouse, and then click the left mouse button in the ladder editing area to determine an element in the editing area, selecting the element in the editing area, you can drag the element to the appropriate position to complete the drawing of the element. 2.3

Ladder Diagram Storage Structure Design

The ladder diagram is composed of various elements that follow the grammatical rules and logic in the ladder diagram, he ladder program is divided into several networks, and these networks are numbered in units of networks, in the network, logical operations are performed from left to right. However, during the editing process, the number of networks and elements in the ladder diagram and the type of elements are uncertain, therefore, in the ladder editing process, a dynamic allocation storage space is needed to save the ladder diagram, by analyzing and studying this feature, the linked list structure can be used to realize dynamic allocation of storage space [6].

Fig. 3. The structure of the three-level doubly linked list.

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This paper proposes to store the ladder diagram by a two-way three-level linked list structure, which is mainly composed of three parts: the element linked list, the row linked list and the network linked list. By using the pointer method, the ladder diagram storage can be completed very efficiently [7]. The structure of the three-level doubly linked list is shown in Fig. 3. 2.4

Ladder Diagram Logic Check Module Design

In ladder diagram programming, we should not only abide by the grammar rules of ladder diagram language but also conform to the logical rules, in addition to the programmer’s self-programming error correction experience, the system can design a set of error classes to summarize these error types by the common syntax and logic errors in ladder diagram programming. When there are syntax and logic errors in the ladder diagram, the software will display the type and location of the error in the form of a block diagram, and the programmer can modify it according to the error information [8]. The logic error detection flowchart of the ladder diagram is shown in Fig. 4.

Fig. 4. The structure of the three-level doubly linked list.

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Ladder Diagram Programming Module Design

Ladder diagram is the most widely used PLC graphics programming language, which is intuitive, easy to understand and easy to learn, however, the ladder language cannot be directly recognized by the PLC in the running system, which needs to be transformed into the target code that can be directly recognized by the PLC. Therefore, the compilation of the ladder diagram is actually the process of converting the ladder diagram into the target code that the PLC device can directly recognize. 3.1

Ladder Diagram Logic Check Module Design

In the IEC 61131-3 programming language, the instruction list is a literal language and has similar features to the assembly language of a microcomputer, It is close to the target code that the CPU can directly recognize, so the process of converting the ladder diagram into the target code can be divided into two steps. First, the ladder diagram can be converted into an instruction list as an intermediate language, and then the intermediate language is converted into the target code, which simplifies the compilation process and enables efficient conversion of target code. There are many conversion methods from ladder diagram to instruction list. This paper mainly studies the conversion method based on binary tree directed graph. Firstly, the ladder diagram needs to be stored in the form of a binary tree directed graph and the positional relationship and the series-parallel relationship of the elements of the ladder diagram can be obtained, then the instruction List is generated by scanning the logical binary tree and using the traversal access method. This method is simple to operate and improves the efficiency of compilation [9,10]. The ladder diagram is transformed into a binary tree directed graph is shown in Fig. 5.

Fig. 5. Ladder diagram and binary tree directed graph.

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Target Code Conversion

Although the instruction list is similar to the assembly language in the machine, it cannot be directly recognized by the PLC device, therefore, the instruction list before running system work, the instruction list source program needs to be converted into machine-recognizable object code. The conversion process of the object code is mainly composed of four stages: lexical, grammar, semantic analysis and generation of object code [11]. The target code conversion flowchart is shown in Fig. 6.

Fig. 6. The target code conversion flowchart.

In the process of target code transformation, Flex tools and Bison tools can be used for lexical and grammatical analysis and interpretation respectively. Flex is a lexical analyzer generator, and Bison is a syntactic analyzer generator, which makes it easy to convert a instruction list into a binary file that the CPU can read [12].

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Communication Module Design

The communication module is the communication bridge between the programming system and the running system of the soft PLC. It is also the only way to

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download the target code and is an important part of the soft PLC, and because most PLC devices now support the Modbus communication protocol, the soft PLC mainly studied in this paper also supports the Modbus communication protocol [13]. When your PLC device belongs to Modbus TCP device, the soft PLC can use Ethernet to communicate with your machine, while the Modbus RCP device can only communicate with the soft PLC through the serial port, usually using RS232 and RS485 serial port line.

5

Conclusions

This paper describes the design method of the soft PLC programming system based on the Visusl Studio 2015 platform, which is divided into modular design, this paper introduces the ladder diagram storage method and the ladder diagram to the instruction list conversion and the instruction list to the target code conversion method, this design method has the advantages of easy operation, high efficiency. Compared with traditional PLC, the soft PLC technology will greatly facilitate the work of programmers, thus improving work efficiency and promoting the development of industrial control. Acknowledgments. This work was supported by the Research on Intelligent Network Controller Based on IPv6(NGII20160319).

References 1. Liao CC (2010) S7–200 PLC programming and application. Machinery Industry Press, Beijing 2. Liao CC (2013) PLC programming and application. Mechanical Industry Press, Beijing 3. Zhang YY (2013) Research and implementation of PLC function block compilation system based on IEC61131-3 standard. Dalian University of Technology 4. Guo SJ (2010) Research and implementation of soft PLC ladder diagram programming system. Graduate University of Chinese Academy of Sciences 5. Zhang K, Yang T (2018) Research and implementation of soft PLC development system based on IEC61131-3 standard. Manuf Autom 11(40):123–126 6. Racchetti L, Fantuzzi C, Tacconi L, et al (2015) Towards an abstraction layer for PLC programming using object-oriented features of IEC61131-3 applied to motion control. In: IECON 2015-41st annual conference of the IEEE industrial electronics society, pp 000298–000303. IEEE 7. Su S (2012) Research and implementation of soft PLC ladder programming system. South China University of Technology 8. Li H (2014) Design and implementation of IL language compiler based on IEC61131-3 standard. Xi’an Electronic Science and Technology University 9. Wang L, Zhang K (2018) Research on the conversion method of soft PLC ladder diagram to instruction list. Sci Technol Innov Appl 7:80–82 10. Wu Y, Lu Y, Xu Q (2015) Research on soft PLC ladder diagram editing and instruction list generation method. J Hefei Univ Technol: Nat Sci Ed 38(10):1353– 1357

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11. Sarker IH, Apu K (2014) MVC architecture driven design and implementation of Java framework for developing desktop application. Int J Hybrid Inf Technol 7(5):317–322 12. Wang Y, Chen Y (2011) Research on soft PLC compilation module based on flex and bison. Manuf Autom 33(17):76–79 13. Goldenberg N, Wool A (2013) Accurate modeling of modbus/TCP for intrusion detection in SCADA systems. Int J Crit Infrastruct Prot 6(2):63–75

Periodic Dynamic Event-Triggered Bipartite Consensus for Multi-agent Systems Associated with Signed Graphs Junjian Li, Xia Chen(B) , and Yanbing Tian School of Information and Control Engineering, Qingdao University of Technology, Qingdao 266520, China [email protected], [email protected], [email protected]

Abstract. In this paper, we propose a periodic dynamic event-triggered control to solve the bipartite consensus problem of first-order multi-agent systems. We design an extra variable for each agent and construct a dynamic event-triggered condition based on it. By using Lyapunov stability method, we prove that all agents can reach bipartite consensus under connected balanced signed graph. The periodic control mechanism excludes the possibility of Zeno behavior. Finally, simulations demonstrate the effectiveness of the obtained results. Keywords: Multi-agent systems · Bipartite consensus · Event-triggered control · Dynamic triggering condition

1 Introduction Multi-agent systems usually represent that multiple agents cooperate to complete the tasks beyond the capacity of a single agent. Consensus, as a basic cooperative behavior of multi-agent systems, means that all agents converge to one common quantity. In the past few years, there has been increasing efforts referring to the consensus problems of multi-agent systems due to its considerable applications in many fields including formation control and distributed robotic systems, see [1–5] and the references therein. In recent years, the event-triggered control method was developed. Event-triggered control was applied to solve the consensus problem due to its superiority in decreasing the frequency of controller updates and unnecessary cost of communication resources, see [6, 7]. In the event-triggered scenario, the information exchange is determined by a designed triggering condition. In [6], an event-triggered consensus mechanism with state-dependent thresholds was proposed. In [7], it discussed the distributed average consensus problem of multi-agent systems based on the triggering condition with exponential decay rate. It is worth pointing out that the references above investigated the continuous event-triggered control. Differently, the periodic event-triggered control scheme was proposed for multi-agent systems in [8], where the triggering condition is verified periodically by a fixed time interval. In [9], the periodic event-triggered control mechanism was provided for multiple non-holonomic wheeled mobile robots to solve the hand c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 310–321, 2020. https://doi.org/10.1007/978-981-32-9682-4_32

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position centroid consensus/formation problems. In [10], it focused on designing a periodic event-triggered consensus protocol for multi-agent systems with communication time delays. Obviously, under the periodic event-triggered control, the inter-event times are lower bounded by the verification period and Zeno behavior is essentially excluded. Recently, a new class of event-triggered mechanism is presented, which is called as dynamic event-triggered control since it introduces an internal dynamic variable into the triggering condition [11]. The dynamic event-triggered control for multi-agent systems was proposed in [12, 13]. The proposed triggering condition of each agent involves extra dynamic variables which plays an essential role to guarantee that the triggering time sequence does not exhibit Zeno behavior. Moreover, some existing triggering conditions are special cases of the dynamic triggering conditions [13]. The multi-agent systems with cooperative relationship are usually discussed. The cooperative interaction networks are normally modeled by graphs whose edge weights are all positive. However, the relationship ‘competition’ is universal in real-world circumstances [14–16]. That is to say, one should examine a more general network containing both cooperative and competitive interactions, which can be presented by a signed graph with both positive and negative adjacency weights. For signed graphs, the bipartite consensus problem was investigated in [17–19], which means that all agents converge to the same value but opposite sign. However, works on bipartite consensus with event-triggered control are still rare. The centralized event-triggered control of multi-agent systems for bipartite consensus was researched in [20] and [21]. However, the centralized control methods are detrimental to resource conservation. In [22], the authors considered the distributed event-triggered control for multi-agent systems but only proved that at least one agent does not have Zeno behavior. Thus, for bipartite consensus with the event-triggered control, the in-depth investigations are needed. Motivated by the analysis above, this paper proposes a periodic dynamic eventtriggered mechanism to solve the bipartite consensus problem for multi-agent systems with signed graphs. The contributions of this paper are as follows: (i) The bipartite consensus, instead of consensus, can be achieved by using an event-triggered scheme for agents with both cooperative and competitive relationships. (ii) The proposed control law is based on local sampled states and the triggering condition contains an extra dynamic variable. (iii) By applying Lyapunov stability theory, we derive the sufficient ranges of the verification period and the triggering condition parameters to guarantee bipartite consensus. Moreover, the periodic control mechanism excludes Zeno behavior for all agents. The remainder of this paper is organised as follows. In Sect. 2, some preliminaries are summarized, and the main aim of our paper is formulated. In Sect. 3, the periodic dynamic event-triggered control is presented to solve the bipartite consensus problem. In Sect. 4, the numerical example is given to illustrate our theoretical results. Finally, the paper is concluded in Sect. 5.

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2 Preliminaries 2.1

Signed Graphs

The interaction network among agents is described by a signed graph G = (V , E , A ), where V = {1, 2, ..., n} is the node set, E ⊆ V ×V is the edge set, and A = [ai j ] ∈ Rn×n is the adjacency matrix of the signed weights of G where ai j = 0 ⇔ ( j, i) ∈ E and / E . Assume there is no self-loops, i.e., aii = 0, ∀i ∈ V . An edge ( j, i) ai j = 0 ⇔ ( j, i) ∈ indicates node i can receive information from node j. The edge set E = E + ∪ E − , where E + = {( j, i)|ai j > 0} and E − = {( j, i)|ai j < 0} are the sets of positive and negative edges, respectively. The neighbor set of node i is denoted by Ni = { j ∈ V | ( j, i) ∈ E , j = i}. Denote |Ni | as the number of neighbors of agent i. A path from node i to node j is a sequence of edges in the form of (i, ik1 ), (ik1 , ik2 ),..., (ikl , j) with distinct nodes ikm , m = 1, ..., l, i = j. For undirected graph G, if there is a path between any two nodes of G , G is called connected. The Laplacian matrix L of a signed graph is defined as L = diag(

∑

k∈N 1

|a1k |,

∑

k∈N 2

|a2k |, ...,

∑

|ank |) − A .

(1)

k∈N n

For a signed graph G , if there exists a bipartition V1 , V2 of nodes, where V1 ∪V2 = V and V1 ∩ V2 = , such that ai j ≥ 0 for i, j ∈ Vl (l ∈ {1, 2}) and ai j ≤ 0 for i ∈ Vl , j ∈ Vq , l = q(l, q ∈ {1, 2}), then it is said that G is structurally balanced. For many practical scenarios, structurally balanced graphs generally exist. Moreover, we have the following well-known property of structurally balanced signed graphs. Lemma 1. If a signed graph G is structurally balanced, there exists a diagonal matrix D = diag(σ1 , σ2 , . . . , σn ), the entries of DA D are all nonnegative, i.e., DA D ≥ 0, where σi ∈ {1, −1}, ∀i ∈ V . The corresponding matrix D is called Gauge Transformation in [15]. 2.2

Bipartite Consensus

Consider a group of n single-integrator agents under a signed graph G . Each agent is regarded as a node in G and modeled as x˙i (t) = ui (t),

i = 1, 2, 3, . . . , n,

(2)

where xi (t) ∈ R is the state of agent i and ui (t) ∈ R is the control law, respectively. Assume the signed graph G is undirected in this paper. We aim to construct an eventtriggered control scheme to achieve bipartite consensus, i.e., lim xi (t) = σi d, ∀i ∈ V , t→∞ where σi is defined in Lemma 1, d is the same absolute value of the final consensus states of all agents and d = 1n ∑i σi xi (t).

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3 Main Results In this section, we will design a periodic dynamic triggering mechanism to determine the event time sequence, where the triggering condition is verified periodically. And we will prove that the system can achieve bipartite consensus. Under the periodic event-triggered control mechanism, choose a constant h as the verification period and let h, 2h, 3h, · · · be the verification time instants. At the verification time instants when the triggering condition of the agent is satisfied, the state of the agent is sampled and transmitted to its neighbors without delays. The control laws of the corresponding agent and its neighbors are updated. Denote the event time sequence of agent i as {kli h}, where l represents the lth event time instant. Obviously, the event time sequence is a sub-sequence of the verification time sequence. Then, we get the control law of agent i: ui (t) = −

∑

j∈N i

i |ai j |(xi (kli h) − sign(ai j )x j (klj h)),t ∈ [kli h, kl+1 h),

(3)

where klj h represents the last event time of agent j before t (including t). Obviously, the i h). control law is piecewise constant during the time interval [kli h, kl+1 To simplify the analysis, we substitute xi (k) for xi (kh) and denote xˆi (k) = i h), i = 1, 2, ..., n. xi (kli h), kh ∈ [kli h, kl+1 Combining the system (2) with the control law (3), the original system (2) is discretized into: xi (k + 1) = xi (k) − h

∑

|ai j |(xˆi (k) − sign(ai j )xˆ j (k)).

(4)

j∈N i i

i ) (Substitute k and k for Denote ei (k) = xˆi (k) − xi (k) for k ∈ [kli , kl+1 l i kh and kl h, respectively). Define the augmented vectors x(k) = [x1 (k), . . . , ˆ = [xˆ1 (k), . . . , xˆn (k)]T and e(k) = [e1 (k), . . . , en (k)]T = x(k) ˆ − x(k). Then, xn (k)]T , x(k) we have

x(k + 1) = x(k) − hLx(k). ˆ

(5)

Moreover, some intermediate variables are introduced. Denote zi (k) = σi xi (k) and i ). Similarly, define the augmented vectors z(k) = ezi (k) = zˆi (k) − zi (k) for k ∈ [kli , kl+1 T [z1 (k), . . . , zn (k)] , zˆ(k) = [ˆz1 (k), . . . , zˆn (k)]T and ez (k) = [ez1 (k), . . . , ezn (k)]T . Then we obtain z(k) = Dx(k), so that z(k + 1) = z(k) − hLD zˆ(k),

(6)

where LD = DLD, D−1 = D. The consensus analysis of x(k) can be transformed into the consensus analysis of z(k), since x(k) achieves bipartite consensus while z(k) achieves consensus. In the following, we present two useful lemmas. Lemma 2. Assume G is structurally balanced. The average quantity of zi (k), i.e., z¯ = 1 n ∑i zi (k) is time-invariant. z¯ also represents the signed average quantity of xi (k), i.e., z¯ = 1n ∑i σi xi (k).

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Proof. Due to z¯ = 1n ∑i zi (k) = 1n 1T z(k), it yields Δ z¯ = z(k + 1) − z(k) = 1n 1T z(k + 1) − 1 T 1 T T n 1 z(k) = n 1 (−hLD zˆ(k)). As 1 LD = 0, one has Δ z¯ ≡ 0. The proof is completed. Lemma 3. Assume G is structurally balanced, undirected and connected. For the Laplacian matrix LD defined in (6), the vector δ (t) satisfying 1T δ (t) = 0, the propδ T (t)LD δ (t) T δ (t)=0 δ (t)δ (t)

erty min

= λ2 (LD ) holds, where λ2 (LD ) is the smallest positive eigenvalue

of LD . Proof. From the Example 1 in [15], by the Gauge Transformation D, the original structurally balanced signed graph G can be converted to a connected undirected graph, the Laplacian matrix of which is LD . Then, from Courant-Fischer Theorem in [23], the proof is completed. We propose the following extra dynamic variable ηi to agent i:

ηi (k + 1) = (1 − βi )ηi (k),

(7)

where ηi (k + 1) = ηi ((k + 1)h), ηi (k) = ηi (kh), β ∈ (0, 1), ηi (0) > 0 and ηi (k + 1) = (1 − βi )k ηi (0) > 0. Note that σi σi = σ j σ j = 1, zi (k) = σi xi (k) and ezi (k) = σi ei (k). The triggering condition is constructed as

ηi (k) + (

ρi qˆi (k) − e2i (k)) < 0, 4 ∑ |ai j |

(8)

j∈N i

where ρi ∈ (0, 1) and qˆi (k) = ∑ |ai j |(xˆi (k) − xˆ j (k))2 . Then, we have j∈N i

i = inf{t > rli , ηi (k) + ( rl+1

ρi qˆi (k) − e2i (k)) < 0}, 4 ∑ |ai j | j∈N i

which indicates

ηi (k) + (

ρi qˆi (k) − e2i (k)) ≥ 0. 4 ∑ |ai j |

(9)

j∈N i

Based on the above analysis, the main results are proposed in the following theorem. Theorem 1. Suppose the communication signed graph G is structurally balanced. Then, the bipartite consensus of the system (4) can be achieved by the control law (3) and the triggering condition (8), if the verification period h satisfies

0 < h < min{

1 − ρmax βmin , }, 2|Nmax |a˜max 2|Nmax |a˜max

(10)

where ρmax = maxi {ρi }, βmin = mini {βi }, |Nmax | = maxi {|Ni |} and a˜max is the maximum element of DA D.

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Proof. In order to prove the bipartite consensus of the system (5), we firstly prove the consensus of the system (6). For the system (6), we choose a candidate Lyapunov function as n

W (k) = V (k) + ∑ ηi (k), i=1

where V (k) = (z(k)− z¯ ·1)T (z(k)− z¯ ·1). Then the difference of V (k) along the trajectory of the system (6) and (7) is n

Δ W (k) = W (k + 1) −W (k) = Δ V (k) + ∑ Δ ηi (k) i=1

n

= zT (k + 1)z(k + 1) − zT (k)z(k) + ∑ (ηi (k + 1) − ηi (k)) i=1

n

= (z(k) − hLD zˆ(k))T (z(k) − hLD zˆ(k)) − zT (k)z(k) + ∑ (ηi (k + 1) − ηi (k)) i=1

n

T = −2hzT (k)LD zˆ(k) + h2 zˆT (k)LD LD zˆ(k) + ∑ (ηi (k + 1) − ηi (k)) i=1

n

T = −2h(ˆz(k) − ez (k))T LD zˆ(k) + h2 zˆT (k)LD LD zˆ(k) + ∑ (ηi (k + 1) − ηi (k)) i=1

n

n 1 = −2h ∑ ∑ |ai j |(ˆzi (k) − zˆ j (k))2 + 2h ∑ ∑ |ai j |ezi (k)(ˆzi (k) − zˆ j (k)) i=1 j∈N i 2 i=1 j∈N i n

+ h2 ∑ (

∑

i=1 j∈N i

n

|ai j |(ˆzi (k) − zˆ j (k)))2 + ∑ (−βi ηi (k)).

(11)

i=1

Note that ezi (k) = σi ei (k), qˆzi (k) = ∑ [|ai j |σi2 xˆi2 (k) − 2σi σ j ai j σi xˆi (k)σ j xˆ j (k) + j∈N i

|ai j |σ 2j xˆ2j (k)] = ∑ |ai j |(ˆzi (k) − zˆ j (k))2 = σi2 qˆi (k). Recalling the inequality (9), it is j∈N i

equivalent to

ηi (k) ≥ σi2 e2i (k) −

ρi σ 2 qˆi (k) 4 ∑ |ai j | i j∈N i

= e2zi (k) −

ρi qˆzi (k). 4 ∑ |ai j |

(12)

j∈N i

Applying the inequality ab ≤ a2 + 14 b2 and recalling the inequality (11), one has

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J. Li et al. n

1 |ai j |(ˆzi (k) − zˆ j (k))2 i=1 j∈N i 2

Δ W (k) ≤ −2h ∑ n

∑

1 |ai j |[e2zi (k) + (ˆzi (k) − zˆ j (k))2 ] 4 i=1 j∈N i

+ 2h ∑ n

+ h2 ∑ (

∑

n

∑

|ai j |(ˆzi (k) − zˆ j (k)))2 + ∑ (−βi ηi (k)).

i=1 j∈N i

(13)

i=1

p p yk )2 ≤ p ∑k=1 y2k From the Cauchy-Schwarz inequality, we can deduce that (∑k=1 for any positive integer p and any real number y p . Then, it obtains

(

∑

|ai j |(ˆzi (k) − zˆ j (k)))2 ≤ |Ni |

j∈N i

∑

|ai j |2 (ˆzi (k) − zˆ j (k))2

j∈N i

≤ |Ni |a˜m i

∑

|ai j |(ˆzi (k) − zˆ j (k))2 ,

j∈N i

where a˜m i is the maximum element in the ith row of DA D. Combining (14) with (13), it has n

1 |ai j |(ˆzi (k) − zˆ j (k))2 i=1 j∈N i 2

Δ W (k) ≤ −2h ∑ n

∑

1 |ai j |[e2zi (k) + (ˆzi (k) − zˆ j (k))2 ] 4 i=1 j∈N i

+ 2h ∑

∑

n

+ h2 ∑ |Ni |a˜m i i=1 n

∑

n

j∈N i n

a˜i j (ˆzi (k) − zˆ j (k))2 + ∑ (−βi ηi (k))

≤ −h ∑ qˆzi (k) + 2h ∑

i=1

∑

i=1 j∈N i

i=1

|ai j |e2zi (k)

n

n n 1 + h ∑ qˆzi (k) + h2 |Nmax |a˜max ∑ qˆzi (k) + ∑ (−βi ηi (k)). 2 i=1 i=1 i=1

Considering (12), one has n

n

Δ W (k) ≤ −h ∑ qˆzi (k) + 2h ∑

∑

|ai j |[ηi (k) +

i=1 j∈N i

i=1

n

n

ρi qˆzi (k)] 4 ∑ |ai j | j∈N i n

1 + h ∑ qˆzi (k) + h2 |Nmax |a˜max ∑ qˆzi (k) + ∑ (−βi ηi (k)) 2 i=1 i=1 i=1 n 1 ≤ [−h + h(ρmax + 1) + h2 |Nmax |a˜max ] ∑ qˆzi (k) + (2ha˜max |Nmax | 2 i=1 n

− βmin ) ∑ ηi (k) i=1

n

= −2ζ zˆT (k)LD zˆ(k) − ξ ∑ ηi (k), i=1

(14)

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where βmin = mini {βi }, ζ = h − 12 h(ρmax + 1) − h2 |Nmax |a˜max , ξ = −2ha˜max |Nmax | + n

βmin and ∑ qˆzi (k) = 2ˆzT (k)LD zˆ(k). i=1

Furthermore, it obtains n

Δ W (k) ≤ −2ζ zˆT (k)LD zˆ(k) − ξ ∑ ηi (k) i=1

n

≤ −2ζ (ˆz(k) − z¯ · 1)T LD (ˆz(k) − z¯ · 1) − ξ ∑ ηi (k) i=1

(i)

n

≤ −2ζ λ2 (LD )Vˆ (k) − ξ ∑ ηi (k), i=1

where Vˆ (k) = (ˆz(k) − z¯ · 1)T (ˆz(k) − z¯ · 1), (i) is due to the fact 1T (ˆz(k) − z¯ · 1) = 0 and Lemma 3. Then, we have W (k + 1) = W (k) + Δ W (k) ≤ W (0) + Δ W (0) + · · · + Δ W (k − 1) + Δ W (k) n

= V (0) + ∑ ηi (0) − 2ζ λ2 (LD )Vˆ (0) i=1

n

− ξ ∑ ηi (0) − · · · − 2ζ λ2 (LD )Vˆ (k − 1) i=1 n

n

i=1

i=1

− ξ ∑ ηi (k − 1) − 2ζ λ2 (LD )Vˆ (k) − ξ ∑ ηi (k).

(15)

From the condition (10), we know that in the inequality (15) −2ζ λ2 (LD )Vˆ (k) ≤ 0, −ξ

n

∑ ηi (k) < 0.

i=1

Due to V (k) ≥ 0 and ηi (k) > 0, it implies W (k) ≥ 0. Considering the equality (7), ηi (k) → 0 when k → ∞. From W (k) ≥ 0, Vˆ (k) ≥ 0 and lim ηi (k) = 0, it can obtain k→∞

that lim Vˆ (k) = 0 by the inequality (15). Hence, it means that lim zˆ(k) = z¯. Then, k→∞

k→∞

lim LD zˆ(k) = 0. By the system (6) and the triggering condition (8), one has lim ez (k) =

k→∞

k→∞

0, i.e., lim z(k) = lim zˆ(k) = z¯ · 1. Recalling the equality z(t) = z(k) − (t − kh)LD zˆ(k), k→∞

k→∞

t ∈ (kh, (k + 1)h), we get lim z(t) = lim z(k) = z¯ · 1. In other words, xi (t) → σi z¯. The t→∞

proof is completed.

k→∞

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Remark 1. We call (8) a dynamic triggering condition since it involves the extra dynamic variables ηi (k). We have a static triggering condition when ηi (k) = 0, i.e., ρi 2 4 ∑ a˜i j qˆi (k) − ei (k) ≤ 0. Therefore, we can consider the static triggering condition as j∈N i

a specific case of the dynamic triggering condition (8). Moreover, for a given state of the system, the next event time instant decided by the dynamic triggering condition is larger than that decided by the static triggering condition. This can be easily explained as the existence of ηi (k) > 0 in the dynamic triggering condition.

4 Simulations In this section, a numerical example is given to illustrate the effectiveness of theoretical results. For the interaction graph shown in Fig. 1, the Laplacian matrix L is ⎤ ⎡ 7 −2 0 4 1 0 ⎢ −2 6 −1 0 3 0 ⎥ ⎥ ⎢ ⎢ 0 −1 1 0 0 0 ⎥ ⎥ L=⎢ ⎢ 4 0 0 4 0 0 ⎥ ⎥ ⎢ ⎣ 1 3 0 0 6 −2 ⎦ 0 0 0 0 −2 2

Fig. 1. Communication topology.

We choose h = 0.008 as the verification period. The initial states of agents are chosen as x(0) = [3 − 1 2 1 − 1.5 0.3]T . The parameters of the equality (7) and the triggering condition (8) are selected with ηi (0) = 7, βi = 0.1, ρi = 0.7. Figure 2 illustrates the states evolution of each agent. It is shown that the states of all agents achieve bipartite consensus, i.e., the states of agent 1, 2, 3 converge to 0.7 while the states of agent 4, 5, 6 converge to −0.7. Figure 3 demonstrates the corresponding inter-event times and it shows that there is no Zeno behavior.

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3 2.5

The state trajectories

2 1.5 1 0.5 0 -0.5 -1 -1.5 0

1

2

3

4

5

6

Time (sec)

Fig. 2. State responses under the control law (3) and the triggering condition (8). 1

2

0.5

1

Sampling periods (sec)

0

0 0

2

4

6

0

Time (sec)

2

4

6

Time (sec)

0.5

0.4 0.2

0

0 0

2

4

6

0

Time (sec)

2

4

6

Time (sec)

1

0.4

0.5

0.2

0

0 0

2

4

Time (sec)

6

0

2

4

6

Time (sec)

Fig. 3. The inter-event times of agents (The dynamic periodic event-triggered control).

When ηi (0) = 0, we obtain a static triggering condition. The initial states of agents and other parameters are set the same as those in the dynamic event-triggering control. State responses and the inter-event times of agents under the control law (3) and the static triggering condition are shown in Figs. 4 and 5, respectively. The average inter-time instants of the system under dynamic and static periodic event-triggered control are 0.3015 and 0.2888, respectively. It implies that the dynamic event-triggered control can save more communication resources to some extend.

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The state trajectories

2 1.5 1 0.5 0 -0.5 -1 -1.5 0

1

2

3

4

5

6

Time (sec)

Fig. 4. State responses under the control law (3) and the static triggering condition. 1

2

0.5

1

Sampling periods (sec)

0

0 0

2

4

6

0

Time (sec)

2

4

6

Time (sec)

0.5

0.4 0.2

0

0 0

2

4

6

0

Time (sec)

2

4

6

Time (sec)

1

0.4

0.5

0.2

0

0 0

2

4

Time (sec)

6

0

2

4

6

Time (sec)

Fig. 5. The inter-event times of agents (The static periodic event-triggered control).

5 Conclusions In this paper, we investigate the bipartite consensus problem of multi-agent systems via a periodic dynamic event-triggered control mechanism. We design an extra dynamic variable for each agent to construct a dynamic triggering condition. The proposed control scheme is based on local information and sampled states of neighbors. On the basis of Lyapunov theory, we prove that the system can reach bipartite consensus. Finally, numerical simulations illustrate the effectiveness of the theoretical results. Acknowledgments. This work is supported by National Nature Science Foundation of China under Grant 61703225, Shandong Provincial Natural Science Foundation under Grant ZR2017BF033, and A Project of Shandong Province Higher Educational Science and Technology Program under Grants J17KA060 and J16LN29.

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References 1. Ren W, Beard RW, Atkins EM (2007) Information consensus in multivehicle cooperative control. IEEE Control Syst 27(2):71–82 2. Shen B, Wang Z, Hung YS (2010) Distributed H∞ -consensus filtering in sensor networks with multiple missing measurements: the finite-horizon case. Automatica 46(10):1682–1688 3. Abdessameud A, Tayebi A (2011) Formation control of VTOL unmanned aerial vehicles with communication delays. Automatica 47(11):2383–2394 4. Wu Y, Su H, Shi P, Lu R (2017) Output synchronization of nonidentical linear multiagent systems. IEEE Trans Cybern 47(1):130–141 5. Lu R, Shi P, Su H, Wu ZG, Lu J (2018) Synchronization of general chaotic neural networks with nonuniform sampling and packet missing: a switched system approach. IEEE Trans Neural Netw Learn Syst 29(3):523–533 6. Dimarogonas DV, Frazzoli E, Johansson KH (2012) Distributed event-triggered control for multi-agent systems. IEEE Trans Autom Control 57(5):1291–1297 7. Seyboth GS, Dimarogonas DV, Johansson KH (2013) Event-based broadcasting for multiagent average consensus. Automatica 49(1):245–252 8. Meng X, Chen T (2013) Event-based agreement protocols for multi-agent networks. Automatica 49(7):2125–2132 9. Chen X, Hao F, Ma B (2017) Periodic event-triggered cooperative control of multiple nonholonomic wheeled mobile robots. IET Control Theory Appl 11(6):890–899 10. Wang A (2016) Event-based consensus control for single-integrator networks with communication time delays. Neurocomputing 173:1715–1719 11. Girard A (2015) Dynamic triggering mechanisms for event-triggered control. IEEE Trans Autom Control 60(7):1992–1997 12. Ge X, Han Q (2017) Distributed formation control of networked multi-agent systems using a dynamic event-triggered communication mechanism. IEEE Trans Ind Electr 64(10):8118– 8127 13. Yi X, Liu K, Dimarogonas DV, Johansson KH (2017) Distributed dynamic event-triggered control for multi-agent systems. IEEE Trans Autom Control. https://doi.org/10.1109/TAC. 2018.2874703 14. Facchetti G, Iacono G, Altafini C (2011) Computing global structural balance in large-scale signed social networks. Proc National Acad Sci USA 108(52):20953–20958 15. Fiedler B, Gedeon T (1999) A Lyapunov function for tridiagonal competitive-cooperative systems. SIAM J Math Anal 108:469–478 16. Easley D, Kleinberg J (2010) Networks, crowds, and markets: reasoning about a highly connected world. Cambridge University Press, Cambridge 17. Hou Y, Li J, Pan Y (2003) On the Laplacian eigenvalues of signed graphs. Linear Multilinear Algebra 51(1):21–30 18. Altafini C (2013) Consensus problems on networks with antagonistic interactions. IEEE Trans Autom Control 58(4):935–946 19. Valcher ME, Misra P (2014) On the consensus and bipartite consensus in high-order multiagent dynamical systems with antagonistic interactions. Syst Control Lett 66:94–103 20. Zhou Y, Hu J (2013) Event-based bipartite consensus on signed networks. In: Proceedings of IEEE international conference on cyber technology in automation, control and intelligent systems, Nanjing, China, pp 326–330 21. Ma C, Sun W (2018) Bipartite consensus for multiagent systems via event-based control. Math Problems Eng. https://doi.org/10.1155/2018/8046576 22. Zeng J, Li F, Qin J, Zheng W (2015) Distributed event-triggered bipartite consensus for multiple agents over signed graph topology. In: Proceedings of the 34th Chinese control conference, Hangzhou, China, pp 6930–6935 23. Horn RA, Johnson CR (1987) Matrix Analysis. Cambridge University Press, Cambridge

Random Multi-scale Gaussian Kernels Based Relevance Vector Machine Yinhe Gu, Xuemei Dong(B) , Jian Shi, and Xudong Kong School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China [email protected]

Abstract. This paper establishes random multi-scale Gaussian kernels based relevance vector machine (RMGK-RVM) for regression learning problems. The mixture of multiple Gaussian kernels with different scale parameters sampled from some predefined distribution is used as the model. Under the Bayesian inference framework, RMGK-RVM can learn the whole distribution of the prediction variable. In this way, the uncertainties of input data are fully considered and the prediction accuracy of the target variable is improved for complicated data. The experimental results on one simulation data and three real-life data sets show that the proposed method performs favorably. Keywords: Relevance vector machine (RVM) · Kernel methods · Bayesian learning · Multi-scale kernels

1 Introduction Kernel methods have been widely studied in the machine learning community [1], whose best-known member is the support vector machine (SVM, [2]). The main characteristic of a kernel method is that it avoids the explicit mapping which is needed to get linear learning algorithms to learn a nonlinear function or decision boundary. Generally, its generalization performance is controlled by the choice of the kernel, namely, different kernel functions have different learning abilities. When facing with complicated application scenarios, such as multi-task learning [3], multiple kernel learning (MKL) algorithms are needed to make up the limited flexibility caused by a single kernel [4, 5]. However, the selecting of the basis kernels and the realization of optimization algorithms are two difficult problems in the previous MKL research. On the other hand, the robust data modeling techniques have received considerable attention in the field of applied statistics and machine learning [6], since sample data collected from the various sensors may be contaminated by some noises or outliers. Inspired by the Bayesian methods, which introduce uncertainty measures into the model, and combine the prior knowledge with observed data, Tipping [7] proposed the relevance vector machine (RVM). Instead of finding a single pointwise estimator of the target value according to some given criterion, the author defined the prior distributions over the combined weights of the kernel functions and found an entire predictive c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 322–329, 2020. https://doi.org/10.1007/978-981-32-9682-4_33

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distribution over them. However, as mentioned earlier, a single kernel in RVM cannot guarantee a satisfying prediction for solving regression problems with complicated data. In this paper, we propose the random multi-scale Gaussian kernels based relevance vector machine (RMGK-RVM), which combines organically the multiple kernels method and the Bayesian framework. The scale parameters for different Gaussian kernels are sampled from a uniform distribution, whose characteristic parameters can be determined by the cross-validation technique through the learning process. This provides an effective solution for the difficult and significant setting problem in MKL learning. Moreover, by setting the automatic relevance determination (ARD) [8] priors on the regression coefficients and using the Bayesian inference theory, we can obtain a model with more sparsity as well as improved learning performance than the classical RVM. The following notation is used throughout this paper. Vectors are denoted by boldface lowercase letters, e.g., x, while matrices are denoted by boldface uppercase letters, e.g., A. All vectors are assumed to be the column vectors unless specifically stated. Symbol xi denotes the i-th element of the vector x, and Ai, j the (i, j) entry of the matrix A.

2 Related Work Considering a regression problem, for the given data D = {(xi ,ti )|i = 1, . . . , N}, where the input variable xi ∈ X ⊂ Rd and its output ti ∈ T = [−M, M] for some M > 0, our goal is to find an appropriate model from some pre-chosen hypothesis function space H to learn the regressor defined as t¯ = Et|x [t],

x ∈ X,

under the assumption that the samples in D are independent and identically distributed. Different choices of H essentially determine different approximate models. For the classical single kernel based SVM algorithm, H =



f : f (x) =

N

∑ w j K(x, x j ) + w0

 .

(1)

j=1

Here K : X ×X → R is a Mercer kernel [9] and the combination parameters {wi } need to be learned from training data by the algorithm. SVM has been widely applied in many fields. However, a single kernel limits its learning ability and it can only provide a point estimation for the prediction variable, which usually result in poor generalization performance when dealing with complex data. Under the Bayesian inference framework, RVM was proposed in [7]. It assumes that the observations are corrupted by simple, independently drawn Gaussian white noises with the same variance σ 2 , that is p(ti |xi , w , σ 2 ) = N(ti | f (xi ), σ 2 ),

(2)

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and

N

p(t|X, w , σ 2 ) = ∏ p(ti |xi , w , σ 2 ), i=1

where X := (x1 , · · · , xN )T , t := (t1 , · · · ,tN ). At the same time, the combination parameters {wi } are considered as random variables in RVM and the automatic relevance determination (ARD) Gaussian with the same parameter can be chosen as their prior distribution, namely,   wi ∼ N wi |0, α −1 . This prior can help to obtain a sparse solution with most of the components of w tend to be zero. Compared with SVM, RVM can obtain a smaller size of relevance vectors than that of support vectors. Moreover, it can deduce the whole posterior distribution of the output t, which reduces the influence of noises and outliers. When predicting new samples, we can use either the posterior expectation or the maximum of the posterior probability. Unfortunately, RVM is also established on a single kernel, for complex data, such as nonstationary or heterogeneous data, it performs poorly.

3 The Proposed Algorithm In the following, we will introduce our proposed RMGK-RVM method in detail. Firstly, different from the model in (1), our model is N Gaussian kernels with different scales. Namely, N

g(x) = ∑ wi Kθi (x, xi ) + w0 , i=1

(3)

  where Kθi (x, xi ) = exp −θi ||x − xi ||2 and the wights {wi : i = 0, · · · , N} will be learned under the Bayesian framework. Note that in the model (3), each training sample corresponds to a different kernel scale. Therefore, the number of scale parameters is too large to select them one by one. Our approach is to randomly select them from a uniform distribution U(0, Ω ). The choice of Ω directly affects the expressing ability of the model. In order to obtain an appropriate Ω , we use the cross-validation (CV) technique at the training stage. Once all the scale parameters are determined, the entire kernel function space as well as its complexity and ability to approximate the function are determined. When all the scale parameters take the same value, it becomes the original RVM. Similar to RVM, we assume that t = g(x) + ε , with ε being Gaussian white noise. Therefore, N   p(t|w, X, σ 2 ) = ∏ N t j |g(x j ), σ 2 j=1

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For w = (w0 , w1 , . . . , wN ), we assume that each of its components is independent and has different ARD Gaussian prior, we get N

p(w|α ) = ∏ N(wi |0, αi−1 ) i=0

Denote Φ = (φ (x1 ), · · · , φ (xN ))T with φ (x) = (1, Kθ1 (x, x1 ), · · · , KθN (x, xN )) and A = diag(α0 , α1 , . . . , αN ). Since the priors are Gaussian distributions, according to the Bayesian rule, we can get the posterior distribution of the weight vector w given by p(w|t, X, α , σ 2 ) = N(w|μ , Σ −1 ), with

μ= and

1 Σ Φ T t. σ2

1 T Φ Φ + A, σ2 At the same time, the predicted distribution, defined as

Σ −1 =

p(t∗ |X, t, x∗ , α , σ 2 ) =



(4)

(5)

p(t∗ |x∗ , β , σ 2 )p(w|t, X, α , σ 2 )dw,

is also a Gaussian distribution, namely,   p(t∗ |X, t, x∗ , α , σ 2 ) = N t∗ |μ T φ (x∗ ), β −1 . with β −1 = σ 2 + φ (x∗ )T Σ φ (x∗ ). To obtain the above distribution, we need to find the α , σ 2 . Here we use an iterative method as in [7] to get their approximation solutions. Firstly, we initialize the value of the two hyperparameters and calculate the mean (4) and the covariance matrix (5), then, iteratively update the parameters,

αi new = (σ 2 )new =

γi , μi2 1 ||t − μ T Φ ||, N − ∑i γi

where γi = 1 − αi Σii and Σii is the i-th diagonal element of covariance matrix (5). In this process, we can find that many components of the hyperparameter α tend to infinity, which results in the corresponding wi tending to zero. In other words, we can get a sparse solution similar to the Lasso method. Following these efforts, given a new input x∗ , we can use Bayesian estimators, such as posterior expectation μ T φ (x∗ ), as its prediction.

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4 Experimental Validation This section reports some simulation results on one simulation data set and three real data sets, Airfoil, Energy and Concrete from UCI1 . The input features are scaled into [−1, 1] and the output values are normalized into [0, 1]. We randomly choose 75% data set as training set and 25% as test set. To compare the performance with different algorithms comprehensively, we consider two accuracy measures, based on the test data T = {(xi , yi )|i = 1, . . . , T }. The first metric is the Mean Squared Error (MSE) given by MSE =

1 T

T

∑ (yi − g(xi ))2 .

i=1

The second measure is the coefficient of determination, denoted as R2 , which provides a measure of how well the observed outcomes are replicated by the model, R2 = 1 −

∑Ti=1 (yi − g(xi ))2 . ¯2 ∑Ti=1 (yi − y)

Comparisons are carried out for the proposed RMGK-RVM algorithm, Bayesian linear regression (BLR) [10], the classical machine learning algorithms-SVM [2] and Relevance Vector Machine (RVM) which based on single Gaussian kernel (SK-RVM) [7]. Since for different data set, except for the BLR algorithm, which has no model parameters, the optimal parameters included in other three algorithms are different, we give the selection ranges of them used in our experiments in Table 1. And for every algorithm, we use 10-fold cross-validation to determine the parameters. We carry out the experiments with different parameters for each data set and only present the best performance among them. Table 1. Parameter settings Algorithms

Parameters

Parameters range

SVM

RBF scale σ

σ ∈ {2−8 , 2−7 , . . . , 27 , 28 }

SK-RVM

RBF scale σ

σ ∈ {2−8 , 2−7 , . . . , 27 , 28 }

RMGK-RVM Uniform distribution parameter Ω Ω ∈ {2−8 , 2−7 , . . . , 27 , 28 }

4.1

Function Approximation

We firstly consider approximation results on the following simulation function, y = 0.2e−(10x−4) + 0.5e−(80x−40) + 0.2e−(80x−20) + ε , 2

2

2

x ∈ [0, 1].

Here, the noise ε has the distribution N(0, 0.01). We randomly select 500 samples as our learning set. From its figure shown in Fig. 1, we can see that the distribution has some peaks and is not smooth.

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Fig. 1. The distribution of Data set 1 Table 2. Experimental results of five algorithms on Dataset 1 Algorithms

Training MSE Test MSE R2

BLR

0.0069

0.0077

Active number

0.0221

SVM

0.0039

0.0051

0.3559 28

SK-RVM

0.0042

0.0052

0.3430 3

RMGK-RVM 0.0020

0.0023

0.7082 4.2

The experimental comparisons on DataSet 1 are shown in Table 2, in which, the active number represents the number of samples with non-zero weights in the final model. The results in this table depict that the simple linear Bayesian method cannot achieve satisfactory performance in fitting the data with a complex distribution. Two single kernel based methods, SVM and SK-RVM, can reduce the test MSEs to 0.0051, 0.0052, respectively. However, our proposed RMGK-RVM can significantly improve the generalization abilities with a test MSE 0.0023. Considering the active number, our proposed model has a similar value as SK-RVM, while far less than SVM. 4.2 Real-Life Data Sets Regression The sets we use in the following experiments are the Airfoil, which consists of 1,503 samples, each with 5 input features and 1 output value; the Energy, that contains 768 samples, each with 8 features and 2 output values (we choose the second output value as the label value); and the Concrete which consists of 103 samples, each with 7 features and 3 output values and we choose the last compressive strength as the final output of the model. Next, we will compare our algorithm with SVM and RVM on these three data sets. Specific measures include training error, test error, and R2 . The results are shown in Table 3. 1

http://archive.ics.uci.edu/ml/datasets.html.

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Table 3. Experimental results of SVM, SK-RVM and RMGK-RVM in three real data sets Data

SVM Train

Test

R2

SK-RVM Train Test

R2

RMGK-RVM Train Test

R2

Airfoil

0.0060 0.0085 0.7360 0.0087 0.0110 0.6599 0.0029 0.0054 0.8324

Energy

0.0054 0.0073 0.8921 0.0054 0.0075 0.8889 0.0015 0.0035 0.9469

Concrete 0.0045 0.0107 0.7026 0.0025 0.0058 0.8374 0.0006 0.0026 0.9272

Figures in Table 3 show the good performance of our proposed RMGK-RVM algorithm on three realistic data sets. Compared with SVM and SK-RVM, RMGK-RVM has different degrees of improvement on the three evaluation indicators. This result indicates the efficiency of multiple kernels. To further analyze the approximation ability of the four algorithms, we do a series of comparative experiments on Airfoil and Energy. Specifically, for a fixed test set, we gradually increase the number of training samples to dynamically analyze the performance of each algorithm. We choose the test MSE as the evaluation index. The specific results are given in Figs. 2 and 3. As we can see, the prediction effect of BLR is the worst in all cases. While the other three kernel-based algorithms have significant improvements as the sizes of training samples increase, especially for the proposed RMGK-RVM algorithm.

Fig. 2. Test MSEs on Airfoil

Fig. 3. Test MSEs on Energy

5 Conclusions This paper aims to propose a learning framework which combines the multiple kernel model and Bayesian inference. To handle the selection of the basis kernels, we propose a randomized multi-scale method. The characteristic parameter in the random distribution is determined by the cross-validation technique during the learning process. This results in low computation complexity as well as diversity models. Furthermore, based on Bayesian inference theory, the whole distribution of the output variable can be obtained. Experimental results on four data sets show good performance of the proposed method compared with other existing three algorithms.

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Acknowledgments. This work is supported partly by First Class Discipline of Zhejiang-A (Zhejiang Gongshang University - Statistics), National Natural Science Foundation of China under grant 11571031.

References 1. Hofmann T, Sch¨olkopf B, Smola AJ (2008) Kernel methods in machine learning. Ann Stat 36(3):1171–1220 2. Cortes C, Vapnik V (1995) Support-vector networks. Mach Learn 20(3):273–297 3. Xu YL, Li XX, Chen DR (2018) Learning rates of regularized regression with multiple gaussian kernels for multi-task learning. IEEE Trans Neural Netw Learn Syst 29(11):5408– 5418 4. Bach FR, Lanckriet GR, Jordan MI (2004) Multiple kernel learning, conic duality, and the SMO algorithm. In: Proceedings of the twenty-first international conference on machine learning, vol 69. ACM, New York 5. G¨onen M, Alpaydm E (2011) Multiple kernel learning algorithms. J Mach Learn Res 12:2211–2268 6. Hampel FR, Ronchetti EM, Rousseeuw PJ (2011) Robust statistics: the approach based on influence functions. Wiley, New York 7. Tipping ME (2001) Sparse bayesian learning and the relevance vector machine. J Mach Learn Res 1:211–244 8. Wipf D, Nagarajan S (2007) A new view of automatic relevance determination. In: International conference on neural information processing systems, pp 1625–1632, Curran Associates Inc 9. Cuker F, Smale S (2011) On the mathematical foundations of learning. Bull Am Math Soc 39(1):1–49 10. Xin Y, Xiao GS (2009) Bayesian linear regression. Secur Ticket Control 15(1):1052–1056

Research on Image Segmentation Based on CBCT Roots Zhengguang Xu, Shijie Che(B) , and Zhaohui Zhou School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China [email protected] http://www.ustb.edu.cn/

Abstract. Intraoral scanning devices can only scan the crown portion outside the gums, and two-thirds of the roots cannot reconstruct the 3D model. For dentists, the 3D model of the root is very important. In the past, the root model of the root was reconstructed by manually drawing the root edge of the CBCT. But this method is inefficient and costly. Therefore, how to establish a faster and more accurate root segmentation method is the key to reconstructing the root 3D model. Aiming at the problems of traditional medical image segmentation technology for CBCT root segmentation accuracy, this paper optimizes and improves the whole convolution network, and proposes a CBCT root image segmentation method based on Res Unet network. It also integrates multi-model blending and data enhancement methods. The experimental results of CBCT root image are verified by experiments. The results show that the proposed method not only solves the problem of deep learning in the small sample data of CBCT root image, but also significantly improves the segmentation accuracy of CBCT root image. Keywords: CBCT root · Medical image segmentation Multi-model blending · Small sample

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Introduction

In recent years, visualization techniques for human tissues and organs have become an important tool for computer-aided diagnosis. Since the Computer To mography (CT) image of the tooth contains both rich information on the crown and root, it provides reliable data for reconstructing a complete tooth model. The segmentation of teeth is an important step in the reconstruction of tooth models, and it is of great significance for the study of tooth segmentation methods [1]. Information such as the shape of the teeth and the specific location of the roots is important for clinical operations such as orthodontic surgery, implant surgery, and root canal treatment [2]. This information is often manually measured and acquired prior to surgery, which is often time consuming and does not meet very high accuracy requirements. Therefore, the accuracy of the diagnosis and c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 330–339, 2020. https://doi.org/10.1007/978-981-32-9682-4_34

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surgery of the oral cavity and the success rate of the surgery can be obtained by obtaining a three-dimensional digital model of the tooth. In order to obtain accurate three-dimensional information, accurate segmentation of the dental CT image must be required. However, tooth segmentation based on Cone Beam Computer Tomography (CBCT) images is still a very challenging task. It can be seen from Fig. 1 that the CBCT tooth image segmentation has the following difficulties [3]: in the crown portion, adjacent teeth adhere, the tooth gap is small, causing the common boundary of adjacent teeth to disappear; in the root portion, the image noise Stronger, the contrast between the root and alveolar bone is low, causing the boundary of the tooth to be blurred and discontinuous; the contour of the tooth in the root and crown section is flexible and may split into several parts.

Fig. 1. CBCT root image

At present, convolutional neural networks (CNN) have performed well in a variety of computer vision tasks [4–7]. This powerful ability to learn advanced features from raw data prompted us to use deep learning methods to solve CBCT tooth image segmentation problems.

2

Related Work

A Benefited from the rapid development of convolution neural networks, the problem of semantic segmentation has made great progress. Different from the early semantic segmentation approaches based on CNN that were passing image patches through the network [8], most of existing segmentation models use fully convolutional network architectures. This is far more efficient, since it avoids repeated storage and redundant convolutional computation caused by using overlapping image patches. In the typical convolution network architectures, a convolution layer is often followed by a pooling layer, which can increase the receptive field size rapidly after several layers but reduce the resolution of the output maps. To solve this problem, the bilinear interpolation or learned deconvolution filters can be used to make the network produce corresponding output as same

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size as input [9,10]. And then, some methods also use skip connections [11] or multi-scale connections [12] with earlier high-resolution layers to output more accurate segmentation result. For medical image segmentation, U-Net [11], that consists of a contracting path and an expansive path, can be trained end-to-end from very few medical images. At present, many theories and experiments have shown that a deeper network will have better performance [13]. However, as depth increases, network training becomes more difficult. Kaiming He et al. [13] proposed a residual network, which uses the identity mapping to promote training and effectively solves the training problem brought by network depth. Ioffe, Szegedy et al. [14] proposed a standardized normalization method (Batch Normalization), which proved to be equally difficult to overcome the problem of neural network training. In this paper, we propose a Res Unet network that combines the advantages of both U-Net networks and residual networks, supplemented by effective data enhancement methods and different learning strategies. Experiments show that compared with the existing methods, the proposed method not only effectively solves the problem of less medical image samples, but also significantly improves the segmentation accuracy of CBCT tooth images.

Fig. 2. The architecture of the Res Unet

3 3.1

Approach Network Architectures

Our Res Unet network is also an encoder-decoder structure, as shown in Fig. 1, which consists of three parts: an encoder, a connector, and a decoder. The

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encoder encodes and compresses the input image, and consists of a convolution module (Conv-BN-ReLu-Pool) and four residual blocks. Each residual module is composed as shown in Fig. 2(b). The difference between each module is the setting of the convolution layer parameters in each residual module. The connector is mainly used for the path of information propagation between the encoder and the decoder, allowing information to be propagated between the deep and shallow layers in an easier manner, which is advantageous for the decoder to better recover the image. The decoder is used to recover the image and classify each pixel in the image. The part is composed of five decoder blocks, each of which includes a BN layer, a ReLu active layer and a deconvolution layer [15] (Fig. 3).

Fig. 3. Blocks of convolution neural networks. (a) Traditional block; (b) Residual block

The Res Unet network proposed in this paper is different from the U-Net network. First, Batch Normalization is performed on the input of each layer. Secondly, the “Convolution-Relu” module in U-Net is replaced by the residual module. Then, subsampling with a convolution of step size 2 instead of some pool pooling layers in the U-Net network. Finally, unnecessary clipping operations in the U-Net network are removed, so that the image size of the network output is consistent with the input image size, and the depth of the network is deepened. 3.2

Loss Function

We map the output values of the network via the softmax function. The segmentation network is given a CBCT root image capable of outputting a probability

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map of the same size root. The value of the probability map ranges from 0 to 1, indicating the probability that the pixel is a root, that is, the larger the value, the greater the probability that the pixel is the root. The most desirable result is that all root corresponding values in the CBCT root image are 1 and the non-root is 0. To do this, we implement this constraint by building and minimizing the loss function. We use the cross entropy function to construct the loss function and minimize the loss function with Adam. Softmax Function:

exp(ac (x)) Pc (x) = K k=1 exp(ak (x))

(1)

Where x is the pixel value of the image, ak (x) represents the output value of the k-channel feature map, K represents the number of categories (K = 2), and Pc (x) represents the probability that the pixel belongs to the class k. Cross Entropy Loss Function: LSEG = −

K 

yk log(pk (x))

(2)

k=1

Where, yk Indicates the indicator variable, if the category and the sample have the same category, it is 1, otherwise it is 0. Pk (x) represents the probability that the pixel belongs to the class k. 3.3

Data Enhancement

In the case where more image data is not available, in addition to improving the network structure, it is also important to augment data through data enhancement methods. We used a combination of multiple data enhancement methods to generate more new transformed images from the original image. In the specific implementation, we use online data enhancement methods. The image transformation is done by the CPU, while the GPU is training the previous batch of images. Therefore, these data enhancement methods do not increase training time. The first data enhancement methods include common data expansion methods such as cropping, flipping, and mirroring; the second data enhancement methods include transmission transformation and elastic transformation. We perform random transmission transformation or elastic transformation on the transformed image to further enhance the data. The third method of data enhancement includes changing the brightness value and contrast of the image, enabling the network to learn CBCT root image in different illumination environments, and has better robust performance.

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335

Learning Strategy

In the process of network model training, we can’t determine how many times the network model can be optimized after training. Therefore, we divide the training set into a training set and a verification set. The training set is used for training, and the verification set is used to select the optimal model. We divide the above training set D into five mutually similar reciprocal subsets, and then use the union of 4 subsets as the training set each time, and the remaining subsets as the verification set; thus, we can get 5 sets of training/verification sets. Therefore, 5 trainings and verifications can be performed, and the 5 trained optimal models are obtained for model fusion. Although the validation set is not learned on its corresponding single training model, it is fully utilized as a whole. Experiments show that this method contributes to the improvement of network performance.

4

Experiments and Results

Our experiments were based on the PyTorch 0.4.0 open source framework and trained on the GPU of the Geforce GTX TITAN X. 4.1

Datasets and Detail

Our experiments were validated on the CBCT root dataset, which contained a total of 100 images with an image size of 576 × 576. According to the consistency of the image category distribution, we divide the data set into stratified samples, 90 of which are used as training sets and 10 are used as test sets. The Res Unet network model is trained using the above data set, and the training parameters are shown in Table 1. Where base lr represents the underlying learning rate; lr decay represents the learning decay rate, which is used to control the rate at which the learning rate slows down as the number of training increases; batch size represents the number of batch training images; weight decay represents weight decay, Prevent overfitting. iter nums indicates how many iterations are specified, the learning rate is updated, and the maximum number of iterations. Table 1. Training parameters Name base lr lr decay batch size weight decay iter nums Value 0.0001 5

12

0.0005

[100,50,50]

Cross Entropy Loss Function: Where, Indicates the indicator variable, if the category and the sample have the same category, it is 1, otherwise it is 0. represents the probability that the pixel belongs to the class k.

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4.2

Evaluation

In this experiment, the network accuracy is evaluated by the Mean Intersection over Union (MIoU) evaluation index commonly used in image segmentation.  n 1  i ii (3) MIoU = k ti + j nji − nii ti =



nij

(4)

j

Where nij represents the number of pixels in category i predicted to be category j.

5

Results and Analysis

In order to verify the performance of our proposed Res Unet network, we tested the accuracy and training time of the Res Unet network on the superalloy microstructure image dataset, and compared it with other methods such as FCN and U net, as shown in Fig. 4.

Fig. 4. The segmentation results with different networks. (a) Original image; (b) Ground truth; (c) Output of FCN; (d) Output of U net; (e) Output of Res Unet

In terms of accuracy, the Res Unet network can obtain very good segmentation results for CBCT root images with different illumination environments

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and differ-ent morphological distributions, and even for complex and noisecontaminated images. The processing results of such images by FCN and U net networks are not ideal, as shown in Fig. 4. Especially fcn, the segmentation result is very poor, the big reason is that there are few samples that resulting in the fcn network not being fully studied and there are few root targets that disappeared.

Fig. 5. Loss function contrast

From Fig. 5, we can see that our proposed Res Unet can converge faster and obtain lower loss values in the training process. From Table 2, we can also see our proposed Res Unet gets the hightest MIoU in three networks. In the case that the test time is not much different, the segmentation result is greatly improved. Table 2. Comparison of models with different adversarial networks Architecture Mean intersection Model Test time (s) of model over union parameters (M) FCN

0.709

13.43

0.188

U net

0.911

3.10

0.106

Res Unet

0.940

3.24

0.118

Finally, we migrated the decoder part of the Res Unet network and trained our network. Although the accuracy is limited, it improves the training efficiency

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of the network. In addition, we train multiple models for fusion on this basis, and the accuracy is improved by 1%, as shown in Table 3. Table 3. Comparison of Res Unet with different training strategies Training strategy Mean intersection Test time (s) over union Res Unet

0.940

0.188

Pretrained

0.946

0.118

Blending

0.954

0.352

Through the above experimental verification and comparative analysis, the Res Unet proposed in this paper shows excellent performance in the CBCT root image segmentation task, especially for the root boundary adhesion and the boundary blurred image, it can also obtain very good segmentation effect and solve at the same time. The problem of small sample image segmentation provides a useful exploration for deep learning in the application of small sample images. Compared with the existing methods, the deep learning method based on Res Unet network improves the segmentation accuracy of microstructure images, and also shortens the training time of the network to a large extent, and provides a more efficient and accurate method for 3D model reconstruction.

References 1. Kan W, Li C, Jing L, Yanheng Z (2014) Tooth segmentation on dental meshes using morphologic skeleton. Comput Graph 38:199–211 2. Li J, Bowman C, Fazel-Rezai R et al (2009) Speckle reduction and lesion segmentation of OCT tooth images for early caries detection. In: Annual international conference of the IEEE engineering in medicine and biology society. Annual Conference on IEEE engineering in medicine and biology society, vol 2009, pp 1449– 1452 3. Gao H, Chae O (2010) Individual tooth segmentation from CT images using level set method with shape and intensity prior. Pattern Recogn 43(7):2406–2417 4. Krizhevsky A, Sutskever I, Hinton GE (2017) ImageNet classification with deep convolutional neural networks. Commun ACM 60(6):84–90 5. Zeiler MD, Fergus R (2014) Visualizing and understanding convolutional networks. In: European conference on computer vision (ECCV), pp 818–833 6. Simonyan K , Zisserman A (2015) Very deep convolutional networks for large-scale image recognition. In: Computer vision and pattern recognition (ICLR) 7. Szegedy C, Liu W, Jia Y et al (2015) Going deeper with convolutions. In: Computer vision and pattern recognition(CVPR), pp 1–9 8. Sermanet P, Eigen D, Zhang X et al (2013) OverFeat: integrated recognition, localization and detection using convolutional networks. Eprint Arxiv 9. Badrinarayanan V, Kendall A, Cipolla R (2017) Segnet: a deep convolutional encoder-decoder architecture for image segmentation. IEEE Trans Pattern Anal 39(12):2481–2495

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10. Long J, Shelhamer E, Darrell T (2014) Fully convolutional networks for semantic segmentation. IEEE Trans Pattern Anal Mach Intell 39(4):640–651 11. Ronneberger O, Fischer P, Brox T (2015) U-Net: convolutional networks for biomedical image segmentation. In: International conference on medical image computing and computer-assisted intervention. Springer, Cham, pp 234–241 12. Zhao H, Shi J, Qi X et al (2017) Pyramid scene parsing network. In: Computer vision and pattern recognition (CVPR), pp 2881–2890 13. Girshick R, Donahue J, Darrell T et al (2014) Rich feature hierarchies for accurate object detection and semantic segmentation. In: Computer vision and pattern recognition (CVPR), pp 580–587 14. Ioffe S, Szegedy C (2015) Batch normalization: accelerating deep network training by reducing internal covariate shift. In: Machine learning 15. Dumoulin V, Visin F (2016) A guide to convolution arithmetic for deep learning. In: Machine learning

Efficient and Fast Expression Recognition with Deep Learning CNN-ELM Yiping Zou and Xuemei Ren(B) School of Automation, Beijing Institute of Technology, Beijing 100000, China [email protected]

Abstract. Facial expression recognition is a significant direction in facial computer version. Although convolutional neural networks (CNNs) have received great attention in recognition task especially for images, they require considerable time in computation and are easily to be trapped in over-fitting due to kinds of reasons. This paper suggests a fast and efficient network for expression recognition, which takes full advantages of CNN and ELM (Extreme Learning Machine). Facial expressions can be learned well and calculated fast with satisfying accuracy through it. Experimental results on real-life expression database prove that our proposed approach can effectively reduce the calculation time and improve the performance. Keywords: Facial expression recognition · Facial analysis Extreme Learning Machine · Computer version

1

·

Introduction

Facial expression recognition occupies a significant position not only in the study of human’s behaviour and intention, but also in the computer version, especially in the field of facial recognition. With the help of successful facial expression recognition, many progress can be obtained in machines’ cognition and deduction for human, making computer acting more like a person. Commonly, facial expression consists of seven basic emotions which include happy, normal, sad, surprised, anger, fear and disgust. Nevertheless, there is a high error rate sometimes even if by artificial recognition, not to mention by computers. In the past few years, automation expression recognition has always been a challenging study. Some research has been dedicated to it. Among conventional methods, handcrafted features such as PCA and LBP have been broadly used with framework of classification. Calder et al. [1] proposed a method based on PCA and showed that there is a huge difference be-tween recognition for expression and for only human faces. LBP has been applied to expression issue [2,3] and achieved nice performance in view of it is a good local texture feature. Building on SVM, Gabor wavelet [4,5] also acts effectively. c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 340–348, 2020. https://doi.org/10.1007/978-981-32-9682-4_35

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With the fast developing of neural network, CNN has becoming a useful solution in classification, recognition and detection, especially for large image datasets. Having thousands shared parameters, CNN is capacity to fit lots of complex training samples and extract useful features to feature maps. There are a lot of recognition problems which CNN can be applied to, like facial detection, recognition, posture estimation and so on [6–9]. The finer the functions that need to be extracted, the more layers of CNN that need to be increased. However, a deep CNN needs huge amount of expression data to train, otherwise it will fall into over-fitting. And as the network deepens, training time also increases for adjusting the parameters constantly. Besides, too many layers lead to egradation easily [10]. Therefore, this paper applies ELM to a deep CNN for expression recognition. ELM [11,12] is an efficient learning method for SLFNs (Single-hidden Layer Feed forward Networks), which has a better generalization performance than the learning approach basing on gradient-tuning, like backpropagation. At the same time, it ensures and rises the learning of speed dramatically. Thus it has been previously applied to different areas. However, ELM is hard to expand to deep neural networks due to the fixed input and target, and also lacks a capacity of learning complex features [13,14]. In this paper, we proposed an improved deep convolutional neural network that combines the advantage of ELM and the application in facial expression recognition. In the proposed network, the convolutional part is adjusted to be more suitable to recognize facial expression datasets, and the original fully connected layers are optimized by ELM algorithm. Thus not only can the superior features be extracted sufficiently but also the computational speed and capability of generalization can be rose. The trained model is used to predict a real-life facial emotion database, consequently showing that it achieves higher accuracy and efficiency with faster speed compared to the benchmark methods. The paper is organized as follows. The details of the proposed network are illustrated in Sect. 2. And Sect. 3 presents experimental results and analysis its performance specifically. The conclusions are shown in Sect. 4.

2 2.1

Method Description Method Pipeline

The flow chart of the proposed method is shown in Fig. 1. Before formal training for facial expression recognition, this paper uses transfer learning to learn the features of facial emotion better. In the process of formal training, pictures can be automatically read features without manual labelling of key points. CNN-ELM can not only learn facial expressions well, but also accurately classify expressions. 2.2

CNN-ELM Design for Expression Recognition

CNN-ELM (Convolutional Neural Network with Extreme Learning Machine) will be given more details in this section. Our network is inspired by VGG16 [8]

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Fig. 1. System pipeline of this paper

model. With repeated stacking of 3 × 3 convolutional layers and 2 × 2 pooling layers, a great improvement can be acquired by prolonging the length of network. The architecture has 13 convolutional layers and two fully connected layers adopting ELM algorithm. Before the fully connected layers, the optimized methods of weights are based on gradient decent learning. Instead of the 224 × 224 input image used by VGG, we simply use 48 × 48 size of input considering that the size of facial expression images are usually small and it is inconvenient to make the size bigger forcibly because of pixel loss. Therefore we change the stride for pooling layer a little because decimals are not desirable. The full network is shown in Fig. 2.

Fig. 2. The architecture

ELM is a fast and efficient algorithm proposed for SLFNs (single-hidden layer feedforward neural networks) [11,12]. Random values can be assigned to the weights between the input layer and hidden layer as long as the activation functions of hidden layer are infinitely differentiable in any interval. Since that generalized in-verse of the output matrices of hidden layer can calculate the output weights, all the weights in the SLFNs are not necessarily tuned by gradient decent during many iterations. Supposed that the network has N examples, L hidden nodes and O categories, there are matrices of input X = [x1 , x2 , . . . , xN ]

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343 T

and labels T = [t1 , t2 , . . . , tO ] respectively, where xi = [xi1 , xi2 , . . . , xim ] , T ti = [ti1 , ti2 , . . . , tim ] . g(xi ) represents for the activation functions (functions like Sigmoid, Fourier, Hardlimit, Gaussian and Multiquadrics are all suitable) in hidden nodes. Thus the output matrices G of hidden layer can be expressed as ⎤ ⎡ g1 (x1 ) . . . gL (x1 ) ⎥ ⎢ .. .. .. G=⎣ (1) ⎦ . . . g1 (xN ) . . . gL (xN ) If β = [β1 , β2 , . . . , βO ] is regarded as the output weights, there exist an equation such that L  βi g (xj ) = tj j = 1, . . . , L (2) i=1

It can be rewritten as Gβ = T

(3)

Actually, not only does ELM aim to find the minimum training loss but also the minimum norm of weights: Minimize : βσ1 + CGβ − T 

(4)

C is a parameter balancing these two terms. In most cases, N is unequal with L, as a result, the optimal solution of the linear system Gβ = T can be obtained by calculating the generalized inverse matrices of G. βˆ = G † T

(5)

Here are two common formulas to solve βˆ when σ1 = σ2 = p = q = 2 [11,15,16]:

−1 T I GT G G T N >L C + ˆ (6) G=

−1 T N ≤L GT CI + GT G Since that ELM can achieve an expected capability of learning without iteratively tuning weights, fast is also a remarkable feature of it. It is proved that ELM is many times faster than the same type of network, and is ideal for scenarios that require fast detection. For facial expression recognition, in view that human expressions are very complicated, we need a deep convolutional network to learn the characteristics of different expressions better. It can be said that CNN mainly undertakes the work of feature picking, which is convenient for the latter two fully connected layers to use ELM for calculation.

3

Experiments

Currently, for the reason that it is hard and extremely time consuming to collect real-life facial expression in different situations and make labels for each of

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them, the available facial expression datasets are not rich. In this paper, Facial Expression Recognition 2013 (FER-2013) is chosen as the dataset for network. Compared to other datasets of the same type, FER2013 has several features: the samples are rich and vary widely. And it is collected by Internet and contains different races, ages, postures and so on. The expression of each image is natural rather than deliberately posed in laboratory. However, even human can only reach 65 ± 5% accuracy in this dataset, so it can been seen that FER-2013 is a very difficult dataset. 3.1

Transfer Learning

Transfer learning aims to apply the learned experience to the target study. Due to the complexity and approximation of expressions and in order to rise the performance, we pretrained our network on the ImageNet classification task (224 × 224 input image) firstly, and then downscaled the resolution to finetune the network on FER2013. For finetuning we used the convolutional layers and a fully connected layer, consequently 60% accuracy was achieved. In the later training, weights of convolutional layers were keep. 3.2

Recognition Performance

The parameter C is finally set at 1 after we tried {0.01, 0.1, 1, 10, 100}. As we can see in Table 1, 85.91% of accuracy has been achieved by our network, which is far higher than other leading models [17,18] on FER-2013. Table 2 shows more details about each expression. We note that there exists a high accuracy for “happy” and “surprise”. It is also noteworthy that although “disgust” has the smallest sample set (far smaller than other expression), it still ranks third according to the table. Observing the dataset, the primitive reason is that the network is more prone to recognize the expressions that pull more facial muscles, which is same as the human eyes. Table 1. Performance comparison with other models on FER-2013 Model

ACC(%)

Our network

85.91%

VGG19

71.49%

ResNet18

71.19%

RBM

69.77%

Unsuperviesed team 68.15% Maxim Milakov

67.48%

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Table 2. The details of each expression of the model Expression Precision Recall

F-Score

Anger

95.36%

75.51% 84.28%

Disgust

93.62%

80.00% 86.27%

Fear

82.11%

73.86% 77.77%

Happy

98.02%

96.01% 97.01%

Normal

72.98%

89.73% 80.49%

Sad

79.31%

77.44% 78.36%

Surprised

84.93%

95.43% 89.82%

Table 3. Performance comparison for speed Model

Time(s)/epoch

Our network

35

Conventional CNN 155 Table 4. Presentation of correct recognized images’ results in probability scores (%) Expression

Anger

Disgust

Fear

Happy

Sad

Surprised

Normal

0

0

0

0

0

0

100

5

1

2

1

84

1

6

0

0

100

0

0

0

0

0

0

0

100

0

0

0

2

1

1

0

0

95

0

99

0

1

0

0

0

0

1

99

0

0

0

0

0

Besides, as ELM provides a simpler, more stable solution for the output weights in a deterministic approach, speed is improved greatly. For equality, we compared the speed of our network with another network called conventional CNN which has the same convolutional layers and same weights but two different fully connected layers (use gradient descent). The comparison is shown in Table 3. Both of networks were run at CPU E5-1620 v4, 3.50 GHz. Considering time consuming, we only sample the time of 1 epoch. As shown below, our network is extremely fast. The network of this paper runs nearly 5 times faster than conventional CNN.

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Table 5. Presentation of incorrect recognized images’ results in probability scores (%) Expression

Anger

Disgust

Fear

Happy

Sad

Surprised

Normal

1

1

1

0

39

1

57

23

2

12

2

4

1

56

3

40

7

2

46

1

1

18

2

17

2

28

1

32

30

3

24

8

16

2

17

12

2

9

11

43

2

21

13

2

12

33

15

22

3

In order to giving a more intuitive feeling about the results of recognition. Tables 4 and 5 illustrate the score of each picked images from the dataset whose category is different, where Table 4 shows the scores of several correct recognized images while Table 5 is the opposite. The bold parts in tables represent the correct category. The tables below reflect that our network has mastered the salient features of each expression and are very confident so that it can give a score of 100% when an image can be recognized correctly, while the network seems hesitantly when making an error decision. On account of the features of the FER-2013 dataset, we suppose that the similarity and subtlety of expressions and wrong labels may be the main reasons for the error.

4

Conclusion

Facial expression recognition, as one of the significant direction in facial expression, has a lot of potential applications in different areas. As real-life expressions change fast, both of accuracy and speed should be concerned. In this paper, an efficient and fast method for facial expression recognition is proposed, which base on a designed and improved deep convolutional neural network combining extreme learning machine. Making best of CNN and ELM, experiments on FER database show that complex facial expressions are fully learned, extracted and recognized in a high efficient way which greatly reduces computational time while rises accuracy significantly compared to similar types of conventional convolutional neural networks, not to mention human eyes. The proposed approach is very competitive and applicable in the direction of facial recognition. Universal capability is expected. In addition to solving the problem of facial expression recognition, in the future, we also intend to apply the method to a wider range

Efficient and Fast Expression Recognition

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of facial issues or other areas such as speech recognition, which will be under our consideration and investigation. Acknowledgments. This work is supported by National Natural Science Foundation (NNSF) of China under Grant No. 61433003, 61973036.

References 1. Calder AJ, Burton AM, Miller P, Young AW, Akamatsu S (2001) A principal component analysis of facial expressions. Vis Res 41(9):1179–1208. https://doi. org/10.1016/S0042-6989(01)00002-5 2. Marasamy P, Sumathi S (2012) Automatic recognition and analysis of human faces and facial expression by LDA using wavelet transform. In: IEEE international conference on computer communication & informatics. IEEE Press. https://doi. org/10.1109/ICCCI.2012.6158798 3. Shan C, Gong S, Mcowan PW (2009) Facial expression recognition based on local binary patterns: a comprehensive study. Image Vis Comput 27(6):803–816. https:// doi.org/10.1016/j.imavis.2008.08.005 4. Liu WF, Wang ZF (2006) Facial expression recognition based on fusion of multiple gabor features. In: 18th international conference on pattern recognition, vol 3, pp 536–539. IEEE. https://doi.org/10.1109/ICPR.2006.538 5. Buciu I, Kotropoulos C, Pitas I (2003) ICA and Gabor representation for facial expression recognition. In: International conference on image processing, vol 2, pp II–855. IEEE. https://doi.org/10.1109/ICIP.2003.1246815 6. Taigman Y, Yang M, Ranzato MA, Wolf L (2014) Deepface: closing the gap to human-level performance in face verification. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 1701–1708 7. Sun Y, Wang X, Tang X (2015) Deeply learned face representations are sparse, selective, and robust. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 2892–2900. https://doi.org/10.1109/CVPR. 2015.7298907 8. Simonyan K, Zisserman A (2014) Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556 9. Kim B-K, Roh J, Dong S-Y, Lee S-Y (2016) Hierarchical committee of deep convolutional neural networks for robust facial expression recognition. J Multimodal User Interfaces 10(2):173–189. https://doi.org/10.1007/s12193-015-0209-0 10. He K, Zhang X, Ren S, Sun J (2016) Deep residual learning for image recognition. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 770–778 11. Huang G-B, Zhou H, Ding X, Zhang R (2011) Extreme learning machine for regression and multiclass classification. IEEE Trans Syst Man Cybern Part B (Cybernetics) 42(2):513–529. https://doi.org/10.1109/tsmcb.2011.2168604 12. Huang G-B, Zhu Q-Y, Siew C-K (2006) Extreme learning machine: theory and applications. Neurocomputing 70(1–3):489–501. https://doi.org/10.1016/j. neucom.2005.12.126 13. An L, Yang S, Bhanu B (2015) Efficient smile detection by extreme learning machine. Neurocomputing 149:354–363. https://doi.org/10.1016/j.neucom.2014. 04.072

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14. Kim J, Kim J, Jang G-J, Lee M (2017) Fast learning method for convolutional neural networks using extreme learning machine and its application to lane detection. Neural Netw 87:109–121. https://doi.org/10.1016/j.neunet.2016.12.002 15. Huang G-B, Ding X, Zhou H (2010) Optimization method based extreme learning machine for classification. Neurocomputing 74(1–3):155–163. https://doi.org/10. 1016/j.neucom.2010.02.019 16. Banerjee KS (1973) Generalized inverse of matrices and its applications. Taylor & Francis Group, Milton Park 17. Kuang L (2016) Facial expression recognition method integrated in convolutional network. Zhejiang University 18. Facial Expression Recognition. https://github.com/WuJie1010/Facial-ExpressionRecognition.Pytorch

On Collective Behavior of a Group of Autonomous Agents on a Sphere Yi Guo, Jinxing Zhang, and Jiandong Zhu(B) School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, China [email protected], [email protected], [email protected]

Abstract. In this paper, we investigate collective behavior of a group of autonomous agents on a sphere. For the first order nonlinear multi-agent system model with repulsive gains and an m regular graph, it is proved that the system state converges to an non-synchronization equilibrium with centroid at the origin. For the second order nonlinear multi-agent system model with attractive gains and general undirected graphs, it is proved that the system state converges to an equilibrium. Finally, some simulations are given to validate the obtained theoretical results.

Keywords: Nonlinear multi-agent system Graph theory

1

· Collective behavior ·

Introduction

In nature, there are many kinds of collective behavior such as synchronization of flocks, coherence resonance of oscillators and biological rhythms of cells. Many mathematical models were proposed to simulate the collective behavior such as Kuramoto model [1], Boid model [3] and Vicsek model [4]. Kuramoto model has attracted much attention because of a wide range of applications in chemistry, biology, neuroscience and so on. The mathematical expression of the original Kuramoto model dynamics system is x˙ i = ωi +

N k  sin(xi − xj ), N j=1

i = 1, 2, · · · , N,

(1)

where xi ∈ R is the phase of the ith individual oscillator, ωi ∈ R is the natural frequency, and k > 0 is the coupling gain between the individuals of the dynamical network, N is the number of individuals. In [5], the network structure of the Kuramoto model is generalized from a complete graph to a general graph x˙i = ωi +

k  sin(xi − xj ), N

i = 1, 2, · · · , N,

j∈Ni

c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 349–356, 2020. https://doi.org/10.1007/978-981-32-9682-4_36

(2)

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where Ni is the neighborhood of the individual i. By the idea of [8] and [6], the Kuramoto model is extended to the unit sphere of the high dimensional space as follows:  (xj − xi , xj xi ), i = 1, 2, · · · , N, (3) x˙i = Ωi xi + k j∈Ni

where xi ∈ Rn is limited on the unit sphere and Ωi is an N × N antisymmetric matrix. In [7], the coordinated motion of multi-agents on the sphere is investigated, especially the synchronization behavior of the first-order Kuramoto-like oscillators that couple each other in the network structure. In [2], a nonlinear model for a class of multiple agents moving on a spherical surface is considered. The first-order dynamic model and the second-order dynamic model of multiagents in spherical motion are proposed. The rejection coefficient can better describe the repulsion between individuals compared with the existing highdimensional Kuramoto model. Most of the existing papers on the high-dimensional Kuramoto model mainly consider the network topology of complete graph. This paper considers a nonlinear multi-agent system that moves on a spherical surface. We first achieve the collective behavior of the first order high-dimensional Kuramoto model under the topology of regular graphs. Secondly, for the second order high-dimensional Kuramoto model, a convergence result is obtained under the topology of general undirect graphs. Finally, the obtained theoretical results are verified by some simulations.

2 2.1

Main Result First-Order Model

Consider a directed graph G = (V, E, W ). The vertex set is V = {1, 2, . . . , N }, the edge set is E ⊂ V × V and the nonnegative weighted adjacency matrix is W = (wij ). A directed edge (j, i) ∈ E means that the information of j can be received by i. Assume that wij = 1 if and only (j, i) ∈ E. We consider the agents governed by the fist-order nonlinear multi-agent system limited on a sphere x˙ i =

N 

wij (ka −

j=1

kr xi , xj  )(xj − xi ), 2 xi − xj  xi 2

(4)

where xi ∈ Rn and i = 1, 2, . . . , N . In [2], the high-dimensional Kuramoto model under the complete graph has been addressed. The main work of this subsection is to generalize a result of [2] into the case of directed regular graphs. Definition 2.1.3. The graph G = (V, E, W ) is said to be an m regular graph if N  l=1

wil =

N  l=1

wli = m, ∀ i = 1, 2, 3, . . . , N.

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Theorem 2.1.1. Consider m the high-dimensional Kuramoto model (4) limited 1 on a sphere. Let x = m i=1 xi and suppose that the directed graph G is an m regular graph. If ka = 0 and kr > 0, then x → 0 (t → ∞). Proof. Since the high-dimensional Kuramoto model (4) is limited on a sphere, we let ri (0) = r for i = 1, 2, . . . , N . Then ri (t) = r for all t ≥ 0. As ka = 0 and kr > 0, we have  kr wij xi , xj  x˙i = − (xj − xi ), i = 1, 2, . . . , N. (5) xi − xj 2 r2 j∈Ni

Write the N equations shown in (5) into the compact form as follows: ⎛⎡  k x ,x  kr w1j ( x1r −x1 2j r2 ) −w12 x1 −x 2 2 j ⎜⎢ j∈N1 ⎜⎢  k x ,x  kr ⎜⎢ −w21 w2j ( x2r −x2 2j r2 ) x2 −x1 2 ⎜⎢ j j∈N2 ⎢ x˙ =⎜ ⎜⎢ .. .. ⎜⎢ . . ⎜⎢ ⎝⎣ kr r −w −wN 1 x k−x N 2 2 x −x2 2 1 N

Since

 wij

N



kr xi , xj  xi − xj 2 r2

− wij

⎤

···

kr −w1N x1 −x

2 N

···

kr −w2N x2 −x

2 N

⎞

⎥ ⎟ ⎥ ⎟ ⎥ ⎟ ⎥ ⎟ ⎥⊗ In ⎟ x. (6) ⎥ ⎟ .. .. ⎥ ⎟ . . ⎥ ⎟  ⎠ kr xN ,xj  ⎦ ··· wN j ( x −x 2 r2 ) N

j∈Nn

kr = wij xi − xj 2



j

kr (xi , xj  − r2 xi − xj 2 r2

 (7)

and xi − xj 2 = 2r2 − 2xi , xj , we have

 wij

Let

kr xi , xj  xi − xj 2 r2 ⎡

⎢ ⎢ ⎢ ⎢ A=⎢ ⎢ ⎢ ⎣

− 12





w1j kr2r

j∈N1

0 .. . 0

− wij

(8)

kr = −wij xi − xj 2

0 ···  − 12 w2j kr2r · · · j∈N2

.. . 0



kr 2r2

0 0

.. . .  wN j kr2r · · · − 12 ..

 .

(9)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(10)

j∈NN

and

⎡  kr kr w1j x1 −x −w12 x1 −x 2 2 j 2 ⎢ j∈N1  ⎢ −w kr kr w2j x2 −xj 2 21 x2 −x1 2 ⎢ ⎢ j∈N2 L=⎢ .. .. ⎢ ⎢ . . ⎣ kr r −w −wN 1 xN k−x 2 N 2 xN −x2 2 1

kr · · · −w1N x1 −x 2 N

⎤

⎥ ⎥ kr · · · −w2N x2 −x 2 ⎥  N ⎥ ⎥. .. .. ⎥ . ⎥ .  ⎦ r ··· wN j xN k−x 2 j j∈NN

(11)

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Then (6) becomes x˙ = ((A + L) ⊗ In )x. Let β T = (1, 1, · · · , 1), we have x ¯=

1 T (β ⊗ In )x, N

 wij j∈Ni

 kr kr = wji , 2 xi − xj  xi − xj 2

β T L = 0.

j∈Ni

So 1 x˙ = (β T ⊗ In )x˙ N 1 = (β T ⊗ In )((A + L) ⊗ In ) x N 1 = (β T A ⊗ In )x. N

(12)

We calculate the derivative of V (x) = 12 x2 with respect to t as follows: V˙ = xT x˙ 1 = 2 xT (β ⊗ In )(β T A ⊗ In )x N 1 = 2 xT (ββ T A ⊗ In )x N ⎛⎛ 1   −2 w1j kr2r − 12 w2j kr2r j∈N1 j∈N2 ⎜⎜ ⎜⎜ − 1  w kr − 1  w kr ⎜ 1j r 2 2j r 2 2 1 T⎜ ⎜⎜ 2 j∈N1 j∈N2 = 2 x ⎜⎜ .. .. ⎜⎜ N ⎜⎜ . .  ⎝⎝ 1  kr 1 w1j r2 − 2 w2j kr2r −2 j∈N1

j∈N2

· · · − 12



wN j kr2r

⎞

⎞

⎟ ⎟ ⎟ wN j kr2r ⎟ ⎟ ⎟ ⎟ ⎟ j∈NN ⊗ In ⎟ x ⎟ .. .. ⎟ ⎟ . ⎟ ⎟ .  ⎠ ⎠ kr 1 · · · −2 wN j r2 · · · − 12

j∈NN



j∈NN

1 kr = − m 2 2 xT (ββ T ⊗ In )x 2 r N 1 kr = − m 2 xT x 2 r 1 kr = − m 2 x2 ≤ 0. 2 r

(13)

Suppose that the equality of (13) holds. Then − 12 m kr2r x2 = 0 {x|V˙ (x) = 0} = {x = (x1 , x2 · · · , xN )T |

N 1  xi = 0}. N i=1

(14)

Let Ωε = {x = (β T ⊗ In )x}|0 ≤ x2 ≤ nr2 − ε ∀ε > 0}.

(15)

Then Ωε is an invariant compact set of (5). Moreover, from (14), it follows that x → 0.

Collective Behavior

2.2

353

Second-Order Model

Let the dynamic equation of the i-th agent in the second-order model be  x˙ i = vi , v˙ i = k1 xi + k2 vi + ui .

(16)

Let C = 12 xTi xi . C˙ = xTi vi  W , ˙ = vi 2 + xT (k1 xi + k2 vi + ui ) W i = vi 2 + k1 xTi xi + k2 xTi vi + xTi ui . Assume k1 = ⎧ ⎨x˙i = vi , ⎩v˙i =

−vi 2 , xT i xi

−vi 2 xi xT i xi

(17)

k2 = −kv , kv > 0 and xTi ui = 0. In [2]

− kv vi +

 j∈Ni

wij (ka −

kr xi −xj 2 )(xj

−

xi ,xj  r 2 xi ).

(18)

The main work of this section is to investigate whether the high-dimensional Kuramoto model under a tree can reach equilibrium. Lemma 2.2.1. Let S = {(xi , vi ) | xi  = r, vi  ≤ Nkkva r +1, i = 1, 2, 3, · · · , N }. Then S is an invariant set of (18) in high-dimensional Kuramoto model. N  wij viT xj when vi  ≥ Proof. Let K = 12 viT vi and K˙ = v˙i T vi = −kv vi 2 +ka j=1

N ka r kv ,

K˙ ≤ 0. K is a non-increasing function. Thus, if (xi (0), vi (0)) ∈ S, we have xi (t) = r, vi (t) ≤ Nkkva r + 1(∀t > 0, i = 1, 2, 3, · · · , N ). Therefore, S is an invariant set of (18). Theorem 2.2.1. Consider the high-dimensional Kuramoto model (18) defined on the unit sphere with ka > 0, kr = 0, W T = W . If the graph is a tree, then x = (xT1 , xT2 , · · · , xTN )T converges to an equilibrium. Proof. Let H

=

1 2

N  i=1

viT vi −

ka T 2 x W x.

We calculate its derivative by

Lemma 2.2.1 as follows

N N 2   i H˙ = ( −v v T x − kv viT vi + wij ka (viT xj − xT xi i i

= = =

i=1 N 

i

(−kv viT vi



j=1

xi ,xj  T r 2 vi xi ))

− ka v T W x

wij (ka (viT xj )) − ka v T W x j∈Ni N  N  N N   − kv vi 2 + ka wij (viT xj ) − ka wij (viT xj ) i=1 i=1 j=1 i=1 j=1 N  kv vi 2 − i=1 +

i=1 N 

≤ 0.

(19)

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Let E = {(xi , vi ) | xi  = r, vi  = 0, i = 1, 2, 3, · · · , N }

(20)

and let M be the largest constant set of E. Assume that the initial value T T T T T T T T T T T (xT (0), v T (0))T = (xT 1 (0), x2 (0), x3 (0), · · · , xN (0), v1 (0)) , v2 (0)) , v3 (0)) , · · · , vN (0)) ∈ M ⊂ E,

(21) and the solution is (xT (t), v T (t))T . Since M is an invariant set, (xT (t), v T (t))T ∈ M. So v(t) = 0, v(t) ˙ = 0. By LaSalles Invariance Principle, if t → ∞, (xi (t)vi (t)) → M , we have vi  = 0, v˙ i  = 0. Since (18) N N   xi , xj  wij xj = wij xi , (22) r2 j=1 j=1 we conclude that xi 

N  j=1

wij xj . Thus if the graph is tree, then x1  x2  x3 · · · 

xN , and the result holds.

3

Simulations

This chapter mainly verifies the theoretical results through the following examples and trajectory curves. Example 1. Consider the graph is regular circles, including N = 4, n = 2, kr = 1, r = 1. and the matching equation for the first order system ⎧ −1 ⎪ x˙ 1 = x1 −1 ⎪ −x2 2 (x2 − x2 , x1 x1 ) + x1 −x4 2 (x4 − x4 , x1 x1 ), ⎪ ⎪ −1 −1 ⎨x˙ = 2 x2 −x3 2 (x3 − x3 , x2 x2 ) + x1 −x2 2 (x1 − x2 , x1 x2 ), (23) −1 ⎪ x˙ 3 = x3 −1 ⎪ −x4 2 (x4 − x4 , x3 x3 ) + x2 −x3 2 (x2 − x3 , x2 x3 ), ⎪ ⎪ ⎩x˙ = −1 −1 (x − x , x x ) + (x − x , x x ). 4

x1 −x4 2

1

1

4

4

x3 −x4 2

4

4

3

4

We take the initial value of the system state: x1 (0) = [0.5547, 0.8321], x2 (0) = [−0.5547, 0.8321], x3 (0) = [−0.4472, 0.8944], x4 (0) = [0.8575, 0.5145]. Figure 1 shows the trajectory curve on the unit circle, and we can see that two points are approaching two synchronization points respectively, and the two synchronization points are symmetric. Figures 2 and 3 give the individual time response curve. Example 2. Consider the graph is a tree, including N = 3, n = 2, ka = 1, kv = 1, r = 1. And the matching equation for the first order system ⎧ 2 ⎪ ⎨v˙ 1 = −v1  x1 − v1 + x2 − x1 , x2 x1 , (24) v˙ 2 = −v2 2 x2 − v2 + x1 + x3 − x1 , x2 x2 − x2 , x3 x2 , ⎪ ⎩ 2 v˙ 3 = −v3  x3 − v3 + x2 − x3 , x2 x3 .

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1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Fig. 1. Trajectory curve on the unit circle 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

0

5

10

15

20

25

30

Fig. 2. The time response curve of the first component of each individual state 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

0

5

10

15

20

25

30

Fig. 3. The time response curve of the second component of each individual state 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

-1

-0.5

0

0.5

1

Fig. 4. Trajectory curve on the unit circle 1.5

1

0.5

0

-0.5

-1

-1.5

0

2

4

6

8

10

12

14

16

18

20

Fig. 5. The time response curve of the first component of each individual state 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

0

2

4

6

8

10

12

14

16

18

20

Fig. 6. The time response curve of the second component of each individual state

We take the initial value of the system state: x1 (0) = [0.60000.8000], x2 (0) = [−0.80000.6000], x3 (0) = [−0.6000 − 0.8000] Figure 4 shows the trajectory curve is on the unit circle, and we can see that two of the points are approaching two synchronization points respectively, and

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the two synchronization points are symmetric. Figures 5 and 6 give the individual time response curve.

4

Conclusions

This paper has investigated the collective behavior of the high-dimensional Kuramoto model on the spherical surface and generalizes the existing related results. In the first-order dynamic system, when the network transmitting interindividual information is a regular graph, all individuals on the sphere can tend to an equilibrium. In the second-order dynamic system, when the network transmitting inter-individual information is a tree, all individuals on the sphere can achieve partial synchronization.

References 1. Kuramoto Y (1975) Self-entrainment of a population of coupled nonlinear oscillators. In: Proceedings of international symposium on mathematical problems in theoretical physics, vol 39, pp 420–422 2. Li W, Spong W (2014) Unified cooperative control of multiple agents on a sphere for different spherical patterns. IEEE Trans Automatic Control 59(5):1283–1289 3. Reynolds C (1987) Flocks, herds, and schools: a disturbuted behavioral model. Comput Graph 21(4):25–34 4. Vocsek T et al (1995) Novel type of phase transitions in a system of self-driven particles. Phys Rev 75(6):1226–1229 5. Jadbaie A, Motee N, Barahona M (2004) On the stability of the Kuramoto model of coupled nonlinear oscillators. In: Proceeding of the 2004 American control conference, pp 4296–4301 6. Lohe MA (2009) Non-abelian Kuramoto model and synchronization. J Phys A: Math Theor 42:395101 7. Sepulchre R, Paley DA, Leonard NE, Joosten D (2007) Stabilization of planar collective motion: all-to-all communication. IEEE Trans Autom Control 52(5):811–824 8. Olfati-Saber R (2006) Swarms on sphere: a programmable swarm with synchronous behaviors like oscillator networks. In: Proceedings of IEEE conference on decision control, pp 5060–5066

A Novel Single-Input Rule Module Connected Fuzzy Logic System and Its Applications to Medical Diagnosis Qiye Zhang(B) and Chunwei Wen School of Mathematics and Systems Science, LMIB of the Ministry of Education, Beihang University, Beijing 100191, China [email protected]

Abstract. The fuzzy logic system is an expert system based on fuzzy logic and fuzzy reasoning. It consists of a series of fuzzy rules, which can better represent expert knowledge and effectively treat uncertainty and inaccuracy existing in the field of medical diagnosis. To overcome the rules explosion problem, this paper proposes a novel single-input rule module connected fuzzy logic system, in which the rules adopt single-input antecedents and multi-output consequents with the form of Fuzzy Weighted Linear Function (SIRM-FWLF for short). The updating formulas using the steepest descent algorithm are also deduced to optimize the system parameters. At last, numerical simulations are conducted in the MATLAB environment to compare our SIRM-FWLF model with Mamdani, TSK fuzzy logic systems and fuzzy functional SIRM inference model for medical diagnosis of breast cancer. The simulation results show that the proposed SIRMFWLF FLS can provide higher diagnostic accuracy and recall rates than other fuzzy logic systems or fuzzy inference models. Keywords: Fuzzy logic system · Single-input rule module · Triangular fuzzy number · Graded mean integration representation · Steepest descent algorithm

1 Introduction Fuzzy sets were first introduced by Zadeh in 1965 [1] to describe the implicity and ambiguity we often met with. Since then the theory of fuzzy sets has been developed rapidly, including fuzzy arithmetic, fuzzy logic, fuzzy control, etc. [2]. Fuzzy logic system was first proposed by Mamdani in 1974 [3] for controlling of simple dynamic plant, and then Takagi-Sugeno-Kang (TSK) fuzzy logic systems (FLS) was brought forward for system identification [4, 5]. Both of them are characterized by IF-THEN rules, and have the same antecedent form, but they are different in the form of consequents. Regardless of the differences between Mamdani and TSK FLSs, they all have been developed and widely applied to many fields, such as system identification, pattern recognition, communication network and signal processing, etc, due to their powerful c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 357–366, 2020. https://doi.org/10.1007/978-981-32-9682-4_37

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abilities of tackling with uncertainties [6–10]. In recent years, FLSs have been also utilized as computer-aided system for medical diagnosis [11–15]. For instance, in [11] Lee and Wang proposed a fuzzy expert system for diabetes decision support application in 2011. They built a five-layer fuzzy ontology to model the diabetes knowledge, and developed a semantic decision support agent (SDSA) for knowledge construction, fuzzy ontology generating, and semantic fuzzy decision making. In [12] Lukmantoa and Irwansyah designed a fuzzy hierarchical architecture to perform early detection against Diabetes Mellitus (DM) in 2015, which is based on how a medical doctor concluded according to indication that someone has the potential against DM. In 2016, Norouzi et al. put forward an adaptive neuro fuzzy inference system (ANFIS) for predicting the renal failure time-frame of chronic kidney disease (CKD) based on real clinical data [13]. They used the ANFIS model to predict glomerular filtration rate (GFR) values. Reddy et al. [14] and Paul et al. [15] employed a hybrid OFBAT with rule-based fuzzy logic model and a fuzzy rule-based system, respectively, for the heart disease prediction or the risk level assessment of heart disease more recently. As we know, the performance of a fuzzy logic system depends on its inference rules. However, both Mamdani and TSK FLSs have network-type rule structure, which often causes the rules explosion problem when a FLS has multiple inputs. This also makes the setting and adjustment of rule parameters complicated. Thus Yubazaki et al. proposed the single input rule modules (SIRMs for short) connected fuzzy inference model in 1997 [16] to overcome the issue on “curse of dimensionality”. SIRMs can reduce the number of fuzzy rules drastically as it has one input type “IF-THEN” rule form. And then SIRMs inference model has been often used for stabilization control of inverted pendulum system or other fuzzy control systems [17–19]. Due to the rules reduction, many extensions of the SIRMs connected fuzzy inference model have been reported. In order to identify nonlinear function, Seki et al. brought forward the functional-type SIRMs [20], in which the consequents are generalized to functions from real numbers. Mitsuishi and Shidama designed a optimal control of the functional-type SIRMs in [21]. In 2010, Seki further built fuzzy functional SIRMs model (FFSIRM for short) [22], in which the consequents are generalized to fuzzy sets. He also applied the FFSIRM model to medical diagnosis and show its superiority over the conventional SIRMs model and Mamdani FLSs [23], and considered the issue on nonlinear function approximation of FFSIRM in [24]. Lately, Seki and Nakashima have proposed the SIRMs connected fuzzy inference model with functional weights (SIRM-FW for short) for medical diagnosis [25], in which they replaced the single-variable crisp weights of SIRMs by the functional ones. Furthermore, Li et al. have developed the SIRMs connected fuzzy inference system with multi-variable functional weights (FWSIRM-FIS for short), and conducted analysis and optimization design of FWSIRM-FIS in [26]. In this paper we propose a novel SIRMs model with fuzzy weighted linear functional consequents (SIRM-FWLF for short). The proposed rules establish some relationship between consequents and antecedents in advance, so that our SIRM-FWLF model could reduce the times of training, and optimize parameters faster. We also deduce the updating formulas by using the steepest descent algorithm to optimize the system parameters. To show the superiority of our proposed SIRM-FWLF model, we conduct numerical experiments in the MATLAB environment to compare our SIRM-

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FWLF FLS with FFSIRM model, Mamdani and TSK FLSs for medical diagnosis of breast cancer. The rest of this paper is organized as follows. Section 2 provides some preliminaries of fuzzy set theory and FLSs. In Sect. 3, the proposed SIRM-FWLF are described in detail. Section 4 conducts numerical simulations for breast cancer diagnosis to compare the proposed SIRM-FWLF system with FFSIRM model, Mamdani and TSK FLSs. In Sect. 5, conclusions and some suggestions for future research are given.

2 Preliminaries In this section, we recall some concepts and results about triangular fuzzy numbers [2, 27, 28], the structures of Mamdani and TSK FLSs [7, 9], and the frameworks of SIRMs and FFSIRM model [16, 22]. Without loss of generality, let R denote the set of the real numbers; for a non-empty set X and a fuzzy set A on X, let μA (x) denote the membership function (MF for short) of A. Let a, b, c ∈ R, a triangular fuzzy number A is determined by the following MF μA (x) [28]: ⎧ x−a ⎪ ⎪ , a ≤ x ≤ b, ⎪ ⎨b−a (1) μA (x) = x − c , b ≤ x ≤ c, ⎪ ⎪ b−c ⎪ ⎩ 0, otherwise, which can be denoted by A = (a, b, c). In this paper, we will use the width-equal triangular fuzzy number for each antecedent fuzzy set in all involved FLSs, its MF is determined by the center a and the width d, and can be expressed as: ⎧ ⎨ 1 − |x − a| , a − d ≤ x ≤ a + d, (2) μA (x) = d ⎩0, otherwise. For a triangular fuzzy number A = (a, b, c), the graded mean integration representation a + 4b + c [27] (GMIR for short) of A can be computed as [28]: P(A) = . 6 2.1 Mamdani Fuzzy Logic Systems A Mamdani FLS is composed of four modules: fuzzier, rules, inference and defuzzifier, see [7] for details. Fuzzifier: In this paper we use singleton fuzzifier. It maps a crisp point x = (x1 , x2 , . . . , x p ) ∈ X1 × X2 × . . . × Xp ≡ X into a fuzzy singleton Ax = μx1 × · · · × μx p with support x = (x1 , x2 , . . . , x p ), where μxi (i = 1, . . . , p) is defined by  1, xi = xi ,  μxi (xi ) = 0, xi = xi .

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Rules: Suppose that the system have p inputs x1 ∈ X1 , . . . , x p ∈ Xp , one output and M rules, so that the l-th rule in a Mamdani FLS has the form: Rl : IF x1 is Al1 and . . . x p is Alp , THEN y is Gl , where Ali , Gl (i = 1, . . . , p; l = 1, . . . , M) are fuzzy numbers. Center of Sets Defuzzifier: Center-of-sets (COS for short) defuzzifier merges the inference engine and defuzzifier into one module. Thus the final output of a Mamdani FLS using COS defuzzifier can be calculated as: p

yCOS (x) =

l ∑M l=1 c Ti=1 μAl (xi ) i

p

∑M l=1 Ti=1 μAl (xi ) i



l l ∑M l=1 c f (x) , M ∑l=1 f l (x)

(3)

where cl is the centroid of the l-th consequent Gl , T stands for a t-norm operation, and p μAl (xi ) is called firing level (or firing strength) of the l-th rule. f l (x) = Ti=1 i

2.2

TSK Fuzzy Logic Systems

For a TSK FLS with p inputs, one output and M rules, the l-th (l = 1, . . . , M) rule has the following form: Rl : IF x1 is Al1 and . . . x p is Alp , THEN yl (x) = c10 + c11 x1 + · · · + c1p x p , where Fil (i = 1, 2, . . . , p) are antecedent fuzzy sets, and cli (i = 1, 2, . . . , p) are consequent parameters, which are crisp numbers. Since the consequents yl (x) (l = 1, . . . , M) of a TSK FLS are all crisp numbers, its final output can be easily computed as: yTSK (x) =

l l ∑M l=1 y (x) f (x) , l ∑M l=1 f (x)

(4)

p where f l (x)  Ti=1 μAl (xi ) is still the firing level of the l-th rule. i

2.3

SIRMs Inference Model

Next let us introduce the rule structure of SIRMs model [16]. Assume that the system has p inputs x1 ∈ X1 , . . . , x p ∈ Xp and one output. Then the i-th single-input rule module in the SIRMs model has the following form [16]: Rules-i : IF xi is Aij , THEN yi is yij , where xi corresponds to the i-th input item is the sole variable of the antecedent part of Rules-i, and yi is the variable of its consequent part. Aij and yij are, respectively, fuzzy sets and real number of the j-th rule of the Rules-i, where i = 1, 2, . . . , p; j = 1, 2, . . . , mi , and mi stands for the number of rules in Rules-i.

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Given an input xi0 to Rules-i, the firing level of the j-th rule in Rules-i is hij = Aij (xi0 ), the inference result y0i of Rules-i can be calculated as: m

y0i

=

i hij yij ∑ j=1

m

i hij ∑ j=1

.

(5)

The final output of the SIRMs inference model will be p

ySIRM = ∑ wi y0i ,

(6)

i=1

where wi , (i = 1, . . . , p) is the weight of the i-th input item xi , stands for the importance degree of xi . 2.4 FFSIRM Inference Model Like a SIRMs model, we assume that the system has p inputs x1 ∈ X1 , . . . , x p ∈ Xp and one output. Then the i-th single-input rule module in the FFSIRM model has the form [22]: Rules-i : IF xi is Aij , THEN yi is Fji , FFSIRM and SIRMs inference models have the same antecedent forms. However the consequent of FFSIRM model is a fuzzy set Fji , while the consequent of SIRMs model is a crisp number. Given an input xi0 to Rules-i, the firing level of the j-th rule in Rules-i is hij = Aij (xi0 ). In order to get a crisp output, one can choose a proper defuzzifier to replace the consequent fuzzy set Fji with a crisp value pij . Then the inference result y0i of input xi0 is mi hij pij ∑ j=1 . The final output of the FFSIRM inference model is the same with that y0i = mi hij ∑ j=1 of the SIRMs model, and also computed by (6).

3 The Proposed SIRM-FWLF Inference Model and Its Parameter Updating In this section, we will introduce our SIRM-FWLF inference model in detail, and deduce the updating formulas using steepest descent algorithm. 3.1 The Proposed SIRM-FWLF Inference Model We also assume that the system has p inputs x1 ∈ X1 , . . . , x p ∈ Xp and one output. Then the i-th single-input rule module in the proposed SIRM-FWLF model has the following form: Rules-i : IF xi is Aij , THEN yi is Fji (xi ) = Dij xi + E ij ,

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where xi is the i-th variable of antecedents, and yi is the variable of its consequents. Aij , Dij and E ij are all fuzzy numbers, (i = 1, 2, . . . , p; j = 1, 2, . . . , mi ), here mi stands for the number of rules in Rules-i. Such rules take into consideration the connections between antecedents and its consequents, which may reduce the training times, and optimize the system parameters fast. One can see that our proposed SIRM-FWLF model is a special case of the FFSIRM inference model. So given an input xi0 to Rules-i, the firing level of the j-th rule in Rules-i is still hij = Aij (xi0 ). In order to get a crisp output, we also need choose a proper defuzzifier to replace the consequent fuzzy set Fji (xi0 ) with a crisp value pij . For this, we need to compute Fji (xi0 ) = Dij xi0 + E ij first. We will use the center-of-sets (COS for short) defuzzifier, and compute the GMIR [27] of Fji (xi0 ) for its centriod. In [29], Tian has proved that linear operation keeps shapes of triangular, trapezoidal and Gaussian fuzzy numbers. And in [30], Zhang et al. have shown that linear operation keeps GMIRs of triangular, trapezoidal fuzzy numbers. Thus for computation simplicity, in this paper we would like to use triangular fuzzy numbers for antecedent and consequent fuzzy sets of all rules. Denote that triangular fuzzy numbers Dij = [daij , dbij , dcij ] and E ij = [eaij , ebij , ecij ], then the GMIR f ji of the j-th consequents Fji (xi0 ) will be f ji = d ij xi0 + eij =

daij + 4dbij + dcij 6

xi0 +

eaij + 4ebij + ecij 6

.

(7)

And the inference result y0i of the input xi0 can be calculated as m

y0i =

i hij f ji ∑ j=1

m

i hij ∑ j=1

.

(8)

The final output of the SIRM-FWFL inference model is still the same with that of the SIRMs model, and also computed by (6). 3.2

Parameters Optimization for SIRM-FWFL Model Based on the Steepest Descent Method

In this subsection, we will deduce the parameters’ updating formulas of our proposed SIRM-FWFL model by using the steepest descent algorithm. These parameters include the centers aij and widths d ij of width-equal triangular fuzzy numbers for antecedents Aij , six parameters daij , dbij , dcij and eaij , ebij , ecij of triangular fuzzy numbers for the coefficients Dij and E ij of consequents, respectively, (i = 1, . . . , p; j = 1, . . . , mi ). For the training input-output data (x, y) = (x1 , x2 , . . . , x p ; y) of some FLS, we use the square error objective function EFLS = 12 (yDES − yFLS )2 to evaluate the error between the desired output value yDES and the actual FLS output value yFLS . We rewrite the objective function for our proposed SIRM-FWFL model as ESIRM-FWFL = 12 (yDES − ySIRM-FWFL )2 . Take the antecedent parameter aij , d ij as an example. From (6), we get that

∂ y0i ∂ ESIRM-FWFL = (y − y ) · (−w ) · . DES SIRM-FWFL i ∂ aij ∂ aij

(9)

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f ji − y0i ∂ hij ∂ y0i = · . According to (2), the derivation of hij to mi ∂ aij hij ∂ aij ∑ j=1 the center aij and the width d ij of the antecedent Aij are respectively:

And from (8) we have

∂ hij ∂ aij

=

sgn(xi0 − aij ) d ij

,

∂ hij ∂ d ij

=

|xi0 − aij | (d ij )2

,

(10)

where sgn denotes the well-known sign function. Combine the above formulas, we get the derivative of the objective function ESIRM-FWFL on the parameter aij : f ji − y0i sgn(xi0 − aij ) ∂ ESIRM-FWFL = (y − y ) · (−w ) · · . DES SIRM-FWFL i mi ∂ aij hij d ij ∑ j=1

(11)

The learning algorithms at t + 1 step of parameter aij and d ij are obtained as follows: aij (t + 1) = aij (t) + α (yDES − ySIRM-FWFL ) · wi ·

f ji − y0i sgn(xi0 − aij ) · , mi hij d ij ∑ j=1

(12)

f ji − y0i |xi0 − aij | · . mi hij (d ij )2 ∑ j=1

(13)

d ij (t + 1) = d ij (t) + β (yDES − ySIRM-FWFL ) · wi ·

Similarly, we can obtain the learning algorithms at t + 1 step of the consequent coefficients as follows: hij mi hij ∑ j=1

1 · xi0 , 6

(14)

dbij (t + 1) = daij (t) + γ2 (yDES − ySIRM-FWFL ) · wi ·

hij mi ∑ j=1 hij

2 · xi0 , 3

(15)

dcij (t + 1) = dcij (t) + γ3 (yDES − ySIRM-FWFL ) · wi ·

hij mi ∑ j=1 hij

1 · xi0 , 6

(16)

daij (t + 1) = daij (t) + γ1 (yDES − ySIRM-FWFL ) · wi ·

eaij (t + 1) = eaij (t) + δ1 (yDES − ySIRM-FWFL ) · wi · ebij (t + 1) = ebij (t) + δ2 (yDES − ySIRM-FWFL ) · wi · ecij (t + 1) = ecij (t) + δ3 (yDES − ySIRM-FWFL ) · wi ·

hij mi ∑ j=1 hij

1 · , 6

(17)

2 · , mi hij 3 ∑ j=1

(18)

hij

hij

1 · , mi hij 6 ∑ j=1

(19)

where α , β , γ1 , γ2 , γ3 , δ1 , δ2 , δ3 are learning rates, t represents the number of learning iterations.

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4 Numerical Simulations for Medical Diagnosis of Beast Cancer In this section, we will conduct several numerical simulations for medical diagnosis of some breast cancer data, which is from the machine learning UCI data sets and can be download from the web site: http://archive.ics.uci.edu/ml/datasets/, to compare our proposed SIRM-FWLF FLS with FFSIRM model [22], and the most commonly used Mamdani and TSK FLSs. In the breast cancer data, there are 699 patient cases with 10 indicator items. The items includes ID number, and 9 diagnostic features with the value between 0 to 10. The last column stands for class attribute: benign or malignant. We first preprocess the datas in the original data set as follows: (1) replace the missing values using the mean value of the corresponding indicator term; (2) to reduce the complexity, we employ the decision tree algorithm to reduce the indicators to four items: uniform of Cell Size (x1 ), uniform of Cell Shape (x2 ), Bare Nuclei (x3 ) and Normal Nucleoli (x4 ), which will be used as the four inputs of the involved FLSs, and normalized into [0, 1]. Furthermore, the datas will be classified to Group 1 (benign) and Group 2 (malignant). The desired outputs in Group 1 and Group 2 are set to be 0.25, 0.75, respectively, and the inference result yFLS of some FLS will be classified as follows: Group 1 (benign): yFLS ≤ 0.3; Group 2 (malignant): yFLS > 0.3. We will adopt the steepest descent algorithm based on the square error function to optimize the system parameters. For each FLS or inference model, we will use the width-equal triangular fuzzy number for each antecedent fuzzy set. For Mamdani and TSK FLSs, each antecedent is divided into two width-equal triangular fuzzy sets Aij , determined by its centers aij and its widths d ij , (i = 1, . . . , 4; j = 1, 2). For FFSIRM model and the proposed SIRM-FWLF FLS, each antecedent is divided into four widthequal triangular fuzzy numbers Aij , (i = 1, . . . , 4, j = 1, . . . , 4). General triangular fuzzy numbers Fji = ( f aij , f bij , f cij ) are used for FFSIRM consequents, while in SIRM-FWFL, its consequents Fji (xi0 ) = Dij xi0 + E ij are determined by Dij = (daij , dbij , dcij ), and E ij = (eaij , ebij , ecij ). The importance degree for each input item is set to be 0.5. In the simulations, we utilize five-fold cross validation to test all the FLSs, which means that 5 simulations are run for each FLS, and 4 in 5 (558) datas are used for training and 1 in 5 (141) datas for testing in each simulation. And learning iterations of each simulation are executed 10 times. We take accuracy and recall rates as the performance index for each FLS or inference model, which are defined as follows [11]. Let the true positive (TP) and the true negative (TN) denote the correct classification. The false positive (FP) is when the outcome is not accurately predicted as yes, however, it is no. Still, the false negative (FN) is when the outcome is not accurately predicted as no, however, it is yes. Then accuracy rate is the proportion of the total number of predictions that were correct, recall rate is the proportion of the positive cases that were correctly identified. Equation (20) show the formulas for the accuracy and recall rates, respectively: accuracy =

TN +TP × 100%, T N + FN + T P + FP

recall =

TP × 100%. T P + FN

(20)

Table 1 shows the accuracy and recall rate of different models using the same training and testing data for beast cancer medical diagnosis. From Table 1 we can see the pro-

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posed SIRM-FWFL FLS have the best accuracy than FFSIRM model, TSK and Mamdani FLSs. Moreover, our SIRM-FWFL model can reach 0.9916 recall rate in malignant cases. This is significant in medical diagnostics because it can ensure timely treatment of diseases. In benign cases, SIRM-FWFL reachs 0.9671 recall rate, which is also the highest one than the other FLSs. Table 1. Medical diagnosis results (means and STDs of 5-fold cross validation) for four FLSs Performance index

SIRM-FWFL

FFSIRM

TSK

Mamdani

Accuracy rate

0.9742 ± 0.0074

0.9556 ± 0.0115

0.9485 ± 0.0183

0.9384 ± 0.0135

Recall rate (benign)

0.9671 ± 0.0159

0.9561 ± 0.0198

0.9585 ± 0.0222

0.9338 ± 0.0334

Recall rate (malignant)

0.9916 ± 0.0168

0.9549 ± 0.0238

0.9145 ± 0.0397

0.9625 ± 0.0359

5 Conclusions In this paper, a novel fuzzy logic system named SIRM-FWFL has been proposed, which adopts single-input antecedents and multi-output consequents with the form of fuzzy weighted linear function. The proposed SIRM-FWFL inference model does not only reduce greatly the number of rules like other SIRM models, bus also establishes some relationship between consequents and antecedents in advance, so that it can reduce the times of training, and optimize parameters faster. Several simulations for beast cancer diagnosis have been conducted, and the simulation results show that our SIRM-FWFL system have better recall rate and accuracy than other FLSs or SIRM models, which indicates that our SIRM-FWFL model would be beneficial in the medical fields. In the future work we would like to add expert advice in our SIRM-FWFL FLS to increase the credibility of the model.

References 1. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353 2. Klir G, Yuan B (1995) Fuzzy sets and fuzzy logic: theory and applications. Prentice-Hall, Upper Saddle River 3. Mamdani EH (1974) Applications of fuzzy algorithms for simple dynamic plant. Proc IEEE 121(12):1585–1588 4. Takagi T, Sugeno M (1985) Fuzzy identification of systems and its applications to modeling and control. IEEE Trans Syst Man Cybern 15(1):116–132 5. Sugeno M, Kang GT (1988) Structure identification of fuzzy model. Fuzzy Sets Syst 28:15–33 6. Jang JSR (1993) ANFIS: adaptive-network-based fuzzy inference system. IEEE Trans Syst Man Cybern 23(3):665–685 7. Wang LX (1996) A course in fuzzy systems and control. Prentice-Hall, Upper Saddle River 8. Tanaka K, Wang HO (2001) Fuzzy control systems design and analysis. Wiley, New York 9. Mendel JM (2001) Uncertain rule-based fuzzy logic systems: introduction and new directions. Prentice-Hall International (UK) Ltd., London 10. Castillo O, Melin P (2008) Type-2 fuzzy logic: theory and applications. Springer, Berlin 11. Lee CS, Wang MH (2011) A fuzzy expert system for diabetes decision support application. IEEE Trans Syst Man Cybern-Part B: Cybern 40(1):139–153

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12. Lukmanto RB, Irwansyah E (2015) The early detection of diabetes mellitus (DM) using fuzzy hierarchical model. Procedia Comput Sci 59:312–319 13. Norouzi J, Yadollahpour A, Mirbagheri SA, Mazdeh MM, Hosseini SA (2016) Predicting renal failure progression in chronic kidney disease using integrated intelligent fuzzy expert system. Comput Math Methods Med 2016(3):1–9 14. Reddy GT, Khare N (2017) An efficient system for heart disease prediction using hybrid OFBAT with rule-based fuzzy logic model. J Circuits Syst Comput 26(4):1–21 15. Paul AK, Shill PC, Rabin MRI, Murase K (2018) Adaptive weighted fuzzy rule-based system for the risk level assessment of heart disease. Appl Intell 48:1739–1756 16. Yubazaki N, Yi J, Hirota K (1997) SIRMs (single input rule modules) connected fuzzy inference model. J Adv Comput Intell Intell Inf 1(1):23–30 17. Yi J, Yubazak N, Hirota K (2001) Stabilization control of series-type double inverted pendulum systems using the SIRMs dynamically connected fuzzy inference model. Artif Intell Eng 15(3):297–308 18. Yi J, Yubazaki N, Hirota K (2001) Upswing and stabilization control of inverted pendulum system based on the SIRMs dynamically connected fuzzy inference model. Fuzzy Sets Syst 122(1):139–152 19. Yi J, Yubazaki N, Hirota K (2002) A proposal of SIRMs dynamically connected fuzzy inference model for plural input fuzzy control. Fuzzy Sets Syst 125(1):79–92 20. Seki H, Ishii H, Mizumoto M (2008) On the generalization of single input rule modules connected type fuzzy reasoning method. IEEE Trans Fuzzy Syst 16(5):1180–1187 21. Mitsuishi T, Shidama Y (2011) Optimal control using functional type SIRMs fuzzy reasoning method. In: Honkela T, Duch W, Girolami M, Kaski S (eds) Artificial neural networks and machine learning (ICANN 2011-Part II), vol 6792. Lecture Notes in Computer Science. Springer, Berlin, pp 237–244 22. Seki H (2010) Fuzzy functional SIRMs inference model. In: Proceedings of joint 5th international conference on soft computing and intelligent systems and 11th international symposium on advanced intelligent systems (SCIS & ISIS2010), Okayama, Japan, pp 512–516 23. Seki H (2010) An expert system for medical diagnosis based on fuzzy functional SIRMs inference model. In: Proceedings of joint 5th international conference on soft computing and intelligent systems and 11th international symposium on advanced intelligent systems (SCIS & ISIS2010), Okayama, Japan, pp 517–521 24. Seki H (Feburary 2012) Nonlinear function approximation using fuzzy functional SIRMs inference model. In: WSEAS international conference on artificial intelligence, pp 201–206 25. Seki H, Nakashima T (2014) Medical diagnosis and monotonicity clarification using SIRMs connected fuzzy inference model with functional weights. In: IEEE international conference on fuzzy systems 26. Li CD, Gao JL, Yi JQ, Zhang GQ (2017) Analysis and design of functionally weighted single-input-rule-modules connected fuzzy inference systems. IEEE Trans Fuzzy Syst 99:1– 15 27. Chen SH, Li GC (2000) Representation, ranking, and distance of fuzzy number with exponential membership function using graded mean integration method. Tamsui Oxford J Math Sci 16(2):123–131 28. Chou CC (2003) The canonical representation of multiplication operation on triangular fuzzy numbers. Comput Math Appl 45(10):1601–1610 29. Tian X (2018) The design of fuzzy Convolutional neural network and its applications. In: Master’s thesis of Beihang Unversity, Beijing 30. Zhang QY, Liu YQ, Tian X (2018) A novel fuzzy logic system with consequents as fuzzy weighted averages of antecedents. In: Proceedings of 2018 Chinese Intelligence Systems Conference, Wenzhou, China. Lecture Notes in Electrical Engineering, vol 529, pp 571–582

Adaptive Finite-Time Neural Consensus Tracking for Second-Order Nonlinear Multiagent Systems with Unknown Control Directions Guoqing Liu, Lin Zhao(B) , and Jinpeng Yu Qingdao University, Qingdao 266000, China [email protected]

Abstract. This paper mainly studies the problem of consensus tracking for second-order nonlinear multiagent system with unknown control directions. The backstepping technology based on command filtered can deal with the problem that the output of each agent can track the leaders state with acceptable accuracy in finite-time and the issue of computational explosion has also been well solved at the same time. The unknown nonlinear dynamics in the system are approximated using neural network. We define the error compensation signal to compensate the error generated by filtering. The Nussbaum type function is introduced to solve the problem of unknown control directions. The virtual control signal, control input signal and adaptive law are also designed respectively. A numerical example is given in the simulation to show the superiority of this scheme. Keywords: Unknown control directions · Backstepping technology Nussbaum type function · Multiagent systems

1

·

Introduction

With the rapid development of technology, the control of consensus of multiagent systems (MASs) is now receiving more and more attention because of the powerful applications of it in unmanned aerial vehicles, unmanned ship and unmanned boats [1–3]. Two viewpoints have been considered in the control of consensus, that is the leader-less consensus and the leader-followers consensus. Because of the consensus problems with finite-time convergence have many advantages, and the consensus tracking problem of MASs with unknown nonlinear dynamics is complex and difficult, so many scholars research the finite-time consensus tracking with unknown nonlinear MASs [4–6]. Backstepping is an effective technique for tracking problems of nonlinear systems. The procedure of backstepping was settled in [7] and the virtual control signal with the explosion of complexity problem can be solve by the dynamic surface control in [8–11]. In the backstepping design process, the output by the c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 367–374, 2020. https://doi.org/10.1007/978-981-32-9682-4_38

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filter with the input function as the virtual control signal is applied to the command filtered backstepping to eliminate the explosive term [12,13]. And the unknown dynamics of nonlinear is approximated by neural network approximation technology [14]. However, the unknown control directions are not considered in them. We research the problem of consensus tracking for second-order nonlinear MASs with unknown control directions. The finite-time command filtered backstepping method combined with the technique of Nussbaum-type gains is proposed, which can overcome the unknown control directions and guarantee the tracking output errors of the closed-loop system arrive into the desired neighborhood. 1.1

Problem Formulation and Preliminaries

This paper we study the communications between N agents and use a weighted directed graph G = (v, ε) to describe it, where v = {1, 2, ..., N } represent the nodes and ε ∈ v × v represent the edges. Nodes i to j is described as (i, j) ∈ ε, Ni = {j|(j, i) ∈ ε} describe the node i s neighbors. The matrix A = [aij ] ∈ / RN ×N describe the weights, where aij > 0 when (j, i) ∈ ε, aij = 0 when (j, i) ∈ N ε and aii = 0 for ∀i. D = diag{d1 , d2 , ..., dN } with di = a and L = ij j=1 D − A is the Laplacian matrix. If there has a form of continuous edge sequence {(i, k), (k, l), ..., (m, j)}, we say that there exists a direct path from node i to note j. The communications between N agents and a leader agent 0, use a extension ¯ = (¯ graph G v , ε¯) to describe it, B = diag{b1 , b2 , ..., bN } is the adjacency matrix where bi > 0 if from nodes 0 to i has an edge, otherwise, bi = 0. The root node is defined that exists a node has a directed path from it to the other node, if the root node exists then G has a spanning tree.

2

System Description

The communications between N agents and a leader agent 0 of the MAS use a ¯ to describe it. We assume that G ¯ contains a spanning tree, and directed graph G the leader node is root node. The functions about the follower are ith given as x˙ i,1 = fi,1 (xi,1 ) + gi,1 (xi,1 )xi,2 x˙ i,2 = fi,2 (xi ) + gi,2 (xi )Ki ui yi = xi,1 , i ∈ v

(1)

where xi = [xi,1 , xi,2 ]T ∈ R2 is the state vector, and the systems output signal is denote as yi ∈ R and the function of ui ∈ R is the control input with the gain Ki = 1 or Ki = −1. fi,m (·) of the system is unknown smooth nonlinear function, The transfer coefficient and the sign of Ki ∈ R are assumed to be unknown. r(t) ∈ R represents the leaders signal and assume that r(t) and r(t) ˙ are smooth, bounded and known functions.

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Assumption 1. gi,m (·) is a known smooth nonlinear function and ρ1 < |gi,m | < ρ2 , m = 1, 2, where ρ1 and ρ2 are known positive constants. ¯ contains s spanning tree and the leader Assumption 2. The extension graph G node is the root node.

3

Main Results

The tracking errors of the ith agent are defined as zi,1 =

N  j=1

aij (yi − yj ) + bi (yi − r)

(2)

zi,2 = xi,2 − πi,2 ,i ∈ v The command filter is chosen as φ˙ i,1,1 = li,1,1 1 li,1,1 = −ri,1,1 |φi,1,1 − αi,1 | 2 sign(φi,1,1 − αi,1 ) + φi,1,2 φ˙ i,1,2 = −ri,1,2 sign(φi,1,2 − li,1,1 )

(3)

in which, πi,2 (t) = φi,1,1 (t) and π˙ i,2 (t) = li,1,1 (t) is the output of the command filter. Lemma 1 [4]. For any real numbers λ1 > 0, λ2 > 0, 0 < γ < 1, the Lyapunov function V˙ (x) + λ1 V (x) + λ2 V γ (x) ≤ 0, that the system can be stable in finitetime, where finite-time is Tr ≤ t0 + [1/(λ1 (1 − γ))] ln[(λ1 V 1−γ (t0 ) + λ2 )/λ2 ]. Now, we designed the virtual control signals αi,1 , ui as follows: αi,1 =

1 {−ki,1 zi,1 (di +bi )gi,1

i) {−ki,2 zi,2 − ui = − ℵ(ζ gi,2

−

T vi,1 θˆi Si,1 Si,1

2h2 i,1

T vi,2 θˆi Si,2 Si,2

2h2 i,2

− 12 vi,1 −

N  j=1

γ aij gj,1 xj,2 + bi r˙ − si,1 vi,1 }

γ − 12 vi,2 + π˙ i,2 − (di + bi )gi,1 zi,1 − si,2 vi,2 }

(4) where ki,1 , ki,2 and si,1 , si,2 are designed positive constants, γ ∈ (0, 1), ℵ(ζ) is a smooth Nussbaum-type function, and the ζi is defined as ζ˙i = −vi,2 {−ki,2 zi,2 −

T Si,2 1 vi,2 θˆi Si,2 γ − vi,2 + π˙ i,2 −(di +bi )gi,1 zi,1 −si,2 vi,2 } (5) 2 2hi,2 2

The system tracking error is defined as vi,m = zi,m − ξi,m ,i ∈ v, m = 1, 2

(6)

In order to eliminate the error πi,2 − αi,1 .Therefore, the following error compensation signal ξi,m is further defined as ξ˙i,1 = −ki,1 ξi,1 + (di + bi )gi,1 (πi,2 − αi,1 ) + (di + bi )gi,1 ξi,2 − li,1 sign(ξi,1 ) (7) ξ˙i,2 = −ki,2 ξi,2 − (di + bi )gi,1 ξi,1 − li,2 sign(ξi,2 )

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Now, in order to prove that the designed control signal is correct we establish the Lyapunov function equation as Vi,1 =

1 2 v 2 i,1

(8)

Derivation of this can be obtained as ⎛ ⎞ N  ˙ ¯ a g x − bi r˙ − ξi,1 + (di + bi )gi,1 zi,2 ⎠ f − V˙ i,1 = vi,1 ⎝ i,1 j=1 ij j,1 j,2 +(di + bi )gi,1 αi,1 + (di + bi )gi,1 (πi,2 − αi,1 )

(9)

N where f¯i,1 = (di + bi )fi,1 − j=1 aij fj,1 , and it is unknown but it can be approximated by neural networks T f¯i,1 = Wi,1 Si,1 (Zi,1 ) + δi,1

(10)

where zi,1 = [xi,1 , xj,1 ]T , |δi,1 | ≤ εi,1 , εi,1 is a positive constant. We can get the following results: 2 T Wi,1  Si,1 Si,1 vi,1 1 1 2 1 + h2i,1 + vi,1 + ε2i,1 2 2hi,1 2 2 2 2

vi,1 f¯i,1 ≤

(11)

where hi,1 > 0 is a constant. Now substituting (4), (5), (7), (10) and (11) into (9), we can get  V˙ i,1 ≤ vi,1

2 T vi,1 (Wi,1 2 −θˆi )Si,1 Si,1 2h2i,1

+ (di + bi ) gi,1 vi,2 −



− ki,1 vi,1

γ si,1 vi,1

+ li,1 sign(ξi,1 )

1 1 + h2i,1 + ε2i,1 (12) 2 2

Continue to construct another Lyapunov function: 1 2 Vi,2 = Vi,1 + vi,2 2

(13)

V˙ i,2 = V˙ i,1 + vi,2 (fi,2 + gi,2 Ki ui − π˙ i,2 − ξ˙i,2 ) + ζ˙i − ζ˙i

(14)

T T fi,2 = Wi,2 Si,2 (Zi,2 ) + δi,2

(15)

Then we get

Similarly,

and εi,2 is a positive constant. We can get the following results:

2 2    γ+1 1 2 1 2 2 V˙ i,2 ≤ −ki,m vi,m − si,m vi,m + vi,m li,m sign(ξi,m ) + 2 hi,m + 2 εi,m m=1

+

2  m=1

2 T (Wi,m 2 −θˆi )Si,m Si,m vi,m 2h2i,m

m=1

+ (Ki ℵ(ζi ) + 1) ζ˙i (16)

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where hi,2 is a constant, from the Yangs inequality, we can get li,m 2 li,m v + 2 i,m 2

vi,m li,m sign(ξi,m ) ≤ which means 2  V˙ i,2 ≤ − ki,m − m=1

+

2  m=1

li,m 2



2 vi,m −

2  m=1

γ+1 si,m vi,m +

2 T (Wi,m 2 −θˆi )Si,m Si,m vi,m 2h2i,m

(17)

2  li,m m=1

2

+ 12 h2i,m + 12 ε2i,m



+ (Ki ℵ(ζi ) + 1) ζ˙i

(18)   2 2 we denote θi = max Wi,1  , Wi,2  , the constants κi , ρi are all positive. And we construct the adaptive function of θi as 2 2 T  κi vi,m Si,m Si,m ˆθ˙i = −2κi ρi θˆi + 2 2hi,m m=1

(19)

Consider the Lyapunov function as: N 

N  1 ˜2 Vi,2 + θi V = 2κ i i=1 i=1

(20)

where θ˜i = θi − θˆi , then we get   N  N  2 2   l li,m 1 2 1 2 2 2 V˙ ≤ − ki,m vi,m v − i,m + + h + ε i,m 2 2 2 i,m 2 i,m −

i=1 m=1 N  2  i=1 m=1

γ+1 si,m vi,m

+

N  i=1

2ρi θ˜i θˆi +

i=1 m=1 N  i=1

(21)

(Ki ℵ(ζi ) + 1) ζ˙i

By further deduction we can get  N  N  2 2   l γ+1 2 2 V˙ ≤ − ki,m vi,m vi,m − i,m − si,m vi,m 2 i=1 m=1 i=1 m=1   γ+1 γ+1 N  N N 2   li,m ςi ˜2 2 ςi ˜2 2 1 2 1 2 − + + h + ε ( ) + ( ) θ θ i,m i,m i i 2 2 2 κi κi −

i=1 m=1 N N   ςi ˜2 2 ρi oi θi2 κi θi + i=1 i=1

+

N  i=1

i=1

(22)

i=1

(Ki ℵ(ζi ) + 1) ζ˙i

Applying these to push and we can get V˙ ≤ −aV − bV

γ+1 2

+η+

N  i=1

(Ki ℵ(ζi ) + 1) ζ˙i

(23)

  (γ+1)/2 , and η = where a = min {2ki,m − li,m , 2ςi } b = min si,m · 2(γ+1)/2 , (2ςi )  N N 2 2 2 2 i=1 m=1 (1/2)li,m + (1/2)hi,m + (1/2)εi,m + N + i=1 ρi oi θi . From (22), we know that N  V˙ ≤ −aV + η + (Ki ℵ(ζi ) + 1) ζ˙i (24) i=1

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N From [11], we know i=1 (Ki ℵ(ζi ) + 1) ζ˙i is bounded in [0, tf ), so we let ηmax = N maxt∈[0,tf ] i=1 (Ki ℵ(ζi ) + 1) ζ˙i and η¯ = η + ηmax . So the inequation (24) can be changed to γ+1 (25) V˙ ≤ −aV − bV 2 + η¯ So from [4] that we know  in finite time vi,m will converge to the region |vi,m | ≤  2/(γ+1) η /[(1 − υ)a], 2(¯ η /[(1 − v)b]) }, υ ∈ (0, 1). We further construct min{ 2¯ the Lyanunov function of ξi N 2 1 2 V¯ = ξ 2 i=1 m=1 i,m

(26)

We can get its derivative N  N  2 2   2 V¯˙ = − ki,m ξi,m − ξi,m li,m sign(ξi,m )

+

i=1 m=1 N  i=1

i=1 m=1

(27)

(di + bi )gi,1 ξi,1 (πi,2 − αi,1 )

From [4], we know |πi,2 − αi,1 | ≤  in finite time T and combine with ρ1 < |gi,m (·)| < ρ2 can get √ 1 V¯˙ ≤ −k0 V¯ − (l0 − 2 · 2N ρ ¯ 2 )V¯ 2 (28) √

where k0 = 2 min {ki,m } , l0 = 2 min {li,m }, √|(di + bi )| |(πi,2 − αi,1 )| ≤ . ¯ Then, we can choose suitable li,m such that l0 − 4N ρ ¯ 2 > 0, then limt→T ξi,m = 0,  2/(γ+1) η /[(1 − υ)a], 2(¯ η /[(1 − v)b]) } We have the consequence |zi1 | ≤ min{ 2¯ T

T

r] and we can Denote Z1 = [z1,1 , z2,1 , ..., zN,1 ] , Θ1 = [y1 − r, y2 − r,√..., yN −  −1 get Θ = (H ⊗ I ) Z . In finite time |y − r| ≤ [ N min{ 2¯ η /[(1 − υ)a], 1 N ×N 1 1  2(¯ η /[(1 − v)b])

2/(γ+1)

}/σmin (H)] and σmin (H) in which is minimum singular.

Remark 1. The control parameters will affect the consensus tracking errors of agents. In order to get the small tracking errors, the system parameters ki,m , li,m , κi,m need to be large and system parameters li,m , γ need to be small.

4

Numerical Results

¯ and we Consider the system with one leader and three followers in the graph G use the Fig. 1 in [4] to illustrate the communication. The system dynamics f1,1 = cos(x1,1 ), g1,1 = 1, f1,2 = x1,1 x1,2 , g1,2 = 1 f2,1 = sin(0.5x2,1 ), g2,1 = 1, f2,2 = x2,1 e−0.3x2,2 , g2,2 = 1 f3,1 = cos(−0.5x3,1 ), g3,1 = 1, f3,2 = x3,1 x3,2 , g3,2 = 1. T

(29) T

r(t) = sin(0.5t) with the initial value of x1 (0) = [1.5, −0.5] x2 (0) = [−0.7, 0.3] , T x3 (0) = [1.4, −0.4] . The parameters ki,m = 10, si,m = 20, li,m = 8, γ = (3/5), hi,m = 1, κi = 1, ρi = 1 ri,1,1 = 450, ri,1,2 = 2 × 103 , Ki = −1. The Nussbaum type function is chosen as ζ 2 cos(ζ), the responses of xi,1 and r are showed in Fig. 1.

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1.5 r x1,1 x2,1

1

x i,1,i=1,2,3;r

x3,1

0.5

0

-0.5

-1

0

1

2

3

4

5

6

7

8

9

10

Times(s)

Fig. 1. Time response xi,1 (t), i = 1, 2, 3, of and r(t)

5

Conclusion

This paper designs the protocol to solve the problem of consensus tracking for nonlinear multiagent system with unknown control directions. In the designed protocol, the command filter is introduced to deal with the computational explosion problem, and the error compensation mechanism is also introduced and it can converge in finite-time. The virtual control signal is designed and it can ensure that the state of all agents converges to the state of leaders in finite-time, and the tracking error is desired small neighborhood. The uncertain nonlinear dynamic of the system can be solved by the neural network, and the Nussbaum type function can deal with the unknown control directions. Acknowledgments. This work was supported by the National Natural Science Foundation of China (61603204, 61573204), the China Postdoctoral Science Foundation (2017M612206) and the Shandong Province Outstanding Youth Fund (ZR2018JL020).

References 1. Olfati-Saber R et al (2004) Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Autom Control 49(9):1520–1533 2. Ma HW, Wang Z et al (2016) Neural-network-based distributed adaptive robust control for a class of nonlinear multiagent systems with time delays and external noises. IEEE Trans Syst Man Cybern Syst 46(6):750–758 3. Tang Y, Xing X et al (2016) Tracking control of networked multi-agent systems under new characterizations of impulses and its applications in robotic systems. IEEE Trans Ind Electron 63(2):1299–1307

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4. Zhao L, Yu J, Lin C, Ma Y (2018) Adaptive neural consensus tracking for nonlinear multiagent systems using finite-time command filtered backstepping. IEEE Trans Syst Man Cybern Syst 48(11):2003–2012 5. Zhao L, Jia Y, Yu J (2017) Adaptive finite-time bipartite consensus for secondorder multi-agent systems with antagonistic interactions. Syst Control Lett 102:22– 31 6. Zhao L, Yu J, Shi P (2019) Command filtered backstepping based attitude containment control for spacecraft formation. IEEE Trans Syst Man Cybern Syst. https://doi.org/10.1109/TSMC.2019.2896614 7. Krstic M, Kanellakopoulos I et al (1995) Nonlinear and adaptive control design. Wiley, New York 8. Swaroop D et al (2000) Dynamic surface control for a class of nonlinear systems. IEEE Trans Autom Control 45(10):1893–1899 9. Tong S, Li Y, Feng G, Li T (2011) Observer-based adaptive fuzzy backstepping dynamic surface control for a class of MIMO nonlinear systems. IEEE Trans Syst Man Cybern B Cybern 41(4):1124–1135 10. Zhang T, Ge SS (2008) Adaptive dynamic surface control of nonlinear systems with unknown dead zone in pure feedback. Automatica 44(7):1895–1903 11. Ge SS, Hong F, Lee T (2004) Adaptive neural control of nonlinear time-delay systems with unknown virtual control coefficients. IEEE Trans Syst Man Cybern B Cybern 34(1):499–516 12. Zhao L, Yu J, Lin C (2018) Command filter based adaptive fuzzy bipartite output consensus tracking of nonlinear coopetition multi-agent systems with input saturation. ISA Trans 80:187–194 13. Zhao L, Yu J, Lin C (2019) Distributed adaptive output consensus tracking of nonlinear multi-agent systems via state observer and command filtered backstepping. Inf Sci 478:355–374 14. Zhao L, Yu J, Yu H, Lin C (2019) Neuroadaptive containment control of nonlinear multi-agent systems with input saturations. Int J Robust Nonlinear Control 29(9):2742–2756

Adaptive Finite-Time Bipartite Output Consensus Tracking of Second-Order Nonlinear Multi-agent Systems with Input Saturation Xiao Chen, Lin Zhao(B) , and Ruiping Xu Qingdao University, Qingdao 266071, China [email protected]

Abstract. This paper considers the finite-time adaptive consensus tracking of second-order nonlinear multi-agent systems with input saturation. The uncertain nonlinear multi-agent systems can achieve adaptive neural consensus tracking by using the fuzzy logic system to approximate the unknown nonlinear dynamics. In the control scheme, we design a differentiator to obtain both the intermediate signal and its differentiator in each step of the distributed backstepping for each agent. Construct errors compensation to eliminate filtering errors. The result and simulation demonstrate that this approach is effective. Keywords: Nonlinear multi-agent systems · Bipartite output consensus tracking · Input saturation Finite-time convergence

1

·

Introduction

Recently the study of multi-agent systems has gained more and more attention because of the development and analysis of artificial intelligence problem solving and control architectures [1–5]. Multi-agent systems are composed of agents and their environment and divided into two main categories: leaderless consensus and leader-follower consensus. Compared with leaderless consensus, leader-follower consensus has a given desired common value. Due to the existence of the leader, leader-follower consensus’s tracking process can save the energy and have better communication and orientation among flock. Based on previous research, adaptive tracking of nonlinear multi-agents has multi-dimensional research value [3,4], as [5] considered the bipartite output tracking of nonlinear multi-agent systems. But the systems studied above are asymptotically stable and do not consider finite-time convergence. Compared with the asymptotic control method, the finite-time control technology has the advantages of fast response speed, high tracking accuracy and c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 375–383, 2020. https://doi.org/10.1007/978-981-32-9682-4_39

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strong anti-interference ability [6–8]. [6,7] used command filter based backstepping and [8] considered the adaptive finite-time attitude tracking of the spacecraft. [9–11] studied some issues of consensus tracking. But none of them use the command filter based backstepping or the adaptive finite-time attitude tracking method to apply to nonlinear coopetition multi-agent systems. Moreover, in practical applications, the input saturation unconsidered not only affects system performance, but also destroys system stability. Many application systems and control methods have begun to consider the impact of input saturation on the system, such as bilateral teleoperation, robust spacecraft, and uncertain mechanical systems [12]. This paper considers the adaptive fuzzy tracking of second-order nonlinear coopetition multi-agent system (CMAS). The backstepping method based on finite-time command filtered can guarantee the finite time convergence of closedloop system. Consider input saturation when constructing the controller, making it more versatile.

2

System Description

The CMAS can be described as G = (ν, ε, A) under the knowledge of graph theory. ν= {1, · · · , M}, ε and A indicate nodes, edges and the matrix of G. The cooperative network studied in this paper consists of a leader agent 0 and M ¯ = (¯ followers, and is represented as G ν , ε¯) with graph, where ν¯ = ν ∪{0} and ε¯ = ¯ contains a spanning ν¯ × ν¯. More detail describes can refer to [5], like the graph G tree with the root node being the leader node. We assume that ν = {ν1 , ν2 } have the bipartition ν1 ∪ ν2 = ν and ν1 ∩ ν2 = ∅, and as,j ≥ 0 for ∀s, j ∈ νl (l = 1, 2), as,j ≤ 0 for ∀s ∈ νl , ∀j ∈ νk (l, k ∈ {1, 2}). s ∈ ν1 , σs = 1, s ∈ ν2 , σs = −1. The sth follower has the following dynamics: x˙ s,1 =fs,1 (xs,1 ) + gs,1 (xs,1 )xs,2 x˙ s,2 =fs,2 (xs ) + gs,2 (xs )us

(1)

ys =xs,1 , s ∈ ν T

where xs = [xs,1 , xs,2 ] is the state vector of the sth follower with xs(0) = xs0 ,and ys is the output. fs,l (·) (l = 1, 2) is smooth and unknown. gs,l (·) (l = 1, 2) is a know and smooth function with ρs,1 < |gs,l | < ρs,2 , where ρs,1 , ρs,2 are known scalars greater than zero. In the system, we consider the input us ∈ R to be saturated nonlinear as (2) (us max > 0, us min < 0) are unknown constants) according to [5]:  s us max ∗ tanh( usωmax ), ωs ≥ 0 (2) h(ωs ) = ωs us min ∗ tanh( us min ), ωs < 0 According to mean-value theorem, there is a constant χs (0 < χs < 1) such that h(ωs ) = h(ωs0 ) + hωχs (ωs − ωs0 ). If ωs0 = 0, we get that h(ωs ) = hω χs ωs , then ¯ s ). Further we have |us − hs | ≤ Ds (Ds is the same we get us = hω χs ωs + h(ω

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as its definition in [5]). Considering that the control input ωs is not infinite, so there is a constant to make 0 < ζs ≤ hω χs ≤ 1. Represents yd ∈ R as the leader’s output, which with its derivative y˙d are assumed to be smooth continuous and bounded. We design ωs in order to achieve the bipartite output consensus tracking:  lim ys = yd , s ∈ ν1 t→∞ (3) lim ys = −yd , s ∈ ν2 t→∞

We use the radial basis function in the fuzzy logic system (FLS) to approxiT mate the unknown fs,l , that is, there is a FLS Ws,l Ss,l (x) to make:   T sup fs,l (x) − Ws,l Ss,l (x) ≤ εs,l

(4)

x∈Ω

Ws,l is the ideal constant weight vector. The ps,l (x) in the basic function vector Ss,l (x) are chosen as Gaussian functions.

3

Main Results

In this paper, we achieve finite time tracking by using both the tracking signal and the tracking error signal to be implemented in a limited time. The tracking errors are constructed as follows: zs,1 = bs (ys,1 − σs yd ) +

M 

|as,j |(ys,1 − sign(as,j )yj,1 )

j=1

zs,2 = xs,2 − Πs,2 s ∈ ν

(5)

The control functions χs,l are constructed as followers: χs,1 =

χs,2

N T  Ss,1 vs,1 θˆs Ss,1 1 ( as,j gj,1 xj,2 + bs σs y˙ d − ks,1 zs,1 − 2 (bs + ds )gs,1 j=1 2hs,1

1 γ − vs,1 − ss,1 vs,1 ) 2 T vs,2 θˆs Ss,2 Ss,2 1 γ = (−ks,2 zs,2 − − ss,2 vs,2 ) 2 gs,2 2hs,2

(6)

where ks,m and ss,m are positive constants, γ is assumed to be 0 < γ < 1. We designed θˆs to estimate the unknown adaptive parameter θs : 2  ˙ θˆs = −2κs ρs θˆs +

κs 2 vs,m 2 ST S 2h s,m s,m s,m m=1

(7)

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    2 2 where θs = max Ws,1 ζs , Ws,2 ζs , κs > 0, ρs > 0 are constants, and θˆs (0) > 0. In order to avoid calculating the explosion, we use the following differentiator to get the virtual signal: 1/2

ϕ˙ s,1,1 = νs,1,1 = −γs,1,1 |ϕs,1,1 − χs,1 | ϕ˙ s,1,2 = −γs,1,2 sign(ϕs,1,2 − νs,1,1 )

sign(ϕs,1,1 − χs,1 ) + ϕs,1,2 (8)

with Πs,2 (t) = ϕs,1,1 (t) , and Π˙ s,2 (t) = νs,1,1 (t). Regardless of the effects of noise, the solutions of (8) are finite-time stable. By choosing the γs,1,1 and γs,1,2 , the following values can also be achieved in finite-time: ϕs,1,1 = χs,1,0 , νs,1,1 = χ˙ s,1,0

(9)

When we take the noise into account, we assume the input meets |χs,1 − χs,1,0 | ≤ κs,1 , then (10) can be realized in finite-time: |ϕs,1,1 − χs,1,0 | ≤ μs,1,1 κs,1 = s,1,1 1

|νs,l,1 − χ˙ s,1,0 | ≤ λs,1,1 κ 2 s,1 = s,1,2

(10)

where μs,1,1 > 0, λs,1,1 > 0. Construct the error compensating signal ξs,l : ξ˙s,1 = − ks,1 ξs,1 +(bs + ds )gs,1 (Πs,2 − χs,1 ) + (bs + ds )gs,1 ξs,2 − ls,1 sign(ξs,1 ) ξ˙s,2 = − ls,2 sign(ξs,2 ) (11) with ξs,l (0) = 0 (l = 1, 2), ls,m is a designed constant. Then we can give νs,l by vs,l = zs,l − ξs,l , s ∈ ν

(12)

We compensate for the filtering error at each step of the finite- time command filtering. If we can prove that the vs,l and error compensation signals in (12) can all achieve finite time convergence, then the bipartite output consensus tracking can be achieved in finite-time. The proof are as follows: step 1: Define: Vs,1 =

1 2 v 2 s,1

(13)

We get: V˙ s,1 =vs,1 v˙ s,1 =vs,1 (f¯s,1 −

M 

as,j gj,1 (xj,1 )xj,2 − bs σs y˙ d − ξ˙s,1 + (bs + bs )gs,1 zs,2

j=1

+ (bs + ds )gs,1 χs,1 + (bs + ds )gs,1 (Πs,2 − χs,1 ))

(14)

M where f¯s,1 = (bs + ds )fs,1 (xs,1 ) − as,j fj,1 (xj,1 ). According to the FLS, we j=1

have T f¯s,1 = Ws,1 Ss,1 (Ys,1 ) + δs,1

(15)

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where Ys,1 = [xs,1 , xj,1 ]T , |δs,1 | ≤ εs,1 . According to the basic inequality, we have: 2

vs,1 f¯s,1 ≤

2 T

Ws,1 Ss,1 Ss,1 vs,1

2h2s,1

1 1 2 1 + h2s,1 + vs,1 + ε2s,1 2 2 2

(16)

where hs,1 is a scalar greater than zero. Bring ξs,1 , χs,1 and (16) into (14), we have: 2

1

Ws,1

2 T 2 V˙ s,1 ≤ 2 ζs vs,1 ( − θˆs )Ss,1 Ss,1 − ks,1 vs,1 + ls,1 vs,1 sign(ξs,1 ) 2hs,1 ζs γ+1 − ζs ss,1 vs,1 + (bs + ds )gs,1 vs,1 vs,2 +

h2s,1 2

+

ε2s,1

(17)

2

step 2: Define: 1 2 Vs,2 = Vs,1 + vs,1 2

(18)

V˙ s,2 = V˙ s,1 + vs,2 v˙ s,2 = V˙ s,1 + vs,2 (z˙s,2 − ξ˙s,1 ) = V˙ s,1 + vs,2 (fs,2 (xs ) + gs,2 us − Π˙ s,2 + ls,2 sign(ξs,2 ))

(19)

Define f¯s,2 = (bs + ds )gs,1 vs,1 + fs,2 (xs ) − Π˙ s,2 , and it can be approximated as: T f¯s,2 = Ws,2 Ss,2 (Ys,2 ) + δs,2

(20)

where |δs,2 | ≤ εs,2 , Ys,2 = [xTs , νs,1 , Πs,2 ]T . Further, we can get 2 − ζs vs,2 gs,2 hχωs ωs ≤ ks,2 ζs vs,2

2 ˆ T vs,2 θs Ss,2 Ss,2

2h2s,2

γ − ζs ss,2 vs,2

¯ s ) ≤ 1 v 2 + 1 ρ2 D2 vs,2 gs,2 h(ω 2 s,2 2 s,2 s

(21)

(22)

Then (19) can be rewritten as: 2  ε2 h2 ls,l ls,1 2 ls,2 2 V˙ s,2 ≤ − [(ks,1 − )vs,1 + ζs ks,2 − − 1)vs,2 ) + ( s,l + s,l + 2 2 2 2 2 l=1

2

+

ρs,2 Ds2 2

+

2  l=1

 1

Ws,l

γ+1 ˆs )S T Ss,l − ζ v ( − θ ζs ss,l vs,l (23) s s,l s,l 2h2s,l ζs 2

2

l=1

Construct the Lyapunov function to approve the vs,l , ξs,l in (12) is finite time convergence. Construct V as: V =

2  l=1

Vs,l +

2  ζs ˜2 θ 2κs s l=1

(24)

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where θ˜s = θs − θˆs . And it can be approximated to be: V˙ ≤ −

M  2 

2 avs,l +

s=1 l=1

−

2 M  

M  2  ε2 h2 ρ2 Ds2 ls,l + s,2 ) ( s,l + s,l + 2 2 2 2 s=1 l=1

γ+1 ζs ss,l vs,l +

s=1 l=1

2 

2ζs ρs θ˜s θˆs

(25)

l=1

  l ls,2 where a = min ks,1 − s,1 ζ k − − 1 , and ρs θ˜s θˆs ≤ (ρs os /2)θs2 − s s,2 2 2 [(ρs (2os − 1))/2os ] θ˜2 , where os > (1/2). Then we get: s

V˙ ≤ − +

2 M   s=1 l=1

2 M  2 M    ε2 h2 ρ2 Ds2 ls,l γ+1 ) + s,2 )− ( ( s,l + s,l + ζs ss,l vs,l 2 2 2 2 s=1 s=1

2 

M 

2 avs,l +

l=1

ζs ρs os θs2 +

ζs (

s=1

l=1

l=1

M 

M 

γ+1 ςs ςs ˜2 γ+1 ςs ζs ( θ˜s2 ) 2 − 2 ζs θ˜s2 (26) θ ) 2 − κs s κ κ s s s=1 s=1

where ςs = κs [(ρs (2os − 1))/2os ]. According to the conclusion in [13], we get: V˙ ≤ −aV − bV

γ+1 2

+η

(27) γ+1

γ+1

where a = min {2ks,1 − ls,1 , 2ζs ks,2 − ls,1 − 2, 2ςs }, b = min{ζs ss,l 2 2 , (2ςs ) 2 }, M 2 h2 2 ε2 ρ2s,2 Ds2 ls,l s,l η = ( ( s,l + + ) + ) + 1 + ζs ρs os θs2 . According to the 2 2 2 2 s=1 l=1

l=1

derivation of [13], the vs,l , θ˜s will reach the region in finite-time. Now we are going to prove the error compensating signal ξs,l can be abounded in finite-time. Construct the Lyapunov function as: 1 2 2 V¯ = (ξ + ξs,2 ) 2 s=1 s,1 M

(28)

Then we get: V¯˙ =

M  s=1

+

M 

2 (−ks,1 ξs,1 )−

M  2 

(ls,l ξs,l sign(ξs,l ))+

s=1 l=1

M 

(bs + ds )gs,1 ξs,1 ξs,2

s=1

(bs + ds )gs,1 ξs,1 (Πs,2 − χs,1 )

(29)

s=1

According to (10), we have:

 M   1 ˙ ¯ V¯ ≤ −(k0 − k0 )V¯ − (l0 − 2 2 ¯ 1 ρ2 )V¯ 2 s=1

(30)

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√ where k0 = 2 min{ks,l }, k¯0 = 2 max{ (bs +d2s )gs,1 +ks,2 , (bs +d2s )gs,1 }, l0 = 2 min{ls,l },  ¯ 1 = max{s,2 , (bs + ds )s,1 }. Choose the right ls,l that meets l0 −

M 2 s=1 2 ¯ 1 ρ2 > 0, k0 − k¯0 > 0 , then ξi,l can reach 0 in finite-time. According to the analysis and calculation in [13], the system can convergence in (31) in finite-time.  

 2/(γ+1) 2η/(1 − ϑ)a, 2(η/(1 − ϑ)b) (31) |zs,1 | ≤ min From Θ1 = (H ⊗ IN ×N )−1 Z1 , (32) can be achieved in finite-time.  

 √ 2/(γ+1) M min 2η/(1 − ϑ)a, 2(η/(1 − ϑ)b) |ys − σs yd | ≤ σmin (H)

4

(32)

Simulations

The bipartite system of Fig. 1 was constructed according to the system description. The initial value of the system is selected as x1,1 (0) = 1.5, x1,2 (0) = −0.5, x1,2 (0) = −0.5, x2,1 (0) = −0.7, x2,2 (0) = 0.3, x3,1 (0) = 1.4, x3,2 (0) = −0.4. The system dynamics are chosen as: f1,1 = cos(x1,1 ), f1,2 = x1,1 x1,2 , g1,1 = g1,2 = 1, f2,1 = sin(0.5x2,1 ), f2,2 = x2,1 e−0.3x2,2 , g2,1 = g2,2 = 1, f3,1 = cos(−0.5x3,1 ), f2,2 = x3,1 x3,2 , g2,1 = g2,2 = 1. And ks,l = 10, ss,l = 20, ls,l = 8, γ = 3/5, hs,l = κs = ρs = 1, rs,l,1 = 450, rs,l,2 = 2000. For the FLS, η = 4, μs,l are distributed evenly in the range [−2, 2] × · · · × [−2, 2], M = 10. The output is yd = sin(0.5t).

Fig. 1. Interactions among agents

Figure 2 shows the output response for each agent in the system.

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1 y1 y2 y3 yd

yd y1 y2 y3

0.5

0

−0.5

−1

−1.5

0

1000

2000

3000

4000 5000 6000 Time£¨ms£©

7000

8000

9000

10000

Fig. 2. The output response for each agent in the system.

5

Conclusions

This paper focus on the bipartite output tracking of second-order multi-agent coopetition systems with input saturation. By using the finite-time commend filtered backstepping, the virtual control signals and the error compensation signals are proofed to be finite-time convergence, which means the tracking errors converge into the needed neighborhood in finite-time. Based on the system above, we take input saturation into consider, and then verified the feasibility of the system. Acknowledgments. The work was supported by the National Natural Science Foundation of China (61603204, 61573204), the Shandong Province Outstanding Youth Fund (ZR2018JL020), and the Shandong Province Natural Science Foundation (ZR2017MF055).

References 1. Olfati-Saber R et al (2004) Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Autom Control 49(9):1520–1533 2. Taheri M et al (2017) Adaptive fuzzy wavelet network control of second order multi-agent systems with unknown nonlinear dynamics. ISA Trans 69:89–101 3. Zhao L, Yu J, Yu H, Lin C (2019) Neuroadaptive containment control of nonlinear multi-agent systems with input saturations. Int J Robust Nonlin 29(9):2742–2756 4. Zhao L, Yu J, Lin C (2019) Distributed adaptive output consensus tracking of nonlinear multi-agent systems via state observer and command filtered backstepping. Inform Sci 478:355–374 5. Zhao L, Yu J, Lin C (2018) Command filter based adaptive fuzzy bipartite output consensus tracking of nonlinear coopetition multi-agent systems with input saturation. ISA Trans 80:187–194

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6. Zhao L, Yu J, Shi P (2019) Command filtered backstepping based attitude containment control for spacecraft formation. IEEE Trans Syst Man Cybern Syst. https://doi.org/10.1109/TSMC.2019.2896614 7. Zhang Y, Cui G et al (2017) Command filtered backstepping tracking control of uncertain nonlinear strict-feedback systems under a directed graph. Trans Inst Meas Control 39(7):1027–1036 8. Zhao L, Yu J, Yu H (2018) Adaptive finite-time attitude tracking control for spacecraft with disturbances. IEEE Trans Aerosp Electron Syst 54(3):1297–1305 9. Polycarpou MM (1996) Stable adaptive neural control scheme for nonlinear systems. IEEE Trans Autom Control 41(3):447–451 10. Altafini C (2013) Consensus problems on networks with antagonistic interactions. IEEE Trans Autom Control 58(4):935–946 11. Yoo S (2013) Distributed consensus tracking for multiple uncertain nonlinear strict feedback systems under a directed graph. IEEE Trans Neural Netw Learn 24(4):666–672 12. Cui G, Xu S et al (2016) Distributed consensus tracking for non-linear multi-agent systems with input saturation: a command filtered backstepping approach. IET Control Theory A 10(5):509–516 13. Zhao L, Yu J et al (2018) Adaptive neural consensus tracking for nonlinear multiagent systems using finite-time command filtered backstepping. IEEE Trans Syst Man Cybern Syst 48(11):2003–2012

Neural Network Based Adaptive Backstepping Control of Uncertain Flexible Joint Robot Systems Dongdong Wang, Lin Zhao(B) , and Jinpeng Yu Qingdao University, Qingdao 266000, China [email protected]

Abstract. For flexible joint (FJ) robotic systems with uncertainties, a command filter based backstepping control is proposed in this paper. Through the control scheme, an adaptive controller is constructed to track desired position. In order to overcome complex computation problem in backstepping technology, a command filter is used, and the filtering error compensation is further defined. To deal with the uncertain dynamics of flexible joint robot system, the neural network approximation technology is adopted. The simulation results of FJ robot are given to show the effectiveness. Keywords: Flexible joint robot · Command filtering backstepping Error compensation · Neural network

1

·

Introduction

With the development of science and technology, robots have been widely used in various fields of human society [1–3]. However, traditional rigid robots often fail to accomplish tasks under some special conditions due to their own materials and structures. Therefore, more and more attention has been paid to the flexible joint robot system, and corresponding control methods also need higher requirements [4–6]. The backstepping as a famous tool for solving tracking control of nonlinear systems, however, there is a “computational explosion” problem [7]. The reason is that the derivative of virtual control signal is required in every step of the computational process. Dynamic surface control (DSC) is proposed to solve this problem [5,6], but the filtering errors are not compensated. The command filter based backstepping method brings new idea to solve the computation and errors compensation errors [8–10]. Moreover, there are usually uncertain parameters in the actual dynamic equations, especially for flexible joint robot system [11,12]. In [13], the adaptive command filter based backstepping method for uncertain SISO nonlinear systems are studied. But as far as we know, for flexible joint robot systems with uncertainties, the adaptive backstepping control based on command filtering has not been studied. c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 384–392, 2020. https://doi.org/10.1007/978-981-32-9682-4_40

Adaptive

385

Based on these analyses, the output tracking problem of flexible joint robot system with uncertain parameters under adaptive command filtering backstepping control method is proposed. The command filter is used to solve the computational complexity of backstepping method, and the error compensation signal is proposed to solve the error caused by filtering. The uncertain dynamics in the system are approximated by the neural network function approximation technology.

2

System Description

The model for flexible joint robot is given as follows: M (q) q¨ + C (q, q) ˙ q˙ + G (q) + F q˙ + K (q − qm ) = 0 J q¨m + B q˙m + K (qm − q) = u y=q

(1)

where q ∈ Rn , q˙ ∈ Rn and q¨ ∈ Rn represent joint position, velocity and acceleration vectors respectively. qm ∈ Rn , q˙m ∈ Rn and q¨m ∈ Rn represent the brake position, velocity and acceleration vectors respectively. M (q) ∈ Rn×n is the symmetric inertia matrix, C (q, q) ˙ ∈ Rn×n is the centripetal and Coriolis n torques matrix, G (q) ∈ R is the gravity term, F ∈ Rn×n is a diagonally positive definite matrix of friction coefficient at joints. Positive definite constants, diagonal matrices K ∈ Rn×n , J ∈ Rn×n and B ∈ Rn×n represent the joint flexibility, inertia and inherent damping terms, respectively. The control vector u ∈ Rn is the torque input of the actuator and y ∈ Rn is the system output. C (q, q) ˙ and G (q) contain uncertainties. Let q = x1 , q˙ = x2 , qm = x3 , q˙m = x4 , we get the following equation: x˙ 1 = x2 x˙ 1 = M −1 (x1 ) [−C (x1 , x2 ) x2 − G (x1 ) − F x2 − K (x1 − x3 )] x˙ 3 = x4

(2)

x˙ 4 = J −1 [−Bx4 − K (x3 − x1 ) + u] y = x1 And then, the Eq. (2) can be rewritten as follows: x˙ 1 = x2 x˙ 2 = f2 + g2 x3 x˙ 3 = x4

(3)

x˙ 4 = f4 + g4 u y = x1 where f2 = −M −1 (x1 ) [C (x1 , x2 ) x2 + G (x1 ) + F x2 + Kx1 ] ; g2 = M −1 (x1 ) K; f4 = −J −1 [Bx4 + K (x3 − x1 )] ; g4 = J −1 .   We assume that the term g2 is bounded, that is M −1 (x1 ) K  ≤ λ1 .

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Main Results

Choose the following filter: φ˙ i,s,1 = li,s,1 1

li,s,1 = −ri,s,1 |φi,s,1 − αi,s | 2 sign (φi,s,1 − αi,s ) + φi,s,2 φ˙ i,s,2 = −ri,s,2 sign (φi,s,2 − li,s,1 )

(4)

For the above filters, according to αi,s (i = 1, 2, 3 s = 1, 2, . . . , n) defined in (7), if the input signal αi,s (i = 1, 2, 3 s = 1, 2, . . . , n) is not affected by noise, the command filter is finite time stable; if αi,s (i = 1, 2, 3 s = 1, 2, . . . , n) is affected by noise, and |αi,s − αi0,s | ≤ ρi,s , we have |ϕi,s,1 − αi0,s | ≤ i,s ; |li,s,1 − α˙ i0,s | ≤ i+1,s . Denote z1 z2 z3 z4

= x1 − xd = x2 − π2 = x3 − π3 = x4 − π4

(5)

where xd is a known reference signal, xd and x˙ d are smooth, known and bounded signals. The output of the command filter is πi+1 (i = 1, 2, 3) and the input is αi (i = 1, 2, 3). Aiming at the tracking error (πi+1 − αi ) produced by the command filter, we define the error compensation mechanism: ζ˙1 ζ˙2 ζ˙3 ζ˙4

= −k1 ζ1 + ζ2 + π2 − α1 = −k2 ζ2 + g2 (π3 − α2 ) − ζ1 + g2 ζ3 = −k3 ζ3 − g2 ζ2 + ζ4 + π4 − α3 = −k4 ζ4 − ζ3

(6)

Among them, proportional gain ki (i = 1, 2, 3, 4) > 0, ζi (0) = 0. The virtual control function is defined as: α1 = −k1 z1 + x˙ d ⎡

⎛

T v2,1 θˆ2 S2,1 S2,1 T ˆ ⎢ ⎜ v S θ 1 1 ⎜ 2,2 2 2,2 S2,2 ⎢ α2 = g2−1 ⎢−k2 z2 − z1 + π˙ 2 − v2 − 2 ⎜ .. 2 2h ⎝ ⎣ . v2,n θˆ2 S T S

⎞⎤ ⎟⎥ ⎟⎥ ⎟⎥ ⎠⎦

2,n 2,n

α3 = −k3 z3 − g2 z2 + π˙ 3 ⎡

⎛

⎢ 1 1 ⎜ ⎢ ⎜ u = g4−1 ⎢−k4 z4 − z3 + π˙ 4 − v4 − 2 ⎜ 2 2h ⎝ ⎣

T v4,1 θˆ4 S4,1 S4,1 T ˆ v4,2 θ4 S4,2 S4,2

⎞⎤

⎟⎥ ⎟⎥ ⎟⎥ .. ⎠⎦ . T v4,n θˆ4 S4,n S4,n

(7)

Adaptive

where vi is given by: ν1 ν2 ν3 ν4

= z1 − ζ1 = z2 − ζ2 = z3 − ζ3 = z4 − ζ4

387

(8)

For the error compensation mechanism, we establish Lyapunov equation:   V¯ = 12 ζ1T ζ1 + ζ2T ζ2 + ζ3T ζ3 + ζ4T ζ4

(9)

And then, we have: V¯˙ = ζ1T ζ˙1 + ζ2T ζ˙2 + ζ3T ζ˙3 + ζ4T ζ˙4 = − k1 ζ1T ζ1 − k2 ζ2T ζ2 − k3 ζ3T ζ3 − k4 ζ4T ζ4 + ζ1T (π2 − α1 ) + ζ2T g2 (π3 − α2 ) + ζ3T (π4 − α3 )

ζ1T ζ1 2 (π3 −α2 )T g2T g2 (π3 −α2 ) 2

≤ − k1 ζ1T ζ1 − k2 ζ2T ζ2 − k3 ζ3T ζ3 − k4 ζ4T ζ4 + + + ≤−

ζT ζ (π2 −α1 )T (π2 −α1 ) + 22 2 2 ζ3T ζ3 (π4 −α3 )T (π4 −α3 ) 2 + 2 λ2  k0 V¯ + 22 + 12 3 + 24

+

(10)

 2 +λ21 3 +4 where k0 = min (2k1 − 1, 2k2 − 1, 2k3 − 1, k4 ), lim ζi  ≤ . k0 t→∞ The following four steps show the stability of the compensation error tracking signal. Step 1: Set the Lyapunov equation as: V1 = 12 v1T v1

(11)

And then, we have:     V˙ 1 = v1T v˙ 1 = v1T z˙1 − ζ˙1 = v1T x˙ 1 − x˙ d − ζ˙1   = v1T f1 + g1 α1 + g1 (x2 − α1 ) − x˙ d − ζ˙1   = v1T f1 + g1 α1 + g1 (x2 − π2 ) + g1 (π2 − α1 ) − x˙ d − ζ˙1   = v1T α1 + z2 + (π2 − α1 ) − x˙ d − ζ˙1

(12)

Substituting α1 , ζ˙1 into (12) yields: V˙ 1 = v1T [−k1 z1 + x˙ d + z2 + (π2 − α1 ) − x˙ d + k1 ζ1 − ζ2 − (π2 − α1 )] = v1T [−k1 (z1 − ζ1 ) + (z2 − ζ2 )] = −k1 v1T v1 + v1T v2

(13)

Step 2: Set the Lyapunov equation as: V2 = V1 + 12 v2T v2

(14)

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And then, we have:     V˙ 2 = V˙ 1 + v2T v˙ 2 = V˙ 1 + v2T z˙2 − ζ˙2 = V˙ 1 + v2T x˙ 2 − π˙ 2 − ζ˙2     f2 + g2 α2 + g2 (x3 − π3 ) = V˙ 1 + v2T f2 + g2 x2 − π˙ 2 − ζ˙2 = V˙ 1 + v2T +g2 (π3 − α2 ) − π˙ 2 − ζ˙2

(15)

Since the function f2 = [f2,1 , · · · , f2,n ]T , the neural network function approximation technology is used to approximate it. f2,s (s = 1, · · · , n) can be approximated by: T S2,s + δ2,s f2,s = W2,s (16) where S2,s is vector of basis function, W2,s is ideal weight matrix, and δ2,s  ≤ ε2 , ε2 > 0 is approximation error. We got from Young’s inequality:   n 2 T 2  v2,s W2,s 2 S2,s S2,s v2,s ε22 h2 + + + v2T f2 ≤ (17) 2 2h 2 2 2 s=1

where h > 0 is a constant. Substituting α2 , ζ˙2 and (17) into (15) yields:  2 2  n T    2 v2,s (W2,s 2 −θˆ2 )S2,s S2,s V˙ 2 ≤ − ki viT vi + g2 v2T v3 + + h2 + 2h2 s=1

i=1

ε22 2

 (18)

Step 3: Set the Lyapunov equation as: V3 = V2 + 12 v3T v3

(19)

And then, we have:     V˙ 3 = V˙ 2 + v3T v˙ 3 = V˙ 2 + v3T z˙3 − ζ˙3 = V˙ 2 + v3T x˙ 3 − π˙ 3 − ζ˙3   (20) = V˙ 2 + v3T f3 + g3 α3 + g3 (x4 − α3 ) − π˙ 3 − ζ˙3   = V˙ 2 + v3T f3 + g3 α3 + g3 (x4 − π4 ) + g3 (π4 − α3 ) − π˙ 3 − ζ˙3   = V˙ 2 + v3T α3 + z4 + (π4 − α3 ) − π˙ 3 − ζ˙3 Substituting α3 , ζ˙3 into (20) yields: V˙ 3 = V˙ 2 − k3 v3T v3 − g2 v3T v2 + v3T v4

(21)

Step 4: Set the Lyapunov equation as: V4 = V3 + 12 v4T v4

(22)

    V˙ 4 = V˙ 3 + v4T v˙ 4 = V˙ 3 + v4T z˙4 − ζ˙4 = V˙ 3 + v4T x˙ 4 − π˙ 4 − ζ˙4   = V˙ 3 + v4T f4 + g4 u − π˙ 4 − ζ˙4

(23)

And then, we have:

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f4,s (s = 1, · · · , n) can be approximated by: T S4,s + δ4,s f4,s = W4,s

By Young’s inequality, we have:  n 2 T  v4,s W4,s 2 S4,s S4,s + v4T f4 ≤ 2h2 s=1

h2 2

(24)

+

2 v4,s 2

+

ε24 2

 (25)

where h > 0 is a constant. Substituting α4 , ζ˙4 and (25) into (23), we can see that:   ⎞ ⎛ 2 2 T 4 n W2,s  − θˆ2 S2,s v2,s S2,s 2 2   T   ε h ⎝ + 2⎠ V˙ 4 ≤ − ki vi vi + + 2 2h 2 2 s=1 i=1   ⎞ ⎛ 2 2 T n W4,s  − θˆ4 S4,s v4,s S4,s  ε2 h2 ⎝ + 4⎠ (26) + + 2 2h 2 2 s=1 2

Define θi,s = Wi,s  , (i = 2, 4 s = 1, 2, · · · , n) ; θi = max {θi,s }. The updating rules of the estimation are given as follows: n  1 ˙ 2 T v2,s S2,s S2,s θˆ2 = −r2 ρ2 θˆ2 + 2 r2 2h s=1 n  1 ˙ 2 T v4,s S4,s S4,s θˆ4 = −r4 ρ4 θˆ4 + 2 r4 2h s=1

(27)

where r2 > 0 ; r4 > 0 and ρ2 > 0 ; ρ4 > 0 are constants. Definitions θ˜i = θi − θˆi we have: (28) θ˜2 = θ2 − θˆ2 , θ˜4 = θ4 − θˆ4 Set the Lyapunov equation as: V˜4 = V4 +

1 ˜2 2r2 θ2

+

1 ˜2 2r4 θ4

(29)

And then, we have: 4    ˙ ˙ ki viT vi V˙ 4 + 2r12 θ˜2 θ˜2 + 2r14 θ˜4 θ˜4 ≤ − i=1   n 2 ˜ T 2 ˜ T  v2,s S2,s v4,s S4,s θ2 S2,s θ4 S4,s ε22 ε24 h2 h2 + + + + + + 2h2 2 2 2h2 2 2 s=1     ˙ ˙ + r12 θ˜2 θ˙2 − θˆ2 + r14 θ˜4 θ˙4 − θˆ4 4        ki viT vi + 12 n h2 + ε22 + ρ2 θ˜2 θˆ2 + ρ4 θ˜4 θˆ4 + 12 n h2 + ε24 ≤−

V˜˙ 4 =

≤−

i=1 4   i=1

   ki viT vi + 12 n ε22 + ε24 + nh2 + ρ2 θ˜2 θˆ2 + ρ4 θ˜4 θˆ4 (30)

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By Young’s inequality, we have θ˜i θˆi ≤ − 12 θ˜i2 + 12 θi2 , and then: θ˜2 θˆ2 ≤ − 12 θ˜22 + 12 θ22 ; θ˜4 θˆ4 ≤ − 12 θ˜42 + 12 θ42

(31)

Substituting (31) into (30) yields: 4   1    ki viT vi + n ε22 + ε24 + nh2 + ρ2 θ˜2 θˆ2 + ρ4 θ˜4 θˆ4 V˜˙ 4 ≤ − 2 i=1

≤−

4   ρ θ˜2 + ρ θ˜2   ρ2 θ22 + ρ4 θ42 1  2 2 4 4 ki viT vi − + + n ε22 + ε24 + nh2 (32) 2 2 2 i=1

≤ −aV˜4 + b

  ρ θ 2 +ρ θ 2 where a = min (2ki , ρ2 r2 , ρ4 r4 ), b = 2 2 2 4 4 + 12 n ε22 + ε24 + nh2 . Further, we can see that:   1 T ˜4 (t) ≤ V˜4 (t0 ) − b e−a(t−t0 ) + b ≤ V˜4 (t0 ) + b , ∀t ≥ t0 v v ≤ V 4 2 4 a a a In the end, we can see that:  v1  ≤  lim z1 ≤

t→∞

(33)

 2b , a 2b + a

ζ1  ≤ 

12 + λ21 22 + 32 k0

2 + λ21 3 + 4 k0

(34)

And, we know that all control signals are bounded.

4

Simulation

We define M (q) = [Mmn ] ∈ R2×2 , C (q, q) ˙ = [Cmn ] ∈ R2×2 as: M11 = a1 + 2a2 cos (q2 ) , M12 = M21 = a3 + a2 cos (q2 ) , M22 = a3 , C11 = −a2 sin (q2 ) q˙2 , C12 = −a2 sin (q2 ) (q˙1 + q˙2 ) , C21 = a2 sin (q2 ) q˙1 , 2 2 2 C22 = 0, a1 = I1 + m1 lc1 + m2 l12 + I2 + m2 lc2 , a2 = m2 l1 lc2 , a3 = I2 + m2 lc2 (35)

with m1 and m2 are the masses of the connecting rod, lc1 and lc2 are the mass centers of the Connecting rod, I1 and I2 are the moments of inertia. We assume that G (q) and F are zero. J = diag[25, 25], B = diag[12, 12], K = diag[25, 25]

(36)

The parameters are chosen as L1 = 1.4, L2 = 1.4; Lc1 = 1, Lc2 = 1; I1 = 0.2, I2 = 0.2; m1 = 1, m2 = 1 (37) The desired trajectory xd = [2sin (t) , 2cos (t)]T . The control parameters are chosen ki = 25 (i = 1, 2, 3, 4), r1 = 50, r2 = 50. As shown in the Fig. 1, the trajectory under the action of the control method is given.

Adaptive

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q2 xd1

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x

d2

Position

1 0.5 0 -0.5 -1 -1.5 -2

0

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8

10

12

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Time(s) Fig. 1. Signal trajectories of q and xd

5

Conclusion

The flexible joint robot system is controlled by using the method of command filtered backstepping in this paper. Command filter is applied to avoid differentiate of virtual signal. To deal with the uncertain dynamics of flexible joint robot system, the neural network approximation technology is adopted. We proved that the position tracking error is convergent, and the convergence region is within a neighbourhood small enough at the origin. Acknowledgements. This work was supported by the National Natural Science Foundation of China (61603204, 61573204), and the Shandong Province Outstanding Youth Fund (ZR2018JL020).

References 1. Rodriguez-Angeles A (2004) Mutual synchronization of robots via estimated state feedback: a cooperative approach. IEEE Trans Control Syst Technol 12(4):542–554 2. Nuno G, Ortega R, Basanez L, Hill D (2011) Synchronization of networks of nonidentical EulerCLagrange systems with uncertain parameters and communication delays. IEEE Trans Autom Control 56(4):935–941 3. Hl W (2016) Consensus of networked mechanical systems with communication delays: a unified framework. IEEE Trans Autom Control 59(6):1571–1576

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4. Nanos K, Papadopoulos EG (2015) On the dynamics and control of flexible joint space manipulators. Control Eng Pract 45:230–243 5. Yoo SJ, Park JB, Choi YH (2008) Adaptive output feedback control of flexiblejoint robots using neural networks: dynamic surface design approach. IEEE Trans Neural Netw 19(10):1712–1726 6. Yoo SJ (2014) Distributed adaptive containment control of networked flexible-joint. Expert Syst Appl 41(2):470–477 7. Farrell JA, Polycarpou M, Sharma M et al (2009) Command filtered backstepping. IEEE Trans Autom Control 54(6):1391–1395 8. Dong WJ, Marios M et al (2012) Command filtered adaptive backstepping. IEEE Trans Control Syst Technol 20(3):566–580 9. Zhao L, Yu JP, Yu HS et al (2019) Neuroadaptive containment control of nonlinear multi-agent systems with input saturations. Int J Robust Nonlinear Control 29(9):2742–2756 10. Zhao L, Yu JP, Lin C (2019) Distributed adaptive output consensus tracking of nonlinear multi-agent systems via state observer and command filtered backstepping. Inf Sci 478:355–374 11. Jia YM (2003) Robust control with decoupling performance for steering and traction of 4WS vehicles under velocity-varying motion. IEEE Trans Control Syst Tech 8(3):554–569 12. Zhao L, Yu JP, Yu HS (2018) Adaptive finite-time attitude tracking control for spacecraft with disturbances. IEEE Trans Aerosp Electron Syst 54(3):1297–1305 13. Zhao L, Yu JP, Lin C (2018) Adaptive neural consensus tracking for nonlinear multi-agent systems using finite-time command filtered backstepping. IEEE Trans Sys Man Cybern: Syst 48(11):2003–2012

RSSI Localization Algorithm Based on Electromagnetic Spectrum Detection Mingming Ma1,2 , Yonghui Zhang1,2(B) , Zhenjia Chen1,2 , and Chao He1,2 1

Hainan University, Haikou 570228, Hainan, China School of Information and Communication Engineering, Hainan University, No. 58 Renmin Avenue, Meilan District, Haikou, Hainan, China [email protected] 2

Abstract. Wireless signal location technology is one of the key technologies of the electromagnetic spectrum detection system. This paper proposes an RSSI (Received signal strength indication) localization algorithm based on electromagnetic spectrum detection, which uses selforganizing network to realize data acquisition and data transmission of detection nodes; In order to reduce the error of the RSSI ranging technology on the positioning accuracy, the number of appropriate detection nodes is selected, the RSSI value is averaged, and a large amount of sample data is analyzed to establish an optimal signal attenuation model; The least squares method is used to transform the nonlinear function described by the RSSI localization problem into a linear estimation problem for precise positioning. The simulation results show that the verification results of simulation data are consistent with those of measured data in positioning accuracy. Keywords: RSSI · Electromagnetic Spectrum Detection Detection Node · The Least Squares Method

1

·

Introduction

With the rapid development of modern communication, global positioning system, ubiquitous computing, distributed information processing and other technologies, location-aware computing and location-based services are becoming more and more important in practical applications. GPS (Global Position System) is currently the most widely used and successful positioning technology [1]. However, GPS needs to add extra hardware in the positioning process. The ranging between nodes is easily interfered by various factors such as environment, node hardware, network attack, etc. As a result, nodes are large in size and have high energy consumption and cost. The wireless sensor networks are composed of a large number of inexpensive small or micro sensor nodes deployed in the monitoring area. Each node forms a multi-hop, self-organized communication network in a wireless manner. In the monitoring area, real-time monitoring of c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 393–400, 2020. https://doi.org/10.1007/978-981-32-9682-4_41

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various environments or monitoring objects, perception and collaborative processing of information are carried out, and the perceived information is sent to the user terminal. It has the characteristics of low power consumption and low cost [2]. According to the positioning algorithm, the current positioning algorithms are mainly divided into two categories: range-based algorithm and range-free algorithm. According to the positioning algorithm, the current positioning algorithm is mainly divided into two categories: range-based algorithm and rangefree algorithm [3]. The basic principle of range-based algorithm is to convert the received information into distance value, and then use triangulation, triangulation, maximum likelihood estimation method or least square method to calculate the node position according to the distance value [4–6]. Commonly used ranging technologies are RSSI, TOA (Time of Fight) and AOA (Angle of Arrival). Comparing various ranging methods, the RSSI-based positioning algorithm has simple detection equipment, and the detection node does not need to add additional hardware, low power consumption, low cost, and high precision. [7] proposed a localization algorithm based on least squares estimation, which is based on the RSSI signal attenuation model to accurately locate the nodes to be measured. The simulation results show that the algorithm has strong adaptive ability for complex and variable environments. When the positioning environment changes, the positioning accuracy of the positioning algorithm can still reach a higher level. This paper proposes an RSSI localization algorithm based on electromagnetic spectrum detection system. The detection system refreshes the position information, power, center frequency and other environmental parameters of the detection node in real time to establish the optimal signal attenuation model. In order to reduce the influence of multipath and masking effect, this paper selects the appropriate number of detection nodes and takes the average value of multiple measurements for each node to obtain more accurate signal strength value. Using the least squares method, the position coordinates of the nodes to be measured are expressed by algebraic calculation. This algorithm eliminates the influence of various interferences on the measurement data, has strong adaptive ability, high ranging accuracy, low energy consumption and high robustness.

2

Spectrum Detection System Architecture

The system is composed of detection node, backbone network and upper computer, realizing real-time signal recognition, signal source positioning, spectrum monitoring, data analysis and other functions. The user can send instructions to the main control module through the upper computer. When the spectrum data is detected by the detection node, the signal fluctuation method is used to identify the target frequency band. The data of the target frequency band is obtained by the form of frequency sweep, that is, the characteristic information such as the center frequency, bandwidth, latitude and longitude, and power of the signal source. The detection data is shared by the multicast to the detection

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network. The main control module analyzes the spectrum database according to the instructions of the upper computer, and feeds the result back to the upper computer in real time. 2.1

Backbone Network

The self-organizing network does not need preset network facilities, and can be quickly deployed and automatically networked at any time and anywhere. The distributed characteristics of the network make the system highly resistant to damage [8]. The 700 MHZ frequency band has good penetration performance, small penetration attenuation, wide coverage, high RF (Radio frequence) signal transmission quality, suitable for large-scale network coverage, and low networking cost. Therefore, the system selects the blank TV frequency band 700 MHz as the communication frequency band of the electromagnetic spectrum detection network backbone network, and builds the electromagnetic spectrum detection system with the self-organizing network as the backbone network. Each node in the system is composed of 700 MHz network card, master control module and RF module. The 700 MHz network segment provides IP services for the detection nodes. Any one of the detection nodes forms a wireless mesh network and can communicate with each other. Multiple detection nodes form the entire spectrum detection system. Schematic diagram of the backbone network model is shown in Fig. 1.

Fig. 1. Backbone network model.

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Detection Node

The system selects the embedded equipment as the main control module. The RF module measures the spectrum data. The main control module processes the spectrum data to realize spectrum detection, data storage, data analysis and other functions. The upper computer adopts Client/Server (C/S) architecture to realize the interaction between the main control module. In addition, the main control module is connected with the sensor through the I/O bus to obtain a large amount of sensor data in real time. When the electromagnetic spectrum detection system is initialized, the main control module acts as the server first to advertise its own IP address through multicast. At this time, the listening process transitions from the running state to the blocking state, and waits for the client to send the connection request in real time. As the client, the host computer sends a request to create a socket connection to the server. The server receives the request and the response state transitions from the blocked state to the running state. The server feeds back the socket description. After receiving the description, the client sends a confirmation message, and the server receives the confirmation message to mark the connection creation success [9]. After the connection is successful, the detecting node performs data detection on the spectrum data of the target frequency band according to the instruction of the upper computer. The system builds many detection nodes, each of which forms a Mesh network and can communicate with each other. A plurality of detection nodes form a distributed wireless mesh network. Each detection node has independent functions and can realize electromagnetic spectrum detection, data storage, spectrum occupancy calculation and other functions. 2.3

Multicast-Datenformat

Multicast technology implements one-to-multipoint data transmission between the sender and all receivers in the multicast group, which has the advantages of saving network spectrum resources, reducing network load, and alleviating the shortage of spectrum resources to improve network throughput. When the system obtains the spectrum data, the detecting node sends multicast to the network through the multicast technology. Other nodes in the network receive the data packet, parse out the IP address and multicast data of the sending end, and implement data interaction. Multicast data format shown in Fig. 2. When the detection node uses the method of signal fluctuation to identify the target frequency band, the longitude and latitude, center frequency, power, bandwidth and a large number of sensor data of the detection node are acquired and recorded in real time. Detection data is transmitted to the network in the form of multicast for data interaction. After obtaining the parameters of the center frequency, bandwidth, power and other parameters of multiple detection nodes, the RSSI positioning of the multi-band multi-band can be performed.

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Fig. 2. Multicast data format.

3 3.1

RSSI Localization Algorithm Using Least Square Method RSSI Ranging Model

RSSI localization algorithm is a positioning algorithm that measures the distance between the signal and the receiving point according to the strength of the received signal. The distance between the detection node and the node to be tested was calculated by the known attenuation model and the RSSI value of the transmitted signal, and then the distance equation is listed to solve the coordinates of the node to be tested. In the actual environment, electromagnetic wave transmission is often anisotropic due to multipath effect, weather change, obstacles and many other reasons. The size of RSSI decreases with the increase of the distance between transceiver and receiver, but it is non-linear. The traditional attenuation model of wireless signal propagation is RSSI (d) = RSSI (d0 ) − 10nlog (d/d0 ) + ξσ

(1)

where: RSSI (d) is the intensity value of RSSI received at distance d, in dBm. RSSI (d0 ) is the RSSI signal strength value received by the node to be tested at the corresponding d0 meters, and it is expressed in dBm. d0 is the reference distance in m, when d0 = 1 m; RSSI (d0 ) is A. d is the distance between the detection node and the node to be tested. n is the path attenuation index, which is closely related to the surrounding environment and obstacles. The normal random variable indicating the standard deviation is ξσ , in dBm and it depends on the multipath environment. When the RSSI value of the detection node is known, the distance d between the detection node and the node to be tested can be calculated according to the signal attenuation model, and then the coordinate of the node to be tested can be precisely positioned. However, since n and d0 are closely related to the environment in which the wireless signal is transmitted, the A and n parameters are different in different actual environments, and the ranging models are different. 3.2

RSSI Positioning Method

The specific positioning algorithm process is as follows: 1. Initialize to obtain the GPS and RSSI values of the detection node. 2. The node to be tested periodically

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transmits a signal. 3. The detecting node receives the information of the signal to be tested, records the RSSI measurement value of the same node to be tested, and shares the detected data into the network through multicast. 4. Sort the RSSI values by size, average the three measured values with the largest value, and then convert to the distance d by the ranging model. 5. Select the three detection nodes closest to the node to be tested for positioning. 6. List the distance equation and use the least squares method to obtain the coordinates of the signal to be tested.

4 4.1

Experimental Simulation The Test Environment

The test relies on the electromagnetic spectrum detection platform, and the portable embedded system is selected as the main controller and server. The test frequency band is 700 MHz and the sampling rate is 2 MHz. When the detecting node measures the spectrum data, the PC terminal can access the center frequency, bandwidth of the single node and power data for positioning. 4.2

Experimental Results and Analysis

In actual applications, wireless signals are affected by many uncertain factors. In the process of RSSI positioning, the number of detection nodes, the processing mode of RSSI value and other factors will cause problems of low accuracy and high cost. Therefore, we need to establish an optimal attenuation model and select appropriate parameters for signal source localization. In the experiment, when the number of detection nodes is increased successively, the positioning accuracy of the detection nodes is higher, but when the number of detection nodes reaches a certain level, the average positioning error of the nodes to be tested tends to be stable. Considering the system cost and the average error of specific positioning, we set the number of detection nodes to 5 during the simulation. The processing mode of RSSI value greatly influences the positioning accuracy. In the experiment, we respectively carried out three processing methods of single measurement, weighted measurement and average measurement for the RSSI value. Nodes with similar performance were selected to carry out ranging experiment with an interval of 5 m. The experimental results show that when ranging is within 15 m, the average ranging error is the smallest, with a range of 1.4 m. The single ranging error is the largest, ranging error is about 2 m. In the distance between 15 m and 30 m, the ranging error gradually increased, but is still the average ranging error is the smallest. Therefore, in the simulation process, we average the RSSI value. In this experiment, the Matlab simulation tool is used to simulate the RSSI positioning algorithm on the platform of electromagnetic spectrum detection to verify the performance of the algorithm. Assume that in the square region of

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100 m * 100 m, five detection nodes are selected, and the infinite signal carrier frequency is 700 MHz. Simultaneously perform multiple simulation positioning on different signal sources. The simulation results are shown in Fig. 3:

Fig. 3. The simulation results of the least squares method.

Fig. 4. The simulation results of trilateral positioning

In the Fig. 3, the triangle represents the position of the detection node. The circle represents the true position of the node to be tested, and the “+” repre-

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sents the specific location based on the RSSI positioning. Compared with the Fig. 4, it can be seen from the Fig. 3 that the verification result of the simulation data and the verification result of the measured data show consistency in the positioning accuracy. The normalized average positioning error is 0.0615 during the positioning process, which further demonstrates the effectiveness and reliability of the algorithm.

5

Conclusions

In this paper, an RSSI algorithm based on electromagnetic spectrum detection system is proposed. By detecting the mutual cooperation between nodes, the functions of environmental parameter acquisition, node coordinate query and update of each detection node are completed. Based on the actual environment, the optimal ranging model of the wireless sensor is established, and the least squares fitting algorithm is used for positioning. The simulation results show that the algorithm proposed in this paper reduces the positioning error caused by signal instability, and has high adaptability and good positioning accuracy. Since the spectrum data of all the detection nodes are transmitted to the network through multicast, in the future research, we can further improve the positioning accuracy by improving the environmental parameters in the multicast data structure. Acknowledgments. This work was supported by the National Natural Science Foundation of China (61561018).

References 1. Wang Y, Jia X, Lee HK (2003) An indoors wireless positioning system based on wireless local area network infrastructure, pp 22–25 2. Sun L (2005) Wireless sensor network, pp 131–140 3. Lessmann S (2008) Benchmarking classification models for software defect prediction: a proposed framework and novel findings. IEEE Trans Softw Eng 34(4):485–496 4. Harter A, Hopper A, Steggles P et al (2002) The anatomy of a context-aware application, pp 59–68 5. Girod L, Estrin D (2001) Robust range estimation using acoustic and multimodal sensing, June, pp 1312–1320 6. Niculescu D, Nath B (2003) Ad hoc positioning system (APS) using AoA, pp 1734– 1743 7. Shamantha RB, Shirshu V (2016) An algorithmic approach to wireless sensor networks localization using rigid graphs, pp 1–11 8. Luo T, Zhao M, Li J, Le G, Wang X (2013) Cognitive radio ad hoc network MAC protocol, pp 1337–1348 9. Sun M, Zhao C, Salous S (2017) A spectrum detection method for cognitive radio networks with dynamic noise properties, pp 1–3

Distributed H∞ Consensus Control with Nonconvex Input and Velocity Constraints Jiahui Shi(B) and Hongqiu Zhu School of Automation, Central South University, Changsha 410083, Hunan, People’s Republic of China [email protected]

Abstract. This paper investigates the distributed H∞ consensus problem for second-order multi-agent systems of directed networks. We propose a nonlinear protocol to ensure the consistency of all agents, while input and velocity states stay in certain nonconvex constraint sets. Using this protocol, the stability of the transformed system is demonstrated. Some sufficient conditions are obtained to guarantee the consistency of system under the effect of interference. Furthermore, we provided a simulation result to demonstrate the feasibility of the proposed theory. Keywords: Distributed H∞ control · Multi-agent systems · Nonconvex constraint consensus

1 Introduction In recent years, the distributed coordination problems in multi-agent systems (MAS) have attracted widespread attention [1–3]. As a significant issue in this field, the distributed H∞ consensus problem has been studied frequently by scholars [4–13]. Lin et al. first introduced the distributed H∞ consensus problem and presented sufficient conditions to let the linear multi-agent systems reach consensus under the desired disturbance and time-delay [6]. After that, the authors also developed a linear algorithm with nonuniform time-delays for second-order systems subjected to interference in [7]. Mo et al. introduced a neighbor-based protocol and take the factor of time-delay and external disturbances into account [9]. In addition, Li et al. presented two distributed algorithms for multi-agent systems with time-delay, and derived sufficient conditions such that the system reach asymptotically stable with the desired H∞ performance [10]. In practical situations, the states of vehicle are always constrained in nonconvex sets, such as saturation and dead zone. For example, the quad-rotor UAVs can change the direction and speed flexibly through the motors speed adjustment, but the largest speed is limited by the performance of motors. The area formed by the driving force is not convex. However, most previous work assumed that agent states can be arbitrarily large without any constraint. The existing results on convex or nonconvex constraints, e.g., [14–24], only considered the consensus convergence of the system and rarely considered its performance, e.g., the H∞ performance and the finite-time convergence. c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 401–412, 2020. https://doi.org/10.1007/978-981-32-9682-4_42

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The main contribution of this paper is to extend the distributed H∞ consensus problem of second-order continuous-time multi-agent systems to the situation where the input and velocity states are constrained in nonconvex sets. Authors propose a nonlinear distributed algorithm based on local information of each agent. In analysis process, a coordinate transformation is adopted such that the original nonlinear system apply to the Metzler matrix theory. Then a model transformation is introduced into the system, and the stability of transformed system under the desired H∞ performance is investigated by Lyapunov function. As a result, we obtain the sufficient conditions to make system achieve consensus.

2 Problem Formulation 2.1

Preliminaries

Consider there is a system that contains n agents. Define the connected graph as G (t) = (V , E (t)), where V = {1, 2, . . . , n} is the aggregate of nodes, E (t) ⊆ V × V is the aggregate of edges. The aggregate of neighbors is defined as Ni = { j ∈ V : ( j, i) ∈ E (t)}, and τi represents the number of neighbors. If edge ( j, i) ∈ E (t), then its weight ai j > 0, otherwise ai j = 0. Define the adjacency matrix as A(t) = (ai j (t))n×n and the degree matrix as D(t) = diag{d1 , d2 , . . . , dn }, where di = ∑ j∈Ni ai j . The Laplacian matrix can be described as L(t) = [L(t)]n×n = D(t) − A(t). If there is a sequence (vik−1 , vik ), (vik−2 , vik−1 ), . . . , (v2 , v1 ), it can be regarded as a directed route from vi1 to vik . The graph is considered to be strongly connected if it has a route from each node to each other node, and it is regarded to have a directed spanning tree if the graph has at least one node with the paths to all other nodes. Besides, the matrix with nonnegative off-diagonal elements are called Metzler matrix. Lemma 1 [2]. Consider a directed graph. If one of eigenvalues of Laplacian matrix is 0 and its corresponding eigenvector is 1n , then this graph is considered to have a directed spanning tree. Meanwhile, the other eigenvalues have strictly positive real parts. Lemma 2 [1]. Consider a linear system x˙ = A(t)x where A(t) is always a Metzler matrix with zero row sums. If there is a interval T satisfying tt+T A(t)ds jointly have a directed spanning tree for any t > 0, then all elements will come to a common value for any solution x. Lemma 3 [6]. For the matrix φ2N = 2NIN − EN , one of its eigenvalues is 0 and the others are 2N with multiplicity 2N −1. Furthermore, there is an orthogonal matrix U  2NI 0 2N−1 2N and the last column of U is 12N . That is, U = satisfying U T φ2N U = 0 0 [U1 U¯ 1 ] where U¯ 1 = 12N and U1 is the rest part. 2N

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2.2 Problem Formulation To investigate the constrained consensus problem, we first make the assumption as follows: Assumption 1 [14]. Let Ui , Vi ⊆ Rr , i = 1, . . . , N be two nonempty bounded closed sets satisfying sup SUi (x) = ρ¯ i > 0, inf SUi (x) = ρ i > 0 and sup SVi (y) = ν¯ i > 0, x∈Ui

x∈U / i

y∈Vi

inf SVi (y) = ν i > 0 for all i, where 0 ∈ Ui , 0 ∈ Vi , SUi (·) and SVi (·) be two constraint

y∈V / i

operators such with ⎧ αβ x ⎪ ⎨ x sup {β : ∈ Ui , 0 < α < β }, x = 0 x SUi (x) = x 0≤β ≤x ⎪ ⎩ 0, x=0 and

⎧ αβ y ⎪ ⎨ y sup {β : ∈ Vi , 0 < α < β }, y = 0 y y 0≤β ≤y SVi (y) = ⎪ ⎩ 0, y=0

Under Assumption 1, the dynamics function of system is: x˙i (t) = vi (t) i ∈ {1, 2, . . . , N}, v˙i (t) = ui (t) + ωi (t),

(1)

where xi , vi , ui and ωi (t) ∈ Rr represent the position, velocity, input and external interference, respectively. Operators SUi (·), SVi (·) are used to find vectors with largest magnitude in the directions of x, y. These vectors satisfy that SUi (x) ≤ x, kSUi (x) ∈ Ui and SVi (x) ≤ y, kSVi (y) ∈ Vi for all k ∈ [0, 1]. ρ¯ i , ρ i denotes the supremum and infimum of the distance from any point to origin point, meaning that the acceleration of agent cannot be arbitrarily large, but its direction is not limited. Similarly, ρ¯ i , ρ i means each agent can move toward any certain direction and cannot exceed the upper bound of velocity. This paper is intended to develop appreciate algorithm and get sufficient conditions such that the position states of agents converge to a common vector, the velocity converge to 0 within nonconvex constraint set Vi , and the control input is constrained in nonconvex set Ui . In the coming sections, only the condition of r = 1 is discussed. For the case of r = 1, we can infer the similar results in the same manner.

3 Main Results 3.1 Control Law and Dynamic Description To solve the consensus problem, we adopt the following control law: ui (t) = −qi vi (t) + αi (t), i ∈ {1, 2, . . . , N}

(2)

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where qi (t) > 0 denotes the feedback gain and αi (t) =

∑ ai j [x j (t) − xi (t)] +

j∈Ni (t)

∑ ai j [e j v j (t) − ei vi (t)], ei is a positive parameter determined later. Then system (1)

j∈Ni (t)

with velocity and input constraint is written as: x˙i (t) = SVi [vi (t)] i ∈ {1, 2, . . . , N}. v˙i (t) = SUi [ui (t) + ωi (t)], Denote that

(3)

⎧ ⎨ SUi [ui (t) + ωi (t)] , ui (t)) = 0 ui (t) + ωi (t) bi (t) = ⎩ 1, ui (t) = 0,

and

⎧ ⎨ SVi (vi (t)) , vi (t) = 0 vi (t) ci (t) = i = 1, . . . , N. ⎩ 1, vi (t) = 0

It can be inferred that SUi [ui (t) + ωi (t)] = bi (t)[ui (t) + ωi (t)] and SVi (vi (t)) = ci (t)vi (t). Note that bi (t) and ci (t) are scalings of input and velocity, respectively. The dynamic function can be written as x˙i (t) = ci (t)vi (t) v˙i (t) = bi (t)[−qi vi (t) + αi (t) + ωi (t)].

(4)

Define yi (t) = xi (t) + ei vi (t). Therefore, we have ci (t) ci (t) xi (t) + yi (t) ei ei ci (t) y˙i (t) = [bi (t)qi − ](xi (t) − yi (t)) ei + ei bi (t)Σ j∈Ni ai j [y j (t) − yi (t)] + ei bi (t)ωi (t), x˙i (t) = −

(5)

Let x(t) = [x1T (t), . . . , xNT (t)]T , y(t) = [yT1 (t), . . . , yTN (t)]T . Define z(t) = [zT1 (t), . . . , zT2N (t)] = [xT (t), yT (t)]T where z(t) ∈ Rr , and ω (t) = [0TN , ω1T (t), . . . , ωNT (t)]. Then the system (5) can be written as z˙(t) = δ (t)z(t) + ω¯ (t), 

where δ (t) =

A1 (t)

− A1 (t)

A2 (t)

− A2 (t) − A3 (t)L

diag{ f1 (t), . . . , fN (t)}, fi (t) = bi (t)qi −

(6)

 c (t) , A1 (t) = diag{− c1e(t) , . . . , − NeN }, A2 (t) = 1

ci (t) ¯ (t) = K(t)× ei , A3 (t) = diag{e1 b1 (t), . . . , eN bN (t)}, ω

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ω (t), K(t) = [0 A3 (t)] , i ∈ {1, 2, . . . , N}. We can easily infer that δ (t) is a Metzler matrix

with zero row sums for any t > 0. Next, to investigate the constrained consensus problems with external disturbances, consider the following output function: Z(t) = Cz(t), where the initial condition z(s) = z(0), s ∈ (−∞, 0), C =

(7) 

A4 0



0 ∈ R2r×2r and A4 = 0

η IN − Nη EN , η is a positive constant. Thus, we can rewrite the system dynamic function as: z˙(t) = δ (t)z(t) + ω¯ (t), Z(t) = Cz(t),

(8)

It is noted that the position state of each agent will reach the same value x¯ as time t → ∞ if the output function Z(t) converges to zero. In addition, according to x˙i (t) = vi (t), it is satisfied that limt→∞ vi (t) = 0 for all i. Thus, the system (1) can reach consensus if limt→∞ Zi (t) = 0. In order to research the influence of external disturbances ω (t) on the output function Z(t), we define a performance index with a prescribed scalar γ > 0: J(ω ) =

∞ 0

[Z T (t)Z(t) − γ 2 ω T (t)ω (t)]dt < 0

(9)

i.e., Tω Z (s)∞ = sup Z2 < γ . ω 2 ≤1

3.2 Main Theorem In this paper, due to the nonlinearity of the system, we will only analyze the H∞ performance in the local area around the consensus point. Before the presentation of results, we make some assumptions and lemmas first. Assumption 2. For any i = 1, 2, . . . , N and j ∈ Ni , there are positive constants ω¯ i , ki such that |ωi (t)| ≤ ω¯ i , max{ai j } ≤ ki . And |vi (0)| < 2qωi¯ , |x j (0) − xi (0) + e j v j (0) − ei vi (0)|
≥ bi . qi |vi (0)| + ∑ j∈Ni ai j |y j (0) − yi (0)| + ω¯ i (3 + τi )ω¯

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If bi (t) ≥ bi does not hold for some t = t , there must be a t ∗ ∈ (0,t ) satisfying t ∗ = ∗ inf{t ∗ > 0|bi (t ∗ ) = bi } and bi (t) < bi for t ∈ (t ∗ ,t ). It is obvious that qi bi (t ∗ ) − ci (tei ) > qi bi − e1i > 0. Since system (6) is a nonlinear system, we define that the system matrix of system (6) is δ . Then we will prove that δ is a Metzler matrix at time t ∗ in three cases. Case 1: If the sign of external disturbance ω (t ∗ ) is same as xi (t ∗ ) − yi (t ∗ ), there must ∗ exist a positive constant β1 such that (bi (t ∗ )qi − ci (tei ) )(xi (t ∗ ) − yi (t ∗ )) + ei bi ωi (t ∗ ) = ∗

β1 (bi (t ∗ )qi − ci (tei ) )(xi (t ∗ ) − yi (t ∗ )). In this way, the system matrix of system (6) is   β1 A1 (t ∗ ) −A1 (t ∗ ) ∗ . δ (t ) = A2 (t ∗ ) −A2 (t ∗ ) − A3 (t ∗ )L

It can be inferred that δ is a Metzler matrix. Meanwhile, all of the matrix row sums are zero. Case 2: If the sign of external disturbance ω (t ∗ ) is same as y j (t ∗ ) − yi (t ∗ ), similar to the Case 1, there must exist a positive constant β2 such that Σ j∈Ni ai j [y j (t ∗ ) − yi (t ∗ )] + ωi (t ∗ ) = (L + β2 I)[y j (t ∗ ) − yi (t ∗ )]. Then the system matrix is   −A1 (t ∗ ) A1 (t ∗ ) ∗ . δ (t ) = A2 (t ∗ ) −A2 (t ∗ ) − A3 (t ∗ )(L + β2 I) Similarly to Case 1, it can be inferred that δ is a Metzler matrix with zero row sums. Case 3: If the sign of external disturbance ωi (t ∗ ) is different from both xi (t ∗ ) − yi (t ∗ ) and y j (t ∗ ) − yi (t ∗ ), there must be a constant β3 ∈ (0, 1) such that ∗ β3 (bi (t ∗ )qi − cie(t∗ ) )|yi (t ∗ ) − xi (t ∗ )| ≥ ei bi (t ∗ )ωi (t ∗ ) or β3 max{ei bi (t ∗ )ai j |y j (t ∗ ) − i yi (t ∗ )} ≥ ei bi (t ∗ )ωi . In this way, δ is still a Metzler matrix. It is proved that the system (6) can be written as linear system z˙(t ∗ ) = δ (t ∗ )z(t ∗ ), where δ is a Metzler matrix in all cases. According to the definition of derivative, we have z(t ∗ + Δ t) = (I + δ (t ∗ ))z(t ∗ ) + o(Δ t), which means the solution of transformed system function has convexity at t ∗ . Therefore, |yi (t) − xi (t)| and |y j (t) − yi (t)| will decrease and bi (t) will increase. It contradicts that bi (t) < bi when t ∈ (t ∗ ,t ), which means that t ∗ does not exist. Thus, bi ≤ bi (t) ≤ 1 is satisfied for all t. Moreover, if vi (t) = SVi [vi (t)], we have |viν(t)| ≤ ci (t) < 1. Then we can get ci (0) ≥

qi ν ν > ≥ ci . |vi (0)| 2ω¯

Therefore, it can be obtained that ci ≤ ci (t) < 1 is satisfied for all t. Theorem 1. Consider the system (8) where Assumptions 1 and 2 hold. The system under control law (2) can reach consensus with Twz (s)∞ < λ γ (λ = max{ei bi (t)}) if the following inequality is satisfied for any t PU1T δ (t)U1 +U1T δ T (t)U1 P +U1T CT CU1 + where P is a positive definite matrix.

1 PU T K(t)K T (t)U1 P < 0, γ2 1

(10)

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Proof. Before the analysis of stability, we introduce a model transformation into the 12N 2N t system (8). Define that zˆ(t) = z(t) − ∑ 0 ω¯ (s)ds, θ (t) = U1T zˆ(t) and θ¯ (t) = 2N i=1 √ U¯ 1T zˆ(t). From Lemma 3, it can be inferred U1T U¯ 1 = 0, U1T 12N = 0, U¯ 1T 12N = 2N. Therefore, 12N 2N U T z˙(t) = U T zˆ˙(t) +U T ∑ ω¯ i (t) 2N i=1 ⎤ ⎡   0 θ˙ (t) ⎥ ⎢ = + ⎣ 1 2N ⎦ ˙ ¯ ¯ √ ω (t) θ (t) ∑ i 2N i=1 ⎤ ⎡   U1T ω¯ (t) (11) θ (t) ⎥ ⎢ T + ⎣ 1 2N = U δ (t)U ⎦ θ¯ (t) √ ∑ ω¯ i (t) 2N i=1 ⎤ ⎡    U1T ω¯ (t) ¯ δ (t) 0 θ (t) ⎥ ⎢ + ⎣ 1 2N = ⎦ ς (t) 0 θ¯ (t) ¯ √ ωi (t) ∑ 2N i=1 where δ¯ (t) = U1T δ (t)U1 , ς (t) = U¯ 1T δ (t)U1 , ς (t) ∈ R1×(n−1) . θ¯ (t) can be regarded as the average of at instant t. From (11), it is obvious that θ˙¯ (t) = ς (t)θ (t), which means that θ˙¯ (t) completely depends on θ (t). Now the system turns into the following form: ⎧˙ θ (t) = δ¯ (t)θ (t) +U1T ω¯ (t) ⎪ ⎨

 θ (t) ⎪ = CU1 θ (t). ⎩ Z(t) = Cz(t) = Cˆz(t) = CUU zˆ(t) = [CU1 0] ¯ θ (t) 

T

(12)

Evidently, if limt→∞ θ (t) = 0, then limt→∞ Z(t) = 0, e.g., the consensus of system is arrived. To discuss the stability of (12) when ω ≡ 0, we first consider the Lyapunov function V (t) = θ T (t)Pθ (t) where P is a positive definite matrix, and the derivation is V˙ = θ˙ T (t)Pθ (t) + θ T (t)Pθ˙ (t) = θ T (t)(δ¯ (t)P + Pδ¯ T (t))θ (t). Since ω (t) ≡ 0, we have V˙ = θ T (t)(Pδ¯ (t) + δ¯ T (t)P)θ (t) + θ T (t)PU1T K ω (t) + ω T (t)K T (t)U1 Pθ (t). To analyze the effect of external interference on the output function, suppose that θ (0) = 0, which means V (0) = 0. Then, we introduce the function into the performance

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index JT =

T 0

[Z T (t)Z(t) − γ 2 ω T (t)ω (t)]dt:

JT = =

T 0

T 0

[Z T (t)Z(t) − γ 2 ω T (t)ω (t)]dt [θ T (t)U1T CT CU1 θ (t) − γ 2 ω T (t)ω (t) + V˙ (t)]dt

− V (T ) +V (0)

=

T 0

[θ T (t)U1T CT CU1 θ (t) − γ 2 ω T (t)ω (t)

+ θ T (t)(Pδ¯ (t) + δ¯ T (t)P)θ (t) + θ T (t)PU1T K(t)ω (t) + ω T (t)K T (t)U1 Pθ (t)]dt −V (T )

≤

T 0

ξ T (t)Γ ξ (t)dt, 

where

Γ=

Pδ¯ + δ¯ T P +U1T CT CU1

 PU1T K(t)

K T (t)U1 P

−γ 2 IN

,

ξ T (s) = [θ T (t) ω T (s)]. It is evident that if Γ < 0, then JT < 0. When T → +∞, Z(t)22 < γ 2 ω (t)22 , thus Tω Z (s)∞ = sup Z2 < γ . ω 2 ≤1

By Schur Complement Lemma, we can derive that Pδ¯ (t) + δ¯ T (t)P +U1T CT CU1 < 0 if Γ < 0. Since U1T CT CU1 > 0, Pδ¯ (t) + δ¯ T (t)P < 0 is satisfied which means that the nonconvex constrained H∞ consensus is achieved when Γ < 0 holds for all t. By Schur Complement Lemma, we have 1 Pδ¯ + δ¯ T P +U1T CT CU1 + 2 PU1T K(t)K T (t)U1 P < 0 γ if Γ < 0, then the proof is complete. Remark 1. Note that matrix inequality (10) is supposed to be solvable to ensure that the graph is a direct connected graph. For P is a positive definite matrix, we can transform the equality (10) into U1T δ (t)U1 P−1 + P−1U1T δ T (t)U1 + P−1U1T CT CU1 P−1 +

1 T U K(t)K T (t)U1 < 0. γ2 1

Note that δ (t) is a Metzler matrix with zero row sums, therefore −δ (t) can be regarded as the Laplacian matrix of a graph G . Since [−δ (t)]i,i+n < 0 and [−δ (t)]i+n,i < 0, there is an undirected path from node i to node i + n. Therefore, G has at least one spanning tree. Meanwhile, matrix −δ (t) has a simple zero eigenvalue with eigenvector 1N while the other eigenvalues have strictly positive real parts, which means that U1T δ (t)U1 is a Hurwitz matrix. By Lyapunov Theory, there must be positive definite matrices P and Ψ such that U1T δ (t)U1 P−1 + P−1UqT δ T (t)U1 = −Ψ .

Consensus

⎡ For matrix C = ⎣

η IN −

409

⎤

η EN N

0

⎦ and K(t) = [0 A3 (t)], there must exist appropriate

0 0 parameter η , qi to satisfy the Eq. (10). Corollary 1. Consider an undirected connected network of agents. Under Assumptions 1 and 2, the nonconvex constrained consensus for system (3) with control law (2) is achieved with Tω Z (s)∞ < γ if ⎡ U1T ⎣

R(IN − B) − 2DT −1 +CT C

⎤

RB

− 2RB + 2T −1 − T BL − LT B +

RB

1 2 ⎦ U1 < 0 T γ2

(13)

is satisfied, where B = diag{b1 , . . . , bN }, D = diag{c1 , . . . , cN }, R = diag{q1 , . . . , qN }, T = diag{e1 , . . . , eN }. Proof. From Theorem (1), the sufficient condition to achieve constrained H∞ consensus is given. Let P = I2N and define that B = diag{b1 , . . . , bN }, D = diag{c1 , . . . , cN }, R = diag{q1 , . . . , qN }, S = diag{b1 (t), . . . , bN (t)}, F = diag{c1 (t), . . . , cN (t)}, T = diag{e1 , . . . , eN }. Then Γ (t) < (t) can be transformed by Schur Complement Lemma into the following form Γ (t): 1 Γ (t) = δ¯ + δ¯ T +U1T CT CU1 + 2 U1T K(t)K T (t)U1 γ ⎡ ⎤ CT C 0 ⎢ ⎥ =U1T (δ + δ T )U1 +U1T ⎣ ⎦ U1 1 A3 (t)2 0 2 γ ⎡ ⎤   RS − 2FT −1 +CT C 0 − RS RS ⎥ T T⎢ U1 +U1 ⎣ =U1 ⎦ U1 1 RS − RS 0 − RS + 2FT −1 − T SL − LT S + 2 (T S)2 γ < 0.



−RS

RS



≤ 0 holds for all bi (t) > 0. Replace bi (t) with bi in − RS     −R(S − B) R(S − B) −RS RS R, then we can get that ≤ 0. Therefore, ≤ R(S − B) − R(S − B) RS − RS   −RB RB for all t > 0. Furthermore, for −L ≤ 0, −T SL − LT S ≤ −T BL − LT B. In RB − RB this way, we have Γ ≤ Ω where It is obvious that

RS

⎡

Ω = U1T ⎣

R(IN − B) − 2DT −1 +CT C RB

RB

− 2RB + 2T −1 − T BL − LT B +

Thus, Γ < 0 holds for all t if Ω < 0.

⎤ 1 2 ⎦ U1 . T γ2

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4 Simulations In this section, we present a simulation example verify the feasibility of the theory. Consider a system containing six agents where the control input ui of agents remain in √   3 2 2 a nonconvex set Ui = {x ∈ R | x ≤ 1} {x ∈ R | x + [ 2 0]T  ≤ 12 } {x ∈ R2 | x −

√ 3 1 T velocity vi is constrained in a nonconvex set Vi = {v ∈ R2 | ||v|| ≤ 2 0]  ≤ 4 } and the √ √ 3 3 1  1  1 2 T 2 T 2 } {v ∈ R | ||v + [ 4 0] || ≤ 2 } {v ∈ R | v − [ 4 0]  ≤ 4 }. Let qi = 10, and

[

Assumptions (1) and (2) are satisfied. In Fig. 1(a), it shows four graphs with six nodes, and it is obvious that all graphs jointly have a spanning tree. It denotes four different switching topologies of network, and the interval of switching is 0.01 s. All the weights of edges are 1.

(a) Four directed graphs

(b) A finite state machine

Fig. 1. Topological structures of agents

 T Suppose that the initial values of system are given as x1 (0) = 3 4 , x2 (0) =  T  T  T  T  T 7 3 , x3 (0) = −3 4 , x4 (0) = 4 − 2 , x5 (0) = 6 3 , x6 (0) = −5 5 , and  T the initial velocity of each agent is vi (0) = 0 0 , i = 1, 2, . . . , 6. Let the disturbance ω = [0.05 0.03 0.05 0 − 0.08 − 0.05]T . Meanwhile, choose the performance index γ = 1 as an appropriate solution of Theorem 1, and let the weights of all edges to be 0.2. Figure 2 show that both input and the velocity of each agent are remained in the given nonconvex constraint sets, respectively. And in Fig. 3, it presents the position trajectory of agents under the control law (2). The image suggests that the consistence of positions can be achieved and the velocities will tend to zero. Finally, in Fig. 4, it presents the energy trajectories of Z(t) and disturbance signal, which indicates that performance Tω z (s)∞ < 1.

Consensus 1.5

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0.8 0.6

1 0.4 0.5

vi2

ui2

0.2 0

0 -0.2

-0.5 -0.4 -1 -0.6 -1.5 -1

-0.8

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-0.2

0

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-0.8 -0.5

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-0.1

ui1

0

0.1

0.2

0.3

0.4

0.5

vi1

(a) The input of all agents

(b) The velocity of all agents

Fig. 2. The input and velocity of agents 8 15 6 the energy of the disturbance signal

x

i2

4

10

2

xi

x

i1

0 the energy of Z(t) -2

5

-4

-6 0

100

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400

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600

Time (t)

Fig. 3. The position trajectories of agents

0 0

100

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Fig. 4. The energy trajectories of Z(t) and disturbance signal

5 Conclusions The authors investigate the distributed H∞ consensus problem of second-order multiagent systems on directed graphs with nonconvex input and velocity constraints, and present sufficient conditions to ensure the convergency of system under the required disturbance. A simulation is given to examine the availability of the theory. In addition, only the case of continuous-time is discussed in this paper, and we assume that the timedelay does not exist. Therefore, it is worth trying to apply this method to the discretetime system and time-delay situation in the future work.

References 1. Moreau L (2004) Stability of continuous-time distributed consensus algorithm. In: Proceedings of the IEEE conference decision and control, pp 3998–4003 2. Ren W, Beard RW (2005) Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Trans Autom Control 50(5):655–661 3. Wu Z, Xu Y, Pan Y, Su H, Tang Y (2018) Event-triggered control for consensus problem in multi-agent systems with quantized relative state measurements and external disturbance. IEEE Trans Circuits Syst I-Regul Pap 65(7):2232–2243

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4. Jia Y (2000) Robust control with decoupling performance for steering and traction of 4WS vehicles under velocity-varying motion. IEEE Trans Control Syst Technol 8(3):554–569 5. Jia Y (2003) Alternative proofs for improved LMI representations for the analysis and the design of continuous-time systems with polytopic type uncertainty: a predictive approach. IEEE Trans Autom Control 48(6):1413–1416 6. Lin P, Jia Y, Li L (2008) Distributed robust H∞ consensus control in directed networks of agents with time-delay. Syst Control Lett 57(8):643–653 7. Lin P, Jia Y (2010) Robust H∞ consensus analysis of a class of second-order multi-agent systems with uncertainty. IET Control Theory Appl 4(3):487–498 8. Liu Y, Jia Y (2010) H∞ consensus control of multi-agent systems with switching topology: a dynamic output feedback protocol. Int J Control 83(3):527–537 9. Mo L, Jia Y (2011) H∞ consensus control of a class of high-order multi-agent systems. IET Control Theory Appl 5(1):247–253 10. Li P, Qin K, Shi M (2015) Directed networks of second-order agents with mixed uncertainties time-delay. Neurocomputing 148:332–339 11. Cao W, Zhang J, Ren W (2015) Leader-follower consensus of linear multi-agent systems with unknown external disturbances. Syst Control Lett 82:64–70 12. Zhang H, Yang R, Yan H, Yang F (2016) H∞ consensus of event-based multi-agent systems with switching topology. Inf Sci 370:623–635 13. Lin P, Ren W (2017) Distributed H∞ constrained consensus problem. Syst Control Lett 140:45–48 14. Mo L, Lin P (2018) Distributed consensus of second-order multiagent systems with nonconvex input constraints. Int J Robust Nonlinear Control 28(11):3657–3664 15. Ren W (2008) On consensus algorithms for double-integrator dynamics. IEEE Trans Autom Control 53:1503–1509 16. Nedic A, Ozdaglar A, Parrilo PA (2010) Constrained consensus and optimization in multiagent networks. IEEE Trans Autom Control 55(4):922–938 17. Lee U, Mesbahi M (2011) Constrained consensus via logarithmic barrier functions. In: 2011 50th IEEE conference on decision and control and European control conference 18. Cao Y, Ren W (2012) Distributed coordinated tracking with reduced interaction via a variable structure approach. IEEE Trans Autom Control 57:33–48 19. Lin P, Ren W (2014) Constrained consensus in unbalanced networks with communication delays. IEEE Trans Autom Control 59(3):775–781 20. Lin P, Ren W, Gao H (2017) Distributed velocity-constrained consensus of discrete-time multi-agent systems with nonconvex constraints, switching topologies, and delays. IEEE Trans Autom Control 62(11):5788–5794 21. Lin P, Ren W, Farrell JA (2017) Distributed continuous-time optimization: nonuniform gradient gains, finite-time convergence, and convex constraint set. IEEE Trans Autom Control 62(5):2239–2253 22. Lin P, Ren W, Yang C, Gui W (2018) Distributed consensus of second-order multi-agent systems with nonconvex velocity and control input constraints. IEEE Trans Autom Control 63(4):1171–1176 23. Lin P, Ren W, Yang C, Gui W (2019) Distributed optimization with nonconvex velocity constraints, nonuniform position constraints and nonuniform stepsizes. IEEE Trans Autom Control 64(6):2575–2582 24. Lin P, Ren W, Yang C, Gui W (2019) Distributed continuous-time and discrete-time optimization with nonuniform unbounded convex constraint sets and nonuniform stepsizes. IEEE Trans Autom Control (accepted). 10.11.9/TAC.2019.2910946

Correction-Based Diffusion LMS Algorithms for Secure Distributed Estimation Under Attacks Huining Chang1(B) , Wenling Li1 , and Junping Du2 1

School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China {ZY1703102,lwlmath}@buaa.edu.cn 2 Key Laboratory of Intelligent Telecommunications Software and Multimedia, School of Computer Science and Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China [email protected]

Abstract. In this paper, we mainly study the distributed estimation problem under attacks, which is mainly used to estimate the position parameters. To solve this problem, a correction-based secure diffusion least mean square (CS-dLMS) algorithm, which is a hybrid algorithm that composes of a non-cooperative LMS (nc-LMS) algorithm and a correction-based diffusion least-mean squares (CdLMS) algorithm, is proposed for distributed estimation. The nc-LMS algorithm is used to provide a reliable reference system, which can detect reliable neighbor nodes by setting a threshold under network attacks. The correction-based least mean square algorithm can estimate unknown parameters by interacting with neighbor nodes. In order to guarantee the mean performance of the CS-dLMS algorithm under attracks, a sufficient condition is proposed. Finally, Simulation results are provided to verify the effectiveness of the proposed algorithm and it outperforms the C-dLMS algorithm and nc-LMS algorithm. Keywords: Distributed estimation · CS-dLMS algorithm · Sensor attacks · Communication attacks

1 Introduction Distributed estimation, where nodes are used to estimate an interesting parameter under cyber attacks, has received increasing attention at present. The distributed algorithm, which is obtained by combining different cooperation modes with different adaptive algorithms such as incremental least mean square algorithm, diffusion least mean square algorithm [1] and consistent distributed least mean square algorithm, is widely applied to sensor networks, smart grids, machine learning and biological networks. In the past few years, a large number of distributed estimation algorithms have been proposed. In [2], in order to compensate for the shortcomings of centralized algorithms, distributed dLMS was proposed to improve estimated performance of the algorithm. After the algorithm is proposed, a diffusion least mean square algorithm based on local equality constraints is used to solve the problem with constraints between network c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 413–421, 2020. https://doi.org/10.1007/978-981-32-9682-4_43

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nodes [3]. In [4], the variable step size dLMS algorithm, that accelerates the convergence speed by dynamically changing the step size in real time, is proposed to solve the problem of slow convergence of algorithm. In order to improve the steady-state performance, researchers have proposed a precise diffusion strategy for deterministic problems [5]. It is worth pointing out that these algorithms only converge well under secure network conditions. However, In practice, there is often a phenomenon in which communications and sensors are attacked, that is, the network is not always secure because wireless transmission media are more susceptible to eavesdropping. Therefore, it is possible for an attacker to redesign the sensor, or extract information from the transmission communication [6, 7], and inject false information to attack the entire system [8]. If these attacks occur, existing distributed estimation algorithms do not provide good estimates because they do not identify the sensor or communication being attacked. Therefore, it is important to develop secure distributed algorithms. Recently, secure distributed estimation algorithms have received much attention [9, 10]. For example, in order to make better estimates under malicious attacks, the literature [11] proposed a reputation-based diffusion LMS (R-dLMS) algorithm. The authors propose a weight that varies over time, which is inversely proportional to the distance between the reference and the local estimate in this algorithm. However, if the damaged neighbor has multiple large or small values, the robustness of the algorithm is poor. In order to provide good estimation performance in the presence of damaged sensors and communication impairment, this paper proposes a correction-based secure diffusion least mean square (CS-dLMS) algorithm. The chapters in this paper are distributed as follows: In the second part, we present the problem of parameter estimation under attack. Section 3 introduces the CS-dLMS algorithm to solve the attack problem. In Sect. 4, we give the simulation results. Finally, a summary is made.

2 Problem Formulation 2.1

Distributed Estimation Problem

In a sensor network consisting of n nodes, the unknown parameters through the noisy observation data of each node can be estimated via a linear model as follows. dk∗ (i) = uk (i)wo + vk (i)

(1)

where uk (i) is a M-dimensional row regression vector with the covariance matrix Ru,k = E{uk (i)T uk (i)}, dk∗ (i) is a zero-mean observation, vk (i) is the measure2 , and w o is ment noise of the zero-mean Gaussian distribution with the variance σv,k an M-dimensional unknown parameter column vector. In addition, we let Rd ∗ u,k = E{dk∗ (i)uk (i)T }. However, we cannot estimate the parameters directly from Eq. (1), so we estimate the parameters by minimizing the cost function as follows: wo = arg min J glob (w) = w ∈RM

N

∑ E{|dk∗ (i) − uk (i)w|2 }

k=1

(2)

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415

We can directly get the value of wo by deriving the cost function and letting the derivative function be zero. The result is as follows:  −1   N

∑ Ru,k

wo =

k=1

N

∑ Rd∗ u,k

(3)

k=1

However, it is worth noting that this centralized algorithm cannot process data in real time, and network transmission is prone to failure. Therefore, we use a distributed network topology to estimate parameters. The algorithm will be described in the third part. 2.2 Network Attacks Problem In this paper, we consider both attack issues including sensor attacks and communication attacks.  The sensor attack means that the attacker sends some false information  dk (i), uk,i that satisfies the following equation to destroy the sensor. dk (i) = dk∗ (i) + yk (i)

(4)

where dk∗ (i) is the real data as shown in Eq. (1), and yk (i) satisfying the following equation is a linear combination of the regression vector uk (i) and the attack vector qk , i.e. yk (i) = uk,i qk

(5)

In order to simplify the formula, no matter whether or not the node k receives the attack, its measurement equation is written in the following form. dk (i) = uk,i (wo + qk ) + vk (i)

(6)

where qk = 0M if k is a normal sensor, but qk = 0M if k is a attacked sensor. For sensor attacks, once a sensor in the network is attacked, the estimated values obtained from previous algorithms may not converge to wo . Therefore, we need to propose a new algorithm to estimate the parameters under sensor attacks. Considering that the communication attacks occur in the process of node k communicating with neighboring nodes, we supposed that node k accepts information φ,i and w,i from its neighbor node , then if the communication line between k and  is controlled by the attacker and the information is damaged, the parameter estimated by node k may be different from the real value. Considering that there are normal communications and attacked communications in the network, the information received by node k is as follows: φr,i = φ,i + zk,i

(7)

= w,i + zk,i

(8)

r w,i

where zk,i is the attack vector maliciously injected by the attacker. It equals to an zero vector for a normal communication from  to k, but a non-zero vector for a attacked communication  to k.

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Normal communication and data transmission are considered ideal and noise free. Nevertheless, attacks are often encountered in practice, so our goal is to develop a distributed estimation system that can be well estimated in the case of sensor attacks and communication attacks. Before proceeding with this paper, we introduce the following assumptions for our attacked network environment: independent distributed Assumption 1: The regressor vector uk,i , which is identically   in time and spatially independent and is independent of w, j for all  and for j ≤ i, arises from a zero-mean random process. Assumption 2: The measuring noise vk,i that is temporally stationary, temporally white, and independent in time and space is zero-mean Gaussian, and is independent of other signals. Assumption 3: For each node k, there is no neighbor that is attacked on both the sensing and the communication to node k. In addition, the number ofattack events (including sensor attacks and communication attacks) is less than n2k , where nk is  nk  the number of neighbor nodes and 2 represents the smallest integer greater than or equal to n2k .

3 Secure Distributed Estimstion Solutions 3.1

C-dLMS Algorithm

In order to estimate the unknown parameter wo more accurately, a dLMS algorithm is proposed. However, we propose an correction-based diffusion least mean square algorithm that performs better than the lms algorithm, where the “correction” step is added to the algorithm between the adaptation and combination step. The algorithm is as follows: ⎧ ∗ ϕk,i = wk,i−1 + μ uTk,i (dk,i − uk,i wk,i−1 ) ⎪ ⎪ ⎨ φk,i = ϕk,i + λ (wk,i−1 − ϕk,i−1 ) ⎪ ⎪ ⎩ wk,i = ∑ a,k φ,i

(9a) (9b) (9c)

∈N k

where μ is a constant learning rate parameter, wk,i is the estimated value at i iteration for each node k, ϕk,i and φk,i are the M-dimension intermediate variables, λ is a positive gain factor, and Nk is a collection of neighbor nodes of k. The combination coefficient a,k that is the element of the matrix A, satisfies, / Nk 1T A = 1T , A1 = 1, a,k = 0 if  ∈

(10)

When the gain factor is in a certain range, the correction-based dLMS algorithm shows better estimated performance than the dLMS algorithm, that is, the mean square error of the correction-based dLMS is smaller than the mean square error of dLMS.

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When the communication and sensor in the network are attacked at the same time, the modified dlms algorithm becomes ⎧ ϕk,i = wk,i−1 + μ uTk,i (dk,i − uk,i wk,i−1 ) (11a) ⎪ ⎪ ⎪ ⎪ ⎪ (11b) ⎨ φk,i = ϕk,i + λ (wk,i−1 − ϕk,i−1 ) φr,i = φ,i + zk,i ⎪ ⎪ ⎪ r ⎪ ⎪ ⎩ wk,i = ∑ a,k φ,i

(11c) (11d)

∈N k

where dk and zk,i are defined in the second part. Because there are communication attacks and sensor attacks in Eq. (11), the algorithm cannot accurately estimate the position parameters, and then we need to propose a secure algorithm that can be robust to attacks. 3.2 CS-dLMS Algorithm In this section, a correction-based secure diffusion LMS (CS-dLMS) is proposed to achieve reliable distributed estimation in a network attack environment. The CS-dLMS can be thought of as a hybrid algorithm consisting of an nc-LMS algorithm and a CdLMS algorithm. The nc-LMS algorithm is used to provide a reliable reference estimate for detecting the trusted neighbors of each node, and the C-dLMS algorithm is used to estimate parameter for distributed estimation problem. The nc-LMS algorithm used to find reliable estimates is as follows:

wˇ k,i = wˇ k,i−1 + μ uTk,i dk (i) − uk,i wˇ k,i−1

(12)

The reason we choose the reference estimate from the nc-LMS algorithm is that the estimators of the different sensors are independent of each other. Therefore, attacks on sensors or communications cannot be propagated. Our design idea is that for each node k, we receive the wˇ ,i estimated from the neighbor nodes according to the nc-LMS algorithm, and then sort each component m of all vectors from small to large.   ˇ (m) = wˇ (m) , · · · , wˇ (m) , · · · , wˇ (m) W (13) k,i  ,i s ,i n ,i 1

wˇ (m)

< wˇ (m)

k

< wˇ (m) ,

and 1 , s , t , nk ∈ Nk . where  ,i s ,i nk ,i 1 From the attacker’s point of view, the purpose of the attack is to drive the estimated parameters of the estimator differ greatly from the actual values, so that is, the estimator ˇ (m) . Then the of the attack is likely to appear in the left or right hand of the set W k,i 2 intermediate n -th estimator is very reliable in large probability. Considering these, we select a reliable reference estimate below. (m) w¯ k,i = wˇ (m) [ 2n ],i

(14)

Thus, we compare the intermediate estimator of the neighbor node with the reference estimate, and if the intermediate vector is within a certain neighborhood of the reference estimate, then the neighbor is determined to be a reliable neighbor.

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We use random error variable

  r(m) (m) s˘k,i  max w ˘,i − wk,i

(15)

Similarly we define random error variable   r(m) (m) sk,i  max φ,i − wk,i

(16)

Intuitively, an effective attack causes the intermediate vector estimated by the estimator to be away from its corresponding true value. Considering the nc-LMS and CdLMS subsystems, we can define a random variable   (17) bk,i = max sk,i , s˘k,i Then through the following threshold test, we can detect attacks on the sensor and communication.   (18) Nk,i =  ∈ Nk |Tk,i = b2k,i < γk,i In our proposed design, we found a reference estimate based on the nc-LMS algorithm, which is always reliable under Hypothesis 3. Designing a threshold based on a reliable reference estimate, when the intermediate estimate of the received neighbor satisfies the threshold test (18), the neighbor is a reliable neighbor, such a damaged sensor and compromised communication can be detected. Then we choose the threshold [12] 2 γk,i = 4σˆ k,i 2 of w where the sample mean μˆ k,i and variance σˆ k,i ¯ k,i are as follows [12]:   μˆ k,i+1 = iμˆ k,i + wk,i /(i + 1)    2 2 2 σˆ k,i+1 = (i − 1)σˆ k,i + wk,i − μˆ k,i  /M /i

(19)

(20) (21)

2 = 0. where μˆ k,1 = 0M×1 and σˆ k,1 Through the threshold test, the instantaneous trust neighbor set of each node can be detected. We can then design the combined weights based on the detected secure network topology. Note that because each node’s trusted neighbors are dynamic, the combination matrix dynamically changes over time. Therefore, based on the Metropolis rule, the modified combination matrix is designed as follows:

⎧ ⎨ 1/ max nk,i , n,i if  ∈ Nk,i \k ak,i = 1 − ∑∈N k,t \k ck,i if  = k (22) ⎩ 0 otherwise

where nk,i and n,i represent the number of neighbors of nodes k and  at time i, and Nk,i \k denotes the neighbor of node k except itself. After getting the combination coefficient, we have wk,i =

∑

∈N k,i

ck,i φr,i

(23)

Intuitively, the implementation of the CS-dLMS algorithm is given in Algorithm 1.

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Algorithm 1. CS-dLMS Algorithm Initialization: Let wk,0 = 0, wˇ k,0 = 0, ϕk,0 = 0, μˆ k,1 = σˆ k,1 = 0 Compute:For each node k at time i > 0 1) Adaption: Adapt estimates wˇ k,i and φk,i at each node k by (6), (11a), (11b), and (12). 2) Communication: r and φr from the neighbor  of the node k. Receive w,i ,i 3)Detection: (m) = w ˇ (m) Reference estimate:w¯ k,i [ 2n ],i Threshold test: 2 γk,i = 4 σˆ k,i  Nk,i =  ∈ Nk |Tk,i = b2k,i < γk,i where      r(m)

(m)

r(m)

(m)

bk,i = max max w ˘,i − wk,i , max φ ,i − wk,i Update threshold: 2 by (20) and (21). Update the statistics μˆ k,i and σˆ k,i+1 4)Combination: wk,i = ∑ a,k φr,i ∈Nk

where combination weight a,k is computed by (22)

4 Simulation Results In this section, we demonstrate the effectiveness of the proposed CS-dLMS algorithm in the case of network attack, and compare it with the LMS, nc-LMS and C-dLMS algorithms by simulation. First, we choose a network composed of 20 nodes with a combination coefficient that satisfies Metropolis rules. The network topology is shown in Fig. 1. Parameter initialization is set as follows: the estimated value wk,0 = 0, the estimated value produced by the nc-LMS system wˇ k,0 = [0, 0]T , ϕk,0 = [0, 0]T , and the sample mean and variance μˆ k,1 = [0, 0]T , σˆ k,1 = 0. In addition, we choose the learning rate is μ = 0.02, and the gain factor is selected as λ = 0.8. The regression uk,i is a 1 × 2 zeromean Gaussian distributed with covariance Ru,k = I2 . The mean and covariance of the measured noise vk,i are 0 and 0.25. In this paper, we consider that attacks in the network occur simultaneously on sensors and communications. Among them, the damaged communication is randomly selected and changes with time. In addition, sensor 11 is selected as a damaged sensor that does not change over time and qk = [5, 5]T . Figure 2 depicts transient learning curves for different algorithms. As can be seen from the figure, the nc-LMS and the dLMS algorithms has the worst estimation performance. The performance of CS-dLMS is the best among these algorithms.

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Fig. 1. A network composed of 20 nodes

20

10

0 dLMS with attacks CS−dLMS reference nc−dLMS dLMS without attacks

−10

−20

−30

−40

0

500

1000

Fig. 2. Transient learning curves for different algorithms

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5 Conclusions In this paper, we mainly study the adaptive estimation under the condition of network attacks, and propose a correction-based secure diffusion least mean square(CS-dLMS) algorithm. Then, we prove that the reference system is reliable, that is, we can detect reliable neighbors through the reference system. Finally, we also proved that the CSdLMS algorithm also showed good performance under network attack conditions by simulation. Acknowledgments. This work was supported by the Science and Technology Major Project of Guangxi (GuikeAA18118054), NSFC (61573031, 61573059) and BJNSF (4162070).

References 1. Di Lorenzo P, Sayed AH (2013) Sparse distributed learning based on diffusion adaptation. IEEE Trans Signal Process 61(6):1419–1433 2. Sayed AH, Lopes CG (2007) Adaptive processing over distributed networks. IEICE Trans Fundam Electron 90(8):1504–1510 3. Nassif R, Richard C, Ferrari A, Sayed AH (2017) Diffusion LMS for multitask problems with local linear equality constraints. IEEE Trans Signal Process 65(19):4979–4993 4. Omer Bin Saeed M, Zerguine A, Zummo SA (2010) Variable step-size least mean square algorithms over adaptive networks. In: 10th international conference on information science, signal processing and their applications (ISSPA 2010), Kuala Lumpur, pp 381–384 5. Yuan K, Ying B, Zhao X, Sayed AH (2018) Exact diffusion for distributed optimization and learning - Part I: algorithm development. IEEE Trans Signal Process 67:708–723. https://doi. org/10.1109/TSP.2018.2875898 6. Wu SC, Liu B, Bai X, Hou YG (2015) Eavesdropping-based gossip algorithms for distributed consensus in wireless sensor networks. IEEE Signal Process Lett 22(9):1388–1391 7. Vempaty A, Ray P, Varshney P (2004) False discovery rate based distributed detection in the presence of Byzantines. IEEE Trans Aerosp Electron 50(3):1826–1840 8. Marano S, Matta V, Tong L (2009) Distributed detection in the presence of Byzantine attacks. IEEE Signal Process Mag 57(1):16–29 9. Vempaty A, Tong L, Varshney P (2013) Distributed inference with Byzantine data: state of the art review on data falsifification attacks. IEEE Signal Process Mag 30(5):65–75 10. Kailkhura B, Brahma S, Han YS, Varshney PK (2014) Distributed detection in tree topologies with Byzantines. IEEE Trans Signal Process 62:3208–3219 11. Lu G, Chen W, Huang D (2015) Distributed diffusion least mean square algorithm based on the reputation mechanism. J Electron Inf Technol 37(5):1234–1240 (in Chinese) 12. Liu Y, Li C (2018) Secure distributed estimation over wireless sensor networks under attacks. IEEE Trans Aerosp Electron Syst 54(4):1815–1831

Personal Credit Scoring via Logistic Regression with Elastic Net Penalty Juntao Li1 , Mingming Chang1(B) , Pengjie Tian1 , Liuyuan Chen2 , and Xiaoxia Mu3 1

College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China [email protected] 2 Journal Editorial Department, Henan Normal University, Xinxiang 453007, China 3 College of Computer and Information Engineering, Henan Normal University, Xinxiang 453007, China

Abstract. Credit scoring is the risk assessment of customers. A reliable credit scoring model can provide decision support for financial institutions. In this paper, the logistic regression model with elastic net penalty (LR-EN) is proposed to assess personal credit score. Results on German bank personal credit data show that the proposed method can greatly improve classification precision of “bad” customer compared with other three methods. In addition, the attributes selected by LR-EN are well interpreted. Keywords: Credit scoring · Logistic regression · Elastic net penalty · Cost sensitive

1 Introduction With the rapid development of economy and the change of the mass consumption concept, the scale of credit is expanding rapidly, which is accompanied by the growth of credit risk [1]. Particularly, personal credit has the characteristics of large amount, high cost and asymmetric information thus attracts extensive attention in recent years [2]. For example, Oskarsdottir et al. included call nets as a new big data source to improve the statistical performance of credit scoring models [3]. Liberati and Camillo improved the classification accuracy of credit scoring model by incorporating personality characteristics into the model [4]. For financial institutions, they need a quantitative risk management approach to help them make decisions and minimize credit losses. Therefore, it is necessary to build a reliable personal credit scoring model. There are many studies on credit scoring models, which can be divided into two main categories: one is based on machine learning methods, such as support vector machine [5], decision tree [6], neural network [7]; The other is based on statistical models [8], such as multiple linear regression, linear discriminant analysis, logistic regression. Machine learning methods can usually fit data well and improve the accuracy of prediction, but there will be problems such as large amount of calculation and inconvenience of interpretation. Statistical model method is simple in structure and easy to c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 422–428, 2020. https://doi.org/10.1007/978-981-32-9682-4_44

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explain, so it is favored in practical application. As a classical statistical model, logistic regression model has been widely used. Traditional logistic regression model has the advantage of simple calculation and easy interpretation, but it can not solve the multiple collinearity problem very well. Credit scoring generally involves many attributes, and some of attributes are often related. In order to improve the traditional logistic regression model, penalty technique can be introduced. For example, Hoerl and Kennard proposed ridge regression [9], which minimizes the residual sum of squares subject to a bound on the L2 -norm of the coefficients. However, ridge regression always keeps all the features in the model so that lacks of conciseness. Tibshirani proposed lasso [10], which is a penalized least squares method imposing an L1 -penalty on the regression coefficients. Zou and Hastie putted forward the elastic net [11], which combines ridge regression penalty and lasso penalty. It can not only achieve variable selection but also eliminate the degradation and oversimplification caused by extreme correlation. It is suitable for explaining the problem of multiple collinearity with more variables. This paper proposes the logistic regression model with elastic net penalty (LR-EN) for personal credit scoring. The validity of the model is demonstrated by taking the German bank personal credit data as an example. Compared with other three methods, the proposed method not only decreases a lot in the proportion of “bad” customers misclassified as “good” customers, but also is interpretable in attributes selection. The rest of this paper is organized as follows: Sect. 2 provide the problem statement and preliminaries, Sect. 3 presented the LR-EN based personal credit scoring, the conclusions are given in Sect. 4.

2 Problem Statement and Preliminaries 2.1 Data Description German bank persona credit data is provided by Hans Hofmann, which is publicly available in the UCI machine learning database. It consists of 1000 personal credit records with 20 attributes, including 700 “good” customers and 300 “bad” customers. Among 20 attributes, there are 7 continuous numerical attributes and 13 categorical attributes. 7 numerical attributes are: duration in month (A2), credit amount (A5), installment rate in percentage of disposable income (A8), present residence since (A11), age in years (A13), number of existing credits at this bank (A16) and number of people being liable to provide maintenance for (A18). 13 categorical attributes are: status of existing checking account (A1), credit history (A3), purpose (A4), savings account/bonds (A6), present employment since (A7), personal status and sex (A9), other debtors/guarantors (A10), property(A12), other installment plans (A14), housing (A15), job (A17), telephone (A19), whether they are foreign workers or not (A20). To make the data suitable for algorithms that cannot handle categorical variables, Strathclyde University edited the original data by encoding several orderly categorical attributes into integers and adding several indicative variables. Take housing (A15) as an example, rental housing is marked as (1,0), private housing is marked as (0,1), and free housing is marked as (0,0). The edited data contains 24 attributes and it is also available in the UCI machine learning database. For notation convenience, we make

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the following denotations. Let X = (x1 , x2 , . . . , x1000 )T be 1000 × 24 data matrix, where xi = (xi1 , xi2 , · · · , xi24 )T be the attributes of the i-th sample, i ∈ {1, 2, . . . , 1000}. Y = (y1 , y2 , . . . , y1000 )T be the label vector, where yi ∈ {0, 1}, i.e., “good” customers and “bad” customers are respectively denoted as 0 and 1. The purpose of personal credit scoring is to evaluate whether a customer is “good” customer or “bad” customer. The so-called “good” customers refer to those who could repay loans on time. The “bad” customers refer to those who can not repay loans on time. Effective credit scoring model help more “good” customers apply for loans to increase profits for financial institutions, but also reject “bad” customers and reduce losses. Credit scoring is essentially a binary classification problem. We devoted to build a model according to the given data set {(xi , yi )}Ni=1 , so as to obtain the potential relationship between the customer’s credit and characteristics. Then the model can be used to predict whether a new customer is “good” or “bad”, i.e. for a new input xi , the output of the model is yi ∈ {0, 1}. In addition, selecting important attributes that affect personal credit risk should be taken into account. 2.2

Logistic Regression and Elastic Net Penalty

Logistic regression is a classical classification model of statistical learning [8], it is defined by conditional probability distribution as follows: eβ x+β0 , T 1 + eβ x+β0 T

p(y = 1|x) = p(y = 0|x) =

1 T 1 + eβ x+β0

(1)

,

(2)

where x = (x1 , x2 , · · · , x j )T ∈ Rn is input, y ∈ {0, 1} is output, β ∈ Rn is coefficient vector, β0 ∈ R is offset. For a given input x, p(y = 1|x) and p(y = 0|x) can be calculated by (1) and (2), then logistic regression model output the value of y with larger conditional probability. In order to determine logistic regression model, we estimate parameters β and β0 by maximum likelihood method, for data set {(xi , yi )}Ni=1 , xi ∈ Rn , yi ∈ {0, 1}, let p(y = 1|x) = π (x), p(y = 0|x) = 1 − π (x),

(3)

The likelihood function is N

∏[π (xi )]yi [1 − π (xi )]1−yi ,

(4)

i=1

From (1)–(4), we can get the logarithmic likelihood function N

(β , β0 ) = ∑ [yi (β T xi + β0 ) − ln(1 + eβ

Tx

i +β0

)],

(5)

i=1

Maximization (5) is equivalent to minimization −(β , β0 ), i.e., (β , β0 ) = arg min{−(β , β0 )}.

(6)

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The elastic net penalty is proposed by Zou et al. [11], which can be defined as follows: (7) Pα = 1/2(1 − α )  β 22 +α  β 1 , where α ∈ [0, 1] is regularization parameter, Pα is a compromise between the ridge regression penalty (α = 0) and the lasso penalty (α = 1). Ridge regression reduces the coefficients of correlation factors and allows them to work together. Lasso does not pay attention to those factors with strong correlation and tends to choose one of them and ignores the others. When α increases from 0 to 1, the sparsity of the penalty coefficient varies from ridge regression to lasso.

3 LR-EN Based Personal Credit Scoring 3.1 Model and Algorithm By introducing elastic net penalty into logistic regression, the logistic regression model with elastic net penalty (LR-RN) is obtained, which can be described as follows: (β , β0 ) = arg min{−

T 1 N [yi (β T xi + β0 ) − ln(1 + eβ xi +β0 )] + λ Pα }, ∑ N i=1

(8)

where β , β0 , (xi , yi ) and Pα are defined the same as Subsect. 2.2, N is the number of samples, λ is regularization parameter, which control the intensity of penalty term. As the value of λ increases, the penalty term has greater influence on the coefficient β . To build models and evaluate performances, we randomly select 70% of the data as training set Xtrain and the remaining 30% as testing set Xtest . To ensure the class balance in division, Xtrain consist of 490 “good” customers and 210 “bad” customers. Xtest consist of 210 “good” customers and 90 “bad” customers. Ytrain and Ytest can be divided corresponding to Xtrain and Xtest . The above random dividing process are carried out 5 times to avoid the occasionality of the experiment. The process of classification and attributes selection are described as follows: Step 1. Input training set Xtrain , testing set Xtest and corresponding label vector Ytrain , Ytest ; Step 2. Fit LR-EN model on training data (Xtrain , Ytrain ) by a publicly available R package glmnet [12]; – Standardize Xtrain for purpose of eliminating the influence of dimension; – Give a few values of parameter α i.e. α = 0.2, 0.5, 0.8; – Determine the corresponding optimal λ by 10 fold cross-validation for each α ; – Determine the optimal parameter pairs (αo , λo ) and fit the optimal model; Step 3. Construct binary classifier: f = sign[(β T x + β0 ) − 0.5]; Step 4. Predict customers are “good” or “bad” on Xtest and record the result Yp ; Step 5. Extract the non-zero coefficients of the optimal model, attributes corresponding to non-zero coefficients are considered as important attributes related to classification.

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Result Analysis

In this section, we compare LR-EN with logistic regression with ridge regression penalty (LR-RR), logistic regression with lasso penalty (LR-L) and traditional logistic regression (LR). Three criteria are used to evaluate their performances, i.e. total classification precision (Pt ), classification precision of “good” customers (P1 ) and classification precision of “bad” customers (P2 ), defining as follows: P1 = T P1 /(T P1 + FN1 ),

(9)

P2 = T P2 /(T P2 + FN2 ),

(10)

Pt = (T P1 + T P2 )/(T P1 + FN1 + T P2 + FN2 ),

(11)

where T P1 (FN1 ) and T P2 (FN2 ) are respectively the number of “good” customers and “bad” which are correctly (mistakenly) predicted. The average and the standard deviation of evaluation criteria (9)–(11) for 5 experiments are shown in Table 1. Table 1. Accuracies of four methods LR-EN

LR-RR

LR-L

LR

Pt 0.720(0.01) 0.761(0.02) 0.757(0.02) 0.759(0.02) P1 0.698(0.02) 0.877(0.02) 0.862(0.03) 0.859(0.03) P2 0.772(0.03) 0.490(0.07) 0.512(0.10) 0.525(0.04)

Table 1 shows the Pt of four methods are not distinctly different, all of them are above 70%. LR-RR achieves the best Pt , which is 76.1%. Pt of LR-EN is 72%, which is 4.1% lower than that of LR-RR. However, there are significant difference in P1 and P2 among four methods. P1 of LR-RR, LR-L and LR are relatively high, all reaching above 85%. P1 of LR-EN is 69.8%, which is 17.9% lower than that of LR-RR. Table 1 also shows that LR-EN achieves the best P2 among four methods, achieving 72.2%, which is 28.2%, 26%, 24.7%, higher than that of LR-RR, LR-L and LR, respectively. Besides, LR-EN has the minimum standard deviation of three criteria among four methods. This shows that the proposed model has outstanding stability. Note that personal credit scoring is a cost sensitive problem, the cost of misclassifying “good” customers into “bad” customers is different from the cost of misclassifying “bad” customers into “good” customers. The former may shut out customers with good credit and bring bad experiences to customers, financial institutions also reduce certain returns. While the latter may bring serious financial losses to financial institutions. Obviously, it is more expensive to misclassify “bad” customers into “good” ones. Therefore, on the premise of guaranteeing the Pt , we should pay more attention to improving the P2 . Although the total accuracy of LR-EN model is slightly lower than that of the other three models, it is about 25% higher than the other three models in P2 . This shows that the performance of the proposed model on credit scoring is satisfactory.

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Table 2. Attributes selected via LR-EN Original attribute Current attribute Specific meaning A1

X1

Status of existing checking account

A2

X2

Duration in month

A3

X3

Credit history

A6

X5

Savings account/bonds

A7

X6

Present employment since

A9

X7

Personal status and sex

A12

X9

Property

A13

X10

Age in years

A14

X11

Other installment plans

A20

X15

Foreign worker

A4

X16

Purpose

A4

X17

Purpose

To illustrate the interpretability of the model, we provide the attributes selected by LR-EN in all five experiments in Table 2. Table 2 shows that 12 attributes are selected by LR-EN, which are considered as the main factors affecting customer credit risk. Among these attributes, “status of existing checking account”, “savings account/bonds”, “other installment plans” and “property” represent the current economic level of customers. Customers with high economic level are more likely to repay loans on time, and vice versa. “Credit history” is a more intuitive attribute of credit risk, as financial institutions tend to reject customers with a history of default. “Age in years” indirectly affects the repayment ability of customers. Generally older customers may have more stable incomes, thus with better loan repayment ability. The listed attributes are closely related to personal credit risk, which shows that the model is easy to interpret.

4 Conclusions In this paper, the logistic regression with elastic net penalty is proposed for personal credit scoring. The model is applied to German bank personal credit data for empirical analysis. Results show that the number of “bad” customers misclassified as “good” customers has been greatly reduced, which can help financial institutions reject customers with poor credit and reduce losses. In addition, the attributes selected by LR-EN are highly associated with personal credit risk, which reveals that the proposed model is reliable in attributes selection. Acknowledgments. This work was supported by the Natural Science Foundation of China (61203293, 61702164, 31700858), Scientific and Technological Project of Henan Province (172102210047, 162102310461, 172102310535), Natural Science Foundation of Henan Province (162300410184), Foundation of Henan Educational Committee (18A520015), Scientific Research Project of Zhengzhou (153PKJGG128), Foundation for University Young Key Teacher of Henan Province (2016GGJS-079).

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References 1. Crook JN, Edelman DB, Thomas LC (2007) Recent developments in consumer credit risk assessment. Eur J Oper Res 183(3):1447–1465 2. Lessmann S, Baesens B, Seow HV, Thomas LC (2015) Benchmarking state-of-the-art classification algorithms for credit scoring: an update of research. Eur J Oper Res 247(1):124–136 3. Oskarsdottir M, Bravo C, Sarraute C, Vanthienen J, Baesens B (2019) The value of big data for credit scoring: enhancing financial inclusion using mobile phone data and social network analytics. Appl Soft Comput 74:26–39 4. Liberati C, Camillo F (2018) Personal values and credit scoring: new insights in the financial prediction. J Oper Res Soc 69(12):1994–2005 5. Bellotti T, Crook J (2009) Support vector machines for credit scoring and discovery of significant features. Expert Syst Appl 36(2):3302–3308 6. Xia YF, Liu CZ, Li YY, Liu NN (2017) A boosted decision tree approach using Bayesian hyper-parameter optimization for credit scoring. Expert Syst Appl 78:225–241 7. Beque A, Lessmann S (2017) Extreme learning machines for credit scoring: an empirical evaluation. Expert Syst Appl 86:42–53 8. Hastie T, Tibshirani R, Friedman J (2001) The elements of statistical learning. Springer series in statistics. Springer, New York 9. Hoerl A, Kennard R (1988) Ridge regression. In: Encyclopedia of statistical sciences, vol 8, pp 129–136 10. Tibshirani R (1996) Regression shrinkage and selection via the lasso. J Roy Stat Soc Ser B-Stat Methodol 58:267–288 11. Zou H, Hastie T (2005) Regularization and variable selection via the elastic net. J Roy Stat Soc Ser B-Stat Methodol 67(2):301–320 12. Friedman J, Hastie T, Tibshirani R (2010) Regularization paths for generalized linear models via coordinate descent. J Stat Softw 33(1):1–22

Application of Wireless Sensor Networks in Offshore Electromagnetic Spectrum Monitoring Scenarios Chao He1,2 , Yonghui Zhang1,2(B) , Zhenjia Chen1,2 , and Xia Guo1,2 1

2

Hainan University, Haikou 570228, Hainan, China [email protected] School of Information and Communication Engineering, Hainan University, No. 58 Renmin Avenue, Meilan District, Haikou, Hainan, China

Abstract. For the vast sea area, the data of offshore electromagnetic spectrum monitoring is very scarce, and it is still in a relatively blank stage. The main reason is that the offshore electromagnetic spectrum monitoring system is affected by waves, currents, tides and winds, and will undulate, roll and longitudinal. Shaking, three-dimensional movement along the route, the complexity and instability of the marine environment, the offshore electromagnetic spectrum monitoring has been greatly hindered, the monitoring data is fluctuating, accurate and low, and the real-time performance is poor. The traditional electromagnetic spectrum monitoring method can only collect spectral data of discrete points, which is difficult to apply to analyzing spatial spectrum distribution. The system applies the wireless sensor network to the offshore electromagnetic spectrum monitoring, and monitors the antenna attitude (course angle, pitch angle, roll angle), latitude and longitude, node movement speed, parameters such as wind speed, wind direction, an effective way of dealing with complex sea environment, guarantee the stability of the monitoring system in offshore electromagnetic spectrum monitoring. Keywords: Offshore electromagnetic spectrum monitoring Wireless sensor network · Complex marine environment · Antenna attitude

1

·

Introduction

As the core of wireless broadband communication technology, wireless spectrum resources are a kind of non-renewable scarce resources. How to allocate and utilize them efficiently has always been an important factor affecting the performance of wireless communication technologies. With the rapid development of wireless radio frequency technology, wireless spectrum resources are becoming less and less. A wide variety of radios occupy different frequency bands, and spectrum usage becomes complicated, which brings great inconvenience to c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 429–435, 2020. https://doi.org/10.1007/978-981-32-9682-4_45

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relevant regulatory authorities. There are many problems with the monitoring, management, distribution and use of the electromagnetic spectrum in the region. In order to ensure the rational and orderly use of spectrum resources and the normal operation of wireless communication networks, advanced technical means and equipment are needed to analyze and evaluate the electromagnetic environment of space. At present, research status and development prospects of electromagnetic spectrum monitoring include optimization of electromagnetic spectrum monitoring communication network, design of electromagnetic spectrum monitoring wireless sensor network, multi-source spectrum sensing data fusion, and spectrum identification. At present, the method of electromagnetic spectrum monitoring is to use fixed monitoring stations and mobile monitoring equipment (such as monitoring vehicles and portable monitoring equipment), which has played an important role in the daily spectrum monitoring work and the spectrum monitoring guarantee of major activities in recent years. However, the existing monitoring method is limited by its coverage, usually only for spectrum monitoring of urban areas, and it is difficult to effectively cover such as the sea area, it is difficult to realize the seamless and interested in all areas of the whole monitoring period, which seriously affected the electromagnetic spectrum management effectively implementation of frequency spectrum resource planning and coordination. In the offshore electromagnetic spectrum monitoring system independently developed and designed, this paper focuses on the research and design of wireless sensor network, explores the influence of antenna attitude on electromagnetic spectrum detection, and is committed to maximizing the monitoring coverage of the monitoring system, so that it can effectively serve the electromagnetic spectrum monitoring system and achieve better organic integration.

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Main Parameters and Difficulties of Spectrum Monitoring

Radio monitoring in China is one of the important tasks assigned to the radio regulatory department by the Radio Regulations of the People’s Republic of China. The radio management department uses advanced technical means and facilities to basic parameters of radio transmission, such as frequency, frequency error, and radio frequency. Radio, transmission bandwidth and other indicators are systematically measured, the signal is monitored, the identification of the transmission is identified, the frequency band utilization and channel occupancy are counted, and the signal usage is analyzed to fully grasp the electromagnetic environment and find illegal radio stations. The illegal radio station eliminates interference by locating the interference source direction; performs electromagnetic environment test for establishing and maintaining the nationally approved

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transmitter database, establishing and maintaining the national spectrum occupancy database, and providing a technical basis for effective assignment frequency. The scope of radio monitoring includes radio services and engineering, science and medicine. The frequency bands to be monitored should include the frequency bands that have been used and developed. For the city, the spectrum monitoring of electromagnetic spectrum detection is gradually perfecting. However, for the vast sea area, the offshore electromagnetic spectrum detection activity is very scarce and still in a relatively blank stage. The main reason is due to the instability of the complex sea environment. The electromagnetic spectrum monitoring activity at sea has been greatly hindered, and the fluctuation of detection data is large and inaccurate, and the real-time performance is low. The traditional electromagnetic spectrum detection method can only collect spectral data of discrete points, and it is difficult to apply to the analysis of spatial spectrum distribution. In response to the above problems, our self-designed marine distributed electromagnetic spectrum detection system (shown in Fig. 1) realizes real-time detection of wireless communication frequency bands, realizing real-time detection of channel occupancy, frequency band occupancy and spatial spectrum occupancy, based on electromagnetic spectrum. The database establishes a spectrum map. The software radio module is used as a radio detection module to perform perceptual analysis on the electromagnetic spectrum environment of the current communication environment. The embedded main control module stores the collected electromagnetic spectrum data and the auxiliary parameter data locally and performs local electromagnetic spectrum data processing. The software radio module and the embedded main control module form a detection node. The self-organizing network based on the blank TV frequency band serves as the backbone network of the detecting node communication network, which facilitates the dynamic access of the upper computer and the sharing of data. The self-organizing network as the backbone network can not only be applied to the detection environment of fixed points, but also is suitable for the ground and marine electromagnetic spectrum detection environment. Use wireless sensor network to obtain and record the parameters of latitude, longitude, temperature, humidity, altitude, light intensity, heading angle, pitch angle, roll angle, node moving speed, wind direction and wind speed of the current monitoring node, which has better parameters for sea surface parameters. Control, make spectrum data more accurate and efficient analysis. Each monitoring node functions independently and builds multiple monitoring nodes. Multiple monitoring nodes can sense the resources of multi-dimensional spectrum space. Its low cost and high flexibility enable a wide range of distributed electromagnetic spectrum monitoring networks to be effectively built. The data is fused by using the optimized cooperative spectrum sensing algorithm for channel occupancy and spectrum usage rules.

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Fig. 1. Offshore electromagnetic spectrum monitoring system

3

The Role of Wireless Sensors in Offshore Spectrum Monitoring

The ship sails on the sea according to its own power. It is affected by waves, currents, tides and winds. The monitoring system is up and down, rolling, pitching, and three-dimensional movement along the course. Electromagnetic waves are transmitted during the sea. It is the reflection and absorption of irregular sea surface. The signal source energy of the island reef or the coast is mainly affected by the loss of free space propagation and the energy of the sea surface reflection. The signal source of the marine vessel is affected by the waves, causing the antenna tilt angle to change in real time, and the energy distribution of the radio wave continues to change. The most fundamental factor that causes changes in the energy of radio waves is the mobility of the waves. In order to overcome the obstacles caused by the characteristics of the sea to the electromagnetic spectrum detection, we must monitor the impact of the waves on the hull in real time. We use the wireless sensor to cooperate with the electromagnetic spectrum detection system for detection. The main parameters to be detected are the attitude of the antenna (heading angle, pitch angle, and horizontal). Rolling angle), latitude and longitude, wind speed, wind direction, wave height, temperature, hull moving speed, etc. There are two main methods for measuring antenna attitude: GPS attitude measurement technology and inertial navigation system attitude measurement technology; GPS attitude measurement technology is that GPS satellites continuously transmit navigation and positioning signals, and user equipment is mainly based on GPS receivers. The navigation data determines the position of the satellite when transmitting the signal, and determines the delay of the signal during transmission through the received ranging code, obtains the necessary information and observation, and realizes different positioning and attitude measurement through data processing

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task. However, the GPS update frequency is low, and the GPS signal cannot be guaranteed to be received continuously and without failure, so that GPS positioning and attitude measurement are affected. Inertial navigation system attitude measurement technology is composed of two parts of the gyroscope and accelerometer inertial components, gyroscope and accelerometer can accurately measure the angular velocity and acceleration specific force relative to the inertial space. Ingeniously design and use the gyroscope to measure the angular velocity of the Earth’s rotation, and use the accelerometer to calculate the acceleration to measure the attitude. Conventionally, inertial devices mainly include gyro and accelerometers to measure the attitude information of the carrier, but the reliability is not guaranteed, and the high-precision inertial device cost is also high, and the error accumulation disadvantage cannot be avoided. Considering that the GPS does not accumulate over time with respect to the inertial navigation system, the GPS/INS integrated navigation system is selected, and the data output by the sensor is combined to improve the accuracy on the one hand and the cost on the other. The position, speed, acceleration and attitude information provided is more accurate. The wind speed and wind direction of the sea surface affect the size of the wave fluctuation and the real-time flow direction of the current, and grasp the wind speed and direction information on the sea surface in real time. The sudden situation that occurs during the spectrum monitoring process can be pre-processed, and the wind speed is too large and exceeds When the threshold is set, a warning will be issued and the person can return in time to avoid unnecessary losses.

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Analysis of Monitoring Results of Offshore Electromagnetic Spectrum

A vertically polarized dipole antenna is used as the test antenna, the maximum gain is 2.1 dBi, the antenna height is 2 m, and the maximum angular deviation of the antenna is 15.0. The above antenna is used in the offshore electromagnetic spectrum monitoring system (Fig. 2 is the offshore electromagnetic spectrum monitoring). The window interface of the system) uses 700 MHz as the carrier test frequency. It is assumed that the transmitting end of the experimental group is located at the shore, the receiving end is located at the sea surface, and the hull is swaying in three types (up and down, left and right roll, front and rear roll), and the test distance is 0. −500 m, three scenarios were tested at the same time to compare the stability of the electromagnetic spectrum monitoring system and the influence of the antenna attitude change on the measurement accuracy. Experiments show that in the three forms of hull swaying, up and down, left and right roll, front and rear roll, the hull forward and backward sway has the greatest influence on the sea surface electromagnetic spectrum monitoring, which plays a leading role (Fig. 3 is the antenna forward tilt test data, Fig. 4 is the antenna back test data).

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Fig. 2. Offshore electromagnetic spectrum monitoring system

Fig. 3. Antenna forward tilt test data

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Fig. 4. Antenna back test data

Conclusion

Ocean exploration and ocean informationization have become national strategies. The monitoring and control of offshore electromagnetic spectrum resources is an urgent problem to be solved. The technical advantages of wireless sensors will be able to make up for the shortcomings of current electromagnetic spectrum monitoring equipment and bring revolutionary changes to the completion of electromagnetic spectrum monitoring tasks. But at the same time, there are still some problems that require a lot of effort to solve.

References 1. Geng D, Zhang Z, Liu X (2018) Study on polarization stabilization technology of shipborne triaxial antenna. Radio Commun Technol 44(264(4)):93–97 2. Liu X (2010) Research on automatic spectrum monitoring system. Xidian University 3. Zhang Y, Wang H, Wen J et al (2011) Beam pointing correction of one-dimensional phased array antenna in swing state. Telecommun Technol 51(6):94–97 4. Wang J, Yu F, Dai H et al (2010) Application of wireless sensor networks in electromagnetic spectrum monitoring. In: Proceedings of the 7th China communications society academic annual meeting 5. Wang S (2003) Research on anti-swaying scheme of shipborne antenna servo system. Radiocommun Technol 29(2):47–49

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6. Ji S, Wang J, Liu Y et al (2011) Research status and development direction of radio monitoring technology. In: Symposium proceedings on spectrum management and monitoring system construction 7. Fang H (2015) Research on the characteristics of marine radio wave propagation considering hull sway in the blank TV frequency band *. Telev Technol 39(13):140– 144 8. Li Y (2012) Transmission design and implementation of electromagnetic spectrum monitoring system for wireless sensor network. Xidian University

Modified ELM-RBF with Finite Perception for Multi-label Classification Peng Chen1 , Qing Li1(B) , Zitong Zhou2 , Ziyi Lu1 , Hao Zhou1 , and Jiarui Cui1 1

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School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China [email protected] Beijing Municipal Branch, Industrial and Commercial Bank of China, Beijing 100080, China

Abstract. An improved multi-label classification algorithm based on ELM-RBF (Extreme Learning Machine for RBF Networks) is proposed in this article. On the one hand, different clustering analyses are applied to improve the stability of ELM-RBF model; on the other hand, a finite perception method is presented so as to keep the main feature, randomness, of ELM models in a reasonable degree. The main advantage of this F-ELM-RBF (ELM-RBF with Finite Perception) model is an adaptive ability to determine label thresholds which can distinguish the relevant labels from irrelevant labels more scientifically. Statistical experiments show that this algorithm has a good performance on different data sets.

Keywords: Multi-label classification Threshold

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· ELM-RBF · Finite perception ·

Introduction

Multi-label classification is widespread in industrial production and internet, associated with real-valued data, images and texts. Different from single-label classification, samples in multi-label tasks have more than one category labels which may even be correlated with each other. When the tags are increasing, computational complexity of multi-label problems surges significantly. These years, Multi-label data are growing all the time and advanced deep networksbased representation learning methods can effectively extract features from data and build distributed vectors. It is a valuable topic to design or improve a faster classifier to deal with these “Big Data”. Extreme learning machine (ELM) is one of the most popular algorithms in recent years. This kind of algorithms are based on random projection and have a fast computing speed. Extreme learning machine for radial basis function networks (ELM-RBF) is a smart variation of ELM. This kind of algorithm extents ELM from single layer forward neural networks to radio basis function (RBF) networks [1]. Similar to ELM (input weights W and biases b of hidden layer c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 436–445, 2020. https://doi.org/10.1007/978-981-32-9682-4_46

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are initialized randomly), in ELM-RBF, centers C and spreads σ of RBF kernels are randomly determined. Then, the output weights of hidden layer are analytically calculated by finding the Moore-Penrose generalized inverse and the minimum norm least-squares solution of a general linear system. Compared with traditional neural networks which are trained during iterations, algorithms like ELM and ELM-RBF are faster and more efficient, so they are good choices for complex machine learning problems such as multi-label classification problems. This paper is organized as follows. In Sect. 2, existing works are described briefly. Section 3 illustrates the modified ELM-RBF algorithm proposed in this article. Experimental analysis and conclusion are given in Sects. 4 and 5 respectively.

2 2.1

Related Works ELM-RBF

As written in Sect. 1, ELM-RBF is a smart extension of ELM from SLFN case to RBF case [1]. Figure 1 shows the structure of ELM-RBF network with 3 input nodes, 5 hidden nodes, and 4 output nodes. The number of hidden nodes, K, is a parameter which should be determined by experimenters. The number of input nodes is equal with the dimension of input samples while the number of output nodes equates the number of categories for classification, or the number of labels for multi-label classification.

Fig. 1. The structure of original ELM-RBF network

We use RBF kernel in hidden layer, the RBF function is written as Eq. (1),   x − ck 2 hk (x) = h (ck , σk , x) = exp − , k = 1, 2, . . . , K (1) σk2

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where x is n-dimension input vector; ck is n-dimension vector, also the centers of the k th node of RBF; σk the impact width of this node. Like ELM, in this kind of model, ck and σk are determined arbitrarily. Thus, the output of ELM-RBF can be illustrated as Eq. (2), f (x) =

L 

βi hi (x) = h (x) β

(2)

i=1

where βi is the output weight between hidden layer and output layer. The training procession of ELM-RBF is similar to ELM, simply to solve linear equations: Hβ = T (3) where H is the output matrix of hidden layer and T is the output label matrix. The basic steps of ELM-RBF are written in Algorithm 1.

Algorithm 1. ELM-RBF Input: Training set D; testing set D ; the number of hidden nodes; Output: A trained ELM-RBF model and its outputs for test set D ; 1: Initialize the ck and σk of RBF kernels randomly; 2: Compute the output matrix H as equation (3); 3: Compute the weights β of output layer as equation (4); 4: Compute the output values of test set D using the trained ELM-RBF; 5: return Outputs, Accuracy.

2.2

Threshold Setting Strategies for Multi-label ELM

When the models like ELM-RBF and ELM are extended to multi-label tasks, the essential issue is to determine a group of threshold values which can distinguish the relevant labels from irrelevant labels. Traditional method is to preprocess the training dataset, and convert the single-polar labels into bi-polar labels (changing the labels form [0, 1] to [−1, 1]). Commonly, 0 is selected as a threshold (in reference [2]). Then, the positive output values represent the relevant labels and vice versa. Luo etc. proposed a multiple linear regression method to determine threshold value and applied it in K-ELM to deal with multi-label classification issues [3]. This method designed a linear function to generate the threshold values, as shown in formula (4). t (x) = a∗ × f (x) + b∗

(4)

where a∗ is n dimension row vector and f (x) is n dimension column vector which is constituted of output values of network and also has a relationship with labels; b∗ is offset.

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The structures of models like ELM or ELM-RBF are initialized randomly. This is the main reason why the performance of these algorithms is unstable. In order to enhance the stability of ELM-RBF, two methods are applied into the F-ELMRBF (ELM-RBF with finite perception) proposed in this article. Firstly, centers of RBF kernels are determined by clustering analysis on training instances. After that, a finite perceptual range is defined according to the dispersion degree of input instances. 3.1

The Determination of Centers of RBF

Considering that the determination of centers C is our main work in the first step, centroid-based clustering methods like K-means, K-medoids, etc. are selected for this issue. In addition, we also use hierarchical clustering method as a comparison. K-means algorithm is an effective clustering strategy. This algorithm is also one of the most commonly used training methods for RBF to find centers C. The main steps of K-medoids algorithm are similar to K-means. The only one difference is that means (centers) C are defined as centroids of each group in K-means algorithm while in K-medoids, the medoid (center) of each cluster is the vector which has the minimum sum distance between this vector and other vectors in the cluster. Hierarchical clustering methods are also called Connectivity-based clustering methods. We use typically hierarchical clustering strategy, agglomerative clustering as the comparison. 3.2

The Determination of Limited Perception Range

It should be mentioned that randomness is the main feature of ELM and ELMRBF. σk (k = 1, 2, . . . , K, K is the number of centers, also the number of hidden nodes) is determined randomly in our proposed F-ELM-RBF. Compared with ELM-RBF, the range of σk is designed according to the input instances near the center ck . In other words, σk is selected randomly between 0 and σ ∗ . The equations for these parameters is written in Eq. (9) to Eq. (11), σ ∗ = Q × σ ∗∗ × exp (D)

(5)

cmax σ ∗∗ = √ (6) 2K sum (D1 + D2 + ..Dn ) (7) D= n where cmax is the largest distance between two centers which were generated by a clustering algorithm. Dn is the distance between the center ci and the No.n instance which was clustered to center ck . Q is new parameter which should be defined by user.

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When σk is confined to a constant value such as σ ∗∗ (this is also the empirical formula for RBF network), this F-ELM-RBF model equates to a RBF network [4]. The details of F-ELM-RBF are written in Algorithm 2. Algorithm 2. F-ELM-RBF Input: Training set D; testing set D ; the number of hidden nodes; perception parameter Q; Output: F-ELM-RBF model and its outputs for test set D ; 1: Initialize the ck by defined clustering strategy; 2: Determine σk randomly by formula (16); 3: Compute the output matrix H as equation (3); 4: Compute the weights β of output layer as equation (4); 5: Compute the output values of test set D using the trained F-ELM-RBF; 6: return Outputs, Accuracy.

3.3

Experiment on Single-Label Classification

In order to determine suitable clustering method and evaluate the performance of modified ELM-RBF, algorithms including ELM, K-ELM (Kernel ELM) [5], RBF network, ELM-RBF, F-ELM-RBF are used to handle binary classification task, diabetes diagnosis application [6]. F-ELM-RBF1, F-ELM-RBF2 and F-ELMRBF3 refer to F-ELM-RBF models using K-means, K-medoids and agglomerative clustering methods respectively. The number of hidden nodes for ELM, RBF, ELM-RBF, F-ELM-RBF are set to 30. The activation function for ELM is RBF function. In K-ELM, regularization coefficient C = 1, kernel type is RBF kernel and σ = 1. In F-ELM-RBF, the value of Q is 10. In RBF, σ is determined as mentioned in Sect. 3.2. In this paper, out RBF model is only trained using unsupervised least squares loss in order to compare all the models in similar time overhead. All the experiments are conducted on a PC with an Intel core(TM) i74720HQ CPU @ 2.6 GHz processor and 8 GB RAM. Development environment is MATLAB R2016a on 64 bit windows 8 system. Each group of experiments are repeated 5 times. Table 1 shows the average results of these experiments using different models (ELM, K-ELM, ELM-RBF, RBF, F-ELM-RBF) for solving binary classification. Remarkably, in single-label tasks, the performance of ELM-RBF is a little worse than other ELM models. The classification ability of F-ELM-RBF is enhanced significantly. F-ELM-RBF model with K-medoids clustering method is more likely to have a better solution and average results. 3.4

Threshold Setting Strategies

In this paper, three threshold setting methods are designed for F-ELM-RBF. All these methods are designed under the assumption that samples close to each other in one feature space are more likely to have a similar threshold value.

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Table 1. Experimental results on binary classification (Diabetes diagnosis [6]) Algorithms

Training accuracy (%) Testing accuracy (%) Best accuracy (%)

ELM

76.91

77.60

78.65

K-ELM

78.99

78.13

—–

RBF

76.39

74.10

74.48

ELM-RBF

77.69

76.56

77.43

F-ELM-RBF1 78.79

77.53

78.65

F-ELM-RBF2 79.64

78.13

79.17

F-ELM-RBF3 77.86

76.92

77.42

The initial phrase in the presented F-ELM-RBF algorithm is to determine the centers. In this phrase, input samples can be assigned into different clusters. For M instances clustered near one center, the threshold value for them can be determined in different strategies. Average Value. According to the comparison between network output values fj (xi ) and yj (xi is the No.i input training instance, fj (xi ) is the No.j output value of network, yj is the corresponding label in the label set of xi ), the threshold value ti of xi can be defined in formula (8).   1 / Yi , r ≥ fj (xi )} ti = arg min {yj |yj ∈ Yi , r ≤ fj (xi )} + arg min {yj |yj ∈ 2 r∈R r∈R (8) The threshold value tk for these M instances can be determined as the average of ti (i = 1, 2, m) as written in formula (9). tk =

1 (t1 + t2 + . . . tm ) m

(9)

Maximum/Minimum Value. Maximum/minimum values are selected as a replacement of average values. Similar to “average value strategy”, the threshold of xi can be defined as Eq. (10).   / Yi , r ≥ fj (xi )} ti = max arg min {yj |yj ∈ Yi , r ≤ fj (xi )} , arg min {yj |yj ∈ r∈R

r∈R

(10) We can also choose the smaller value in the above set, as shown in formula (11).   ti = min arg min {yj |yj ∈ Yi , r ≤ fj (xi )} , arg min {yj |yj ∈ / Yi , r ≥ fj (xi )} r∈R

r∈R

(11) Then the threshold value of each cluster can be determined by formula (9), (12) or (13). tk = max {t1 , t2 , . . . tm } (12)

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tk = min {t1 , t2 , . . . tm }

(13)

Create Liner Function by Multiple Regress in Clusters (C-Function). This strategy is inspired by the threshold setting strategy in reference [3]. This strategy designed a linear function to distinguish the relevant labels from irrelevant labels. Considering the limitation of the classification ability and global approximation ability of linear function, for input samples with relatively huge instances, an improved and more targeted threshold setting method should be researched further for F-ELM-RBF model. Due to the fact that input samples can be assigned into several clusters, it is reasonable to believe that the linear function for each cluster can be more effective, adaptive and robust. Thus the multiple regress model written in Sect. 2.2 can be applied in these instance clusters respectively. After the thresholds for all the clusters are determined one by one, the test instances can be classified by KNN and matched into one cluster with tk . Algorithm details of multi-label F-ELM-RBF can be described as Algorithm 3.

Algorithm 3. F-ELM-RBF algorithm for multi-label classification Input: Training set D; testing set D ; the number of hidden nodes; perception parameter Q; Output: F-ELM-RBF model and its outputs for test set D ; 1: Preprocess the input instances by changing the (0,1) labels to (-1,1) labels; 2: Initialize the ck by defined clustering strategy; 3: Determine σk randomly by formula (16); 4: Compute the output matrix H as equation (3); 5: Compute the weights β of output layer as equation (4); 6: Compute the output values of test set D using the trained F-ELM-RBF; 7: Obtain the final outputs by comparing the output value and threshold of each instance; 8: return Outputs, Accuracy.

We also conduct experiments on emotions data set [7] to evaluate the performance of all the threshold setting strategies. In addition to those strategies mentioned above, the method in reference [3], “Function” method, is also applied in our experiments as a comparison. The performance of F-ELM-RBF with different threshold setting strategies is shown in Table 2. Table 2. Experimental results about threshold setting strategies for F-ELM-RBF on emotions [7] data set Indicator

Threshold setting strategies for F-ELM-RBF 0 Average Minimum Maximum C-Function Function

Hamming loss 0.2283 0.2215

0.2537

0.2743

0.2412

0.2781

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As shown in Table 2, “Average Value” performs best; 0 and “C-Function” also have a good performance. It is reasonable to believe that threshold setting in clusters can have a competitive effect on multi-label tasks.

4

Experiments and Analyses

In this article, 4 excellent algorithms are selected as comparisons so as to evaluate the performance of F-ELM-RBF. They are MLKNN [8], original ELM for multilabel classification (threshold is 0) [2], Kernel ELM for multi-label classification [5] and original ELM-RBF [1] with 0 as threshold. We use 3 real-world data sets to assess the performance of different multi-label classification models. They are “Emotions data set” [7], “Yeast data set” [9] and “Scene data set” [10]. 5 evaluation indicators are used to measure the performance of models. They are Hamming Loss, One-Error, Coverage, Ranking Loss and Average Precision. Apart from “Average Precision” which is the higher the metric value is, the better the algorithm behaves, others are the lower the values are, the better the performance will be. We set the parameters as Table 3. For ELM models except K-ELM, clustering centers, also the numbers of hidden nodes K are determined according to different data sets. Q is the new added parameter in F-ELM-RBF. All the experiments are conducted in a PC with an Intel core(TM) i7-4720HQ CPU @ 2.6 GHz processor and 8 GB RAM. Development environment is MATLAB R2016a on 64 bit Windows8 system. Each group of experiments are repeated 5 times. Table 3. Parameter setting Models

Data sets Emotions

MLKNN [8]

N um = 10 Smooth = 1

ELM [2]

K = 60

K-ELM [5]

Regularization coefficient C = 1, RBF Kernel, σ = 1

ELM-RBF [1] K = 60 F-ELM-RBF

Yeast K = 173 K = 173

Scene K = 250 K = 250

K = 60 Q = 10 K = 173 Q = 10 K = 250 Q = 2

The performance of these models is evaluated and analyzed in 2 aspects: behaviors in algorithm indicators and stabilities. Statistical data in different data sets are shown in Tables 4, 5 and 6 respectively. From the perspective of indicators, in emotions data set, except coverage, F-ELM-RBF has a better performance than other models except in coverage. KELM also performs well. In yeast data set, F-ELM-RBF is inferior to ML-KNN, but it is also gain a competitive performance in hamming loss, one-error and ranking loss than other models. K-ELM is better than F-ELM-FBF in coverage in this data set. In scene data sets, F-ELM-RBF shows the best performance.

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Indicators

Multi-label algorithms MLKNN

ELM

K-ELM

ELM-RBF

Average precision 0.7574 ± 0 0.7582 ± 0.00692 0.7706 ± 0 0.7714 ± 0.0106

F-ELM-RBF 0.7768 ± 0.0053

2.0618 ± 0 2.1000 ± 0.00635 2.0560 ± 0 2.0022 ± 0.0461 2.0045 ± 0.0156

Coverage Hamming loss

0.2313 ± 0 0.2391 ± 0.00635 0.2360 ± 0 0.2255 ± 0.0078

0.2215 ± 0.0048

One-error

0.3428 ± 0 0.3461 ± 0.00307 0.3258 ± 0 0.3371 ± 0.0191

0.3169 ± 0.0102

Ranking loss

0.2080 ± 0 0.2089 ± 0.00769 0.1999 ± 0 0.1943 ± 0.0085

0.1935 ± 0.0043

Table 5. Performance of different models on yeast [9] data set Indicators

Multi-label algorithms MLKNN

ELM

K-ELM

ELM-RBF

F-ELM-RBF

Average precision 0.7641 ± 0 0.7323 ± 0.00473 0.7467 ± 0 0.7474 ± 0.00401 0.7513 ± 0.00391 6.3773 ± 0 6.8815 ± 0.04680 6.6423 ± 0 6.6527 ± 0.00473 6.6493 ± 0.04283

Coverage Hamming loss

0.2008 ± 0 0.2158 ± 0.00374 0.2268 ± 0 0.2045 ± 0.00146 0.2044 ± 0.00214

One-error

0.2334 ± 0 0.2774 ± 0.01692 0.2574 ± 0 0.2595 ± 0.00840 0.2458 ± 0.00649

Ranking loss

0.1683 ± 0 0.1991 ± 0.00130 0.1864 ± 0 0.1850 ± 0.00201 0.1843 ± 0.00341

Table 6. Performance of different models on scene [10] data set Indicators

Multi-label algorithms MLKNN

ELM

K-ELM

ELM-RBF

F-ELM-RBF

Average precision 0.8512 ± 0 0.8225 ± 0.00541 0.8482 ± 0 0.8658 ± 0.00445 0.8704 ± 0.00300 Coverage

0.5685 ± 0 0.6679 ± 0.02622 0.5385 ± 0 0.5072 ± 0.01687 0.4895 ± 0.01042

Hamming loss

0.1005 ± 0 0.1146 ± 0.00283 0.0992 ± 0 0.0870 ± 0.00207 0.0858 ± 0.00222

One-error

0.2425 ± 0 0.2845 ± 0.00671 0.2525 ± 0 0.2199 ± 0.00700 0.2140 ± 0.00551

Ranking loss

0.0931 ± 0 0.1116 ± 0.00523 0.0875 ± 0 0.0796 ± 0.00308 0.0764 ± 0.00184

Apart from yeast data sat, the performance of F-ELM-RBF is significantly better than other models in hamming loss. This can also illustrate that our threshold setting method is effective and better than traditional method. From the perspective of stability, ML-KNN and K-ELM perform best and their standard deviations are all equal to 0. This is the main room that should be improved in F-ELM-RBF model.

5

Conclusion

F-ELM-RBF is proposed in this paper. By clustering and limited perception, the classification ability and stability of F-ELM-RBF are enhanced. Statistical experiments are applied to analyze the performance of different clustering methods. In single-label classification task, F-ELM-RBF gains a similar performance to K-ELM. The presented F-ELM-RBF is extended to multi-label tasks. Because of the special structures such as centers of F-ELM-RBF, more targeted threshold

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setting methods can be used in this model. F-ELM-RBF provides a new threshold setting thinking for models like ELM in multi-label tasks. Comparative experiments show that F-ELM-RBF can gain more excellent performance.

References 1. Huang GB, Siew CK (2005) Extreme learning machine: RBF network case. In: Proceedings of the 8th international conference on control, automation, robotics and vision, ICARCV 2004, Kunming, China. IEEE 2. Venkatesan R, Meng JE (2015) Multi-label classification method based on extreme learning machines. In: Proceedings of the 13th international conference on control, automation, robotics and vision, ICARCV 2014, Singapore. IEEE, pp 619–624 3. Luo FF, Guo WZ, Yu YL, Chen GL (2017) A multi-label classification algorithm based on kernel extreme learning machine. Neurocomputing 260:313–320 4. Zhang ML (2009) ML-RBF: RBF neural networks for multi-label learning. Neural Process Lett 29(2):61–74 5. Huang GB, Zhou H, Ding X, Zhang R (2012) Extreme learning machine for regression and multiclass classification. IEEE Trans Syst Man Cybern Part B Cybern 42(2):513–529 6. Blake C, Merz C (1998) UCI repository of machine learning databases. University of California, Irvine. http://www.ics.uci.edu/∼mlearn/MLRepository.html 7. Trohidis K, Tsoumakas G, Kalliris G, Vlahavas I (2011) Multi-label classification of music into emotions. EURASIP J Audio Speech Music Process 2011, Article ID 4 8. Zhang ML, Zhou ZH (2007) ML-KNN: a lazy learning approach to multi-label learning. Pattern Recogn 40(7):2038–2048 9. Elisseeff A, Weston J (2001) A kernel method for multi-labelled classification. In: Proceedings of the 14th international conference on neural information processing systems: natural and synthetic, NIPS 2001, Vancouver, Canada. MIT Press 10. Boutell MR, Luo J, Shen XP, Brown CM (2004) Learning multi-label scene classification. Pattern Recogn 37(9):1757–1771

An Iterative Learning Scheme-Based Fault Estimator Design for Nonlinear Systems with Quantised Measurements Xiaoyu Liu1,2 , Shanbi Wei1,2(B) , and Yi Chai1,2 1

2

College of Automation, Chongqing University, Chongqing 400044, China [email protected] State Key Laboratory of Power Transmission Equipment and System Security and New Technology, Chongqing University, Chongqing 400044, China

Abstract. This paper deals with fault estimation problem for a class of nonlinear system with quantised measurements. In this paper, a logarithmic quantiser is introduced and an iterative learning observer scheme is constructed, meanwhile the number of quantisation levels of output signals are finite. Compared with the existing approaches of observerbased fault estimation, the proposed iterative learning observer in this paper considers both state error and fault estimation which generated by previous iteration and use them to improve the fault estimation performance in the current iteration. Simultaneously, Lyapunov stability theory is employed to achieve the stability and convergence of the designed observer. Furthermore, the extension from nominal system to system with parameter uncertainties subjecting to Bernoulli-distributed white sequences with known conditional probabilities is also addressed. Finally, an illustrative example are presented to demonstrate the theoretical results. Keywords: Fault estimation · Iterative learning scheme Nonlinear uncertain system · Linear matrix inequality

1

·

Introduction

The ideal situation of system data measurement and transmission is infinite precision. While, in practical situations, the ideal situation may not be implementable, considering the presence of signal quantisation or capacity-limited feedback, such as digital computers with A/D and D/A converters, network medium with limited capacities. The presence of these factors may cause low resolution of the transmitted data and large quantisation errors. It is, therefore, the main purpose of this paper is to consider the fault estimation problem for a class of nonlinear quantitative measurement systems [1–3].

c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 446–456, 2020. https://doi.org/10.1007/978-981-32-9682-4_47

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The main contributions as follows: (1) An iterative learning scheme-based fault estimator design for nonlinear systems with quantised measurements is proposed, in which a lmi conditions to constraints in spite of satisfy measurement signal with a number of quantisation levels. (2) The proposed method considers the discrete-time system and inherits the advantages of extension to network systems with networked delay, data loss, multi-sensors, etc. Nonlinear system with quantised measurements and the problem formulation is introduced in Sect. 2. In Sect. 3, iterative learning scheme-based fault estimation is proposed to achieve desired fault estimation results. In Section 4, convergence analysis is used to solve the problem. Then, iterative learning observer is realized by an optimal problem and is extended to general case with randomly occurring parameter uncertainties in Sect. 5. Simulation results is provided to illustrating the effectiveness of the proposed method in Sect. 6, and in Sect. 7 concluding remarks are given.

2

Problem Statement and Preliminaries

Consider the following nonlinear system: x (t + 1) = Ax (t) + Bu (t) + Bg g (x (t) , t) + Bf f (t) y (t) = Cx (t)

(1)

where t ∈ [0, T ], the state vector x (t) ∈ n , the output vector y (t) ∈ p , u (t) ∈ m is the input vector, f (t) ∈ q stands for the fault signal. g (x (t) , t) ∈ r is nonlinear term. A ∈ n×n , B ∈ n×r , Bg ∈ n×r , Bf ∈ n×q and C ∈ p×n , n > p ≥ q. Fault function f (t) ∈ q could represent various types of faults. If B = Bf , f (t) ∈ q represents actuator faults, otherwise f (t) ∈ q represents sensor faults or process faults. The system (1) is a repetitive system with a running cycle T , which has periodic characteristics, xk (t) = x (t), yk (t) = y (t), k ∈ Z+ . Each cycle is regarded as a repetitive operation cycle. f (t) ∈ q is considered as intermittent fault signal, f (t) = f (t + T ). Definition 1. The quantizers is modeled by qy = q(y(t))

(2)

where q(y(t)) = [y1 (t) · · · yn (t)]. q(·) is logarithmic quantizers, and satisfies ⎧ 1 1 ⎨ us , 1+θ us < v < 1−θ us q(v) = (3) 0, v=0 ⎩ −q(−v), v 0,

(4)

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By using the sector bound approach, q(y(t)) can be written as q(y(t)) = (I + q (t))y(t)

(5)

where q (t) = diag[1q (t), 2q (t), · · · , nq (t)], and |iq (t)| < θ, i ∈ 1, 2, · · · , n. If iq (t) = 0, the system output is not affected by data quantization. Assumption 1. The pairs (A, B) and (A, C) are stabilizable and detectable, respectively. Assumption 2. Initial state value is xk (0) = x (0). Assumption 3. The nonlinear term g (x (t) , t) satisfies the Lipschitz conditions if there exist a positive parameter δ. ||g (x1 (t) , t) − g (x2 (t) , t) || ≤ δ||x1 (t) − x2 (t) ||, ∀x1 (t) , x2 (t) ∈ n

(6)

When the set S = n is globally Lipschitz, g (0, t) = 0. Time delay, noise, unknown input, model uncertainties and sensor faults are all exist in reality, the complex environment and cumbersome process makes this factors come into the system inadvertently. Analyze the impact of this factors which called randomly occurring parameter uncertainties is addressed in this paper.

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Iterative Learning Observer Design

The observer-based fault estimator is constructed as follow: xk (t) + Bu (t) + Bg g (ˆ xk (t) , t) + Bf fˆk (t) + L [q(y (t)) − yˆk (t)] x ˆk (t + 1) = Aˆ yˆk (t) = C x ˆk (t) (7) In Eq. (7), x ˆk (t) and yˆk (t) is the state estimation and output estimation. The parameter matrix L stands for the gain of state observer. fˆk (t) represents the estimation of fault signal f (t). xk (t), t), ek (t) = xk (t) − Meanwhile, defining that Δgk (t) = g(xk (t), t) − g(ˆ ˆ x ˆk (t), rk (t) = fk (t) − fk (t). The dynamic error system is defined as follow. Δek (t) = ek (t + 1) − ek (t) = (A − I − LC)ek + Bf rk (t) + Bg Δgk (t) − Lq (t)Cxk (t)

(8)

Δyk (t) = yk (t) − yˆk (t) Then, the fault estimation law: fˆk+1 (t) = fˆk (t) + K1 ek (t) + K2 Δek (t)

(9)

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In which, K1 and K2 represents gain matrices. The error of fault estimation is given as follow. rk+1 (t) = f (t) − fˆk+1 (t) = rk (t) − K1 ek (t) − K2 Δek (t)

(10)

= M1 ek (t) + M2 rk (t) + M3 Δgk (t) + M4 x(t) In Eq. (10), the matrices are defined as M1 = −[K1 + K2 (A − I − LC)], M2 = (I − K2 Bf ), M3 = −K2 Bg and M4 = K2 LCq (t).

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Convergence Analysis

Assuming that the initial state has been accurately reset. A novel optimal function is proposed. Theorem 1. Consider nonlinear system (1) as well as Assumptions 1, 2 and 3 hold. The sufficient condition of the error dynamic system (8), which satisfy asymptotically stable and the fault estimation error convergence, is that scalar ϕ ∈ [0, 1] , λ > 0, ς1 > 0, ς2 > 0, εi > 0, i = 1, 2, and positive-definite matrices P = P T , Q = QT exists, also the symmetric definite matrix Π satisfies ⎡ ⎤ −I 0 Π13 Π14 −K2 Bg 0 0 0 K2 0 ¯ 0 ⎢ ∗ −P Π23 P Bf P Bg 0 0 −L 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ Π33 0 0 0 0 0 0 0 ⎥ ⎢ ⎥ ¯T ⎥ ⎢ ∗ ∗ ∗ −ϕ 0 0 0 0 0 CT L ⎢ ⎥ ⎢ ∗ ∗ ∗ ∗ −λ 0 √ 0 0 0 0 ⎥ ⎢ ⎥ 0

(12)

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From (8) and (12), the derivative of V (t) with respect to time can be calculated as ΔVk (t) = eTk (t) [(A − LC)T P (A − LC) − P ]ek (t) + rkT (t) BfT P Bf rk (t) + ΔgkT (t)BgT P Bg Δgk (t) + xT (t)[(LCΔq (k))T P LCΔq (t) + AT QA − Q]x(t) + 2rkT (t) BfT P (A − LC)ek (t) + 2ΔgkT (t)BgT P (A − LC)ek (t) + 2xT (t)(LCΔq (t))T P (A − L1 C)ek (t) + 2ΔgkT (t)BgT P1 Bf rk (t) + 2xT (t)(LCΔq (t))T P Bf rk (t) + 2xT (t)(LCΔq (t))T P Bg Δgk (t) + 2uTk (t)B T QAx (t) + 2ftT (t) BfT QAx (t) + 2g T (x (t) , t) BgT QAx (t)

 

 

  Θ1

Θ2

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+ 2f T (t) BfT QBu (t) + 2g T (x (t) , t) BgT QBu (t) + 2g T (x (t) , t) BgT QBf f (t)

 

 

  Θ4

Θ5

Θ6

+ uT (t)B T QBu (t) + f T (t) BfT QBf f (t) + g T (x (t) , t) BgT QBg g (x (t) , t)

 

 

  Θ7

Θ8

Θ9

(13) (14) ΔV (t) ≤ ξkT (t)Π1 ξ(t) + δ1 + δ2 + δ3 + δ4 + δ5 + δ6 + δ7 + δ8 T  ek (t) rk (t) g (x (t) , t) x (t) , δ1 = u ¯2 λ1max (B T QQB), where ξk (t) = 2 T 2 T ¯2 λ4max (B T B), δ2 = f¯ λ2max (Bf QQBf ), δ3 = f¯ λ3max (Bf QQBf ), δ4 = u ¯2 λ5max (B T B), δ6 = f¯2 λ6max (BfT Bf ), δ7 = u ¯2 λ7max (B T QB), δ8 = δ5 = u 2 T ¯ ¯ ¯ = |u(t)|∞ , f = |f (t)|∞ . Based on Lyapunov stability thef λ2 max (Bf QBf ), u ory, the error dynamic system is stable and the designed observer is converged if the inequations V (t) > 0 and V (t) < 0 hold. The inequation V (t) < 0 holds 5 +δ6 +δ7 +δ8 is true and Π1 < 0. only if the inequation ||ξkT (t)||2 > δ1 +δ2 +δ3 +δ4 +δ φ A performance index is introduced, so that the convergence and H∞ robustness performance is attainable, γ ∈ [0, 1] at k ∈ Z+ . J1 =

Td 

T [rk+1 (t)rk+1 (t) − γ 2 rkT (t)rk (t)] ≤ 0

(15)

t=0

There exists a positive scalar λ ∈ [0, 1] by using Assumption 3, which satisfies J2 =

Td    λδeTk (t) ek (t) − λΔgkT (t) Δgk (t) ≥ 0

(16)

t=0

The optimal function J1 is rewrote as follow. J1 < J1 +J2    = J1 + J2 + ΔV (t) − [V (Td ) − V (0)]  T d  ξkT (t)Π2 ξk (t)dt − [V (τ ) − V (0)] ≤ 0 = t=0

(17)

Fault Estimation

¯ If the following inequalities holds ¯ = P L, then L = P −1 L. Denoting L ⎡ ⎤ −I 0 −K1 − K2 (A − I) I − K2 Bf −K2 Bg 0 0 ⎢ ∗ −P ⎥ 0 0 0 P A − LC P Bf ⎢ ⎥ ⎢ ∗ ∗ −P + C T C + λ ⎥ 0 0 0 0 ⎢ ⎥ 2 ⎢ ⎥ 0 0 0 ∗ −γ I Π1 = ⎢ ∗ ∗ ⎥ ⎢ ∗ ∗ ⎥ ∗ ∗ −λ 0 √ 0 ⎢ ⎥ T ⎣ ∗ ∗ ∗ ∗ ∗ −ε1 3δBg Q ⎦ ∗ ∗ ∗ ∗ ∗ ∗ Π2 77

  Π2 ⎡ ⎤ ¯ 0 0 0 K2 P −1 LΔ ¯ q (k)C 0 0 K2 P −1 LC ¯ ⎢∗ 0 0 0 0 0 −LΔq (k)C ⎥ ⎢ ⎥ ⎢∗ ∗ ⎥ 0 000 0 ⎢ ⎥ ⎢ ⎥ 0 and ε3 > 0, then the inequality (18) holds, such that Π4 = ⎡ ⎤ −I 0 Π4 13 Π4 14 −K2 Bg 0 0 0 K2 0 ¯ ⎢ ∗ −P Π4 23 P Bf P Bg 0 0 −L 0 0 ⎥ ⎢ ⎥ T ¯T ⎥ ⎢ ∗ ∗ Π4 33 0 L ⎥ 0 0 0 0 0 C ⎢ ⎢ ∗ ∗ ∗ −γ 2 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ ∗ −λ 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ 0 0 ⎥ ∗ ∗ ∗ −ε1 Π4 67 0 ⎢ ⎥ ⎢ ∗ ∗ 0 0 ⎥ ∗ ∗ ∗ ∗ Π4 77 0 ⎢ ⎥ ¯T ⎥ ⎢ ∗ ∗ 0 L ∗ ∗ ∗ ∗ ∗ −ε2 I ⎢ ⎥ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −ε−1 0 ⎦ 3 P ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −ε3 P Π4 13 = −K1 − K2 (A − I) Π4 14 = I − K2 Bf Π4 23 = P A − LC −P + C T C + λδ Π4 33 = √ Π4 67 = 3δBgT Q Π4 77 = AT (Q + 4ε1 )A − Q + δ 2 Bg T QBg + ε2 θ2 C T C

(19)

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Considering the nonlinear terms ε−1 3 P and ε3 P of equality (19), the following constraint and approximation is used. P > ς1 I, ε3 +

1 ≥2 ε3

(20)

then a new LMI is constructed of α and β=αε3 . Moreover, from the Eq. (11), there exists ε13 P ≥ 2P − ε3 P = (2ς1 − ς2 )I and ε3 P ≥ ε3 ς1 I = ς2 I. This completes the proof.

5

Iterative Learning Observer Realization and Extension

Considering the parameter uncertainties in reality, which come into the system inadvertently, the following nonlinear uncertain system is introduced: x (t + 1) = (A + α (t) ΔA)x (t) + Bu (t) + Bf f (t) + Bg g (x (t) , t) y (t) = (C + β (t) ΔC)x (t)

(21)

α (t) ΔA (t) represents the uncertain state parameter matrix and β (t) ΔC(t) denotes output parameter matrix. Then, α (t) and β (t) is defined to represents the parametric variation of random property. For system (21), the following definition and assumptions are available. The corresponding observer-based fault estimator is proposed as follow: x (t) , t) + L(q(y (t)) − yˆ (t)) x ˆ (t + 1) = Aˆ x (t) + Bu (t) + Bf fˆ (t) + Bg g (ˆ (22) yˆ (t) = C x ˆ (t) Assumption 4. The matrix ΔA (t) and ΔC (t) represents the uncertainty of norm-bounded parameters of the following structure. ΔA (t) = G1 F1 (t) N1 , ΔC (t) = G2 F2 (t) N2

(23)

where Gi and Ni is sufficiently dimensional known matrices, and, matrix Fi (t) is unknown but satisfy the conditions that Fi (t) FiT (t) ≤ I, i = 1, 2. α (t) and β (t) is white sequences of Bernoulli distribution, which values are either zero or one. P rob {α (t) = 1} = υ1 , P rob {α (t) = 0} = 1 − υ1 (24) P rob {β (t) = 1} = υ2 , P rob {β (t) = 0} = 1 − υ2 In which, ν1 ∈ [0, 1] and ν2 ∈ [0, 1], and α (t), β (t) are independent of each other. Then (8), the dynamic error system is rewrote as follow: Δek (t) = ek (t + 1) − ek (t) = (A − I − LC)ek (t) + Bf rk (t) + Bg Δg (x (t) , t) + L[α(t)ΔA − Δq (t)C − β(t)ΔC − Δq (t)ΔC]xk (k) Δyk (t) = yk (t) − yˆk (t)

(25)

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Corollary 1. If there exists symmetric positive definite matrixes P = P T > 0, Q = QT > 0, scalars ς1 > 0, ς2 > 0, ϕ ∈ [0, 1] , εi > 0, i = 1, 2, Theorem 1 can satisfy the optimal functions (26). M in {ϕ} , s.t.Π < 0

(26)

Π ⎡ = ⎤ −I 0 Π13 Π14 Π15 0 0 0 0 0 0 K2 0 ¯ Π210 LG ¯ 2 ⎢ ∗ −P Π23 P Bf 0 0 0 Π28 L 0 0 ⎥ ⎢ ⎥ T ¯T ⎥ ⎢ ∗ ∗ Π33 0 L ⎥ 0 0 0 0 0 0 0 0 C ⎢ ⎢ ∗ ∗ ∗ −γ 2 I 0 0 0 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ ⎥ ∗ −λ 0 0 0 0 0 0 0 0 ⎢ ⎥ ⎢ ∗ ∗ ∗ ⎥ Π 0 0 0 0 0 0 ∗ ∗ −ε 1 67 ⎢ ⎥ ⎢ ∗ ∗ ∗ ⎥ 0 0 0 0 0 0 ∗ ∗ ∗ Π 77 ⎢ ⎥ ⎢ ∗ ∗ ∗ ⎥ I 0 0 0 0 Π ∗ ∗ ∗ ∗ −σ 1 813 ⎥ ⎢ T ⎥ ¯ ⎢ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −σ I 0 0 0 L 2 ⎢ ⎥ ⎢ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −σ3 I 0 0 Π1013 ⎥ ⎢ ⎥ ¯ ⎥ ⎢ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −σ4 I 0 GT2 L ⎢ ⎥ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Π1212 0 ⎦ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −ζ2 I ¯ 1 G1 , Π210 = Lυ ¯ 2 G2 , Π13 = −K1 − K2 (A − I), Π23 = P A − LC, Π28 = Lυ √ Π33 = P + C T C + λ, Π15 = −K2 Bg , Π14 = I − K2 Bf , Π67 = 3δBgT Q Π77 = AT (Q + 3ε1 )A + δ 2 Bg T QBg − Q + σ1 N1T N1 + σ2 θ2 C T C + σ3 N2T N2 + σ4 θ2 N2T N2 ¯ Π1013 = υ2 GT L ¯ T , Π1212 = −(2ζ1 − ζ2 )I Π813 = υ1 GT1 L, 2 (27) As the discussion above, the proposed fault estimator design (21) can employs Corollary 1.

6

Illustrative Example

A numerical example is used in this section, in order to demonstrate the effectiveness and validity. Considering the nonlinear systems (21), then the parameter uncertainties randomly occurring according to (23). xk (t) = [x1,k (t), x2,k (t), x3,k (t), x4,k (t)]T . The sampling period is T = 0.1. ⎡ ⎡ ⎤ ⎤ 1 0.1 0 0 0 ⎢ 0 ⎢ 0.0247 ⎥ 1 0.1 0 ⎥ ⎢ ⎥ ⎥ A=⎢ ⎣ −2.16 −1.36 0.58 1.6 ⎦ B = ⎣ ⎦ 0 −0.1 0 0 1 −0.0476

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⎡

⎡ ⎤ ⎤ 0 0.1 ⎢ 0.05 ⎥ ⎢ 0.1 ⎥ ⎢ ⎥ ⎥ Bf = ⎢ ⎣ 0 ⎦ Bg = ⎣ 0.1 ⎦ , g [xk (t) , t] = sin (x1,k (t)) . −0.05 0.1 ⎧ 0.5 sin (2πt) , t ∈ [0, 1s) ⎪ ⎪ ⎪ ⎪ ⎨ sin (2πt) , t ∈ [1s, 2s) f (t) = 1.5 sin (2πt) , t ∈ [2s, 3s) ⎪ ⎪ 2 sin (2πt) , t ∈ [3s, 4s) ⎪ ⎪ ⎩ 2.5 sin (2πt) , t ∈ [4s, 5s) The matrices ΔA (t) and ΔC (t) are given as   T 10 M1 = 0 0.1 0 0.1 , M2 = , 01     0.2 0.1 0.1 0.2 N1 = 0.02 0.01 0.01 0.01 , N2 = . 0.1 0.2 0.2 0.1 And P rob {α (t) = 1} = 0.6, P rob {α (t) = 0} = 0.4, t ∈ [0, Td ], P rob {β (t) = 1} = 0.9, P rob {β (t) = 0} = 0.1, t ∈ [0, Td ]. In which, α (t) = 1 or β (t) = 1 stands for the occupation of fault and α (t) = 0 or β (t) = 0 means no fault. Considering that F1 (t) = 0.5 sin (2πt) and F2 (t) = cos (πt).  T  T Then, xd (0) = 0 0 0 0 and uk (t) = 1 1 1 1 . The effectiveness is demonstrating by addressing the randomly occurring uncertainties (23) and the probability distribution (24). Introducing 28 to further evaluate the performance of fault estimation in different iterations.     Ek = sup f (t) − fˆk (t) (28) t∈[0,Td ]

We do simulation with ρ = 0.7. The fault estimation results are shown as the following two figures, Figs. 1 and 2. In Fig. 1, the fault estimation results and actual fault signals with time-varying fault is showed. With the increase iterations, the fault estimation results and actual fault are more close to each other. As the proof above shows, the performance of proposed fault estimation observer and algorithm is excellent, therefore it can be directly employed in the estimation of the actual fault. The maximum absolute error decreases with iterations increase and converges to zero, the variation trend is exhibited in Fig. 2. The satisfactory estimation performance has been achieved as Fig. 2 shows. Table 1. The tracking trajectory of abrupt fault for nonlinear system with different quantization density ρ

0.7

θ

0.1765 0.1111 0.0811 0.0526 0.0256 0.005

Ek 0.046

0.8 0.039

0.85 0.036

0.9 0.026

0.95 0.023

0.99 0.009

Fault Estimation Actual fault value Fault estimating value

14 12

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Fig. 2. The tracking trajectory with time-varying parameter uncertainties

Table 1 shows the performance of the maximal tracking error with different quantization density of quantizer is presented in Table 1. It shows that the maximal tracking error decreases with quantization density from 0.046 to 0.009. It is obvious that tracking error is determined by quantisation level. It should be noted that the tracking residual error originates from measuring link and cannot be eliminated by the iterative learning scheme with the integral calculus, completely.

7

Conclusion

In this paper, we have investigated the observer-based iterative learning method for fault estimation of nonlinear systems, where the data quantization are coex-

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isting in measurement. For analysis of iterative leaning scheme, considering the effects of data quantization is more practical. Furthermore, a new scheme has been proposed to describe the parametric uncertainty of random occurrence and data quantization in a unified framework. The problem of observer-based iterative learning method for fault estimation has been transformed into an optimal problem. Acknowledgment. This work was funded by the National Natural Science Foundation of China (61374135, U1637107). All data generated or analysed during this study are included in this published article.

References 1. Tabatabaeipour SM, Bak T (2014) Robust observer-based fault estimation and accommodation of discrete-time piecewise linear systems. J Franklin Inst 351(1):277–295 2. Hu Z, Zhao G, Zhang L, Zhou D (2016) Fault estimation for nonlinear dynamic system based on the second-order sliding mode observer. Circ Syst Signal Process 35(1):101–115 3. Jun H, Shanbi W, Yi C (2018) An iterative learning scheme-based fault estimator design for nonlinearsystems with randomly occurring parameter uncertainties. Complexity 1–12. https://doi.org/10.1155/2018/7280182

Smooth Trajectory Generation for Linear Paths with Optimal Polygonal Approximation Kai Zhao1,2 , Shurong Li3(B) , and Zhongjian Kang1 1

3

College of Information and Control Engineering, China University of Petroleum (East China), Qingdao 266580, China 2 College of Computer Science and Information Engineering, Anyang Institute of Technology, Anyang 455000, China School of Automation, Beijing University of Posts and Telecommunications, Beijing 100876, China [email protected]

Abstract. Previous methods cannot give consideration to both the efficient fitting and overlapping phenomenon of sensitive corners known as “ripple effect”. In this paper, a NURBS interpolation scheme is developed, which consists of two stages. A NURBS curve is fitted efficiently through optimal polygonal approximation based on Mixed Integer Programming (MIP) in stage one. The S-type feedrate profile is planned according to proposed overlap-free method integrating kinematic and chord error limitations (OFIKC). The presented scheme can obtain a NURBS curve by fitting G01 short lines, which is very useful for achieving higher processing speed, and ensure that the acceleration, the jerk and the chord error are within allowable range. The simulation results show the feasibility and effectiveness of proposed scheme for the continuous short line tool paths. Keywords: Curve fitting · MIP · Feedrate planning · Polygonal approximation

1 Introduction Many machining tools and CAD/CAM/CNC systems can only support line and circular interpolations, which may lead to a larger number of short lines to meet high accuracy requirements. And what’s more, numerous tiny lines inevitably affect the efficiency of processing as a result of frequent acceleration and deceleration. One method to solve the problem above is to convert short lines into parametric curves by approximation or interpolation approaches. The cubic spline curve is used to fit the multiple short lines, considering continuities of the slope and curvature at each junction [1]. A path smoothing method based on dual NURBS path for 5-Axis machine tools is presented, which fits the linear trajectories of tool tip and tool orientation with two spline curves respectively [2]. Transforming short lines into a parametric curve, satisfactory performances are yielded because the fitted curve is not less than G2 continuity [3]. During scheduling feedrate, the constraints such as continuous jerk and chord error can be added to ensure c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 457–465, 2020. https://doi.org/10.1007/978-981-32-9682-4_48

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machining quality [4]. But in previous methods, overlapping situations known as “ripple effect” are commonly neglected. In this paper, a two-stage interpolation scheme is proposed. In the first stage, a NURBS fitting method is adopted, which can determine the control points with the optimal polygonal approximation. In the second stage, to eliminate “ripple effect”, an overlap-free method integrating kinematic and chord error limitations (OFIKC) is presented. The rest of the paper is organized as follows. Section 2 simply introduces the optimal polygonal approximation. Section 3 proposes the curve fitting method. The OFIKC method is elaborated in detail in Sect. 4. A NC file, is processed with the proposed scheme to demonstrate the effectiveness and practicability in Sect. 5. Finally, conclusions are given in Sect. 6.

2 Optimal Polygonal Approximation Based on MIP The problem for polygonal approximation of a digital planar curve can be solved as an optimization problem with Dynamic Programming (DP) method, whereas the DP method becomes more computationally expensive with the discrete points increasing. To overcome this shortcoming, a novel approach is proposed, depending on Mixed Integer Programming (MIP) technique [5]. For a digital planar curve, the optimal polygonal approximation aims to redraw the curve with the least discrete points. MIP-based model defines a decision matrix U with (N + 1) × (N + 1) elements, where N + 1 denotes number of discrete points in the pattern. Each element ui j represents a decision variable, and if ui j = 1, the line segment between discrete points i and j is selected as a part of the approximation pattern, otherwise is not. Meanwhile, a widely used distortion measure, namely Integer Square Error (ISE), is introduced to indicate deformation degree of the approximation pattern. Assuming segment Li j is used to approximate curve Ci j shown in Fig. 1, the distortion measure is written as j

Γ (i, j) = ∑ t(ck , ci c j )2

(1)

k=i

For closed curves, Γi j is redefined as ⎧ j ⎪ 2 ⎪ i< j ⎨ ∑ t(ck , ci c j ) Γi j = k=i j N ⎪ ⎪ ⎩ ∑ t(ck , ci c j )2 + ∑ t(ck , ci c j )2 k=0

k=i

(2) i> j

In order to balance the points number and distortion measure, the objective function can be defined as N

N

f = min ∑ ∑ Γi j ui j

(3)

i=0 j=0 i= j

Actually, the cost function is equivalent to the ISE of the approximation curve defined by decision matrix U, noting that decision variables on the principal diagonal are not involved, because uii only denotes a points not a segment.

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Fig. 1. Distortion of segment Li j .

Three linear constraints are added to ensure that the optimization problem has a unique optimal solution. These constraints are described by Eq. 4, which fix K + 1 points. The optimal approximation polygon is obtained by connecting these points in counterclockwise direction with segments. ⎧ N N ⎪ ⎪ K + 1 = ⎪ ∑ ∑ ui j ⎪ ⎪ ⎪ i=0 j=0 ⎪ ⎪ ⎪ ⎨ N N uir = ∑ uli , ∀i ∈ [0, N] (4) ∑ ⎪ ⎪ r=0 l=0 ⎪ ⎪ ⎪ N i ⎪ ⎪ ⎪ ⎪ ⎩ ∑ ∑ ui j = 1 i=0 j=0

3 Curve Fitting with Optimal Polygonal Approximation Although the discrete points are efficiently reduced by optimal polygonal approximation, the planer curve is still composed of segments. So the machining is still timeconsuming. Curve fitting is employed to overcome the drawback. One of widely used parametric curves is the NURBS curve expressed as

α (s) =

Br,p (s)wr Tr r=0 Br,p (s)wr K

∑

(5)

The NURBS curve α (s) is determined by basis function Br,p (s), weight coefficient wr and control point Tr . To avoid introducing nonlinearity, wr is set to 1 by default and then NURBS curve degenerates to B-spline curve. Moreover, 3-order B-spline is commonly used in CNC fitting. As a result, once the optimal control point sequence T is found, the optimal NURBS curve α (s) can be achieved. And the problem of selecting optimal control point sequence is transformed into searching optimal interpolated points, which is generalized as two steps. Step one initially selects a few points as seed points, and step two adds new points gradually until the optimization goal is achieved. Considering a digital planer curve consisting of N + 1 points {Q0 , Q1 , · · · , QN }, the objective function can be written as    N  K   (6) min ∑ Qi − ∑ Br,p (s)Tr  K,Tr   i=0 r=0

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Control points are calculated with the following equation, ⎤⎡ ⎡ T0 1 0 ··· 0 ⎥ ⎢ T1 ⎢ B0,p (s1 ) B1,p (s1 ) · · · 0 ⎥⎢ ⎢ ⎥ ⎢ .. ⎢ .. ⎥⎢ . ⎢ . ⎥⎢ ⎢ ⎣ 0 ··· BK−1,p (sK−1 ) BK,p (sK−1 ) ⎦ ⎣ TK−1 0 ··· TK 0 1

⎤

⎡

Q0 Q1 .. .

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎦ ⎣ QK−1 QK

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(7)

where QK = [Q0 , Q1 , · · · , QK ]T are the vertexes of optimal approximate polygon gained with MIP. These vertexes are parameterized with chord length method which is found to be quite adequate for engineering applications [6]. And the corresponding knots are computed with the average technique. If the objective function can not be minimized by current points, new points should be selected to add into QK. The whole process of curve fitting is illustrated as Algorithm 1. Algorithm 1. Curve fitting algorithm with optimal polygonal approximation Input: G01 points set {Qi ∈ R2 : i = 0, · · · , N}, tolerance ε Output: A 3-order B-spline curve α (s) with the fewest control points 1: Set le f t = 2, right = N + 1, p = 3, Mid = (le f t + right)/2. 2: Search the optimal approximate polygon consisting of Mid points based on the MIP. 0 0 0 3: Get the  fitted curve α (s)  = B (s)T according to Eq. 5 and Eq. 7. N  K  4: if ∑ Qi − ∑ Br,p (s)Tr  < ε i=0 r=0 5: right = Mid and let α (s) = α 0 (s) 6: else 7: le f t = Mid 8: end 9: check if right − le f t > 1. If the condition is satisfied, goto step 2. Otherwise, return the curve α (s).

4 Feedrate Planning In this section, OFIKC algorithm is presented, which can eliminate the “ripple effect”. And the influence of accelerations, jerks and the chord error are simultaneously considered. 4.1

Adaptive Feedrate Adjustment

One widely used NURBS interpolation is the first-order approximation interpolation technique which is written as F(si ) · Tip

si+1 = si +

d α (s)

ds

s=si

(8)

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where Tip is interpolation period. F(si ) can be adjusted according to the chord error to keep accuracy. One adaptive feedrate adjustment law [7] is written as follows,  ⎧ ⎨ Fc , 2 ρi 2 − (ρi − δ )2 > Fc Tip  (9) Fa = ⎩ 2 ρ 2 − (ρ − δ )2 , others i i Tip where Fa denotes the adaptive feedrate with confined error, Fc the command feedrate, ρi the radius of curvature, and δ the allowable chord error. 4.2 Feedrate Smoothing

Jm

Jerk

Jerk

V2 V1

Acceleration

Am

Feedrate

V2 V1

Acceleration

Feedrate

If the feedrate is adaptively adjusted according to Eq. 9, the lowest speed appears where the radius of curvature is the smallest. This adjustment law actually finds sensitive corners where the speed needs to be reduced in the geometric path. The acceleration and the jerk may be beyond the ability of the machine tool when cutter enters or leaves a sensitive corner. So the feedrate needs to be re-planed when the cutter encounters a sensitive corner. The bell-shape profile are commonly used to smooth the feedrate. For simplicity and flexibly adapting different command speeds, two structures of Acceleration/Deceleration are employed. Take the acceleration process for example shown in Fig. 2, it includes either three sub-phases: increasing acceleration phase, constant acceleration phase and decreasing acceleration phase or two sub-phases: increasing acceleration phase and decreasing acceleration phase. Let Am denote the maximal acceleration, and Jm the maximal jerk.

A'm

Jm -Jm

-Jm Time

Time

(a)

(b)

Fig. 2. Two bell shape feedrate profiles: (a) Type 1; (b) Type 2.

4.3 Feedrate Scheduling Scheme: OFIKC Generally, the larger command feedrate is given, the “ripple effect” shown in Fig. 3 more easily appears when confined chord error is used to change the feedrate adaptively. We divide the sensitive corners into three types according to the severity of “ripple effect”: the severe overlap (such as corner 1, 2 and 3), the single corner (such as corner 4), the slight overlap (such as corner 5 and 6). When the cutter encounters the

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#1 #2 #3

single corner #4

#5

Feedrate

Smin(5)

severe overlapping corners

#6

slight overlapping corners

C(4,5)

Time

Fig. 3. The feedrate profile planed with OFIKC (solid line).

corners of severe overlap, it cannot finish Acc/Dec process according to Fig. 2, because the distance between the two adjacent corners is so short. The single corner is not influenced by the adjacent corners at all. The slight overlap generally consists of two sensitive corners, and the distance between the two corners is long enough for the cutter to accelerate or slow down from one corner to another. But the cutter cannot reach the command feedrate. The presented OFIKC searches the sensitive corners, distinguishes the types of “ripple effect” and re-plans the feedrate near them. The method is elaborated in detail as follows, 1. Calculate si+1 according to Eq. 8 with the feedrate equal to the command feedrate, that is, Vi = Fc . 2. Check if the chord error at si is larger than the maximal tolerance δ . If it is true, adjust the feedrate Vi according to Eq. 9 and recalculate si+1 until the feedrate satisfies allowable value. Otherwise, jump step 1 and continue to get next interpolation point. 3. Continue two steps above until the entire curve is traversed. Then store the feedrate at each interpolation point into an array V. 4. Detect the sensitive corners indicated by serial numbers like that in Fig. 3. – Assuming there are P+1 sensitive corners, calculate the arc length between each two adjacent sensitive corners with adaptive Simpson’s method and obtain arc length sequence C = [C(0, 1), · · · ,C( j, j + 1), · · · ,C(P − 1, P)]. – Obtain the minimum feedrate V j and corresponding parameter value s¯ j at each corner, where j denotes the jth sensitive corner. 5. Detect the severe overlap. Calculate the minimum distance Samin ( j, j + 1) which is needed for the cutter accelerating or slowing down from the sensitive corner j to the j + 1, according to following formula, Samin ( j, j + 1) = ⎧ ⎨ V j+1 +V j ( Am + |V j+1 −V j | ), |V j+1 −V j | > 2 Jm  Am ⎩ (V j+1 +V j ) |V j+1 −V j | , otherwise Jm

A2m Jm

(10)

Test if the condition Samin ( j, j + 1) < C( j, j + 1) is satisfied. If the condition at some corners are not satisfied, such as corner 1, 2 and 3 in Fig. 3, combine these adjacent sensitive corners by setting the feedrate as the lowest speed of these corners. After all

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the sensitive corners are checked, the new minimum feedrate V j  and corresponding parameter value s¯j are obtained again, and the number of sensitive corners is reduced to P + 1. Otherwise, let P = P and go to next step. 6. Compute the minimum distance which is required to increase feedrate from V j  to Fc or decrease from Fc to V j  according to the bell shape profile illustrated in Subsect. 4.2. Store these minimum distances in an array Smin = [Smin (0), · · · , Smin ( j), · · · , Smin (P + 1)], which are computed according to the method in literature [8]. 7. Test if there is any slight overlapping situation between two arbitrary and adjacent corners. Criteria of judgment is easily understood by following description, – Recalculate the arc length C ( j, j + 1) between parameters s¯j and s¯j+1 .  ( j) + S ( j + 1) ≤ C (i, j), the two sensitive corners has slight overlap. – If Smin min Let Vlow take the larger one in V j  and V j+1  , and search the new optimal feedrate Vc between Vlow and Fc with a bisection searching method. Then Vc is used to replace Fc as the command feedrate, which will not lead to overlap. otherwise, jump next step. 8. Re-plan the feedrate profile of each new sensitive corner according to Fig. 2.

5 A Numerical Example In Fig. 4, the car shape pattern with 242 tiny segments is read from a NC file generated by the CAM system. The length of each line segment is between 0.0545 mm and 2.3447 mm. The whole length of original path is 204.0523 mm. This G01 contour is fitted by method of feature points section (FPS) [9] and Algorithm 1. The length of fitted path with two methods is 204.0697 mm and 204.0584 mm respectively. Figure 4a and b shows the fitting results of two methods with different control points. The final fitted curves are shown in Fig. 4c, which are determined by 83 control points. From these results, we can see that better performance is always got by Algorithm 1. During machining, the parameters for simulation are set as following: the interpolation period Tip = 1 ms, the acceleration bound Am = 500 mm/s2 , the jerk bound Jm = 10000 mm/s3 , the tolerance for chord error δ = 1 µm and the command feedrate Fc = 150 mm/s. Besides the proposed OFIKC method, another three machining methods are also implemented for comparison, such as constant feedrate, adaptive feedrate with confined chord error and point-point. The detailed information of machining results are listed in Table 1. The point-point strategy can satisfy the requirements of machining accuracy perfectly, but it takes so long time, that is, 34.847 s. It is because this strategy make machine tool stop at each vertex of the G01 contour. However, these vertexes are not needed, since the surface of the workpiece is smoother without these sharp corners. Although the constant and adaptive scheme greatly reduce the processing time, the contour errors for the two methods are increased. The proposed OFIKC method finds the tradeoff between the accuracy and the machining efficiency. And more importantly, OFIKC ensures that kinematics indexes and the chord error are within allowable limits, which is favorable for actual processing. Figure 4b shows the profiles of feedrate, chord error, acceleration and jerk machining with OFIKC.

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40

Proposed FPS

Proposed FPS

35

30

30

25

25

Y (mm)

Y (mm)

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15

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Fig. 4. Simulation results for the car pattern: (a) Fitted curves with 25 control points; (b) Fitted curves with 40 control points; (c) Local figure of final fitted curve (83 control points); (d) The profiles of feedrate, chord error, acceleration and jerk. Table 1. Comparison of performances for four different machining methods Method Constant Adaptive feedrate Point to point Proposed

Machining time (s) Contour error (µm) Mean Max 1.362

6.6

29.3

1.382

6.5

27.2

0

0

5.1

25.4

34.847 1.596

6 Conclusions In this paper, a complete CNC processing scheme is elaborated, including curve fitting and a novel interpolator. Curve fitting focuses on transforming the tiny line segments from a NC file into a NURBS curve. During the stage of curve fitting, the optimal polygonal approximation based on the MIP is employed to fit the G01 lines. In addition, the bisection method helps to decide the optimal number of control points. With the fitted curve, an overlap-free interpolator is designed based on first order interpolation technique integrating kinematics and chord error limitations. Through applying the proposed scheme to a NC file, satisfactory simulation results are obtained.

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Acknowledgments. This work is supported by the National Natural Science Foundation of China [grant number 61573378]. This work is partially supported by the Key Technologies R&D Program of Henan province [grant number 182102210197].

References 1. Tsai MS, Nien HW, Yau HT (2010) Development of a real-time look-ahead interpolation methodology with spline-fitting technique for high-speed machining. Int J Adv Manuf Technol 47(5–8):621–638. https://doi.org/10.1007/s00170-009-2220-7 2. Li DD, Zhang WM, Zhou W et al (2018) Dual NURBS path smoothing for 5-axis linear path of flank milling. Int J Precis Eng Manuf 19(12):1811–1820. https://doi.org/10.1007/s12541018-0209-6 3. Jahanpour J, Alizadeh MR (2015) A novel acc-jerk-limited NURBS interpolation enhanced with an optimized S-shaped quintic feedrate scheduling scheme. Int J Adv Manuf Technol 77(9–12):1889–1905. https://doi.org/10.1007/s00170-014-6575-z 4. Zhang Y, Ye P, Zhang H et al (2018) A local and analytical curvature-smooth method with jerk-continuous feedrate scheduling along linear toolpath. Int J Precis Eng Manuf 19(10):1529–1538. https://doi.org/10.1007/s12541-018-0180-2 5. Aguilera-Aguilera EJ et al (2015) Novel method to obtain the optimal polygonal approximation of digital planar curves based on Mixed Integer Programming. J Vis Commun Image Represent 30:106–116. https://doi.org/10.1016/j.jvcir.2015.03.007 6. Piegl LA, Tiller W (2000) Least-squares B-spline curve approximation with arbitrary end derivatives. Eng Comput 16(2):109–116. https://doi.org/10.1007/PL00007188 7. Yeh SS, Hsu PL (2002) Adaptive-feedrate interpolation for parametric curves with a confined chord error. Comput Aided Des 34(3):229–237. https://doi.org/10.1016/s00104485(01)00082-3 8. Du D, Liu Y, Guo X et al (2010) An accurate adaptive nurbs curve interpolator with realtime flexible acceleration/deceleration control. Robot Comput-Integr Manuf 26(4):273–281. https://doi.org/10.1016/j.rcim.2009.09.001 9. Park H, Lee JH (2007) B-spline curve fitting based on adaptive curve refinement using dominant points. Comput Aided Des 39(6):439–451. https://doi.org/10.1016/j.cad.2006.12.006

Single Image Defogging Method Combined with Multi-exposure Fusion Approach Based on Parameter Dynamic Selection Yuanyuan Li1 , Hexu Hu1 , Hongyan Wei1 , Xinhua Huang2(B) , Penghua Li1 , and Zhiqin Zhu1 1

College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065, China 2 College of Automation, Chongqing University, Chongqing 400044, China [email protected]

Abstract. A single image defogging method combined with multi-exposure fusion approach based on parameter dynamic selection is proposed in this paper. Gamma correction was used to improve the contrast of the haze source image, and spatial linear saturation stretch was used to enhance the color saturation. Then, multi-exposure fusion scheme based on dynamic parameter selection is used to collect the patches with the best contrast, saturation and texture structure from each image and fuse them into a single fog-free image. The experimental results show that the proposed method is effective in defogging. Keywords: Image defogging · Gamma correction · Spatial linear saturation stretch · Multi-exposure image fusion

1 Introduction Haze is an atmospheric phenomenon caused by the absorption or reflection of light by particles floating in the atmosphere. Under severe weather conditions, the scene radiation obtained decreases sharply and pollutes due to atmospheric scattering. On one hand, haze can lead to the decline of image quality and affect the visual quality of images. On the other hand, it can reduce the efficiency of subsequent image processing and computer vision tasks, such as feature extraction, recognition and classification [1–3]. Therefore, haze removal or dehazing is necessary before performing following vision tasks in applications, including outdoor surveillance, traffic monitoring, air navigation and remote sensing. The removal of haze is a challenging problem because the spread of smog depends on the unknown depth at different locations [4]. In the imaging scene, haze degradation changes as depth increases. There are many researches for image defogging, and the mainstream methods can be divided into two categories: algorithms based on image restoration and image enhancement. In image restoration method, atmospheric degradation model is used for image defogging. And on this basis, He et al. [5] proposed an effective image prior-dark channel prior to remove haze from a single input image. c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 466–470, 2020. https://doi.org/10.1007/978-981-32-9682-4_49

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Wang et al. [6] presented a multiscale depth fusion (MDF) method for defog from a single image. However, since haze is a depth-dependent phenomenon, the resulting image degradation is spatially different with different areas of the image being more affected [7]. In this circumstance, the unavailable depth information can be obtained via the physical model of haze formation. In image processing [15, 17], image enhancement based defogging algorithm has become a very popular research direction. Among them, the most classical and widely used image enhancement algorithms include histogram equalization algorithm [9, 10], Retinex algorithm [11], wavelet transform [12] and so on. The algorithms are calculated simply and performed in real time. Human visual characteristics or computer vision system can be satisfied with the images after defogging. However, the cause of smog image degradation has not been fully considered. As a result, the method of image enhancement improves the effect of image to a certain extent, but the overall performance of image defogging effect is general. In this paper, the local contrast of the image is improved by reducing the exposure of the fuzzy image through a series of gamma corrections, and the color saturation is enhanced by stretching the spatial linear saturation. Thus, a group of underexposed image sequences containing the contrast and saturation enhancement regions are generated. In order to further explain the spatial variation characteristics of weather degradation, multi-exposure image fusion [16] approach based on parameter dynamic selection to collect the regions with the best contrast, saturation and texture structure from each image to fuse into a single haze-free image. The main contributions of this paper are to apply the multi-exposure image fusion algorithm in the spatial domain to the image defogging for the first time, and optimize the image visual quality locally and globally by combining gamma correction and spatial linear saturation stretch. The rest of this paper is organized in following section. The framework of our proposed method is shown in Sect. 2, Sect. 3 presents experimental results and we conclude this paper in Sect. 4.

2 Proposed Method for Image Dehazing For image defogging, there are three main steps: (1) reduce the exposure of fuzzy image through a series of gamma correction [7] to improve the local contrast of the image; (2) use spatial linear saturation stretch to enhance color saturation; (3) the multi-exposure fusion scheme based on dynamic parameter selection [14] is used to collect the areas with the best contrast, saturation and texture structure from each image and fuse them into a single fog-free image. The algorithm steps in this paper are shown in Fig. 1. The calculation of gamma correction on image intensity is shown in Eq. (1): I(x) = α · I  (x)γ

(1)

Where α and γ are positive numbers. I(x) is the image after gamma correction, and its value is between 0 to 1. The relationship curve of input and output show that the curve generated by the value of γ > 1 is completely opposite to that generated by the value γ < 1. For haze image, the brightness is relatively high, in order to improve the contrast

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SSIM

FADA

GPR

0.7415 0.6528

Proposed 0.7563 0.3256

need to reduce the exposure. Therefore, the value of γ has to be set above 1. Six lowexposure images with enhanced local contrast were generated by selecting six values of γ greater than 1. The color saturation of each image is then enhanced by stretching the spatial linear saturation, which is calculated by Eq. (2).

Sk (x) =

∑

c∈{R,G,B}

 Ekc (x) −

EkR (x) + EkG (x) + EkB (x) 3

 (2)

Six low-exposure images with enhanced contrast and saturation are obtained by the above approach. The best contrast and saturation patches in the image are extracted by using the multi-exposure fusion scheme based on parameter dynamic selection. Texturecartoon decomposition is carried out on the enhanced image to obtain the texture component. The dynamic parameters of the image are obtained by calculating the entropy of the image texture. The structural patch decomposition is used to decompose the image patch into three independent components: signal strength, signal structure, signal intensity. The image patches with the best contrast, saturation and texture structure quality are fused into a single fog-free image.

3 Experimental Results We used a total of 84 sets of images in this experiment. In this paper, we analyze the defogging effect of source image through subjective and objective evaluation. The objective indexes used in our experiments are SSIM [13] and FADA [8]. SSIM measures the similarity between the fogging image and the source image from three aspects: brightness, contrast and structure. FADA is an index to measure the defogging ability of an image, a smaller value indicates a smaller amount of fog. The results of GPR [18] were selected as the comparison experiment. As shown in Fig. 2, (a) and (d) are two haze images, (b) and (e) are fusion results of GPR, and (c) and (f) are the corresponding two defogging images of our proposed algorithm. (a) and (d), both of which are covered with thick fog. The overall contrast is low, and the original color of the object cannot be seen clearly. From visual observation, the fog of the GPR results is significantly weakened, but the color is unnatural. The brightness of (b) and (e) is reduced, the contrast and saturation are improved, and the color of image is softer and more natural. The average objective evaluation results of the source image (a) and (d) are shown in Table 1. The FADA value is small, indicating that the proposed method does achieve the effect of fog removal. The SSIM value is high, proving that the distortion rate of the image is low. The SSIM of GPR is smaller and FADA is larger than the proposed method, which

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proves that the proposed method has better effect than GPR. Therefore, from the subjective and objective evaluation, the defogging ability of our proposed method is pretty well.

Fig. 1. Image defogging algorithm framework

Fig. 2. Image defogging result. (a) and (d) are haze image. (b) and (e) are defogging image of GPR. (c) and (f) are defogging image of proposed method

4 Conclusion This paper proposed a single image defogging method combined with multi-exposure fusion approach based on parameter dynamic selection, which used the gamma correction and spatial linear saturation stretch to improve the contrast and saturation of original haze images. And then the multi-exposure fusion approach based on parameter dynamic selection is applied to collect the image patches with the best saturation and contrast in each image and fuse these patches into a single fog-free image. The defogging effect of our proposed method was evaluated from subjective and objective aspects. From the evaluation results, the image quality has been greatly improved, the color saturation and contrast have been enhanced. Therefore, the image defogging method proposed by us is feasible. However, the calculation time is a little too long, we hope it can be improved in the future work.

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Acknowledgments. This work is jointly supported by National Natural Science Foundation of China under Grant 61803061, 61703347; Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJQN201800603); the Chongqing Natural Science Foundation Grant cstc2016jcyjA0428; the Common Key Technology Innovation Special of Key Industries of Chongqing science and Technology Commission under Grant No. cstc2017zdcy-zdyfX0067 and cstc2017zdcy-zdyfX0055; the Artificial Intelligence Technology Innovation Significant Theme Special Project of Chongqing science and Technology Commission under Grant No. cstc2017rgzn-zdyfX0014 and cstc2017rgzn-zdyfX0035.

References 1. Xu C, Li M, Sun X (2013) An edge-preserving variational method for image decomposition. Chin J Electron 22(1):109–113 2. Li X, Zhang S, Liu Q, Zhang B, Liu D, Lu B, Na X (2009) Fast segmentation method of high-resolution remote sensing image. J Infrared Millimeter Waves 28(2):146–150 3. Zhu Y, Zhu Y, Wen Z, Chen W, Huang Q (2012) Detection and recognition of abnormal running behavior in surveillance video. Math Probl Eng 2012:1–14 4. Cai B, Xu X, Jia K (2016) DehazeNet: an end-to-end system for single image haze removal. IEEE Trans Image Process 25(11):5187–5198 5. He K, Sun J, Tang X (2009) Single image haze removal using dark channel prior. In: Conference on computer vision and pattern recognition. IEEE 6. Wang Y, Fan C (2014) Single image defogging by multiscale depth fusion. IEEE Trans Image Process 23(11):4826–4837 7. Galdran A (2018) Image dehazing by artificial multiple-exposure image fusion. Signal Process 149:135–147 8. Choi LK, You J, Bovik AC (2015) Referenceless prediction of perceptual fog density and perceptual image defogging. IEEE Trans Image Process 24(11):3888–3901 9. Yu Z, Bajaj C (2004) A fast and adaptive method for image contrast enhancement. In: IEEE international conference on image processing, vol 2, pp 1001–1004 10. Thomas G, Flores-Tapia D, Pistorius S (2011) Histogram specification: a fast and flexible method to process digital images. IEEE Trans Instrum Meas 60(5):1565–1578 11. Rahman Z, Jobson DJ, Woodell GA (2004) Retinex processing for automatic image enhancement. J Electron Imaging 13(1):100–110 12. Russo F (2002) An image enhancement technique combining sharpening and noise reduction. IEEE Trans Instrum Meas 51(4):824–828 13. Hu Q (2013) Multi-sensor image fusion algorithm based on SSIM. J SE Univ 43:159–162 14. Li Y, Zheng M, Hu H, Wang H, Zhu Z (2018) A novel multi-exposure image fusion approach based on parameter dynamic selection. In: Proceedings of 2018 Chinese intelligent systems conference, vol 20, no 12, pp 935–951 15. Li Y, Sun Y, Huang X, Qi G, Zheng M, Zhu Z (2018) An image fusion method based on sparse representation and sum modified-Laplacian in NSCT domain. Entropy 20(7):522–539 16. Zhu Z, Zheng M, Qi G, Wang D, Xiang Y (2019) A phase congruency and local Laplacian energy based multi-modality medical image fusion method in NSCT domain. IEEE Access 7:20812–20814 17. Liu Z, Chai Y, Yin H, Zhou J, Zhu Z (2017) A novel multi-focus image fusion approach based on image decomposition. Inf Fusion 35:102–116 18. Ullah E, Nawaz R, Iqbal J (2013) Single image haze removal using improved dark channel prior. In: International conference on modelling

Finite-Time Consensus for Second-Order Leader-Following Multi-agent Systems with Disturbances Based on the Event-Triggered Scheme Yan Cui and Xiaoshan Wang(B) College of Physics and Information Engineering, Shanxi Normal University, Linfen 041000, China xiaoshan [email protected]

Abstract. In this paper, the finite-time distributed event-triggered consensus control is considered for second-order multi-agent systems with the external disturbances. Firstly, sufficient event-triggered conditions are proposed to ensure the system achieve finite-time consensus for leader-following networks. Then, for the external disturbance, the original system is transformed into a sliding mode variable structure system to reduce it, and the finite-time consensus condition is obtained. In addition, zeno phenomenon is eliminated by proving that there is a positive lower bound between the two event-triggered moments of the agent. Finally, the effectiveness of the theoretical results is verified by numerical simulation. Keywords: Finite-time consensus · Second-order systems · Integral sliding mode · External disturbances

1 Introduction A multi-agent system is a collection of computable agents, each of which is a physical or abstract entity that not only acts on itself and the environment, but communicates with other agents. Multi-agent systems are widely applied in many scientific research fields, including robot formation control [1–3], intelligent transportation [4], unmanned aerial vehicle [5] and so on. In the research process of the consensus problem, the tracking convergence rate of the multi-agent system has gradually become an important performance index. Some practice systems, such as intelligence robot system [6, 7], the satellites system [8, 9] and so on, are generally required to achieve finite-time consensus [10–14]. Thus the research of finite-time consensus is extremely necessary. Although experts have done a lot of researches on the problem of finite-time consensus of the multi-agent systems, there are few researches on the finite-time consensus with event-triggered strategy. Eventtriggered strategy communicates with other agents through non-equal-period sampling control. It can reduce the number of updates to the single agent controller and the computation of the processor. c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 471–486, 2020. https://doi.org/10.1007/978-981-32-9682-4_50

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Some results have been involved in [15–17]. A distributed synchronization control algorithm and an event-triggered mechanism were proposed in [15], Jiang et al. adopted the feedback domination technique method to figure out consensus problem for a kind of nonlinear multi-agent systems. Using event-triggered linear feedback laws and event-triggered strategy for each follower agent, Xie et al. [16] solved the problem of global leader-following consensus for a multi-agent system. In addition, to deal with the problem including the external disturbances, Zero behavior and staying on the sliding surface designed and so on, a new series of two-layer event-triggered sliding-mode control were presented and the finite-time consensus was guaranteed in [17]. The above papers basically solved the problem of asymptotically consensus under event triggered condition. But in practice, some systems need to converge to the consensus state in a finite time. In general, it is necessary to study the finite-time of second-order multiagent system under event-triggered condition. Moreover, in order to prevent interference from affecting the system, the sliding mode variable structure method be employed to work out this issue. There are some results that have been applied to other systems [18, 19]. Wang et al. [18] adopted a novel fuzzy integral sliding manifold function for nonlinear stochastic systems, and combined sufficient conditions to ensure that the stochastic stability of the closed-loop system was derived. In [19], the distributed consensus tracking problem for the fractionalorder multi-agent systems was studied. And a controller was proposed by Bai et al., which could solve the two cases–the followers described by the fractional-order linear dynamics and intrinsic nonlinear dynamics. Thus combining the integral sliding mode function and the finite-time protocol in this paper, the sliding mode control law is developed, which allows the system to continue to slide along the fixed sliding surface when disturbed by external environment. The contributions of this paper are as follows: (1) A new second-order nonlinear leader-following control protocol employed eventtriggered condition is presented. Thus the convergence time of agents state is dramatically decreased. Compared with the nominal homogeneous protocol in [20], this paper not only ensures the robustness of the ultimate boundedness, but also the exact value of the finite time is obtained. (2) Finite-time protocol with disturbances by integral sliding mode is introduced in this paper. Compared with the system in [21], a new sliding mode system is showed, which enables the system to reduce chattering and slide better on the sliding surface. The general structure of this paper is as follows: the second part is the basic theory and some necessary Lemma. The new finite-time protocol with applied the event-triggered conditions and integral sliding mode control protocol are respectively given in the third part. The fourth part gives some examples and simulations to prove the validity of the protocol, and the fifth part gives the conclusion.

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2 Preliminaries and Problem Statement 2.1 Graph Theory If there is a vertex set V = {v1 , v2 , · · · , vn } and an edge set E = {e1 , e2 , · · · , em }, if for any edge ek in E, there is a vertex pair (vi , v j ) and its correspondence in V , we call the group composed of V and E a graph, denoted as G, defined as G = (V, E). Edge ek can be represented by its two endpoints, vi and v j , as (vi , v j ) or vi v j . If vi is an endpoint of edge ek , the vertex vi is associated with ek . If vi , v j is the two endpoints of the same edge ek , then the vertex vi is called adjacent to the v j . If the edge ei , e j is associated with the same vertex, two edges ei and e j are called adjacent. When an edge e = uv is related to the order of its two endpoints u and v, that is, the edge uv and vu are different, e is called a directed edge. An arrow points geometrically in the direction of a directed edge: the vertex pointed to by the arrow is called the end point of the edge, the other endpoint is the starting point of the edge, and the directed edge is associated with the two endpoints. A graph with all edges oriented is called a directed graph, often denoted as D = (v, a), where V is a set of vertices, A is a set of directed edges, and a directed edge is usually called an arc. If the direction of the arcs of a directed graph is removed, an undirected graph is obtained, which is called the base graph of the directed graph. 2.2 Problem Statement The weighted undirected communication topology graph G(V, E, A), each agent can be regarded as a vertex in the topology graph, the information transfer between agents can be regarded as the edge of the graph, and the multi-agent system with n agents is considered. The Leader-following dynamic model of the second order multi-agents with N followers and one leader is ⎧ x˙i (t) = vi (t), ⎪ ⎪ ⎪ ⎨ v˙ (t) = u (t) + d , i = 1, 2, · · · , n i i i (1) ⎪ x ˙ = v 0 ⎪ ⎪ 0 ⎩ v˙0 = 0 where xi (t), vi (t) and di are the position state, velocity state and disturbance of agent i, respectively, and ui (t) is the control input, and xo , vo are the position and velocity of the leader respectively. And ui (t) = un (t) + ust (t), un (t) = ai j (γ1 ∗ sig(x j − xi )ai j + γ2 ∗ sig(v j − vi )ai j ) + bi (sig(xi − x0 )ai j + sig(vi − v0 )ai j )

(2a)

1 2

ust = −k1 sig (si ) + ρi ρ˙ i = −k2 sgn(si ) si = vi − vi (0) +

 t 0

−un d τ = 0, i = 1, 2, . . . , N

(2b) (2c)

where 0 < ai j < 0, γ1 , γ2 are control gains, sig(x) = sign(x) · |x|α , ust is related to the integral sliding mode control parameter s, si (t) is designed function by the integral mode.

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Definition 1. Finite-time consensus: A multi-agent system (1) is called a finite-time consensus, if there is a time T0 ∈ [0, +∞) determined that allows the state of each agent to meet the following condition: lim ||xi (t) − x0 (t)|| = 0, lim ||vi (t) − v0 (t)|| = 0

t→T0

t→T0

and xi (t) = x0 (t), vi (t) = v0 (t), for any t ≥ T0 , i, j ∈ I. Lemma 1. Suppose the function V (t) : [0, ∞) → [0, ∞) is differentiable, if the right derivative of V (t) at the point t = 0 is differentiable and there is dVdt(t) ≤ −KV (t)α , 1−α

(0) where: K > 0; 0 < α < 1, then V (t) will tend to 0 at the finite time t ∗ (0 < t ∗ ≤ VK(1− α ) ).

Lemma 2 [10]. If x1 , x2 , · · · , xn ≥ 0, and 0 < p ≤ 1, then (∑ni=1 xi ) p ≤ ∑ni=1 xip . Lemma 3. Suppose a, b, c > 0, if b + c ≥ a, then

a b+c

≥

a−b c .

Lemma 4 [10]. If the Laplacian matrix of graph G is L, then (1) If G has a spanning tree, then 0 is one of its eigenvalues, and all the other eigenvalues are in its positive real part; (2) if G is strongly connected, then there exists a positive column vector ω ∈ Rn such that ω T L(A) = 0; (3) if G is connected, then for any x = (x1 , x2 , · · · , xn )T ∈ Rn , there is xT L(A)x = 1 n 2 ∑i=1 ai j (xi − x j ), and L(A) is positive definite; (4) The second minimum eigenvalue of L(A) is called the algebraic connectivity of G(A), denoted as λ2 (L(A)) > 0. If graph G is connected, then the algebraic x connectivity of G(A) is equivalent to minx∈0.1 / T x=0 T T x L(A)x ≥ λ2 (L(A))x x.

T L(A)x

xT x

, and if 1T x = 0, then

Lemma 5 [20]. Suppose that Assumption holds and di = 0, the consensus tracking can be achieved in finite time T > 0 by the protocol.

3 Main Results Firstly, it is proved that two control gains are added in protocol un (t). By making full use of all state and velocity information of the multi-agent in the system, and constructing a appropriate lyapunov function, the leader has been tracked by the followers and the second-order finite-time consensus can be achieved under the condition of eventdriven. Secondly, the protocol is combined with integral sliding mode, and the parameters of ust (t) are adjusted, that is, all the control inputs u(t) are obtained. Under the proper construction of lyapunov function, the system can achieve finite-time consensus of second-order multi-agent under matching perturbation.

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For the distributed event-triggered control strategy, each agent has its own eventtriggered time which is used to decide whether to update control input, and this means that each agent is updated asynchronous. The event times for each agent i are represented by the t0i ,t1i , · · · . sequence. Define the position measurement error for agent i as i ) ex,i = xi (tki ) − xi ,t ∈ [tki ,tk+1 And the velocity measurement error for agent i is denoted as i ev,i = vi (tki ) − vi ,t ∈ [tki ,tk+1 ) i ), t i t ∈ [tki ,tk+1 k+1 (k = 0, 1, . . .) is the kth trigger moment of the agent i. Thus, in the agent control algorithm, each agent should take into account the last update state of its neighbors. The control law for agent i is updated at its own event times t0i ,t1i , · · · , meanwhile the number of events in its neighbors t0j ,t1j , · · · , j ∈ N. It is worth noting that the definition of k indicates x j (tkj ) = x j (t) + ex, j (t), v j (tkj ) = v j (t) + ev, j (t). And let x(t) ˆ = x j (t) − xi (t)

eˆx (t) = ex, j (t) − ex,i (t) v(t) ˆ = v j (t) − vi (t)

(3)

eˆv (t) = ev, j (t) − ev,i (t) Hence, we can obtain ⎧˙ xˆi (t) = vˆi (t), ⎪ ⎪ ⎪ ⎨ v˙ˆ (t) = un (t) i ⎪ = ai j (γ1 ∗ sig(x(t) ˆ + eˆx (t))ai j + γ2 ∗ sig(v(t) ˆ + eˆv (t))ai j ) ⎪ ⎪ ⎩ ai j + bi ∗ (sig(xi (t) − x0 (t)) + sig(vi (t) − v0 (t))ai j ),

(4)

Forming an event-driven control protocol, which enables multi-agent systems to achieve finite-time consensus, is the crucial part of this paper. In the following portion, sufficient condition are designed to allow the agents to attain finite-time consensus. And the control input of each agent is not updated at all times, but is updated based on the event-triggered scheme. 3.1 Finite-Time Consensus with Event-Triggered Scheme The event condition for agent i express as follows: |eˆx,i (t)| ≤ Mi |xˆi (t)| |eˆv,i (t)| ≤ Ni |vˆi (t)|

(5)

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Theorem 1. Assuming that the topology graph G is strongly undirected and the control gain satisfies R1 , if for any i, j,t, there is ai j (t) = a ji (t), then the protocol (2a) can solve the finite-time consensus problem of the system under the event-triggered condition (5). Furthermore, according to the constructed Lyapunov function, an upper bound of time t is obtained to make all the states of the system uniform. 2

R1 ≤ ≤ R1 ≥

1 n n 1+a0 (vˆ j − vˆi )2 2 ∑i=1 ∑ j=1 (ai j R2 ) 1 n n 2 2 ∑i=1 ∑ j=1 ai j (maxk xk (0) − mink xk (0)) 1 n n 2 2 ∑i=1 ∑ j=1 ai j R2 (maxk vk (0) − mink vk (0)) 1 n n 2 2 ∑i=1 ∑ j=1 ai j (maxk xk (0) − mink xk (0)) ∑ni=1 ∑nj=1 (ai j R2 )2 (maxk vk (0) − mink vk (0))2 1 n n 2 2 ∑i=1 ∑ j=1 ai j (maxk xk (0) − mink xk (0))

(6a) (6b)

Proof. Construct the Lyapunov function candidate V (t) V (t) =

1 n n ∑∑ 4 i=1 j=1 n

+∑

 t

 xˆ j −xˆi 0

γ1 ∗ (1 + M ai j ) ∗ ai j ∗ sig(s)ai j ds + n

∑

 t

1 n 2 ∑ vˆi (t) 2 i=1

bi ∗ vˆi ∗ sig(x0 − xi ) ds + bi ∗ vˆi ∗ sig(v0 − vi )ai j ds 0 0 i=1 i=1 n n 1 V˙ (t) = γ1 ∗ (1 + M ai j ) ∗ ai j (vˆ j − vˆi ) ∗ sig(xˆ j − xˆi )ai j 4 i=1 j=1 n n + vˆi ai j (γ1 ∗ sig(x(t) ˆ + eˆx (t))ai j + γ2 ∗ sig(v(t) ˆ + eˆv (t))ai j ) i=1 j=1 n + vˆi bi ∗ (sig(xi (t) − x0 (t))ai j + sig(vi (t) − v0 (t))ai j ) i=1 n n + bi ∗ vˆi ∗ sig(x0 − xi )ai j + bi ∗ vˆi ∗ sig(v0 − vi )ai j i=1 i=1 1 n n ≤ γ1 ∗ (1 + M ai j ) ∗ ai j (vˆ j − vˆi ) ∗ sig(xˆ j − xˆi )ai j 4 i=1 j=1 1 n n − ∗ai j γ1 (vˆ j (t) − vˆi (t))(1 + M ai j ) ∗ ai j |xˆ j (t) − xˆi (t)|ai j 4 i=1 j=1 1 n n − ∗ai j γ2 (vˆ j (t) − vˆi (t))(1 + N ai j ) ∗ ai j |vˆ j (t) − vˆi (t)|ai j 4 i=1 j=1 ai j

∑∑

∑ ∑ ∑ ∑

∑∑

∑∑ ∑∑

∑

(7)

Finite-Time Consensus

=−

477

1 n n ∑ ∑ ∗ai j γ2 (1 + N ai j ) ∗ ai j (vˆ j (t) − vˆi (t))1+2∗ai j 4 i=1 j=1

1+a0  1+2∗ai j  2 2 1 n n 2 1+a0 1+a0 = − ∑ ∑ (ai j ∗ R2 ) [(vˆ j (t) − vˆi (t)) ] 4 i=1 j=1

⎡

1+2∗ai j

2

n n 2 1+a0 1+a ⎢ ∑i=1 ∑ j=1 (ai j ∗ R2 ) 0 [(vˆ j (t) − vˆi (t)) ]

1 =− ⎣ 4

2

∑ni=1 ∑nj=1 (ai j ∗ R2 ) 1+a0 [(vˆ j (t) − vˆi (t))2 ]

⎡ ∗

V (t)

2

⎥ ⎦

(8)

0 ⎤ 1+a 2

2

∑n ∑n (a ∗ R ) 1+a0 [(vˆ j (t) − vˆi (t))2 ] ⎣ i=1 j=1 i j 2

⎤ 1+a0

∗V (t)⎦

where ao = maxi j ai j , R2 = γ2 (1 + N ai j ) vˆ j − vˆi ≤ maxk vk (t) − mink vk (t) ≤ maxk vk (0) − mink vk (0) 2

V1 (t) =

∑ni=1 ∑nj=1 (ai j ∗ R2 ) 1+a0 [(vˆ j (t) − vˆi (t))2 ]

1+2∗ai j 1+a0

2

∑ni=1 ∑nj=1 (ai j ∗ R2 ) 1+a0 (vˆ j (t) − vˆi (t))2

(9)

2

V2 (t) =

∑ni=1 ∑nj=1 (ai j ∗ R2 ) 1+a0 (vˆ j (t) − vˆi (t))2 V (t)

Suppose i0 − j0 = arg max(vˆi − vˆ j

)2 , i,

j ∈ In , ai j ∈ /0 1+2∗ai j 1+a0

2

V1 (t) ≥

∑ni=1 ∑nj=1 (ai j ∗ R2 ) 1+a0 [(vˆ j0 (t) − vˆi0 (t))2 ] 2

∑ni=1 ∑nj=1 (ai j ∗ R2 ) 1+a0 (vˆ j0 (t) − vˆi0 (t))2 1+2∗ai j 1+a0

2

≥

≥

∑ni=1 ∑nj=1 (ai j ∗ R2 ) 1+a0 [(vˆ j0 (t) − vˆi0 (t))2 ]

−1

2

∑ni=1 ∑nj=1 (ai j ∗ R2 ) 1+a0 2   1+2∗ai j −1 mini, j∈In ,ai j =0 (ai j ∗ R2 ) 1+a0 (maxkVk (0) − minkVk (0))2 1+a0 2

∑ni=1 ∑nj=1 (ai j ∗ R2 ) 1+a0

we will define K1 =

2   1+2∗ai j −1 mini, j∈In ,ai j =0 (ai j ∗ R2 ) 1+a0 (maxkVk (0) − minkVk (0))2 1+a0 2

∑ni=1 ∑nj=1 (ai j ∗ R2 ) 1+a0

(10)

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where R1 = γ1 (1 + M ai j ),

a b+c

>

a−b c , 2

V2 (t) = A=

∑ni=1 ∑nj=1 (ai j ∗ R2 ) 1+a0 (vˆ j (t) − vˆi (t))2 A + B +C + 12 vˆTi ∗ vˆi |xˆ j − xˆi |2 1 n n ∗a ∗ R i j 1 ∑∑ 4 i=1 ai j + 1 j=1 n

n

B = ∑ ∑ bi i=1 j=1 n

n

C = ∑ ∑ bi A≤

i=1 j=1 n n

|xi − x0 |2 bi + 1 |vi − v0 |2 bi + 1

(11)

1 ∑ ∑ ∗ai j ∗ R1 (maxk xk (0) − mink xk (0))2 = A1 4 i=1 j=1 n

n

B ≤ ∑ ∑ bi (xk (0) − x0 )2 = B1 i=1 j=1 n n

C ≤ ∑ ∑ bi (vk (0) − v0 )2 = C1 i=1 j=1

2

V2 (t) ≥

∑ni=1 ∑nj=1 (ai j ∗ R2 ) 1+a0 (vˆ j (t) − vˆi (t))2 A1 + B1 +C1 + 12 vˆTi vˆi 2

≥ A1 =

∑ni=1 ∑nj=1 (ai j ∗ R2 ) 1+a0 (vˆ j (t) − vˆi (t))2 − A1 − B1 −C1 1 T 2 vˆi vˆi

1 n n ∑ ∑ ∗ai j ∗ R1 (maxk xk (0) − mink xk (0))2 4 i=1 j=1

≥

1 n n ∑ ∑ (maxk vk (0) − minvk (0))2 2 i=1 j=1

≥

2 1 n n (ai j R2 ) 1+a0 (vˆ j − vˆi )2 ∑ ∑ 2 i=1 j=1

n

B1 = ∑

n

n

i=1 j=1 n n

i=1 j=1

≥ − ∑ ∑ (bi ∗ R2 ) n

2 1+a0

i=1 j=1 n

C1 = ∑

(vˆ j − vˆi )2 n

n

∑ bi (maxk vk (0))2 ≥ ∑ ∑ bi (maxk vk (0) − mink vk (0))2

i=1 j=1 n n

≥∑

n

∑ bi (maxk xk (0))2 ≥ ∑ ∑ bi (maxk xk (0) − mink xk (0))2

i=1 j=1

2

∑ (bi ∗ R2 ) 1+a0 (vˆ j − vˆi )2

i=1 j=1

(12)

Finite-Time Consensus

V2 (t) ≥ =

1 2

479

2

∑ni=1 ∑nj=1 (ai j R2 ) 1+a0 (vˆ j − vˆi )2 1 T 2 vˆi vˆi

(13)

Vˆ L(B)Vˆ 1 ˆT ˆ 2V V

≥ 2λ2 (L(B)) where B = (R2 ∗ A)

1+a0 2 1+a0 1 dV (t) ≤ − (2 ∗ K1 λ2 (L(B))V (t)) 2 dt 4

K2 = minK1 (t)λ2 (L(B)) then 1+a0 a0 −3 dV (t) ≤ −2 2 (K2V (t)) 2 dt

By Lemma 1, the exact value of finite time t ∗ =

5−a0

1−a0

2 2 V (0) 2

1+a0

can be obtained, so that

(1−a0 )K2 2

V (t) = 0,t ≥ t ∗ . Then xˆ j (t) = xˆi (t), vˆ j (t) = vˆi (t), that is x j (t) = xi (t), v j (t) = vi (t). It can be seen that the proposed protocol (2a) can solve the problem of finite time consensus of the systems (4). Next, we will prove the event-triggered condition proposed can exclude the zeno behaviour. Zeno behavior refers to the existence of an infinite number of triggers by T   T an agent in a finite period of time. Suppose ρ (t) = eˆx eˆv , δ (t) = xˆ vˆ , The event-triggered condition can be rewritten as follow:       1 0 eˆx M 0 xˆ f (t) = − i 0 1 eˆv 0 Ni vˆ = F ∗ ρ (t) − P ∗ δ (t)     10 Mi 0 where F = ,P= . 01 0 Ni Theorem 2. For multi-agent system (1), if the event trigger condition (5) is established, the interval tk+1 − tk between any two consecutive event trigger times in the system is P·F −1 not less than τ = (||A||+||B||)·(P·F −1 +||A||+||B||) under the action of consistency control protocol (4). Proof. According to the principle of event trigger control, the interval between any two (t)|| continuous time trigger times in the system is ||||ρδ (t)||

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The time, which takes to grow from 0 to P · F −1 , is reworded as τ . Then the time (t)|| derivative of ||||ρδ (t)|| , we have 1

d(ρ T (t)ρ (t)) 2 d||ρ (t)|| = dt||δ (t)|| dt(δ T (t)δ (t)) 21 ρ (t) · δ˙ (t) δ T (t) · δ˙ (t) · ||ρ (t)|| − = ||ρ (t)|| · ||δ (t)|| ||δ (t)|| · ||δ (t)|| ˙ ˙ δ (t) δ (t) · ||ρ (t)|| δ˙ (t) ||ρ (t)|| + )· ≤ = (1 + 2 ||δ (t)|| ||δ (t)|| ||δ (t)|| ||δ (t)||   0 0 || N×N N×N · δ (t)|| 0N×N IN ||ρ (t)|| )· ≤ (1 + ||δ (t)|| ||δ (t)||   0N×N 0N×N || · (δ α1 (t) + ρ α2 (t))|| γ1 (1 + M)α1 γ2 (1 + N)α2 ||ρ (t)|| − (1 + )· ||δ (t)|| ||δ (t)|| ||ρ (t)|| ||A · δ (t) − B · (δ (t) + ρ (t))|| ) ≤ (1 + ||δ (t)|| ||δ (t)|| ||ρ (t)|| ||ρ (t)|| 2 ) + ||A|| · (1 + ) = ||B|| · (1 + ||δ (t)|| ||δ (t)|| ||ρ (t)|| 2 ≤ (||B|| + ||A||) · (1 + ) ||δ (t)||     0 0 0N×N 0N×N (t)|| where A = N×N N×N , B = , then . Suppose y = ||||ρδ (t)|| 0N×N IN γ1 (1 + M)α1 γ2 (1 + N)α2 there (14) y˙ ≤ (||A|| + ||B||) · (1 + y)2 In this formula, y satisfies y ≤ ψ (t, ψ0 ), where ψ (t, ψ0 ) is the solution of equation ψ˙ = (||A|| + ||B||) · (1 + y)2 , ψ (0, ψ0 ) = ψ0 . From the triggered condition (5), it can seen that the solution of the equation satisfies ψ (τ , 0) = P · F −1 . To solve Eq. (14), we P·F −1 can get τ = (||A||+||B||)·(P·F −1 +||A||+||B||) . 3.2

Finite-Time Consensus Protocol for Second-Order Integral Sliding Mode Systems

Next, the system (1) is showed as follows:  x˙i (t) = vi (t) v˙i (t) = ui (t) + di

(15)

˙ ∞ ≤ η , applying Theorem 3. Suppose that there exists a constant η > 0, such that ||d|| the protocol (2a) and (2b) yield to the system (15), and combining the integral sliding

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mode function (2c), we can guarantee that the multi-agent system sequentially slide along it to reach the accurate consensus tracking in finite time. 1

s˙i = −k1 sig 2 (si ) + ρi + di ρ˙ i = −k2 sgn(si ), i = 1, 2, . . . N Let zi = ρi + di 1

s˙i = −k1 sig 2 (si ) + zi z˙i = −k2 sgn(si ) + d˙i , i = 1, 2, . . . N.

(16)

Proof. The Lyapunov function candidate is chosen as V = ∑Ni=1 Vi , Vi = ∑ni=1 ξiT Pi ξi ,  1  2 2×2 > 0 and ξi = sig si where Pi ∈ R zi 

ξ˙i =  =

1 − 21 s˙ i 2 |si | z˙i

 

1 − 21 [−k sig 12 (s ) + z ] i i 1 2 |si | −k2 sgn(si ) + d˙i

(17)

  1 1 −1 −k1 sig 2 (si ) + zi 2 = |si | 1 2 −2[k2 − d˙i sgn(si )]sig 2 (si ) 1

= |si |− 2 Ai ξi 

 1 − 12 k1 2 so, we can get the first time derivative of V along the −[k2 − d˙i sgn(si )] 0 trajectories of system (15) is

where Ai =

N

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i=1

V˙ = ∑ V˙i = ∑ |si |− 2 ξiT (ATi Pi + Pi Ai )ξi 1

(18)

almost everywhere. Since the matrix A − i is Hurwitz if and only if k1 > 0, k2 > 0 and Pi is positive definite, one has N

N

i=1

i=1

V˙ = ∑ V˙i = − ∑ |si |− 2 ξiT Qi ξi < 0 1

(19)

where Pi and Qi are related by the algebraic Lyapunov equation ATi Pi + Pi Ai = −Qi , i = 1

1

1

1

1, 2, . . . , N. In light of |si | 2 = |sig 2 (si )| ≤ ||ξi ||2 ≤ λm in− 2 (Pi )Vi 2 , the inequality (19) can be further deduced to

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1

N

1

≤ −β ∑ Vi 2 i=1 N

1

(20)

≤ −β ∑ V 2 1

i=1

1 2

i )λm in(Qi ) where β = mini=1,...N { λm in λm(Pax(P } > 0. By Lemma 1, the exact value of finite time i) 1 2

t ∗ = 2V (0) can be obtained, so that V (t) = 0,t ≥ t ∗ . Then x j (t) = xi (t), v j (t) = vi (t). β It can be seen that the proposed protocol ui (t) can solve the problem of finite time consensus of the systems (15).

4 Simulation In this section, consensus sliding mode protocols with nonlinear event-triggered scheme will be applied to second-order leader-following multi-agents systems to verify the validity of the proposed results. Consider a network topology diagram of the four agents. We choose that γ1 = −1.5; γ2 = −0.55; k1 = −0.35; k2 = 7; the control gain Mi = Ni = [8; 6; 7; 5]T and the initial states x(0) = [−197; −172; 197; 170]T , v(0) = [95; 120; −120; −75]T , x0 = 35; v0 = 3, taking sample interval t = 0.1, the disturbances are selected as d1 = 0.3 ∗ cos(0.3 ∗ t + pi/4), d2 = 0.4∗sin(0.3∗t), d3 = 0.3∗sin(0.2∗t + pi/3), d4 = 0.4∗cos(0.2∗t). And the relationship between followers and leader is presented by bi . bi = [3.28; 3.5; 3.3; 3.5]. The network between the agents is directed based the adjacency matrix ai j . ⎡ ⎤ 0 1 0.7 1 ⎢0.7 0 0.8 1.1 ⎥ ⎥ ai j = ⎢ ⎣0.6 0.8 0 1.12⎦ 1.1 0.5 0.6 0 Evolution curve of position state, velocity state, the sliding mode protocol are shown in Figs. 1, 2 and 3, respectively. As shown in the figures, the leader can be traced by the followers and the second-order system achieve finite-time consensus under the protocol (2a), (2b) and (2c), which verify the correctness of Theorems 1 and 3. And the convergence time of the velocity is shorter than of the position, because the proposed protocol acts firstly on the velocity, and the position will slowly reach the consensus after the velocity state reaches agreement. In addition, Figs. 4 and 5 described the trigger state of each agent. Figure 4 shows the value of the agent’s trigger time, Fig. 5 depicts the event interval of agent. In short, there is a positive lower bound between the two event-triggered moments of the agent, so the zeno phenomenon is excluded, which is consistent with the results of Theorem 2.

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5 Conclusions In this paper, the finite-time distributed event-triggered consensus control is considered for second-order multi-agent systems with disturbances. The distributed event-triggered consensus strategy is proposed, which only rely on the information between the agent and its neighbor states. Thus the second-order leader-following system attain finite-time consensus under the protocol control. In addition, this paper also shows the exclusion of zeno phenomenon in the second order system with distributed event-triggered scheme. Moreover, based on the nonlinear event-triggered finite-time consensus protocol, a integral sliding mode protocol is developed, which can achieve accurate finite-time in the presence of disturbances. The above results in second-order multi-agents systems are extended to the high-order agents network with disturbances in the future work. Acknowledgments. This work was supported by the NSFC (61503231, 61473015).

References 1. Xiao F, Wang L, Chen J, Gao YP (2009) Finite-time formation control for multi-agent systems. Automatica 45(11):2605–2611. https://doi.org/10.1016/j.automatica.2009.07.012 2. Yang QK, Cao M, Marina HGD, Fang H, Chen J (2018) Distributed formation tracking using local coordinate systems. Syst Control Lett 111:70–78. https://doi.org/10.1016/j.sysconle. 2017.11.004 3. Du HB, Zhu WW, Wen GH, Duan ZS (2017) Distributed formation control of multiple quadrotor aircraft based on nonsmooth consensus algorithms. IEEE Trans Cybern 1–12. https://doi.org/10.1109/tcyb.2017.2777463

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4. Thien RTY, Kim Y (2018) Decentralized formation flight via PID and integral sliding mode control. Aerosp Sci Technol 81:322–332. https://doi.org/10.1016/j.ifacol.2018.12.003 5. Jiang T, Lin DF, Song T (2019) Novel integral sliding mode control for small-scale unmanned helicopter. J Franklin Inst 356(5):2668–2689. https://doi.org/10.1016/j.jfranklin. 2019.01.035 6. Boysen N, Schwerdfeger S, Weidinger F (2017) Scheduling last-mile deliveries with truckbased autonomous robots. Eur J Oper Res 217(3):1085–1099. https://doi.org/10.1016/j.ejor. 2018.05.058 7. Amigoni F, Luperto M, Schiaffonati V (2017) Toward generalization of experimental results for autonomous robots. Robot Auton Syst 90:4–14. https://doi.org/10.1016/j.robot.2016.08. 016 8. Haghighi R, Pang CK (2017) Robust concurrent attitude-podition control of a swarm of underactuated satellites. IEEE Transl Control Syst Technol 99:1–12. https://doi.org/10.1109/ TCST.2017.2656025 9. Varma S, Kumar KD (2012) Multiple satellite formation flying using defferential aerodynamic drag. J Spacecr Rockets 49(2):325–336. https://doi.org/10.2514/1.52395 10. Wang L, Xiao F (2010) Finite-time consensus problems for networks of dynamic agents. IEEE Trans Autom Control 55(4):950–955. https://doi.org/10.1109/TAC.2010.2041610 11. Zhang SX, Xie DS, Yan WS (2017) Decentralized event-triggered consensus control strategy for leader-follower networked systems. Phys A Stat Mech Appl 479:498–508. https://doi.org/ 10.1016/j.physa.2017.02.063 12. Tian XH, Liu HL, Liu HT (2018) Robust finite-time consensus control for multi-agent systems with disturbances and unknown velocities. ISA Trans 80:73–80. https://doi.org/10. 1016/j.isatra.2018.07.032 13. Meng XY, Xie LH, Soh YC (2017) Asynchronous periodic event-triggered consensus for multi-agent systems. Automatica 84:214–220. https://doi.org/10.1016/j.automatica.2017.07. 008 14. Hu ZL, Yang JY (2018) Distributed finite-time optimization for second order continuous-time multiple agents systems with time-varying cost function. Neurocomputing 287(26):173–184. https://doi.org/10.1016/j.neucom.2018.01.082 15. Jiang CR, Du HB, Zhu WW, Yin LS, Jin XZ, Wen GH (2018) Synchronization of nonlinear networked agents under event-triggered control. Inf Sci 459:317–326. https://doi.org/10. 1016/j.ins.2018.04.058 16. Xie YJ, Lin ZL (2018) Global leader-following consensus of a group of discrete-time neutrally stable linear systems by event-triggered bounded controls. Inf Sci 459:302–316. https:// doi.org/10.1016/j.ins.2018.02.061 17. Wang LM, Ge MF, Zeng ZG, Hu JH (2018) Finite-time robust consensus of nonlinear disturbed multiagent systems via two-layer event-triggered control. Inf Sci 466:270–283. https://doi.org/10.1016/j.ins.2018.07.039 18. Wang YY, Xia YQ, Li HY, Zhou PF (2018) A new integral sliding mode design method for nonlinear stochastic systems. Automatica 90:304–309. https://doi.org/10.1016/j.automatica. 2017.11.029 19. Bai J, Wen GG, Rahmani A, Yu YG (2017) Distributed consensus tracking for the fractionalorder multi-agent systems based on the sliding mode control method. Neurocomputing 235:210–216. https://doi.org/10.1016/j.neucom.2016.12.066 20. Yu SH, Long XJ (2015) Finite-time consensus for second-order multi-agent systems with disturbances by integral sliding mode. Automatica 54:158–165. https://doi.org/10.1016/j. automatica.2015.02.001 21. Wang GD, Wang XY, Li SH (2018) Sliding-mode consensus algorithms for disturbed secondorder multi-agent systems. J Franklin Inst 355(15):7443–7465. https://doi.org/10.1016/j. jfranklin.2018.07.027

An Entropy-Based Inertia Weight Krill Herd Algorithm Chen Zhao1,2 , Zhongxin Liu1,2(B) , Zengqiang Chen1,2 , and Yao Ning1,2 1

College of Artificial Intelligence, Nankai University, Tianjin 300350, China [email protected] 2 Tianjin Key Laboratory of Intelligent Robotics, Nankai University, Tianjin 300350, China

Abstract. The krill herd (KH) algorithm is an emerging meta-heuristic algorithm for solving complex problems. Though it is robust in optimization, there are some parameters that need to be fine-tuned for improved performance. This paper proposes an entropy-based inertia weight krill herd (EBIWKH) algorithm, which could adaptively adjust the inertia weight according to the variance of population entropy. It is tested on CEC2017 benchmark functions and compared with other inertia weight adjustment strategies. Experimental results show that the EBIWKH algorithm is more robust and stable than other compared algorithms. Keywords: The krill herd algorithm · Inertia weight adjustment · Population entropy

1 Introduction With the development of science and technology, practical optimization problems become more and more complex, some of which are even NP-hard in nature. In the face of these problems’ challenge, a great number of meta-heuristic optimization algorithms have been proposed by researchers over the past few decades. Among these intelligent algorithms, the krill herd (KH) algorithm [1] is very attractive due to its excellent computational stability and robust search ability. Since the inception of KH algorithm for tackling optimization problems, there have been a lot of work to improve its performance [2]. In [3], a stud selection and crossover operator is introduced into the standard KH algorithm to enhance its reliability and accurateness. In [4], an improved KH algorithm is proposed with the combination of opposition-based learning strategy to guide the global search process. In [5], a new krill herd migration operator is utilized to search optimal solutions for the regular KH algorithm. In [6], a global exploration operator is incorporated into the KH algorithm for solving data clustering problems. Although the KH algorithm perform well on optimization problems, there are some control parameters can be fine-tuned to further improve its performance. Among these parameters, inertia weight is critical due to its influence on the trade-off between exploration and exploitation [1]. However, few work have been done on the adjustment of c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 487–498, 2020. https://doi.org/10.1007/978-981-32-9682-4_51

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inertia weight in the KH algorithm. In this paper, several inertia weight adjustment strategies applied in the particle swarm optimization (PSO) [7] are introduced into the KH algorithm because inertia weight plays the same role in the two swarm intelligent algorithm. Moreover, this paper proposes an entropy-based inertia weight (EBIW) adjustment mechanism, in which inertia weight is adaptively changed according to the variation of population entropy. And this strategy is applied into the KH algorithm, yielding the entropy-based inertia weight krill herd (EBIWKH) algorithm. The effectiveness of the novel algorithm is tested on CEC2017 testing functions. It is illustrated that the novel algorithm performs more efficiently compared to other strategies. The mainframe of this paper is organized as follows. The standard KH algorithm is reviewed in Sect. 2. The present work is described in Sect. 3, which contains the introduction of several existing inertia weight adjustment strategies and a detailed description of the EBIWKH algorithm. Then, the superiority of the EBIWKH algorithm is verified on CEC2017 benchmark functions in Sect. 4. Finally, the present research and the future work are concluded in Sect. 5.

2 The Standard KH Algorithm The KH algorithm is a novel population-based evolutionary algorithm for global optimization problems [1]. When the KH algorithm is applied to find an optimal solution, each krill is influenced by three motions: (1) movement induced by other krill individuals; (2) foraging action; (3) random diffusion. In general, an optimization problem is d–dimensional, and the position of the ith krill can be described as a vector, Xi = (xi1 , xi2 , . . . , xid ) ∈ Rd , i = 1, 2, . . . , n. For a clear and simple description of the KH algorithm, the following Lagrangian model is adopted [8]: dXi = Ni + Fi + Di , (1) dt where Ni , Fi , Di ∈ Rd correspond to the three motions respectively. For the ith krill, the first motion, Ni , dependents on two parts: the attractive force offered by others and a repulsive effect. The movement can be modeled as: target

Ni = ω1 Niold + N max αi , αi = αilocal + αi

,

(2)

where Niold is the previous motion which offers a repulsive effect, ω1 ∈ (0, 1) is the inertia weight of the repulsive effect, and N max is the maximum induced speed. The direction of the attraction, αi , is determined by two factors: a local effect αilocal (protarget (supplied by the best krill individual). vided by neighbors) and a target effect αi For the ith krill, the second motion, Fi , is influenced by two parts: the current location of food and its previous information. The movement can be expressed as: Fi = ω2 Fiold +V f βi , βi = βifood + βibest ,

(3)

where Fiold contains the previous information about food, ω2 ∈ (0, 1) is the inertia weight of the second motion, and V f is the foraging speed. The direction, βi , is relied

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on two parts: an attractive effect βifood (provided by the current food location) and the effect βibest (supplied by the previous best position of the ith krill). The third motion, Di , is a stochastic process, which is defined as: Di = Dmax (1 − t/tmax )δ ,

(4)

where Dmax is the maximum diffusion speed of each krill, and the effect of Di gradually decreases with the increasing of iterations. The direction of the movement is decided by δ ∈ Rd , a random vector with elements in (−1, 1). During the update process, the position vector of the ith krill in [t,t + Δ t] is given as: dXi n , Δ t = c ∑ j=1 (b j − b j ), Xi (t + Δ t) = Xi (t) + Δ t (5) dt where b j and b j are the upper and lower bound of the jth element in Xi , and c is a constant scalar in (0, 2). It is noted that inertia weights (ω1 and ω2 ) are important to keep the balance between exploration and exploitation. In [1], both parameters are modified as:

ω (t) = ωinit − (ωinit − ωend ) × t/tmax ,

(6)

where ωinit = 0.9, ωend = 0.1. In the strategy, ω (t) decreases linearly from ωinit to ωend with iterations. In this way, exploration is emphasized at the beginning of searching process and exploitation is encouraged at the end of searching process. Due to its excellent performance, the improved version is considered as the standard KH algorithm in this paper.

3 Inertia Weight Adjustment Strategies and EBIWKH In this section, various existing inertia weight adjustment strategies are introduced into the KH algorithm for improved performance. Then, an entropy-based inertia weight (EBIW) adjustment mechanism is described in detail. The EBIWKH algorithm is proposed by applying the new strategy into the KH algorithm, the pseudo code of which is given for a clear description. 3.1 Several Existing Strategies PSO [7] is a classical population-based evolutionary algorithm, the searching process of which is defined as: Vi = ω Viold + c1 (Pi − Xi ) + c2 (Pg − Xi ),

(7)

where ω ∈ (0, 1) is the inertia weight, Viold is the previous velocity of the ith individual, Pi is the best previous position of the ith individual, and Pg is best position found so far by the whole population. The comparison between (2), (3) and (7) apparently shows that the inertia weight plays the same role in both algorithms, namely to keep a balance between the current information and history information. Therefore, it is a reasonable

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choice to introduce several existing strategies in PSO to the KH algorithm. Several existing inertia weight adjustment strategies are presented as follows. In [9], an exponential inertia weight (EIW) adjustment strategy is proposed, which is defined as: ω (t) = ωinit exp[−a(t/tmax )]. (8) In the strategy, ω is initialized to ωinit and its value is exponentially changed according to iterations. a is a scalar which could determines the final value of ω (t). In [10], an adaptive inertia weight (AIW) adjustment strategy is proposed, which is expressed as: ω (t) = (ωinit − ωend ) × s + ωend . (9) In the strategy, the inertia weight is modified according to the success rate, s, and it is defined as:  1, f (Pi (t)) < f (Pi (t − 1)) ∑ni=1 sˆi s= , sˆi = (10) 0, otherwise, n where n is the number of population, f (Pi (t)) is the fitness value of Pi (t). In [11], an adaptive chaotic inertia weight (ACIW) strategy is proposed, which is calculated by:

ω (t) = [(ωinit − ωend )(tmax − t)/tmax + ωend ] × z, z = 4s(1 − s).

(11)

In the strategy, the inertia weight is influenced by a chaotic sequence and the success rate. In [12], an oscillatory (OSC) strategy is proposed, and it is formulated as:

ω (t) ∼ N (μ (t), (σ (t))2 ), μ (t) = σ (t) = ωˆ (t),

(12)

where ωˆ (t) is adjusted with iterations:

ωˆ (t) = ωinit − (ωinit − ωend ) × t/tmax

(13)

In the strategy, the searching process are driven along oscillatory trajectories to explore and exploit the searching space thoroughly. 3.2

The EBIWKH Algorithm

Shannon’s information theory could measure the uncertainty of one system and it has been applied into various fields [13, 14]. To some extent, a high entropy indicates the diversity while a low entropy implies the similarity [14]. Due to its characteristic, entropy is utilized in this paper to reflect the population distribution and to tune the inertia weight better. Assume that G j = (x1j , x2j , . . . , xnj )T , j = 1, 2, . . . , d, is the jth gene sequence of the whole population. The jth-dimentional space is divided equally into M parts and the length of each part is Δ l = (b j − b j )/M. Let kmj , m = 1, 2, . . . , M, denote the number of elements in G j which lies in [b j + (m − 1)Δ l, b j + mΔ l]. Then, the entropy of the jth gene is calculated by: M hˆ j = − ∑m=1 pmj log2 pmj , pmj = kmj /n.

(14)

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The population entropy is estimated by: h=

∑dj=1 hˆ j . d

(15)

Based on the population entropy mentioned above, an EBIW adjustment mechanism is proposed: ωinit − ωend 20h + ωend , u = − 10. (16) ω (t) = 1 + exp(−u) log2 M The new strategy is incorporated into the KH algorithm to keep the balance between exploration and exploitation, yielding the EBIWKH algorithm. For a clear understanding of the proposed algorithm, its pseudo code is presented in Algorithm 1.

Algorithm 1. Entropy-based inertia weight krill herd algorithm 1: Initialize the position of n krill and evaluate their fitness. Assign M a value and calculate the population entropy h using formula (15). 2: while t < tmax do 3: Calculate ω (t) according to (16). 4: for i = 1 : n do 5: Calculate Ni , Fi , Di and update its position to Xi (t + 1). 6: end for 7: Update h. 8: t = t + 1. 9: end while 10: Output the global optimal solution.

In the EBIWKH algorithm, the inertia weight is adjusted with the variance of entropy. At the beginning phase, krill individuals are dispersed homogeneously in the whole searching space, h is close to log2 M and ω (t) is equal to ωinit . A large value of inertia at the beginning stage could help the KH algorithm explore the searching space thoroughly. At the later phase, krill individuals are focused on certain areas, h is close to 0 and ω (t) is equal to ωend . A small value of inertia weight at the later stage could encourage individuals exploit target areas carefully. Compared to the standard KH algorithm, the variance of inertia weight is based on the population entropy, not on iterations. The usage of population information to finetune parameter may be more reasonable. The effectiveness of the EBIWKH algorithm will be tested on several optimization problems in the next section.

4 Simulation and Analysis A series of experiments are given as follows, which are implemented in Matlab R2018a running on a PC with Windows 10 64-bit OS, Intel i5-6500 CPU at 3.20 GHz, and 8 GB RAM.

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The effectiveness of the proposed algorithm is evaluated through experiments on CEC2017 testing functions [15]. The testing set is categorized into four parts: unimodal functions f1 − f3 , multimodal functions f4 − f10 , hybrid functions f11 − f20 , composition functions f21 − f30 . The searching range of all dimensions is set to [−100, 100], and the optimal fitness values are: fi (X ∗ ) = i × 100, i = 1, 2, . . . , 30, for example, f1 (X ∗ ) = 100. To verify the superiority of the EBIW strategy, several existing strategies described in Sect. 3.1 are incorporated into the KH algorithm for comparison, which contains EIW [9], AIW [10], ACIW [11], OSC [12]. For all strategies, the same parameters are used: ωinit = 0.9, ωend = 0.1, the population size is set to 50, the dimension is set to 30 and the maximum number of iteration is 3000. Besides, a is set to 2 for the EIW strategy and M is set to 50 for the EBIW strategy. Owing to the randomness of meta-heuristic algorithms, the performance of one algorithm cannot be judged by the result of a single run. Therefore, each algorithm for different benchmark functions is executed 50 times independently. In order to compare the performance of different methods quantitatively, the original results are normalized. An example of normalization is described as follows. As shown in Table 1, the first row shows the original average values of f22 , the maximum of which is 2532.338 (AIWKH) and the minimum of which is 2367.598 (EBIWKH). Then, the value of KH is normalized to (2496.567 − 2367.598)/(2532.338 − 2367.598) = 0.783. Similarly, the normalized average values of f22 can be obtained, which are listed in the second row of Table 1. After normalization, a performance index Σ of each methods is provided, namely a cumulative sum of all normalized value found on each functions. The results of 50 trials are shown in Tables 2, 3 and 4. Table 1. The original and normalized average value of f22 Values

KH

Original values

2496.567 2367.598 2532.338 2528.905

Normalized values

EBIWKH AIWKH 0.783

0.000

1.000

ACIWKH EIWKH 0.979

OSCKH

2427.478 2508.462 0.363

0.855

From Table 2, it is seen that the EBIWKH algorithm performs the best on twenty of thirty benchmark functions. Besides, the values of Σ show that the EBIWKH algorithm is more effective at solving CEC2017 problems compared to other five algorithms. Though the OSCKH algorithm only performs the best on f2 , its performance index ranks the second, which means the OSCKH algorithm is possessed of balanced capability in tackling CEC2017 problems. The standard KH algorithm performs well on two multimodal functions, f6 and f10 , and its performance index ranks the third. As shown in Table 3, the EBIWKH algorithm surpasses other algorithms on fifteen functions and its Σ is minimum. The OSCKH algorithm ranks second and the basic KH algorithm ranks third. The EIWKH algorithm ranks fourth while it performs well on five functions. In combination of the results in Table 2, the EIWKH algorithm could find the optimal value of f1 , f10 and f15 , but its average performance is not the best on the three functions. That means the performance of the EIWKH algorithm for solving CEC2017 problems is instable.

An Entropy-Based Inertia Weight KH Algorithm Table 2. Mean normalized optimization results Functions

EIWKH

OSCKH

f1

0.441

0.000

0.518

1.000

0.484

0.393

f2

0.193

0.077

0.192

0.602

1.000

0.000

f3

0.711

0.000

0.879

1.000

0.931

0.725

f4

0.598

0.000

0.847

1.000

0.682

0.439

f5

0.079

0.228

0.970

1.000

0.000

0.371

f6

0.000

0.017

0.734

1.000

0.184

0.219

f7

0.176

0.079

0.219

1.000

0.000

0.053

f8

0.140

0.000

1.000

0.614

0.186

0.298

f9

0.238

0.000

0.532

1.000

0.309

0.293

f10

0.000

0.657

0.574

1.000

0.557

0.670

f11

0.686

0.000

0.964

1.000

0.712

0.526

f12

0.342

0.000

0.864

1.000

0.718

0.537

f13

0.483

0.000

0.755

1.000

0.522

0.372

f14

0.582

0.000

0.800

1.000

0.663

0.411

f15

0.355

0.000

0.835

1.000

0.643

0.521

f16

0.472

0.000

0.871

1.000

0.609

0.395

f17

0.277

0.000

0.921

1.000

0.909

0.456

f18

0.456

0.000

0.797

1.000

0.653

0.438

f19

0.512

0.000

1.000

0.925

0.733

0.570

f20

0.127

0.150

0.983

1.000

0.000

0.094

f21

0.442

0.591

0.000

1.000

0.456

0.740

f22

0.783

0.000

1.000

0.979

0.363

0.855

f23

0.600

0.341

0.000

1.000

0.401

0.454

f24

0.582

0.723

0.000

1.000

0.989

0.154

f25

0.580

0.000

0.911

1.000

0.651

0.564

f26

0.662

0.313

0.813

0.000

1.000

0.077

f27

0.499

0.000

0.626

1.000

0.436

0.348

f28

0.528

0.000

0.956

1.000

0.717

0.712

f29

0.504

0.000

0.942

1.000

0.630

0.130

f30

0.504

0.000

0.808

1.000

0.683

0.538

12.550

3.175

21.310

28.121

16.821

12.353

Σ

KH

EBIWKH

Fig. 1. Performance comparison for f4

AIWKH

ACIWKH

Fig. 2. Performance comparison for f9

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EIWKH

OSCKH

f1

0.535

0.161

0.236

1.000

0.000

0.412

f2

0.404

1.000

0.000

0.129

0.157

0.005

f3

0.652

0.000

0.795

1.000

0.712

0.539

f4

0.476

0.000

0.663

1.000

0.511

0.324

f5

0.172

0.541

0.759

1.000

0.000

0.297

f6

0.278

0.420

0.000

1.000

0.350

0.455

f7

0.590

0.298

0.000

1.000

0.118

0.960

f8

0.160

0.000

1.000

0.803

0.357

0.518

f9

0.513

0.000

0.523

1.000

0.062

0.442

f10

0.031

0.158

1.000

0.838

0.000

0.425

f11

0.755

0.000

0.965

1.000

0.772

0.647

f12

0.321

0.000

1.000

0.750

0.399

0.353

f13

0.373

0.000

0.667

0.613

1.000

0.628

f14

0.000

0.193

0.131

1.000

0.769

0.265

f15

0.229

0.086

0.163

1.000

0.000

0.159

f16

0.768

0.000

0.709

1.000

0.807

0.515

f17

1.000

0.000

0.340

0.413

0.621

0.098

f18

0.000

0.217

0.070

1.000

0.262

0.395

f19

0.663

0.000

1.000

0.695

0.675

0.512

f20

0.847

0.137

1.000

0.658

0.000

0.020

f21

1.000

0.000

0.933

0.126

0.404

0.551

f22

0.000

0.197

0.135

1.000

0.238

0.418

f23

0.869

0.897

0.037

1.000

0.815

0.000

f24

0.545

1.000

0.508

0.000

0.017

0.135

f25

0.690

0.000

1.000

0.862

0.554

0.552

f26

0.951

0.000

1.000

0.107

0.630

0.726

f27

0.487

0.313

0.549

1.000

0.322

0.000

f28

0.428

0.000

1.000

0.581

0.843

0.464

f29

0.422

0.000

0.452

1.000

0.241

0.328

f30

0.000

0.159

0.121

0.481

1.000

0.391

14.159

5.776

16.755

23.057

12.635

11.533

Σ

KH

EBIWKH

Fig. 3. Performance comparison for f11

AIWKH

ACIWKH

Fig. 4. Performance comparison for f16

An Entropy-Based Inertia Weight KH Algorithm Table 4. The normalized variance of different methods Functions

EIWKH

OSCKH

f1

0.440

0.000

0.510

1.000

0.954

0.393

f2

0.008

0.001

0.010

0.128

1.000

0.000

f3

0.237

0.000

1.000

0.818

0.439

0.695

f4

0.523

0.000

0.836

1.000

0.773

0.369

f5

0.992

0.000

0.996

1.000

0.519

0.367

f6

0.006

0.000

1.000

0.627

0.015

0.037

f7

0.750

0.000

0.964

0.438

1.000

0.190

f8

0.000

0.493

1.000

0.879

0.316

0.564

f9

0.086

0.000

1.000

0.674

0.542

0.630

f10

0.308

0.625

0.000

0.878

0.523

1.000

f11

0.672

0.000

1.000

0.722

0.587

0.440

f12

0.406

0.000

1.000

0.928

0.790

0.027

f13

0.438

0.000

0.769

1.000

0.222

0.388

f14

0.657

0.000

0.789

1.000

0.294

0.238

f15

0.293

0.000

0.936

1.000

0.591

0.598

f16

0.647

0.000

1.000

0.765

0.794

0.574

f17

0.000

0.326

0.467

1.000

0.169

0.231

f18

0.338

0.000

0.545

1.000

0.540

0.301

f19

0.145

0.000

1.000

0.840

0.652

0.406

f20

0.000

0.348

0.453

1.000

0.820

0.319

f21

0.625

0.934

0.000

1.000

0.539

0.237

f22

0.619

0.000

1.000

0.845

0.262

0.724

f23

0.000

0.206

0.415

0.465

0.528

1.000

f24

0.506

0.000

0.343

1.000

0.684

0.235

f25

0.139

0.000

0.960

1.000

0.338

0.348

f26

0.000

0.850

0.469

1.000

0.314

0.363

f27

1.000

0.000

0.183

0.953

0.412

0.753

f28

0.320

0.000

1.000

0.745

0.437

0.402

f29

0.693

0.720

1.000

0.698

0.000

0.536

f30

0.670

0.000

0.859

1.000

0.890

0.377

11.516

4.503

21.507

25.404

15.946

12.743

Σ

KH

EBIWKH

Fig. 5. Performance comparison for f22

AIWKH

ACIWKH

Fig. 6. Performance comparison for f28

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The data in Table 4 illustrate the stability and reliability of the EBIWKH algorithm. It has a good performance on twenty-one testing functions and its performance index is minimum. In other words, the EBIWKH algorithm can always find optimal solutions on CEC2017 benchmark functions. Besides, the KH algorithm ranks second and the OSCKH algorithm ranks third. In addition, convergence graphs are drawn to give a detailed description of the proposed algorithm. The results taken for drawing the graphs are the original average values obtained from 50 independent runs, not normalized. Due to the limitation of paper, only six representative problems are provided in Figs. 1, 2, 3, 4, 5 and 6. Among the six problems, f4 and f9 are multimodal, f11 and f16 are hybrid functions, f22 and f28 belong to composition functions set. The convergence graphs of simple unimodal functions are not given because these functions are too simple to clarify the performance of different strategies. From the six graphs, it is easy to conclude that the EIWKH algorithm is superior to all others. At the beginning searching phase, the EIWKH algorithm spends less time to find candidate areas owing to its excellent convergence. At the later searching phase, the EIWKH algorithm could search the candidate areas carefully to find an optimal solution. Therefore, the EBIW adjustment strategy could significantly improve the performance of the KH algorithm. Based on the analysis above-mentioned, it can be seen that the EBIWKH is able to obtain good result on most CEC2017 benchmark functions. The no free lunch theorems [16] have taught us that expecting an algorithm to perform the best on all functions is unrealistic. Therefore, it can be concluded that EBIWKH algorithm, to some extent, is the best among six algorithms on CEC2017 problems. And the comparison with other methods shows that the EBIW adjustment strategy is capable of enough competitiveness on the KH algorithm.

5 Conclusion The krill herd (KH) algorithm is a robust swarm intelligent algorithm for optimization. In the novel algorithm, there is a critical parameter which influence its performance, namely the inertia weight. It is required to fine-tune the parameter for improved performance. Firstly, several existing inertia weight adjustment strategies are introduced into the KH algorithm, the effectiveness of which have been verified in the particle swarm optimization algorithm. Then, an entropy-based inertia weight (EBIW) adjustment mechanism is proposed for solving this problem. In the new strategy, the inertia weight is modified with the variance of population entropy, which could reflect the diversity of population. Due to the entropy reduction in optimization process, the strategy could help the KH algorithm to explore space at the beginning phase and to exploit thoroughly in the final phase. Compared to other strategy, the EBIW method is relatively reasonable with the introduction of population information. Finally, the experiment results implemented on CEC2017 problems have seen the effectiveness and reliability of the entropy-based inertia weight krill herd (EBIWKH) algorithm.

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Future research will focus on two issues. Firstly, for more verification, the EBIWKH algorithm would be applied into solving practical engineering problems, such as testsheet composition problem and sensor selection problem. Secondly, the KH algorithm will be hybridized with other strategies for improved performance. Acknowledgments. This work is supported by National Natural Science Foundation of China under Grant No.61573200, 61573199.

References 1. Gandomi AH, Alavi AH (2012) Krill herd: a new bio-inspired optimization algorithm. Commun Nonlinear Sci Numer Simul 17(12):4831–4845. https://doi.org/10.1016/j.cnsns.2012. 05.010 2. Wang GG, Gandomi AH, Alavi AH, Gong D (2019) A comprehensive review of krill herd algorithm: variants, hybrids and applications. Artif Intell Rev 51(1):119–148. https://doi.org/ 10.1007/s10462-017-9559-1 3. Wang GG, Gandomi AH, Alavi AH (2014) Stud krill herd algorithm. Neurocomputing. 128(5):363–370. https://doi.org/10.1016/j.neucom.2013.08.031 4. Wang GG, Deb S, Gandomi AH, Alavi AH (2016) Opposition-based krill herd algorithm with Cauchy mutation and position clamping. Neurocomputing. 177:147–157. https://doi. org/10.1016/j.neucom.2015.11.018 5. Wang GG, Gandomi AH, Alavi AH (2014) An effective krill herd algorithm with migration operator in biogeography-based optimization. Appl Math Model 38(9):2454–2462. https:// doi.org/10.1016/j.apm.2013.10.052 6. Jensi R, Jiji GW (2016) An improved krill herd algorithm with global exploration capability for solving numerical function optimization problems and its application to data clustering. Appl Soft Comput 46:230–245. https://doi.org/10.1016/j.asoc.2016.04.026 7. Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of ICNN 1995 - international conference on neural networks, vol 4, pp 1942–1948. https://doi.org/10.1109/ ICNN.1995.488968 8. Hofmann EE, Haskell A, Klinck JM, Lascara CM (2004) Lagrangian modelling studies of Antarctic krill ( Euphausia superba ) swarm formation. ICES J Mar Sci 61(4):617–631. https://doi.org/10.1016/j.icesjms.2004.03.028 9. Ting TO, Shi Y, Cheng S, Lee S (2012) Exponential inertia weight for particle swarm optimization. In: Advances in swarm intelligence, vol 1, no 8, pp 83–90. https://doi.org/10.1007/ 978-3-642-30976-2 10 10. Nickabadi A, Ebadzadeh MM, Safabakhsh R (2011) A novel particle swarm optimization algorithm with adaptive inertia weight. Appl Soft Comput 11(4):3658–3670. https://doi.org/ 10.1016/j.asoc.2011.01.037 11. Arasomwan MA, Adewumi AO (2013) On adaptive chaotic inertia weights in particle swarm optimization. In: 2013 IEEE symposium on swarm intelligence (SIS), pp 72–79. https://doi. org/10.1109/SIS.2013.6615161 12. Shi H, Liu S, Wu H, Li R, Liu S, Kwok N, Peng Y (2018) Oscillatory particle swarm optimizer. Appl Soft Comput 73:316–327. https://doi.org/10.1016/j.asoc.2018.08.037 13. Hinojosa S, Dhal KG, Elaziz MA, Oliva D, Cuevas E (2018) Entropy-based imagery segmentation for breast histology using the Stochastic Fractal Search. Neurocomputing. 321:201– 215. https://doi.org/10.1016/j.neucom.2018.09.034 14. Zhang H, Xie J, Ge J, Lu W, Zong B (2018) An entropy-based PSO for DAR task scheduling problem. Appl Soft Comput 73:862–873. https://doi.org/10.1016/j.asoc.2018.09.022

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15. Awad NH, Ali MZ, Suganthan PN, Liang JJ, Qu BY (2016) Problem definitions and evaluation criteria for the CEC 2017 special session and competition on single objective realparameter numerical optimization. Technical report 16. Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1(1):67–82. https://doi.org/10.1109/4235.585893

State Estimation for One-Sided Lipschitz System with Markovian Jump Parameters Zhenkun Zhu, Jun Huang(B) , and Ming Yang School of Mechanical and Electrical Engineering, Soochow University, Suzhou 215021, China [email protected]

Abstract. This paper deals with the state estimation for one-sided Lipschitz system with Markovian jump parameters. The stochastic observer frame is constructed by Luenberger observer theory. In the following sections, the observer is designed and sufficient conditions are given so that the error system is exponentially stable in mean square. Finally, one example is simulated to demonstrate the proposed methods are effective.

Keywords: One-sided Lipschitz Markovian jump parameters

1

· Full-order observer ·

Introduction

In recent years, one-sided Lipschitz system (OLS) has become a hot topic [1– 4]. This sort of systems can represent more plants in reality since one-sided Lipschitz function is more general than Lipschitz function. In [5], Hu first put forward nonlinear observer design for one-sided Lipschitz condition (OLC). On the basis of [5,6] presented quasi-OLC to improve the result of [5]. [7] proposed OLS with unified observer framework after introducing the quadratically innerbounded condition, which has been adopted by most of works. [8] derived the unified observer framework by S-procedure, which makes the condition for the existence of observer less conservative than that in [7]. Following this line, [9] and [10] designed the full-order and reduced-order observers respectively. When it turns to the discrete OLS, several results can be found in [11–13]. More recently, under OLC and quadratically inner-bounded condition, [14] designed full-order observer with unknown input by using linear matrix inequality (LMI), and [15] studied OLS with uncertain parameter and admissible external disturbance. However, when the nonlinear function only satisfies OLC, few researches have been completed except for [16]. Furthermore, this kind of systems sometimes includes several operation modes with Markovian jump parameters, which need more advanced methods to treat. Motivated by above discussions, we try to develop new approaches to cope with the problems mentioned above. The rest of the paper is organized as follows. Section 2 presents the problem and gives some important preparations. The main c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 499–507, 2020. https://doi.org/10.1007/978-981-32-9682-4_52

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results, including sufficient conditions that make the error systems exponentially stable in mean square (ESMS), are given in Sect. 3. Section 4 reveals that the designed full-order observer is effective through a numerical example. In the text, ·, · means the inner product. W < (>)0 stands for the negative(positive) definite matrix W with W = W T . λmin (W ) denotes the minimal eigenvalues of the matrix W and λmax (W ) represents the maximal eigenvalues of the matrix W .

2

Problem Formulation and Preparations

Start with system as follows  x(t) ˙ = A(s(t))x(t) + D(s(t))f (H(s(t))x(t), u(t)), y(t) = C(s(t))x(t),

(1)

where x(t) ∈ Rm represents the state of system, u(t) ∈ Rs , y(t) ∈ Rk are the control input and output respectively. f (·, ·) : Rm × Rs → Rm is a nonlinear function. A(s), D(s), H(s) and C(s) are given matrices with appropriate dimensions. s(t), t ≥ 0 is a right-continuous Markov chain, which takes values on the probability space in a finite state space S = {1, 2, · · · , N }, where Π = (πiq ) (i, q ∈ S) is the generator given by  1 + πii Δ + o(Δ), q = i, P {s(t + Δ) = q|s(t) = i} = q = i, πiq Δ + o(Δ), o(Δ) = 0, πiq ≥ 0 represents the transformation rate Δ  πiq . from mode i of time t to mode q of time t + Δ if q = i and πii = − where Δ > 0, and lim

Δ→0

q=i

In this paper, we investigate the condition where the transformation rates i i of s(t) are partly unknown. Denote that S = Suk ∪ Ski , where Suk = {q : i πiq is unknown} and Sk = {q : πiq is known}. The variable t is omitted for simplicity. Then, the system (1) can be represented by  x(t) ˙ = A(s)x + D(s)f (H(s)x, u), (2) y(t) = C(s)x. Firstly, let us introduce some basic knowledge about one-sided Lipschitz nonlinear function. Definition 1. If the following holds, ≤ α||w − z||2 , ∀ w, z ∈ D,

(3)

where α is a constant. Then, in the region D concerned with x, f (x, u) is called one-sided Lipschitz function. If the inequality (3) holds everywhere, the function is globally one-sided Lipschitz. The following assumptions are needed to obtain the main result.

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Assumption 1. The function f (x, u) is only one-sided Lipschitz in D, i.e. f (x, u) satisfies the condition (3). Remark 1. If the following holds, (f (w, u) − f (z, u))T (f (w, u) − f (z, u)) ¯ ≤ ρ||w − z||2 + ψ , ∀ w, z ∈ D,

(4)

¯ concerned with x, the f (x, u) where ρ and ψ are constants. Then, in the region D satisfies quadratically inner-bounded condition. The relationship among the three kinds of functions is shown in Fig. 1. It can be seen that if a function satisfies OLC, it may not be quadratically innerbounded. Therefore, the type of the system considered here is more general.

One−Sided Lipschitz

Lipschitz

Quadratically Inner−Bounded

Fig. 1. The one-sided Lipschitz, Lipschitz and quadratically inner-bounded function sets

Then, the full-order observer is constructed as follows: x ˆ˙ = A(s)ˆ x + D(s)f (H(s)ˆ x + F (s)(y − Cx), u) + L(s)(y − C(s)ˆ x),

(5)

where L(s) ∈ Rm×k is the observer gain. The error system can be achieved by subtracting (5) from (2): e˙ = (A(s) − L(s)C(s))e + D(s)Δf (s),

(6)

where e = x − x ˆ and Δf (s) = f (H(s)x, u) − f (H(s)ˆ x + F (s)(y − Cx), u). According to the error system (6), we introduce the following definitions. Definition 2. Let C 2 (Rm × S; R) be the set of non-negative functions G(e, s) on Rm × S, for each G(e, s), the weak infinitesimal operator LG is defined by LG(e, i) = Ge (e, s) ·

 de  + πiq G(e, q), dt s=i q∈S

where Ge (e, s) = (

∂G(e, s) ∂G(e, s) ,··· , ). ∂e1 ∂en

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Definition 3. Denote the initial state of error system by e(0) = e0 and s(0) = s0 . If there are positive constants b1 , b2 satisfying ∀ t ≥ 0, E[||e||2 ] ≤ b1 exp{−b2 t}E[||e0 ||2 ]. The error system (6) is said to be ESMS. In what follows, any matrix N (s)|s=i is denoted by Ni . For instance, B(s)|s=i and f (H(s)x, u, s)|s=i are denoted by Bi and f (Hi x, u, i), respectively. In the next section, we will present sufficient conditions, where the error system (6) is ESMS.

3

Main Result

Theorem 1. Let Assumption 1 hold. If there exist matrices Pi ∈ Rm×m > 0, Wi ∈ Rm×m > 0, Yi ∈ Rm×k , Fi ∈ Rm×k , and constants γi > 0 such that for any i ∈ S,  πiq (Pq − Wi ) + γi I ≤ 0, (7) Γi + 2α(Hi − Fi Ci )T (Hi − Fi Ci ) + q∈Ski

where Γi = Pi Ai + ATi Pi − Yi Ci − CiT YiT , DiT Pi = Hi − Fi Ci ,

(8)

i Pq − Wi ≤ 0, ∀ q ∈ Suk , q = i,

(9)

Pq − Wi ≥ 0, ∀ q ∈

i Suk ,

q = i,

(10)

then the error system (6) is ESMS. Meanwhile, the observer gains can be designed by Li = Pi−1 Yi . Proof. It is derived from (8) that eT Pi Di Δfi = eT (Hi − Fi Ci )T Δfi = eT (Hi − Fi Ci )T (f (Hi x, u) − f (Hi x ˆ + Fi (y − Cx), u)).

(11)

By applying Assumption 1, we can get ˆ + Fi (y − Cx), u)) eT (Hi − Fi Ci )T (f (Hi x, u) − f (Hi x ≤ αeT (Hi − Fi Ci )T (Hi − Fi Ci )e. Thus,

eT Pi Di Δfi ≤ αeT (Hi − Fi Ci )T (Hi − Fi Ci )e.

(12)

(13)

Consider the Lyapunov function candidate as follows: G(e, s) = eT P (s)e.

(14)

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According to Definition 2, along with the error system (6), LG(e, i) can be calculated as LG(e, i) =2eT Pi (Ai − Li Ci )e + 2eT Pi Di Δfi   (15) + πiq eT Pq e + πiq eT Pq e. q∈Ski

i q∈Suk

Combining (13) with (15) results in LG(e, i) ≤2eT Pi (Ai − Li Ci )e + 2αeT (Hi − Fi Ci )T (Hi − Fi Ci )e   πiq eT Pq e + πiq eT Pq e. + Since

 q∈S

q∈Ski

(16)

i q∈Suk

πiq = 0, we have −



πiq eT Wi e = 0.

(17)

q∈S

Then, adding (17) to (15) yields LG(e, i) =2eT Pi (Ai − Li Ci )e + 2αeT (Hi − Fi Ci )T (Hi − Fi Ci )e   πiq eT (Pq − Wi )e + πiq eT (Pq − Wi )e. + q∈Ski

Let us turn to the term indicates  i q∈Suk



i q∈Suk πiq eT (Pq

 i q∈Suk

(18)

i q∈Suk

πiq eT (Pq − Wi )e. If q = i, then πiq ≥ 0, thus (9)

πiq eT (Pq − Wi )e ≤ 0. If q = i, then πiq < 0, thus (10) means that − Wi )e ≤ 0. Hence, the following inequality holds 

πiq eT (Pq − Wi )e ≤ 0.

(19)

i q∈Suk

Substituting (19) into (18) yields LG(e, i) ≤2eT Pi (Ai − Li Ci )e + 2αeT (Hi − Fi Ci )T (Hi − Fi Ci )e  πiq eT (Pq − Wi )e. +

(20)

q∈Ski

In view of (7), we can obtain LG(e, i) ≤ −γi eT e. Let γ = min{γi }, λ1 = max{λmax (Pi )}, λ2 = min{λmin (Pi )} and γm = i∈S

i∈S

i∈S

follows from (21) that LG(e, s) < −γeT e < −γm G(e, s).

(21) γ . It λ1 (22)

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Then, we obtain  t LG(e(v), s(v))dv]. E[G(e, s)] − E[G(e0 , s0 )] = E[

(23)

0

Combining (22) with (23) yields  E[G(e, s)] − E[G(e0 , s0 )] < −γm

t

E[G(e(v), s(v))]dv.

(24)

0

By Gronwall-Bellman inequality, we have E[G(e, s)] < exp{−γm t}E[G(e0 , s0 )],

(25)

which implies that E[||e||2 ]
0, Wi ∈ Rm×m > 0, Yi ∈ Rm×k , Fi ∈ Rm×k , and constants γi > 0 which satisfy that for any i ∈ S   Φi (Hi − Fi Ci )T < 0, (27) 1 ∗ − 2|α| I where Φi = Pi Ai + ATi Pi − Yi Ci − CiT YiT +

 q∈Ski

πiq (Pq − Wi ) + γi I, Yi = Pi Li ,

DiT Pi = Hi − Fi Ci ,

(28)

i , q = i, Pq − Wi ≤ 0, ∀ q ∈ Suk

(29)

i , q = i, Pq − Wi ≥ 0, ∀ q ∈ Suk

(30)

then the error system (6) is ESMS. The proof for the above theorem is obvious since the condition (27) is just the sufficient conditions for (7).

State Estimation

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Numerical Example

We study the system (2) with: 

  

0.1 −0.21 0.5 0.3 A1 = , D1 = , C1 = 1 0 , 0.1 −9.5 −0.5 −4.1     u 2 1 , f (H1 x, u) = H1 = , 1 1 −3 sin(2x1 + x2 ) + (3x2 − x1 ) 3    

0.2 −0.401 0.6 0 , D2 = , C2 = 1 0 , A2 = 0.2 −9.3 1 −2.1     u 1.9 3.8 , f (H2 x, u) = H2 = , 1 3 −2 sin(1.9x1 + 3.8x2 ) + (2x2 − 3x1 ) 3    

0.3 −1 0.75 2 , D3 = , C3 = 1 0 , A3 = −0.5 −7.2 3 −3.43    u 2 1.5 1 , u = sin t. , f (H3 x, u) = H3 = 0 −1 sin(2x1 + 1.5x2 ) + x23 From Theorem 2, we can solve that     26.8 2.95 11600 2.69 P1 = , W1 = , 2.95 0.947 2.69 5.46       11400 −9.93 648 , F1 = , L1 = , Y1 = −8.92 5.04 −2030     849 4.75 526 4.1 , W2 = , P2 = 4.75 0.952 4.1 0.95       136000 −512 155 , F2 = , L2 = , Y2 = 1580 13 888     53.6 0.25 11600 3.88 , W3 = , P3 = 0.25 0.438 3.88 2.49       14100 −25.5 399 , F3 = , L3 = . Y3 = −63.2 −70.4 −372 In the sequel, we use Monte Carlo method to simulate the state of the system and the observer. The initial state of x and x ˆ are given by [1 2]T , [2 2.5]T respectively. And we define the error between the system and observer as e (e = x − x ˆ). Figure 2 along with Fig. 3 shows the error trajectories of the fullorder observer under random switching signals s(t). It is indeed that the designed full-order observer is effective.

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Fig. 2. Error between the system state x1 and the estimated state of the designed full-order observer x ˆ1

Fig. 3. Error between the system state x2 and the estimated state of the designed full-order observer x ˆ2

5

Conclusion

To summarize, we research the observer design for the OLS with Markovian jump parameters. What makes our work different from current works is that we only consider OLC. After giving the sufficient conditions by LMI forms, we simulate the relevant full-order observer to show the presented methods are effective. Acknowledgement. Thanks to China Postdoctoral Science Foundation (2017M61 1903) and the National Natural Science Foundation of China (61403267).

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References 1. Zhang W, Su H, Su S, Wang D (2014) Nonlinear H∞ observer design for one-sided Lipschitz systems. Neurocomputing 145:505–511 2. Nguyen M, Trinh H (2016) Observer design for one-sided Lipschitz discrete-time systems subject to delays and unknown inputs. SIAM J Control Optim 54:1585– 1601 3. Nguyen M, Pathirana N, Trinh H (2018) Robust observer design for uncertain one-sided Lipschitz systems with disturbances. Int J Robust Nonlinear Control 28:1366–1380 4. Abbas R, Mohammad A (2019) Robust H∞ sliding mode observer-based faulttolerant control for one-sided Lipschitz nonlinear systems. Asian J Control 21:114– 129 5. Hu G (2006) Observers for one-sided Lipschitz non-linear systems. IMA J Math Control Inf 23:395–401 6. Hu G (2012) A note on observer for one-sided Lipschitz non-linear systems. IMA J Math Control Inf 17:4968–4977 7. Abbaszadeh M, Marquez H (2010) Nonlinear observer design for one-sided Lipschitz systems. In: Proceedings of the American Control Conference 8. Zhang W, Su H, Liang Y, Han Z (2012) Non-linear observer design for one-sided Lipschitz systems: an linear matrix inequality approach. IET Control Theory Appl 6:1297–1303 9. Zhang W, Su H, Wang H, Han Z (2012) Full-order and reduced-order observers for one-sided Lipschitz nonlinear systems using Riccati equations. Commun Nonlinear Sci Numer Simul 17:4968–4977 10. Lan Y, Wang L, Ding L (2016) Full-order and reduced-order observer design for a class of fractional-order nonlinear systems. Asian J Control 18:1467–1477 11. Mohamed B, Mohamedand B, Michel Z (2012) Observer design for one-sided Lipschitz discrete-time systems. Syst Control Lett 61:879–886 12. Minh N, Trinh H (2016) Unknown input observer design for one-sided Lipschitz discrete-time systems subject to time-delay. Appl Math Comput 286:57–71 13. Yang Y, Lin C, Chen B (2019) Nonlinear H∞ observer design for one-sided Lipschitz discrete-time singular systems with time-varying delay. Int J Robust Nonlinear Control 9:252–267 14. Tian J, Ma S, Zhang C (2019) Unknown input reduced-order observer design for one-sided Lipschitz nonlinear descriptor Markovian jump systems. Asian J Control 21:952–964 15. Badreddine E, Hicham E, Abdelaziz H (2019) New approach to robust observerbased control of one-sided Lipschitz non-linear systems. IET Control Theory Appl 13:333–342 16. Zhang W, Su H, Zhu F (2016) Improved exponential observer design for one-sided Lipschitz nonlinear systems. Int J Robust Nonlinear Control 26:3958–3973

Model Reference Adaptive Control with Unknown Gain Sign Heqing Liu, Tianping Zhang(B) , Ziwen Wu, and Yu Hua College of Information Engineering, Yangzhou University, Yangzhou 225127, China [email protected]

Abstract. In this paper, a model reference adaptive control scheme is proposed by using Nussbaum function for first order linear system with unknown gain sign. Only one self-tuning parameter is designed. Based on Lyapunov method, the Nussbaum parameter is determined. Using Nussbaum function, unknown gain sign is handled. According to Babalat’s lemma, the tracking error is proved to converge to zero. A numerical example is provided to demonstrate the effectiveness of the proposed method.

Keywords: Model reference adaptive control Nussbaum function

1

· Unknown gain sign ·

Introduction

It is well known that model reference adaptive control and self-tuning control are two conventional control methods. However, their design needs to know the control gain sign. In order to solve the unknown gain sign problem, Nussbaum function was introduced in [1–4]. Adaptive neural control was developed for a class of output feedback nonlinear systems with unknown high-frequency gain sign in [5]. In [6], adaptive dynamic surface control was investigated for constrained nonlinear systems with unknown gain sign. Adaptive control was addressed based on backstepping design in [7]. However, unknown gain sign had not been discussed for model reference control in the existing adaptive control teaching books [7,8]. In this paper, the problem of model reference adaptive control is discussed for first order linear system with unknown gain sign. The adaptive laws of Nussbaum parameter and an adjustable unknown parameter are designed by using Lyapunov approach. Unknown control gain sign is effectively dealt with by introducing Nussbaum function. Using Babalat’s lemma, the tracking error is proved to asymptotically converge to zero and all signals in the closed-loop system are bounded.

c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 508–514, 2020. https://doi.org/10.1007/978-981-32-9682-4_53

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Problem Statement and Assumptions

Consider the following first order linear time invariant system, and its transformation function is defined as follows P (s) =

YP (s) kp = U (s) s + ap

(1)

where YP (s), U (s) denote the Laplace transforms of the output yp and input u in P (s), respectively, kp and ap are unknown constants. The transformation function of reference model is described as follows: M (s) =

Ym (s) km = R(s) s + am

(2)

where km and am > 0 are determined by the designer, Ym (s), R(s) denote the Laplace transforms of the output and input in M (s), respectively. The control objective is to design adaptive control signal u for system (1) such that the output yp follows the specified desired trajectory ym . Assumption 1: The sign of kp is unknown. Assumption 2: The input r(t) of reference model is bounded. To handle the unknown control gain sign, the Nussbaum gain technique is employed in this paper. A function N (ζ) is called a Nussbaum-type function if it has the following properties: s (i) lim sup 1s 0 N (ζ)dζ = +∞ s→∞  s (ii) lim inf 1s 0 N (ζ)dζ = −∞ s→∞

Commonly used Nussbaum functions include: ζ 2 cos(ζ), ζ 2 sin(ζ), exp(ζ 2 ) 2 cos( π2 ζ) [1,3,4]. For clarity, the even Nussbaum function, N (ζ) = eζ cos((π/2)ζ) is used throughout this paper. Lemma 1 (Barbalat Lemma): If f (t) is uniform continuous function defined t over [0, ∞), and lim 0 |f (τ )|dτ < +∞, then lim f (t) = 0. t→∞

t→∞

Lemma 2 [2]: Let V (·) and ζ(·) be smooth functions defined on [0, tf ) with V (t) ≥ 0, ∀t ∈ [0, tf ), and N (·) be an even smooth Nussbaum-type function. If the following inequality holds:  t ˙ V (t) ≤ c0 + (gN (ζ) + 1)ζdτ, ∀t ∈ [0, tf ) (3) 0

where g is a nonzero constant and c0 represents some suitable constant, then t ˙ must be bounded on [0, tf ). V (t), ζ(t) and 0 (gN (ζ) + 1)ζdτ

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Model Reference Adaptive Control Design and Stability Analysis

From (1) and (2), we have the state equations of plant and reference model are described as follows: (4) y˙ p = −ap yp (t) + kp u(t) y˙ m = −am ym (t) + km r(t)

(5)

Define the tracking error e0 as follows: e0 = yp − ym

(6)

e˙ 0 = −ap yp + am ym + kp u − km r = −am e0 + (am − ap )yp + kp u − km r

(7)

d 1 2 ( e ) = −am e20 + e0 [(am − ap )yp + kp u − km r] dt 2 0 = −am e20 + kp e0 u + ζ˙ + e0 (am − ap )yp − km e0 r − ζ˙

(8)

It yields

Therefore, we have

Design the Nussbaum parameter ζ and the control law u as follows: ζ˙ = e0 [−km r + cˆ0 (t)yp ]

(9)

u = N (ζ)[−km r + cˆ0 (t)yp ]

(10)

where cˆ0 (t) is the estimate of c0 , c0 = (am −ap ). Define the parameter estimation error as follows: (11) ϕy = cˆ0 (t) − (am − ap ) From (9) and (10), we have e0 u = N (ζ) · ζ˙

(12)

Substituting (12) into (8), we obtain d 1 2 ( e ) = −am e20 + kp N (ζ)ζ˙ + ζ˙ − cˆ0 (t)e0 yp + e0 (am − ap )yp dt 2 0 = −am e20 + (kp N (ζ) + 1) ζ˙ − ϕy e0 yp Let V =

1 2 1 e0 + ϕ2y 2 2g

(13)

(14)

The adaptive law of cˆ0 is designed as follows: cˆ˙0 = ge0 yp

(15)

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where g > 0 is a design constant. Using (13) and (14), the derivative of V with respect to time t is V˙ = −am e20 + (kp N (ζ) + 1) ζ˙ (16) Integrating (16) over [0, t], we have 

t

V (t) = V (0) +

 ˙ − kp (N (ζ) + 1) ζdτ

0

 ≤ V (0) +

t

am e20 dτ

0 t

˙ kp (N (ζ) + 1) ζdτ

(17)

0

According to Lemma 2, we have V (t) is bounded, furthermore, we have e0 , ζ, t ˙ ϕy ∈ L∞ , and .yp , ζ, ˙ u, e˙ 0 ∈ L∞ . Noting (17), we get k (N (ζ)+ 1)ζdτ, 0 p 

t

 am e20 dτ

= V (0) − V (t) +

0

t

˙ < +∞ kp (N (ζ) + 1) ζdτ

0

i.e., e0 ∈ L2 . Due to e0 , e˙ 0 ∈ L∞ , according to Barbalat’s lemma, as t → ∞, we have e0 (t) → 0. From the above the discussion, we have all signals are bounded, and the tracking error converges to zero. Theorem 1. Consider system (1) and reference model (2) under Assumptions 1 and 2, the controller (10) and adaptive laws (9), (15). For any bounded initial conditions, we have all signals in the closed-loop system are bounded, and the tracking error asymptotically converges to zero.

4

Simulation Results

In order to demonstrate the effectiveness of the proposed control approach, a numerical example is provided. Assume that M (s) =

0.5 s+2

1.5 s+4 i.e., y˙ m = −2ym + 0.5u, y˙ p = −4yp − 1.5u, the reference input is selected as r(t) = sin(0.5t). The control objective is to design adaptive control u(t) such that the system output yp (t) asymptotically tracks the desired reference model output ym (t). Select ζ˙ = e0 [−0.5r + c0 (t)yp ], N (ζ) = ζ 2 cos ζ, the adaptive law c˙0 (t) = ge0 yp the control law u = N (ζ)[−0.5r+c0 (t)yp ]; Choose the initial values y p (0) = 0.1, y m (0) = 0, c0 (0) = 0.1, ζ(0) = 0.5, g = 37. Simulation results are shown in Figs. 1, 2, 3 and 4. P (s) = −

H. Liu et al.

0.15 0.1 0.05

e

0

0 −0.05 −0.1 −0.15 −0.2

0

10

20

30 40 Time (sec)

50

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50

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Fig. 1. Tracking error e0

0.4

y ,y

p m

0.2

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−0.2

−0.4

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10

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Fig. 2. Output yp (solid line) and desired trajectory ym (dashed line)

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u

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−0.5

−1

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40

Fig. 3. Control signal u

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ζ

0.8 0.7 0.6 0.5 0.4

0

10

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30 Time (sec)

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Fig. 4. Curve of Nussbaum parameter ζ

5

Conclusion

This paper focuses on the problem of model reference adaptive control. Using Nussbaum function, unknown gain sign is handled. Based on Lyapunov method, the adaptive laws of Nussbaum parameter and unknown constant are designed. It is proved that all signals in the closed-loop system are bounded, and tracking error asymptotically converges to zero using Babalat’s lemma. Theoretical analysis and simulation results demonstrate the effectiveness of the proposed control strategy. Acknowledgments. This work was partially supported by the National Natural Science Foundation of China (61573307), the Natural Science Foundation of Jiangsu Province (BK20181218) and Yangzhou University Top-level Talents Support Program (2016).

References 1. Nussbaum RD (1983) Some remarks on the conjecture in parameter adaptive control. Syst Control Lett 3(3):243–246 2. Ye XD, Jiang JP (1998) Adaptive nonlinear design without a priori knowledge of control directions. IEEE Trans Autom Control 43(11):1617–1621 3. Ge SS, Hong F, Lee TH (2004) Adaptive neural control of nonlinear time-delay system with unknown virtual control coefficients. IEEE Trans Syst Man CybernPart B: Cybern 34(1):499–516. https://doi.org/10.1109/TSMCB.2003.817055 4. Ryan EP (1991) A universal adaptive stabilizer for a class of nonlinear systems. Syst Control Lett 16(3):209–218 5. Xia XN, Zhang TP (2014) Adaptive output feedback dynamic surface control of nonlinear systems with unmodeled dynamics and unknown high-frequency gain sign. Neurocomputing 143:312–321. https://doi.org/10.1016/j.neucom.2014.05.061

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6. Zhang TP, Xia MZ, Yi Y, Shen QK (2017) Adaptive neural dynamic surface control of pure-feedback nonlinear systems with full state constraints and dynamic uncertainties. IEEE Trans Syst Man Cybern Syst 47(8):2378–2387. https://doi.org/10. 1109/TSMC.2017.2675540 7. Krstic M, Kanellakopoulos I, Kokotovic PV (1995) Nonlinear and Adaptive Control Design. Wiley, New York 8. Han CJ (1990) Adaptive Control. Qinghua University Press, Beijing

Adaptive Neural Network Control of Uncertain Systems with Full State Constraints and Unknown Gain Sign Heqing Liu, Tianping Zhang(B) , Meizhen Xia, and Ziwen Wu College of Information Engineering, Yangzhou University, Yangzhou 225127, China [email protected]

Abstract. In this paper, adaptive neural network control is proposed based on improved dynamic surface control (DSC) method and the approximation capability of radial basis function (RBF) neural networks (NNs) for a class of uncertain constrained pure-feedback nonlinear systems with unmodeled dynamics and unknown gain sign. By constructing a one to one nonlinear mapping, the pure-feedback system with full state constraints is transformed into a novel pure-feedback system without state constraints. The dynamic uncertainties are handled using an auxiliary dynamic signal. Using mean value theorem and Nussbaum function, an adaptive NN control scheme is developed based on the transformed system. The designed control strategy removes the conditions that the upper bound of the control gain is known, and the lower bounds and upper bounds of the virtual control coefficients are known. By theoretical analysis, all the signals in the closed-loop system are shown to be semi-globally uniformly ultimately bounded (SGUUB), and the full state constraints are not violated. A numerical example is provided to demonstrate the effectiveness of the proposed method. Keywords: State constraints · Unmodeled dynamics · Adaptive control · Neural networks · Dynamic surface control Pure-feedback systems

1

·

Introduction

It is well known that unmodeled dynamics widely exists in many practical nonlinear systems due to modeling errors and modeling simplifications. They can severely degrade the closed-loop system performance. Sometimes, they can even make system be unstable. Therefore, many methods were proposed to deal with such systems with unmodeled dynamics based on backstepping design or dynamic surface control in [1–6]. An auxiliary dynamic signal in [1,4,5] or a Lyapunov function description in [6] or a normalization signal were usually used to handle unmodeled dynamics in [7]. In the past decade, adaptive control of constrained nonlinear systems has been received much attention. Based on backstepping [8,9] and DSC [10–13], several adaptive control schemes were proposed for constrained nonlinear systems in c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 515–525, 2020. https://doi.org/10.1007/978-981-32-9682-4_54

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[14–28]. Output constraint or time-varying output constraint or partial state constraints and known virtual control gains were carried out using barrier Lyapunov function (BLF) in [14–17]. By introducing a nonlinear mapping, adaptive tracking control was developed for a class of uncertain non-affine systems with timevarying asymmetric output constraints in [21]. In [22], backstepping DSC control was proposed by using BLF for constrained uncertain strict-feedback nonlinear dynamic systems with deadzone. Using DSC, an adaptive control strategy was proposed based on iBLF and the mean value theorem for completely non-affine pure-feedback systems with output constraint in [23]. By introducing a symmetric nonlinear mapping, a class of strict-feedback nonlinear system with unit control gain and output constraint in [24] was transformed an unconstrained strict-feedback nonlinear system. The controller did not need to redesign based on [9]. Using the above method, adaptive control problem with output constraint was discussed for a class of stochastic switched non-affine nonlinear systems in [25]. The symmetric constraints were carried out in [24,25]. Furthermore, by constructing a novel one to one nonlinear mapping, three adaptive control schemes were proposed for strict-feedback or pure-feedback nonlinear systems with full state constraints and unmodeled dynamics in [26–28]. Approximationbased adaptive DSC was developed using integral BLF (iBLF)and mean value theorem for uncertain systems in the form of nonlinear pure-feedback with full state constraints in [29]. But every virtual control gain and its derivative were assumed to be bounded. In this paper, by introducing a one to one asymmetric nonlinear mapping and using the property of Nussbaum function, adaptive NN control is proposed for a class of uncertain pure-feedback nonlinear systems with unknown gain sign based on improved DSC. The full state constraints are carried out, and unmodeled dynamics is effectively handled. The design method does not need to use mean value theorem except the final an equation. The proposed design approach removes the conditions that the approximation errors are assumed to be bounded before the stability is proved by using backstepping.

2

Problem Description and Basic Assumptions

Consider the following constrained pure-feedback nonlinear systems with unmodeled dynamics ⎧ ξ˙ = q(ξ, x, t) ⎪ ⎪ ⎨ xi , xi+1 ) + di (ξ, x, t), 1 ≤ i ≤ n − 1 x˙ i = fi (¯ (1) = f xn ) + gn (¯ xn )u + dn (ξ, x, t), n ≥ 2 x ˙ ⎪ n (¯ ⎪ ⎩ n y = x1 , where x = [x1 , x2 , . . . , xn ]T ∈ Rn is the state vector, ξ ∈ Rn0 is the unmodx2 , x3 ), eled dynamics, u ∈ R is the input, y ∈ R is the output, f1 (x1 , x2 ), f2 (¯ xn ), gn (¯ xn ) are the unknown smooth functions; x ¯i = [x1 , x2 . . . , xi ]T , . . ., fn (¯ i = 1, . . . , n, d1 (ξ, x, t), d2 (ξ, x, t), . . ., dn (ξ, x, t) are the unknown uncertain disturbances.

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The control objective is to design adaptive control u(t) for system (1) such that the output y follows the specified desired trajectory yd , and every state xi ∈ Ωxi = {xi : − kbi1 < xi < kbi2 } is not violated for i = 1, . . . , n, where kbi1 , kbi2 are known positive design constants. Assumption 1 [1]. The unmodeled dynamics ξ is said to be exponentially inputstate-practically stable (exp-ISpS), i.e., for system ξ˙ = q(ξ, x, t), if there exists ¯ 2 of class K∞ and a Lyapunov function V (ξ) such that functions α ¯1, α α ¯ 1 (ξ) ≤ V (ξ) ≤ α ¯ 2 (ξ)

(2)

and there exist two constants c > 0, d ≥ 0 and a class K∞ function γ such that ∂V (ξ) q(ξ, x, t) ≤ −cV (ξ) + γ(|x1 |) + d, ∀t ≥ 0 ∂ξ

(3)

where c and d are known positive constants, γ(·) is a known function of class K∞ . Assumption 2 [1,2]. There exist unknown nonnegative continuous functions ϕi1 (·) and non-decreasing continuous functions ϕi2 (·) such that |di (ξ, x, t)| ≤ ϕi1 (¯ xi ) + ϕi2 (ξ), ∀(ξ, x, t) ∈ Rn0 × Rn × R+

(4)

where ϕi2 (0) = 0, i = 1, . . . , n. Assumption 3 [2].   There exist positive constants gi0 such that system (1) sat ∂fi (¯xi ,xi+1 )  isfies  ∂xi+1  ≥ gi0 > 0, i = 1, . . . , n − 1, ∀[x1 , . . . , xn ]T ∈ Rn . xn ) is unknown, and there exists a known Assumption 4 [2]. The sign of gn (¯ xn )|, ∀¯ xn ∈ Rn . positive constant gn0 such that 0 < gn0 ≤ |gn (¯ Assumption 5 [12]. The desired trajectory vectors are continuous and available, [yd , y˙ d , y¨d ]T ∈ Ωd with known compact set Ωd = {[yd , y˙ d , y¨d ]T : yd2 +y˙ d2 +¨ yd2 ≤ 3 B0 } ⊂ R , and |yd | < B1 < min{kb11 , kb12 }, where B0 , B1 are two known positive constants. Lemma 1 [1]. If V is an exp-ISpS Lyapunov function for a system ξ˙ = q(ξ, x, t), i.e. (2) and (3) hold, then, for any constant c¯ ∈ (0, c), any initial instant t0 > 0, any initial condition ξ0 = ξ(t0 ), r0 > 0, for any continuous function γ¯ such that γ¯ (|x1 |) ≥ γ(|x1 |), there exist a finite T0 = max{0, log[ V r(ξ00 ) ]/(c − c¯)} ≥ 0, a nonnegative function D(t0 , t), defined for all t ≥ t0 and a signal described by r˙ = −¯ cr + γ¯ (|x1 |) + d, r(t0 ) = r0 such that D(t0 , t) = 0 for t ≥ t0 + T0 , and V (ξ) ≤ r(t) + D(t0 , t) with D(t0 , t) = max{0, e−c(t−t0 ) V (ξ0 ) − e−¯c(t−t0 ) r0 }, where log(•) stands for the natural logarithm of •.

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In this paper, we employ the Nussbaum gain technique to handle the unknown control gain sign. Nussbaum functions ζ 2 cos(ζ), ζ 2 sin(ζ), and exp(ζ 2 ) cos((π/2)ζ) are usually used in [30]. For clarity, the even Nussbaum function 2 N (ζ) = eζ × cos(( π2 )ζ) is employed. Lemma 2 [30]. Let V (·), ζ(·) be smooth functions defined on [0, tf ) with V (t) ≥ 0, ∀t ∈ [0, tf ), and N (·) be an even smooth Nussbaum-type function. If the following inequality holds  t ˙ c1 τ dτ V (t) ≤ c0 + e−c1 t g(x(τ ))N (ζ)ζe 0  t ˙ c1 τ dτ , ∀t ∈ [0, tf ) (5) ζe + e−c1 t 0

where c0 represents some suitable constant, c1 is a positive constant, and g(x(τ )) is a time–varying parameter which takes values in the unknown closed intervals t ˙ must be bounded on / I, then V (t), ζ(t), 0 g(x(τ ))N (ζ)ζdτ I = [l− , l+ ], with 0 ∈ [0, tf ).

3

Adaptive NN Control with Full State Constraints

To carry out full state constraints, choose the change of coordinates as follows: k +x k si = log kbbi1 −xii − log kbbi1 i2 i2 (6) i = 1, . . . , n From (6), we get that its inverse mapping is xi = kbi2 −

kbi2 + kbi1

, i = 1, . . . , n

(7)

esi + e−si + 2 x˙ i , i = 1, . . . , n kbi1 + kbi2

(8)

kbi1 s i kbi2 e

+1

Therefore, we obtain s˙ i =

System (1) can be rewritten as follows: ⎧ ξ˙ = q(ξ, x, t) ⎪ ⎪ ⎪ ⎪ s˙ 1 = F1 (s1 , s2 ) + s2 + k1 (s1 )D1 (ξ, s¯n , t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ... sn−1 , sn ) + sn s˙ n−1 = Fn−1 (¯ ⎪ ⎪ ⎪ ⎪ + k (s ¯n , t) n−1 n−1 )Dn−1 (ξ, s ⎪ ⎪ ⎪ ⎪ s ˙ = F (¯ s ) + k (s )G (¯ s n n n n n n )u ⎪ ⎩ n + kn (sn )Dn (ξ, s¯n , t)

(9)

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where s¯i = [s1 , . . . , si ]T , i = 1, . . . , n, esi + e−si + 2 , i = 1, . . . , n kbi1 + kbi2 Fi (¯ si+1 ) = ki (si )fi (¯ xi , xi+1 ) − si+1 , i = 1, . . . , n − 1 ki (si ) =

k

(10)

sn ) = kn (sn )fn (¯ xn , 0) Fn (¯

(11)

Gn (¯ sn ) = gn (¯ xn )

(12)

Di (ξ, s¯n , t) = di (ξ, x, t), i = 1, . . . , n.

(13)

+y

k

Define yˆd = log kbb11 −ydd − log kbb11 . We choose the change of coordinates as 12 12 follows: z1 = s1 − yˆd , zi = si − ωi , i = 2, . . . , n, where ωi is the output of a firstorder filter with αi−1 as the input, and αi−1 is an intermediate control which shall be developed for the corresponding (i − 1)th subsystem. Finally, control signal u is designed at step n. ∗T Si (Zi ) be the approxSuppose ΩZi ⊂ Ri+4 be a given compact set, and Whi imation of RBF NNs over the compact set ΩZi to hi (Zi ) as discussed in [31], where unknown continuous function hi (Zi ) will be given later. Then, we have ∗T hi (Zi ) = Whi Si (Zi ) + εhi (Zi )

(14)

sTi+1 , zi , ω˙ i , r]T ∈ Ri+4 , i = 1, . . . , n−1, Zn = [¯ sTn , zn , ω˙ n , r]T ∈ Rn+3 where Zi = [¯ the basis function vector Si (Zi ) = [si1 (Zi ), . . . , sili (Zi )]T ∈ Rli , and sij (Zi ) chosen as the commonly used Gaussian functions, which have the form sij (Zi ) = exp[−

(Zi − μij )T (Zi − μij ) ] φ2ij

(15)

where j = 1, . . . , li , i = 1, . . . , n, μij = [μij1 , μij2 , . . . , μijqij ]T is the center of the receptive field with qij = i + 4, 1 ≤ i ≤ n − 1, qnj = n + 3 and φij is the width ∗ are ideal constant weights, which are defined as of the Gaussian function; Whi follows: ∗ T = arg min [ sup |Whi Si (Zi ) − hi (Zi )|]. Whi Whi ∈Rli Zi ∈ΩZ i

(16)

For clarity, define some notations as follows: z¯i = [z1 , . . . , zi ]T

(17)

y¯j = [y2 , . . . , yj ]T

(18)

θi =

∗ 2  Whi g0

(19)

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¯ θˆi = [θˆ1 , . . . , θˆi ]T Vzi =

1 2 z , i = 1, . . . , n 2 i

(20) (21)

where θˆi is the estimate of θi at time t, i = 1 . . . , n, yj = ωj − αj−1 , j = 2, . . . , n, g0 = min{1, gn0 }. ˜ = (·) ˆ − (·), and  ·  denotes the 2-norm. Throughout this paper, let (·) The virtual control laws α1 , αi and control law u are designed as follows: α1 = −c1 z1 −

1 z1 θˆ1 S1 (Z1 )2 2a21

(22)

αi = −ci zi −

1 ˆ zi θi Si (Zi )2 2a2i

(23)

u=

N (ζ)

1 cn zn + 2 zn θˆn Sn (Zn )2 ] kn (sn ) 2an

(24)

1 ζ˙ = cn zn2 + 2 zn2 θˆn Sn (Zn )2 2an

(25)

where ci > 0 is a design constant, θˆi , which will be given later, is the estimate of θi at time t, N (ζ) = ζ 2 cos(ζ). The adaptation law of the unknown parameter θi is designed as follows: 1 ˙ θˆi = γi [ 2 zi2 Si (Zi )2 − σi θˆi ] 2ai

(26)

where γi , ai and σi are strictly positive constants, θˆi is the estimate of θi at time t, θˆi (0) ≥ 0. Define ωi+1 as follows: τi+1 ω˙ i+1 + ωi+1 = αi ,

ωi+1 (0) = αi (0)

(27)

where τi+1 > 0 is a positive design constant, i = 1, . . . , n − 1. Step 1: Let ω1 = yˆd . Then, the derivative of Vz1 is g0 a2 1 V˙ z1 ≤ z1 [z2 + y2 + α1 ] + 2 z12 θ1 S1 (Z1 )2 + 1 + z1 ε1 (Z1 ) + 2a1 2g0 4

(28)

where a1 is a positive design constant, Z1 = [¯ sT2 , z1 , ω˙ 1 , r]T ∈ R5 . Furthermore, we obtain 1 1 g0 θ˜1 z12 S1 (Z1 )2 V˙ z1 ≤ (−c1 + 2)z12 + z22 + y22 − 4 4 2a21 1 a2 + 1 + + |z1 |η1 (¯ z2 , y2 , θˆ1 , r, yd , y˙ d ) 2 4

(29)

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where continuous function η1 (¯ z2 , y2 , θˆ1 , r, yd , y˙ d ) satisfies |ε1 (Z1 )| ≤ η1 (¯ z2 , y2 , θˆ1 , r, yd , y˙ d )

(30)

From Young’s inequality, we have 1 |z1 |η1 ≤ z12 + η12 4 Therefore, we obtain 1 1 1 1 g0 θ˜1 z12 S1 (Z1 )2 a2 V˙ z1 ≤ (−c1 + 3)z12 + z22 + y22 − + 1 + + η12 (31) 2 4 4 2a1 2 4 4 Noting Assumption 2, we have y2

z˙1 θˆ1 S1 (Z1 )2 + c1 z˙1 + τ2 2a21 ˙ z1 θˆ1 S1 (Z1 )2 z1 θˆ1 dS1 (Z1 )2  + + 2 2a1 2a21 dt

y˙ 2 = −

|y˙ 2 +

y2 ¯ | ≤ ξ2 (¯ z3 , y¯3 , θˆ2 , r, yd , y˙ d , y¨d ) τ2

(32)

(33)

¯ where ξ2 (¯ z3 , y¯3 , θˆ2 , r, yd , y˙ d , y¨d ) is a continuous function. From (32) and (33), we obtain y22 ¯ + |y2 |ξ2 (¯ z3 , y¯3 , θˆ2 , r, yd , y˙ d , y¨d ) τ2 1 y2 ≤ − 2 + y22 + ξ22 τ2 4

y2 y˙ 2 ≤ −

(34)

Step i (2 ≤ i ≤ n − 1): The derivative of Vzi with respect to t is 1 V˙ zi ≤ zi [zi+1 + yi+1 + αi + hi (Zi )] + 4 1 g0 2 a2 ≤ zi [zi+1 + yi+1 + αi ] + 2 zi θi Si (Zi )2 + i + + zi εi (Zi ) 2ai 2 4 where ai is a positive design constant,

2 si+1 ) + zi ki2 (si ) ϕi1 (¯ xi ) + ϕi2 (¯ α1−1 (r + D0 )) − ω˙ i hi (Zi ) = Fi (¯

(35)

(36)

with Zi = [¯ sTi+1 , zi , ω˙ i , r]T ∈ Ri+4 . i+1 . Since Noting yi+1 = ωi+1 − αi , i = 2, . . . , n − 1, we know that ω˙ i+1 = − yτi+1 zi+1 = si+1 − ωi+1 , it follows that si+1 = zi+1 + yi+1 − ci zi −

1 ˆ zi θi Si (Zi )2 2a2i

(37)

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Substituting (14) and (37) into (35), using Young’s inequality, and by induction ¯ for some continuous function ηi (¯ zi+1 , y¯i+1 , θˆi , yd , y˙ d ), we obtain 1 2 1 2 g0 θ˜i zi2 Si (Zi )2 V˙ zi ≤ (−ci + 2)zi2 + zi+1 + yi+1 − 4 4 2a2i 1 a2 ¯ + i + + |zi |ηi (¯ zi+1 , y¯i+1 , θˆi , r, yd , y˙ d ) 2 4

(38)

where continuous function ηi satisfies ¯ |εi (Zi )| ≤ ηi (¯ zi+1 , y¯i+1 , θˆi , r, yd , y˙ d )

(39)

Using Young’s inequality, we have 1 1 1 2 1 2 g0 θ˜i zi2 Si (Zi )2 a2i V˙ zi ≤ (−ci + 3)zi2 + zi+1 + + ηi2 (40) + yi+1 − + 2 4 4 2ai 2 4 4 From Assumption 2, we get yi+1

z˙i θˆi Si (Zi )2 + ki z˙i + τi+1 2a2i ˙ zi θˆi Si (Zi )2 zi θˆi dSi (Zi )2  + + 2a2i 2a2i dt

y˙ i+1 = −

(41)

Noting (41) and by induction for some continuous function ξi+1 , we have |y˙ i+1 +

yi+1 ¯ | ≤ ξi+1 (¯ zi+2 , y¯i+2 , θˆi+1 , r, yd , y˙ d , y¨d ) τi+1

(42)

here, if l ≥ n, then zl = zn , yl = yn , θˆl = θˆn . From (41) and (42), we obtain yi+1 y˙ i+1 ≤ −

2 yi+1 1 2 2 + yi+1 + ξi+1 τi+1 4

(43)

Step n: The derivative of Vzn is 1 V˙ zn ≤ zn [kn (sn )Gn (¯ sn )u + hn (Zn )] + 4 1 g0 2 a2 sn )u + 2 zn θn Sn (Zn )2 + n + + zn εn (Zn ) (44) ≤ zn kn (sn )Gn (¯ 2an 2 4 where an is a positive design constant,

2 sn ) + zn kn2 (sn ) ϕn1 (¯ xn ) + ϕn2 (¯ α1−1 (r + D0 )) − ω˙ n (45) hn (Zn ) = Fn (¯ with Zn = [¯ sTn , zn , ω˙ n , r]T ∈ Rn+3 .

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Similar to the discussion at the ith step, we have g0 zn2 θ˜n V˙ zn ≤ Gn (¯ sn )N (ζ)ζ˙ + ζ˙ − cn zn2 − Sn (Zn )2 2a2n 1 a2 + n + + zn εn (Zn ) 2 4 ¯ where continuous function ηn (¯ zn , y¯n , θˆn−1 , yd , y˙ d ) satisfies ¯ |εn (Zn )| ≤ ηn (¯ zn , y¯n , θˆn−1 , r, yd , y˙ d ) Define the compact set Ωn as follows: ¯ Ωn = {[¯ znT , y¯nT , θˆnT ]T : Vn ≤ p} ⊂ Rpn where p is any given positive constant, pn = 3n − 1, and n

n  g0 ˜2  1  2 Vzj + y Vn = θj + 2γj 2 j=2 j j=1

(46)

(47)

(48)

(49)

Theorem 1. Consider the closed-loop system consisting of system (1) under Assumptions 1–5, the nonlinear mapping (6), the inverse mapping (7), the controller (24) with Nussbaum parameter ζ determined by (25), and adaptation law (26). For bounded initial conditions, satisfying Vn (0) ≤ p, and xi (0) ∈ Ωxi , there exist constants ci > 0, τi > 0, γi > 0, σi > 0 such that all of the signals in the closed–loop system are SGUUB, and the full state constraints are not violated, i.e., xi ∈ Ωxi , ∀t ≥ 0, in addition, ci and τi satisfy ⎧ α ⎨ ci ≥ 4 + 20 , i = 1, . . . , n 1 1 ≥ 1 4 + α20 , i = 2, . . . , n (50) ⎩ τi α0 = min{γ1 σ1 , . . . , γn σn } Proof. To save space, the proof is omitted.

4

Conclusions

By introducing a one to one asymmetric nonlinear mapping, the uncertain nonaffine nonlinear system with full state constraints has been transformed into a novel unconstrained non-affine nonlinear system. Using the transformed system and a modified DSC method, adaptive NN control has been developed. The proposed design method does not assume the control gain sign is known. By designing an auxiliary dynamical signal, unmodeled dynamics is effectively handed. Using Young’s inequality, only a parameter is adjusted online at each recursive step. Using the property of Nussbaum function, the problem of unknown gain sign is solved. It is shown that all the signals in the closed-loop system are SGUUB. The full states can abide by the desired constraints. Acknowledgments. This work was partially supported by the National Natural Science Foundation of China (61573307), the Natural Science Foundation of Jiangsu Province (BK20181218) and Yangzhou University Top-level Talents Support Program (2016).

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References 1. Jiang ZP, Praly L (1998) Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties. Automatica 34(7):825–840. https://doi.org/10.1016/ S0005-1098(98)00018-1 2. Zhang TP, Shi XC, Zhu Q, Yang YQ (2013) Adaptive neural tracking control of pure-feedback nonlinear systems with unknown gain signs and unmodeled dynamics. Neurocomputing 121:290–297. https://doi.org/10.1016/j.neucom.2013.04.023 3. Xia XN, Zhang TP (2014) Adaptive output feedback dynamic surface control of nonlinear systems with unmodeled dynamics and unknown high-frequency gain sign. Neurocomputing 143:312–321. https://doi.org/10.1016/j.neucom.2014.05.061 4. Zhang TP, Xia XN (2015) Decentralized adaptive fuzzy output feedback control of stochastic nonlinear large-scale systems with dynamic uncertainties. Inf Sci 315:17– 38. https://doi.org/10.1016/j.ins.2015.04.002 5. Zhang TP, Xia XN, Zhu JM (2014) Adaptive neural control of state delayed nonlinear systems with unmodeled dynamics and distributed time-varying delays. IET Control Theory Appl 8(12):1071–1082. https://doi.org/10.1049/iet-cta.2013.0803 6. Jiang ZP, Hill DJ (1999) A robust adaptive backstepping scheme for nonlinear systems with unmodeled dynamics. IEEE Trans Autom Control 44(9):1705–1711. https://doi.org/10.1109/9.788536 7. Arcak M, Kokotovic PV (2000) Robust nonlinear control of systems with input unmodeled dynamics. Syst Control Lett 41(2):115–122. https://doi.org/10.1016/ S0167-6911(00)00044-X 8. Kanellakipoulos I, Kokotovic PV, Morse AS (1991) Systematic design of adaptive controllers for feedback linearizable systems. IEEE Trans Autom Control 36(11):1241–1253 9. Krstic M, Kanellakopoulos I, Kokotovic PV (1995) Nonlinear and Adaptive Control Design. Wiley, New York 10. Swaroop D, Hedrick JK, Yip PP, Gerdes JC (2000) Dynamic surface control for a class of nonlinear systems. IEEE Trans Autom Control 45(10):1893–1899. https:// doi.org/10.1109/TAC.2000.880994 11. Wang D, Huang J (2005) Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strict-feedback form. IEEE Trans Neural Netw 16(1):195–202. https://doi.org/10.1109/TNN.2004.839354 12. Zhang TP, Ge SS (2008) Adaptive dynamic surface control of nonlinear systems with unknown dead zone in pure feedback form. Automatica 44(7):1895–1903. https://doi.org/10.1016/j.automatica.2007.11.025 13. Zhang TP, Zhu Q, Yang YQ (2012) Adaptive neural control of non-affine purefeedback nonlinear systems with input nonlinearity and perturbed uncertainties. Int J Syst Sci 34(4):375–388. https://doi.org/10.1080/00207721.2010.519060 14. Tee KP, Ge SS, Tay EH (2009) Barrier Lyapunov functions for the control of output-constrained nonlinear systems. Automatica 45(4):918–927. https://doi.org/ 10.1016/j.automatica.2008.11.017 15. Tee KP, Ren BB, Ge SS (2011) Control of nonlinear systems with timevarying output constraints. Automatica 47(11):2511–2516. https://doi.org/10. 1016/j.automatica.2011.08.044 16. Tee KP, Ge SS (2011) Control of nonlinear systems with partial state constraints using a barrier Lyapunov function. Int J Control 84(12):2008–2013. https://doi. org/10.1080/00207179.2011.631192

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17. Qiu YN, Liang XG, Dai ZY, Cao JX, Chen YQ (2015) Backstepping dynamic surface control for a class of nonlinear systems with time-varying output constraints. IET Control Theory Appl 9(15):2312–2319. https://doi.org/10.1049/iet-cta.2015. 0019 18. Ren BB, Tee KP, Lee TH (2010) Adaptive neural control for output feedback nonlinear systems using a barrier Lyapunov function. IEEE Trans Neural Netw 21(8):1339–1345. https://doi.org/10.1109/TNN.2010.2047115 19. He W, Dong YT, Sun CY (2015) Adaptive neural network control of unknown affine systems with input deadzone and output constraint. ISA Trans 58:96–104. https://doi.org/10.1016/j.isatra.2015.05.014 20. Liu Z, Lai GY, Zhang Y, Philip Chen CL (2015) Adaptive neural output feedback control of output-constrained nonlinear systems with unknown output nonlinearity. IEEE Trans Neural Netw Learn Syst 26(8):1789–1802. https://doi.org/10.1109/ TNNLS.2015.2420661 21. Meng WC, Yang QM, Si SN, Sun YX (2016) Adaptive neural control of a class of output-constrained non-affine systems. IEEE Trans Cybern 46(1):85–95. https:// doi.org/10.1109/TCYB.2015.2394797 22. Han SI, Lee JM (2012) Adaptive fuzzy backstepping dynamic surface control for output constrained non-smooth nonlinear dynamic system. Int J Control Autom Syst 10(4):684–696. https://doi.org/10.1007/s12555-012-0403-8 23. Kim BS, Yoo SG (2015) Adaptive control of nonlinear pure-feedback systems with output constraints: integral barrier Lyapunov functional approach. Int J Control Autom Syst 13(1):249–256. https://doi.org/10.1007/s12555-014-0018-3 24. Guo T, Wu XW (2014) Backstepping control for output-constrained nonlinear systems based on nonlinear mapping. Neural Comput Appl 25(7–8):1665–1674. https://doi.org/10.1007/s00521-014-1650-9 25. Yin S, Yu H, Shahnazi R, Haghani A (2017) Fuzzy adaptive tracking control of constrained nonlinear switched stochastic pure-feedback systems. IEEE Trans Cybern 47(3):579–588. https://doi.org/10.1109/TCYB.2016.2521179 26. Zhang TP, Xia MZ, Yi Y (2017) Adaptive neural dynamic surface control of strictfeedback nonlinear systems with full state constraints and unmodeled dynamics. Automatica 81:232–239. https://doi.org/10.1016/j.automatica.2017.03.033 27. Zhang TP, Xia MZ, Yi Y, Shen QK (2017) Adaptive neural dynamic surface control of pure-feedback nonlinear systems with full state constraints and dynamic uncertainties. IEEE Trans Syst Man Cybern Syst 47(8):2378–2387. https://doi. org/10.1109/TSMC.2017.2675540 28. Zhang TP, Wang NN, Wang Q, Yang Y (2018) Adaptive neural control of constrained strict-feedback nonlinear systems with input unmodeled dynamics. Neurocomputing 272:596–605. https://doi.org/10.1016/j.neucom.2017.07.034 29. Kim BS, Yoo SG (2014) Approximation-based adaptive control of uncertain nonlinear pure-feedback systems with full state constraints. IET Control Theory Appl 8(17):2070–2081. https://doi.org/10.1049/iet-cta.2014.0254 30. Ge SS, Hong F, Lee TH (2004) Adaptive neural control of nonlinear time-delay system with unknown virtual control coefficients. IEEE Trans Syst Man Cybern Part B Cybern 34(1):499–516. https://doi.org/10.1109/TSMCB.2003.817055 31. Ge SS, Hang CC, Lee TH, Zhang T (2001) Stable Adaptive Neural Network Control. Kluwer Academic, Boston

Smooth Globally Convergent Velocity Observer Design for Uncertain Robotic Manipulators Yuan Liu(B) and Shangzheng Liu Nanyang Institute of Technology, Nanyang 473004, China [email protected]

Abstract. In this paper, we give a solution for global joint velocities estimation of the robotic manipulators with parametric uncertainty. A new adaptive observer of joint velocity is proposed to handle parametric uncertainties in robot dynamics. Different from most existing results, we don’t need an upper bound of velocity magnitude. Compared with nonsmooth observers, smoothness of the dynamics of the observer we design guarantees that it is easy to implement. Finally, We prove that the global asymptotic convergence of the state estimates to their true values can finally be acquired. Keywords: Global convergence · Robot manipulator Parametric uncertainty · Velocity estimation

1

·

Introduction

Over the past few decades, the problem of precisely controlling robot manipulators has attracted much attention due to its broad applications. To this end, one interesting problem is the estimation of joint-space velocities of the manipulators, which is very important since many commercial robots aren’t usually equipped with the precise velocity sensors. Even if this kind of velocity sensors are equipped, the output is often contaminated with some noise. Considering the practical restrictions, lots of efforts has been made to the velocity estimating problem in mechanical systems in recent years. The existing methods of designing velocity observers can be divided to smoothness and non-smoothness considering the closed loop system dynamics. In [1], the authors designed a kind of non-smooth observer which is capable of acquiring semi-global convergence of the estimation errors of velocity to zero by using model based approaches. Considering global convergence, a non-smooth velocity observer was proposed in [2] by employing dynamic model of a class of rigid body. Note that [3] presented a non-smooth sliding mode observer and finite-time convergence was achieved. In [4] and [5], by using non-model based of observer dynamics, the authors achieved local or global convergence of the errors of velocity estimation to zero. However, even though the non-smooth observers c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 526–534, 2020. https://doi.org/10.1007/978-981-32-9682-4_55

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have simple structures, they are actually limited in practical applications because they are usually sensitive to the chattering phenomenon and noise. Smooth observers were proposed with achieving local or semi-global convergent problems, such as [6] and [7], to just name a few. It is worth noting that a kind of globally convergent velocity observers with smooth system dynamics was presented in [8]. Yet the uncertainty of the manipulator system model mentioned in [8] was not global and part of the parameters still have to be exactly known, which would potentially limit its application. Recently, a fullorder global velocity observer was proposed for a kind of mechanical systems in [9], in which both with or without constraints are taken into consideration. The mentioned approach was based on the notions of I&I (Immersion and Invariance), which aimed at finding a certain manifold. Based on I&I theory, a state-feedback passivity-based controller was studied in [10], which allocates to the closed-loop a port-Hamiltonian structure with a desired energy function. Unfortunately, the system model had to be exactly known. In this short paper, we aim to design a adaptive velocity observer for a kind of robot manipulators with uncertain parameters, and we extend the existing results to the more general case of uncertain parameters in system dynamics. In this paper, we proved the stability of the propose observer by using the method of Lyapunov function. Furthermore, we give the explicit procedure of getting the controllers’ gain matrices. The structure of this short paper is designed as follows, the system model and the properties of Euler-Lagrange system is presented in Sect. 2. In Sect. 3, the adaptive observer is designed, and stability analysis of the proposed observer is conducted in Sect. 4.

2 2.1

Problem Formulation System Dynamics

The robotic manipulators we take into consideration can be described by EulerLagrange equation M (q)¨ q + C(q, q) ˙ q˙ + g(q) = u (1) where q(t) ∈ Rm denotes the coordinates vector, M (q) ∈ Rm×m is the symmetric inertial matrix, C(q, q) ˙ ∈ Rm is the coriolis and centrifugal torques vector, g(q) is the gravitational terms and τ (t) is the torques vector. Before proceeding, some fundamental properties for Euler-Lagrange system are presented in the following. Property 1. The inertia matrix M (q) has the property of positive definiteness, and there also exists two positive constants lm and lm satisfy the following inequality (2) 0 < lm I ≤ M (q) ≤ lm I Property 2. With regard to a differentiable variable vector ζ ∈ Rm , the manipulator system dynamics can be linearly parameterized as ˙ M (q)ζ˙ + C(q, q)ζ ˙ + g(q) = Y (q, q, ˙ ζ, ζ)θ

(3)

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in which θ ∈ Rk denotes a constant vector and Y (·) ∈ Rm×k is the regressor matrix. Property 3. M˙ (q) − 2C(q, q) ˙ is skew-symmetric, and in another word, for any variable vector ζ ∈ Rm , it follows that   ˙ ζ=0 (4) ζ T M˙ (q) − 2C(q, q) Property 4. The following equations with respect to q˙ is satisfied by the coriolis matrix C(q, q) ˙ C(ε, ζ + η) = C(ε, ζ) + C(ε, η)

(5)

C(ε, ζ)η = C(ε, η)ζ

(6)

C(ε, ζ) ≤ lc1 ζ

(7)

where lc1 is a positive constant, and vector ε, ζ and η ∈ Rm . For the convenience of the following observer design, let us define vector function tanh(·) ∈ Rm and the matrix function sech(·) ∈ Rm×m as Tanh(ζ) = [tanh(ζi1 ), · · ·, tanh(ζim )]T

(8)

sech(ζ) = diag{sech(ζi1 ), · · ·, sech(ζim )}

(9)

and where ζ = [ζi1 , · · ·, ζi1 ] ∈ Rm and similar notations hold for cosh(ζ). Also cosh(ζ) = diag{1/sech(ζi1 ), · · ·, 1/sech(ζim )}

(10)

Assumption 1. There exists positive constants lg and lc2 that for all ζ, υ ∈ Rm the following equations are satisfied g(ζ) − g(υ) ≤ lg  tanh(ζ − υ)

(11)

C(ζ, q) ˙ − C(υ, q) ˙ ≤ lc2 q ˙ tanh(ζ − υ)

(12)

For detail proof of Assumption 1, one can reference [11].

3

Adaptive Observer Design

Suppose that the constant parameters in C(q, q) ˙ and g(q) can not be precisely achieved. We aim to develop a new observer for q and q, ˙ and the observer designed should be smooth. Let qˆ˙ and qˆ represent the estimate of q˙ and q. Moreover, Let us define the estimation errors of position and velocity as q˜ = q − qˆ

(13)

q˜˙ = q˙ − qˆ˙

(14)

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Inspired by [8], a velocity observer is designed as the following equatioin M (q)(q¨ˆ + γ) + Y (ˆ q , qˆ˙)θˆ + δ = u where γ and δ are designed as ⎧ γ = sech2 (ϑ)[tanh(˜ q ) − tanh(ϑ)] ⎪ ⎪ ⎨ q ) + tanh(ϑ)] − sech2 (q)[tanh(˜ 2 ⎪ δ = tanh(˜ q ) − κ(t)cosh (ϑ) tanh(ϑ) ⎪ ⎩ + ι(t)cosh(˜ q)

(15)

(16)

and according Property 2, the term Y (ˆ q , qˆ˙)θˆ in (15) satisfies ˆ q , qˆ˙)qˆ˙ + gˆ(ˆ Y (ˆ q , qˆ˙)θˆ = C(ˆ q) where θˆ is the estimate of θ, and is updated as ⎧  q , qˆ˙)T q + ϕ ⎨ θˆ = Γ Y (ˆ d ϕ˙ = dt Y (ˆ q , qˆ˙)T q − Y (ˆ q , qˆ˙)T [qˆ˙ − tanh(˜ q) ⎩ − tanh(ϑ)]

(17)

(18)

with a positive definite matrix Γ . In (16), κ(t) > 0 and ι(t) > 0 denote the scalar functions which will be defined later. ϑ in (16) and (18) is designed as ⎧ ϑ = arctanh[ψ − κ(t)q] ⎪ ⎪ ⎨ ˙ ψ = sech2 (ϑ)[tanh(˜ q ) − tanh(ϑ)] (19) + κ(t)ν + κ(t)q ˙ ⎪ ⎪ ⎩ ν = qˆ˙ − tanh(˜ q ) − tanh(ϑ) By the definition of ϑ in (19), ψ + κ(t)q should be bounded by |ψ + κ(t)q| < 1. We will show this in the following lemma. Lemma 1. Denote  = ψ − κ(t)q. If ϑ, ψ and ν is defined in (19), then  can be bounded by || < 1 under the condition that ε(0) = ϑ(0). Proof. Define a variable vector ε with the following differential equation ε˙ = tanh(˜ q ) − tanh(ε) − κ(t)cosh2 (ε)ξ

(20)

ξ = q˜˙ + tanh(˜ q ) + tanh(ϑ)

(21)

where ξ given as It can be obtained that the derivative of tanh(ε) satisfies 

q ) − tanh(ε)] − κ(t)ξ [tanh(ε)] = sech2 (ε)[tanh(˜

(22)

Next, differentiating tanh(ϑ) along with the second equation of (19), yields  ˙ + κ(t)q˙ [tanh(ϑ)] = ψ˙ + κ(t)q

q ) − tanh(ϑ)] − κ(t)ξ = sech2 (ϑ)[tanh(˜

(23)

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Note that tanh(ε) and tanh(ϑ) have identical differential equations by comparing (22) with (23), thus if the initial values are identical, i.e. ε(0) = ϑ(0), then tanh(ε) and tanh(ϑ) remain equal for all the time. Therefore,  can be bounded by || < 1. Remark 1: We choose the initial condition ψ(0) = κ(0)q(0), i.e. ϑ(0) = 0. Thus, if ε(0) is chosen to be zero, then tanh[ε(t)] = tanh[ϑ(t)] for all t > 0. Furthermore, we can conclude that ϑ˙ = tanh(˜ q ) − tanh(ϑ) − κ(t)cosh2 (ϑ)ξ based on the characteristics of tanh(·) function. With the aid of Lemma 1, we can present the stability analysis of the velocity observer.

4

Stability Analysis

By virtue of (14), (21) and the last equation of (19), it follows that ξ = q˙ − ν

(24)

Subtracting (15) from the above manipulator system equation (1), we have M (q)(q¨˜ − γ) + C(q, q) ˙ q˙ + g(q) − Y (ˆ q , qˆ˙)θˆ + δ = 0

(25)

Next, we replace q¨˜ of (25) by the term ξ, ξ˙ and ν and it yields that M (q)ξ˙ + C(q, q)ξ ˙ + Y (ˆ q , qˆ˙)θ˜ + x = 0

(26)

in which θ˜ = θ − θˆ and x is given by x =κ(t)M (q)ξ − M (q)sech2 (˜ q )ξ + C(q, ν)ξ + C(q, ν)ν − C(ˆ q , qˆ˙) − g(ˆ q ) + g(q) + δ

(27)

According to Property 4, we have C(q, ν)ν − C(ˆ q , qˆ˙)qˆ˙ = C(q, ν)ν − C(q, ν)qˆ˙ + C(q, ν)qˆ˙ − C(ˆ q , qˆ˙)qˆ˙ = C(q, ν − qˆ˙)(ν − qˆ˙) + C(q, qˆ˙)qˆ˙ − C(ˆ q , qˆ˙)qˆ˙

(28)

Now it is time to present the main results in the form of the following theorem. Theorem 1. Considering the observer given by (15) for Euler-Lagrange system (1), the estimation errors q˜˙ and q˜ defined in (13) and (14) would globally converge to zero.

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Proof. Let us define a nonnegative function in the following form ˜ = V (˜ q , ϑ, ξ, θ)

m

ln[cosh(˜ qk )] +

k=1

m

ln[cosh(ϑk )]+

k=1

(29)

1 1 T ξ M (q)ξ + θ˜T Γ −1 θ˜ 2 2 ˜ yields Taking the derivative of V (˜ q , ϑ, ξ, θ) ˜ = tanh(˜ V˙ (˜ q , ϑ, ξ, θ) q )q˜˙ + tanh(ϑ)ϑ˙ + ξ T M (q)ξ˙ 1 ˙ + ξ T M˙ (q)ξ − θ˜T Γ −1 θˆ 2

(30)

˙ ˙ where the fact that θ˜ = −θˆ is used. From the proof of Lemma 1, it follows that q˜˙ = ξ − tanh(˜ q ) − tanh(ϑ) and ϑ˙ = tanh(˜ q ) − tanh(ϑ) − κ(t)cosh2 (ϑ)ξ. Next, differentiating θˆ in (18) yields ˙ θˆ = −Γ Y (ˆ q , qˆ˙)T ξ

(31)

In addition, by making use of (26) and the skew-symmetricity of M˙ (q)−2C(q, q), ˙ (30) can be transformed into V˙ = − tanhT (˜ q ) tanh(˜ q ) − tanhT (ϑ) tanh(ϑ) − κ(t)ξ T M (q)ξ + ξ T [M (q)sech2 (˜ q) T − C(q, ν)]ξ − ξ [C(q, ν − qˆ˙)(ν − qˆ˙)

(32)

+ C(q, qˆ˙)qˆ˙ − C(ˆ q , qˆ˙)qˆ˙ + g(q) − g(ˆ q) − ιi (t)cosh(˜ q )] In light of (7) in Property 4 and Assumption 1, V˙ is bounded by V˙ ≤ −  tanh(˜ q )2 −  tanh(ϑ)2 − κ(t)lm ξ2 + (lm + lc1 ν)ξ2 + lc1 ξν − qˆ˙2

√ + ξ[lc2  tanh(˜ q )qˆ˙2 + lg  tanh(˜ q ) − ι(t) m]

≤ − tanh(˜ q )2 −  tanh(ϑ)2 − κ(t)lm ξ2 √ √ + [lm + lc1 (qˆ˙ + 2 m)]ξ2 + [4mlc1 + 2 m √ (lc2 qˆ˙2 + lg ) − ιi (t) m]ξ Now we can design the scalar function κ(t) and ι(t) as  √ κ(t) = lm + lc1 ( qˆ˙2 + α + 2 m) + β √ ι(t) = 4 mlc1 + 2(lc2 qˆ˙2 + lg )

(33)

(34) (35)

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From (34) and (35), it is obvious that (33) can be written in the following form V˙ ≤ − tanh(˜ q )2 −  tanh(ϑ)2 − βξ2

(36)

˜ is globally Lipschitz. And it is obvious that Finally, recalling that V (˜ q , ϑ, ξ, θ) ˜ is unbounded with regard to q˜, ϑ, ξ, θ, ˜ and similar to the analysis in V (˜ q , ϑ, ξ, θ) [8], we can conclude that q˜, ϑ, ξ, θ˜ asymptotically converges to zero by invoking the LaSalle-Yoshizawa theorem. Remark 2: Compared with [8], we extend the global convergent velocity observer design to the much more general scenario. In [8], the parametric uncertainty is modeled as partially known, and the exact Centrifugal and Coriolis forces matrix C(q, v) have to be used in the observer design. Different from that, we remove this limit and achieve a less conservative result. It is worth noting that the proposed globally velocity observer can be used in the consensus problem of multiple mechanical systems.

5

Numerical Simulation

Let us consider a 2DOF manipulator with the following dynamics



q˙1 M11 M12 q¨1 C11 C12 u1 + = M21 M22 q¨2 C21 0 q˙2 u2 where

M11 = a1 + 2a3 cos q2 + 2a4 sin q2 M12 = M21 = a2 + a3 cos q2 + a4 sin q2 M22 = a2 C11 = −hq˙2 C12 = −h(q˙1 + q˙2 ) C21 = hq˙1 h = a3 sin q2 − a4 cos q2

with

2 2 a1 = I1 + ml lc1 + Ie + me lce + me l12 2 a2 = Ie + me lce

a3 = me l1 lce cos δe a4 = me l1 lce sin δe where m1 = 1, l1 = 1, me = 2.5, δe = 0.52, I1 = 0.12, lc1 = 0.5, Ie = 0.25, lce = 0.6, The initial status is set as q(0) = [−0.785, 0] rad, q(0) = [0, 0] rad/s, qˆ˙(0) = [2.7, 4] rad/s and Γ in (18) is identity matrix with appropriate dimension. The velocity estimation is shown in Fig. 1, where the solid curve denotes q˜˙1 and dash one represents q˜˙2 .

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Fig. 1. Velocity estimation error

6

Conclusion

Through this short paper, the authors design a new globally convergent joint velocity estimator for a kind of robotic manipulators modeled by Euler-Lagrange equation with uncertain parameters. Different from most existing methods, the velocity observer mentioned above doesn’t rely on an advance known upper bound for the velocity magnitude. Thus we extend the results to the more general case of uncertain parameters.

References 1. De Wit CC, Fixon N, Astrom KJ (1992) Trajectory tracking in robot manipulators via nonlinear estimated state feedback. IEEE Trans Robot Autom 8(1):138–144. https://doi.org/10.1109/70.127249 2. Salcudean S (1991) A globally convergent angular velocity observer for rigid body motion. IEEE Trans Autom Control 36(12):1493–1497. https://doi.org/10.1109/9. 106169 3. Davila J, Fridman L, Levant A (2005) Second-order sliding-mode observer for mechanical systems. IEEE Trans Autom Control 50(11):1785–1789. https://doi. org/10.1109/TAC.2005.858636 4. Xian B, De Queiroz MS, Dawson DM, Mcintyre ML (2004) A discontinuous output feedback controller and velocity observer for nonlinear mechanical systems. Automatica 40(4):695–700. https://doi.org/10.1016/j.automatica.2003.12.007 5. Su Y, Muller PC, Zheng C (2007) A simple nonlinear observer for a class of uncertain mechanical systems. IEEE Trans Autom Control 52(7):1340–1345. https:// doi.org/10.1109/TAC.2007.900851 6. Pagilla PR, Tomizuka M (2001) An adaptive output feedback controller for robot arms: stability and experiments. Automatica 37(7):983–995. https://doi.org/10. 1016/S0005-1098(01)00048-6 7. Arteaga MA (2003) Robot control and parameter estimation with only joint position measurements. Automatica 39(1):67–73. https://doi.org/10.1177/ 0278364906063830

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8. Namvar M (2009) A class of globally convergent velocity observers for robotic manipulators. IEEE Trans Autom Control 54(8):1956–1961. https://doi.org/10. 1109/TAC.2009.2023960 9. Astolfi A, Ortega R, Venkatraman A (2010) A globally exponentially convergent immersion and invariance speed observer for mechanical systems with nonholonomic constraints. Automatica 46(1):182–189. https://doi.org/10.1109/TAC. 2009.2023960 10. Romero JG, Ortega R, Sarras I (2015) A globally exponentially stable tracking controller for mechanical systems using position feedback. IEEE Trans on Autom Control 60(3):818–823. https://doi.org/10.1109/TAC.2014.2330701 11. Zhang F, Dawson DM, De Queiroz MS, Dixon WE (2000) Global adaptive output feedback tracking control of robot manipulators. IEEE Trans Autom Control 45(6):1203–1208. https://doi.org/10.1109/9.863607

Optimal Sensor Placement for TDOA-Based Source Localization with Distance-Dependent Noises Yueqian Liang(B) and Changqing Liu China Academy of Electronics and Information Technology, Beijing 100041, China [email protected], [email protected]

Abstract. Time-difference-of-arrival (TDOA) sensor networks play an important role in passive source localization scenario. TDOA measurement noises are verified to be distance-dependent. In this letter by maximizing the determinant of the Fisher information matrix (FIM), det(FIM), the optimal TDOA sensor deployment problem is investigated under distance-dependent noise assumption. The general det(FIM) is derived. And the optimal sensor placements are discussed in detail when the sensor noise levels are high, low or mixed from firstly a special case and then the general case. Keywords: Optimal sensor placement · Time-difference-of-arrival · Distance-dependent noise · Source localization · Fisher information

1

Introduction

Source localization, which aims to identify an accurate position for an emitting or a reflective source, covers a broad set of applications in both military and civilian areas. And wireless sensor networks (WSNs), comprised of sensors which can provide noisy range, bearing, time-of-arrival (TOA), time-difference-of-arrival (TDOA), received signal strength (RSS) measurements, their combinations or mixtures, can usually be employed to achieve source localization [1–4]. Although letting a single sensor move along a well-designed path, which can be well guaranteed by some advanced control strategies (See Refs. [5–8] for some examples), can improve the localization performance to some degree. The utilization of multiple low-cost homogenous or heterogeneous sensors to achieve significant localization improvement has become the trend. It is well known that the geometric configuration/placement of WSNs with respect to the true source location can significantly influence the localization performance. Mart´ınez and Bullo studied the optimal configuration problem of range-only sensor networks, and also proposed a technique to maintain the obtained configuration in [2]. In [3,4], the maximization of the determinant of the Fisher information matrix (FIM), whose inverse defines the well-known c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 535–544, 2020. https://doi.org/10.1007/978-981-32-9682-4_56

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Cram´er-Rao lower bound (CRLB), was chosen by Bishop et al. to systematically identify the optimal configuration of multiple homogeneous range, bearing, TOA and RSS sensors. By defining a unified optimality criterion, Zhao et al. recently formulated the optimal homogeneous range, bearing and RSS sensor placement problems as an identical parameter optimization problem and then gave a detailed discussion utilizing the results achieved in the frame theory field [9]. Fang et al. further discussed in detail the influence of the prior information on the optimal range-only sensor placement [10]. Recently, instead of the artificial invariant statistical characteristic assumption for the measurement noises, more realistic distance-dependent noises have been paid more and more attention. Three types of distance-dependent noise models were mainly exploited, the one comprised of additive and multiplicative noises [11,12], the one with a basic Gaussian white noise multiplying by a function of the sensor-source distance [13,14] and the one with a path-loss exponent, a reference distance and a corresponding reference noise level [15–17]. Moreno-Salinas et al. presented the optimal bearing sensor placement conditions in 3-dimensional (3D) space by minimizing the trace of the CRLB [13]. Yan et al. discussed the influence of the distance-dependent noise on the regular sensor placement and linear sensor placement of range sensors [14]. To minimize the position error bound for agent localization using range measurements, Jourdan and Roy designed a RELOCATE algorithm and investigated its convergency [15]. Considering range sensors with different noise covariances, Perez-Ramirez et al. elaborated the optimal landmark placements for 2D and 3D autonomous guided vehicle localization [16]. Huang et al. analyzed the TDOAbased localization performance when distance-dependent noises were involved, concluding that under low noise level assumption, the FIM approaches to a simplified form by abandoning the second-order term about the noise variances, and based on the simplified FIM, they gave out the optimal TDOA sensor placement conditions [17]. In this paper we focus on the optimal placement problem of TDOA sensors with distance-dependent noises. The main contribution is that based on the proposed original FIM of the distance-dependent noise model, the optimal sensor placements are presented according to different noise levels.

2 2.1

Optimality Criterion Evaluation TDOA Measurement with Distance-Dependent Noise

Consider the scenario that a stationary source is to be localized. At least 2 TDOA measurements (3 static TDOA sensors) are needed. The true source location is assumed to be ps = (xs , ys )T , and the location of the i-th TDOA sensor is denoted as pi = (xi , yi )T with i = 1, 2, · · · , N ≥ 3. The true sensor-source distance and the true bearing of the i-th sensor with respect to the source measuring  from the positive x-axis can be computed by ri = (yi − ys )2 + (xi − xs )2 and θi = arctan2(yi − ys , xi − xs ) respectively.

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The noisy TOA measurement is given by [17] ri + εi tˆi = ti + εi = t0 + c0

(1)

where t0 is the unknown signal transmission time, c0 is the signal propagation speed, and εi indicates the distance-dependent Gaussian-distributed measurement noise, satisfying   αi    ri 2 2 εi ∼ N 0, Ri = N 0, σi (2) r0 Here r0 > 0 is the path-loss reference distance (typically setting to be 1), αi > 0 is the path-loss exponent and σi > 0 denotes the standard deviation of the measurement noise at the reference distance r0 . And as usual the measurement noises of different sensors are assumed to be independent. Because of the usual unknown nature of t0 , choosing the first sensor as the reference sensor, we can use the TDOA measurements for source localization, which are stacked as ˆt = [tˆ21 , tˆ31 , · · · , tˆN 1 ]T = t + ε = 1 [r2 − r1 , r3 − r1 , · · · , rN − r1 ]T c0 + [ε2 − ε1 , ε3 − ε1 , · · · , εN − ε1 ]T (3)   It can be seen that ε ∼ N 0(N −1)×1 , R with R = R1 + R2 =  2 ] . Here 0 and 1 are used to denote R12 1(N −1)×(N −1) + diag [R22 , R32 , · · · , RN a zero matrix and an all-ones matrix with appropriate dimensions respectively. 2.2

Optimality Metric: The Determinant of FIM

The CRLB is well known to represent the best estimation of an unbiased estimator and is defined by the inverse of the FIM [3,18]. Different scalar measures of the CRLB and the FIM, such as the trace of the CRLB (tr(CRLB)), the determinant of the FIM (det(FIM)) and the maximum eigenvalue of the CRLB (λmax (CRLB)), can be chosen as the optimality metrics to identify the optimal placement of WSNs. We choose the maximization of det(FIM), which is equivalent to maximizing the amount of information contained in the noisy measurements and minimizing the volume of the uncertainty ellipsoid [3,18], as the optimality criterion to optimally deploy the TDOA sensors. Given ˆt ∼ N (t, R), the (i, j)-th entry of the FIM I is computed by   T    ∂t ∂t 1 −1 −1 ∂R −1 ∂R R R Iij = + tr R (4) ∂xi ∂xj 2 ∂xi ∂xj where i, j is 1 or 2, and x1 = xs , x2 = ys . Define the following notations, N

 1 αi ai , bi = , ci = ai + b2i ai = 2 2 , A = c0 Ri ri i=1



1 ai − 2 A

 , C=

N  i=1

ci

(5)

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Taking r1 = R1 1(N −1)×1 , then we have R1 = r1 rT1 . Using Sherman-Morrison formula, we can get R−1 = R−1 2 −

−1 R12 R−1 2 1(N −1)×(N −1) R2

(6)

1 + R12 11×(N −1) R−1 2 1(N −1)×1

Substituting (6) into (4) and making a full simplification, we obtain   1 X2 + 12 X3 X6 + 12 X4 X5 C + X1 + X3 X4 I= C − X1 + X5 X6 2 X2 + 12 X3 X6 + 12 X4 X5

(7)

where X1 =

N  i=1

ci cos 2θi , X2 =

N 

ci sin 2θi ,

i=1

 √ √ N  2 2 bi bi +√ −√ X3 = ai cos θi , X4 = ai cos θi , A A A A i=1 i=1   √ √ N N   2 2 bi bi +√ −√ X5 = ai sin θi , X6 = ai sin θi A A A A i=1 i=1 N 



(8)

are some formal variables. Then the determinant of I is C(X3 X4 + X5 X6 ) (X3 X6 − X4 X5 )2 C 2 − X12 − X22 + − 4 4 16 X1 (X3 X4 − X5 X6 ) + X2 (X3 X6 + X4 X5 ) − 4

|I|=

(9)

Remark 1. As in Section III-C of [17], a similar discussion can be done to conclude that the localization performance does not depend on the selection of reference sensor.

3

Optimal Sensor Placements

3.1

Case with Identical {ai , bi , ci }

We firstly consider a special case that for all i = 1, 2, · · · , N , we have σi = σ, αi = α and ri = r. This implies identical {ai , bi , ci }. And det(FIM) is now |I| = A0 A1 (N 2 − Y12 − Y22 ) + A0 A2 (Y32 + Y42 ) −

A0 A2 (Y1 Y32 − Y1 Y42 + 2Y2 Y3 Y4 ) N

(10)

where the formal variables are Y1 =

N  i=1

cos 2θi , Y2 =

N  i=1

sin 2θi , Y3 =

N  i=1

cos θi , Y4 =

N  i=1

sin θi

(11)

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and the involved coefficients are A0 =

2N r2 + (N − 2)c20 α2 σ 2 rα >0 16N 3 c40 σ 4 r4+2α

(12)

A1 = 2N 2 r2 + N (N − 2)c20 α2 σ 2 rα > 0 A2 = −4N r + 2 2

(13)

2N c20 α2 σ 2 rα

(14)

Owing to different reference noise level σ (resulting in A2 ’s with different signs), the optimal sensor placement differs, which is discussed from the following three cases. √ Case 1. Critical noise level: σ = σth = 2N r1−α/2 /(c0 α). This very special case implies that A2 = 0 and further |I| = A0 A1 (N 2 − Y12 − 2 Y2 ), from which we can easily find that the det(FIM) is maximized to 

|I|max,1

2N r2 + (N − 2)c20 α2 σ 2 rα = A0 A1 N = 16c40 σ 4 r4+2α 2

2 (15)

if and only if Y1 = Y2 = 0. And therefore the conclusions drawn in [3,9] for range sensors are applicable. Case 2. Low noise level: σ < σth . In this case, we have A2 < 0. Regarding (10) as a multivariate continuous function with respect to the formal variables {Y1 , Y2 , Y3 , Y4 } and according to the extreme theory, we get that det(FIM) is maximized to |I|max,1 if and only if Y1 = Y2 = Y3 = Y4 = 0. This optimal condition is same with that for the TOA sensors given in [3]. A typical solution (regular solution) arranges the sensors on the vertices of a regular polygon. Case 3. High noise level: σ > σth . Since in this case A2 > 0, we can see from (10) that the maximum det(FIM) is larger than |I|max,1 = A0 A1 N 2 if we take Y1 = Y2 = 0 while Y3 and Y4 are not 0. Moreover, we conclude that without loss of generality, (1) if N is even, det(FIM) is maximized when θ1 = θ2 = · · · = θN/2 = φ1 and θN/2+1 = θN/2+2 = · · · = θN = φ2 , and (2) if N is odd, det(FIM) is maximized when θ1 = θ2 = · · · = θ(N −1)/2 = φ1 , θ(N −1)/2+1 = θ(N −1)/2+2 = · · · = θN −1 = φ2 and θN = φ3 . φ1 , φ2 and φ3 are specified as follows. When N is even, the determinant of the FIM becomes |I| = A0 N 3 (N − 1)c20 α2 σ 2 rα sin (φ1 − φ2 )2

(16)

from which we see that the maximum det(FIM) is   α2 (N − 1) 2N r2 + (N − 2)c20 α2 σ 2 rα 3 2 2 2 α |I|max,2 = A0 N (N − 1)c0 α σ r = 16c20 σ 2 r4+α (17) if and only if φ1 − φ2 = ±lπ/2 with l a non-zero integer.

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When N is odd, the determinant of the FIM is A0 (N − 1)2  B0 cos (φ1 − φ2 ) + B0 cos (φ1 − φ3 ) + B0 cos (φ2 − φ3 ) 2 − B0 cos (2φ1 − φ2 − φ3 ) − B0 cos (φ1 − 2φ2 + φ3 ) − B0 cos (φ1 + φ2 − 2φ3 ) + B1  + B2 cos (2φ1 − 2φ2 ) + B3 cos (φ1 − φ2 ) cos (φ1 + φ2 − 2φ3 ) (18)

|I| =

where B0 = A2 /N > 0 B1 = B2 = B3 =

(19)

−3B0 /2 + (N + 3N )c20 α2 σ 2 rα B0 /2 − (N 2 − N )c20 α2 σ 2 rα B0 − 4N c20 α2 σ 2 rα 2

(20) (21) (22)

We find that when φ1 , φ2 and φ3 satisfy φ1 + φ2 − 2φ3 = 2lπ with l = 0, ±1, ±2, · · · ± sin ((φ1 − φ2 )/2) + (1 + B3 /B0 ) sin (φ1 − φ2 )

(23a)

∓ sin (3(φ1 − φ2 )/2) + 2B2 /B0 sin (2φ1 − 2φ2 ) = 0

(23b)

the det(FIM) is maximized. Treating (φ1 − φ2 ) as an unknown variable, we can find from (23b) the solution of (φ1 −φ2 ). Then we can get (φ2 −φ3 ) from relation (23a) and further the maximum det(FIM) |I|max,3 from (18). Note that both the solution of (φ1 − φ2 ) and the maximum det(FIM) |I|max,3 are omitted herein because of their extreme complexity. Remark 2. In Case 3, a solution determined by Y1 = Y2 = Y3 = Y4 = 0 (e.g., the regular solution) corresponds to a local maxima of |I| and thus can be regarded as a suboptimal sensor placement. Remark 3. The above discussion reveals that when the reference noise level σ is larger than the threshold σth , and when N ≥ 4, the optimal sensor placement requires that two or more TDOA sensors should be deployed at the same location. To avoid this abnormal placement and noticing the continuity of the det(FIM) with respect to the sensor locations, practical suboptimal placements can be exploited in real applications to deploy the sensors near the theoretical optimal locations. Remark 4. Note that for a given odd N , different {α, r, σ} results in different {B0 , B1 , B2 , B3 }, and thereby implies different optimal solutions satisfying (23a and 23b). Example 1. Consider the situation when N = 3, α = 4, r = 50 m and 1/(c0 σ) = 18 dB (or, σ = 1.585 × 10−2 /c0 s). The threshold is σth = 1.225 × 10−2 /c0 s, implying σ > σth . Figure 1(a) depicts the normalized det(FIM) with respect to θ2 and θ3 by fixing θ1 = 0, from which we can see that the regular solutions, [2π/3, 4π/3] and [4π/3, 2π/3] (the two small points), are just two local

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Fig. 1. The evolutions of the normalized det(FIM) with respect to θ2 and θ3 with θ1 = 0 (Example 1): (a) 1/(c0 σ) = 18 dB; (b) 1/(c0 σ) = 20 dB.

Fig. 2. Comparison of the theoretical maximum det(FIM) |I|max,4 and the true maximum det(FIM) obtained by numerical methods (Example 2). They are depicted as (red) filled squares and (blue) disks respectively. And the reference noise levels are chosen as (for i = 1, 2): (a) 1/(c0 σi+1 ) = 1/(c0 σi ) dB; (b) 1/(c0 σi+1 ) = 1/(c0 σi ) + 1 dB; (c) 1/(c0 σi+1 ) = 1/(c0 σi ) + 2 dB; (d) 1/(c0 σi+1 ) = 1/(c0 σi ) + 3 dB.

optima, and the true optimal solutions are [1.0142, 2.0284], [2.0284, 1.0142], [1.0142, 5.2690], [5.2690, 1.0142], [4.2547, 5.2690] and [5.2690, 4.2547] (the six large points), which conform to (23a and 23b). As a comparison, when 1/(c0 σ) = 20 dB (σ = 10−2 /c0 s), corresponding to σ < σth , the regular solutions are optimal ones (Fig. 1(b)).

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General Case with Fixed {ri }

Let us now see the general case. Denote rmin = min {ri } , rmax = max {ri } , αmin = min {αi } , αmax = max {αi } , σmin = min {σi } ,   √ √   αmin /2 αmax /2 c0 αmax rmax c0 αmin rmin , σth,2 = 2N rmax σmax = max {σi } , σth,1 = 2N rmin

(24) Inspired by the above discussion for the special case, we have the following conclusions according to different noise levels. Case 1. Low noise level: σmax < σth,1 . √ √ In this case, it can be seen that for all i = 1, 2, · · · , N, bi /A− 2/ A < 0. This corresponds to Case 2 of the special case. Similarly, by regarding (9) as a multivariate continuous function about the formal variables {X1 , X2 , X3 , X4 , X5 , X6 } and applying the extreme theory, we draw the conclusion that the det(FIM) is maximized to |I|max,4 = C 2 /4 if and only if X1 = X2 = X3 = X4 = X5 = X6 = 0

(25)

Remark 5. The mutual dependency of the formal variables X1 , X2 , X3 , X4 , X5 , X6 makes that (25) may be unsolvable, especially under the situations that {ai , bi , ci }’s for different sensors differ much from each other and that the sensor number N is small. And therefore the theoretical maximum det(FIM) |I|max,4 may not be reached under these situations. Case 2. High noise level: σmin √ >√σth,2 . In this case, we have bi /A − 2/ A > 0 for all i. This is analogous to Case 3 of the special case, and the theoretical maximum det(FIM) |I|max,5 is larger than |I|max,4 . Case 3. Mixed noise level: Otherwise. If it is not the first two cases, the theoretical maximum det(FIM) may also √ √ exceed |I|max,4 if the sensors with positive bi /A − 2/ A are enough. √ √ Remark 6. When there exists at least one i making bi /A − 2/ A > 0 (Case 2 and Case 3), the optimal condition and its corresponding maximum det(FIM) cannot be analytically given. Instead, one can only resort to numerical methods providing specific parameters. The obtained maximum det(FIM) can be |I|max,4 . Example 2. Consider N = 3. We set all ri to be 50 m, and the path-loss exponents of the i-th sensor as αi = 3.8 + 0.1(i − 1) for i = 1, 2. Varying the reference noise levels σi , we can get the simulation results as shown in Fig. 2. Under the above setting, we have 1/(c0 σth,1 ) = 19.1195 dB and 1/(c0 σth,2 ) = 17.1918 dB, which indicates that when all 1/(c0 σi ) > 19.1195, it corresponds to Case 1, when all 1/(c0 σi ) < 17.1918, it corresponds to Case 2, and otherwise it corresponds to Case 3. We can see from Fig. 2(a) that when all 1/(c0 σi ) are 16 or 17,

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the maximum det(FIM) is larger than |I|max,4 , and when all 1/(c0 σi ) are larger than 19, the maximum det(FIM) is less than |I|max,4 because of the unsolvability of (25). And when all 1/(c0 σi ) are 18 or 19, the maximum det(FIM) is slightly smaller than |I|max,4 . It can also be observed from Figs. 2(b)–(d) that as the difference of 1/(c0 σi ) increases, which results in {ai , bi , ci }’s with larger difference, the gap between the true maximum det(FIM) with |I|max,4 enlarges. 3.3

About Source Location Uncertainty

In the real sensor placement scenario, the true source position is always not known to us. An effective strategy to address the source location uncertainty is to exploit the average det(FIM) similar to the average tr(CRLB) in [13]. The source is assumed to be in a certain area and the corresponding probability density function of the source locating in the area is appropriately modeled. By employing numerical methods, “optimal” sensor deployment can be obtained. Similar conclusions as that in [13] can be drawn.

4

Conclusions

The optimal sensor placement of multiple TDOA sensors with distancedependent noises for source localization has been investigated in this letter. The results turn out to be very different when the reference noise levels are varying, all high, all low or mixed.

References 1. Mao G, Fidan B, Anderson BDO (2007) Wireless sensor network localization techniques. Comput Netw 51(10):2529–2553 2. Mart´ınez S, Bullo F (2006) Optimal sensor placement and motion coordination for target tracking. Automatica 42(4):661–668 3. Bishop AN, Fidan B, Anderson BD, Do˘ gan¸cay K, Pathirana PN (2010) Optimality analysis of sensor-target localization geometries. Automatica 46(3):479–492 4. Bishop AN, Jensfelt P (2009) An optimality analysis of sensor-target geometries for signal strength based localization. In: Proceedings 5th International Conference ISSNIP, Melbourne, Australia, pp 127–132 5. Jia Y (2000) Robust control with decoupling performance for steering and traction of 4WS vehicles under velocity-varying motion. IEEE Trans Control Syst Technol 8(3):554–569 6. Jia Y (2003) Alternative proofs for improved LMI representations for the analysis and the design of continuous-time systems with polytopic type uncertainty: A predictive approach. IEEE Trans Autom Control 48(8):1413–1416 7. Aguiar AP, Hespanha JP (2007) Trajectory-tracking and path-following of underactuated autonomous vehicles with parametric modeling uncertainty. IEEE Trans Autom Control 52(8):1362–1379 8. Sujit PB, Saripalli S, Sousa JB (2014) Unmanned aerial vehicle path following: a survey and analysis of algorithms for fixed-wing unmanned aerial vehicles. IEEE Control Syst Mag 42(1):42–59

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9. Zhao S, Chen BM, Lee TH (2013) Optimal sensor placement for target localisation and tracking in 2D and 3D. Int J Control 86(10):1687–1704 10. Fang X, Yan W, Zhang F, Li J (2015) Optimal sensor placement for range-based dynamic random localization. IEEE Geosci Remote Sens Lett 12(12):2393–2397 11. Wang X, Fu M, Zhang H (2012) Target tracking in wireless sensor networks based on the combination of KF and MLE using distance measurements. IEEE Trans Mob Comput 11(4):567–576 12. Wang W, Ma H, Wang Y, Fu M (2015) Performance analysis based on least squares and extended Kalman filter for localization of static target in wireless sensor networks. Ad Hoc Netw 25(Part A):1–15 13. Moreno-Salinas D, Pascoal A, Aranda J (2013) Sensor networks for optimal target localization with bearings-only measurements in constrained three-dimensional scenarios. Sensors 13(8):10386–10417 14. Yan W, Fang X, Li J (2014) Formation optimization for AUV localization with range-dependent measurements noise. IEEE Commun Lett 18(9):1579–1582 15. Jourdan DB, Roy N (2008) Optimal sensor placement for agent localization. ACM Trans Sens Netw 4(3):13 16. Perez-Ramirez J, Borah DK, Voelz DG (2013) Optimal 3-D landmark placement for vehicle localization using heterogeneous sensors. IEEE Trans Veh Tech 62(7):2987– 2999 17. Huang B, Xie L, Yang Z (2015) TDOA-based source localization with distancedependent noises. IEEE Trans Wirel Commun 14(1):468–480 18. Uci´ nski D (2004) Optimal Measurement Methods for Distributed Parameter System Identification. CRC Press, Boca Raton

Qualitative Path Reasoning with Incomplete Information in VAR-Space Xiaodong Wang(B) , Yan Zhang, Nan Xiao, and Ming Li Mudanjiang Normal University, Mudanjiang 157012, China [email protected]

Abstract. The method that can deal with incomplete information is of great significance in practical applications. This paper presents a method for the qualitative path reasoning with incomplete information in VARSpace. Specifically, the conceptual neighborhood of qualitative positions in VAR-space is defined, and based on the 1-order (or 2-order) neighbors and the rule of the continuity of movement, the incomplete data problem can be solved. The experimental results indicate that the proposed method is effective. Keywords: Conceptual neighbourhood Path reasoning · Incomplete data

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· Qualitative method ·

Introduction

The qualitative method is suitable for expressing spatial relationships between entities, and it has been used to abstract essential information from the real world. Due to its ability to deal with incomplete (or uncertain) data, this method has attracted increasing attention and has been applied in many fields, for example, spatial data query, qualitative navigation, GIS and high level vision [1–3]. In recent years, the qualitative method has also been expanded to research of the motion of objects. Because the most tasks (such as reasoning tasks) depend on the representation, the basic task is to find a satisfiable formalism for expressing motion knowledge. So far, many spatial representation methods using qualitative information have been proposed, they have their own characteristics and application scope, and some of them have been used to represent motion knowledge. Among them, the metric diagram/place vocabulary (MD/PV) model [4] can abstract the essential attributes from quantitative data of spatial knowledge, and therefore, this model has been widely accepted and is considered as a pioneering work in this field. In essence, the MD/PV model is a spatial decomposition technique. Now, based on the technique of decomposition of space, a series of new qualitative spatial representation methods emerge [5,6], although the standard of decomposition in these methods is different. Due to the advantages of the technique c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 545–552, 2020. https://doi.org/10.1007/978-981-32-9682-4_57

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of decomposition, a qualitative representation method (VAR-Space) based on adjacency relationship in Voronoi diagram is proposed, in which the trajectory of motion can be qualitatively described, and further, the qualitative motion can also be reasoned by the changes in Voronoi diagram [7], but this method is difficult to work normally when the information is incomplete. Because it is often difficult to obtain complete information in real scenarios, reasoning algorithms should be able to deal with data missing. In this paper, we first introduce VAR-Space and its basic concepts. Then we analyze the spatial structure of VAR-Space, and based on it, we establish the reasoning algorithm which can reason motion path with incomplete data. Finally, we conduct experiments to verify proposed method and draw a conclusion.

2

Basic Concepts About VAR-Space

Voronoi diagram is one of the most important structure in computational geometry, and the adjacency relations in Voronoi diagram is concord with the perceptual and linguistic spaces of humans. Taking the adjacency relation as the criterion of decomposition of space, a set of points with the same adjacency relations is regarded as a qualitative position, and all of qualitative positions constitute the VAR-Space. In this section, we first introduce Voronoi diagram [8], then define the spatial adjacency relation in Voronoi diagram, and based on the adjacency relation, define the qualitative position and qualitative space. Definition 1 (Voronoi Diagram). In the plane R2 , let d(·, ·) be distance function and P be a set of points (called site or generator), call the region V R(pi ) V R(pi ) = {p ∈ R2 | d(p, pi ) ≤ d(p, pj )

f or all i = j}

Voronoi region or Voronoi cell, and call V D(P ) =



V R(pi )

pi ∈P

the Voronoi diagram of S. In Voronoi diagram, the common line segment of two Voronoi regions is called a Voronoi edge, and endpoints of line segment are called Voronoi vertices. The circle, which passes through the generators incident to the vertex and whose centre is at Voronoi vertex, is called circumcircle of generators. In V D(S), the edge between the generators si , sj ∈ S is denoted as e(si , sj ), and the sets of all edges and vertices in V D(S) are denoted as E(S) and V (S), respectively. Definition 2 (Voronoi Adjacency Relations). In Voronoi diagram V D(S), for any two generators si , sj ∈ S, let I(si , sj ) = V R(si , S) ∩ V R(sj , S), i = j,

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If I(si , sj ) ∈ E(S), then the spatial relation between generators si , sj is called adjacent, denoted as adj(si , sj ); if I(si , sj ) ∈ V (S), then the spatial relation is called co-circle, denoted as coc(si , sj ); otherwise, i.e., I(si , sj ) = ∅, the spatial relation is called dis-adjacent, denoted as dadj(si , sj ). All generators sj ∈ S, i = j, which are adjacent, co-circle, and dis-adjacent to si , are denoted as Adj(si , S), Coc(si , S), Dadj(si , S), respectively. And V AR(si , S) = {Adj(si , S), Coc(si , S), Dadj(si , S)} is called as Voronoi adjacency relations of si in V D(S), denoted as V AR(si ). If a Voronoi adjacency relations V AR(si ) of si in V D(S) is var, then the sets of generators adjacent to si , circle to si , and dis-adjacent to si are denoted as Adj|var , Coc|var , and Dadj|var . From the qualitative viewpoint, the qualitative position can be viewed as a set of points in plane R2 , in which all points have the same one (or more) of spatial attributes. Based on the above viewpoint, the following definition can be made. Definition 3 (Qualitative Position). In plane R2 , let S be a set of fixed points and p be a movable point. If the R2 is partitioned into a finite set of mutually disjoint subdomains {Q1 , Q2 , . . . , Qm }, i.e., (

m 

Qi = R2 ) ∧ (Qi ∩ Qj = ∅, i = j, 1 ≤ i, j ≤ m),

i=1

and furthermore, p is anywhere in Qi (1 ≤ i ≤ m), and the Voronoi adjacency relations V AR(p, S ∪ {p}) of p in V D(S ∪ {p}) is the same one, denoted as [qi ], then [qi ] is called as qualitative position, and [Q]S =

m 

[qi ]

i=1

is the set of all qualitative positions determined by S. Definition 4 (n-order neighbor, VAR-Space). Let [Q]S be the set of qualitative positions determined by set S of points in R2 , if two qualitative positions [qi ], [qj ] ∈ [Q]S can directly transform into one another, without by the other qualitative positions, then [qi ] and [qj ] are called conceptual neighbouring. If [qi ] and [qj ] are connected directly, then they are called mutual 1-order neighborhoods; if [qi ] can reach [qj ] by the 1-order neighbor of [qj ], then [qi ] is called 2-order neighbor of [qj ]. Similarly, n-order neighbor of [pj ] can be defined. And further, all qualitative positions in [Q]S and all conceptual neighbouring relations between them constitute Voronoi Adjacency Relations Space, denoted as [R].

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Definition 5 (Qualitative Path). Let [R] be Voronoi Adjacency Relations Space, in which, [Q]S is the set of qualitative positions. The sequence of qualitative positions {[q0 ], [q1 ], · · · , [qn ]},

([qi ] ∈ [Q]S , 0 ≤ i ≤ n)

is called qualitative path, denoted as [path]. Here, [qi ] and [qi+1 ] are conceptual neighbouring, n is called the lengthen of [path], denoted as L[path] .

3

The Basic Idea of the Proposed Method

Different from the previously proposed qualitative path reasoning algorithm [7], which requires the complete data, the algorithm in this section can handle the incomplete data concerning moving objects. That is to say, even if the obtained data is incomplete, the proposed method can still give a feasible solution. In Voronoi diagram, the generator moving from one qualitative position to another can causes the Voronoi edge degenerating into one point or new edge generating from one point. As shown in Fig. 1(a), generator p moving from qualitative position 1 to 3 by 2, is accompanied by the degenerating and generating of Voronoi edge. Therefore, the qualitative moving path of generator p can be reasoned by the continuous changes of the edge. Note that, there may be more than one movement between qualitative position causing the same change in Voronoi diagram edges, (e.g., degenerating or generating of Voronoi diagram edge), which can be seen in Fig. 1(b). This situation includes that point 27 moving from the position adjacent to arc2 to arc arc2 can cause the degeneration of the Voronoi diagram edge e(3, 5), and similarly, moving from the position adjacent to arc1 to arc1 do the same. The Algorithm 1 can find out all the movements between the two adjacency qualitative positions associated with the same Voronoi diagram edge.

si

p

1 2

3

(a)

(b)

Fig. 1. Analysis of Voronoi spatial relationship. (a) p moving causing the changes in Voronoi spatial relation. (b) The relation between the edge e(3|5) degenerating and the transition of neighboring qualitative positions.

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Algorithm 1. IdentifyingTransitions Data: e(si , sj ), V D(S), IdxTab(si ), [R]. Result: T (, ). begin search (v1 , v2 , s2 , s4 ) in V D(S) by e(s1 , s3 ) ; [P] [P] [P] [P] initialize qualitative position Set: Sc1 , Sc2 , Sarc1 , Sarc2 by (v1 , v2 , s2 , s4 ) ; [P] forall the [p]i ∈ Sarc2 do [P] [P] work out [p]t ∈ (Sc1 − Sc2 ) and adjacent to [p]i ; T (, ) ← ([p]i , [p]t ); end [P ] forall the [p]i ∈ Sarc1 do [P] [P] work out [p]t ∈ (Sc2 − Sc1 ) and adjacent to [p]i ; T (, ) ← ([p]t , [p]i ); end end

lost data ei [p] [p]

i

e i+1 [p]

j

[p]

j

e i+2

[p]

e i+3 [p]

k

[p]

i

[p]

(a)

k

[p]

l

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[p]

j

[p] m

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[p]

l

: first-order neighbor of [p] j : second-order neighbor of [p]

j

(b)

Fig. 2. Reasoning qualitative path with incomplete information. (a) qualitative path. (b) Neighbor of qualitative position.

As shown in Fig. 2(a), the moving generator is in the qualitative position [p]i at a moment, and its following movement leads to the changes in Voronoi edge: ei , ei+1 , ei+2 , ei+3 . In the sequential connection method, firstly, it need to be identify that all transitions (or called one-step movement) between the qualitative positions associated with the dynamic voronoi diagram edge, and then the one-step movement between the pairs of the qualitative position, of which the starting qualitative position is same as the one in previous moment, is the one that the moving generator passes though and end qualitative position of the transitions is the one where the moving generator is. By repeating the above process, the qualitative path {[p]i , [p]j , [p]k , [p]l , [p]m }, through which moving generator passes, can be inferred. However, in the case of incomplete data, e.g., ei+1 lost, the position [p]k cannot be get by the current position [p]i and dynamic edge ei , and the reasoning process cannot go on. According to the definition of concept neighborhood given above, we can know that if the motion occurs directly between two qualitative positions, they

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are 1-order neighborhoods of each other, and if the motion between two qualitative positions must pass through the 1-order neighborhood, then the two are 2-order neighborhoods. Figure 2(b) is the 1-order neighbors and 2-order ones of qualitative position [P ]j . Let moving generator be in qualitative position [p]i , during the course of movement, the sequence of dynamic Voronoi edges be ei , ei+2 , ei+3 , where ei+1 lost. As mentioned above, using sequential connection method, by the initial position [p]i and the dynamic voronoi edge ei+1 , the position [p]j in the next moment can be inferred, but due to dynamic voronoi edge ei+1 lost, the position [p]k cannot be got. Looking at Fig. 2(b), it can see that the position [p]k must be the 1-order neighborhood of [p]j , and [p]k is also the starting position of transitions of the two qualitative positions associated with the dynamic voronoi edge ei+2 . Thus, the [p]k can be inferred and the case where the length of missing data is 1 can be handled. Similarly, if the length of missing data is 2, e.g., ei+1 , ei+2 lost, the position [p]l should be the 2-order neighborhood of [p]j and is also the starting position of transitions of the two qualitative positions associated with the edge ei+3 . The description of Algorithm 2 for inferring qualitative path with incomplete data is as follows, and the cases where single or two data are lost can be handled.

Algorithm 2. Reasoning for Path with Incomplete Data Data: E = {e1 , e2 , . . . , en }, [p]ini , [R] Result: [P ath] begin [p] ← [p]ini ; add [p] into [P ath]; /* i = 1, 2, . . . , n for ei ∈ E do T ← IdentifyTransitions(ei ) /* T = {([s]1 , [d]1 ), . . . , ([s]k , [d]k )} /* j = 1, 2, . . . , k forall the ([s]j , [d]j ) ∈ T do if [s]j == [p] then /* no data loss add [d]j into [P ath]; else /* data loss if [s]j is 1-order neighbor of [p] then /* single data loss add [s]j into [P ath]; add [d]j into [P ath]; [p] ← [d]j ; else /* two data loss [m] ← common 1-order neighbor of [p] and [s]j ; add [m] into [P ath]; add [s]j into [P ath]; add [d]j into [P ath]; [p] ← [d]j ; end end end end end

*/ */ */ */ */ */

*/

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Experiments and Results

In above sections, we analyze the spatial structure of qualitative location in VAR-space and present an algorithm for reasoning path with incomplete data. In this section, the experiments were conducted to verify the proposed method being feasible and valid. At the same time, the limitations of the method are also pointed out. Firstly, in plane R2 , we randomly choose some points as generators of Voronoi diagram and then construct VAR-space. Then, we let the point s107 move along the trajectory (see Fig. 3(a)), and write down the all degeneration or generation of Voronoi diagram edges (listed in Table 1). Finally, we randomly delete a few data (marked with underline) in Table 1, and use the remaining data as a test case to verify the algorithm proposed in this paper, and the result of the algorithm is shown in Fig. 3(b). We can see that although the data are incomplete, the Algorithm 2 proposed in this paper can still infer out the qualitative path of the point passing. However, we also find that if more than two consecutive data are lost, the algorithm cannot solve the problem. Table 1. Changes of Voronoi diagram edges. e− (i, j): degeneration of Voronoi diagram edge between the points pi and pj , e+ (i, j): generating of Voronoi diagram edge between the points pi and pj , the data marked with underline is deleted. e− (30, 106)

e+ (107, 24)

e+ (107, 5)

e+ (31, 106)

e− (30, 24)

−

+

−

+

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e (107, 31) −

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e (24, 55) −

e (107, 64) +

e (24, 64)

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e (106, 107) e (30, 24)

e (30, 107) e (24, 55)

e (65, 64)

e− (55, 107)

e− (67, 64)

e− (24, 107) e+ (64, 65)

e+ (24, 64)

e+ (107, 23)

Fig. 3. Reasoning for the qualitative path of movable point. (a) point s61 moving along the curve. (b) result of reasoning.

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Conclusions

This paper proposes a method for qualitative path reasoning in VAR-Space, where the Voronoi diagram edges changing (degeneration or generating), caused by the motion of point, are used. Different from the previous method, this method can handle with incomplete information. That is to say, if a few data is lost, the method can still infer the qualitative path of the moving generator passing, which is necessary in the real environment. However, this method can only deal with the case of single data loss or two consecutive data loss now, and more complex cases need to be further studied. Acknowledgement. The work was supported by the Natural Scientific Foundation of Heilongjiang Province (No. F2016039), the project of Educational Commission of Heilongjiang Province (No. 1351MSYYB004), and the project of Mudanjiang Normal University (No. QN2018004 and No. SY201315).

References 1. Bogaert P. (2008) A qualitative calculus for moving point objects constrained by networks. Ph.D. thesis, University of Ghent 2. Cohn A, Hazarika S (2001) Qualitative spatial representation and reasoning: an overview. Fundam Inform 46(1–2):1–29 3. Ibrahim ZM, Tawfik AY (2007) An abstract theory and ontology of motion based on the regions connection calculus. In: Proceedings of the 7th international symposium on abstraction, reformulation and approximation, Whistler, Canada, pp 230–242 4. Forbus KD (1981) A study of qualitative and geometric knowledge in reasoning about motion. Technical report AITR-615, MIT Artificial Intelligence Laboratory, Cambridge, MA 5. Denis M (1997) The description of routes: a cognitive approach to the production of spatial discourse. Curr Psychol Cogn 16(4):409–458 6. Fogliaroni P, Wallgr¨ un J, Clementini E, Tarquini F, Wolter D (2009) A qualitative approach to localization and navigation based on visibility information. In: Hornsby K, Claramunt C, Denis M (eds) Spatial information theory. Lecture notes in computer science, vol 5756. Springer, Berlin, pp 312–329 7. Wang X, Li M, Liao S (2017) Reasoning for qualitative path without initial position in VAR-space. In: Jia Y, Du J, Zhang W (eds)Proceedings of 2017 Chinese intelligent systems conference (CISC 2017), MuDanJiang, China, September 2017. Lecture notes in electrical engineering, vol 460, pp 749–756 8. Aurenhammer F, Klein R (2000) Voronoi diagrams. In: Sack J, Urrutia J (eds) Handbook of computational geometry. Elsevier Science Publishers, Amsterdam, pp 201–290

A Novel Competitive Particle Swarm Optimization Algorithm Based on Levy Flight Yao Ning1,2 , Zhongxin Liu1,2(B) , Zengqiang Chen1,2 , and Chen Zhao1,2 1

2

College of Artificial Intelligence, Nankai University, Tianjin 300350, China [email protected] Key Laboratory of Intelligent Robotics of Tianjin, Tianjin 300350, China

Abstract. The particle swarm optimization is a classical optimization algorithm (PSO) that has been applied to various fields. It implements simple but efficient evolution operators to search for optimums in parameter space. However, it could not reach a good balance between global search and local search when it comes to multimodal’ problem. Levy flight is a kind of stochastic process with scale invariant, turned out to accord to animals’ behavior. This paper utilizes the levy flight operator to enhance the PSO’s global search ability during the iteration, which is based on the hierarchy structure of swarms. Therefore, a novel competitive particle swarm optimization based on levy flight (CLFPSO) is proposed. A number of benchmarks has been tested on the CLPSO with other five typical PSO algorithms. The experimental results show that the proposed algorithm could reach more outstanding and accurate consequences compared with other algorithms. Keywords: Particle swarm optimization algorithm · Hierarchy structure · Levy flight · A number of benchmarks

1

Introduction

Deterministic algorithm and stochastic algorithm are two types of classical optimization algorithm according to their nature [1]. The former follows a rigorous procedure, searching for the optimums mainly with gradient information in parameter space. On the other hand, the stochastic algorithm always conducts the searching procedure with random walk. The random walk is a populationbased action, which is so called swarm intelligence. Furthermore, it is not rootless and is the imitation to nature phenomenon, thus these nature-inspired algorithm is also called heuristic algorithm. Lots of excellent and classical heuristic algorithm have been proposed since last century. The Genetic Algorithm (GA) inspired by the Darwin’s evolution theory was proposed in 1992 [2]. Artificial bee colony (ABC) algorithm was proposed from the inspiration about the behavior of bees’ foraging [3]. The Imperialist Competitive Algorithm (ICA) was inspired c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 553–565, 2020. https://doi.org/10.1007/978-981-32-9682-4_58

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by the behavior of empire and colony [4]. And the whale optimization algorithm (WOA), whose inspiration comes from the foraging behavior of humpbacks, was introduced in [5], and so on. Inspired by the bird flocking and fish schooling, Kennedy and Erberhart proposed the particle swarm optimization (PSO) in 1995 [6]. The PSO algorithm was widely acknowledged by its easiness to implement and high converge rate. It has been applied to various of areas like scheduling [7], feature selection [8]. However, the premature convergence on complex multimodal problems is the main problem of PSO. The poor performance comes from the unbalance between global search and local search. When the algorithm scans the parameter space, numbers of local optima trap the PSO too quickly, and poor global search make PSO unable to escape from them. To improve its performance on multimodal problems, researchers have done lots of works. To balance the global search and local search through parameters, Shi and Eberhart put forward the inertial weight for the original PSO in [9]. They also discussed the parameter selection in [10]. Some researchers improve algorithms via combining two or more similar features from PSO algorithm and another algorithm. In [11], a hybrid algorithm was introduced based on the GA and PSO, which was used to train recurrent neural networks. As the PSO algorithm is poor at global search, whereas the ABC suffers from converging slowly due to the poor local search, the ABC with PSO are combined in [12,13], which had achieved excellent result. Another method is topology structure, which is used to control the balance between global search and local search. Kennedy found that PSO with a small neighborhood performed better on complex problems, whereas PSO with a large neighborhood performed better on simple problems [14,15]. Cheng and Jin proposed a social learning PSO (SLPSO) in [16]. The particles could learn not only from their own best history position, but also from any particle with better fitness. This strategy broadened particles’ diversity and made particles have more chance to find the global optimal. It is necessary to point out that the main common aim of learning strategies mentioned above is to make PSO algorithm improve the accuracy of results and converge speed while reaching a good balance on global search and local search. As the ‘no free lunch’ theory [17,18] goes, any optimization algorithm can’t outperform on all of problems. For example, unimodal problems call for small population diversity, while multimodal problems require larger population diversity to help particles jump out of local optimal. Thus, two or more learning strategies are necessary to improve algorithms’ performance. This paper adopted hierarchy learning structure and levy flight operator to enhance algorithm’s searching ability. For the hierarchy learning structure, it divides swarm into several parts, which is helpful to reach a balance between global search and local search, thus higher accuracy results will be found. As for the levy flight operator, it makes particles adopt more appropriate step, which is of great benefit to global search. The rest of the paper is organized as follow: The classical PSO algorithm is introduced in Sect. 2, and the CLFPSO is depicted in detail in Sect. 3. In Sect. 4,

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the superiority of CLFPSO will be evaluated through comparing with various outstanding PSO algorithms on classical unimodal.

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Particle Swarm Optimization Algorithm

The PSO algorithm searches for the optimum in objective space through multiple random walks. As a population-based algorithm, every particle is a potential solution for the objective problem [6]. The merit of solution depends on the best fitness ever found by PSO. At the beginning of the PSO, individuals in population are random initialized as particles with position Xi = (x1 , x2 , x3 , .., xD ) and Vi = (v1 , v2 , v3 , .., vD ), where D is the dimension of searching space. In PSO, the best solution of swarm and individual are recognized as been remembered by all the group, namely the global best (Gbest) and the personal best (pBest). Both the step length and direction are guided by personal best and global best, which makes the particles move toward to global optimum. The equation representing the element of velocity is given as t+1 t t t = ωVi,j + c1 r1 (pBestti,j − Xi,j ) + c2 r2 (gBesttj − Xi,j ), Vi,j

(1)

where ω denotes the inertial weight and it linearly deceases from 0.9 to 0.4 with t. t denotes the iteration number. c1 and c2 are two constants for social learning and personal learning called acceleration coefficients. Both r1 and r2 are random chosen numbers within [0, 1]. The position of particle i is expressed as t+1 t+1 t = Xi,j + Vi,j . Xi,j

3

(2)

Levy Flight

Levy flight was a type of random walk first proposed by Paul Levy in 1937, which obey the rule of Levy distribution in general. The Levy distribution is a −1−β (0 < β < 2) that extremely power-low distribution with formula L(s) ∼ |s| long jumps occurs in Levy flight. Mathematically speaking, a simple version of Levy distribution can be defined as:   γ γ 1 exp[− ] , 0 < μ < s < ∞ 3/2 2π 2(s−μ) (s−μ) , L(s, γ, μ) = 0, otherwise where μ > 0 is the location and a minimum step of distribution and γ is a scale parameter. In addition, Levy distribution should be defined in terms of Fourier transform F (k) = exp[−α|k|β ], 0 < β ≤ 2, The inverse integral is generally given as  1 ∞ cos(ks) exp[−α|k|β ]dk. L(s) = π 0

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10-4 α=1 α = 1.5 α = 1.8 α=2

10-6

10-8

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10-12 -10

-8

-6

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0

2

4

6

8

10

Fig. 1. The Levy distribution with α = 1, 1.5, 1.8, 2

It can be easily seen from the formula mentioned above that the Levy flight is symmetric with respect to α = 0. Besides this, when the β is 1, the integral corresponding Cauchy distribution and when the β is 2, the integral corresponds to Gaussian distribution. The PDF of Levy flight with different α is given as below (Fig. 1). The variance of levy flight is σ 2 (t) ∼ t3−β (1 ≤ β ≤ 2) [1]. It will increasing very fast as the t goes, It also a reason for the more excellent performance in exploring undiscovered and large-scale search space than Brownian random walk where the variance of Brown motion is σ 2 (t) ∼ t. Thus, the levy flight is more likely to take a larger step, which is beneficial to global search. When it comes to the step length of levy random walk in simulation, it is carried out in Mantegna’s algorithm as u s = 1/β |v| where u ∼ N (0, δu2 ), v ∼ N (0, δv2 ), where  δu =

Γ (1 + β) sin(πβ/2) Γ [(1 + β)/2]β2(β−1)/2

1/β , δv = 1.

Another character is that the value of β decides the shape of distribution. It makes longer jump steps for smaller β values and shorter jump steps for bigger β values.

4

Competitive Particle Swarm Optimization Algorithm Based on Levy Flight (CLFPSO)

Compared with the original PSO, CLFPSO adopts hierarchy structure and levy flights in order to reach a good balance for global search and local search. For the hierarchy structure, it is an effective topology structure from ICA. In ICA, all individuals are divided into empires and colonies according to their fitness after initialization. Every empire randomly dominates some colonies based on

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the fitness. As the competition among empires starts, colonies move toward their empire and exchange their role if the fitness of colony exceeds that of the empire. This hierarchy structure diverse the swarm to a certain degree. The levy flight is another mechanism for stronger global search ability. Random walks with longer steps occurs sometimes make particles have more chance to get rid of local optima, which improves the issue of premature. As the initialization of N individuals, m particles with larger fitness value are chosen as leaders. Automatically, the rest (N − m) particles become followers. Then, the number of followers randomly allocated to leader I(I = 1, 2, .., m) is given as (3) round(pI × (N − m)), where the power proportion pI is denoted by     C   I pI =  m ,  j=1 Cj 

(4)

and the normalized cost CI of leader I is given as CI = cI − max {cj } 1≤j≤m

(5)

and cI is the fitness value of leader I. Finally, the swarm is divided into m groups. In the swarm, followers and leaders remember their personal best, which is denoted as F ollowerPi (i = 1, 2, .., round(pI × (N − m))) and LeaderPI (I = 1, 2, .., m) respectively. In addition, the global best is commonly shared in the whole swarm. For the leaders, they are the best in their group, so they update their velocity and position based on the personal best (LeaderPI (I = 1, 2, .., m)) of their own and the global best (gBest). Specifically, the jth element of velocity is updated according to t+1 t t t t Vi,j = ωVi,j + c1 r1 (LeaderPi,j − Xi,j ) + c2 r2 (gBesttj − Xi,j )

(6)

For the followers, they are the main force of global search. On one hand, they learn from the corresponding leader’s personal experience in their group. On the other hand, they take a random walk in the searching space, which is effective in exploring undiscovered space. Thus the jth element of velocity’s update equation is given as t+1 t t t = ωVi,j + ac1 levy(t) + c2 r3 (LeaderPi,j − Xi,j ), Vi,j

(7)

where a is a scale parameter for levy random walk. c2 is the same as that in original PSO. The levy(t) is a linearly decreasing levy flight generator, which direct particles move on randomly. The β in levy flight generator is increasing from 0.6 to 2. Thus the step length would decrease as iteration numbers increasing, which conforms with that stronger global search is need as first, whereas local search in the end. The hierarchy structure is depicted in Fig. 2.

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Fig. 2. The hierarchy structure for social learning in swarms

The initial weight is updating as ω = ωmax − (

ωmax − ωmin ) × t. iteration

(8)

In our study, the ωmax is set as 0.9, where ωmin is set as 0.2. And the position update equation is as same as in the original PSO. The global best and personal best are updated as particles positions and velocity iterate in (6) and (7). If some particles fly beyond the boundary, they are reinitialized as a new particle still blong to quondam leaders. The pseudo code of CLFPSO is given as follows.

5

Experiments and Results

To test the performance of CLFPSO, six unimodal test functions and ten multimodal functions are chosen on 30 and 50 dimensions. Different from unimodal test functions, there lies numbers of local optimal in multimodal functions. It is difficult for algorithms to get rid of local optimal, whose number will rapidly increase with the dimension. These benchmarks are listed in Table 1. Specifically, dynamic parameter tuning PSO (DPTPSO) [19], adaptive PSO (APSOVI) [20], fully informed PSO (FIPSO) [21], social learning PSO (SLPSO) [16] with original PSO [6] algorithm are compared with the proposed CLPSO. The parameters for each algorithm are the same as in the paper, which are listed in Table 2. In CLFPSO, acceleration coefficients c1 = c2 = 1.49, the maximum of inertial weight ωmax = 0.9 and the minimum of inertial weight ωmin = 0.2. The scale parameter a = 0.01. The initial population size is set as 40 and the number of Leader is set as 4. The number of iterations is 2000 and each algorithm runs 30 times. All the algorithms are implemented in Matlab R2016a using computer with Core i5-4200H CPU, 2.8 GHz, 8 GB RAM. The operating system of the computer is Window 10.

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Algorithm 1. Framework of CLFPSO 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29:

Initialize the swarm with N particles. Position Xi (i = 1, 2, ..., N ) and velocity Vi (i = 1, 2, ..., N ) are generated randomly. Let ωmin = 0.2, ωmax = 0.9, maxiter = 2000, c1 = c2 = 1.49, t = 1. Evaluate the fitness of swarm f it and get the gBest and pBesti (i = 1, 2, ..., N ). Classify the swarm by leaders and followers according to (3) and (4). while t < maxiter do: Calculate ω according to (8) Randomly generate r1 , r2 , r3 . for leaders do: Update velocity and position according to (6) and (2). end for if the leader’s position is beyond the boundary then: Reinitialize the position of the leader. end if Evaluate leaders’s fitness and update LeaderPI (I = 1, 2, ..., m) and gBest. for followers do: Update velocity and position according to (7) and (2). end for if the follower’s position is beyond the boundary then: Reinitialize the position of the follower. end if Evaluate the followers’ fitness. if f it(f olloweri ) < f it(LeaderI ) then: //f olloweri ∈ leaderI Exchange the role of the f olloweri and leaderI . end if Update the F ollowerPi (i = 1, 2, ..., round(pI × (N − m))) and the gBest. t = t + 1. end while Output the best solution.

The best results of experiments are listed in Table 3, from which we could ensure that CLFPSO is capable of searching for the global optimal in most test functions. Further more, CLFPSO is dominant compared with other algorithm. Only in f9 and f14 , it performs worse than FIPSO and SLPSO respectively. In addition, other five algorithms could not reach the similar results on the same benchmarks when it comes to 50 dimensions. PSO, DPTPSO only get 0.024 and 4.71E−05 on f1 on 50-D, far bigger than they get on 30-D. SLPSO could get 3.20 on f2 when it come to 50-D, which is much worse compared with the results on 30-D. However, CLFPSO still keeps its outstanding performance except f14 . The results shows that hierarchy structure and levy flight could help algorithms escape from the trap of local optimal.

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1010

105

100

100

Best score obtained so far

Best score obtained so far

As we can see from Table 4, the mean and std values of experiments of CLFPSO still take the leading position. These two values reflect the average behavior of algorithm on the same test functions. SLPSO could reach the same results as CLFPSO on f16 in 30-D. However, it does not perform well on 50-D. The excellent AVE results show that CLFPSO not only has better performance, but also is robust. The mean converge curves of f5 , f8 , f12 , f15 are depicted in Fig. 3. The quickly converge curves imply that the CLFPSO could reach more accurate results with fewer iterations. The search mechanism of CLFPSO combines the hierarchy structure and levy flight together, which make particles have a clear vision of work. For the leader, they aim at local search. For the Follower, they are the main force of global search. Their role will exchange during the iteration, which make algorithm reach an good balance between global search and local search.

10-10

10-20

10-30

DPTPSO CLFPSO FIPSO SLPSO

0

200

400

600

800

10-15

1000 1200 1400 1600 1800 2000

DPTPSO CLFPSO FIPSO SLPSO

0

200

400

600

800

1000 1200 1400 1600 1800 2000

Iteration

(a) f5

(b) f8 1010

100

100

10-2

DPTPSO CLFPSO FIPSO SLPSO

10-4

0

200

400

600

800

1000 1200 1400 1600 1800 2000

Iteration

(c) f12

Best score obtained so far

Best score obtained so far

10-10

Iteration

102

10-6

10-5

10-10

10-20

DPTPSO CLFPSO FIPSO SLPSO

10-30

10-40

0

200

400

600

800

1000 1200 1400 1600 1800 2000

Iteration

(d) f15

Fig. 3. The mean converge curves of 50-D

Table 1. Benchmarks used in the paper

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Parameters PSO

DPTPSO APSOVI FIPSO

SLPSO

c

2.0, 2.0 1.3, 1.3

1.49, 1.49 2.05, 2.05

ω:

0.9–0.4 0.9–0.4

0.9–0.3

  √ 2   2−(c1 +c2 )− (c1 +c2 )2 −4(c1 +c2 )

1

Δω

–

0.05

–

–

0.1

dim/100 ∗ 0.01

Table 3. The best result of each algorithm

f1 f2

Dimensions CLFPSO

PSO

APSOVI FIPSO

SLPSO

30-D

0

8.00E−13 3.89E−15

DPTPSO

0.20

0.0065

8.44E−94

50-D

0

0.013

4.71E−05

2.51

1.43E−04 3.42E−64

30-D

0

967.98

1.23

60.89

2.49E−04 7.79E−08

50-D

0

4.21E+04 448.16

30-D

9.13E−215 5.87

50-D

1.33E+03 0.046

3.20

0.0072

2.23

0.0024

5.26E−26

1.57E−218 58.11

5.48

7.52

4.51E−04 7.45E−12

30-D

1.07E−216 10.00

30.09

0.54

6.17E−04 6.82E−49

50-D

9.36E−220 40.00

40.67

1.52

1.18E−05 1.85E−33

30-D

0

5.25

3.71

59.20

7.30E−08 18.11

50-D

0

92.50

97.09

159.92

2.86E−04 32.28

30-D

1.89E−05

0.022

0.081

0.0080

0.025

0.0089

50-D

1.38E−05

2.83

3.34

0.03

0.011

0.027

30-D

8.88E−16

1.63E−06 7.32E−05

0.66

4.87E−04 4.44E−15

50-D

8.88E−16

0.020

2.04

0.79

1.75E−04 7.99E−15

30-D

0

73.07

48.71

64.98

0.063

0

50-D

0

81.18

64.95

28.32

0.0019

7.83E−124

30-D

1.00

2.72

1.75

1.13

0.90

1.55

50-D

0.90

4.17

3.54

1.89

0.90

0.90

f10 30-D

0

9.08E−14 2.23E−11

0.76

0.17

0

50-D

0

0.0034

1.54E−06

1.05

2.93E−05 0

f11 30-D

0

34.82

97.51

24.06

1.75E−08 7.96

50-D

0

231.40

236.53

85.36

4.54E−11 18.90

f12 30-D

3.19E−11

4.50

5.22

4.23

2.96E−07 6.68

50-D

6.77E−10

10.11

8.42

10.31

2.49E−08 13.35

f13 30-D

0

0.29

0.39

0.79

0.0012

50-D

0.69

1.09

8.00E−04 0.599

f3 f4 f5 f6 f7 f8 f9

0.29

0

1.60

f14 30-D

1.87E−43

2.21E+03 234.66

2.96E−09 1.77E−07 8.96E−120 2.23

50-D

3.41E−14

8.15E+08 6.92E+12

f15 30-D

1.35E−32

8.79E−10 5.38E−14E 2.24E−04 2.16E−05 1.35E−32 0.024

2.92E−07 2.51E−86

50-D

1.35E−32

2.02

f16 30-D

1.57E−32

1.58E−11 1.16E−14

2.17e−04 5.98e−10 1.57E−32

50-D

9.42E−33

0.86

2.85E−04 1.31E−06 9.42E−33

5.92E−04

7,01E−06 1.30E−05 1.35E−32

Functions Dimensions Criterias 30-D f1 50-D 30-D f2 50-D 30-D f3 50-D 30-D f4 50-D 30-D f5 50-D 30-D f6 50-D 30-D f7 50-D 30-D f8 50-D 30-D f9 50-D 30-D f10 50-D 30-D f11 50-D 30-D f12 50-D 30-D f13 50-D 30-D f14 50-D 30-D f15 50-D 30-D f16 50-D

PSO CAPSO AVE STD AVE STD 0 0 333.33 1.83E+03 1.29E–308 0 1.77E+04 1.07E+04 0 0 2.82E+04 1.15E+04 2.68E–319 0 8.21E+04 2.04E+04 5.87E–203 0 12.89 4.91 1.48E–202 0 71.35 8.04 5.46e–188 0 53.66 18.47 1.22E–200 0 15.67 32.66 0 0 3.06E+04 4.27E+04 0 0 2.69E+06 1.46E+07 2.67E–04 1.93E–04 3.79 4.65 3.06E–04 2.62E–04 32.02 20.32 8.88E–16 0 10.61 9.58 8.88E–16 0 18.31 4.03 0.032 0.17 62.78 16.51 2.22E–15 1.03E–15 118.53 25.73 1.0 5.32E–06 2.86 0.062 0.99 0.018 4.26 0.061 0 0 12.05 31.24 0 0 129.48 99.57 0 0 135.97 44.87 0 0 354.79 51.31 1.56R–06 3.33E–06 6.94 0.99 2.88E–06 3.55E–06 12.38 1.37 0 0 1.13 2.46 0 0 11.28 4.86 2.01E–12 9.59E–12 2.42E+08 5.95E+08 3.41E–14 1.81E–13 5.26E+19 1.99E+20 0 0 2.37E–07 8.42E–07 9.42E–33 2.78E–48 4.22 2.01 1.57E–32 5.56E–48 0.11 0.21 1.35E–32 5.56E–48 2.73E+07 1.04E+08

DPTPSO AVE STD 3.98E–06 4.83 0.030 0.073 1.17E+03 2.84E+03 8.32E+03 8.28E+03 0.31 0.16 10.74 3.34 54.17 21.20 135.98 40.16 2.16E+04 3.84E+04 5.34E+06 2.03E+07 3.52 4.13 41.24 22.07 1.68 2.58 8.98 5.72 55.47 20.12 109.60 22.93 2.73 0.21 4.13 0.14 0.019 0.026 0.011 0.018 165.31 34.79 370.49 54.52 6.93 1.07 12.52 1.48 0.48 0.075 1.34 1.68 8.53E+06 2.65E+03 2.14E+18 5.45E+18 2.00E+03 4.07E+03 0.90 0.77 0.017 0.039 1.77 2.88

APSOVI AVE STD 4.91 4.83 10.07 5.025 1.97E+03 4.01E+03 9.39E+03 9.24E+03 3.44 0.59 13.68 4.08 7.25 8.38 19.95 17.77 3.63E+08 1.63E+04 455.30 308.22 0.11 0.18 0.98 3.09 1.89 0.57 2.43 0.63 41.22 19.01 86.85 26.46 2.03 0.44 3.09 00.67 1.04 0.062 1.14 0.058 73.73 29.58 162.59 38.51 6.79 1.56 13.08 1.82 1.02 0.14 1.83 1.62 328.55 1.01E+03 5.16E+10 2.76E+11 5.17 3.96 0.51 0.72 0.062 0.13 0.12 0.39

FIPSO AVE STD 3.06 2.42 4.93 4.46 225.64 355.38 2.45E+03 7.32E+03 1.25 2.54 2.03 7.03 0.013 0.021 0.013 0.014 35.33 102.00 29.25 104.76 0.18 0.11 0.16 0.10 0.34 1.18 0.47 1.43 1.02 2.13 1.58 3.41 1.63 1.03 1.75 1.76 15.31 42.27 9.51 50.73 7.64E–06 1.17E–05 1.41E–05 1.85E–05 2.61 3.26 6.87 7.01 0.53 0.80 0.73 1.39 8.46E–06 1.55E–05 1.24E–05 1.35E–05 3.40 2.59 0.25 0.48 0.34 0.54 1.20 4.51

Table 4. The ave and std values in all functions on 30-D and 50-D SLPSO AVE STD 3.48E–92 5.94E–92 7.27E–63 7.37E–63 1.57E–06 2.35E–06 168.73 910.27 3.90E–25 3.04E–25 7.01E–04 0.0038 6.0E–48 4.75E–48 5.94E–33 3.65E–33 29.24 21.42 48.55 28.72 0.015 0.0027 0.041 0.0068 7.40E–15 1.34E–15 8.47E–15 1.80E–15 0.27 1.48 0.27 1.48 2.15 1.04 4.26 3.03 7.39E–04 0.0028 5.15E–04 0.0022 14.06 5.46 29.58 5.91 10.24 1.23 19.49 2.09 1.02 0.025 0.67 0.048 2.10E–112 1.06E–111 8.01E–77 4.15E–76 4.17E–92 6.63E–91 0.0083 0.045 1.57E–32 5.56E–48 0.0015 0.0038

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Conclusions

This paper proposed a novel PSO algorithm combined with hierarchy structure and more effective random walks obeying the levy distribution. The hierarchy structure diverses PSO algorithm and make particles have a clear vision of searching work. The levy flight random walk make algorithm have more chance to take a large step, which is of great benefit to escape from the local optimal. This two operators combined together will make algorithm reach a good balance between global search and local search. Six unimodal and ten multimodal benchmarks have been chosen to test the CLFPSO about the searching ability and optimal converge behavior on 30 and 50 dimensions, five outstanding PSO have been adopted to compare with the proposed CLFPSO. The results verify that the CLFPSO outperforms on most of test functions and could reach more accurate results with fewer iterations. Acknowledgments. This work is supported by National Natural Science Foundation of China under Grant No. 61573200, 61573199.

References 1. Yang XS (2010) Nature-Inspired Metaheuristic Algorithm. Luniver Press 2. Goldberg DE, Holland JH (1988) Genetic algrithms and machine learning. Mach Learn 3:95–99. https://doi.org/10.1145/168304.168305 3. Devis K, Basturk B (2005) An idea based on honey bees warm for numerical optimization. Technical report-TR06, Erciyes University, Engineering Faculty, Computation Engineer Department, Kayseri, Turkey 4. Esmaeil AG, Caro L (2007) Imperialist competitive algorithm: an algorithm for optimization inspired by imperialistic competition. In: IEEE congress on evolutionary computation. IEEE Press, Singapore, pp 4661–4667. https://doi.org/10. 1109/CEC.2007.4425083 5. Mirjalili S, Lewis A (2016) The whale optimization algorithm. Adv Eng Softw 95:51–67. https://doi.org/10.1016/j.advengsoft.2016.01.008 6. Kennedy J, Eberhart R (1995) Particle swarm optimization. In: IEEE international conference on neural network. IEEE Press, Perth, pp 1942–1948. https://doi.org/ 10.1109/ICNN.1995.488968 7. Kim HH, Kim DG, Choi JY et al (2017) Tire mixing process scheduling using particle swarm optimization. Comput Ind Eng 110:333–343. https://doi.org/10. 1016/j.cie.2017.06.012 8. Yang HC, Zhang SB, Deng KZ et al (2007) Research into a feature selection method for hyperspectral imagery using PSO and SVM. J China Univ Min Technol 17:473– 478. https://doi.org/10.1016/s1006-1266(07)60128-x 9. Shi YH, Eberhart RC (1998) A modified particle swarm optimizer. In: IEEE world congress on computational intelligence. IEEE Press, Anchorage, pp 69–73. https:// doi.org/10.1109/ICEC.1998.699146 10. Shi YH, Eberhart RC (1998) Parameter selection in particle swarm optimization. In: International conference on evolutionary programming, vol 1447. Springer, Heidelberg, pp 591–600. https://doi.org/10.1007/BFb0040810

Novel Competitive Particle Swarm Optimization

565

11. Juang CF (2004) A hybrid of genetic algorithm and particle swarm optimization for recurrent network design. IEEE Trans Syst Man Cybern B Cybern 34(2):997–1006. https://doi.org/10.1109/TSMCB.2003.818557 12. Shi XH, Li YW, Li HJ et al (2010) An integrated algorithm based on artificial bee colony and particle swarm optimization. In: 2010 sixth international conference on natural computation. IEEE Press, Yantai. https://doi.org/10.1109/ICNC.2010. 5583169 13. Li ZY, Wang WY, Yan YY, Li Z (2015) PS-ABC: a hybrid algorithm based on particle swarm and artificial bee colony for high-dimensional optimization problems. Expert Syst Appl 42(22):8881–8895. https://doi.org/10.1016/j.eswa.2015.07.043 14. Kennedy J (1999) Small worlds and mega-minds: effects of neighborhood topology on particle swarm performance. In: 1999 congress on evolutionary computation, vol 99. IEEE Press, Wasinton, pp 1391–1938. https://doi.org/10.1109/CEC.1999. 785509 15. Kennedy J, Mendes R (2002) Population structure and particle swarm performance. In: IEEE congress on evolutionary computation. IEEE Press, Honolulu, pp 1671–1676. https://doi.org/10.1109/CEC.2002.1004493 16. Cheng R, Jin YC (2015) A social learning particle swarm optimization algorithm for scalable optimization. Inf Sci 291:43–60. https://doi.org/10.1016/j.ins.2014.08. 039 17. Wolpert D, Macready W (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1:67–82. https://doi.org/10.1109/4235.585893 18. Wolpert D (2001) The supervised learning no-free-lunch theorems. In: Soft computing in industrial applications, pp 25–42. https://doi.org/10.1007/978-1-44710123-9 3 19. Iwasaki N, Yasuda K, Ueno G (2006) Dynamic parameter tuning of particle swarm optimizatin. IEEE Trans Electr Electron Eng 1:353–363. https://doi.org/10.1002/ tee.20078 20. Xu G, Qu JP, Yang ZT (2008) An improved adaptive particle swarm optimization algorithm. J South China Univ Technol 36:6–10. https://doi.org/10.1007/978-14471-2386-6 43 21. Kennedy J, Mendes R, Neves J (2004) The fully informed particle swarm: simpler, maybe better. IEEE Trans Evol Comput 8:204–210. https://doi.org/10.1109/ TEVC.2004.826074

Research on Establishment Method of Steel Composition Model Based on High Dimension and Small Sample Data Xin Wei1 , Dongmei Fu1(B) , Mindong Chen2 , and Qiong Yao3 1

2

School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China [email protected] SINOPEC Research Institute of Safety Engineering, Qingdao 266100, China 3 Key Laboratory of Space Launching Site Reliability Technology, Xichang Satellite Launch Center, Haikou 571126, China

Abstract. This paper hopes to predict the corrosion potential of steel by establishing a model of the relationship between the corrosion potential of steel in the marine environment and its component content. However, the number of experimental samples in this paper is very small, only 14, while the experimental sample has a high feature dimension, 22 dimensions. Firstly, by increasing the factor level of seawater environment, the experimental samples are expanded by orthogonal experiment method, and then the dimensionality of the samples is reduced by using locality preserving projections, which improves the problem of highdimensional small samples of data. Finally, the extreme learning machine algorithm and the particle swarm-extreme learning machine optimization algorithm are used to build the model respectively. The results show that the prediction accuracy of the PSO-ELM model is better than that of the ELM model. Keywords: Small sample · Dimensionality reduction · Extreme learning machine · Particle swarm optimization

1 Introduction This paper hopes to predict the corrosion potential of steel by establishing a model of the relationship between the corrosion potential of steel in the marine environment and its component content. However, the number of samples in this paper is very small, only 14 steel chemical composition and corrosive potential data, it is difficult to establish a good prediction accuracy model [1, 2]. After research, it was found that the corrosion potential of steel is the median value of the potential interval after forming a stable oxide film in seawater environment, which is converted from several potential monitoring values. The monitoring value of corrosion potential of steel will change with the change of seawater environment from its exposure test in seawater environment [3]. Orthogonal test is a classic statistical test design and optimization method. For multi-factor systems, multiple factors can be used to optimize multiple factors at the same time, and the influence of different factors on the response value can be compared, which can save time c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 566–573, 2020. https://doi.org/10.1007/978-981-32-9682-4_59

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and labor. Reduce the cost of the test [4]. In this paper, the orthogonal experimental design is used to design the sample data of the relationship between the steel composition, the environment and its corresponding corrosion potential, and the expansion of the experimental sample is realized he characteristic dimension of the experimental sample is high, 22 dimensions, and there are many singular values where the feature value is 0. Locally preserved projection [5] (LPP) is a manifold learning dimension reduction method, which can maintain the local geometry of the original sample data in high-dimensional space well into low-dimensional space by linear projection. Therefore, LPP is used to reduce dimensionality. The relationship between corrosion potential of carbon steel and low alloy steel and its composition and seawater environment is highly non-linear. It is difficult to establish a complete and accurate mathematical model for carbon steel and low alloy steel by traditional statistical regression method. Neural network model is especially suitable for the fitting of non-linear model. It simulates the brain nerve to learn the external environment, and has self-learning, self-organizing and self-adaptive ability and strong non-linear function approximation ability [6, 7]. Artificial neural network (ANN) is one of the typical neural network models. Some scholars [8, 9] used ANN to establish the model of the relationship between corrosion rate and composition and environment, and the fitting result are good. But the artificial neural network needs to set a lot of parameters, such as the design of network topology, the adjustment of weights and thresholds. At the same time, the learning time of the artificial neural network is too long to reach the goal of learning [10, 11]. The main feature of the Extreme Learning Machine (ELM) [12, 13] neural network is that the parameters of the hidden layer nodes can be given randomly or artificially and need not be adjusted. The learning process only needs to calculate the output weight. ELM has the advantages of high learning efficiency and strong generalization ability. Particle swarm optimization (PSO) is a good global search optimization algorithm [14, 15], which can optimize the initial input weights and thresholds of ELM. In this paper, ELM model and PSO-ELM model are established respectively, and their prediction results are compared.

2 Materials and Method 2.1 Experimental Data There are only 14 samples in this paper. Each sample includes chemical composition and corrosion potential of steel. The experimental data are shown in Table 1. The number of experimental data samples in this paper is very small, and the dimensions are relatively high. If the sample size is too small, it will be easy to over-fit and the model expression ability is insufficient. Because of the high dimension of sample features, it is easy to cause dimension disaster and destroy the original data relationship between experimental samples. In this paper, data preprocessing is needed for high-dimensional small sample data.

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X. Wei et al. Table 1. Composition content of steel (%)

2.2

Steels

C

Q235

0.1554 0.0959 0.3193 0.0241 0.0086 0.0145 0.0415 0

Si

Mn

0.0496 ...

−684.5

Q345B

0.17

0.22

0

−684.5

D36

0.072

0.1388 1.2186 0.0124 0.0034 0

0

0

0

...

−688

Q345DZ35

0.1

0.28

1.42

0.01

0.002

0

0

0

0

...

−695.5

921

0.12

0.33

0.37

0.08

0.04

2.72

1.05

0.24

0

...

−653

Q450NQR1

0.0697 0.3257 1.0426 0.0167 0.0079 0.1299 0.6239 0

X70

0.0672 0.181

1.5407 0.0131 0.0027 0

0.2075 0.0575 0

...

−687

X80

0.04

0.3

1.79

0.013

0.001

0

0.025

0

0

...

−665

E690

0.11

0.29

1.12

0.013

0.003

0.41

0.46

0.41

0.27

...

−673

E460

0.06

0.17

1.5

0.014

0.002

0.4

0.25

0.2

0.26

...

−670

Pure Q235

0.042

0.18

0.35

0.008

0.003

0

0

0

0

...

−693.5

grain steel1

0.097

0.26

1.64

0.010

0.006

0

0

0

0.20

...

−678.5

grain steel2

0.091

0.21

0.40

0.013

0.016

0

0

0

0.040

...

−695

Micro-alloy steel 0.064

0.22

1.18

0.008

0.005

0

0

0

0.32

...

−680

0.88

P 0.018

S 0.005

Ni 0

Cr 0

Mo 0

Cu

Others Potential ...

0.2636 ...

−687.5

Small Sample Expansion

The experimental environment of this experiment is Sanya sea water. It was found that the corrosion potential of steel is the median value of the potential interval after forming a stable oxide film in seawater environment, which is converted from several potential monitoring values. The monitoring value of corrosion potential of steel will change with the change of seawater environment from its exposure test in seawater environment. In this paper, the experimental steel was put into Sanya seawater environment for 140 days. The corrosion potential monitoring value of steel and the environmental monitoring value of Sanya seawater were obtained from Sanya corrosion site. The monitoring period was 10 days. The experimental data include 14 levels of component factors and 14 levels of environmental factors. In this paper, the orthogonal table L196 (141 , 141 ) of mixing levels is selected to design the experimental samples of the relationship between the component content of steel, seawater environment and corrosion potential of steel. As a result, the experimental sample size was designed to be 196. The sample characteristics are soaking time, water temperature, dissolved oxygen, conductivity, salinity, pH, seawater velocity, C, P, S, Si, Mn, Ni, Cr, Mo, Cu, V, Nb, Ti, Al, B, N, a total of 22 dimensions, the sample label is the corrosion potential Monitoring value. 2.3

Local Preserved Projection

High-dimensional sparse data may contain many irrelevant and redundant features, which not only increase the computational difficulty, but also reduce the accuracy of modeling. Feature transformation is an effective way to solve this challenge. Local Preserved Projection (LPP) is a manifold learning dimensionality reduction method. It can project the data relations of original samples in high-dimensional space well into lowdimensional space through linear projection.

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2.4 Extreme Learning Machine The algorithm randomly initializes the input weights and biases, and then obtains the output weights by minimizing the least square loss function, and then obtains the optimal solution. The network only needs one parameter: the number of neurons in the hidden layer. In the process of building the network, the parameters need not be updated iteratively, and the calculation cost is very low. The neural network structure of ELM is shown in Fig. 1.

Fig. 1. Structural diagram of extreme learning machine

The ELM neural network model is represented as follows: L

∑ βi g(Wi X j + bi ) = o j , j = 1, . . . , N,

(1)

i=1

Where Wi is the input weight, bi is the bias of hidden layer units, βi is the output weight and g(x) is the activation function L is the number of hidden layer neurons, Wi X j is the inner product of sum. The objective of learning a single hidden layer neural network is to minimize the output error, which can be expressed as N

∑ ||o j − t j || = 0,

(2)

j=1

That is to say, exist βi ,Wi and bi make: L

∑ βi g(Wi X j + bi ) = t j , j = 1, . . . , N,

(3)

i=1

In order to train the single hidden layer neural network, we hope to get βˆi , Wˆ i and bˆi make: L

L

|| ∑ βˆi g(Wˆ i X j + bˆi ) − T || = min || ∑ βi g(Wi X j + bi ) − T ||, i = 1, . . . , L, i=1

β ,W,b

i=1

(4)

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This is equivalent to minimizing the loss function: E=

N

L

∑ ( ∑ βi g(Wi X j + bi ) − t j )2 ,

(5)

j=1 i=1

Traditional gradient descent algorithm can solve this problem, but the basic gradient-based learning algorithm needs to adjust all parameters in the iteration process. In ELM algorithm, once the input weight Wi and the bias of hidden layer bi are randomly determined, the output weight matrix of hidden layer is uniquely determined. The training of single hidden layer neural network can be transformed into solving a linear system, and then the output weight can be obtained. 2.5

Particle Swarm Optimization-Extreme Learning Machine

Particle swarm optimization (PSO) is a good global search algorithm. The input weights and thresholds of the extreme learning machine are taken as the particles of the particle swarm optimization algorithm. For each individual in the population, i.e. the input weights matrix and the hidden layer deviation, the output weights matrix is calculated by the extreme learning machine algorithm. The root-mean-square error of particles is calculated from the training samples of the extreme learning machine. And it will be taken as the fitness of the particle swarm optimization algorithm. In each iteration, the particle updates its speed and position through individual extremum and global extremum to reach the condition to be satisfied and ends the iteration. The formula is updated as follows: vid t+1 = α vid t +U(0, β )(pid − vid t ),

(6)

xid t+1 = xid t + vid t+1 ,

(7)

where i is the target particle’s index, d is the dimension, xi is the particle’s position, vi is the velocity, pi is the best position found so far by i, α and β are constants, and U(0, β ) is a uniform random number generator. 2.6

Evaluation Index

In this paper, the prediction accuracy of corrosion potential monitoring value of steel is evaluated as follows: 1 n (8) MAE = ∑ |yi − yi |, n i=1  1 n RMSE = (9) ∑ (yi − yi )2 , n i=1 MAPE =

1 n |yi − yi | ∑ yi . n i=1

(10)

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3 Result 3.1 Model Prediction Results Based on ELM The training sample set of this experiment is the first to 182 groups of the experimental sample set. The test sample set is from 183 to 196 of the experimental sample set, and the corresponding test steel is D36. The input sequence of training samples is random, and based on the training sample set, the number of optimal hidden layer neurons in ELM is determined by cross validation method, which is 50. When predicting the test steel D36, it is necessary to predict the time series of corrosion potential monitoring values of D36 since when it is put into seawater environment. The prediction order of test samples is according to the soaking time, i.e. the order from 183 groups to 196 groups of real samples. ELM parameters are set to 9-50-1. Based on the ELM model, the time series prediction results of corrosion potential monitoring values of carbon steel D36 in seawater environment are shown in Fig. 2 and the prediction accuracy is shown in Table 2. -620

corrosion potential/mV

-640

-660

-680

-700

-720 actual value predictive value

-740

0

20

40

60

80

100

120

soaking time/days

Fig. 2. Prediction of corrosion potential of D36

Table 2. Prediction accuracy of carbon steel D36 Evaluation index MAE Result

RMSE MAPE

0.1554 0.0959 0.3193

140

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Model Prediction Results Based on PSO-ELM

In this paper, particle swarm optimization (PSO) is used to optimize the initial input weights and thresholds of ELM, and then ELM uses the optimal initial input weights and thresholds to predict the corrosion potential monitoring values of test samples. The parameters of PSO are set to 50 iterations and the population size is 500. The experimental methods and parameters of PSO-ELM model are consistent with those of ELM model. Based on PSO-ELM model, the time series prediction results of corrosion potential monitoring value of carbon steel D36 in seawater environment are shown in Fig. 3 and the prediction accuracy is shown in Table 3. The PSO-ELM model is superior to the ELM model in predicting the accuracy of carbon steel D36. The PSO-ELM algorithm has good model prediction accuracy. -620

corrosion potential/mV

-640

-660

-680

-700

-720 actual value predictive value

-740

0

20

40

60

80

100

120

140

soaking time/days

Fig. 3. Prediction of corrosion potential of D36

Table 3. Prediction accuracy of carbon steel D36 Evaluation index MAE RMSE MAPE Result

3.00

3.92

0.45%

4 Conclusions In this paper, orthogonal experiment and local preservation projection are used to solve the problem of high dimension and small sample Data of carbon steel and low alloy steel. Then the extreme learning machine algorithm and particle swarm optimization algorithm are used to model the model respectively. The results show that the prediction accuracy of PSO-ELM model is better than that of ELM model.

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Acknowledgments. This work was supported by National Key Research and Development Program of China (Grant No. 2017YFB0702104) and Other Projects of the Ministry of Science and Technology of China (Grant No. 2012FY113000).

References 1. Meignen S, Meignen H (2006) On the modeling of small sample distributions with generalized Gaussian density in a maximum likelihood framework. IEEE Trans Image Process 15:1647–1652. https://doi.org/10.1109/tip.2006.873455 2. Seghouane AK, Bekara M (2004) A small sample model selection criterion based on Kullback’s symmetric divergence. IEEE Trans Sig Process 52:3314–3323. https://doi.org/10. 1109/TSP.2004.837416 3. Luo H, Dong CF, Li XG (2012) The electrochemical behaviour of 2205 duplex stainless steel in alkaline solutions with different pH in the presence of chloride. Electrochim Acta 64:211–220. https://doi.org/10.1016/j.electacta.2012.01.025 4. Xuan W, Leung DYC (2011) Optimization of biodiesel production from camelina oil using orthogonal experiment. Appl Energy 88:3615–3624. https://doi.org/10.1016/j.apenergy. 2011.04.041 5. Li JB, Pan JS, Chu SC (2008) Kernel class-wise locality preserving projection. Inf Sci 178:1825–1835. https://doi.org/10.1016/j.ins.2007.12.001 6. Fukushima K (1980) Neocognitron: a self-organizing neural network model for a mechanism of pattern recognition unaffected by shift in position. Biol Cybern 36:193–202. https://doi. org/10.1007/bf00344251 7. Neves G, Cooke SF, Bliss TVP (2008) Synaptic plasticity, memory and the hippocampus: a neural network approach to causality. Nat Rev Neurosci 9:65–75. https://doi.org/10.1038/ nrn2303 8. Cui F, Zhang H, Ghosn M (2018) Seismic fragility analysis of deteriorating RC bridge substructures subject to marine chloride-induced corrosion. Eng Struct 155:61–72. https://doi. org/10.1016/j.engstruct.2017.10.067 9. Huttunen-Saarivirta E, Rajala P, Marja-Aho M (2018) Ennoblement, corrosion, and biofouling in brackish seawater: comparison between six stainless steel grades. Bioelectrochemistry 120:27–42. https://doi.org/10.1016/j.bioelechem.2017.11.002 10. Li W, Wang D, Chai T (2013) Burning state recognition of rotary kiln using ELMs with heterogeneous features. Neurocomputing 102:144–153. https://doi.org/10.1016/j.neucom.2011. 12.047 11. Shrestha S, Bochenek Z, Smith C (2014) Extreme Learning Machine for classification of high resolution remote sensing images and its comparison with traditional Artificial Neural Networks (ANN). EARSeL eProceedings 13:49 12. Ye Y, Ren J, Wu X (2017) Data-driven soft-sensor modelling for air cooler system pH values based on a fast search pruned-extreme learning machine. Asia-Pac J Chem Eng 12:186–195. https://doi.org/10.1002/apj.2064 13. Arellano-Prez JH, Negrn OJR, Escobar-Jimnez RF (2018) Development of a portable device for measuring the corrosion rates of metals based on electrochemical noise signals. Measurement 122:73–81. https://doi.org/10.1016/j.measurement.2018.03.008 14. Kennedy J, Eberhart R (2011) Particle swarm optimization. In: Proceedings of 1995 IEEE international conference neural networks, Perth, Australia, 27 November–4 December 1942– 1948, vol 4. https://doi.org/10.1109/ICNN.1995.488968 15. Da Y, Xiurun G (2005) An improved PSO-based ANN with simulated annealing technique. Neurocomputing 63:527–533. https://doi.org/10.1016/j.neucom.2004.07.002

A Note on Actuator Fault Detection for One-Sided Lipschitz Systems Ming Yang, Jun Huang(B) , and Fei Sun School of Mechanical and Electrical Engineering, Soochow University, Suzhou 215131, China [email protected]

Abstract. This paper investigates the actuator fault detection for one-sided Lipschitz systems. The feature of the nonlinear function lies that the Lipschitz constant is unknown and one-sided Lipschitz constant is known. The aim is to design an adaptive observer for this system. Then, based on the observer, an algorithm to detect the fault occurrence in the actuator is proposed. An example is simulated to verify the feasibility of the approach. Keywords: One-sided Lipschitz · Adaptive observer · Fault detection

1 Introduction Recently, the investigation of one-sided Lipschitz systems (OLSs) has drawn considerable attention in the control community. [1] firstly proposed the definition of onesided Lipschitz in the nonlinear observer design problem. [2] presented the definition of quadratical inner-boundedness (QIB), and the existence conditions of observers were formulated by the form of linear matrix inequalities (LMIs). In order to obtain less conservative conditions, [3] improved the results of [2] by S-procedure method. After that, many works were dedicated to the observer design for OLSs, such as [4–6]. In [4], a stochastic observer design method for OLSs is proposed. [5] considered the unknown input observers for the continuous and discrete systems, respectively. [6] extended the observer design method to singular OLSs. Generally speaking, fault detection (FD) and reconstruction has always been a hot topic in the study of safety for practical systems. If the fault appears in the actuator, the harm caused by is enormous. Thus, it is essential to detect the actuator fault online. To name a few, there exist many methods to deal with the FD, among which the method based on observer is very popular. Indeed, FD based on observer of Lipschitz systems have been studied extensively in the past decades [7–10]. When turns to OLSs, few works have been done except for [11]. In [11], Li et al. considered the nonlinear function which conforms to be both one-sided Lipschitz and QIB. As pointed out in [12], the plants considered in the mentioned references [2–6, 11] are in fact a subclass of OLSs. Motivated by the above analysis, we further consider the FD problem for OLSs, where the nonlinear function only satisfies one-sided Lipschitz constraint but not QIB. The remainder of this paper is organized as follows: Sect. 2 presents necessary basics, main results including the adaptive observer (AO) and FD algorithm are given in Sect. 3. Section 4 simulates one example to show the validation of the proposed approach. c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 574–581, 2020. https://doi.org/10.1007/978-981-32-9682-4_60

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1.1 Problem Formulation and Preliminaries Consider the system with actuator fault:  x˙ = Ax + B f (Hx, u) + Dg(x, u) + Tu f , y = Cx,

(1)

where x ∈ Rn , u ∈ Rm , and y ∈ Rq are the state, the control input, and the output respectively. f (·, ·) ∈ Rr and g(·, ·) ∈ R p are both nonlinear matrix functions. The signal u f ∈ Rs stands for the unknown actuator fault vector. A ∈ Rn×n , B ∈ Rn×r , C ∈ Rq×n , D ∈ Rn×p , H ∈ Rr×n , and T ∈ Rn×s are determined matrices. Assumption 1. f (·, ·) is one-sided Lipschitz with respect to w, i.e., < f (w, u) − f (w, ˆ u), w − w> ˆ ≤ α ||w − w|| ˆ 2,

(2)

where α is the one-sided Lipschitz constant. Assumption 2. g(·, ·) is Lipschitz with respect to w, i.e., ||g(w, u) − g(w, ˆ u)|| ≤ σ ||w − w||, ˆ

(3)

where σ is the Lipschitz constant but unknown. Assumption 3. There exist positive definite matrice P ∈ Rn×n , matrices L ∈ Rn×q , F ∈ Rr×q , W ∈ R p×q and constant ε > 0 such that (A − LC)T P + P(A − LC) + 2α (H − FC)T (H − FC) + 2ε I ≤ 0,

(4)

PB = (H − FC) ,

(5)

D P = WC.

(6)

T

T

Remark 1. Generally, f (Hx, u) is only one-sided Lipschitz in this paper. For the purpose of design, checking the condition of QIB is not so easy. Thus, if we remove this constraint, it will obviously extend the type of the system.

2 Main Result For the system (1), an AO is constructed as follows 1 x˙ˆ = Axˆ + B f (H xˆ + F(y −Cx), ˆ u) + Dg(x, ˆ u) + L(y −Cx) ˆ + ξˆ DW (y −Cx), ˆ 2

(7)

with adaption law ˙ ξˆ = τ ||W (y −Cx)|| ˆ 2,

(8)

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where the constant τ is positive. From (7) and (1), the error system can be obtained 1 e˙ = (A − LC)e + BΔ f + DΔ g − ξˆ DWCe + Tu f , 2

(9)

ˆ u), Δ g = g(x, u) − g(x, ˆ u). where e = x − x, ˆ Δ f = f (Hx, u) − f (H xˆ + F(y −Cx), Theorem 1. Suppose that Assumptions 1–3 hold and u f = 0, the error system (9) is asymptotically stable with lim e(t) = 0, i.e., the system (7) with (8) is an AO for the t→∞ system (1). ˙ Proof. Let ξ˜ = ξ − ξˆ , and then ξ˙˜ = −ξˆ = −τ ||WCe||2 . Here we employ the Lyapunov function candidate as follows 1 V = eT Pe + τ −1 ξ˜ 2 , 2

(10)

Taking the derivative of (10) along (9), one can get V˙ = 2eT Pe˙ + τ −1 ξ˜ ξ˙˜ 1 = 2eT P[(A − LC)e + BΔ f + DΔ g − ξˆ DWCe] − ξ˜ ||WCe||2 2 = 2eT P(A − LC)e + 2eT PBΔ f + 2eT PDΔ g − ξˆ eT PDWCe − ξ˜ ||WCe||2 .

(11)

By Assumption 1, together with (4) and (5), we have 2eT [P(A − LC)e + PBΔ f ] = eT [(A − LC)T P + P(A − LC) + 2α (H − FC)T (H − FC)]e ≤ −2ε eT e.

(12)

It follows from Assumption 2 and (6) that 2eT PDΔ g = 2(DT Pe)T Δ g = 2(WCe)T Δ g ≤ 2||WCe|| ||Δ g|| ≤ 2σ ||WCe|| ||e||

σ2 ||WCe||2 + ε ||e||2 , ≤ 2||σ WCe|| ||e|| ≤ ε

(13)

and

ξˆ eT PDWCe = ξˆ eT (WC)T WCe = ξˆ ||WCe||2 .

(14)

Substituting (12)–(14) into (11), we have

σ V˙ ≤ −ε eT e + ( − ξˆ − ξ˜ )||WCe||2 . ε

(15)

V˙ ≤ −ε eT e.

(16)

2

Let ξ =

σ2 , then ε

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By integrating two sides of (16), we can get  t 0

V˙ (t)dt ≤ −

 t 0

ε eT (s)e(s)ds.

(17)

In view of the fact that V (t) > 0 and V (0) < ∞, it is deduced from (17) that  t 0

ε eT (s)e(s)ds ≤ V (0) < ∞.

(18)

By Barbalat Lemma [13], it is concluded that lim ε eT e(t) = 0, which implies that t→∞

lim e(t) = 0.

t→∞

Remark 2. From the above proof, it is not difficult to notice that the estimation of unknown parameter ξˆ or σˆ may not converge to its original value ξ or σ . To some extend, the error ξ˜ or σ˜ is only Lyapunov stable but not asymptotically stable. It is indeed that Assumption 3 provides the sufficient conditions, under which we can design the AO for the system. However, they are not the standard LMIs or linear matrix equalities (LMEs) when α = 0, and we need derive another forms. Theorem 2. If there exist positive definite matrix P ∈ Rn×n , matrices Y ∈ Rn×q , F ∈ Rr×q , W ∈ R p×q and constant ε > 0 such that  T  A P + PA −YC −CT Y T + 2ε I (H − FC)T ≤ 0, (19) (H − FC) − 2|Iα | PB = (H − FC)T ,

(20)

D P = WC,

(21)

T

then (4)–(6) hold. Proof. Let Y = PL and by using the Schur complement, one can prove that (19)–(21) are the sufficient conditions of (4)–(6). Remark 3. It is obvious that (19) is LMI while (20) and (21) are LMEs. We can use Scilab [14] to solve the feasible solutions of the constraints of (19)–(21). Remark 4. If the actuator has no fault, the designed AO (7) with (8) is valid. In other words, the state x of the above system (1) can be asymptotically recovered by x. ˆ When the fault u f appears, the AO (7) with (8) can be used to detect the fault. Theorem 3. Let ey = y − Cxˆ and μ > 0 be the alarm threshold. If the system (7) with (8) is an AO for the system (1), then the following algorithm  if ||ey || ≤ μ , the actuator has no fault, (22) if ||ey || > μ , the actuator has a fault, can be used to detect the fault u f . The proof is direct and omitted here due to the length of this paper. It is worth noting that the alarm threshold μ is very essential in the design of practical systems and affects the accuracy of the FD algorithm.

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3 Numeric Simulation Consider the system (1), where   0.90 0.21 A= , 0.35 −1.2



   0.10 , C= 10 , 0.72     0.21 H = 1.20 0.81 , D = , 0.46 B=

1

f (Hx, u) = 0.5 sin(Hx) − (Hx) 3 ,

g(x, u) = 2sin(x),

Solving the conditions (19)–(21) yields     833.01 −3.54 23.41 P= , L= , −3.54 1.62 174.02

F = −79.61,

α = 0.5.

W = 173.12.

By Theorem 1, the AO has the following form: ⎧ x˙ˆ1 = 0.9xˆ1 + 0.21xˆ2 + 0.10(0.5sin(1.2xˆ1 + 0.81xˆ2 − 79.61(y − xˆ1 )) − (1.2xˆ1 + 0.81xˆ2 ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎨ −79.61(y − xˆ1 )) 3 ) + 0.42sin(xˆ1 + 23.41(y − xˆ1 ) + 18.1776ξ (y − xˆ1 ) x˙ˆ2 = 0.35xˆ1 − 1.2xˆ2 + 0.72(0.5sin(1.2xˆ1 + 0.81xˆ2 − 79.61(y − xˆ1 )) − (1.2xˆ1 + 0.81xˆ2 ⎪ 1 ⎪ ⎪ −79.61(y − xˆ1 )) 3 ) + 0.92sin(xˆ1 ) + 174.02(y − xˆ1 ) + 39.8176ξ (y − xˆ1 ) ⎪ ⎪ ⎩˙ ξ = 173.12(y − xˆ1 ) Firstly, we show the simulation result of the AO. The condition x(0) and x(0) ˆ are chosen T T ˆ as [0.1 2] and [0.11 0.2] , and the initial value of ξ is 0, the recovered state xˆ of the AO is depicted in Figs. 1 and 2. The estimation of the unknown parameter ξ is presented in Fig. 3. It can be seen that the designed AO is effective. 25 x1 estimation of x 1

20

State

15

10

5

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time(sec)

Fig. 1. The state x1 and the estimation of x1

1.8

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When u f is defined by: ⎧ ⎨ 50, 0.4 < t ≤ 0.6, u f = 80, 1 < t ≤ 1.2, ⎩ 0, else. The alarm threshold μ is chosen as 0.05. Figure 4 shows that the fault occurs during the time intervals [0.4, 0.6] and [1, 1.2], which is consistent to the form of function u f . Thus, the proposed FD algorithm is valid. 14 x2

12

estimation of x 2

State

10

8

6

4

2

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time(sec)

Fig. 2. The state x2 and the estimation of x2

18 16

the estimation of

14 12 10 8 6 4 2 0 0

1

2

3

4

5

6

7

8

9

Time(sec)

Fig. 3. The estimation of the unknown parameter ξ

10

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M. Yang et al. 0.25

||ey||

0.2

line for alarm

0.15

0.1

0.05

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time(sec)

Fig. 4. The time response of ||ey || and the alarm line

4 Conclusion This paper has studied the FD problem of OLSs. First, we design an AO for the OLS where the nonlinear function is only one-sided Lipschitz. Then, we present FD algorithm based on the designed AO. Finally, an example is given to show that the FD algorithm is effective. Acknowledgement. This work is supported by National Natural Science Foundation of China (61403267), China Postdoctoral Science Foundation (2017M611903), and Undergraduate Training Program for Innovation and Entrepreneurship, Soochow University.

References 1. Hu G (2006) Observers for one-sided Lipschitz non-linear systems. IMA J Math Control Inf 23(4):395–401 2. Abbaszadeh M, Marquez H (2010) Nonlinear observer design for one-sided Lipschitz systems. In: Proceedings of the 2010 American control conference 3. Zhang W, Su H, Liang Y, Han Z (2012) Observers for one-sided Lipschitz non-linear systems. IET Control Theory Appl 6(9):1297–1303 4. Barbata A, Zasadzinski M, Ali H, Messaoud H (2015) Exponential observer for a class of one-sided Lipschitz stochastic nonlinear systems. IEEE Trans Autom Control 60(1):259–264 5. Zhang W, Su H, Zhu F, Azar G (2015) Unknown input observer design for one-sided Lipschitz nonlinear systems. Nonlinear Dynam 79(21):1469–1479 6. Zulfiqar A, Rehan M, Abid M (2016) Observer design for one-sided Lipschitz descriptor systems. Appl Math Model 40(3):2301–2311 7. Pertew A-M, Marquez H-J, Zhao Q (2007) LMI-based sensor fault diagnosis for nonlinear Lipschitz systems. Automatica 43(8):1464–1469 8. Zhang X, Polycarpou M-M, Parisini T (2010) Fault-diagnosis of a class of nonlinear uncertain systems with Lipschitz nonlinearities using adaptive estimation. Automatica 46(2):290– 299 9. Chen W, Saif M (2007) A sliding mode observer-based strategy for fault detection, isolation, and estimation in a class of Lipschitz nonlinear systems. Int J Syst Sci 38(12):943–955

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10. Zuo Z, Zhang J, Wang Y (2015) Adaptive fault-tolerant tracking control for linear and Lipschitz nonlinear multi-agent systems. IEEE Trans Ind Electron 62(6):3923–3931 11. Li L, Yang Y, Zhang Y, Ding S (2014) Fault estimation of one-sided Lipschitz and quasione-sided Lipschitz systems. In: Proceedings of the 33rd Chinese control conference 12. Zhang W, Su H, Zhu F, Bhattacharyya SP (2016) Improved exponential observer design for one-sided Lipschitz nonlinear systems. Int J Robust Nonlinear Control 26(18):3958–3973 13. Kristic M, Modestino J, Deng H (1998) Stabilization of nonlinear uncertain systems. Springer, Heidelberg 14. Ghaoui L-E, Nikoukhah R, Delebecque F (1995) LMITOOL: a package for LMI optimization. In: Proceedings of 1995 34th IEEE conference on decision and control

Aperiodic Control Strategy for Multi-agent Systems with Time-Varying Delays Hongguang Zhang(B) Chifeng University, Chifeng 024000, China [email protected] http://www.cfxy.cn

Abstract. This note discusses event-triggered strategies for the multiagent systems(MAS) with time-varying delays. Each agent’s state information will be broadcasted to its neighbors when the error exceeds the threshold which is called “event-triggered”. And we develop two kinds of the event trigger functions, one uses its neighbors’ and its own information, and the other one only need its own information. The linear matrix inequality (LMI) is used to ensure the consistency of the MAS. An example is given to demonstrate the effectiveness of our event-triggered strategies.

Keywords: Event-triggered control Time-varying delays

1

· Multi-agent system · LMI ·

Introduction

Coordination and cooperative control of groups of autonomous has attracted several researchers in recent years. The great developments technology of communication, wireless, embedded devices, enable the development of autonomous air, ground, or underwater vehicles. Some new distributed control mechanisms are proposed for such MASs, one of them is “event-triggered” control [1,2]. Event-based control strategies seem to be appropriate for cooperative control tasks of MASs, since it can be expected decrease the number of control actions, just like control updates or measurement broadcasts, significantly. For this reason, an event-based implementation of the consensus protocol is developed in [4–6]. In [4], the overall system reaches average consensus asymptotically, while Zeno-behavior is excluded [12]. However it is necessary that each agent need to know all its neighbors information at once. This strong requirement is relaxed in [5]. In [6] the assumption that each agent updates its control-law not only at its own event times, but also whenever one of its neighbors triggers an event is not necessary. But then the overall system does not necessarily converge to the average value of all initial states. Recently, [7,10,11] extends the novel event-based c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 582–590, 2020. https://doi.org/10.1007/978-981-32-9682-4_61

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strategies to a large class of multi-agent systems with constant time delay and switching-topology, in which the event only need its own information. The starting point of the present work are the latest results on event-based cooperative control discussed above [4,7]. The goal of this paper is to propose event-based strategies for the MASs with time-varying delays. The event trigger function uses its neighbors and its own information (“state-based”).

2 2.1

Preliminaries Algebraic Graph Theory

Learn from [9]. For an undirected graph G with N vertices, the adjacency matrix H = H(G) is the N × N matrix given by hij = 1, if (i, j) ∈ E, where E is the set of edge, and hij = 0, otherwise. If hij between vertices i and j is non-zero, we called i and j are neighbors. Ni denotes the neighbor vertices set of vertex i, and |Ni | denotes the number of the neighbor vertices. The graph G is called connected when there is a path between any two vertices of the graph G. Let Θ be the n × n diagonal matrix of |Ni | (i = 1, 2 . . . n). Then Θ is called the degree matrix of G. The Laplacian of G is the symmetric positive semidefinite matrix L = Θ − H. For a connected graph, the Laplacian has a single zero eigenvalue and the corresponding eigenvector is the vector of ones, 1, and 0 = λ1 (G) ≤ λ2 (G) ≤ ... ≤ λn (G). If G is connected, then λ2 (G) > 0. 2.2

System Model

We consider N agents, with xi ∈ R denoting the state of agent i. The agent obeys a single integrator model: x˙ i (t) = ui (t), i ∈ N = {1, ..., N }.

(1)

where ui (t) denotes the control input at time t. The state of agent i passes through a communication channel eij with time delay d(t) before getting to agent j when the agent can get its own state without time delay. The time delay d(t) is a time-varying continuous function that satisfies 0 ≤ d(t) ≤ τ,

(2)

˙ ≤ μ < 1, d(t)

(3)

and where τ and μ are constants. In the continuous linear system, we take the control law:  ui (t) = [(xj (t − d(t)) − xi (t))]. j∈Ni

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for all t > τ . The closed-loop equation of the nominal system can be rewritten as:  [(xj (t − d(t)) − xi (t))]. (4) x˙ i (t) = j∈Ni

So that x(t) ˙ = −Θx(t) + Hx(t − d(t)), where x(t) = [x1 (t), ..., xN (t)]T is the stack vector of agents’ states. And the proof of the stability of such MAS can be found in [3]. 2.3

Problem Statement

Consider a network of continuous-time integrator with communication delays. For an undirected graph, all agents’ states converge to an “agreement point” which is the average of the initial states [3]. We suppose that the multi-agent systems with continuous feedback is stable. Assume agent i can be triggered to broadcast by itself at time instants ti0 , ti1 , ..., tin . And agent i Monitoring its information without time-delay. The control law can be rewritten as   [(xj (tjkj (t) ) − xi (t)], t ≥ τ ui (t) = j∈Ni (5) 0, τ > t ≥ 0 where function kj (t) = arg min{t − d(t) − tjl | t − d(t) ≥ tjl }. l∈N

And the dynamic function can be rewritten as x(t) ˙ = −Θx(t) + H x ˆ(t − d(t)), where x ˆ(t − d(t)) denote discrete sample values. The distributed cooperative control problem can be described as follows: derive control laws of form (5), and event times ti0 , ti1 , ..., for each agent i ∈ N that drive system (1) to an agreement point.

3

The Stability of MAS with Time-Varying Delay: LMI Approach

Consider MASs with time-varying delay d(t). If the consistency of MASs can be satisfied, we have the agreement point: lim x(t) = 1

t→∞

N 1  xi (t0 ) = 1¯ x N i=1

We use the decomposition x(t) = 1¯ x + δ(t), where, δ is called the disagreement vector in [3] and 1 is the vector of ones. The dynamic function can be rewritten as ˙ = −Θδ(t) + Hδ(t − d(t)) + (−Θ + H)1¯ δ(t) x,

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notice that (−Θ + H)1 = −L1 = 0. L is Laplacian matrix of the undirected graph. We can get the review of the Laplacian matrix and its properties from [9]. The measurement ei (t) = δi (tik ) − δi (t), t ∈ [tik , tik+1 ), is taken to build the event function fi (t) = α(ei (t)) − β(δ(t − d(t)), where the function α: R → R, β: Rn → R. The agent i’s broadcasting will be triggered whenever fi (t) > 0. The main result of this paper summarized in the following. Theorem 1. For given τ > 0, μ < 1. System (1) with control law (5). Assume time-varying delays satisfying (2), (3) and the graph G is undirected and connected. The MAS (1) is asymptotically stable if there exists symmetry matrix A = AT > 0, B = B T ≥ 0, R = RT ≥ 0, Q = QT ≥ 0, Ki = KiT > 0(i = 1, 2), and diagonal matrix S > 0, T > 0 such that the following LMI holds: ⎡ ⎤ Υ11 Υ12 Υ13 Υ14 ⎢ ∗ Υ22 Υ23 Υ24 ⎥ ⎥ (6) Υ =⎢ ⎣ ∗ ∗ Υ33 0 ⎦ < 0 ∗ ∗ ∗ Υ44 where Υ11 = −AΘ − ΘT AT + τ ΘT (K1 + K2 )Θ 1 + B + R + T − (K1 + K2 ), τ 1 Υ12 = −τ ΘT (K1 + K2 )H + AH + K1 , τ 1 Υ13 = K2 , τ Υ14 = AH − τ ΘT (K1 + K2 )H, 2 Υ22 = −(1 − μ)B − K1 + τ H T (K1 + K2 )H + Q, τ 1 Υ23 = K1 , τ Υ24 = τ H T (K1 + K2 )H + Q, 1 Υ33 = −R − (K1 + K2 ), τ Υ44 = τ H T (K1 + K2 )H − (1 − μ)S + Q.

Proof. Consider a Lyapunov function V as follow:  t  V (δ) = δ T (t)Aδ(t) + δ T (t)Rδ(t)ds + t−τ



t

+ t−d(t)

t

δ T (t)Bδ(t)ds

t−d(t)



0

eT (s)Se(s)ds + −τ



t

t+θ

˙ δ˙ T (s)(K1 + K2 )δ(s)dsdθ

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where A = AT > 0, B = B T ≥ 0, R = RT ≥ 0, Ki = KiT > 0(i = 1, 2). And calculate the derivative: V˙ (δ) ˙ + δ T (t)(B + R)δ(t) − δ T (t − τ )Rδ(t − τ ) = 2δ T (t)Aδ(t)

−

t t−τ

T ˙ ˙ −(1 − d(t))δ (t − d(t))Bδ(t − d(t)) + τ δ˙ T (t)(K1 + K2 )δ(t) T ˙ ˙ + eT (t)Se(t) − (1 − d(t))e (t − d(t))Se(t − d(t)) δ˙ T (s)(K1 + K2 )δ(s)ds

˙ + δ T (t)(B + R)δ(t) − δ T (t − τ )Rδ(t − τ ) ≤ 2δ T (t)Aδ(t)

−

t

˙ −(1 − μ)δ T (t − d(t))Bδ(t − d(t)) + τ δ˙ T (t)(K1 + K2 )δ(t) ˙ δ˙ T (s)K1 δ(s)ds−

t−d(t) T

 t−d(t) t−τ

˙ δ˙ T (s)K1 δ(s)ds−

(1 − μ)e (t − d(t))Se(t − d(t)).

t

t−τ

T ˙ (t)Se(t) δ˙ T (s)K2 δ(s)ds+e

−

From the conclusions of [8] we try to exploit Jensen Inequality:  −

t

˙ ≤− δ˙ T (t)K2 δ(t)

t−τ

 t  t 1 T ˙ ˙ ( K2 ( δ(s)ds) δ(s)ds) τ − d(t) t−τ t−τ

(7)

And  −

t−d(t) t−τ

 t−d(t)  t−d(t) ˙δ T (t)K1 δ(t) ˙ ≤ −1( ˙δ(s)ds)T K2 ( ˙ δ(s)ds) τ t−τ t−τ

1 ≤ − [δ(t − d(t)) − δ(t − τ )]T K1 [δ(t − d(t)) − δ(t − τ )] τ t T ˙ ˙ − t−d(t) δ (t)K1 δ(t) t t 1 T ˙ ˙ ( t−d(t) δ(s)ds) K1 ( t−d(t) δ(s)ds) ≤ − d(t) 1 ≤ − [δ(t) − δ(t − d(t))]T K1 [δ(t) − δ(t − d(t))] τ

(8)

(9)

combine with Eqs. (7), (8) and (9), V˙ (t) < η T (t)Υ˜ η(t) + eT (t)Se(t),

(10)

where η(t) = [δ T (t), δ T (t − d(t)), δ T (t − τ ), eT (t − d(t))]T Υ˜ is the matrix ⎡

Υ˜11 ⎢ ∗ Υ˜ = ⎢ ⎣ ∗ ∗

Υ12 Υ˜22 ∗ ∗

Υ13 Υ23 Υ33 ∗

⎤ Υ14 Υ˜24 ⎥ ⎥ 0 ⎦ Υ˜44

(11)

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where Υ˜11 = −AΘ − ΘT AT + τ ΘT (K1 + K2 )Θ 1 + B + R − (K1 + K2 ), τ 1 Υ12 = −τ ΘT (K1 + K2 )H + AH + K1 , τ 1 Υ13 = K2 , τ Υ14 = AH − τ ΘT (K1 + K2 )H, 2 Υ˜22 = −(1 − μ)B − K1 + τ H T (K1 + K2 )H, τ 1 Υ23 = K1 , τ Υ˜24 = τ H T (K1 + K2 )H, 1 Υ33 = −R − (K1 + K2 ), τ Υ˜44 = τ H T (K1 + K2 )H − (1 − μ)S, With Eq. (12), the following inequality holds ˆ − d(t)) + δ T (t)T δ(t) = (δ(t − d(t)) + e(t − eT (t)Se(t) ≤ δˆT (t − d(t))Qδ(t T d(t))) Q(δ(t − d(t)) + e(t − d(t))) +δ T (t)T δ(t), where Q is a symmetric positive definite matrix and S and T is a positive definite diagonal matrix. There is ⎤ ⎡ Υ11 Υ12 Υ13 Υ14 ⎢ ∗ Υ22 Υ23 Υ24 ⎥ ⎥ V˙ (δ) ≤ Υ = ⎢ ⎣ ∗ ∗ Υ33 0 ⎦ ∗ ∗ ∗ Υ44 where Υ11 = Υ˜11 + T, Υ22 = Υ˜22 + Q, Υ24 = Υ˜24 + Q, Υ44 = Υ˜44 + Q. Therefore, if Υ < 0 then V˙ < 0. Theorem 2. As we see from fi (t) = eT (t)Se(t) − hi (t) that if hi (t) = 0 then e(t) ≤ 0. Zeno Sampling may be inevitable as the upper bound of ei (t) converges to be zero. But we can see from the following equation fi (t) = S(i, i)e2i (t) −

1 ˆT ˆ − d(t)) − T (i, i)δ 2 (t) δ (t − d(t))Qδ(t i N

(12)

ˆ − d(t)) and δ(t) is zero. Assume that hi (t) will never be zero unless both δ(t ∗ ˙ ∗ ) = e(t ˆ ˙ ∗ ) = 0 which δ(t − d(t)) and δ(t) is zero at time t . So that we have δ(t

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means that ei (t∗+ ) = ei (t∗ ). Note that fi (t∗ ) = 0, ei (t∗ ) = 0. There is fi (t∗+ ) = S(i, i)e2i (t∗+ ) + 0 = 0, the event will not be triggered again i.e., Zeno-behavior will not exhibit. Theorem 3. Consider the system (1) with control law (5) and event function (12). Assume time-varying delay delays satisfied (2), (3) and the graph G is undirected and connected. If LMI (6) holds that for each agent, Zeno-behavior will not exhibit. Proof. As the LMI (6) holds for all t ≥ 0. We will prove that the inter-event interval is bounded from below by a certain time tM in . Noting ˆ − d(t)) + T (i, i)δi (t)2 S(i, i)ei (t)2 ≤ δˆT (t − d(t))Qδ(t ˆ − d(t))2 + T (i, i)δ(t)2 , ≤ λmax (Q)δ(t So that ˆ − d(t)) + T (i, i)δ(t), S(i, i)|ei (t)| ≤ λmax (Q)δ(t And we have ˆ − d(t)) + k2 δ(t), |ei (t)| ≤ k1 δ(t where k1 =

λmax (Q) min(S(i,i)) ,

max(T (i,i))

k2 =

i∈N

i∈N

min(S(i,i))

.

i∈N

Assume that event of agent i is triggered at time t∗ (at that time, ei (t∗ ) = 0). In order to derive a positive lower bound on the inter-event times, an upper bound on the measurement error ei (t) for t > t∗ is computed  t |ui (s)|ds. |ei (t)| ≤ t∗

And the upper bound of the input u is: |ui (t)| ≤ u(t) ≤  − Θδ(t) − Hδ(t − d(t)) − He(t − d(t)) ˆ − d(t)) ≤ Θδ(t) + Hδ(t

(13)

This leads to the bound ˆ − d(t))). |ei (t)| ≤ (t − t∗ )(Θδ(t) + Hδ(t ˆ − d(t)) + Noting that the event will not be triggered before |ei (t)| = k1 δ(t k2 δ(t). With the conclusion of the remark 3.2, the tmin is equivalent to t − t∗ =

ˆ − d(t)) + k2 δ(t) k1 δ(t ˆ − d(t)) + Θδ(t) Hδ(t k2 k1 , ) H Θ > 0.

≥ min( = tmin

(14)

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Example

In this section, simulation is given to prove previous results. Now we consider the MAS with four agents whose Laplacian matrix is given by ⎡ ⎤ 1 −1 0 0 ⎢ −1 3 −1 −1 ⎥ ⎥ L=⎢ ⎣ 0 −1 2 −1 ⎦ 0 −1 −1 2 We consider event-triggered MASs with the same initial conditions and set the τ = 0.2, μ = 0.5. The next simulations depicts how the event-triggered strategies worked on agent 1. Figure 1 shows the simulation with state-based event function. The top plot of Fig. 1 shows the evolution of the state which is supposed to converges to be zero.

Fig. 1. State-based event-triggered

And we know from the figure that the MAS is stable under the event-triggered strategy. The approach (State-based) seems to have less updates and a faster convergence rate.

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Conclusions

We consider the impaction of time-varying delays on event-triggered MASs. And we proposed two kinds of event functions for such MASs. In state-independent case each agent required knowledge of the its own information and the other one(state-based) additional needs the neighbors’ information with time-varying delays. The performance using state-based approach seems to be more smooth and stable. Zeno-behavior will not exhibit when our event-triggered strategies execute. Results of this paper are supposed through simulated examples. Future work will focus on more complex cooperative tasks.

References 1. Astrom KJ, Bernhardsson BM (2002) Comparison of Riemann and Lebesgue sampling for first order stochastic systems. In: Proceedings of the 41st IEEE Conference on Decision and Control. Las Vegas, Nevada USA, pp 2011–2016 2. Lemmon M, Chantem T, Hu XS, Zyskowski M (2007) On self-triggered full information H-infinity controllers. In: Proceedings of the 10th International Conference on Hybrid Systems: Computation and Control, Pisa, Italy. Springer Berlin, pp 371–384 3. Olfati-Saber R, Murray RM (2004) Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Autom Control 49(9):1520–1533 4. Dimarogonas DV, Johansson KH (2009) Event-triggered cooperative control. In: European Control Conference, pp 3015–3020 5. Dimarogonas DV, Johansson KH (2009) Event-triggered control for multi-agent systems. In: 48th IEEE Conference Decision and Control and 28th Chinese Control Conference, Shanghan, P.R. China, pp. 16–18 6. Dimarogonas D, FrazKoli E (2009) Distributed event-triggered control strategies for multi-agent systems. In: 47th Annual Allerton Conference on Communications. Control and Computing, USA 7. Seyboth G (2010) Event-based Control for Multi-Agent Systems. Masters Degree Project Stockholm, Sweden 8. He Y, Wang QG, Xie L, Lin C, February 2007 Further improvement of freeweighting matrices technique for systems with time-varying delay. IEEE Trans Autom Control 52(2) 9. Godsil C, Royle G (2001) Algebraic Graph Theory. Springer Graduate Texts in Mathematics, vol 207 10. Chen X, Hao F, Ma B (2017) Periodic event-triggered cooperative control of multiple non-holonomic wheeled mobile robots. IET Contr Theory Appl 11(6):890–899 11. Zhang Z, Hao F, Zhang L, Wang L (2014) Consensus of linear multiagent systems via event-triggered control. Int J Control 87(6):1243–1251 12. Johansson KH, Egerstedt M, Lygeros J, Sastry SS (1999) On the regulariKation of Keno hybrid automata. Syst Control Lett 38:141–150

Robust Convergence of High-Order Adaptive Iterative Learning Control Against Iteration-Varying Uncertainties Zirong Guo1,2 , Deyuan Meng1,2(B) , and Jingyao Zhang1,2 1

School of Automation Science and Electrical Engineering, Beihang University (BUAA), Beijing 100191, China [email protected] 2 The Seventh Research Division, Beihang University (BUAA), Beijing 100191, China

Abstract. In this paper, adaptive iterative learning control (AILC) algorithms using a high-order law for linear time-varying (LTV) systems with iterationvarying uncertainties are proposed. Sufficient conditions are derived to guarantee the robust convergence of tracking error such that the bounded-input-boundedoutput stability of LTV systems can be achieved. Extensions of established AILC results to nonlinear systems are further developed. Numerical simulations are implemented to validate the effectiveness of the theoretical results. Keywords: Adaptive iterative learning control · Linear time-varying system · Iteration-varying uncertainties

1 Introduction Adaptive iterative learning control (AILC) [1] is an effective control technique that combines adaptive control and iterative learning control together. AILC is often applied on plants with repeatability, and whose system parameters are not totally explicit [2, 3]. With wide application prospects, AILC has been used in many industrial areas, such as welding robots, handling robots [4] and rapid heating process in wafer manufacturing [5]. AILC algorithms have been studied by scholars comprehensively [6, 7]. After the optimal ILC algorithm was initially explored for linear systems [8], many AILC algorithms based on optimization have been developed [9–12]. These algorithms use optimal methods to estimate unknown parameters adaptively. For nonlinear discrete-time systems satisfying the global Lipschitz condition, a model-free AILC algorithm can make the tracking error converge to zero asymptotically, which only uses the input and output data without involving any information of the systems [13]. The results are extended to a high-order AILC algorithm [14]. The algorithm applies to the high-order model, and thus the convergence speed of tracking error is proved to be faster than the first-order ones. Unfortunately, the above-mentioned results do not take time-varying systems and the iteration-varying uncertainties into account. With unknown time-varying parameters in system model, traditional optimal c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 591–598, 2020. https://doi.org/10.1007/978-981-32-9682-4_62

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AILC algorithms are invalid, which can not guarantee the stability of the system and convergence of the tracking error. Moreover, if the iteration-varying uncertainties are further considered for time-varying systems, the estimation error of unknown parameters may no longer converge. In this paper, for LTV systems with iteration-varying uncertainties, an AILC algorithm based on high-order linearization model is designed and analyzed. It is shown that the tracking error can be guaranteed within a bound if iteration-varying uncertainties are bounded. If iteration-varying uncertainties converge along iteration axis, the tracking error can converge to zero. Furthermore, the AILC algorithm is also extended to deal with nonlinear systems. Numerical simulation examples are provided to validate the robust convergence results of the proposed AILC approach. The rest of this paper is organized as follows. We propose the concerned problems in Sect. 2. The robust convergence results of LTV system and the extensions to nonlinear systems are presented in Sects. 3 and 4, respectively. Numerical examples are given in Sect. 5 and conclusion is shown in Sect. 6.

2 Problem Statement 2.1

System Description

Consider a LTV system with iteration-varying uncertainties:  x k (t + 1) = A (t)xxk (t) + B (t)uk (t) yk (t) = C (t)xxk (t) + wk (t)

(1)

where t ∈ {0, · · · , N} is the discrete-time index and k ∈ Z+ is the iteration index. xk (t) ∈ Rn , uk (t) ∈ R, yk (t) ∈ R, and wk (t) ∈ R denote the state, the input, the output, and the output disturbance, respectively. Let yd (t),t ∈ {0, · · · , N} be the desired output trajectory and ek (t) = yd (t) − yk (t) be the tracking error. The target is to make the track error robustly converge along iteration axis as k → ∞, i.e., lim supk→∞ |ek (t)| ≤ βe with a relatively small bound βe ≥ 0. 2.2

High-Order Linearization Model

Before proposing the model, two necessary assumptions are introduced as follows. Assumption 1: The initial value xk (0) at every iteration k is identical, namely xk (0) = x0 , ∀k ∈ Z+ . Assumption 2: The output disturbance wk (t) is bounded, i.e., supk∈Z+ ,0≤t≤N |wk (t)| ≤ βw for βw > 0. Then, a high-order linearization model is developed with the following lemma.

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Lemma 1. Consider the LTV system (1) satisfying Assumptions 1 and 2. A high-order linearization model is proposed as follows: Y k−1 (t + 1) = Θ (t)(uuk (t) − uk−1 (t)) + wk (t + 1) − wk−1 (t + 1) Y k (t + 1) −Y

(2)

where Y k (t + 1) = [yk (1), yk (2), · · · , yk (t + 1)]T , u k (t) = [uk (0), uk (1), · · · , uk (t)]T , w k (t + 1) = [wk (1), wk (2), · · · , wk (t + 1)]T , and Θ (t) is a coefficient matrix satisfying ⎡ ⎤ C(1)B(0) 0 ··· 0 ⎢ ⎥ C(2)A(1)B(0) C(2)B(1) ··· 0 ⎢ ⎥ Θ (t) = ⎢ ⎥. .. .. .. .. ⎣ ⎦ . . . . C(t + 1) ∏ti=1 A(i)B(0) C(t + 1) ∏ti=2 A(i)B(1) · · · C(t + 1)B(t) A simpler one-dimensional linearization model function is introduced for the convenience of algorithm derivation, which is also the last line of (2). yk (t + 1) = yk−1 (t + 1) + θ t (t)Δ uk (t) + wk (t + 1)

(3)

where we denote Δ u k (t) = u k (t) − u k−1 (t), wk (t + 1) = wk (t + 1) − wk−1 (t + 1) and

t

t



θ t (t) = C(t + 1) ∏ A(i)B(0),C(t + 1) ∏ A(i)B(1), · · · ,C(t + 1)B(t) i=1

i=2

=[θ (0), θ (1), . . . , θ (t)]. t

t

t

In addition, supk∈Z+ ,0≤t≤N |wk (t)| ≤ βw = 2βw and sup0≤t≤N θ t (t) ≤ Θu are satisfied. Remark 1. Assumption 1 and 2 are common hypotheses which confirm identical initial states and bounded disturbance respectively in ILC, and which are usually satisfied in real control systems. Lemma 1 describes the relationship between input and output, and it is written into matrix form for subsequent processing. 2.3 Algorithm Statement Parameter Estimation Algorithm: Because the parameter θ t (t) in (3) is unknown, a modified projection algorithm [15] is utilized for its iterative estimation:  ˆ t (t)Δ u k−1 (t) η Δ y (t + 1) − θ k−1 k−1 t t θˆ k (t) = θˆ k−1 (t) + Δ u Tk−1 (t) (4) μ + Δ u k−1 (t)2 t t t θˆ k (t) = θˆ 0 (t), if sgn(θˆkt (i)) = sgn(θˆ0t (i)), i = 0, . . . ,t or θˆ k (t) ≤ ε t θˆ k (t)

(5)

where is the estimation of θ t (t), μ > 0, η ∈ (0, 2), and ε denote a weighting factor, a step-size factor, and a small positive number, respectively. The initial value of t model parameter θˆ 0 (t) should be selected to satisfy that its all elements should have t same signs of θ t (t). The boundary of θˆ k (t) has been proved in the literature [15], so we have supk∈Z+ |θˆkt (i)| ≤ βθ , i = 0, 1, . . . ,t for a finite bound βθ ≥ 0.

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Control Algorithms: Consider an objective function with high-order factors:

2

M

∑ αm ek−m+1 (t + 1)

J(uk (t), α ) =

+ λ (uk (t) − uk−1 (t))2

(6)

m=1

where λ > 0 is a weighting factor. α = [α1 , . . . , αM ] are the high-order factors that M ¯ > 0. Rewrite (3) as satisfy ∑M m=1 αm = 1, 0 < αm ≤ 1, and α1 + α2 − ∑m=3 αm = α yk (t + 1) = yk−1 (t + 1) + [θ t (t − 1), θ t (t)][Δ u Tk (t − 1), Δ uk (t)]T + wk (t + 1)

(7)

where θ t (t − 1) = [θ t (0), θ t (1), . . . , θ t (t − 1)]. By substituting (7) into (6) and neglecting the disturbance wk (t + 1), we can obtain   t−1

J(uk (t), α ) = α1 ek−1 (t + 1) − ∑ θ t (i)Δ uk (i) − θ t (t)Δ uk (t) i=0

M

+

2

∑ αm (ek−m+1 (t + 1))

(8) + λ |uk (t) − uk−1 (t)|2

m=2

where Δ uk (i) = uk (i) − uk−1 (i), i = 0, 1, . . . ,t. By calculating ∂ u∂ J(t) = 0 and replacing k t the unknown θ t (t) with θˆ (t), we can propose the following learning control law: k

ˆt ρα12 θˆkt (t) ∑t−1 i=0 θk (i)Δ uk (i) λ + α12 |θˆkt (t)|2 ρα1 θˆkt (t)[α1 ek−1 (t + 1) + ∑M m=2 αm ek−m+1 (t + 1)] + t 2 ˆ λ + α |θ (t)|2

uk (t) = uk−1 (t) −

1

(9)

k

where ρ > 0 is a positive factor which makes the control law more generalized. Remark 2. The aforementioned algorithms only use the input and output information. Thus, they can be applied to linear time-varying process.

3 Main Results Theorem 1. Consider the LTV system (1) satisfying Assumptions 1 and 2 and let the proposed control algorithms (4), (5), and (9) be applied. If the parameters λ and ρ are selected to satisfy λ >

ρ 2Θu2 4 ,

then the following two results hold:

(1) the system is bounded-input-bounded-output stable; (2) the bounded tracking of ILC can be achieved, i.e., lim supk→∞ |ek (t)| ≤ βe is satisfied. If the system (1) satisfies Assumption 3 additionally, a further conclusion can be drawn.

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Assumption 3: The output disturbance wk (t) of the LTV system (1) converges, such that limk→∞ wk (t) = w(t). Theorem 2. Consider the LTV system (1) satisfying Assumptions 1, 2, and 3, and let the proposed control algorithms (4), (5), and (9) be applied. If the parameters λ and ρ ρ 2Θu2 4 ,

are selected to satisfy λ >

then the following two results hold:

(1) the system is bounded-input-bounded-output stable; (2) the perfect tracking of ILC can be achieved, i.e., limk→∞ ek (t) = 0 is satisfied. Proofs of Theorems 1 and 2. We can exploit the extended contraction mapping method [16] and make full use of the property of nonnegative matrix to accomplish the proofs of Theorems 1 and 2, which however are omitted here due to the page limitation.  Remark 3. Theorem 1 shows that the robust convergence of the tracking error can be ensured by the AILC algorithms (9) for the time-varying system in the presence of iteration-varying uncertainties. Further, Theorem 2 shows that the zero-error convergence can be achieved if the iteration-varying uncertainties converge along the iteration axis. These results extend the results of [14], which are established for time-invariant system without iteration-varying uncertainties.

4 Extensions to Nonlinear Systems Consider a time-varying nonlinear discrete system with external disturbances:  x k (t + 1) = f (xxk (t), uk (t),t) yk (t) = g(xxk (t),t) + wk (t)

(10)

where f (·) and g(·) are continuous differentiable nonlinear real functions. An additional basic assumption is made as follows: Assumption 4: For all x k (t) ∈ Rn , uk (t) ∈ R, and 0 ≤ t ≤ N, exist and are bounded, which are expressed by sup

x k (t)∈Rn ,uk (t)∈R,0≤t≤N

   ∂f     ∂ x k (t)  ≤ β fx ,

sup

x k (t)∈Rn ,uk (t)∈R,0≤t≤N

   ∂f     ∂ uk (t)  ≤ β fu ,

∂f ∂f ∂ x k (t) , ∂ uk (t)

sup

x k (t)∈Rn ,uk (t)∈R,0≤t≤N

∂g ∂ x k (t)

and

   ∂g     ∂ x k (t) 

≤ βgx

with some finite bounds β fx , β fu , and βgx . Thus the high-order dynamic model is developed with the following lemma. Lemma 2. Consider the nonlinear system (10) satisfying Assumptions 1, 2 and 4. An optimal gradient vector θ kt (t) exists such that

yk (t + 1) = yk−1 (t + 1) + θ kt (t)Δ u k (t) + wk (t + 1)







(11)



where θ kt (t) = [θkt (0), θkt (1), . . . , θkt (t)]. In addition, supk∈Z+ θ kt (t) ≤ Θu and supk∈Z+ |wk (t)| ≤ βw are satisfied.

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Based on Lemma 2, we can extend the result of Theorem 1 to the nonlinear system (10) in a similar way as the application of adaptive ILC to the LTV system (3). This motivates us to propose the following corollary. Corollary 1. Consider the nonlinear system (10) satisfying Assumptions 1, 2, and 4 and let the proposed control algorithms (4), (5), and (9) be applied. If the parameters

λ and ρ are selected to satisfy λ >

2

ρ 2 Θu 4

, then the following two results hold:

(1) the system is bounded-input-bounded-output stable. (2) the bounded tracking of ILC can be achieved, i.e., lim supk→∞ |ek (t)| ≤ βe is satisfied. If Assumption 3 is additionally satisfied, the perfect tracking performance can be achieved, i.e., limk→∞ ek (t) = 0. Proof. We can develop the proof of Corollary 1 by also exploiting the extended contraction mapping method [16] and making the use of the property of nonnegative matrix, which is omitted due to the page limitation. 

5 Numerical Examples Consider the LTV system with       0 0.5 0 A(t) = B C (t) = 0 1 . , (t) = ,C −3 −3 1 −0.5 − 10 t −0.5 − 10 t Desired trajectory is yd (t) = 10−6 (t −1)3 (4−0.03(t −1)),t ∈ {0, · · · , 200}. The parameter setting is given by θ0t (t) = 0.9,t ∈ {0, · · · , 200}, ρ = 1, η = 1, λ = 0.5, μ = 0, 1, and α [3] = [0.8, 0.14, 0.06]. wk (t),t ∈ {0, · · · , 200} varies arbitrarily within [−βw , βw ]. Two cases are considered for uncertain output disturbance as follows: (a) Boundedness: βw is set constantly to be 0.05; (b) Convergence: βw is set to be 0.05 × 0.5k , which means disturbance converge to zero along iteration axis; It is clear to see that the Assumption 2 holds in the case (a), and both Assumptions 2 and 3 hold in the case (b). Figures. 1, 2 and 3 plot the simulation test results. Figure 1 depicts the tracking error evolution evaluated by max1≤t≤200 |ek (t)| for the first 20 iterations. Specifically, Fig. 1(a) shows that tracking error decreases and varies along the iteration axis within a small bound in the case (a). On the contrary, Fig. 1(b) describes that tracking error converge to zero with increasing iterations in the case (b). Figure 2 presents the evolution of input evaluated by max0≤t≤200 |uk (t)| along the iteration axis, and input boundedness is obvious in both cases (a) and (b). Finally, tracking performance is shown in Fig. 3, where desired trajectory (red) and output (blue) at the 20 iteration are plotted. Clearly, the perfect tracking can be achieved except t = 0 in case (b). The simulations of Figs. 1, 2 and 3 are consistent with the theoretical results of Theorems 1 and 2.

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16

16

14 12

12

max1≤t≤200 |ek (t)|

max1≤t≤200 |ek (t)|

14

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10 8 6

10 8 6 4

4

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2 0

0 0

2

4

6

8

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12

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14

16

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20

0

2

4

6

8

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(a)

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Fig. 1. Tracking error evolution of ILC processes evaluated by max1≤t≤200 |ek (t)| for the first 20 iterations.

1

1.2

0.9 0.8 0.7

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Fig. 2. Control input evolution of ILC processes evaluated by max0≤t≤200 |uk (t)| for the first 20 iterations.

2

2 yd t (k) y20 (t)

yd t (k) y20 (t)

0

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Fig. 3. Tracking of output with desired reference trajectory after 20 iterations.

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6 Conclusion In this paper, a high-order adaptive iteration learning control law has been proposed to address time-varying parameters and unknown model uncertainties in LTV systems. The stability and robustness properties of this law have been studied. Sufficient conditions have been derived to guarantee the robust convergence of tracking error. Moreover, similar results are also given for nonlinear systems as extensions. The validity of the proposed results has been verified through simulation tests. Acknowledgments. This work was supported in part by the National Natural Science Foundation of China (No. 61873013, 61922007, 61520106010) and in part by the Fundamental Research Funds for the Central Universities.

References 1. Shen D, Xu J-X (2019) Adaptive learning control for nonlinear systems with randomly varying iteration lengths. IEEE Transact Neural Netw Learn Syst 30(4):1119–1132 2. Norrlof M (2002) An adaptive iterative learning control algorithm with experiments on an industrial robot. IEEE Transact Robot Autom 18(2):245–251 3. Tayebi A (2004) Adaptive iterative learning control for robot manipulators. Automatica 40(7):1195–1203 4. Arimoto S (1990) Robustness of learning control for robot manipulators. In: IEEE International Conference on Robotics and Automation Robotics and Automation, Cincinnati, OH, USA, 13–18 May 1990, pp 1528–1533 5. Xu J-X, Chen Y, Lee TH, Yamamoto S (1999) Terminal iterative learning control with an application to RTPCVD thickness control. Automatica 35(9):1535–1542 6. Chien C-J, Hsu C-T, Yao C-Y (2001) Fuzzy system-based adaptive iterative learning control for nonlinear plants with initial state errors. IEEE Transact Fuzzy Syst 12(5):724–732 7. Tayebi A, Chien C-J (2007) A unified adaptive iterative learning control framework for uncertain nonlinear systems. IEEE Transact Autom Control 52(10):1907–1913 8. Amann N, Owens DH, Rogers E (1996) Iterative learning control for discrete-time systems with exponential rate of convergence. IEE Proc Control Theory Appl 143(2):217–224 9. Owens DH, Chu B, Songjun M (2012) Parameter-optimal iterative learning control using polynomial representations of the inverse plant. Int J Control 85(5):533–544 10. Bristow DA (2008) Weighting matrix design for robust monotonic convergence in norm optimal iterative learning control. In: proceedings of IEEE American Control Conference. Seattle, WA, USA, 11–13 June 2008, pp 4554–4560 11. Owens DH, Freeman CT, Chu B (2014) An inverse-model approach to multivariable norm optimal iterative learning control with auxiliary optimisation. Int J Control 87(8):1646–1671 12. Volckaert M, Diehl M, Swevers J (2013) Generalization of norm optimal ILC for nonlinear systems with constraints. Mech Syst Sig Process 39:280–296 13. Hou Z, Jin S (2014) Model Free Adaptive Control. CRC Press, Boca Raton 14. Chi R, Hou Z, Huang B, Jin S (2018) Computationally efficient data-driven higher order optimal iterative learning control. IEEE Transact Neural Netw Learn Control 3(12):5971– 5980 15. Chi R, Hou Z, Huang B, Jin S (2015) A unified data-driven design framework of optimalitybased generalized iterative learning control. Comput Chem Eng 77(9):10–23 16. Meng D, Moore KL (2017) Robust iterative learning control for nonrepetitive uncertain systems. IEEE Transact Autom Control 62(2):907–913 17. Rugh WJ (1996) Linear System Theory. Prientice Hall, Upper Saddle River

Distributed Robust Control of Signed Networks Subject to External Disturbances Mingjun Du1 , Baoli Ma1 , Deyuan Meng1(B) , Hua Yang2 , and Hong Jiang2 1

2

The Seventh Research Division, Beihang University (BUAA), Beijing 100191, China [email protected] Beijing Institute of Control Engineering, Beijing 100190, China

Abstract. This paper aims to study the distributed robust control problems of signed networks subject to external disturbances. We perform a model transformation of signed networks, by which the consensus problems can be transformed into the stability problems. With robust H∞ control theory, we can develop the sufficient conditions for signed networks achieving bipartite consensus (respectively, state stability) with the desired resilient performance when its associated signed digraphs are structurally balanced (respectively, unbalanced). In addition, two simulation examples are proposed to demonstrate the correctness of the theoretical results. Keywords: Bipartite consensus · External disturbance · Robust H∞ control · Signed network · Structural balance

1 Introduction Networked system is composed of multiple agents, in which all agents can transmit information with each other. Recently, research in distributed control has been one of the most active areas of networked systems [1, 2]. The study of distributed control concentrates on how to make all agents accomplish a common objective. They also possess the potential value in many fields, such as biology, compute and robotic. In practical applications, almost all networked systems are subject to disturbances since there exist uncertainties in the external circumstance, such as modeling error, aging and strong wind. It is necessary to require the networked system to have good resilient performance when it is in the face of external disturbances. Toward this end, the references [3–7] have studied the resilient performance of networked systems by a robust H∞ analysis method. It is worthwhile pointing out that the references [1–7] mainly focus on traditional networked systems which only contain cooperative interactions among agents. Different from traditional networked systems, a new kind of networked systems has appeared, in which both cooperative interactions and antagonistic interactions among agents are considered. This kind of networked systems is called signed networked system since it can be described as a signed digraph whose nodes denote agents and edges with positive (or negative) weight represent cooperative (or antagonistic) interaction between nodes. c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 599–608, 2020. https://doi.org/10.1007/978-981-32-9682-4_63

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Distributed control of signed networked systems has attracted more and more attentions. Owing to the existence of antagonistic interactions, it may exhibit more plentiful collective behaviors, such as bipartite consensus [8], modulus consensus [9], interval bipartite consensus [10] and bipartite containment tracking [11]. To the best of our knowledge, until now, there have been no studies on robust control of signed networked systems in face of external disturbances. Motivated by the above discussions, this paper explores the robust control of signed networked system subject to external disturbances. We introduce the model transformation of signed networked system by addressing a dynamic output, in which the consensus problem can be equivalently transformed into the stability problem. Using the robust H∞ analysis method, the sufficient condition is proposed to guarantee that the signed networked system realizes consensus objective with desired resilient performance. In addition, simulation examples are provided to illustrate the effectiveness of the developed results. The rest of this paper is organized as follows. The problem statements are proposed in Sect. 2. The model transformation of signed networked systems is investigated in Sect. 3. In Sect. 4, the resilient performance is analysed for signed networked system in face of external disturbances. Simulation examples and conclusions are given in Sects. 5 and 6, respectively. Notations: In this paper, we denote Fn = {1, 2, · · · , n}, 1n = [1, 1, · · · , 1]T ∈ Rn , 0n = [0, 0, · · · , 0]T ∈ Rn and diag{d1 , d2 , · · · , dn } as a diagonal matrix whose diagonal elements are d1, d2 , · · · , dn and non-diagonal elements are zero. For a vector x(t) ∈ Rn , let ||x(t)||2 = [ 0∞ xT (t)x(t)dt]1/2 denote the energy of x(t). We say that x(t) ∈ L2 if and only if ||x(t)||2 < ∞. For a matrix A ∈ Rn×n , A > 0 (or A < 0) represents a positive (or negative) definite matrix. For a real number a ∈ R, let |a| and sgn(a) be its absolute value and sign function of a, respectively.

2 Problem Statements Let G = (V , E , A ) represent a signed digraph, in which V = {v1 , v2 , · · · , vn } is a node set, E ⊆ {(vi , v j ) : vi , v j ∈ V )} is an edge set and A = [ai j ] ∈ Rn×n is an adjacency matrix. The element ai j satisfies ai j = 0 ⇔ (v j , vi ) ∈ E and ai j = 0, otherwise. An edge (v j , vi ) implies that vi can receive information from v j and v j is called the neighbor of vi . All neighbors of vi can be denoted as N(i) = { j : (v j , vi ) ∈ E }. Let P = {(vm0 , vm1 ), (vm1 , vm2 ), · · · , (vmk−1 , vmk ) represent a directed path from vm0 to vmk , in which vm0 , vm1 , · · · , vmk are different nodes. The signed digraph G is said to be strongly connected if there exists a directed path from every node to every other node. Besides, the signed digraph G is structurally balanced if its node set V can be divided   into two subsets V1 and V2 which satisfy V1 V2 = V and V1 V2 = ∅ such that ai j ≥ 0 for vi , v j ∈ Vl , l ∈ {1, 2} and ai j ≤ 0 for vi ∈ V p , v j ∈ Vq , p = q (p, q ∈ {1, 2}). Otherwise, the signed digraph G is structurally unbalanced. The Laplacian matrix L of G is defined as ⎧ ⎨ ∑ |aik |, j = i L = [li j ] ∈ Rn×n with li j = k∈N(i) ⎩ j = i. −ai j ,

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Let G = (V , E , A ) be the induced digraph from G = (V , E , A ), where A = [|ai j |] ∈ Rn×n . The Laplacian matrix L of G is given by L = diag{∑nj=1 |a1 j |, ∑nj=1 |a2 j |, · · · , ∑nj=1 |an j |} − A . A set of all gauge transformations is provided by Dn = {Dn = diag{σ1 , σ2 , · · · , σn } : σi ∈ {−1, 1}, i ∈ Fn }. From [8], if G is structurally balanced, then there exists a gauge transformation such that L = Dn LDn , Dn ∈ Dn . When the signed digraph G is strongly connected, it follows from [8] that • L has a zero eigenvalue and other eigenvalues with positive real parts if and only if G is structurally balanced; • all eigenvalues of L have positive real parts if and only if G is structurally unbalanced. In this paper, we consider a signed network with n agents which can be described by a strongly connected signed digraph G . Agents are represented as nodes and interactions among nodes are regarded as edges with positive weight or negative weight in the signed digraph G . Let xi (t) denote the state of node vi and the dynamics of node vi is given by x˙i (t) = ui (t) + ωi (t), ∀i ∈ Fn

(1)

where ui (t) is the control protocol to be designed and ωi (t) ∈ L2 is the external disturbance. For arbitrary initial states of nodes, the system (1) can achieve • bipartite consensus if and only if limt→∞ [x j (t) − xi (t)] = 0, ∀i, j ∈ Fn ; • state stability if and only if limt→∞ xi (t) = 0, ∀i ∈ Fn . Based on the nearest neighbor rule, the control protocol is provided by ui (t) = −

∑

|ai j | [xi (t) − sgn(ai j )x j (t)] , ∀i ∈ Fn .

(2)

j∈N(i)

Let x(t) = [x1 (t), x2 (t), · · · , xn (t)]T , u(t) = [u1 (t), u2 (t), · · · , un (t)]T and ω (t) = [ω1 (t), ω2 (t), · · · , ωn (t)]T . With L, we can write (1) and (2) as a compact form x(t) ˙ = −Lx(t) + ω (t).

(3)

In the following, we target at studying how to make the system (3) realize consensus objective with the desired resilient performance.

3 Model Transformation In this section, we aim to investigate the model transformation of signed networks by introducing a dynamic output which can be employed to measure the disagreement among nodes.

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Structurally Balanced Signed Networks

It is obvious that there exists an essential difference between bipartite consensus objective and stability (i.e., limt→∞ x(t) = 0n ) of the system (3). However, we find that the bipartite consensus can be reflected by the output stability. To be specific, because G is structurally balanced, there exists a gauge transformation Dn = {σ1 , σ2 , · · · , σn } such that L = Dn LDn . We address a dynamic output z(t) = [z1 (t), z2 (t), · · · , zn (t)]T whose component is defined as zi (t) = xi (t) −

σi n

n

∑ σ j x j (t), ∀i ∈ Fn .

j=1

With (3), the dynamics of vi can be rewritten as  x(t) ˙ = −Lx(t) + ω (t) z(t) = Cx(t) where C satisfies

⎡

n−1 n ⎢ − σ2 σ1 n ⎢

C=⎢ ⎣

.. .

− σ1nσ2 · · · − σ1nσn n−1 · · · − σ2nσn n .. .. .. . . .

− σnnσ1 − σnnσ2 · · ·

(4) ⎤ ⎥ ⎥ ⎥. ⎦

(5)

n−1 n

We can easily see from (5) that the eigenvalues of C are 1 with multiplicity n − 1 and 0 with multiplicity 1. The vector Dn 1n is not only right eigenvector but also left eigenvector of C associated with eigenvalue 0. Since   C is symmetric, there exists an orthogonal In−1 0n−1 T matrix U such that U CU = T holds, in which the last column of U is D√n 1nn . 0n−1 0 Lemma 1. Consider a signed digraph G that is strongly connected and structurally balanced. Then,   U T LU = Φ1 0n holds, where Φ1 ∈ Rn×(n−1) . Proof. Since the last column of U is

D√ n 1n , n

we can directly obtain this result.

Next, we study how to decompose the system (4). We first introduce an auxiliary vector as follows  Dn 1n n t x(t) ˆ = x(t) − ∑ 0 σi ωi (τ )d τ . n i=1   Let U = U1 U2 with U1 ∈ Rn×(n−1) and U2 = D√n 1nn . With U1 and U2 , we define two vectors as δ (t) = U1T x(t) ˆ and δ¯ (t) = U2T x. ˆ

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On one hand, it can directly deduce   δ˙ (t) Dn 1n n T U x(t) ˙ = ˙¯ +U T ∑ σi ωi (t) n i=1 δ (t)     0n−1 δ˙ (t) = ˙¯ + √1 n σ ω (t) δ (t) n ∑i=1 i i

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(6)

where we use U1T U2 = 0n−1 . On the other hand, it follows from (4) that ˙ = −U T Lx(t) +U T ω (t) U T x(t)    Dn 1n n t T ˆ + = −U L x(t) ∑ 0 σi ωi (τ )d τ +U T ω (t) n i=1    T  δ (t) U1 ω (t) T = −U LU ¯ + U2T ω (t) δ (t)     T  U1T ω (t) U1 LU1 0n−1 δ (t) + √1 n =− σ ω (t) U2T LU1 0 δ¯ (t) n ∑i=1 i i in which we use LU2 = 0n from Lemma 1. With (6) and (7), we can obtain ⎧ T T ˙ ⎪ ⎪ δ (t) = −U1 LU1 δ (t) +U1 ω (t) ⎪ ⎪ ⎨ ˙¯ δ (t) = −U2T LU1 δ (t)   ⎪ ⎪  δ (t)  ⎪ ⎪ z(t) = Cx(t) = Cx(t) = U1 δ (t). ˆ = U1 0 ⎩ δ¯ (t)

(7)

(8)

It is obvious from (8) that δ¯ (t) is completely dependent on δ (t). Therefore, we can explore the following reduced-order system  δ˙ (t) = −U1T LU1 δ (t) +U1T ω (t) (9) z(t) = U1 δ (t). instead of the system (8). We can easily see from the system (9) that if limt→∞ δ (t) = 0n−1 holds, then it denotes limt→∞ z(t) = 0n which implies that the networked system (3) can achieve the bipartite consensus. 3.2 Structurally Unbalanced Signed Networks When the signed digraph G is structurally unbalanced, we introduce an output vector z(t) = x(t). Thus, the dynamics of nodes can be rewritten as  x(t) ˙ = −Lx(t) + ω (t) (10) z(t) = x(t). It is immediate to obtain from (10) that the output z(t) converging to zero implies that the networked system (3) can reach the state stability.

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4 Main Results In this section, we target at examining the consensus performance of the networked system (9) through evaluating z(t) ∈ L2 corresponding to ω (t) ∈ L2 . Besides, we introduce an induced transfer matrix norm of Tzω (s) as follows ||Tzω ||2−2 =

sup ||ω (t)||2 ≤1

||z(t)||2 .

(11)

It is worth noticing that ||Tzω ||2−2 can also be considered as H∞ norm of Tzω (s) (see [3–6] for more details). Since (11) can be rewritten as ||z(t)||2 ≤ ||Tzω ||2−2 ||ω (t)||2 , we realize that ||Tzω ||2−2 can be used to measure the resilient performance of the signed networked system with external disturbances satisfying ω (t) ∈ L2 . Theorem 1. Consider the networked system (3) whose communication topology can be described by a strongly connected signed digraph G , and let ωi (t) ∈ L2 , ∀i ∈ Fn . For a prescribed number γ > 0, there exist the following results. (1) When G is structurally balanced, the system (4) can achieve the bipartite consensus with ||Tzω ||2−2 < γ if there exists a positive definite matrix P ∈ R(n−1)×(n−1) satisfying the following Riccati inequality: −U1T LT U1 P − PU1T LU1 +

1 PU T U1 P +U1T U1 < 0. γ2 1

(12)

(2) When G is structurally unbalanced, the system (10) can reach the state stability with ||Tzω ||2−2 < γ if there exists a positive definite matrix P ∈ Rn×n such that − LT P − PL +

1 T P P + In < 0. γ2

(13)

Proof. (1) G is structurally balanced. This proof can be divided into two steps. Step 1. We aim to analyse that the bipartite consensus objective holds. Let Q = −(−U1T LT U1 P − PU1T LU1 + and

1 PU T U1 P +U1T U1 ) γ2 1

1 Q˜ = Q + 2 PU1T U1 P +U1T U1 . γ

With (12), it can be easily found that Q > 0 and Q˜ > 0 hold. We can further obtain ˜ −U1T LT U1 P − PU1T LU1 = −Q. Therefore, we realize that the matrix −U1T LU1 is Hurwitz stable. The external disturbance ω (t) ∈ L2 implies limt→∞ ω (t) = 0n . With the input-to-state stability (ISS), it is immediate to derive that the reduced-order system (9) is asymptotically stable [i.e., limt→∞ δ (t) = 0n−1 ]. Besides, we know limt→∞ z(t) = limt→∞ U1 δ (t) = 0n which denotes that the system (3) can achieve the bipartite consensus.

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Step 2. We investigate how to guarantee the bipartite consensus performance of the system (4). Using the strict inequality (12), we can choose 0 < ε < 1 such that −U1T LT U1 P − PU1T LU1 +

1

γ 2 (1 − ε )

PU1T U1 P +U1T U1 < 0.

(14)

Applying the Schur’s complement formula to (14) can lead to   T   −U1T LT U1 P − PU1T LU1 PU1T U1  U1 0 + 0, gives

δ T (T1 )Pδ (T1 ) +

 T1 0

zT (t)z(t)dt < γ 2 (1 − ε )

 T1 0

ω T (t)ω (t)dt

(17)

in which we use the zero-valued initial condition (i.e., δ (0) = 0n−1 ). Now let T1 → ∞. Since ω (t) ∈ L2 and U1T LU1 is Hurwitz stable, we have limt→∞ δ (t) = 0n−1 . Therefore, we can write (17) as (18) ||z(t)||22 < γ 2 (1 − ε )||ω (t)||22 when T1 → ∞. It follows from (18) that ||z(t)||2 < γ ||ω (t)||2 and ||Tzω ||2−2 < γ hold. Hence, under the condition (12), the system (3) can achieve the bipartite consensus with performance ||Tzω ||2−2 < γ when the signed digraph G is structurally balanced. (2) G is structurally unbalanced. This proof is similar to the proof of structurally balanced case, and thus its proof details are omitted for simplicity. The proof of Theorem 1 is complete. Remark 1. From Theorem 1, we can solve the distributed robust control problems of signed networks in the presence of finite-energy external disturbances, in which the sufficient conditions are provided to ensure that signed networks can achieve the bipartite consensus (respectively, state stability) with desired performance when its associated

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signed digraphs are structurally balanced (respectively, unbalanced). Employing the Schur’s complement formula, the Riccati inequality (12) is equivalent to the following linear matrix inequality (LMI): ⎤ ⎡ U1T −U1T LT U1 P − PU1T LU1 PU1T ⎣ U1 P −γ In 0n×n ⎦ < 0. U1 0n×n −γ In At the same time, the Riccati inequality (13) is equal to ⎡ T ⎤ −L P − PA PT InT ⎣ P −γ In 0n×n ⎦ < 0. In 0n×n −γ In

5 Simulations In this section, we introduce two examples to illustrate the developed results, in which the signed network contains four agents. We consider the initial state x(0) = [1, 2, 3, 4]T and the following external disturbance

ω (t) =

5 [cos(π t), cos(2π t), cos(3π t) cos(4π t)]T . 2t + 1

It can be verified that ω (t) ∈ L2 holds.

Fig. 1. Left: signed digraph G1 . Right: signed digraph G2 . The symbols “+” and “−” denote the positive and negative weight, respectively.

Example 1. Consider the system (4) whose communication topology is described by the signed digraph G1 in Fig. 1(a). It can be easily see that G1 is strongly connected and structurally balanced. Next, we select the performance index γ = 1 and P = 0.5 × diag{1, 1, 1, 1}. It follows from Theorem 1 that the weights of edges (v1 , v2 ), (v2 , v3 ), (v3 , v4 ) and (v4 , v1 ) are 282.2136, −282.2136, 282.2136 and −282.2136, respectively. We plot the simulation results of the system (4) in Fig. 2. From Fig. 2, we can clearly see that the system (4) can achieve the bipartite consensus with consensus performance ||Tzω (s)||2−2 < 1, which coincides with the result of Theorem 1.

Distributed Robust 4

2

The energy of z(t) The energy of ω(t)

100

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Fig. 2. (Example 1). Left: bipartite consensus of the system (4) under the signed digraph G1 . Right: the energy trajectories of the output z(t) and the external disturbance ω (t). 4

2

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Fig. 3. (Example 2). Left: State stability of the system (3) under the signed digraph G2 . Right: The energy trajectories of the output z(t) and the external disturbance ω (t).

Example 2. We employ the signed digraph G2 in Fig. 1(b) to represent the communication topology of the system (3). Different from G1 , the signed digraph G2 is strongly connected and structurally unbalanced. We choose the performance index γ = 1 and P = 0.5 × diag{1, 1, 1, 1}. Using Theorem 1, we can compute the edge weights of G2 by solving the Riccati inequality (13). Specifically, the weights of edges (v1 , v2 ), (v2 , v3 ), (v3 , v4 ) and (v4 , v1 ) are 496.8028, 496.8028, 496.8028 and −496.8028, respectively. The simulation tests are exhibited in Fig. 3. According to Fig. 3, it is shown that the system (3) can reach the state stability with ||Tzω (s)||2−2 < 1. We can see that the simulation result in Fig. 3 is consistent with Theorem 1.

6 Conclusions In this paper, the distributed robust control issues have been investigated for signed networked systems with finite-energy disturbances. Benefitting from model transformations, we have transformed the consensus problems of signed networked systems into the standard robust H∞ control problems. Hence, we have derived the sufficient conditions for signed networked systems achieving consensus objective with desired

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resilient performance. These results have been verified by addressing two numerical examples. It is notable that the developed results can be extended to signed networked systems with switching topologies and communication delays. Acknowledgments. This work was supported in part by the National Natural Science Foundation of China (No. 61873013, No. 61922007, No. 61520106010, No. 61573034) and in part by the Fundamental Research Funds for the Central Universities.

References 1. Olfati-Saber R, Murray RM (2004) Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Autom Control 49(9):1520–1533 2. Ren W, Beard RW (2005) Consensus seeking in multiagent systems under dynamically chaning interaction topologies. IEEE Trans Autom Control 50(5):655–661 3. Lin P, Jia Y (2008) Distributed robust H∞ consensus control in directed networks of agents with time-delay. Syst Control Lett 57(8):643–653 4. Lin P, Jia Y (2010) Robust H∞ consensus analysis of a class of second-order multi-agent systems with uncertainty. IET Control Theory Appl 4(3):487–498 5. Liu Y, Jia Y (2010) H∞ consensus control of multi-agent systems with switching topology: a dynamic output feedback protocol. Int J Control 83(3):527–537 6. Liu Y, Jia Y (2010) Consensus problem of high-order multi-agent systems with external disturbances: an H∞ analysis approach. Int J Robust Nonlinear Control 20(14):1579–1593 7. Mo L, Jia Y (2011) H∞ consensus control of a class of high-order multiagent systems. IET Control Theory Appl 5(1):247–253 8. Altafini C (2013) Consensus problems on networks with antagonistic interactions. IEEE Trans Autom Control 58(4):935–946 9. Meng Z, Shi G, Johansson KH, Cao M, Hong Y (2016) Behaviors of networks with antagonistic interactions and switching topologies. Automatica 73:110–116 10. Meng D, Du M, Jia Y (2016) Interval bipartite consensus of networked agents associated with signed digraphs. IEEE Trans Autom Control 61(12):3755–3770 11. Meng D (2017) Bipartite containment tracking of signed networks. Automatica 79:282–289

Obstacle Avoidance Based on 2D-Lidar in Unknown Environment Kai Yan and Baoli Ma(B) The Seventh Research Division, Beihang University, Beijing 100191, China [email protected]

Abstract. This paper present a new obstacle avoidance algorithm based on 2Dlidar. The algorithm is applied to the problem of reactive navigation for onmidirectional mobile robots in the face of concave obstacles. Two-layer structures are involved in the proposed algorithm. The first layer aims at determining the current exit point. In the second layer, a basic planner is adopted, steering the robot move to the current exit. The core idea of the algorithm is to determine the exit point at each time step. To verify the effectiveness of the proposed algorithm, numerical simulations are presented. The algorithm is compared with potential field method. Results show that the proposed algorithm has good performance especially facing with concave obstacles. Keywords: Obstacle avoidance · 2D-lidar · Exits · Concave obstacles

1 Introduction Obstacle avoidance is of vital importance to successful applications of mobile robot systems. Robots should be able to gain its goal position navigating safely among unknown obstacles. Good performance of obstacle avoidance is the basic premise of robot on a particular task. Obstacle avoidance has been deeply studied and several methods have been developed. These methods can be mainly classified into two categories: global path planning algorithms and local obstacle avoidance methods. Path planning approaches, such as A*, D* [1] and RRT [2], generate complete paths from the robot position to the goal. Usually, the simultaneous location and mapping (SLAM) technology is needed because the global map maybe unknown or the environment is dynamic. While there are two less addressed issues for SLAM: (1) the time-consuming building and updating of the obstacle map, and (2) the high dependence on the precise dense laser sensor for the mapping work [3]. Traditional local obstacle avoidance algorithms mainly include: potential field method [4], the vector field histogram [5], and Dynamic Window approaches [6]. According to the present sensor data, the local obstacle avoidance algorithms obtain just the next input control without generating an complete path. Therefore, the key advantage of local techniques over global ones lies in their low computational complexity, which is particularly important when the world model is updated frequently based on sensor information [6]. However, the approaches above share a common drawback: c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 609–618, 2020. https://doi.org/10.1007/978-981-32-9682-4_64

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they are local methods, and therefore not able to deal with complex obstacles configurations encountered by the robot during navigation [7], e.g., narrow passage, or concave obstacle which could cause the robot trapped in local minima. Therefore, in practical applications path planning algorithms and obstacle avoidance algorithms are usually combined to better solve navigation problems [8]. As most algorithms were proposed more than ten years ago, they are mainly based on ultrasonic sensors, which are inaccurate for environmental detection. Researchers have to come up with various data processing methods, such as certainty grid, to deal with the large amount of data errors. With the development of technology, the emergence of lidar solves the problem of data accuracy [9]. In this paper, we propose an obstacle avoidance algorithm for onmi-directional mobile robots based on 2D-lidar. This algorithm can solve the problem of concave obstacle well by using the idea of looking for exits. There is a two-layer structure in the algorithm. At each time step, the current exit point is determined in the first layer, then the moving direction is computed by a basic planner in the second layer. The conducted simulation scenarios by MATLAB show the strength of the proposed approach. Compared with potential field method, the new algorithm has the advantages of high efficiency and reduced vibration. And the ability of avoiding concave obstacles in most cases is the most important feature. The paper is organized as follows. In Sect. 2, we present a description of the problem. In Sect. 3, each part in the flow diagram of the algorithm is introduced in detail. The key of the algorithm is how to find exits. Section 4 describes experimental results, followed by a discussion of further research issues.

2 Algorithms Description 2.1

Problem Formulation

This paper aims to provide an obstacle avoidance algorithm for omni-directional mobile robots. This algorithm constructs a mapping from lidar data to the motion direction of robot. The input of this algorithm are the real-time information of lidar and the relative position of the target. Output is the moving direction at the next moment. The data of obstacles collected by lidar is based on the polar coordinate system with mobile robot as the center. In this paper, a two-dimensional 360◦ laser scanning ranging radar is used. It can carry out 360◦ laser ranging scanning in two-dimensional plane and generate the plane laser-point cloud information in its space. Assume that angle resolution of lidar is set to 1◦ , we can get 360 lidar points in one scan. The obstacle avoidance algorithm employed in this work, illustrated in Fig. 1, has a two-layer structure. In the first layer, the current exit point is determined. In the second layer, a basic planner steers the robot moving to the current exit. With this structure, the algorithm makes full use of lidar data and can effectively avoid large concave obstacles.

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Fig. 1. Flow diagram of the algorithm

2.2 Find Exits The key of the algorithm is how to find exits. To solve this problem, we designed two sub-programs. One is used to find expectant exits. The output of the sub-program is an exit point, near which the collision-free space is large enough for the robot to pass through. The other is used to determine whether the passageway to the expectant exit point is blocked. Only when the expectant exit is not blocked, the expectant exit can be convert to a real exit. The details of the two subprograms are as follows: (1) Find expectant exits First of all, we assume that there are no obstacles in areas not accessible to lidar. In addition, angle resolution of lidar is set to 1◦ and we can get 360 polar points. The basic idea: among these 360 points, if the distance between two adjacent points is greater than a certain threshold dt , it means that the two points are discontinuous and there is an opening. If the opening is large enough, then there is a expectant exit. The distance between two adjacent points can be calculated by:  dθ ,θ +1 = dθ2 + dθ2 +1 − 2 · dθ · dθ +1 · cos(1◦ ) ≈ dθ − dθ +1

(1)

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where dθ , dθ +1 is the distance corresponding to angle θ and θ + 1 among lidar points, dθ ,θ +1 denotes the distance between polar points (θ , dθ ) and (θ + 1, dθ +1 ). The method of finding a expectant exit is explained by Fig. 2. It requires two conditions: (1) dAB > dt . (2) The half circle shown in Fig. 2 can not contain any lidar points. Therefore, there is an expectant exit in Fig. 2(a), but there are no exits in Fig. 2(b), though dAB satisfy dAB > dt in Fig. 2(b). The exit in Fig. 2(a) corresponds to an exit point C. In addition, The initial value of constant dt should be set: dt > 2dr and the semicircle would be on the left side of the ray RA if the farther point B is on the left side of the nearer point A.

(a) There is an expectant exit.

(b) There are no expectant exits.

Fig. 2. Examples of expectant exits

(2) Determine whether passageways to expectant exits are blocked If the exit is determined, a basic planner will be used to guide the car to the exit point. The basic planner adopted in this paper can guide the robot to the exit, which will be described in detail later. As is shown in Fig. 3(a), point C correspond to one expectant exit, which satisfy two conditions of an expectant exit as mentioned above. However, exit point C is hard to reach by using the basic planner because of the concave obstacle between the current position of robot and point C. Therefore we need to make sure that the passageway to the exit point is not blocked before an expectant exit is converted to a real exit. As we can see in Fig. 3(a), θC , dRC are given by:

θC ≈ θB − arcsin dt /(2dθB )  dRC = (dθB )2 − (dt /2)2

θB , θC ∈ Z

(2) (3)

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(a) The expectant exit is blocked.

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(b) The expectant exit is not blocked.

Fig. 3. Examples of exit

A expectant exit is not blocked should meet the flowing criterias: (1) dθC > dRC

(4)

(2) dMN > dt , ∀M ∈ {(θ , dθ ) ∈ Data|(θ , dθ ) ∈ CDER}, ∀N ∈ {(θ , dθ ) ∈ Data|θC < θ < θC + 90◦ } (3) dM N  > dt ,

(5)



∀M ∈ {(θ , dθ ) ∈ Data|(θ , dθ ) ∈ BCRF},



∀N ∈ {(θ , dθ ) ∈ Data|θC − 90◦ < θ < θC }

(6)

where dθC is the distance corresponding to angle θC , Data is the set of lidar points, dMN   is the distance between M and N, dM N  is the distance between M and N If the three conditions are met at the same time, then the expectant exit turn into a real exit. There is another example in Fig. 3(b), where the expectant exit is not blocked, thus point C in Fig. 3(b) is an exit point. Now we have known how to find exits. But in order for the robot to move towards the target, when determining the current exit, the target should be considered as the first expectant exit first, then we consider whether the target point is blocked. If the target point is not a exit point, we can search exits from the target direction to both sides, and finally select the exit point which is closer to the target point: if dl ≤ dr , if dl > dr ,

θe = θl , dθe = dl θe = θr , dθe = dr

(7) (8)

where (θe , de ), (θl , dl ), (θr , dr ) is the polar coordinates of final exit, the first exit found counterclockwise from the angle of the target and the first exit found clockwise from the angle of the target, dl is the distance between point (θl , dl ) and goal point, dr is the distance between point (θr , dr ) and goal point.

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Reconfirmation of the Current Exit

After finding an exit, the primary obstacle avoidance algorithm will guide the robot moving to the exit until it reaches the exit. But at the next moment, with the environment changing, it could cause the target become directly achievable or the current exit to be blocked. Therefore, before reaching the current exit, it is necessary to re-judge whether the target and the current exit are exit in each control cycle. With the subprograms designed above, the problem is simple. Firstly, we need to determine whether there are obstacles near the current exit point. It can be implemented by the flowing function:  dθ2C + dθ2 − 2 · dθC · dθ · cos(θC − θ ) > dt /2, ∀θ = 1, 2, · · · , 360 (9) where (θC , dθC ), (θ , dθ ) is polar coordinates of exit point C and lidar points respectively. Then, determine whether the current exit is blocked, which can be implemented by the second subprogram designed above. If this condition holds and the passageway to the current exit is not blocked, then the basic planner can be carried out. 2.4

Basic Planner

After determining the present exit, the final steering direction of robots is obtained by the primary obstacle avoidance algorithm: dθo = min{dθ },  θo − 90◦ θr = θo + 90◦

θ ∈ (θe − 90◦ , θe + 90◦ ) and θ ∈ Z − 90◦ − θ

+ 90◦ − θ

i f |θo e | ≤ |θo e| i f |θo − 90◦ − θe | > |θo + 90◦ − θe |

(10) (11)

where (θo , dθo ) is a lidar point, which is nearest to the robot in the half plane of the exit point, θe is the angle of exit point, θr is the output angle, which is perpendicular to θo and toward the target. 2.5

Determine Whether Exit Is Reached

When the global coordinates of the robot are known, the problem is very clear. But in the experiments carried out in this paper, the global coordinates of the robot are unknown. Therefore, we adopt the method of integrating the velocity to estimate the coordinates of the exit relative to the robot:  xet+1 = xet − vx · Δ t (12) yet+1 = yet − vy · Δ t where (xet , yet ), (xet+1 , yet+1 ) are the estimated cartesian coordinates of the exit in time step t and t+1 in the robot coordinate system, respectively. vx , vy is components of linear velocity on x-axis y-axis of robot in time step t, Δ t is the time interval between two control commands.

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3 Experimental Results To verify the effectiveness of the algorithm, a two-dimensional simulation environment in MATLAB is established, which consists of a enclosed space of 6 m × 6 m and three rectangle obstacles of 2 m × 0.5 m. At any point in the environment, the robot can obtain the lidar scanning datas of the environment. As shown in the Fig. 4(a), the boundary of the simulation environment and the black objects are obstacles. Figure 4(b) shows the information scanned by radar when the position of the robot is [3, 4] in Fig. 4(a), and each small red circle represents a data point of lidar.

Fig. 4. Example of lidar datas

Within this simulation environment, the performance of the algorithm proposed in this paper was compared to the performance of the potential field method. In the work of [4], the authors present the concept of the potential field method. In this method, it is assumed that the robot moves under the effect of the forces of two potential fields. The attractive potential field generated by the target produces an attractive force (Fatt ) that attracts the robot to the position of the target all the time. The other field is the repulsive field generated by the obstacles. This field produces a repulsive force (Frep ) that moves the robot away from the obstacles. The robot moves under a virtual force which is the sum of the two forces (Ftotal ) [10]. Ftotal = Fatt + Frep

(13)

In this experiment, Fatt and Frep are defined as: Fatt = ξ (qgoal − q)    η ρ (q,q1 ) − ρ10 ρ (q,q1 )2 ∇ρ (q, qobst ) obst obst Frep = 0

(14) if if

ρ (q, qobst ) ≤ ρ0 ρ (q, qobst ) > ρ0

(15)

where qgoal , q denotes cartesian coordinates of the goal and the robot, ρ (q, qobst ) is the distance between q and qobst , ρ0 is a parameter that represents the impacts scope

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of obstacles, here is set to 1. ξ , η are attractive factor and repulsive factor, and is set to 2 and 1 respectively. Detailed information about the Potential Field Method can be found in [4]. In addition, assume that the diameter of the robot is 0.25 m, the parameters dr , dt of the algorithm proposed in this paper are set to 0.2 m, 0.6 m respectively. In the meanwhile, linear velocity of the robot v is set to 0.5 m/s, time interval of sending control commands Δ t is set to 0.1 s.

Fig. 5. Trajectories of the robot

Firstly, the two obstacle avoidance algorithms are tested in general obstacle environment. The trajectories of the robot is shown in Fig. 5. The yellow and green circles are the start and target positions of the robot. The trajectories consisting of red circles and blue circles are generated by the potential field method and the proposed algorithm respectively. As we can see from the pictures, the robot succeeded in its mission to catch the target with both algorithms in Fig. 5(a) and (b). This proves the obstacle avoidance ability of the algorithm while in scenario (c), potential field method is trapped in local minima but the new algorithm does not have this problem. At the same time, we can find that the path generated by the potential field method in Fig. 5(a) has obvious jitter when passing through narrow channel, but the robot can pass smoothly with the proposed algorithm. In addition, in the first two scenarios, the result shows that the new approach is a very efficient approach and the generated path is close to the optimal path. As a further step, the two obstacle avoidance algorithms are tested when there are concave obstacles in the path. The results are shown in Fig. 6. In these scenarios, the robot can not reach the target with potential field method, but the proposed method can do it well. When the robot is outside the concave obstacle, it will directly bypass the concave area. Even if the starting point is inside the concave area, the robot using the new algorithm can successfully reach the target. This shows the most important advantages of this algorithm over other algorithms. Though the new algorithm has the advantages mentioned above, this does not mean that the robot with the new algorithm can evade any obstacles and will not be trapped in any obstacles. The reason is that when the robot decides its moving direction, it does not make full use of the previous obstacle information. This is the defect of any reactive obstacle avoidance algorithms, which constructs a mapping from sensor information to robot control signal [11]. This can be illustrated by Fig. 7. Because the radar can’t detect

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Fig. 6. Trajectories of the robot with concave obstacles

the environment behind the obstacle, the lidar data at the starting point B in Fig. 7(c) is the same as that in Fig. 7(b). Consequently, the initial moving direction in Fig. 7(c) is identical to that in Fig. 7(b). Similarly, when the robot is close to A, the moving direction will be similar to that in Fig. 7(a), then the robot will come back to B. As a result, the robot will keep repeating its movements and not reach the target. Therefore, this kind of problem may be solved if the previous exit information is taken into account when judging the direction of motion.

Fig. 7. A situation that falls into a dead cycle

4 Conclusions Aiming at the problem that traditional local obstacle avoidance algorithms are easily trapped in concave obstacles, a new obstacle avoidance algorithm based on 2D-lidar is proposed in this paper. By MATLAB simulation, compared with potential field method, the new algorithm has the advantages of high efficiency and avoiding vibration. The most important feature of this algorithm is that it can effectively avoid concave obstacle in most cases. A common problem of reactive obstacle avoidance algorithms, including this algorithm, is presented and more efforts can be made to solve this problem in the future. In short, the conducted simulation scenarios reflects the strength of the proposed approach to be adapted for obstacle avoidance of robots in unknown environments.

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Acknowledgments. This work was supported by National Natural Science Foundation of China (No. 61573034, 61327807).

References 1. Stentz A (1997) Optimal and efficient path planning for partially known environments. Springer, Boston, pp 203–220 2. Kuffner J, LaValle SM (2000) RRT-connect: an efficient approach to single-query path planning, vol 2, pp 995–1001 3. Tai L, Paolo G, Liu M (2017) Virtual-to-real deep reinforcement learning: continuous control of mobile robots for mapless navigation. In: 2017 IEEE/RSJ international conference on intelligent robots and systems (IROS), pp 31–36 4. Khatib O (1990) Real-time obstacle avoidance for manipulators and mobile robots. Springer, New York, pp 396–404 5. Borenstein J, Koren Y (1991) The vector field histogram-fast obstacle avoidance for mobile robots. IEEE Trans Robot Autom 7(3):278–288 6. Fox D, Burgard W, Thrun S (1997) The dynamic window approach to collision avoidance. IEEE Robot Autom Mag 4(1):23–33 7. Sgorbissa A, Zaccaria R (2012) Planning and obstacle avoidance in mobile robotics. Robot Auton Syst 60(4):628–638 8. Choi Y, Jimenez H, Mavris DN (2017) Two-layer obstacle collision avoidance with machine learning for more energy-efficient unmanned aircraft trajectories. Robot Auton Syst 98:158– 173 9. Peng Y, Qu D, Zhong Y, Xie S, Luo J, Gu J (2015) The obstacle detection and obstacle avoidance algorithm based on 2-D lidar. In: 2015 IEEE international conference on information and automation, pp 1648–1653 10. Jaradat MAK, Al-Rousan M, Quadan L (2011) Reinforcement based mobile robot navigation in dynamic environment. Robot Comput-Integr Manufact 27(1):135–149 11. Lamiraux F, Bonnafous D, Lefebvre O (2004) Reactive path deformation for nonholonomic mobile robots. IEEE Trans Robot 20(6):967–977

Fault Tolerant Consensus Control in a Linear Leader-Follower Multi-agent System Xingxia Wang1,2 , Zhongxin Liu1,2(B) , Zengqiang Chen1,2 , and Fuyong Wang1,2 1

College of Artificial Intelligence, Nankai University, Tianjin 300350, China [email protected] 2 Tianjin Key Laboratory of Intelligent Robotics, Nankai University, Tianjin 300350, China

Abstract. Fault tolerant consensus control in a linear leader-follower multi-agent system is investigated in this paper. The communication topology among followers is supposed to be undirected. An loss of effect fault is considered for the system. Firstly, a performance index function is proposed to analyze the consensus of the faulty system. Then, based on the Euler-Lagrange equation, a controller that guarantees the consensus of the faulty system without solving the difficult HJB equation is obtained in this paper. Finally, simulations are utilized to verify the effectiveness of the proposed method. Keywords: Multi-agent system (MAS) · Fault tolerant control (FTC) · Consensus · Performance index function · Optimal

1

Introduction

Recent decades have witnessed the rapid development of multi-agent system (MAS). Inspired by the behavior of biotic community in nature, it can be applied to many aspects, such as formation control [1], satellites [2], mobile robots [3], etc. MAS has the advantage that a single agent cannot catches up, and the consensus problem plays an important role in MAS cooperative control. The existing algorithms of consensus problem can be classified into two kinds: the leaderless consensus and the leader-follower consensus. For the leaderless situation, the consensus is achieved when the states of all the agents converge to the same value that usually is unknown. For the leader-follower case, the consensus is achieved when all the states of the followers converge to the state of the leader. Compared with the leaderless condition, the leader-follower condition can be more flexible, which can be utilized to track some desired trajectory. In most works on consensus of MAS, they don’t involve the situation when fault occurs in the system, which is quite common in reality. When fault occurs, because agents are interconnected, it may generate catastrophic damage to the c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 619–628, 2020. https://doi.org/10.1007/978-981-32-9682-4_65

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consensus of the system. Thus, fault tolerant control (FTC) is of great importance to avoid system breakdown when fault shows up in the system. The concept of fault in linear system is proposed by Beard [4]. Based on whether contains the procedure of fault detection and fault isolation, FTC methods are divided into two kinds: passive FTC and active FTC. For the passive FTC, which can be viewed as a special case of robust control, a fixed controller is proposed to keep the stability of the system when fault occurs. In the process of its controller design, certain kinds of fault are previously considered, which means that the controller is robust and only robust to limited types of faults. For the active FTC, it has the process of fault detection and fault isolation, which provides relatively accurate information about the present fault. Based on the obtained information of the fault, a more promising controller is obtained to maintain the consensus of the faulty system. The passive FTC has the advantage of no needs for fault detection and extra hardware, which makes it easy to be realized and saves costs. However, since its controller is fixed and faults are stochastic, it isn’t sensitive enough to some unknown and unpredicted faults. Compared with the passive FTC, the active FTC has the ability to settle faults case by case. In [5], the authors investigates algorithms used for active FTC in single system. Even though FTC has a relatively long history, not much effort emphasizes on the FTC of MAS. FTC in multi agent system calls for the design of an appropriate controller to guarantee the consensus of the faulty system. In [6], the authors studies consensus of a faulty MAS using an optimal method that contains an HJB equation. In [7], the authors addresses the problem of simultaneous fault detection and consensus control of linear MAS using only relative output information of agents. In [8], the authors introduces a delta operator to unify the consensus of MAS with faults and mismatches. In [9], the authors utilizes a generalized likelihood ratio test to solve the event triggered fault detection problem. In [10], the authors proposes an adaptive FTC protocol to keep the consensus for both linear and Lipschitz nonlinear systems. In [11], the authors uses an integrated design method to tackle the fault estimation and fault tolerant problem. In [12], the authors utilizes an optimal control method that concludes Off-policy Reinforcement Learning to solve the FTC of linear MAS. To the best of author’s knowledge, the existing methods for FTC of MAS are sliding mode method, adaptive control and other integrated methods. Not much research focuses on optimal control methods. Moreover, the common optimal method appeared in literatures needs to solve the HJB equation, which is difficult to be solved. Based on the situation, an optimal method without solving the HJB equation to address fault tolerant control of linear leader-follower MAS is proposed in this paper. The Euler-Lagrange equation is utilized to obtain the optimal controller that can guarantee the consensus of the MAS. Moreover, the obtained controller has the additional advantage of low-calculation-cost, which is verified by simulation results. The rest of the paper is organized as: some preliminary knowledge and problem statement are shown in Sect. 2. The design and analysis of proposed method

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are given in Sect. 3. Simulation and comparison are demonstrated to show the effectiveness of the proposed method in Sect. 4. Finally, a conclusion is summarized in Sect. 5.

2

Preliminary Knowledge and Problem Statement

In this section, some basic knowledge of graph theory and the problem statement are introduced. 2.1

Basic Graph Theory

A linear MAS with one leader and N followers is considered in this paper. A directed or undirected graph is utilized to describe the communication topology of the system. The communication topology among agents is denoted by G = (V, E, A), where V = {υ0 , υ1 , υ2 , . . . , υN } denotes the set of all agents, E ⊆ V ×V represents for the path between agents, A = [aij ] stands for the corresponding adjacency matrix. The component of E is eij , (i, j = 0, 1, 2, . . . , N ), when there exists a / E. For directed path from agent i to agent j, eij = (υi , υj ) ∈ E. Otherwise, eij ∈ the undirected graph, eij = eji (i, j = 0, 1, 2, . . . , N ). If eij = (υi , υj ) ∈ E, the (j, i) element of A is defined as 1. Otherwise, aji = 0. The definition of neighbors set of agent i is defined as: Ni = {υj ∈ V|eij = (υi , υj ) ∈ E}. |Ni | denotes the number of neighbors of agent i. A graph is said to be strongly connected if there exists a path between any two agents. If there exists one agent that has a path to any other agent, the graph is said to have a spanning tree. Throughout the following analysis, the following Assumption 1 is supposed to be held. Assumption 1. The communication topology of the leader-follower system is strongly connected, and there exists at least one edge that connects the leader and follower.   2.2

Problem Statement

For a linear MAS with one leader and N followers, the dynamics of agents can be: (1) x˙ 0 = Ax0 x˙ i = Axi + Bui , i = 1, 2, . . . , N

(2)

where subscript 0 denotes the leader, and subscript i = 1, 2, . . . , N denotes the followers. xi = (xi1 , xi2 , . . . , xin ) ∈ Rn , i = 0, 1, . . . , N denotes the state of the agent i. ui = (ui1 , ui2 , . . . , uiq ) ∈ Rq , i = 1, 2, . . . , N denotes the controller for agent i. A ∈ Rn×n , B ∈ Rn×q are two corresponding matrices that related to the system.

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The consensus is achieved if the states of the followers converge to the state of the leader, which can be expressed as: lim ||xi (t) − x0 (t)|| = 0, i = 1, 2, . . . , N

t→∞

(3)

where the norm || · || denotes the 2-norm. In the following research, the following Assumption 2 is supposed to be held. Assumption 2. The matrix pair (A, B) is controllable.

 

The major failures appeared in MAS can be classified into three kinds: controller failure, actuator failure, sensor failure. And the most common and most investigated kind is actuator failure, which can also be categorized into three kinds: stuck, loss of effect (LOE), outage. The case when a LOE fault occurs in the system is considered in this paper. For simplicity, we assume that the leader doesn’t have fault. In that case, the real control input of agent i on the hth level can be uF ih , which can be expressed through the following equation: uF ih = (1 − ρih )uih , i = 1, 2, . . . , N, h = 1, 2, . . . , q

(4)

where h = 1, 2, . . . , q denotes the dimension of the controller, 0 < ρih < 1 represents the degree of failure. The larger the ρih , the severe the failure. = Rewrite (4) into a vector form, which can be described as: uF i F F , . . . , u ), i = 1, 2, . . . , N . (ui1 , uF i2 iq Thus, the faulty dynamic of followers can be rewritten as: x˙ i = Axi + BuF i , i = 1, 2, . . . , N

(5)

The consensus of the faulty system will be studied in Sect. 3.

3

Proposed Method

This section contains the main result of the research. An optimal method is utilized to guarantee the consensus of the faulty system in this paper. In order to make the states in a neighbors set to be as close as possible, the following performance index function of agent i is defined as: ⎡ ⎤  T  T T ⎣ui F (t) Hui F (t) + (xi (t) − xj (t)) Q(xi (t) − xj (t))⎦dt (6) Ji = 0

j∈Ni

where Q ∈ Rn×n and H ∈ Rq×q are positive definite matrices. If Ji is minimized, then the states in a neighbors set can be as close as possible. Base on the Assumption 1, which means that there exists a path between any two agents, if all Ji , (i = 1, 2, . . . , N ) reach its minimum value, all the states in the MAS can be as close as possible. Next, this paper will focus on minimizing (6) for the followers.

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Lemma 1. [13] The following system (7) with its performance index function (8): y˙ = f (t, y(t), c(t)), y(0) = y0 (7)  T w(y(0), c) = g(t, x(t), c(t))dt + h(y(T )) (8) 0

where y(t) ∈ Rn denotes the state of the system, c(t) ∈ Rq denotes the controller, y(0) denotes the initial state of the system, f (t, y(t), c(t)) and g(t, y(t), c(t)) are continuous functions on R1+n+q . Further, both of them are partial derivable with respect to x, u. h(x) ∈ C 1 . T denotes a positive number. Let H(t, y, c, λ) = g(t, y, c) + λf (t, y, c) be the Hamiltonian function. If c∗ is a control that yields a local minimum for the performance index function, y ∗ (t) and λ∗ (t) are the corresponding state and costate, then it is necessary that ∗

∗

∗

y˙ ∗ (t) = f (t, y ∗ , c∗ ) = ( ∂H(t,y∂λ,c ,λ ) ), y ∗ (0) = y(0) ∗ ∗ ∗ ∗ λ˙ ∗ = − ∂H(t,y∂y,c ,λ ) , λ∗ (T ) = ∂h(y∂y(T )) And, for all t ∈ [0, T ],

∂H(t, y ∗ , c∗ , λ∗ ) =0 ∂c

(9)

(10)  

Lemma 1 is applied to the analysis of the faulty system in this paper. Based on Lemma 1, the following theorem for the faulty MAS is obtained. Theorem 1. Suppose that the Assumptions 1 and 2 holds. For the system described by (1), (5), with (6) as its corresponding performance index function. The optimal controller u∗i , i = 1, 2, . . . , N that minimizes Ji , i = 1, 2, . . . , N can be: |Ni | (I − Λi )−1 H −1 B T λ∗i (t), (i = 1, 2, . . . , N ) (11) u∗i (t) = − 2 where Λi = diag(ρi1 , ρi2 , . . . , ρiq ), i = 1, 2, . . . , N denotes the fault that occurs in each dimension of the controller, I denotes an identity matrix, λ∗i denotes the additional variable to be defined. Moreover, the system can achieve consensus   under the controller u∗i , i = 1, 2, . . . , N . Proof. Step 1. Calculation of ui ∗ , i = 1, 2, . . . , N For the system, define the tracking error for agent i, i = 1, 2, . . . , N as:  ei (t) = (xi − xj ), i = 1, 2, . . . , N (12) j∈Ni

The time derivate of ei (t) is:  e˙ i (t) = (x˙ i − x˙ j ) j∈Ni  F (BuF = Aei (t) + i (t) − Buj (t)) j∈Ni

(13)

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The controllers in a neighbors set are assumed to be independent. For the error system (12), define the corresponding performance index function the same as Ji , and its Hamiltonian function as: T F Hi (t, ei , ui , λi ) = ei T (t)Qei (t) + (uF i (t)) H(ui (t))  T F + λi (t)(Aei (t) + (Bui (t) − BuF j (t)))

(14)

j∈Ni

where λi ∈ Rn is a function to be defined. Do some transform on Hi (t, ei , ui , λi ), which can be rewritten as:  T T F F (BuF Hi (t, ei , ui , λi ) = (uF i (t)) H(ui (t)) + λi (t)(Aei (t) + i (t) − Buj (t))) j∈Ni  T + (xi (t) − xj (t)) Q(xi (t) − xj (t)) j∈Ni   T T T F xi (t) Qxi (t) − 2 xi (t) Qxj (t) + (uF = i (t)) H(ui (t)) j∈Ni j∈Ni   T F xj (t) Qxj (t) + λi T (t)(Aei (t) + (BuF + i (t) − Buj (t))) j∈Ni

j∈Ni

(15) To obtain the optimal controller, calculate the partial derivate of Hi (t, ei , ui , λi ) with respect to ei , ui : ∂Hi (t,ei ,ui ,λi ) ∂ui ∂Hi (t,ei ,ui ,λi ) ∂ei

= 2(I − Λi )T H(I − Λi )ui + |Ni |(I − Λi )T B T λi   = 2Qxi (t) + 2Qxj (t) + AT λi j∈Ni

(16)

j∈Ni

According to Lemma 1, calculating the expression of optimal controller, we can get: ⎧ ⎨ u∗i (t) = − |Ni | (I − Λi )−1 H −1 B T λ∗i (t) 2 ˙ ∗ (t) = −  2Q(x∗ (t) − xj (t)) − AT λ∗ (t), λ∗ (T ) = 0 , i = 1, 2, . . . , N λ i i i i ⎩ j∈Ni

(17) Step 2. Consensus Analysis According to the property of optimal control, the optimal controller u∗i (t), i = 1, 2, . . . , N meets the following equation: Hi (t, e∗i , u∗i , λ∗i ) = 0

(18)

where x∗i , e∗i denote the state and tracking error for agent i under the optimal controller u∗i , respectively. Define a Lyapunov function as: 

1 ∞ T T ei (t)Qei (t) + (ui F (t)) H(ui F (t)) dt, i = 1, 2, . . . , N (19) Vi (ei (t)) = 2 t Note that function Vi (ei (t)) is positive definite. Based on [14], (14) can be transformed into the following equation: Hi (t, ei , ui , λi ) = ei T (t)Qei (t) + (ui F (t))T H(ui F (t)) T  i (t) F (BuF + ∂V i (t) − Buj (t))) ∂ei (t) (Aei (t) + j∈Ni

(20)

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i (ei (t)) In that case, λi (t) = ∂V∂e . i (t) Thus, (18) can be transformed as:

 ∂Vi ∗T (t) ∗ F ∗T ∗ F∗ T F∗ (Ae∗i (t) + (BuF i (t) − Buj (t))) = −(ei (t)Qei (t) + (ui (t)) H(ui (t))) ∗ ∂ei (t) j∈N i

(21) Next, the Lyapunov method is applied to prove the asymptotically stability of the error system (12). Take time derivate of Vi (ei (t)), one can get: V˙ i (e∗i (t)) =

∂ V˙ iT (Ae∗i (t) ∂e∗ i (t)

+

 j∈Ni

∗ ∗ (BuF i (t) − Buj (t)))

∗ F∗ T F∗ = −(e∗T i (t)Qei (t) + (ui (t)) H(ui (t)))

(22)

Because Q, H are positive negative matrices, we can obtain that V˙ i (e∗i (t)) < 0. According to the Lyapunov method, the error system (12) will be asymptotically stable, which further indicates that lim e∗i (t) = 0, i = 1, 2, . . . , N . t→∞ For agent i, any other agent j is either inside or outside its neighbors set. If j ∈ Ni , then because e∗i (t) converges to zero, x∗j (t) will converge to x∗i (t). If j∈ / Ni , due to the connectivity assumption, there exists a path between agent i and agent j. Thus, the final state of agent i will be passed on to one of its neighbors set which in turn passes it to another number in its own neighbors set until it reaches agent j. In this case, agent j will have the same state as agent j. Finally, all the follower agents will have the same state. For arbitrary two agent i and agent j in the system, agent j is either inside or outside Ni . If j ∈ Ni , then because e∗i (t) converges to zero, x∗j (t) will converge / Ni , due to Assumption 1, there exists a path between agent i and to x∗i (t). If j ∈ agent j. Based on the transitivity of the states, agent j will have the same state as agent j. Finally, all the follower agents will have the same state. Because the leader is in one of the Ni , i = 1, 2, . . . , N , all the agents will have the same state as the leader. The proof is completed.   Remark 1. Compared with other methods, the controller is obtained without solving the difficult HJB equation, which saves the calculation cost. In the next part, we will further verify the effectiveness of the proposed method.

4

Simulation

To verify the effectiveness of the proposed method, a MAS system with one leader and five followers is simulated in this section. Moreover, a comparision experiment is utilized to prove the low-calculation-cost property of the proposed method. The communication topology of the system is shown in Fig. 1. In Fig. 1, vertex 0 denotes the leader and vertexes 1–5 denote followers. The communication flow between followers are undirected.

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Fig. 1. Communication topology of the system

Define the state of the system as: xi = (xi1 , xi2 )T , i = 0, 1, . . . , 5. The parameters of the system are set as:         1 −2 1 0 60 0 80 A= ,B = , Q = 600 × ,H = 1 0 1 −1 0 30 08 The fault ρih , i = 1, 2, . . . , 5, h = 1, 2 that occurs in every agent is bounded by (0, 1). The simulation results are as the following two figures.

Fig. 2. Error of xi1 , i = 1, 2, . . . , 5.

Fig. 3. Error of xi2 , i = 1, 2, . . . , 5.

Figures 2 and 3 indicate that the error of each dimension converges to zero, which means that the consensus of the faulty MAS system is achieved. And the trajectory of all the agents is shown in Fig. 4, which shows that the trajectories among agents will converge to the same value. To verify the low-calculation-cost property of the proposed method, a comparison with the method in [10] is performed. This simulation is performanced 30 times, and of each time, the mean square error(MSE) and running time are 5

||x −x ||

2

recorded. The definition of MSE is: M SE = i=1 5i 0 . The next two figures show that the running time and MSE of the two different methods on 30 times.

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Fig. 4. Trajectories of agents

Fig. 5. MSE of different methods.

Fig. 6. Running time of different methods.

As shown in Figs. 5 and 6, the running time of the proposed method is shorter than that in [10] at each iteration, which verifies its advantage of low-calculationcost property. Moreover, the MSE obtained in the proposed method is less than that in [10].

5

Conclusion

Fault tolerant control (FTC) in a linear leader-follower multi-agent system is studied in this paper. Firstly, the dynamic of a faulty MAS system is established. Then a performance index function is introduced to analyze the stability of the system. Based on the Euler-Lagrange equation, the authors obtaines the optimal controller that minimizes the objective function without solving the HJB equation and guarantees the consensus of the system. Moreover, the controller has the advantage of low-calculation-cost. Finally, simulations are used to verify the effectiveness of the proposed method. In the future, other kinds of faults in MAS will be investigated. Acknowledgments. This work is supported by National Natural Science Foundation of China under Grant No. 61573200, 61573199.

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References 1. Alexander F, Richard MM (2002) Information flow and cooperative control of vehicle formations. IFAC Proc Vol 35(1):115–120. https://doi.org/10.3182/200207216-es-1901.00100 2. Nghia TD, Yusuke N (2017) Satellite and central transitions selective 1H/27Al D-HMQC experiments at very fast MAS for quadrupolar couplings determination. Solid State Nucl Magn Reson 84:83–88. https://doi.org/10.1016/j.ssnmr.2016.12. 014 3. Rincon JA, Garcia E, Julian V, Carrascosa C (2018) The JaCalIVE framework for MAS in IVE: a case study in evolving modular Robotics. Neurocomputing 275:608–617. https://doi.org/10.1016/j.neucom.2016.08.160 4. Beard RV (1971) Failure accommodation in linear systems through selfreorganization. Department of Aeronautics and Astronautics, MIT, USA 5. Hwang I, Kim S, Kim Y, Seah CE (2010) A survey of fault detection, isolation, and reconfiguration methods. IEEE Trans Control Syst Technol 18(3):636–653. https://doi.org/10.1109/TCST.2009.2026285 6. Semsar-Kazerooni E, Khorasani K (2010) Team consensus for a network of unmanned vehicles in presence of actuator faults. IEEE Trans Control Syst Technol 18(5):1155–1161. https://doi.org/10.1109/tcst.2009.2032921 7. Davoodi M, Meskin N, Khorasani K (2016) Simultaneous fault detection and consensus control design for a network of multi-agent systems. Autom (J IFAC) 66:185–194. https://doi.org/10.1016/j.automatica.2015.12.027 8. Zheng DH, Zhang HB, Zhang JA, Wang G (2018) Consensus of multi-agent systems with faults and mismatches under switched topologies using a Delta operator method. Neurocomputing 315:198–209. https://doi.org/10.1016/j.neucom.2018.07. 017 9. Chitraganti S, Sid MA, Aberkane S (2018) Event triggered fault detection in linear systems using generalized likelihood ratio test. IFAC-PapersOnLine 51(1):444–449. https://doi.org/10.1016/j.ifacol.2018.05.073 10. Zuo ZQ, Zhang J, Wang YJ (2015) Adaptive fault tolerant tracking control for linear and Lipschitz nonlinear multi-agent systems. IEEE Trans Ind Electron 62(6):3923–3931. https://doi.org/10.1109/TIE.2014.2367034 11. Liu Y, Yang GH (2019) Integrated design of fault estimation and fault-tolerant control for linear multi-agent sytems using relative outputs. Neurocomputing 329:468– 475. https://doi.org/10.1016/j.neucom.2018.11.005 12. Dehshalie ME, Menhaj MB, Karrari M (2019) Fault tolerant cooperative control for affine multi-agent systems: an optimal control approach. J Franklin Inst 356(3):1360–1378. https://doi.org/10.1016/j.franklin.2018.09.038 13. Jacob E (2005) LQ dynamic optimization and differential games. Wiley, Hoboken 14. Vamvoudakis KG, Lewis FL, Hudas GR (2012) Multi-agent differential graphical games: online adaptive learning solution for synchronization with optimality. Automatica 48(8):1598–1611. https://doi.org/10.1016/j.automatica.2012.05.074

A Two Phase Method for Skyline Computation Haipeng Du, Lizhen Shao(B) , Yang You, Zhuolin Li, and Dongmei Fu University of Science and Technology Beijing, Beijing 100083, China [email protected]

Abstract. Skyline computation has been widely used in many research areas such as database visualization, data mining and multi-criteria decision making. In order to process massive data, in this paper, we propose a two phase method to calculate Skyline points. In the first phase, all the supported non-dominated skyline points are calculated with Quickhull algorithm. Then in the second phase, all the unsupported non-dominated points are calculated with Block Nested Loop (BNL) algorithm. We test the algorithm on three different types of synthetic data sets and compare it with BNL algorithm and Sort Filter Skyline (SFS) algorithm. Experimental results show that our proposed algorithm performs better than BNL and SFS on most of the test datasets. Keywords: Skyline

1

· BNL · Quickhull

Introduction

Skyline operator was proposed in [1] and has been widely developed since then. Skyline operator can handle multiple and conflicting objects simultaneously, and return all the objects that are not dominated by others [2–4]. A simple example of skyline is shown in Fig. 1. Tourists always prefer a hotel which is close to the sea and cheaper among all the hotels, however, this is a pair of conflicting goals. The hotels that we most interested in are A, L, M, F, N and C in comparison, because there are no other hotels which are better than them in both price and distance. Therefore, points A, L, M, F, N and C are called Skyline points (or nondominated points) and all the Skyline points form a set, called Skyline. Existing skyline computation algorithms can be classified into two categories, depending on whether pre-computed indexes are used on data or not. Indexbased algorithms include INDEX [5], Bitmap [5], Nearest Neighbor (NN) [6] and Branch and Bound Skyline (BBS) [7], while none index-based algorithms include BNL [1], Divide and Conquer (D&C) [1], SFS [8], Linear Elimination Sort for Skyline (LESS) [9]. In general, index-based algorithms tend to have better performance, since they use specialized structure to avoid accessing entire dataset, but they have limited application due to the indexed dataset. On the other hand, none index-based algorithms are more generic and flexible, easy to be applied in various database system. c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 629–637, 2020. https://doi.org/10.1007/978-981-32-9682-4_66

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Fig. 1. A skyline example.

Among all the none index-based skyline algorithms, BNL is a nested-loop method that compares every pair of data points in the dataset and output the points that are not dominated by others. BNL performs well especially when the number of dimensions of the data is small or the distribution of data is correlated. For the reason of simplicity and universality, BNL has been widely used in many areas. Zhang et al. [10] proposed a MapReduce version of BNL algorithm (MRBNL) for processing skyline query on the MapReduce framework. Abourezq and Idrissi [11] developed an BNL based algorithm and use it in cloud computing. Yakine et al. [12] used skyline in the area of wireless Ad-Hoc Network. Li and Yoo [13] proposed a continuous skyline query processing scheme over skewed dynamic data set with BNL algorithm. Cosgaya-Lozano et al. [14] pushed skyline query to FPGAs and proposed a modified version of BNL for parallel execution. Bikakis et al. [15] applied BNL to an external memory algorithm and analyzed its performance. Min et al. [16] used BNL algorithm to analyze the multidimensional medical data. In [17], BNL method is used to reduce the decision space and it focuses only on interesting Web services that are not dominated by any other service. In this paper, we focus on BNL algorithm. To improve the performance of BNL for processing massive data, we are going to propose a two phase method for skyline computation based on BNL and Quickhull algorithms. The rest of the paper is organized as follow. Section 2 introduces the BNL algorithm. In Sect. 3, we propose the two phase method and experimental results are in Sect. 4. Finally, we draw some conclusions in Sect. 5.

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In this section, we will review BNL algorithm [7]. Dominance is an important concept in Skyline computation. For a given data set, we say that point p dominates another point q if p is as good or better in all dimensions and better in at least one dimension than q, denoted as p ≺ q.

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BNL algorithm uses a list to hold a number of points. Every candidate point is checked if it is dominated by any other points within the list. For each candidate point p, there are three cases: (i) if p is dominated by any point in the list, then p is discarded as it is not part of the skyline; (ii) if p dominates any point in the list, then p is inserted, and all points in the list dominated by p are discarded; and (iii) if p is neither dominated by, nor dominates, any point in the list, then p is simply inserted without discarding any point. Table 1 shows the process of obtaining skyline points. Table 1. Description of BNL algorithm. Action List A

A

B

A, B

C

A, B, C

D

A, B, C

E

A, B, C

F

A, C, F

G

A, C, F

H

A, C, F

I

A, C, F

J

A, C, F

K

A, C, F

L

A, C, F, L

M

A, C, F, L, M

N

A, C, F, L, M, N

S

A, C, F, L, M, N

BNL can be applied directly to any data set with different data distribution and size without any indexing and preprocessing. However, when limiting the size of the lists, it needs a temporary file to store the overflow points.

3

Two Phase Method for Skyline Compuation

For a given point p in Skyline, let P conv denote a convex combination of the skyline excluding p. A point p in Skyline is called a supported nondominated point if and only if there does not exists a point p ∈ P conv such that p ≺ p, otherwise it is called unsupported nondominated point. Take the data set from Fig. 1 as an example, A, F and C are supported nondominated points, and L, M, N are unsupported nondominated points. The two phase skyline computation method computes supported nondominated points and unsupported nondominated points separately. Firstly all the

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supported nondominated points are computed with Quickhull algorithm (for the detail of the algorithm, the reader is referred to [18]). Then the remaining unsupported nondominated points are computed using the original BNL algorithm. The search space for skyline points can be restricted to several simplices given by the supported nondominated facet as indicated in Fig. 2. Here we take the data set from Fig. 1 as an example to show how the two phase method works. In the Figure, A, C and F are supported nondominated points, while L, M, N are unsupported nondominated points. We first use Quickhull algorithm to compute the convexhull points A, C, E, F, I, G and J, see Fig. 3. Furthermore by comparison, we can obtain the supported nondominated points A, C and F.

Fig. 2. Supported nondominated points.

12

10

8

6

4

2

0

0

100

200

300

400

500

Fig. 3. The convex hull.

As it can be seen that each two consecutive supported nondominated points forms a nondominated facet. Each nondominated facet can be used to determine

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an anti-idea point, see Fig. 2. For example, the anti-idea point formed by the nondominated facet of A − F is U , for the nondominated facet of F − C is V. Anti-idea points and their corresponding nondominated facets can be used to form simplices. For example, two simplices in Fig. 2 are AUF and FVC. Now we can apply BNL algorithm only to data points in the simplices. By this way, we actually separate skyline computation into several parts. It can effectively reduce the amount of calculation. Since every point in the candidate should be compared in a repeatedly loop in BNL, the smaller the candidate set is, the less the computational time will be used. Therefore our proposed algorithm can effectively improve the efficiency of BNL, especially when it tackles massive data.

4

Experimental Results

In this section, we report the experimental results for our proposed two-phase algorithm. We compared runtime of two-phase algorithm with original BNL and SFS. The SFS algorithm improves BNL performance by presorting the input dataset, which skip much of redundant computation. All the experiments are carried out on a personal PC with a Inter Core i5-8259U CPU and 16 GB RAM running macOS 10.14.5. And all algorithms are implemented in Python 3.7. 25

20

25

15

20

20 15

10

10

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5 5 0

10

0 −5

−5

5 −10

0

0

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15

(a) Independent

20

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−10

−15 −15

−10

−5

0

5

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(b) Anti-correlated

20

25

−15 −15

−10

−5

0

5

10

15

20

(c) Correlated

Fig. 4. Randomly produced data sets.

The data sets used in our experiments are generated in a similar way as described in [1]. Three types of data sets are generated: (1) Independent where the attribute values of the points are generated using an uniform distribution; (2) Correlated which contain points whose attribute values are good in one dimension and are also good in other dimensions; (3) Anti-correlated which contain points whose attribute values are good in one dimension but are bad in one or all of the other dimensions. Figure 4 shows the three types of two-dimensional generated data sets. For each type, we randomly generate six different data sets (with 102 –107 points) in 2D and 3D space, respectively. To eliminate the uncertainty of randomly generated data sets, we fix the randomly generated data sets, and 30 individual tests are taken on each of the data set, then we record the average computational time.

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Independent Anti-correlated Correlated

100

6

14

2

1000

6

30

2

10000

8

36

3

100000

10

44

1

1000000

23

59

2

10000000 17

72

3

Table 3. Number of skyline points of 3D data sets. Data set 100

Independent Anti-correlated Correlated 14

31

5

1000

28

63

4

10000

51

122

2

100000

80

151

2

1000000

115

274

5

10000000 109

409

6

Table 4. Computational time for different sizes of 2D data sets. Size

Independent SFS

BNL

Anti-correlated Two-phase SFS

BNL

Correlated Two-phase SFS

BNL

Two-phase

100

0.0004 0.0006 0.0008

0.0006 0.0011 0.0008

0.0003 0.0003 0.0006

1000

0.0027 0.0043 0.0010

0.0060 0.0103 0.0026

0.0035 0.0028 0.0008

10000

0.0243 0.0286 0.0035

0.0328 0.0794 0.0009

0.0244 0.0252 0.0031

100000

0.2342 0.2143 0.0424

0.4438 0.5229 0.0541

0.2376 0.2137 0.0288

1000000

2.3793 2.1200 0.5137

7.3826 3.4097 0.6493

2.3769 2.1787 0.3439

10000000 25.0176 24.1960 6.3494

36.8370 39.4128 33.6965

24.2759 21.2006 4.6070

Table 5. Computational time for different sizes of 3D data sets. Size

Independent SFS

BNL

Anti-correlated Two-phase SFS

BNL

Correlated Two-phase SFS

BNL

Two-phase

100

0.0008 0.0011 0.0011

0.0021 0.0033 0.0018

0.0004 0.0005 0.0008

1000

0.0052 0.0104 0.0025

0.0125 0.0275 0.0069

0.0031 0.0034 0.0012

10000

0.0367 0.0670 0.0411

0.0630 0.1852 0.1015

0.0282 0.0295 0.0047

100000

0.3196 0.5367 0.2187

0.4149 1.1782 1.0425

0.2897 0.2662 0.0363

1000000

3.5530 4.4866 1.8448

3.3573 7.2852 7.2929

2.9556 2.6916 0.4616

10000000 31.1175 34.2324 19.7056

33.8572 66.1543 55.0583

29.8124 26.7000 5.9377

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Tables 2 and 3 show the number of skyline points for different types of 2D and 3D data sets. From the two tables we can see that: (1) The number of skyline points in three dimension is larger than that of in two dimension; (2) The number of skyline points increases as the number of total points increase; (3) The number of skyline points for correlated data sets is fairly small while it is fairly large for the anti-correlated data sets. The number of the skyline points for independent data sets is somewhere in between the correlated and anti-correlated data sets. Tables 4 and 5 show the computational time for different types of data sets in 2D and 3D space respectively. As it can be seen that: (1) The two phase method performs better than the original BNL algorithm on all the tested data sets; (2) For 2D cases, as the size of data sets increases, the two phase method performs much better than the original BNL and SFS algorithm; (3) All of the three methods perform better in correlated data set than that in independent and anti-correlated data sets. This is because, compared to independent and anti-correlated data sets, there is less nondominated points to compute than in correlated data set; (4) Compared to BNL, SFS is a very efficient algorithm especially when the data set is 3-dimensional anti-correlated.

5

Conclusions

In this paper, we have proposed a two phase method for skyline computation, which is based on Quickhull and BNL algorithms. Firstly Quickhull algorithm is used to find the supported nondominated points and thus separates the skyline computation into several parts. Then BNL is used to calculated the unsupported nondominated points. We tested our algorithm on three types of randomly produced data sets, i.e., independent, anti-correlated and correlated. Experimental results showed that our two phase method performed better than the original BNL algorithm on all the three types of tested data sets. Our algorithm is also superior than SFS algorithm on most of the tested data sets. In the future, we can consider combine Quickhull with some other algorithm to improve the skyline computational efficiency. Acknowledgments. This work was supported by the Fundamental Research Funds for the Central Universities (Grant No. FRF-GF-18-029B) and Science and Technology on Space Intelligent Control Laboratory for National Defense (Grant No. KGJZDSYS2018-13).

References 1. Borzsony S, Kossmann D, Stocker K (2001) The skyline operator. In: Proceedings 17th international conference on data engineering. IEEE, Heidelberg, pp 421–430. https://doi.org/10.1109/ICDE.2001.914855 2. Tomoiag˘ a B, Chindri¸s M, Sumper A, Sudria-Andreu A, Villafafila-Robles R (2013) Pareto optimal reconfiguration of power distribution systems using a genetic algorithm based on NSGA-II. Energies 6(3):1439–1455. https://doi.org/10.3390/ en6031439

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3. Rodger JA, Pankaj P, Nahouraii A (2014) A petri net pareto ISO 31000 workflow process decision making approach for supply chain risk trigger inventory decisions in government organizations. Intell Inf Manag 6:157–170. https://doi.org/10.4236/ iim.2014.63017 ¨ 4. Ozyer T, Zhang M, Alhajj R (2011) Integrating multi-objective genetic algorithm based clustering and data partitioning for skyline computation. Appl Intell 35(1):110–122. https://doi.org/10.1007/s10489-009-0206-7 5. Tan KL, Eng PK, Ooi BC (2001) Efficient progressive skyline computation. In: VLDB 2001 proceedings of the 27th international conference on very large data bases, vol 1. Morgan Kaufmann, San Francisco, pp 301–310 6. Kossmann D, Ramsak F, Rost S (2002) Shooting stars in the sky: an online algorithm for skyline queries. In: VLDB 2002 proceedings of the 28th international conference on very large data bases, Hong Kong, China, pp 275–286 7. Papadias D, Tao Y, Fu G, Seeger B (2005) Progressive skyline computation in database systems. ACM Trans Database Syst 30(1):41–82. https://doi.org/10. 1145/1061318.1061320 8. Chomicki J, Godfrey P, Gryz J, Liang D (2005) Skyline with presorting: theory and optimizations. In: Klopotek MA, Wierzcho ST, Trojanowski K (eds) Intelligent information processing and web mining. Advances in soft computing, vol 31. Springer, Heidelberg. https://doi.org/10.1007/3-540-32392-9 72 9. Godfrey P, Shipley R, Gryz J (2007) Algorithms and analyses for maximal vector computation. VLDB J-Int J Very Large Data Bases 16(1):5–28. https://doi.org/ 10.1007/s00778-006-0029-7 10. Zhang B, Zhou S, Guan J (2011) Adapting skyline computation to the mapreduce framework: algorithms and experiments. In: International conference on database systems for advanced applications. Springer, Heidelberg, pp 403–414. https://doi. org/10.1007/978-3-642-20244-5 39 11. Abourezq M, Idrissi A (2014) Introduction of an outranking method in the cloud computing research and selection system based on the skyline. In: IEEE eighth international conference on research challenges in information science (RCIS). IEEE, Marrakech, pp 1–12. https://doi.org/10.1109/RCIS.2014.6861067 12. Yakine F, Abourezq M, Idrissi A (2016) Skyline method in wireless ad-hoc networks routing. WSEAS Trans Commun 15:137–146 13. Li H, Yoo J (2014) An efficient scheme for continuous skyline query processing over dynamic data set. In: IEEE international conference on big data and smart computing (BIGCOMP). IEEE, Bangkok, pp 54–59. https://doi.org/10.1109/bigcomp. 2014.6741407 14. Cosgaya-Lozano A, Rau-Chaplin A, Zeh N (2007) Parallel computation of skyline queries. In: 21st international symposium on high performance computing systems and applications (HPCS 2007). IEEE, Saskatoon, p 12. https://doi.org/10.1109/ HPCS.2007.25 15. Bikakis N, Sacharidis D, Sellis T (2014) A study on external memory scan-based skyline algorithms. In: Decker H, Lhotsk L, Link S, Spies M, Wagner RR (eds) Database and expert systems applications. Springer, Cham, pp 156–170. https:// doi.org/10.1007/978-3-319-10073-9 13 16. Che M, Wang L, Jiang Z (2018) An approach to multidimensional medical data analysis based on the skyline operator. In: IEEE international conference on industrial engineering and engineering management (IEEM). IEEE, Bangkok, pp 1806– 1810. https://doi.org/10.1109/IEEM.2018.8607324

A Two Phase Method for Skyline Computation

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17. Ouadah A, Hadjali A, Nader F, Benouaret K (2019) SEFAP: an efficient approach for ranking skyline web services. J Ambient Intell Human Comput 10(2):709–725. https://doi.org/10.1007/s12652-018-0721-7 18. Barber CB, Dobkin DP, Dobkin DP, Huhdanpaa H (1996) The quickhull algorithm for convex hulls. ACM Trans Math Softw 22(4):469–483. https://doi.org/10.1145/ 235815.235821

Synchronization Analysis of Delayed Neural Networks with Stochastic Missing Data Nan Xiao1(B) , Guilai Zhang1 , and Yuan Ma2 1 School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China [email protected] 2 School of Computer and Communication Engineering, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China

Abstract. This article considers the synchronization control problem for a type of delayed neural networks (DNNs) with some structured uncertainty. The system is assumed to be modeled with stochastic missing data in it which may be caused by the packet dropout phenomenon. Based on Lyapunov functional method, by considering the properties of the stochastic variables, the systems can be seen as a type of switch system and the stabilization criterion is established by using switching techniques. The obtained criterion can guarantee the globally asymptotic stable of such systems with unreliable communication links. The criterion we obtained is in the form of Linear Matrix Inequalities (LMIs). Then we can solve the LMIs easily by using the MATLAB LMI Toolbox by giving parameters, and the gain of controller can be obtained. Finally, we employ two numerical examples in order to show the validation and effectiveness of our designed controller. Keywords: Delayed neural networks · Switching method Synchronization · Stochastic missing data

1

·

Introduction

In many practical systems, the phenomenon of time delay is often appeared, such as signal transmission system, evolutionary system, dynamic system and neural network system. For some type of systems, the time delay may be caused by several factors. For example, the device transmits power or a signal through a long distance may cause time delay phenomenon, the unavoidable mistake of some system’s measuring elements may also cause time delay phenomenon. Since the existence of time delay can degrade system’s performance, which can lead system to instability and hard to be controlled with, much work has been done to research the stability and stabilization problem for time-delay systems. Till now, there are mainly two types of stability criteria: delay-independent criteria c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 638–648, 2020. https://doi.org/10.1007/978-981-32-9682-4_67

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and delay-dependent criteria. It is considered that for a small delay the later one has less conservativeness than the former one. For getting a less conservative stability criteria, several methods are proposed, which include free weighting matrices method, Park’s inequalities, Jensen’s inequality, reciprocally convex method, delay dividing method and augmented Lyapunov method. On the other hand, the neural network systems(NNs) have been widely applied in many practical areas, such as speech recognition, image classification, secure communication and natural language processing. There are a lot of research about the stability problem for NNs. Among them, the synchronization problem for neural networks systems with time delay have attracted more attention in recent years [1–3]. Meanwhile, a lot of practical systems are often affected by some unavoidable disturbances. The disturbances may be caused by the systems’ environmental circumstances, in which it is named as additive randomly occurred nonlinear disturbances (RONs). In some recently research, the RONs are assumed to be subject to random abrupt changes of its environmental circumstances. It may occur in a probabilistic way, for example, the failures of systems could occur randomly, or its components need repairs for random times, etc [4– 8]. In the research mentioned above, based on the assumption that the RONs are Bernouli distributed while sequences, some stability criteria are proposed. The obtained stability criteria can guarantee system’s globally exponentially stable in mean square sense. Another method to cope with this problem is by using the switching method, which is a useful tool in the study of stable problem for switched systems. A switched system is a hybrid dynamical system, including many subsystems, which can be continuous subsystems or discrete subsystems. For each subsystems there exist the switching rules that coordinate all the systems. The stability analysis for switched systems is hard since the stability of its subsystems is not equal to the stability of the whole system. For example, each subsystem is unstable, but the hole systems are stable, and it may also hold true on the contrary. The problem is how to design the control law such that the hole system can retain stability for arbitrary switched sequence or for given switched sequence. Literature [9–13] research the stability problem for systems, which subject to some stochastic parameters. In these research, the stability criteria and stabilization criteria are obtained by utilizing the switching technique. Based on this technique, the obtained results can guarantee system’s mean-square exponentially stable. In [10], the interval time-delay system’s stability problem is studied. The time-delay is divided into several parts, by allowing the derivative of Lyapunov function have the positive upper bounds and negative upper bounds for system’s different value of time delay, delay-dependent stability criteria are derived inspired by the switching method. The main idea is that since the system may be unstable for its time delay belong to a big time interval, for the case it belong to a small time interval, the system may be stable. However, this method is different with traditional method which used to cope with the stability problem of time-delay systems. By using intermittent control method, the synchronization problem for chaotic neural networks is investigated in [11], in

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which for every control period there are two switches changed alternatively, and the controller is periodic activated. System’s synchronization is achieved under the designed control law. Literature [12] studies the problem of synchronization of complex-variable delayed chaotic systems, by using Lyapunov functional method and comparison theorem, the synchronization criteria are obtained in case of the two systems with discontinuous coupling and have switch-off phenomenon for sometimes. However, in some case, there were few research about the synchronization problem for DNNs subject to the existence of RONs by using the switching method in the article before, which motivated us to research this problem. Motivated by the above discussion, in this article, we do some research on the synchronization problem for time delayed neural networks. The system is assumed to have some unreliable communication links with it. These links are modeled as stochastic variables with Bernouli distribution. Based on Lyapunov functional method and switching technique, sufficient conditions are established by considering the probability of the missing data. During the time interval that the controller is effective, we let the derivative of Lyapunov function have a negative upper bound. While when the data is missing, which means the controller’s signal is lost, we let the derivative of LKF have a positive upper bound. The systems then can be seen as a switch system and we cope with it by using switch technics. The upper bounds for both cases is choose for getting the controller’s gain matrix. The obtained criterion is in the form of LMIs with the given parameters in it, and by solving it through MATLAB LMI Toobox the gain of the controller is obtained. With the designed controller, the error system’s globally asymptotic stable can be guaranteed. Finally, we give two examples to show the feasibility and effectiveness of our proposed control method. Notation: Throughout this paper, diag{· · ·} denotes a block-diagonal matrix. I stands for identity matrix, and 0 are is zero matrix. The positive definite matrix P is denoted as P > 0. ∗ means symmetric block in symmetric matrices. · is the Euclidean norm. The superscripts T is matrix transposition. The superscripts −1 is matrix inverse.

2

Model Foundation and Preliminaries

In this section, the delayed neural networks that we considered is show as follows: x(t) ˙ = Cx(t) + W1 f (x(t)) + W2 f (x(t − h)), x(t) = φ(t), −h ≤ t ≤ 0.

(1)

where x(t) ∈ Rn is the neuron’s state vector. f (x(t)) = [f1 (x1 (t)), f2 (x2 (t)), · · · , fn (xn (t))]T is a continuous vector function, which denotes the neuron activation function. It is bounded, with initial condition f (0) = 0, and for i = 1, 2, · · · , n, it satisfies the following condition: ki− ≤

fi (σ1 ) − fi (σ2 ) ≤ ki+ , ∀σ1 , σ2 ∈ R, σ1 = σ2 . σ1 − σ2

(2)

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in which ki+ and ki− are constant scalars. The diagonal matrix C satisfies C = diag{c1 , c2 , · · · , cn } < 0. W1 = [w1ij ]n×n is the connection weight matrix, and W2 = [w2ij ]n×n represents the delayed connection weight matrix. φ(t) is system’s initial condition. h > 0 presents system’s time delay. In this paper, the DNNs (1) is considered as the drive system, to synchronize with it, the response system under controlled is as follows: y(t) ˙ = Cy(t) + W1 f (y(t)) + W2 f (y(t − h)) + u(t), y(t) = ϕ(t), −h ≤ t ≤ 0.

(3)

where y(t) ∈ Rn is the corresponding response state vector, u(t) is the control input vectors to be designed, and ϕ(t) is the response system’s initial condition. As we all known, in many practical systems, there exist the unreliable communication links. In this paper, the data loss phenomena are modeled by using the stochastic approach. We assume the state feedback controller u(t) in (3) is affected by some randomly occurred perturbations and we define it as u(t) = γ(t)K(y(t) − x(t)),

(4)

where K ∈ Rn×n is the controller’s gain matrix. The stochastic variable γ(t) in (4), represents the probability of system’s missing data. In this paper, we assume that the variable γ(t) has the type of Bernoulli distribution, and it takes values on zero and one. The variable γ(t) = 0 denotes the controller data is missing and γ(t) = 1 when the data is received, it satisfies Pr{γ(t) = 1} = γ, Pr{γ(t) = 0} = 1 − γ, γ ∈ [0, 1].

(5)

Defining the synchronization error as δ(t) = y(t) − x(t). Then From drive system (1) and response system (3) together with (4) we derive the following closed-loop error system: ˙ = (C + γ(t)K)δ(t) + W1 f˜(δ(t)) + W2 f˜(δ(t − h)), δ(t) δ(t) = ϕ(t) − φ(t), −h ≤ t ≤ 0.

(6)

where f˜(δ(t)) = f (y(t)) − f (x(t)), f˜(δ(t − h)) = f (y(t − h)) − f (x(t − h)). According to (2) we can see that, the nonlinear function f˜i (δi (t)) satisfies some inequalities and it can be rewritten as follows: ki− ≤

f˜i (δi (t)) ≤ ki+ , δi (t)

(i = 1, 2, · · · , n).

(7)

In this paper, we aim to design the synchronization controller (4) for error system (6) under the constraint of (4–5). The designation of controller is obtained by considering the properties of the stochastic variables and using Lyapunov functional method. Under the presence of missing data, the synchronization of systems (1) and (3) can be achieved with our designed controller, as well as the error system (6) can achieve asymptotically stable, which means, δ(t) → 0, for t → +∞.

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Before stating our main results, the following lemma and assumption will be useful in the derivation of our synchronization criterion, and we give them below: Lemma 1 ([1]). Suppose that x = [x1 , x2 , · · · , xn ]T ∈ Rn , S = diag{s1 , s2 , · · · , sn } stand for a positive-semidefinite diagonal matrix, G(x) = [g1 (x1 ), g2 (x2 ), · · · , gn (xn )]T represents a continuous nonlinear function, which satisfies li− ≤

gi (x) ≤ li+ , x = 0, x ∈ R, i = 1, 2, · · · , n. x

in which, li+ and li− are constant scalars. Then 

x G(x)

T 

−ST1 ST2 ∗ −S



 x ≥ 0, G(x)

where l+ +l− l+ +l− l+ +l− T1 = diag{l1+ l1− , l2+ l2− , · · · , ln+ ln− }, T2 = diag{ 1 2 1 , 2 2 2 , · · · , n 2 n }. Assumption 1. We define the time sequence t0 < t1 < · · · < ti < · · · (i ∈ N ) as the switching instants. For each time instants, the transmission data change its state alternatively. We suppose that the transmission data is missing and received at every instant alternatively. For simplicity, we assume that in time interval [t2i−1 , t2i ], i = 1, 2, · · · ,the transmission is successful. i.e., the controller starts receiving data at instances t1 , t3 , · · · , t2i+1 , · · · . We also suppose that, there exist a minimum value for the time length between all the time intervals t2i−1 and t2i+1 , and we denote it as T , which means, t2i+1 − t2i−1 ≥ T, ∀i ∈ N + .

3

Main Results

The asymptotic stability criterion is derived in this section for the error dynamic system (6) based on Lyapunov theory and switching techniques. For simplicity, in the following, we denote ξ(t) = [δ T (t) δ T (t − h) f˜T (δ(t)) f˜T (δ(t − h))]T , k+ +k−

k+ +k−

− k+ +kn

Γ1 = diag{k1+ k1− , k2+ k2− , · · · , kn+ kn− }, Γ2 = diag{ 1 2 1 , 2 2 2 , · · · , n 2 We now state the following result for the error dynamic system (6).

}.

Theorem 1. For given scalars γ > 0 and h > 0, the closed-loop error system (6) is asymptotically stable, if for choosing parameters μ1 > 1, α > 0, β > 0, and T¯ > 0, the inequality (11) is hold, and there exist positive definite matrices Pi , Qi , Ri , positive diagonal matrices Ui , Vi (i = 1, 2), and matrices W , such that the following LMIs (8–10) hold. ⎡ ⎤ Λ1 0 P1 W1 + U1 Γ2 P 1 W2 ⎢ ∗ −eβh Q1 − V1 Γ1 ⎥ 0 V 1 Γ2 ⎥ < 0, (8) Σ1 = ⎢ ⎣ ∗ ⎦ ∗ R1 − U1 0 βh ∗ ∗ ∗ −e R1 − V1

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⎡

⎤ Λ2 0 P2 W1 + U2 Γ2 P 2 W2 ⎢ ∗ −e−αh Q2 − V2 Γ1 ⎥ 0 V 2 Γ2 ⎥ < 0, Σ2 = ⎢ ⎣ ∗ ⎦ ∗ R2 − U2 0 −αh ∗ ∗ ∗ −e R2 − V2

(9)

Pi ≤ μ1 Pj , Qi ≤ μ1 Qj , Ri ≤ μ1 Rj , ∀i, j ∈ {1, 2},

(10)

1

μ T eβ¯γ −αγ < 1.

(11)

where Λ1 = P1 C + C T P1 + Q1 − βP1 − U1 Γ1 , Λ2 = P2 C + C T P2 + W + W T + Q2 + αP2 − U2 Γ1 , μ2 = e(α+β)h , μ = μ21 μ2 , γ¯ = 1 − γ. Moreover, the feedback gain matrix is given by K = P2−1 W . Proof. For the error dynamic system (6) and any t ≥ 0, we construct the Lyapunov-Krasovskii functionals (LKF) as follows:

t eβ(t−s) δ T (s)Q1 δ(s)ds V1 (t) = δ T (t)P1 δ(t) + t−h (12)

t β(t−s) ˜T ˜ e + f (δ(s))R1 f (δ(s))ds, for γ(t) = 0, t−h

V2 (t) = δ T (t)P2 δ(t) +

t

+ t−h



t

t−h

e−α(t−s) δ T (s)Q2 δ(s)ds

−α(t−s) ˜T

e

f (δ(s))R2 f˜(δ(s))ds,

(13) for γ(t) = 1.

Differentiating the LKF (12–13), we obtain ˙ − βδ T (t)P1 δ(t) + δ T (t)Q1 δ(t) − eβh δ T (t − h)Q1 δ(t − h) V˙ 1 (t) = 2δ T (t)P1 δ(t) T + f˜ (δ(t))R1 f˜(δ(t)) − eβh f˜T (δ(t − h))R1 f˜(δ(t − h)) + βV1 (t), (14)

˙ + αδ T (t)P2 δ(t) + δ T (t)Q2 δ(t) − e−αh δ T (t − h)Q2 δ(t − h) V˙ 2 (t) = 2δ T (t)P2 δ(t) + f˜T (δ(t))R2 f˜(δ(t)) − e−αh f˜T (δ(t − h))R2 f˜(δ(t − h)) − αV2 (t). (15)

According to Lemma 1, for i = 1, 2, there exist positive diagonal matrices Ui and Vi , the sector constrains (7) are equal to the following inequalities (16–17):    T  δ(t) δ(t) −Ui Γ1 Ui Γ2 ≥ 0, (16) ∗ −Ui f˜(δ(t)) f˜(δ(t))

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and 

δ(t − h) f˜(δ(t − h))

T 

−Vi Γ1 Vi Γ2 ∗ −Vi



 δ(t − h) ≥ 0. f˜(δ(t − h))

(17)

For the case when γ(t) = 0, which means that the system is without control. In this situation we can allow the derivative of LKF have a positive upper bound. From (14) together with (16–17) we can get the following result: V˙ 1 (t) − βV1 (t) ≤ ξ T (t)Σ1 ξ(t),

(18)

while for the case when γ(t) = 1, which means that the system is under control. In this situation we can allow the derivative of LKF have a negative upper bound. From (15)–(17) we have the following result by setting P2 K = W : V˙ 2 (t) + αV2 (t) ≤ ξ T (t)Σ2 ξ(t),

(19)

Notice that we have the following relation from inequalities (10) and (12–13) V2 (t) ≤ μ1 V1 (t), V1 (t) ≤ μ1 μ2 V2 (t). Suppose that t ∈ [t2n−1 , t2n ], then together with above inequalities we can get the following result: V2 (t) < e−α(t−t2n−1 ) V2 (t2n−1 ) < μ1 e−α(t−t2n−1 ) V1 (t2n−1 ) < μ1 e−α(t−t2n−1 ) eβ(t2n−1 −t2n−2 ) V1 (t2n−2 ) < μ1 (μ1 μ2 )e−α(t−t2n−1 ) eβ(t2n−1 −t2n−2 ) V2 (t2n−2 ) < · · · < (μ21 μ2 )n e−αT1 +βT0 V2 (t0 ), where Ti denotes the time intervals’ total length for variable γ(t) = i, (i = 0, 1). Notice that from (5) we can conclude that lim TT01 = 1−γ γ , thus for given t→+∞

scalar > 0, there ∃ T˜ > t0 , such that T0 1−γ + , for t > T˜. < T1 γ Since T0 + T1 = t − t0 , we have T0
(t − t0 ), 1+ 1+

thus we can get

β¯ γ − αγ + βγ . 1+ By Assumption 1 we have n ≤ (t − t0 )/T , therefore, we can conclude that −αT1 + βT0
m, but the top speed of the pursuers is slower than the top speed of the evaders, that is v p max < vemax . The pursuit-evasion model is described in Fig. 1. The classical control strategy is defined as

Fig. 1. Pursuit-evasion model

⎧ ⎪ ⎨−μi max μi = σi ⎪ ⎩ μi max where

σi < −μi max −μi max ≤ σi ≤ μi max σi > μi max

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ye − y p ) − θi xe − x p Our approach is to let agents learn control strategy itself.

σi = tan−1 (

4 Main Result In this section, we mainly introduce our main research contents of this paper. To test the performance of multi-agent deep reinforcement learning framework, we implement our method in the pursuit-evasion game. In this game, several slow-moving pursuers work together to capture the quick-moving evaders in a limited space. Once the pursuers capture one of the evaders, they will get a reward, and the evaders will be punished. Whichever run out of the limited space, they will be punished. We hope that the pursuers can capture the evaders as soon as possible. Pursuers. Pursuers have a slower speed but the number of pursuers is more than the evaders, so, they can capture the evader via cooperation. Here, agents are pursuers. We define the state of each agent as its partial observation of environment, which is denoted as si , Based on the state information si , each agent has its own policy μi , according to the policy μi (ai |si ), the agent takes ai and get the reward ri at each time step. State si : The state of each agent include its own position Pi = (xi , yi ), other agents position {Pm , ..., Pn }, the angle θi of the direction between pursuer i and the evaders,and the position Pe = (xe , ye ) of the evaders, the environment obstacles’s position Po = (xo , yo ). Then si = {Pi , Pm , ..., Pn , θi , Pe , Po }. Action ai : The action of each agent include the self accelerated velocity, the direction where the agent should go, which is a angle range in [−π , π ]. Evaders. We consider the evader is part of the environment, which has a fast speed, and can move in a random direction. 4.1 Reward Function In our method, each agent has its individual reward function. The reward setting is based on the distance between the pursuer and the evader. The reward ri is defined as ⎧  ⎪ −0.1 (xi − xe )2 + (yi − ye )2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨individual reward(i) + global reward ri = global reward ⎪ ⎪ ⎪−10 ⎪ ⎪ ⎪ ⎩ −10

not capture capture by agent i capture by other agent run out of limited space collide with each other or obstacles

By combining individual reward and global reward, the agent can maximize both individual reward and team reward, which avoids that some agents can make a profit even it does not work when there is only a global reward, or only maximize individual reward and ignore team reward when there is only global reward. Both cases can’t complete task well. As we defined an individual reward function for each agent, so when the agents are heterogeneous, we can also solve the pursuer-evader game problem.

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Bi Directional Recurrent Neural Network

The inputs of Q network of each agent are the states and actions of all the agents. When new agents join the game or there are some agents exit, the dimension of input will be not the same. Aim at this situation, how to address scalability problem is very important. One naive way is re-training the whole network. However, re-training the whole network is computational and time cosuming. Here we resort to the Bi directional recurrent neural network, which have the properties with dynamic inputs and dynamic outputs. With this method, when there are agents that are playing in the game have appeared malfunction or new agents join the game, it can be solved timely. Becouse that one feature of RNN network is parameters sharing, different agents can share the same parameters. So the number of agents is independent of the number of parameters.

Fig. 2. Policy network

Fig. 3. Q network

The whole network consists of a multi-agent policy network and a Q network as illustrated in Figs. 2 and 3, both of the policy network and the Q-network are based on the Bi-directional RNN structure, the policy network, which takes in a individual state, outputs the action for corresponding agent. As the Bi-RNN structure is bidirectional, so the agents can communicate with each other, thereby which can solve the problem of communication between agents and share the information with its cooperative partners. 4.3

Multi-agent Policy Learning and Q Function Learning

We define the objective function of single agent i as Ji (θ ), and Ji (θ ) = ξ Es∼D [Qi (s, μ θ (s))]. It’s same to the objective function introduced in Eq. (1), where each agent aims to maximize their individual Q value function. We try to maximize the mean Q value of all the pursuers. That is, our goal is to maximize n

ξ

J(θ ) = Es∼D [ ∑ Qi (s, μ θ (s))]

(2)

i=1

where θ is the policy network parameters, ξ is the Q network parameters, n is the number of pursuers and set D is the previous experiences which consists of tuples  {s, {ai }ni=1 , {ri }ni=1 , s }. In our method, policy network and is used to learn the optimal policy for each agent, and Q network is used to criticize the policy. At time step t, the Q network loss function is as follows: n

ξ

L(ξ ) = ∑ (yˆti − Qi (st , at ))2

(3)

i=1

yˆti

= ri (st , μ

θ

ξ (st )) + γ Qi



(st+1 , μiθ



(st+1 ))

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where the ξ is the target Q network parameters, the θ is the target policy network parameters, and the rit is the reward of agent i at time step t. We use the policy gradient to optimize the policy network parameters, the object function is defined as Eq. (2). The gradients can be written as: n

n

i

j

ξ

∇θ J(θ ) = Es∼D [∑ ∑ ∇θ μ θj (s).∇a j Qi (s, μ θ (s))]

(4)

To ensure agent can exploration, in every time step, we add Ornstein-Uhlenbeck process noise to the output of policy network. Same as policy learning, we apply Q function gradient to train the Q network, Based on the Q network loss function which is defined in Eq. (3), we can calculate the gradient is as follows: n

ξ





∇ξ L(ξ ) = Es∼D [∑(ri (st , μ θ (st )) + γ Qi (st+1 , μ θ (st+1 )) i ξ ξ θ − Qi (st , μ (st ))).∇ξ Qi (st , μ θ (st ))]

(5)

4.4 Multi-agent Deep Reinforcement Learning Algorithm Specification of the learning algorithm is given as in Algorithm 1 (see the next page).

5 Experiments 5.1 Experiment Settings To verify the performance of our method, we first implement it in the condition where three pursuers work together to catch the evader. The environment can reference Fig. 4. The mean pursuers reward is shown in Fig. 5. We can see that the reward increases and then convergence in our algorithm, it’s show that our algorithm is reasonable and the slow pursuers can work together to capture the fast evader in a limited space. To test the scalability of our method, the experiments setting is that: (1) First, the pursuit-evasion game is that three pursuers to capture an evader, when the trained pursuers can capture the evader, then two more pursuers join the game at 25000 episodes. We compare our result with retraining the whole network and the comparison result is showing at Fig. 6, we can see that the mean reward increases after two new pursuers join the game. In addition, our method converges quicker and has a higher reward than retraining the network. As the reward is designed as negative distance between the pursuers and the evader, it means that the reward is larger, the performance is better. From the experimental result, we can see that our method has a good scalability.

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Algorithm 1. Initial Q network and policy network with ξ and θ   Initial Q and policy target network with ξ ← ξ and θ ← θ Initial replay buffer D for episode=1 to E do initial a random process ϑ for action exploration receive initial observation state s1 for time step t=1 to T do for each agent i, select and execute action ai,t = μiθ (st ) + Nt receive reward [ri,t ]N i=1 and observe new state st+1 N store transition (st , [ai,t ]N i=1 , [ri,t ]i=1 , st+1 ) in D N sample a random minibatch of M transitions (sm,t , [am,i,t ]N i=1 , [rm,i,t ]i=1 , sm,t+1 ) from D compute target value for each agent in each transition using the recurrent neural network compute Q gradient according to Eq. (5)

ξ =

  1 M N ξ [(rm,i + λ Qm,i (sm,t+1 , μ θ (sm,t+1 )) ∑ ∑ M m=1 i=1

ξ

ξ

−Qm,i (sm , μ θ (sm ))).∇ξ Qm,i (sm , μ θ (sm ))] compute policy gradient estimation according to Eq. (4) and replace Q value with the Q network estimation θ =

1 M N N ξ ∑ ∑ ∑ [∇θ aθj (sm ).∇a j Qm,i (sm , μ θ (sm ))] M m=1 i=1 j=1

update the network based on Adam using the above gradient estimators update the target networks:     ξ ← γξ + (1 − γ )ξ , θ ← γθ + (1 − γ )θ end end

(2) In order to better verify the scalability of our algorithm, we set another experiment. First, five pursuers capture the evader, then two pursuers exit the game at 25000 episodes, the experimental result is showing at Fig. 7. It is also found that the reward can converge rapidly.

Multi-agent Deep Reinforcement Learning

Fig. 4. Three pursuers work together to capture an evader in a limited space, the magenta are the pursuers, the blue is the evader and the black is obstacle

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Fig. 5. Three pursuers work together to capture an evader, the pursuers mean reward

5.2 Parameter Tuning In our training, we set the Adam [26] as optimizer with learning rate equal to 0.001, and other parameters are the default values. The max episode is 60k, the max time step of each episode is 1000; The batch size is 64, the experience replay buffer max size is 10k, when the episode is reach 1000 episodes (more than 1000k time steps), we start training the network.

Fig. 6. Three pursuers capture the evader at beginning and two new pursuers join the game at 25000 episodes

Fig. 7. Five pursuers capture the evader at beginning and two pursuers exit the game at 25000 episodes

6 Conclusions In this paper, we proposed a method to solve the scalability of multi-agent deep reinforcement learning in pursuer-evader game, which based on the deep deterministic policy gradient framework and the bi-directional RNN. Both the policy and Q network

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employ Bi-RNN. The communication for information is done by Bi-RNN in the internal layers. In addition, the parameters in Bi-RNN structure is shared, so when there are new agents join or exit, it can use the previous parameters to work. In our method, we design individual reward function for each agent, this increases the flexibility in pursuer-evader game, no matter the agents are homogeneous or heterogeneous. In the experimental part, we can see that this method is suitable in pursuer-evader game.

References 1. Soga S, Kobayashi I (2013) A study on the efficiency of learning a robot controller in various environments. In: 2013 IEEE symposium on adaptive dynamic programming and reinforcement learning (ADPRL), pp 164–169 2. Awheda MD, Schwartz HM (2016) A fuzzy reinforcement learning algorithm using a predictor for pursuit-evasion games. In: 2016 annual IEEE systems conference (SysCon). IEEE, pp 1–8 3. Schwartz HM, Howard M (2014) Multi-agent machine learning: a reinforcement approach. Wiley Publishing, Hoboken, pp 144–199 4. Jouffe L (1998) Fuzzy inference system learning by reinforcement methods. IEEE Trans Syst Man Cybern 28(3):338–355 5. Desouky SF, Schwartz HM (2011) Q (λ )-learning adaptive fuzzy logic controllers for pursuit-evasion differential games. Int J Adapt Control Signal Process 25(10):910–927 6. Awheda MD, Schwartz HM (2015) The residual gradient FACL algorithm for differential games. In: 2015 IEEE 28th Canadian conference on electrical and computer engineering (CCECE). IEEE, pp 1006–1011 7. Mnih V, Kavukcuoglu K, Silver D (2015) Human-level control through deep reinforcement learning. Nature 518(7540):529–533 8. Silver D, Huang A (2016) Mastering the game of go with deep neural networks and tree search. Nature 529(7587):484–489 9. Silver D, Schrittwieser J, Simonyan K (2017) Mastering the game of go without human knowledge. Nature 550(7676):354–359 10. Levine S, Finn C, Darrell T (2016) End-to-end training of deep visuomotor policies. J Mach Learn Res 17(1):1334–1373 11. Mao H, Alizadeh M, Menache I (2016) Resource management with deep reinforcement learning. In: Proceedings of the 15th ACM workshop on hot topics in networks. ACM, pp 50–56 12. Jaques N, Gu S, Turner RE (2017) Tuning recurrent neural networks with reinforcement learning. In: Proceedings of the 34th international conference on machine learning 13. Tan M (1993) Multi-agent reinforcement learning: independent vs. cooperative agents. In: Proceedings of the tenth international conference on machine learning, pp 330–337 14. Enright JJ, Wurman PR (2011) Optimization and coordinated autonomy in mobile fulfillment systems. In: Workshops at the twenty-fifth AAAI conference on artificial intelligence 15. Stephan J, Fink J, Kumar V (2017) Concurrent control of mobility and communication in multirobot systems. IEEE Trans Robot 33(5):1248–1254 16. Foerster JN, Farquhar G, Afouras T (2018) Counterfactual multi-agent policy gradients. In: Thirty-second AAAI conference on artificial intelligence 17. Lowe R, Wu Y, Tamar A, Harb J, Abbeel OP, Mordatch I (2017) Multi-agent actor-critic for mixed cooperative-competitive environments. In: Advances in neural information processing systems, pp 6382–6393

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18. Littman ML (1994) Markov games as a framework for multi-agent reinforcement learning. In: Machine learning proceedings 1994, pp 157–163 19. Bilgin AT, Kadioglu UE (2015) An approach to multi-agent pursuit evasion games using reinforcement learning. In: 2015 international conference on advanced robotics (ICAR). IEEE, pp 164—169 20. Foerster J, Assael YM, Freitas N (2016) Learning to communicate with deep multi-agent reinforcement learning. In: Advances in neural information processing systems, pp 2137– 2145 21. Khan A, Zhang C, Lee DD (2018) Scalable centralized deep multi-agent reinforcement learning via policy gradients. arXiv preprint arXiv 22. Lillicrap TP, Timothy P (2015) Continuous control with deep reinforcement learning. Comput Sci 8(6):187 23. Foerster J, Assael IA, Freitas N (2016) Learning to communicate with deep multi-agent reinforcement learning. In: Advances in neural information processing systems, pp 2137– 2145 24. Tesauro G (2004) Extending q-learning to general adaptive multi-agent systems. In: Advances in neural information processing systems, pp 871–878 25. Silver D, Lever G, Heess N (2014) Deterministic policy gradient algorithms. In: International conference on international conference on machine learning, ICML, pp 387–395 26. Kingma DP, Ba J (2015) Adam: a method for Stochastic Optimization. In: 3rd international conference for learning representations, San Diego

Event-Triggered Adaptive Path-Following Control for Micro Helicopter Mingrun Bai1 , Ming Zhu1(B) , Tian Chen1 , and Zewei Zheng2 1

2

School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, People’s Republic of China [email protected], {zhuming,chen tian}@buaa.edu.cn Seventh Research Division, School of Automation Science and Electrical Engineering, Beijing 100191, People’s Republic of China [email protected]

Abstract. This paper references a kind of path-following problem of a micro helicopter, which bases on the theories of vector field guidance method, adaptive sliding mode control, and event-triggered control. Firstly, to satisfy the requirements of aircraft position tracking, a method for accurate vector field guidance method is present. Secondly, in order to track the desired attitude which is calculated by the yaw after the vector field guidance method with disturbances, an adaptive sliding mode attitude controller is designed. Thirdly, to maintain the desired velocity, an adaptive velocity controller is introduced. Moreover, an event-triggered control method optimizes the controller’s outputting frequency. Stability analysis indicates that all variables are ultimately bounded. At the same time, the simulation results demonstrate the effectiveness of the mentioned control algorithm. Keywords: Micro helicopter · Path-following · Vector field · Sliding mode · Event-trigger

1 Introduction Unmanned micro helicopter is a kind of aircraft which can achieve vertical taking-off, hovering and low-speed flights. Compared with manned aircraft, it has the advantages of low cost, small size, convenient use, strong survivability and low requirements for combat environment. It is a task for micro helicopter to complete the reconnaissance mission and track the path autonomously in the complicated and variable battlefield. While, flight targets are difficult to be obtained at a narrow space. As a result, it’s important to choose a proper guidance method. On the other hand, the batteries which are loaded by Micro helicopters are limited due to size and weight limitations. So, we may design a controller to reduce the frequent changes in actuation system in order to reduce the energy consuming. In [1], an integrated method for developing guidance for autonomous vehicle trajectory following is described. [2] proposes a path tracking method for UAV that provides c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 670–680, 2020. https://doi.org/10.1007/978-981-32-9682-4_70

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a constant line of sight between the UAV and the observation target. The required position is constantly moving due to the high-speed of aircraft. If the interference, such as wind, is considered, the method of tracking the moving point may cause a serious problem. The work proposed in [3] builds on the path concept by constructing a vector field around the path to follow. A vector field around a predefined path is proposed by the VF guidance method, where the directions of the vector field are the direction of flight. The feature of VF guidance method is the connection between the outer loop (guidance) and the inner loop (attitude), which can control the flight path by stabilizing the posture. It is suitable for path tracking of micro helicopters. As an event-driven and aperiodic control signal update strategy, event-triggered control mechanism can reduce the communication burdens [12], effectively. It is triggered whenever a state correlated or output related criterion is met. In [13], an eventtriggered suboptimal tracking control method is proposed to deal with a class of nonlinear discrete-time systems. Indeed, less attention has been paid to the development of event-triggered control algorithms for the tracking problems especially in nonlinear systems. Therefore, combining event-triggered controller with nonlinear helicopter model can be a new innovation. Firstly, We simplify the dynamic model and calculate the desired attitude in three dimensions of the micro helicopter. Then, we choose a vector field guidance method, use the adaptive sliding mode control to stabilize the attitude, and adaptive control to control the velocity. Based on the above control methods, the event-triggered controller is designed to control the output frequency. The rest of the paper is organized as follows. The Sect. 2 shows the modeling of a model-scaled helicopter and the vector field guidance used in this paper. The Sect. 3 introduces the design of the path-following controller, adaptive sliding mode controller and the event-triggered controller. The Sect. 4 performs Lyapunov stability analysis. The Sect. 5 presentes the results in the form of numerical simulations. Final section gives the conclusions.

2 Micro Helicopter Model 2.1 Dynamics of Helicopter As is shown in Fig. 1, the helicopter model is established in the earth reference frame (ERF) and the body reference frame (BRF). Its dynamic equation can be established by the Newton-Euler equation [4]. p˙ = K v Θ˙ = K Ω m˙v = R f − mge3 J Ω˙ = −Ω × J Ω + τ

(1)

where p = [px , py , pz ]T is the position in the ERF; v = [vx , vy , vz ]T is the velocity in the BRF; m is the mass of the micro helicopter; g is the gravity acceleration; Θ = [φ , θ , ψ ]T represents the Euler angle (i.e., roll, pitch, yaw) between BRF and ERF. Ω = [Ωx , Ωy , Ωz ]T is the angular velocity of the helicopter in the body coordinate system,

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f and τ are the acting forces and torques on the helicopter in BRF, respectively, e 3 = [0, 0, 1]T and J = diag{Jx , Jy , Jz } denote the total mass and inertial moment matrix, R is the rotation matrix of the body coordinate system to the earth coordinate system. K is the Euler rotation matrix.

Fig. 1. Helicopter model sketch

2.2

Forces and Torques Simplification

Due to space limitations, according to the literature [4–6], we simplify the micro helicopter force and torques equation as v˙ = −ge3 + τ F /m + τ fd /m ˙ Ω = −J −1 (Ω × J Ω + τ Q + τ qd )

(2)

where τ F and τ Q are controller inputs. Since τ fd and τ qd are small, they are ignored directly in some controller designs. As a result, they are treated as unmodeled dynamics. The nominal states are disturbed by τ fd and τ qd , respectively. 2.3

Vector Field Guidance

This method calculates the vector field around the path to be tracked. The vector in the field points to the path to follow and indicates the desired direction of flight, which is used as the process command for the micro helicopter. Figure 2 indicates an diagrammatic sketch of a vector field for linear and circular paths. From Fig. 2, the tracking error dl of linear path and dc of circular path are expressed in [14]. As a result, we have: d˙ = usin(ψ − ψ p,i ) + vcos(ψ − ψ p,i ) = Usin(ψ − ψ p,i + β )

(3)

√ where U = u2 + v2 , β = arctan(v/u). Figure 2 shows that ψ p,l will be a constant when tracking the linear path, and ψ p,c = arctan(y − y0 , x − x0 ) − π /2 around the circle path.

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Fig. 2. Geometric representation of the vector field guidance method that guides the straight path (left) and the circular path (right).

3 Path-Following Control Method The controller consists of four significant steps, as is present in Fig. 3: the expected yaw angle ψc is calculated from the vector field guidance in the outer loop, which is used to calculate the desired attitude; then the specified attitude is obtained by the secondorder command filter and adaptive sliding-mode attitude control under the disturbances d ω [7, 8]; in order to maintain the proper value of velocity under interference f v , the adaptive velocity controller is designed; event-triggered controller is present to adjust control amount.

Fig. 3. A block diagram of controllers with four control loops: guidance, attitude control, velocity control and event-triggered control.

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3.1

Guidance Law

In this section, to track the desired path, we may obtain the desired yaw ψc by the guidance law, as is shown in Fig. 2. As a result, we set the VV F as: 1 VV F = d 2 2

(4)

And the first-order derivative of VV F is as present: ˙ = Usin(ψ − ψ p + arctan(kd d)) V˙V F = dd

(5)

As d˙ = Usin(ψ − ψ p + β ), the desired yaw is designed as [3]

ψc = ψ p − β + arctan(kd d)

(6)

where kd is a positive constant. Subsequently, using Eq. 5, 6 can be obtained as Ukd V˙V F = −  d2 ≤ 0 1 + (kd d)2 3.2

(7)

Attitude Control

In the former section, we got the heading angle ψc . Then, the main rotor thrust T m , the command roll angle φc , and pitch angle θc are extracted from the controller τ F . According to the literature [10, 11], φc and θc can be obtained by the following equation: ⎧ ux sinψc − uy cosψc π π ⎪ ⎪ ), φc ∈ (− , ) ⎨ φc = arcsin(  τF  2 2 (8) u cosψc + uy sinψc π π ⎪ ⎪ ⎩ θc = arctan( x ), θc ∈ (− , )  τF,z  2 2 According to the real flight situation, the singularity problem when τF,z = 0 is not considered. To track the desired attitude Θ c , an adaptive sliding-mode attitude controller we designed is as follows:  Θ˙ = K Ω (9) Ω˙ = −JJ −1 (Ω × J Ω + τ Q + τ q ) d

Thus, we have:

Θ¨ = K˙ Ω + K Ω˙ = F ω + Bω τ Q + d ω where F ω = K˙ Ω − K J −1 Ω × J Ω , B ω = K J −1 , d ω = K τ qd The sliding surface is shown as: s = Θ˙ e + k ω Θ e

(10)

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where k ω = diag{ks,φ , ks,θ , ks,ψ } is a positive definite diagonal constant matrix, and Θ e = Θ − Θ c . Then, the first derivative of the Eq. 10 is: s˙ = Θ¨ e + k ω Θ˙ e = F ω + k ω Θ˙ + B ω τ Q − (Θ¨ c + k ω Θ˙ c ) + d ω

(11)

Therefore, to obtain the derivatives, a second-order command filter is introduced, which is designed for the convenience of calculations of the first- and second-order derivatives of desired attitude Θ c .  Ξ˙ 1 = Ξ 2 (12) Ξ˙ 2 = −2Λ ω n Ξ 2 − ω 2 (Ξ 1 − Θ c ) n

where ω n = diag{ωn,φ , ωn,θ , ωn,ψ } is the damping frequency, Λ = diag{Λφ , Λθ , Λψ } is the damping ratio. By defining Θˆ˙ c = Ξ 2 , Θˆ¨ c = Ξ˙ 2 , Eq. 11 can be rewritten like s˙ = F ω + k ω Θ˙ + B ω τ Q − (Θˆ¨ c + k ω Θˆ˙ c ) + d ∗ω

(13)

where d ∗ω = d ω − k ω (Θ˙ c − Θˆ˙ c ) − Θ¨ c + Θˆ¨ c . Setting a significantly large ω n and Λ can ensure rapid tracking of the attitude command signal, and the d ∗ω is bounded because of the bounded k ω (Θ˙ c − Θˆ˙ c ) − Θ¨ c , namely  d ∗ω ≤ δd ∗ω . ⎧ s ⎪ k s s − F ω − k ω Θ˙ + k ω Θˆ˙ c + Θ¨ c − δˆd ∗ω ⎨ τ Q = B −1 ) ω (−k s (14) ⎪ ⎩ δ˙ˆ ∗ = k ( s  −k δˆ ∗ ) d,s δω d ω dω where δˆd ∗ω is the estimate of δd ∗ω . k s is positive definite diagonal constant matrices. kd,s , s is the adaptive sliding mode compensator of d ∗ω . kδω , and positive constants. −δˆd ∗ω s ˙ The attitude control inputs and the adaptive control law are τ Q and δˆd ∗ω , respectively. Therefore, we choose the Lyapunov function for the attitude dynamics control loop 1 1 ˜2 δ ∗ VΘ = sT s + 2 2kd,s d ω

(15)

where δ˜d ∗ω = δˆd ∗ω − δd ∗ω . As a result, the first-order derivative of VΘ is as follows: 1 ˜ ˙˜ V˙Θ = s T s˙ + δ ∗δ ∗ kd,s d ω d ω 1 1 ≤ −ssT k s s − kδω δ˜d2∗ + kδω δ 2 d ∗ω ω 2 2

(16)

3.3 Velocity Control In order to ensure that the helicopter tracks the predetermined path as quickly as possible, the velocity control response must be rapid and accurate. Then, to track the required

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velocity and estimate the velocity error v e = v − v c , the velocity controller is designed, in which v c = [uc , vc , wc ]T is the desired velocity, and v = [u, v, w]T is the velocity of micro helicopter in the BRF. The micro helicopter’s velocity model is shown as follow, according to Eq. 2: (17) v˙ = F v + B v τ F + f v τ

f where F v = −gee3 , B v = m1 , f v = md Then, v˙ e = v˙ − v˙ c = F v + B v τ F + d ∗v

(18)

where d ∗v = f v − v˙ c . ⎧ v ⎪ k v v e − F v − δˆdv∗ e ) ⎨ τ F = B −1 v (−k  ve  ⎪ ⎩ δ˙ˆ ∗ = k ( v  −k δˆ ∗ ) e d,v

(19)

δv d v

dv

where δˆd ∗v is the estimation of δd ∗v , kd,v and kδv are positive constants. k v is positive definite diagonal constant matrices. The velocity control inputs is designed as τ F . Then, the adaptive compensator of d ∗v is −δˆd ∗v vvvee  . We choose the Lyapunov function for velocity dynamics control loop as 1 1 ˜2 Vve = v Te v e + δ 2 2kd,v dv

(20)

in which δ˜d v = δˆd v − δd v , and the first-order derivative of Vve is 1 ˜ ˙˜ V˙ve = v Te v˙ e + δ ∗δ ∗ kd,v d v d v = −vvTe k v v e − δˆd ∗v  v e  +vvTe dv∗ + δ˜d ∗v ( v e  −kδv δˆd ∗v ) 1 1 ≤ −vvTe k v v e − kδv δ˜d2∗ + kδv δd2∗ v v 2 2 3.4

(21)

Event-Triggered Control

The event-triggered manner is designed to update the control input for nonlinear systems. Then, the triggering mechanism is designed as [9] w = [τ F , τ Q ]T u = w (tk ), ∀t ∈ [tk ,tk+1 ) tk+1 = inf{t ∈ R || e |≥ kth ,t0 = 0}

(22) (23) (24)

where e = w − v is the measurement error, and tk , k ∈ z+ , is the update time. This means that once Eq. 24 is triggered, the time will be updated as tk+1 , and u (tk+1 ) will be outputted to the system. From Eq. 24, we have | w (t) − u (t) |≤ kth , for t ∈ [tk ,tk+1 ).

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4 Stability Analysis Helicopter models Eqs. 3, 4 under the action of τ F , τ Q input and unknown disturbance d, if the desired yaw satisfies Eq. 6, the controller selects Eqs. 14 and 19, then the following holds: (1) The path tracking and state tracking errors can converge quickly and reach the desired value; (2) All control inputs and state are limited. Proof. The whole Lyapunov function is present as V = VV F +VΘ +Vve 1 1 1 ˜2 1 1 ˜2 = d2 + sT s + δd ∗ + v Te v e + δ ω 2 2 2kd,s 2 2kd,v d v

(25)

From the Eqs. 7, 16, 21, the derivative of V satisfies Ukd 1 1 1 1 V˙ < −  − s T k s s − kδω δ˜2 d ∗ω + kδω δ 2 d ∗ω − v Te k v v e − kδv δ˜d2∗ + kδv δd2∗ v v 2 2 2 2 2 1 + (kd d) (26) Then, we got V˙ ≤ −kV V + Δ (27) where kV = min{− 

Ukd 1 + (kd d)2

, kδω , kδv , kd,s , kd,v , s, ve }

and

1 1 Δ = kδω δ 2 d ∗ω + kδv δd2∗ v 2 2 The event-triggered controller inputs are bounded, because the event-triggered control does not change the range of the control inputs.

5 Simulation In this section, matlab numerical analysis methods are used to verify the adaptive sliding-mode control algorithm. The parameter data of the numerical verification is shown in Table 1. The reference path is a straight line starting from the origin with a slope of 45◦ . The micro helicopter starts with (−5, 10, 0) and the tracking reference path is shown in Fig. 4. The numerical simulation of attitude tracking is shown in Fig. 5. The actual helicopter’s attitude, as shown by the red line, is well tracked to the desired attitude value. It is expected that the pitch and roll will quickly become zero, which means the helicopter attitude can be stabilized quickly. The yaw will stabilize at 45◦ . Figure 6 shows the linear tracking error can converge to a small neighborhood around zero. The numerical simulation of velocity tracking is shown in Fig. 7. u, v, and w can quickly stabilized at the desired velocity, which indicates that the velocity controller has good tracking performance. This article only considers the two-dimensional pathfollowing problem, so the z direction is not discussed.

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7.4(kg) 9.81(m/s2 ) 0.16(kgm2 ) 0.3(kgm2 ) 0.32(kgm2 ) 0.05(kgm2 ) 0.78 250 diag{5.8, 5.8, 5.8} diag{1.8, 1.8, 1.8}

kd kdv kδω kδv kth vc kv ωn kω

0.2 2 5.8 4 0.5 [1, 1, 0]T diag{1.7, 1.7, 1.7} diag{2.8, 2.8, 2.8} diag{5.8, 5.8, 5.8}

Fig. 4. Trajectory of micro helicopter pathfollowing [x, y]

Fig. 5. Expectation and actual posture for roll (top), pitch (middle), yaw (bottom)

Fig. 6. Position error of the helicopter to the desired path

Fig. 7. Expectation and actual velocity for x (top), y (middle), z (bottom) directions

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Fig. 8. Velocity controller input in three directions, τF

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Fig. 9. Attitude controller input in three directions, τQ

The adaptive controller compensates for external disturbances, control inputs and filter estimation errors. The event-triggered control results are shown in Figs. 8 and 9. Under the action of the event-triggered controller, the input control amount will exhibit a gradient change. Event-triggered control can also achieve the desired control performance and reduce the number of actuator changing actions. As a result, the above control method demonstrates that the micro helicopter may track the determined path by inputting an appropriate amount of control.

6 Conclusions Event-triggered adaptive path-following control has proven to follow the desired path of micro helicopters. Based on the vector field guidance, the sliding mode adaptive control can track the variables of the vector field guidance and predict the unmodeled dynamics, precisely. The event-triggered controller can reduce the number of actuator changing actions and reduce energy waste, effectively. However, due to the small magnitude of the model, the changes in the controllers are relatively small, and it is difficult to achieve a one hundred percent tracking effect. The precise tracking that follows will be the focus of research.

References 1. Kaminer I, Pascoal A, Hallberg E, Silvestre C (1998) Trajectory tracking for autonomous vehicles: an integrated approach to guidance and control. AIAA J Guidance Control Dyn 21(1):29–38 2. Rysdyk R (2003) UAV path following for constant line-of-sight. In: Proceedings of the 2nd AIAA unmanned unlimited conference, AIAA-2003-6626 3. Nelson DR, Barber DB, McLain TW, Beard RW (2007) Vector field path following for miniature air vehicles. IEEE Trans Robot 23(3):519–529

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4. Zhu B, Huo W (2014) 3D path-following control for a model-scaled autonomous helicopter. IEEE Trans Control Syst Technol 22(5):1927–1934 5. Ahmed B, Pota HR, Garratt M (2010) Flight control of a rotary wing UAV using backstepping. Int J Robust Nonlinear Control 20(6):639–658 6. Bogdanov A, Wan EA (2007) State-dependent Riccati equation control for small autonomous helicopters. J Guid Control Dyn 30(1):47–60 7. Rahmani M, Ghanbari A, Ettefagh MM (2016) Hybrid neural network fraction integral terminal sliding mode control of an Inchworm robot manipulator. Mech Syst Signal Process 80:117–136 8. Su Z, Wang H, Li N, Yu Y, Wu J (2018) Exact docking flight controller for autonomous aerial refueling with back-stepping based high order sliding mode. Mech Syst Signal Process 101:338–360 9. Zheng Z, Lau G-K, Xie L (2018) Event-triggered control for a saturated nonlinear system with prescribed performance and finite-time convergence. Int J Robust Nonlinear Control 28:5312–5325 10. Zuo ZY, Wang CL (2014) Adaptive trajectory tracking control of output constrained multirotors systems. IET Control Theory Appl 8(13):1163–1174 11. Zuo ZY (2011) Adaptive trajectory tracking control of a quadrotor unmanned aircraft. In: 30th Chinese control conference, pp 2435–2439 12. Dong L, Zhong X, Sun C (2017) Adaptive event-triggered control based on heuristic dynamic programming for nonlinear discrete-time systems. IEEE Trans Neural Netw Learn Syst 28(7):1594 13. Garcia E, Antsaklis PJ, Montestruque LA (2014) Model-based control of networked systems. Springer, New York 14. Chen T, Zhu M, Zheng Z (2019) Asymmetric error-constrained path-following control of a stratospheric airship with disturbances and actuator saturation. Mech Syst Signal Process 119:501–522

Robust SLAM Algorithm in Dynamic Environment Using Optical Flow Yiying Ma and Yingmin Jia(B) School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China [email protected]

Abstract. Visual SLAM(Simultaneous Localization and Mapping) is one of the hottest research areas nowadays. Most of the SLAM methods assume that the scene is stationary. It is difficult to face the factual complex environment. To solve this problem, an improved RGB-D SLAM algorithm combined with Optical Flow and RANSAC (Random Sample Consensus) was proposed in this paper. Optical flow is used to detect moving objects in the scene. RANSAC is used to calculate the homography matrix and optimize matching results. We used a common data set for experimental verification. The results show that this algorithm can effectively detect dynamic objects, improve the accuracy of visual odometry and the robustness of the system. Keywords: SLAM

1

· Dynamic environment · Optical Flow · RANSAC

Introduction

Visual SLAM is one of the hot research fields, which has been extensively used in mobile robots, autonomous driving, and virtual reality. The signature achievement of visual SLAM is MonoSLAM proposed in 2007 [1], which used Extended Kalman Filter (EKF) as the back-end to track sparse feature points in the frontend. PTAM divided positioning and construction into two independent threads, used nonlinear optimization in the back-end [2]. ORB-SLAM, inherited from PTAM, divided the whole system into three threads: tracking, local mapping, and loop-closiong [3]. It is currently the popular open source SLAM system. Most of the SLAM Algorithms assumed that the scene is static and dynamic change only caused by the motion of the camera. The performance of the traditional visual SLAM method in the dynamic scene is difficult to meet the need of robots for precise positioning and mapping. It is necessary to add processing to moving objects. For the above problems, the existing methods can be divided into three categories. The first category mainly optimizes the SLAM system by detecting and tracking dynamic targets through image processing techniques. Tan et al. proposed a novel prior-based adaptive RANSAC to efficiently remove outliers [4], and Lee et al. proposed an algorithm based on Constrained multikernel (CMK) to detect and track dynamic objects [5]. Besides, multi-sensor c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 681–689, 2020. https://doi.org/10.1007/978-981-32-9682-4_71

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information complementarity can be utilized to realize dynamic object detection. CoSLAM using camera poses estimation of multiple cameras to deal with dynamic objects [6]. Hu et al. proposed a dynamic monitoring method based on feature reprojection by integrating cameras with IMU [7]. With the rapid development of deep learning, more and more researchers apply semantic segmentation to the SLAM. For example, MaskFusion introduces instance meaning segmentation in the SLAM technology, which can achieve the recognition, tracking, and reconstruction of multiple moving objects [8]. The main contributions of this paper are as follows: 1. Combining the moving object detection method based on optical flow with camera pose estimation, effectively solve the problem of accurate camera positioning in dynamic environment. 2. Some new rules have been formulated for keyframes selection in dynamic environments to prevent large changes in camera pose estimation after removing dynamic object interference. Validate the effectiveness of the method on an open source data set [9]. Test results demonstrate that the proposed algorithm has better robustness in dynamic environments.

2

System Structure

Traditional visual SLAM framework is composed of visual odometry, optimization, loop closing, and mapping. Based on the ORB-SLAM2 algorithm [11], our algorithm adds dynamic object detection links and real-time map construction threads. The system consists of four threads: Tracking, Local Mapping, Loop Closing, and Mapping. The system framework is shown in Fig. 1. 2.1

Tracking

RGB-D camera can obtain depth information directly from the sensor, without excessive initialization. The main work of this thread is: extract features of the image, optimize the matching of feature points by RANSAC method, and combine the previous frame for pose estimation. Using optical flow method combined with pose estimation to remove dynamic feature points that affect system performance. 2.2

Local Mapping

This thread is utilized to build a local map, which consists of keyframes and map points. Extract features from keyframes, filter and generate map points, optimize with local Bundle Adjustment (BA), and screen newly inserted keyframes. Developed some rules that can effectively prevent large pose changes after removing dynamic objects.

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Fig. 1. System structure

2.3

Loop Closing

It is divided into loop detection and loop correction. The loop detection uses the Bag of Words (BoW) to determine whether the robot motion has reached the previous position and calculates the similarity transformation. Then use the graph optimization method for global BA. 2.4

Mapping

Using pose and image information, a dense point cloud map can be constructed, which will be updated in real time. Using the PCL library to process the point cloud data, and use the voxel filter to filter the obtained point cloud data [10]. Under the premise of ensuring that the original geometry is unchanged, the number of points is reduced, and the speed of subsequent reconstruction is accelerated.

3 3.1

Robust SLAM Algorithm Homography Matrix

Homography matrix describes the mapping relationship between the points of the same plane in different images. P1 , P2 represent the coordinates of a point P on plane α in two different camera coordinate systems (Fig. 2).

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Fig. 2. Plane mapping

P2 = (RP1 + t)

(1)

where R is the rotation matrix between two camera coordinate systems, t is translation vector. nT P1 = d (2) where n is the unit normal vector of plane α in the first camera coordinate system, t is the distance from the origin. Combine Eqs. 1 and 2,  T    n tnT P1 = R + P2 = RP1 + t (3) P1 = H P1 . d d H is the homography matrix in the first camera coordinate system,need to T transform it into the imaging plane coordinate system. p1 = (x1 , y1 , 1) and T p2 = (x2 , y2 , 1) is image point position of point P in two images. p1 = KP1 , p2 = KP2 , K is camera interior parameter matrix. Combined with Eq. 3, there is ⎡ ⎤   h11 h12 h13 tnT ⎣ ⎦ H = h21 h22 h23 = K R + (4) K−1 . d h31 h32 h33 which satisfies

⎡

⎤ ⎡ ⎤⎡ ⎤ x1 h11 h12 h13 x2 ⎣ y1 ⎦ = ⎣ h21 h22 h23 ⎦ ⎣ y2 ⎦ . 1 h31 h32 h33 1

(5)

The homography equation is generally solved by Direct Linear Transformation (DLT). If the selected matching point is located in the actual dynamic region, will get a wrong result, which makes the difference between the estimated pose and the actual pose too large. RANSAC algorithm is used to calculate the homography matrix of the image to reduce the interference caused by the dynamics. RANSAC can use an iterative method to find the point of the optimal parametric model in a set of external data sets. In order to simplify the calculation in actual processing, it is usually setting h33 =1. RANSAC algorithm randomly extracts four pairs of matching feature

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points in the matching data set and calculates the homography matrix. Then, the model is utilized to test all the data, calculate the number of data points and satisfy the projection error of the model. If the model is the optimal model, Then the corresponding cost function is minimal. Its cost function is:  2  2

n i i i i  i i i h11 x1 + h12 y1 + h13 i h21 x1 + h22 y1 + h23 x2 + y2 J(x , y ) = h31 xi1 + h32 y1i + h33 h31 xi1 + h32 y1i + h33 i=1 (6) 3.2

Dynamic Object Detection

The common detection methods of moving objects are mainly divided into three categories: Background Subtraction, Temporal Difference, and Optical Flow. The first two detection methods are suitable for static background model but perform poorly in dynamic background. In the SLAM process, the camera position is also constantly changing. The moving object detection based on optical flow method is more suitable when the camera is moving. Optical flow is described as the motion information of the image brightness. The optical flow method uses the change of the pixel in the image sequence in the time domain and the linear relationship between adjacent frames to find the correspondence between the previous frame and the current frame, thereby the motion information of the object between adjacent frames is calculated. In this paper, Lucas-Kabade(LK) optical flow is used to detect the dynamic objects of the scene. LK Optical flow is based on the following three assumptions: (1) Constant brightness: the brightness of tracked pixels don’t vary with time. (2) Small motion: the motion in the image changes slowly. (3) Spatial consistency: this pixel has a similar motion to pixels in the domain. Suppose the image has a pixel point A (x, y), whose brightness at time t is I (x, y, t).The basic optical flow constraint equation can be written as

 u  Ix Iy (7) = −It v dy ∂I ∂I dx where ∂I ∂t = It , ∂x = Ix , ∂y = Iy , u = dt , v = dt . If all the pixels in the neighborhood of the feature point do the similar motion, the following basic equations can be obtained by several basic optical flow equations ⎤ ⎤ ⎡ ⎡ I1x I1y I1t ⎢ I2t ⎥ ⎢ I2x I2y ⎥  ⎥ u ⎥ ⎢ ⎢ (8) = −⎢ . ⎥. ⎢ .. .. ⎥ ⎣ .. ⎦ ⎣ .. ⎦ v Int Inx Iny

For convenience, the above formula is abbreviated as Ax = z.

(9)

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Using the least squares method to solve the above equation can get : −1 T  A z. x = AT A

(10)

The final optical flow equation is: ⎡ n ⎤−1 ⎡ n ⎤ n  2  

 Iix Iix Iiy ⎥ ⎢ − Iix It ⎥ ⎢ u i=1 i=1 ⎥ ⎢ i=1 ⎥. =⎢ n n n  2 ⎦ ⎣  ⎣ ⎦ v Iix Iiy Iiy − Iiy It i=1

i=1

(11)

i=1

The optical flow field of each point can be obtained through the above calculation. Then the foreground and background can be separated by threshold segmentation. The dynamic feature points extracted by this method also contain the moving points generated by the camera motion. Using the homography matrix obtained in Sect. 3.1, the feature points of the actual motion in the scene are selected out, and the external points caused by camera motion are eliminated to further improve the applicability of the algorithm. 3.3

Keyframe and Optimization

The local map is comprised of keyframes and map points. The purpose of constructing the local map is to strengthen the constraint relationship between the camera pose and map points while saving computing resources and memory as much as possible. Optimizing the pose of keyframes and map points can improve the output accuracy of the system. Keyframe selection follows the following rules: If there are fewer map points currently being tracked, insert keyframes in the following cases: 1. No keyframes are inserted after 30 consecutive normal frames. 2. The local mapping thread is idle. 3. The trace is about to fail. Local mapping and loop detection threads filter keyframes. The keyframes that are ultimately retained need to meet the following rules: 1. Track at least 20 map points. 2. More than 10 normal frames between the previous keyframes. 3. The map points can be observed in other common frames below 90%. Using the g2o library for nonlinear optimization. Our design chooses the Levenberg-Marquardt method, which is a widely used least squares optimization algorithm with the following iteration formula:  −1 xn−1 = xn − JTf Jf + μI JTf f (xn )

4

(12)

Experiment and Analysis

Our hardware configuration is: Intel i7-8700, 3.2 GHz, 6G, GTX1060. The system is Ubuntu 16.04 and test data set is TUM RGB-D data sets. TUM RGB-D data sets used in this experiment can be divided into three categories: static scene, low dynamic scene, and high dynamic scene. Objects of

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static scene are absolutely stationary. In low dynamic scenes, the motion area only occupies a small part of the scene, such as a person sitting on a chair waving arms. High dynamic scenes contain multiple moving people, which not only affect the accuracy of the SLAM system, but also can cause the entire SLAM system failed to track. The TUM data set records the actual motion trajectory of the camera and provides a range of convenient inspection tools for performance evaluation of the SLAM system. Relative Pose Error (RPE) is commonly used to evaluate the performance of visual odometry, which calculates the relative motion error at different times. The relative pose errors (including Translation Error and Rotation Error) of the system in three different scenarios are presented in Table.1. The original algorithm has excellent data correlation characteristics, and it has certain adaptability to the dynamic environment. Therefore, it performs better in static environment and low dynamic environment, and our improvement is not obvious. In high dynamic environment, the new algorithm has a more targeted treatment of dynamic objects, which can greatly improve the accuracy of the visual odometry. Table 1. Relative pose error Scene

Seq

TE.RMSE(m) Original Ours

Static

f 2/lnl 0.657599 0.700121

RE.RMSE(*) Original Ours 3.234633

2.479384

Low f 3/sr 0.028211 Dynamic f 3/sh 0.036132

0.026459 0.840689 0.026206 0.936475

0.811690 0.797823

High f 3/ws 0.578755 Dynamic f 3/wh 0.833958

0.040278 10.447460 0.739342 0.337578 19.756841 6.774932

Figure 3 shows the position output accuracy of the visual odometer in high dynamic scene. It can be observed that the mean and peak value of the RPE significant reduction after algorithm improvement. The accuracy of the visual odometer has been significantly improved. Absolute Trajectory Error(ATE) is used to measure the error between the actual trajectory and the estimated trajectory. It can be used to evaluate the performance of the entire SLAM system. Table. 2 and Fig. 4 show the performance of the system in three different scenes. It can be seen that these two algorithms have good performance in static environment. In low dynamic environment, the improved new algorithm can more effectively detect whether the camera motion reaches the loop and improve the accuracy of the overlap. In the high dynamic environment, the original algorithm is easy to track failure, and the resulting motion trajectory is quite different from the actual trajectory. New algorithm tracks static map points, reduces the possibility of tracking failure, and improves track accuracy.

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Fig. 3. Plot of RPE for the high dynamic sequences

Fig. 4. ATE in different sequences

Table 2. Absolute Trajectory Error Scence

Seq

Original Ours

Static

f 2/lnl 0.538424 0.167706

Low f 3/sr 0.019517 0.018394 Dynamic f 3/sh 0.023159 0.018081 High f 3/ws 0.406473 0.043436 Dynamic f 3/wh 0.549013 0.224724

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Conclusions

In this paper, an RGB-D SLAM algorithm for dynamic environment optimization is proposed. In the visual odometer, optical flow method combined with camera pose calculation is integrated to extract moving objects of the scene and eliminate its impact on the system. RANSAC algorithm is used to optimize the accuracy of feature matching. Using the TUM data set for validation. Experimental results demonstrate that our algorithm can effectively eliminate the influence of dynamic objects of the scene and improve the accuracy of the visual odometer. Acknowledgments. This work was supported by the NSFC (61327807, 6152 1091, 61520106010, 61134005), and the National Basic Research Program of China (973 Program: 2012CB821200, 2012CB821201).

References 1. Davison AJ, Reid ID, Molton ND, Stasse O (2007) MonoSLAM: real-time single camera SLAM. IEEE Trans Pattern Anal Mach Intell 29(6):1052–1067 2. Klein G, Murray D (2009) Parallel tracking and mapping on a camera phone. In: 2009 8th IEEE international symposium on mixed and augmented reality. IEEE 3. Mur-Artal R, Montiel JMM, Tardos JD (2015) ORB-SLAM: a versatile and accurate monocular SLAM system. IEEE Trans Robot 31(5):1147–1163 4. Tan W, Liu H, Dong Z, Zhang G, Bao H (2013) Robust monocular SLAM in dynamic environments. In: 2013 IEEE International symposium on mixed and augmented reality (ISMAR). IEEE Computer Society 5. Lee KH, Hwang JN, Okapal G, Pitton J: Driving recorder based on-road pedestrian tracking using visual SLAM and constrained multiple-kernel. In: IEEE international transportation systems. IEEE (2014) 6. Zou D, Tan P (2012) CoSLAM: collaborative visual slam in dynamic environments. IEEE Trans Pattern Anal Mach Intell 35(2):354–366 7. Hu JS, Tseng CY, Chen MY (2014) Detection of moving features using IMUcamera without knowing both the initial conditions and gravity direction. In: 2013 CACS international automatic control conference (CACS). IEEE 8. R¨ unz M, Agapito L (2018) Maskfusion real-time recognition, tracking and reconstruction of multiple moving objects 9. J¨ urgen S, Engelhard N, Endres F, Burgard W, Cremers, D (2012) A benchmark for the evaluation of RGB-D SLAM systems. In: 2012 IEEE/RSJ international conference on intelligent robots and systems. IEEE 10. Papon J, Abramov A, Schoeler M, Florentin W (2013) Voxel cloud connectivity segmentation - supervoxels for point clouds. In: Computer vision and pattern recognition. IEEE 11. Mur-Artal R, Tard´ os JD (2016) ORB-SLAM2: an open-source SLAM system for monocular, stereo, and RGB-D cameras. IEEE Trans Robot 33(5):1255–1262

Velocity Tracking Control Based on Throttle-Pedal-Moving Data Mapping for the Autonomous Vehicle Mingxing Li(B) and Yingmin Jia The Seventh Research Division and the Center for Information and Control, School of Automation Science and Electrical Engineering, Beihang University (BUAA), Beijing 100191, People’s Republic of China [email protected]

Abstract. In this paper, the velocity tracking problem is studied. Without employing a complex longitudinal dynamic model, a numerical modeling method is established based on the off-line data of the vehicle motion. And a controller which not only makes the tracking error of the velocity smaller and also guarantees the input of the braking/accelerating smoothly changing is designed. To verify the tracking performance, the simulating based on the Electric Vehicle Reference Application of Matlab 2018 is implemented and results conclude that the new control strategy is very effective. Keywords: Velocity tracking Autonomous vehicle

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· Numerical method ·

Introduction

Automatic driving technology has attracted a growing attention in the last years with better road utilization and performances of safety and comfort. Velocity tracking is the basis of the automatic driving technology with the subject to make the vehicle moving with the desired velocity. To achieve this goal, many control methods are established based on the analysis model which need to know the engine parameters of the controlled vehicle [1]. However, these parameters are not easy to obtain for an autonomous driving engineers [2,3]. Thus the PID control [4], fuzzy control [5,6], or model predictive control [7] are usually used to achieve the velocity tracking object. But these methods can not given a satisfy tracking results [8]. Different from above results, some other researchers such as Xu, etc. in [9] given an numerical modeling to control the velocity. Based on the speedthrottle-acceleration mapping and a first-order linear time-invariant approach, an optimal tracking controller is designed to tracking the desired velocity. And this first-order linear time-invariant approach of the vehicle dynamics is not reasonable. Different from the above works, an reasonable numerical model of the vehicle is established and a parameter-varying PD type controller is designed in this c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 690–698, 2020. https://doi.org/10.1007/978-981-32-9682-4_72

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paper. It is organized as follows: modeling and problem formulation are introduced in Sect. 2. Modeling the previous dynamics of diving force is shown in Sect. 3. And the designing process of the controller is described in Sect. 4. The simulation results are described in Sect. 5 and the conclusion is made at last in Sect. 6.

2

Modeling and Problem Formulation

Fig. 1. Longitudinal dynamics of the autonomous vehicle

As shown in Fig. 1, the longitudinal dynamics of the autonomous vehicle by using Newton’s second law is [1]: v˙ = (Fv − Ff )/m

(1)

where m is the vehicle mass, v is the longitudinal velocity, Fv is the sum force of the accelerating force and braking force, Ff is the sum of the external disturbance force which includes the aerodynamic drag force Fa , the rolling resistance force Fr , the gravitational force Fg and other external forces Fe which has not been considered. And the following equation is established: Ff = Fa + F r + F g + F e

(2)

If let α denote the road slope angle, then it is easy to obtain that: Fg = mg sin(α)

(3)

the aerodynamic force can be expressed as follows: Fa = sign(v + vwind )ρCd AF (v + vwind )2 /2

(4)

where vwind is the environmental wind velocity, and the rolling resistance force is usually modeled by Fr = kr mg. Thus, system (1) can be rewritten as: v˙ = (Fv − mg sin(α) − sign(v + vwind )ρCd AF (v + vwind )2 /2 − kr mg − Fe )/m (5)

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According to the works [2–8], force Fv is a nonlinear function of the throttle angle and the braking pressure. The throttle angle is a function of the accelerator pedal ψt , and the throttle angle is a function of the brake pedal ψb . Thus, Fv = Fv (v, ψt , ψb ), and to a given vehicle, this relationship is almost not changing with the external environmental, the load and other external factors. However, to model this relationship accurately is not easy and not all of the model parameters could be obtained by an autonomous driving engineer. This problem makes it is almost impossible to track the desired velocity precisely. Furthermore, to system (5), if Fe = 0, vwind = 0 and α = 0 then we have: v˙ = (Fv − sign(v)ρCd AF v 2 /2 − kr mg)/m

(6)

Let Ca = ρCd AF , Ca > 0, and assume kr is a slow varying parameter which means kr ∈ [kr,min , kr,max ], k˙ r = 0, where 0 < kr,min < kr,max , then we have:    ∂Fv ∂Fv ˙ ∂Fv ˙ − |v|Ca v˙ + v¨ = ψt + ψb /m (7) ∂v ∂ψt ∂ψb To any given vehicle, ψt ψb ≡ 0, thus we have: ⎧ ∂F ⎪ ⎨ ∂ψvt ψ˙ t , while the vehicle is accelerating ∂Fv ˙ ∂Fv ˙ ψt + ψb = ⎪ ∂ψt ∂ψb ⎩ ∂Fv ψ˙ b , while the vehicle is braking ∂ψb

(8)

Thus, let u = ψ˙ t , ψ = ψt if the vehicle is accelerating, and u = ψ˙ b , ψ = ψb if the vehicle is braking, then: ∂Fv ˙ ∂Fv ˙ ∂Fv u ψt + ψb = ∂ψt ∂ψb ∂ψ System (7) can be rewritten as follows:  ψ˙ = u 

v v¨ = ∂F ∂v − |v|Ca v˙ +

∂Fv ∂ψ u

(9)

/m

(10)

∂Fv v In system (5) and (10), Fv is only depend on v, ψ. Thus if ∂F ∂v and ∂ψ are obtained more exactly, then the desired velocity is tracked more precisely. Hence, to an actual vehicle, system (10) is used to obtain the dynamics of Fv previously, and then, according this previous dynamics, design feedback controller to track the desired velocity precisely.

3

Modeling the Previous Dynamics of Diving Force

It is easy to obtained that the Lipschitz conditions of Fv to ψ is satisfied. To system (10), let Fvm = Fv /m − kr g, Fvma = Fv /m − kr g − |v|Cam v/2 and Cam = Ca /m, then the following differential equations are established: v˙ = Fvm − |v|Cam v/2 = Fvma ψ˙ = u

(11) (12)

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To the above equations, parameter Cam , states v and v, ˙ input u are all measurable and computable. Thus, if let y = v˙ + |v|Cam v/2, then from Eq. (11), we can get that   ∂Fvma ∂Fvma u (13) F˙ vma = v˙ + ∂v ∂ψ   ∂Fvma ∂Fvma u (14) = Fvma + ∂v ∂ψ and let a = v˙ express the vehicle acceleration, obtain the following equations: ∂a ∂Fvm ∂a ∂Fvm = , = ∂v ∂v ∂ψ ∂ψ

(15)

According to above results and to obtained the approaching dynamics of Fvma , take any suitable points (v0 , ψ0 ) and use the following two sets of parallel lines to cover the work plane of the vehicle: vi = v0 + ihi , i = 0, ±1, ±2, · · ·

(16)

ψj = ψ0 + jτj , j = 0, ±1, ±2, · · ·

(17)

Then we have the following approaching system: ψ˙ = u F˙vma = fij Fvma + gij u a(vi , ψj ) − a(vi−1 , ψj ) fij = vi − vi−1 a(vi , ψj ) − a(vi , ψj−1 ) gij = ψi − ψi−1

(18) (19) (20) (21)

Thus to any time step ti = ti−1 + hi and t ∈ [ti−1 , ti ] the following equation can be obtained from the above equations: Fvma (t) = Fvma (vi−1 , ψi−1 ) + (fij Fvma (t) + gij u)(t − ti−1 ) = (1 − fij (t − ti−1 ))−1 (Fvma (vi−1 , ψi−1 ) + (t − ti−1 )gij u) = (1 − fij (t − ti−1 ))−1 (Fvma (vi−1 , ψi−1 ) + gij (ψ(t) − ψi−1 ) (22) where ψ(t) ∈ [ψi−1 , ψi ] if ψi−1 < ψi and ψ(t) ∈ [ψi , ψi−1 ] if ψi−1 > ψi . From now on, we get the approaching dynamics model of Fvma and u. This dynamics can be used to model the dynamics of Fvma and u. From Eq. (11), we get that Fvm = a + |v|Cam v/2 and if ψ is not changing then Fvm is a decrease function of v , thus fij < 0 in Eq. (20) should be always established. Furthermore, if u = 0 which means ψ is a constant, then lim Fvma = t→∞ 0. To gij , gij > 0 while the vehicle is accelerating, and gij < 0 while the vehicle is braking. Relationships of ψ, a and v are shown in Fig. 2. Since there are inner dynamics, thus not all the values of fij and gij can be used which are calculated from (20) and (21). To solve this problem, fij and gij are smoothed which are denoted by fijs and gijs .

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Fig. 2. The relationships of ψ, a and v

4

Controller Designing

As shown in Sect. 2, the velocity tracking control is to track the desired velocity vd and desired acceleration ad . From Sect. 3 and Eqs. (1)–(4), the controlled system can be rewritten into: v˙ = Fvma − g sin α − Fe /m − Fa + |v|Cam v/2

(23)

Let the lumped disturbance ω = |v|Cam v/2−Fe /m−Fa +Fvma −Fvmas −g sin α, then we have the following controlled system:

v˙ = Fvmas − ω (24) F˙ vmas = fijs Fvmas + gijs u. To this system, velocity v and acceleration a are measurable. Let e = v − vd then e˙ = a−ad where vd is the desired velocity and ad is the desired acceleration. The control process is shown as Fig. 3. In this figure, the error equation is expressed as following:

e˙ = Fe − ω (25) F˙ e = fijs Fe + gijs u.

Fig. 3. Velocity tracking process

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where e(t0 ) = 0 and Fe (t0 ) = 0. Since only e and e˙ are measurable, the following controller is taken: (26) u = Kp e + Kd e˙ such that for R > 0, the following inequality is established:  ∞ (eT e + uT Ru − γω T ω)dτ < 0

(27)

0

From Eq. (26), Fe can be rewritten into as: F˙e = fijs Fe + gijs u = fijs Fe + gijs (Kp e + Kd e) ˙ Thus if let ef = Fe − gijs Kd e and Kp = −fijs Kd , then we have: e˙ f = F˙ e − gijs Kd e˙ = fijs Fe + gijs Kp e = fijs ef + (gijs Kp + fijs gijs Kd )e = fijs ef e˙ = ef + gijs Kd e − ω To ef , ef (t0 ) = Fe (t0 ) − gijs Kd e(t0 ) = 0, thus ef ≡ 0, which means that: e˙ = gijs Kd e − ω

(28)

To the above system, if there is P > 0 such that: V˙ + eT e + uT Ru − γω T ω = 2eT P e˙ + eT e + uT Ru − γω T ω = 2eT P (gijs Kd e − ω) + eT e − γω T ω + ((gijs Kd −fijs )e − ω)T Kd RKd ((gijs Kd − fijs )e − ω)) = (2P gijs Kd + 1 + (gijs Kd − fijs )2 Kd2 R)e2 +(Kd2 R − γ)ω 2 − 2(P + (gijs Kd − fijs )Kd2 R)eω If let R = γKd−2 and P = γ(fijs − gijs Kd ), then we have: V˙ + eT e + uT Ru − γω T ω = (2γ(fijs − gijs Kd )gijs Kd + 1 + γ(gijs Kd − fijs )2 )e2 2 2 = (1 − γ(gijs Kd2 − fijs ))e2 2 2 2 If 1 − γ(gijs Kd2 − fijs ) < 0, i.e. gijs Kd < −((γ −1 + fijs ))1/2 then:

V˙ + eT e + uT Ru − γω T ω < 0 2 1/2 and P > γ(fijs + (γ −1 + fijs ) ) > 0 which means inequality (27) is established. 2 1/2 ) , and the controller is taken as: Thus, while gijs Kd < −(γ −1 + fijs

u = Kd (e˙ − fijs e) the following inequality is established:  ∞ (eT e + γ(e − fijs e) ˙ T (e − fijs e) ˙ − γω T ω)dτ < 0 0

for any given γ.

(29)

(30)

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Simulation

The vehicle model in the example Electric Vehicle Reference Application of Matlab 2018 is used to verify our control strategy. The desired velocity is taken as Drive Cycle Source FTP75. According results in the Sect. 4, the parameters fijs and gijs are shown in the following Figs. 4 and 5.

Fig. 4. Values of gij

Fig. 5. Values of fij

And the tracking results are shown by Fig. 6. It is shown that the tracking error is almost in the rank [−0.2, 0.2] m/s, and the RMS error is 0.0414 which is much less than the PI 0.2351 and 0.1726 of the method which is the proposed in [1].

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Fig. 6. Tracking result of our control method

6

Conclusions

In this paper, the velocity tracking problem is studied. A numerical modeling method is established and a parameter-varying PD type controller is designed. From the simulating results, the high accurate tracking result is obtained. However, the disturbances such as wind and different road surface conditions, and the coupling performance of the lateral dynamics are not considered in this paper. It should be studied further in the future. Acknowledgement. This work was supported by the NSFC (61327807,61521091, 61520106010, 61134005, 61703020) and the National Basic Research Program of China (973 Program: 2012CB821200, 2012CB821201).

References 1. Kim H, Kim D, Shu I, Yi K (2016) Time-varying parameter adaptive vehicle speed control. IEEE Trans Veh Technol 65(2):581–588 2. Na J, Chen AS, Herrmann G, Burke R, Brace C (2018) Vehicle engine torque estimation via unknown input observer and adaptive parameter estimation. IEEE Trans Veh Technol 67(1):409–422 3. Milan´es V, Villagr´ a J, P´erez J, Gonz´ alez C (2012) Low-speed longitudinal controllers for mass-produced cars: a comparative study. IEEE Trans Ind Electron 59(1):620– 628 4. De Santis RM (1994) A novel PID configuration for speed and position control. J Dyn Syst Meas Control 116(3):542–549 5. Tian L, Hao W, Zhu X (2014) Research on cruise control system based on fuzzy adaptive method. Autom Instrum 9:11–14 6. Naranjo JE, Gonzalez C (2006) ACC+Stop&Go maneuvers with throttle and brake fuzzy control. IEEE Tans Intell Transp Syst 7(2):213–225 7. Li S, Li K, Rajamani R, Wang J (2011) Model predictive multi-objective vehicular adaptive cruise control. IEEE Trans Control Syst Technol 19(3):556–566

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8. Li SE, Gao F, Cao D, Li K (2016) Multiple-model switching control of vehicle longitudinal dynamics for platoon-level automation. IEEE Trans Veh Technol 65(6):4480–4492 9. Xu S, Peng H, Song Z, Chen K, Tang Y (2018) Accurate and smooth speed control for an autonomous vehicle. In: IEEE intelligent vehicles symposium (IV), pp 19761982, June 2018

Consensus Tracking Control for Switched Multiple Non-holonomic Mobile Robots Lixia Liu1 , Xiaohua Wang2 , Lan Xiang3(B) , Zhonghua Miao2 , and Jin Zhou1 1

Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China 2 School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200072, China 3 Department of Physics, School of Science, Shanghai University, Shanghai 200444, China [email protected]

Abstract. This brief focuses on the cooperative consensus tracking control for a network of switched multiple-nonholonomic mobile robots (MNMRs), where MNMRs are formulated as the hybrid dynamical systems with the switched behaviors occurring in either the system model parameters or communication network topology. A cooperative switched consensus tracking strategy is developed for MNMRs based on backstepping technique and sliding mode technique. Specially, a switched dynamics torque controller is designed to track a given consensus trajectory by integrating kinematic models, and its asymptotic stability is then fully guaranteed by using Lyapunov-like analysis with average dwell time method. Finally, the effectiveness of the proposed consensus tracking methodology is illustrated by numerical simulation example. Keywords: Switched consensus tracking · Nonholonomic mobile robots · Hybrid dynamical systems · Backstepping technique · Average dwell time method

1 Introduction Cooperative control problem of multiple-robot systems has become an active research topic in the robotics community for the past few decades since they possess some distinguished advantages over single-robot counterpart [1, 2]. In particular, MNMRs, as an important and representative of multi-robot systems, have recently drawn substantially attentions due to their potential and extensive engineering applications, especially in complex unstructured task environments, such as automatic vehicles manufacturing, disaster rescue exploration, deliver dangerous materials, and so forth [3, 4]. In general, the design of cooperative control scheme for MNMRs becomes more complex and difficult with an increase of the number of robots [5–8]. This is because even the tracking control or stabilization of single mobile robot over nonholonomic constraint is also a quite difficult issue based on the well-known Brockett’s Theorem [9, 10]. As a result, c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 699–707, 2020. https://doi.org/10.1007/978-981-32-9682-4_73

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a variety of nonlinear control approaches have been proposed and analyzed for singlenonholonomic robot in recent years, including sliding modetechnique, backstepping scheme, neural network and fuzzy control, etc [11–13]. As is generally known, the cooperation control of multiple-robot systems can be rephrased as the problem of consensus (or synchronization) tracking control for multiagent systems. With the great advance of information technique, computer science, artificial intelligence and biological engineering over the past decade, a variety of consensus or synchronization protocols (or algorithms) have been proposed and developed for various types of multi-agent systems, especially for single or double integrator dynamics, or Lagrangian dynamics [14, 15]. The cooperative consensus tracking control of MNMRs is more challenging than multi-agent systems with linear or nonlinear holonomic agent dynamics. This is mainly because in this case trajectory tracking and posture stabilization must be simultaneously considered. Accordingly, most of current research on consensus or synchronization tracking control has mainly focused on the design of kinematic controller for MNMRs. It is quite common that the switched phenomenon is encountered in real multiplerobot control systems due to a wide variety of causes such as uncertain robotic dynamics, intermittent unavailability of controllers or actuators, temporal failures of communication links between robots and inevitably external disturbances [?]. Switched system is a typical hybrid dynamical system, which is composed of subsystems and switching rules [16]. Accordingly, switched control strategy is often regarded as an effective technique to deal with a variety of complex hybrid dynamics including varying structures, abruptly parameter jumping, etc [17]. However, to our knowledge, up to now just few works are concerned with cooperative control problems of multiple-robot systems, in particular when the switching effects of the system model and the communication interaction among agents are taken into account at the same time [18]. For practical reasons, it will be a strong motivation to develop the cooperation algorithms for switched consensus tracking control for multiple mobile robots under nonholonomic constraints. Inspired by aforementioned backgrounds, this brief is concerned with the coordinated consensus tracking control for MNMRs with the switched behaviors occurring in either the system model parameters or the communication network topology. The main objective is to design the suitable switched controller for a network of MNMRs such that each of robots can track a desired common reference trajectory. By using backstepping technique in combination with Lyapunov-like analysis with average dwell time method, a switched dynamics torque controller is then developed to fully ensure the switched consensus tracking for MNMRs. Subsequently, an illustrate simulation example is provided to visualize the theoretical results.

2 Preliminaries 2.1

Graph Theory

Assume that the communication interaction for a network of n robots corresponds a directed graph G = (V , E , A ), where V = {1, · · · , n} denotes n robots, E ⊆ V × V describes the information flow among the robots. The communication topology G can be expressed by a adjacency matrix A = [ai j ] ∈ Rn×n , where ai j = 1 if (i, j) ∈ E , and

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ai j = 0 otherwise. A directed tree in a directed graph is that every node has one parent except the root node. If there are more than one roots node have directed paths to other node, then the directed graph has a directed spanning tree. Assumed that self-edges are not permitted, i.e., aii = 0. Furthermore, the Laplician matrix LA = [li j ] ∈ Rn×n is defined as lii =

n

∑

j=1,i= j

ai j and li j = −ai j , i = j. As usual, the directed communication

topology G of the network with n robots is assumed to possess a spanning tree [?]. 2.2 Switched Stability For a switched nonlinear system x(t) ˙ = fσ (t) (x), where x ∈ Rn is the state, σ (t) : [0, ∞) → Λ = {1, 2, · · · d} is the switching law, and σ (t) = p means that the pth subsystem starts work [16]. Then the following lemma regarding average dwell time (ADT) is introduced. Lemma 1. [19] If there exist function Vi = xT Pi x i, j ∈ Λ and two class K∞ functions αi1 and αi2 , a positive number λ such that

αi1 x(t)2 ≤ Vi (x) ≤ αi2 x(t)2 , ∂ Vi (x) fi (x) ≤ −λ Vi (x), ∂x Vi (x) ≤ μ V j (x),

(1) (2) (3)

for some positive number μ , then the nonlinear switched system x(t) ˙ = fσ (t) (x) is globally asymptotically stable under any switching signal with the ADT:

τˇ > τˇ ∗ =

ln μ , λ

(4)

is average dwell time. where τkˆ = tkˆ − tk−1 ˆ 2.3 Switched MNMR Models Consider a network of d NMRs, and the ith mobile robot subjected to m constraints is described as switched Euler-Lagrangian dynamics [11]: Miσ (t) (qi )q¨i +Viσ (t) (qi , q˙i )q˙i + Giσ (t) (qi ) = Biσ (t) (qi )τiσ (t) + ATiσ (t) (qi )λi , i ∈ Ψ = {0, 1, 2, · · · , d},

(5)

where qi , q˙i ∈ Rn are the vectors of generalized coordinates and velocities, respectively. ˙ ∈ Rn×n is a coriolis and centrifugal Miσ (t) (qi ) ∈ Rn×n is a inertia matrix, Ciσ (t) (q, q) n force matrix and Giσ (t) (qi ) ∈ R is the gravitational vectors, Aiσ (t) (q) ∈ Rm×n is the matrix concerning the m constraints, and Biσ (t) (q) ∈ Rn×(n−m) is the input transformation matrix. τiσ (t) ∈ R(n−m) is the control input vector, and λi is the vector of constraint force. In general, the nonholonomic kinematic constraints for ith mobile robot

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is: Aiσ (t) (qi )q˙i = 0 with Aiσ (t) (qi )ηiσ (t) (qi ) = 0, where ηiσ (t) (qi ) ∈ Rn×(n−m) is formed by linearly independent vector in the null space of Aiσ (t) (qi ). Substituting the Eq. (6) and its derivative into Eq. (5), then the d NMRs (5) can be modeled as following switched systems: q˙i = ηiσ (t) (qi )zi ¯ ¯ Miσ (t) (qi )˙zi + Viσ (t) (qi , zi ) + Giσ (t) (qi ) = τ¯iσ (t) .

(6) (7)

where qi = (xi , yi , θi ) , zi = (vi , ωi ) , and M¯ iσ (t) (qi ) = ηiTσ (t) (qi )Miσ (t) (qi )ηiσ (t) (qi ), V¯iσ (t) (qi , zi ) = ηiσ (t) (qi )T [{Miσ (t) (qi )η˙ iσ (t) (qi )+Viσ (t) (qi , q˙i )ηiσ (t) (qi )}, G¯ iσ (t) (qi ) = ηiσ (t) (qi )T Giσ (t) (qi ),τ¯iσ (t) = ηiTσ (t) (qi )Biσ (t) (qi )τiσ (t) . As usual, the motion models of MNMRs are composed of kinematic steering system (6) and additional dynamics system (7). It is usually assumed that each robot is pure rolling and non-slipping on the ground with the constraint x˙i sin θi − y˙i cos θi = 0. In this case, the kinematic equation for the network of NMRs is given by: ⎡ ⎤ ⎡ ⎤ x˙i vi cos θi ⎣ y˙i ⎦ = ⎣ vi sin θi ⎦ , (8) ωi θ˙i where xi , yi are position states, θi is a direction angle, vi , ωi are forward linear velocity and angular velocity of the ith robot, respectively. The following properties derived from Euler-Lagrangian dynamics (7) are introduced for sequel analysis: p Property 1 (Boundedness). M¯ ip (qi ) are positive defined matrix, and mi1 xi 2 ≤ p p p 2 ¯ xi Mip xi ≤ mi2 xi  where mi1 , mi2 are positive constants.

Property 2 (Skew symmetric property). The matrix M˙¯ ip − 2V¯ip is skew symmetric.

3 Switched Consensus Tracking 3.1

Switched Control Objective

The primary objective of this brief is to design the suitable switched controller for a network of MNMRs such that each of robots can track a desired common reference trajectory. Sliding-mode subcontroller is given to guarantee the stability of the subsystem. Accordingly, the motion models of MNMR switches to corresponding sub-controllers when sudden changes appear. In this paper, suppose that the switched times are limited, and the designed sub-controllers switched synchronize the subsystems. Therefore, the concept of switched consensus tracking control for MNMRs is defined as follows: Definition 1. The switched dynamics control protocols τ¯iσ (t) , (i = 1, 2, · · · , d) (or the switched torque inputs) for each robot i are said to achieve consensus tracking for a

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network of NMRs (5) with respect to a given consensus reference trajectory being given by ⎧ ⎨ x˙r = vr cos θr y˙r = vr sin θr (9) ⎩˙ θr = wr where qr = [xr , yr , θr ]T , with zr = [vr , wr ]T ,vr > 0, wr > 0. If each robot i can realize a smooth velocity ziR = fiR (ei , zr , Ki ) by the switched kinematics sub-controller, such that zi → ziR , then lim (qi − qr ) = 0, i = 1, 2, · · · , d, as t → 0, where ei is the tracking t→∞ consensus errors, zr is the reference velocity vector, and Ki is the control gain vector. 3.2 Switched Controller Design By Defnition 1, in order to realize an ideal time-varying velocity for each robot i, a tracking error vector with respect to reference trajectory is introduced [13] ⎤⎡ ⎤ ⎡ ⎤ ⎡ xr − xi cos φi sin φi 0 ei1 ei = ⎣ ei2 ⎦ = ⎣ − sin φi cos φi 0 ⎦ ⎣ yr − yi ⎦ . (10) ei3 φr − φi 0 0 1 Accordingly, an auxiliary velocity control input ziR = [v f i , w f i ] for robot i is given by





vfi vr cos ei3 + k1 ei1 = , wfi wr + k2 vr ei2 + k3 vr sin ei3

where k1 , k2 are the positive parameters to be designed. We define Lyapunov function 1 1 − cos ei3 Vi = (e2i1 + e2i2 ) + , 2 ki2 its derivative is

(11)

2

ki3 sin ei3 V˙i = −ki1 e2i2 − ≤ 0. (12) ki2 By Barbalate’s lemma, ei1 , ei2 and ei3 asymptotically converge to zero. This indicates that if the kinematics input of ith robot reach a smooth velocity ziR = fic (ei , zr , Ki ), then all the robots can track a desired trajectory, i.e., lim (qr − qi ) = 0 as t → ∞. t→∞ Next, a torque controller need to be designed to make real velocity zi converges to ideal velocity ziR eventually. Thus, an auxiliary reference velocity zri is designed beforehand, n

 t

j=1

0

zrip = ziR − ∑ aipj

(zi (r) − z j (r))dr − σip

 t 0

(zi (r) − ziR (r))dr − Δi .

(13)

their derivatives are n

z˙ri = z˙iR − ∑ aipj (zi − z j ) − σip (zi − ziR ), j=1

(14)

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where Δi = ziR (0) − zri (0) and σip > 0 is the feedback control gain with respect to an ideal velocity input viR . Then, a sliding vector is defined as sip = zi − zrip , the corresponding switched dynamics controller (or switched torque input) for each robot is given by

τ¯iσ (t) = Mip (qi )˙zrip +Cip (qi , q˙i )zrip + Gip (qi ) − Kip sip ,

(15)

where Kip is a positive definite matrix, i ∈ Ψ , σ (t) = p ∈ Λ . In view of Eqs. (7) and (15), the closed loop controlled system of the switched dynamics (7) can be expressed as M¯ ip s˙ip = −Cip sip − Kip sip .

(16)

Remark 1. It can be seen that the designed switched torque input (15) is directly derived from the kinematics input ziR . It can be divided into two main parts. The first part denotes the hybrid dynamics of system model, including varying structures, abruptly parameter jumping, etc, while the second part describes the switching effects of communication interaction among agents, such as network-induced packet losses, communication obstacles, etc. Therefore, the proposed control strategy has a clear physical meaning. 3.3

Convergence Analysis

With the above preparations, the main results on the switched consensus tracking control for MNMRs kinematics (6) and dynamics (7) are introduced below. Theorem 1. Consider a directed graph network of MNMRs (6), (7) with a switched spanning tree. Then the switched dynamics control protocols τ¯iσ (t) (i = 1, 2, · · · , d) (or the switched torque inputs) with respect to the switched kinematics sub-controller can always solve the cooperative consensus tracking problem in the sense of Definition 1 if the ADT satisfies ln μi∗ τˇia > τˇia∗ = , (17) λi min where μi =

max (uipq ), λi min =

i∈Ψ ,p,q∈Λ

1 min λip . 2 i∈Ξ ,p∈Λ

Proof. Consider following candidate Lyapunov function for closed loop system (16) 1 Vip (si ) = sip M¯ ip sip , 2

(18)

V˙ip (sip ) = −sip Kip sip ,

(19)

and its derivative is calculated as

for ∀t ∈ [tkˆ ,tk+1 ˆ ). In combination with Property 1 and Eq. (18), we have, p min mi2

i∈Ψ ,p∈Λ

2

sip 2

mp mp ≤ i2 sip 2 ≤ Vip (sip ) ≤ i1 sip 2 ≤ 2 2

p max mi1

i∈Ψ ,p∈Λ

2

sip 2 .

(20)

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By the Eqs. (19) and (20) 2λmin (Kip ) V˙ip (sip ) ≤ −λmin (Kip )sip 2 ≤ − Vip (sip ) = λipVip (sip ), p mi2

(21)

2λmin (Kip ) . p mi2

where i ∈ Ψ , p ∈ Λ , λip =

It follows from the Eq. (20), one can get: Vip ≤ where μi =

p mi2 pq p max (uipq )Viq (sip ) = μiViq (sip ) q Viq (si ) = ui Viq ≤ i∈Ψ ,p,q∈Λ mi1

(22)

max (uipq ).

i∈Ψ ,p,q∈Λ

All in all, when the subcontrollers switching synchronously with the subsystems under the known σkˆ (t), the in Eqs. (20), (21) and (22) hold which satisfy the conditions (1), (2) and (3) in Lemma 1, where λi min = 12 min (λip ). Accordingly, the switched Lagrange i∈Ξ ,p∈Λ

systems are global asymptotic stability, and si → 0 as t → 0. In what follows, we prove that zi → ziR to confirm qi → qr as t → ∞. For convenience, the sliding-mode variables can be rewritten as matrix form: Ec = −

 t 0

(L p + A0p )Ec dt +

 t 0

(zirp − z pjr )dt + s p − Δ

(23)

where Ec = [(zi − ziR )T ]T their derivative are E˙c = −(L p + A0p )Ec dt + (zirp − z pjr ) + s˙p

(24)

We have proved that si → 0 in previous part. Since there is a spanning tree in the switched communication topology, −(L p + A0p ) is Hurwitz in Eq. (24). Further more, lim ziR −z jR  = 0, Ec exponentially converges to zero, that is, lim zi −ziR  = 0, which t→∞ t→∞ leads to the position states tracking the reference trajectories. Remark 2. The proof of Theorem 1 is mainly based on Lyapunov-like analysis and backstepping technique. When the switched kinematics sub-controller is utilized to realize an ideal time-varying velocity for each robot, then the switched torque controller can be designed to track a given consensus reference trajectory. As a result, a switched torque controller in combination with the switched kinematics sub-controller is then developed to fully ensure the switched consensus tracking for MNMRs.

4 Illustrated Examples In this section, a representative example is performed to show the validity of the proposed method. We here assume a network of 6 mobile robots with two switched subsystems, where the detailed model parameters are given below [13]. ⎤ ⎡ rip rip 2 +I )+w 2 −I ) (m b (m b ip ip ip ip ip 2 2 ip ip 4b 4bip ⎦, M¯ ip = ⎣ ip rip (25) rip 2 −I ) (m b (mip b2ip + Iip ) + wip ip ip 2 2 ip 4b 4b ip

ip

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and

⎡ V¯ip = ⎣

0 r

− 2bip2 cd θ˙i

⎤ rip ˙i c d θ ip ip 2 2bip

⎦,

0

(26)

ip

where Iip = cip dip + 2wip b2ip + Icip + 2Imip , aip = 2p, bip = 0.75 + 0.2p, dip 0.08p, rip = 0.15 + 0.04p, wip = 0.005, Imip = 0.0025 + 0.12p, p = {1, 2}.

= 0.3 +

Accordingly, the shifted communication topology is stated in Fig. 1(a). It is obviously that all the conditions in Theorem 1 are satisfied. The simulation results with the proposed switched consensus tracking control algorithm are presented by Fig. 1(b) to (d). As shown in Fig. 1(b) , all the vehicles converge to a reference path with the switched parameter and communication structure. Correspondingly, Fig. 1(c) shows that the linear and angle velocities also converge to desired velocities. Figure 1(d) shows that the trajectory error states xei , yei , respectively. In conclusion, the simulation results reveal that the proposed switched consensus tracking controller is valid. agent1 agent2 agent3 agent4 agent5 agent6 agent

5 4 3

R

y

2 1 0 −1 −2 −3

(a) Switched communication topology

4

10

12

x

ei

x

31 41

0 5

10

61

x x x

−1

51

0

x

0

−2

15

x 0

5

t

e1 e2 e3 e4 e5 e6

10

15

t

4

2 y

12

2

1 ei

22

0

32

y

i2

8

1

21

2

42

−2 −4

6 x

2 11

4 i1

2

(b) Trajectories of six robots in x − y plane

6

−2

0

0

5

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62

y

15

t

(c) The velocities vi1 , vi2 of six robots

−2

e3

ye4 y

−1

52

e1

ye2

0

e5

ye6 0

5

10

15

t

(d) x,y trajectories error regarding time t, respectively

Fig. 1. Switched communication topology and trajectory states

5 Conclusions This paper has studied the problem of consensus tracking problem of switched MNMRs in presence of changed model parameters and shifted communication struc-

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ture. A switched consensus controller for MNMRs has been presented to reach specific tracking performance. Besides, networked topology has been introduced to described the interaction among robots. Finally, typical example has been performed to verify the effectiveness of the proposed control law.

References 1. Chung SJ, Slotine JJE (1998) Cooperative robot control and concurrent synchronization of Lagrangian systems. IEEE Trans Robot 25(3):686–700 2. Wang Q, Chen Z, Yi Y (2017) Adaptive coordinated tracking control for multi-robot system with directed communication topology. Int J Adv Robot Syst 14(2):1–10 3. Chuwa D (2004) Sliding mode tracking control of nonholonomic wheeled mobile robots in polor coordniates. IEEE Trans Control Syst Technol 12(4):637–644 4. Cheng L, Hou ZG, Tan M (2010) Neural-network-based adaptive leader-following control for multiagent systems with uncertainties. IEEE Trans Neural Netw 21(8):1351–1358 5. Miao Z, Yu J, Ji J, Zhou J (2019) Multi-objective region reaching control for a swarm of robots. Automatica 103:81–87 6. Liu J, Miao Z, Ji J (2017) Group regional consensus of networked Lagrangian systems with input disturbances. J Dyn Syst Meas Control 139(9):094501 7. Zhou J, Cai J, Ma M (2015) Adaptive practical synchronisation of Lagrangian networks with a directed graph via pinning control. IET Control Theory Appl 9(14):2157–2164 8. Liu L, Yu J, Ji J, Miao Z, Zhou J (2019) Cooperative adaptive consensus tracking for multiple nonholonomic mobile robots. Int J Syst Sci 50:1556–1567. https://doi.org/10.1080/ 00207721.2019.1617366 9. Chen G, Lewis FL (2011) Distributed adaptive tracking control for synchronization of unknown networked Lagrangian systems. IEEE Trans Syst Man Cybern-Part B Cybern 41(3):805–816 10. Liu YC, Chopra N (2012) Controlled synchronization of heterogeneous robotic manipulators in the task space. IEEE Trans Robot 28(1):268–275 11. Fierro FR, Lewis FL (1995) Control of a nonholonomic mobile robot: backstepping kinematics into dynamics. In: Proceeding of the 34th conference on decision & control, New Orleans, pp 805–3810 12. Fierro FR, Lewis FL (1998) Control of a nonholonomic mobile robot using neural networks. IEEE Trans Neural Netw 9(4):589–600 13. Fukao T, Nakagawa H, Adachi N (2000) Adaptive tracking control a nonholonomic mobile robot. IEEE Trans Robot Autom 16(5):609–615 14. Mehrabian AR, Tafazoli S, Khorasani KI (2010) Cooperative tracking control of EulerLagrange systems with switching communication network topologies. In: IEEE/ASME international conference on advanced intelligent mechatronics montr´eal, pp 6–9 15. Mei J, Ren W, Ma G (2011) Distributed coordinated tracking with a dynamic leader for multiple Euler-Lagrange systems. IEEE Trans Autom Control 56(6):1415–1421 16. Guo RW, Wang Y (2012) Stability analysis for a class of switched linear systems. Asian J Control 14(3):817–826 17. Sankaranarayanan V, Mahindrakar AD (2009) Switched control of a nonholonomic mobile robot. Commun Nonlinear Sci Numer Simul 14:2319–2327 18. Ni W, Cheng DZ (2010) Leader-following consensus of multi-agent systems under fixed and switching topologies. Syst Control Lett 59:209–217 19. Wang X, Zhao J (2017) Autonomous switched control of load shifting robot manipulators. IEEE Trans Ind Electron 64(9):7161–7170

Consensus for a Class of Nonlinear Multi-Agent Systems with Switching Graphs and Arbitrarily Bounded Communication Delays Yali Liao(B) , Peng Xiang, Mengmeng Duan, and Hongqiu Zhu School of Automation, Central South University, Changsha 410083, China yali [email protected], {xp282828,duanmengmeng,hqcsu}@csu.edu.cn

Abstract. A distributed algorithms are devoted to studying consensus problem for a class of strong nonlinear multi-agent systems with switching graphs and arbitrarily bounded communication delays. The communication graphs are directed and changing dynamically, and we assume that the union of the graphs is connected strongly among each certain time interval. A consensus algorithm is introduced for switching graphs and zero communication delays, which guarantees that all agents’ states ultimately achieve a consensus convergence. Then, the other consensus algorithm is given for switching graphs and arbitrarily bounded communication delays. we show that the distributed continuous-time consensus problem is able to be addressed as time evolves. Keywords: Consensus · Multi-agent systems · Delays

1 Introduction Distributed consensus problems for multi-agent systems have caused extensive concern from control research fields [1–18]. For example, article [1] presented second-order consensus algorithms and also derived abundant essential conditions of reaching a consensus with the unidirectional information exchange topologies. Article [2] studied the continuous-time stability problems about linear time-varying system, whose system can be transformed into a Metzler matrix and present sufficient conditions to tackle with the consensus problem. In [6], a Lyapunov function and related spatial decomposition methods were presented even though the graphs are not necessarily connected. Although articles [10–15] have been presented to make all agents’ states reach a consensus when the changing dynamically topologies and the communication time-delays coexist, these extant works were either based on double integrator multiagent systems or high-order linear multiagent systems. Hence, the analytical approaches there are unable to be directly applied to the consensus problem for a class of nonlinear multiagent systems with switching graphs and arbitrarily bounded communication delays. Our purpose is to investigate the distributed consensus for a class of high-order nonlinear systems with the consideration of dynamically changing graphs and communication delays in this paper. In the traditional control, this system can be stabilized c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 708–715, 2020. https://doi.org/10.1007/978-981-32-9682-4_74

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by applying the backstepping approach, e.g., [19–21]. However, when it is taken into account as a separate agent under the setting of distributed coordination control, it is hard to find appropriate algorithms to address the stability problem. There are two parts to the dynamics in this system, i.e., the linear high-order integral dynamics and the nonlinear function. For linear high-order integral dynamic multiagent systems, most of the extant results are able to address consensus problem by making a general model transformation and then using the properties of nonnegative matrix, which is unable to be applied directly to this paper on account of the existence of strong nonlinear term and the integral coupling of the states. The analytical approach is to make a novel model transformation in n steps to transform the original system into a linear-like system, whose system is able to be considered as the Laplacian matrix of some graph. Then, we present a distributed consensus algorithm and use the Metzler matrix theory to enable all agents achieve a consensus convergence. The communication graphs are directed and changing dynamically, and we assume that union of the graphs is connected strongly within each certain time interval. We show that the consensus is able to be achieved as time evolves and the communication delays have on effect on the stability of the systems. Notations: Let IR denote The set of 1 dimensional real column vectors. The set of all r × l real matrices is represented as IRr×l . The transpose of the vector x is defined as xT and the matrix H is defined as HT ; ⊗ is used to represent the Kronecker product; 1 is used to denote the column vector that all of entries is one in the compatible size.

2 Preliminaries In a graph G (I , A ), each agent is deemed as a node, where I = {1, 2, · · · , N} denotes the set of N agents, and A = [ai j ] ∈ IRn×n is used to denote the weighted adjacency matrix. The weighted adjacency matrix A is used to be defined by aii = 0, ai j > μ for a certain constant μ > 0 and i = j when agent i is able to receive valid information from agent j, and ai j = 0 otherwise. We define L as the Laplacian of the directed graph G , where Lii = ∑nj=1 ai j and Li j = −ai j . And let the set of neighbors of node i be defined as Ni = { j ∈ I | ai j > μ }. A directed path is a sequence of the form ai1 ,i2 > μ , ai2 ,i3 > μ , · · · . The directed graph G is considered to be connected strongly when there is a directed path from every node to every other node. Define the union of a team of the graphs G1 , · · · , Gn with the same nodes as the graph G , in which edge set is the union of the graphs G1 , · · · , Gn .

3 Model Consider an agent network consisting of N agents. We assume that the kinematics of each agent has the following form:

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r˙i1 (t) = ri2 (t) + f1 (ri1 (t)) r˙i2 (t) = ri3 (t) + f2 (ri1 (t), ri2 (t)) .. .

(1)

r˙i(n−1) (t) = rin (t) + fn−1 (ri1 (t), . . . , ri(n−1) (t)) r˙in (t) = ui (t), where ri j (t) ∈ IR, ui (t) ∈ IR are used to represent the state and the control input linked with agent i, and each f j (ri1 (t), ri2 (t), ri3 (t), . . . , ri j (t)) : IR j → IR is a certain nonlinear function which might be used to denote the uncertainties or other nonlinearities. Assumption 1. For all i ∈ {1, · · · , N} and j ∈ {1, · · · , n−1}, each

∂ n−1 fi j (ri1 ,··· ,ri j ) α

α

αj

∂ ri11 ∂ ri22 ···∂ ri j

exists

for all ri1 , · · · , ri j ∈ IR and every α1 , α2 , · · · , αn non-negative integers, such that n − 1 = α1 + α2 + · · · + α j . Our main purpose is to introduce distributed algorithms to enable all agents cooperatively achieve a consensus convergence as time evolves, i.e., limt→+∞ [ri (t)−r j (t)] = 0.

4 Main Results 4.1

Model Transformations

Because of the existence of strong nonlinearities caused by fi1 (ri1 (t)), · · · , fi(n−1) (ri1 (t), . . . , ri(n−1) (t)) and the integral coupling of states, performing the analysis directly on the system (1) is of great difficulty and some transformation needs to be made to find the system nature according to the idea of the backstepping approach. In the following, we make a model transformation in n steps to convert the original system into a linear-like system. Step 1: Let r˘i1 (t)  ri1 (t)

(2)

ζi1 = 0.

(3)

and

Step 2: Let r˘i2 (t) 

1 σi ri2 (t) + ζi2 (ri1 (t)),

(4)

where σi > 0 is a constant and

ζi2 (ri1 (t)) =

1 γi f i1 (ri1 (t)) + r˘i1 (t).

(5)

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Then, r˙˘i1 (t) = ri2 (t) + fi1 (ri1 (t)) = ri2 (t) + fi1 (ri1 (t)) + σi r˘i1 (t) − σi r˘i1 (t) = σi (˘ri2 (t) − r˘i1 (t)). That is, r˙˘i1 (t) = σi (˘ri2 (t) − r˘i1 (t)). Step p (2 ≤ p ≤ n): Let r˘ip (t) 

1 p−1

σi

rip (t) + ζip (ri1 (t), · · · , ri(p−1) (t)),

(6)

where

ζip (ri1 (t), · · · , ri(p−1) (t)) 1 = p−1 fi(p−1) (ri1 (t), · · · , ri(p−1) (t)) σi (ri1 (t), · · · , r (t)) + r˘ + 1 ζ˙ σi i(p−1)

i(p−2)

(7)

i(p−1) (t).

It follows that r˘˙i(p−1) (t) 1 r˙i(p−1) (t) + ζ˙i(p−1) (ri1 (t), · · · , ri(p−2) (t)) = p−2 =

σi

fi(p−1) (ri1 (t), · · · , ri(p−1) (t)) ˙ +ζi(p−1) (ri1 (t), · · · , ri(p−2) (t)) +σi r˘i(p−1) (t) − σi r˘i(p−1) (t) = σi (˘rip (t) − r˘i(p−1) (t)). 1

p−2

σi

rip (t) +

1

p−2

σi

That is, r˙˘i(p−1) (t) = σi (˘rip (t) − r˘i(p−1) (t)).

(8)

From the definition in (6), we have that r˘in (t) =

1 r (t) + ζin (ri1 (t), · · · , ri(n−1) (t)). σin−1 in

(9)

Under Assumption 1, it follows that ζip (ri1 (t), · · · , ri(p−1) (t)) and ζ˙ip (ri1 (t), · · · , ri(p−1) (t)) both exist for all p ∈ {2, · · · , n} and can be obtained by recursively calculating (2)–(7). Moreover, from (2)–(7), it is easy to see that the values of ζih (ri1 (t), · · · , ri(p−1) (t)) and ζ˙ip (ri1 (t), · · · , ri(p−1) (t)) are only related to the first p − 1 and the first p components of ri (t) for all p ∈ {2, · · · , n}. Calculating r˙˘in (t), then r˙˘in (t) = ζ˙n (ri1 (t), · · · , ri(n−1) (t)) +

1 u (t). rin−1 i

(10)

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Let r˘i (t) = [˘ri1 (t), r˘i2 (t), r˘i3 (t), · · · , r˘in (t)]T , ⎡ −σi σi 0 · · · ⎢ 0 −σi σi · · · ⎢ ⎢ .. .. . . Ai = ⎢ ⎢ 0 0 ⎢ . . . .. . . −σ ⎣ .. i 0 0 ··· 0

⎤ 0 0⎥ ⎥ .. ⎥ n×n .⎥ ⎥∈R ⎥ σi ⎦ 0

and  T B = 0 0 · · · 0 1 ∈ Rn×1 . From (6), (8) and (10), the system (1) can be transformed into r˙˘i (t) = Ai r˘i (t) + Bζ˙n (ri1 (t), · · · , ri(n−1) (t)) + Bui (t)

(11)

for all i. 4.2

Algorithms and Theorems

Observing the form of the Eq. (11), for the sake of addressing the consensus problem mentioned in Sect. 3, we propose two algorithms as follows. (I) Consensus algorithm for dynamically changing graphs and zero communication delays. ui (t) = −σin−1 ζ˙n (ri1 (t), · · · , ri(n−1) (t)) + σin ri1 (t) −σi rin (t) − σin ζin (ri1 (t), · · · , ri(n−1) (t))) −σin−1 ∑ j∈Ni (t) ai j (t)(ri1 (t) − r j1 (t)).

(12)

(II) Consensus algorithm for dynamically changing graphs and arbitrarily bounded communication delays. ui (t) = −σin−1 ζ˙n (ri1 (t), · · · , ri(n−1) (t)) + σin ri1 (t) −σi rin (t) − σin ζin (ri1 (t), · · · , ri(n−1) (t))) −σin−1 ∑ j∈Ni (t) ai j (t)(ri1 (t) − r j1 (t − τi j )),

(13)

where 0 ≤ τi j ≤ η1 is used to represent the communication delays from agent j to agent i for some constant η1 . It should be noted that the constant η1 can be arbitrarily large, i.e., the communication delays can be arbitrarily bounded large. Assumption 2. Let s0 , s1 , s2 , · · · , be a sequence of switching times of the directed graph G (t) such that s0 = 0, 0 < sk+1 − sk ≤ η2 for some constant η2 > d. Suppose that the union of all the graphs in [sk , sk+1 ) is strongly connected. Theorem 1. Under Assumptions 1 and 2, if ∑ j∈Ni (t) ai j (t) ≤ ρ < σi for a certain constant ρ > 0 and all i, for the system (1) with algorithm (12), all agents achieve cooperatively a consensus state as time evolves, i.e., limt→+∞ [ri (t) − r j (t)] = 0 for all i, j.

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Theorem 2. Under Assumptions 1 and 2, if ∑ j∈Ni (t) ai j (t) ≤ ρ < σi for a certain constant ρ > 0 and all i, for the system (1) with algorithm (13), all agents achieve cooperatively a consensus state as time evolves, i.e., limt→+∞ [ri (t) − r j (t)] = 0 for all i, j. Next, detailed proofs of the Theorems 1 and 2 will be given. Proof of Theorem 1. From (2), (9), (11) and (12), we have that r˙˘in (t) = σi r˘i1 (t) − σi r˘in (t) − ∑ j∈Ni (t) ai j (t)(˘ri1 (t) − r˘ j1 (t)). Let

⎡ −σi σi 0 · · · ⎢ 0 −σi σi · · · ⎢ ⎢ .. .. . . Ei = ⎢ ⎢ 0 0 ⎢ . . . .. . . −σ ⎣ .. i σi 0 · · · 0

0 0 .. .

(14)

⎤

⎥ ⎥ ⎥ ⎥ ∈ Rn×n . ⎥ ⎥ σi ⎦ −σi

Let r˘i (t) = [˘ri1 (t)T , r˘i2 (t)T , r˘i3 (t)T , · · · , r˘in (t)T ]T . From (11) and (14), we have that r˙˘i (t) = Ei r˘i (t) − B ∑ j∈Ni (t) ai j (t)(˘ri1 (t) − r˘ j1 (t))

(15)

for all i. Let Z(t) = [˘r1 (t)T , r˘2 (t)T , r˘3 (t)T , · · · , r˘N (t)T ]T , E = diag{E1 , · · · , EN } and ⎤ ⎡ 0 ··· 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ .. . . . . ⎢ . . ⎥ B˜ = ⎢ . ⎥ ∈ Rn×n . ⎥ ⎢ ⎥ ⎢ ⎣0 0⎦ 1 0 ··· 0 The Eq. (15) is able to be written as ˙ = Ψ (t)Z(t), Z(t)

(16)

˜ That is, the system (1) with the algorithm (13) is able to be where Ψ (t) = E − L(t) ⊗ B. expressed in the form of (16). Since ∑ j∈Ni (t) ai j (t) ≤ ρ < σi for some constant ρ > 0 and all i, then the coefficient of the component r˘i1 (t) in Ψ (t) is nonnegative and lower bounded by a certain common positive constant. we observe the form of Ψ (t), which is able to be deemed as the Laplacian matrix of some graph, denoted by G¯(t). Recall that each nonzero ai j (t) satisfies that ai j (t) ≥ μ . Then, in accordance with the proofs of Theorem 1 in [2], limt→+∞ (˘ri (t) − r˘ j (t)) = 0 for all i, j can be proved eventually. This implies from (2) that limt→+∞ (ri1 (t) − r j1 (t)) = 0 and hence limt→+∞ ( f1 (ri1 (t)) − f1 (r j1 (t))) = 0 for all i, j. In consequence, the Eq. (4) yields that limt→+∞ (ri2 (t) − r j2 (t)) = 0 and hence

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limt→+∞ ( f2 (ri2 (t), ri1 (t)) − f2 (r j2 (t), r j1 (t))) = 0 for all i, j. By analogy, we can draw the conclusion that limt→+∞ (ri (t) − r j (t)) = 0 for all i, j. Proof of Theorem 2. Combining the analysis in Theorem 1 and the approach in Theorems 1 and 2 in [2], this theorem can be similarly proved.

5 Conclusions In this paper, the distributed cooperation consensus problem for a class of nonlinear multiagent systems with the consideration of changing dynamically graphs and bounded arbitrarily communication delays is investigated. A new distributed algorithm has been presented to enable all agents’ states eventually converge to a common point. We show that the consensus problem is able to be tackled with when the union of all the graphs is connected strongly and the communication delays have no effect on the stability of the systems. Acknowledgments. This work was supported by the Fundamental Research Funds for the Central Universities of Central South University (2019zzts564).

References 1. Ren W, Atkins E (2010) Distributed multi-vehicle coordinated control via local information exchange. Int J Robust Nonlinear Control 17(10–11):1002–1033 2. Moreau L (2004) Stability of continuous-time distributed consensus algorithms. arXiv:math.OC/0409010v1 3. Jia Y (2000) Robust control with decoupling performance for steering and traction of 4WS vehicles under velocity-varying motion. IEEE Trans Control Syst Technol 8(3):554–569 4. Jia Y (2003) Alternative proofs for improved LMI representations for the analysis and the design of continuous-time systems with polytopic type uncertainty: a predictive approach. IEEE Trans Autom Control 48(8):1413–1416 5. Cao M, Morse AS, Anderson BDO (2006) Reaching an agreement using delayed information. In: Proceedings of IEEE conference on decision and control, pp 3375-3380 6. Hong Y, Gao L, Cheng D, Hu J (2007) Lyapunov-based approach to multiagent systems with switching jointly connected interconnection. IEEE Trans Autom Control 52(5):943–948 7. Hu J, Cao J, Yu J, Hayat T (2014) Consensus of nonlinear multi-agent systems with observerbased protocols. Syst Control Lett 72:71–79 8. Li X, Wu H, Yang Y (2017) Consensus of heterogeneous multi-agent systems with arbitrary bounded communication delay. Math Probl Eng 2017:1–6 9. Lin P, Jia Y, Li L (2008) Distributed robust H∞ consensus control in directed networks of agents with time-delay. Syst Control Lett 57(8):643–653 10. Lin P, Jia Y (2009) Consensus of second-order discrete-time multi-agent systems with nonuniform time-delays and dynamically changing topologies. Automatica 45(9):2154– 2158 11. Lin P, Jia Y (2010) Consensus of a class of second-order multi-agent systems with time-delay and jointly-connected topologies. IEEE Trans Autom Control 55(3):778–784 12. Lin P, Li Z, Jia Y, Sun M (2011) High-order multi-agent consensus with dynamically changing topologies and time-delays. IET Control Theory Appl 5(8):976–981

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13. Yang T, Jin Y, Wang W, Shi Y (2011) Consensus of high-order continuous-time multi-agent systems with time-delays and switching topologies. Chin Phy B 20(2):020511 14. Lin P, Ren W, Gao H (2017) Distributed velocity-constrained consensus of discrete-time multi-agent systems with nonconvex constraints, switching topologies, and delays. IEEE Trans Autom Control 62(11):5788–5794 15. Wang Y, Li Q, Xiong Q (2018) Distributed consensus of high-order continuous-time multiagent systems with nonconvex input constraints, switching topologies, and delays. Neurocomputing. https://doi.org/10.1016/j.neucom 16. Lin P, Ren W, Farrell JA (2017) Distributed continuous-time optimization: nonuniform gradient gains, finite-time convergence, and convex constraint set. IEEE Trans Autom Control 62(5):2239–2253 17. Lin P, Ren W, Yang C, Gui W (2019) Distributed optimization with nonconvex velocity constraints, nonuniform position constraints and nonuniform stepsizes. IEEE Trans Autom Control 64(6):2575–2582 18. Lin P, Ren W, Yang C, Gui W (2019) Distributed continuous-time and discrete-time optimization with nonuniform unbounded convex constraint sets and nonuniform stepsizes. IEEE Trans Autom Control. https://doi.org/10.1109/TAC.2019.2910946, 19. Kanellakopoulos I, Kokotovic PV, Morse AS (1991) Systematic design of adaptive controllers for feedback linearizable systems. IEEE Trans Autom Control 36(11):1241–1253 20. Krstic M, Kanellakopoulos I, Kokotovic PV (1992) Adaptive nonlinear control without overparametrization. Syst Control Lett 19:177–185 21. Freeman RA, Kokotovic PV (1992) Backstepping design of robust controllers for a class of nonlinear systems. In: Proceedings of the IFAC nonlinear control systems design, Bordeaux, France, pp 431–436 22. Godsil C, Royle G (2001) Algebraic Graph Theory. Springer-Verlag, New York 23. Horn RA, Johnson CR (1987) Matrix Analysis. Cambridge University Press, Cambridge

Cooperative Control of High-Speed Train Systems With Velocity Constraints and Parameter Uncertainties Yi Huang1 , Shuai Su2 , Hailiang Hou1,3 , and Yonggang Li1(B) 1

School of Automation, Central South University, Changsha 410083, China {yihcsu,liyonggang}@csu.edu.cn, [email protected] 2 State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China [email protected] 3 School of Information, Hunan University of Humanities, Science and Technology, Loudi 417000, China

Abstract. This paper investigates a cooperative control problem of high-speed train systems with velocity constraints and parameter uncertainties. A nonlinear control algorithm using local communication is adopted. The control algorithm includes two parts: one is to completely compensate the certain nonlinear parts of system dynamics and the other is to deal with the uncertain nonlinear terms by using the dynamic feedback damping gain. Based on mode transformation technique and convexity analysis method, the system conditions are deduced to guarantee that all trains eventually stop in a coordinated manner and maintain a distance from their neighbour trains. Keywords: Cooperative control · High-speed train systems · Velocity constraints · Parameter uncertainties

1 Introduction Recently, the advanced automatic high-speed train control has received much attention from the researchers [1–18]. The early train operation control works were focused on the single train control [1, 2]. However, with the increase in the operating density and velocity of the high-speed trains, single train operation control is difficult to meet the needs of high efficiency and safety. Hence, some researchers have turned their attention to cooperative control for multiple high-speed trains. For example, articles [3] and [4] investigated the coordination operation control problem of multiple high-speed trains with consideration of nonlinear operation resistance, where each train adjusts its states by using the neighbouring trains’ information. However, the works in [1–4] mainly focused on the train operation control under ideal environment without the velocity constraints. In the practical situation, restrict by the practical factors, such as limitation of traction system of high-speed train and so on, the velocity of high-speed train is usually limited in a range. To our knowledge, c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 716–723, 2020. https://doi.org/10.1007/978-981-32-9682-4_75

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there are no available results on coordination control of multiple high-speed trains in a railway line under the restrict of velocity. Many works on constrained cooperative control problem can be found in the field of multi-agent systems [7–13]. For example, the consensus problem for multi-agent systems with input constraints was studied in [7] and [8], while the case of position constraints was investigated in [9–11]. Recently, the nonconvex velocity constraints for consensus problem have been considered in [12] and [13], where the novel nonconvex constraint operator is used to deal with the nonlinearity of nonconvex constraints, but the results are applicable to linear dynamics and did not consider the situation of parameter uncertainties. It is an effective way to consider the efficiency and safety of the multiple high-speed trains coordination operation through describing the high-speed train system as a multi-agent system. Based on the multiagent method, the scheduling optimization problem of high-speed train system without constraints and parameter uncertainties were solved in [14]. In this paper, we devote to solve the cooperative control problem of multiple highspeed trains with the consideration of velocity constraints in the presence of nonlinear resistance and resistance coefficient uncertainty. A nonlinear distributed control law is proposed to drive all high-speed trains move in a coordination way. Through a model transformation, the origin system is transformed into an equivalent time-varying system whose system matrix is a stochastic matrix. Finally, by using convexity analysis methods, some conditions are deduced to guarantee that all trains stop in a coordinated manner. Notation: IR denotes the set of real numbers. In represents the n × n dimensional unit matrix. For a diagonal matrix diag{Q1 , · · · , Qm }, each diagonal entry is a matrix Qi , i = 1, · · · , m. diag{Q} is used to represent a diagonal matrix with its all diagonal entries being Q. Qi j is used to denote the i jth element of the matrix Q. The Kronecker product is denoted as ⊗. 1 represents a column vector of suitable dimension where all the entries are equal to 1, i.e., [1, 1, · · · , 1]T .

2 Preliminaries and Problem Statement Suppose that a high-speed train system consists of n high-speed trains. Each train is treated as a particle and the relationship of trains is described by using algebraic graph theory as follows [19]. Let G (I , E , A ) represents a graph including n orders, where I = {1, · · · , n} denotes a set of n nodes, E ⊆ I × I denotes the set of edges, and A = [ai j ] represents a weighted adjacency matrix. ei j = (i, j) is used to denote an edge of G . If ei j ∈ E , ai j ≥ μ > 0 and ai j = 0 otherwise, where μ is a positive constant. Ni = { j ∈ I : (i, j) ∈ I } is used to denote the neighbor set for i. Let L = [li j ] be the Laplacian matrix of the graph G , and lii = ∑nj=1 ai j and li j = −ai j , i = j. L(k)ii = Σ nj=1 ai j for all i = j. Therefore, the train corresponds to a node in the graph G (kT ). For the edge (i, j) ∈ E , it represents that train i and train j can communicate with each other. Usually, for a matrix C = [ci j ] ∈ IRRn×n , if all the elements of C are positive, i.e., ci j ≥ 0, and C1 = 1, then the matrix is a stochastic matrix [20].

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Suppose that the dynamics of each high speed train with velocity constraints has the following form xi (k + 1) = xi (k) + vi (k)T, (1) vi (k + 1) = Sat[vi (k) + ui (k)T − fi (k)T ], where xi (k) ∈ IR, vi (k) ∈ IR and ui (k) ∈ IR denote the operating position, velocity and control input of train i, respectively, fi (k) denotes the running resistance of train i. Sat[vi ] denotes the saturation operation of velocity such that

Sat[vi ]

=

⎧ ⎪ ⎨vi , 0 ≤ vi ≤ α i , α i , vi > α i , ⎪ ⎩ 0, vi < 0,

(2)

where α i > 0 denote the maximum velocity of train i. It is clear that vi (k + 1) =Sat[vi (k) + ui (k)T − (c0 + c¯v vi (k) + c¯a v2i (k))T ] ∈ [0, α i ]. Here, we assume that the initial conditions of xi (k) and vi (k) satisfy the system dynamics of (1) for all k ≤ 0 and all i. In this paper, we consider the operation resistance consist of rolling mechanical resistance, aerodynamic resistance and so on, which is given by Davis formula as follows [21] fi (k) = c0 + c¯v vi (k) + c¯a v2i (k), where c0 , c¯v , c¯a are coefficients of the resistance and satisfy the following assumption. Assumption 1. Suppose that c¯v = cv + Δ cv and c¯a = ca + Δ ca , where cv and ca are all constants, |Δ cv | ≤ β1 and |Δ ca | ≤ β2 denote the uncertainties of the corresponding coefficients for positive constants β1 and β2 . The control target is to keep a distance from the front train, i.e., xi − x j >= d¯i j , where d¯i j = di − d j , di and d j denote the position deviation constants. The constraint of the train operation is assumed that successive tracking trains stop with a safety margin and the speed of all trains are 0.

3 Main Results To solve this problem, we propose the following control algorithm ui (k) = ui1 (k) + ui2 (k), ui1 (k) = c0 + cv vi (k) + ca v2i (k), ui2 (k) = −hi (k)vi (k) + ∑ ai j (x j (k)−xi (k)+d¯i j ),

(3)

j∈N i (k)

for all k ≥ 0, where hi (k) is the dynamic feedback coefficient which will be designed later, ai j ≥ μ if j ∈ Ni (k) and ai j = 0 otherwise. The control algorithm is divided into two parts, ui1 (k) is used to compensate the running resistance fi (k), while ui2 (k) is used to guarantee the control objective. Note that

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the system dynamics (1) still has a uncertain nonlinear term Δ ca v2i (k). From Assumption 1, since the velocity is subject to constraints, we have |Δ ca vi (k)| ≤ β2 α¯ i , for all k ≥ 0. Define x˜i (k) = xi (k) − di and qi (k) = hi (k) + Δ cv + Δ ca vi (k) for all i and all k. Since the velocity of each train is bounded in its own closed and nonempty constraint interval, qi (k) is also bounded. Then, the system (1) with (3) has the following form. x˜i (k + 1) = x˜i (k) + vi (k)T, vi (k + 1) = Sat[vi (k) − qi (k)vi (k)T +

∑

ai j (x˜ j (k) − x˜i (k))T ].

(4)

j∈N i (k)

Define Sat[vi (k)−qi (k)vi (k)T +

pi (k) =

∑

ai j (x˜ j (k) − x˜i (k))T ]

j∈N i (k)

∑

vi (k)−qi (k)vi (k)T +

,

ai j (x˜ j (k) − x˜i (k))T

(5)

j∈N i (k)

for all k ≥ 0. Let pi (k) = 1 when vi (k) − qi (k)vi (k)T +

∑

ai j (k)(x˜ j (k) − x˜i (k))T j∈N i (k) 1−pi (k)(1−qi (k)T ) . Then, we have pi (k)(1 T

0. Clearly, 0 < pi (k) ≤ 1. Let ri (k) = qi (k)T ) = 1 − ri (k)T . From (4) and (5), we have Sat[vi (k) − qi (k)vi (k)T + = (1 − ri (k)T )vi (k) + pi (k)

∑

−

ai j (x˜ j (k) − x˜i (k))T ]

j∈N i (k)

∑

=

ai j (x˜ j (k) − x˜i (k))T.

(6)

j∈N i (k)

Assumption 2. Suppose that there exists a time sequence of k0 , k1 , k2 , · · · in the operation process of multiple high-speed trains, where k0 = 0, 0 < kn+1 − kn ≤ η for all n, where η is a positive integer, and the union of the communication graphs G (kn ), G (kn + 1), · · · , G (kn+1 − 1) has directed spanning trees. Assumption 3. Suppose that β1 + β2 α¯ i < hi (k) < T1 − β1 − β2 α¯ i , 0 < ri (k) ≤ qi (k + 1) < T1 and ∑ j∈I ai j ≤ ci < (hi (k) − β1 − β2 α¯ i )2 /4, for all i, k, where ci > 0 is a constant. 2 Let v˜i (k) = x˜i (k) + ri (k) vi (k). Then the system (4) is equivalent to ri (k)T x˜i (k + 1) = ri (k)T 2 v˜i (k) + (1 − 2 )x˜i (k) ri (k)T ri (k) v˜i (k + 1) = (1 − 2 − ri (k+1) (1 − ri (k)T ))x˜i (k) i (k) (1 − ri (k)T ) + ri (k)T +( rir(k+1) 2 )v˜i (k) 2 pi (k) ∑ ai j (x˜ j (k) − x˜i (k))T + ri (k+1)

(7)

j∈N i (k)



r (k)

1− i 2 T i 2 T with B21 (i, k) B21 (i,k) B22 (i,k) ri (k) ri (k)T ¯ ri (k+1) (1 − ri (k)T ) + 2 , E(k)

Let Ei (k) = and B22 (i, k) =

r (k)

 = 1−

ri (k)T 2

−

ri (k) ri (k+1) (1

− ri (k)T )

= diag{E1 (k), · · · , En (k)}, Di (k) =

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Y. Huang et al. 0

2T ri (k+1)



0 0

¯ , D(k) = diag{D1 (k), · · · , Dn (k)} and P(k) = diag{p1 (k), p2 (k), · · · , pn (k)}.

Denote ξ (k) = [x˜1 (k), v˜1 (k), · · · , x˜n (k), v˜n (k)]T . Then, (7) can be rewritten as ¯ − (P(k)L(k) ⊗ I2 )D(k)) ¯ ξ (k + 1) = (E(k) ξ (k).

(8)

¯ − (P(s)L(s) ⊗ I2 )D(s). ¯ Let Θ (k, l) = ∏ks=l E(s) It can be deduced the relation between ξ (k + 1) and ξ (l) for all k ≥ l ≥ 0 is as follows

ξ (k + 1) = Θ (k, l)ξ (l).

(9)

¯ ¯ In the following, the non-negative property of the system matrices E(k), E(k) − ¯ (P(k)L(k) ⊗ I2 )D(k) and Θ (k, l) were investigated in Lemma 1. Lemma 2 studies the lower bound of saturation operator pi (k). Lemma 3 studies the lower boundedness of ¯ ¯ − (P(k)L(k) ⊗ I2 )D(k). each valid entry in E(k) ¯ ¯ − (P(k)L(k) ⊗ I2 )D(k) ¯ Lemma 1. Under Assumption 3, for all k ≥ l ≥ 0, E(k), E(k) and Θ (k, l) are stochastic matrices. ¯ ¯ Proof. From the definition of E(k), we have E(k)1 = 1. Note that L(k)1 = 0. ¯ ¯ ¯ ¯ Thus, (E(k) − (P(k)L(k) ⊗ I2 )D(k))1 = E(k)1 = 1. Hence, each row sum of E(k) − ¯ is 1. It follows that Θ (k, l)1 = 1. (P(k)L(k) ⊗ I2 )D(k) According to the definition of qi (k) and bi (k), from Assumption 3, it is easy to see that (10) 0 < qi (k) ≤ ri (k) ≤ qi (k + 1) < T1 . ri (k) 1 ¯ Then, 1 − ri (k) 2 T > 2 and 2 T > 0. Thus, all entries of E(k) are nonnegative. In addition, by selecting proper nonzero ai j , it can be obtained that ∑ j∈I ai j ≤ ci for some constant ci < (hi (k) − β1 − β2 α¯ i )2 /4. From (10), it follows that ∑ j∈I ai j
2 − ri (k) ci . Note that ri (k) ≥ (hi (k)+ β1 + β2 αi ) 2T β1 − β2 α¯ i )2 > 4ci . It follows that ri (k)T 2 − ri (k) ci > 0. Hence, all entries

> (hi (k)−

¯ of E(k) − ¯ (P(k)L(k) ⊗ I2 )D(k) are nonnegative. Hence, all entries of the matrices Θ (k, l) are nonnegative. ¯ ¯ ¯ − (P(k)L(k) ⊗ I2 )D(k) and Θ (k, l) are stochastic matrices, In summary, E(k), E(k) for all k ≥ l ≥ 0. Lemma 2. For all i, k,

|α i | ( T1 +2T ci ){|ξ j (0)|}

≤ pi (k) ≤ 1.

Proof. From (9), we can deduce that ξ (k) = Θ (k − 1, 0)ξ (0). From Lemma 1, it is easy to see that the relationship ξi (k) with ξ j (0) is a convex combination. Then, it can be obtained that ξi (k) ≤ max j ξ j (0), for all i. It follows that 1 |vi (k) − qi (k)vi (k)T | ≤ | max j {ξ j (0)}|, T

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and

∑

|

ai j (x˜ j (k) − x˜i (k))T | ≤ |2T ci max j {ξ j (0)}|.

j∈N i (k)

Then, we have

∑

|vi (k) − qi (k)vi (k)T +

ai j (x˜ j (k) − x˜i (k))T | ≤ |(

j∈N i (k)

1 + 2T ci )max j {ξ j (0)}|. T

From (2), we have Sat[vi (k) − qi (k)vi (k)T +

∑

ai j (x˜ j (k) − x˜i (k))T ] ≥ α i .

j∈N i (k)

Thus, Sat[vi (k) − qi (k)vi (k)T + ∑ ai j (x˜ j (k) − x˜i (k))T ] pi (k) =

j∈N i (k)

vi (k) − qi (k)vi (k)T +

∑

ai j (x˜ j (k) − x˜i (k))T

j∈N i (k)

≥

( T1

|α i | . + 2T ci )max j {|ξ j (0)|}

Particularly, pi (k) = 1 when vi (k) − qi (k)vi (k)T + ∑ j∈N i (k) ai j (x˜ j (k) − x˜i (k))T = Sat(vi (k) − qi (k)vi (k)T + ∑ j∈N i (k) ai j (x˜ j (k) − x˜i (k))T ). ¯ ¯ Lemma 3. Under Assumption 3, each nonzero entry of E(k)−(P(k)L(k)⊗I has 2 )D(k) a positive lower bound, for all i and all k. 1 Proof. From Lemma 2 and Assumption 3, we have 1 − ri (k) 2 T > 2 , ri (k) ≤ qi (k + 1) i (k) − rir(k+1) (1 − ≤ ri (k + 1) and ri2 (k) ≥ (hi (k) + β1 + β2 α¯ )2 > 4ci . Then, 1 − ri (k)T 2

ri (k)T ) >

ri (k)T 2

>

1 2

and

ri (k)T 2

− r2T ci ≥ i (k)

(hi (k)+β1 +β2 α¯ )T 2

ci − (hi (k)+2T β +β 1

¯ )T 2α

.

¯ ¯ − (P(k)L(k) ⊗ I2 )D(k) and Note that ai j > μ . Hence, each nonzero entry of E(k) Θ (k, l) has a positive lower bound. Lemma 4 ([13]). Under Assumptions 2 and 3, there exist two constants 0 ≤ ei (l) ≤ 1 for any l ≥ 0 and 0 < σ ≤ 1 such that limk→+∞ Θ (k, l)hi = ei (l) for all h and k

Θ (k, l)i j − e j (l) ≤ C(1 − σ ) nˆ μ for all k ≥ l, where i ∈ {1, · · · , 2n}, ∑2n i=1 ei (l) = 1, k −2 C = (1 − σ ) , nˆ ≥ 4n, h ∈ {1, · · · , 2n} is a positive integer and h > nˆη − 1. Theorem 1. Under Assumptions 1–3, the control algorithm (3) can achieve the cooperative control of multiple high-speed trains with velocity constraints and parameters uncertainties (1), i.e., lim |xi (k) − x j (k)| = |d¯i j | and lim |vi (k)| = 0. k→+∞

k→+∞

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Proof. From Lemma 3, we have Θ (k, l) is a stochastic matrix for any k ≥ l. Let xi (0) and vi (0) satisfy the dynamic of (1). Let l = 0 and xˆ = ∑2n j=1 e j (0)ξ j (0). From 2n ˆ = | ∑ j=1 Θ (k − 1, 0)i j ξ j (0) − ∑2n lemma 4, it follows that |ξi (k) − x| j=1 e j (0)ξ j (0)| = k−1

nˆ η | ∑2n ∑2n j=1 (Θ (k − 1, 0)i j − e j (0))ξ j (0)| ≤ |C(1 − σ ) j=1 ξ j (0)|, where xˆ ∈ R is the position when the state x¯i reach a consensus as k → +∞, for all i. Hence, ˆ = 0 and as a result limk→+∞ |x¯i (k) − x| ˆ = limk→+∞ |v¯i (k)| = 0. By limk→+∞ |ξi (k) − x| simple calculation, we have limk→+∞ |xi (k) − x j (k)| = |di j | and lim |vi (k)| = 0. Thus

k→+∞

the proof is completed. Note that the gain hi (k) in (3) is dynamic and can be adjusted in real time according to its own state and the information of the neighbour train. In fact, the cooperative control objective also can be achieved if the gain is a constant. By selecting hi (k) = hi , we have the following controller ui (k) = ui1 (k) + ui2 (k), ui1 (k) = c0 + cv vi (k) + ca v2i (k), ui2 (k) = −hi vi (k) + ∑ ai j (x j (k)−xi (k)+d¯i j ).

(11)

j∈N i (k)

Let v˜i (k) = x˜i (k) + ωi vi (k), where ωi = qi2(k) is a constant. Then, the corresponding result can be obtained in the following corollary. Corollary 1. Consider the high-speed train systems (1) with velocity constraints and parameters uncertainties. Assume the union of the communication graphs has spanning tree. Then, the controller (11) solves cooperative problem of (1) if the parameters hi and ci satisfy the following conditions:

β1 + β2 α¯ i < hi
ds ,

t ≥ 0,

where ds is the safe distance. For convenience, the circumnavigation problem of a group of targets with the safe-distance constraint can be called as safe-circumnavigation. We now formally state the safe-circumnavigation problem for multiple moving targets studied in this paper. Problem 1. Consider n non-stationary targets with unknown position, and a single agent with only bearing measurements of each target. The task is to establish targets’ position estimators and a control protocol to drive the agent to achieve safecircumnavigation around the targets. Remark 1. Comparing with the case of single target [13], there is new characteristic for the safe-circumnavigation problem of multiple targets. For single target, the circling center is the target, and its bearing information can directly be used by the agent. While for multiple targets, the bearing information of the targets’ center x (t) can not be directly used, where the estimated center of the targets is used as the circling center. Let xˆi (t) be the estimated position of each target and denote xˆ (t) as the estimated center of the targets, that is xˆ (t) 

1 n ∑ xˆ i (t) n i=1

and denote ϕ (t) as the unit vector in the direction of the line going from y (t) to xˆ (t), that is

ϕ (t) 

xˆ (t) − y (t) . ˆx (t) − y (t)

(2)

Let ϕ¯ (t) be the unit vector obtained by π /2 clockwise rotation of ϕ (t) (see Fig. 1). Also, let x˜ (t) be the estimation error for the center of the target, that is x˜ (t)  xˆ (t) − x (t). The assumptions are stated as bellow. Assumption 1. It is assumed that the velocities of each target are bounded so there exists a positive scalar ε such that ˙x i (t) < ε for all t ≥ 0. Assumption 2. There exists a positive scalar r such that sup xxi (t) − x (t) ≤ r for all t ≥ 0.

i

Safe-circumnavigation

x xi

r

727

d xˆ

i

y i

Fig. 1. The relationship among y , x , x i and xˆ at time t.

Remark 2. The condition sup xxi (t) − x (t) ≤ r guarantees that the distance between i

the targets is bounded, which is a necessary condition for the agent to circumnavigate the targets. The following proposition is also needed, which is borrowed from [11]. Proposition 1. A time-varying matrix ω (t): R+ → Rn×m ∀n, r ≥ 1 is persistently exciting if there exist some positive ε1 , ε2 , T such that

ε1 ≤

 t0 +T t0

(ω T (t)ξ )2 dt ≤ ε2 ,

∀t0 ∈ R+ .

(3)

where ξ ∈ Rn can be any constant unit vector. Consider the differential equation x˙ (t) = −ω (t)ω T (t)xx(t), where ω (t) is an n × m time-varying regulated matrix. Then x is exponentially asymptotically stable iff ω (t) is persistently exciting.

3 Proposed Solution In order to solve Problem 1, we adopt estimators and a control protocol like [12] to perform circumnavigation and find some conditions to guarantee safety. The estimator is   (4) x˙ˆ i (t) = I − ϕ i (t)ϕ Ti (t) (yy(t) − xˆ i (t)) . Here I is the 2 × 2 identity matrix, and ϕ i (t) is given by (1), then ϕ i (t) ϕ Ti (t) is a projection matrix onto the vector ϕ i (t). The control protocol is y˙ (t) = (yy(t) − xˆ (t) − d) ϕ (t) + λ ϕ¯ (t).

(5)

Here, λ is a positive scalar, which represents the tangential velocity of the agent. For convenience, we define ρ (t)  xx(t) − y (t), ρi (t)  xxi (t) − y (t), ρˆ (t)  ˆx (t) − y (t), and ρˆ i (t)  ˆx i (t) − y (t),

and denote ρˆ 0  ρˆ (0).

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S. Cao et al.

Theorem 1. Suppose Assumption 2 holds. If ˜x i (t) < δ , ρˆ i (0) ≥ δ + r¯ and d ≥ 2δ + r¯, then ρi (t) > ds and ρˆ i (t) = 0 for all t ≥ 0, where r¯ = r + ds . Proof. To prove the theorem, we only need to demonstrate that: (i) ρ (t) − ds > r; (ii) ρˆ (t) > 0 for all t ≥ 0. Considering (2), one has   T ρ˙ˆ (t) = x˙ˆ (t) − y˙ T (t) ϕ (t). Note that x˙ˆ (t) = x˙˜ (t) + x˙ (t) and recall the definition of ϕ¯ i (t). Substituting y˙ (t) of (5) into above equation, we obtain that T ρ˙ˆ (t) = −ρˆ (t) + d + x˙˜ (t)ϕ (t) + x˙ T (t)ϕ (t).

By the definition of ϕ¯ i (t), (4) can be written as x˙ˆ i (t) = ϕ¯ i (t)ϕ¯ Ti (t) (yy(t) − xˆ i (t)) . Considering (1), one gets 0 = ϕ¯ i (t)ϕ¯ Ti (t) (yy(t) − x i (t)) . Take difference between the above two equations, and note that x˙˜ (t) = x˙ˆ (t) − x˙ (t), it follows that x˙˜ i (t) = −ϕ¯ i (t)ϕ¯ Ti (t)˜x i (t) − x˙ i (t).

(6)

According to x˙˜ (t) = 1n ∑ni=1 x˙˜ i (t), then the solution is

ρˆ (t) − d = e−t (ρˆ 0 − d) −

1 n ∑ n i=1

 t 0

e−(t−τ ) x˜ Ti (t)ϕ¯ i (t)ϕ¯ Ti (t)ϕ (t)dτ .

(7)

When |ρˆ 0 − d| ≥ δ , we know that |ρˆ (t) − d| < e−t |ρˆ 0 − d| + δ

 t 0

e−(t−τ ) dτ ≤ |ρˆ 0 − d| .

(8)

   and Ur+ (xx)  shown in  the Fig. 2, let Ur (xx)  z zz − x  < r  As  z zz − x  ≤ r (where x may be replaced by xˆ and also r may be another positive constant). Suppose Assumption 1 holds, the condition max xxi (t)−xx(t) ≤ r means that i

there may exist targets in the red background Ur+ (xx). Therefore, to prove the theorem we only need to find proper initial condition ρˆ 0 and parameter d for differential equation (5). To achieve the goal, since |ρˆ (t) − d| ≤ δ when |ρˆ 0 − d| = δ , we choose ρˆ 0 ≥ δ + r¯ and d ≥ 2δ + r¯. Obviously, by such a choice, the agent can not enter Uδ +¯r (ˆx ) from the dashed area, that is, (i) and (ii) hold. Then the proof of this theorem is completed.

 Remark 3. According to Theorem 1, ϕ (t), ϕ¯ (t) and ϕ i (t), ϕ¯ i (t) are always definable if collision never happens.

Safe-circumnavigation

δ d

ˆ

ds

r¯

729

δ

r

Fig. 2. The relationship among r, δ , d, x and xˆ .

Theorem 2. Suppose Assumption 1 and the conditions in Theorem 1 hold, then ρ (t)−d converges to a neighborhood of zero. Proof. According to Theorem 1, ρˆ (t) is bounded. Then with the condition that ˜x i (t) < δ , ˜x (t) < δ also holds. Since

ρ (t) = xx(t) − y (t) = [xx(t) − xˆ (t)] + [ˆx (t) − y (t)] ≤ xx(t) − xˆ (t) + ˆx (t) − y (t) = ˜x (t) + ρˆ (t), we obtain that ρ (t) is also bounded, which means that ρ (t) − d converges to a neighborhood of zero. 

Theorem 3. Suppose Assumptions 1, 2 and conditions in Theorem 1 hold, then for i = 1, 2, · · · , n, the solution of the homogeneous differential function of (6) z˙ i (t) = −ϕ¯ i (t)ϕ¯ Ti (t)zzi (t) exponentially goes to zero, where ε in Assumption 1 satisfies (i) if d < ρˆ 0 < 2d,

ε < min λ

(ii) if ρˆ 0 ≤ d,

ε < min λ



 2  2

2 R R R(ρˆ 0 − d) R R(ρˆ 0 − d)  , 1− ,λ 1− − ,λ 1− − d ρˆ 0 ρˆ 0 2d − ρˆ 0 2d − ρˆ 0



 2  2

2 R R R(d − ρˆ 0 ) R R(d − ρˆ 0 )  . 1− ,λ 1− − ,λ 1− − d ρˆ 0 ρˆ 0 2d − ρˆ 0 2d − ρˆ 0

Here R  δ + r. Proof. Similar to the proof of Lemma 4 in [13], for i = 1, 2, · · · , n, assume γu (t) is the angle from the unit vector U to the vector ϕ¯ i (t) (as shown in Fig. 4). Hence, U T ϕ¯ i (t) = cos γu (t). Then (3) can be written as

ε1 ≤

 t0 +T t0

cos2 γu (t)dt ≤ ε2 ,

∀t0 ∈ R+ .

(9)

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Noting that cos2 (·) ≤ 1, the upper bound ε2 obviously exists. We can think of μi as a stationary target. Consider the relative velocity v of the agent to the ith target. Then we have d γu (t) ϕ¯ Ti (t)vv = . dt ρ (t) According to (8), ρˆ (t) has an upper bound, so ρi (t) also has an upper bound ρi max , consequently ϕ¯ T (t)vv γu (t + t0 ) ≥ γu (t0 ) + i t. ρi max Therefore, if inf{ϕ¯ Ti (t)vv} > 0, γu (t) is monotone increasing and can not converge to a constant value, meaning that there always exist some positive ε1 and T that satisfy (9). Using Proposition 1, ϕ¯ i (t) is persistently exciting, then z i exponentially goes to zero. Hence, we only need to demonstrate that inf{ϕ¯ Ti (t)vv} > 0, which is equivalent to inf{ϕ¯ Ti (t)˙y } > ε . Note that

ϕ¯ Ti (t)˙y = (ρˆ (t) − d)ϕ¯ Ti (t)ϕ (t) + λ ϕ¯ Ti (t)ϕ¯ (t).

Denote θ as the acute angle between ϕ i (t) and ϕ (t). Then we will discuss the following two cases. 1. When ρˆ (t) > d, the limiting case is shown in Fig. 3. In this case, let f1 (θ )  ϕ¯ Ti (t)˙y = λ cos θ + d sin θ − R. 1 , θ 1 ), then Suppose θ ∈ (θmin max 1 1 inf{ϕ¯ Ti (t)˙y } = min{ f1 (θmin ), f1 (θmax )}. 1 = arcsin R . Considering (8), we know Obviously, θmax d

ρˆ (t) < d + |ρˆ 0 − d|. 1 > arcsin R ; if ρˆ ≤ d, then ρ ˆ (t) < 2d − ρˆ 0 Thus, if ρˆ 0 > d, then ρˆ (t) < ρˆ 0 and θmin 0 ρˆ 0 R 1 > arcsin and θmin . Hence, 2d−ρˆ 0



inf{ϕ¯ Ti (t)˙y} ≥ min λ

and inf{ϕ¯ Ti (t)˙y } ≥ min

λ



1−

 1−



R ρˆ 0

2

R 2d − ρˆ 0

R(ρˆ 0 − d) − ,λ ρˆ 0

2



R(d − ρˆ 0 ) − ,λ 2d − ρˆ 0

R  1 − ( )2 (when ρˆ 0 > d), d



R  1 − ( )2 (when ρˆ 0 ≤ d). d

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2. When ρˆ (t) < d, the limiting case is shown in Fig. 4. In this case, let f2 (θ )  ϕ¯ Ti (t)˙y = λ cos θ − d sin θ + R. 2 , θ 2 ), then Noting that f2 (θ ) is monotone decreasing, suppose θ ∈ (θmin max 2 inf{ϕ¯ Ti (t)˙y } = f2 (θmax ).

According to (8), we know

ρˆ (t) > d − |ρˆ 0 − d|. 2 < arcsin R ; if ρ ˆ 0 ≤ d, then Thus, if d < ρˆ 0 < 2d, then ρˆ (t) > 2d − ρˆ 0 and θmax 2d−ρˆ 0 R 2 ρˆ (t) > ρˆ 0 and θmax < arcsin ρˆ 0 . Hence,





R 1− 2d − ρˆ 0

inf{ϕ¯ Ti (t)˙y } ≥ λ



and inf{ϕ¯ Ti (t)˙y }

≥λ

1−

R ρˆ 0

2 −

R(ρˆ 0 − d) (when d < ρˆ 0 < 2d), 2d − ρˆ 0

−

R(d − ρˆ 0 ) (when ρˆ 0 ≤ d). ρˆ 0

2

Therefore, if ε satisfies the inequality in this theorem, then inf{ϕ¯ Ti (t)˙y } > ε holds and the conclusion of this theorem follows readily. 

xˆ (t)

xˆ (t)

x(t)

x(t)

xi (t)

(t) U

θ

y(t)

xi (t)

i (t)

γu (t)

(t)

i (t)

ε

α

i (t)

Fig. 3. The relationship between ϕ (t), ϕ i (t), U and γu (t) in limiting conditions when ρˆ (t) > d.

U

(t)

θ

γu (t)

y(t)

i (t)

(t)

ε

α

Fig. 4. The relationship between ϕ (t), ϕ i (t), U and γu (t) in limiting conditions when ρˆ (t) < d.

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4 Simulation In this section, result from Matlab simulation is presented. The parameters used in the simulation are given as follows. First, we consider the case where there are three non-stationary targets such that x 1 = [3 + 0.25t, 2 + sin(0.03t) + 0.025t]T , x 2 = [1 + 0.03t, 2 + sin(0.025t) + 0.03t]T , x 3 = [3 + 0.025t, 3 + sin(0.035t) + 0.025t]T . Then we assume the initial value of estimation of multiple targets are xˆ 1 (0) = [3, 2]T , xˆ 2 (0) = [3, 1]T and xˆ 3 (0) = [6, 3]T . We also assume that λ = 5, r¯ = 1.5 and δ = 2.24. Adopting the restrict condition in Theorem 1, d is not less than 2δ + r¯, thus we assume d = 7. We can see the simulation result shown in Fig. 5. It shows that the agent converges to the circle with desired radius and circumnavigates the moving targets. As shown in Fig. 6, the estimation error x˜ i (t) for each target converges to a neighborhood of zero and ρ (t) − d has the same property. 18

14

Y (m)

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2

-2

-6 -8

-4

0

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Fig. 5. The trajectory of the agent and multiple targets. 2.5

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ρ(t) −d

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-2

0

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(a)

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0

0

20

40

60

80

time (sec)

(b)

Fig. 6. The figure (a) shows ρ (t) − d and the figure (b) shows ˜x (t).

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5 Conclusions and Future Works In this paper we investigated the safe-circumnavigation problem of multiple nonstationary targets with only bearing measurements. We first assume the targets are moving at a suitable velocity and introduce a position estimator to localize the targets and a control protocol to guarantee the agent moves on the circular trajectory around multiple targets. In order to avoid collision, we propose several preconditions to keep the agent and the targets in safe distance. Future directions of research include solving the localization and circumnavigation problem with multiple targets in 3D space; extending the problem to the case where there are more than one agent; consider the estimation of the speed of targets if they are moving at an unknown constant speed.

References 1. Aranda M, Lopez-Nicolas G, Sagues C, Zavlanos MM (2014) Three-dimensional multirobot formation control for target enclosing. In 2014 IEEE/RSJ international conference on intelligent robots and systems, (Chicago. IL, USA), pp 357–362 2. Kim T-H, Sugie T (2007) Cooperative control for target-capturing task based on a cyclic pursuit strategy. Automatica 43:1426–1431 3. Kim T-H, Hara S, Hori Y (2010) Cooperative control of multi-agent dynamical systems in target-enclosing operations using cyclic pursuit strategy. Int J Control 83(10):2040–2052 4. Matveev AS, Semakova AA, Savkin AV (2016) Range-only based circumnavigation of a group of moving targets by a non-holonomic mobile robot. Automatica 65:76–89 5. Sato K, Maeda N (2010) Target-enclosing strategies for multi-agent using adaptive control strategy. In: 2010 IEEE international conference on control applications (Yokohama, Japan), pp 1761–1766 6. Shames I, Dasgupta S, Fidan B, Anderson BDO (2012) Circumnavigation using distance measurements under slow drift. IEEE Trans Autom Control 57(4):889–903 7. Sharma R, Kothari M, Taylor CN, Postlethwaite I (2010) Cooperative target-capturing with inaccurate target information In: 2010 American control conference, (Marriott Waterfront, Baltimore, MD, USA), pp 5520–5525 8. Shi Y, Li R, Teo KL (2015) Cooperative enclosing control for multiple moving targets by a group of agents. Int J Control 88(1):80–89 9. Shi Y, Li R, Wei T (2015) Target-enclosing control for second-order multi-agent systems. Int J Syst Sci 46(12):2279–2286 10. Cao Y (2015) UAV circumnavigating an unknown target under a GPS-denied environment with range-only measurements. Automatica 55:150–158 11. Anderson BDO (1977) Exponentiai stability of linear equations arising in adaptive identification. IEEE Trans Autom Control 22(1):83–88 12. Deghat M, Xia L, Anderson BDO, Hong Y (2015) Multi-target localization and circumnavigation by a single agent using bearing measurements. Int J Robust Nonlinear Control 25:2362–2374 13. Deghat M, Shames I, Anderson BDO, Yu C (2010) Target localization and circumnavigation using bearing measurements in 2D. In: 49th IEEE conference on decision and control, (Hilton Atlanta Hotel. Atlanta, GA, USA), pp 334–339 14. Li R, Shi Y, Song Y (2018) Localization and circumnavigation of multiple agents along an unknown target based on bearing-only measurement: a three dimensional solution. Automatica 94:18–25

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15. Shi Y, Li R, Teo KL (2019, to be published) Safe-circumnavigation of one single agent around a group of targets with only bearing information teacher, it need changing. J Franklin Inst 16. Daingade S, Sinha A (2016) Fail-safe encircling of multiple unmanned aerial vehicles with bearing only measurements. Int J Micro Air Veh 8(4):240–251 17. Zheng R, Liu Y, Sun D (2015) Enclosing a target by nonholonomic mobile robots with bearing-only measurements. Automatica 53:400–407 18. Miao Z, Wan Y, Fierro R (2017) Cooperative circumnavigation of a moving target with multiple nonholonomic robots using backstepping design. Syst Control Lett 103:58–65 19. Shi Y, Li R, Teo KL (2017) Rotary enclosing control of second-order multi-agent systems for a group of targets. Int J Syst Sci 48(1):13–21

Dissipative Formation Control for Fuzzy Multi-Agent Systems Under Switching Topologies Jiafeng Yu1(B) , Wen Xing2 , Jian Wang3 , and Qinsheng Li1 1

Marine Engineering College, Jiangsu Maritime Institute, Nanjing 211170, China [email protected], [email protected] 2 College of Automation, Harbin Engineering University, Harbin 150001, China [email protected] 3 School of Mathematics and Physics, Bohai University, Jinzhou 121001, China [email protected]

Abstract. In this paper, the formation control with strictly dissipativity for first-order nonlinear multi-agent systems (MASs) under switching topologies is investigated. Firstly, formation control protocols are designed to guarantee that MASs with switching topologies can reach the prespecified desired formation. Then, based on the polynomial Lyapunov function method, sufficient conditions are presented to ensure maintaining the formation of agents and strictly dissipativity for MASs subject to external disturbances. Finally, illustrative example is given to demonstrate the effectiveness of obtained theoretical results. Keywords: Fuzzy modeling · Dissipativity Multi-agent systems · Switching topologies

1

· Formation control ·

Introduction

The Takagi-Sugeno (T-S) fuzzy model has received much attention because of approximating a nonlinear system with any accuracy, for example [2,3,7–9,11]. Recently, polynomial fuzzy model is presented for modeling of nonlinear systems [10]. This model can be regarded as a generalization of the T-S fuzzy model. It can be observed that analysis and synthesis for polynomial fuzzy systems are in terms of sum of squares (SOS), which has lots of improvements over the LMI method for T-S fuzzy systems. For instance, in [6], nonlinear controller designs are presented by SOS optimization. Formation control of multi-agent systems (MASs) has received much attention owing to its widespread applications, including unmanned aerial vehicles, unmanned Helicopters, mobile robots. Various control methods are adopted to realize the formation control, such as H∞ control [1], adaptive control [14], eventtriggered control [5], neural networks [4]. Dissipativity theory is presented in [12] based on input-output energy related consideration. Dissipativity theory is a more flexible and less conservative robust c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 735–741, 2020. https://doi.org/10.1007/978-981-32-9682-4_77

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control design, since it provides a better trade-off between gain and phase performances in control system analysis. Motivated by the fact stated above, this paper concentrates on the formation control problem with strict dissipativity for fuzzy MASs under switching directed topologies. The formation control protocol is devised to guarantee that all followers’ states converge to the state trajectory of the leader. The stability analysis is completed by employing the polynomial Lyapunov function. Notation: Let Rn be n-dimensional Euclidean space. N is the set of natural numbers. Symbol ⊗ denotes the Kronecker product. The superscript T denotes matrix transpose. He (M ) is defined as M + M T .

2 2.1

Graph Theory and Polynomial Fuzzy Model Graph Theory

Let G = (V, E, A) be a directed graph with a nonempty set of nodes V = {υ1 , υ2 , ..., υN } , a set of edge E = {(υi , υj ) : υi , υj ∈ V}, and a weighted adjacency matrix A = [aij ] ∈ RN ×N . In a directed graph G, element aij = 1 if (υj , υi ) ∈ E, otherwise aij = 0. The neighboring set of node υi is denoted by Ni = {υj ∈ V : (υi , υj ) ∈ E} . The Laplacian matrix L = [Li,j ] ∈ RN ×N corresponding to graph G is defined as ⎧ N ⎨  a , i = j, ik Li,j = k=1,k=i (1) ⎩ − aij , i = j. Denote G¯ as a directed graph consisting of G and the leader. Define σ(t) : [0, +∞) → {1, 2, . . . , m} as a switching signal whose value at time t is the index of communication topology. 2.2

Polynomial Fuzzy Model

The dynamics of agent i can be described by x˙ i = f (xi ) + ui ,

(2)

where xi ∈ Rn is the state of agent i. ui ∈ Rn denotes the control input. f (xi ) ∈ Rn is the nonlinear function. i = 1, 2, . . . , N. N is the number of agents. The virtual leader’s system has a similar structure of (2) s˙ = f (s),

(3)

where s ∈ Rn denotes the state of leader. Define the error state as ei = xi −s−hi . The error dynamics can be described by (4) e˙ i = a(xi , ei )ei + q(xi , ei , hi ) + ui ,

Formation Control

737

where a(xi , ei ) and q(xi , ei , hi ) are the polynomial matrix. The fuzzy rules are developed as follows Rp : If φi,1 (t) is M1p and · · · and φi,q (t) is Mqp , then e˙ i = ap (ei )ei + qp (xi , ei , hi ) + ui ,

(5)

where φi (t) is the premise variable. Mkp refers to the fuzzy sets. p = 1, 2, . . . , r, and r is the number of fuzzy rules. The defuzzification process of the model (5) can be written as e˙ i =

r 

hp (φi (t)){ap (ei )ei + qp (xi , ei , hi ) + ui },

(6)

p=1

where r  ωp (φi (t)) hp (φi (t)) =  Mjp (φi,j (t)). , ωp (φi (t)) = r p=1 ωp (φi (t)) p=1

3

Formation Control with Strictly Dissipativity

This section concerns with the formation control issue for first-order nonlinear MASs under switching interaction topologies. The augmented system can be expressed as e˙ i = zi =

r  p=1 r 

hp (φi (t)){ap (ei )ei + qp (xi , ei , hi ) + ui + dp (ei )wi (t)}, hp (φi )(czp (ei )ei + dzp (ei )wi (t)),

p=1

where dp (ei ), czp (ei ) and dzp (ei ) denote polynomial matrices. Design the following formation control protocol  σ(t) σ(t) σ(t) σ(t) σ(t) ui = −c aij Γ (e)(xi − xj − hij ) − cdi Γ i (e)(xi − s − hi ), σ(t)

j∈Ni

i = 1, 2, . . . , N, σ(t)

where c is the coupling strength. ui can ensure that all followers approach to σ(t) achieve desired formation. Ni represents the neighboring set of the ith agent at σ(t). hji = (hj − hi ). The overall multi-agent network is described by e˙ =

z=

r 

¯ e, h)e + c(L¯ ⊗ Γ σ(t) (e))e + Dp w(t)}, hp (φ(t)){Ap (e)e + Q(x,

p=1 r  p=1

hp (φ(t))(Czp e + Dzp w(t)),

(7)

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where σ(t) σ(t) σ(t) L¯σ(t) = −Lσ(t) −Dσ(t) , Dσ(t) = diag{d1 d2 . . . dN }, Dp = IN ⊗ dp (ei ), Dzp = IN ⊗ dzp (ei ), Czp = IN ⊗ czp (ei ). Theorem 1. Given a scalar δ > 0, symmetric matrices Q and R, positive definite matrix Y . The fuzzy error closed-loop system in (7) is asymptotically stable and strictly dissipativity, that is, all the follower agents in (1) can achieve the desired formation, if graph G¯σ(t) has at least one directed spanning tree, σ(t) and there exist symmetric positive definite polynomial matrices p (e) ∈ Rn×n and Γ¯ (e)σ(t) ∈ Rn×n , arbitrary vectors η1 , η2 and η3 , nonnegative polynomials ε1 (e), ε2 (e) and ε3 (e), such that the following SOS conditions hold, σ(t)

η1T (p (e) − ε1 (e)I)η1 is SOS, σ(t) η2T (Γ¯ (e) − ε2 (e)I)η2 is SOS, −η3T

(Θ + ε3 (e)I) η3 is SOS,

p = 1, 2. . .r,

(8) (9) (10)



Ω + Ψ1 Ψˆ2 , ∗ Ψ3 ¯ + He(L¯σ(t) ⊗ Γ¯ σ(t) (e)) Ω = He(P (e)Ap (e)) + He(P (e)Q) n  ∂P (e(t)) ¯ k + c(L¯σ(t) ⊗ Γ σ(t) (e))k )e, (Ak (e) + Q +

where

Θ=

σ(t)

k=1

∂ek

p

σ(t)

T L¯ = −L − Dσ(t) , Ψ1 = −Czp (IN ⊗ Q)Czp , σ(t) T T Ψˆ2 = −Czp (IN ⊗ Q)Dzp − Czp (IN ⊗ Y ) + P (e)Dp , T T Ψ3 = −Dzp QDzp − X − Dzp (IN ⊗ Y ) − (IN ⊗ Y )T Dzp − δI, σ(t) σ(t) σ(t) σ(t) σ(t) P (e) = IN ⊗ p (e), Γ¯ (e) = cp (e)Γ (e). σ(t) σ(t) σ(t) ⊗ Γ (e))k represent the kth row of Ap (e) and c(L¯ ⊗ Akp (e) and c(L¯

Γ

4

σ(t)

(e)), respectively. Space is limited, the proof process is ignored.

Illustrative Example

This section provides a example to demonstrate the correctness of the obtained theoretical results. A nonlinear MASs consisting of four followers and one virtual leader with switching topologies is shown in Fig. 1. The dynamics of agent i is described by chaotic equation [13] (11) x˙ i (t) = f (xi ) + ui , where

⎤ α(xi2 − xi1 ) f (xi ) = ⎣ βxi1 − xi1 xi3 − xi2 ⎦ , xi1 xi2 − λxi3 ⎡

(12)

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Fig. 1. Switching topologies.

in which α = 10, β = 28, λ = 8/3. The error dynamics is ⎤ ⎡ −α α 0 −1 ei1 − xi1 ⎦ e + e˙ i = ⎣ β − xi3 xi2 xi1 − ei1 −λ ⎡ ⎤ α(hi2 − hi1 ) ⎣ hi3 ei1 + hi1 ei3 − xi1 hi3 − hi1 xi3 + hi1 hi3 + (βhi1 − hi2 ) ⎦ + ui (13) −hi1 ei2 − hi2 ei1 + xi1 hi2 + xi2 hi1 − hi1 hi2 − λhi3 The polynomial fuzzy model of error dynamics (13) is constructed by Rqkm : If xi1 is M1q and xi2 is M2k and xi3 is M3m , then e˙ i = aqkm (ei )ei + ui , q, k, m = 1, 2, where xi1 , xi2 and xi3 (i = 1, 2, 3, 4) are the premise variables. Suppose φ1 (t) ∈ [M11 , M12 ], φ2 (t) ∈ [M21 , M22 ], and φ3 (t) ∈ [M31 , M32 ], where M11 = −23, M12 = 23, M12 = 23, M22 = 32, M31 = −61, M32 = 61. The parameters are as follows  T   dp = 1 1 1 , czp = 1 0 0 , dzp = 0.1, γ = 1.  T   Dp = I4 ⊗ 1 1 1 , Czp = I4 ⊗ 1 0 0 , Dzp = I4 ⊗ 0.1. Choose ¯ = I4 ⊗ diag(2, 2, 3), Y = −0.5, Q R = 0.8, Q = −0.6, δ = 0.9, c = 2,  T  T h1 = −1 1 0 , h2 = 1 1 0 ,  T  T h3 = 1 −1 0 , h4 = −1 −1 0 .

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The external disturbance is



w(t) =

1, 0.5s ≤ t ≤ 1s, 0, otherwise.

(14)

Applying the SOSTOOLS for solution of SOS conditions. It can be seen from Figs. 2 and 3, the followers track virtual leader in a short time, and maintain the prespecified square desired formation. 20

follower 1

xi1, s1

follower 2 follower 3

0

follower 4 leader 0

−20

0

0.5

1 5 0 −5

xi2, s2

20 0 −20

0

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2

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4.5 follower 1 follower 2 follower 3

0

0.5

0.02 1

follower 4

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5 follower 1

40 xi3, s3

5

follower 2 follower 3

20

follower 4 leader 0

0 0

0.5

1

1.5

2

2.5 time (s)

3

3.5

4

4.5

5

Fig. 2. The state trajectory for four agents and one virtual leader.

10 t=3s

xi3

5

t=2.5s

0

t=2s

−5 15

xi2

10 5

0

1

2

3

4 xi1

Fig. 3. Formations snapshots of the four followers.

5

6

Formation Control

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741

Conclusion

This paper focuses on formation control issue with strictly dissipativity for MASs under switching directed topologies. Formation control protocol is designed to guarantee that all the agents can achieve the desired formation. Simulation example has verified the theoretical results obtained.

References 1. Dong R, Xiong W (2018) Event-triggered H∞ consensus of nonlinear Multi-agent Systems with Markovian switching topologie. In: Proceedings of the 37th Chinese control conference, China, pp 6499–6504 2. Fei Z, Shi S, Wang T, Ahn CK (2018) Improved stability criteria for discrete-time switched T-S fuzzy systems. IEEE Trans Syst Man Cybern: Syst. https://doi.org/ 10.1109/TSMC.2018.2882630 3. Han C, Zhang G, Wu L, Zeng Q (2012) Sliding mode control of T-S fuzzy descriptor systems with time-delay. J Franklin Inst 349:1430–1444 4. Kuo CW, Tsai CC, Lee CT (2019) Intelligent leader-following consensus formation control using recurrent neural networks for small-size unmanned helicopters. IEEE Trans Syst Man Cybern: Syst. https://doi.org/10.1109/TSMC.2019.2896958 5. Li X, Dong X, Li Q, Ren Z (2018) Event-triggered time-varying formation control for general linear multi-agent systems. J Franklin Inst. https://doi.org/10.1016/j. jfranklin.2018.01.025 6. Prajna S, Parrilo P, Parrilo P (2004) Nonlinear control synthesis by convex optimization. IEEE Trans Autom Control 49:310–314 7. Shi P, Su X, Li F (2016) Dissipativity-based filtering for fuzzy switched systems with stochastic perturbation. IEEE Trans Autom Control 61:1694–1699 8. Shi P, Zhang Y, Chadli M, Agarwal R (2016) Mixed H-infinity and passive filtering for discrete fuzzy neural networks with stochastic jumps and time delays. IEEE Trans Neural Netw Learn Syst 27:903–909 9. Tanaka K, Wang H (2001) Fuzzy control systems design and analysis: a linear matrix inequality approach. Wiley, New York 10. Tanaka K, Yoshida H, Ohtake H, Wang HO (2009) A sum of squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems. IEEE Trans Fuzzy Syst 17:911–922 11. Wang Y, Xia Y, Ahn CK, Zhu Y (2019) Exponential stabilization of Takagi-Sugeno fuzzy systems with aperiodic sampling: an aperiodic adaptive event-triggered method. IEEE Trans Syst Man Cybern: Syst 49:444–454 12. Willems JC (1972) Dissipative dynamical systems-part 1: general theory. Arch Ration Mech Anal 45:321–351 13. Zhao Y, Li B, Qin J, Gao H, Karimi HR (2013) H∞ consensus and synchronization of nonlinear systems based on a novel fuzzy model. IEEE Trans Cybern 43:2157– 2169 14. Zuo S, Song Y, Lewis LF, Davoudi A (2018) Adaptive output formation-tracking of heterogeneous Multi-Agent Systems using time-varying L2-Gain design. IEEE Control Syst Lett 2:236–241

One-Shot Chinese Character Recognition Based on Deep Siamese Networks Huichao Li, Xuemei Ren(B) , and Yongfeng Lv School of Automation, Beijing Institute of Technology, Beijing 100081, China [email protected]

Abstract. This paper applies deep siamese network to one-shot Chinese handwritten character recognition. Different from common image classification tasks, the CASIA HWDB1.1 dataset used here contains more than 3000 categories, with only few samples in each one. We propose a basic deep siamese model as well as an improved model with multi-layer features mechanism and batch normalization for extracting the similarity of the input pairs, and implement one-shot recognition by categorizing the test example to the class where the support sample is the most similar. Experiments prove that our model is able to recognize Chinese characters of unseen classes in training with only one support example efficiently. Keywords: Siamese network · One-shot learning Convolutional neural network · CASIA dataset

1

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Introduction

Most recognition and classification tasks of image is aimed to solve the problem of several categories with a large amount of samples in each class. However, humans are able to recognize images quickly with imagination and association even though they saw only one instance before. The problem corresponding to this mechanism is a learning task with few samples, one of which is defined as one-shot learning [1]. Moreover, resulting from focusing on the association and transfer of the visual features, this model can recognize not only the images from classes learned before, but also ones from unseen classes without re-training. Metric learning is one of representative methods in study of one-shot learning, which get many impressive improvements in recent years. Koch et al. propose a deep siamese model and apply it on one-shot recognition of images basing on similarity [2]. Sung et al. present relation network and improve one-shot recognition performance by replacing the pre-specified distance metric with learnable non-linear layers [3]. Vinyals et al. propose matching networks based on memory augmentation and creative training method related to support set [4]. Snell et al. view mean value of support set samples in embedding space as the prototype [5]. Besides, some researchers apply siamese networks on other scenarios. Like Swati c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 742–750, 2020. https://doi.org/10.1007/978-981-32-9682-4_78

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et al. use siamese model to extract the visual features of preprocessed chromosome images [6]. Martin et al. apply siamese networks on humanity activities classification [7]. Chinese is one of the languages with the largest number of characters, and most Chinese characters have complex structures, there are many researches on Chinese character recognition [8,9], while study on one-shot recognition for few samples dataset of Chinese character is rare. In this paper we apply deep siamese architecture on Chinese handwriting character one-shot recognition. Concretely, we introduce the dataset we use in Sect. 2, in Sect. 3 we propose a basic model as well as an improved model of deep siamese networks which fuse features from each convolutional layers for distance metric, in Sect. 4 we analyze the process of one-shot Chinese character recognition, and in Sect. 5 we train siamese models with samples from dataset HWDB1.1 [10] in the way of one-shot learning and apply it on character recognition. The experiment results prove that our model is able to recognize Chinese characters efficiently.

2 2.1

Dataset and Preprocessing CASIA HWDB1.1 Dataset

Aiming at solving the recognition of multiple categories with few examples, we conduct research with offline Chinese handwriting character dataset named HWDB1.1 of CASIA-OLHWDB, which is provided by the National Laboratory of Pattern Recognition of Institute of Automation of Chinese Academy of Sciences [10]. There are 3755 categories of Chinese handwritten isolated characters in this dataset, the examples’ number of each class is 240 and 60 in training set and test set respectively, the samples of each class is produced by different writers. 2.2

Chinese Character Images Preprocessing

The original images of HWDB1.1 are gray-scale images with various sizes, in order to extract features of the samples by convolution network in siamese model, we preprocess the raw images by resizing and binarizing, as shown in Fig. 1. Specifically, we resize the images to 82 × 82 pixels, making it appropriate for convolutional calculation. Considering the color related information has nothing to do with Chinese character recognition, we set threshold as 200 to binarize the raw images, in which way we avoid the influence of color and the weight of writing on recognition of Chinese characters images.

Fig. 1. Preprocessing of Chinese character images

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Model of Deep Siamese Networks

For the task of one-shot recognition, we build the model with metric-based method. We use siamese convolutional networks as our basic architecture, subnetworks share the weights and each of them extracts features of preprocessed input pairs respectively. Then both of the high-dimensional features are transmitted to distance layers for L1 distance calculation, the resulting distance vectors are passed through a fully connected layer and activated by sigmoid function, the final output is a value representing similarity of the input pairs, which varies within the range between 0 and 1. The structure of our siamese model is shown in Fig. 2. Additionally, we make some improvements in terms of structure for extracting similarity representation more efficiently.

Fig. 2. Architecture of deep siamese model

3.1

Convolutional Siamese Networks with Multi-layer Features

In siamese networks, the structures of each subnet are the same, and they share the parameters. Each sub-network of our siamese model has four convolutional layers and first three of them are followed by max-pooling layers. The stride and padding of all the convolutional kernels and max-pooling filters are set as 2 and 0 respectively. Image of 82 × 82 pixels is fed to a sub-network and processed by each layers for feature extraction, specific parameter settings and details of each layer’s feature map are shown in the Fig. 3. He et al. replace the last fully connection layer with global average pooling in their ResNet model [11]. Liu et al.combine the predictions of feature maps from different layers for dealing with different sizes of the same object in their SSD detection model [12]. Inspired by these works, we add global average pooling information of features extracted by each convolutional layers to the embedding space and concatenate them with high level features which get from the last convolutional layer, in order to obtain better feature representation for distance calculation. The structure improvement is also shown in the Fig. 3. After subnetworks we get a concatenated vector with length of 6720 in embedding space

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and it is fed to a fully connected layer with sigmoid active function, the output is a final feature vector to be passed to distance layer.

Fig. 3. Structure of each sub-network in deep siamese model

3.2

Batch Normalization

We add batch normalization [13] after each convolutional layer and before ReLU activate function. The purpose of batch normalization (abbreviated as BN) here is to change the input of convolution layer into standard normal distribution, so that the input signals fall in the non-linear region of activation function which is more sensitive to input, avoiding the situation that the gradient vanishes as the number of network layers increases, and raise the efficiency of signal transmission in the network. The process of BN can be represented as xi − μB xi =  2 σB +  yi = γ xi + β

(1)

where B = {x1 , x2 , ..., xm } is the input batch, μB is the mean value of the batch, 2 is the batch variance, xi is the normalized value, γ and β are the learned σB

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parameters to scale and shift the normalized vector. γ and β are updated with other parameters of the network simultaneously, which counteract the role of BN in to some extent, making BN and the model adapt to each other in the training process.

4

One-Shot Learning for Chinese Character Recognition

Difference from traditional classification, one-shot model is aiming for metalearning [14]. Most classification model’s parameters contain feature knowledge of the trained data, while one-shot model learns how to learn, making the model be able to recognize objects without large-scale data. The number of categories of HWDB1.1 is over 3000, which is more than Omniglot, the frequently used dataset in one-shot learning researches. Moreover, the structure of Chinese characters is relatively complex and the handwriting style varies from person to person, thus one-shot recognition of HWDB1.1 dataset is a challenging task. 4.1

Chinese Character Data Arrangement for One-Shot Learning

The raw data of HWDB1.1 has been divided into two sets, both of them include all 3755 categories i.e. 3755 Chinese handwritten characters, and the two original sets have 240 samples and 60 samples in each category respectively. We randomly select 60 samples from each class in the first original set to ensure the few-samples characteristic of one-shot task. Then we arrange the two original sets in three parts: set for training, set for validating after each epoch and set for the final test. As we know the samples in the same category are produced by different writers, we choose first 2500 characters of the first original set as training set, and the following 600 characters in first original set as validation set, then we select the last 655 characters of the second original set as test set, making the classes of these three sets have no intersection, the training set and the test set have no intersection in terms of writers. We randomly selected 40000, 500 and 600 sample pairs from three sets respectively for training, validating and testing. 4.2

One-Shot Recognition

Refer to study of Koch et al. [2], we implement one-shot recognition by categorizing the test example to the class where the it have the nearest similarity with the support sample, utilizing the verification ability of the deep siamese network. The mechanism of one-shot recognition can be represented as Cp = arg max S (c) C

(2)

where C is the number of support categories, one of which the test example x belongs to. We randomly select one sample each from these categories and view (C) is the output of the deep siamese all of them as support set {xc }C c=1 . While S model with test sample x and the support sample xC as input, representing

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the similarity of the input image pairs. Cp is the prediction result, we feed each support sample xc along with the test sample x into the twin model and feedforward to get the similarity of them. The class of support sample which has the highest similarity with the test example is viewed as the prediction Cp . The process of one-shot Chinese character recognition is shown in Fig. 4.

Fig. 4. One-shot Chinese character recognition

5

Experiments

We conduct experiments with our basic deep siamese model and the improved multi-layer feature siamese BN model. All of our experiments are implemented in Pytorch framework on an Nvidia GeForce GTX 1080Ti GPU. 5.1

Training Setup

As mentioned in Sect. 4.1, we divide the HWDB1.1 dataset into three parts for training, validating and testing respectively. We use pairs of validation set to evaluate the model after each epoch during training and set the maximum of epoch for training to 185. In our experiments the network weights are initialized to normal distribution before training, and we use a global learning rate that decays with the increase of epoch and set batch size to 64. We train our model with binary cross entropy loss with L2 regularization. The loss used in our experiments can be presented as      (3) Ji (ω) = yi logSω x1i , x2i + (1 − yi ) log 1 − Sω x1i , x2i + λω22 yi is the label vector of the i-th batch, where label is 0 for input pairs  from different categories and 1 for input pairs from the same categories. Sω x1i , x2i is the output of deep siamese model with two images of i-th batch as input. λω22 is the L2 regularization term. We implement one-shot recognition in the way of five-way one-shot, this is, in each trial we set five support samples, which randomly chosen from different five categories for one-shot character recognition.

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Experimental Results

The max validation accuracy and test accuracy is shown in Table 1, our basic siamese model achieve an advanced test accuracy on one-shot Chinese character recognition task, and we promote the performance by improving the siamese model with batch normalization and multi-layer feature structure. Table 1. Experiment results Model

Max val accuracy (%) Test accuracy (%)

Siamese network

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The curves of training loss and validating accuracy are shown in Figs. 5 and 6. In Fig. 5, the loss of our model decreases gradually as the epoch increases. The loss curves of models is similar except for that the loss of the improved siamese network start at a lower value, and the final loss of two models are 0.0849 and 0.0796 respectively. In Fig. 6, the accuracy on validation set of improved siamese model increases faster than that of basic siamese model in the early stages of convergence, and the validation accuracy of improved model is always higher than that of basic siamese network in the stable period. Observing the final test accuracy and the curve of training loss and validation accuracy, it is clear that we realize the one-shot Chineses character recognition using our basic siamese network and we improve the performance of our model with multi-layer features mechanism and batch normalization.

Fig. 5. One-shot recognition training loss

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Fig. 6. One-shot recognition validation accuracy

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Conclusions

This paper applies deep siamese networks to one-shot recognition on Chinese handwritten character of HWDB1.1 dataset. We use preprocessed character images to train the model in the way of one-shot learning. We utilize deep siamese model to extract the similarity of the test example and each support sample, in each one-shot trial the class of the support sample which has the highest similarity to the test sample is seen as the recognition result. In experiments, we build a basic deep siamese networks and an improved model expended with multilayer features mechanism and batch normalization, achieving advanced results on one-shot Chinese character recognition task. Acknowledgments. This work was supported by National Natural Science Foundation of China (Nos. 61433003 and 61973036).

References 1. Feifei L, Fergus R, Perona P (2006) One-shot learning of object categories. IEEE Trans Pattern Anal Mach Intell 28(4):594–611. https://doi.org/10.1109/TPAMI. 2006.79 2. Koch G, Zemel R, Salakhutdinov R (2015) Siamese neural networks for one-shot image recognition. In: Proceedings of the 32nd international conference on machine learning. JMLR.org, France 3. Flood S, Yongxin Y, Li Z, et al (2018) Learning to compare: relation network for few-shot learning. In: Proceedings of the IEEE conference on computer vision and pattern recognition. IEEE, USA, pp 1199–1208. https://doi.org/10.1109/CVPR. 2018.00131

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4. Vinyals O, Blundell C, Lillicrap T et al (2016) Matching networks for one shot learning. In: Advances in neural information processing systems, Spain, pp 3630– 3638 5. Snell J, Swersky K, Zemel R (2017) Prototypical networks for few-shot learning. In: Advances in neural information processing systems, USA, pp 4080–4090 6. Jindal S, Gupta G, Yadav M, Sharma M, Vig L (2017) Siamese networks for chromosome classification. In: International conference on computer vision workshops (ICCVW). IEEE, Italy, pp 72–81. https://doi.org/10.1109/ICCVW.2017.17 7. Martin K, Wiratunga N, Sani S et al (2017) A convolutional Siamese network for developing similarity knowledge in the SelfBACK dataset. In: Proceedings of the international conference on case-based reasoning workshops. CEUR-WS, Norway, pp 85–94 8. Chenglin L, Fei Y, Dahan W et al (2013) Online and offline handwritten Chinese character recognition: benchmarking on new databases. Pattern Recogn 46(1):155– 162 9. Tianfu G, Chenglin L (2008) High accuracy handwritten Chinese character recognition using LDA-based compound distances. Pattern Recogn 41(11):3442–3451 10. Chenglin L, Fei Y, Dahan W et al (2011) CASIA online and offline Chinese handwriting databases. In: International conference on document analysis and recognition. IEEE, China, pp 37–41 11. Kaiming H, Xiangyu Z, Shaoqing R et al (2016) Deep residual learning for image recognition. In: Proceedings of the IEEE conference on computer vision and pattern recognition. IEEE, USA, pp 770–778 12. Wei L, Dragomir A, Dumitru E et al (2016) SSD: single shot multibox detector. In: 14th European conference on computer vision. Springer, Netherlands, pp 21–37. https://doi.org/10.1007/978-3-319-46448-0 2 13. Sergey I, Christian S (2015) Batch normalization: accelerating deep network training by reducing internal covariate shift. In: Proceedings of the 32nd international conference on machine learning, pp 448–456. JMLR.org, France 14. Schweighofer N, Doya K (2003) Meta-learning in reinforcement learning. Neural Netw 16(1):5–9

Fuzzy-Model-Based Consensus for Multi-Agent Systems Under Directed Topology Jiafeng Yu1 , Jian Wang2 , Wen Xing3(B) , Chunsong Han4 , and Qinsheng Li1 1

Marine Engineering College, Jiangsu Maritime Institute, Nanjing 211170, China [email protected], [email protected] 2 School of Mathematics and Physics, Bohai University, Jinzhou 121001, China [email protected] 3 College of Automation, Harbin Engineering University, Harbin 150001, China [email protected] 4 School of Mechanical and Electrical Engineering, Qiqihar University, Qiqihar 161002, China [email protected]

Abstract. This paper is concerned with the consensus control problem for nonlinear multi-agent systems (MASs) with an arbitrary directed communication network. Consensus control protocols are designed for the MASs to enforce all the followers to track the trajectory of a leader. Based on the Lyapunov function method, sufficient conditions are presented to ensure the global consensus. Finally, illustrative examples are given to demonstrate the effectiveness of the obtained theoretical results.

Keywords: Fuzzy modeling Directed topology

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Introduction

Recently, polynomial fuzzy model is presented for modeling of nonlinear plants [1,2]. The polynomial fuzzy model is regarded as a generalization of the T-S fuzzy model. The stability conditions for the polynomial fuzzy system are expressed as sum of squares (SOS), and can be numerically solved by SOSTOOLS [3–5]. The SOS approach is utilized for polynomial fuzzy control system design [6–10]. Consensus problem of nonlinear multi-agent systems (MASs) has received a great deal attention due to its widespread applications in cooperative control of robots, formation control of unmanned aerial vehicles, synchronization of complex networks. Various control methods have been proposed, such as adaptive control [11–13], sliding mode control [14,15], H∞ control [9,16], and so on. Motivated by [9,16], this paper focuses on the consensus control problem for multi-agent systems by polynomial fuzzy model under an arbitrary directed topological structure, which may be a directed spanning tree, or not a directed c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 751–757, 2020. https://doi.org/10.1007/978-981-32-9682-4_79

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spanning tree. Firstly, the consensus control protocol is designed that guarantees the error states converging to zero. Stability analysis is addressed by employing the Lyapunov function. The rest of this paper is organized as follows. In Sect. 2, basic graph theory and polynomial fuzzy model are introduced. In Sect. 3, a consensus protocol for nonlinear multi-agent system is designed. Simulations are given to illustrate the effectiveness of proposed design method in Sect. 4. Finally, concluding remarks are given in Sect. 5. Notations: Let Rn be n-dimensional Euclidean space. Symbol ⊗ stands for the Kronecker product. He (M ) is defined as M + M T .

2 2.1

Graph Theory and Polynomial Fuzzy Model Graph Theory

Let G = (V, E, A) be a directed graph with a nonempty set of nodes V = {υ1 , υ2 , ..., υN }, a set of edge E = {(υi , υj ) : υi , υj ∈ V}, and a weighted adjacency matrix A = [aij ] ∈ RN ×N . In a directed graph G, element aij = 1 if (υj , υi ) ∈ E, otherwise aij = 0. The neighboring set of node υi is denoted by Ni = {υj ∈ V : (υi , υj ) ∈ E}. The Laplacian matrix L = [Li,j ] ∈ RN ×N corresponding to graph G is defined as ⎧ N ⎨  a , i = j, ik Li,j = k=1,k=i (1) ⎩ −aij , i = j. Denote G¯ as a directed graph consisting of G and the leader. 2.2

Polynomial Fuzzy Model

Consider the nonlinear multi-agent systems consisting of one leader and N identical follower agents. The dynamics of agent i is described by x˙ i = f (xi ) + ui ,

(2)

where xi ∈ Rn is the state of agent i. ui ∈ Rn denotes the control input. f (xi ) ∈ Rn represents the nonlinear function. i = 1, 2, . . . , N . N is the number of agents. The virtual leader’s dynamics has a similar structure as (2) s˙ = f (s),

(3)

where s ∈ Rn denotes the state of leader. Define the error state as ei = xi − s. The error dynamic system is described by (4) e˙ i = a(xi , ei )ei + ui ,

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where a(xi , ei )ei = f (xi ) − f (s). The following fuzzy rules are developed: Rp : IF φi,1 (t) is M1p and · · · and φi,q (t) is Mqp , Then e˙ i = ap (ei )ei + ui ,

(5)

where φi (t) denotes the premise variable. M1p , . . . , Mqp are the fuzzy sets for p = 1, 2, . . . , r. Constant r stands for the number of fuzzy rules. ap (ei ) is the polynomial matrix. The defuzzification process of model (5) can be written as r  e˙ i = hp (φi (t)){ap (ei )ei + ui },

(6)

p=1

where

r  ωp (φi (t)) hp (φi (t)) =  (φ (t)) = Mjp (φi,j (t)), , ω p i r p=1 ωp (φi (t)) p=1

Polynomial fuzzy error dynamic system for all agents is expressed as e˙ =

r 

hp (φ(t)) {Ap (e)e + u} ,

(7)

p=1

where hp (φ(t)) = diag{hp (φ1 (t)), hp (φ2 (t)), . . . , hp (φN (t))}, Ap (e) = diag{ap (e1 ), ap (e2 ), . . . , ap (eN )}, e = [eT1 eT2 . . . eTN ]T , u = {uT1 uT2 . . . uTN }T .

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Consensus Design

In this section, our objective is to design consensus protocol. We will give our main result as follows. Theorem 1. The follower agents in (2) can achieve consensus with the leader in (3). If G¯ has at least one directed spanning tree, and there exists nonnegative polynomials ε1 (e(t)), ε2 (e(t)) and ε3 (e(t)), positive symmetric polynomial matrices p(e(t)) ∈ Rn×n and Γ¯ (e(t)) ∈ Rn×n , such that the following SOS conditions are satisfied, η1T (p(e(t)) − ε1 (e(t))I1 )η1 is SOS, η2T (Γ¯ (e(t)) − ε2 (e(t))I2 )η2 is SOS, −η3T (Π + ε3 (e(t))I3 )η3 p = 1, 2, . . . , r,

is SOS,

(8) (9) (10)

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where

Π = He(P (e(t))AP (e(t))) + He(L¯ ⊗ Γ¯ (e(t)) n  ∂P (e(t)) + {Ak (e(t))e(t) + c(L¯ ⊗ −(e(t)))k e(t)}, k=1

∂ek

P

P (e(t)) = IN ⊗ p(e(t)), Γ¯ (e(t)) = c · p(e(t))Γ (e(t)), L¯ = −L − D. AkP (e(t)) is the kth row of AP (e(t)). η1 , η2 and η3 are arbitrary vector that are independent variables in e(t). I1 , I2 and I3 are appropriate dimensions identity matrices. Space is limited, the proof process is ignored.

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Illustrative Example

Here, the illustrative examples are given to demonstrate the effectiveness of the proposed theoretical results. Example 1. Consider consensus problem with a directed spanning tree. Suppose the nonlinear multi-agent network consisting of four followers and one leader as shown in Fig. 1(a). The dynamics of each follower agent is described by the following chaotic equation [16]

where

x˙ i (t) = f (xi (t), t) + ui (t),

(11)

⎤ α(xi2 − xi1 ) f (xi (t), t) = ⎣ βxi1 − xi1 xi3 − xi2 ⎦ , xi1 xi2 − λxi3

(12)

⎡

in which α = 10, β = 28, λ = 8/3. The error dynamics can be represented as ⎤ ⎡ αei2 (t) − αei1 e˙ i (t) = ⎣ βei1 − ei2 + ei1 ei3 − xi3 ei1 − xi1 ei3 ⎦ + ui (t) −ei1 ei2 − λei3 + xi1 ei2 + xi2 ei1 ⎤ ⎡ −α α 0 −1 ei1 (t) − xi1 (t) ⎦ e(t) + ui (t) = ⎣ β − xi3 (t) −λ xi2 (t) xi1 (t) − ei1 (t) (13) The polynomial fuzzy model of error system (13) can be constructed as follows: Rqkm : If xi1 (t) is M1q and xi2 (t) is M2k and xi3 (t) is M3m , then e˙ i (t) = aqkm (ei (t))ei (t) + ui (t), i = 1, 2, . . . , N ; q, k, m = 1, 2,

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Fig. 1. Communication topology, (a) denotes agents graph with directed spanning tree; (b) denotes agents graph without directed spanning tree.

where xi1 (t), xi2 (t) and xi3 (t) (i = 1, 2, . . . , N ) are the premise variables. Suppose φ1 (t) ∈ [M11 , M12 ], φ2 (t) ∈ [M21 , M22 ], and φ3 (t) ∈ [M31 , M32 ], where M11 = −23, M12 = 23, M21 = −32 , M22 = 32, M31 = −61, M32 = 61. The four follower agents have tracked the leader as shown in Fig. 2.

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Example 2. Consider consensus without directed spanning tree in directed graph as shown in Fig. 1(b). In Example 1, assume that d1 = 0, d2 = 0, d3 = 1, d4 = 1, The four follower agents have tracked the leader as shown in Fig. 3.

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Conclusion

This paper focuses on consensus control problem of nonlinear leader-follower multi-agent systems. Consensus protocol is designed so that each follower achieve consensus with the leader. Polynomial Lyapunov function can be applied the stability analysis. Finally, simulation examples have verified the theoretical results obtained. Acknowledgements. This work is partially supported by the National Natural Science Foundation of China (No. 61463001 and No. 61672304), Heilongjiang Province Postdoctoral Science Foundation (No. LBH-Z15043), China Postdoctoral Science Foundation (No. 2016M591514), Qing Lan Project of Jiangsu Province (No. 1602-2), Key Subject of Jiangsu Province Modern Education (No. 61980), PH.D Work Station of Jiangsu Maritime Institute (No. BS1602).

References 1. Tanaka K, Yoshida H, Ohtake H, Wang HO (2007) Learning entity and relation embeddings for knowledge graph completion. In: Proceedings of the 2007 American control conference, USA, pp 4071–4076 2. Tanaka K, Yoshida H, Ohtake H, Wang HO (2009) A sum of squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems. IEEE Trans Fuzzy Syst 17:911–922 3. Prajna S, Papachristodoulou A, Seiler P, Parrilo P (2004) New developments in sum of squares optimization and SOSTOOLS. In: Proceedings of the 2004 American control conference, USA, pp 5606–5611 4. Prajna S, Papachristodoulou A, Parrilo P (2002) Introducing SOSTOOLS: a general purpose sum of squares programming solver. In: Proceedings of the 41st conference on decision and control, USA, pp 741–746 5. Prajna S, Papachristodoulou A, Seiler P, Parrilo P (2004) SOSTOOLS: sum of squares optimization toolbox for MATLAB, Version 2.00. California Institute of Technology

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6. Furqon R, Chen Y, Tanaka M, Tanaka K, Wang HO (2017) An SOS-based control Lyapunov function design for polynomial fuzzy control of nonlinear systems. IEEE Trans Fuzzy Syst 25:775–787 7. Zhang F, Li L, Wang W (2011) Introducing SOSTOOLS: stability and stabilization for a class of polynomial discrete fuzzy system with time delay by sum of squares optimization. In: Proceedings 2011 eighth international conference on fuzzy systems and knowledge discovery, China, pp 713–717 8. Lam H, Tsai S (2014) Stability analysis of polynomial-fuzzy-model-based control systems with mismatched premise membership functions. IEEE Trans Fuzzy Syst 22:223–229 9. Tabarisaadi P, Mardani M, Shasadeghi M, Safarinejadian B (2017) Stability analysis of polynomial-fuzzy-model-based control systems with mismatched premise membership functions. J Franklin Inst PP:8398–8420 10. Gassara H, Hajjaji A, Krid M, Chaabane M (2018) Stability analysis and memory control design of polynomial fuzzy systems with time delay via polynomial lyapunov-krasovskill functional. Int J Control Autom Syst 16:2011–2020 11. Wen G, Chen CLP, Liu Y, Liu Z (2017) Eural network-based adaptive leaderfollowing consensus control for a class of nonlinear multiagent state-delay systems. IEEE Trans Cybern 47:2151–2160 12. Shi P, Shen Q (2015) Cooperative control of multi-agent systems with unknown state-dependent controlling effects. IEEE Trans Autom Sci Eng 12:827–834 13. Shi P, Shen Q (2017) Observer-based leader-following consensus of uncertain nonlinear multi-agent systems. Int J Robust Nonlinear Control 27:3794–3811 14. Zhang J, Lyu M, Shen T, Liu L, Bo Y (2018) Sliding mode control for a class of nonlinear multi-agent system with time delay and uncertainties. IEEE Trans Cybern 65:865–875 15. Ren C, Chen CLP (2015) Sliding mode leader-following consensus controllers for second-order nonlinear multiagent systems. IET Control Theory Appl 9:1544–1552 16. Zhao Y, Li B, Qin J, Gao H, Karimi HR (2013) H∞ consensus and synchronization of nonlinear systems based on a novel fuzzy model. IEEE Trans Cybern 43:2157– 2169

Fuzzy Control and Non-contact Free Loop for an Intermittent Web Transport System Yimin Zhou(B) and Yi Zhang University of Chinese Academy of Sciences, Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China [email protected]

Abstract. In this paper, a control method is designed for the unwinding unit connected with a discontinuous transport system in the digital printing line. Discontinuous unwinding is a nonlinear system with high frequency of on and off which is quite a complex control system. In the method, a non-contact free paper loop is used to connect the unwinding unit and the back-end equipment which can accommodate the motors speed difference with the paper buffer. Moreover, a fuzzy PD speed control system with the feedback of the paper loop position is designed to filter the disturbances of the discontinuous unwinding such as the intermittent on and off or the acceleration variation. The proposed control system can smooth and stabilize the speed and the tension, while no overshoot and better dynamic response can be achieved for the unwinding system, with simplified requirements of the driving motor, less control code and less investment. Simulation experiments have been performed to prove the efficiency of the proposed controller. Keywords: Fuzzy control Printing press

1

· Free loop · Multi-motor synchronization ·

Introduction

High-speed inkjet continuous feed-printing is the fastest-growing sector in the digital printing field, which consists of cutter unit or puncher unit that requires the web transport system periodically switching on and off with high frequency [1,2]. In the process of the unwinding, the diameter and rotational inertia of the unwinding roll are time-variant, so that the speed and tension control are nonlinear and strong coupling. Discontinuous unwinding will add more external disturbance into the system, i.e., high frequency 300–600 times/min on and off representing 5–10 g acceleration, which makes the system control extremely complex and the traditional PID control is quite difficult to meet the requirements. There are two kinds of methods to solve this problem. The first type is to use varied PID or adaptive PID intelligent algorithm to solve the time-varying c Springer Nature Singapore Pte Ltd. 2020  Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 758–767, 2020. https://doi.org/10.1007/978-981-32-9682-4_80

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problem [3]. Ji [4] has studied the tension control of the gravure printer and has proposed a variable structure tension control strategy, the “series PI + parallel PD” correction. Compared with the PI strategy, it can increase the damping ratio and reduce the overshoot for the small roll, while it can reduce the damping ratio and improve the dynamic response ability for the large roll. Sheng [5] has adopted the fuzzy-PID control method in the tension control of high speed paper cutter. Chen [6] has studied the tension control of the gravure printing machine by using ITAE function as the objective function and proposed a pseudo-parallel genetic algorithm to optimize the PID control parameters, which can improve the static and dynamic performance of the system. The second type is the feed forward compensation algorithms, which consist of establishing mathematical model and calculating compensation by the moment of inertia, friction compensation and deceleration compensation. Shin [7] has proposed a new compensation method for tension disturbance due to the unknown roll shape. Nevaranta [8] has validated the mathematical model of the discontinuous web transport system and developed a new web tension control structure. Huang [9] has studied the adaptive speed control in the crimping machine, and improved the tension control during deceleration with proportion coefficient adjustment of PID based on calculating the added value of moment of inertia. Meng [10] has analyzed the rewinding machine in the process of steady speed and variable speed, and put forward the tension control strategy that the moment of inertia of the compensation should be divided into two parts, one is the paper roll radius decrease to cause the rotating speed increase, the other part is the rewinding acceleration. In general, the adaptive PID intelligent algorithm can solve the time-varying problem caused by the variation of the roll radius but cannot remove periodic disturbance [11,12]. The performance of the feed forward compensation algorithm highly depends on the accuracy of the mathematical models for discontinuous unwinding and various disturbances, and it has high requirement on the servo motor. In this paper, a closed-loop fuzzy PD speed control system is proposed which can not only eliminate the feedback of discontinuous unwinding and disturbance, but also ensure fast response of the unwinding system. The reminder of the paper is organized as follows. Section 2 describes the mathematical model of the tension control and free-loop control. Section 3 discusses the simulation experiments and results. Section 4 analyzes the fuzzy control and comparison. The conclusion is given in Sect. 5.

2 2.1

Problem Formulation Tension Control Model

Generally, the unwinding system consists of the traction motor and the unwinding motor [13]. As shown in Fig. 1, the traction motor is in the speed mode, and the unwinding motor is a magnetic powder brake working in the torque mode, which can provide a reverse torque to maintain the tension of the paper.

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Fig. 1. The tension control model

Based on the torque balance, the paper tension model can be developed as,  F R − T = βω + J dω dt (1) ω = VR1 where F is the paper tension, T is the braking moment, β is the friction coefficient, V1 is the traction linear speed, ω is the unwinding speed, R is the roll radius, and J is the total moment of the inertia of the roll [14]. From Eq. 1, the paper tension can be rewritten as, F =

T + βω + J dω dt R

(2)

dω If the linear velocity V1 is assumed to be stable, dω dt → 0, J dt is a small quantity. Then the accuracy of the tension F can be obtained ideally with the intelligent PID algorithm, or feed forward algorithm to compensate for the variation of the roll radius R. If the system is working under the discontinuous unwinding condition, the linear velocity V1 is unstable and J dω dt would be a large variable. At this time, it is necessary to investigate how to reduce the impact of the acceleration and deceleration on the control system.

2.2

Free-Loop Velocity Control Model

As shown in Fig. 2, the structure of the free-loop mode is quite simple, which is mainly used for connecting the unwinder with the back-end equipment. The paper web is connected by free hanging, then the paper tension F can be expressed in another form, F = hAρ

(3)

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Fig. 2. Free loop control model

where h is the height, A is the sectional area, and ρ is the material density. Here the tension is generated by the weight of the suspended material. The advantage of free-loop is that the inertia of the paper is much smaller than that of the idle roller, meaning less shock and quicker response, when the system is in either accelerating or decelerating state. In this mode, the speed synchronization is adjusted by measuring the paper web height with a position sensor. The speed control algorithm generally adopts the proportional control, V1 = V2 + K(X − X0 )

(4)

where V1 is the following speed, V2 is the main speed, K is the proportional coefficient, X is the distance between the sensor and the paper, and X0 is the reference distance. The motor speed difference can be absorbed by the paper buffer, even if the dynamic performance of the unwinding unit is not so good but the buffer is enough to balance the speed automatically. Then the paper buffer length can be designed as,   (5) L = max( (V2 − V1 ) dt) − min( (V2 − V1 ) dt) Among them L is the length of the total paper web buffer, which relates to the two parts: first, there should be enough paper to process when accelerating; second, there should be enough space to accommodate the redundant paper web when decelerating. 2.3

Multiple Free-Loop Models

The free-loop has the advantages of simple structure and small inertia with acceleration and deceleration. It can also make more paper buffer by increasing the length of the paper web, as shown in Fig. 3, where serval loops can form

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Fig. 3. Multiple free loops model

Fig. 4. Multiple units free loops model

multiple free-loop modes. The following units are also connected with the freeloop, which forms the multiple units free-loops mode, seen in Fig. 4. In fact, the control problem of the unwinding unit is a problem of multi-group motor synchronization [15]. Since there is a buffer between each motor, no strong coupling of the speed and the tension would exist. Hence, parallel control is a simple and reliable solution. The buffer length can be treated as the speed feedback signal and its relationship is as follows,    (6) (V3 − V1 ) dt = (V3 − V2 ) dt + (V2 − V1 ) dt If the main velocity V3 is set, the free-loop length between V3 and V2 is treated as the feedback signal of the velocity V2 , and the feedback signal of the velocity V1 is the sum of the two free-loop lengths: between V3 and V2 , and between V2 and V1. So the change of the velocity V2 will not affect the feedback signal of V1 , as (V3 − V1 ) dt. Therefore, the parallel control strategy functions in high quality.

3

Simulation Experiments and Analysis

The proposed parallel control of fuzzy control and non-contact free-loop are performed in the simulation experiments. Considering that the feeding motor V2 and the traction motor V1 are both first-order inertia systems, the acceleration time is set as 0.4s and 1.6s based on the experimental calculation. The transfer 1 1 and 1.6s+1 , and the control function is calculated function is simplified to 0.4s+1 from Eq. 4,  (7) V1 = V3 + K (V3 − V1 ) dt Figure 5 depicts the flow chart of the simulation. The upper part of Fig. 5 is for the feeding motor V2 and the lower part is for the traction motor V1 . The running simulation curve is shown in Fig. 6. It can be seen from the above three curves that after the change of the step signal generated by the main speed V3 , the traction motor V1 and the feeding motor V2 follow quickly and adjust to the

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Fig. 5. The simulation flow chart of the multi-motor synchronous parallel control

Fig. 6. The results of the multi-motor synchronous parallel control

balance position in about 10 s. Meanwhile, it can be seen from the below three curves in Fig. 6 that the position adjustment is the same and converges to the zero point fast, which meets the design requirements with satisfied performance. In summary, the dynamic performance of the unwinder has been improved due to the free-loop increasing of the buffer length, and it is easy to control when multiple loops are all under the same position control mode. Because multiple loops can be treated as a whole, it is possible to use parallel control in the motor synchronization to transfer many strong coupling problems into weak coupling and simple problem.

4

Fuzzy Control Analysis

Discontinuous unwinding generally consists of intermittent on and off, existing in the system of paper web cutting, punching. If the unwinding unit follows the signal, the motor will always be in the cycle of acceleration and deceleration. Frequent acceleration and deceleration is obviously worse for the motor, especially the traction motor working with a large inertia paper roll, which puts forward some requirements as follow:

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(1) It is expected that the traction motor does not follow the acceleration and deceleration in the process of intermittent operation, and the vibration generated by the intermittent operation should be absorbed within the free loops; (2) System should response quickly to achieve speed synchronization in the first start or final stop, remarkable acceleration or deceleration stages; As for the control model, these requirements can be explained as: (1) The system will not make any action if the deviation is within the allowable range; (2) The system will adjust immediately if it exceeds the scope; (3) If it is found that the deviation is too large and out of range, large correction operation should be carried out immediately. The most appropriate solution to this kind of problem is the fuzzy control. The following is the simulation experiments of the proportional control method and fuzzy control method under intermittent traction. 4.1

Proportional Control Simulation

By adjusting Eq. 7, the equation of the proportional control adjustment under intermittent feeding can be obtained as,   1 t+τ V3 dt + K (V3 − V1 ) dt (8) V1 = τ t where V1 is the following speed, V3 is the main speed and in the intermittent motion. Selecting its average value as the signal, the Simulink flow chart is illustrated in Fig. 7.

Fig. 7. The simulation flow chart of the intermittent feed-forward proportional control

The simulation curves shown in Fig. 8 demonstrate that the velocity following curve is stable around the average value of the main velocity, but is slightly up and down. The reason is that the position signal cannot filter the oscillation generated by the intermittent movement, and these oscillation signals will be used as the input and affect the following velocity.

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Fig. 8. The curves of the intermittent feed proportional control

4.2

Fuzzy Control Simulation

Fuzzy control is a widely used intelligent control method, which is especially suitable for complex systems with nonlinear, large hysteresis and strong coupling characteristics. Compared with the traditional PID control, the fuzzy control does not require to establish the mathematical model accurately, only require master the operator control experience and transfer into control rules to be implemented by the machine. The two-dimensional fuzzy controller selects the given deviation E and the deviation rate EC as the input, and generally adopts the fuzzy conditional statements of “if E = X and EC = Y then U = Z” to form the inference rules. The simulation flow chart is illustrated in Fig. 9 and the fuzzy control surface is shown in Fig. 10.

Fig. 9. The simulation flow chart of intermittent feed fuzzy control

From the simulation curve in Fig. 11, it can be seen that after adopting the fuzzy control, the position still shocks with the intermittent traction curve, but the traction motor speed signal has been stable after a short adjustment, no shock, and the adjusting time is short, while overshoot amplitude is rather small.

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Fig. 10. The fuzzy control surface for intermittent feed

Fig. 11. The curves of the intermittent feed fuzzy control

5

Conclusions

The fuzzy controller designed in this paper is combined with free-loop, which can filter the acceleration and deceleration of the discontinuous feeding and all kinds of interference, to ensure the excellent dynamic performance of the system, and improve the stability of the tension system. It can be extended to the multiple free-loops systems to provide more paper buffer, or the connection between multiple work units in the whole assembly line, especially the discontinuous processing line, which has a bright application prospect. Acknowledgments. This work is supported under the Shenzhen Science and Technology Innovation Commission Project Grant Ref. JCYJ20160510154736343 and Ref. JCYJ20170818153635759, and Science and Technology Planning Project of Guangdong Province Ref. 2017B010117009, and Guangdong Provincial Engineering Technology Research Center of Intelligent Unmanned System and Autonomous Environmental Perception.

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References 1. Chen J, Yin Z, Xiong Y, Quan J (2009) A hybrid control method of tension and position for a discontinous web transport system. In: Proceedings of the IEEE international conference on information and automation, pp 265–270 2. Zhou X, Wang D, Wang J, Chen S (2016) Precision design and control of a flexurebased roll-to roll printing system. Precis Eng 45:332–341 3. Wang Z, Nan H, Shi T et al (2018) No-tension sensor closed-loop control method with adaptive PI parameters for two-motor winding system. Math Prob Eng 38(3):41–53 4. Ji T (2003) Research on tension control system of gravure printing machine. Master’s thesis of Zhejiang University 5. Shen X (2009) Application of fuzzy-PID in tension control of high-speed paper cutter. Master thesis, Shandong institute of light industry 6. Chen Z (2005) Research on constant tension control strategy for unwinding system of gravure printing machine. Master thesis, Huazhong University of science and technology 7. Shin K, Jang J, Kang H, Song S (2003) Compensation method for tension disturbance due to an unkown roll shape in a web transport system. IEEE Trans Ind Appl 39(5):1422–1428 8. Nevaranta N, Niemela M, Pyrhonen J et al (2012) Indirect tension control method for an intermittent web transport system. In: 15th international power electronics and motion control conference EPE-PEMC, Novi Sad, Serbia 9. Huang B, Zhou H, Qi Q (2004) Application of adaptive speed control in tension winder. J Wuhan Univ (Eng) 37(5):101–103 10. Meng Y, Chen Z (2009) Analysis of rotary inertia compensation for paper rewinding machine. China Pap J 24(2):89–91 11. Yang M, Zhang S (2014) The research of tension control system in web press based on the fuzzy adaptive PID controller. 2014 IEEE 9th conference on industrial electronics and applications (ICIEA), pp 1204–1208 12. Chen Q, Li W (2017) Fuzzy PID controller for a constant tension winch in a cable laying system. IEEE Trans Ind Electron 64(4):2924–2932 13. Chen D, Lu L, Tu H (2011) A V-F coordinated control system for two motors without any tension sensors. In: Proceedings of the IEEE 4th international conference on intelligent computation technology and automation, pp 576–579 14. An L, Yang Y, Tang J, Cai Z (2008) A new method for measuring the moment of inertia of winder. Electr Transm 38(3):41–53 15. Wang Q, He F (2016) The synchronous control of multi-motor drive control system with floating compensation. In: 2016 12th world congress on intelligent control and automation (WCICA), pp 1949–1954

Author Index

A An, Cuijuan, 185

Du, Mingjun, 599 Duan, Mengmeng, 708

B Bai, Mingrun, 670

F Fang, Shuang, 73 Feng, Jianlin, 150 Feng, Peng, 649 Fu, Dongmei, 39, 566, 629

C Cao, Jinlei, 167 Cao, Shida, 724 Cao, Xianbing, 101 Chai, Yi, 446 Chang, Huining, 413 Chang, Mingming, 422 Che, Shijie, 330 Chen, Liuyuan, 422 Chen, Mindong, 566 Chen, Mou, 64 Chen, Peng, 436 Chen, Tian, 670 Chen, Xia, 310 Chen, Xiao, 375 Chen, Zengqiang, 48, 254, 487, 553, 619 Chen, Zhenjia, 393, 429 Cheng, Ju, 101, 111 Cheng, Xinming, 658 Cui, Jiarui, 436 Cui, Yan, 471 D Ding, Dawei, 185 Dong, Xuemei, 322 Du, Haipeng, 629 Du, Junping, 413

G Gu, Guiding, 11 Gu, Yinhe, 322 Guan, Zhihong, 658 Guo, Fumin, 159 Guo, Xia, 429 Guo, Xuemei, 1, 276 Guo, Yi, 349 Guo, Zirong, 591

H Han, Chunsong, 751 Hao, Fei, 150 He, Chao, 393, 429 He, Peng, 649 He, Ruijiao, 167 Hou, Hailiang, 716 Hu, Bin, 658 Hu, Hexu, 466 Hua, Yu, 508 Huang, Jun, 499, 574 Huang, Xinhua, 466 Huang, Yi, 716

© Springer Nature Singapore Pte Ltd. 2020 Y. Jia et al. (Eds.): CISC 2019, LNEE 592, pp. 769–771, 2020. https://doi.org/10.1007/978-981-32-9682-4

770 J Jia, Yingmin, 681, 690 Jiang, Hong, 599 K Kang, Zhongjian, 457 Kong, Xudong, 322 L Li, Huichao, 742 Li, Jingyi, 111 Li, Jun, 85 Li, Junjian, 310 Li, Juntao, 422 Li, Longlong, 131, 236 Li, Ming, 545 Li, Mingxing, 690 Li, Penghua, 122, 466 Li, Qing, 436 Li, Qinsheng, 735, 751 Li, Rui, 724 Li, Shurong, 457 Li, Tao, 658 Li, Wenling, 413 Li, Xiaodong, 85 Li, Yaoyao, 73 Li, Yifeng, 195 Li, Ying, 39 Li, Yinguo, 122 Li, Yonggang, 716 Li, Yongliang, 245 Li, Yuanyuan, 122, 466 Li, Zhiwei, 140 Li, Zhuolin, 39, 629 Liang, Shuang, 254 Liang, Yueqian, 535 Liao, Yali, 708 Liao, Yuxin, 85 Liu, Changqing, 535 Liu, Guoqing, 367 Liu, Heqing, 508, 515 Liu, Junjie, 48 Liu, Lixia, 699 Liu, Shangzheng, 526 Liu, Shuai, 30 Liu, Xiaoyu, 446 Liu, Yuan, 526 Liu, Zhongxin, 254, 487, 553, 619 Long, Zourong, 649 Lu, Na, 73 Lu, Ziyi, 436 Luo, Shibin, 85 Lv, Siyao, 131, 236 Lv, Yongfeng, 742

Author Index M Ma, Baoli, 599, 609 Ma, Mingming, 393 Ma, Yinjin, 649 Ma, Yiying, 681 Ma, Yuan, 638 Meng, Deyuan, 591, 599 Miao, Zhonghua, 699 Mo, Lipo, 101, 111 Mu, Xiaoxia, 422 N Ning, Yao, 487, 553 P Pan, Yuedou, 245, 266 Pei, Zibo, 39 R Ren, Xuemei, 159, 340, 742 Ren, Yuanhong, 140 S Shao, Lizhen, 629 Shi, Jiahui, 401 Shi, Jian, 322 Shi, Yingjing, 724 Su, Shuai, 716 Su, Xiaoli, 292 Sun, Fei, 574 Sun, Liang, 208 Sun, Mingwei, 48 Sun, Qinglin, 48 Sun, Yuqing, 140 T Tian, Hao, 30 Tian, Pengjie, 422 Tian, Yanbing, 310 W Wang, Dongdong, 384 Wang, Fuyong, 254, 619 Wang, Guoli, 1, 276 Wang, Jian, 735, 751 Wang, Sizhe, 24 Wang, Weigang, 167, 176 Wang, Xiaodong, 545 Wang, Xiaohua, 699 Wang, Xiaoshan, 471 Wang, Xingxia, 619 Wang, Zhigang, 24 Wei, Biao, 649 Wei, Hongyan, 466

Author Index Wei, Linbing, 58 Wei, Shanbi, 446 Wei, Xin, 566 Wei, Xinjiang, 218, 227 Wei, Yongli, 227 Wen, Chunwei, 357 Wu, Bei, 64 Wu, Xia, 85 Wu, Xiaoyang, 292 Wu, Yingying, 73 Wu, Ziwen, 508, 515

X Xia, Meizhen, 515 Xiang, Lan, 699 Xiang, Peng, 708 Xiao, Jiangwen, 658 Xiao, Li, 724 Xiao, Nan, 545, 638 Xie, Yongsheng, 58 Xing, Wen, 735, 751 Xu, Gengxin, 167, 176 Xu, Lin, 658 Xu, Pengfei, 30 Xu, Ruiping, 375 Xu, Zhengguang, 30, 330 Xu, Zimin, 276

Y Yan, Kai, 609 Yang, Hua, 599 Yang, Ming, 499, 574 Yang, Shengjie, 167 Yao, Qiong, 39, 566 Yin, Yixin, 292 You, Yang, 629 Yu, Jiafeng, 735, 751 Yu, Jinpeng, 367, 384

771 Z Zhang, Guilai, 638 Zhang, Hongguang, 582 Zhang, Jie, 185 Zhang, Jingyao, 591 Zhang, Jinxing, 349 Zhang, Qiye, 357 Zhang, Sen, 292 Zhang, Sijing, 73 Zhang, Tianping, 508, 515 Zhang, Yan, 545 Zhang, Yi, 758 Zhang, Yihong, 131, 236 Zhang, Yonghui, 301, 393, 429 Zhao, Chen, 487, 553 Zhao, Fen, 122 Zhao, Hanxu, 218 Zhao, Jiaxing, 266 Zhao, Jinbin, 39 Zhao, Kai, 457 Zhao, Kun, 301 Zhao, Lin, 367, 375, 384 Zhao, Xin, 73 Zheng, Zewei, 670 Zhong, Wei, 1 Zhou, Hao, 436 Zhou, Jin, 699 Zhou, Wuneng, 131, 140, 236 Zhou, Xiaohui, 11 Zhou, Yimin, 758 Zhou, Zhaohui, 330 Zhou, Zitong, 436 Zhu, Bing, 208 Zhu, Hongqiu, 401, 708 Zhu, Jiandong, 195, 349 Zhu, Ming, 670 Zhu, Ronggang, 64 Zhu, Zhenkun, 499 Zhu, Zhiqin, 466 Zou, Yiping, 340