Control of Nonlinear Systems via PI, PD and PID: Stability and Performance [1 ed.] 9781138317642, 9780429455070, 9780429847639, 9780429847622, 9780429847646

Table of contents :

Introduction. Classical PID Control. Adaptive PI Control for SISO Affine Systems. Generalized PI Control for SISO Nonaffine Systems. Adaptive PI Control for MIMO Nonlinear Systems. Adaptive PI Control for Strict Feedback Systems. Adaptive PID Control for MIMO Nonlinear Systems. PD Control Application to High Speed Trains. PID Control Application to Robotic Systems.

Citation preview

i

i “FM” — 2018/9/19 — 16:44 — page 1 — #1

i

i

Control of Nonlinear Systems via PI, PD and PID Stability and Performance

i

i i

i

i

i “FM” — 2018/9/19 — 16:44 — page 2 — #2

i

i

AUTOMATION AND CONTROL ENGINEERING A Series of Reference Books and Textbooks Series Editors FRANK L. LEWIS, Ph.D., Fellow IEEE, Fellow IFAC

Professor The Univeristy of Texas Research Institute The University of Texas at Arlington

SHUZHI SAM GE, Ph.D., Fellow IEEE

Professor Interactive Digital Media Institute The National University of Singapore

STJEPAN BOGDAN

Professor Faculty of Electrical Engineering and Computing University of Zagreb

RECENTLY PUBLISHED TITLES Discrete-Time Recurrent Neural Control: Analysis and Applications, Edgar N. Sánchez Electric and Plug-in Hybrid Vehicle Networks: Optimization and Control, Emanuele Crisostomi, Robert Shorten, Sonja Stüdli, and Fabian Wirth Adaptive and Fault-Tolerant Control of Underactuated Nonlinear Systems, Jiangshuai Huang and Yong-Duan Song Optimal and Robust Scheduling for Networked Control Systems, Stefano Longo, Tingli Su, Guido Herrmann, and Phil Barber Deterministic Learning Theory for Identification, Recognition, and Control, Cong Wang and David J. Hill Networked Control Systems with Intermittent Feedback, Domagoj Toli´c and Sandra Hirche Doubly Fed Induction Generators: Control for Wind Energy, Edgar N. Sanchez and Riemann Ruiz-Cruz Optimal Networked Control Systems with MATLAB®, Jagannathan Sarangapani and Hao Xu Cooperative Control of Multi-agent Systems: A Consensus Region Approach, Zhongkui Li and Zhisheng Duan Nonlinear Control of Dynamic Networks, Tengfei Liu, Zhong-Ping Jiang, and David J. Hill Modeling and Control for Micro/Nano Devices and Systems, Ning Xi, Mingjun Zhang, and Guangyong Li Linear Control System Analysis and Design with MATLAB®, Sixth Edition, Constantine H. Houpis and Stuart N. Sheldon Real-Time Rendering: Computer Graphics with Control Engineering, Gabriyel Wong and Jianliang Wang Anti-Disturbance Control for Systems with Multiple Disturbances, Lei Guo and Songyin Cao Tensor Product Model Transformation in Polytopic Model-Based Control, Péter Baranyi, Yeung Yam, and Péter Várlaki

i

i i

i

i

i “FM” — 2018/9/19 — 16:44 — page 3 — #3

i

i

Control of Nonlinear Systems via PI, PD and PID Stability and Performance

Yong-Duan Song

i

i i

i

i

i “FM” — 2018/9/19 — 16:44 — page 4 — #4

i

i

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2019 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper Version Date: 20180919 International Standard Book Number-13: 978-1-138-31764-2 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Song, Yong-Duan, author. Title: Control of nonlinear systems via PI, PD and PID : stability and performance / Yong-Duan Song. Description: First edition. | Boca Raton, FL : CRC Press/Taylor & Francis Group, 2018. | Includes bibliographical references. Identifiers: LCCN 2018026890 | ISBN 9781138317642 (hardback : acid-free paper) Subjects: LCSH: Adaptive control systems. | Nonlinear control theory. Classification: LCC TJ217 .S67 2018 | DDC 629.8/36--dc23 LC record available at https://lccn.loc.gov/2018026890

Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

i

i i

i

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page vii — #3

✐

✐

To my family for the understanding, support and love.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page vi — #4

✐

✐

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page vii — #5

✐

✐

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Preview of Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3

2

Classical PID Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Three Actions of PID Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Proportional Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Integral Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Derivative Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Tuning Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 5 6 7 8 8

3

Adaptive PI Control for SISO Affine Systems . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Design Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 PI Control Design for First-order Nonlinear Systems . . . . . . 3.3.2 PI Control Design for High-order Nonlinear Systems . . . . . . 3.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 10 12 12 14 20 21 23

4

Generalized PI Control for SISO Nonaffine Systems . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 System Description and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 27 29

vii

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page viii — #6

✐

✐

viii

Contents

4.3.1 PI Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adaptive Fault-tolerant PI Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 PI Control under Actuator Failures . . . . . . . . . . . . . . . . . . . . . . 4.4.2 PI Control under Actuator and Sensor Faults . . . . . . . . . . . . . 4.5 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30 35 35 36 38 41

5

Adaptive PI Control for MIMO Nonlinear Systems . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Neural Networks and Function Approximation . . . . . . . . . . . 5.3 PI Control Design and Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Neuoadaptive PI Control for Square Systems . . . . . . . . . . . . . 5.3.2 Neuoadaptive PI Control for Non-square Systems . . . . . . . . . 5.4 Modified PI Control Based on BLF . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Neuro-adaptive PI Control for Square Systems . . . . . . . . . . . . 5.4.2 Neuro-adaptive PI Control for Non-square Systems . . . . . . . . 5.5 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43 44 44 47 48 49 53 55 56 58 61 65 65

6

Adaptive PI Control for Strict Feedback Systems . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 System Description and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 PI-like Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 67 68 71 81 84

7

Adaptive PID Control for MIMO Nonlinear Systems . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Problem Formulation and Error Dynamics . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Error Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Nussbaum Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 PID-like Control Design and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 PID Control for Square Systems . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 PID Control for Non-square Systems . . . . . . . . . . . . . . . . . . . 7.3.3 Analysis and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 87 88 89 90 91 94 96 97

8

PD Control Application to High-Speed Trains . . . . . . . . . . . . . . . . . . . . . 99 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 8.2 Modeling and Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 8.2.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 8.2.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.4

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page ix — #7

✐

✐

Contents

9

ix

8.3 Control Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Structural Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Robust Adaptive PD-like Control Design . . . . . . . . . . . . . . . . 8.3.3 Low-Cost Adaptive Fault-tolerant PD Control . . . . . . . . . . . . 8.3.4 Comparison and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 105 109 111 112 114

PID Control Application to Robotic Systems . . . . . . . . . . . . . . . . . . . . . . 9.1 Robotic Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 PID Control for Robotic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Square System (joint-space tracking) . . . . . . . . . . . . . . . . . . . . 9.2.2 Non-square System (task-space tracking) . . . . . . . . . . . . . . . . 9.3 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 117 117 117 118 118

10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page x — #8

✐

✐

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/25 — 16:35 — page xi — #9

✐

✐

Preface

The proportional integral derivative (PID) controller in the form we know it today emerged between 1915 and 1940. Since PID control is simple in structure and inexpensive in implementation, it has been undoubtedly the most widely employed controller in industry. In fact, PID controllers are sufficient for many control problems, particularly when process dynamics are benign and the performance requirements are modest. However, there are still problems that limit the applications of the PID controller. One of those is how to determine the appropriate PID gains to ensure system stability and desirable performance. The Ziegler–Nichols tuning procedure, published in 1942, is simple and intuitive, but it creates a closed-loop system that is very poorly damped and that has poor stability margins. Since then, various methods for tuning PID gains have been suggested, but a systematic means is yet to be established and the information about those methods is scattered in the control theory. With the increasing control demands for various practical systems which are generally nonlinear, uncertain and with abnormal actuation (such as asymmetric saturation, dead-zone module, loss of effectiveness, etc.), traditional PID control seems to lack theoretical support and is losing efficiency. Thus, a series of control strategies is proposed to tackle the control problems for all kinds of nonlinear systems. Although various control schemes have successfully addressed the control problems for nonlinear systems, the resultant solutions seem quite sophisticated — not only complicated in structure, but also expensive in computation. As a consequence, these complex control methods are not much appreciated in practical applications. By this token, the PID control seems still the most favorable choice for the control of practical systems if it could be made effective in dealing with system nonlinearities and uncertainties. Industrial experience has clearly indicated that automatic tuning of PID gains is a highly desirable and useful feature. Therefore, the focus of this book is on PI/PD/PID controller for nonlinear systems with self-tuning gains, wherein an exposition of adaptive PI/PD/PID control methods developed recently for numerous nonlinear systems is provided. All these PI/PD/PID controllers are able to adaptively update the gains through analytic algorithms and there is no need for xi

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/25 — 16:35 — page xii — #10

✐

✐

xii

Preface

human tuning or trial and error process. Besides, the stability condition (the primary concern for any control system) is established for the corresponding systems with PI/PD/PID controller in the loop. Furthermore, in order to make the control scheme more reliable in practical applications, in this book, the proposed PID control strategies are equipped with fault-tolerant capabilities to accommodate the abnormal actuation characteristics which may occur during system operation. Constraints (due to physical saturation, safety specifications, etc.) imposed on system outputs or states, together with the issue of prescribed control performance, are also considered in control design. In the last chapters of the book, the PI/PD/PID control scheme is applied to practical systems such as high-speed trains and robotic systems. The effectiveness of the proposed adaptive PI/PD/PID controller is demonstrated and validated via computer simulations. Several books on PID controllers are available on the market, but this book exclusively focuses on PI/PD/PID control with gain auto-tuning mechanisms for nonlinear systems. While efforts have been made on PI/PD/PID control for nonlinear systems, there is still much room for further research and development. We hope that this book will aid in understanding the essence of PID control, providing readers with alternative perspectives concerning the development of PI/PD/PID controllers for typical nonlinear systems. Yongduan Song Chongqing, China

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page xiii — #11

✐

✐

Acknowledgments

There are numerous individuals without whose help this book would not have been completed. My sincere thanks go to Dr. Yujuan Wang, Dr. Danyong Li, Qing Chen, Zhirong Zhang, Ziyun Shen, Ye Cao, Shuyan Zhou, Xiucai Huang, Kai Zhao. In particular, I would like to express my gratitude to the following authors for allowing me to use the materials of the papers for compiling this book. • Y. D. Song, Y. J. Wang, and C. Y. Wen, “Adaptive fault-tolerant PI tracking control with guaranteed transient and steady-state performance,” IEEE Trans. Autom. Control, vol. 62, no. 1, pp. 481-487, 2017. • Q. Song, Y. D. Song, “Generalized PI control design for a class of unknown nonaffine systems with sensor and actuator faults,” Syst. Control Lett., vol. 64, no. 1, pp. 86-95, 2014. • Y. D. Song, J. X. Guo, and X. C. Huang, “Smooth neuroadaptive PI tracking control of nonlinear systems with unknown and nonsmooth actuation characteristics,” IEEE Trans. Neural Netw. Learn. Syst., vol. 28, no. 9, pp. 2183-2195, 2017. • Y. D. Song, Z. Y. Shen, L. He, and X. C. Huang, “Neuroadaptive control of strict feedback systems with full-state constraints and unknown actuation characteristics: an inexpensive solution”, IEEE Trans. Cybern., DOI: 10.1109/TCYB.2017.2759498. • Y. D. Song, X. C. Huang, and C. Y. Wen, “Robust adaptive fault-tolerant PID control of MIMO nonlinear systems with unknown control direction,” IEEE Trans. Ind. Electron., vol. 64, no. 6, pp. 4876-4884, 2017. • Y. D. Song, X. C. Yuan, “Low-cost adaptive fault-tolerant approach for semiactive suspension control of high-speed trains,” IEEE Trans. Ind. Electron., vol. 63, no. 11, pp. 7084-7093, 2016. I also would like to thank the Research Institute of Intelligent Systems at Chongqing University for their support. The writing of this book was supported in part by the National Natural Science Foundation of China.

xiii

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page xiv — #12

✐

✐

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page xv — #13

✐

✐

Acronyms

Abbreviations UUB MAS

uniformly ultimately bounded multi-agent systems

Notations R Σ |a| kxk max min sup inf ∀ ∈ → < (>) ≤ (≥) ≪ (≫) Rn y˙ y¨ y(i) w.r.t.

field of real numbers summation the absolute of a scalar a the norm of a vector x maximum minimum supremum, the least upper bound infimum, the greatest lower bound for all belongs to tends to less (greater) than less (greater) than or equal to much less (greater) than the n−dimensional Euclidean space the first derivative of y with respect to time the second derivative of y with respect to time the i−th derivative of y with respect to time with respect to

xv

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page xvi — #14

✐

✐

xvi

sat(·) sgn(·) diag{a1 , · · · , an } P>0 P≥0 AT (xT ) λmax (P) (λmin (P)) f −1 (·)

Acronyms

the saturation function the signum function a diagonal matrix with diagonal elements a1 to an a positive definite matrix P a positive semi-definite matrix P the transpose of matrix A (a vector x) the maximum (minimum) eigenvalue of a symmetric matrix P the inverse of a function f

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 1 — #15

✐

✐

Chapter 1

Introduction

1.1 Motivation A proportional integral derivative (PID) controller is a control loop feedback mechanism widely used in industrial control systems. Fig. 1.1 is a block diagram of the classical PID controller. A PID controller continuously calculates an error value e(t) as the difference between a desired setpoint r(t) and a measured process variable y(t) and applies a correction based on proportional, integral, and derivative terms (denoted P, I, and D, respectively) which give their name to the controller.

Fig. 1.1 Classical PID controller in a feedback loop. r(t) is the desired process input or setpoint, y(t) is the measured process output, u(t) is the control input, and e(t) = r(t)− y(t) is the discrepancy between the setpoint and the output.

In PID control, the P-control (proportional control) is the action based on current behavior of the system, I-control (integral control) is the accumulated effort using the experience information of bygone state, whereas D-control (derivative control) is the predictive effort based on the tendency information for the ongoing state. Since the Ziegler and Nichols’ PID tuning rules were published in 1942, the PID control has survived the challenge of advanced control theories. The PID’s long life comes from its clear meaning and effectiveness in practice; thus the PI/PD/PID control has been widely accepted in industry. Especially, it is suitable for control systems that can establish accurate mathematical models. However, most practical 1

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 2 — #16

✐

✐

2

1 Introduction

industrial processes are nonlinear, uncertain, and are difficult to establish accurate mathematical models for. Thus, the traditional PID gains tuning methods cannot obtain optimal control performance. To deal with this problem, various PID gains self-tuning methods have been suggested. Though the PI/PD/PID control has been widely accepted in industry, PI/PD/PID control itself is still short of the theoretical basis, e.g., the optimality of PI/PD/PID control, performance tuning rules of PI/PD/PID control, and automatic performance tuning methods have not been clearly presented especially for the trajectory tracking control of nonlinear systems. Our interest in revisiting PI/PD/PID control is largely motivated by the fact that, although various advanced control methods have been developed during the past decades, the preferred one in engineering practice is still the PI/PD/PID control, due to its simplicity in structure and intuitiveness in concept. It has gained wide application in practical engineering systems. However, the well-known PI/PD/PID control exhibits two major drawbacks that restrict its application to more general systems. The first one is the determination of the PI/PD/PID gains for a given system is an ad hoc and painstaking process. Thus far, there exists no systematic means to guide the determination of such gains that ensure system stability and performance, although various methods for tuning PI/PD/PID gains have been suggested in the literature. The second one is that although PI/PD/PID control has been demonstrated to be quite effective in dealing with certain linear time-invariant systems, its applicability to nonlinear systems remains unclear and lacks theoretical insurance for closed-loop system stability and performance. Furthermore, it is desirable or required to equip such PI/PD/PID schemes with adaptive and fault-tolerant capabilities yet guaranteeing transient performance.

1.2 Objectives Firstly, this book attempts to provide readers with an overview of the basic principle of PID control. Traditional PID control is characterized with constant PID gains and is oriented for set point regulation; thus it seldom works satisfactorily for general nonlinear systems with uncertain dynamics and unpredictable disturbances. Besides, stability has always been the major concern with traditional PID control due to the lack of the systematic procedure for determining the proper stability-ensured PID gains for a given dynamic system. Secondly, through detail theoretical analysis and technical development, this book intends to show how conventional PI/PD/PID controllers could be extended and generalized to deal with various systems, such as SISO nonlinear systems, SISO nonaffine systems, and MIMO nonlinear systems. The emphasis is on how to enable these controllers with the capabilities of tuning their gains automatically to compensate for system uncertainties and reject external disturbances. Furthermore, as nonsmooth actuation characteristics or actuation failures (partial loss of effectiveness (PLOE) or total loss of effectiveness (TLOE) ) might occur during

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 3 — #17

✐

✐

1.3 Preview of Chapters

3

system operation, effort is also made on designing PI/PD/PID control with adaptive and fault-tolerant capabilities.

1.3 Preview of Chapters In Chapter 2, a brief review of the traditional PID control with fixed gains is presented. In Chapter 3, a generalized PI control with adaptively adjusting gains is presented for single input single output (SISO) nonlinear systems. We consider two control schemes: one is for the first-order nonlinear system; the other is for the high-order nonlinear system. Besides, the developed PI controller is suitable for nonlinear systems with undetectable disturbances and actuation failures. Meanwhile, the pre-scribed transient and steady-state performances are dynamically maintained. In Chapter 4, a generalized adaptive PI control is developed for unknown nonaffine dynamic systems. As the control inputs enter into and influence the dynamic behavior of the nonaffine system in a nonlinear and implicit way, control design for such systems becomes quite challenging. The proposed control is able to accommodate both sensor and actuator faults. In Chapter 5, neuro-adaptive PI control algorithms with self-tuning gains are developed for a class of multi-input multi-output (MIMO) normal-form nonlinear systems subject to unknown actuation characteristics and external disturbances. It is shown that the proposed neuro-adaptive PI control is continuous and smooth everywhere and ensures the uniform ultimate boundedness of all the signals of the closed-loop system. Furthermore, the crucial compact set precondition for a neural network (NN) to function properly is guaranteed with the barrier Lyapunov function (BLF), allowing the NN unit to play its learning/approximating role during the entire system operation. In Chapter 6, a neuro-adaptive PI control for a class of uncertain nonlinear strict feedback systems with full-state constraints and unknown actuation characteristics is presented. In order to deal with the modeling uncertainties and the actuation characteristics impact, the neural networks are utilized at each step of the back stepping design procedure. In Chapter 7, it is shown that the structurally simple and computationally inexpensive PID control, popular with SISO linear time-invariant systems, can be generalized and extended to control nonlinear MIMO systems with nonparametric uncertainties and actuation failures. By utilizing the Nussbaum-type function and the matrix decomposition technique, non-square systems with unknown control direction are also considered. In Chapter 8, the PD-like controller is designed for a high-speed train system. The situation is further complicated if actuation faults occur. The resultant control scheme is capable of automatically generating the intermediate control parameters

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 4 — #18

✐

✐

4

1 Introduction

and literally producing the PD-like controller. The whole process does not require precise information regarding system model or system parameter. In Chapter 9, the robust adaptive PID controller is applied to a robotic system. Under the proposed PID-like control the vibrations are effectively suppressed in the presence of parametric uncertainties and varying operation conditions.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 5 — #19

✐

✐

Chapter 2

Classical PID Control

In this chapter, the structure of the PID controller and the roles of the three (proportional, integral, and derivative) terms of the PID controller, together with the tuning of the PID gains, are discussed.

2.1 The Three Actions of PID Control A typically PID controller involves three types of control actions: a proportional action, an integral action, and derivative action, which can be mathematically expressed as Z t

de(t) (2.1) dt where K p , Ki , and Kd denote the proportional, integral, and derivative gain, respectively. The role of each term is described and discussed briefly in what follows. u(t) = K p e(t) + Ki

0

e(τ )d τ + Kd

2.1.1 Proportional Action The proportional control action is proportional to the current control error, which can be expressed as u(t) = K p e(t) = K p (r(t) − y(t)) (2.2) where K p is the proportional gain. The role of such control is quite obvious since it implements the typical operation of increasing the control effort when the control error is large (with appropriate sign). The transfer function of a proportional controller can be derived trivially as

5

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 6 — #20

✐

✐

6

2 Classical PID Control

C(s) = K p

(2.3)

If K p is made large, the steady-state error would be small. But the dynamic response would become worse because the damping is too low. Apparently, a proportional controller has the advantage of providing a small control input when the control error is small and therefore can avoid excessive control efforts. The main drawback of using a pure proportional controller is that it produces a steady-state error. It is worth noting that this would still occur even if the process bears an integrating dynamics (i.e., its transfer function has a pole at the origin of the complex plane), in case a constant load disturbance occurs. This motivates the addition of a bias (or reset) term ub , namely, [1, 2, 3] u(t) = K p e(t) + ub (2.4) The value of ub can be fixed at a constant level (usually at (umax + umin )/2) or can be adjusted manually until the steady-state error is reduced to zero, where umax and umin denote the maximum and minimum value of the control input, respectively.

2.1.2 Integral Action The integral action is proportional to the integral of the control error, i.e., u(t) = Ki

Z t

e(τ )d τ

(2.5)

0

where Ki is the integral gain. With the integral action, the resultant control makes use of the past values of the control error to generate its control signal. The corresponding transfer function is: Ki C(s) = (2.6) s The presence of a pole at the origin of the complex plane allows the steady-state error to be reduced to zero when a step reference signal is applied or a step load disturbance occurs. In other words, the integral action is able to set automatically the correct value of ub in (2.4) so that the steady-state error is zero [1]. This actually results in a PI controller with the following transfer function C(s) = K p (1 +

1 ) Ti s

(2.7)

where Ti is integration time constant. The block diagram in Fig. 2.1 shows how integral action is implemented using positive feedback with a first-order system. The controller output is low-pass-filtered and feed back with positive gain. The integral action is used to generate the bias term ub in (2.4) in the proportional controller automatically, often called automatic reset. For this reason the integral action is also often called automatic reset. Thus, the use of a proportional action in conjunction to an integral action, i.e., a PI controller, solves the main problems of

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 7 — #21

✐

✐

2.1 The Three Actions of PID Control

7

the oscillatory response associated to an on-off controller and of the steady-state error associated to a pure proportional controller.

Fig. 2.1 PI controller involving P and I actions.

2.1.3 Derivative Action The derivative control takes the following form, u(t) = Kd

de(t) dt

(2.8)

where Kd is the derivative gain, which makes use of the predicted future values of the control error. The corresponding controller transfer function is C(s) = Kd s

(2.9)

Upon using Euler formula to approximate the derivative in (2.8), it is derived that e(t + Td ) ≃ e(t) + Td

de(t) dt

(2.10)

where Td is the sampling period, implying that e(t) + Td de(t) dt is able to reflect the value of the control error at time t + Td . So if a control law proportional to this expression is considered, i.e., u(t) = K p (e(t) + Td

de(t) ) dt

(2.11)

then the control input at time t is actually based on the predicted value of the control error at time t +Td . In order words, the controller (2.11) consisting of P and D terms is able to enhance the transient response of the closed-loop system. For this reason the derivative action is also called anticipatory control, or rate action, or pre-act [1]. It should be stressed that although the derivative action has great potential in improving the control performance as it can anticipate an incorrect trend of the control error and

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 8 — #22

✐

✐

8

2 Classical PID Control

counteract it, it also creates some critical issues that should be carefully addressed in control design in practice.

2.2 Tuning Methods As the structure of the PID controller is fixed, the tuning of its PID gains should be carefully considered for different applications with various requirements. By properly tuning the three parameters, a PID controller can deal with specific process requirements. There is no uniform method for tuning the PID gains. The most commonly used one is the “trial-and-error” tuning, which starts with determining the proportional gain K p first, then trying to find the integral time constant Ti and the derivative time constant Td , with which the integral gain Ki is obtained by Ki = K p /Ti and the derivative gain Kd is set as Kd = Td K p . The Ziegler-Nichols (ZN) tuning rule is also a popular method used in practice. The tuning rule is simple and needs only the ultimate information, which can be estimated easily by simple identification methods, such as the continuous-cycling method and relay feedback identification method [4]. The ZN tuning rule works satisfactorily for certain processes. However, because the ZN tuning rule uses only the ultimate data of the process, its performance is uncertain for those systems with unusual frequency response characteristics. For more complicated systems, manual calculation methods are no longer practical. Software based PID tuning and loop optimization is a must. There are some software packages that gather the data, develop process models, and suggest optimal tuning. Some software packages can even develop tuning procedures by gathering data from reference changes [1, 2].

2.3 Conclusion A brief overview of traditional PID controller is presented in this chapter, starting with the three actions of PID control. In order to make the PIDcontrol function satisfactorily with some specific process requirements, proper methods for tuning PID gains should be utilized. Although a number of PID tuning methods (e.g., trial-and-error tuning, Ziegler—Nichols method and PID tuning software) are available in literature, the traditional PID controllers with fixed gains are apparently ineffective in dealing with the increasingly sophisticated systems. Several PI/PD/PID control design methods for complex nonlinear systems are presented in the following chapters.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 9 — #23

✐

✐

Chapter 3

Adaptive PI Control for SISO Affine Systems

It is a long lasting open problem to synthesize a general PI control for nonlinear systems with its gains analytically determined, yet ensuring stability and transient performance. The problem is further complicated if modeling uncertainties and external disturbances as well as actuation failures are involved in the systems. In this chapter, a generalized PI control with adaptively adjusting gains is presented, which gracefully obviates the ad hoc and time-consuming “trial and error” process for determining the gains as involved in traditional PI control; collectively accommodates modeling uncertainties, undetectable disturbances, and undetectable actuation failures that might occur in the systems; and dynamically maintains prespecified transient and steady-state performance.

3.1 Introduction The problem addressed in this chapter is: would PI (proportional and integral) control be applicable to uncertain nonlinear systems? Our interest in revisiting PI control is largely motivated by the fact that, although various advanced control methods have been developed during the past decades, the preferred one in engineering practice is still the PID/PI control, due to its simplicity in structure and intuitiveness in concept. Therefore it has gained wide application in practical engineering systems [5, 6, 7]. However, the well-known PI control exhibits two major drawbacks that restrict its application to more general systems. The first one is that the determination of the PI gains for a given system is an ad hoc and painstaking process. Thus far there exists no systematic means to guide the determination of such gains that ensure system stability and performance, although various methods for tuning PI gains have been suggested in the literature [2, 3, 5, 8]. The second one is that although PI control has been demonstrated quite effective in dealing with certain linear time-invariant systems, its applicability to nonlinear systems remains unclear and lacks theoretical insurance for closed-loop system stability and performance. While some efforts have been made in developing 9

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 10 — #24

✐

✐

10

3 Adaptive PI Control for SISO Affine Systems

algorithms for tuning/adjusting PID/PI gains by utilizing generic algorithms, neural networks, and/or fuzzy system techniques [9, 10, 11, 12, 13] (to name a few), there still leaves much to be desired in most existing methods in terms of simplicity, affordability, and effectiveness. The interesting issue to address is therefore: would it be possible to construct PI-like control capable of dealing with nonlinear uncertain systems where the PI gains are systematically and adaptively determined by the control algorithm itself? Furthermore, is it possible to equip such a PI scheme with adaptive and fault-tolerant capabilities yet guarantee transient performance? The purpose of this chapter is to present a solution to address these issues.

3.2 Problem Formulation Consider the following class of uncertain nonlinear systems, x˙k = xk+1 , k = 1, 2, · · · , n − 1 x˙n = g(X,t)ua + f (X,t)

(3.1)

where xk ∈ R (k = 1, · · · , n) is the kth state with x1 = x; X = [x1 , · · · , xn ]T ; ua ∈ R is the actual control input of the system (the output of the actuator); g(·) ∈ R is the time-varying and uncertain control gain; f (·) ∈ R denotes the lumped uncertainties and external disturbances. As unanticipated actuator faults may occur, we additionally include such scenarios in the model, where the actual control input ua and the designed input u are no longer the same in that ua = ρ (tρ ,t)u + ur (tr ,t)

(3.2)

where 0 ≤ ρ (·) ≤ 1, known as the “healthy indicator” [12], indicates the actuation effectiveness, ur (·) is the uncontrollable portion of the control signal, tρ and tr denote, respectively, the time instant at which the loss of actuation effectiveness fault and the additive actuation fault occur. In this chapter, we consider the case that 0 < ρ (·) ≤ 1, i.e., although losing its effectiveness, the actuation is still functional such that ua can be influenced by the control input u all the time. In addition, tρ and tr are assumed completely unknown; this fact, together with the unknown and time varying ρ and ur , literally implies that the occurrence instant and the magnitude of the actuation faults are unpredictable. The dynamic model considering actuation failures then becomes x˙k = xk+1 , k = 1, 2, · · · , n − 1 x˙n = g(X,t)ρ (tρ ,t)u + f (X,t) + g(X,t)ur(tr ,t)

(3.3)

The objective is to design a PI-like tracking controller for the system with lumped uncertainties and disturbances as well as actuator faults as described by

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 11 — #25

✐

✐

3.2 Problem Formulation

11

(3.3) such that not only stable tracking is achieved, but also pre-described performance is ensured, yet all the internal signals are continuous and bounded. More specifically, the PI-like control ensures that: 1) the tracking error E = X − X ∗ = [ε1 , ε2 , · · · , εn ]T (ε = ε1 ) converges to a small residual set containing the origin for any given desired trajectory X ∗ = [x∗ , x˙∗ , · · · , x∗(n−1) ]T ; 2) the tracking error is confined within a pre-given bound all the time, i.e., there exist performance functions µ1k (t) and µ2k (t) such that µ1k (t) ≤ εk (t) ≤ µ2k (t) (k = 1, · · · , n) for all t ≥ 0. In addition, the convergence rate is controlled by e−a0t for some pre-specified constant a0 > 0; and 3) all the internal signals in the system are ensured to be continuous and bounded. To proceed, the following assumptions are in order. Assumption 3.1 The control gain g(·) is unknown and time-varying but bounded away from zero, i.e., there exist some unknown constants g and g¯ such that 0 < g ≤ |g(·)| ≤ g¯ < ∞, and g(·) is sign-definite (in this note sgn(g) = +1 is assumed without loss of generality). Assumption 3.2 The desired state x∗ and its derivative up to (n − 1)th are assumed to be smooth and bounded. In addition, x∗(n) , the nth derivative of x∗ , is bounded by an unknown constant xm , i.e., |x∗(n) | ≤ xm < ∞, ∀t ≥ t0 . Assumption 3.3 For uncertain nonlinearities f (·), there exist an unknown constant c f ≥ 0 and a known scalar function ϕ (X,t) ≥ 0 such that | f (·)| ≤ c f ϕ (·). If X is bounded, so is ϕ (X,t). Assumption 3.4 ρ (·) and ur (·) are unknown, possibly fast time-varying and unpredictable, but bounded in that there exist some unknown constants ρm and r¯ such that 0 < ρm ≤ ρ (·) ≤ 1 and |ur (·)| ≤ r¯ < ∞. Remark 3.1 Assumptions 3.1—3.2 are commonly imposed in most existing works in addressing the tracking control problem of system (3.1) [10, 14, 15, 16, 17]. Assumption 3.3 is related to the extraction of the core information from the nonlinearities of the system, which can be readily done for any practical system with only crude model information. As for Assumption 3.4, it is noted that most FDD/FDI based fault tolerant control implicitly assumes that the faults vary with time slowly enough to allow for timely fault identification and diagnosis [18, 19] or that one has enough information on the faults to carry out parametric decomposition [16], while Assumption 3.4 imposes no such restriction, and thus seems more practical. Remark 3.2 Note that in practice it would be very difficult, if not impossible, to obtain the exact values of those bounds involved in Assumptions 3.1—3.4. The developed PI-like control in this chapter, however, is independent of those bound parameters, thus there is no need for analytical estimation of such bounds despite the fact that those bounds do exist in stability analysis.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 12 — #26

✐

✐

12

3 Adaptive PI Control for SISO Affine Systems

3.3 Design Details To help understand the fundamental idea and the technical development of the proposed method, we start with controller design for the first-order nonlinear systems, followed by the extension to the high-order case.

3.3.1 PI Control Design for First-order Nonlinear Systems In this subsection we develop the generalized PI control law for first-order nonlinear systems with actuation failures as described by (3.2). In this case (3.3) with (3.2) becomes x(t) ˙ = g(x,t)ρ (·)u(t) + g(x,t)ur (·) + f (x(t),t) (3.4) where x ∈ R denotes the system state. To facilitate the PI controller design, we first introduce a filtered variable s as, s = ε +β

Z t

ε dτ

(3.5)

0

where ε = x − x∗ is the tracking error, and β > 0 is a free parameter chosen by the designer. To establish the main results, the following lemma is needed. Lemma 3.1 R Consider the filtered variable s defined in (3.5). If limt→∞ s = 0, then ε (t) and 0t ε d τ converge asymptotically to zero as t → ∞Rwith the same decreasing rate as that of s. In addition, if s is bounded, so are ε and 0t ε d τ . Proof. The proof can be readily done by using the L’Hopital’s rule, so is omitted here. The proposed generalized PI control is of the form u = −(k p1 + ∆ k p1 (t))ε (t) − (kI1 + ∆ kI1 (t))

Z t

ε (τ )d τ

(3.6)

0

Different from the traditional PI control that involves constant gains, the PI gains here consist of two parts: 1) constant gains k p1 > 0 and kI1 = β k p1 > 0, with k p1 and β being chosen freely by the designer and 2) time-varying gains ∆ k p1 (t) and ∆ kI1 (t) determined automatically and adaptively by the following algorithm,

∆ k p1 = with

cˆψ 2 , ψ |s| + ι

∆ kI1 = β ∆ k p1

σ1 ψ 2 s2 c˙ˆ = −σ1 γ1 cˆ + ψ |s| + ι

(3.7)

(3.8)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 13 — #27

✐

✐

3.3 Design Details

13

where cˆ is the estimation of c with c being a virtual parameter to be defined later, ψ (·) = 1 + ϕ (·) + |ε | is a scalar and readily computable function, β , σ1 and γ1 are positive design parameters chosen by the designer, ι > 0 is a small constant. At this point, it is worth stressing the essential difference between the traditional PI control and the proposed one. Firstly, unlike the traditional PI control where the PI gains are case-based, hand-tuned by trial and error, and remain constant during the entire system operation, here the PI gains have two components, one is constant and the other is time-varying. Furthermore, the constant part is determined by the designer quite flexibly and the time-varying part is consistently tuned automatically and adaptively by the algorithm. In addition, the P-gains (k p1, ∆ k p1 ) and I-gains (kI1 , ∆ kI1 ) are determined correlatively through the parameter β , rather than independently as in traditional PI control. Most importantly, the proposed PI control with the gains so determined ensures system stability, despite modeling uncertainties and actuator faults, as stated in the following theorem. Theorem 3.1. Consider the nonlinear uncertain system (3.4) with actuation failures as described by (3.2) under Assumptions 3.1—3.4. If controlled by the generalized PI controller (3.6) with the PI gains updated by (3.7) and (3.8), then modeling uncertainties and actuation faults are accommodated automatically without the need for fault detection and ultimately uniformly bounded stable tracking is ensured. Furthermore, all the internal signals in the system Rare guaranteed to be continuous and bounded everywhere. More specifically, u, ε , 0t ε d τ , ε˙ , s, s, ˙ cˆ and c˙ˆ are bounded and continuous everywhere. Proof. By utilizing the filtered variable s as given in (3.5), we re-express (3.4) as s˙ = gρ u + +gur + f − x˙∗ + β ε (t)

(3.9)

Noting that we have purposely linked k p and kI through β by kI = β k p , which not only reduces the design degree of complexity from 2 to 1, but also allows for (3.6) to be expressed as u = −(k p1 + ∆ k p1)s, facilitating stability analysis as seen shortly. Defining V1 = 21 s2 , and taking the time derivative of V1 along (3.9) yields V˙1 = −k p1 gρ s2 − gρ∆ k p1s2 + s(gur + f − x˙∗ + β ε (t)) It is straightforward from Assumptions 3.1—3.4 that |gur − x˙∗ + f + β ε (t)| ≤ g¯ ¯r + xm + c f ϕ (·) + β |ε | ≤ cψ (·) with c = max{g¯ ¯r + xm , c f , β } < ∞ and ψ (·) = 1 + ϕ (·) + |ε |. Here c is an unknown virtual (bearing no physical meaning) parameter. Upon inserting ∆ k p1 as given in (3.7), we get 2 2

gρ cˆψ s V˙1 ≤ −k p1 gρm s2 + [− + |s|cψ ] ψ |s| + ι ≤ −k p1gρm s2 + [−gρm

cˆψ 2 s2 cψ 2 s2 + cψ |s|ι + ] ψ |s| + ι ψ |s| + ι

≤ −k p gρm s2 + [(c − gρm c) ˆ

ψ 2 s2 + cι ] ψ |s| + ι

(3.10)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 14 — #28

✐

✐

14

3 Adaptive PI Control for SISO Affine Systems

|s| where the facts that cˆ ≥ 0 for any c(0) ˆ ≥ 0 and ψψ|s|+ ι ≤ 1 for any ι > 0 have been used. Note that there appears a parameter estimation error of the form •˜ = • − gρm •ˆ in (3.10), we thus introduce the error of the form c˜ = c − gρm c, ˆ named virtual parameter estimation error here, and blend such error into the second part of the Lyapunov function candidate, V2 = 2σ11 g c˜2 . Such treatment allows the unknown and time-varying control gain g(·) and the unknown actuation effectiveness fault ρ (·) to be processed gracefully. By considering the Lyapunov function candidate V = V1 +V2 , it thus follows from (3.10) that c˙ˆ ψ 2 s2 − ) + cι ] V˙ ≤ −k p1 gρm s2 + [(c − gρm c)( ˆ ψ |s| + ι σ1

By inserting the adaptive law for cˆ given in (3.8), we then have V˙ ≤ −k p1gρm s2 + cι + γ1 c˜cˆ

Note that c˜cˆ = c˜ gρ1m (c − c) ˜ ≤

1 2 2 2gρm (c − c˜ ),

then we have

γ1 2 γ1 2 V˙ ≤ −k p1 gρm s2 − c˜ + c + cι ≤ −l1V + l2 2gρm 2gρm where l1 = min{2k p1gρm , γ1 σ1 } > 0, and l2 =

γ1 c2 2gρm

(3.11)

+ cι < ∞. From (3.11), it can be

l2 l1 }

˜ ≤ is globally attractive. Once (s, c) ˜ ∈ / Ω, concluded that the set Ω = {(s, c)|V ˙ V < 0. Therefore, there exists a finite ˜ ∈ Ω for ∀t > T0 . This q time T0 such that (s, c) √ 2l2 further implies that |s| ≤ 2V1 ≤ l1 for ∀t > T0 . That is, s is Ultimately Uniformly Bounded (UUB), and thus the tracking error ε is also UUB according to Lemma 3.1. In the sequel we prove all the internal signals in the system are continuous and bounded. By solving (3.11), it is derived that, V (t) ≤ exp−l1t V (0) + ll12 ∈ L∞ for all t ≥ 0, which then implies Rthat s ∈ L∞ and cˆ ∈ L∞ . According to Lemma 3.1, s ∈ L∞ implies that ε ∈ L∞ and 0t ε d τ ∈ L∞ , where ε ∈ L∞ further implies that x ∈ L∞ (because x∗ is bounded from Assumption 3.2) and then ϕ (x,t) ∈ L∞ by Assumption 3.3. From the definition of ψ (·), i.e., ψ (·) = 1 + ϕ (·) + |ε |, it follows that ψ (·) ∈ L∞ , which ensures that both ∆ k p1 and ∆ kI1 are bounded. Then u ∈ L∞ and c˙ˆ ∈ L∞ from (3.6)—(3.8). Finally, one can conclude from (3.9) that s˙ ∈ L∞ . Therefore, all the internal signals in the system are continuous and bounded.

3.3.2 PI Control Design for High-order Nonlinear Systems In this subsection we extend the previous PI-like control to the high-order nonlinear system (3.3) involving modeling uncertainties and actuation faults. The goal is to develop a PI-like control that not only exhibits robust, adaptive, and fault-tolerant capabilities, but also guarantees pre-specified transient and steady-state performance.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 15 — #29

✐

✐

3.3 Design Details

15

To proceed, we introduce a filtered variable ω as follows,

ω = β1 ε1 + β2 ε2 + · · · + βn−1εn−1 + εn

(3.12)

with ε1 = ε (t) and ε˙k = εk+1 (k = 1, 2, · · · , n − 1), where βk (k = 1, · · · , n − 1) is a constant chosen by the designer such that the polynomial sn−1 + βn−1 sn−2 + · · · + β1 is Hurwitz. The following Lemma is needed in order to establish the boundedness relation between the filtered error ω and the tracking error εk (k = 1, · · · , n). Lemma 3.2 Consider the filtered error ω given in (3.12); if ω → 0 as t → ∞, then the tracking error ε (t) and its derivative εk (k = 2, · · · , n) converge asymptotically to zero as t → ∞ with the same decreasing rate as that of ω . Furthermore, if ω is preserved within a pre-specified performance bound all the time, i.e., there exist performance functions µ1 (t) and µ2 (t) such that

µ1 (t) ≤ ω (t) ≤ µ2 (t), ∀t ≥ 0

(3.13)

then similar bounds also hold for εk (k = 1, · · · , n). Proof. The proof is given in the Appendix. According to Lemma 3.2, to guarantee that the tracking error εk (k = 1, · · · , n) satisfies the pre-described performance requirements, it is sufficient to design a PIlike control to ensure that the filtered error ω (t) is confined within the bound (3.13) all the time. With this in mind, we choose the performance functions µ1 (t) and µ2 (t) as in the earlier work [17, 20],

µ1 (t) = −δ η (t), ¯

µ2 (t) = δ¯ η (t)

(3.14)

where 0 < δ ≤ δ¯ are lower-upper bound constants and η (t) is a strictly decreasing ¯ smooth function of time that determines the convergence rate of ω (t). The rate function η (t) in this chapter is chosen as η (t) = (η0 − η∞ )e−a0t + η∞ , where η0 > η∞ > 0 and a0 is an arbitrarily positive constant. It is noted that δ¯ η0 and −δ η0 characterize the upper bound of the overshoot of ω (t) and the lower bound of ¯the undershoot of ω (t), respectively. In addition, the transient performance can be adjusted and improved by changing the design parameters η0 , η∞ , a0 , δ and δ¯ ¯ properly. We know from (3.13) that both pre-required transient performance and steady-state control precision can be achieved if the tracking error is controlled to behave and evolve according to (3.13). On the other hand, however, technical challenge arises in control design and stability analysis if we directly deal with (3.13) due to such constraint. To circumvent this difficulty, we carry out the following performance transformation

ω (t) = T (ν )η (t)

(3.15)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 16 — #30

✐

✐

16

3 Adaptive PI Control for SISO Affine Systems

where ν is the new transformed error and T (ν ) is a smooth and strictly increasing function which is thus invertible. It is interesting to note that if T (ν ) exhibits the following properties: i). −δ < T (ν ) < δ¯ ; ii). limν →−∞ T (ν ) = −δ , limν →∞ T (ν ) = δ¯ ; iii). limν →0 T (ν ) = 0, in¯ particular, T (0) = 0, then it can be readily verified that (3.13) is naturally ensured as long as ν is controlled to be bounded for t ≥ 0 (because T (ν ) in (3.15) is thus bounded according to its properties). Therefore, the problem of ensuring pre-specified performance bound can be addressed by stabilizing ν . To this end, we explicitly express ν as defined in (3.15) as

ν = T −1 (

ω (t) ) η (t)

(3.16)

which is well defined because T (ν ) is invertible and η (t) is strictly positive. It should be mentioned that the inversion in (3.16) can be implemented if and only if −δ < ω (t) ¯ η (t) < δ is satisfied for all t ≥ 0. To ensure this, we need to select initial value η (0), −δ and δ¯ such that −δ < ω (0) < δ¯ for any arbitrary initial tracking error ω (0), and η (0)

also we must design an appropriate control to ensure signal ν satisfying ν (t) ∈ L∞ , (t) < δ¯ is and therefore T (ν ) is confined within (−δ , δ¯ ), then from (3.15), −δ < ωη (t) satisfied all the time. The above analysis indicates that the problem of guaranteeing prescribed performance boils down to selecting η (0) properly according to the maximum possible initial tracking error ω (0) and designing control u to steer ν (t) into a compact set and confine ν (t) within such set ever after (i.e., ν (t) is UUB). As the first step, we choose T (ν ) as [17], T (ν ) =

δ¯ e(ν +r) − δ e−(ν +r) ¯ e(ν +r) + e−(ν +r)

(3.17)

with r = 12 ln(δ /δ¯ ). Note that T (ν ) satisfies the above conditions (i)—(iii), and the ¯ variable r is well defined because δ /δ¯ is always positive. The important implication of the¯ above transformation is that if T (ν ) is ensured to be bounded such that −δ < T (ν ) < δ¯ , the lower and upper bounds imposed on ω (t) as in (3.13) are satisfied thanks to (3.15). In other words, the constrained tracking error relation (3.13) can be converted into the unconstrained relation (3.15) as long as the signal ν is controlled to be bounded, all the time. Therefore, we only need to focus on stabilizing ν such that ν is ensured to be UUB. To this end, note that

ν = T −1 ( with λω (t) =

ω (t) η (t) .

ω (t) 1 1 ) = ln(δ¯ λω (t) + δ¯ δ ) − ln(δ¯ δ − δ λω (t)) η (t) 2 2 ¯ ¯ ¯

(3.18)

It then follows that

ν˙ =

∂ T −1 ˙ ω η˙ λω = kω (ω˙ − ) ∂ λω η

(3.19)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 17 — #31

✐

✐

3.3 Design Details

17

where

1 1 1 − ( ) (3.20) 2η λω + δ λω − δ¯ ¯ Note that kω is strictly positive as η ∈ (η0 , η∞ ) and λω ∈ (−δ , δ¯ ). This is crucial to ¯ the following controller design. Note that in (3.15) the filtered error ω (t) rather than the tracking error ε (t) itself is used. It is such treatment that allows the problem of guaranteeing transient performance for the high-order nonlinear system to be addressed gracefully, as seen shortly. As discussed earlier, UUB control of ω (t) is sufficient to ensure prescribed performance bounded control of εk (t) (k = 1, · · · , n). We now present the following PI-like control to stabilize and meanwhile ensure the pre-specified transient performance of system (3.3), kω =

u = −(k p2 + ∆ k p2 )ν (t) − (kI2 + ∆ kI2 )

Z t

ν (τ )d τ

(3.21)

0

Here the PI gains consist of two parts: 1) k p2 and kI2 given by k p2 = kω−1 k¯ p2 ,

kI2 = β k p2

(3.22)

where kω is given as in (3.20) and k¯ p2 > 0 is user-defined constant, and 2) ∆ k p2 and ∆ kI2 , adjusted automatically and adaptively by the following algorithm

∆ k p2 = with the adaptive law

bˆ χ 2 , χ |ν + β 0 ν (τ )d τ | + kω−1 ι Rt

R

∆ kI2 = β ∆ k p2

σ2 kω χ 2 (ν + β 0t ν (τ )d τ )2 b˙ˆ = −σ2 γ2 bˆ + R χ |ν + β 0t ν (τ )d τ | + kω−1 ι

(3.23)

(3.24)

where bˆ is the estimation of the virtual parameter b to be defined later, χ (·) = ϕ (·) + ˙ 1 + |ε2 | + · · ·+ |εn | + | ηη ω | + kω−1 |ν (t)| is a scalar and readily computable function, β , σ2 and γ2 are positive design parameters chosen by the designer, and ι > 0 is a small constant. It is seen that in constructing the PI gains the rate function η (t) and the tracking error ω (t) are explicitly utilized and both k p2 and kI2 are now time-varying rather than constant. Theorem 3.2. Consider the high-order uncertain nonlinear system with actuation failures as described by (3.3). Suppose that Assumptions 3.1—3.4 hold. If the control algorithms (3.21)—(3.23) with the adaptive law (3.24) are applied, then it is established that 1) UUB stable tracking is ensured; 2) the tracking error εk (k = 1, · · · , n) converges to small residual sets containing the origin at the rate of e−a0t ; 3) the tracking error εk (k = 1, · · · , n) remains within the bounds related to

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 18 — #32

✐

✐

18

3 Adaptive PI Control for SISO Affine Systems

(3.13) all the time; and 4) all the internal signals are bounded and continuous everywhere. Proof. From the definition of the filtered error ω , the high-order system (3.3) can be rewritten as ω˙ = gρ u + f + gur − x˙∗n + β1 ε2 + · · · + βn−1εn (3.25) which, upon using (3.19), can be further expressed in terms of the transformed variable ν as ν˙ = kω (gρ u + L(·)) (3.26) where L(·) = f + gur − x˙∗n + β1 ε2 + · · · + βn−1εn − ηη ω . Define a filtered variable S as ˙

S = ν +β

Z t

ν (τ )d τ

(3.27)

0

Then the control law given in (3.21) becomes u = −kω−1 k¯ p2 S − ∆ k p2S, and the error dynamic (3.27) is rewritten as S˙ = kω (gρ u + L(·) + kω−1β ν (t))

(3.28)

From Assumptions 3.1—3.4, it is readily verified that |L(·) + kω−1 β ν (t)| ≤ c f ϕ (·) + ˙ ξ + β1 |ε2 | + · · · + βn−1|εn | + | ηη ω | + kω−1 β |ν (t)| ≤ bχ (·), where b = max{c f , ζ , β1 , · · · , βn−1 , 1, β } < ∞ with ζ = g¯ ¯r + xm < ∞, and χ (·) = ϕ (·) + 1 + |ε2 | + · · · + |εn | + η˙ −1 ω | η | + kω |ν (t)| (a computable and implementable signal). A newly defined virtual ˆ with which the Lyapunov parameter estimate error is introduced as, b˜ = b − gρm b, function candidate is constructed as, V = V3 +V4 , with V3 = 1 S2 and V4 = 1 b˜ 2 . 2

2σ2 gρm

Taking the time derivative of V3 along (3.28) yields V˙3 = SS˙ = Skω (gρ u + L(·) + kω−1β ν (t))

(3.29)

By using the control law u as equivalently expressed as u = −kω−1 k¯ p2 S − ∆ k p2S and plugging in (3.23), one readily gets that V˙3 = −k¯ p2 gρ S2 + Skω (−gρ∆ k p2S + L(·) + kω−1β ν (t)) kω gρ bˆ χ 2 S2 ≤ −k¯ pgρm S2 + [− + |S|kω bχ ] χ |S| + kω−1ι 2 2 ˆ kω χ S + b ι ] ≤ −k¯ p2gρm S2 + [(b − gρm b) χ |S| + kω−1ι

(3.30)

The time derivative of V4 is computed, by using the adaptive law for bˆ given in (3.24), as ˆ γ2 bˆ − V˙4 = (b − gρm b)(

kω χ 2 S 2 ) χ |S| + kω−1ι

(3.31)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 19 — #33

✐

✐

3.3 Design Details

19

Combining (3.30) and (3.31) yields that V˙ = V˙3 + V˙4 ≤ −k¯ p2 gρm S2 + γ2 b˜ bˆ + bι By using the fact that b˜ bˆ ≤

1 2 ˜2 2gρm (b − b ),

(3.32)

one gets

γ2 ˜ 2 γ2 2 b + b + bι ≤ −l3V + l4 V˙ ≤ −k¯ p2 gρm S2 − 2gρm 2gρm

(3.33)

2

γ2 b with l3 = min{2k¯ p2gρm , γ2 σ2 } > 0, and l4 = 2g ρm + bι < ∞. By following the same line as in the proof of Theorem 3.1, it is readily derived from (3.33) that S is Ultimately Uniformly Bounded (UUB) and then the transformed error ν (t) is also UUB according to Lemma 3.1. It thus follows from the definition of T (ν ) given in (3.17) that T (ν ) is UUB. Note that ω (t) = T (ν )η (t), it is then concluded that the filtered error ω (t) converges to a small residual set containing the origin with the rate of e−a0t as defined in η . According to Lemma 3.2, the tracking error εk (k = 1, 2, · · · , n) has the same decreasing rate as that of ω and similar bounds also hold for εk if ω (t) is bounded. Thus it is established that εk (k = 1, 2, · · · , n) converges to a small residual set containing the origin at the rate controlled by η (t). Namely, the transient and steady-state performances are guaranteed with the proposed PI-like control. Furthermore, the maximum overshoot of the tracking error εk (k = 1, 2, · · · , n) is confined within a pre-given bound and the transient performance and steady-state tracking precision can be improved by adjusting the design parameters of η (t), δ and δ¯ . In the following, we prove that all the internal signals are bounded and continuous everywhere. According to (3.33), it can be derived that V ∈ L∞ , and then S ∈ L∞ and bˆ ∈ L R ∞ . By the definition of S as in (3.27), it follows from Lemma 3.1 that ν ∈ L∞ and 0t ν d τ ∈ L∞ , thus T (ν ) is bounded, which further implies that ω (t) is bounded. From Assumption 3.2 and the definition of ω (t), it follows that xk (k = 1, · · · , n) is bounded, then χ (t) is bounded because ϕ (t) is bounded according to Assumption 3.3. Thus ∆ k p2 and ∆ kI2 are bounded, and therefore, u is bounded. It then can be concluded that all the internal signals are bounded and continuous everywhere.

Remark 3.3 It is seen that in the proposed PI-like control, there is no need for ad hoc process for PI gains determination and the transient performance is guaranteed without the need for FDD/FDI to provide the fault occurrence instant, fault type, or fault magnitude. Remark 3.4 In developing the control schemes, a number of virtual parameters b, c, and upper/lower bounds such as ρm , g, ¯ g, etc. are defined and used in stability analysis, but these parameters are not involved in the control algorithms, thus analytical estimation of those parameters (a nontrivial task) is not needed in setting up and implementing the proposed PI-like control strategies.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 20 — #34

✐

✐

20

3 Adaptive PI Control for SISO Affine Systems

3.4 Numerical Examples To verify the effectiveness of the proposed PI-like control, we use the well-known inverted pendulum example taken from [10]: x˙1 = x2 , x˙2 = g(x1 , x2 )ua (t) + f (x1 , x2 ) + d(t)

(3.34)

with f (x1 , x2 ) =

9.8 sin x1 − mlx22 cos x1 sin x1 /(mc + m) l[4/3 − m cos2 x1 /(mc + m)]

(3.35)

cos x1 /(mc + m) l[4/3 − m cos2 x1 /(mc + m)]

(3.36)

and g(x1 , x2 ) =

where m is the mass of the pole, mc is the mass of the cart, l is the half length of the pole, and ua is the actual control input. The simulation is conducted with m = 0.1kg, mc = 1kg and l = 0.5m; d(t) = 5 cos(2t); ua = ρ (·)u(t) + ur (·) with ρ (·) and ur (·) π being shown in Fig.3.1; x∗ = 30 sin(t). For comparison, both traditional PI control with constant PI gains and the proposed PPB-based PI control with adaptive gains as given in Theorem 3.2 are tested. The procedure for the proposed PPB-based PI controller design is as follows: 1) choose free design parameters: σ2 = 0.1, γ2 = 1, and β1 = 1 (defined in (3.12)); 2) derive the computable scalar function ϕ (·) in Assumption 3.3 to get ϕ (·) = 1 + x22 ; 3) pre-specify the performance function µ1 (t) and µ2 (t) given in (3.14) with δ = 0.25, δ¯ = 0.5 and η (t) = 0.385e−2t + 0.015; 4) determine the gains: the first part for ‘P’ gain is chosen as k p2 = kω−1 k¯ p2 with k¯ p2 = 10; the adaptive part for ‘P’ gain, i.e., ∆ k p2 , is computed automatically by the adaptive algorithms (3.23)—(3.24) with no need for “trial and error” process; the ˆ gains for ‘I’ are calculated by kI2 = β k p2 and ∆ kI2 = β ∆ k p2 . In addition, b(0) = 0. For the traditional PI control with constant gains, to guess the “right” PI gains, by referring the proposed PI gains self-tuning within 1 to 20 for ‘P’ and 10 to 500 for ‘I’, we set k p = 10 and kI = 100 for the traditional PI control in the simulation. The results are shown in Figs. 3.2—3.5. It is clearly seen that the proposed PPBbased PI control has much better transient and steady-state performance as compared with the traditional one. In particular, the PPB-based PI-like control ensures that the tracking error is confined within the pre-specified bound, as theoretically predicted. Furthermore, by comparing the control results in the simulation as reflected by the tracking error ε1 , ε2 , and the control input ui as well as the PI gains k p and kI , it is seen that under the same condition the proposed PI control with adaptive self-tuning (time-varying) PI gains leads to much better control results as compared with the traditional PI control with constant PI gains.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 21 — #35

✐

✐

3.5 Conclusion

21 1.1 1 0.9

ρ

0.8 0.7 0.6 0.5 0.4 0

50

100

150

100

150

(a) ρ 0.15 0.1

ur

0.05 0

−0.05 −0.1 0

50

(b) ur Fig. 3.1 The actuation faults simulated 0.3

Proposed PPB PI control Traditional PI control Lower bound for 1 Upper bound for 1

0.2 0.1

1

0.2

0.1

0 −0.1 0

5

10

0

−0.1 0

50

t

100

150

Fig. 3.2 Position tracking performance comparison between traditional PI and the proposed one.

3.5 Conclusion Although PI/PID control has been widely utilized in practice, no framework for analytically specifying the “correct” PI/PID gains for general nonlinear systems is available. In this chapter, PI-like tracking control design for nonlinear systems subject to uncertainties and external disturbances as well as actuation faults is studied. The PI gains are analytically derived from stability and performance consideration. Furthermore, modeling uncertainties, external disturbances and

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 22 — #36

✐

✐

22

3 Adaptive PI Control for SISO Affine Systems 0.5

0.2

Proposed PPB PI control Traditional PI control Lower bound for 2 Upper bound for 2

0.4 0

0.3

2

0.2

−0.2 0

5

10

0.1 0 −0.1 −0.2 0

50

t

100

150

Fig. 3.3 Velocity tracking performance comparison between traditional PI and the proposed one. 150

P gain under proposed PPB PI control P gain under traditional PI control I gain under proposed PPB PI control I gain under traditional PI control

PI gains

100

50

0 0

50

100

t

150

Fig. 3.4 The evolution of self-tuning P and I gains. 20

Proposed PPB PI control

Tranditional PI control

ui

10

0

−10

−20 0

50

t

100

150

Fig. 3.5 Control signals.

actuation faults are collectively accommodated with the proposed PI-like control law. It should be pointed out that to ensure the prescribed performance bound, certain information on system initial condition must be known a priori. Removing such constraint and extending the results to MIMO nonlinear systems or even nonaffine systems represent an interesting topic for future research.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 23 — #37

✐

✐

3.6 Appendix

23

3.6 Appendix Proof of Lemma 3.2: To facilitate the proof, we first transform (3.12) into the following form:

ω (t) = ω˙ n−1 + αn−1 ωn−1 ωn−1 = ω˙ n−2 + αn−2ωn−2 .. .

.. .. . .

(3.37)

ω2 = ω˙ 1 + α1 ω1 = ε˙1 + α1 ε1 ω1 = ε1 Solving the first differentiate equation in (3.37) yields

ωn−1 (t) = e−αn−1t ωn−1 (0) + e−αn−1t

Z t

ω (τ )eαn−1 τ d τ

(3.38)

0

R

From (3.38) it is not hard to check that if 0t ω (τ )eαn−1 τ d τ is bounded, then ωn−1 → 0 Rt as t → ∞. If, however, 0 ω (τ )eαn−1 τ d τ is unbounded, we then apply the L’Hopital’s rule to (3.38) and obtain

ω (t)eαn−1t ω (t) = lim t→∞ αn−1 eαn−1 t t→∞ αn−1

lim ωn−1 (t) = 0 + lim

t→∞

(3.39)

which implies that if ω (t) → 0 as t → ∞, then ωn−1 → 0 as t → ∞ with the same decreasing rate as ω , and so is ω˙ n−1 by the first equation in (3.37). Similarly, from the second equation in (3.37), we have

ωn−1 (t) ω (t) = lim t→∞ αn−2 t→∞ αn−1 αn−2

lim ωn−2 (t) = lim

t→∞

(3.40)

This means that ωn−2 (t) → 0 as t → ∞ with the same decreasing rate as ω (t), and so is ω˙ n−2 (t). By carrying out the same procedure for the rest of the equations in (3.37), we can conclude that both ωk (t) and ω˙ k (t) (k = 1, · · · , n − 1) converge to zero at the same decreasing rate as ω (t). From the definition of ωk (t) (k = 1, · · · , n − 1) in (3.37), it is straightforward that εk (t) → 0 (k = 1, · · · , n) at the same decreasing rate as ω (t) when ω (t) → 0 as t → ∞. Further, if there exist performance functions µ1 (t) = −δ η (t) and µ2 (t) = δ¯ η (t) such that −δ η (t) < ω (t) < δ¯ η (t), then we get ¯ from (3.37)¯ that

αn−1 ωn−1 (0) + δ η0 δ η0 < ωn−1 (t) ¯ − ¯ αn−1 eαn−1t αn−1 αn−1 ωn−1 (0) − δ¯ η0 δ¯ η0 < + αn−1 eαn−1t αn−1

(3.41)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 24 — #38

✐

✐

24

3 Adaptive PI Control for SISO Affine Systems

in which we have used the fact that the maximum of η (t) is η0 . Note that ω˙ n−1 = ω (t) − αn−1 ωn−1 from the first equation in (3.37). To establish the lower bound for ω˙ n−1 , we use the lower bound on ω (t) (−δ η0 ) minus the upper bound on αn−1 ωn−1 ¯ of (3.38)). And to establish the upper (which is obtained from the right-hand side bound for ω˙ n−1 , we use the upper bound on ω (t) (δ¯ η0 ) minus the lower bound on αn−1 ωn−1 (which is obtained from the left-hand side of (3.38)). Thus we have

αn−1 ωn−1 (0) − δ¯ η0 ¯ − δ η0 − ( + δ η0 ) < ω˙ n−1 (t) eαn−1t ¯ αn−1 ωn−1 (0) + δ η0 < δ¯ η0 − ( ¯ − δ η0 ) eαn−1t ¯

(3.42)

Denote the minimum and maximum bounds of ω by ωmin and ωmax , and the minimum and maximum bounds of ωk by ωk,min and ωk,max (k = 1, 2, · · · , n), respectively. By using the relation of ωmin ≤ ω ≤ ωmax and ωk,min ≤ ωk ≤ ωk,max , we can establish the following general lower and upper bounds for ωn−1 from (3.38):

αn−1 ωn−1 (0) − ωmin ωmin + < ωn−1 (t) αn−1 eαn−1t αn−1 αn−1 ωn−1 (0) − ωmax ωmax + < αn−1 eαn−1t αn−1

(3.43)

and the lower and upper bounds for ω˙ n−1 from (3.39):

ωmin − αn−1 ωn−1,max < ω˙ n−1 (t) < ωmax − αn−1ωn−1,min

(3.44)

Then the lower and upper bounds for ωk and ω˙ k (k = 1, · · · , n − 1) are established by computing step by step:

αk ωk (0) − ωk+1,min ωk+1,min + < ωk (t) αk eαk t αk αk ωk (0) − ωk+1,max ωk+1,max < + αk eαk t αk ωk+1,min − αk ωk,max < ω˙ k (t) < ωk+1,max − αk ωk,min

(3.45)

Finally, the bounds for εk (k = 1, · · · , n) can be readily derived as

α1 ε1 (0) − ω2,min ω2,min α1 ε1 (0) − ω2,max ω2,max + < ε1 < + α1 eα1t α1 α1 eα1t α1 ω2,min − α1 ε1,max < ε2 (t) < ω2,max − α1 ε1,min

(3.46)

and the bounds for other εk (k = 3, · · · , n) can be obtained by applying the previous method to the equations in (3.37) step by step.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 25 — #39

✐

✐

Chapter 4

Generalized PI Control for SISO Nonaffine Systems

This chapter is concerned with the tracking control problem of a class of unknown nonaffine dynamic systems that involve unpredictable sensor and actuation failures. As the control inputs enter the system and influence its behavior through a nonlinear and implicit way, control design for such systems becomes quite challenging. The underlying problem becomes even more complex if the system dynamics is unavailable for control design yet involves unanticipated sensor and/or actuator faults. In this chapter, a generalized PI control is developed for a class of nonaffine systems in the presence of sensor and actuator faults.

4.1 Introduction Most practical systems are nonaffine in that the control inputs impact the dynamic behavior of the systems through a nonlinear and implicit way. In such systems no control gains are explicitly defined and the control inputs are no longer proportional to the control gains in contrast to the affine systems. As such, the problem of control design for nonaffine systems becomes an interesting yet challenging topic of research that has received considerable attention during the past decades [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. Various attempts have been made in the literature on control design for nonaffine systems, ranging from linearization of nonaffine systems into affine systems, to direct compensation of nonaffine systems by integrating implicit function theory with the universal approximation capabilities of neural networks (NN) and fuzzy systems. For instance, adaptive neural design approaches are proposed for uncertain SISO nonaffine nonlinear systems in [21, 22] and a decentralized adaptive neural network controller is introduced in [23] for a class of nonaffine systems, where a radial basis function neural network is utilized to represent the controller’s structure. NN and linearization methods are used in [24, 25] also. Instead of modeling the unknown systems directly, the T-S fuzzy-neural model is employed to approximate a virtual linearized system of a real system with modeling errors and 25

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 26 — #40

✐

✐

26

4 Generalized PI Control for SISO Nonaffine Systems

external disturbances in [26], and a T-S fuzzy model-based control integrated with an observer is presented in [27]. In [28], an indirect adaptive fuzzy controller is proposed, where the nonaffine nonlinear system is first transformed into an affine form by using a Taylor series expansion around an operation trajectory. However, the indirect adaptive approach has the drawback of singularity, i.e., division by zero may occur in the control law [28, 29]. The works of [30, 31, 32] studied direct adaptive fuzzy control approaches for nonaffine systems. An observer-based direct adaptive fuzzy-neural control scheme is presented in [33] using the implicit function theorem and the Taylor series expansion. Adaptive NN control has also been developed in the current literature [34, 35]. It is noted that the linearization-based methods could leave out important nonlinear dynamic information of the systems, whereas the NN-fuzzy based methods heavily rely on the approximation capabilities of the NN/fuzzy units, and construction of which demands additional caution in design and implementation of NN-fuzzy based control methods. There are several interesting works attempting different control design methods for nonaffine systems. In [36, 37], time-varying smooth state feedback control strategies are developed for a class of polynomial systems. A class of nonlinear nonaffine systems is considered in [38], where the state vector is not completely available. The proposed methodology introduces integrators in the input channel and combines sliding mode and Luenbergerlike observers. In the work of [13], through equivalence with the approximate dynamic inversion method, it was shown that a stabilizing tracking proportional-integral (PI) controller exists for minimum-phase nonaffine-in-control systems. Although fruitful results on nonaffine control systems have been reported in the literature, to our best knowledge, very few have explicitly considered the situation of modeling/parameter perturbations as well as actuation/sensor failures in nonaffine systems simultaneously. As actuator and/or sensor failure could cause serious safety problems to engineering systems if no proper action is taken in time, fault-tolerant control (FTC) has been counted as one of the most promising control technologies for maintaining specified safety performance of a system in the presence of unexpected faults. Various FTC methods have been proposed in the literature and books during the past decade (e.g., [39, 40, 41, 42], to just name a few). Among most existing FTC approaches, the one that does not rely on fault detection and diagnosis (FDD) is of particular interest in practice due to the fact that it does not demand timely and precise fault detection and diagnosis for implementation [43, 44, 45]. However, most methods turn out fairly complicated and can only deal with affine nonlinear systems. In this chapter, we propose a generalized control method for a class of nonaffine systems in the presence of modeling uncertainties and unexpected actuator and sensor failures. The structure of the proposed control scheme is motivated by [11, 13] for affine systems. It also has its origin from [6, 46], and is inspired by our recent work on generalized fault-tolerant control of a class of nonaffine systems [47]. To some extent, the results of this chapter extend and complement [6, 11, 13, 46, 47].

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 27 — #41

✐

✐

4.2 System Description and Preliminaries

27

4.2 System Description and Preliminaries Consider the following nonaffine system (n)

(n−1)

xa = g(xa , x˙a , . . . , xa

, ua )

(4.1)

(i)

where xa ∈ R represents the actual system state, xa ∈ R(i = 0, 1, . . . , n) denotes the ith derivative of xa with respect to time, ua ∈ R is the actual control input and g(·) ∈ R is a smooth nonlinear function of its argument. While (4.1) is general enough to describe most practical engineering systems, the inevitable situation in practice has not been well reflected therein. That is, since unanticipated sensor and actuation faults may occur during system operation, it is therefore important to include such a scenario explicitly in the model. • Modeling actuation failures When the actuator fails to function normally, the actual control input ua and the designed input u are not identical anymore; instead, they are related through ua = ρ u + ε

(4.2)

where 0 ≤ |ε | ≤ εm < ∞ is the uncertain partition of the control that is completely out of control, ρ (·) is the “healthy indicator” reflecting the effectiveness of the actuator. The typical faulty scenarios are listed in Table 4.1. In this chapter, we consider the case that the actuator suffers from losing actuation effectiveness in that 0 < ρ ≤ 1, as considered in several studies [14, 48]. Table 4.1 Typical faulty scenario. Healthy indication ρ =1 0 T , where e = x − xr denotes (n+1) the tracking error, emin > 0 is an any small constant. Here, xr , x˙r , x¨r , . . . , xr are assumed to be smooth and bounded. Our focus will be on developing a structurally simple control, generalized control that does not need any explicit information about f (·), nor any other parameter estimation or analysis on uncertain bound in the control scheme. To facilitate the control design, we define a filtered variable s in terms of tracking errors: s = e(n−1) + αn−1 e(n−2) + · · · + α1 e (4.6)

α1 , . . . , αn−1 are some constants determined such that the characteristic polynomial of h(p) = p(n−1) + αn−1 p(n−2) + · · · + α1 p is Hurwitz. It can be proved that s → 0 (or bounded) as t → ∞ can guarantee e(t) and its derivatives up to (n − 1)th converge asymptotically to zero (or bounded) as t → ∞ [54]. Based on (4.4), it is straightforward to get

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 30 — #44

✐

✐

30

4 Generalized PI Control for SISO Nonaffine Systems (n−1)

s˙ = g(xa , x˙a , . . . , xa with

, ua ) + ξ

(n)

ξ = −xr + αn−1 e(n−1) + · · · + α1 e˙

(4.7) (4.8)

The rest of the development will be focused on deriving the generalized control schemes which are computationally inexpensive and structurally simple yet capable of accommodating unknown and nonaffine systems with both actuator and sensor faults. Three sets of generalized control schemes are developed and we start with the first one to deal with healthy sensoring and actuation devices, the second one to accommodate fading actuators, and the third one to cope with both sensoring and actuation failures.

4.3.1 PI Control In such a case, we have from (4.4) and (4.5) that x(n) = f (x, x, ˙ . . . , x(n−1) , u)

(4.9)

For later technical treatment, we rewrite (4.7) as s˙ = f (x, x, ˙ . . . , x(n−1) , u) + ξ

(4.10)

which, by using the idea from [11, 50], can be further expressed as s˙ = ν + γ u + ξ

(4.11)

ν = f (x, x, ˙ . . . , x(n−1) , u) − γ u

(4.12)

with where γ > 0 is a design parameter chosen by the designer. As f (·) is assumed to be completely unavailable, ν as defined in (4.12) is incomputable. For this reason, we construct the following simple estimator to estimate ν , w˙ = −ξ − γ u − k1(w + s), νˆ = k1 (w + s) (4.13) with k1 > 0. By taking the initial estimate νˆ (0) = 0, w can be initialized as w(0) = −s(0). Therefore, upon embedding νˆ into control design, we propose the following tracking control for the nonaffine system (4.9), 1 u = (−νˆ − ξ − k0 s) γ

(4.14)

in which k0 > 0 isRthe control gain. From (4.13) and (4.14), it is obtained that w˙ = −k0 s, thus w = k0 0t s(σ )d σ − s(0). Then the explicit expression for νˆ can be found from the second equation of (4.13) as

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 31 — #45

✐

✐

4.3 Control Design

31

 Zt  νˆ = k1 k0 s(σ )d σ + s − s(0)

(4.15)

0

Now the following theorem is in order. Theorem 4.1. Consider the nonaffine system (4.9). Let the PI control strategy be 1 1 u = − (k1 + k0 )s − k0 k1 γ γ

Z t 0

1 1 s(σ )d σ + k1 s(0) − ξ γ γ

(4.16)

where t ≥ 0, γ > 0 is defined as before, k0 and k1 are control parameters selected by the designer, and ξ is computable signal as defined in (4.8). Then the tracking error s is ensured to be ultimately uniformly bounded (UUB) by choosing k0 and k1 large enough. Meanwhile, s can be made sufficiently small by selecting k0 sufficiently large. Remark 4.2 The motivation and significance of the proposed control scheme is twofold: (1) it is of PI-form and thus exhibits the well-known features of robustness and simplicity; (2) it is low cost and able to deal with unknown nonaffine systems without the need for lengthy experiment or tedious “trial and error” process to determine the PI gains as usually required in traditional PI control. Proof. To proceed with the proof of Theorem 4.1, we define the estimation error of ν as e0 = ν − νˆ (4.17) Substituting (4.16)—(4.17) in (4.11) leads to s˙ = −k0 s + e0

(4.18)

It is noted that the following description can be obtained from (4.15)

ν˙ˆ = k0 k1 s(t) + k1 s˙

(4.19)

Thus, it can be seen from the above description and (4.17) as well as (4.18) that e˙0 = −k1 e0 + ν˙

(4.20)

Also, from (4.16) it is straightforward to show that 1 1 1 u˙ = − (k1 + k0 )s˙ − k0 k1 s − ξ˙ γ γ γ

(4.21)

By the definition of e = x − xr , it can be noted that x = e + xr and x(i) = e(i) + = 1, . . . , n). Then with (4.12), it is derived that

(i) xr (i

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 32 — #46

✐

✐

32

4 Generalized PI Control for SISO Nonaffine Systems

ν˙ = f˙(x, x, ˙ . . . , x(n−1) , u) − γ u˙ ∂f ∂f ∂f ∂f x˙ + x¨ + · · · + (n−1) x(n) + u˙ − γ u˙ = ∂x ∂ x˙ ∂u ∂x ∂f ∂f = (e˙ + x˙r ) + (e¨ + x¨r ) ∂x ∂ x˙ ∂f ∂f (n) u˙ − γ u˙ + · · · + (n−1) (e(n) + xr ) + ∂u ∂x

(4.22)

Substituting (4.9), (4.18), and (4.21) into (4.22) yields,

∂f ∂f ∂f (n) (e˙ + x˙r ) + (e¨ + x¨r ) + · · · + (n−1) (e(n) + xr ) ∂x ∂ x˙ ∂x   1 1 ∂f 1 + − (k1 + k0 )s˙ − k0 k1 s − ξ˙ ∂u γ γ γ   1 1 1 −γ − (k1 + k0 )s˙ − k0 k1 s − ξ˙ γ γ γ 1∂f 1∂f 2 1∂f =− k1 e0 + k s− k0 e0 − k02 s + (k1 + k0 )e0 γ ∂u γ ∂u 0 γ ∂u 1 ∂ f (n+1) ∂f ∂f ∂f (n) (n+1) + x˙r + x¨r + · · · + (n−1) xr + xr − xr ∂x ∂ x˙ γ ∂u ∂x ∂f ∂f ∂f 1∂f + e˙ + e¨ + · · · + (n−1) e(n) − αn−1 e(n) ∂x ∂ x˙ γ ∂u ∂x 1∂f −··· − α1 e¨ + αn−1 e(n) + · · · + α1 e¨ γ ∂u

ν˙ =

(4.23)

Based on (4.18) and (4.23), and upon using (4.20), we have the following error dynamic equations associated with system (4.4) s˙ = −k0 s + e0 ,

e˙0 = −

1∂f k1 e0 + Γ γ ∂u

(4.24)

with

Γ =

1∂f 2 1∂f ∂f ∂f k0 s − k0 e0 − k02 s + k0e0 + x˙r + x¨r γ ∂u γ ∂u ∂x ∂ x˙ ∂f 1 ∂ f (n+1) (n) (n+1) + · · · + (n−1) xr + xr − xr γ ∂u ∂x ∂f ∂f ∂f 1∂f + e˙ + e¨ + · · · + (n−1) e(n) − αn−1 e(n) ∂x ∂ x˙ γ ∂u ∂x 1∂f −··· − α1 e¨ + αn−1e(n) + · · · + α1 e¨ γ ∂u

(4.25)

To carry out the analysis, the following two propositions should be given first.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 33 — #47

✐

✐

4.3 Control Design

33

Proposition 4.1. By Assumption 4.1 (i.e., system dynamic equation (4.9) satisfies the Lipschitz condition), it can be shown that Γ as expressed in (4.25) satisfies |Γ | ≤ d0 |s| + d1 |e0 | + d2, where d0 , d1 , and d2 are some non-negative constants. Proof. From the assumption that f (x, x, ˙ . . . , x(n−1) , u) satisfies the Lipschitz (n−1) condition for (x, x, ˙ ...,x , u) ∈ G0 ⊆ Rn+1 (domain of interest under healthy actuation condition), i.e., for any (x, x, ˙ . . . , x(n−1) , u1 ) ∈ G0 , (n−1) (n−1) (x, x, ˙ ...,x , u2 ) ∈ G0 , (x1 , x, ˙ ...,x , u) ∈ G0 , (x2 , x, ˙ . . . , x(n−1) , u) ∈ G0 , . . ., (n−1) (n−1) (x, ˙ . . . , x1 , u) ∈ G0 , (x, x, ˙ . . . , x2 , u) ∈ G0 , the following inequalities hold x, (n−1) , u ) − f (x, x, (n−1) , u ) ≤ l |u − u |, f (x, x, ˙ . . . , x ˙ . . . , x 1 2 1 1 2 (n−1) (n−1) ˙ ...,x , u) − f (x2 , x, ˙ ...,x , u) ≤ l2 |x1 − x2 |, . . ., f (x1 , x, (n−1) (n−1) (n−1) (n−1) ˙ . . . , x1 , u) − f (x, x, ˙ . . . , x2 , u) ≤ ln+1 x1 − x2 f (x, x, li (i = 0, 1, . . . , n) is a non-negative constant. It can obtained that the partial be derivatives of f to u, x, . . . , x(n−1) , are bounded, i.e., ∂∂ uf ≤ c1 , ∂∂ xf ≤ c2 , ∂∂ xf˙ ≤ c3 , f · · · , ∂ x∂(n−1) ≤ cn+1 with ci (i = 1, . . . , n + 1) is a non-negative constant. Meanwhile, (n+1)

x˙r , x¨r , . . . , xr are bounded as assumed earlier, γ and k0 are positive constants chosen by the designer. Therefore, it follows from (4.25) that |Γ | ≤ j1 e˙ + j2 e¨ + · · · + jn e(n) + h1 |s| + h2 |e0 | + h3 j1 j2 (n) = α1 e˙ + α2 e¨ + · · · + jn e + h1 |s| + h2 |e0 | + h3 α1 α2 ≤ jmax α1 e˙ + α2 e¨ + · · · + e(n) + h1 |s| + h2 |e0 | + h3 = jmax |−k0 s + e0| + h1 |s| + h2 |e0 | + h3 ≤ d0 |s| + d1 |e0 | + d2

in which ji (i = 1, . o . . , n), h1 , h2 , and h3 are some non-negative constants, jmax = n j1 j2 max α1 , α2 , · · · , jn .

Proposition 4.2. There exists a minimum constant kmin such that, for all k1 > kmin > 0, the tracking error s and estimation error e0 are bounded with initial conditions in Hs × He0 . Here Her and He0 are two compact sets and always exist. Proof. From Lemma 2.4 in [55] and Proposition 4.2, we define the function W (s, e0 ) = c

e20 Vs (s) +µ c + 1 − Vs(s) µ + 1 − e20

where the positive constants c and µ are chosen such that Hs × He0 ⊂ S1 , where S1 = (s, e0 ) : W (s, e0 ) ≤ c2 + µ 2 + 1 . Such positive constants  c and µ always exist since W (s, e0 ) is proper in the set S2 = {s : Vs (s) < c + 1} × e0 : e20 ≤ µ + 1 .

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 34 — #48

✐

✐

34

4 Generalized PI Control for SISO Nonaffine Systems

Then, for each strictly positive real number ϑ , there exists a minimum constant kmin such that, for all k1 > kmin > 0, W (s, e0 ) satisfies W˙ ≤ −Φ (s, e0 ), where Φ (s, e0 ) is continuous on S2 and positive definite on the set  S3 = (s, e0 ) : b0 + ϑ ≤ W (s, e0 ) ≤ c2 + µ 2 + 1

In other words, all trajectories of the closed-loop plant (4.24) with initial conditions in Hs × He0 are bounded and captured by the set S1 \S3 (elements in S1 but not in S3 ), i.e., the tracking error s and estimation error e0 are bounded with initial conditions in Hs × He0 . Now, we can move on with the proof. According to Proposition 4.2, since S1 \S3 is a compact set, we have |Γ | ≤ δ0 (0 < δ0 < ∞) with initial conditions in S1 \S3 and 2

e2

all t ≥ 0. Consider the Lyapunov function candidate V = s2 + 20 . It can be shown with (4.24) and (4.25) that   1∂f V˙ = e0 −k1 e0 + Γ + s (−k0 s + e0 ) γ ∂u 1 ≤ −k1 β |e0 |2 − k0|s|2 + |e0 | δ0 + se0 γ     1 1 1 2 ≤ − k1 β − |e0 | − k0 − |s|2 + |e0 | δ0 γ 2 2      2 3 1 |e0 | 1 ≤ − k1 β − |e0 |2 − k0 − |s|2 − − δ0 + δ02 (4.26) γ 4 2 2

By choosing the control parameters k0 and k1 such that k1 > 3γ /2β and k0 > 1/2 and completing square and with certain computation, it is not difficult to further express V˙ as V˙ ≤ −λvV + λ0

(4.27)

with 

1 3 1 λv = min k1 β − , k0 − γ 4 2



> 0,

λ0 = δ02 < ∞

Meanwhile, because k1 > 3γ /2β , the following inequalities hold,       1 3 1 δ0 2 δ02 |e0 |2 − k0 − |s|2 − |e0 | − + V˙ ≤ − k1 β − γ 2 2 2 4   2 δ 1 ≤ − k0 − |s|2 + 0 2 4

(4.28)

In view of (4.26) and (4.28), inequalities
one actually has the following
simultaneously V˙ ≤ − k0 − 12 |s|2 + λ0 and V˙ ≤ − k0 − 21 |s|2 + λ1 with λ1 = δ02 /4. From which it holds that V˙ < 0 if s is outside of either the compact

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 35 — #49

✐

✐

4.4 Adaptive Fault-tolerant PI Control

35

    q q 1 1 regions E1 = s| |s| ≤ λ0 /k0 − 2 or E2 = s| |s| ≤ λ1 /k0 − 2 . Because

λ0 > λ1 , E1 encloses E2 , the system error trajectory may move in or out of E2 (the small region), but once inside the set E1 , it cannot go out of it. Thus UUB tracking is ensured with the proposed control scheme [51]. Note that s can be made sufficiently small by selecting k0 large enough. Remark 4.3 If f (·) is known, β can be then readily obtained, thus it is straightforward to get the lower bound on k1 . Even if f (·) is unknown, as β is a positive constant and γ is selected by the control designer, it is not difficult to get a proper k1 (large enough) so that k1 > 3γ /2β . Thus far, the development of the control algorithm has been carried out under healthy actuation and sensoring conditions. It is interesting to examine if the generalized control method is also effective when the system suffers from actuation and sensoring faults. This is addressed in subsequent sections.

4.4 Adaptive Fault-tolerant PI Control 4.4.1 PI Control under Actuator Failures The system dynamic behavior under actuation faults is governed by x(n) = f (x, x, ˙ · · · , x(n−1) u, ρ , ε ) + ξ

(4.29)

By using the definition of s as defined in (4.7) we further have from (4.29) that s˙ = f (x, x, ˙ · · · , x(n−1) u, ρ , ε , p) + ξ (n)

ξ = −xr + αn−1 e(n−1) + · · · + α1 e˙

(4.30)

Note that while (4.30) looks similar to (4.10), the essential difference is that here in (4.30) we have that (ρ , ε ) ∈ {(ρ , ε ) | 0 < ρ ≤ 1, ε 6= 0}, rather than (ρ , ε ) ∈ {(ρ , ε ) | ρ = 1 and ε = 0} as in (4.10). Thus it is worth examining the applicability of the previously established generalized control to the case with actuation faults. Theorem 4.2. Consider the nonaffine system (4.29) with the Assumptions 4.1—4.3. Let the generalized control strategy be 1 1 u = − (k1 + k0 )s − k0 k1 γ γ

Z t 0

1 1 s(σ )d σ + k1 s(0) − ξ γ γ

(4.31)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 36 — #50

✐

✐

36

4 Generalized PI Control for SISO Nonaffine Systems

where γ > 0, k0 > 0 and k1 > 0 are chosen as properly by control parameters, ξ is defined as before. Then UUB tracking control is ensured in the presence of actuator faults. Proof. Following the same line as above, it can be derived that s˙ = −k0 s + e0 ,

e˙0 = −

1∂f k1 e0 γ ∂u

(4.32a)

in which

Γ1 =

1∂f 2 1∂f ∂f k0 s − k0 e0 − k02 s + k0e0 + x˙r γ ∂u γ ∂u ∂x ∂f ∂f 1 ∂ f (n+1) (n) + x¨r + · · · + (n−1) xr + xr ∂ x˙ γ ∂u ∂x ∂f ∂f ∂f (n+1) −xr + e˙ + e¨ + · · · + (n−1) e(n) ∂x ∂x ∂x 1∂f 1∂f αn−1 e(n) − · · · − − α1 e¨ + αn−1 e(n) − γ ∂u γ ∂u ∂f ∂f + · · · + α1 e¨ + ρ˙ + ε˙ ∂ρ ∂ε

(4.32b)

With the domain of interest under actuation failure, the boundedness of Γ1 is still ensured by the relation as in Propositions 4.1—4.2 in spite of the involvement of ρ , ρ , ε˙ , ε˙ ; therefore the result of Theorem 4.2 can be established by following the same line as in the proof of Theorem 4.1.

4.4.2 PI Control under Actuator and Sensor Faults When both actuator and sensor failures are involved, the error dynamics of the nonaffine systems (4.4) and (4.5) can be rewritten as s˙ = f (x, x, ˙ . . . , x(n−1) , δ0 , δ1 , . . . , δn−1 , δn , u, ρ , ε ) + ξ (n)

ξ = − xr + αn−1 e(n−1) + . . . + α1 e˙

(4.33)

It is interesting to show that without altering the structure of the generalized control, UUB stable tracking is still ensured in the face of sensor and actuator failures, as stated in the following theorem. Theorem 4.3. Consider the nonaffine system (4.4) with Assumptions 4.1— 4.3. Let the generalized control strategy be 1 1 u = − (k1 + k0 )s − k1 k0 γ γ

Z t 0

1 1 s(σ )d σ + k1 s(0) − ξ γ γ

(4.34)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 37 — #51

✐

✐

4.4 Adaptive Fault-tolerant PI Control

37

where t ≥ 0, γ , k0 and k1 are defined as before, ξ is shown in the second term of (4.30). Then UUB tracking control is ensured in the presence of actuator faults. Proof. In this case the tracking error and estimation error dynamic equations associated with system (4.4) become s˙ = − k0 s + e0,

e˙0 = −

1∂f k1 e0 + Γ2 γ ∂u

(4.35)

in which

Γ2 =

1∂f 2 1∂f ∂f ∂f ∂f (n) k s− k0 e0 − k02 s + k0 e0 + x˙r + x¨r + · · · + (n−1) xr γ ∂u 0 γ ∂u ∂x ∂ x˙ ∂x 1 ∂ f (n+1) ∂ f ˙ ∂f ˙ ∂f ˙ ∂f ˙ (n+1) + xr + δ0 + δ1 + · · · + δn−1 + δn − xr γ ∂u ∂ δ0 ∂ δ1 ∂ δn−1 ∂ δn (4.36) ∂f ∂f ∂f 1∂f 1∂f + αn−1 e(n) − · · · − α1 e¨ e˙ + e¨ + · · · + (n−1) en − ∂x ∂ x˙ γ ∂u γ ∂u ∂x ∂ f ∂ f ρ˙ + ε˙ + αn−1 e(n) + · · · + α1 e¨ + ∂ρ ∂ε

With the domain of interest under actuation and sensor failures, the boundedness on Γ2 is still ensured by the relation as in Proposition 4.1 in spite of the involvement of ρ , ρ˙ , ε , ε˙ , δ , and δ˙ . Therefore, following the same line as in the proof of Theorem 4.1, UUB of e can be established. Note that when sensor faults occur, the actual state and the measure done are linked with xa = x + δ (·); here δ (·) denotes the error due the sensoring faults. Consequently, the actual tracking error is ea = xa − xr , and | ea |=| xa − xr |=| x + δ (·) − xr |≤| e | + | δ (·) |< ∞. Thus it is established that bounded actual tracking error is ensured in the presence of both sensor and actuator failures. Remark 4.4 It is interesting to note that if the system experiences some nonlinear (n−1) (n−1) perturbation such that xna = g(xa , x˙a , . . . , xa , ua ) + ∆ g1 (xa , . . . , xa ) or (n−1) (n−1) (n−1) n xa = g(xa , x˙a , . . . , xa , ua ) + ∆ g2 (xa , . . . , xa , ua ) where ∆ g1 (xa , . . . , xa ) and (n−1) ∆ g2 (xa , . . . , xa , ua ) represent the nonlinear perturbations, the control problem associated with such perturbed systems can be addressed similarly. For instance, one can define the following new (perturbed) nonaffine functions (n−1) (n−1) (n−1) F1 (xa , . . . , xa , ua ) = g(xa , . . . , xa , ua ) + g1(xa , . . . , xa ) and (n−1) (n−1) (n−1) F2 (xa , . . . , xa , ua ) = g(xa , . . . , xa , ua ) + g2(xa , . . . , xa , ua ) and then by examining if the lumped (perturbed) functions F1 and F2 satisfy the conditions as imposed previously on the nonaffine function f (·). If so, similar stability results can be established for the perturbation case. In fact, since the precise information of the nonaffine function f (·) is not involved in designing and implementing the proposed generalized control, any nonlinear perturbation (affine or nonaffine) from such function can be collectively handled as long as the perturbed function still satisfies the conditions as imposed.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 38 — #52

✐

✐

38

4 Generalized PI Control for SISO Nonaffine Systems

Table 4.2 Sensor faults in different time interval (Ts1 = 10, Ts2 ).

δi δ1 δ2

Time(s) [0, Ts1 ] 0 0

[Ts1 , Ts2 ] √ 0.4 exp(−t 2 /2)/ 2π + 0.06 sin(0.2t) 0

[Ts2 , 30] √ −0.4 exp(−t 2 /2)/ √ 2π + 0.06 sin(0.2t) 2 0.5 exp(−t /2)/ 2π + 0.03 cos(0.1t)

Some related works (i.e., [37, 46]) addressed the perturbation problem but the scenarios and conditions involved are quite different. For instance, in [37] the nonaffine system studied is of the form of polynomial system with known model information. The control is not PI but a model-based one in which precise system information is required, whereas the system considered in [46] is an affine rather than nonaffine system. Furthermore, none have considered the sensing and actuation failures. The significance of the result is fourfold: (1) The resultant control scheme for the nonaffine system is of PI structure with faulttolerant capability in which no fault detection and diagnosis (FDD) unit is needed and no trial and error process is required to determine the PI gains in contrast to traditional PI control method, thus the control strategy is quite user-friendly and cost-effective. (2) Both sensor and actuator failures are considered and accommodated simultaneously with the proposed control scheme. (3) There is no need for making any linearization or approximation to the nonaffine system in deriving the control scheme. With only the Lipschitz condition, the control scheme ensures ultimately uniformly bounded (UUB) tracking stability. Furthermore, the generalized control structure remains unaltered even if unnoticeable actuation faults and sensor faults occur during system operation. (4) The proposed generalized control scheme can be easily set up without the need for any explicit information of the system except for its control direction. The independence of the PI controller from the nonaffine system model renders it relatively insensitive to system model uncertainties and perturbations, actuation faults, and sensor faults.

4.5 Illustrative Examples To demonstrate the ideas for control design and to verify the effectiveness of the control scheme, the simulation example considered is a double inverted pendulum [13, 52]. The motivations for selecting it were (i) it is a second-order system, (ii) it is a practical problem that is nonaffine in the control input (motor force versus magnetic current), (iii) the system is a MIMO nonlinear system with actuator and sensor faults. These characteristics make this problem a sufficiently challenging one to demonstrate benefits of the proposed technique. Considering the parameter uncertainty, external disturbance, and actuator/sensor faults simultaneously, the

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 39 — #53

✐

✐

4.5 Illustrative Examples

39

(a) The tracking processes of xa1 and tracking error of s1 .

(b) The tracking processes of xa2 and tracking error of s2 . Fig. 4.1 The tracking process and tracking error.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 40 — #54

✐

✐

40

4 Generalized PI Control for SISO Nonaffine Systems

models in [13] are represented as x¨a1 =(α1 + ∆ α1 ) sin(xa1 ) + β1 + ξ1 tanh(ua1 ) + σ1 sin(x˙a2 ) + d1(t) x¨a2 =(α2 + ∆ α2 ) sin(xa2 ) + β2 + ξ2 tanh(ua2 ) + σ2 sin(x˙a2 ) + d2(t) x˙a1 =x˙1 + δ1 , x˙a2 = x˙2 + δ2

(4.37)

where δi (i = 1, 2) denotes the uncertain portion of the measurement due to sensing faults. The parameters involved in the model are defined as

αi = (

mi gr kr2 − ), Ji 4Ji

βi =

kr (l − b), 2Ji

ξi =

uimax , Ji

σi =

kr2 4Ji

(4.38)

where the parametric uncertainty ∆ αi and the external disturbance di (i = 1, 2) are the same as in [52]. For simulation, the sensor fault δi is chosen as in Table 4.2. The actuation healthy indicators for the two actuators simulated are as follows: (a) For the first actuator, ua1 = ρ1 u1 + ε1 with ρ1 = 0.8 + 0.2 sin(π t/4) and ε1 = 0.02 cos(t). (b) For the second one, ua2 = ρ2 u2 + ε2 with ρ2 = 0.5 + 0.5 sin(π t/4) and ε2 = 0.02 sin(t). As seen clearly, the control inputs enter into the system in a nonlinear way and are corrupted with significant undetectable actuation faults so that control of such a system becomes nontrivial. However, the developed generalized control scheme can be readily applied to this case because the only required Lipschitz condition holds with this system. The objective is to make the system state xai (i = 1, 2) track the desired trajectory xr with sufficient precision. To this end we introduce the tracking error ei = xi − xr and define si = e˙i + λi ei . Correspondingly, the generalized control is of the form (i = 1, 2) 1 1 u = − (ki1 + ki0 )si − ki1 ki0 γi γi

Z t 0

1 1 si (σ )d σ + ki1 si (0) − (−x¨r + λiei ). γ γi

(4.39)

The desired trajectory is xr = sin(2π t/10). The control parameters of u1 and u2 are chosen as, γi = 1, λi = 1, k10 = 8, k11 = 8, k20 = 8, k21 = 8. Tracking process and tracking error are shown in Fig. 4.1. One can observe that good tracking control performance is achieved for this system with the proposed lowcost generalized control scheme (it is low-cost in the sense that there is no need for lengthy experiment or a tedious “trial and error” process to determine the PI gains as usually needed in traditional PI control). Remark 4.5 As the system under consideration is nonaffine and subject to actuator and sensor failures, it is quite challenging to design the corresponding control scheme to ensure stable tracking. The problem is made even more difficult due to model uncertainties (with little available information on its nonlinearities). For comparison, several other control schemes can be considered. The first one is the traditional PI control for which one has to use the “trial and error” process to pre-determine the proportional and integral gains. The other popular method is the

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 41 — #55

✐

✐

4.6 Conclusion

41

radial basis function (RBF) based neural network method, whose performance largely depends on the choice of NN parameters (centers and widths, as well as the number of neurons—no general guidelines are available for such choice in the literature, especially for the nonaffine systems with actuation failures). We tested both methods and it turns out that the tracking errors actually diverge from the example considered (not shown here due to page limit). In particular, for the traditional PI control scheme the “smart” gains cannot be found—the tracking errors failed to converge even though extensive attempts have been made to selecte the gains.

4.6 Conclusion As most practical systems are both nonlinear and nonaffine, control of such systems becomes extremely challenging if the information available for control design is limited. In this chapter, the problem of control design for a class of nonaffine nonlinear systems with actuator and sensor failures is studied and a fault-tolerant generalized control strategy is developed in which no linearization or approximation is made. Formative stability analysis is conducted based on nonlinear singular perturbation theory. The results are also verified via numerical simulation. The proposed generalized controller is a linear realization of a nonlinear control law, and is independent of the nonlinear function f (x, x, ˙ x, ¨ . . . , x(n−1) , δ0 , δ1 , . . . , δn−1 , δn , u, ρ , ε ) in (4.5), except for the sign of the (n−1) ,δ ,δ ,...,δ ∂ f (x,x, ˙ x,...,x ¨

,δ ,u,ρ ,ε )

0 1 n−1 n ). It should be pointed control effectiveness, sign ( ∂u out that the conditions imposed on the nonaffine function f (·) are somewhat strong in general, but for the class of important practical systems under consideration, those conditions do validate. Relaxing such conditions to allow for the application of the results to a larger class of nonaffine systems represents an interesting topic for further study. Also, note that the development for the generalized control is based on SISO nonaffine systems, a natural question to ask is whether or not the results can be extended to MIMO nonaffine systems. Intuitively, such extension seems possible, although non-trivial. In fact, the example utilized for simulation verification in this chapter is actually a MIMO nonaffine system. The control performance indicates that the proposed method is indeed applicable for such MIMO nonaffine systems. Nevertheless, cautions have to be taken in several aspects in carrying out the extension, including: (For easy description, the square nonaffine system, i.e., m = n, is considered, and the nonaffine function is denoted as X˙a = F(Xa ,Ua ) ∈ Rn , Xa ∈ Rn , Ua = Γ U + d ∈ Rn , here Γ ∈ Rn×n is a diagonal matrix associated with the actuator health condition, d ∈ Rn is the uncontrollable portion in the actual control input.)

• A filtered variable vector S in terms of tracking error E = X − Xr ∈ Rn must be utilized S = E (n−1) + An−1 E (n−2) + · · · + A1 E. Here

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 42 — #56

✐

✐

42

• •

•

•

•

4 Generalized PI Control for SISO Nonaffine Systems


i , α i , . . . , α i (i = 1, . . . , n − 1) is a diagonal matrix composed of Ai = diag α11 nn 22 some constants that have the same properties as α1 , . . . , αn−1 defined in (4.6). ∆ As ∂ F(X∂aU,U,P) = B(·) ∈ Rn×n is a matrix for the case of MIMO, the “sign” condition for the SISO case now should be related to the symmetry and positivity (or negativity) of the matrix, i.e., B = BT > 0 or B = BT < 0 (for square system). Operation on matrix or vector rather than scalar or single variable has to be used  T in stability analysis. And the nth derivative of the vector H = h1 · · · hn should h iT (n) be defined as the nth derivative of each element in H, i.e., H (n) = h(n) . 1 · · · hn As F is vector function in the MIMO system, the partial derivatives of F with (n−1) respect to ul , x j , . . . , x j are vector. Then the Lipschitz condition for the MIMO case should be expressed as the norm





of each partial derivative, i.e.,

∂F

∂F



∂ u1 ≤ c1 , . . . , ∂ ul ≤ cl , . . . , ∂∂uFn ≤ cn ;





∂F

∂F

∂F

≤ d , ≤ d , . . . ,

∂ x1 11 ∂ x˙1 12

∂ x(n−1) ≤ d jn , . . . , j





∂F

∂F

≤ d jn , . . . ,

∂∂xFn ≤ dn1 ,

∂ x˙ j ≤ d j2 , . . . ,

∂ x(n−1) j



∂F

∂F

≤ dnn . ≤ d , . . . ,

∂ x˙n n2

∂ x(n−1)

n The structure of the generalized controller takes the following form, R U = − 1γ (K1 + K0 )S − 1γ K0 K1 0t S(σ )d σ + 1γ K1 S(0) − 1γ Z. U = [u1 , . . . , un ]T is the control vector, Z is the vector composed of corresponding variables, K0 and K1 are the control parameter matrices (ideally diagonal). The Lyapunov function candidate used for stability proof is of the form V = ST S/2 + E0T E0 /2; here E0 is the vector of estimation error.

Clearly, the corresponding development for MIMO nonaffine systems is somewhat more involved as compared to the SISO case. However, following the same lines as in the case of SISO, similar results can be established for MIMO nonaffine systems. In fact, such extension is possible, at least for a certain class of MIMO nonaffine systems as already confirmed by the MIMO example utilized. Formal derivation and analysis, beyond the scope of this chapter though, is certainly an interesting and important topic for further study.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 43 — #57

✐

✐

Chapter 5

Adaptive PI Control for MIMO Nonlinear Systems

This chapter considers the tracking control problem for a class of multi-input multi-output (MIMO) nonlinear systems subject to unknown actuation characteristics and external disturbances. Neuro-adaptive PI control with self-tuning gains is proposed, which is structurally simple and computationally inexpensive. Different from traditional PI control, the proposed one is able to adjust its PI gains online using stability-guaranteed analytic algorithms without involving manual tuning or the trial and error process. It is shown that the proposed neuro-adaptive PI control is continuous and smooth everywhere and ensures the uniformly ultimately boundedness of all the signals of the closed-loop system. Furthermore, the crucial compact set precondition for a neural network (NN) to function properly is guaranteed with the barrier Lyapunov function (BLF), allowing the NN unit to play its learning/approximating role during the entire system operation.The salient feature also lies in its low complexity in computation and effectiveness in dealing with modeling uncertainties and nonlinearities. Both square and non-square nonlinear systems are addressed. The benefits and feasibility of the developed control are also confirmed by simulations.

5.1 Introduction Dynamical systems with actuation constraints, such as non-smooth input saturation, hysteresis, or dead-zone, etc. (encountered frequently in industrial applications), pose significant challenge for control design. Although various methods have been suggested to address this issue (see, for instance, [56, 57, 58, 59, 60, 61, 62, 63], to name just a few), the resultant solutions are not only structurally complicated and computationally expensive, but also heavily dependent on system model and very few have considered the MIMO case. Our interest in PI control stems from its simplicity and intuitiveness. Justifications for the use of PI/PID controllers on various systems have been furnished in [5, 6, 8, 9, 10, 12, 46, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74]. The key for PI control design 43

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 44 — #58

✐

✐

44

5 Adaptive PI Control for MIMO Nonlinear Systems

is the determination of its gains. Early efforts on tuning P and I parts collectively can be found in [64] and [65]. However, these methods require some experimental and graphical procedures which are rather time consuming. Subsequently, methods based on online parameter estimation without the need for experimental procedure have gained much attention. By using adaptive parameter estimation, methods for self-adjusting PID gains are developed in [66, 67]. In [68], a knowledge-based PID auto-tuner is proposed. Nevertheless, many of these approaches are only applicable or effective for linear time-invariant (LTI) systems [5, 6, 8, 69]. It has firstly been proved that a PI/PID controller can be tuned for nonlinear systems in [46]. An auto-tuning PID control based on the Lyapunov approach for a class of continuous time single-input single-output (SISO) nonlinear systems is developed in [70]. Adaptive robust PID control with sliding mode is pursued in [71]. The arrival of the neural network and fuzzy system has brought a new avenue for control design under various conditions [57, 59, 72, 73, 75, 76, 77, 78, 79, 80]. For example in [78], an adaptive neural network control strategy based on linear matrix inequalities is proposed for multiple mobile manipulators. In [79], to handle the full-state constraints of the robotic system, the adaptive neural network is designed. Further in [80], adaptive neural network control for a robotic manipulator based on barrier Lyapunov function is developed. In particular, efforts on using neural network and fuzzy system for PI/PID control design have also been made in the literature. For instance, methods based on generic algorithm, neural network, and fuzzy system technique are proposed in [9, 10, 12, 72, 73, 74]. In [10], a self-tuning fuzzy PID method for a class of MIMO systems is developed, where the fuzzy rules depend largely on designer skills. Approximating an unknown ideal controller by designing the PID controller for a class of nonlinear dynamic systems is developed in [74]; its drawback is the lack of a general stability-guaranteed mechanism for its gain tuning. In this chapter, we attempt to construct a neural adaptive control scheme that bears proportional and integral (PI) structure for MIMO uncertain nonlinear systems with asymmetric and nonsmooth actuation dynamics.

5.2 Problem Formulation 5.2.1 System Description Consider the following MIMO nonlinear dynamic system with unknown and constrained actuation described by:

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 45 — #59

✐

✐

5.2 Problem Formulation

45

x˙i =xi+1 ,

i = 1, 2, . . . , n − 1

x˙n =F(x,t) + B(x,t)u + Fd (t) y =x1

(5.1)

u =H(v) T

where xi = [xi1 , . . . , xim ]T ∈ Rm , i = 1, . . . , n and x = [xT1 , . . . , xTn ] ∈ Rmn is the system state vector with initial conditions x(0) = x0 ; y ∈ Rm is the system output; F(x,t) = [ f1 (x,t), . . . , fm (x,t)]T ∈ Rm represents unknown nonlinear function vector; B(x,t) ∈ Rm×r is the control gain matrix and Fd (t) ∈ Rm denotes all the possible external disturbances; u = H(v) = [h1 (v1 ), . . . , hr (vr )]T represents the control vector of the system with unknown actuation characteristics, where v is the actual control design. In this chapter, we consider two typical actuation models as shown in Fig. 5.1. Model 1: Asymmetric non-smooth saturation with unknown slope [60, 81].

3

H(v) Γ(v)

sat(v)

3

2

2

1

1

sat(v)

H(v) Γ(v)

0

0

v

v

-1

-1

-2

-2

-3 -3 -6

-4

-2

0

2

4

6

-5

0

5

Fig. 5.1 Left: asymmetric non-smooth saturation function and the smooth approximating function. Right: asymmetric non-smooth saturation function with dead-zone and the smooth approximating function.

  σ¯ , hi (vi ) = l(vi ),  −σ ,

vi > vma1 −vma2 ≤ vi ≤ vma1 vi < −vma2

i = 1, . . . , r

(5.2)

Model 2: Asymmetric non-smooth saturation with dead-zone [58].  σ¯ ,      l1 (vi − b1), hi (vi ) = 0,    l2 (vi + b2),   −σ ,

vi > vmb1 b1 ≤ vi ≤ vmb1 −b2 ≤ vi ≤ b1 i = 1, . . . , r −vmb2 ≤ vi ≤ −b2 vi < −vmb2

(5.3)

where l1 and l2 are the slope of the dead-zone characteristic, b1 > 0, −b2 < 0, vmb1 > 0, and −vmb2 < 0 represent the break points. To facilitate the controller construction, a smooth function is used to approximate the saturation function [60, 63], which

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 46 — #60

✐

✐

46

5 Adaptive PI Control for MIMO Nonlinear Systems

takes the form as ui =hi (vi ) = Γi (vi ) + εi (vi ),

Γi (vi ) =

i = 1, . . . , r

σ¯ eι vi − σ e−ι vi eι vi + e−ι vi

(5.4) (5.5)

where ι > 0 is a design parameter, and εi (vi ) = hi (vi ) − Γi (vi ) is an unknown function bounded by (5.6) | εi (vi ) |=| hi (vi ) − Γi(vi ) |≤ Di where Γi (vi ) is known smooth function, and εi (vi ) is the approximate error with Di being a positive constant. With the help of Γ (v), the original system (5.1) can be expressed as x˙i =xi+1 , i = 1, 2, . . . , n − 1 x˙n =F(x,t) + B(x,t)(Γ (v) + ε (v)) + Fd (t)

(5.7)

y =x1 The control objective of this chapter is to develop a neuro-adaptive PI control scheme for system (5.1), or equivalently (5.7), such that the output y closely follows the desired trajectory yd and all the internal signals in the system are ensured to be continuous and uniformly ultimately bounded (UUB). We define e = y − yd as the output tracking error with yd = [yd1 , . . . , ydm ]T ∈ Rm being the desired trajectory. To continue, we need the following assumptions, which are quite standard in the literature. Assumption 5.1 The desired tracking trajectory yd j , j = 1, . . . , m, as well as its derivative up to nth-order are known smooth functions of time and bounded. And the state vector x is available for control design. Assumption 5.2 There exists a positive constant ρ , such that max Di ≤ ρ < ∞, i = 1, . . . , r. Since Γi (vi ) is a smooth function of vi , we use the mean value theorem with respect to v to get,

Γ (v) = Γ (0) + L(ξ )v

(5.8)

with 

l11 |v1 =ξ11 l12 |v2 =ξ12  l21 |v =ξ l22 |v =ξ 1 21 2 22  L(ξ ) =  .. ..  . . lr1 |v1 =ξr1 lr2 |v2 =ξr2

 · · · l1r |vr =ξ1r · · · l2r |vr =ξ2r    .. ..  . . · · · lrr |vn =ξrr

(5.9)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 47 — #61

✐

✐

5.2 Problem Formulation

47 ∂Γ (v)

where ξ = [ξ1T , . . . , ξrT ] ∈ Rr×r and li j = ∂jvi , i = 1, . . . , r, j = 1, . . . , r. For later development, the following filtered error is introduced.

ε = λn−1 e + . . . + λ1e(n−2) + e(n−1)

(5.10)

where λi , i = 1, . . . , n − 1 is a positive constant determined such that the polynomial λn−1 c + . . . + λ1 c(n−2) + c(n−1) is Hurwitz; thus it holds that the boundedness of ε implies the boundedness of output tracking error e and its derivative up to (n−1)thorder [70]. Differentiating (5.10) with respect to time and using (5.7) and (5.8), we obtain

ε˙ =λn−1 e˙ + . . . + λ1 e(n−1) + e(n) =Ψ (x,t) + F(x,t) + B(x,t)ε (v) + B(x,t)Γ (0) + B(x,t)L(ξ )v + Fd (t) =Ψ (x,t) + F(x,t) + B(x,t)ε (v) + B(x,t)Γ (0) + G(x, ξ ,t)v + Fd (t)

(5.11)

where G(x, ξ ,t) =B(x,t)L(ξ ) (n)

Ψ (x,t) = − yd + λn−1e˙ + . . . + λ1e(n−1)

(5.12)

Remark 5.1 In view of Assumption 5.1, yd and its up to n-th order derivatives are known and bounded, thus Ψ (x,t) and ε can be calculated from the state x and the derivatives of yd , and therefore can be used in the control design. Our focus will be on developing a neuro-adaptive PI control strategy with simple structure and without requiring precise information on F(x,t), B(x,t), Fd (t), and L(ξ ). To proceed, we define the generalized error E as, E = ε +δ

Z t

ε dτ

(5.13)

0

where δ is a positive design parameter.

5.2.2 Neural Networks and Function Approximation In this chapter, widely used radial basis function neural networks (RBFNNs) are utilized to approximate the unknown nonlinear function [77, 78, 79, 80, 81, 82, 83], so that the output of the RBFNN is given by O(Z) = W T S(Z)

(5.14)

where Z ∈ Rl and O ∈ R denote the NNs input and output, respectively, W ∈ R p denotes the adjustable weight. Based on the well-known approximation property, given a continuous nonlinear scalar function ϕ (Z), there exists an ideal RBFNN

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 48 — #62

✐

✐

48

5 Adaptive PI Control for MIMO Nonlinear Systems

capable of approximating the smooth nonlinear function, on a compact set Ωz , with sufficient accuracy, i.e.,

ϕ (Z) = W ∗T S(Z) + η (Z)

(5.15)

∗

where W is the optimal constant weight, η (Z) denotes the approximate error and S(Z) = [s1 (Z), . . . , s p (Z)]T is the basis function vector. One of the typical choices for S(Z) is ! (Z − µ j )T (Z − µ j ) s j (Z) = exp − (5.16) φ 2j where µ j = [µ j1 , . . . , µ jq ]T , j = 1, . . . , p, and φ j is the width of the Gaussian function and for all Z ∈ Ωz , W ∗ obeys
W ∗ = arg minp sup |W T S(Z) − ϕ (Z)| (5.17) W ∈R

Widespread practical application of NNs shows that, if the NN node number p is chosen large enough, |η (Z)| can be reduced to an arbitrarily small value over a compact set [77, 78, 79, 80], namely, there exists an unknown constant such that |η (Z)| ≤ ηN < ∞

(5.18)

A neural adaptive PI control scheme, as conceptually depicted in Fig. 5.2, is developed to solve the tracking control problem in the next section.

Fig. 5.2 Neuro-adaptive PI tracking control with self tuning gains.

5.3 PI Control Design and Stability Analysis For the dynamic system (5.1), equivalently expressed as in (5.7) in terms of the approximate error ε , it is nontrivial to build a control scheme without using F(x,t), Fd and G(x, ξ ,t) directly. The problem is further complicated if G(x, ξ ,t) is

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 49 — #63

✐

✐

5.3 PI Control Design and Stability Analysis

49

non-square, unknown and time-varying. In this chapter, the following two cases on G(x, ξ ,t) are considered. Case 1: The matrix G(x, ξ ,t) is square, unknown and time-varying, and unnecessarily symmetric. Define G∗ (x, ξ ,t) = (GT (·) + G(·))/2 and assume that G∗ (x, ξ ,t) is either positive definite or negative definite. Remark 5.2 The condition imposed on G∗ (·) guarantees that the system as described in (5.1) is controllable, which has been commonly used by most existing works in addressing MIMO systems [82, 83, 84]. Without loss of generality, we consider that G∗ (·) is positive definite, such that for all x in the domain of interest, there exists an unknown positive constant g0 satisfying 0 < g0 ≤ min{eig(G∗ (·))}

(5.19)

Case 2: The matrix G(x, ξ ,t) is non-square (i.e., m < r) and partially known. Assume that there exists a known and bounded matrix A(x) ∈ Rm×r with full row rank and an unknown matrix M(x, ξ ,t) ∈ Rr×r , such that G(x, ξ ,t) = A(x)M(x, ξ ,t)

(5.20)

Let M ∗ = A(M + M T )AT /2. Assume that M ∗ is positive definite, which is necessary for the system under consideration to be controllable. In such case, for all x in the domain of interest, there exists an unknown positive constant m0 satisfying 0 < m0 ≤

1 min{eig(M ∗ )} kAk

(5.21)

5.3.1 Neuoadaptive PI Control for Square Systems In this chapter, we propose the following control scheme which is of PI structure: v = −(kP1 + ∆ kP1 (·))ε − (kI1 + ∆ kI1 (·))

Z t

ε dτ

(5.22)

0

Compared with traditional PI control with constant gains, in the proposed PI controller, both proportional gain and integral gain consist of two components: one is constant (kP1, kI1 ) and the other is time-varying (∆ kP1 , ∆ kI1 ), and the time-varying parts are self-tuned via the analytical algorithms developed in the paper, described briefly here for your convenience: 1) choosing kP1 > 0 (no need for tuning); 2) setting kI1 = δ kP1 (no need for tuning), with δ > 0 chosen freely; 3) self-tuning ∆ kP1 adaptively and automatically by

∆ kP1 (·) =c1 aˆΦ 2 (Z)

(5.23)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 50 — #64

✐

✐

50

5 Adaptive PI Control for MIMO Nonlinear Systems

with the adaptive law a˙ˆ = −γ aˆ + c1 Φ 2 (Z)kEk2

(5.24)

where Φ (z) = 1 + ||S(Z)|| with E and S(Z) being defined as in (5.13) and (5.16), respectively, γ > 0 and c1 > 0 are chosen by the designer. 4) self-tuning ∆ kI1 via ∆ kI1 (·) = δ ∆ kP1 (·) (tuning at the same pace as that of ∆ kP1 but with different scale as specified by δ ). With the proposed neuro-adaptive PI control scheme, the following result is obtained. Theorem 5.1. Consider the nonlinear system (5.1) with the error dynamics governed by (5.11). Under the conditions as imposed in Assumptions 5.1—5.2 and Case 1, if the PI control as presented on (5.22) is applied, then the tracking error e is ensured to be ultimately uniformly bounded. Besides, all the internal signals in the system are bounded and the control signal is continuous and smooth everywhere. Remark 5.3 The proposed control bears the general PI form, thus is simple in structure and inexpensive in computation. Unlike traditional PI control, however, the proposed one exhibits several silent features: 1) PI gains are not constant but time varying, although there is a constant part; the time varying part is consistently updated automatically without the need for hand tuning or trial and error process; the constant part can be chosen freely by the designer; 2) note that the P-gain and I-gain are purposely linked to each other through the parameter δ , rather than being determined independently as in traditional PI control, and it is such treatment that simplifies the gain tuning process and facilitates stability analysis; 3) it is robust against external disturbance and adaptive to modeling uncertainties; and 4) the control is model-independent in that even if F(x,t), B(x,t) and Fd (t) change, there is no need for human interference for resetting the PI gains. Instead, the proposed algorithm will automatically adjust the gains to accommodate the uncertainties and reject the disturbances. Proof. By the definition of E as given in (5.13) and using (5.11) we have E˙ =Ψ (x,t) + F(x,t) + Fd (t) + B(x,t)ε (v) + B(x,t)Γ (0) + G(x, ξ ,t)v + δ ε

(5.25)

=Q(x, v,t) + G(·)v where Q(x, v,t) =Ψ (x,t) + F(x,t) + Fd (t) + B(x,t)ε (v) + B(x,t)Γ (0) + δ ε

(5.26)

A feasible way to reconstruct the lumped uncertain term Q(·) as defined in (5.26) is the key for establishing the stability result. Here in the proposed PI control scheme, the NN unit is used to approximate the upper bound of Q(·), rather than Q(·) itself. Such treatment circumvents the technical difficulty that would ensue if the NN were used to approximate Q(·) directly. More specifically, as Q(·) contains ε (v), directly

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 51 — #65

✐

✐

5.3 PI Control Design and Stability Analysis

51

approximating Q(·) would require v as part of the training input signals to the NN unit, creating the notorious algebra-loop problem. Interestingly, if we consider the upper bound of Q(·), such a problem disappears because we have ||Q(x, v,t)|| ≤||Ψ (x,t)|| + ||F(x,t) + Fd (t)|| + ||B(x,t)||ρ + δ ||ε || = ϕ (x, Ψ ) (5.27) which is interdependent of v (where Γ (0) = 0 and max Di ≤ ρ have been used). Now, although completely unknown, ϕ (·) is a nonlinear and scalar function depending on Z = [xT , Ψ T ]T , and it can be approximated by NN as follows

ϕ (Z) = kQ(·)k =W ∗T S(Z) + η (Z)

≤kW ∗T kkS(Z)k + ηN (Z)

(5.28)

≤aΦ (Z)

where Φ (Z) = kS(Z)k + 1, a = max{kW ∗ k, ηN }. Choose the following Lyapunov function candidate 1 1 2 V = ET E + a˜ 2 2g0

(5.29)

where a˜ = a − g0aˆ is estimation error, and g0 is defined as in (5.19). Differentiating V with respect to time and using (5.25) and (5.28), yields V˙ =E T E˙ − a˙ˆa˜

=E T Q(·) + E T G(·)v − a˙ˆa˜

≤kQ(·)kkEk + E T G(·)v − a˙ˆa˜ ≤ϕ (Z)kEk + E T G(·)v − a˙ˆa˜

(5.30)

≤aΦ (Z)kEk + E T G(·)v − a˙ˆa˜

Bearing in mind (5.13) and (5.22), the actual control input v can be expressed as v = −(kP1 + ∆ kP1 (·))E, then we further have V˙ ≤aΦ (Z)kEk − (kP1 + ∆ kP1 (·))E T G(·)E − a˙ˆa˜

(5.31)

Notice that although G is unnecessarily symmetric, (G(·) + GT (·))/2 is symmetric and (G(·) − GT (·))/2 is skew symmetric, thus, E T (G(·) − GT (·))E/2 = 0 for any E ∈ Rm . Upon using the condition on G as imposed in Case 1, it is readily shown that G(·) + GT (·) G(·) − GT (·) E − ET E 2 2 = − E T G∗ (·)E

−E T G(·)E = − E T

(5.32)

≤ − g0 kEk2

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 52 — #66

✐

✐

52

5 Adaptive PI Control for MIMO Nonlinear Systems

Substituting (5.32) into (5.31), we obtain V˙ ≤ aΦ (Z)kEk − (kP1 + ∆ kP1 (·))g0 kEk2 − a˙ˆa˜

(5.33)

Using Young’s inequality, one immediately gets that for any c1 > 0

Φ (Z)kEk ≤ c1 Φ 2 (Z)kEk2 +

1 4c1

(5.34)

Inserting (5.34) into (5.33) generates a V˙ ≤ − kP1g0 kEk2 + c1 aΦ 2 (Z)kEk2 + − ∆ kP1 (·)g0 kEk2 − a˙ˆa˜ 4c1

(5.35)

Note that 1 1 a(a ˜ − a) ˜ = (aa˜ − a˜2 ) g0 g0 1 1 1 ≤ ( a2 + a˜2 − a˜2 ) g0 2 2 1 2 ≤ (a − a˜2) 2g0

a˜aˆ =

(5.36)

In view of (5.23), (5.24), and (5.36), we derive from (5.35) that V˙ ≤ − kP1g0 kEk2 + c1 aΦ 2 (Z)kEk2

− c1g0 aˆΦ 2 (Z)kEk2 + γ aˆa˜ − c1 a˜Φ 2 (Z)kEk2 +

≤ − kP1g0 kEk2 + γ a˜aˆ + ≤ − kP1g0 kEk2 −

a 4c1

a 4c1 (5.37)

γ 2 γ 2 a a˜ + a + 2g0 2g0 4c1

≤ − Λ1V + Θ1 with

Λ1 = min{2kP1 g0 , γ },

Θ1 =

γ 2 a a + 2g0 4c1

(5.38)

being some constants. Consequently, from (5.37), it is concluded that V enters the set 1 ˙ Ω1 = {V | kV k ≤ Θ Λ1 } as time goes by. Once V is outside the set Ω 1 , V < 0, therefore, there exists √ a finite time T0 such that V ∈ Ω1 for ∀t > T0 , which further implies that kEk ≤ 2V ≤ 2ΛΘ11 for ∀t > T0 . Thus, we can conclude that kEk is ultimately uniformly bounded (UUB) according to [63]. Thus ε and also the tracking error e are UUB. In the sequel we prove all the internal signals in the system are continuous and bounded. From (5.37), it is derived that

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 53 — #67

✐

✐

5.3 PI Control Design and Stability Analysis

53

V ≤ exp(−Λ1t)V (0) +

Λ1 Θ1

(5.39)

Therefore, V ∈ R ℓ∞ , which implies that E ∈ ℓ∞ and aˆ ∈ ℓ∞ . E ∈ ℓ∞ guarantees that ε ∈ ℓ∞ and 0t ε d τ ∈ ℓ∞ . Note that ε ∈ ℓ∞ implies that e(i) ∈ ℓ∞ , i = 1, . . . , n which further implies that x ∈ ℓ∞ and Ψ (x,t) ∈ ℓ∞ , hence it holds that ∆ kP1 (·) ∈ ℓ∞ , ∆ kI1 (·) ∈ ℓ∞ , a˙ˆ ∈ ℓ∞ , u ∈ ℓ∞ via (5.22) to (5.24). Finally, one can conclude that ε˙ ∈ ℓ∞ and E ∈ ℓ∞ . Therefore, all the internal signals in the system are bounded. Remark 5.4 In order to address the time varying control gain in the system and the unknown weights, we choose not to directly estimate the unknown weights. Instead, we estimate the normalized version of a, and such treatment significantly simplifies the stability analysis as seen shortly.

5.3.2 Neuoadaptive PI Control for Non-square Systems For the case that G(·) is non-square satisfying the condition (5.20) as stated in Case 2, the corresponding PI controller is proposed as follows v = −(kP2 + ∆ kP2 (·))ε − (kI2 + ∆ kI2 (·))

Z t

ε dτ

(5.40)

0

The tuning rules for PI gains are given as follows: 1) selecting k0 > 0; AT 2) computing kP2 by kP2 = k0 kAk ; 3) setting kI2 = δ kP2 ; 4) computing ∆ kP2 by

∆ kP2 (·) =c2 aˆΦ 2 (Z)

AT kAk

(5.41)

with aˆ being updated by a˙ˆ = −ω aˆ + c2 Φ 2 (Z)kEk2

(5.42)

5) setting ∆ kI2 (·) = δ ∆ kP2 (·). where ω , δ , c2 and k0 are positive design constants chosen by the designer. We are ready to state the following results. Theorem 5.2. For MIMO nonlinear system (5.1) with the error dynamics governed by (5.11) under the conditions as imposed in Assumptions 5.1—5.2 and Case 2, the PI control scheme (5.40) with the tuning rule described therein guarantees that all the internal signals in the closed-loop system are bounded. Moreover, the tracking error is ensured to be UUB.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 54 — #68

✐

✐

54

5 Adaptive PI Control for MIMO Nonlinear Systems

Proof. Consider the following Lyapunov function candidate 1 2 1 a˜ V = ET E + 2 2m0

(5.43)

where a˜ = a − m0 aˆ and m0 is defined as in (5.21). Taking time derivative of (5.43) and using the fact that v = −(kP2 + ∆ kP2 (·))E, and inserting (5.25), (5.28), and (5.41), one has V˙ =E T E˙ − a˙ˆa˜

=E T Q(·) + E T G(·)v − a˙ˆa˜

AMAT ≤aΦ (Z)kEk − (c2aˆΦ (Z) + k0 )E E − a˙ˆa˜ kAk 2

(5.44)

T

Under condition as imposed in Case 2, it is straightforward to show that −E T

AMAT A(M + M T )AT A(M − M T )AT E = − ET E − ET E kAk 2kAk 2kAk M∗ E = − ET kAk

(5.45)

≤ − m0 kEk2

Applying Young’s inequality, it holds that for any c2 > 0

Φ (Z)kEk ≤ c2 Φ 2 (Z)kEk2 +

1 4c2

(5.46)

Thus, V˙ becomes a − c2 m0 aˆΦ 2 (Z)kEk2 − a˙ˆa˜ V˙ ≤ − k0 m0 kEk2 + c2 aΦ 2 (Z)kEk2 + 4c2

(5.47)

Note that 1 1 a(a ˜ − a) ˜ = (aa˜ − a˜2) m0 m0 1 1 1 ≤ ( a2 + a˜2 − a˜2 ) m0 2 2 1 ≤ (a2 − a˜2) 2m0

a˜aˆ =

(5.48)

Using (5.42) and (5.48), we get from (5.47) that

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 55 — #69

✐

✐

5.4 Modified PI Control Based on BLF

55

a V˙ ≤ − k0m0 kEk2 + ω a˜aˆ + 4c2 ω 2 ω 2 a ≤ − k0m0 kEk2 − a˜ + a + 2m0 2m0 4c2 ≤ − Λ2V + Θ2

(5.49)

with

Λ2 = min{2k0 m0 , ω },

Θ2 =

ω 2 a a + 2m0 4c2

(5.50)

being some constants. Then by following the same analysis as in the proof of Theorem 2, it can be established that V˙ < 0 if V is outside of the compact region 2 Ω2 = {V | kV k ≤ Θ Λ2 } E is confined in the set Ω 2 and the UUB of E is ensured. Then the uniformly ultimately boundedness of ε and therefore e are ensured. The proof is complete.

5.4 Modified PI Control Based on BLF Motivated by the established PI control scheme with well-explained analytical tuning algorithms, we now present a modified version to ensure the full functionality of the method. Note that to use NN for function approximation, the selected training input vector Z must remain in a compact set. As the NN input vector Z is linked with E, it is thus necessary to ensure the boundedness of kEk. To this end, we make use of the unique feature of barrier Lyapunov function to develop strategies for confining/constraining the NN input (refer [85, 86, 87] for the fundamental properties and practical application of BLF in infinite dimensional systems with constraints). As the first step, we construct the following BLF VB =

1 β2 ln 2 2 β − kEk2

(5.51)

where β > 0 is a pre-specified constant. Note that E is the generalized error as defined in (5.13), and VB is a valid Lyapunov function, which is continuous and positive definite in the set Ω := {kEk | kEk < β }. We will show that the developed control method ensures that 0 ≤ VB < ∞; consequently, it always holds that kEk < β . Therefore, the selected training input vector for the NN (i.e., Z) will always remain in a compact set. As a result, the NN unit in the proposed PI scheme is able to maintain its learning or approximating capability during the entire process of system operation.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 56 — #70

✐

✐

56

5 Adaptive PI Control for MIMO Nonlinear Systems

5.4.1 Neuro-adaptive PI Control for Square Systems The neuro-adaptive PI control method is as follows v = − (kP3 + ∆ kP3(·)) ε − (kI3 + ∆ kI3 (·))

Z t

ε dτ

(5.52)

0

where kP3 , kI3 and ∆ kP3 , ∆ kI3 are not tuned independently. Instead, they are determined according to the following rules: 1) choosing kP3 > 0; 2) setting kI3 = δ kP3 , with δ > 0 chosen freely; 3) computing ∆ kP3 by

∆ kP3 (·) = with

c3 aˆΦ 2 (Z) β 2 − kEk2

c3 Φ 2 (Z)kEk2 a˙ˆ = −γ aˆ + 2 (β − kEk2)2

(5.53)

(5.54)

4) setting ∆ kI3 (·) = δ ∆ kP3 (·). where again γ , δ , β , and c3 > 0 are free design parameters. Theorem 5.3. Consider the nonlinear system (5.1) with the error dynamics governed by (5.11). Under the conditions as imposed in Assumptions 5.1—5.2 and Case 1, if the PI control as presented in (5.52) is applied, then the tracking error e is ensured to be UUB. Besides, all the internal signals in the system are bounded, and for kE(0)k < β , the NN input always remains within a compact set. Proof. Consider the following Lyapunov function candidate V = VB +

1 2 a˜ 2g0

(5.55)

where a˜ = a − g0a, ˆ g0 is defined as (5.19), and VB is defined as (5.51). Differentiating V with respect to time, and using (5.25) yields V˙ =V˙B − a˙ˆa˜ 1 = 2 E T E˙ − a˙ˆa˜ β − ET E 1 1 = 2 E T Q(·) + 2 E T G(·)v − a˙ˆa˜ 2 β − kEk β − kEk2 1 1 ≤ 2 kEkkQ(·)k + 2 E T G(·)v − a˙ˆa˜ β − kEk2 β − kEk2

(5.56)

Bearing in mind (5.13) and (5.52), the actual control input v can be expressed as

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 57 — #71

✐

✐

5.4 Modified PI Control Based on BLF

57

v = − (kP3 + ∆ kP3(·)) E

(5.57)

Substituting (5.57) into (5.56), and using (5.28) and (5.32), we get from (5.56) that k p3 + ∆ kP3 (·) T ϕ (Z)kEk V˙ ≤ 2 − E G(·)E − a˙ˆa˜ 2 β − kEk β 2 − kEk2 aΦ (Z)kEk k p3 + ∆ kP3(·) ≤ 2 − g0 kEk2 − a˙ˆa˜ β − kEk2 β 2 − kEk2 ≤− where

ϒ1 =

(5.58)

kP3 g0 kEk2 + ϒ1 β 2 − kEk2

∆ kP3 (·) aΦ (Z)kEk − g0 kEk2 − a˙ˆa˜ β 2 − kEk2 β 2 − kEk2

(5.59)

By using the Young’s inequality, one immediately gets that for any c3 > 0

Φ (Z)kEk ≤

c3 Φ 2 (Z)kEk2 β 2 − kEk2 + β 2 − kEk2 4c3

(5.60)

Inserting (5.53), (5.54), and (5.60) into (5.59), we have c3 a˜Φ 2 (Z)kEk2 a c3 g0 aˆΦ 2 (Z)kEk2 c3 aΦ 2 (Z)kEk2 γ a ˆ a ˜ − + − + (β 2 − kEk2)2 4c3 (β 2 − kEk2)2 (β 2 − kEk2)2 a ≤γ aˆa˜ + 4c3

ϒ1 ≤

(5.61)

By virtue of (5.36), we further have

ϒ1 ≤

γ a (a2 − a˜2) + 2g0 4c3

(5.62)

Inserting (5.62) into (5.58), one has V˙ ≤ − kP3 g0

kEk2

β 2 − kEk2

−

γ 2 γ 2 a a˜ + a + 2g0 2g0 4c3

(5.63)

Note that for all kEk < β , it holds that −

kEk2

β 2 − kEk2

1 β2 < − ln 2 2 β − ET E

(5.64)

Thus, we get kP3 g0 β2 γ 2 γ 2 a V˙ ≤ − ln 2 − a˜ + a + T 2 β − E E 2g0 2g0 4c3 ≤ − Λ3V + Θ3

(5.65)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 58 — #72

✐

✐

58

5 Adaptive PI Control for MIMO Nonlinear Systems

with

Λ3 = min{kP3 g0 , γ },

Θ3 =

a γ 2 a + 2g0 4c3

(5.66)

being some constants. It thus follows from (5.65) thatR V ∈ ℓ∞ , which implies that E ∈ ℓ∞ and aˆ ∈ ℓ∞ . E ∈ ℓ∞ guarantees that ε ∈ ℓ∞ and 0t ε d τ ∈ ℓ∞ . Note that ε ∈ ℓ∞ implies that e(i) ∈ ℓ∞ , i = 1, . . . , n which further implies that x ∈ ℓ∞ and Ψ (x,t) ∈ ℓ∞ , hence it holds that ∆ kP3 (·) ∈ ℓ∞ , ∆ kI3 (·) ∈ ℓ∞ , a˙ˆ ∈ ℓ∞ , u ∈ ℓ∞ via (5.52) to (5.54). Finally, one can conclude that ε˙ ∈ ℓ∞ and E ∈ ℓ∞ . Therefore, all the internal signals in the system are bounded. From the inequality (5.65), it is seen that VB is bounded for all t ≥ 0, thus kEk remains in Ω3 := {kEk | kEk < β }. Thus E is confined in the set Ω3 . Then the UUB of ε and thus e are ensured. Furthermore, the control action v and v˙ generated by the control scheme are continuous and smooth everywhere, as shown in the Appendix.

5.4.2 Neuro-adaptive PI Control for Non-square Systems For this case, the corresponding PI control law is proposed as follows, v = − (kP4 + ∆ kP4(·)) ε − (kI4 + ∆ kI4 (·))

Z t

ε dτ

(5.67)

0

the rules for tuning PI gains are determined as follows 1) selecting k1 > 0; AT ; 2) computing kP4 by kP4 = k1 kAk 3) setting kI4 = δ kP4 ; 4) computing ∆ kP4 by c4 aˆΦ 2 (Z) AT β 2 − kEk2 kAk

(5.68)

c4 Φ 2 (Z)kEk2 a˙ˆ = −ω aˆ + 2 (β − kEk2)2

(5.69)

∆ kP4 (·) = where aˆ is updated by

5) setting ∆ kI4 (·) = δ ∆ kP4 (·). with ω , δ , c4 , and k1 being positive design constants chosen by the designer. We are ready to state the following results. Theorem 5.4. Consider the MIMO nonlinear system (5.1) with the error dynamics governed by (5.11). Under the conditions as imposed in Assumptions 5.1—5.2 and Case 2, if the PI control as presented on (5.67) is applied, then the tracking error e

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 59 — #73

✐

✐

5.4 Modified PI Control Based on BLF

59

is ensured to be UUB, and all the internal signals in the system are bounded, and for kE(0)k < β , there always exists a compact set for NN input. Proof. Consider the following Lyapunov function candidate V = VB +

1 2 a˜ 2m0

(5.70)

where a˜ = a − m0 a, ˆ m0 is defined as (5.21), and VB is defined as (5.51). Taking time derivative of (5.70) and inserting (5.25) and (5.28), we have 1 E T E˙ − a˙ˆa˜ β 2 − kEk2 1 1 aΦ (Z)kEk + 2 E T Gv − a˙ˆ a˜ ≤ 2 β − kEk2 β − kEk2

V˙ =

(5.71)

Employing (5.13), (5.67), and (5.68), the actual control design v can be expressed as v = − (kP4 + ∆ kP4 (·)) E = − (k1 +

c4 aΦ 2 (Z) AT ) E β 2 − kEk2 kAk

(5.72)

Thus, one has E T Gv = − (k1 +

c4 aΦ 2 (Z) T AT )E G E β 2 − kEk2 kAk

c4 aΦ 2 (Z) T AMAT = − (k1 + 2 )E E β − kEk2 kAk

(5.73)

Under the condition as imposed in Case 2, it is straightforward to show that −E T

A(M + M T )AT A(M − M T )AT AMAT E = − ET E − ET E kAk 2kAk 2kAk M∗ = − ET E kAk

(5.74)

≤ − m0 kEk2

By using Young’s inequality, we have

Φ (Z)kEk ≤

c4 Φ 2 (Z)kEk2 β 2 − kEk2 + β 2 − kEk2 4c4

(5.75)

Substituting (5.72)—(5.75) into (5.71), it is readily derived that k1 m0 kEk2 a + ω a˜aˆ V˙ ≤ − 2 + β − kEk2 4c4

(5.76)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 60 — #74

✐

✐

60

5 Adaptive PI Control for MIMO Nonlinear Systems

Utilizing the inequalities (5.48) and (5.64), we have from (5.76) that V˙ ≤ − k1m0

a kEk2 ω 2 ω 2 a˜ + a + − 2 2 β − kEk 2m0 2m0 4c4

k 1 m0 β2 ω 2 ω 2 a ln 2 − a˜ + a + T 2 β − E E 2m0 2m0 4c4 < − Λ4V + Θ4 ≤−

(5.77)

with

Λ4 = min{k1 m0 , ω },

Θ4 =

ω 2 a a + 2m0 4c4

(5.78)

being some constants. Then by following the same analysis as in the proof of Theorem 5.3, and noting the inequality (5.77), we see the boundedness of VB , thus kEk remains in Ω4 := {kEk | kEk < β }. Thus E is confined in the set Ω4 and the uniformly ultimately boundedness of E is ensured. Then the UUB of ε and thus e are ensured. Remark 5.5 Although the underlying tracking control problem is quite complicated when the system is MIMO nonlinear with unknown and constrained non-smooth actuation characteristics, the proposed solution is structurally simple, computationally inexpensive and functionally effective. Note that a number of bounded parameters such as ηN , g0 , m0 , etc., are defined and used in stability analysis; however these parameters are not involved in the developed control algorithms. Thus analytical estimation on those parameters (a nontrivial task) is not needed in setting up and implementing the proposed PI control methods. Remark 5.6 The significance of this result is threefold: 1) The PI controller realization is independent of the nonlinear function F(x,t), Fd (·) and B(x) except for the sign of the control effectiveness; 2) PI gains are determined analytically and are updated automatically and adaptively without the need for trial and error process; and 3) while the system is nonlinear and subject to significant uncertainties, the resultant control remains simple in structure and inexpensive in computation. Remark 5.7 By using BLF, the NN unit in the control scheme maintains its approximation capability during the whole process of system operation. Thus the proposed neuro-adaptive PI control scheme is fully functional over the entire system operation envelope. Also control action is continuous and smooth everywhere. Remark 5.8 Note that in setting up the modified PI control scheme, one needs to choose β such that β > kE(0)k = kε (0)k. As the maximum possible initial states of the system are normally available, such β can be trivially determined accordingly. In practice, it can be simply chosen as a relatively large constant.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 61 — #75

✐

✐

5.5 Illustrative Examples

61

5.5 Illustrative Examples To validate the effectiveness of the proposed neuro-adaptive PI control scheme, we conduct numerical simulations. The particular objective is to verify if stable tracking in the presence of modeling uncertainties and constrained non-smooth actuation characteristics is achieved by the proposed neuro-adaptive PI control. To this end, two examples are tested. Example 5.1. To examine the applicability and feasibility of the proposed method to a real-world system. We consider a two-joint rigid-link robotic manipulator system described by: M(q)q¨ + C(q, q) ˙ q˙ + G(q) + τd =τ

(5.79)

where state vector q ∈ R2 denotes the joint position, τd is the external disturbances, M(q) ∈ R2×2 is the inertia matrix of the manipulator, C(q, q) ˙ ∈ R2×2 is the centripetal 2 Coriolis matrix of the manipulator, G(q) ∈ R is gravitational vector and τ ∈ R2 is the manipulator’s torque input vector. Where   ρ + ρ2 + 2ρ3 cos(q2 ) ρ2 + ρ3 cos(q2 ) M(q) = 1 (5.80) ρ2 + ρ3 cos(q2 ) ρ2 

−ρ3q˙2 sin(q2 ) −ρ3 (q˙1 + q˙2) sin(q2 ) C(q, q) ˙ = −ρ3q˙2 sin(q2 ) 0   ρ4 cos(q1 ) + ρ5 cos(q1 + q2 ) G(q) = ρ5 cos(q1 + q2)   0.2 sin(t) τd = 0.2 sin(t)



ρ =[ρ1 , ρ2 , ρ3 , ρ4 , ρ5 ] = [3.5, 0.76, 0.87, 3.04, 0.87]

(5.81) (5.82) (5.83) (5.84)

Comparison with traditional PI control with constant gains is made in the simulation. For the proposed PI control, the ‘P’ gain is produced by kP3 + ∆ kP3 with kP3 = 2 and the self-tuning part ∆ kP3 is updated by (5.53). For the traditional PI control, we set k p1 = 15, ki1 = 0.31 which are obtained by a number of trial and error tests. It can be observed from Fig. 5.3(b) that the proposed PI control leads to higher tracking accuracy as compared with traditional PI control. Also Fig. 5.3(e) clearly indicates that the PI gains are self adjusting continuously during system operation, rather than fixed as with the traditional PI control, thus giving rise to better performance.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 62 — #76

✐

✐

62

5 Adaptive PI Control for MIMO Nonlinear Systems 2

q1

0.6

qd1

1

10-3

10

0.4

e 1 under the proposed PI control e 1 under the tradtional PI control

5

0

0

0.2

-1

-5 4

6

8

10

12

14

0

-2 0

5

10

15

20

25

30

35

40

45

2

50

qd2

1

0

5

10

15

20

25

30

35

40

45

50

0.1

q2

0

10-3 5

-0.1

0

0

-0.2

-5

-1 -0.3 -2 0

5

10

15

20

25

30

35

40

45

50

e 2 under the proposed PI control

-10 4

-0.4 0

5

6

10

8

10

15

12

e 2 under the tradtional PI control

14

20

25

30

35

40

45

50

Time (sec)

(a) Tracking process.

(b) Tracking errors.

u1 under the proposed PI control

20

20

u1 under the traditional PI control u2 under the traditional PI control

u2 under the proposed PI control

10

10 0

0

-10 0

5

10

15

20

25

30

35

40

45

50

-10 0

Time(sec)

1

0.1

Dead-zone 21

21.1

21

21.05

21.1

(c) Saturation function input signal u from the proposed PI controller.

25

30

35

40

45

50

21.05

21.1

Dead-zone

Dead-zone

0.2

0

0

-1 21

21.05

-0.2 20.9

21.1

21.15

21.2

20.95

21

(d) Saturation function input signal u from the traditional PI controller. 0.8

KP under the proposed PI control

400

20

Dead-zone

-0.1 21.3 20.95

21.2

15

0.4

1

0

-2 20.9

10

2

0 -1

5

Time(sec)

0.2

2

KP under the traditional PI control

300

0.7 0.6

200 100

0.5

0 0

5

10

15

20

25

10

30

35

40

45

50

KI under the proposed PI control

0.4 0.3

KI under the traditional PI control

0.2

5

0.1 0

0 0

5

10

15

20

25

30

35

40

45

50

0

5

10

Time (sec)

(e) P and I gains.

15

20

25

30

35

40

45

50

Time(sec)

(f) Evolution of the virtual parameter a. ˆ

Fig. 5.3 The simulation results.

Example 5.2. Consider the following non-square system: x˙1 =x2 x˙2 =0.2x21 + 0.2x3 + sin(x3 ) + (2 + 0.2 sin(t))u1 + u2 + fd1 x˙3 =x4 x˙4 =

(5.85)

exp(x4 ) − 1 + cos(x1 ) + (2 + 0.2 sin(t))u2 + u3 + fd2 exp(x4 ) + 1

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 63 — #77

✐

✐

5.5 Illustrative Examples

63

with the initial conditions x1 (0) = 0.4, x2 (0) = 0.6, x3 (0) = 1.1, x4 (0) = 1.15, the desired trajectories are yd1 = 1.5 sin(t) and yd2 = 1.5 cos(t), the external disturbances fd1 = 1.2 cos(t), fd2 = sin(t). The proposed neural-adaptive PI control with the adaptive laws (5.52), (5.53), and (5.54) in the presence of non-smooth asymmetric saturation as described in Theorem 5.4 are tested. We consider the Model 1 for the actuation with the actuation parameters being chosen as: σ¯ = 8, σ = −10, ι = 1 and l(v) = v. This corresponds to non-smooth and asymmetric actuation. Note that in this case the gain matrix G as defined in (5.12) can be obtained as 

G =

(72+7.2 sint)e−2ξ11 (1+e−2ξ11 )2

0

36e−2ξ22 0 (1+e−2ξ22 )2 −2 ξ −2 22 (72+7.2 sint)e 36e ξ33 (1+e−2ξ22 )2 (1+e−2ξ33 )2

 

(5.86)

Although G is unknown and non-square, it can be decomposed as AM(·), with   2 + 0.2 sin(t) 1 0 A= (5.87) 0 2 + 0.2 sin(t) 1   m1 0 0 M(·) =  0 m2 0  (5.88) 0 0 m3 where

m1 =

36e−2ξ11 >0 (1 + e−2ξ11 )2

(5.89)

m2 =

36e−2ξ22 >0 (1 + e−2ξ22 )22

(5.90)

m3 =

36e−2ξ33 >0 (1 + e−2ξ33 )2

(5.91)

It is readily verified that A is matrix with full row rank and M(·) is symmetric and positive definite; thus the conditions imposed in Case 2 all hold, and the control scheme as described in Theorem 5.4 applies to this case. The matrix norm of A is defined as q (5.92) kA(·)k = max χ (AT A) where χ is eigenvalues of the matrix AT A. It is observed that the PI control as designed in Theorem 5.4 ensures good tracking performance in the presence of modeling nonlinearities and uncertain actuation characteristics. It is also noted from Fig. 5.4(e)—Fig. 5.4(f) that the PI gains in the proposed method are indeed consistently self-tuning rather than remaining constant during system operation.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 64 — #78

✐

✐

64

5 Adaptive PI Control for MIMO Nonlinear Systems 0.4

x1

2

e 1 under the proposed PI control

0.2

x d1

e 1 under the traditional PI control

0.2

0.1

0 0

0

0.5

1

1.5

5

10

2

2.5

3

-2 0

5

10

15

20

25

30

35

40

45

50

-0.2 0

15

20

25

30

35

40

45

50

0.1

x2

2

0

x d2

-0.1

0

-0.2

-0.1

0 e 2 under the proposed PI control

-0.3

-2 0

5

10

15

20

25

30

35

40

45

50

-0.2 0.5

-0.4 0

5

1

10

1.5

2

15

2.5

20

e 2 under the traditional PI control

3

25

30

35

40

45

50

Time (sec)

(a) Tracking process. 10

(b) Tracking errors.

u1 under the proposed PI control u3 under the proposed PI control

0 -5

u1 under the traditional PI control

10 Control inputs

u2 under the proposed PI control

5

u2 under the traditional PI control

5

u3 under the traditional PI control

0 -5 -10

-10 0

5

10

15

20

25

30

35

40

45

0

50

Time(sec)

10

5

10

15

20

25

30

10

2

5

0.5

5

1

0

0

0

-0.5

-5

-5

saturation boundary

-10 0

0.2

0.4

0.6

0.8

40

45

50

0 saturation boundary

-1

-10

-1 20

25

30

35

0

40

(c) Saturation function input signal u from the proposed PI controller. KP under the proposed PI control

100

35

Time (sec)

1

0.2

0.4

0.6

0.8

20

25

30

35

40

(d) Saturation function input signal u from the traditional PI controller. 0.6

KP under the traditional PI control

0.5

80 60

0.4

40 0

5

10

15

20

25

30

35

40

45

50

0.3

KI under the proposed PI control

1

KI under the traditional PI control

0.2

0.8 0.6

0.1

0.4 0.2 0

5

10

15

20

25

30

35

Time(sec)

(e) P and I gains.

40

45

50

0 0

5

10

15

20

25

30

35

40

45

50

Time(sec)

(f) Evolution of the virtual parameter a. ˆ

Fig. 5.4 The simulation results.

For comparison, both traditional PI control with constant PI gains and the proposed neural-adaptive PI control with adaptive gains as given in Theorem 5.4 are tested. In the simulation we set k p2 = 80, ki2 = 1.6 for the traditional PI control. The proposed PI control has better transient performance and faster rate of convergence compared with the traditional one as seen from Fig. 5.4(b), while the control input of the proposed control is smoother and less saturated (thus, less wearing to the actuator) than that of the traditional PI control as observed from Fig. 5.4(c) and Fig.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 65 — #79

✐

✐

5.7 Appendix

65

5.4(d), implying less wear to the actuator. The neuro-adaptive PI gains depend on the virtual parameter a, ˆ which is smooth and bounded as illustrated in Fig. 5.4(f).

5.6 Conclusion Neuro-adaptive PI tracking control for a class of nonlinear uncertain systems with unknown actuation characteristics is studied in this chapter. The developed method allows the PI gains to be updated automatically without the need for manual tuning or trial and error process. The salient feature of the proposed control is that it exhibits the simple PI structure with analytical algorithms for PI gains determination. Using the BLF and Young’s inequality, the developed adaptive control guarantees that all closed-loop signals are UUB and that the NN input vector is within a compact set, ensuring the full functionality of the NN unit (thus the whole neuro-adaptive PI scheme) during the entire operation process of the system. Extending the design method to a wider class of nonlinear systems (such as strict feedback systems) with partial state feedback represents an interesting topic for future research.

5.7 Appendix Proof. According to (5.13) and (5.52), the actual control input v can be expressed as v = −(kP + ∆ kP (·))E

(5.93)

Differentiating it with respect to time, yields v˙ = − (kP + ∆ kP (·))E˙ −

d (∆ kP (·)))E dt

(5.94)

In view of (5.53), we have c3 a˙ˆΦ 2 (Z) + 2c3 aˆΦ (Z) dtd (Φ (Z)) 2c3 aˆΦ 2 (Z)E T E˙ d (∆ kP (·)) = + dt β 2 − kEk2 (β 2 − kEk2)2

(5.95)

Note that Φ (Z) = 1 + kS(Z)k, we get d d (Φ (Z)) = (kS(Z)k) dt dt

(5.96)

From (5.16) and (5.53), it is seen that s˙j (Z) = −

(Z − µ j )T (Z − µ j ) 2 (Z − µ j )T Z˙ exp[− ] 2 φj φ 2j

(5.97)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 66 — #80

✐

✐

66

5 Adaptive PI Control for MIMO Nonlinear Systems

c3 Φ 2 (Z)kEk2 a˙ˆ = −γ aˆ + 2 (β − kEk2)2

(5.98)

as Z = [xT , Ψ T ]T , we have Z˙ = [x˙T , Ψ˙ T ]T Note that x is the state vector which is bounded under the proposed control scheme, F(x,t) and B(x,t) are smooth functions, and the input u has been shown to be bounded; therefore we can draw a conclusion that x˙n ∈ ℓ∞ from (5.1) and (n) Assumption 5.2, which implies that x1 ∈ ℓ∞ . Moreover, owing to the ε ∈ ℓ∞ through the proof of Theorem 5.3, which implies that e as well as their derivatives up to (n − 1)th-order are bounded from (5.10), which further implies that ε˙ ∈ ℓ∞ from (5.11) and Assumption 5.1. Apparently, E˙ ∈ ℓ∞ . Thus it is not difficult to show ˙ that x˙ ∈ ℓ∞ and Ψ˙ ∈ ℓ∞ , which implies Z˙ ∈ ℓ∞ , thus we can easy to get S(Z) ∈ ℓ∞ . ˙ Moreover, it is seen that aˆ ∈ ℓ∞ . Consequently, we can conclude that v˙ ∈ ℓ∞ according to (5.96)—(5.98), which implies that v is smooth. It is thus established that the control signal v is bounded and smooth.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 67 — #81

✐

✐

Chapter 6

Adaptive PI Control for Strict Feedback Systems

In this chapter, we present a neuro-adaptive control for a class of uncertain nonlinear strict-feedback systems with full-state constraints and unknown actuation characteristics where the break points of the dead-zone model are considered as time-variant. In order to deal with the modeling uncertainties and the impact of the non-smooth actuation characteristics, neural networks are utilized at each step of the backstepping design. By using the barrier Lyapunov function (BLF), together with the concept of virtual parameter, we develop a neuro-adaptive control scheme ensuring tracking stability and at the same time maintaining full-state constraints. The proposed control strategy bears the structure of proportional-integral (PI) control, with the PI gains being automatically and adaptively determined, making its design less demanding and its implementation less costly.

6.1 Introduction Nonlinearities and uncertainties are the most common factors encountered in modern engineering systems that make the corresponding control problem rather challenging, enticing increasing attention among the control community, resulting in significant progress [88]. To handle the nonlinear unknown functions, neural network (NN) has been widely used because of its approximate ability [89, 90, 91]. Recently, the results on adaptive NN control technique have been extended from SISO systems to MIMO systems. The adaptive NN controllers are designed for nonlinear MIMO systems with unmodeled uncertainties [92] and with time delay [93, 94]. To deal with the problem of state constraints, barrier Lyapunov functions are designed to guarantee that state constraints are not violate. In [95], an adaptive NN control is presented for a class of uncertain nonlinear systems with full-state constraints and time delays by using BLFs. In [96], BLFs are adopted to address the full-state constraints for nonlinear systems under unknown control direction, while actuation constraints are not considered.

67

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 68 — #82

✐

✐

68

6 Adaptive PI Control for Strict Feedback Systems

In practice, actuation constraints and state constraints are both inevitable. For nonlinear systems with both actuation constraints (input dead zone, saturation [97, 98, 99, 100]) and output or state constraints (physical saturation, safety specifications [82, 95, 96, 101, 102, 103, 104, 105, 106, 107]), the underlying control problem becomes even more challenging. At the same time, there is a rich collection of results on the control problems of such systems [79, 80, 95, 96, 97, 98, 101, 102, 103, 104, 105], the proposed solutions still leave much to be desired in terms of simplicity and effectiveness. Note that PI (proportional and integral) control is simple in structure and inexpensive in implementation, and thus has been widely used in practical applications. In [108], a PI tracking control solution for normal nonlinear systems is proposed without considering the issue of state constraints. To our best knowledge, effort on PI-based control of uncertain strict-feedback systems under both full-state constraints and unknown actuation characteristics has not been reported in the literature. In this chapter, we attempt to solve the control problem under both unknown actuation characteristics and full-state constraints for a class of uncertain nonlinear strict-feedback systems. An inexpensive neuro-adaptive control scheme is presented to tackle such control problems. The proposed control strategy is of PI structure with self-tuned gains.

6.2 System Description and Preliminaries Consider a class of nonlinear systems:   x˙i = fi (x¯i ) + gi(x¯i )xi+1 , i = 1, · · · , n − 1 x˙n = fn (x¯n ) + gn(x¯n )D(u)  y = x1

(6.1)

where x¯i = [x1 , · · · xn ]T ∈ Rn is the vector of state variable, fi (x¯i ) and gi (x¯i ) are the unknown smooth nonlinear functions, D(u) is the unknown actuation characteristic where u ∈ R is the actual control input and y ∈ R is the output of the system. In the system (6.1) under consideration, each of the system states is constrained in a compact set, i.e., there exists a positive constant ci such that |xi | < ci . Also, the system exhibits actuation characteristics in the input channel that is expressed as  σ, u > umb1     l1 (u − b1(t)), b1 (t) ≤ u ≤ umb1 −b2 (t) < u < b1 (t) D(u) = 0, (6.2)   l (u + b (t)), −u ≤ u ≤ −b (t)  m 2 2 2 b2   σ, u < −umb2

where u ∈ R is the actual control input, l1 and l2 are the functions of the dead-zone characteristic, b1 (t) > 0, b2 (t) > 0, umb1 > 0 and umb2 > 0 represent the unknown

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 69 — #83

✐

✐

6.2 System Description and Preliminaries

69

break points where b1 (t) and b2 (t) are functions of t. Shown in Fig. 6.1 is the structure of the actuation model.

Fig. 6.1 Asymmetric non-smooth saturation function with dead-zone model.

Remark 6.1 Note that the break points b1 and b2 of the dead-zone are treated as time-variant, which is more effective in reflecting the practical scenarios where, for example, the friction coefficient and the smooth degree vary with different contact time instants. To deal with the asymmetric non-smooth saturation function with dead-zone, a smooth function is introduced to approximate the saturation function [60, 108, 109], which can be expressed as D(u) = Γ (u) + η (u)

(6.3)

σ eκ u − σ e− κ u eκ u + e− κ u

(6.4)

Γ (u) =

where κ > 0 is a constant and η (u) is the approximate error bounded by |η (u)| = |D(u) − Γ (u)| ≤ ρ

(6.5)

with ρ being a positive and unknown constant. Since Γ (u) is a smooth function of u, we employ the mean value theorem on Γ (u) to get

Γ (u) = Γ (0) +

∂Γ (u) u ∂u

(6.6)

and define b(·) =

∂Γ (u) ∂u

(6.7)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 70 — #84

✐

✐

70

6 Adaptive PI Control for Strict Feedback Systems

Due to the fact that Γ (0) = 0, we can rewrite D(u) as D(u) = bu + η (u)

(6.8)

The control objective of this chapter is to design a performance guaranteed neuroadaptive PI control for system (6.1) so that the output y closely follows the desired trajectory yd while the full-state constraints are not violated. To achieve our control objective, the following assumptions are imposed on system (6.1). Assumption 6.1 [105] The desired tracking trajectory yd and its derivative up to ( j) nth order are bounded. That is to say, yd (t) and yd (t), j = 1, . . . , n satisfy |yd (t)| ≤ ( j) A0 < c1 and yd (t) ≤ A j , respectively. Meanwhile, the state variable xi is available for control design. Assumption 6.2 [109] The sign of the unknown function gi (x¯i ), i = 1, · · · , n is known, and there exists a positive constant gi0 , such that gi (x¯i ) satisfies 0 < gi0 ≤ gi (x¯i ) < ∞.

Assumption 6.3 [109] The sign of b(·) is definite, and there exist unknown positive constants b0 and bm such that 0 < b0 ≤ b(·) ≤ bm < ∞. Definition 6.1. [104] A barrier Lyapunov function is a scalar function V (x), defined with respect to the system x˙ = f (x) which is continuous, positive definite on an open region D and has continuous first-order partial derivatives at every point of D, has the property V (x) → ∞ as x approaches the boundary of D, satisfies V (x) ≤ p, for any t ≥ t0 along the solution of x˙ = f (x) for x(t0 ) ∈ D and positive constant p. Lemma 6.1 [105] For a dynamic system as follows v(t) ˙ ≤ −av(t) + bw(t)

(6.9)

where a, b are positive constants and w(t) is a positive function, then for any given bounded initial condition v(t0 ) ≥ 0, we have v(t) ≥ 0 for ∀t ≥ t0 . Lemma 6.2 [80] For any positive constant d ∈ R, the following inequality holds for any x ∈ R in the interval |x| < |d|: −

x2 d2 ≤ − log 2 2 2 d −x d − x2

(6.10)

Lemma 6.3 [108] Consider an intermediate variable Z defined as Z = e+ε

Z t

ed τ

(6.11)

0

where ε is a positive design parameter. Then, ifR Z → 0, as t → ∞, then e → 0 and Rt t 0 ed τ → 0 as t → ∞; if Z ∈ l∞ , then e ∈ l∞ and 0 ed τ ∈ l∞ .

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 71 — #85

✐

✐

6.3 PI-like Control Design

71

In order to achieve the control objective, we choose to use NNs to deal with the unknown functions because of its good capabilities in function approximation. For any given continuous function f (X) : Rn → R on a compact set U ⊂ Rn , there exists the NNs = W T S(X), so that the continuous nonlinear function f (X) can be approximated with sufficient accuracy by choosing ideal NNs as f (X) = φ (X) = W ∗ T S(X) + ξ (X)

(6.12)

where W ∗ ∈ Rm denotes the ideal neural weight vector with m > 1 being the NN node number, X ∈ Rn is the NN input vector, and ξ (X) is the approximation error. If the NN node number m is large enough, |ξ (X)| can be reduced to an arbitrary ¯ and ξ¯ . S(X) = small value. In general, W ∗ and ξ (X) are bounded by the constants W T [s1 (X), s2 (X), · · · , sm (X)] is the Gaussian basis function vector. One typical form of Gaussian basis function can be express as kX − µi k2 si (X) = exp − ωi2

!

, i = 1, · · · , m

(6.13)

where µi = [µi1 , · · · µin ]T and ωi are the center and the width of the Gaussian function, respectively.

6.3 PI-like Control Design In this section, we will apply a backstepping design procedure to design the neuroadaptive controller. To proceed, we define a compact set ΩZ : {|zi | < di , i = 1, · · · , n}, where di > 0 is some prespecified constant and zi denotes the generalized error to be specified shortly. Now by using the backstepping design procedure, our design consists of the following steps. Step 1: We R define the tracking error as e1 = x1 − yd and the generalized error as z1 = e1 + ε1 0t e1 d τ where ε1 is a positive design parameter. Thus the time derivative of z1 is z˙1 = e˙1 + ε1 e1 = f1 (x1 ) + g1(x1 )x2 − y˙d + ε1 (x1 − yd )

(6.14)

For later development, we define the unknown function as

φ1 (Z1 ) = f1 (x1 ) − y˙d + ε1 (x1 − yd ) − g1(x1 )ε2

Z t 0

e2 d τ

(6.15)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 72 — #86

✐

✐

72

6 Adaptive PI Control for Strict Feedback Systems

 T where Z1 = x1 , x2 , x˙1 , yd , y˙d , θˆ1 , ε2 , e2 and θˆ1 will be defined later. In order to deal with the unknown nonlinear function, the NNs are used to approximate the unknown function φ1 (Z1 ) as

φ1 (Z1 ) = W1∗ T S1 (Z1 ) + ξ1 (Z1 )

(6.16)

where S1 (Z1 ) ∈ Rm1 is the Gaussian basis function vector with m1 > 1 being the NN node number, the ideal weight vector W1∗ and the approximation error ξ1 (Z1 ) are bounded by positive constants W¯ 1 and ξ¯1 , respectively. Choosing the following symmetric BLF V1 =

1 d2 1 log 2 1 2 + g10 θ˜12 2 d1 − z1 2

(6.17)

¯2 where θ˜1 = θˆ1 − θ1 and θˆ1 > 0 is the estimation of θ1 = g−1 10 W1 . Then we can derive V˙1 as z1 z˙1 V˙1 = 2 2 + g10 θ˜1 θ˙ˆ 1 d1 − z1

(6.18)

Substituting (6.14), (6.15), and (6.16) into (6.18), we obtain V˙1 =

1 z1W1∗ T S1 (Z1 ) + 2 d1 − z21 + +

1

z1 ξ1 (Z1 ) d12 − z21

1

˜ ˙ˆ

z1 g1 (x1 )x2 + g10θ1 θ 1 d12 − z21 Z t 1 d12 − z21

z1 g1 (x1 )ε2

0

e2 d τ

(6.19)

By using Young’s inequality, we can have the fact that 1 z1W1∗ T S1 (Z1 ) 2 d1 − z21

1

z1 ξ1 (Z1 ) ≤ d12 − z21

≤

1 2 1 1 a + 2 1 2a21 (d 2 − z2 )2 1

1

×z21W¯ 12 kS1 (Z1 )k2 1

2 2(d12 − z21 )

g10 z21 +

1 ¯2 ξ 2g10 1

(6.20)

(6.21)

where a1 > 0 is a design constant. RDefine the virtual error as e2 = x2 − α1 and the generalized error as z2 = e2 + ε2 0t e2 d τ , where ε2 is a positive design parameter and α1 is the intermediate control (virtual control) designed as the following PI-like form

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 73 — #87

✐

✐

6.3 PI-like Control Design

73

α1 = −(kP1 + ∆ kP1 (·))e1 − (kI1 + ∆ kI1 (·))

Z t 0

e1 d τ

(6.22)

where kP1 and kI1 are the constant components which are expressed as kP1 = λ1 and kI1 = λ1 ε1 with design constants λ1 > 0 and ε1 > 0. ∆ kP1 and ∆ kI1 are the timevarying parts, which are respectively expressed as   1 1 1 ∆ kP1 (·) = 2 2 + 2 θˆ1 kS1 (Z1 )k2 (6.23) d1 − z1 2 2a1

∆ kI1 (·) = ε1 ∆ kP1 (·)

(6.24)

with θˆ1 being the parameter estimation. Then, we can get 1 1 z1 g1 (x1 )x2 + 2 2 z1 g1 (x1 )ε2 2 2 d1 − z1 d1 − z1 1 = 2 2 z1 g1 (x1 )(α1 + z2 ) d1 − z1 1 1 1 ≤ − 2 2 λ1 g10 z21 − g10 z21 2 2 (d − z2 )2 d1 − z1 1

Z t 0

e2 d τ

1

1 1 g10 z21 θˆ1 kS1 (Z1 )k2 − 2 2a1 (d 2 − z2 )2 1

1

1 + 2 2 z1 g1 (x1 )z2 d1 − z1

(6.25)

Substituting (6.20), (6.21), and (6.25) into (6.19), we can further obtain 1 1 1 ¯2 ξ + z1 g1 (x1 )z2 V˙1 ≤ a21 + 2 2g10 1 d12 − z21 " # 1 1 2 ˙ 2 −θ˜1 g10 z kS1 (Z1 )k − θˆ 1 2a21 (d 2 − z2 )2 1 1 1 1 − 2 2 λ1 g10 z21 d1 − z1 where the coupling term

1 z g (x )z d12 −z21 1 1 1 2

(6.26)

will be cancelled in the following step by

selecting the appropriate controller. Step i (2 ≤ i ≤ n −R 1): Define the virtual error as ei = xi − αi−1 and the generalized error as zi = ei + εi 0t ei d τ where εi is a positive design parameter. Then the time derivative of zi is

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 74 — #88

✐

✐

74

6 Adaptive PI Control for Strict Feedback Systems

z˙i = e˙i + εi ei = fi (x¯i ) + gi(x¯i )xi+1 − α˙ i−1 +εi (xi − αi−1 )

(6.27)

where α˙ i−1 can be derived as i−1

α˙ i−1 =

 ∂ αi−1  f j (x¯ j ) + g j (x¯ j )x j+1 j=1 ∂ x j

∑

i−1

+∑

∂ αi−1

( j) j=0 ∂ yd

( j+1)

yd

i−1

∂ αi−1 ˙ˆ θj ˆ j=1 ∂ θ j

+∑

(6.28)

The unknown function is defined as

φi (Zi ) = fi (x¯i ) + εi (xi − αi−1 ) − gi(xi )εi+1 −α˙ i−1 +

Z t 0

ei+1 d τ

di2 − z2i gi−1 (x¯i−1 )zi−1 2 di−1 − z2i−1

(6.29)

From (6.27) and (6.28) we get φi (Zi ) is a function of x1 , · · · , xi , yd , y˙d , · · · , h iT (i) , θˆi−1 , so that, Zi = x1 , · · · , xi , yd , y˙d , · · · y , θˆ1 , · · · , θˆi−1 ∈ Ωi . It is

(i) yd , θˆ1 , · · ·

d

mentioned before that xn , n = 1, . . . , i is constrained in the compact sets |xn | < cn , ( j) yd (t) and yd (t), j = 1, . . . , i are respectively constrained in the compact sets ( j) |yd (t)| ≤ A0 < c1 and yd (t) ≤ A j , and θˆm , m = 1, . . . , i − 1 is also bounded since ¯2 ¯ they’re the estimations of θi = g−1 i0 Wi where Wi is a constant, so it is clear to see that the training input vector Zi of the NN remains in a compact set; thus it is appropriate to use the NNs to approximate the unknown function φi (Zi ) as

φi (Zi ) = Wi∗ T Si (Zi ) + ξi (Zi )

(6.30)

where Si (Zi ) ∈ Rmi is the Gaussian basis function vector with mi > 1 being the NN node number, Wi∗ denotes the ideal weight vector, ξi (Zi ) denotes the approximation error, and |Wi∗ | ≤ W¯ i , |ξi (Zi )| ≤ ξ¯i with constants W¯ i , ξ¯i > 0. Choosing the following Lyapunov function candidate 1 d2 1 Vi = Vi−1 + log 2 i 2 + gi0 θ˜i2 2 2 di − zi

(6.31)

¯2 where θ˜i = θˆi − θi and θˆi > 0 is the estimation of θi = g−1 i0 Wi . Differentiating Vi with respect to time yields V˙i = V˙i−1 +

zi z˙i + gi0 θ˜i θ˙ˆ i di2 − z2i

(6.32)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 75 — #89

✐

✐

6.3 PI-like Control Design

75

Substituting (6.27), (6.29), and (6.30) into (6.32), we obtain 1 1 ziWi∗ T Si (Zi ) + 2 2 zi ξi (Zi ) + gi0θ˜i θ˙ˆ i di2 − z2i di − zi Z t 1 1 + 2 2 zi gi (x¯i )xi+1 + 2 2 zi gi (xi )εi+1 ei+1 d τ di − zi di − zi 0 1 − 2 zi−1 gi−1 (x¯i−1 )zi di−1 − z2i−1

V˙i = V˙i−1 +

(6.33)

Using Young’s inequality, we immediately get 1 ziWi∗ T Si (Zi ) 2 di − z2i

1

zi ξi (Zi ) ≤ di2 − z2i

≤

1 1 2 1 ai + 2 2 2ai (d 2 − z2 )2

i i 2 2 ¯2 ×zi Wi kSi (Zi )k

1

g z2 + 2 i0 i 2 2 2(di − zi )

1 ¯2 ξ 2gi0 i

(6.34)

(6.35)

where ai > 0 is a design constant. The virtual error is defined as ei+1 = xi+1 − αi where αi is the intermediate control designed as the following PI form

αi = −(kPi + ∆ kPi (·))ei − (kIi + ∆ kIi (·))

Z t 0

ei d τ

(6.36)

where the corresponding PI gains are given as follows: The constant components are kPi = λi and kIi = λi εi where λi > 0, εi > 0 are design constants, the time-varying parts are ∆ kPi and ∆ kIi , which are computed as   1 ˆ 1 1 2 + ∆ kPi (·) = 2 2 θi kSi (Zi )k (6.37) di − zi 2 2a2i

∆ kIi (·) = εi ∆ kPi (·)

(6.38)

with θˆi being the parameter estimation which has been defined before. Then, we can further derive that

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 76 — #90

✐

✐

76

6 Adaptive PI Control for Strict Feedback Systems

Z

t 1 1 z g ( x ¯ )x + z g (x ) ε ei+1 d τ i i i i i i i+1 i+1 di2 − z2i di2 − z2i 0 1 = 2 2 zi gi (x¯i )(αi + zi+1 ) di − zi 1 1 1 ≤ − 2 2 λi gi0 z2i − 2 gi0 z2i θˆi kSi (Zi )k2 di − zi 2ai (d 2 − z2 )2 i

1

− g z2 + 2 i0 i 2(di2 − z2i )

i

1 zi gi (x¯i )zi+1 2 di − z2i

(6.39)

Substituting (6.34), (6.35), and (6.39) into (6.33), V˙i can be further bounded as 1 1 1 ¯2 ξ + 2 2 zi gi (x¯i )zi+1 V˙i ≤ V˙i−1 + a2i + 2 2gi0 i di − zi " # 1 1 2 ˙ 2 −θ˜i gi0 z kSi (Zi )k − θˆ i 2a2i (d 2 − z2 )2 i i i 1 1 − 2 zi−1 gi−1 (x¯i−1 )zi − 2 2 λi gi0 z2i 2 di−1 − zi−1 di − zi

(6.40)

As in step n − 1, it has already been obtained that 1 i−1 1 V˙i−1 ≤ ∑ a2j + 2 j=1 2

i−1

1

∑ g j0 ξ¯ j2

j=1

1 + 2 zi−1 gi−1 (x¯i−1 )zi di−1 − z2i−1 " # i−1

2 ˙ 1 1 2 − ∑ θ˜ j g j0 z S j (Z j ) − θˆ j 2a2j (d 2 − z2 )2 j j=1 j j i−1

−∑

1

2 2 j=1 d j − z j

λ j g j0 z2j

(6.41)

Then, (6.40) can be rewritten as 1 V˙i ≤ 2

i

i

1

1

1

∑ a2j + 2 ∑ g j0 ξ¯ j2 + d 2 − z2 zi gi (x¯i )zi+1

j=1 i

− ∑ θ˜ j g j0 j=1 i

−∑

"

1

2 2 j=1 d j − z j

where the coupling term

i

j=1

2 1 1 z2j S j (Z j ) − θ˙ˆ j 2 2 2a j (d 2 − z2 ) j

j

(6.42)

i

#

λ j g j0 z2j

1 z g (x )z di2 −z2i i i i i+1

will be compensated in the following step.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 77 — #91

✐

✐

6.3 PI-like Control Design

77

Step n: Define the virtual error as en = xn − αn−1 and the generalized error as R zn = en + εn 0t en d τ where εn is a positive design parameter. Thus the time derivative of zn is z˙n = e˙n + εn en = fn (x¯n ) + gn(x¯n )D(u) − α˙ n−1 + εn (xn − αn−1 ) = fn (x¯n ) + gn(x¯n )bu + gn(x¯n )η (u) −α˙ n−1 + εn (xn − αn−1 )

(6.43)

and α˙ n−1 can be derived as n−1

α˙ n−1 =

 ∂ αn−1  f j (x¯ j ) + g j (x¯ j )x j+1 j=1 ∂ x j

∑

n−1

+∑

∂ αn−1

( j) j=0 ∂ yd

( j+1)

yd

n−1

∂ αn−1 ˙ˆ θj ˆ j=1 ∂ θ j

+∑

(6.44)

Choosing the following symmetric BLF 1 1 d2 Vn = Vn−1 + log 2 n 2 + gn0 b0 θ˜n2 2 dn − zn 2

(6.45)

where θ˜n = θˆn − θn and θˆn > 0 is the estimation of θn = (gn0 b0 )−1W¯ n2 . Differentiating Vn with respect to time yields zn z˙n V˙n = V˙n−1 + 2 2 + gn0b0 θ˜n θ˙ˆ n dn − zn

(6.46)

Define the unknown function

φn (Zn ) = fn (x¯n ) − α˙ n−1 + εn (xn − αn−1 ) d 2 − z2n + 2 n gn−1 (x¯n−1 )zn−1 dn−1 − z2n−1 1 + 2 2 g2 ρ 2 zn 2an(dn − z2n ) n0

(6.47)

where ρ is the bound of η (u) and an > 0 is a design constant. As we can see from (6.43) and (6.44), φn (Zn ) is a function of (n) x1 , · · · , xn , yd , y˙d , · · · , yd , θˆ1 , · · · , θˆn−1 , so that, h iT (n) ˆ ˆ Zn = x1 , · · · , xn , yd , y˙d , · · · yd , θ1 , · · · , θn−1 ∈ Ωn . Similar to step i, the training input vector Zn of the NN also remains in a compact set, thus the NNs can be utilized to approximate the unknown function φn (Zn ) as

φn (Zn ) = Wn∗ T Sn (Zn ) + ξn (Zn )

(6.48)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 78 — #92

✐

✐

78

6 Adaptive PI Control for Strict Feedback Systems

where Sn (Zn ) ∈ Rmn is the Gaussian basis function vector with mn > 1 being the NN node number, Wn∗ denotes the ideal weight vector, ξn (Zn ) denotes the approximation error, and |Wn∗ | ≤ W¯ n , |ξn (Zn )| ≤ ξ¯n with constants W¯ n , ξ¯n > 0. From (6.43), (6.46), (6.47), and (6.48), we can further get 1 1 znWn∗ T Sn (Zn ) + 2 2 zn ξn (Zn ) dn2 − z2n dn − zn 1 1 + 2 2 zn gn (x¯n )bu + 2 2 zn gn (x¯n )η (u) dn − zn dn − zn 1 +gn0 b0 θ˜n θ˙ˆ n − 2 zn−1 gn−1 (x¯n−1 )zn dn−1 − z2n−1 1 g2n0 ρ 2 z2n − 2 2 2an (dn − z2n )2

V˙n = V˙n−1 +

(6.49)

Using Young’s inequality, we can obtain 1 1 1 1 znWn∗ T Sn (Zn ) ≤ a2n + 2 dn2 − z2n 2 2an (dn2 − z2n )2 ×z2nW¯ n2 kSn (Zn )k2

1 1 ¯2 1 zn ξn (Zn ) ≤ g b z2 + ξn 2 n0 0 n 2 2 dn2 − z2n 2g n0 b0 2(dn − zn ) 1

zn gn (x¯n )η (u) ≤ dn2 − z2n

1 2 1 1 g2 ρ 2 z2 an + 2 2 2an (dn2 − z2n )2 n0 n

(6.50)

(6.51)

(6.52)

We design the actual neuro-adaptive PI controller u as u = −(kPn + ∆ kPn (·))en − (kIn + ∆ kIn (·))

Z t 0

en d τ

(6.53)

where the PI parameters are computed as follows: The constant parts kPn and kIn are designed as kPn = λn and kIn = λn εn with design constants λn > 0 and εn > 0. The time-varying parts ∆ kPn and ∆ kIn are chosen as   1 1 1 ∆ kPn (·) = 2 2 + 2 θˆn kSn (Zn )k2 (6.54) dn − zn 2 2an

∆ kIn (·) = εn ∆ kPn (·)

(6.55)

with θˆn being the parameter estimation and its updating law will be given later. Then, we have

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 79 — #93

✐

✐

6.3 PI-like Control Design

79

1 zn gn (x¯n )bu dn2 − z2n 1 1 ≤ − 2 2 λn b0 gn0 z2n − g b z2 2 n0 0 n 2 2 dn − zn 2(dn − zn ) −

1 1 gn0 b0 z2n θˆn kSn (Zn )k2 2a2n (dn2 − z2n )2

(6.56)

Substituting (6.50), (6.51), (6.52), and (6.56) into (6.49), we can further derive that 1 ¯2 1 V˙n ≤ V˙n−1 + a2n + ξn − 2 2 λn gn0 b0 z2n 2gn0 b0 dn − zn " # 1 1 2 ˙ z2 kSn (Zn )k − θˆ n −θ˜n gn0 b0 2a2n (dn2 − z2n )2 n 1 − 2 zn−1 gn−1 (x¯n−1 )zn dn−1 − z2n−1

(6.57)

At this step we can design the updating law for θˆ j , j = 1, . . . , n as

2 1 1 θ˙ˆ j = −γ j θˆ j + 2 z2 S j (Z j ) 2a j (d 2 − z2 )2 j j j

(6.58)

where γ j > 0, j = 1, . . . , n is a design constant. Using (6.42) with i = n − 1 and (6.58), we can rewrite (6.57) as 1 n−1 1 ¯2 1 n−1 1 ¯ 2 V˙n ≤ ∑ a2j + a2n + ∑ ξj + ξ 2 j=1 2 j=1 g j0 2gn0 b0 n n−1

−∑

1

2 2 j=1 d j − z j

n−1

λ j g j0 z2j − ∑ γ j g j0 θ˜ j θˆ j j=1

1 − 2 2 λn gn0 b0 z2n − γn gn0 b0 θ˜n θˆn dn − zn

(6.59)

By employing Young’s inequality, we have −γ j g j0 θ˜ j θˆ j = −γ j g j0 θ˜ j2 − γ j g j0 θ˜ j θ j 1 1 ≤ − γ j g j0 θ˜ j2 + γ j g j0 θ j2 2 2 −γn gn0 b0 θ˜n θˆn = −γu gn0 b0 θ˜n2 − γu gn0 b0 θ˜n θn 1 1 ≤ − γn gn0 b0 θ˜n2 + γn gn0 b0 θn2 2 2

(6.60)

(6.61)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 80 — #94

✐

✐

80

6 Adaptive PI Control for Strict Feedback Systems

Then, using (6.10), (6.60), and (6.61), (6.59) can be rewritten as 1 n−1 1 n−1 1 ¯ 2 1 ¯2 ξj + ξ V˙n ≤ ∑ a2j + a2n + ∑ 2 j=1 2 j=1 g j0 2gn0 b0 n +

1 n−1 1 1 n−1 γ j g j0 θ j2 + γn gn0 b0 θn2 − ∑ γ j g j0 θ˜ j2 ∑ 2 j=1 2 2 j=1

n−1 d 2j 1 − γn gn0 b0 θ˜n2 − ∑ λ j g j0 log 2 2 2 dj − zj j=1

−λn gn0 b0 log

dn2 dn2 − z2n

(6.62)

Let  β1 = min 2g j0 λ j , 2gn0 λn b0 , γ j , γn

β2 =

(6.63)

1 n−1 2 1 ¯2 1 n−1 1 ¯ 2 a j + a2n + ∑ ξ + ξ ∑ 2 j=1 2 j=1 g j0 j 2gn0b0 n +

1 1 n−1 γ j g j0 θ j2 + γn gn0 b0 θn2 ∑ 2 j=1 2

(6.64)

together with (6.45) and (6.62) , we can finally obtain V˙n ≤ −β1Vn + β2

(6.65)

Remark 6.2 The proposed control schemes (6.22), (6.36), (6.53) are all of PI structure with the proportional component and the integral component. It is worth noting that the P-gain and the I-gain are composed of two parts, respectively: one is the constant part (kPi and kIi , i = 1, · · · , n) and the other one is the time-varying part (∆ kPi and ∆ kIi , i = 1, · · · , n). The time-varying parts of the proposed neuro-adaptive control are self-tuned via the laws (6.23), (6.24), (6.37), (6.38), (6.54), and (6.55). Remark 6.3 Although several unknown bounds such as gi0 , ξ¯i etc. are used in the stability analysis, these parameters are not involved in the control scheme as seen from (6.22), (6.36), (6.53), thus there is no need for analytical estimation of these parameters, making the proposed neuro-adaptive algorithm easy to design and simple to implement. Theorem 6.1. Consider the nonlinear system (6.1) with unknown functions and both actuation characteristics (6.2) under Assumptions 6.1 and 6.2; if the proposed neuroadaptive laws (6.22), (6.36), (6.53) and the updating law (6.58) are applied, then all the internal signals are bounded and the full-state constraints are not violated.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 81 — #95

✐

✐

6.4 Illustrative Examples

81

Proof. From (6.45) and the inequality (6.65), we can see the boundedness of Vn , thus d2 |zi | remains in the set |zi | < di , i = 1, · · · , n; also, it holds that log 2 j 2 and θ˜ j , j = d j −z j

1, . . . , n are bounded. Since θˆi = θ˜i + θi , it is seen that θˆi is bounded. Moreover, according to Lemma 3, the boundedness of zn , i = 1, · · · , n ensures the boundedness of en , i = 1, · · · , n. From (6.36), (6.37), and (6.38), it is seen that αi , i = 1, . . . , n − 1 (i) is a function of θˆi , xi , yd , yd , i = 1, · · · , n − 1 which are all bounded; hence, αi is bounded. Similarly, from (6.53), (6.54), and (6.55), we find that u is a function of (i) θˆi , xi , yd , yd , i = 1, · · · , n which implies that u is bounded. Since e1 = x1 − yd and |yd (t)| ≤ A0 , we have the bound of x1 is |x1 | ≤ |e1 | + |yd | < 2d1 + A0 ≤ c1 which will be further proved in the Appendix. With the boundedness of αi , i = 2, . . . , n − 1 and ei = xi − αi−1 , i = 1, · · · , n, we can further prove that |xi | ≤ ci . Therefore, all the internal signals in the system are bounded.

6.4 Illustrative Examples To verify the effectiveness of the proposed neuro-adaptive control scheme, we conduct numerical simulations. In order to show the advantages of the proposed control scheme, a comparison is made between the proposed neuro-adaptive control scheme and the traditional PI control. Consider the following nonlinear system:   x˙1 = x21 sin(x1 ) + (2 + x21)x2 x˙ = x1 x22 + (3 + cos(x1 x2 ))D(u) (6.66)  2 y = x1

where x1 and x2 are to be constrained in |x1 | ≤ 1.5 and |x2 | ≤ 1, u is the control input and y is the output. The initial trajectory conditions are: x1 (0) = 0.2, x2 (0) = −1. The parameters of the actuation characteristics function D(u) are chosen as: l1 = 1, l2 = 1, umb1 = 1.8, umb2 = 1, σ = 1.7, σ = −0.8, b1 (t) = |sin(t)| and b2 (t) = |0.8 sin(t)|. 0.25

0.6

x1(proposed)

e1(proposed)

x1(PI)

0.4

0.15

Tracking error e1

Tr acking pro cess

0.2

0

−0.2

0.1

0.05

−0.4

0

−0.6

−0.05

−0.8

e1(PI)

0.2

yd

0

5

10

15 Time(sec)

20

25

Fig. 6.2 System output and reference signal yd .

30

−0.1

0

5

10

15 Time(sec)

20

25

30

Fig. 6.3 The tracking error e.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 82 — #96

✐

✐

82

6 Adaptive PI Control for Strict Feedback Systems

The simulation purpose is to test if the developed neuro-adaptive control is able to ensure the output y tracking the reference signal yd (t) = 0.5 sin(t) and all the signals in the closed-loop system being bounded. The design parameters are chosen as a1 = 0.3, a2 = 0.5, γ1 = 10, γ2 = 5, λ1 = 4, λ2 = 5, ε1 = 0.01 and ε2 = 0.01. The initial values of the adaptive laws are θˆ1 (0) = 0.1 and θˆ2 (0) = 0.1. The bounds of z1 and z2 are set as d1 = 0.5 and d2 = 2. The node numbers of neurons are m1 = m2 = 20, the centers of the NNs are µ1 = µ2 = 0, and the width of the Gaussian functions are chosen as ω12 = ω22 = 4. The control gains of the traditional PI control, via a lengthy trial and error process, are determined as P = 5, I = 0.01. The simulation results are shown in Figs. 6.2—6.10. Fig. 6.2 shows the system output tracking progress, which confirms that indeed the system output closely tracks the desired trajectory. Fig. 6.3 is the corresponding output tracking error e1 . Fig. 6.4 indicates that the PI gains are self-tuning continuously. The boundness of the controller u and the adaptation laws are shown in Figs. 6.5 and 6.7, respectively. The control with both saturation and dead-zone D(u) is shown in Fig. 6.6. Fig. 6.8 shows the phase portraits of (x1 , x2 ), from which we can see that x1 and x2 start from the initial state [0.2, −1] and then constrain in a range within |x1 | ≤ 1.5 and |x2 | ≤ 1 which implies that the bounds for x1 , x2 are not overstepped; thus the full state constraints are not violate. To further test the robustness and adaptivity of the

20

2

P gain I gain

18

1.5

16

Control input u

14

PI gains

12 10 8

1

0.5

0

6 4

−0.5

2 0

0

5

10

15 Time(sec)

20

25

Fig. 6.4 Adaptive P gain and adaptive I gain.

30

−1

0

5

10

15 Time(sec)

20

25

30

Fig. 6.5 Control input u.

proposed control method, we add additional disturbance 2sin(2t) to the system and compare the control performance with traditional PI control with fixed PI gains. As can be seen from Figs. 6.9 and 6.10, the proposed neuro-adaptive PI control performs much better than the traditional PI control, mainly because of the adaptive gain-tuning capability of the proposed method.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 83 — #97

✐

✐

6.4 Illustrative Examples

83

2

0.18 θˆ1 θˆ2

0.16 1.5

0.12 Estimation of θ

C o ntrol input v

0.14

input saturation

2 1

1

dead−zone

0 0.5

0

0.2

0.4

0.6

0.1 0.08 0.06 0.04

0

0.02 −0.5

0

5

10

15 Time(sec)

20

25

0

30

0

5

10

15 Time(sec)

20

25

30

Fig. 6.7 Estimations of θˆ1 , θˆ2 .

Fig. 6.6 The saturation control input D(u) with dead-zone.

0.6

0.5

x1(proposed) x1(PI)

0.4

yd

Tra cking pro cess

0.2

2

x (t)

0

0

−0.2

−0.5

−0.4

−0.6

−1 −0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x (t) 1

Fig. 6.8 The phase portraits of (x1 , x2 ).

−0.8

0

5

10

15 Time(sec)

20

25

30

Fig. 6.9 Tracking error e1 under disturbance.

0.25 e1(proposed) e (PI) 1

0.2

Tracking error e1

0.15

0.1

0.05

0

−0.05

−0.1

0

5

10

15 Time(sec)

20

25

30

Fig. 6.10 Tracking error e1 under disturbance.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 84 — #98

✐

✐

84

6 Adaptive PI Control for Strict Feedback Systems

6.5 Conclusion In this chapter, a neuro-adaptive PI control is introduced for nonlinear systems with full-state constraints and input dead-zone by using the barrier Lyapunov function. The proposed control is of PI structure with PI gains being self-tuned continuously and adaptively through the established analytical algorithms. Simulation results have shown satisfactory tracking performance of the developed control scheme. Extension of the proposed method to more general MIMO strict feedback nonlinear systems represents an interesting topic for future research.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 85 — #99

✐

✐

Chapter 7

Adaptive PID Control for MIMO Nonlinear Systems

In this chapter we show that the structurally simple and computationally inexpensive PID control, popular with SISO linear time-invariant systems, can be generalized and extended to control nonlinear MIMO systems with nonparametric uncertainties and actuation failures. By utilizing the Nussbaum-type function and the matrix decomposition technique, non-square systems with unknown control direction are also considered. Furthermore, with the proposed analytic algorithms for adaptively tuning PID gains, the resultant PID control can be made robust, adaptive and fault-tolerant, and applicable to nonlinear systems with non-vanishing uncertainties and unexpected actuation faults.

7.1 Introduction The ubiquitous PID controller has continued to be the most widely used process control technique for many decades. Our interest in revisiting PID control is largely motivated by the fact that the PID controller is a key part of almost every control loop and has significantly untapped capability, offering the simplest and yet most efficient solution to many real-world control problems. For a PID controller to function satisfactorily for the vast majority of industrial control loops, its PID gains have to be properly designed and tuned. The widespread application of PID has stimulated and sustained the development of various PID tuning techniques as briefly summarized in the following. 1) Review and motivation: The key to PID design is its control parameter determination. Early methods for tuning PID gains, surveyed by [5], were characterized with constant gains and oriented for set point regulation, most of which, such as analytical methods [110] and optimization-based methods [111] (to name a few), impose particular conditions regarding the plant models. Alternative tuning methods, which do not require a priori knowledge of the system under control are iterative feedback tuning (IFT) [69] and extremum seeking (ES) [112]. These strategies are proven effective only for certain linear time-invariant systems. 85

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 86 — #100

✐

✐

86

7 Adaptive PID Control for MIMO Nonlinear Systems

As most practical industrial systems are time-varying and nonlinear, exploration of the applicability of PID control to nonlinear systems has received considerable attention. The authors in [46] first proved that PI/PID controllers can be tuned for a class of nonlinear systems. In [70], a self-tuning PID control was developed for a class of continuous-time SISO nonlinear systems based on the Lyapunov approach. Adaptive robust PID controller design based on a sliding mode was pursued in [71]. Effort on applying PID to MIMO nonlinear systems, although much more challenging due to intrinsic loop interactions and couplings, has also been made. For instance, by using singular perturbation technique, the authors in [113] investigated the PID-based control of certain nonlinear MIMO systems. Auto-tuning fuzzy PID control methods for a class of MIMO systems was investigated in [9] and [10]. The works in [114] and [115] developed self-tuning/learning PID controllers based on adaptive neural network (NN) for multivariable nonlinear systems. It is noted that, however, when applying neural/fuzzy based PID control to nonlinear MIMO systems, control engineers are confronted with new issues and challenges, such as difficulty in selecting the vast amount of the required neural/fuzzy parameters, acquisition typically of local results, and the notorious loss of modeling controllability issue, etc. [83]. Thus far, although a vast amount of research results are published in the literature, there still lacks a systematic means for determining the PID gains properly. As PID regulators are the backbone of most industrial control systems, the problem of determining their parameters is thus of great importance, motivating our effort to develop a new method for constructing PID control and endow it with an auto-tuning mechanism capable of computing the “correct” PID gains automatically when connected to the field. 2) Contributions of this chapter: In this chapter we tackle the problem of tuning PID gains from a different angle, leading to an analytical and user-friendly solution for equipping PID control with auto-tuning gains plus robust adaptive and faulttolerant capabilities. The contribution of this chapter is fourfold. • Firstly, new PID control design ensuring stable tracking control for square and non-square MIMO nonlinear systems is presented: Traditional PID control is characterized by constant PID gains and is oriented for set point regulation. Thus seldom are works satisfactory for general nonlinear systems with uncertain dynamics and unpredictable disturbances. In this chapter, new PID tracking control algorithms applicable to both square and non-square MIMO nonlinear systems are developed. The appeal of the proposed PID control is that it has the capability of automatically tuning its gains to compensate system uncertainties, reject external disturbances, and accommodate actuation failures. • Secondly, rigorous stability proof for nonlinear systems with PID in the loop is provided. Stability has always been the major concern with traditional PID control due to the lack of a systematic procedure for determining the “correct” stabilityensured PID gains for a given system. By borrowing the concept of virtual parameter proposed in [116] and the artfully chosen Lyapunov function candidate, a PID-like tracking strategy is developed for the considered nonlinear systems with rigorous proof of stability. It turns out that the PID gains herein are not invariant, but rather

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 87 — #101

✐

✐

7.2 Problem Formulation and Error Dynamics

87

consistently self-adjusted according to the analytic algorithms derived from stability consideration. • Thirdly, the proposed PID control is able to tolerate unexpected actuation faults. Because actuation failures partial
loss of effectiveness (PLOE) [117, 118] or total loss of effectiveness (TLOE) [17] might occur during system operation, it is thus crucial to enable PID control with adaptive and fault-tolerant capabilities. In [114], an auto-tuning PID controller is proposed for a multivariable process with additive actuation faults, which, however, can be treated as additional disturbances, thus can be easily accommodated by any NN-based or robust method. In this chapter both additive and ramping faults [12, 16, 119] are addressed by the proposed PID control without any fault detection and diagnosis (FDD) or fault detection and identification (FDI) unit to monitor whether an actuation failure occurs or not [18, 114, 120, 121]. Instead, the fault impact is automatically and adaptively accommodated by the developed PID algorithms. • Fourthly, the proposed PID control does not require a priori knowledge of the control direction. When there is no a priori knowledge about the control direction, PID control of such systems becomes much more difficult. The first solution for the unknown control direction problem was developed in [122], where the Nussbaum-type gain was originally proposed. In [123], an indirect adaptive control scheme was developed for a class of uncertain MIMO nonlinear systems with nonsymmetric control gain matrix and unknown control direction. However, only square control gain matrix is considered and the control algorithms are fairly complicated. In this chapter, a structurally simple PID-like control for both square and non-square uncertain systems with unknown control direction is developed. One might question that since there exists a rich collection of results in the literature on tuning PID controllers, what is the rationale behind exploring another one? For this question we have at least three justifications. First, PID controller is by far the most widely used control algorithm in the process industry, thus improvements in tuning PID controllers will have a significant practical impact. Second, the useful but simple rules and insights presented in this chapter may contribute to a significantly improved understanding into how the controller should be tuned. Third, effective and systematic PID tuning is not available for nonlinear systems with modeling uncertainties and actuation failures. The method presented in this chapter provides a fully user-favorable self-tuning solution that enables PID control with robust, adaptive, and fault-tolerant capabilities.

7.2 Problem Formulation and Error Dynamics Consider the following MIMO nth order nonlinear system:

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 88 — #102

✐

✐

88

7 Adaptive PID Control for MIMO Nonlinear Systems

x˙i =xi+1 ,

i = 1, . . . , n − 1

x˙n =F(x) + G(x,t)ua + D(x,t) y =x1

(7.1)

where xi = [xi1 , . . . , xim ]T ∈ Rm , i = 1, . . . , n and x = [xT1 , . . . , xTn ]T ∈ Rmn are state vectors; ua = [ua1 , . . . , ual ]T ∈ Rl is the input vector of the system; y = [y1 , . . . , ym ] ∈ Rm is the output vector; F(x) = [ f1 (·), . . . , fm (·)] ∈ Rm represents a smooth nonlinear function vector, G(x,t) ∈ Rm×l (m ≤ l) is a time-varying matrix and D(x,t) ∈ Rm denotes all the other modeling uncertainties and external disturbances. As unanticipated actuation faults are inevitable for long-term operation, the following abnormal actuator input-output model [117, 118] is explicitly considered as part of the system model in conjunction with (7.1). ua (t) = ρ (t)u + ε (t)

(7.2)

where u = [u1 , . . . , ul ] ∈ Rl is the input vector of the actuator and ε (t) = [ε1 , . . . , εl ] ∈ Rl represents the uncontrollable portion of actuation input, ρ = diag{ρ j } ∈ Rl×l , j = 1, . . . , l, is a diagonal matrix with ρ j being the “healthy indicator” reflecting the effectiveness of the jth actuator. The PLOE actuation faults together with bounded uncontrollable actuation faults are considered in this chapter (i.e., 0 < ρ j < 1 and kε k ≤ ε¯ < ∞).

7.2.1 Error Dynamics (i−1)

Define ei = x1 − y∗ (i−1) , i = 1, . . . , n. For later technical development, the following filtered error is introduced. z(t) = λn−2 e1 + . . . + λ1en−2 + en−1

(7.3)

z˙(t) = λn−2 e2 + . . . + λ1 en−1 + en

(7.4)

thus with proper choice of λi , i = 1, . . . , n − 2, it holds that λn−2 + λn−3s + . . . + λ1 sn−3 + sn−2 is Hurwitz. Thus it is readily shown that the boundedness of z and z˙ guarantees the boundedness of the output tracking error e1 and its derivative up to n − 1th [46]. Therefore, we will focus on designing PID control to drive z and z˙ to a small residual set containing the origin. Based upon (7.2), (7.3), and (7.4), it is derived that z¨ = F(·) + G(·)(ρ u + ε ) + D(·) − y∗(n) + λn−2e3 + . . . + λ1 en

(7.5)

The special attention of this chapter is on the development of a PID-like control strategy with robust adaptive and fault tolerant capabilities so that the tracking objective is achieved without using F(·), G(·), D(·), ρ (·), and ε (·) explicitly. To this end, the following assumptions are made.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 89 — #103

✐

✐

7.2 Problem Formulation and Error Dynamics

89

∗ th Assumption 7.1 The desired tracking trajectory

∗ (n) y and its derivative up to n − 1

≤ y¯ < ∞ and the state vector x are order are known and bounded. Besides, y available for control design.   Assumption 7.2 0 < ρ j0 ≤ ρ j , max |ρ˙ j | ≤ cρ < ∞ and max |ε˙ j | ≤ cε < ∞ for j = 1, . . . , l, where ρ j0 , cρ and cε are some unknown non-negative constants.

Assumption 7.3 There exists a non-negative constant a f and non-negative scalar function ϕ f (x), such that kF(x) + G(x,t)ε + D(x,t)k ≤ a f ϕ f (x)

(7.6)

Remark 7.1 Assumption 7.1 is commonly imposed in most existing works; see [83] for example. Assumption 7.2 imposes a bounded condition on the failure variation rate, which is necessary for a feasible fault accommodation solution to be developed. Still, it is much more complex than that in [114] as the faults hinder the controller design and stability analysis, which cannot be easily formulated as constraints. In addition, there is no need to diagnose the time and pattern of the fault occurrence. Assumption 7.3 is related to the extraction of the deep-rooted information from the nonlinearities of the system, which can be readily fulfilled for any practical system with only crude model information [116]. It is noted that (7.6) is sufficient to guaranteed continuous and bounded control. In order to achieve continuous control with bounded variation, one can apply “soft 2-norm” to ϕ f (x) such that it is derivable with respect to x.

7.2.2 Nussbaum Function A function N(ζ ) can be treated as a Nussbaum-type function if it has the following useful properties [122]: lim sup

s→∞

Z s

N(ζ )d ζ = +∞,

s0

lim inf

s→∞

Z s s0

N(ζ )d ζ = −∞

(7.7)

Throughout this paper, the even Nussbaum function N(ζ ) = ζ 2 cos(ζ ) with the property of N(0) = 0 is considered. Lemma 7.1 [124] Let V (t) and ζ (t) be smooth functions defined on [0,t f ) with V (t) > 0, ∀t ∈ [0,t f ). For any t ∈ [0,t f ), if the following inequality holds: V (t) < c0 + e−c1t

Z t 0


g(τ )N(ζ ) + 1 ζ˙ ec1 τ d τ

(7.8)

where c0 > 0 and c1 > 0 are suitable constants and g(τ ) is a time-varying parameter which takes values in the unknown closed intervals L = [l − , l + ] with 0 ∈ / L, then Rt V (t), ζ (t) and 0 g(τ )N(ζ )ζ˙ ec1 τ d τ must be bounded on [0,t f ).

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 90 — #104

✐

✐

90

7 Adaptive PID Control for MIMO Nonlinear Systems

Lemma 7.2 [123] Let M be n × n symmetric matrix and x¯ ∈ Rn be a nonzero vector, T if α = x¯x¯TMx¯x¯ , then there is at least one eigenvalue of M in the interval (−∞, α ] and at least one in [α , +∞).

7.3 PID-like Control Design and Analysis The proposed PID control is of the form Z t

u = kP0 + κP (·) N(ζ )Λ z + kI0 + κI (·) N(ζ )Λ z(·)d τ | {z } | 0 {z } P

I

(7.9)


dz(·) + kD0 + κD (·) N(ζ )Λ dt } | {z D

where N(ζ ) is the Nussbaum-type function and Λ ∈ Rl×m is a known matrix to be designed. Note that there
are six sets of parameters to be determined in (7.9), three of which kP0 , kI0 and kD0 are constants while the other three κP (·), κI (·), κD (·) are time-varying. If these parameters were selected independently, one would have six degrees of freedom for parameter determination, making the process for tuning the PID gains rather complicated and time-consuming. Here we propose to link these parameters through a common coefficient γ as follows: kP0 =2γ kD0 ,

kI0 = γ 2 kD0 ,

κP (·) =2γκD (·),

(7.10)

κI (·) = γ 2 κD (·)

which allows the PID controller to be expressed as   Z t
dz(·) u = kD0 +κD (·) N(ζ )Λ 2γ z+γ 2 z(·)d τ + dt 0

(7.11)

where γ > 0 is a user-designed parameter such that z2 + 2γ
z + γ 2 is Hurwitz. PID control in (7.11) now has only two gains i.e., kD0 and κD (·) to determine, which is addressed in what follows. To proceed, we define the following generalized error. E(P, I , D) = 2γ z + γ 2

Z t 0

z(·)d τ +

dz(·) dt

(7.12) R

It can be proved that the boundedness of E 1 ensures the boundedness of z, 0t z(·)d τ and dz(·) dt [46]. Thus the original tracking control task now boils down to stabilizing 1 To simplify the subsequent notation, we sometimes use E or E(·) to denote the generalized error E(P, I, D) whenever no confusion arises.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 91 — #105

✐

✐

7.3 PID-like Control Design and Analysis

91

E(·). More specifically, the task is to develop a systematic strategy to determine kD0 > 0 and to tune κD (·) automatically and adaptively, such that E is globally ultimately uniformly bounded (GUUB). To bridge the generalized error (7.12) with the error dynamics (7.5), we take time derivative of (7.12) to get ˙ P, I , D) =G(x,t)ρ (·)u + F(x) + G(x,t)ε + D(x,t) − y∗(n) E(

+ λn−2e3 + · · · + λ1en + 2γ z˙ + γ 2 z = B(x,t)u + H(x,t)

(7.13)

where B(x,t) = G(x,t)ρ (·) and H(x,t) = F(x)+ G(x,t)ε + D(x,t)− y∗ (n) + λn−2 e3 + · · · + λ1 en + 2γ z˙ + γ 2 z. According to Assumptions 7.1 and 7.3, H(x,t) can be upper bounded as kH(x,t)k ≤ a f ϕ f (x)+ y¯ + λn−2 ke3 k + · · ·+ λ1 ken k + 2γ k˙zk + γ 2 kzk ≤ aϕ (·) (7.14) with  a = max a f , y, ¯ λ1 , . . . , λn−2 , 2γ , γ 2

ϕ (·) =ϕ f (x) + ke3 k + · · · + kenk + kzk + k˙zk + 1

(7.15) (7.16)

where a > 0 is an unknown constant and ϕ (x) is a readily computable scalar function, called a “core function” in [116]. Note that we can apply “soft 2-norm” to ϕ (·) such that continuous control with bounded control rate can be achieved. Now we develop a PID-like control to cope with the situation involving square or non-square B(·), respectively.

7.3.1 PID Control for Square Systems Firstly we consider the case that B(·) is square. In this case, the following assumption is needed. Assumption 7.4 The matrix BT (·) + B(·) is either positive or negative definite.
Based on Assumption 7.4, let α (t) = 2E1T E E T BT (·) + B(·) E for any E 6= 0, we can readily conclude from Lemma 7.2 that

λ ≤ λm (t) ≤ α (t) ≤ λM (t) ≤ λ

(7.17)

where λ and λ are some constants, λ
m (t) and λM (t) are the minimum and maximum eigenvalues of matrix 12 BT (·) + B(·) , respectively. Then for any E 6= 0, it holds that
1 T T E B (·) + B(·) E = α1 (t)kEk2 2

(7.18)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 92 — #106

✐

✐

92

7 Adaptive PID Control for MIMO Nonlinear Systems

If E = 0, (7.18) also holds. As BT (·) + B(·) is either positive or negative definite, λm (t) and λM (t) are both positive or both negative, i.e., λm (t) · λM (t) > 0, which implies that the sign of α1 (t) is strictly positive or strictly negative but unknown. Remark 7.2 Since B(·) is completely unavailable, it is nontrivial to get the parameters α1 (t), λ , λ , λm (t) and λM (t) as defined in (7.18); those parameters, although existing, are only used for control analysis and will not be included in the control scheme so that actual estimation or calculation of each of them is not needed as seen later. For this case we employ the PID-like controller as given in (7.9)—(7.11) with

Λ =I, κD (·) =aˆϕ 2 (·)

(7.19)

and the updating laws for ζ and aˆ are given by
ζ˙ = kD0 + κD (·) kEk2

a˙ˆ = −σ0 aˆ + σ1 ϕ 2 (·)kEk2

(7.20) (7.21)

where σ0 and σ1 are some positive design constants, aˆ is estimation of a, I ∈ Rm×m is a unit matrix, and ϕ (·) is the “core function” as defined in (7.16). Theorem 7.1. Consider the MIMO system (7.1) subject to actuation faults (7.2). Under Assumptions 7.1—7.4, if the PID control strategy as governed by (7.9)—(7.11), (7.19)—(7.21) is applied, then i) the control action is continuous and smooth everywhere; ii) all the internal signals are globally ultimately uniformly bounded (GUUB), and iii) globally uniformly and ultimately bounded (GUUB) full-state tracking error is ensured. Proof. Consider the following Lyapunov function candidate 1 2 1 V1 = E T E + a˜ 2 2σ1

(7.22)

where the estimate error is defined as a˜ = a − a. ˆ With (7.11), (7.13), and (7.14), the time derivative of V1 is derived as 1 1 V˙1 =E T E˙ − a˜a˙ˆ = E T H(·) + E T B(·)u − a˜a˙ˆ σ1 σ1
1 ≤aϕ (·)kEk+ kD0 +κD (·) N(ζ )E T B(·)E− a˜a˙ˆ σ1

(7.23)

By using (7.18), it is derived that

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 93 — #107

✐

✐

7.3 PID-like Control Design and Analysis

93



1 1 E T B(·)E = E T B(·)+BT (·) E + E T B(·)−BT (·) E 2 2 | {z } skew symmetric

1 = ET 2

(7.24)


B(·) + BT (·) E = α1 (t)kEk2

Integrating (7.21) and (7.24) into (7.23) yields


σ0 V˙1 ≤ aϕ (·)kEk + kD0 +κD (·) N(ζ )α1 (t)kEk2 − a˜ϕ 2 (·)kEk2 + a˜aˆ σ1

By using the following facts aϕ (·)kEk ≤ aϕ 2 (·)kEk2 + 4a and V˙1 can be further bounded as

σ0 σ0 2 σ0 2 σ1 a˜aˆ ≤ 2σ1 a − 2σ1 a˜ ,



V˙1 ≤ kD0 + aˆϕ 2 (·) kEk2 + kD0 + κD (·) N(ζ )α1 (t)kEk2 − kD0 kEk2 σ0 2 σ0 2 a − a˜ + a + 2σ1 2σ1 4 By substituting (7.19) and (7.20) into (7.26), it is deduced that
V˙1 ≤ α1 (t)N(ζ ) + 1 ζ˙ + ϑ1 , α1 (t) 6= 0

with ϑ1 = 2σσ01 a2 + a4 . Let ς1 = ec1t , which yields

ϑ1 c1

(7.25)

(7.26)

(7.27)

with c1 > 0 being constant and multiply (7.27) by

dV1 (t)ec1t ≤ α1 (t)N(ζ )ζ˙ ec1t + ζ˙ ec1t + ϑ1 ec1t dt

(7.28)

Integrating (7.28) over [0,t] yields Z
V1 (t) ≤ς1 + V1 (0) −ς1 e−c1t +e−c1t

≤c0 + e

−c1t

Z t 0

t 0


α1 (τ )N(ζ )+1 ζ˙ ec1 τ d τ


α1 (τ )N(ζ ) + 1 ζ˙ ec1 τ d τ

(7.29)

where c0 = ς1 +V1(0) is positive constant.
By using Lemma 7.1, it is concluded from R (7.29) that V1 (t), ζ (t), 0t α1 (τ )N(ζ ) + 1 ζ˙ ec1 τ d τ are bounded on [0,t f ). By (7.22), it holds that E and a˜ are bounded. According to Proposition 2 in [125], if the solution of the closed-loop is bounded, then t f = +∞. As a˜ = a − aˆ and a is bounded, then R aˆ is bounded. From (7.12), the boundedness of E ensures that z, z˙ and 0t z(·)d τ are bounded [46]. Subsequently, it is obtained from (7.3) and (7.4) that ei , i = 1, . . . , n (i−1) is bounded. Since ei = x1 − y∗ (i−1) and y∗ (i−1) is bounded, x is bounded and thus ϕ f (x) in (7.6) is bounded. Therefore, it is concluded from (7.11), (7.13), (7.19), and ˙ˆ E, ˙ κD (·) are bounded and further derived from (7.1) and (7.5) that x˙ (7.21) that u, a, and z¨ are bounded, respectively. Besides, it holds from (7.12) that

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 94 — #108

✐

✐

94

7 Adaptive PID Control for MIMO Nonlinear Systems

u˙ =

∂u ˙ ∂ u ∂ N(ζ ) ˙ ∂ u ∂ ϕ (·) ∂ u ˙ E+ + aˆ ζ+ ∂E ∂ N(ζ ) ∂ ζ ∂ ϕ (·) ∂ t ∂ aˆ

(7.30)

with
∂u ∂u ∂u = kD0 +κD (·) N(ζ ), = aN( ˆ ζ )E, = ϕ 2 (·)N(ζ )E, ∂E ∂ ϕ (·) ∂ aˆ

∂ u ∂ N(ζ ) = kD0 +κD (·) 2ζ cos(ζ ) − ζ 2 sin(ζ ) E ζ˙ , ∂ N(ζ ) ∂ ζ ∂ ϕ (·) ∂ ϕ (·) ∂ ϕ (·) ∂ ϕ (·) ∂ ϕ (·) = x˙ + z˙ + z¨ + e˙i ∂t ∂x ∂z ∂ z˙ ∂ ei

(7.31)

Note that ϕ (·) is differentiable with respect to its arguments and that all the signals ˙ˆ z, z˙, z¨, and ϕ (·) are all bounded and ˙ N(ζ ), ζ , ζ˙ , a, including E, E, ˆ ei , e˙i , a, continuous, then it is obvious that u˙ is bounded and continuous, which implies that u is uniformly continuous with bounded
control variation. R Since V1 (0) and 0t α1 (τ )N(ζ ) + 1 ζ˙ ec1 τ d τ are bounded, it is concluded from (7.22) and (7.29) that 1 σ0 2 a lim kEk2 ≤ ς1 = a + 2 2σ1 c1 4c1

t→∞

(7.32)

which implies that, as t → ∞, kEk converges into the residual set r   a σ0 2 Θ1 = kEk|kEk ≤ a + σ1 c1 2c1

7.3.2 PID Control for Non-square Systems Secondly, we consider the more general case that B(·) is non-square. Under this case the control problem becomes more complex as the control scheme developed above is no longer valid. To tackle this case, the following assumption is introduced. Assumption 7.5 The matrix B(·) is non-square and partially known, in that it can be decomposed as B(·) = B0 (·)M(·) (7.33) where B0 (·) ∈ Rm×l is a known bounded matrix with full row rank and M(·) ∈ Rl×l is completely unknown and unnecessarily symmetric. The only information available
for control design is that B0 M T (·) + M(·) BT0 is negative or positive definite. With this assumption, by Lemma 7.2, it holds that


1 E T B0 M T (·) + M(·) BT0 E = α2 (t)kEk2 2kB0 k

(7.34)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 95 — #109

✐

✐

7.3 PID-like Control Design and Analysis

95

where α2 (t) 6= 0 for any t ≥ 0. Now we construct the PID-like controller as in (7.9)—(7.11) with

Λ=

BT0 , kB0 k

κD (·) = aˆϕ 2 (·)

and the updating laws for ζ and aˆ are given by
ζ˙ = kD0 + κD (·) kEk2, a˙ˆ = −σ0 aˆ + σ1 ϕ 2 (·)kEk2

(7.35)

(7.36)

where σ0 and σ1 are some positive design constants, aˆ is estimation of a and ϕ (·) is a readily computable scalar function as defined in (7.16). Theorem 7.2. Consider the MIMO system (7.1) subject to actuation faults (7.2). Under Assumptions 7.1—7.3, 7.5, if the PID control strategy as governed by (7.9)—(7.11), (7.35)—(7.36) is applied, then all the signals q in the closed-loop being bounded with kEk are ultimately bounded as kEk ≤ σσ1 0c2 a2 + 2ca2 . Furthermore, the control action is continuous with bounded control variation. Proof. Consider the Lyapunov function candidate 1 1 2 V2 = E T E + a˜ 2 2σ1

(7.37)

with a˜ = a − a. ˆ By using (7.11), (7.12), (7.14), (7.33), and (7.35), the time derivative of V2 is derived as 1 1 V˙2 =E T E˙ − a˜a˙ˆ = E T H(·) + E T B(·)u − a˜a˙ˆ σ1 σ1
E T B0 MBT0 E 1 ˙ ≤aϕ kEk+ kD0 +κD (·) N(ζ ) − a˜aˆ kB0 k σ1

(7.38)

By using (7.34), it holds that


T  E T B0 MBT0 E 1 E = E T B0 MBT0 + B0 MBT0 kB0 k 2kB0 k 
T  1 + E T B0 MBT0 − B0 MBT0 E 2kB0 k | {z }

(7.39)

skew symmetric

=α2 (t)kEk2

Integrating (7.36) and (7.39) into (7.38) yields
σ0 V˙2 ≤ aϕ (·)kEk + kD0 +κD (·) N(ζ )α2 (t)kEk2 − a˜ϕ 2 (·)kEk2 + a˜aˆ σ1

(7.40)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 96 — #110

✐

✐

96

7 Adaptive PID Control for MIMO Nonlinear Systems

Following the same lines as in the proof of Theorem 7.1, it is straightforward to show that (7.41) V˙2 ≤ α2 (t)N(ζ )ζ˙ + ζ˙ + ϑ2 , α2 (t) 6= 0

with ϑ2 = 2σσ01 a2 + a4 . Then let ς2 = ϑc22 with c2 > 0 being constant, by following the same lines as that from (7.28) to (7.29), we can directly get that V2 (t) ≤ c0 + e−c2t

Z t 0


α2 (τ )N(ζ ) + 1 ζ˙ ec2 τ d τ

(7.42)

with c0 = V2 (0) + ς2 . Then by following the same analysis as that used in Theorem 7.1, n the stability qresults that:o 1) kEk converges into the residual set

Θ2 = kEk|kEk ≤ σσ1 0c2 a2 + 2ca2 ultimately; 2) all the signals in the closed loop are bounded, and 3) the control input is continuous with bounded control variation, can be established.

7.3.3 Analysis and Discussion 1) Note that, although the underlying tracking control problem is quite complicated because the system considered is MIMO with non-vanishing uncertainties and unpredictable actuation faults as well as unknown control direction, the proposed solution is structurally simple and computationally inexpensive. Furthermore, in setting up and implementing the proposed PID control, one only needs to derive the core function readily obtainable with certain crude information on the system as discussed in [116] and select γ > 0, kD0 > 0, σ0 > 0 and σ1 > 0, all of which have clear direction for the choice. No trial and error process is needed. 2) As reflected in the error residual sets Θ1 and Θ2 , the developed PID algorithms offer the obvious recipe for improving the control precision by enlarging c1 , c2 , σ1 and reducing σ0 properly. 3) For MIMO systems with B(·) being symmetric and positive definite and known control direction, several other methods can also be used to achieve UUB tracking for the system (see for instance, [83, 123, 126]) under the healthy actuation assumption. But, as it turns out, the proposed PID control bears much simpler structure and functions effectively under faulty actuation and unknown control direction. Furthermore, when B(·) is nonsquare, most existing methods are no longer applicable, while the proposed PID control scheme is able to cope with such systems gracefully. 4) In developing the control schemes, a number of virtual parameters such as a f , a, r, λ p and upper/lower bounds such as y, ¯ ρ j0 , cρ , cε , p0 , pm , etc., are defined and used in stability analysis, but these parameters are not involved in the control laws; thus additional analytical effort to estimate those parameters (a nontrivial task) is not needed in setting up and implementing the proposed PID control strategies,

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 97 — #111

✐

✐

7.4 Conclusion

97

consequently simplifying control design and rendering the proposed PID control user-friendly. 5) The developed control schemes bear PID structure, but unlike traditional PID control, the proposed one does not need an ad hoc and time consuming trial and error process for PID gain determination. Furthermore, it is robust against system uncertainties, adaptive to unknown system parameters, and fault-tolerant to unpredictable actuation faults.

7.4 Conclusion This chapter has proposed a set of PID control schemes for uncertain MIMO nonlinear systems with unknown control direction as well as actuation failures. In contrast to traditional PID control that lacks a systematic way for determining the PID gains, the proposed method allows the PID gains to be adaptively and automatically tuned by the developed algorithms without the need for ad hoc and frustrating trial and error process. Both square and nonsquare systems without a priori knowledge of the control direction are considered. Furthermore, the resultant PID control exhibits robust adaptive and fault-tolerant capabilities. Extension of the PID method to nonaffine systems with state estimation [116, 127] or multi-agent systems presents an interesting topic for future research [128].

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 98 — #112

✐

✐

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 99 — #113

✐

✐

Chapter 8

PD Control Application to High-Speed Trains

Excessive vertical and lateral motion of a train body could endanger the operation safety of a high-speed train. The situation is further complicated if actuation faults occur. This chapter investigates a semi-active approach for suppressing such motions. By using the structural properties of the system model, a new control scheme is proposed to account for various factors such as input nonlinearities, actuator failures, and external disturbances in the system simultaneously. The resultant control scheme is capable of automatically generating the intermediate control parameters and literally producing the PD-like controller - the whole process does not require precise information on the system model or system parameters. Furthermore, unlike traditional PD control, the one proposed has the stability-guaranteed algorithms to self-adjust its PD gains and there is no need for human tuning or a trial and error process. Such user-friendly features are deemed favorable for practical implementation. The effectiveness of the proposed controller is tested using computer simulations in the presence of parametric uncertainties and varying operation conditions.

8.1 Introduction As the speed of the train increases, the vibration of the train body becomes increasingly noticeable, which negatively impacts the operational safety of the high-speed train (HST) [129]. For this reason, suspension control plays an important role in suppressing such vibration. Suspension can be broadly classified into passive, active, and semi-active suspension systems [130]. Passive suspension consists of spring and damper, active suspension needs outside energy. However, the semi-active suspension does not need an outside power supply, and is able to produce an adjustable damping force to control the system. Thus, semi-active suspension is widely used in many fields. The sky-hood control approach [131, 132, 133] is the most popular one due to its simple structure. Such a model independent scheme is economical and convenient in practical application. 99

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 100 — #114

✐

✐

100

8 PD Control Application to High-Speed Trains

However, the damping force can only be regulated to the neighborhood of the critical point, which may cause self-oscillation when the vibration frequency is high enough. Linear optimal control algorithm achieves good control performance in a certain range. This method ignores nonlinear dynamics so that control parameters are calculated according to the simplified model, which can only ensure satisfactory performance under certain operation conditions, rendering it a local optimal method [134, 135]. Robust control (such as H2 /H∞ control) [136, 137, 138] could maintain the system’s stability and reliability, but the control design process is quite complicated. Thus far, most existing suspension control methods are either based on the linear system model, thus impractical, or are too complicated and too costly to implement. In this chapter, by making use of the predictive capability of the differential operation, we include the derivative part D, together with the proportional part P, to build an adaptive PD control, which is simpler than widely used PID, thus less demanding in design and less expensive in implementation. The rest of the chapter is organized as follows. In Section 8.2 we first establish a more comprehensive model capable of reflecting 9-DOF motions in HST. Based on the structural properties of the model, we develop a new PD control strategy ensuring stabilization of MIMO nonlinear systems in Section 8.3. Note that traditional PD control, although simple and effective for certain linear systems, seldom works satisfactorily for general nonlinear systems with uncertain dynamics and unpredictable disturbances. The proposed PD control not only offers the simplicity in structure, but also maintains its effectiveness in dealing with nonlinear systems with modeling uncertainties and external disturbances simultaneously. Using the Lyapunov stability theory we derive analytic algorithms to tune the PD gains automatically and adaptively, avoiding the undesirable trial and error process normally required in traditional PD method, making it a favorable and affordable solution to vibration suppression of HST. As actuation faults are inevitable for systems under long-time operation, it is crucial to enable PD control to have adaptive and fault-tolerant capabilities; therefore in this section we also derive a PD-like fault-tolerant control scheme to cope with this situation. It is shown that there is no need for fault detection and diagnosis (FDD) or fault detection and identification (FDI) [121, 139, 140] to monitor whether an actuation failure occurs or not. Instead, the fault impact is automatically and adaptively accommodated by the developed algorithms. Finally, we conduct numerical simulation verification on the proposed method in Section 8.4.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 101 — #115

✐

✐

8.2 Modeling and Problem Statement

101

(a) The aerial views of the train

(b) The front view of the train Fig. 8.1 9-DOF vehicle dynamic model.

8.2 Modeling and Problem Statement 8.2.1 Modeling In this section we derive the dynamic model for the suspension system in HST. Note that every single railway vehicle normally consists of a vehicle body, two bogies, and two wheel sets, as conceptually shown in Fig.8.1 [141], where yw , yt denote the lateral displacement of the wheel sets, and bogie, ψw , ψt , ψc represent the shake angle of the wheel sets, bogie and body, φt , φc denote the roll angle of the bogie and body, Mw , Mt , Mc are the wheel set mass, bogie mass and body mass, ds , dw are the secondary and primary vertical suspension semi-spacing, Hcb , Hbt , Htw denote the secondary lateral suspension height of the body and bogies and the primary lateral suspension height, respectively.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 102 — #116

✐

✐

102

8 PD Control Application to High-Speed Trains

By using Newton’s law, we establish the following dynamics for wheel sets, bogies, and the body: • The wheel sets lateral dynamics Mw y¨w1 + 2[ f py1 + k py (·)(yw1 − yt + Htw φt − lt ψt )] = D(l)

(8.1)

Mw y¨w2 + 2[ f py2 + k py(·)(yw2 − yt + Htw φt + lt ψt )] − D(l) = 0

(8.2)

• The wheel sets shake dynamics

Jwz ψ¨ w1 + 2dw2 [ f px1 + k px (·)(ψw1 − ψt )] − D(l) = 0

(8.3)

Jwz ψ¨ w2 + 2dw2 [ f px2 + k px (·)(ψw2 − ψt )] − D(l) =

(8.4)

0

• The bogies lateral dynamics Mt y¨t − 2{[ f py1 + k py (·)(yw1 − yt + Htw φt − lt ψt )] +[ f py2 + k py(·)(yw2 − yt + Htw φt + lt ψt )]} + 2[ fsy + ksy

(8.5)

(·)(yt − yc + Hbt φt + Hcb φc − lc ψc )] = 0

• The bogies roll dynamics

Jtx φ¨t + 2Htw {[ f py1 + k py (·)(yw1 − yt + Htw φt − lt

ψt )] + [ f py2 + k py (·)(yw2 − yt + Htw φt + lt ψt )]} + 2 Hcb [ fsy + ksy (·)(yt − yc + Hbt φt + Hcb φc − lc ψc )]+

(8.6)

4dw2 (k pz (·)φt + f pz ) + 2ds2[ksz (·)(φt − φc ) + fsz1 ] = 0 • The bogies shake dynamics Jtz ψ¨ t − 2lt {[ f py1 + k py(·)(yw1 − yt + Htw φt

−lt ψt )] − [ f py2 + k py (·)(yw2 − yt + Htw φt + lt ψt )]} +2dw2 [k px (·)(ψt − ψw1 ) − f px1 + k px (·)(ψt − ψw2 )

(8.7)

− f px2 ] + 2ds2[ksx (·)(ψt − ψc ) + fsx ] = 0

• The body roll dynamics

Jcx φ¨c + 2Hcb[ fsy + ksy (·)(yt − yc + Hbt φt

+Hcb φc − lc ψc ) − 2ds2[ksz (·)(φt − φc ) + fsz1 ] = 0

(8.8)

• The body shake dynamics Jcz ψ¨ c − 2lc [ fsy + ksy (·)(yt − yc + Hbt φt

+Hcb φc − lc ψc ) + 2ds2[ksz (·)(ψc − ψt ) + fsz2 ] = 0

(8.9)

The stiffness and damping force in the above equations are defined as follows: k pz , ksz represent the primary and secondary vertical stiffness, k py , ksy denote the primary and secondary lateral stiffness, k px , ksx represent the primary and secondary longitudinal stiffness, f px , fsx are the primary and secondary longitudinal damping force, f py , fsy denote the primary and secondary lateral damping force, f pz , fsz

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 103 — #117

✐

✐

8.2 Modeling and Problem Statement

103

represent the primary and secondary vertical damping force and D(l) is external disturbing force. For convenience, we define xm = [yw1 , yw2 , ψw1 , ψw2 , yt , φt , ψt , φc , ψc ]T ∈ R9 , vm = [y˙w1 , y˙w2 , ψ˙ w1 , ψ˙ w2 , y˙t , φ˙t , ψ˙ t , φ˙c , ψ˙ c ]T ∈ R9 and x = (xm , vm )T ∈ R18 as the state vectors. Together with the control input being u = [ f py1 , f py2 , f px1 , f px2 , fsy , fsz1 , f pz , fsx , fsz2 ]T ∈ R9 and the output vector y ∈ R9 , the above dynamic equations can be expressed as:    x˙m = vm (8.10) v˙m = Bu + f (x) + C(·)   y = xm

where B, f (x) and C are given in (8.11), and the other related elements are defined as follows.  −2  0 0 0 0 0 0 0 0 Mw −2  0 0 0 0 0 0 0 0    Mw 2   −2dw  0 0 0 0 0 0 0 0  Jwz   2   w   0 0 0 −2d 0 0 0 0 0 Jwz   2 2 2   0 0 − 0 0 0 0 B =  Mt Mt Mt  ∈ R9×9   2Hcb 2ds2  0 0 0 0 − Jcx Jcx 0 0 0    2ds2 4dw2  − 2Htw − 2Htw 0 2Hbt 0 − − − 0 0   Jtx J J J J tx tx tx tx   2dw2 2dw2 2ds2 2lt   2lt − Jtz Jtz 0 0 0 0  Jtz  Jtz Jtz 2 2ds 2lc 0 0 0 0 0 0 0 − Jcz Jcz  r1 (·)yw1 + r2 (·)yt + r7 (·)φt + r8 (·)ψt   r3 (·)yw2 + r4 (·)yt + r9 (·)φt + r10(·)ψt     r5 (·)ψw1 + r11(·)ψt     r6 (·)ψw2 + r12(·)ψt   9  f (x) =  r13 (·)yw1 + r14 (·)yw2 + r15 (·)yt + r25(·)φt + r26(·)φc + r27 ψc  ∈R  r16 (·)yw1 + r17 (·)yw2 + r18 (·)yt + r28 (·)φt + r29 (·)φc + r30 (·)ψc    r19 (·)yw1 + r20 (·)yw2 + r21 (·)ψw1 + r22 (·)ψw2 + r31 (·)ψt + r32 (·)ψc      r23 (·)yt + r33 (·)φt + r34 (·)φc + r35 (·)ψc r24 (·)yt + r36 (·)φt + r37 (·)ψt + r38 (·)φc + r39 (·)ψc 

C=[ r1 = r3 =

−2K py (·) Mw ,

r7 = r9 =

−2K py (·)Htw , Mw

r5 = r6 =

−2dw2 K px (·) , Jwz

T D(l) D(l) D(l) D(l) , , , , 0, 0, 0, 0, 0] ∈ R9 Mw Mw Jwz Jwz

r2 = r4 =

(8.11)

2K py (·) Mw

r8 = −r10 = r11 = r12 =

2K py (·)lt Mw

2dw2 K px (·) Jwz

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 104 — #118

✐

✐

104

8 PD Control Application to High-Speed Trains

r13 = r14 =

2K py (·) Mt ,

r15 =

−4K py (·)−2Ksy (·) Mt

r25 =

4K py (·)Htw −2Ksy (·)Hbt , Mt

r27 =

2Ksy (·)lc , Mt

r18 =

4K py (·)Htw −2Ksy (·)Hbt , Jtx

r28 =

−4K py (·)H 2 tw −2Ksy (·)H 2 bt −4dw2 K pz (·)−2ds2 Ksz (·) Jtx

r29 =

−2Ksy (·)Hbt Hcb +2ds2 Ksz (·) , Jtx

r24 =

2Ksy (·)lc Jcz ,

r32 =

2Ksx (·)ds2 , Jtz

r33 =

−2Ksy (·)Hcb Hbt +2ds2 Ksz (·) , Jcx

r34 =

−2Ksy (·)H 2 cb −2ds2 Ksz (·) , Jcx

r36 =

2Ksy (·)Hbt lc , Jcz

r37 =

2Ksz (·)ds2 Jcz

r38 =

2Ksy (·)lc Hcb , Jcz

r39 =

−2Ksy (·)lc2 −2ds2 Ksz (·) Jcz

r26 =

r16 = r17 =

r31 =

−2Ksy (·)Hcb Mt

−2K py (·)Htw Jtx

r30 =

2Ksy (·)Hbt lc Jtx

r19 = −r20 =

2K py (·)lt Jtz

−4K py (·)lt2 −2Ksx (·)ds2 −4dw2 K px (·) Jtz

r23 = −

2Ksy (·)Hcb Jcx

r21 = r22 = r35 =

2K px (·)dw2 Jtz

2Ksy (·)Hcb lc Jcx

Remark 8.1 Compared with the existing 4-DOF model [142], the established 9-DOF dynamic model is more effective in reflecting the dynamic behavior of HST, which, however, complicates the suppression control problem. Furthermore, it is worth mentioning that unlike most existing methods that assume all the spring stiffnesses are constant, here time-varying stiffness coefficients and external disturbances are considered, as is usually the case in practice; thus the resultant dynamic model becomes rather complicated.

8.2.2 Problem Statement As the state vector xm reflects the rolling, shaking, and lateral behavior of the vehicle, the suspension control objective is to generate suitable damping force u, so that the xm converges to zero or to the small neighborhood of zero as t → ∞. Note that (8.10) is an MIMO system with nonlinearities and uncertainties. In particular, B, C(·) and f (x) are all quite complex and time varying. Our objective is to develop a low-cost yet effective control solution to this problem, which does not need specific information on system model and parameters. Meanwhile, the proposed solution is able to accommodate actuation failures, as detailed in the following section.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 105 — #119

✐

✐

8.3 Control Scheme

105

8.3 Control Scheme 8.3.1 Structural Properties Although f (x) and C(·) are unknown and complex, with the bounded assumption on the unknown and time-varying parameters, it is readily shown that k f (x) + C(·)k ≤ µ1 (|yw1 | + |yw2 | + |ψw1 | + |ψw2 | + |yt | + |φt | + |ψt | + |φc | + |ψc |) + µ2 ≤ a f (|yw1 | + |yw2 | + |ψw1 | + |ψw2 | + |yt | + |φt | + |ψt | + |φc | + |ψc | + 1) = a f ϕ f (x) where a f > 0 is some unknown constant and ϕ f (x) is a computable scalar function which is bounded if x is bounded. Note that in practice it would be very difficult, if not impossible, to obtain the exact values of a f . The control in this chapter, however, as detailed later, is independent of this parameter, thus there is no need for analytical estimation of such a parameter except the fact that this parameter is used in stability analysis.

8.3.2 Robust Adaptive PD-like Control Design The special attention of this chapter is on the development of a control strategy with simple structure that demands no or little information on f (x) and C(·) explicitly. Therefore, we construct the following PD-like control with adaptive PD gains so that it is adaptive to known system parameters and robust against external disturbances. u = (k p0 + k p(·))Nxm + (kD0 + kD (·))N

dxm dt

(8.12)

where N is a sign matrix related to the system actuation direction, k p0 , kD0 are constants, while k p (·), kD (·) are time varying. To reduce the number of parameters to be tuned, we propose to link these parameters through a common coefficient γ as follows: k p0 = 2γ kD0 , k p (·) = 2γ kD (·) (8.13) which allows the PD controller to be expressed as   dxm u = (kD0 + kD (·))N 2γ xm + dt

(8.14)

from which it is seen that the PD control (8.12) with four control gains now reduces to the one with only two gains to determined. Namely, only kD0 and kD (·) need to be determined.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 106 — #120

✐

✐

106

8 PD Control Application to High-Speed Trains

Remark 8.2 The proposed controller bears the general PD form, thus is simple in structure and inexpensive in computation. Unlike traditional PD control, however, PD gains are not constant but time varying, although there is a constant part; the time varying part is consistently updated automatically without the trial and error process; the constant part can be chosen freely by the designer. Besides, note that the P-gain and D-gain are purposely linked to each other through the parameter γ , rather than being determined independently as in traditional PD control, it is such treatment that simplifies the gain tuning process and facilitates stability analysis. To proceed, we define the generalized error E as: E(P, D) = 2γ xm +

dxm dt

(8.15)

where γ > 0 is a free parameter chosen by the designer. To simplify the subsequent notation, we use E to denote the generalized error E(P, D) whenever no confusion arises. To bridge the generalized error (8.15) with the dynamics model (8.10), we take a time derivative of (8.15) to get E˙ = 2γ x˙m + x¨m = 2γ x˙m + Bu + f (x) + C(·) = Bu + Γ (·)

(8.16)

where Γ (·) = f (x) + C(·) + 2γ x˙m . According to (8.12), Γ (·) can be upper bounded as kΓ (x,t)k ≤ a f ϕ f (x) + 2γ kx˙m k ≤ aϕ (x) (8.17) with

(

a = max{a f , 2γ } ϕ (x) = ϕ f (x) + kx˙m k

(8.18)

where a is an unknown constant and ϕ (x) is a readily computable scalar function, called the core or deep-rooted function. Note that the matrix B is non-Hurwitz nor symmetric positive definite, but B is a invertible upper triangular, and the sign of the diagonal element is certain and known, thus the sign matrix N is set as N = −diag{sign{b j}} = diag{1, 1, 1, 1, 1, −1, 1, −1, 1}

(8.19)

It is clear that the matrix NB is Hurwitz, which is expressed in (8.20). Therefore NB meets the condition as stated in the following lemma. Lemma 8.1 Consider the system x(t) ˙ = A(t)x(t), t ≥ 0, where A(t) is continuously differentiable. Suppose A(t) and its first derivative are uniformly bounded and A(t) is Hurwitz. Then there exist symmetric and positive definite matrices P(t) and Q(t) such that A(t)T P(t) + P(t)A(t) = −Q(t)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 107 — #121

✐

✐

8.3 Control Scheme 

−2 Mw

107 0 −2 Mw

 0    0 0    0 0  2 2  NB =  Mt Mt   0 0   2Htw 2Htw − Jtx − Jtx   2lt t − 2l  Jtz Jtz 0 0

0 0 −2dw2 Jwz

0 0

0 0

0 0

0 0

0 0

0 0

0

0

0

0

0

0 − M2t

0 0

0 0

0 0

0

0

4dw2 Jtx

0

−2dw2 Jwz

0

2d 2

cb s − 2H Jcx − Jcx

0

0

0

0

2dw2 Jtz

2dw2 Jtz

0

0

2

2ds bt − 2H Jtx − Jtx −

2

0

0

0

2lc Jcz

s − 2d Jtz

0

0

0

 0 0    0    0   0   ∈ R9∗9  0    0   0  

(8.20)

2

s − 2d Jcz

where P(t) is continuously differentiable and satisfies

˙ ≤ pd kP(t)k ≤ p0 P(t)

where p0 > 0 and pd > 0 are some unknown constants. The proof of Lemma 8.1 can be obtained in [143, 144]. Assumption 8.1 NB is bounded with finite variation rate, i.e., kNBk ≤ B0 < ∞ and kd(NB)/dtk ≤ B p < ∞. Then upon applying Lemma 8.1 to NB, it holds that for a symmetric and positive definite matrix Q, there always exists a symmetric and positive definite matrix P with kPk ≤ pn < ∞ such that (NB)T P + P(NB) = −Q (8.21) apparently, since Q is symmetric and positive definite, there exists an unknown positive constant wQ such that 0 < wQ ≤ min{eig{Q}}

(8.22)

with eig{Q} being the eigenvalue of Q. Now we are ready to state the first result on robust adaptive PD-like control. Theorem 8.1. : Consider the MIMO nonlinear system (8.10). Under Assumption 8.1 and the condition that B(·) is an unknown invertible triangular matrix with known sign of each diagonal element. If the PD-like control law (8.12) is applied with N being defined as in (8.19) and

with sˆ being updated by

kD (·) = sˆϕ (x)2

(8.23)

s˙ˆ = −σ0 sˆ + σ1 ϕ (x)2 kEk2

(8.24)

where sˆ is the estimation of the virtual parameters to be defined later, σ0 and σ1 are positive design constants and ϕ (x) is the computable function given in (8.18),

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 108 — #122

✐

✐

108

8 PD Control Application to High-Speed Trains

then all the internal signals, including the system state are ensured to be uniformly ultimately bounded. Proof. Choosing the following Lyapunov candidate function V1 = E T PE +

1 s˜2 2σ1 wQ

(8.25)

where P is a symmetric and positive definite matrix like defined in (8.21). Let s˜ be defined as s˜ = s − wQ s, ˆ where wQ > 0 is some unknown constant defined as in (8.22). Note that the definition of the parameter error s˜ is in contrast to common definition of the form “s− s”. ˆ It is such treatment that facilitates the stability analysis, as seen shortly. This technique, borrowed from [116], will be used in the subsequent development. With (8.14), (8.15), and (8.16) the time derivation of V can be written as ˙ ˙ + E T PE˙ − s˜sˆ V˙1 = E˙ T PE + E T PE σ1 = uT BT PE + E T PBu + 2E T PΓ (·) −

s˜s˙ˆ σ1

= kD0 E T ((NB)T P + P(NB))E + kD (·)E T ((NB)T P s˜s˙ˆ +P(NB))E + 2E T PΓ (·) − σ1 s˜s˙ˆ ≤ −wQ kD0 kEk2 − wQ kD (·)kEk2 + 2E T PΓ (·) − σ1

(8.26)

In view of (8.17), it holds that k2PΓ (·)k ≤ 2pn aϕ (x) = sϕ (x) where s = 2pn a is some unknown constant (or virtual parameter for it bears no physical meaning). Now (8.26) can be further expressed as s˜s˙ˆ V˙1 ≤ −wQ kD0 kEk2 − wQ kD (·)kEk2 + sϕ (x) kEk − σ1 with the fact that sϕ (x) kEk ≤ sϕ (x)2 kEk2 + 4s and straightforward to derive that V˙1 ≤ −kD0 wQ kEk2 + s + σ0 s2 − σ0 s˜2 4

2σ1 wQ

σ0 σ0 σ0 2 2 σ1 s˜sˆ ≤ 2σ1 wQ s − 2σ1 wQ s˜ ,

it is

2σ1 wQ

= −kD0 wQ kEk2 − 2σσ10wQ s˜2 + 4s + 2σσ10wQ s2 ≤ −α1V1 + θ1 where α1 = min{

kD0 wQ wP , σ0 }

(8.27) > 0 and θ1 =

that will enter into the set Ω = {V1 | |V1 | ≤

s 4

+ 2σσ10wQ s2 > 0. It follows from (8.27)

θ1 + µ1 min{2kD0 wQ ,σ0 } }

with µ1 > 0 being a small

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 109 — #123

✐

✐

8.3 Control Scheme

109

constant in finite time, which implies that E ∈ ℓ∞ and s˜ ∈ ℓ∞ . From (8.15), it follows that xm ∈ ℓ∞ , x˙m ∈ ℓ∞ , which further implies that x ∈ ℓ∞ and ϕ (x) is bounded. Finally, ˙ˆ E, ˙ kD (·) are bounded . From (8.25) it can be shown that it is bounded we get that u, s, q

within the set {E| kEk ≤ 2(kθD01 +wµQ1 ) }. Thus the stability is ensured and all the internal signals are bounded and smooth.

8.3.3 Low-Cost Adaptive Fault-tolerant PD Control In the real situation, unanticipated actuation faults are inevitable for long-term operation; the actual forces acting on the secondary and primary suspensions are then determined by: ua (t) = ρ (t)u(t) + o(t) (8.28) which leads to the following dynamics  x˙m = vm     v˙ = Bρ (t)u(t) + Bo(t) + f (x) + C(·) m ′  = B (·)u(t) + Bo(t) + f (x) + C(·)    y = xm

(8.29)

where B′ = Bρ (t), ρ = diag{ρ1, ρ2 , ρ3 , ρ4 , ρ5 , ρ6 , ρ7 , ρ8 , ρ9 } is a time-varying and diagonal matrix and o = [o1 , o2 , o3 , o4 , o5 , o6 , o7 , o8 , o9 ]T ∈ R9 represents the uncontrollable portion of actuation input. Assumption 8.2 0 < ρ i ≤ 1, i = 1, 2, ..., 9, which is related to the actuator deficiency or “health indicator” [15, 145], ko(t)k ≤ c0 ≤ ∞, where c0 is a non-negative constant. Remark 8.3 1) Compared with the additive faults considered in [114], which can be simply treated as additional disturbances, these faults are much more challenging to tackle because such faults render the control gain unknown and time varying. 2) Assumption 8.2 imposes a bounded condition on the failure variation rate, which is necessary for a feasible fault accommodation solution to be developed. From (8.29), we can establish the following generalized state dynamics E˙ = 2γ x˙m + x¨m = 2γ x˙m + B′ (·)u + Bo(t) + f (x) + C(·) ′

(8.30)

′

= B (·)u + Γ (·) where Γ ′ (·) = f (x) + C(·) + 2γ x˙m + Bo(t). According to (8.17), Γ ′ (·) can be upper bounded as

′

Γ (x,t) ≤ kΓ (x,t)k + kBk c0 ≤ a′ ϕ ′ (x) (8.31)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 110 — #124

✐

✐

110

8 PD Control Application to High-Speed Trains

with

(

a′ = max{a f , c0 kBk , 2γ } ϕ ′ (x) = ϕ f (x) + 1 + kx˙m k

(8.32)

where a′ is an unknown constant and ϕ ′ (x) is core or deep-rooted function. It should be stressed that because of the involvement of actuation faults, the ′ resultant control gain B becomes unknown and time-varying (because ρ (t) is time-varying), this brings about additional challenge for suspension control design. However, it is interesting to note that ρ = diag{ρ1, ρ2 , ρ3 , ρ4 , ρ5 , ρ6 , ρ7 , ρ8 , ρ9 } with ′ ρi ∈ (0, 1] [12]; it holds that NB = NBρ (t) is Hurwitz, too. ′

Assumption 8.3 NB is bounded with finite variation rate, i.e., kNB′ k ≤ Bm < ∞ and kd(NB′ )/dtk ≤ Bq < ∞. Then we can apply Lemma 8.1 to NB′ to get that for a symmetric and positive define matrix Q′ (t), there always exists a time-varying positive definite matrix P′ (t) with (kP′ (t)k ≤ pm < ∞) such that ′

T

′

(NB ) P′ (t) + P′ (t)(NB ) = −Q′ (t)

(8.33)

and there exists an unknown constant w′Q > 0 such that 0 < wQ′ ≤ min{eig{Q′ (t)}} (eig{Q′ (t)} represents the eigenvalue of the matrix Q′ (t) and kdP′ (t)/dtk ≤ pq < ∞ ( pq is a constant). Theorem 8.2. Consider the MIMO nonlinear system (8.10) subject to actuation faults (8.28) with Assumptions 8.2—8.3 and the condition that B(·) is an unknown invertible triangular matrix with known sign for each diagonal element. If the PD-like control law (8.12) is applied with N being defined as in (8.19) and kD (·) = rˆφ 2 (·)

(8.34)

r˙ˆ = −σ2 rˆ + σ3 φ 2 (·)kEk2

(8.35)

φ (·) = ϕ (x) + kxm k + kx˙m k

(8.36)

with ′

where rˆ is the estimation of the virtual parameter r = max{2pq γ , pq , 2pm a′ }; σ2 > 0, σ3 > 0 are positive design constants. Then for any initial condition, uniformly ultimately bounded (UUB) stabilization of lateral, roll and shake motions are ensured. Furthermore, all the signals are continuous and bounded everywhere. Proof. Consider the Lyapunov candidate function V2 = E T P′ E +

1 r˜2 2σ3 wQ′

(8.37)

where P′ is a time-varying positive definite matrix like that in (8.33) and r˜ is defined as r˜ = r − wQ′ rˆ. Then it can be readily shown that

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 111 — #125

✐

✐

8.3 Control Scheme

111

˙ V˙2 = E˙ T P′ E + E T P˙′ E + E T P′ E˙ − σr˜r3ˆ ′

T

′

′

= kD0 E T ((NB ) P′ + P′ (NB ))E + kD (·)E T ((NB )

T

′ ′ ˙ P′ + P (NB ))E + E T (P˙′ E + 2P′Γ (·)) − σr˜r3ˆ

˙

≤ −wQ′ kD0 kEk2 − wQ′ kD (·)kEk2 + rφ (·) kEk − σr˜r3ˆ ≤ −kD0 wQ′ kEk2 − 2σσ32wQ r˜2 + 4r + 2σσ32wQ r2 ≤ −α2V2 + θ2 kD0 wQ′ wP′ , σ2 }

(8.38)

σ2 r 2 4 + 2σ3 wQ′ r > 0, which implies that E r will converge into a compact set {E| kEk ≤ 2(kθD02 +wµ2′ ) }, and all the signals are UUB

where α2 = min{

> 0 and θ2 =

Q

and continuous everywhere.

8.3.4 Comparison and Analysis (1) While PID has been widely used for many industrial applications, the applicability of PD control has not been well addressed. By using the predictive capability of the differentiator, we integrate the derivative (D) part into the proportional (P) part to build an adaptive PD control, bearing an even simpler structure than that of PID, rendering the overall control scheme inexpensive in computation and realization. (2) Unlike existing 4-DOF model in [116], the established dynamic model has 9DOF, thus is more effective in reflecting the dynamic behavior of HST. (3) This proposed control, although exhibiting similar PD structure, does not need ad hoc and time consuming trial and error processes for PD gain determination, in contrast to traditional PD control. Furthermore, it is robust against system uncertainties, adaptive to unknown system parameters, and fault-tolerant to unpredictable actuation faults. (4) As reflected in the error residual set θ2 , these new PD algorithms offer the obvious recipe for improving the control precision by enlarging kD0 and σ3 , and reducing σ2 properly. (5) In developing the control schemes, a number of virtual parameters such as a f , r, wQ′ and upper/lower bounds such as pm , pq , etc., are defined and used in stability analysis, but these parameters are not involved in the control laws, thus additional analytical effort to estimate those parameters (a nontrivial task) is not needed in setting up and implementing the new PD control strategies, consequently simplifying control design and rendering the new PD control user-friendly.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 112 — #126

✐

✐

112

8 PD Control Application to High-Speed Trains

8.4 Simulation Examples To validate the proposed low-cost adaptive fault-tolerant PD control, numerical simulations are performed. The physical parameters of the vehicle are given as [146] Mt = 2280 kg, Mw = 1650 kg, Jtz = 3000 kg · m2 Jcz = 2485756 kg · m2 , Jcx = 89396 kg · m2 Jwz = 1200 kg · m2 , Jtx = 2650 kg · m2 Hcb = 0.98 m, Htw = 0.1745 m, Hbt = 0.0945 m lt = 1.25 m, lc = 9 m, dw = 1 m, ds = 1 m Ksz = 3.5 × 105 (1 + 0.08 sin(t + 0.4)) N/m K px = 1.45 × 107 (1 + 0.08 sin(t + 0.3)) N/m K py = 7.5 × 106 (1 + 0.13 sin(4t + 0.7)) N/m K pz = 6.65 × 105 (1 + 0.06 sin(3t + 0.6)) N/m Ksy = 2.08 × 105 (1 + 0.09 sin(3t + 0.4)) N/m the external disturbing forces D(l) = 0.2 sin(2t) the function ϕ f (x) can be extracted as

ϕ f (x) = |yw1 | + |yw2 | + |ψw1 | + |ψw2 | + |yt | + |φt | + |ψt | + |φc | + |ψc | + 1

(8.39)

the ϕ ′ (x) and φ (·) are expressed as

ϕ ′ (x) = ϕ f (x) + 1 + kx˙m k

(8.40)

φ (·) = ϕ ′ (x) + kxm k + kx˙m k

(8.41)

Also the actuator efficiency variables for each of the nine control channels simulated are as follows:  0 12  1, 0 11.5  0 14 ✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 113 — #127

✐

✐

8.4 Simulation Examples

113

Fig. 8.2 Profile of the time varying actuator efficiency variables ρ1 -ρ9 .

Fig. 8.3 The lateral displacement of the wheel sets, body, and bogie.

ρ4 =

 

1, 0.6 + 0.025(t − 7)2 ,  0.7,

0 < t ≤ 10 10 < t ≤ 12 t > 12

ρ9 = ρ5 = ρ1 ρ6 = ρ2 ρ7 = ρ3 ρ8 = ρ4

as illustrated in Fig. 8.2. The fault-tolerant control algorithms are tested and the simulation results are presented in Figs. 8.3—8.6, where Figs. 8.3—8.4 are the lateral displacement and rate of the wheel sets, body, and bogie, respectively, from which it is observed that the proposed adaptive fault-tolerant control scheme performs well even if some of the actuators lose their effectiveness during the system operation. The PD control

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 114 — #128

✐

✐

114

8 PD Control Application to High-Speed Trains

Fig. 8.4 The rate of the wheel sets, body, and bogie.

signals u are shown in Fig. 8.5. Fig. 8.6 is the self-regulating process of the PD gains. As predicted theoretically, the self-adjusted PD control exhibits robust adaptive and fault-tolerant capabilities and no ad hoc process is needed to accomplish the task. It is observed that under the proposed PD-like control the vibrations are effectively suppressed in the presence of parametric uncertainties and varying operation conditions. For comparison, we examine the performance of the proposed control with commonly used PD control. The integral absolute error (IAE) for both position and velocity is used as the performance index, which is expressed as S0 =

Z t 0

|xm |d τ

or

S0 =

Z t 0

|vm |d τ

(8.42)

Fig. 8.7 is the control performance by the traditional PD method and the proposed method. As is clearly shown in the performance index curve, the traditional PD control with fixed gains is unable to maintain satisfactory performance when actuation faults occur, whereas the proposed method is able to deal with this situation effectively in that the integral absolute error (IAE) index remains almost constant, confirming its fault-tolerant capabilities.

8.5 Conclusion The low-cost adaptive fault-tolerant PD control for the 9-DOF vehicle semi-active suspension system is developed in this chapter. The control design and convergence analysis only depend on the input and output information of dynamical systems, and the monotonic convergence can be guaranteed. With the theoretical analysis and simulation, it can be concluded that the proposed approach is able to deal with various uncertainties and nonlinearities or even actuation faults, which suppresses

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 115 — #129

✐

✐

8.5 Conclusion

115

Fig. 8.5 The designed control inputs u.

Fig. 8.6 The adaptively and automatically adjusted PD gains.

excessive motion of train body in vertical and lateral directions effectively. The salient feature of the proposed PD-like control also lies in its PD gains being tuned adaptively and automatically without the need for time-consuming trial and error process. Extending the method to a more general MIMO nonlinear systems is an interesting topic for future research.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 116 — #130

✐

✐

116

8 PD Control Application to High-Speed Trains

(a) The lateral displacement of the wheel sets, body and bogie.

(b) The rate of the wheel sets, body and bogie. Fig. 8.7 Control performance (IAE) comparison between traditional PD control with fixed gains and the proposed PD-like control with adaptive gains.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 117 — #131

✐

✐

Chapter 9

PID Control Application to Robotic Systems

This chapter illustrates and analyzes the application of adaptive PID to robotic systems. Detail design, analysis and simulation are provided.

9.1 Robotic Modelling Consider an n-joint rigid-link robotic manipulator with the following joint-space dynamics ˙ = ua (9.1) Dq (q)q¨ + Cq (q, q) ˙ q˙ + Gq (q) + τ (q,t) where q ∈ Rn is the joint displacement; D(q) ∈ Rn×n , Cq (q, q) ˙ ∈ Rn×n and n Gq (q) ∈ R are the symmetric positive definite inertial matrix, Coriolis and ˙ ∈ Rn centrifugal matrix, and gravitational force vector, respectively; τ (q,t) represents the non-parametric frictional and the modeling uncertainties; ua ∈ Rn denotes the control vector of joint torque/force. Then for different tasks, we need to deal with different type of systems (i.e., square or non-square systems).

9.2 PID Control for Robotic Systems 9.2.1 Square System (joint-space tracking) Let y = x1 = q ∈ R3 and z = e1 = x1 − y∗ , it is straightforward to derive the following filtered error dynamics (9.2) z¨ = e¨1 = q¨ − y¨∗ = G(·)ρ (t)u +F(·) +G(·)ε +D(·) − y¨∗
2 −1 ˙ q˙ + Gq (q) , and D(·) = −D−1 with G(·) = D−1 ˙ q τ (q,t). q , F(·) = q˙ − Dq Cq (q, q) 117

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 118 — #132

✐

✐

118

9 PID Control Application to Robotic Systems

Let B(·) = G(·)ρ (t), it is seen that (9.2) is a square error dynamic system. It is straightforward to establish that the PID control governed by (7.9)—(7.11), (7.19)— (7.21) for square systems applies to (9.2).

9.2.2 Non-square System (task-space tracking) When operating in 3D space described by the Cartesian coordinates X ∈ R3 , the dynamics is subject to the following constraint X = ϖ (q). Let y∗ = X ∗ ∈ R3 , y = x1 = X, and z = e1 = X − X ∗ ; then it is straightforward to derive the following filtered error dynamics z¨ =e¨1 = ϖ¨ (q) − y¨∗ = J˙q˙2 + J(·)q¨ − y¨∗ =G(·)ρ (t)u + F(·) + G(·)ε + D(·) − y¨∗

(9.3)

G(·) = J(·)D−1 q

(9.4)

with F(·) = J˙q˙2 − J(·)D−1 q


Cq (q, q) ˙ q˙ + Gq (q)

D(·) = −J(·)D−1 ˙ q τ (q,t)

(9.5) (9.6)

3×n is the Jacobian matrix. Let B(·) = J(·)D−1 ρ (t); then where J(·) = ∂ E(·) q ∂q ∈ R 3×n B(·) ∈ R (n > 3) is the unknown and non-square gain matrix, which can be decomposed as B(·) = B0 (·)M(·) with B0 (·) = J(·) and M(·) = D−1 q ρ (t). To see if the proposed control scheme is applicable to non-square system (9.3), we need to examine if all the conditions imposed hold. Firstly, the “core” function ϕ f (·) under Assumption 7.3 for F(·) + G(·)ε + D(·) − y¨∗ can be extracted as [116]

ϕ f (·) = kqk ˙ 2 + kqk ˙ +1

(9.7)

2 Then the core function for (9.2) and (9.3) is readily derived as ϕ (·) = kqk ˙ +
˙ + kqk −1 T 1 + kzk + k˙zk. Also, under the condition that 12 D−1 ρ (·) + (D ρ (·)) is positive q q or negative definite, the proposed PID control as governed by (7.9)—(7.11), (7.35)— (7.36) for non-square systems immediately applies to (9.3).

9.3 Case Studies In order to validate the effectiveness and feasibility of the proposed PID control schemes, a 3-link robotic arm with 3 revolute joints, borrowed from [147], is used for simulation verification (see [147] for detailed model expression and definition). The three control channels of the system are subject to both additive and actuation effectiveness faults, which are determined by

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 119 — #133

✐

✐

9.3 Case Studies

119

ua1 =ρ1 u1 +ε1 , ρ1 = 1 − 0.2 tanh(t), ε1 = 0.02 cos(2t)

ua2 =ρ2 u2 +ε2 , ρ2 = 0.9 + 0.1 sin(π t), ε2 = 0.02 sin(3t) ua3 =ρ3 u3 +ε3 , ρ3 = 0.8 + 0.2 cos(π t), ε3 = 0.02 tanh(t)

(9.8)

The parameters for the model are chosen as that in [147] for simulation. Case 1: B(·) is square. In this case, the task is to regulate the output y (i.e., q) to track the desired trajectory y∗ , which is set as: y∗ = [1.2 sin(π t/4), 1.2 cos(π t/4), sin(π t/4)]T . The PID control scheme takes the same form as that in (7.9)—(7.11) with the gains being adjusted by (7.19) and (7.21), where the scalar function is extracted as ϕ (·) = kqk ˙ 2 + kqk ˙ + 1 + kzk + k˙zk. The simulation is conducted with the parameters chosen as: kD0 = 200, γ = 1.5, δ0 = 0.2, δ1 = 50 and the initial conditions given as: a(0) ˆ = 0, ζ (0) = 0, q1 (0) = q2 (0) = q3 (0) = 0.6, q˙1 = q˙2 = q˙3 = 0. The simulation results are given in Fig. 9.1. It is seen from Fig. 9.1 (a) that the tracking process is fairly good. Fig. 9.1 (b) illustrates the continuity and boundedness of the control signal u, and Fig. 9.1 (c) depicts the adaptively and automatically adjusted PID gains. Case 2: The matrix B(·) is non-square. When operating in the 2D space described by the Cartesian coordinates X1 ∈ R2 , the dynamics is subject to the constraint X1 = ϖ (q). Let y = X1 , e1 = y − y∗ and z = e1 , the following task-space filtered error dynamics can be derived as z¨ = G(·)ρ (t)u + F(·) + G(·)ε + D(·) − y¨∗

(9.9)

where G(·), F(·), and D(·) are given in (9.3)-(9.6), in which the Jacobian matrix J(·) is   ∂ ϖ (q) J J J J(·) = (9.10) = 11 12 13 J21 J22 J23 ∂q with J11 = − l1 sin(q1 ) − l2 sin(q1 + q2) − l3 sin(q1 + q2 + q3)

J12 = − l2 sin(q1 + q2 ) − l3 sin(q1 + q2 + q3 ), J13 = −l3 sin(q1 + q2 + q3) J21 =l1 cos(q1 )+l2 cos(q1 + q2)+l3 cos(q1 + q2 + q3)

(9.11)

J22 =l2 cos(q1 + q2)+l3 cos(q1 + q2 + q3), J23 = l3 cos(q1 + q2 + q3 ) where l1 > 0, l2 > 0 and l3 > 0 denote the length of the robotic arms, then it is readily obtained that J(·) is a full row rank matrix. Let B(·) = J(·)D−1 q ρ (t); then 2×3 B(·) ∈ R is the unknown and non-square gain matrix, which can be decomposed as B(·) = B0 (·)M(·) with B0 (·) = J(·) and M(·) = D−1 q ρ (t). Thus this is a non-square MIMO system satisfying all the imposed conditions in Theorem 7.2. Furthermore, the deep-rooted information ϕ f (·) under Assumption 7.3 for F(·) + G(·)ε + D(·) can be extracted as ϕ f (·) = kqk ˙ 2 + kqk ˙ +1 (9.12)

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 120 — #134

✐

✐

9 PID Control Application to Robotic Systems

y3 and y3

y2 and y2

y1 and y1

120 2

y1 y1

0 −2 2

0

5

10

15

y2 y2

0 −2 2

0

5

10

15

y3 y3

0 −2

0

5

10

15

Time(sec)

(a) The tracking processes of y1 , y2 and y3 .

Th e contr ol inputu

30

u1 u2 u3

20

10

0

−10

0

5

10

15

Time(sec)

(b) The input signal u(t).

Th e time-va r yin g gains

10000 κP (·) κI (·)

8000

κD (·)

6000 4000 2000 0

0

5

10

15

Time(sec)

(c) The time-varying input gains. Fig. 9.1 The simulation results for the case that B(·) is square.

which is computable and independent of system parameters. Then it is readily ˙ 2 + kqk ˙ + 1 + kzk + k˙zk. The desired obtained from (7.16) that ϕ (·) = kqk ∗ trajectories are set as: y1 = cos(t/π ), y∗2 = sin(t/π ), and the initial conditions are given by: a(0) ˆ = 0, ζ (0) = 0, q1 (0) = π /20, q1 (0) = q1 (0) = π /5 and q˙1 = q˙2 = q˙3 = 0. The parameters are chosen as: kD0 = 500, γ = 1.5, δ0 = 0.005, δ1 = 0.02. The corresponding simulation results are presented in Fig. 9.2, where Fig. 9.2 (a) shows the circling tracking process, Fig. 9.2 (b) depicts the continuity and boundedness of the control signals provided, and Fig. 9.2 (c) presents the adaptively and automatically adjusted procedure of PID gains. As predicted theoretically, the

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 121 — #135

✐

✐

9.3 Case Studies

121

Tracking process of the end -effector

self-adjusted PID control exhibits robust adaptive and fault-tolerant capabilities and no ad hoc process for determining PID gains is needed to accomplish the task.

Tracking circle Desired circle

1 0.5 0 −0.5 −1 −2

−1

0 Time(sec)

1

2

(a) 2D tracking process under the proposed PID control.

Th e contr ol inputu

40 20 0

u1 u2 u3

−20 −40 −60 −80

0

5

10 Time(sec)

15

20

(b) The input signal u(t).

Th e time-va r yin g gains

2500 κP (·) κI (·) κD (·)

2000

1500

1000

500

0

5

10 Time(sec)

15

20

(c) The time-varying input gains. Fig. 9.2 The simulation results for the case that B(·) is non-square.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 122 — #136

✐

✐

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 123 — #137

✐

✐

Chapter 10

Conclusion

In this book, we have introduced a number of approaches for designing self-tuning PI/PD/PID controller for nonlinear systems. The main contents can be summarized as follows: • In Chapter 3, a PI control with self-tuning gains is presented for a class of single input single output (SISO) nonlinear system. In order to help understand the proposed method, the controller design starts with the first-order nonlinear systems, and then extends to high-order systems. The developed PI control ensures the stable tracking of nonlinear systems. Meanwhile, rigorous stability conditions for nonlinear systems under the proposed PI-like scheme are established. It is shown that the derived PI-like control exhibits fault-tolerant capability without the need for FDD/FDI and transient performance is ensured despite system nonlinearities, modeling uncertainties, and actuation faults. However, it should be mentioned that, to ensure the prescribed performance, the initial conditions of the system must be known a priori. • In Chapter 4, a generalized PI control design is developed for a class of unknown nonaffine systems. The resultant control is able to deal with unknown nonaffine systems without the need for any trial and error process to determine the PI gains. Besides, the proposed generalized control scheme can be easily set up without the need for detailed system information except for its control direction. That is to say, the PI controller is insensitive to system model uncertainties and perturbations. The control scheme is also robust, adaptive, and fault-tolerant to modeling uncertainties and sensor/actuator failures. Such features are achieved by a generalized PI control with fixed structure. • In Chapter 5, a neuro-adaptive PI control is proposed for a class of multi-input multi-output (MIMO) nonlinear systems with unknown and nonsmooth actuation characteristics and external disturbances. The external disturbance is assumed to be immeasurable but bounded and it is not necessary to know the bound itself in the scheme. There is a NN unit involved in the scheme and the precondition for the NN unit to function properly is that the training (input) signal of the NN unit must remain within a compact set, which is explicitly assumed by using the

123

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 124 — #138

✐

✐

124

•

•

•

•

10 Conclusion

barrier Lyapunov function. The PI gains are self-tuned by the analytical algorithms established without the need for human interference or trial and error processes. MIMO systems with both square and non-square gain matrix are addressed therein. In Chapter 6, a neuro-adaptive PI control for a class of uncertain nonlinear strict feedback systems is presented. Both state constraints and unknown actuation characteristics are considered by using the barrier Lyapunov function at each step. In order to deal with the modeling uncertainties and the actuation characteristics impact, the neural networks are utilized at each step of the back-stepping design. The proposed neuro-adaptive control also exhibits simple structure and demands inexpensive online computations, rendering an inexpensive and non-complex solution. In Chapter 7, new PID tracking control algorithms applicable to both square and non-square MIMO nonlinear systems are developed. It shows that the structurally simple and computationally inexpensive PID control can be generalized and extended to control nonlinear MIMO systems with nonparametric uncertainties and actuation failures. By utilizing the concept of virtual parameter with Lyapunov function candidate, rigorous stability proof is provided for the nonlinear systems with PID in the loop. Besides, by utilizing the Nussbaum-type function and the matrix decomposition technique, non-square systems with unknown control direction are also considered. In Chapter 8, the PD-like adaptive controller is designed for high-speed train systems. Firstly, a more comprehensive model capable of reflecting 9-DOF motions in HST is established. Secondly, the proposed PD control not only remains simple in structure, but also maintains its effectiveness in dealing with nonlinear systems with modeling uncertainties and external disturbances simultaneously. Thirdly, the control scheme also has robust adaptive capabilities so that the stabilization is achieved without using the information on system model and parameters explicitly. Finally, the proposed PD control is able to tolerate unexpected actuation faults and there is no need for fault detection and diagnosis (FDD) or fault detection and identification (FDI) to monitor whether an actuation failure occurs or not. In Chapter 9, the robust adaptive PID controllers are applied to a robotic system. The proposed PID-like control ensures stable trajectory tracking in the presence of parametric uncertainties and varying operation conditions, validating the effectiveness of the method.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 125 — #139

✐

✐

Bibliography

1. A. Visioli, Practical PID Control, Springer-Verlag, London, 2006. 2. K. J. Astrom, T. Hagglund, “PID controllers: theory, design, and tuning,” Research Triangle Park, NC: Instrument Soc. Amer., pp. 699-714, 1995. 3. A. O’Dwyer, Handbook of PI and PID Controller Tuning Rules, London, U.K.: Imperial College Press, 2003. 4. J. G. Ziegler, N. B. Nichols, “Optimum settings for automatic controllers,” Trans. ASME, vol. 64, no. 11, 1942. ˚ 5. K. J. Astrom, T. H¨agglund, C. C. Hang, and W. K. Ho, “Automatic tuning and adaptation for PID controllers–a survey,” Control Eng. Pract., vol. 1, no. 4, pp. 699-714, 1993. 6. C. A. Desoer, C. A. Lin, “Tracking and disturbance rejection of MIMO nonlinear systems with PI controller,” IEEE Trans. Autom. Control, vol. 30, no. 9, pp. 861-867, 1985. 7. Caio F. de Paula, Lu´ıs H. C. Ferreira, “An improved analytical PID controller design for Non-Monotonic phase LTI systems,” IEEE Trans. Control Syst. Tech., vol. 20, no. 5, pp. 1328-1333, 2012. 8. K. H. Ang, G. Chong, and Y. Li, “PID control system analysis, design, and technology,” IEEE Trans. Control Syst. Tech., vol. 13, no. 4, pp. 559-576, 2005. 9. M. Cheng, Q. Sun, and E. Zhou, “New self-tuning fuzzy PI control of a novel doubly salient permanent-magnet motor drive,” IEEE Trans. Ind. Electron., vol. 53, no. 3, pp. 814-821, 2006. 10. R. Shahnazi, Mohammad-R. Akbarzadeh-T., “PI adaptive fuzzy control with large and fast disturbance rejection for a class of uncertain nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 16, no. 1, pp. 187-197, 2008. 11. J. Alvarez-Ramirez, G. Fernandez, and R. Suarez, “A PI controller configuration for robust control of a class of nonlinear systems,” J. Franklin Inst., vol. 339, no. 1, pp. 29-41, 2002. 12. Q. Song, Y. D. Song, “Generalized PI control design for a class of unknown nonaffine systems with sensor and actuator faults,” Syst. Control Lett., vol. 64, no. 1, pp. 86-95, 2014. 13. J. Teo, J. P. How, and E. Lavretsky, “Proportional-integral controllers for minimum-phase non-affine-in-control systems,” IEEE Trans. Autom. Control, vol. 55, no. 6, pp. 1477-1482, 2010. 14. W. C. Cai, X. Liao, and Y. D. Song, “Indirect robust adaptive fault-tolerant control for attitude tracking of spacecraft,” J. Guid., Control, Dyn., vol. 31, no. 5, pp. 1456-1463, 2008. 15. Y. D. Song, H. N. Chen, and D. Y. Li, “Virtual-point-based fault-tolerant lateral and longitudinal control of 4W-steering vehicles,” IEEE Trans. Intell. Transp. Syst., vol. 12, no. 4, pp. 1343-1351, 2011. 16. X. Tang, G. Tao, and S. M. Joshi, “Adaptive actuator failure compensation for nonlinear MIMO systems with an aircraft control application,” Automatica, vol. 43, no. 11, pp. 18691883, 2007.

125

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 126 — #140

✐

✐

126

Bibliography

17. W. Wang, C. Y. Wen, “Adaptive actuator failure compensation control of uncertain nonlinear systems with guaranteed transient performance,” Automatica, vol. 46, no. 12, pp. 2082-2091, 2010. 18. P. Kabore, H. Wang, “Design of fault diagnosis filters and fault-tolerant control for a class of nonlinear systems,” IEEE Trans. Autom. Control, vol. 46, no. 11, pp. 1805-1810, 2001. 19. S. N. Huang, K. K. Tan, and T. H. Lee, “Automated fault detection and diagnosis in mechanical systems,” IEEE Trans. Syst., Man, Cybern. C, Appl. Rev., vol. 37, no. 6, pp. 1360-1364, 2007. 20. C. P. Bechlioulis, and G. A. Rovithakis, “Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems,” Automatica, vol. 45, no. 2, pp. 532-538, 2009. 21. C. Wang, D. J. Hill, S. S. Ge, and G. R. Chen, “An ISS-modular approach for adaptive neural control of pure-feedback systems,” Automatica, vol. 42, no. 5, pp. 723-731, 2006. 22. H. B. Du, H. H. Shao, and P. J. Yao, “Adaptive neural network control for a class of lowtriangular-structured nonlinear systems,” IEEE Trans. Neural Netw., vol. 17, no. 2, pp. 509514, 2006. 23. B. Karimi, M. B. Menhaj, “Non-affine nonlinear adaptive control of decentralized large-scale systems using neural networks,” Inf. Sci., vol. 180, pp. 3335-3347, 2010. 24. H. Deng, H. X. Li, and Y. H. Wu, “Feedback-linearization-based neural adaptive control for unknown nonaffine nonlinear discrete-time systems,” IEEE Trans. Neural Netw., vol. 19, no. 9, pp. 1615-1625, 2008. 25. Q. M. Yang, J. B. Vance, and S. Jagannathan, “Control of nonaffine nonlinear discretetime systems using reinforcement-learning-based linearly parameterized neural networks,” IEEE Trans. Syst., Man, Cybern., vol. 38, no. 4, pp. 994-1001, 2008. 26. W. Y. Wang, Y. H. Chien, Y. G. Leu, and T. T. Lee, “Adaptive T-S fuzzy-neural modeling and control for general MIMO unknown nonaffine nonlinear systems using projection update laws,” Automatica, vol. 46, no. 5, pp. 852-863, 2010. 27. W. Y. Wang, Y. H. Chien, and T. T. Lee, “Observer-based T-S fuzzy control for a class of general nonaffine nonlinear systems using generalized projection-update laws,” IEEE Trans. Fuzzy Syst., vol. 19, no. 3, pp. 493-504, 2011. 28. J. Wang, S. S. Ge, and T. H. Lee, “Adaptive fuzzy sliding mode control of a class of nonlinear systems,” in: Proc. Third Asian Control Conf., pp. 599-604, 2000. 29. S. Labiod, T. M. Guerra, “Indirect adaptive fuzzy control for a class of nonaffine nonlinear systems with unknown control directions,” Int. J. Control, Autom. Syst., vol. 8, no. 4, pp. 903-907, 2010. 30. S. S. Ge, J. Zhang, “Neural-network control of non-affine nonlinear system with zero dynamics by state and output feedback,” IEEE Trans. Neural Netw., vol. 14, no. 4, pp. 900918, 2003. 31. J. H. Park, G. T. Park, S. H. Kim, and C. J. Moon, “Direct adaptive self-structuring fuzzy controller for non-affine nonlinear system,” Fuzzy Sets and Systems, vol. 153, no. 3, pp. 429-445, 2005. 32. S. Labiod, T. M. Guerra, “Adaptive fuzzy control of a class of SISO non-affine nonlinear systems,” Fuzzy Sets and Systems, vol. 158, no. 10, pp. 1126-1137, 2007. 33. Y. G. Leu, W. Y. Wang, and T. T. Lee, “Observer-based direct adaptive fuzzy-neural control for non-affine nonlinear systems,” IEEE Trans. Neural Netw., vol. 16, no. 4, pp. 853-861, 2005. 34. H. Du, S. S. Ge, “Output feedback adaptive neural control for a class of non-affine non-linear systems with a dynamic gain observer,” IET Control Theory Appl., vol. 5, no. 3, pp. 486-497, 2011. 35. H. Du, S. S. Ge, and J. K. Liu, “Adaptive neural network output feedback control for a class of non-affine non-linear systems with unmodelled dynamics,” IET Control Theory Appl., vol. 5, no. 3, pp. 465-477, 2011. 36. L. Wei, “Time-varying feedback control of nonaffine nonlinear systems without drift,” Syst. Control Lett., vol. 29, no. 2, pp. 101-110, 1996.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 127 — #141

✐

✐

Bibliography

127

37. E. Moulay, W. Perruquetti, “Stabilization of nonaffine systems: a constructive method for polynomial systems,” IEEE Trans. Autom. Control, vol. 50, no. 4, pp. 520-526, 2005. 38. G. Bartolini, E. Punta, “Reduced-order observer in the sliding-mode control of nonlinear non-affine systems,” IEEE Trans. Autom. Control, vol. 55, no. 10, pp. 2368-2373, 2010. 39. A. S. Willsky, “A survey of design method for failure detection in dynamic systems,” Automatica, vol. 12, no. 6, pp. 601-611, 1976. 40. D. D. Siljak, “Reliable control using multiple control system,” Int. J. Control, vol. 31, no. 2, pp. 303-329, 1980. 41. R. J. Patton, “Fault tolerant control systems: the 1997 situation (survey),” IFAC Safe Process’97, vol. 2, pp. 1033-1054, Hull, UK, 1997. 42. F. Caccavale, L. Villani, Fault Diagnosis and Fault Tolerance for Mechatronic Systems, Springer-Verlag, Berlin, 2003. 43. L. L. Fan, Y. D. Song, “Fault-tolerant control and disturbance attenuation of a class of nonlinear systems with actuator and component failures,” Acta Automat. Sinica, vol. 37, no. 5, pp. 623-628, 2011. 44. P. Panagi, M. M. Polycarpou, “Decentralized fault tolerant control of a class of interconnected nonlinear systems,” IEEE Trans. Autom. Control, vol. 56, no. 1, pp. 178-184, 2011. 45. J. H. Richter, W. P. M. H. Heemels, N. van de Wouw, J. Lunze, “Reconfigurable control of piecewise affine systems with actuator and sensor faults: stability and tracking,” Automatica, vol. 47, no. 4, pp. 678-691, 2011. 46. H. K. Khalil, “Universal integral controllers for minimum-phase nonlinear systems,” IEEE Trans. Autom. Control, vol. 45, no. 3, pp. 490-494, 2000. 47. Q. Song, Y. D. Song, “PI-like fault-tolerant control of nonaffine systems with actuator failures,” Acta Automat. Sinica, vol. 38, no. 6, pp. 1033-1040, 2012. 48. H. Alwi, C. Edwards, “Fault tolerant control using sliding modes with on-line control allocation,” Automatica, vol. 44, no. 7, pp. 1859-1866, 2008. 49. Jang-Hyun Park, Gwi-Tae Park, “Robust adaptive fuzzy controller for non-affine nonlinear systems with dynamic rule activation,” Int. J. Robust Nonlinear Control, vol. 13, no. 2, pp. 117-139, 2003. 50. R. Padhi, N. Unnikrishnan, and S. N. Balakrishnan, “Model-following neuro-adaptive control design for non-square, non-affine nonlinear systems,” IET Control Theory Appl., vol. 1, no. 6, pp. 1650-1661, 2007. 51. Y. D. Song, “Adaptive motion tracking control of robot manipulators-nonregressor based approach,” Int. J. Control, vol. 63, no. 1, pp. 41-54, 1996. 52. J. T. Spooner, K. M. Passino, “Decentralized adaptive control of nonlinear systems using radial basis neural networks,” IEEE Trans. Autom. Control, vol. 44, no. 11, pp. 2050-2057, 1999. 53. Q. Song, Y. D. Song, T. Tang, and B. Ning, “Computationally inexpensive tracking control of high-speed trains with traction/braking saturation,” IEEE Trans. Intell. Transp. Syst., vol. 12, no. 4, pp. 1116-1125, 2011. 54. J. E. Slotine, W. Li, Applied Nonlinear Control, Prentice Hall, Englwood Cliffs, NJ, 1991. 55. A. Teel, L. Praly, “Tools for semiglobal stabilization by partial state and output feedback,” SIAM J. Control Optim., vol. 33, no. 5, pp. 1443-1448, 1995. 56. A. Boulkroune, M. M’Saad, and H. Chekireb, “Design of a fuzzy adaptive controller for MIMO nonlinear time-delay systems with unknown actuator nonlinearities and unknown control direction,” Inf. Sci., vol. 180, no. 24, pp. 5041-5059, 2010. 57. D. R. Seidl, S. L. Lam, J. A.Putman, and R. D. Lorenz, “Neural network compensation of gear backlash hysteresis in position-controlled mechanisms,” IEEE Trans. Industry Applica., vol. 31, no. 6, pp. 1475-1483, 1995. 58. J. Chen, X. P. Wang, and R. F. Ding, “Gradient based estimation algorithm for Hammerstein systems with saturation and dead-zone nonlinearities,” Applied Mathematical Modelling, vol. 36, no. 1, pp. 238-243, 2012.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 128 — #142

✐

✐

128

Bibliography

59. J. Peng, R. Dubay, “Identification and adaptive neural network control of a DC motor system with dead-zone characteristics,” IEEE Trans. Industry Applica., vol. 50, no. 4, pp. 588-598, 2011. 60. C. Y. Wen, J. Zhou, Z. T. Liu, and H. Y. Su, “Robust adaptive control of uncertain nonlinear systems in the presence of input saturation and external disturbance,” IEEE Trans. Autom. Control, vol. 56, pp. 1672-1678, 2011. 61. F. L. Lewis, W. K. Tim, L. Z. Wang, and Z. X. Li, “Deadzone compensation in motion control systems using adaptive fuzzy logic control,” IEEE Trans. Control Syst. Tech., vol. 7, no. 6, pp. 731-742, 1999. 62. J. J. Ma, B. Chen, S. S. Ge, Z. Q. Zheng, and K. F. Liu, “Adaptive NN control of a class of nonlinear systems with asymmetric saturation actuators,” IEEE Trans. Neural Netw. Learn. Syst., vol. 26, np. 7, pp. 1532-1538, 2015. 63. H. Modares, F. L. Lewis, and M. B. Naghibi-Sistani, “Adaptive optimal control of unknown constrained-input systems using policy iteration and neural networks,” IEEE Trans. Neural Netw. Learn. Syst., vol. 24, pp. 1513-1525, 2013. 64. E. J. Davidson, “Multivariable tuning regulators: the feedforward and robust control of general servomechanism problem, ”IEEE Trans. Autom. Control, vol. 21, no. 1, pp. 35-47, 1975. 65. J. Pentinnen, H. N. Koivo, “Multivariable tuning regulators for unknown systems,” Automatica, vol. 16, no. 4, pp. 393-398, 1980. 66. P. J. Gawthrop, “Self-tuning PID controller: algorithms and implementation,” IEEE Trans. Autom. Control, vol. 31, no. 3, pp. 201-209, 1986. 67. F. Radke, R. Isermann, “A parameter adaptive PID controller with stepwise parameter optimization,” Automatica, vol. 23, no. 4, pp. 449-457, 1987. 68. T. H. Lee, C. C. Hang, W. K. Ho, and P. K. Yue, “Implementation of a knowledge-based PID auto-tuner,” Automatica, vol. 29, no. 4, pp. 1107-1113, 1993. 69. O. Lequin, M. Gevers, M. Mossberg, E. Bosmans, and L. Triest, “Iterative feedback tuning of PID parameters: comparison with classical tuning rules,” Control Eng. Pract., vol. 11, no. 9, pp. 1023-1033, 2003. 70. W. D. Chang, R. C. Hwang, and J. G. Hsieh, “A self-tuning PID control for a class of nonlinear systems based on the Lyapunov approach,” Journal of Process Control, vol. 12, no. 2, pp. 233-242, 2002. 71. W. D. Chang, J. J. Yan, “Adaptive robust PID controller design based on a sliding mode for uncertain chaotic systems,” Chaos, Solitons and Fractals, vol. 26, no. 1, pp. 167-175, 2005. 72. J. Chen, T. Huang, “Applying neural networks to on-line updated PID controllers for nonlinear process control,” Journal of Process Control, vol. 14, no. 2, pp. 211-230, 2004. 73. C. L. Chen, F. Y. Chang, “Design and analysis of neural/fuzzy variable structural PID control systems,” IEE Proc. Control Theory and Applica., vol. 143, no. 2, pp. 200-208, 1996. 74. A. Boubakir, S. Labiod, and F. Boudjema, “A stable self-tuning proportional-integralderivative controller for a class of multi-input-multi-output nonlinear systems,” Journal of Vibration and Control, vol. 18, no. 2, pp. 228-239, 2012. 75. J. O. Jang, G. J. Jeon, “A parallel neuro-controller for DC motors containing nonlinear friction,” Neurocomputing, vol. 30, no. 1-4, pp. 233-248, 2000. 76. Z. J. Zhang, Z. J. Li, Y. N. Zhang, Y. M. Luo, and Y. Q. Li, “Neural-dynamic-method-based dual-arm CMG scheme with time-varying constraints applied to humanoid robots,” IEEE Trans. Neural Netw. Learn. Syst., vol. 26, no. 12, pp. 3251-3262, 2015. 77. C. L. Philip Chen, G. X. Wen, Y. J. Liu, and F. Y. Wang, “Adaptive consensus control for a class of nonlinear multiagent time-delay,” IEEE Trans. Neural Netw., vol. 25, no. 6, pp. 1217-1226, 2014. 78. Z. J. Li, C. Y. Su, “Neural-adaptive control of single-master-multipleslaves teleoperation for coordinated multiple mobile manipulators with time-varying communication delays and input uncertainties,” IEEE Trans. Neural Netw. Learn. Syst., vol. 24, no. 9, pp. 1400-1413, 2013. 79. W. He, Y. H. Chen, and Z. Yin, “Adaptive neural network control of an uncertain robot with full-state constraints,” IEEE Trans. Cybern., vol. 46, no. 3, pp. 620-629, 2016.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 129 — #143

✐

✐

Bibliography

129

80. W. He, A. O. David, Z. Yin, and C. Y. Sun, “Neural network control of a robotic manipulator with input deadzone and output constraint,” IEEE Trans. Syst., Man, Cybern., vol. 46, no. 6, pp. 759-770, 2016. 81. M. Chen, S. Z. S. Ge, and B. B. Ren, “Adaptive tracking control of uncertain MIMO nonlinear systems with input constraints,” Automatica, vol. 47, no. 3, pp. 452-465, 2011. 82. W. C. Meng, Q. M. Yang, and Y. X. Sun, “Adaptive neural control of nonlinear MIMO systems with time-varying output constraints,” IEEE Trans. Neural Netw. Learn. Syst., vol. 26, no. 5, pp. 1074-1085, 2015. 83. A. Theodorakopoulos, G. A. Rovithakis, “A simplified adaptive neural network prescribed performance controller for uncertain MIMO feedback linearizable systems,” IEEE Trans. Neural Netw. Learn. Syst., vol. 26, pp. 589-600, 2015. 84. H. J. Xu, P. A. Ioannou, “Proportional-integral controllers for minimum-phase nonaffine in control systems,” IEEE Trans. Autom. Control, vol. 48, pp. 728-742, 2003. 85. W. He, S. S. Ge, “Vibration control of a flexible beam with output constraint,” IEEE Trans. Ind. Electron., vol. 62, no. 8, pp. 5023-5030, 2015. 86. W. He , S. Zhang, and S. S. Ge, “Adaptive control of a flexible crane system with the boundary output constraint,” IEEE Trans. Ind. Electron., vol. 61, pp. 4126-4133, 2014. 87. W. He, S. S. Ge, “Cooperative control of a nonuniform gantry crane with constrained tension,” Automatica, vol. 66, pp. 146-154, 2016. 88. M. Krstic, ´ I. Kanellakopoulos, and P. V. Kokotovic, ´ Nonlinear and Adaptive Control Design, New York, NY, USA: Wiley, 1995. 89. S. S. Ge, C. Wang, “Adaptive NN control of uncertain nonlinear pure-feedback systems,” Automatica, vol. 38, no. 4, pp. 671-682, 2002. 90. X. Zhao, S. Peng, X. Zheng, and J. Zhang, “Intelligent tracking control for a class of uncertain high-order nonlinear systems,” IEEE Trans. Neural Netw. Learn. Syst., vol. 27, no. 9, pp. 1976-1982, 2016. 91. H. Wang, W. Sun, and P. X. Liu, “Adaptive intelligent control of nonaffine nonlinear timedelay systems with dynamic uncertainties,” IEEE Trans. Syst., Man, Cybern. Syst., vol. 47, no. 7, pp. 1474-1485, 2017. 92. F. G. Rossomando, C. Soria, and R. Carelli, “Adaptive neural sliding mode compensator for a class of nonlinear systems with unmodeled uncertainties,” Engineering Applications of Artificial Intelligence, vol. 26, pp. 2251-2259, 2013. 93. B. Chen, X. Liu, K. Liu, and C. Lin, “Novel adaptive neural control design for nonlinear MIMO time-delay systems,” Automatica, vol. 45, no. 6, pp. 1554-1560, 2009. 94. X. Zhao, H. Yang, H. R. Karimi, and Y. Zhu, “Adaptive neural control of MIMO nonstrictfeedback nonlinear systems with time delay,” IEEE Trans. Cybern., vol. 46, no. 6, pp. 13371349, 2016. 95. D. P. Li, D. J. Li, “Adaptive neural tracking control for nonlinear time-delay systems with full state constraints,” IEEE Trans. Syst., Man, Cybern. Syst., vol. 47, no. 7, pp. 1590-1601, 2017. 96. Y. J. Liu, S. Tong, “Barrier Lyapunov functions for Nussbaum gain adaptive control of full state constrained nonlinear systems,” Automatica, vol. 76, pp. 143-152, 2017. 97. T. P. Zhang, S. S. Ge, “Adaptive neural control of MIMO nonlinear state time-varying delay systems with unknown dead-zones and gain signs,” Automatica, vol. 43, no. 6, pp. 10211033, 2007. 98. T. Zhang, S. S. Ge, “Adaptive neural network tracking control of MIMO nonlinear systems with unknown deadzones and control directions,” IEEE Trans. Neural Netw., vol. 20, no. 3, pp. 483-497, 2009. 99. M. Chen, G. Tao, “Adaptive fault-tolerant control of uncertain nonlinear large-scale systems with unknown dead zone,” IEEE Trans. Cybern., vol. 46, no. 8, pp. 1851-1862, 2016. 100. C. Chen, Z. Liu, Y. Zhang, and C. L. P. Chencand S. Xie, “Saturated Nussbaum function based approach for robotic systems with unknown actuator dynamics,” IEEE Trans. Cybern., vol. 46, no. 10, pp. 2311-2322, 2016. 101. K. P. Tee, S. S. Ge, “Control of nonlinear systems with partial state constraints using a barrier Lyapunov function,” Int. J. Control, vol. 84, no. 12, pp. 2008-2023, 2011.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 130 — #144

✐

✐

130

Bibliography

102. Y. J. Liu, S. Tong, “Barrier Lyapunov functions-based adaptive control for a class of nonlinear pure-feedback systems with full state constraints,” Automatica, vol. 64, pp. 7075, 2016. 103. B. Ren, S. S. Ge, K. P. Tee, and T. H. Lee, “Adaptive neural control for output feedback nonlinear systems using a barrier Lyapunov function,” IEEE Trans. Neural Netw., vol. 21, no. 8, pp. 1339-1345, 2010. 104. K. P. Tee, S. S. Ge, and E. H. Tay, “Barrier Lyapunov functions for the control of outputconstrained nonlinear systems,” Automatica, vol. 45, no. 4, pp. 918-927, 2009. 105. Y. J. Liu, J. Li, and S. C. Tong, “Neural network control-based adaptive learning design for nonlinear systems with full-state constraints,” IEEE Trans. Neural Netw. Learn. Syst., vol. 27, no. 7, pp. 1562-1571, 2016. 106. K. P. Tee, B. Ren, and S. S. Ge, “Control of nonlinear systems with time-varying output constraints,” Automatica, vol. 47, no. 11, pp. 2511-2516, 2011. 107. B. Kim, S. Yoo, “Approximation-based adaptive tracking control of nonlinear pure-feedback systems with time-varying output constraints,” Int. J. Control, Autom. Syst., vol. 13, no. 2, pp. 257-265, 2015. 108. Y. D. Song, J. X. Guo, and X. C. Huang, “Smooth neuroadaptive PI tracking control of nonlinear systems with unknown and nonsmooth actuation characteristics,” IEEE Trans. Neural Netw. Learn. Syst., vol. 28, no. 9, pp. 2183-2195, 2017. 109. K. Zhao, Y. D. Song, and C. Y. Wen, “Computationally inexpensive fault tolerant control of uncertain nonlinear systems with non-smooth asymmetric input saturation and undetectable actuation failures,” IET Control Theory Appl., vol. 10, no. 15, pp. 1866-1873, 2016. 110. A. J. Isaksson, S. F. Graebe, “Analytical PID parameter expressions for higher order systems,” Automatica, vol. 35, no. 6, pp. 1121-1130, 1999. 111. J. Zhang, J. Zhuang, and H. Du, “Self-organizing genetic algorithm based tuning of PID controllers,” Inf. Sci., vol. 179, no. 7, pp. 1007-1018, 2009. 112. N. J. Killingsworth, M. Krstic, “PID tuning using extremum seeking: online, model-free performance optimization,” IEEE Control Syst. Mag., vol. 26, no. 1, pp. 70-79, 2006. 113. K. K. Tan, Q. G. Wang, and C. C. Hang, Advances in PID Control. Springer Science & Business Media, 2012. 114. D. L. Yu, T. K. Chang, and D. W. Yu, “Fault tolerant control of multivariable processes using auto-tuning PID controller,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 35, no. 1, pp. 32-43, 2005. 115. D. L. Yu, T. K. Chang, and D. W. Yu, “A stable self-learning PID control for multivariable time varying systems,” Control Eng. Pract., vol. 15, no. 12, pp. 1577-1587, 2007. 116. Y. D. Song, X. C. Huang, and C. Y. Wen, “Tracking control for a class of unknown nonsquare MIMO nonaffine systems: a deep-rooted information based robust adaptive approach,” IEEE Trans. Autom. Control, vol. 61, no. 10, pp. 3227-3233, 2015. 117. Y. J. Wang, Y. D. Song, and F. L. Lewis, “Robust adaptive fault-tolerant control of multiagent systems with uncertain nonidentical dynamics and undetectable actuation failures,” IEEE Trans. Ind. Electron., vol. 62, no. 6, 3978-3988, 2015. 118. Y. J. Wang, Y. D. Song, M. Krstic, and C. Y. Wen, “Fault-tolerant finite time consensus for multiple uncertain nonlinear mechanical systems under single-way directed communication interactions and actuation failures,” Automatica, vol. 63, pp. 374-383, 2016. 119. Y. D. Song, Y. J. Wang, and C. Y. Wen, “Adaptive fault-tolerant PI tracking control with guaranteed transient and steady-state performance,” IEEE Trans. Autom. Control, vol. 62, no. 1, pp. 481-487, 2017. 120. S. Yin, G. Wang, H. Gao, “Data-driven process monitoring based on modified orthogonal projections to latent structures,” IEEE Trans. Control Syst. Tech., vol. 24, no. 4, pp. 481-487, 2016. 121. X. Zhang, T. Parisini, and M. M. Polycarpou, “Adaptive fault-tolerant control of nonlinear uncertain systems: an information-based diagnostic approach,” IEEE Trans. Autom. Control, vol. 49, no. 8, pp. 1259-1274, 2004. 122. R. D. Nussbaum, “Some remarks on the conjecture in parameter adaptive control,” Syst. Control Lett., vol. 3, no. 5, pp. 243-246, 1983.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 131 — #145

✐

✐

Bibliography

131

123. W. X. Shi, “Adaptive fuzzy control for MIMO nonlinear systems with nonsymmetric control gain matrix and unknown control direction,” IEEE Trans. Fuzzy Syst., vol. 22, no. 5, pp. 1288-1300, 2014. 124. S. S. Ge, F. Hong, and T. H. Lee, “Adaptive neural control of nonlinear time-delay systems with unknown virtual control coefficients,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 34, no. 1, pp. 499-516, 2004. 125. E. R. Ryan, “A universal adaptive stabilizer for a class of nonlinear systems,” Syst. Control Lett., vol. 16, no. 3, 209-218, 1991. 126. H. Xu, P. A. Ioannou, “Robust adaptive control for a class of MIMO nonlinear systems with guaranteed error bounds,” IEEE Trans. Autom. Control, vol. 48, no. 5, pp. 728-742, 2003. 127. S. Yin, X. Zhu, J. Qiu, and H. Gao, “State estimation in nonlinear system using sequential evolutionary filter,” IEEE Trans. Ind. Electron., vol. 63, no. 6, pp. 3786-3794, 2016. 128. J. Qin, C. Yu, and H. Gao, “Coordination for linear multi-agent systems with dynamic interaction topology in the leader-following framework,” IEEE Trans. Ind. Electron., vol. 61, no. 5, pp. 2412-2422, 2014. 129. A. Orvnas, S. Stichel, and R. Persson, “Ride comfort improvement in a high-speed train with active secondary suspension,” J. Mech. Syst. Tansp. Logistics, vol. 3, no. 1, pp. 206215, 2010. 130. R. S. Prabakar, C. Sujatha, and S. Narayanan, “Optimal semi-active preview control response of a half car vehicle model with magnetorheological damper,” Journal of Sound and Vibration, vol. 326, no. 3, pp. 400-420, 2009. 131. X. Liu, J. Luckel, and K.-P. Jaker, “An active suspension/tilt system for a mechatronic railway carriage,” Control Eng. Pract., vol. 10, no. 9, pp. 991-998, 2002. 132. W. Ding, J. Bu, and Y. Liu, “Strategy and method of high-speed train suspensions lateral semi-active control in China,” China Railway Sci., vol. 23, no. 4, pp. 1-7, 2002. 133. Q. Li, Y. Zhao, “Review of the research of the semi-suspension control of the high-speed train,” Diesel Locomotives, vol. 1, pp. 7-11, 2008. 134. L. Palkovies, P. J. Th. Venhovens, “Investigation on stability and Possible chaotic motions in the controlled wheel suspension system,” Veh. Syst. Dyn., vol. 21, no. 1, pp. 269-296, 1992. 135. Z. F. Fang, Z. X. Deng, and L. Zheng, “The research progress of semi-active suspension system,” Journal of Chongqing University, vol. 27, no. 1, pp. 46-48, 2003. 136. S. Chantranuwathana, H. Peng, “Adaptive robust control for active suspensions,” in Proc. Amer. Control Conf., San Diego, CA, USA, pp. 1702-1706, 1999. 137. Y. Lin, C. Lin, and C. C. Yang, “Robust active vibration control for rail vehicle pantograph,” IEEE Trans. Veh. Technol., vol. 56, no. 4, pp. 1994-2004, 2007. 138. L. H. Zong, X. L. Gong, S. H. Xuan, and C. Y. Guo, “Semi-active H∞ control of high-speed railway vehicle suspension with magnetorheological dampers,” Veh. Syst. Dyn., vol. 51, no. 5, pp. 600-625, 2013. 139. P. Kabore, H. Wang, “Design of fault diagnosislters and fault-tolerant control for a class of nonlinear systems,” IEEE Trans. Autom. Control, vol. 46, no. 11, pp. 1805-1810, 2001. 140. X. Zhang, M. M. Polycarpou, and T.Parisini, “Fault diagnosis of a class of nonlinear uncertain systems with Lipschitz nonlinearities using adaptive estimation,” Automatica, vol. 46, no. 2, pp. 290-299, 2010. 141. C. J. Chen, “The research for the active and semi-active suspension system of high-speed train,” Southwest Jiaotong University, 2006. 142. D. Y. Li, Y. D. Song, and W. C. Cai, “Neuro-adaptive fault-tolerant approach for active suspension control of high-speed trains,” IEEE Trans. Intell. Transp. Syst., vol. 16, no. 5, pp. 2446-2456, 2015. 143. F. Amato, G. Celentano, and F. Garofalo, “New sufficient conditions for the stability of slowly varying linear systems,” IEEE Trans. Autom. Control, vol. 38, no. 9, pp. 1409-1411, 1993. 144. H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ, USA:Prentice-Hall, 2002. 145. Q. Song and Y. D. Song, “Data-based fault-tolerant control of high-speed trains with traction/braking notch nonlinearities and actuator failures,” IEEE Trans. Neural Netw., vol. 22, no. 12, pp. 2250-2261, 2011.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 132 — #146

✐

✐

132

Bibliography

146. G. D. Jin, “The research of adaptive control method for train’s semi-active suspension,” Xinan Jiaotong University, 2011. 147. X. Xin, M. Kaneda, “Swing-up control for a 3-dof gymnastic robot with passive first joint: design and analysis,” IEEE Trans. Robotics, vol. 23, no. 6, pp. 1277-1285, 2007.

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 133 — #147

✐

✐

Index

A

F

Actuator effectiveness, 35, 122 fading, 38 failures, PI control under, 43–44 output, 16 simulated, 48

Fault detection and diagnosis (FDD), 34, 95 high-speed trains, 108, 132 MIMO systems, 95 nonaffine system, 46 Fault detection and identification (FDI), 95 high-speed trains, 108 MIMO systems, 95 Fault-tolerant control (FTC), 34 First-order nonlinear systems, PI control design for, 18–20 Fuzzy systems, 33

B Barrier Lyapunov function (BLF), 3, 132 definition of, 78 nonlinear systems and, 51 strict feedback systems and, 75, 78

C Classical PID control, 5–13 anticipatory control, 8 automatic reset, 7 derivative action, 7–8 integral action, 6–7 non-steady state scenario, 12 PID tuning software, 12–13 proportional action, 5–6 three actions of PID control, 5–8 trial-and-error tuning, 8–11 tuning methods, 8–13 Ziegler—Nichols methods, 11

D Deep-rooted function, 114, 118 Derivative action, 7–8

E Error dynamics (MIMO), 96–97 Extremum seeking (ES), 94

G Gaussian basis function vector, 79 Globally ultimately uniformly bounded (GUUB), 99, 100

H High-order nonlinear systems, PI control design for, 20–26 High-speed trains (HSTs), PD control application to, 107–124 comparison and analysis, 119–120 control scheme, 113–120 deep-rooted function, 114 dynamics, 110 fault detection and diagnosis, 108, 132 fault detection and identification, 108, 132 integral absolute error, 122, 123 low-cost adaptive fault-tolerant PD control, 117–119 modeling and problem statement, 109–113 Newton’s law, 110 problem statement, 113 robust adaptive PD-like control design, 113–117

133

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 134 — #148

✐

✐

134

Index iterative feedback tuning, 94 neural network, 94 Nussbaum function, 97–98 partial loss of effectiveness, 95 PID control for non-square systems, 102–104 PID control for square systems, 99–102 PID-like control design and analysis, 98–105 problem formulation and error dynamics, 95–98 soft 2-norm, 99 total loss of effectiveness, 95

simulation examples, 120–123 structural properties, 113 uniformly ultimately bounded stabilization, 119 HSTs, see High-speed trains (HSTs), PD control application to

I Integral absolute error (IAE), 122, 123 Integral action, 6–7 Iterative feedback tuning (IFT), 94

L Linear time-invariant (LTI) systems, 52 Lipschitz condition, 36

M Multi-input multi-output (MIMO) nonlinear systems, 3, 131 Multi-input multi-output (MIMO) nonlinear systems, adaptive PI control for, 51–74 appendix (proof), 73–74 closed-loop system, 61 illustrative examples, 69–72 linear time-invariant systems, 52 modified PI control based on BLF, 63–68 neural networks and function approximation, 55–56 neuro-adaptive PI control for non-square systems (modified PI control), 66–68 neuroadaptive PI control for non-square systems (PI control), 61–63 neuro-adaptive PI control for square systems (modified PI control), 64–66 neuroadaptive PI control for square systems (PI control), 57–61 PI control design and stability analysis, 56–63 problem formulation, 52–56 radial basis function neural networks, 55 single-input single-output nonlinear systems, 52 system description, 52–55 Multi-input multi-output (MIMO) nonlinear systems, adaptive PID control for, 93–105 analysis and discussion, 104–105 core function, 99 error dynamics, 96–97 extremum seeking, 94 fault detection and diagnosis, 95 fault detection and identification, 95 globally ultimately uniformly bounded, 99, 100

N Neural network (NN), 3, 33 MIMO nonlinear systems, 51, 94 RBF-based, 49, 55 strict feedback systems, 75 Non-steady state (NSS) scenario, 12 Nussbaum function, 97–98

P Partial loss of effectiveness (PLOE), 3, 95 PID control, classical, see Classical PID control PID tuning software, 12–13 PI/PD/PID control, 2 Proportional action, 5–6 Proportional-integral (PI) control, strict feedback systems, 75 Proportional-integral (PI) controller automatic reset configuration, 7 design, 18, 26 non-square systems, 61 SISO nonaffine systems, 34 square systems, 57 Proportional integral derivative controller (PID controller) definition of, 1 self-tuning, 94 trial-and-error tuning, 10 tuning parameters, 8 ZN tuning rule, 11

R Radial basis function (RBF), 49 Radial basis function neural networks (RBFNNs), 49, 55 Robotic systems, PID control application to, 125–129 case studies, 126–129 non-square system (task-space tracking), 126 PID control for system, 125–126

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 135 — #149

✐

✐

Index robotic systems description, 125 square system (joint-space tracking), 125–126

S Sensor faults, PI control under, 44–46 Single input single output (SISO) affine systems, adaptive PI control for, 15–31 appendix (proof of lemma), 29–31 design details, 18–26 first-order nonlinear systems, PI control design for, 18–20 healthy indicator, 16 high-order nonlinear systems, PI control design for, 20–26 numerical examples, 26–27 problem formulation, 16–17 Ultimately Uniformly Bounded, 20, 25 Single input single output (SISO) nonaffine systems, generalized PI control for, 33–50 actuation failures, 35 actuator and sensor faults, PI control under, 44–46 adaptive fault-tolerant PI control, 43–46 control design, 37–43 fault detection and diagnosis, 34, 46 fault-tolerant control, 34 faulty scenario, 35 illustrative examples, 46–49 Lipschitz condition, 36 Lyapunov function candidate, 42 PI control, 38–43 PI control under actuator failures, 43–44 proportional-integral controller, 34 RBF based neural network method, 49 sensor failures, 35

135 system description and preliminaries, 35–37 “trial and error” process, 39 T-S fuzzy-neural model, 34 ultimately uniformly bounded, 39 uniformly ultimately bounded tracking stability, 33, 46 Single input single output (SISO) nonlinear systems, 3, 52, 131 Soft 2-norm, 99 Strict feedback systems, adaptive PI control for, 75–92 barrier Lyapunov function, 75 Gaussian basis function vector, 79 illustrative examples, 89–91 neural network, 75 PI-like control design, 79–89 system description and preliminaries, 76–79

T Three-term controllers, 5 Total loss of effectiveness (TLOE), 3, 95 Trial-and-error tuning, 8–11 T-S fuzzy-neural model, 34

U Ultimately uniformly bounded (UUB) MIMO nonlinear systems, 60 SISO affine systems, 20, 25 SISO nonaffine systems, 33, 39, 46 stabilization, high-speed trains, 119, 132

Y Young’s inequality, 60, 80, 87

Z Ziegler—Nichols (ZN) tuning rule, 11

✐

✐ ✐

✐

✐

✐ “pidbook20180916” — 2018/9/19 — 17:14 — page 136 — #150

✐

✐

✐

✐ ✐

✐