Output Regulation and Cybersecurity of Networked Switched Systems 3031309715, 9783031309717

Table of contents :
Acknowledgements
Contents
1 Introduction
1.1 Background
1.2 Status of Research
1.2.1 Current Status of Switching Rules Design for Switched Systems
1.2.2 Research Status of Event-Triggered Control in Switched System
1.2.3 Research Status of Security Control in Switched Systems
1.2.4 Research Status of the Bumpless Transfer Control of Switched Systems
1.2.5 Research Status of Output Regulation Problems in Switched Systems
1.3 Proposed Problem Solving
1.4 Basic Definitions and Lemmas
1.5 Organization of the Book
References
2 Event-Triggered Output Regulation for Networked Flight Control System Based on an Asynchronous Switched System Approach
2.1 Introduction
2.2 Flight Control System Modelling
2.3 Problem Formulation and Preliminaries
2.3.1 Error Feedback Controller
2.3.2 Alternate Event-Triggered Communication Scheme
2.3.3 Error Feedback Controller with Alternate Event-Triggered Mechanism
2.3.4 Modeling of the Closed-Loop System
2.3.5 Event-Triggered Asynchronous Output Regulation Problem
2.4 Main Results
2.5 Illustrative Example
2.6 Conclusion
References
3 Output Regulation for Networked Switched Systems with Alternate Event-Triggered Control Under Transmission Delays and Packet Losses
3.1 Introduction
3.2 Problem Formulation
3.2.1 System Model and Asynchronous Controller Structure
3.2.2 Alternate Event-Triggered Mechanism with the Alternate Property
3.2.3 The Error Feedback Controller Based on the Alternate Event-Triggered Mechanism
3.2.4 The Resultant Closed-Loop Switched System
3.2.5 The Event-Triggered Output Regulation Problem Formulation
3.3 Main Results
3.3.1 Event-Triggered Output Regulation Problem with Transmission Delays
3.3.2 Event-Triggered Output Regulation Problem with Both Transmission Delays and Packet Losses
3.4 Comparison with Existing Results
3.5 Application to an F-18 Aircraft Model
3.6 Conclusion
References
4 Memory-Based Event-Triggered Output Regulation for Networked Switched Systems with Severely Unstable Dynamics
4.1 Introduction
4.2 Problem Formulation
4.2.1 System Modelling
4.2.2 Event-Triggered Mechanisms
4.2.3 Dynamic Output Feedback Controller with Event-Triggered Mechanisms
4.2.4 The Closed-Loop System
4.3 Main Results
4.3.1 Event-Triggered Asynchronous Output Regulation Problem with Only Network-Induced Delays
4.3.2 Event-Triggered Asynchronous Output Regulation Problem in the Presence of Network-Induced Delays, Packet Disorders, Packet Losses
4.4 Simulation Example
4.5 Conclusion
References
5 Almost Output Regulation for Switched Linear Systems with Bumpless Transfer Control Under Destabilizing Behaviors
5.1 Introduction
5.2 Problem Formulation
5.3 Main Results
5.4 Numerical Example
5.5 Conclusion
References
6 Mixed Event-Triggered Output Regulation for Networked Switched Systems with Severely Unstable Dynamics Under Long-Duration DoS Attacks
6.1 Introduction
6.2 Problem Formulation
6.2.1 System Model and Dynamic Output Feedback Controller
6.2.2 Long-Duration Denial-of-Service Attacks
6.2.3 Mixed Event-Triggered Mechanisms
6.2.4 The Closed-Loop Networked Switched System
6.2.5 Event-Triggered Output Regulation Problem
6.3 Main Results
6.4 Simulation Example
6.5 Conclusion
References
7 Dissipative Event-Triggered Output Regulation for Networked Switched Systems with Severely Unstable Dynamics Against Integrity Deception Attacks
7.1 Introduction
7.2 Problem Formulation
7.2.1 System Model and Dynamic Output Feedback Controller
7.2.2 Integrity Deception Attacks
7.2.3 Resilient Event-Triggered Mechanisms
7.2.4 Dynamic Output Feedback Controller Based on Resilient Event-Triggered Mechanisms
7.2.5 The Closed-Loop Networked Switched System
7.2.6 Event-Triggered Output Regulation Problem with Dissipative Property
7.3 Main Results
7.4 Simulation Example
7.5 Conclusion
References
8 Event-Triggered Multi-source Bumpless Transfer Control for Networked Switched Systems with Almost Output Regulation Against Switching Deception Attacks
8.1 Introduction
8.2 Problem Formulation
8.2.1 System Modelling
8.2.2 Event-Triggered Mechanism
8.2.3 Controllers Under Switching Deception Attacks
8.2.4 The Closed-Loop Switched System
8.2.5 Multi-source Bumpless Transfer Performance
8.2.6 Control Objectives
8.3 Main Results
8.3.1 Solvable Conditions for Almost Output Regulation of Unstable Switched Systems
8.3.2 Solvable Conditions for Bumpless Transfer Problems of Unstable Switched Systems
8.4 Simulation Example
8.5 Conclusion
References
9 Conclusions

Citation preview

Studies in Systems, Decision and Control 475

Lili Li Jun Fu

Output Regulation and Cybersecurity of Networked Switched Systems

Studies in Systems, Decision and Control Volume 475

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

Lili Li · Jun Fu

Output Regulation and Cybersecurity of Networked Switched Systems

Lili Li College of Marine Electrical Engineering Dalian Maritime University Dalian, China

Jun Fu State Key Laboratory of Synthetical Automation for Process Industries Northeastern University Shenyang, China

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-031-30971-7 ISBN 978-3-031-30972-4 (eBook) https://doi.org/10.1007/978-3-031-30972-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Acknowledgements

There are numerous individuals without whose constructive comments, useful suggestions and wealth of ideas this monograph could not have been completed. Special thanks go to Profs. Tianyou Chai and Dan Ma at Northeastern University, Prof. Tieshan Li at the University of Electronic Science and Technology of China and Prof. Albertos at the Polytechnic University of Valencia, for their valuable suggestions, constructive comments and support. Thanks also go to our students, Linyang Song, Yu Zhang, Mingzhe Ju, Yalin Chen, Zhilin Yu and Xiaowei Zhao for their commentary. The authors are especially grateful to their families for their encouragement and never-ending support when it was most required. Finally, we would like to thank the editors at Springer for their professional and efficient handling of this project. The writing of this book was supported in part by the National Natural Science Foundation of China (61825301, 62273068, 51939001), the National Key Research and Development Program of China (2018AAA0101603) and the Natural Science Foundation of Liaoning Province (2023-MS-120). Dalian, China Shenyang, China June 2023

Lili Li Jun Fu

v

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Status of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Current Status of Switching Rules Design for Switched Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Research Status of Event-Triggered Control in Switched System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Research Status of Security Control in Switched Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Research Status of the Bumpless Transfer Control of Switched Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Research Status of Output Regulation Problems in Switched Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Proposed Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Basic Definitions and Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Event-Triggered Output Regulation for Networked Flight Control System Based on an Asynchronous Switched System Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Flight Control System Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Problem Formulation and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Error Feedback Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Alternate Event-Triggered Communication Scheme . . . . . . . 2.3.3 Error Feedback Controller with Alternate Event-Triggered Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Modeling of the Closed-Loop System . . . . . . . . . . . . . . . . . . . 2.3.5 Event-Triggered Asynchronous Output Regulation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 10 10 12 13 15 15 16 19 20 23

29 29 31 33 33 34 36 37 37

vii

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2.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Output Regulation for Networked Switched Systems with Alternate Event-Triggered Control Under Transmission Delays and Packet Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 System Model and Asynchronous Controller Structure . . . . 3.2.2 Alternate Event-Triggered Mechanism with the Alternate Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 The Error Feedback Controller Based on the Alternate Event-Triggered Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 The Resultant Closed-Loop Switched System . . . . . . . . . . . . 3.2.5 The Event-Triggered Output Regulation Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Event-Triggered Output Regulation Problem with Transmission Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Event-Triggered Output Regulation Problem with Both Transmission Delays and Packet Losses . . . . . . . . 3.4 Comparison with Existing Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Application to an F-18 Aircraft Model . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Memory-Based Event-Triggered Output Regulation for Networked Switched Systems with Severely Unstable Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 System Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Event-Triggered Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Dynamic Output Feedback Controller with Event-Triggered Mechanisms . . . . . . . . . . . . . . . . . . . . . 4.2.4 The Closed-Loop System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Event-Triggered Asynchronous Output Regulation Problem with Only Network-Induced Delays . . . . . . . . . . . . . 4.3.2 Event-Triggered Asynchronous Output Regulation Problem in the Presence of Network-Induced Delays, Packet Disorders, Packet Losses . . . . . . . . . . . . . . . . . . . . . . . .

38 45 48 48

51 51 53 53 54 56 57 58 58 58 65 68 69 72 73

75 75 77 77 78 79 81 81 82

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ix

4.4 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92 97 98

5 Almost Output Regulation for Switched Linear Systems with Bumpless Transfer Control Under Destabilizing Behaviors . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 103 105 112 115 116

6 Mixed Event-Triggered Output Regulation for Networked Switched Systems with Severely Unstable Dynamics Under Long-Duration DoS Attacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 System Model and Dynamic Output Feedback Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Long-Duration Denial-of-Service Attacks . . . . . . . . . . . . . . . 6.2.3 Mixed Event-Triggered Mechanisms . . . . . . . . . . . . . . . . . . . . 6.2.4 The Closed-Loop Networked Switched System . . . . . . . . . . . 6.2.5 Event-Triggered Output Regulation Problem . . . . . . . . . . . . . 6.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Dissipative Event-Triggered Output Regulation for Networked Switched Systems with Severely Unstable Dynamics Against Integrity Deception Attacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 System Model and Dynamic Output Feedback Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Integrity Deception Attacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Resilient Event-Triggered Mechanisms . . . . . . . . . . . . . . . . . . 7.2.4 Dynamic Output Feedback Controller Based on Resilient Event-Triggered Mechanisms . . . . . . . . . . . . . . . 7.2.5 The Closed-Loop Networked Switched System . . . . . . . . . . . 7.2.6 Event-Triggered Output Regulation Problem with Dissipative Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 119 121 121 122 123 124 126 127 139 145 145

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7.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Event-Triggered Multi-source Bumpless Transfer Control for Networked Switched Systems with Almost Output Regulation Against Switching Deception Attacks . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 System Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Event-Triggered Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Controllers Under Switching Deception Attacks . . . . . . . . . . 8.2.4 The Closed-Loop Switched System . . . . . . . . . . . . . . . . . . . . . 8.2.5 Multi-source Bumpless Transfer Performance . . . . . . . . . . . . 8.2.6 Control Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Solvable Conditions for Almost Output Regulation of Unstable Switched Systems . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Solvable Conditions for Bumpless Transfer Problems of Unstable Switched Systems . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157 170 176 176

179 179 181 181 182 183 185 185 186 187 187 193 195 199 200

9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Chapter 1

Introduction

1.1 Background With the rapid development of transportation, aerospace, industrial manufacturing and other industries, as well as the increasing scale and complexity of all kinds of control systems, the complexity, automation and performance requirements of modern control systems are also increasing, which challenges the traditional control methods and theories. The proposal and development of switched system control theory and method have a profound engineering background. The switched system has important applications in effectively describing the typical switched system models with multi-mode, multi-controller, multi-level and other characteristics in engineering practice [1–3], such as network control system, aircraft control system, robot system. The research on it can not only promote the rapid development of modern control theory, but also provide effective methods to solve many practical problems. Based on this, scholars continue to study the switched system deeply and extensively. In fact, examples of switched systems can be found everywhere in daily life. Here are two examples of switched systems. Example 1.1 The two-tank system [4] As shown in Fig. 1.1, a two-tank system consists of two tanks. Among them, the water is injected into the tank at a fixed speed w through the hose, and the two water tanks are drained out at a fixed speed v1 and v2 respectively, and the hose can only inject water into one tank at each instant. The control objective is to keep the water levels of the two tanks above the pre-specified water levels r1 and r2 . Therefore, it is necessary to design a switched controller to realize the switching of the water supply hose between the two tanks. It is assumed that the switching is completed instantaneously, and the initial water level is above the specified water level, that is, x1 ≥ r1 , x2 ≥ r2 . When the water level in a tank drops below the specified water level, the controller switches the hose to fill the tank with water.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Li and J. Fu, Output Regulation and Cybersecurity of Networked Switched Systems, Studies in Systems, Decision and Control 475, https://doi.org/10.1007/978-3-031-30972-4_1

1

2

1 Introduction

Fig. 1.1 The two-tank system

The two-tank system can be described by a switched system. The two continuous states x1 and x2 of the system represent the water levels of the two tanks, respectively, and the two discrete states q1 and q2 of the system represent the injection of water into tank 1 and tank 2, respectively. When filling tank 1 (mode q1 ) with water, the continuous dynamics are |

x˙1 x˙2

|

|

| w − v1 = , −v2

When filling tank 2 (mode q2 ) with water, the continuous dynamics are |

x˙1 x˙2

|

|

| −v1 = . w − v2

The discrete dynamics is the{ switching of q1 |→ q2 and } q2 → q1 . The switching condition of q1 → q2 is{ A12 = [x1 x2|]T ∈ R 2 |}x2 ≤ r2 . The switching conditions for q2 → q1 is A21 = [x1 x2 ]T ∈ R 2 |x1 ≤ r1 . The switched system is shown in Fig. 1.2. Example 1.2 [5] Spherical Inverted Pendulum (SIP) system. As shown in Fig. 1.3, the spherical pendulum is mounted on a cart through an unactuated joint (θ, ϕ). The cart can move in the x − y plane by two orthogonal planar forces u x and u y . Then the dynamics of the SIP can be described as x2 ≤ r2

q1

q2 .x = −v 1 1

.x = w − v 1 1 .x = −v

.x = w − v 2 2

x2 ≥ r2

x1 ≥ r1

2

2

x1 ≤ r1 Fig. 1.2 Two-tank system as a switched system

1.1 Background

3

Fig. 1.3 Spherical inverted pendulum [5]

x¨ = u x , y¨ = u y , θ¨ = sin θ cos θ − ϕ˙ 2 sin θ cos θ − u x cos θ + u y sin θ sin ϕ, 1 ϕ¨ = (cos θ sin θ + 2θ˙ ϕ˙ sin θ cos θ − u y cos θ cos ϕ), cos2 θ where (x, y) is the normalized planar location of the center of the mass of the cart and angles θ and ϕ denote rotation about the x-axis and y-axis. The nonlinear SIP dynamics can be described by the switching of two subsystems: q˙ ∈ { f 1 (q, u), f 2 (q, u)}, where f 1 (q, u) = f (q, u)|u y =0 , f 2 (q, u) = f (q, u)|u x =0 , q = (x, x, ˙ y, y˙ , θ, θ˙ , ϕ, ϕ) ˙ is the state of the system and indices 1 and 2 represent the SIP under x and y axises actuation separately. The switched nonlinear system has only one continuous input variable u, denoting the composed force magnitude, while the force direction is determined by the active subsystem index. Linearizing the nonlinear SIP around its vertical equilibrium point (q, u) = 0, the switched linear system is obtained as follows ⎧ ˙ = A p x(t) + B p u(t) + Adp d(t), ⎨ x(t) y(t) = C p x(t) + Ddp d(t), ⎩ z(t) = E p x(t) + F p u(t) + Fdp d(t), where x ∈ R x is the system state, u ∈ R u is the control input, y ∈ R y is the measured output, z ∈ R z is the controlled output, d ∈ R d is the disturbance input, p ∈ {1, 2} means that the p-th subsystem is activated, and

4

1 Introduction

⎤ ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ 000 0 0 0 ⎢1 ⎥ ⎢0 ⎥ ⎢0 ⎥ 0 0 0⎥ ⎥ ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ ⎢ ⎢ ⎢0 ⎥ 0 0 0⎥ ⎥ ⎢0 ⎥ ⎢0 ⎥ ⎢ ⎥ ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ 0 0 0⎥ ⎢0 ⎥ ⎢1 ⎥ ⎢0 ⎥ ⎥, B1 = ⎢ ⎥, B2 = ⎢ ⎥, Bd1 = Bd2 = ⎢ ⎥, ⎢0 ⎥ ⎢0 ⎥ ⎢ 0.5 ⎥ 1 0 0⎥ ⎥ ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ ⎢ −1 ⎥ ⎢0 ⎥ ⎢0 ⎥ 0 0 0⎥ ⎥ ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ ⎣0 ⎦ ⎣0 ⎦ ⎣ 0.2 ⎦ 0 0 1⎦ 010 0 −1 0 ⎤ ⎤ ⎡ 10000000 10000000 ⎢0 0 1 0 0 0 0 0⎥ ⎢0 0 1 0 0 0 0 0⎥ ⎥ ⎥ ⎢ E1 = E2 = ⎢ ⎣ 0 0 0 0 1 0 0 0 ⎦, C 1 = C 2 = ⎣ 0 0 0 0 1 0 0 0 ⎦, 00000010 00000010 | }T | }T | }T = Dd2 = 0 0 0 0 , F1 = F2 = 0 0 0 0 , Fd1 = Fd2 = 0 0 0 0 . ⎡

0 ⎢0 ⎢ ⎢0 ⎢ ⎢ ⎢0 A1 = A2 = ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎣0 0

Dd1

1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 0 0 1 0 0 ⎡

Switched systems are a special class of hybrid systems that can be seen as consisting of a set of continuous (or discrete) time systems and a rule that determines how each system switches to the other, and the operation of the entire switched system is determined by this switching rule [6, 7]. The operation of the whole switched system is determined by this switching rule. This rule is also known as a switching law, switching signal or switching function, which is usually a segmented constant-value function dependent on state or time. A continuous-time switched system consisting of m subsystems is described by the following mathematical model [8] {

x(t) ˙ = f σ (t, x(t), u(t), ω(t)), x(t0 ) = x0 , y(t) = h σ (t, x(t), ω(t)),

(1.1)

where x ∈ R x is the system state, u ∈ R u is the control input, y ∈ R y is the output of the system, ω ∈ Rω represents external signals (e.g. interference), the segmented constant value function σ taken from M /\ {1, 2, . . . , m} represents the switching rule, and for each i ∈ M, f i , gi is a smooth vector field. A simple block diagram of the switched system is given in Fig. 1.4. Fig. 1.4 Schematic diagram of the switched system

Subsystem 1

Subsystem 2 ... Subsystem m

Switching rules

Output

1.1 Background

5

As can be seen in Fig. 1.4, the switched system is essentially a multi-model structure, with each of its sub-models {

x(t) ˙ = f i (t, x(t), u(t), ω(t)), x(t0 ) = x0 , y(t) = h i (t, x(t), ω(t)),

(1.2)

called a subsystem of the switched system (1.1), i ∈ M. The dynamics of the switched system are determined by the dynamics of the subsystems together with the switching rules. The switching rules can depend on the time, the state variables of the system, the output variables, the external signals of the system, and their past values, which can be expressed in general form as σ (t + ) = ϕ(t, σ (t − ), x(t), y(t), w(t)), t ≥ 0 with σ (t + ) = lim+ σ (t + h), σ (t − ) = h→0

lim− σ (t + h). σ (t) = i indicates that the i-th subsystem is activated. Clearly, there

h→0

is one and only one subsystem that is activated at any given instant. Furthermore, it is assumed that the switching rule σ has a finite number of switches over a finite time interval. For the same switched system, the implementation of different switching rules may produce completely different system behavior, so the design of the switching rules plays a pivotal role in the synthesis of the switched system. The following two examples are given to illustrate that the selection of reasonable switching rules will make the switched system stable, while inappropriate switching rules will also lead to the instability of the whole switched system. Example 1.3 Consider the following switched linear system. x(t) ˙ = Aσ x(t), σ ∈ {1, 2}, |

(1.3)

| | | −0.5 −0.4 −0.5 −3 where A1 = , A2 = . 3 −0.5 0.4 −0.5 It is easy to verify that both subsystems of the switched system are stable, and their state trajectories are shown in Fig. 1.5a, b. In the following, two different switching rules are chosen so that the switched system obtains two very different results, i.e. the system is stable and unstable. Starting from an arbitrary initial state, first take the switching rule (I) as follows: when the system trajectory enters the first and third quadrants, the switched system switches to the second subsystem; when the system trajectory enters the second and fourth quadrants, the switched system switches to the first subsystem. The phase plane of its state trajectory is shown in Fig. 1.5c, and the switched system obtained is asymptotically stable. The switching rule (II) is then chosen as follows: when the system trajectory enters the first and third quadrants, the switched system switches to the first subsystem when the system trajectory enters the second and fourth quadrants, the switched system switches to the second subsystem. The phase plane of the state trajectory is shown in Fig. 1.5d, and the obtained switched system is unstable. Example 1.4 Consider the following switched linear system.

6

1 Introduction

Fig. 1.5 Phase trajectories of the switched system (1.3)

x(t) ˙ = Aσ x(t), σ ∈ {1, 2},

(1.4)

|

| | | −1.9 0.6 0.1 −0.9 , A2 = . 0.6 −0.1 0.1 −1.4 The eigenvalues of A1 are λ1.1 = 0.0817 and λ1.2 = −2.0817, and those of A2 are λ2.1 = 0.0374 and λ2.2 = −1.3374. It is clear that both subsystems are unstable, and their state trajectories are shown in Fig. 1.6a, b. Starting from an arbitrary initial state, a reasonable switching rule (I) is firstly selected, that is, the switching signal σ meets the dwell time (DT) requirement τmin = τmax = 2s. The state trajectory is shown in Fig. 1.6c. Clearly, the switched system can be stabilized by the switching signal. Then, a switching rule (II) that does not meet the requirements is selected, and its state trajectory is shown in Fig. 1.6d. The resulting switched system is unstable. where A1 =

As can be seen from the above examples, the stability of the switched system is not inherited from the stability of the subsystems. Even if all subsystems are stable, inappropriate switching rules can lead to instability of the whole switched system. On the contrary, even if the subsystems are unstable, the stability of the switched system can be achieved through a reasonable design of the switching rules, where the

1.1 Background

7

Fig. 1.6 State trajectories of the switched system

decrement of the Lyapunov function at the switching instant is used to compensate for the increment of the Lyapunov function when each subsystem is operated individually. However, in practical engineering, part of the switching instants may also be unstable due to faults, external disturbances and other factors, that is, the Lyapunov function at the switching instants increase, which is a severely unstable dynamic (SUD), then only through the decrement of the Lyapunov function at stabilizing switching instants to offset the increment of the Lyapunov function at destabilizing switching instants and when each subsystem operates separately. The design of the switching rules, therefore, plays an important role in ensuring the stability of the switched system. As the area of controlled systems grows, it is often necessary to exchange information between controllers, actuators and sensors via shared communication networks. The open network structure between switched system subsystems and controllers in a networked environment increases productivity and at the same time raises problems such as network-induced delays, packet loss and other network-induced phenomena due to bandwidth limitations, as well as communication limitations and cyber attacks.

8

1 Introduction

The interaction between switching behavior, SUDs, cyber attacks, communication constraints and external disturbances makes the analysis and control of switched systems very difficult. On the one hand, not all sampled data provides a significant improvement in system performance, and large amounts of data transmission can lead to network congestion and data leakage. Event-triggered mechanisms (ETMs) can effectively mitigate communication constraints and actuator wear and tear, save limited network bandwidth and reduce energy consumption, as network transmissions only take place when triggering conditions are met [9, 10]. The ETM is also effective in reducing energy consumption. When event triggering is designed in conjunction with switching rules, the transmission of system data can be adjusted to the actual state of each subsystem, allowing event triggering control of switched systems to achieve these benefits while ensuring the desired performance of the switched system. However, for severely unstable switched systems, the ETM can induce asynchronous switching and even destroy the performance of the switched system, so its event-triggered control synthesis deserves further study. On the other hand, the information security incident represented by Stuxnet [11] shows that open network environments are inevitably threatened by cyber attacks, which have serious implications for the security operation of the system [12]. In recent years, there have been incidents in which property was lost and national security was threatened due to cyber attacks. In 2017, the “Wanna Cry” virus ravaged the world, and the attack was estimated to have affected more than 200,000 computers across 150 countries, with total damages ranging from hundreds of millions to billions of dollars [13]. Since 14 February 2022, Ukraine’s critical military, government, education and financial sectors have been subjected to numerous large-scale distributed denial-of-service (DoS) attacks on their network systems. The distributed DoS attacks paralyzed many critical infrastructures and important network systems in Ukraine, severely affecting the social order and the operational command and control of the Ukrainian military, and significantly weakening Ukraine’s wartime operational capabilities [14]. Due to the open nature of the network between subsystems and controllers, switched systems are vulnerable to the threat of cyber attacks. Most of the current research focuses on typical DoS attacks, deception attacks, etc. However, in reality, compared to non-switched systems, the switching signals and control signals transmitted via the network are affected by the attack at the same time, resulting in consecutive asynchronous switching of the subsystems as well as the controllers. Therefore, it is necessary to study the event-triggered security control of switched systems to improve their ability to cope with cyber attacks. Furthermore, because of physical limitations of the system or signal switching, the control input signal of the system is usually discontinuous at certain instants in the actual industrial process, as shown in Fig. 1.7. The phenomenon of abrupt signal jumps is known as bumps. Considering the characteristics of the switched system, as well as the ETM and cyber attacks mentioned above, whenever a series of events such as switching, data updating, and cyber attacks occur, abrupt and large variations in the control signal will appear. As a result, the control input signal may change frequently, causing bumps to occur on a regular basis. However, many industrial process control fields, such as aerospace, power

1.1 Background

9

Fig. 1.7 Schematic diagram of control signal bump

systems, robotics, nuclear industry, coal mining, do not want signal bumps because they are an undesirable transient behavior that will cause the transient performance of switched systems to decline and even destroy stability. As a result, it is necessary to investigate how to suppress control signal bumps. Bumpless transfer control (BTC) is an effective method for suppressing the bump phenomenon, which can improve the system’s transient performance [15]. The level of suppression of control signal bump is called bumpless transfer performance (BTP) [16]. BTC has received a lot of attention in recent years, and many research results have achieved good results [17–19]. It is worth noting that the majority of articles concentrate on the bump caused by switching behavior. The bump phenomenon will be caused not only by the switching behavior of the unstable switched system in the cyber environment but also by the data update of subsystems and controllers and cyber attacks. The increased number of bump sources causes problems with conflict and superposition of bumps from different sources, posing significant challenges in the research of BTC of destabilizing switched systems. As a result, research into the BTC problem has high research value for improving the reliability, security, and transient performance of destabilized switched systems. In addition, in order to deal with the increasing accuracy requirements of various control systems, the increasingly complex operating environment and the increased sources of system interference, control methods that can suppress interference should be used in the system design process [20]. The output regulation problem (ORP) is a mathematical formulation of practical control problems such as vibration suppression in high-speed trains, speed control of car engines, interference suppression in aircraft, take-off and landing of aircraft in bad weather conditions, coordination and manipulation of robots. The ORP is one of the central problems in the field of control theory and applications and has attracted the attention of the control community for decades and has been a driving force in the development of modern control theory and applications. While tracking and/or disturbance suppression problems are common tasks in control system design and can be handled by a variety of methods, the ORP differs from other approaches to tracking and disturbance suppression problems in that it aims to deal with the reference input signal and external disturbances generated by a system of differential equations. This system of differential equations is called the outer system and the signal generated by the outer system is called the external

10

1 Introduction

signal, which simulates the reference input and disturbance signals. The control system stabilization problem, tracking problem, disturbance attenuation problem and so on can be considered as special cases of the ORP. For example, if the external disturbance to the system is zero, then the ORP can be reduced to the system’s calming problem; if the external system is constant, then the ORP can be reduced to the system’s output tracking problem. Due to the special structural nature of the switched system, even if the output regulation (OR) of each subsystem is solvable, inappropriate switching rules cannot guarantee the OR of the whole switched system. On the contrary, if the OR of each subsystem is not solvable, the OR of the switched system can be achieved by designing a reasonable switching rule. Therefore, it is very meaningful to study the OR of the switched systems.

1.2 Status of Research 1.2.1 Current Status of Switching Rules Design for Switched Systems Due to the special composition of the switched system, if the switching rules are not designed properly, even if all the subsystems are stable, it is not possible to ensure the stability of the switched system. On the contrary, if the switching rules are designed properly, even if all the subsystems are unstable, the switched system can achieve stability. Therefore, the design of the switching rules is essential to achieve the stability of the switched system. At present, the main design methods of switching rules include the common Lyapunov function method [21, 22], the multiple Lyapunov function method [23], the switching Lyapunov function method [24], the average dwell time (ADT) method [25], and the mode-dependent average dwell time (MDADT) method [26], etc. It is worth noting that the switching rules design based on the ADT framework has become the most widely used approach due to its simplicity and flexibility. When all the subsystems are stable, the switched system can be guaranteed to be stable as long as the switching is guaranteed to be slow enough. In the literature [25], the definition of ADT is proposed to achieve slow switching by limiting the mean interval between any two consecutive switching instants to not less than a constant, which is determined by the common incremental coefficient of Lyapunov functions with adjacent modes at switching instants and the common decay rate coefficient of all subsystems. In the literature [26], the concept of MDADT is proposed, which is characterized by the fact that the lower bound constant of the ADT interval, the incremental coefficient and the decay rate coefficient are dependent on each subsystem. By constraining the upper bound on the ratio of the total activation time of the unstable/ stable subsystem, the above results are extended in the literature [27] to the presence of partial subsystem instability, using the decrease in the Lyapunov function when the stable subsystem is activated and at the stability switching instant to offset the

1.2 Status of Research

11

increase in the Lyapunov function due to the activation of the unstable subsystem. In the literature [28], slow switching rules and fast switching rules are designed for the stable and unstable subsystems, respectively, to ensure the stability of the whole switched system. For the switched system in which all subsystems are unstable, the core idea is to compensate for the state divergence of the unstable subsystem by using the state convergence at the switching instant. In recent research results, the restriction on the subsystems is further relaxed to the case where all subsystems are unstable, where the idea of the literature [25] is no longer applicable and only the decrease of the Lyapunov function at the stabilizing switching instant can be used to offset the increase of the Lyapunov function when each subsystem is running. When all subsystems are unstable but all switching instants are stable, the literature [29] firstly increases the number of switching with the help of discretized Lyapunov functions in the framework of DT by constraining the maximum and minimum DT of each subsystem while avoiding frequent switching and achieved the purpose of offsetting the influence of unstable subsystems with the help of stabilizing switching instants. On this basis, some literature has carried out in-depth discussions on systems and problems around switched positive systems [30–32], impulsive stabilization [33, 34], and fault detection [35]. The literature [36] simplifies the form of Lyapunov functions and extends the results to the stabilization problem and H∞ control. A periodic switching signal design scheme is proposed in the literature [37]. However, the requirement that the switching behavior at all switching instants is stable is also a very strict constraint. When all subsystems are unstable and some switching instants are unstable (severely unstable), the literature [38] balances the increment of the Lyapunov function at the SUDs and the decrement at the stabilizing switching instants by limiting the ratio of destabilizing and stabilizing switching instants in each switching period, where the destabilizing and stabilizing switching behaviors need to be periodically arranged according to certain constraints and each subsystem also need to be grouped by the type of switching behavior. In the literature [39], by constructing piecewise Lyapunov functions, the Lyapunov function decreasing intervals and decreasing points are added to the activated intervals of each subsystem to offset the Lyapunov function growth in other activated intervals and at the switching instants, which reduces the computational complexity, but a stable activated interval must exist for each subsystem. The literature [40] uses the properties of a two-dimensional matrix associated with a given parameter to estimate the state dispersion at two consecutive switching instants, constraining the number of destabilizing and stabilizing switching instants in a pre-given, fixed-length subsequence of switching instants, relaxing to some extent the restriction on the periodic arrangement of destabilizing and stabilizing switching instants in the literature [38] and avoiding the use of discretized Lyapunov functions.

12

1 Introduction

1.2.2 Research Status of Event-Triggered Control in Switched System Using a time-triggered control strategy in networked control systems (NCSs) will take up too much communication and computational resources, and the wear and tear of the actuator will be more severe. To overcome these drawbacks, an ETM is proposed in the literature [41] to ensure the system control performance while judging whether the triggering condition holds according to certain real-time information of the system, and then adjusting the sampling interval for on-demand control. With the close integration and development of network communication technology and modern control technology, ETMs are gradually favored by scholars in the field of network control systems, and many important research results have emerged [42, 43]. By applying the ETM to the switched system and making it designed in cooperation with the switching rule, the system data transmission can be adjusted in time according to the actual state of each subsystem, which enables the switched system to ensure the expected performance while overcoming the above-mentioned drawbacks. The ETMs of switched systems can be divided into two categories: ETMs with a single triggering condition and ETMs with multiple triggering conditions. In terms of ETM of switched systems with a single triggering condition, the literature [44] analyzes the event-triggered stabilization problem of switched linear systems based on consecutive ETM with state information using the ADT method. The literature assumes that there is no subsystem switching or only one subsystem switching within the adjacent event trigger time interval, which makes the results conservative to a certain extent. The literature [45] firstly gives the minimum interval time of two consecutive event triggers to exclude the Zeno phenomenon and then proposes a consecutive ETM based on the output information to consider the event triggering stabilization problem of the switched linear system by discussing the relationship between the switching instants and the triggering instants. The literature [46] investigates the stability of the switched system based on consecutive ETM by considering the case of multiple events occurring during the active period of a switched subsystem and gives a lower bound on the time interval between the occurrence of two consecutive events to avoid the Zeno phenomenon. To avoid the phenomenon from the design of the triggering mechanism, the finite-time bounded and finite-time H∞ performance of networked switched systems (NSSs) is considered in the literature [47] based on the periodic ETM associated with the previous trigger instant. However, the ETM of the switched system passes both the system data and the system mode to the controller, and if the trigger instant does not match the switching instant, asynchronous switching will inevitably occur, and it may even result in a bad asynchronous situation where the controller mode cannot match the subsystem mode for the whole subsystem activation time. Considering the asynchronous situation caused by switching, the literature [48] designs a consecutive ETM with data transfer and mode transfer functions to give the stability of the switched system under asynchronous switching. In the literature [49], a periodic ETM related to the current

1.2 Status of Research

13

instant state is designed to consider the finite time control problem of NSSs under asynchronous switching. Since the event triggering of switched systems with a single triggering condition as mentioned above cannot cope well with the impact of asynchronous switching, cyber attacks, and unstable subsystems on the switched system. Thus, It is necessary to propose an ETM with multiple triggering conditions [50–53]. First, in order to weaken the effect of asynchronous switching as much as possible, it is necessary to add mode comparison triggering conditions to the ETM, e.g., the literature [51] proposes an ETM incorporating DT, which ensures that the system data must be updated once during the activation time of the subsystem. The ETMs in the literature [50, 52] add mode comparison triggering conditions to ensure that the data is transmitted as soon as the mode of the switched system changes or at the nearest sampling instant, which minimizes asynchronous switching triggered by the mismatch between the trigger instant and the switching instant. Second, when the switched system is attacked by DoS, the system cannot communicate normally, and thus the performance of the switched system is seriously affected. In order to mitigate the impact of communication interruption caused by DoS attack, the literature [53] adds a triggering condition that is triggered once at the end instant of the impact of DoS attack, which weakens the adverse impact of DoS attack on the switched system. Finally, for the switched system containing partially unstable subsystems, the literature [50] designs two different triggering conditions at the activation time of stable and unstable subsystems, respectively, to improve the performance of the switched system containing unstable subsystems.

1.2.3 Research Status of Security Control in Switched Systems With the development of network technology, NSSs have become a hot research topic. Owing to the strong openness of the network, cyber attacks such as DoS attacks can easily be injected into NSSs. The normal operation of the system is seriously damaged. Due to the mutual influence of switching behaviors and cyber attacks, there are complicated relations between switching instants and starting/ending instants of cyber attacks, which leads to more challenges. The majority of current research in the area of security control of switched systems has been directed at the most typical cyber attacks: DoS attacks and deception attacks. DoS attack is a highly destructive network attack. It causes network interruption by consuming network resources, so that the remote controller and actuator cannot receive feedback signals or control signals normally, thus destroying the stability of the system. Firstly, for the switched system that suffers from DoS attack and loses the trigger signal, the literature [54] analyzes the security of the switched system in which all subsystems are stable under synchronous switching by designing an output feedback controller oriented to the random DoS attack satisfying Bernoulli

14

1 Introduction

probability distribution in sensor-to-controller (S2C) channels. In the literature [55], a discrete uncertain switched system under random DoS attack in the S2C channel is analyzed. By designing the integrated event trigger mechanism, the dynamic output feedback sliding mode controller of DoS attack information and the synchronous switching rule, the mean square exponential stability of uncertain switched system is ensured that all subsystems are stable, so as to effectively reduce the impact of DoS attack on the system performance. In the literature [50], a constrained DoS attack model is established to limit the frequency and duration of DoS attacks, and the corresponding state observer, event trigger mechanism, and synchronous switching rules are designed for the switched system containing some unstable subsystems to realize the security control of the switched system under DoS attacks in the S2C channel. Secondly, for the switched system that suffers from DoS attack and loses the trigger signal as well as the switching signal thus causing asynchronous switching, the literature [56] portrays a dual-terminal periodic DoS attack model, establishes a collaborative design method of ADT and switching trigger parameters for the switched system in which all subsystems are stable, and clarifies the coupling relationship between DoS attack, synchronous and asynchronous periods to cope with DoS attack and the complex stability analysis brought by its induced asynchronous switching. Here the dual terminals refer to the S2C channel and the controller-to-actuator (C2A) channel. The literature [57] discusses in detail the multi-case asynchronous switching due to dual-terminal DoS attacks and network-induced delays, providing a joint design scheme of controller gain, switching rules, and event-triggered parameters. The literature [53, 58] consider the DoS attack model with constrained attack frequency and duration, and the literature [53] proposes a corresponding event-triggered scheme to deal with DoS attack-induced transmission delay by combining DoS attack information, and establishes a control criterion for event triggering H∞ control of the switched system that is stable for all subsystems under DoS attack using the ADT approach. As an important kind of cyber attacks with strong concealment, the deception attack affects the operation of sensors, controllers, and actuators in NSSs [59–63] by tampering with data packets, which is often difficult to be detected. For switched systems threatened by deception attacks, which may result in the controller receiving wrong trigger signals, the literature [59, 62, 63] consider deception attacks in S2C channels and investigate the security control of switched systems with all stable subsystems under synchronous switching. For the dual-terminal deception attacks, the literature [60] proposes a security control of switched system method with all stable subsystems under synchronous switching. However, the switched system subject to deception attack, the controller side may receive not only the wrong trigger signal but also the wrong switching signal, which leads to asynchronous switching. The literature [61] considers the asynchronous switching of the system under the deception attack in the S2C channel, and achieves the security control of the switched system with all stable subsystems by designing an adaptive ETM.

1.2 Status of Research

15

1.2.4 Research Status of the Bumpless Transfer Control of Switched Systems The BTC strategy was initially proposed to suppress the damage to the transient performance of the system caused by a sudden signal change caused by controller switching [64], and then gradually evolved linear interpolation pattern and L 2 norm constraint technique and other methods in combination with the needs of engineering applications [15, 16, 65], which ensure the smoothness and continuity of the control signal by adopting an appropriate constraint/compensation strategy. The majority of switched system analysis and synthesis focuses on steady-state performance, and reducing or suppressing the undesirable transient behavior caused by discontinuous changes in the control signal is crucial to enhance the overall performance of the system. However, the traditional BTC techniques for switched systems rely on the modification of pre-designed controllers under pre-known switching sequences [66–68]. A linear interpolation pattern with the bumpless transfer (BT) property, which does not require the switching sequence to be known, is proposed in the literature [69] since the switching sequence of switched systems is often difficult to be known in advance. To realize a smooth transition of the control signal, the approach performs the linear interpolation of previously built controllers within the minimal DT. Subsequently, the literature [17] proposes a BTC strategy that limits the amplitude of some control signals during the total running interval of the system and achieves smooth switching by imposing parameter limits [17, 70] on all subsystem controller gains or designing compensators and filters [71, 72]. Following that, switched system scholars have continued to push for improvements to this control signal amplitude limiting method by incorporating switched system characteristics. The literature [70, 73, 74] designs new indexes of BTP by constraining the control signal amplitude only during the activation period of subsystems. As a result, the essence of BTC for a switched system is to suppress control signal bumps at the switching instant; it is more reasonable to reduce or suppress control signal amplitude only at the switching instant [75–80]. A BTP index is proposed in the literature [75], which restricts the control signal amplitude only at the switching instant to reduce the restriction on the controller gain. Due to the influence of the network, the networked switching controller will also appear as the control signal bump when updating the data for NSSs. By suppressing both the control signal bump at the controller switching instant and the controller data update instant, the literature [75] improves the BTP index.

1.2.5 Research Status of Output Regulation Problems in Switched Systems The ORP is a classical problem in control theory, which aims to design a feedback controller to make the output asymptotically track a predetermined set of orbits

16

1 Introduction

of a class while ensuring the stability of the closed-loop system under undesired perturbations in the system. The interaction of the continuous and discrete dynamics of the switched system makes the ORP of the switched system more difficult, but at the same time makes the problem more meaningful. A lot of research results have been obtained on the ORP of switched systems [81–86]. In the ORP of switched systems, the regulator equation is the key to transforming the ORP into a stabilization problem, where a general framework is established in the literature [87]. The literature [88] solves the ORP of a switched system under the assumption that the ORP of a convex combinatorial system is solvable. However, it is usually difficult to find such convex combinatorial systems. The literature [89] gives solvability conditions for the ORP of switched nonlinear systems using the ADT method. In the framework of common coordinate transformation, i.e., the regulator equations of each subsystem obtained by the same coordinate transformation have a common solution, the solvability condition of the ORP of a stochastic time-delay switched system is given in the literature [81] based on dissipativity. The literature [82] studies the ORP of the discrete switched system based on the incremental passivity, and solves the ORP of the controlled object by designing an error feedback controller. In the literature [83], the switching internal mode is used to solve the ORP of a switched nonlinear system with both stable and unstable subsystems. In the framework of non-common coordinate transformation. The literature [84] designs the state feedback controller and dynamic error feedback controller for the switched system respectively, and establishes almost output regulation (AOR) solvability criterion for the switched positive system. The literature [85] focuses on the switched system in the network environment, and proposes a joint design scheme of ETM, dynamic error feedback controller, switching rules based on MDADT and ORP solvability criteria. The literature [86] gives sufficient conditions for the solvability of the ORP of a discrete switched system by jointly designing the controller and switching rules without assuming the solvability of the subsystem ORP. In some severe operating conditions, the actual physical system may also be affected by other unknown disturbances, and the anti-disturbance effect of the OR control strategy is often unsatisfactory. Based on this, an AOR control strategy is proposed in the literature [84, 90]. This control strategy converts the almost ORP into a robust control problem by solving the regulator equations, while suppressing the external and unknown disturbances to the system. The literature [91] investigates the ORP under external disturbances, unknown disturbances, and unpredictable disturbances by designing a disturbance observer to observe unpredictable disturbances.

1.3 Proposed Problem Solving At present, the research of the OR and cyber security of NSSs mainly focuses on the switched system with stable subsystems, the situation of system output, as well as the control signals, suffered cyber attacks and the BTC technology for the control signal fluctuation caused by a single inducement. The above-mentioned research

1.3 Proposed Problem Solving

17

results have enriched the progress of research on the steady-state and transient performance of switched systems security control. However, with further research, some new problems which are NSSs with all modes unstable and partial switching instants destabilizing and the special impact of cyber attacks on the switched system such as switching signals under attack are found. This has generated many difficult and unsolved key problems for the existing research on the OR and cyber security problems of the NSSs, as follows. (1) Existing switching rules based on MDADT method require pre-designed relevant parameters. Due to such limitations, the design of ETM has a great conservatism. To better integrate event-triggered control into the design of switched systems, this book designs an event-triggered switching signal that satisfies the MDADT requirements and can be dynamically adjusted according to the triggering situation instead of being scheduled in advance, i.e., the switching instants would be affected by the event-triggered instants. Furthermore, in the known literature, the relevant achievements under the DT frame ensure the stability of the switched system with SUDs under the synchronous switching of subsystems and controllers by grouping the switching sequences by a fixed number and establishing the ratio relationship between stabilizing switching behavior and destabilizing switching behavior in each group. In fact, the occurrence of destabilizing switching behavior is not fixed, and the switching rule that limits the number of destabilizing switching instants within each group has some limitations. The asynchronous switching problem of NSSs with SUDs is not to be neglected because the asynchronous switching of subsystems and controllers will inevitably occur due to factors such as network-induced delay. In addition, the discrete Lyapunov function, which is widely used in the switched system with all modes unstable, is complicated to derive. Therefore, it is a difficult problem to establish switching rules which rely on a more relaxed relationship between the total number of stabilizing and destabilizing switching instants under asynchronous switching of subsystems and controllers, and to explore a new method that does not rely on discrete Lyapunov functions to ensure the stability of switched systems with SUDs. (2) In the existing literature, the periodic ETM has the disadvantage of many samples and wasting communication resources, while the consecutive ETM reduces the number of samples and effectively saves communication resources, but it may cause the Zeno phenomenon. According to the special structure of the switched system, this book proposes an alternate event-triggered mechanism (AETM) that can transmit both data and mode information. This ETM combines the advantages of consecutive ETM which can continuously detect data and avoid missing valid information, and periodic ETM which can exclude the Zeno phenomenon. At the same time, it can significantly reduce the number of data transmissions. Currently, most of the studies on event triggering for switched systems do not consider the network-induced constraints in NCSs, which will reduce the accuracy of the research results. Therefore, based on the AETM, it is important to study the ORP of switched systems in network

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transmission environments with time delay and packet loss. In addition, the ETM based on a single triggering condition is often difficult to meet the control performance under cyber attacks in the switched system with SUDs. Hence, according to the mode information, destabilizing switching behavior and cyber attacks of the switched system, it is necessary to design an ETM with multiple event triggering conditions for the switched system with SUDs to weaken the adverse effects of the above factors on the system. (3) At present, the research on DoS attacks mainly focuses on the case that all subsystems are stable and the duration of a single DoS attack is shorter than the DT of a subsystem. In fact, under some extreme conditions, the object of study may be a switched system with SUDs, and the single duration of DoS attacks may be longer than the total DT of several successively activated subsystems. Therefore, under the above type of DoS attacks, it is very challenging to study new switching rules and realize the security control of the switched system with SUDs. For the security control of switched systems under deception attacks, the existing researches focus on the case of a single attack function. Since the switched system has many different modes, the attacker may use different attack functions for subsystems in different modes to achieve the most serious attack effect. Therefore, considering only the case of a single attack function cannot guarantee the security of each subsystem, and it is necessary to characterize a deception attack model with multiple attack functions for the switched system. Secondly, when NSSs are subjected to deception attacks, most studies consider the tampering of system output and control signals. However, in practice, deception attacks for switched systems not only tamper with the system output and control signals, but also inevitably tamper with the switching signals, which may cause continuous asynchronous switching phenomena and seriously damage the system performance. Therefore, the security control problem of switched systems with SUDs under the above deception attacks has high research value and important significance. In addition, most of the existing research only considers a single type of cyber attacks, and due to the constant emergence of new cyber attacks, only considering one type of attack cannot guarantee the overall security of the switched system. How to characterize a cyber attack model with multiple attack types for switched systems and solve the security control problem of unstable switched systems under the influence of this attack is a key problem worth studying. (4) Most interpolation-type BTC strategies in the literature use the linear interpolation method to modify pre-designed controllers, which have the shortcomings of inflexible controller design and long controller switching transition times. A key problem worth studying is how to design a new controller interpolation method that directly designs the controller under pre-unknown switching sequences and shortens the transition time of controller switching. When considering cyber effects, the BTC method with the control signal constrained is more effective. This type of method can only suppress the bump of control signals caused by controller switching and data updating, according to the existing results. However, cyber attacks will also cause the bump of control signals,

1.4 Basic Definitions and Lemmas

19

and there will be conflicts and superposition between the bump of the three sources, causing larger damage to the transient performance of systems. As a result, a BTP index must be designed to balance the conflict and superposition of different sources of the bump in order to suppress the control signal bump caused by controller switching, data update, and cyber attacks.

1.4 Basic Definitions and Lemmas Several definitions are introduced that will be used throughout the book. Definition 1.1 Given the switching signal σ (t) for the closed-loop system of the system (1.1) is exponentially stable if for u(t) and initial conditions x(t0 ), there exist constants a > 0,b > 0 such that the solution of the system satisfies ||x(t)|| ≤ ae−b(t−t0 ) ||x(t0 )||, ∀t ≥ t0 . Definition 1.2 Given the switching signal σ (t) for the closed-loop system of the system (1.1) is mean-square exponentially stable if for u(t) and initial conditions x(t0 ), there exist constants a˜ > 0,b˜ > 0 such that the expectation of the solution of { } ˜ the system satisfies E ||x(t)||2 ≤ ae ˜ −b(t−t0 ) ||x(t0 )||2 , ∀t ≥ t0 . Definition 1.3 For the closed-loop system of the system (1.1) and the switched exosystem w(t) ˙ = Sσ (t) w(t), if there exists an appropriately designed switching signal σ (t), an ETM, and feedback controllers such that (i) When w = 0, the corresponding closed-loop system is exponentially stable. (ii) When w /= 0, with the zero initial condition, the solution of the corresponding closed-loop system satisfies lim e(t) = lim (Cσ (t) χ (t) + Q σ (t) w(t)) = 0. t→∞

t→∞

Then the event-triggered ORP for switched system (1.1) is said to be solvable. Definition 1.4 ([25]) For a switching signal σ (t) and each t2 ≥ t1 ≥ 0, let Nσ (t2 , t1 ) denote the number of discontinuities of σ (t) in the open interval (t2 , t1 ). We say that σ (t) has an ADT τa if there exist two positive numbers N0 (we call N0 the chatter bound here) and τa such that Nσ (t2 , t1 ) ≤ N0 + t2τ−ta 1 , ∀t2 ≥ t1 ≥ 0. Definition 1.5 For a switching signal σ (t) and any T ≥ t ≥ 0, let Nσ i (T, t) be the switching numbers that the i-th subsystem is activated over the interval [t, T ] and Ti (T , t) denote the total running time of the i-th subsystem over the interval [t, T ], i ∈ M. We say that σ (t) has an MDADT τai if there exists positive numbers N0i (we call N0i the mode-dependent chatter bounds here) and τai such that Nσ i (T, t) ≤ , ∀T ≥ t ≥ 0. N0i + Ti (T,t) τa To achieve our control objective, we introduce the following lemmas. Lemma 1.6 ([92]) For any matrices X, Y , with appropriate dimensions, inequality X T Y + Y T X ≤ X T X + Y T Y always holds.

20

1 Introduction

| S11 S12 , S11 ∈ Rn×n and Lemma 1.7 ([93]) For given symmetric matrices S = ∗ S22 S22 ∈ Rm×m , the following three inequalities are equivalent: |

(1) S < 0, T −1 (2) S11 < 0, S22 − S12 S11 S12 < 0, −1 T (3) S22 < 0, S11 − S12 S22 S12 < 0. Lemma 1.8 ([94]) For any positive definite matrix Z T = Z with appropriate dimensions, the scalar 0 ≤ d(t) ≤ d and the {vector function y(·) : t [−d(t), 0] → Rn such that the following inequality −d(t) t−d(t) y T (s)Z y(s) ds ≤ { {t t − t−d(t) y T (s) ds Z t−d(t) y(s) ds always holds. Lemma 1.9 ([95]) For any matrix Y T = Y positive definite and symmetric matrix U and constant λ, inequality −Y U −1 Y ≤ λ2 U − 2λY always holds.

1.5 Organization of the Book This book studies the ORPs and the cybersecurity for NSSs. The structure of the book is summarized as follows. This chapter has introduced the system description and some background knowledge, and also addressed the motivations of the book. In Chap. 2, the event-triggered ORP is investigated for a networked flight control system (NFCS) with a switched system approach. By properly scheduling triggered times of periodic sampling associated with continuous event-triggering and switching instants of each subsystem, an AETM based on the subsystem model is proposed to transmit the triggered information to the candidate controller. Since the subsystem may switch between two adjacent events while the corresponding controller does not switch, asynchronous switching may occur. Meanwhile, an event-triggered switching signal caused by asynchronous switching is designed with the MDADT approach and event-triggered instants. By constructing multiple Lyapunov functions in the framework of the input delay approach and designing an error feedback controller, the sufficient condition for event-triggered asynchronous ORP is solved. Finally, the proposed methods are proved to be effective by the F-18 aircraft model. In Chap. 2, the event-triggered ORP is investigated for a networked flight control system (NFCS) with a switched system approach. By properly scheduling triggered times of periodic sampling associated with continuous event-triggering and switching instants of each subsystem, an AETM based on the subsystem model is proposed to transmit the triggered information to the candidate controller. Since the subsystem may switch between two adjacent events while the corresponding controller does not switch, asynchronous switching may occur. Meanwhile, an event-triggered switching signal caused by asynchronous switching is designed with the MDADT approach and event-triggered instants. By constructing multiple Lyapunov functions in the

1.5 Organization of the Book

21

framework of the input delay approach and designing an error feedback controller, the sufficient condition for event-triggered asynchronous ORP is solved. Finally, the proposed methods are proved to be effective by the F-18 aircraft model. In Chap. 3, the event-triggered ORP is investigated for a class of linear NSSs. A mode-dependent AETM is integrated into NSSs with transmission delays and packet losses for the first time based on continuous detection and periodic sampling, and a joint-designed switching signal with the triggering information is presented. Under transmission delays and packet losses, a series of resultant error feedback controllers are synthesized in four cases based on the relationships between switching instants and event-triggered instants. Then, a criterion ensuring asynchronous OR performance is formed with transmission delays by the input delay method. With the presence of packet losses, a set of sufficient conditions is also dedicated to the proposed control issue by adding the restraint in the number of successive packet losses associated with a modified AETM. Eventually, a comparison and an F-18 aircraft model reveal the effectiveness of the obtained results in the simulation. In Chap. 4, the event-triggered asynchronous output regulation problem (EAORP) of an NSS with SUDs is studied, including all mode instabilities and partial switching moment instabilities, that is, the Lyapunov function increases the activation intervals of all subsystems and some switching instants. Firstly, a memory-based modecompared event-triggered mechanism (MMETM) for switched systems is proposed to effectively shorten asynchronous intervals, which employs historical sampled outputs and compares the mode of the current sampled instant and the adjacent sampled instant. Then, the maximum ADT for a novel switching signal is derived with a constraint on the ratio of total destabilizing switchings to total stabilizing switchings, which relaxes the requirement that the regular arrangement of destabilizing and stabilizing switchings. Moreover, with the help of different coordinate transformations in the EAORP, the discretized Lyapunov functions are no longer needed when synthesizing NSSs with USDs, and the asynchronous switching situation is also discussed. Afterward, by designing a dynamic output feedback controller (DOFC), sufficient conditions are given to solve the EAORP for NSSs with USDs subject to network-induced delays, packet disorders, and packet losses. Finally, the effectiveness of the proposed methods is verified via a switched RLC circuit. In Chap. 5, the AOR with the BTC is studied for switched linear systems with destabilizing behaviors, whose all subsystems are unsolvable and some switching instants are unstable with finite increments of the Lyapunov function, Switching instants with decrements and increments of the Lyapunov function are described as stable and unstable switching instants respectively. Firstly, a hybrid average dwell time (HADT) strategy is proposed to restrict the DT strategy is proposed to restrict the occurrence ratio of unstable/stable switching instants and reasonably arranges the number of stable switching instants to offset the increment of the Lyapunov function caused by unstable switching instants and unsolvability of the involved problem for Then, by interpolating the gains of adjacent controllers within the minimum DT, a dynamic error feedback controller with Then, by interpolating the gains of adjacent controllers within the minimum DT, a dynamic error feedback controller with BT property is designed to suppress the control bumps induced by controller switching.

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1 Introduction

Finally, a numerical example is given to demonstrate the effectiveness of the proposed method. In Chap. 6, the event-triggered output regulation problem (EORP) under the DoS attacks is considered for NSSs with USDs. The USDs here refer to the unsolvable OR of each subsystem and the destabilization at partial switching instants, which indicates that the Lyapunov function does not decrease monotonically in activation intervals of each subsystem and increases at partial switching instants. First, long-duration DoS attacks (LDDAs) are considered, where LDDAs imply that their duration may be longer than the total DT of several adjacent activated subsystems. By imposing constraints at switching instants, consecutive asynchronous subsystem switching caused by LDDAs and USDs is allowed, that is, the subsystem switches several times but the controller switching is blocked by LDDAs and controllers fail to switch correspondingly. Second, mixed ETMs, combining event-triggered conditions and periodic sampling conditions, are designed to reduce network burden under LDDAs and improve system performance subject to destabilizing switching. Then, an improved DT for switching signal permits irregular arrangement of destabilizing and stabilizing switching and is more suitable for NSSs subject to LDDAs. Moreover, sufficient conditions ensure the solvability of EORP for NSSs with USDs under LDDAs, network-induced delays, random packet losses, and packet disorders. Finally, a switched RLC circuit shows the feasibility of the proposed method. In Chap. 7, the resilient EORP with dissipativity subject to deception attacks for NSSs with SUDs is investigated, which includes the unsolvable OR of each subsystem and some destabilizing switchings and implies that the Lyapunov function rises in activated intervals of all subsystems and at partial switching instants. First, an integrity deception attack (IDA) is delineated for the first time in NSSs for the tampered switching signal, system output, and control, where the consecutive asynchronous switching caused by the deceived switching signal is permitted by modifying inequality conditions at switching instants, which means switching from one asynchronous case to another at a switching instant. Second, two resilient ETMs for sensor-to-controller and controller-to-actuator channels are devised in conjunction with dissipative parameters in triggering thresholds to achieve the balance between limited network resources and system performance against destabilizing switching and IDAs. Third, the enhanced DT constraint for switching signal collected attack parameters not only overcomes the conventional arrangement of destabilizing and stabilizing switchings but is also more appropriate for NSSs impacted by IDAs. Furthermore, the solvability condition of EORP with dissipativity is deduced for NSSs with SUDs subject to IDAs, network-induced delays, random packet losses, and packet disorders via designing a DOFC. Finally, an F-18 aircraft model is used to demonstrate the feasibility of the proposed methods. In Chap. 8, the event-triggered multi-source bumpless transfer (MSBT) control problem is investigated for NSSs with the AOR performance subject to switching deception attacks (SDAs). Firstly, an improved DT approach is developed to derive a proper switching rule and ensure the feasibility of the AOR for NSSs allowing the AOR of each subsystem unsolvable and some switching instants destabilizing (i.e. with Lyapunov function increments), where destabilizing and stabilizing switching

References

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instants are subsystem-independent. Secondly, a novel description of deception attacks called SDAs is established that contains multiple attack functions switching based on the currently activated subsystem, which considers the characteristics of NSSs and can cause the most severe damage to system control performance. Due to the networks and the event-triggered scheme, the switching of SDAs does not coincide with the switching of subsystems and controllers, which pose a challenge to system analysis. Thirdly, MSBT controllers are designed to suppress the control bumps induced by multiple sources including asynchronous controller switching, event triggering mechanism, and SDAs. Finally, an application to a switched RLC circuit is given to verify the effectiveness of the proposed method. In Chap. 9, we conclude the monograph by briefly summarizing the main theoretical findings presented in our book, and proposing unsolved problems for further investigations.

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67. Ge, S.S., Sun, Z.: Switched controllability via bumpless transfer input and constrained switching. IEEE Trans. Autom. Control 53(7), 1702–1706 (2008) 68. Cheong, S.Y., Safonov, M.G.: Slow-fast controller decomposition bumpless transfer for adaptive switching control. IEEE Trans. Autom. Control 57(3), 721–726 (2011) 69. Zhao, Y., Zhao, J., Fu, J.: Bumpless transfer control for switched positive linear systems with L1-gain property. Nonlinear Anal. Hybrid Syst 33, 249–264 (2019) 70. Yang, D., Huang, C., Zong. G.: Finite-time H∞ bumpless transfer control for switched systems: A state-dependent switching approach. Int. J. Robust Nonlinear Control. 30(4), 1417–1430 (2020) 71. Zhao, Y., Ma, D., Zhao, J.: L2 bumpless transfer control for switched linear systems with almost output regulation. Syst. Control Lett. 119, 39–45 (2018) 72. Shi, Y., Sun, X.: Bumpless transfer control for switched linear systems and its application to aero-engines. IEEE Trans. Circuits Syst. I Regul. Pap. 68(5), 2171–2182 (2021) 73. Zhao, Y., Zhao, J.: H∞ reliable bumpless transfer control for switched systems with state and rate constraints. IEEE Trans. Syst. Man Cybern. Syst. 50(10), 3925–3935 (2020) 74. Yang, D., Zong, G., Nguang, S.K., Zhao, X.: Bumpless transfer H∞ anti-disturbance control of switching markovian LPV systems under the hybrid switching. IEEE Trans Cybern 52(5), 2833–2845 (2022) 75. Zhao, Y., Zhao, J.: Event-triggered bumpless transfer control for switched systems with its application to switched RLC circuits. Nonlinear Dyn. 98(3), 1615–1628 (2019) 76. Ma, Y., Li, Z., Zhao, J.: Output consensus for switched multi-agent systems with bumpless transfer control and event-triggered communication. Inf. Sci. 544, 585–598 (2021) 77. Zhang, S., Zhao, J.: Dwell-time-dependent H∞ bumpless transfer control for discrete-time switched interval type-2 fuzzy systems. IEEE Trans. Fuzzy Syst. 30(7), 2426–2437 78. Y. Shi, J. Zhao, X.M. Sun. A bumpless transfer control strategy for switched systems and its application to an aero-engine. IEEE Trans. Ind. Inform. 17(1), 52–62 (2021, 2022) 79. Shi, J., Zhao, J.: State bumpless transfer control for a class of switched descriptor systems. IEEE Trans. Circuits Syst. I Regul. Pap. 68(9), 3846–3856 (2021) 80. Zong, G., Huang, C., Yang, D.: Bumpless transfer fault detection for switched systems: a state-dependent switching approach. Sci. China Inf. Sci. 64, 172208 (2021) 81. Li, L., Jin, C., Ge, X.: Output regulation control for switched stochastic delay systems with dissipative property under error-dependent switching. Int. J. Syst. Sci. 49(2), 383–391 (2018) 82. Li, J., Zhao, J.: Incremental passivity and incremental passivity-based output regulation for switched discrete-time systems. IEEE Trans Cybern 47(5), 1122–1132 (2017) 83. Long, L., Zhao, J.: Robust and decentralized output regulation of switched non-linear systems with switched internal model. IET Control Theory Appl. 8(8), 561–573 (2014) 84. Wang, P., Zhao, J.: Almost output regulation for switched positive systems with different coordinates transformations and its application to a positive circuit model. IEEE Trans. Circuits Syst. I, Regul. Pap. 66(10), 3968–3977 (2019) 85. Li, L., Song, L., Li, T., Fu, J.: Event-triggered output regulation for networked flight control system based on an asynchronous switched system approach. IEEE Trans. Syst. Man Cybern: Syst. 51(12), 7675–7684 (2021) 86. Li, J., Zhao, J.: Output regulation for switched discrete-time linear systems via error feedback: an output error-dependent switching method. IET Control Theory Appl. 8(10), 847–854 (2014) 87. Huang, J., Chen, Z.: A general framework for tackling the output regulation problem. IEEE Trans. Autom. Control 49(12), 2203–2218 (2004) 88. Liu, Y., Zhao, J.: Error feedback output regulation problem for a class of linear switching systems with perturbations. Control and Decision 16(Suppl.), 815–817 (2001) 89. Dong, X., Zhao, J.: Output regulation for a class of switched nonlinear systems: an average dwell-time method. Int. J. Robust Nonlinear Control 23(4), 439–449 (2013) 90. Zattoni, E., Perdon, A.M., Conte, G.: The output regulation problem with stability for linear switching systems: a geometric approach. Automatica 49(10), 2953–2962 (2013) 91. Zhao, Y., Liu, Y., Ma, D.: Output regulation for switched systems with multiple disturbances. IEEE Trans. Circuits Syst. I Regul. Pap. 67(12), 5326–5335 (2020)

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92. Chen, X., Hao, F.: Observer-based event-triggered control for certain and uncertain linear systems. IMA J. Math. Control. Inf. 30(4), 527–542 (2013) 93. Yaz, E.E.: Linear matrix inequalities in system and control theory. Proc. IEEE 86(12), 2473– 2474 (1994) 94. Gu, K., Kharitonov, V.L., Chen, J.: Stability of time-delay systems. Birkhäuser Boston (2003) 95. Qi, Y., Zhao, X., Huang, J.: H∞ filtering for switched systems subject to stochastic cyberattacks: a double adaptive storage event triggering communication. Appl. Math. Comput. 394, 125789 (2021)

Chapter 2

Event-Triggered Output Regulation for Networked Flight Control System Based on an Asynchronous Switched System Approach

In this chapter, the EORP is investigated for an NFCS with a switched system approach. By properly scheduling triggered times of periodic sampling associated with continuous event-triggering and switching instants of each subsystem, an AETM based on the subsystem model is proposed to transmit the triggered information to the candidate controller. Since the subsystem may switch between two adjacent events while the corresponding controller does not switch, asynchronous switching may occur. Meanwhile, an event-triggered switching signal caused by asynchronous switching is designed with the MDADT approach and event-triggered instants. By constructing multiple Lyapunov functions in the framework of the input delay approach and designing an error feedback controller, the sufficient condition for EAORP is solved. Finally, the proposed methods are proved to be effective by the F-18 aircraft model.

2.1 Introduction With the continuous development and maturity of aerospace technology, more and more aircraft adopt data bus technology to overcome the drawbacks of traditional point-to-point connection ways, such as complex wiring, high maintenance cost, huge volume and so on. Real-time networks such as data bus are applied to connect sensors, controllers, and actuators of aircraft to form a closed loop, which constitutes an NFCS [1, 2]. This solves the increasingly complex problem of data sharing and data transmission among the components of the system. Especially, for aircraft attitude control systems, the elevator, aileron, engine and other components connected via network cooperate and communicate with each other to complete entire flight targets and better meet the requirements of aircraft intelligence and autonomy in the information age.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Li and J. Fu, Output Regulation and Cybersecurity of Networked Switched Systems, Studies in Systems, Decision and Control 475, https://doi.org/10.1007/978-3-031-30972-4_2

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2 Event-Triggered Output Regulation for Networked Flight Control …

On the other hand, switched systems are widely used as a modeling method to describe the dynamic behavior of modern flight vehicles [2–4]. Each operating point within the flight full envelope can be viewed as a corresponding subsystem of a switched system and the dynamic behaviors of a flight can be regarded as switchings among adjacent operating points under an appropriately designed switching signal. Therefore, the overall dynamic of a flight can be modeled as a switched system. Meanwhile, the design of switching signals is presented as an important issue in the study of switched systems, which mainly depends on system state [5, 6], time [7, 8], or both. Especially, the MDADT method [8] is decided by incremental parameter and attenuation rate of Lyapunov function corresponding to each subsystem. Under switching signals based on this method, the stability analysis of switched systems and the design of controllers are more flexible, and the results obtained are less conservative than the ADT method. Recently, event-triggered control, instead of periodic sampling control, has been gradually applied to switched systems for data transmission. In this way, the most needed system state or output can be transmitted through the ETM to reduce resource utilization. Due to the complex structure of switched systems, data transmission with an ETM will bring new challenges. On the one hand, the ETM in a switched system not only transfers data but also system mode. The intermittent transmission of event-triggered control will inevitably lead to the delay of controller mode updating. As shown in [9, 10], asynchronous event-triggered control is studied for switched systems. On the other hand, different design of the ETM has different influences on switched systems. In [9, 11, 12], the authors adopt continuous ETMs to study the stability of switched systems. However, it is necessary to prove that there exists a lower bound of an event-triggered interval to avoid the infinite number of triggers. To eliminate the Zeno phenomenon, a periodic ETM is considered for switched systems in the sense that the events just happen at sampling instants, in which a sampling period can be seen as the lower bound of the inter-event interval [10, 13]. Obviously, the effective information may be between adjacent periodic sampling instants, it will inevitably affect the system performance under this mechanism. Meanwhile, there are still other types of ETMs proposed for non-switched systems [14–18]. In [14], a novel approach that combines periodic sampling and continuous event-triggering is presented for an NCS. The approach not only excludes the Zeno phenomenon by taking a sampling period as a waiting time for sensors but also avoids missing information beneficial to system performance improvement by continuous detection of the event-triggered condition. However, the ETM proposed in [14] only applies to a single model system. When exploring to the research of switched systems, the complexity of switching behaviors and the alternation of the two sampled mechanisms will make the analysis process more difficult. Inspired by the idea in [14], it is of great significance to construct an AETM that is more closely related to the dynamical behavior of switched systems. The ORP is a hot issue in the field of control theory. The purpose is to design a feedback controller to guarantee that the system output asymptotically tracks a class of reference inputs and/or the disturbances generated by the exosystem. Great progress

2.2 Flight Control System Modelling

31

has been made in the output regulation for a single model system. Some representative methods, such as the internal model principle, central manifold theorem, system immersion [19, 20] and so on, have been well developed. Output regulation is a kind of anti-disturbance control method derived from practical control problems of aircraft disturbance rejection, aircraft takeoff and landing in extreme weather conditions. Recently, the theory and technology of switched systems have been gradually applied in aerospace and other fields, which also promotes the research of ORP of switched systems [21–26]. Among them, the different coordinate transformation method [21, 25] reduces the conservativeness of the research on output regulation for switched linear systems. At the same time, switched internal model [22], dissipativity [23] and incremental passivity theory [24, 26] are also proposed to solve the ORP of switched nonlinear systems. It is worth noting that none of these works focus on switched systems with networked controllers. This prompts us to study the output regulation of switched linear systems under an event-triggered network transmission mechanism. Motivated by the above discussion, the event-triggered ORP is investigated for NFCS via AETM, whose main contributions are summarized as follows. Firstly, a mode-dependent AETM is proposed for switched linear systems, wherein periodic sampling and continuous event-triggering are alternatively active, and triggered times of the compound triggering scheme and switching instants of each subsystem are scheduled cooperatively. Compared with the literature [9–13], our method can not only exclude the Zeno phenomenon but also extract qualified information more accurately by constantly checking the event-triggered conditions. Meanwhile, the mode-dependent waiting time of sensors can be adjusted according to the ADT of each subsystem, which makes the analysis of output regulation more flexible. The design can also be used to solve the information transmission problem of NFCS in different working statuses. Secondly, a switching signal that satisfies the MDADT condition is designed to be related to the event-triggered instants, so that the event-triggered control is well integrated into the switched system. Finally, the transmission of the system mode through AETM generates a time delay for the mode update of the corresponding controller, where an asynchronous error feedback controller is designed to solve the event-triggered ORP.

2.2 Flight Control System Modelling F-18 aircraft [27–29] is a mainstream aircraft of the U.S. Navy due to its strong firepower and good performance in air electronic warfare. The longitudinal short-period motion of aircraft with the characteristics of short period and fast change, which hinders pilots to take corrective actions in time and affects flight safety and firing accuracy, thus it’s worth studying. To cope with the increasing accuracy requirements, complex operating environments and sources of disturbance, the issue of output regulation for F-18 aircraft should be involved in the design of the flight control system. The nonlinear dynamic model of F-18 aircraft [28] is given by

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2 Event-Triggered Output Regulation for Networked Flight Control …

⎧ ⎪ ⎪ ⎨mU1 (α˙ − q) = −mg sin θ1 − q 1 S(C L α + C D1 )α

qc αc +q 1 S(−C L α˙ 2U − C L q 2U − C L δe δe ), 1 1 ⎪ ⎪ ⎩ I q˙ = q Sc(C + C )α + q Sc(C αc + C qc + C δ ). yy m q 2U1 m δe e mα m Tα m α˙ 2U1 1 1

(2.1)

The actual description of all symbols in (2.1) is shown in [28]. Generally, the longitudinal short period motion within the flight full envelope can be divided into multiple operating points. By using the Jacobian linearization approach on each operating point [28, 29], the nonlinear dynamic model (2.1) can be transformed into the following linearized dynamic model {

x(t) ˙ = Ax(t) + Bu(t) + Dw(t),

e(t) = C x(t) + Qw(t),

(2.2)

where x /\ [α q]T denotes the state vector, u /\ [δ E δ P T V ]T denotes the control input, w denotes the exogenous input signal, e denotes the measured output tracking error, which is the difference between the actual output and desired output, A = [Z α Z q ; Mα Mq ] and B = [Z δ E Z δ P T V ; Mδ E Mδ P T V ] are system matrices, Z α , Z q , Mα , Mq are the longitudinal stability derivatives, Z δ E , Z δ P T V , Mδ E , Mδ P T V are the longitudinal control derivatives, δ E , δ P T V are the symmetric horizontal tail deflection and pitch thrust vectoring nozzle deflection; D and Q show how the exogenous input add to the system or output tracking error; C is the output tracking error system matrix. NFCS is a single system with a large envelope. Its envelope flight dynamics can be divided into different regions according to different operating points (i.e. different Mach and altitude ranges). Thus, a different linear model at an operating point is proposed to describe the dynamic behavior in the vicinity of the corresponding operating point. We assume that 12 models corresponding to 12 operating points [29] cover the entire dynamic behavior of F-18 aircraft. The longitudinal short-period motion can be seen as a linear dynamic switching between adjacent operating points. Thus, (2.1) can be modeled as the following switched linear systems with an external reference signal {

x(t) ˙ = Aσ (t) x(t) + Bσ (t) u(t) + Dσ (t) w(t) e(t) = Cσ (t) x(t) + Q σ (t) w(t)

(2.3)

with a system state x ∈ Rn , control input u ∈ Rq , exogenous input w ∈ Rl , measured output tracking error e ∈ R p , switching signal σ : [t0 , +∞) → M = {1, 2, · · · , m}, Ai ,Bi ,Ci ,Di ,Q i with i ∈ M are constant matrices of appropriate dimensions. m is the number of subsystems. Corresponding to the switching signal σ (t), there exists the switching sequence {(i 0 , t0 ), · · · , (i k , tk ), · · · |k = 1, 2, · · · }, where t0 is the initial time, (i k , tk ) denotes that the i k -th subsystem is active on [tk , tk+1 ). w is the exogenous input representing the reference signal, which is assumed to be generated by the switched exosystem

2.3 Problem Formulation and Preliminaries

w(t) ˙ = Sσ (t) w(t).

33

(2.4)

The following assumptions are necessary to solve the EAORP for NFCS. Assumption 2.1 ([30] Assumption 1) All the eigenvalues of Sσ (t) have nonnegative real parts. Assumption 2.2 ([31] Assumption 2) The matrix pairs (Ai , Bi ) are stabilizable, i = 1, 2, · · · , m. {| | | Ai Di Assumption 2.3 ([31] Assumption 3) The matrix pairs , [Ci Q i ] are 0 Si detectable, i = 1, 2, · · · , m. Assumption 2.4 ([21]) There exist matrices ||i , Ei , Hi and E i that satisfy the following matrix equations. ⎧ ⎨ ||i Si = Ai ||i + Bi Hi Ei + Di , E E = Ei Si , ⎩ i i 0 = Ci ||i + Q i .

(2.5)

Remark 2.1 These assumptions are quite standard for the ORP of both non-switched systems [ 15, 19, 31] and switched systems [21, 23–25]. Especially, the existence of the solutions of the regulator equations in Assumption 2.4, which provides feedforward information to eliminate the steady-state error, guarantees that the ORP is solvable. For detailed proofs and properties of center manifolds see Theorem 8.4.4, Lemma 8.4.1 and Section B.1 in [19].

2.3 Problem Formulation and Preliminaries 2.3.1 Error Feedback Controller As shown in Fig. 2.1, both the triggered information and the mode information are transmitted to the controller under AETM. The zero-order holder (ZOH) is introduced to keep the control input signal and the controller mode continually. The controller may not receive the mode information while the system mode switches, which may result in asynchronous switching between the subsystem and the corresponding controller. Moreover, we just use the measured output tracking error to design controllers. Thus, the error feedback controller is given as follows: {

u(t) = Hσ (t) ξ(t), ξ˙ (t) = E σ (t) ξ(t) + Fσˆ (t) e(t),

(2.6)

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Fig. 2.1 Diagram for event-triggered error feedback control of NSSs

where E σ (t) and Hσ (t) are appropriate dimensions matrices satisfying Assumption 2.4, Fσˆ (t) is the controller gain to be devised, ξ ∈ Rs is the internal state and σˆ (t) = σ (t − dk ) ∈ M is the switching signal of the controller, where d0 = 0, dk represents the period that the controller u i lags with the subsystem i, that is dk = tre+1 − tk with 0 ≤ dk < d Mi , d Mi is the maximal delay period of the controller u i .

2.3.2 Alternate Event-Triggered Communication Scheme The AETM is inspired by the conjunction of periodic sampling and continuous eventtriggering. Different from [14], this chapter aims to design a mode-dependent AETM for a switched system, in which the information of system mode is mainly reflected in the waiting times and the event-triggered thresholds. It means that different waiting times and thresholds are employed when different subsystems are activated, which admits different data update rates for each subsystem according to different operating conditions. For NFCS, compared with operating points during the smooth flight, setting shorter waiting times and smaller event-triggered thresholds for operating points under severe conditions, such as avoiding obstacles, enables the aircraft to update data faster so that it can be faster to achieve a desired system performance more precisely. A well-designed ETM can reduce the data transmission rate and thereby reduce the loss of aircraft actuators while ensuring better control performance. Therefore, it is necessary to design appropriate event-triggered parameters for different operation points. Analyzing the relationship between switching instants and event-triggered instants as shown in Fig. 2.2 results in: (1) For the event-triggered instants {tre }r∞=0 with tre < tre+1 . When the subsystem j runs, the sensor needs to wait for a sampling period h j after an event is triggered at the instant tre , and then the event-triggered condition starts to be continuously

2.3 Problem Formulation and Preliminaries

35

Fig. 2.2 The relationship between asynchronous switching and sampling instant under AETM (2.7)

detected until the next event occurs at the instant tre+1 . Specifically, the next event-triggered instant is described by tre+1 = inf { t ≥ tre + h σ (tre ) | ||eˆr (t)||2 > ησ (tre ) ||e(t)||2 } , t∈M

(2.7)

where eˆr (t) = [e T (tre ) − e T (t) ξ˜ T (tre ) − ξ˜ T (t)]T , ξ˜ (t) = ξ(t) − Ei w(t), h i > 0 is the waiting time of the sensor and ηi > 0 is a threshold when the subsystem i = σ (tre ) works. To ensure that at least one event trigger occurs between two consecutive switching instants, it is assumed that h i is less than the minimal MDADT. ||eˆr (t)||2 > ηi ||e(t)||2 denotes the event-triggered condition. Especially, the following inequality can be derived from the AETM (2.7) on the interval [tk , tre+1 ) ∪ [tre+υ + h i , tre+υ+1 ), ||eˆr +υ (t)||2 ≤ ησ (tre ) ||e(t)||2 , υ ∈ {0, 1, · · · , n − 1},

(2.8)

where eˆr +υ (t) = [e T (tre+υ ) − e T (t) ξ˜ T (tre+υ ) − ξ˜ T (t)]T . N (2) For the switching instants {tk }k=0 with tk < tk+1 . We assume that at most n events happen when the subsystem i is active, where n ≥ 1. Based on the AETM proposed above, to demonstrate that ETM is reasonably applied to data transmission in switched systems, the switching instants here will be determined by certain event-triggered instants, that is, the switching occurs after the sensor waits for a mode-dependent sampling interval h i . Remark 2.2 To better integrate event-triggered control into the design of switched systems, an event-triggered switching signal will be designed to meet the MDADT requirement and could be dynamically adjusted according to the triggering situations, instead of being scheduled in advance. In short, the switching instants would be affected by the event-triggered instants. In the forthcoming switching signal design, the instant tre+n + h i will be selected as the next switching instant tk+1 by following the steps of forthcoming Algorithm 2.1 when the operating time of the subsystem i is longer than the minimal MDADT. A detailed analysis will be described later.

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2 Event-Triggered Output Regulation for Networked Flight Control …

2.3.3 Error Feedback Controller with Alternate Event-Triggered Mechanism Devoted to the cooperation of the AETM (2.7) and the event-triggered switching signal illustrated in Fig. 2.2, the discussion of error feedback controllers can be divided into the following three cases. When t ∈ [tk , tre+1 ), the controller mode holds j since the event with the system mode has not yet been transmitted to the controller, which causes the asynchronous switching. Meanwhile, the controller input keeps the event at the last triggered instant tre until the measured output tracking error satisfies the event-triggered condition at tre+1 . Accordingly, the actual error feedback controller is described as {

u(t) = Hi ξ(tre ), ξ˙ (t) = E i ξ(t) + F j e(tre ).

(2.9.1)

When t ∈ [tre+υ , tre+υ + h i ) with υ ∈ {1, · · · , n}, the controller has updated its inner state and switched its mode from j to i at the triggered instant tre+υ , and then waits for a sampling period h i . So these n intervals are called the waiting intervals. Utilizing the input delay method [32], we represent the input as a continuous-time control with a time-varying delay τ (t) = t − tre+υ ≤ h i . Resultingly, the error feedback controller is described as { u(t) = Hi ξ(t − τ (t)), (2.9.2) ξ˙ (t) = E i ξ(t) + Fi e(t − τ (t)). When [tre+υ + h i , tre+υ+1 ) with υ ∈ {1, · · · , n − 1}, the controller mode still holds the latest transmitted information at the triggered instant tre+υ and (2.8) starts to be detected until a new event occurs at tre+υ+1 . So these n −1 intervals are called the detection intervals. Consequently, the error feedback controller is described as {

u(t) = Hi ξ(tre+υ ), ξ˙ (t) = E i ξ(t) + Fi e(tre+υ ).

(2.9.3)

Remark 2.3 Taking tre+n + h i = tk+1 into account, the operating interval [tk , tk+1 ) of i can | the subsystem ) be completely covered by the above 2n subintervals, i.e. | tk , tk+1 ) = tk , tre+1 ∪ [tre+1 , tre+1 + h i ) ∪ [tre+1 + h i , tre+2 ) ∪ · · · ∪ [tre+n−1 + h i , tre+n ) ∪ [tre+n , tk ). Besides, (2.9.2) shows the rewritten form of controller (2.6) on waiting intervals where the input delay method [10, 13, 14, 31] allows us to represent the piecewise continuous event-triggered control input as a continuous variable with a time-varying delay 0 < τ (t) = t − tre+υ < h i . However, (2.9.1) and (2.9.3) correspond to the intervals that only the event-triggered condition needs to

2.3 Problem Formulation and Preliminaries

37

be persistently monitored and the controller doesn’t need to update the data, so there is no need to rewrite the controller by the input delay method.

2.3.4 Modeling of the Closed-Loop System Associated with (2.3) and Assumption 2.4, the closed-loop system under eventtriggered controller (2.9) is given as follows: {

χ(t) = ( A˜ i + B˜ i j C i )χ (t) + B˜ i j eˆr (t), t ∈ [tk , tre+1 ), ˙ e(t) = C˜ i χ (t),

(2.10.1)

χ(t) = A˜ i χ (t) + B˜ i C i χ (t − τ (t)), t ∈ [tre+υ , tre+υ + h i ), ˙ e(t) = C˜ i χ (t),

(2.10.2)

χ˙ (t) = ( A˜ i + B˜ i C i )χ (t) + B˜ i eˆr+υ (t), t ∈ [tre+υ + h i , tre+υ+1 ), e(t) = C˜ i χ (t),

(2.10.3)

{ {

where ˜ = x(t) − ||i w(t), χ (t) = [ x˜ T (t) ξ˜ T (t) ]T , x(t) T e T eˆr +υ (t) = [ e (tr +υ ) − e (t) ξ˜ T (tre+υ ) − ξ˜ T (t) ]T ,υ ∈ {1, · · · , n − 1}, A˜ i =

|

| | | | | | | | | Ai 0 0 Bi Hi 0 Bi Hi Ci 0 , B˜ i = , C˜ i = Ci 0 . , B˜ i j = , Ci = 0 Ei 0 I Fi 0 Fj 0

2.3.5 Event-Triggered Asynchronous Output Regulation Problem Now, a definition of the event-triggered ORP for NFCS is given based on an asynchronous switching with an event-triggered error feedback control. Definition 2.1 If there exists an appropriately designed asynchronous switching signal σ (t), an AETM (2.7), and error feedback controllers (2.9.1)–(2.9.3) such that (i) When w = 0, the corresponding closed-loop system (2.10.1)–(2.10.3) is exponentially stable. (ii) When w /= 0, with the zero initial condition, the solution of the corresponding closed-loop system (2.10.1)–(2.10.3) satisfies lim e(t) = lim C˜ i χ (t) = 0. t→∞

t→∞

Then the event-triggered ORP for switched system (2.3) is said to be solvable.

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2 Event-Triggered Output Regulation for Networked Flight Control …

2.4 Main Results In this section, a sufficient condition for event-triggered output regulation for switched system (2.3) is presented based on an asynchronous switched system approach with an event-triggered error feedback control. Theorem 2.1 Under Assumptions 2.1–2.4, for given scalars h i > 0, ηi > 0, λi > 0, λi j < 0, μi > 1, μˆ i > 1, d Mi > 0. If there exist appropriate dimensions matrices Pl > 0, Rl > 0, Ui > 0, X i ,, X 1i , Mϑi , Nρi , ||i , Ei , Hi , E i , Fi and Ti j with ϑ ∈ {1, 2}, ρ ∈ {1, 2, 3}, l ∈ {i, i j}, i, j ∈ M, i /= j such that 0i > 0, 0l < 0, ui < 0, ei < 0, | /\i =

TiTj −μˆ i P j ∗ Pi j − 2I

(2.11)

| < 0, Pi < μi Pi j .

(2.12)

The ORP for switched system (2.3) is solvable by the error feedback controller (2.6) together with the AETM (2.7) under the event-triggered switching signal with MDADT subjected to {τai > τai∗ =

ln(μi μˆ i ) + 2(λi − λi j )d Mi } ∧ {τai = tre+n + h i − tk }, 2λi

(2.13)

where tre+n is the first event-triggered instant after tk + τai∗ , which can be calculated by the forthcoming Algorithm 2.1, the new switching instant can be set as tk+1 = tre+n + h i , | 0i = | 0l =

| h i X 1i − h i X i Pi + h i X˜ i , ∗ −h i X 1i − h i X 1iT + h i X˜ i

| T Pl A˜ i + A˜ iT Pl + C i RlT + Rl C i + 2λl Pl + ηi C˜ iT C˜ i Rl , ∗ −I

φi13 = N1iT − N3i , φi22 = −M2iT − M2i , φi23 = N2iT , φi33 = N3i + N3iT , ui11 = φi11 + (2βi h i − 1) X˜ i , ui12 = φi12 + h i X˜ i , ui13 = φi13 + (2βi h i − 1)(X 1i − X i ),

ui14 = M1iT , ui22 = φi22 + h i Ui , ui23 = φi23 + h i (X 1i − X i ), ui25 = M2iT , T ), u36 = ui33 = φi33 + (2βi h i − 1)( X˜ i − X 1i − X 1i i

√

T 2C i B˜ iT , ui44 = ui55 = ui66 = −I,

ei11 = φi11 − X˜ i , ei12 = φi12 , ei13 = φi13 − (X 1i − X i ), ei14 = h i N1iT , ei15 = M1iT , T , e26 = M T , e33 = φ 33 − ( X˜ − X − X T ), ei22 = φi22 , ei23 = φi23 , ei24 = h i N2i i 1i i 2i i i 1i

2.4 Main Results

39

ei34 = h i N3iT , ei37 =

√

T 2C i B˜ iT , ei44 = −h i e−2βi h i Ui , ei55 = ei66 = ei77 = −I,

Rl = Pl B˜ l , λi = βi , λi j = −κi , X˜ i = 0.5(X i + X iT ). Proof Since the interval between two consecutive switching instants is divided into three cases, the closed-loop system (2.10.1)–(2.10.3) under three corresponding forms of controllers can be obtained accordingly. More specifically, the following Case 1-Case 3 can be conducted to obtain the exponential decay of the Lyapunov function when the subsystem i is activated. Case 1: On the interval [tk , tre+1 ), the subsystem i runs asynchronously with the corresponding controller, and the event-triggered condition is continuously monitored. Thus, we construct a Lyapunov candidate function V i j (t) = χ T (t)Pi j χ (t). By using Lemma 2.2 in [33] and (2.8), and taking the time derivative of the Lyapunov candidate function by introducing the decay rate κi of V i j (t) yields V˙ i j (t) − 2κi V i j (t) ≤ χ T (t)( A˜ iT Pi j + Pi j A˜ i − 2κi Pi j + Pi j B˜ i j C i T +C i B˜ iTj Pi j + Pi j B˜ i j B˜ iTj Pi j + ηi C˜ iT C˜ i )χ (t)

(2.14)

= χ T (t)0i j χ (t). Combining the condition 0i j < 0, Lemma 1.7 and Ri j = Pi j B˜ i j , we can imply 0i j < 0. Thus, for t ∈ [tk , tre+1 ), we have V i j (t) < e2κi (t−tk ) V i j (tk ). Case 2: On the interval [tre+υ , tre+υ + h i ), the subsystem i runs synchronously with the corresponding controller, and the sensor is in a waiting phase. Construct the following Lyapunov candidate function Vi (t) = V Pi (t) + VUi (t) + VX i (t), where V Pi = χ T (t)Pi χ (t), { t VUi = (h i − τ (t)) |

t−τ (t)

e2βi (s−t) χ˙ T (s)Ui χ(s)ds, ˙

χ (t) χ (t − τ (t)) | | −X i + X 1i X˜ i . Xi = ∗ −X 1i − X 1iT + X˜ i

VX i = (h i − τ (t))

Denote γ (t) =

1 τ (t)

}t t−τ (t)

|

|T Xi

| χ (t) , χ (t − τ (t))

χ˙ (s)ds, associated with Lemma 1.8 yields

(2.15)

40

2 Event-Triggered Output Regulation for Networked Flight Control …

−e−2βi h i

{

t

t−τ (t)

χ˙ T (s)Ui χ˙ (s)ds ≤ −τ (t)e−2βi h i γ T (t)Ui γ (t),

(2.16)

where βi denotes the decay rate of V i (t). For any appropriately dimensioned matrices M1i , M2i , N1i , N2i , and N3i , the following identities hold ⎧ 2[χ T (t)M1iT + χ˙ T (t)M2iT ] ⎪ ⎪ ⎨ ˜ ·[ Ai χ (t) + B˜ t C i χ (t − τ (t)) − χ˙ (t)] = 0, T T T T T T ⎪ ⎪ 2[χ (t)N1i + χ˙ (t)N2i + χ (t − τ (t))N3i ] ⎩ ·[−χ (t) + χ (t − τ (t)) + τ (t)γ (t)] = 0.

(2.17)

Taking the time derivative of (2.15), we have V˙i + 2βi Vi ≤ ζ T (t)ψi (τ (t))ζ (t),

(2.18)

where ζ (t) = [ χ T (t) χ˙ T (t) χ T (t − τ (t)) γ T (t) ]T , ψi (τ (t)) = {ψiιν }, ι, ν ∈ {1, 2, 3, 4}, ψi11 = φi11 + [2βi (h i − τ (t)) − 1] X˜ i + M1iT M1i , ψi12 = φi12 + (h i − τ (t)) X˜ i , ψi14 = τ (t)N1iT , ψi13 = φi13 + [2βi (h i − τ (t)) − 1](X 1i − X i ), ψi22 = φi22 + (h i − τ (t))Ui + M2iT M2i , ψi23 = φi23 + (h i − τ (t))(X 1i − X i ), ψi24 = τ (t)N2iT , T T ψi33 = φi33 + [2βi (h i − τ (t)) − 1][ X˜ i − X 1i − X 1i ] + 2C i B˜ iT B˜ i C i , ψi34 = τ (t)N3iT ,

ψi44 = −τ (t)e−2βi h i Ui .

The matrix-function ψi (τ (t)) is affine in τ (t). Applying the Schur complement lemma, the conditions ui < 0 and ei < 0 imply ψi (0) < 0 and ψi (h i ) < 0 respectively. Thus, ψi (τ (t)) < 0 can be obtained by ζ T (t)ψi (τ (t))ζ (t) h i − τ (t) T τ (t) T = ζ (t)ψi (h i )ζ (t) < 0, ζ (t)ψi (0)ζ (t) + hi hi where ζ (t) = [χ T (t) χ˙ T (t) χ T (t − τ (t)) 0]T . For ∀τ (t) ∈ [0, h i ), thus Vi (t) < e e−2βi (t−tr +υ ) Vi (tre+υ ). Case 3: On the interval [tre+υ + h i , tre+υ+1 ), the subsystem i runs synchronously with the corresponding controller, and the event-triggered condition is continuously detected. Construct the Lyapunov candidate function V Pi = χ T (t)Pi χ (t). Similar to ( ) e Case 1, we have V Pi (t) < e−2βi (t−tr +v −h i ) Vi tre+v + h i .

2.4 Main Results

41

Combining Case 2 and Case 3, we define a new Lyapunov function V i (t) during the synchronous operating interval [tre+1 , tk+1 ) { V i (t) =

e , tre+υ + h i ), υ = 1, · · · , n Vi (t), t ∈ [tt+υ . e V Pi (t), t ∈ [tr +υ + h i , tre+υ+1 ), υ = 1, · · · , n − 1

(2.19)

When τ (t) = h i , VUi and VX i in (2.15) are both equal to zero. Therefore, we can imply Vi (t) = V Pi (t) at instant tre+υ + h i . When τ (t) = 0, VUi and VX i in (2.15) are still equal to zero, it also implies Vi (t) = V Pi (t) at instant tre+υ . To sum up, the function V i (t) is continuous on [tre+1 , tk+1 ). Meanwhile, according to the switching signal (2.13), we know tre+n + h i = tk+1 . Therefore, on the interval [tk , tk+1 ), we obtain V i j (t) < e2κi (t−tk ) V i j (tk ), t ∈ [tk , tre+1 ), V i (t) < e−2βi (t−tr +1 ) V i (tre+1 ), t ∈ [tre+1 , tk+1 ). e

Next, we need to consider the Lyapunov function relation before and after instants tk and tre+1 . Owing to the existence of different coordinate transformations and asynchronous switching, V j (tk− ) = χ T (tk− )P j χ (tk− ) is not equal to V i j (tk+ ) = χ T (tk+ )Pi j χ (tk+ ). So, we take χ (tk+ ) = Ti j χ (tk− ), it gives V i j (tk+ ) − μˆ i V j (tk− ) = χ T (tk− )[TiTj Pi j Ti j − μˆ i P j ]χ (tk− ). Taking /\i < 0 into account, applying Schur complement and −Pi−1 j < Pi j − 2I −1 −1 T implied by (Pi j − I )Pi j (Pi j − I ) > 0, one can derive Ti j Pi j Ti j − μˆ i P j < 0. Therefore, V i j (tk+ ) ≤ μˆ i V j (tk− ) is true. Similarly, it is easy to obtain V i (tre+1+ ) ≤ μi V i j (tre+1− ). Therefore, with MDADT approach [8], for ∀t ∈ [tre+1 , tk+1 ), we have V σ (t) (t) = V i (t) ≤ cμi e

Em i=1

[

ln(μi μˆ i )+2(βi +κi )d Mi τ ai

−2βi ]T fi (0, t)

V (t0 ),

(2.20)

Em

where c = e i=1 [ln(μi μˆ i )+2(βi +κi )d Mi ]N0i (0,t) , T fi (0, t) denotes the total running time of the i-th subsystem on [0, T f ). Similarly, for t ∈ [tk , tre+1 ), we have V σ (t) (t) = V i j (t) ≤ ce

Em i=1

[

ln(μi μˆ i )+2(βi +κi )d Mi τ ai

−2βi ]T fi (0, t)

V (t0 ).

(2.21)

With (2.20)–(2.21) and 0i > 0, there exist a = min (λmin {Pi , Pi j , 0i (h)}), i, j∈M

b = max (λmax {Pi , Pi j }) such that a||χ (t)||2 ≤ V σ (t) (t), V σ (t) (t0 ) ≤ b||χ (t0 )||2 . i, j∈M

Then, it is obtained that ||χ (t)||2 ≤

2 ln(μi μˆ i )+(βi +κi )d Mi bcμi max 1 { −2βi }(t−t0 ) τai e i∈M V σ (t) (t) ≤ ||χ (t0 )||2 . a a

42

2 Event-Triggered Output Regulation for Networked Flight Control …

So the closed-loop system (2.10.1)–(2.10.3) is exponentially stable when w(t) = 0. When w(t) /= 0, applying for the center manifold theory [19], there exist W > 0 and ε > 0 such that ||x(t) − ||i w(t)|| + ||ξ(t) − Ei w(t)|| ≤ W e−εt (||x(0) − ||i w(0)|| + ||ξ(0) − Ei w(0)||),

thus lim (||x(t) − ||i w(t)|| + ||ξ(t) − Ei w(t)||) = 0 which leads to lim e(t) = t→∞

lim C˜ i χ (t) = 0. t→∞ Hence, the EAORP is solvable for the switched systems (2.3).

t→∞

Remark 2.4 To get the solvability of output regulation for switched linear systems, a group of regulator equations need to be solved. The different coordinate transformations approach proposed by [21] breaks the restrictions of a common solution for regulator equations of all subsystems. Nevertheless, there was no discussion about calculating the transformation matrices that describe the relationship between the states of the closed-loop system before and after the switching instants in [21]. In this sense, Theorem 2.1 provides a method in linear matrix inequalities (LMIs) form to get transformation matrices Ti j , which makes the result is of much less conservativeness. The following algorithm is helpful to find event-triggered switching instants under (2.13). Algorithm 2.1 Step 1: Choose parameters μi , μˆ i , βi , κi , d Mi , ηi and h i according to conditions in Theorem 2.1. The minimum MDADT τai∗ will be obtained via (2.13). The initial values are set by σ (t0 ) = 1, t0 = t0e = 0, r = 1 and k = 1. Step 2: Generate an event-triggered instant tre by continually monitoring the AETM (2.7) and then stop detecting the inequality in (2.7) for a sampling period h i . If tre+n + h i < τai∗ , set r = r + 1, repeat this step; otherwise, go to Step 3. Step 3: Let tk = tre+n + h i be a switching instant according to (2.13), set k = k +1, go back to Step 2. Remark 2.5 To obtain the minimum number of event transfers (NET), the eventtriggered parameters (h i , ηi ) would be properly chosen by the following procedure. First, a small ηi is chosen on (0, 1), the maximum of h i < τai∗ that makes the LMIs (2.11)–(2.12) in Theorem 2.1 feasible is calculated and denoted as h i . Then, decrease h i by a very small step size such as 0.01 from h i . For the fixed h i , increase ηi step by step up to 1, meanwhile, calculate the corresponding NET for each step, thereby a minimal NET can be obtained with a certain pair of parameters. After that, decrease h i further, repeating a similar idea of increasing ηi to compute another minimal NET corresponding to a new pair of parameters until h i is small. Finally, the optimal values of parameters (h i , ηi ) are determined to achieve the minimum of all minimal NETs.

2.4 Main Results

43

For h i = 0, there is no need to sample in a fixed period in the AETM (2.7), which degenerates into the continuous event triggering that is similar to the one in [9] as follows 2 ˆ > ηi ||e(t)||2 } . tre+1 = inf { t ≥ tre | ||e(t)|| t∈M

(2.22)

Next, we give a corollary to solve the EAORP for system (2.3) utilizing the above mechanism (2.22). Corollary 2.1 Under Assumption 2.1–2.4, for given scalars ηi > 0, λi > 0, λi j < 0, μi > 1, μˆ i > 1, d Mi > 0. If there exist appropriate dimensions matrices Pl > 0, Rl > 0, ||i , Ei , Hi , E i , Fi and Ti j with l ∈ {i, i j}, i, j ∈ M, i /= j such that the following inequalities hold 0l < 0, /\i < 0, Pi < μi Pi j . Then the EAORP for switched system (2.3) is solvable by the error feedback controller (2.6) together with the error-based continuous ETM (2.22) under the asynchronous switching signal satisfying MDADT defined by τai > τai∗ =

ln(μi μˆ i ) + 2(λi − λi j )d Mi . 2λi

(2.23)

/\i is shown in (2.12), where | 0l =

| T Pl A˜ i + A˜ iT Pl + C i RlT + Rl C i + 2λi Pl + ηi C˜ iT C˜ i Rl , ∗ −I

Rl = Pl B˜ l , λi = βi , λi j = −κi j . Proof Following a similar procedure of Case 1 in Theorem 2.1, the result can be easily derived. Remark 2.6 The continuous event-triggering is wildly applied since it can reduce both the NET and the updated frequency of the controller. Then, there is likely to generate an infinite number of events in finite time (Zeno phenomenon). To exclude it, [9] gives a proof that there is a positive lower bound of the inter-event interval. In our proposed AETM (2.7), the length of the waiting interval h i for individual subsystems can be viewed as the lower bound on the inter-event interval to avoid the Zeno phenomenon. When ηi = 0, the inequality in (2.7) always holds, the AETM (2.7) can be simplified into a periodic sampling, where each subsystem has an individual fixed sampling period. The EAORP for system (2.3) with periodic sampling h i is given in the following corollary.

44

2 Event-Triggered Output Regulation for Networked Flight Control …

Corollary 2.2 Under Assumptions 2.1–2.4, for given scalars h i > 0, ηi > 0, λi > 0, λi j < 0, μi > 1, μˆ i > 1. If there exist appropriate dimensions matrices Pl > 0, Ul > 0, X l , X 1l , Mϑl ,Nρl , ||i , Ei , Hi , E i , Fi ,Ti j with ϑ ∈ {1, 2}, ρ ∈ {1, 2, 3}, l ∈ {i, i j}, i, j ∈ M, i /= j such that 0l > 0, ul < 0, el < 0, /\i < 0, Pi < μi Pi j . The asynchronous ORP for system (2.3) employing periodic sampling is solvable by the error feedback controller (2.6), the switching signal with MDADT and sampling instants satisfies {τai > τai∗ =

ln(μi μˆ i ) + 2(λi − λi j )h i } ∧ {τai = trs+n+1 − tk }, 2λi

(2.24)

where trs+n+1 is the first sampling instant after tk + τai∗ and can be obtained by a procedure similar to Algorithm 2.1. {trs }r∞=0 with trs < trs+1 denotes sampling time, h i denotes the asynchronous period of the subsystem i. The next switching instant is set as tk+1 = trs+n+1 . /\i is shown in (2.12), where | 0l =

| Pl + h i X˜ l h i X 1l − h i X l T A˜ + A˜ T M − N − N T , , φl11 = 2λl Pl + M1l i 1l 1l i 1l T + h X˜ ∗ −h i X 1l − h i X 1l i l

T + A˜ T M − N , φ 13 = N T − N , φ 22 = −M T − M , φ 23 = N T , φl12 = Pl − M1l 2l 2l 3l 2l i l 1l l 2l l 2l T , u11 = φ 11 + (2λ h − 1) X˜ , u12 = φ 12 + h X˜ , φl33 = N3l + N3l i i l i l l l l l T , u22 = φ 22 + h U , ul13 = φl13 + (2λi h i − 1)(X 1l − X l ), ul14 = M1l i l l l

T ul23 = φl23 + h i (X 1l − X l ), ul25 = M2lT , ul33 = φl33 + (2λi h i − 1)[ X˜ l − X 1l − X 1l ],

ul36 =

√

T 2C i B˜ iT , ul44 = ul55 = ul66 = −I, el11 = φl11 − X˜ l , el12 = φl12 ,

el13 = φl13 − (X 1l − X l ), el14 = h i N1lT , el22 = φl22 , el23 = φl23 , el24 = h i N2lT , T ], e34 = h N T , e44 = −h λ −2βi h i U , el33 = φl33 − [ X˜ l − X 1l − X 1l i 3l i ˜le l l l ul55 = ul66 = ul77 = −I, Rl = Pl B˜ l , λi = βi , λi j = −κi , λ˜ i = 1, λ˜ i j = 2βi + 2κi + 1,

X˜ l = 0.5[X l + X lT ].

Proof On the interval [tk , trs+1 ), the subsystem i is active, while the controller u j still works. We give the following Lyapunov candidate function. Vi j (t) = V Pi j (t) + VUi j (t) + VX i j (t), }t where VUi j = (h i − τ (t)) t−τ (t) e−2κi (s−t) χ˙ T (s)Ui j χ˙ (s)ds, it is easy to get V Pi j (t) and VX i j (t) from (2.16). When t = trs+1 , τ (t) = 0 implies Vi j (trs+1− ) = V Pi j (trs+1− )

2.5 Illustrative Example

45

s + and Vi (tr+1 ) = V Pi (trs+1+ ), we can obtain Vi j (tk+ ) ≤ μˆ i V j (tk− ). Similarly, Vi (trs+1+ ) ≤ s − μi Vi j (tr+1 ). The rest of the proof is similar to that of Theorem 2.1.

Remark 2.7 A periodic event-triggering mechanism based on periodic sampling for a switched system is proposed in [10], where data is transmitted at sampling instants. In this way, the periodic event-triggering mechanism may manage less effective information, therefore, control performance may be worsened if an effective event occurs between adjacent sampling instants. Our proposed AETM (2.7) can prevent this problem by checking continually.

2.5 Illustrative Example Two operating points are selected within the flight envelope depicted in [29] to solve the ORP for system (2.3) and (2.4). The real system matrices are as follows |

| −0.2423 0.9964 , −2.342 −0.1737 | | −0.0416 −0.01141 m5h40 B1 = Blong = , −2.595 −0.8161 | | −0.5088 0.994 m6h30 A2 = Along = , −1.131 −0.2804 | | −0.09277 −0.01787 m6h30 B2 = Blong = , −6.573 −1.525 | | | | −0.2423 0.4978 0.5088 0.0107 , D2 = , D1 = −1.8420 −0.0877 0.1310 0.6219 | | | | | | 0.2 0 0.1 0 −0.2 0 C1 = , C2 = , Q1 = , −0.1 0.2 0 0.2 0.1 −0.2 | | | | | | 0.1 0 0 0.5 0 1 Q2 = , S1 = , S2 = . 0.3 0.2 −0.5 0 −1 0 m5h40 = A1 = Along

m5h40 The nomenclature, for example Along , is the longitudinal state matrix at Mach0.5 and 40kft. As stated earlier, the linear model at two adjacent operating points can be considered as two successively switched subsystems. We select the two operating points to define the switched systems.

{ σ =

1, if altitude ∈ [35, 45)kft, 2, if altitude ∈ [25, 35)kft.

46

2 Event-Triggered Output Regulation for Networked Flight Control …

Fig. 2.3 State responses of the system (2.3)

There exist the following matrices satisfying Assumption 2.4 |

| | | | | | | 10 −1 0 10 0.5 0 ||1 = , ||2 = , E1 = , E2 = , 01 0 −1 01 0 1.5 | | | | | | | | 0 −0.03 0 −0.03 0 0.5 0 0.3 H1 = , H2 = , E1 = , E2 = . 0 −0.01 0 −0.02 −0.5 0 −3 0 Choose parameters μ1 = 5, μ2 = 3.2,μˆ 1 = 5.1,μˆ 2 = 3.2, β1 = 1, β2 = 1.3, κ1 = 0.6, κ2 = 0.5, d M1 = d M2 = 1.6 in Theorem 2.1. To compare with the other ETM, set η = η1 = η2 = 0.451 and h = h 1 = h 2 = 0.33. The simulation time is T f = 60. Solving the conditions (2.11)–(2.12) in Theorem 2.1, one can obtain |

| | | −3.897 −1.38 −2.786 −0.945 F1 = , F2 = , −3.554 −4.221 −0.395 −1.876 ⎡ ⎡ ⎤ 0.05 0.32 0.07 0.06 0.1 0.01 −0.28 ⎢ 0.18 −0.04 0.27 0.18 ⎥ ⎢ 0.19 0.14 −0.15 ⎢ ⎥ T12 = ⎢ ⎣ −0.36 0.12 0.13 0.01 ⎦, T21 = ⎣ 0.26 0.05 −0.16 0.08 0.22 −0.14 0.13 0.05 −0.12 0.1

⎤ 0.16 0.09 ⎥ ⎥. −0.31 ⎦ 0.17

For simplicity, we assume that the variety of altitudes from one interval to another ∗ ∗ = 4.1793 and τa2 = 5.2561 respectively. In this sense, condition requires at least τa1 (2.13) in Theorem 2.1 can be used to decide switching instants. The asynchronous event-triggered ORP for NFCS is solved under the controller (2.6). Under the initial state χ (0) = [ 4 2.5 −2.5 −4 ]T , Fig. 2.3 is the trajectory of the state responses of the system (2.3). The convergence curves in Fig. 2.4 show that the sampled output tracking error approaches to zero. Figure 2.5 reflects the asynchronously switching signal of the system (2.3) and the relationship between switching instants and the event-triggered instants. Figure 2.6 describes the inter-event intervals and the minimum interval is equal to h = 0.33. In what follows, under preceding system matrices and parameters, the NET related to stability with optimally different ηi and h i will be compared with four different

2.5 Illustrative Example

47

Fig. 2.4 Sampled output tracking errors of the system (2.3)

Fig. 2.5 Switching signal for the system (2.3) and event-triggered samplings

Fig. 2.6 Inter-event intervals

mechanisms within time interval [0, T f ). In the AETM (2.7), with the help of Remark 2.5 and Algorithm 2.1, the minimum NET with the optimal (η∗ , h ∗ ) = (0.451, 0.33). When η = 0, (2.7) is simplified into the periodic sampling, the NET is calculated by [T f / h] + 1 with the maximum sampling period h = 0.56 solving by the LMIs in Corollary 2.2. When h = 0, (2.7) is degenerated into continuous event-triggering, we can obtain the minimum NET is 224 with the optimal η∗ = 0.089 With the same threshold η = 0.451, the system is unstable even though the NET is lower. Meanwhile, under the above system matrices and parameters with any positive η and h, the system (2.3) is unstable by using the periodic event-triggering in [10]. From Table 2.1, compared with the periodic sampling, the continuous event-triggering does

48

2 Event-Triggered Output Regulation for Networked Flight Control …

Table 2.1 The NET related to stability with optimally different η and h ETMs

η

h

NET

Stability

Periodic sampling

0

0.56

108

Yes

Continuous event-triggering [9]

0.089

0

224

Yes

Continuous event-triggering [9]

0.451

0

76

No

Periodic event-triggering [10]

–

–

–

No

AETM (7)

0.451

0.33

96

Yes

not give a significant improvement in NET, while the AETM can reduce 11% NET. Therefore, the proposed AETM is an effective way to reduce the NET.

2.6 Conclusion We have studied the EAORP for NFCS based on the switched system approach. The proposed AETM can effectively reduce the NET and naturally avoid the Zeno phenomenon. Meanwhile, a switching signal based on MDADT and AETM is presented in the case of asynchronous switching. By constructing multiple Lyapunov functions in the framework of input delay, the sufficient conditions for the EAORP have been solved with the error feedback controller. Finally, the illustrative example proves the effectiveness and feasibility of the proposed approach. The future direction will be the event-triggered security control for switched systems in a network environment [34, 35].

References 1. Tee, K.P., Ge, S.S., Tay, F.E.H.: Adaptive neural network control for helicopters in vertical flight. IEEE Trans. Control Syst. Technol. 16(4), 753–762 (2008) 2. Xu, L., Wang, Q., Li, W., Hou, Y.: Stability analysis and stabilisation of full-envelope networked flight control systems: Switched system approach. IET Control Theory Appl. 6(2), 286–296 (2012) 3. Shi, Y., Zhao, J., Liu, Y.: Switching control for aero-engines based on switched equilibrium manifold expansion model. IEEE Trans. Ind. Electron. 64(4), 3156–3165 (2017) 4. Lian, J., Li, C., Xia, B.: Sampled-data control of switched linear systems with application to an F-18 aircraft. IEEE Trans. Ind. Electron. 64(2), 1332–1340 (2017) 5. Branicky, M.S.: Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Autom. Control 43(4), 475–482 (1998) 6. Wang, R., Liu, G., Wang, W., Rees, D., Zhao, Y.: H∞ control for networked predictive control systems based on the switched Lyapunov function method. IEEE Trans. Ind. Electron. 57(10), 3565–3571 (2010) 7. Zhang, L., Gao, H.: Asynchronously switched control of switched linear systems with average dwell time. Automatica 46(5), 953–958 (2010)

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8. Zhao, X., Zhang, L., Shi, P., Liu, M.: Stability and stabilization of switched linear systems with mode-dependent average dwell time. IEEE Trans. Autom. Control 57(7), 1809–1815 (2012) 9. Li, T., Fu, J., Deng, F., Chai, T.: Stabilization of switched linear neutral systems: an eventtriggered sampling control scheme. IEEE Trans. Autom. Control 63(10), 3537–3544 (2018) 10. Ren, H., Zong, G., Li, T.: Event-triggered finite-time control for networked switched linear systems with asynchronous switching. IEEE Trans. Syst. Man Cybern. Syst. 48(11), 1874–1884 (2018) 11. Li, T., Fu, J.: Event-triggered control of switched linear systems. J. Franklin Inst. 354(15), 6451–646 (2017) 12. Qi, Y., Zeng, P., Bao, W.: Event-triggered and self-triggered H∞ control of uncertain switched linear systems. IEEE Trans. Syst. Man Cybern. Syst. 50(4), 1442–1454 (2020) 13. Wang, S., Zeng, M., Park, J.H., Zhang, L.: Finite-time control for networked switched linear systems with an event-driven communication approach. Int. J. Syst. Sci. 48(2), 236–246 (2017) 14. Selivanov, A., Fridman, E.: Event-triggered H∞ control: A switching approach. IEEE Trans. Autom. Control 61(10), 3221–3226 (2016) 15. Yang, R., Zhang, H., Feng, G., Yan, H., Wang, Z.: Robust cooperative output regulation of multi-agent systems via adaptive event-triggered control. Automatica 102, 129–136 (2019) 16. Yan, H., Hu, C., Zhang, H., Karimi, H.R., Jiang, X., Liu, M.: H∞ output tracking control for networked systems with adaptively adjusted event-triggered scheme. IEEE Trans. Syst. Man Cybern. Syst. 49(10), 2050–2058 (2019) 17. Wang, Y., Xia, Y., Ahn, C.K., Zhu, Y.: Exponential stabilization of Takagi–Sugeno fuzzy systems with aperiodic sampling: an aperiodic adaptive event-triggered method. IEEE Trans. Syst. Man Cybern. Syst. 49(2), 444–454 (2019) 18. Yan, H., Sun, J., Zhang, H., Zhan, X., Yang, F.: Event-triggered H∞ state estimation of 2-DOF quarter-car suspension systems with nonhomogeneous Markov switching. IEEE Trans. Syst. Man Cybern. Syst. 50(9), 3320–3329 (2020) 19. Isidori, A., Byrnes, C.I.: Output regulation of nonlinear systems. IEEE Trans. Autom. Control 35(2), 131–140 (1990) 20. Byrnes, C.I., Priscoli, F.D., Isidori, A., Kang, W.: Structurally stable output regulation of nonlinear systems. Automatica 33(3), 369–385 (1997) 21. Dong, X., Sun, X., Zhao, J., Dimirovski, G.M.: Output regulation for switched linear systems with different coordinate transformations. In: Proceedings of UKACC International Conference Control, pp. 92–95. Cardiff, U.K (2012) 22. Long, L., Zhao, J.: Robust and decentralised output regulation of switched non-linear systems with switched Internal model. IET Control Theory Appl. 8(8):561–573 (2014) 23. Jin, C., Li, L., Wang, R., Wang, Q.: Output regulation for stochastic delay systems under asynchronous switching with dissipativity. Int. J. Control 94(2), 548–557 (2021) 24. Li J, Zhao, J.: Incremental passivity and incremental passivity-based output regulation for switched discrete-time systems. IEEE Trans. Cybern. 47(5), 1122–1132 (2017) 25. Li, J., Zhao, J.: Output regulation for switched discrete-time linear systems via error feedback: An output error-dependent switching method. IET Control Theory Appl. 8(10), 847–854 (2014) 26. Pang, H., Zhao, J.: Output regulation of switched nonlinear systems using incremental passivity. Nonlinear Anal. Hybrid Syst. 27, 239–257 (2018) 27. Jafarov, E.M., Tasaltin, R.: Robust sliding-mode control for the uncertain MIMO aircraft model F-18. IEEE Trans. Aerosp. Electron. Syst. 36(4), 1127–1141 (2000) 28. Roskam, J.: Airplane flight dynamics and automatic flight controls. Lawrence, KS, USA: DAR Corp. (2001) 29. Adams, R.J., Buffington, J.M., Sparks, A.G., Banda, S.S.: Robust Multivariable Flight Control. Springer, London, U.K. (1994) 30. Knobloch, H.W., Isidori, A., Flockerzi, D.: Topics in Control Theory. Birkhäuser, Berlin, Germany, 1993 31. Fridman, E.: Output regulation of nonlinear systems with delay. Syst. Control Lett. 50(2), 81–93 (2003)

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32. Fridman, E.: A refined input delay approach to sampled-data control. Automatica 46(2), 421– 427 (2012) 33. Chen, X., Hao, F.: Observer-based event-triggered control for certain and uncertain linear systems. IMA J. Math. Control Inf. 30(4), 527–542 (2013) 34. Tian, E., Chen, P.: Memory-based event-triggering H∞ load frequency control for power systems under deception attacks. IEEE T. Cybern. 50(11), 4610–4618 (2020) 35. Wang, K., Tian, E., Liu, J., Wei, L., Yue, D.: Resilient control of networked control systems under deception attacks: a memory-event-triggered communication scheme. Int. J. Robust Nonlinear Control 30(4), 1534–1548. (2020)

Chapter 3

Output Regulation for Networked Switched Systems with Alternate Event-Triggered Control Under Transmission Delays and Packet Losses

This chapter is concerned with the event-triggered ORP for a class of linear NSSs. A mode-dependent AETM is integrated into the NSSs with transmission delays and packet losses for the first time based on continuous detection and periodic sampling, and a joint-designed switching signal with the triggering information is presented. Under transmission delays and packet losses, a series of resultant error feedback controllers are synthesized in four cases based on the relationships between switching instants and event-triggered instants. Then, a criterion ensuring asynchronous output regulation performance is formed with transmission delays by the input delay method. With the presence of packet losses, a set of sufficient conditions is also dedicated to the proposed control issue by adding the restraint in the number of successive packet losses associated with a modified AETM. Eventually, a comparison and an F-18 aircraft model reveal the effectiveness of the obtained results in the simulation.

3.1 Introduction NSSs, as a kind of NCSs, have many practical applications such as F-18 aircraft [1] and unmanned marine vehicle [2]. To save the network resources and reduce the actuator wear of these practical systems, ETMs are applied [2, 3] while guaranteeing the desired control performance of NSSs. Due to the inherent complex characteristics of triggering and switching behaviors, the use of ETM for NSS is not a simple application. For example, the trigger instants and the switching instants are not independent and need joint analysis. Besides, the mismatch between trigger instants and switching instants will affect the performance of NSSs [3]. Therefore, the ETM for NSSs is more complex than that of NCSs. In recent years, a variety of ETMs for NSSs have been developed, which can be roughly classified as continuous [4, 5], periodic [2, 6], and alternate types [1]. Most of them target at ideal network environments and switch the modes of subsystems and controllers according to a preset ADT which is not © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Li and J. Fu, Output Regulation and Cybersecurity of Networked Switched Systems, Studies in Systems, Decision and Control 475, https://doi.org/10.1007/978-3-031-30972-4_3

51

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3 Output Regulation for Networked Switched Systems with Alternate …

affected by the triggering information. Later, literature such as [3, 7] adopt the preset ADT without triggering information to solve the filtering problem for NSSs under asynchronous or synchronous switching and a periodic ETM subjected to networkinduced phenomena. However, affected by network transmission delays and packet losses, the modes of a subsystem and its corresponding controller will evolve under a more changeable asynchronous switching leading to a more complicated relationship between the triggering and switching instants of successful transmission. To this end, preset switched signals without triggering information may not be able to guarantee the performance of NSSs in real time. From this point of view, the ETM for NSSs under transmission delays and packet losses has not been adequately addressed and needs further investigation. The ORP [8] is one of the core issues in control theory. For non-switched systems, some efforts on ORP have been made [9–12]. Based on the internal model method, [11, 12] solve the ORP of nonlinear and linear systems with dynamic ETMs respectively. For switched systems, there exist a few attempts to solve the ORP using different methods such as switched internal model [13, 14], passivity [15], and dissipativity [16]. To break the limitation of one common solution for regulator equations, different coordinate transformations are utilized for individual subsystems in [1, 17] for ORP of switched systems. Using this method, [1] has solved the event-triggered ORP for NSSs in an ideal network environment. Although some improvements have been achieved, the challenge of how to incorporate network-induced phenomena into the study of ORP for switched systems is of great significance, which is the motivation for this current research. In this chapter, event-triggered ORP for NSSs under transmission delays without and with packet losses respectively, are investigated. The co-design of two AETMs and a cohered switching signal puts forward two solvable conditions by synthesizing an event-triggered error feedback controller. The main contributions and novelties are listed as follows: (1) The AETMs with the alternate property are integrated into NSSs under transmission delays and packet losses. They can not only transmit switching signals but also tolerate bounded transmission delays and limit the number of successive packet losses by adjusting their parameters. (2) A switching signal is jointly designed to achieve the output regulation performance associated with the AETM. The switching instants are also allowed to be dynamically adjusted by the nearest event-triggered instants of successful transmission. By combining the persistent detection/waiting in AETMs with the asynchronism/synchronism of switching signal, the error feedback controller synthesis will be divided into four cases based on the relationships between switching instants and event-triggered instants. (3) The limitation in the requirements on a common solution for all regulator equations is conquered by selecting a different coordinate transformation for each subsystem to cope with the event-triggered ORP for NSSs. The remainder of this chapter is arranged as follows. The model of the eventtriggered NSSs is established in Sect. 3.2. The main results about the event-triggered ORP under transmission delays without and with packet losses are developed in

3.2 Problem Formulation

53

Sect. 3.3. A comparison with existing results is reported in Sect. 3.4 and the effectiveness of the developed control strategy applied to an aircraft model is shown in Sect. 3.5. Some conclusions are outlined in Sect. 3.6. Notation: N denotes the set of natural numbers. |•| denotes the maximum integer which is less than or equal to •.

3.2 Problem Formulation The configuration of the switched systems synthesized by event-triggered error feedback controllers over networked communication is illustrated in Fig. 3.1, where the detailed components will be described in the following.

3.2.1 System Model and Asynchronous Controller Structure The dynamics of the NSSs in Fig. 3.1 are formulated as {

˙ = Aσ (t) x(t) + Bσ (t) u(t) + Dσ (t) w(t), x(t) e(t) = Cσ (t) x(t) + Q σ (t) w(t),

(3.1)

where x ∈ Rn , u ∈ Rm , e ∈ R p denote the state vector, control input and the output tracking error, and Aσ (t) , Bσ (t) , Cσ (t) , Dσ (t) , Q σ (t) are known constant matrices of appropriate dimensions, σ (t) : [0, ∞) → M = {1, · · · , m} is the piecewise constant switching signal. w ∈ Ro is the exogenous input generated by the switched exosystem w(t) ˙ = Sσ (t) w(t).

Fig. 3.1 Diagram of event-triggered control for NSSs

(3.2)

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3 Output Regulation for Networked Switched Systems with Alternate …

Throughout this chapter, it is assumed that: (1) The system state is unmeasurable; (2) All the eigenvalues of Sσ (t) have non-negative real parts [18]; (3) The pairs (| | ) | Aσ (t) Dσ (t) | , Cσ (t) Q σ (t) are stabilizable and detectable (Aσ (t) , Bσ (t) ) and 0 Sσ (t) respectively [18]. For simplicity of technical analysis, we describe the following three timesequences: T1 = {tk }k∈N denotes the set of switching instants, where the system mode switches from σ (tk− ) = j to σ (tk ) = i /= j on tk , i, j ∈ M; T2 = {tre }r ∈N denotes the set of triggered instants on the event-triggered device side; with the occurrence of transmission delays and packet losses, T3 = {sr }r ∈N denotes the subset of successfully transmitted instants in T2 , where T3 ⊆ T2 . Especially, if all eventtriggered sampling data are successfully transmitted over the network, then T3 = T2 , otherwise T3 ⊂ T2 . In this context, the asynchronous error feedback controller is devised as: {

u(t) = Hσ (t) ξ(t), ξ˙ (t) = E σ (t) ξ(t) + Fσˆ (t) e(t),

(3.3)

where Hσ (t) , E σ (t) and Fσˆ (t) are controller gains to be determined, ξ(t) is the controller’s internal state, σˆ (t) = σ (t − dk ) ∈ M is the controller’s switching signal, dk is the period that the controller u i (with mode i) lags behind the corresponding i-th subsystem, i.e. dk = tre+1 − tk with 0 ≤ dk < d Mi , d Mi is the maximal delay period of u i which is prespecified but difficult to know in advance. The forthcoming standard assumption is needed. Assumption 3.1 ([17]): There exist matrices ||i , Ei , E i and Hi satisfying linear regulator equations. ⎧ ⎨ ||i Si = Ai ||i + Bi Hi Ei + Di , E E = Ei Si , ⎩ i i 0 = Ci ||i + Q i .

(3.4)

3.2.2 Alternate Event-Triggered Mechanism with the Alternate Property To facilitate the discussion, the effect of transmission delays in the absence of packet losses will be taken into account first, which implies that all events are successfully transmitted. In this sense, a mode-dependent AETM is developed to generate the event-triggered instants based on successful transmissions { } 2 sr +1 = inf t ≥ sr + h|||e(t)|| ˜ > δ˜σ (sr ) ||e(t)||2 , i∈M

(3.5)

3.2 Problem Formulation

55

where sr , sr +1 ∈ T3 are any two adjacent event-triggered instants, e˜ T (t) = E T T T T [ e (sr ) − e (t) ξ˜ (sr ) − ξ˜ (t) ], ξ˜ (t) = ξ(t) − σ (sr ) w(t), δ˜σ (sr ) > 0 is a modedependent threshold, h > 0 is the length of the waiting time, which is assumed to be not greater than the minimal MDADT to ensure at least one event occurs between two consecutive switching instants. Remark 3.1 AETM (3.5) characterizes an alternate property of the triggering behav2 ˜ > δ˜σ (sr ) ||e(t)||2 is monitored iors so that the alternate triggering condition ||e(t)|| persistently until an event happens and then the triggering device waits for a given period h thereafter. When δ˜i = 0, AETM (3.5) could be simplified as periodic sampling. When h = 0, AETM (3.5) degrades to the continuous ETM. With the help of h, AETM (3.5) could exclude the occurrence of the Zeno phenomenon and reduce the number of triggered events. Furthermore, compared with [19], a mode-dependent threshold is adopted in (3.5) to meet the different performance requirements of each subsystem so that AETM (3.5) involves switching characteristics. Suppose a total of n' events occur and all are successfully transmitted over [tk , tk+1 ). For event transfer instant sr , let ηr be the total transmission delay time from the sensor to the actuator, where ηr ≤ η M ≤ h, η M = supr ∈N {ηr }. Denote the delayed time sequence of T3 by T4 = {tˆr }r ∈N owing to transmission delay, where tˆr = sr + ηr ≤ sr +1 + ηr +1 = tˆr +1 . To analyze the AETM (3.5) with transmission delays, let us focus on [tˆr +υ , tˆr +υ+1 ), υ = 1, 2, · · · , n ' . If tˆr +υ+1 − tˆr +υ ≤ h, then the sensor will wait for the whole interval [tˆr +υ , tˆr +υ+1 ), which implies that there is no persistent detection of the event-triggered condition on it. As illustrated in Fig. 3.2, if tˆr +υ+1 − tˆr +υ > h, then [tˆr +υ , tˆr +υ+1 ) can be divided into waiting interval [tˆr +υ , tˆr +υ + h) and detection interval [tˆr +υ + h, tˆr +υ+1 ). The triggering condition over [tˆr +υ + h, tˆr +υ+1 ) is deduced as: ||e(t ˜ − η(t))||2 − δ˜σ (sr ) ||e(t − η(t))||2 ≤ 0,

(3.6)

where the piecewise delay function η(t) satisfies η(t) =

t − tˆr +1 − h tˆr +1 − t ηr + ηr +1 . tˆr +1 − tˆr − h tˆr +1 − tˆr − h

(3.7)

Apparently, η(t) ∈ [0, η M ], η(tˆr + h) = ηr , η(tˆr +1 ) = ηr +1 , and (3.6) holds for t − η(t) ∈ [sr +υ + h, sr +υ+1 ). Fig. 3.2 The relationship between event-triggered instants and switching instants with transmission delays

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3 Output Regulation for Networked Switched Systems with Alternate …

Remark 3.2 In Fig. 3.2, the switching instant tk+1 is selected as sr +n ' + h with the MDADT constraint, which allows system mode switching to be dynamically adjusted based on triggering information. Detailed analysis devoted to designing the desired switching law will be described later.

3.2.3 The Error Feedback Controller Based on the Alternate Event-Triggered Mechanism Combined with Fig. 3.2, the AETM coheres with the asynchronous switching signals so that the design of error feedback controllers (3.3) is divided into four cases. (i) For the waiting interval [tk , min{tˆr +h, tˆr +1 }), the controller mode does not match the system mode, which leads to asynchronous switching. By using the input delay method [20], the control input during the waiting interval is expressed as a continuous-time function with time-varying delay τ (t) = t −sr ≤ h +η M = τ M . The resultant controller is written as { u(t) = Hi ξ(t − τ (t)), (3.8.1) ξ˙ (t) = E i ξ(t) + F j e(t − τ (t)). (ii) For the detection interval [min{tˆr + h, tˆr +1 }, tˆr +1 ), the control input holds the transmission and mode information at tˆr until the next event would occur and be successfully transmitted at tˆr +1 . The resultant controller is depicted as {

u(t) = Hi ξ(sr ), ξ˙ (t) = E i ξ(t) + F j e(sr ).

(3.8.2)

'

n −1 ˆ (iii) For the waiting interval ∪υ=1 [tr +υ , min{tˆr +υ + h, tˆr +υ+1 }) ∪ [tˆr +n ' , tk+1 ), the controller runs synchronously with the corresponding subsystems, and the control input holds the value at tˆr +υ when the event is successfully transmitted. The resultant controller is described as { u(t) = Hi ξ(t − τ (t)), (3.8.3) ξ˙ (t) = E i ξ(t) + Fi e(t − τ (t)),

where τ (t) = t − sr +υ ≤ h + η M = τ M . '

−1 (iv) For the detection interval ∪nυ=1 [min{tˆr +υ +h, tˆr +υ+1 }, tˆr +υ+1 ), the control input maintains the transmission information and mode information at tˆr +υ until the next event is successfully transmitted at tˆr +υ+1 . The resultant controller is

{

u(t) = Hi ξ(sr ), ξ˙ (t) = E i ξ(t) + Fi e(sr ).

(3.8.4)

3.2 Problem Formulation

57

Remark 3.3 Compared with [1], transmission delays and packet losses are considered in this chapter, which is more in line with the actual network situation. The transmission delays make more complex the study of asynchronous intervals. Specifically, the asynchronous interval [tk , tˆr +1 ) is divided into the waiting interval [tk , min{tˆr + h, tˆr +1 }) and the continuous detection interval [min{tˆr + h, tˆr +1 }, tˆr +1 ), while the asynchronous interval [tk , tre+1 ) is the continuous detection interval in [1]. Due to packet losses, the number of successive packet losses needs to be given, which will be discussed in Theorem 3.2.

3.2.4 The Resultant Closed-Loop Switched System Combining NSSs (3.1) and Assumption 3.1, the closed-loop system is analyzed under the error feedback controller (3.3) based on the AETM (3.5) when the subsystem i is active. ⎧ ⎨ χ˙ (t) = A˜ i χ (t) + π(t) B˜ i j C i χ (t − τ (t)) (3.9.1) ˜ − η(t))], +(1 − π(t))[ B˜ i j C i χ (t − η(t)) + B˜ i j e(t ⎩ e(t) = C˜ i χ (t), t ∈ [tk , tˆr +1 ), ⎧ ⎨ χ˙ (t) = A˜ i χ (t) + π (t) B˜ i C i χ (t − τ (t)) (3.9.2) ˜ − η(t))], +(1 − π (t))[ B˜ i C i χ (t − η(t)) + B˜ i e(t ⎩ e(t) = C˜ i χ (t), t ∈ [tˆr +1 , tk+1 ), where χ T (t) = [ x˜ T (t) ξ˜ T (t) ], x(t) ˜ = x(t) − ||i w(t), { 1, t ∈ [tk , min{tˆr + h, tˆr +1 }), π(t) = 0, t ∈ [min{tˆr + h, tˆr +1 }, tˆr +1 ), { n ' −1 ˆ [tr +υ , min{tˆr +υ + h, tˆr +υ+1 }) ∪ [tˆr +n ' , tk+1 ), 1, t ∈ ∪υ=1 π(t) = n ' −1 0, t ∈ ∪υ=1 [min{tˆr +υ + h, tˆr +υ+1 }, tˆr +υ+1 ), ⎧ T T T T T ⎪ ⎪ [ e (sr ) − e (t − η(t)) ξ (sr )− ξ (t − η(t))] , ⎪ ⎨ t ∈ [min{tˆ + h, tˆ }, tˆ ); r r +1 r +1 e(t ˜ − η(t)) = ⎪ [ e T (sr +υ ) − e T (t − η(t)) ξ T (sr +υ )− ξ T (t − η(t))]T , ⎪ ⎪ ⎩ n ' −1 [min{tˆr +υ + h, tˆr +υ+1 }, tˆr +υ+1 ); υ ∈ [1, . . . , n ' − 1), t ∈ ∪υ=1 { t − sr , t ∈ [tk , min{tˆr + h, tˆr +1 }) τ (t) = n ' −1 ˆ [tr +υ , min{tˆr +υ + h, tˆr +υ+1 }) ∪ [tˆr +n ' , tk+1 ) t − sr +υ , t ∈ ∪υ=1 | | | | | | | | | | Ai 0 0 Bi Hi 0 Bi Hi Ci 0 , B˜ i j = , Ci = , C˜ i = Ci 0 . A˜ i = , B˜ i = Fi 0 Fj 0 0 I 0 Ei

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3 Output Regulation for Networked Switched Systems with Alternate …

3.2.5 The Event-Triggered Output Regulation Problem Formulation Definition 3.1 Given the NSSs (3.1) and the switched exosystem (3.2), if there exists a switching signal σ (t) to be designed and the error feedback controller as shown in (3.8.1)–(3.8.4) with AETM (3.5), the event-triggered ORP with transmission delays and packet losses is stated as follows: (a) The closed-loop system (3.9.1)–(3.9.2) with w = 0, composed of NSSs (3.1) and controller (3.8.1)–(3.8.4), is exponentially stable under some desired switching signal. (b) Under any initial condition, the solution of the closed-loop system (3.9.1)– (3.9.2) with w /= 0 satisfies lim e(t) = lim C˜ i χ (t) = 0.

t→∞

t→∞

(3.10)

3.3 Main Results In what follows, by jointly devising the error feedback controllers, the switching signals and the AETMs, two series of sufficient conditions are developed to achieve the performance of the event-triggered output regulation for NSSs affected by transmission delays without and with packet losses respectively.

3.3.1 Event-Triggered Output Regulation Problem with Transmission Delays Theorem 3.1 Consider the NSSs (3.1) and the switched exosystem (3.2) with transmission delays and Assumption 3.1. For given scalars h > 0, 0 < δ˜l < 1, η M > 0, τ M > 0, d Mi > 0, λi > 0, λi j < 0, μˆ l > 1 and matrices Ji j , J˜i j , Ti j , if there exist appropriate dimensional matrices Pl > 0, Sl0 > 0, Sl1 > 0, Rl0 > 0, Rl1 > 0, G l0 , G l1 , where l ∈ {i, i j}, i /= j, i, j ∈ M satisfying.

where

0l < 0, 0l < 0, ϒl0 ≥ 0, ϒl1 ≥ 0,

(3.11)

[Pi , Si0 , Si1 , Ri0 , Ri1 ] ≤ μˆ i [Pi j , Si0j , Si1j , Ri0j , Ri1j ],

(3.12)

[Pi j , Si0j , Si1j , Ri0j , Ri1j ] ≤ μˆ i j [P j , S 0j , S 1j , R 0j , R 1j ],

(3.13)

3.3 Main Results

59

pv

pv

0l = {φl }, p, v ∈ {1, . . . , 7}, 0l = {ψl }, p, v ∈ {1, . . . , 5}, φl11 = Pl A˜ i + A˜ iT Pl + Sl0 + 3 A˜ iT Rl A˜ i − ϑl Rl0 + 2(λl + 1)Pl , φl12 = ϑl G l0 , φl14 = ϑl (Rl0 − G l0 ), φl24 = ϑl (Rl0 − G l0T ), φl22 = e2λl η M (Sl1 − Sl0 ) − ϑl Rl0 − ϑ l Rl1 , φl23 = ϑ l Rl1 , φl33 = −ϑ l Rl1 − e−2λl τ M Sl1 , φl44 = ϑl (G l0 + G l0T − 2Rl0 ) + δ˜l C i C i , T

T φl46 = C i B˜ lT , φl55 = −I, φl57 = B˜ lT , φl66 = φl77 = Pl + 3Rl − 2I, ψl11 = Pl A˜ i + A˜ iT Pl + Sl0 + 2 A˜ iT Rl A˜ i − ϑl Rl0 + (2λl + 1)Pl ,

ψl12 = ϑl Rl0 , ψl22 = φl22 , ψl23 = ϑ l G l1 , ψl24 = ϑ l (Rl1 − G l1 ), ψl33 = φl33 , ψl34 = ϑ l (Rl1 − G l1T ), T ψl44 = ϑ l (G l1 + G l1T − 2Rl1 ), ψl45 = C i B˜ lT ,

ψl55 = Pl + 2Rl − 2I, Rl = η2M Rl0 + h 2 Rl1 . and other items of 0l and 0l are zero matrices with |

ϒl0

| 1 1| | Rl0 G l0 Rl G l 1 = , ϒl = , G l0T Rl0 G l1T Rl1

then, the asynchronous ORP of the system (3.1) is solvable under the error feedback controller (3.3) based on the mode-dependent AETM (3.5) and the following asynchronous switching signal based on the event-triggered instants and the system mode { } ln(μi μi j ) + 2(λi − λi j )d Mi ∗ Tai > Tai = ∧ {Tai = sr +n ' + h − tk }, (3.14) 2λi where sr +n ' is the first triggered instant after tk + Tai∗ and can be calculated by the forthcoming Remark 3.5 by setting the next switching instant as tk+1 = sr +n ' + h with ϑi = e−2βi η M , ϑi j = ϑ i j = 1, ϑ i = e−2βi τ M , λi = βi , λi j = −κi , μi = μˆ i μ˜ i , μi j = μˆ i j μ˜ i j , δ˜i j = δ˜ j , {|| || || || 1 1 ||2 || 1 1 ||2 || μ˜ i = max ||(Si0 ) 2 Ji j (Si0j )− 2 || , ||(Ri0 ) 2 J˜i j (Ri0j )− 2 || , i, j∈M

|| } ||2 || ||2 || 1 || − 21 ||2 || 1 21 1 − 21 || || 1 21 ˜ 1 − 21 || || 2 ||(Si ) Ji j (Si j ) || , ||(Ri ) Ji j (Ri j ) || , ||Pi Ti j Pi j || ,

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3 Output Regulation for Networked Switched Systems with Alternate …

{|| || || || 1 1 ||2 − 1 ||2 || || 1 μ˜ i j = max ||Pi 2j Ti j P j 2 || , e2(κi +βi )η M ||(Si0j ) 2 Ji j (S 0j )− 2 || , i, j∈M || || 1 ||2 1 || η M e2(κi +βi )η M ||(Ri0j ) 2 J˜i j (R 0j )− 2 || , || || 1 ||2 1 || e2(κi +βi )η M ||(Si1j ) 2 Ji j (S 1j )− 2 || , || || } 1 1 ||2 || he2(κi +βi )η M ||(Ri1j ) 2 J˜i j (R 1j )− 2 || . Proof For the convenience of analysis, the interval [tk , tk+1 ) is divided into two parts corresponding to the closed-loop system (3.9.1) and (3.9.2), denoted as asynchronous and synchronous switchings in the following. Case A: For t ∈ [tk , tˆr +1 ), i = σ (t) /= σˆ (t) = j, choose the following Lyapunov functional for asynchronous switching Vi j = V Pi j + VSi0j + VSi1j + VRi0j + VRi1j ,

(3.15)

where V Pi j = χ T (t)Pi j χ (t), { t 0 e−2κi (s−t) χ T (s)Si0j χ (s)ds, VSi j = t−η M

{

VRi0j = η M { VSi1j =

{

0

t

e−2κi (s−t) χ˙ T (s)Ri0j χ˙ (s)dsdθ,

−η M t+θ t−η M −2κi (s−t)

e

t−τ M { −η M

VRi1j = h

−τ M

{

t

t+θ

χ T (s)Si1j χ (s)ds,

e−2κi (s−t) χ˙ T (s)Ri1j χ˙ (s)dsdθ.

Taking the derivative of (3.15) with respect to t yields V˙ S 0 = 2κi VS 0 + χ T (t)Si0j χ(t) − e2κi η M χ T (t − η M )Si0j χ(t − η M ), ij

ij

V˙ S 1 = 2κi VS 1 + e2κi η M χ T (t − η M )Si1j χ(t − η M ) − e2κi τ M χ T (t − τ M )Si1j χ(t − τ M ), ij ij { t V˙ R 0 = 2κi V R 0 + η2M χ˙ T (t)Ri0j χ(t) ˙ − ηM e−2κi (s−t) χ˙ T (s)Ri0j χ(s)ds, ˙ ij

ij

V˙ R 1 = 2κi V R 1 + h 2 χ˙ T (t)Ri1j χ˙ (t) − h ij

ij

{

t−η M t−η M

t−τ M

In the sequel, the proof will be twofold:

e−2κi (s−t) χ˙ T (s)Ri1j χ(s)ds. ˙

3.3 Main Results

61

(A1) For t ∈ [min{tˆr + h, tˆr +1 }, tˆr +1 ), the closed-loop system (3.9.1) with π(t) = 1 and η(t) ∈ [0, η M ] is considered, where the event-triggering condition is continuously detected. Thus, along the system (3.9.1) with π(t) = 0, we have ˜ − η(t))]. V˙ Pi j = 2χ T (t)Pi j [ A˜ i χ (t) + B˜ i j C i χ (t − η(t)) + B˜ i j e(t With the help of Lemma 1.8 and Park theorem [21], it follows { − ηM

t t−η M

e−2κi (s−t) χ˙ T (s)Ri0j χ(s)ds ˙

| |T | χ (t) − χ (t − η(t)) χ (t) − χ (t − η(t)) 0 , ϒi j χ (t − η(t)) − χ (t − η M ) χ (t − η(t)) − χ (t − η M ) { t−η M −h e−2κi (s−t) χ˙ T (s)Ri1j χ(s)ds ˙

| ≤−

t−τ M

≤ −[χ (t − η M ) − χ (t − τ M )]T Ri1j [χ (t − η M ) − χ (t − τ M )]. Then utilizing Lemma 2.2 in [22] with Cholesky decomposition and (3.6) gives V˙i j − 2κi Vi j ≤ ζ T (t)0i' j ζ (t), where ζ (t) = [χ T (t) χ T (t − η M ) χ T (t − τ M ) χ T (t − η(t)) e˜ T (t − η(t))]T , 0i' j = {φ ' i j }, p, q ∈ {1, . . . , 5}, φ ' i12j = G i0j , φ ' i14j = Ri0j − G i0j , pq

φ ' i11j = Pi j A˜ i + A˜ iT Pi j + Si0j + 3 A˜ iT Ri j A˜ i − Ri0j − 2(κi − 1)Pi j , φ ' i22j = e2K i η M (Si1j − Si0j ) − Ri0j − Ri1j , φ ' i23j = Ri1j , φ ' i24j = −(G i0j )T + Ri0j , φ ' i33j = −Ri1j − e2κi τ M Si1j , T T φ ' i44j = G i0j + (G i0j )T − 2Ri0j + δ˜ j C i C i + C i B˜ iTj (Pi j + 3Ri j ) B˜ i j C i ,

φ ' i55j = B˜ iTj (Pi j + 3Ri j ) B˜ i j − I, Ri j = η2M Ri0j + h 2 Ri1j . From −(Pi j + 3Ri j )−1 < (Pi j + 3Ri j ) − 2I = (Pi j + 3Ri j ) − 2I , it holds that [(Pi j + 3Ri j )−1 − I ](Pi j + 3Ri j )[(Pi j + 3Ri j )−1 − I ] > 0. Associated with Lemma 1.7 for 0i j < 0 in (3.11), it yields 0i' j < 0, which implies ( { }) Vi j (t) ≤ e2κi (t−min{tˆr +h,tˆr +1 }) Vi j min tˆr + h, tˆr +1 .

(3.16)

(A2) For t ∈ [tk , min{tˆr + h, tˆr +1 }), the closed-loop system (3.9.1) with π(t) = 1 and τ (t) ∈ [0, τ M ], τ M = η M + h is under consideration, where the sensor is in the waiting stage and the event-triggering condition (3.6) remains inactive. If τ (t) ∈ [0, η M ] and e(t ˜ − η(t)) = 0, the system dynamics are the same as (A1), then (3.16) also holds by following similar arguments. So we just take the system (3.9.1) with π(t) = 1 and τ (t) ∈ [η M , τ M ] into account, then

62

3 Output Regulation for Networked Switched Systems with Alternate …

V˙ Pi j = 2χ T (t)Pi j [ A˜ i χ (t) + B˜ i j C i χ (t − τ (t))].

(3.17)

By using Lemma 1.8 and Park theorem [21] yields {

t

− ηM

t−η M

e−2κi (s−t) χ˙ T (s)Ri0j χ(s)ds ˙

≤ −[χ (t) − χ (t − η M )]T Ri0j [χ (t) − χ (t − η M )], { −h

t−η M

t−τ M

(3.18)

e−2κi (s−t) χ˙ T (s)Ri1j χ˙ (s)ds

| |T | | χ (t − η M ) − χ (t − τ (t)) 1 χ (t − η M ) − χ (t − τ (t)) . ϒi j ≤− χ (t − τ (t)) − χ (t − τ M ) χ (t − τ (t)) − χ (t − τ M )

(3.19)

Applying Lemma 2.2 in [22] with Cholesky decomposition and (3.15), (3.17)– T (3.19), it is easy to get V˙i j − 2κi Vi j ≤ ζ (t)0i' j ζ (t), where ζ (t) = [χ T (t) χ T (t − η M ) χ T (t − τ M ) χ T (t − τ (t))]T , p˜ q˜

0i' j = {ψ ' i j }, p, ˜ q˜ ∈ {1, . . . , 4}, ψ ' i11j = Pi j A˜ i + A˜ iT Pi j + Si0j + 2 A˜ iT Ri j A˜ i − Ri0j − (2κi − 1)Pi j , ψ ' i12j = Ri0j , ψ ' i22j = e2κi η M (Si1j − Si0j ) − Ri0j − Ri1j , ψ ' i23j = G i1j , ψ ' i24j = Ri1j − G i1j , ψ ' i33j = −Ri1j − e2κi τ M Si1j , ψ ' i34j = Ri1j − (G i1j )T , ψ ' i44j = G i1j + (G i1j )T T − 2Ri1j + C i B˜ iTj (Pi j + 2Ri j ) B˜ i j C i .

Similarly, it holds that Vi j (t) ≤ e2κi (t−tk ) Vi j (tk ).

(3.20)

Case B: For t ∈ [tˆr +1 , tk+1 ), σ (t) = σˆ (t) = i, construct a Lyapunov functional for the closed-loop system (3.9.2) with synchronous switching Vi = V Pi + VSi0 + VSi1 + VRi0 + VRi1 , where V Pi = χ T (t)Pi χ (t), { t e2βi (s−t) χ T (s)Si0 χ (s)ds, VSi0 = t−η M

(3.21)

3.3 Main Results

63

{ V

Ri0

= ηM

t

e2βi (s−t) χ˙ T (s)Ri0 χ(s)dsdθ, ˙

−η M t+θ t−η M 2βi (s−t)

{ VSi1 =

{

0

e

t−τ M { −η M

VRi1 = h

{

−τ M

t

t+θ

χ T (s)Si1 χ (s)ds,

e2βi (s−t) χ˙ T (s)Ri1 χ˙ (s)dsdθ.

Similar to the proof of Case A, it is easy to verify that V˙i + 2βi Vi ≤ n ' −1 n ' −1 ˆ [min{tˆr +υ + h, tˆr +υ+1 }, tˆr +υ+1 ), and on ∪υ=1 [tr +υ , min{tˆr +υ + ζ (t)0i' ζ (t) on ∪υ=1 T ' ˙ ˆ ˆ ˆ h, tr +υ+1 })∪[tr +n ' , tk+1 ), Vi + 2βi Vi ≤ ζ (t)0i ζ (t). For t ∈ [tr +1 , tk+1 ), applying Lemma 1.7 for 0i < 0 and 0i < 0 in (3.11) gives 0' i < 0 and 0' i < 0, so that T

Vi (t) ≤ e−2βi (t−tˆr +1 ) Vi (tˆr +1 ).

(3.22)

Owing to different coordinate transformations, the existence of asynchronous switching signals makes V j (tk− ) /= Vi j (tk+ ). Noticing χ (tk+ ) = Ti j χ (tk− ) together with (3.12), (3.13), (3.16), (3.20) and (3.22), it can be deduced that Vi j (tk+ ) {|| |||| ||2 − 1 |||| 1 || 1 || ≤μˆ i j || Pi 2j Ti j P j 2 |||| P j2 χ (tk− )|| +e

2(κi +βi )η M

|| { ||| 1 || || 0 21 ||(Si j ) Ji j (S 0j )− 2 ||

|| || { 1 1 || || + η M ||(Ri0j ) 2 J˜i j (R 0j )− 2 ||

0

{

tk− tk− −η M

tk−

|| { ||| 1 1 || || + e2(κi +βi )τ M ||(Si1j ) 2 Ji j (S 1j )− 2 || || { || 1 1 || || + h ||(Ri1j ) 2 J˜i j (R 1j )− 2 ||

−η M

−τ M

{

||2 || 1 || ˙ || dsdθ ||(R 0j ) 2 χ(s)

2βi (s−tk− ) ||

tk− +θ

−η M

|| ||2 − || 1 || e2βi (s−tk ) ||(S 0j ) 2 χ (s)|| ds

e

tk− −η M

tk− −τ M

tk− tk− +θ

e

|

||2 || − || 1 || e2βi (s−tk ) ||(S 1j ) 2 χ (s)|| ds ||2 || 1 || ˙ || dsdθ ||(R 1j ) 2 χ(s)

2βi (s−tk− ) ||

|}

≤μˆ i j μ˜ i j V j (tk− ) = μi j V j (tk− ). Hereby, it results in Vi j (tk+ ) ≤ μi j V j (tk− ).

(3.23)

Vi (tr++1 ) ≤ μi Vi j (tˆr−+1 ).

(3.24)

Similarly,

64

3 Output Regulation for Networked Switched Systems with Alternate …

For t ∈ [tk , tˆr +1 ), applying (3.20), (3.22)–(3.24) and the Definition 1.5 yields Vσ (t) (t) = Vi j (t) ≤ ce

Em | ln(μi μi j )+2(βi +κi )d Mi i=1

Tai

| −2βi T fi (0, t)

V (t0 ),

(3.25)

Em

where c = e i=1 [ln(μi μi j )+2(βi +κi )d Mi ]N0i (0, t) , T fi (0, t) denotes the total running time of the subsystem i on (0, t). Similarly, for t ∈ [tˆr +1 , tk+1 ), one has Vσ (t) (t) = Vi (t) ≤ cμi e

Em | ln(μi μi j )+2(βi +κi )d Mi i=1

Tai

| −2βi T fi (0, t)

V (t0 ).

(3.26)

In addition, combining (3.15), (3.21), (3.25) and (3.26) gives a||χ (t)||2 ≤ Vσ (t) (t), Vσ (t) (t0 ) ≤ b||χ (t0 )||2 , where a = min {λmin (Pi ), λmin (Pi j )}, i, j∈M

b = max {λmax (Pi ), λmax (Pi j )} + η M max {λmax (Si0 ), λmax (Si0j )} i, j∈M

+h max {λmax (Si1 ), λmax (Si1j )} + +

i, j∈M 2 τM max {λ (Ri1 ), λmax (Ri1j )}. 2 i, j∈M max

i, j∈M η2M max {λ (Ri0 ), λmax (Ri0j )} 2 i, j∈M max

Thus, it results in | max

||χ (t)||2 ≤ a −1 Vσ (t) (t) ≤ a −1 bc max{μσ (t) , 1}e i∈M

2 ln(μi μi j )+(βi +κi )d Mi Tai

| −2βi (t−t0 )

||χ (t0 )||2 .

The closed-loop system (3.9.1) and (3.9.2) is exponentially stable when w = 0. When w /= 0, there exist W1 > 0, c1 > 0 with the center manifold theory [23], such that ||x(t) − ||i w(t)|| + ||ξ(t) − Ei w(t)|| ≤ W1 e−c1 t (||x(0) − ||i w(0)|| + ||ξ(0) − Ei w(0)||),

which implies lim (||x(t) − ||i w(t)|| + ||ξ(t) − Ei w(t)||) = 0. Then lim e(t) = t→∞

lim C˜ i χ (t) = 0. This completes the proof.

t→∞

t→∞

||

Remark 3.4 Combining Lemma 2.2 in [22] with Cholesky decomposition Q = Z T Z leads to. X T QY + Y T Q X ≤ X T Z T Z Y + Y T Z T Z X ≤ X T Z T Z X + Y T Z T Z Y ≤ X T Q X + Y T QY with Q > 0 and X, Y, Z of suitable dimensions. Utilizing this inequality instead of Lemma 2.2 in [22], simpler forms of φl14 = ϑl (Rl0 − G l0 ), φl15 = 0 and ψl14 = 0 in (3.11) are obtained. This greatly simplifies the calculation process and enables it to be solved directly using the LMI toolbox in Matlab.

3.3 Main Results

65

Remark 3.5 The following iterative steps will be followed to find switching instants with proper event-triggered parameters h and δ˜i to achieve the minimum number of transmitted data (NTD) for practical use. Step 1: Give initial values and parameters. Choose parameters μˆ i , μˆ i j , βi , κi , d Mi satisfying conditions in Theorem 3.1, and set the initial values t0 = s0 = 0, σ (t0 ) = 1, r = 1, k = 1. Get Tai∗ via (3.14). Step 2: Find proper event-triggered parameters (h, δ˜i ). First, a small 0 < δ˜i < 1 is chosen and the maximum h ∗ could be found by solving (3.11)–(3.13) in Theorem 3.1. Then, h will be decreased at a small step size from h ∗ , for a fixed δ˜i , an NTD is calculated by Theorem 3.1, increasing δ˜i step by step until a minimal NTD is got corresponding to a pair of parameters (h, δ˜i ). After that, repeating similar ideas to decrease h ∗ and increase δ˜i to compute another minimal NTD corresponding to a new pair of (h, δ˜i ) until h is small. Finally, the minimum NTD is obtained by comparing all minimal NTDs, and the corresponding parameters (h, δ˜i ) are optimal. Step 3: Generate an event-triggered instant sr by AETM (3.5), and then stop monitoring the event-triggered condition therein during the waiting interval with length h. If sr + h < Tai∗ , then set r = r + 1, and repeat this step; else go to Step 4. Step 4: If sr + h ≥ Tai∗ , then let tk = sr + h be the next switching instant, and k = k + 1, go back to Step 3. Remark 3.6 Compared with [3, 4, 6, 7], the asynchronous switching signal (3.14) including event-triggering behaviors not only satisfies the MDADT but also enables the switching of system mode to be related to some event-triggered instant and the waiting interval in the AETM (3.5). Due to the waiting time h, the minimum interval of two adjacent event-triggered instants is h. Then, Zeno behavior is naturally excluded.

3.3.2 Event-Triggered Output Regulation Problem with Both Transmission Delays and Packet Losses From a practical point of view, transmission delays and packet losses may coexist, which motivates us to modify (3.5) to guarantee the event-triggered output regulation performance under the allowable transmission delays and the number of consecutive packet losses as follows: } { tre+1 = inf t ≥ tre + h|||e(t)||2 > δ σ (tre ) ||e(t)||2 , i∈M

(3.27)

where tre , tre+1 ∈ T2 denote any two adjacent triggered instants, e(t) = [ e T (tre ) − e T (t) ξ˜ T (tre ) − ξ˜ T (t) ]T , the scalar δ σ (tre ) > 0 is a mode-dependent threshold. Similar to (3.5), the event-triggered condition can be expressed as follows: ||e(t − η(t))||2 − δ σ (tre ) ||e(t − η(t))||2 ≤ 0.

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3 Output Regulation for Networked Switched Systems with Alternate …

Theorem 3.2 Consider NSSs (3.1) with (3.2) satisfying Assumption 3.1. For given scalars h > 0, ε > 0, 0 < δ˜l < 1, 0 < δl < 1, η M > 0, τ M > 0, d Mi > 0, λi > 0, λi j < 0, μˆ l > 1, and matrices Ji j , J˜i j , Ti j , if there exist matrices of appropriate dimensions Pl > 0, Sl0 > 0, Sl1 > 0, Rl0 > 0, Rl1 > 0, G l0 , G l1 , where l ∈ {i, i j}, i /= j, i, j ∈ M so that (3.11)–(3.13) hold, the number of successive packet losses. | dr ≤ d M AN S P L /\ logγ (1+w )

| ϕ . γ

(3.28)

Then, the ORP is solvable under the error feedback controller (3.3) based on the AETM (3.27) and the asynchronous switching signal (3.14) which satisfies the event-triggered instants and MDADT condition, where || || || || || || ||} {|| || || || || || || || || w = max ||εC i A˜ i || + ||εC i B˜ i j C i ||, ||εC i A˜ i || + ||εC i B˜ i C i || , δ i j = δ j , i, j∈M / / / / γ = 1 + min{γi }, ϕ = 1 + min{ϕi }, γi = δ i /(1 − δ i ), ϕi = δ˜i /(1 − δ˜i ). i∈M

i∈M

Proof For a successful data transmitted interval [sr +1 , sr +2 ), suppose that there exist dr unsuccessfully transmitted packets on this interval: sr +1 = tde0 < tde1 < tde2 < · · · < tder < tder +1 = sr +2 . For ι = d0 , d1 , . . . , dr , combining the event-triggered condition in (3.27) and e(t) = C i χ (tιe ) − C i χ (t) yields || / || |||| || / || || / || || || || ˜|||| || ||C i χ (t e ) − C i χ (t)|| ≤ δ i || ˜ ||Ci χ (t)|| ≤ δ i || I || C i χ (t)|| = δ i ||C i χ (t)||, ι (3.29) where I˜ = [ I2×2 0 ]. Therefore, it results in || || || || ||C i χ (t) − C i χ (t e )|| ≤ γi ||C i χ (t e )||, t ∈ [t e , t e ). ι ι ι ι+1

(3.30)

Further, if a sufficiently small ε > 0 is chosen such that || || || || ||C i χ (t e − ε) − C i χ (t e )|| ≤ γi ||C i χ (t e )||, ι+1 ι ι

(3.31)

|| || || || e − ε)|| ≤ (1 + γi )||C i χ (tιe )||. Meanwhile, from the AETM (3.27), thus, ||C i χ (tι+1 the following inequality holds || || || || ||C i χ (sr +1 )|| = ||C i χ (t e − ε) − C i χ (t e − ε) + C i χ (sr +1 )|| ι+1 ι+1 || || || || ≤ ||C i χ (t e − ε)|| + ϕi ||C i χ (sr +1 )|| ι+1

Combining (3.31), it leads to

(3.32)

3.3 Main Results

67

|| || || || ||C i χ (t e ) − C i χ (t e − ε)|| ≤ w ||C i χ (t e − ε)||. ι+1 ι+1 ι+1

(3.33)

By uniting (3.31) and (3.32), it is easy to obtain that || || || || ||C i χ (t e ) − C i χ (t e − ε)|| ≤ w (1 + γi )||C i χ (t e )||. ι+1 ι+1 ι

(3.34)

Accordingly, for ∀t ∈ [tder , tder +1 ), one has || || ||C i χ (t) − C i χ (sr +1 )|| ≤ ||C i χ (t) − C i χ (tder )|| + ||

dr −1 E

e [C i χ (tι+1 − ε) − C i χ (tιe )]||

ι=d0

|| d || r −1 ||E || || || e e + || [C i χ (tι+1 ) − C i χ (tι+1 − ε)]|| || || ι=d0

dr −1 dr E || E || || || e || || ≤ γi C i χ (tι ) + w (1 + ϕi )||C i χ (tιe )||. ι=d0

(3.35)

ι=d0

From (3.31) and (3.34), it gives || || || || ||C i χ (t e )|| ≤ ||C i χ (t e ) − C i χ (t e − ε)|| ι+1 ι+1 ι+1 || || || || e + ||C i χ (tι+1 − ε) − C i χ (tιe )|| + ||C i χ (tιe )|| || || ≤ [(1 + w )(1 + γi )]ι+1 ||C i χ (sr +1 )||.

(3.36)

|| || Substituting (3.36) into (3.35) to solve ||C i χ (tιe )||, one can get || || || || ||C i χ (t) − C i χ (sr +1 )|| ≤ [(1 + w )dr (1 + γi )dr +1 − 1]||C i χ (sr +1 )||.

(3.37)

Combining (3.28) and (3.37) yields || || || || ||C i χ (t) − C i χ (sr +1 )|| ≤ ϕi ||C i χ (sr +1 )||.

(3.38)

Hence, it is known that AETM (3.27) can guarantee AETM (3.5) in Theorem 3.1. It is also verified that Theorem 3.2 can be deduced from Theorem 3.1 when AETM (3.27) is used. Remark 3.7 In Theorem 3.2, the maximum allowable number of successive packet losses (MANSPL) d M AN S P L in (3.28) is a non-negative integer with δ i ≤ δ˜i . If δ i = δ˜i , one can obtain d M AN S P L = 0, which implies that no packet is lost and Theorem 3.2 degenerates to Theorem 3.1. Otherwise δ i < δ˜i , the event-triggered threshold is lowered, which may cause more packet transmissions.

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3 Output Regulation for Networked Switched Systems with Alternate …

3.4 Comparison with Existing Results To facilitate the comparison, no transmission delays and packet losses are considered and w = 0. System matrices in (3.1) are taken from [6] |

| | | | | | | 0 1 0 0 1 0 A1 = , B1 = , A2 = , B2 = . 0 −3.2 0.6 0 −0.5 0.1 To degenerate our ORP into the stabilization problem in [6], we set D1 = D2 = 0, Q 1 = Q 2 = 0, | | | −0.68 0.11 −0.57 −1.67 C1 = , C2 = , 0.5 −0.16 −0.67 −0.44 | | | | 0 −0.9 0 1.4 S1 = , S2 = . 0.5 0 −0.9 0 |

For the above, matrices satisfying Assumption 3.1 are: ||1 = ||2 = 0, E1 = E2 = 0, H1 = [1.5 − 0.6], H2 = [0.8 − 0.7], | | | | 2.9 1.6 −0.5 −0.4 E1 = , E2 = . 1.7 1.5 3.2 −0.4 According to Theorem 3.1, let positive parameters d M1 = d M2 = 3.2, β1 = 0.24, β2 = 0.09, κ1 = 0.3, κ2 = 0.18, parameters μˆ 1 = 4.3, μˆ 2 = 4.7, μˆ 21 = 5, μˆ 12 = 6.2 greater than 1, and the coordinate transformation matrices J12 = J21 = J˜12 = J˜21 = T12 = T21 = I4×4 are assumed. ∗ ∗ Based on the above parameters, Ta1 = 0.7829, Ta2 = 0.2295 can be obtained via (3.14), and the controller gains would be |

| | | 0.1001 0.2012 −0.2999 −0.1023 F1 = , F2 = . 0.0011 0.311 0.0002 0.3031 Following the procedure outlined in Remark 3.5, the event-triggered parameters h = 0.2, δ˜1 = 0.379 and δ˜2 = 0.388 are selected to obtain the minimum NTD. In this special case, the trajectories of the system states are shown in Fig. 3.3. Meanwhile, NTD under three pairs (h, δ˜i ) obtained by two sampled mechanisms of [6, 24], and our proposed AETM, respectively, are shown in Table 3.1. The results reflect that the proposed approach can significantly reduce NTD more than the two other approaches (Fig. 3.4).

3.5 Application to an F-18 Aircraft Model

69

Fig. 3.3 State responses of the closed-loop system

Table 3.1 The NTD for three different mechanisms

h

δ˜i (i = 1, 2)

NTD

Periodic sampling in [24]

0.35

–

58

ETM in [6]

0.05

δ˜1 = 0.15, δ˜2 = 0.1

72

AETM

0.20

δ˜1 = 0.379, δ˜2 = 0.388

47

Fig. 3.4 Sampled output tracking errors and control input of the closed-loop system

3.5 Application to an F-18 Aircraft Model The longitudinal short-period motion of the F-18 aircraft [25] can be thought of as achieved by the aircraft model switching between some operating points. Thus, the decoupling longitudinal short-period linear equation can be modeled as NSSs in the form of (3.1) [1]. We set the switching signal σ = 1 and 2 by two operating points with the different flight altitudes at the same Mach, where Mach number is 0.8 and

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3 Output Regulation for Networked Switched Systems with Alternate …

altitude is 12kft and 10kft. The corresponding system matrices selected from [25] are listed below: | | | | −1.562 0.9862 −0.2316 −0.04349 A1 = , B1 = , −14.96 −1.132 −26.48 −5.323 | | | | −1.675 0.9853 −0.2449 −0.04649 A2 = , B2 = . −16.16 −1.212 −28.34 −5.742 Other system matrices in ORP are given correspondingly |

| | | | | | | 0 −0.9 0 1.4 −2.6 −5.2 −5.9 −3.6 S1 = , S2 = , C1 = , C2 = , 0.5 0 −0.9 0 −5.5 −5.4 −6 −3.6 | | | | 3.0378 1.4990 0.1964 13.1042 D1 = , D2 = , 44.9108 72.7271 66.2478 80.5459 | | | | −21.996 −33.384 −40.375 −43.499 Q1 = , Q2 = , −20.338 −48.108 −40.872 −44.052 It is easy to verify matrices below satisfy Assumption 3.1 |

| | | −2.32 −4.8 −4.97 −5.53 ||1 = , ||2 = , −3.07 −4.02 −3.07 −3.02 | | | | −9.2454 −9.6355 −7.2665 −8.2971 E1 = , E2 = , −8.85 −7.6952 −7.8136 −7.4222 | | | | 0.05 −0.03 −0.04 0 H1 = , H2 = , −0.01 0 −0.11 −0.05 | | | | −7.8359 8.7304 12.3810 −12.4698 E1 = , E2 = . −7.0846 7.8359 12.3939 −12.3810 In the light of Theorem 3.2, choose positive parameters h = 0.22, d M1 = d M2 = 2.2, κ1 = 0.02, κ2 = 0.01, β1 = β2 = 0.1, η M = 0.22, δ˜1 = 0.581 and δ˜2 = 0.974 in the interval (0, 1), parameters μˆ 1 = 1.2, μˆ 2 = 2, μˆ 21 = 1.4, μˆ 12 = 1.5 greater than 1, ε = 0.001. The coordinate transformation matrices are: ⎡

⎡ ⎡ ⎤ ⎤ ⎤ 2 −2 −2 2 1 1 −2 2 2 2 −2 2 ⎢ −1 0 0 0 ⎥ ⎢ ⎢ ⎥ ⎥ ⎥, J21 = ⎢ 1 0 0 0 ⎥, J˜12 = ⎢ −1 −1 −1 0 ⎥, J12 = ⎢ ⎣ −2 2 1 −1 ⎦ ⎣ 1 1 −2 1 ⎦ ⎣ −1 −2 0 1 ⎦ 1 0 −1 −1 −2 2 −2 −1 −1 1 1 1 ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ 1 0 0 0 −1 0 0 2 −1 0 −1 2 ⎢ 0 0 1 2⎥ ⎢ ⎢ ⎥ ⎥ ⎥, T12 = ⎢ −1 2 1 1 ⎥, T21 = ⎢ 0 1 1 1 ⎥. J˜21 = ⎢ ⎣ −2 0 −1 2 ⎦ ⎣ −1 0 0 −1 ⎦ ⎣ 0 2 0 2 ⎦ −1 −1 0 2 −1 0 1 1 1 −2 0 −1

3.5 Application to an F-18 Aircraft Model

71

Fig. 3.5 State responses of the closed-loop system under the transmission delays and packet losses

Fig. 3.6 Sampled output tracking errors of the closed-loop system under the transmission delays and packet losses

For given η M = 0.22 and d M AN S P L = 2, δ 1 = 0.547 and δ 2 = 0.4 in (3.27) are obtained by solving (3.11)–(3.13) in Theorem 3.1 and (3.28) in Theorem 3.2. In ∗ ∗ = 0.7235 and Ta2 = 0.7059. Controller gains are addition, Ta1 | F1 =

| | | 0.5249 −0.6325 −0.3642 −0.2908 , F2 = . 0.6754 0.6084 0.8276 0.4654

The initial state χ (0) = [ 8 3 −3 −8 ]T and the simulation time T f = 25s are chosen. In Fig. 3.5, the state responses of the closed-loop system are given based on the switching signals in Fig. 3.7. The convergent curves of the sampled output tracking errors of the closed-loop system, in Fig. 3.6, indicate that the ORP is solvable under the maximum transmission delay and packet losses. In Fig. 3.8, the transmission delay period for sampled measurement and packet losses condition is represented.

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3 Output Regulation for Networked Switched Systems with Alternate …

Fig. 3.7 Switching signals of NSSs under the transmission delays and packet losses

Fig. 3.8 The transmission delay period of sampled measurement and the packet losses condition

3.6 Conclusion Two novel ETMs with alternate properties have been investigated to deal with the ORP for NSSs, which can tolerate bounded transmission delays and limit the number of successive packet losses by adjusting their parameters. Associated with the AETM, a switching signal with the MDADT constraints involving the triggering information is presented. It is allowed to be adjusted dynamically according to the utilized triggering behavior. Two series of sufficient conditions with the upper bound of transmission delays, with and without the MANSPL, to achieve the event-triggered output regulation for NSSs under different coordinate transformations are derived. One of the potential extensions of this work in the future is the distributed output regulation for switched sensor networks under more complex network-induced phenomena.

References

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References 1. Li, L., Song, L., Li, T., Fu, J.: Event-triggered output regulation for networked flight control system based on an asynchronous switched system approach. IEEE Trans. Syst. Man Cybern.: Syst. 51(12), 7675–7684 (2021) 2. Ma, L., Wang, Y., Han, Q.: Event-triggered dynamic positioning for mass-switched unmanned marine vehicles in network environments. IEEE Trans. Cybern. 52(5), 3159–3171 (2022) 3. Ren, H., Zong, G., Karimi, H.R.: Asynchronous finite-time filtering of networked switched systems and its application: an event-driven methodIEEE Trans. Circuits Syst. I: Regul. Pap. 66(1), 391–402 (2019) 4. Li, T., Fu, J., Deng, F., Chai, T.: Stabilization of switched linear neutral systems: an eventtriggered sampling control scheme. IEEE Trans. Automatic Control 63(10), 3537–3544 (2018) 5. Wang, Y., Gao, Y., Wu, D., Niu, B.: Interval observer-based event-triggered control for switched linear systems. J. Franklin Inst. 357(10), 5753–5772 (2020) 6. Xiao, X., Zhou, L., Ho, D.W.C., Lu, G.: Event-triggered control of continuous-time switched linear systems. IEEE Trans. Autom. Control. 64(4), 1710–1717 (2019) 7. Qi, Y., Liu, Y., Niu, B.: Event-triggered H∞ filtering for networked switched systems with packet disorders. IEEE Trans. Syst., Man, Cybern.: Syst. 51(5), 2847–2859 (2021) 8. Huang, J.: Nonlinear Output Regulation: Theory and Applications. SIAM, Philadelphia (2004) 9. Hu, W., Liu, L., Feng, G.: Cooperative output regulation of linear multi-agent systems by intermittent communication: a unified framework of time- and event-triggering strategies. IEEE Trans. Automatic Control 63(2), 548–555 (2018) 10. Khan, G.D., Chen, Z., Zhu, L.: A new approach for event-triggered stabilization and output regulation of nonlinear systems. IEEE Trans. Automatic Control 65(8), 3592–3599 (2020) 11. Liu, W., Huang, J.: Event-triggered global robust output regulation for a class of nonlinear systems. IEEE Trans. Automatic Control 62(11), 5923–5930 (2017) 12. Liu, W., Huang, J.: Robust practical output regulation for a class of uncertain linear minimumphase systems by output-based event-triggered control. Int. J. Robust Nonlinear Control 27(18), 4757–4590 (2017) 13. Li, J., Zhao, J.: Incremental passivity and incremental passivity-based output regulation for switched discrete-time systems. IEEE Trans. Cybern. 47(5), 1122–1132 (2017) 14. Long, L., Zhao, J.: Robust and decentralised output regulation of switched non-linear systems with switched internal model. IET Control Theory Appl. 8(8), 561–573 (2014) 15. Pang, H., Zhao, J.: Output regulation of switched nonlinear systems using incremental passivity. Nonlinear Anal. Hybrid Syst. 27, 239–257 (2018) 16. Jin, C., Li, L., Wang, R., Wang, Q.: Output regulation for stochastic delay systems under asynchronous switching with dissipativity. Int. J. Control 94(2), 548–557 (2021) 17. Li, J., Zhao, J.: Output regulation for switched discrete-time linear systems via error feedback: An output error-dependent switching method. IET Control Theory Appl. 8(10), 847–854 (2014) 18. Fridman, E.: Output regulation of nonlinear systems with delay. Syst. Control Lett. 50(2), 81–93 (2003) 19. Selivanov, A., Fridman, E.: Event-triggered H∞ control: A switching approach. IEEE Trans. Automatic Control 61(10), 3221–3226 (2016) 20. Fridman, E.: A refined input delay approach to sampled-data control. Automatica 46(2), 421– 427 (2010) 21. Park, P., Ko, J.W., Jeong, C.: Reciprocally convex approach to stability of systems with timevarying delays. Automatica 47(1), 235–238 (2011) 22. Chen, X., Hao, F.: Observer-based event-triggered control for certain and uncertain linear systems. IMA J. Math. Control. Inf. 30(4), 527–542 (2013)

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23. Chen, Z., Huang, J.: Attitude tracking and disturbance rejection of rigid spacecraft by adaptive control. IEEE Trans. Automatic Control 54(3), 600–605 (2009) 24. Ma, D., Zhao, J.: Stabilization of networked switched linear systems: an asynchronous switching delay system approach. Syst. Control Lett. 77, 46–54 (2015) 25. Lian, J., Li, C., Xia, B.: Sampled-data control of switched linear systems with application to an F-18 aircraft. IEEE Trans. Ind. Electron. 64(2), 1332–1340 (2017)

Chapter 4

Memory-Based Event-Triggered Output Regulation for Networked Switched Systems with Severely Unstable Dynamics

This chapter studies EAORP for NSSs with SUDs including all modes unstable and partial switching instants destabilizing, which means that the Lyapunov function increases both on the activation intervals of all subsystems and at some switching instants. Firstly, an MMETM for switched systems is proposed to effectively shorten asynchronous intervals, which employs historical sampled outputs and compares the mode of the current sampled instant and the adjacent sampled instant. Then, the maximum ADT for a novel switching signal is derived with a constraint on the ratio of total destabilizing switchings to total stabilizing switchings, which relaxes the requirement that the regular arrangement of destabilizing and stabilizing switchings. Moreover, with the help of different coordinate transformations in the EAORP, the discretized Lyapunov functions are no longer needed when synthesizing the NSSs with SUDs, and the asynchronous switching situation is also discussed. Afterward, by designing a DOFC, sufficient conditions are given to solve the EAORP for NSSs with SUDs subject to network-induced delays, packet disorders, and packet losses. Finally, the effectiveness of the proposed methods is verified via a switched RLC circuit.

4.1 Introduction Switched systems, as a significant part of hybrid systems, could be widely applied in many practical fields [1, 2]. To characterize actual systems more effectively, unstable dynamics of switched systems such as unstable subsystems and destabilizing switchings are taken into consideration, which makes the study more challenging [3–7]. Based on the DT method, ADT method, or periodical switching law, the stability analyses of switched systems with all unstable subsystems are addressed in combination with the discretized Lyapunov function or time-varying Lyapunov function approach respectively in [3–6], while all switching behaviors need to be stabilizing at switching © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Li and J. Fu, Output Regulation and Cybersecurity of Networked Switched Systems, Studies in Systems, Decision and Control 475, https://doi.org/10.1007/978-3-031-30972-4_4

75

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4 Memory-Based Event-Triggered Output Regulation for Networked …

instants. Associated with the DT method with the discretized Lyapunov function and a ratio constraint of destabilizing switchings to stabilizing switchings, [7] studies the stability problem of switched systems with not only all unstable subsystems but also partial destabilizing switchings. Many existing results with SUDs, which means all unstable subsystems and partial destabilizing switching, only focus on synchronous switching via discretized Lyapunov function with a large amount of calculation, while both asynchronous switching and mathematical deduction simplification in this issue need to be investigated in depth. In recent years, different ETMs keep emerging to reduce the bandwidth requirement and actuator abrasion [8], which can be classified into memoryless [9, 10] and memory-based [11–14] ETMs from the perspective of whether or not the previous information is contained. Compared with memoryless types, memory-based types making use of previous information can reduce the redundant triggering or improve transient performance. When it comes to switched systems, the joint design of the ETM and switching scheme brings not only more challenges but also more design freedom [15–18]. The ETM in a switched system transmits both data and system mode to the controller. Due to the mismatch between triggered instants and switching instants, asynchronous switching caused by ETM will occur, which may deteriorate system performance. To weaken these effects, [17] proposes a novel ETM combined with DT, which ensures that data must be transmitted once within a switching interval so that shorten asynchronous intervals. Besides, improper selection of historical values for memory-based ETM in switched systems may cause historical information used by ETM to come from subsystems that are not currently activated. Based on the above, it is of great significance to study memory-based ETMs for switched systems with the function of shortening asynchronous intervals under network-induced delays, packet disorders, and packet losses. The ORP is an important part of the control theory. For switched systems, for whether the solvability of the OR of all subsystems is guaranteed or not, powerful tools such as the ADT method [18, 19] and multiple Lyapunov functions method [20] are utilized to design switching signals for solving the ORP of switched systems. Different coordinate transformation techniques [18, 19] are employed to reduce the limitation that the regulator equations have common solutions. More efforts yet are needed to further improve the ORP of the switched systems in the presence of SUDs. Motivated by the above discussion, this chapter aims to solve the EAORP for NSSs with SUDs. The contributions are threefold: (1) To overcome the limitation of the periodic arrangement of stabilizing switching and destabilizing switching in existing literature, the improved ADT method is proposed under aperiodic switching arrangement only by restricting the ratio of total stabilizing switching to total destabilizing switching. Noted that the discretized Lyapunov functions are no longer utilized when synthesizing the NSSs with SUDs with the aid of different coordinate transformations in the EAORP. Different from the existing results, an asynchronous switching strategy is jointly designed under the consideration of SUDs.

4.2 Problem Formulation

77

(2) By accumulating errors between sampled outputs, including historical and current outputs, and the latest triggered output, an MMETM for NSSs is developed to reduce redundant transmissions caused by highly oscillatory of SUDs, which shortens the length of asynchronous intervals by comparing with modes of the current sampled instant and the adjacent sampled instant. (3) Associated with the improved ADT method, the study is extended to the situation where ORPs of all subsystems are unsolvable and partial switching instants are destabilizing. The EAORP of NSSs with SUDs is solved thereby under the framework of the improved ADT based on the co-design of the DOFC and the MMETM. Meanwhile, network-induced delays, packet disorders, and packet losses are all considered.

4.2 Problem Formulation The configuration of EAORP of NSSs with SUDs consists of a system model, an exosystem, a sampler, two ETMs, a DOFC, and an actuator, as shown in Fig. 4.1 and described as follows.

4.2.1 System Modelling Consider a class of NSSs with SUDs governed by ⎧ ˙ = Aσ (t) x(t) + Bσ (t) u(t) + Cσ (t) w(t), ⎨ x(t) y(t) = Dσ (t) x(t) + E σ (t) w(t), ⎩ w(t) ˙ = Sσ (t) w(t),

(4.1)

with state vector x ∈ Rn x , control input u ∈ Rn u , regulated output y ∈ Rn y , and exogenous input w ∈ Rn w . Aσ (t) , Bσ (t) , Cσ (t) , Dσ (t) , E σ (t) , Sσ (t) are known constant

Fig. 4.1 Diagram of the EAORP for NSSs with SUDs

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4 Memory-Based Event-Triggered Output Regulation for Networked …

matrices with proper dimensions. Sσ (t) has no eigenvalues with negative real parts [19]. σ (t) : [0, ∞) → M = {1, 2, · · · , m} is the switching signal, and its corresponding switching sequence is expressed as T = {tk }k∈N with t0 = 0. On [0, t) with t ∈ [tk , tk+1 ), there exists a stabilizing and destabilizing switching instant sequence T = {tms }m∈N+ and T = {tnd }n∈N with T ∪ T = T . N (0, t) =m + n is the number of switching instants on [0, t). As shown in Fig. 4.1, the regulated output y(t) and the switching signal σ (t) are sampled as a fixed interval h > 0. The buffer is used to store historical sampledoutputs y(sr,v−ι h) and the last sampled mode σ (sr,v h − h), r, ι, v ∈ N. The detection mechanism decides whether to send the sampled y(sr,v h) and σ (sr,v h) to the controller. The triggered output y(εr +1 h) and switching signal σ (εr +1 h) are transmitted from ETM to the controller via a network with a network-induced delay τrsc+1 . The controller output u(tr˜e ) is transmitted to the actuator via a network with a network-induced delay τrca+1 .

4.2.2 Event-Triggered Mechanisms To save the limited network resources, two ETMs are designed for S2C channel and C2A channel respectively. For convenience, denote T1 = {gr h}r ∈N and T2 = {t˜r˜e }r˜ ∈N as instant sequences satisfying triggered conditions of the left-side and the right-side ETM respectively. Denote T3 = {εr h}r ∈N and T4 = {tr˜e }r˜ ∈N as instant sequences that the data transmitted successfully via the left-side and the right-side network respectively. Obviously, T3 ⊆ T1 , T4 ⊆ T2 . For simplicity, the MMETM in the S2C channel is described as εr +1 h = min{min{εr +1 h /\ sr,v h|E(sr,v h) > 0}, v∈N

min{˜εr +1 h /\ sr,v h|σ (sr,v h) /= σ (sr,v h − h)}}, v∈N

(4.2)

by considering network-induced delays in the absence of packet losses and packet disorders, where E(sr,v h) =

min{n,v} E ι=0

γ˜ι e T (sr,v−ι h)oσ (sr,v h) e(sr,v−ι h) − δ σ (sr,v h) y T (εr h)oσ (sr,v h) y(εr h),

e(sr,v−ι h) = y(sr,v−ι h) − y(εr h), sr,v−ι h /\ [εr + (v − ι)]h,

oσ (sr,v h) > 0 is a weight matrix and δ σ (sr,v h) > 0 is a threshold. n is the maximum sampled number of employed historical sampled outputs, v means (that the current Ev−1 ) instant is the v-th after εr h. If n > v and ι = v, γ˜ι =1 − ι=0 γι , else γ˜ι = γι . En γι ∈ [0, 1] is a weight parameter with ι=0 γι =1, v, n, ι ∈ N. To shorten the length of asynchronous intervals caused by the MMETM as much as possible, the ETM in the C2A channel is added as

4.2 Problem Formulation

79

tr˜e+1 = inf{t ≥ tr˜e |δ(t) /= δ(t − ε)},

(4.3)

where ε is a sufficiently small positive scalar, δ(t) = r with t ∈ [εr h,εr +1 h) and δ(0) = 0 is the triggering counter, which increases 1 after each event-triggered instant of ETM (4.2). Remark 4.1 The detection condition in (4.2) consists of two parts. In the first part, cumulative errors between the min{n, v} historical sampled-outputs y(sr,v−ι h) and the latest triggered output y(εr h) are utilized to avoid redundant triggering caused by the highly oscillatory of SUDs. The second part is designed to shorten asynchronous intervals induced by ETM, which ensures that once the subsystem mode switches, the data is updated at the nearest sampled instant. Besides, (4.2) only detects triggered conditions at sampled instants, so the Zeno phenomenon can be avoided. Remark 4.2 Compared with [11–14], the cooperation of two parts in (4.2) ensures all used historical sampled outputs belong to the currently activated subsystem, i.e. [εr h,sr,v h] ⊆ [tk ,sr,v h]. Therefore, (4.2) is more suitable for NSSs. Remark 4.3 If n = 0, the first part of the MMETM (4.2) degenerates into the one in [16].

4.2.3 Dynamic Output Feedback Controller with Event-Triggered Mechanisms It can be seen from (4.2) and Fig. 4.2 that there is at most one switching on an eventtriggered interval. For convenience, it is then assumed that the switching instants tk and tk+1 occur on [εr h, εr +1 h) and [εr +n 1 −1 h, εr +n 1 h), and n 1 − 1 triggering occur sc ca sc , τmca ≤ τrca+l ≤ τ M with τmsc = min {τrsc+l }, τ M = on [tk , tk+1 ). Set τmsc ≤ τrsc+l ≤ τ M r ∈N,l∈N1

ca max {τrsc+l }, τmca = min {τrca+l } and τ M = max {τrca+l }, l ∈ N1 ={0, 1, · · · , n 1 }.

r ∈N,l∈N1

r ∈N,l∈N1

r ∈N,l∈N1

Let τr +l /\ τrsc+l +τrca+l satisfy τm ≤ τr +l ≤ τ M . For t ∈ [εr +l h+τr +l ,εr +l+1 h+τrsc+l+1 ), the input of the controller is y(εr +l h). For t ∈ [εr +l+1 h+τrsc+l+1 , εr +l+1 h +τr +l+1 ), the input of the controller is y(εr +l+1 h). Then, combined with (4.3), |r +l = [εr +l h+τr +l , εr +l+1 h + τr +l+1 ) can be divided into subintervals, i.e. |r +l =

lr || v=0

|r +l,v ∪

lr sc ||

|rsc+l+1,v ,

v=0

where { [sr +l,v h + τr +l , sr +l,v+1 h + τr +l ), v = 0, . . . , lr − 1, |r +l,v = [sr +l,lr h + τr +l , εr +l+1 h + τrsc+l+1 ), v = lr , { [sr +l+1,v h + τrsc+l+1 , sr +l+1,v+1 h + τrsc+l+1 ), v = 0, . . . , lr cs − 1, |rsc+l+1,v = [sr +l+1,lr sc h + τrsc+l+1 , εr +l+1 h + τr +l+1 ), v = lr cs

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4 Memory-Based Event-Triggered Output Regulation for Networked …

Fig. 4.2 Time diagram of data transmission with only network-induced delay

with lr = min{v|sr +l,v+1 h + τr +l ≥ εr +l+1 h lr sc = min{v|sr +l+1,v+1 h+τrsc+l+1 ≥ εr +l+1 h + τr +l+1 }. By using the input delay method [23], we introduce

+

τrsc+l+1 },

and

{

t − sr +l,v h, t ∈ |r +l,v , t − sr +l+1,v h, t ∈ |rsc+l+1,v , η2 (t) = t − tr˜e = t − (εr h + τrsc ), t ∈ |r +l , η1 (t) =

which yields η1m ≤ η1 (t) ≤ η1M and η2m ≤ η2 (t) ≤ η2M with η1M = τ M + h, η1m = τmsc , η2M = τ M − τmsc + h + Tmax , η2m = τmca , and Tmax is the maximum DT. Define { y(sr +l,v h) − y(εr +l h), t ∈ |r +l,v , e(t) = y(sr +l+1,v h) − y(εr +l+1 h), t ∈ |rsc+l+1,v . Thus, the controller associated with (4.2) is set to {

x˙c (t) = Acφ(t) xc (t) + Bcφ(t) xc (t − η1 (t)) + Ccφ(t) y˜ (t), u(t) ˆ = Hcφ(t) xc (t),

(4.4)

where { y˜ (t) =

y(εr +l h) = y(sr +l,v h) − e(t) = y(t − η1 (t)) − e(t), t ∈ |r +l,v , y(εr +l+1 h) = y(sr +l+1,v h) − e(t) = y(t − η1 (t)) − e(t), t ∈ |rsc+l+1,v .

φ(t) = σ (εr +l h) is the switching signal of the controller, t ∈|r +l , xc (t) ∈ Rxc is the controller state. Acφ(t) , Bcφ(t) , Ccφ(t) , Hcφ(t) are matrices to be determined with proper dimensions.

4.3 Main Results

81

4.2.4 The Closed-Loop System For simplicity, the following assumption is very general in the ORP [18–20], yet adds an exponential decay term with time-delay information in (4.4). Assumption 4.1 There exist matrices ||i , Ei , Aci , Bci and Hci satisfying Ai ||i + Bi Hci Ei + Ci − ||i Si = 0, Di ||i + E i = 0, Bci Ei e−Si η1M + Aci Ei = Ei Si .

(4.5)

Denote t˜k /\ εr +1 h + τrsc+1 . Substituting (4.4) into NSSs (4.1) with Assumption 4.1, the closed-loop system is deduced as follows: {

~ci j χ (t − η1 (t)) + C ci j e(t), χ(t) = A˜ i j χ (t) + B˜ i j χ (t − η2 (t))+C ˙ ˜ y(t) = Di χ (t), t ∈ [tk , t˜k ) { ~ci χ (t − η1 (t)) + C ci e(t), χ˙ (t) = A˜ i χ (t) + B˜ i χ (t − η2 (t))+C ˜ ˜ y(t) = Di χ (t), t ∈ [tk , tk+1 )

(4.6.1)

(4.6.2)

where ˜ = x(t) − ||i w(t), χ T (t) = [ x˜ T (t) xcT (t) ], x(t) x˜c (t) = xc (t) − Ei w(t), D˜ i = [ Di 0 ], | | | | | | | | Ai 0 Ai 0 0 Bi Hcj 0 Bi Hci , A˜ i = , B˜ j = , B˜ i = A˜ i j = 0 0 0 0 0 Acj 0 Aci | | | | | | | | 0 0 0 0 0 0 ~ ~ , Cci = , C ci = , C ci j = . Cci j = Ccj Di Bci Cci Di Bci −Ci −C j The initial condition of χ (t) is ϕ(t), where ϕ(t) is a continuous function on [−η M , 0] with η M = max{η1M , η2M }.

4.3 Main Results In this section, the goal is to jointly design a switching signal σ (t) and an output feedback controller (4.4) with ETM (4.2) and (4.3) to solve EAORP for NSSs (4.1) with SUDs, i.e. (i) When w(t) = 0, the closed-loop system (4.6.1) and (4.6.2) is exponentially stable. (ii) When w(t) /= 0, the solution of the closed-loop system (4.6.1) and (4.6.2) with any initial condition satisfies y(t) = D˜ σ (t) χ (t)→ 0, t → ∞. We first develop sufficient conditions for EAORP of (4.6.1) and (4.6.2) with network-induced delays involved only in the networked environment. Then, packet

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4 Memory-Based Event-Triggered Output Regulation for Networked …

disorders and packet losses are taken into account, sufficient conditions for EAORP are given.

4.3.1 Event-Triggered Asynchronous Output Regulation Problem with Only Network-Induced Delays Theorem 4.1 For NSSs (4.1), given the positive constants α, β, λ, h, η1m , η1M , η2m , η2M , n, ε, ρˆϑ > 1, 0 < δ i , γι , μˆ ϑ < 1 and suppose that the subsystem ς ς j switches to subsystem i at tk . If there exist positive definite matrices Pϑ , Sϑ , Rϑ , ς˜ ς ς oi , and matrices Ccϑ , Mϑ , Ti j , Ji j , J˜i j with ϑ ∈ {i, i j}, i /= j, i, j ∈ M, ι ∈ {0, 1, . . . , n}, ς ∈ {0, 1, 2, 3}, ς˜ ∈ {1, 3} such that | ψϑ < 0, ς

ς

ς

ς

ς˜

ς˜

R ϑ Mϑ ς˜ ∗ Rϑ

| > 0,

[ P i j eaη S i j eaη R i j ] ≤ ρˆi j [ P j S ςj R ςj ], ∀tk ∈ T ,

(4.7) (4.8.1)

[ P i j eaη S i j eaη R i j ] ≤ μˆ i j [ P j S ςj R ςj ], ∀tk ∈ T ,

(4.8.2)

[ Pi Siς Riς ] ≤ ρˆi [ Pi j Siςj Riςj ], ∀tk ∈ T ,

(4.9.1)

[ Pi Siς Riς ] ≤ μˆ i [ Pi j Siςj Riςj ], ∀tk ∈ T ,

(4.9.2)

under the switching signal σ (t) with the DT and destabilizing switching conditions / ˜ ρ) ˜ ln(1 μμρ η1M + h ≤ Tmin ≤ Tmax < , κ / / m n > N ∗ = − ln(μμ) ˜ ln(ρ ρ). ˜

(4.10) (4.11)

Then the EAORP of the closed-loop system (4.6.1) and (4.6.2) is solvable under (4.4) with the ETM (4.2) and (4.3), where the 11×11 matrix ψϑ = {ψϑ } is composed of matrices ψϑ11 = A˜ iT Pϑ + Pϑ A˜ i + (2 − ϑ ϑ )Pϑ + Sϑ0 + Sϑ2 + 4 A˜ iT Rϑ A˜ iT − Rϑ0 − Rϑ2 , ψϑ12 = Rϑ0 , ψϑ15 = Rϑ2 , ψϑ17 = Pϑ B˜ iT , ψϑ22 = eϑη1m (Sϑ1 − Sϑ0 ) − Rϑ0 − Rϑ1 , ψϑ23 = Mϑ1 , ψϑ24 = Rϑ1 − Mϑ1 , ψϑ33 = −eϑη1M Sϑ1 − Rϑ1 , ψϑ34 = Rϑ1 − Mϑ1 T , ψϑ44 = Mϑ1 + Mϑ1 T − 2Rϑ1 + δ i D˜ iT oi D˜ i , T ψϑ48 = −δ i D˜ ϑ oi , ψϑ4,10 = C˜ cϑ , ψϑ55 = eϑη2m (Sϑ3 − Sϑ2 ) − Rϑ2 − Rϑ3 , ψϑ56 = Mϑ3 ,

4.3 Main Results

83

ψϑ57 = Rϑ3 − Mϑ3 , ψϑ66 = −eϑη2M Sϑ3 − Rϑ3 , ψϑ67 = Rϑ3 − Mϑ3 T , T ψϑ77 = Mϑ3 + Mϑ3 T − 2Rϑ3 + 4 B˜ iT Rϑ B˜ i , ψϑ88 = δ i oi − γ0 oi , ψϑ8,11 = C cϑ ,

ψϑ99 = diag{−γ1 oi · · · − γn oi }, ψϑ10,10 = ψϑ11,11 = λ2 (Pϑ + 4Rϑ ) − 2λI, 2 2 Rϑ = η1m Rϑ0 + (η1M − η1m )2 Rϑ1 + η2m Rϑ2 + (η2M − η2m )2 Rϑ3 , ς ς ς ς P i j = TiTj Pi j Ti j , S i j = JiTj Si j Ji j , R i j = J˜iTj Ri j J˜i j , a = α − β, ς

ς

ς

ς

(P i j )1/ 2 = (Pi j )1/ 2 Ti j , (S i j )1/ 2 = (Si j )1/ 2 Ji j , (R i j )1/ 2 = (Ri j )1/ 2 J˜i j , ρ = max{ρˆi },μ = max{μˆ i }, ρ˜ = max {ρˆi j }, μ˜ = max {μˆ i j }, i∈M

i∈M

i, j∈M

i, j∈M

ϑ i = α, ϑ i j = β, (η, ς ) ∈ {(η1m , 0), (η1M , 1), (η2m , 2), (η2M , 3)}.

Proof The switching interval [tk , tk+1 ) is divided into the asynchronous interval [tk , t˜k ) and the synchronous interval [t˜k , tk+1 ). The evolution of the corresponding Lyapunov function is thus present in the following two cases. Case 1: On the interval t ∈ [tk , t˜k ), j = φ(t) /= σ (t) = i. Thus, the Lyapunov candidate function is constructed as Vi j (t) = V Pi j (t) + VSi j (t) + VRi j (t),

(4.12)

where V Pi j (t) = χ T (t)Pi j χ (t), { VSi j (t) = +

t

t−η1m { t

eβ(t−s) χ T (s)Si0j χ (s)ds +

t−η2m

{

eβ(t−s) χ T (s)Si2j χ (s)ds + {

VRi j (t) = η1m

{

t

t−η1m

t

θ

+ η2m

t t−η2m

eβ(t−s) χ T (s)Si1j χ (s)ds

t−η1M { t−η2m t−η2M

eβ(t−s) χ T (s)Si3j χ (s)ds,

eβ(t−s) χ˙ T (s)Ri0j χ˙ (s)dsdθ {

t−η1m

+ (η1M − η1m ) {

t−η1m

t−η1M

{

t θ

+ (η2M − η2m )

{

t θ

eβ(t−s) χ˙ T (s)Ri1j χ(s)dsdθ ˙

eβ(t−s) χ˙ T (s)Ri2j χ˙ (s)dsdθ

{

t−η2m t−η2M

{ θ

t

eβ(t−s) χ˙ T (s)Ri3j χ˙ (s)dsdθ.

Taking the time derivative of Vi j (t) along the trajectory of the system (4.6.1) yields V˙ pi j (t) = 2χ T (t)Pi j [ A˜ i χ (t) + B˜ i χ (t − η2 (t)) + C˜ ci j χ (t − η1 (t)) + C¯ ci j e(t)],

84

4 Memory-Based Event-Triggered Output Regulation for Networked …

V˙ Si j (t) = βVSi j (t) + χ T (t)(Si0j + Si2j )χ (t) + eβη1m χ T (t − η1m )(Si1j − Si0j )χ (t − η1m ) − eβη1M χ T (t − η1M )Si1j χ (t − η1M ) + eβη2m χ T (t − η2m )(Si3j − Si2j )χ (t − η2m ) − eβη2M χ T (t − η2M )Si3j χ (t − η2M ), 2 χ˙ T (t)Ri0j χ˙ (t) + (η1M − η1m )2 χ˙ T (t)Ri1j χ(t) ˙ V˙ Ri j (t) = βVRi j (t) + η1m 2 + η2m χ˙ T (t)Ri2j χ˙ (t) + (η2M − η2m )2 χ˙ T (t)Ri3j χ˙ (t) { t eβ(t−s) χ˙ T (s)Ri0j χ˙ (s)ds − η1m t−η1m

{

t−η1m

− (η1M − η1m ) { − η2m

t−η1M t t−η2m

eβ(t−s) χ˙ T (s)Ri1j χ˙ (s)ds

eβ(t−s) χ˙ T (s)Ri2j χ(s)ds ˙ {

− (η2M − η2m )

t−η2m t−η2M

eβ(t−s) χ˙ T (s)Ri3j χ(s)ds. ˙

Applying Lemma 1.8 and Park theorem [21] gets { − ης12 m

t

ς

t−ης12 m

eβ(t−s) χ˙ T (s)Ri j02 χ˙ (s)ds ς

≤ −[χ (t) − χ (t − ης12 m )]T Ri j02 [χ (t) − χ (t − ης12 m )], { −(ης12 M − ης12 m )

t−ης12 m t−ης12 M

ς

eβ(t−s) χ˙ T (s)Ri j13 χ˙ (s)ds ≤ χηT (t)

where (ς02 , ς12 , ς13 ) ∈ {(0, 1, 1), (2, 2, 3)} | | χ (t − ης12 m ) − χ (t − ης12 (t)) . χ (t − ης12 (t)) − χ (t − ης12 M ) Combining (4.13) and (4.14) with (4.2) gives

|

(4.13)

ς ς | Rϑ13 Mϑ 13 χη (t), ς ∗ Rϑ13 (4.14)

and

χη (t)

V˙i j (t) − βVi j (t) ≤ ζ T (t)ψ i j ζ (t),

=

(4.15)

where the 9 × 9 matrix ψ i j = {ψ i j } is composed of matrices 11

12

15

17

22

23

24

12 15 17 22 23 24 ψ i j = ψi11 j , ψ i j = ψi j , ψ i j = ψi j , ψ i j = ψi j , ψ i j = ψi j , ψ i j = ψi j , ψ i j = ψi j , 33 34 44 48 55 34 44 48 55 ˜T ˜ ψ i j = ψi33 j , ψ i j = ψi j , ψ i j = ψi j + C ci j (Pi j + 4Ri j )C ci j , ψ i j = ψi j , ψ i j = ψi j , 56

57

66

67

77

57 66 67 77 ψ i j = ψi56 j , ψ i j = ψi j , ψ i j = ψi j , ψ i j = ψi j , ψ i j = ψi j ,

4.3 Main Results

85 T

88

99

99 ψ i j = ψi88 j + C ci j (Pi j + 4Ri j )C ci j , ψ i j = ψi j ,

ζ T (t) = [χ T (t) χ T (t − η1m ) χ T (t − η1M ) χ T (t − η1 (t)) χ T (t − η2m ) χ T (t − η2M ) χ T (t − η2 (t)) e T (t) e T (t − h) · · · e T (t − nh)].

We thus obtain ψ i j < 0 from ψi j < 0 by applying Lemma 1.7 and −(Pi j + 4Ri j )−1 < λi2j (Pi j + 4Ri j )−2λi j Ii j from Lemma 1.9. Adding this into (4.15) and integrating it over the interval [tk , t˜k ) gives Vi j (t) ≤ eβ(t−tk ) Vi j (tk ).

(4.16)

Case 2: On the interval t ∈ [t˜k , tk+1 ), φ(t) = σ (t) = i. Then, the Lyapunov candidate function is chosen as: Vi (t) = V Pi (t) + VSi (t) + VRi (t)

(4.17)

where V Pi (t) = χ T (t)Pi χ (t), { VSi (t) = +

t t−η1m { t

eα(t−s) χ T (s)Si0 χ (s)ds +

t−η2m

{

eα(t−s) χ T (s)Si2 χ (s)ds + {

VRi (t) = η1m

{

t

t

{

t−η1m

+ (η1M − η1m ) { + η2m

{

t

t−η2m

eα(t−s) χ T (s)Si1 χ (s)ds

t−η1M { t−η2m t−η2M

eα(t−s) χ T (s)Si3 χ (s)ds,

eα(t−s) χ˙ T (s)Ri0 χ(s)dsdθ ˙

θ

t−η1m

t−η1m

θ

+ (η2M − η2m )

t−η1M t

{

t

θ

eα(t−s) χ˙ T (s)Ri1 χ(s)dsdθ ˙

eα(t−s) χ˙ T (s)Ri2 χ˙ (s)dsdθ

{

t−η2m t−η2M

{

t θ

eα(t−s) χ˙ T (s)Ri3 χ˙ (s)dsdθ,

Similar to Case 1, over the interval [t˜k , tk+1 ), it can be readily implied that Vi (t) ≤ eα(t−t˜k ) Vi (t˜k ).

(4.18)

Next, we analyze the relationship between Lyapunov functions before and after tk and t˜k . For any tk ∈ T , we have χ (tk+ ) = Ti j χ (tk− ) due to different coordinate transformations. Combing with (4.12) and (4.17) yields

86

4 Memory-Based Event-Triggered Output Regulation for Networked …

V Pi j (tk+ ) − μˆ i j V P j (tk− ) = χ T (tk+ )Pi j χ (tk+ ) − μˆ i j χ T (tk− )P j χ (tk− ) = χ T (tk− )TiTj Pi j Ti j χ (tk− ) − μˆ i j χ T (tk− )P j χ (tk− ). We then obtain (TiTj Pi j Ti j − μˆ i j P j ) < 0 from (4.8.1) and (4.8.2), which implies V Pi j (tk+ ) ≤μˆ i j V P j (tk− ). Similar analysis shows { Vi j (tk+ )

≤

μˆ i j V j (tk− ) ≤ μV ˜ j (tk− ), tk ∈ T , − ρˆi j V j (tk ) ≤ ρV ˜ j (tk− ), tk ∈ T .

(4.19)

For t˜k , (4.9.1) and (4.9.2) implies that Vi (t˜k+ ) ≤

{

μˆ i Vi j (t˜k− ) ≤ μVi j (t˜k− ), tk ∈ T , ρˆi Vi j (t˜k− ) ≤ ρ Vi j (t˜k− ), tk ∈ T .

(4.20)

Besides, for ∀t ∈ [t˜k , tk+1 ), one can get from (4.16), (4.18) and (4.20) that Vσ (t) (t) = Vi (t) ≤ κeα(t−t˜k ) Vσ (tk )σ (tk−1 ) (t˜k− ) ≤ κ κe ˜ α(t−t˜k ) eβ(t˜k −tk ) Vσ (tk−1 ) (tk− ) ˜ m (ρ ρ) ˜ n e[αTs (0,t)+βTas (0,t)] Vσ (t0 ) (t0 ) ≤ . . . ≤ (μμ) ˜ ˜ κ T (0,t) ln(ρ ρ) e Vσ (t0 ) (t0 ) ≤ em ln(μμ)+n ln(ρ ρ) ˜ ρ)] ˜ −n ln(μμ)−m ˜ ˜ κ N (0,t)Tmax e e Vσ (t0 ) (t0 ) ≤ e N (0,t)[ln(μμ)+ln(ρ

=e

| | ln(ρ ρ) ˜ ˜ N (0,t) ln(μμ)+ln(ρ ˜ ρ)+κ ˜ Tmax + −n ln(μNμ)−m (0,t)

Vσ (t0 ) (t0 ),

then one has Vσ (t) (t) = Vi (t) ≤ e M T (0,t) Vσ (t0 ) (t0 ),

(4.21)

˜ ρ}, ˜ Ts (0, t), Tas (0, t) and T (0, t) denote the total where κ ∈ {μ, ρ}, κ˜ ∈ {μ, synchronous time, asynchronous time, and running time on [0, t), Tmin is the minimum DT, | | n ln(μμ) ˜ + m ln(ρ ρ) ˜ N (0, t) κ Tmax + ln(μμ) . M= ˜ + ln(ρ ρ) ˜ − Tmin (m + n) N (0, t) From (4.10) and (4.11), it gives M < 0. Similar to (4.26), for ∀t ∈ [tk , t˜k ), it holds that Vσ (t) (t) = Vi j (t) ≤ ρe M T (0,t) Vσ (t0 ) (t0 ).

(4.22)

4.3 Main Results

87

2 2 From (4.21), / /(4.22) and a||χ (t)|| ≤ V (t), V (t0 ) ≤ b||χ (t0 )|| , one can obtain ||χ (t)|| ≤ bρ ae M T (0,t) ||χ (t0 )||, where

a = min {λm (Pi j ), λm (Pi )}, i, j∈M

b = max {λ M (Pi j ), λ M (Pi )} + i, j∈M

+

2 E

2 E ς12 =1

+

2 E ς12 =1

ς

ς

ης12 m max {λ M (Si j02 ), λ M (Si 02 )} i, j∈M

ς12 =1 ς

ς

(ης12 M − ης12 m ) max {λ M (Si j13 ), λ M (Si 13 )} i, j∈M

ς12 =1

+

2 E

/ ς ς (ης212 m 2) max {λ M (Ri j02 ), λ M (Ri 02 )} i, j∈M

/ ς ς [(ης12 M − ης12 m )2 2] max {λ M (Ri j13 ), λ M (Ri 13 )}. i, j∈M

Therefore, system (4.6.1) and (4.6.2) is exponentially stable when w(t) = 0. When w(t) /= 0, there exist W > 0, c > 0 with the center manifold theory [24] such that ||x(t) − ||i w(t)|| + ||xc (t) − Ei w(t)|| ≤ W e−ct (||x(0) − ||i w(0)|| + ||xc (0) − Ei w(0)||).

Then lim (||x(t) − ||i w(t)|| + ||xc (t) − Ei w(t)||) = lim χ (t) = 0, that is, t→∞

t→∞

lim y(t) = D˜ i χ (t) = 0. Thus the EAORP of the closed-loop system (4.6.1) and t→∞ [] (4.6.2) is solved. Remark 4.4 Compared with [18–20], the EAORP for NSSs in Theorem 4.1 considers that the EAORPs of all subsystems are unsolvable, and partial switching instants are destabilizing. Remark 4.5 Compared with [3–7], asynchronous switchings are involved in Theorem 4.1 for switched systems with SUDs. Besides, the arrangement of switching instants in [7] is periodic for the synchronous case, that of Theorem 4.1 are arranged aperiodically as long as (4.11) is satisfied. Remark 4.6 To simply the mathematical deduction, the discretized Lyapunov function method is not utilized as in [3–7]. Without loss of generality, V Pi j and V Pi in the Lyapunov function are taken to explain the motivation. In fact, χ (tk+ )= χ (tk− ) is required as in [3–7], V Pi j (tk+ ) ≤ μˆ i j V P j (tk− ) and V Pi (t˜k+ ) ≤ μˆ i V Pi j (t˜k− ) i.e. Pi j ≤ μˆ i j P j and Pi ≤ μˆ i Pi j is impossible to be satisfied at the same time without the discretized Lyapunov function method. However, thanks to different coordinate transformations, there exists Ti j such that χ (tk+ ) = Ti j χ (tk− ) which means that χ (tk+ ) = χ (tk− ) does not have to be true. (4.8.1), (4.8.2)-(4.9.1), (4.9.2) can thus

88

4 Memory-Based Event-Triggered Output Regulation for Networked …

hold even for 0 < μˆ ϑ < 1, then Vi j (tk+ ) ≤μˆ i j V j (tk− ), V Pi (t˜k+ ) ≤ μˆ i V Pi j (t˜k− ) can be ensured without the discretized Lyapunov function method. When n = 0, there is no destabilizing switching. We provide the following sufficient conditions for this particular situation. Corollary 4.1 For NSSs (4.1), given the positive constants α, β, λ, h, η1m , η1M , η2m ,η2M , n, ε, 0 < δ i , γι , μˆ ϑ < 1. If there exist positive definite matrices ς ς ς˜ ς ς Pϑ , Sϑ , Rϑ , oi , and matrices Mϑ , Ti j , Ji j , J˜i j with ϑ ∈ {i, i j}, i /= j, i, j ∈ M, ι ∈ {0, 1, . . . , n}, ς ∈ {0, 1, 2, 3}, ς˜ ∈ {1, 3} such that (4.7), (4.8.2), and (4.9.2) hold under the switching signal with DT condition. η1M + h ≤ Tmin ≤ Tmax

/ ˜ ln(1 μμ) , < κ

(4.23)

and the controller (4.4) with (4.2) and (4.3). Then the EAORP for NSSs (4.1) is solvable, where a, η, μ, ˜ μ are defined as Theorem 4.1. Proof Similar to the proof of Theorem 4.1.

4.3.2 Event-Triggered Asynchronous Output Regulation Problem in the Presence of Network-Induced Delays, Packet Disorders, Packet Losses In the actual network environment, packet disorders and packet losses are not be neglected. If it is assumed that the disordered packets will be discarded, the new MMETM is derived by recalling triggered conditions in (4.2): ˜ sr,v h) > 0}, gr +1 h = min{min{gr +1 h /\ s˜r,v h|E(˜ v∈N

min{g˜r +1 h /\ s˜r,v h|σ (˜sr,v h) /= σ (˜sr,v h − h)}}, v∈N

(4.24)

in the presence of packet losses, where ˜ sr,v h) = E(˜

min{n,v} E

γ˜ι e T (˜sr,v−ι h)oσ (˜sr,v h) e(˜sr,v−ι h) − δ˜σ (˜sr,v h) y T (gr h)oσ (˜sr,v h) y(gr h),

ι=0

e(˜sr,v−ι h) = y(˜sr,v−ι h) − y(gr h), s˜r,v−ι h /\ [gr + (v − ι)]h, oσ (˜sr,v h) > 0 is a weight matrix, δ˜σ (˜sr,v h) > 0 is a threshold, γ˜ι is defined as (4.2). The ETM from the controller to the actuator is modified as ˜ /= δ(t ˜ − ε)}, t˜r˜e+1 = inf{t ≥ t˜r˜e |δ(t)

(4.25)

4.3 Main Results

89

Fig. 4.3 Time diagram of data transmission with network-induced delay, packet disorders, and packet losses

˜ = r , t ∈ [gr h,gr +1 h) and δ(0) ˜ = 0. where δ(t) From Fig. 4.3, the analysis of packet disorders and packet losses is the same as that of network-induced delay only, except the upper bound of η2 (t) is different. Affected by packet disorders and packet losses, η2 (t) ≤ η˜ 2M = τ M − τmsc + h+ (d + 1)Tmax , where d is the maximum number of successive packet losses. We then focus on the EAORP for NSSs (4.1) based on the ETM (4.24) and (4.25). Theorem 4.2 For NSSs (4.1), given the positive constants α, β, λ, h, η1m , η1M , η2m , η2M , n, d, d M , ε, 0 < δ i , γι , μˆ ϑ < 1, ρˆϑ > 1. If there exist positive definite matrices ς ς ς˜ ς ς Pϑ , Sϑ , Rϑ , oi , and matrices Mϑ , Ti j , Ji j , J˜i j with ϑ ∈ {i, i j}, i /= j, i, j ∈ M, ι ∈ {0, 1, . . . , n}, ς ∈ {0, 1, 2, 3}, ς˜ ∈ {1, 3} such that (4.7)–(4.9.1), (4.9.2) hold and δ˜i satisfies δ˜i ≤

1 (((1 + n+1

/ / 1 δ i (n + 1))(1 + o )−d ) d+1 − 1)2 .

(4.26)

Then the EAORP for NSSs (4.1) is solvable under the controller (4.4) based on ETM (4.24) and (4.25) and switching signal (4.11) and η1M + h ≤ Tmin ≤ Tmax where γ = h

/

1−

/

/ ˜ ρ) ˜ ln(1 μμρ , < κ

(4.27)

(n + 1)δ i , γ = max γι , d M is the maximal delay of the ι∈{0,...n}

controller and || || || { ||| || || || ~cϑ || o = (n + 1) · max (n + 1) ||h D˜ i A˜ i || + ||h D˜ i C || i, j∈M || || || / || || / |||} || || || || || ~cϑ || + ||h D˜ i C cϑ || + ||γ γ D˜ i B˜ i || + ||γ γ D˜ i C || . Proof For any successfully transmitted interval [εr h, εr +1 h), it is assumed that there exist dr unsuccessfully transmitted packets, i.e. εr h = g0 h < g1 h < · · · < gdr h < gdr +1 h = εr +1 h. For κ = 0, 1,. . . , dr , applying (4.24) yields

90

4 Memory-Based Event-Triggered Output Regulation for Networked … min{n,v} E

|| || / || / || || || || || γ˜ι || D˜ i χ (gκ+1 h − (ι + 1)h) − D˜ i χ (gκ h)|| ≤ (n + 1)δ˜i || D˜ i χ (gκ h)||.

ι=0

(4.28) Together with the triangle inequality gives min{n,v} E

|| / || || || γ˜ι || D˜ i χ (gκ+1 − (ι + 1)h)||

ι=0

≤

min{n,v} E

|| min{n,v} || E / || / || || || || || γ˜ι || D˜ i χ (gκ+1 − (ι + 1)h) − D˜ i χ (gκ h)||+ γ˜ι || D˜ i χ (gκ h)|| ι=0

ι=0

/ || || || || ≤ (n + 1)( (n + 1)δ˜i + 1)|| D˜ i χ (gκ h)||.

(4.29)

Similar to (4.28) and (4.29), one can get from (4.2) that min{n,v} E

|| √ || || || γι || D˜ i χ (εr h)||

ι=0

≤

min{n,v} E

|| || || / √ || || || || || γι || D˜ i χ (gκ+1 − (ι + 1)h)|| + (n + 1)δ i || D˜ i χ (εr h)||.

(4.30)

ι=0

Then, it is easy to get (min{n,v} E √

) / || || min{n,v} || E √ || || || || || γι − (n + 1)δ i || D˜ i χ (εr h)|| ≤ γι || D˜ i χ (gκ+1 − (ι + 1)h)||. ι=0

ι=0

(4.31) From (4.6.1), (4.6.2), (4.29), and (4.31), it leads to min{n,v} E

|| √ || || || γι || D˜ i χ (gκ+1 h) − D˜ i χ (gκ+1 h − (ι + 1)h)||

ι=0

/ || || || || ≤ o ( (n + 1)δ˜i + 1)|| D˜ i χ (gκ h)||.

For t ∈ [gdr h, gdr +1 h), considering (4.28) and (4.32) yields min{n,v} E ι=0

|| √ || || || γι || D˜ i χ (t − (ι + 1)h) − D˜ i χ (εr h)||

(4.32)

4.3 Main Results

≤

min{n,v} E

91

|| √ || || || γι || D˜ i χ (t − (ι + 1)h) − D˜ i χ (gdr h)||

ι=0

||d −1 || r || √ || || E ˜ || + γι || ( Di χ (gκ+1 h − (ι + 1)h) − D˜ i χ (gκ h))|| || || κ=0 ι=0 || || min{n,v} dr −1 || E √ || || E ˜ || + γι || ( Di χ (gκ+1 h) − D˜ i χ (gκ+1 h − (ι + 1)h))|| || || min{n,v} E

κ=0

ι=0

≤

dr / E

/ r −1 || || dE || || || || || || (n + 1)δ˜i || D˜ i χ (gκ h)|| + o ( (n + 1)δ˜i + 1)|| D˜ i χ (gκ h)||. (4.33) κ=0

κ=0

From (4.28) and (4.33), one can have min{n,v} E

|| || || || γι || D˜ i χ (gκ+1 h)||

ι=0

≤

min{n,v} E

|| √ || || || γι || D˜ i χ (gκ+1 h) − D˜ i χ (gκ+1 h − (ι + 1)h)||

ι=1

+

min{n,v} E

|| || || √ || || || || || γι || D˜ i χ (gκ+1 h − (ι + 1)h) − D˜ i χ (gκ h)|| + || D˜ i χ (gκ h)||

ι=1

≤ [(1 +

/ || || || || (n + 1)δ˜i )(1 + o )]κ+1 || D˜ i χ (εr h)||.

(4.34)

|| || || || Then substituting (4.34) into (4.33) to solve || D˜ i χ (gκ h)|| results in min{n,v} E ι=0

(/

≤

|| √ || || || γι || D˜ i χ (t − (ι + 1)h) − D˜ i χ (εr h)|| )dr +1 || || || || ˜ (n + 1)δi + 1 (1+o )dr − 1)|| D˜ i χ (εr h)||.

(4.35)

Combining (4.26) and (4.35) brings min{n,v} E

|| / / || || √ || || || || || γι || D˜ i χ (t − (ι + 1)h) − D˜ i χ (εr h)|| ≤ δ i (n + 1)|| D˜ i χ (εr h)||.

ι=0

Then it is known that ETM (4.24) can ensure ETM (4.2) in Theorem 4.1. It verifies that Theorem 4.2 can be concluded from Theorem 4.1 when ETM (4.24) is applied.

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4 Memory-Based Event-Triggered Output Regulation for Networked …

Remark 4.7/ The d in (4.26) is a/ non-negative integer which implies (n+1)δ˜i ≤δ i (n+1). If (n+1)δ˜i = δ i (n+1), then d = 0, thereby no packet losses occur, Theorem 4.2 is reduced to Theorem 4.1.

4.4 Simulation Example We employ the switched RLC circuit shown in Fig. 4.4 [22] to verify the effectiveness of Theorem 4.2. Affected by external disturbances, its model is as (4.1), where x = [ qc i L ]T , qc is the charge of the capacitor, i L denotes the flux in the inductance, | Ai =

⎡ ⎤ | | | 0 0 1 10 ⎣ ⎦ , Bi = 1 , Di = − Lc1 i − LR 01 L

Without generality, choose i = 1, 2, 3. Select parameters as c1 = 2, c2 = 0.5, c3 = 1, R = 1, L = 1 then |

| | | | | 0 1 0 1 0 1 , A2 = , A3 = , −0.5 −1 −2 −1 −1 −1 | | | | 0 10 B1 = B2 = B3 = , D1 = D2 = D3 = . 1 01

A1 =

To solve the EAORP, other system matrices are given below |

| | | | | 0.002 0.02 0.01 0.03 0.12 0.02 C1 = , C2 = , C3 = , 0 −0.2 0.4 −1.3 0.06 −2.1 | | | | | | −0.2 0 −0.2 0 −0.2 0 E1 = , E2 = , E3 = , 0 0.2 0 0.2 0.1 0.2 | | | | | | 0.01 −0.9 0.08 −0.836 0.1 −0.9 S1 = , S2 = , S3 = . 0.5 0.01 0.01 0.17 0.15 0.12

Fig. 4.4 Switched RLC circuit

4.4 Simulation Example

93

Obviously, the eigenvalues of S1 , S2 , S3 all have positive real parts, which implies that the injection of the corresponding exogenous input would disable the EAORP of each subsystem as can be seen from Fig. 4.5. Based on the above parameters, the matrices that meet Assumption 4.1 are as follows |

Ac1

| | | | | 0 −0.9 −0.1 −0.41 −0.18 0.19 = , Ac2 = , Ac3 = , 0.5 0 0.02 −0.01 0.15 −0.16 | | | | −0.1441 0.1667 −0.6054 0.3541 Bc1 = , Bc2 = , 0.1332 −0.1441 2.1796 −0.9788 | | | | −0.6919 0.4008 −1 0 Bc3 = , E1 = , 2.3731 −1.0864 0 −1 | | | | | | 0.5 0 10 0.2 0 E2 = , E3 = , ||1 = , 0 1.1 01 0 −0.2 | | | | 0.2 0 0.2 0 ||2 = , ||3 = . 0 −0.2 −0.1 −0.2

According to Theorem 4.2, choose positive parameters α = 2, β = 0.7, λ = 0.2, n = 1, h = 0.01, η1m = η2m = 0.005, η1M =0.05, η2M = 0.945, d M = 0.5, γ0 = 0.9, γ1 = 0.1, δ 1 = 0.9, δ 2 = 0.8, δ 3 = 0.7, μˆ i = 0.2, μˆ i j = 0.4, ρˆi = ρˆi j = 2, i, j ∈{1, 2, 3}, ε= 0.001. Based on the above parameters, solving (4.7)–(4.9.1), (4.9.2), (4.11) and (4.27) yields Ta ≤ 0.8947, N ∗ =1.8219 and controller gains |

Cc1

| | | −0.1690 0.1837 −0.1931 0.2012 = , Cc2 = , 0.1551 −0.1012 −0.2162 0.1131 | | 0.3130 −0.4210 Cc3 = . −0.5154 0.3055

For given d = 1, δ˜1 = 0.0152, δ˜2 = 0.023, δ˜3 = 0.021 are obtained by (4.26). Fig. 4.5 State trajectories of three subsystems

94

4 Memory-Based Event-Triggered Output Regulation for Networked …

A switching signal with Tmin = Tmax = 0.45s is presented in Fig. 4.6, and the value of Lyapunov function has an increment at switching instants when subsystem 3 is activated. These switching instants are the so-called destabilizing switchings. It is / obtained that the number of stabilizing and destabilizing switchings is subject to m n ≈ 1.94 > N ∗ , which implies the sequences of them need not be arranged periodically. Inter-event intervals of ETM (4.24) and (4.25) are shown in Figs. 4.7 and 4.8 respectively. As observed from the first subgraph of Figs. 4.7 and 4.8, the interevents intervals don’t exceed the DT, that is, there exists at least one triggering on a switching interval. The second subgraph of Figs. 4.7 and 4.8. reflects inter-event intervals with network-induced delays and packet losses where the maximum total delay is 0.04s > h and d = 1. Corresponding to Figs. 4.7 and 4.8, the asynchronous switching signal and asynchronous interval are given in Fig. 4.9. If the triggered data at the nearest sampled instant after switching is transmitted successfully, then the asynchronous interval is no more than the maximum total network-induced delay 0.04s, which proves the proposed ETM (4.24) can shorten the length of asynchronous intervals caused by MMETM. Fig. 4.6 Evolution of Lyapunov function

Fig. 4.7 Inter-event intervals of ETM (4.24) in the S2C channel

4.4 Simulation Example

95

Fig. 4.8 Inter-event intervals of ETM (4.25) in the C2A channel

Fig. 4.9 Asynchronous switching signal and asynchronous interval

With χ (0) = [1 0.5 0.1 0.2]T , the state responses and regulated output of the closed-loop system shown in Fig. 4.10 converge to zero with the asynchronous switching signal in Fig. 4.9, which reveals the EAORP of (4.1) with SUDs is solvable. Next, to facilitate the verification of performance differences between various ETMs, simulation comparisons need to be carried out under conditions without packet losses. To prove the effectiveness of memory-based ETM, the comparison between (4.2) and ETM in [15] with σ (sr,v h) /=σ (sr,v h − h) is shown. From Fig. 4.11, the settling time under (4.2) is shorter than that of [15]. This is because more parameters γι ,n,δi in (4.2) provide more flexibility to improve system performance. From Fig. 4.12, there are 81 triggerings in (4.2) and 91 triggerings in ETM in [15], which verifies the memory-based ETM (4.2) can further reduce triggered numbers.

96

4 Memory-Based Event-Triggered Output Regulation for Networked …

Fig. 4.10 The state responses and the regulated output of the closed-loop system

Fig. 4.11 State responses under (4.2) and ETM in [15] with σ (sr,v h) /= σ (sr,v h − h)

To verify the ability of ETM (4.2) to shorten the length of asynchronous intervals, the comparisons between (4.2) and ETMs in [15] and [17] are given. From Table 4.1, although [15] has the fewest triggered numbers, the maximum length of its asynchronous interval is the largest, meanwhile, the stability of the closed-loop system is not achieved. Compared with [17], our proposed ETM can significantly improve the triggered numbers, the maximum length of asynchronous intervals. The above discussion shows that the proposed MMETM can not only reduce triggered numbers but also effectively shorten asynchronous intervals and accelerate convergence.

4.5 Conclusion

97

Fig. 4.12 Inter-event intervals of (4.2) and ETM in [15] with σ (sr,v h) /= σ (sr,v h − h)

Table 4.1 The comparison result with several existing ETMs (T = 25s) ETMs

Triggered numbers

The maximum length of asynchronous intervals

Stability

(2)

81

0.04

Yes

[15]

72

1.66

No

[17]

92

0.39

Yes

4.5 Conclusion The EAORP of NSSs with SUDs subject to network-induced phenomenon has been investigated under asynchronous switching in this chapter. The improved average DT with the ratio constraint of stabilizing switchings to destabilizing switchings allows the aperiodic arrangement of destabilizing switchings and stabilizing switchings. The discretized Lyapunov function is no longer needed in the synthesis of NSSs with SUDs with the aid of different coordinate transformations in the ORP. By using historic sampled outputs and comparing the mode of the current sampled instant and the last sampled instant, the proposed MMETM can reduce the redundant triggering, shorten asynchronous switching intervals and accelerate convergence. Then, sufficient conditions are derived to ensure the solvability of the EAORP for NSSs with SUDs. The security control problems of NSSs might be potential extensions in the future.

98

4 Memory-Based Event-Triggered Output Regulation for Networked …

References 1. Ma, L., Wang, Y., Han, Q.: Event-triggered dynamic positioning for mass-switched unmanned marine vehicles in network environments. IEEE Trans. Cybern. 52(5), 3159–3171 (2022) 2. Xu, L., Wang, Q., Li, W., Hou, Y.: Stability analysis and stabilisation of full-envelope networked flight control systems: switched system approach. IET Control Theory Appl. 6(2), 286–296 (2012) 3. Xiang, W., Xiao, J.: Stabilization of switched continuous-time systems with all modes unstable via dwell time switching. Automatica 50(3), 940–945 (2014) 4. Su, Q., Fan, Z., Li, J.: H∞/H-fault detection for switched systems with all subsystems unstable. IET Control Theory Appl. 13(12), 1796–1803 (2019) 5. Fu, J., Ma, R., Chai, T., Hu, Z.: Dwell-time-based standard H∞ control of switched systems without requiring internal stability of subsystems. IEEE Trans. Autom. Control 64(7), 3019– 3025 (2019) 6. Lu, A., Yang, G.: Stabilization of switched systems with all modes unstable via periodical switching laws. Automatica 122(2020) 7. Wang, Y., Karimi, H.R., Wu, D.: Conditions for the stability of switched systems containing unstable subsystems. IEEE Trans. Circuits Syst. II, Exp. Briefs 66(4), 617–621 (2019) 8. Zhang, X., Han, Q., Ge, X., Ding, D., Ding, L., Yue, D., Peng, C.: Networked control systems: a survey of trends and techniques. IEEE/CAA J. Autom. Sinica 7(1), 1–17 (2020) 9. Zhang, X., Han, Q.: A decentralized event-triggered dissipative control scheme for systems with multiple sensors to sample the system outputs. IEEE Trans. Cybern. 46(12), 2745–2757 (2016) 10. Wang, J., Zhang, X., Han, Q.: Event-triggered generalized dissipativity filtering for neural networks with time-varying delays. IEEE Trans. Neural Netw. Learning Syst. 27(1), 77–88 (2016) 11. Wang, K., Tian, E., Liu, J., Wei, L., Yue, D.: Resilient control of networked control systems under deception attacks: a memory event-triggered communication scheme. Int. J. Robust Nonlin. 30(4), 1534–1548 (2020) 12. Davó, M.A., Prieur, C., Fiacchini, M.: Stability analysis of output feedback control systems with a memory-based event-triggering mechanism. IEEE Trans. Autom. Control 62(12), 6625–6632 (2017) 13. Tian, E., Peng, C.: Memory-based event-triggering H∞ load frequency control for power systems under deception attacks. IEEE Trans. Cybern. 50(11), 4610–4618 (2020) 14. Gu, Z., Shi, P., Yue, D., Yan, S., Xie, X.: Memory-based continuous event-triggered control for networked T-S fuzzy systems against cyber-attacks. IEEE Trans. Fuzzy Syst. 29(10), 3118– 3129 (2021) 15. Ren, H., Zong, G., Karimi, H.R.: Asynchronous finite-time filtering of networked switched systems and its application: an event-driven method. IEEE Trans. Circuits Syst. I, Reg. Papers 66(1), 391–402 (2019) 16. Xiao, X., Park, J.H., Zhou, L.: Event-triggered control of discrete-time switched linear systems with packet losses. Appl. Math. Comput. 333, 344–352 (2018) 17. Xiao, X., Zhou, L., Ho, D.W.C., Lu, G.: Event-triggered control of continuous-time switched linear systems. IEEE Trans. Autom. Control 64(4), 1710–1717 (2019) 18. Li, L., Song, L., Li, T., Fu, J.: Event-triggered output regulation for networked flight control system based on an asynchronous switched system approach. IEEE Trans. Syst., Man, Cybern., Syst. 51(12), 7675–7684 (2021) 19. Wang, P., Zhao, J.: Almost output regulation for switched positive systems with different coordinates transformations and its application to a positive circuit model. IEEE Trans. Circuits Syst. I, Reg. Papers 66(10), 3968–3977 (2019) 20. Li, J., Zhao, J.: Incremental passivity and incremental passivity-based output regulation for switched discrete-time systems. IEEE Trans. Cybern. 47(5), 1122–1132 (2017) 21. Park, P.G., Ko, J.W., Jeong, C.: Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47(1), 235–238 (2011)

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Chapter 5

Almost Output Regulation for Switched Linear Systems with Bumpless Transfer Control Under Destabilizing Behaviors

5.1 Introduction The switched system is a special hybrid system that has been widely concerned, which is composed of a group of continuous (or discrete) time systems and a switching rule [1]. Many excellent results of switched systems can be found in [2–5]. Initially, the research of switched systems focused on the condition that all subsystems were stable [6–9]. Under this circumstance, improper switching signals would still damage system performance. For switched systems with some unstable subsystems, how to design an appropriate switching law making the closed-loop system stable has more practical significance, [10] adopts the ADT strategy by limiting the ratio of DT between stable and unstable subsystems, where the increment of the Lyapunov function during the activation of unstable subsystems is compensated by the decrement of the Lyapunov function when stable subsystems are active. The above idea is only applicable to switched systems with at least one stable subsystem, which will fail if all subsystems are unstable. Recently, the DT approach has been improved to deal with the scenario of all subsystems unstable and all switching instants stable. In [11], the maximum and minimum DT of subsystems are both confined, where stable switching instants (SSIs) are exploited to offset the increased Lyapunov function of unstable subsystems. [12] further simplifies the form of the Lyapunov function, where the problems discussed are extended to stabilization and control. To overcome the requirement of SSIs, [13] allows the presence of some unstable switching instants (USIs) by adding a number ratio restriction and periodic arrangement of SSIs and USIs, which might be an undesired limit that deserves us to further explore the proper design of switching signals for switched systems under destabilizing behaviors. On the other hand, the controller switching inevitably causes the discontinuity of the control signal at switching instants which are called bumps that are an undesirable transient process and may damage system performance [14]. Bumpless transfer control is one of the effective approaches to suppress the bumps in transient performance generated by controller switching [15]. Initially, most of the BTC strategies © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Li and J. Fu, Output Regulation and Cybersecurity of Networked Switched Systems, Studies in Systems, Decision and Control 475, https://doi.org/10.1007/978-3-031-30972-4_5

101

102

5 Almost Output Regulation for Switched Linear Systems with Bumpless …

were constructed to modify pre-designed controllers under pre-known switching signals [16–19]. For pre-unknown switching sequences, a linear interpolation pattern with BT property is proposed in [20] by calculating the interpolation between two adjacent controllers for a while after switching instants, which has a significant inhibitory effect on bumps and yet still needs to modify the controller already designed. [21, 22] suppressed the bumps by limiting the control signal. Recently, by restricting control signals before and after switching instants, the BT property of switched systems is achieved via a direct design of a corresponding controller for each subsystem [23, 24], which requires all switching instants stable and may also lead to frequent switching. It is a pity that in the existing results, the stability of switching behaviors is considered to be single (stable or unstable), which is not in line with reality. Therefore, based on the above discussion, the BT property of switched systems needs to be further investigated for allowing the direct design of the controller and the presence of USIs and SSIs, and how to suppress the control bumps at these two different switching instants is a challenge. OR is one of the basic control problems in the field of control, which ensures that system output asymptotically tracks and/or inhibits a class of reference inputs and/ or external disturbance from the exosystem [25, 26]. Unfortunately, the output error is regulated precisely in a typical OR, while the actual systems and exosystem are inevitably affected by unknown disturbances. Therefore, it is necessary to consider the ORP under unknown disturbances, that is the AOR problem [27]. At present, some relevant results of the AOR problem for switched systems are shown in [28, 29]. However, none of these studies have concentrated on switched systems with destabilizing behaviors, which inspires our third motivation. Both steady-state and transient performances are concerned in this chapter for solving the AOR problem of switched systems with destabilizing behaviors under dynamic error feedback BT controller, where the AOR problems of all subsystems are unsolvable and some switching instants are unstable. The main contributions are as follows: (i) To achieve steady-state performance, a HADT strategy is proposed to solve the AOR problem of switched systems. Compared with [13], the proposed HADT strategy does not need to strictly limit the maximum DT of each subsystem and does not require a periodic arrangement of SSIs and USIs. (ii) To improve the transient performance, an exponential interpolation type of controller is proposed to interpolate adjacent controllers within the minimum DT, which not only allows the direct design of the controller but also ensures faster and smoother transitions to the corresponding controller compared with the linear interpolation type in [20]. (iii) The AOR problem for switched systems with destabilizing behaviors is solved under the HADT framework. The previous results assume that all subsystems or some switching behaviors are stable, which is difficult to achieve in practical applications due to the unknown disturbance. Therefore, the situation considered in this chapter is more general and practical.

5.2 Problem Formulation

103

5.2 Problem Formulation Consider the switched linear system under destabilizing behaviors described by ⎧ ˙ = Aσ (t) x(t) + Bσ (t) u(t) + E σ (t) w(t) + Fσ (t) d(t), ⎨ x(t) e(t) = Cσ (t) x(t) + Dσ (t) w(t) + G σ (t) d(t), ⎩ w(t) ˙ = Hσ (t) w(t) + Jσ (t) d(t),

(5.1)

with the system state x ∈ Rx , control input u ∈ Ru , unknown disturbance input d ∈ Rd belonging to L2 [0, ∞), regulated output error e ∈ Re , reference input and/ or external disturbance signal w ∈ Rw produced by the exosystem, switching signal σ (t) : [0, ∞) → S = {1, 2, · · ·, s} corresponding to its switching sequence {x0 : (i 0 , t0 ), (i 1 , t1 ), ···, (i k , tk ), ···|i k ∈ S, k ∈ N}, s is the number of subsystems, x0 and t0 are the initial state and the initial time, tk denotes the kth switching instant, σ (t) = i k means the i k - th subsystem is active. The forthcoming standard assumptions are needed to discuss the AOR problem of the system (5.1): (i) There are a finite number of switches within a finite time [11]. (ii) All eigenvalues of Hi have non-negative real parts { for any i| ∈ S which |} ensures w(t) is always imposed on (5.1) [30]. (iii) The | | Ai E i are detectable for any i ∈ S [31]. Since it is difficult pairs Ci Di , 0 Hi to measure the system states in practice, we only involve the output error in control synthesis. For convenience, we denote that the system (5.1) switches from subsystem j to subsystem i at the switching instant tk (k ∈ N) where the control input u(t) varies from K j x(tk− ) to K i x(tk+ ). Due to K i /= K j (i, j ∈ S, i /= j), u(t) may have a sudden change at tk , which is called control bumps and may damage the system performance. To suppress bumps and achieve the BT property, an exponential interpolation method, improved from the linear interpolation method in [20], will be proposed as follows via interpolating two adjacent controllers within the minimum DT τmin ≤ tk − tk−1 . The dynamic error feedback controller is devised as ⎧ ⎨ x˙c (t) ={L i xc (t) + Mi e(t), K ji (t)xc (t), t ∈ [tk , tk + τmin ), ⎩ u(t) = K i xc (t), t ∈ [tk + τmin , tk+1 ),

(5.2)

with the controller internal state xc ∈ Rc , the pending controller gains L i , Mi , K i τmin 1+ and K ji (t) = ς (t)K j + (1 − ς (t))K i , ς (t) = e t−(tk +τmin ) , i, j ∈ S, i /= j. Remark 5.1 On the interval [tk , tk +τmin ), we know from (5.2) that ς (tk ) = 1, which means K ji (tk ) = K j and the control bump at the switching instant tk is reduced to zero. In addition, due to ς (t) → 0 as t → tk + τmin , K ji (t) gradually tends to K i and switches to K i at t = tk + τmin . Therefore, the control signal u(t) achieves a smooth transition. It should be noted that the employ of an exponential interpolation for K ji (t) can achieve the corresponding control signal faster and smoother than [20].

104

5 Almost Output Regulation for Switched Linear Systems with Bumpless …

On the interval [tk + τmin , tk+1 ), the control input signal is provided by the controller corresponding to the currently activated subsystem instead of interpolation. Then, substituting the dynamic error feedback controller (5.2) into the system (5.1) obtains the augmented dynamics { ⎧ A ji (t)χ (t) + E i w(t) + F i d(t), t ∈ [tk , tk + τmin ), ⎪ ⎪ ⎨ χ˙ (t) = Ai χ (t) + E i w(t) + F i d(t), t ∈ [tk + τmin , tk+1 ), ⎪ C χ (t) + Di w(t) + G i d(t), e(t) = i ⎪ ⎩ w(t) ˙ = Hi w(t) + Ji d(t),

(5.3)

where ∗

χ T (t) = [ x T (t) xcT (t) ]; A ji (t) = ς (t)A ji + (1 − ς (t))Ai ; C i = [ Ci 0 ], | Ai =

| | | | | | | ∗ Ai Bi K i Ai Bi K j Ei Fi , A ji = , Ei = , Fi = . Mi C i L i Mi C i L i Mi Di Mi G i

The HADT strategy is proposed for switched systems under destabilizing behaviors. According to SSIs and USIs, subsystems are stipulated to belong to the set S− ={1, · · ·, m} when activated from SSIs and the set S+ ={s − l + 1, s − l + 2, · · ·, s} when activated from USIs without loss of generality with m + l = s, m, l ∈ N+ . Obviously, S− ∪ S+ = S, S− ∩ S+ = ∅. According to the slow ADT technology in [32] and the fast ADT technology in [33], we propose the following HADT method. Definition 5.1 For switching signal σ (t) and any t2 > t1 ≥ 0. Let Nσ− (t1 , t2 ) and Nσ+ (t1 , t2 ) denote the number of SSIs and USIs on [t1 , t2 ). Let Tσ− (t1 , t2 ) and Tσ+ (t1 , t2 ) represent the total running time of subsystems in set S− and S+ on [t1 , t2 ). If there exist N0− ≥ 0, τa− ≥ 0, N0+ ≥ 0, τa+ ≥ 0 satisfying {

Nσ− (t1 , t2 ) ≥ N0− +

Nσ+ (t1 , t2 ) ≤ N0+ +

Tσ− (t1 ,t2 ) , τa− Tσ+ (t1 ,t2 ) , τa+

σ (t) ∈ S− , σ (t) ∈ S+ ,

then τa− , τa+ are called the fast and slow ADT, where N0− and N0+ are chattering bounds of subsystems belonging to S− and S+ respectively. Remark 5.2 Among the existing literature, the fast ADT strategy is often exploited to stabilize / switched systems with all subsystems unstable [12], which requires T0− (t1 , t2 ) (Nσ− (t1 , t2 ) − N0− ) ≤ τa− . In this case, the average running time of each subsystem does not exceed τa− , and SSIs are arranged properly to offset the increment of the Lyapunov function caused by unstable subsystems. However, in the presence of USIs, the Lyapunov function may also increase at some

5.3 Main Results

105

switching /instants in our proposed HADT strategy. For this reason, the slow ADT Tσ+ (t2 , t1 ) (Nσ+ (t1 , t2 ) − N0+ ) ≥ τa+ is required for subsystems in S+ , which means that their average running time is limited to no less than τa+ to reduce the impact of USIs. The goal of this chapter is to give the solvability conditions of the AOR problem to ensure the BTP of the unstable switched systems (5.1). For this purpose, we jointly design an exponential interpolation controller (5.2) and switching rules to achieve the following performance. (i) OR property: For w(t) = 0 and d(t) = 0, the system (5.3) is globally asymptotically stable (GAS). For w(t) /= 0 and d(t) = 0, the regulated output e(t) of (5.3) satisfies lim e(t) = 0. t→∞

(ii) L 2 -gain property: Under zero initial condition [ χ T (0) w T (0) ]T = [ 0 0 ]T , for any w(t) and d(t), the regulated output e(t) of (5.3) satisfies { ∞ { ∞ T 2 e (s)e(s)ds ≤ γ d T (s)d(s)ds, (5.4) 0

0

with the L 2 -gain level γ > 0.

5.3 Main Results In this section, sufficient conditions of the AOR problem are deduced for the system (5.1) under the framework of HADT strategy with the occurrence of some USIs and the insolvabilities of the AOR problems for all subsystems, and the jointly-designed dynamic error feedback controller (5.2) with BT property. Theorem 5.1 For the closed-loop system (5.3) and given scalars λ > 1, η > 0, μ ≥ 1,0 < ρ < 1, τmin > 0, q ∈ U = {0, 1, · · · , l − 1}, l ∈ N+ . If (i) there exist matrices T , E, Hi and L i for i ∈ S such that ⎧ ⎨ Ai T + Bi K i E + E i − T Hi = 0, E Hi − L i E = 0, ⎩ Ci T + Di = 0,

(5.5)

(ii) there exist matrices Pi,q˜ = diag{Pi,11q˜ , Pi,22q˜ } > 0 and M˜ i with proper dimensions for q˜ ∈ {q, q + 1}, υ ∈ {i, j}, i, j ∈ S− such that

106

5 Almost Output Regulation for Switched Linear Systems with Bumpless …

⎡

⎤ 11 ui,11q˜ + ui,q Pi,11q˜ Bi K υ Pi,11q˜ (Fi − T Ji ) + CiT G i CiT M˜ iT 0 ⎢ 22 ∗ ui,22q˜ + ui,q −Pi,22q˜ E Ji 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ∗ ∗ G iT G i − γ 2 I 0 G iT M˜ iT ⎥ ≤ 0, (5.6) ⎢ ⎥ 22 ⎣ ∗ ∗ ∗ −Pi,l 0 ⎦ 22 ∗ ∗ ∗ ∗ −Pi,l ⎡ ⎤ 11 11 11 ui,l Pi,l Bi K i Pi,l (Fi − T Ji ) + CiT G i CiT M˜ iT 0 ⎢ ∗ 22 22 −Pi,l E Ji 0 0 ⎥ ui,l ⎢ ⎥ ⎢ ⎥ (5.7) 0 G iT M˜ iT ⎥ ≤ 0, ∗ G iT G i − γ 2 I ⎢ ∗ ⎢ ⎥ 22 ⎣ ∗ ∗ ∗ −Pi,l 0 ⎦ 22 ∗ ∗ ∗ ∗ −Pi,l Pi,0 ≤ ρ P j,l ,

(5.8)

(iii) there exist matrices Q i = diag{Q i11 , Q i22 } > 0 and M˜ i with proper dimensions for υ ∈ {i, j}, i, j ∈ S+ such that ⎡

⎤ []i11 Q i11 Bi K υ Q i11 (Fi − T Ji ) + CiT G i CiT M˜ iT 0 ⎢ ∗ −Q i22 E Ji 0 0 ⎥ []i22 ⎢ ⎥ ⎢ T T ˜ T ⎥ ≤ 0, 2 0 G i Mi ⎥ ∗ Gi Gi − γ I ⎢ ∗ ⎢ ⎥ ⎣ ∗ ∗ ∗ −Q i22 0 ⎦ ∗ ∗ ∗ ∗ −Q i22 Q i ≤ μQ j ,

(5.9)

(5.10)

(iv) for i ∈ S− and j ∈ S+ , Pi,0 ≤ ρ Q j and Q j ≤ μPi,l ,

(5.11)

(v) the HADT meets τa− ≤

(λ − 1) ln ρ + (1 − λ) ln μ ,τ ≥ , −η(1 + λ) a −η(1 + λ)

(5.12)

and the total running time of subsystems in S− and S+ fulfilling Tσ− (t1 , t2 )/Tσ+ (t1 , t2 ) > λ. Then AOR problem of the system (5.3) is solvable with BT property where

5.3 Main Results

107

11 11 11 22 22 22 ui,q = l(Pi,q+1 − Pi,q )/τmin , ui,q = l(Pi,q+1 − Pi,q )/τmin ,

ui,11q˜ = AiT Pi,11q˜ + Pi,11q˜ Ai − η Pi,11q˜ + CiT Ci , ui,22q˜ = L iT Pi,22q˜ + Pi,22q˜ L i + (2 − η)Pi,22q˜ , []i11 = AiT Q i11 + Q i11 Ai − ηQ i11 + CiT Ci , 22 22 []i22 = L iT Q i22 + Q i22 L i + (2 − η)Q i22 , Pi,l ≥ Pi,q , ∀q ∈ U,

{ and the controller gains are Mi =

−1 ˜ Pi,q Mi , i ∈ S− , −1 ˜ Q i Mi , i ∈ S+ .

Proof Combining the coordinate transformation χ˜ (t) = χ (t)−[ T T E T ]T w(t) with (5.5), the closed-loop system (5.3) can be rewritten as { ⎧ ˜ + ϒi d(t), t ∈ [tk , tk + τmin ), ⎨ χ˙˜ (t) = A ji (t)χ(t) Ai χ˜ (t) + ϒi d(t), t ∈ [tk + τmin , tk+1 ), ⎩ e(t) = C i χ˜ (t) + G i d(t),

(5.13)

where ϒiT = [ FiT − JiT T T G iT MiT − JiT E T ], i, j ∈ S. Then, the AOR problem of the system (5.3) is converted to the H∞ performance of the system (5.13). To this end, the following part will show that system (5.13) is GAS for d(t) = 0, and meets the L 2 -gain property for d(t) /= 0 and zero initial condition. For USIs, the Lyapunov function is constructed as { Vi (χ(t)) ˜ =

˜ i ∈ S− , χ˜ T (t)Pi (t)χ(t), T χ˜ (t)Q i χ˜ (t), i ∈ S+ ,

(5.14)

where Pi (t) is the time-varying matrix to be determined later based on the discrete Lyapunov function (DLF) method. For i ∈ S− , the interval [tk , tk + τmin ) is divided into l segments with subdivided intervals Nk,q = [tk + θq , tk + θq+1 ), θq = qh, q ∈ U , h = τmin /l. Then Pi (t) is correspondingly represented by the following linear combination of Pi,q , (q)

Pi (t) = Pi (tk + θq + αh) = (1 − α)Pi,q + α Pi,q+1 /\ Pi (α),

(5.15)

/ where α = (t − tk − θq ) h, t ∈ Nk,q , q ∈ U . On t ∈ [tk + τmin , tk+1 ), Pi (t) is fixed as a constant matrix Pi,l . In summary, for i ∈ S− , { Pi (t) =

(q)

Pi (α), t ∈ ∪ Nk,q , q∈U

Pi,l ,

t ∈ [tk + τmin , tk+1 ).

(5.16)

108

5 Almost Output Regulation for Switched Linear Systems with Bumpless …

Denote 0i (t) = V˙i (χ˜ (t))−ηVi (χ˜ (t))+\(t) and \(t) = eT (t)e(t)−γ 2 d T (t)d(t). First of all, we verify that system (5.13) is GAS for d(t) = 0. For i ∈ S− and t ∈ Nk,q , differentiating (5.14) with (5.16) along the trajectories of system (5.13) yields 0i (t)

|

| ¯ Tji (t)Pi (t) + Pi (t) A ¯ ji (t) + ui,q − η Pi (t) + C¯ iT C¯ i Pi (t)ϒi + C¯ iT G i A ξ(t) = ξ (t) ∗ G iT G i − γ 2 I T

= ξ T (t)[(1 − α)(ς (t)0iji,q + (1 − ς(t))0ii,q ) + α(ς (t)0iji,q+1 + (1 − ς(t))0ii,q+1 )]ξ(t)

(5.17) 11 22 where ξ T (t) = [ χ˜ T (t) d T (t) ], ui,q = diag{ui,q ,ui,q } / l(Pi,q+1 − Pi,q ) τmin , κ ∈ {i, ji}, q˜ ∈ {q, q + 1} and

| 0iκ,q˜ =

P˙i (t)

=

=

| ∗T ∗ T T Aκ Pi,q˜ + Pi,q˜ Aκ + ui,q − η Pi,q˜ + C i C i Pi,q˜ ϒi + C i G i . ∗ G iT G i − γ 2 I

For i ∈ S− , t ∈ [tk + τmin , tk+1 ), taking the time derivation of (5.14) with (5.16) leads to | | T T T Ai Pi,l + Pi,l Ai − η Pi,l + C i C i Pi,l ϒi + C i G i T 0i (t) = ξ (t) ξ(t). (5.18) ∗ G iT G i − γ 2 I For i ∈ S+ , similar deduction gives |

| T T T A Q + Q A − ηQ + C C Q ϒ + C G i ji i i i i ji i i i i 0i (t) = ξ T (t) ξ(t) ∗ G iT G i − γ 2 I =ξ

T

(t)[ς (t)0iji

+ (1 −

(5.19)

ς (t))0ii ]ξ(t)

with κ ∈ {i, ji} and | 0iκ

=

∗T

∗

T

T

Aκ (t)Q i + Q i Aκ (t) − ηQ i + C i C i Q i ϒi + C i G i ∗ G iT G i − γ 2 I

|

on t ∈ [tk , tk + τmin ), whereas on t ∈ [tk + τmin , tk+1 ), it results in |

| T T T Ai Q i + Q i Ai − ηQ i + C i C i Q i ϒi + C i G i 0i (t) = ξ (t) ξ(t). ∗ G iT G i − γ 2 I T

(5.20)

5.3 Main Results

109

22 22 22 According to Pi,l ≥ Pi,q for all q ∈ U , substituting MiT Pi,l for M˜ iT in (5.6) generates

⎡

⎤ 11 + C T M T P 22 M C 11 Pi,11q˜ (Fi − T Ji ) + CiT G i ui,11q˜ + ui,q i i i,q˜ i i Pi,q˜ Bi K j ⎢ ⎥ 22 ∗ ui,22q˜ + ui,q −Pi,22q˜ E Di ⎣ ⎦≤0 ∗ ∗ G iT G i − γ 2 I + G iT MiT Pi,22q˜ Mi G i

via Lemma 1.7. From Lemma 1.6, it holds that 22 22 x T (t)CiT MiT Pi,q Mi Ci x(t) + xcT (t)Pi,q xc (t) 22 22 xc (t) + xcT (t)Pi,q Mi Ci x(t), ≥ x T (t)CiT MiT Pi,q 22 22 x T (t)G iT MiT Pi,q Mi G i x(t) + xcT (t)Pi,q xc (t) 22 22 ≥ x T (t)G iT MiT Pi,q xc (t) + xcT (t)Pi,q Mi G i x(t).

Thus, we have ⎡ ⎤ 11 ui,11q˜ + ui,q CiT MiT Pi,22q˜ + Pi,11q˜ Bi K j Pi,11q˜ (Fi − T Ji ) + CiT G i ⎢ ⎥ 22 ∗ ui,22q˜ + ui,q − 2Pi,22q˜ Pi,22q˜ (Mi G i − E Di ) ⎦ ≤ 0, ⎣ ∗ ∗ G iT G i − γ 2 I which implies 0i (t) ≤ 0 in conjunction with (5.17) for i ∈ S− and t ∈ Nk,q . Besides, performing similar inference on (5.7) and (5.9) gives 0i (t) ≤ 0 combining ˜ − ηVi (χ(t)) ˜ ≤ −eT (t)e(t) ≤ 0 with (5.17)–(5.19). Thus, it is easy to get V˙i (χ(t)) for d(t) = 0. On account of (5.8), (5.10) and (5.11), one drives Vσ (tk ) (χ(t)) ˜ ≤ eη(t−tk ) Vσ (tk ) (χ(t ˜ k )) ˜ k− )) ≤ εeη(t−tk ) Vσ (tk−1 ) (χ(t ≤ · · · ≤ εk eη(t−t0 ) Vσ (t0 ) (χ˜ (t0 )),

(5.21)

where ε = ρ for i ∈ S− or ε = μ, i ∈ S+ . According to the HADT with Tσ− (t1 , t2 )/Tσ+ (t1 , t2 ) > λ, we have Vσ (tk ) (χ(t)) ˜ ≤ εk eη(t−t0 ) Vσ (t0 ) (χ˜ (t0 )) −

+

= ρ Nσ (t0 ,t) μ Nσ (t0 ,t) eη(t−t0 ) Vσ (t0 ) (χ˜ (t0 )) −

+

= eη(t−t0 )+Nσ (t0 ,t) ln ρ+Nσ (t0 ,t) ln μ Vσ (t0 ) (χ˜ (t0 )) ≤e

) ( ) ( T − (t ,t) T + (t ,t) ln ρ+ N0+ + σ +0 ln μ η(t−t0 )+ N0− + σ −0

≤ρ

τa

N0−

μ

N0+

eN

∗

·(t−t0 )

τa

Vσ (t0 ) (χ(t ˜ 0 )),

Vσ (t0 ) (χ(t ˜ 0 )) (5.22)

110

5 Almost Output Regulation for Switched Linear Systems with Bumpless …

ln ρ μ where N ∗ = η + τ −λ(1+λ) + τ +ln(1+λ) ≤ 0 is ensured by (5.12). It implies that a a lim Vσ (tk ) (χ˜ (t)) = 0. Therefore, we can conclude that system (5.13) is GAS for t→∞

d(t) = 0. That is to say, system (5.3) is GAS for w(t) = 0 and d(t) = 0. Moreover, for w(t) /= 0 and d(t) = 0, the globally asymptotical stability of system (5.13) also guarantees that lim e(t) = lim [C σ (t) χ (t) + Dσ (t) w(t)] = lim C σ (t) χ˜ (t) = 0,

t→∞

t→∞

t→∞

i.e. system (5.3) achieves the OR property. Next, for d(t) /= 0, the L 2 -gain property of the system (5.13) will be revealed. In conjunction with 0i (t) ≤ 0, (5.21) and (5.22), one has from the system (5.13) that Vσ (tk ) (χ (t)) ≤ Vσ (tk ) (χ(t ˜ k ))eη(t−tk ) −

{

t tk

≤ εVσ (tk−1 ) (χ(t ˜ k ))eη(t−tk ) −

eη(t−s) \(s)ds

{

t

eη(t−s) \(s)ds.

tk

By induction, the above inequality generates that Vσ (tk ) (χ (t)) ≤ εk eηt Vσ (t0 ) (χ˜ (t0 )) − εk −

{

t1

eη(t−s) \(s)ds − · · · − ε0

{

t

eη(t−s) \(s)ds.

tk

t0

+

In view of εk = ρ Nσ (0,t) μ Nσ (0,t) , one can further get { t − + − + Vσ (tk ) (χ (t)) ≤ ρ Nσ (0,t) μ Nσ (0,t) eηt Vσ (t0 ) (χ˜ (t0 )) − ρ Nσ (s,t) μ Nσ (s,t) eη(t−s) \(s)ds 0 { t − + eη(t−s)+Nσ (s,t) ln ρ+Nσ (s,t) ln μ \(s)ds. =− 0

{t − + Due to Vσ (tk ) (t) ≥ 0, it is obtained that 0 ρ Nσ (s,t) μ Nσ (s,t) eη(t−s) \(s)ds ≤ 0 under zero initial condition. Then integrating both sides {of this inequality { ∞ from t = 0 to ∞ ∞ and rearranging the double integral area yield 0 \(s)ds s F(t)dt ≤ 0 with − + F(t) = e Nσ (s,t) ln ρ+Nσ (s,t) ln μ+η(t−s) . Taking Definition 5.1, τmin ≤( τa− , (5.12) and ) −

ln ρ

+η (t−s)

Nσ (s,t)lnρ+η(t−s) 0 < ρ < 1 into consideration ≥ e τmin and )leads to F(t) > e /( {∞ ln ρ > 0. F(t)dt > −1 η + s { ∞τmin {∞ Thus, one can get 0 \(s)ds = 0 (||e(s)||2 − γ 2 ||d(s)||2 )ds ≤ 0, which implies that system (5.13) owns the L 2 -gain property for d(t) /= 0. Owing to the above two aspects, the H∞ performance of the system (5.13) is [] achieved, which ensures that the AOR problem of the system (5.3) is solved.

5.3 Main Results

111

Remark 5.3 It can be seen from (5.16) that the number of segments l is greater than or equal to 1 under the framework of the DLF. If l = 0, the Lyapunov function in (5.16) will be reduced to the ordinary form of piecewise Lyapunov functions. Remark 5.4 Theorem 5.1 depicts a HADT switching strategy (5.12) for system (5.13) that deals with the occurrence of both USIs and instability of all subsystems. Compared with [13], our proposed strategy only limits the total running time of subsystems belonging to S+ and S− , rather than requires the alternation of SSIs and USIs in a fixed number. Remark 5.5 The forthcoming optimization problem min

γ

s.t.

(5.5) − (5.12)

Pi,q ,Q i ,Mi ,L i ,K υ , q∈{0,1,··· ,l},υ∈{i, ji},i, j∈S

evaluates the anti-disturbance capability of the closed-loop system (5.13) under the BT controller (5.3) and the HADT switching rule (5.12). Remark 5.6 For given scalars μ, ρ, η, λ satisfying Theorem 5.1, the lower bound of the admissible minimal DT τmin can be calculated by ∗ τmin = min{τmin ≤ τa− |(5.5) − (5.12) hold }.

(5.23)

The detailed steps are devised in the following algorithm. ∗ with given scalars μ, ρ, η,λ Algorithm 5.1 Computation on τmin

1: Initialize μ = μ0 > 1, ρ = ρ0 < 1, variations /\μ > 0, /\ρ > 0 and loop counter δ=0; 2: while ρ < 1 do 3: Set δ = δ + 1 and μ = μ0 + /\μ, ρ = ρ0 + /\ρ; ∗ ; 4: Solve (5.23) to obtain τmin 5: if (5.23) is feasible then ∗ ; 6: Record R(0) = τmin 7: else 8: Record R(0) = τa− + 1; 9: end if 10: end while 11: if ∃θ = 1, 2, · · · , 0 such that R(0) ≤ τa− then ∗ =min 12: τmin θ =1,2,···0 {R(0)}; 13: else 14: The GAS cannot be established; 15: end if

112

5 Almost Output Regulation for Switched Linear Systems with Bumpless …

5.4 Numerical Example To verify the AOR problem with BT property, system matrices of (5.1) with three subsystems S = {1, 2, 3} (S− = {1, 2} and S+ = {3}) are considered below |

| | | | | −7.8 3 −1.8 2 −10 2 A1 = , A2 = , A3 = , −5 2 2.5 −2.5 −4 1 | | | | | | 0 −1 −1 B1 = , B2 = , B3 = , 1 −1 0 | | | | | | 23 42 33 H1 = , H2 = , H3 = , 21 21 21 | | | | | | 0.2 0.5 0.4 J1 = , J2 = , J3 = , 0.1 0.3 −0.2 | | | | | | 9.8 0 4.8 1 14 0 E1 = , E2 = , E3 = , 6 −2 −1.5 4.5 6 0 | | | | | | 1 0.5 1 F1 = , F2 = , F3 = , −0.5 0.5 0.2 C1 = [ 1 −1 ], C2 = [ 0.5 −0.5 ], C3 = [ −1 1 ], D1 = C1 , D2 = C2 , D3 = C3 , G 1 = −0.1, G 2 = 0.2, G 3 = 0.1, d(t) = 5te−5t . Based on these parameters, the matrices that satisfy (5.5) are |

| | | | | | | | | 10 −5 0.1 −6 0 −8 0.1 00 T = ,E = , L2 = , L3 = , , L1 = 01 −2 −5 −2 −1 −3 −2 00 K 1 = [ 1 1 ], K 2 = [ −1 1 ], K 3 = [ 1 −1 ]. Selecting parameters λ = 10, η = 1.3, ρ = 0.68, μ = 1.3, τmin = 0.2 that meet the conditions of Theorem 5.1 and solving (5.6)–(5.12), we can get the controller gain matrices |

| | | | | 0.0030 0.0115 0.0150 M1 = , M2 = , M3 = . −0.0024 −0.0101 −0.0105 By solving the optimization problem in Remark 5.5, the optimal L 2 -gain level is γ = 1.7320. To intuitively demonstrate that our method is effective when all subsystems and some switching instants are unstable, we choose γ = 2.4494 in this chapter.

5.4 Numerical Example

113

According to the aforementioned parameters, the external disturbance w(t) as shown in Fig. 5.1 has always affected the system dynamics. Figure 5.2 shows the state response of each subsystem when it runs separately without switching and the divergent curve in Fig. 5.2 illustrates that the AOR problem of each subsystem in (5.1) is unsolvable. In the light of condition (iv) in Theorem 5.1, one can get the HADT fulfilling τa− ≤ 0.2427, τa+ ≥ 0.1651 and Tσ− (t1 , t2 )/Tσ+ (t1 , t2 ) = 11.5 > λ = 10 for Fig. 5.1 External disturbance w(t)

Fig. 5.2 States of subsystems 1, 2, 3

114

5 Almost Output Regulation for Switched Linear Systems with Bumpless …

N0+ = 0 and N0− = 0. A switching signal that satisfies these requirements is shown in Fig. 5.3. In Figs. 5.4 and 5.5, the state responses and the regulated output error of the closed-loop system (5.3) converge to zero under the switching signal in Fig. 5.3 and the initial state χ (0) = [1100.1 0.1]T , which indicates the AOR problem of the system (5.1) is solved. It is worth noting that the Lyapunov function does not decrease monotonically over the running intervals of each subsystem and at instants when the third subsystem is activated, as can be seen from the evolution curve of the Lyapunov function in Fig. 5.6. Moreover, different from [13], it can be observed intuitively from Figs. 5.1 and 5.6 that we do not need to group SSIs and USIs in order and constrain their ratio, which is more flexible in practical applications. Figure 5.7 compares three controllers, i.e. the exponential interpolation form in (5.2), the linear interpolation form in [20], and the ordinary bumpy controller, which illustrates that (5.2) is a bumpless controller with smoother curves at switching instants.

Fig. 5.3 Switching signal

Fig. 5.4 The state responses of the closed-loop system

5.5 Conclusion

115

Fig. 5.5 The regulated output error of the closed-loop system

Fig. 5.6 Evolution of Lyapunov function

Fig. 5.7 The comparison of three control signals

5.5 Conclusion This chapter solves the AOR problem of switched systems with the occurrence of destabilizing behaviors in all subsystems and some switching instants. The proposed HADT switching strategy breaks through the limitation of existing achievements that SSIs and USIs are arranged in some fixed order with their quantity ratio constraints.

116

5 Almost Output Regulation for Switched Linear Systems with Bumpless …

The dynamic error feedback controller is jointly devised in a form of exponential interpolation to realize the BT property which does not require the switching instants and switching sequence to be known in advance. Finally, the effectiveness of the method is proved by simulation. The results of this chapter are worth extending to the cyber security problems in the future.

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Chapter 6

Mixed Event-Triggered Output Regulation for Networked Switched Systems with Severely Unstable Dynamics Under Long-Duration DoS Attacks

In this chapter, the EORP under the DoS attacks is considered for NSSs with SUDs. The SUDs here refer to the unsolvable OR of each subsystem and the destabilization at partial switching instants, which indicates that the Lyapunov function does not decrease monotonically in activation intervals of each subsystem and increases at partial switching instants. First, LDDAs are considered, where LDDAs imply that their duration may be longer than the total DT of several adjacent activated subsystems. By imposing constraints at switching instants, consecutive asynchronous subsystem switching caused by LDDAs and SUDs is allowed, that is, the subsystem switches several times but the controller switching is blocked by LDDAs and controllers fail to switch correspondingly. Second, mixed ETMs, combining event-triggered conditions and periodic sampling conditions, are designed to reduce network burden under LDDAs and improve system performance subject to destabilizing switching. Then, an improved DT for switching signal permits irregular arrangement of destabilizing and stabilizing switching and is more suitable for NSSs subject to LDDAs. Moreover, sufficient conditions ensure the solvability of EORP for NSSs with SUDs under LDDAs, network-induced delays, random packet losses, and packet disorders. Finally, a switched RLC circuit shows the feasibility of the proposed method.

6.1 Introduction Switched systems, as a part of hybrid systems, have attracted more and more attention. In [1, 2], Wang et al. propose an efficient robust control method for switched systems with two time scales for the first time which promotes the development of switched system theory to a certain extent. With the development of network technology, NSSs have become a hot research topic. Owing to the strong openness of the network, cyber attacks such as DoS attacks can easily be injected into NSSs. Due to the mutual © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Li and J. Fu, Output Regulation and Cybersecurity of Networked Switched Systems, Studies in Systems, Decision and Control 475, https://doi.org/10.1007/978-3-031-30972-4_6

119

120

6 Mixed Event-Triggered Output Regulation for Networked Switched …

influence of switching behaviors and DoS attacks, there are complicated relations between switching instants and starting/ending instants of DoS attacks, which leads to more challenges. Under synchronous switching and DoS attacks, [3, 4] study the secure control for NSSs without and with unstable subsystems, [5–7] consider the asynchronous switching caused by DoS attacks and solve the stability of NSSs with all stable subsystems under DoS attacks. Most of the efforts focus on NSSs without SUDs and DoS attack durations not exceeding the DT of active subsystems during attacks. However, in actual NSSs, not only SUDs are not negligible, but also DoS attack duration may be longer than the total DT of several adjacent activated subsystems, which causes consecutive asynchronous switching, that is, the switching of candidate controllers is hindered at several switching instants. These attacks are called LDDAs. For NSSs with SUDs, there is an upper bound on the DT to maintain the stability of switched systems. Therefore, when NSSs with SUDs suffer from LDDAs, it is difficult for existing switching methods to make each subsystem dwell until the end of LDDAs, and thus consecutive asynchronous switching cannot be avoided. Consequently, it is challenging to explore new switching methods for NSSs with SUDs under LDDAs. In NCSs, some mixed ETMs [8–11] are constructed by mixing different types of ETMs to fully utilize their respective advantages to enhance the event triggering performance, being further applied to secure control [12]. When it comes to NSSs, several single types of ETMs, such as discrete ETM [3], continuous ETM [4, 6], adaptive ETM [14], and distributed ETM [15], are designed under cyber attacks, while SUDs are not taken into consideration. However, when both SUDs and LDDAs are harmful to system performance, single types of ETMs may be difficult to maintain the balance between reducing the network burden and ensuring system performance. Thus, how to design a mixed ETM for NSSs with SUDs against LDDAs is a valuable problem. The ORP is an important part of the control field. In recent years, the issue of the EORP has attracted a lot of attention. For non-switched systems, [16] solves the EORP for linear cases by using the internal model method. For NSSs, by using different coordinate transformation methods, [10–12] reduce the limitation that all regulator equations need to have a common solution and solve the EORP. However, the above-mentioned results do not adapt to the EORP under cyber attacks. When the network suffers from attacks, the EORP for NSSs with SUDs is more challenging, which motivates the current study. In summary, this chapter studies the EORP for NSSs with SUDs under LDDAs, network-induced delays, random packet losses, and packet disorders. The main contributions are as follows: (1) LDDAs are considered, where attack durations may be longer than the total DT of several adjacent activated subsystems. Meanwhile, the consecutive asynchronous subsystem switching caused by LDDAs is permitted by placing inequality restrictions at asynchronous switching instants. (2) Mixed ETMs are developed for NSSs with SUDs against LDDAs. These ETMs utilize eventtriggered conditions after stabilizing switching to save limited network resources and employ periodic sampling conditions after destabilizing switching to improve system performance. (3) A new DT constraint is proposed with the parameters of

6.2 Problem Formulation

121

LDDAs and the ratio of stabilizing to destabilizing switching, which not only breaks the regular arrangement of destabilizing and stabilizing switching in existing results but also helps NSSs to resist SUDs and LDDAs. (4) By synthesizing the improved DT, mixed ETMs, and DOFC, sufficient conditions for EORP of NSSs with SUDs are obtained under LDDAs and network-induced phenomenon, without requiring solvability of EORP for each subsystem and stabilization of all switching instants.

6.2 Problem Formulation The framework of EORP for NSSs with SUDs under LDDAs is shown in Fig. 6.1. There is a switched system with the disturbance w(t) generated by an exosystem. A sensor is connected to the network under an ETM. The control is generated by a DOFC, also connected to the network, and sends the control signal to the actuator under the second ETM. An attacker is acted on two-channel networks generating LDDAs. These components are described in detail as follows.

6.2.1 System Model and Dynamic Output Feedback Controller The dynamics of NSSs with SUDs are depicted by ⎧ ˙ = Aσ (t) x(t) + Bσ (t) u(t) + Cσ (t) w(t), ⎨ x(t) y(t) = Dσ (t) x(t) + E σ (t) w(t), ⎩ w(t) ˙ = Sσ (t) w(t),

Fig. 6.1 The framework of the EORP for NSSs with SUDs under LDDAs

(6.1)

122

6 Mixed Event-Triggered Output Regulation for Networked Switched …

where x ∈ Rx , u ∈ Ru , y ∈ R y and w ∈ Rw are state vector, control input, regulated output, and exogenous input. Aσ (t) , Bσ (t) , Cσ (t) , Dσ (t) , E σ (t) , Sσ (t) are constant matrices with proper dimensions and all eigenvalues of Sσ (t) have nonnegative real part [11]. σ (t) : [0, ∞)→ M = {1, 2, . . . , m} is the switching signal, and its corresponding switching sequence is expressed as T = {tk }k∈N with t0 = 0. On [0, t) with t ∈ [tk , tk+1 ), there exist stabilizing and destabilizing switching instant and T = {tn¯d }n∈N sequences T = {tms¯ }m∈N ¯ ¯ + with T ∪ T = T . N (0, t) = m + n is the number of switching instants on [0, t). The DOFC is designed as {

x˙c (t) = Acσ (t) xc (t) + Bcσ (t) xc (sh) + Ccφ(t) y˜ (t), u(t) ˆ = Hcσ (t) xc (t),

(6.2)

where xc (t) ∈ Rxc is the controller’s state, y˜ (t) ∈ R y˜ is the input of the controller ˆ is the output of the controller, φ(t) is the switching signal of the controller. and u(t) sh is the sampling instants, the desired matrices Acσ (t) , Bcσ (t) , Ccφ(t) , Hcσ (t) are with proper dimensions.

6.2.2 Long-Duration Denial-of-Service Attacks In this chapter, LDDAs are simultaneously activated on two-channel networks and satisfy the following assumptions. Assumption 6.1 (LDDAs Frequency[17]) There exists ω ≥ 0 and τ D ≥ 0 such that / N D (t0D , t1D ) ≤ ω + (t1D − t0D ) τ D holds, where N D (t0D , t1D ) is the number of DoS attacks on [t0D , t1D ). Assumption 6.2 (LDDAs Duration) The upper bound of activated intervals of DoS attacks bmax and the lower bound of sleeping intervals of DoS attacks L min satisfy that bmax ≥ supl∈N {dl } and L min ≤ inf l∈N {h l+1 − h l −dl }, where {h l }l∈N denotes instant sequences when LDDAs launch, / Hl = [h l , h l + / dl ) is l-th LDDAs interval with a length dl > 0, bmax N D (t0D , t1D ) (t1D − t0D )< 1 T, T > 1. In particular, different from [3–7], bmax in this chapter may be longer than the total DT of several adjacent activated subsystems. Remark 6.1 The upper bound of DT for each subsystem needs to be constrained to stabilize NSSs with SUDs, so existing switching schemes under the DT framework are unlikely to keep a subsystem dwell until the end of LDDAs to eliminate consecutive asynchronous switching of subsystems. From Fig. 6.2, the subsystem switches at tk+1 , tk+2 , tk+3 , but the controller is blocked by LDDAs and cannot switch correspondingly, which will result in consecutive asynchronous switching on [tk+1 , tk+3 ). To this end, a new switching scheme will be proposed to handle this situation for NSSs with SUDs against LDDAs.

6.2 Problem Formulation

123

Fig. 6.2 Diagram of LDDAs and switching instants

6.2.3 Mixed Event-Triggered Mechanisms Mixed ETMs are devised to ease the network burden in the S2C and the C2A channels. 1. S2C Channel The mixed ETM in the S2C channel is described as y

y

y y εr +1 h = {h l } ∪ min{min{εr +1 h /\ sr,v h∈ / Hl |[] y (sr,v h) > 0}, y

min{˜εr +1 h /\ v∈N

v∈N y sr,v h

(6.3)

y y ∈ / Hl |σ (sr,v h) /= σ (sr,v h − h)}},

where h l is the first sampling instant after the ending instant of the l-th LDDAs,

y [] y (sr,v h) =

⎧ min{n,v} E ⎪ ⎪ y y y ⎪ γι e Ty (sr,v−ι h)uσ (s y h) e y (sr,v−ι h) ⎪ ⎪ r,v ⎨ ι=0

y y y y y ⎪ h)uσ (s y h) y(sr,v h), sr,v h ∈ T 1k , −δσ (s y h) (t)y T (sr,v ⎪ ⎪ r,v r,v ⎪ ⎪ ⎩ y 1 y y sr,v h − εr,v h, sr,v h ∈ T k,

1

y

with T 1k = [tk , tk+1 ) for tk ∈ T , and T k = [tk , tk+1 ) for tk ∈ T , e y (sr,v−ι h) = y y y y y0 y1 y(sr,v−ι h) − y(εr h), sr,v−ι h /\ [εr + (v − ι)]h, θ y ,δσ (s y h) , δσ (s y h) are positive y

constants, δσ (s y

y0

(t) = δσ (s y h)

y1

+ δσ (s y

y

r,v

r,v

e−θ y ||y(sr,v h)|| , δσ (s y h) y

y

y0

y1

(t) ≤ δ i = δi +δi , r,v r,v h) r,v r,v h) E y and the weighted matrix uσ (s y h) > 0. If n > v and ι /= 0, then γι =1 − ιι−1 ˜=0 γ˜ι˜ , else r,v En γι = γ˜ι˜, ι = ι˜. γ˜ι˜ ∈ [0, 1] is a weighted parameter with ι˜=0 γ˜ι˜=1, ι, ι˜, v, n ∈ N. 2. C2A Channel The mixed ETM in the C2A channel is: u u h∈ / Hl |[]u (sr,v h) > 0}, εru+1 h = {h l } ∪ min{min{εru+1 h /\ sr,v v∈N

u u u h∈ / Hl |φ(sr,v h) /= φ(sr,v h − h)}}, min{˜εru+1 h /\ sr,v v∈N

where ⎧ T u u u ⎪ u h) eu (sr,v h) ⎨ eu (sr,v h)uσ (sr,v u u u u h ∈ T 2k , ˆ T (sr,v h)uuσ (sr,v ˆ r,v h), sr,v −δσu (sr,v u h) (t)u u h) u(s []u (sr,v h) = ⎪ 2 ⎩ u u u h − εr,v h, sr,v h ∈ T k, sr,v

(6.4)

124

6 Mixed Event-Triggered Output Regulation for Networked Switched … 2

1

with T 2k = [t˜k , t˜k+1 ) for t˜k ∈ T 1k , and T k = [t˜k , t˜k+1 ) for t˜k ∈ T k , t˜k is the controller u u u h /\ (εru + v)h, eu (sr,v h) = u(s ˆ r,v h) − u(ε ˆ ru h), δσu (sr,v switching instant, sr,v u h) (t) = u

u1 u1 −θu ||u(sr,v h)|| ≤ δ i = δiu0 + δiu1 , constants θu , δσu0(sr,v δσu0(sr,v u h) + δσ (s u h) ,e u h) , δσ (s u h) are r,v r,v positive, and the weighted matrix uuσ (sr,v u h) > 0. u

Remark 6.2 The ETMs (6.3) and (6.4) consist of three parts: (1) The first parts h l relieve the impact of communication interruption induced by LDDAs via triggering once at the first sampling instant after the ending instant of LDDAs. (2) The second parts in (6.3) and (6.4) develop mixed event-triggered conditions to keep the balance between easing the network burden after stabilizing switching and adjusting the y system performance quickly after destabilizing switching: (a) When sr,v h ∈ T 1k y u and sr,v h ∈ T 2k , sr,v h is in the switching interval [tk , tk+1 ) after stabilizing switching u h is in the corresponding controller activated interval [t˜k , t˜k+1 ), instant tk ∈ T and sr,v y u h) > 0 are event-triggered conditions, which are then [] y (sr,v h) > 0 and []u (sr,v 1

y

2

y

u exploited to ease the network burden; (b) When sr,v h ∈ T k and sr,v h ∈ T k , sr,v h is in the switching interval [tk , tk+1 ) after destabilizing switching instant tk ∈ T and u sr,v h is in the corresponding controller activated interval [t˜k , t˜k+1 ). In this situation, y u [] y (sr,v h) > 0 and []u (sr,v h) > 0 are periodic sampling conditions, which can help to adjust the system performance quickly after destabilizing switching. Besides, y time-varying thresholds δσ (s y h) (t) and δσu (sr,v u h) (t) can adjust triggers according to r,v y u y(sr,v h) and u(s ˆ r,v h). (3) The third parts are subsystem and controller mode matching y y u u h) /=φ(sr,v h − h) in (6.4), conditions, i.e. σ (sr,v h) /=σ (sr,v h − h) in (6.3) and φ(sr,v which shorten asynchronous durations caused by second parts in ETMs via exerting a triggering at the nearest sampling instant after the subsystem or controller mode switch. Since (6.3) and (6.4) trigger at sampling instants, the Zeno phenomenon is excluded.

Remark 6.3

min{n,v} E ι=0

y

y

γι e Ty (sr,v−ι h)uσ (s y

r,v h)

y

e y (sr,v−ι h) in (6.3) is not memoryless,

which reduces unnecessary transmissions induced by the rapid and large oscillation of SUDs. Moreover, due to the combination of the second and third parts in (6.3), all employed previous sampled outputs come from the currently activated subsystem, y y y i.e. [εr h, sr,v h]⊆ [tk , sr,v h], which makes the memory-based strategy more suitable for NSSs.

6.2.4 The Closed-Loop Networked Switched System y

Owing to network-induced delays τrsc and τrca , the triggered data at εr h (εru h) reaches y sc , τmca ≤ τrca ≤ the controller (actuator) at εr h + τrsc (εru h+τrca ). Set τmsc ≤ τrsc ≤ τ M ca sc ca sc sc sc ca ca τ M with τm = min{τr }, τ M = max{τr }, τm = min{τr }, τ M = max{τrca }. r ∈N

r ∈N

r ∈N

r ∈N

y

sc ca Let τr /\ τrsc + τrca satisfy τmsc +τmca =τm ≤τr ≤τ M = τ M + τM . Then, |r =[εr h +

6.2 Problem Formulation

125

r τrsc , εr +1 h+τrsc+1 ) can be divided as |r = ∪lv=0 |r,v , where

y

{ |r,v =

y

y

[sr,v h + τrsc , sr,v+1 h + τrsc ), v = 0, . . . , lr − 1, y y [sr,lr h + τrsc , εr +1 h + τrsc+1 ), v = lr ,

y

y

with lr =min{v|sr,v+1 h+τrsc ≥ εr +1 h + τrsc+1 }. By utilizing the input delay method v∈N

y

[18], the piecewise functions are denoted as η(t) = t − sr,v h, t ∈ |r,v , η(t) = y ˜ = t− εru h, t ∈ |r with εru h = max{εru h|εru h + τrca ≤ t}, t−εr h, t ∈ |r , and η(t) r ∈N

which meet boundaries ηm ≤ η(t) ≤ η M , ηm ≤ η(t) ≤ η M , ηm ≤ η(t)≤ ˜ η˜ M with sc ηm = τmsc , η M = τ M + h, η M = τmsc + h + Tmax , η˜ M =τmsc +Tmax , and Tmax is the y y maximum DT. Define e y (t) = y(sr,v h) − y(εr h), t ∈ |r,v . Then, the input of (6.2) is y˜ (t) = α(t)y(εry h) = α(t)[λ y (t)(y(t − η(t)) − e y (t)) + (1 − λ y (t))y(t − η(t)), (6.5) where disordered packets are supposed to be lost, and the Bernoulli-distributed variable α(t) with E{α(t)} = α depicts the packet losses in the S2C channel. If tk ∈ T , then λ y (t) = 1, otherwise λ y (t) = 0. Considering random packet losses and ETM (6.4), the control input u(t) in NSSs is described as ˆ − η(t)) − eu (t)) + (1 − λu (t))u(t ˆ − η(t)], ˜ u(t) = β(t)u(ε ˆ ru h) = β(t)[λu (t)(u(t (6.6) where the Bernoulli-distributed variable β(t) with E{β(t)} = β shows the packet u losses in the C2A channel. eu (t) = y(sr,v h) − y(εru h), t ∈ |r,v , λu (t) = 1, if u t˜k ∈ [tk , tk+1 ) for tk ∈ T ; λ (t) = 0, if t˜k ∈ [tk , tk+1 ) for tk ∈ T . The following assumption is needed to study EORP for NSSs. Assumption 6.3 There exist matrices ||i , Ei , Aci , Bci and Hci satisfying the following regulator equations:

Ai ||i + β Bi Hci Ei + Ci − ||i Si = 0, Di ||i + E i = 0, Bci Ei e

−Si η1M

(6.7)

+ Aci Ei = Ei Si .

The closed-loop systems of NSSs (6.1) under control (6.6) can be divided into the following two situations according to whether the subsystem undergoes consecutive asynchronous switching. (1) When consecutive asynchronous switching does not occur at tk and tk−1 , the closed-loop system is as follows:

126

6 Mixed Event-Triggered Output Regulation for Networked Switched …

{

χ˙ (t) = A˜ i χ (t) + B˜ ci χ (t − η(t)) + /\i1j + /\i2j ,

{

y(t) = D˜ i χ (t), t ∈ [tk , t˜k ); χ(t) ˙ = A˜ i χ (t) + B˜ ci χ (t − η(t)) + /\i1 + /\i2 , y(t) = D˜ i χ (t), t ∈ [t˜k , tk+1 );

(6.8.1)

(6.8.2)

(2) When consecutive asynchronous switching occurs at tk and tk−1 , the closedloop system is as follows: {

χ˙ (t) = A˜ j χ (t) + B˜ cj χ (t − η(t)) + /\1j p + /\2j p ,

{

y(t) = D˜ j χ (t), t ∈ [tk−1 , tk ); χ˙ (t) = A˜ i χ (t) + B˜ ci χ (t − η(t)) + /\i1p + /\i2p , y(t) = D˜ i χ (t), t ∈ [tk , tk+1 ),

(6.8.3)

(6.8.4)

where the initial condition χ (t) = ϕ(t) for t ∈ [−η M , 0], | χ(t) =

x(t) ˜ x˜c (t)

|

| =

| | | | | Aϑ˜ 0 0 0 x(t) − ||i w(t) ˜ ˜ , Bcϑ˜ = , D˜ ϑ˜ = [ Dϑ˜ 0 ], , Aϑ˜ = 0 Bcϑ˜ xc (t) − Ei w(t) 0 Acϑ˜

ϑ˜ ∈ {i, j}, ϑ ∈ {i, i j, i p, j p}, i /= j, i /= p, j /= p, i, j, p ∈ M, /\1ϑ = α(t)[λ y (t)(C˜ cϑ χ (t − η(t)) + C cϑ e y (t)) + (1 − λ y (t))(C˜ cϑ χ (t − η(t))], ˜ /\2ϑ = β(t)[λu (t)( B˜ ϑ χ (t − η(t)) + B ϑ eu (t)) + (1 − λu (t))( B˜ ϑ χ (t − η(t))], | | | | | | | 0 0 0 0 0 0 0 0 , C˜ ci j = , C˜ ci p = , C˜ cj p = , Cci Di 0 Ccj Di 0 Ccp Di 0 Ccp D j 0 | | | | | | | | 0 0 0 0 Bi Hci C ci = , , C ci j = , C ci p = C cj p = , B˜ i = B˜ i j = B˜ i p = −Cci −Ccj −Ccp 0 0 | | | | | | −Bi −B j 0 B j Hcj B˜ j p = , Bi = Bi j = Bi p = , B jp = . 0 0 0 0 |

C˜ ci =

6.2.5 Event-Triggered Output Regulation Problem Our goal is to jointly design a DOFC (6.2), mixed ETMs (6.3) and (6.4), and a switching signal under consecutive asynchronous subsystem switching caused by LDDAs to solve the EORP for NSSs (6.1) with SUDs subject to LDDAs, networkinduced delays, packet losses, and packet disorders. That is, the control aim is

6.3 Main Results

127

twofold: (a) To get an undisturbed (w(t) = 0) mean-square exponentially stable (6.8.1)–(6.8.4). (b) To satisfy lim y(t) = 0 as the solution of (6.8.1)–(6.8.4) with t→∞

w(t) /= 0 and any initial conditions.

6.3 Main Results In this section, sufficient conditions are developed for EORP of NSSs (6.1) with SUDs under LDDAs, and network-induced phenomenon by jointly designing the DOFC (6.2), ETMs (6.3) and (6.4), and a switching signal. Theorem 6.1 Under Assumptions 6.1–6.3, given positive constants λ, κ, λ, h, ηm , y u η M , η M , η˜ M , n, θ y , θu , 0 0, Uϑ > 0, Rϑ > 0, ui > 0, uiu > 0, Mϑ , Ti j , ς ς Ji j , J˜i j and Cci , with ϑ ∈ {i, i j, i p, j p},ϑ ∈ {i j, j p}, i /= j, i /= p, j /= p, i, j, p ∈ M, ι ∈ {0, . . . , n}, ς ∈ {0, 1, 2, 3}, ς˜ ∈ {1, 2, 3} satisfying | ψ ϑ < 0, []ϑ < 0, []0ϑ < 0, []1ϑ < 0,

ς˜

ς˜

Rϑ Mϑ ς˜ ∗ Rϑ

| > 0.

(6.9)

(ii) When the switching from the j-th subsystem to the i-th subsystem is destabilizing, given ρϑ > 1, ς

ς

ς

ς

/\ P j < 0, /\ S j < 0, /\ R j < 0,

(6.10)

/\ P j p < 0, /\ S j p < 0, /\ R j p < 0,

(6.11)

[ Pi Uiς Riς ] N ∗ = − ln(μ) ˜ ln(ρ ρ),

(6.17)

where the 11 × 11 matrix ψ ϑ = {ψ ϑ } in (6.9) consists of matrices 11 12 ψ ϑ = A˜ ϑT˜ Pϑ + Pϑ A˜ ϑ˜ + a1 Pϑ + Uϑ0 + (1 + a1 ) A˜ ϑT˜ Rϑ A˜ ϑT˜ − Rϑ0 , ψ ϑ = Rϑ0 ,

ψ ϑ = Pϑ B˜ cϑ˜ + A˜ ϑT˜ Rϑ B˜ cϑ˜ , ψ ϑ = β Pϑ B ϑ + β A˜ ϑT˜ Rϑ B ϑ , ( ) 22 23 24 ψ ϑ = eϑ ϑ ηm Uϑ1 + Uϑ2 + Uϑ3 − Uϑ0 − Rϑ0 − Rϑ1 − Rϑ2 − Rϑ3 , ψ ϑ = Mϑ1 , ψ ϑ = Mϑ2 , 16

19

ψ ϑ = Mϑ3 , ψ ϑ = Rϑ1 − Mϑ1 , ψ ϑ = Rϑ2 − Mϑ2 , ψ ϑ = Rϑ3 − Mϑ3 , ψ ϑ = −eϑ ϑ η M Uϑ1 − Rϑ1 , 25

26

27

28

33

ψ ϑ = Rϑ1 − Mϑ1T , ψ ϑ = −eϑ ϑ η M Uϑ2 − Rϑ2 , ψ ϑ = Rϑ2 − Mϑ2T , ψ ϑ = −eϑ ϑ η˜ M Uϑ3 − Rϑ3 , 36

44

47

55

ψ ϑ = Rϑ3 − Mϑ3T , ψ ϑ = β B˜ cTϑ˜ Rϑ B ϑ , ψ ϑ = Mϑ1 + Mϑ1T − 2Rϑ1 + δ ϑ˜ D˜ ϑT˜ u ˜ D˜ ϑ˜ + 58

69

66

y

u T (1 + 2α + β) B˜ cTϑ˜ Rϑ B˜ cϑ˜ + δ ϑ˜ H˜ ϑ˜T uuϑ˜ H˜ ϑ˜ + C˜ cϑ [α Pϑ + (4α + 2αβ)Rϑ ]C˜ cϑ

y ϑ

77 88 ψ ϑ = Mϑ2 + Mϑ2T − 2Rϑ2 + B˜ ϑT [2β Pϑ + (2β + 2αβ)Rϑ ] B˜ ϑ , ψ ϑ = Mϑ3 + Mϑ3T − 2Rϑ3 , T

99

10,10

ψ ϑ = 2(αβ + β)B ϑ Rϑ B ϑ − uuϑ˜ , ψ ϑ } { 11,11 y y ψϑ = diag −γ1 u ˜ · · · − γn u ˜ , ϑ

y

T

= −γ0 ui + C cϑ [α Pϑ + (4α + 2αβ)Rϑ ]C cϑ ,

ϑ

the 8 × 8 matrix []ϑ = {θ ϑ } in (6.9) consists of matrices p1 q1

θϑ

p1 q1

= ψϑ

for p1 ∈ {1, . . . , 5}, q1 ∈ {1 . . . , 8},

66 θ ϑ = Mϑ1 + Mϑ1T − 2Rϑ1 + (1 + α + β) B˜ cTϑ˜ Rϑ B˜ cϑ˜ 77 T θ ϑ = Mϑ2 + Mϑ2T − 2Rϑ2 + C˜ cϑ [α Pϑ + (4α + β)Rϑ ]C˜ cϑ ,

θ ϑ = ψ ϑ + a2 B˜ cTϑ˜ Rϑ B˜ cϑ˜ + B˜ ϑT [β Pϑ + (2 + αβ + β)Rϑ ] B˜ ϑ , 88

88

the 9 × 9 matrix []0ϑ = {θ 0ϑ } in (6.9) consists of matrices p2 q2

p2 q2

θ 0ϑ = ψ ϑ

69

69

for p2 ∈ {1, . . . , 5}, q2 ∈ {1 . . . , 9}, θ 0ϑ = ψ ϑ ,

u 66 θ0ϑ = Mϑ1˜ + Mϑ1T − 2Rϑ1 + a1 B˜ cTϑ˜ Rϑ B˜ cϑ˜ + δ ϑ˜ H˜ ϑ˜T uuϑ˜ H˜ ϑ˜ | | 77 T θ 0ϑ = Mϑ2 + Mϑ2T − 2Rϑ2 + C˜ cϑ α Pϑ + (3α + 2αβ)Rϑ C˜ cϑ | | T 88 88 99 = ψ ϑ + B˜ ϑT˜ β Pϑ + (αβ + 4β)Rϑ B˜ ϑ˜ , θ0ϑ θ0ϑ = (αβ + 2β)B ϑ˜ Rϑ B ϑ˜ − uuϑ˜ ,

6.3 Main Results

129

the 10 × 10 matrix []1ϑ = {θ 1ϑ } in (6.9) consists of matrices p3 q3

p3 q3

θ 1ϑ = ψ ϑ

for p3 ∈ {1, . . . , 5}, q3 ∈ {1 . . . , 8},

66 θ 1ϑ = Mϑ1 + Mϑ1T − 2Rϑ1 + (1 + 2α + β) B˜ cTϑ˜ Rϑ B˜ cϑ˜ | | y y T α Pϑ + (3α + αβ)Rϑ C˜ cϑ + δ ϑ˜ D˜ ϑT˜ uϑ˜ D˜ ϑ˜ +C˜ cϑ | | 88 θ 1ϑ = Mϑ3 + Mϑ3T − 2Rϑ3 + B˜ ϑT˜ β Pϑ + (1 + α + 2β)Rϑ B˜ ϑ˜ | T | 77 99 y θ 1ϑ = −γ0 uϑ˜ + C cϑ α Pϑ + (3α + αβ)Rϑ C cϑ , θ 1ϑ = Mϑ2 + Mϑ2T − 2Rϑ2 | | 10,10 11,11 ˜ ∈ {(i p, i), ( j, j p)}, a = κ − λ, θ 1ϑ = ψ ϑ , H˜ ϑ˜ = 0 Hcϑ˜ , (ϑ, ϑ)

a1 = 2α + β, a2 = αβ + β, Rϑ = ηm2 Rϑ0 + {

3 E

ς

(η − η1m )2 Rϑ ,

ς=1

(

) } (η, ς ) ∈ (ηm , 0), (η M , 1), η M , 2 , (η˜ M , 3)

| | | | | ς JiTj −μi j Pυ −ρi j Pυ TiTj TiTj ς aη −ρi j Sυ /\ Pυ = , , /\ Pυ = , /\ Sυ = e ς ∗ Sυ˜ − 2I ∗ Pυ˜ − 2I ∗ Pυ˜ − 2I | | | | ς ς J˜iTj JiTj −μi j Sυ ς ς aη −ρi j Rυ /\ = e /\ Sυ = eaη , , Rυ ς ς ∗ Sυ˜ − 2I ∗ Rυ˜ − 2I | | ς J˜iTj ς aη −μi j Rυ /\ Rυ = e , υ ∈ { j, j p}, υ ∈ { j, j p}, υ˜ ∈ {i j, i p}, ς ∗ Rυ˜ − 2I |

˜ ∈ {( j, j, i j ), ( j p , j p, i p)}, ϑ˜ i = i, ϑ˜ i j = j, ϑ i = λ, ϑ i j = ϑ j p = ϑ i p = κ, (υ, υ, υ) ρ˜ = max {ρi j }, μ˜ = max {μi j } , ρ = max {ρi }, μ = max {μi }. i, j∈M

i, j∈M

i∈M

i∈M

Proof Due to LDDAs, the switching on the interval [tk , tk+1 ) possesses two cases. In the first case, the subsystem switches and the controller switches correspondingly on [tk , tk+1 ), then [tk , tk+1 ) is divided into [tk , t˜k ) with asynchronous mode between the subsystem and the controller, and [t˜k , tk+1 ) with synchronous mode between the subsystem and the controller. In the second case, the subsystem switches but the controller is hindered by LDDAs and does not switch correspondingly on [tk , tk+1 ), then [tk , tk+1 ) is the asynchronous interval. Hence, the proof is threefold. Case A: For t ∈ [tk , t˜k ), j = φ(t) /= σ (t) = i. The Lyapunov candidate function is thus chosen as Vi j (t) = V Pi j (t) + VUi j (t) + VRi j (t), where

(6.18)

130

6 Mixed Event-Triggered Output Regulation for Networked Switched …

{ VUi j (t) =

t

t−ηm

V Pi j (t) = χ T (t)Pi j χ (t), 3 { t−ηm E ς eλ(t−s) χ T (s)Ui0j χ (s)ds + eλ(t−s) χ T (s)Ui j χ (s)ds, {

VRi j (t) = ηm

t

t−ηm

+

3 E

ς =1

{

t θ

eλ(t−s) χ˙ T (s)Ri0j χ˙ (s)dsdθ

{

t−ηm

(η − ηm )

{

t

θ

t−η

ς=1

t−η

ς

eλ(t−s) χ˙ T (s)Ri j χ(s)dsdθ ˙ .

Taking the time derivative and mathematical expectation of Vi j (t) along the system (6.8.1) yields E{V˙ pi j (t)} = E{2χ T (t)Pi j χ˙ (t)}, E{V˙Ui j (t)} = χ T (t)Ui0j χ (t) +

3 E

ς

eληm χ T (t − ηm )Ui j χ (t − ηm )

ς =1

−

3 E

ς

eλη χ T (t − η)Ui j χ (t − η) + λVUi j (t),

ς=0

E{V˙ Ri j (t)} =E{ηm2 χ˙ T (t)Ri0j χ(t)+ ˙ t

+ λVRi j (t) − η1m 3 E ς =1

{ (η − ηm )

ς

(η − ηm )2 χ˙ T (t)Ri j χ˙ (t)}

ς =1

{

−

3 E

t−η1m t−ηm t−η

eλ(t−s) χ˙ T (s)Ri0j χ(s)ds ˙ ς

eλ(t−s) χ˙ T (s)Ri j χ(s)ds. ˙

On the asynchronous interval [tk , t˜k ), there exist four situations for two-channel ETMs, then the following proof of Case A is divided into four subcases. 1 Case (A1): When tk ∈ T and t˜k−1 ∈ T k−1 , the two-channel ETMs are both event-triggered conditions in the second part of (6.3) and (6.4), then (6.8.1) with λ y (t) = λu (t) = 1 is obtained. Thus, E{V˙ pi j (t)} = E{2χ T (t)Pi j χ(t)} ˙ = 2χ T (t)Pi j [ A˜ i χ (t) + α(C˜ ci j χ (t − η(t)) + C ci j e y (t)) + B˜ i χ (t − η(t)) + β( D˜ ci j χ (t − η(t)) + B i j eu (t))]. Utilizing Lemma 1.8 and Park theorem [23] results in

6.3 Main Results

131

{ − (η˜ − ηm )

t−ηm

t−η˜

ς˜

eλ(t−s) χ˙ T (s)Ri j χ(s)ds ˙

~ ~ ≤ [ χ T (t − ηm ) − χ T (t − η(t)) χ T (t − η(t)) − χ T (t − η) ] | || | ~ ς˜ ς˜ χ (t − ηm ) − χ (t − η(t)) R i j Mi j · , ς˜ ~ ∗ Ri j χ (t − η(t)) − χ (t − η)

{

− ηm

t t−ηm

(6.19)

eλ(t−s) χ˙ T (s)Ri0j χ˙ (s)ds

≤ −[χ (t) − χ (t − ηm )]T Ri0j [χ (t) − χ (t − ηm )],

(6.20)

~

where (η, ˜ ς˜ , η(t)) ∈ { (η M , 1, η(t)), (η M , 2, η(t)), (η˜ M , 3, η(t))}. ˜ Combining (6.19) and (6.20) with (6.3) and (6.4) leads to E{V˙i j (t) − λVi j (t)} ≤ ζ T (t)ψ i j ζ (t),

(6.21)

where ζ T (t) = [ χ T (t) χ T (t − ηm ) χ T (t − η M ) χ T (t − η M ) χ T (t − η˜ M ) χ T (t − η(t)) euT (t) e Ty (t) e Ty (t − h) · · · e Ty (t − nh) ]. χ T (t − η(t)) χ T (t − η(t)) ˜ From (6.9), ψ i j < 0. Then, integrating (6.23) over [tk , t˜k ) gives E{Vi j (t)} ≤ eλ(t−tk ) E{Vi j (tk )}.

(6.22)

Case (A2): When tk ∈ T and t˜k−1 ∈ T 1k−1 , the two-channel ETMs are both periodic sampling conditions in the second part of (6.3) and (6.4), then λ y (t) = λu (t)= 0 in (8.1). Thus, similar to (6.23), it is readily obtained that E{V˙i j (t) − λVi j (t)} ≤ ζ (t)[]i j ζ (t), T

(6.23)

where ζ T (t) = [ χ T (t) χ T (t − ηm ) χ T (t − η M ) χ T (t − η M ) ]. ˜ χ T (t − η˜ M ) χ T (t − η(t)) χ T (t − η(t)) χ T (t − η(t)) One has []i j < 0 from (6.9). Similar to Case (A1), (6.22) holds. Case (A3): When tk ∈ T and t˜k−1 ∈ T 1k−1 , the ETM is the form of periodic sampling conditions in the S2C channel and event-triggered conditions in the C2A channel, thus (6.8.1) is got with λ y (t) = 0, λu (t)=1. Then, it gets []0i j < 0 from (6.9). A similar deduction shows that (6.22) holds.

132

6 Mixed Event-Triggered Output Regulation for Networked Switched … 1

Case (A4): When tk ∈ T and t˜k−1 ∈ T k−1 , the ETMs in S2C and C2A channels are in the situations of event-triggered conditions and periodic sampling conditions, then (6.8.1) is concluded with λ y (t) = 1, λu (t) = 0. From (6.9), []1i j < 0. Similarly, (6.22) holds. Case B: For t ∈ [t˜k , tk+1 ), φ(t) = σ (t) = i. Then, construct the Lyapunov candidate function as Vi (t) = V Pi (t) + VUi (t) + VRi (t),

(6.24)

where V Pi (t) = χ T (t)Pi χ (t), { VUi (t) =

t

e

κ(t−s)

t−ηm

{

VRi (t) =ηm

t−ηm

+

3 E ς =1

(s)Ui0 χ (s)ds

+

3 { E ς =1

{

t

χ

T

t θ

t−ηm

t−η

ς

eκ(t−s) χ T (s)Ui χ (s)ds

eκ(t−s) χ˙ T (s)Ri0 χ(s)dsdθ ˙ {

(η − ηm )

t−ηm t−η

{

t θ

ς

eκ(t−s) χ˙ T (s)Ri χ˙ (s)dsdθ

Following a similar procedure to Case A, it yields E{Vi (t)} ≤ eκ(t−t˜k ) E{Vi (t˜k )}.

(6.25)

Case C: Affected by LDDAs, the mode of the activated controller and subsystem is asynchronous on the whole interval [tk , tk+1 ) and [tk−1 , tk ). It is assumed that σ (t) = j for t ∈ [tk−1 , tk ), σ (t) = i for t ∈ [tk , tk+1 ), and φ(t) = p for t ∈ [tk−1 , tk+1 ). For t ∈ [tk−1 , tk ), the corresponding closed-loop system is (6.8.3). From (6.9), it has ψ i p < 0,[]i p < 0. Then, similar to Case A, one can obtain E{V j p (t)} ≤ eλ(t−tk−1 ) E{V j p (tk ), t ∈ [tk−1 , tk ).

(6.26)

For t ∈ [tk , tk+1 ), the corresponding closed-loop system is (6.8.4). From (6.9), ψ j p < 0,[] j p < 0, []0 j p < 0, []1 j p < 0 hold. Similar to Case A, one has E{Vi p (t)} ≤ eλ(t−tk ) E{Vi p (tk )}, t ∈ [tk , tk+1 ).

(6.27)

The next step is to discuss the evolution of the Lyapunov functions at switching instants tk and t˜k . For inconsecutive asynchronous switching instant tk ∈ T , owing to different coordinate transformations, it is obvious that χ (tk+ ) /= χ (tk− ). Together with χ (tk+ )= Ti j χ (tk− ), (6.18) and (6.24), we have

6.3 Main Results

133

E{V Pi j (tk+ ) − μi j V P j (tk− )} =χ T (tk+ )Pi j χ (tk+ ) − μi j χ T (tk− )P j χ (tk− ) =χ T (tk− )TiTj Pi j Ti j χ (tk− ) − μi j χ T (tk− )P j χ (tk− ). −1 −1 Then, it gets −Pi−1 j < Pi j − 2I from (Pi j − I )Pi j (Pi j − I ) > 0. According to /\ P j < 0 in (6.13), by applying Lemma 1.7 and substituting −Pi−1 j for Pi j − 2I , + T one can conclude that Ti j Pi j Ti j − μi j P j < 0. Thus E{V Pi j (tk )} ≤μi j E{V P j (tk− )}. In the same way, it holds that E{VUi j (tk+ )} ≤ μi j E{VU j (tk− )} and E{VRi j (tk+ )} ≤ μi j E{VR j (tk− )}. Therefore, when the switching from j-th subsystem to the i-th subsystem is stabilizing, i.e., tk ∈ T , it is readily obtained that − E{Vi j (tk+ )} ≤ μi j E{V j (tk− )} ≤ μE{V ˜ j (tk )}.

(6.28)

Similarly, for inconsecutive switching instant tk ∈ T , that is the switching from j-th the subsystem to the i-th subsystem is destabilizing, it leads to − E{Vi j (tk+ )} ≤ ρi j E{V j (tk− )} ≤ ρE{V ˜ j (tk )}.

(6.29)

Next, for the controller switching instant t˜k corresponding to inconsecutive switching instant tk ∈ T , (6.15) yields E{Vi (t˜k+ )} ≤ μi E{Vi j (t˜k− )} ≤ μE{Vi j (t˜k− )}.

(6.30)

For the controller switching instant t˜k corresponding to inconsecutive switching instant tk ∈ T , (6.12) gives E{Vi (t˜k+ )} ≤ ρi E{Vi j (t˜k− )} ≤ ρE{Vi j (t˜k− )}.

(6.31)

Then, affected by LDDAs, consecutive asynchronous switching occurs. For the consecutive asynchronous switching instant tk ∈ T , similar to (6.28), it can be derived from (6.14) that − E{Vi p (tk+ )} ≤ μi j E{V j p (tk− )} ≤ μE{V ˜ j p (tk )}.

(6.32)

For the consecutive asynchronous switching instant tk ∈ T , similar to (6.29), (6.11) results in − E{Vi p (tk+ )} ≤ ρi j E{V j p (tk− )} ≤ ρE{V ˜ j p (tk )}.

(6.33)

In summary, for ∀t ∈ [t˜k , tk+1 ), from (6.22), (6.25)–(6.33), it is clear that E{Vσ (t) (t)} = E{Vi (t)} ≤ ce M T (0,t) E{Vσ (t0 ) (t0 )},

(6.34)

134

6 Mixed Event-Triggered Output Regulation for Networked Switched …

where c = ebmax ω , M = −n ln(μ)−m ˜ ln(ρ ρ) ˜ ,T (0, t) and N (0,t)

N (0,t) [ln(μ) ˜ Tmin (m+n)

+ ln(ρ ρ) ˜ + (κ +

bmax ), τD

·Tmax +

Tmin are the total running time on [0, t) and the minimum DT respectively, and (6.16)–(6.17) show that M < 0. For ∀t ∈ [tk , t˜k ) or ∀t ∈ [tk , tk+1 ) with inconsistent modes between the activated controller and subsystem, similar procedures to (6.34) gives E{Vσ (t) (t)} = E{Vi j (t)} ≤ ρce ˜ M T (0,t) E{Vσ (t0 ) (t0 )}.

(6.35)

Combining (6.18), (6.24) (6.34) and (6.35), aE{||χ (t)||2 } ≤ E{Vσ (t) (t)}, and E{Vσ (t) (t0 )}≤bE{||χ (t0 )||2 } concludes / E{||χ (t)||2 } ≤ (bcρ˜ a)e M T (0,t) E{||χ (t0 )||2 },

(6.36)

where a = min {λm (Pi j ), λm (Pi ), λm (Pi p ), λm (P j p )}, i, j, p∈M

b = max {λ M (Pi j ), λ M (Pi ), λm (Pi p ), λm (P j p )} i, j∈M

+ ηm max {λ M (Ui0j ), λ M (Ui0 ), λ M (Ui0p ), λ M (U 0j p )} i, j∈M

+

3 E

ς

ς

ς

ς

(η − ηm ) max {λ M (Ui j ), λ M (Ui ), λ M (Ui p ), λ M (U j p )} i, j∈M

ς =1

/ + (ηm2 2) max {λ M (Ri0j ), λ M (Ri0 ), λ M (Ri0p ), λ M (R 0j p )} i, j∈M

+

3 E ς =1

/ ς ς ς ς (η2 2) max {λ M (Ri j ), λ M (Ri ), λ M (Ri p ), λ M (R j p )}. i, j∈M

Thus, the closed-loop system (6.8.1)–(6.8.4) with w(t) = 0 is mean-square exponentially stable. When w(t) /= 0, the center manifold theory [24] reveals that there exists W > 0, c > 0 satisfying ||x(t) − ||i w(t)|| + ||ξ(t) − Ei w(t)|| ≤ W e−ct (||x(0) − ||i w(0)|| + ||ξ(0) − Ei w(0)||). So lim (||x(t) − ||i w(t)|| + ||ξ(t) − Ei w(t)||) = 0, which brings lim y(t) = 0. t→∞ t→∞ Therefore, the solvability of EORP for NSSs (6.1) based on DOFC is ensured. This completes the proof.

6.3 Main Results

135

Remark 6.4 Theorem 6.1 considers SUDs in NSSs, which is different from existing results on the security control for NSSs, such as [3–7] and [14, 15]. Since the interaction of SUDs and LDDAs can affect the system performance, it is challenging to jointly design appropriate switching laws and ETMs to suppress these impacts. To deal with SUDs, the second parts of mixed ETMs (6.3) and (6.4) ensure that system performance can be adjusted quickly after destabilizing switching, the number of which is limited by the switching law (6.17). The parameters of LDDAs in (6.3) and (6.4) guarantee that one trigger happens at the nearest sampling instant after each attack, which is contained in (6.16) to adjust DT accordingly. Besides, consecutive asynchronous subsystem switching is permitted in our switching scheme by imposing (6.11) and (6.14) compared with existing results such as [3–7] and [14, 15]. In fact, the DT of each subsystem needs to be restricted by an upper bound like (6.16) for NSSs with SUDs. Thus, when NSSs with SUDs are threatened by LDDAs, the consecutive asynchronous subsystem switching induced by LDDAs cannot be avoided completely by prolonging the DT of individual subsystems. Therefore, the key to solving the above-mentioned asynchronous switching for NSSs with SUDs is to discuss the relationship of Lyapunov functions before and after asynchronous switching instants. Remark 6.5 Asynchronous controller switching and consecutive asynchronous subsystem switching have not been considered in existing results on switched systems with SUDs such as [19–21]. Theorem 6.1 not only considers asynchronous switching but also permits irregular arrangement of destabilizing and stabilizing switching under the constraint (6.17), which overcomes the regular arrangement in [19]. Furthermore, with the help of different coordinate transformations of EORP for NSSs, the discretized Lyapunov function is not utilized as in [19–21], which simplifies the mathematical derivation process. Moreover, compared with [12], LDDAs’ parameters are added in (6.16) to make it more suitable for (6.1) under LDDAS. In what follows, we deal with the nonlinear terms in Theorem 6.1 and derive LMI conditions. Theorem 6.2 Under Assumptions 6.1–6.3, given positive constants λ, κ, λ, h, ηm , y u η M , η M , η˜ M , n, θ y , θu , 0 0, Uϑ > 0, Rϑ > 0, ui > 0, uiu > 0, Mϑ , Ti j , Ji j , J˜i j and Cci , with ϑ ∈ {i, i j, i p, j p}, ϑ ∈ {i j, j p}, i /= j, i, j, p ∈ M, ι ∈ {0, . . . , n}, ς ∈ {0, 1, 2, 3}, ς˜ ∈ {1, 2, 3} satisfying | ψϑ < 0, []ϑ < 0, []0ϑ < 0, []1ϑ < 0,

ς˜

where the 13 × 13 matrix ψϑ = {ψϑ } consists of matrices p q4

ψϑ 4

p4 q4

= ψϑ

for p4 ∈ {1, . . . , 5}, q4 ∈ {1 . . . , 9},

ς˜

R ϑ Mϑ ς˜ ∗ Rϑ

| >0

(6.37)

136

6 Mixed Event-Triggered Output Regulation for Networked Switched …

ψϑ66 = Mϑ1 + Mϑ1T − 2Rϑ1 + δ ϑ˜ D˜ ϑT˜ uϑ˜ D˜ ϑ˜ + (1 + 2α + β) B˜ cTϑ˜ Rϑ B˜ cϑ˜ + δ ϑ˜ H˜ ϑ˜T uuϑ˜ H˜ ϑ˜ y

u

y

T ψϑ69 = ψ ϑ , ψϑ6,12 = C˜ cϑ , ψϑ77 = ψ ϑ , ψϑ88 = ψ ϑ ,ψϑ99 = ψ ϑ , ψϑ10,10 = −γ0 uϑ˜ , 69

77

T

11,11

ψϑ10,13 = C cϑ , ψϑ11,11 = ψ ϑ

88

99

y

2

,ψϑ12,12 = ψϑ13,13 = λ [α Pϑ + (4α+2αβ)Rϑ ]−2λI ,

the 9 × 9 matrix []ϑ = {θϑ } consists of matrices p q p q θϑ 5 5 = ψϑ 5 5 for p5 ∈ {1, . . . , 5},q5 ∈ {1 . . . , 8},, θϑ77 = Mϑ2 + Mϑ2T − 2Rϑ2 , 66 88 2 T , θϑ88 = θ ϑ , θϑ99 = λ [α Pϑ + (4α + β)Rϑ ] − 2λI , θϑ66 = θ ϑ , θϑ79 = C˜ cϑ the 10 × 10 matrix []0ϑ = {θ0ϑ } consists of matrices p q

p q

66

66 69 = θ 0ϑ , θ0ϑ θ0ϑ6 6 = ψϑ 6 6 for p6 ∈ {1, . . . , 5}, q6 ∈ {1 . . . , 9},θ0ϑ = ψϑ69 , 77 = ψϑ77 , θ0ϑ 99 2 7,11 10,10 T 88 99 θ0ϑ = C˜ cϑ , θ0ϑ = ψϑ88 ,θ0ϑ = θ 0ϑ , θ0ϑ = λ [α Pϑ + (3α + 2αβ)Rϑ ] − 2λI , the 12 × 12 matrix []1ϑ = {θ1ϑ } consists of matrices T p q y p q 6,11 9,13 T 99 = C˜ cϑ , θ1ϑ = −γ0 uϑ˜ , θ1ϑ = C cϑ , θ1ϑ7 7 = ψϑ 7 7 for p7 , q7 ∈ {1, . . . , 8},θ1ϑ 11,11

10,10 θ1ϑ = ψϑ

2

12,12 11,11 ,θ1ϑ = θ1ϑ = λ [α Pϑ + (3α + αβ)Rϑ ] − 2λI .

Proof In Case A of the proof in Theorem 6.1: Case (A1): By combining Lemma 1.7 with 2

−[α Pi j + (4α + 2αβ)Ri j ]−1 < λ [α Pi j + (4α + 2αβ)Ri j ] − 2λ Ii j from Lemma 1.9, it leads to ψi j < 0 from ψ i j < 0. Case (A2): By using Lemma 1.7 together with 2

−[α Pi j + (4α + β)Ri j ]−1 < λ [α Pi j + (4α + β)Ri j ] − 2λI derived from Lemma 1.9 it yields []i j < 0 from []i j < 0. Case (A3): By combining Lemma 1.7 with 2

−[α Pi j + (3α + 2αβ)Ri j ]−1 < λ [α Pi j + (3α + 2αβ)Ri j ] − 2λ Ii j from Lemma 1.9, one can obtain []0i j < 0 from []0i j < 0. Case (A4): By using Lemma 1.7 together with 2

−[α Pi j + (3α + αβ)Ri j ]−1 < λ [α Pi j + (3α + αβ)Ri j ] − 2λI from Lemma 1.9, it gets []1i j < 0 from []1i j < 0. Then, (6.22) can be deduced in Case A following a procedure similar to the proof of Theorem 6.1.

6.3 Main Results

137

In Case B and Case C of Theorem 6.1, similar to the proof of Case A, one can get ψi < 0 ψi j < 0, ψi p < 0 and ψ j p < 0 from ψ i < 0, ψ i j < 0, ψ i p < 0 and ψ j p < 0 respectively. Thus, (6.22), (6.25)–(6.27) hold. Next, if (6.10)–(6.17) and (6.37) hold, similar to Theorem 6.1, the EORP for NSSs with SUDs is solvable. This completes the proof. For n = 0, there is no destabilizing switching. Then, the forthcoming corollary ensures the solvability of EORP for (6.1). Corollary 6.1 Under Assumptions 6.1–6.3, given positive constants λ, κ, λ, h, ηm , y u η M , η M , η˜ M , n, θ y , θu , 0 ε then u∗

u

y∗

y

9: SST = ST , F T N = T N , δ i = δ i and δ i = δ i ; 10: else if |SST − ST | ≤ ε then 11: if T N < F T N then u∗

u

y∗

y

12: SST = ST , F T N = T N , δ i = δ i and δ i = δ i ; 13: else (continued)

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6 Mixed Event-Triggered Output Regulation for Networked Switched …

(continued) 14: SST = SST and F T N = F T N ; 15: end if 16: else 17: SST = SST and F T N = F T N ; 18: end if 19: end while 20: end while 21: while θu > 0 do 22: Set θu = θu + /\θu ; 23: while θ y > 0 do 24: Set θu = θu + /\θu ; 25: Calculate ST and TN; 26: if ST < SST and SST − ST > ε then 27: SST = ST , F T N = T N , θu∗ = θu and θ y∗ = θ y ; 28: else if |SST − ST | ≤ ε then 29: if T N < F T N then 30: SST = ST , F T N = T N , θu∗ = θu and θ y∗ = θ y ; 31: else 32: SST = ST and F T N = F T N ; 33: end if 34: else 35: SST = SST and F T N = F T N ; 36: end if 37: end while 38: end while u

u∗

y

y∗

39: return δ i = δ i , δ i = δ i , θu = θu∗ , θ y = θ y∗ , SST and FTN;

Remark 6.6 SST is used to store the shortest ST in each loop, so that appropriate y u parameters δ i , δ i , θ y and θu in ETMs (6.3)–(6.4) can be selected according to SST. To store the ST when the system runs for the first time so that Algorithm 6.1 can work properly, we select a value greater than the total running time of the system, such as 1000, as the initial value of SST. Lines 8–18 and 26–36 in Algorithm 6.1 achieve y u the selection of δ i , δ i , θ y and θu based on SST and FTN, where lines 8–9, 26–27 y u are used to compute SST and record corresponding δ i , δ i and θ y , θu by comparing SST and ST in each loop, i.e. ST < SST , and lines 10–14, 28–32 aim to achieve that if there exist SSTs with little difference, i.e. |SST − ST | ≤ ε, the one with FTN y u is chosen and corresponding δ i , δ i and θ y , θu are recorded.

6.4 Simulation Example

139

6.4 Simulation Example In this section, the feasibility of Theorem 6.2 is illustrated via a switched RLC circuit [22]. Its model can be governed by (6.1), where x = [ qc i L ]T , qc is the charge of the capacitor, and i L is the flux in the inductance, | Ai =

⎡ ⎤ | | | 0 0 1 10 ⎣ ⎦ = = , B , D i i 1 − Lc1 i − LR 01 L

with i = 1, 2, 3. Let c1 = 1F, c2 = 2F, c3 = 1.5F, R = 2u, L = 1H , then the system matrices are | | | | given as | | 0 1 0 1 0 1 A1 = , A2 = , A3 = , −1 −2 −0.5 −2 −0.67 −2 | | | | 0 10 B1 = B2 = B3 = , D1 = D2 = D3 = 01 1 Other system matrices for EORP of NSSs and partial controller gains that satisfy Assumption 6.3 are as follows: |

| | | | | 0.66 0.78 0.74 0.3 0.64 0.77 E1 = , E2 = , E3 = , 0.41 0.25 0.49 0.11 0.39 0.15 | | | | | | 0.12 −4 0.1 −2.8 −0.11 −3.1 S1 = , S2 = , S3 = , 0.15 0.12 0.23 0.22 0.12 0.11 | | | | | | 0.21 2.79 0.34 2.11 0.35 0.87 C1 = , C2 = , C3 = , −1.57 0.32 −1.42 0.98 −0.98 −0.97 | | | | | | −0.34 −0.32 −0.75 −0.13 −0.46 −0.11 Ac1 = , Ac2 = , Ac3 = , −0.33 −0.37 −0.11 −0.72 −0.12 −0.49 | | | | 1.0448 −0.7094 −0.6814 0.4834 Bc1 = , Bc2 = , 1.4743 −1.2654 1.6577 −0.3177 | | | | −0.5504 0.2840 Bc3 = , Hc1 = −0.01 −0.02 , 1.3492 −0.2661 | | | | Hc2 = −0.02 −0.01 , Hc3 = −0.01 −0.03 . In the light of Theorem 6.2, select the parameters as λ = 1.82, κ = 2, λ = 0.9, h = 0.1, ηm = 0.01, η M = 0.5, η M = 1.5, η˜ M = 1.4, n = 1, θu = θ y = 0.5, y y u μi = 0.9, μi j = 0.1, ρi = ρi j = 2, i, j ∈ {1, 2, 3}, δ 1 = 0.5, δ 2 = 0.1, δ 1 =0.2,

140

6 Mixed Event-Triggered Output Regulation for Networked Switched …

u

δ 2 = 0.1, γ˜0 = 0.8 γ˜1 = 0.2, α = 0.9, β = 0.8, ω = 3, τ D = 1.1, bmax = 6. Solving (6.10)–(6.17) and (6.37) gets Tmin ≥ 0.6, Tmax ≤ 1.0358, N ∗ = 1.7248 and controller gains |

Cc1

| | | −0.0316 0.0686 0.0769 0.3951 = , Cc2 = , 0.0048 0.0476 0.5185 −0.1008 | | −0.0134 −0.0099 Cc3 = 0.0045 0.0027

It can be seen from Figs. 6.3 and 6.4 that the state trajectories and regulated outputs of three subsystems are all oscillating, which shows the unsolvability of OR for all subsystems. Figure 6.5 shows the random packet losses, LDDAs, and the switching signal with Tmax = Tmin = 1s. Obviously, the duration of LDDAs can be longer than the DT of a subsystem. Under LDDAs in Fig. 6.5, the inter-event intervals and networkinduced delays in two channels are presented in Fig. 6.6. Figure 6.7 describes the asynchronous switching signal and asynchronous intervals in two-channel networks. It can be observed that consecutive asynchronous subsystem switching caused by LDDAs occurs during 2 s ~ 4 s, 9 s ~ 14 s, and 18 s ~ 20 s. If the switching interval is not affected by LDDAs, the length of asynchronous intervals will not exceed the

Fig. 6.3 State trajectories of three subsystems

6.4 Simulation Example

141

Fig. 6.4 Regulated outputs of three subsystems

maximum network-induced delay, which verifies that (6.3) and (6.4) can shorten the length of asynchronous intervals caused by event-triggered conditions. With the switching signal shown in Fig. 6.7, the state responses and regulated output of (6.8.1)–(6.8.4) with the initial state χ (0) = [2.511.30.8]T are shown in Fig. 6.8, proving the solvability of EORP for NSSs (6.1) with SUDs. Figure 6.8 also depicts the evolution of the Lyapunov functions (6.18) and (6.24), which shows that destabilizing switching occurs when subsystem 3 is activated. From Fig. 6.8, the arrangement of stabilizing switching and destabilizing switching are arranged irregularly. In what follows, two comparisons are facilitated to verify the advantage of mixed ETMs in this chapter. First, the second parts in mixed ETMs (6.3) and (6.4) can update data timely and adjust system performance quickly by using periodic sampling conditions after destabilizing switching. To reflect this feature, a comparison with non-mixed ETMs in [12] is implemented in the same situation. It can be found from Fig. 6.9 that mixed ETMs use periodic sampling during 1 ~ 2 s, 4 ~ 5 s, 14 ~ 15 s, 17 ~ 18 s, 20 ~ 21 s and 23 ~ 24 s, namely switching intervals after destabilizing switching without LDDAs. Figure 6.10 indicates that the convergence rate of the system with mixed ETMs during these intervals, especially during 14 s ~ 15 s, is faster than that of the system with ETMs in [12], which leads to a better system performance over the entire running time. Therefore, mixed ETMs are helpful to adjust the system

142

6 Mixed Event-Triggered Output Regulation for Networked Switched …

Fig. 6.5 The random packet losses in two-channel networks, LDDAs and switching signal

Fig. 6.6 Inter-event intervals and network-induced delays in S2C and C2A channels

6.4 Simulation Example

143

Fig. 6.7 Asynchronous switching signals and asynchronous intervals in S2C and C2A channels

Fig. 6.8 The state responses and the regulated output of the closed-loop system and the evolution of the Lyapunov function

144

6 Mixed Event-Triggered Output Regulation for Networked Switched …

Fig. 6.9 Inter-event intervals with mixed ETMs and ETMs in [12] in S2C and C2A channels

Fig. 6.10 The state responses of the closed-loop system with mixed ETMs and ETMs in [12]

References

145

Table. 6.1 The comparison result with existing ETM (T = 25s) ETMs

The maximum length of asynchronous intervals caused by ETM

Settling time

(6.3)

0.04 s

16 s

[8]

0.7 s

22 s

performance. Next, the comparison between (6.3) and the ETM in [8] is given to demonstrate the feature of the third part in ETM (6.3). As illustrated in Table 6.1, the ETM in this chapter can greatly shorten the length of asynchronous intervals, which helps to reduce the settling time of the closed-loop system.

6.5 Conclusion The EORP has been solved for NSSs with SUDs subject to LDDAs, network-induced delays, packet losses, and packet disorders. A more realistic switching situation in NSSs with SUDs has been considered by adding the inequality conditions for consecutive asynchronous subsystem switching caused by LDDAs. Two mixed ETMs for two-channel networks under LDDAs have been devised to fully utilize network resources while adjusting the system performance in time. A novel switching scheme based on a DT with LDDA parameters and a ratio of total stabilizing switching and total destabilizing switching has been developed and oriented toward SUDs and LDDAs in NSSs. Sufficient conditions have been provided to solve the EORP for NSSs under DOFC using the input delay method and the different coordinate transformations. The future study will be EORP for NSSs under other kinds of attacks.

References 1. Wang, J., Huang, Z., Wu, Z., Cao, J., Shen, H.: Extended dissipative control for singularly perturbed PDT switched systems and its application. IEEE Trans. Circuits Syst. I-Regul. Pap. 67(12), 5281–5289 (2020) 2. Wang, J., Xia, J., Shen, H., Xing, M., Park, J.H.: H synchronization for fuzzy Markov jump chaotic systems with piecewise-constant transition probabilities subject to PDT switching rule. IEEE Trans. Fuzzy Syst. 29(10), 3082–3092 (2021) 3. Yang, F., Gu, Z., Tian, E., Yan, S.: Event-based switching control for networked switched systems under nonperiodic DoS jamming attacks. IET Contr. Theory Appl. 14(19), 3097–3106 (2020) 4. Lian, J., Huang, X., Han, Y.: Observer-based stability of switched system under jamming attack and random packet loss. IET Contr. Theory Appl. 14(9), 1183–1192 (2020) 5. Qu, H., Zhao, J.: Stabilisation of switched linear systems under denial of service. IET Contr. Theory Appl. 14(11), 1438–1444 (2020) 6. Han, Y., Lian, J., Huang, X.: Event-triggered H∞ control of networked switched systems subject to denial-of-service attacks. Nonlinear Anal.-Hybrid Syst. 38, 100930 (2020)

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7. Qi, Y., Xing, N., Fu, J., Guan, W.: Adaptive dynamic optimal control for triggered networked switched systems under dual-ended denial-of-service attacks. Int. J. Robust Nonlinear Control 31(9), 4397–4415 (2021) 8. Xiao, X., Zhou, L., Ho, D.W.C., Lu, G.: Event-triggered control of continuous-time switched linear systems. IEEE Trans. Autom. Control 64(4), 1710–1717 (2019) 9. Li, F., Fu, J., Du, D.: An improved event-triggered communication mechanism and L∞ control co-design for network control systems. Inf. Sci. 370–371, 743–762 (2016) 10. Li, L., Song, L., Li, T., Fu, J.: Event-triggered output regulation for networked flight control system based on an asynchronous switched system approach. IEEE Trans. Syst. Man Cybern. -Syst. 51(12), 7675–7684 (2021) 11. Li, L., Fu, J., Zhang, Y., Chai, T., Song, L., Albertos, P.: Output regulation for networked switched systems with alternate event-triggered control under transmission delays and packet losses. Automatica 131, 109716 (2021) 12. Li, L., Zhang, Y., Li, T.: Memory-based event-triggered output regulation for networked switched systems with unstable switching dynamics. IEEE T. Cybern., early access (2021). https://doi.org/10.1109/TCYB.2021.3081927 13. Cao, J., Ding, D., Liu, J., Tian, E., Hu, S., Xie, X.: Hybrid-triggered-based security controller design for networked control system under multiple cyber attacks. Inf. Sci. 548, 69–84 (2021) 14. Qi, Y., Zhao, X., Huang, J.: H∞ filtering for switched systems subject to stochastic cyber attacks: a double adaptive storage event-triggering communication. Appl. Math. Comput. 394, 125789 (2021) 15. Qi, Y., Tang, Y., Ke, Z., Liu, Y., Xu, X., Yuan, S.: Dual-terminal decentralized event-triggered control for switched systems with cyber attacks and quantization. ISA Trans. 110, 15–27 (2021) 16. Liang, D., Huang, J.: Robust output regulation of linear systems by event-triggered dynamic output feedback control. IEEE Trans. Autom. Control 66(5), 2415–2422 (2021) 17. Liu, J., Yang, M., Tian, E., Cao, J., Fei, S.: Event-based security control for state-dependent uncertain systems under hybrid-attacks and its application to electronic circuits. IEEE Trans. Circuits Syst. I-Regul. Pap. 66(12), 4817–4828 (2019) 18. Fridman, E.: A refined input delay approach to sampled-data control. Automatica 46(2), 421– 427 (2010) 19. Wang, Y., Karimi, H.R., Wu, D.: Conditions for the stability of switched systems containing unstable subsystems. IEEE Trans. Circuits Syst. II-Express Briefs 66(4), 617–621 (2019) 20. Xiang, W., Xiao, J.: Stabilization of switched continuous-time systems with all modes unstable via dwell time switching. Automatica 50(3), 940–945 (2014) 21. Fu, J., Ma, R., Chai, T., Hu, Z.: Dwell-time-based standard H∞ control of switched systems without requiring internal stability of subsystems. IEEE Trans. Autom. Control 64(7), 3019– 3025 (2019) 22. Liu, L., Liu, Y., Li, D., Tong, S., Wang, Z.: Barrier Lyapunov function-based adaptive fuzzy FTC for switched systems and its applications to resistance–inductance–capacitance circuit system. IEEE T. Cybern. 50(8), 3491–3502 (2020) 23. Park, P.G., Ko, J.W., Jeong, C.: Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47(1), 235–238 (2011) 24. Chen, Z., Huang, J.: Attitude tracking and disturbance rejection of rigid spacecraft by adaptive control. IEEE Trans. Autom. Control 54(3), 600–605 (2009)

Chapter 7

Dissipative Event-Triggered Output Regulation for Networked Switched Systems with Severely Unstable Dynamics Against Integrity Deception Attacks

This chapter investigates the resilient EORP with dissipativity subject to deception attacks for NSSs with SUDs which includes the unsolvable OR of each subsystem and some destabilizing switchings and implies that the Lyapunov function rises in activated intervals of all subsystems and at partial switching instants. First, an IDA is delineated for the first time in NSSs for the tampered switching signal, system output, and control, where the consecutive asynchronous switching caused by the deceived switching signal is permitted by modifying inequality conditions at switching instants, which means switching from one asynchronous case to another at a switching instant. Second, two resilient ETMs for S2C and C2A channels are devised in conjunction with dissipative parameters in triggering thresholds to achieve the balance between limited network resources and system performance against destabilizing switching and IDAs. Third, the enhanced DT constraint for switching signal collected attack parameters not only overcomes the conventional arrangement of destabilizing and stabilizing switchings but is also more appropriate for the NSSs impacted by IDAs. Furthermore, the solvability condition of EORP with dissipativity is deduced for NSSs with SUDs subject to IDAs, network-induced delays, random packet losses, and packet disorders via designing a DOFC. Finally, F-18 aircraft model is used to demonstrate the feasibility of the proposed methods.

7.1 Introduction NSSs play an important role in NCSs, which have been applied in a variety of practical scenarios such as unmanned marine and vehicles F-18 aircraft [1, 2]. The emergence of networks will give rise to network-induced phenomena as well as threats of cyber attacks. Since switching behaviors have to be arranged in the absence of real system information, it is more difficult for NSSs to achieve security control under deception attacks than conventional NCSs, which has inspired several remarkable research © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Li and J. Fu, Output Regulation and Cybersecurity of Networked Switched Systems, Studies in Systems, Decision and Control 475, https://doi.org/10.1007/978-3-031-30972-4_7

147

148

7 Dissipative Event-Triggered Output Regulation for Networked …

recently [3–7]. Taking one-sided deception attacks into account, [3–5] study the event-triggered control for NSSs via synchronous switching. With two-sided deception attacks, security control for NSSs is implied under synchronous switching in [6]. Reference [7] characterizes an event-triggered control criterion for NSSs via asynchronous switching subject to one-sided data injection attacks. Notably, the current research progresses in security control for NSSs under deception attacks concentrate on two scenarios: The subsystem and controller data (states or outputs) are attacked when transmitted via networks; Switched systems possess admirable stabilization capabilities (all stable subsystems or stabilizing switchings). Nevertheless, to be more in line with the practical situation for NSSs, not only switching signals would inevitably be distorted during transmission via networks suffering from deception attacks, which lead to complicated consecutive asynchronous switching, that is switching from one asynchronous case to another asynchronous case, but also there is a great probability for switched systems to encounter with severely unstable dynamics (SUDs) including all unstable subsystems and partial destabilizing switchings from the application perspective. Consequently, one of our motivations is to figure out how to ensure system performance of NSSs with SUDs against two-sided IDAs, which tamper with all transmitted system data. The ETM, as a criterion for judging the update time of control input, has been witnessed tremendous progress in conserving the limited network bandwidth, alleviating actuator abrasion and lowering energy consumption since proposed for NCSs, which also promotes the emergence of abundant research for NSSs in conjunction with switching behaviors. Multiple triggering conditions (rather than one) are executed collaboratively in some of these ETMs for more effective and reasonable data updates [8–10], which are regarded as resilient ETMs and have recently been utilized in response to cyber attacks [11]. Quite a few unitary ETMs [3–7, 12] are proposed to facilitate the security control for NSSs with competent switching dynamics under cyber attacks. However, the security control synthesis of NSSs based on unitary ETMs is often difficult to fulfill performance requirements due to the adverse effects of both SUDs and cyber attacks. It is thereby non-negligible to explore some well-designed resilient ETMs for eliminating consequential excessive asynchrony and triggering. Meanwhile, the dissipativity, which depicts the relationship between the consumed and supplied energy, has been widely recognized as a less conservative and more advantageous tool concerning system stability and Lyapunov function construction for switched systems [13, 14], which could be naturally involved in the design of ETMs to directly adjust the control performance. Trigger thresholds of ETMs in [15, 16] are associated with dissipative parameters enabling an explicit relevance of the event-triggered control and dissipativity for NCSs with band limits. When it comes to NSSs, it is necessary and reasonable to devise such dissipativity-based ETMs that promote the performance of switching and triggering under attacks. On this account, how to properly develop dissipativity-based ETMs for NSSs with SUDs and attacks deserves an investigation. The OR has been fully investigated from the standpoint of the dissipative or passive frameworks [20–25], with the goal of tracking the desired reference and rejecting the non-vanishing disturbance [17–19]. For non-switched systems, the

7.1 Introduction

149

incremental passivity is used to solve the ORP in [20], and [21] achieves both dissipative performance and OR, where dissipativity guarantees that the energy consumption is less than the energy supply, resulting in better OR performance. When it comes to switched systems without SUDs and networks, [22, 23] develop incremental passivity of switched systems and design incremental passive switched internal model to achieve OR, which allows internal model switches with the plant asynchronously. [24, 25] ensure dissipativity of switched systems rather than stability, which gives greater freedom for Lyapunov functional construction under the impact of external input, then the solvability of ORP is readily ensured based on dissipativity. While these prior works are elegant, one prime concern in NSSs with the presence of SUDs and attacks is the trade-off between OR, security, and communication burden, which is still mostly subtle and unexplored. Therefore, the EORP based on the dissipative property for NSSs with SUDs is desirable yet challenging. Inspired by these facts, this chapter is devoted to solving dissipativity-based EORP of NSSs with SUDs and two-sided networks suffering from IDAs, network-induced delays, random packet losses, and packet disorders. The following four perspectives summarize the primary innovative contributions. (1) An IDA that concurrently tampers with the switching signal, as well as system output and control, is proposed for NSSs, noting that all information transmitted through susceptible networks is inevitably impacted by cyber attacks. For the consecutive asynchronous switching caused by the attacked switching signal, this chapter presents a solution by imposing a modest restriction at asynchronous switching instants. (2) Two resilient ETMs for two-sided networks are devised respectively, which activate the sampling mechanism in switching intervals after destabilizing switching and the event detection mechanism with dissipative parameters in switching intervals after stabilizing switching. Dissipative parameters involved in triggering thresholds establish an intuitive link between the degree of system dissipation and the event-triggering performance. Meanwhile, the resilient feature of these ETMs can effectively overcome the performance deterioration of NSS under SUDs and IDAs with reasonable utilization of limited network resources. (3) To ensure the desired performance of NSSs against IDAs, a modified DT constraint is jointly created with closely integrated attack parameters. Subsequently, the proportion of total stabilizing switching to total destabilizing switching rather than the periodic arrangement is introduced into the suggested DT condition as a compromise with SUDs. (4) For EORP of NSSs, sufficient conditions are deduced by the appropriate co-design of switching signal with DT and ratio conditions, two resilient ETMs based on dissipative property and DOFC taking SUDs, two-sided IDAs, and network-induced phenomena into consideration.

150

7 Dissipative Event-Triggered Output Regulation for Networked …

7.2 Problem Formulation With the architecture illustrated in Fig. 7.1, a switched system with an exosystem, a sampler/sensor, two ETMs, a DOFC, and an actuator constitute the framework of dissipativity-based EORP for NSSs with SUDs under IDAs, where the transmitted data containing the switching signal and the outputs of the switched system and DOFC is intentionally tampered with by an attacker. Each component will be described in detail.

7.2.1 System Model and Dynamic Output Feedback Controller The dynamics of NSSs with SUDs in Fig. 7.1 are depicted by ⎧ ˙ = Aσ (t) x(t) + Bσ (t) u(t) + Cσ (t) w(t), ⎨ x(t) y(t) = Dσ (t) x(t) + E σ (t) w(t), ⎩ w(t) ˙ = Sσ (t) w(t),

(7.1)

where SUDs include the unsolvable OR of each subsystem and some destabilizing switchings and mean the Lyapunov function is rising in activated intervals of all subsystems and at partial switching instants. x ∈ Rx , u ∈ Ru , y ∈ R y and w ∈ Rw are state vector, control input, regulated output and exogenous input. σ (t) : [0, ∞) →M = {1, 2, · · · , m} is the switching signal, its corresponding switching sequence is depicted as T = {tk }k∈N with t0 = 0. N (0, t) = m + n is the number of switchings on [0, t), there exist sequences of stabilizing and destabilizing switching instants T = {tms }m∈N+ , T = {tnd }n∈N+ with T ∪ T = T . i = σ (t) ∈ M, Ai , Bi , Ci ,

Fig. 7.1 Diagram of the EORP with dissipativity for NSSs with SUDs and IDAs

7.2 Problem Formulation

151

Di , E i , Si are constant matrices with compatible dimensions, and all eigenvalues of Si have non-negative real parts [26]. The DOFC is designed as {

x˙c (t) = Acφ(t) xc (t) + Bcφ(t) xc (sh) + Ccφ(t) y˜ (t), u(t) ˆ = Hcφ(t) xc (t),

(7.2)

where xc (t) ∈ Rxc , y˜ (t) ∈ R y˜ , u(t) ˆ ∈ Ruˆ and φ(t) ∈ M are state vector, input, output and switching signal of the controller, sh is the sampling instant, Aci , Bci , Cci , Hci are matrices to be determined with proper dimensions. Furthermore, the following propositions are necessary. Definition 7.1 NSSs (7.1) under DOFC (7.2) are said to be strict (Qi , Si , Ri )δi -dissipative under σ (t), if there exist matrices QiT = Qi , Si , negative definite RiT = Ri of compatible dimensions, a parameter δi > 0 and a nonnegative function ςi (·) with ςi (0) = 0 satisfying. { E{

t

[w T (s)Qi w(s) + 2w T (s)Si y(s) + y T (s)Ri y(s)]ds} + ςi (x0 ) 0 { t w T (s)w(s)ds, ∀t ≥ 0, i ∈ M. ≥ δi

(7.3)

0

Assumption 7.1 There exist matrices ||i , Ei , Aci , Bci and Hci such that. Ai ||i + α u Bi Hci Ei + Ci = ||i Si , Di ||i + E i = 0, Bci Ei e−Si η1M + Aci Ei = Ei Si .

(7.4)

7.2.2 Integrity Deception Attacks For cyber physical systems, [27] reported a queuing type of deception attack in terms of frequency and duration that is closer to attackers’ true intention in analogy with the frequently utilized stochastic form. Here, this form of deception attack will be transferred to the network of NSSs. Different from [27], all data passed across the network, such as switching signal σ (t) and system outputs y(t) in the S2C ˆ in the C2A channel, would be tampered with channel and controller output u(t) during attacking, hence we term it IDA. Define the sequence {tld }l∈N as launched instants of IDAs and dl > 0 as the length of its l-th attack. Assumption 7.2 Energy signals of f iσ (t), f y (t) and f u (t) of data injection for ˆ launched by switching signal σ (t), system output y(t), and controller output u(t)

152

7 Dissipative Event-Triggered Output Regulation for Networked … sampling instants

switching instants of subsystems switching instants of controllers affected by network-induced delays

The activated or ended instants of IDAs IDAs duration intervals

Fig. 7.2 Diagram of switching instants sequence affected by IDAs

IDAs satisfy f iσ (t) ∈ Z with i + f iσ (t) ∈ M, || f y (t)||2 ≤ ||G y y(t)||2 , || f u (t)||2 ≤ ||G u u(t)||2 , where real constant matrices G y and G u are determined by characteristics of IDAs, u(t) = Hci x˜c (t), x˜c (t) = xc (t) − Ei w(t). Assumption 7.3 (Attack Frequency) There exist ω I ≥ 0 and τ I ≥ 0 satisfying / N I (t0I , t1I ) ≤ ω I + (t1I − t0I ) τ I , where N I (t0I ,t1I ) is the number of occurring of IDAs on [t0I , t1I ). Assumption 7.4 (Attack Duration) It is assumed that b ≥ supl∈N {dlI } and I b ≤inf l∈N {tl+1 − tlI − dlI }, where b and b denote the upper bound of activated intervals bound of sleeping intervals of IDAs respectively, / and the lower / bN I (t0I , t1I ) (t1I − t0I )< 1 T, T > 1. Remark 7.1 IDAs will cause two abnormal switching conditions. (1) If an IDA is launched at tld and is ended at tld + dl during [tk+2 , tk+3 ), i.e. tk+2 < tld < tld + dl < tk+3 , then there may exist more than one asynchronous intervals [tk+2 , t˜k+2 ) and k˜ k˜ [t˜k+2 , t k+2 ) on [tk+2 , tk+3 ). (2) If the duration of an IDA dl+1 is greater than the total length of one or adjacent switching intervals, e.g. dl+1 > tk+6 − tk+4 in Fig. 7.2, then the consecutive asynchronous switchings will occur on [tk+4 , tk+6 ) because the controller always receives the modal inconsistent with the currently activated subsystem.

7.2.3 Resilient Event-Triggered Mechanisms For networks in S2C and C2A channels, two resilient ETMs are proposed to adjust the communication policies in time according to the severity of IDAs and SUDs. (1) ETM in S2C channel The resilient ETM in the S2C channel is as follows y

d

y

y y h|[] y (sr,v h) > 0}, εr +1 h = {t l } ∪ min{min{εr +1 h /\ sr,v y

min{˜εr +1 h v∈N

v∈N y y /\ sr,v h|σ (sr,v h)

y /= σ (sr,v h − h)}},

(7.5)

7.2 Problem Formulation

153

d

where t l is the first sampling instant after the l-th attacks,

y [] y (sr,v h) =

⎧ min{n,v} E ⎪ ⎪ y y y ⎪ γι e Ty (sr,v−ι h)uσ (s y h) e y (sr,v−ι h) ⎪ ⎪ r,v ⎨ ι=0

⎪ ⎪ ⎪ ⎪ ⎪ ⎩

y

− δσ (s y

r,v h)

y

y (t)y T (sr,v h)(uσ (s y

r,v h)

y y y − Rσ (sr,v )y(sr,v h), sr,v h ∈ T 1k , h) 1

y y sr,v h − εr,v h,

y sr,v h ∈ T k, 1

with T 1k = [tk , tk+1 ) for tk ∈ T , T k = [tk , tk+1 ) for tk ∈ T , y y y y y y e y (sr,v−ι h) = y(sr,v−ι h)−y(εr h), sr,v−ι h /\ [εr + (v − ι)]h, δσ (s y h) (t) = r,v

y

δσ (s y h) +δσ (s y h) e−θ y ||y(sr,v h)|| , constants θ y , δσ (s y h) , δσ (s y h) are positive, the weighted r,v r,v r,v r,v y matrix uσ (s y h) > 0, and Rσ (s y h) is the dissipative matrix. If n > v and ι /= 0, then r,v r,v E γι =1E − ι−1 ι˜=0 γ˜ι˜ , else γι = γ˜ι˜ with ι = ι˜, where γ˜ι˜ ∈ [0, 1] is a weighted parameter with nι˜=0 γ˜ι˜ = 1, ι, ι˜, v, n ∈ N. y0

y1

y0

y1

(2) ETM in C2A Channel The resilient ETM in the C2A channel is depicted as: d

u u εru+1 h = {t l } ∪ min{min{εru+1 h /\ sr,v h|[]u (sr,v h) > 0},

min{˜εru+1 h v∈N

v∈N u u /\ sr,v h|φ(sr,v h)

u /= φ(sr,v h − h)}},

(7.6)

where ⎧ T u u e (s h)uuφ(sr,v u h) eu (sr,v h) ⎪ ⎪ ⎨ u r,v u u u u 2 u (t)u T (sr,v h)uuφ(sr,v − δφ(s u h) u(sr,v h), sr,v h ∈ T k , u []u (sr,v h) = r,v h) ⎪ ⎪ ⎩ u 2 u u h, sr,v h ∈ T k, sr,v h − εr,v 2

1

with T 2k = [t˜k , t˜k+1 ) for t˜k ∈ T 1k , T k = [t˜k , t˜k+1 ) for t˜k ∈ T k , t˜k is the u u u h)−u(εru h), sr,v h /\ (εru + v)h, h) = u(sr,v controller switching instant, eu (sr,v u u u1 u0 u0 u1 , constants δφ(s , δφ(s , θu are positive, δφ(s (t) =δφ(s e−θu ||u(sr,v h)|| + δφ(s u u u u u r,v h) r,v h) r,v h) r,v h) r,v h) u and the weighted matrix uφ(sr,v u h) > 0. Remark 7.2 Resilient ETMs (7.5) and (7.6) are composed of three main parts to realize the timely and dynamic adjustment of communication policies. The first part d {t l } ensures data transmission at the first sampling instant after each IDA, which can alleviate the consecutive asynchronous switching induced by the deceived switching signal in time. The second parts of ETMs (7.5) and (7.6) are divided into two cases y u h ∈ T 2k , that is to according to the severity of SUDs: (1) When sr,v h ∈ T 1k and sr,v y u h say, sr,v h belongs to [tk , tk+1 ) after stabilizing switching instant tk ∈ T and sr,v ˜ ˜ is within the corresponding controller working interval [tk , tk+1 ), event detections

154

7 Dissipative Event-Triggered Output Regulation for Networked … y

u [] y (sr,v h) > 0 and []u (sr,v h) > 0 take on the task of dynamically adjusting triggered y y u h) with the help of time-varying thresholds δσ (s y h) (t) rate in light of y(sr,v h) and u(sr,v r,v

and

u δφ(s (t). u r,v h)

(2) When

y sr,v h

∈

1 Tk

2 y u and sr,v h ∈T k , sr,v h belongs to [tk , tk+1 ) after u T and sr,v h is within the corresponding controller y u [] y (sr,v h) > 0 and []u (sr,v h) > 0 are preserved

destabilizing switching instant tk ∈ interval [t˜k , t˜k+1 ), event detections as periodic sampling conditions, quickly improving system performance after destay y u u h)/= φ(sr,v h − h) bilizing switching. The last part σ (sr,v h) /=σ (sr,v h − h) or φ(sr,v specifies that a trigger must be occurred once at the first sampling instant after each switching instant of subsystem or controller, shortening the length of asynchronous intervals governed by the second part of (7.5) or (7.6). In addition, the Zeno phenomenon can be naturally excluded since triggers are only generated at sampling instants. Remark 7.3 Dissipativity, which refers to the fact that the consumption of system energy is less than the supply, presents a more general interpretation for the system stability, thus making dissipative parameters of the system closely related to stability. The dissipative parameter Rσ (s y h) in ETM (7.5) is dedicated to maintaining the good r,v performance of NSSs based on event-triggered in bandwidth-constrained Emin{n,v} T control y y y γ˜ι e y (sr,v−ι h)uσ (s y h) e y (sr,v−ι h) in (7.5) networks. The cumulative errors ι=0 r,v overcome undesirable releases owing to the large oscillation of SUDs. It is worth mentioning that the synthesis of the second and third parts in (7.5) makes all historical information used by (7.5) belong to the currently activated subsystem, i.e. y y y [εr h, sr,v h]⊆ [tk , sr,v h], which ensures the effectiveness of memory-based ETM (7.5) for NSSs.

7.2.4 Dynamic Output Feedback Controller Based on Resilient Event-Triggered Mechanisms Affected by the network-induced delay τrsc (τrca ) in S2C (C2A) channel, the data y released at the triggering instant εr h (εru h) in ETM (7.5) (ETM (7.6)) will reach the y sc sc , controller (actuator) at εr h + τr (εru h + τrca ). It is assumed that τmsc ≤ τrsc ≤ τ M ca sc ca ca ca sc sc sc ca ca τm ≤ τr ≤ τ M with τm = min{τr }, τ M = max{τr }, τm = min{τr }, τ M = r ∈N

r ∈N

r ∈N

sc ca max{τrca }. Define τr = τrsc + τrca , τm = τmsc +τmca , τ M = τ M + τM , then it holds that r ∈N

y

y

τm ≤ τr ≤ τ M . Thus, the interval |r = [εr h + τrsc ,εr +1 h + τrsc+1 ) can be divided into r |r = ∪Lv=0 |r,v , where { |r,v =

y

y [sr,v h + τrsc , sr,v+1 h + τrsc ), v = 0, . . . , Lr − 1, y

y

[sr,Lr h + τrsc , εr +1 h + τrsc+1 ), v = Lr ,

7.2 Problem Formulation

155

y

y

with Lr = min{v|sr,v+1 h + τrsc ≥ εr +1 h + τrsc+1 }. In light of the input delay method v∈N

y

y

[28], we define η(t) = t − sr,v h, t ∈ |r,v , η(t) = t − εr h, t ∈ |r , and η(t) ˜ = t − εru h, t ∈ |r with εru h = max{εru h|εru h + τrca ≤ t} which yields ηm ≤η(t)≤ η M , r ∈N

sc ηm ≤ η(t) ≤ η M , ηm ≤ η(t) ˜ ≤ η˜ M with ηm = τmsc , η M =τ M +h, η M = τmsc +h+Tmax , y y η˜ M = τmsc +Tmax . Define e y (t) =y(sr,v h)−y(εr h), t ∈ |r,v . Then, the DOFC under ETM (5) and IDAs is

{

x˙c (t) = Acφ(t) xc (t) + Bcφ(t) xc (t − η(t)) + Ccφ(t) y˜ (t), u(t) ˆ = Hcφ(t) xc (t), t ∈ |r ,

where y˜ (t) = α y (t)y(εr h) = α y (t)[λ y (t)(y(t − η(t)) − e y (t)) + (1 − λ y (t))y(t − η(t))+γ (t) f y (t)], t ∈ |r,v . It is assumed that disordered packets are discarded. The Bernoulli-distributed variable α y (t) with E{α y (t)}= α y depicts the packet losses in the S2C channel. If tk ∈ T , then λ y (t) = 1, else λ y (t) = 0. And if IDAs happen, then γ (t) = 1, otherwise γ (t) = 0.

7.2.5 The Closed-Loop Networked Switched System Based on ETM (7.6), the control input of NSSs (7.1) under attacks and random packet losses can be expressed as u(t) = α u (t)u(εru h) = α u (t)[λu (t)(u(t − η(t)) − eu (t)) + (1 − λu (t))u(t − η(t) ˜ + γ (t) f u (t)],

(7.7)

u h) − u(εru h), t ∈ |r,v , the Bernoulli-distributed variable α u (t) where eu (t) =u(sr,v with E{α u (t)}= α u depicts the packet losses in the C2A channel. For t ∈ T 2k = 1 [t˜k , t˜k+1 ), λu (t) = 1 if t˜k ∈ T 1k , λu (t) = 0 if t˜k ∈ T k . Affected by IDAs, the closed-loop systems of NSSs (7.1) under control (7.7) can be divided into the following three situations. (1) When the subsystem is not violated by IDAs within [tk , tk+1 ), the closed-loop system is as follows

{

χ(t) ˙ = A˜ i χ (t) + B˜ i χ (t − η(t)) + /\i1j + /\i2j ,

{

y(t) = D˜ i χ (t), t ∈ [tk , t˜k ); χ˙ (t) = A˜ i χ (t) + B˜ i χ (t − η(t)) + /\i1 + /\i2 , y(t) = D˜ i χ (t), t ∈ [t˜k , tk+1 );

(7.8.1)

(7.8.2)

(2) When the subsystem encounters IDAs within [tk , tk+1 ) all the time, the closedloop system is as follows

156

7 Dissipative Event-Triggered Output Regulation for Networked …

{

{

1 2 + /\iq , χ(t) ˙ = A˜ iq χ (t) + B˜ iq χ (t − η(t)) + /\iq

y(t) = D˜ i χ (t), t ∈ [tk , t˜k ); χ˙ (t) = A˜ i p χ (t) + B˜ i p χ (t − η(t)) + /\i1p + /\i2p , y(t) = D˜ i χ (t), t ∈ [t˜k , tk+1 );

(7.8.3)

(7.8.4)

(3) When the subsystem encounters more than once IDAs within [tk , tk+1 ), the closed-loop system is as follows {

{

1 2 + /\iq , χ(t) ˙ = A˜ iq χ (t) + B˜ iq χ (t − η(t)) + /\iq

y(t) = D˜ i χ (t), t ∈ [tk , t˜k ); χ˙ (t) = A˜ i p χ (t) + B˜ i p χ (t − η(t)) + /\i1p + /\i2p , ˜ y(t) = D˜ i χ (t), t ∈ |˜ kk ;

⎧ ⎨ χ(t) ˙ = A˜ i χ (t) + B˜ i χ (t − η(t)) + /\i1 + /\i2 , ⎩ y(t) = D˜ χ (t), t ∈ [t˜ , t˜0 ) or | k˜ or [t n d −1 , t ), i k k k+1 k k

(7.8.5)

(7.8.6)

(7.8.7)

| | ˜ where the initial condition is χ (t) = ϕ(t) for t ∈ −η M , 0 , t˜kk is the controller k˜

switching instant after IDA occurs, t k is the controller switching instant after IDA ˜ ˜ k˜ ends, n d is the number of occurrences of IDAs on [tk , tk+1 ), |˜ kk = [t˜kk , t k ) (k˜ ∈ ˜

k k˜ ˜ {0, 1, . . . , n d − 1}), | k = [t k , t˜kk+1 ) (k˜ ∈ {0, 1, · · · , n d − 2}), ϑ ∈ {i, i j, i p, iq}, ~

~

ϑ ∈ {i, j, p, q}, i /= j, i /= p, i /= q, (ϑ, ϑ) ∈ {(i, i), (i j, j ), (i p, p), (iq, q)}, |

| | | | | Ai 0 0 0 0 Bi H ~ ˜ ˜ cϑ , , Bϑ = , Bcϑ = 0B~ 0 A~ 0 0 cϑ cϑ | | | | | | 0 0 0 −Bi , C˜ cϑ = Bi = , D˜ i = [ Di 0 ], , C cϑ = C ~ Di 0 −C ~ 0 A˜ ϑ =

cϑ

cϑ

/\1ϑ = α y (t)[λ y (t)(C˜ cϑ χ (t − η(t)) + C cϑ e y (t)) + (1 − λ y (t))(C˜ cϑ χ (t − η(t)) − γ (t)C cϑ f y (t)], /\2ϑ = α u (t)[λu (t)( B˜ cϑ χ (t − η(t)) + B i eu (t)) + (1 − λu (t))( B˜ cϑ χ (t − η(t)) ˜ − γ (t)B i f u (t)], i, j, p, q ∈ M.

7.3 Main Results

157

7.2.6 Event-Triggered Output Regulation Problem with Dissipative Property We aim to solve EORP with (Qi , Si , Ri )-δi -dissipative property for NSSs with SUDs against IDAs, network-induced delays, random packet losses, and packet disorders by co-designing the DOFC, ETMs (7.5)–(7.6) and a switching signal with abnormal switching characteristics caused by IDAs, that is: (a) The closed-loop system (7.8.1)– (7.8.7) with w(t) = 0 is mean-square exponentially stable. (b) The solution of (7.8.1)–(7.8.7) with w(t) /= 0 satisfies lim E{y(t)} = 0. (c) The NSS (7.8.1)–(7.8.7) t→∞

is strict (Qi , Si , Ri )-δi -dissipative, i.e. (7.3) holds.

7.3 Main Results This section provides a joint design scheme of DOFC based on ETMs (7.5)–(7.6), a switching signal under improved DT constraints, and sufficient conditions for the solvability of the ORP with the help of the (Qi , Si , Ri )-δi -dissipative framework for NSSs with SUD against IDAs and network-induced phenomena. Theorem 7.1 Under Assumptions 7.1–7.4, positive constants y u λ, κ, λ, h, ηm , η M , η M , η˜ M , n, θ y , θu , 0 < δ i , δ i ,γ˜ι˜ < 1, α y , α u , δi and matrices G y , G u are given. The dissipativity-based EORP for NSSs (7.1) under DOFC (7.2) with ETMs (7.5)–(7.6) is achieved if the following conditions hold: ς

ς

y

(i) There exist matrices Pϑ > 0, Uϑ > 0, Rϑ > 0, ui > 0, uiu > 0, Ri < 0 and ς˜ ς ς Ccϑ , Qi , Si , Mϑ ,Ti j , Ji j , J˜i j with ϑ ∈ {i, i j, i p, iq}, ϑ ∈ {i j, j p, i p}, i /= j, i /= p, i /= q, i, j, p, q∈ M, ι˜ ∈ {0,· · · , n}, ς ∈ {0, 1, 2, 3}, ς˜ ∈ {1, 2, 3} such that | | ς˜ ς˜ R ϑ Mϑ ψ ϑ < 0, []ϑ < 0, []0ϑ < 0, []1ϑ < 0, > 0. (7.9) ς˜ ∗ Rϑ ˜ ϑ˜ ∈{ρ˜ϑ˜ , μ˜ ϑ˜ }, (ii) Given ρϑ > 1, ρ˜ϑ˜ > 1, 0 < μϑ , μ˜ ϑ˜ < 1, wϑ ∈ {ρϑ , μϑ }, w ϑ˜ ∈ {iq, i p}, p, q ∈ M, it yields that ς

ς

/\ P j < 0, /\U j < 0, /\ R j < 0,

(7.10)

[ Pi Uiς Riς ] 0, c > 0 such that ||x(t) − ||i w(t)||+||ξ(t)−w(t)|| ≤ W e−ct (||x(0) − ||i w(0)|| + ||ξ(0) − Ei w(0)||). Thus lim (||x(t) − ||i w(t)|| + ||ξ(t) − Ei w(t)||) = 0, which implies lim χ(t) = t→∞

0, then lim y(t) = D˜ i χ(t) = 0.

t→∞

t→∞

Remark 7.4 Theorem 7.1 considers deception attacks capable of tampering with all data transmitted through networks, which we call it IDAs. For NSSs in Theorem 7.1, IDAs will result in anomalous switching during attacks as indicated in Remark 7.1, this is a noticeable contrast from the existing literature, e.g., [3–7], which does not include the tampering with the switching signal. Theorem 7.1 addresses these anomalous switching issues by imposing (7.12)–(7.15) at asynchronous switching instants. Remark 7.5 Theorem 7.1 addresses for the first time the security control of NSSs with SUDs in which the switching signals are tampered with compared to the existing literature such as [3–7]. The interaction of SUDs and IDAs will cause substantial

7.3 Main Results

167

impairment to the system’s performance. Therefore, we conduct an appropriate joint design of a modified switching law and resilient ETMs to counteract these impacts. Specifically, for SUDs, the second parts of ETMs (7.5)–(7.6) enable system performance to adjust rapidly after destabilizing switching, and the DT switching law adds a proportion condition (7.16) to suppress the divergence of system dynamics due to SUDs by limiting the proportion of stabilizing and destabilizing switchings. For IDAs, the first parts of ETMs (7.5)–(7.6) guarantee a trigger occurs after the end of each IDA, and the activation time of each subsystem is calculated by the DT condition (7.15) in the switching law and adjusted according to IDAs parameters. Remark 7.6 Different from OR with dissipative property in [24, 25], SUDs and IDAs are not neglected. Remark 7.7 Compared with the earlier research on switched systems with all unstable subsystems or/and partial destabilizing switchings [29–31], asynchronous switching is addressed in this chapter. Different from [29], Theorem 7.1 breaks the limitation on the regular arrangement of destabilizing and stabilizing switchings owing to (7.16), and it is not required to assume that only certain subsystems are activated during switching intervals after destabilizing switching. Thanks to different coordinate transformations for EORP of NSSs, there is no need to apply discretized Lyapunov functions like [29–31]. Moreover, the DT (7.15) in Theorem 7.1 comprises the parameters of attacks, which is more suited for NSSs under IDAs than the DT condition in [32]. Next theorem handles nonlinear terms in Theorem 7.1 and provides LMI conditions. Theorem 7.2 Under Assumptions 7.1–7.4, positive constants λ, κ, λ, h, ηm ,η M , η M , y u η˜ M , n, θ y , θu , 0 0, Rϑ > 0, ui > 0, uiu > 0, Ri < 0 and Ccϑ , Qi , Si , ς˜ ς ς Mϑ , Ti j , Ji j , J˜i j with ϑ ∈ {i, i j, i p, iq}, ϑ ∈ {i j, j p, i p}, i /= j, i /= p, i /= q, i, j, p, q ∈ M, ι˜ ∈ {0, . . . , n}, ς ∈ {0, 1, 2, 3}, ς˜ ∈ {1, 2, 3} fulfilling (ii)–(iii) and. | ψϑ < 0, []ϑ < 0, []0i j < 0, []1i j < 0,

ς˜

ς˜

R ϑ Mϑ ς˜ ∗ Rϑ

| > 0,

(7.46)

where the 18 × 18 matrix ψϑ = {ψϑ } consists of matrices p q4

ψϑ 4

p4 q4

= ψϑ

for p4 ∈ {1, . . . , 12}, q4 ∈ {2, . . . , 5},

1,14 ψϑ11 = /\ϑ , ψϑ1,13 = G˜ uT = G˜ ϑ , ϑ , ψϑ yT

y y T ψϑ66 = Mϑ1 + Mϑ1T − 2Rϑ1 + δ i D˜ iT (ui − Ri ) D˜ i , ψϑ6,15 = B˜ cϑ , ψϑ6,16 = C˜ ciT ,

168

7 Dissipative Event-Triggered Output Regulation for Networked …

/ T 2 u y ψϑ6,17 = H˜ ϑT , ψϑ10,10 = −γ0 ui , ψϑ10,18 = C cϑ , ψϑ16,16 = λ δ ~ uu~ − 2λI, ϑ

ϑ

2

ψϑ13,13 = λ [α u Pϑ + (4α u + 3α y α u )Rϑ ] − 2λI, 2

ψϑ14,14 = λ [α y Pϑ + (4α y + 3α y α u )Rϑ ] − 2λI, 2

ψϑ15,15 = λ [(5α u + 2α y α u )Pϑ + (2α u + α y α u )Rϑ ] − 2λI, 2

ψϑ17,17 = λ [(5α y + 2α y α u )Pϑ + (α y + α y α u )Rϑ ] − 2λI, 2

ψϑ18,18 = λ [α y Pϑ + (5α y + 3α y α u )Rϑ ] − 2λI ; the 13 × 13 matrix []ϑ = {θϑ } consists of matrices p q5

θϑ 5

p5 q5

= θϑ

for p5 ∈ {1, . . . , 5}, q5 ∈ {2, . . . , 9},

θϑ11 = A˜ ϑT Pϑ + Pϑ A˜ ϑ + a1 Pϑ + Uϑ0 − Rϑ0 − D˜ iT Ri D˜ i + (1 + a1 ) A˜ ϑT Rϑ A˜ ϑ , 1,11 T θϑ1,10 = G˜ uT = G˜ ϑ , θϑ66 = θ ϑ , θϑ77 = ψ ϑ , θϑ7,12 = C˜ cϑ , θϑ88 = ψ ϑ , ϑ , θϑ yT

66

77

88

99 2 T θϑ8,13 = B˜ cϑ , θϑ99 = θ ϑ , θϑ10,10 = θϑ13,13 = λ [α u Pϑ + (4α u + 2α u α y )Rϑ ] − 2λI, 2

θϑ11,11 = λ [α y Pϑ + (4α y + 2α y α u )Rϑ ] − 2λI, 2

θϑ12,12 = λ [α y Pϑ + (4α u + 2α y α u )Rϑ ] − 2λI ; the 15 × 15 matrix []0ϑ = {θ0ϑ } consists of matrices p q6

θ0ϑ6

p q6

= ψϑ 6

for p6 ∈ {2, . . . , 5}, q6 ∈ {1, . . . , 10},

11 θ0ϑ = A˜ ϑT Pϑ + Pϑ A˜ ϑ + a1 Pϑ + Uϑ0 − Rϑ0 − D˜ iT Ri D˜ i + (1 + a1 ) A˜ ϑT Rϑ A˜ ϑ , 1,11 66 θ0ϑ = G˜ uT , θ0ϑ = Mϑ1 + Mϑ1T − 2Rϑ1 + (2α + β) B˜ ϑT Rϑ B˜ ϑ , ϑ 77 yT 1,12 6,13 6,14 7,15 69 T 77 T θ0ϑ = G˜ ϑ , θ0ϑ = ψϑ69 , θ0ϑ = B˜ cϑ , θ0ϑ = H˜ ϑT , θ0ϑ = ψ ϑ , θ0ϑ = C˜ cϑ ,

7.3 Main Results

169 88

99

12,12

10,10 88 99 θ0ϑ = ψ ϑ , θ0ϑ = θ 0ϑ , θ0ϑ = ψϑ

14,14 , θ0ϑ =λ

2

/

u

δ ~ uu~ − 2λI, ϑ

ϑ

2

11,11 13,13 θ0ϑ = θ0ϑ = λ [α u Pϑ + (4α u + 2α u α y )Rϑ ] − 2λI, 2

12,12 15,15 θ0ϑ = θ0ϑ = λ [α y Pϑ + (4α y + 2α u α y )Rϑ ] − 2λI ;

the 16 × 16 matrix []1ϑ = {θ1ϑ } consists of matrices p q7

θ1ϑ7

p q7

= ψϑ 7

for p7 ∈ {1, . . . , 5}, q7 ∈ {2, . . . , 10},

11 θ1ϑ = A˜ ϑT Pϑ + Pϑ A˜ ϑ + a1 Pϑ + Uϑ0 − Rϑ0 − D˜ iT R D˜ i + (1 + a1 ) A˜ ϑT Rϑ A˜ ϑ , y y 66 θ1ϑ = Mϑ1 + Mϑ1T − 2Rϑ1 + δ i D˜ iT (ui − Ri ) D˜ i + (1 + a1 ) B˜ ϑT Rϑ B˜ ϑ , 1,12 1,13 6,14 T 77 88 θ1ϑ = G˜ uT , θ1ϑ = G˜ ϑ , θ1ϑ = C˜ cϑ , θ1ϑ = ψ ϑ , θ1ϑ = ψϑ , ϑ yT

77

88

T 10,10 y 8,15 9,16 10,10 T 99 θ1ϑ = B˜ cϑ , θ1ϑ = −γ0 ui , θ1ϑ = C cϑ , θ1ϑ = θ 1ϑ , 2

12,12 15,15 θ0ϑ = θ0ϑ = λ [α u Pϑ + (4α u + 2α u α y )Rϑ ] − 2λI, 2

13,13 14,14 θ0ϑ = θ0ϑ = λ [α y Pϑ + (4α y + 2α u α y )Rϑ ] − 2λI, 2

16,16 θ0ϑ = λ [α y Pϑ + (5α y + α u + α u α y )Rϑ ] − 2λI.

Proof Utilizing Lemma 1.7 with. −[(5α y + 2α y α u )Pi j + (α y + α y α u )Ri j ]−1 2 < λ [(5α y + 2α y α u )Pi j + (α y + α y α u )Ri j ] − 2λI from [12], it leads to ψi j < 0 from ψ i j < 0. Similarly, one can obtain []0i j < 0 and []1i j < 0 from []0i j < 0 and []1i j < 0. Then, (7.22) can be deduced in Case A via a similar procedure in Theorem 7.1. In Case B, C and D of Theorem 7.1, similar to the proof of Case A, one can get ψi < 0, ψi j < 0, ψi p < 0 and ψiq < 0 from ψ i < 0, ψ i j < 0, ψ i p < 0 and ψ iq < 0. Therefore, if (7.10)–(7.17) and (7.46) hold, similar to Theorem 7.1, the EORP with dissipativity for NSSs (7.8.1)–(7.8.7) is ensured.

170

7 Dissipative Event-Triggered Output Regulation for Networked …

7.4 Simulation Example The longitudinal short-period motion of F-18 aircraft can be regarded as switching between models corresponding to each operation point. Therefore, the F-18 aircraft can be modeled as NSSs (7.1). We choose three operating points with different fight states, where Mach numbers are 0.5, 0.6, 0.7, and altitudes are 30kft, 40kft, 14kft. Some system matrices are listed below |

| | | | | −0.5088 0.994 −0.2423 0.9964 −1.175 0.9871 , A2 = , A3 = , −1.131 −0.2804 −2.342 −0.1732 −8.458 −0.8776 | | | | | | −0.0927 −0.0178 −0.0416 −0.0114 −0.194 −0.0359 B1 = , B2 = , B3 = −6.573 −1.525 −2.595 −0.8161 −19.29 −3.803

A1 =

are selected from [33]. Other system matrices for EORP of NSSs are preset accordingly |

C1 =

| | | | | 0.1853 0.8124 0.3105 1.5031 0.3564 1.661 , C2 = , C3 = , 1.5167 0.3232 1.9321 0.9854 1.2482 0.6471 | | | | | | 0.54 0.89 0.63 0.41 0.14 0.89 E1 = , E2 = , E3 = , 0.46 0.36 0.49 0.23 0.35 0.74 | | | | | | 0.21 −3 0.2 −3.7 −0.11 −4.2 S1 = , S2 = , S3 = . 0.11 0.21 0.24 0.12 0.13 0.24

Then the following controller gains can readily be verified to fulfill Assumption 7.1 | | | | | | −3.33 −2.24 −2.23 −0.31 −2.33 −3.21 Ac1 = , Ac2 = , Ac3 = −1.94 −3.16 −0.64 −1.64 −3.09 −3.58 | | | | | | 2.155 −3.612 −1.234 1.564 −2.031 2.413 Bc1 = , Bc2 = , Bc3 = . 2.242 −1.213 2.248 −1.846 3.562 −2.889 The IDAs signals are selected as f y (t) = tanh(0.14y(t)), f u (t) = tanh(0.05u(t)), then it is clear that G y = 0.14 and G u = 0.05. In light of Theorem 7.2, choose parameters as λ = 1.15, κ = 1.1, λ = 0.9, h = 0.1, n = 1, ηm = 0.01, η M = 0.5, η M = 1.5, η˜ M = 1.4, θu = θ y = 0.5, μi = 0.9, μi j = 0.1, y y μ˜ i j = 0.8, ρˆi = 1.8, ρˆi j = 2, ρ˜i j = 1.8, i, j ∈ {1, 2, 3}, δ 1 = 0.015, δ 2 = 0.01, u u δ 1 = 0.022, δ 2 = 0.02, γ0 = 0.9, γ1 = 0.1, α y = 0.9, α u = 0.8, τ D = 5. Solving (7.10)–(7.17), one can obtain 0.6 ≤ Tmin , Tmax ≤ 0.8035, N ∗ = 1.7976 and controller gains

7.4 Simulation Example

171

|

Cc1

| | | | | −0.537 0.459 0.787 0.554 −0.342 −0.457 = , Cc2 = , Cc3 = . 0.672 0.434 0.335 −0.177 0.993 0.726

Figure 7.3 shows the random packet losses in S2C and C2A channels, IDAs, and the switching signal with Tmin = Tmax = 0.8s. Figure 7.4 reflects the networkinduced delays and inter-event intervals in two channels affected by IDAs in Fig. 7.3. The switching signals and asynchronous intervals in the two channels are shown in Fig. 7.5. One can observe that consecutive asynchronous switchings caused by IDAs occur during 2 s ~ 3.1 s, 10 s ~ 13.5 s, and there exist two asynchronous switching intervals on 16.8 s ~ 17.4 s due to IDAs. If the switching interval is not affected by IDAs and packet losses, then the length of asynchronous intervals will be less than the maximum network-induced delay, which demonstrates that ETMs (7.5)–(7.6) are effective to shorten the asynchronization duration caused by ETMs. Under the asynchronous switching signal shown in Fig. 7.5, the state responses, the regulated output of the closed-loop system (7.8.1)–(7.8.7) and the evolution of the Lyapunov function (7.46) and (7.23) are presented in Fig. 7.6 with the initial state χ (0) = [531.8 1.6]T , which illustrates the solvability of dissipativity-based is destabilizing when the 3-th EORP for NSSs (7.1) with SUDs, and the switching / subsystem is activated. It can be found that m n ≈ 3.625 > N ∗ and the arrangement of stabilizing and destabilizing switchings is irregular. In what follows, we will implement two comparisons under non-resilient and resilient ETMs in the presence or absence of dissipative parameters respectively

Fig. 7.3 The packet losses in S2C and C2A channels, IDAs and switching signal

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7 Dissipative Event-Triggered Output Regulation for Networked …

Fig. 7.4 Network-induced delays, inter-event intervals in S2C and C2A channels

Fig. 7.5 The switching signal in S2C and C2A channels, and asynchronous intervals

7.4 Simulation Example

173

Fig. 7.6 The state responses and the dissipativity-based regulated output of the closed-loop system and Evolution of the Lyapunov function

to verify that the proposed resilient ETMs improve the system performance. First, Figs. 7.7 and 7.8 present the comparisons of inter-event intervals and closed-loop state responses with or without resilient ETM in the presence of dissipative parameters. From Fig. 7.7, one can see that the resilient ETM is preserved as periodic sampling during 0.6 s ~ 1.2 s, 3.6 s ~ 4.2 s, 6 s ~ 6.6 s, 13.8 s ~ 14.4 s, 16.8 s ~ 17.4 s, 18.6 s ~ 19.2 s, that is switching intervals after destabilizing switching. Comparing the two subgraphs in Fig. 7.8, we can obtain that state responses of the closed-loop system converge faster under the resilient ETM. Then, Figs. 7.9 and 7.10 show the comparison under resilient ETMs with or without dissipative parameters. Figure 7.9 reveals that resilient ETM with dissipative parameters leads to fewer transmissions, and Fig. 7.10 indicates better system performance is ensured under resilient ETM with dissipative parameters. In summary, dissipative parameters in ETM are very helpful in improving system performance.

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7 Dissipative Event-Triggered Output Regulation for Networked …

Fig. 7.7 Inter-event intervals with and without resilient ETMs in the presence of dissipative parameters in S2C and C2A channels

Fig. 7.8 The state responses of the closed-loop system with and without resilient ETMs in the presence of dissipative parameters

7.4 Simulation Example

175

Fig. 7.9 Inter-event intervals under resilient ETMs with and without dissipative parameters in S2C and C2A channels

Fig. 7.10 The state responses of closed-loop system under resilient ETMs with and without dissipative parameters

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7 Dissipative Event-Triggered Output Regulation for Networked …

7.5 Conclusion Based on the dissipative theory, this chapter addresses the EORP for NSSs with SUDs in the presence of IDAs and network-induced delays, packet losses, and packet disorders in two-channel networks. Compared with the conventional deception attacks, this chapter analyzes the circumstance when the switching signal is tampered with by IDAs, and deals with the consecutive asynchronous controller switching issue by adding inequality requirements at switching instants. The resilient ETMs are meant to fully conserve network resources while maintaining system performance against IDAs and destabilizing switchings by adjusting triggering rate via dynamic thresholds, specifically, the ETM in the S2C channel contains dissipative parameters. Novel DT criteria are proposed, including parameters of IDAs and the ratio of total destabilizing switchings to total stabilizing switchings, to adapt to NSSs suffering from IDAs and relax the strict restrictions on switchings. Future study will be extended to the field of distributed switched sensor networks.

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Chapter 8

Event-Triggered Multi-source Bumpless Transfer Control for Networked Switched Systems with Almost Output Regulation Against Switching Deception Attacks

This chapter investigates the MSBT control problem of NSSs with the AOR performance subject to SDAs. Firstly, an improved DT approach is developed to derive a proper switching law and ensure the feasibility of the AOR for NSSs allowing the AOR of each subsystem unsolvable and some switching instants destabilizing (i.e. with Lyapunov function increments), where destabilizing and stabilizing switching instants are subsystem-independent. Secondly, a novel description of deception attacks called SDAs is established that contains multiple attack functions switching based on the currently activated subsystem, which considers the characteristics of NSSs and can cause the most severe damage to system control performance. Due to the networks and the event-triggered scheme, the switching of SDAs does not coincide with the switching of subsystems and controllers, which pose a challenge to system analysis. Thirdly, MSBT controllers are designed to suppress the control bumps induced by multiple sources including asynchronous controller switching, event triggering mechanism, and SDAs. Finally, an application to a switched RLC circuit is given to verify the effectiveness of the proposed method.

8.1 Introduction NSSs are a type of switched systems that consists of a family of subsystems, a switching rule, and transmitted data over networks, which have been extensively investigated [1, 2] due to their simple structures and long transmission distances. In recent years, the DT framework has taken into consideration a more practical situation in which all subsystems of switched systems are unstable and only adopt stabilizing switching instants to compensate for the divergence of unstable subsystems [3–6]. Accordingly, the stability analysis and property of switched systems with all subsystems unstable are respectively addressed in [3, 4] while requiring all switching instants stabilizing. However, in the practical switching of switched © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Li and J. Fu, Output Regulation and Cybersecurity of Networked Switched Systems, Studies in Systems, Decision and Control 475, https://doi.org/10.1007/978-3-031-30972-4_8

179

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systems, it is difficult to guarantee that all switching instants are stabilizing. To this end, the requirement of switching instants is alleviated in [5], which allows a certain proportion of destabilizing switching to appear periodically. This periodic restriction of destabilizing and stabilizing switching instants in [5] is further released in [6], which extends the stability analysis to the ORP characterizing how to drive the system output tracking and/or suppressing the reference signal and/or disturbances generated by the exosystem. Notably, the output error is regulated precisely in typical OR, while the actual systems and exosystem are inevitably affected by unknown disturbances. Therefore, it is necessary to consider the ORP under unknown disturbances, that is the AOR problem [7, 8]. However, the unstable subsystems will lead to the unsolvable AOR of each subsystem. It is a challenge to design a switching rule to achieve the AOR of destabilizing switched systems in the presence of unknown disturbance. ETMs for NSSs, which determine whether to perform data transmission through networks, have recently become popular due to their advantages such as effectively implementing system and controller switching within limited network bandwidth and significantly reducing actuators abrasion in the absence of continuous communication [9–11]. Unfortunately, data transmission through networks is vulnerable to cyber-attacks. The deception attack, an important kind of cyber-attacks with strong concealment, affects the operation of sensors, controllers, and actuators in NSSs [12– 14] by tampering with data packets, which is often difficult to be detected. Therefore, the control synthesis of NSSs under deception attacks is important and even more challenging. In [12], the resilient control of NSSs is addressed under deception attacks on sensors, controllers, and actuators. The event-driven finite-time control for NSSs is considered in [13] under deception attacks in the network from the sensor to the controller. The decentralized event-triggered control for NSSs is discussed in [14] under deception attacks in communication networks both from the sensor to the controller and from the controller to the actuator. Notably, deception attacks mentioned above are considered to have been imposing on systems, which is inconsistent with the actual characteristics of attackers randomly performing attacks. A stochastic deception attack is adopted in [15] for the H∞ filtering of NSSs. However, due to the multi-mode and multi-controller characteristics of NSSs, the deception attack considered in the above research may not be effective against it, and attackers may be more intelligent to impose different deception attacks on different subsystems, which is difficult to depict effectively by using a single attack function. How to properly describe deception attacks for NSSs and investigate the corresponding security control problems is more challenging. Bumps in control signals are undesired transient phenomena that damage system performance and may be effectively restrained via the BTC method [16]. Bumps in NSSs can be caused by changes in control signals induced by many factors such as controller switching, event triggering, and attacks. Early results only consider bumps produced by controller switching. A method to reduce control bumps is concerned in [17] by restricting the control amplitude in the whole state space, which is too stringent for BTC. To overcome this limitation, [18–20] improve the method in [17] by restricting the control amplitude only in subsystem activation intervals.

8.2 Problem Formulation

181

The above restrictions are further alleviated to limit the control amplitude only at switching instants in [21, 22]. Furthermore, a limitation on the control amplitude at triggering instants is proposed to reduce bumps caused by ETM in [22]. For NSSs, it is worth mentioning that transmission data tampered by deception attacks also yields control bumps at the start and end instants of attacks. In this account, how to design BT controllers for NSSs to suppress bumps generated by multiple sources such as controller switching, event triggering, and attacks to achieve the transient performance with AOR also serves as the motivation of our work. An ETM, a switching rule, and MSBT controllers are jointly designed to achieve the AOR for NSSs under SDAs. The main contributions are threefold: (i) An improved DT method is proposed to solve the AOR problem of NSSs, which permits the AOR problem of individual subsystem unsolvable and some switching instants destabilizing. Furthermore, our method eases the restriction that stabilizing/destabilizing switching instants depend on subsystems compared with [5, 6]. (ii) The stochastic SDAs are described for the first time in the presence of multiple attack functions, in which the attacker may select a corresponding attack function to tamper with the associated controller based on the currently activated subsystem. The network-induced delay and attack delay then eventuate a variety of complex asynchronous switching behaviors among the modes of subsystems, SDA functions, and attacked controllers. We analyze these complicated dynamic behaviors for each subsystem during any activated interval by combining switching strategies under the improved DT method. (iii) The MSBT performance is proposed, which conjointly attenuates the control bumps caused by multiple incentives at instants of controller switching, event-triggering, and SDAs via adding the norm restriction compared with single-source BTP index [17–22]. The remainder of this article is organized as follows. Section 2 formulates the preliminaries. The conditions for AOR and MSBT performance of NSSs with SDAs are shown in Sect. 3. Section 4 provides an example. Section 5 concludes this chapter.

8.2 Problem Formulation 8.2.1 System Modelling Consider the switched linear system described by {

˙ = Aσ (t) x(t) + Bσ (t) u(t) + E σ (t) w(t) + Fσ (t) d(t), x(t) e(t) = Cσ (t) x(t) + Dσ (t) w(t) + G σ (t) d(t), x(t0 ) = x0 ,

(8.1)

where x ∈ R x is the system state, u ∈ R u refers to the control input, d ∈ R d is the unknown disturbance belonging to L 2 [0, ∞), e ∈ R e is the regulated output error, x0 is the initial state at the initial time t0 . σ (t) : [0, ∞) → S = {1, 2, · · ·, s} represents

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the switching signal with the number of subsystems s. For convenience, σ (t) = i indicates the i-th subsystem is activated and tk denotes the k-th subsystem switching instant. The reference input w ∈ R w is produced by the exosystems w(t) ˙ = Mσ (t) w(t) + Nσ (t) d(t),

(8.2)

where Ai , Bi , Ci , Di , E i , Fi , G i , Mi , Ni are constant matrices with proper dimensions and eigenvalues Mi are assumed to be distinct with zero real parts for all i ∈ S [8].

8.2.2 Event-Triggered Mechanism As shown in Fig. 8.1, an ETM generates event-trigger instants based on the sampled values ξ T (t) = [x T (t) w T (t)] and the switching signal σ (t) with a fixed period h > 0, where the released ξ(trk h) is transmitted through the unreliable network subject to network-induced delays and SDAs. The ETM that mitigates the impact of asynchronous switching is shown below. ⎧ k k k k min+ {vh|erT (tr,v h)er (tr,v h) ⎪ ⎨ tr +1 h = tr h + v∈N T k k k k ˜ r,v h)}, if σ (tr,v h − h) = σ (tr,v h), > ασ (t) x˜ (tr,v h)x(t ⎪ ⎩ k+1 k else t1 h = tr,v h,

(8.3)

k where trk h represents the r-th triggering time after tk , tr,v h = trk h + vh is the v-th k k k k ˜ r h) − x(t ˜ r,v h), ασ (t) ∈ [0, 1) is the trigger sampling time after tr h, er (tr,v h) = x(t + k ˜ = [I − ||]. The ˜ rk h) with || threshold, k ∈ N , v, r ∈ N and x(t ˜ r h) = ||ξ(t coordinate transformation matrix || is subject to the following general assumption for the ORP.

Fig. 8.1 A diagram of NSSs under SDAs

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183

Assumption 8.1 ([23]) There exists a matrix || with proper dimensions satisfying Ai || − ||Mi + E i = 0, Ci || + Di = 0, i ∈ S. Remark 8.1 Zeno phenomenon can be excluded because the ETM (8.3) is detected k k h) = σ (tr,v h), the inequality condition only at sampling times. When σ (tr,v−1 in the ETM (8.3) is verified to determine whether to trigger or not. Otherwise k k σ (tr,v−1 h) /= σ (tr,v h), trigger immediately to shorten asynchronous intervals between the subsystem and its corresponding controller. Thus, at least one trigger occurs between two switches.

8.2.3 Controllers Under Switching Deception Attacks The controller is described as u(t) = [ K σˆ (t) L σˆ (t) ]ξ(trk h), t ∈ [t˜rk , t˜rk+1 ).

(8.4)

where K σˆ (t) is the controller gain to be designed, L σˆ (t) = −K σˆ (t) ||, σˆ (t) is the switching signal of controllers. Since the controller updates according to the sequence of triggering instants generated by ETM (8.3), t˜rk = trk h+τrk is the controller updating {τrk } < h, and the controllers’ instant, τrk is the network-induced delay with max + r ∈N ,k∈N

switching instant is t˜1k . Therefore, when t ∈ [tk , tk+1 ), the switching signal σˆ (t) of { σ (tk−1 ), t ∈ [tk , t˜1k ) controllers satisfies that σˆ (t) = . Figure 8.2 explicitly shows σ (tk ), t ∈ [t˜1k , tk+1 ) the relationship among switching instants, triggering instants, controller updating instants, and network-induced delay. k In the light of the input delay method [24], we define τ (t) = t − tr,v h and its k {τr } + h with 0 ≤ τ (t) ≤ τ M . It is easy to deduce that maximum τ M /\ max + r ∈N ,k∈N

k k x(t ˜ r,v h) = x(t ˜ − τ (t)) and x(t ˜ rk h) = er (tr,v h) + x(t ˜ − τ (t)). Then, the controller (8.4) can be rewritten as k u(t) = K σˆ (t) [er (tr,v h) + x(t ˜ − τ (t))].

Fig. 8.2 Time diagram of data transmission with network-induced delays

(8.5)

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Next, we consider the impact of SDAs on the controller (8.5). Notably, it inevitably takes time for attackers to obtain NSSs information and impose a matched attack function, which leads to the switching of SDAs lagging behind the switching of the corresponding subsystem, whose asynchronous interval is denoted as [tk , t˜sk ) with the attack switching instant t˜sk = tk + ςk and the lag time ςk subject to max{ςk } < h. k∈N

Therefore, the controller (8.5) under SDAs can be expressed as k ˜ σ (t) (ξ(t − ς (t))), (8.6) ˜ u(t) = β(t)K ˜ − τ (t))] + β(t)K σˆ (t) ||h σˆ (t) [er (tr,v h) + x(t

where h σ (t) (ξ(t − ς (t))) is the attack signal of SDAs with a switching signal { σ (tk−1 ), t ∈ [tk , t˜sk ), σ (t) = ς (t) ∈ [0, ς M ) stands for the attack delay of σ (tk ), t ∈ [t˜sk , tk+1 ), ˜ = 1 − β(t), β(t) takes values in {0, 1} SDAs with its maximum ς M ∈ [0, h), β(t) with prob{β(t) = 1} = β and β(t) = 1 means SDAs occur. To avoid being detected by security detection mechanisms, attackers usually take the initiative to limit the attack energy, so the following assumption is reasonable. Assumption 8.2 For real constant matrices Hσ (t) with proper dimensions, SDAs with a switching signal σ (t) satisfy || || || || ||h σ (t) (ξ(t − ς (t)))||2 ≤ || Hσ (t) ξ(t − ς (t))||2 . Remark 8.2 The meaning of SDA covers the following three aspects. First, SDA refers to a deception attack with multiple attack functions, where the attacker may select a corresponding attack function to tamper with the associated controller based on the currently activated subsystem. Since the NSSs comprise multiple subsystems and multiple controllers, it is difficult to characterize the tampering and destruction of different control signals corresponding to different systems by using a common attack function. For example, the attack function that destroys the control performance of some subsystems may potentially become a controller correction term that enhances the control performance of some other subsystems. As a result, considering the deception attack with a single attack function commonly used in literature such as [12–15] cannot guarantee the security control of NSSs. Assumption 8.2 intends to utilize the SDA model to depict the scenario of deception attacks that cause severe harm to the control performance of a NSS. Second, the SDA can characterize a class of time-varying deception attacks under some circumstances. This leads to an extensive study on the features of the attacks. Third, the ordinary deception attack in [12–15] is a special type of the SDA. The SDA degenerates into the ordinary deception attack when all attack functions are the same or the attack function is fixed.

8.2 Problem Formulation

185

8.2.4 The Closed-Loop Switched System Substituting (8.6) into (8.1) with Assumption 8.1 yields the following closed-loop switched system ⎧ k ˙˜ = Aσ (t) x(t) ⎪ x(t) ˜ − τ (t)) + er (tr,v h)] ˜ + (1 − β)Bσ (t) K σˆ (t) [x(t ⎪ ⎪ ⎪ k ⎪ ˜ − τ (t)) + er (tr,v h)] + (β − β(t))Bσ (t) K σˆ (t) [x(t ⎨ ˜ −1 x(t ˜ σ (t) (|| + Fσ (t) d(t) + β Bσ (t) K σˆ (t) ||h ˜ − ς (t))) ⎪ ⎪ −1 ˜ ˜ ⎪ β)B K ( || x(t ˜ − ς (t))) + (β(t) − ||h σ (t) σ (t) σˆ (t) ⎪ ⎪ ⎩ e(t) = C x(t) ˜ + G d(t), x(θ ˜ ) = ϕ(θ ), θ ∈ [−τ M , 0), σ (t) σ (t)

(8.7)

where the vector-valued initial function ϕ(θ ) is differentiable on [−τ M , 0) with ϕ(0) = x˜0 .

8.2.5 Multi-source Bumpless Transfer Performance For t ∈ [tk , tk+1 ), the definition of the MSBT performance for (8.7) is proposed as follows. Definition 8.1 For given constants δ1 , δ2 , the closed-loop system (8.7) achieves the MSBT performance if || || ˜ + ˜ σ (t + ) (ξ(tb+ − ς (tb+ ))) ˜ rk h) + β(tb+ )K σˆ (tb+ ) ||h ||β(tb )K σˆ (tb+ ) x(t b ˜ b+ )x(t ˜ b− )K σˆ (t − ) [β(t ˜ r p h) + β(tb+ )x(t ˜ rk h)] − β(t b ||2 ˜ σ (t − ) (ξ(tb− − ς (tb− )))|| −β(tb− )K σˆ (tb− ) ||h || b ≤ δ12 ||x(tb )||2 + δ22 ||w(tb )||2 holds, where tb is the instant when the control bump appears and tr p represents the triggering time before trk h. If tb ∈ [tk , t˜1k ), then tr p = tlk−1 ; Otherwise, tr p = tlk−1 , where tlk−1 is the last triggering time before tk . Remark 8.3 Due to the ETM (3), each switching is accompanied by a triggering. Thus, control bumps appear at the controller update time t˜rk , the SDAs’ start time tas and end time tae , as well as the attack switching instant t˜sk . In contrast to the MSBT performance index proposed in Definition 8.1, the existing definitions of BTP (such as [18–21]) often utilize norm restrictions with respect to a single incentive only at switching instant tk to attenuates control signal fluctuations. Definition 8.1 degrades into BTP index of [22] in the absence of network-induced delays and SDAs. In particular, the MSBT performance index in Definition 8.1 can be simplified into four ˜ b+ ) and β(t ˜ b− ): cases due to different values of β(t

186

8 Event-Triggered Multi-source Bumpless Transfer Control …

Table 8.1 Bumpy instants

β(tb+ )

β(tb− )

Bumpy instants

MSBT performance

1

0

SDAs’ start time tas

(8.8a)

0

1

SDAs’ end time tae (8.8b)

1

1

Attack switching instant t˜sk

(8.8c)

0

0

Controller update time t˜rk

(8.8d)

(a) For β(tb+ ) = 1 and β(tb− ) = 0, the control bump occurs at the SDAs’ start time tas , where the MSBT performance is written as || ||2 || + + k || ˜ σ (t + ) (ξ(tas − x(t − ς (t ))) − K ˜ h) || K σˆ (tas+ ) ||h || σˆ (tas ) as r as ≤ δ12 ||x(tas )||2 + δ22 ||w(tas )||2 .

(8.8a)

(b) When β(tb+ ) = 0 and β(tb− ) = 1, the control bump appears at the SDAs’ end time tae , where the MSBT performance is obtained as || ||2 || || − − ˜ σ (t − ) (ξ(tae ˜ rk h) − K σˆ (tae− ) ||h − ς (t ))) || K σˆ (tae+ ) x(t || ae ae ≤ δ12 ||x(tae )||2 + δ22 ||w(tae )||2 .

(8.8b)

(c) If β(tb+ ) = 1 and β(tb− ) = 1, the control bump arises at the attack switching instant t˜sk , where the MSBT performance turns into || ||2 || ˜ σ (t˜k− ) (ξ(t˜sk− − ς (t˜sk− )))|| ˜ σ (t˜k+ ) (ξ(t˜sk+ − ς (t˜sk+ ))) − K σˆ (t˜k− ) ||h ||K σˆ (t˜sk+ ) ||h || s s s || || || || 2 2 ≤ δ12 ||x(t˜sk )|| + δ22 ||w(t˜sk )|| . (8.8c) (d) For β(tb+ ) = 0 and β(tb− ) = 0, the control bump is generated at the controller update time t˜rk , where the MSBT performance becomes the ordinary BTP || || K

˜ rk h) σˆ (t˜rk+ ) x(t

||2 ||2 ||2 || || − K σˆ (t˜rk+ ) x(t ˜ r p h)|| ≤ δ12 ||x(t˜rk )|| + δ22 ||w(t˜rk )|| .

(8.8d)

For simplicity, the above discussion is summarized in Table 8.1.

8.2.6 Control Objectives In what follows, we aim to establish sufficient conditions that ensure the AOR and MSBT performance of the destabilizing NSSs (8.1) under stochastic SDAs. To this

8.3 Main Results

187

end, a proper switching signal σ (t), an ETM (8.3), and a security controller (8.6) with the MSBT performance are jointly designed to achieve the following specifications: (i) OR property: For w(t) = 0 and d(t) = 0, the system (8.7) is asymptotically meansquare stable. For w(t) /= 0 and d(t) = 0, e(t) of (8.1) satisfies lim E{e(t)} = 0. t→∞

(ii) L 2 -gain property: Under zero initial condition [ x˜ T (0) w T (0) ] = [ 0 0 ], for {∞ {∞ any w(t) and d(t), e(t) satisfies E{ 0 ||e(s)||2 ds} ≤ γ 2 E{ 0 ||d(s)||2 ds} with the L 2 -gain level γ > 0.

8.3 Main Results In this section, we provide solvable conditions on the AOR with MSBT performance of the closed-loop switched system (8.7) suffering from stochastic SDAs. To facilitate the discussion, suppose that the subsystem j switches to the subsystem i at the switching instant tk . Now, we are ready to first derive the AOR result of NSSs (8.7).

8.3.1 Solvable Conditions for Almost Output Regulation of Unstable Switched Systems Theorem 8.1 For the closed-loop switched system (8.7) with Assumptions 8.1–8.2, scalars η > 0, h > 0, γ > 0, λ > 1, 0 < ρ < 1, μ ≥ 1, αi ≥ 0 and β ≥ 0 are given. If. (i) there exist positive definite matrices Pε , Q oε , Roε (ε ∈ { ji, i}, o ∈ {1, 2}) and matrices K υˆ , S1ε , S2ε (υˆ ∈ { j, i}) of proper dimensions and such that 0ε ≤ 0,

(8.9) 1

[ P ji Q oji Roji ] > μ[ P j Q oj Roj ], ∀ j ∈ Si ,

(8.10)

[ P ji Q oji Roji ] > ρ[ P j Q oj Roj ], ∀ j ∈ Si1 ,

(8.11)

2

[ Pi Q oi Roi ] > μ[ P ji Q oji Roji ], ∀ j ∈ Si ,

(8.12)

[ Pi Q oi Roi ] > ρ[ P ji Q oji Roji ], ∀ j ∈ Si2 ,

(8.13)

188

8 Event-Triggered Multi-source Bumpless Transfer Control …

where we denote j ∈ Si1 when the Lyapunov function decreases at tk , otherwise 1 j ∈ Si , and denote j ∈ Si2 when the Lyapunov function decreases at t˜1k , otherwise 2 1 2 j ∈ Si , Si ∪ Si1 = Si ∪ Si2 = S/{i}; (ii) the number of stabilizing switching instants n − and the number of destabilizing switching instants n + satisfy n − /n + > λ, where n = n − + n + is the total switching number of Lyapunov function, n − , n + ∈ N+ ; (iii) the switching signal σ (t) satisfies the minimum and maximum DT limits Tmin ≥ ln ρ λ μ 2h and Tmax < −η(1+λ) . Then, the AOR problem of the closed-loop switched system (8.7) is solvable under controller (8.6) with ETM (8.3), where the 11 × 11 matrix 0ε = {φε } consists of the following block matrices φε11 = AiT Pε + Pε Ai + (2 − β − η)Pε + Q 1ε + Q 2ε + R1ε + R2ε 2 T 2 T + (3 − β)(τ M Ai R1ε Ai + ς M Ai R2ε Ai ) + CiT Ci ,

φε12 = −S1ε , φε13 = −R1ε + S1ε , φε14 = −S2ε , φε15 = S2ε − R2ε , 2 T 2 T φε17 = Pε Fi + CiT G i + τ M Ai R1ε Fi + ς M Ai R2ε Fi , φε22 = −eητ M Q 1ε + R1ε , φε23

= S1ε − R1ε , φε33 = 2R1ε − 2S1ε + αi2 I, φε44 = −eλς M Q 2ε + R2ε , φε45 = −R2ε + S2ε , ˜ −1 with υ ∈ { j, i}, φε66 = −I ˜ −T HυT Hυ || φε55 = 2R2ε − 2S2ε + || 2 T 2 T Fi R1ε Fi + ς M Fi R2ε Fi ), φε88 = −I, φε39 = K iT BiT , φε77 = G iT G i − γ 2 I + (3 − β)(τ M

2 2 ˜ T K iT BiT , R1ε + ς M R2ε ) + Pε ], φε6,10 = K iT BiT , φε8,11 = || φε99 = (β − 1)[4(τ M

2 2 2 2 φε10,10 = (β − 1)[4(τ M R1ε + ς M R2ε ) + Pε ], φε11,11 = β[3(τ M R1ε + ς M R2ε ) + Pε ].

Proof The deduction of the AOR of (8.7) is divided into two steps: the OR and the L2-gain properties. First, let us show how the OR property of (8.7) is achieved. The following Lyapunov function candidate is considered.

Vε (t) = V1ε (t) + V2ε (t) + V3ε (t), V1ε (t) = x˜ T (t)Pε x(t), ˜

8.3 Main Results

189

{

t

V2ε (t) =

e

{

η(s−t) T

x˜ (s)Q 1ε x(s)ds ˜ +

t−τ M

t

eη(s−t) x˜ T (s)Q 2ε x(s)ds, ˜

t−ς M

{ V3ε (t) = τ M + ςM

0

{

t

˙˜ eη(ϕ−t) x˙˜ T (ϕ)R1ε x(ϕ)dϕds

−τ M t+s { 0 { t −ς M

˙˜ eη(ϕ−t) x˙˜ T (ϕ)R2ε x(ϕ)dϕds.

t+s

The value of /\ε = E{V˙ε (t) − ηVε (t) + ||e(t)||2 − γ 2 ||d(t)||2 } is deduced from two intervals: the asynchronous interval [tk , t˜1k ) and the synchronous interval [t˜1k , tk+1 ). Next, let us consider the corresponding subsystem on each interval. Case 1: On the interval [tk , t˜1k ), ε = ji, the i-th subsystem is activated, while the j-th controller is still running. Taking the time derivative of V ji (t) along the system (8.7) yields E{V˙1 ji (t)} ⎧ T k 2 x˜ (t)P ji { Ai x(t) ˜ + (1 − β)Bi K j [x(t ˜ − τ (t)) + er (tr,v h)] ⎪ ⎪ ⎪ ⎪ ˜ ||h +F d(t) + β B K (ξ(t − ς (t)))}, ⎪ i j j i ⎪ ⎨ tk + ςk < tk + τk ,t ∈ [tk , tk + ςk ), or tk + ςk > tk + τk , = k ˜ − τ (t)) + er (tr,v h)] ˜ + (1 − β)Bi K j [x(t 2 x˜ T (t)P ji { Ai x(t) ⎪ ⎪ ⎪ ⎪ ˜ ⎪ ||h + F β B K (ξ(t − ς (t)))}, d(t) + i i j i ⎪ ⎩ tk + ςk < tk + τk ,t ∈ [tk + ςk , t˜1k ), 2 T 2 T E{V˙2 ji (t)} = ηV2 ji (t) + τ M x˜ (t)Q 1 ji x(t) ˜ + ςM x˜ (t)Q 2 ji x(t) ˜

− e−ητ M x˜ T (t − τ M )Q 1 ji x(t ˜ − τM ) − e−ης M x˜ T (t − ς M )Q 2 ji x(t ˜ − ς M ), 2 ˙T ˙˜ + ς 2 x˙˜ T (t)R2 ji x(t) ˙˜ E{V˙3 ji (t)} = ηV3 ji (t) + τ M x˜ (t)R1 ji x(t) M { t ˙˜ e−η(s−t) x˙˜ T (s)R1 ji x(s)ds − τM t−τ M t

{ − ςM

˙˜ e−η(s−t) x˙˜ T (s)R2 ji x(s)ds.

t−ς M

Applying Lemma 1.8 and Park theorem [25], we obtain { −τ M

t t−τ M

˙˜ e−η(s−t) x˙˜ T (s)R1 ji x(s)ds ≤ χ T (t)

|

| R1 ji S1 ji χ (t) ∗ R1 ji

where χ T (t) = [ x˜ T (t) − x˜ T (t − τ (t)) x˜ T (t − τ (t)) − x˜ T (t − τ M ) ]. Moreover, we || ||2 k ˜ − τ (t))||2 − ||er (tr,v h)|| > infer from the ETM (8.3) and Assumption 8.2 that αi ||x(t

190

8 Event-Triggered Multi-source Bumpless Transfer Control …

|| ||2 || || ˜ −1 x(t 0 and ||Hυ x(t ˜ − ς (t))||2 − ||h υ (|| ˜ − ς (t)))|| > 0. Thus /\ ji < ζ T (t)0 ji ζ (t), where ζ T (t) = [ x˜ T (t) x˜ T (t − τ M ) x˜ T (t − τ (t)) x˜ T (t − ς M ) k ˜ −1 x(t x˜ T (t − ς (t)) erT (tr,v h) d T (t) h υT (|| ˜ − ς (t))) ]

and the matrix 0 ji is composed of the following block matrices T 2 T θ 11 ji = Ai P ji + P ji Ai − η P ji + Q 1 ji + Q 2 ji + R1 ji + R2 ji + τ M Ai R1 ji Ai 2 T + ςM Ai R2 ji Ai + CiT Ci , 2 T θ 13 ji = (1 − β)P ji Bi K j + τ M (1 − β) Ai R1 ji Bi K j − R1 ji + S1 ji 2 + ςM (1 − β)AiT R2 ji Bi K j , 14 15 θ 12 ji = −S1 ji , θ ji = −S2i , θ ji = S2 ji − R2 ji , T 2 T 2 θ 16 ji = τ M (1 − β)Ai R1 ji Bi K j + ς M (1 − β)Ai R2 ji Bi K j + (1 − β)P ji Bi K j , T 2 T 2 T θ 17 ji = P ji Fi + C i G i + τ M Ai R1 ji Fi + ς M Ai R2 ji Fi , 2 T 2 T ˜ ˜ ˜ θ 18 ji = β P ji Bi K j || + τ M β Ai R1 ji Bi K j || + ς M β Ai R2 ji Bi K j ||, ητ M θ 22 Q 1 ji + R1 ji , θ 23 ji = −e ji = S1 ji − R1 ji , T T 2 T T 2 2 θ 33 ji = 2R1 ji − 2S1 ji + τ M (1 − β)Bi K j R1 ji Bi K j + ς M (1 − β)Bi K j R2 ji Bi K j + αi I,

T T 2 T T 2 θ 36 ji = τ M (1 − β)K j Bi R1 ji Bi K j + ς M (1 − β)K j Bi R2 ji Bi K j ,

T T 2 T T 44 λς M 2 Q 2 ji + R2 ji , θ 37 ji = τ M (1 − β)K j Bi R1 ji Fi + ς M (1 − β)K j Bi R2 ji Fi , θ ji = −e 55 ˜ −1 ˜ −1 T T θ 45 ji = −R2 ji + S2 ji , θ ji = 2R2 ji − 2S2 ji + (|| ) Hυ Hυ || , T T 2 T T 2 θ 66 ji = τ M (1 − β)Bi K j R1 ji Bi K j + ς M (1 − β)Bi K j R2 ji Bi K j − I, T T 2 T T 2 θ 67 ji = τ M (1 − β)K j Bi R1 ji Fi + ς M (1 − β)K j Bi R2 ji Fi ,

8.3 Main Results

191

T 2 2 T 2 T θ 77 ji = G i G i − γ I + τ M F ji R1 ji Fi + ς M Fi R2 ji Fi , T 2 T 2 ˜ ˜ θ 78 ji = τ M β Fi R1 ji Bi K j || + ς M β Fi R2 ji Bi K j ||, 2 2 ˜ ˜ ˜T T T ˜T T T θ 88 ji = τ M β || K j Bi R1 ji Bi K j || + ς M β || K j Bi R2 ji Bi K j || − I.

In virtue of Lemma 1.7, it can be derived from (8.9) that 0 ji ≤ 0, which implies that /\ ji ≤ 0, or equivalently, E{V˙ ji (t) − ηV ji (t)} ≤ −E{||e(t)||2 } ≤ 0 for d(t) = 0. Integrating both sides of this inequality over the interval t ∈ [tk , t˜1k ) gives E{V ji (t)} ≤ eη(t−tk ) E{V ji (tk )}.

(8.14)

Case 2: On the interval [t˜1k , tk+1 ), ε = i, the i-th subsystem and controller are activated. Similar to Case 1, we are ready to obtain that /\i < 0, which means E{V˙i (t) − ηVi (t)} ≤ −E{||e(t)||2 } ≤ 0 for d(t) = 0 resulting in E{Vi (t)} ≤ eη(t−t˜1 ) E{Vi (t˜1k )}, t ∈ [t˜1k , tk+1 ). k

(8.15)

Recalling that the j-th subsystem switches to the i-th subsystem at tk and the Lyapunov function switches at tk and t˜1k on the interval [tk , tk+1 ), it follows from (8.9)–(8.12) that the values of Lyapunov functions before and after tk and t˜1k satisfy { E{V ji (tk+ )}

≤

1

μE{V j (tk− )}, j ∈ Si , E{Vi (t˜1k+ )} ≤ ρE{V j (tk− )}, j ∈ Si1 ,

{

1

μE{V ji (t˜1k− )}, j ∈ Si , ρE{V ji (t˜1k− )}, j ∈ Si2 . (8.16)

Together with (8.14) and (8.15) infers that E{Vσ (t) (t)} + − μn ρ n eη(t−t0 ) E{Vσ (t0 ) (t0 )} for t ∈ [tk , tk+1 ). Consequently, it holds that n+

≤

n−

E{Vσ (t) (t)} ≤ eη(t−t0 )+ln μ +ln ρ E{Vσ (t0 ) (t0 )} n + − ≤ e( 2 +1)ηTmax +n ln μ+n ln ρ E{Vσ (t ) (t0 )} ≤e

ηTmax +n

(

ηTmax 2

λ

μρ + ln1+λ

0

)

E{Vσ (t0 ) (t0 )}

(8.17)

for t ∈ [t˜1k , tk+1 ). Similarly, 3

E{Vσ (t) (t)} ≤ e 2

ηTmax +n

(

ηTmax 2

λ

μρ + ln1+λ

)

E{Vσ (t0 ) (t0 )}

(8.18)

is true for t ∈ [tk , t˜1k ). Thus, for d(t) = 0 and w(t) = 0, the system (8.7) is asymptotically mean-square stable; For w(t) /= 0 and d(t) = 0, it is clear that

192

8 Event-Triggered Multi-source Bumpless Transfer Control …

lim E{e(t)} = lim E{Cσ (t) x(t) + Dσ (t) w(t)} = lim E{Cσ (t) x(t)} ˜ = 0, which t→∞ t→∞ indicates the OR property of the system (8.7). Next, for d(t) /= 0, the L 2 -gain property of the system (8.7) will be derived. In conjunction with uε (t) ≤ 0, (8.17) and (8.18), one can obtain

t→∞

E{Vσ (tk ) (x(t))} ˜ ≤ E{Vσ (tk ) (x( ˜ t˜rk ))}eη(t−t˜r ) − E{ k

{

t˜rk

˜ t˜rk ))}eη(t−t˜r ) − E{ ≤ κ E{{Vσ (tk ) (x( k

{ where κ =

t

eη(t−s) |(s)ds}

{

t

eη(t−s) |(s)ds},

t˜rk

ρ, j ∈ Si1 ∪ Si2 , 1 2 The above inequality yields that μ, j ∈ Si ∪ Si . {

k ηt

E{Vσ (tk ) (x(t))} ˜ ≤ E{κ e Vσ (t0 ) (x(t ˜ 0 )) − κ { − κ k−1

t1

k

eη(t−s) |(s)ds

t0 t˜11

eη(t−s) |(s)ds − · · · − ε0

{

t˜rk

t1 −

t

eη(t−s) |(s)ds}.

+

Since κ k = ρ n μn , one can further obtain n−

n + ηt

{

t

−

+

E{Vσ (tk ) (x(t))} ˜ ≤ ρ μ e E{Vσ (t0 ) (x(t ˜ 0 )) − ρ n μn eη(t−s) |(s)ds} 0 { t − + = −E{ eη(t−s)+n ln ρ+n ln μ |(s)ds}. 0

{t It follows from E{Vσ (tk ) (x(t))} ˜ ≥ 0 that E{ 0 F(s)|(s)ds} ≤ 0 with F(t) = − + en ln ρ+n ln μ+η(t−s) under zero initial condition. Then integrating both sides of inequality {this { ∞ from t = 0 to ∞ and rearranging the double integral area yield ∞ |(s)ds (8.17), (8.18) and 0 < ρ < 1 gives 0 s F(t)dt ≤ (0. Recalling ) ) ( 2 ln ρ {∞ +η (t−s) n − lnρ+η(t−s) and s F(t)dt > −1/ η + 2Tlnminρ > 0. F(t) > e ≥ e Tmin {∞ {∞ Thus, it is concluded that 0 |(s)ds = 0 (||e(s)||2 − γ 2 ||d(s)||2 )ds ≤ 0, which [] implies that the system (8.7) achieves the L 2 -gain property for d(t) /= 0. Remark 8.4 Asynchronous switching and destabilizing switching instants may diverge the trajectories of NSSs, which brings great restrictions to the design of the 1 2 switching rule. To this end, Theorem 8.1 depicts four novel partitions Si , Si1 , Si , Si2 for destabilizing/stabilizing switching instants of each subsystem associated with LMI conditions (8.10)–(8.13). Through this novel description, we not only overcome the shortcoming that a certain subsystem is only activated after a fixed type of

8.3 Main Results

193

switching instants (stabilizing or destabilizing) in [5, 6], but also avoid the utilization of discrete Lyapunov functions, which offers greater flexibility for system and controller switching, and greatly reduces calculation amount. Theorem 8.1 gives the solvable conditions for the AOR problem of the closed-loop switched system (8.7). To achieve the AOR with MSBT performance of the system (8.7) suffering from stochastic SDAs, we provide the conditions for achieving the MSBT performance on the premise that Theorem 8.1 is solvable.

8.3.2 Solvable Conditions for Bumpless Transfer Problems of Unstable Switched Systems Theorem 8.2 Based on Theorem 8.1, for the closed-loop system (8.7) with Assumptions 8.1–8.2, scalars δ1 > 0, δ2 > 0 are given. If there exist matrices K υˆ (υˆ ∈ { j, i}) such that up ≤ 0

(8.19)

where the 6 × 6 matrix u p = {ϕ p } ( p ∈ {1, 2, 3}) consists of the following block 2 2 22 33 44 55 66 matrices: ϕ 11 p = −I , ϕ p = −I , ϕ p = −δ1 I , ϕ p = −δ2 I , ϕ p = −2I , ϕ p = −2I , 15 26 15 T 26 T ˜ K + 0.5I , ϕ2 = K υˆ + 0.5I , ϕ1 = K υˆ + 0.5I , ϕ1 = K υˆ + 0.5I , ϕ2 = || υˆ 15 26 T T T T ˜ ˜ ϕ3 = || K υˆ + 0.5I , ϕ3 = || K υˆ + 0.5I , then the AOR problem of the system (8.7) achieves the MSBT performance. Proof This proof contains four cases in terms of different sources of bumps according to Definition 8.1 and Remark 8.3. Case a: Bumps are caused by the beginning of SDAs if β(tb+ ) = 1 and β(tb− ) = 0. In this scenario, four situations are discussed below. For t ∈ [tk , t˜1k ) and t ∈ [tk , t˜sk ), the j-th attack function and the j-th controller are active, then the MSBT performance (8.8a) is achieved if ⎡

⎤ ˜ T K Tj K j || ˜ −|| ˜ T K Tj K j 0 || 0 ⎢ ⎥ ∗ K Tj K j 0 0 ⎥ ⎢ χ1T (t)⎢ ⎥χ1 (t) ≤ 0, ⎣ ∗ ∗ −δ12 I 0 ⎦ ∗ ∗ ∗ −δ22 I

(8.20)

˜ −1 x(t holds where χ1T (t) = [ h Tj (|| ˜ as − ς (tas ))) x˜ T (trk h) x˜ T (tas ) w T (tas )]. Moreover, other situations that t˜1k > t˜sk with t ∈ [t˜sk , t˜1k ), t˜sk > t˜1k with t ∈ [t˜1k , t˜sk ) and t ∈ [max{t˜1k , t˜sk }, tk+1 ) can be handled in the same way as above. Thus, (8.20) can be obtained by using the Lemma 1.7 for (8.19) ( p = 2), which implies that bumps caused by the start of SDAs can be reduced.

194

8 Event-Triggered Multi-source Bumpless Transfer Control …

Case b: Bumps are caused at the end of SDAs if β(tb+ ) = 0 and β(tb− ) = 1. Using the same deduction as Case a for (8.19) ( p = 2) infers that bumps caused by the end of SDAs can be reduced with the MSBT performance (8.8b). Case c: Bumps are caused by the switching of SDAs if β(tb+ ) = β(tb− ) = 1. To ensure the MSBT performance (8.8c), the following inequality is needed that ⎡ ˜T T ˜ −|| ˜ T K T K υˆ || ˜ 0 || K υˆ K υˆ || υˆ T T ⎢ ˜ ˜ 0 ∗ || K υˆ K υˆ || χ2T (t)⎢ ⎣ ∗ ∗ −δ 2 I ∗

∗

1

∗

⎤ 0 0 ⎥ ⎥χ2 (t) ≤ 0 0 ⎦ −δ22 I

(8.21)

˜ −1 x( ˜ −1 x( where χ2T (t) = [ h iT (|| ˜ t˜sk − ς (t˜sk ))) h Tj (|| ˜ t˜sk − ς (t˜sk ))) x˜ T (t˜sk ) w T (t˜sk ) ]. Then, (8.21) can be derived by applying the Lemma 1.7 for (8.19) ( p = 3), which means that bumps caused by the switching of SDAs can be suppressed. Case d: Bumps are caused by the event-triggering if β(tb+ ) = β(tb− ) = 0. For t = t˜1k , the following inequality ⎤ 0 K iT K i −K iT K j 0 T ⎢ ∗ 0 0 ⎥ Kj Kj ⎥χ3 (t) ≤ 0 χ3T (t)⎢ 2 ⎣ ∗ ∗ −δ1 I 0 ⎦ ∗ ∗ ∗ −δ22 I ⎡

(8.22)

is required to ensure the MSBT performance (8.8d), where χ3T (t) = [ x˜ T (trk h) x˜ T (tr p h) x˜ T (t˜rk ) w T (t˜rk ) ]. Similarly, for t ∈ [t˜1k , tk+1 ), one can obtain ⎤ 0 K iT K i −K iT K i 0 ⎢ ∗ 0 0 ⎥ K iT K i ⎥χ3 (t) ≤ 0. χ3T (t)⎢ 2 ⎣ ∗ ∗ −δ1 I 0 ⎦ ∗ ∗ ∗ −δ22 I ⎡

(8.23)

Then, (8.22) and (8.23) can be derived by applying the Lemma 1.7 for (8.19) ( p = 1), which indicates that bumps caused by the event-triggering can be suppressed. Therefore, the MSBT performance of the AOR problem is guaranteed for the [] system (8.7). Remark 8.5 The forthcoming optimization problem min

γ

s.t.

(8.9) − (8.13), (8.19)

Pε ,Q ε ,Rε ,K υˆ , υ∈{i, ˆ j},ε∈{i, ji},i, j∈S

8.4 Simulation Example

195

evaluates the anti-disturbance capability of the system (8.1) under SDAs with the jointly designed MSBT controller (8.6), ETM (8.3) and switching signal σ (t) in Theorem 8.1.

8.4 Simulation Example Consider the switched resistance-inductor-capacitance (RLC) circuit in [26] with three capacitors, whose dynamic equation is as follows: 1 x2 + rw1 + rd1 , L 1 R x˙2 = − x1 − x2 + u i + rw2 + rd2 , ci L 1 e = x2 + rw3 + rd3 , i ∈ S0 = { 1,2,3} , L x˙1 =

(8.24)

where x T = [ x1 x2 ] = [ qc i L ] is the system state with the charge qc in the capacitor and the flux i L in the inductance, the control input u i is the voltage. According to (8.1), [ rw1 rw2 ]T = E i w and rw3 = Di w with w T = [ w1 w2 ] denote the reference signal generated by the exosystem w(t) ˙ = Mσ (t) w(t) + Nσ (t) d(t) with |

| | | | | 23 42 30 , M2 = , M3 = , 21 21 01 | | | | | | 0.2 0 0.5 0 0.2 0.2 N1 = , N2 = , N3 = . 0 0.1 0 0.3 0 0.1 M1 =

Also, [ rd1 rd2 ]T = Fi d and rd3 = G i d with d T = [ d1 d2 ] denote the unknown disturbance. Selecting parameters of the resistance L = 1H, the inductance R = 0.8i and the capacitances c1 = 250μF, c2 = 125μF, c3 = 50μF, then |

| | | | | 0 1 0 1 0 1 , A2 = , A3 = , A1 = −0.004 −0.8 −0.008 −0.8 −0.02 −0.8 BiT = [ 0 1 ], Ci = Di = [ 0 1 ]. To solve the AOR problem, other system matrices are given as |

| | | | | 2 2.6 4.32 1.6 2.6 −0.3 E1 = , E2 = , E3 = , 2.01 1.8 1.84 0.24 3.84 3.62

196

8 Event-Triggered Multi-source Bumpless Transfer Control …

|

| | | 1 0 0 0 F1 = , F2 = F3 = . −0.5 1 0.5 1 Based on the above parameters, there exist the following matrices satisfying Assumptions 8.1 and 8.2: |

| | | | | | | 10 0.1 0.2 0.1 0.2 0.1 0 || = , H1 = , H2 = , H3 = . 01 0 0.1 0.3 0.4 0 0.1 Selecting parameters h = 0.1, |λ = 3, η = 0.2, ρ = 0.75, μ = 1.2, disturbance | input d T (t) = 0.5e−0.5t 0.2e−0.2t and solving LMIs (8.9)–(8.13), (8.19) yield the DT bounds Tmax = Tmin = 0.2s in the light of Theorem 8.1. The controller gain matrices are obtained as K 1 = [ −0.0952 0.0950 ], K 2 = [ −0.0883 0.0671 ], K 3 = [ −0.1220 0.0648 ]. Based on Theorem 8.2, and the L 2 -gain level γ = 1.7667 according to Remark 8.5. It is not difficult to find from the diverging curves in Fig. 8.3 that the AOR problem of each subsystem in (8.24) is unsolvable. Next, we show the effectiveness of our proposed method. The probability of SDAs is captured in Fig. 8.4. Figure 8.5 reflects triggering instants generated by the ETM (8.3) with transmission delays. Figure 8.6 displays asynchronous switching signals of systems and controllers according to Theorem 8.1 for Fig. 8.3 States of subsystems 1, 2, 3

8.4 Simulation Example

197 1

2

S11 = ∅, S1 = {2, 3}, S21 = ∅, S1 = {2, 3}, 1

2

S12 = {1}, S2 = {3}, S22 = {1}, S2 = {3}, 1

2

S13 = {1, 2}, S3 = ∅, S23 = {1, 2}, S3 = ∅, It can be seen from Fig. 8.6 that the ratio of stabilizing and destabilizing switching instants is 164/35, which satisfies n − /n + > 3. In addition, unlike the method in [5], the novel DT method proposed in this chapter does not need to impose periodic restrictions on the switching instants. | |T With the initial state x(0) ˜ = 1 2 , Fig. 8.7 shows the state responses and the regulated output error of the NSSs under the SDAs, which indicates the AOR problem of the system (8.24) is solved and it is easy to see that the controllers with MSBT performance improve the transient performance of the NSSs.

Fig. 8.4 Bernoulli stochastic variable of β(t)

Fig. 8.5 Triggering instants of ETM (8.3)

198

8 Event-Triggered Multi-source Bumpless Transfer Control …

Fig. 8.6 Switching signals of systems and controllers Fig. 8.7 State responses and output error of the closed-loop system (8.7)

It is worth noting that the Lyapunov function does not decrease monotonically over the running intervals of each subsystem and at instants when the third subsystem is activated, as depicted in Fig. 8.8. Figure 8.9 shows the difference between bumpy and bumpless control signals, where our MSBT controller owns better transient behaviors.

8.5 Conclusion

199

Fig. 8.8 Evolution of Lyapunov function Fig. 8.9 The comparison of bumpy and bumpless control signals

8.5 Conclusion This chapter solves the event-triggered MSBT control problem with AOR performance for the destabilizing NSSs which suffer from SDAs with the asynchronous switching. The improved DT method breaks through the binding of switching instants stability and subsystems. The considered SDAs well describe the switching characteristics of deception attacks in NSSs for the first time. The MSBT control strategy is proposed that can effectively inhibit the bumps caused by the multi-sources, including controller switching, event-triggering, and SDAs. The results of this chapter are worth extending to the other types of attacks in the future.

200

8 Event-Triggered Multi-source Bumpless Transfer Control …

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Chapter 9

Conclusions

The research of OR and cyber security of NSSs is extremely important and valuable work. In recent years, NSSs have received considerable attention in many fields. Then, in practice, NSSs may encounter some bad conditions, such as all subsystems being unstable and control signals and switching signals suffering various cyber attacks, which makes the research of steady-state and transient performance for OR and cyber security of NSSs difficult. Several related theorems proposed in this book can solve these problems, which makes its research content of important theoretical and practical significance. The main contributions of this book are concluded as follows: In Chap. 2, the event-triggered ORP is investigated for an NFCS with a switched system approach. By properly scheduling triggered times of periodic sampling associated with continuous event-triggering and switching instants of each subsystem, an AETM based on the subsystem model is proposed to transmit the triggered information to the candidate controller. Since the subsystem may switch between two adjacent events while the corresponding controller does not switch, asynchronous switching may occur. Meanwhile, an event-triggered switching signal caused by asynchronous switching is designed with the MDADT approach and event-triggered instants. By constructing multiple Lyapunov functions in the framework of the input delay approach and designing an error feedback controller, the sufficient condition for EAORP is solved. Finally, the proposed methods are proved to be effective by the F-18 aircraft model. In Chap. 3, the event-triggered ORP is investigated for a class of NSSs. A modedependent AETM is integrated into the NSSs with transmission delays and packet losses for the first time based on continuous detection and periodic sampling, and a joint-designed switching signal with the triggering information is presented. Under transmission delays and packet losses, a series of resultant error feedback controllers are synthesized in four cases based on the relationships between switching instants and event-triggered instants. Then, a criterion ensuring asynchronous ORP is formed with transmission delays by the input delay method. With the presence of packet © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 L. Li and J. Fu, Output Regulation and Cybersecurity of Networked Switched Systems, Studies in Systems, Decision and Control 475, https://doi.org/10.1007/978-3-031-30972-4_9

203

204

9 Conclusions

losses, a set of sufficient conditions is also dedicated to the proposed control issue by adding the restraint in the number of successive packet losses associated with a modified AETM. Eventually, a comparison and an F-18 aircraft model reveal the effectiveness of the obtained results in the simulation. In Chap. 4, the EAORP of an NSS with SUDs is studied, including all mode instabilities and partial switching moment instabilities, that is, the Lyapunov function increases the activation intervals of all subsystems and some switching moments. Firstly, an MMETM for switched systems is proposed to effectively shorten asynchronous intervals, which employs historical sampled outputs and compares the mode of the current sampled instant and the adjacent sampled instant. Then, the maximum ADT for a novel switching signal is derived with a constraint on the ratio of total destabilizing switchings to total stabilizing switchings, which relaxes the requirement that the regular arrangement of destabilizing and stabilizing switchings. Moreover, with the help of different coordinate transformations in the EAORP, the discretized Lyapunov functions are no longer needed when synthesizing the NSSs with SUDs, and the asynchronous switching situation is also discussed. Afterward, by designing a DOFC, sufficient conditions are given to solve the EAORP for NSSs with SUDs subject to network-induced delays, packet disorders, and packet losses. Finally, the effectiveness of the proposed methods is verified via a switched RLC circuit. In Chap. 5, the AOR with the BTC is studied for switched linear systems with destabilizing behaviors, whose all subsystems are unsolvable and some switching instants are unstable with finite increments of the Lyapunov function, Switching instants with decrements and increments of the Lyapunov function are described as stable and unstable switching instants respectively. Firstly, a HADT strategy proposed to restrict DT strategy is proposed to restrict the occurrence ratio of unstable/stable switching instants and reasonably arranges the number of stable switching instants to offset the increment of the Lyapunov function caused by unstable switching instants and unsolvability of the involved problem for Then, by interpolating the gains of adjacent controllers within the minimum DT, a dynamic error feedback controller with Then, by interpolating the gains of adjacent controllers within the minimum DT, a dynamic error feedback controller with BT property is designed to suppress the control bumps induced by controller switching. Finally, a numerical example is given to demonstrate the effectiveness of the proposed method. In Chap. 6, the EORP under DoS attacks is considered for NSSs with SUDs. The SUDs here refer to the unsolvable OR of each subsystem and the destabilization at partial switching instants, which indicates that the Lyapunov function does not decrease monotonically in activation intervals of each subsystem and increases at partial switching instants. First, LDDAs are considered, where LDDAs imply that their duration may be longer than the total DT of several adjacent activated subsystems. By imposing constraints at switching instants, consecutive asynchronous subsystem switching caused by LDDAs and SUDs is allowed, that is, the subsystem switches several times but the controller switching is blocked by LDDAs and controllers fail to switch correspondingly. Second, mixed ETMs, combining event-triggered conditions and periodic sampling conditions, are designed to reduce

9 Conclusions

205

network burden under LDDAs and improve system performance subject to destabilizing switching. Then, an improved DT for switching signal permits irregular arrangement of destabilizing and stabilizing switching and is more suitable for NSSs subject to LDDAs. Moreover, sufficient conditions ensure the solvability of EORP for NSSs with SUDs under LDDAs, network-induced delays, random packet losses, and packet disorders. Finally, a switched RLC circuit shows the feasibility of the proposed method. In Chap. 7, the resilient EORP with dissipativity subject to deception attacks for NSSs with SUDs is investigated, which includes the unsolvable OR of each subsystem and some destabilizing switchings and implies that the Lyapunov function is rises in activated intervals of all subsystems and at partial switching instants. First, IDA is delineated for the first time in NSSs for the tampered switching signal, system output, and control, where the consecutive asynchronous switching caused by the deceived switching signal is permitted by modifying inequality conditions at switching instants, which means switching from one asynchronous case to another at a switching instant. Second, two resilient ETMs for S2C and C2A channels are devised in conjunction with dissipative parameters in triggering thresholds to achieve the balance between limited network resources and system performance against destabilizing switching and IDAs. Third, the enhanced DT constraint for switching signal collected attack parameters not only overcomes the conventional arrangement of destabilizing and stabilizing switchings but is also more appropriate for the NSSs impacted by IDAs. Furthermore, the solvability condition of EORP with dissipativity is deduced for NSSs with SUDs subject to IDAs, network-induced delays, random packet losses, and packet disorders via designing a DOFC. Finally, an F-18 aircraft model is used to demonstrate the feasibility of the proposed methods. In Chap. 8, the event-triggered MSBT control problem is investigated for NSSs with the AOR performance subject to SDAs. Firstly, an improved DT approach is developed to derive a proper switching law and ensure the feasibility of the AOR for NSSs allowing the AOR of each subsystem unsolvable and some switching instants destabilizing (i.e. with Lyapunov function increments), where destabilizing and stabilizing switching instants are subsystem-independent. Secondly, a novel description of deception attacks called SDAs is established that contains multiple attack functions switching based on the currently activated subsystem, which considers the characteristics of NSSs and can cause the most severe damage to system control performance. Due to the networks and the event-triggered scheme, the switching of SDAs does not coincide with the switching of subsystems and controllers, which pose a challenge to system analysis. Thirdly, MSBT controllers are designed to suppress the control bumps induced by multiple sources including asynchronous controller switching, event triggering mechanism, and SDAs. Finally, an application to a switched RLC circuit is given to verify the effectiveness of the proposed method.