Event-Triggered Control of Switched Linear Systems (Studies in Systems, Decision and Control, 365) 3030716031, 9783030716035

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Studies in Systems, Decision and Control 365

Jun Fu Tai-Fang Li

Event-Triggered Control of Switched Linear Systems

Studies in Systems, Decision and Control Volume 365

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.

More information about this series at http://www.springer.com/series/13304

Jun Fu Tai-Fang Li •

Event-Triggered Control of Switched Linear Systems

123

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

Tai-Fang Li College of Control Science and Engineering Bohai University Jinzhou, China

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-030-71603-5 ISBN 978-3-030-71604-2 (eBook) https://doi.org/10.1007/978-3-030-71604-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 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

Preface

Switched systems have received tremendous research attention since they possess both important theoretical significance and practical engineering applications. Many important methodologies and techniques have been developed for the analysis and synthesis of switched systems including stability and performance analysis, robust control, robust filtering, and fault detection problems. This book presents the research work on some different types of switched systems including switched linear systems, switched linear delay systems, switched linear neutral systems and switched nonlinear systems with event-triggered control, time-triggered control, hysteresis switching control, reliable control and fault-tolerant control. Some sufficient conditions are established, respectively, for stability and performances of these kinds of switched systems in terms of solutions of linear matrix inequalities. Synthesis problems including control design, fault detection and model approximation are addressed according to the derived analysis conditions. The present book is not really a book on systematic theoretical knowledge of switched systems but rather a book on switched systems written from up-to-date research developments and novel methodologies on switched systems. This book is based on the papers in which some have been published and some are still undiscovered. The main body of the book consists of eight chapters. These methodologies provide a framework for stability and performance analysis, including event-triggered control, time-triggered control, robust filter design, and model approximation for switched systems. In addition, this book provides valuable reference material for researchers who wish to explore the area of switched systems. The compendious frame and description of the book are given as follows. Chapter 2 studies observer-based event-triggered sampled-data control of switched linear systems. Chapter 3 studies both state-based and observer-based event-triggered sampled-data control for switched linear systems under an improved event-triggered sampling mechanism proposed based on the one in Chap. 2. Chapters 4 and 5 study event-triggered sampled-data control for switched linear delay systems and switched linear neutral systems, respectively. Chapter 6 considers periodic sampled-data control for switched linear neutral systems. Chapter 7 studies hysteresis switching control for switched linear neutral systems. Chapter 8 v

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investigates reliable control of switched nonlinear systems. Chapter 9 is concerned with fault-tolerant control of switched nonlinear systems. Chapter 10 summarizes the results of the book finally. Shenyang, China Jinzhou, China October 2020

Jun Fu Tai-Fang Li

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, Jun Zhao and Yuanwei Jing from Northeastern University, Profs. Chunyi Su and Yongmin Zhang from Concordia University, and Prof. Fang Deng from Beijing Institute of Technology, for their valuable suggestions, constructive comments and supports. Our acknowledgements also go to our fellow colleagues who have offered invaluable support and encouragement throughout this research effort. In particular, we would like to acknowledge the contributions of Tao Yang and Ying Jin. 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 under Grants 61825301, 61873041 and 61503041, and National Key Research and Development Program of China under Grant 2018AAA0101603. Shenyang, China Jinzhou, China October 2020

Jun Fu Tai-Fang Li

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Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Switched System . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Switched Linear/Nonlinear System . . . . 1.1.2 Switched Delay/Neutral System . . . . . . 1.2 Stability and Stabilization . . . . . . . . . . . . . . . . 1.3 Sampled-Data Control . . . . . . . . . . . . . . . . . . . 1.3.1 Periodic Sampled-Data Control . . . . . . . 1.3.2 Event-Triggered Sampled-Data Control . 1.4 Reliable and Fault-Tolerant Control . . . . . . . . . 1.5 Outline of the Book . . . . . . . . . . . . . . . . . . . .

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Observer-Based Event-Triggered Control for Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2.2 Problem Statement and Preliminaries . . . . 2.3 Main Results . . . . . . . . . . . . . . . . . . . . . 2.3.1 Observer Design . . . . . . . . . . . . . 2.3.2 Sampling Mechanism . . . . . . . . . . 2.3.3 Stability Analysis . . . . . . . . . . . . . 2.3.4 Control Design . . . . . . . . . . . . . . 2.4 Illustrative Example . . . . . . . . . . . . . . . . 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . .

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Switched Linear . . . . . . . . . .

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Improved Event-Triggered Control for Switched Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Problem Statement and Preliminaries . . . . . . . . . . . . . 3.3 State-Based Event-Triggered Control . . . . . . . . . . . . . 3.3.1 Sampling Mechanism . . . . . . . . . . . . . . . . . . . 3.3.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . .

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3.4 Observer-Based Event-Triggered Control 3.4.1 Observer Design . . . . . . . . . . . . 3.4.2 Sampling Mechanism . . . . . . . . . 3.4.3 Stability Analysis . . . . . . . . . . . . 3.5 Minimum Inter-Event Interval . . . . . . . . 3.6 Illustrative Example . . . . . . . . . . . . . . . 3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . .

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Event-Triggered Control for Switched Linear Delay Systems . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Problem Statement and Preliminaries . . . . . . . . . . . . . . . . . 4.3 State-Based Event-Triggered Control . . . . . . . . . . . . . . . . . 4.3.1 Sampling Mechanism . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Observer-Based Event-Triggered Control . . . . . . . . . . . . . . 4.4.1 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Sampling Mechanism . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Event-Triggered Control of Switched Linear Neutral 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Problem Statement and Preliminaries . . . . . . . . . . 5.2.1 Switched Linear Neutral System . . . . . . . . 5.2.2 Observer Design . . . . . . . . . . . . . . . . . . . 5.2.3 Event-Triggered Sampling Mechanism . . . 5.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Stability Analysis . . . . . . . . . . . . . . . . . . . 5.3.2 Minimum Inter-Event Interval . . . . . . . . . 5.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Periodic Sampled-Data Control for Switched Linear Neutral Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Problem Statement and Preliminaries . . . . . . . . . . . . . . . . 6.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Sampling Interval with No Switch . . . . . . . . . . . . 6.3.2 Sampling Interval with One Switch . . . . . . . . . . . . 6.3.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Hysteresis Switching Control for Switched Linear Neutral Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 System Description and Preliminaries . . . . . . . . . . . . . . 7.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8

Reliable Control for a Class of Switched Nonlinear Systems 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Problem Statement and Preliminaries . . . . . . . . . . . . . . . . 8.3 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Fault-Tolerant Control for a Class of Uncertain Switched Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Problem Statement and Preliminaries . . . . . . . . . . . . . . 9.3 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Description of the Haptic Display System . . . . . 9.4.2 Formulated Problem . . . . . . . . . . . . . . . . . . . . . 9.4.3 Simulation and Analysis . . . . . . . . . . . . . . . . . . 9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Notations and Acronyms

■ ♦ ▲ , 2 8 P R Rþ Rn Rnm N max min inf ‚min ðÞ ‚max ðÞ rðÞ I In 0 0nm PT P1 diag P[0 P\0 P0 P0 jj

End of proof End of remark End of example Is defined as Belongs to For all Sum Field of real numbers Field of positive real numbers Space of n-dimensional real vectors Space of n  m real matrices Field of nonnegative integral numbers Maximum Minimum Infimum Minimum eigenvalue of a matrix Maximum eigenvalue of a matrix Switching signal Identity matrix n  n identity matrix Zero matrix Zero matrix of dimension n  m Transpose of matrix P Inverse of matrix P Block diagonal matrix with blocks fP1 ; . . .; Pm g P is real symmetric positive definite P is real symmetric negative definite P is real symmetric positive semi-definite P is real symmetric negative semi-definite Euclidean vector norm

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‖‖  LMI LMIs ZOH PWM CSTR HþD

Notations and Acronyms

Euclidean matrix norm (spectral norm) Symmetric terms in a symmetric matrix Linear matrix inequality Linear matrix inequalities Zero-order holder Pulse width modulation Continuous stirred tank reactor Human þ device

List of Figures

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

1.1 1.2 1.3 1.4 1.5 1.6 1.7 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 3.1 3.2

Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6 Fig. 3.7 Fig. 3.8

Switched system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Switching control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PWM-driven boost converter . . . . . . . . . . . . . . . . . . . . . . . . . Schematic diagram of the process . . . . . . . . . . . . . . . . . . . . . . Architecture of Hopfield neutral network . . . . . . . . . . . . . . . . Architecture of switched Hopfield neutral network . . . . . . . . . Drilling system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diagram of observer-based event-triggered switched system . . State responses of the error system . . . . . . . . . . . . . . . . . . . . . State responses of the observer . . . . . . . . . . . . . . . . . . . . . . . . State responses of the closed-loop system (2.39) . . . . . . . . . . Switching signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Triggering condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Triggering instants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diagram of state-based event-triggered switched system . . . . . State responses of system (3.44) under triggering condition (3.3) with g ¼ 1 and e ¼ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results under triggering condition (3.3) with g ¼ 1 and e ¼ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . State responses of system (3.44) under triggering condition (3.3) with g ¼ 1 and e ¼ 0:1 . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results under triggering condition (3.3) with g ¼ 1 and e ¼ 0:1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . State responses of system (3.44) under triggering condition (3.3) with g ¼ 1 and e ¼ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results under triggering condition (3.3) with g ¼ 1 and e ¼ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results under triggering condition (3.25) with e ¼ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Figures

Fig. 3.9 Fig. 3.10 Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.1 6.2 6.3 6.4 7.1 7.2 7.3 7.4

Fig. Fig. Fig. Fig. Fig. Fig. Fig.

7.5 7.6 8.1 8.2 8.3 9.1 9.2

Fig. 9.3 Fig. 9.4

Simulation results under triggering condition (3.25) with e ¼ 0:1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results under triggering condition (3.25) with e ¼ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diagram of asynchronous switching . . . . . . . . . . . . . . . . . . . . Diagram of the piecewise Lyapunov function . . . . . . . . . . . . . State responses of the closed-loop system (4.47) . . . . . . . . . . Control input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Event-triggering condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . Switching signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Triggering instants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sampling states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of asynchronous switching . . . . . . . . . . . . . . . . . . State responses of the observer . . . . . . . . . . . . . . . . . . . . . . . . State responses of the error system . . . . . . . . . . . . . . . . . . . . . State responses of the closed-loop system (5.51) . . . . . . . . . . Sampling states under the ZOH . . . . . . . . . . . . . . . . . . . . . . . Switching signals of the system and the controller . . . . . . . . . Triggering condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Triggering instants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . State response of the closed-loop system (6.41) . . . . . . . . . . . Switching signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sampled state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Lyapunov functions xT Nij x\0 . . . . . . . . . . . . . . . . . Multiple Lyapunov functions xT Nij x ¼ 0. . . . . . . . . . . . . . . . . Multiple Lyapunov functions xT Nij x [ 0. . . . . . . . . . . . . . . . . State trajectories of system (7.29) under switching law (7.32) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control input under switching law (7.32) . . . . . . . . . . . . . . . . Switching signal (7.32) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . System state responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Switching sequences of gain matrices . . . . . . . . . . . . . . . . . . . Switching signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A system with a haptic device . . . . . . . . . . . . . . . . . . . . . . . . State responses (case where faults considered in controller design) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zoomed-in unstable state responses (case where faults not considered in controller design) . . . . . . . . . . . . . . . . . . . . Switching signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

50 55 55 73 73 74 74 75 75 81 94 95 95 96 96 97 97 98 113 114 114 115 120 121 121

. . . . . . .

. . . . . . .

127 128 128 138 139 140 148

. . 151 . . 151 . . 152

Chapter 1

Introduction

1.1 Switched System Switched systems as a special class of hybrid systems are developed and modeled arising from many engineering systems since there are many parameters would change to adapt different system operating status during the running process [4, 58, 63, 82]. Switched systems have attracted increasing attention of a large number of researchers from all over the world. Switched systems are regarded as consisting of a family of continuous-time or discrete-time systems with discrete switching rules. Dynamic performances of a swtiched system are determined by both of subsystems and a switching rule. Switched systems are divided into different classes according to their different inherent characters, such as switched linear systems [2, 3, 35–37, 88–90], switched nonlinear systems [31, 45, 69, 77, 133], continuous-time switched systems [29, 96, 130], discrete-time switched systems [64, 118, 119, 121], switched delay systems [86, 87, 110], switched neutral systems [52, 99, 104, 105, 126], switched stochastic systems [56, 75], switched positive systems [132] and so on. The inherent hybrid dynamic behavior of switched systems makes stability analysis and control synthesis more complex than non-switched systems. The specific analysis tools are henceforth developed for different types of switched systems, see [13, 43, 58, 60, 62, 130] and the references therein. In this book, we are interested in continuous-time switched linear/nonlinear systems and continuous-time switched linear delay/neutral systems as our focus of study. In order to make the expression more concise, we name them switched linear/nonlinear systems and switched delay/neutral systems instead if without special emphasis.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. Fu and T.-F. Li, Event-Triggered Control of Switched Linear Systems, Studies in Systems, Decision and Control 365, https://doi.org/10.1007/978-3-030-71604-2_1

1

2

1 Introduction

Fig. 1.1 Switched system

1.1.1 Switched Linear/Nonlinear System Generally speaking, a switched system is composed of a family of subsystems and a logic rule that governs the switching among subsystems, and is mathematically described by x(t) ˙ = f σ (x(t)), x(t0 ) = x0

(1.1)

where x(t) is the system state, σ : [0, ∞) → M = {1, 2, . . . , m} is a rightcontinuous piecewise constant function called switching signal that orchestrates switching between subsystems. x0 is the initial state. A diagram of a switched system is shown schematically in Fig. 1.1. Switched system (1.1) is called a switched nonlinear system if at least one subsystem is nonlinear. If all subsystems are linear, then system (1.1) is called a switched linear system which can be further described by x(t) ˙ = Aσ x(t), x(t0 ) = x0

(1.2)

where Ai , i ∈ M, is a constant matrix which defines subsystem i of switched system (1.2). In addition to switched systems, switching control appears as an effective control strategy in switched system field [44, 58]. For a given process, in some situations, we cannot find a controller such that the closed-loop system displays a desired behavior. In other cases, a continuous feedback law that solves the problem may not exist. A possible alternative in such situations is to incorporate logic-based decision into the control law and implement switching among a family of controllers. This may yield a switched closed-loop system [58]. A diagram of switching control is shown in Fig. 1.2.

1.1 Switched System

3

Decision Maker

Controller 1 Controller 2

Switching Signal

Process

Fig. 1.2 Switching control

Switched systems have attracted increasingly attention since the 1990s, especially for switched linear systems. Switched systems deserve investigating for both theoretical and practical reasons. Many operating modes, such as abrupt parameters variation and sudden structure change, can be modelled by switched systems. Here, we show two examples of system modelling to present the background of the switched linear system and the switched nonlinear system. Example 1.1 ([133] (Switched linear system)) Consider a simplified PWM-driven boost converter shown in Fig. 1.3. There are two storage elements in the circuit: inductor L and capacitor C. In addition, the source voltage and load are, respectively, represented by E and R. The PWM-driven switching signal s(t) that controls the on 1 and off 0 state of the switch is generated by comparing a reference signal Vr e f and a repetitive triangular waveform. That is, s(t) ∈ {0, 1}. Then, the differential equations for the boost converter are given as follows: 1 1 νC (t) + (1 − s(t)) i L (t), RC C 1 1 i˙L (t) = −(1 − s(t)) νC (t) + s(t) E. L L

ν˙C (t) = −

(1.3)

Define x1 (t) = νC (t), x2 (t) = i L (t), u(t) = E, σ(t) = s(t) + 1, and  1 1    1    0 − RC C 0 − RC 0 , B1 = A1 = , B2 = 1 . , A2 = 0 0 0 − L1 0 L Then the boost converter can be described as a switched linear system by x(t) ˙ = Aσ x(t) + Bσ u(t), σ ∈ {1, 2}.

(1.4) 

4

1 Introduction

Fig. 1.3 PWM-driven boost converter

Fig. 1.4 Schematic diagram of the process

Example 1.2 ([7] (Switched nonlinear system)) Consider a CSTR fed by a single inlet stream through a selector valve which is connected to two different source streams, shown in Fig. 1.4.

1.1 Switched System

5

The CSTR at each operating mode can be described by  qi  E C A fi − C A − a0 e− RT C A , C˙ A = V  qi  E ˙ T fi − T − a1 e− RT C A + a2 (Tc − T ), T = V

(1.5)

where i ∈ {1, 2} represents the arbitrary signal which determines the position of the selector valve or equivalently the reactor mode, C A is the reactant A concentration, T is the reactor temperature, qi is the feed flow rate, V is the volume of the reactor, C A fi is the concentration, T fi is the temperature, E is the activation energy, Tc is a variable temperature of the coolant steam, R is a gas constant, a0 , a1 and a2 are constant coefficients. In this model, it is assumed that the position of the selector valve is determined by an arbitrary signal and this signal which determines the mode of the reactor is known at each time. Defining the states x1 = C A − C A , x2 = T − T  and the control input u = Tc − Tc , then system (1.5) can be written in the form of switched nonlinear system: x˙ = f i (x) + g i (x)u, i ∈ {1, 2}

(1.6)

where x = (x1 , x2 )T , f i (x) = ( f 1i (x), f 2i (x))T , g i (x) = (g1i (x), g2i (x))T , and qi V qi i f2 = V g1i = 0, f 1i =

  − E/R C A fi − C A − x1 − a0 (x1 + C A )e x2 +T  ,     − E/R T f i − T  − x2 − a1 e x2 +T  (x1 + C A ) + a2 TC − x2 − T  ,

g2i = a2 .  Up to now, stability and control synthesis of switched linear systems have obtained systematic results [2, 3, 8, 24, 62, 107, 108]. Despite switched linear systems are relatively easy to handle and the rapid progress has been made for them, many fundamental problems are still either unexplored or less well understand. While for switched nonlinear systems, there are still no systematic method for control synthesis, even for stability and stabilization [111, 129, 130]. Many existing analysis approaches are proposed for switched nonlinear systems with special structures [73, 95]. Therefore, switched systems, no matter switched linear systems or switched nonlinear systems, need to be further studied.

6

1 Introduction

1.1.2 Switched Delay/Neutral System Switched delay systems and switched neutral systems are both special classes of switched systems arising from practical processes. A switched delay (neutral) system is defined that at least one subsystem is a delay (neutral) system in a switched system [86] ([85]). In fact, a switched neutral system can be regarded as a special case of a switched delay system since a neutral system is actually a delay system which contains delays not only in state but also in derivative of the state [9, 10, 19, 30, 38, 39]. Delay exists almost everywhere for a control system due to the time it takes to acquire the information needed for decision-making, to create control decisions, and to execute these decisions. Delay terms arise from many different control systems, including process control, network control, biology control and intelligent control [79]. Due to the complexity of internal structure of switched systems and the universal existence of delays, it is necessary to study switched delay systems which can not only complete the theoretical knowledge of switched systems but also provide methodology for applications of switched delay systems. In the followings, we mainly focus on switched linear delay systems and switched linear neutral systems. A switched linear delay system can be described by x(t) ˙ = Aσ x(t) + Bσ x(t − h(t)) + Cσ u(t), t > t0 x(t0 ) = x(t0 + θ) = φ(θ), − h¯ < θ < 0

(1.7)

where x(t) is the system state, u(t) is the control input, σ : [0, ∞) → M = {1, 2, . . . , m} is the switching signal, {(Ai , Bi , Ci )|i ∈ M} is a pair of constant matrices which defines subsystem i of system (1.7). h(t) denotes a discrete time-varying delay with boundary condition ¯ h < h(t) < h,

(1.8)

where h and h¯ are constants. t0 is the initial time. φ(θ) is the initial continuous function. A switched linear neutral system can be described by x(t) ˙ − Bσ2 x(t − h 2 (t)) = Aσ x(t) + Bσ1 x(t − h 1 (t)) + Cσ u(t), t > t0 x(t0 ) = x(t0 + θ) = φ(θ), − max{h¯ 1 , h¯ 2 } < θ < 0 (1.9) where {(Ai , Bi1 , Bi2 , Ci )|i ∈ M} is a pair of constant matrices defined subsystem i of system (1.9). h 1 (t) and h 2 (t) denote a discrete time-varying delay and a neutral time-varying delay, respectively, with boundary conditions h 1 < h 1 (t) < h¯ 1 , h 2 < h 2 (t) < h¯ 2 ,

(1.10)

1.1 Switched System

7

Fig. 1.5 Architecture of Hopfield neutral network

where h 1 , h¯ 1 , h 2 and h¯ 2 are constants. In the followings, we present two examples to show the background of the switched delay system and the switched neutral system. Example 1.3 ([47] (Switched delay system)) A Hopfield neural network with timevarying delays, shown in Fig. 1.5, can be modelled by x(t) ˙ = −C x(t) + B f (x(t − τ (t))) + J,

(1.11)

where x(t) is the state vector associated with neurons, B is the interconnection matrix, C is a positive diagonal matrix, f (·) is the neuron activation function vector, J is a constant external input vector, τ (t) is the transmission delay. Let u(t) = x(t) − x  , where x  the unique equilibrium point which satisfying −C x  + B f (x  ) + J = 0. System (1.11) can be written by u(t) ˙ = −Cu(t) + Bg(u(t − τ (t))),

(1.12)

where g(u(·)) = (g1 (u 1 ), g2 (u 2 ), . . . , gn (u n ))T and gi (u i (·)) = f i (u i (·) + xi ) − f i (xi ). Therefore, the prototypical architecture of switched Hopfield neural networks with parametric uncertainty, shown in Fig. 1.6, can be described by u(t) ˙ = −(Cσ + ΔCσ (t))u(t) + (Bσ + ΔBσ (t))g(u(t − τ (t))),

(1.13)

where σ is a switching signal which takes its values in the finite set I = {1, 2, . . . , N }, ΔCi (t) and ΔBi (t), i ∈ I, are parametric uncertainties and continuous matrix-valued functions of t. 

8

1 Introduction

Fig. 1.6 Architecture of switched Hopfield neutral network

Example 1.4 ([80, 81] (Switched neutral system)) A drilling mechanical system, shown in Fig. 1.7, can be described by the partial differential equation: GJ

∂2θ ∂θ ∂2θ (ξ, t) − I (ξ, t) − β (ξ, t) = 0, ξ ∈ (0, L), t > 0 2 2 ∂ξ ∂t ∂t

(1.14)

with boundary conditions   ∂θ ∂θ G J (0, t) = ca (0, t) − Ω(t) , ∂ξ ∂t   ∂2θ ∂θ ∂θ (L , t) , G J (L , t) + I B 2 (L , t) = −T ∂ξ ∂t ∂t

(1.15)

where θ(ξ, t) is the angle of rotation, I is the inertia, G is the shear modulus and J is the geometrical moment of inertia. The drilling behaviour can be described by ω(t) ¨ − Υ ω(t ¨ − 2Γ ) + Ψ ω(t) ˙ + Ψ Γ ω(t ˙ − 2Γ )   1 ca 1 Ω(t − Γ ), ˙ + Υ T ω(t ˙ − 2Γ ) + 2Ψ = − T ω(t) √ IB IB ca + I G J

(1.16)

where ω(t) ˙ is the angular velocity at the bottom extremity, and Υ =

ca − ca +

√ √

IGJ IGJ

√ , Ψ =

IGJ , Γ = IB



I L. GJ

Setting x1 = ω, x2 = ω, ˙ x = (x1 , x2 )T , u(t) = Ω(t), τ1 = 2Γ, τ2 = Γ , then the behaviour of the oil well drilling system at the bottom extremity can be rewritten by x(t) ˙ − C x(t ˙ − τ1 ) =Ax(t) + Bx(t − τ1 ) + Du(t − τ2 ) + f 1i (t, x2 (t)) + f 2i (t, x2 (t − τ1 )), i ∈ {1, 2},

(1.17)

1.1 Switched System

9

Fig. 1.7 Drilling system

where x1 (t), x2 (t) are the angular position and velocity of the drill string at the bottom end, respectively, and A, B, C and D are given by  0 1 A= 0 −Ψ −

 cb IB

0 , B= 0

Υcb IB

    0 0 0 0 ,C = ,D = , 0Υ Π −ΥΨ

 √ √ 2Ψca I √I G J , Ψ = I G J , Π = √ , τ = L , τ1 = 2τ2 . with Υ = cca − 2 I GJ B ca + I G J a+ I G J System (1.17) is a switched neutral system and functions f 1i (t, x2 (t)), f 2i (t, x2 (t − τ1 )) switch according to the rule ⎧ ⎪ ⎨ f or x2 = 0 : f 11 (t, x2 (t)) = f 21 (t, x2 (t −γbτ1 )) = 0 − x2 (t) f or x2 > 0 : f 12 (t, x2 (t)) = −c1 − c2 e ν f γb ⎪ ⎩ f 22 (t, x2 (t − τ1 )) = c1 Υ + c2 Υ e ν f x2 (t − τ1 ) with c1 =

Wob Rb μcb IB

and c2 =

Wob Rb (μsb IB

− μcb ).

(1.18)



10

1 Introduction

1.2 Stability and Stabilization Stability is one of the most basic and important properties of control systems, especially for switched systems [36, 37, 58, 129]. The distinguish of stability analysis between switched systems and non-switched systems is that the dynamical behavior of switched systems is determined by the interaction of a finite number of subsystems and a discrete switching law, and therefore three basic problems on stability and design of switched systems are proposed in [60] which are described below: Problem A. Find conditions that guarantee that the switched system is asymptotically stable for any switching signal. Problem B. Identify some classes of switching signals for which the switched system is asymptotically stable. Problem C. Construct a switching signal that makes the switched system asymptotically stable. Problem A is of great importance for the situation when a given plant is being controlled by means of switching among a family of stabilizing controllers, each of which is designed for a specific task. The general existing analysis approach for solving this problem is to find whether a common Lyapunov function exists or not. The existence of a common Lyapunov function for all subsystems has been proved as a sufficient and necessary condition for a switched system to be asymptotically stable under arbitrary switching, and some works have devoted on this problem, see for example [73, 83]. Although Problem A is increasingly being concerned, a necessary condition for stability under arbitrary switching is that all of the individual subsystems are asymptotically stable. However, because of the existing of switching law, a switched system may not stable even all individual subsystems are stable and what’s more interesting is that a switched system may stable under a suitable switching law even all individual subsystems are not stable. Problem B is henceforth proposed for that one can identify some switching signals to make the switched systems asymptotically stable which allows some individual subsystems unstable. Subsequently, multiple Lyapunov function approach which targets at this special characters of switched systems is proposed to be an effective way to solve this problem [13, 28, 130]. Problem C is proposed for that how to design a suitable switching law to make the switched system stable even there are some unstable subsystems. This problem reflects the characteristics of switched system since the discrete dynamic of switching mechanisms can determine the global performance of the switched system. An average dwell time switching strategy is proposed arising from dwell time approach since a switched system could be stable if make the stable subsystem active enough long time to dissipate the transient effects of switching and restrict the unstable subsystems to be active in a relatively short time. It has been turned out that the average dwell time approach is useful in stability analysis and control synthesis of switched systems, see for example [115, 116, 119, 120, 134] and the references therein. Besides the time-dependent switching approach, such as dwell time and average dwell time approaches, state-dependent switching is another kind of important

1.2 Stability and Stabilization

11

switching strategy. When all the individual subsystems are unstable, it is possible to find a state-dependent switching law that renders the switched system asymptotically stable. A typical switching law is given by the state-dependent switching strategy such as the convex combination method [54, 58] and the min-projection strategy [58]. In recent years, there is enormous growth of literature in analysis and design of switched systems. The reader can refer to [88] for surveys of recent developments. Stability analysis of switched delay systems is another popular issue and have attracted lots of researchers’ attention. Switched delay systems can be seen as a special case of switched systems from the view of the definition, and switched delay systems can also be seen as a more general case of switched systems from the view of the system structure. Since a switched delay system consists of a family of delay subsystems and a switching law, multiple Lyapunov–Krasovskii functional method is developed by considering the special characters of switched delay systems [50, 53, 55, 56]. Average dwell time technique and state-based switching strategies have also been adopted to deal with switched delay systems, respectively, and synthesis problem of switched delay systems, such as stabilization, H∞ control, sampled-data control, have been studied, see the Refs. [65–67, 71, 72, 109, 117] and the references therein.

1.3 Sampled-Data Control Due to the rapid progress of computer and digital technologies, most practical industrial systems are continuous but with digital sensors and computing controllers, the feedback signals are discrete (sampled), that is to say, not all information of state can be obtained in a continuous manner. Sampled-data control is hence proposed and becomes one of interesting topics in control theory, which has attracted increasing attention in recent years [6, 17, 18, 41, 46]. In sampled-data control field, periodic sampling and event-triggered sampling are two popular sampling mechanisms, which we introduce separately in the following.

1.3.1 Periodic Sampled-Data Control Periodic sampled-data control is based on a sampling mechanism that all the sampling times are determined by a fixed period. Periodic sampled-data control is of simple structure and is easy to design, analyze and realize. It can reduce the execution cost compared with continuous feedback. From a practical implementation point of view, sampled-data controllers are more favourable in practical applications due to the rapid progress of computer and digital technologies and non-approximate treatment of the intersample behavior from sampled-data control’s own features, which results in no degradation of the closed-loop performance [17], and therefore, much more attention have been paid on periodic sampled-data control [33].

12

1 Introduction

Periodic sampled-date control of switched systems has gradually arised more and more concern following the development of switched systems [33, 34, 61, 124]. Due to the special structure of switched systems, it is necessary to consider asynchronous switching problem when one studies sampled-data control for switched systems, which stems from the mismatch between the subsystem and its matched controller caused by the sampling mechanism. In [120], asynchronous switching control of switched systems with average dwell time technique is studied in both continuous-time and discrete-time situations. In [56, 131], the authors deal with asynchronous switching control for deterministic switched linear systems and stochastic ones, respectively. Since most practical systems are continuous-time, for which one usually either directly designs continuous-time controllers or first discretizes the continuous systems and then develops the corresponding discrete-time controllers, see for example [99, 106]. The above mentioned references are concerned with asynchronous switching control of switched systems assuming that asynchronous switching is caused by transmitted delays, and hence no discrete state yields during the feedback process. However, asynchronous switching induced by sampled-data control is different since discrete states are brought by sampling mechanisms. The authors of [94] propose a finite-time stabilizer for a class of switched linear systems under asynchronous switching. The authors of [59] present an important result on sampled-data quantized state feedback stabilization of switched linear systems by using an encoding and control strategy. In [29], the authors investigate the stabilization problem of switched linear systems for both the known switching process and the unknown switching setting, for which cases the dwell time technique and the online adaptive estimation method are combined with the sampled-date control, respectively. In [61], the authors study the robust delay-dependent H∞ control for a class of switched linear delay systems. However, the authors convert the sampleddata control problem into a time-delay one and used time-varying delay approach instead of sampled-date feedback control input. Furthermore, impacts from sampling or from asynchronous switching are not considered on stabilization of switched systems. Moreover, although these references give sampled-data control strategies for switched systems with their own spectacular features, they are not able to cope with stabilization of switched neutral systems under either asynchronous or synchronous switching because time delays appear not only in states but also in state derivative of the considered system (see (1.9)). Thus, sampled-data control for switched delay systems, especially for switched neutral systems needs to be further studied.

1.3.2 Event-Triggered Sampled-Data Control Despite periodic sample-data control is with the advantages of simple structure and is easy to analyze, design and realize, it leads to unnecessary waste of communication and computation resources. Event-triggered sampled-data control is hence developed in the digital implementation of real time control systems, by which the control task is executed after an occurrence of an external event, generated by some well-

1.3 Sampled-Data Control

13

designed event-triggered sampling mechanisms. Compared with periodic sampleddata control, event-triggered control is with a more flexible and practical sampling strategy. The conspicuous advantage of event-triggered control is that it can significantly reduce the number of control task executions while retaining a satisfactory closed-loop performance. Several event-triggered control strategies have been developed in the early works, see [5, 6, 11, 16, 25, 27, 42, 51, 70, 84, 91, 92, 97, 98, 123, 125]. However, these works all focus on non-switched systems and few theoretical results focus on studying event-triggered control of switched delay systems, even on switched linear systems. For event-triggered control of switched systems, there are two situations for the active function of sampling mechanism need to be considered, one is that the sampler does not transmit the information of the switching signal when an event is triggered, and therefore no asynchronous switching problem yields. The other situation is that the sampler transmits the information of the switching signal when an event is triggered, and asynchronous switching problem has to be considered which brought difficulty on stability analysis. Moreover, in event-triggered control, the execution of control tasks occurs in a varying period, and therefore an infinite number of events may generate in finite time (the so called zeno behavior). Zeno behavior needs to be avoided. In this book, we present some results on event-triggered control of switched systems and switched delay systems.

1.4 Reliable and Fault-Tolerant Control Uncertainties are inevitable in real-world applications. It may destabilize a closedloop system if it was ignored when the controller is designed [78]. Uncertainties are roughly classified into parametric uncertainties and structural uncertainties [68, 135]. For parametric uncertainties, adaptive estimators are usually designed. For structural uncertainties, robust controllers are usually used to compensate for it when it is norm-bounded and also stabilize the systems [49]. When uncertainties are involved in switched systems, much attention has been paid on robust control of switched systems, see [48, 107, 113, 119, 122, 129]. Since the nature of hybrid dynamic systems is inherently nonlinear, studies of switched nonlinear systems have become one of the key topics in the control community. On the other hand, maintenances or repairs in the highly automated industrial systems cannot be always achieved immediately, for preserving safety and reliability of the systems, the possibility of occurrence and presence of uncertain faults must be taken into account during the system analysis and control design to avoid life-threatening prices and heave economic costs caused by faults [23, 40, 95, 103, 127], which makes fault-tolerant control attract more and more attention [57, 76, 93, 100]. Fault-tolerant control of switched nonlinear systems is one of promising research interests due to the fact that many practical systems can be cast into hybrid dynamic systems. Compared with available results on switched systems and traditional fault-tolerant control, there are very few results on fault-tolerant control of

14

1 Introduction

switched nonlinear system [26, 48, 112]. Reference [48] adopts the way in [93] to deal with fault actuators. In addition, structural uncertainties of input matrices were not considered for this class of nonlinear switched systems in [48]. By uniting safe-parking and reconfiguration-based approaches for a class of switched nonlinear systems, [26] proposes two switching strategies to realize fault-tolerant controls against actuator faults. But when actuator faults occur, the two methods both need to determine reparation time for faulty actuators, which is not easy to acquire in many real-world scenarios. Reference [112] studies observer-based fault-tolerant control of a class of switched nonlinear system with external disturbances. From a system structure point of view, the nonlinear item is connected to the system in a parallel way, which can be directly compensated for by control signals. However, it is worth noting that fault-tolerant control for nonlinear cascade systems is much more complicated, where the nonlinear term, as a nonlinear subsystem, is cascaded to the other subsystem. Thus, [32] proposes a fault-tolerant controller for nonlinear cascade systems where the uncertainties in the linear subsystem are norm-bounded. Since the norm-boundedness is used, it makes upper bound of the uncertainties are too large, thus it makes gains of designed controllers deviate from what it could be, and it also may consume more energy than what really would need. Chapters 8 and 9 study the problem of robust fault-tolerant control of switched cascade nonlinear systems with structural uncertainties existing in both system matrices and input matrices, and proposes a fault-tolerant control method for this class of switched systems by using average dwell-time techniques.

1.5 Outline of the Book Structure of the book is summarized as follows. Chapter 2 studies the observer-based event-triggered sampled-data control problem for a class of continuous-time switched linear systems. First, an exponential observer is designed for the switched linear system to support to realize feedback control. Then, an event-triggered sampling mechanism is proposed, under which a sampled-data-based switching controller is constructed to achieve stabilization combining with the average dwell time technique. A sufficient condition for exponential stability of the closed-loop switched linear system is obtained by using multiple Lyapunov function method. Different from the existing work, multiple switches are allowed to happen between arbitrary two consecutive triggering instants in this study, which relaxes the constraint that at most one switching allowed within every consecutive sampling interval adopted in the existing references. A numerical example is given to show the effectiveness of the proposed method. Chapter 3 studies both the state-based and the observer-based output feedback event-triggered sampled-data control problems for a class of continuous-time switched linear systems. An improved event-triggered sampling mechanism is introduced, under which a sampled-data switching controller is set to achieve Lyapunov stability of the closed-loop switched system while satisfying the average dwell time

1.5 Outline of the Book

15

switching strategy. Comparing with the triggering condition proposed in Chap. 2, a constant ε is added into the proposed event-triggered sampling mechanism such that the sampling period and the system performance can be improved by adjusting the value of ε. Moreover, if we let ε = 0, the improved event-triggered sampling mechanism would degenerate into the one considered in Chap. 2. Zeno behavior in the sampling process of event-triggered control can be excluded from the theoretical analysis under the improved triggering condition. The advantage of the proposed method is presented by a simulation example. Chapter 4 studies both the state-based and the observer-based event-triggered sampled-data control for a class of switched linear delay systems with asynchronous switching. Compared with the switched non-delay systems, switched delay systems are with more complex system structure in which at least one subsystem is a delay system. We extend the event-triggered sampled-data control approach proposed in Chap. 2 to study switched linear delay systems, in which an event is triggered whenever an error exceeds a dynamic threshold. A sampled-data-based switching controller is built to achieve stabilization combining with the average dwell time technique. Two sufficient conditions for exponential stability of the closed-loop switched delay system are obtained by using multiple Lyapunov–Krasovskii functional method. Some free weighting matrices are introduced to help achieving the sufficient conditions. In Chap. 5, we study the observer-based output feedback stabilization problem for a class of switched linear neutral systems under an event-triggered sampling mechanism and for the first time address event-triggered control of switched neutral systems. Switched neutral systems are a special kind of switched delay systems but with more complexity structure in which at least one subsystem is a neutral system. As we all know that a neutral system is a more general delay system that delay is contained not only in its state but also in the derivation of its state which bring more difficulty in stability analysis, and certainly more complexity for switched neutral systems. Furthermore, we study the asynchronous switching control caused by the constructed event-triggered sampling mechanism. An event will be triggered if the condition is satisfied and then the current state and the active mode will be transmitted to controller to decide the switching of controller which lead to asynchronous switching between the switched system and the switching controller. Under the observer-based controller, combining with switching policy provided that average dwell time conditions of subsystems are satisfied, we propose a sufficient condition which guarantees exponential stability of the closed-loop switched neutral system. Chapter 6 studies the periodic sampled-data control problem for a class of switched linear neutral systems. A sampled-data-based switching controller is set up for stabilization. By analyzing the relationship between the sampling period and the dwell time, a bond between the sampling period and the average dwell time is revealed to form a switching condition. Subject to this switching condition and certain control gains related constraints, exponential stability of the closed-loop switched neutral system is guaranteed. We also consider asynchronous switching which is caused by the sampling link. Moreover, delays, including state delay and neutral delay, are all time-varying ones with boundary, and thus the result of this work is less conservative

16

1 Introduction

than the ones with constant delays. Some free weighting matrices are introduced to obtain the decoupled constraints, and therefore technically reduces the computational complexity. The proposed method is verified by a numerical example. Chapter 7 studies stabilization of switched linear neutral systems based on a statebased switching strategy. An improved state-based switching strategy which is called hysteresis-based switching strategy is proposed by generalizing the well-known minswitching strategy. A multiple generalized Lyapunov–Krasovskii functional is constructed which allows some increase of connecting adjacent Lyapunov–Krasovskii functional at switching points. A delay-dependent sufficient criterion is obtained to guarantee asymptotic stability of switched neutral systems. A numerical example is given to show the effectiveness of the proposed method. Chapter 8 considers the problem of robust fault-tolerant control of a class of switched cascade nonlinear systems with structural uncertainties existing in both system matrices and input matrices, and proposes a fault-tolerant control method for this class of switched systems by using average dwell time techniques. The proposed control design works on both the switched nonlinear systems with actuator faults and its nominal systems without necessarily changing any structures and/or parameters of the proposed controllers. The proposed method treats all actuators without necessarily classifying all actuators into faulty actuators and robust ones in a unified way for easy and practical applications. Chapter 9 presents a fault-tolerant control method for switched cascade nonlinear systems where the nonlinear subsystem contains structural uncertainties and also whose actuators are allowed to partially fail. We first define the representation of partial actuator fault models, and then find a common Lyapunov function to design a controller to asymptotically stabilize switched systems. The proposed method is invariant to actuator fault modes in the sense that the proposed controller is completely independent from the fault modes; as a result, the proposed method works well for all the possible actuator fault modes (except the case that all actuators completely fail) without needing to modify the controller. The proposed method is independent of switching policies provided that a common Lyapunov function is found. The proposed method is verified using a simulated haptic display system with switched virtual environments. Chapter 10 summarizes the results of the book.

Chapter 2

Observer-Based Event-Triggered Control for Switched Linear Systems

2.1 Introduction In this chapter, we aim to study the event-triggered sampled-data control problem for a class of switched linear systems. Since system states cannot be measured directly in some special situations of practical systems, see [20, 21], a state observer is first designed and then an event-triggered sampling mechanism is constructed according to the observer state. The feature of this chapter lies in three aspects: (1) An exponential observer is designed for a switched linear system to estimate the system state which helps to realize feedback control. (2) An event-triggered sampling mechanism is proposed for the switched linear system based on the designed observer state, under which a sampled-data-based switching controller is constructed for the switched system to achieve stabilization while satisfying an average dwell time technique. (3) Multiple switches are allowed to happen between arbitrary two consecutive triggering instants, which relaxes the constraint that at most one switch is allowed within every consecutive sampling interval adopted in the existing references. (4) A sufficient condition is obtained for exponential stability of the closed-loop switched system by using multiple Lyapunov function method. The rest of this chapter is organized as follows. In Sect. 2.2, we describe the control system and give some preliminaries. Main results, including the design of an exponential observer, an event-triggered sampling mechanism and a sampleddata-based switching controller, and stability analysis of the closed-loop system, are presented in Sect. 2.3. In Sect. 2.4, a numerical example is presented to show the effectiveness of the proposed method. Section 2.5 concludes the whole chapter.

2.2 Problem Statement and Preliminaries Consider the continuous-time switched linear system © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. Fu and T.-F. Li, Event-Triggered Control of Switched Linear Systems, Studies in Systems, Decision and Control 365, https://doi.org/10.1007/978-3-030-71604-2_2

17

18

2 Observer-Based Event-Triggered Control for Switched Linear Systems



x(t) ˙ = Aσ x(t) + Bσ u(t), x(t0 ) = x0 y(t) = Cσ x(t),

(2.1)

where x(t) ∈ Rn is the system state, u(t) ∈ Rm is the control input, y(t) ∈ Rq is the measurable output, x0 is the initial state, σ : [0, ∞) → M = {1, 2, . . . , m} is a right-continuous piecewise constant function called switching signal that orchestrates switching between the subsystems, M is a finite index set, {(Ai , Bi , Ci ) : i ∈ M} is a collection of matrix pairs defining the individual subsystem of system (2.1). It is assumed that the pairs (Ai , Bi ) and (Ai , Ci ) are controllable and observable, respectively. Corresponding to switching signal σ, there exists a switching sequence {(l0 , t0 ), (l1 , t1 ), . . . , (li , ti ), . . . |li ∈ M, i ∈ N} ,

(2.2)

∞ with which means that the li th subsystem is active when t ∈ [ti , ti+1 ), where {ti }i=0 ti < ti+1 denotes a switching time sequence. We denote the number of discontinuities of switching signal σ on a semi-open interval (t, s] by Nσ (s, t). Without loss of generality, we assume that solution x(t) of system (2.1) is continuous everywhere. In this chapter, our task is to obtain a sufficient condition under which exponential stability of the closed-loop system (2.1) can be guaranteed with an event-triggered sampling mechanism and a sampled-data-based switching controller. Some useful definitions and lemmas are introduced before the main results.

Definition 2.1 ([13]) If there exists a number τd > 0 such that any two switches are separated by at least τd , then τd is called the dwell time. Definition 2.2 ([13]) If there exists positive numbers τa and N0 , which satisfy τa > τd and N0 > 1, respectively, such that Nσ (t, s) ≤ N0 +

t −s , ∀t ≥ s ≥ 0, τa

(2.3)

then τa is called the average dwell time. Remark 2.3 From Definition 2.2, we know that the switching of the system will not be too fast. N0 is the chatter bound.  Definition 2.4 Equilibrium x  = 0 of system (2.1) is said to be exponentially stable under switching law σ, if solution x(t) of system (2.1) satisfies x(t) ≤ κe−λ(t−t0 ) x(t0 ), ∀t ≥ t0 where κ > 0 and λ > 0 are constants. Lemma 2.5 ([15]) For any real vectors u and ν, and a symmetric positive matrix R of compatible dimensions, the following inequality holds u T ν + ν T u ≤ u T Ru + ν T R −1 ν.

2.2 Problem Statement and Preliminaries

19

Lemma 2.6 ([12]) Let P ∈ Rn×n be a symmetric matrix and x ∈ Rn . The following inequality holds λmin (P)x T x ≤ x T P x ≤ λmax (P)x T x. Lemma 2.7 ([12] (Schur complement lemma)) For the given matrix  S=

S11 S12 ∗ S22

 < 0,

T T where S11 = S11 , S22 = S22 , the followings are equivalent: T −1 S11 S12 < 0; (1) S11 < 0, S22 − S12 −1 T S12 < 0. (2) S22 < 0, S11 − S12 S22

2.3 Main Results In this section, we first design a state observer for each individual subsystem and form a switching observer which adopts a same switching law with switched system (2.1), then we develop an event-triggered sampling mechanism based on the observer state and a sampled-data-based switching controller. At last, we present stability analysis of the closed-loop system. A diagram of observer-based event-triggered control of a switched system is shown in Fig. 2.1, in which a ZOH is introduced to keep the control signal continuous.

2.3.1 Observer Design For a switched system, design of an observer is different from a non-switched system, we have to consider the synchronization of the switching between the system and the observer. First, we construct observer ˙ˆ = Ai x(t) ˆ + Bi u(t) + L i (y(t) − Ci x(t)), ˆ ∀i ∈ M x(t)

(2.4)

for subsystem i of system (2.1), where x(t) ˆ ∈ Rn is the observer state, L i is the observer gain. Let error be e(t) = x(t) − x(t). ˆ From (2.1) and (2.4), an error system can be easily derived by ˙ˆ = (Ai − L i Ci )e(t). e(t) ˙ = x(t) ˙ − x(t)

(2.5)

20

2 Observer-Based Event-Triggered Control for Switched Linear Systems

Fig. 2.1 Diagram of observer-based event-triggered switched system

It is reasonable that we make the switching observer to be designed possessing the same switching strategy with system (2.1) under dwell time technique. A switching observer and a switching error system are hence obtained from (2.4) and (2.5), which can be described by ˙ˆ = Aσ x(t) ˆ + Bσ u(t) + L σ Cσ e(t), x(t)

(2.6)

e(t) ˙ = (Aσ − L σ Cσ )e(t).

(2.7)

and

The following lemma is proposed to guarantee that observer (2.6) exponentially estimates the state of system (2.1). Lemma 2.8 Consider error system (2.7). If there exist matrices Ri > 0, R j > 0, X i and X j of appropriate dimensions, and constants μ > 1, δ > 0 such that AiT Ri − CiT X iT + Ri Ai − X i Ci < 0, ∀i ∈ M

(2.8)

Ri ≤ μR j , R j ≤ μRi , ∀i, j ∈ M, i = j,

(2.9)

and

then system (2.7) is exponentially stable with any switching signals satisfying Ta > ln μ , which implies that observer (2.6) is an exponential observer of system (2.1), and δ observer gain L i can be obtained by L i = Ri−1 X i .

2.3 Main Results

21

Proof Consider error system (2.7). Construct the piecewise quadratic Lyapunov function Vi (t) = e T (t)Ri e(t), ∀i ∈ M,

(2.10)

where Ri > 0. Differentiating (2.10) along solutions of system (2.7), we have ˙ V˙i (t) = e˙ T (t)Ri e(t) + e T (t)Ri e(t)   T T = e (t) (Ai − L i Ci ) Ri + Ri (Ai − L i Ci ) e(t) = e T (t)Q i e(t),

(2.11)

where Q i = AiT Ri − CiT L iT Ri + Ri Ai − Ri L i Ci . Let X i = Ri L i . Then, LMI (2.8) is equivalent to Q i < 0 which implies that V˙i (t) is degenerative. We hence have V˙i (t) ≤ −λmin (−Q i )e(t)2 λmin (−Q i ) T e (t)Ri e(t) ≤− λmax (Ri ) = −δi Vi (t), where δi =

λmin (−Q i ) λmax (Ri )

(2.12)

> 0. Therefore, for t ∈ [ti , ti+1 ), we have Vi (t) ≤ e−δi (t−ti ) Vi (ti ).

(2.13)

We assume that Vli (ti ) ≤ μVli−1 (ti− ) for ∀li , li−1 ∈ M, where μ > 1. Let 0 and δ = min {δi }. Then, we have l = Nσ (t, t0 ) < t−t Ta ∀i∈M

Vli (t) ≤ e−δ(t−ti ) μVli−1 (ti− ) − ) ≤ e−δ(t−ti−1 ) μ2 Vli−2 (ti−1 − ≤ e−δ(t−ti−2 ) μ3 Vli−3 (ti−2 )

≤ ··· ≤ e−δ(t−t0 ) μl Vl0 (t0 ) ≤ e−δ(t−t0 ) μ ≤e

t−t0 Ta

Vl0 (t0 )

  − δ− lnTaμ (t−t0 )

Vl0 (t0 ).

(2.14)

Furthermore, from (2.10), we have Vli (t) ≥ λmin (Rli )e(t)2 ≥ min {λmin (Rli )}e(t)2 = αe(t)2 , ∀li ∈M

(2.15)

22

2 Observer-Based Event-Triggered Control for Switched Linear Systems

and Vl0 (t0 ) ≤ λmax (Rl0 )e(t0 )2 ≤ max {λmax (Rli )}e(t0 )2 = βe(t0 )2 , ∀li ∈M

(2.16)

where α = min {λmin (Rli )}, β = max {λmax (Rli )}. Therefore, from (2.14), (2.15) ∀li ∈M

∀li ∈M

and (2.16), we have 

β − e(t) ≤ e α 2

 δ− lnTaμ (t−t0 )

e(t0 )2 .

(2.17)

We know from (2.17) and Definition 2.4 that system (2.7) is exponential stability, which implies observer (2.6) is an exponential observer of system (2.1). This completes the proof. 

2.3.2 Sampling Mechanism As shown in Fig. 2.1, a detector and a sampler are embedded in one module. In this subsection, we first propose an event-triggered sampling mechanism and then set up a sampled-data-based switching controller under the proposed sampling mechanism. The event-triggered sampling mechanism is set up based on the observer state by 2 2 ≥ ηx(t) ˆ , e(t) ˆ

(2.18)

where e(t) ˆ = x(t) ˆ − x( ˆ tˆk ), tˆk is the kth sampling time and η > 0 is a constant. The module including a detector and a sampler possesses two functions: monitoring and sampling. The detector detects condition (2.18) continuously to determine whether a sampling is generated or not. Once condition (2.18) is satisfied, the sampler samples the latest state information of observer immediately, and then memories and transmits the information to the controller. We denote the time when a sampling happens by ˆ ˆ ˆ {tˆk }∞ k=0 with tk < tk+1 and assume the first sampling is generated at time t0 = t0 . With the state x( ˆ tˆk ) sampled at time tˆk , the next sampling time tˆk+1 can be determined by

2 2 . ˆ ≥ ηx(t) ˆ tˆk+1 = inf t > tˆk |e(t)

(2.19)

When condition (2.18) is satisfied, an event happens, and e(t) ˆ will be reset to zero and start growing until it triggers a new measurement update. Suppose that there are n samplings happening on the interval [ti , ti+1 ). According to switching sequence (2.2), without loss of generality, we assume that subsystem i is active on the interval [ti , ti+1 ). The controller is set up by

2.3 Main Results

23

⎧ K i x( ˆ tˆk ), t ∈ [ti , tˆk+1 ) ⎪ ⎪ ⎪ ⎨ K x( i ˆ tˆk+1 ), t ∈ [tˆk+1 , tˆk+2 ) uσ = ui = ⎪ · · · ⎪ ⎪ ⎩ ˆ tˆk+n ), t ∈ [tˆk+n , ti+1 ) K i x(

(2.20)

where K i is the control gain which matches with subsystem i for ∀i ∈ M. The controller receives state x( ˆ tˆk ) at sampling time tˆk and holds it until the next event happens at time tˆk+1 . On the interval [tˆk , tˆk+1 ), the controller only computes at the sampling time. Remark 2.9 In this chapter, we assume that the sampler only transmits the sampled state and thus no asynchronous switching happens. It is rational to assume that the switching controller possesses a same switching rule with the switched system since we adopt the time-dependent switching strategy which can be specified in advance. 

2.3.3 Stability Analysis Suppose that there are n samplings happening on the interval [ti , ti+1 ) and the (k + 1)th sampling on [t0 , ti→∞ ) is the first sampling on [ti , ti+1 ). For ∀t ∈ {[ti , tˆk+1 ), ˆ = x(t) ˆ − x( ˆ tˆk+ j ) always holds for [tˆk+1 , tˆk+2 ), . . . , [tˆk+n , ti+1 )}, error e(t) j = 0, 1, . . . , n. Moreover, according to switching sequence (2.2), subsystem i is active throughout the interval [ti , ti+1 ). We hence obtain from (2.6) and (2.20) that ˙ˆ = Ai x(t) ˆ + Bi K i x( ˆ tˆk+ j ) + L i Ci e(t) x(t) = Ai x(t) ˆ + Bi K i (x(t) ˆ − e(t)) ˆ + L i Ci e(t) = (Ai + Bi K i )x(t) ˆ + L i Ci e(t) − Bi K i e(t). ˆ

(2.21)

Combining with (2.7), for t ∈ [ti , ti+1 ), we have 

x(t) ˆ˙ = (Ai + Bi K i )x(t) ˆ + L i Ci e(t) − Bi K i e(t) ˆ e(t) ˙ = (Ai − L i Ci )e(t).

(2.22)

Recall that e(t) = x(t) − x(t), ˆ system (2.1) is stable if and only if the augmented switched system 

˙ˆ = (Aσ + Bσ K σ )x(t) x(t) ˆ + L σ Cσ e(t) − Bσ K σ e(t) ˆ e(t) ˙ = (Aσ − L σ Cσ )e(t)

is stable. The compact form of (2.23) can be written as

(2.23)

24

2 Observer-Based Event-Triggered Control for Switched Linear Systems

˙ = A¯ σ ξ(t) + B¯ σ e(t), ˜ ξ(t) 

where ξ(t) =

 x(t) ˆ , e(t)

A¯ σ =



(2.24)

 L σ Cσ Aσ + Bσ K σ , 0 Aσ − L σ C σ

    e(t) ˆ −Bσ K σ 0 , e(t) ˜ = . B¯ σ = 0 0 0 Consider system (2.24). Construct the piecewise quadratic Lyapunov function Vi (t) = ξ T (t) P¯i ξ(t), ∀i ∈ M,

(2.25)

where P¯i = diag{Pi , Pi }, in which Pi > 0 implies that P¯i > 0. When subsystem i is active, differentiating (2.25) along solutions of system (2.24), we have   T  ˜ ˜ P¯i ξ(t) + ξ T (t) P¯i A¯ i ξ(t) + B¯ i e(t) V˙i (t) = A¯ i ξ(t) + B¯ i e(t) = ξ T (t)( A¯ iT P¯i + P¯i A¯ i )ξ(t) + e˜ T (t) B¯ iT P¯i ξ(t) + ξ T (t) P¯i B¯ i e(t) ˜ = ξ T (t)( A¯ iT P¯i + P¯i A¯ i )ξ(t) − eˆ T (t)K iT BiT Pi x(t) ˆ − xˆ T (t)Pi Bi K i e(t). ˆ (2.26) From triggering condition (2.18), and using Lemma 2.5 by letting R = I , we have ˆ + xˆ T (t)Pi Bi K i K iT BiT Pi x(t) ˆ V˙i (t) ≤ ξ T (t)( A¯ iT P¯i + P¯i A¯ i )ξ(t) + eˆ T (t)e(t) T T T T T ≤ ξ (t)( A¯ i P¯i + P¯i A¯ i )ξ(t) + xˆ (t)(Pi Bi K i K i Bi Pi + η I )x(t) ˆ = ξ T (t) Q¯ i ξ(t), (2.27) where Q¯ i =



 Pi L i Ci Q¯ i11 , ∗ AiT Pi − CiT L iT Pi + Pi Ai − Pi L i Ci

Q¯ i11 = AiT Pi + K iT BiT Pi + Pi Ai + Pi Bi K i + Pi Bi K i K iT BiT Pi + η I. In order to make sure that V˙i (t) is degenerative, we assume that Q¯ i < 0. Then, we have V˙i (t) ≤ −λmin (− Q¯ i )ξ(t)2 λmin (− Q¯ i ) T ≤− ξ (t) P¯i ξ(t) λmax ( P¯i ) = −δ¯i Vi (t), where δ¯i =

λmin (− Q¯ i ) λmax ( P¯i )

> 0. From (2.28), on the interval [ti , ti+1 ), we have

(2.28)

2.3 Main Results

25 ¯

Vi (t) < e−δi (t−ti ) Vi (ti ).

(2.29)

Suppose that Vli (ti ) ≤ μVli−1 (ti− ) for ∀li , li−1 ∈ M, where μ > 1. Let l = Nσ (t, t0 ) ≤ t−t0 and δ¯ = min {δ¯i }. We have τa ∀i∈M

¯

Vli (t) ≤ e−δ(t−ti ) μVli−1 (ti− ) ¯

− ≤ e−δ(t−ti−1 ) μ2 Vli−2 (ti−1 ) ≤ ··· ¯

≤ e−δ(t−t0 ) μl Vl0 (t0 ) ¯

≤ e−δ(t−t0 ) μ ≤e

t−t0 τa

Vl0 (t0 )

  ¯ ln μ (t−t0 ) − δ− τa

Vl0 (t0 ).

(2.30)

From (2.25), we have Vli (t) ≥ λmin ( P¯li ) ξ(t)2

≥ min λmin ( P¯li ) ξ(t)2 = αξ(t)2 ,

(2.31)

Vl0 (t0 ) ≤ λmax ( P¯li ) ξ(t0 )2

≤ max λmax ( P¯li ) ξ(t0 )2 = βξ(t0 )2 ,

(2.32)

∀li ∈M

and

∀li ∈M





where α = min λmin ( P¯li ) and β = max λmax ( P¯li ) . Thus, from (2.30), (2.31) ∀li ∈M

∀li ∈M

and (2.32), we have 

β − ξ(t) ≤ e α 2

 ¯ ln μ (t−t0 ) δ− τa

ξ(t0 )2 .

(2.33)

From (2.33) and Definition 2.4, exponential stability of system (2.24) is guaranteed when δ¯ − lnτaμ > 0, which implies system (2.1) is exponentially stable. The following theorem concludes the main result of this section. Theorem 2.10 Consider system (2.24) with sampling instants determined by (2.19). For the given gains K i , K j and scalars μ > 1 and δ > 0, if there exist matrices Pi > 0, P j > 0 of appropriate dimensions such that ⎡

⎤ Qˆ i11 Pi L i Ci Pi Bi K i ⎣ ∗ A T Pi − C T L T Pi + Pi Ai − Pi L i Ci 0 ⎦ < 0, ∀i ∈ M, i i i ∗ ∗ −I

(2.34)

26

2 Observer-Based Event-Triggered Control for Switched Linear Systems

and Pi ≤ μP j , P j ≤ μPi , ∀i, j ∈ M, i = j, where

(2.35)

Qˆ i11 = AiT Pi + K iT BiT Pi + Pi Ai + Pi Bi K i + η I,

then system (2.24) is exponentially stable for any switching signals with average dwell time τa satisfying τa > lnδμ , which implies that system (2.1) is exponentially stable, where observer gains L i can be obtained from Lemma 2.5. Proof From Schur complement lemma, inequality (2.34) is equivalent to Q¯ i < 0 which guarantees V˙i (t) in (2.27) is degenerative. The other part of proof is obvious from the above analysis and thus omitted. 

2.3.4 Control Design In the process of stability analysis, we assume R = I in (2.27) when using Lemma 2.5. In fact, an arbitrary matrix Ri > 0 can be adopted to obtain a less conservative result ⎤ ⎡ 11 Q˜ i Pi L i Ci Pi Bi K i ⎣ ∗ A T Pi − C T L T Pi + Pi Ai − Pi L i Ci (2.36) 0 ⎦ < 0, i i i ∗ ∗ −Ri where

Q˜ i11 = AiT Pi + K iT BiT Pi + Pi Ai + Pi Bi K i + ηλmax (Ri )I,

Pre- and post-multiplying both sides of matrix inequality (2.36) by Pi−1 , applying Schur complement lemma, and letting P¯i = Pi−1 , R¯ i = Pi−1 Ri Pi−1 , and Mi = K i Pi−1 , then inequality (2.36) can be transformed into the following inequality ⎤ L i Ci P¯i Bi Mi P¯i Qˇ i11 ⎥ ⎢ ∗ P¯i A T − P¯i C T L T + Ai P¯i − L i Ci P¯i 0 0 i i i ⎥ < 0, ⎢ ⎦ ⎣ ∗ ¯ ∗ − Ri 0 1 ∗ ∗ ∗ − ηλmax (Ri ) I ⎡

where

(2.37)

Qˇ i11 = P¯i AiT + MiT BiT + Ai P¯i + Bi Mi .

It is obvious that (2.37) is not a LMI since λmax {Ri } is coupled with R¯ i However, we can choose a scalar δˆ large enough first instead of λmax {Ri } in (2.37), and then if

2.3 Main Results

27

we can obtain feasible solutions R¯ i for ∀i ∈ M such that δˆ > max {λmax {Ri }}, then ∀i∈M

inequality (2.37) holds, and control gain K i can be obtained by K i = Mi P¯i−1 . The result is concluded in the following theorem. Theorem 2.11 Consider system (2.26) with sampling instants determined by (2.21). For the given scalars μ > 1, δˆ > 0, δ > 0, and gains L i , if there exist matrices P¯i > 0, P¯ j > 0, R¯ i , Mi for ∀i, j ∈ M, i = j of appropriate dimensions such that ⎡

⎤ Qˇ i11 L i Ci P¯i Bi Mi P¯i ⎢ ∗ P¯ A T − P¯ C T L T + A P¯ − L C P¯ 0 0 ⎥ i i i i i i i i i ⎢ ⎥ i ⎢ ∗ ⎥ < 0, ¯ ∗ − R 0 i ⎣ ⎦ 1 ∗ ∗ ∗ − ˆI

(2.38)

ηδ

P¯i ≤ μ P¯ j , P¯ j ≤ μ P¯i , where

(2.39)

Qˇ i11 = P¯i AiT + MiT BiT + Ai P¯i + Bi Mi ,

and verifying max {λmax { P¯i−1 R¯ i P¯i−1 }} < δˆ satisfied, then system (2.24) is exponen∀i∈M

tially stable for any switching signals with average dwell time τa satisfying τa > lnδμ , which implies that system (2.1) is exponentially stable, and control gain can be obtained by K i = Mi P¯i−1 .

2.4 Illustrative Example In this section, we present a numerical example to show the effectiveness of the proposed method. Example 2.12 Consider the switched linear system 

x(t) ˙ = Aσ x(t) + Bσ u(t) y(t) = Cσ x(t),

where σ ∈ {1, 2}, and system matrix parameters are given by 

     −2.5 0.3 1 , B1 = , C1 = −0.1 0.1 , 0 −1.5 1       −1.2 0 1 A2 = , B2 = , C2 = 0.1 −0.1 . 0.1 −1.6 1 A1 =

Let μ = 1.1. Solving the conditions in Lemma 2.8, we obtain

(2.40)

28

2 Observer-Based Event-Triggered Control for Switched Linear Systems

 L1 =

   16.7739 −1.2000 , L2 = . −1.8107 9.9214

Set up the triggering condition by 2 2 ≥ x(t) ˆ . e(t) ˆ

(2.41)

  ˆ u 1 = 0.5 0.2 x(t),   u 2 = 0.3 −0.2 x(t). ˆ

(2.42)

Let the controller be

Then, by solving the inequalities in Theorem 2.10, we obtain the following feasible solution     5.8151 −4.8065 5.9832 −4.7668 , P2 = , P1 = −4.8065 8.0952 −4.7668 7.7928 ⎡ ⎤ −26.7718 23.7784 −10.6245 10.6245 ⎢ 23.7784 −21.7174 9.5281 −9.5281 ⎥ ⎥ Q¯ 1 = ⎢ ⎣ −10.6245 9.5281 −7.8265 0.8178 ⎦ , 10.6245 −9.5281 0.8178 −8.1131 ⎡ ⎤ −13.3908 15.2694 −5.4473 5.4473 ⎢ 15.2694 −23.9570 8.3036 −8.3036 ⎥ ⎥ Q¯ 2 = ⎢ ⎣ −5.4473 8.3036 −4.4183 0.3755 ⎦ . 5.4473 −8.3036 0.3755 −8.3298 Thus, δ is computed by   0.2668 0.2585 λmin (− Q¯ i ) , = 0.0220 = min ¯i ) i∈{1,2} λmax ( P i∈{1,2} 11.8950 11.7399

δ = min

and τa∗ = lnδμ = 4.3323. Construct a switching sequence satisfying τa = 5 > τa∗ and choose the initial states xˆ0 = e0 = [−3 2]T . The state responses of the error system and the observer are shown in Figs. 2.2 and 2.3, respectively. Therefore, state responses of system (2.40) is obtained in Fig. 2.4. Switching signal, triggering condition, triggering instants and control input are illustrated in Figs. 2.5, 2.6, 2.7 and 2.8, respectively. The simulation result illustrates that switched system (2.40) is stable under triggering condition (2.41) and the sampled-data-based controller with gains  K 1 and K 2 .

2.4 Illustrative Example

29

State responses of the error system

3

2

1

0

-1

-2

-3 0

5

10

15

20

25

30

35

40

45

50

30

35

40

45

50

Time (sec) Fig. 2.2 State responses of the error system

State responses of the observer

2

1

0

-1

-2

-3 0

5

10

15

20

25

Time (sec) Fig. 2.3 State responses of the observer

30

2 Observer-Based Event-Triggered Control for Switched Linear Systems 4

State responses of the system

3 2 1 0 -1 -2 -3 -4 -5 -6 0

5

10

15

20

25

30

35

40

45

50

30

35

40

45

50

Time (sec) Fig. 2.4 State responses of the closed-loop system (2.39)

Switching signal

2

1

0

5

10

15

20

25

Time (sec) Fig. 2.5 Switching signal

2.4 Illustrative Example

31

14

12

Triggering condition

10 10 -4

20

8 15 10

6

5

4 0 10

2

12

14

16

18

20

Time (sec)

0 0

5

10

15

20

25

30

35

40

45

50

Time (sec) Fig. 2.6 Triggering condition

Triggering instant

1

0

5

10

15

20

25

Time (sec) Fig. 2.7 Triggering instants

30

35

40

45

50

32

2 Observer-Based Event-Triggered Control for Switched Linear Systems 0.4 0.2 0

Control input

-0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 0

5

10

15

20

25

30

35

40

45

50

Time (sec) Fig. 2.8 Control input

2.5 Conclusion In this chapter, we have presented a result on observer-based event-triggered sampleddata control of switched linear systems. A sufficient condition has been obtained for exponential stability of the closed-loop switched linear system with an eventtriggered sampling mechanism and a sampled-data-based controller under multiple Lyapunov function method and the average dwell time technique. In this chapter, the asynchronous switching problem and the zeno behavior in event-triggered sampling process are not considered, and thus stability analysis is simplified. We will discuss them in the following chapters.

Chapter 3

Improved Event-Triggered Control for Switched Linear Systems

3.1 Introduction In this chapter, we study both state-based and observer-based output feedback eventtriggered sampled-data control for a class of continuous-time switched linear systems. Compared with the result of Chap. 2, an improved event-triggered sampling mechanism is introduced, under which the controller is triggered to achieve Lyapunov stability of the closed-loop switched system combining with the average dwell time technique. A constant threshold ε is added into the proposed event-triggered sampling mechanism, the sampling period and the system performance can be improved by adjusting the value of ε. Moreover, if we let ε = 0, the improved event-triggered sampling mechanism would degenerate into the one considered in Chap. 2. Zeno behavior can be excluded from theoretical analysis in sampling process under the improved triggering condition. Two sufficient conditions are obtained for stability of the closed-loop system by using multiple Lyapunov function method under two different feedback control approaches. The effectiveness of the proposed method is verified by a numerical example finally. The rest of this chapter is organized as follows. In Sect. 3.2, the problem statement and preliminaries are presented. In Sect. 3.3, a state-based event-triggered sampling mechanism is designed and a sufficient condition is obtained for stability of the closed-loop system under the proposed sampling mechanism and a sampled-databased switching controller by using multiple Lyapunov function method with the average dwell time technique. In Sect. 3.4, an exponential observer is designed first, and then an observer-based event-triggered sampling mechanism and a sampled-databased switching controller are set up, under which a sufficient condition is obtained for stability of the closed-loop system by using multiple Lyapunov function method and the average dwell time technique. Section 3.5 discusses the minimum inter-event internal to exclude the zeno behavior of the sampling process. Section 3.6 presents an illustration example to show the effectiveness of the proposed method and Sect. 3.7 concludes the chapter. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. Fu and T.-F. Li, Event-Triggered Control of Switched Linear Systems, Studies in Systems, Decision and Control 365, https://doi.org/10.1007/978-3-030-71604-2_3

33

34

3 Improved Event-Triggered Control for Switched Linear Systems

3.2 Problem Statement and Preliminaries Consider the continuous-time switched linear system 

x(t) ˙ = Aσ x(t) + Bσ u(t), x(t0 ) = x0 y(t) = Cσ x(t),

(3.1)

where x(t) ∈ Rn is the system state, u(t) ∈ Rm is the control input, y(t) ∈ Rq is the measurable output, x0 is the initial state, σ : [0, ∞) → M = {1, 2, . . . , m} is the switching signal that orchestrates switching between subsystems, M is a finite index set, {Ai , Bi , Ci , i ∈ M} is a collection of matrix pairs defining subsystem i of system (3.1). It is assumed that the pairs (Ai , Bi ) and (Ai , Ci ) are controllable and observable, respectively. Corresponding to switching signal σ, there always exists a switching sequence {(l0 , t0 ), (l1 , t1 ), . . . , (li , ti ), . . . |li ∈ M, i ∈ N} ,

(3.2)

∞ with which means that the li th subsystem is active when t ∈ [ti , ti+1 ), where {ti }i=0 ti < ti+1 is a switching time sequence. Without loss of generality, we assume that there are no state jumps at switching instants and solution x(·) of system (3.1) is continuous everywhere. We denote the number of discontinuities of σ on a semi-open interval (t, s] by Nσ (s, t), and use τa to denote an average dwell time which is defined in Definition 2.2. The following definition is closely relevant with the main results.

Definition 3.1 Switched system (3.1) is said to be Lyapunov stable with exponential convergence rate if there exist some constants κ > 0, λ > 0, δ > 0 and ε > 0 such that for switching signal σ, solutions of (3.1) with x(t0 ) ≤ δ satisfy x(t) ≤ κe−λ(t−t0 ) (x(t0 ) − δ) + ε, ∀t ≥ t0 .

3.3 State-Based Event-Triggered Control In this section, we will study the state-based event-triggered sampled-data control for switched  ∞ system (3.1). The structure of the control system is shown in Fig. 3.1, where tˆk k=0 with tˆk < tˆk+1 denotes a sampling time sequence.

3.3 State-Based Event-Triggered Control

u (t )

Switched system

35

x(t )

Sensor

ZOH

u (tˆk )

Controller

x(tˆk )

Continuous signal

Event detector Discrete signal

Fig. 3.1 Diagram of state-based event-triggered switched system

3.3.1 Sampling Mechanism Based on the sampling mechanism discussed in Chap. 2, we propose an improved triggering mechanism by adding a constant ε into the right-side hand of (2.20), which turns into e(t)2 ≥ ηx(t)2 + ε,

(3.3)

where e(t) = x(t) − x(tˆk ), ε > 0 and η > 0 are both constants. See Fig. 3.1, we assume that a detector and a sampler are both embedded in the module of event detector. In this module, the detector detects condition (3.3) continuously by receiving state from the sensor to determine whether the sampler is active or not. Once condition (3.3) is satisfied, the sampler samples the latest state information immediately, and transmits it to the controller. If condition (3.3) is triggered, an event happens, and error e(t) is reset to zero and starts growing until it triggers a new measurement update. Without loss of generality, we assume that the first event is generated at time tˆ0 = t0 . If state x(tˆk ) is sampled at time tˆk , then the next sampling instant tˆk+1 can be determined by tˆk+1 = inf{t > tˆk |e(t)2 ≥ ηx(t)2 + ε}.

(3.4)

Suppose that there are n samplings happening on the interval [ti , ti+1 ) and tˆk+1 is the first sampling instant on this interval. According to switching sequence (3.2), we assume that subsystem i is active on the interval [ti , ti+1 ). Therefore, the piecewise continuous controller can be set by

36

3 Improved Event-Triggered Control for Switched Linear Systems

⎧ K i x(tˆk ), t ∈ [ti , tˆk+1 ) ⎪ ⎪ ⎪ ⎨ K x(tˆ ), t ∈ [tˆ , tˆ ) i k+1 k+1 k+2 uσ = ui = ⎪ · · · ⎪ ⎪ ⎩ K i x(tˆk+n ), t ∈ [tˆk+n , ti+1 )

(3.5)

where K i is the control gain for ∀i ∈ M. The controller receives the latest state x(tˆk ) at the sampling time tˆk and holds it until the next event generates. On sampling interval [tˆk , tˆk+1 ), the controller only computes at sampling instant tˆk . A ZOH is introduced to keep the control signal continuous.

3.3.2 Stability Analysis Suppose that there are n samplings happening on the interval [ti , ti+1 ) and tˆk+1 is the first sampling time on this interval. For ∀t ∈ {[ti , tˆk+1 ), [tˆk+1 , tˆk+2 ), . . . , [tˆk+n , ti+1 )}, error e(t) = x(t) − x(tˆk+ j ) always holds for j ∈ {0, 1, . . . , n}. Substituting controller (3.5) into system (3.1), we have x(t) ˙ = Aσ x(t) + Bσ K σ x(tˆk+ j ) = Aσ x(t) + Bσ K σ (x(t) − e(t)) = (Aσ + Bσ K σ )x(t) − Bσ K σ e(t).

(3.6)

The following theorem presents the main result of this section. Theorem 3.2 Consider system (3.1) with controller (3.5) determined by triggering condition (3.4). For the given gains K i , K j , and scalars η > 0, μ > 1, ε > 0, δ > 0, N0 ≥ 1, if there exist matrices Pi > 0 and P j > 0 of appropriate dimensions such that T

Ai Pi + Pi Bi K i + Pi Ai + K iT BiT Pi + η I Pi Bi K i < 0, ∀i ∈ M (3.7) ∗ −I and Pi ≤ μP j , P j ≤ μPi , ∀i, j ∈ M, i = j

(3.8)

then system (3.1) is Lyapunov stable for any switching signals with average dwell time τa satisfying τa > lnδμ , and the state of system (3.1) exponentially converges to the bounded region ⎧ ⎨

⎫   δτa  εe (μ − 1)eδτa N0 + 1 − εμ ⎬ B(ε) := x(t)| x(t) ≤ . ⎩ min (λmin (Pi ))δ(eδτa − μ) ⎭ ∀i∈M

(3.9)

3.3 State-Based Event-Triggered Control

37

Proof Construct the piecewise quadratic Lyapunov function Vi (t) = x T (t)Pi x(t), ∀i ∈ M,

(3.10)

where Pi > 0. Differentiating (3.10) along solutions of system (3.6), we have V˙i (t) = ((Ai + Bi K i )x(t) − Bi K i e(t))T Pi x(t) + x T (t)Pi ((Ai + Bi K i )x(t) − Bi K i e(t)) = x T (t)(Ai + Bi K i )T Pi x(t) − e T (t)K iT BiT Pi x(t) + x T (t)Pi (Ai + Bi K i )x(t) − x T (t)Pi Bi K i e(t).

(3.11)

Applying Lemma 2.6 and considering triggering condition (3.3), we have V˙i (t) ≤ x T (t)(Ai + Bi K i )T Pi x(t) + x T (t)Pi (Ai + Bi K i )x(t) + e T (t)e(t) + x T (t)Pi Bi K i K iT BiT Pi x(t) ≤ x T (t)Q i x(t) + ε,

(3.12)

where Q i = AiT Pi + K iT BiT Pi + Pi Ai + Pi Bi K i + Pi Bi K i K iT BiT Pi + η I. From Schur complement lemma, we know that inequality (3.7) is equivalent to Q i < 0. Thus, we have V˙i (t) ≤ −λmin (−Q i )x(t)2 + ε λmin (−Q i ) T x (t)Pi x(t) + ε ≤− λmax (Pi ) = −δi Vi (t) + ε, where δi =

λmin (−Q i ) λmax (Pi )

(3.13)

> 0. Integrating (3.13) from ti to t, we have Vi (t) ≤ e

−δi (t−ti )



t

Vi (ti ) + ε

e−δi (t−s) ds.

(3.14)

ti

Moreover, we assume that Vli (ti ) ≤ μVli−1 (ti− ) for ∀li , li−1 ∈ M. Let lti = l(ti , t) = i + N0 and δ = min δi > 0. From (3.14), we have Nσ (ti , t) ≤ t−t τa ∀i∈M

38

3 Improved Event-Triggered Control for Switched Linear Systems

 ε 1 − e−δ(t−ti ) Vli (t) ≤ e−δ(t−ti ) μVli−1 (ti− ) + δ  ≤ e−δ(t−ti ) μ e−δ(ti −ti−1 ) Vli−1 (ti−1 )  ε   ε + 1 − e−δ(ti −ti−1 ) + 1 − e−δ(t−ti ) δ δ ≤ e−δ(t−ti−1 ) μVli−1 (ti−1 )  ε  εμ  −δ(t−ti ) e 1 − e−δ(t−ti ) − e−δ(t−ti−1 ) + + δ δ − ≤ e−δ(t−ti−1 ) μ2 Vli−2 (ti−1 )  ε  εμ  −δ(t−ti ) e 1 − e−δ(t−ti ) − e−δ(t−ti−1 ) + + δ δ ≤ ···  εμlt1  −δ(t−t2 ) e − e−δ(t−t1 ) ≤ e−δ(t−t0 ) μlt0 Vl0 (t0 ) + δ  εμl−2  −δ(t−t3 ) e + − e−δ(t−t2 ) δ + ···  εμ2  −δ(t−ti−1 ) + e − e−δ(t−ti−2 ) δ  ε  εμ  −δ(t−ti ) e 1 − e−δ(t−ti ) − e−δ(t−ti−1 ) + + δ δ l t2   ε(μ − 1)  ε ε −δ(t−t0 ) lt0 Vl0 (t0 ) − + ≤e μ μm e−δ(t−ti−m ) + δμ δ δ m=0   ln μ ε ≤ μ N0 e−(δ− τa )(t−t0 ) Vl0 (t0 ) − δμ l t2 ε ε(μ − 1) δτa N0  e em(ln μ−δτa ) + , + δ δ m=0

(3.15)

where τa is the minimum dwell time. From (3.10), we have Vli (t) ≥ λmin (Pli )x(t)2   ≥ min λmin (Pli ) x(t)2 = αx(t)2 ,

(3.16)

Vl0 (t0 ) ≤ λmax (Pl0 )x(t0 )2   ≤ max λmax (Pli ) x(t0 )2 = βx(t0 )2 ,

(3.17)

∀li ∈M

and

∀li ∈M

3.3 State-Based Event-Triggered Control

39

    where α = min λmin (Pli ) > 0 and β = max λmax (Pli ) > 0. Thus, combining ∀li ∈M

∀li ∈M

with (3.15), (3.16) and (3.17), we have 

β − x(t) ≤ μ N0 e α 2

 δ− lnτaμ (t−t0 )

  ε 2 x(t0 ) − δμβ

l t2 ε ε(μ − 1) δτa N0  em(ln μ−δτa ) + + e . αδ αδ m=0

The condition τa > lnδμ in Theorem 3.2 implies that δ − Then from (3.18), we have

ln μ τa

(3.18)

> 0 and ln μ − δτa < 0.

    β N0 − δ− lnτaμ (t−t0 ) ε x(t0 )2 − μ e α δμβ δτa (N0 +1) ε(μ − 1)e ε + + . αδ(eδτa − μ) αδ

x(t)2 ≤

(3.19)

Therefore, Lyapunov stability of system (3.6) is guaranteed from (3.19) and Definitions 3.1, which completes the proof.  Remark 3.3 Control gains K i can be obtained if we adapt the similar transformation method like the one in Chap. 2. It can be derived easily and we omit it here. 

3.4 Observer-Based Event-Triggered Control In this section, we study observer-based event-triggered control problem for switched system (3.1) under the improved triggering condition. First, we design a state observer, and then propose an observer-based sampling mechanism and a sampleddata-based switching controller to achieve a closed-loop switched system. At last we present stability analysis of the obtained closed-loop system.

3.4.1 Observer Design For each individual subsystem i, we construct observer based on the output of system (3.1) ˙ˆ = Ai x(t) ˆ + Bi u i (t) + L i (y(t) − Ci x(t)), ˆ x(t)

(3.20)

where x(t) ˆ ∈ Rn is the observer state, L i is the observer gain. Suppose that the switching observer is collocated with the sensor which has a same switching law

40

3 Improved Event-Triggered Control for Switched Linear Systems

with system (3.1). Then a switching observer can be formed as ˙ˆ = Aσ x(t) ˆ + Bσ u σ (t) + L σ Cσ (x(t) − x(t)). ˆ x(t)

(3.21)

Let error be e(t) = x(t) − x(t). ˆ Then the error system e(t) ˙ = (Aσ − L σ Cσ )e(t)

(3.22)

can be established from (3.1) and (3.21). The following lemma gives the design of an exponential observer. Lemma 3.4 Consider error system (3.22). If there exist matrices Ri > 0, R j > 0, and X i , X j of appropriate dimensions, and scalars μ > 1, δ > 0 such that AiT Ri − CiT X iT + Ri Ai − X i Ci < 0, ∀i ∈ M

(3.23)

Ri ≤ μR j , R j ≤ μRi , ∀i, j ∈ M, i = j

(3.24)

and

then system (3.20) is exponentially stable for any switching signals with average dwell time τa satisfying Ta > lnδμ , which also implies that observer (3.21) is an exponential observer of system (3.1), and observer gain L i can be obtained by L i = Ri−1 X i . Proof Refer to the proof of Lemma 2.5 in Chap. 2.



3.4.2 Sampling Mechanism ˆ ˆ Let a detector and a sampler be in a same module and {tˆk }∞ k=0 with tk < tk+1 denotes a sampling time sequence. Construct the triggering condition 2 2 ≥ ηx(t) ˆ + ε, e(t) ˆ

(3.25)

ˆ tˆk ) where e(t) ˆ = x(t) ˆ − x( ˆ tˆk ), ε > 0 and η > 0 are both constants. With the state x( sampled at time tˆk , the next sampling time tˆk+1 can be determined by   2 2 ˆ ≥ ηx(t) ˆ +ε . tˆk+1 = inf t > tˆk |e(t)

(3.26)

Under sampling condition (3.25), we construct a sampled-data-based controller for feedback. Suppose that there are n samplings happening on the interval [ti , ti+1 ) and tˆk+1 is the first sampling instant on this interval. According to switching sequence (3.2), we set the piecewise continuous controller

3.4 Observer-Based Event-Triggered Control

41

⎧ K i x( ˆ tˆk ), t ∈ [ti , tˆk+1 ) ⎪ ⎪ ⎪ ⎨ K x( i ˆ tˆk+1 ), t ∈ [tˆk+1 , tˆk+2 ) uσ = ui = ⎪ · · · ⎪ ⎪ ⎩ ˆ tˆk+n ), t ∈ [tˆk+n , ti+1 ) K i x(

(3.27)

where K i is the control gain for ∀i ∈ M. The controller receives state x( ˆ tˆk ) at sampling time tˆk and holds it until the next event happens at time tˆk+1 . On the interval [tˆk , tˆk+1 ), the controller only computes at sampling time tˆk .

3.4.3 Stability Analysis According to switching sequence (3.2), we assume that subsystem i is active on [ti , ti+1 ). Suppose that there are n samplings happening on the interval [ti , ti+1 ) and tˆk+1 is the first sampling instant on this interval. For ∀t ∈ {[ti , tˆk+1 ), [tˆk+1 , tˆk+2 ), . . . , ˆ = x(t) ˆ − x( ˆ tˆk+ j ) always holds for j = 0, . . . , n. Substituting [tk+n , ti+1 )}, error e(t) controller u i into (3.20), we have ˙ˆ = Ai x(t) ˆ + Bi K i x( ˆ tˆk+ j ) + L i Ci e(t) x(t) = Ai x(t) ˆ + Bi K i (x(t) ˆ − e(t)) ˆ + L i Ci e(t) = (Ai + Bi K i )x(t) ˆ + L i Ci e(t) − Bi K i e(t). ˆ

(3.28)

Together with (3.22) and (3.28), for t ∈ [ti , ti+1 ), we have 

˙ˆ = (Ai + Bi K i )x(t) x(t) ˆ + L i Ci e(t) − Bi K i e(t) ˆ e(t) ˙ = (Ai − L i Ci )e(t).

(3.29)

We know from e(t) = x(t) − x(t) ˆ that system (3.1) is stable under the feedback control (3.27) if and only if the augmented system 

x(t) ˆ˙ = (Aσ + Bσ K σ )x(t) ˆ + L σ Cσ e(t) − Bσ K σ e(t) ˆ e(t) ˙ = (Aσ − L σ Cσ )e(t)

(3.30)

is stable. The compact form of (3.30) can be written as ˙ = A¯ σ ξ(t) + B¯ σ e(t), ˜ ξ(t) where



x(t) ˆ ξ(t) = , e(t)

A¯ σ =



L σ Cσ Aσ + Bσ K σ , 0 Aσ − L σ C σ

(3.31)

42

3 Improved Event-Triggered Control for Switched Linear Systems



e(t) ˆ −Bσ K σ 0 , e(t) ˜ = . B¯ σ = 0 0 0 We thus focus the study on system (3.31). The following theorem gives the main result of this section. Theorem 3.5 Consider system (3.31) with sampling instants determined by (3.25). For given gains K i , K j and scalars η > 0, μ > 1, ε > 0, δ > 0 and N0 ≥ 1, if there exist matrices Pi > 0, P j > 0 and L i , L j of appropriate dimensions such that ⎡

⎤ Qˆ i11 Pi L i Ci Pi Bi K i ⎣ ∗ Pi Ai − Pi L i Ci + A T Pi − C T L T Pi 0 ⎦ < 0, ∀i ∈ M i i i ∗ ∗ −I

(3.32)

and Pi ≤ μP j , P j ≤ μPi , ∀i, j ∈ M, where

(3.33)

Qˆ i11 = AiT Pi + K iT BiT Pi + Pi Ai + Pi Bi K i + η I,

then system (3.31) is Lyapunov stable for any switching signals with average dwell time τa satisfying τa > lnδμ , and the state of system (3.31) exponentially converges to the bounded region ⎧ ⎫   δτa  ⎨ εe (μ − 1)eδτa N0 + 1 − εμ ⎬ B(ε) := ξ(t)|ξ(t) ≤ . ⎩ min (λmin (Pi ))δ(eδτa − μ) ⎭

(3.34)

∀i∈M

Proof The proof is analogous to the proof of Theorem 3.2.



Remark 3.6 In fact, control gains K i can also be obtained if we adapt the same transformation method like the one in Chap. 2. 

3.5 Minimum Inter-Event Interval The event-triggered sampling mechanism brings more complicated dynamic behaviour to a switched system than the time-triggered one. In this section, we will prove that there always exists a non-zero lower bound of the minimum inter-event interval for triggering condition (3.3) to avoid zeno behavior. We just consider the case of observer-based event-triggered control. The minimum inter-event interval in state-based event-triggered control can be easily deduced following the analysis below.

3.5 Minimum Inter-Event Interval

43

Theorem 3.7 With triggering condition (3.25), the minimum inter-event interval is lower bounded by a positive scalar. Proof Suppose that n samplings occur on the interval [ti , ti+1 ) and without loss of generality, we assume that tˆk+1 , . . . , tˆk+n are sampling instants on interval [ti , ti+1 ). On intervals [ti , tˆk+1 ), [tˆk+1 , tˆk+2 ), . . . , [tˆk+n , ti+1 ), no matter which sampling interˆ = x(t) ˆ − x( ˆ tˆk+l ) holds, where val t belongs to, x( ˆ tˆk+l ) are constants and e(t) l = 0, 1, . . . , n. Then, for ∀t ∈ [ti , ti+1 ), we have ˙ˆ = x(t) ˙ˆ e(t) ˆ + Bli K li x( ˆ tˆk+l ) + L li Cli e(t) = Ali x(t) = Ali e(t) + (Ali + Bli K li )x( ˆ tˆk+l ) + L li Cli e(t).

(3.35)

Hence, ˆ tˆk+l ) e(t) ˆ = e Ali (t−tˆk+l ) e(  t   e Ali (t−s) (Ali + Bli K li )x( ˆ tˆk+l ) + L li Cli e(s) ds. + tˆk+l

(3.36)

Since e( ˆ tˆk+l ) = x( ˆ tˆk+l ) − x( ˆ tˆk+l ) = 0, then  e(t) ˆ =

t tˆk+l

  e Ali (t−s) (Ali + Bli K li )x( ˆ tˆk+l ) + L li Cli e(s) ds.

(3.37)

Therefore,  t      Ali (t−s)  e(t) ˆ = (Ali + Bli K li )x( e ˆ tˆk+l ) + L li Cli e(s) ds   tˆk+l  t   ≤ eAli (t−s) (Ali + Bli K li )x( ˆ tˆk+l ) + L li Cli e(s) ds 

tˆk+l t



tˆk+l t

≤

   eAli (t−s) (Ali + Bli K li )x( ˆ tˆk+l ) + L li Cli e(s) ds

 eAli (t−s) (Ali + Bli K li )x( ˆ tˆk+l ) tˆk+l  l   i   (Am −L m Cm )s    e(0))ds. m=1 + L li Cli  e   

≤

(3.38)

Noticing that Am − L m Cm are Hurwitz for ∀m = 1, . . . , li with λmax (Am − L m Cm ) 0 for any given sampling instant tˆk+l . If Ali  = 0, we have T φ(tˆk+l ) =

 2 ηx(t) ˆ + ε,

(3.43)

which also indicates that T > 0. With the above discussion, it can be concluded that there always exists a positive lower bound of the minimum inter-event interval. 

3 Improved Event-Triggered Control for Switched Linear Systems 4

Triggering condition

State responses of the system

48

2 0 -2 0.1

-4

10

20 Time (sec)

30

10

5

0

-6 0

10

20

30

40

0

50

10

20

30

40

50

40

50

Time (sec)

Time (sec)

Control input

Switching signal

0.5

2

1

0

10

20

30

40

50

0 -0.5 -1 -1.5 0

10

Time (sec)

20

30

Time (sec)

Fig. 3.8 Simulation results under triggering condition (3.25) with ε = 1

3.6 Illustrative Example In this section, we present two numerical examples to verify the effectiveness of the proposed methods based on state-based event-triggered sampled-data control and observer-based event-triggered sampled-data control, respectively. Moreover, we present the advantages of the proposed sampling mechanism by comparing with the one proposed in Chap. 2. Example 3.8 Consider the switched linear system 

x(t) ˙ = Aσ x(t) + Bσ u(t) y(t) = Cσ x(t),

(3.44)

where σ ∈ {1, 2}, and system parameters are the same with Example 2.9. Case 1. (State-based event-triggered sampled-data control) We set up triggering condition (3.3) with η = 1 and take different values of ε to show its influence on system dynamic. Let μ = 1.1. By solving the conditions in Theorem 3.2, we obtain the feasible solution

49

4

Triggering condition

State responses of the system

3.6 Illustrative Example

2 0 0.15

-2

0.1 0.05

-4

15

20 25 Time (sec)

30

10 0.2 0.1

-0.1 15

10

20

30

20 25 Time (sec)

30

0

-6 0

0

5

40

0

50

10

20

30

40

50

40

50

Time (sec)

Time (sec) 0.5

Control input

Triggering instant

1

0

10

20

30

40

50

0 -0.5 -1 -1.5 0

10

Time (sec)

20

30

Time (sec)

Fig. 3.9 Simulation results under triggering condition (3.25) with ε = 0.1



1.8721 0.0664 P1 = , 0.0664 1.7475

−5.3321 2.6103 Q1 = , 2.6103 −2.5229



1.8277 0.0901 P2 = , 0.0901 1.7721

−1.7397 0.5643 Q2 = . 0.5643 −4.9648

δ is computed by $ δ = min

i∈{1,2}

λmin (−Q i ) λmax (Pi )

%

$ = min

i∈{1,2}

0.9633 1.6438 , 1.9008 1.8942

% = 0.5068

and τa∗ = lnδμ = 0.1881. Choose an initial state x0 = [−3 2]T . In order to show the advantage of the proposed method, we take ε = 1, 0.1, 0, successively. Under sampling condition (3.3) with different values of ε and a switching signal satisfying τa = 0.5 > τa , we obtain simulation results shown in Figs. 3.2, 3.3, 3.4, 3.5, 3.6 and 3.7, respectively. From which we can see that sampling times reduce when ε becomes larger and the state trajectories are influenced by the change of ε. Therefore, by adjusting and selecting the appropriate parameter ε, one can reduce the unnecessary sampling times while maintaining the performance of the closed-loop system. 

50

3 Improved Event-Triggered Control for Switched Linear Systems

Triggering condition

State of the system

4 2 0 0.01

-2

0.005

-4

0 15

-6 0

10

20 25 Time (sec)

20

30

30

40

10

10

10-3

5

5 0 10

12

14 16 Time (sec)

18

0 0

50

10

20

30

40

50

40

50

Time (sec)

Time (sec) 0.5

Control input

Triggering instant

1

0

10

20

30

Time (sec)

40

0 -0.5 -1 -1.5 0

10

20

30

Time (sec)

Fig. 3.10 Simulation results under triggering condition (3.25) with ε = 0

Case 2. (Observer-based event-triggered sampled-data control) Set up triggering condition (3.25) with η = 1 and take ε = 1, 0.1, 0, respectively. We obtain simulation results shown in Figs. 3.5, 3.6 and 3.7, respectively. If ε = 0 in (3.25), then sampling mechanism (3.25) is the same with (2.20) and the result naturally degenerates to the one in Chap. 2. Moreover, from simulation results, we can see that sampling times can be reduced when ε becomes larger and the closedloop system is still stable. The improved sampling mechanism provides a more flexible space of design which can help reducing the unnecessary sampling times while maintaining the performance of the closed-loop system. 

3.7 Conclusion We have obtained the results on event-triggered sampled-data control for switched linear systems under an improved event-triggered sampling mechanism which adds a constant threshold ε to increase elasticity of designing. Sufficient conditions have been achieved for stability of the formed closed-loop systems under multiple Lyapunov function method and the average dwell time technique. Furthermore, a lower bound has been derived for the minimum inter-event interval to avoid zeno behavior in event-triggered sampling process (Figs. 3.8, 3.9 and 3.10).

Chapter 4

Event-Triggered Control for Switched Linear Delay Systems

4.1 Introduction In this chapter, we study state-based and observer-based event-triggered control problems for a class of switched linear delay systems. Compared with switched non-delay systems, switched delay systems are with more complex system structure in which at least one subsystem is a delay system. The feature of this chapter lies in three aspects: (1) The event-triggered control method proposed in Chap. 2 is extended to study switched linear delay systems. Based on the different designed sampling mechanisms, two sufficient conditions of exponential stability for the closed-loop switched delay systems are obtained by using multiple Lyapunov-Krasovskii functional method and the average dwell time technique. (2) Design of the sampled-databased switching controller is proposed to stabilize the switched delay system and asynchronous switching is considered in the event-triggered control processes. (3) Some free weighting matrices are introduced for achieving stability condition for the closed-loop switched delay systems. The rest of this chapter is organized as follows. In Sect. 4.2, we define the switched linear delay system and give preliminaries. Section 4.3 presents stability analysis of the closed-loop switched delay system under a state-based event-triggered sampling mechanism and a sampled-data-based switching controller. In Sect. 4.4, an exponential observer for the switched delay system is designed first, and then we analyze stability of the closed-loop switched delay system under an observer-based eventtriggered sampling mechanism and a sampled-data-based switching controller. A simple numerical example is presented in Sect. 4.5 to show the effectiveness of the developed result, and Sect. 4.6 concludes the whole chapter.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. Fu and T.-F. Li, Event-Triggered Control of Switched Linear Systems, Studies in Systems, Decision and Control 365, https://doi.org/10.1007/978-3-030-71604-2_4

51

52

4 Event-Triggered Control for Switched Linear Delay Systems

4.2 Problem Statement and Preliminaries Consider the continuous-time switched linear delay system ⎧ ⎪ ˙ = Aσ x(t) + Bσ x(t − τ ) + Cσ u(t), t > t0 ⎨x(t) y(t) = Dσ x(t), ⎪ ⎩ xt0 = x(t0 + θ) = φ(θ), θ ∈ [−τ , 0]

(4.1)

where x(t) ∈ Rn is the system state, u(t) ∈ Rm is the control input, y(t) ∈ Rq is the measurable output. σ : [0, ∞) → M = {1, 2, . . . , m} is the switching signal, where M is a finite index set. {(Ai , Bi , Ci , Di ) : i ∈ M} is a collection of matrix pairs defining individual subsystem of system (4.1), τ is a constant delay, φ(θ) is a continuously differential vector initial function on [−τ , 0], t0 is the initial time. Corresponding to switching signal σ, there exists a switching sequence {xt0 : (l0 , t0 ), . . . , (li , ti ), . . . |li ∈ M, i ∈ N},

(4.2)

∞ with which means that the li th subsystem is active when t ∈ [ti , ti+1 ), where {ti }i=0 ti < ti+1 is a switching time sequence. Without loss of generality, we assume that there is no state jumping on switching surfaces and state trajectory x(·) is continuous everywhere. Moreover, we use Nσ (t, s) to denote the number of discontinuities of switching signal σ on a semi-open interval (s, t] and assume that the switching satisfies assumption of the average dwell time. The structure of the event-triggered control system is the same with Fig. 2.1 but instead the switched system with the switched delay system which is described by (4.1). The following definition and lemma are useful for the main results.

Definition 4.1 Equilibrium x  = 0 of system (4.1) is said to be globally uniformly exponentially stable under switching signal σ, if solution x(t) of system (4.1) satisfies x(t) ≤ κe−λ(t−t0 ) x(t0 )r , ∀t ≥ t0 for positive constants κ and λ, where x(t0 )r = sup x(t0 + θ). −r ≤θ≤0

Lemma 4.2 ([136]) For any constant matrix M > 0, scalars r1 , r2 satisfying r1 < r2 , and a vector function ω : [r1 , r2 ] → Rn such that the integrations concerned are well defined, then  −(r2 − r1 )

r2

r1

 ω T (s)Mω(s)ds ≤ −

r2

r1

 ω T (s)ds M

r2

r1

ω(s)ds.

4.3 State-Based Event-Triggered Control

53

4.3 State-Based Event-Triggered Control 4.3.1 Sampling Mechanism In this subsection, an event-triggered sampling mechanism is set up first based on the system state, and then a switching controller is set up based on the sampling mechanism. Construct the event-triggered sampling mechanism e(t)2 ≥ η x(t)2 ,

(4.3)

ˆ ˆ where e(t) = x(t) − x(tˆk ), η > 0 is a constant, and {tˆk }∞ k=0 with tk < tk+1 denotes a sampling time sequence. The controller updates the current state information and switching signal when an event occurs, and holds until the next event occurs. Asynchronous switching between the system and the controller is thus caused. A long asynchronous period would lead to instability of the closed-loop system. With the state x(tˆk ) sampled at instant tˆk , the next sampling instant tˆk+1 is determined by   tˆk+1 = inf t > tˆk |e(t)2 ≥ ηx(t)2 .

(4.4)

Without loss of generality, we suppose tˆ0 = t0 and n samplings occur on non-switched interval [ti , ti+1 ) and tˆk+1 is the first sampling instant on this interval. Moreover, in order to simplify the analysis process, we assume that 0 < τm < τd , where τd and τm denote dwell time and the maximally asynchronous period, respectively. Furthermore, we suppose from (4.2) that subsystem j is the li−1 th subsystem, which is active on the interval [ti−1 , ti ) and subsystem i is the li th subsystem which is active on [ti , ti+1 ), where i, j ∈ M possess arbitrariness. For t ∈ [ti , ti+1 ), controller can be set by ⎧ K j x(tˆk ), t ∈ [ti , tˆk+1 ) ⎪ ⎪ ⎪ ⎨ K x(tˆ ), t ∈ [tˆ , tˆ ) i k+1 k+1 k+2 uσ = ⎪ · · · ⎪ ⎪ ⎩ K i x(tˆk+n ), t ∈ [tˆk+n , ti+1 ),

(4.5)

where K i and K j are controller gains. Between the consecutive samplings, controller only updates the state information and the switching information at sampling instants. We introduce a ZOH to keep control signal continuously. For t ∈ {[ti , tˆk+1 ), . . . , [tˆk+n , ti+1 )}, e(t) = x(t) − x(tˆk+ j ) holds for all j = 0, 1, . . . , n. Thus, when subsystem i is active on [ti , ti+1 ), the closed-loop form of (4.1) can be written as

54

4 Event-Triggered Control for Switched Linear Delay Systems

x(t) ˙ =

(Ai + Ci K j )x(t) + Bi x(t − τ ) − Ci K j e(t), t ∈ [ti , tˆk+1 ) (Ai + Ci K i )x(t) + Bi x(t − τ ) − Ci K i e(t), t ∈ [tˆk+1 , ti+1 ).

(4.6)

4.3.2 Stability Analysis In this section, we first present a theorem and then give the analysis process. Theorem 4.3 Consider system (4.1) with sampling instants determined by (4.4). For given positive scalars τ , μ > 1, λs , λu , η, τm and gains K i , K j , if there exist matrices Pi > 0, Q i > 0, Pi j > 0, Q i j > 0, Mi and M j of appropriate dimensions for ∀i, j ∈ M, i = j such that ⎤ Πi11j Pi j − M j + AiT M Tj M j Bi 0 M j Ci K j 0 ⎢ ∗ −M j − M Tj M j Bi 0 0 M j Ci K j ⎥ ⎥ ⎢ −λs τ ⎥ ⎢ ∗ ∗ −e Qi j 0 0 0 ⎥ < 0, ⎢ 44 ⎥ ⎢ ∗ 0 0 ∗ ∗ Πi j ⎥ ⎢ ⎦ ⎣ ∗ ∗ ∗ ∗ −I 0 ∗ ∗ ∗ ∗ ∗ −I ⎡

(4.7)

⎡

⎤ Πi11 Pi − Mi + AiT MiT Mi Bi Mi Ci K i 0 ⎢ ∗ Mi Bi 0 Mi C i K i ⎥ −Mi − MiT ⎢ ⎥ −λs τ ⎢ ∗ ⎥ < 0, ∗ −e Q 0 0 i ⎢ ⎥ ⎣ ∗ ⎦ ∗ ∗ −I 0 ∗ ∗ ∗ ∗ −I

(4.8)

Pi ≤ μP ji , Q i ≤ μQ ji , P ji ≤ μP j , Q ji ≤ μQ j ,

(4.9)

where Πi11j = M j Ai + AiT M Tj + Q i j + 2η I − λu Pi j , (λs + λu )e−λs τ Qi j , τ = Mi Ai + AiT MiT + Q i + 2η I + λs Pi ,

Πi44j = − Πi11

then system (4.1) is exponentially stable for any switching signals with average dwell time τa satisfying τa >

2 ln μ + τm (λu + λs ) . λs

(4.10)

4.3 State-Based Event-Triggered Control

55

Fig. 4.1 Diagram of asynchronous switching

Fig. 4.2 Diagram of the piecewise Lyapunov function

Proof Since we assume that τm < τd , at most one switch occurs during arbitrary a sampling interval. Figure 4.1 presents the relationship between the switching time and the sampling time in asynchronous switching situation. As shown in Fig. 4.1, for t ∈ [ti , tˆk+1 ), system (4.1) switches from subsystem j to subsystem i. However, sampler does not work on this interval, and therefore controller u j is still working. Since subsystem i is active under control u j , system (4.1) is unstable. As shown in Fig. 4.2, we choose the piecewise Lyapunov-Krasovskii functional for t ∈ [ti , tˆk+1 ) as  Vi j (t) = x (t)Pi j x(t) +

t

T

x T (s)eλs (s−t) Q i j x(s)ds,

(4.11)

t−τ

where Pi j > 0, Q i j > 0. Taking the time derivative of (4.11) along solutions of system (4.1), we have ˙ + x T (t)Q i j x(t) V˙i j (t) − λu Vi j (t) ≤ x˙ T (t)Pi j x(t) + x T (t)Pi j x(t) − x T (t − τ )e−λs τ Q i j x(t − τ ) − λu x T (t)Pi j x(t)  t −λs τ x T (s)Q i j x(s)ds. (4.12) − (λs + λu )e t−τ

56

4 Event-Triggered Control for Switched Linear Delay Systems

From Lemma 4.2, we have −(λs + λu )e−λs τ



t

x T (s)Q i j x(s)ds

t−τ

(λs + λu )e−λs τ ≤− τ





t

x (s)ds Q i j

t

T

t−τ

x(s)ds.

(4.13)

t−τ

From (4.6), for any invertible matrix M j with appropriate dimensions, we have ˙ − Ai x(t) − Bi x(t − τ ) − Ci K j e(t)] = 0. (4.14) −2[x T (t)M j + x˙ T (t)M j ][x(t) Remark 4.4 In (5.14), we introduce the free weighting matrix M j to relate items x(t), ˙ x(t), x(t − τ ) and e(t). M j can be obtained by solving LMIs (5.7)–(5.9). Please refer to [101, 102] for the free weighting matrix method.  According to Lemma 2.6, we have 2x T (t)M j Ci K j e(t) ≤ x T (t)M j Ci K j K Tj CiT M Tj x(t) + e T (t)e(t),

(4.15)

˙ + e T (t)e(t). 2 x˙ T (t)M j Ci K j e(t) ≤ x˙ T (t)M j Ci K j K Tj CiT M Tj x(t)

(4.16)

and

Adding the left-hand side of (4.14) into (4.12) and using (4.13), (4.15) and (4.16) and taking into account triggering condition (4.13), we have V˙i j (t) − λu Vi j (t) ≤ ς T (t)Θi j ς(t),

(4.17)

where 



ς (t) = x (t) x˙ (t) x (t − τ ) T

T

T

t

T

 x (s)ds , T

t−τ

⎤ M j Bi 0 Θi11j Pi j − M j + AiT M Tj ⎥ ⎢ ∗ Θi22j M j Bi 0 ⎥, Θi j = ⎢ −λs τ ⎦ ⎣ ∗ Qi j 0 ∗ −e −λs τ (λs +λu )e ∗ ∗ ∗ − Qi j τ ⎡

Θi11j = M j Ai + AiT M Tj + Q i j + M j Ci K j K Tj CiT M Tj + 2η I − λu Pi j , Θi22j = −M j − M Tj + M j Ci K j K Tj CiT M Tj . From (4.17), we know that Θi j < 0 implies V˙i (t) < λu Vi (t). Integrating this inequality from ti to t, we have Vi j (t) ≤ eλu (t−ti ) Vi j (ti ).

(4.18)

4.3 State-Based Event-Triggered Control

57

As shown in Fig. 4.1, for t ∈ [tˆk+1 , ti+1 ), no switch occurs on this interval, the sampler only transmits switching signal at time instant tˆk+1 , controller u i is active until the next switching time ti+1 , and thus subsystem i and controller u i are active synchronously, which implies that the state of the closed-loop system is convergent. We choose the piecewise Lyapunov-Krasovskii functional  Vi (t) = x (t)Pi x(t) +

t

T

x T (s)eλs (s−t) Q i x(s)ds,

(4.19)

t−τ

where Pi > 0, Q i > 0. From (4.19), we have ˙ + λs x T (t)Pi x(t) V˙i (t) + λs Vi (t) = x˙ T (t)Pi x(t) + x T (t)Pi x(t) + x T (t)Q i x(t) − x T (t − τ )e−λs τ Q i x(t − τ ).

(4.20)

Moreover, according to (4.6), for arbitrary an invertible matrix Mi with appropriate dimensions, we have ˙ − Ai x(t) − Bi x(t − τ ) − Ci K i e(t)] = 0. (4.21) −2[x T (t)Mi + x˙ T (t)Mi ][x(t) According to Lemma 2.6, we have 2x T (t)Mi Ci K i e(t) ≤ x T (t)Mi Ci K i K iT CiT MiT x(t) + e T (t)e(t),

(4.22)

˙ + e T (t)e(t). 2 x˙ T (t)Mi Ci K i e(t) ≤ x˙ T (t)Mi Ci K i K iT CiT MiT x(t)

(4.23)

and

Adding the left-hand side of (4.21) to (4.20) and taking (4.22), (4.23) and triggering condition (4.3) into account, we have V˙i (t) + λs Vi (t) ≤ Γ T (t)Θi Γ (t),

(4.24)

where   Γ T (t) = x T (t) x˙ T (t) x T (t − τ ) , ⎡ 11 ⎤ Θi Pi − Mi + AiT MiT Mi Bi Θi = ⎣ ∗ −Mi − MiT + Mi Ci K i K iT CiT MiT Mi Bi ⎦ , ∗ ∗ −e−λs τ Q i Θi11 = Mi Ai + AiT MiT + Mi Ci K i K iT CiT MiT + Q i + λs Pi + 2η I. From (4.24), we know that Θi < 0 implies that V˙i (t) < −λs Vi (t).

(4.25)

58

4 Event-Triggered Control for Switched Linear Delay Systems

Integrating inequality (4.25) from tˆk+1 to t, we have Vi (t) < e−λs (t−tˆk+1 ) Vi (tˆk+1 ).

(4.26)

From (4.9), we have Vli (ti ) ≤ μVli−1 (ti− ) for ∀li , li−1 ∈ M with μ > 1. Note that 0 . Then, for t ∈ [ti , tˆk+1 ), we have i = Nσ (t0 , t) ≤ N0 + t−t τa Vσ (t) = Vli (t) ≤ eλu (t−ti ) Vli (ti ) ≤ μeλu (t−ti ) Vli−1 (ti− ) ≤ μeλu (t−ti ) e−λs (ti −tˆk+1− j ) Vli−1 (tˆk+1− j ) − ≤ μ2 eλu (t−ti ) e−λs (ti −tˆk+1− j ) Vli−1 (tˆk+1− j)

≤ μ2 eλu (t−ti +tˆk+1− j −ti−1 ) × e−λs (ti −tˆk+1− j ) Vli−1 li−2 (ti−1 ) ≤ ··· ≤ μ2i−1 eλu [t−ti +(i−1)τm ] e−λs [ti −(i−1)τm ] Vl0 (t0 ) ≤ μ2i−1 eiλu τm −λs (t−iτm ) Vl0 (t0 ) 1 = ei[2 ln μ+τm (λu +λs )]−λs (t−t0 ) Vl0 (t0 ) μ 1 ≤ e[2 ln μ+τm (λu +λs )]N0 μ 

×e

2 ln μ τm (λu +λs ) −λs τa + τa



(t−t0 )

Vl0 (t0 ),

(4.27)

where tˆk+1− j denotes the (k + 1 − j)th sampling which is also the first sampling instant on the interval [ti−1 , ti ). For t ∈ [tˆk+1 , ti+1 ), we have Vσ (t) = Vli (t) ≤ e−λs (t−tˆk+1 ) Vli (tˆk+1 ) − ≤ μe−λs (t−tˆk+1 ) Vli−1 (tˆk+1 )

≤ μe−λs (t−tˆk+1 ) eλu (tˆk+1 −ti ) Vli−1 (ti ) ≤ μ2 e−λs (t−tˆk+1 ) eλu (tˆk+1 −ti ) Vli−2 (ti− ) ≤ μ2 e−λs (t−tˆk+1 +ti −tˆk+1− j ) eλu (tˆk+1 −ti ) Vli−2 (tˆk+1− j ) ≤ ··· ≤ μ2i e−λs [t−(tˆk+1 −ti )−(tˆk+1− j −ti−1 )−···−(tˆk+1−m −t1 )−tˆ0 ] × eλu (tˆk+1 −ti +tˆk+1− j −ti−1 +···+tˆk+1−m −t1 ) Vl0 (t0 ) ≤ μ2i eiλu τm −λs (t−iτm −t0 ) Vl0 (t0 ) ≤ e[2 ln μ+τm (λu +λs )]N0

4.3 State-Based Event-Triggered Control 

×e

59

2 ln μ τm (λu +λs ) −λs τa + τa



(t−t0 )

Vl0 (t0 ).

(4.28)

From (4.11) and (4.19), we have Vσ (t) ≥

min {λmin {Pli , Pli li+1 }}x(t)2 = ax(t)2 ,

∀li ,li+1 ∈M

(4.29)

and Vσ (t0 ) ≤

max {λmax {Pli , Pli li+1 }}φ2

∀li ,li+1 ∈M

+τ

max {λmax {Q li , Q li li+1 }}φ2 = bφ2 ,

∀li ,li+1 ∈M

(4.30)

where a= b=

min {λmin {Pli , Pli li+1 }},

∀li ,li+1 ∈M

max {λmax {Pli , Pli li+1 }} + τ

∀li ,li+1 ∈M

max {λmax {Q li , Q li li+1 }}.

∀li ,li+1 ∈M

Combining (4.27)–(4.30), we have 1 Vli (t) a   1 b [2 ln μ+τm (λu +λs )]N0 e ≤ 1, μ a

x(t)2 ≤



×e

2 ln μ τm (λu +λs ) −λs τa + τa

 (t−t0 )

φ2 .

(4.31)

By using Schur complement lemma, inequalities (4.7) and (4.8) are equivalent to Θi j < 0 and Θi < 0. From Definition 4.1, exponential stability of system (4.1) is guaranteed from (4.31) when condition (4.10) holds. This completes the proof. 

4.3.3 Control Design In fact, Theorem 4.3 does not give the design of control. However, control gain K i can be derived by using a transform technique. The following theorem presents the result of control design directly. Theorem 4.5 Consider system (4.1) with sampling instants determined by (4.4). For given positive scalars τ , μ > 1, λs , λu , η, τm , δ, if there exist matrices P¯i > 0, Q¯ i > 0, S¯i > 0, M¯ i > 0, P¯i j > 0, Q¯ i j > 0, S¯i j > 0, M¯ j > 0 of appropriate dimensions for ∀i, j ∈ M, i = j such that

60

4 Event-Triggered Control for Switched Linear Delay Systems

⎡

Π¯ i11j P¯i j − M¯ j + M¯ j AiT Bi M¯ j ⎢ ∗ T − M¯ j − M¯ j Bi M¯ j ⎢ ⎢ ∗ −λs τ ¯ Qi j ∗ −e ⎢ ⎢ ∗ ∗ ∗ ⎢ ⎢ ⎢ ∗ ∗ ∗ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗

⎤ 0 Ci R¯ j 0 M¯ j 0 0 Ci R¯ j 0 ⎥ ⎥ 0 0 0 0 ⎥ ⎥ ⎥ 0 0 ⎥ < 0, Π¯ i44j 0 ⎥ 0 ⎥ ∗ − S¯i j 0 ⎥ ∗ ∗ − S¯i j 0 ⎦ 1 I ∗ ∗ ∗ − 2ηδ

⎡ ¯ 11 ¯ ⎤ Πi Pi − M¯ i + M¯ i AiT Bi M¯ i Ci R¯ i 0 M¯ i ⎢ ∗ − M¯ i − M¯ iT Bi M¯ i 0 Ci R¯ i 0 ⎥ ⎢ ⎥ −λs τ ¯ ⎢ ∗ ∗ −e 0 0 ⎥ Qi 0 ⎢ ⎥ < 0, ⎢ ∗ ∗ ∗ − S¯i 0 0 ⎥ ⎢ ⎥ ⎣ ∗ ∗ ∗ ∗ − S¯i 0 ⎦ 1 ∗ ∗ ∗ ∗ ∗ − 2ηδ I P¯i ≤ μ P¯ ji , Q¯ i ≤ μ Q¯ ji , P¯ ji ≤ μ P¯ j , Q¯ ji ≤ μ Q¯ j ,

(4.32)

(4.33)

(4.34)

where Π¯ i11j = Ai M¯ j + M¯ j AiT + Q¯ i j − λu P¯i j , (λs + λu )e−λs τ ¯ Qi j , τ = Ai M¯ i + M¯ i AiT + Q¯ i + λs P¯i ,

Π¯ i44j = − Π¯ i11 and verifying  max

∀i, j∈M

 −1 ¯ ¯ λmax { M¯ i−1 S¯i M¯ i−1 }, λmax { M¯ −1 } 0, Si > 0. Based on Lemma 2.6 and recall triggering condition (4.4), we have e T (t)Si j e(t) ≤ λmax (Si j )e(t)2 ≤ ηλmax (Si j )x(t)2 , e T (t)Si e(t) ≤ λmax (Si )e(t)2 ≤ ηλmax (Si )x(t)2 . Therefore, adapting the similar deriving technique, we can obtain ⎡

⎤ Πˆ i11j Pi j − M j + AiT M Tj M j Bi 0 M j Ci K j 0 ⎢ ∗ −M j − M Tj M j Bi 0 0 M j Ci K j ⎥ ⎢ ⎥ −λs τ ⎢ ∗ ⎥ ∗ −e Qi j 0 0 0 ⎢ ⎥ < 0, 44 ⎢ ∗ ⎥ 0 0 ∗ ∗ Πi j ⎢ ⎥ ⎣ ∗ ⎦ ∗ ∗ ∗ −Si j 0 ∗ ∗ ∗ ∗ ∗ −Si j

(4.39)

⎡

⎤ Πˆ i11 Pi − Mi + AiT MiT Mi Bi Mi Ci K i 0 ⎢ ∗ Mi Bi 0 Mi C i K i ⎥ −Mi − MiT ⎢ ⎥ −λs τ ⎢ ∗ ⎥ < 0, ∗ −e Qi 0 0 ⎢ ⎥ ⎣ ∗ ⎦ 0 ∗ ∗ −Si ∗ ∗ ∗ ∗ −Si where

(4.40)

Πˆ i11j = M j Ai + AiT M Tj + Q i j + 2ηλmax {Si j }I − λu Pi j , Πˆ i11 = Mi Ai + AiT MiT + Q i + 2ηλmax {Si }I + λs Pi .

−1 on both side of inequalities (4.39) and (4.40), respecMultiplying M −1 j and Mi ¯ tively, and applying Schur complement lemma, then letting M¯ j = M −1 j , Qi j = −1 −1 ¯ −1 −1 ¯ −1 −1 ¯ −1 ¯ M j Q i j M j , Pi j = M j Pi j M j , Si j = M j Si j M j , R j = K j M j , Mi = Mi−1 , Q¯ i = Mi−1 Q i Mi−1 , P¯i = Mi−1 Pi Mi−1 , S¯i = Mi−1 Si Mi−1 , R¯ i = K i Mi−1 , and intro-

62

4 Event-Triggered Control for Switched Linear Delay Systems

ducing a scalar δ large enough satisfying δ > max {λmax (Si j ), λmax (Si )}, we can ∀i, j∈M

obtain that matrix inequalities (4.39) and (4.40) are inequivalent with LMIs (4.32) and (4.40). 

4.4 Observer-Based Event-Triggered Control 4.4.1 Observer Design The objective of this subsection is to design an exponential observer for system (4.1). We construct the observer ˙ˆ = Ai x(t) ˆ + Bi x(t ˆ − τ ) + Ci u(t) + L i (y(t) − Di x(t)) ˆ x(t)

(4.41)

for subsystem i, ∀i ∈ M, where x(t) ˆ ∈ Rn is the observer state, L i is the observer gain of subsystem i. Let error be e(t) = x(t) − x(t). ˆ From (4.1) and (4.41), we have e(t) ˙ = (Ai − L i Di )e(t) + Bi e(t − τ ).

(4.42)

Definition 4.6 If system (4.42) is exponentially stable, then (4.41) is an exponential observer of subsystem i. Based on Definition 4.6, we propose a lemma to design observer (4.41) such that (4.41) exponentially estimates the state of subsystem i. Lemma 4.7 For given scalars α > 0 and τ > 0, if there exist matrices Pi > 0, Q i > 0, Wi and Si such that ⎤ Ωi11 Ωi12 Si Bi ⎣ ∗ −Si − SiT Si Bi ⎦ < 0, ∀i ∈ M ∗ ∗ −e−ατ Q i ⎡

(4.43)

where Ωi11 = αPi + Q i + Si Ai − Wi Di + AiT SiT − DiT WiT , Ωi12 = Pi − Si + AiT SiT − DiT WiT , then system (4.42) is exponentially stable, and from Definition 4.6, (4.41) is an exponential observer of subsystem i and observer gain L i is given by L i = Si−1 Wi . Proof Choose the Lyapunov-Krasovskii functional  Vi (t) = e T (t)Pi e(t) +

t t−τ

e T (s)eα(s−t) Q i e(s)ds.

(4.44)

4.4 Observer-Based Event-Triggered Control

63

Taking the time derivative of Vi (t) along solutions of system (4.42), we have ˙ V˙i (t)+αVi (t) = e˙ T (t)Pi e(t) + e T (t)Pi e(t) + e T (t)(αPi + Q i )e(t) − e T (t − τ )e−ατ Q i e(t − τ ).

(4.45)

According to (4.42), for arbitrary an invertible matrix Si with appropriate dimensions, we have ˙ − (Ai − L i Di )e(t) − Bi e(t − τ )] = 0. −2[e T (t)Si + e˙ T (t)Si ][e(t)

(4.46)

Adding the left-hand side of (4.46) into (4.45), we have V˙i (t) + αVi (t) = ζ T (t)Ωi ζ(t),

(4.47)

where   ζ T (t) = e T (t) e˙ T (t) e T (t − τ ) , ⎡ 11 ⎤ Ω˜ i Ω˜ i12 Si Bi Ωi = ⎣ ∗ −Si − SiT Si Bi ⎦ , ∗ ∗ −e−ατ Q i Ω˜ i11 = αPi + Q i + Si (Ai − L i Di ) + (Ai − L i Di )T SiT , Ω˜ i12 = Pi − Si + (Ai − L i Di )T SiT . Let Wi = Si L i . V˙i (t) + αVi (t) < 0 follows from inequality (4.43). Integrating inequality V˙i (t) + αVi (t) < 0 from t0 to t, we have Vi (t) < e−α(t−t0 ) Vi (t0 ), which guarantees that system (4.42) is exponentially stable.

(4.48) 

From (4.41) and (4.42), we obtain the switching observer ˙ˆ = Aσ x(t) x(t) ˆ + Bσ x(t ˆ − τ ) + Cσ u(t) + L σ Dσ e(t),

(4.49)

and the error system e(t) ˙ = (Aσ − L σ Dσ )e(t) + Bσ e(t − τ ) with the same switching rule of system (4.1).

(4.50)

64

4 Event-Triggered Control for Switched Linear Delay Systems

4.4.2 Sampling Mechanism An observer-based event-triggered sampling mechanism is introduced with the form 2  e(t) ˆ  ≥ η ξ(t)2 ,

(4.51)

where e(t) ˆ = x(t) ˆ − x( ˆ tˆk ), ξ(t) = [xˆ T (t) e T (t)]T , η > 0 is a constant, and {tˆk }∞ k=0 with tˆk < tˆk+1 denotes a sampling time sequence. An event detector and a sampler are located in a same module. The detector detects the triggering condition (4.51) continuously and triggers the sampler when condition (4.51) is satisfied. When an event occurs, the sampler samples the current the information of observer state and the switching signal, and transmits the information to controller. The controller updates its current state information and switching signal with the information received from sampler and holds until the next event occurs. With the state x( ˆ tˆk ) sampled at instant ˆtk , the next sampling instant tˆk+1 is determined by   2 ˆ ≥ ηξ(t)2 . tˆk+1 = inf t > tˆk |e(t)

(4.52)

As shown in Fig. 4.1, we assume that tˆ0 = t0 and n samplings occur on the interval [ti , ti+1 ), in which tˆk+1 is the first sampling instant. Asynchronous switching is inevitable and a long asynchronous period would lead to instability of the closed-loop system. In order to simplify the analysis process, we assume that τm < τd , where τd and τm denote dwell time and the maximally asynchronous period, respectively. Furthermore, we assume that subsystem j is active on the interval [ti−1 , ti ) and subsystem i is active on the interval [ti , ti+1 ), where i, j ∈ M possess arbitrariness. For t ∈ [ti , ti+1 ), the controller can be set by ⎧ K j x( ˆ tˆk ), t ∈ [ti , tˆk+1 ) ⎪ ⎪ ⎪ ⎨ K x( i ˆ tˆk+1 ), t ∈ [tˆk+1 , tˆk+2 ) u= ⎪ ··· ⎪ ⎪ ⎩ ˆ tˆk+n ), t ∈ [tˆk+n , ti+1 ), K i x(

(4.53)

where K i and K j are controller gains. Between the consecutive samplings, controller only updates the state and the switching signal at the sampling instants. We thus introduce a ZOH to keep control signal continuously. For t ∈ {[ti , tˆk+1 ), . . . , [tˆk+n , ti+1 )}, e(t) ˆ = x(t) ˆ − x( ˆ tˆk+ j ) holds for all j = 0, 1, . . . , n. Thus, when subsystem i is active on [ti , ti+1 ), the closed-loop system can be written as ⎧ (Ai + Ci K j )x(t) ˆ + Bi x(t ˆ − τ) ⎪ ⎪ ⎪ ⎨ ˆ + L i Di e(t), t ∈ [ti , tˆk+1 ) −Ci K j e(t) ˙ˆ = x(t) ⎪ ˆ + Bi x(t ˆ − τ) (Ai + Ci K i )x(t) ⎪ ⎪ ⎩ ˆ + L i Di e(t), t ∈ [tˆk+1 , ti+1 ). −Ci K i e(t)

(4.54)

4.4 Observer-Based Event-Triggered Control

65

Recalling e(t) = x(t) − x(t) ˆ and ξ(t) = [xˆ T (t) e T (t)]T , we have ˙ = ξ(t)

A¯ i j ξ(t) + B¯ i ξ(t − τ ) + C¯ i j e(t), ˜ t ∈ [ti , tˆk+1 ) ˜ t ∈ [tˆk+1 , ti+1 ), A¯ i ξ(t) + B¯ i ξ(t − τ ) + C¯ i e(t),

(4.55)

where    L i Di Ai + C i K j −Ci K j 0 , C¯ i j = , 0 Ai − L i Di 0 0     Bi 0 −Ci K i 0 , , C¯ i = B¯ i = 0 Bi 0 0     L i Di e(t) ˆ Ai + C i K i , e(t) ˜ = . A¯ i = 0 Ai − L i Di 0

A¯ i j =



From (4.55), the closed-loop system can be rewritten by ˙ = A¯ σi σ j ξ(t) + B¯ σi ξ(t − τ ) + C¯ σi σ j e(t), ˜ ξ(t)

(4.56)

where   L σi Dσi Aσi + Cσi K σ j , A¯ σi σ j = 0 Aσi − L σi Dσi     −Cσi K σ j 0 Bσi 0 . C¯ σi σ j = , B¯ σi = 0 0 0 Bσi For system (4.56), if i = j, then A¯ σi σi = A¯ σi and C¯ σi σi = C¯ σi .

4.4.3 Stability Analysis In this section, we analyze stability of the closed-loop system (4.56). The following theorem present the main result of this section. Theorem 4.8 Consider system (4.1) with sampling instants determined by (4.42). For given positive scalars τ , μ > 1, λs , λu , η, τm and gains K i , K j , if there exist matrices Pi > 0, Q i > 0, Pi j > 0, Q i j > 0, Mi and M j of appropriate dimensions for ∀i, j ∈ M, i = j such that

66

4 Event-Triggered Control for Switched Linear Delay Systems

⎡

Θ˜ i11j ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

Θ˜ i12j Θ˜ i13j 0 M j Bi 0 Θ˜ i22j Θ˜ i23j Θ˜ i24j 0 M j Bi 33 ˜ ∗ Θi j 0 M j Bi 0 ∗ ∗ Θ˜ i44j 0 M j Bi ∗ ∗ ∗ Θ˜ i55j 0 ˜ ∗ ∗ ∗ ∗ Θi66j ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

⎡ ˜ 11 Θi ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

Θ˜ i12 Θ˜ i22 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

Θ˜ i13 Θ˜ i23 Θ˜ i33 ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

0 Θ˜ i24 0 ˜ Θi44 ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 0 Θ˜ i77j ∗ ∗ ∗ ∗ ∗

0 0 0 0 0 0 0 Θ˜ i88j ∗ ∗ ∗ ∗

Θ˜ i19j 0 0 0 0 0 0 0 −I ∗ ∗ ∗

0 0 0 0 0 Θ˜ i311 j 0 0 0 0 0 0 0 0 0 0 0 0 −I 0 ∗ −I ∗ ∗

⎤ 0 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ < 0, ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ −I

⎤ Mi Bi 0 Θ˜ i17 0 0 0 0 Mi Bi 0 0 0 0 ⎥ ⎥ Mi Bi 0 0 0 Θ˜ i39 0 ⎥ ⎥ 0 Mi Bi 0 0 0 0 ⎥ ⎥ 0 0 0 0 ⎥ Θ˜ i55 0 ⎥ < 0, ∗ Θ˜ i66 0 0 0 0 ⎥ ⎥ ∗ ∗ −I 0 0 0 ⎥ ⎥ ∗ ∗ ∗ −I 0 0 ⎥ ⎥ ∗ ∗ ∗ ∗ −I 0 ⎦ ∗ ∗ ∗ ∗ ∗ −I

Pi ≤ μP ji , Q i ≤ μQ ji , P ji ≤ μP j , Q ji ≤ μQ j ,

(4.57)

(4.58)

(4.59)

where Θ˜ i11j = M j Ai + M j Ci K j + AiT M Tj + K Tj CiT M Tj + Q i j + 2η I − λu Pi j , Θ˜ i12j = M j L i Di , Θ˜ i13j = Pi j − M j + AiT M Tj + K Tj CiT M Tj , Θ˜ i22j = M j Ai − M j L i Di + AiT M Tj − DiT L iT M Tj + Q i j + 2η I − λu Pi j , Θ˜ i23j = DiT L iT M Tj , Θ˜ i24j = Pi j − M j + AiT M Tj − DiT L iT M Tj , Θ˜ i33j = Θ˜ i44j = −M j − M Tj , Θ˜ i55j = Θ˜ i66j = −e−λs τ Q i j , Θ˜ i77j = Θ˜ i88j = −

(λs + λu )e−λs τ Qi j , τ

4.4 Observer-Based Event-Triggered Control

67

Θ˜ i19j = Θ˜ i311 j = −M j C i K j , Θ˜ i11 = Mi Ai + Mi Ci K i + AiT MiT + K iT CiT MiT + Q i + 2η I + λs Pi , Θ˜ i12 = Mi L i Di , Θ˜ i13 = Pi − Mi + AiT MiT + K iT CiT MiT , Θ˜ i22 = Mi Ai − Mi L i Di + AiT MiT − DiT L iT MiT + Q i + 2η I + λs Pi , Θ˜ i23 = DiT L iT MiT , Θ˜ i24 = Pi − Mi + AiT MiT − DiT L iT MiT , Θ˜ i33 = Θ˜ i44 = −Mi − MiT , Θ˜ i55 = Θ˜ i66 = −e−λs τ Q i , Θ˜ i17 = Θ˜ i39 = −Mi Ci K i . then system (4.1) is exponentially stable for any switching signals with average dwell time τa satisfying τa >

2 ln μ + τm (λu + λs ) . λs

(4.60)

Proof As shown in Fig. 4.1. For t ∈ [ti , tˆk+1 ), system (4.1) switches from subsystem j to subsystem i, but controller u j is still working. We choose the piecewise Lyapunov-Krasovskii functional Vi j (t) = ξ T (t) P¯i j ξ(t) +



t

ξ T (s)eλs (s−t) Q¯ i j ξ(s)ds,

(4.61)

t−τ

where P¯i j = diag{Pi j , Pi j } > 0, and Q¯ i j = diag{Q i j , Q i j } > 0. Taking the time derivative of (4.61) along solutions of system (4.56), we have ˙ + ξ T (t) Q¯ i j ξ(t) V˙i j (t) − λu Vi j (t) ≤ ξ˙T (t) P¯i j ξ(t) + ξ T (t) P¯i j ξ(t) − ξ T (t − τ )e−λs τ Q¯ i j ξ(t − τ ) − λu ξ T (t) P¯i j ξ(t)  t −λs τ ξ T (s) Q¯ i j ξ(s)ds. − (λs + λu )e

(4.62)

t−τ

From Lemma 4.2, we have −(λs + λu )e−λs τ



t

ξ T (s) Q¯ i j ξ(s)ds

t−τ

≤−

(λs + λu )e−λs τ τ



t

t−τ

ξ T (s)ds Q¯ i j



t t−τ

ξ(s)ds.

(4.63)

68

4 Event-Triggered Control for Switched Linear Delay Systems

According to (4.55), for arbitrary an invertible matrix M¯ j = diag{M j , M j } with appropriate dimensions, we have ˙ − A¯ i j ξ(t) − B¯ i ξ(t − τ ) − C¯ i j e(t)] ˜ = 0. −2[ξ T (t) M¯ j + ξ˙T (t) M¯ j ][ξ(t)

(4.64)

According to Lemma 2.6, we have ˜ ≤ ξ T (t) M¯ j C¯ i j C¯ iTj M¯ Tj ξ(t) + e˜ T (t)e(t), ˜ 2ξ T (t) M¯ j C¯ i j e(t)

(4.65)

˙ + e˜ T (t)e(t). ˜ ≤ ξ˙T (t) M¯ j C¯ i j C¯ iTj M¯ Tj ξ(t) ˜ 2ξ˙T (t) M¯ j C¯ i j e(t)

(4.66)

and

Adding the left-hand side of (4.64) into (4.62), and combining (4.63), (4.65) and (4.66), and considering triggering condition (4.51), we have V˙i (t) − λu Vi (t) ≤ ς T (t)Θi j ς(t),

(4.67)

where 

ς (t) = ξ (t) ξ˙T (t) ξ T (t − τ ) T



t

T

 ξ (s)ds , T

t−τ

⎤ 0 M¯ j B¯ i Θi11j P¯i j − M¯ j + A¯ iTj M¯ Tj ⎥ ⎢ ∗ 0 M¯ j B¯ i Θi22j ⎥, Θi j = ⎢ −λs τ ¯ ⎦ ⎣ ∗ 0 Qi j ∗ −e (λs +λu )e−λs τ ¯ ∗ ∗ ∗ − Qi j τ Θi11j = M¯ j A¯ i j + A¯ iTj M¯ Tj + Q¯ i j + 2η I + M¯ j C¯ i j C¯ iTj M¯ Tj − λu P¯i j , Θi22j = − M¯ j − M¯ Tj + M¯ j C¯ i j C¯ iTj M¯ Tj . ⎡

From (4.67), we know that Θi j < 0 implies V˙i (t) < λu Vi (t).

(4.68)

Integrating inequality (4.68) from ti to t, we have Vi j (t) ≤ eλu (t−ti ) Vi j (ti ).

(4.69)

For t ∈ [tˆk+1 , ti+1 ), no switch occurs on this interval, the sampler only transmits switching signal at time instant tˆk+1 , and controller u i is active until the next switching time ti+1 . Therefore, subsystem i and controller u i are active synchronously on [tˆk+1 , ti+1 ). We choose the Lyapunov-Krovskii functional

4.4 Observer-Based Event-Triggered Control

Vi (t) = ξ T (t) P¯i ξ(t) +

69



t

ξ T (s)eλs (s−t) Q¯ i ξ(s)ds,

(4.70)

t−τ

where P¯i = diag{Pi , Pi } > 0, Q¯ i = diag{Q i , Q i } > 0. Taking the time derivative of (4.70) along solutions of system (4.56), we have ˙ + λs ξ T (t) P¯i ξ(t) + ξ T (t) Q¯ i ξ(t) V˙i (t) + λs Vi (t) = 2ξ T (t) P¯i ξ(t) − ξ T (t − τ )e−λs τ Q¯ i ξ(t − τ ).

(4.71)

Moreover, for arbitrary an invertible matrix M¯ i = diag{Mi , Mi } with appropriate dimensions, we have ˙ − A¯ i ξ(t) − B¯ i ξ(t − τ ) − C¯ i e(t)] ˜ = 0. −2[ξ T (t) M¯ i + ξ˙T (t) M¯ i [ξ(t)

(4.72)

According to Lemma 2.6, we have ˜ ≤ ξ T (t) M¯ i C¯ i C¯ iT M¯ iT ξ(t) + e˜ T (t)e(t), ˜ 2ξ T (t) M¯ i C¯ i e(t)

(4.73)

˙ + e˜ T (t)e(t). ˜ ≤ ξ˙T (t) M¯ i C¯ i C¯ iT M¯ iT ξ(t) ˜ 2ξ˙T (t) M¯ i C¯ i e(t)

(4.74)

and

Adding the left-hand side of (4.72) into (4.71) and taking (4.73), (4.74) and triggering condition (4.51) into account, we have V˙i (t) + λs Vi (t) ≤ Γ T (t)Θi Γ (t),

(4.75)

where   Γ T (t) = ξ T (t) ξ˙T (t) ξ T (t − τ ) , ⎡ 11 ⎤ Θi P¯i − M¯ i + A¯ iT M¯ iT M¯ i B¯ i Θi = ⎣ ∗ − M¯ i − M¯ iT + M¯ i C¯ i C¯ iT M¯ iT M¯ i B¯ i ⎦ , ∗ ∗ −e−λs τ Q¯ i Θi11 = M¯ i A¯ i + A¯ iT M¯ iT + M¯ i C¯ i C¯ iT M¯ iT + Q¯ i + λs P¯i + 2η I. Therefore, Θi < 0 implies that V˙i (t) < −λs Vi (t). Integrating this inequality from tˆk+1 to t, we have Vi (t) < e−λs (t−tˆk+1 ) Vi (tˆk+1 ).

(4.76)

From (4.59), we have Vli (ti ) ≤ μVli−1 (ti− ) for ∀li , li−1 ∈ M with μ > 1. Note that 0 . Then, for t ∈ [ti , tˆk+1 ), we have i = Nσ (t0 , t) ≤ N0 + t−t τa

70

4 Event-Triggered Control for Switched Linear Delay Systems

Vσ (t) = Vli (t) ≤ eλu (t−ti ) Vli (ti ) ≤ μeλu (t−ti ) Vli−1 (ti− ) ≤ μeλu (t−ti ) e−λs (ti −tˆk+1− j ) Vli−1 (tˆk+1− j ) ≤ ··· 1 ≤ e[2 ln μ+τm (λu +λs )]N0 μ 

×e

2 ln μ τm (λu +λs ) −λs τa + τa

 (t−t0 )

Vl0 (t0 ),

(4.77)

where tˆk+1− j denotes the (k + 1 − j)th sampling which is also the first sampling instant on the interval [ti−1 , ti ). For t ∈ [tˆk+1 , ti+1 ), we have Vσ (t) = Vli (t) ≤ e−λs (t−tˆk+1 ) Vli (tˆk+1 ) − ) ≤ μe−λs (t−tˆk+1 ) Vli−1 (tˆk+1

≤ μe−λs (t−tˆk+1 ) eλu (tˆk+1 −ti ) Vli−1 (ti ) ≤ ··· ≤ μ2i eiλu τm −λs (t−iτm −t0 ) Vl0 (t0 ) ≤ e[2 ln μ+τm (λu +λs )]N0 

×e

2 ln μ τm (λu +λs ) −λs τa + τa



(t−t0 )

Vl0 (t0 ).

(4.78)

Moreover, we have Vσ (t) ≥ λmin { P¯li , P¯li li+1 } min {λmin { P¯li , P¯li li+1 }}ξ(t)2 = aξ(t)2 ,

≥

∀li ,li+1 ∈M

(4.79)

and Vσ (t0 ) ≤

max {λmax { P¯li , P¯li li+1 }}φ2

∀li ,li+1 ∈M

+τ

max {λmax { Q¯ li , Q¯ li li+1 }}φ2 = bφ2 ,

∀li ,li+1 ∈M

where a= b=

min {λmin { P¯li , P¯li li+1 }},

∀li ,li+1 ∈M

max {λmax { P¯li , P¯li li+1 }} + τ

∀li ,li+1 ∈M

Combining (4.77)–(4.80), we have

max {λmax { Q¯ li , Q¯ li li+1 }}.

∀li ,li+1 ∈M

(4.80)

4.4 Observer-Based Event-Triggered Control

71

1 Vl (t) a i   1 b [2 ln μ+τm (λu +λs )]N0 ≤ 1, e μ a

ξ(t)2 ≤



×e

2 ln μ τm (λu +λs ) −λs τa + τa

 (t−t0 )

φ2 .

(4.81)

According to Definition 4.1, exponential stability of system (4.1) is guaranteed when condition (4.60) holds.  Remark 4.9 Note that inequalities Θi j < 0 and Θi < 0 are both nonlinear matrix inequalities. Applying Schur complement lemma, inequalities Θi j < 0 and Θi < 0 can be converted into ⎡ ¯ 11 ¯ Θi j Pi j − M¯ j + A¯ iTj M¯ Tj M¯ j B¯ i T ⎢ ∗ M¯ j B¯ i − M¯ j − M¯ j ⎢ −λs τ ¯ ⎢ ∗ ∗ −e Qi ⎢ ⎢ ∗ ∗ ∗ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗

⎤ 0 M¯ j C¯ i j 0 0 0 M¯ j C¯ i j ⎥ ⎥ 0 0 0 ⎥ ⎥ < 0, 0 0 ⎥ Θ¯ i44j ⎥ ∗ −I 0 ⎦ ∗ ∗ −I

(4.82)

and ⎡

⎤ Θ¯ i11 P¯i − M¯ i + A¯ iT M¯ iT M¯ i B¯ i M¯ i C¯ i 0 ⎢ ∗ − M¯ i − M¯ iT 0 M¯ i C¯ i ⎥ M¯ i B¯ i ⎢ ⎥ −λs τ ¯ ⎢ ∗ ∗ −e 0 ⎥ Qi 0 ⎢ ⎥ < 0, ⎣ ∗ ∗ ∗ −I 0 ⎦ ∗ ∗ ∗ ∗ −I

(4.83)

where Θ¯ i11j = M¯ j A¯ i j + A¯ iTj M¯ Tj + Q¯ i j + 2η I − λu P¯i j , (λs + λu )e−λs τ ¯ Qi j , τ = M¯ i A¯ i + A¯ iT M¯ iT + Q¯ i + 2η I + λs P¯i .

Θ¯ i44j = − Θ¯ i11

Substituting matrix parameters of (4.56) into (4.82) and (4.83), we obtain LMIs (4.57) and (4.58) which can be solved by LMI toolbox directly.  Remark 4.10 In this theorem, the control gain K i is given in advanced. In fact, one can obtain the control gain by referring to the similar transform technique presented in Sect. 4.3.3. 

72

4 Event-Triggered Control for Switched Linear Delay Systems

4.5 Illustrative Example In this section, we present a numerical example for state-based event-triggered control method proposed in this chapter. Example 4.11 Consider the switched linear delay system x(t) ˙ = Aσ x(t) + Bσ x(t − τ ) + Cσ u(t),

(4.84)

where σ ∈ {1, 2}, and system parameters are given by 

    −2 0.1 −0.1 0.1 1 , B1 = , C1 = 0 −1.3 0 0.1 0      −3.5 0.1 0.1 0 1 , B2 = , C2 = A2 = 0.2 −1.3 0.1 0.1 0 A1 =

 0 , 1  0 , 1

and τ = 0.2. Set up an triggering condition by e(t)2 ≥ 0.5x(t)2 .

(4.85)

Let λs = 0.1, λu = 2, μ = 1.1, τm = 0.2, and the control be 

   1 0.1 0.5 0 K1 = , K2 = . 0.2 −1 0.1 −0.8 By solving inequalities (4.7), (4.8) and (4.9) in Theorem 4.3, we obtain the feasible solutions     5.0471 −0.6519 5.4755 −0.6813 , P2 = . P1 = −0.6519 2.7022 −0.6813 2.8385     2.1299 −0.3794 2.2442 −0.3996 , Q2 = . Q1 = −0.3794 0.3432 −0.3996 0.3572     5.2785 −0.6715 5.5676 −0.6805 P12 = , P21 = . −0.6715 2.7898 −0.6895 2.8204     2.2048 −0.3928 2.2419 −0.3999 , Q 21 = . Q 12 = −0.3928 0.3533 −0.3999 0.3545     1.6673 −0.1813 1.9888 −0.3611 , M2 = . M1 = −0.1939 1.1882 −0.1068 1.3783 According to (4.10), we have τa >

2 ln μ + τm (λs + λu ) = 6.1062. λs

4.5 Illustrative Example

73

State responses of the system

2

1

0

-1

-2

-3 0

1

2

3

4

5

6

7

8

9

10

7

8

9

10

Time (sec) Fig. 4.3 State responses of the closed-loop system (4.47) 0 -0.2 -0.4

Control input

-0.6 -0.8 -1 -1.2 -1.4 -1.6 -1.8 -2 0

1

2

3

4

5

Time (sec) Fig. 4.4 Control input

6

74

4 Event-Triggered Control for Switched Linear Delay Systems

12

Triggering condition

10

8

10 -6

10

6 5

4 0 2

2

3

4

Time (sec)

0 0

1

2

3

4

5

Time (sec)

Switching signal

Fig. 4.5 Event-triggering condition

2

1

0

5

10

15

Time (sec) Fig. 4.6 Switching signals

20

25

30

4.5 Illustrative Example

75

Triggering instant

1

0

1

2

3

4

5

Time (sec) Fig. 4.7 Triggering instants 2

Sampling state

1

0

-1

-2

-3 0

1

2

3

Time (sec) Fig. 4.8 Sampling states

4

5

76

4 Event-Triggered Control for Switched Linear Delay Systems

Choose the initial state x0 = [−3 2]T . We obtain the system state responses and the controller responses in Figs. 4.3 and 4.4, respectively. Triggering condition, switching signal, triggering instants and sampling states are presented in Figs. 4.5, 4.6, 4.7 and 4.8, respectively. From the simulation results, we can see that system (4.84) is stable under control input (4.15) determined by sampling mechanism (4.85). Figure 4.6 shows that asynchronous switching is allowed in control process. 

4.6 Conclusion In this chapter, we have presented the results on stabilization of the switched linear delay system under a state-based event-triggered sampling mechanism and an observer-based event-triggered sampling mechanism. Asynchronous switching caused by the sampling mechanism has been studied. Two sufficient conditions have been obtained to guarantee globally exponential stability of the closed-loop switched linear delay system subject to the average dwell time technique. Some free weighting matrices have been introduced to help achieving the stability condition which can be solved by LMI toolbox.

Chapter 5

Event-Triggered Control of Switched Linear Neutral Systems

5.1 Introduction In this chapter, we study the observer-based output feedback control problem for a class of switched linear neutral systems under an event-triggered sampling mechanism and for the first time address event-triggered control of switched linear neutral systems. Switched neutral systems are a special kind of switched delay systems but with more complexity structure in which at least one subsystem is a neutral system. As we all know that a neutral system is a more general delay system that delay is contained not only in its state but also in the derivation of its state which bring more difficulty in stability analysis. Furthermore, we also study asynchronous switching problem which is caused by the event-triggered sampling mechanism in the control process. An event occurs when a triggering condition is satisfied. Under an observerbased controller combining with switching policy provided that average dwell time conditions of subsystems are satisfied, we obtain a sufficient condition in LMIs form which guarantees exponential stability of the closed-loop switched neutral system by using multiple Lyapunov–Krasovskii functional method. Some free weighting matrices are introduced which decrease the conversation of solving the LMIs in the sufficient condition. The rest of this chapter is organized as follows. Section 5.2 defines the switched linear neutral system, and designs a state observer for the system. Based on the observer, an event-triggered sampling mechanism ans a sampled-data-based switching controller are set up to form a closed-loop system. In Sect. 5.3, we present stability analysis of the closed-loop system and exclude zeno behavior from theoretical analysis. A numerical example is presented in Sect. 5.4 to show the proposed result, and Sect. 5.5 concludes the chapter finally.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. Fu and T.-F. Li, Event-Triggered Control of Switched Linear Systems, Studies in Systems, Decision and Control 365, https://doi.org/10.1007/978-3-030-71604-2_5

77

78

5 Event-Triggered Control of Switched Linear Neutral Systems

5.2 Problem Statement and Preliminaries 5.2.1 Switched Linear Neutral System Consider the continuous-time switched linear neutral system ⎧ ⎪ ˙ − Cσ x(t ˙ − h(t)) = Aσ x(t) + Bσ x(t − τ (t)) + Dσ u(t), t > t0 ⎨x(t) y(t) = E σ x(t), ⎪ ⎩ xt0 = x(t0 + θ) = ϕ(θ), θ ∈ [−r, 0]

(5.1)

where x(t) ∈ Rn is the system state, u(t) ∈ Rm is the control input, y(t) ∈ R p is the measurable output, σ : [0, ∞) → M = {1, 2, . . . , m} is the switching signal that orchestrates switching between subsystems, Ai , Bi , Ci , Di and E i , i ∈ M are real matrices of appropriate dimensions, which define subsystem i, each subsystem is controllable and detectable, and the matrix Ci satisfies Ci  < 1 and Ci = 0, τ (t) and h(t) are both time-varying delays, which satisfy 0 < τ (t) ≤ τ , τ˙ (t) ≤ τˆ < 1, ˙ ≤ hˆ < 1, 0 < h(t) ≤ h, h(t)

(5.2)

where τ , τˆ , h and hˆ are constants. ϕ(θ) is a continuously differential vector initial function on [−r, 0], r = max{τ , h}, t0 is the initial time. Corresponding to switching signal σ, there exists a switching sequence {xt0 : (l0 , t0 ), (l1 , t1 ), . . . , (li , ti ), . . . |li ∈ M, ∀i ∈ N },

(5.3)

which means that the li th subsystem is active when t ∈ [ti , ti+1 ), where ti is the switching instant. Without loss of generality, we assume the state trajectory x(·) is continuous everywhere. Moreover, we use Nσ (t, s) to denote the number of discontinuities of switching signal σ on a semi-open interval (s, t], and τd and τa to denote a dwell time and an average dwell time, respectively. In this chapter, we mainly study event-triggered control of switched neutral system with the same system structure of Fig. 2.1.

5.2.2 Observer Design We first consider subsystem i and construct the observer ˙ˆ − h(t)) = Ai x(t) ˙ˆ − Ci x(t ˆ + Bi x(t ˆ − τ (t)) x(t) + Di u(t) + L i (y(t) − E i x(t)), ˆ

(5.4)

5.2 Problem Statement and Preliminaries

79

where x(t) ˆ ∈ Rn is the observer state, L i is the observer gain. Let e(t) = x(t) − x(t). ˆ From (5.1) and (5.4), a dynamic sub-error system is given by ˙ − h(t)) = (Ai − L i E i )e(t) + Bi e(t − τ (t)). e(t) ˙ − Ci e(t

(5.5)

The following definitions are given for designing an exponential observer for system (5.1). Definition 5.1 If system (5.5) is exponentially stable, then (5.4) is an exponential observer of subsystem i. Definition 5.2 Equilibrium x  = 0 of system (5.1) is said to be globally uniformly exponentially stable under σ, if solution x(t) of system (5.1) satisfies x(t) ≤ κe−λ(t−t0 ) x(t0 )r , ∀t ≥ t0 for positive constants κ and λ, where ˙ 0 + θ)} . x(t0 )r = sup {x(t0 + θ), x(t −r ≤θ≤0

Based on Definition 5.1, we propose a lemma to guarantee that (5.4) exponentially estimates the state of subsystem i. Lemma 5.3 For given positive scalars α, τ , h, τˆ < 1 and hˆ < 1, system (5.5) is exponentially stable if there exist matrices Pi > 0, Q i > 0, Ri > 0, Wi and Si of appropriate dimensions for ∀i ∈ M such that ⎡

Ωi11 ⎢ ∗ ⎢ ⎣ ∗ ∗

⎤ Ωi12 SiT Bi SiT Ci ⎥ Ωi22 SiT Bi SiT Ci ⎥ < 0, −ατ ⎦ ∗ −(1 − τˆ )e Qi 0 ˆ −αh Ri ∗ ∗ −(1 − h)e

(5.6)

where Ωi11 = αPi + Q i + SiT Ai − Wi E i + AiT Si − E iT WiT , Ωi12 = Pi − SiT + AiT Si − E iT WiT , Ωi22 = −SiT − Si + Ri , then from Definition 5.1, we know that (5.4) is an exponential observer of subsystem i and observer gain L i can be obtained by L i = Si−T Wi . Proof Choose the piecewise Lyapunov–Krasovskii functional

80

5 Event-Triggered Control of Switched Linear Neutral Systems

Vi (t) = e T (t)Pi e(t) + +

t

t t−τ (t)

e T (s)eα(s−t) Q i e(s)ds

e˙ T (s)eα(s−t) Ri e(s)ds, ˙

(5.7)

t−h(t)

where Pi > 0, Q i > 0 and Ri > 0. Taking the time derivative of Vi (t) along solutions of system (5.5), we have ˙ V˙i (t) + αVi (t) ≤e˙ T (t)Pi e(t) + e T (t)Pi e(t) + e T (t)(αPi + Q i )e(t) + e˙ T (t)Ri e(t) ˙ − (1 − τˆ )e T (t − τ (t))e−ατ Q i e(t − τ (t)) ˆ e˙ T (t − h(t))e−αh Ri e(t − (1 − h) ˙ − h(t)).

(5.8)

From (5.5), for any invertible matrix Si of appropriate dimensions, we have 

T ˙ + Ci e(t ˙ − h(t)) 2e (t)SiT + 2e˙ T (t)SiT (−e(t) +(Ai − L i E i )e(t) + Bi e(t − τ (t))) = 0.

(5.9)

Adding the left-hand side of (5.9) into the right-hand side of (5.8), we have V˙i (t) + αVi (t) ≤ ζ T (t)Ωi ζ(t),

(5.10)

where   ζ T (t) = e T (t) e˙ T (t) e T (t − τ (t)) e˙ T (t − h(t)) , ⎤ ⎡ 11 12 Ω˜ i Ω˜ i SiT Bi SiT Ci ⎥ ⎢ ∗ Ω 22 SiT Bi SiT Ci i ⎥, Ωi = ⎢ −ατ ⎦ ⎣ ∗ ∗ −(1 − τˆ )e Qi 0 −αh ˆ ∗ ∗ ∗ −(1 − h)e Ri 11 T ˜ Ωi = αPi + Q i + Si (Ai − L i E i ) + (Ai − L i E i )T Si , Ω˜ i12 = Pi − SiT + (Ai − L i E i )T Si . Let Wi = SiT L i . Then, V˙i (t) + αVi (t) < 0 follows from inequality (5.6). Integrating inequality V˙i (t) + αVi (t) < 0 from t0 to t gives Vi (t) ≤ e−α(t−t0 ) Vi (t0 ), which guarantees that system (5.5) is exponentially stable. From Definition 5.1, we know that Lemma 5.3 guarantees (5.4) is an exponential observer of subsystem i.  From (5.4) and (5.5), we form a switching observer ˙ˆ − h(t)) ˙ˆ − Cσ x(t x(t) ˆ + Bσ x(t ˆ − τ (t)) + Dσ u(t) + L σ E σ e(t) = Aσ x(t)

(5.11)

5.2 Problem Statement and Preliminaries

81

Fig. 5.1 Illustration of asynchronous switching

and an error system ˙ − h(t)) = (Aσ − L σ E σ )e(t) + Bσ e(t − τ (t)) e(t) ˙ − Cσ e(t

(5.12)

for switched neutral system (5.1).

5.2.3 Event-Triggered Sampling Mechanism Based on the observer, we develop an event-triggered sampling mechanism 2 ≥ ηξ(t)2 , e(t) ˆ

(5.13)

where e(t) ˆ = x(t) ˆ − x( ˆ tˆk ), ξ(t) = [xˆ T (t) e T (t)]T , η > 0 is a threshold, and {tˆk }∞ k=0 with tˆk < tˆk+1 denotes a sampling time sequence. The sampling mechanism receives the state from the observer and monitors triggering condition (5.13) continuously to determine whether an event is generated or not. Once condition (5.13) is satisfied, an event occurs, the sampler samples the current state information and the switching signal and transmits them to the controller. The controller updates the newest state and switching information when it receives information from sampler and holds them until the next event happens. Since the sampling time and the switching time are asynchronous, switching of the controller and the system are certainly asynchronous. A long mismatched period would led to instability on the closed-loop system, we hence introduce an assumption that ˆ tˆk ) τm < τd , where τm demotes the maximum mismatch period. With the state x( sampled at the time tˆk , the next sampling instant tˆk+1 is determined by   2 ˆ ≥ ηξ(t)2 . tˆk+1 = inf t > tˆk |e(t)

(5.14)

Let tˆ0 = t0 . Suppose that n times samplings occur on the interval [ti , ti+1 ) and tˆk+1 is the first sampling instant on this interval, see Fig. 5.1. Without loss of generality, we suppose from (5.3) that subsystem j is active on the interval [ti−1 , ti ) and subsystem i is active on the interval [ti , ti+1 ). Thus, for ∀t ∈ [ti , ti+1 ), the observer-based controller is set to

82

5 Event-Triggered Control of Switched Linear Neutral Systems

⎧ K j x( ˆ tˆk ), t ∈ [ti , tˆk+1 ) ⎪ ⎪ ⎪ ⎨ K x( i ˆ tˆk+1 ), t ∈ [tˆk+1 , tˆk+2 ) u= ⎪ · · · ⎪ ⎪ ⎩ ˆ tˆk+n ), t ∈ [tˆk+n , ti+1 ), K i x(

(5.15)

where K i and K j are the controller gains. On the consecutive sampling interval, the controller only updates the state information at sampling instants, and a ZOH is introduced to keep the control signal continuously. For ∀t ∈ {[ti , tˆk+1 ), . . . , [tˆk+n , ti+1 )}, e(t) ˆ = x(t) ˆ − x( ˆ tˆk+ j ) holds for j = 0, 1, . . . , n. Thus, when subsystem i is active on the interval [ti , ti+1 ), the closed-loop form of (5.4) can be written as ˙ˆ − h(t)) = ˙ˆ − Ci x(t x(t) ⎧ (Ai + Di K j )x(t) ˆ + Bi x(t ˆ − τ (t)) ⎪ ⎪ ⎪ ⎨ ˆ + L i E i e(t), t ∈ [ti , tˆk+1 ) −Di K j e(t) ⎪ + D K ˆ + Bi x(t ˆ − τ (t)) (A i i i ) x(t) ⎪ ⎪ ⎩ ˆ + L i E i e(t), t ∈ [tˆk+1 , ti+1 ). −Di K i e(t)

(5.16)

Recall e(t) = x(t) − x(t) ˆ and ξ(t) = [xˆ T (t) e T (t)]T , we have ˙ − C¯ i ξ(t ˙ − h(t)) = ξ(t)  A¯ i j ξ(t) + B¯ i ξ(t − τ (t)) + D¯ i j e(t), ˜ t ∈ [ti , tˆk+1 ) ¯ ¯ ¯ ˜ t ∈ [tˆk+1 , ti+1 ), Aii ξ(t) + Bi ξ(t − τ (t)) + Dii e(t),

(5.17)

where    L i Ei Ai + Di K j −Di K j 0 ¯ , Di j = , 0 Ai − L i E i 0 0     L i Ei Ai + Di K i −Di K i 0 , D¯ ii = , A¯ ii = 0 Ai − L i E i 0 0       e(t) ˆ Bi 0 Ci 0 , C¯ i = , e(t) ˜ = B¯ i = . 0 Bi 0 Ci 0 A¯ i j =



Now, our task is to analyze stability of the generalized system ˙ − h(t)) = A¯ σi σ j ξ(t) + B¯ σi ξ(t − τ (t)) + D¯ σi σ j e(t), ˙ − C¯ σi ξ(t ˜ ξ(t) which is equivalent to system (5.1) under control input (5.15), where A¯ σi σ j =



L σi E σi Aσi + Dσi K σ j 0 Aσi − L σi E σi

 ,

(5.18)

5.3 Main Results

83

      −Dσi K σ j 0 Bσi 0 Cσi 0 , B¯ σi = , C¯ σi = . D¯ σi σ j = 0 0 0 Bσi 0 Cσi

5.3 Main Results 5.3.1 Stability Analysis In this section, we mainly analyze stability of the closed-loop system (5.18). The following theorem presents a sufficient condition for exponential stability of system (5.18). Theorem 5.4 Consider system (5.18) with sampling instants determined by (5.14). For given positive scalars h, τ , hˆ < 1, τˆ < 1, μ > 1, λs , λu , η and τm , if there exist matrices Pi j > 0, Q i j > 0, Ri j > 0, Pi > 0, Q i > 0, Ri > 0, Mi , M j , L i , K i and K j of appropriate dimensions for ∀i, j ∈ M, i = j such that  Πi j =

Πi11j Πi12j ∗ Πi22j



Πi11 Πi12 Πi = ∗ Πi22

 < 0,

(5.19)

< 0,

(5.20)



Pi j ≤ μP j , Q i j ≤ μQ j , Ri j ≤ μR j , Pi ≤ μPi j , Q i ≤ μQ i j , Ri ≤ μRi j , where ⎡

Πi11j

Π¯ i11j M Tj L i E i Π¯ i13j 22 T T ⎢ ∗ ¯ Πi j Ei L i M j ⎢ ⎢ ∗ ∗ Π¯ i33j ⎢ ⎢ ∗ ∗ ∗ ⎢ =⎢ ∗ ∗ ⎢ ∗ ⎢ ⎢ ∗ ∗ ∗ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗

⎤ 0 M Tj Bi 0 M Tj Ci 0 Π¯ i24j 0 M Tj Bi 0 M Tj Ci ⎥ ⎥ T 0 M j Bi 0 M Tj Ci 0 ⎥ ⎥ Π¯ i44j 0 M Tj Bi 0 M Tj Ci ⎥ ⎥ ⎥, ∗ Π¯ i55j 0 0 0 ⎥ ⎥ ∗ ∗ Π¯ i66j 0 0 ⎥ ⎥ ∗ ∗ ∗ Π¯ i77j 0 ⎦ ∗ ∗ ∗ ∗ Π¯ i88j

(5.21)

84

5 Event-Triggered Control of Switched Linear Neutral Systems

Πi12j

Πi22j

⎡ ¯ 19 ⎤ Πi j 0 0 0 −M Tj Di K j 0 0 0 ⎢ 0 Π¯ 210 0 0 0 0 0 0⎥ ij ⎢ ⎥ T ⎢ 0 0 00 0 0 −M j Di K j 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ 0 0 0 0 0 0 0 ⎢ ⎥, =⎢ 0 00 0 0 0 0⎥ ⎢ 0 ⎥ ⎢ 0 0 00 0 0 0 0⎥ ⎢ ⎥ ⎣ 0 0 00 0 0 0 0⎦ 0 0 00 0 0 0 0 ⎡ ¯ 99 ⎤ Πi j 0 0 0 0 0 0 0 ⎢ ∗ Π¯ i1010 0 0 0 0 0 0 ⎥ j ⎢ ⎥ 1111 ⎢ ∗ ¯ ∗ Πi j 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ Π¯ i1212 0 0 0 0 ⎥ ⎥, j =⎢ ⎢ ∗ ∗ ∗ ∗ −I 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ ∗ ∗ −I 0 0 ⎥ ⎢ ⎥ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ −I 0 ⎦ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I

Π¯ i11j = M Tj Ai + M Tj Di K j + AiT M j + K Tj DiT M j + Q i j (λs + λu )e−λs h Ri j , h = M Tj Ai − M Tj L i E i + AiT M j − E iT L iT M j + Q i j

− λu Pi j + 2η I − Π¯ i22j

(λs + λu )e−λs h Ri j , h = Pi j − M Tj + AiT M j + K Tj DiT M j ,

− λu Pi j + 2η I −

Π¯ i13j Π¯ i24j = Pi j − M Tj + AiT M j − E iT L iT M j , Π¯ i33j = Π¯ i44j = Ri j − M j − M Tj , Π¯ i55j = Π¯ i66j = −(1 − τˆ )e−λs τ Q i j , ˆ −λs h Ri j , Π¯ i77j = Π¯ i88j = −(1 − h)e

(λs + λu )e−λs h Ri j , h (λs + λu )e−λs h Π¯ i99j = Π¯ i1010 Ri j , =− j h (λs + λu )e−λs τ Π¯ i1111 Qi j , = Π¯ i1212 =− j j τ ⎤ ⎡ 11 T Π¯ i Mi L i E i Π¯ i13 0 MiT Bi 0 ⎢ ∗ Π¯ i22 E iT L iT Mi Π¯ i24 0 MiT Bi ⎥ ⎥ ⎢ ⎢ ∗ ∗ Π¯ i33 0 MiT Bi 0 ⎥ 11 ⎥, ⎢ Πi = ⎢ ∗ ∗ Π¯ i44 0 MiT Bi ⎥ ⎥ ⎢ ∗ ⎣ ∗ ∗ ∗ ∗ Π¯ i55 0 ⎦ ∗ ∗ ∗ ∗ ∗ Π¯ i66

Π¯ i19j = Π¯ i210 j =

5.3 Main Results

85

⎡

Πi12

Πi22

MiT Ci 0 −MiT Di K i 0 0 ⎢ 0 M T Ci 0 0 0 i ⎢ T ⎢ M Ci 0 0 0 −MiT Di K i i =⎢ ⎢ 0 M T Ci 0 0 0 i ⎢ ⎣ 0 0 0 0 0 0 0 0 0 0 ⎡ −λs h ˆ −(1 − h)e Ri 0 0 −λs h ⎢ ˆ ∗ −(1 − h)e R 0 i ⎢ ⎢ ∗ ∗ −I ⎢ =⎢ ∗ ∗ ∗ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗

⎤ 0 0⎥ ⎥ 0⎥ ⎥, 0⎥ ⎥ 0⎦ 0 0 0 0 −I ∗ ∗

0 0 0 0 −I ∗

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥, 0 ⎥ ⎥ 0 ⎦ −I

Π¯ i11 = MiT Ai + MiT Di K i + AiT Mi + K iT DiT Mi + Q i + λs Pi + 2η I, Π¯ i22 = MiT Ai − MiT L i E i + AiT Mi − E iT L iT Mi + Q i + λs Pi + 2η I, Π¯ i13 = Pi − MiT + AiT Mi + K iT DiT Mi , Π¯ i24 = Pi − MiT + AiT Mi − E iT L iT Mi , Π¯ i33 = Π¯ i44 = Ri − Mi − MiT , Π¯ i55 = Π¯ i66 = −(1 − τˆ )e−λs τ Q i then system (5.18) is globally exponentially stable for any switching signals with average dwell time τa satisfying τa >

2 ln μ + τm (λu + λs ) . λs

(5.22)

Proof On the interval [ti , tˆk+1 ), subsystem i is active, but controller u j is still working. Choose the piecewise Lyapunov–Krasovskii functional Vi j (t) = ξ T (t) P¯i j ξ(t) + +

t



t

t−τ (t)

ξ T (s)eλs (s−t) Q¯ i j ξ(s)ds

˙ ξ˙T (s)eλs (s−t) R¯ i j ξ(s)ds,

(5.23)

t−h(t)

where P¯i j = diag{Pi j , Pi j } > 0, Q¯ i j = diag{Q i j , Q i j } > 0 and R¯ i j = diag {Ri j , Ri j } > 0. By taking the time derivative of (5.23) along solutions of system (5.18), we have ˙ + ξ T (t) Q¯ i j ξ(t) V˙i j (t) − λu Vi j (t) ≤ 2ξ T (t) P¯i j ξ(t) − (1 − τˆ )ξ T (t − τ (t))e−λs τ Q¯ i j ξ(t − τ (t)) ˙ − λu ξ T (t) P¯i j ξ(t) + ξ˙T (t) R¯ i j ξ(t)

86

5 Event-Triggered Control of Switched Linear Neutral Systems

˙ − h(t)) ˆ ξ˙T (t − h(t))e−λs h R¯ i j ξ(t − (1 − h) t ξ T (s) Q¯ i j ξ(s)ds − (λs + λu )e−λs τ − (λs + λu )e−λs h

t−τ (t) t



˙ ξ˙T (s) R¯ i j ξ(s)ds.

(5.24)

t−h(t)

From Lemma 4.2, we have − (λs + λu )e ≤−

−λs τ

(λs + λu )e τ



t

t−τ (t) −λs τ t

ξ T (s) Q¯ i j ξ(s)ds

t−τ (t)

ξ T (s)ds Q¯ i j



t

t−τ (t)

ξ(s)ds,

(5.25)

and − (λs + λu )e−λs h



t

˙ ξ˙T (s) R¯ i j ξ(s)ds

t−h(t) t

(λs + λu )e−λs h ≤− h =−

t−h(t)

ξ˙T (s)ds R¯ i j



t

˙ ξ(s)ds

t−h(t)

(λs + λu )e−λs h [ξ(t) − ξ(t − h(t))]T R¯ i j [ξ(t) − ξ(t − h(t))] . h

(5.26)

From (5.18), for any invertible matrix M¯ j = diag{M j , M j } of appropriate dimensions, we obtain the identity ˙ + C¯ i ξ(t ˙ − h(t)) (2ξ T (t) M¯ Tj + 2ξ˙T (t) M¯ Tj )(−ξ(t) ˜ = 0. + A¯ i j ξ(t) + B¯ i ξ(t − τ (t)) + D¯ i j e(t))

(5.27)

From Lemma 2.6, we have ˜ ≤ ξ T (t) M¯ Tj D¯ i j D¯ iTj M¯ j ξ(t) + e˜ T (t)e(t), ˜ 2ξ T (t) M¯ Tj D¯ i j e(t)

(5.28)

˙ + e˜ T (t)e(t). ˜ ≤ ξ˙T (t) M¯ Tj D¯ i j D¯ iTj M¯ j ξ(t) ˜ 2ξ˙T (t) M¯ Tj D¯ i j e(t)

(5.29)

and

Combining (5.25)–(5.29) together with (5.24) and taking into account the sampling condition (5.13), we have V˙i j (t) − λu Vi j (t) ≤ ς T (t)Θi j ς(t), where

(5.30)

5.3 Main Results

87

 ς T (t) = ξ T (t) ξ˙T (t) ξ T (t − τ (t)) ξ˙T (t − h(t)) ξ(t − h(t)) ⎡

Θi11j ⎢ ∗ ⎢ ⎢ ∗ Θi j = ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

Θi12j M¯ Tj B¯ i M¯ Tj C¯ i Θi15j Θi22j M¯ Tj B¯ i M¯ Tj C¯ i 0 ∗ Θi33j 0 0 ∗ ∗ Θi44j 0 ∗ ∗ ∗ Θi55j ∗ ∗ ∗ ∗

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥, 0 ⎥ ⎥ 0 ⎦ Θi66j

t

 ξ(s)ds

t−h(t)

(λs + λu )e−λs h ¯ Θi11j = M¯ Tj A¯ i j + A¯ iTj M¯ j + Q¯ i j − Ri j h + 2η I + M¯ Tj D¯ i j D¯ iTj M¯ j − λu P¯i j , Θi12j = P¯i j − M¯ Tj + A¯ iTj M¯ j , Θi22j = R¯ i j − M¯ Tj − M¯ j + M¯ Tj D¯ i j D¯ iTj M¯ j , Θi33j = −(1 − τˆ )e−λs τ Q¯ i j , ˆ −λs h R¯ i j , Θi44j = −(1 − h)e (λs + λu )e−λs h ¯ Ri j , h (λs + λu )e−λs τ ¯ =− Qi j . τ

Θi55j = −Θi15j = − Θi66j

Thus Θi j < 0 implies V˙i j (t) ≤ λu Vi j (t). Integrating this inequality from ti to t, we have Vi j (t) ≤ eλu (t−ti ) Vi j (ti ). For t ∈ [tˆk+1 , ti+1 ), since there is no switch occurring on this interval, the controller only updates the state information, subsystem i and controller u i are active on the interval [tˆk+1 , ti+1 ). Choose the piecewise Lyapunov–Krasovskii functional Vi (t) = ξ T (t) P¯i ξ(t) + +

t



t

t−τ (t)

ξ T (s)eλs (s−τ ) Q¯ i ξ(s)ds

˙ ξ˙T (s)eλs (s−h) R¯ i ξ(s)ds,

(5.31)

t−h(t)

where P¯i = diag{Pi , Pi } > 0, Q¯ i = diag{Q i , Q i } > 0 and R¯ i = diag{Ri , Ri } > 0. Hence, we have ˙ + ξ T (t) P¯i ξ(t) ˙ V˙i (t) + λs Vi (t) = ξ˙T (t) P¯i ξ(t) T T ˙ + λs ξ (t) P¯i ξ(t) + ξ (t) Q¯ i ξ(t) + ξ˙T (t) R¯ i ξ(t) − (1 − τˆ )ξ T (t − τ (t))e−λs τ Q¯ i ξ(t − τ (t)) ˙ − h(t)). ˆ ξ˙T (t − h(t))e−λs h R¯ i ξ(t − (1 − h)

(5.32)

88

5 Event-Triggered Control of Switched Linear Neutral Systems

From (5.18), for the invertible matrix M¯ i = diag{Mi , Mi } of appropriate dimensions, we have ˙ + C¯ i ξ(t ˙ − h(t)) [2ξ T (t) M¯ iT + 2ξ˙T (t) M¯ iT ][−ξ(t) ˜ = 0. + A¯ i ξ(t) + B¯ i ξ(t − τ (t)) + D¯ i e(t)]

(5.33)

From Lemma 2.6, we have ˜ ≤ ξ T (t) M¯ iT D¯ i D¯ iT M¯ i ξ(t) + e˜ T (t)e(t), ˜ 2ξ T (t) M¯ iT D¯ i e(t)

(5.34)

˙ + e˜ T (t)e(t). ˜ ≤ ξ˙T (t) M¯ iT D¯ i D¯ iT M¯ i ξ(t) ˜ 2ξ˙T (t) M¯ iT D¯ i e(t)

(5.35)

and

Adding the left-hand side of (5.33) to the right-hand side of (5.32) and considering (5.34), (5.35) and sampling condition (5.13), we have V˙i (t) + λs Vi (t) ≤ Γ T (t)Θi Γ (t),

(5.36)

where ˙ − h(t))], Γ T (t) = [ξ T (t) ξ˙T (t) ξ(t − τ (t)) ξ(t ⎤ ⎡ 11 Θi P¯i − M¯ iT + A¯ iT M¯ i M¯ iT B¯ i M¯ iT C¯ i ⎢ ∗ Θi22 M¯ iT B¯ i M¯ iT C¯ i ⎥ ⎥, Θi = ⎢ ⎣ ∗ ∗ Θi33 0 ⎦ ∗ ∗ ∗ Θi44 Θi11 = A¯ iT M¯ i + M¯ iT A¯ i + M¯ iT D¯ i D¯ iT M¯ i + Q¯ i + λs P¯i + 2η I, Θi22 = R¯ i + M¯ iT D¯ i D¯ iT M¯ i − M¯ iT − M¯ i , Θi33 = −(1 − τˆ )e−λs τ Q¯ i , ˆ −λs h R¯ i . Θi44 = −(1 − h)e Therefore, Θi < 0 implies that V˙i (t) ≤ −λs Vi (t). Integrating this inequality from tˆk+1 to t, we have Vi (t) ≤ e−λs (t−tˆk+1 ) Vi (tˆk+1 ). From (5.21), we have Vli (ti ) ≤ μVli−1 (ti− ) for ∀li , li−1 ∈ M with μ > 1. Note that 0 i = Nσ (t0 , t) ≤ N0 + t−t . Then, for t ∈ [ti , tˆk+1 ), we have τa

5.3 Main Results

89

Vσ (t) = Vli li−1 (t) ≤ eλu (t−ti ) Vli li−1 (ti ) ≤ μeλu (t−ti ) Vli−1 (ti− ) ≤ μeλu (t−ti ) e−λs (ti −tˆk+1− j ) Vli−1 (tˆk+1− j ) − ≤ μ2 eλu (t−ti ) e−λs (ti −tˆk+1− j ) Vli−1 (tˆk+1− j)

≤ μ2 eλu (t−ti +tˆk+1− j −ti−1 ) e−λs (ti −tˆk+1− j ) Vli−1 li−2 (ti−1 ) ≤ ··· ≤ μ2i−1 eλu (t−ti +(i−1)τm ) e−λs (ti −(i−1)τm ) Vl0 (t0 ) ≤ μ2i−1 eiλu τm −λs (t−iτm ) Vl0 (t0 ) 1 = ei(2 ln μ+τm (λu +λs ))−λs (t−t0 ) Vl0 (t0 ) μ   1 [2 ln μ+τm (λu +λs )]N0 2 τlna μ + τm (λτua+λs ) −λs (t−t0 ) ≤ e e Vl0 (t0 ), μ

(5.37)

where tˆk+1− j denotes the (k + 1 − j)th sampling which is also the first sampling instant on the interval [ti−1 , ti ). For t ∈ [tˆk+1 , ti+1 ), we have Vσ (t) = Vli (t) ≤ e−λs (t−tˆk+1 ) Vli (tˆk+1 ) − ) ≤ μe−λs (t−tˆk+1 ) Vli li−1 (tˆk+1

≤ μe−λs (t−tˆk+1 ) eλu (tˆk+1 −ti ) Vli li−1 (ti ) ≤ μ2 e−λs (t−tˆk+1 ) eλu (tˆk+1 −ti ) Vli−1 (ti− ) ≤ μ2 e−λs (t−tˆk+1 +ti −tˆk+1− j ) eλu (tˆk+1 −ti ) Vli−1 (tˆk+1− j ) ≤ ··· ≤ μ2i e−λs (t−(tˆk+1 −ti )−(tˆk+1− j −ti−1 )−···−(tˆk+1−m −t1 )−tˆ0 ) × eλu (tˆk+1 −ti +tˆk+1− j −ti−1 +···+tˆk+1−m −t1 ) Vl0 (t0 ) ≤ μ2i eiλu τm −λs (t−iτm −t0 ) Vl0 (t0 ) 

≤ e[2 ln μ+τm (λu +λs )]N0 e

2 ln μ τm (λu +λs ) −λs τa + τa



(t−t0 )

Vl0 (t0 ).

(5.38)

 λmin { P¯li , P¯li l j } ξ(t)2 = a ξ(t)2

(5.39)

Moreover, from (5.23) and (5.31), we have Vσ (t) ≥ min

∀li ,l j ∈M



and 

     λmax { P¯li , P¯li l j } + τ max λmax { Q¯ li , Q¯ li l j } ϕ2 ∀li ,l j ∈M ∀li ,l j ∈M   ¯ ¯ + h max λmax { Rli , Rli l j } ϕ ˙ 2

Vσ (t0 ) ≤

max

∀li ,l j ∈M

90

5 Event-Triggered Control of Switched Linear Neutral Systems

≤ b max{ϕ , ϕ} ˙ 2,

(5.40)

where 

 λmin { P¯li , P¯li l j } , ∀li ,l j ∈M   b = max λmax { P¯li , P¯li l j } + τ max {λmax { Q¯ li , Q¯ li l j }} ∀li ,l j ∈M ∀li ,l j ∈M   + h max λmax { R¯ li , R¯ li l j } .

a = min

∀li ,l j ∈M

Combining (5.37)–(5.40), we have 1 ξ(t)2 ≤ {Vli (t), Vli l j (t)} a  1 b [ln μ+τm (λu +λs )]N0 e ≤ 1, μ a 

×e

2 ln μ τm (λu +λs ) −λs τa + τa

 (t−t0 )

max{ϕ, ϕ} ˙ 2.

(5.41)

Hence, according to Definition 5.2, exponential stability of system (5.18) is guaranteed from (5.41) when τa satisfies condition (5.22). It is obvious that inequalities Θi j < 0 and Θi < 0 are both nonlinear. Applying Schur complement lemma to Θi j < 0 and Θi < 0, we have ⎡ ¯ 11 Θi j ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

Θ¯ i12j M¯ Tj B¯ i M¯ Tj C¯ i Θi15j Θ¯ i22j M¯ Tj B¯ i M¯ Tj C¯ i 0 ∗ Θi33j 0 0 ∗ ∗ Θi44j 0 ∗ ∗ ∗ Θi55j ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

⎤ 0 M¯ Tj D¯ i j 0 0 0 M¯ Tj D¯ i j ⎥ ⎥ 0 0 0 ⎥ ⎥ 0 0 0 ⎥ ⎥ < 0, 0 0 0 ⎥ ⎥ Θi66j 0 0 ⎥ ⎥ ∗ −I 0 ⎦ ∗ ∗ −I

(5.42)

and ⎡

⎤ Θ¯ i11 P¯i − M¯ iT + A¯ iT M¯ i M¯ iT B¯ i M¯ iT C¯ i M¯ iT D¯ i 0 ⎢ ∗ R¯ i − M¯ iT − M¯ i M¯ iT B¯ i M¯ iT C¯ i 0 M¯ iT D¯ i ⎥ ⎢ ⎥ ⎢ ∗ ∗ Θi33 0 0 0 ⎥ ⎢ ⎥ < 0, ⎢ ∗ 0 0 ⎥ ∗ ∗ Θi44 ⎢ ⎥ ⎣ ∗ ∗ ∗ ∗ −I 0 ⎦ ∗ ∗ ∗ ∗ ∗ −I where

(5.43)

5.3 Main Results

91

(λs + λu )e−λs h ¯ Θ¯ i11j = M¯ Tj A¯ i j + A¯ iTj M¯ j + Q¯ i j − Ri j + 2η I − λu P¯i j , h Θ¯ i12j = P¯i j − M¯ Tj + A¯ iTj M¯ j , Θ¯ i22j = R¯ i j − M¯ Tj − M¯ j , Θ¯ i11 = M¯ iT A¯ i + A¯ iT M¯ i + Q¯ i + 2η I + λs P¯i . Substitute the matrix parameters of (5.18) into (5.42) and (5.43). We thus obtain the equivalent LMIs (5.19) and (5.20) of Θi j < 0 and Θi < 0, which can be solved by Matlab LMI toolbox directly. 

5.3.2 Minimum Inter-Event Interval In this section, we prove that there always exists a lower bound on the inter-event interval to exclude zeno behavior. For system (5.1), the switching instants are denoted by t0 , t1 , t2 , . . ., and the implementation of the feedback controller is done by sampling the state at time instants tˆ0 , tˆ1 , tˆ2 , . . . . Suppose t0 = tˆ0 . Since τm < τd , there must be more than one sampling on the non-switching interval [ti , ti+1 ). Let T = tˆk+1 − tˆk denote the consecutive sampling interval. We prove from two aspects: (i) One sampling occurs on [ti , ti+1 ). We assume that tˆk is the sampling instant on [ti , ti+1 ). If tˆk = ti , then it is obvious that T ≥ τd > 0. If tˆk ∈ (ti , ti+1 ), then T > τd − τm > 0. (ii) Multiple samplings occur on [ti , ti+1 ). Suppose that tˆk and tˆk+1 are any two consecutive sampling instants on ˆ = x(t) ˆ − x( ˆ tˆk ), we thus obtain [ti , ti+1 ). For ti ≤ tˆk < t ≤ tˆk+1 < ti+1 , recall e(t) from (5.16) that ˙ˆ = x(t) ˙ˆ = Ai e(t) ˆ + Bi x(t ˆ − τ (t)) e(t) ˙ + Ci x(t ˆ tˆk ) + L i E i e(t). ˆ − h(t)) + (Ai + Di K i )x(

(5.44)

Since e( ˆ tˆk ) = 0, the response of (5.42) is e(t) ˆ =

t

tˆk

˙ˆ − h(s)) e Ai (t−s) (Bi x(s ˆ − τ (s)) + Ci x(s

+ (Ai + Di K i )x( ˆ tˆk ) + L i E i e(s))ds.

(5.45)

Therefore, we have e(t) ˆ ≤

tˆk

t

˙ˆ − h(s)) eAi (t−s) (Bi x(s ˆ − τ (s)) + Ci x(s

ˆ tˆk ) + L i E i e(s))ds. + (Ai + Di K i )x(

(5.46)

92

5 Event-Triggered Control of Switched Linear Neutral Systems

From Theorem 5.4, we know e(t) and x(t) ˆ are convergent on [tˆk , tˆk+1 ), which implies that there exist positive constants κ1 , κ2 and λ1 such that e(t) ≤ κ1 e−λ1 (t−tˆk ) e(tˆk )r . Thus e(t) ˆ ≤

t tˆk

eAi (t−s) (ϕ1 x( ˆ tˆk )r + ϕ2 x( ˆ tˆk )

+ κ1 e−λ1 (s−tˆk ) L i E i e(tˆk )r )ds t eAi (t−s) ds − (tˆk ), ≤ φ(tˆk )

(5.47)

tˆk

ˆ tˆk )r + ϕ2 x( ˆ tˆk ) + κ1 L i E i e(tˆk )r , where φ(tˆk ) = ϕ1 x( κ2 (Bi  + Ci ) and ϕ2 = Ai + Di K i . If Ai  = 0, then we have e(t) ˆ ≤

(tˆk ) > 0, ϕ1 =

 φ(tˆk )  Ai (t−tˆk ) e − 1 − (tˆk ). Ai 

(5.48)

Recall triggering condition (5.13), the next event will not be generated before √ e(t) ˆ = ηξ(t). Thus, the inter-event interval can be lower bounded by  √  ( ηξ(t) + (tˆk ))Ai  1 ln T ≥ t − tˆk = + 1 > 0. Ai  φ(tˆk )

(5.49)

If Ai  = 0, then we have T ≥ t − tˆk =

√

ηξ(t) + (tˆk ) > 0. φ(tˆk )

(5.50)

From the above analysis, it can be concluded that there exists a positive lower bound of the minimum inter-event interval, which means that zeno behavior is theoretically excluded.

5.4 Illustrative Example In this section, we illustrate the effectiveness of the proposed control strategy by a numerical example. Example 5.5 Consider the switched linear neutral system 

x(t) ˙ − Cσ x(t ˙ − h(t)) = Aσ x(t) + Bσ x(t − τ (t)) + Dσ u(t) y(t) = E σ x(t),

(5.51)

5.4 Illustrative Example

93

where σ ∈ {1, 2}, and system parameters are given by 

     −2 0 −0.1 0.1 0.1 0.1 , B1 = , C1 = , A1 = 0 −1.3 0 0.1 0 −0.1 

     −1 −3.5 0 D1 = , E 1 = 1 1 , A2 = , 1 0 −1.3  B2 =

       0.1 0 0.1 0.1 1 , C2 = , D2 = , E2 = 1 1 , 0.1 0.1 0 −0.1 −1 h(t) = 0.2sin t + 0.1, τ (t) = 0.1sin t + 0.2.

Let α = 1. By solving inequality (5.6) in Lemma 5.3, we obtain the feasible solutions 

   115.4022 7.4798 123.8488 −2.1675 P1 = , P2 = , 7.4798 115.7881 −2.1675 120.9144     43.7758 −1.8515 36.1510 0.0999 , R2 = , R1 = −1.8515 42.7324 0.0999 40.6694     34.7166 5.1290 44.5302 −0.7698 , Q2 = , Q1 = 5.1290 35.9813 −0.7698 39.8371     42.2558 −27.6963 31.9717 −19.9763 S1 = , S2 = , 19.9477 40.2358 20.1367 37.1833     6.1803 −9.1500 , W2 = . W1 = 39.1977 49.1144 Therefore, we obtain the observer gains  L1 =

   −0.2367 −0.8354 , L2 = . 0.8113 0.8720

Take η = 1, λs = 0.5, λu = 1, μ = 1.1 and τm = 1. Immediately, we obtain τa∗ = 3.1906 from (5.22) in Theorem  5.4. Choose  a fixed switching period τa = 3.5 and controller gains K 1 = 0.1 0.2 , K 2 = 0.2 0.5 . By solving the conditions in Theorem 5.4, we obtain the feasible solutions     15.5109 7.1858 15.9955 6.8658 P1 = , P2 = , 7.1858 10.4511 6.8658 10.7505     3.1080 1.8485 3.1181 1.8440 R1 = , R2 = , 1.8485 2.2763 1.8440 2.2261     5.8974 1.8913 5.8822 1.9055 , Q2 = , Q1 = 1.8913 1.5501 1.9055 1.4864

94

5 Event-Triggered Control of Switched Linear Neutral Systems

State responses of the observer

3

2

1

0

-1

-2 0

2

4

6

8

10

12

14

16

18

20

Time (sec) Fig. 5.2 State responses of the observer



   3.0754 1.8444 3.1162 1.8437 , R21 = , 1.8444 2.2558 1.8437 2.2669     15.1524 7.3514 15.7995 7.0505 P12 = , P21 = . 7.3514 10.3764 7.0505 10.6165     5.9080 1.9063 5.9445 1.9104 , Q 21 = . Q 12 = 1.9063 1.5296 1.9104 1.5551     6.3657 1.9649 4.5359 1.3755 , M2 = . M1 = 5.8187 6.8594 5.5282 5.1711 R12 =

Let initial state be xˆ0 = e0 = [−2 3]T . We obtain the simulation results in Fig. 5.2 system state responses and sampled-state responses in Fig. 5.2. The triggering instants are presented in Fig. 5.3. Switching signals of the controlled system and controller are illustrated in Fig. 5.4. From the simulation results, we can see that system (5.51) is stabilized under control input (5.15) determined by the sampled state triggered by condition (5.13). 

5.4 Illustrative Example

95

State responses of the error system

3

2

1

0

-1

-2 0

2

4

6

8

10

12

14

16

18

20

12

14

16

18

20

Time (sec) Fig. 5.3 State responses of the error system 6

State responses of the system

5 4 3 2 1 0 -1 -2 -3 -4 0

2

4

6

8

10

Time (sec) Fig. 5.4 State responses of the closed-loop system (5.51)

96

5 Event-Triggered Control of Switched Linear Neutral Systems 3

Sampling states

2

1

0

-1

-2 0

2

4

6

8

10

12

14

16

18

20

Time (sec) Fig. 5.5 Sampling states under the ZOH

Switching signals

Switching signal of the system Switching signal of the controller

2

1

0

2

4

6

8

10

12

Time (sec) Fig. 5.6 Switching signals of the system and the controller

14

16

18

20

5.4 Illustrative Example

97

30

Triggering condition

25

20

10 -5

5 4

15

3 2

10 1 0

5

5

6

7

8

9

10

Time (sec)

0 0

2

4

6

8

10

12

14

16

18

20

Time (sec) Fig. 5.7 Triggering condition

Triggering instant

1

0

2

4

6

8

10

Time (sec) Fig. 5.8 Triggering instants

12

14

16

18

20

98

5 Event-Triggered Control of Switched Linear Neutral Systems 1.2

1

Control input

0.8

0.6

0.4

0.2

0

-0.2 0

2

4

6

8

10

12

14

16

18

20

Time (sec) Fig. 5.9 Control input

5.5 Conclusion We have presented a result on event-triggered control of switched linear neutral systems. The event is triggered only when the defined error exceeds a given dynamic threshold condition. The global exponential stability of the closed-loop system under the event-triggered control is guaranteed subject to an average dwell time switching strategy (Figs. 5.5, 5.6, 5.7, 5.8 and 5.9).

Chapter 6

Periodic Sampled-Data Control for Switched Linear Neutral Systems

6.1 Introduction In this chapter, we focus on studying the periodic sampled-data control for a class of switched linear neutral systems. By analyzing the relationship between the sampling period and the dwell time of switched system, a bond between sampling period and average dwell time is revealed to form a switching condition. Subject to this switching condition and certain control gains related constraints, exponential stability of the closed-loop switched neutral system can be guaranteed. The main feature of this chapter lies in four aspects: (1) A sample-data-based stabilization method is proposed for the switched linear neutral system under asynchronous switching situation. (2) Delays in the switched neutral system, including state delay and neutral delay, are all time-varying and with certain constraint conditions. The result of this work is less conservative than the one for switched linear neutral system with constant delays. (3) A bond between the sampling period and the average dwell time is revealed to form a switching condition which is a key point to guarantee system stability. A sufficient condition is obtained for stability of the closed-loop switched neutral systems by using multiple Lyapunov–Krosovskii functional method and the average dwell time technique. (4) Some free weighting matrices are introduced to obtain the decoupled constraints, and therefore technically reduces the computational complexity. The rest of this chapter is organized as follows. Section 6.2 describes the switched linear neutral system and the periodic sampling mechanism. Section 6.3 analyzes stability of the closed-loop switched neutral system. A numerical example is presented in Sect. 6.4 to show the developed result, and Sect. 6.5 concludes the chapter.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. Fu and T.-F. Li, Event-Triggered Control of Switched Linear Systems, Studies in Systems, Decision and Control 365, https://doi.org/10.1007/978-3-030-71604-2_6

99

100

6 Periodic Sampled-Data Control for Switched Linear Neutral Systems

6.2 Problem Statement and Preliminaries Consider the continuous-time switched linear neutral system 

x(t) ˙ − Cσ x(t ˙ − h(t)) = Aσ x(t) + Bσ x(t − τ (t)) + Dσ u(t) x(t0 + θ) = ϕ(θ), θ ∈ [−r, 0],

(6.1)

where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the control vector, σ : [0, ∞) → M = {1, 2, . . . , m} is the switching signal, M is a finite index set, {(Ai , Bi , Ci , Di ) : i ∈ M} is a collection of matrix pairs defining the individual subsystem of system (6.1), and all the eigenvalues of matrix Ci are inside the unit circle, τ (t) and h(t) denote the discrete time-varying delay and the neutral time-varying delay, respectively, which satisfy 0 < τ (t) ≤ τ , τ˙ (t) ≤ τˆ < 1, ˙ ≤ hˆ < 1, 0 < h(t) ≤ h, h(t)

(6.2)

where τ , τˆ , h and hˆ are constants. ϕ(θ) is a continuously differential vector initial function on [−r, 0], r = max{τ , h}. We denote the number of discontinuities of switching signal σ on the interval (s, t] by Nσ (t, s). For all i ∈ M, we assume that subsystem i is stabilizable and there is no jump at all switching instants. Definition of the average dwell time is presented in Definition 2.2, and exponential stability of switched linear neutral system (6.1) is given below. Definition 6.1 Equilibrium x  = 0 of system (6.1) is said to be globally uniformly exponentially stable under σ, if solution x(t) of system (6.1) satisfies x(t) ≤ κe−λ(t−t0 ) x(t0 )r , ∀t ≥ t0 for positive constants κ and λ, where ˙ 0 + θ)} . x(t0 )r = sup {x(t0 + θ), x(t −r ≤θ≤0

Suppose that state measurements are taken at times tk := kτs , k ∈ N , where τs is a fixed sampling period. When the sampler gets the state information, it also transmits the switching signal to the controller. However, the sampling time and the switching time do not happen synchronously. The delayed time will last to the next sampling time. During this period, the active subsystem would be unstable. Thus the control objective is to stabilize system (6.1) by designing a sampled-data controller meanwhile taking into account asynchronous switching. In order to simplify the analysis process, we assume the sampling period τs is no larger than the dwell time τd .

6.2 Problem Statement and Preliminaries

101

Without loss of generality, we assume that subsystem i is active on the interval [tk , tk+1 ). The control law produced by the sampler with a ZOH on [tk , tk+1 ) can be represented as u(t) = u(tk ) = u(t − (t − tk )) = u(t − η(t)) = K i x(t − η(t)),

(6.3)

˙ = 1 for t = tk . Moreover, where η(t) = t − tk is piecewise with derivative η(t) η(t) < tk+1 − tk = τs .

6.3 Main Results We aim to stabilize system (6.1) under assumption τs ≤ τd . This assumption guarantees that at most one switch occurs within each sampling interval. If no switch occurs within a sampling interval, then the active subsystem is synchronous with its matched controller. If one switch happens within a sampling interval, then the active subsystem is synchronous with its corresponding controller before the switch, but becomes asynchronous after the switch until the next sampling time begins. Thus, we analyse stability of system (6.1) in two cases: sampling interval with no switch and sampling interval with one switch.

6.3.1 Sampling Interval with No Switch Consider the case when σ(tk ) = σ(tk+1 ) = i ∈ M. Subsystem i is active on the whole interval [tk , tk+1 ). The closed-loop system is formed by ˙ − h(t)) = Ai x(t) + Bi x(t − τ (t)) + Di K i x(t − η(t)). x(t) ˙ − Ci x(t Construct the piecewise Lyapunov–Krasovskii functional  Vi (t) = x T (t)Pi x(t) +  + + +

x T (s)Q i eλs (s−t) x(s)ds

t−τs t

t−τ (t)  t



t

x T (s)Ri eλs (s−t) x(s)ds x˙ T (s)Mi eλs (s−t) x(s)ds ˙

t−h(t) t t−αη(t)

x T (s)Ni eλs (s−t) x(s)ds

(6.4)

102

6 Periodic Sampled-Data Control for Switched Linear Neutral Systems

 +

0



−τs

t

x˙ T (s)Si eλs (s−t) x(s)dsdθ, ˙

(6.5)

t+θ

where Pi > 0, Q i > 0, Ri > 0, Mi > 0, Ni > 0, Si > 0, and α is a constant satisfying 0 < α < 1. Remark 6.2 From the construction of (6.5), we expect to obtain a sufficient stability condition for system (6.1) depending on the bounds of the discrete time-varying delay, the neutral time-varying delay, the derivative of the discrete time-varying delay, the derivative of the neutral time-varying delay and the sampling period.  When subsystem i is active, for ∀λs > 0, taking the time derivative of (6.5) along solutions of system (6.4), we have ˙ V˙i (t) + λs Vi (t) ≤ x˙ T (t)Pi x(t) + x T (t)Pi x(t) + λs x T (t)Pi x(t) + x T (t)Q i x(t) − e−λs τs x T (t − τs )Q i x(t − τs ) + x T (t)Ri x(t) − (1 − τˆ )e−λs τ x T (t − τ (t))Ri x(t − τ (t)) + x˙ T (t)Mi x(t) ˙ + x T (t)Ni x(t) + τs x˙ T (t)Si x(t) ˙ ˆ −λs h x˙ T (t − h(t))Mi x(t − (1 − h)e ˙ − h(t)) − (1 − α)e−αλs τs x T (t − αη(t))Ni x(t − αη(t))  t −λs τs x˙ T (s)Si x(s)ds ˙ −e − e−λs τs − e−λs τs

t−αη(t) t−αη(t)



t−η(t) t−η(t)

x˙ T (s)Si x(s)ds ˙



x˙ T (s)Si x(s)ds. ˙

(6.6)

t−τs

From Lemma 4.7, we have −e−λs τs



t

x˙ T (s)Si x(s)ds ˙

t−αη(t)

−e−λs τs ≤ ατs





t

x˙ (s)ds Si

t

x(s)ds ˙

T

t−αη(t)

t−αη(t)

 −e−λs τs  T x (t) − x T (t − αη(t)) Si [x(t) − x(t − αη(t))] , ατs  t−αη(t) −e−λs τs x˙ T (s)Si x(s)ds ˙ =

t−η(t)

−e−λs τs ≤ (1 − α)τs



t−αη(t)

 x˙ (s)ds Si

t−αη(t)

T

t−η(t)

t−η(t)

x(s)ds ˙

(6.7)

6.3 Main Results

103

 −e−λs τs  T x (t − αη(t)) − x T (t − η(t)) (1 − α)τs × Si [x(t − αη(t)) − x(t − η(t))] ,  t−η(t) −λs τs −e x˙ T (s)Si x(s)ds ˙ < 0. =

(6.8) (6.9)

t−τs

According to (6.4), for any invertible matrix Wi of appropriate dimensions, we have   ˙ − Ai x(t) − Bi x(t − τ (t)] −2 x˙ T (t)WiT + x T (t)WiT [x(t) −Ci x(t ˙ − h(t)) − Di K i x(t − η(t))) = 0.

(6.10)

Adding the left-hand side of (6.10) into (6.6), and using (6.7)–(6.9), we have V˙i (t) + λs Vi (t) < ξ T (t)Ωi ξ(t),

(6.11)

where  ξ T (t) = x T (t) x T (t − τ (t)) x˙ T (t − h(t)) x T (t − η(t))  x T (t − τs ) x T (t − αη(t)) x˙ T (t) , ⎡ −λs τs Ω 11 WiT Bi WiT Ci WiT Di K i 0 e ατs Si Pi + AiT Wi − WiT ⎢ i 0 0 0 0 BiT Wi ⎢ ∗ Ωi22 ⎢ 33 T ∗ ∗ Ω 0 0 0 C ⎢ i i Wi ⎢ −λs τs e 44 T Ωi = ⎢ ∗ ∗ ∗ Ωi 0 (1−α)τs Si Di K iT Wi ⎢ 55 ⎢ ∗ ∗ ∗ ∗ Ωi 0 0 ⎢ ⎣ ∗ 0 ∗ ∗ ∗ ∗ Ωi66 ∗ ∗ ∗ ∗ ∗ ∗ Ωi77 Ωi11 = Ri + Ni + λs Pi + Q i −

e−λs τs Si + WiT Ai + AiT Wi , ατs

Ωi22 = −(1 − τˆ )e−λs τ Ri , ˆ −λs h Mi , Ωi33 = −(1 − h)e e−λs τs Si , (1 − α)τs = −e−λs τs Q i ,

Ωi44 = − Ωi55

e−λs τs e−λs τs Si − Si − (1 − α)e−αλs τs Ni , (1 − α)τs ατs = Mi + τs Si − Wi − WiT .

Ωi66 = − Ωi77

From (6.11), we know that Ωi < 0 implies

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦

104

6 Periodic Sampled-Data Control for Switched Linear Neutral Systems

V˙i (t) + λs Vi (t) < 0.

(6.12)

Integrating inequality (6.12), we have Vi (t) < e−λs (t−tk ) Vi (tk ) ≤ νVi (tk ),

(6.13)

where ν := max e−λs (t−tk ) , 0 ≤ t − tk < τs .

6.3.2 Sampling Interval with One Switch Suppose that one switch has occurred on the interval [tk , tk+1 ), that is σ(tk ) = i and σ(tk+1 ) = j = i. The controller knows that one switch occurs from subsystem i to subsystem j on the interval [tk , tk+1 ) but does not know when it happens. Assume that the time of the switch from i to j be tk + t¯, where t¯ ∈ (0, τs ]. When t ∈ [tk , tk + t¯), subsystem i is active. We derive as above and obtain that Vi (t) < e−λs (t−tk ) Vi (tk ).

(6.14)

When t ∈ [tk + t¯, tk+1 ), the resulting closed-loop system becomes ˙ − h(t)) = A j x(t) + B j x(t − τ (t)) + D j K i x(t − η(t)). x(t) ˙ − C j x(t

(6.15)

Construct the piecewise Lyapunov–Krasovskii functional  V j (t) =x (t)P j x(t) +

t

T

 + + + +

t

t−τ (t)  t

x T (s)R j eλs (s−t) x(s)ds x˙ T (s)M j eλs (s−t) x(s)ds ˙

t−h(t)  t



x T (s)Q j eλs (s−t) x(s)ds

t−τs

x T (s)N j eλs (s−t) x(s)ds

t−αη(t) 0  t −τs

x˙ T (s)S j eλs (s−t) x(s)dsdθ, ˙

(6.16)

t+θ

where P j > 0, Q j > 0, R j > 0, M j > 0, N j > 0 and S j > 0. When subsystem j is active, for ∀λu > 0, taking the time derivative of (6.16) along solutions of system (6.15), we have

6.3 Main Results

105

˙ V˙ j (t) − λu V j (t) ≤ x˙ T (t)P j x(t) + x T (t)P j x(t) − λu x T (t)P j x(t) + x T (t)Q j x(t) − e−λs τs x T (t − τs )Q j x(t − τs ) + x T (t)R j x(t) − (1 − τˆ )e−λs τ x T (t − τ (t))R j x(t − τ (t)) + x˙ T (t)M j x(t) ˙ + x T (t)N j x(t) + τs x˙ T (t)S j x(t) ˙ ˆ −λs h x˙ T (t − h(t))M j x(t − (1 − h)e ˙ − h(t)) − (1 − α)e−αλs τs x T (t − αη(t))N j x(t − αη(t))  t −λs τs x˙ T (s)S j x(s)ds ˙ −e − e−λs τs − e−λs τs

t−αη(t) t−αη(t)



t−η(t)  t−η(t)

x˙ T (s)S j x(t)ds ˙

x˙ T (s)S j x(t)ds ˙

t−τs

− (λs + λu )e

−λs τs



− (λs + λu )e−λs h

t−τ (t)  t

− (λs + λu )e−αλs τs − (λs + λu )

0

x T (s)Q j x(s)ds

t−τs  t

− (λs + λu )e−λs τ



t



−τs

x T (s)R j x(s)ds x˙ T (s)M j x(s)ds ˙

t−h(t)  t

x T (s)N j x(s)ds

t−αη(t) t

x˙ T (s)S j eλs (s−t) x(s)dsdθ. ˙

(6.17)

t+θ

From Lemma 4.7, we have −e

−λs τs



t

x˙ T (s)S j x(s)ds ˙

t−αη(t)  t

−e−λs τs ≤ ατs

 x˙ T (s)ds S j

t−αη(t)

t

x(s)ds ˙

t−αη(t)

−e−λs τs T (x (t) − x T (t − αη(t)))S j (x(t) − x(t − αη(t))), ατs  t−αη(t) − e−λs τs x˙ T (s)S j x(s)ds ˙ =

t−η(t)

−e−λs τs ≤ (1 − α)τs



t−αη(t)

t−η(t)

 x˙ T (s)ds S j

t−αη(t)

t−η(t)

x(s)ds ˙

(6.18)

106

6 Periodic Sampled-Data Control for Switched Linear Neutral Systems

−e−λs τs (x T (t − αη(t)) − x T (t − η(t)))S j (x(t − αη(t)) − x(t − η(t))), (1 − α)τs (6.19)  t−η(t) − e−λs τs x˙ T (s)S j x(t)ds ˙ < 0, (6.20) =

t−τs

− (λs + λu )e−λs τs ≤

−(λs + λu )e τs

− (λs + λu )e ≤

−λs τ

−(λs + λu )e τ

− (λs + λu )e−λs h



t

x T (s)Q j x(s)ds

t−τs −λs τs  t



t−τs



t



t−τ (t) t

x T (s)ds R j



t

x˙ (s)ds M j T

t−h(t)

(6.22)

x˙ T (s)ds,

(6.23)

t−h(t)

x T (s)N j x(s)ds  x T (s)ds N j

t

x(s)ds,

t−αη(t) t

x(s)ds, t−τ (t)



t

t−αη(t) −αλs τs  t

−τs

t

x˙ T (s)M j x(s)ds ˙

t−h(t)

−(λs + λu )e τs  0  − (λs + λu )

(6.21)

x T (s)R j x(s)ds

−(λs + λu )e−λs h ≤ h  t −αλs τs − (λs + λu )e ≤

x(s)ds, t−τs

t−τ (t) −λs τ  t



t

x T (s)ds Q j

(6.24)

t−αη(t)

x˙ T (s)S j eλs (s−t) x(s)dsdθ ˙ < 0.

(6.25)

t+θ

From (6.15), for any invertible matrices Wi of appropriate dimensions, we have   ˙ − A j x(t) − B j x(t − τ (t)) −2 x˙ T (t)WiT + x T (t)WiT x(t)

 −C j x(t ˙ − h(t)) − D j K i x(t − η(t)) = 0.

(6.26)

Utilizing (6.18)–(6.26), we have V˙ j (t) − λu V j (t) < ζ T (t)Ψ j ζ(t),

(6.27)

where ζ T (t) =   T ξ (t)

t

t−τ (t)

 x (s)ds

t

T

 x (s)ds

t

T

t−αη(t)

 x (s)ds

t

T

t−h(t)

 x (s)ds , T

t−τs

6.3 Main Results

107

⎡

⎤ Ψ j11 0 0 0 0 ⎢ ∗ Ψ j22 0 0 0 ⎥ ⎢ ⎥ 33 ⎥ Ψj = ⎢ ⎢ ∗ ∗ Ψj 0 0 ⎥ , ⎣ ∗ ∗ ∗ Ψ 44 0 ⎦ j ∗ ∗ ∗ ∗ Ψ j55 ⎡ 11 T Π j Wi B j WiT C j WiT D j K i ⎢ ∗ Π 22 0 0 ⎢ j ⎢ ∗ 33 ∗ Π 0 ⎢ j ⎢ 11 Ψj = ⎢ ∗ ∗ ∗ Π 44 j ⎢ ⎢ ∗ ∗ ∗ ∗ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Π 11 j = R j + N j − λu P j + Q j −

0 0 0 0 Π 55 j ∗ ∗

e−λs τs ατs

S j P j + A Tj Wi − WiT 0 B Tj Wi 0 C Tj Wi −λs τs e Sj K iT D Tj Wi ατs 0 0 Π 66 0 j ∗ Π 77 j

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦

e−λs τs S j + WiT A j + A Tj Wi , ατs

−λs τ Π 22 Rj, j = −(1 − τˆ )e

ˆ −λs h M j , Π 33 j = −(1 − h)e e−λs τs Sj, (1 − α)τs = −e−λs τs Q j ,

Π 44 j =− Π 55 j

e−λs τs e−λs τs Sj − S j − (1 − α)e−αλs τs N j , (1 − α)τs ατs = M j + τs S j − Wi − WiT ,

Π 66 j =− Π 77 j

(λs + λu )e−λs τ Rj, τ (λs + λu )e−αλs τs =− Nj, τs (λs + λu )e−λs h Mj, =− h (λs + λu )e−λs τs =− Q j. τs

Ψ j22 = − Ψ j33 Ψ j44 Ψ j55

From (6.17), we know that Ψ j < 0 implies V˙ j (t) − λu V j (t) < 0.

(6.28)

Integrating inequality (6.28), we have V j (t) < eλu (t−tk −t¯) V j (tk + t¯).

(6.29)

108

6 Periodic Sampled-Data Control for Switched Linear Neutral Systems

From (6.14) and (6.29), for t ∈ [tk , tk+1 ), we have V j (t) < eλu (t−tk −t¯) V j (tk + t¯) ≤ [λmax (P j ) + τ λmax (R j ) + hλmax (M j ) + ατs λmax (N j ) + τs λmax (Q j ) + τs2 λmax (S j )]eλu (t−tk −t¯) Vi (tk + t¯)− /λmin (Pi ) ≤ [λmax (P j ) + τ λmax (R j ) + hλmax (M j ) + ατs λmax (N j ) + τs λmax (Q j ) + τs2 λmax (S j )]e−λs (tk +t¯−tk ) eλu (t−tk −t¯) Vi (tk )/λmin (Pi ) ≤ μi j Vi (tk ) ≤ μVi (tk ),

(6.30)

where μi j := max{((λmax (P j ) + τ λmax (R j ) + hλmax (M j ) + ατs λmax (N j ) + τs λmax (Q j ) + τs2 λmax (S j ))e−(λs +λu )t¯+λu (t−tk ) )/λmin (Pi )}, and μ := max μi j . i, j∈M

6.3.3 Stability Analysis 0 Definition 2.2 implies that Nσ (tk0 , tk ) ≤ N0 + k−k for every p such that τa ≥ pτs . p We now derive a lower bound on p to guarantee convergence. From Definition 2.1, we know that Nσ (tk0 , tk ) equals the number k − k0 of intervals of the form (tl , tl+1 ), k0 ≤ l ≤ k − 1, which contain a switch. Combining the conclusions of Sects. 6.3.1 and 6.3.2, we have the following bound for all k ≥ k0

V (tk ) = Vσ (tk ) k−k0

k−k0

≤ μ N0 + p ν k−k0 −N0 − p Vσ (t0 )  μ  N0  1 p−1 k−k0 μpν p Vσ (t0 ) = ν    μ  N0 (k−k0 ) ln μ 1p ν p−1 p e Vσ (t0 ). = ν 1

We want to ensure that μ p ν

p−1 p

(6.31)

< 1, which is equivalent to p >1+

log μ . log ν1

(6.32)

6.3 Main Results

109

Thus if 

 log μ τa > 1 + τs , log ν1

(6.33)

then V (tk ) ≤

 μ  N0 ν

e

 1 (k−k0 ) ln μ p ν

p−1 p



Vσ (t0 ).

(6.34)

From the defined piecewise Lyapunov–Krasovskii functional, we have V (tk ) ≥ λmin (Pi )x(t)2 ≥ min {λmin (Pi )} x(t)2 = ax(t)2 , ∀i∈M

(6.35)

and  Vσ (t0 ) ≤ max {λmax (Pi )} + τ max {λmax (Ri )} ∀i∈M

∀i∈M

+ h max {λmax (Mi )} + ατs max {λmax (Ni )} ∀i∈M ∀i∈M  2 +τs max {λmax (Q i )} + τs max {λmax (Si )} xt0 2c ∀i∈M

=

∀i∈M

bxt0 2c ,

(6.36)

where a = min {λmin (Pi )} , ∀i∈M

b = max {λmax (Pi )} + τ max {λmax (Ri )} + h max {λmax (Mi )} ∀i∈M

∀i∈M

∀i∈M

+ ατs max {λmax (Ni )} + τs max {λmax (Q i )} + τs2 max {λmax (Si )} . ∀i∈M

∀i∈M

∀i∈M

From (6.34), (6.35) and (6.36), we have b  μ  N0 (k−k0 ) ln x(t)2 ≤ e a ν

 1 μpν

p−1 p



xt0 2c ,

(6.37)

which means that system (6.1) is exponentially stable. The following theorem concludes the main result. Theorem 6.3 Consider system (6.1). For given positive constants α < 1, λs , λu , τs , h, τ , τˆ < 1 and hˆ < 1, if there exist matrices P¯i > 0, R¯ i > 0, M¯ i > 0, N¯ i > 0, S¯i > 0, Q¯ i > 0, W¯ i > 0, P¯ j > 0, R¯ j > 0, M¯ j > 0, N¯ j > 0, S¯ j > 0, Q¯ j > 0 and Ji , J j of appropriate dimensions such that

110

6 Periodic Sampled-Data Control for Switched Linear Neutral Systems

⎡

Ω¯ i11 Bi W¯ i Ci W¯ i Di Ji ⎢ ∗ Ω¯ 22 0 0 ⎢ i ⎢ ∗ Ω¯ i33 0 ⎢ ∗ ⎢ ⎢ ∗ ∗ ∗ Ω¯ i44 ⎢ ⎢ ∗ ∗ ∗ ∗ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 0 Ω¯ i55 ∗ ∗

⎤ S¯i P¯i + W¯ iT AiT − W¯ i , ⎥ 0 W¯ iT BiT ⎥ ⎥ 0 W¯ iT CiT ⎥ ⎥ −λs e τs ¯ T T ⎥ < 0, J D S i i i (1−α)τs ⎥ ⎥ 0 0 ⎥ 66 ⎦ ¯ 0 Ωi 77 ¯ ∗ Ωi e−λs τs ατs

(6.38)

and ⎡ ¯ 11 Ψj ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

0 Ψ¯ j22 ∗ ∗ ∗

0 0 Ψ¯ j33 ∗ ∗

0 0 0 Ψ¯ j44 ∗

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ < 0, ∀i, j ∈ M, i = j 0 ⎦ Ψ¯ j55

where e−λs τs ¯ Ω¯ i11 = R¯ i + N¯ i + λs P¯i + Q¯ i − Si + Ai W¯ i + W¯ iT AiT , ατs Ω¯ i22 = −(1 − τˆ )e−λs τ R¯ i , ˆ −λs h M¯ i , Ω¯ i33 = −(1 − h)e e−λs τs ¯ Si , (1 − α)τs = −e−λs τs Q¯ i ,

Ω¯ i44 = − Ω¯ i55

e−λs τs ¯ e−λs τs ¯ Si − Si − (1 − α)e−αλs τs N¯ i , (1 − α)τs ατs Ω¯ i77 = M¯ i + τs S¯i − W¯ iT − W¯ i , ⎤ ⎡ 11 −λs τs Π¯ j B j W¯ i C j W¯ i D j Ji 0 e ατs S¯ j P¯ j + W¯ iT A Tj − W¯ i ⎥ ⎢ ∗ Π¯ 22 0 0 0 0 W¯ iT B Tj ⎥ ⎢ j ⎥ ⎢ ∗ 33 T T ¯ ¯ ∗ Πj 0 0 0 Wi C j ⎥ ⎢ ⎥ ⎢ −λs τs 11 e 44 T T Ψ¯ j = ⎢ ∗ ⎥, ∗ ∗ Π¯ j 0 (1−α)τs S¯ j Ji D j ⎥ ⎢ 55 ⎥ ⎢ ∗ ¯ ∗ ∗ ∗ Πj 0 0 ⎥ ⎢ 66 ⎦ ⎣ ∗ ¯ ∗ ∗ ∗ ∗ Πj 0 77 ∗ ∗ ∗ ∗ ∗ ∗ Π¯ j

Ω¯ i66 = −

e−λs τs ¯ ¯ ¯ ¯ ¯ Π¯ 11 = R + N − λ + Q − P S j + A j W¯ i + W¯ iT A Tj , j j u j j j ατs −λs τ ¯ Rj, Π¯ 22 j = −(1 − τˆ )e ˆ −λs h M¯ j , Π¯ 33 j = −(1 − h)e

(6.39)

6.3 Main Results

111

e−λs τs ¯ Sj, (1 − α)τs = −e−λs τs Q¯ j ,

Π¯ 44 j =− Π¯ 55 j

e−λs τs ¯ e−λs τs ¯ Sj − S j − (1 − α)e−αλs τs N¯ j , (1 − α)τs ατs = M¯ j + τs S¯ j − W¯ iT − W¯ i ,

Π¯ 66 j =− Π¯ 77 j

(λs + λu )e−λs τ ¯ Rj, τ (λs + λu )e−αλs τs ¯ =− Nj, τs (λs + λu )e−λs h ¯ =− Mj, h (λs + λu )e−λs τs ¯ =− Q j, τs

Ψ¯ j22 = − Ψ¯ j33 Ψ¯ j44 Ψ¯ j55

then system (6.1) is exponentially stable for any switching signals with average dwell time τa satisfying 

 log μ τs , τa > 1 + log ν1

(6.40)

where μ and ν are constants, with control gain K i = Ji W¯ i−1 . Proof It is omitted.



Remark 6.4 Note that controller gain K i cannot be obtained by solving Ωi < 0 and Ψ j < 0 directly. If, however, we let W¯ i = Wi−1 , multiply diag{W¯ iT , . . . , W¯ iT } and diag{W¯ i , . . . , W¯ i } on the left-hand side and on the right-hand side of the inequalities Ωi < 0 and Ψ j < 0, respectively, and also let R¯ i = W¯ iT Ri W¯ i , N¯ i = W¯ iT Ni W¯ i , P¯i = W¯ iT Pi W¯ i , Q¯ i = W¯ iT Q i W¯ i , S¯i = W¯ iT Si W¯ i , M¯ i = W¯ iT Mi W¯ i , Ji = K i W¯ i , R¯ j = W¯ iT R j W¯ i , N¯ j = W¯ iT N j W¯ i , P¯ j = W¯ iT P j W¯ i , Q¯ j = W¯ iT Q j W¯ i , S¯ j = W¯ iT S j W¯ i , M¯ j = W¯ iT M j W¯ i , then we can obtain inequalities (6.39) and (6.40), which are equivalent to Ωi < 0 and Ψ j < 0. As a result, easily solving (6.39) and (6.40)  directly gives K i . Remark 6.5 In (6.40), τs can be chosen in advance. ν and μ are computed according to (6.13) and (6.30).  Remark 6.6 In order to obtain control gain K i , we introduce free weighting matrix Wi when the ith controller is active, (see Eqs. (6.13) and (6.29)), so that control gain K i can be obtained by solving conditions (6.39) and (6.40) simultaneously as described in Remark 6.4, which overcomes the technical challenge of solving the coupled variables of matrix inequalities in (6.30). 

112

6 Periodic Sampled-Data Control for Switched Linear Neutral Systems

Remark 6.7 In this chapter, our work is based on the assumption τs ≤ τd which simplifies the analysis process into two cases, that is at most one switch may occur on arbitrary a sampling interval. In fact, the method can be extended to deal with the case of multiple switches occur on a sampling interval. We just need analyze the monotony property of Lyapunov–Krasovskii functional on each individual nonswitching interval during the sampling interval, and then analyze the overall descend property of Lyapunov–Krasovskii functional to guarantee stability. 

6.4 Illustrative Example A numerical example is given to illustrate the effectiveness of the proposed method in this section. Example 6.8 Consider the switched linear neutral system ˙ − h(t)) = Aσ x(t) + Bσ x(t − τ (t)) + Dσ u(t), x(t) ˙ − Cσ x(t

(6.41)

where σ ∈ {1, 2}, and system parameters are given by 

       −2.5 0.3 −0.2 0.2 −0.1 0.1 1 , B1 = , C1 = , D1 = , 0 −1.5 0.1 −0.1 0 0.2 1         −1.2 0 0.3 0 0.1 0 1 A2 = , B2 = , C2 = , D2 = , 0.1 −1.6 0.1 0.1 0 −0.1 1 A1 =

τ (t) = 0.1sin t + 0.3, h(t) = 0.1sin t + 0.1. Let parameters be λs = 1, λu = 1, α = 0.4 and τs = 0.5. Solving conditions in Theorem 6.3, we obtain the feasible solution     0.3756 0.0049 0.2054 0.0043 ¯ ¯ P1 = , P2 = , 0.0049 0.2920 0.0043 0.3087     0.0903 −0.0022 0.0509 0.0052 ¯ ¯ , R2 = , R1 = −0.0022 0.0628 0.0052 0.0620     0.0946 0.0066 0.0331 0.0024 , Q¯ 2 = , Q¯ 1 = 0.0066 0.0587 0.0024 0.0649     0.0627 0.0019 0.0227 0.0003 , N¯ 2 = , N¯ 1 = 0.0019 0.0390 0.0003 0.0445     0.0385 0.0045 0.0461 0.0050 , S¯2 = , S¯1 = 0.0045 0.0442 0.0050 0.0394     0.1458 0.0106 0.1606 0.0082 W¯ 1 = , W¯ 2 = , 0.0106 0.1663 0.0082 0.1744

6.4 Illustrative Example

113

3

State response

2

1

0

-1

-2

0

10

20

30

40

50

60

Time (sec) Fig. 6.1 State response of the closed-loop system (6.41)

   0.0561 −0.0001 0.0464 −0.0004 , M¯ 2 = , −0.0001 0.0617 −0.0004 0.0567     J1 = −0.0191 −0.0406 , J2 = −0.0325 −0.0173 ,

M¯ 1 =



and thus we have   K 1 = J1 W¯ 1−1 = −0.1136 −0.2367 ,   K 2 = J2 W¯ 2−1 = −0.1979 −0.0902 . From (6.39), we obtain the average dwell time τa∗ = 1.9860. Construct a certain switching sequence satisfying τa = 2.2 > τa∗ . Let the initial state be x0 = [−3 2]T . The state response of the closed-loop system is shown in Fig. 6.1. Switching signals of the system and the controller are shown in Fig. 6.2, and the control input and sampled state are shown in Figs. 6.3 and 6.4, respectively. From the figures, we can see that the closed-loop switched system is stabilized under the sampled-data control input.

114

6 Periodic Sampled-Data Control for Switched Linear Neutral Systems 5 Switching signal of the system Switching signal of the controller

Switching signal

Switching signal

4

3

2

1 0

1

2

3

4

5

6

Time (sec)

2

1

0

10

20

30

40

50

60

40

50

60

Time (sec) Fig. 6.2 Switching signals

Control input

0.1

0

-0.1

-0.2

0

10

20

30

Time (sec)

Fig. 6.3 Control input

6.5 Conclusion

115

Fig. 6.4 Sampled state

6.5 Conclusion We have presented a new result on sampled-data-based state feedback stabilization of a class of switched linear neutral systems under asynchronous switching. A relationship between the average dwell time and the sampling period has been revealed to form a switching condition to guarantee exponential stability. Furthermore, by introducing free weighting matrices, we have obtained controller gains by solving the resulting LMIs.

Chapter 7

Hysteresis Switching Control for Switched Linear Neutral Systems

7.1 Introduction In this chapter, we study the stabilization problem for a class of switched linear neutral systems based on a state-dependent switching strategy. A state-dependent switching strategy, which is also called hysteresis-based switching, is proposed by generalizing the min-switching strategy. A multiple generalized Lyapunov–Krasovskii functional method is proposed which allows some increase of connecting adjacent Lyapunov– Krasovskii functionals at switching points. A delay-dependent sufficient condition is obtained for asymptotic stability of the closed-loop switched linear neutral system. Some free weighting matrices are introduced to reduce the conservative of the stability condition. The rest of chapter is organized as follows. Section 7.2 describes the switched linear neutral system. Section 7.3 analyzes stability of the switched linear neutral system under a state-dependent switching law. A numerical example is presented in Sect. 7.4 to show the developed result, and Sect. 7.5 concludes the chapter.

7.2 System Description and Preliminaries Consider the continuous-time switched linear neutral system 

x(t) ˙ − Cσ x(t ˙ − h) = Aσ x(t) + Bσ x(t − τ ) + Dσ u(t), t > t0 xt0 = ϕ(θ), θ ∈ [− max{h, τ }, 0],

(7.1)

where x(t) ∈ Rn is the system state, u(t) ∈ Rm is the control input, h and τ are both constants which denote neutral delay and discrete delay, respectively. ϕ(θ) is a continuously differentiable initial function on [− max{h, τ }, 0], σ : [0, ∞) → M = {1, 2, . . . , m} is the switching signal. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. Fu and T.-F. Li, Event-Triggered Control of Switched Linear Systems, Studies in Systems, Decision and Control 365, https://doi.org/10.1007/978-3-030-71604-2_7

117

118

7 Hysteresis Switching Control for Switched Linear Neutral Systems

Corresponding to switching signal σ, there always exists a switching sequence {xt0 : (l0 , t0 ), . . . , (li , ti ), . . . |li ∈ M, i ∈ N},

(7.2)

∞ with ti < ti+1 which means that the li th subsystem is active on [ti , ti+1 ), where {ti }i=0 denotes a switching time sequence. Ai , Bi , Ci and Di , i ∈ M are known real matrices which define subsystem i and Ci satisfies Ci  < 1. Without loss of generality, we assume that t0 = 0 and the state trajectory x(·) of system (7.1) is continuous everywhere. The controller is set as

u(t) = u σ (t) = K 1σ x(t) + K 2σ x(t − τ ),

(7.3)

where K 1i and K 2i are controller gains of subsystem i. From (7.3), the closed-loop system (7.1) can be written as ˙ − h) = (Aσ + Dσ K 1σ )x(t) + (Bσ + Dσ K 2σ )x(t − τ ). x(t) ˙ − Cσ x(t

(7.4)

The objective of this chapter is to design a state-dependent switching rule and a switching controller in the form of (7.3) to stabilize system (7.1). A useful lemma is introduced for the main result. Lemma 7.1 ([58]) For any symmetric matrices T0 and T1 with x T T1 x ≥ 0 subject to x = 0, if there exist a scalar β ≥ 0, such that T0 − βT1 > 0, then x T T0 x > 0.

7.3 Main Result It is well known that the min-switching strategy is a classical switching strategy in the switched system field. Reference [54] proposes the min-switching strategy for switched linear systems by partitioning the state space Rn into m regions:   Ωi = x ∈ Rn |x T (Pi − P j )x ≤ 0, ∀i, j ∈ M ,

(7.5)

  Ωi j = x ∈ Rn |x T (Pi − P j )x = 0, ∀i, j ∈ M ,

(7.6)

and

where Pi > 0, P j > 0. Switches only occur on switching surfaces x T Pi x = x T P j x according to the switching law σ := argmin {Vi (x) : ∀i ∈ M} ,

(7.7)

7.3 Main Result

119

where Vi (x) = x T Pi x. From (7.5) and (7.6), we know that Lyapunov function Vσ (x) is continuous at all switching points. Furthermore, although the stability property can be retained by adopting switching strategy (7.7), sliding modes and chattering behavior cannot be avoided. In order to relax the continuous constraint on Vσ (x) and avoid sliding modes, we redefine m regions: Ωi = {x ∈ Rn |x T (Pi − P j + Ni j )x ≤ 0, ∀i, j ∈ M, i = j},

(7.8)

Ωi j = {x ∈ Rn |x T (Pi − P j + Ni j )x = 0, ∀i, j ∈ M, i = j},

(7.9)

and

where Pi > 0, P j > 0, and Ni j are symmetric matrices satisfying Ni j + N jk ≤ Nik and Nii = 0 for ∀i, j, k ∈ M. It is obvious that Ωi j is the boundary of Ωi and m 

Ωi = Rn .

(7.10)

i=1

In fact, if (7.10) does not hold, there exists x ∈ Rn satisfying x ∈ / Ωi , then we have an integer q and a sequence i 1 , i 2 , . . . , i q , i k = i k+1 , k = 1, . . . , q, i q+1 = i 1 such that x T (Pik − Pik+1 + Nik ik+1 )x > 0, ∀i k ∈ M.

(7.11)

Taking the sum over k and noticing Ni j + N jk ≤ Nik , and Nii = 0, we have q  k=1

x (Pik − Pik+1 + Nik ik+1 )x = T

q 

x T Nik ik+1 x ≤ 0,

(7.12)

k=1

which contradicts (7.11). Hence, (7.10) holds. We design a hysteresis-based switching strategy based on (7.8) and (8.9) ⎧   ⎪ i, if x(0) ∈ Ωi /Ωi (Ω j | j ∈ M, j = i) ⎪ ⎪ ⎪ ⎨t = 0, σ(0) = minarg{Ω , Ω }, if x(0) ∈ Ω (Ω | j ∈ M, j = i) i j i j  − ⎪ i, if x(t) ∈ Ωi and σ(t ) = i ⎪ ⎪ ⎪ ⎩t > 0, σ(t) = minarg{Ω | j ∈ M}, if x(t) ∈ Ω and σ(t − ) = i. j ij (7.13) Switching law (7.13) can be described as follows. Let σ(0) = i if x(0) ∈ Ωi and σ(0) = minarg{Ωi , Ω j } if x(0) belongs to the overlapping area of some partition regions. For t > 0, if σ(t − ) = i and x(t) ∈ Ωi , the state trajectory remains in Ωi until it hits the boundary Ωi j . This means that switching only takes place on the boundary Ωi j . If σ(t − ) = i and x(t) ∈ Ωi j , then x(t) enters Ω jmin , jmin = minarg{Ω j }. In fact,

120

7 Hysteresis Switching Control for Switched Linear Neutral Systems

Fig. 7.1 Multiple Lyapunov functions x T Ni j x < 0

x(t) ∈ Ωi j means that x T (Pi − P jmin + Ni jmin )x = 0 and x T (Pi − P j + Ni j )x = 0 for j = jmin . Thus, we have x T P jmin x = x T (Pi + Ni jmin )x. This in turn gives x T (P jmin − P j + N jmin j )x = x T (Pi + Ni jmin − P j + N jmin j )x ≤ x T (Pi − P j + Ni j )x = 0.

(7.14)

Remark 7.2 In order to design a hysteresis-based switching strategy avoiding sliding modes and chattering behavior, we introduce x T Ni j x when partitioning the state space Rn and thus obtain m overlapping regions. However, the condition x T (Ni j + N jk )x ≤ x T Nik x is necessary to guarantee that the partition regions cover  the entire state space Rn . Remark 7.3 From (7.8) and (7.9), x T Ni j x determines the monotony of Vσ at switching points. Especially for x T Ni j x = 0, x = 0, the connecting adjacent Lyapunov functions are continuous at switching points, under which switching strategy (7.13) degenerates into the min-switching strategy (7.7) exactly. The influencing behaviour is illustrated in Figs. 7.1, 7.2 and 7.3.  We now analyze stability of the closed-loop system (7.4). The following theorem gives the main result. Theorem 7.4 Consider system (7.4). If there exist scalars αi j < 0, τ > 0 and h > 0, matrices P¯i > 0, P¯ j > 0, ∀ j ∈ M, j = i, R¯ 1 > 0, R¯ 2 > 0, Q¯ 1 > 0, Q¯ 2 > 0, Z 1i , Z 2i , M¯ > 0 and N¯ i j of appropriate dimensions such that

7.3 Main Result

121

Fig. 7.2 Multiple Lyapunov functions x T Ni j x = 0

Fig. 7.3 Multiple Lyapunov functions x T Ni j x > 0

⎡

⎤ Ψi11 Ψi12 Bi M¯ + Di Z 2i h1 R¯ 2 Ci M¯ 0 j j ⎢ ∗ R¯ 1 + h R¯ 2 − 2 M¯ Bi M¯ + Di Z 2i 0 Ci M¯ 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ − Q¯ 1 0 0 0 ⎥ ⎢ ⎥ < 0, ⎢ ∗ ∗ ∗ − h1 R¯ 2 0 0 ⎥ ⎢ ⎥ ⎣ ∗ ∗ ∗ ∗ − R¯ 1 0 ⎦ ∗ ∗ ∗ ∗ ∗ − τ1 Q¯ 2

(7.15)

122

⎡

Πi11j ⎢ ∗ ⎢ ⎣ ∗ ∗

7 Hysteresis Switching Control for Switched Linear Neutral Systems

Πi12j M¯ AiT + Z 1iT DiT + Ci M¯ M¯ AiT + Z 1iT DiT − M¯ + N¯ iTj Πi22j M¯ BiT + Z 2iT DiT + Ci M¯ M¯ BiT + Z 2iT DiT − M¯ T ¯ iT ¯ i − M¯ + MC ∗ Ci M¯ + MC ∗ ∗ − M¯ − M¯ T

⎤ ⎥ ⎥ < 0, (7.16) ⎦

N¯ i j + N¯ jk ≤ N¯ ik , ∀i, j, k ∈ M,

(7.17)

where 1 ¯ ¯ ¯ R2 + Ai M¯ + Di Z 1i + M¯ AiT + Z 1iT DiT Ψi11 j = Q1 + τ Q2 − h m  + αi j ( P¯i − P¯ j + N¯ i j ), j=1

Ψi12 j Πi11j Πi12j Πi22j

¯ = M¯ AiT + Z 1iT DiT + P¯i − M, T = Ai M¯ + Di Z 1i + M¯ Ai + Z 1iT DiT , = M¯ AiT + Z 1iT DiT + Bi M¯ + Di Z 2i , = Bi M¯ + Di Z 2i + M¯ BiT + Z 2iT DiT ,

then system (7.4) is asymptotically stable under switching law (7.13) where Pi = M¯ −1 P¯i M¯ −1 , P j = M¯ −1 P¯ j M¯ −1 and Ni j = M¯ −1 N¯ i j M¯ −1 , and the controller gains can be obtained by K 1i = Z 1i M¯ −1 and K 2i = Z 2i M¯ −1 . Proof Construct the piecewise Lyapunov–Krasovskii functional  t Vi (t) =x T (t)Pi x(t) + x T (s)Q 1 x(s)ds t−τ  t x˙ T (s)R1 x(s)ds ˙ +  +  +

t−h 0  t −τ 0 −h

t+θ t

x T (s)Q 2 x(s)dsdθ



x˙ T (s)R2 x(s)dsdθ, ˙

(7.18)

t+θ

where Pi > 0, Q 1 > 0, Q 2 > 0, R1 > 0 and R2 > 0. When subsystem i is active on the interval [tk , tk+1 ), differentiating (7.18) along solutions of system (7.4), we have ˙ V˙i (t) =x˙ T (t)Pi x(t) + x T (t)Pi x(t) + x T (t)Q 1 x(t) − x T (t − τ )Q 1 x(t − τ ) + x˙ T (t)R1 x(t) ˙ − x˙ T (t − h)R1 x(t ˙ − h)  t x T (s)Q 2 x(s)ds + τ x T (t)Q 2 x(t) − t−τ

7.3 Main Result

123



t

+ h x˙ T (t)R2 x(t) ˙ −

x˙ T (s)R2 x(s)ds. ˙

(7.19)

t−h

From Lemma 4.2, we have  −

t

x T (s)Q 2 x(s)ds ≤ −

t−τ

1 τ





t

x T (s)ds Q 2

t−τ

t

x(s)ds,

(7.20)

t−τ

and  −

t

x˙ T (s)R2 x(s)ds ˙ ≤−

t−h

=−

1 h



t

 x˙ T (s)ds R2

t−h

t

x(s)ds ˙

t−h

 1 T x (t) − x T (t − h) R2 [x(t) − x(t − h)] . h

(7.21)

Moreover, from system (7.4), for an arbitrary matrix M of appropriate dimensions, we have   2 x T (t)M + x˙ T (t)M [(Ai + Di K 1i )x(t) +(Bi + Di K 2i )x(t − τ ) + Ci x(t ˙ − h) − x(t)] ˙ = 0.

(7.22)

Combining (7.20) and (7.21) and adding the left-hand side of (7.22) into (7.19), we have V˙i (t) ≤ ξ T (t)Φi ξ(t),

(7.23)

where    t x T (s)ds , ξ T (t) = x T (t) x˙ T (t) x T (t − τ ) x T (t − h) x˙ T (t − h) t−τ ⎤ ⎡ 11 1 12 Φi Φi M(Bi + Di K 2i ) h R2 MCi 0 ⎢ ∗ R1 + h R2 − 2M M(Bi + Di K 2i ) 0 MCi 0 ⎥ ⎥ ⎢ ⎢ ∗ 0 0 0 ⎥ ∗ −Q 1 ⎥, Φi = ⎢ ⎢ ∗ ∗ ∗ − h1 R2 0 0 ⎥ ⎥ ⎢ ⎣ ∗ ∗ ∗ ∗ −R1 0 ⎦ ∗ ∗ ∗ ∗ ∗ − τ1 Q 2 1 Φi11 = Q 1 + τ Q 2 − R2 + M(Ai + Di K 1i ) + (Ai + Di K 1i )T M, h Φi12 = (Ai + Di K 1i )T M + Pi − M.  When subsystem i is active on the interval [tk , tk+1 ), mj=1 αi j x T (t)(Pi − P j + Ni j )x(t) ≥ 0 follows from switching law (7.13) and condition αi j < 0. Then, Lemma 7.1 allows us to replace inequality Φi < 0 by

124

7 Hysteresis Switching Control for Switched Linear Neutral Systems

⎡

⎤ Φ¯ i11 Φ¯ i12 M(Bi + Di K 2i ) h1 R2 MCi 0 ⎢ ∗ R1 + h R2 − 2M M(Bi + Di K 2i ) 0 MCi 0 ⎥ ⎢ ⎥ ⎢ ∗ 0 0 0 ⎥ ∗ −Q 1 ⎢ ⎥ < 0, ⎢ ∗ 0 ⎥ ∗ ∗ − h1 R2 0 ⎢ ⎥ ⎣ ∗ ∗ ∗ ∗ −R1 0 ⎦ ∗ ∗ ∗ ∗ ∗ − τ1 Q 2

(7.24)

where Φ¯ i11j = Q 1 + τ Q 2 − +

m 

1 R2 + M(Ai + Di K 1i ) + (Ai + Di K 1i )T M h

αi j (Pi − P j + Ni j ),

j=1

Φ¯ i12 = (Ai + Di K 1i )T M + Pi − M. Hence, inequality (7.24) implies V˙i (t) < 0, which means Vi (t) is strictly decreasing on the interval [tk , tk+1 ). Obviously, condition (7.24) is a nonlinear matrix inequality. Pre- and post-multiplying both sides of inequality (7.24) by diag{M −1 , M −1 , ¯ Q¯ 2 = M¯ Q 2 M, ¯ M −1 , M −1 , M −1 , M −1 }, and letting M¯ = M −1 , Q¯ 1 = M¯ Q 1 M, ¯ P¯ j = M¯ P j M, ¯ R¯ 1 = M¯ R1 M, ¯ R¯ 2 = M¯ R2 M, ¯ Z 1i = K 1i M, ¯ Z 2i = P¯i = M¯ Pi M, ¯ and N¯ i j = M¯ Ni j M, ¯ we obtain the equivalent LMI (7.15) which can be solved K 2i M, by using LMI Toolbox directly. Moreover, from system (7.4), we have   2 x T (t)M + x T (t − τ )M + x˙ T (t − h)M + x˙ T (t)M [(Ai + Di K 1i )x(t) +(Bi + Di K 2i )x(t − τ ) + Ci x(t ˙ − h) − x(t)] ˙ = 0. (7.25) Let Vi j (t) = x T (t)Ni j x(t). When subsystem i is active on the interval [tk , tk+1 ), differentiating Vi j (t) along solutions of system (7.4) and adding the left-hand side of (7.25) into V˙i j (t), we have ¯ V˙i j (t) = ς¯T (t)Γ¯i j ς(t),

(7.26)

where   ς¯T (t) = x T (t) x T (t − τ ) x˙ T (t − h) x˙ T (t) , ⎡ 11 12 Γ¯i j Γ¯i j (Ai + Di K 1i )T M + MCi (Ai + Di K 1i )T M − M + NiTj ⎢ ∗ Γ¯ 22 (Bi + Di K 2i )T M + MCi (Bi + Di K 2i )T M − M ij Γ¯i j = ⎢ ⎣ ∗ ∗ MCi + CiT M −M + CiT M ∗ ∗ ∗ −M − M T T Γ¯i11 j = M(Ai + Di K 1i ) + (Ai + Di K 1i ) M, T Γ¯i12 j = M(Bi + Di K 2i ) + (Ai + Di K 1i ) M,

⎤ ⎥ ⎥, ⎦

7.3 Main Result

125

T Γ¯i22 j = M(Bi + Di K 2i ) + (Bi + Di K 2i ) M.

¯ M, ¯ M, ¯ M}. ¯ We know Pre- and post-multiplying both sides of Γ¯i j < 0 by diag{ M, ˙ ¯ that (7.16) is equivalent to Γi j < 0, which implies Vi j (t) < 0. According to switching law (7.13), at each switching instant, we have Vik+1 (tk+1 ) − Vik (tk+1 ) = x T (tk+1 )(Pik+1 − Pik )x(tk+1 ) = x T (tk+1 )Nik+1 ik x(tk+1 ).

(7.27)

From (7.14) and (7.27), we have Vik+1 (tk+1 ) − Vik (tk+1 ) + Vik+2 (tk+2 ) − Vik+1 (tk+2 ) = x T (tk+1 )(Pik+1 − Pik )x(tk+1 ) + x T (tk+2 )(Pik+2 − Pik+1 )x(tk+2 ) = x T (tk+1 )Nik ik+1 x(tk+1 ) + x T (tk+2 )Nik+1 ik+2 x(tk+2 ) ≤ x T (tk+1 )Nik ik+1 x(tk+1 ) + x T (tk+1 )Nik+1 ik+2 x(tk+1 ) ≤ x T (tk+1 )Nik ik+2 x(tk+1 ). Thus, k  (Vi p+1 (x(t p+1 )) − Vi p (x(t p+1 ))) p=0

≤ x T (t1 )Ni0 ik+1 x(t1 ) ≤ x0T Ni0 ik+1 x0 .

(7.28)

Choose a class GK function β(x0 ) = maxx≤x0  {|x T Ni j x|, 1 ≤ i, j ≤ m}. Then x0T Ni0 ik+1 x0 ≤ β(x0 ) follows. Condition (7.28) guarantees that the overall increment of Lyapunov–Krasovskii functional Vσ (t) at switching points is decreasing. With condition (7.13), asymptotic stability of system (7.4) follows.  Remark 7.5 In order to simplify the tedious computation and highlight the design of switching law (7.13), we only consider constant delays in this chapter. In fact, our proposed method can be extended to deal with the case of the considered systems with time-varying delays by choosing different Lyapunov–Krasovskii functional.  Remark 7.6 If x T (tk+1 )Nik+1 ik x(tk+1 ) ≤ 0, then we know from (7.28) that Vik+1 (tk+1 ) ≤ Vik (tk+1 ) holds naturally at switching instant tk+1 . Thus conditions (7.16) and (7.17) hold naturally and condition (7.15) guarantees asymptotic stability of system (7.4). 

126

7 Hysteresis Switching Control for Switched Linear Neutral Systems

7.4 Illustrative Example A numerical example is given to illustrate the effectiveness of the proposed method in this section. Example 7.7 Consider the switched linear neutral system ˙ − h) = Aσ x(t) + Bσ x(t − τ ) + Dσ u(t), x(t) ˙ − Cσ x(t

(7.29)

where σ ∈ {1, 2}, and coefficient matrices are given by 

       −5 −0.5 −0.4 0 −0.1 0.5 1 , B1 = , C1 = , D1 = , 0 1 0 −0.6 0.1 0 0         1 0 −0.4 0 −0.1 0.5 0 , B2 = , C2 = , D2 = . A2 = −1 −5 −1 −0.3 0.1 0 1 A1 =

Suppose τ = h = 0.1. Choose parameters α12 = α21 = −10. By solving conditions in Theorem 7.4, we have     3.1026 −0.2902 4.1236 −0.2940 P1 = , P2 = , −0.2902 7.1135 −0.2940 5.6144     −0.3919 0.0741 −0.4436 0.0752 , N21 = , N12 = 0.0741 −1.0959 0.0752 −1.1319 and u 1 (t) = [4.4139 0.6726]x(t) + [0.4001 − 0.0365]x(t − 0.1), u 2 (t) = [1.1032 4.5679]x(t) + [0.9853 0.3001]x(t − 0.1).

(7.30) (7.31)

According to (7.13), the switching law is given by ⎧ 1, if x(0) ∈ Ω1 or (x(t) ∈ Ω1 and σ(t − ) = 1) ⎪ ⎪ ⎪ ⎨ or (x(t) ∈ Ω21 and σ(t − ) = 2) σ(t) = ⎪ 2, if x(0) ∈ Ω2 or (x(t) ∈ Ω2 and σ(t − ) = 2) ⎪ ⎪ ⎩ or (x(t) ∈ Ω12 and σ(t − ) = 1)

(7.32)

7.4 Illustrative Example

127

4 x1 x2

3

System states

2

1

0

−1

−2

0

10

20

30

40

50 Time (sec)

60

70

80

90

100

Fig. 7.4 State trajectories of system (7.29) under switching law (7.32)

where      T −1.4130 0.0778  Ω1 = x(t) ∈ R x (t) x(t) ≤ 0 , 0.0778 0.4032       0.5775 0.0714 2 T x(t) ≤ 0 , Ω2 = x(t) ∈ R x (t) 0.0714 −2.6310 

2

Ω12 and Ω21 are the boundaries of Ω1 and Ω2 , respectively. State trajectories of the closed-loop system with x0 = [4 − 2] are depicted in Fig. 7.4. From Fig. 7.4, we can see that the state is convergent. The control input and the switching signal are shown in Figs. 7.5 and 7.6, respectively.

128

7 Hysteresis Switching Control for Switched Linear Neutral Systems 20

15

Control input

10

5

0

−5

−10

0

10

20

30

40

50 Time (sec)

60

70

80

90

100

70

80

90

100

Fig. 7.5 Control input under switching law (7.32)

Switching signal

2

1

0

10

20

30

Fig. 7.6 Switching signal (7.32)

40

50 Time (sec)

60

7.5 Conclusion

129

7.5 Conclusion In this chapter, we have presented a stabilization result for a class of continuous-time switched linear neutral systems by designing a hysteresis-based switching strategy which improves the min-switching strategy. A switching controller is designed for stabilization and a delay-dependent stability criterion has been obtained by using multiple generalized Lyapunov–Krasovskii functional method.

Chapter 8

Reliable Control for a Class of Switched Nonlinear Systems

8.1 Introduction In this chapter, we study the problem of robust fault-tolerant control of a class of switched cascade nonlinear systems with structural uncertainties existing in both system matrices and input matrices, and proposes a fault-tolerant control method for this class of switched systems by using average dwell-time techniques. The main features and contributions of this chapter are highlighted as follows: (1) The proposed control design works on both the switched nonlinear systems with actuator faults and its nominal systems (i.e., without actuator faults) without necessarily changing any structures and/or parameters of the proposed controllers. Therefore, it is classified as a reliable control, or passive fault-tolerant control strategy [128]. (2) The proposed method unlike [57, 93, 100], in a unified way for easy and practical applications, treats all actuators without necessarily classifying all actuators into faulty actuators and robust ones. (3) The proposed method is independent of arbitrary switching signals provided that the average switching time is greater than some constant time. The layout of this chapter is as follows. Section 8.2 presents the problem formulation and preliminaries. The details about designing the controllers of the nonlinear switched systems and its stability analysis are presented in Sect. 8.3. A numerical example is given in Sect. 8.4. Section 8.5 concludes this chapter.

8.2 Problem Statement and Preliminaries Consider a class of uncertain switched nonlinear systems described by 

z˙ = gi (z, x) x˙ = (Ai + ΔAi )x + (Bi + ΔBi )u i ,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. Fu and T.-F. Li, Event-Triggered Control of Switched Linear Systems, Studies in Systems, Decision and Control 365, https://doi.org/10.1007/978-3-030-71604-2_8

(8.1)

131

132

8 Reliable Control for a Class of Switched Nonlinear Systems

where x ∈ Rr and z ∈ Rn−r are the system states, u i ∈ Rqi is the control input, i(t) : [0, ∞) → M = {1, 2, . . . , m} is the switching signal, gi (z, x) is a known nonlinear function, Ai and Bi are known constant matrices, and ΔAi and ΔBi are matrix functions representing structural uncertainties. Some assumptions are introduced here for system (8.1). Assumption 8.1 (Ai , Bi ) is controllable and all the states are available for feedback. Assumption 8.2 ΔAi and ΔBi are the structure uncertainties with bounded norms, i.e., ΔAi  ≤ δi , ΔBi  ≤ θi ,

(8.2)

where δi and θi are constants. Assumption 8.3 gi (z, x) satisfies global Lipschitz condition, i.e., there exists constant L i > 0 such that gi (z, x1 ) − gi (z, x2 ) ≤ L i x1 − x2 , ∀z, x1 , x2 .

(8.3)

Assumption 8.4 There exist a smooth positive-definite function W (z) with W (0) = 0, positive constants k1 , k2 , βi and γ for i ∈ M such that dW (z) gi (z, 0) ≤ −βi z2 , dz    dW (z)     dz  ≤ γz,

(8.4)

k1 z2 ≤ W (z) ≤ k2 z2 . Then, we design state feedback controllers as follows: u i = K i x,

(8.5)

where K i ∈ Rq1 ×r are constant matrices. Given whether a fault occurs on each actuator or not, a matrix L is is introduced to represent fault situation of the actuators of the ith subsystem as follows: L is = diag(l1i , l2i , . . . , lqi ),

(8.6)

where if l ij = 1 actuator j, j ∈ {1, 2, . . . , q}, is normal and if l ij = 0 actuator j is faulty and L is = 0. Therefore, the closed-loop switched nonlinear systems involving uncertain structures and actuator faults are given as follows:

8.2 Problem Statement and Preliminaries



133

z˙ = gi (z, x)   x˙ = (Ai + ΔAi ) + (Bi + ΔBi )L is K i x.

(8.7)

The control objective is to design feedback gain matrices K i , i ∈ M, such that switched nonlinear system (8.7) are globally exponentially stable for all uncertain matrices ΔAi , ΔBi and the actuator faults.

8.3 Control Design This section will present the main results on the robust fault-tolerant control of nonlinear switched cascaded system (8.1) and stability analysis of the closed-loop system (8.7). Before presenting the main theorem, we need the following lemma. Lemma 8.1 For ∀x, y ∈ Rr and constant ε > 0 and symmetric positive matrix Π , the following inequalities hold: x T y + yT x ≤

xT Πx xT Πx yT y + εy T Π −1 y ≤ +ε . ε ε λmin (Π )

(8.8)

Theorem 8.2 Suppose that Assumptions 8.1–8.4 are satisfied. If there exist constants ε > 0 and μ ≥ 1, and symmetric positive matrices Hi , Ui and Q¯ i such that ε δ 2 Ir + λ0 Pi − Q¯ i = AiT Pi + Pi Ai + λmin (Hi ) i    1 1 ε 1 2 Hi + εθi Ir + Bi Ui + Iq + Iqi BiT Pi , + Pi ε ε λmin (Ui ) i ε Pi ≤ μP j , P j ≤ μPi ,

∀i, j ∈ M

(8.9)

(8.10)

have positive definite solution Pi . Then (i) If average dwell time τa ≥ τa∗ =

ln μ , λ ∈ [0, λ0 ), λ

(8.11)

the closed-loop system (8.7) is globally exponentially stable under arbitrary switching rules with controller gain K i = −BiT P, i.e., u i = K i x = −BiT P x are faulttolerant feedback controllers which stabilize switched system (8.1) globally and exponentially. (ii) State estimation of system (8.7)

134

8 Reliable Control for a Class of Switched Nonlinear Systems

x(t) ¯ ≤

b2 −(λ0 −λ)t e 2 x(0), ¯ b1

(8.12)

where x¯ = (x z)T , b1 = min[k1 , min {λmin (Pi )}], ∀i∈M

b2 = min[k2 , min {λmax (Pi )}], K i = −BiT Pi . ∀i∈M

Proof Consider the piecewise Lyapunov function V (x, z) = Vi = x T Pi x + W (z),

(8.13)

where Pi > 0, and W (z) is a smooth positive definite function with W (0) = 0. Along the trajectory of system (8.7), the time derivation of V (x, z) is V˙ = x T (AiT Pi + Pi Ai + ΔAiT Pi + Pi ΔAi − 2Pi Bi L is BiT Pi dW (z) gi (z, x). − Pi Bi L is ΔBiT Pi − Pi ΔBi L is BiT Pi )x + dz

(8.14)

According to Lemma 8.1, for the constant ε > 0 and symmetric positive definite matrices Hi and Ui , also Assumption 8.2, we have  x T ΔAiT Pi + Pi ΔAi x   1 ≤ x T εΔAiT Hi−1 ΔAi + Pi Hi Pi x ε   ε 1 ΔAiT ΔAi + Pi Hi Pi x ≤ xT λmin (Hi ) ε   ε 1 δ 2 Ir + Pi Hi Pi x, ≤ xT λmin (Hi ) ε    x T −2Pi Bi L is BiT Pi x = x T 2Pi Bi (−L is )BiT Pi x  1 ≤ x T εPi Bi (−L is )Ui−1 (−L is )BiT Pi + Pi Bi Ui BiT Pi x ε  ε 1 Pi Bi (−L is )(−L is )BiT Pi + Pi Bi Ui BiT Pi x ≤ xT λmin (Ui ) ε  i i 1 T ε(−L s )(−L s ) T T Pi Bi Bi Pi + Pi Bi Ui Bi Pi x, ≤x λmin (Ui ) ε and

(8.15)

(8.16)

8.3 Control Design

135

 x T −Pi B L is ΔBiT Pi − Pi ΔBi L is BiT Pi x   = x T Pi Bi (−L is )ΔBiT Pi + Pi ΔBi (−L is )BiT Pi x  1 T T i i T ≤ x εPi ΔBi ΔBi Pi + Pi Bi (−L s )(−L s )Bi Pi x ε  1 T 2 i i T ≤ x εθi Pi Pi + (−L s )(−L s )Pi Bi Bi Pi x ε   1 T 2 T = x εθi Pi Pi + Pi Bi Bi Pi x. ε

(8.17)

  Note that the last steps of (8.16) and (8.17) follow because of (−L is )(−L is ) = 1. Using (8.13)–(8.17), we have

ε AiT Pi + Pi Ai + δ 2 Ir λmin (Hi ) i     1 1 ε 2 BiT x Hi + εθi L r + Bi Ui + +Pi ε ε λmin (Ui )Iqi dW (z) + gi (z, x), dz

V˙i ≤ x T

(8.18)

and thus

ε δ 2 Ir + λ0 Pi V˙i + λ0 Vi ≤ x T AiT Pi + Pi Ai + λmin (Hi ) i     1 1 ε 1 x Hi + εθi2 L r + Bi Ui + +Pi + Iqi BiT ε ε λmin (Ui )Iqi ε dW (z) dW (z) + gi (z, 0) + [gi (z, x) − gi (z, 0)] + λ0 W (z) dz dz dW (z) dW (z) = −μx T Q¯ i x + gi (z, 0) + [gi (z, x) − gi (z, 0)] + λ0 W (z). dz dz (8.19) According to the global Lipschitz condition in Assumption 8.3, we have V˙i + λ0 Vi ≤ −λmin ( Q¯ i ) x2 − βi z2 + γ L i zx + λ0 k2 z2  2 γ Li ¯ = −λmin ( Q i ) x − z 2λmin ( Q¯ i )  γ 2 L i2 z2 . − βi − λ0 k2 − (8.20) 4λmin ( Q¯ i ) Choosing βi − λ0 k2 −

γ 2 L i2 4λmin ( Q¯ i )

≥ 0, we have

136

8 Reliable Control for a Class of Switched Nonlinear Systems

V˙i ≤ −λ0 Vi .

(8.21)

Vi (t) ≤ μV j (t), ∀i, j ∈ M.

(8.22)

Using (8.10) and (8.13), we have

To an arbitrarily given t > 0, let 0 = t0 < t1 < t2 < · · · < tk = t Ni (t0 ,t) be the switching constants during (0, t). From (8.21) and (8.22), we know that V (t) ≤ V (tk )e−λ0 (t−tk ) ≤ μV (tk− )e−λ0 (t−tk ) ≤ μV (tk−1 )e−λ0 (t − tk−1 ) ≤ ··· ≤ μ Ni (t,0) e−λ0 t V (0) = e−λ0 t+Ni (t,0) ln μ V (0),

(8.23)

if τa satisfies (8.11), i.e. for any t > 0 Ni (t, 0) ≤

t ln μ , , τa∗ = τa∗ λ

(8.24)

then Ni (t, 0) ln μ ≤ λt holds, thus V (t) ≤ e−(λ0 −λ)t V (0).

(8.25)

According to Assumption 8.4, there exist two positive constants b1 and b2 so that   b1 x2 + z2 ≤ V (t) ≤ b2 x2 + z2 .

(8.26)

From (8.25) and (8.26), we have 2 x(t) ¯ ≤

1 b2 2 V (t) ≤ e−(λ0 −λ)t x(0) ¯ . b1 b1

(8.27)

Hence,

x(t) ¯ ≤ which completes the proof.

b2 −(λ0 −λ)t e 2 x(0), ¯ b1

(8.28) 

8.3 Control Design

137

Remark 8.3 The controllers designed in Theorem 8.2 are suitable to both the actuators with faults (L is = Iqi ) and those without faults (L is = Iqi ) with an advantage that no changes are needed to make on structures and/or parameters of the controllers to guarantee globally asymptotical stability and satisfy other desired performances of system (8.1).  Remark 8.4 Pre- and post-multiplying both sides of matrix inequality (8.9) by Γ = Pi−1 , applying Schur complement lemma, and letting σ be a sufficiently small constant, inequality (8.9) can be transformed into the following LMIs: ⎡ ⎢ ⎣

Γ AiT + λ0 Ir + Ai Γ + Λi ∗

−

∗



Γ εi δ2 λmin (Hi ) i

∗

−σ

−1

Γ Ir

0

⎤ ⎥ ⎦ < 0,

(8.29)

−Q i−1

where  1 1 ε 1 2 Ui + Iq + Iqi BiT . Λi = Hi + εθi Ir + Bi ε ε λmin (Ui ) i ε  Remark 8.5 Note that the proposed method in this chapter can be easily extended to a wider class of nonlinear switched systems which has the following form: 

z˙ = gi (z, x) x˙ = f i (x) +  f i (x) + (Bi (x) + Bi (x))u i .

(8.30)

However, it has to be in this cascade structure. Stabilization of a class of non-cascade nonlinear systems is out of the scope of this chapter. 

8.4 Illustrative Example In this section, we use a numerical example to validate the proposed method. Example 8.6 Consider the uncertain nonlinear switched system 

z˙ = gσ (z, x) x˙ = (Aσ + ΔAσ )x + (Bσ + ΔBσ )u σ ,

where σ ∈ {1, 2}, and system matrix parameters are given by

(8.31)

138

8 Reliable Control for a Class of Switched Nonlinear Systems

Fig. 8.1 System state responses

⎡

⎤ ⎡ ⎤ −4 −2 0 1 0 A1 = ⎣ 0 −4 0 ⎦ , B1 = ⎣ 0.5 0.5 ⎦ , −0.4 1 −4 −0.5 1 ⎡

⎤ ⎡ ⎤ −2 −2 0 1 0 A1 = ⎣ 0 −2.5 0 ⎦ , B1 = ⎣ 0.6 0.5 ⎦ , −0.4 1 −2 −0.6 0.9 ⎡

⎤  0.1 0.002 0 10 H1 = ⎣ 0.002 0.1 0 ⎦ , U1 = , 01 0 0 0.1 ⎡

⎤  0.2 0.002 0 20 ⎣ ⎦ 0.002 0.2 0 H2 = , U1 = , 02 0 0 0.2 ⎡

⎤  0.7244 0.4067 0.1153 00 1 ⎣ ⎦ 0.4067 0.6398 −0.3002 , L s = Q1 = , 01 −0.1153 −0.3002 0.6398

8.4 Illustrative Example

139

Fig. 8.2 Switching sequences of gain matrices

⎡

⎤  0.8244 0.4067 0.1153 10 Q 2 = ⎣ 0.4067 0.6398 −0.3002 ⎦ , L 2s = , 00 0.1153 −0.3002 0.7363 g1 (z, x) = −z 3 + x1 sin(z), g2 (z, x) = −z 3 + x1 cos(z), ε1 = 1.2, ε2 = 1.5, δ1 = δ2 = 0.1, θ1 = θ2 = 0.2. Solving Riccati equations (8.9) and (8.10) with the parameters above gives a positive definite matrix solution ⎡ ⎤ ⎡ ⎤ 0.8662 0.0209 −0.0696 2.1576 0.3710 −0.1492 P1 = ⎣ 0.0209 0.8227 0.2560 ⎦ , P2 = ⎣ 0.3710 0.2467 0.3710 ⎦ . −0.0696 0.2560 0.7889 −0.1492 0.3710 1.9508 Then, according to u i = K i x = −BiT Pi x, i = 1, 2, the controller can be designed. Let initial conditions be x0 = [−3 3 − 2] and z 0 = 2. Also let λ = 0.06, μ = 1.1 and λ0 = 0.1. We have τa∗ = lnλμ = 1.5. Take τa∗ ≤ τa = 2 and choose the switching rule:

140

8 Reliable Control for a Class of Switched Nonlinear Systems

Fig. 8.3 Switching signal

 i=

1, k = 0, 2, 4, 6, · · · 2, k = 1, 3, 5, 7, · · ·

tk = 2k.

As shown in Fig. 8.1, the state feedback control law guarantees that system (8.19) are still asymptotically stable when the second actuator of the first subsystem and the first actuator of the second subsystem have faults, as indicated by L 1s and L 2s . Figure 8.1 is the time history of the states, where the stars represent switching points. Figures 8.2 and 8.3 represent switching sequences of controller gains and switching signal, respectively. From the figures, the effectiveness of the proposed control method is shown by this numerical example. 

8.5 Conclusion This chapter has introduced a fault-tolerant control method for a class of uncertain switched nonlinear systems by using average dwell-time techniques. A sufficient condition is carried out on globally exponential stabilization of the switched nonlinear systems against actuator faults under arbitrarily switching signals provided that

8.5 Conclusion

141

switching is on-the-average slow enough. This method can also be easily applied to robust fault-tolerant control problems of uncertain switched systems with sensor faults, or with both sensor and actuator faults.The simulation result son a numerical example have shown the effectiveness of the proposed method.

Chapter 9

Fault-Tolerant Control for a Class of Uncertain Switched Nonlinear Systems

9.1 Introduction In this chapter, we present a fault-tolerant control method for the switched cascade nonlinear systems where the nonlinear subsystem contains structural uncertainties and also whose actuators are allowed to partially fail. We first define the representation of partial actuator fault models, and then find a common Lyapunov function to design a controller to asymptotically stabilize the switched systems. The contributions of this chapter lies in three aspects: (1) The proposed method is invariant to actuator fault modes in the sense that the proposed controller is completely independent from the fault modes; as a result, the proposed method works well for all the possible actuator fault modes (except the case that all actuators completely fail) without needing to modify the controller. (2) The proposed method is independent of switching policies provided that a common Lyapunov function is found. (3) The proposed method is verified using a simulated haptic display system with switched virtual environments. The rest of this chapter is as follows. In Sect. 9.2, the problem statement is given. In Sect. 9.3, the detailed design is presented. In Sect. 9.4, the proposed method is applied to a simulated haptic display system with switched virtual environments. Section 9.5 concludes with brief remarks.

9.2 Problem Statement and Preliminaries Consider the system 

z˙ = gi (z, x) + Δgi (z, x) x˙ = (Ai + ΔAi )x + (Bi + ΔBi )u i ,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. Fu and T.-F. Li, Event-Triggered Control of Switched Linear Systems, Studies in Systems, Decision and Control 365, https://doi.org/10.1007/978-3-030-71604-2_9

(9.1)

143

144

9 Fault-Tolerant Control for a Class of Uncertain Switched Nonlinear Systems

where x ∈ Rr and z ∈ Rn−r are system states, u i ∈ Rqi is the control input, i(t) : [0, ∞) → M = {1, 2, . . . , m} is the switching signal, gi (z, x) is a known nonlinear function, Δgi (z, x) is unknown structural uncertainty, Ai and Bi are constant matrices, ΔAi and ΔBi are norm-bounded structural uncertainties, i.e., there exist positive constants δi and θi such that ΔAi  ≤ δi and ΔBi  ≤ θi . States (z, x) are assumed to be available. Assumption 9.1 Δgi (z, x) satisfies uniformly linear growth condition for ∀x, i.e., for ∀x, there exists a positive constant η such that Δgi (z, x) ≤ ηz.

(9.2)

To describe the situation where the actuators have partial failures, a matrix L is (L is = 0) is introduced L is = diag{l1i , l2i , . . . , lqi i },

(9.3)

where 0 ≤ L ij ≤ 1( j ∈ 1, 2, . . . , qi ). For example, l ij = 0.3 indicates that the jth actuator fails and only 30% of the complete signal can pass through it during the whole operation time or its time sub-intervals, l ij = 1 means that the jth actuator works normally, and l ij = 0 represents that the jth actuator has a complete failure. Assume that the controller is in the form of u i = K i x,

(9.4)

where K i ∈ Rqi ×r is a constant matrix. The closed-loop switched nonlinear system is then  z˙ = gi (z, x) + Δgi (z, x) (9.5) x˙ = [(Ai + ΔAi ) + (Bi + ΔBi )L is K i ]x. Our objective is to seek the feedback gain matrix K i (i ∈ M) such that the closedloop system (9.5) undergoing partial actuator faults is asymptotically stable.

9.3 Stabilization In this section, we will stabilize system (9.1) using the common Lyapunov function method. The main theorem and its proof are given below. Theorem 9.1 Suppose that Assumption 9.1 is satisfied and gi (z, x) satisfies uniformly global Lipschitz condition for ∀z i.e., there exists E i > 0 such that gi (z, x1 ) − gi (z, x2 ) ≤ E i x1 − x2 , ∀z, x1 , x2 .

(9.6)

9.3 Stabilization

145

Further assume that there exist a smooth positive definite W (z) with W (0) = 0 and constants βi > 0, γ > 0, i ∈ M satisfying dW (z) gi (z, 0) ≤ −βi z2 , dz

(9.7)

and    dW (z)     dz  ≤ γz.

(9.8)

If there exist ε > 0 and symmetric positive definite matrices Hi , Ui and Q i such that the following Riccati equations ε AiT P + P Ai + δ 2 Ir + Q i λmin (Hi ) i     1 1 ε 1 2 Hi + εθi Ir + Bi Ui + Iq + Iqi BiT P = 0 +P ε ε λmin (Ui ) i ε

(9.9)

have a symmetric positive definite matrix P, then K i = −BiT P renders the closedloop switched system (9.5) globally asymptotically stable under arbitrary switching policies. Proof Consider the Lyapunov function V (x, z) = μx T P x + W (z),

(9.10)

where μ > 0 is a constant. Then along the trajectory of (9.5), the time derivative of V (x, z) is V˙ = μx T (AiT P + P Ai + ΔAiT P + PΔAi − 2P Bi L is BiT P − P B L is ΔBiT P − PΔBi L is BiT P)x dW (z) dW (z) gi (z, x) + Δgi (z, x). + dz dz

(9.11)

Thus we have  1 T ˙ V ≤ μx AiT P + P Ai + εΔAiT Hi−1 ΔAi + P Hi P ε 1 −1 + εP Bi (−L is )Ui (−L is )BiT P + P Bi Ui BiT P ε  1 T i i T +εPΔBi ΔBi P + P Bi (−L s )(−L s )Bi P x ε dW (z) dW (z) gi (z, x) + Δgi (z, x). + dz dz

(9.12)

146

9 Fault-Tolerant Control for a Class of Uncertain Switched Nonlinear Systems

1  i i T i i 2 (L s L s ) L s L s ≤ 1, we have Because (−L is )(−L is ) = λmin

 V˙ ≤ μx T AiT P + P Ai +

ε 1 ΔAiT ΔAi + P Hi P λmin (Hi ) ε  1 1 ε P Bi BiT P + P Bi Ui BiT P + εθi2 P P + P Bi BiT P x + λmin (Ui ) ε ε dW (z) dW (z) + gi (z, x) + Δgi (z, x). (9.13) dz dz

By using ΔAi  ≤ δi and ΔBi  ≤ θi , we have  T ˙ V ≤ μx AiT P + P Ai +

ε 1 δ 2 + P Hi P λmin (Hi ) i ε  1 1 ε T T 2 T P Bi Bi P + P Bi Ui Bi P + εθi P P + P Bi Bi P x + λmin (Ui ) ε ε dW (z) dW (z) gi (z, x) + Δgi (z, x). + (9.14) dz dz

Furthermore,

ε δ 2 Ir V˙ ≤ μx T AiT P + P Ai + λmin (Hi ) i      1 1 ε 1 Hi + εθi2 L r + Bi Ui + Iqi + Iqi BiT P x +P ε ε λmin (Ui ) ε dW (z) dW (z) dW (z) + gi (z, 0) + [gi (z, x) − gi (z, 0)] + Δgi (z, x). (9.15) dz dz dz By using (9.9), we have dW (z) V˙ ≤ −μx T Q i x + [gi (z, x) − gi (z, 0)] dz dW (z) dW (z) + gi (z, 0) + Δgi (z, x). dz dz By Assumption 9.1 and (9.6)–(9.8), we have V˙ ≤ −μλmin (Q i )x T x − βi z2 + γ E i zx + γηz2 2  γ Ei 2 x = −μλmin (Q i )x − (βi + γη) z − 2(βi + γη) γ 2 E i2 + x2 4(βi + γη)2

(9.16)

9.3 Stabilization

147

 γ 2 E i2 x2 = − μλmin (Q i ) − 4(βi + γη) 2  γ Ei x . − (βi + γη) z − 2(βi + γη) 

Since βi > 0 and (βi + γη) z −

2

γ Ei x 2(βi +γη)

 ˙ V < − μλmin (Q i ) −

(9.17)

≥ 0, we have

 γ 2 E i2 x2 . 4(βi + γη)

(9.18)

γ 2 E i2 > 0, i ∈ M 4λmin (Q i )4(βi + γη)

(9.19)

So, choosing μ>

yields that the closed-loop system (9.5) is globally asymptotically stable under arbitrary switching laws.  Corollary 9.2 After pre- and post-multiplying (9.9) with X = P −1 , and applying the Schur complement lemma, (9.9) in Theorem 9.1 can be converted into the following linear matrix inequalities with a sufficiently small positive constant σ ⎡

Δi

⎢ ⎣ ∗ − ∗



X εi δ2 λmin (Hi ) i

∗

−σ

−1

X Ir

0

⎤ ⎥ ⎦ < 0, i ∈ M,

(9.20)

−Q i−1

where 1 Δi = X AiT + Ai X + Hi + εθi2 Ir ε   1 ε 1 Ui + Iq + Iqi BiT . + Bi ε λmin (Ui ) i ε Proof Simple and omitted.



Remark 9.3 For better performance, the parameters and positive definite matrices in Assumption 9.1 and in Theorem 9.1 can be chosen in a “trial and error” design manner but subject to satisfaction of all the assumptions made.  Remark 9.4 The advantage is that there is no need to update the parameters or change the structure of the proposed controller, i.e., the proposed method can achieve reliable control in the sense that the method works well for all the possible actuator faults (except the case that all actuators completely fail) without needing to modify the controller. The effect of faults is dominated by making the time derivative of

148

9 Fault-Tolerant Control for a Class of Uncertain Switched Nonlinear Systems

the Lyapunov function candidate negative definite, the distinguishing feature of the proposed method. 

9.4 Application In this section, we first give a description of the simulated haptic display system with switched virtual environments, then formulate the system into a fault-tolerant control problem of the switched system given practical considerations. Finally, we provide simulation results and qualitative analysis.

9.4.1 Description of the Haptic Display System A system with a haptic device can be represented by Fig. 9.1. As shown in Fig. 9.1, through a haptic interface bi-directionally and a visual and audio display mono-directionally, a human communicates with a computer. Thus the nature of this interaction depends highly on the configuration of the remote site. Usually the configurations are characterized by the interaction between a human participant and a computer interface. Since the interaction is both at the signal level and symbolic level, the overall system is hybrid [114]. For such systems, stability is an issue due to the interaction of the human with the virtual environment through a mechanical haptic display, as such, this issue has received much attention [1, 14, 22, 74, 114]. When a haptic interaction with a switched virtual environment is considered (e.g., a user touches different objects in the virtual environment), the system switches between different dynamic behavior, studied in [74] specifically for two different environments. In addition, aging of actuator components is a continuous and slow process, thus, considering partial faults is reasonable and fits practical application. To our best knowledge, there have not been any published results considering haptic systems with switched virtual environments where partial faults occur in its actuators.

Fig. 9.1 A system with a haptic device

9.4 Application

149

The next subsection will illustrate the application of the proposed method to a haptic system scenario.

9.4.2 Formulated Problem This section applies the developed method to the haptic display system in [74] where the input of H+D subsystem depends only on the states of the virtual environments. The whole system is ⎧ x1 b+b ⎪ ⎨z˙ = − m z + m x˙1 = − BK2 x1 − K x2 + K u ⎪ ⎩ x˙2 = xM1 − BM3 x2 ,

(9.21)

where z is the velocity of the haptic device, x1 and x2 are states respectively representing the force generated by the spring and the velocity of mass M of an equivalent 2nd order mechanical system, and are damping parameters, is spring stiffness coefficient, and are damping parameters. This system is equivalent to a haptic system with switched virtual environments [74, 114]. According to the nature of the dynamics of the switched virtual environment of [74, 114] (values of the parameters below taken from these references), for the situation where the velocity of the mass is also to be directly controlled, (9.21) can be reformulated as: 

where

z˙ = gi (z, x) + Δgi (z, x) x˙ = (Ai + ΔAi )x + (Bi + ΔBi )u i ,

(9.22)



   −1 −20 −0.5 −50 A1 = , A2 = , 50 −0.5 20 −1 

   2 0 5 0 , B2 = , B1 = 0 1.5 0 2.5 g1 = g2 = −

x1 b + b z + , Δg1 = Δg2 = sin(x1 )z, m m

and ΔA1 , ΔA2 , ΔB1 and ΔB2 are zero matrices. The objective is to stabilize system (9.22) without requiring update to the parameters or changing the structure of the proposed controller; i.e., to achieve reliable control in the sense that the proposed method works well for all possible actuator faults (except the case that all actuators completely fail) without needing to modify the controller.

150

9 Fault-Tolerant Control for a Class of Uncertain Switched Nonlinear Systems

9.4.3 Simulation and Analysis We can easily verify Assumption 9.1 since the uncertainties Δg1 , Δg2 is affine in z, sin(x1 ) is bounded and conditions (9.6)–(9.9) in the theorem can also be easily satisfied with W (z) = 21 z 2 and g1 , g2 being linear with respect to z and x. We consider the following fault situation: for the first subsystem, the first actuator allows only eighty percent of the desired control signal to pass, and the second actuator completely fails; for the second subsystem, the first actuator completely fails and the second actuator only allows forty percent of the desired control signal to pass, i.e.,  L 1s =

 0.8 0 , 0 0

 L 2s =

 0 0 . 0 0.4

Solving the LMIs gives a common solution matrix P as follows:  P=

 1.5889 −0.1478 . −0.1478 0.4741

Initial values are x(0) =

  1 , z(0) = 0. 1

We have carried out the simulation studies, under the same switching signal, for both the proposed method and the case in which the actuator faults are the same but not considered during the course of controller design. From Fig. 9.2, we can see that the proposed controller can stabilize the system. However, if the faults were not considered in the controller design the states of the system will diverge as shown in Fig. 9.3. Figure 9.4 is the switching signal. From the figures, we can conclude that the proposed method is effective in terms of stabilization as shown by this practical example.

9.4 Application

151

Fig. 9.2 State responses (case where faults considered in controller design)

Fig. 9.3 Zoomed-in unstable state responses (case where faults not considered in controller design)

152

9 Fault-Tolerant Control for a Class of Uncertain Switched Nonlinear Systems

Fig. 9.4 Switching signal

9.5 Conclusion In this chapter, we have presented a fault-tolerant control method for a class of switched cascade nonlinear systems with structural uncertainties by utilizing the common Lyapunov function method, and then applied the proposed method to haptic display systems. This method aims to deal with partial failure of actuators. The proposed method is independent from switching policies under certain conditions.

Chapter 10

Conclusions

The focus of the book has been placed on event-triggered control, time-triggered control, hysteresis switching control, fault-tolerant control and reliable control for some classes of switched systems (including switched linear systems, switched delay systems, switched neutral systems, switched nonlinear systems). Specifically, several research problems have been investigated in detail. 1. The event-triggered control problem has been investigated for continuous-time switched linear systems. First, an exponential switching observer has been designed for the switched linear system to estimate system state which supports to realize feedback control. Second, based on the switching observer, an eventtriggered sampling mechanism is set up through which the sampling instants are determined. Third, under the event-triggered sampling mechanism, a sampleddata-based switching controller is built, in which the state and the switching signal are updated according to the information transmitted from the sampler. At last, a sufficient condition for stability of the closed-loop switched linear system is obtained by using the multiple Lyapunov function method combining with average dwell time technique. Multiple switches are allowed to happen between arbitrary two consecutive triggering times. 2. An improved event-triggered sampling mechanism has been investigated for continuous-time switched linear systems. So called “improved” means that a constant threshold is added into the right-hand side of the original dynamical inequality condition of the event-triggered sampling mechanism. The advantage of introducing the constant threshold is that the sampling frequency can be changed by adjusting the value of the constant threshold. Especially, the improved eventtriggered sampling mechanism includes the original one as a special case if let the constant threshold be equal to zero. Furthermore, the zeno phenomenon can be excluded easily since the constant threshold can guarantee the existence of minimum positive lower bound between two continuous sampling interval. Under the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. Fu and T.-F. Li, Event-Triggered Control of Switched Linear Systems, Studies in Systems, Decision and Control 365, https://doi.org/10.1007/978-3-030-71604-2_10

153

154

3.

4.

5.

6.

10 Conclusions

improved event-triggered sampling mechanism, both state-based and observerbased output feedback event-triggered control for continuous-time switched linear systems have been studied. Sufficient conditions of guaranteeing stability of switched linear systems have been proposed under multiple Lyapunov function approach combining with average dwell time switching strategy. An event-triggered control problem has been investigated for switched linear delay systems. A state-based event-triggered sampling mechanism and an observerbased event-triggered sampling mechanism are proposed, respectively. Since the sampling mechanism transmits not only the state information but also the switching signal of switched delay systems, asynchronous switching is caused. Two sufficient stability criteria are obtained under the two different kinds of sampling mechanisms by using multiple Lyapunov–Krasovskii functional method and the average dwell time technique. Some free weighting matrices are introduced to help achieving stability conditions. An event-triggered sampled-data control problem has been investigated for switched linear neutral systems. Switched neutral systems are a special kind of switched delay system but with more complexity structure in which at least one subsystem is a neutral system. As we all know that a neutral system is a more general delay system that delay is contained not only in its state but also in the derivation of its state which bring more difficulty in stability analysis, and certainly more complexity for switched neutral systems. Furthermore, asynchronous switching caused by the constructed sampling mechanism has also been investigated. An event will be triggered if the condition is satisfied and then the current state and the active mode will be transmitted to the controller to decide which subcontroller is active which leads to asynchronous switching between the system and the controller. Under the observer-based controller, combining with switching rule provided that average dwell time conditions of subsystems are satisfied, a sufficient stability criterion has been proposed which guarantees exponential stability of the switched neutral system. A periodic sampled-data control problem has been studied for switched linear neutral systems. Delays, including state delay and neutral delay, considering in switched neutral system are all time-varying ones with boundary. The periodic sampling control means that the controller is active according to the information transmitted by the sampler in which the sampling frequency is fixed. By analyzing the relationship between the sampling period and the dwell time, a bond between the sampling period and average dwell time has been revealed to form a switching condition. Subject to this switching condition and certain control gains related constraints, exponential stability of the switched neutral system has been guaranteed. We propose a sample-data-based control method for switched linear neutral systems under asynchronous switching. Some free weighting matrices have been introduced to obtain the decoupled constraints which technically reduces the computational complexity. A hysteresis-based switching control problem has been investigated for switched linear neutral systems. A state-based switching strategy has been designed by generalizing the min-switching strategy depending on state-space partition. A mul-

10 Conclusions

155

tiple generalized Lyapunov–Krasovskii functional has been constructed which allows some increase of connecting adjacent Lyapunov–Krasovskii functional at the switching points. A delay-dependent sufficient condition is proposed to guarantee asymptotic stability of the closed-loop switched linear neutral system by using multiple generalized Lyapunov–Krasovskii functional approach combining. Some free weighting matrices have been introduced to help achieving the stability condition. 7. A robust fault-tolerant control problem of a class of switched cascade nonlinear systems with structural uncertainties existing in both system matrices and input matrices, and a fault-tolerant control approach for this class of switched systems by using average dwell-time techniques has been proposed. The proposed control design has been proved efficiently on both the switched nonlinear systems with actuator faults and its nominal systems (i.e., without actuator faults) without necessarily changing any structures and/or parameters of the proposed controllers. Therefore, it has been classified as a reliable control, or passive fault-tolerant control strategy. The proposed approach, in a unified way for easy and practical applications, has been treated all actuators without necessarily classifying all actuators into faulty actuators and robust ones. The proposed approach is independent of arbitrary switching signals provided that the average switching time is greater than some constant time. 8. A fault-tolerant control problem has been investigated for the switched cascade nonlinear systems where the nonlinear subsystem contains structural uncertainties and also whose actuators are allowed to partially fail. The representation of partial actuator fault models has been defined, and a common Lyapunov function has been proposed to design a controller to asymptotically stabilize the switched systems. The proposed approach is invariant to actuator fault modes in the sense that the proposed controller is completely independent from the fault modes; as a result, the proposed method works well for all the possible actuator fault modes (except the case that all actuators completely fail) without needing to modify the controller. The proposed method is independent of switching policies provided that a common Lyapunov function is found.

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