Robust control for nonlinear time-delay systems 978-981-10-5131-9, 9811051313, 978-981-10-5130-2

Table of contents :
Front Matter....Pages i-xii
Introduction....Pages 1-10
Front Matter....Pages 11-11
Robust Stabilization of Single Nonlinear Time-Delay System....Pages 13-26
Robust Model Reference Adaptive Control for Interconnected Time-Delay Systems....Pages 27-39
Front Matter....Pages 41-41
Decentralized Adaptive Control for Interconnected Time-Delay Systems....Pages 43-59
Memoryless State Feedback Control for Uncertain Nonlinear Time-Delay System....Pages 61-74
Exponential Stabilization for Interconnected Time-Delay Systems....Pages 75-91
Robust Adaptive Control for Time-Delay System via T-S Fuzzy Approach....Pages 93-112
Front Matter....Pages 113-113
Adaptive Tracking of Time-Delay System with Unknown Dead-Zone Input....Pages 115-128
Decentralized Fuzzy Networked Control Systems Design with Sector Input....Pages 129-155
Front Matter....Pages 157-157
Robust Control for a Class of Time-Delay System via Razumikhin Lemma....Pages 159-171
Backstepping Control for Nonlinear Time-Delay System via L-K Function....Pages 173-191
NN-Based Output Feedback Tracking of Nonlinear Time-Delay System....Pages 193-214
Output Feedback Stabilization for Interconnected Time-Delay Systems....Pages 215-246
Robust Control of Time-Delay System with Unknown Control Direction....Pages 247-270
Decentralized Prescribed Performance Tracking of Stochastic Time-Delay System....Pages 271-290
Back Matter....Pages 291-300

Citation preview

Changchun Hua · Liuliu Zhang Xinping Guan

Robust Control for Nonlinear Time-Delay Systems

Robust Control for Nonlinear Time-Delay Systems

Changchun Hua Liuliu Zhang Xinping Guan •

Robust Control for Nonlinear Time-Delay Systems

123

Xinping Guan Department of Automation Shanghai Jiao Tong University Shanghai China

Changchun Hua Institute of Electrical Engineering Yanshan University Qinhuangdao China Liuliu Zhang Institute of Electrical Engineering Yanshan University Qinhuangdao China

ISBN 978-981-10-5130-2 DOI 10.1007/978-981-10-5131-9

ISBN 978-981-10-5131-9

(eBook)

Library of Congress Control Number: 2017943106 © Springer Nature Singapore Pte Ltd. 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express 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. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Time delay appears in many physical processes due to the period of time it takes for the signals to transmit. Time-delay systems are largely encountered in modeling propagation phenomena, population dynamics, interconnected systems, supply chains, and systems controlled over communication networks. It is well known that time delay in control systems may lead to deterioration of the closed-loop performance or even destabilize the systems; therefore, specific analysis techniques and design methods are needed to be developed for control systems in the presence of time delay. The time-delay systems can be divided into linear time-delay systems and nonlinear time-delay systems. Recently, the stability analysis and control design of linear time-delay systems have been extensively studied with the popular tools—Lyapunov–Krasovskii functional method and Lyapunov–Razumikhin method. The stability and stabilization conditions can be transformed into solvable linear matrix inequalities (LMIs) with the help of Schur complement lemma. Compared with linear time-delay systems, the study of nonlinear time-delay systems is more important for control theory and control applications, as most of practical systems have nonlinear dynamics and nonlinear uncertainties generally exist in practical engineering systems due to the modeling error and external disturbances. However, the analysis and synthesis of nonlinear time-delay systems are more difficult and challenging. The main reasons are as follows: (i) It is not easy to select Lyapunov functional for nonlinear time-delay systems. The Lyapunov functional for linear time-delay systems is generally chosen to be quadratic. But for nonlinear time-delay systems, we should construct Lyapunov functional based on the specific system structure, which increases the difficulties of stability analysis and control design. (ii) It is difficult to compensate for the time-delay effect while designing nonlinear controllers. For the control design of time-delay systems, we aim to design memoryless controllers independent of time delay, because the time delay is variable in practical systems and it is impossible to obtain the exact values of time delay. Hence, the controller design of nonlinear time-delay systems is different from that of the nonlinear systems free of time delay, as how to systematically design controllers to overcome the effect of time delay is very difficult.

v

vi

Preface

This book is devoted to report the latest results on nonlinear time-delay control theory and applications, especially the robust control of time-delay systems with nonlinear uncertainties. This book collects some novel works related to commonly encountered nonlinear time-delay systems, such as the nonlinear systems in the form of high-order polynomial dynamics, systems with nonlinear input, and triangular nonlinear systems. Undoubtedly, the results in this book will enrich nonlinear system theory and time-delay system theory. Our treatment is theoretically oriented, although some illustrative examples are included in this book. The reader is assumed to have some background in nonlinear systems and control. Although this book is primarily intended for students and practitioners of control theory, it is also a valuable reference for those in fields such as communication engineering and economics. Moreover, we believe that this book should be suitable for certain advanced courses or seminars. In Chap. 1, the background and some descriptions of nonlinear time-delay system are provided. Then, the rest of this book will be presented under the following parts: Part I: The first part of this book is concerned with nonlinear time-delay systems with uncertainties in high-order polynomial form. In Chap. 2, based on the Lyapunov–Krasovskii functional and Razumikhin lemma, two classes of memoryless control design methods are proposed for single nonlinear time-delay system. In Chap. 3, the results are extended to a class of large-scale time-delay systems with interrelated N subsystems and the decentralized robust model reference adaptive controller is constructed. Part II: The second part of this book focuses on some new results on nonlinear time-delay systems with general uncertainties. In Chap. 4, the decentralized adaptive state feedback control strategy is proposed for interconnected systems. In Chap. 5, the stabilization problem is investigated for a class of single uncertain mismatched systems with multiple time delays and a memoryless state feedback controller is constructed. In Chap. 6, the result is extended to a class of large-scale systems and the solution of the resulting closed-loop system can exponentially converge to a ball with adjustable radius. In Chap. 7, the control problem of nonlinear time-delay systems is studied via the T-S fuzzy approach. Part III: The third part of this book is dedicated to nonlinear time-delay systems with two kinds of nonlinear inputs. In Chap. 8, the adaptive tracking control law for nonlinear time-delay system with non-symmetric dead-zone input is presented. In Chap. 9, based on T-S fuzzy approach, the decentralized networked control problem is investigated for large-scale time-delay systems with sector input. Part IV: The last part of this book is devoted to controller design for nonlinear time-delay systems with triangular structure. In Chap. 10, the robust control problem is investigated for nonlinear time-delay systems with the form of triangular structure via Razumikhin lemma. In Chap. 11, the state feedback control problem is addressed for a class of nonlinear time-delay systems via Lyapunov–Krasovskii function. In Chap. 12, the robust output tracking control problem is investigated for a class of nonlinear time-delay systems with unmodeled dynamics. In Chap. 13, decentralized dynamic output feedback control problem is considered for a class of

Preface

vii

nonlinear interconnected systems with time delay. In Chap. 14, the output feedback problem is investigated for a class of uncertain nonlinear time-delay systems with unknown control direction. In Chap. 15, dynamic output feedback tracking control strategy is presented for stochastic interconnected time-delay systems with prescribed performance requirement. The support from the National Natural Science Foundation of China (61673335, 61290322, 61322303, 61273222, 60974018, 60604004), Nature Science Foundation of Hebei Province (F2016203467, F2014203267, F2011203110, 15967629D, GCC2014033), Program for the Outstanding Young Innovative Talent of China is gratefully acknowledged. Qinhuangdao, China Qinhuangdao, China Shanghai, China January 2017

Changchun Hua Liuliu Zhang Xinping Guan

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Description of Nonlinear Time-Delay Systems . . . . . . . . . . 1.2.1 Quasi Nonlinear Time-Delay Systems . . . . . . . . . . 1.2.2 Pure Nonlinear Time-Delay Systems . . . . . . . . . . . 1.2.3 Interconnected Nonlinear Time-Delay Systems . . . 1.3 Problems Studied in This Book . . . . . . . . . . . . . . . . . . . . . 1.3.1 High-Order Polynomial Uncertainties . . . . . . . . . . 1.3.2 General Uncertainties . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Nonlinear Input . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 System with Lower Triangular Structure . . . . . . . . 1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I

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

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

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

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

1 1 3 3 4 5 6 6 7 7 8 9

. . . . . . .

. . . . . . .

. . . . . . .

High-Order Polynomial Nonlinear Uncertainties

2

Robust Stabilization of Single Nonlinear Time-Delay System . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 System Formulation and Preliminaries . . . . . . . . . . . . . . . . 2.3 Adaptive Robust State Feedback Controller . . . . . . . . . . . . 2.4 Novel Nonlinear Feedback Controller . . . . . . . . . . . . . . . . . 2.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

13 13 14 16 19 22 26

3

Robust Model Reference Adaptive Control for Interconnected Time-Delay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 System Formulation and Preliminaries . . . . . . . . . . . . . . . . . . . . 3.3 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 28 30 35 39

ix

x

Contents

Part II 4

5

6

7

Decentralized Adaptive Control for Interconnected Time-Delay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 System Formulation and Preliminaries . . . . . . . . . . . 4.3 Decentralized Feedback Control . . . . . . . . . . . . . . . . 4.4 Application to Decentralized Control for a Class of Interconnected Systems . . . . . . . . . . . . . . . . . . . . 4.5 Illustrative Examples. . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

43 43 44 46

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

50 54 59

Memoryless State Feedback Control for Uncertain Nonlinear Time-Delay System . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 System Description . . . . . . . . . . . . . . . . . . . . . . 5.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . 5.4 Simulation Example . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

61 61 62 63 71 73

Exponential Stabilization for Interconnected Time-Delay Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 System Formulation and Preliminaries . . . . . . . . . . . . . . . . 6.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

75 75 76 78 88 90

Robust Adaptive Control for Time-Delay System via T-S Fuzzy Approach . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 System Formulation and Assumptions . . . . . . . 7.3 Virtual Control Design . . . . . . . . . . . . . . . . . . . 7.4 Controller Design . . . . . . . . . . . . . . . . . . . . . . . 7.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

93 93 94 98 101 107 112

Adaptive Tracking of Time-Delay System with Unknown Dead-Zone Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

115 115 116 118 125 128

Part III 8

General Nonlinear Uncertainties

. . . . . .

. . . . . . .

. . . . . .

. . . . . . .

. . . . . .

. . . . . . .

. . . . . .

. . . . . . .

. . . . . . .

. . . . . .

. . . . . . .

. . . . . .

. . . . . . .

. . . . . .

. . . . . . .

Nonlinear Input

Contents

9

xi

Decentralized Fuzzy Networked Control Systems Design with Sector Input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 System Formulation and Assumptions . . . . . . . . . . . . . . . . 9.3 Virtual Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Parameters Known Case . . . . . . . . . . . . . . . . . . . . 9.4.2 Parameters Unknown Case . . . . . . . . . . . . . . . . . . 9.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

129 129 130 133 136 137 144 147 155

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

159 159 160 162 171

11 Backstepping Control for Nonlinear Time-Delay System via L-K Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 System Description and Preliminaries . . . . . . . . . . . . . . . . . 11.3 Controller Design for the Second-Order System . . . . . . . . . 11.4 Extension to the nth-Order System . . . . . . . . . . . . . . . . . . . 11.5 Application to Chemical Reactor Systems . . . . . . . . . . . . . 11.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

173 173 174 176 180 185 187 191

12 NN-Based Output Feedback Tracking of Nonlinear Time-Delay System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 System Formulation and Some Assumptions . . . . . . . . . . . . . . . 12.3 Preliminary Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Choosing Proper Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193 193 194 197 198 200 208 211 214

Part IV

Time-Delay System with Lower Triangular Structure

10 Robust Control for a Class of Time-Delay System via Razumikhin Lemma . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Problem Formulation and Preliminaries . . . . . . 10.3 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

13 Output Feedback Stabilization for Interconnected Time-Delay Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 13.2 System Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

xii

Contents

13.3 Robust Controller Design . . . . . . . . . 13.3.1 Observer Design . . . . . . . . . 13.3.2 Controller Design . . . . . . . . 13.4 Adaptive Neural Network Control . . 13.5 Simulation Investigation . . . . . . . . . . 13.6 Conclusion . . . . . . . . . . . . . . . . . . . .

. . . . . .

220 220 222 232 240 246

14 Robust Control of Time-Delay System with Unknown Control Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Controller Design: Known Bound Functions . . . . . . . . . . . . . . . 14.6 Controller Design: Unknown Bound Functions . . . . . . . . . . . . . 14.7 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

247 247 248 249 251 253 260 264 270

15 Decentralized Prescribed Performance Tracking of Stochastic Time-Delay System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 System Formulation and Preliminaries . . . . . . . . . . . . . . . . 15.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 15.2.2 Basic Knowledge on Stochastic System . . . . . . . . 15.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 Reduced-Order Observer Design . . . . . . . . . . . . . . 15.3.2 Prescribed Performance Transformation . . . . . . . . . 15.3.3 Adaptive Neural Network Controller Design . . . . . 15.4 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

271 271 272 272 275 276 276 278 278 287 290

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . . . . . . .

. . . . . .

. . . . . . . . . . .

. . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

Chapter 1

Introduction

1.1 Background Time delay is an inherent characteristic of physical system when materials or energy is transmitting through a certain route. The phenomenon of time delay exists in various engineering systems such as chemical process, power system, rolling systems, long transmission lines in pneumatic systems, and systems controlled by communication networks. The existence of time delay may lead to deterioration of the closed-loop performance, and even destabilize the systems. Therefore, the stability analysis and control design of time-delay systems are significant for practical engineering applications. It is well known that most of the systems encountered in engineering processes are nonlinear in essence. For example, in a mechatronic system, the actuators cannot increase its power infinitely, and there always exist saturation nonlinearities. In general, linear model is an approximation of a real nonlinear system and modeling errors always exist. Early studies of control theory mainly focused on linear system models, because the system is relatively simple and high accuracy performance is not required. However, with the rapid development of science and technology, the industrial application systems are becoming much more complex and the accuracy requirement is also increasing. It is not easy to achieve expected control objectives based on linearized models of industrial processes, because the global stability could not be achieved as the linearized model is only locally feasible. Therefore, direct research work on practical nonlinear system models should be launched in order to obtain effective nonlinear controllers. In addition, the uncertainties in practical systems cannot be avoided due to the inaccuracy of modeling, external disturbances, change of working conditions, and component aging. The uncertainties could affect systems’ performance and even destabilize the systems. It is well known that the development of modern control theory based on state-space model has made a big breakthrough in 1960–1970s; however, it has been proved that these theories are seriously dependent on accurate mathematical models that result in the invalidness of modern control theory when it is applied for practical engineering applications. Thus, the study of uncertainties in systems is of importance in both theoretical research and practical applications.

© Springer Nature Singapore Pte Ltd. 2018 C. Hua et al., Robust Control for Nonlinear Time-Delay Systems, DOI 10.1007/978-981-10-5131-9_1

1

2

1 Introduction

Fig. 1.1 Structure diagram of hydraulic automatic gauge control system in cold rolling mill

Fig. 1.2 Multi-slave networked teleoperation systems

From the above observations, it is known that the nonlinear time-delay systems with uncertainties are largely encountered in practical engineering applications. Two examples are given in the following. The first one is a hydraulic automatic gauge control system in cold rolling mill. The structure diagram is shown in Fig. 1.1, where h outd is the desired thickness of rolled piece, u is control input, i is electric current, xv is the valve core displacement, x p is the hydraulic cylinder piston displacement, h in and h out are the entry and export thickness, respectively. It is known that the thickness gauge is usually installed at the place far from the roller that produces the thickness of the piece, which can protect the thickness gauge from damage in accidents during real productions. The time delay of system measurements appears between the thickness gauge and the roller, and it is hard to obtain a precise mathematic model because of the complex structure and various external disturbances. Therefore, the appropriate control design based on feedback control theory to keep the rolling mill system operating steadily and precisely is an important topic that deserves to be pursued. The second example is a master–slave teleoperation system. Networked teleoperation system has numerous applications such as space exploration, underwater operation, nuclear accident, and minimally invasive surgery. A multiple slave teleoperation system is shown in Fig. 1.2. The master and slaves exchange information via the network communication channel. The human operates the master, and the master information is sent to the multiple slave side through wireless/wired communication network. The environmental force and state information of the slaves are sent back to the master via the same or different channel. It is well known that the robots are nonlinear Lagrangian systems, and the external disturbances and modeling

1.1 Background

3

inaccuracy always exist in networked multiple robots teleoperation systems. Therefore, the networked teleoperation system is a typical nonlinear time-delay system with uncertainties.

1.2 Description of Nonlinear Time-Delay Systems 1.2.1 Quasi Nonlinear Time-Delay Systems Consider the following nonlinear time-delay systems x˙ (t) = Ax (t) + Ad x (t − d) + Bu (t) + f (t, x (t − d))

(1.1)

where A, Ad , B are known matrices, d is time delay, f (x (t − d)) is the time-delay nonlinear term, which satisfies the Lipschitz condition or first-order bounded condition, i.e.,  f (t, x (t − d)) ≤ α x (t − d) (1.2) where α is a positive constant. Quasi nonlinear time-delay system is modeled as (1.1) that satisfies Lipschitz condition (1.2). Such nonlinear time-delay systems have been well studied in the existing literature. By choosing Lyapunov–Krasovskii (L-K) functional [2, 162] or employing Razumikhin lemma, we can obtain delay-independent and delay-dependent stability conditions and controller design criteria. The stability results are described in the form of Riccati equation or LMIs that can be easily solved by using MATLAB toolbox. Next, we simply introduce L-K method related to obtain the delay-independent results. Firstly, choose Lyapunov–Krasovskii functional for system (1.1) as  V (x (t)) = x (t) P x (t) + (ε1 + ε2 )

t

x T (ξ) x (ξ) dξ

T

(1.3)

t−d

Consider the case of u = 0, and compute the derivative of V (x (t)) along (1.1), we can obtain that if there exist positive definite matrix P and positive constants ε1 , ε2 that satisfy the following Riccati inequation −1 2 T P A + A T P + ε−1 1 α P P + (ε1 + ε2 ) I + ε2 P Ad Ad P < 0

(1.4)

  which implies that V˙ ≤ 0 when x = 0, V˙ = 0 , and the system is asymptotic stable. Similarly, for the case u (t) = K x (t) , it is easy to obtain that if there exist positive definite matrix P and positive constants ε1 , ε2 satisfying −1 2 T P (A + B K )+( A + B K )T P +ε−1 1 α P P +(ε1 + ε2 ) I +ε2 P Ad Ad P < 0 (1.5)

4

1 Introduction

then, the controller u (t) = K x (t) renders the closed-loop system asymptotically stable. Suppose that the time-delay term of quasi nonlinear time-delay system (1.1) has the following form f (t, x (t − d)) = M F (t) N x (t − d)

(1.6)

where M, N are appropriate constant matrices, F (t) is the uncertainty satisfying F (t) ≤ 1. Obviously, F (t) is the uncertainty of time-delay matrix of linear systems, then the problems of stability analysis and control of system (1.1) can be easily dealt with by using existing linear time-delay theories. The above analysis presents the L-K method for quasi nonlinear time-delay systems, which attributes to calculating the inequalities (1.4), (1.5), and the final results can be obtained in LMI forms by using Schur complement lemma. The key of the L-K method is how to choose suitable L-K functional, which will affect the conservatism of results directly. Moreover, the results of quasi nonlinear time-delay systems are the natural extension of linear time-delay systems, and the analysis methods are similar as for linear time-delay systems.

1.2.2 Pure Nonlinear Time-Delay Systems Pure nonlinear time-delay systems have the following form x˙ = F (x, x (t − d) , u)

(1.7)

where F (·) is a nonlinear function. Furthermore, the affine form is represented as x˙ = f (x, x (t − d)) + g (x, x (t − d)) u

(1.8)

where d is the time delay, f (·) and g (·) are nonlinear functions. Most of the existing literatures consider the pure nonlinear time-delay systems with certain structures and assumptions. For instance, [184] considered the control problems of a class of first-order nonlinear systems with single time delay by using PI controller and a time-delay compensator. [59] considered a class of affine nonlinear time-delay systems by using exact linearization and Smith predictor method. The control problem of SISO nonlinear time-delay systems by using the differential geometry method was considered in [173]. The systems with triangular structures are also popular, which have attracted a lot of researchers’ attention. For nonlinear time-delay systems in upper triangular form, based on the forwarding and saturation design method, the problem of asymptotic stabilization by state feedback was studied in [203], and a decentralized adaptive state feedback control scheme was proposed in [201]. For nonlinear time-delay systems in lower triangular form, based on the backstepping design method, the problem of

1.2 Description of Nonlinear Time-Delay Systems

5

adaptive fuzzy tracking control was investigated in [15, 178, 179] and the problem of adaptive fuzzy stabilization was considered in [42, 171], respectively. In addition, some intelligent control methods have been applied for nonlinear time-delay systems. For example, with the T-S fuzzification method, a nonlinear timedelay system can be converted to a T-S fuzzy time-delay system. The transformed system consists of piecewise linear time-delay systems, and then the existing results on linear time-delay systems can be applied.

1.2.3 Interconnected Nonlinear Time-Delay Systems Many practical systems contain subsystems, such as power systems, complex grid control systems, and electromechanical systems. The systems consisting of many subsystems are called large-scale systems or interconnected systems. That each subsystem is stable cannot guarantee the stability of the overall large-scale systems; on the other hand, the stable interconnected systems cannot guarantee the stability of its subsystems. Moreover, time-delay is very common in large-scale systems because of the time it takes for the information to be transmitted among subsystems. Therefore, the study of stable analysis and control problems of nonlinear time-delay interconnected systems is a challenging issue that has drawn many researchers’ attention. There are two control methods for nonlinear interconnected time-delay systems: centralized control and decentralized control. Centralized control uses the global information of large-scale systems to design controllers, and each controller of a subsystem is not only dependent on its own state variables but also dependent on other subsystems’ information. Since large-scale systems possess some characteristics, such as huge scale, various structures or parameters, complicated coupling between subsystems, the signals and information are over-abundant in centralized control and it is not easy to be applied for practical engineering systems. On the other hand, decentralized control is a method that each controller of a subsystem only depends on its own information but the design also takes the global information of the large-scale system into consideration. Therefore, decentralized control problems of the large-scale systems have been widely studied because of its simple structure and easy application compared with centralized control. Up to now, there have been many results related to the research of interconnected nonlinear time-delay systems. For instance, state feedback controllers had been designed to solve the decentralized control problem of nonlinear systems in [19, 48, 151, 154, 190]. For the case that the state could not be completely obtained, the output feedback controllers were used to solve the stabilization problem in [78, 83, 197]. In addition, based on backstepping method, the decentralized delay-independent controller was designed for large-scale interconnected stochastic systems in [34, 186].

6

1 Introduction

1.3 Problems Studied in This Book Four major issues of nonlinear time-delay systems are presented in this book. First, since delay perturbations are usually bounded by high-order polynomial form, we relax the restrictive conditions in the existing results where time-delay uncertainties are bounded by first-order linear function in Part I. Then, as for some practical systems whose time-delay uncertainties cannot be described as high-order polynomial form, Part II focuses on a more general nonlinear uncertainty condition. Furthermore, in order to deal with the problem of deterioration of system performance induced by nonlinear inputs, several feedback controller design methods are presented in Part III. Finally, based on backstepping approach, some efforts are devoted to controller design for nonlinear time-delay systems in the form of triangular structure in Part IV for that numerous mathematical models of practical systems are in triangular form or they can be transformed into triangular form under certain conditions.

1.3.1 High-Order Polynomial Uncertainties In the early studies, the controller design for systems with time-delay uncertainties was based on the uncertainties bounds and these bounds could be estimated. However, sometimes, the uncertainties bounds are over-estimated that may lead to large initial values of the controllers, and the induced actuator saturation further results in system performance degradation. In order to relax the restrictive conditions on uncertainties, many results have been proposed, see [110, 122, 132, 137, 141, 175]. For a class of uncertain systems with multiple time delays, the robust stabilization via linear control was studied in [110]. In [137], the problem of adaptive control was investigated for a class of uncertain dynamic delay systems. In [175], by employing a Razumikhin-type theorem, a robust stability criterion was given for a class of linear systems subject to mismatched delayed time-varying nonlinear perturbations. In all of the abovementioned results, the uncertainties were assumed to be bounded by first-order linear functions, and the bounds can be divided into known bounds and unknown bounds. But it is known that using first-order linear function to bound the uncertainties is still a strong restrictive condition for time-delay systems. Hence, some assumptions are proposed to relax the restrictive first-order linear function condition to high-order form. In Part I, we consider the robust adaptive control problems for nonlinear timedelay systems, the nonlinear system dynamics are described by the sum of linear nominal systems and nonlinear time-delay uncertainties, and the time-delay functions are assumed to be bounded by high-order polynomial functions with unknown gains. First, we investigate a class of single systems, based on Lyapunov–Krasovskii functional and Razumikhin lemma, and delay-dependent and delay-independent controllers are proposed to guarantee the uniformly ultimate bound stability of the closed-loop system. Then, the results are extended to a class of large-scale time-delay

1.3 Problems Studied in This Book

7

systems with N subsystems, and the robust model reference adaptive controller is constructed. The techniques investigated in this part are expected to be applied in wide practical systems.

1.3.2 General Uncertainties In previous section, we discussed the controller design of nonlinear systems with uncertainties that are bounded by high-order polynomial form, such uncertainty bound is reasonable as they can cover a large class of practical nonlinear time-delay systems. However, it is worth to be mentioned that there are also considerable practical nonlinear time-delay systems whose nonlinear uncertainties cannot be described by high-order polynomial form. Due to the wide applications of such nonlinear systems with general uncertainties, more general nonlinear bound conditions than high-order polynomial form are required. In Part II, the adaptive control problem is considered for several classes of nonlinear time-delay systems with uncertainties that are bounded by smooth nonlinear functions. First, for interconnected matching systems, adaptive state feedback control strategy is proposed to render the closed-loop system stable. The obtained controllers are delay-independent, and the result can also be applied to a class of interconnected systems with linear nominal systems. Second, the stabilization problem is investigated for a class of single uncertain mismatched systems with multiple time delays and a memoryless state feedback controller is constructed by choosing appropriate Lyapunov–Krasovskii functional. Then, the result is extended to a class of large-scale systems and the closed-loop system state variables can be guaranteed to converge to a ball with adjustable radius. Finally, the T-S fuzzy approach is applied for nonlinear time-delay systems and different memoryless state feedback controllers are proposed according to whether the parameters of the bound function are known or not. In this part, the nonlinear uncertainties are bounded by known nonlinear functions with possibly unknown parameters, and there are also some other bound conditions for nonlinear uncertainties in the literature that will be further studied in following parts.

1.3.3 Nonlinear Input Due to the physical limitations of the actuator, mechanical design and manufacturing, some input constraints such as dead zone, sector input, input saturation, and hysteresis cannot be avoided in practical systems. The appearance of the input constraints may seriously deteriorate the system performance, and even destabilize the closed-loop system. Hence, the nonlinear dynamical systems with input constraints have attracted lots of attention and many significative research results have been obtained. In Part

8

1 Introduction

III, we mainly aim at the analysis and design of nonlinear time-delay systems with dead-zone and sector input. Dead-zone input is a class of non-smooth nonlinear inputs that characterize certain non-sensitivity of small control inputs, and the dead-zone input is largely encountered in a variety of dynamic systems such as valves and DC servo motors [5]. The existing results related to dead-zone input can be classified into the following: (i) employing the inverse dead-zone nonlinearity to minimize the effects of the dead zone; (ii) simplifying the dead-zone model as a linear term and a disturbance term. How to deal with dead-zone input for nonlinear time-delay systems is important, which is a challenging issue for researchers. The sector input is a typical nonlinear input that satisfies the following condition u T (t)Θ(u(t)) ≥ γu T (t)u(t) where u(t) is the control input, γ is a positive scalar, and Θ(·) is the nonlinear input function. The main research results of sector input include [92, 114, 189, 199]. Specifically, in [92, 189], the synchronization and control problems of chaotic systems with sector input were considered. Reference [199] studied decentralized adaptive control problems of a class of time-varying delay systems with sector input. [114] designed an adaptive sliding mode observer for a class of uncertain nonlinear systems with sector input based on measurable output. Many research results for nonlinear systems with dead-zone input and sector input have been obtained in the literature; however, some certain limitations still exist, such as characteristic parameters of the dead-zone input and sector input are known or partly known and specific system structures are required to be satisfied. In Part III, we first design the adaptive tracking control law for nonlinear time-delay system with non-symmetric dead-zone input based on the simplified model whose precise information is not required. Then, we present the decentralized networked control for large-scale time-delay systems with the sector input by using T-S fuzzy approach.

1.3.4 System with Lower Triangular Structure The development of nonlinear control theory has witnessed unprecedented progress over recent years with substantial insights and trends for control of real-world systems. Triangular structural nonlinear systems are particularly investigated over the past decade. One kind of the triangular structural nonlinear systems are feedforward systems, which are also called upper triangular systems. Many researchers have paid their attention to feedforward systems in the past few years. For instance, [128] gave the first global asymptotic stabilization result for feedforward systems with a homogeneous chain of integrators as an approximation around the origin. Reference [200] proposed a common global state feedback controller design for general feedforward systems that did not require priori knowledge of system nonlinearities.

1.3 Problems Studied in This Book

9

On the other hand, feedback systems, which are also called lower triangular systems, are another important class of triangular structural nonlinear systems. In this book, we mainly investigate the output feedback control of nonlinear time-delay systems with lower triangular structure. The lower triangular system dynamics can be described as ⎧ x˙i (t) = xi+1 (t) + f i (x i (t) , x¯i (t − d)) , ⎪ ⎪ ⎨ i = 1, . . . , n − 1 (1.9) = u (t) + f n (x n (t) , x n (t − d)) , x ˙ ⎪ n ⎪ ⎩ y (t) = x1 (t) where xi , u, y are the state, control input and output of the system, respectively, f i (·) are nonlinear functions and x¯i = [x1 , x2 , . . . , xi ]. It is well known that numerous mathematical models of the practical systems are in triangular form or can be transformed into triangular form under certain conditions. Furthermore, it has been shown that the control of lower triangular systems through backstepping method is always possible. Consequently, the lower triangular systems have been widely studied in the literature. In [61], the authors employed the approximation idea of neural network to construct a state feedback controller and the resulting closed-loop system was stable in the sense of uniform ultimate boundedness (UUB). In [71], the output feedback controller was constructed for a class of time-delay systems with triangular structure via backstepping method. The neural network-based nonlinear decentralized adaptive controllers were proposed for nonlinear time-delay systems in lower triangular form in [16, 94, 105, 129] considered the output feedback stabilization and finite-time stabilization for the stochastic strict feedback systems. In Part IV, some typical control problems of nonlinear time-delay systems are presented. First, a general backstepping design method is presented for lower triangular structural systems. After that, taking the uncertainties of practical systems into consideration, some approximation approaches (e.g., RBF neural network, fuzzy logic systems) are employed in controller design procedure for both single and interconnected dynamic systems. Finally, much more complex conditions originating from practical systems and performance requirement are considered, in particular, the output feedback control problems for uncertain nonlinear systems with unknown control direction and prescribed performance are solved.

1.4 Summary As it is well known that nonlinear systems exist in a wide range of real-world applications, time delay and uncertainties are also inherent and unavoidable in practice; thus, it is important to investigate the stability analysis and control of uncertain nonlinear systems with time delay. In this book, we present some work related to several commonly encountered nonlinear systems, such as the nonlinear systems that can

10

1 Introduction

be represented as high-order polynomial form or general nonlinear form and the systems with nonlinear input or in triangular structure form. The results in this book will undoubtedly advance the study of nonlinear control theory, enrich the content of nonlinear time-delay system theory, and lay the foundation for decentralized control theory in nonlinear large-scale time-delay systems.

Part I

High-Order Polynomial Nonlinear Uncertainties

Chapter 2

Robust Stabilization of Single Nonlinear Time-Delay System

Abstract The problem of robust stabilization for a class of uncertain dynamic systems with multiple delayed state perturbations is considered. It is assumed that perturbations of the time-delay sections are not bounded by first-order linear functions, but bounded by high-order functions with unknown gains. And the time delay considered is time varying. Two classes of controllers are proposed. When the time derivative of each time-varying time delay is less than one, a class of adaptive state feedback controllers are proposed based on Lyapunov–Krasovskii method, which can render the closed-loop systems uniformly ultimately bounded stable. Novel nonlinear feedback controllers are developed by employing Razumikhin lemma, and the controller also can render the closed-loop systems stable in the sense of uniform ultimate boundedness. Finally, several examples are given to show the potential of the proposed techniques.

2.1 Introduction In the control literature, for dynamic systems with delayed state perturbations, the delayed state perturbations are generally supposed to be bounded by first-order linear functions. The existing results can be divided into two classes: The bounds are known and the bounds are unknown. With the bounds known, one can employ the bounds to construct some types of stabilizing state feedback controllers or to develop some stability conditions [21, 31, 54, 55, 95, 110, 132, 141, 175, 205], and the results often come to solving LMI. If the bounds are partially known or not known, some papers appeared. In [23], the authors developed a saturation-type robust adaptive controller for a class of uncertain dynamic systems, where the uncertainty bounds were partially known. In [137], a robust adaptive feedback controller was developed for a class of uncertain linear dynamic delay systems where the bounds of the uncertainty were unknown, but the results needed to examine whether the perturbation of the delayed state can satisfy a bound. In [72], an adaptive feedback controller and a novel nonlinear feedback controller were proposed to solve the control problem of time-delay systems with the bounds of uncertainties completely unknown. However, in practical control problems, the delay perturbations may not be bounded by © Springer Nature Singapore Pte Ltd. 2018 C. Hua et al., Robust Control for Nonlinear Time-Delay Systems, DOI 10.1007/978-981-10-5131-9_2

13

14

2 Robust Stabilization of Single Nonlinear …

first-order function of time delay, but they are bounded by high-order polynomial and the time delay is often time varying. Therefore in this chapter, we will investigate the problem of robust control for a class of dynamic systems with multiple time-varying delays, and the uncertain sections are bounded by high-order polynomial functions with unknown gains. Employing our former idea [72], we propose two classes of controllers in this chapter. When the derivative of each time-varying time delay is less than one, a class of adaptive feedback controllers are proposed, and we prove the closed-loop systems are uniformly ultimately bounded stable based on Lyapunov stability theory and Lyapunov–Krasovskii function. If the time delay is bounded, by employing Razumikhin-type lemma, we construct a class of novel nonlinear feedback controllers, which can also render the corresponding closed-loop systems stable in the sense of uniform ultimate boundedness.

2.2 System Formulation and Preliminaries Consider a class of dynamic systems described by the following differentialdifference equations ⎧ r ⎨ d x(t) = Ax (t) + Bu (t) +  E x t − h (t) , t  (2.1a) j j dt j=1 ⎩ x (t) = ψ (t) , t ∈ [t0 − τ , t0 ] (2.1b) where t ∈ R is the time, x (t) ∈ R n is the state, u (t) ∈ R m is the control input, A and B are the known constant matrices of appropriate dimensions, E j (·) : R n × R → R n , j ∈ {1, 2, . . . , r } , is nonlinear continuous vector function which represents delayed state perturbations for the system. In addition, the time delay h j (t) , j = 1, 2, . . . , r, are assumed to be time varying. The initial condition is given by (2.1b) where ψ (t) is a continuous function on [t0 − τ , t0 ], and  τ := max h j (t) , j = 1, 2, . . . r . For proposing our controllers, we introduce the following standard assumptions which will be used in the following sections. Assumption 2.1 The pair {A, B} given in (2.1a) is completely controllable. Assumption 2.2 There exists the continuous vector function η j (·) : R n × R → R m , j ∈ {1, 2, . . . , r } such that for all (x, t) ∈ R n × R         E j x t − h j (t) , t = Bη j x t − h j (t) , t

(2.2)

and the following inequalities are satisfied s



      i

η j x t − h j (t) , t ≤ βi j x t − h j (t)

i=1

where · denotes the Euclidean norm, βi j is unknown positive constant.

(2.3)

2.2 System Formulation and Preliminaries

15

Remark 2.1 Assumption 2.1 is standard and denotes the internally stability of the nominal system. It is well known that Assumption 2.2 is a matching condition assumption. Different from the assumptions in existing literatures investigating robust control for time-delay systems, we assume that the uncertain section is not bounded by a linear function (for (2.3) s = 1), but bounded by a nonlinear function (2.3). Reference [72] proposed different control strategy to investigate the robust stabilization for system (2.1a)–(2.1b) with uncertainties bounded by a linear function, and the bounds were not known. In this chapter, we will investigate the control problem for a large class of time-delay systems and βi j is also not required to be known. Before giving our main results, the following lemmas are given, which will be needed in this chapter. Lemma 2.1 ([57]) Consider the retarded functional differential equation d x (t) = f (t, xt ) dt

(2.4)

with the initial condition x (t) = ψ (t) , t ∈ [t0 − h, t0 ] Suppose that the functions γi (·) , i = 1, 2, 3,are of K ∞ -class. If there is a continuous function V (·) : [t0 − h, ∞) × R n → R + such that (1) γ1 (x) ≤ V (t, x) ≤ γ2 (x) , t ∈ [t0 − h, ∞), x ∈ R n . (2) There exists a continuous nonincreasing function p (s) > s for s > 0 V˙ (t, x) ≤ −γ3 (x) + ν

(2.5)

V (ξ, x (ξ)) < p ((V (t, x (t)))) , t − h ≤ ξ ≤ t, t ≥ t0

(2.6)

if where if ν > 0 is a constant, the solutions of (2.5) are uniformly ultimately bounded, ∞ and if ν > 0 is a time-varying parameter and satisfies 0 ν (t) dt < ∞, the system is asymptotically stable. Lemma 2.2 For any positive scalars a, b, and c, the following inequality holds ay c − by c+1 ≤

b c

ac b (c + 1)

c+1

where y is a positive variable parameter. Proof Define function f (y) = ay c − by c+1 , then we have d f (y) = acy c−1 − b (c + 1) y c dy

16

2 Robust Stabilization of Single Nonlinear …

and let

d f (y) dy

= 0 (y > 0) , then we can obtain that acy c−1 − b (c + 1) y c = 0

= y ∗ , and we want to prove that f (y ∗ ) is the maximum  2  y=y ∗ < 0 : value, so we need to examine that d f (y) 2 further that y =

ac b(c+1)

dy

  d 2 f (y)  c−2 − b (c + 1) cy c−1  y=y ∗ y=y ∗ = ac (c − 1) y 2 dy  c−2

ac ac = ac (c − 1) − b (c + 1) c b (c + 1) b (c + 1) c−2

ac = −ac b (c + 1) 0 there exists T = T (,δ) > 0 independent of t0 such that e (t, t0 , ψ) ≤  for all t ≥ t0 + T when xt0  < δ.

3.3 Main Results From Assumption 3.3, we further obtain the following inequalities with the help of Assumption 3.4 N       H i j x j , x j t − di j (t) , t  j=1

≤

pi j N  

ij N   s    l   αi js x j + βi jl x j t − di j (t) 

q

j=1 s=1

=

pi j N  

j=1 l=1 ij N   s      l αi js xm j + e j  + βi jl xm j t − di j (t) + e j t − di j (t) 

q

j=1 s=1

≤

pi j N   j=1 s=1

=

N  j=1

j=1 l=1

 s αi js e j  +

qi j N  

  l β i jl e j t − di j (t)  + δi

j=1 l=1

N       T αiTj Ui j e j  + β i j Wi j e j t − di j (t)  + δi

(3.7)

j=1

where αi js , β i jl , and δi are unknown positive scalars, and

T T  αi j = αi j1 , αi j2 , . . . αi j pi j , β i j = β i j1 , β i j2 , . . . β i jqi j ,       p T 2 Ui j (·) = e j  , e j  , . . . e j  i j ,        2 q T Wi j (·) = e j t − di j (t)  , e j t − di j (t)  , . . . e j t − di j (t)  i j Since the states xmi of reference model system are bounded, there always exist positive scalars αi js , β i jl , and δi such that inequality (3.7) holds. Now, we are ready to present our main result in this chapter.

3.3 Main Results

31

Theorem 3.1 For system (3.1), the following adaptive feedback controller u i = u i1 + u i2 + u i3

(3.8)

where u i1 = −Ni xi + Mi ri + K i ei ∂V T (ei ) u i2 = −θi (t) BiT i ∂ei

(3.9)

∂V T (ei )

u i3

i −ϑi (t) BiT ∂e i =  T  T ∂Vi (ei )   Bi ∂ei 

in which θi (t) and ϑi (t) are adaptive parameters with adaptive laws  2 T   ˙θi = 1 Γi  B T ∂Vi (ei )  − 1 Γi σ1i θi i  2 ∂ei  2    T ∂ViT (ei )  1 1  − Φi σ2i ϑi B ϑ˙ i = Φi  2  i ∂ei  2

(3.10)

where Γi , Φi , σ1i and σ2i are positive scalars, and Vi (ei ) =

hi  k  1 T ei Pi ei , h i = max p ji , q ji ( j ∈ [1, N ]) k k=1

(3.11)

K i and Pi are matrices satisfying (3.5) will render the closed-loop error system uniformly ultimately bounded stable. Proof Define the following Lyapunov function candidate  (e, θ, t) = V

N  i=1

N   2 Vi (e, θ, t) = {Vi (ei ) + Γi−1 θi −  θi i=1

+Φi−1 (δi − ϑi )2 +

N  j=1

νi j

t t−di j (t)

  e j (ξ)2k dξ}

(3.12)

where νi j are sufficiently small positive scalars, and  θi are also positive scalars defined in (3.15) (below). In [77], similar Lyapunov function was employed to investigate the control problem of nonlinear systems free of time delay. In this chapter, we will prove the stability of closed-loop system based on the function.  (·) along the trajectories of the closed-loop By taking the time derivative of V system, we have

32

3 Robust Model Reference Adaptive Control …

˙ (e, θ, t) =  V

N 

V˙ i (e, θ, t)

i=1

≤

hi N    T k−1 T   ei Pi ei ei (Ami + Bi K i )T Pi + Pi (Ami + Bi K i ) ei i=1 k=1

⎛

⎞ N      ∂V (e ) i i i j x j , x j t − di j (t) , t ⎠ ⎝ + Bi H ∂e i i=1 j=1 N 

+

N  N 

     2k

2k νi j e j  − 1 − di∗j e j t − di j (t) 

i=1 j=1

 N     ∂Vi (ei ) 2Γi−1 θi −  θi θ˙i − 2Φi−1 (δi − ϑi ) ϑ˙ i + Bi u i3 ∂ei i=1   N   T ∂ViT (ei ) 2  − θi  (3.13)  Bi  ∂e i i=1

+

Also, the following inequality holds N    ∂Vi (ei )    Bi Hi j x j , x j t − di j (t) , t ∂ei j=1

≤

N 

αiTj Ui j

j=1

   T      T ∂Vi (ei )T   + δi  B T ∂Vi (ei )  e j   B i  i    ∂e ∂e i

i

  N      T ∂Vi (ei )T  T     + β i j Wi j e j t − di j (t)  Bi  ∂e i

j=1

  2   N αi j   ∂Vi (ei )T   2   2 T B  + εi j Ui j e j   ≤  4ε  i ∂e j=1

ij

i

⎛   N T   T ∂Vi (ei )  ⎜ + +δi  ⎝  Bi  ∂e

 2     β i j   T ∂Vi (ei )T 2  

Bi  ∗  ∂e i i 1 − d 4ν j=1 ij ij     2

+νi j 1 − di∗j Wi j e j t − di j (t)  

(3.14)

Let ⎛  ⎞  2 αi j 2 βi j   ⎝

⎠ θi = + 4εi j 4 1 − di∗j νi j j=1 N 

(3.15)

3.3 Main Results

33

Then, we have N    ∂Vi (ei )    Bi Hi j x j , x j t − di j (t) , t ∂ei j=1    T  T ∂Vi (ei )T 2    + δi  B T ∂Vi (ei )  B ≤ θi  i  i   ∂ei ∂ei    qi j pi j N    2k     2k  ∗ + 1 − di j νi j e j t − di j (t)  + εi j e j  j=1

k=1

(3.16)

k=1

We know    T ∂ViT (ei )  ∂Vi (ei )   Bi u i3 = −ϑi (t)  Bi ∂ei ∂ei 

(3.17)

Substituting (3.16) and (3.17) into (3.13), we further have ˙ (e, θ, t) =  V

N 

V˙ i (ei , θ, t)

i=1

⎡  qi j hi N N     2k  T k−1  T   ⎣ ei Pi ei −ei Q i ei + νi j e j  ≤ i=1

k=1

 pi j

+

k=1

−

N  i=1

 2k εi j e j 

j=1

 +

N 

k=1

 −1    θi θ˙i − 2Φi−1 (δi − ϑi ) ϑ˙ i 2Γi θi − 

i=1

    N  T ∂ViT (ei )     T ∂ViT (ei ) 2   +   B B θi −  θi  − ϑ (δ (t)) i i    i  i ∂e ∂e i

i=1

i

By applying (3.10), one has ˙ (x, θ, t) ≤  V

N 

 −

i=1

hi   T k−1  T  ei Pi ei ei Q i ei k=1

  θi (−σ1i θi ) + (δi − ϑi (t)) (σ2i ϑi ) + θi −   qi j ⎫ pi j N    2k   2k ⎬ + νi j e j  + εi j e j  ⎭ j=1

k=1

k=1

If we choose parameters   ν j = max νi j , ε j = max εi j , for i ∈ [1, N ] i

i

(3.18)

34

3 Robust Model Reference Adaptive Control …

the following inequality will hold ⎧  qi j ⎫ pi j hi N ⎨  N    2k   2k ⎬  T k−1  T   ei Pi ei ei Q i ei + νi j e j  + εi j e j  − ⎭ ⎩ i=1 k=1 j=1 k=1 k=1 ⎧ ⎛ ⎞⎫ hj hj hi N ⎨  N    2k   2k ⎬  T k−1  T   ⎝ ≤ ei Pi ei ei Q i ei + ν j e j  + ε j e j  ⎠ − ⎭ ⎩ i=1

=

k=1

j=1

k=1

k=1

hi ! N  

"  k−1  T  − eiT Pi ei ei Q i ei + N νi ei 2k + N εi ei 2k

i=1 k=1

    where ν j = 0 qi j < j ≤ h j and ε j = 0 pi j < j ≤ h j . Considering the following equality   θi (−σ1i θi ) + (δi − ϑi ) (σ2i ϑi ) θi −   2  2 1 1 1 1 = −σ1i θi −  θi2 − σ2i δi − ϑi + σ2i δi2 θi + σ1i  2 4 2 4 we further have ˙ (e, θ, t) ≤  V

hi N    −λmin (Pi )k−1 λmin (Q i ) ei 2k i=1 k=1

1 1 +N νi ei 2k + N εi ei 2k + σ2i δi2 + σ1i  θi2 4 4

# (3.19)

For Pi , and Q i are positive matrices, parameters νi j and εi j can be selected to be small enough to render that the following inequality holds − λk−1 min (Pi ) λmin (Q i ) + N νi + N εi = −Πi < 0

(3.20)

where Πi are positive scalars. Furthermore, one has ˙ (e, θ, t) ≤  V

% N $  1 1 −h i Πi ei 2k + σ2i δi2 + σ1i  θi2 4 4 i=1

(3.21)

Based on Lyapunov stability theory, the proposed decentralized state feedback controller (3.8)–(3.10) will guarantee the closed-loop error system is uniformly ultimately bounded stable. Remark 3.2 For the controller u i3 in (3.9), we can choose the following candidate to avoid zero appearing in the denominator

3.3 Main Results

35 ∂V T (e )

i i −ϑi (t) BiT ∂e i   u i3 =  T ∂ViT (ei )  Bi ∂ei  + e−r t

where  and r are positive scalars. With the controller above, it is easy for us to obtain the closed-loop system is uniformly ultimately bounded stable. Remark 3.3 For inequality (3.21), it is easy for us to obtain that the stable bounds of error e can be rendered sufficiently small by reducing the values of parameters σ1i and σ2i . Remark 3.4 In this section, we have investigated the control problem for interconnected time-delay systems with the uncertainties bounded by high-order polynomials. With the gains unknown, we employed adaptive control idea and designed the controllers. Specially, if the uncertainties are bounded by first-order polynomials, that is, pi j = qi j = 1, the model reference adaptive control problem was considered in [27]. In [27], the time-delay interconnections were needed to be precisely known and the controllers designed were dependent of the time delay. In this chapter, the conditions imposed on the interconnected systems are looser and the controllers constructed are independent of the time delay. Therefore, the results obtained in this part are expected to solve the decentralized model reference control problem for a larger class of interconnected time-delay systems.

3.4 Numerical Example In this section, we give a numerical example to verify the validity of the controller designed in previous section. Consider the following interconnected time-delay system: subsystem I:  x˙1 =

−1 1 3 1



x11 x12



⎞ ⎛   0 0 2 + u + ⎝ δ11 x11 + δ12 x22 + δ13 x12 x21 (t − 0.5) + ⎠ 1 2 +δ14 x11 (t − 0.25 (1 + sin (t)))

subsystem II:  x˙2 =

−1 1 −3 3



x21 x22



⎛ ⎞   0 0 2 2 + u + ⎝ δ21 x21 + δ22 x12 + δ24 x12 (t − 1) ⎠ 1 +δ23 x12 x22 (t − 0.5 (1 + cos (t)))

where the parameters δi j are unknown parameters. Based on the theorem proposed in this chapter, we will design the decentralized adaptive feedback controller.

36

3 Robust Model Reference Adaptive Control …

The desired reference models are selected as % $ % $ %$ 0 −1 1 xm11 + x˙m1 = r (t) xm12 1 1 −6 −5 $

x˙m2

−1 1 = −4 −5

%$

% $ % 0 xm21 + r (t) xm22 1 2

r1 (t) = 100 sin (t) , r2 (t) = 100 cos (t) Therefore, we obtain the following controller from Theorem 3.1    2  2 u 1 = − (9x11 + 6x12 ) − 20θ1 (t) e12 + e11 e12 + e12    2  2 2ϑ1 (t) e12 + e11 + e12 e12     +r1 (t) − 10e12 − e12 + e2 + e2 e12  11 12    2  2 u 2 = − (x21 + 8x22 ) − 20θ2 (t) e22 + e21 e22 + e22    2  2 2ϑ2 (t) e22 + e21 + e22 e22     +r2 (t) − 10e22 − e22 + e2 + e2 e22  21

22

and the adaptive laws    2  2  2  2 2 2 e12  − 0.01θ1 , θ˙2 = e22 + e21 e22  − 0.01θ2 + e12 + e22 θ˙1 = e12 + e11        2  2 2 2 e12  − 0.01ϑ1 , ϑ˙ 2 = 2 e22 + e21 e22  − 0.01ϑ2 + e22 ϑ˙ 1 = 2 e12 + e11 + e12

Fig. 3.1 Control result of state x11 (The solid line is response of x11 and the dot line is xm11 )

10

x11

xm11

8 6 4 2 0 −2 −4 −6 −8 −10

0

5

10

15

20

25

30

35

40

3.4 Numerical Example Fig. 3.2 Control result of state x12 (The solid line is response of x12 and the dot line is xm12 )

37 15

x

12

x

m12

10

5

0

−5

−10

−15

Fig. 3.3 Control result of state x21 (The solid line is response of x21 and the dot line is xm21 )

0

5

10

15

20

25

30

35

15

40

x

21

xm21

10

5

0

−5

−10

−15

0

5

10

15

20

25

30

35

40

We choose δi j = 1 and the initial conditions are x11 = 8, x12 = 4, x21 = −4, x22 = −8 xm11 = 1, xm12 = 3, xm21 = −1, xm22 = −2 The simulation results are shown in Figs. 3.1, 3.2, 3.3, and 3.4, from which we can see that the decentralized feedback controller can render the states of the controlled system quickly track the states of model reference system.

38 Fig. 3.4 Control result of state x22 (The solid line is response of x22 and the dot line is xm22 )

3 Robust Model Reference Adaptive Control … ʱÐòͼ: 20

x x

22

m22

15

10

5

0

−5

−10

−15

Fig. 3.5 Control result of the system states

0

5

10

15

20

25

30

35

8

40

x11 x12

6

x x

21 22

4 2 0 −2 −4 −6 −8

0

5

10

15

20

25

30

35

40

Specially, we consider that the signals r1 (t) and r2 (t) are chosen as r1 (t) = r2 (t) = 0. It is easy to get that the reference model is asymptotically stable. With the designed controller, the state responses of the closed-loop system are shown in Fig. 3.5, from which we can see that the closed-loop system is uniformly ultimately bounded stable.

3.5 Conclusion

39

3.5 Conclusion In this chapter, model reference adaptive control problem for a class of large-scale time-delay systems is investigated. The decentralized feedback controllers and corresponding adaptive laws are designed. Based on Lyapunov stability theory, we prove the resulting closed-loop error system is uniformly ultimately bounded stable. A numerical example is given to verify the feasibility and validity of the main results. Different from some existing literatures, in this chapter the uncertain interconnections with time-varying time delay are bounded by high-order nonlinear functions and the gains need not to be known. Therefore, the results obtained are expected to apply to a large class of interconnected systems.

Part II

General Nonlinear Uncertainties

Chapter 4

Decentralized Adaptive Control for Interconnected Time-Delay Systems

Abstract The problem of robust adaptive stabilization is considered for a class of time-varying nonlinear large-scale systems subject to multiple time-varying delays in the interconnections. The interconnections satisfy the match condition and are bounded by nonlinear functions that may contain a high-order polynomial with time delay. Without the knowledge of these bounds, we present adaptive state feedback controllers that are continuous and independent of time delay. Based on the Lyapunov stability theorem, we prove that the controllers can render the closed-loop systems uniformly ultimately bounded stable. In addition, the results are applied to stabilize a class of interconnected systems whose nominal systems are linear. Finally, several examples are given to show the potential of the proposed techniques.

4.1 Introduction The decentralized control schemes, different from the classic centralized information structure, have been considered with significant interests for the control of interconnected systems in recent years. The main objectives of decentralized control are to find some feedback laws for adapting the intersections from the other subsystems where no state information is transferred. The advantage of decentralized control design is to reduce the complexity and therefore allows the control implementation to be more feasible. Furthermore, time delay is frequently encountered in various engineerings and can be a cause of instability [11, 54, 72, 123, 126]. There are instances where delays in the interconnections for many physical systems must be included in the model to account for transmission or information delays. Lee and Radovic [100] and Alimi and Derbel [1] considered the control problem of the class of time-invariant large-scale interconnected systems free of uncertainties subject to multiple constant delays. Hu [65] considered a class of large-scale interconnected systems with delays and assumed that all delays were equal and constant, while the result only applies to system for which the number of inputs and outputs is equal to the number of states [174]. Oucheriah [139] considered the problem of robust stabilization of a class of time-varying large-scale systems subject to multiple time-varying delays in the interconnections. In Chap. 3, we consider that the © Springer Nature Singapore Pte Ltd. 2018 C. Hua et al., Robust Control for Nonlinear Time-Delay Systems, DOI 10.1007/978-981-10-5131-9_4

43

44

4 Decentralized Adaptive Control for Interconnected Time-Delay Systems

uncertain time-delay interconnections are bounded by high-order nonlinear functions and the gains are unknown. However, the time-delay interconnected systems considered above are mainly linear, and the interconnections of the systems are accurately known or bounded by first-order polynomial or high-order polynomial. In this chapter, we will consider a class of time-varying nonlinear large-scale systems subject to multiple time-varying delays in the interconnections. The interconnections satisfy the so-called matching condition, and the uncertainties are bounded by nonlinear functions which are partly known. Adaptive state feedback control strategy is proposed and controllers obtained are independent of time delay. Based on Lyapunov stability theorem, it has been shown that the proposed controllers can render the closed-loop system globally uniformly ultimately bounded stable. The result is also applied to stabilize a class of interconnected time-delay systems whose nominal systems are linear. Finally, several examples are included to illustrate the theoretic results developed in this chapter.

4.2 System Formulation and Preliminaries Consider the nonlinear time-varying composite system S with multiple delays in interconnections defined by N interconnected subsystems Si , i = 1, 2, . . . , N : ·

Si : x i = f i (xi , t) + gi (xi , t) u i +

N 

      Hi j xi , x j , xi t − di j (t) , x j t − di j (t) , t

j=1

(4.1) where xi ∈ R ni and u i ∈ R m i represent the state and control vectors of the subsyswith appropriate tem Si , f i (xi , t)and gi (xi, t) are assumed to be known   functions  dimensions, Hi j xi , x j , xi t − di j (t) , x j t − di j (t) , t is an uncertain nonlinear interconnection, which indicates the interconnections among the current states and the delayed states of system Si and S j , while di j (t) is the bounded time-varying delay and differentiable satisfying 0 ≤ di j (t) ≤ d i j < ∞, d˙i j (t) ≤ di∗j < 1 where d i j , di∗j are positive scalars, and initial condition is given as follows   xi (t) = Ωi (t) , t ∈ t0 − d i j , t0 , i = 1, 2, . . . , N

(4.2)

where Ωi (t) is a continuous initial function. The following assumptions are imposed on system (4.1).       Assumption 4.1 Interconnection Hi j xi , x j , xi t − di j (t) , x j t − di j (t) , t satisfies the so-called matching condition, that is,

4.2 System Formulation and Preliminaries

45

      Hi j xi , x j , xi t − di j (t) , x j t − di j (t) , t       i j xi , x j , xi t − di j (t) , x j t − di j (t) , t = gi (xi , t) H       i j xi , x j , xi t − di j (t) , x j t − di j (t) , t is bounded by where uncertain part H    N           Hi j xi , x j , xi t − di j (t) , x j t − di j (t) , t     j=1  ≤

pi j N  

ij N        αi js Ui js x j + βi jl Wi jl x j t − di j (t)

q

j=1 s=1

=

N  j=1

  αiTj Ui j x j +

j=1 l=1 N 

   βiTj Wi j x j t − di j (t)

(4.3)

j=1

in which functions Ui js (·) and Wi jl (·) are known, pi j and qi j are proper known scalars, αi js and βi jl are unknown scalars, and T  αi j = αi j1 , αi j2 , · · · αi j pi j  T βi j = βi j1 , βi j2 , · · · βi jqi j  T Ui j (·) = Ui j1 (·) , Ui j2 (·) , · · · Ui j pi j (·)  T Wi j (·) = Wi j1 (·) , Wi j2 (·) , · · · Wi jqi j (·)

(4.4)

Remark 4.1 The scalars pi j , qi j and functions Ui js (·) , Wi jl (·) are chosen according i j . In the existing literature investigating decentralized to the structure of functions H control problem of interconnected system, matching conditions were often assumed and many practical systems satisfy this assumption [19, 43, 151]. In the matching parts, condition (4.3) is imposed, in which the interconnections Ui j (·) and Wi j (·) can be nonlinear functions. Assumption 4.2 There exist continuous function ki (xi ) , positive function Vi (xi , t) and functions γi1 , γi2 and γi3 of class κ (zero at zero, positive and increasing) such that for all xi and t, the following inequalities hold γi1 (xi ) ≤ Vi (xi , t) ≤ γi2 (xi ) ∂Vi (xi , t) ∂Vi (xi , t) + ( f i (xi , t) − gi (xi , t) ki (xi )) ≤ −γi3 (xi ) ∂t ∂xi

(4.5)

Remark 4.2 Assumption 4.2 guarantees that the nominal subsystems of system (4.1) are stabilizable using state feedback. If the nominal subsystems are exponentially stabilizable using state feedback, Assumption 4.2 is also satisfied.

46

4 Decentralized Adaptive Control for Interconnected Time-Delay Systems

Before proposing our main results, we set an example of system (4.1) as follows

x˙1 = −x1 + x13 e4| x1 | + u 1 + δ11 x12 + δ12 x1 + δ13 x1 x2 + δ14 x2 x1 e|x1 | x˙2 = −x2 + u 2 + δ21 x12 + δ22 x2 (t − 2) + δ23 x1 x2 + δ24 x2 x1 (t − 0.5)

where δ1 j and δ2 j ( j = 1, 2, 3, 4) are bounded parameters which may be time varying, and the bounds are not known. We will show how to obtain pi j , qi j , and Ui js , Wi jl . For this system, the interconnections satisfy matching condition, and 2 

      1 j xi , x j , xi t − di j (t) , x j t − di j (t) , t H

j=1

= δ11 x12 + δ12 x1 + δ13 x1 x2 + δ14 x2 x1 e|x1 | ≤ |δ11 | x12 + |δ12 | |x1 | + |δ131 | x12 + |δ132 | x22 + |δ141 | x22 + |δ142 | x12 e2|x1 | ≤ α111 |x1 | + α112 x12 + α113 x12 e2|x1 | + α121 x22 where α111 , α112 , α113 , and α121 are unknown positive scalars. From Assumption 4.1, we can see p11 = 3, p12 = 1, q11 = q12 = 0, and select   U11 (·) = |x1 | x12 x12 e2|x1 | , U12 (·) = x22 , W11 (·) = 0, W12 (·) = 0 In the same way, the following bounds can also be obtained as U21 (·) = x12 , U22 (·) = x22 W21 (·) = x12 (t − 0.5) , W22 (·) = |x2 (t − 2)| We can see that the example satisfies Assumption 4.1. Assumption 4.2 also holds since the two nominal subsystems are completely controllable and can be easily stabilized with Lyapunov function V1 = x12 and V2 = x22 . In this chapter, decentralized adaptive feedback controller will be constructed to stabilize this class of interconnected systems.

4.3 Decentralized Feedback Control In this section, we will propose a decentralized state feedback controller which can render the closed-loop system stable in the sense of uniform ultimate boundedness.

4.3 Decentralized Feedback Control

47

Theorem 4.1 For system (4.1) satisfying Assumptions 4.1 and 4.2 if the following inequality ⎧ ⎫ N ⎨ N N     2    2 ⎬ εi j Ui j x j  + νi j Wi j x j  −γi3 (xi ) + (4.6) ⎩ ⎭ i=1

j=1

j=1

≤ −γ (x) is satisfied, where εi j and νi j are positive scalars, γ (·) is a class κ function, then the feedback control law u i = −ki (xi ) − θi gi (xi , t)T

∂Vi (xi , t)T ∂xi

(4.7)

with adaptive law  T  2 ˙θi = 1 Γi gi (xi , t)T ∂Vi (xi , t)  − Γi ηi θi  2  ∂xi

(4.8)

where Vi (xi , t) satisfies Assumption 4.2, Γi and ηi are positive scalars, will render the closed-loop system uniformly ultimately bounded stable. Proof We first define a Lyapunov function candidate for the closed-loop system as follows  (x, θ, t) = V

N 

Vi (x, θ, t)

i=1

=

⎧ N ⎨  i=1

⎩

Vi (xi , t) +

N 

 νi j

j=1

⎫ ⎬   2 + Γi−1 θi −  θi ⎭

t

t−di j (t)

   Wi j x j (ξ) 2 dξ

(4.9)

 θi is defined as follows  θi =

 2 N   αi j  j=1

4εi j

+

N  j=1

 2 βi j    4νi j 1 − di∗j

(4.10)

θi .Then, by taking the derivative of V (·) where adaptive scalar θi is used to estimate  along the trajectories of the closed-loop system, we obtain N N   (x, θ, t) dV ∂Vi (xi , t) · d Vi (x, θ, t)  ∂Vi (xi , t) = ≤ + xi dt dt ∂t ∂xi i=1 i=1

48

4 Decentralized Adaptive Control for Interconnected Time-Delay Systems

+

N         2 2  νi j Wi j x j (t)  − νi j 1 − di∗j Wi j x j t − di j (t)  j=1

+ 2Γi

−1

  dθi θi −  θi dt



Applying (4.3), (4.5) and (4.7), we obtain ⎧ N ⎨   (x, θ, t) dV ∂Vi (xi , t) ∂Vi (xi , t) ≤ + ( f i (xi , t) − gi (xi , t) ki (xi )) ⎩ dt ∂t ∂xi i=1

  ∂Vi (xi , t)T  −θi gi (xi , t)T  ∂xi +

2    dθi  θi  + 2Γi−1 θi −   dt

N  

      2 2  νi j Wi j x j (t)  − νi j 1 − di∗j Wi j x j t − di j (t) 

j=1

 N   ∂Vi (xi , t)T  + gi (xi , t)T  ∂xi j=1

     T  αi j Ui j x j (t) 

⎫   N  ⎬ T     ∂V , t) (x   i i + gi (xi , t)T  βiTj Wi j x j t − di j(t)   ⎭ ∂xi j=1 ⎧  2 N ⎨   ∂Vi (xi , t)T    T ≤ −γ (xi ) − θi gi (xi , t)    ⎩ i3 ∂xi i=1

+

N  

      2 2  νi j Wi j x j (t)  − νi j 1 − di∗j Wi j x j t − di j (t) 

j=1

 N   ∂Vi (xi , t)T  + gi (xi , t)T  ∂xi

     dθi   T θi  αi j Ui j x j (t) + 2Γi−1 θi −   dt j=1 ⎫   N  T  ⎬  T ∂Vi (xi , t)  β T W  x t − d g + , t) (4.11) (x  i i  ij ij j i j(t)   ⎭ ∂xi j=1

Since N  N  T      gi (xi , t)T ∂Vi (xi , t)  αT Ui j x j (t)   ij ∂x i i=1 j=1  N  N  N N  T αi j 2    2   2  T ∂Vi (x i , t)     g εi j Ui j x j (t) + , t) ≤ (x i i  4εi j  ∂xi i=1 j=1 i=1 j=1

4.3 Decentralized Feedback Control

49

and N  N  T       gi (xi , t)T ∂V (xi , t)  β T Wi j x j t − di j (t) ij   ∂xi i=1 j=1    N N  T βi j 2   2 T ∂V (x i , t)    g , t) ≤ (x i i  ∗  ∂xi i=1 j=1 4νi j 1 − di j

+

N N  

    2 νi j 1 − di∗j Wi j x j t − di j (t) 

i=1 j=1

so we further obtain N  (x, θ, t)    dθi dV −γi3 (xi ) + 2Γi−1 θi −  ≤ θi dt dt i=1

+

N  j=1

(4.12)

N     2  2 εi j Ui j x j (t)  + νi j Wi j x j (t)  j=1

 2  T αi j   2 gi (xi , t)T ∂Vi (xi , t)  +  4εi j  ∂xi j=1  2  N T βi j    2 T ∂Vi (x i , t)     gi (xi , t) +  ∗  ∂xi j=1 4νi j 1 − di j   T  2 ∂V , t) (x i i T  −θi  gi (xi , t)  ∂xi N 

By using (4.10), we can get ⎧ N ⎨ N N   (x, θ, t)      2  2 dV ≤ εi j Ui j x j (t)  + νi j Wi j x j (t)  −γi3 (xi ) + ⎩ dt i=1 j=1 j=1  T  2   dθi T ∂Vi (x i , t)  g + 2Γi−1 θi −  θi , t) + θi  (x  i i  dt ∂xi   T  2 T ∂Vi (x i , t)  g −θi  , t) (x  i i  ∂xi

50

4 Decentralized Adaptive Control for Interconnected Time-Delay Systems

Applying (4.6) and (4.8), one obtains   (x, θ, t)   dV ≤ −γ (x) − 2ηi θi −  θi θi dt i=1 N

≤ −γ (x) −

N  

(4.13)

  2ηi θi2 − 2ηi |θi |  θi 

i=1

≤ −γ (x) −

N 

ηi θi2 +

i=1

N 

ηi  θi2

i=1

From (4.10), we know that  θi is bounded, so the closed-loop system is uniformly ultimately bounded stable based on Lyapunov stability theory. Remark 4.3 From (4.13), we know that one can obtain the upper bound on the steadystate as small as desired by decreasing the value of ηi . So the system designers can turn the size of the residual set by adjusting properly parameter ηi . To obtain good transient performance, we should choose the function ki (xi ) properly. In practical systems, the function ki (xi ) should be selected to render function γi3 (xi ) positive enough, so that function γ (x) is sufficiently positive. The good transient performance will be obtained based on (4.13). Remark 4.4 In Theorem 4.1, the key problem is how to get positive function Vi (xi , t) and ki (xi ) to obtain −γi3 (xi ) satisfying inequality (4.6). We should confirm the class of −γi3 (xi ) according to the given functions Ui j (·) and Wi j (·) firstly, then select  proper  Vi (xi , t) and ki (x) to satisfy inequality (4.5). Particularly, if Ui j x j = U j x j and Wi j x j = W j x j for all i ∈ [1, N ], we can select ε j = εi j , ν j = νi j , then if the following inequality −γi3 (xi ) + N εi Ui (xi )2 + N νi Wi (xi )2 < 0 is satisfied, inequality (4.6) will hold.

4.4 Application to Decentralized Control for a Class of Interconnected Systems Let us consider the following class of interconnected systems with time delay Si : x˙i = Ai (t) xi + Bi (t) u i (t) + Bi (t)

N  j=1

    i j x j , x j t − di j (t) , t H

(4.14)

4.4 Application to Decentralized Control for a Class of Interconnected Systems

51

where Ai (t) and Bi (t) are linear time-varying matrices, while the interconnections satisfy the following inequalities      N      i j x j , x j t − di j (t) , t  H     j=1 ≤

pi j N  

ij N   s    l   αi js x j + βi jl x j t − di j (t) 

q

j=1 s=1

=

N  j=1

j=1 l=1

N       αiTj Ui j x j  + βiTj Wi j x j t − di j (t)  j=1

    in which pi j and    qi j are known scalars representing the highest order of x j and x j t − di j (t) , respectively, the parameters αi js and βi jl are unknown scalars similar to those of system (4.1). Based on Theorem 4.1, we will propose decentralized feedback controllers for system (4.14) to render the closed-loop system stable in the sense of uniform ultimate boundedness. Before proposing the adaptive state feedback controller, we introduce the following standard assumption. Assumption 4.3 There exists a positive parameter matrix σi (t) satisfying the following Riccati inequality holds .

P i (t) + Ai (t)T Pi (t) + Pi (t) Ai (t) − Pi (t) Bi (t) σi (t) Bi (t)T Pi ≤ −Q i (t) (4.15) where Pi (t) and Q i (t) are positive matrices satisfying λmin (Pi (t)) > ai and λmin (Q i (t)) > ai ; here, ai is a sufficiently small positive scalar. Corollary 4.1 When system (4.14) satisfies above two inequalities, the following adaptive feedback controller will render the closed-loop system stable in the sense of uniform ultimate boundedness. ∂Vi (xi ) T 1 u i = − σi (t) Bi (t)T Pi (t) xi (t) − θi BiT 2 ∂xi

(4.16)

k h i 1  T where σi (t) and Pi (t) satisfy Assumption 4.3, Vi (xi ) = k=1 xi Pi xi , h i = k   max p ji , q ji ( j ∈ [1, N ]) , and θi is an adaptive parameter whose adaptive law is  2  T ∂V 1 (x ) i i   T θ˙i = Γi  Bi (t)  − Γi ηi θi  2  ∂xi in which Γi and ηi are positive scalars.

(4.17)

52

4 Decentralized Adaptive Control for Interconnected Time-Delay Systems

Proof Based on Theorem 4.1, define Lyapunov function for system (4.14) as follows  (x, θ, t) = V

N 

Vi (x, θ, t)

i=1

=

⎧ hi N ⎨  1 i=1

⎩

k

k=1

xiT Pi xi

 2 θi +Γi−1 θi − 

k

+

qi j N  

 νi j

t−di j (t)

j=1 k=1



t

  x j (ξ)2k dξ

⎛  ⎞  2 αi j 2 βi j   ⎝  ⎠ +  θi = ∗ 4ε i j 4 1 − di j νi j j=1

where

N 

 (·) along the trajectories of the closed-loop Then, by taking the derivative of V system, similar to the proof of Theorem 4.1, we have ˙ (x, θ, t) =  V

N 

·

V i (x, θ, t)

i=1

≤

h N i    i=1

+

xiT Pi xi

2  k−1  T   −xi Q i xi − θi −  θi 2BiT Pi xi 

k=1

qi j N   j=1

ij  2k   2k νi j x j  + εi j x j 

p

k=1

! +2Γi

−1



· θi −  θi θi

k=1

By applying (4.17), we can obtain ˙ (x, θ, t) ≤  V

N 

 −

i=1

+

hi   k=1

N 

qi j 

j=1

k=1

xiT Pi xi

k−1 

 2  xiT Q i xi − ηi θi2 + ηi  θi 

!⎫ pi j  2k   2k ⎬ νi j x j  + εi j x j  ⎭ k=1

If we choose parameters     ν j = max νi j , ε j = max εi j , for i ∈ [1, N ]



4.4 Application to Decentralized Control for a Class of Interconnected Systems

53

the following inequality holds ⎧ !⎫ qi j pi j hi N ⎨  N    2k   2k ⎬  T k−1  T   xi Pi xi xi Q i xi + νi j x j  + εi j x j  − ⎭ ⎩ i=1 k=1 j=1 k=1 k=1 ⎧ ⎛ ⎞⎫ hj hj hi N ⎨  N    2k   2k ⎬  T k−1  T   ⎝ ≤ xi Pi xi xi Q i xi + ν j x j  + ε j x j  ⎠ − ⎭ ⎩ i=1

=

k=1

j=1

k=1

k=1

hi " N  

#  k−1  T  − xiT Pi xi xi Q i xi + N νi xi 2k + N εi xi 2k

i=1 k=1

Then, we have that ˙ (x, θ, t) ≤  V

hi N   

−λmin (Pi )k−1 λmin (Q i ) xi 2k

i=1 k=1 N   2    ηi θi2 − ηi  +N νi xi 2k + N εi xi 2k − θi  i=1

Based on Assumption 4.3, there exist ν j and ε j small enough to render the following inequality satisfied k −λk−1 min (Pi ) λmin (Q i ) + N νi + N εi ≤ −ai + N νi + N εi

= −Πi < 0 where Πi is positive scalar. Further, we can obtain ˙ (x, θ, t) ≤ −  V

N  i=1

h i Πi xi 2k −

N  

 2  ηi θi2 − ηi  θi 

i=1

Based on the Lyapunov stability theorem, the proposed feedback controller (4.16) with adaptive law (4.17) can render the closed-loop system uniformly ultimately bounded stable. In the above, the time-varying interconnected systems are investigated. For timeinvariant case, we have the following corollary: Corollary 4.2 For system (4.14) with Ai and Bi being constant matrix, the adaptive feedback controller ∂V (xi ) T 1 (4.18) u i = − σi BiT Pi xi − θi BiT 2 ∂xi

54

4 Decentralized Adaptive Control for Interconnected Time-Delay Systems

with adaptive law

   ∂V (x ) T 2 1 i   θ˙i = Γi  BiT  − Γi ηi θi   2 ∂xi

(4.19)

will render the closed-loop system uniformly ultimately bounded stable. In (4.18) and (4.19), Γi and ηi are positive scalars, V (xi ) =

hi  1 k=1

k

xiT Pi xi

k

  , h i = max p ji , q ji ( j ∈ [1, N ])

σi and Pi are positive scalar and positive matrix, respectively, satisfying the following inequality (4.20) AiT Pi + Pi Ai − σi Pi Bi BiT Pi = −Q i < 0 We know if (4.20) holds, there always exists scalar ai satisfying λmin (Pi ) > ai and λmin (Q i ) > ai which means Assumption 4.3 is satisfied. The proof is similar to that of Corollary 4.1, so it is omitted here. Remark 4.5 Many authors considered system (4.14) based on Riccati inequalities and LMIs (linear matrix inequalities) with the interconnections known or bounded by a known linear function [65], [125] and [139]. In this chapter, the interconnections may be bounded by high-order polynomial. Furthermore, we adopt adaptive method and do not have to know the bounds.

4.5 Illustrative Examples In this section, we will present two examples to demonstrate the validity of our results. Example 4.1 Consider the following nonlinear interconnected systems with time delay ! $ % $ % · · x 11 0 −x11 + u1 x1 = · (4.21) = 3 2x12 x12 x 12 ⎛ ⎞ 0 2 2 ⎠ |x22 |1/2 δ11 x12 x21 + δ12 x12 +⎝ +δ13 x12 |x11 (t − 0.5 |sin (t)|)|1/2 x21 (t − 0.25 |sin (t)|) ! ! $ % · 0 · x 21 −x21 + u2 x2 = · = 2 x22 1 x 22 % $ 0 + δ21 |x11 |1/2 x21 + δ22 x12 |x22 |1/2 + δ23 x12 (t − 2) |x22 (t − 1)|1/2

4.5 Illustrative Examples

55

where δi j (i = 1, 2; j = 1, 2, 3) are bounded unknown parameters. The interconnections satisfy matching condition, and the bounds are 2 U11 (x1 ) = x12 , W11 (x1 (t − d11 (t))) = |x11 (t − 0.5 |sin (t)|)| T  2 2 U12 (x2 ) = x21 |x22 | , W12 (x1 (t − d12 (t))) = x21 (t − 0.25 |sin (t)|)   2 2 T U21 (x1 ) = |x11 | x12 , W21 (x1 (t − d21 (t))) = x12 (t − 2) T  2 |x22 | , W22 (x2 (t − d22 (t))) = x22 (t − 1) U22 (x2 ) = x21

1 = 2 = 0 2 2 4 2 2 + x12 , V2 (x2 , t) = x21 + x22 , k1 (x1 ) = 3x12 , k2 (x2 ) = when V1 (x1 , t) = x11 2 x22 + 2x22 , inequality (4.5)

∂V1 (x1 , t) ∂V1 (x1 , t) + ( f 1 (x1 , t) − g1 (x1 , t) k1 (x1 )) ∂t ∂x1 2 4 4 2 4 + 4x12 − 6x12 = −2x11 − 2x12 = γ13 (x1 ) = −2x11 ∂V2 (x2 , t) ∂V2 (x2 , t) + ( f 2 (x2 , t) − g1 (x2 , t) k2 (x2 )) ∂t ∂x2 2 3 3 2 4 2 + 2x22 − 2x22 − 4x22 = −4x21 − 4x22 = γ23 (xi ) = −4x21 is satisfied. For inequality (4.6) ⎧ N ⎨ 

=

i=1

⎩

N 



−γi3 (xi ) +

N  j=1

⎫ N   2    2 ⎬ εi j Ui j x j  + νi j Wi j x j  ⎭ j=1

 4   2 2 4 4 2 4 2 −2x11 + ε21 x11 − 4x12 − 4x21 − 4x22 + ε11 x12 + ε12 x21 + x22

i=1

  4   4 2 2 4 4 4 + ε22 x21 + ν11 x11 +x12 + x22 + ν12 x21 + ν21 x12 + ν22 x22

if we select εi j = νi j = 0.1, (i = 1, 2, j = 1, 2) , it is easy to see that (4.6) is satisfied. Based on Theorem 4.1, the feedback controllers are 2 2 − 2θ1 x12 u 1 = −3x12 2 u 2 = −x22 − 2x22 − 2θ2 x22

with the adaptive law  2 T   ˙θ1 = Γ1 g1 (x1 , t) ∂V1 (x1 , t)  − Γ1 η1 θ1 = 2x 4 − 0.01θ1 12   ∂x1

56

4 Decentralized Adaptive Control for Interconnected Time-Delay Systems

Fig. 4.1 The states response curves of system (4.21) with δi j = 1

3

x11 x12 x21 x22

2.5 2 1.5 1 0.5 0 -0.5 -1

0

1

2

3

4

5

6

7

8

9

10

 2  ∂V2T (x2 , t)   − Γ2 η2 θ2 = 2x 2 − 0.01θ2 g , t) θ˙2 = Γ2  (x 22  2 2  ∂x2 For simulation, we give the following initial conditions: θ1 (0) = 1, θ2 (0) = 1    T  T x1 (t) = 3 2 , x2 (t) = 1 −1 , t ∈ t0 − 2, t0 The simulations are done via the Simulink toolbox in MATLAB 6.5. We use the fixed step size 0.01 and ode4 (Runge-Kutta). When the unknown parameters δi j = 1 (i = 1, 2; j = 1, 2, 3), the simulation result is shown in Fig. 4.1. From the figure, we can see that the adaptive controllers render the closed-loop system uniformly ultimately bounded stable. Based on Theorem 4.1, the controllers are obtained without the knowledge of the bounds of the interconnections, which means that the bounds of interconnections can be arbitrary. Further, let us make simulations when the controller and initial conditions are the same, but δi j = 5 and δi j = 10. The state response trajectories are shown in Figs. 4.2 and 4.3, respectively. From the figures, we can see the controllers render the corresponding system stable, which further shows that the proposed controllers are valid and the conclusions are feasible. Example 4.2 Consider the following interconnected time-delay system $ x˙1 =

−1 0 3 1 ⎛

%$

x11 x12

%

$ % 0 + u 1

(4.22) ⎞

0 2 + δ13 x12 x21 (t − 0.5) + ⎠ + ⎝ δ11 x11 + δ12 x22 2 +δ14 x11 (t − 0.25 |sin (t)|) % $ % $ %$ 0 −2 0 x21 + x˙2 = u x22 1 −3 3

4.5 Illustrative Examples

57

Fig. 4.2 The states response curves of system (4.21) with δi j = 5

3

x11 x12 x21 x22

2.5 2 1.5 1 0.5 0 -0.5 -1

Fig. 4.3 The states response curves of system (4.21) with δi j = 10

0

1

2

3

4

5

6

7

8

9

3

10

x11 x12 x21 x22

2.5 2 1.5 1 0.5 0 -0.5 -1

0

1

2

3

4

5

6

7

8

9

10

⎞ 0 2 ⎠ δ21 x21 + δ22 x12 +⎝ 2 +δ23 x12 x22 (t − 0.5 |sin (t)|) + δ24 x12 (t − 1) ⎛

where δi j (i = 1, 2; j = 1, 2, 3, 4) are unknown scalars. The interconnections satisfy matching conditions, and the bounds are   2 2 T , W11 (x1 (t − d11 (t))) = x11 U11 (x1 ) = |x11 | x12 (t − 0.25 |sin (t)|) 2 2 , W12 (x1 (t − d12 (t))) = x21 U12 (x2 ) = x22 (t − 0.5) 2 2 , W21 (x1 (t − d21 (t))) = x12 U21 (x1 ) = x12 (t − 1) U22 (x2 ) = |x21 | , W22 (x2 (t − d22 (t))) = x22 (t − 0.25 |sin (t)|)

For Assumption 4.3, if we select $ Q1 =

% $ % 2 −3 43 , σ1 = 12I, Q 2 = , σ2 = 14I −3 10 38

58

4 Decentralized Adaptive Control for Interconnected Time-Delay Systems

Fig. 4.4 The states response curves of system (4.22) with δi j = 1

3

x11 x12 x21 x22

2.5 2 1.5 1 0.5 0 -0.5 -1

Fig. 4.5 The states response curves of system (4.22) with δi j = 5

0

1

2

3

4

5

6

7

8

10

9

3

x11 x12 x21 x22

2.5 2 1.5 1 0.5 0 -0.5 -1

0

1

2

3

4

5

6

7

8

9

10

then P1 = P2 = I is a solution. So based on Corollary 4.1 or Corollary 4.2, we have the following feedback controller   2  2 + x12 u 1 = −6x12 (t) − 2θ1 x12 + x12 x11   2  2 + x22 u 2 = −7x22 (t) − 2θ2 x22 + x22 x21 with adaptive law .  2 2  2 θ1 = 4 x12 + x12 x11 + x12 − 0.1θ1 .  2   2 2 θ2 = 4 x22 + x22 x21 + x22 − 0.1θ2

The initial conditions are chosen as follows: θ1 (0) = 10, θ2 (0) = −5    T  T x1 (t) = 8 5 , x2 (t) = 3 1 , t ∈ t0 − h, t0 The simulation circumstance is the same of Example 4.1. When the unknown parameters δi j = 1, 5, 10 (i = 1, 2; j = 1, 2, 3, 4), respectively, the simulation results

4.5 Illustrative Examples Fig. 4.6 The states response curves of system (4.22) with δi j = 10

59 3

x11 x12 x21 x22

2.5 2 1.5 1 0.5 0 -0.5 -1

0

1

2

3

4

5

6

7

8

9

10

are shown in Figs. 4.4, 4.5 and 4.6. From the figures, we see that the controllers render the closed-loop systems stable in the sense of uniform ultimate boundedness, which further shows that Corollaries 4.1 and 4.2 are feasible.

4.6 Conclusion In this chapter, we consider the control problem for a class of time-varying nonlinear large-scale systems with time delay in the interconnections, and the interconnections can be nonlinear. An adaptive state feedback controller is proposed which is independent of time delay, and render the closed-loop system uniformly ultimately bounded stable. The result is also applied to control a class of interconnected systems whose nominal system is linear, and corresponding state feedback controller and adaptive law are obtained. Finally, numerical examples are given to demonstrate the validity of the results developed in this chapter. It is shown from the example that the results obtained are effective and feasible. Therefore, our results can be expected to have some applications to practical control problems of uncertain dynamic interconnected systems with time delay.

Chapter 5

Memoryless State Feedback Control for Uncertain Nonlinear Time-Delay System

Abstract The robust control problem for a class of uncertain time-delay systems is investigated. The systems include multiple time delays and uncertain nonlinear functions. According to the input matrix, the system is decomposed into two subsystems. The linear virtual control law is designed for the first subsystem by solving the LMI. Then, by constructing a proper Lyapunov–Krasovskii functional, the memoryless state feedback controller is developed and the closed-loop system is proved to be asymptotically stable. Simulations are presented to verify the effectiveness of the proposed method.

5.1 Introduction For uncertain nonlinear systems, sliding mode control (SMC) method is known to be an efficient way to tackle the robust stabilization problems [32, 156]. Sliding mode control has a number of attractive features such as fast response, good transient performance, order reduction, and invariance to certain external disturbances. From both theoretical research and practical application points of view, SMC method has been extended to dealing with the stabilization problem of uncertain time-delay systems [27, 33, 50, 68, 107, 121, 134, 188]. In [33], the time-delay-independent sliding mode surface was designed based on Razumikhin lemma, and the corresponding controller was constructed. In [134], the robust integral sliding mode surface was constructed for time-delay stochastic system in the form of LMI and then the controller was designed based on the integral sliding mode surface. Although the resulting closed-loop system has good dynamic performance and steady-state performance via SMC, it has a crucial fundamental limitation that the designed controllers have to be memorial. A large controller memory is required to store a large amount of past information, and also the precise delay information must be obtained for controller implementation. However, the controller equipped with a large memory is difficult to implement in practical system, and the precise delay time is difficult for us to obtain, especially when it is time varying. Therefore, it is desirable to construct memoryless controller for time-delay system. By assuming that the delayed state is bounded by a function of current state, [107, 188] constructed the memoryless controller. However, © Springer Nature Singapore Pte Ltd. 2018 C. Hua et al., Robust Control for Nonlinear Time-Delay Systems, DOI 10.1007/978-981-10-5131-9_5

61

62

5 Memoryless State Feedback Control for Uncertain …

it seems restrictive to make assumptions on the state trajectory of the closed-loop system before the controller design. In this chapter, we investigate a class of nonlinear time-delay systems with a mismatched term. Inspired by the sliding mode control idea, we also decompose the system into two subsystems according to the input matrix. The backstepping method is used to construct the state feedback controller for the cascade subsystems. For the first subsystem, the exponentially stable virtual control law is designed. Then, a memoryless state feedback controller is constructed based on the virtual control law. By choosing Lyapunov–Krasovskii functional, we show that the resulting closedloop system is asymptotically stable. Compared with sliding mode controller, the designed backstepping controller is continuous and memoryless, which is easier for implementation. Finally, simulations illustrate the effectiveness of the proposed method.

5.2 System Description Consider the following time-delay system x˙ (t) = Ax (t) +

r 

Adi x (t − di (t))

i=1

+B (u (t) + ΔH (x (t − d1 (t)) , x (t − d2 (t)) , · · · x (t − dr (t)))) ,   x (t) = ϕ (t) , t ∈ −d 0 (5.1) where x ∈ Rn and u ∈ Rm (m ≤ n) are the state and control input of the system, ˙ respectively,   di (t) are the time-varying delays satisfying di (t) ≤ μi < 1 and di (t) ≤ d i , max d i = d, r is a positive integer representing the number of multiple delays. A ∈ Rn×n , B ∈ Rn×m , and Adi ∈ Rn×n are known system matrices. Without loss of generality, we assume Rank(B) = m. Adi x (t − di (t)) is the mismatched part, and ΔH (·) is matched uncertain nonlinear function. For system (5.1), we impose the following assumption. Assumption 5.1 The uncertain function ΔH (·) satisfies ΔH (x (t − d1 (t)) , x (t − d2 (t)) , · · · x (t − dr (t)))2 r    ≤ ψi x (t − di (t))2

(5.2)

i=1

where · denotes the Euclidean norm, and ψi (·) is class-k function (strictly increasing and ψi (0) = 0). Remark 5.1 For system (5.1), if the bound functions ψi (·) are linear, we may use the quadratic Lyapunov function or functional to design the state feedback controller

5.2 System Description

63

[51, 124, 133, 145]. If the functions ψi (·) are nonlinear, following the idea of [27, 33, 50, 68, 107, 121, 134, 188], we may design the switching sliding mode controller (See (5.13) below). But it can be found that the controller must include the delayed state x (t − di (t)) , i ∈ [1, r ] , which is not desirable for practical applications. In this chapter, we propose a new method to design a continuous and memoryless controller for the system (5.1).

5.3 Controller Design For system (5.1), we choose the coordinate transformation z (t) =

z 1 (t) = Γ x (t) , z 2 (t)

0(n−m)×m where the matrix Γ satisfies Γ Γ = I and Γ B = , in which I is the B m×m identity matrix and B is a nonsingular matrix. The transformed system is

T

z˙ 1 (t) = A11 z 1 (t) + A12 z 2 (t) +

r  

 Adi11 z 1 (t − di (t)) + Adi12 z 2 (t − di (t)) ,

i=1

z˙ 2 (t) = A21 z 1 (t) + A22 z 2 (t) +

r  

Adi21 z 1 (t − di (t)) + Adi22 z 2 (t − di (t))



i=1

+Bu (t) + BΔH (z (t − d1 (t)) , z (t − d2 (t)) , · · · z (t − dr (t)))

(5.3)

where ΔH (z (t − d1 (t)) , · · · z (t − dr (t))) = ΔH (x (t − d1 (t)) , · · · x (t − dr (t))) and



A11 A12 A21 A22

= Γ AΓ

−1



Adi11 Adi12 , Adi21 Adi22



= Γ Adi Γ −1 .

For system (5.3), we consider the following state transformation 

y1 (t) = z 1 (t) , y2 (t) = z 2 (t) − K y1 (t)

(5.4)

where K is the virtual control input matrix to be determined below. Then, the new system is obtained as

64

5 Memoryless State Feedback Control for Uncertain …

⎧ r      ⎪ ⎪ ⎪ y ˙ A + A K y Adi11 + Adi12 K y1 (t − di (t)) = + (t) (t) 1 11 12 1 ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ r ⎪  ⎪ ⎪ ⎪ Adi12 y2 (t − di (t)) , ⎨ +A12 y2 (t) + i=1

⎪ y˙2 (t) = A21 y1 (t) + A22 (y2 (t) + K y1 (t)) + Bu (t) ⎪ ⎪ ⎪ r ⎪    ⎪ ⎪ ⎪ + Adi21 y1 (t − di (t)) + Adi22 y2 (t − di (t)) + Adi22 K y1 (t − di (t)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ i=1  +BΔ H (y (t − d1 (t)) , y (t − d2 (t)) , · · · y (t − dr (t))) − K y˙1 (t) (5.5) where  (y (t − d1 (t)) , · · · y (t − dr (t))) = ΔH (z (t − d1 (t)) , · · · z (t − dr (t))) . ΔH Noting that

= = ≤ ≤

  ψi x (t − di (t))2   ψi z (t − di (t))2   ψi y1 (t − di (t))2 + y2 (t − di (t)) + K y1 (t − di (t))2    ψi 1 + 2 K 2 y1 (t − di (t))2 + 2 y2 (t − di (t))2       ψi 2 1 + 2 K 2 y1 (t − di (t))2 + ψi 4 y2 (t − di (t))2

we have   Δ H  (y (t − d1 (t)) , · · · y (t − dr (t)))2 r       ψ1i y1 (t − di (t))2 + ψ2i y2 (t − di (t))2 ≤

(5.6)

i=1

where ψ1i (·) and ψ2i (·) are continuously differentiable class-k functions satisfying       ψ1i y1 (t − di (t))2 ≥ ψi 2 1 + 2 K 2 y1 (t − di (t))2     ψ2i y2 (t − di (t))2 ≥ ψi 4 y2 (t − di (t))2 . In view that ψ1i (0) = ψ2i (0) = 0, there exist functions ψ 1i (·) and ψ 2i (·) such that ψ1i (χ) = χψ 1i (χ) and ψ2i (χ) = χψ 2i (χ). For y1 -subsystem of system (5.5), choose the following Lyapunov functional V1 =

y1T

(t) P y1 (t) + e

−ω1 t

t r  i=1t−d (t) i

  eω1 ς y1T (ς) Q i y1 (ς) + ϑ y2 (ς)2 dς (5.7)

5.3 Controller Design

65

where ω1 and ϑ are positive scalars, P and Q i are positive matrices, then we have the following lemma. Lemma 5.1 If there exist positive matrices H, Ti and matrix W such that the following LMI holds ⎡

Π11 Ad111 H + Ad112 W Ad211 H + Ad212 W ⎢ ∗ − (1 − μ ) e−ω1 d 1 T 0 ⎢ 1 1 ⎢ −ω1 d 2 ⎢ T2 ∗ ∗ − − μ (1 2) e Π =⎢ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ 21 ω1 V1 , using 2ω1−1 ϑ (r + 1) y2 (t)2 instead of V1 gives inequality (5.23). For the whole system (5.5), select the Lyapunov functional (5.24) U = W + V2 It follows that 1 U˙ ≤ − ω1 ρ (V1 ) V1 − ω2 V2 + 2

r  i=1

y1 (t)2 +

r    ψ1i y1 (t)2

(5.25)

i=1

In view that y1T P y1 ≤ V1 and ρ (·) is a positive nondecreasing function, with (5.5) one has 1 (5.26) U˙ ≤ − ω1 ρ (V1 ) V1 − ω2 V2 4 then the closed-loop system is asymptotically stable based on Lyapunov stability theory. The proof is completed. Remark 5.3 Note that matrix P is positive definite and function ψ1i (·) is class-k, then there always exists function ρ (·) such that inequality (5.15) holds. Moreover, the memoryless state feedback controller can always be constructed to render the resulting closed-loop system asymptotically stable. Remark 5.4 It should be pointed out that the gain matrix K of controller (5.14) is dependent on the bounds of time delays and their derivatives, which is perfectly reasonable and understandable, and does not post any problem with physical realization of feedback law. Before the controller design, we should first estimate the bounds of time delays and their derivatives. If the delays are constant, then parameters μi = 0. The upper bounds of time delays can be chosen large enough to include all the possible delays. With the bounds, one may solve LMI (5.8) to obtain K , then further design the controller (5.14). According to above analysis, we may design the controller via the following procedure: (i) Check Assumption 5.1 and obtain function ψi (·); (ii) Choose coordinate transformation Γ to get system (5.3), then confirm functions ψ1i (·) and ψ2i (·); (iii) Solve LMI (5.8) to obtain matrix K and P, then obtain function ρ (·) based on (5.15); (iv) Choose parameters η and ϑ, then design the controller (5.14).

5.4 Simulation Example

71

5.4 Simulation Example In this section, the sliding mode controller and the backstepping controller are designed, and the comparisons between them are done for the following time-delay system ⎡

⎤ ⎡ ⎤ −3 1 0 0.2 0.1 0 x˙ = ⎣ 1 −2 0.2 ⎦ x + ⎣ 0 0.2 1 ⎦ x (t − d1 (t)) 1 1 2 1 4 1 ⎡ ⎤ ⎡ ⎤ 0.1 0.1 0 0 + ⎣ 0 0.2 1 ⎦ x (t − d2 (t)) + ⎣ 0 ⎦ 1 4 2 1 (u + ΔH (x (t − d1 (t)) , x (t − d2 (t))))

(5.27)

Let the time delays and nonlinear function ΔH (·) be d1 (t) = 0.3 (1 + sin t) , d2 (t) = 0.4 (1 + sin t) , ΔH (·) = δ1 x (t − d1 (t))2 + δ2 x (t − d1 (t))3 + δ3 x (t − d2 (t)) where δi ∈ [0, 1] are uncertain parameters. For system (5.27), we can simply take Γ = I. Let parameter ω1 = 1. Solving LMI (5.8) gives   K = −0.0594 −0.2453 , P =



1.1784 −0.0279 . −0.0279 1.0375

Note that  ΔH (·)2 ≤ 3 x (t − d1 (t))4 + x (t − d1 (t))6  + x (t − d2 (t))2

(5.28)

  Then, ψ1 (χ) = 3 χ2 + χ3 and ψ2 (χ) = 3χ. In view that K 2 = 0.0637, we choose ψ11 (χ) = 27χ2 + 81χ3 , ψ12 (χ) = 9χ ψ21 (χ) = 48χ2 + 192χ3 , ψ22 (χ) = 12χ In view that P > I, by (5.15), we select ρ (χ) = 44 + 108χ + 324χ2 . First, let us design the sliding mode controller. Choose the sliding mode surface S = z 2 (t) − K z 1 (t) . Based on Remark 5.2, one may design the switching and

72

5 Memoryless State Feedback Control for Uncertain …

Fig. 5.1 State responses with sliding mode controller (5.29)

3

x1 x2 x3

2

1

0

-1

-2

-3

0

1

2

3

4

5

6

7

8

memorial controller (5.13) as     u (t) = −1.0670 −3.5689 z 1 − 1.0491z 2 + −1.0119 −1.0550 z 1 (t − d1 (t))   − x (t − d1 (t))2 + x (t − d1 (t))3 + x (t − d2 (t)) sign (S (t)) −2sign (S (t)) − 2.2453z 2 (t − d1 (t)) − 2.2453z 2 (t − d2 (t))   + −1.0059 −4.0550 z 1 (t − d2 (t))

(5.29)

Then, based on Theorem 5.1, by letting ω2 = 1, ϑ = 0.5, η = 100 we design the continuous and memoryless controller   u (t) = −1.0047 −3.3116 y1 (t) − 1.0491y2 (t) − y2 (t) − 50y2 (t)   2    y2 (t) −0.75 44 + 108 ∗ 3y22 + 324 ∗ 3y22   2 4 −0.5 12 + 48y2 (t) + 192y2 (t) y2 (t) (5.30)   For simulation, choose the initial value as x (0) = 3 0 −3 and uncertain parameters δi = 1. The state responses are shown in Figs. 5.1 and 5.2 with controller (5.29) and (5.30), respectively. From the responses, we see that the sliding mode controller and backstepping controller are both effective and can render the state response curve converging to zero quickly. We know that the time delays of system often change due to the variational environment, and the delays in the controller could not follow the changes immediately. For example, we consider the case that d1 (t) is changed into 0.1 (1 + cos t) for system (5.27). We use the controller (5.29) with the delays as the same as before, and the state response curve is shown in Fig. 5.3, from which we can find that the closed-loop

5.4 Simulation Example Fig. 5.2 State responses with backstepping controller (5.30)

73 3

x1 x2 x3

2 1 0 -1 -2 -3

Fig. 5.3 State responses with the variable time-delay sliding mode controller (5.29)

7

0

1

2

3

4

5

6

7

8

x 106

6 5 4 3 2 1 0 -1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

system is emanative, which indicates that this controller is not effective for this case. With the memoryless controller (5.30), the response is shown in Fig. 5.4, from which one can see that good dynamic and steady-state performance remains as before.

5.5 Conclusion In this chapter, the stabilization problem is investigated for a class of uncertain systems with multiple time delays. Based on input matrix, the system is decomposed into two subsystems. The linear virtual control law is designed for the first subsystem,

74 Fig. 5.4 State responses with the variable time-delay backstepping controller (5.30)

5 Memoryless State Feedback Control for Uncertain … 3

x1 x2 x3

2

1

0

-1

-2

-3

0

1

2

3

4

5

6

7

8

and the gain matrix can be obtained by solving LMI. Then, the memoryless state feedback controller is constructed by choosing proper Lyapunov–Krasovskii functional. Based on Lyapunov stability theory, it is shown that the closed-loop system is asymptotically stable. Compared with the sliding mode controller, the controller designed via our proposed method is continuous and does not include the delayed state, which is an important and desirable feature for practical applications.

Chapter 6

Exponential Stabilization for Interconnected Time-Delay Systems

Abstract Decentralized exponential stabilization problem is investigated for a class of large-scale systems with time-varying delays. The considered systems have mismatches in time-delay functions. A state coordinate transformation is first employed to change the original system into a cascade system. Then, the virtual linear state feedback controller is developed to stabilize the first subsystem. Based on the virtual controller, a memoryless state feedback controller is constructed for the second subsystem. By choosing new Lyapunov–Krasovskii functional, the designed decentralized continuous adaptive controller makes the solutions of the closed-loop system exponentially convergent to a ball, which can be rendered arbitrary small by adjusting design parameters. Finally, a numerical example is given to show the feasibility and effectiveness of the proposed design techniques.

6.1 Introduction The information transmission among the subsystems often induces appearance of time delay in the interconnected systems. For large-scale systems with linear interconnection, the stability analysis and control problem have been extensively investigated. Reference [76] first introduced time delay into decentralized control of largescale systems and investigated the exponential stabilization problem. The stabilization problem of large-scale stochastic systems with time delay was studied in [191], while stabilization of a class of time-varying large-scale systems subject to multiple time-varying delays in the interconnections was investigated in [139]. Reference [160] proposed variable structure control method for large-scale systems with known linear time-delay interconnection. Based on linear matrix inequality approach, the stability criteria were presented in [113]. For the case that the uncertain interconnections are bounded by linear functions with unknown coefficients, [185] presented the adaptive controller design methodology. Reference [140] further considered the input nonlinearity case and the closed-loop system was shown to be exponentially stable. By analysis on the above-cited literatures, there are the following restrictions: (i) The subsystems should be linearly interconnected; (ii) The systems considered often satisfy the matching condition. Obviously, these conditions will limit the application © Springer Nature Singapore Pte Ltd. 2018 C. Hua et al., Robust Control for Nonlinear Time-Delay Systems, DOI 10.1007/978-981-10-5131-9_6

75

76

6 Exponential Stabilization for Interconnected Time-Delay Systems

of the achievements of the former literatures. With the interconnections bounded by polynomial functions, [69] proposed the decentralized control design method. In this chapter, the above restricted conditions are removed on the interconnected time-delay systems. We consider a class of interconnected time-delay systems with mismatched time-delay functions and general nonlinear interconnections. By changing the subsystem into a cascade system, we successfully dispose of the mismatched function. With the help of proposed novel nonlinear Lyapunov–Krasovskii functional, the uncertain nonlinear time-delay interconnections are well dealt with. The decentralized memoryless state feedback controller is designed such that the solutions of the resulting closed-loop system are uniformly ultimately bounded and exponentially convergent toward a ball with adjustable radius. Finally, numerical simulation is presented to show the potential of the proposed techniques.

6.2 System Formulation and Preliminaries Consider an interconnected system with the ith subsystem described by Si : x˙i (t) = Ai xi (t) + Adi xi (t − τi (t)) + Bi (u i + Hi (t, x1 (t) , x2 (t) , . . . , x N (t) , x1 (t − di1 (t)) , x2 (t − di2 (t)) , . . . , x N (t − di N (t))))

(6.1)

where N is the total number of subsystems in the large-scale system, xi ∈ ni and u i ∈ m i represent the states and control vectors of the ith subsystem, respectively. Ai , Adi ∈ ni ×ni and Bi ∈ ni ×m i are known constant matrices. Without loss of generality, we assume Rank (Bi ) = m i . Hi (·) are uncertain nonlinear interconnections, which indicate the coupling among the current state and the delayed state of subsystem Si and other subsystems. The time delay τi (t) represents the delay time of state xi in xi subsystem, while di j (t) denotes the delay time of state x j (t) in xi (t) subsystem. τi (t) and di j (t) are bounded and differentiable satisfying ·

0 ≤ τi (t) ≤ τ i < ∞, τ i (t) ≤ τi∗ < 1, 0 ≤ di j (t) ≤ d i j < ∞, d˙i j (t) ≤ di∗j < 1

(6.2)

where τ i , τi∗ , d i j and di∗j are known positive scalars. The initial condition of ith subsystem is given as follows xi (t) = Υi (t) , t ∈ [t0 − h i , t0 ] , i = 1, 2, . . . N ,   where h i = max τ i , d 1i , d 2i , . . . , d N i , Υi (t) are continuous functions.

6.2 System Formulation and Preliminaries

77

Assumption 6.1 The nonlinear interconnected functions Hi satisfy Hi  ≤

N  

     T T θi j αi j x j (t) + ϑi j β i j x j t − di j (t) 

(6.3)

j=1

where θi j ∈ ri j and ϑi j ∈ si j are unknown parameter vectors, ri j and si j are

T known positive integers, αi j (·) = αi j1 (·) , αi j2 (·) , . . . , αi jri j (·) , β i j (·) =

T β i j1 (·) , β i j2 (·) , . . . , β i jsi j (·) where αi jl (·) and β i j p (·) are smooth class-κ functions with known structure and there exist functions αi jl (·) and βi j p (·) such that αi jl (χ) = χαi jl (χ) and β i j p (·) = χβi j p (χ) . Remark 6.1 With functions αi jl (χ) = β i jl (χ) = χ (linear interconnection), the robust control problem of system (6.1) were investigated extensively. With the coefficients of the bound functions unknown and Adi = 0, the control problem has been considered in [140, 160, 185]. However, the proposed methods of above-cited literature are not suitable for the nonlinear interconnection case. In this chapter, a methodology is proposed to deal with the decentralized control problem of the system. In view that Rank (Bi ) = m i , there always exists a non-singular matrix Γi ∈ ni ×ni such that   Oi((ni −m i )×m i ) , (6.4) Γi Bi = B i(m i ×m i ) where B i is a non-singular matrix. Choosing the coordinate transformation z i (t) = Γi xi (t) gives ·

z i1 (t) = Ai11 z i1 (t) + Ai12 z i2 (t) + Adi11 z i1 (t − τi (t)) + Adi12 z i2 (t − τi (t)) ·

z i2 (t) = Ai21 z i1 (t) + Ai22 z i2 (t) + Adi21 z i1 (t − τi (t)) + Adi22 z i2 (t − τi (t)) + B i u i (t) + Hi

(6.5)

where z i1 (t) ∈ ni −m i , z i2 (t) ∈ m i and 

Ai11 Ai12 Ai21 Ai22

 = Γi Ai Γi

−1



Adi11 Adi12 , Adi21 Adi22



= Γi Adi Γi−1 .

By choosing the state transformation, we obtain the cascade nonlinear time-delay system (6.5). In the following, we first construct a linear virtual feedback controller to stabilize the z i1 -subsystem, then the decentralized state feedback controller is further designed based on the linear virtual controller.

78

6 Exponential Stabilization for Interconnected Time-Delay Systems

For system (6.5), we choose the following state transformation 

yi1 (t) = z i1 (t) , yi2 (t) = z i2 (t) − K i yi1 (t)

(6.6)

where K i yi1 (t) is the virtual controller to be designed for stabilizing the z i1 subsystem. With (6.6), the new system arises ·

y i1 (t) = (Ai11 + Ai12 K i ) yi1 (t) + (Adi11 + Adi12 K i ) yi1 (t − τi (t)) + Ai12 yi2 (t) + Adi12 yi2 (t − τi (t)) ·

y i2 (t) = Ai21 yi1 (t) + Ai22 yi2 (t) + Adi21 yi1 (t − τi (t)) + Adi22 yi2 (t − τi (t)) + Ai22 K i yi1 (t) + Adi22 K i yi1 (t − τi (t)) ·

− K i y i1 (t) + B i u i (t) + Hi

(6.7)

Our aim of this chapter is to construct a memoryless controller such that the solutions of the closed-loop system exponentially converge to an adjustable bounded region.

6.3 Controller Design First, we show how to determine the virtual control law K i yi1 (t). For yi1 -subsystem, choose the following Lyapunov functional T Vi = yi1 (t) Pi yi1 (t) + Wi

(6.8)

with Wi = +

1 1 − τi∗



t

T e−γi (t−ξ) yi1 (ξ) Q i yi1 (ξ) dξ

t−τi (t) −1 γi τ i  t εi2 e e−γi (t−ξ) 1 − τi∗ t−τi (t)

yi2 (ξ)2 dξ

(6.9)

where Pi and Q i are positive matrices, γi and εi2 are positive scalars. With Lyapunov functional (6.8), we have the following preliminary result: Lemma 6.1 For system (6.7), if there exist positive matrices Mi , L i and matrix Ni such that the following LMI holds for i = 1, 2, . . . , N

6.3 Controller Design

79

 Ψi =

Adi11 Mi + Adi12 Ni Ψi11 −e−γi τ i L i (Adi11 Mi + Adi12 Ni )T

 γi , ψi (yi2 (t)) in which li and   = 2i yi2 (t) Φi ςi−1 i yi2 (t)2 where ςi is a positive scalar satisfying ςi < γi − κi . Proof For yi2 -subsystem, choose the Lyapunov functional as the following form T V i = yi2 (t) yi2 (t) + W i +

2 1  ∗ θi − θi (t) 2li

(6.21)

6.3 Controller Design

where θi∗ = μ−1

81

N   T T θi j θi j + ϑi j ϑi j + μ−1 μ1 + μ2 +

1 2

 γi +

eγi τ i 1−τi∗

with scalars

j=1

μ1 and μ2 defined below, and Wi =

si j  t N   μeγi d i j e−γi (t−ξ) ∗ 1 − d t−d (t) ij ij j=1 l=1   2         × β i jl 2 1 +  K j  Γ j−1   y j1 (ξ)     eγi τ i 2   +β i jl 2 Γ j−1   y j2 (ξ) dξ + 1 − τi∗  t   × e−γi (t−ξ) μ yi1 (ξ)2 + yi2 (ξ)2 dξ

(6.22)

t−τi (t)

T The time derivative of yi2 (t) yi2 (t) along system (6.7) satisfies ·

T T 2yi2 (t) y i2 (t) = 2yi2 (t) [Ai21 yi1 (t) + Ai22 yi2 (t) + u i (t)

+ Ai22 K i yi1 (t) − K i (Ai11 + Ai12 K i ) yi1 (t) − K i Ai12 yi2 (t) + Hi + (Adi22 K i + Adi21 − K i (Adi11 + Adi12 K i )) × yi1 (t − τi (t)) + (Adi22 − K i Adi12 ) yi2 (t − τi (t))]

(6.23)

By direct verification, one has T 2yi2 (t) Hi ≤

N 

 T   2 yi2 (t) θi j αi j x j (t)

j=1

T +ϑi j β i j

≤

   x j t − di j (t) 

N   T  T T μ−1 θi j θi j + ϑi j ϑi j yi2 (t) yi2 (t) j=1

  N  2      2         μ αi j x j (t) + β i j x j t − di j (t)  +

(6.24)

j=1

and T 2yi2 (t) ((Adi22 K i + Adi21 − K i (Adi11 + Adi12 K i )) ×yi1 (t − τi (t)) + (Adi22 − K i Adi12 ) yi2 (t − τi (t))) T ≤ μ−1 μ1 yi2 (t)2 + μyi1 (t − τi (t)) yi1 (t − τi (t)) T + μ2 yi2 (t)2 + yi2 (t − τi (t)) yi2 (t − τi (t))

(6.25)

82

6 Exponential Stabilization for Interconnected Time-Delay Systems

where   μ1 = λmax (Adi22 − K i Adi12 ) (Adi22 − K i Adi12 )T , μ2 = λmax ((Adi22 K i + Adi21 − K i (Adi11 + Adi12 K i ))  × (Adi22 K i + Adi21 − K i (Adi11 + Adi12 K i ))T , here, λmax (X ) denotes the maximum eigenvalue of matrix X. T

In view that αi j = αi j1 , αi j2 , . . . αi jri1 and αi2jl are class-κ functions, one has ri j         αi j x j (t) 2 = αi2jl Γ j−1 z j (t) l=1

              ≤ αi2jl 1 +  K j  Γ j−1   y j1 (t) + Γ j−1   y j2 (t) ri j

l=1 ri j

≤



(6.26)

               αi2jl 2 1 +  K j  Γ j−1   y j1 (t) + αi2jl 2 Γ j−1   y j2 (t)

l=1

Here, we used the inequality αi2jl (a + b) ≤ αi2jl (2a) + αi2jl (2b) for positive a and b. Similarly, we have     2  β i j x j t − di j (t)   ≤

si j  

l=1 2 + β i jl

        2   β i jl 2 1 +  K j  Γ j−1   y j1 t − di j (t) 

       2 Γ j−1   y j2 t − di j (t) 

(6.27)

Substituting (6.26) and (6.27) into (6.24) gives T 2yi2

ri j N          μ αi2jl 2 Γ j−1   y j2 (t) (t) Hi ≤ j=1 l=1

si j N           −1   2     + αi jl 2 1 + K j Γ j  y j1 (t) + μ j=1 l=1

   2         × β i jl 2 1 +  K j  Γ j−1   y j1 t − di j (t)       2   + β i jl 2 Γ j−1   y j2 t − di j (t)  + μ−1

N   j=1

T T T θi j θi j + ϑi j ϑi j yi2 (t) yi2 (t)

(6.28)

6.3 Controller Design

83

The time derivative of W i satisfies ·

Wi ≤

 eγi τ i  T T μyi1 (t) yi1 (t) + yi2 (t) yi2 (t) 1 − τi∗ T −μyi1 (t − τi (t)) yi1 (t − τi (t)) − γi W i T −yi2

(t − τi (t)) yi2 (t − τi (t)) −

si j N  

μ

j=1 l=1

   2         × β i jl 2 1 +  K j  Γ j−1   y j1 t − di j (t)       2   + β i jl 2 Γ j−1   y j2 t − di j (t)  si j N     μeγi d i j  2    −1    y j2 (t) + 2 β Γ  i jl j 1 − di∗j j=1 l=1         2  + β i jl 2 1 +  K j  Γ j−1   y j1 (t)

(6.29)

It is easy to verify that the following equalities hold ri j N  N          μ αi2jl 2 Γ j−1   y j2 (t) i=1 j=1 l=1

         + αi2jl 2 1 +  K j  Γ j−1   y j1 (t)

r ji N  N        = μ α2jil 2 Γi−1  yi2 (t) i=1 j=1 l=1

    + α2jil 2 (1 + K i ) Γi−1  yi1 (t)

(6.30)

and si j N  N     μeγi d i j  2    −1     ∗ β i jl 2 Γ j  y j2 (t) 1 − di j i=1 j=1 l=1        2   + β i jl 2 1 +  K j  Γ j−1   y j1 (t) s ji N  N     μeγ j d ji  2   β jil 2 Γi−1  yi2 (t) ∗ 1 − d ji i=1 j=1 l=1    2  + β jil 2 (1 + K i ) Γi−1  yi1 (t)

=

(6.31)

Substituting (6.19), (6.25), and (6.28) into (6.23) and noting (6.17), (6.20) and (6.29)–(6.31), one obtains

84

6 Exponential Stabilization for Interconnected Time-Delay Systems

 N · N  ·   · 1 ∗ · T θi − θi (t) θi (t) Vi = 2yi2 (t) y i2 (t) + W i − li i=1 i=1 ≤

N     −γi V i + θi∗ − θi (t) yi2 (t)2 i=1

2 1  ∗ · γi  ∗ θi − θi (t) − θi − θi (t) θi (t) + 2li li  T −2yi2 (t) ψi (yi2 (t)) + ϕi (yi1 (t)) N   2 γi  ∗ −γi V i + θi − θi (t) + ϕi (yi1 (t)) = 2l i i=1  ∗  T + σi θi − θi (t) θi (t) − 2yi2 (t) ψi (yi2 (t))

(6.32)

For the whole system (6.7), we choose the Lyapunov functional as the following form   Vi N   Vi + Φi (ζ) dζ (6.33) U= 0

i=1

Then, with (6.11) and (6.32) the time derivative of U satisfies ·

U≤

N     −γi V i + σi θi∗ − θi (t) θi (t) + ϕi (yi1 (t)) i=1

2 γi  ∗ T θi − θi (t) − 2yi2 (t) ψi (yi2 (t)) 2li   + Φi (Vi ) −γi Vi + i yi2 (t)2

+

(6.34)

By direct verification, the following inequalities hold   2 σi  ∗ 2 σi  ∗ θi − θi (t) + θ σi θi∗ − θi (t) θi (t) ≤ − 2 2 i

(6.35)

and   Φi (Vi ) −γi Vi + i yi2 (t)2

  = − (γi − ςi ) Φi (Vi ) Vi + Φi (Vi ) −ςi Vi + i yi2 (t)2

≤ − (γi − ςi ) Φi (Vi ) Vi   + i yi2 (t)2 Φi ςi−1 i yi2 (t)2 By considering that σi li − γi ≥ 0,

 Vi 0

Φi (ζ) dζ ≤ Φi (Vi ) Vi and

 T  T κi Φi yi1 (t) Pi yi1 (t) yi1 (t) Pi yi1 (t) ≤ κi Φi (Vi ) Vi ,

(6.36)

6.3 Controller Design

85

we substitute (6.18), (6.35), and (6.36) into (6.34) and get N   U≤ −γi V i − (γi − ςi − κi ) Φi (Vi ) Vi ·

i=1

 2 σi li − γi  ∗ +δ − θi − θi (t) 2li ≤ −γU + δ

(6.37)

where γ = min {γi − ςi − κi } for i = 1, 2, . . . , N and δ =

N 

σi 2

 ∗ 2 θi . With

i=1

(6.37), it gives U (t) ≤ e−γt U (0) +

δ γ

(6.38)

where U (0) is a positive scalar defined as U (0) =

N  

 V i (0) +

 Φi (ζ) dζ ,

0

i=1

with

 T Vi (0) = yi1 (0) Pi yi1 (0) +

ε−1 eγi τ i + i2 ∗ 1 − τi

Vi (0)



0 −τi (0)

0 −τi (0)

eγi ξ T y (ξ) Q i yi1 (ξ) dξ 1 − τi∗ i1

eγi ξ yi2 (ξ)2 dξ

and 2 eγi τ i 1  ∗ θi − θi (0) + 2li 1 − τi∗   eγi ξ μ yi1 (ξ)2 + yi2 (ξ)2 dξ

T V i (0) = yi2 (0) yi2 (0) +

 ×

0

−τi (0)

si j  0 N   μeγi d i j + eγi ξ ∗ 1 − d −d (0) i j i j j=1 l=1   2         × β i jl 2 1 +  K j  Γ j−1   y j1 (ξ)     2   +β i jl 2 Γ j−1   y j2 (ξ) dξ.

In view that

 Vi 0

Φi (ζ) dζ ≥ Φi (0) Vi , from (6.33) and (6.38) we have

86

6 Exponential Stabilization for Interconnected Time-Delay Systems T yi2 (t)2 + Φi (0) yi1 (t) Pi yi1 (t)

≤

N   i=1

 δ V i + Φi (0) Vi ≤ e−γt U (0) + γ

(6.39)

then   1 δ −γt yi1 (t) ≤ , e U (0) + Φi (0) λmin (Pi ) γ δ yi2 (t)2 ≤ e−γt U (0) + γ 2

(6.40)

where λmin (Pi ) denotes the minimum eigenvalue of matrix Pi . Furthermore, (6.40) gives z i 2 = yi1 (t)2 + yi2 (t) + K i yi1 2   ≤ 1 + 2 K i 2 yi1 (t)2 + 2 yi2 (t)2   δ , ≤ ai U (0) e−γt + γ with ai =

1+2K i 2 Φi (0)λmin (Pi )

+ 2. Considering that xi (t) = Γi−1 z i (t) , one obtains γ

xi (t) ≤ bi e− 2 t U (0) + ci

(6.41)

     1/2 in which bi = Γi−1  (ai U (0))1/2 and ci = Γi−1  aγi δ . Thus the solutions are uniformly ultimately bounded and exponentially convergent to a ball Ωi = { xi | xi  ≤ ci } . Since that ci can be rendered arbitrary small by choosing small parameters σl , the converging region Ωi can be rendered arbitrary small. The proof is completed. Remark 6.2 The key of controller design is how to obtain function Φi (·) . We may determine it via the following approach: With (6.17) one has T i (yi1 (t)) ϕi (yi1 (t)) = yi1 (t) yi1 (t) ϕ

where ϕ i (·) satisfies  2 μeγi τ i K i )2 Γi−1  ∗ + 4μ (1 + 1 − τi  r ji N       ∗ α2jil 2 (1 + K i ) Γi−1  yi1 (t) j=1

l=1

(6.42)

6.3 Controller Design

87

 s ji   −1   μeγ j d ji 2  + β 2 (1 + K i ) Γi  yi1 (t) 1 − d ∗ji jil l=1 ≤ϕ i (yi1 (t)) . Here we use the equalities α jil (χ) = χα jil (χ) and β ji p (χ) = χβ ji p (χ) . In view that T i (yi1 ) yi1 (t) yi1 (t) ϕ   1 T ϕ i (yi1 (t)) ≤ κi yi1 (t) Pi yi1 (t) κi λmin (Pi )

(6.43)

then one may easily choose function Φi (·) satisfying  T  Φi yi1 (t) Pi yi1 (t) ≥

1 κi λmin (Pi )

ϕ i (yi1 (t))

(6.44)

Note that Pi is a positive definite matrix, then there always exists a non-decreasing function Φi such that inequality (6.44) holds.  eγi τ i is in the parameter Remark 6.3 The known parameter μ−1 μ1 + μ2 + 21 γi + 1−τ ∗ i ∗ θi . This operation is used to render the controller more concise. The initial value θi (0) of adaptive parameter can adopt the known term, thus the control performance is the same and controller is more simple. Based on (6.41), one knows that the exponential decay rate is determined by parameter γ. Bigger γi and smaller ςi , κi lead to better transient performance of the closed-loop system. With it, one may choose big γi and small parameters εi1 and εi2 to solve LMI (6.10), and then obtain matrices Pi and K i . We choose small parameters ςi and κi to get functions Φi (·) and ψi (yi2 (t)). In fact, the parameters ςi and κi can be arbitrary small; thus, γ is mainly determined by parameter γi , whose maximum value can be obtained by solving the optimization problem of LMI (6.10). With the matrices and functions, the controller (6.18) can be constructed and the resulting closed-loop system will have good transient performance. Remark 6.4 For dealing with the mismatched function, we transform each subsystem into a cascade nonlinear time-delay system. The linear virtual control law is designed for z i1 -subsystem. For the uncertain nonlinear time-delay interconnections with unknown coefficients, the linear quadratic Lyapunov–Krasovskii functionals of former literatures are not applicable. We propose a novel nonlinear Lyapunov– Krasovskii functional (6.33) to solve the problem. With the help of inequality (6.11) and functional (6.33), the memoryless decentralized adaptive state feedback controller is successfully constructed, which renders that the resulting closed-loop system has good transient and steady-state performances. Remark 6.5 In this chapter, the matched interconnection case is considered. In fact, the proposed method can be applied to mismatched interconnection case, while the

88

6 Exponential Stabilization for Interconnected Time-Delay Systems

mismatched interconnection should be bounded by linear functions with known coefficients. For this case, we may first decompose the interconnections into mismatched part and matched part. Obviously, the mismatched part appears in z i1 -subsystem (6.5). With the mismatched interconnections bounded by linear function, we may design the linear virtual control law to stabilize the z 1 = [z 11 , z 21 , . . . z N 1 ]T subsystem with z 2 = [z 21 , z 22 , . . . z N 2 ]T = 0 under a similar LMI condition (containing the coefficients of linear interconnection bounds). With the proposed method of choosing nonlinear Lyapunov functional, a new functional can be constructed and the memoryless decentralized controller can be designed to ensure the good performances of the closed-loop system.

6.4 Numerical Example In this section, we give a numerical example to show the validity of the controllers designed in previous section. Consider an interconnected system (6.1) composed of two subsystems with ⎡

⎤ ⎡ ⎤ −1 1 0 −0.1 0 0 A1 = ⎣ 0 1 2 ⎦ , Ad1 = ⎣ 0 −0.1 0.1 ⎦ , 1 11 1 2 1 ⎡

⎤ ⎡ ⎤ −0.1 1 1 0.1 0 0 A2 = ⎣ 1 −1 1 ⎦ , Ad2 = ⎣ 0.1 0.1 0.1 ⎦ , 1 1 1 1 2 3

T and B1 = B2 = 0 0 1 , the interconnections are H1 = c11 (t) +

x2l (t − d12 (t)) l=1 c12 (t) x1T (t) x2 (t − d12 (t))

H2 = c21 (t) +

3 

3 

x1l (t − d21 (t)) l=1 c22 (t) x2T (t) x1 (t − d21 (t)) ,

where cil (t) are bounded time-varying parameters with bounds unknown. Now, we employ the proposed method to construct the memoryless controller.

T Consider Bi = 0 0 1 , then Γi = Ii . By solving LMI (6.10) with τ1∗ = τ2∗ = di∗j = 0.5, τ 1 = τ 2 = d i j = 1, εi1 = εi2 = 1 and γi = 2, we have

6.4 Numerical Example

89



   2.5839 1.2708 2.0419 0.2626 , P2 = , 1.2708 2.6715 0.2626 1.5366



 K 1 = −2.9321 −6.2822 , K 2 = −3.5329 −2.2376 . P1 =

With Assumption 6.1, one knows α11 (x1 (t)) = x1 (t)2 , α22 (x2 (t)) = x2 (t)2 , T

β 12 = x2 (t − d12 (t)) x2 (t − d12 (t))2 , T

β 21 = x1 (t − d21 (t)) x1 (t − d21 (t))2 , and α12 (x2 (t)) = α21 (x1 (t)) = β 11 (·) = β 22 (·) = 0. By letting  μ = 0.0001, we choose function ϕi (yi1 (t)) = 0.01 yi1 (t)2 + yi1 (t)4 from (6.17). Based on (6.18) with κi = 0.5, in view that Pi > Ii , we select Φi (χ) = 0.02 (χ + 1) . −1 With i = εi1 +

−1 γi τ i εi2 e 1−τi∗

= 15.7781 and ςi = 1, we choose ψi (yi2 (t)) =

0.157781yi2 + 2.489484 yi2 2 yi2 . Based on Theorem 6.1, we construct the decentralized memoryless controller

 u 1 (t) = 41.7050 75.0009 y11 (t) − 13.5645y12 (t) 1 − θ1 (t) y12 (t) − y12 (t) − 3 y12 (t)2 y12 (t) 2

 u 2 (t) = 21.0348 12.8538 y21 (t) − 6.7704y22 (t) 1 − θ2 (t) y22 (t) − y22 (t) − 3 y22 (t)2 y22 (t) 2

(6.45)

(6.46)

with the adaptive law ·

θ1 (t) = 50 y12 (t)2 − 2θ1 (t) , ·

θ2 (t) = 50 y22 (t)2 − 2θ2 (t) . The time delays of systems are chosen as τ1 (t) = τ2 (t) = di j (t) = 0.5 (1 + sin t)

T for ξ ∈ [−1, 0] , and and the initial values are x1 (ξ) = x2 (ξ) = −1 0 1 θ1 (0) = θ2 (0) = 10. The simulation results are shown in Figs. 6.1 and 6.2 with cil (t) = 1 + sin t. From the two figures, we can see that the proposed method is effective and can stabilize the closed-loop system quickly.

90

6 Exponential Stabilization for Interconnected Time-Delay Systems 1

0.5

0

−0.5

−1

−1.5

0

1

2

3

4

5

6

7

8

9

10

6

7

8

9

10

Fig. 6.1 The response curve of the first subsystem 1

0.5

0

−0.5

−1

−1.5

−2

−2.5

0

1

2

3

4

5

Fig. 6.2 The response curve of the second subsystem

6.5 Conclusion It is a challenging problem for designing decentralized controller for interconnected time-delay systems with mismatched function and uncertain nonlinear interconnections. A new methodology is proposed to deal with the controller design problem

6.5 Conclusion

91

in this chapter. With the developed new nonlinear Lyapunov–Krasovskii functional, we show that the designed controller renders that the closed-loop system has good transient state performance (exponentially converge) and good steady-state performance (arbitrary small converging region). Finally, a numerical example is given to verify the feasibility and validity of the main results.

Chapter 7

Robust Adaptive Control for Time-Delay System via T-S Fuzzy Approach

Abstract The robust control problem is investigated for a class of uncertain nonlinear time-delay systems. Via the Takagi–Sugeno (T-S) fuzzification, the T-S fuzzy systems are obtained with each local model in the form of time-delay systems with uncertain nonlinear functions. The mismatched nonlinear functions satisfy the Lipschitz condition, while the matched parts are bounded by nonlinear functions with unknown coefficients. Based on the input matrix, the system is decomposed into two cascade subsystems. The virtual controller is designed for the first subsystem, and then, a memoryless adaptive controller is presented. By employing a new Lyapunov–Krasovskii functional, we show that the resulting closed-loop system is exponentially stable and the solutions are uniformly ultimately bounded. Finally, simulation examples are given to show the effectiveness of the proposed methods.

7.1 Introduction During the last two decades, fuzzy logic technique has been widely used for nonlinear system modeling, especially for systems with incomplete plant information [36, 211, 212]. Fuzzy logic systems serve well as universal approximators [96, 146]. The wellknown Takagi–Sugeno fuzzy model is a popular and convenient tool in functional approximations [166]. This model is based on using a set of fuzzy rules to describe a global nonlinear system in terms of a set of local linear models which are smoothly connected by fuzzy membership functions. A great number of theoretical results on function approximation, stability analysis, and controller synthesis have been developed for T-S fuzzy models during the last decade or so [36]. The T-S modelbased fuzzy control method appears to be an effective tool for controller design of complex nonlinear systems. In [3, 6, 12–14, 35, 37, 41, 52, 112, 172, 182], the authors employed T-S fuzzy time-delay systems to approximate nonlinear time-delay system. In [6, 12–14, 112, 182], the stability analysis was conducted for T-S fuzzy time-delay systems. H∞ control problem and guaranteed cost control problem were considered in [3, 37, 41, 52]. The filtering problem was investigated for T-S fuzzy time-delay systems in [4, 12]. The controller design method for large-scale T-S fuzzy system is reported in [172]. In the above literatures cited on T-S fuzzy system, the local model is required to be in the form of linear time-delay systems. © Springer Nature Singapore Pte Ltd. 2018 C. Hua et al., Robust Control for Nonlinear Time-Delay Systems, DOI 10.1007/978-981-10-5131-9_7

93

94

7 Robust Adaptive Control for Time-Delay System via T-S Fuzzy Approach

However, the local mode under fuzzification may contain nonlinear functions. The reasons are as follows. (i) The uncertainties are difficult for us to obtain for practical industrial plants, which often appear as nonlinear functions. (ii) Using few fuzzy rules for practical nonlinear systems often induces the existence of nonlinear functions in the local model. (iii) To render the approximation error smaller between the fuzzy model and practical system model, we should employ some nonlinear functions in the local model. The control problem of this class of systems is more difficult than that of the linear form. Via the variable structure control method, [216, 217] considered the uncertain T-S fuzzy system free of time delay. In [49, 111], the T-S fuzzy time-delay systems were investigated and the memorial switching controllers were successfully constructed. Since the discontinuous control schemes not only induce the problems of the existence and uniqueness of solutions, but also may cause the chattering phenomena and excite the high-frequency phenomena, we should try to employ the smooth controller in practical systems if possible. On the other hand, memorial controller needs a large controller memory to store a large amount of past information, and also the precise delay information must be available for controller implementation. In practical systems, a controller equipped with a large memory is costly, and the precise delay time is difficult to obtain, especially when it is of time varying. In view of these observations, we aim to design the continuous and memoryless state feedback controllers for nonlinear time-delay systems in this chapter. We use a set of fuzzy rules to describe a global nonlinear system into a set of local time-delay systems with uncertain nonlinear functions. The uncertainties are bounded by nonlinear functions with unknown coefficients. First, based on the control input matrix, we decompose the system into two subsystems. The linear virtual control law is designed such that the first subsystem is exponentially stable. With the virtual control law, a memoryless adaptive state feedback controller is constructed. By choosing a Lyapunov–Krasovskii functional, we show that the solutions of the resulting closed-loop system exponentially converge to a region with an adjustable bound. The controllers constructed in this chapter are memoryless, which are easier for implementation than the memory controller presented in the literature [49, 111]. Finally, simulations are carried out to show the effectiveness of the theoretic results obtained.

7.2 System Formulation and Assumptions Consider the following nonlinear time-delay system ·

x (t) = F (x (t) , x (t − d (t))) + ΔF (x (t) , x (t − d (t))) + G (x (t)) u,   x (t) = ϕ (t) , t ∈ −d, 0

(7.1)

7.2 System Formulation and Assumptions

95

where x (t) ∈ n and u (t) ∈ m are the state and control input of the system, respectively, d (t) is the delay time of the system state with d (t) ≤ d, F (·), G (·) are two continuous nonlinear functions, and ΔF (·) is an uncertain nonlinear function. A fuzzy dynamic model was developed by the pioneering work [166] to represent local input/output relations of nonlinear systems. This dynamical model is described by IF-THEN rules, which will be employed to handle the control design problem of the nonlinear time-delay system (7.1). The ith rule of this fuzzy model is shown below: Plant Rule i : IF θ1 is μi1 and · · · and θ p is μi p , THEN ·

x (t) = Ai x (t) + Adi x (t − d (t)) + Bi u (t) + Δf i (x (t) , x (t − d (t)))

(7.2)

where Ai ∈ n×n , Adi ∈ n×n , and Bi ∈ n×m are constant real matrices. θ j (x) ( j = 1, . . . , p) are the premise variables, which are the functions of state variables, μi j (i = 1, . . . , r, j = 1, . . . , p) are the fuzzy sets, here r is the number of IF-THEN rules and p is the number of the premise variables. It is assumed that the premise variables are independent of the input variable. Then, the overall fuzzy model is achieved by fuzzy blending (aggregation) of each individual rule (model) as follows: ·

x (t) =

r 

h i (θ) (Ai x (t) + Adi x (t − d (t))

i=1

+Bi u (t) + Δf i (x (t) , x (t − d (t))))

(7.3)

r  where h i (θ) = μi (θ) / μi (θ) . i=1

Some assumptions as made on system (7.3) as follows. Assumption 7.1 Matrix Bi = B, i = 1, 2, . . . , r, and Rank (B) = m. Without loss of generality, we assume   O(n−m)×m . B= Im×m Assumption 7.2 With the nonlinear function Δf i (x (t) , x (t − d (t))) expressed in the following form, 

δ (x (t) , x (t − d (t))) Δf i (x (t) , x (t − d (t))) = i vi (x (t) , x (t − d (t)))

 (7.4)

where δi (x (t) , x (t − d (t))) ∈ (n−m)×1 and vi (x (t) , x (t − d (t))) ∈ m×1 are matched function and mismatched function, respectively, there hold

96

7 Robust Adaptive Control for Time-Delay System via T-S Fuzzy Approach

δi (x (t) , x (t − d (t)))2 ≤ αi1 x (t)2 + αi2 x (t − d (t))2

(7.5)

and vi (x (t) , x (t − d (t))) ≤ i∗T ρi (x (t)) + ϑi∗T σi (x (t − d (t)))

(7.6)

where αi1 and αi2 are known positive scalars, and    ∗  ∗ ∗ ∗ ∗ T ∗ T , i2 , . . . , iq , ϑi∗ = ϑi1 , ϑi2 , . . . , ϑis i∗ = i1  T ρi (·) = ρi1 (·) , ρi2 (·) , . . . , ρiq (·) , σi (·) = [σi1 (·) , σi2 (·) , . . . , σis (·)]T , in which i∗j and ϑil∗ are unknown positive scalars, functions ρi j (·) and σil (·) are known smooth class-k functions (strictly increasing and ρi j (0) = σil (0) = 0). ·

Assumption 7.3 The time delay d (t) satisfies d (t) ≤ d < 1. Remark 7.1 In Assumption 7.1, we assume that the Bi = B. This assumption can also be seen in [111, 216]. If G (x (t)) = B in (7.1), always holds.  this assumption  O(n−m)×m Without loss of generality, it is assumed that B = . If the system matrix Im×m B is not the case, we may choose nonsingular coordinate transformation such that transformed B is in the standard form. Based on the rank m of matrix B, we make the matrix decomposition (7.4). For the general case that Rank(B) = r < m, one can choosenonsingular coordinate transformations such that B (the changed B) as  O(n−r )×r . Then, the uncertain nonlinear function Δf i is accordingly changed and Ir ×r the matrix decomposition (7.4) can be done based on r in a similar way. Remark 7.2 Assumption 7.2 is imposed on system (7.3), in which the mismatched part is bounded by a linear function (7.5) and the matched part is bounded by nonlinear functions with unknown coefficients (7.6). The stability analysis and stabilization methods for T-S fuzzy methods proposed in [3, 4, 12–14, 35, 37, 41, 52, 112, 182] assume linear time delays, and thus not suitable for dealing with the nonlinear case. Assumption 7.3 is a general condition, which is often needed for constructing the Lyapunov–Krasovskii functional for time-varying delay systems. In this chapter, we will propose a new method to construct a continuous and memoryless controller such that the closed-loop system is exponentially stable. Based on the matrix B, we decompose system (7.3) into the following form

7.2 System Formulation and Assumptions ·

x 1 (t) =

r 

97

h i (θ) (Ai11 x1 (t) + Ai12 x2 (t)

i=1

+ Adi11 x1 (t − d (t)) + Adi12 x2 (t − d (t)) + δi (x (t) , x (t − d (t)))) , ·

x 2 (t) =

r 

h i (θ) (Ai21 x1 (t) + Ai22 x2 (t)

i=1

+ Adi21 x1 (t − d (t)) + Adi22 x2 (t − d (t)) + u (t) + vi (x (t) , x (t − d (t))))

(7.7)

For system (7.7), we choose the following coordinate transformation 

z 1 (t) = x1 (t) , z 2 (t) = x2 (t) − K x1 (t)

(7.8)

where matrix K x1 (t) is the virtual control input to be designed for stabilizing z 1 subsystem. With the transformation, we have the following new system, ·

z 1 (t) =

r 

h i (θ) ((Ai11 + Ai12 K ) z 1 (t)

i=1

+ (Adi11 + Adi12 K ) z 1 (t − d (t)) + Ai12 z 2 (t) + Adi12 z 2 (t − d (t)) + δ i (z (t) , z (t − d (t))) , ·

z 2 (t) =

r 

h i (θ) ((Ai21 + Ai22 K ) z 1 (t) + Ai22 z 2 (t)

i=1

+ (Adi21 + Adi22 K ) z 1 (t − d (t)) + Adi22 z 2 (t − d (t)) ·

+u (t) + v i (z (t) , z (t − d (t)))) − K x 1 (t)

(7.9)

where δ i (z (t) , z (t − d (t))) = δi (x (t) , x (t − d (t))) and v i (z (t) , z (t − d (t))) = vi (x (t) , x (t − d (t))) . If we design a controller such that system (7.9) is exponentially stable, then the former system (7.3) is also exponentially stable from the linear coordinate transformation (7.8). In the next section, we will consider system (7.9) directly instead of system (7.3) and construct the memoryless controller.

98

7 Robust Adaptive Control for Time-Delay System via T-S Fuzzy Approach

7.3 Virtual Control Design In this section, we will show how to choose the virtual control input K x1 (t) to stabilize the first subsystem. For z 1 -subsystem of (7.9), choose the following Lyapunov functional V = z 1T (t) Pz 1 (t) + VC

(7.10)

with

VC =

t

 e−γ(t−ξ)

t−d(t)

+

−1 γd e υ z 2 (ξ)2 dξ z 1T (ξ) Qz 1 (ξ) + 1 − d

r

 eγd  t 2 2 e−γ(t−ξ) i−1 αi2 z 2 (ξ)2 + ε−1 2 αi2 z 2 (ξ) dξ  1 − d i=1 t−d(t)

where P and Q are positive matrices, i , υ, and ε2 are positive scalars. Then, we have the following result: Lemma 7.1 If there exist positive matrices M, L, matrix N and positive scalars i such that the following LMIs hold for i = 1, 2, . . . , r ⎤ Ψi11 Adi11 M + Adi12 N Ψi13

 Ψi = ⎣ ∗ − 1 − d e−γd L Ψi23 ⎦ < 0 ∗ ∗ Ψi33 ⎡

(7.11)

where γ, υ, ε1 and ε2 are given small parameters, ∗ represents the transpose of the T T + N T Ai12 + L + i I + corresponding matrices, Ψi11 = Ai11 M + Ai12 N + M Ai11 T , and γ M + υ Ai12 Ai12   Ψi13 = M N T N T 0 0 0 ,   Ψi23 = 0 0 0 N T M N T ,  −1 −1 Ψi33 = diag − i αi1 I − i αi1 I − i ε−1 1 I  −1 −1 −1 − i αi2 I − i αi2 I − i ε2 I , then it follows under the virtual input matrix K = N M −1 that ·

V ≤ −γV + ς z 2 (t)2 where ς is a positive scalar satisfying

(7.12)

7.3 Virtual Control Design

ς≥

r 

99



i−1 αi1

+

2

i−1 ε−1 1 αi1

i=1

+ υ −1 +

 γd 2

i−1 αi2 + ε−1 2 αi2 e + 1 − d

υ −1 eγd 1 − d

(7.13)

Proof The time derivative of V satisfies ·

V ≤

r 

h i (θ) 2z 1T (t) P ((Ai11 + Ai12 K ) z 1 (t)

i=1

+ (Adi11 + Adi12 K ) z 1 (t − d (t)) + Ai12 z 2 (t) +Adi12 z 2 (t − d (t)) + δ i (z (t) , z (t − d (t))) υ −1 eγd T z 2 (t) z 2 (t) + z 1T (t) Qz 1 (t) + 1 − d

 − 1 − d e−γd z 1T (t − d (t)) Qz 1 (t − d (t)) − υ −1 z 2T (t − d (t)) z 2 (t − d (t)) − γVC r  eγd  −1 2 2 +

i αi2 z 2 (t)2 + ε−1 2 αi2 z 2 (t)  1 − d i=1 −

r 

 2 2

i−1 αi2 z 2 (t − d (t))2 + ε−1 2 αi2 z 2 (t − d (t))

(7.14)

i=1

By direct verification, the following inequalities hold r  h i (θ) 2z 1T (t) P (Ai12 z 2 (t) + Adi12 z 2 (t − d (t))) i=1

≤

r 

T h i (θ) υz 1T (t) P Ai12 Ai12 Pz 1 (t) +

i=1 −1 T T υ z 2 (t) z 2 (t) + υz 1T (t) P Adi12 Adi12 Pz 1  −1 T + υ z 2 (t − d (t)) z 2 (t − d (t))

(t) (7.15)

and 2z 1T (t) P

r 

h i (θ) δ i (z (t) , z (t − d (t)))

i=1

≤

r 

h i (θ) i z 1T (t) P Pz 1 (t)

i=1

 2    δ i (z (t) , z (t − d (t)))

+ i−1

(7.16)

100

7 Robust Adaptive Control for Time-Delay System via T-S Fuzzy Approach

where  2   δ i (z (t) , z (t − d (t)))

 ≤ αi1 z 1 (t)2 + z 2 (t) + K z 1 (t)2

 + αi2 z 1 (t − d (t))2 + z 2 (t − d (t)) + K z 1 (t − d (t))2

 ≤ αi1 z 1 (t)2 + K z 1 (t)2

 + αi2 z 1 (t − d (t))2 + K z 1 (t − d (t))2 2 2 T T + αi1 z 2 (t)2 + ε−1 1 αi1 z 2 (t) + ε1 z 1 (t) K K z 1 (t) 2 2 + αi2 z 2 (t − d (t))2 + ε−1 2 αi2 z 2 (t − d (t))

+ ε2 z 1T (t − d (t)) K T K z 1 (t − d (t))

(7.17)

Substituting (7.15)–(7.17) into (7.14) gives ·

V ≤

r  i=1

 h i (θ) 

z 1 (t) z 1 (t − d (t))

T

 Φi

z 1 (t) z 1 (t − d (t))



r υ −1 eγd  −1 2

i αi1 + i−1 ε−1 + 1 αi1 1 − d i=1

 γd 2 e α

i−1 αi2 + ε−1 2 i2 z 2 (t)2 + 1 − d

− γV + υ −1 +

where

⎡ Φi = ⎣

Φi11 ∗

(7.18)

⎤ P (Adi11 + Adi12 K )  −γd − 1 − d e Q + i−1 αi2 I ⎦ , + i−1 αi2 K T K + i−1 ε2 K T K

with Φi11 = P (Ai11 + Ai12 K ) + (Ai11 + Ai12 K )T P + Q + i P P + i−1 αi1 I + T P.

i−1 αi1 K T K + γ P + i−1 ε1 K T K + υ P Ai12 Ai12 −1 −1 P P on both sides of Φi and letting M = P −1 , Multiplying diag N = K P −1 , L = P −1 Q P −1 , one sees that that Φi < 0 is equivalent to Ψi < 0. Then, from (7.13) and (7.18), inequality (7.12) holds. Thus, the proof is completed. ·

Remark 7.3 Letting z 2 (t) = 0 in inequality (7.12), we have V ≤ −γV , thus z 1 subsystem is exponentially stable. If we use the sliding mode control methodology to construct the controller for system (7.7), the sliding mode surface can be chosen as S (t) = x2 (t) − K x1 (t) . On the sliding mode surface S (t) = 0, the system is exponentially stable. Based on the sliding mode surface, we may easily design the state feedback controller via the similar approach to that of [111]. But the sliding mode controller must be discontinuous and memorial, which needs the precise delayed state for implementation. In the sequel, we aim to design the continuous and memoryless state feedback controller.

7.4 Controller Design

101

7.4 Controller Design Based on the virtual control input above, we will construct a memoryless state feedback controller in this section such that the closed-loop system has the good transient and steady-state performances. Theorem 7.1 For system (7.7), if LMI (7.11) holds, the solution of the closed-loop system is uniformly ultimately bounded and exponentially converges to the bounded region by the following memoryless controller 1 1 u (t) = − γz 2 (t) − ζ (t) z 2 (t) − z 2 (t) Θ (z 2 (t)) 2 2  ς 2ς z 2 (t)2 z 2 (t) − Φ 2 γ with the adaptive law

·

ζ (t) = l z 2 (t)2 − μlζ (t)

(7.19)

(7.20)

where l and μ are positive scalars, Φ (·) is a class-k function and Θ (·) is a continuous function, which are chosen such that Θ (z 2 (t)) ≥ 2

ϕ (z 1 (t)) ≤

r r −γd     ρi (2 z 2 (t))2 + 2e σ i (2 z 2 (t))2 ,  1 − d i=1 i=1

  γ T z 1 (t) Pz 1 (t) Φ z 1T (t) Pz 1 (t) 4

(7.21)

in which functions σ i (·) and ρi (·) are obtained from σi (χ) = χσ i (χ) and ρi (χ) = χρi (χ) , and ϕ (z 1 (t)) is defined in (7.33). Proof For z 2 -subsystem, we choose the Lyapunov–Krasovskii functional as W = z 2T (t) z 2 (t) +

r

eγd  t e−γ(t−ξ) Ωi (ξ) dξ 1 − d i=1 t−d(t)

with Ωi (ξ) = ν (Adi21 + Adi22 K ) z 1 (ξ)2 + Adi22 z 2 (ξ)2 + νσi2 (2 (1 + K ) z 1 (ξ)) + νσi2 (2 z 2 (ξ)) + ν (Adi11 + K Adi12 ) z 1 (ξ)2 + ν Adi12 z 2 (ξ)2

(7.22)

102

7 Robust Adaptive Control for Time-Delay System via T-S Fuzzy Approach

 + ναi2 z 1 (ξ)2 + K z 1 (ξ)2 + ναi2 z 2 (ξ)2 2 2 2 + νε−1 2 αi2 z 2 (ξ) + νε2 K z 1 (ξ)

(7.23)

where ν is a positive scalar. The time derivative of z 2T (t) z 2 (t) is ·

2z 2T (t) z 2 (t) r  = 2z 2T (t) h i (θ) ((Ai21 + Ai22 K ) z 1 (t) i=1

+ (Adi21 + Adi22 K ) z 1 (t − d (t)) + Ai22 z 2 (t) +Adi22 z 2 (t − d (t)) + v i (z (t) , z (t − d (t))))  · + 2z 2T (t) u (t) − K z 1 (t)

(7.24)

One can easily verify the following inequalities ·

− 2z 2T (t) K z 1 = −2z 2T (t) K

r 

h i (θ) ((Ai11 + Ai12 K ) z 1 (t)

i=1

+ (Adi11 + K Adi12 ) z 1 (t − d (t)) + Ai12 z 2 (t) +Adi12 z 2 (t − d (t)) + δ i (z (t) , z (t − d (t))) r   2

(Ai11 + Ai12 K ) z 1 (t)2 ≤ 5ν −1  K T z 2 (t) + ν i=1

+ Ai12 z 2 (t) + (Adi11 + K Adi12 ) z 1 (t − d (t))2  2    + Adi12 z 2 (t − d (t))2 + δ i (z (t) , z (t − d (t))) 2

2z 2T

(t)

r 

(7.25)

h i (θ) ((Ai21 + Ai22 K ) z 1 (t)

i=1

+ ( Adi21 + Adi22 K ) z 1 (t − d (t))) ≤ 2ν −1 z 2 (t)2 + ν

r 

(Ai21 + Ai22 K ) z 1 (t)2 i=1

+ (Adi21 + Adi22 K ) z 1 (t − d (t))2



(7.26)

7.4 Controller Design

103

and 2z 2T

(t)

r 

h i (θ) (Ai22 z 2 (t) + Adi22 z 2 (t − d (t)))

i=1

≤

r 

 Ai22 z 2 (t)2 + Adi22 z 2 (t − d (t))2 i=1

+ 2 z 2 (t)2

(7.27)

From the coordinate transformation (7.8), we have ρi j (x (t)) = ρi j (z 1 (t) + z 2 (t) + K z 1 (t)) ≤ ρi j ((1 + K ) z 1 (t) + z 2 (t)) ≤ ρi j (2 (1 + K ) z 1 (t)) + ρi j (2 z 2 (t)) , and σil (x (t − d (t))) ≤ σil (2 (1 + K ) z 1 (t − d (t))) + σil (2 z 2 (t − d (t)))

(7.28)

With the above two inequalities, it follows that 2z 2T (t)

r 

h i (θ) v i (z (t) , z (t − d (t)))

i=1

≤

r 

 h i (θ) 2 z 2 (t) i∗T ρi (x (t)) + ϑi∗T σi (x (t − d (t)))

i=1 r      2 2 ≤ 2ν −1 i∗  + ϑi∗  z 2 (t)2 i=1

+ ν ρi (2 (1 + K ) z 1 (t))2 + ν ρi (2 z 2 (t))2 + ν σi (2 (1 + K ) z 1 (t − d (t)))2  +ν σi (2 z 2 (t − d (t)))2

(7.29)

The time derivative of W is given by ·

W ≤ −

2z 2T

·

(t) z 2 (t) +

γd

r t 

γe 1 − d i=1

r  i=1

t−d(t)



eγd Ωi (t) − Ωi (t − d (t)) 1 − d

e−γ(t−ξ) Ωi (ξ) dξ



(7.30)

104

7 Robust Adaptive Control for Time-Delay System via T-S Fuzzy Approach

Substituting (7.17), (7.19), (7.21), (7.24)–(7.27) and (7.29) into (7.30) gives ·

 W ≤ −γW + ζ ∗ − ζ (t) z 2 (t)2   2 2 + ϕ (z 1 (t)) − ςΦ ς z 2 (t) z 2 (t)2 γ

(7.31)

where ζ ∗ meets ζ ∗ I ≥ 2ν −1 I + 5ν −1 K K T + 2I  r  eγd T T Ai22 + Adi22 Adi22 + Ai22 1 − d i=1    2 2 + 2ν −1 i∗  + ϑi∗  I

 2 T + ν αi1 I + ε−1 1 αi1 I + Ai12 Ai12

 νeγd T −1 2 + Adi12 Adi12 + αi2 I + ε2 αi2 I 1 − d

(7.32)

and ϕ (z 1 (t)) is defined as ϕ (z 1 (t)) 

eγd ε2 K z 1 (t)2 = ν ε1 + 1 − d  r  eγd (Adi21 + Adi22 K ) z 1 (t)2 +ν 1 − d i=1

+ (Ai21 + Ai22 K ) z 1 (t)2 + (Ai11 + Ai12 K ) z 1 (t)2 eγd (Adi11 + K Adi12 ) z 1 (t)2 1 − d

 + αi1 z 1 (t)2 + K z 1 (t)2 +

 eγd αi2 z 1 (t)2 + K z 1 (t)2 1 − d + ρi (2 (1 + K ) z 1 (t))2

+

eγd σi (2 (1 + K ) z 1 (t))2 + 1 − d

(7.33)

For the whole system (7.9), we select the Lyapunov functional as follows

U=W+ 0

V

Φ (ξ) dξ +

2 1 ∗ ζ − ζ (t) 2l

(7.34)

7.4 Controller Design

105

Then, the time derivative of U is ·

 U ≤ −γW + ζ ∗ − ζ (t) z 2T (t) z 2 (t)   2ς 2 z 2 (t)2 z − ςΦ 2 (t) γ · · 1 ∗ ζ − ζ (t) ζ (t) + ϕ (z 1 (t)) + Φ (V ) V − l

(7.35)

Note that · 1 ϕ (z 1 (t)) + Φ (V ) V ≤ ϕ (z 1 (t)) − γΦ (V ) V 2   1 2 + Φ (V ) − γV + ς z 2 (t) 2

(7.36)

 In view of class-k function Φ (·) , we have Φ x1T P x1 ≤ Φ (V ), then it follows from (7.21) that 1 (7.37) ϕ (z 1 (t)) − γΦ (V ) V ≤ 0 4 It can be easily verified that 

 1 2 Φ (V ) − γV + ς z 2 (t) 2   2ς 2 z 2 (t) z 2 (t)2 ≤ ςΦ γ

(7.38)

Here, the variable change is used: if 21 γV ≥ ς z 2 (t)2 , (7.38) holds; if 21 γV < ς z 2 (t)2 , we use 2ςγ z 2 (t)2 instead of V and obtain (7.38). Substituting (7.36)–(7.38) and adaptive law (7.20) into (7.35) gives ·

 γ Φ (V ) V + μ ζ ∗ − ζ (t) ζ (t) 4 2 μ γ μ ∗ ≤ −γW − Φ (V ) V − ζ − ζ (t) + ζ ∗2 4 2 2

U ≤ −γW −

(7.39)

Note that ζ ∗ and μ are positive scalars. Then, the solution of the closed-loop system is uniformly ultimately bounded based on the Lyapunov stability theory. Now, we V show that the resulting system is exponentially stable. In view of 0 Φ (ξ) dξ ≤ V Φ (V ) , we have · μ U ≤ −ωU + ζ ∗2 (7.40) 2

106

where ω = min

7 Robust Adaptive Control for Time-Delay System via T-S Fuzzy Approach

γ 4

 , μl . Inequality (7.40) gives U (t) ≤ e−ωt U (0) +

μ ∗2 ζ 2ω

(7.41)

 V (0) where U (0) = W (0) + 0 Φ (ξ) dξ + 2l1 (ζ ∗ − ζ (0))2 . From (7.33), there exists a positive scalar c such that ϕ (z 1 (t)) ≥ cz 1T (t) Pz 1 (t) . Then, with (7.37) we , further with the help of (7.34) we have choose Φ (ξ) ≥ 4c γ U (t) ≥ z 2T (t) z 2 (t) +

4c T z (t) Pz 1 (t) γ 1

(7.42)

With (7.41) and (7.42), it follows that 4c T μ ∗2 z 1 (t) Pz 1 (t) ≤ e−ωt U (0) + ζ , γ 2ω μ ∗2 ζ z 2T (t) z 2 (t) ≤ e−ωt U (0) + 2ω

(7.43)

Considering (7.8), we further have  μ ∗2 ζ e−ωt U (0) + , 4cλmin (P) 2ω x2 (t)2 ≤ 2 z 2 (t)2 + 2 K x1 (t)2   K 2 γ e−ωt U (0) ≤ 2+ 2cλmin (P)   K 2 γ μ ∗2 ζ + 2+ 2cλmin (P) 2ω x1 (t)2 ≤

γ

(7.44)

It is implied by (7.44) that the solution of the resulting closed-loop system is uniformly ultimately bounded and exponentially converges to a bounded region. The proof is completed. Remark 7.4 To avoid the high gain of adaptive parameter ζ (t) , we have employed the σ-modification adaptive law (7.20). The solution of the resulting closed-loop system is shown to be uniformly ultimately bounded and converges to a bounded region with adjustable radius. The radius can be rendered arbitrary small by choosing small parameter μ based on (7.44). If one chooses the parameter μ = 0 in the adaptive ·

law, then the time derivative of U satisfies U ≤ −γW − 41 γΦ (V ) V. Based on the Lyapunov stability theory, the system state x (t) will converge to zero asymptotically. Remark 7.5 For system (7.7), we have investigated the case that the coefficients of bound functions are not known. If the coefficients are available, we may use ζ ∗ instead of ζ (t) in the controller. Via the similar argument, one may show that the closed-loop system is exponentially asymptotically stable.

7.4 Controller Design

107

Remark 7.6 As the development so far, the disturbance function Δf i is supposed to be time invariant. If Δf i is of time-varying and satisfies Δf i (x (t) , x (t − d (t)) , t) = Bvi (x (t) , x (t − d (t)) , t) + δi (x (t) , x (t − d (t))) , where function δi (x (t) , x (t − d (t))) yields (7.5) and vi (x (t) , x (t − d (t)) , t) ≤ i∗T ρi (x (t)) + ϑi∗T σi (x (t − d (t))) +  (t)

(7.45)

in which  (t) is a known time-varying function. We may design the controller of the following form 1 1 u (t) = − γz 2 (t) − ζ (t) z 2 (t) − z 2 (t) Θ (z 2 (t)) 2 2  2ς z 2 (t)2 z 2 (t)2 −  (t) sign (z 2 (t)) − ςΦ γ

(7.46)

where  (t) sign (z 2 (t)) is the new added robustification term to overcome the effect of time-varying function  (t) . Via the similar approach to the one proposed above, it can be shown that the closed-loop system is also exponentially stable.

7.5 Simulation In this section, we will give two examples to show the validity of the proposed controller design methodology. Example 7.1 The first example is a continuous stirred tank reactor (CSTR) used in [12, 111]: ·

yi =

3 

h i (Ai y (t) + Adi y (t − d (t)) + Bi u i + Δf (y (t) , y (t − d (t))))

i=1

(7.47)  T in which Bi = 0 1 and 

   −1.4274 0.0757 −2.0508 0.3958 , A2 = , −1.4189 −0.9442 −6.4066 1.6168     −4.5279 0.3167 0.25 0 , Adi = . A3 = −26.2228 0.9837 0 0.25

A1 =

108

7 Robust Adaptive Control for Time-Delay System via T-S Fuzzy Approach

In [111], the variable structure controller was presented to stabilize the system. Note that the controller designed in [111] must use the delayed state. We now employ the proposed method in this chapter to construct the memoryless controller. The disturbance function Δf i (y (t) , y (t − d)) is assumed to be of the following form  δi (y (t) , y (t − d (t))) , Δf i (y (t) , y (t − d (t))) = vi (y (t) , y (t − d (t))) 

with δi  ≤ 0.1 (y (t) + y (t − d (t))) and vi (y (t) , y (t − d (t))) = a sin (t) y (t)2 + b cos (t) y (t − d (t))2 . With the parameters γ = 0.5, d = 0.2, d = 0.4, υ = 0.5, ε1 = ε2 = 0.1, and αi1 = αi2 = 0.1, solving LMI (7.11) gives P = 0.0633, K = −0.0394. Then, the controller is constructed as the following form 1 u (t) = −3z 2 (t) − ζ (t) z 2 (t) − 2z 23 (t) 2 with the adaptive law

(7.48)

·

ζ (t) = 100 z 2 (t)2 − 2ζ (t)

(7.49)

For simulation, we choose d (t) = 0.2 (1 + sin (t)) , δi (y (t) , y (t − d (t))) = 0.1 sin (t) y1 (t) + 0.1 cos (t) y2 (t − d (t)) , vi (y (t) , y (t − d (t))) = 2 sin (t) y (t)2 + 2 cos (t) y (t − d (t))2 . The simulation is performed on the former nonlinear system ·

x 1 (t) = f 1 (x (t)) + 0.25x1 (t − d (t)) , ·

x 2 (t) = f 2 (x (t)) + 0.25x2 (t − d (t)) + u (t)

(7.50)

T  where x (t) = x1 (t) x2 (t) , and  f 1 (x) = −1.25x1 (t) + 0.072 (1 − x1 (t)) exp

x2 (t) 1+

x2 (t) 20

,

7.5 Simulation

109

1 y

1

y2

0.5

0

−0.5

−1

−1.5

−2

0

1

2

3

4

5

6

7

8

9

10

Fig. 7.1 The state response

 f 2 (x) = −1.55x2 (t) + 0.576 (1 − x1 (t)) exp

x2 (t) 1+



x2 (t) 20

.

 T The system state y (t) of (7.47) is x (t)−xd (t) , where xd (t) = 0.1440 0.8862 is a steady state for system (7.50) with u (t) = 0. It should be noted that our controller is designed based on the T-S fuzzy system (7.47) which is the fuzzification of system (7.50). The simulation is directly done on former system (7.50). With x1 (0) = 1, x2 (0) = −1 and ζ (0) = 2, the state response is shown in Fig. 7.1, from which we can see that the constructed controller provides the closed-loop system with the good dynamics and steady-state performances. The responses of the adaptive parameter ζ (t) and control input u (t) are depicted in Fig. 7.2, which show that ζ (t) and u (t) converge to zero quickly. Contrast to the variable structure controller in [111], the designed controller (7.48) is memoryless and the chattering phenomenon is avoided. Example 7.2 Consider the following nonlinear time-delay system [49] ⎡

⎤ −3 2 1 ⎦ x (t) 1 x (t) = ⎣ 2 1 + sin (x3 (t)) 1 1 (x2 (t))2 + 1 ⎡ ⎤ 0.5 0 0 1 0.2 ⎦ x (t − d (t)) +⎣ 0 −0.2 −0.5 1 ⎡ ⎤ ⎡ ⎤ 0 β (t) x1 (t) + ⎣ 0 ⎦ u (t) + ⎣ −0.5β (t) x1 (t − d (t)) ⎦ 1 Δh (x (t − d (t))) + f (t) ·

110

7 Robust Adaptive Control for Time-Delay System via T-S Fuzzy Approach 25

ζ u

20

15

10

5

0

−5 0

1

2

3

4

5

6

7

8

9

10

Fig. 7.2 Adaptive parameter and control input

with 0 ≤ β (t) ≤ 2, d (t) is the time-varying delay, f (t) ∈ [−1, 1] is a disturbance function ( f (t) = cos (100t) in the simulation), and Δh (x (t − d (t))) = sin (t) x (t − d (t))2 . The various matrices used for transforming the original system into a polytopic formulation are     1 0 Ai12 = , Adi12 = , 1 0.2   0.5 0 Ad111 = Ad311 = , 0 1   0.5 0 Ad211 = Ad411 = , −1 1     −3 2 −1 2 , A211 = , A111 = 2 1 2 1     −3 2 −1 2 , A411 = . A311 = 2 0 2 0 By solving LMI (7.11) with parameters γ = 0.5, d = 0.2, υ = 0.5 and d = 0.4, one has     2.6401 0.7094 K = −3.5163 −4.6151 , P = . 0.7094 1.9476

7.5 Simulation

111

1 x1 x2

0.5

x3

0 −0.5 −1 −1.5 −2 −2.5 −3

0

1

2

3

4

5

6

7

8

9

10

Fig. 7.3 The state response

Following Theorem 7.1 and Remark 7.6, we design the memoryless adaptive controller as 1 u (t) = −4z 2 (t) − ζ (t) z 2 (t) − 2z 23 (t) − sign (z 2 (t)) , 2 with the adaptive law

·

ζ (t) = 400 z 2 (t)2 − 2ζ (t) . The delay time is chosen as d (t) = 0.2 (1 + sin (t)) , the initial values are x (0) = T 1 0 −1 and ζ (0) = 2. The state response is shown in Fig. 7.3, while Fig. 7.4 exhibits the adaptive parameter and control input. From the two figures, we see that the resulting closed-loop system has good transient and steady-state performance. In this section, two examples are presented to demonstrate the controller design procedure via our method. Compared with the memory controllers designed via the variable structure control method, the designed controllers in this chapter are memoryless, which are easier for practical applications. If the uncertain disturbance is bounded by functions of system state (Example 7.1), the continuous state feedback controller can be designed from Theorem 7.1. If the system disturbance is bounded by time-varying nonlinear function (Example 7.2), we should add a robust term in the system (See Remark 7.7), then the designed controller has chattering effect. 

112

7 Robust Adaptive Control for Time-Delay System via T-S Fuzzy Approach 50 ζ u

40 30 20 10 0 −10 −20 −30 −40 −50

0

1

2

3

4

5

6

7

8

9

10

Fig. 7.4 Adaptive parameter and control input

7.6 Conclusion In this chapter, the control problem of nonlinear time-delay systems is studied via the T-S fuzzy approach. The local subsystem of the fuzzy system includes the uncertain nonlinear functions with time delay. The mismatched uncertainties satisfy the Lipschitz condition and the matched parts are bounded by nonlinear functions with unknown coefficients. We decompose the system into two subsystems. The virtual control input is designed for the first subsystem. Then, the continuous controller is constructed with the help of a new Lyapunov–Krasovskii functional. If the parameters of bound functions are known, the resulting closed-loop system is shown to be asymptotically stable. If the parameters are not known, the constructed adaptive controller renders the solution of the closed-loop system exponentially convergent to a region with an adjustable bound. Finally, simulations are given to show the effectiveness of the proposed method.

Part III

Nonlinear Input

Chapter 8

Adaptive Tracking of Time-Delay System with Unknown Dead-Zone Input

Abstract This chapter focuses on the tracking control problem for a class of nonlinear system with time delay and dead-zone input. The non-symmetric dead-zone input case is considered without the knowledge of the dead-zone parameters. The time-delay uncertainties are bounded by a nonlinear function with unknown coefficients. By constructing a novel Lyapunov functional, we design a simple and smooth adaptive state feedback controller. It is shown that the solution of the resulting closedloop error system converges to an adjustable region exponentially. Finally, numerical examples are included to show the effectiveness of the theoretical results.

8.1 Introduction Dead zone, saturation, backlash, and hysteresis are the most common actuator nonlinearities in practical control system applications. The presence of such nonlinearities in the feedback control systems may induce severe deterioration of the system performances. Many controller design methods are presented to deal with the non-smooth nonlinearity case, for example, [5, 144, 167, 220] and the references therein. Dead-zone input nonlinearity is a non-smooth function that characterizes certain non-sensitivity for small control inputs. It is well known that dead zone can severely limit the system performances. Control of systems with dead-zone nonlinearity is an important area of control system research and typically challenging. To deal with this problem, many techniques are presented to design the feedback controllers. In [5, 22, 144, 168, 169], the adaptive method was extensively used to construct the state feedback controller and output feedback controller. With the help of the neural network approximation method, an adaptive neural network controller was constructed to deal with the dead-zone case in [150]. Via backstepping method, [219] constructed the adaptive output feedback controller for a class of nonlinear systems with dead-zone input nonlinearity. The inverse dead-zone nonlinearity was employed to minimize the effects of the dead zone, such as [168, 219]. By modeling the input dead zone as a combination of a line and a disturbance-like term, [180] designed an adaptive feedback controller for the symmetric dead-zone input case. Reference [75] further relaxed the assumption © Springer Nature Singapore Pte Ltd. 2018 C. Hua et al., Robust Control for Nonlinear Time-Delay Systems, DOI 10.1007/978-981-10-5131-9_8

115

116

8 Adaptive Tracking of Time-Delay System …

and designed the state feedback controller with the requirement of the maximum and the minimum values of the characteristic slopes known. It is well known that time delay often exists in the practical world extensively, such as power systems and chemical reactor systems. Due to the effect of time delay, these systems may be unstable and the control performances are often hardly assured. The Lyapunov-Krasovskii functional method and Razumikhin lemma are often employed to deal with the stability analysis and controller design problem. For nonlinear time-delay systems with triangular structure, the backstepping method was used to construct the state feedback controller [46] and output feedback controller [70]. For nonlinear time-delay system with dead-zone input, constructive result was proposed in [213] with the bounds of uncertain delay functions available. In this chapter, the control design problem is considered for a class of nonlinear systems with time delay and dead-zone input. With the uncertain nonlinear time-delay functions unknown, a novel adaptive state feedback controller is presented. With the help of Lyapunov-Krasovskii functional, we show that the solution of the resulting closed-loop system exponentially converges to an adjustable small region. The proposed controller design scheme is also effective for the time-delay-free case. The precise maximum and minimum values of the characteristic slopes are not required to be known, which were needed in [75]. Also, the delay functions in [213] are relaxed, whose bounds are in the form of nonlinear functions with unknown coefficients in this chapter. Finally, simulations are done to show the effectiveness of the main results.

8.2 System Description Consider the following time-delay system with unknown dead-zone input 

·

x i (t) = xi+1 (t) , i = 1, 2, . . . , n − 1 , · x n (t) = f (t, x1 (t − d1 (t)) , x2 (t − d2 (t)) , . . . , xn (t − dn (t))) + Γ (u (t)) (8.1)   x (t) = ϕ (t) , t ∈ −d 0

where xi (t) ∈ R and u (t) ∈ R are the state variable and control input of system, respectively, x (t) = [x1 (t) , x2 (t) , . . . , xn (t)]T , f (·) is the uncertain smooth nonlinear function, di (t) are the delay parameters satisfying di (t) ≤ di∗ and ·

d i (t) ≤ d i < 1, Γ (u (t)) is a single non-symmetric dead-zone input nonlinearity defined as ⎧ if u (t) ≥ br ⎨ m r (u (t) − br ) if − bl < u (t) < br Γ (u (t)) = 0 (8.2) ⎩ m l (u (t) + bl ) if u (t) ≤ −bl

8.2 System Description

117

Fig. 8.1 Dead-zone nonlinearity

The non-symmetric dead-zone input is shown in Fig. 8.1. The parameters m r and m l stand for the right and the left slopes of the dead-zone characteristic. The parameters br and bl represent the break points of the input nonlinearity. The following assumptions are imposed on system (8.1): Assumption 8.1 Parameters m r , m l , bl , and br are positive and unknown. Assumption 8.2 The uncertain function f (·) satisfies the following inequality  f (t, x1 (t − d1 (t)) , x2 (t − d2 (t)) , . . . , xn (t − dn (t))) n  ≤ θi αi (|xi (t − di (t))|) + γ

(8.3)

i=1

where θi and γ are unknown positive scalars and functions αi (·) are known smooth class-κ function (strictly increasing and αi (0) = 0). Similarly to [75, 180], the dead-zone input can be expressed in the following form Γ (u (t)) = m (t) u (t) + d (t) where m (t) =

⎧ if u (t) ≥ br ⎨ −m r br m l if u (t) ≤ 0 , d (t) = −m (t) u (t) if − bl < u (t) < br . m r if u (t) > 0 ⎩ m l bl if u (t) ≤ −bl

Then, one knows that there exists a positive scalar η such that η ≤ m l and η ≤ m r .

118

8 Adaptive Tracking of Time-Delay System …

Problem For system (8.1) with Assumptions 8.1 and 8.2, given signal

T · xd (t) = (xd1 (t) , xd2 (t) , . . . , xdn (t))T = yd (t) , y d (t) , . . . , yd(n−1) (t) , ·

where yd (t) is sufficiently smooth and x d (t) ∈ L ∞ , design a smooth and memoryless state feedback controller such that the system state x (t) exponentially tracks the signal xd (t) and the resulting tracking error can be rendered arbitrary small by adjusting design parameters. Remark 8.1 For system (8.1) free of time delay, the adaptive state feedback control problem was considered in [180] with symmetric dead-zone nonlinearity and [75] with non-symmetric dead-zone nonlinearity. However, the maximum and the minimum values of the characteristic slopes are required to be known precisely. In engineering practice, it may be difficult for us to obtain the precise values of them. Therefore, it is significant to investigate the unknown parameters case. In this chapter, the case is considered and a simple adaptive state feedback control scheme is proposed. Remark 8.2 Assumption 8.2 is imposed on nonlinear function f (·). If the function f (·) only contains the state variables as [75, 180], we can always find functions αi (·) satisfying (8.3). The time delay is often encountered in a practical system, and the existence of time delay renders the controller design more complex and challenging. In [213], the MIMO nonlinear time-delay system was considered via the backstepping method and a novel neural network controller was constructed. But the upper bound of uncertain delay functions is required to be known. In this chapter, this assumption is relaxed with Assumption 8.2 on the delay functions.

8.3 Controller Design In this section, we will design the adaptive tracking controller for system (8.1) with Assumptions 8.1 and 8.2. Defining the tracking error ei (t) = xi (t)−xdi (t) gives the following error system ⎧· ⎪ ⎨ ei (t) = ei+1 (t) , i = 1, 2, . . . , n − 1 · en (t) = f (t, x1 (t − d1 (t)) , x2 (t − d2 (t)) , . . . , xn (t − dn (t))) ⎪ · ⎩ + Γ (u (t)) − x dn (t)

(8.4)

With (8.4), one has

 · · e (t) = (A + B L) e + B f − Le + Γ (u (t)) − x dn (t) T  where e (t) = e1 (t) e2 (t) · · · en (t) , and

(8.5)

8.3 Controller Design

A=

119



O(n−1)×1 I(n−1)×(n−1) ,B = O1×1 O1×(n−1)

O(n−1)×1 1



⎡

⎤T L1 ⎢ ⎥ , L = ⎣ ... ⎦ , Ln

where the elements of matrix O are zeros, I denotes the identity matrix, and matrix L is determined by the following inequality P (A + B L) + (A + B L)T P ≤ − P

(8.6)

in which P is a positive matrix,  is a given positive scalar. In view of the structure of matrices A and B, one knows that there always exists matrix L such that A + B L is stable, then inequality (8.6) will stand with a scalar . With the knowledge of the above, we now present our main result: Theorem 8.1 For system (8.1) with Assumptions 8.1 and 8.2, the solution of the closed-loop error system exponentially converges to an adjustable region with the following state feedback controller 1 1 u (t) = − θ (t) B T Pe (t) ρ (W ) − γ (t) tanh 2 2



B T Pe (t) ρ (W ) ε

 (8.7)

in which ·  2 θ (t) = l1 B T Pe (t) ρ (W ) − l1 σ1 θ (t) , θ (0) > 0,   T B Pe (t) ρ (W ) · T − l2 σ2 γ (t) , γ (0) > 0 γ (t) = l2 B Pe (t) ρ (W ) tanh ε

(8.8) where W = e T (t) Pe (t) , ε, l1 , l2 , σ1 and σ2 are positive scalars, function ρ (·) is a positive non-decreasing function which yields δ

 n  eωdi∗ i=1

1 − di



αi2

  (2 ei (t)) + ce (t) e (t) ≤ νρ e (t)T Pe (t) e (t)T Pe (t) T

(8.9) in which δ is a positive parameter, c, ω, and ν are positive scalars satisfying cI ≥ L T L , ω ≤ l1 σ1 , ω ≤ l2 σ2 , and ν < . Proof For system (8.5), we choose the following Lyapunov functional V = V1 + V2 with

(8.10)

120

8 Adaptive Tracking of Time-Delay System …



W

V1 =

ρ (ξ) dξ,

0

2 2 1  ∗ 1  ∗ θ − ηθ (t) + γ − ηγ (t) 2ηl1 2ηl2 n ωdi∗  t  δe + e−ω(t−ξ) αi2 (2 |ei (ξ)|) dξ, 1 − d t−d (t) i i i=1

V2 =

where θ∗ and γ ∗ are positive scalars defined in (8.16) below. The time derivative of Lyapunov function V1 yields ·   V 1 = ρ (W ) e T (t) P (A + B L) + (A + B L)T P e (t)

 · + 2ρ (W ) e T (t) P B m (t) u (t) + d (t) − Le (t) + f − x dn (t)

(8.11)

In view that αi (|xi (t − di (t))|) ≤ αi (|ei (t − di (t))| + |xdi (t − di (t))|) ≤ αi (2 |ei (t − di (t))|) + αi (2 |xdi (t − di (t))|)

(8.12)

we have 2ρ (W ) e T (t) P B f

 n     T   θi αi (|xi (t − di (t))|) + γ ≤ 2ρ (W ) e (t) P B i=1

  ≤ 2ρ (W ) e T (t) P B  ×  n   θi (αi (2 |ei (t − di (t))|) + αi (2 |xdi (t − di (t))|)) + γ i=1 n 

≤

 −1 2 2  δ θi ρ (W ) e T (t) P B B T Pe (t) + δαi2 (2 |ei (t − di (t))|)

i=1

  n    T θi αi (2 |xdi (t − di (t))|) + 2ρ (W ) e (t) P B  γ +

(8.13)

i=1

In addition, the following inequality holds −2ρ (W ) e T (t) P B Le (t) ≤ δ −1 ρ2 (W ) e T (t) P B B T Pe (t) + δe T (t) L T Le (t) ≤ δ −1 ρ2 (W ) e T (t) P B B T Pe (t) + δce T (t) e (t) (8.14) Substituting (8.6), (8.13), and (8.14) into (8.11) gives

8.3 Controller Design

121

·   V 1 ≤ −ρ (W ) e T (t) Pe (t) + δce T (t) e (t) + γ ∗ e T (t) P B  ρ (W )

+ 2ρ (W ) e T (t) P Bm (t) u (t) + θ∗ ρ2 (W ) e T (t) P B B T Pe (t) n  + δαi2 (2 |ei (t − di (t))|)

(8.15)

i=1

where θ∗ and γ ∗ are positive scalars satisfying θ∗ = δ −1 + 

n 

δ −1 θi2 ,

i=1

 n ·     θi αi (2 |xdi (t − di (t))|) γ ≥ 2 γ + |d (t)| + x dn (t) + ∗

(8.16)

i=1

The time derivative of V2 is ·

· · 1 ∗ 1 ∗ θ (t) − ηθ (t) θ (t) − γ − ηγ (t) γ (t) l1 l2  n  ωd ∗  i e 2 2 +δ αi (2 |ei (t)|) − αi (2 |ei (t − di (t))|) 1 − di i=1 ∗  t n  δeωdi −ω e−ω(t−ξ) αi2 (2 |ei (ξ)|) dξ 1 − d t−d (t) i i i=1

V2 ≤ −

(8.17)

With θ (0) > 0 and γ (0) > 0, it follows from adaptive law (8.8) that the solutions satisfy θ (t) > 0 and γ (t) > 0 for t > 0. With (8.7) and η ≤ m l , η ≤ m r , one has 2ρ (W ) e T (t) P Bm (t) u (t)  = −ρ (W ) e T (t) P Bm (t) θ (t) B T Pe (t) ρ (W )  T  B Pe (t) ρ (W ) + γ (t) tanh ε  2  T ≤ −m (t) θ (t) B Pe (t) ρ (W )   T B Pe (t) ρ (W ) T + γ (t) B Pe (t) ρ (W ) tanh ε  T 2 ≤ −ηθ (t) B Pe (t) ρ (W )   T B Pe (t) ρ (W ) − ηγ (t) B T Pe (t) ρ (W ) tanh ε Substituting (8.18) into (8.15) gives

(8.18)

122

8 Adaptive Tracking of Time-Delay System … ·   V 1 ≤ −ρ (W ) e (t)T Pe (t) + γ ∗ − ηγ (t) B T Pe (t) ρ (W ) tanh ×  T   2  B Pe (t) ρ (W ) + δce T (t) e (t) + θ∗ − ηθ (t) B T Pe (t) ρ (W ) ε   T   B Pe (t) ρ (W ) ∗ T ∗ T  + γ B Pe (t) ρ (W ) − γ B Pe (t) ρ (W ) tanh ε n  + δαi2 (2 |ei (t − di (t))|) (8.19) i=1

With (8.8), (8.17), and (8.19), one has ·   V ≤ −ρ (W ) e (t)T Pe (t) + δce T (t) e (t) + θ∗ − ηθ (t) σ1 θ (t) − ωV2     + γ ∗ − ηγ (t) σ2 γ (t) + γ ∗ e T P B  ρ (W )   T B Pe (t) ρ (W ) − γ ∗ B T Pe (t) ρ (W ) tanh ε ∗ n ωd  δe i 2 2 ω  ∗ ω  ∗ θ − ηθ (t) + γ − ηγ (t) + αi2 (2 ei (t)) + 2ηl1 2ηl2 1 − di i=1

(8.20) By considering the inequality |φ| − φ tanh positive scalar , we have



  γ e T (t) P B  ρ (W ) − γ ∗ B T Pe (t) ρ (W ) tanh ∗

φ 



≤ 0.2785 for variable φ and

B T Pe (t) ρ (W ) ε



≤ 0.2785γ ∗ ε (8.21)

It is easy to verify that  2 σ1  ∗ 2  ∗ ησ1 2 σ1  ∗ θ (t) − θ − ηθ (t) + θ θ − ηθ (t) σ1 θ (t) = − 2 2η 2η

(8.22)

 2 σ2  ∗ 2  ∗ ησ2 2 σ2  ∗ γ (t) − γ − ηγ (t) + γ γ − ηγ (t) σ2 γ (t) = − 2 2η 2η

(8.23)

and

Substituting (8.21)–(8.23) into (8.20) gives ·

2 l 1 σ1 − ω  ∗ θ − ηθ (t) − ωV2 2ηl1 2 σ1  ∗ 2 σ2  ∗ 2 l 2 σ2 − ω  ∗ θ + γ + 0.2785γ ∗ ε − γ − ηγ (t) + 2ηl2 2η 2η

V ≤ − ( − ν) ρ (W ) e (t)T Pe (t) −

(8.24)

8.3 Controller Design

123

Consider that ρ (W ) is a non-decreasing function, then With l1 σ1 ≥ ω and l2 σ2 ≥ ω, we have

W 0

ρ (ξ) dξ ≤ ρ (W ) W .

·

V ≤ −μV + ζ where μ = min { − ν, ω} and ζ = Based on (8.25), one obtains

σ1 2η

(θ∗ )2 +

σ2 2η

(8.25) (γ ∗ )2 + 0.2785γ ∗ ε.

V (t) ≤ e−μt V (0) +

ζ μ

(8.26)

where V (0) is defined as follows with the help of (8.10) 

e T (0)Pe(0)

V (0) =

ρ (ξ) dξ +

0

+

∗  n  δeωdi

1 − di i=1

0 −di (0)

2 2 1  ∗ 1  ∗ θ − ηθ (0) + γ − ηγ (0) 2ηl1 2ηl2

eωξ αi2 (2 |ei (ξ)|) dξ.

With (8.9), one has   δc T e (t) e (t) ≤ ρ e (t)T Pe (t) e (t)T Pe (t) , ν and then

thus, ρ (ξ) ≥

  ρ e (t)T Pe (t) ≥ δc . νλ max(P)

δc νλ max (P)

(8.27)

With (8.27), it follows that

δc ζ e (t)T Pe (t) ≤ V1 ≤ V ≤ e−μt V (0) + , νλ max (P) μ and further

 e (t) ≤

 1/2 νλ max (P) ζ −μt e V (0) + δcλ min (P) μ

(8.28)

then the solution e (t) of the tracking error system exponentially converges to a

1/2   νζλ max(P) bounded region Ω := X X  ≤ δcμλ min(P) . The proof is completed. Remark 8.3 For controller design, we need matrices P and L . If there exist matrix N and positive matrix M such that the following LMI holds AM + B N + M A T + N T B T + M ≤ 0

(8.29)

124

8 Adaptive Tracking of Time-Delay System …

then the matrices P and L are obtained with P = M −1 and L = N M −1 . We can see that inequality (8.6) is equivalent to the above LMI (8.29). Remark 8.4 Since that αi (·) is a smooth class-κ function, then there always exists a smooth function ρ (·) such that (8.9) holds, which can be determined as the following method: For αi (·) is a smooth class-κ function, there exists function αi (χ) such that αi (χ) = χαi (χ), then δ

 n  eωdi∗ i=1

1 − di

 αi2

(2 |ei (t)|) + ce (t) e (t)



T

∗ n  4eωdi



(2 |ei (t)|) + c 1 − di  n   4eωdi∗ δ T 2 ≤ e (t) Pe (t) αi (2 |ei (t)|) + c . λmin (P) 1 − di i=1 ≤ δe (t) e (t) T

αi2

i=1

One may choose 



ρ e (t) Pe (t) ≥ T

δ νλmin (P)

 n  4eωdi∗ i=1

1 − di

 αi2

(2 |ei (t)|) + c

(8.30)

In view of the smoothness of function αi (·) , there always exists the nondecreasing function ρ (·) such that (8.30) holds. With (8.30), the inequality (8.9) holds. Remark 8.5 If the controller is designed as the following form   1 u (t) = − θ (t) B T Pe (t) ρ (W ) − γ (t) sign B T Pe (t) 2

(8.31)

with adaptive law ·    2 · θ (t) = l1 B T Pe (t) ρ (W ) , γ (t) = l2  B T Pe (t) ρ (W ) .

Choosing the same Lyapunov functional (8.10), via the similar proof of Theorem 8.1, one may get ·

V ≤ − ( − ν) ρ (W ) e (t)T Pe (t) ∗  t n  δeωdi −ω e−ω(t−ξ) αi2 (2 |ei (ξ)|) dξ 1 − d i t−di (t) i=1

(8.32)

With  > ν, the error state e (t) → 0 as t → ∞. Since the discontinuous adaptive control schemes not only induce the problems of existence and uniqueness of

8.3 Controller Design

125

solutions, but also display chattering phenomena and excite high-frequency phenomena, we should try to employ the smooth controller in practical systems if possible. To render the controller smooth, we use tanh function instead of the sign function and employ the inequality (8.21). The smooth controller renders that the closed-loop system is stable in the sense of uniform ultimate boundedness. Remark 8.6 Based on (8.28), the transient state performance of the closed-loop system is determined by parameter μ and the steady-state performance is related to parameters μ, σ1 , σ2 , η, ε, ν, δ and matrix P. We may choose big parameters ω,  and small ν to get big exponential decay rate μ. Also, choose small parameters σ1 , σ2 , νλ max(P) ζ is small, then the good steady-state ε, ν/δ and proper matrix P such that δcλ min(P) μ performance is guaranteed. Since σ1 , σ2 , and ε can be chosen freely, one knows that the tracking error can be rendered arbitrary small. Remark 8.7 Adaptive control problem was considered for linear systems with input constraints in [28, 136, 142, 218]. In [28, 136, 142], the stable and unstable discrete systems were considered. The adaptive control problem of continuous system was investigated in [218], and a novel global stable controller was presented. However, the method proposed is not applicable for nonlinear case. This chapter presents a new adaptive controller design methodology for the problem. With di (t) = 0, it is easy to see that the constructed controller is also effective for the delay-free case. In the proof of Theorem 8.1, we introduce the parameter η in (8.18). With the help of this parameter, we prove the effectiveness of the designed adaptive controllers. This parameter is not required to be known for controller design; thus, the proposed scheme does not need the condition imposed in [75] that the maximum and the minimum values of the slopes should be available.

8.4 Simulation Examples In this section, two examples are presented to show the effectiveness of the proposed controller design method. Example 8.1 Consider the following nonlinear time-delay system 

·

x 1 (t) = x2 (t) · x 2 (t) = f (t, x1 (t − d1 (t)) , x2 (t − d2 (t))) + Γ (u (t))

(8.33)

with the dead-zone input Γ (u (t)) and f (t, x1 (t − d1 (t)) , x2 (t − d2 (t))) = σ1 (t) x12 (t − d1 (t)) + σ2 (t) x2 (t − d2 (t)) ≤ σ 1 x12 (t − d1 (t)) + σ 2 |x2 (t − d2 (t))|

(8.34)

where σ1 (t) and σ2 (t) are uncertain time-varying disturbances with the bounds σ 1 and σ 2 , respectively.

126

8 Adaptive Tracking of Time-Delay System …

Obviously, from (8.34), Assumption 8.2 holds with α1 (χ) = χ2 and α2 (χ) = |χ|. Now, we employ the proposed scheme to design the tracking controller. With  = 1, solving inequality (8.6) gives

P=



T 3.1194 1.4476 −3.3513 . ,L = 1.4476 1.6718 −2.7154

  We obtain λmax L T L = 18.6044, then choose c = 20. Based on (8.9), choosing δ = 0.1, d = 0.5, d ∗ = 2, and ν = 0.1 gives   ρ e (t)T Pe (t) = 100 ∗ e (t)T Pe (t) + 40. Further, the controller is designed as 1 1 u (t) = − θ (t) B T Pe (t) ρ (W ) − γ (t) tanh 2 2



B T Pe (t) ρ (W ) ε

 (8.35)

with adaptive law ·  2 θ (t) = 100 B T Pe (t) ρ (W ) − 2θ (t) ,   T B Peρ (W ) · T − 2γ (t) γ (t) = 100B Pe (t) ρ (W ) tanh ε

(8.36)

Choose the tracking signal xd1 (t) = sin (t) and xd2 (t) = cos (t) , the time delay di (t) = 0.5 (1 + sin (t)) , σ1 (t) = sin (t) , σ2 (t) = cos (t) , the dead-zone input parameters m l = 1, m r = 0.7, br = 1, bl = 3, and the initial values as follows: x1 (0) = 0.8, x2 (0) = −0.2, θ (0) = 1, γ (0) = 1. The simulation result is shown in Figs. 8.2 and 8.3. The tracking error curves are shown in Fig. 8.2, from which we can see that the constructed controller can render that the closed-loop error system has good transient and steady-state performances.

Fig. 8.2 Tracking error response

8.4 Simulation Examples

127

Example 8.2 Consider the following system 

·

x 1 (t) = x2 (t)   · −x1 x 2 (t) = θ1 1−e + θ2 x22 (t) + 2x1 (t) sin (x2 (t)) + Γ (u (t)) 1+e−x1

(8.37)

where θ1 and θ2 are unknown scalars. In [75], the tracking control problem of system (8.37) is investigated. Now, we use the proposed method of this chapter to construct the controller. It is easy to verify that there exist positive scalars θ1 , θ21 and θ22 such that    1 − e−x1   2  θ 1   1 + e−x1 + θ2 x2 (t) + 2x1 (t) sin (x2 (t)) ≤ θ1 + θ21 |x1 | + θ22 x22 (t)

(8.38)

Considering (8.38), we can use the same controller (8.35) with the adaptive law (8.36). Choose θ1 = θ2 = 1, the same dead-zone parameters of the system (8.33) and the initial values as x1 (0) = 1, x2 (0) = 0, θ (0) = 1, γ (0) = 1,

Fig. 8.3 Dead-zone input

Fig. 8.4 Tracking error response

128

8 Adaptive Tracking of Time-Delay System …

Fig. 8.5 Dead-zone input

the error response curve and control input are shown in Figs. 8.4 and 8.5, respectively. From the two figures, one can find that the proposed controller is effective and can quickly track the signal given.

8.5 Conclusion Tracking control problem is considered for a class of nonlinear time-delay systems with non-symmetric dead-zone input. The time-delay uncertainties are bounded by nonlinear function with unknown coefficients. A simple smooth adaptive controller is constructed such that the solution of the resulting closed-loop system exponentially converges to an adjustable region. The proposed adaptive control scheme does not require the knowledge of maximum and minimum values of the characteristic slopes, which is desirable for practical applications.

Chapter 9

Decentralized Fuzzy Networked Control Systems Design with Sector Input

Abstract The robust control problem has been investigated for a class of largescale nonlinear networked control systems with nonlinear sector input. The time delays have been inherent for the systems because of the information transmission through the communication networks. By T-S fuzzyfication for each subsystem, the interconnected T-S fuzzy subsystems are obtained. When the bound parameters are known, the decentralized memoryless state feedback controller is constructed. When the parameters of bound functions are not available, the adaptive method is used, and the memoryless decentralized adaptive state feedback controller is developed. By the construction of the proper Lyapunov–Krasovskii functional, the exponential stabilization of the resultant closed-loop system is proved for the both cases. Finally, the theoretic results are applied to the decentralized controller design of networked interconnected chemical reactor systems.

9.1 Introduction Large-scale systems consist of a set of interconnected subsystems, which can be found in the diverse fields, such as power systems, digital communication networks, economic systems, and urban traffic networks. The subsystem in the large-scale system may be far away from the others, then the wireless/wired communication networks are used to connect them, and the mutual information is exchanged through the communication channel. The presence of the networks renders the control problem difficult and challenging due to the transmission delays. Recently, the analysis and control design for networked control system (NCS) have received considerable attentions [60, 176]. The research for the time-delay systems [40, 51, 58, 210, 225] made a solid foundation for analyzing the time-delay effect in NCSs. The time-varying delays, possible packet dropouts, and the quantization effects were considered in the constructed model. The Lyapunov–Krasovskii functional was employed, and the results were given in the form of linear matrix inequality (LMI). In this chapter, we consider the decentralized control problem of a class of nonlinear large-scale systems with the subsystems exchanging information through the communication networks. The subsystems are in the nonlinear forms, and they are © Springer Nature Singapore Pte Ltd. 2018 C. Hua et al., Robust Control for Nonlinear Time-Delay Systems, DOI 10.1007/978-981-10-5131-9_9

129

9 Decentralized Fuzzy Networked Control Systems . . .

130

nonlinearly interconnected. The nonlinear input case for each subsystem is considered. By T-S fuzzyfication, each subsystem is described in the form of T-S fuzzy model with nonlinear interconnections. Based on the input matrix, we express each subsystem in a cascade form. First, a virtual control input is constructed and a lemma is proposed. Based on the lemma, we design a decentralized state feedback controller for the case that the upper bound parameters of interconnections are available. By employing the new Lyapunov–Krasovskii functional, we show that the resultant closed-loop system is exponentially asymptotically stable. For the case that the parameters of bound functions are not known, a decentralized adaptive state feedback controller is developed and it is shown that the solution of the corresponding closedloop system converges exponentially to a bounded region. Simulations confirm the effectiveness of the proposed controller design schemes.

9.2 System Formulation and Assumptions In this chapter, we consider a class of large-scale systems composed of N subsystems. The subsystems exchange the state information through communication networks, see Fig. 9.1. The ith subsystem is described by x˙i (t) = Fi (xi (t) , xi (t − τi )) + G i (xi ) (Θi (u i (t)) +Ψi (x1 (t − di1 ) , x2 (t − di2 ) , . . . , x N (t − di N ))) ,   x (t) = ϕ (t) , t ∈ −d, 0

(9.1)

where xi ∈ ni and u i ∈ m i are the state and control input, respectively, n i and m i are positive scalars; Fi (·) and G i (·) are deterministic smooth nonlinear functions; Ψi (·) are uncertain nonlinear interconnections; Θi (u i (t)) are nonlinear inputs; τi are the unknown time delays of the ith subsystem; di j are the unknown time delays denoting the time used for transmitting the information of jth subsystem to xi subsystem, and we assume dii = τi . The time delays τi and di j satisfy τi ≤ τ i and di j ≤ d i j with the bounds τ i and d i j known. We show the interconnections of the subsystems in Fig. 9.2. From the figure, one can see that the ith subsystem receives the delayed information from other subsystems because of the network transmission delays. For system (9.1), we make the following assumptions:

Fig. 9.1 Large-scale system with subsystems exchanging information through networks

9.2 System Formulation and Assumptions

131

Fig. 9.2 Large-scale system with subsystems in the form of 9.1

Assumption 9.1 The nonlinear input Θi (u i (t)) satisfies the following condition u iT (t) Θi (u i (t)) ≥ γi u iT (t) u i (t)

(9.2)

where γi is a positive scalar. Assumption 9.2 The uncertain interconnection function Ψi satisfies Ψi (x1 (t − di1 ) , x2 (t − di2 ) , . . . , x N (t − di N )) ≤

N     ϑi∗j σi j x j t − di j 

(9.3)

j=1

where ϑi∗j are positive scalars, σi j (·) are known smooth class-k functions (strictly increasing and σi j (0) = 0). Further σi j (χ) = χσ i j (χ). Remark 9.1 There may exist the disturbances and possible faults in the control input of a practical system, so we consider the nonlinear input case in this chapter. Obviously, the nonlinear input includes the traditional sector input case. Moreover, the upper bound of the sector input is not needed. In this chapter, the interconnections Ψi are matched with the control input function Θi (u i (t)). This match condition is often imposed on the time-delay systems with uncertain nonlinearities, which can

9 Decentralized Fuzzy Networked Control Systems . . .

132

be seen in [24, 25, 84, 86, 90, 111]. Different from the above literatures, the system considered is with large-scale and the interconnections are nonlinear. The control problem is more challenging. The uncertain interconnections are assumed to satisfy Assumption 9.2. With the information exchange through the communication networks, the time delays are inherent in the model. The functions σi j (·) are nonlinear, then the functions Ψi are very general. The problem of this chapter is formulated as follows: (i) If the parameters γi and ϑi∗j are available, design a decentralized controller such that the closed-loop system is exponentially stable; (ii) If the parameters γi and ϑi∗j are not known, construct an adaptive controller to render the solution of the closed-loop system to converge to a bounded region exponentially. In this chapter, we will employ the T-S fuzzy approach to describe the large-scale networked system (9.1), and then propose a new decentralized controller design method for resultant model. A fuzzy dynamic model was developed in the pioneering work [166] to represent local input/output relations of nonlinear systems. This dynamical model is described by IF-THEN rules. For ith subsystem of the large-scale j system (9.1), the jth rule li of this fuzzy model is shown below: j j j Plant Rule li : IF θi1 is μi1 and · · · and θi pi is μi pi , THEN x˙i (t) = Ail xi (t) + Dil xi (t − τi ) + Bil (Θi (u i (t)) + Ψi ) where Ail ∈ ni ×ni , Dil ∈ ni ×ni , and Bil ∈ ni ×m i are constant real matrices. Without loss of the generality, we assume Rank (Bil ) = m i . θi h (h = 1, . . . , pi ) are the premise variables, which are the functions of state variables. They are completely j measurable. μil ( j = 1, . . . , ri , l = 1, . . . , pi ) are the fuzzy sets, where ri is the number of IF-THEN rules and pi is the number of the premise variables. It is assumed that the premise variables are independent of the input variable. Then, the overall fuzzy model is achieved by fuzzy blending of each individual rule as follows: x˙i (t) =

ri 

h il (θi ) (Ail xi (t) + Dil xi (t − τi )

l=1

+ Bil (Θi (u i (t)) + Ψi ))

(9.4)

ri  where h il (θi ) = μil (θi ) / μil (θi ) . l=1

Assumption 9.3 For all l ∈ [1, ri ], matrices Bil = Bi . For simplicity, we assume  O(ni −m i )×m i . Bi = Im i ×m i Remark 9.2 Assumption 9.3 is imposed on the input matrix Bil . If the control input matrix G i (xi ) = Bi , this condition always holds. Although this condition is a little restrictive, it is often assumed for dealing with the uncertain nonlinearites, see [24, 25,

9.2 System Formulation and Assumptions

133

111] and the references therein. If matrix Bi is not the form, we can choose coordinate transformation to render it into the form. For a matrix Bi with Rank (Bi ) = m i , therealways exist matrix Γi ∈ R ni ×ni and non-singular matrix i ∈ R m i ×m i such O(ni −m i )×m i = Γi Bi i . Further by letting x i = Γi xi and Θ i (u i (t)) = that Im i ×m i −1 i Θi (u i (t)) , we can obtain the similar form of (9.4). Remark 9.3 For the form of large-scale system (9.1), the stability analysis and control problem have been considered in [84, 86, 90, 99, 119, 130, 143, 209]. In [84, 86, 90, 130, 143], the nominal system was assumed to be linear function (Fi (xi (t) , xi (t − τi )) = Hi1 xi (t) + Hi2 xi (t − τi )) and the interconnection was known precisely or restricted to be linear form (σi j (χ) = χ in (9.3)). In [99, 119, 209], the nonlinear nominal subsystems were considered and T-S fuzzy method was employed for the controller design, but the interconnections among the subsystems were still required to be linear or bounded by linear functions. It is still an open problem to design the decentralized controllers for large-scale time-delay systems with nonlinear nominal subsystems and nonlinear interconnections. Remark 9.4 How to deal with the uncertain nonlinearities is a challenging issue for the control of time-delay systems. In [24, 25, 86, 99, 111], the sliding mode control (SMC) method was proposed for the time-delay systems with nonlinear uncertainties. Although the resultant closed-loop system has good dynamic performance and steady-state performance via SMC, it has a crucial fundamental limitation that the designed controllers have to be with memory. The controllers must use the delayed state of the system, so that the precise time delays have to be known, and also a large memory is needed for controller implementation. In practical systems, the controller equipped with a large memory is not desirable, and the precise delay time is difficult to obtain. This chapter aims to overcome the shortcomings and design the smooth and memoryless state feedback controller which is more suitable for practical applications.

9.3 Virtual Control Design For the appearance of the time-delay nonlinear interconnections Ψi , we could not construct a Lyapunov–Krasovskii functional for system (9.4) directly and then design the decentralized controller. Now, we propose a new method to solve this problem. We decompose the system (9.4) into two subsystems, a virtual control input is designed for the first subsystem and then the controller is designed for the second subsystem with the use of the information of the first subsystem. In this section, a preliminary lemma is proposed, which is useful for our controller design in Sect. 9.4. With Assumption 9.3, system (9.4) is rewritten in the following form

9 Decentralized Fuzzy Networked Control Systems . . .

134

⎧ ri  ⎪ ⎪ ⎪ x˙i1 (t) = h il (θi ) (Ail11 xi1 + Ail12 xi2 ⎪ ⎪ ⎪ ⎪ l=1 ⎪ ⎨ +Dil11 xi1 (t − τi ) + Dil12 xi2 (t − τi )) ri  ⎪ ⎪ ⎪ x ˙ h il (θi ) (Ail21 xi1 + Ail22 xi2 + Dil21 xi1 (t − τi ) = ⎪ i2 (t) ⎪ ⎪ ⎪ l=1 ⎪ ⎩ +Dil22 xi2 (t − τi ) + Ψi + Θi (u i (t)))

(9.5)

  T T (t) xi2 (t) = xiT (t), matrices Ailpq and Dilpq where xi1 ∈ ni −m i , xi2 ∈ m i , xi1 are correspondingly obtained by decomposing Ail and Dil . For system (9.5), we choose the following coordinate transformation 

z i1 (t) = xi1 (t) z i2 (t) = xi2 (t) − K i xi1 (t)

(9.6)

where matrix K i will be determined later. With the transformation, the new system is obtained ⎧ ri  ⎪ ⎪ ⎪ z ˙ h il (θi ) ((Ail11 + Ail12 K i ) z i1 (t) = (t) ⎪ i1 ⎪ ⎪ ⎪ l=1 ⎪ ⎪ ⎪ +Ail12 z i2 (t) + (Dil11 + Dil12 K i ) z i1 (t − τi ) ⎪ ⎪ ⎪ ⎪ ⎨ +Dil12 z i2 (t − τi )) ri  (9.7) ⎪ z ˙ h il (θi ) ((Ail21 + Ail22 K i ) z i1 (t) = (t) ⎪ i2 ⎪ ⎪ ⎪ l=1 ⎪ ⎪ ⎪ + (Dil21 + Dil22 K i ) z i1 (t − τi ) ⎪ ⎪ ⎪ ⎪ +Θi (u i (t)) + Ail22 z i2 (t) + Dil22 z i2 (t − τi ) ⎪ ⎪  ⎩ +Ψ i (z 1 (t − di1 ) , . . . , z N (t − di N )) − K i z˙ i1 (t) where Ψ i (z 1 (t − di1 ) , . . . , z N (t − di N )) = Ψi (x1 (t − di1 ) , . . . , x N (t − di N )) . The first subsystem of (9.7) is expressed as follows: z˙ i1 (t) = Ai (t) z i1 (t) + Di (t) z i1 (t − τi ) + gi where



ri   Ai (t) Di (t) gi = h il (θi ) [Ail11 + Ail12 K i l=1

Dil11 + Dil12 K i Ail12 z i2 (t) + Dil12 z i2 (t − τi ) ] For z i1 subsystem, choose the following Lyapunov functional

9.3 Virtual Control Design

135 T i Vi = z i1 (t) Pi z i1 (t) + V

(9.8)

with i = V +



t

T e−ωi (t−ξ) z i1 (ξ) Q i z i1 (ξ) dξ

t−τi  0 t

−τi t+θ

T z˙ i1 (ξ) Hi z˙ i1 (ξ) dξdθ

(9.9)

where Pi , Q i , and Hi are positive definite matrices, ωi are positive scalars. Lemma 9.1 For given positive scalars ωi and εi , if there exist matrices Γi p , Υi p , p ∈ [1, 4] and positive definite matrices Pi , Q i , and Hi such that the following matrices Ωi(5×5) < 0 hold for all i ∈ [1, N ], and the elements of the matrices Ωi are shown below: Ωi(1,1) = Pi Ai (t) + AiT (t) Pi + Γi1 + Γi1T − Υi1 Ai (t) − AiT (t) Υi1T + Q i + ωi Pi , Ωi(1,2) = Pi Di (t) − Γi1 + Γi2T − Υi1 Di (t) − AiT (t) Υi2T , Ωi(1,3) = Γi3T + Υi1 − AiT (t) Υi3T , Ωi(1,4) = Pi + Γi4T − Υi1 − AiT (t) Υi4T , Ωi(1,5) = −τi Γi1 , Ωi(2,2) = −Γi2 − Γi2T − Υi2 Di (t) − DiT (t) Υi2T − e−ωi τi Q i , Ωi(2,3) = −Γi3T + Υi2 − DiT (t) Υi3T , Ωi(2,4) = −Γi4T − Υi2 (t) − DiT (t) Υi4T , Ωi(2,5) = −τi Γi2 , Ωi(3,3) = Υi3 + Υi3T + τi Hi , Ωi(3,4) = −Υi3 + Υi4T , Ωi(3,5) = −τi Γi3 , Ωi(4,4) = −Υi4 − Υi4T − εi Pi Pi , Ωi(4,5) = −τi Γi4 , Ωi(5,5) = −τi Hi + ωi τi2 Hi , then the time derivative of Lyapunov–Krasovskii functional Vi satisfies (9.10) V˙i ≤ −ωi Vi + εi Pi gi 2 Proof With (9.8), the first-order time derivative of Vi satisfies ˙ + 2z T (t) P (A (t) z (t) + D (t) z (t − τ ) + g )  V˙i = V i i i i1 i i1 i i i1 With (9.9), we have ˙ = −ω V   V i i i + ωi + +



0



t

T z˙ i1 (ξ) Hi z˙ i1 (ξ) dξdθ −τi t+θ T T z i1 (t) Q i z i1 (t) − e−ωi τi z i1 (t − τi ) Q i z i1 (t − τi )  t · T T τi z˙ i1 z˙ i1 (t) Hi z i1 (t) − (ξ) Hi z˙ i1 (ξ) dξ. t−τi

 Because z i1 (t) − z i1 (t − τi ) =

t

t−τi

z˙ i1 (ξ) dξ, one has

(9.11)

9 Decentralized Fuzzy Networked Control Systems . . .

136

  T T T 0 = 2 z i1 (t) Γi1 + z i1 (t − τi ) Γi2 + z˙ i1 (t) Γi3 + giT Γi4 ∗    t z i1 (t) − z i1 (t − τi ) − z˙ i1 (ξ) dξ , t−τi   T T T 0 = 2 z i1 (t) Υi1 + z i1 (t − τi ) Υi2 + z˙ i1 (t) Υi3 + giT Υi4 ∗ (˙z i1 (t) − Ai (t) z i1 (t) − Di (t) z i1 (t − τi ) − gi (t)) . Letting θ = −τi , we have  ωi

0



t

−τi t+θ

 T z˙ i1

(ξ) Hi z˙ i1 (ξ) dξdθ ≤ ωi τi

t t−τi

T z˙ i1 (ξ) Hi z˙ i1 (ξ) dξ.

Using the above two equations and inequality, we have 1 V˙i ≤ −ωi Vi + τi



t

t−τi

χiT (t, ξ) Ωi χi (t, ξ) dξ + εi Pi gi 2 ,

  where χiT (t, ξ) = z i1 (t) z i1 (t − τi ) z˙ i1 (t) gi z˙ i1 (ξ) . With Ωi < 0, we obtain (9.10). Now the proof is completed. Remark 9.5 It is obvious that the virtual control input design condition is delay dependent, which is looser than that of delay independent one in [73]. The free weighting matrices Γi p and Υi p are employed to derive the less conservative conditions. The details on the use of weight matrices were shown in [58]. To solve inequality Ωi < 0, we may convert it into the standard LMI form by the following simple method. We choose Υi p = λi p Pi for p ∈ [1, 4] where λi p are design parameters. We choose the parameters λi3 < 0 and λi4 > 0 to render Ωi(3,3) < 0 and  Ωi(4,4) < 0. The parameters λi1 and λi2 can be freely chosen. Multiply diag Pi−1 Pi−1 · · · Pi−1 ri  h il (θi ) Ω i(5×5) , then Ω i(5×5) on both sides of Ωi . Further we have Ωi(5×5) = l=1

can be changed into the strict LMIs via the following operations: Let P i = Pi−1 , Q i = Pi−1 Q i Pi−1 , Γ i p = Pi−1 Γi p Pi−1 , H i = Pi−1 Hi Pi−1 and Mi = K i Pi−1 .

9.4 Controller Design In this section, we consider two cases: The parameters (γi and ϑi∗j ) are available and not available. For the parameters known case, the decentralized memoryless state feedback controller is constructed such that the closed-loop system is exponentially asymptotically stable. If the parameters are not available, we design the decentralized adaptive state feedback controller such that the solution of the closed-loop system converges exponentially to a bounded region.

9.4 Controller Design

137

Before designing the controllers, we have the following result with inequality ri  t εi eτ i t−τi e−(t−ωi ) 2ri Pi Dil12 z i2 (ξ)2 dξ, then (9.10) (9.10). Choose V i = Vi + l=1

gives V˙ i ≤ −ωi V i + εi Pi gi 2 + εi

ri  2ri l=1

  ∗ eτ i Pi Dil12 z i2 (t)2 − Pi Dil12 z i2 (t − τi )2 . With gi =

ri 

h il (θi ) (Ail12 z i2 (t) + Dil12 z i2 (t − τi )), one has

l=1

Pi gi 2 ≤

ri ri   2ri Pi Ail12 z i2 (t)2 + 2ri Pi Dil12 z i2 (t − τi )2 l=1

(9.12)

l=1

Further the following inequality holds V˙ i ≤ −ωi V i + εi

ri  2ri l=1

  × Pi Ail12 z i2 (t)2 + eτ i Pi Dil12 z i2 (t)2 ≤ −ωi V i + i z i2 (t)2 where i =

(9.13)

ri   T  T 2εi ri λmax Ail12 Pi Pi Ail12 + eτ i Dil12 Pi Pi Dil12 , here λmax (χ) l=1

denotes the maximum eigenvalue of matrix χ. With above knowledge, we are ready to present the new controller design method.

9.4.1 Parameters Known Case If the parameters γi and ϑi∗j are available, the information can be used for the controller design. We have the following main result: Theorem 9.1 For system (9.4) with Ωi < 0 holding in Lemma 9.1, the following decentralized state feedback controller

9 Decentralized Fuzzy Networked Control Systems . . .

138

u i (t) = −2γi−1

N  j=1

−

i γi−1 2



φi

ζi∗ γi−1 z i2 2

eμ j d ji σ 2ji (2 z i2 ) z i2 −



i 2 z i2  z i2 ωi − κi − ω i

(9.14)

renders the closed-loop system asymptotically stable with an exponential decay rate, in which κi and ω i are positive scalars satisfying κi + ω i < ωi , μ j is a positive scalar, ζi∗ is a positive parameter satisfying ζi∗ Ii ≥ eμi τ i

ri  

T T Dil12 Dil12 + Dil22 Dil22



l=1 −1

+ 2 +

K i K iT + μi Ii + 2−1 Ii + 2Ii + 2K i K iT

ri  

N   T T Ail12 Ail12 + Ail22 Ail22 + −1 ϑi∗2j Ii

l=1

(9.15)

j=1

where Ii is an identity matrix, μi and  are positive scalars, and φi (·) is a nondecreasing function satisfying  T  T ϕi (z i1 (t)) ≤ κi z i1 (t) Pi z i1 (t) φi z i1 (t) Pi z i1 (t)

(9.16)

in which function ϕi (·) is defined in (9.32) below. Proof For z i2 -subsystem, choose the Lyapunov–Krasovskii functional as the following form T (9.17) Wi = z i2 (t) z i2 (t) + Ωi where Ωi =

 ri  eμi τ i l=1

t

e−μi (t−ξ) (Dil12 z i2 (ξ)2

t−τi

+ Dil22 z i2 (ξ)2 +  (Dil11 + Dil12 K i ) z i1 (ξ)2 +  (Dil21 + Dil22 K i ) z i1 (ξ)2 )dξ  t N        μi d i j + e e−μi (t−ξ) (σi2j 2 1 +  K j  z j1 (ξ) j=1

+

σi2j

t−di j

   2 z j2 (ξ) dξ

Then, the time derivative of Wi along (9.7) is

(9.18)

9.4 Controller Design

139

T W˙ i = 2z i2

ri 

h il (θi ) ((Ail21 + Ail22 K i ) z i1 + Ail22 z i2

l=1

+ (Dil21 + Dil22 K i ) z i1 (t − τi ) + Dil22 z i2 (t − τi )  T +Θi (u i (t)) + Ψ i + Ω˙ i − 2z i2 (t) K i z˙ i1 (t)

(9.19)

We know that T − 2z i2 (t) K i z˙ i1 (t)

=

T −2z i2 Ki

ri 

h il (θi ) ((Ail11 + Ail12 K i ) z i1 + Ail12 z i2

l=1

+ (Dil11 + Dil12 K i ) z i1 (t − τi ) + Dil12 z i2 (t − τi ))

(9.20)

and the time derivative of Ωi satisfies Ω˙ i ≤ −μi Ωi +

ri  (eμi τ i Dil12 z i2 2 + eμi τ i Dil22 z i2 2 l=1

− Dil12 z i2 (t − τi )2 − Dil22 z i2 (t − τi )2 + eμi τ i (Dil11 + Dil12 K i ) z i1 (t)2 + eμi τ i (Dil21 + Dil22 K i ) z i1 (t)2 −  (Dil11 + Dil12 K i ) z i1 (t − τi )2 −  (Dil21 + Dil22 K i ) z i1 (t − τi )2 ) +

N  eμi d i j j=1

          ∗ σi2j 2 1 +  K j  z j1 (t) + σi2j 2 z j2 (t) −

N 

       (σi2j 2 1 +  K j  z j1 t − di j 

j=1

    + σi2j 2 z j2 t − di j 

(9.21)

It is easy to verify that the following inequalities hold −

ri 

T h il (θi ) 2z i2 (t) K i (Ail11 + Ail12 K i ) z i1 (t)

l=1 −1

≤

ri   T   K z i2 (t)2 +  (Ail11 + Ail12 K i ) z i1 (t)2 i l=1

(9.22)

9 Decentralized Fuzzy Networked Control Systems . . .

140

−

ri 

T h il (θi ) 2z i2 (t) K i (Dil11 + Dil12 K i ) z i1 (t − τi )

l=1 ri  2   (Dil11 + Dil12 K i ) z i1 (t − τi )2 ≤ −1  K iT z i2  +

(9.23)

l=1

T − 2z i2 (t) K i

ri 

h il (θi ) (Ail12 z i2 (t) + Dil12 z i2 (t − τi ))

l=1 ri  2    Ail12 z i2 2 + Dil12 z i2 (t − τi )2 ≤ 2  K iT z i2  +

(9.24)

l=1

T 2z i2

(t)

ri 

h il (θi ) ((Ail21 + Ail22 K i ) z i1 (t)

l=1

+ (Dil21 + Dil22 K i ) z i1 (t − τi )) ≤

ri  ((Ail21 + Ail22 K i ) z i1 (t)2 l=1

+ (Dil21 + Dil22 K i ) z i1 (t − τi )2 ) + 2−1 z i2 (t)2

(9.25)

and T 2z i2 (t)

ri 

h il (θi ) (Ail22 z i2 (t) + Dil22 z i2 (t − τi ))

l=1 ri    Ail22 z i2 2 + Dil22 z i2 (t − τi )2 ≤ 2 z i2  + 2

(9.26)

l=1

In addition, by considering that functions σi j (·) are class-k, one has    σi2j x j t − di j          ≤ σi2j z j1 t − di j  + z j2 t − di j + K j z j1 t − di j           ≤ σi2j 1 +  K j  z j1 t − di j  + z j2 t − di j             ≤ σi2j 2 1 +  K j  z j1 t − di j  + σi2j 2 z j2 t − di j  then,

(9.27)

9.4 Controller Design

T 2z i2 (t)

141 ri 

h il (θi ) Ψ i ≤

N 

l=1

N  T −1 ϑi∗2j z i2 (t) z i2 (t) +  (σi2j

j=1

j=1

           2 1 +  K j  z j1 t − di j  + σi2j 2 z j2 t − di j  )

(9.28)

The nonlinear control can be expressed as the following form u i (t) = −Λi (z i2 ) z i2 , and obviously Λi (z i2 ) > 0. Then with Assumption 9.1, one has T z i2 (t) Θi (u i (t)) T = Λi−1 (z i2 ) Λi (z i2 ) z i2 (t) Θi (−Λi (z i2 ) z i2 ) T ≤ −γi Λi−1 (z i2 ) Λi (z i2 ) z i2 (t) z i2 Λi (z i2 ) T z i2 = −γi Λi (z i2 ) z i2

(9.29)

Substituting (9.14), (9.20)–(9.26), (9.28), and (9.29) into (9.19) and using the following equations N N         2    eμi d i j σi2j 2 1 +  K j  z j1  + σi2j 2 z j2  i=1 j=1

=

N  N 

  eμ j d ji σ 2ji (2 (1 + K i ) z i1 ) + σ 2ji (2 z i2 )

(9.30)

i=1 j=1

we have W˙ =

N 

W˙ i ≤

i=1

N  (−μi Wi + ϕi (z i1 (t)) i=1

− i z i2 (t)2 φi (

i z i2 (t)2 )) ωi − κi − ω i

(9.31)

in which ϕi (z i1 (t)) is defined as ri  ϕi (z i1 (t)) =  ((Ail11 + Ail12 K i ) z i1 (t)2

+e

l=1 μi τ i

(Dil11 + Dil12 K i ) z i1 (t)2

+ (Ail21 + Ail22 K i ) z i1 (t)2 + eμi τ i (Dil21 + Dil22 K i ) z i1 (t)2 ) N  +  eμ j d ji σ 2ji (2 (1 + K i ) z i1 (t)) j=1

(9.32)

9 Decentralized Fuzzy Networked Control Systems . . .

142

To deal with the nonlinear function ϕi (z i1 (t)) in (9.31), for the whole system (9.7) we select the following Lyapunov functional U=

N 





Vi

Wi +

 φi (ξ) dξ

(9.33)

0

i=1

With (9.31), the time derivative of U satisfies U˙ ≤

N    · (−μi Wi + ϕi (z i1 (t)) + φi V i V i i=1



i z i2 (t)2 ωi − κi − ω i

− i z i2 (t) φi 2

 (9.34)

By verification, with (9.13) and (9.16) the following inequality holds   · ϕi (z i1 (t)) + φi V i V i      ≤ κi V i φi V i + φi V i −ωi V i + i z i2 (t)2      = −ω i V i φi V i + φi V i i z i2 (t)2 − (ωi − κi − ω i ) V i    

i 2 2 z z ≤ −ω i V i φi V i + i i2 (t) φi i2 (t) ωi − κi − ω i With (9.35) and

 Vi

(9.35)

  φi (ξ) dξ ≤ φi V i V i , one has

0

U˙ ≤ −κU

(9.36)

where κ = min {μi , ω i } for all i ∈ [1, N ], then U ≤ e−κt U (0) where U (0) =

N  

T z i2 (0) z i2 (0) + Ωi (0) +

(9.37)

 V i (0) 0

φi (ξ) dξ



with V i (0) and

i=1

Ωi (0) defined at t = 0 for V i (t) and Ωi (t) . With (9.6), (9.17), and (9.33), inequality (9.37) gives T z i2 (t) z i2 (t) ≤ Wi ≤ U ≤ e−κt U (0)

and

 T φi (0) z i1 (t) Pi z i1 (t) ≤

0

Vi

φi (ξ) dξ ≤ U ≤ e−κt U (0)

(9.38)

(9.39)

9.4 Controller Design

143

From (9.16) and (9.32), one knows that φi (0) > 0. With (9.38) and (9.39), we have e−κt U (0) z i1 (t)2 ≤ , z i2 (t)2 ≤ e−κt U (0) , λmin (Pi ) φi (0) where λmin (Pi ) denotes the minimum eigenvalue of matrix Pi , then with (9.6) it gives e−κt U (0) , λmin (Pi ) φi (0)   K i 2 2 xi2 (t) ≤ 2 1 + e−κt U (0) λmin (Pi ) φi (0) xi1 (t)2 ≤

(9.40)

Then, the closed-loop system is asymptotically stable with exponential decay rate. The proof is completed. Remark 9.6 From (9.40), we know that the closed-loop system is asymptotically stable with exponential decay rate 21 κ. Considering κ = min {μi , ω i }, we may choose large μi and ω i to get large κ, then the good transient performance is achieved. The controller (9.14) is a nonlinear controller, and we need parameters ζi∗ and functions φi (·) for control implementation. The parameters ζi∗ are chosen based on (9.15), and functions φi (·) are determined by (9.16). The controller (9.14) is memoryless, which does not need the precise delay values of the information transmission in the networks. Compared with the sliding mode controller, the controller designed via our proposed method is continuous and does not include the delayed state, which is an important and desirable feature for practical applications. For the parameter , there is no limitation. The parameter  and functions φi (·) should be chosen to achieve small control gain based on (9.23)–(9.25) and (9.16). Remark 9.7 The key of the control design is the selection of the functions φi (·)for all i ∈ [1, N ] . The functions can be chosen by the following method: with (9.32), one knows that there exist positive scalars ci such that N  ϕi (z i1 (t)) ≤  eμ j d ji σ 2ji (2 (1 + K i ) z i1 (t)) j=1

+ ci z i1 (t)2 , then, we choose functions φi (·) such that  T  φi z i1 (t) Pi z 1 (t) ≥

ci κi λmin (Pi )

+

4 (1 + K i )2 κi λmin (Pi )

9 Decentralized Fuzzy Networked Control Systems . . .

144

∗

N 

eμ j d ji σ 2ji (2 (1 + K i ) z i1 (t)) .

j=1

Obviously, (9.16) holds with the selected functions.

9.4.2 Parameters Unknown Case In this part, we consider the case that parameters γi and ϑi∗j are unknown. Since they are not available, the controller design parameters γi−1 and ζi∗ could not be obtained. The adaptive control method is employed to design the decentralized state feedback controller in this section. We have the following main result: Theorem 9.2 For system (9.4) with Ωi < 0 holding in Lemma 9.1, the following controller ⎛ N  1 u i (t) = − βi (t) z i2 (t) ⎝ eμ j d ji σ 2ji (2 z i2 (t)) 2 j=1  

i z i2 (t)2 (9.41) +1 + φi ωi − κi − ω i renders that the solution of the closed-loop system converges exponentially to a bounded region, in which parameters , κi , ω i , μ j and function φi (·) are the same as in Theorem 9.1, and βi (t) is the adaptive parameter with initial value βi (0) > 0 and the tuning law is as follows: ⎛ β˙i (t) = ai z i2 (t)2 ⎝  +1 + φi

N  eμ j d ji σ 2ji (2 z i2 (t)) j=1

i z i2 (t)2 ωi − κi − ω i

 − ai bi βi (t)

(9.42)

where ai and bi are positive scalars. Proof For system (9.7), we select the Lyapunov functional U =U+

N  2 1  ∗ βi − βi (t) 2a i i=1

(9.43)

  where U is defined in (9.34), and βi∗ is defined as βi∗ ≥ max 4γi−1 , ζi γi−1 , i γi−1 which is not known.

9.4 Controller Design

145

Then, the time derivative of U satisfies ⎛ N    βi∗ − βi (t) z i2 (t)2 ⎝1+ U˙ ≤ −κU + i=1

 φi

−

⎞   N

i z i2 2 + eμ j d ji σ ji (2 z i2 )2 ⎠ ωi − κi − ω i j=1

N   1  ∗ βi − βi (t) β˙i (t) a i=1 i

(9.44)

With the adaptive law (9.42), (9.44) gives U˙ ≤ −κU +

 N   2 bi ∗2 bi  ∗ β − βi (t) + βi − 2 i 2 i=1

≤ −κU + b

(9.45)

where κ = min {κ, a1 b1 , a2 b2 , . . . , a N b N } and b =

N 

bi 2

βi∗2 .

i=1

From (9.45), we know U ≤ e−κt U (0) +

b κ

with U (0) = U (0) +

 ∗ 2 βi − βi (0) . Similarly, we have

N 

1 2ai

i=1

e−κt U (0) + κb , z i2 (t)2 λmin (Pi ) φi (0) b ≤ e−κt U (0) + , κ

z i1 (t)2 ≤

and subsequently e−κt U (0) + κb , λmin (Pi ) φi (0)   K i 2 2 xi2 (t) ≤ 2 1 + e−κt U (0) λmin (Pi ) φi (0)   K i 2 2b 1+ + κ λmin (Pi ) φi (0) xi1 (t)2 ≤

then, the solution of the closed-loop system converges to a bounded region

(9.46)

9 Decentralized Fuzzy Networked Control Systems . . .

146

⎧ ⎫ b κ ⎨ xi (t)| xi (t)2 ≤ ⎬ λmin (Pi )φi (0)   Ωxi = 2 i ⎩ ⎭ 1 + λminK + 2b κ (Pi )φi (0) with exponential decay rate 21 κ. The proof is completed. Remark 9.8 To avoid the high gain of adaptive parameter βi (t), we employ the σ-modification adaptive law (9.42). From (9.46), one knows the exponentially decaying rate is determined by parameter κ, thus we may choose big κ and big ai such that the closed-loop system has good transient performance. With (9.46), one knows that as t → ∞, xi (t) converges to the region Ωxi . Thus, we may choose small parameters bi in the adaptive law to obtain small b such that the converging region is small, then the good steady-state performance of the closed-loop system is achieved. If one chooses the parameter bi = 0 in the adaptive law, then the time derivative of ·

U satisfies U ≤ −κU from (9.45). Based on Lyapunov stability theory, we obtain that system state z i (t) will converge to zero asymptotically, and then xi (t) will also converge to zero asymptotically. Remark 9.9 In this chapter, we consider the case that the delays in the nominal subsystem and interconnections are constants. In fact, the proposed method is also applicable to the time-varying delay case. If the delays τi and di j are τi (t) and di j (t), respectively, we can also use the proposed for the controller design. By assuming ·

·

that τi (t) ≤ τ i , τ i (t) ≤ τi∗ < 1, di j ≤ d i j and d i j (t) ≤ di∗j < 1, we can construct the similar Lyapunov functional (9.8), (9.17), and (9.33) with the information τi∗ and di∗j . Further, the controller can be constructed by the proposed method.       By letting σi j x j t − di j  = x j t − di j  (linear interconnection case) in Assumption 9.2, we can obtain the corollaries from Theorems 9.1 and 9.2 directly as follows: Corollary 9.1 For system (9.4) satisfying Assumption 9.1, 9.2 with σi j (χ) = χ (linear interconnection) and Ωi < 0 in Lemma 9.1, the following decentralized state feedback controller u i (t) = −2γi−1

N  ζ ∗ γ −1 γ −1 i h i z i2 (t) eμ j d ji z i2 (t) − i i z i2 (t) − i 2 2 j=1

renders the closed-loop system exponentially asymptotically stable, μ j is a positive scalar, ζi∗ is a positive parameter satisfying (9.15), and νi is a positive scalar such that N 4 (1 + K i )2  μ j d ji 4 (1 + K i )2 N ci + . e + νi ≥ κi λmin (Pi ) κi λmin (Pi ) j=1 κi λmin (Pi ) where κi is a positive scalar satisfying κi < ωi .

9.4 Controller Design

147

Corollary 9.2 For system (9.4) satisfying Assumption 9.1, 9.2 with σi j (χ) = χ and Ωi < 0 in Lemma 9.1, if the parameters γi and ϑi∗j are not available, the following decentralized controller u i (t) = − 21 βi (t) z i2 (t) renders that the solution of the closed-loop system converges exponentially to a bounded region, where βi (t) is the adaptive parameter with βi (0) > 0 and the tuning law of βi (t) is β˙i (t) = ai z i2 (t)2 − ai bi βi (t), where ai and bi are positive scalars. The corollaries have the following merits: (i) The corollaries can deal with the nonlinear nominal system with the use of T-S fuzzy method. (ii) The controller designed is delay dependent which results in less conservative condition for control design. Based on the above discussions on Theorems 9.1 and 9.2, we can design the decentralized controller by the following design steps: Step 1: Use T-S fuzzyfication for system (9.1) and obtain the matrices Ail , Dil , and Bil . Step 2: Check Assumption 9.1–9.3, then have the bound functions σi j (·) . Step 3: Solve matrices Ωi(5×5) < 0 and obtain K i . Step 4: If the parameters γi and ϑi∗j are known, use Theorem 9.1 to design the controller; If they are not available, use Theorem 9.2 to construct the controller.

9.5 Simulations In this section, two examples are presented to show the effectiveness and feasibility of the proposed control schemes. Example 9.1 Consider an interconnected time-delay system which is composed of two subsystems as follows:

and

⎧ 3 x˙11 (t) = 2x12 (t) − 0.1x11 (t) + x12 (t − τ1 ) ⎪ ⎪ ⎪ 3 ⎪ (t) ⎨ x˙12 (t) = x13 (t) + x11 (t − τ1 ) + 0.1x11 x˙13 (t) = x11 (t) + x12 (t − τ1 ) + x13 (t − τ1 ) ⎪ ⎪ +Θ1 (u 1 (t)) + αx21 (t − d12 ) x11 (t − d11 ) ⎪ ⎪ ⎩ +αx12 (t − d11 ) x22 (t − d12 )

(9.47)

⎧ 3 x˙21 (t) = 3x22 (t) − 0.1x21 (t) + x22 (t − τ2 ) ⎪ ⎪ ⎪ 3 ⎪ (t) ⎨ x˙22 (t) = 2x23 (t) + x21 (t − τ2 ) + 0.1x21 x˙23 (t) = x21 (t) − x21 (t − τ2 ) ⎪ ⎪ +Θ2 (u 2 (t)) + αx21 (t − d22 ) x11 (t − d21 ) ⎪ ⎪ ⎩ +αx12 (t − d21 ) x22 (t − d22 )

(9.48)

  where xiT = xi1 xi2 xi3 is the state of ith subsystem, and α is an uncertain parameter, and nonlinear input Θi (u i (t)) = (2 + sin (u i (t))) u i (t) .

9 Decentralized Fuzzy Networked Control Systems . . .

148

For the large-scale system, the interconnections satisfy αx21 (t − d12 ) x11 (t − d11 ) + αx12 (t − d11 ) x22 (t − d12 ) α 2 α 2 ≤ x21 (t − d12 ) + x11 (t − d11 ) 2 2 α 2 α 2 + x12 (t − d11 ) + x22 (t − d12 ) 2 2 α α ≤ x1 (t − d11 )2 + x2 (t − d12 )2 , 2 2 and αx21 (t − d22 ) x11 (t − d21 ) + αx12 (t − d21 ) x22 (t − d22 ) α 2 α 2 ≤ x21 (t − d22 ) + x11 (t − d21 ) 2 2 α 2 α 2 + x12 (t − d21 ) + x22 (t − d22 ) 2 2 α α ≤ x1 (t − d21 )2 + x2 (t − d22 )2 2 2 From (9.3), one knows that σi j (χ) = χ2 , ϑi∗j = 21 α. Now, we employ the developed T-S fuzzy method to solve the problem. For the first subsystem, we use the following fuzzy rules Rule 1: If X 11 (t) is μ11 , then x˙1 (t) = A11 x1 (t) + D11 x1 (t − τ1 ) + B (Θ1 (u 1 (t)) + Ψ1 ) ; Rule 2: If X 11 (t) is μ21 , then x˙1 (t) = A12 x1 (t) + D12 x1 (t − τ1 ) + B (Θ1 (u 1 (t)) + Ψ1 ) ; x2 where μ11 = 1011 , μ21 = 1 − μ11 , Ψ1 = αx21 (t − d12 ) x11 (t − d11 ) + αx12 (t − d11 ) x22 (t − d12 ) and ⎡

A11

D11

⎤ ⎡ ⎤ −1 2 0 020 = ⎣ 1 0 1 ⎦ , A12 = ⎣ 0 0 1 ⎦ , 1 00 100 ⎡ ⎤ ⎡ ⎤ 010 0 = D12 = ⎣ 1 0 0 ⎦ , B = ⎣ 0 ⎦ . 011 1

For the second subsystem, the following fuzzy rule is employed: Rule 1: If X 21 (t) is μ12 , then x˙2 (t) = A21 x2 (t) + D21 x2 (t − τ2 ) + B (Θ2 (u 2 (t)) + Ψ2 ) ; Rule 2: If X 21 (t) is μ22 , then x˙2 (t) = A22 x2 (t) + D22 x2 (t − τ2 ) + B (Θ2 (u 2 (t)) + Ψ2 ) ; x2 where μ12 = 1021 , μ22 = 1 − μ12 , Ψ2 = αx21 (t − d22 ) x11 (t − d21 ) + αx12 (t − d21 ) x22 (t − d22 ) and

9.5 Simulations

149 1

Fig. 9.3 State response of the first subsystem (9.47) with control (9.49)

x11 x12 x13

0.5 0 −0.5 −1 −1.5 −2 −2.5 −3

0

1

2

3

4

5

6

7

8

Time

Fig. 9.4 State response of the second subsystem (9.48) with control (9.49)

0.6

x21 x22 x23

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8

0

1

2

3

4

5

6

7

8

Time

⎡

A21

D21

⎤ ⎡ ⎤ −1 3 0 030 = ⎣ 1 0 2 ⎦ , A22 = ⎣ 0 0 2 ⎦ , 1 00 100 ⎡ ⎤ 0 10 = D22 = ⎣ 1 0 0 ⎦ . −1 0 0

Based on Lemma 9.1, we first construct the virtual controller. We choose ωi = 1, τ i = 0.2, εi = 1, λi1 = λi3 = −1, and λi2 = λi4 = 1. Solving Ωi < 0 gives 

  2.8520 1.2602 , K 1 = −2.9715 −4.5059 , P1 = 1.2602 2.3751    2.9984 1.6599 , K 2 = −1.3306 −2.1042 . P2 = 1.6599 4.8566

150

9 Decentralized Fuzzy Networked Control Systems . . .

Now, we design the decentralized state feedback controller. First, with the bound functions of uncertain interconnections known, we employ Theorem 9.1 to design the controller. With α = 1, γi = 1, μi = 1, ζi∗ = 20, κi = ω i = 0.25, the controller is constructed as follows: 3 u i (t) = −174z i2 (t) − 350z i2 (t) .

(9.49)

Choose the initial values of system state x11 (ξ) = 1, x12 (ξ) = 0, x13 (ξ) = −1, x21 (ξ) = 0.5, x22 (ξ) = 0, x23 (ξ) = −0.5, where ξ ∈ [−1, 0] . Choose the time delays τi = di j = 0.2 for i, j = 1, 2. With the controller (9.49), the state response is shown in Figs. 9.3 and 9.4. From the figures, we find that the corresponding closed-loop system has good transient and steady-state performances. Based on Theorem 9.2, the controller is designed as follows:   1 2 + 20 u i (t) = − βi (t) z i2 (t) 100z i2 2

(9.50)

 2 2 β˙i (t) = z i2 (t)2 100z i2 (t) + 20 − βi (t)

(9.51)

with adaptive law

With the controller (9.50) and adaptive law (9.51), the state response curves are shown in Figs. 9.5 and 9.6 with uncertain parameter α = 50. From the figures, one can see that the designed decentralized adaptive controller is effective. Because the adaptive parameters are tuned online, they can approximate the desirable parameters. The control structure of (9.49) and (9.50) is the same, and the parameters βi (t) are adaptively tuned online, so the responses of the state variables are similar with the two controllers. Example 9.2 In chemical industry, the chemical reactor recycle system is very popular. It is well known that a reactor recycle not only increases the overall conversion but also reduces the reaction cost. For the recycling, the input to be recycled must be separated from the yields, then do the separation operation and finally travel through pipes. This set of operations introduces delays in the recycle system. We consider two continuous reactor subsystems, and they exchange information through the communication networks. Figure 9.7 shows the interconnected chemical reactor systems. The chemical reactor subsystem includes two reactors, and the state-space equation is of two dimension. Consider an interconnected system composed of two continuous stirred tank reactor nonlinear subsystems, which are described as follows:

9.5 Simulations

151 1

x11 x12 x13

0.5 0 −0.5 −1 −1.5 −2 −2.5 −3

0

1

2

3

4

5

6

7

8

Time

Fig. 9.5 State response of the first subsystem (9.47) with control (9.50) 0.6

x21 x22 x23

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8

0

1

2

3

4

5

6

7

8

Time

Fig. 9.6 State response of the second subsystem (9.48) with control (9.50)

⎧ x˙11 (t) = −1 x11 (t) + Dα (1 − x11 (t)) ⎪ ⎪  λ1  ⎪   ⎪ ⎪ x12 (t) 1 ⎪ ⎪ + ∗ exp − 1 x11 (t − τ1 ) 1+x (t) ⎪ 12 λ1 ⎪ ⎨  γ0 x˙12 (t) = λ11 + β1 x12 (t) + H Dα (1 − x11 (t)) ⎪    ⎪  ⎪ ⎪ x12 (t) ⎪ ⎪ + λ11 − 1 x12 (t − τ2 ) ∗ exp 1+x12 (t) ⎪ ⎪ γ0 ⎪ ⎩ +Θ1 (u 1 (t)) + φ1 (x1 (t − d) , x2 (t − d))

(9.52)

9 Decentralized Fuzzy Networked Control Systems . . .

152

x11 (t )

u1 (t )

x12 (t )

x21 (t )

u2 (t )

x22 (t )

Fig. 9.7 Large-scale chemical reactor systems with two subsystems

and

⎧   ⎪ x22 (t) −1 ⎪ x ˙ x = + D − x exp (t) (t) (1 (t)) ⎪ 21 α 21 1+x22 (t) ⎪ λ2 21 ⎪ γ0 ⎪  ⎪ 1 ⎪ ⎪ ⎪ ⎨ + λ2 − 1 x21 (t − τ2 ) x˙ (t) = λ12 + β2 x22 (t) + H Dα (1 − x21 (t)) ⎪ ⎪ 22    ⎪  ⎪ ⎪ x22 (t) ⎪ ∗ exp 1+x + λ12 − 1 x22 (t − τ2 ) ⎪ 22 (t) ⎪ ⎪ γ0 ⎪ ⎩ +Θ2 (u 2 (t)) + φ2 (x1 (t − d) , x2 (t − d))

(9.53)

where xi1 and xi2 are the conversion rates of the reaction and the dimensionless temperature, τ1 and τ2 are the recycle delay time. γ0 = 20, H = 8, Dα = 0.072, λ1 = λ2 = 0.8, β = 0.3, Θ1 (u 1 (t)) = (2 + cos (u 1 (t))) u 1 (t), Θ2 (u 2 (t)) = (2 + sin (u 2 (t))) u 2 (t), φ1 and φ2 are the uncertain interconnections with φ1 (x1 (t − d) , x2 (t − d)) = δ1 (x11 (t − d) − x21 (t − d)) + δ2 (x1 (t − d) − x2 (t − d))T (x1 (t − d) + x2 (t − d)) , φ2 (x1 (t − d) , x2 (t − d)) = δ3 (x12 (t − d) − x22 (t − d)) + δ4 (x1 (t − d) − x2 (t − d))T (x1 (t − d) + x2 (t − d)) . where δ1 , δ2 , δ3 , and δ4 are uncertain parameters. Using the T-S fuzzyfication [111] for each subsystem, we have ⎧ 3  ⎪ ⎪ ⎪ y ˙ = h 1i (Ai1 y1 + Di1 y1 (t − τ1 ) + B (u 1 + φ1 )) ⎪ ⎨ 1 i=1

3  ⎪ ⎪ ⎪ ⎪ = h 2i (Ai2 y2 + Di2 y (t − τ2 ) + B (u 2 + φ2 )) y ˙ 2 ⎩ i=1

(9.54)

9.5 Simulations

153

Fig. 9.8 State response of the first subsystem (9.52) with control (9.55)

1.5

y11 y12

1 0.5 0 −0.5 −1 −1.5 −2

0

5

10

15

Time

Fig. 9.9 State response of the second subsystem (9.53) with control (9.55)

1

y21 y22

0.5

0

−0.5

−1

−1.5

0

5

10

15

Time

T   T in which yi = xi − xdi = yi1 yi2 , here xdi = 0.1440 0.8862 is the equilib T rium of each subsystem, B = 0 1 , and 

 −1.4274 0.0757 −2.0508 0.3958 Ai1 = , Ai2 = , −1.4189 −0.9442 −6.4066 1.6168   −4.5279 0.3167 0.25 0 , Di j = . Ai3 = −26.2228 0.9837 0 0.25 Choosing the parameters ωi = 1, τ i = 0.2, εi = 1, λi1 = λi3 = −1, and λi2 = λi4 = 1, solving the LMI gives

9 Decentralized Fuzzy Networked Control Systems . . .

154 Fig. 9.10 State response of the first subsystem (9.52) with control (9.56)

1.5

y11 y12

1 0.5 0 −0.5 −1 −1.5 −2

0

5

10

15

Time

Fig. 9.11 State response of the second subsystem (9.53) with control (9.56)

1

y21 y22

0.5

0

−0.5

−1

−1.5

0

5

10

15

Time

Pi = 6.0261, K i = 1.5229. Based on Theorem 9.1, with δi = 1 we design the following controller 3 u i (t) = −50z i2 (t) − 50z i2 (t)

(9.55)

According to Theorem 9.2, we construct the adaptive controller as follows:   2 1 + 20 u i (t) = − βi (t) z i2 (t) 20z i2 2

(9.56)

9.5 Simulations

155

where  2 2 + 20 − βi (t) β˙i (t) = z i2 (t)2 20z i2

(9.57)

The controllers (9.55) and (9.56) are applied to the interconnected system (9.52)– (9.53). The delays are chosen as τi = d = 0.2. The simulation results are shown T  in Figs. 9.8, 9.9, 9.10, and 9.11 with yi = xi − xd = yi1 yi2 . Figures 9.8 and 9.9 show the state responses of two subsystems with controller (9.55), from the figures we see that the designed controller is effective and the asymptotical stability performance is achieved. With the adaptive controller (9.56) and the tuning law (9.57), the state responses are shown in Figs. 9.10 and 9.11. The figures show that the system state is rendered to converge to a bounded small region quickly, thus the designed decentralized controller achieves the good transient and steady-state performances.

9.6 Conclusion The control problem is investigated for a class of large-scale nonlinear systems with the subsystems exchanging information through networks. The time delays are inherent for the systems because of the information transmission through the computer networks. By T-S fuzzyfication for each subsystem, the interconnected T-S fuzzy systems are obtained. If the bound parameters are known, the memoryless state feedback controller is constructed such that the closed-loop system is asymptotically stable with an exponential decay rate. For the case that the parameters are not available, we design the memoryless decentralized adaptive state feedback controller and the solution of the closed-loop system is shown to converge exponentially to a bounded region.

Part IV

Time-Delay System with Lower Triangular Structure

Chapter 10

Robust Control for a Class of Time-Delay System via Razumikhin Lemma

Abstract The robust control problem is investigated for nonlinear time-delay systems with the form of triangular structure. The uncertain delay disturbances are bounded by nonlinear functions with unknown coefficients. Via the backstepping method, the state feedback time-delay-independent controller is constructed with the help of Razumikhin lemma. Based on Lyapunov stability theory, it is showed that the resulting closed-loop system is UUB stable.

10.1 Introduction Backstepping method is one of the most popular techniques of nonlinear control design [88, 93, 98]. However, it is not an easy task to apply the backstepping method to design controllers for time-delay nonlinear systems with triangular structure [38, 66, 131, 223]. The existence of time delay renders the control problem much more complex and difficult. For a special class of nonlinear time-delay systems, observerbased output feedback controllers were designed in [70]. Although the output feedback control problem was solved for the systems with specific assumptions, state feedback control for general nonlinear time-delay systems with triangular structure is still a difficult problem. In [46], a neural network controller was successfully constructed to control a class of nonlinear time-delay systems. The results in [38, 46, 70, 131] were obtained via the Lyapunov–Krasovskii functional method; thus, it is required that the time delay is either constant or the derivative of the time-varying delays is known and less than one. However, the delays of practical systems may be time varying and their derivative information might be unknown. In [79], a framework was proposed on controlling nonlinear time-delay systems via Razumikhin lemma. In the work, a control Lyapunov-Razumikhin function (CLRF) was proposed and a time-delay-dependent controller was developed. In [80], a delay-independent controller design method was also proposed based on the CLRF idea, but the virtual controller design conditions were not easy to verify. In this chapter, the control problem is considered for a class of triangular structure nonlinear time-delay systems with the coefficients of bound functions unknown. By employing backstepping method, we construct a delay-independent state feedback © Springer Nature Singapore Pte Ltd. 2018 C. Hua et al., Robust Control for Nonlinear Time-Delay Systems, DOI 10.1007/978-981-10-5131-9_10

159

160

10 Robust Control for a Class of Time-Delay System …

control law. The resulting closed-loop control system is shown to be stable in the sense of uniform ultimate boundedness (UUB) based on the Razumikhin lemma.

10.2 Problem Formulation and Preliminaries Consider the following time-delay system ⎧· x i (t) = xi+1 (t) + Fi (x i (t)) ⎪ ⎪ ⎪ ⎪ ⎨ +Hi (x i (t) , x i (t − di (t)) , δi (t)) , i = 1, . . . , n − 1 ⎪ ⎪ ⎪ x· n (t) = u (t) + Fn (x n (t)) ⎪ ⎩ +Hn (x n (t) , x n (t − dn (t) , δn (t)))

(10.1)

where xi ∈  and u ∈  are the state and the control input of the system, respectively. di (t) is the time-varying time delay in xi -subsystem, which satisfies di (t) ≤ τ , where τ is a positive scalar, δi (t) is the uncertain time-varying parameter. The following notation is used: x i (t) = [x1 (t) , x2 (t) , . . . , xi (t)]T , then x (t) = x n (t) and x1 (t) = x 1 (t). Fi (·) are known smooth nonlinear functions and Hi (·) are unknown uncertain nonlinear functions. For system (10.1), we impose the following assumption: Assumption 10.1 The uncertain nonlinear functions δi (t)) yield

Hi (x i (t), x i (t−di (t)),

|Hi (x i (t) , x i (t − di (t)) , δi (t))| ≤ θi  ηi (x i (t)) +

i 



 i j x j (t − di (t)) + εi , ϑi j β

j=1

ηi (·) where εi are known positive scalars, θi and ϑi j are unknown positive scalars,  i j (·) are class-k∞ functions are known positive and smooth nonlinear functions, β i j (0) = 0. and  ηi (0) = β For the system (10.1), we choose the following state transformation 

z 1 (t) = x1 (t) z i (t) = xi (t) − αi−1 (x i−1 (t)) , i = 2, 3, . . . , n

(10.2)

where αi−1 (·) are the smooth virtual control inputs with αi−1 (0) = 0, which will be determined in Sect. 10.3. Under the above state transformation, we can obtain the new system

10.2 Problem Formulation and Preliminaries

⎧· z 1 (t) = z 2 (t) + α1 (x1 (t)) + F1 (x 1 (t)) ⎪ ⎪ ⎪ ⎪ +H ⎪ 1 (x 1 (t) , x 1 (t − d1 (t)) , δ1 (t)) ⎪ ⎪ · ⎪ ⎪ (t)) ⎪ ⎨ z i (t) = z i+1 (t) + αi (x i (t)) + Fi (x i ∂αi−1 · +Hi (x i (t) , x i (t − di (t)) , δi (t)) − i−1 j=1 ∂x j x j ⎪ ⎪ ⎪ i = 2, 3, . . . , n − 1 ⎪ ⎪ · ⎪ ⎪ ⎪ z n = u + Hn (x n (t) , x n (t − dn (t)) , δn (t)) ⎪ ⎪ ⎩ ∂αn−1 · +Fn (x n (t)) − n−1 j=1 ∂x j x j

161

(10.3)

Remark 10.1 System (10.1) is a typical cascade nonlinear system with time delays. For the form of system (10.1) free of time delays, we know there always exists a stabilizing state feedback controller based on the results in [93, 98]. Furthermore, it is not difficult for us to conclude that there always exists a time-delay-dependent controller to stabilize the time-delay system (10.1). However, delays in practical systems are often variant and difficult to be known precisely. Therefore, time-delayindependent controllers are warranted. The ISS nonlinear small gain theorem was used for nonlinear time-delay systems via the Razumikhin lemma in [170], but the results are not easy to be extended to dealing with the composite system (10.3). The reason is that the Lyapunov function chosen in the Razumikhin lemma is for the overall system, but not for a single z i -subsystem. In this note, we will design a timedelay-independent controller for system (10.1) to render the closed-loop system UUB stable. Some researchers have focused on the stabilization problem of the system (10.1). In [38, 131], the Lyapunov–Krasovskii functional method was used to construct a controller with the help of the backstepping method. From the transformed system (10.3), it is easily observed that the delay part z i (t − di (t)) exists not only in z i -subsystem, but also in zl -subsystems (l > i) due to the state transformation. If the Lyapunov–Krasovskii method is used, one needs to design the virtual control input αi (·) in the ith step to dominate (or cancel) the delay part z i (t − di (t)). Therefore, it is difficult to design αi (·) for it should be determined according to zl subsystems (l > i), but zl -subsystems are obtained according to the virtual control input αi (·). This leads to the difficulty in designing the virtual control inputs and the system controller. In this chapter, we will use Razumikhin lemma to construct a state feedback controller for the system and the above-mentioned problem of designing virtual control inputs will be overcome. Remark 10.2 In this chapter, we impose Assumption 10.1 on the uncertain delay functions of system (10.1). In [89], constructive results were presented to stabilize the nonlinear time-delay systems for the case that the parameter is unknown and the delay functions are required to be bounded by |x1 (t − di )|. Compared with that of [89], the system considered in this chapter is more general. Remark 10.3 In view of state transformation (10.2) and αi−1 (0) = 0, from Assumption 10.1, we easily obtain

162

10 Robust Control for a Class of Time-Delay System …

|Hi (x i (t) , x i (t − di (t)) , δi (t))| ≤ θi ηi (z i (t)) +

i 



 ϑi j βi j z j (t − di (t)) + εi

(10.4)

j=1

where ηi (·) and βi j (·) are smooth functions obtained based on the virtual control i j (·). Functions ηi (·) are positive definite, ηi (·) and β inputs αi−1 (·) and functions  and βi j (·) are class-k functions with βi j (χ) = χβ i j (χ). In the controller design procedure, we will employ (10.4) to construct the state feedback controller. Now, we revisit the useful Razumikhin lemma [56]. Lemma 10.1 Suppose f :  × C → n takes (bounded sets of C) into bounded sets of n and consider the retarded functional differential equation (RFDE) ·

x (t) = f (t, xt ) . Suppose that u (s) , v (s), and w (s) are continuous nondecreasing functions, u (s) → ∞ as s → ∞. If there are a continuous function V :  × n → , a continuous nondecreasing function p : + → + , p (s) > s for s > 0 and a constant σ ≥ 0 such that (1) u (x) ≤ V (t, x) ≤ v (x) ·

(2) V (t, x (t)) ≤ −w (x (t)) + σ, if V (t + θ, x (t + θ)) < p (V (t, x (t))) ∀θ ∈ [−τ , 0] then the solutions of the RFDE are uniformly ultimately bounded. In this case, it is said that the system is UUB stable. If σ = 0, the system is said to be asymptotically stable.

10.3 Main Results In this part, we will construct a state feedback controller based on Razumikhin lemma. Choose the following quadratic Lyapunov function for the system (10.3) V =

n 

z 2j (t)

(10.5)

j=1

Obviously, condition (10.1) of lemma 1 is satisfied. Further, we choose p (V (x (t))) = q 2 V ; here, q is a positive scalar satisfying q > 1. So if the following condition holds for 0 ≤ d j (t) ≤ τ 



z j t − d j (t) ≤ z t − d j (t) < q z (t)

(10.6)

10.3 Main Results

163

and if we can design a controller such that condition (10.2) of lemma 1 is satisfied, the closed-loop system will be UUB stable. Now, we present the controller design procedure. Step 1: For z 1 -subsystem, consider the function V1 = z 12 , its time derivative satisfies ·

V 1 = 2z 1 (t) (z 2 (t) + α1 (x1 (t)) + F1 (x 1 (t)) +H1 (x 1 (t) , x 1 (t − d1 (t))) , δ1 (t)) ≤ 2z 1 (t) z 2 (t) + 2z 1 (α1 (x1 (t)) + F1 (x 1 (t))) + 2 |z 1 (t)| (θ1 η1 (z 1 (t)) + ϑ11 β11 (|z 1 (t − d1 (t))|)) + 2 |z 1 (t)| ε1

(10.7)

In view of η1 (0) = 0, there always exists function η 1 (χ) such that η1 (χ) = χη 1 (χ), then further, we have 2 |z 1 (t)| θ1 η1 (z 1 (t)) ≤ θ12 z 12 (t) + z 12 (t) η 21 (z 1 (t))

(10.8)

With inequality (10.6), one has 2 |z 1 (t)| ϑ11 β11 (|z 1 (t − d1 (t))|) ≤ 2 |z 1 (t)| ϑ11 β11 (q z (t)) Considering the following inequality which holds for k∞ function β11 (·) β11 (q z (t)) ≤

n  β11 (nq |zl (t)|) , l=1

one can obtain 2 |z 1 (t)| ϑ11 β11 (|z 1 (t − d1 (t))|) ≤ 2 |z 1 (t)|

n  ϑ11 β11 (nq |zl (t)|) l=1

≤ nϑ211 z 12 (t) +

n  2 n 2 q 2 zl2 (t) β 11 (nq |zl (t)|) l=1

Substituting (10.8) and (10.9) into (10.7) gives ·

V 1 ≤ 2z 1 (t) (z 2 (t) + α1 (x1 (t)) + F1 (x 1 (t))) 

ε2 + θ12 + nϑ211 z 12 (t) + h 1 z 12 + 1 h1

(10.9)

164

10 Robust Control for a Class of Time-Delay System …

+

n 

2

n 2 q 2 zl2 (t) β 11 (nq |zl (t)|)

(10.10)

l=1

where h 1 is a positive scalar. Choose 1 1 1 α1 (x1 (t)) = − kz 1 (t) − h 1 z 1 − F1 (x 1 (t)) − a1 z 1 (t) 2 2 2 1 1 2 − b1 z 13 (t) − n 2 q 2 z 1 (t) β 11 (nq |z 1 (t)|) 2 2

(10.11)

where a1 , b1 , and k are positive scalars. Substituting (10.11) into (10.10) leads to ·

V 1 ≤ −kV1 + 2z 1 (t) z 2 (t) 

 − a1 − θ12 + nϑ211 z 12 (t) − b1 z 14 (t) +

n  ε2 2 n 2 q 2 zl2 (t) β 11 (nq |zl (t)|) + 1 h1 l=2

(10.12)

Step i: For z i -subsystem, the time derivative of z i2 (t) is ·

2z i (t) z i (t) = 2z i (t) (z i+1 (t) + αi (x i (t)) + Fi (x i (t)) +Hi (x i (t) , x i (t − di (t)) , δi (t))) i−1  ∂αi−1 



x j+1 (t) + F j x j (t) ∂x j j=1  



+H j x j (t) , x j t − d j (t) , δ j (t)

− 2z i (t)

With (10.4), it follows 2z i (t) (Hi (x i (t) , x i (t − di (t)) , δi (t)) −

i−1  ∂αi−1 j=1

∂x j ⎛

Hj



⎞ 



x j (t) , x j t − d j (t) , δ j (t) ⎠

≤ 2 |z i (t)| ⎝θi ηi (z i (t)) +

i 

⎞

 ϑi j βi j z j (t − di (t)) ⎠

j=1

+ 2 |z i (t)| εi + 2

i−1   j=1

   

z i (t) ∂αi−1  θ j η j z j (t) + ε j  ∂x  j

(10.13)

10.3 Main Results

+

165

j 

  



ϑ jl β jl z l t − d j (t)

(10.14)

l=1

With ηi (0) = 0, there exist functions ηil (·) such that ηi (z i (t)) = (z l (t)). Moreover, the following inequalities hold

i

l=1 z l

(t) ηil

2 |z i (t)| θi ηi (z i (t)) =

i 

2θi |z i (t)| zl (t) ηil (z l (t))

l=1

≤

i  

z i2 (t) ηil2 (z l (t)) + θi2 zl2 (t)

(10.15)

l=1

and 2

 i−1     

z i (t) ∂αi−1  θ j η j z j (t)   ∂x j

j=1

  j i−1    ∂αi−1   θ j zl (t) η jl (z l (t)) 2 z i (t) = ∂x  j

j=1 l=1

≤

 j i−1   j=1 l=1

∂αi−1 z i (t) η jl (z l (t)) ∂x j



2 +

θ2j zl2

(t)

(10.16)

With (10.6), we have 2 |z i (t)|

i 



 ϑi j βi j z j (t − di (t))

j=1

=

i i  

2 |z i (t)| ϑi j βi j (nq |zl (t)|)

j=1 l=1

+

i n  

2 |z i (t)| ϑi j βi j (nq |zl (t)|)

j=1 l=i+1

≤

i  i    2 n 2 q 2 z i2 (t) β i j (nq |zl (t)|) + ϑi2j zl2 (t) j=1 l=1

+

i n     2 n 2 q 2 zl2 (t) β i j (nq |zl (t)|) + ϑi2j z i2 (t) j=1 l=i+1

(10.17)

166

10 Robust Control for a Class of Time-Delay System …

and   j i−1     



∂αi−1  ϑ jl β jl z l t − d j (t) 2 z i (t)  ∂x j

j=1 l=1

 i   j i−1      ∂α i−1  ϑ jl ≤ 2 z i (t) β jl (nq |z m (t)|)  ∂x j

j=1 l=1 n 

+

β jl (nq |z m (t)|)

m=i+1 j i−1  i  

≤

 2 2

n q

j=1 l=1 m=1

+

j i i−1   

∂αi−1 z i (t) ∂x j

ϑ2jl z m2 (t) +

j=1 l=1 m=1

+

m=1



j i−1  

2

2

β jl (nq |z m (t)|)

j n i−1   

ϑ2jl z i2 (t)

j=1 l=1 m=i+1

n 

 n 2 q 2 z m2 (t)

j=1 l=1 m=i+1

∂αi−1 ∂x j

2

2

β jl (nq |z m (t)|)

(10.18)

In addition, the following inequalities hold  i−1    ∂αi−1   2 |z i (t)| εi + 2 z i (t) ∂x  ε j j

j=1

≤ h i z i2 (t) +

  i 2 ∂αi−1 2  ε j h i z i (t) + ∂x j h j=1 j=1 i

i−1 

(10.19)

Substituting (10.14)–(10.19) into (10.13) gives ·

2z i (t) z i (t) ⎛

⎞ i−1   

∂α i−1 x j+1 (t) + F j x j (t) ⎠ ≤ 2z i (t) ⎝z i+1 (t) + αi (x i (t)) + Fi (x i (t)) − ∂x j j=1

+

i 

θi2 zl2 (t) +

l=1

+

j=1 l=1 m=1

+

j=1 l=1

θ2j zl2 (t) +

j=1 l=1

j  i−1  i 

j  i−1  

j i−1  

2 (t) + ϑ2jl z m

j i−1  

(n − i) ϑ2jl z i2 (t)

j=1 l=1

i  j=1

⎛ ⎞ i  ⎝ ϑi2j zl2 (t) + (n − i) ϑi2j z i2 (t)⎠ l=1

2  i ∂αi−1 z i (t) η jl (zl (t)) + (z i (t) ηil (z l (t)))2 ∂x j l=1

10.3 Main Results

+

i n  

167 2

n 2 q 2 zl2 (t) β i j (nq |zl (t)|) +

j=1 l=i+1

+

j=1 l=1 m=1

+

j=1

+

2

n 2 q 2 z i2 (t) β i j (nq |zl (t)|)

j=1 l=1

j  i−1  i  

i−1 

i  i 

 ∂αi−1 2 nqz i (t) β jl (nq |z m (t)|) ∂x j

  i  ∂αi−1 2 h i z i (t) + h i z i2 (t) + ∂x j

j=1

 j i−1  n  

nqz m (t) β jl (nq |z m (t)|)

j=1 l=1 m=i+1

ε2j hi  ∂αi−1 2 ∂x j

(10.20)

By choosing the virtual control input as 1 1 αi (x i (t)) = − kz i (t) − h i z i (t) − z i−1 (t) 2 2 i−1  

1 ∂αi−1  1 − ai z i (t) − bi z i3 (t) − Fi (x i (t)) + x j+1 + F j x j (t) 2 2 ∂x j j=1

  i−1 j i ∂αi−1 2 1  z i (t) nqβ jl (nq |z m (t)|) − 2 j=1 l=1 m=1 ∂x j −

1  2 2 1 2 n q z i (t) β pj (nq |z i (t)|) − z i (t) ηil2 (z l (t)) 2 p=1 j=1 2 l=1

−

  i−1 p−1 j ∂α p−1 2 2 1  2 2 n q z i (t) β jl (nq |z i (t)|) 2 p=2 j=1 l=1 ∂x j

i−1

p

i

  2  i−1 i−1 j ∂αi−1 2 1   ∂αi−1 1 − h i z i (t) − z i (t) η jl (z l (t)) 2 j=1 ∂x j 2 j=1 l=1 ∂x j 1  2 2 2 n q z i (t) β i j (nq |zl (t)|) 2 j=1 l=1 i

− we have

i

(10.21)

168

10 Robust Control for a Class of Time-Delay System … ·

2z i (t) z i (t) ≤ −kz i2 (t) − 2z i−1 (t) z i (t) + 2z i (t) z i+1 (t) − ai z i2 (t) p  i−1  2  nqz i (t) β pj (nq |z i (t)|) − bi z i4 (t) − p=1 j=1

+

i n  

2

n 2 q 2 zl2 (t) β i j (nq |zl (t)|)

j=1 l=i+1

 p−1 j  i−1    ∂α p−1 2 nqz i (t) β jl (nq |z i (t)|) − ∂x j p=2 j=1 l=1 +

 j n i−1   

nqz m (t) β jl (nq |z m (t)|)

j=1 l=1 m=i+1

+

i 

θi2 zl2 (t) +

j i−1  

l=1

+

 i i   j=1

+

θ2j zl2 (t) +

j=1 l=1

ϑi2j zl2

2

i  ε2j j=1

(t) + (n −

∂αi−1 ∂x j

i) ϑi2j z i2

hi 

(t)

l=1

j i−1  

(n − i) ϑ2jl z i2 (t) +

j i−1  i  

ϑ2jl z m2 (t)

(10.22)

j=1 l=1 m=1

j=1 l=1

Step n: By direct verification, the following inequalities hold 2 |z n (t)| θn ηn (z n (t)) ≤

n  

2 z n2 (t) ηnl (z l (t)) + θn2 zl2 (t)



(10.23)

l=1

 n−1    

∂αn−1   2 z n (t) ∂x  θ j η j z j (t) j=1

≤

 j n−1   j=1 l=1

j

∂αn−1 z n (t) η jl (z l (t)) ∂x j



2 +

θ2j zl2

(t)

(10.24)

10.3 Main Results

169

2 |z n (t)|

n 

ϑn j βn j (z n (t − di (t)))

j=1

 n  n  2  2 2 nqz n (t) β n j (nq |zl (t)|) + ϑn j zl (t) ≤

(10.25)

j=1 l=1

and   j n−1    ∂αn−1  ϑ jl β jl (z l (t − dl (t))) 2 z n (t) ∂x  j

j=1 l=1

j n−1  n   

≤

nqz n (t)

j=1 l=1 m=1

+

j n−1  n  

∂αn−1 β (nq |z m (t)|) ∂x j jl

2

ϑ2jl z m2 (t)

(10.26)

j=1 l=1 m=1

We choose the following controller 1 1 1 1 u (t) = − kz n (t) − h n z n (t) − z n−1 (t) − an z n (t) − bn z n3 (t) 2 2 2 2 n−1  

∂αn−1  x j+1 + F j x j (t) − Fn (x n (t)) + ∂x j j=1 −

 2 n−1 j ∂αn−1 1  z n (t) η jl (z l (t)) 2 j=1 l=1 ∂x j

−

1 1  2 2 2 2 z n (t) ηnl n q z n (t) β n j (nq |zl (t)|) (z l (t)) − 2 l=1 2 j=1 l=1

−

 2 n−1 j n ∂αn−1 1  z n (t) nq β jl (nq |z m (t)|) 2 j=1 l=1 m=1 ∂x j

−

1  2 2 2 n q z n (t) β i j (nq |z n (t)|) 2 i=1 j=1

n

n−1

n

n

i

 2 n−1 i−1 j ∂αi−1 1  − z n (t) nq β (nq |z n (t)|) 2 i=2 j=1 l=1 ∂x j jl With (10.23)–(10.27), one easily obtains

(10.27)

170

10 Robust Control for a Class of Time-Delay System … ·

2z n (t) z n (t) ≤ −kz n2 (t) − 2z n−1 (t) z n (t) − an z n2 (t) − bn z n4 (t) −

i n−1  

2

n 2 q 2 z n2 (t) β i j (nq |z n (t)|) +

i=1 j=1

−

 n 2 q 2 z n2 (t)

j n−1  

θ2j zl2

(t) +

∂αi−1 ∂x j

n n  

j=1 l=1

+

θn2 zl2 (t)

l=1

j n−1  i−1   i=2 j=1 l=1

+

n 

2

2

β jl (nq |z n (t)|)

ϑ2n j zl2 (t)

j=1 l=1

j n−1  n  

n  ε2j

ϑ2jl z m2 (t) +

j=1 l=1 m=1

j=1

(10.28)

hn

With (10.12), (10.22), and (10.28), it is easy for us to get ·

V =

n  · 2z i (t) z i (t) j=1

n n  i  ε2j 

 2 4 ≤ −kV + (ci − ai ) z i − bi z i + h i=1 i=1 j=1 i

(10.29)

where n n m−1  

  c1 = θ12 + nϑ211 + θ2j + θ2j m=2 j=1

j=1

+

j n  

ϑ2jm +

j=1 m=1

ci =

n 

θ2j +

j=i

j n m−1   m=2 j=1 l=1

j n−1  

θl2 +

j=i l=i

+

ϑ2jl ,

j i−1  

j n  

ϑ2jl +

j=i l=1

(n − i) ϑ2jl +

i 

(n − i) ϑi2j

j=1

j n−1  l  

2 ϑlm ,

j=i l=1 m=1

j=1 l=1

i = 2, 3, . . . , n − 1, cn = θn2 +

n  j=1

ϑ2n j +

j n−1   j=1 l=1

ϑ2jl

(10.30)

10.3 Main Results

171

If parameter ai ≥ ci , we have ·

V ≤ −kV +

i n   ε2j i=1 j=1

hi

(10.31)

and if parameter ai < ci , one has ·

V ≤ −kV +

n n i 2  (ci − ai )2   ε j + 4bi h i=1 i=1 j=1 i

(10.32)

From (10.31) and (10.32), the resulting closed-loop system is UUB stable based on Lemma 10.1. With the above analysis, we have the following main result: Theorem 10.1 For system (10.1) satisfying Assumption 10.1, the state feedback controller (10.20) renders the resulting closed-loop system UUB stable. Remark 10.4 In engineering system, the control requirement often appears in a range, which means that the solutions of the resulting closed-loop system should belong to a suitable bound. Our aim is to design a controller such that the system state converges to a required region. Therefore, UUB is a significant concept for engineering applications. For a practical system, we may estimate the unknown parameters θi and ϑi j at first. Then, based on the estimated values, we obtain ci from (10.30). The parameters ai can be chosen to be a little larger than the estimated ci . If the parameters εi = 0 and the parameters ai ≥ ci , it can be seen from (10.31) that the resulting closed-loop system is asymptotically stable. The parameters bi are further to guarantee the boundedness of the solutions of closed-loop system when ai < ci . In practical application, we can adjust the parameters k, ai , bi , and h i to render that the closed-loop system has good transient and steady-state performances.

10.4 Conclusion In this chapter, we have studied the state feedback control problem for a class of nonlinear time-delay systems. The systems considered are in the form of triangular structure with unknown time-varying delays. Based on the Razumikhin lemma, a time-delay-independent feedback controller is designed via the backstepping method. The resulting closed-loop system is proved to be UUB stable.

Chapter 11

Backstepping Control for Nonlinear Time-Delay System via L-K Function

Abstract The state feedback control problem is addressed for a class of nonlinear time-delay systems. The time delays appear in all state variables of the nonlinear system, which brings a challenging issue for controller design. With an introduced new Lyapunov–Krasovskii functional and the help of backstepping method, we develop memoryless state feedback controller, which does not need the precise knowledge of time delay. It is rigorously proved that the closed-loop system is asymptotically stable. Chemical reactor plants are typical nonlinear systems with time delay. We apply the developed method to the control design of a two-stage chemical reactor with delayed recycle streams, and the simulation results verify the effectiveness of the main results.

11.1 Introduction Backstepping method has proved to be powerful for the controller design of nonlinear systems with strict-feedback form, see [93, 98] and the references therein. Many practical industrial systems are founded to be the strict-feedback form. In [159], the authors successfully designed the state feedback controller for piezoactuator-driven stages via backstepping method. The output feedback backstepping control method was proposed in [198] for the magnetic levitation systems. How to apply this method to the controller design of the nonlinear systems with time delay is a challenging subject. With the time delay only appearing in system output, the dynamic output feedback controller was presented in [71]. For the state feedback problem of nonlinear time-delay systems, the delays appear in all state variables. To deal with the time delay, we should construct the Lyapunov–Krasovskii functional in each step. This is different from the output feedback controller design, in which we only construct the Lyapunov functional in the first step to deal with the delays because of all the delay uncertainties bounded by the functions of the delayed output. Thus, the state feedback control is a challenging problem for the nonlinear time-delay systems. To deal with the problem, some research achievements have been made. In [61, 213], the neural networks were used to approximate the unknown nonlinear functions and the novel delay independent controllers were designed such that the closed-loop system © Springer Nature Singapore Pte Ltd. 2018 C. Hua et al., Robust Control for Nonlinear Time-Delay Systems, DOI 10.1007/978-981-10-5131-9_11

173

174

11 Backstepping Control for Nonlinear Time-Delay System via L-K Function

was semiglobally stable in the sense of uniform ultimate boundedness. A state feedback controller was designed such that the closed-loop system was asymptotically stable in [131], but it was lately shown in [102, 224] that the virtual control input of each step was difficult to design using the proposed recursive design method. It was reported in [102, 103] that the memorial controller can be designed for such systems; however, a memorial controller requires the precise value of time delay and a large number of past states should be stored for controller implementation. In Chap. 10, we developed a control design method based on Razumikhin lemma. But the controller design procedure is complicated and not applicable to many nonlinear time-delay systems. Furthermore, when a new subsystem is added, the former virtual control input needs to be redesigned. Compared with Razumikhin method, Lyapunov–Krasovskii functional method is more flexible for the stability analysis and controller design of time-delay systems. In [90], the control Lyapunov functional was developed for nonlinear systems with time delay and corresponding control problem was discussed. In this chapter, we consider the controller design problem for a class of strictfeedback nonlinear systems with time delay. By choosing new Lyapunov–Krasovskii functional, we develop a memoryless state feedback controller via the backstepping method, which does not require the precise knowledge of the time delay. It is proved that the resultant closed-loop system is asymptotically stable. In chemical industry, the chemical reactor recycle system [101, 130, 143] is often employed to fully utilize the raw materials. It is well known that the chemical reactor system is nonlinear for the complex behavior. Time delay is an inherent phenomenon in the recycling system. Therefore, it is a typical nonlinear system with time delay. We applies the theoretic achievements to the controller design of chemical reactor recycling systems. The simulation results show the effectiveness of the proposed method.

11.2 System Description and Preliminaries Consider the following time-delay system ⎧ x˙i (t) = xi+1 (t) + gi (x i (t)) ⎪ ⎪ ⎪ ⎪ + f i (x1 (t − d1 ) , x2 (t − d2 ) , . . . , xi (t − di )) ⎪ ⎪ ⎨ i = 1, 2, . . . , n − 1 x˙n (t) = u (t) + gn (x n (t)) ⎪ ⎪ ⎪ ⎪ + ⎪ ⎪ f n (x1 (t − d1 ) , x2 (t − d2 ) , . . . , xn (t − dn )) ⎩ x (t) = ϕ (t) , t ∈ −d 0

(11.1)

where xi ∈ R and u ∈ R are the state variable and control input, respectively; x i (t) = (x1 (t) , x2 (t) , . . . , xi (t))T , x (t) = x n (t); di is unknown delay time, ϕ (t) is the initial continuous function, d ≥ max {d1 , d2 , . . . , dn } ; gi (·) and f i (·) are known and unknown smooth nonlinear functions, respectively, and gi (0) = f i (0) = 0.

11.2 System Description and Preliminaries

175

For system (11.1), we choose the following state transformation 

x1 (t) = z 1 (t) , xi+1 (t) = z i+1 (t) − αi (x i (t)) , i = 1, 2, . . . , n − 1

(11.2)

where αi (·) are the smooth virtual control inputs to be determined. With the transformation (11.2), it gives ⎧ ⎪ ⎪ z˙ 1 (t) = z 2 (t) − α1 (x1 (t)) + g1 (x1 (t)) ⎪ ⎪ ⎪ ⎪ + f 1 (x1 (t − d1 )) , ⎪ ⎪ z˙ i (t) = z i+1 (t) − αi (x i (t)) ⎪ ⎪ ⎪ ⎪ + f i (x1 (t − d1 ) , x2 (t − d2 ) , . . . , xi (t − di )) ⎪ ⎪ ⎪ i−1 ⎪ ⎪ ⎨ ∂αi−1 (x i−1 (t)) x˙ j (t) , + gi (x i (t)) + ∂x j (t) ⎪ j=1 ⎪ ⎪ ⎪ i = 2, 3, . . . n − 1, ⎪ ⎪ ⎪ ⎪ z ˙ ⎪ n (t) = u (t) + f n (x 1 (t − d1 ) , . . . , x n (t − dn )) ⎪ ⎪ n−1 ⎪ ⎪ ⎪ ∂αn−1 (x n−1 (t)) ⎪ + gn (x n (t)) + ⎪ x˙ j (t) ⎪ ∂x j (t) ⎩

(11.3)

j=1

For system (11.3), we impose the following assumption: Assumption 11.1 The nonlinear functions (t − di )) satisfy

f i (x1 (t − d1 ) , x2 (t − d2 ) , . . . , xi

| f i (x1 (t − d1 ) , x2 (t − d2 ) , . . . , xi (t − di ))| i ≤ ϕl (xl (t − dl )) l=1

where ϕl are known smooth functions with ϕl (0) = 0. Remark 11.1 Assumption 11.1 is very general for nonlinear function fi , which is satisfied in many systems. Considering the smoothness of functions ϕl and the designed virtual control inputs αi , we have f i2 (x1 (t − d1 ) , x2 (t − d2 ) , . . . , xi (t − di )) ≤ (ϕ1 (z 1 (t − d1 )) +

i ϕl (zl (t − dl ) − αl−1 (z l−1 (t − dl−1 ))))2 l=2

≤ ρi (z i (t − di )) +

i i−1

 βi jl z j (t − dl ) j=1 l= j

(11.4)

176

11 Backstepping Control for Nonlinear Time-Delay System via L-K Function

where αl−1 (z l−1 (t)) = αl−1 (x l−1 (t)) , ρi (·), and βi jl (·) are known positive definite and sufficiently smooth functions with ρi (χ) = χ2 ρi (χ) , where ρi (χ) are also smooth functions. For the transformed system (11.3), if the delayed state is available, one may easily construct a state feedback controller via the backstepping method. But it is not easy to obtain the precise information of delays for many practical systems. Therefore, it is desirable to construct the memoryless state feedback controller for these systems. In this chapter, we use new Lyapunov–Krasovskii functional to design the controller such that the resultant closed-loop system is asymptotically stable.

11.3 Controller Design for the Second-Order System In this section, we show the procedure of designing a memoryless controller for the second-order cascade time-delay system. Let us consider the transformed second-order cascade system ⎧ z˙ 1 (t) = z 2 (t) − α1 (z 1 (t)) ⎪ ⎪ ⎨ + g (x (t)) + f (x (t − d )) , 1 1 1 1 1 ∂α1 (x1 (t)) z ˙ x˙1 (t) = u + g + (t) (t) (x (t)) ⎪ 2 2 2 ∂x1 (t) ⎪ ⎩ + f 2 (x1 (t − d1 ) , x2 (t − d2 ))

(11.5)

Firstly, for z 1 -subsystem of (11.5), we construct the Lyapunov functional as V1 = z 12 (t) + U1

(11.6)

t in which U1 = t−d1 ε11 eγ1 (ξ−t) z 12 (ξ) ρ1 (z 1 (ξ)) dξ with parameters ε11 > 0 and γ1 > 2. In (11.6), we use an exponential function in U1 , which is useful for the design of the memoryless controller. Along system (11.5), the derivative of V1 satisfies V˙1 = 2z 1 (t) (z 2 (t) − α1 (x1 (t)) + g1 (x1 (t)) + f 1 (x1 (t − d1 ))) + ε11 z 12 ρ1 (z 1 (t)) − ε11 e−γ1 d1 z 12 (t − d1 ) ρ1 (z 1 (t − d1 )) − γ1 U1 . Note that di ≤ d, it follows γ1 d 2 z 1 (t) V˙1 ≤ 2z 1 (t) (z 2 (t) − α1 (z 1 ) + g1 (x1 (t))) + ε−1 11 e

+ ε11 e−γ1 d f 12 (x1 (t − d1 )) + ε11 z 12 (t) ρ1 (z 1 (t)) − ε11 e−γ1 d z 12 (t − d1 ) ρ1 (z 1 (t − d1 )) − γ1 U1

(11.7)

11.3 Controller Design for the Second-Order System

177

We select the virtual control input 1 −1 γ1 d ε e z 1 (t) + g1 (x1 (t)) 2 11 1 1 1 + ε11 z 1 (t) ρ1 (z 1 (t)) + γ1 z 1 (t) + z 1 (t) 2 2 2

α1 (x1 (t)) =

(11.8)

Substituting (11.4) and (11.8) into (11.7) gives V˙1 ≤ −γ1 V1 + z 22 (t)

(11.9)

Then, for z 2 -subsystem we choose the following Lyapunov functional V2 = z 22 (t) + U2

(11.10)

with

U2 = +

t

t−d2

t t−d1

ε22 eγ2 (ξ−t) (z 22 (ξ) ρ2 (z 2 (ξ)) + β212 (z 1 (ξ)))dξ eγ2 (ξ−t) (ε21 z 12 (ξ) ρ1 (z 1 (ξ)) + ε22 β211 (z 1 (ξ)))dξ,

where ε21 , ε22 , and γ2 are positive scalars. Along system (11.5), the time derivative of z 22 (t) is 2z 2 (t) z˙ 2 (t) = 2z 2 (t) (u (t) + g2 (x 2 (t)) ∂α1 (x1 (t)) + f 2 (x1 (t − d1 ) , x2 (t − d2 )) + x˙1 (t)) ∂x1 (t) ∂α1 (x1 (t)) ≤ 2z 2 (t) (u (t) + (x2 (t) + g1 (x1 )) ∂x1 (t) 1 ∂α1 (x1 (t)) 2 + g2 (x 2 (t)) + eγ2 d (ε−1 ) 21 z 2 (t) ( 2 ∂x1 (t) −γ2 d + ε−1 (ε21 f 12 (x1 (t − d1 )) 22 z 2 (t)) + e + ε22 f 22 (x1 (t − d1 ) , x2 (t − d2 ))) ∂α1 (x1 (t)) ≤ 2z 2 (t) (u + g2 (x 2 ) + (x2 + g1 (x1 )) ∂x1 (t) 1 ∂α1 (x1 (t)) 2 −1 + eγ2 d (ε−1 ) ) 22 z 2 (t) + ε21 z 2 (t) ( 2 ∂x1 (t) + e−γ2 d (ε22 z 22 (t − d2 ) ρ2 (z 2 (t − d2 )) + ε22 β211 (z 1 (t − d1 )) + ε22 β212 (z 1 (t − d2 )) + ε21 z 12 (t − d1 ) ρ1 (z 1 (t − d1 )))

(11.11)

178

11 Backstepping Control for Nonlinear Time-Delay System via L-K Function

and the time derivative of U2 is U˙ 2 = −γ2 U2 + ε21 z 12 (t) ρ1 (z 1 (t)) + ε22 β211 (z 1 (t)) + ε22 β212 (z 1 (t)) + ε22 z 22 (t) ρ2 (z 2 (t)) − e−γ2 d1 (ε21 z 12 (t − d1 ) ρ1 (z 1 (t − d1 )) + ε22 β211 (z 1 (t − d1 ))) − ε22 e−γ2 d2 (β212 (z 1 (t − d2 )) + z 22 (t − d2 ) ρ2 (z 2 (t − d2 )))

(11.12)

Note that di ≤ d, with (11.11) and (11.12) the time derivative of V2 yields ∂α1 (x1 (t)) V˙2 ≤ 2z 2 (t) (u (t) + g2 (x 2 ) + (x2 (t) + g1 (x1 )) ∂x1 (t) 1 ∂α1 (x1 (t)) 2 −1 ) )) + eγ2 d (ε−1 22 z 2 (t) + ε21 z 2 (t) ( 2 ∂x1 (t) − γ2 U2 + ε21 z 12 (t) ρ1 (z 1 (t))

 + ε22 β211 (z 1 (t)) + β212 (z 1 (t)) + z 22 (t) ρ2 (z 2 (t))

(11.13)

For system (11.5), we design the controller ∂α1 (x1 (t)) (x2 (t) + g1 (x1 )) − g2 (x 2 ) ∂x1 (t) 1 1 1 γ2 d z 2 (t) − γ2 z 2 (t) − ε22 z 2 (t) ρ2 (z 2 (t)) − ε−1 22 e 2 2 2   ∂α1 (x1 (t)) 2 1 −1 γ2 d − u s (t) − ε21 e z 2 (t) 2 ∂x1 (t)

u (t) = −

(11.14)

where u s is a compensatory part to be defined below. Substituting (11.14) into (11.13) gives V˙2 ≤ −γ2 V2 + ε21 z 12 (t) ρ1 (z 1 (t)) + ε22 (β211 (z 1 (t)) + β212 (z 1 (t))) − 2z 2 (t) u s (t)

(11.15)

Now, we present the preliminary result of this paper: Theorem 11.1 For cascade system (11.5), if there exists non-decreasing positive function η1 (·) with η1 (0) > 0 such that ε21 z 12 (t) ρ1 (z 1 (t)) + ε22 (β211 (z 1 (t)) + β212 (z 1 (t)))

 ≤ η1 z 12 z 12 then the state feedback controller (11.14) with

(11.16)

11.3 Controller Design for the Second-Order System

1 u s (t) = z 2 (t) η1 2

179



2z 22 γ1

 (11.17)

renders the closed-loop system (11.5), (11.8), (11.14) and (11.17) asymptotically stable.

V1 η1 (χ) dχ for z 1 -subsystem, Proof We choose the Lyapunov functional V 1 = 0

then

 V˙ 1 ≤ η1 (V1 ) −γ1 V1 + z 22 (t)

  1 1 2 ≤ − γ1 η1 (V1 ) V1 + η1 (V1 ) z 2 (t) − γ1 V1 2 2  2  2z 1 (t) 2 ≤ − γ1 η1 (V1 ) V1 + z 22 (t) η1 2 γ1

(11.18)

The above inequality can be checked as follows: if z 22 (t) ≤ 21 γ1 V1 , the above inequal2z 2

ity holds; if z 22 (t) > 21 γ1 V1 , using γ12 instead of V1 gives above inequality. For system (11.5), choosing the Lyapunov functional V 2 = V2 + V 1 gives   1 ˙ V2 ≤ − γ1 − 1 η1 (V1 ) V1 − γ2 V2 2 ≤ −γ 2 V 2

(11.19)



  where γ 2 = min 21 γ1 − 1 , γ2 . Based on (11.19) with γ1 > 2, the closed-loop system is asymptotically stable. With the smoothness of the virtual control input (11.8), the designed controller (11.14) renders system (11.1) asymptotically stable. Remark 11.2 The compensatory part u s (t) (11.17) plays an important role in the designed controller (11.14), which is used to dominate the effect of delayed z 1 in z 2 subsystem. This operation helps us design the memoryless controller and guarantees the stability of the closed-loop system. From Theorem 11.1, we know that the key of the controller design is to determine function η1 (·) . Based on (11.4), we obtain functions ρi (·) and βi jl (·) . For the smoothness of functions f i , there exist smooth 2   functions

2 βi jl such that βi jl (χ) χ ≥ βi jl (χ) . We choose function η1 (·) such that  212 (z 1 (t)). Obviously, the choη1 z 1 (t) ≥ ε21 ρ1 (z 1 (t)) + ε22 β211 (z 1 (t)) + ε22 β sen function will satisfy (11.16). Remark 11.3 Different from the Lyapunov functional used in [102, 103, 131, 224], an exponential function is adopted in (11.25). With the proposed new nonlinear Lyapunov functional, the memoryless controller is successfully designed such that the closed-loop system is asymptotically stable. From the controller design procedure, we can see that the precise delay information di is not used and the upper bound d of di is employed instead. It is easy and feasible to estimate the upper bound d for many

180

11 Backstepping Control for Nonlinear Time-Delay System via L-K Function

practical time-delay systems. In this chapter, it is considered that the time delays di are constants. If they (di are di (t)) and satisfy d˙i (t) ≤ ϑi < 1, we t are time-varying 1 γ1 (ξ−t) 2 ε e z choose U 1 = 1−ϑ 1 (ξ) ρ1 (z 1 (ξ)) dξ and t−d1 (t) 11 1 U2 =

1 1 − ϑ2

t t−d2 (t)

ε22 eγ2 (ξ−t) (z 22 (ξ) ρ2 (z 2 (ξ))

t 1 + β212 (z 1 (ξ)))dξ + eγ2 (ξ−t) 1 − ϑ1 t−d1 (t)

 ∗ ε21 z 12 (ξ) ρ1 (z 1 (ξ)) + ε22 β211 (z 1 (ξ)) dξ, instead of the former U1 and U2 . With the functional, we can also use the proposed method to design the corresponding controller.

11.4 Extension to the nth-Order System In this section, the proposed controller design idea is applied to an n-order system. We have the following theorem: Theorem 11.2 If there exist non-decreasing functions ηi−1 (·) for 2 ≤ i ≤ n such that i−1 m=1

εim z m2 (t) ρm (z m (t)) +

i m−1 m

 εim βi jl z j (t)

m=1 j=1 l= j



≤ ηi−1 V i−1 V i−1

(11.20)

where εim are positive scalars, V i−1 = Vi−1 +

V i−2

ηi−2 (χ) dχ with Vi−1 obtained

0

from (11.24) and (11.25) below, then the closed-loop system is asymptotically stable with the following control law u (t) = −

n−1 ∂αn−1 (x n−1 (t))

m=1

∂xm (t)

(xm+1 (t) + gm (x m (t)))

1 1 eγn d z n (t) − γn z n (t) − gn (x n (t)) − ε−1 2 nn 2  2  2z n 1 1 − εnn z n (t) ρn (z n (t)) − z n (t) ηn−1 2 γ n−1 2   n−1 ∂αn−1 (x n−1 (t)) 2 1 ε−1 − eγn d z n (t) nm 2 ∂xm (t) m=1

(11.21)

11.4 Extension to the nth-Order System

181

where γn > 0 and γ n−1 > 2. Proof Step 1: For z 1 -subsystem of (11.3), we choose Lyapunov functional V1 as (11.6) and the virtual control input α1 (x1 ) as (11.8) with parameter γ1 determined by (11.32). We obtain V˙1 ≤ −γ1 V1 + z 22 (t). Step 2: For z 2 -subsystem, we select the Lyapunov functional V2 as (11.10). The virtual control input is chosen as ∂α1 (x1 (t)) (x2 (t) + g1 (x 1 )) + g2 (x 2 ) ∂x1 (t)  2  2z 2 1 −1 γ2 d 1 ε e + γ2 + 1 z 2 (t) + z 2 (t) η1 + 2 22 2 γ1  2 ∂α1 (x1 (t)) 1 γ2 d + ε−1 z 2 (t) 21 e 2 ∂x1 (t) 1 + ε22 z 2 (t) ρ2 (z 2 (t)) 2

α2 (x 2 ) =

(11.22)

where parameter γ2 is determined by (11.32) and function η1 (·) satisfies (11.16). With the virtual control input, it follows V˙2 ≤ −γ2 V2 + z 32 (t) + ε21 z 12 (t) ρ1 (z 1 (t)) + ε22 β211 (z 1 (t))  2 2z 2 . + ε22 β212 (z 1 (t)) − z 22 (t) η1 γ1 Choosing Lyapunov functional V 2 = V2 + V 1 for (z 1 , z 2 ) subsystem of (11.3) gives V˙ 2 ≤ −γ 2 V 2 + z 32 (t) (11.23)  

 where γ 2 = min 21 γ1 − 1 , γ2 . Step i: We choose the Lyapunov functional as Vi = z i2 (t) + Ui

(11.24)

with Ui =

i

t

m=1 t−dm

+

εim eγi (ξ−t) z m2 (ξ) ρm (z m (ξ)) dξ

i m−1 m m=1 j=1 l= j

t t−dl

 εim eγi (ξ−t) βm jl z j (ξ) dξ

(11.25)

182

11 Backstepping Control for Nonlinear Time-Delay System via L-K Function

where parameter γi is determined by (11.32). The time derivative of z i2 satisfies 2z i (t) z˙ i (t) ≤ 2z i (t) (z i+1 (t) − αi (x i (t)) + gi (x i (t)) + f i (x1 (t − d1 ) , x2 (t − d2 ) , . . . xi (t − di )) + 2z i (t)

i−1 ∂αi−1 (x i−1 (t)) m=1

∂xm (t)

x˙m (t))

1 ≤ 2z i (t) (z i+1 − αi (x i ) + gi (x i ) + εii−1 eγi d z i (t) 2 i−1 ∂αi−1 (x i−1 (t)) + (xm+1 (t) + gm (x m ))) ∂xm (t) m=1 + εii e−γi d f i2 (x1 (t − d1 ) , x2 (t − d2 ) , . . . xi (t − di ))   i−1 ∂αi−1 (x i−1 (t)) 2 −1 γi d 2 + εim e z i (t) ∂xm (t) m=1 +

i−1

εim e−γi d f m2 (x1 (t − d1 ) , . . . xm (t − dm )) .

m=1

With Assumption 11.1, it follows 2z i (t) z˙ i (t) ≤ 2z i (t) (z i+1 (t) − αi (x i (t)) + gi (x i (t)) i−1 ∂αi−1 (x i−1 (t))

1 (xm+1 + gm (x m )) + εii−1 eγi d z i (t) 2 m=1  2  i−1 1 γi d −1 ∂αi−1 (x i−1 (t)) εim + e z i (t) 2 ∂xm (t) m=1 +

+ e−γi d

∂xm (t)

i

εim z m2 (t − dm ) ρm (z m (t − dm ))

m=1

+ e−γi d

i m−1 m

 εim βm jl z j (t − dl ) m=1 j=1 l= j

The time derivative of functional Ui satisfies

(11.26)

11.4 Extension to the nth-Order System

U˙ i = −γi Ui +

i

183

εim (z m2 (t) ρm (z m (t)) − e−γi dm

m=1

∗ z m2 (t − dm ) ρm (z m (t − dm ))) +

m i m−1 εim m=1 j=1 l= j





  ∗ βm jl z j (t) − e−γi dl βm jl z j (t − dl )

(11.27)

We choose the following virtual control input αi (x i (t)) =

i−1 ∂αi−1 (x i−1 (t))

∂xm (t)

m=1

(xm+1 (t) + gm (x m (t)))

1 1 + εii−1 eγi d z i (t) + gi (x i ) + (γi + 1) z i (t) 2 2  2 2z i 1 1 + εii z i (t) ρi (z i (t)) + z i (t) ηi−1 2 γ i−1 2  2 i−1 1 γi d −1 ∂αi−1 (x i−1 (t)) εim + e z i (t) 2 ∂xm (t) m=1

(11.28)

where function ηi−1 (·) satisfies (11.20). With (11.26)–(11.28), choosing the Lya V i−1 ηi−1 (χ) dχ for (z 1 , z 2 , . . . z i )-subsystem gives punov functional V i = Vi + 0 2 V˙ i ≤ −γ i V i + z i+1 (t)

(11.29)



  where γ i = min 21 γ i−1 − 1 , γi . Step n: Using the backstepping method, we construct the state feedback controller

V n−1 (11.21). Choosing V n = Vn + ηn−1 (χ) dχ, one obtains 0

V˙ n ≤ −γ n V n where γ n = min

 1

γ 2 n−1

(11.30)

  − 1 , γn > 0. Furthermore, inequality (11.30) gives V n ≤ e−γ n t V n (0)

where V n (0) = Vn (0) + 0

V n−1 (0)

ηn−1 (χ) dχ, Vn (0) is defined as

(11.31)

184

11 Backstepping Control for Nonlinear Time-Delay System via L-K Function

Vn (0) = z i2 (0) + +

i

0

m=1 −dm

n m−1 m 0 m=1 j=1 l= j

−dl

eγi (ξ−t) εim z m2 (ξ) ρm (z m (ξ)) dξ

 eγi (ξ−t) εim βm jl z j (ξ) dξ,

V1 (0)

and V n−1 (0) can be obtained via the recursive method with V 1 (0) =

η1 (χ)

0

dχ. From (11.31), one can see that the resulting closed-loop system is asymptotically stable. The proof is completed. Remark 11.4 In the proof, it is required that γ n > 0. Consider that γ i = min  1 γ − 1 , γi , one may determine the parameters γi via the following proi−1 2 cedure: First, we select a proper parameter γ n > 0, then choose γn ≥ γ n , γn−1 ≥ 2 (γn + 1) , γi−1 ≥ 2 (γi + 1) , it is obtained that γn−i ≥ 2i γn + 2i + 2i−1 + 2i−2 + · · · + 22 + 2 = 2i γn + 2i+1 − 2

(11.32)

V i−2

Remark 11.5 Consider V i−1 = Vi−1 + 0

2 ηi−2 (χ) dχ, we have V i−1 ≥ z i−1

(t) + ηi−1 (0) V i−2 . By using the recursive method, one obtains V i−1 ≥ Φi−1 (z i−1 (t)) with Φi−1 (·) defined as Φi−1 (z i−1 (t)) 2 2 2 = z i−1 (t) + ηi−1 (0) z i−2 (t) + ηi−1 (0) ηi−2 (0) z i−3 (t)

+ ··· +

i−1 

ηi− j (0) z 12 (t) .

j=1

We choose function ηi−1 (·) such that i−1 m=1

εim z m2 (t) ρm (z m (t)) +

m i m−1

 εim βi jl z j (t)

m=1 j=1 l= j

≤ ηi−1 (Φi−1 (z i−1 (t))) Φi−1 (z i−1 (t))

(11.33)

then inequality (11.20) holds. Considering the form of Φi−1 (z i−1 (t)) and sufficiently smooth functions ρi (·), βi jl (·), we see that the condition (11.33) can be easily satisfied.

11.4 Extension to the nth-Order System

185

Remark 11.6 The control problem is considered for the strict-feedback nonlinear system with time delays. In fact, the proposed methodology can be extended into the systems with stable unmodeled dynamics. Consider the following system ⎧ η˙ (t) = μ (η (t) , x1 (t) , x1 (t − τ )) , ⎪ ⎪ ⎪ ⎪ x˙i (t) = xi+1 (t) + gi (x i (t)) ⎪ ⎪ ⎨ + f i (η (t) , η (t − τi ) , x1 (t − d1 ) , . . . , xi (t − di )) i = 1, 2, . . . , n − 1 ⎪ ⎪ ⎪ ⎪ x˙n (t) = u (t) + gn (x n (t)) ⎪ ⎪ ⎩ + f n (η (t) , η (t − τn ) , x1 (t − d1 ) , . . . , xn (t − dn ))

(11.34)

where η ∈ Rr is the unmodeled state (not available for feedback) and the unmodeled subsystem is Input-to-State Stable (ISS) with x1 (t) and x1 (t − τ ) as the inputs, τ and τi are time delays of state η. Similar to Assumption 11.1, f i (·) of (11.34) satisfies f i2 (η (t) , η (t − τi ) , x1 (t − d1 ) , . . . , xi (t − di )) ≤ ρi (z i (t − di )) + κi1 (η (t)) + κi2 (η (t − τi )) +

i−1



 ϕj xj t − dj ,

j=1

where κi1 and κi2 are k∞ functions, ρi and ϕ j are the same as defined before. For the ISS η subsystem, we choose a new Lyapunov function to overcome the effect of functions κi1 (·) and κi2 (·) by using the changing supply function idea, which is similar to that of [84, 86]. Then, the virtual control input of z 1 subsystem is designed by additionally considering η subsystem. By employing the proposed method, one may further design the memoryless controller.

11.5 Application to Chemical Reactor Systems In chemical industry, the chemical reactor recycle system [101, 130, 143] is very popular. It is well known that a reactor recycle not only increases the overall conversion but also reduces the reaction cost. For the recycling, the input to be recycled must be separated from the yields, then do the separation operation and finally travel through pipes. This set of operations introduces delays in the recycle system. Let us consider a cascade chemical system with two reactors A and B shown in Fig. 11.1. The compositions C A , C B of produce streams from the reactors are the system state, which are to be controlled. A and B are time-delay systems themselves, and there exists the time delay on recycling some compositions of A to B. The input of system A comes from the system B and the external disturbances, while the input

186

11 Backstepping Control for Nonlinear Time-Delay System via L-K Function

Fig. 11.1 A cascade chemical reactor system

of system B is the delayed system state of A, control input, and external disturbances. The whole plant is described by the following model ⎧ 1 ˙ ⎪ ⎪ C A = −k A C A − θ A H A (C A , C A (t − d A )) ⎪ 1−R B ⎪ ⎪ ⎨ + VA C B + δ A (t, C A (t − d A )) C˙ B = −k B C B − θ1B H B (C B , C B (t − d B )) ⎪ ⎪ ⎪ + VR A C A (t − d A ) + VR B C B (t − d B ) ⎪ B ⎪ ⎩ + FB u + δ B (t, C B (t − d B )) VB (t)

(11.35)

where Ri are the recycle flow rates, θi are the reactor residence times, ki are the reaction constants, F is the feed rate, Vi are reactor volumes, Hi are nonlinear functions representing the complex behavior of the systems, and δi are nonlinear functions for describing the system uncertainties and external disturbances. The controller design problem for chemical system (11.35) has received considerable attentions [101, 130, 143]. Because there is no effective method developed for nonlinear time-delay systems, the existing proposed methods are restricted to the linear models [101, 130, 143]. However, the reactor system is often nonlinear for the complex behavior; then, using the linear system to describe the nonlinear plant leads to the imprecision. Also the time-delay effect is inherent for the chemical systems. In this chapter, we use the proposed method to design the controller for the nonlinear plant with time delays, which makes the closed-loop system globally stable. With δ A = 0 and δ B = 0, we can compute the equilibrium point of the system. Assume H A = C A + C A (t − d A ) and H B = C B2 (t) . With (11.35), one knows the equilibrium point C ∗A and C B∗ satisfy  2 1 − RB ∗ + k A C ∗A = CB θA VA 1 ∗2 RA ∗ RB ∗ k B C B∗ + C = C + C . θB B VB A VB B 

11.5 Application to Chemical Reactor Systems

187

Further letting x1 = C A (t) − C ∗A and x2 = C B (t) − C B∗ gives ⎧ x˙1 = −k A x1 − θ1A x1 − θ1A x1 (t − d A ) ⎪ ⎪ ⎪ ⎨ + 1−R B x + δ (t, C (t − d )) 2 A A A VA 2C ∗ RA 1 2 ⎪ = −k x − x x1 (t − d A ) − θ BB x2 (t) x ˙ + (t) 2 B 2 ⎪ 2 θ V B B ⎪ ⎩ + RB x − d B ) + VFB u (t) + δ B (t, C B (t − d A )) VB 2 (t

(11.36)

Obviously, we can see that (11.36) is a typical nonlinear time-delay system belonging to system (11.1). The controller is designed based on Theorem 11.1, and the corresponding simulations are performed in the following section.

11.6 Simulations For system (11.36), we choose the following parameters θi = 2, ki = 0.5, Ri = 0.5, Vi = 0.5, F = 0.5. The equilibrium point is C ∗A = 14/9, C B∗ = 7/3. Further, one has ⎧ x˙1 = −0.5x1 − 0.5x1 − 0.5x1 (t − d A ) ⎪ ⎪ ⎨ + x2 + δ A (t, x1 (t − d A )) x˙ = −0.5x2 − 0.5x22 (t) + x1 (t − d A ) − 73 x2 (t) ⎪ ⎪ ⎩ 2 + x2 (t − d B ) + u (t) + δ B (t, x2 (t − d B ))

(11.37)

where δ A (t, x1 (t − d A )) = δ A (t, C A (t − d A )), δ B (t, x2 (t − d B )) = δ B (t, C B × (t − d B )). The uncertainties are as the following functions: δ A (t, x1 (t − d A )) = 0.5ϑ1 (t) x1 (t − d A ) and δ B (t, x2 (t − d B )) = 0.5ϑ2 (t) x22 (t − d B ) e0.01x2 (t−d) , where |ϑi (t)| ≤ 1. First, we employ the linear method [101, 130, 143] to design the linear controller. By linearizing the system (11.37) on the zero equilibrium point, we have the following linear model:  x˙1 = −x1 (t) + x2 (t) + 0.5 (Δ1 − 1) x1 (t − d A ) (11.38) x˙2 = −2.8333x2 + x1 (t − d A ) + x2 (t − d B ) + u (t) where Δ1 and Δ2 represent the uncertainties with |Δ1 | ≤ 1 and |Δ2 | ≤ 1. The system (11.38) is expressed as

188

11 Backstepping Control for Nonlinear Time-Delay System via L-K Function

  x1 (t − d A ) 0.5 (Δ1 − 1) 0 1 1 x2 (t − d B )     −1 1 0 + x (t) + 0 −1.3571 u (t) 

x˙ (t) =

(11.39)

 T Based on the linear method, we design u (t) = K x (t) . Let B = 0 1 and 

   −1 1 0.5 0 , E1 = , 0 −2.8333 0 0     00 00 , E3 = . E2 = 10 01 A=

For system (11.39), choose the Lyapunov functional

V = xT Px +

t

x T (ξ) Q 1 x (ξ) dξ +

t−d A

t

x T (ξ) Q 2 x (ξ) dξ t−d B

where P, Q 1 and Q 2 are positive matrices, then we have the following result: If there exists a matrix K such that ⎡ ⎤ P (E 2 − 0.5E 1 ) P E 3 (1, 1) ⎣ (E 2 − 0.5E 1 )T P −Q 1 + ε−1 (11.40) 0 ⎦ 1, and S (z) = [s1 (z) , . . . , sl (z)]T , with si (z) chosen as the commonly used Gaussian functions, which is in the following form  − (z − μi )T (z − μi ) , si (z) = exp ηi2 

198

12 NN-Based Output Feedback Tracking of Nonlinear Time-Delay System



where μi = μi1 , μi2 , . . . , μiq is the center of the receptive field and ηi is the width of the Gaussian function. It has been proven that the neural network can approximate any continuous function over a compact set Ωz ⊂ q to arbitrary accuracy as Θ (z) = θ∗T S (z) + z , ∀z ∈ Ωz , where θ∗ is an ideal constant weight, and z is the approximation error. The ideal weight vector θ∗ is an artificial quantity required for analytical purpose. ∗ θ is defined as the value of θ that minimizes | z | for all z ∈ Ωz in a compact region, that is,    θ∗ := arg min sup Θ (z) − θ T S (z) , z ∈ Ωz . θ∈l

Neural network approximation idea has been applied to the controller design for uncertain nonlinear systems extensively [164]. In [44, 45, 61], state feedback controllers are constructed for time-delay nonlinear systems and nonlinear MIMO systems via neural network method. Observer-based output feedback controller is designed in [26]. In [207], neural network controller is designed for strict-feedback nonlinear systems via backstepping method. In this chapter, we use the neural networks to approximate unknown nonlinear functions and construct the dynamic output feedback controller. The corresponding neural network controller with the adaptive law is proposed in Sect. 12.5.

12.4 Observer Design We propose the observer of the following form ⎧·  ⎪  x i (t) =  x i (t) xi+1 (t) + Fi t,  ⎪ ⎪ ⎨ x1 (t)) , i = 1, . . . , n − 1 +ki (x1 (t) −  ·  ⎪ ⎪  x (t) = u (t) + Fn t,  x n (t) ⎪ ⎩ n x1 (t)) +kn (x1 (t) − 

(12.5)

where parameters ki (i ∈ [1, n]) are chosen to render (A + C)T P + P (A + C) + β −1 P D D T P + β E T E < −Q

(12.6)

in which matrices P and Q are positive definite, β is a positive scalar, I is a unit matrix, and matrix A is defined as

12.4 Observer Design

199

⎡

−k1 1 0 ⎢ −k2 0 1 ⎢ ⎢ ⎢ −k3 0 0 ⎢ ⎢ A = ⎢ −k4 0 0 ⎢ ⎢ . .. .. ⎢ .. . . ⎢ ⎣ −kn−1 0 0 −kn 0 0

⎤ 0 ··· 0 0 0 ··· 0 0⎥ ⎥ . ⎥ 1 · · · .. 0 ⎥ ⎥ . . .. .. ⎥ . 0 . . .⎥ ⎥ ⎥ .. .. .. ⎥ . . . 0⎥ 0 ··· 0 1⎦ 0 0 00

(12.7)

It is easy to check that if inequality (12.6) holds, the observer system state  x will approximate system state x with Hi (·) = 0. Then, with (12.1) and (12.5) we obtain the composite system ⎧· ⎪ z (t) = Q (t, z (t) , x (t) , x (t − d (t))) ⎪ ⎪ ⎪ · ⎪  (t, x1 (t) , x2 (t) , . . . , xn (t)) ⎪ e (t) = Ae (t) + F ⎪ ⎪ ⎪  (t, z (t) , z (t − h 1 (t)) ,  ⎪ x2 (t) , . . . ,  xn (t)) + H − F (t, x1 (t) ,  ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ . . . , z (t − h n (t)) , x (t) , x (t − d1 (t)) , . . . , x (t − dn (t))) ·

·

 y (t) = x2 (t) + F1 (t, x1 (t)) − y d (t) ⎪ ⎪ +H1 (t, z (t) , z (t − h 1 (t)) , x, x (t − d1 (t))) ⎪ ⎪ ⎪ ·  ⎪ ⎪ ⎪ x i (t) =  x i (t) xi+1 (t) + Fi t,  ⎪ ⎪ ⎪ +k (x (t) −  ⎪ x1 (t)) , i = 2, . . . , n − 1 i 1 ⎪ ⎪ ⎪·  ⎩  x n (t) = u (t) + Fn t,  x n (t) + kn (x1 (t) −  x1 (t))

(12.8)

 (·) = [H1 (·) , H2 (·) , . . . , Hn (·)]T . From inequality where e (t) = x (t) −  x (t), H (12.4), we have n n   2  H  ≤ κi (z (t)) + κi (z (t − h i (t))) i=1

+ +

n  i=1 n 

i=1

εi +

n 

Hi1 ( y (t))  y (t)

i=1

Hi2 ( y (t − di (t)))  y (t − di (t))

(12.9)

i=1

This inequality will be used in the controller design procedure. Remark 12.4 To solve inequality (12.6), we decompose A = A + k B with k = T

k1 · · · kn and 

 0 I(n−1)×(n−1) , B = −1 0 · · · 0 . A= 0 0 

200

12 NN-Based Output Feedback Tracking of Nonlinear Time-Delay System

Then, inequality (12.6) can be solved via the following LMI 

(1, 1) P D D T P −β I

 0, δ = δ2 ε1 + −1

n 

εi ,

i=1

 1 κ (z (t)) i 1 − h i∗ i=1   1 + δ2 κ1 (z (t)) + κ . (z (t)) 1 1 − h ∗1

ψ (z (t)) =

n 

 −1 κi (z (t)) +

Proof The time derivative of Mi along system (12.16) satisfies

 · · M 1 = 2 y (t) v2 (t) − α1 + F1 (t, y (t)) − y d (t) y (t) (α1 − α1 ) + 2 y (t) (e2 (t) + H1 ) + 2

 · ≤ 2 y (t) v2 (t) − α1 + F1 (t, y (t)) − y d (t)   n δ1 δ2 n  y 2 (t) + 2 + y (t) (α1 − α1 ) + e22 (t) + H12 + δ1 δ2 n n

(12.19)

With the equalities (12.12)–(12.14), it gives ·

δ1 2 δ2 e2 (t) + H12 − λ1 y 2 (t) + 2 y (t) v2 (t) n n   n  1 −1 y (t) −  Hi2 ( y (t)) + Hi1 ( y (t))  1 − di∗ i=1   1 − δ2 y (t) H12 ( y (t)) + H11 ( y (t))  1 − d1∗  T − y (t) ρ ( y (t)) + 2 y (t) θ∗ − θ (t) S ( y (t))

M1 ≤

+ 2 y (t) − κ y 2 (t)

(12.20)

204

12 NN-Based Output Feedback Tracking of Nonlinear Time-Delay System

Similarly, one obtains   ∂αi−1 M i = 2vi (t) v(i+1) (t) − αi + ∂t    ∂αi−1 · + 2vi (t) y (t) + Fi t,  x i (t) ∂ y (t)  i−1  ∂αi−1 ·  x l (t) + ki (x1 (t) −  x1 (t)) + 2vi (t) ∂ xl (t) l=1  i  ∂αi−1 · ∂αi−1 θ (t) + + 2vi (t) yd(l) (t) (l−1) ∂θ (t) (t) l=1 ∂ yd   ∂αi−1 = 2vi (t) v(i+1) (t) − αi + ∂t  i  ∂αi−1 + 2vi (t) yd(l) (t) + ki (x1 (t) −  x1 (t)) (l−1) ∂ y (t) d l=1    ∂α(i−1) · + 2vi (t) + y (t) + Fi t,  x i (t) ∂ y (t)  i−1  ∂αi−1 · ∂αi−1 · θ (t) +  x l (t) + 2vi (t) ∂θ (t) ∂ xl (t) l=1 ·

(12.21)

In addition, we have ∂α(i−1) · y (t) ∂ y (t) ∂α(i−1) = 2vi (t) x2 (t) + F1 (t, y (t)) + H1 ) (e2 (t) +  ∂ y (t) ∂α(i−1) x2 (t) + F1 (t, y (t))) = 2vi (t) ( ∂ y (t) ∂α(i−1) + 2vi (t) (e2 (t) + H1 ) ∂ y (t) ∂α(i−1) x2 (t) + F1 (t, y (t))) ≤ 2vi (t) ( ∂ y (t)    n n ∂α(i−1) 2 δ1 2 δ2 vi (t) + + + e2 (t) + H12 δ1 δ2 ∂ y (t) n n

2vi (t)

(12.22)

12.5 Controller Design

205

According to (12.21) and (12.22), we choose the virtual control inputs  1 ∂αi−1 λi vi (t) + + v(i−1) (t) + Fi t,  x i (t) 2 ∂t ∂αi−1 x2 (t) + F1 (t, y (t))) + ki (x1 (t) −  x1 (t)) + ( ∂ y (t)     ∂α(i−1) 2 ∂αi−1 · 1 n n + + + θ (t) vi (t) 2 δ1 δ2 ∂ y (t) ∂θ (t)

αi =

+

i−1 i   ∂αi−1 · ∂αi−1  x l (t) + yd(l) (t) (l−1) ∂ x (t) l ∂ y (t) d l=1 l=1

(12.23)

Then, one obtains ·

M i ≤ −λi vi2 (t) − 2v(i−1) (t) vi (t) + 2vi (t) v(i+1) (t) δ1 δ2 + e22 (t) + H12 . n n Based on the backstepping method, one can design the controller 

 1 ∂αn−1 λn vn (t) + v(n−1) (t) + + Fn t,  x n (t) 2 ∂t ∂αn−1 x2 (t) + F1 (t, y (t))) x1 (t)) + + kn (x1 (t) −  ( ∂ y (t)     ∂αn−1 2 ∂αn−1 · 1 n n + + + θ (t) vn (t) 2 δ1 δ2 ∂ y (t) ∂θ (t) # n−1 n   ∂αn−1 · ∂αn−1 +  x l (t) + yd(l) (t) (l−1) ∂ x (t) l (t) l=1 l=1 ∂ yd

u (t) = −

(12.24)

Then, one obtains ·

M n ≤ −λn vn2 (t) − v(n−1) (t) vn (t) +

δ1 2 δ2 e2 (t) + H12 n n

(12.25)

With Assumption 12.2, the time derivative of V along the system (12.16) satisfies ·  V ≤ e (t)T (A + C)T P + P (A + C) e (t)  + e (t)T β −1 P D D T P + β E T E e (t)  2  ·  − 2Γ −1 θ∗ − θ (t) θ (t) . + e T (t) P Pe (t) + −1  H

With (12.9), (12.10), and (12.15), it follows

206

12 NN-Based Output Feedback Tracking of Nonlinear Time-Delay System ·

V ≤ −e (t)T (Q − P P) e (t)  T − 2 θ∗ − θ (S ( y (t))  y (t) − lθ (t)) −1

+

+ −1 + −1

n  i=1 n  i=1 n 

−1

κi (z (t)) + 

n 

κi (z (t − h i (t)))

i=1

εi + −1

n 

Hi1 ( y (t))  y (t)

i=1

Hi2 ( y (t − di (t)))  y (t − di (t))

(12.26)

i=1

The time derivative of W satisfies ·

W ≤

n 

−1



i=1

1  y (t) Hi2 ( y (t)) 1 − di∗

− y (t − di (t)) Hi2 ( y (t − di (t))))  1 + δ2  y (t) H12 ( y (t)) 1 − d1∗ − y (t − d1 (t)) H12 ( y (t − d1 (t))))   n  1 −1 +  κi (z (t)) − κi (z (t − h i (t))) 1 − h i∗ i=1   1 + δ2 κ1 (z (t)) − κ1 (z (t − h 1 (t))) . 1 − h ∗1 Then, we obtain ·

U ≤ −e (t)T (Q − P P) e (t) + δ1 e22 (t)  T − y (t) ρ ( y (t)) + 2 θ∗ − θ (t) lθ (t) + 2 y (t) n n   λi vi2 (t) + δ2 ε1 + −1 εi − κ y 2 (t) − i=1

i=1

 1 + κi (z (t)) 1 − h i∗ i=1   1 + δ2 κ1 (z (t)) + κ1 (z (t)) 1 − h ∗1  T y (t) ρ ( y (t)) + 2 θ∗ − θ (t) lθ (t) ≤ −e (t)T Qe (t) −  n 

−

n  i=1

 −1 κi (z (t)) + α

λi vi2 (t) + ψ (z (t)) + δ + 2 y (t) − κ y 2 (t)

(12.27)

12.5 Controller Design

207

Note that T  2l θ∗ − θ (t) θ (t)  2  2 = −l θ∗ − θ (t) − l θ (t)2 + l θ∗   2  2 ≤ −l θ∗ − θ (t) + l θ∗  and

(12.28)

2 y (t) − κ y 2 (t) ≤ κ −1 2 ,

one has ·  2 U ≤ −e (t)T Qe (t) −  y (t) ρ ( y (t)) − l θ∗ − θ (t) n  2  + l θ∗  − λi vi2 (t) + ψ (z (t)) + δ + κ −1 2

(12.29)

i=1

The proof is completed. If there is no unmodeled dynamics in the considered system, by choosing ρ ( y (t)) = 0 and ψ (z (t)) = 0, we can obtain the following inequality with (12.29) n ·  2  2  U ≤ −e (t)T Qe (t) − l θ∗ − θ (t) + l θ∗  − λi vi2 (t) + δ + κ −1 2 , i=1

then the closed-loop system is UUB stable. In the next section, we will show how to choose proper function ρ ( y (t)) such that the closed-loop system with the unmodeled dynamics is stable. Remark 12.5 From the above procedure, we can see that the key of controller design is to construct the virtual control α1 (·) in the first step. With α1 (·) obtained, it is not difficult to design α j (·) ( j = 2, . . . , n − 1) and construct the control u (t) via the step by step method. Different from the classic backstepping method, here the virtual control α1 (·) designed in the first step should not only include the information of  y (t) subsystem, but also include the information of vi (t) subsystem. Remark 12.6 By employing the developed method, we only use one neural network in this chapter. Different from the results reported in the literature, one neural network is utilized to approximate a composite function (12.13), which will render the control design procedure much simpler. As a special case, we consider the following system ⎧· ⎪ x 1 (t) = x2 (t) + f o1 (y (t)) + f 1 (y (t)) ⎪ ⎪ · ⎪ ⎪ ⎨ x i (t) = xi+1 (t) + f oi (y (t)) + f i (y (t)) , i = 2, . . . , n − 1 ⎪ · ⎪ ⎪ ⎪ x (t) = u (t) + f on (y (t)) + f n (y (t)) ⎪ ⎩ n y (t) = x1 (t)

208

12 NN-Based Output Feedback Tracking of Nonlinear Time-Delay System

where f oi (y (t)) and f i (y (t)) (i = 1, . . . , n) are known and unknown smooth functions, respectively, with f i (0) = 0. The control problem of this system was investigated in [26], in which the n neural networks were used to approximate n unknown functions f i (y (t)) , and the observer-based neural network controller was successfully constructed. We can easily see that the above system is a special case of system (12.1) and satisfies Assumptions 12.1–12.4, then the controller design idea can be used. For the system, we may use only one neural network to approximate the bound function (12.13) and construct the corresponding observer-based controller.

12.6 Choosing Proper Functions In this section, we will show the constructed output feedback controller can render the closed-loop system UUB stable by choosing proper function ρ ( y). We can now formulate our main theorem of this chapter. Theorem 12.2 Under Assumptions 12.1–12.4, the constructed observer (12.5) based controller (12.24) will render the resulting closed-loop system UUB stable. Proof With the help of inequality (12.3), one has ·

y (t)|) + γ (| y (t − d (t))|) + τ U 0 ≤ −α0 (z (t)) + γ (|

(12.30)

By employing the changing supply functions idea [163], we define new function " U0 (t,z(t)) 1 U0 = ϕ (ξ) dξ + 1 − d∗ 0 " t  y (ξ)|)) γ (| y (ξ)|) dξ ϕ ◦ α ◦ α0−1 ◦ (6γ (| ∗

(12.31)

t−d(t)

where ϕ ◦ α (X ) = ϕ (α (X )) , ϕ : + → + is a smooth nondecreasing function such that ϕ (ξ) > 0 and ϕ (0) = 0. It is clear that U 0 is a positive definite function. Further, we can obtain the time derivative of U 0 satisfies ·

y (t)|) + γ (| y (t − d (t))|) + τ ) U 0 ≤ ϕ (U0 (t, z (t))) (−α0 (z (t)) + γ (| 1  + y (t)|)) γ (| y (t)|) ϕ ◦ α ◦ α0−1 ◦ (6γ (| 1 − d∗  y (t − d (t))|)) γ (| y (t − d (t))|) (12.32) − ϕ ◦ α ◦ α0−1 ◦ (6γ (|

12.6 Choosing Proper Functions

209

With Assumption 12.1, it follows that y (t)|) ϕ (U0 (t, z (t))) (−α0 (z (t)) + γ (| +γ (| y (t − d (t))|) + τ ) 1 ≤ − ϕ ◦ α (z (t)) α0 (z (t)) 2  1 y (t)|) + ϕ (U0 (t, z (t))) − α0 (z (t)) + γ (| 2 + γ (| y (t − d (t))|) + τ ) . y (t)|) ≤ Let us consider the following two cases. The first case is that γ (| y (t − d (t))|) ≤ 16 α0 (z (t)), and τ ≤ 16 α0 (z (t)) ; under (z (t)), γ (| this case, we obtain 1 α 6 0

y (t)|) ϕ (U0 (t, z (t))) (−α0 (z (t)) + γ (| + γ (| y (t − d (t))|) + τ ) 1 ≤ − ϕ ◦ α (z (t)) α0 (z (t)) 2

(12.33)

y (t)|) > 16 α0 (z (t)), γ (| y (t − d (t))|) > The second case is that γ (| 1 (z (t)), and τ > 6 α0 (z (t)) ; under this case, it is obtained that

1 α 6 0

y (t)|) + γ (| y (t − d (t))|) + τ ) ϕ (U0 (t, z (t))) (−α0 (z (t)) + γ (|  1 ≤ − ϕ ◦ α (z (t)) α0 (z (t)) + ϕ ◦ α ◦ α0−1 ◦ 6τ 6τ . 2  y (t)|) y (t)|)) γ (| + ϕ ◦ α ◦ α0−1 ◦ (6γ (|  −1 + ϕ ◦ α ◦ α0 ◦ (6γ (| y (t − d (t))|)) γ (| y (t − d (t))|) (12.34) Based on (12.33) and (12.34), inequality (12.34) always holds for the two cases. If the functions do not satisfies the above two cases, it is easy to check that inequality (12.34) still holds. Further, one can obtain · 1 U 0 ≤ − ϕ ◦ α (z (t)) α0 (z (t))  2 y (t)|) + ϕ ◦ α ◦ α0−1 ◦ (6γ (| y (t)|)) γ (|  1 + y (t)|)) γ (| y (t)|) ϕ ◦ α ◦ α0−1 ◦ (6γ (| ∗ 1−d  + ϕ ◦ α ◦ α0−1 ◦ 6τ 6τ .

Based on Assumption 12.4, there exists a desired function ϕ (·) such that

210

12 NN-Based Output Feedback Tracking of Nonlinear Time-Delay System

1 ϕ ◦ α (z (t)) α0 (z (t)) ≥ ψ (z (t)) 4 Similarly, we can select a desired function ρ ( y) such that   yρ ( y) ≥ ϕ ◦ α ◦ α0−1 ◦ (6γ (| y (t)|) y (t)|)) γ (| 1  + y (t)|)) γ (| y (t)|) ϕ ◦ α ◦ α0−1 ◦ (6γ (| ∗ 1−d

(12.35)

For the whole system, choose the following Lyapunov function U = U + U 0; then, the time derivative of U along the composite system (12.16) is ·

·

·

U = U + U0

 2 ≤ −e (t)T Qe (t) − l θ∗ − θ (t)

−

n 

λi vi2 (t) −

i=1

1 ϕ ◦ α (z (t)) α0 (z (t)) + δ 4

(12.36)

 where δ = δ + κ −1 2 + ϕ ◦ α ◦ α0−1 ◦ 6τ 6τ + l θ∗ 2 . From (12.36), we obtain that as t → ∞  e (t) ≤

δ  λmin Q

1/2

  , θ∗ − θ (t) ≤



δ l

1/2

,

 where λmin Q denotes the minimum eigenvalue of matrix Q, and  |vi (t)| ≤

δ λi

1/2

 , ϕ ◦ α (z (t)) α0 (z (t)) ≤ 4δ.

Note the boundedness of v1 (t) , we obtain the output signal y (t) is bounded. x1 (t) is bounded. Consider the With the boundedness of e1 (t) , one obtains that  x2 (t) is bounded based on boundedness of virtual control input α1 (·) , we obtain  x3 (t) , . . . ,  xn (t) are (12.11). Via the recursive method, one can obtain that  x2 (t) ,  bounded. According to the boundedness of error signal e (t) and observer state  x (t), we obtain that the state x (t) is bounded. Based on above analysis, it is obtained that the resulting closed-loop system is UUB stable. Remark 12.7 From (12.36), the tracking error satisfies  |y (t) − yd (t)|
0. Here, parameter δ1 can be sufficiently small, while parameter  should be big since the large parameter  will result in the small tracking error. Further, obtain function ψ (z (t)). Step 4: According to (12.35), determine the function ρ ( y (t)). Based on the proposed backstepping method, design the virtual control input α1 (·) (12.14) with the corresponding adaptive law, here parameters δ2 and l should be selected small enough, while parameter λ1 should be big enough. Via the step by step method, the output feedback controller u (t) (12.24) is finally constructed.

12.7 Simulation Example In this section, the following time-delay nonlinear system is considered ·

z (t) = −z (t) + x1 (t − d (t)) sin x2 (t) , ·

x 1 (t) = x2 (t) + x12 (t) +

x13 (t − d1 (t)) x1 (t) − x15 (t) + + z (t) , 4 1 + x1 (t) 1 + x12 (t − d1 (t))

·

x 2 (t) = u (t) + x13 (t) + 10x2 (t) + (sin 5t) x2 (t) + +

x1 (t) + x15 (t) 1 + x14 (t)

x14 (t − d2 (t)) sin x2 (t) + z (t) , 1 + x12 (t − d2 (t))

y (t) = x1 (t)

(12.37)

212

12 NN-Based Output Feedback Tracking of Nonlinear Time-Delay System ·

where d1 (t) and d2 (t) are the time-varying delays satisfying d 1 (t) ≤ d1∗ < 1 and ·

d 2 (t) ≤ d2∗ < 1. It is easy for us to obtain that system (12.37) is in the form of system (12.1) where F1 = x12 (t) , F2 = x13 (t) + 10x2 (t) + (sin 5t) x2 (t) , H1 =

x13 (t − d1 (t)) x1 (t) − x15 (t) + , 1 + x14 (t) 1 + x12 (t − d1 (t))

H2 =

x1 (t) + x15 (t) x14 (t − d2 (t)) sin x2 (t) + , 1 + x14 (t) 1 + x12 (t − d2 (t))

d (t) = d1 (t) = d2 (t) = 0.5 (1 + sin t)

(12.38)

For system (12.37), choosing Lyapunov function W = z 2 for z-subsystem gives ·

W = 2z (t) (−z (t) + x1 (t − d (t)) sin x2 (t)) ≤ −z 2 (t) + x12 (t − d (t)) , then Assumption 12.1 holds. Further, it is easy to check that system (12.37) also satisfies Assumptions 12.2–12.4 from (12.38). Matrices C, D, G (t) , and E are as follows     0 0 00 C= ,D = , 0 10 01   0 0 G (t) = , E = I. 0 sin 5t Therefore, the dynamic output feedback controller can be designed to solve the tracking control problem. First, the observer is designed ⎧· ⎪ x 1 (t) =  x2 (t) + x12 (t) + k1 (x1 (t) −  x1 (t)) ⎨ ·

 x (t) = u (t) + x1 (t) + (sin 5t)  x2 (t) + 10 x2 (t) ⎪ ⎩ 2 x1 (t)) +k2 (x1 (t) −  3

(12.39)

With Q = I, we obtain the gain parameters via LMI (12.10) k1 = 434.7, k2 = 320.99

(12.40)

Further, the virtual control is constructed ·

α1 = 10 y (t) + x12 (t) + θ T (t) S ( y (t)) − y d (t)

(12.41)

12.7 Simulation Example

with adaptive law

213

·

θ (t) = S ( y (t))  y (t) − 0.01θ (t)

(12.42)

Here, function S ( y) is chosen as S ( y (t)) = [S1 , S2 , S3 , S4 , S5 ]T with



 y (t) − 4)2 /2 , S2 = exp − ( y (t) − 2)2 /2 , S1 = exp − (

2

  S3 = exp − y (t) /2 , S4 = exp − ( y (t) + 2)2 /2 ,

 S5 = exp − ( y (t) + 4)2 /2 . Based on observer (12.39) and virtual control (12.41), we design the controller

y (t) + x13 (t) + (10 + sin 5t)  x2 (t) u = − v2 (t) +  ∂α1 x2 (t) + F1 (y (t))) + k2 (x1 (t) −  x1 (t)) + ( ∂ y (t) #   ∂α1 2 ∂α1 · ∂α1 ·· ∂α1 ·· + 4v2 + θ+ y (t) + · y d (t) ∂y ∂θ ∂ yd (t) d ∂ y d (t)

(12.43)

The signal to be tracked is chosen as sine waves, that is, yd = sin t. The initial values are chosen as z (0) = 0.1, x1 (0) = 1, x2 (0) = 0.2,  x1 (0) =  x2 (0) = 0, θ (0) = 0. The simulation results are shown in Figs. 12.1, 12.2, and 12.3 with the horizontal axis as the time. The response curve and the error curve are shown in Figs. 12.1 Fig. 12.1 The response curve of output y and the signal to be tracked

1.5

y sin(t)

1 0.5 0 −0.5 −1 −1.5

0

5

10

15

214

12 NN-Based Output Feedback Tracking of Nonlinear Time-Delay System

Fig. 12.2 The tracking error curve

1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

Fig. 12.3 The response curves of variables z and x2

0

5

10

10

15

z x2

5 0 −5 −10 −15 −20

0

5

10

15

and 12.2, from which one can see that the output y (t) can approximate the tracking signal yd (t). Figure 12.3 shows the response curves of variables z (t) and x2 (t), from which we can see that the states are uniformly ultimately bounded.

12.8 Conclusion In this chapter, the problem of robust output tracking control for a class of time-delay nonlinear systems is considered. The systems are in the form of triangular structure with unmodeled dynamics. First, we construct an observer whose gain matrix is scheduled via linear matrix inequality approach. For the case that the information of uncertainties bounds is not completely available, we design an observer-based neural network controller by employing the backstepping method. The resulting closed-loop system is ensured to be stable in the sense of uniform ultimate boundedness with the help of changing supplying function idea. The observer and the controller designed are both independent of the time delay. Finally, numerical simulations are conducted to verify the effectiveness of the main theoretic results obtained.

Chapter 13

Output Feedback Stabilization for Interconnected Time-Delay Systems

Abstract In this chapter, dynamic output feedback control problem is investigated for a class of nonlinear interconnected systems with time delay. Decentralized observer independent of the time delay is first designed. Then, we employ the bounds information of uncertain interconnections to construct the decentralized output feedback controller via backstepping design method. Based on Lyapunov stability theory, we show that the designed controller can render the closed-loop system asymptotically stable with the help of the changing supplying function idea. Furthermore, the corresponding decentralized control problem is considered under the case that the bounds of uncertain interconnections are not precisely known. By employing the neural network approximation theory, we construct the neural network output feedback controller with corresponding adaptive law. The resulting closed-loop system is stable in the sense of semiglobal boundedness. The observers and controllers constructed in this paper are independent of the time delay. Finally, simulations are done to verify the effectiveness of the theoretic results obtained.

13.1 Introduction The decentralized control problem for interconnected systems has received considerable attention over the past years, see [10, 161] and the references therein. State feedback controllers have been designed to solve the decentralized control problem of nonlinear systems in [19, 43, 48, 74, 151, 157, 190]. The adaptive state feedback controllers were constructed in [1, 43]. Based on backstepping method, the state feedback controller was designed in [190]. With the interconnections unknown, the decentralized neural network controller was presented in [74]. Hierarchical control problem was investigated for interconnected systems via recurrent neural network in [63]. Under the case that the state variables cannot be totally obtained, the output feedback controllers were used to solve the stabilization problem in [78, 83, 149, 196, 197]. However, the systems investigated in the above quoted literature are free of the time delay. The existence of time delay will render the control problem more complicated and difficult. For large-scale nonlinear systems with delays, in [38, 131], the authors considered the robust control problem for time-delay nonlinear systems © Springer Nature Singapore Pte Ltd. 2018 C. Hua et al., Robust Control for Nonlinear Time-Delay Systems, DOI 10.1007/978-981-10-5131-9_13

215

216

13 Output Feedback Stabilization for Interconnected Time-Delay Systems

via backstepping method. But the results are not desirable [53, 66]. In [70], we have investigated the control problem for a class of time-delay nonlinear systems via backstepping method and constructed the corresponding output feedback controller. In [46], state feedback controller was successfully constructed to control a class of time-delay nonlinear systems via neural network method. In this chapter, we will investigate the dynamic output feedback control problem for a class of large-scale interconnected systems with time delay. Each subsystem is in the form of the triangular form and includes the internal dynamics. The observer is designed at first. With the bounds information of uncertain interconnections known, we construct the observer-based output feedback controller via backstepping method. By employing Lyapunov–Krasovskii functional method and changing supplying function idea, we prove that the closed-loop system is asymptotically stable. Furthermore, the output feedback control problem is investigated under the case that the bounds of uncertain interconnections are not available. Employing neural network approximation idea, we construct the decentralized output feedback controller. The resulting closed-loop system is stable in the sense of semiglobal boundedness. Finally, computer simulations are done to show the feasibility and effectiveness of the main results.

13.2 System Formulation In this chapter, we investigate a class of large-scale dynamic systems whose ith subsystem can be put into the following normal form ⎧· ⎪ z i (t) = Q i (t, z i (t) , x1 (t) , . . . , x N (t) , ⎪ ⎪ ⎪ ⎪ − ri1 (t)) , . . . , x N (t − ri N (t))), x ⎪ ⎪ · 1 (t   ⎪ ⎪ ⎪ + φi j x ij (t) ⎪ x i j (t) = xi( j+1) (t)  ⎪ ⎪ ⎪ +ϕi j t, z i (t) , z i t − τi j (t) ⎪  , x1 (t) ,  ⎨ . . . , x N (t) , x1 t − di j1 (t) , . . . , x N t − di j N (t) , ⎪ ⎪ j = 1, . . . , n i − 1,  ⎪  · ⎪ ⎪ ⎪ x ini (t) = u i (t) + φini x ini (t) ⎪ ⎪ ⎪ ⎪ +ϕini t, z i (t) ,z i t − τini (t) , x1 (t) , . . . , x N (t) , ⎪ ⎪ ⎪ ⎪ x1 t − dini 1 (t) , . . . , x N t − dini N (t) , ⎪ ⎩ yi (t) = xi1 (t)

(13.1)

where 1 ≤ i ≤ N , N is total number of the subsystems which is known for controller  T design, xi (t) = xi1 (t) , . . . , xini (t) is the state of the subsystem and x i j (t)= T  xi1 (t) , . . . , xi j (t) , z i (t) ∈ ai represents the state of zero dynamic system (not available for feedback), u i (t) ∈  and yi (t) ∈  are the control input and the output of subsystem, respectively. Q i (·) , φi j (·), and ϕi j (·) are smooth nonlinear functions with φi j (t, 0, 0, . . . , 0) = ϕi j (t, 0, 0, . . . , 0) = Q i (t, 0, 0, . . . , 0) = 0. The

13.2 System Formulation

217 ·

·

time-varying delays are bounded and satisfy d i jl (t) ≤ di∗jl < 1, τ i j (t) ≤ τi∗j < 1 ·

and r i j (t) ≤ ri∗j < 1. The initial values of system (13.1) are z i (t) = 1i (t) , t ∈ [t0 − τi , t0 ] , xi (t) = 2i (t) , t ∈ [t0 − di , t0 ] .  where 1i (t) and 2i (t) are given continuous function vectors, τi = max τi∗j and  di = max dl∗ji . The solution of system (13.1) is z i (t; t0 , 1i ) and xi (t; t0 , 2i ) , for simplicity, we use z i (t) and xi (t) to represent the solution in the rest part. In view that the nonlinear functions of system (13.1) are smooth, then the local Lipschitz condition is satisfied; thus, there exists a unique solution for system (13.1). The structure involved in (13.1) free of time delay has been investigated by several earlier papers in the nonlinear control literatures such as [83, 87]. System (13.1) can be used to represent many important physical systems; for example, cold rolling mills, wind tunnel, and water resources systems (see, for example, [127] and the references therein). The system can be seen as a model of the global normal form of input/output feedback linearization for large-scale time-delay system. For system (13.1), the following additional conditions are assumed. Assumption 13.1 There exists an ISS (Input-to-State Stable)   Lyapunov function Πi (t, z i ) for the z i -system in (13.1) with y j and y j t − ri j (t) as the inputs. Namely, there are known class-k∞ functions αi , αi , αi0 , γi j and γ i j satisfying the following inequalities αi (z i (t)) ≤ Πi (t, z i (t)) ≤ αi (z i (t)) , ∂Πi (t, z i ) ∂Πi (t, z i ) + Q i ≤ −αi0 (z i (t)) ∂t ∂z i N

      γi j  y j (t) + γ i j  y j t − ri j (t)  +

(13.2)

j=1

Assumption 13.2 Function vector   T φi (xi (t)) = φi1 (xi1 (t)) , φi2 (x i2 (t)) , . . . , φini x ini (t) yields φi (xi (t)) = φi (xi1 ) + (Ci + Di Fi (xi1 (t)) E i ) xi (t)

(13.3)

218

13 Output Feedback Stabilization for Interconnected Time-Delay Systems

where φi (·) is a known function vector, Ci , Di , Fi (xi1 (t)), and E i are known matrices with FiT (xi1 (t)) Fi (xi1 (t)) ≤ Ii ; here, Ii is an identity matrix. Assumption 13.3 Nonlinear interconnection part ϕi j (·) satisfies the following inequality   ϕi2j (t, bi (t) , bi t − τi j (t) , c1 (t) , . . . , c N (t) ,     c1 t − di j1 (t) , . . . , c N t − di j N (t) N   

≤ κi j (bi (t)) + κi j bi t − τi j (t)  + (ck1 (t) ϕi jk (ck1 (t))

      + ck1 t − di jk (t) ϕi jk ck1 t − di jk (t)

k=1

(13.4)

where bi (t) ∈ ai and ci (t) ∈ ni are the arbitrary time-varying parameter vectors, κi j (·) and κi j (·) are known positive definite functions with κi j (0) = κi j (0) = 0, ϕi jk (·) and ϕi jk (·) are proper smooth nonlinear functions. Assumption 13.4 The functions αi0 , κi j , κi j , γi j and γ i j satisfy the following local property γ i j (t) γi j (t) < +∞, lim+ sup < +∞, 2 t→0 t→0 t t2 κi j (t) κi j (t) lim sup < +∞, lim+ sup < +∞ t→0+ t→0 αi0 (t) αi0 (t) lim+ sup

(13.5)

Remark 13.1 Assumption 13.1 is to guarantee the nominal model of uncertain system (13.1) is minimum phase. implies that z i -subsystem is Input This assumption  to-State Stable with y j and y j t − ri j (t) as the inputs. The function φi (·) is required to satisfy Assumption 13.2, which includes measurable output xi1 and the unmeasured state xil with l ∈ [2, n i ] . This assumption is looser than that of [26, 70]. In [26], function φi (·) only contained the measurable output y (t). In [70], the assumption on function φi (·) did not use the good characteristic of the triangular structure and large ρ (upper bounds of uncertainties) induced no solution of inequality (13.6) of [70]. In Assumption 13.2, we employ the matrices Ci , Di , Fi (xi1 ), and E i to express the function φi (·) and the triangular structure characteristic is used. Compared with that of [70], the observer design condition of this chapter (see (13.10) below) is looser. Remark 13.2 For Assumption 13.3 with bi (t) = z i (t) and ci (t) = xi (t) , we can find that the interconnections ϕi j (·) among the N subsystems should be bounded by nonlinear  functions  which include the unmeasured state variable z i (t), delayed output yk t − di jk (t) , and the measured output yk (t). We impose Assumption 13.4 on the functions to guarantee that one can choose functions ρi (yi (t)) in the virtual control law (13.28) to render the whole system stable, which can be seen in Sect. 13.3.

13.2 System Formulation

219

Remark 13.3 Instead of Assumption 13.3, we may assume that function ϕi j satisfies the following inequality    ϕi j (t, bi (t) , bi t − τi j (t) , c1 (t) , . . . , c N (t) ,     c1 t − di j1 (t) , . . . , c N t − di j N (t)  N   

≤ κi j (bi (t)) +  κi j bi t − τi j (t)  + (ck1 (t) ϕ i jk (ck1 (t))

     + ck1 t − di jk (t)  ϕi jk ck1 t − di jk (t) )

k=1

(13.6)

where  κi j (·) and  κi j (·) are known positive functions, ϕ ϕi jk (·) are noni jk (·) and  linear functions. This condition is often assumed on investigating the decentralized control problem of interconnected systems in existing literature such as [83, 87]. It is easy to find that if inequality (13.6) holds, inequality (13.4) will stand by choosing κi2j (bi (t)) , κi j (bi (t)) = (2N + 2) ϕi jk (ck1 (t)) = (2N + 2) ck1 (t) ϕ i2jk (ck1 (t)) ,     2   κi j bi t − τi j (t)  = (2N + 2)  κi j bi t − τi j (t)  , and    ϕi jk ck1 t − di jk (t)   2    ϕi jk ck1 t − di jk (t) = (2N + 2) ck1 t − di jk (t) 

(13.7)

Since that inequality (13.4) holds for arbitrary variables bi (t) and ci (t), then functions ϕi jl (·) satisfy χϕi jl (χ) ≥ 0 for arbitrary variable χ. In this chapter, the following problem will be investigated: Problem: For system (13.1) satisfying above assumptions, design the decentralized output feedback controller to render the closed-loop system stable. In detail, we will consider to design the following decentralized dynamic output feedback controller · ℵi (t) = ζi (ℵi (t) , yi (t)) , u i (t) = μi (ℵi (t) , yi (t)) (13.8) (i) With the functions ϕi jk (·) and ϕi jk (·) known, the decentralized output feedback controller will be designed such that the closed-loop system is asymptotically stable. (ii) With the functions ϕi jk (·) and ϕi jk (·) unknown, the decentralized adaptive neural network output feedback controller will be constructed to render the closedloop system stable in the sense of semiglobal boundedness.

220

13 Output Feedback Stabilization for Interconnected Time-Delay Systems

In this chapter, the following lemma [163] is used to prove our main result:  ∈ k∞ are such that β (s) = O [β (s)] Lemma 13.1 Assume that the functions β, β as s → 0+ . Then, there exists a smooth nondecreasing function q (s) such that (s) ≤ q (s) β (s) for all s ∈ [0, ∞) . β

13.3 Robust Controller Design In this section, we design a robust output feedback control law for the control task stated in Sect. 13.2. Firstly, an observer is designed. Then, based on the new augmented system, it is shown how to recursively construct a dynamic output feedback law. Finally, an appropriate choice of the design parameters and functions gives a solution of the aforementioned stabilization problem.

13.3.1 Observer Design The following standard decentralized observer is proposed ⎧·   ⎪ x i j (t) =  x i j (t) + ki j (xi1 (t) −  xi( j+1) (t) + φi j  xi1 (t)) , ⎨ j = 1, . . . , n i − 1, ⎪   ⎩·  x ini (t) = u i (t) + φini  x in (t) + kini (xi1 (t) −  xi1 (t))

(13.9)

T where  x i j (t) = xi1 ,  xi2 ,  xi3 , . . . ,  xini and parameters ki j ( j ∈ [1, n i ]) are chosen such that −Q i > (Ai + Ci )T Pi + Pi (Ai + Ci ) + βi−1 Pi Di DiT Pi + βi E iT E i ,

(13.10)

where matrices Pi and Q i are positive definite, βi is a positive scalar, and matrix Ai is defined as ⎡ ⎤ −ki1 1 0 0 · · · 0 0 ⎢ −ki2 0 1 0 · · · 0 0 ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ −ki3 0 0 1 · · · .. 0 ⎥ ⎢ ⎥ ⎢ ⎥ (13.11) Ai = ⎢ −ki4 0 0 0 . . . ... ... ⎥ . ⎢ ⎥ ⎢ ⎥ . . . . . . .. .. .. .. .. .. 0 ⎥ ⎢ ⎢ ⎥ ⎣ −ki(n −1) 0 0 0 · · · 0 1 ⎦ i −kini 0 0 0 0 0 0

13.3 Robust Controller Design

221

Further with (13.1) and (13.9), the composite system is obtained ⎧· z i (t) = Q i (t, z i (t) , x1 (t) , . . . , x N (t) , ⎪ ⎪ ⎪ ⎪ x1 (t − ri1 (t)) , . . . , x N (t − ri N (t))), ⎪ ⎪ ⎪ · ⎪ ⎪ e xi (t)) + ϕi , ⎪ i (t) = Ai ei (t) + φi (x i (t)) − φi ( ⎪ ⎪ ⎨ y· (t) = x (t) + φ (x (t)) + ϕ , i2 i1 i1 i1 i ·   ⎪ ⎪  x i j (t) =  x i j (t) xi( j+1) (t) + φi j  ⎪ ⎪ ⎪ ⎪ −  x , j = 2, . . . , n i − 1, +k (x (t) (t)) i j i1 i1 ⎪ ⎪ ·   ⎪ ⎪ ⎪ x ini (t) ⎪ ⎩ x ini (t) = u i (t) + φini  xi1 (t)) +kini (xi1 (t) − 

(13.12)

T where ei (t) = xi (t) −  xi (t) , ϕi = ϕi1 , ϕi2 , . . . , ϕini . With the help of inequality (13.4), one has the following inequality ϕi 2 ≤

ni

    κi j (z i (t)) + κi j z i t − τi j (t)  j=1

+

ni

N

(yk (t) ϕi jk (yk (t)) j=1 k=1

      + yk t − di jk (t) ϕi jk yk t − di jk (t)

(13.13)

Therefore, if we design a controller to render the composite system (13.12) stable, the controller will also render the system (13.1) stable. Remark 13.4 To solve inequality (13.10), we decompose Ai = Ai + ki Bi with  Ai =

0 I(ni −1)×(ni −1) 0 0



⎤ ki1 ⎥ ⎢ , ki = ⎣ ... ⎦ , kini ⎡



and Bi = −1 0 · · · 0 , then inequality (13.10) can be solved via the following LMI  Πi =

Πi11 Pi Di DiT Pi −βi Ii

 < 0,

(13.14)

   T where Πi11 = Pi Ai + Ci + Υi Bi + BiT ΥiT + Ai + Ci Pi + βi E iT E i + Q i , Pi and Q i are positive matrices, Υi is an unknown vector. For the given matrices Ai , Bi , Ci , Di , E i and positive matrix Q i , one can solve (13.14) to get matrices Pi ,Υi and scalar βi simultaneously via LMI toolbox of MATLAB. Furthermore, the observer gain vector ki can be obtained by ki = Pi−1 Υi .

222

13 Output Feedback Stabilization for Interconnected Time-Delay Systems

13.3.2 Controller Design For the composite system (13.12), we consider the following state coordinate transformation ⎧ ⎨ vi1 (t) = xi1 (t) ,   xi1 , . . . ,  xi( j−1) , vi j (t) =  xi j (t) + αi( j−1) yi ,  (13.15) ⎩ j = 2, . . . , n i where functions αi( j−1) (·) are the virtual control laws satisfying αi( j−1) (0, 0, . . . , 0) = 0. Under the transformation (13.15), system (13.12) is changed into ⎧· ⎪ z i (t) = Q i (t, z i (t) , x1 (t) , . . . , x N (t) , ⎪ ⎪ ⎪ ⎪ x1 (t − ri1 (t)) , . . . , x N (t − ri N (t))), ⎪ ⎪ ⎪· ⎪ ⎪ e xi (t)) + ϕi , ⎪ ⎪ · i (t) = Ai ei (t) + φi (xi (t)) − φi ( ⎪ ⎪ ⎪ + ϕi1 , ⎪ ⎨ y· i (t) = xi2 (t) + φi1 (yi (t)) ∂α j−1) · v i j (t) = vi( j+1) (t) − αi j + ∂ yi(i (t) y i (t) ⎪    j−1 ∂αi( j−1) · ⎪ ⎪ ⎪ x il (t) + φi j  x i j (t) + l=1 ∂xil (t)  ⎪ ⎪ ⎪ ⎪ +k −  x , j = 2, . . . , n i − 1, (x (t) (t)) ⎪ ij i1 i1 ⎪ ⎪  · ∂αi (ni −1) · ∂α · ⎪ i n −1 n −1 ( ) i i ⎪ v in (t) = ⎪ y  x il (t) + (t) i i ⎪ l=1 ∂ y ∂ xil (t) (t) i  ⎪  ⎩ x ini (t) + kini (xi1 (t) −  xi1 (t)) +u i (t) + φini 

(13.16)

In the following part, we will consider how to choose the virtual control law αi( j−1) (·) and design the decentralized control input u i for the composite system (13.16). Firstly, we have the following lemma: Lemma 13.2 For system (13.16), choose the Lyapunov–Krasovskii functional U=

N

Ui =

i=1

N

(Vi + Wi )

i=1

where Vi = eiT (t) Pi ei (t) +

ni

vi2j (t) ,

j=1

Wi =

nk N

k=1 j=1

N



σk 1 − dk∗ji

1 + δ2k ∗ 1 − dk1i k=1

t

t−dk ji (t)



t

t−dk1i (t)

yi () ϕk ji (yi ()) d

yi () ϕk1i (yi ()) d

(13.17)

13.3 Robust Controller Design

+

ni

 σi

t t−τi j (t)

j=1

+

223

δ2i 1 − τi1∗



1 κi j (z i ()) d 1 − τi∗j

t

t−τi1 (t)

κi1 (z i ()) d

(13.18)

where σi and δ2i are positive scalars. Then, there exist virtual control inputs αi( j−1) (·) such that the time derivative of U along system (13.16) satisfies ·

U≤

ni N



(−eiT (t) Q i ei (t) − λi j vi2j (t) i=1

j=1

− yi (t) ρi (yi (t)) + ψi (z i (t))), in which Q i = Q i − σi−1 Pi Pi − δ1i I > 0,   1 κ ψi (z i (t)) = δ2i κi1 (z i (t)) + (z (t)) i1 i 1 − τi1∗   ni

1 + σi κi j (z i (t)) + κi j (z i (t)) , 1 − τi∗j j=1 where ρi (yi (t)) is the nonlinear function to be defined, λi j and δ1i are positive scalars. Proof According to Assumption 13.3, one can see that Wi is positive definite, then U is a Lyapunov–Krasovskii functional. Now, let us consider the time derivative of U along system (13.16). First, the time derivative of eiT (t) Pi ei (t) along (13.16) satisfies ·

2eiT (t) Pi ei (t)   = eiT (t) Pi Ai + AiT Pi ei (t) + 2eiT (t) Pi (φi (xi ) − φi ( xi ) + ϕi )   −1 T T T T ≤ ei (t) Pi Ai + Ai Pi ei (t) + βi ei (t) Pi Di Di Pi ei (t) + σi ϕi 2 + βi eiT (t) E iT E i ei (t) + σi−1 eiT (t) Pi Pi ei (t) . With (13.10) and (13.13), it yields ·

2eiT (t) Pi ei (t) ≤ −eiT (t) Q i ei (t) + σi−1 eiT (t) Pi Pi ei (t) + σi

ni

    κi j (z i (t)) + κi j z i t − τi j (t)  j=1

224

13 Output Feedback Stabilization for Interconnected Time-Delay Systems

+ σi

ni

N

(yk (t) ϕi jk (yk (t)) j=1 k=1

     + yk t − di jk (t) ϕi jk yk t − di jk (t) )

(13.19)

2 Along system (13.16), the time derivative of vi1 (t) satisfies ·

2vi1 (t) v i1 (t) = 2yi (t) (ei2 (t) + vi2 (t) − αi1 (vi1 (t)) + φi1 (yi ) + ϕi1 ) = 2yi (t) (vi2 (t) − αi1 (vi1 (t)) + φi1 (yi (t))) + 2yi (t) (ei2 (t) + ϕi1 ) ≤ 2yi (t) (vi2 (t) − αi1 (vi1 (t)) + φi1 (yi (t)))   ni ni δ1i 2 δ2i 2 yi2 (t) + + + ei2 (t) + ϕ δ1i δ2i ni n i i1

(13.20)

and time derivative of vi2j (t) yields for j = 2, . . . , n i − 1, ·

2vi j (t) v i j (t)  ∂αi( j−1) · = 2vi j (t) vi( j+1) (t) + y (t) ∂ yi (t) i  j−1

∂αi( j−1) ·  x il (t) − αi j + ∂ xil (t) l=1     + 2vi j (t) φi j  x i j (t) + ki j (xi1 (t) −  xi1 (t))

(13.21)

and ·

2vini (t) v ini (t) 

n −1

i

∂αi(ni −1) · ∂αi(ni −1) · = 2vini (t) u i + y i (t) +  x il (t) ∂ yi (t) ∂ xil (t) l=1     + 2vini (t) φini  x ini (t) + kini (xi1 (t) −  xi1 (t))



Note that ∂αi( j−1) · y (t) ∂ yi (t) i ∂αi( j−1) = 2vi j (t) xi2 (t) + φi1 (yi (t)) + ϕi1 ) (ei2 (t) +  ∂ yi (t)

2vi j (t)

(13.22)

13.3 Robust Controller Design

225

∂αi( j−1) δ1i 2 xi2 (t) + φi1 (yi (t))) + e ( ∂ yi (t) n i i2    ∂αi( j−1) 2 δ2i 2 ni ni vi j (t) + + + ϕ δ1i δ2i ∂ yi n i i1 ≤ 2vi j (t)

(13.23)

Based on (13.19)–(13.23), we get ·

·

V i = 2eiT (t) Pi ei (t) + 2

ni

·

vi j (t) v i j (t)

j=1

≤ −eiT (t) Q i ei (t) + σi−1 eiT (t) Pi Pi ei (t) + σi

ni

(κi j (z i (t))

j=1 ni

N

    (yk (t) ϕi jk (yk (t)) + κi j z i t − τi j (t)  + σi j=1 k=1

     + yk t − di jk (t) ϕi jk yk t − di jk (t) ) + 2yi (t) (vi2 (t) − αi1 (vi1 (t)) + φi1 (yi (t)))   ni ni 2 2 yi2 (t) + δ1i ei2 + + (t) + δ2i ϕi1 δ1i δ2i  j−1 n

i −1

∂αi( j−1) · + 2vi j (t) vi( j+1) (t) − αi j +  x il (t) ∂ xil (t) j=2 l=1       1 ni ∂αi( j−1) 2 ni vi j (t) + φi j  x i j (t) + + 2 δ1i δ2i ∂ yi  ∂αi( j−1) xi2 (t) + φi1 (yi (t))) + ki j (xi1 (t) −  + xi1 (t)) ( ∂ yi (t)  ∂αi(ni −1) + 2vini (t) u i + xi2 (t) + φi1 (yi (t))) ( ∂ yi (t) n

i −1   ∂αi(ni −1) · + φini  x ini (t) + kini (xi1 (t) −   x il (t) xi1 (t)) + ∂ xil (t) l=1      ∂αi(ni −1) 2 1 ni ni + + vini (t) 2 δ1i δ2i ∂ yi

The time derivative of Wi satisfies ·

Wi ≤

nk N

k=1 j=1



 σk  yi (t) ϕk ji (yi (t)) 1 − dk∗ji

     − σk yi t − dk ji (t) ϕk ji yi t − dk ji (t)



(13.24)

226

13 Output Feedback Stabilization for Interconnected Time-Delay Systems

+

N 

k=1

δ2k ∗ yi (t) ϕk1i (yi (t)) 1 − dk1i

 − δ2k yi (t − dk1i (t)) ϕk1i (yi (t − dk1i (t)))   ni

   σi + κi j (z i (t)) − σi κi j z i t − τi j (t)  1 − τi∗j j=1   1 + δ2i κi1 (z i (t)) − κi1 (z i (t − τi1 (t))) 1 − τi1∗

(13.25)

Note that ni

N

N

σi (yk (t) ϕi jk (yk (t))

i=1 j=1 k=1

     + yk t − di jk (t) ϕi jk yk t − di jk (t) ) =

nk N

N



σk (yi (t) ϕk ji (yi (t))

i=1 k=1 j=1

      + yi t − dk ji (t) ϕk ji yi t − dk ji (t)

(13.26)

and N N



δ2i (yk (t) ϕi1k (yk (t))

i=1 k=1

+ yk (t − di1k (t)) ϕi1k (yk (t − di1k (t)))) =

N

N

δ2k (yi (t) ϕk1i (yi (t))

i=1 k=1

+ yi (t − dk1i (t)) ϕk1i (yi (t − dk1i (t)))



(13.27)

Based on (13.24)–(13.27), we choose the following virtual control inputs   1 ni 1 1 ni yi (t) + αi1 = ρi (yi (t)) + λi1 yi (t) + φi1 (yi (t)) + 2 2 2 δ1i δ2i   N nk 1 1

+ σk ϕ (yi (t)) + ϕk ji (yi (t)) 2 k=1 j=1 1 − dk∗ji k ji 1

δ2k + 2 k=1 N



 1 ∗ ϕk1i (yi (t)) + ϕk1i (yi (t)) 1 − dk1i

and for j = 2, 3, . . . , n i − 1.

(13.28)

13.3 Robust Controller Design

227

  1 λi j vi j (t) + vi( j−1) (t) + φi j  x i j (t) + ki j (xi1 (t) −  xi1 (t)) 2    ∂αi( j−1) 2 ∂αi j ni 1 ni xi2 (t) + φi1 (yi (t))) + + vi j (t) + ( ∂ yi (t) 2 δ1i δ2i ∂ yi

αi j =

+

j−1

∂αi( j−1) l=1

∂ xil (t)

( xi(l+1) (t) + φil ( xil (t)) + kil (xi1 (t) −  xi1 (t)))

(13.29)

Further, the decentralized output feedback controller is designed as 

  1 λin vin (t) + vi(ni −1) (t) + φini  x ini (t) + kini (xi1 (t) −  xi1 (t)) 2 i i    ∂αi(ni −1) 2 ni 1 ni ∂αi(ni −1) xi2 (t) + φi1 (yi (t))) + vini (t) + + ( 2 δ1i δ2i ∂ yi (t) ∂ yi (t)

ui = −

+

n

i −1 l=1

  ∂αi(ni −1) ( xi(l+1) (t) + φil  x il (t) + kil (xi1 (t) −  xi1 (t))) ∂ xil

(13.30)

With virtual control inputs defined in (13.28), (13.29), and control input (13.30) for system (13.16), one has  N 

· · U= Vi + Wi ·

i=1 N

2 ≤ (−eiT (t) Q i ei (t) + σi−1 eiT (t) Pi Pi ei (t) + δ1i ei2 (t)

−

i=1 ni

λi j vi2j (t) − yi (t) ρi (yi (t)) + ψi (z i (t)))

j=1

≤

ni N



(−eiT (t) Q i ei (t) − λi j vi2j (t) i=1

j=1

− yi (t) ρi (yi (t)) + ψi (z i (t)))

(13.31)

then the proof is completed for the lemma. In the following section, we will show how to obtain function ρi (yi ) and analyze the stability problem of the closed-loop system. Remark 13.5 The existence of time delay renders the control problem more difficult than that of controlling the nonlinear systems free of the time delay. Different from the classic backstepping method, here the virtual control law αi1 (y) designed in the first step should not only include the yi information of vi1 subsystem but also include the yi information of vl j (l = 1, 2 . . . , N , j = 2, . . . , n i ) subsystem. With αi1 (y)

228

13 Output Feedback Stabilization for Interconnected Time-Delay Systems

obtained in the first step, we can further get αi j ( j = 2, . . . , n i − 1) and controller u i via step-by-step method. In the first step, we add the function ρi (yi ), which is used to eliminate the impact from the z i -subsystem. In particular, if there is no zero dynamics in the system, we choose ρi (yi ) = 0, then the time derivative of Lyapunov functional U yields ·

U ≤−

N

⎛ ⎝eiT (t) Q i ei (t) +

ni

i=1

⎞ λi j vi2j (t)⎠ .

j=1

It is easy for us to see that the closed-loop system is asymptotically stable. We will formulate one of our main results in this chapter. Theorem 13.1 Under Assumptions 13.1–13.4, the dynamic output feedback controller (13.28)–(13.30) renders the resulting closed-loop system asymptotically stable. Proof With the help of Assumption 13.1, one has ·

Π i ≤ −αi0 (z i (t)) +

N

N 

    γi j  y j (t) + γ i j  y j t − ri j (t)  .

j=1

j=1

By employing the changing supply functions idea proposed by [163], we define 

 t 1 Πi = ωi (ζ) dζ + 1 − ri∗j t−ri j (t) 0 j=1       −1 ωi ◦ αi ◦ αi0 ◦ 4N γ i j  y j (ζ) γ i j  y j (ζ) dζ Πi (t,z i (t))

N

(13.32)

where ◦ means that ωi ◦ αi (π) = ωi (αi (π)), function ωi : + → + is a positive smooth nondecreasing function. It is clear that Π i is a positive definite function. We know ωi (Πi (t, z i (t))) (−αi0 (z i (t)) +

N

j=1

+

N

   γ i j  y j t − ri j (t)  )

j=1

1 = − ωi (Πi (t, z i (t))) αi0 (z i (t)) 2

  γi j  y j 

13.3 Robust Controller Design

229

 + ωi (Πi (t, z i (t))) +

  1 γi j  y j  − αi0 (z i (t)) + 2 j=1 N

    γ i j  y j t − ri j (t)  .

N

j=1

  Now let us consider the following three cases: The first case is that γi j  y j  <    1 1 α αi0 (z i (t)) hold for all i and j. and γ i j  y j t − ri j (t)  < 4N 4N i0 (z i (t)) For this case, one has ωi (Πi (t, z i (t))) (−αi0 (z i (t)) +

N

  γi j  y j 

j=1

+

N

   γ i j  y j t − ri j (t)  )

j=1

1 ≤ − ωi (Πi (t, z i (t))) αi0 (z i (t)) (13.33) 2      1 αi0 (z i (t)) and γ i j  y j t − ri j (t)  ≥ The second case is that γi j  y j  ≥ 4N 1 α hold for all i and j. Then, the following two inequalities hold 4N i0 (z i (t))   ωi (Πi (t, z i (t))) γi j  y j    ≤ ωi (αi (z i (t))) γi j  y j        −1 ≤ ωi ◦ αi ◦ αi0 ◦ 4N γi j y j γi j  y j  , and    ωi (Πi (t, z i (t))) γ i j  y j t − ri j (t)       −1 ≤ ωi ◦ αi ◦ αi0 ◦ 4N γ i j  y j t − ri j (t)     × γ i j  y j t − ri j (t)  . With the two inequalities, one has ⎛ ωi (Πi (t, z i (t))) ⎝−αi0 (z i (t)) +

N

⎞ N  

   γi j  y j  + γ i j  y j t − ri j (t)  ⎠

j=1

1 ≤ − ωi (Πi (t, z i (t))) αi0 (z i (t)) 2 N

      −1 ωi ◦ αi ◦ αi0 + ◦ 4N γi j  y j  γi j  y j  j=1

j=1

230

+

13 Output Feedback Stabilization for Interconnected Time-Delay Systems N

        −1 ωi ◦ αi ◦ αi0 ◦ 4N γ i j  y j t − ri j (t)  γ i j  y j t − ri j (t)  j=1

(13.34)      1 The third case is that γi j  y j  ≥ 4N αi0 (z i (t)) or γ i j  y j t − ri j (t)  ≥ 1 α holds for some i and j. One can see that inequality (13.34) also 4N i0 (z i (t)) stands for this case. Therefore, inequality (13.34) always holds for the three cases. With (13.34), the time derivative of Π i yields · 1 Π i ≤ − ωi ◦ αi (z i (t)) αi0 (z i (t)) 2 N

      −1 + ωi ◦ αi ◦ αi0 ◦ 4N γi j  y j  γi j  y j  j=1

+

N

j=1

     1  −1 y j  γi j y j  . ∗ ωi ◦ αi ◦ αi0 ◦ 4N γ i j 1 − ri j

Based on the Lemma 13.1 and Assumption 13.4, there exists a smooth nondecreasing function qi (s) such that ψi (s) ≤ qi (s) αi0 (s) . In view of k∞ function αi (·) , one knows that there exists a nondecreasing function ωi (·) such that 1 ω ◦ αi (s) ≥ qi (s) . Noting that αi0 (s) is k∞ function, we have 4 i 1 ωi ◦ αi (z i (t)) αi0 (z i (t)) ≥ ψi (z i (t)) . 4 Similarly, there exist nondecreasing functions q 1i j (·) and q 2i j (·) satisfying γ ji (|yi (t)|) ≤ q 1i j (|yi (t)|) yi2 (t) and γ ji (|yi (t)|) ≤ q 2i j (|yi (t)|) yi2 (t), then we can choose smooth function ρi (yi ) such that yi (t) ρi (yi (t)) N %

  & ω j ◦ α j ◦ α−1 q 1i j (|yi (t)|) yi2 (t) ≥ j0 ◦ 4N γ ji (|yi (t)|) j=1

+

N

j=1

  & 1 % ω j ◦ α j ◦ α−1 q 2i j (|yi (t)|) yi2 (t) , j0 ◦ 4N γ ji (|yi (t)|) ∗ 1 − r ji

and then yi ρi (yi ) ≥

N %

  & |) ω j ◦ α j ◦ α−1 ◦ 4N γ γ ji (|yi |) (|y ji i j0 j=1

13.3 Robust Controller Design

+

N

j=1

231

  & 1 % ω j ◦ α j ◦ α−1 γ ji (|yi (t)|) . j0 ◦ 4N γ ji (|yi |) ∗ 1 − r ji

Noting the following equality N N

       −1 γi j  y j  ωi ◦ αi ◦ αi0 ◦ 4N γi j  y j  i=1 j=1

+

N N

i=1 j=1

=

      1  −1 ωi ◦ αi ◦ αi0 ◦ 4N γ i j  y j  γi j y j  1 − ri∗j

N

N %

  & |) ω j ◦ α j ◦ α−1 ◦ 4N γ γ ji (|yi |) (|y ji i j0 j=1 i=1

+

N N

j=1 i=1

=

  & 1 % −1 |) ω ◦ α ◦ α ◦ 4N γ γ ji (|yi |) (|y j j i ji j0 1 − r ∗ji

N

N %

  & ω j ◦ α j ◦ α−1 γ ji (|yi |) j0 ◦ 4N γ ji (|yi |) i=1 j=1

+

N N

i=1 j=1

  & 1 % ω j ◦ α j ◦ α−1 γ ji (|yi |) , j0 ◦ 4N γ ji (|yi |) ∗ 1 − r ji

we have N

yi ρi (yi )

i=1 N

N

       −1 γi j  y j  ωi ◦ αi ◦ αi0 ◦ 4N γi j  y j  ≥ i=1 j=1

+

N N

i=1 j=1

      1  −1 y j  γi j y j  . ∗ ωi ◦ αi ◦ αi0 ◦ 4N γ i j 1 − ri j

 N  Choosing the Lyapunov function U I = i=1 Π i + Ui for the whole interconnected system (13.16), with (13.32), one obtains ·

UI =

 N  ·

· Πi + Ui i=1

232

13 Output Feedback Stabilization for Interconnected Time-Delay Systems N 

1 − ωi ◦ αi (z i (t)) αi0 (z i (t)) 4 i=1 ⎞ ni

λi j vi2j (t)⎠ . − eiT (t) Qei (t) −

≤

j=1

Based on Lyapunov stability theorem, the closed-loop system is asymptotically stable. Remark 13.6 In the controller design procedure, we tend to choose small parameter σk in virtual control input αi1 to avoid the high control gain. But small σk may induce that the inequality Q i − σi−1 Pi Pi − δ1i I > 0 cannot hold. To overcome the drawback, we may first choose a small parameter σk , then solve the following inequality i > (Ai + Ci )T Pi + Pi (Ai + Ci ) −Q + βi−1 Pi Di DiT Pi + βi E iT E i + σi−1 Pi Pi , i is a given positive matrix satisfying Q i − δ1i I > 0. This inequality is where Q equivalent to the following LMI ⎡

⎤ i11 Pi Di Pi Π i = ⎣ DiT Pi −βi Ii 0 ⎦ < 0. Π 0 −σi Ii Pi     i . i11 = Pi Ai + Ci + Υi Bi + BiT ΥiT + Ai + Ci T Pi + βi E iT E i + Q where Π i , solving the above LMI gives With the given matrices Ai , Bi , Ci , Di , E i , and Q parameter βi and matrices Pi , Υi . In this section, we have investigated a large class of interconnected time-delay systems with zero dynamics in each subsystem. The observer is designed, and the gain parameters are obtained by solving LMI. Using the recursive method, we design the observer-based output feedback controller. Finally, by employing the changing supplying function idea, we show that the closed-loop system is asymptotically stable.

13.4 Adaptive Neural Network Control In the above, we have illustrated the design procedure of decentralized output feedback controller for the case that the functions ϕk ji (·) and ϕk ji (·) are known. However, in practical systems, they may be unavailable. For this case, the adaptive neural network can be a good choice as an approximator.

13.4 Adaptive Neural Network Control

233

For system (13.16) with functions ϕk ji (·) and ϕk ji (·) known, we have chosen virtual control input αi1 as 1 1 ρi (yi (t)) + λi1 yi (t) + φi1 (yi (t)) 2 2  1 ni ni yi (t) + Θi (yi (t)) , + + 2 δ1i δ2i

αi1 =

where  N nk 1 1

Θi (yi (t)) = σk ϕ (yi (t)) 2 k=1 j=1 1 − dk∗ji k ji 1

δ2k (ϕk1i (yi (t)) 2 k=1  ϕk1i (yi (t)) . N

+ ϕk ji (yi (t))) + +

1 ∗ 1 − dk1i

If functions ϕk ji (·) and ϕk ji (·) are unknown, Θi (yi (t)) is not available, then Θi (yi (t)) will not be used in the first step of the recursive design procedure. From the knowledge of RBF neural network, there exists an optimal parameter θi∗ such that Θi (yi (t)) = θi∗T ξi (yi (t)) + i in a given compact set Ω yi (t) ⊂ . We select αi1 as 1 i (θi , yi (t)) λi1 yi (t) + φi1 (yi (t)) + Θ 2  1 ni ni 1 yi (t) + ρi (yi (t)) + + 2 δ1i δ2i 2

αi1 =

(13.35)

i (θi (t) , yi (t)) = θiT (t) ξi (yi (t)) is used to approximate nonlinear function where Θ Θi (yi (t)) , and θi is the tuned parameter with adaptive law ·

θi (t) = Γi ξi (yi (t)) yi (t) − Γi li θi (t)

(13.36)

where Γi is a positive matrix and li is a positive scalar. Similar to the recursive method in Sect. 13.3, for subsystem (13.12), we choose the following state transformation 

vi1 (t) = xi1 (t) , xi j (t) + αi( j−1) , j = 2, . . . , n i vi j (t) = 

(13.37)

where functions αi( j−1) are the virtual control inputs, which will be determined in the following parts. With (13.37), subsystem (13.12) is changed into

234

13 Output Feedback Stabilization for Interconnected Time-Delay Systems

⎧· z i (t) = Q i (t, z i (t) , x1 (t) , . . . , x N (t) , ⎪ ⎪ ⎪ ⎪ x1 (t − ri1 (t)) , . . . , x N (t − ri N (t))), ⎪ ⎪ ⎪ · ⎪ ⎪ ei = Ai ei (t) + φi (xi (t)) − φi ( xi (t)) + ϕi , ⎪ ⎪ ⎪ · ⎪ ⎪ + ϕi1 , ⎪ ⎪ y· i (t) = xi2 (t) + φi1 (yi (t)) ∂α ⎪ i( j−1) · ⎪ ⎪ v α + y = v − (t) (t) i( j+1) ij ⎪ ∂ yi (t) i (t) ⎨ ij ·  j−1 ∂αi( j−1) · ∂αi( j−1) x il (t) + ∂θi (t) θi (t) + l=1 ∂xil (t)  ⎪   ⎪ ⎪ +φ  x + k −  xi1 (t)) , (t) (x (t) ⎪ ij ij ij i1 ⎪ ⎪ ⎪ − 1, j = 2, . . . , n ⎪ i ⎪ ⎪ ∂α i −1) · · ⎪ ⎪ v y i (t) = u + ∂iy(in(t) (t) (t) ⎪ in i i ⎪ ⎪ ⎪ ·  ∂α ∂αi (ni −1) · ⎪ i (n i −1) n −1 i ⎪+ ⎪ θ  x + (t) i il (t) l=1 ⎪ ∂θ ∂ x (t) (t) i il  ⎩ +φini  x ni (t) + kini (xi1 (t) −  xi1 (t))

(13.38)

For system (13.38), we choose the following Lyapunov functional U =V +W +H

(13.39)

where V, W , and H are defined as V = H=

N

Vi =

eiT (t) Pi ei (t) ,

i=1

i=1

N

ni N



Hi =

i=1

W =

N

N

Hi j ,

i=1 j=1

Wi

i=1

=

nk N

N

i=1 k=1 j=1

σk 1 − dk∗ji



t t−dk ji (t)

yi (ξ) ϕk ji (yi (ξ)) dξ

 t δ2k yi (ξ) ϕk1i (yi (ξ)) dξ + ∗ 1 − dk1i t−dk1i (t) i=1 k=1 ni  t N

σi κi j (z i (ξ)) dξ + 1 − τi∗j i=1 j=1 t−τi j (t) N

N

+

N

i=1

δ2i 1 − τi1∗



t t−τi1 (t)

κi1 (z i (ξ)) dξ

(13.40)

13.4 Adaptive Neural Network Control

235

with   T  Hi1 = yi2 (t) + θi∗ − θi (t) Γi−1 θi∗ − θi (t) , Hi j = vi2j (t) , j = 1. The time derivative of Hi j along system (13.38) is ·

H i1 = 2yi (t) (ei2 (t) + vi2 (t) − αi1 (vi1 (t)) ·  T + φi1 (yi (t)) + ϕi1 ) − 2 θi∗ − θi Γi−1 θi (t) ≤ 2yi (t) (vi2 (t) − αi1 (vi1 (t)) + φi1 (yi (t)))   ni ni δ1i 2 yi2 (t) + + + e (t) δ1i δ2i n i i2 δ2i 2 + ϕ + 2yi (t) (αi1 (vi1 (t)) − αi1 (vi1 (t))) n i i1 · T  − 2 θi∗ − θi Γi−1 θi (t)

(13.41)

With (13.35) and (13.36), it yields ·

H i1 ≤ −λi1 yi2 (t) + 2yi (t) vi2 (t) − yi (t) ρi (yi (t)) + N

 T + 2li θi∗ − θi (t) θi (t) −



δ1i 2 δ2i 2 e (t) + ϕ n i i2 n i i1

δ2k ∗ ϕk1i (yi (t)) yi (t) 1 − dk1i k=1 

ni  N

σk ϕ (yi (t)) yi (t) + δ2k ϕk1i (yi (t)) yi (t) − 1 − dk∗ji k ji k=1 j=1  + σk ϕk ji (yi (t)) yi (t)

(13.42)

Similarly, for j = 2, . . . , n i − 1, we have   j−1

· ∂αi( j−1) · ∂αi( j−1) · H i j = 2vi j (t) vi( j+1) (t) − αi j + y i (t) +  x il (t) ∂ yi (t) ∂ xil (t) l=1     ∂αi( j−1) · (13.43) + 2vi j (t) φi j  x i j (t) + ki j (xi1 (t) −  θi (t) xi1 (t)) + ∂θi (t) Note that 2vi j (t)

∂αi( j−1) · y (t) ∂ yi (t) i

236

13 Output Feedback Stabilization for Interconnected Time-Delay Systems

∂αi( j−1) xi2 (t) + φi1 (yi (t))) ( ∂ yi (t)    ∂αi( j−1) 2 ni ni vi j (t) + + δ1i δ2i ∂ yi (t) δ1i 2 δ2i 2 + e (t) + ϕ n i i2 n i i1 ≤ 2vi j (t)

(13.44)

We choose the virtual control inputs   1 x i j (t) λi j vi j (t) + vi( j−1) (t) + φi j  2 j−1

∂αi( j−1) ·  x il (t) + ki j (xi1 (t) −  xi1 (t)) + ∂ xil (t) l=1

αi j =

∂αi( j−1) xi2 (t) + φi1 (yi (t))) ( ∂ yi (t)     ∂αi( j−1) 2 ni ni vi j (t) + + δ1i δ2i ∂ yi (t) ∂αi( j−1) + (Γi ξi (yi (t)) yi (t) − Γi li θi (t)) ∂θi (t) +

(13.45)

Further, one has ·

H i j ≤ −λi j vi2j (t) − 2vi( j−1) vi j (t) + 2vi j (t) vi( j+1) (t) +

δ1i 2 δ2i 2 e (t) + ϕ n i i2 n i i1

(13.46)

Employing the recursive method, we design the following controller  u i (t) = −

  1 λini vini (t) + vi(ni −1) (t) + φini  x i j (t) 2

xi1 (t)) + + kini (xi1 (t) − 

n

i −1 l=1

∂αi(ni −1) ·  x il (t) ∂ xil (t)

∂αi(ni −1) + xi2 (t) + φi1 (y (t))) ( ∂ yi (t)     ∂αi(ni −1) 2 ni ni vi j (t) + + δ1i δ2i ∂ yi (t) ' ∂αi(ni −1) + (Γi ξi (yi (t)) yi (t) − Γi li θi (t)) ∂θi (t) then

(13.47)

13.4 Adaptive Neural Network Control

237

·

2 H ini ≤ −λini vin (t) − 2vi(ni −1) vini (t) + i

δ1i 2 δ2i 2 ei2 (t) + ϕ ni n i i1

(13.48)

It follows from (13.46) and (13.48) that ·

H=

ni N



·

Hij

i=1 j=1

≤−

ni N



λi j vi2j (t) +

i=1 j=1

N

  2 2 δ1i ei2 (t) + δ2i ϕi1

(13.49)

i=1

Based on (13.19), (13.25), and (13.49), the time derivative of U yields ·

U≤

N

 T    2 −ei (t) Q i − σi−1 Pi Pi ei (t) + δ1i ei2 (t) i=1

+

N % &

 T 2li θi∗ − θi (t) θi (t) + 2yi (t) i i=1

−

N

⎛ ⎝ yi (t) ρi (yi (t)) +

i=1

+

ni N

i=1 j=1

+

N

N

 σi

⎞ λi j vi2j (t)⎠

j=1

1 κi j (z i (t)) κi j (z i (t)) + 1 − τi∗j

 δ2i κi1 (z i (t)) +

i=1

≤

ni

1 κi1 (z i (t)) 1 − τi1∗





 T (−eiT (t) Q i ei (t) + 2yi (t) i + 2li θi∗ − θi (t) θi (t))

i=1 ni N

  ψi (z i (t)) − λi j vi2j (t) − yi (t) ρi (yi (t)) +

(13.50)

i=1 j=1

where Q i and ψi (z i (t)) are the same as defined in Sect. 13.3. Considering that 2yi (t) i ≤ h i yi2 (t) + h i−1 i2 , where h i is a positive scalar satisfies h i < λi1 , and  T 2li θi∗ − θi (t) θi (t)   2 2 = −li θi∗ − θi (t) − li θi (t)2 + li θi∗  we obtain

(13.51)

238

13 Output Feedback Stabilization for Interconnected Time-Delay Systems ·

U ≤−

N

⎛ ⎝eiT (t) Q i ei (t) +

ni

i=1

+

⎞ λi j vi2j (t)⎠

j=1

N %

&  2  2 li θi∗  − li θi∗ − θi (t) + h i yi2 (t) + h i−1 i2

i=1

+

N

i=1

=−

N

(ψi (z i (t)) − yi (t) ρi (yi (t))) ⎛ ⎝eiT

(t) Q i ei (t) +

ni

i=1

+

⎞ λi j vi2j

(t)⎠

j=2

N % &

 2 li θi∗ − θi (t) − yi (t) ρi (yi (t)) i=1

+

N

ψi (z i (t)) −

N

i=1

λi1 yi2 (t) + δ

(13.52)

i=1

where λi1 = λi1 − h i > 0, δ =

N % &

 2 li θi∗  + h i−1 i2

(13.53)

i=1

For the whole system, we choose the following Lyapunov functional  = U + Πi U

(13.54)

where Π i is the same defined in Sect. 13.3. Similarly, there exist functions ωi (·) and ρi (·) such that the following inequality holds ·

·

·

 = U + Πi U ni N %

2

 eiT (t) Q i ei (t) + li θi∗ − θi (t) + ≤− λi j vi2j (t) i=1

j=2

 1 + λi1 yi2 (t) + ωi ◦ αi (z i (t)) αi0 (z i (t)) + δ 4

(13.55)

Then, one has the following Theorem: Theorem 13.2 For the system (13.38) satisfying Assumption 13.1–13.4 with the functions ϕi jk and ϕi jk unknown, the designed dynamic output feedback controller (13.35), (13.36), (13.45), (13.47) renders the corresponding closed-loop system stable in the sense of semiglobal boundedness.

13.4 Adaptive Neural Network Control

239

Based on Lyapunov stability theory, from (13.55), we know that the system state will converge to the following region N %

eiT

ni  ∗ 2

  λi j vi2j (t) (t) Q i ei (t) + li θi − θi (t) +

i=1

j=2

 1 2 + λi1 yi (t) + ωi ◦ αi (z i (t)) αi0 (z i (t)) ≤ δ 4

(13.56)

Furthermore, as t → ∞, one has δ 

ei (t)2 ≤

 2 δ  , θi∗ − θi (t) ≤ li Qi

λmin δ δ yi2 (t) ≤ , vi2j (t) ≤ for j = 2, . . . , n i − 1 λ λi1 ij

(13.57)

  where λmin Q i denotes the minimum eigenvalue of matrix Q i . According to inequality (13.57), we know that yi (t) is bounded, then the continuous virtual control xi2 (t) is also bounded input αi1 (·) (13.35) is bounded. Further, vi2 (t) is bounded, so  with the state transformation (13.37). By the recursive method, we find that  xi j (t) is bounded. For the bounded ei (t) in (13.57), it is easy for us to see that the state xi (t) of system (13.1) is also bounded. In addition, based on inequality (13.57), we have 1 ωi ◦ αi (z i (t)) αi0 (z i (t)) ≤ δ, 4 then the internal state z i is also bounded. Therefore, the solution of the resulting closed-loop system is bounded. Remark 13.7 For neural network approximation, we may first estimate the domain of the approximation function, then choose the center and width of NN node. In the above, we use the neural network to approximate function Θi (yi (t)) , which includes design parameters σk and δ2k . If the parameters σk and δ2k are large, then more NN nodes are needed for good approximation, which will render the controller structure more complex. We may first choose small parameters σk and δ2k . With σk , one may solve the LMI in Remark 13.6 to obtain the observer gain matrix ki . Then, the decentralized output feedback neural network controller with fewer nodes can  will be less be constructed. From (13.55), one knows that the time derivative of U than zero outside the compact set ⎧ ⎪ ⎪ ⎨

  2 δ  e (t)2 ≤ ,  θi  ≤ lδi , i  λmin ( Q i )  Ωi = X i (t)  yi2 (t) ≤ λδ , vi2j (t) ≤ λδi j , ⎪ i1  ⎪ ⎩  ω ◦ α (z (t)) α (z (t)) ≤ 4δ i

i

i

i0

i

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

,

240

13 Output Feedback Stabilization for Interconnected Time-Delay Systems



T where  θi = θi∗ − θi (t) and X i (t) = z iT (t) , eiT (t) , xiT (t) ,  θiT (t) . This compact set can be rendered small by choosing adjustable parameters λi j , li , h i and matrices Q i . We choose a big compact Ω i such that Ωi ⊂ Ω i at first. The initial values of system state are in this compact Ω i . Based on the compact Ω i , we then choose the center and width of NN node. With the NN controller (13.47), from (13.55), we , see that the state must move in a direction of decreasing the Lyapunov function U thus, the state will stay in the compact set Ω i all the time and finally converge to the compact set Ωi . Remark 13.8 In [26], the central output feedback control problem is investigated for a class of nonlinear systems via neural network, and at each step, one NN is used to approximate a nonlinear function. For an n-order nonlinear system, n neural networks are used for controller design. In this chapter, we consider the decentralized control problem for a large-scale nonlinear systems including N subsystems with i th subsystem in the n i order. We use one NN to approximate a composite function at the design of the first virtual control input; thus, the controller design complexity is reduced. In the whole chapter, we only use N neural networks on controlling the interconnected systems with N subsystems. The NN structure of this chapter should be differentiated for n i times which is in contrast to a single differentiation due to over-parameterized NN structures in [46].

13.5 Simulation Investigation In this section, the following interconnected system will be investigated which includes two subsystems. First subsystem is ·

z 1 (t) = −z 1 (t) + y1 (t) y2 (t − r12 (t)) , ·

x 11 (t) = x12 (t) + φ11 + ϕ11 , ·

x 12 (t) = u 1 (t) + φ12 + ϕ12 , y1 (t) = x11 (t)

(13.58)

and the second subsystem is ·

z 2 (t) = −z 2 (t) + y1 (t − r21 (t)) y2 (t) , ·

x 21 (t) = x22 (t) + φ21 + ϕ21 , x22 (t) = u 2 (t) + φ22 + ϕ22 , y2 (t) = x21 (t)

(13.59)

here, for simplicity, we use φi j and ϕi j (i, j = 1, 2) to express the functions. In the simulation, the functions are chosen as follows:

13.5 Simulation Investigation

241

φ11 = 0, φ12 = x12 (t) + 2x12 sin (x11 (t)) , 2 ϕ11 = x11 (t − d111 (t)) sin t + x21 (t − d121 (t)) + z 1 (t) , 2 ϕ12 = 0.5x11 (t − d121 (t)) sin t + x21 (t − d122 ) + z 1 (t) , φ21 = 0, φ22 = x22 (t) + 2x22 cos (x21 (t)) , 2 ϕ21 = x21 (t − d212 (t)) cos t + x11 (t − d211 (t)) + z 2 (t) , 2 ϕ22 = 0.5x21 (t − d222 (t)) cos t + x11 (t − d221 (t)) + z 2 (t)

(13.60)

and the time delays are chosen as di jl (t) = r21 (t) = 0.5 (1 + sin(t)). From (13.58)– (13.60), we know that the two subsystems are interconnected. For the two subsystems, choosing Πi (t, z i (t)) = z i2 (t) gives ·

Π 1 = 2z 1 (t) (−z 1 (t) + y1 (t) y2 (t − r12 (t))) 1 1 ≤ −z 12 (t) + y14 (t) + y24 (t − r12 (t)) 2 2

(13.61)

and ·

Π 2 = 2z 2 (−z 2 (t) + y1 (t − r21 (t)) y2 (t)) 1 1 ≤ −z 22 (t) + y14 (t − r21 (t)) + y24 (t) 2 2

(13.62)

so Assumption 13.1 holds. From the functions φi j in (13.60), we choose  C1 = C2 =

   00 0 , D1 = D2 = , 01 2



F1 = sin (x11 (t)) , F2 = cos (x21 (t)) , E 1 = E 2 = 0 1 , then Assumption 13.2 holds. Further, we can see that the uncertain interconnections yield   2 x (t − d111 (t)) sin t + x21 (t − d121 (t)) + z 1 (t)2 11 4 2 ≤ 3x11 (t − d111 (t)) + 3x21 (t − d121 (t)) + 3z 12 (t)

(13.63)

  0.5x11 (t − d121 (t)) sin t + x 2 (t − d122 ) + z 1 (t)2 21 2 4 ≤ 1.5x11 (t − d121 (t)) + 3x21 (t − d122 ) + 3z 12 (t)

(13.64)

  2 x (t − d212 (t)) cos t + x11 (t − d211 (t)) + z 2 (t)2 21 4 2 ≤ 3x21 (t − d212 (t)) + 3x11 (t − d211 (t)) + 3z 22 (t)

(13.65)

242

13 Output Feedback Stabilization for Interconnected Time-Delay Systems

  0.5x21 (t − d222 (t)) cos t + x 2 (t − d221 (t)) + z 2 (t)2 11 2 4 ≤ 1.5x21 (t − d222 (t)) + 3x11 (t − d221 (t)) + 3z 22 (t)

(13.66)

then Assumption 13.3 holds. It is not difficult for us to check that Assumption 13.4 stands based on the above inequalities. Therefore, the control design method proposed in this chapter can be applied to the stabilization problem. We design the following observer ·

 x 11 (t) =  x12 (t) + k11 (x11 (t) −  x11 (t)) , ·

 x 12 (t) = u 1 (t) +  x12 (t) + 2 x12 sin (x11 (t)) + k12 (x11 (t) −  x11 (t)) , ·

 x 21 (t) =  x22 (t) + k21 (x21 (t) −  x21 (t)) , ·

 x 22 (t) = u 2 (t) +  x22 (t) + 2 x22 cos (x21 (t)) + k22 (x21 (t) −  x21 (t))

(13.67)

Choose σ1 = σ2 = 2 and Q i = I, then solve LMI (13.57) and obtain the observer

T gain k1 = k2 = 44.7122 557.3833 . First, we assume that the bound functions are precisely known. The controller will be constructed based on Theorem 13.1. With the inequalities (13.61)–(13.67), we have ψi (z i (t)) = 9z i2 by choosing parameter δ2i = 0.5. Noting that αi0 (z i (t)) = z i2 (t), we select function ωi (·) = 36 from the proof of Theorem 13.1. With it, one can further choose ρi (yi (t)) = 54yi3 (t) . With the functions, we construct the virtual control inputs as the following form 1 1 ρi (yi (t)) + λi1 yi (t) 2 2   nk N

1 1 + σk ϕ (yi (t)) + ϕk ji (yi (t)) 2 k=1 j=1 1 − dk∗ji k ji   ni 1 ni yi (t) + φi1 (yi (t)) + + 2 δ1i δ2i   N 1 1

+ δ2k ∗ ϕk1i (yi (t)) + ϕk1i (yi (t)) 2 k=1 1 − dk1i

αi1 =

=

 1 81yi3 (t) + 27yi (t) 2

∗ in which dk1i = 0.5, λi1 = 2, σi = 2, δ1i = δ2i = 0.5 for i = 1, 2.

(13.68)

13.5 Simulation Investigation

243

Further, the decentralized output feedback controller is designed  u 1 (t) = −  x12 (t) + 2 x12 sin (x11 (t)) + k12 (x11 (t) −  x11 (t)) + x11 (t)     + 2 40.5y13 (t) + 13.5y1 (t) +  x12 (t) + 121.5y12 (t) + 13.5  x12 (t)  2   2 3 + 4 121.5y1 (t) + 13.5 40.5y1 (t) + 13.5y1 (t) +  x12 (t) (13.69) and  x22 (t) + 2 x22 cos (x21 (t)) + k22 (x21 (t) −  x21 (t)) + x21 (t) u 2 (t) = −      x22 (t) + 121.5y22 (t) + 13.5  x22 (t) + 2 40.5y23 (t) + 13.5y2 (t) +   2   2 3 + 4 121.5y2 (t) + 13.5 40.5y2 (t) + 13.5y2 (t) +  x22 (t) (13.70) The initial values of the simulation parameters are chosen as shown below z 1 (0) = 2, z 2 (0) = 3, x12 (0) = 2, x22 (0) = 2,  x11 (0) =  x12 (0) =  x21 (0) =  x22 (0) = 1, x11 (ς) = 0.5, x21 (ς) = 0.5, ς ∈ [−1, 0] . The simulation results are shown in Figs. 13.1 and 13.2, in which the state response curves of the first subsystem and the second subsystem are shown, respectively. From the figures, we can see that the constructed dynamic output feedback controller renders the closed-loop system asymptotically stable. 4 z2 x

2

21

x22

0 −2 −4 −6 −8 −10 −12

0

5

Fig. 13.1 Response curves of subsystem (13.58)

10

15

244

13 Output Feedback Stabilization for Interconnected Time-Delay Systems 4 z

2

x

2

21

x22

0 −2 −4 −6 −8 −10 −12

0

5

10

15

Fig. 13.2 State response curves of subsystem (13.59)

If the information of bound function is not precisely known, adaptive neural network control method will be employed to cope with this stabilization problem. Based on Theorem 13.2, we design the virtual control inputs 1 (y1 (t)) , α11 = 27y13 + 3y1 + Θ 2 (y2 (t)) , αi1 = 27y23 + 3y2 + Θ 1 (y1 (t)) = θ1T ξ (y1 (t)) and Θ 2 (y2 (t)) = θ2T ξ (y2 (t)) , in which θ1 (t) and with Θ θ2 (t) are the parameters to be tuned via the following adaptive laws ·

θ1 (t) = ξ1 (y1 (t)) y1 (t) − 0.001θ1 (t) , ·

θ2 (t) = ξ2 (y2 (t)) y2 (t) − 0.001θ2 (t) , we choose ξi (yi ) = [ξi1 (yi ) , ξi2 (yi ) , ξi3 (yi ) , ξi4 (yi ) , ξi5 (yi )]T , in which

ξi1 (yi (t)) = exp − (yi (t) − 8)2 /2 ,

ξi2 (yi (t)) = exp − (yi (t) − 4)2 /2,

ξi3 (yi (t)) = exp − (yi (t))2 /2 ,

ξi4 (yi (t)) = exp − (yi (t) + 4)2 /2 ,

13.5 Simulation Investigation

245

ξi5 (yi (t)) = exp − (yi (t) + 8)2 /2 . Further, we can construct the decentralized adaptive output feedback controller  u 1 (t) = −  x12 (t) + α11 + y1 (t) +  x12 (t) + 2 x12 sin (x11 (t)) ∂α11  x12 (t) ∂ y1 (t)  '  ∂α11 2 ∂α11 · + 4 ( x12 (t) + α11 ) + θ1 (t) , ∂ y1 (t) ∂θ1 (t) + k12 (x11 (t) −  x11 (t)) +

 u 2 (t) = −  x22 (t) + α21 + y2 (t) +  x22 (t) + 2 x22 cos (x21 (t)) ∂α21  x22 (t) ∂ y2 (t)   ' ∂α21 2 ∂α21 · . + θ2 (t) + 4 ( x22 (t) + α21 ) ∂θ2 (t) ∂ y2 (t) + k22 (x21 (t) −  x11 (t)) +

With the decentralized output feedback controller designed above, state response curves are shown in Figs. 13.3 and 13.4. From the figures, we can see that the constructed time-delay-independent output feedback controller can render the closedloop system stable. 2 z1 x11

1

x

12

0

−1

−2

−3

−4 0

5

10

Fig. 13.3 State response curves of subsystem (13.58) via NN control

15

246

13 Output Feedback Stabilization for Interconnected Time-Delay Systems 3 z

2

x

2

21

x22

1 0 −1 −2 −3 −4 −5

0

5

10

15

Fig. 13.4 State response curves of subsystem (13.59) via NN control

13.6 Conclusion Dynamic output feedback control problem for nonlinear interconnected systems with time delay is investigated. Adaptive neural network is employed to approximate unknown nonlinear function. First, we design the dynamic output feedback controller with the bounds of nonlinear functions known based on backstepping design method. With the help of changing Lyapunov function idea, we prove that the constructed controllers can render the closed-loop asymptotically stable. Further, we consider the case that the bounds of uncertain interconnections cannot be totally obtained. Decentralized adaptive neural network controllers are proposed. Based on Lyapunov– Krasovskii functional method, we also show that the resulting closed-loop system is stable in the sense of semiglobal boundedness. Finally, a numerical example is given to illustrate the feasibility and effectiveness of the main results obtained.

Chapter 14

Robust Control of Time-Delay System with Unknown Control Direction

Abstract The robust control problem is investigated for a class of uncertain nonlinear time-delay systems via dynamic output feedback approach. The considered system is in the strict-feedback form with unknown control direction. A fullorder observer is constructed with the gains computed via LMI at first. Then, with the bounds of uncertain functions known, we design the dynamic output feedback controller such that the closed-loop system is asymptotically stable. Furthermore, when the bound functions of uncertainties are not available, the adaptive fuzzy logic system is employed to approximate the uncertain function and the corresponding output feedback controller is designed. It is shown that the resulting closed-loop system is stable in the sense of semiglobal uniform ultimate boundedness. Finally, simulations are done to verify the feasibility and effectiveness of the obtained theoretic results.

14.1 Introduction The control direction may be unknown for practical systems [46, 88, 116, 148, 151, 202]. The Nussbaum function was proposed to cope with the problem in [135]. Then, the function was used to investigate the servomechanism problem [148], adaptive feedback control problem [148, 202], and output regulation problem [88, 116]. In [213], the state feedback control problem was considered for a class of time-delay systems with unknown control direction and the delay independent controller design methodology was presented. In this chapter, we consider the dynamic output feedback control problem for strict-feedback nonlinear time-delay systems with unknown control direction. The observer is designed with the gain matrix obtained by solving LMI at first. By using the Nussbaum-type function detecting the control direction, we construct an observer-based output feedback controller under the case that the bounds of uncertain nonlinear functions are known. It is shown that the resulting closed-loop system is asymptotically stable based on Lyapunov stability theory. Considering that the precise mathematical model of nonlinear functions are difficult to obtain in practical systems, we also investigate the controller design problem for this case. Adaptive fuzzy logic system is employed to approximate the functions, and the adaptive fuzzy © Springer Nature Singapore Pte Ltd. 2018 C. Hua et al., Robust Control for Nonlinear Time-Delay Systems, DOI 10.1007/978-981-10-5131-9_14

247

248

14 Robust Control of Time-Delay System with Unknown Control Direction

output feedback controller is designed such that the closed-loop system is stable in the sense of uniform ultimate boundedness (UUB). Finally, simulations are done to show the validity of the theoretic results.

14.2 Problem Formulation In this chapter, a class of uncertain time-varying nonlinear systems with time delay is considered, which can be put into the following form ⎧· x i (t) = xi+1 (t) + φi (x i (t)) + gi (t, x (t) , x (t − τi )) , ⎪ ⎪ ⎨ i = 1, . . . , n − 1 ⎪ x· n (t) = δu (t) + φn (x n (t)) + gn (t, x (t) , x (t − τn )) ⎪ ⎩ y (t) = x1 (t)

(14.1)

where xi ∈ , u ∈ , and y ∈  are the state variable, control input, and output T  of system, respectively. x i (t) = x1 (t) x2 (t) · · · xi (t) and then x (t) = x n (t). τi is the delay time of state x (t) in xi -subsystem. δ is an unknown nonzero scalar, whose sign determines the control direction. φi (·) and gi (·) are smooth functions with φi (0) = 0 and gi (t, 0, 0) = 0. For system (14.1), we make the following assumptions: Assumption 14.1 Nonlinear function φi (·) satisfies φi (x i (t)) = E i x i (t) + ηi (x i (t))

(14.2)

where E i and ηi (x i (t)) are known matrix and function, respectively. Further, let T  η (·) = η1 (·) η2 (·) · · · ηn (·) , function vector η (·) yields        1  η (x (t)) − η (ξ (t)) ≤ ρ  1 x (t) − ξ (t) δ   δ

(14.3)

  T



where ρ is a positive scalar, η (ξ (t)) = η1 (ξ1 (t)) η2 ξ 2 (t) · · · ηn ξ n (t) T  with ξ i (t) = ξ1 (t) ξ2 (t) · · · ξi (t) and then ξ n (t) = ξ (t) . Assumption 14.2 Nonlinear function gi (·) yields |gi (t,  (t) ,  (t − τi ))| ≤ Hi1 (1 (t)) + Hi2 (1 (t − τi ))

(14.4)

for i = 1, 2, . . . , n, where  (t) = [h 1 (t) , h 2 (t) , . . . , h n (t)]T ∈ n , Hi1 (·) and Hi2 (·) are positive functions with Hi1 (χ) = χH i1 (χ) and Hi2 (χ) = χH i2 (χ). Assumption 14.3 Unknown parameter δ meets δ ≤ |δ| ≤ δ, where δ and δ are known positive scalars.

14.2 Problem Formulation

249

Remark 14.1 Function φi (·) in this chapter is required to satisfy Assumption 14.1 and the system is more general than that of [82, 88], where functions φi (·) are in the form of φi (y). Assumption 14.3 is very common, and one can easily obtain the bounds of uncertain parameter δ for practical systems. Remark 14.2 Assumption 14.2 is imposed on the time-delay functions gi (·), and this assumption can also be seen in [71]. Using x (t) instead of  (t) gives |gi (t, x (t) , x (t − τi ))| ≤ Hi1 (y (t)) + Hi2 (y (t − τi )) . This assumption implies that the uncertain function gi may include unmeasured state variables xi (i ∈ [2, n]), but the bounds of gi should only contain y (t) and y (t − τi ) , for example, gi = sin (xi ) y (t) y (t − τi ) is a function satisfying Assumption 14.2. In fact, if the function gi is time invariant and only contains the output variable y (t) or delayed output y (t − τi ) [26, 88], function (14.1) always holds. With Assumption 14.2, we further obtain |gi (t, x (t) , x (t − τi ))|2 ≤ 2Hi12 (y (t)) + 2Hi22 (y (t − τi )) 2

2

= 2y 2 (t) H i1 (y (t)) + 2y 2 (t − τi ) H i2 (y (t − τi )) . We will use the above inequality to design the output feedback controller. In this chapter, the robust control problem is investigated for system (14.1) under Assumptions 14.1–14.3. The following two cases are considered: (i) If the functions Hi1 (·) and Hi2 (·) are known, the memoryless dynamic output feedback controller is constructed such that the resulting closed-loop system is asymptotically stable. (ii) If functions Hi1 (·) and Hi2 (·) are not available, we design the adaptive fuzzy output feedback controller such that the closed-loop system is UUB stable.

14.3 Preliminaries Before proposing our main results, we introduce the knowledge about Nussbaumtype function and adaptive fuzzy logic systems. Definition 14.1 N (·) is an even smooth Nussbaum-type function, if the function has the following properties [135]

1 s lim sup N (ζ) dζ = ∞, s→∞ s 0

s 1 lim inf N (ζ) dζ = −∞. s→∞ s 0

250

14 Robust Control of Time-Delay System with Unknown Control Direction

From the definition, one knows that Nussbaum functions should have infinite gains and infinite switching  frequency. There are many functions satisfying the above conditions, such as ex p ζ 2 cos (ζ) , ζ 2 cos (ζ) . In this chapter, an even Nussbaum function ζ 2 cos (ζ) is used. For a Nussbaum-type function, we have the following property [46]:   Property 14.1 Let  V (·) and ζ (·) be smooth functions defined on t0 , t f with V (t) ≥ 0, ∀t ∈ t0 , t f , N (·) an even smooth Nussbaum-type function. If the following inequality holds for ∀t ∈ t0 , t f

V (t) ≤ h 1 +

t

·

e−h 2 (t−ϑ) (δ N (ζ (ϑ)) + 1) ζ (ϑ) dϑ

(14.5)

t0

where h 1 and h 2 are positive scalars, then V (t) , ζ (t) and ·   ζ (ϑ) dϑ must be bounded on t0 , t f .

t t0

(δ N (ζ (ϑ)) + 1)

The Nussbaum-type function has been applied to dealing with the control problem of nonlinear systems with control direction unknown [46, 88, 116, 148, 202]. In this paper, we use the function to construct the observer-based output feedback controller for system (14.1) with the control direction unknown. During the past years, fuzzy logic systems have been extensively used as universal approximators for controllers design of dynamic systems with precise model unknown. A fuzzy system is a collection of fuzzy IF-THEN rules of the form: R ( j) : I F z 1 (t) is A1 and · · · and z n (t) is Anj j

T H E N y (t) is B j . By using the strategy of singleton fuzzification, product inference, and centeraverage defuzzification, the output of the fuzzy system is l y (z (t)) =

j=1

yj

l j=1

 n n

i=1

μ A j (z i (t)) i

i=1 μ A j (z i (t))

 ,

i

where μ A j (z i (t)) is the membership function of linguistic variable z i (t) , and y j is i   the point in R at which μ B j achieves its maximum value (assume that μ B i y i = 1). Introducing the concept of the fuzzy basic function vector ς (z (t)) gives y (z (t)) = θ (t)T ς (z (t)) , where T  θ (t) = y 1 (t) , y 2 (t) , . . . , y l (t) , ς (z (t)) = (ς1 (z (t)) , ς2 (z (t)) , . . . , ςl (z (t)))T ,

14.3 Preliminaries

251

and ς j (z (t)) is defined as n

μ A j (z i (t)) i . n j=1 i=1 μ A j (z i (t))

ς j (z (t)) = l

i=1

i

Based on the universal approximation theorem [177], there exists the optimal approximation parameter θ∗ such that θ∗T ς (z (t)) can approximate a nonlinear function G (z (t)) to any desired degree over a compact set Ωz . The parameter θ∗ is defined as follows    T  ∗ θ = arg min sup θ (t) ς (z (t)) − G (z (t)) , θ∈Ωθ

z∈Ωz

where Ωθ and Ωz denote the sets of suitable bounds on θ (t) and z (t) , respec tively. The minimum approximation error satisfies θ∗T ς (z (t)) − G (z (t)) ≤ μ over z (t) ∈ Ωz , where μ is a positive scalar. Adaptive fuzzy logic system is an efficient approximator for unknown nonlinear function. With the help of adaptive fuzzy logic systems, [211, 212] considered the state feedback controller design problem and [104, 165] investigated the output feedback control problem. In [222], an adaptive fuzzy controller was constructed to stabilize a class of nonlinear systems in strict-feedback form. For nonlinear timedelay systems with function unknown, the memorial adaptive fuzzy controller was presented in [69]. In this chapter, we consider designing a memoryless adaptive fuzzy controller for nonlinear time-delay system via dynamic output feedback control approach.

14.4 Observer Design For system (14.1), we design the delay independent observer of the following form ⎧·

 ⎪ ⎪ ⎨ ξ i (t) = ξi+1 (t) + E i ξ i (t) + ηi ξ i (t) − ki ξ1 (t) , i = 1, . . . , n − 1

 ⎪ ⎪ ⎩ ξ· (t) = u (t) + E ξ (t) + η ξ (t) − k ξ (t) n

n n

n

n

(14.6)

n 1

where ξi (t) ∈  is the state variable of observer, ξ i (t) = [ξ1 (t) , ξ2 (t) , . . . , ξi (t)]T , parameters ki (i ∈ [1, n]) are chosen to render the following inequality holds (A + E)T P + P (A + E) + β −1 P P + βρ2 I < −Q

(14.7)

252

14 Robust Control of Time-Delay System with Unknown Control Direction

where matrices P and Q are positive definite, β is a positive scalar, I is an identity matrix, and matrices A and E are defined as ⎡

⎤ −k1 1 0 0 · · · 0 0 ⎢ −k2 0 1 0 · · · 0 0 ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ −k3 0 0 1 · · · .. 0 ⎥ ⎢ ⎥ ⎢ ⎥ A = ⎢ −k4 0 0 0 . . . ... ... ⎥ , ⎢ ⎥ ⎢ . .. .. .. .. .. ⎥ ⎢ .. . . . . . 0⎥ ⎢ ⎥ ⎣ −kn−1 0 0 0 · · · 0 1 ⎦ −kn 0 0 0 0 0 0 ⎡ ⎤ E 11 0 0 · · · 0 ⎢ E 21 E 22 0 · · · 0 ⎥ ⎢ ⎥ ⎢ ⎥ E = ⎢ E 31 E 32 E 33 · · · 0 ⎥ ⎢ .. .. .. . . .. ⎥ ⎣ . . . ⎦ . .

(14.8)

E n1 E n2 E n3 · · · E nn

  where Er 1 Er 2 · · · Err = Er . With system (14.1) and observer (14.6), the composite system is obtained ⎧· ⎪ e (t) = Ae (t) + Ee (t) + Kδ y (t) + 1δ η (x (t)) ⎪ ⎪ ⎪ ⎪ −η (ξ (t)) + 1δ g ⎪ ⎪ · ⎪ ⎪ ⎪ y (t) = x2 (t) + E 11 x1 (t) + η1 (x1 (t)) ⎪ ⎨ +g1 (t, x (t) , x (t − τ1 ))

 · ⎪ ⎪ ξ ξ ξ = ξ + E + η (t) (t) (t) (t) ⎪ i+1 i i i i i ⎪ ⎪ ⎪ ⎪ −ki ξ1 (t) , i = 2, . . . , n − 1 ⎪ ⎪

 ⎪ · ⎪ ⎩ ξ (t) = u (t) + E ξ (t) + η ξ (t) − k ξ (t) n n n n 1 n n where e =

x δ

(14.9)

− ξ, g (·) = [g1 (·) , g2 (·) , . . . , gn (·)]T and K = [k1 , k2 , . . . , kn ]T .

Remark 14.3 To solve inequality (14.7), we decompose A = A + K B with    0 I(n−1)×(n−1) , B = −1 0 · · · 0 A= 0 0 

(14.10)

Then, inequality (14.7) can be solved via the following LMI  =

(1,1) P P −β I