Intelligent Control, Filtering and Model Reduction Analysis for Fuzzy-Model-Based Systems: 385 (Studies in Systems, Decision and Control, 385) [1st ed. 2022] 3030812138, 9783030812133

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Studies in Systems, Decision and Control 385

Xiaojie Su Yao Wen Yue Yang Peng Shi

Intelligent Control, Filtering and Model Reduction Analysis for Fuzzy-ModelBased Systems

Studies in Systems, Decision and Control Volume 385

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

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

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Xiaojie Su · Yao Wen · Yue Yang · Peng Shi

Intelligent Control, Filtering and Model Reduction Analysis for Fuzzy-Model-Based Systems

Xiaojie Su College of Automation Chongqing University Chongqing, China

Yao Wen College of Automation Chongqing University Chongqing, China

Yue Yang College of Automation Chongqing University Chongqing, China

Peng Shi School of Electrical and Electronic Engineering The University of Adelaide Adelaide, SA, Australia

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-030-81213-3 ISBN 978-3-030-81214-0 (eBook) https://doi.org/10.1007/978-3-030-81214-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To My Family X. Su To My Family Y. Wen To My Family Y. Yang To My Family P. Shi

Preface

Problem formulations of physical systems and processes can often lead to complex nonlinear systems, which may cause analysis and synthesis difficulties. Study of nonlinear systems is often problematic due to their complexities. One effective way of representing a complex nonlinear dynamic system is the so-called Takagi-Sugeno (T-S) fuzzy model, which is governed by a family of fuzzy IF-THEN rules that represent local linear input-output relations of the system. It incorporates a family of local linear models that smoothly blend together through fuzzy membership functions. This, in essence, is a multi-model approach in which simple sub-models (typically linear models) are fuzzily combined to describe the global behavior of a nonlinear system. Fuzzy logic method has been studied and developed for decades. It is known to be an effective control approach to some ill-defined and complex control processes. Thanks to fuzzy logic, expert knowledge about the control processes can be employed to heuristically design fuzzy controllers with some linguistic IF-THEN rules. Practically, human knowledge can be represented as linguistic statements and incorporated into the fuzzy logic controller. As a result, the design method can operate with intelligence. Analysis and synthesis including state-feedback control, output-feedback control, tracking control, optical control, filtering, fault detection, and model reduction for a class of T-S fuzzy systems are all thoroughly studied. Fresh novel techniques including the Linear Matrix Inequality (LMI) techniques, the slack matrix method, and so on, are applied to such systems. This monograph is divided into four sections. First, we focus on stabilization synthesis and controller design for T-S fuzzy systems. The following problems are investigated in this book: (1) the problem of stability analysis and stabilization for T-S fuzzy systems with the time-varying delay; (2) the problem of Hankel-norm output feedback controller design for a class of T-S fuzzy stochastic systems; (3) the problem of L2 –L∞ dynamic output feedback controller design for nonlinear switched systems with nonlinear perturbations. Secondly, the reliable filtering and fault detection problems are solved for fuzzy systems. The below problems are studied: (1) the problem of the dissipativity-based filtering problem for fuzzy switched systems with stochastic perturbation; (2) the fault detection filtering problem for nonlinear switched stochastic system; (3) the problem of reliable filter design with strictly dissipativity for discrete-time T-S fuzzy time-delay systems. vii

viii

Preface

Then the theories and techniques developed in the previous part are extended to the model reduction and model approximation of T-S fuzzy systems. The below problems are studied: (1) the reduced-order model approximation problem for discretetime hybrid switched nonlinear systems; (2) the model approximation problem for dynamic systems with time-varying delays under the fuzzy framework; (3) the model approximation problem for T-S fuzzy switched systems with stochastic disturbance; (4) the H∞ reduced-order filter design problem for discrete-time fuzzy delayed systems with stochastic perturbation. Finally, two real applications are proposed to demonstrate the feasibility and effectiveness of the fuzzy control design presented in the previous parts. The first application is the dissipative event-triggered fuzzy control of truck-trailer system. In view of the fuzzy model, the stability of the resulting system is analyzed in terms of Lyapunov stability theory. Additionally, the explicit expression of the desired controller is given in view of Linear Matrix Inequalities (LMIs), which ensures the resulting closed-loop system is asymptotically stable and strictly (X , Y , Z) –θ –dissipative . The second one is the event-triggered fuzzy control of inverted pendulum systems. By employing the parallel distributed compensation law, sufficient conditions for the resulting fuzzy system and the event-triggered fuzzy controller are presented for the nonlinear inverted pendulum system. The main contents are suitable for a one-semester graduate course. This publication is a research reference whose intended audience includes researchers, postgraduate students. Chongqing, China Chongqing, China Chongqing, China Adelaide, Australia July 2021

Xiaojie Su Yao Wen Yue Yang Peng Shi

Acknowledgements

There are numerous individuals without whose help this book will not have been completed. Special thanks go to Prof. Ligang Wu from Harbin Institute of Technology, Prof. Yong-Duan Song from Chongqing University, Prof. Michael V. Basin from the Autonomous University of Nuevo Leon, Prof. Hamid Reza Karimi from University of Agder, Dr. Hak Keung Lam from King’s College London, Prof. Rongni Yang from Shandong University, Prof. Jianxing Liu from Harbin Institute of Technology, for their valuable suggestions, constructive comments, and support. Our acknowledgments also go to our fellow colleagues who have offered invaluable support and encouragement throughout this research effort. Thanks go to our students, Hongying Zhou, Fengqin Xia, Xinxin Liu, Bingna Qiao, Yaoyao Tan, Feng Hu, and Qianqian Chen for their commentary. The authors are especially grateful to their families for their encouragement and never-ending support when it was most required. Finally, we would like to thank the editors at Springer for their professional and efficient handling of this project. The writing of this book was supported in part by the National Key R&D Program of China under Grant (2019YFB1312002), the Key-Area Research and Development Program of Guangdong Province under Grant (2020B0909020001), the National Natural Science Foundation of China (617 72095), and Chongqing Science Fund for Outstanding Young Scholars (cstc2019jcyjjqX0015).

ix

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Fuzzy-Model-Based Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 T-S Fuzzy Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Fuzzy-Model-Based Control System . . . . . . . . . . . . . . . . . 1.2.3 Stability Analysis of Fuzzy Control Systems . . . . . . . . . . 1.3 Intelligent Control of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . 1.3.1 Intelligent Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Fuzzy Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Reduced-Order Method Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Reduced Filtering and Control . . . . . . . . . . . . . . . . . . . . . . 1.5 Event-Triggered Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Publication Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Publication Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I

1 1 2 3 5 6 8 8 10 11 11 12 14 16 17

Stability Analysis and Fuzzy Control

2

Stabilization Synthesis of T-S Fuzzy Delayed Systems . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 System Description and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 2.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 State Feedback Fuzzy Control . . . . . . . . . . . . . . . . . . . . . . 2.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 25 28 28 34 37 41

3

Output Feedback Control of Fuzzy Stochastic Systems . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 System Description and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 3.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 State-Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Hankel-Norm Output Feedback Control . . . . . . . . . . . . . .

43 43 43 46 46 49 xi

xii

Contents

3.4 3.5 4

Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54 58

L2 –L∞ Output Feedback Control of Fuzzy Switching Systems . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 System Description and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 4.3 System Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Dynamic Output Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Reduced-Order Controller Design . . . . . . . . . . . . . . . . . . . 4.4.2 Full-Order Controller Design . . . . . . . . . . . . . . . . . . . . . . . 4.5 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 59 59 63 69 69 72 74 86

Part II

Fuzzy Filtering and Fault Detection

5

Dissipative Filtering of Fuzzy Switched Systems . . . . . . . . . . . . . . . . . . 89 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2 System Description and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 89 5.2.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2.2 Dissipativity Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.3.1 Dissipativity Performance Analysis . . . . . . . . . . . . . . . . . . 93 5.3.2 Dissipativity-Based Filter Design . . . . . . . . . . . . . . . . . . . 98 5.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6

Fault Detection for Switched Stochastic Systems . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 System Description and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 6.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 System Performance Analysis . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Fault Detection Filter Design . . . . . . . . . . . . . . . . . . . . . . . 6.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 105 109 109 112 116 118

7

Reliable Filtering for T-S Fuzzy Time-Delay Systems . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 System Description and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Dissipativity Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Reciprocally Convex Approach . . . . . . . . . . . . . . . . . . . . . 7.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Reliable Dissipativity Analysis . . . . . . . . . . . . . . . . . . . . . 7.3.2 Reliable Filter Design with Dissipativity . . . . . . . . . . . . . 7.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121 121 121 121 124 125 126 126 134 140 145

Contents

xiii

Part III Model Reduction and Reduced-Order Synthesis 8

Reduced-Order Model Approximation of Switched Systems . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 System Description and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 8.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Pre-specified Performance Analysis . . . . . . . . . . . . . . . . . 8.3.2 Model Approximation by Projection Technique . . . . . . . 8.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

149 149 149 152 152 156 159 162

9

Model Reduction of Time-Varying Delay Fuzzy Systems . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 System Description and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 9.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Performance Analysis via Reciprocally Convex Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Model Approximation via Projection Technique . . . . . . . 9.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165 165 165 168

10 Model Approximation of Fuzzy Switched Systems . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 System Description and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 10.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Hankel-Norm Performance Analysis . . . . . . . . . . . . . . . . . 10.3.2 Model Approximation by the Hankel-Norm Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185 185 185 188 188

11 Reduced-Order Filter Design of Fuzzy Stochastic Systems . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 System Description and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 11.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 H∞ Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Reduced-Order Filter Design . . . . . . . . . . . . . . . . . . . . . . . 11.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215 215 215 219 219 225 231 241

168 176 181 184

200 206 214

Part IV Event-Triggered Fuzzy Control Application 12 Dissipative Event-Triggered Fuzzy Control of Truck-Trailer Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 System Description and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Truck-Trailer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

245 245 245 245

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Contents

12.2.2 T-S Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Dissipative Performance Analysis . . . . . . . . . . . . . . . . . . . 12.3.2 Fuzzy Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

247 251 251 258 261 265

13 Event-Triggered Fuzzy Control of Inverted Pendulum Systems . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 System Description and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Inverted Pendulum System . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 T-S Fuzzy System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Fuzzy Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Stability of the Nonlinear Inverted Pendulum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Fuzzy Control of Inverted Pendulum Systems . . . . . . . . . 13.3.3 Event-Triggered Fuzzy Control . . . . . . . . . . . . . . . . . . . . . 13.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

267 267 267 267 270 272 272 279 282 286 291

14 Conclusion and Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 14.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 14.2 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

Notations and Acronyms

 ∈ ∀  R Rn Rn×m Z Z+ E{·} He(A) lim max min sup inf rank(·) trace(·) λmin (·) λmax (·) I In 0 0n×m XT X∗ X −1 X > ( 0, ∀t ≥ 0.

i=1

Therefore, h i (x(t)) ≥ 0, i = 1, 2, . . . , r ;

r 

h i (x(t)) = 1.

(1.2)

i=1

In general, the T-S fuzzy model can be established using two approaches: (1) By using certain system identification algorithms [235, 272] based on the input–output data. This approach is suitable for nonlinear systems for which mathematical models are not available, but input–output data are available. (2) If the mathematical model of the nonlinear system is available, the T-S fuzzy model can be derived from the mathematical model by using the concept of sector nonlinearity or local approxi-

1.2 Fuzzy-Model-Based Systems

5

mation [275, 299]. Notably, in the second approach, the grades of the membership may be uncertain if they are in terms of uncertain system parameters. In this case, a nonlinear plant subject to parameter uncertainties can be represented as a T-S fuzzy model with uncertain grades of membership.

1.2.2 Fuzzy-Model-Based Control System The most widely used approach to design the fuzzy controller pertains to the statefeedback fuzzy controller [122, 226, 338], which has a structure similar to the T-S fuzzy model and is a weighted sum of several linear state-feedback sub-controllers. The control action is described by certain linguistic rules. Let us consider the following state-feedback fuzzy controller:  Controller Form: Rule j: IF ϑ1 (x(t)) is N j1 and . . . and ϑq (x(t)) is N jq , THEN u(t) = K j x(t),

j = 1, 2, . . . , s,

where N jn is the fuzzy set of rule j corresponding to the function ϑn (x(t)), j = 1, 2, . . . , s, n = 1, 2, . . . , q, and s is the number of IF-THEN rules. K j ∈ Rm×n is the gain matrix of the state feedback controller in each rule, and a compact form of the controller is given by u(t) =

s 

g j (x(t))K j x(t),

(1.3)

j=1

where g j (x(t)) =

q    ν j (x(t)) , ν (x(t)) = μN jn θn (x(t)) , j s  n=1 ν j (x(t)) j=1

with g j (x(t)) ≥ 0,

j = 1, 2, . . . , s;

s 

g j (x(t)) = 1,

(1.4)

j=1

  g j (x(t)) are the normalized grades of membership function, and μN jn ϑn (x(t)) are the membership functions corresponding to the fuzzy set N jn . A fuzzy-model-based control system consists of a nonlinear plant represented by the T-S fuzzy model (1.1) and fuzzy controller (1.3) connected in a closed loop.

6

1 Introduction

Throughout this book, as derived from (1.2) and (1.4), the following property is used during the system performance analysis: r 

h i (x(t)) =

s 

i=1

g j (x(t)) =

j=1

r  s 

h i (x(t))g j (x(t)) = 1.

(1.5)

i=1 j=1

Consider (1.1), (1.3) and (1.5), the closed-loop fuzzy-model-based control system can be described by δx(t) =

r 

s    h i (x(t)) Ai x(t) + Bi g j (x(t))K j x(t)

i=1

=

s r  

j=1

h i (x(t))g j (x(t))Ai j x(t),

i=1 j=1

where Ai j  Ai + Bi K j , x(t) ∈ Rn is thestate vector, δ denotes the derivative operator in ˙ and the shift forward operator in discrete  continuous time i.e.,δx(t) = x(t) time i.e., δx(t) = x(t + 1) .

1.2.3 Stability Analysis of Fuzzy Control Systems The stability analysis and control synthesis are essential aspects in fuzzy-modelbased control problems. The most popular approach to investigate the stability of fuzzy-model-based control systems is based on the Lyapunov method [151, 225, 324]. The stability analysis process can be realized using the following steps: 1. Construct a fuzzy model representing the nonlinear plant. 2. Select the type of fuzzy controller for the control process. 3. Formulate a fuzzy-model-based control system by connecting the fuzzy model and fuzzy controller in a closed loop, as displayed in Fig. 1.2. 4. Define a Lyapunov function candidate, which is a scalar positive function. 5. Set the stability conditions based on the Lyapunov stability method. Moreover, in the stability analysis of fuzzy-model-based control systems, the conservativeness is related to several factors: 1. Types of Lyapunov Functions: A Lyapunov function is a mathematical tool to investigate the stability problem for fuzzy-model-based control systems. By employing different types or forms of Lyapunov function candidates to approximate the domain of the feasible solution, different stability conditions pertaining to different levels of conservativeness are obtained. 2. Types of Stability Analyses: The type of stability analysis determines the information of the membership functions to be considered, which influences

1.2 Fuzzy-Model-Based Systems

7

the conservativeness of the stability conditions. In particular, in membershipfunction-independent and membership-function-dependent stability analyses, the information of the membership functions is not considered and considered to set the stability conditions, respectively. In the latter case, the stability analysis results depend on the considered nonlinear model and often correspond to relaxed stability conditions. 3. Methods of Stability Analyses: The methods used to realize the stability analysis, such as those for managing and incorporating the membership functions influence the degree of conservativeness for the stability conditions. For T-S fuzzy-model-based control systems, as shown in Fig. 1.3, the approaches to realize the stability analysis based on the Lyapunov stability theory can be classified into two types, specifically, membership-function-independent and membershipfunction-dependent approaches depending on whether the information of the membership functions is considered in the stability analysis. As the membership-functionindependent stability analysis does not consider the membership functions, the stability analysis results are often more conservative than those based on the membershipfunction-dependent approach, in which the membership functions are considered in the stability analysis.

1.2.3.1

Membership-Function-Independent Stability Analysis

In the membership-function-independent stability analysis, the information of the membership functions is not considered, and only the local control subsystems of the fuzzy-model-based control systems are managed. Thus, the stability conditions do not involve any membership functions. Once there exists a feasible solution to the stability conditions, the fuzzy-model-based control system is guaranteed to be stable for any shape of the membership functions. However, when membership functions are not considered in the stability analysis, certain information of the nonlinearity is ignored. Therefore, the membership-function-independent stability results are potentially conservative.

1.2.3.2

Membership-Function-Dependent Stability Analysis

The membership-function-dependent stability conditions take into account the membership functions of the fuzzy model and fuzzy controller in the stability analysis. The obtained stability conditions include the information of the membership functions. In general, more relaxed stability conditions are obtained compared with those in the membership-function-independent stability analysis as more information of the fuzzy-model-based control system is considered. However, as the information of the membership functions is implemented in the form of slack matrices in the stability analysis and the number of stability conditions is generally high, the computational demand to determine a feasible solution for the stability conditions is high. Moreover,

8

1 Introduction

Fig. 1.3 Stability analysis approaches for T-S fuzzy-model-based control systems

the obtained stability conditions are specific to the fuzzy-model-based control system to be controlled and not generalized for all shapes of the membership functions.

1.3 Intelligent Control of Nonlinear Systems 1.3.1 Intelligent Control Since the 20th century, the requirements for control systems have evolved with the development of science and technology [107, 112, 114, 309]. From linear to nonlinear systems and from single-input single-output (SISO) control systems to multipleinput multiple-output (MIMO) control systems, multiple control approaches have been integrated to complement the strengths of different techniques [46, 68, 178, 271, 313]. Intelligent control, as a novel technology, can help realize the preset control tasks of a system autonomously without human intervention [60, 75, 85, 201, 236]. Through this control implementation, the system control mode evolves from

1.3 Intelligent Control of Nonlinear Systems

9

ordinary automatic control to a more advanced intelligent control mode. Intelligent control strategies transform the control model from certain to uncertain and provide a more convenient path for the information exchange between the input and output devices of the control system and the external environment. Moreover, when using intelligent control schemes, the control task of the system changes from a single task to a more complex control task. Thus, a more ideal solution exists for the control problem of nonlinear systems, which cannot be easily solved using ordinary automatic control systems. Intelligent control strategies enable an automatic control system to achieve self-adaptation, self-organization, self-learning, and self-coordination [197, 279, 301, 317]. Intelligent control represents the development trend of the control theory, which can effectively solve complex control problems, and several related techniques can be applied to industry, agriculture, service industry, military aviation, and other fields beyond control pertaining to finance, management, civil engineering, and design, among other domains. Most control systems are designed under the assumption of perfect data transmission in both the sensor-to-controller and controller-to-actuator channels. This assumption is valid for most point-to-point control structures, but not for the widely used networked control systems (NCSs), in which the control loop is closed through a certain form of communication networks. Compared with traditional point-topoint control systems, the main advantages of NCSs are the low cost, flexibility and easy reconfigurability, inherent reliability and robustness to failure, and adaptation capability [20, 101, 108, 155, 265]. Consequently, NCSs have been applied in a broad range of areas such as power grids, water distribution networks, transportation networks, haptics collaboration over the Internet, mobile sensor networks, and unmanned aerial vehicles [78, 160, 234, 344, 348]. However, the introduction of communication channels in the control loop induces several network-induced critical issues or constraints such as variable transmission delays, data-packet dropouts, packet disorder, and quantization errors, which can significantly degrade the system performance and even destabilize the system in certain conditions. In recent years, the issues induced by the NCSs have posed considerable challenges to conventional control and communication theory and have attracted considerable attention from researchers of multiple disciplines including control, communication, and mathematics [65, 92, 174, 354]. The typical research topics pertaining to NCSs include the stability of NCSs under various network constraints, state estimation over lossy networks, controller/filter design of NCSs with guaranteed stability, and performance optimization [134, 198, 263, 339]. Moreover, recently, the benefit of using wireless communication technology in large-scale industrial processes has become evident [64, 145, 282, 283, 314], especially in the form of cyber-physical systems. The utilization of wireless networks in industrial process control enables the realization of new system architectures and designs. Nevertheless, many industrial control processes involve severe nonlinear characteristics, which renders the analysis and design highly challenging. In recent decades, fuzzy-logic-control (FLC) has received considerable attention from both academic and industrial communities. Notably, the FLC has been demonstrated to be a simple and powerful strategy for the analysis and synthesis of many com-

10

1 Introduction

plex nonlinear systems and nonanalytic systems [224, 227, 233, 336]. Significant research efforts have been devoted for both theoretical advances and implementation techniques for fuzzy controllers, and many industrial applications of the FLC have been reported in the existing literature [203, 228, 274]. Among various model-based fuzzy control methods, the approach based on the T-S model is well suited to design model-based nonlinear system controller [80, 126, 149, 244]. The research interest in the systematic analysis and design of networked nonlinear systems via T-S fuzzy dynamic models has increased, and multiple significant results have been reported [88, 165, 215, 346, 356].

1.3.2 Fuzzy Control The mathematical modelling of physical systems and processes often generates complex nonlinear systems, the synthesis and analysis of which is highly difficult. The research on nonlinear systems is often problematic due to their complexities. One effective way of representing a complex nonlinear dynamic system is by using the T-S fuzzy model [5, 16, 27, 126, 142], which is governed by a family of fuzzy IF-THEN rules that represent the local linear input–output relations of the system. This model incorporates a family of local linear models smoothly blended through fuzzy membership functions. This approach, in essence, is a multi-model approach, in which simple sub-models (typically linear models) are fuzzily combined to describe the global behaviour of a nonlinear system [7, 21, 23, 260]. Within these fuzzy models, the local dynamics in different state space regions are represented by linear models [24, 29, 37, 41, 48]. An overall fuzzy model of the system is created by fuzzily ‘blending’ these linear models. Based on the fuzzy model, the control design can be realized based on the parallel distributed compensation (PDC) scheme. In particular, a linear state-feedback controller is designed for each local linear model. The obtained overall controller is usually nonlinear and represents a fuzzy ‘blending’ of each individual linear controller [33, 44, 56, 61, 69, 72]. Nevertheless, because only a part of the state information may be known in realworld engineering, the state-feedback control is not entirely effective to ensure the desired performance level [11, 12, 189, 257]. Consequently, intensive research has been conducted in the area of output feedback control (OFC) design. Although the state-output feedback control (SOFC) technique (see, for instance, [100, 110, 188]) can be used to address the dynamic output feedback control (DOFC) problem, the process involves several analytical difficulties. Nonetheless, several solutions have been obtained for DOFC problems. For instance, the DOFC problems for T-S fuzzy systems were addressed in [40, 128, 147, 191, 345], the corresponding results for Markovian hybrid jump systems were reported in [9, 156, 177, 240? ], and the feasibility conditions for stochastic switched systems were defined in [54, 90, 91, 175, 361]. In addition, the mixed H2 /H∞ fuzzy OFC design techniques were introduced in [49, 81, 221, 305]. However, to the best of our knowledge, only a few studies have focused on the fuzzy output feedback controller design with the L2 –L∞ perfor-

1.3 Intelligent Control of Nonlinear Systems

11

mance for nonlinear switched systems. Furthermore, the previously obtained results in this domain must be further investigated in terms of multiple aspects, for instance, the mechanism of selecting the fuzzy piecewise Lyapunov functions for nonlinear complex systems to reduce conservativeness and the design strategy for switched DOFC to ensure the system stability and achieve L2 –L∞ performance level. These problems are the motivation for the current research. Practical systems, especially those pertaining to chemical processes and communication, commonly involve time delays, which reduce the system performance and may lead to instability. The prevalent use of stochastic systems has led to the widespread application of stochastic modelling in science and engineering domains [247, 303]. Many key results have been reported for the T-S fuzzy model [42, 239, 262, 360], switched systems [34, 158, 252, 291], and Markovian jumping systems [14, 36, 241, 248, 332]. The general control synthesis methodologies cannot satisfy the requirements for T-S fuzzy systems that incorporate intelligent control method, filtering, and model reduction analysis. Considering these aspects, this monograph presents the innovative research developments and methodologies pertaining to the synthesis and analysis of T-S fuzzy systems in a unified matrix inequality setting. Researchers exploring the problems of intelligent control, filtering, and model reduction for fuzzy-model-based systems can find valuable reference material in this text. The aspects of stability analysis and stabilization, dynamic output feedback (DOF) control, full- and reduced-order filter design, fault detection, and model reduction problems for a class of T-S fuzzy systems are thoroughly investigated. Moreover, novel techniques are applied to systems, including the delay-partitioning method, slack matrix method, reciprocally convex approach, and event-triggered strategy, [15, 22, 31, 288, 349].

1.4 Reduced-Order Method Synthesis 1.4.1 Model Reduction With the increasing demands of higher security, reliability, and performance in several engineering domains, fault detection techniques have attracted considerable attention. Model-based fault diagnosis represents an effective approach to solve the fault detection problems in technical processes [45, 87, 102, 231, 269, 277]. The key strategy for fault detection involves two parts, specifically, the construction of a residual signal and computation of a residual evaluation function to be compared against a predefined threshold. A fault alarm is generated when the residual evaluation function exceeds the threshold. To detect faults in a timely manner and avoid false alarms, the residual signal should be sensitive to faults and robust against modelling errors or disturbances for a fault detection problem [26, 35, 38, 79, 280]. Recently, several model-based approaches for fault detection problems have been reported for dynamic systems, along with several significant results. For example, the authors of

12

1 Introduction

[192] used a dynamic event-triggered strategy to address the fault detection problem for nonlinear stochastic systems. Certain researchers proposed a novel reliable decentralized control method for interconnected discrete delay systems, in [176]. Moreover, a novel model-based fault detection and prediction scheme for dynamic T-S fuzzy systems was established in [267]. In [95], the sensor fault reconfigurable control problem was investigated for switched systems. In [304], the fault detection problem for two-dimensional Markovian jump systems was addressed. Furthermore, the fault detection issue for uncertain sampled-data systems using deterministic learning was solved in [50]. Despite these achievements, only a few results pertaining to nonlinear switched models with stochastic disturbances have been reported, although considerable effort was expended in the past decade to examine filtering problems due to their theoretical and practical significance in control engineering and signal processing [47, 105, 199, 259, 292]. The nonlinear dynamic systems in various engineering fields commonly correspond to high-order mathematical models [113, 144, 214], and thus, the performance evaluation and system stability analysis are highly difficult and complex. Hence, methods to simplify the original system with lower-order filters based on certain criteria have been focused on [6, 96, 111, 157]. The goal of reduced-order filtering is to incorporate a filter of a lower order than that of the original system based on specific standards. In recent years, many techniques have been introduced to process mathematical models by using reduced-order filters, such as H∞ filtering [3, 67, 138], H2 filtering [84, 185, 246], and L2 − L∞ filter designs [32, 173, 211]. In [238], the reduced-order model approximation problem for discrete-time hybrid switched nonlinear systems was addressed via T-S fuzzy modelling. In [211], the problem of L2 − L∞ filtering for discrete-time T-S fuzzy systems with stochastic perturbation was considered. Model order reduction for an electronic circuit design was described in [270]. Moreover, in [221], the problem of stabilization and mixed H2 /H∞ reduced-order dynamic OFC of discrete systems was solved. The authors in [327] addressed the robust fuzzy L2 − L∞ filtering problem for uncertain discretetime Markov jump systems with nonhomogeneous jump processes. Furthermore, the dissipative filtering issues for a class of nonlinear delayed systems approximated by T-S fuzzy models were considered in [17]. In addition, the authors of [255] investigated the model approximation for switched systems with stochastic perturbation. In general, the reduced-order filter design can be flexibly and simply implemented in practical applications, which is a motivation for the current study.

1.4.2 Reduced Filtering and Control Among different filtering techniques, the H∞ filter has garnered notable interest as it does not involve any statistical assumptions regarding the exogenous noises and can consider the uncertainty in the system model [116, 330, 335, 362]. The H∞ filter design approach considers the worst case from the process noise to the minimization of the estimation error and is thus of considerable value in many practical applica-

1.4 Reduced-Order Method Synthesis

13

tions. Significant results pertaining to H∞ filtering have been reported. Specifically, the H∞ filtering problem for linear systems with uncertain disturbances ([43, 94, 97]), hybrid systems ([57, 168, 308]), singular systems ([172, 187, 289]), and T-S fuzzy systems ([106, 193, 254]) was solved. However, the measurement outputs of a dynamic system contain incomplete observations in practice, as contingent failures of all sensors may occur in a system [40, 130, 152], which may deteriorate the filter performances and pose certain hazard risks, see for example, [163, 164, 290, 318]. In this regard, the problem of reliable filter design is a key issue and has attracted considerable attention, for instance, in [249, 315, 331], reliable filtering problems for discrete-time-delay systems were investigated. Furthermore, for T-S fuzzy systems ([127, 151, 296, 297]), problems of reliable filtering in the presence of sensor faults have attracted research interest [171, 276, 306]. Moreover, the event-driven reliable dissipative filtering issue for a class of T-S fuzzy systems was considered in [140]. The problem of dissipativity-based reliable fuzzy filter design for uncertain nonlinear systems was investigated in [364] using the interval type-2 fuzzy method. The authors in [281] focused on the reliable L2 − L∞ filter design for continuoustime Markov jump systems, based on the T-S fuzzy model. In addition, the reliable exponential H∞ filtering problem for singular Markovian jump systems with timevarying delays subject to sensor failures was resolved in [167]. The authors of [123] investigated the robust passive reliable filtering issue for T-S fuzzy systems with time-varying delays, random uncertainties, and missing measurements. In practice, many modern industrial systems involve complex features, such as system parameter jumps and the chattering phenomenon [28, 243, 266, 304]. To accurately describe such phenomena, a concept of switched systems has been introduced to overcome the drawbacks of discrete or continuous dynamics derived from the traditional control theory. Switched systems are composed of a finite number of continuous or discrete dynamic subsystems, as well as a switching strategy, indicating the active subsystem at each time instant [98, 124, 300]. Moreover, hybrid stochastic switched systems, which represent a significant component of stochastic jump systems, are composed of a finite number of independent control subsystems, including discrete-time or continuous-time dynamics, and a switching signal governing the activation of these concerned subsystems [200, 206, 208, 212]. A large class of practical systems and processes, including advanced transportation management systems, automated highway systems, communication systems, and network control systems, can be characterized as hybrid stochastic switched systems [154, 319, 322]. Moreover, several intelligent control strategies have been introducing by hybrid switching controllers, which can effectively overcome the limitations of the traditionally adopted single controller and considerably enhance the resulting closed-loop control system performance. In this manner, the corresponding closedloop control systems can be transformed to typical hybrid stochastic switched systems [210, 223, 261]. Recently, several researchers focused on hybrid stochastic switched systems and achieved notable achievements. For example, the switched controller design problems of hybrid switched systems were addressed in [166, 307, 328], model reduction/approximation approaches for stochastic switched time-delay systems were established in [39, 305, 316], mixed H∞ and passive filtering issues were

14

1 Introduction

Fig. 1.4 A structure diagram of the event-triggered control system

investigated in [52, 262, 363], and stability analysis and stabilization problems for switched linear systems were discussed in [141, 196, 302].

1.5 Event-Triggered Strategy At present, two primary communication schemes, specifically, the time-triggered scheme and event-triggered scheme dominate the given research field. In general, the time-triggered communication scheme, proposed for stability analysis and system modelling, may result in transmitting certain “unnecessary” sampling data and wasting the limited space and communication bandwidth in the network [179, 310, 325]. Consequently, the event-triggered scheme was established to gradually replace the time-triggered scheme. The event-triggered technique can mitigate the unnecessary wastage of limited resources while ensuring a satisfactory system performance and has thus garnered increasing research interest [285, 326, 334]. The overall concept of the event-triggered communication scheme is that the control process is completed only if the pre-specified triggering criteria is satisfied, owing to which the state signals are not constantly triggered and transmitted in each sampling period. In this manner, the requirement for the communication bandwidth and computation resources during transmission can be considerably reduced. The structure of the event-triggered control system is shown in Fig. 1.4. Moreover, NCSs have attracted considerable research attention owing to their advantages of reduced weight, low cost, high reliability, and flexible installation and maintenance [253, 283, 287]. Nevertheless, in NCSs, the sharing of the limited network bandwidth of the communication channels often leads to network time delays, stochastic processes, data-packet dropouts, and the quantization effect, which may

1.5 Event-Triggered Strategy

15

lead to the instability and divergence of the NCSs [53, 77, 218]. Time delays generally lead to oscillation, divergence, and even instability in many practical dynamic systems such as electronic circuits, telecommunication, chemical processes, and neural networks [195, 253, 311]. The recently proposed techniques of stability analysis can be classified as delay-dependent [176, 251, 358] and delay-independent approaches [1, 139, 242]. The former approach, which considers the size of the time-delay information, is widely preferred over delay-independent technique. Accordingly, the establishment of methods to reduce the conservativeness by adopting a delaydependent technique is of key interest in the research on time-varying delay systems. In addition, it is essential to alleviate the negative impacts of these relevant issues to achieve the ideal system performance, and the primary task to reduce the data transmission. In this regard, it is necessary to employ the event-triggered communication scheme. Notable achievements has been realized the research on eventtriggered approach research. In particular, solutions for event-triggered H∞ filtering issues were introduced in [333, 342]; the fault detection problems were considered in [133, 295]; H∞ stabilization problems were presented in [104, 268]; optimal control methods for discrete event systems were proposed in [180, 258]; and maximum likelihood state estimation approaches were described in [230, 347]. In general, T-S fuzzy systems and dissipativity have been widely recognized and utilized [137, 351]. As another active research domain, many developed methods have been expounded to enhance the effectiveness of the delay-dependent stability condition. Among these advanced methods, the delay-partitioning method, as specified in [184, 217, 245], has been recognized as an effective approach to reduce the conservativeness of the stability conditions for T-S fuzzy delayed systems. This approach initially divides the lower bound of the time-varying delay into several uniform components. Subsequently, the corresponding Lyapunov–Krasovskii function (LKF) is constructed for each delay subinterval. The obtained stability conditions are less conservative than those obtained previously [55, 323]. However, these conditions lead to an increased computational complexity during optimization. Conversely, the reciprocally convex approach discussed in [205] established a lower bound lemma, the key concept of which is to address the information of each delay subinterval and achieve results identical to those of other inequality lemmas with considerably less conservativeness and reduced requirements for state decision variables. This approach has been extensively applied to different dynamic time-varying delay systems such as nonlinear dynamic systems with time-varying delay constraints [249, 343, 353] and continuous-time/discrete-time linear delayed systems [74, 190, 209]. However, there remains considerable scope to enhance the existing results for T-S fuzzy dynamic systems involve uncertain time-delay constraints [109, 350]. A complex and challenging task pertains to further reducing the conservativeness and optimizing the results affected by time-varying delays, particularly when the interval of the timevarying delay increases and becomes uncertain. Thus, based on the considered fuzzy control method, the control approach with an efficient event-triggered scheme [135, 161, 253] is proposed to reduce the burden of communication and save bandwidth resources.

16

1 Introduction

1.6 Publication Contribution This book is aimed at highlighting the state-of-the-art research for realizing stability/performance analysis and addressing optimal synthesis problems for fuzzymodel-based systems. The contents of this book can be divided into four parts. The book describes the stability analysis and controller design of fuzzy systems. Certain sufficient conditions for the stability and controller design for the considered fuzzy systems with different performances are established. The adopted methodologies include the Lyapunov stability approach and LMI technique. The key objective of using these advanced approaches is to reduce the conservativeness of the obtained results, thereby facilitating the subsequent design. Furthermore, several optimal synthesis problems, including the stabilization, OFC with different system performances, L2 –L∞ OFC, dissipative fuzzy control, switched OFC, switched dissipative filtering, fault detection in the presence of sensor nonlinearities, and reduced-order model approximation, are investigated based on the analysis results. Moreover, relevant simulation examples and applications are described to validate the effectiveness and applicability of the design methods. The key features of this book can be summarized as follows. (1) A unified framework is established for the analysis and synthesis of T-S fuzzy systems involving parameter uncertainties such as external perturbations and faults. (2) A series of problems are solved using novel approaches to analyse and synthesize continuousand discrete-time fuzzy systems, including stability/performances analysis and stabilization, OFC, tracking control, filtering, fault detection, and model reduction. (3) A set of newly developed techniques (e.g., the Lyapunov stability theory, LMI technique, and convex optimization) are exploited to address the emerging mathematical/computational challenges. This publication is a timely reflection of the developing new domain of system analysis and synthesis theories for T-S fuzzy systems. The book can be considered to be a collection of a series of latest research results and therefore serves as a useful textbook for senior and/or graduate students who are interested in gaining knowledge regarding the (1) state-of-the-art of fuzzy systems and fuzzy control area; (2) recent advances in nonlinear systems; (3) recent advances in stability/performance analysis, stabilization, OFC, tracking control, fault-tolerant control, filtering, fault detection, and reduced-order model approximation problems. The readers can also familiarize themselves with several new concepts, models, and methodologies with theoretical significance in system analysis and control synthesis. Moreover, the book can be used as a practical research reference for engineers working on stabilization, intelligent control, and filtering problems for T-S fuzzy systems. The objective is to fill the gaps in the literature by providing a unified and structured framework for stability/performances analysis and synthesis of T-S fuzzy systems. Generally, this advanced publication is aimed at 3rd/4th-year undergraduates, postgraduates and academic researchers. Prerequisite knowledge includes that of fuzzy sets, linear algebra, matrix analysis, and linear control system theory.

1.6 Publication Contribution

17

The target readership includes (1) control engineers working on nonlinear control, fuzzy control and optimal control; (2) system engineers working on intelligent control systems; (3) mathematicians and physicians working on uncertain systems; (4) postgraduate students majoring in control engineering, system sciences and applied mathematics. This publication is also a useful reference for (1) mathematicians and physicians working on intelligent systems and nonlinear systems; (2) computer scientists working on algorithms and computational complexities; (3) 3rd/4th-year students who are interested in advanced control theories and its applications.

1.7 Publication Outline The content of this monograph is divided into four parts. Part one focuses on the stability analysis and controller design problem for T-S fuzzy systems. The second part corresponds to the filter design and fault detection problems for fuzzy switched systems. Part three focuses on the model reduction problem for fuzzy systems. Part four presents several applications of fuzzy control approaches. The organization structure of this monograph is shown in Fig. 1.5, and the main contents of this monograph are illustrated in Fig. 1.6. This chapter describes the research backgrounds, motivations, and research problems, including stabilization synthesis, output feedback control design, filter design, fault detection design, and reduced-order model approximation for fuzzy systems. Subsequently, the outline of the monograph is presented. Part One focuses on the stability analysis and controller design problem for TS fuzzy systems. Part One begins with Chap. 2 and consists of three chapters. Chapter 2 investigates the issues regarding the stability analysis and stabilization for T-S fuzzy systems with time-varying delays. By appropriately choosing the LKF, together with the reciprocally convex method, a new sufficient stability condition employing the delay-partitioning approach is proposed for the delayed T-S fuzzy systems, which can significantly reduce the conservativeness compared to that pertaining to the existing results. Based on the obtained stability condition and PDC law, the state-feedback fuzzy controller is designed for the overall fuzzy systems. Furthermore, the parameters of the designed fuzzy controller are derived in terms of LMIs, which can be easily obtained by applying optimization techniques. Chapter 3 focuses on the issue of the Hankel-norm output feedback controller design for T-S fuzzy stochastic systems. The full-order controller design scheme with the Hankel-norm performance is established by using the fuzzy-basisdependent Lyapunov function method and transforming the Hankel-norm controller parameters. Sufficient conditions are presented to design the controllers

18

1 Introduction

Fig. 1.5 Organizational structure of this publication

such that the resulting closed-loop system is stochastically stable and satisfies the specified performance. By solving a convex optimization issue, we can obtain the desired controller, which can be promptly resolved using standard numerical algorithms. Chapter 4 address the L2 –L∞ output feedback controller design problem for switched systems with nonlinear perturbations in the T-S fuzzy framework. First, the average dwell time (ADT) method is considered to stabilize the non-

1.7 Publication Outline

19

Fig. 1.6 Main contents of this publication

linear switched system exponentially through an arbitrary switching law. Subsequently, based on the piecewise Lyapunov functions, a fuzzy-rule-dependent output feedback controller is proposed to ensure that the closed-loop system is exponentially stable with a weighted L2 –L∞ performance (γ , α). The solvable conditions of the desired dynamic controller are derived by employing the linearization approach. The controller matrices can be obtained in terms of several strict LMIs, which can be resolved numerically through efficient standard software.

20

1 Introduction

Part Two focuses on the filter design and fault detection problem for fuzzy switched systems. Part Two begins with Chap. 5 and consists of three chapters. Chapter 5 investigates the dissipativity-based filtering problem for T-S fuzzy switched systems with stochastic perturbation. First, sufficient conditions are established to ensure the mean-square exponential stability of T-S fuzzy switched systems. Next, we design a filter for the system of interest, subject to Brownian motion. The piecewise Lyapunov function approach and ADT method are combined, and valid fuzzy filters are presented such that the filter error dynamics exhibit mean-square exponential stability and the prescribed dissipative property. Moreover, the solvability conditions of the designed filter are specified through linearization. Chapter 6 considers the problem of fault detection filtering for the nonlinear switched stochastic system in the form of the T-S fuzzy model. The objective is to establish a robust fault detection approach for a nonlinear switched system with Brownian motion. Based on the observer-based fuzzy filter as a residual generator, the fault detection problem can be reformulated as a fuzzy filtering problem. By using the piecewise Lyapunov functions and ADT approach, a fuzzy-rule-dependent fault detection filter is established to ensure that the overall dynamic system is mean-square exponential stable and exhibits weighted H∞ performance. Moreover, the solvable conditions of the fuzzy filter are established through linearization. Chapter 7 addresses the reliable filtering problem with the requirement of strict dissipative performance for discrete-time T-S fuzzy delayed systems. A reliable filter design is prepared to ensure strict dissipativity for dynamic filtering systems. First, sufficient conditions for the reliable dissipativity analysis are established using a reciprocally convex method for T-S fuzzy systems subject to sensor failures. In addition, a reliable filter is established based on the convex optimization technique, which can be readily resolved through the standard numerical toolbox. Part Three focuses on the model reduction problem for fuzzy switched systems and fuzzy stochastic systems. Part Three begins with Chap. 8 and consists of four chapters. Chapter 8 considers the reduced-order model approximation issue for discrete hybrid switched nonlinear systems through T-S fuzzy modelling. We attempt to establish a reduced-order model for a high-dimension hybrid switched system, which can approximate the original high-order model subject to the prescribed system performance index. First, the mean-square exponential stability

1.7 Publication Outline

21

conditions are formulated to ensure the specific H∞ performance for the resulting dynamic error system via the efficient Lyapunov stability method and ADT approach. The solutions of the relevant model reduction problems are derived using the projection technique, through which the algorithms of the reducedorder model parameters are established using a cone complementary linearization (CCL) method. Chapter 9 presents a novel solution for the model approximation problem for dynamic time-varying delay systems based on fuzzy modelling. We attempt to establish a reduced-order model, which can approximate the original high-order model under a specific system performance index and ensure the asymptotic stability for the overall system. A less conservative stability condition for the overall error system with a specific performance level is derived using the reciprocally convex strategy. The reduced-order model can be ultimately established via the projection strategy, which transfers the model approximation work to a sequential minimization issue under LMI constraints based on the CCL algorithms. Chapter 10 examines the model approximation issue of a T-S fuzzy switched system with stochastic disturbances. Considering a high-order system, we constructing a reduced-order model, which can approximate the original system with a Hankelnorm performance and convert it to a lower-order switched system. Sufficient conditions are derived using the ADT method and piecewise Lyapunov function to ensure the resulting error system is mean-square exponentially stable and satisfies the Hankel-norm performance sense. Based on a linearization process, the preceding model approximation can be transformed to a convex optimization issue. Chapter 11 reports on the design of the H∞ reduced-order filter for T-S fuzzy delayed systems with stochastic perturbation. Using a novel Lyapunov function and the reciprocally convex approach, the fuzzy-dependent conditions are established to ensure that the resulting filtering system is mean-square asymptotically stable and satisfies the prescribed H∞ performance. Next, feasible solutions of the designed reduced-order filter are specified, which can be converted to a convex optimization problem via the convex linearization strategy. Finally, simulation examples, including that of an inverted pendulum, are explained to demonstrate the validity and superiority of the presented H∞ reduced-order filtering scheme. Part Four is aimed at examining two applications of fuzzy control approaches. Part Four begins with Chap. 12 and consists of two chapters. Chapter 12 addresses the dissipative fuzzy control issue for fuzzy dynamic systems using an event-triggered approach, which is adopted to reduce the transmissions and ensure the stability of the closed-loop system. Furthermore, the dissipative

22

1 Introduction

performance is considered in the design of the fuzzy controller to ensure that the resulting closed-loop system is asymptotically stable and strictly (X, Y, Z )– θ –dissi pative. Based on the fuzzy model, the stability is analysed by using the Lyapunov stability theory. In addition, the designed controller is compactly described in the form of LMIs. An illustrative example is presented to validate the proposed design approach. Chapter 13 focuses on the event-based fuzzy control design and its application to an inverted pendulum system. First, the time-varying delays in the inverted pendulum system and event-based method are considered in the system stability analysis. The interval of the time-delay is segmented to l non-uniform sub-intervals by applying an efficient delay-partition approach. The information in every subinterval is handled using the reciprocally convex technique. The stability conditions of the inverted pendulum model are established to be less conservative than existing research achievements. The reduction in the conservativeness is more notable when the number of delay partitions l is smaller. In addition, through the PDC rule, feasible conditions for the designed eventtriggering fuzzy controller are derived for the considered inverted pendulum system. Chapter 14 presents the concluding remarks and highlights the future research directions for the presented work.

Part I

Stability Analysis and Fuzzy Control

Chapter 2

Stabilization Synthesis of T-S Fuzzy Delayed Systems

2.1 Introduction This chapter investigates the issues regarding the stability analysis and stabilization for T-S fuzzy systems with time-varying delays. By appropriately choosing the LKF, together with the reciprocally convex method, a new sufficient stability condition employing the delay-partitioning approach is proposed for the delayed T-S fuzzy systems, which can significantly reduce the conservativeness compared to that pertaining to the existing results. Based on the obtained stability condition and PDC law, the state feedback fuzzy controller is designed for the overall fuzzy systems. Furthermore, the parameters of the designed fuzzy controller are derived in terms of LMIs, which can be easily obtained by applying optimization techniques.

2.2 System Description and Preliminaries Introduce a class of continuous-time nonlinear systems which can be expressed by the following T-S fuzzy time-varying delay model:  Plant Form: Rule i: IF θ1 (t) is μi1 and θ2 (t) is μi2 and · · · and θ p (t) is μi p , THEN 

x(t) ˙ = Ai x(t) + Adi x(t − d(t)) + Bi u(t), ¯ 0 , i = 1, 2, . . . , r, x(t) = ϕ(t), t ∈ −d,

where x(t) ∈ Rn denotes the state vector; u(t) ∈ Rq denotes the control input vector; ˙  d(t) denotes the time-varying delay which satisfies 0  d  d(t)  d¯ < ∞, d(t) ¯ τ are real constant scalars. Partition the delay interval into m τ , where d, d and  m ¯ = i=1 ¯ Let ηi be the length of the fractions, [d, d] [di−1 , di ], with d0 = d, dm = d. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 X. Su et al., Intelligent Control, Filtering and Model Reduction Analysis for Fuzzy-Model-Based Systems, Studies in Systems, Decision and Control 385, https://doi.org/10.1007/978-3-030-81214-0_2

25

26

2 Stabilization Synthesis of T-S Fuzzy Delayed Systems

subinterval, ηi = di − di−1 , i = 1, . . . , m, with d−1 = 0. Ai , Adi , Bi are system matrices with suitable dimensions; θ1 (t), θ2 (t), . . . , θ p (t) are the premise variables; μi1 , . . . , μi p denote the fuzzy sets; r is the number of IF-THEN rules; ϕ(t) represents the initial condition. Given a pair of (x(t), u(t)), the complete T-S fuzzy time-varying delay system is described as x(t) ˙ =

r 

h i (θ(t)) [Ai x(t) + Adi x(t − d(t)) + Bi u(t)] ,

(2.1)

i=1

with  νi (θ(t)) , ν (θ(t)) = μi j (θ j (t)), i r  j=1 νi (θ(t)) p

h i (θ(t)) =

i=1

where μi j (θ j (t)) represents the grade of membership of θ j (t) in μi j . Thus, for all t, we can see νi (θ(t))  0, i = 1, 2, . . . , r , ri=1 h i (θ(t)) = 1. For notational simplicity, define A(t) 

r  i=1

h i (θ(t))Ai ,

Ad (t) 

r 

h i (θ(t))Adi , B(t) 

i=1

r 

h i (θ(t))Bi .

i=1

Suppose the premise variable of the fuzzy model θ(t) is available for feedback, which means h i (θ(t)) is available for feedback control. Assume the controller’s premise variables are the same as those in the plant. The PDC strategy is applied and the state feedback fuzzy controller follows the rules as below:  Fuzzy Controller: Rule i: IF θ1 (t) is μi1 and θ2 (t) is μi2 and · · · and θ p (t) is μi p , THEN u(t) = K i x(t),

i = 1, 2, . . . , r,

where K i denotes the gain matrix of the state feedback controller. The controller can be further constructed as follows: u(t) =

r 

h i (θ(t))K i x(t).

i=1

The compact form of the fuzzy controller is expressed as u(t) = K (t)x(t),  where K (t) = ri=1 h i (θ(t))K i . Therefore, the closed-loop system can be obtained as

(2.2)

2.2 System Description and Preliminaries

x(t) ˙ =

r  r 

27

  h i (θ(t))h j (θ(t)) Ai x(t) + Adi x(t − d(t)) + Bi K j x(t) , (2.3)

i=1 j=1

or expressed in the compact form as x(t) ˙ = [A(t) + B(t)K (t)] x(t) + Ad (t)x(t − d(t)).

(2.4)

To propose the main results, the following lemmas are introduced. Lemma 2.1 ([206]) Let f 1 , f 2 , . . . , f N : Rm → R have positive values in an open subset D of Rm . And the reciprocally convex combination of f i over D meets  1   f i (t) = f i (t) + max gi, j (t), gi, j (t) βi i βi =1} i i i= j

min 

{βi |βi >0,

which subject to 

gi, j : Rm → R, g j,i (t) = gi, j (t),

 f i (t) gi, j (t) 0 . g j,i (t) f j (t)

Lemma 2.2 ([238]) Given any constant matrix M > 0, scalars b > a > 0, vector function w : [a, b] → Rn , then −(b − a)

t−a



t−a

w T (s)Mw(s)ds  −

t−b

t+θ



−b

t

t−a

M

t−b

b2 − a 2 −a t T w (s)Mw(s)dsdθ − 2 −b t+θ

−a t T −a − w(s)dsdθ M −b

T w(s)ds

 w(s)ds ,

t−b

 w(s)dsdθ .

t+θ

Remark 2.3 During the subsequent derivation of the LKF, the key is how to handle the amplification derived from the negative integral terms of quadratic quantities. Thus, Lemma 2.1 is an effective approach to deal with these terms for the concerned delayed systems. This processing method has been used for time-varying delay systems in [74] and [206], which will be utilized in our later derivation.

28

2 Stabilization Synthesis of T-S Fuzzy Delayed Systems

2.3 Main Results 2.3.1 Stability Analysis We partition the delay interval into two parts in this section to describe our scheme more clearly. By using the reciprocally convex method to handle the Lyapunov function, we present a new stability criterion for fuzzy systems with time-varying delays. Theorem 2.4 Assume that the controller gains in (2.2) are known in advance. Given scalars d0 , d2 and τ , the delayed fuzzy system (2.3) is asymptotically stable, if there exist matrices S N > 0, R N > 0, Q N > 0, N = 0, 1, 2, P > 0, Y > 0, M(t), Z 1 and Z 2 which meet the following conditions: Φ(t) + Φ1 (t) Φ(t) + Φ2 (t)

S1 Z 1  S1

S2 Z 2  S2

< 0, < 0,

(2.5) (2.6)

 0,

(2.7)

 0,

(2.8)

where Φ(t)  d02 W9T S0 W9 − (W1 − W2 )T S0 (W1 − W2 ) + η12 W9T S1 W9 d04 T W R0 W9 − (η0 W1 − W6 )T R0 (η0 W1 − W6 ) 4 9 (d 2 − d02 )2 T (d 2 − d12 )2 T W9 R 1 W9 + 2 W9 R 2 W9 + W1T Q 0 W1 + 1 4 4 − (η1 W1 − W7 )T R1 (η1 W1 − W7 ) − W2T Q 0 W2 + W1T Y W1 + η22 W9T S2 W9 +

+ (τ − 1)W5T Y W5 + sym(W1T P W9 ) − (η2 W1 − W8 )T R2 (η2 W1 − W8 ) + sym(M(t)W A (t)) − W3T Q 1 W3 + W3T Q 2 W3 − W4T Q 2 W4 + W2T Q 1 W2 , Φ1 (t)  −(W5 −W2 )T S1 (W5 −W2 )−(W5 −W3 )T Z 1T (W5 −W2 ) −(W5 −W2 )T Z 1 (W5 −W3 )−(W5 −W3 )T S1 (W5 −W3 ) −(W3 −W4 )T S2 (W3 −W4 ), Φ2 (t)  −(W5 −W3 )T S2 (W5 −W3 )−(W5 −W4 )T Z 2T (W5 −W3 ) −(W5 −W3 )T Z 2 (W5 −W4 )−(W5 −W4 )T S2 (W5 −W4 ) W1 W3 W5 W7

−(W2 − W3 )T S1 (W2 − W3 ),    [In 0n,8n ], W2  0n,n In 0n,7n ,      0n,2n In 0n,6n , W4  0n,3n In 0n,5n ,      0n,4n In 0n,4n , W6  0n,5n In 0n,3n ,        0n,6n In 0n,2n , W8  0n,7n In 0n,n , W9  0n,8n In ,

W A (t)  [A(t) + B(t)K (t) 0n,3n Ad (t) 0n,3n − In ].

2.3 Main Results

29

Proof Firstly, we construct the Lyapunov function as follows: V (t) 

5 

Vi (t),

(2.9)

i=1

where V1 (t)  x T (t)P x(t), 2 t−di−1  V2 (t)  x T (s)Q i x(s)ds, t−di

i=0 t

V3 (t) 

x (s)Y x(s)ds+d0

t−d(t)

V4 (t) 

2 

ηi

i=1

V5 (t) 



0



t

T



−di−1 −di



x˙ T (s)S0 x(s)dsdθ, ˙

x˙ T (s)Si x(s)dsdθ, ˙

t+θ

2 2  di2 − di−1 i=0

t

−d0 t+θ

2

−di−1



0



θ

−di

t

x˙ T (s)Ri x(s)dsdλdθ. ˙

t+λ

The time derivative of the Lyapunov function is obtained as ˙ + x˙ T (t)P x(t), V˙1 (t)  x T (t)P x(t) 2   T  x (t−di−1 )Q i x(t−di−1 )−x T (t−di )Q i x(t−di ) , V˙2 (t)  i=0

˙ V˙3 (t)  d02 x˙ T (t)S0 x(t)−d 0



t

t−d0

T x˙ T (s)S0 x(s)ds+x ˙ (t)Y x(t)

T ˙ − (1 − d(t))x (t − d(t))Y x(t − d(t)),   t−di−1 2  2 T T ˙ ηi x˙ (t)Si x(t) V4 (t)  ˙ − ηi x˙ (s)Si x(s)ds ˙ , t−di

i=1

V˙5 (t)  −

2  i=0

−

2  i=0

2 di2 − di−1 2



−di−1 −di

t

x˙ T (s)Ri x(s)dsdθ ˙

t+θ

2 (di2 − di−1 )2 T x˙ (t)Ri x(t). ˙ 4

Based on Lemma 2.2, it follows that

(2.10)

30

2 Stabilization Synthesis of T-S Fuzzy Delayed Systems

˙ − d0 ))T S0 (x(t)−x(t − d0 )) V˙3 (t) d02 x˙ T (t)S0 x(t)−(x(t)−x(t + x T (t)Y x(t) − (1 − τ )x T (t − d(t))Y x(t − d(t)), V˙5 (t) 

2 2  (di2 − di−1 )2

4

i=0

x˙ T (t)Ri x(t) ˙

2  ηi x(t) − −

T x(s)ds Ri ηi x(t) −

t−di−1

t−di

i=0

(2.11)

t−di−1



x(s)ds .

(2.12)

t−di

For V˙4 (t), if d0 < d(t) < d1 , according to Lemmas 2.1 and 2.2, it can obtain that ˙ − η1 V˙4 (t) = η12 x˙ T (t)S1 x(t) − η1

t−d(t)



t−d0

t−d(t)

x˙ T (s)S1 x(s)ds ˙ + η22 x˙ T (t)S2 x(t) ˙

x˙ T (s)S1 x(s)ds ˙ − η2

t−d1



t−d1

x˙ T (s)S2 x(s)ds ˙

t−d2

η12 x˙ T (t)S1 x(t) ˙ + η22 x˙ T (t)S2 x(t) ˙ − (x(t − d1 ) − x(t − d2 ))T S2 (x(t − d1 ) − x(t − d2 )) η1 (x(t −d(t))−x(t −d0 ))TS1 (x(t −d(t))−x(t −d0 )) − d(t)−d0 η1 (x(t −d(t))−x(t −d1 ))TS1 (x(t −d(t))−x(t −d1 )) − d1 −d(t) η12 x˙ T (t)S1 x(t) ˙ + η22 x˙ T (t)S2 x(t) ˙ − (x(t − d1 ) − x(t − d2 ))T S2 (x(t − d1 ) − x(t − d2 ))

T

S1 Z 1 x(t −d(t))−x(t −d0 ) x(t −d(t))−x(t −d0 ) . (2.13) −  S1 x(t −d(t))−x(t −d1 ) x(t −d(t))−x(t −d1 ) When d(t) = d0 or d(t) = d1 , (2.13) still holds because of x(t − d(t)) − x(t − d0 ) = 0 or x(t − d(t)) − x(t − d1 ) = 0. In view of the T-S fuzzy systems in (2.3), it has ˙ = 0,  = 2ζ T (t)M(t)[(A(t) + B(t)K (t))x(t) + Ad (t)x(t − d(t)) − x(t)]  where M(t) = ri=1 h i (θ(t))Mi . Hence, considering (2.10), (2.11), (2.12) and (2.13), when ζ(t) = 0, it is easy to obtain V˙ (t) = V˙ (t) +  ≤ ζ T (t)(Φ(t) + Φ1 (t))ζ(t) < 0, where

2.3 Main Results

31

 ζ(t)  x T (t) x T (t − d0 ) x T (t − d1 ) x T (t − d2 ) x T (t − d(t)) t−d0 t−d1 t T x T (s)ds x T (s)ds x T (s)ds x˙ T (t) . t−d0

t−d1

t−d2

For V˙4 (t), if d1 < d(t) < d2 , utilizing the same method as above, it is not difficult to get that ˙ − η1 V˙4 (t) = η12 x˙ T (t)S1 x(t)



t−d0

˙ − η2 +η22 x˙ T (t)S2 x(t) −η2

t−d(t)

x˙ T (s)S1 x(s)ds ˙

t−d1 t−d1

x˙ T (s)S2 x(s)ds ˙

t−d(t)

x˙ T (s)S2 x(s)ds ˙

t−d2

 η12 x˙ T (t)S1 x(t) ˙ + η22 x˙ T (t)S2 x(t) ˙ −(x(t − d0 ) − x(t − d1 ))T S1 (x(t − d0 ) − x(t − d1 ))

T

S2 Z 2 x(t −d(t))−x(t −d1 ) x(t −d(t))−x(t −d1 ) . (2.14) −  S2 x(t −d(t))−x(t −d2 ) x(t −d(t))−x(t −d2 ) When d(t) = d1 or d(t) = d2 , (2.14) still holds due to x(t − d(t)) − x(t − d1 ) = 0 or x(t − d(t)) − x(t − d2 ) = 0. Likewise, if d1 < d(t) < d2 , considering (2.10), (2.12) and (2.14), when ζ(t) = 0, we can obtain V˙ (t) = V˙ (t) +   ζ T (t)(Φ(t) + Φ2 (t))ζ(t) < 0. Therefore, it is shown that the fuzzy system (2.3) is asymptotically stable. It is noticed that Theorem 2.4 cannot be directly solved by LMIs for stability analysis. The next goal is to transform the inequalities (2.5) and (2.6) to finite LMIs, which can be readily solved by the standard numerical software. Then we propose the following theorem. Theorem 2.5 Assuming the controller gains K i in (2.2) are known in advance. Given scalars d0 , d2 and τ , the delayed fuzzy system (2.3) is asymptotically stable, if there exist matrices S N > 0, R N > 0, Q N > 0, N = 0, 1, 2, P > 0, Y > 0, Mi , Z 1 and Z 2 which satisfy (2.7), (2.8) and the following inequalities for i, j, k = 1, 2, . . . , r : Φiik + Φ1iik < 0, + Φ jik + Φ1 jik < 0, 1  i < j  r,

(2.15) (2.16)

Φiik + Φ2iik < 0, Φi jk + Φ2i jk + Φ jik + Φ2 jik < 0, 1  i < j  r,

(2.17) (2.18)

Φi jk + Φ1i jk

where

32

2 Stabilization Synthesis of T-S Fuzzy Delayed Systems

Φi jk d02 W9T S0 W9 − (W1 − W2 )T S0 (W1 − W2 ) d04 T W R 0 W9 4 9 (d 2 − d02 )2 T − (η0 W1 − W6 )T R0 (η0 W1 − W6 ) + 1 W9 R 1 W9 4 (d 2 − d12 )2 T W9 R 2 W9 − (η1 W1 − W7 )T R1 (η1 W1 − W7 ) + 2 4 + sym(W1T P W9 ) − (η2 W1 − W8 )T R2 (η2 W1 − W8 ) + η12 W9T S1 W9 + η22 W9T S2 W9 +

+ W1T Y W1 + (τ − 1)W5T Y W5 + W1T Q 0 W1 − W2T Q 0 W2 + W2T Q 1 W2 − W3T Q 1 W3 + W3T Q 2 W3 − W4T Q 2 W4 + sym(Mi W A jk ), Φ1i jk  − (W5 − W2 )T S1 (W5 − W2 ) − (W5 − W3 )T Z 1T (W5 − W2 ) − (W5 − W2 )T Z 1 (W5 − W3 ) − (W5 − W3 )T S1 (W5 − W3 ) − (W3 − W4 )T S2 (W3 − W4 ), Φ2i jk  − (W5 − W3 )T S2 (W5 − W3 ) − (W5 − W4 )T Z 2T (W5 − W3 ) − (W5 − W3 )T Z 2 (W5 − W4 ) − (W5 − W4 )T S2 (W5 − W4 ) − (W2 − W3 )T S1 (W2 − W3 ), W A jk  [A j + B j K k 0n,3n Ad j 0n,3n − In ]. Proof On account of the fuzzy basis functions, the inequalities (2.5) and (2.6) can be expressed as Φ(t) + Φ1 (t)  Φ(t) + Φ2 (t) 

r 

h i (θ(t))

r 

h j (θ(t))

r 

i=1

j=1

k=1

r 

r 

r 

i=1

h i (θ(t))

h j (θ(t))

j=1

h k (θ(t))(Φi jk + Φ1i jk ) < 0, (2.19) h k (θ(t))(Φi jk + Φ2i jk ) < 0. (2.20)

k=1

It is obvious that if the conditions (2.15), (2.16), (2.17) and (2.18) are met, the above-mentioned inequalities (2.19) and (2.20) hold. In other words, the T-S fuzzy time-delay system (2.3) is asymptotically stable. For the sake of reducing the conservatism, the results are extended to the case whose delay interval is partitioned into arbitrarily m parts and the Lyapunov function is chosen as follows: V (t) 

5  i=1

where

Vi (t),

(2.21)

2.3 Main Results

33

V1 (t)  x T (t)P x(t), m t−di−1  x T (s)Q i x(s)ds, V2 (t)  i=0 t

V3 (t) 

t−di

t−d(t)

V4 (t) 

m 

ηi

i=1

V5 (t) 



x T (s)Y x(s)ds + d0

−di−1

−di



t

i=0



t

x˙ T (s)S0 x(s)dsdθ, ˙

−d0 t+θ

x˙ T (s)Si x(s)dsdθ, ˙

t+θ

m 2  di2 − di−1 2

0

−di−1

−di



0 θ



t

x˙ T (s)Ri x(s)dsdλdθ. ˙

t+λ

On the basis of the aforementioned Lyapunov function, it can easily obtain the following results, which can be proved with similar arguments as those in the proof of Theorems 2.4 and 2.5. Theorem 2.6 Assuming the controller gains K i in (2.2) are known. Given scalars d0 , dm , m ≥ 2 and τ , the delayed fuzzy system (2.3) is asymptotically stable, if there exist matrices S N > 0, R N > 0, Q N > 0, N = 0, 1, . . . , m, Z l , l = 1, . . . , m, P > 0, Y > 0 and Mi which satisfy the following conditions for i, j, k = 1, 2, . . . , r : Φiik + Φliik < 0, Φi jk + Φli jk + Φ jik + Φl jik < 0, 1  i < j  r,

Sl Z l  0, l = 1, . . . , m,  Sl where Φi jk 

m  (d N2 − d N2 −1 )2 T W2m+5 R N W2m+5 4 N =0

−

m 

(η N W1 − Wm+4+N )T R N (η N W1 − Wm+4+N )

N =0 m  + (W NT +1 Q N W N +1 − W NT +2 Q N W N +2 ) + W1T Y W1

+

N =0 m 

T η 2N W2m+5 S N W2m+5 + sym(W1T P W2m+5 )

N =0 T − (W1 − W2 )T S0 (W1 − W2 ) + (τ − 1)Wm+3 Y Wm+3 + sym(Mi W A jk ),

T



Sl Z l Wm+3 − Wl+1 Wm+3 − Wl+1 Φli jk  − Wm+3 − Wl+2  Sl Wm+3 − Wl+2

(2.22) (2.23) (2.24)

34

2 Stabilization Synthesis of T-S Fuzzy Delayed Systems

−

m 

(Wh+1 − Wh+2 )T Sh (Wh+1 − Wh+2 ),

h=1, h=l

 W A jk  A j + B j K k 0n,(m+1)n Ad j 0n,(m+1)n − In ,   Wt  0n,(t−1)n In 0n,(2m+5−t)n , t = 1, 2, . . . , 2m + 5. 

Remark 2.7 The presented stability conditions do not require the limited upper bound for the delay derivative, which are more common than the results with strict restraints τ < 1. It is noticeable that our approach is are better suited for real systems. Remark 2.8 In Theorem 2.6, we utilized the reciprocally convex approach combined with the delay partitioning method and a novel Lyapunov function (2.21) to analyze the stability of the T-S fuzzy delayed system. The key point is on reducing the conservativeness so that the controller design issues have workable solutions. To demonstrate the effectiveness of our proposed delay division method, the sufficient conditions are proposed firstly when the delay interval is partitioned into 2 parts. Then, to further reduce the conservatism, this case is extended to a general one, and we partition the delay interval into arbitrary m(m ≥ 2) parts.

2.3.2 State Feedback Fuzzy Control In this section, we focus on the state feedback controller design for the concerned T-S fuzzy systems (2.3). Theorem 2.9 Given scalars λi1 , λi2 , . . . , λi(2m+5) , d0 , dm , m ≥ 2 and τ , a state feedback controller (2.2) exists such that the closed-loop fuzzy system in (2.3) is asymptotically stable, if there exist matrices Sˆ N > 0, Rˆ N > 0, Qˆ N > 0, N = 0, 1, . . . , m, X , Zˆ l , l = 1, 2, . . . , m, Pˆ > 0, Yˆ > 0 and G i which satisfy the following conditions for i, j = 1, 2, . . . , r : Πii + Πlii < 0, Πi j + Πli j + Π ji + Πl ji < 0, 1  i < j  r,

Sˆl Zˆ l  0, l = 1, . . . , m,  Sˆl where

(2.25) (2.26) (2.27)

2.3 Main Results

35

Πi j 

m  (d N2 − d N2 −1 )2 T W2m+5 Rˆ N W2m+5 4 N =0

− +

m  N =0 m 

(η N W1 − Wm+4+N )T Rˆ N (η N W1 − Wm+4+N ) T η 2N W2m+5 Sˆ N W2m+5 +

N =0

− −

Πli j

m 

(W NT +1 Qˆ N W N +1

N =0

W NT +2 Qˆ N W N +2 )+ W1T Yˆ W1 +sym(W1T Pˆ W2m+5 ) T (W1 − W2 )T Sˆ0 (W1 − W2 ) + (τ −1)Wm+3 Yˆ Wm+3

+ sym( Mˆ i Wˆ Ai j ),

T



Sˆl Zˆ l Wm+3 − Wl+1 Wm+3 − Wl+1 − Wm+3 − Wl+2 Wm+3 − Wl+2  Sˆl −

m 

(Wh+1 − Wh+2 )T Sˆh (Wh+1 − Wh+2 ),

h=1, h=l

  Wˆ Ai j  Ai X + Bi G j 0n,(m+1)n Adi X 0n,(m+1)n − X , T  Mˆ i  λi1 In λi2 In . . . λi(2m+5) In . If the above conditions are solvable, the feedback gains of the fuzzy controllers can be represented as K i = G i X −1 , i = 1, 2, . . . , r.

(2.28)

Proof For the sake of brevity, define Z = X −T and introduce the following matrix: E  diag{Z , Z , . . . , Z }.    2m+5

Pre- and post-multiplying (2.25) and (2.26) with E and E T respectively, which yields E(Πii + Πlii )E T < 0, E(Πi j + Πli j )E T + E(Π ji + Πl ji )E T < 0, 1  i < j  r, where

36

2 Stabilization Synthesis of T-S Fuzzy Delayed Systems

EΠi j E T 

m  (d N2 − d N2 −1 )2 T W2m+5 Z Rˆ N Z T W2m+5 4 N =0

+ −

m  N =0 m 

T η 2N W2m+5 Z Sˆ N Z T W2m+5 + W1T Z Yˆ Z T W1

(η N W1 − Wm+4+N )T Z Rˆ N Z T (η N W1 − Wm+4+N )

N =0

+

m 

(W NT +1 Z Qˆ N Z T W N +1 − W NT +2 Z Qˆ N Z T W N +2 )

N =0

+ sym(W1T Z Pˆ Z T W2m+5 )−(W1 −W2 )T Z Sˆ0 Z T (W1 −W2 ) T + (τ − 1)Wm+3 Z Yˆ Z T Wm+3 + sym(E Mˆ i Wˆ Ai j E T ),

T



Z Sˆl Z T Z Zˆ l Z T Wm+3 −Wl+1 Wm+3 −Wl+1 EΠli j E T  − Wm+3 −Wl+2  Z Sˆl Z T Wm+3 −Wl+2

−

m 

(Wh+1 − Wh+2 )T Z Sˆh Z T (Wh+1 − Wh+2 ).

h=1, h=l

Define S N Z Sˆ N Z T , R N  Z Rˆ N Z T , Q N  Z Qˆ N Z T , Y  Z Yˆ Z T , P Z Pˆ Z T ,

Z l  Z Zˆ l Z T ,

Mi  E Mˆ i .

Then we have T  E Mˆ i Wˆ Ai j E T = λi1 Z T λi2 Z T . . . λi(2m+5) Z T   × Ai X + Bi G j 0n,(m+1)n Adi X 0n,(m+1)n − X E T   = Mi A j + B j K k 0n,(m+1)n Ad j 0n,(m+1)n − In , i, j, k = 1, 2, . . . , r. Thus, it can see that the results obtained by pre- and post-multiplying (2.25) and (2.26) are equivalent to (2.22) and (2.23). In a similar way, pre- and post-multiplying (2.27) by diag{Z , Z } and diag{Z T , Z T }, we can get (2.24). Therefore, from the results in Theorem 2.6, the T-S fuzzy delayed system with our designed controller is asymptotically stable. The proof is then completed.

2.4 Illustrative Example

37

2.4 Illustrative Example Example 2.10 Consider the inverted pendulum system in [132] and its schematic diagram is shown in Fig. 2.1. To design a fuzzy controller, we set up a T-S fuzzy model for the corresponding nonlinear system. So first of all, construct a T-S fuzzy model of the pendulum system with approximation methods, and then stabilize the inverted pendulum system with the proposed controller. Some parameters in the pendulum system are given as M

mass of the cart, 1.378 kg;

m l

mass of the pendulum, 0.051 kg; length of the pendulum, 0.325 m;

gr

coefficient of the delayed resonator, 0.7 kg/s;

g

acceleration due to gravity, 9.8 m/s2 ;

cr θ(t) y(t)

coefficient of the damper, 5.98 kg/s; angle the pendulum makes with the top vertical; displacement of the cart;

d(t) u(t)

time-varying delay; force applied to the cart.

Fig. 2.1 Inverted pendulum on a cart with a delayed resonator

38

2 Stabilization Synthesis of T-S Fuzzy Delayed Systems

For alignbrevity’s sake, the notation “(t)” in some places will be omitted. Suppose the pendulum can be modeled as a thin rod and consider Newton’s law, the following equations of the system motion can be obtained: d2 d2 y + m (y + l sin θ) = u − Fr , dt 2 dt 2 d2 m 2 (y + l sin θ)l cos θ = mgl sin θ, dt M

where Fr (t) = gr y˙ (t − d(t)) + cr y˙ (t) represents the force of the damper and delayed resonator. Based on the above equations and the state variables x1 = y, x2 = θ, ˙ the achieved state-space equations of the concerned system are as x3 = y˙ , x4 = θ, below: x˙1 = x3 , x˙2 = x4 , −mg sin x2 cr x3 + gr x3 (t − d(t)) − u x˙3 = , − M cos x2 M (M + m)g sin x2 x42 sin x2 cr x3 +gr x3 (t − d(t)) − u x˙4 = + + . Ml cos2 x2 cos x2 Ml cos x2 Then introduce the following T-S fuzzy model which represents the inverted pendulum system.  Plant Form: Rule 1: IF x2 is about 0 rad, THEN x(t) ˙ = A1 x(t) + Ad1 x(t − d(t)) + B1 u(t). ˙ = A2 x(t) + Ad2 x(t − Rule 2: IF x2 is near γ (0 < |γ| < 1.57rad), THEN x(t) d(t)) + B2 u(t), ⎤ ⎤ ⎤ ⎡ ⎡ 0 0 1 0 0 00 0 0 ⎥ ⎥ ⎢0 ⎢ ⎢ 0 0 1⎥ ⎥ , Ad1 = ⎢0 0 0gr 0⎥ , B1 = ⎢ 01 ⎥ , A1 = ⎢ cr ⎦ ⎦ ⎦ ⎣ 0 − mg ⎣ ⎣ −M 0 0 0 −M 0 M M gr (M+m)g cr 1 0 0 Ml 0 − Ml 0 0 Ml Ml ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ 0 0 1 0 0 00 0 0 ⎢0 ⎢0 0 0 0⎥ ⎢ 0 ⎥ 0 0 1⎥ ⎥ ⎥ ⎥ ⎢ ⎢ A2 = ⎢ ⎣ 0 − mgβ − cr 0⎦ , Ad2 = ⎣0 0 − gr 0⎦ , B2 = ⎣ 1 ⎦ , Mα M M M gr 1 cr 0 0 0 Mlα − Mlα 0 0 (M+m)gβ Mlα2 Mlα ⎡

where x(t) = [x1T (t) x2T (t) x3T (t) x4T (t)]T , set |γ| = 0.52 rad, α = cos γ, β = (sin γ)/γ. The fuzzy basis functions h 1 (x2 (t)) and h 2 (x2 (t)) are chosen as triangular ones: h 1 (x2 (t)) = 1 −

|x2 (t)| |x2 (t)| , h 2 (x2 (t)) = . |γ| |γ|

2.4 Illustrative Example

39

Table 2.1 Comparison of upper bounds and controller feedback gains for different cases d¯ Method d0 K1 K2 T heor em 2.9, m=2

T heor em 2.9, m=3

T heor em 2.9, m=4

0.1

1.4527

0.3

1.5321

0.5

1.3224

0.1

1.4575

0.3

1.5797

0.5

1.4074

0.1

1.5456

0.3

1.5985

0.5

1.4615

[0.0013 30.2859 7.5837 5.1625] [0.0026 31.6369 7.6595 5.3417] [0.0028 29.7307 7.4063 5.0375] [0.0134 30.7909 7.6601 5.2958] [0.0059 31.5425 7.6834 5.3664] [0.0054 30.7601 7.6066 5.2111] [0.1696 31.2491 7.7618 5.4203] [0.0103 31.8430 7.6205 5.4085] [0.0107 30.8056 7.4986 5.2148]

[0.0011 29.5536 7.3600 4.4054] [0.0022 30.6965 7.4303 4.5661] [0.0026 29.0849 7.2264 4.3056] [0.0112 30.0020 7.4255 4.4940] [0.0052 30.6082 7.4267 4.5654] [0.0048 29.9372 7.3605 4.4352] [0.1422 30.2531 7.4861 4.5755] [0.0088 30.8357 7.3804 4.6171] [0.0095 29.9472 7.2809 4.4471]

Firstly, we make a comparison on the upper bounds obtained from the inverted pendulum system with different lower bound d0 . Table 2.1 lists the upper bounds and the fuzzy controller gains, it’s easily to conclude that the allowable upper bound of the controller is enhanced along with the number of partitioned segments m raising. That is to say, the conservatism of our designed scheme is decreased when the number of fractioning is increased. In the mean time, the unknown variables increase. Thus, there is a tradeoff between conservatism reduction and computation complexity. Set the partitioning fractions m = 2, d0 = 0.1, τ = 0.3, and the initial condition is set as x(t) = [0 0.4 0 0]T . The dynamics of the original inverted pendulum system is presented in Fig. 2.2, from which we can see the system is unstable. By employing Theorem 2.9, we stabilize the system with the allowable upper bound of time-varying delay d¯ = 1.3894. Letting d¯ = 1.2, the following solutions can be obtained which satisfy the conditions in Theorem 2.9.

40

2 Stabilization Synthesis of T-S Fuzzy Delayed Systems

Fig. 2.2 States of the original system without control

  G 1 = −0.0265 0.0132 −0.0291 0.0370 ,   G 2 = −0.0272 0.0189 −0.0301 0.0093 , ⎡ ⎤ 0.0664 −0.0011 0.0008 −0.0011 ⎢ 0.0000 0.0017 −0.0035 −0.0040 ⎥ ⎥ X =⎢ ⎣ −0.0053 0.0018 0.0084 −0.0108 ⎦ , 0.0014 −0.0109 0.0020 0.0498   K 1 = 0.0983 30.0079 7.7666 4.8405 ,   K 2 = 0.0851 29.4028 7.5489 4.1888 . Consequently, the proposed fuzzy controller in (2.2) is given by   u(t) =h 1 (x2 (t)) 0.0983 30.0079 7.7666 4.8405 x(t)   + h 2 (x2 (t)) 0.0851 29.4028 7.5489 4.1888 x(t). Figure 2.3 shows that the obtained fuzzy state feedback system makes the closedloop states converge to zero combined with the above controller.

2.5 Conclusion

41

Fig. 2.3 States of the controlled fuzzy system

2.5 Conclusion The stability and stabilization issues for continuous-time T-S fuzzy systems with time-varying delays were considered. Using the delay-partitioning approach and reciprocally convex method, new sufficient conditions, which could reduce the conservativeness against that of the existing results, were established. Next, an effective primary domain controller was developed to stabilize the fuzzy closed-loop systems. Finally, an illustrative example was given to verify the effectiveness of the approaches presented in this chapter.

Chapter 3

Output Feedback Control of Fuzzy Stochastic Systems

3.1 Introduction This chapter focuses on the issue of the Hankel-norm output-feedback controller design for T-S fuzzy stochastic systems. The full-order controller design scheme with the Hankel-norm performance is established by using the fuzzy-basis-dependent Lyapunov function method and transforming the Hankel-norm controller parameters. Sufficient conditions are presented to design the controllers such that the resulting closed-loop system is stochastically stable and satisfies the specified performance. By solving a convex optimization issue, we can obtain the desired controller, which can be promptly resolved using standard numerical algorithms.

3.2 System Description and Preliminaries Some stochastic nonlinear systems can be described with a set of linear systems in local regions. The T-S fuzzy stochastic models with r fuzzy rules are given by  Plant Form: Rule i: IF θ1 (k) is Mi1 and θ2 (k) is Mi2 and . . . and θ p (k) is Mi p , THEN ⎧ ⎨ x(k + 1) = Ai x(k)+ Bi u(k)+ Di ω(k)+ L i x(k)(k), y(k) = Ci x(k)+ Fi ω(k)+ Ni x(k)(k), ⎩ z(k) = E i x(k)+G i u(k)+ Hi ω(k),

(3.1)

where x(k) ∈ Rn denotes the state vector; y(k) ∈ R p denotes the measured output; u(k) ∈ Rs denotes the control input; z(k) ∈ Rq denotes the controlled output; and ω(k) ∈ Rl denotes the disturbance input vector which is assumed to belong to 2 [0, ∞). Mi j represents the fuzzy set; r is the number of IF-THEN rules; © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 X. Su et al., Intelligent Control, Filtering and Model Reduction Analysis for Fuzzy-Model-Based Systems, Studies in Systems, Decision and Control 385, https://doi.org/10.1007/978-3-030-81214-0_3

43

44

3 Output Feedback Control of Fuzzy Stochastic Systems

  θ(k) = θ1 (k), θ2 (k), . . . , θ p (k) , denoted by θ for simplicity, represents the premise variables vector. (k) is a scalar Brownian motion defined on the probability space (, F, P) relative to an increasing family (Fk )k∈N of σ-algerbras Fk ⊂ F generated by ((k))k∈N . The stochastic process (k) is independent, and it’s supposed that E {(k)} = 0, E {(k)2 } = δ 2 . Ai , Bi , Ci , Di , E i , Fi , G i , Hi , L i and Ni are known constant matrices subject to  suitable dimensions. The fuzzy membership functions p

M i j (θ j ) p , i = 1, . . . , r , with Mi j θ j representing are expressed as h i (θ)  r j=1 M θ ( ) ı j j ı=1 j=1 the grade of membership of θ j in Mi j . Thus, for all k we have h i (θ) ≥ 0, i = 1, 2, . . . , r,

r 

h i (θ) = 1.

(3.2)

i=1

Let  be a set of basis functions which satisfy (3.2). A more compact presentation of the nonlinear systems in the discrete-time T-S fuzzy model is written as ⎧ x(k + 1) = A(h)x(k) + B(h)u(k) + D(h)ω(k) ⎪ ⎪ ⎨ +L(h)x(k)(k), y(k) = C(h)x(k) + F(h)ω(k) + N (h)x(k)(k), ⎪ ⎪ ⎩ z(k) = E(h)x(k) + G(h)u(k) + H (h)ω(k),

(3.3)

where   A(h)  ri=1 h i (θ)Ai , B(h)  ri=1 h i (θ)Bi ,   C(h)  ri=1 h i (θ)Ci , E(h)  ri=1 h i (θ)E i ,  r D(h)  i=1 h i (θ)Di , F(h)  ri=1 h i (θ)Fi ,  r G(h)  i=1 h i (θ)G i , H (h)  ri=1 h i (θ)Hi , r r L(h)  i=1 h i (θ)L i , N (h)  i=1 h i (θ)Ni , with h  (h 1 , h 2 , . . . , h r ) ∈ . The fuzzy-basis-dependent output feedback controller is designed as

(K ) :

xc (k + 1) = Ac (h)xc (k) + Bc (h)y(k), u(k) = Cc (h)xc (k) + Dc (h)y(k),

(3.4)

where xc (k) ∈ Rn denotes the state vector of the controller; the fuzzy-basis-functiondependent matrices Ac (h), Bc (h), Cc (h) and Dc (h) are controller parameters with appropriate dimensions. The resulting closed-loop system which combines the proposed controller (3.4) with original system (3.3) is represented as

¯ ¯ ¯ ξ(k + 1) = A(h)ξ(k) + D(h)ω(k) + L(h)ξ(k)(k), ¯ ¯ z(k) = E(h)ξ(k) + H¯ (h)ω(k) + M(h)ξ(k)(k),

(3.5)

3.2 System Description and Preliminaries

45

T  where ξ(k)  x T (k) xcT (k) , and   A(h) + B(h)Dc (h)C(h) B(h)Cc (h) ¯ A(h)  , Ac (h) Bc (h)C(h)   D(h) + B(h)Dc (h)F(h) ¯ , D(h)  Bc (h)F(h)   ¯ E(h)  E(h) + G(h)Dc (h)C(h) G(h)Cc (h) , H¯ (h) H (h) + G(h)Dc (h)F(h),   L(h) + B(h)Dc (h)N (h) 0 ¯ , L(h)  0 Bc (h)N (h)   ¯ M(h)  G(h)Dc (h)N (h) 0 .

Definition 3.1 The closed-loop system in (3.5) is mean-square asymptotically stable if ω(k) = 0, limk→∞ E {|ξ(k)|} = 0. For a mean-square asymptotically stable closed-loop system in (3.5), we have z = {z(k)} ∈ 2 [0, ∞) when ω = {ω(k)} ∈ 2 [0, ∞). Definition 3.2 Given a scalar γ > 0, the closed-loop system in (3.5) is meansquare with a Hankel-norm error performance level γ if  stable ∞asymptotically T −1 T 2 T z (i)z(i) < γ E i=T i=0 ω (i)ω(i), for all ω ∈ 2 [0, ∞) with ω(k) = 0, ∀k ≥ T and satisfy the zero initial condition. Remark 3.3 In this chapter, the Hankel-norm measure for the desired stable systems is proposed. This criterion possesses a series of important characteristics. Firstly, it is well known that the Hankel-norm is to lie between the L2 –L2 (for continuous-time systems) or 2 –2 (for discrete-time systems) norm and the L2 –L∞ (for continuoustime systems) or 2 –∞ (for discrete-time systems) norm in the frequency domain and thus appears as a tradeoff between the two common error criteria. Secondly, this norm is aimed to the singular values of Hankel matrices, and the singular values of such matrices are rather insensitive to perturbations, which lead to the obtained model fairly robust to handle the uncertainties. Thirdly, the principal advantage of the Hankel-norm index is its ability to quantify frequency dependent interactions and it can be utilized for input-output pairing. In consideration of these features, the Hankel-norm measure has been widely applied to system control and analysis during some practical applications.

46

3 Output Feedback Control of Fuzzy Stochastic Systems

3.3 Main Results 3.3.1 State-Feedback Control In this section, by employing fuzzy-basis-dependent Lyapunov functions, some new sufficient conditions are proposed for the resulting closed-loop system in (3.5), which is to be mean-square asymptotically stable with Hankel-norm error performance level γ. First of all, the fuzzy-basis-dependent Lyapunov functions are constructed:

V (ξ(k), k)  ξ T (k)[P −1 (h)]ξ(k), V(ξ(k), k)  ξ T (k)[P −1 (h)]ξ(k),

(3.6)

where P(h) 

r  i=1

h i (θ)Pi , P(h) 

r 

h i (θ)Pi .

(3.7)

i=1

On the strength of fuzzy-basis-dependent Lyapunov functions in (3.6), we propose the following results. Theorem 3.4 The closed-loop system in (3.5) is mean-square asymptotically stable with Hankel-norm error performance level γ, if there exist fuzzy-basis-dependent matrices P(h) and P(h) which satisfy 0 < P(h) < P(h),

(3.8)

for any h ∈  and a constant matrix  such that for any h, h + ∈ , ⎡

⎤ ¯ 0 δ L(h) 0 −P(h + ) ⎢ ⎥ ¯ ¯  −P(h + ) A(h) D(h) ⎢ ⎥ < 0, ⎣   P(h) −  − T 0 ⎦    −γ 2 I ⎡ ⎤ ¯ −P(h + ) 0 0 0 δ L(h) ⎢ ⎥ ¯  −P(h + ) 0 0 A(h) ⎢ ⎥ ⎢ ⎥ < 0, ¯   −I 0 δ M(h) ⎢ ⎥ ⎣ ⎦ ¯    −I E(h) T     P(h) −  − 

(3.9)

(3.10)

where h +  {h 1 [θ(k +1)] , h 2 [θ(k +1)] , . . . , h r [θ(k +1)]}. Proof Assume there exist a constant matrix  and the matrices P(h), P(h) satisfying (3.8) such that inequalities (3.9) and (3.10) hold. Based on (3.9) and (3.10), there exist scalars β2 > β1 > 0 and

3.3 Main Results

47

β1 I ≤ P(h) <  + T , P(h) <  + T ≤ β2 I .

(3.11)

On account of (3.8), the matrix P(h) is invertible and for any h ∈ , 0
V(k).

(3.13)

Based on (3.12), the following inequalities

[P(h) − ]T P −1 (h) [P(h) − ] ≥ 0, [P(h) − ]T P −1 (h) [P(h) − ] ≥ 0,

signify that

T P −1 (h) ≥  + T − P(h), T P −1 (h) ≥  + T − P(h).

(3.14)

Combining with (3.9) and (3.10), we obtain ⎤ ¯ 0 δ L(h) 0 −P(h + ) ⎥ ⎢ ¯ ¯  −P(h + ) A(h) D(h) ⎥ < 0, ⎢ −1 ⎣   −P (h) 0 ⎦    −γ 2 I ⎡ ⎤ ¯ −P(h + ) 0 0 0 δ L(h) ⎢ ⎥ ¯  −P(h + ) 0 0 A(h) ⎢ ⎥ ⎢ ⎥ < 0. ¯   −I 0 δ M(h) ⎢ ⎥ ⎣ ⎦ ¯    −I E(h)     −P −1 (h) ⎡

(3.15)

(3.16)

Along the trajectory of closed-loop system in (3.5) and in view of the Lyapunov functions in (3.6), it follows that E {ΔV (ξ(k + 1), k + 1)}  E {V (ξ(k + 1), k + 1)−V (ξ(k), k)}  T −1 + ¯ ¯ ¯ = E A(h)ξ(k)+ D(h)ω(k)+ L(h)ξ(k)(k) P (h )   ¯ ¯ ¯ × A(h)ξ(k)+ D(h)ω(k)+ L(h)ξ(k)(k)  T −1 − ξ (k)P (h)ξ(k)    T  Π1 (k) Π2 (k) ξ(k) ξ(k) , =  Π3 (k) ω(k) ω(k)

(3.17)

48

3 Output Feedback Control of Fuzzy Stochastic Systems

where ¯ ¯ − P −1 (h) + δ L¯ T (h)P −1 (h + ) L(h), Π1 (k)  A¯ T (h)P −1 (h + ) A(h) T −1 + ¯ ¯ Π2 (k)  A (h)P (h ) D(h), ¯ Π3 (k)  D¯ T (h)P −1 (h + ) D(h). On the basis of the inequalities (3.15) and ω(k) = 0, we have ΔV (k) < 0, and the resulting closed-loop system in (3.5) is mean-square asymptotically stable. Based on Definition 3.2, introduce the performance index as E

∞ 

 z (i)z(i) < γ 2 T

i=T

T −1 

ω T (i)ω(i),

(3.18)

i=0

for all ω ∈ 2 [0, ∞) with ω(k) = 0, ∀k ≥ T . Then establish two inequalities:

E

∞

 z T (i)z(i) < V(ξ(T ), T ), T −1 T ω (i)ω(i). V (ξ(T ), T ) < γ 2 i=0

i=T

(3.19)

From (3.17), we can obtain 

ξ(k) E {ΔV (ξ(k + 1), k + 1)} − γ ω (k)ω(k) = ω(k) 2

T

T 

Π¯ 1 (k) Π¯ 2 (k)  Π¯ 3 (k)



 ξ(k) , ω(k)

where ¯ ¯ − P −1 (h) + δ L¯ T (h)P −1 (h + ) L(h), Π¯ 1 (k) A¯ T (h)P −1 (h + ) A(h) T −1 + ¯ Π¯ 2 (k) A¯ (h)P (h ) D(h), ¯ Π¯ 3 (k)  D¯ T (h)P −1 (h + ) D(h) − γ 2 I. We can obtain the first inequality in (3.19) by summing up two sides of the inequality from 0 to T − 1. Then propose V(k) in (3.6). Similarly, it’s easy to get ΔV(k) < 0 with zero initial condition. Consider ω(k) = 0, ∀k ≥ T and ΔV(k) < 0, for any k ≥ T , (3.16) ensures ˆ < 0, E {ΔV(ξ(k +1), k +1)}+z T (k)z(k) = ξ T (k)Π(k)ξ(k) where ¯ ¯ Πˆ (k) A¯ T (h)P −1 (h + ) A(h) + δ L¯ T (h)P −1 (h + ) L(h) −1 T T ¯ ¯ − P (h) + E¯ (h) E(h) + δ M¯ (h) M(h).

(3.20)

3.3 Main Results

49

Summing up two sides of (3.20) from T to ∞ that results in V(ξ(∞), ∞) − V(ξ(T ), T ) +

∞ 

z T (i)z(i) < 0.

(3.21)

i=T

Since V(∞) ≥ 0, we have the second inequality in (3.19). By considering (3.13) and (3.19), it yields (3.18), and the proof is completed. Remark 3.5 The obtained results of T-S fuzzy stochastic systems further reduce the conservativeness because of the universality of the Lyapunov functions utilized which include the fuzzy-basis-independent one as a particular case [7, 191, 250]. Next, we are going to demonstrate the full-order output feedback Hankel-norm controller design issues, and these issues can be converted into linear matrices inequalities optimization issues, which can be resolved numerically readily.

3.3.2 Hankel-Norm Output Feedback Control Theorem 3.6 Based on the overall closed-loop system in (3.5), and given a constant scalar γ > 0, if there exist fuzzy-basis-dependent matrices P1 (h), P2 (h), P3 (h), P1 (h), P2 (h), P3 (h), R(h), S(h), A (h), B(h), C (h), D(h), and constant matrices X , Y and S such that for any h, h + ∈ , ⎡

⎤ ˜ +) −P(h 0 δ Π˜ L (h) 0 ⎢ ˜ +)  −P(h Π˜ A (h) Π˜ D (h) ⎥ ⎢ ⎥ 0, P3i > 0, P2i , P2i , Ri , Si , Ai , Bi , Ci and Di for all i, j, g, l ∈ {1, . . . , r } such that

52

3 Output Feedback Control of Fuzzy Stochastic Systems

⎡

⎤ −P˜ j 0 δ Π˜ Lig 0 ⎢  −P˜ j Π˜ Aig Π˜ Dig ⎥ ⎢ ⎥< ⎣  T ˜ ˜ ˜  Pi − Π − Π 0 ⎦    −γ 2 I ⎡ ˜ ⎤ −P j 0 0 0 δ Π˜ Lig ⎢  −P˜ j 0 0 ⎥ Π˜ Aig ⎢ ⎥ ⎢  ⎥< ˜ Mig  −I 0 δ Π ⎢ ⎥ ⎣  ⎦ ˜   −I Π Eig T ˜ ˜ ˜     Pi − Π − Π     P1i P2i P1 j P2 j − <  P3i  P3 j   −P1i −P2i <  −P3i

0,

(3.30)

0,

(3.31)

0,

(3.32)

0,

(3.33)

where     P1i P2i P1i P2i , P˜ i  , P˜ i   P3i  P3i   A X + Bi Cg Ai + Bi Dg Ci Π˜ Aig  i , Ag Y Ai + Bg C i     L X + Bi Rg 0 Di + Bi Dg Fi , Π˜ Lig  i , Π˜ Dig  0 Y Di + Bg Fi Sg     Π˜ Eig  E i X +G i Cg E i +G i Dg Ci , Π˜ Mig  G i Rg 0 , where Π˜  is given in Theorem 3.6. There exists a Hankel-norm output feedback controller as (3.4) such that the closed-loop system in (3.5) is mean-square asymptotically stable with Hankel-norm performance level γ. In addition, there are two nonsingular matrices U and V such that VU = S − YX , and the designed fuzzy controller can be obtained as

3.3 Main Results

53

Ac (h) 

r  r 

h i (θ)h g (θ) V −1 Y Bi Dg Ci X U −1

i=1 g=1

+ V −1 Ag U −1 − V −1 Y Bi Cg U −1

−V −1 Y Ai X U −1 − V −1 Bg Ci X U −1 ,

Bc (h) 

r r  

h i (θ)h g (θ) V −1 Bg − V −1 Y Bi Dg ,

i=1 g=1

Cc (h) 

r r  

h i (θ)h g (θ) Cg U −1 − Dg Ci X U −1 ,

i=1 g=1

Dc (h) 

r 

h g (θ)Dg .

(3.34)

g=1

Proof For the given fuzzy-based function h ∈  and matrices P1i , P2i , P3i , P1i , P2i , P3i , Ri , Si , Ai , Bi , Ci and Di which satisfy (3.22)–(3.25), define   ⎧ P1 (h)  ri=1 h i (θ)P1i , P2 (h)  ri=1 h i (θ)P2i , ⎪ ⎪   ⎪ ⎪ P1 (h)  ri=1 h i (θ)P1i , P2 (h)  ri=1 h i (θ)P2i , ⎪ ⎪   ⎨ S(h)  ri=1 h i (θ)Si , R(h)  ri=1 h i (θ)Ri , r  ⎪ P3 (h)  i=1 h i (θ)P3i , P3 (h)  ri=1 h i (θ)P3i , ⎪   ⎪ ⎪ ⎪ A (h)  ri=1 h i (θ)Ai , B(h)  ri=1 h i (θ)Bi , ⎪ ⎩ r r D(h)  i=1 h i (θ)Di . C (h)  i=1 h i (θ)Ci ,

(3.35)

It is not difficult to have r  r r  

h i (θ)h j (θ)h g (θ)

i=1 i= j i=g

⎤ −P˜ j 0 δ Π˜ Lig 0 ⎢  −P˜ j Π˜ Aig Π˜ Dig ⎥ ⎥ < 0, ×⎢ ⎣  T ˜ ˜ ˜  Pi − Π − Π 0 ⎦    −γ 2 I ⎡

r  r r  

h i (θ)h j (θ)h g (θ)

i=1 i= j i=g

⎡

(3.36)

−P˜ j 0 ⎢  −P˜ j ⎢ ×⎢  ⎢  ⎣    

0 0 −I  

⎤ 0 δ Π˜ Lig ⎥ 0 Π˜ Aig ⎥ ⎥ < 0, 0 δ Π˜ Mig ⎥ ⎦ ˜ −I Π Eig T ˜ ˜ ˜  Pi − Π − Π

(3.37)

54

3 Output Feedback Control of Fuzzy Stochastic Systems r  r  i=1 i= j

 P1i − P1 j P2i − P2 j < 0, h i (θ)h j (θ)  P3i − P3 j

(3.38)

 −P1i −P2i < 0. h i (θ)  −P3i

(3.39)



r  i=1



It is clear that (3.36)–(3.39) are satisfied if the conditions in (3.30)–(3.33) hold. Hence, it completes the proof. Remark 3.8 Based on the fuzzy basis functions, Theorem 3.7 is presented from parameter-dependent matrix inequalities’ conditions in Theorem 3.6, which cannot be directly implemented for the output feedback controller design. This converts the obtained sufficient conditions to some finite LMIs, which can be readily solved using standard numerical software. Remark 3.9 Theorem 3.7 provides the sufficient solvability conditions for Hankelnorm output feedback controller design issue of T-S fuzzy stochastic system, therefore, a desired controller can be obtained by resolving the optimization issue as follow: min ϕ subject to ((3.30) − (3.33)) with ϕ = γ 2 .

3.4 Illustrative Example Example 3.10 Consider the modified Henon mapping system with stochastic disturbances: ⎧ ⎨ x1 (k + 1) = −[εx1 (k)]2 +[0.01x1 (k)+0.02x2 (k)] (k) (3.40) +0.3x2 (k) + 1.4, ⎩ x2 (k + 1) = εx1 (k) + 0.01x2 (k)(k), where (k) denotes the stochastic process and δ 2 = E {(k)2 }. The constant ε ∈ [0, 1] is the retarded coefficient. Let θ = εx1 (k). Suppose θ(k) ∈ [− , ], > 0. With similar methods in [275], the nonlinear term θ2 can be described as θ2 = h 1 (θ)(− )θ + h 2 (θ) θ, where h 1 (θ), h 2 (θ) ∈ [0, 1], and h 1 (θ) + h 2 (θ) = 1. Then the fuzzy basis functions h 1 (θ) and h 2 (θ) are expressed by h 1 (θ) =

    θ θ 1 1 1− , h 2 (θ) = 1+ . 2 2

It is noted that h 1 (θ) = 1 and h 2 (θ) = 0 when θ is − and that h 1 (θ) = 0 and h 2 (θ) = 1 when θ is . And for the given fuzzy control input u(k), the nonlinear stochastic system in (3.40) is constructed with the following T-S fuzzy model:

3.4 Illustrative Example

55

 Plant Form: Rule 1: IF θ is − , THEN x(k + 1) = A1 x(k) + B1 u  (k) + L 1 x(k)(k). Rule 2: IF θ is , THEN x(k + 1) = A2 x(k) + B2 u  (k) + L 2 x(k)(k), where u  (k) = 1.4 + u(k) and 

     ε 0.3 0.01 0.02 1 , L1 = , B1 = , ε 0 0 0.01 0       −ε 0.3 0.01 0.02 0 A2 = , L2 = , B2 = . ε 0 0 0.01 1 A1 =

The regulated output and disturbance terms are added, then (3.40) turns as  Plant Form: Rule 1: IF θ(k) is − , THEN ⎧ ⎨ x(k + 1) = A1 x(k)+ B1 u  (k)+ D1 ω(k)+ L 1 x(k)(k), y(k) = C1 x(k)+ F1 ω(k)+ N1 x(k)(k), ⎩ z(k) = E 1 x(k). Rule 2: IF θ(k) is , THEN ⎧ ⎨ x(k + 1) = A2 x(k)+ B2 u  (k)+ D2 ω(k)+ L 2 x(k)(k), y(k) = C2 x(k)+ F2 ω(k)+ N2 x(k)(k), ⎩ z(k) = E 2 x(k), where      T   C1 = 1 − ε 0 , N1 = 0.01 0.02 , D1 = 1 0 , E 1 = 1 0 , F1 = 1,      T   C2 = ε 0 , N2 = 0.03 0.06 , D2 = 1 0 , E 2 = 1 0 , F2 = 0.5.   x (k) , ε = 0.9 and = 0.9. Assumed the initial condition as Moreover, x(k) = 1 x2 (k)   1 ϕ(k) = . Here, the Hankel-norm error performance level is given by γ = −1 25.2684, and the corresponding controller parameters are obtained as

56

3 Output Feedback Control of Fuzzy Stochastic Systems

Fig. 3.1 States of the open-loop system



   9.5385 62.4944 10.5735 61.9773 2 Ac (h) = h 1 (θ)h 2 (θ) + h 1 (θ) −1.6038 −10.5558 −1.7835 −10.4730   −1.0350 0.5171 , + h 21 (θ) 0.1797 −0.0828       −17.1856 3.3442 −13.8414 2 2 + h 2 (θ) + h 1 (θ)h 2 (θ) , Bc (h) = h 1 (θ) 3.0213 −0.5781 2.4432     Cc (h) = h 1 (θ)h 2 (θ) −0.0764 −0.5225 + h 21 (θ) −1.0974 −5.9260   + h 22 (θ) 1.0210 5.4035 , Dc (h) = −1.8821. The disturbance input ω(k) is set as ω(k) = 2e(−0.18k) sin(5k). The simulation results are shown in Figs. 3.1, 3.2, 3.3. Thereinto, Fig. 3.1 draws the state response x1 (k) and x2 (k) of the open-loop system, and Fig. 3.2 plots the state response x1 (k) (solid line) and x2 (k) (dash-dot line) of the closed-loop system. The control input u(k) is plotted in Fig. 3.3. It can be seen that the proposed controller guarantees the mean-square asymptotic stability with a Hankel-norm performance level of the concerned system.

3.4 Illustrative Example

Fig. 3.2 States of the closed-loop system

Fig. 3.3 Control input u(k)

57

58

3 Output Feedback Control of Fuzzy Stochastic Systems

3.5 Conclusion In this chapter, the DOF Hankel-norm controller design issue was examined for T-S fuzzy stochastic systems. First, the Hankel-norm controller design scheme was established using certain additional matrix variables, which decoupled the Lyapunov functions and rendered the controller design feasible. Sufficient conditions with less conservativeness were obtained through the use of fuzzy-basis-dependent Lyapunov functions. Furthermore, a full-order OFC problem could be transformed to an optimization problem through the parameter transformation. Finally, a numerical simulation was provided to verify the validity of our controller design method.

Chapter 4

L2 –L∞ Output Feedback Control of Fuzzy Switching Systems

4.1 Introduction This chapter address the L2 –L∞ output-feedback controller design problem for switched systems with nonlinear perturbations in the T-S fuzzy framework. First, the ADT method is considered to stabilize the nonlinear switched system exponentially through an arbitrary switching law. Subsequently, based on the piecewise Lyapunov functions, a fuzzy-rule-dependent output-feedback controller is proposed to ensure that the closed-loop system is exponentially stable with a weighted L2 – L∞ performance (γ, α). The solvable conditions of the desired dynamic controller are derived by employing the linearization approach. The controller matrices can be obtained in terms of several strict LMIs, which can be resolved numerically through efficient standard software.

4.2 System Description and Preliminaries By applying the following T-S fuzzy modelling, a dynamic nonlinear system can be described by a class of switched fuzzy linear systems:  Plant Form: [ j] [ j] [ j] [ j] [ j] [ j] [ j] Rule Ri : IF θ1 (t) is μi1 and θ2 (t) is μi2 and · · · and θ p (t) is μi p , THEN   f x(t), t ,  [ j] [ j] [ j] [ j]  y(t) = Ci x(t) + D1i u(t) + D2i ω(t) + G i g x(t), t ,

(4.1b)

[ j] z(t) = L i x(t)

(4.1c)

[ j]

[ j]

[ j]

[ j]

x(t) ˙ = Ai x(t) + B1i u(t) + B2i ω(t) + Fi +

[ j] K i u(t),

i = 1, 2, . . . , r,

(4.1a)

where x(t) ∈ Rn represents the state vector; u(t) ∈ Rm represents the control input; ω(t) ∈ Rl represents an exogenous disturbance that L2 [0, ∞); y(t) ∈ R p denotes the measurement output; z(t) ∈ Rq represents the controlled output; r © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 X. Su et al., Intelligent Control, Filtering and Model Reduction Analysis for Fuzzy-Model-Based Systems, Studies in Systems, Decision and Control 385, https://doi.org/10.1007/978-3-030-81214-0_4

59

4 L2 –L∞ Output Feedback Control of Fuzzy Switching Systems

60

[ j]

[ j]

is the number of IF-THEN rules; μi1 , . . . , μi p denote the fuzzy sets; θ[ j] (t) =  [ j]  [ j] [ j] θ1 (t), θ2 (t), . . . , θ p (t) are premise variables; the positive integer N denotes the number of subsystems. σ j (t) : [0, ∞) → {0, 1},

N 

σ j (t) = 1, t ∈ [0, ∞),

j=1

j ∈ N = {1, 2, . . . , N }, is a switching signal assigning which subsystem is activated at a switching instant. 

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

Ai , B1i , B2i , Fi , Ci , D1i , D2i , G i , L i , K i



j ∈ N is a set of matrices parameterized by an index set N = {1, 2, . . . , N }. 

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

Ai , B1i , B2i , Fi , Ci , D1i , D2i , G i , L i , K i



    are real constant matrices. f x(t), t ∈ R f and g x(t), t ∈ Rg are known nonlinear functions which satisfy the assumption as below. Remark 4.1 This chapter focuses on the nonlinear switched system which is modeled by T-S fuzzy rules such as (4.1a)–(4.1c). It is observed that Karer [120] has proposed the hybrid fuzzy systems in some cases, i.e., as a hybrid fuzzy model satisfying particular conditions as (4.1a)–(4.1c). Then employing the ADT method and piecewise Lyapunov functions, DOFC is proposed to guarantee the closed-loop system is exponentially stable subject to a weighted L2 –L∞ performance level. Hence, we need additional theoretical verifications to see whether the results for switched fuzzy systems can be generalized to hybrid fuzzy systems, which prompts us to carry on the research in this area in the future.     Assumption 4.1 The functions f x(t), t and g x(t), t satisfy the zero  nonlinear  initial condition ( f 0, 0 = 0 and the Lipschitz condition, that is there exist known real matrices M and N which satisfy

   



f x(t), t − f y(t), t  M(x − y) ,



  



g x(t), t − g y(t), t  N (x − y) . [ j] Suppose the control  input u(t) cannot influence the premise variables θ (t). For a pair of x(t), u(t) , the resulting system output is described by

4.2 System Description and Preliminaries

x(t) ˙ =

N 

σ j (t)

j=1

+ y(t) =

σ j (t)

j=1

z(t) =

+

r 



[ j] Fi [ j]



hi

θ[ j] (t)

σ j (t)

j=1

+

r 

θ[ j] (t)

[ j] G i g(t)

[ j] hi





[ j]

[ j]

Ai x(t) + B1i u(t)

 f (t) ,

i=1

[ j] D2i ω(t)

N 

[ j]

hi

i=1

[ j] B2i ω(t)

N 

+

r 

61

[ j]

(4.2a)

 [ j] [ j] Ci x(t) + D1i u(t)

 ,

θ (t)



(4.2b) [ j] L i x(t)

+

[ j] K i u(t)

 ,

(4.2c)

i=1

where  [ j] p     νi θ[ j] (t) [ j] [ j] [ j] [ j] h i θ[ j] (t) = r μil θl (t) , , νi θ[ j] (t) =   [ j] [ j] l=1 νi θ (t) i=1

 [ j] [ j] [ j] θl (t) represents the grade of membership of θl (t) in μil . Hence, it fol  [ j] [ j] lows that νi θ[ j] (t)  0, h i θ[ j] (t)  0 for i = 1, 2, . . . , r , and  r [ j] [ j] θ h (t) = 1 for all t. i=1 i [ j]

and μil

It is assumed that the premise variables θ[ j] (t) are available for the controller design. Thus, the structure of designed controllers can be described as follows by utilizing the PDC technique.  Dynamic Output Feedback Control Form: [ j] [ j] [ j] [ j] [ j] [ j] [ j] Rule Ri : IF θ1 (t) is μi1 and θ2 (t) is μi2 and · · · and θ p (t) is μi p , THEN [ j]

[ j]

x˙c (t) = Aci xc (t) + Bci y(t), [ j] u(t) = Cci xc (t),

(4.3a)

i = 1, 2, . . . , r,

(4.3b) [ j]

[ j]

[ j]

where xc (t) ∈ Rr is the controller state vector with r  n, Aci , Bci and Cci are controller parameters to be determined later. A complete form of the dynamic output feedback controller (4.3) is expressed as x˙c (t) =

N 

σ j (t)

j=1

u(t) =

N  j=1

r 

[ j]

   [ j] [ j] θ[ j] (t) Aci xc (t) + Bci y(t) ,

(4.4a)

[ j]

 [ j] θ[ j] (t) Cci xc (t).

(4.4b)

hi

i=1

σ j (t)

r  i=1

hi

4 L2 –L∞ Output Feedback Control of Fuzzy Switching Systems

62

Thus, combining the system model (4.2) and designed controller (4.4), the resulting closed-loop system can be given by ˙ = ξ(t)

N 

σ j (t)

j=1



× z(t) =

N  j=1

 where ξ(t) 

r 

[ j] hi



[ j]

θ (t)

r 

i=1

r 

 θ[ j] (t)

l=1

[ j] A˜ il ξ(t)

σ j (t)

[ j]

hl

 [ j] ˜ + Fil η(t) ,

+

[ j] B˜ il ω(t)

[ j]

r    [ j] [ j] θ[ j] (t) h l θ[ j] (t) C˜ il ξ(t),

hi

i=1

(4.5a) (4.5b)

l=1

    f  x(t), t x(t) and , η(t)  xc (t) g x(t), t

  ⎧ [ j] [ j] [ j] ⎪ A B C [ j] i 1i cl ⎪ ⎪ A˜ il  [ j] [ j] [ j] [ j] [ j] [ j] , ⎪ ⎪ Bcl Ci Acl + Bcl D1i Ccl ⎪ ⎪ ⎨     [ j] [ j] F B 0 [ j] [ j] i 2i ˜ F˜il  ⎪ [ j] [ j] , Bil  [ j] [ j] , ⎪ ⎪ 0 Bcl G i Bcl D2i ⎪ ⎪   ⎪ ⎪ ⎩ C˜ [ j]  [ j] [ j] [ j] . K C L il i i cl

(4.6)

Define ⎧    N r r     ⎪ [ j] [ j] [ j] ⎪ ⎪ A˜ t, σ j (t)  σ j (t) h i θ[ j] (t) h l θ[ j] (t) A˜ il , ⎪ ⎪ ⎪ j=1 i=1 l=1 ⎪ ⎪    N r r ⎪     ⎪ [ j] [ j] [ j] [ j] [ j] ⎪ ˜ ⎪ θ θ (t)  σ (t) h (t) h (t) B˜ il , B t, σ j j ⎨ i l j=1

i=1

l=1

j=1

i=1

l=1

   N r r     ⎪ [ j] [ j] [ j] ⎪ ⎪ σ j (t) h i θ[ j] (t) h l θ[ j] (t) C˜ il , C˜ t, σ j (t)  ⎪ ⎪ ⎪ j=1 i=1 l=1 ⎪ ⎪   r N r ⎪ ⎪ ⎪ F˜ t, σ j (t)   σ j (t)  h [ j] θ[ j] (t)  h [ j] θ[ j] (t) F˜ [ j] . ⎪ ⎩ i l il Figure 4.1 plots the compact presentation of the closed-loop system (4.5). Before proposing the main results, the following definitions are introduced. Definition 4.2 [306] When ω(t) = 0, the equilibrium ξ  (t) = 0 of the closed-loop system (4.5) is exponentially stable under the switching parameter σ j (t), if ξ(t) satisfies ξ(t)2  μ ξ(t0 )2 e−λ(t−t0 ) , ∀t  t0 , for any constants μ  1 and λ > 0.

4.2 System Description and Preliminaries

63

Fig. 4.1 Block diagram of the resulting closed-loop system

Definition 4.3 [306] For the scalars γ > 0 and α > 0, the closed-loop system in (4.5) is exponentially stable with a weighted L2 –L∞ performance level (γ, α), if it is exponentially stable  for any switching signal σ j (t) when ω(t) = 0, and under zero initial condition ξ(0) = 0 , for all nonzero ω(t) ∈ L2 [0, ∞), the following inequality holds:

sup e ∀t

∞

−αt T

z (t)z(t) < γ

ω T (t)ω(t)dt.

2 0

4.3 System Performance Analysis [ j]

[ j]

[ j]

Assume that for given matrices Aci , Bci , and Cci in (4.4), the sufficient criteria are established to ensure the dynamic closed-loop system (4.5) is exponentially stable with a weighted L2 –L∞ performance level (γ, α). Theorem 4.4 Given scalars γ > 0, α > 0, if there exist a scalar ε > 0 and matrices P [ j] > 0 such that the following inequalities hold for j ∈ N : [ j]

φii  1 1 [ j] [ j] [ j] φii + φil + φli r −1 2 [ j] ϕii  1 [ j] 1 [ j] [ j] ϕii + ϕil + ϕli r −1 2

< 0, i = 1, 2, . . . , r,

(4.7)

< 0, 1  i < l  r,

(4.8)

< 0, i = 1, 2, . . . , r,

(4.9)

< 0, 1  i < l  r,

(4.10)

4 L2 –L∞ Output Feedback Control of Fuzzy Switching Systems

64

with ⎡

⎤ [ j] [ j] [ j] φ11il P [ j] B˜ il P [ j] F˜il ⎦,  −I 0   −εI   T  [ j] M  MT M + NT N, [ j] −P [ j] C˜ il   ϕil  , K I 0 ,  −γ 2 I T  [ j] [ j] [ j] P [ j] + αP [ j] + εK T MK. φ11il P [ j] A˜ il + A˜ il [ j] φil ⎣

For any switching signal, the dynamic system (4.5) is exponentially stable with a weighted L2 –L∞ performance level (γ, α) if Ta > Ta = lnαρ , ρ  1, and the following inequality holds: P [ j]  ρP [s] , ∀ j, s ∈ N .

(4.11)

In addition, an estimate of the state decay is described as ξ(t)2  μe−λt ξ(0)2 ,

(4.12)

where 

  λ = α − lnTaρ > 0, τ = min∀ j∈N λmin P [ j] ,   μ = ϑτ  1, ϑ = max∀ j∈N λmax P [ j] .

(4.13)

Proof The Lyapunov function is chosen as follow:    V ξ(t), σ j (t) = ξ T (t)P σ j (t) ξ(t),

(4.14)

N    σ j (t)P [ j] ( j ∈ N ) are to be determined later. And along the where P σ j (t)  j=1

trajectory of (4.5), the derivative of (4.14) is equivalent to N r r       [ j] [ j] V˙ ξ(t), σ j (t) = 2 σ j (t) h i θ[ j] (t) h l θ[ j] (t) j=1

× ξ (t)P T

=

N  j=1

i=1

[ j]

σ j (t)



[ j] [ j] A˜ il ξ(t) + F˜il η(t)

r  i=1

l=1

[ j]

hi



θ[ j] (t)

r  l=1



[ j]

hl

 θ[ j] (t)

4.3 System Performance Analysis

65

   T  [ j] T [ j] ˜ [ j] [ j] ˜ ξ(t) × ξ (t) P Ail + Ail P + 2ξ (t)P T



N  j=1

σ j (t)

[ j]

[ j] F˜il η(t)

r 

[ j]

hi



r    [ j] θ[ j] (t) h l θ[ j] (t)

i=1

l=1

  T  [ j] [ j] T T P [ j] × εη (t)η(t) + ξ (t) P [ j] A˜ il + A˜ il 

 T [ j] [ j] + ε−1 P [ j] F˜il F˜il P [ j] ξ(t) .

(4.15)

Consider Assumption 4.1, and define M, N that



  







f x(t), t  M x(t) , g x(t), t  N x(t) . It can be seen that

     

2

f x(t), t = f T x(t), t f x(t), t  M x(t)2 = x T (t)M T M x(t),

     

2

g x(t), t = g T x(t), t g x(t), t  N x(t)2 = x T (t)N T N x(t). Thus, we can get             η T x(t), t η x(t), t = f T x(t), t f x(t), t + g T x(t), t g x(t), t  ξ T (t)K T MKξ(t).

(4.16)

Based on (4.14)–(4.16), it follows that N r r       [ j] [ j] σ j (t) h i θ[ j] (t) h l θ[ j] (t) ξ T (t) V˙ ξ(t), σ j (t)  j=1



i=1

l=1

T [ j] F˜il P [ j]  T  [ j] + A˜ il P [ j] + εK T MK ξ(t). [ j]

[ j]

× P [ j] A˜ il + ε−1 P [ j] F˜il



(4.17)

Based on (4.7)–(4.8) and (4.17), employing the Schur complement method, which yields     V˙ ξ(t), σ j (t) < −αξ T (t)P σ j (t) ξ(t) = −αV ξ(t), σ j (t) .

(4.18)

66

4 L2 –L∞ Output Feedback Control of Fuzzy Switching Systems

As for the piecewise switching signal σ j (t) (t > 0), let 0 = t0 < t1 < · · · < tk < · · · < t, (k = 0, 1, . . .), denote switching points of σ j (t) under the interval (0, t).  Thus, the jk th subsystem is activated when t ∈ tk , tk+1 ). Starting with t   tk in (4.18), then   V ξ(t), σ j (t) < e−α(t−tk ) V ξ(tk ), σ j (tk ) .

(4.19)

Utilizing (4.11) and (4.14), at switching instant tk , we have   V ξ(tk ), σ j (tk ) < ρV ξ(tk− ), σ j (tk− ) . Consequently, from (4.19)–(4.20) and φ = Nσ j (0, t) 

t−0 , Ta

(4.20)

it follows that

  V ξ(t), σ j (t)  e−α(t−tk ) ρV ξ(tk− ), σ j (tk− )   · · ·  e−α(t−0) ρφ V ξ(0), σ j (0)  ln ρ  e−(α− Ta )t V ξ(0), σ j (0)  ln ρ = e−(α− Ta )t V ξ(0), σ j (0) .

(4.21)

On the basis of (4.14), it yields   V ξ(t), σ j (t)  τ ξ(t)2 , V ξ(0), σ j (0)  ϑξ(0)2 ,

(4.22)

where τ and ϑ are given in (4.13). Combining (4.21) and (4.22), we have

2

ϑ ln ρ 1 

ξ(t)  V ξ(t), σ j (t)  e−(α− Ta )t ξ(0)2 . τ τ

(4.23)

When ω(t) = 0, the closed-loop system in (4.5) is exponentially stable on account of Definition 4.2 with t0 = 0. When ω(t) = 0, the L2 –L∞ performance of the overall system is analyzed then. Introduce       J ξ(t), σ j (t)  V˙ ξ(t), σ j (t) +αV ξ(t), σ j (t) −ω(t)ω(t)    ψ T (t)φ t, σ j (t) ψ(t), where

(4.24)

4.3 System Performance Analysis

67

         φ¯ t, σ j (t) P σ j (t) B˜ t, σ j (t) φ t, σ j (t)  ,  −I           φ¯ t, σ j (t)  P σ j (t) A˜ t, σ j (t) + A˜ T t, σ j (t) P σ j (t)         + ε−1 P σ j (t) F˜ t, σ j (t) F˜ T t, σ j (t) P σ j (t)     ξ(t) T + αP σ j (t) + εK MK, ψ(t)  . ω(t)

(4.25)

  Hence, in view of ψ(t) = 0 and (4.7), (4.8), we have J ξ(t), σ j (t) < 0. Let ζ(t) = −ω T (t)ω(t), then     V˙ ξ(t), σ j (t)  −αV ξ(t), σ j (t) − ζ(t).

(4.26)

Employing a similar way in the proof of exponential stability, (4.26) yields   t −α(t−tk ) V ξ(t), σ j (t) < e V ξ(tk ), σ j (tk ) − e−α(t−s) ζ(s)ds.

(4.27)

tk

Consider φ = Nσ j (0, t) 

t−0 Ta

and (4.20), (4.27), it follows that

 V ξ(t), σ j (t)  ρe

−α(t−tk )

 t − − V ξ(tk ), σ j (tk ) − e−α(t−s) ζ(s)ds tk

φ −α(t−0)

ρ e

t1  φ V ξ(0), σ j (0) − ρ e−α(t−s) ζ(s)ds 0

φ−1



−ρ

t2

e

−α(t−s)

t ζ(s)ds − · · · − ρ

0

t1

e−α(t−s) ζ(s)ds

tk

 = e−αt−Nσ j (0,t) ln ρ V ξ(0), σ j (0) −

t

e−α(t−s)+Nσ j (s,t) ln ρ ζ(s)ds.

(4.28)

0

Provided that ξ(0) = 0, (4.28) leads to  t V ξ(t), σ j (t)  e−α(t−s)+Nσ j (s,t) ln ρ ω T (s)ω(s)ds. 0

Multiplying two sides of (4.29) by e−Nσ j (0,t) ln ρ , it yields

(4.29)

4 L2 –L∞ Output Feedback Control of Fuzzy Switching Systems

68

e

−Nσ j (0,t) ln ρ

 t V ξ(t), σ j (t)  e−α(t−s)−Nσ j (0,s) ln ρ ω T (s)ω(s)ds 0

t ω T (s)ω(s)ds.



(4.30)

0

Note that Nσ j (0, t)  (4.30) means

t Ta

e

and Ta > Ta =

−αt



ln ρ , α



then Nσ j (0, t) ln ρ  αt. Therefore,

t

V ξ(t), σ j (t) 

ω T (s)ω(s)ds.

(4.31)

0

Based on (4.14) and (4.31), it can get

e

  ξ (t)P σ j (t) ξ(t) 

−αt T

t

∞ ω (s)ω(s)ds 

ω T (t)ω(t)dt.

T

0

(4.32)

0

Since t = T   0 is an arbitrary time moment, then e

  ξ (T )P σ j (T  ) ξ(T  ) 

−αT  T



∞ ω T (t)ω(t)dt.

(4.33)

0

On account of (4.9)–(4.10),       γ −2 C˜ T t, σ j (t) C˜ t, σ j (t) < P σ j (t) . Combining (4.33) and (4.34) yields      γ −2 e−αT ξ T (T  )C˜ T T  , σ j (T  ) C˜ T  , σ j (T  ) ξ(T  ) ∞   −αT  T    e ξ (T )P σ j (T ) ξ(T )  ω T (t)ω(t)dt. 0

For any T   0, e

−αT  T





z (T )z(T ) ≤ γ

∞ ω T (t)ω(t)dt.

2 0

Taking the supremum over T   0, which signifies

(4.34)

4.3 System Performance Analysis

sup e ∀t

−αt T

z (t)z(t) < γ

69

∞ ω T (t)ω(t)dt.

2 0

Thus, the closed-loop system satisfies a given weighted L2 –L∞ performance level. Remark  4.5 In Theorem 4.4, a fuzzy-basis-dependent Lyapunov function V (t)  ξ T (t)P σ j (t) ξ(t) is established over the switching signal σ j (t). It is proven  to be less conservative compared with usual Lyapunov functions (when P σ j (t) = P). Remark 4.6 If ρ = 1 in Ta > lnαρ , then Ta > Ta = 0, which means the switching signal σ j (t) is arbitrary. It implies a general Lyapunov function is demanded for all subsystems. If ρ > 1 and α → 0 in Ta > lnαρ , the closed-loop system can be operated at one of the subsystems continuously   as Ta →  ∞. In  addition, based on Assumption 4.1, the nonlinearities f x(t), t and g x(t), t satisfy Lipschitz conditions. Therefore, the controller design method is useful in some actual applications.

4.4 Dynamic Output Feedback Control 4.4.1 Reduced-Order Controller Design In this subsection, a criterion is given to handle the reduced-order controller issue for nonlinear switched systems in (4.5). Theorem 4.7 For the given scalars γ > 0, α > 0, if there exist a scalar ε > 0 and [ j] [ j] [ j] matrices P [ j] > 0, Q [ j] > 0, Acil , Bcl , Ccl , which satisfy the following conditions for j ∈ N , [ j] φ˜ ii  1 ˜ [ j] 1 ˜ [ j] ˜ [ j] φ + φli φ + r − 1 ii 2 il [ j] ϕ˜ ii 1  [ j] 1 [ j] [ j] ϕ˜ ii + ϕ˜ il + ϕ˜ li r −1 2

with

< 0, i = 1, 2, . . . , r,

(4.35)

< 0, 1  i < l  r,

(4.36)

< 0, i = 1, 2, . . . , r,

(4.37)

< 0, 1  i < l  r,

(4.38)

4 L2 –L∞ Output Feedback Control of Fuzzy Switching Systems

70

⎡ T ⎤  ⎤ [ j] [ j] [ j] [ j] Li −P [ j] −I φ˜ 11il φ˜ 12il φ˜ 13il ⎥ ⎢ [ j] [ j] φ˜ il ⎣  −I 0 ⎦ , ϕ˜ il  ⎣  −Q [ j] ϕ˜ [ j] ⎦ , 23il   −εI   −γ 2 I     [ j] [ j] [ j] [ j] ˜ [ j] ˜ [ j] φ P B + HB D φ [ j] [ j] 2i cl 2i φ˜ 11il  111il ˜ 112il , φ˜ 12il  , [ j] [ j]  φ113il B2i   [ j] [ j] [ j] P [ j] Fi HBcl G i [ j] ˜ φ13il  , [ j] Fi 0 T  [ j] [ j] [ j] [ j] [ j] [ j] [ j] + HBcl Ci + αP [ j] + εM, φ˜ 111il P [ j] Ai + P [ j] Ai + HBcl Ci T T   [ j] [ j] [ j] [ j] [ j] [ j] [ j] + αI, ϕ˜ 23il  L i Q [ j] + K i Ccl HT , φ˜ 112il  Acil + Ai T  [ j] [ j] [ j] [ j] [ j] [ j] [ j] φ˜ 113il Ai Q [ j] + Ai Q [ j] + B1i Ccl HT + B1i Ccl HT + αQ [ j] . ⎡

Then the output feedback controller in (4.4) guarantees that the closed-loop system in (4.5) is exponentially stable subject to a specific weighted L2 –L∞ performance level. And the controller parameters are given by ⎧  T  T [ j] [ j] [ j] [ j] [ j] [ j] [ j] ⎪ ⎪ Acil  P [ j] B1i Ccl HQ2 + HP2 Acl HQ2 ⎪ ⎪ ⎪ ⎪ [ j] [ j] [ j] [ j] ⎨ +P [ j] Ai Q [ j] + HP2 Bcl Ci Q [ j]  T [ j] [ j] [ j] [ j] [ j] ⎪ HQ +HP B D C , ⎪ 2 2 cl 1i cl ⎪ ⎪  ⎪ T ⎪ ⎩ B [ j]  P [ j] B [ j] , C [ j]  C [ j] Q[ j] . cl

2

cl

cl

(4.39)

2

cl

Proof Firstly, P [ j] is constructed as  P

[ j]



[ j]

P1 

so, Q[ j]

−1   P [ j] 

[ j]

HP2 [ j] P3



[ j]

Q1 

 ,

[ j]

HQ2 [ j] Q3

 ,

 T where H = Ir ×r 0r ×(n−r ) , P1 ∈ Rn×n , P2 ∈ Rr ×r , and P3 ∈ Rr ×r . Without loss [ j] [ j] [ j] [ j] of generality, suppose that P2 and Q2 are nonsingular (if not, then P2 and Q2 [ j] [ j] can be added with matrices P2 and Q2 , having sufficiently small norms, such [ j] [ j] [ j] [ j] that P2 + P2 and Q2 + Q2 are still nonsingular and satisfy (4.7)–(4.10). Next, define the following matrices:

4.4 Dynamic Output Feedback Control

⎡ JP  ⎣ 

[ j]

P1

[ j]

[ j]

HP2

71

⎤ T

I 0

⎡

⎦ , JQ[ j]  ⎣

I 0



[ j]

Q1

[ j]

HQ2

⎤ T ⎦ .

(4.40)

Note that [ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

P [ j] JQ = JP , Q[ j] JP = JQ , P1 Q1 + HP2

 T [ j] HQ2 = I.

[ j]

[ j]

Performing the congruence transformation to φil < 0 with diag{JQ , I, I }, it follows that ⎡ ⎤ T  T  T [ j] [ j] [ j] [ j] [ j] [ j] [ j] JQ φ11il JQ JQ P [ j] B˜ il P [ j] F˜il JQ ⎢ ⎥ ⎣ ⎦ < 0. (4.41)  −I 0   −εI [ j]

[ j]

Performing the congruence transformation to ϕil < 0 with diag{JQ , I }, we have 

T  [ j] [ j] P [ j] JQ − JQ 

[ j]



T  T  [ j] C˜ il < 0. −γ 2 I

[ j]

JQ

(4.42)

[ j]

Let P [ j] = P1 , Q [ j] = Q1 , and from (4.39) that 

[ j] JQ

T

P

[ j]

[ j] JQ 



 P [ j] I , I Q [ j]

  [ j] [ j] [ j] [ j]  T P [ j] Ai + HBcl Ci Acil [ j] [ j] ˜ [ j] [ j] JQ P Ail JQ  , [ j] [ j] [ j] [ j] Ai Ai Q [ j] + B1i Ccl HT   [ j] [ j] [ j] T  P [ j] B2i + HBcl D2i [ j] [ j] ˜ [ j] JQ P Bil  , [ j] B2i   [ j] [ j] [ j] T  P [ j] Fi HBcl G i [ j] [ j] ˜ [ j] P Fil  JQ , [ j] Fi 0 ⎤ ⎡ T  [ j]  T  T L i ⎥ ⎢ [ j] [ j] T ⎦ . JQ C˜ il (4.43) ⎣  [ j] [ j] [ j] [ j] T L i Q + K i Ccl H On the basis of (4.40)–(4.43), it can be observed that (4.35)–(4.38) hold. Consequently, in consideration of Theorem 4.4, the overall system is exponentially stable subject to a weighted L2 –L∞ performance level (γ, α). Besides, the gains of the reduced-order controller can be obtained by resolving the conditions (4.39).

72

4 L2 –L∞ Output Feedback Control of Fuzzy Switching Systems

Remark 4.8 The dynamic output feedback controller design approach can be achieved with independent or dependent fuzzy-rules. The premise variable θ(k) is fully available in fuzzy-rule-dependent method, but the fuzzy-rule-independent method is employed at instances when θ(k) is inaccessible. That is to say, we can [ j] [ j] [ j] let [Aci , Bci , Cci ]  [Aci , Bci , Cci ] or select [Aci , Bci , Cci ] = [A, B, C] in (4.4), which result in different non-parameterized controllers with different computational complexity and conservativeness. In this chapter, the fuzzy-rule-dependent method is utilized to design the controller, which is less conservative. Remark 4.9 The obtained results employing the projection lemma are usually expressed with LMIs plus an additional rank constraint. Nevertheless, due to the rank constraint is non-convex, the obtained conditions are not easy to solve by the numerical software. In this chapter, a linearization technique is put forward to resolve the output feedback controller design issue, which can be implemented [ j] [ j] with simulation toolbox. In addition, owing to the matrices P2 and Q2 can be available in advance, the controller matrices in (4.39) can be obtained by letting  T [ j] [ j] [ j] [ j] HP2 HQ2 = I − P1 Q1 .

4.4.2 Full-Order Controller Design The results of full-order L2 –L∞ dynamic output feedback controller are derived employing the similar methods in Theorem 4.7. Theorem 4.10 Given scalars γ > 0 and α > 0, if there exist a scalar ε > 0 and [ j] [ j] [ j] matrices P [ j] > 0, Q [ j] > 0, Acil , Bcl , Ccl , which satisfy the following conditions for j ∈ N : [ j] φˆ ii 1 ˆ [ j] 1  ˆ [ j] ˆ [ j] φ + φli φ + r − 1 ii 2 il [ j] ϕˆ ii  1 [ j] 1 [ j] [ j] ϕˆ ii + ϕˆ il + ϕˆ li r −1 2

with

< 0, i = 1, 2, . . . , r,

(4.44)

< 0, 1  i < l  r,

(4.45)

< 0, i = 1, 2, . . . , r,

(4.46)

< 0, 1  i < l  r,

(4.47)

4.4 Dynamic Output Feedback Control

73

⎡ T ⎤  ⎤ [ j] [ j] [ j] [ j] Li −P [ j] −I φˆ 11il φˆ 12il φˆ 13il ⎥ ⎢ [ j] [ j] φˆ il  ⎣  −I 0 ⎦ , ϕˆ il  ⎣  −Q [ j] ϕˆ [ j] ⎦ , 23il   −εI   −γ 2 I     [ j] [ j] [ j] [ j] ˆ [ j] φ˜ [ j] φ P B + B D [ j] [ j] 111il 112il 2i cl 2i φˆ 11il  , φˆ 12il  , [ j] [ j] B2i  φˆ 113il   [ j] [ j] [ j] T  P [ j] Fi Bcl G i [ j] [ j] [ j] [ j] [ j] ˆ φ13il  , , ϕˆ 23il  L i Q [ j] + K i Ccl [ j] Fi 0 T  [ j] [ j] [ j] [ j] [ j] [ j] [ j] φˆ 111il  P [ j] Ai + Bcl Ci + P [ j] Ai + Bcl Ci + αP [ j] + εM, T  [ j] [ j] [ j] [ j] [ j] [ j] [ j] φˆ 113il  Ai Q [ j] + B1i Ccl + Ai Q [ j] + B1i Ccl + αQ [ j] , ⎡

where φ˜ 112il are given in Theorem 4.7, and the full-order controller matrices are expressed as [ j]

[ j]

 T  T [ j] [ j] [ j] [ j] [ j] Q2 + P2 Acl Q2 + P [ j] Ai Q [ j]  T [ j] [ j] [ j] [ j] [ j] [ j] [ j] [ j] + P2 Bcl Ci Q [ j] + P2 Bcl D1i Ccl Q2 ,  T [ j] [ j] [ j] [ j] [ j]  P2 Bcl , Ccl  Ccl Q2 . [ j]

[ j]

Acil  P [ j] B1i Ccl

[ j]

Bcl

 Proof In the first place, set the matrix P

Q[ j]

[ j]

−1  = P [ j] 

 

 [ j] P2 [ j] , then P3

[ j]

P1  [ j]

Q1 

 [ j] Q2 [ j] . Q3

Define the matrices ⎡ JP  ⎣  [ j]

⎤

[ j]

P1

[ j]

P2

T

I 0

⎡

⎦ , JQ[ j]  ⎣

I 0

[ j]

Q1

⎤

T ⎦ ,  [ j] Q2

(4.48)

and notice that [ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

P [ j] JQ = JP , Q[ j] JP = JQ , P1 Q1 + P2

 T [ j] Q2 = I. [ j]

Performing congruence transformations to (4.7) and (4.9) with diag{JQ , I, I } and [ j] diag{JQ , I }, respectively, it follows that (4.44)–(4.47). Therefore, on the basis of

4 L2 –L∞ Output Feedback Control of Fuzzy Switching Systems

74

Theorem 4.4, the resulting closed-loop system is exponentially stable subject to a weighted L2 –L∞ performance level (γ, α). Remark 4.11 The feasible conditions for the DOFC design issue are given in Theorem 4.7 (or Theorem 4.10), which can be validly calculated by using the standard optimization toolbox in terms of strict LMIs. The computational complexity mainly depends on the number of fuzzy rules r and the proposed algorithm asks for 2n 2 + mn + n free variables. If r is 3 or less, the corresponding complexity is smaller and the result is simpler. Accordingly, if r is larger, the more iterations happen during the simulation, which increases the algorithm complexity. Therefore, it is important to select appropriate fuzzy-based rules and reduce the number of fuzzy rules properly without influencing system performance. The designed reduced-order/full-order DOFC in (4.4) can be obtained by dealing with the convex optimization issue as below: min δ subject to (4.35)–(4.38) or (4.44)–(4.47) with δ = γ 2 .

4.5 Illustrative Example Two examples are provided to illustrate the effectiveness of the proposed design technique in this section. The first one gives numerical results of reduced-order/fullorder DOFC. The second one is presented to demonstrate its applicability in cognitive radio (CR) systems. Example 4.12 Consider the switched system (4.1) with two subsystems and the parameters are given as follows: Subsystem 1. ⎡

A[1] 1

⎤ ⎡ ⎤ ⎡ ⎤ −1.8 −0.3 −0.5 1.2 0.3 [1] [1] = ⎣ 0.2 −1.8 0.3 ⎦ , B11 = ⎣ 0.7 ⎦ , B21 = ⎣ 0.3 ⎦ , 0.3 0.6 −1.5 0.7 0.4

[1] [1] [1] G [1] 1 = 0.4, K 1 = 1.2, D11 = 0.4, D21 = −0.3, ⎡ ⎤   0.2 0.1 0.1 C1[1] = 1.2 0.6 1.5 , [1] ⎣ ⎦ F1 = 0.1 0.1 0.0 ,   L [1] 1 = 1.2 −0.8 −1.2 , 0.0 0.2 0.2 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −2.1 0.2 0.4 0.8 −0.3 [1] [1] [1] A2 = ⎣ 0.3 0.6 0.2 ⎦ , B12 = ⎣ 0.7 ⎦ , B22 = ⎣ 0.6 ⎦ , −0.2 0.2 −2.2 0.5 −0.5 [1] [1] [1] G [1] 2 = 0.3, K 2 = 0.3, D12 = 0.5, D22 = 0.4, ⎡ ⎤   0.2 0.1 0.1 C2[1] = 1.3 0.6 1.1 , [1] ⎣ ⎦ F2 = 0.1 0.2 0.1 ,   L [1] 2 = 0.7 0.8 1.2 . 0.1 0.0 0.1

4.5 Illustrative Example

75

Subsystem 2. ⎡

A[2] 1

⎤ ⎡ ⎤ ⎡ ⎤ −1.8 −0.3 −0.5 1.0 0.3 [2] [2] = ⎣ 0.2 −1.8 0.3 ⎦ , B11 = ⎣ 0.7 ⎦ , B21 = ⎣ 0.3 ⎦ , 0.3 0.6 −1.5 0.7 −0.4

[2] [2] [2] G [2] 1 = 0.4, K 1 = 1.2, D11 = 0.3, D21 = −0.3, ⎡ ⎤   0.2 0.0 0.1 C1[2] = 1.1 0.6 1.5 , [2] F1 = ⎣ 0.1 0.3 0.2 ⎦ ,   L [2] 1 = 1.2 −0.8 −1.2 , 0.1 0.1 0.2 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −2.1 0.2 0.4 0.8 −0.3 [2] [2] ⎣ 0.3 0.6 0.2 ⎦ , B12 = ⎣ 0.7 ⎦ , B22 = ⎣ 0.6 ⎦ , A[2] 2 = −0.2 0.2 −2.2 0.5 −0.5 [2] [2] [2] G [2] 2 = 0.3, K 2 = 0.3, D12 = 0.3, D22 = 0.4, ⎡ ⎤   0.2 0.1 0.2 C2[2] = 1.3 0.6 1.1 , F2[2] = ⎣ 0.1 0.2 0.1 ⎦ ,   L [2] 2 = 0.7 0.8 1.2 . 0.1 0.2 0.1

    The nonlinearities f x(t), t and g x(t), t in (4.1) are set as ⎡

⎤ 0.2x1 (t) + 0.1x2 (t) f x(t), t = ⎣ 0.2x1 (t) + 0.3x2 (t) + 0.2x3 (t) ⎦ sin(t), 0.1x1 (t) + 0.1x3 (t)     g x(t), t = 0.1x1 (t) + 0.2x2 (t) + 0.2x3 (t) sin(t), 



which satisfy Assumption 4.1 with ⎡

⎤ 0.2 0.1 0.0 M  ⎣ 0.2 0.3 0.2 ⎦ , 0.1 0.0 0.1

  N  0.1 0.2 0.2 .

Case 1. Firstly, consider the full-order DOFC design problem with r = 3. In view of the sufficient conditions (4.44)–(4.47) in Theorem 4.10, the minimum feasible γ can be obtained as γmin = 0.8904. The corresponding parameters of the full-order DOFC are calculated as

4 L2 –L∞ Output Feedback Control of Fuzzy Switching Systems

76

⎡

A[1] c1

[1] Bc1

A[1] c2

[1] Bc2

A[2] c1

[2] Bc1

A[2] c2

[2] Bc2

⎤ −57.7172 −2.2637 −1.3706 = ⎣ 7.2168 −4.4209 1.7200 ⎦ , 7.5700 −3.0131 −4.2822 ⎡ ⎤ ⎡ ⎤T −5.4582 −1.3376 [1] = ⎣ −6.4293 ⎦ , Cc1 = ⎣ −0.1185 ⎦ , −2.0298 −0.1315 ⎡ ⎤ 0.3481 8.4100 10.5366 = ⎣ 3.0083 −7.8703 0.6900 ⎦ , 9.2645 −0.8920 −2.9582 ⎡ ⎤ ⎡ ⎤T −5.8390 1.4527 [1] = ⎣ −1.2570 ⎦ , Cc2 = ⎣ 0.8489 ⎦ , −7.6154 0.5020 ⎡ ⎤ −83.6564 −5.3496 13.5642 = ⎣ −9.6521 −16.9328 5.8525 ⎦ , −18.8543 −6.5568 −7.4303 ⎡ ⎤ ⎡ ⎤T −21.0503 −1.9514 [2] = ⎣ 0.4475 ⎦ , Cc1 = ⎣ −0.6286 ⎦ , 6.6054 0.3450 ⎡ ⎤ −69.0979 35.8456 −18.1179 = ⎣ −4.6730 −4.4102 −9.2470 ⎦ , −24.6190 −3.7769 −11.4027 ⎡ ⎤ ⎡ ⎤T −20.3258 −1.1406 [2] = ⎣ 0.3074 ⎦ , Cc2 = ⎣ 1.5259 ⎦ . 8.7067 −1.7735

Case 2. Introduce the reduced-order DOFC design issue with r = 2. Considering (4.35)–(4.38) in Theorem 4.7, we have the minimum feasible γ as γmin = 0.9030. Accordingly, the gains of the reduced-order DOFC can be obtained as     T −92.8590 −5.6818 2.1114 −1.5374 [1] [1] = = , , Bc1 , Cc1 7.5781 −9.3000 −7.1962 −0.4716      T −58.6042 34.0300 −7.2397 −0.0400 [1] [1] = , Bc2 = , Cc2 = , 14.2169 0.9743 −8.5207 1.8118      T −66.5137 −10.5625 −9.8210 −2.1134 [2] [2] = , Bc1 = , Cc1 = , 10.0811 −7.7937 −3.3116 −0.5735      T −66.4444 29.6925 −4.3730 −1.3128 [2] [2] = , Bc2 = , Cc2 = . 20.1712 −3.8907 −7.4381 2.4636

A[1] c1 = A[1] c2 A[2] c1 A[2] c2



It can be seen from the results that the performance level γ increases when the dimension of reduced-order controllers decreases, as shown in Table 4.1.

4.5 Illustrative Example

77

Table 4.1 Values of γ vs. reduced-order dimension Reduced-order dimension γ Full-order r = 3 Reduced-order r = 2

0.8904 0.9030

ε 10.9745 10.9119

Fig. 4.2 Structure of closed-loop system over CR networks

The initial condition is selected as x(t) = 0, xc (t) = 0, the simulation time T  = 6s, and the switching signal randomly changes from ‘1’ and ‘2’, where ‘1’ and ‘2’ stand for the first and second subsystems, as shown in Fig. 4.2. The fuzzy basis functions are chosen by      1   1  h 1 x1 (t) = 1 − sin x12 (k) , h 2 x1 (t) = 1 + sin x12 (k) . 2 2 The disturbance input ω(t) belongs to L2 [0, ∞), which makes it energy-bounded (Fig. 4.3). And we choose ω(t) =

5 sin(0.9t) . (0.75t)2 + 3.5

78

4 L2 –L∞ Output Feedback Control of Fuzzy Switching Systems

Fig. 4.3 Switching signal

In Case 1, the simulation results of full-order L2 -L∞ DOFC are exhibited in Figs. 4.4, 4.5, 4.6, 4.7. Figure 4.4 plots the states of the closed-loop system, and the DOFC states are drawn in Fig. 4.5. The control input u(k) and the controlled output z(k) are depicted in Figs. 4.6 and 4.7, respectively. In Case 2, the simulation results of reduced-order L2 –L∞ DOFC are shown in Figs. 4.8, 4.9, 4.10, 4.11. Example 4.13 In this example, the CR system from [157, 184] is investigated to verify the effectiveness of our proposed scheme, which is plotted schematically in Fig. 4.2. First, suppose that every channel in CR system owes two states (busy and idle), and the channel sojourn times in each state are independent and identically distributed random variables, which follow certain probability distribution functions, that have probable connections to the states to be switched. Then, the sensor first chooses one channel to sense using the sensing strategy; if the channel is idle, the signal is transmitted through it. If not, it stops transmission to avoid collision. Hence, the CR model is approximated by a group of switched fuzzy systems, which denotes the switch between idle and busy states. Introduce the CR system with following parameters:

4.5 Illustrative Example

Fig. 4.4 The states of the closed-loop system in Case 1

Fig. 4.5 The states of the DOF controller in Case 1

79

80

4 L2 –L∞ Output Feedback Control of Fuzzy Switching Systems

Fig. 4.6 Control input u(t) in Case 1

Fig. 4.7 Controlled output z(t) in Case 1

4.5 Illustrative Example

Fig. 4.8 The states of the closed-loop system in Case 2

Fig. 4.9 The states of the DOF controller in Case 2

81

82

4 L2 –L∞ Output Feedback Control of Fuzzy Switching Systems

Fig. 4.10 Control input u(t) in Case 2

Fig. 4.11 Controlled output z(t) in Case 2

4.5 Illustrative Example

83

Subsystem 1.      [1] G [1] = 0.4, D11 = 0.4, −1.8 −0.3 1.2 0.5 [1] [1] , B11 = , B21 = , 1[1] [1] 0.3 −1.8 0.5 0.3 K 1 = 1.2, D21 = 0.3,       [1] C1 = 1.2 0.6 , −1.5 1.0 −2.1 0.2 [1]  A = = , [1]  , −1.3 −2.0 0.5 −2.1 L 1 = 1.2 −0.8 , 2     [1] = 0.5, G [1] = 0.2, D12 0.8 −0.3 [1] = , B22 = , 2[1] [1] 0.7 0.6 K 2 = 0.3, D22 = 0.3,     [1] C2 = 1.3 0.6 , −2.2 1.2   = , −2.5 −1.2 L [1] 2 = 0.7 1.2 .

A[1] 1 = F1[1] [1] B12

F2[1]



Subsystem 2.      [2] = 0.3, G [2] = 0.4, D11 −1.8 −0.3 1.0 0.5 [2] [2] = = , B11 , B21 , 1[2] [2] 0.6 −1.5 0.7 0.3 = −0.3, K 1 = 1.2, D21       [2] C1 = 1.1 0.6 , −2.1 1.5 −2.1 0.1   A[2] = , , 2 = 1.2 −2.1 0.2 −2.5 1.2 −1.2 = , L [2] 1     [2] = 0.3, G [2] = 0.3, D12 0.8 −0.3 [2] = , B22 = , 2[2] [2] 0.7 0.6 K 2 = 0.3, D22 = −0.3,     [2] C2 = 1.3 0.6 , −2.1 2.5   = , 2.2 −2.1 L [2] 2 = 0.7 1.2 .

A[2] 1 = F1[2] [2] B12

F2[2]



Matrices M and N under the Assumption 1 are set to 

 0.2 0.1 M , 0.3 0.2

  N  0.1 0.2 .

Then we can calculate the minimum feasible γmin = 0.6655. The initial condition is x(0) = [1.1 − 0.5]T , and other system settings are the same as Example 4.12. The simulation results of full-order DFOC are plotted in Figs. 4.12, 4.13, 4.14, 4.15. Figure 4.12 draws the states x1 (t)–x2 (t) of the closed-loop system, and the controller states xc1 (t)–xc2 (t) are depicted in Fig. 4.13. The control input u(k) and the controlled output z(k) are displayed in Figs. 4.14 and 4.15, respectively. It can be seen that the system is exponentially stable during every run of the simulation.

84

4 L2 –L∞ Output Feedback Control of Fuzzy Switching Systems

Fig. 4.12 The states of the CR system

Fig. 4.13 The states of the DOF controller

4.5 Illustrative Example

Fig. 4.14 Control input u(t)

Fig. 4.15 Controlled output z(t)

85

86

4 L2 –L∞ Output Feedback Control of Fuzzy Switching Systems

4.6 Conclusion In this chapter, the DOFC issue for T-S fuzzy switched systems was investigated. The ADT technique was applied to stabilize the switched systems exponentially with an arbitrary switching rule. Next, we constructed a piecewise Lyapunov function and derived the sufficient conditions to ensure that the corresponding closed-loop system was exponentially stable with a specific L2 –L∞ performance level γ. Moreover, the feasible conditions of the fuzzy-rule-dependent DOFC were derived through linearization, which could be promptly resolved using the standard toolbox. In the end, two numerical simulations were proposed to illustrate advantages of the developed scheme.

Part II

Fuzzy Filtering and Fault Detection

Chapter 5

Dissipative Filtering of Fuzzy Switched Systems

5.1 Introduction The dissipativity-based filtering problem is investigated for T-S fuzzy switched systems with stochastic perturbation in this chapter. First, sufficient conditions are established to ensure the mean-square exponential stability of T-S fuzzy switched systems. Next, we design a filter for the system of interest, subject to Brownian motion. The piecewise Lyapunov function approach and ADT method are combined, and valid fuzzy filters are presented such that the filter error dynamics exhibit mean-square exponential stability and the prescribed dissipative property. Moreover, the solvability conditions of the designed filter are specified through linearization.

5.2 System Description and Preliminaries 5.2.1 System Description In this chapter, we introduce nonlinear switched systems with stochastic perturbation, which are expressed by the T-S fuzzy switched stochastic models as follows: [ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

Rule Ri : IF θ1 (t) is μi1 and θ2 (t) is μi2 and · · · and θ p (t) is μi p , THEN   [ j] [ j] [ j] d x(t) = Ai x(t) + Bi ω(t) dt + E i x(t)d,   [ j] [ j] [ j] dy(t) = Ci x(t) + Di ω(t) dt + Fi x(t)d, [ j]

z(t) = L i x(t), i = 1, 2, . . . , r,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 X. Su et al., Intelligent Control, Filtering and Model Reduction Analysis for Fuzzy-Model-Based Systems, Studies in Systems, Decision and Control 385, https://doi.org/10.1007/978-3-030-81214-0_5

(5.1a) (5.1b) (5.1c)

89

90

5 Dissipative Filtering of Fuzzy Switched Systems

where x(t) ∈ Rn represents the state vector; ω(t) ∈ Rq represents the disturbance input, and ω(t) belongs to L2 [0, ∞); y(t) ∈ R p represents the output; z(t) ∈ Rl denotes the signal to be estimated;  (t) denotes the scalar Brownian motion based  probability space (, F, {Ft }t≥0 , P) satisfying E {d (t)} = 0 and  on the E d 2 (t) = dt; the positive integer N is the number of subsystems; σ j (t) : N j ∈ N = {1, 2, · · · , N }, [0, ∞) → {0, 1}, and j=1 σ j (t) = 1, t ∈ [0, ∞), stands for the switching signal specifying which subsystem is activated at the [ j] [ j] switching moment; r is the number of IF-THEN rules; μi1 , . . . , μi p denote the fuzzy [ j]

j

j

sets; θ1 (t), θ2 (t), . . . , θ p (t) represent the premise variables;

 [ j] [ j] [ j] [ j] [ j] [ j] [ j] : j ∈ N are a group of matrices paramAi , Bi , E i , Ci , Di , Fi , L i [ j] [ j] [ j] [ j] [ j] [ j] [ j] eterized by N = {1, 2, . . . , N }, and Ai , Bi , E i , Ci , Di , Fi , L i are real constant matrices. Suppose that the premise variables don’t depend on the disturbance input ω(t). Given a set of (x(t), ω(t)), the eventual output of the concerned system is given by

d x(t) =

N j=1

+ dy(t) =

σ j (t)

r

[ j] E i x(t)d

N j=1

σ j (t)

[ j] h i=1 i

[ j]

θ (t)



 [ j] [ j] Ai x(t) + Bi ω(t) dt

,

r

[ j]



i=1

(5.2a)    [ j] [ j] [ j] [ j] h i θ (t) Ci x(t) + Di ω(t) dt

+ Fi x(t)d , z(t) =

N j=1

σ j (t)

r i=1

(5.2b) [ j]

hi





[ j]

θ [ j] (t) L i x(t),

(5.2c)

where  [ j] p     νi θ [ j] (t) [ j] [ j] [ j] [ j] [ j] [ j]  , νi θ (t) = h i θ (t) =  μil θl (t) , [ j] [ j] r θ (t) l=1 i=1 νi  [ j] [ j] [ j] [ j] and μ(il) θl (t) stands for the grade of membership of θl (t) in μil . Assume    r  [ j] [ j] [ j] νi θ [ j] (t)  0, i = 1, 2, . . . , r, νi θ [ j] (t) > 0 for all t. Hence, h i θ [ j] (t) i=1  r  [ j] h i θ [ j] (t) = 1 for all t.  0 for i = 1, 2, . . . , r and i=1

It is assumed the premise of the fuzzy system θ [ j] (t) is available for feed variable [ j] back, which means h i θ (t) is available for feedback. Provided that the premise variable of the fuzzy filter is identical to the system. On the basis of PDC tech-

5.2 System Description and Preliminaries

91

nique, the fuzzy-dependent filter is proposed to share the same IF-THEN sections. Subsequently, we propose a fuzzy filter described by [ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

Rule Ri : IF θ1 (t) is μi1 and θ2 (t) is μi2 and · · · and θ p (t) is μi p , THEN [ j]

[ j]

d xc (t) = Aci xc (t)dt + Bci dy(t), [ j]

z c (t) = L ci x f (t), i = 1, 2, · · · , r,

(5.3a) (5.3b)

where xc (t) ∈ Rk denotes the state vector of the fuzzy filter with k  n; z c (t) ∈ Rl [ j] [ j] [ j] denotes the estimation of z(t); Aci , Bci , and L ci are fuzzy filter parameters to be designed. The fuzzy filter (5.3) can be further expressed as d xc (t) = z c (t) =

N

σ j (t)

j=1

N

σ j (t)

r

  [ j] [ j] [ j] h i (θ [ j] (t)) Aci x f (t)dt + Bci dy(t) ,

i=1

r

j=1

[ j] [ j] h (θ [ j] (t))L ci xc (t). i=1 i

(5.4a) (5.4b)

T  Denoting ec (t)  z(t) − z c (t), ξ(t)  x T (t) xcT (t) , and augmenting the system (5.2) to contain the information in (5.4), thus the overall fuzzy filtering error dynamic is obtained as   r N r [ j] [ j] dξ(t) = σ j (t) h i θ [ j] (t) h l θ [ j] (t) j=1 i=1 l=1 

 [ j] [ j] [ j] A˜ il ξ(t) + B˜ il ω(t) dt + E˜ il ξ(t)d (t) , (5.5a) ec (t) =

N 

σ j (t)

j=1

r  i=1

[ j]

hi

r    [ j] [ j] θ [ j] (t) h l θ [ j] (t) L˜ il ξ(t),

(5.5b)

l=1

where 

   [ j] [ j] Ai B 0 [ j] i ˜  [ j] [ j] [ j] , Bil  [ j] [ j] , Bcl Di Bcl Ci Acl   [ j]   Ei 0 [ j] [ j] ˜ E il  , L˜ il  L i[ j] −L [clj] . [ j] [ j] Bcl Fi 0 [ j] A˜ il

Remark 5.1 Actually, there are two methods in the fault detection filter design. One is fuzzy-parameter-independent method, which is fit for the premise variables of the fuzzy model are unavailable. And the other one is fuzzy-parameter-dependent method, which is applied in the case that is available. We solve the fault detection filtering problem with fuzzy-parameter-dependent method in this chapter. Owing to the use of the fuzzy-parameter-dependent approach, information of premise variables are fully taken into account, thus the obtained results are less conservative.

92

5 Dissipative Filtering of Fuzzy Switched Systems

Definition 5.2 For given T2 > T1  0, Nσ j (T1 , T2 ) represents the number of switchings of σ j (t) over (T1 , T2 ). If Nσ j (T1 , T2 )  N0 + (T2 − T1 )/Ta holds for Ta > 0, N0  0. Then Ta is named as the ADT. Definition 5.3 When ω(t) = 0, the equilibrium ξ (t) = 0 of fault detection system in (5.5) is said to be mean-square exponentially stable with σ j (t) if ξ(t) satisfies the following condition:   E ξ(t)2  η ξ(t0 )2 e−λ(t−t0 ) , ∀t  t0 , where η  1 and λ > 0.

5.2.2 Dissipativity Definition Consider [99] and the T-S fuzzy switched system in (5.5), S(ω(t), ec (t)) is a real valued function which denotes the following supply rate. Definition 5.4 (Supply Rate) The supply rate is a real valued function, S(ω(t), ec (t)) :  × Z → R, which is locally Lebesgue integrable independently of the initial condition and the input, that is for any ω(t) ∈ , ec (t) ∈ Z , t  0, it satisfies  t that 0 |S(ω(t), ec (t))|dt < +∞. Definition 5.5 (Dissipative System) The T-S fuzzy switched system (5.5) under supply rate S(ω(t), ec (t)) is known as dissipative if there is a nonnegative function V (x(t)) : X → R, namely the storage function, such that the following inequality satisfies:  T

  S(ω(t), ec (t))dt , (5.6) E V (x(t )) − V (x(0))  E 0

for x0 ∈ X , input ω(t) ∈  and t ≥ 0 (or, in other words as differently: for admissible inputs ω(t) that make the state from x(0) to x(t ) during [0, t ], where x(t ) denotes the state variable when t = t ). Definition 5.6 Given matrices Z ∈ Rl×l , X ∈ Rq×q , Y ∈ Rl×q with Z and X being symmetric, the T-S fuzzy switched system (5.5) is (Z , Y , X )-dissipative if there exists the real function ψ(·) with ψ(0) = 0, and ∀t  0, 

t 

E 0

ec (t) ω(t)

T 

Z Y X



  ec (t) dt +ψ(x0 )  0. ω(t)

Moreover, if the following condition holds for δ > 0,

(5.7)

5.2 System Description and Preliminaries



t

E 0



ec (t) ω(t)

93

   Z Y ec (t) dt + ψ(x0 ) ω(t) X  t ω T (t)ω(t)dt, ∀t  0, δ

T 

(5.8)

0

the system (5.5) is called as strictly (Z , Y , X )-δ-dissipative. 1

1

Remark 5.7 Assume that Z  0, it follow that −Z = (Z−2 )2 , for some Z−2  0. Hence, the index in Definition 5.6 contains some particular cases, such as H∞ performance, the positive real performance, and the sector bounded performance.

5.3 Main Results 5.3.1 Dissipativity Performance Analysis Theorem 5.8 Given matrices 0  Z ∈ Rl×l , X ∈ Rq×q , Y ∈ Rl×q with symmetric Z and X , and scalars δ > 0, α > 0, if there exist 0 < P [ j] ∈ R(n+k)×(n+k) for j ∈ N, [ j]

[ j] Πil

+

Πii < 0, i = 1, 2, . . . , r,

(5.9a)

[ j] Πli

(5.9b)

< 0, 1  i < l  r,

where ⎡

[ j]

[ j]

Π Π12il ⎢ 11il ⎢ δI − X [ j] Πil  ⎢ ⎣



 T 1 [ j] P [ j] L˜ il Z−2 0 0 0 −P [ j] −I

[ j] E˜ il

T

⎤ ⎥ ⎥ ⎥, ⎦

(5.10)

⎧ T  ⎪ ⎨ Π [ j]  P [ j] A˜ [ j] + A˜ [ j] P [ j] + α P [ j] , 11il il il T  with [ j] [ j] [ j] ⎪ [ j] ˜ ˜ ⎩ Π12il  P Bil − L il Y T. Then the concerned system in (5.5) is mean-square exponentially stable subject to the (Z , Y , X )-δ-dissipativity for any switching signal with the ADT satisfying ln μ Ta > Ta∗ = , where μ  1 and P [ j]  μP [s] , ∀ j, s ∈ N . α Proof On account of the switching signal and fuzzy membership functions, it follows from (5.9a)–(5.9b) that

94

5 Dissipative Filtering of Fuzzy Switched Systems

N j=1

⎡

  r [ j] [ j] h i θ [ j] (t) h l θ [ j] (t) i=1 l=1  T  T 1 ⎤ [ j] [ j] [ j] E˜ il Π12il P [ j] L˜ il Z−2 ⎥ ⎥ δI − X 0 0 ⎥ < 0. [ j] ⎦ 0 −P −I

σ j (t) [ j]

Π ⎢ 11il ⎢ ⎢ ⎣

r

(5.11)

The Lyapunov functional is selected as V (ξt , σ j )  ξ T (t)

 N j=1

σ j (t)P [ j] ξ(t),

(5.12)

where P [ j] > 0, j ∈ N are to be decided. Along the system in (5.5) for a fixed σ j (t), by using the Itô formula, we can get the stochastic differential as follows:  N r [ j] d V (ξt , σ j ) =L V (ξt , σ j )dt + 2 σ j (t) h i θ [ j] (t) j=1 i=1  r [ j] [ j] T [ j] ˜ [ j] h θ (t) ξ (t)P E il ξ(t)d (t), (5.13a) l=1 l   r N r [ j] [ j] σ j (t) h θ [ j] (t) h θ [ j] (t) L V (ξt , σ j ) = j=1 i=1 i l=1 l   T  [ j] [ j] P [ j] ξ(t) + ξ T (t) ξ T (t) P [ j] A˜ il + A˜ il 

[ j] E˜ il

T P

[ j]

[ j] T [ j] ˜ [ j] ˜ E il ξ(t) + 2ξ (t)P Bil ω(t) .

(5.13b)

Consider (5.13b) and for any nonzero ω(t) ∈ L2 [0, ∞), we have Π (ξt , σ j ) L V (ξt , σ j ) + αV (ξt , t) − e Tf (t)Z e f (t) − 2ω T (t)Y e f (t) − ω T (t) [X − δ I ] ω(t) N r r      [ j] [ j]  σ j (t) h i θ [ j] (t) h l θ [ j] (t) j=1



ξ(t) ω(t)

i=1

T 

[ j] Π˜ 11il



[ j] Π˜ 12il [ j] Π˜ 22il



l=1

 ξ(t) , ω(t)

(5.14)

where ⎧ T T   [ j] ⎪ [ j] ˜ [ j] ˜ il[ j] P [ j] + E˜ il[ j] P [ j] E˜ il[ j] ˜ 11il ⎪ Π  P + A A ⎪ il ⎪ ⎨ T  [ j] [ j] +α P [ j] − L˜ il Z L˜ il , ⎪ ⎪ T  ⎪ ⎪ ⎩ Π˜ [ j]  P [ j] B˜ [ j] − L˜ [ j] Y T , Π˜ [ j]  δ I − X . 12il il il 22il

(5.15)

5.3 Main Results

95

Based on (5.11) and employing Schur complement method, it yields Π (ξt , t) < 0, that is L V (ξt , σ j ) < −αV (ξt , σ j ) + Ψ (t), with Ψ (t)  e Tf (t)Z e f (t) + 2ω T (t)Y e f (t) + ω T (t) [X − δ I ] ω(t). Therefore, d V (ξt , σ j ) < − αV (ξt , σ j )dt + 2

N  j=1

r 

[ j]

hl



σ j (t)

r 

[ j]

hi

 θ [ j] (t)

i=1

[ j] θ [ j] (t) ξ T (t)P [ j] E˜ il ξ(t)d (t) + Ψ (t)dt.

l=1

It can be observed that d[eαt V (ξt , σ j )] =αeαt V (ξt , σ j )dt + eαt d V (ξt , σ j )   N r r     [ j] [ j] 0 to t and make expectations. Then it is not difficult to obtain  t





E V (ξt , σ j ) < e−α(t−tk ) E V (ξtk , σ j ) + E e−α(t−s) Ψ (s)ds . tk

Considering P [ j]  μP [s] and (5.12), at switching instant tk , it has 

  E V (ξtk , σ j )  μE V (ξtk− , σ j ) .

(5.17)

Consequently, based on (5.13)–(5.17) and ϑ = Nσ j (0, t)  (t − 0)/Ta ,  t



E V (ξt , σ j )  E e−α(t−s)+Nσ j (s,t) ln μ Ψ (s)ds 0  −αt+Nσ j (0,t) ln μ V ξ0 , σ j . +e Under the zero initial condition ξ(0) = 0, (5.18) signifies

(5.18)

96

5 Dissipative Filtering of Fuzzy Switched Systems







E V (ξt , σ j )  E

t

e

−α(t−s)+Nσ j (s,t) ln μ

Ψ (s)ds .

(5.19)

0

Next, multiplying two sides of (5.18) by e−Nσ j (0,t) ln μ , we can get  t



E e−Nσ j (0,t) ln μ V (ξt , σ j )  E Ψ (s)ds .

(5.20)

0

Due to Nσ j (0, t) ≤ t/Ta and Ta > Ta∗ = ln μ/α, we have Nσ j (0, t) ln μ ≤ αt. Therefore,  t



E e−αt V (ξt , σ j )  E Ψ (s)ds .

(5.21)

0

It is easily seen that for arbitrary t  0, 

t

E



Ψ (s)ds  E e−αt V (ξt , σ j )  0,

0

Then the proof is complete. On the basis of Theorem 5.8, the following result on the estimation of the state decay can be obtained. Theorem 5.9 For α > 0, assume there are matrices 0 < P [ j] ∈ R(n+k)×(n+k) such that for j ∈ N , T T   [ j] [ j] [ j] [ j] P [ j] + α P [ j] + E˜ ii P [ j] E˜ ii < 0, P [ j] A˜ ii + A˜ ii i = 1, 2, . . . , r, T  [ j] [ j] + P [ j] + E˜ il P [ j] E˜ il 2α P [ j] + T T   [ j] [ j] [ j] [ j] +P [ j] A˜ li + A˜ li P [ j] + E˜ li P [ j] E˜ li < 0, [ j] P [ j] A˜ il



[ j] A˜ il

T

1  i < l  r.

(5.22a)

(5.22b)

Then the dynamic system in (5.5a) is mean-square exponentially stable under any ln μ , where μ  1 and P [ j]  switching signal with ADT which satisfies Ta > Ta∗ = α μP [s] , ∀ j, s ∈ N . In addition, the state decay estimate is expressed as   E ξ(t)2  ηe−λt ξ(0)2 , where

(5.23)

5.3 Main Results

97

⎧ # $ ⎨ a = min λmin P [ j] , λ = α − lnTaμ > 0, ∀ j∈I # $ ⎩ b = max λmax P [ j] , η = ab  1.

(5.24)

∀ j∈I

Proof Consider a piecewise Lyapunov function as (5.12). Based on (5.22), we have   r r [ j] [ j] σ j (t) h i θ [ j] (t) h l θ [ j] (t) j=1 i=1 l=1   T T   [ j] [ j] [ j] [ j] P [ j] A˜ ii + A˜ ii P [ j] + α P [ j] + E˜ ii P [ j] E˜ ii < 0. N

(5.25)

For ω(t) = 0 and a fixed σ j (t), it yields from (5.5a) that L V (ξt , σ j ) < −αξ T (t)P [ j] ξ(t) = −αV (ξt , σ j ).

(5.26)

Hence, d V (ξt , σ j ) 0, δ > 0 , and matrices 0  Z ∈ Rl×l , X ∈ Rq×q , Y ∈ Rl×q with symmetric Z and X , assume there are matrices 0 < U [ j] ∈ [ j] [ j] [ j] Rn×n , 0 < V [ j] ∈ Rk×k , Aci ∈ Rk×k , Bci ∈ Rk× p , and Lci ∈ Rl×k such that for j ∈ N , i = 1, 2, . . . , r , l = 1, 2, . . . , r , [ j]

Φii < 0, i = 1, 2, . . . , r, [ j] [ j] Φil + Φli < 0,  [ j] [ j] 

U

1  i < l  r,

V > 0, V [ j]

⎢ ⎢ ⎢ ⎢ [ j] Φil  ⎢ ⎢ ⎢ ⎣

[ j]

[ j]

Φ11il Φ12il

[ j]

Φ22il

⎤ T  T [ j] [ j] Φ15il Φ16il ⎥ [ j] [ j] Φ23il 0 0 Φ26il ⎥ ⎥ δI − X 0 0 0 ⎥ ⎥, −U [ j] −V [ j] 0 ⎥ ⎥ −V [ j] 0 ⎦ −I [ j]

Φ13il



(5.33b) (5.33c)

where ⎡

(5.33a)

[ j]

Φ14il

with T #  $T [ j] [ j] [ j] [ j] [ j] Φ11il U [ j] Ai + Bcl Ci + Ai U [ j] + αU [ j] T  T T   1 [ j] [ j] [ j] [ j] Bcl + Ci , Φ16il  L i Z−2 , T T  T   [ j] [ j] [ j] [ j] [ j] Bcl V [ j] + Ci + αV [ j] , Φ12il Acl + Ai

5.3 Main Results

99

T T   1 [ j] [ j] [ j] [ j] [ j] Φ22l Acl + Acl + αV [ j] , Φ26il  − Lcl Z−2 , T  [ j] [ j] [ j] [ j] [ j] , Φ13il U [ j] Bi + Bcl Di − Y L i T  [ j] [ j] [ j] [ j] [ j] , Φ23il V [ j] Bi + B f l Di + Y Lcl [ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

Φ14il U [ j] E i + Bcl Fi , Φ15il  V [ j] E i + Bcl Fi . In that way, there exists a fuzzy-parameter-dependent filter as (5.3) such that the resulting dynamic system in (5.5) is mean-square exponentially stable subject to a (Z , Y , X )-δ-dissipative performance under any switching signal with ADT which satisfies Ta > Ta∗ = lnαμ . Furthermore, if above-mentioned conditions have feasible

 [ j] [ j] [ j] solutions U [ j] , V [ j] , Aci , Bci , Lci , then the parameters of our designed fuzzy filter (5.3) can be reformulated as −1 −1   [ j] [ j] [ j] [ j] [ j] [ j] Aci = V [ j] Aci , Bci = V [ j] Bci , L ci = Lci .

(5.34)

Proof Partition the matrix P [ j] as  P [ j]  [ j]

[ j]

[ j]

P1 P2 [ j] P3

[ j]



 > 0,

[ j] P2



[ j]



P4

0(n−k)×k

,

(5.35)

[ j]

where P1 ∈ Rn × Rn , P2 ∈ Rn × Rk , and P3 ∈ Rk × Rk . Without loss of gen[ j] erality, P4 is assumed to be nonsingular. The following nonsingular matrices are introduced:   ⎧ I 0 ⎪ ⎪   [ j] −1 T ⎨F  , [ j] [ j] P2 0 P3   ⎪ ⎪ ⎩ U [ j]  P [ j] , V [ j]  P [ j] P [ j] −1 P [ j] T , 1 2 3 2

(5.36)

and ⎧  −1  T ⎪ [ j] ⎨ A[ j]  P [ j] A[ j] P [ j] P2 , 2 3 ci ci  −1  T [ j] [ j] [ j] [ j] ⎪ ⎩ L[cij]  L [cij] P3[ j] P2 , Bci  P2 Bci .

(5.37)

Performing a congruence transformation to (5.9) with diag{F [ j] , I, F [ j] , I }, we have

100

5 Dissipative Filtering of Fuzzy Switched Systems



T   [ j] [ j] [ j] diag F , I, F , I Πii diag F [ j] , I, F [ j] , I < 0,

(5.38a)



 T   [ j] [ j] diag F [ j] , I, F [ j] , I Πil + Πli diag F [ j] , I, F [ j] , I < 0.

(5.38b)

In consideration of (5.36)–(5.37), we can get

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩



 [ j] [ j] [ j] U [ j] E i + Bcl Fi 0 , [ j] [ j] [ j] V [ j] E i + Bcl Fi 0   [ j] [ j] Y L i −Y Lcl ,   [ j] [ j] [ j] [ j] U [ j] Ai + Bcl Ci Acl [ j] [ j] [ j] [ j] , V [ j] Ai + Bcl Ci Acl   [ j] [ j] [ j] [ j] # [ j] $T [ j] [ j] U B + B D i cl i P B˜ il = , F [ j] [ j] [ j] V [ j] Bi + Bcl Di  [ j] [ j]  # [ j] $T [ j] [ j] U V P F = , F V [ j]    1 T  1 T  1 T [ j] [ j] [ j] 2 [ j] 2 2 ˜ Z− L il F = Z− L i − Z− Lcl .

⎧ # [ j] $T [ j] [ j] [ j] ⎪ ⎪ ⎪ F P E˜ il F = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [ j] ⎪ ⎪ Y L˜ il F [ j] = ⎪ ⎪ ⎪ ⎪ ⎪ # [ j] $T [ j] [ j] [ j] ⎪ ⎪ ⎪ P A˜ il F = ⎪ ⎨ F

(5.39)

Considering (5.39), it yields from (5.38) that (5.33). Besides, (5.37) is equal to ⎧ −1   −T −1  −T ⎪ [ j] [ j] [ j] [ j] [ j] [ j] ⎪ V [ j] P3 Aci P2 P3 , ⎨ Aci = P2 −1   −T −1 ⎪ [ j] [ j] [ j] [ j] ⎪ ⎩ Bci[ j] = P2[ j] V [ j] P3 Bci , L ci = Lci . [ j]

[ j]

(5.40)

[ j]

Notice that the filter parameters Aci , Bci and L ci in (5.3) can be described by (5.40),  −T [ j] [ j] which means P2 P3 can be considered as a similarity transformation on the state-space realization of the filter, which makes no influence in the filter mapping −T  [ j] [ j] from y to e f . Without loss of generality, set P2 P3 = I , then (5.34). As a result, the dissipativity-based fuzzy filter in (5.3) can be obtained with (5.34). This completes the proof then. Remark 5.11 It can be observed that the obtained conditions in Theorem 5.10 are all in the form of LMIs, and the dissipative fuzzy filter parameters of nonlinear switched stochastic systems can be calculated via resolving the convex optimization issue.

5.4 Illustrative Example

101

5.4 Illustrative Example Consider the nonlinear switched stochastic model with N = 2, which can be approximated by the T-S fuzzy model in (5.1). And the model parameters are set as below: Subsystem 1. ⎡

⎤ ⎡ ⎤ −2.4 −0.1 0.2 0.4 ⎣ 0.2 −1.5 0.2 ⎦ , B1[1] = ⎣ 0.3 ⎦ , D1[1] = 0.3, A[1] 1 = −0.3 0.0 −1.8 0.6 ⎡ ⎤ ⎡ ⎤ −2.1 0.2 −0.1 0.7 ⎣ 0.4 −1.6 −0.2 ⎦ , B2[1] = ⎣ 0.8 ⎦ , D2[1] = 0.1, A[1] 2 = 0.3 0.2 −2.8 0.4 ⎡ ⎤ ⎡ ⎤ 1.2 0.1 0.1 1.1 0.2 0.1 E 1[1] = ⎣ 0.1 1.3 0.2 ⎦ , E 2[1] = ⎣ 0.0 1.2 0.2 ⎦ , 0.1 0.1 1.1 0.3 0.1 1.3     [1] [1] F1 = 0.2 0.1 0.3 , F2 = 0.3 0.2 0.1 ,     C1[1] = 1.0 0.6 1.5 , C2[1] = 1.0 1.0 2.0 ,     [1] L [1] 1 = 1.0 0.5 1.3 , L 2 = 0.8 1.5 1.0 . Subsystem 2. ⎡

⎤ ⎡ ⎤ −1.9 −0.3 0.2 0.3 ⎣ 0.4 −2.1 0.2 ⎦ , B1[2] = ⎣ 0.4 ⎦ , D1[2] = 0.2, A[2] 1 = −0.2 0.0 −1.6 0.5 ⎡ ⎤ ⎡ ⎤ −1.9 0.3 −0.2 0.6 ⎣ 0.3 −1.7 −0.2 ⎦ , B2[2] = ⎣ 0.7 ⎦ , D2[2] = 0.3, A[2] 2 = 0.2 0.1 −2.3 0.3 ⎡ ⎤ ⎡ ⎤ 1.2 0.2 0.3 1.1 0.1 0.2 E 1[2] = ⎣ 0.2 1.2 0.2 ⎦ , E 2[2] = ⎣ 0.0 1.2 0.2 ⎦ , 0.1 0.2 1.3 0.2 0.3 1.3     [2] [2] F1 = 0.5 0.2 0.4 , F2 = 0.2 0.3 0.2 ,     C1[2] = 1.1 0.7 1.2 , C2[2] = 0.9 0.8 2.2 ,     [2] L [2] 1 = 1.1 0.4 1.2 , L 2 = 0.8 1.4 1.2 . The dissipative performance parameters in Definition 5.6 are chosen as Z = −0.9, Y = 0.3 and X = 0, 7. By solving the conditions in Theorem 5.10, the fuzzy filter matrices can be computed as

102

5 Dissipative Filtering of Fuzzy Switched Systems

⎡ ⎤ −4.4135 −1.1553 −2.4555 ⎣ ⎦ [1] A[1] c1 = −5.1192 −4.6978 −7.6770 , Bc1 −2.0791 −1.1566 −5.1215   [1] Cc1 = −0.9768 −0.5022 −1.2007 , ⎡ ⎤ −5.0918 −1.3716 −3.8062 [1] ⎣ 0.0109 −6.0679 −6.7309⎦ , Bc2 A[1] c2 = −1.6008 −1.6673 −7.9648   [1] Cc2 = −0.5810 −1.4839 −1.0604 , ⎡ ⎤ −3.6636 −1.0860 −1.6620 ⎣ ⎦ [2] A[2] c1 = −1.0052 −3.6918 −1.4547 , Bc1 −1.7457 −0.9609 −3.7415   [2] Cc1 = −1.1016 −0.3927 −1.1612 , ⎡ ⎤ −4.3988 −1.2216 −3.3446 ⎣ ⎦ [2] A[2] c2 = −1.2884 −3.6354 −5.0169 , Bc2 −1.2536 −1.7072 −8.4814   [2] Cc2 = −0.7637 −1.3552 −1.2269 .

⎡ ⎤ −1.8014 = ⎣−5.1877⎦ , −2.0073 ⎡ ⎤ −2.0518 = ⎣−3.1718⎦ , −2.1770 ⎡ ⎤ −1.4592 = ⎣−1.5834⎦ , −1.5698 ⎡ ⎤ −1.8432 = ⎣−2.1572⎦ , −2.4045

Figure 5.1 shows the switching signal, which is generated randomly, where ‘1’ and ‘2’ denote the first and the second subsystem, respectively. The fuzzy basis functions

Fig. 5.1 Switching signal σ j (t), j ∈ N = {1, 2}

5.4 Illustrative Example

103

Fig. 5.2 Signal z(t) and its estimation z c (t)

are given by   (x1 (t) − ϑ)2 , h 2 (x1 (t))  1 − h 1 (x1 (t)). h 1 (x1 (t))  exp − 2σ 2 When x(t) = 0 (x f (t) = 0), that is the zero initial condition, and the disturbance input ω(t) is set to ω(t) = sin(0.1t) . The simulation results of the dissipative fuzzy 4t 2 +5 filter design scheme are exhibited in Figs. 5.2, 5.3. The signal z(t) (solid line) and its estimation z c (t) (dash-dot line) are plotted in Fig. 5.2. Figure 5.3 draws the corresponding estimation error ec (t).

5.5 Conclusion The dissipativity-based filtering issue was resolved for T-S fuzzy switched systems with stochastic perturbations. Using the ADT approach and piecewise Lyapunov functions, sufficient conditions were established to ensure the mean-square exponential stability with dissipative performance for the corresponding filtering error dynamics. Furthermore, the solvable conditions for the dissipativity-based fuzzy filter were derived through linearization. In the end, a simulation example was presented to demonstrate the validity of the proposed scheme.

104

Fig. 5.3 Estimation error ec (t)

5 Dissipative Filtering of Fuzzy Switched Systems

Chapter 6

Fault Detection for Switched Stochastic Systems

6.1 Introduction In this chapter, the problem of fault detection filtering is settled for the nonlinear switched stochastic system in the form of T-S fuzzy model. The objective is to establish a robust fault detection approach for a nonlinear switched system with Brownian motion. Based on the observer-based fuzzy filter as a residual generator, the fault detection problem can be reformulated as a fuzzy filtering problem. By using the piecewise Lyapunov functions and ADT approach, a fuzzy-rule-dependent fault detection filter is established to ensure that the overall dynamic system is meansquare exponential stable and exhibits weighted H∞ performance. Moreover, the solvable conditions of the fuzzy filter are established through linearization.

6.2 System Description and Preliminaries The nonlinear switched stochastic systems are considered as  σ j (t) F j (x(t), u(t), ω(t), f (t)) dt j=1  + G j (x(t), u(t), ω(t), f (t)) d(t) ,  N σ j (t) H j (x(t), u(t), ω(t), f (t)) dt dy(t) = j=1  + J j (x(t), u(t), ω(t), f (t)) d(t) ,

d x(t) =

N

(6.1a)

(6.1b)

where x(t) ∈ Rn denotes the state vector; u(t) ∈ Rm denotes the known input; ω(t) ∈ Rq is the disturbance input; f (t) ∈ Rl represents the fault to be detected; u(t), ω(t) and f (t) belong to L2 [0, ∞); y(t) ∈ R p denotes the system output; (t) represents © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 X. Su et al., Intelligent Control, Filtering and Model Reduction Analysis for Fuzzy-Model-Based Systems, Studies in Systems, Decision and Control 385, https://doi.org/10.1007/978-3-030-81214-0_6

105

106

6 Fault Detection for Switched Stochastic Systems

the Brownian motion defined on the probability space (Ω, F, {Ft }t≥0 , P), which  satisfies E {d(t)} = 0 and E d2 (t) = dt; the positive integer N stands for the  number of subsystems; σ j (t) : [0, ∞) → {0, 1}, and N j=1 σ j (t) = 1, t ∈ [0, ∞), j ∈ N = {1, 2, · · · , N }, represents the switching signal specifying which subsystem is activated at the switching instant; F j (·), G j (·), H j (·) and J j (·) are nonlinear regular functions. For given t, σ j (t) is denoted as σ j for simplicity, which may depend on t or x(t), or both, or other hybrid schemes. Suppose σ j is unknown, while its momentary value is obtainable. Based on the switching signal σ j , the switching sequence is given by {( j0 , t0 ), ( j1 , t1 ), . . . , ( jk , tk ), . . . , | jk ∈ N , k = 0, 1, . . .} with t0 = 0, which implies the jk th subsystem is activated during t ∈ tk , tk+1 ). The nonlinear switched system is approximated by a T-S fuzzy model, which is utilized to deal with the fault detection filter design issue for the nonlinear interconnected system. [ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

Rule Ri : IF θ1 (t) is μi1 and θ2 (t) is μi2 and · · · and θ p (t) is μi p , THEN   [ j] [ j] [ j] [ j] d x(t) = Ai x(t)+ B0i u(t)+ Bi ω(t)+ B1i f (t) dt + E i x(t)d(t),   [ j] [ j] [ j] [ j] [ j] dy(t) = Ci x(t)+ D0i u(t)+ Di ω(t)+ D1i f (t) dt + Fi x(t)d(t), [ j]

where i = 1, 2, . . . , r , and r denotes the number of IF-THEN fuzzy rules; μi1 , . . . , [ j] j [ j] j μi p represent fuzzy sets; θ1 (t), θ2 (t), . . . , θ p (t) are premise variables, denoted by 

j [ j] [ j] [ j] [ j] [ j] [ j] [ j] [ j] [ j] [ j] : j ∈ N is a class of θ p ; Ai , Bi , B0i , B1i , E i , Ci , Di , D0i , D1i , Fi [ j]

[ j]

[ j]

[ j]

[ j]

[ j]

matrices parameterized by N = {1, 2, . . . , N }, and Ai , B0i , Bi , B1i , E i , Ci , [ j] [ j] [ j] [ j] D0i , Di , D1i and Fi represent some constant real matrices. Provided that the premise variables don’t depend on the input u(t). For a group of (x(t), u(t)), the resulting output of the fuzzy switched systems is expressed as

  [ j] [ j] θ[ j] Ai x(t) + B0i u(t) σj d x(t) = j=1   [ j] [ j] + Bi ω(t) + B1i f (t) dt + E i x(t)d(t) ,

  r N [ j] [ j] [ j] Ci x(t) + D0i u(t) σj h θ[ j] dy(t) = j=1 i=1 i   [ j] [ j] [ j] +Di ω(t) + D1i f (t) dt + Fi x(t)d(t) , N

r

[ j] h i=1 i

(6.2a)

(6.2b)

     [ j]    p [ j]  [ j]  [ j]  [ j] [ j] where h i θ[ j] = νi θ[ j] / ri=1 νi θ[ j] , νi θ[ j] = l=1 μil θl , with

  [ j] [ j] [ j] [ j] [ j]  be the grade of membership of θl in μil . If νi θ[ j] ≥ 0, i = 1, 2, μil θl

6.2 System Description and Preliminaries

107

   [ j]  [ j]  . . . , r, ri=1 νi θ[ j] > 0 for all t, then h i θ[ j] ≥ 0 for i = 1, 2, . . . , r and r [ j]  [ j]  θ = 1 for all t. i=1 h i In this note, for the plant (6.2), a fault detection filter is proposed as a residual generator. Assumed that the premise variable of the fault detection filter is the same as the one of the plant. By using the PDC technique, the fuzzy-rule-dependent filter is designed to use the same IF-THEN sections as the following form: [ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

Rule Ri : IF θ1 (t) is μi1 and θ2 (t) is μi2 and · · · and θ p (t) is μi p , THEN [ j]

[ j]

d x f (t) = A f i x f (t)dt + B f i dy(t), χ f (t) =

[ j] C f i x f (t),

i = 1, 2, · · · , r,

(6.3a) (6.3b)

where x f (t) ∈ Rn stands for the state of the filter; χ f (t) ∈ Rl denotes the residual [ j] [ j] [ j] signal; A f , B f and C f are filter matrices to be determined. The filter (6.3) can be reformulated as   N r [ j] [ j] [ j] d x f (t) = σj h i (θ[ j] ) A f i x f (t)dt + B f i dy(t) , (6.4a) j=1 i=1 r N [ j] [ j] σj h i (θ[ j] )C f i x f (t). (6.4b) χ f (t) = j=1

i=1

To improve the performance in fault detection, a weighting function is combined with the fault f (s), that is to say, f w (s) = W (s) f (s), where f (s) and f w (s) represent the Laplace transformations of f (t) and f w (t), respectively. A state space realization of f w (s) = W (s) f (s) is x˙w (t) = Aw xw (t) + Bw f (t),

(6.5a)

f w (t) = Cw xw (t),

(6.5b)

where xw (t) ∈ Rk stands for the state vector with xw (0) = 0, and Aw , Bw , Cw are constant matrices. Setting e f (t)  χ f (t) − f w (t) and expanding the system (6.2) to contain the information in (6.4) and (6.5), then the resulting fault detection system is reformulated as

 r

 r N [ j] [ j] σj h i θ[ j] h l θ[ j] dζ(t) = j=1 i=1 l=1

   [ j] [ j] [ j] ˜ ˜ ˜ Ail ζ(t) + Bil υ(t) dt + E il K ζ(t)d(t) , (6.6a) e f (t) = where

N j=1

σj

r i=1

[ j]

hi

 r θ[ j]

l=1

[ j]

hl

 [ j] θ[ j] C˜ f l ζ(t),

(6.6b)

108

6 Fault Detection for Switched Stochastic Systems

⎡

[ j] A˜ il

[ j] B˜ il

[ j] E˜ il

⎤ ⎡ ⎤ [ j] Ai 0 0 x(t)  ⎣ B [f j]l Ci[ j] A[fj]l 0 ⎦ , ζ(t)  ⎣ x f (t) ⎦ , xw (t) 0 0 Aw ⎡ ⎤ ⎡ ⎤T [ j] [ j] [ j] B0i Bi B1i I [ j] [ j] [ j] [ j] [ j]  ⎣ B [f j]l D0i B f l Di B f l D1i ⎦ , K  ⎣ 0 ⎦ , 0 0 0 Bw ⎤T ⎡ ⎡ ⎤ ⎡ [ j] ⎤ 0 Ei u(t) T ⎥ ⎢

[ j]  ⎣ B [f j]l Fi[ j] ⎦ , υ(t)  ⎣ ω(t) ⎦ , C˜ f l  ⎣ C [f j]l ⎦ . f (t) 0 −CwT

Definition 6.1 The equilibrium ζ  (t) = 0 of fault detection system in (6.6) with υ(t)  = 0 ismean-square exponentially stable under σ j (t) if its solution ζ(t) satisfies E ζ(t)2 ≤ η ζ(t0 )2 e−λ(t−t0 ) , ∀t ≥ t0 , for constants η ≥ 1 and λ > 0. Definition 6.2 [125] Given T2 > T1 ≥ 0, let Nσ j (T1 , T2 ) represent the number of switchings of σ j (t) over (T1 , T2 ). If Nσ j (T1 , T2 ) ≤ N0 + (T2 − T1 )/Ta holds for Ta > 0, N0 ≥ 0. And Ta is named as an ADT. Definition 6.3 For a scalar γ > 0, the fault detection system in (6.6) is meansquare exponentially stable subject to a weighted H∞ performance level (γ, α) if it is mean-square exponentially stable when υ(t) ≡ 0. Moreover, for all nonzero υ(t) ∈ L2 [0, ∞) and zero initial condition, the following equality satisfies: ∞ ∞ E 0 e−αt e Tf (t)e f (t)dt < γ 2 0 υ T (t)υ(t)dt. Consequently, the concerned fault detection problem in this chapter can be denoted by two steps: 1. Generate a residual signal: for the system (6.2), propose a H∞ fuzzy-ruledependent filter (6.4) to produce a residual signal. In the mean time, the filter is given to make sure that the overall fault detection system (6.6) is mean-square exponentially stable under a weighted H∞ performance level γ > 0. 2. Establish a fault detection scheme: choose a threshold and an evaluation function. In this chapter, the residual evaluation function J (χ f ) and threshold Jth are given by 

t0 +t 

J (χ f )  t0

Jth 

χTf (t)χ f (t)dt,

sup

0 =ω∈L2 ,0 =u∈L2 , f =0

J (χ f ),

(6.7) (6.8)

where t0 is the initial evaluation instant, t  denotes the evaluation time. Furthermore, the appearance of faults can be detected by making a comparison between J (χ f ) and Jth on account of the following standard:

6.2 System Description and Preliminaries

109

J (χ f ) > Jth

⇒

Faults

Jth

⇒

No Faults.

J (χ f ) ≤

⇒

Alarm,

6.3 Main Results 6.3.1 System Performance Analysis Theorem 6.4 For the given scalars α > 0, γ > 0, suppose there are matrices P [ j] > 0 such that for j ∈ N , [ j]

Πii < 0, i = 1, 2, . . . , r,

 1 1 [ j] [ j] [ j] Πii + Πil + Πli < 0, 1  i < l  r, r −1 2

(6.9a) (6.9b)

where ⎡

[ j]

Πil

T

T [ j] [ j] [ j] [ j] Π11il P [ j] B˜ il K T E˜ il P [ j] C˜ f l ⎢ ⎢ −γ 2 I 0 0 ⎢  ⎣  0  −P [ j]    −I

⎤ ⎥ ⎥ ⎥, ⎦

(6.10)

T

[ j] [ j] [ j] with Π11il  P [ j] A˜ il + A˜ il P [ j] + αP [ j] . The resulting dynamic system in (6.6) is mean-square exponentially stable with a weighted H∞ error performance (γ, α) under any switching signal with Ta > Ta∗ = lnαμ and for μ ≥ 1, P [ j] ≤ μP [s] , ∀ j, s ∈ N .

(6.11)

Furthermore, an estimate of the state decay satisfies   E ζ(t)2 ≤ ηe−λt ζ(0)2 ,

(6.12)

where

 [ j]  , λ = α − ln μ/Ta > 0, b = max∀ j∈I  [ j]λmax P η = b/a ≥ 1, a = min∀ j∈I λmin P .

(6.13)

Proof On the basis of the switching signal and fuzzy basis functions, it yields from (6.9a)–(6.9b) that

110

6 Fault Detection for Switched Stochastic Systems

N ⎡

j=1

σj

[ j]

r

 r

 [ j] θ[ j] h l θ[ j] l=1

T

T ⎤ [ j] [ j] K T E˜ il P [ j] C˜ il ⎥ ⎥ 0 0 ⎥ < 0. [ j] ⎦ 0 −P  −I

[ j]

i=1

hi

[ j]

P [ j] B˜ il Π ⎢ 11il ⎢  −γ 2 I ⎢ ⎣    

(6.14)

Introduce the following Lyapunov functional: V (ζt , t)  ζ T (t)

 N j=1

 σ j P [ j] ζ(t),

(6.15)

where P [ j] > 0, j ∈ N are to be decided. For a fixed σ j (t) and υ(t) = 0, along the trajectory of (6.6) and employing the Itô formula, we have

 N r [ j] d V (ζt , t) = L V (x˜t , t)dt + 2 σj h i θ[ j] j=1 i=1

 r [ j] [ j] T [ j] ˜ [ j] h θ ζ (t)P E il K ζ(t)d(t), l=1 l

 r

 r N [ j] [ j] σj h i θ[ j] h l θ[ j] ζ T (t) L V (ζt , t) = j=1 i=1 l=1

 T

[ j] [ j] [ j] 2P [ j] A˜ il + K E˜ il P [ j] E˜ il K ζ(t).

(6.16a)

(6.16b)

Using the Schur complement approach and considering (6.14) and (6.16b), it yields L V (ζt , t) < −αζ T (t)P [ j] ζ(t) = −αV (ζt , t).

(6.17)

Notice that d[eαt V (ζt , t)] < 2

N  j=1

αt T

σj

r  i=1

e ζ (t)P

[ j]

[ j]

hi

r



 [ j] θ[ j] h l θ[ j] l=1

[ j] E˜ il K ζ T (t)d(t).

(6.18)

Integrate two sides of (6.18) from t  > 0 to t and make expectations. It is not hard to get    E {V (ζt , t)} < e−α(t−t ) E V (ζt  , t  ) .

(6.19)

For any t > 0 and an arbitrary piecewise switching signal σ j (t), denote 0 = t0 < t1 < · · · < tk < · · · (k = 0, 1, . . .) and the switching points of σ j (t) over the  interval (0, t). As we mentioned before, the jk th subsystem is activated during t ∈ tk , tk+1 ). Denoting t  = tk in (6.19) yields

6.3 Main Results

111

  E {V (ζt , t)} < e−α(t−tk ) E V (ζtk , tk ) .

(6.20)

At the switching moment tk , it follows from (6.11) and (6.15) that   E V (ζtk , tk ) ≤ μE V (ζtk− , tk− ) .

(6.21)

Hence, based on (6.20)–(6.21) and ϑ = Nσ j (0, t) ≤ (t − 0)/Ta , we can obtain E {V (ζt , t)} ≤ e−(α−ln μ/Ta )t V (ζ0 , 0).

(6.22)

It yields from (6.15) that   E {V (ζt , t)} ≥ aE ζ(t)2 , V (ζ0 , 0) ≤ bζ(0)2 ,

(6.23)

where a and b are denoted in (6.13). Combining (6.22)–(6.23), then we get  1  b E ζ(t)2 ≤ E {V (ζt , t)} ≤ e−(α−ln μ/Ta )t ζ(0)2 . a a

(6.24)

Consider Definition 6.1 when t0 = 0, the overall dynamics in (6.6) with υ(t) = 0 is mean-square exponentially stable. Next, the weighted H∞ performance for (6.6) defined in Definition 6.3 is introduced. For any nonzero υ(t) ∈ L2 [0, ∞), along the solution of dynamic error system with a fixed β, we can get L V (ζt , t) + αV (ζt , t) + e Tf (t)e f (t) − γ 2 u T (t)u(t)

 r

 N r [ j] [ j] ≤ σj h i θ[ j] h l θ[ j] j=1 i=1 l=1  T  [ j] [ j] [ j]    ζ(t) ζ(t) Π˜ 11il P B˜ il , υ(t) υ(t)  −γ 2 I T





T



[ j] [ j] [ j] [ j] [ j] [ j] where Π˜ 11il  P [ j] A˜ il + P [ j] A˜ il + αP [ j] + E˜ il K P [ j] E˜ il K + C˜ il the Schur’s complement, it can be seen that (6.14) equals to

N  j=1

σj

r  i=1

[ j] hi



θ

[ j]

r  l=1

[ j] hl

(6.25) T

[ j] C˜ il . Using

  ˜ [ j] [ j] ˜ [ j] Π11il P Bil < 0, θ[ j]  −γ 2 I

and L V (ζt , t) + αV (ζt , t) + e Tf (t)e f (t) − γ 2 u T (t)u(t) < 0. Denote Γ (t)  e Tf (t)e f (t) − γ 2 u T (t)u(t), then (6.26) can be converted to

(6.26)

112

6 Fault Detection for Switched Stochastic Systems

L V (ζt , t) < −αV (ζt , t) − Γ (t),

(6.27)

and d V (ζt , t) < 2

N 

σj

j=1

P

[ j]

r 

[ j]

hi

r



 [ j] θ[ j] h l θ[ j] ζ T (t)

i=1

l=1

[ j] E˜ il K ζ T (t)d(t)

− αV (ζt , β, t)dt − Γ (t)dt.

Notice that αt

d[e V (ζt , t)] < e

αt

 − Γ (t)dt + 2

N 

σj

j=1

r 

r 

[ j]

hi

 θ[ j]

i=1



 [ j] [ j] T [ j] ˜ [ j] T h l θ ζ (t)P E il K ζ (t)d(t) .

(6.28)

l=1

Utilizing the similar strategy as the proof of the previous exponential stability, we have   t −α(t−tk ) −α(t−s) E V (ζtk , tk ) −E e Γ (s)ds . (6.29) E V (ζt , t) < e tk

Under the zero initial condition, based on (6.21), (6.29) and ϑ = Nσ j (0, t) ≤ (t − 0)/Ta , it gives  0

t

e−α(t−s)−Nσ j (0,s) ln μ e Tf (s)e f (s)ds ≤ γ 2



t

e−α(t−s) u T (s)u(s)ds.

(6.30)

0

Due to Nσ j (0, t) ≤ t/Ta and Ta > Ta∗ = ln μ/α, we can get Nσ j (0, t) ln μ ≤ αt. Therefore, (6.30) signifies  0

t

e−α(t−s)−αs e Tf (s)e f (s)ds

 ≤γ

t

2

e−α(t−s) u T (s)u(s)ds.

(6.31)

0

Integrate the above inequality from t = 0 to ∞ and it’s easy to obtain the weighted H∞ performance level (γ, α). Thus this completes the proof.

6.3.2 Fault Detection Filter Design On the basis of proposed weighted H∞ performance conditions and the convex linearization technique, we are going to deal with the fault detection issue for fuzzy switched stochastic systems.

6.3 Main Results

113

Theorem 6.5 Given scalars α > 0 and γ > 0, assume there are matrices 0 < U [ j] ∈ [ j] [ j] [ j] Rn×n , 0 < V [ j] ∈ Rn×n , 0 < V [ j] ∈ Rk × Rk , A f i ∈ Rn×n , B f i ∈ Rn× p , and C f i ∈ Rl×n such that for j ∈ N , i = 1, 2, . . . , r , l = 1, 2, . . . , r , [ j]

ii < 0, i = 1, 2, . . . , r,

 1 1 [ j] [ j] [ j] il + li < 0, 1  i < l  r, ii + r −1 2  [ j] [ j]  U V > 0,  V [ j]

(6.32a) (6.32b) (6.32c)

where

[ j]

il

[ j]

3il

[ j]

4il

⎡ ⎡ [ j] [ j] ⎡ ⎤ ⎤ ⎤ [ j] [ j] [ j] [ j] [ j] [ j] 11il 12il 0 1il 2il 3il 14il 15il 16il ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ ⎢ ⎥ [ j] ⎢ ⎥ [ j] ⎢ [ j] [ j] [ j] [ j] ⎥  ⎢  −γ 2 I 0 ⎥, 1il  ⎢  22l ⎥, 0 ⎥, 2il  ⎢24il   26il ⎦ 25il ⎣ ⎣ ⎣ ⎦ ⎦ [ j] [ j]   4il 0 0 V [ j] Bw   33 ⎤ ⎡

T

T [ j] [ j] 18il 0 0 ⎥ ⎢ 17il  ⎥

⎢ [ j] T ⎥ , ⎢ 0 0 0 Cfl ⎦ ⎣ T 0 0 0 −Cw 

  −U [ j] −V [ j] [ j] , −I , , −V  diag  −V [ j]

with [ j]



T





[ j] [ j] [ j] T [ j] T [ j] T U [ j] + Ci Bfl + B f l C i + Ai + αU [ j] ,  





[ j] [ j] T [ j] [ j] T [ j] T Bfl  A f l + Ai V + Ci + αV [ j] , 

[ j] [ j] T [ j] [ j] [ j] [ j]  Afl + Afl + αV [ j] , 17il  U [ j] E i + B f l Fi , [ j]

11il  U [ j] Ai [ j]

12il [ j]

22l [ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j] [ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

14il  U [ j] B0i + B f l D0i , 15il  U [ j] Bi 18il  V [ j] E i

[ j]

[ j]

+ B f l Di ,

T V [ j] + αV [ j] , + B f l Fi , 33  V [ j] Aw + Aw

16il  U [ j] B1i + B f l D1i , 24il  V [ j] B0i + B f l D0i , 25il  V [ j] Bi

+ B f l Di , 26il  V [ j] B1i + B f l D1i .

There exists a fuzzy-basis-dependent filter as (6.3) to assure the overall error system in (6.6) is mean-square exponentially stable with a weighted H∞ performance (γ, α) under any switching signal over the ADT satisfying Ta > Ta∗ = lnαμ . In addition, if [ j] [ j] [ j] the above conditions have feasible solutions U [ j] , V [ j] , V [ j] , A f i , B f i , C f i , the desired fault detection filter matrices in (6.3) can be described as

114

6 Fault Detection for Switched Stochastic Systems



[ j]

[ j]

Afi Bfi [ j] Cfi 0





−1   [ j] [ j]  Afi Bfi 0 V [ j] = . [ j] Cfi 0 0 I

(6.33)

Proof Based on Theorem 6.4, define P [ j]  diag {U [ j] , V [ j] } in (6.9), with U [ j] ∈ R2n × R2n and V [ j] ∈ Rk × Rk . Particularly, for a set of (γ, α), the resulting system (6.6) is mean-square exponentially stable subject to a weighted H∞ performance (γ, α) if there are some suitable matrices and the following equations satisfy: [ j]

Φii < 0, i = 1, 2, . . . , r,

 1 1 [ j] [ j] [ j] Φii + Φil + Φli < 0, 1  i < l  r, r −1 2

(6.34a) (6.34b)

where ⎡

[ j]

Φil

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

[ j]

Φ11il     

T T

[ j] [ j] [ j] 0 U [ j] Bˆ il Eˆ il U [ j] 0 Cˆ f l [ j] Φ22 V [ j] Bˆ w 0 0 −CwT 2  −γ I 0 0 0 0 0   −U [ j] 0    −V [ j]     −I

⎤ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

with T 

 [ j] [ j] [ j] Φ11il  U [ j] Aˆ il + Aˆ il U [ j] + αU [ j] , Cˆ f l  0 C [f j]l ,   [ j] Φ22  V [ j] Aw + AwT V [ j] + αV [ j] , Bˆ w  0 0 Bw ,     [ j] [ j] 0 0 A E [ j] [ j] i i ˆ , Aˆ il  [ j] [ j] [ j] , E il  [ j] [ j] B f l Ci A f l B f l Fi 0   [ j] [ j] [ j] B B B [ j] 0i i 1i Bˆ il  [ j] [ j] [ j] [ j] [ j] [ j] . B f l D0i B f l Di B f l D1i  Then U [ j] is partitioned as U [ j]  [ j]

[ j]

[ j]

U1 U2 [ j]  U3

 [ j]

[ j]

> 0, where U1 ∈ Rn × Rn , U2 ∈ [ j]

Rn × Rn , and U3 ∈ Rn × Rn . Owing to a full-order filter is proposed, U2 is a [ j] square matrix. Without loss of generality, provided that U2 is nonsingular (If not, [ j] [ j] [ j] [ j] U2 may be perturbed by U2 with sufficiently small norm such that U2 + U2 ) and satisfies (6.34). Introduce several nonsingular matrices as

6.3 Main Results

 F

[ j]



115

I

0 −1

T

[ j] [ j] U2 0 U3



[ j]

V [ j]  U2

,

−1

T [ j] [ j] U3 U2 ,

(6.35)

[ j]

U [ j]  U1 ,

and 

[ j]

[ j]

Afi Bfi [ j] Cfi 0





[ j]

U2 0  0 I



[ j]

[ j]

Afi Bfi [ j] Cfi 0



−1

T  [ j] [ j] U2 U3 0 . 0 I

(6.36)

Perform a congruence transformation to (6.34) with Γ  {F [ j] , I, I, F [ j] , I, I } and we can get [ j]

Γ T Φii Γ < 0, 

 1 1 [ j] [ j] [ j] Φii + Φil + Φli Γ < 0. ΓT r −1 2

(6.37a) (6.37b)

On account of (6.35)–(6.36), it yields 

F

 [ j] T

 [ j] U [ j] Aˆ il F [ j]

=

 [ j] T [ j] [ j] [ j] U Eˆ il F = F  [ j] T [ j] [ j] U Bˆ il = F

[ j]

[ j]

[ j]

Afl

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

[ j]

U [ j] Ai + B f l Ci

[ j]

V [ j] Ai + B f l Ci

Afl   [ j] [ j] [ j] [ j] U E i + B f l Fi 0 

V [ j] E i + B f l Fi

U [ j] B0i + B f l D0i

0

,

 ,

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪  ⎪ [ j] [ j] [ j] [ j] ⎪ [ j] [ j] [ j] [ j] ⎪ U Bi + B f l Di U B1i + B f l D1i ⎪ ⎪ ⎪ , ⎪ [ j] [ j] [ j] [ j] [ j] [ j] [ j] [ j] ⎪ ⎪ V Bi + B f l Di V B1i + B f l D1i ⎪ ⎪   ⎪ ⎪ [ j] [ j]   ⎪ U V  [ j] T [ j] [ j] ⎪ [ j] [ j] [ j] ˆ ⎪ U F = [ j] [ j] , C f l F = 0 C f l . ⎪ F ⎪ ⎪ ⎭ V V V [ j] B0i + B f l D0i

(6.38)

On the basis of (6.38), it follows from (6.32) that (6.38). In addition, (6.36) is equal to   

 −1   [ j] [ j]  

−T [ j] [ j] [ j] [ j] [ j] Afi Bfi Afi Bfi U2 0 U2 U3 0 = × [ j] [ j] Cfi 0 Cfi 0 0 I 0 I ⎡ $

%−1

−T −1 ⎤  [ j] [ j]  [ j] [ j] U2 V [ j] U3 0⎦ Afi Bfi =⎣ [ j] Cfi 0 0 I 

 −T [ j] [ j] U2 U3 0 × . (6.39) 0 I

116

6 Fault Detection for Switched Stochastic Systems [ j]

[ j]

[ j]

It can be observed that the filter matrices A f i , B f i and C f i in (6.3) can be rewritten as −T

[ j] [ j] (6.39), which means U2 U3 can be regarded as the similarity transformation on the state-space realization for the designed filter, and has no influence in the filter −T

[ j] [ j] mapping from y to χ f . Without loss of generality, denote U2 U3 = I , thus it gives (6.33). As a consequence, the fuzzy-basis-dependent fault detection filter in (6.3) is established via (6.33). The proof is then completed.

6.4 Illustrative Example Consider the nonlinear switched stochastic system in (6.1) with N = 2, which can be formulated as the T-S fuzzy model and the relevant matrices are given as follows: Subsystem 1. A[1] 1 A[1] 2

[1] B11

E 1[1] C1[1] C2[1]

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −3.0 0.2 0.4 0.5 0.3 [1] = ⎣ 0.3−1.7 0.5⎦ , B1[1] = ⎣0.8⎦ , B01 = ⎣0.6⎦ , 0.2 0.5−2.5 0.6 0.5 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −2.7 0.3 0.6 0.5 0.4 [1] = ⎣ 0.2−1.5 0.8⎦ , B2[1] = ⎣0.6⎦ , B02 = ⎣0.6⎦ , 0.3 0.4−2.4 0.3 0.7   ⎡ ⎤ ⎡ ⎤ [1] 0.4 0.6 F1 = 0.1 0.2 0.4 , [1] = ⎣ 0.5 ⎦ , B12 = ⎣ 0.4 ⎦ ,   0.4 0.3 F2[1] = 0.1 0.2 0.2 , ⎡ ⎤ ⎡ ⎤ 0.2 0 0.1 0.3 0.1 0.2 = ⎣ 0.3 0.1 0.2 ⎦ , E 2[1] = ⎣ 0.1 0.2 0.2 ⎦ , 0.0 0.1 0.2 0.0 0.3 0.1   [1] [1] = 1.5 0.6 1.3 , D01 = 0.6, D1[1] = 0.3, D11 = 0.4,   [1] [1] [1] = 1.0 1.3 0.7 , D02 = 0.4, D2 = 0.3, D12 = 0.4.

Subsystem 2. ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −3.2 0.1 0.3 0.4 0.2 [2] ⎣ 0.3−1.8 0.4 ⎦ , B1[2] = ⎣ 0.9 ⎦ , B01 = ⎣ 0.5 ⎦ , A[2] 1 = 0.3 0.4−2.6 0.5 0.4 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −2.9 0.4 0.7 0.4 0.5 [2] ⎣ 0.3−1.3 0.6 ⎦ , B2[2] = ⎣ 0.5 ⎦ , B02 A[2] = ⎣ 0.3 ⎦ , 2 = 0.4 0.3−2.2 0.4 0.8

6.4 Illustrative Example

117

  ⎤ ⎡ ⎤ 0.3 0.2 F1[2] = 0.2 0.3 0.5 , [2] [2] B11 = ⎣ 0.4 ⎦ , B12 = ⎣ 0.1 ⎦ ,   0.5 0.5 F2[2] = 0.2 0.1 0.2 , ⎡ ⎤ ⎡ ⎤ 0.1 0 0.3 0.1 0.2 0.1 E 1[2] = ⎣ 0.4 0.2 0.1 ⎦ , E 2[2] = ⎣ 0.4 0.1 0.3 ⎦ , 0.0 0.2 0.1 0.0 0.4 0.2   [2] [2] [2] C1 = 1.6 0.4 1.4 , D01 = 0.5, D1[2] = 0.2, D11 = 0.5,   [2] [2] C2[2] = 0.9 1.1 0.8 , D02 = 0.2, D2[2] = 0.4, D12 = 0.5. ⎡

The weighting matrix W (s) is assumed as W (s) = 5/(s + 5). And the corresponding parameters in (6.5) are set as Aw = −5, Bw = 5 and Cw = 1. Solving the LMIs in Theorem 6.5, the minimized feasible γ can be computed as γ ∗ = 1.0876, and ⎡

⎤ ⎡ ⎤ −5.9191−0.3430−0.1660 −1.0772 ⎣ −1.9168−2.0822−1.7281 ⎦ , B [1] ⎣ −1.3602 ⎦ , A[1] f1 = f1 = −1.2755 0.6872−5.6867 −1.1693 ⎡ ⎤ ⎡ ⎤ −5.8700−1.8943 0.0918 −1.7928 ⎣ −1.0856−2.5724−0.5889 ⎦ , B [1] ⎣ −1.1643 ⎦ , A[1] f2 = f2 = −0.2762−0.0162−4.6259 −0.9090 ⎡ ⎤T ⎡ ⎤T −0.0323 −0.2410 [1] ⎣ ⎦ ⎣ −0.0497 −0.0741 ⎦ , = , C = C [1] f1 f2 −0.3092 0.0016 ⎡ ⎤ ⎡ ⎤ −4.4854−0.2213−0.7589 −0.7733 ⎣ −1.8817−2.6268−0.3243 ⎦ , B [2] ⎣ −1.1491 ⎦ , A[2] f1 = f1 = −0.8301 0.1435−5.2591 −1.2526 ⎡ ⎤ ⎡ ⎤ −3.3415−0.2332 0.3802 −0.5826 ⎣ −0.2229−2.1478 1.1747 ⎦ , B [2] ⎣ −0.3967 ⎦ , A[2] f2 = f2 = 0.1982−1.1113−4.3050 −1.4342 ⎡ ⎤T ⎡ ⎤T −0.1938 −0.0016 [2] ⎣ ⎦ ⎣ −0.0701 0.0452 ⎦ . C [2] = , C = f1 f2 −0.2003 −0.3281 In this example, the fuzzy basis functions are chosen as h 1 (x1 (t))   2 (x1 (t)−ϑ)2 , h , with ϑ = 5 and σ = 1. The (x exp − (x1 (t)−ϑ) 2 1 (t))  1 − exp − 2σ 2 2σ 2 disturbance signal ω(t) is assumed to be random noise and the known input is set to

1, 2.5 ≤ t ≤ 5, u(t) = sin(t), 0 ≤ t ≤ 10; and the fault signal is given by f (t) = . 0, otherwise. Figure 6.1 plots the weighting fault signal f w (t). The evaluation function and the threshold are chosen as (6.7)–(6.8). And we simulate the standard Brownian motion via employing the discretization technique

118

6 Fault Detection for Switched Stochastic Systems

Fig. 6.1 Weighting fault signal f w (t)

in [89]. In addition, the simulation time t ∈ [0, T ∗ ] with T ∗ = 10, the normally ∗ distributed variance δt = NT ∗ with N ∗ = 211 , step size Δt = ρδt with ρ = 2. For the proposed filter with fuzzy-fuzzy-dependent situation, simulation results are shown in Figs. 6.2–6.3. Among them, Fig. 6.2 draws the residual signal χ f (t) and Fig. 6.3 plots the evaluation function of J (χ f ) for the fault-free case (dash-dot line) and fault case (solid line). When the residual signal is produced, then establish the fault detection measure. For the fuzzy-rule-dependent situation, with a threshold given as Jth = 0.1136, Fig. 6.2 displays J (χ f ) > Jth for t = 2.7, which implies the fault signal f (t) can be detected 0.2s after its appearance.

6.5 Conclusion Weighted H∞ fault detection filtering has been a key focus for nonlinear switched systems with stochastic disturbances, and the corresponding issue was addressed using the fuzzy-rule-dependent approach. Based on the piecewise Lyapunov functions and ADT method, sufficient conditions were established such that the mean-square exponential stability with weighted H∞ performance for the dynamic error system could be ensured. Moreover, the solvable conditions of the proposed filter were derived based on the linearization procedure. In the last, to demonstrate the availability of our proposed scheme, a simulation has been provided.

6.5 Conclusion

Fig. 6.2 Residual signal χ f (t)

Fig. 6.3 Evaluation function of J (χ f )

119

Chapter 7

Reliable Filtering for T-S Fuzzy Time-Delay Systems

7.1 Introduction In this chapter, the reliable filtering problem with strict dissipative performance is considered for discrete-time T-S fuzzy systems with time-delays. A reliable filter design is prepared to ensure strict dissipativity for dynamic filtering systems. First, sufficient conditions for the reliable dissipativity analysis are established using a reciprocally convex method for T-S fuzzy systems subject to sensor failures. In addition, a reliable filter is established based on the convex optimization technique, which can be readily resolved through the standard numerical toolbox.

7.2 System Description and Preliminaries 7.2.1 System Description Consider the nonlinear system, which can be approximated with the T-S fuzzy model with the time-varying delay as follows:  Plant Form: Rule i: IF θ1 (k) is Mi1 and . . . and θ p (k) is Mi p , THEN ⎧ x(k + 1) = Ai x(k) + Adi x(k − d(k)) + Bi ω(k), ⎪ ⎪ ⎪ ⎪ y(k) = Ci x(k) + Cdi x(k − d(k)) + Di ω(k), ⎨ z(k) = L i x(k) + L di x(k − d(k)) + Fi ω(k), ⎪ ⎪ x(ι) = ψ(ι), ι = −d2 , −d2 + 1, . . . , 0, ⎪ ⎪ ⎩ i = 1, 2, . . . , r,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 X. Su et al., Intelligent Control, Filtering and Model Reduction Analysis for Fuzzy-Model-Based Systems, Studies in Systems, Decision and Control 385, https://doi.org/10.1007/978-3-030-81214-0_7

(7.1)

121

122

7 Reliable Filtering for T-S Fuzzy Time-Delay Systems

where x(k) ∈ Rn denotes the state vector; y(k) ∈ Rl denotes the measured output signal; ω(k) ∈ R p represents the disturbance input which is belong to 2 [0, ∞); z(k) ∈ Rq denotes the signal to be estimated; the system delay d(k) is a positive integer, which is supposed to be time-varying and satisfies 1  d1  d(k)  d2 , where d2 and d1 are positive constant scalars denoting the maximum and minimum set; r is the quantity of IF-THEN fuzzy delay, respectively. Mi j represents the fuzzy   rules; θ(k) = θ1 (k), θ2 (k), . . . , θ p (k) denotes the premise variables. Ai , Adi , Bi , condition. Ci , Cdi , Di , L i , L di and Fi are constant matrices; ψ(k) stands for the initial p j=1 Mi j (θ j (k)) The fuzzy membership functions are expressed as h i (θ(k))  r  p M θ (k) , ) ij( j i=1 j=1

 where Mi j θ j (k) denotes the grade of membership of θ (k) in M . Consequently, j i j we can obtain h i (θ(k)) ≥ 0, i = 1, 2, . . . , r and ri=1 h i (θ(k)) = 1 for all k. A more complete expression of the T-S fuzzy model with time-varying delay can be formulated as (Σ): ⎧ r ⎪ ⎪ x(k + 1) = h i (θ(k)) [Ai x(k) + Adi x(k − d(k)) +Bi ω(k)] , ⎪ ⎪ ⎪ i=1 ⎪ ⎪ r ⎪ ⎨ y(k) = h i (θ(k)) [Ci x(k) + Cdi x(k − d(k)) +Di ω(k)] , (7.2) i=1 ⎪ r ⎪ ⎪ ⎪ z(k) = h i (θ(k)) [L i x(k) + L di x(k − d(k)) +Fi ω(k)] , ⎪ ⎪ ⎪ i=1 ⎪ ⎩ x(ι) = ψ(ι), ι = −d2 , −d2 + 1, . . . , 0. In this note, an efficient filter with sensor failures is proposed for the estimation of z(k): (Σ f ) :

ˆ + B f yˆ (k), x(k ˆ + 1) = A f x(k) ˆ + D f yˆ (k), zˆ (k) = C f x(k)

(7.3)

where x(k) ˆ ∈ Rk denotes the state vector of our designed filter (7.3) with k ≤ n; q zˆ (k) ∈ R denotes the estimation of z(k); A f , B f , C f and D f stand for some suitably T  dimensioned matrices to be decided, and yˆ (k)  yˆ1 (k) . . . yˆm (k) represents the signal from the sensor which might be faulty. Remark 7.1 On account of the fact that the fuzzy rule is taken into account, thus the fuzzy-rule-dependent filter has less conservativeness than a fuzzy-ruleindependent one. In this chapter, we design the fuzzy-rule-independent filters as (7.3) and moreover, the developed techniques can be broaden into the design for fuzzy-rule-dependent filters. The failure form in [73] is utilized as yˆ j (k) = βε j y j (k), where

j = 1, 2, . . . , m,

7.2 System Description and Preliminaries

123

0 ≤ β ε j ≤ βε j ≤ β¯ε j ,

j = 1, 2, . . . , m,

with 0 ≤ βε j ≤ 1 in which the variables βε j quantify the faults of sensors. And it follows that yˆ (k) = Bε y(k),

Bε = diag{βε1 , βε2 , . . . , βεm }.

Remark 7.2 Considering the above model, when β ε j = β¯ε j , it is a normal fully

operating case, yiF (k) = yi (k); when β ε j = 0, it includes the outage case in [279]; when β = 0 and β¯ε j = 1, it becomes the case that the intensity of the feedback εj

signal from actuator may variate. Define B¯ ε  diag{β¯ε1 , β¯ε2 , . . . , β¯ε j , . . . , β¯εm }, B ε  diag{β ε1 , β ε2 , . . . , β ε j , . . . , β εm }, Bε0  diag{βε01 , βε02 , . . . , βε0 j , . . . , βε0m }, Λ  diag{α1 , α2 , . . . , α j , . . . , αm }, E ε  diag{ ε1 , ε2 , . . . , ε j , . . . , εm }, where βε0 j 

β¯ ε j +β ε j 2

and α j 

β¯ ε j −β ε j 2

. It yields

Bε = Bε0 + E ε , | ε j | ≤ α j .

(7.4)

Extending the model (Σ) to contain our proposed filter (Σ f ), the following dynamic filtering system can be obtained, which is denoted as (Σe ): ⎧ r   ⎪ ⎪ ξ(k + 1) = h i (θ(k)) A¯ i ξ(k) + A¯ di ξ(k − d(k)) + B¯ i ω(k) , ⎪ ⎪ ⎨ i=1 r   e(k) = h i (θ(k)) L¯ i ξ(k) + L¯ di ξ(k − d(k)) + F¯i ω(k) , ⎪ ⎪ ⎪ i=1 ⎪ ⎩ ξ(ι) = ϕ(ι), ι = −d2 , −d2 + 1, . . . , 0,

(7.5)

T  where ξ(k)  x T (k) xˆ T (k) , e(k)  z(k) − zˆ (k) and ⎫  

0 0 Ai Adi ⎪ ⎪ , A¯ di  , ⎪ ⎪ B C A B C 0 B B ⎬ f ε i f f ε di 

  B i , L¯ di  L di − D f Bε Cdi 0 , ⎪ B¯ i  ⎪ ⎪ ⎪  B f Bε Di  ¯L i  L i − D f Bε Ci −C f , F¯i  Fi − D f Bε Di . ⎭ A¯ i 

Furthermore, define

(7.6)

124

7 Reliable Filtering for T-S Fuzzy Time-Delay Systems

⎫ ⎪ h i (θ(k)) A¯ di , ⎪ ⎪ ⎪ ⎪ i=1 i=1 ⎪ ⎬ r r ¯ ¯ B(k)  h i (θ(k)) B¯ i , L(k)  h i (θ(k)) L¯ i , ⎪ i=1 i=1 ⎪ ⎪ r r ⎪ ¯ ⎪ F(k)  h i (θ(k)) F¯i , L¯ d (k)  h i (θ(k)) L¯ di . ⎪ ⎭ ¯ A(k) 

r

h i (θ(k)) A¯ i ,

i=1

A¯ d (k) 

r

(7.7)

i=1

Definition 7.3 The dynamic filtering system in (7.5) is asymptotically stable if ω(k) = 0, lim |ξ(k)| = 0. k→∞

For a system (Σe ), which is asymptotically stable, it follows that e = {e(k)} ∈ 2 [0, ∞) when ω = {ω(k)} ∈ 2 [0, ∞).

7.2.2 Dissipativity Definition In this section, discussions on dissipative systems are introduced. Dissipative systems can be regarded as a generalization of passive systems with more general internal and supplied energies [341]. A system is called “dissipative” if “power dissipation” exists in the system. Dissipative systems are those that cannot store more energy than that supplied by the environment and/or by other systems connected to them, i.e., dissipative systems can only dissipate but not generate energy [220]. Based on [99], associated with the discrete-time T-S fuzzy time-varying delay system in (7.5) is a real valued function G(ω(k), e(k)) called the supply rate which is formally defined as follows. The classical form of dissipativity in [99] is obviously applicable to the discretetime T-S fuzzy time-varying delay system in (7.5) in the following. Remark 7.4 Passive systems are a kind of particular dissipative systems, which owes a bilinear supply rate, i.e. G(ω, e) = e T ω. If a system with a constant positive feed forward of X is passive, then the procedure is dissipative in regard to the supply rate G(ω, e) = e T ω + ω T X ω, where X = X T ∈ R p× p . In a similar way, if a system with a constant negative feedback of Z is passive, then the procedure is dissipative in regard to the supply rate G(ω, e) = e T Ze + e T ω, where Z = Z T ∈ R p× p . Definition 7.5 For the given matrices Z ∈ Rq×q , X ∈ R p× p , Y ∈ Rq× p , where Z and X are symmetric, for the real function (·) with (0) = 0, the discrete T-S fuzzy system with time-varying delay as (7.5) is dissipative if the following condition satisfies: T   T  e(k) Z Y e(k) + (ψ(0)) ≥ 0, ∀T ∗ ≥ 0. ω(k)  X ω(k) k=0

(7.8)

7.2 System Description and Preliminaries

125

Moreover, for δ > 0, if the following equality holds: T   T T   e(k) Z Y e(k) + (ψ(0)) ≥ δ ω T (k)ω(k), ∀T ∗ ≥ 0, ω(k)  X ω(k) k=0

(7.9)

k=0

then the resulting system (7.5) is called as strictly dissipative. 1

1

Provided that Z ≤ 0, then it follows −Z = (Z−2 )2 , for Z−2 ≥ 0. Remark 7.6 The dissipative theory generalizes some different system theories, containing small gain theorem, passivity theorem, circle criterion, and so forth. Some special circumstances can be obtained by choosing different (Z, Y, X ), which are shown as follows: • If Z = −I , Y = 0 and X = γ 2 I (γ > 0), the strict dissipative performance becomes H∞ performance. • If Z = 0, Y = I and X = 0, the strict dissipative performance changes to the positive real performance. • If Z = −θ I , Y = 1 − θ and X = θγ 2 I (γ > 0, θ ∈ [0, 1]), it reduces to the mixed performance. • Z = −I , Y = 21 (K1 + K2 )T and X = − 21 (K1T K2 + K2T K1 ) (γ > 0, for constant matrices K1 , K2 ), it converts into the sector bounded performance.

7.2.3 Reciprocally Convex Approach This part discusses a specialized class of function compositions, which can be utilizing to deal with the double summation terms in Lyapunov functions for T-S fuzzy systems with time-varying delay. Definition 7.7 Set Ψ1 , Ψ2 , . . ., and Ψ N : Rm → Rn as a limited number of functions such that they owe positive values among an open subset D of Rm . And the reciprocally convex combination over D is of the form as 1 1 1 Ψ1 + Ψ2 + · · · + Ψ N : D → Rn , ϑ1 ϑ2 ϑN where ϑi > 0 and i ϑi = 1.

(7.10)

A lower bound for the reciprocally convex method of scalar positive functions Ψi = f i is introduced in the following lemma. Lemma 7.8 [206] Let f 1 , f 2 , . . ., and f N : Rm → R have positive values in an open subset D of Rm . The reciprocally convex combination of f i over D satisfies the following condition:

126

7 Reliable Filtering for T-S Fuzzy Time-Delay Systems

 1   f i (θ) = f i (θ) + max gi, j (θ), gi, j (θ) ϑi i ϑi =1} i i i= j

min

{ϑi |ϑi >0,

where

gi, j : Rm → R, g j,i (θ) = gi, j (θ),

  f i (θ) gi, j (θ) ≥0 . g j,i (θ) f j (θ)

We focus on designing a reliable filter (Σ f ) for system (7.2) and achieve the following goals simultaneously: • The dynamic filtering system in (7.5) with ω(k) = 0 is asymptotically stable. • The dynamic filtering system in (7.5) is strictly dissipative. Assumption 7.1 System (Σ) in (7.2) is asymptotically stable. Remark 7.9 In fact, we have omitted the control input in system (Σ), then the original system to be estimated must be asymptotically stable, which is a precondition for the dynamic filtering system (Σe ) to be asymptotically stable.

7.3 Main Results 7.3.1 Reliable Dissipativity Analysis In this part, the Lyapunov functions and reciprocally convex approach are utilized to investigate the asymptotic stability with strict dissipativity for the augmented filtering system in (7.5). Denote d = d2 − d1 and ⎫ ⎪ h i (θ(k))Q 1i , ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎬ r Q 2 (k)  h i (θ(k))Q 2i , ⎪ i=1 ⎪ ⎪ r ⎪ ⎪ Q 3 (k)  h i (θ(k))Q 3i , ⎪ ⎭ Q 1 (k) 

r

(7.11)

i=1

where Q 1i > 0, Q 2i > 0, Q 3i > 0, i = 1, 2, . . . , r , are all (n + k) × (n + k) matrices. Theorem 7.10 For matrices 0 ≥ Z ∈ Rq×q , X ∈ R p× p , Y ∈ Rq× p with Z and X being symmetric, and scalar δ > 0, if there are matrices 0 < P ∈ R(n+k)×(n+k) , 0 < Q 1i ∈ R(n+k)×(n+k) , 0 < Q 2i ∈ R(n+k)×(n+k) , 0 < Q 3i ∈ R(n+k)×(n+k) , 0 < S1 ∈ R(n+k)×(n+k) , 0 < S2 ∈ R(n+k)×(n+k) and M ∈ R(n+k)×(n+k) such that for i, j, s, t = 1, . . . , r ,

7.3 Main Results

127

Ξi jst < 0,  S2 M T ≥ 0,  S2

(7.12) (7.13)

where ⎡

Ξi jst

Ξ11i Ξ12 0 0 Ξ15i ⎢  Ξ22 j Ξ23 Ξ24 0 ⎢ ⎢   Ξ33s Ξ34 Ξ35i ⎢ ⎢    Ξ44t 0 ⎢ ⎣     Ξ55i     

⎤ Ξ16i 0 ⎥ ⎥ Ξ36i ⎥ ⎥, 0 ⎥ ⎥ Ξ56i ⎦ Ξ66

Ξ11i  −P + Q 1i + Q 2i + (d + 1)Q 3i − S1 , Ξ12  S1 , Ξ15i  − L¯ iT Y, T Ξ35i  − L¯ di Y, Ξ22 j  −Q 1 j − S1 − S2 , Ξ23  −M + S2 , Ξ33s  −Q 3s − 2S2 + M + M T , Ξ34  −M + S2 , Ξ44t  −Q 2t − S2 , Ξ55i  − F¯iT Y − Y T F¯i − X + δ I, Ξ24  M,   1 Ξ16i  A¯ iT d1 ( A¯ iT − I ) d( A¯ iT − I ) L¯ iT Z−2 ,     1 1 T T T ¯T Ξ36i  A¯ di d1 A¯ di d A¯ di L di Z−2 , Ξ56i  B¯ iT d1 B¯ iT d B¯ iT F¯iT Z−2 , Ξ66  diag{−P −1 , −S1−1 , −S2−1 , −I }, then the dynamic filtering system in (7.5) with sensor failure is asymptotically stable with strict dissipativity based on Definition 7.5. Proof Considering the fuzzy membership functions, it follows from (7.12) that r 

h i (θ(k))

i=1

r  j=1

h j (θ(k − d1 ))

r 

h s (θ(k − d(k)))

s=1

r 

h t (θ(k − d2 ))Ξi jst < 0.

t=1

A more complete expression can be presented as Ξ (k) < 0, where

(7.14)

128

7 Reliable Filtering for T-S Fuzzy Time-Delay Systems

⎤ Ξ11 (k) Ξ12 0 0 Ξ15 (k) Ξ16 (k) ⎢  Ξ22 (k) Ξ23 Ξ24 0 0 ⎥ ⎥ ⎢ ⎢   Ξ33 (k) Ξ34 Ξ35 (k) Ξ36 (k) ⎥ ⎥, Ξ (k)  ⎢ ⎢  0 ⎥   Ξ44 (k) 0 ⎥ ⎢ ⎣     Ξ55 (k) Ξ56 (k) ⎦      Ξ66 ⎡

Ξ11 (k)  −P + Q 1 (k) + Q 2 (k) + (d + 1)Q 3 (k) − S1 , Ξ15 (k)  − L¯ T (k)Y, Ξ22 (k)  −Q 1 (k − d1 ) − S1 − S2 , Ξ44 (k)  −Q 2 (k − d2 ) − S2 , Ξ33 (k)  −Q 3 (k − d(k)) − 2S2 + M + M T , Ξ35 (k)  − L¯ dT (k)Y, ¯ Ξ55 (k)  − F¯ T (k)Y − Y T F(k) − X + δ I,   1 Ξ16 (k)  A¯ T (k) d1 ( A¯ T (k)− I ) d( A¯ T (k)− I ) L¯ T (k)Z−2 ,   1 Ξ36 (k)  A¯ dT (k) d1 A¯ dT (k) d A¯ dT (k) L¯ dT (k)Z−2 ,   1 Ξ56 (k)  B¯ T (k) d1 B¯ T (k) d B¯ T (k) F¯ T (k)Z−2 .

Employing the Schur complement method, (7.14) signifies Ξ¯ (k) < 0, where

(7.15)

⎡

⎤ Ξ¯ 11 (k) Ξ12 Ξ¯ 13 (k) 0 Ξ¯ 15 (k) ⎢  Ξ22 (k) Ξ23 Ξ24 0 ⎥ ⎢ ⎥ ¯ 33 (k) Ξ34 Ξ¯ 35 (k) ⎥ , Ξ¯ (k)  ⎢   Ξ ⎢ ⎥ ⎣    Ξ44 (k) 0 ⎦     Ξ¯ 55 (k) T ¯ ¯ Ξ¯ 11 (k)  A¯ (k)P A(k) − P + Q 1 (k) − L¯ T (k)Z L(k) + (d + 1)Q 3 (k) 2 ¯ T 2 ¯ ¯ ¯ +d1 [ A(k) − I ] S1 [ A(k) − I ] + d [ A(k) − I ]T S2 [ A(k) − I] +Q 2 (k) − S1 , ¯ ¯ Ξ13 (k)  d12 [ A(k) − I ]T S1 A¯ d (k) − L¯ T (k)Z L¯ d (k) + A¯ T (k)P A¯ d (k) 2 ¯ +d [ A(k) − I ]T S2 A¯ d (k),

¯ ¯ ¯ ¯ Ξ¯ 15 (k)  d12 [ A(k) − I ]T S1 B(k) − I ]T S2 B(k) + A¯ T (k)P B(k) + d 2 [ A(k) T T ¯ ¯ ¯ − L (k)Z F(k) − L (k)Y,

Ξ¯ 33 (k)  A¯ dT (k)P A¯ d (k) − Q 3 (k − d(k)) − 2S2 + M T + d 2 A¯ dT (k)S2 A¯ d (k) +M − L¯ dT (k)Z L¯ d (k) + d12 A¯ dT (k)S1 A¯ d (k), ¯ ¯ Ξ¯ 35 (k)  A¯ dT (k)P B(k) + d12 A¯ dT (k)S1 B(k) − L¯ dT (k)Y + d 2 A¯ dT (k)S2 B(k) ¯ − L¯ dT (k)Z F(k),

¯ ¯ ¯ Ξ¯ 55 (k)  B¯ T (k)P B(k) + d12 B¯ T (k)S1 B(k) + δ I − F¯ T (k)Z F(k) T T 2 T ¯ − F¯ (k)Y − Y F(k) − X + d B¯ (k)S2 B(k). Next, the fuzzy Lyapunov functions are considered as follows:

7.3 Main Results

129

V (k) 

5 

Vi (k),

(7.16)

i=1

where ⎧ V1 (k)  ξ T (k)Pξ(k), ⎪ ⎪ ⎪ 2 k−1 ⎪ T ⎪ ⎪ ⎪ V2 (k)  ξ (i)Q j (i)ξ(i), ⎪ ⎪ ⎪ j=1 i=k−d j ⎪ ⎪ ⎪ k−1 ⎪ ⎪ ⎪ ξ T (i)Q 3 (i)ξ(i) + ⎨ V3 (k)  i=k−d(k) −1 k−1

⎪ ⎪ ⎪ d1 ζ T (i)S1 ζ(i), V4 (k)  ⎪ ⎪ ⎪ j=−d1 i=k+ j ⎪ ⎪ ⎪ −d 1 −1 k−1 ⎪ ⎪ ⎪ dζ T (i)S2 ζ(i), V5 (k)  ⎪ ⎪ ⎪ j=−d i=k+ j 2 ⎪ ⎩ ζ(k)  ξ(k + 1) − ξ(k).

−d 1

k−1

j=−d2 +1 i=k+ j

ξ T (i)Q 3 (i)ξ(i),

Along the trajectories of the dynamic filtering system (Σe ) in (7.5), and taking the difference of the Lyapunov functions in (7.16), it yields that ΔV (k)  V (k + 1) − V (k) =

5 

ΔVi (k),

(7.17)

i=1

where ΔV1 (k) = ξ T (k + 1)Pξ(k + 1) − ξ T (k)Pξ(k), ΔV2 (k) = ξ T (k)Q 1 (k)ξ(k) + ξ T (k)Q 2 (k)ξ(k) −ξ T (k − d1 )Q 1 (k − d1 )ξ(k − d1 ) −ξ T (k − d2 )Q 2 (k − d2 )ξ(k − d2 ), k−1  ΔV3 (k) = (d + 1)ξ T (k)Q 3 (k)ξ(k) +

ξ T (i)Q 3 (i)ξ(i)

i=k−d(k+1)+1 k−1 

−

ξ T (i)Q 3 (i)ξ(i) −

k−d 1

ξ T (i)Q 3 (i)ξ(i)

i=k−d2 +1

i=k−d(k)+1

−ξ (k − d(k))Q 3 (k − d(k))ξ(k − d(k)) T

≤ −ξ T (k − d(k))Q 3 (k − d(k))ξ(k − d(k)) +(d + 1)ξ T (k)Q 3 (k)ξ(k), k−1  ΔV4 (k) = d12 ζ T (k)S1 ζ(k) − d1 ζ T (i)S1 ζ(i)  ≤−

k−1 

i=k−d1

i=k−d1

T ζ(i)

 S1

k−1 

 ζ (i) + d12 ζ T (k)S1 ζ(k) T

i=k−d1

= − [ξ(k) − ξ(k − d1 )] S1 [ξ(k) − ξ(k − d1 )] + d12 ζ T (k)S1 ζ(k).

130

7 Reliable Filtering for T-S Fuzzy Time-Delay Systems

Due to

S2 M T  S2

 ≥ 0, the following condition satisfies:

⎡  ⎣

ϑ1 ζ1 (k) ϑ2

−

ϑ2 ζ (k) ϑ1 2

⎤T ⎦

S2 M T  S2



⎡  ⎣

ϑ1 ζ1 (k) ϑ2

−

ϑ2 ζ (k) ϑ1 2

⎤ ⎦ ≥ 0,

where d2 − d(k) , d d(k) − d1 . ζ2 (k)  ξ(k − d1 ) − ξ(k − d(k)), ϑ2  d ζ1 (k)  ξ(k − d(k)) − ξ(k − d2 ), ϑ1 

Applying Lemma 7.8, it follows for d1 ≤ d(k) ≤ d2 that ΔV5 (k) = d ζ (k)S2 ζ(k) − d 2 T

k−d(k)−1 

ζ T (i)S2 η(i)

i=k−d2

−d

k−d 1 −1 

ζ T (i)S2 ζ(i)

i=k−d(k)

d ≤− d2 − d(k)

k−d(k)−1  i=k−d2

T k−d(k)−1   ζ(i) S2 ζ(i) ⎤T ⎡

i=k−d2

⎤ k−d k−d 1 −1 1 −1   d ⎣ − ζ(i)⎦ S2 ⎣ ζ(i)⎦ + d 2 ζ T (k)S2 ζ(k) d(k) − d1 i=k−d(k) i=k−d(k)

 T  T ζ (k) S2 M ζ1 (k) ≤− 1 + d 2 ζ T (k)S2 ζ(k)  S2 ζ2 (k) ζ2 (k) 

T  S2 −S2 ξ(k − d1 ) ξ(k − d1 ) =− ξ(k − d(k)) ξ(k − d(k))  S2 

T  S2 −S2 ξ(k − d(k)) ξ(k − d(k)) −  S2 ξ(k − d2 ) ξ(k − d2 ) ⎤T ⎡ ⎤ ⎡ ⎤ ⎡ 0 M −M ξ(k − d1 ) ξ(k − d1 ) − ⎣ ξ(k − d(k)) ⎦ ⎣  −M − M T M ⎦ × ⎣ ξ(k − d(k)) ⎦   0 ξ(k − d2 ) ξ(k − d2 ) ⎡

+ d 2 ζ T (k)S2 ζ(k). When d(k) = d1 or d(k) = d2 , it has ζ1 (k) = 0 or ζ2 (k) = 0. Thus, the inequality in ΔV5 (k) still satisfies. And we can get the following condition:

7.3 Main Results

131

¯ ΔV (k) = ζ¯T (k)Ξˆ (k)ζ(k),

(7.18)

where T  ¯ ζ(k)  ξ T (k) ξ T (k − d1 ) ξ T (k − d(k)) ξ T (k − d2 ) ω T (k) , ⎡ ⎤ Ξˆ 11 (k) Ξ12 Ξˆ 13 0 Ξˆ 15 (k) ⎢  Ξ24 0 ⎥ Ξ22 (k) Ξ23 ⎢ ⎥ ⎢ ˆ ˆ ˆ Ξ (k)  ⎢   Ξ33 (k) Ξ34 Ξ35 (k) ⎥ ⎥, ⎣  0 ⎦   Ξ44 (k)     Ξˆ 55 (k) ¯ Ξˆ 11 (k)  A¯ T (k)P A(k) − P + Q 1 (k) + Q 2 (k) − S1 + (d + 1)Q 3 (k) ¯ ¯ ¯ ¯ + d12 [ A(k) − I ]T S1 [ A(k) − I ] + d 2 [ A(k) − I ]T S2 [ A(k) − I ], T 2 T 2 ¯ ¯ Ξˆ 13 (k)  A¯ (k)P A¯ d (k) + d1 [ A(k) − I ] S1 A¯ d (k) + d [ A(k) − I ]T S2 A¯ d (k), ¯ ¯ ¯ ¯ Ξˆ 15 (k)  A¯ T (k)P B(k) + d12 [ A(k) − I ]T S1 B(k) − I ]T S2 B(k), + d 2 [ A(k) Ξˆ 33 (k)  A¯ dT (k)P A¯ d (k) + d12 A¯ dT (k)S1 A¯ d (k) − 2S2 + M − Q 3 (k − d(k)) + d 2 A¯ dT (k)S2 A¯ d (k) + M T , 2 ¯T 2 ¯T ¯ ¯ Ξˆ 35 (k)  A¯ dT (k)P B(k)+d 1 Ad (k)S1 B(k)+d Ad (k)S2 B(k), 2 ¯T 2 ¯T ¯ ¯ Ξˆ 55 (k)  B¯ T (k)P B(k)+d 1 B (k)S1 B(k)+d B (k)S2 B(k).

On the basis of (7.13), (7.15), (7.18) and the zero input signal ω(k) = 0, we have ΔV (k) < 0, hence the dynamic filtering system (7.5) is asymptotically stable. In the following, the strict dissipativity of system (7.5) is discussed. Define T   T T   e(k) Z Y e(k) ω T (k)ω(k), ∀T ∗ ≥ 0. J (T )  −δ ω(k)  X ω(k) 

k=0

k=0

When ξ(k) = 0, that is the zero initial condition, for k = −d2 , −d2 + 1, . . . , 0 and any non-zero ω(k) ∈ 2 [0, ∞), it is observed that V (T  + 1) − V (0) − J (T  ) 

 T  T  Z Y e(k) e(k) = ΔV (k) −  X − δI ω(k) ω(k) k=0

¯ < 0. = ζ¯T (k)Ξ¯ (k)ζ(k)

(7.19)

Considering (7.19) and V (T  + 1) > 0, we can get J (T  ) > 0.

(7.20)

132

7 Reliable Filtering for T-S Fuzzy Time-Delay Systems

On the basis of Definition 7.5, the system in (7.5) is asymptotically stable with strict dissipativity. Then this completes the proof. Notice that when ω(k) = 0 in the system (Σ), it has x(k + 1) = A(k)x(k) + Ad (k)x(k − d(k)),

(7.21)

where A(k) 

r 

h i (θ(k))Ai ,

Ad (k) 

i=1

r 

h i (θ(k))Adi .

i=1

Then we will provide a stability condition for the concerned system. Consider the system in (7.21) and let ⎫ ⎪ h (θ(k))Q , ⎪ 1i ⎪ i=1 i ⎪ ⎪ i=1 ⎪ ⎪ r ⎪ r ⎪ S1 (k)  h i (θ(k))S1i , Q2 (k)  i=1 h i (θ(k))Q2i , ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎬ r r S2 (k)  h i (θ(k))S2i , Q3 (k)  h i (θ(k))Q3i , ⎪ i=1 i=1 ⎪ ⎪ r r ⎪ ¯ 2, ⎪ ⎪ S¯1 (k)  h i (θ(k))S¯1i , S¯2 (k)  h i (θ(k))M ⎪ ⎪ ⎪ i=1 i=1 ⎪ ⎪ r ⎪ ¯ ¯ i, ⎪ ⎪ M(k)  h i (θ(k))M ⎭ P(k) 

r

h i (θ(k))Pi , Q1 (k) 

r

(7.22)

i=1

where Pi > 0, S1i > 0, S2i > 0, Q1i > 0, Q2i > 0, Q3i > 0, S¯1i > 0, S¯2i > 0 and Mi (i = 1, 2, . . . , r ) are all n × n matrices. Based on Theorem 7.10, the additional conditions for (7.21) are proposed. Corollary 7.11 For 1 ≤ d1 ≤ d2 , the system in (7.21) with time-varying delay d(k) satisfying d1 ≤ d(k) ≤ d2 is asymptotically stable provided that there are matrices Pi > 0, S1i > 0, S2i > 0, Q1i > 0, Q2i > 0, Q3i > 0, S¯1i > 0, S¯2i > 0 and Mi such that the following conditions satisfy: Γostlii < 0, Γostli j + Γostl ji < 0, o, s, t, l, i, j

 S¯2i MiT ≥ 0,  S¯2i S1i − S¯1 j ≥ 0, S2i − S¯2 j ≥ 0, where

o, s, t, l, i ∈ (1, 2, . . . , r ),  1 ≤ i = j ≤ r, ∈ (1, 2, . . . , r ),

(7.23) (7.24)

i ∈ (1, 2, . . . , r ),

(7.25)

i, j ∈ (1, 2, . . . , r ), i, j ∈ (1, 2, . . . , r ),

(7.26) (7.27)

7.3 Main Results

133

⎤ 0 Γ11tl ji Γ12t Γ13l ji ⎢  Γ22ti Γ23i Γ24i ⎥ ⎥, ⎢ ⎣   Γ33sl ji Γ34i ⎦    Γ44oi ⎡

Γostl ji

with Γ11tl ji  A Tj Pl A j +Q1i +Q2i +d12 [A j − I ]T S1i [A j − I ] + (d +1)Q3i − S¯1t +d 2 [A j − I ]T S2i [A j − I ]−Pi , Γ13l ji  A Tj Pl Ad j +d12 [A j − I ]T S1i Ad j +d 2 [A j − I ]T S2i Ad j , Γ22ti  −Q1t − S¯1t − S¯2i , Γ12t  S¯1t , Γ23i  −Mi + S¯2i , Γ33sl ji  AdT j Pl Ad j − Q3s + d12 AdT j S1i Ad j − 2S¯2i + Mi + MiT + d 2 AdT j S2i Ad j , Ξ44oi  −Q2o − S¯2i , Γ34i  −Mi + S¯2i , Γ24i  Mi . Proof The fuzzy Lyapunov functions are selected as V(k) 

5 

Vi (k),

i=1

where ⎧ T ⎪ ⎪ V1 (k)  x (k)P(k)x(k), η(k)  x(k + 1) − x(k), ⎪ 2 ⎪ k−1 T ⎪ ⎪ ⎪ V2 (k)  x (i)Q j (i)x(i), ⎪ ⎪ ⎪ j=1 i=k−d j ⎪ ⎪ ⎪ −d k−1 k−1 T ⎪ ⎨ V (k)  x T (i)Q (i)x(i) + 1 x (i)Q3 (i)x(i), 3 3 j=−d2 +1 i=k+ j i=k−d(k) ⎪ ⎪ −1 k−1 ⎪ ⎪ ⎪ d1 η T (i)S1 (i)η(i), V4 (k)  ⎪ ⎪ ⎪ j=−d i=k+ j 1 ⎪ ⎪ ⎪ −d 1 −1 k−1 ⎪ ⎪ ⎪ V5 (k)  dη T (i)S2 (i)η(i). ⎩ j=−d2 i=k+ j

Using the similar ways as the proof in Theorem 7.10, we can get the corresponding results readily. Remark 7.12 In Corollary 7.11, sufficient stability conditions for (7.21) are established via reciprocally convex method. There exist some techniques to handle the T-S fuzzy systems with time-delays, such as the delay partition method [299], circumventing the bounding inequalities approach [82], the input-output method [251]. Compared with these methods, our proposed results have advantages as twofold. Firstly, our proposed stability conditions don’t include the interval delay item in Lyapunov functions and reduce the computational complexity in simulations. Sec-

134

7 Reliable Filtering for T-S Fuzzy Time-Delay Systems

ondly, the conservativeness is further reduced via utilizing the reciprocally convex approach. Furthermore, compare Corollary 7.11 with the existing results [82, 251, 299], see Table 7.1 in Example 7.17, it is observable that the results we obtained are more efficient than the results in [82, 251, 299].

7.3.2 Reliable Filter Design with Dissipativity On the basis of Theorem 7.10, the following theorems present the reliable filter with strict dissipativity for T-S fuzzy system with time-varying delays in (7.2). Firstly, consider the dynamic filtering system in (7.5) owes known sensor failure parameters. Theorem 7.13 For the matrices 0 ≥ Z ∈ Rq×q , X ∈ R p× p , Y ∈ Rq× p with Z and X being symmetric, and scalars δ > 0, provided that there are matrices 0 < O ∈ Rn×n , 0 < L ∈ Rk×k , 0 < Q¯ 1i ∈ R(n+k)×(n+k) , 0 < Q¯ 2i ∈ R(n+k)×(n+k) , 0 < Q¯ 3i ∈ (n+k)×(n+k) ¯ , R(n+k)×(n+k) , 0 < S¯1 ∈ R(n+k)×(n+k) , 0 < S¯2 ∈ R(n+k)×(n+k) , M∈R n×n n×k k×k k× p q×k and D f ∈ Rq× p such W1 ∈ R , W2 ∈ R , A f ∈ R , B f ∈ R , C f ∈ R that for i, j, s, t = 1, . . . , r , Υi jst < 0,

(7.28)

Π ≥ 0,

(7.29)

where ⎡

S21 ⎢  Π ⎢ ⎣   ⎡ Υ11i ⎢  ⎢ ⎢  Υi jst  ⎢ ⎢  ⎢ ⎣  

S22 M1T S24 M2T  S21  

⎤ M3T M4T ⎥ ⎥, S22 ⎦ S24

Υ12 0 0 Υ15i Υ22 j Υ23 Υ24 0  Υ33s Υ34 Υ35i   Υ44t 0    Υ55i    

⎤ Υ16i 0 ⎥ ⎥ Υ36i ⎥ ⎥, 0 ⎥ ⎥ Υ56i ⎦ Υ66

with   

Q 1i1 Q 2i2 Q 2i1 Q 2i2 Q 3i1 Q 3i2 ¯ ¯ , Q 2i  , Q 3i  ,  Q 1i4  Q 2i4  Q 3i4  

Υ11i1 Υ11i2 −Q 1 j1 − S11 − S21 −Q 1 j2 − S12 − S22 , Υ22 j  ,   Υ11i4  −Q 1 j4 − S14 − S24 



 −L iT Y + CiT BεT D Tf Y M1 M2 Υ33s1 Υ33s2 , M¯  , Υ15i   , C Tf Y  Υ33s4 M3 M4

Q¯ 1i  Υ11i Υ33s

7.3 Main Results

Υ55i  −(FiT Y − DiT BεT D Tf Y) − X + δ I − (FiT Y − DiT BεT D Tf Y)T ,   

T T T T −L di Y + Cdi Bε D f Y S11 S12 M1 M2 , Υ12  , Υ24  , Υ35i  T S12 S14 M3 M4 0 Υ11i1  −O + Q 1i1 + Q 2i1 + (d + 1)Q 3i1 − S11 , Υ11i2  −IL + Q 1i2 + Q 2i2 + (d + 1)Q 3i2 − S12 , Υ11i4  −LT + Q 1i4 + Q 2i4 + (d + 1)Q 3i4 − S14 ,   T + M1 + M1T , Υ56i  Υ56i1 Υ56i2 Υ56i3 Υ56i4 , Υ33s1  −Q 3s1 − S21 − S21     Υ16i  Υ16i1 Υ16i2 Υ16i3 Υ16i4 , Υ36i  Υ36i1 Υ36i2 Υ36i3 Υ36i4 , T + M2 + M3T , Υ33s2  −Q 3s2 − S22 − S22 T + M4 + M4T , Υ33s4  −Q 3s4 − S24 − S24 

S11 − W1 − W1T S12 − W2 − (LT I T )T , Υ662   S14 − LT − L

 S21 − W1 − W1T S22 − W2 − (LT I T )T Υ663  ,  S24 − LT − L 

T Ai O + CiT BεT B Tf I T AiT IL + CiT BεT B Tf , Υ16i1  ATf I T ATf 

T T T T T T T T T Adi O + Cdi Bε B f I Adi IL + Cdi Bε B f Υ36i1  , 0 0   T Υ56i1  Bi O + DiT BεT B Tf I T BiT IL + DiT BεT B Tf ,

T  Ai W1 + CiT BεT B Tf I T − W1 AiT W2 + CiT BεT B Tf − W2 , Υ16i2  d1 ATf I T − LT I T ATf − LT 

T T T T T T T T T A W +Cdi Bε B f I Adi W2 +Cdi Bε B f Υ36i2  d1 di 1 , 0 0   Υ56i2  d1 BiT W1 + DiT BεT B Tf I T BiT W2 + DiT BεT B Tf , 

T Ai W1 + CiT BεT B Tf I T − W1 AiT W2 + CiT BεT B Tf − W2 , Υ16i3  d ATf I T − LT I T ATf − LT 

T T T T T T T T T Adi W1 +Cdi Bε B f I Adi W2 +Cdi Bε B f Υ36i3  d , 0 0   Υ56i3  d BiT W1 + DiT BεT B Tf I T BiT W2 + DiT BεT B Tf ,  

−M1 + S21 −M2 + S22 S21 S22 ¯ Υ23  , S ,  2 T −M3 + S22 −M4 + S24  S24  

−M1 + S21 −M2 + S22 S11 S12 ¯ , S , Υ34   1 T −M3 + S22 −M4 + S24  S14 

−Q 2t1 − S21 −Q 2t2 − S22 , Υ44t   −Q 2t4 − S24

135

136

7 Reliable Filtering for T-S Fuzzy Time-Delay Systems

 Υ16i4 

1



−C Tf Z−2 1

1

T T T T L di Z−2 − Cdi Bε D f Z−2 0 1

, Υ661 

1

 Υ36i4 

1

L iT Z−2 −CiT BεT D Tf Z−2



 −O −IL ,  −LT

, I



Ik×k 0(n−k)×k

,

1

Υ56i4  FiT Z−2 − DiT BεT D Tf Z−2 , Υ66i  diag{Υ661 , Υ662 , Υ663 , −I }, the dynamic filtering system with sensor failures in (7.5) is asymptotically stable with strict dissipativity based on Definition 7.5. In addition, if the conditions have feasible solutions (O, L, A f , B f , C f , D f , W1 , W2 , Q 1i1 , Q 1i2 , Q 1i4 , Q 2i1 , Q 2i2 , Q 2i4 , Q 3i1 , Q 3i2 , Q 3i4 , S11 , S12 , S14 , S21 , S22 , S24 , M1 , M2 , M3 , M4 ), and the corresponding parameters of our proposed filter (Σ f ) in (7.3) are described as A f = L−1 A f , B f = L−1 B f , C f = C f , D f = D f .

(7.30)

Proof Based on Theorem 7.10, it is not difficult to conclude the dynamic filtering system in (7.5) is asymptotically stable and strictly dissipative if there are matrices 0 < P ∈ R(n+k)×(n+k) , 0 < Q 1i ∈ R(n+k)×(n+k) , 0 < Q 2i ∈ R(n+k)×(n+k) , 0 < Q 3i ∈ R(n+k)×(n+k) , (i = 1, . . . , r ), 0 < S1 ∈ R(n+k)×(n+k) , 0 < S2 ∈ R(n+k)×(n+k) , M ∈ R(n+k)×(n+k) and W ∈ R(n+k)×(n+k) , which satisfy (7.13) and Ξ¯ i jst < 0,

(7.31)

where  1 A¯ iT P d1 ( A¯ iT − I )W d( A¯ iT − I )W L¯ iT Z−2 ,   1 T T T T  A¯ di P d1 A¯ di W d A¯ di W L¯ di Z−2 ,   1  B¯ iT P d1 B¯ iT W d B¯ iT W F¯iT Z−2 ,

Ξ¯ 16i  Ξ¯ 36i Ξ¯ 56i



Ξ¯ 66  diag{−P, S1 − W − W T , S2 − W − W T , −I }, ⎤ ⎡ Ξ11i Ξ12 0 0 Ξ15i Ξ¯ 16i ⎢  Ξ22 j Ξ23 Ξ24 0 0 ⎥ ⎥ ⎢ ⎢  ¯  Ξ33s Ξ34 Ξ35i Ξ36i ⎥ ⎥. Ξ¯ i jst  ⎢ ⎢    Ξ44t 0 0 ⎥ ⎥ ⎢ ⎣     Ξ55i Ξ¯ 56i ⎦      Ξ¯ 66

Partition P as P

P1 P2  P3



> 0,

P2 

P4 0(n−k)×k

 ,

(7.32)

7.3 Main Results

137

where 0 < P1 ∈ Rn×n , 0 < P3 ∈ Rk×k and P4 ∈ Rk×k . Without loss of generality, P4 is assumed to be nonsingular. Denote N  P + T , with being a positive scalar and 

 

I N1 N2 N4 0 ,N  , N2  . T  n×n  N3 0(n−k)×k  0k×k Due to P > 0, it yields N > 0 for > 0 near the origin. Then it can be seen that there exists an arbitrarily small > 0 such that N4 is nonsingular and (7.31) is feasible with P replaced by N . Because of N4 being nonsingular, it follows that P4 is nonsingular. Define the following nonsingular matrices as  I 0 ,  0 P3−1 P4T −1 T  P4 P3 P4 ,  F −T S¯1 F −1 ,  F −T Q¯ 1i F −1 ,  P1 ,

F L S1 Q 1i O

 ⎫ W1 W2 P4−T P3 ⎪ W ,⎪ ⎪ ⎪ (I P4 )T P3 ⎪ ⎪ ⎬ ¯ −1 , M  F −T MF S2  F −T S¯2 F −1 , ⎪ ⎪ ⎪ ⎪ ⎪ Q 3i  F −T Q¯ 3i F −1 , ⎪ ⎭ −T ¯ −1 Q 2i  F Q 2i F ,

and A f  P4 A f P3−1 P4T , B f  P4 B f , Df  Df. C f  C f P3−1 P4T ,

(7.33)

 (7.34)

Then it yields F A¯ iT PF =

T

T F T A¯ di PF =

B¯ iT PF = F T A¯ iT WF = T F A¯ di WF =

B¯ iT WF = 1

F T L¯ iT Z−2 = 1

T F T L¯ di Z−2 =

F T L¯ iT Y = 1

AiT O + CiT BεT B Tf I T AiT IL + CiT BεT B Tf ATf I T ATf

 , 

T T T T T T T T T Adi O + Cdi Bε B f I Adi IL + Cdi Bs B f , 0 0  T  Bi O + DiT BεT B Tf I T BiT IL + DiT BεT B Tf , 

T Ai W1 + CiT BεT B Tf I T AiT W2 + CiT BεT B Tf , ATf I T ATf 

T T T T T T T T T Bε B f I Adi W2 + Cdi Bε B f Adi W1 + Cdi , 0 0   T Bi W1 + DiT BεT B Tf I T BiT W2 + DiT BεT B Tf ,  1 1  

L iT Z−2 − CiT BεT D Tf Z−2 W1 W2 , , F T WF = 1 LT I T LT −C Tf Z−2   

1 1 T T T T O IL L di Z−2 − Cdi Bε D f Z−2 , F T PF = , LT I T LT 0  

T

T T T T L i Y − CiT BεT D Tf Y Bε D f Y L di Y − Cdi T ¯T , , F L di Y = −C Tf Y 0 1

1

F¯iT Z−2 = FiT Z−2 − DiT BεT D Tf Z−2 , F¯iT Y = FiT Y − DiT BεT D Tf Y.

(7.35)

138

7 Reliable Filtering for T-S Fuzzy Time-Delay Systems

Performing congruence transformations to (7.13) and (7.31) with diag(F, F) and diag (F, F, F, F, I, F, F, F, I ), respectively, and based on (7.33)–(7.35), we can get (7.28)–(7.29). Furthermore, (7.30) is equal to ⎫ A f  P4−1 A f P4−T P3 = (P4−T P3 )−1 L−1 A f P4−T P3 , ⎬ B f  P4−1 B f = (P4−T P3 )−1 L−1 B f , ⎭ C f  C f P4−T P3 , D f  D f .

(7.36)

Actually, A f , B f , C f and D f in (7.3) can be expressed by (7.36), which means P4−T P3 can be regarded to be a similarity transformation on the state-space realization for the filter and has no influence in the filter mapping from y to e. ˆ Without loss of generality, denote P4−T P3 = I , then we get (7.30). Hence, the filter (Σ f ) in (7.3) can be established via (7.30). The proof is completed. Next, on the basis of Theorem 7.13, we design a reliable filter with strict dissipativity when the sensor failure parameter is unknown but satisfies the condition in (7.4). Theorem 7.14 For the given matrices 0 ≥ Z ∈ Rq×q , X ∈ R p× p , Y ∈ Rq× p with Z and X being symmetric, and scalar δ > 0, if there exist matrices 0 < O ∈ Rn×n , 0 < L ∈ Rk×k , 0 < Q¯ 1i ∈ R(n+k)×(n+k) , 0 < Q¯ 2i ∈ R(n+k)×(n+k) , 0 < Q¯ 3i ∈ R(n+k)×(n+k) , 0 < S¯1 ∈ R(n+k)×(n+k) , 0 < S¯2 ∈ R(n+k)×(n+k) , M¯ ∈ R(n+k)×(n+k) , W1 ∈ Rn×n , W2 ∈ Rn×k , A f ∈ Rk×k , B f ∈ Rk× p , C f ∈ Rq×k D f ∈ Rq× p , and π > 0 such that for i, j, s, t = 1, . . . , r , (7.29) and ⎡

⎤ Υˆi jst Υˆ1i Υˆ2 ⎣  −πΛ−2 0 ⎦ < 0,   −π I where 

 T T T T D Tf Y −L iT Y + CiT Bε0 Y + Cdi Bε0 D Tf Y −L di ˆ ,  , Υ Υˆ15i  35i C Tf Y 0 T T Υˆ55i  −(FiT Y − DiT Bε0 D Tf Y)−(FiT Y − DiT Bε0 D Tf Y)T − X + δ I, 

 T T T  T T T  1  T T T T T 2 ˆ Bf I Bf Bf I Bf Υ3  B f I B f −D f Z− ,

    Υˆ16i  Υˆ16i1 Υˆ16i2 Υˆ16i3 Υˆ16i4 , Υˆ36i  Υˆ36i1 Υˆ36i2 Υˆ36i3 Υˆ36i4 , 1 1   T Υˆ56i  Υˆ56i1 Υˆ56i2 Υˆ56i3 Υˆ56i4 , Υˆ56i4  FiT Z−2 − DiT Bε0 D Tf Z−2 , 

T T T T T T B f I AiT IL+CiT Bε0 Bf Ai O +CiT Bε0 , Υˆ16i1  ATf I T ATf 

T T T T T T T T T Bε0 B f I Adi IL+Cdi Bε0 B f Adi O +Cdi ˆ , Υ36i1  0 0

(7.37)

7.3 Main Results

139

  T T T T T B f I BiT IL+ DiT Bε0 Bf , Υˆ56i1  BiT O + DiT Bε0

T  T T T T T Ai W1 + CiT Bε0 B f I − W1 AiT W2 + CiT Bε0 B f − W2 ˆ Υ16i2  d1 , ATf I T − LT I T ATf − LT 

T T T T T T T T T A W +Cdi Bε0 B f I Adi W2 +Cdi Bε0 B f , Υˆ36i2  d1 di 1 0 0   T T T T T B f I BiT W2 + DiT Bε0 Bf , Υˆ56i2  d1 BiT W1 + DiT Bε0 

T T T T T T B f I − W1 AiT W2 + CiT Bε0 B f − W2 Ai W1 + CiT Bε0 ˆ Υ16i3  d , ATf − LT ATf − LT 

T T T T T T T T T Bε0 B f I Adi W2 +Cdi Bε0 B f Adi W1 +Cdi Υˆ36i3  d , 0 0   T T T T T B f I BiT W2 + DiT Bε0 Bf , Υˆ56i3  d BiT W1 + DiT Bε0    1 1  1 1 T 2 T T T 2 T 2 T T T 2 L Z − C B D Z − − L Z − C B D Z ε0 i i f di − di ε0 f − , , Υˆ36i4  Υˆ16i4  1 0 −C Tf Z−2 ⎡ T ⎤ Ci ⎤ ⎡ ⎢ 0 ⎥ Υ11i Υ12 0 0 Υˆ15i Υˆ16i ⎢ ⎥ ⎢  Υ22 j Υ23 Υ24 Υ25 Υ26 ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎢ T ⎥ ⎢  ⎢ ⎥  Υ33s Υ34 Υˆ35i Υˆ36i ⎥ ⎥ , Υˆ1i  ⎢ Ci ⎥ , Υˆi jst  ⎢ ⎥ ⎢  ⎢ ⎥   Υ44t Υ45 Υ46 ⎥ ⎢ ⎢ 0 ⎥ ⎢ ⎥ ⎦ ⎣     Υˆ55i Υˆ56i ⎢ 0T ⎥ ⎣ Di ⎦      Υ66 0 T  Υˆ2  0 0 0 0 π(D Tf Y)T π Υˆ3T , and Υ11i , Υ12 , Υ22 j , Υ23 , Υ24 , Υ33s , Υ34 , Υ44t , Υ66 are noted in Theorem 7.13. Then the dynamic filtering system in (7.5) with sensor failure is asymptotically stable with strict dissipativity on account of Definition 7.5. And the filter matrices in the form of (7.3) can be obtained by (7.30). Proof Substituting Bε by Bε0 + E ε in (7.28), then Υ¯i jst = Υˆi jst + Υˆ1i E ε Υˆ2T + Υˆ2 E ε Υˆ1iT < 0.

(7.38)

Employing x T y + y T x ≤ πx T x + π −1 y T y for π > 0, consider (7.4) and we have Υ¯i jst ≤ Υˆi jst + π Υˆ1i Λ2 Υˆ1iT + π −1 Υˆ2 Υˆ2T .

(7.39)

Therefore, (7.39) is true if (7.37) holds via using the Schur complement method. Thus it completes the proof. Remark 7.15 It can be observed that the conditions in Theorem 7.14 are strict LMIs. Consequently, the reliable filter design issue can be settled via convex optimization

140

7 Reliable Filtering for T-S Fuzzy Time-Delay Systems

algorithms and the filter matrices can be readily computed with the standard software in MATLAB. Remark 7.16 Many practical systems can be modeled as T-S fuzzy systems with time delay, such as automotive systems, robotics systems, chemical procedures and so on. In some engineering fields, to estimate system state of the corresponding system with noise inputs and prevent the occurrence of contingent failures and uninterrupted signal measurements, it is necessary to design an efficient filter. Such as the Henon mapping system in [276] can be approximated with the T-S fuzzy system with timevarying delay, and system states are estimated via filter design. In the next section, the reliable filter method based on reciprocally convex strategy can be applied into the Henon mapping system in Example 7.18 to estimate system states.

7.4 Illustrative Example In this section, Example 7.17 is given to illustrate the validity and advantage of our proposed methods. Example 7.18 is presented to apply the reliable filter design scheme to the Henon mapping system in [276], which is a representative system exhibiting chaotic behavior and usually used as the proving ground in the theory of dynamic systems, such as the Ising model in statistical mechanics. Example 7.17 Consider the T-S fuzzy system with time-varying delay in (7.21), and the relevant system parameters are set as



 −0.291 1 0.012 0.014 , Ad1 = , A1 = 0 0.95 0 0.015

  −0.1 0 0.01 0 A2 = , Ad2 = , 1 −0.2 0.01 0.015

(7.40)

which has been introduced in [82, 299]. In this example, d(k) stands for the time-varying delay, and the upper delay bound can be obtained via employing proposed approaches in Corollary 7.11. Table 7.1 shows the specific comparison, where the obtained upper bounds of time delay are given as for their corresponding lower bounds. It can be observed that the developed techniques in this chapter are better than the results shown in [82, 299]. Example 7.18 The Henon mapping system with time-varying delay is considered: ⎧ x1 (k + 1) = − [μx1 (k) + (1 − μ)x1 (k − d(k))]2 ⎪ ⎪ ⎪ ⎪ +0.3x2 (k) + ω(k), ⎨ x2 (k + 1) = μx1 (k) + (1 − μ)x1 (k − d(k)), ⎪ ⎪ y(k) = μx1 (k) + (1 − μ)x1 (k − d(k)) + ω(k), ⎪ ⎪ ⎩ z(k) = x1 (k),

(7.41)

7.4 Illustrative Example

141

Table 7.1 Allowable upper bound of d2 For different d1 d1 = 3 Corollary 7.11 of d2 = 13 [251] Theorem 7.10 of d2 = 14 [82] Theorem 7.14 of d2 = 23 [299] Corollary 7.11 d2 = 26

for different values of d1 d1 = 5 d1 = 10 d2 = 14 d2 = 19

d1 = 12 d2 = 22

d2 = 16

d2 = 20

d2 = 21

d2 = 25

d2 = 29

d2 = 32

d2 = 28

d2 = 33

d2 = 35

where ω(k) denotes the disturbance input and μ ∈ [0, 1] denotes the retarded coefficient. Set θ(k) = μx1 (k) + (1 − μ)x1 (k − d). It is assumed that θ(k) ∈ [−ν, ν], ν > 0. Utilizing the similar process in [276], the nonlinear term θ2 (k) can be completely described as θ2 (k) = h 1 (θ(k))(−ν)θ(k) + h 2 (θ(k))νθ(k), where h 1 (θ(k)), h 2 (θ(k)) ∈ [0, 1], and h 1 (θ(k)) + h 2 (θ(k)) = 1. It is easy to get the fuzzy basis functions as h 1 (θ(k)) =

! " ! " θ(k) θ(k) 1 1 1− , h 2 (θ(k)) = 1+ . 2 ν 2 ν

From the above descriptions, we have h 1 (θ(k)) = 1 and h 2 (θ(k)) = 0 when θ(k) is −ν, h 1 (θ(k)) = 0 and h 2 (θ(k)) = 1 when θ(k) is ν. The nonlinear system in (7.41) can be approximated by the T-S fuzzy model as follows:  Plant Form: Rule 1: IF θ(k) is −ν, THEN ⎧ ⎨ x(k + 1) = A1 x(k) + Ad1 x(k − d(k)) + B1 ω(k), y(k) = C1 x(k) + Cd1 x(k − d(k)) + D1 ω(k), ⎩ z(k) = L 1 x(k). Rule 2: IF θ(k) is ν, THEN ⎧ ⎨ x(k + 1) = A2 x(k) + Ad2 x(k − d(k)) + B2 ω(k), y(k) = C2 x(k) + Cd2 x(k − d(k)) + D2 ω(k), ⎩ z(k) = L 2 x(k), where

142

7 Reliable Filtering for T-S Fuzzy Time-Delay Systems





 μν 0.3 (1 − μ)ν 0 1 , Ad1 = , B1 = , μ 0 1−μ 0 0

   −μν 0.3 −(1 − μ)ν 0 1 A2 = , Ad2 = , B2 = , C 0 1−μ 0 0       C1 = μ 0 , Cd1 = 1 − μ 0 , D1 = 1, L 1 = 1 0 ,       C2 = μ 0 , Cd2 = 1 − μ 0 , D2 = 0.5, L 2 = 1 0 . A1 =

T  In this example, x(k) = x1T (k) x2T (k) , μ = 0.8, ν = 0.2, 1 ≤ d(k) ≤ 3. Resolving the conditions in Theorems 7.13 and 7.14, we can obtain following results for different filtering cases: • H∞ performance case: Z = −I , Y = 0, X = γ 2 I , Bε = 1. Resolving the conditions in Theorem 7.13, it yields γmin = 1.3774, and the relevant filter parameters are given by ⎫

  1.1457 0.3504 0.7705 ⎬ Af = , Bf = , −0.3086 0.4303 −1.0511   ⎭ C f = 10−3 × −0.7191 −0.2939 , D f = 0.5592.

(7.42)

• Strictly dissipative case: Z = −0.25, Y = −0.2, X = 1, B ε = 0.8, B¯ ε = 0.9. Resolving the conditions in Theorem 7.14, the relevant filter parameters are obtained as

  ⎫ 0.8147 0.0378 0.6635 ⎬ , Bf = , Af = −0.0847 0.6813 −0.9081 (7.43)   ⎭ C f = −0.0258 −0.0031 , D f = 0.7645.

 Set the zero initial condition, that is x(0) = 0 x(0) ˆ = 0 , and the disturbance 3sin(0.9k) signal ω(k) is assumed as ω(k) = (0.75k) 2 +3.5 . The corresponding simulation results for the designed filters are shown in Figs. 7.2, 7.3, 7.4 and 7.5. Figure 7.1 plots the time-varying delay d(k) changing randomly between d1 = 1 and d2 = 3. Figures 7.2 and 7.4 draw the signal z(k) (solid line) and its estimations zˆ (k) (dash-dot line), separately. The estimation errors e(k) are displayed in Figs. 7.3 and 7.5. It can be seen that the estimation error of the reliable dissipative filter as (7.43) is smaller than the H∞ filter as (7.42). In fact, this is obvious due to the fact that the H∞ performance is a particular situation in strict dissipative performance and thus owes more limiting conditions.

7.4 Illustrative Example

Fig. 7.1 Time-varying delays d(k)

Fig. 7.2 Signal z(k) and its estimation zˆ (k) of the H∞ filter

143

144

7 Reliable Filtering for T-S Fuzzy Time-Delay Systems

Fig. 7.3 Estimation error e(k) for the H∞ performance case

Fig. 7.4 Signal z(k) and its estimation zˆ (k) of the dissipative reliable filter

7.5 Conclusion

145

Fig. 7.5 Estimation error e(k) for the dissipative case

7.5 Conclusion A reliable filtering technique with a dissipative performance for discrete-time T-S fuzzy delayed systems was designed. First, the sufficient conditions were formulated to ensure the asymptotical stability and strict dissipativity for the dynamic filtering system by using the reciprocally convex method. The reliable filter design issue could be transformed to a convex optimization issue. Finally, some illustrative examples were put forward to show the availability of the developed techniques.

Part III

Model Reduction and Reduced-Order Synthesis

Chapter 8

Reduced-Order Model Approximation of Switched Systems

8.1 Introduction This chapter considers the reduced-order model approximation issue for discrete hybrid switched nonlinear systems through T-S fuzzy modelling. We attempt to establish a reduced-order model for a high-dimension hybrid switched system, which can approximate the original high-order model subject to the prescribed system performance index. First, the mean-square exponential stability conditions are formulated to ensure the specific H∞ performance for the resulting dynamic error system via the efficient Lyapunov stability method and ADT approach. The solutions of the relevant model reduction problems are derived using the projection technique, through which the algorithms of the reduced-order model parameters are established using a CCL method.

8.2 System Description and Preliminaries Consider the following discrete-time high-order nonlinear hybrid stochastic switched systems: x(k + 1) =

N 

      ρ j (k) C j x(k), ω(k) + G j x(k), ω(k) (k) ,

(8.1a)

      ρ j (k) H j x(k), ω(k) + J j x(k), ω(t) (k) ,

(8.1b)

j=1

y(k) =

N  j=1

where the state variable x(•) ∈ Rn denotes the state vector; ω(•) ∈ Rm denotes the disturbance input belonging to 2 [0, ∞); ω(•) is supposed as energy bounded © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 X. Su et al., Intelligent Control, Filtering and Model Reduction Analysis for Fuzzy-Model-Based Systems, Studies in Systems, Decision and Control 385, https://doi.org/10.1007/978-3-030-81214-0_8

149

150

8 Reduced-Order Model Approximation of Switched Systems

 ∞ p T and ω(•)2  k=0 ω (•)ω(•); y(•) ∈ R represents the measure output; (•) denotes a stochastic process on the probability space (Ω, F, to an increas P) related  ing family (Fk )k∈N of σ-algebras Fk ⊂ F produced by (k) k∈N . The stochastic process (•) is independent and E{(k)} = 0, E{(k)2 } = μ; N is a positive integer representing the quantity of subsystems. ρ j (k) : [0, ∞) → {0, 1}, and

N 

ρ j (k) = 1, k ∈ [1, ∞), j ∈ N = {1, 2, · · · , N },

j=1

stands for the stochastic switching signal implying which subsystem is attainable at the switching instant, ρ j is introduced for simplicity; C j (•), G j (•), H j (•) and J j (•) represent a group of nonlinear regular functions. At the discrete sampling time k, ρ j (k) may be established by x(•) or k, or both, or other hybrid schemes. Based on [186], the real-time value of ρ j is assumed to be accessible. For switching signal ρ j , the switching sequence is

( j0 , k0 ), ( j1 , k1 ), . . . , ( jκ , kκ ), . . . , | jκ ∈ N , κ = 0, 1, . . . with k0 = 0,  which means the jκ th subsystem is activated during k ∈ kκ , kκ+1 ). The T-S fuzzy modelling method is utilized to deal with the reduced-order model approximation issue for nonlinear hybrid stochastic switched systems. [ j] [ j] [ j] [ j] [ j] [ j] Fuzzy Rule Ri : IF ϑ1 (k) is Mi1 and ϑ2 (k) is Mi2 and · · · and ϑ p (k) is [ j] Mi p , THEN [ j]

[ j]

[ j]

[ j]

[ j]

x(k + 1) = Ai x(k) + Bi ω(k) + E i x(k)(k), y(k) = Ci x(k) + Di ω(k), [ j]

[ j]

where i = 1, 2, . . . , r , and r stands for the quantity of fuzzy rules; Mi1 , . . . , Mi p

[ j] [ j] [ j] represent the fuzzy sets; ϑ1 (•), ϑ2 (•), . . . , ϑ p (•) represent the premise variables,

[ j] [ j] [ j] [ j] [ j] [ j] : j ∈ N are a group of matrices denoted as ϑ p ; Ai , Bi , Ci , Di , E i [ j] [ j] [ j] [ j] [ j] parameterized by N = {1, 2, . . . , N }, and Ai , Bi , Ci , Di and E i are given

system matrices. that the premise variables are independent on ω(•). As for a set of  It is assumed  x(•), ω(•) , the final output of hybrid switched fuzzy models can be expressed as x(k + 1) = y(k) =

N 

ρj

r 

j=1

i=1

N 

r 

j=1

ρj

i=1

hi

  [ j] [ j] [ j] ϑ[ j] Ai x(k)+ Bi ω(k)+ E i x(k)(k) ,

(8.2a)

[ j] hi

  [ j] [ j] [ j] ϑ Ci x(k)+ Di ω(k) ,

(8.2b)

[ j]

8.2 System Description and Preliminaries

151



 [ j]      p [ j]  [ j]  [ j]  [ j] [ j] ϑ = Mi ϑ[ j] / ri=1 Mi ϑ[ j] , Mi ϑ[ j] = l=1 Mil ϑl , [ j] [ j] [ j] [ j] being the grade of membership of ϑl in Mil . It is assumed that with Mil ϑl       [ j] [ j] [ j]  Mi ϑ[ j]  0, i = 1, 2, . . . , r, ri=1 Mi ϑ[ j] > 0 for all k, then h i ϑ[ j]  0   [ j]  for i = 1, 2, . . . , r and ri=1 h i ϑ[ j] = 1 for all k. In this chapter, for the hybrid switched system in (8.2), the original high-order system is approximated by the reduced-order model as follows: [ j]

where h i

x(k ˜ + 1) =

N  j=1

y˜ (k) =

N 

  ρ j A˜ r[ j] x(k) ˜ + B˜ r[ j] ω(k) + E˜ r[ j] x(k)(k) ,

(8.3a)

  ρ j C˜ r[ j] x(k) ˜ + D˜ r[ j] ω(k) ,

(8.3b)

j=1

where x(k) ˜ ∈ Rk denotes the state vector of designed reduced-order model, and [ j] [ j] k < n; y˜ (k) ∈ R p represents the output of designed reduced-order model; A˜ r , B˜ r , [ j] [ j] [ j] C˜ r , D˜ r and E˜ r are  matrices  with appropriate dimensions to be decided lately. x(k) Introduce x(k) ˘ = , er (k)  y(k) − y˜ (k), the resulting dynamic error sysx(k) ˜ tem can be reformulated as x(k ˘ + 1) = er (k) =

N 

ρj

r 

j=1

i=1

N 

r 

j=1

ρj

[ j] hi

[ j] hi



ϑ

[ j]



 [ j] [ j] [ j] ˘ ˘ ˘ ˘ Bi ω(k)+ E i x(k)(k) ˘ , (8.4a) Ai x(k)+



 [ j] [ j] ˘ ˘ ϑ ˘ Di ω(k) , Ci x(k)+ [ j]

(8.4b)

i=1

where ⎧       [ j] [ j] [ j] ⎪ Bi Ei 0 0 ⎪ [ j] [ j] ⎨ A˘ [ j]  Ai ˜ ˘  [ j] , Bi [ j] , E i  [ j] , i 0 A˜ r 0 E˜ r B˜ r   ⎪ ⎪ ⎩ C˘ [ j]  C [ j] −C˜ r[ j] , D˘ [ j]  D [ j] − D˜ r[ j] . i i i i

(8.5)

Definition 8.1 The overall dynamic error system in (8.4) with ω(k) = 0 is called as ˘ satisfies the following condimean-square exponentially stable under ρ j (k) if x(k) tion:

E x(k) ˘  η x(k ˘ 0 ) (k−k0 ) , ∀k  k0 , for η  1 and 0 < < 1. Definition 8.2 For γ > 0 and 0 < β < 1, the dynamic error system in (8.4) is referred to as owing a given H∞ performance index (γ, β) if it is mean-square exponentially stable with ω(k) = 0, and when x(k) = 0, the following condition

152

8 Reduced-Order Model Approximation of Switched Systems

holds for all nonzero ω(k) ∈ 2 [0, ∞):  E

∞ 

 β s erT (s)er (s)

s=k0

< γ2

∞ 

ω T (s)ω(s).

(8.6)

s=k0

8.3 Main Results 8.3.1 Pre-specified Performance Analysis Firstly, sufficient conditions of the prescribed H∞ performance for the concerned error system (8.4) are given. Theorem 8.3 Given scalars 0 < β < 1, γ > 0 and σ  1, if there exists matrix P [ j] ∈ R(n+k)×(n+k) and P [ j] > 0 such that for j ∈ N , i = 1, 2, . . . , r, ⎡

[ j]

Ξi

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

T

T

T ⎤ [ j] [ j] C˘ i E˘ i ⎥

T

T ⎥ [ j] [ j] ⎥ D˘ i −γ 2 I B˘ i 0 ⎥  [ j] −1 ⎥ < 0, ⎥  − P 0 0 ⎥ ⎦   −I 0  [ j] −1    − μP

−β P [ j] 0    



[ j] A˘ i

(8.7)

the dynamic error system in (8.4) owns the H∞ performance index (γ, β) with meansquare exponential stability for any stochastic switching signal under the ADT satisfying Ta > Ta = lnβσ , where σ  1 and P [ j]  σ P [s] , ∀ j, s ∈ N .

(8.8)

Moreover, an upper bound of the state decay estimate function is given by E {x(k)} ˘  η x(k ˘ 0 ) (k−k0 ) ,

(8.9)

where  

 1 Ta

βσ , η 

    b , a  min λmin P [ j] , b  max λmax P [ j] . ∀ j∈N ∀ j∈N a

(8.10)

Proof Consider the stochastic switching signal ρ j (k) and fuzzy basis functions, it follows from (8.7) that

8.3 Main Results

153 N 

ρj

j=1

r 

[ j]

hi

[ j] ϑ[ j] Ξi < 0.

(8.11)

i=1

The piecewise smooth Lyapunov function is constructed as ! N

 V x(k), ˘ ρ j  x˘ T (k) ρ j P [ j] x(k), ˘

(8.12)

j=1

where R(n+k)×(n+k)  P [ j] > 0, j ∈ N are to be determined. For k ∈ [kl , kl+1 ), define " "

#



#  E V x(k ˘ + 1), ρ j − V x(k), ˘ ρj E V x(k), ˘ ρj =

N 

ρj

j=1



r 

[ j]

hi



ϑ[ j] x˘ T (k)

i=1

[ j] A˘ i

T



T [ j] [ j] [ j] ˘ P [ j] A˘ i − P [ j] + μ E˘ i P [ j] E˘ i x(k).

Then we have " "

#

# + E (1 − β)V x(k), ˘ ρj E V x(k), ˘ ρj =

N 

ρj

j=1



[ j] A˘ i

r 

[ j]

hi



ϑ[ j] x˘ T (k)

i=1

T



T [ j] [ j] [ j] ˘ P [ j] A˘ i − β P [ j] + μ E˘ i P [ j] E˘ i x(k).

(8.13)

Based on (8.11), it yields "



# E V x(k), ˘ ρ j + (1 − β)V x(k), ˘ ρj < 0, ∀k ∈ [kl , kl+1 ), ∀ j ∈ N . (8.14) As for any switching signal and k > 0, set k0 < k1 < · · · < kl < · · · < k N , (l = 1, . . . , N ) which denote the piecewise transition points of ρ j during the space of time (0, k). Due to the jl th subsystem is activated over k ∈ [kl , kl+1 ). Hence, for k ∈ [kl , kl+1 ), it follows from (8.14) that

154

8 Reduced-Order Model Approximation of Switched Systems

" "

#

# < β k−kl E V x(k E V x(k), ˘ ρj ˘ l ), ρ j (kl ) .

(8.15)

Considering (8.8) and (8.12), we can get " "

#

# ˘ l ), ρ j (kl−1 ) . E V x(k ˘ l ), ρ j (kl ) < σE V x(k Then it can be concluded from (8.15)–(8.16) and Na (k0 , k) 

k−k0 Ta

(8.16) that

" "

#

# 1 k−k0 T a E V x(k), ˘ ρj  (βσ ) E V x(k ˘ 0 ), ρ j (k0 ) .

(8.17)

From (8.12), it is workable to seek positive scalars a and b, with a  b, which are shown as (8.10), then " # "

# 2 E V x(k), ˘ ρj ,  aE x(k) ˘ "

# E V x(k ˘ 0 ), ρ j (k0 )  bx(k ˘ 0 )2 .

(8.18) (8.19)

Combine (8.17) and (8.19), then we have #

" E x(k) ˘

2

 Define 

"

# b 1 1  E V x(k), ˘ ρj  (βσ Ta )k−k0 x(k ˘ 0 )2 . a a

1

βσ Ta and we can get " #  b k−k0 x(k ˘ 0 ). E x(k) ˘  a

ln σ It yields from Definition 8.1 that if 0 < < 1, that is Ta > Ta = ceil(− ln ), the β resulting dynamic system in (8.4) with ω(k) = 0 is mean-square exponentially stable, where the function ceil( f ) represents the rounding real scalar f to the nearest integer, which is equal to or greater than f . Next, the H∞ performance index (γ, β) for concerned system is analyzed then. An index is introduced as "



˘ ρ j + erT (k)er (k) J (k)  E V x(k), ˘ ρ j + (1 − β)V x(k), # 2 T − γ ω (k)ω(k) .

8.3 Main Results

155

Then it yields J (k) =

N 

ρj

r 

j=1

[ j] hi



ϑ

[ j]

T

 x(k) ˘



ω(k)

i=1

[ j]

[ j]

Ξ11i Ξ12i [ j]  Ξ22i



 x(k) ˘ , ω(k)

where

T

T

T [ j] [ j] [ j] [ j] [ j] [ j] [ j] Ξ11i  A˘ i P [ j] A˘ i + C˘ i P [ j] E˘ i − β P [ j] , C˘ i + μ E˘ i

T

T [ j] [ j] [ j] [ j] [ j] P [ j] B˘ i + C˘ i D˘ i , Ξ12i  A˘ i

T

T [ j] [ j] [ j] [ j] [ j] P [ j] B˘ i + D˘ i D˘ i −γ 2 I. Ξ22i  B˘ i On account of (8.11) and Schur’s complement method, for k ∈ [kl , kl+1 ), it is easy to obtain J (k) < 0. Set Ω(k)  erT (k)er (k) − γ 2 ω T (k)ω(k), then "



E V x(k), ˘ ρj

#

"



< E − (1 − β)V x(k), ˘ ρj

# − Ω(k) .

(8.20)

Hence, for k ∈ [kl , kl+1 ), we can obtain from (8.20) that " "

#

# < β k−kl E V x(k ˘ l ), ρ j (kl ) E V x(k), ˘ ρj −E

" k−1

# β k−1−s Ω(s) .

(8.21)

s=kl

Considering (8.16) and (8.21), it yields " " " # k−1

#

# E V x(k), ˘ ρj < β k−kl E V x(k ˘ l ), ρ j (kl ) − E β k−1−s Ω(s) , s=kl

.. .

" "

#

# k1 −k0 σE V x(k ˘ 0 ), ρ j (k0 ) E V x(k ˘ 1 ), ρ j (k1 ) < β − σE

" k 1 −1

β

k1 −1−s

# Ω(s) .

s=k0

Then on the basis of the above conditions and Na (k0 , k) 

k−k0 , Ta

it has

156

8 Reduced-Order Model Approximation of Switched Systems

" "

#

# < β k−k0 σ Na (k0 ,k) E V x(k E V x(k), ˘ ρj ˘ 0 ), ρ j (k0 ) −E

" k−1

# β k−1−s σ Na (s,k) Ω(s) .

(8.22)

s=k0

It can be seen from the zero initial condition and (8.22) that E

" k−1

β

k−1−s

σ

Na (s,k)



erT (s)er (s)

− γ ω (s)ω(s) 2

T

#

< 0.

s=k0

Then we pre- and post-multiply the above inequality by σ −Na (0,k) , and we can obtain E

" k−1

 # β k−1−s σ −Na (0,s) erT (s)er (s) − γ 2 ω T (s)ω(s) < 0.

(8.23)

s=k0

Owing to Na (0, s)  signifies E

" k−1

β

β ln σ and Ta > − ln , it has Na (0, s)  −s ln . Therefore, (8.23) β ln σ

s Ta

k−1−s

ln β σ s ln σ erT (s)er (s)

#