Control Synthesis for Semi-Markovian Switching Systems 9819903165, 9789819903160

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Studies in Systems, Decision and Control 465

Wenhai Qi Guangdeng Zong

Control Synthesis for Semi-Markovian Switching Systems

Studies in Systems, Decision and Control Volume 465

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 worldwide 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.

Wenhai Qi · Guangdeng Zong

Control Synthesis for Semi-Markovian Switching Systems

Wenhai Qi School of Engineering Qufu Normal University Rizhao, Shandong, China

Guangdeng Zong School of Control Science and Engineering Tiangong University Tianjin, China

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-981-99-0316-0 ISBN 978-981-99-0317-7 (eBook) https://doi.org/10.1007/978-981-99-0317-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Most of the physical systems in the actual operation may be affected by random parameters, which will lead to sudden changes in the system structure. These changes may suffer from sudden events, such as random component failures, changing subsystem interconnection, unexpected events, and uncontrolled configuration changes. Then, the above system models cannot be accurately described by the traditional system, which can be characterized by Markovian switching systems. As a popular type of hybrid system, Markovian switching systems consist of some subsystems described by differential equations or difference equations and random switching rules among them. Because of a good engineering background, Markovian switching systems find wide applications in many complex dynamic systems including energy systems, macroeconomic models, networked control systems, faulttolerant systems, manufacturing systems, and sensor network systems. In these circumstances, more and more experts have begun to study Markovian switching systems from different disciplines, thus promoting the rapid development of the corresponding theory. It is well known that the transition rate plays an important role in the dynamic characteristics and is subject to the probability distribution function of the sojourn time. For Markovian switching systems, the sojourn time follows a unique memoryless exponential distribution. In such circumstances, the transition rate becomes time-invariant only related to the latest mode and is independent of sojourn time, which greatly limits the practical application scope of Markovian switching systems. Compared with Markovian switching systems, the sojourn time in semi-Markovian switching systems obeys a more general nonexponential distribution including Weibull distribution, phase-type distribution, Gaussian distribution, and so on, which leads to the time-varying characteristic of the transition rate matrix and brings both challenges and chances to the analysis and synthesis of dynamics. It should be pointed out that the traditional Markovian switching systems are known as a special class of semi-Markovian switching systems that can better model the dynamic systems subject to inevitable stochastic changes. Semi-Markovian switching systems have many applications in various fields of practical engineering, such as sliding

v

vi

Preface

mode control, adaptive control, event-triggered control, finite-time control, and fault detection. In the past decades, the control synthesis for Markovian switching systems has been intensively investigated and has attracted increasing attention. Although a large number of the corresponding works have been developed from various disciplines, there still exist many fundamental problems with less well understanding. In particular, there still lacks a unified framework to cope with the issue of control synthesis for semi-Markovian switching systems. This motivates us to write related work. The monograph aims to present up-to-date research developments and references on the control synthesis for stochastic switching systems subject to the semiMarkovian process. Owing to the particularity of semi-Markovian switching systems, many previous approaches for traditional Markovian switching systems cannot be extended to semi-Markovian switching systems, which makes analysis and synthesis of semi-Markovian switching systems full of challenges. By using multiple semiMarkovian Lyapunov function approaches, a basic theoretical framework is formed towards the issue of control synthesis for semi-Markovian switching systems. The book can be used for researchers to carry out studies on semi-Markovian switching systems and is suitable for graduate students of control theory and engineering. It may also be a valuable reference for control design of stochastic switching systems by engineers. The contents of the book are divided into 12 chapters which contain several independent yet related topics, and they are organized as follows. Chapter 1 introduces some basic background knowledge on semi-Markovian switching systems, and also describes the main work of the book. Chapter 2 considers the problem of sliding mode control law for semi-Markovian switching systems with signal quantization. Chapters 3 and 4 address the problems of stochastic stability and finite-time reachability for stochastic semi-Markovian switching systems and stochastic singular semi-Markovian switching systems via sliding mode control approach. Chapter 5 gives theoretical developments in detail for finite-time sliding mode control of semiMarkovian switching systems with quantized measurement. Chapters 6 and 7 study adaptive event-triggered sliding mode control for semi-Markovian switching systems and finite-time synchronization for delayed semi-Markovian switching neural networks with quantized measurement. Sliding mode control problem for fuzzy semi-Markovian switching systems is discussed in Chaps. 8 and 9. The specified discrete-time sliding mode control is synthesized for discrete-time semi-Markovian switching systems with denial-of-service attacks through a discrete-time semiMarkovian kernel in Chap. 10. The sliding mode control issue is addressed for networked semi-Markovian switching systems with deception attacks in Chap. 11. Finally, Chap. 12 concludes some future study directions related to the contents of the book. Rizhao, China November 2022

Wenhai Qi Guangdeng Zong

Acknowledgements

There are numerous individuals without whose constructive comments, useful suggestions, and wealth of ideas this monograph could not have been completed. Special thanks go to Prof. Wei Xing Zheng, Western Sydney University; Prof. Yang Shi, University of Victoria; Prof. Ju H. Park, Yeungnam University; Prof. Jinde Cao, Southeast University; Prof. Hamid Reza Karimi, Politecnico di Milano and Prof. Xianwen Gao, Northeastern University, for their valuable suggestions, constructive comments, and support. Next, our acknowledgments go to many colleagues who have offered support and encouragement throughout this research effort. Finally, the authors would like to express their sincere gratitude to the editors of the book for their time and kind help. The monograph was supported in part by the National Natural Science Foundation of China (62073188), the Postdoctoral Science Foundation of China (2022T150374), and the Natural Science Foundation of Shandong (ZR2021MF083). Rizhao, China November 2022

Wenhai Qi Guangdeng Zong

vii

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Stability Analysis for S-MSSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Stability for Stochastic S-MSSs . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Stability for Continuous-Time Linear S-MSSs . . . . . . . . . 1.2.3 Stability for Discrete-Time Linear S-MSSs . . . . . . . . . . . . 1.3 Control Synthesis for S-MSSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Control and Filtering for S-MSSs . . . . . . . . . . . . . . . . . . . . 1.3.2 Sliding Mode Control for S-MSSs . . . . . . . . . . . . . . . . . . . . 1.3.3 Finite-Time Control for S-MSSs . . . . . . . . . . . . . . . . . . . . . 1.3.4 Event-Triggered Control for S-MSSs . . . . . . . . . . . . . . . . . 1.4 Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 3 6 7 9 11 13 15 17 19 20

2

Quantized Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 2.3 Stochastic Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Reachability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 30 32 36 37 39 40

3

Sliding Mode Control Under Stochastic Disturbance . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 3.3 Sliding Mode Control Law Design . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Stochastic Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Reachability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43 44 46 46 53

ix

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Contents

3.6 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56 58 59

Sliding Mode Control Under Stochastic Disturbance and Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 4.3 Sliding Mode Control Law Design . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Stochastic Admissibility Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Reachability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 62 63 64 71 73 76 76

5

Finite-Time Sliding Mode Control Under Quantization . . . . . . . . . . . 79 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 80 5.3 Sliding Mode Control Law Design . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.4 Finite-Time Boundedness Analysis over Reaching Phase Within [0, T ∗ ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.5 Finite-Time Boundedness Analysis over Sliding Motion Phase Within [T ∗ , T ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.6 Gain Matrix Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.7 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6

Adaptive Event-Triggered Sliding Mode Control . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 6.3 Stochastic Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Reachability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 103 104 109 115 116 120 120

7

Finite-Time Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 7.3 Finite-Time Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 123 124 127 136 140 141

Contents

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8

Fuzzy Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 8.3 Sliding Surface Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Stochastic Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Reachability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 143 144 146 147 154 156 160 161

9

Fuzzy H∞ Sliding Mode Control Under Phase-Type Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 9.3 Sliding Mode Control Law Design . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Stochastic Stability with H∞ Performance Index . . . . . . . . . . . . . . 9.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 163 164 167 170 177 181 181

10 Sliding Mode Control Under Denial-of-Service Attacks . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 10.3 Sliding Surface Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 σ -Error Mean Square Stability Analysis . . . . . . . . . . . . . . . . . . . . . 10.5 Reachability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183 183 184 187 187 194 196 201 202

11 Sliding Mode Control Under Deception Attacks . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 11.3 Stochastic Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Reachability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205 205 206 208 212 213 217 217

12 Conclusion and Future Research Direction . . . . . . . . . . . . . . . . . . . . . . 221 12.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 12.2 Future Research Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

Symbols

R Rn Rn×m AT A−1 A>0 A≥0 A 0, such that P{|| x(t : t0 , x0 , ϑ0 ) ||< ρ, for all t ≥ t0 } ≥ 1 − ε.

(1.6)

(ii) System (1.1) is said to be stochastically asymptotically stable in the large if for any initial conditions ϑ0 , x0 , and t0 ≥ 0, it is stochastically stable with P{ lim x(t : t0 , x0 , ϑ0 ) = 0} = 1. t→∞

(1.7)

The following theorem shows the stochastic asymptotic stability in the large. Theorem 1.1 ([40]) If we find Vi (x, t) ∈ C 2 (Rn × S ; R+ ), α1 , α2 ∈ K∞ , and scalars μ ≥ 1, λi ∈ R, such that α1 (|| x(t) ||) ≤ Vi (x, t) ≤ α2 (|| x(t) ||), ∀i ∈ S , jVi (x, t) ≤ λi Vi (x, t), ∀i ∈ S , Vi (x, t) ≤ μV j (x, t), ∀i, j ∈ S , Σ μ E (eλ j τ j ) pi j ≤ 1, ∀i ∈ S , j∈S

(1.8) (1.9) (1.10) (1.11)

where C 2 (Rn × S ; R+ ) is the set of nonnegative functions Vi (x, t) : Rn × S |→ R+ , then system (1.1) realizes stochastic asymptotic stability in the large. Remark 1.2 If the condition ||x(t)|| → ∞ ⇒ Vi (x, t) → ∞ is satisfied, then the Lyapunov function Vi (x, t) is radially unbounded. The radial unboundedness is one of the important conditions to guarantee the global stability of equilibrium point. From the condition (1.8), we can see that Lyapunov function Vi (x, t) is positive definite and radially unbounded, for i ∈ S . The condition (1.9) provides the quantitative estimation of the stability degree for each subsystem, in which the larger λi means the larger degree of instability for the ith subsystem. For the deterministic switched systems, the condition (1.10) is a standard condition, from which, one has Vi (t) ≤ μV j (t) ≤ μ2 Vi (t), for i, j ∈ S , and reasonability of μ ≥ 1. The condition (1.11) plays a key role to realize stability for S-MSSs and is related to the sojourn time τ j and the semi-Markovian transition probability pi j .

6

1 Introduction

Remark 1.3 The structure of semi-Markovian process is described by the transition rate matrix and the distribution function of sojourn time. In Theorem 1.1, it does not require to constrain the transition rate within a finite interval implying that the transition rate matrix is more general. In [44], the stabilization problem is studied for stochastic S-MSSs with sojourn time subject to phase distribution. The main idea is to transform phase-type stochastic S-MSSs into stochastic MSSs through supplementary vector and model transformation approach.

1.2.2 Stability for Continuous-Time Linear S-MSSs Consider the continuous-time linear S-MSSs [45]: x(t) ˙ = A (ϑ(t))x(t), x(t0 ) = x0 , rt0 = r0 , t0 = 0,

(1.12)

where x(t) ∈ R n is the state and {ϑ(t), t ≥ 0} is the semi-Markovian process in S = {1, 2, . . . , N }. The transition rate qi j (h) is bounded as q i j ≤ qi j (h) ≤ q¯ i j . Theorem 1.2 ([45]) If we find matrix Pi > 0, i ∈ S , such that Ai T Pi + Pi Ai +

ΣN j=1

qi j (h)P j < 0,

(1.13)

then system (1.12) is stochastically stable. In [45], the authors propose to partition the time h into M sections in every working mode. Due to the time-varying property of the transition rate qi j (h), they denote q i j,m and q¯ i j,m as the lower and upper bounds of the transition rate during the mth section, respectively. The following corollary is established. Corollary 1.1 ([45]) If we find matrix Pi,m > 0, i ∈ S , m ∈ M , such that Ai T Pi,m + Pi,m Ai + Ai T Pi,m + Pi,m Ai +

ΣN j=1

q¯ i j,m P j,m < 0,

(1.14)

j=1

q i j,m P j,m < 0,

(1.15)

ΣN

then system (1.12) is stochastically stable. Here, m ∈ M = {1, 2, . . . , M}. Remark 1.4 Following [45], there are also some other works on dealing with the transition rate qi j (h). In Ref. [46], the authors have proposed a reasonable ΣM ΣM εm qi j,m , m=1 εm = 1, εm ≥ 0, m = 1, 2, . . . , M, assumption that qi j (h) = m=1

1.2 Stability Analysis for S-MSSs

7

⎧ ⎨ q + (m − 1) q¯ i j −q i j , i /= j, j ∈ S , m−1 ij and qi j,m = In [47], qi j (h) has the follow⎩ q¯ − (m − 1) q¯ i j −q i j , i = j, j ∈ S . ij m−1 ing form qi j (h) = qi j + Δqi j , where qi j = 21 (q¯ i j + q i j ) and |Δqi j | ≤ κi j with κi j = .∞ 1 (q¯ i j − q i j ). In [48], it is assumed that E {qi j (h)} = 0 qi j (h) f i (h)dh, in which the 2 mathematical expectation E {·} is adopted and there is no bounded constraint about the transition rate qi j (h). Characterized by the positivity of their state signals, output signals, or input signals, positive systems find potential applications in absolute temperature of physics and price of economics [49, 50]. When a plant is subject to positive constraints, the following stability condition is given. Consider the continuous-time linear positive S-MSSs [51]: x(t) ˙ = A (ϑ(t))x(t), x(t0 ) = x0 , ϑt0 = ϑ0 , t0 = 0,

(1.16)

where x(t) ∈ R n is the state and {ϑ(t), t ≥ 0} is the semi-Markovian process in S = {1, 2, . . . , N }. System (1.16) is positive if and only if Ai is Metzler matrix, for i ∈ S . Definition 1.3 ([50, 51]) System (1.16) {. ∞ is said to be } stochastically stable if for ϑ0 and non-negative initial state x0 , E 0 ||x(t)||1 dt| < ∞ holds. Theorem 1.3 ([51]) System (1.16) is stochastically stable if and only if there exists νi ∈ R+n , ∀i ∈ S , such that Ai νi +

ΣN j=1

q ji (h)ν j 0, T i ∈ Z≥1 , if we find matrix Pi (t) > 0, such that ∀i ∈ S , t ∈ Z[1,T i −1] (T i ∈ Z≥2 ), (Ai T )t Pi (t)Ai t − h i Pi (0) < 0,

(1.23)

and ∀i ∈ S , ∀ j ∈ S˜i , τ ∈ Z[1,T i ] (T i ∈ Z≥1 ), ΣT i τ =1

(Ai T )τ

Pi (τ ) τ Ai + (1 − θ¯ i )(Ai T )τ P j (0)Ai τ − Pi (0) < 0, ςi

(1.24)

Σ i Σ Σ where ςi =θ¯ i + (1 − p¯ i )φ, θ¯ i = T τ =1 j∈S¯ i qi j (τ ), Pi (τ ) = j∈S¯ i qi j (τ )P j (0), Σ ΣT i p¯ i = j∈S¯ i pi j , and φ satisfies τ =1 f i j (τ ) ≥ φ, ∀i ∈ S , ∀ j ∈ S˘i ∪ S`i . Remark 1.6 In [60], sufficient conditions are developed for ensuring the meansquare stability analysis of S-MSSs with incomplete semi-Markovian kernel. Meantime, the mean-square stability analysis criteria for S-MSSs with incomplete jump information and transition information are presented in [61]. In [62], the authors address the stability analysis for discrete-time non-homogeneous S-MSSs with the sojourn-time probability distribution functions subject to independent jump instants.

1.3 Control Synthesis for S-MSSs This section recalls and summarizes some recent developments for control synthesis of S-MSSs, in which the robust control theory plays an important role against model uncertainties and external disturbances. Recent works are mainly devoted to

10

1 Introduction

continuous-time S-MSSs, such as repetitive control [63], anti-disturbance control [64], reliable control [65–67], passivity [68, 69], dissipative control [70, 71], and non-fragile control [72]. Other literature related to discrete-time S-MSSs can be founded in [73–79]. Consider the continuous-time and discrete-time S-MSSs [45, 59]: x(t) ˙ = A (ϑ(t))x(t) + B(ϑ(t))u(t), x(k + 1) = A (g(k))x(k) + B(g(k))u(k),

(1.25) (1.26)

where x(t) ∈ R n and x(k) ∈ R n are the system states; u(t) ∈ R m and u(k) ∈ R m are the control inputs. {ϑ(t), t ≥ 0} and {g(k), k ∈ Z+ } are the finite continuous-time semi-Markovian process and discrete-time semi-Markovian chain. For the controllers u(t) = K (ϑ(t))x(t) and u(k) = K (g(k))x(k), one has x(t) ˙ = (A (ϑ(t)) + B(ϑ(t))K (ϑ(t)))x(t),

(1.27)

x(k + 1) = (A (g(k)) + B(g(k))K (g(k)))x(k).

(1.28)

Then, the stochastic stabilization problem is formulated as: For given systems (1.25) and (1.26), construct the feedback controllers such that the continuous-time and discrete-time closed-loop systems (1.27) and (1.28) are stochastically stable, respectively. For the control systems with model uncertainties, it needs to take the robust performance into consideration. For uncertain neutral systems with semiMarkovian switching parameters, dissipative performance is derived by resorting to new Lyapunov-Krasovskii functionals [70]. Remark 1.7 It is noted that the above aforementioned works are based on the implicit assumption of fully accessible information of system modes, i.e., the controller mode and system mode are synchronized. Unfortunately, this ideal assumption is difficult to be satisfied in practice. For example, some complex factors in networked control systems, such as communication delays and packet losses, may result in the asynchrony between the controller mode and the system mode. When this phenomenon is neglected, the designed control law usually fails to achieve an expected performance and may even make the system unstable. In order to achieve a good performance and desired goal, the existence of asynchronous behavior needs to be considered during the controller design process. For the continuous-time and discrete-time cases, the asynchronous controllers [80, 81] are designed as: ¯ u(t) = K (ϑ(t))x(t), u(k) = K (g¯ (k))x(k),

(1.29) (1.30)

¯ = {1, 2, . . . , N¯ } sub¯ where the observation modes ϑ(t) and g¯ (k) take values in S ¯ = [λ˜˜ iα ] with respect ject to the pre-known conditional probability Ω = [λ˜ iα ] and Ω ¯ to ϑ(t) or g¯ (k), the probabilities of which are defined by:

1.3 Control Synthesis for S-MSSs

11

¯ Pr {ϑ(t) = α|ϑ(t) = i} = λ˜ iα , Pr {g¯ (k) = α|g(k) = i} = λ˜˜ , iα

(1.31) (1.32)

ΣN¯ ΣN¯ ˜ where λ˜ iα ≥ 0, λ˜˜ iα ≥ 0, λ˜ i α = 1, and λ˜ iα = 1. α=1 α=1 In [80, 81], asynchronous controllers are constructed to realize stochastic stability for the corresponding continuous-time and discrete-time S-MSSs. Furthermore, an asynchronous observer is designed for the discrete-time S-MSSs [82] and an asynchronous feedback controller is proposed for non-homogeneous discrete-time S-MSSs with limited sojourn time information [83].

1.3.1 Control and Filtering for S-MSSs In order to ensure the performance of S-MSSs, many researchers focus upon H∞ control theory [58, 66, 67, 80, 84, 85], H∞ filter theory [46, 86], L1 filter theory [87], and L∞ control theory [51]. Among these control theories, H∞ control theory becomes one of the most attractive and dominated research topics. Consider the S-MSSs: x(t) ˙ = A (ϑ(t))x(t) + B(ϑ(t))u(t) + G (ϑ(t))w(t), z(t) = C (ϑ(t))x(t) + D(ϑ(t))w(t),

(1.33)

where x(t) ∈ R n , u(t) ∈ R m , z(t) ∈ R p , and w(t) ∈ R q are the state, control input, output, and disturbance input, respectively. {ϑ(t), t ≥ 0} is the finite continuous-time semi-Markovian process. Definition 1.6 ([58, 67]) For a certain there exists a controller } γ , if {. } {. ∞ positive scalar ∞ u(t) = K (ϑ(t))x(t), such that E 0 ||z(s)||2 ds ≤ γ 2 E 0 ||w(s)||2 ds under zero initial conditions, then system (1.33) with controller u(t) = K (ϑ(t))x(t) is said to achieve H∞ performance. In [58], a Luenberger observer is constructed to estimate the system state and sufficient conditions are established for stochastic stability of singular S-MSSs. For networked nonlinear S-MSSs subject to actuator fault, parameter uncertainties and partially unknown transition rate, an improved dynamic event-triggered scheme is proposed for the observer-based finite-time H∞ control [67]. Remark 1.8 Besides H∞ performance, L1 performance plays a vital role in studying controller design for S-MSSs subject to positive constraint [87–89]. Due to the non-negative property of positive systems, it is natural to apply L1 -norm to measure the input and output variables. To facilitate the system analysis, researchers turn to construct the appropriate linear co-positive Lyapunov function. Such an approach makes full use of the features of positive systems and makes it convenient to evaluate the derivative of the Lyapunov function.

12

1 Introduction

It is noted that, filtering techniques play an important role in signal estimation and can be employed to suppress the effect of the external noise without exactly known statistics. Up to now, many important developments on the filter design have been made for S-MSSs [46, 69, 86, 90, 91]. Consider the S-MSSs: x(t) ˙ = A (ϑ(t))x(t) + G (ϑ(t))w(t), y(t) = E (ϑ(t))x(t) + F (ϑ(t))w(t), z(t) = C (ϑ(t))x(t) + D (ϑ(t))w(t),

(1.34)

where x(t) ∈ R n , y(t) ∈ R l , z(t) ∈ R p , and w(t) ∈ R q are the state, output, estimated signal, and disturbance input, respectively. {ϑ(t), t ≥ 0} is the finite continuous-time semi-Markovian process. For system (1.34), construct a mode-dependent filter as below: ˙ˆ = A f (ϑ(t))x(t) ˆ + B f (ϑ(t))y(t), x(t) zˆ (t) = C f (ϑ(t))x(t), ˆ

(1.35)

where x(t) ˆ is the estimated state of the state x(t); the matrices A f (ϑ(t)), B f (ϑ(t)), and C f (ϑ(t)) are to be designed. Defining x(t) ˜ = x(t) − x(t) ˆ and e(t) = z(t) − zˆ (t), one has the following augmented system: ˜ ζ˙ (t) = A˜(ϑ(t))ζ (t) + B(ϑ(t))w(t), ˜ e(t) = C˜f (ϑ(t))x(t) ˆ + D(ϑ(t))w(t),

(1.36)

where ]T [ ζ (t) = x T (t) x˜ T (t) , ] [ A (ϑ(t)) 0 ˜ , A (ϑ(t)) = A (ϑ(t)) − B f (ϑ(t))E (ϑ(t)) − A f (ϑ(t)) A f (ϑ(t)) ] [ G (ϑ(t)) ˜ , B(ϑ(t)) = G (ϑ(t)) − B f (ϑ(t))F (ϑ(t)) [ ] ˜ C˜(ϑ(t)) = C (ϑ(t)) − C f (ϑ(t)) C f (ϑ(t)) , D(ϑ(t)) = D (ϑ(t)). Then, the H∞ filter problem is formulated as follows: For given system (1.34), design that {. system (1.36) }is stochastically stable and satisfies {. ∞ a filter such } ∞ E 0 ||e(s)||2 ds ≤ γ 2 E 0 ||w(s)||2 ds under zero initial conditions where γ is a positive scalar. For stochastic neutral-type semi-Markovian switching neural networks [69], a Luenberger-type observer is proposed to make the filter error dynamics mean-square exponentially stable with a prescribed attenuation level, in which the filter parameters

1.3 Control Synthesis for S-MSSs

13

are solved by the well developed optimization techniques. A mode dependent H∞ filter is set up for S-MSSs with sojourn time dependent transition rate [86]. Based on the event-triggered mechanism, a sufficient condition is established for the existence of dissipative filter under the framework of S-MSSs with time-varying delay [90]. Based on Lyapunov-Krasovskii formulation of the small gain condition, a delaydependent H∞ memory filter is given for S-MSSs [91]. In addition, a new numerical approximation of the Kalman-Bucy filter for S-MSSs is derived [92] by using an optimal quantization technique. In [93], the fault estimation problem is investigated for S-MSSs subject to partially unknown transition rate and output quantization. For S-MSSs with sensor fault, an observer-based sliding mode control scheme is proposed to stabilize the fault system [94].

1.3.2 Sliding Mode Control for S-MSSs Sliding mode control is widely regarded as an effective robust control method especially affected by parameter uncertainty and external disturbances [95–98]. This approach has received extensive attention due to its fast response and good transient response, and has successfully made many breakthroughs in solving complex systems and engineering control problems. Consider the S-MSSs described by x(t) ˙ = (A (ϑ(t)) + ΔA (ϑ(t), t))x(t) + B(ϑ(t))(u(t) + f (t, ϑ(t), x(t))), (1.37) y(t) = C (ϑ(t))x(t), where x(t) ∈ R n and u(t) ∈ R m are the state and control input. {ϑ(t), t ≥ 0} is the finite continuous semi-Markovian process. f (t, ϑ(t), x(t)) satisfies || f (t, ϑ(t), x(t))|| ≤ χ (ϑ(t))||x(t)|| with χ (ϑ(t)) > 0. ΔA (ϑ(t), t)) is the uncertainty. For system (1.37), consider the following linear sliding surface [104]: s(t) = − C x1 (t) + x2 (t),

(1.38)

[ ]T where x(t) = x1T (t) x2T (t) , x1 (t) is not related to the input matrix B(ϑ(t)), x2 (t) is dependent of the input matrix B(ϑ(t)), and C is the parameter to be chosen. In order to make full use of the multi-modal information of the system and reduce the conservativeness, a mode-dependent integral-type sliding surface [95] is constructed as: . t T BiT Xi (Ai + Bi Ki )x(s)ds, (1.39) s(t) = Bi Xi x(t) − 0

where Xi and Ki are the real matrices to be designed such that BiT Xi Bi is nonsingular.

14

1 Introduction

In the existing works, the proposed sliding surface for sliding mode control of S-MSSs is mostly mode-dependent. In such a case, there always exist the repetitive jumps in sliding surface to cause potential instability. Then, a mode-independent sliding surface [47] is chosen as: s(t) = D x(t),

(1.40)

Σ T where D . N j=1 h j B j . Then, the sliding mode control problem is formulated as: For the given system (1.37), construct an appropriate sliding mode control law to realize finite-time reachability of artificially specified sliding surface. For the sliding mode control law design, when the state trajectories arrive at the sliding surface, one has s(t) = 0 and s˙ (t) = 0. In such a case, we can get the equivalent controller. Next, stochastic stability is built for the closed-loop sliding dynamics by the Lyapunov function approach. Finally, designing an appropriate sliding mode control law realizes the reachability of the sliding surface. Remark 1.9 For the sliding mode control law design, a mode-dependent Lyapunov function is chosen according to the multi-modal information of the system. Next, the sliding mode control law is related to the controller gains and semi-Markovian switching. Finally, the chattering effect often exists under the presented sliding mode s(t) , where ε > 0 is a very small control law. The symbol sgn(s(t)) is replaced by ||s(t)||+ε constant. Recently, with the help of stochastic switching systems theory, a large number of results for sliding mode control strategy have been put forward concerning SMSSs [47, 48, 95–113]. For passive sliding mode control of uncertain singular SMSSs with actuator failures, sufficient conditions are derived to realize the stochastic admissibility and robust passivity [47], in which a common sliding surface is designed to weaken the jumping effect. Considering the immeasurability of the system state, an output-feedback sliding mode control law is proposed for uncertain continuous-time S-MSSs in a descriptor system setup [48]. With the plant transformation technique, the phase-type S-MSSs are transformed into the associated MSSs and an observerbased sliding mode control law is synthesized such that the associated MSSs satisfy the reaching condition [95]. Furthermore, this approach is extended to investigate integral sliding mode control law design for singular phase-type S-MSSs [96] and H∞ synchronization for master-slave phase-type S-MSSs [97]. In [100], an observer-based adaptive sliding mode control design is presented for nonlinear uncertain singular S-MSSs. Taking account of the influence of quantized factor, sliding mode control is used to stabilize S-MSSs and singular S-MSSs [101– 103]. Based on the T-S fuzzy approach, sliding mode control is proposed to realize finite-time reachability for S-MSSs [104–107]. Under the framework of asynchronization between controller mode and system mode, an asynchronous sliding mode

1.3 Control Synthesis for S-MSSs

15

control law is designed for S-MSSs [108, 109]. However, the aforementioned works described above are built subject to infinite-time region. For sliding mode control law design with specified finite-time region, it is required to consider both the reaching phase and the sliding motion phase. For given finite-time region [0, T ], finite-time sliding mode control law is investigated to discuss the reaching phase [0, T ∗ ] and the sliding motion phase [T ∗ , T ] for S-MSSs [110, 111]. In [113], the sliding mode control problem is discussed for networked S-MSSs with the measurement channel subject to random delay.

1.3.3 Finite-Time Control for S-MSSs In many practical control applications, it is expected to improve the transient performance and to drive the system state to the steady-state within a desired finite time. This practical requirements necessitate the study of the so-called finite-time stability. Consider the S-MSSs: x(t) ˙ = (A (ϑ(t)) + ΔA (ϑ(t), t))x(t) + B(ϑ(t))(u(t) + f (t, ϑ(t), x(t))), (1.41) where x(t) ∈ R n and u(t) ∈ R m are the state and input. {ϑ(t), t ≥ 0} is the finite continuous-time semi-Markovian process. f (t, ϑ(t), x(t)) satisfies || f (t, ϑ(t), x(t))|| ≤ χ (ϑ(t))||x(t)||, where χ (ϑ(t)) > 0. ΔA (ϑ(t), t)) is the uncertainty. Definition 1.7 ([84]) For a given time constant T , system (1.41){ is said to be} finite-time stable about (ε1 , ε2 , R, T ), if x T (0)Rx(0) ≤ ε1 ⇒ E x T (t)Rx(t) < ε2 , ∀t ∈ [0, T ], where 0 < ε1 < ε2 and R > 0. Here, the finite-time stability problem is formulated as: For given system (1.41), construct an appropriate controller such that the corresponding system is finite-time stable. It is noted that the stochastic stability is based on Lyapunov stability that describes the steady-state behavior over an infinite-time region. Finite-time stability describes transient performance during the finite-time level. For example, the carrier rocket will launch the satellite to a predetermined orbit within given time region. The vehicle speed cannot exceed the specified speed during the rush hours of the urban transportation systems. For finite-time control of S-MSSs, the performance is affected by many complex factors, such as initial condition, initial time, terminal condition, terminal time, and semi-Markovian process. To date, quite a few applications of finite-time stability theory for S-MSSs are proposed; for details, see [67, 75, 84, 110, 111, 114–119]. Based on the Abel Lemma, a finite-time non-fragile state estimator is designed for discrete-time semi-Markovian switching neural networks with unreliable communication links [75]. For nonlinear S-MSSs subject to immeasurable premise variables, T-S fuzzy approach is adopted to investigate the sliding mode control problem in finite-time region, in which both the

16

1 Introduction

reaching phase and the sliding motion phase are taken into consideration [110]. Inputoutput finite-time stability focuses on the output performance within a finite time, that is, for given finite-time interval and initial condition, the output response is expected to be limited within a specified threshold. In [116], with the help of a novel LyapunovKrasovskii functional, the input-output finite-time stabilization is investigated for nonlinear S-MSSs with time-varying delay, in which both the transition rate and the state delay are bounded. The robust reliable control for uncertain S-MSSs with input saturation is considered within a specified finite-time region [117]. For linear S-MSSs with generally uncertain transition rate [118], sufficient condition is established for ensuring the stochastic finite-time stability. For finite-time stability of S-MSSs subject to time delay, in contrast with Definition 1.7, a different definition is proposed. Consider the delayed S-MSSs: x(t) ˙ = (A (ϑ(t)) + ΔA (ϑ(t), t))x(t) + Ad (ϑ(t))x(t − δ(t)) + B(ϑ(t))(u(t) + f (t, ϑ(t), x(t), x(t − δ(t)))), x(t0 + e) = ψ(e), ∀e ∈ [−δ, 0],

(1.42)

where x(t) ∈ R n and u(t) ∈ R m are the state and input. δ(t) is given as 0 < δ(t) ≤ δ, ˙ ≤ h 1 . {ϑ(t), t ≥ 0} is the finite continuous-time semi-Markovian process. The δ(t) nonlinearity f (t, ϑ(t), x(t), x(t − δ(t))) satisfies || f (t, ϑ(t), x(t), x(t − δ(t)))|| ≤ χ1 (ϑ(t))||x(t)|| + χ2 (ϑ(t))||x(t)||, where χ1 (ϑ(t)) > 0, χ2 (ϑ(t)) > 0. Definition 1.8 ([114, 115]) For a given constant T , system (1.42) is said to be finite-time stable about (ε1 , ε2 , R, T ), if { } { } E sup−δ≤ϑ≤0 [x T (ϑ)Rx(ϑ), x˙ T (ϑ)R x(ϑ)] ˙ ≤ ε1 ⇒ E x T (t)Rx(t) < ε2 , ∀t ∈ [0, T ], where 0 < ε1 < ε2 and R > 0. Remark 1.10 Finite-time stabilization problem is considered for S-MSSs with time delay [114, 115], in which a different finite-time stability definition is inspired by [84]. It relaxes the restricted constraint of the derivatives of time delay and makes full use of the delay information, where some extra variables are introduced to enlarge the feasible region of the parameter solution. Compared with finite-time stability, fixed-time stability is concerned with steady state characteristics under a fixed-time interval. There exists a time constant T such that the state trajectories can converge to the equilibrium point within a fixed-time interval [0, T ]. Fixed-time stability has no constraints about the initial conditions and the system state [120]. Fixed-time global synchronization is investigated, where the sliding mode control law is designed to drive the state trajectory onto the prescribed sliding manifold in fixed time [121].

1.3 Control Synthesis for S-MSSs

17

1.3.4 Event-Triggered Control for S-MSSs The traditional control approach is generally based on time-triggered mechanism. It always needs to periodically perform the output sampling, control quantity calculation and other control actions of the controlled object, and the time interval of control operation is determined by the overall design index of the system in advance. Then, the system is open-loop operation in the sampling period and is required to execute control instructions frequently enough to ensure the worst-case performance. Since the time interval of control action can not be adjusted in real time according to the control requirement, the time-triggered mechanism may cause unnecessary waste of limited computing and communication resources. Unlike the traditional periodical sampling scheme in computer control systems, the event-triggered mechanism is proposed in the late 1990s [122, 123]. Under the event-triggered mechanism, the sampled signal is transmitted only when the pre-designed triggering condition is satisfied, thus significantly saving the communication resource. Recent works have been devoted to event-triggered control for S-MSSs (see, e.g., [64, 66, 67, 77, 84, 85, 90, 107, 109, 124–132]). In [65, 90], an event-triggered sampling scheme is designed as: tk+1 = inf{t > tk || [x(t) − x(tk )]T Πi [x(t) − x(tk )] ≥ δi x T (t)Πi x(t)},

(1.43)

where δi ∈ (0, 1) and Πi > 0 are the event-triggered parameters, and tk (k ∈ N) denotes the triggering instant. In the event-triggered sampling scheme (1.43), the input and output of the event trigger are continuous-time signal x(t) and discrete-time signal x(tk ), respectively. Through the continuous check of state signal x(t), the state is sampled only when the scheme (1.43) is satisfied. The event-triggered sampling scheme can save network bandwidth and energy consumption. However, due to the continuous check of the event-triggered sampling scheme, the computational burden is increased, which, on the other hand, makes the implementation difficult. In order to overcome the drawback of event-triggered sampling scheme (1.43), a sampled-data based discrete event-triggered scheme [77, 107, 124, 128, 130, 131, 133–135] is proposed as: tk+1 h = tk h + min j∈N { j h|e T ((tk + j )h)Πi e((tk + j)h) > δi x T ((tk + j)h)Πi x((tk + j )h)},

(1.44)

where δi ∈ (0, 1), Πi > 0, e((tk + j )h) = x((tk + j)h) − x(tk h), h is sampling period, tk h(tk , k ∈ N) and (tk + j)h denote the latest triggering instant and current sampling instant. Different from the continuous-time event-triggered scheme [64, 90], the discrete event-triggered scheme does not require the extra hardware, in which Zeno behavior is absolutely excluded, since the minimum event-trigger interval is h.

18

1 Introduction

In [67, 84, 85], an improved discrete event-triggered communication scheme is given as: tk+1 h = tk h + min j∈N { j h|e T ((tk + j )h)Πi e((tk + j)h) > δi x T ((tk + j)h)Πi x((tk + j )h)},

(1.45)

k h) . where e((tk + j)h) = x¯ (tk h) − x(tk h), x¯ (tk h) = x((tk + j )h)−x(t 2 To further reduce the triggering frequency, the following adaptive event-triggered sampling scheme is proposed in [132, 136]:

tk+1 h = tk h + min j∈N { j h|e T ((tk + j )h)Πi e((tk + j)h) > υ(tk h)x T (tk h)Πi x(tk h)},

(1.46)

where the adaptive parameter υ(tk h) represents a triggered threshold. Here, υ(t) is updated by υ(t) ˙ =

1 1 ( − υ0 )e T (tk, j h)Πi e(tk, j h), υ(t) υ(t)

(1.47)

where υ0 > 0, tk, j = tk + j, and υ(tk h) is the sampled value of υ(t) at t = tk h. In [129], a dynamic event-triggered sampling scheme is determined as: ˜ tk+1 h = tk h + min{ j h|e T ((tk + j )h) Fe((t k + j )h) j∈S

− σ x ((tk + j )h) F˜ x((tk + j )h) ≥ F((tk + j)h)}, T

(1.48)

where e((tk + j )h) = x(tk h) − x((tk + j )h), {( jk + j)h} ⊆ [tk h, tk+1 h), F˜ > 0 is a weighting matrix, and σ ∈ [0, 1] represents the predefined threshold. Then, the dynamic rule is determined by the function . F(tk, j h) =

f0 , x(tk, j h) ≤ x(tk, j−1 h), f (tk, j h), otherwise, f (t

h)

j−1 where f 0 > 0 is a positive constant. f (tk, j h) = 1+qeT (t k, h) ˜ k, j h) is a monotonically Fe(t k, j decreasing function, where q > 0 is a threshold parameter, and the initial values of F(tk, j h) and f (tk, j h) are given as F(0) = f (0) = f 0 . In addition, for a class of Takagi-Sugeno fuzzy S-MSSs [66], different eventtriggered schemes are adopted to compare their efficacy in network resource saving. Considering the asynchronous phenomenon between system mode and controller mode [109], a novel event-triggered detector condition is developed based on the quantized state vector. For the stability of semi-Markovian switching networked control systems with time-varying delay and actuator faults [127], an improved static event-triggered mechanism is introduced by employing the Bessel-Legendre inequality approach.

1.4 Organization of the Book

19

Remark 1.11 For event-triggered control system, there are mainly four types of system modeling: hybrid system model, piecewise system model, perturbed system model, and time-delay system model. Due to the dynamics of the event-triggeredinduced error, the hybrid system model truly describes behaviors of an event-triggered control system, which makes it less conservative than the perturbed system model. The piecewise system model has the least conservatism, while computational complexity of the perturbed system model is the lowest. In the time-delay system model, it is convenient to study the effects of network-induced factors and co-design the event-triggered scheme and controller. Then, the time-delay theory is adopted to solve the corresponding event-triggered control issue. Recently, many works focus on the time-delay system model [67, 77, 84, 85, 107, 114, 119, 124, 128–134]. For the other three types of system modeling, please refer to the recent work in [137].

1.4 Organization of the Book This book studies the control synthesis for semi-Markovian switching systems. Structure of the book is summarized as follows. This chapter has introduced the system description and some background knowledge, and also addressed the motivations of the book. Chapter 2 investigates the problem of sliding mode control for a class of S-MSSs with signal quantization. By using the weak infinitesimal operator theory, sufficient conditions are given for the corresponding stochastic stability criteria. Furthermore, an appropriate sliding mode control law is proposed to drive the state signals onto the predefined manifold and the effect of quantization error can be effectively attenuated. Chapter 3 deals with the problem of sliding mode control design for nonlinear stochastic S-MSSs. By using the stochastic semi-Markovian Lyapunov function, sojourn-time-dependent sufficient conditions are developed to guarantee the closedloop sliding mode dynamics stochastically stable. The sliding mode control law is constructed to ensure reachability of the sliding mode dynamics in a finite-time level. Chapter 4 considers the problem of sliding mode control design for nonlinear stochastic singular S-MSSs. A set of sufficient conditions are developed such that the closed-loop sliding mode dynamics are stochastically admissible. Then, the sliding mode control law is proposed to ensure the reachability in a finite-time region. In Chap. 5, we address the sliding mode control for S-MSSs with quantized measurement in finite-time level. By the key points of stochastic semi-Markovian Lyapunov function and observer design theory, a desired sliding mode control law is constructed to guarantee that the system trajectories can arrive at the specified sliding surface within an assigned finite-time level. Sojourn-time-dependent sufficient conditions are established to ensure the required finite-time boundedness performance including both reaching phase and sliding motion phase. The problem of sliding mode control for stochastic switching systems subject to semi-Markovian process via an adaptive event-triggered mechanism is investigated in Chap. 6. An adaptive event-triggered mechanism is adopted to effectively reduce the

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number of triggering than static event-triggered mechanism. Sojourn-time-dependent sufficient conditions are established for stochastic stability. A suitable sliding mode control law is designed such that the system state can be driven onto the specified sliding surface in a finite-time region. Chapter 7 addresses the issue of finite-time synchronization for delayed semiMarkovian switching neural networks with quantized measurement. Sufficient conditions are constructed to realize finite-time synchronization of the resulting error system over a finite-time interval. Then, the solvability conditions for the desired finite-time controller can be determined under a linear matrix inequality framework. Chapter 8 investigates the problem of fuzzy sliding mode control for T-S fuzzy S-MSSs in the absence of quantization. Robust stochastic stability criteria are given for the corresponding system. Then, the desired sliding mode control is constructed to depend on the quantizer level. Chapter 9 focuses on the sliding mode control for phase-type stochastic nonlinear S-MSSs via the T-S fuzzy strategy. By using the plant transformation method and the supplementary variable technique, phase-type S-MSSs are equivalent to associated MSSs. By using Lyapunov functions and inequality optimization problems, sufficient conditions are provided for stochastic stability with a prescribed H∞ performance index. A fuzzy sliding mode control law is synthesized to guarantee that the associated T-S fuzzy MSSs fulfill the reaching condition in bounded time. Chapter 10 considers the problem of discrete-time sliding mode control for networked uncertain S-MSSs subject to random denial-of-service attacks. Under the discrete-time semi-Markovian kernel, the specified discrete-time sliding mode control law is synthesized to guarantee the finite-time reachability of the sliding region. Chapter 11 addresses the sliding mode control issue for networked S-MSSs under deception attacks. In the fact of an appropriate sliding mode control law, the states are driven onto the sliding region despite semi-Markovian switching. Finally, in Chap. 12, the perspectives of control synthesis for S-MSSss are concluded and predicated.

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114. Jin, Y.J., Qi, W.H., Zong, G.D.: Finite-time synchronization of delayed semi-Markov neural networks with dynamic event-triggered scheme. Int. J. Control, Autom., Syst. 19(6), 2297– 2308 (2021) 115. Shan, Y.N., She, K., Zhong, S.M., Cheng, J., Zhao, C., Fu, Q.H.: Finite-time boundedness of state estimation for semi-Markovian jump systems with distributed leakage delay and linear fractional uncertainties. Int. J. Syst. Sci. 50(12), 2362–2384 (2019) 116. Yan, H.C., Tian, Y.X., Li, H.Y., Zhang, H., Li, Z.C.: Input-output finite-time mean square stabilization of nonlinear semi-Markovian jump systems. Automatica 104, 82–89 (2019) 117. Aravindh, D., Sakthivel, R., Kong, F.C., Anthonid, S.M.: Finite-time reliable stabilization of uncertain semi-Markovian jump systems with input saturation. Appl. Math. Comput. 384, Article Id 125388 (2020) 118. Cheng, G.F., Ju, Y.Y., Mu, X.W.: Stochastic finite-time stability and stabilisation of semiMarkovian jump linear systems with generally uncertain transition rates. Int. J. Syst. Sci. 52(1), 185–195 (2021) 119. Qi, W.H., Hou, Y.K., Zong, G.D., Ahn, C.K.: Finite-time event-triggered control for semiMarkovian switching cyber-physical systems with FDI attacks and applications. IEEE Trans. Circuits Syst. I: Regul. Pap. 68(6), 2665–2674 (2021) 120. Zhao, W., Wu, H.Q.: Fixed-time synchronization of semi-Markovian jumping neural networks with time-varying delays. Adv. Differ. Equ. 2018(1), 1–21 (2018) 121. Wang, Z.B., Wu, H.Q.: Global synchronization in fixed time for semi-Markovian switching complex dynamical networks with hybrid couplings and time-varying delays. Nonlinear Dyn. 95(3), 2031–2062 (2019) 122. Arzen, K.E.: A simple event-based PID controller. In: Proceeding of 14th World Congress of IFAC, vol. 18, pp. 423–428 (1999) 123. Astrom, K.J., Bernhardsson, B.M.: Comparison of periodic and event based sampling for first order stochastic systems. In: Proceeding of 14th World Congress of IFAC, vol. 11, pp. 301–306 (1999) 124. Wu, X.H., Mu, X.W.: Event-triggered control for networked nonlinear semi-Markovian jump systems with randomly occurring uncertainties and transmission delay. Inf. Sci. 487, 84–96 (2019) 125. Syed Ali, M., Vadivel, R., Kwon, O.M.: Decentralised event-triggered impulsive synchronisation for semi-Markovian jump delayed neural networks with leakage delay and randomly occurring uncertainties. Int. J. Syst. Sci. 50(8), 1636–1660 (2019) 126. Pradeep, C., Yang, C., Murugesu, R., Rakkiyappan, R.: An event-triggered synchronization of semi-Markov jump neural networks with time-varying delays based on generalized freeweighting-matrix approach. Math. Comput. Simul. 155, 41–56 (2019) 127. Lu, H.Q., Guo, C.Q., Hu, Y., Zhou, W.N.: Event-triggered stability analysis of semi-Markovian jump networked control system with actuator faults and time-varying delay via BesselLegendre inequalities. Complexity 2019, Article Id 6927528 (2019) 128. Zhang, H.Y., Qiu, Z.P., Cao, J.D., Abdel-Aty, M., Xiong, L.L.: Event-triggered synchronization for neutral-type semi-Markovian neural networks with partial mode-dependent timevarying delays. IEEE Trans. Neural Netw. Learn. Syst. 31(11), 4437–4450 (2020) 129. Zong, G.D., Ren, H.L.: Guaranteed cost finite-time control for semi-Markov jump systems with event-triggered scheme and quantization input. Int. J. Robust Nonlinear Control 29(15), 5251–527 (2019) 130. Wang, J., Chen, M.S., Shen, H.: Event-triggered dissipative filtering for networked semiMarkov jump systems and its applications in a mass-spring system model. Nonlinear Dyn. 87(4), 2741–2753 (2017) 131. Shen, H., Chen, M.S., Wu, Z.G., Cao, J.D., Park, J.H.: Reliable event-triggered asynchronous extended passive control for semi-Markov jump fuzzy systems and its application. IEEE Trans. Fuzzy Syst. 28(8), 1708–1722 (2020) 132. Qi, W.H., Zong, G.D., Zheng, W.X.: Adaptive event-triggered SMC for stochastic switching systems with semi-Markov process and application to boost converter circuit model. IEEE Trans. Circuits Syst. I: Regul. Pap. 68(2), 786–796 (2021)

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133. Peng, C., Han, Q.L.: A novel event-triggered transmission scheme and L2 control co-design for sampled-data control systems. IEEE Trans. Autom. Control 58(10), 2620–2626 (2013) 134. Yue, D., Tian, E., Han, Q.L.: A delay system method for designing event-triggered controllers of networked control systems. IEEE Trans. Autom. Control 58(2), 475–481 (2013) 135. Zong, G.D., Ren, H.L., Karimi, H.R.: Event-triggered communication and annular finite-time H∞ filtering for networked switched systems. IEEE Trans. Cybern. 51(1), 309–317 (2021) 136. Gu, Z., Tian, E.G., Liu, J.L.: Adaptive event-triggered control of a class of nonlinear networked systems. J. Frankl. Inst. 354(9), 3854–3871 (2017) 137. Peng, C., Li, F.Q.: A survey on recent advances in event-triggered communication and control. Inf. Sci. 457, 113–125 (2018)

Chapter 2

Quantized Sliding Mode Control

In this chapter, the problem of sliding mode control design for nonlinear semi-Markovian switching systems with quantization is discussed. First, a mode-independent sliding surface is designed to avoid the potential repetitive jumping effects. Then, sufficient conditions are obtained for stochastic stability through the weak infinitesimal operator theory. Furthermore, an appropriate sliding mode control law is designed to force the system states onto the predefined sliding surface to attenuate the effect of quantization error. Finally, a single-link robot arm model is shown to illustrate the effectiveness.

2.1 Introduction Due to the powerful model ability, Markovian switching systems (MSSs) have received more and more attention (see e.g., [1–3]). As an important factor, the sojourn time of transition rate is generally assumed to satisfy the exponential distribution. As the sojourn time may conform to a more general non-exponential distribution in many practical applications, the assumption is ideal. Different from MSSs, the sojourn time in continuous-time S-MSSs do not depend on the exponential distributions, whose transition rates possess “memory” property. Hence, it can be said that MSSs are recognized as a particular case of S-MSSs. In recent years, many results of S-MSSs have been reported (see e.g., [4–10]). As an effective robust control strategy, sliding mode control approach [11, 12] has attracted much increasing. Under the sliding mode control framework, the state signals can be forced onto an artificially specified hyper-surface (i.e. the sliding mode surface) in a finite-time interval. Recently, a few results of sliding mode control approach for stochastic switching systems have been reported (see e.g., [7–9, 13–17]).

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 W. Qi and G. Zong, Control Synthesis for Semi-Markovian Switching Systems, Studies in Systems, Decision and Control 465, https://doi.org/10.1007/978-981-99-0317-7_2

29

30

2 Quantized Sliding Mode Control

With the rapid development of digital computers and digital communication facilities, the signal quantization has drawn much attention. Moreover, it is an effective strategy to save the network resources [18, 19]. For digitized network communication systems, the output signals of the controller must be quantized due to the finite rate network. Very recently, the dynamical behaviors of quantized control systems have attracted a high amount of attention and many significant results have been reported; see for instance, [7, 10, 20–22]. For Markovian process, there is a constraint that the sojourn time follows the memoryless exponential distribution [1–3, 13–17, 21, 22], limiting its scope of application. Next, the quantized constraint is not considered in [2, 3, 13–17]. Moreover, due to semi-Markovian process, uncertainty, signal quantization, and nonlinearity, it is challenging to investigate such system, which motivates our study. In this chapter, we will discuss the quantized sliding mode control design for nonlinear S-MSSs. The contributions are summarized as: (i) The assumption that the sojourn time in stochastic switching systems satisfies an exponential distribution [1–3, 13–17, 21, 22], is removed. Then, a mode-independent sliding surface is constructed to avoid the repetitive jumping effects. (ii) Based on semi-Markovian Lyapunov function and logarithmic quantizer, sufficient conditions for stochastic stability are proposed. (iii) Through a bounded area around the sliding surface, the sliding mode control law is constructed to force the system states onto the sliding surface within a finite time.

2.2 Problem Statements and Preliminaries Consider the nonlinear S-MSSs as z˙ (t) = (A (αt ) + ΔA (αt ))z(t) + B(αt )(u(t) + f (t, αt , z(t))),

(2.1)

where z(t) ∈ Rn and u(t) ∈ Rm are state vector and input vector. {αt , t ≥ 0} means the semi-Markovian process in Θ = {1, 2, . . . , ℘} with the probability transition given as . Pr {αt+Δ¯ = ρ|αt = ξ } =

¯ + o(Δ), ¯ κξρ (l)Δ ξ /= ρ, ¯ ¯ ξ = ρ, 1 + κξ ξ (l)Δ + o(Δ),

where l means the sojourn time, κξρ (l) ≥ 0 stands for the transition rate from ξ .℘ κξρ (l) = −κξ ξ (l). The transition rate is given as to ρ for ξ /= ρ, and ρ=1,ρ/=ξ

κ ξρ ≤ κξρ (l) ≤ κ¯ ξρ with real constant scalars κ ξρ and κ¯ ξρ . For αt = ξ ∈ Θ, A (αt ), ΔA (αt ), B(αt ), and f (t, αt , z(t)) are, respectively, denoted as Aξ , ΔAξ (t), Bξ , and f ξ (t, z(t)). The nonlinear function and the uncertainty satisfy

2.2 Problem Statements and Preliminaries

31

|| f ξ (t, z(t))|| ≤ ρξ ||z(t)||, ΔAξ (t) = Mξ Fξ (t)Nξ , where ρξ denotes known scalar, Mξ and Nξ stand for known matrices with Fξ (t) satisfying FξT (t)Fξ (t) ≤ I . [ Consider the control]signal u(t) quantized via a logarithmic quantizer Q(·) = Q1 (·) Q2 (·) · · · Qm (·) , where Q(·) is assumed to be symmetric, that is, Q. (−u . (t)) = −Q. (u . (t)), 1 ≤ . ≤ m. The set of quantized levels of Q. (·) takes { } . { (0) } . {0} , ±ϑ. .. = ±ϑ.(ι) : ϑ.(ι) = (φ. )ι ϑ.(0) , ι = ±1, ±2, . . . 0 < φ. < 1, ϑ.(0) > 0, where φ. and ϑ.(0) mean the quantizer density and the initial quantization values of the sub-quantizer Q. (·) given as ⎧ ϑ (ι) ⎨ ϑ.(ι) , if 1+λ. . < u . (t) < Q. (u . (t)) = 0, if u . (t) = 0, ⎩ −Q. (−u . (t)), if u . (t) < 0,

(ι) ϑ. 1−λ.

, (2.2)

. with λ¯ = max1≤. ≤m {λ. }, λ = min1≤. ≤m [λ. ], λ. = 1−φ , 1 ≤ . ≤ m, ι = 1+φ. ±1, ±2, . . . , which means that 0 < λ. < 1, 0 < λ¯ < 1. Then, one has Q(u(t)) = (I + Λ)u(t), where Λ = diag{Λ1 , Λ2 , . . . , Λm }, and Λ. ∈ [−λ. , λ. ], 1 ≤ . ≤ m. Also, one has −1 < Λ. < 1. Replacing u(t) with (I + Λ)u(t) in (2.1) yields

z˙ (t) = (Aξ + ΔAξ (t))z(t) + Bξ ((I + Λ)u(t) + f ξ (t, z(t))).

(2.3)

Definition 2.1 ([1]) System{.(2.1) is said to } be stochastically stable if for α0 ∈ Θ ∞ and z 0 ∈ Rn , there holds E 0 ||z(t)||2 dt < ∞. Lemma 2.1 ([4]) For any matrices D ∈ Rn×n f , E ∈ Rn×n f and F(t) ∈ Rn f ×n f with F T (t)F(t) ≤ I , one has D F(t)E + E T F T (t)D T ≤ ε D D T + ε−1 E T E, where ε is any positive scalar. For mode-dependent sliding surface [7–9, 13–16], the repetitive jumps exist in sliding surface to cause potential instability of sliding mode motion. To avoid this issue, a mode-independent sliding surface is proposed as s(t) = D z(t), where D .

.℘

ρ=1

h ρ BρT [12].

(2.4)

32

2 Quantized Sliding Mode Control

Next, the sliding mode control law is designed as u(t) = BξT Pξ z(t) − ηξ sgn((Xξ DBξ )T s(t)),

(2.5)

where nonsingular matrix Pξ , matrix Xξ , and positive scalar ηξ will be given later. Remark 2.1 Different from the traditional control approach [4–6, 10], the sliding mode control law can deal with the system uncertainty with strong robustness to disturbance and unmodeled dynamics, especially for nonlinear systems. Owing to simple algorithm and fast response, the sliding mode control strategy shows robustness to external noise and parameter perturbation. Combining (2.5) and (2.3) yields z˙ (t) = (Aξ + ΔAξ (t) + Bξ (I + Λ)BξT Pξ )z(t) − Bξ ((I + Λ)ηξ sgn((Xξ DBξ )T s(t)) − f ξ (t, z(t))).

(2.6)

The aim of this chapter is to construct an appropriate sliding mode control law such that the system trajectories can reach the specified sliding surface within a finite time and make the closed-loop system stochastic stable.

2.3 Stochastic Stability Analysis Applying the Lyapunov function and probability theory, sufficient conditions are proposed to realize stochastic stability for the sliding dynamics and solve the deterministic matrices Pξ and Xξ in Theorem 2.1. Theorem 2.1 If there exist symmetric matrix Pξ > 0, matrix Xξ , and scalars ε1ξ > 0, ε2ξ > 0, ∀ξ ∈ Θ, such that Pξ Bξ = D T Xξ DBξ , ¯ ξ1 Π ξ Π1

< 0,

(2.8)

< 0,

(2.9)

where ¯ ξ1 Π ¯ ξ11 Π ξ

Π 11

(2.7)

[ ] [ ξ ξ ] ξ ¯ ξ11 Π21 Π 11 Π21 Π ξ = Π = , , ξ ξ 1 ∗ Π31 ∗ Π31 .℘ = Pξ Aξ + AξT Pξ + κ¯ ξρ Pρ , ρ=1 .℘ κ ξρ Pρ , = Pξ Aξ + AξT Pξ + ρ=1

2.3 Stochastic Stability Analysis

33

ξ

Π21 = [Pξ Bξ , Pξ Mξ , ε1ξ Nξ T , Pξ Bξ , ε2ξ ρξ I ], 1 ξ Π31 = −diag{ (I + λ¯ I )−1 , ε1ξ I , ε1ξ I , ε2ξ I , ε2ξ I }, 2 then system (2.6) achieves stochastic stability.

Proof For Lyapunov function S1 (z(t), ξ ) = z T (t)Pξ z(t),

(2.10)

one has jS1 (z(t), ξ ) 1 ¯ αt+Δ¯ )|αt = ξ } − S1 (z(t), ξ )] = lim [E {S (z(t + Δ), ¯ ¯ Δ→0 Δ 1 .℘ ¯ ¯ Pr {αt+Δ¯ = ρ|αt = ξ }z T (t + Δ)P = lim [ ρ z(t + Δ) ¯ ρ=1,ρ/=ξ ¯ Δ Δ→0 T ¯ ¯ + Pr {αt+Δ¯ = ξ |αt = ξ }z T (t + Δ)P ξ z(t + Δ) − z (t)Pξ z(t)] Pr {αt+Δ¯ = ρ, αt = ξ } T 1 .℘ ¯ ¯) = lim [ z (t + Δ)P ρ z(t + Δ ¯ ρ=1,ρ/=ξ ¯ Pr {αt = ξ } Δ Δ→0 Pr {αt+Δ¯ = ξ, αt = ξ } T T ¯ ¯ z (t + Δ)P + ξ z(t + Δ) − z (t)Pξ z(t)] Pr {αt = ξ } ¯ − Vξ (l)) T 1 .℘ λξρ (Vξ (l + Δ) ¯ ¯ = lim [ z (t + Δ)P ρ z(t + Δ) ¯ ρ=1,ρ/ = ξ ¯ 1 − Vξ (l) Δ Δ→0 ¯ T 1 − Vξ (l + Δ) ¯ ¯ − z T (t)Pξ z(t)]. + (2.11) z (t + Δ)P ξ z(t + Δ) 1 − Vξ (l) Considering the expansion of Taylor formula leads to ¯ + o(Δ), ¯ ¯ = z(t) + z˙ (t)Δ z(t + Δ)

(2.12)

¯ → 0. where Δ Then, one has jS1 (z(t), ξ ) ¯ − Vξ (l)) 1 .℘ λξρ (Vξ (l + Δ) ¯ + o(Δ)] ¯ T = lim [ [z(t) + z˙ (t)Δ ¯ ρ=1,ρ/ = ξ ¯ 1 − Vξ (l) Δ→0 Δ ¯ 1 − Vξ (l + Δ) ¯ + o(Δ)] ¯ T ¯ + o(Δ)] ¯ + [z(t) + z˙ (t)Δ Pρ [z(t) + z˙ (t)Δ 1 − Vξ (l) ¯ + o(Δ)] ¯ − z T (t)Pξ z(t)]. Pξ [z(t) + z˙ (t)Δ (2.13)

34

2 Quantized Sliding Mode Control ¯ ¯ 1−V ξ (l+Δ) V (l)−V (l+Δ) = 1, lim ξ 1−V ξξ(l) 1−V ξ (l) ¯ ¯ Δ→0 Δ→0 = λξρ κξ (l), ξ /= ρ [4], we have

¯ V ξ (l+Δ)−V ξ (l) ¯ (1−V ξ (l)) Δ

= κξ (l),

¯ − Vξ (l)) T 1 .℘ λξρ (Vξ (l + Δ) z (t)Pρ z(t) ¯ ρ=1,ρ/=ξ ¯ 1 − Vξ (l) Δ Δ→0 .℘ .℘ λξρ κξ (l)z T (t)Pρ z(t) = κξρ (l)z T (t)Pρ z(t). =

(2.14)

Based on lim and κξρ (l)

= 0, lim

¯ Δ→0

lim

ρ=1,ρ/=ξ

ρ=1,ρ/=ξ

Combining (2.11)–(2.14) yields E {jS1 (z(t), ξ )} = 2z T (t)Pξ [(Aξ + ΔAξ (t) + Bξ (I + Λ)BξT Pξ )z(t) + Bξ (−(I + Λ)ηξ sgn((Xξ DBξ )T s(t)) + f ξ (t, z(t)))] .℘ + z T (t) κξρ (l)Pρ z(t). ρ=1

(2.15)

By Lemma 2.1, we have 2z T (t)Pξ Δ Aξ (t)z(t) −1 T z (t)Pξ Mξ MξT Pξ z(t) + ε1ξ z T (t)Nξ T Nξ z(t), ≤ ε1ξ

2z T (t)Pξ Bξ f ξ (t, z(t)) −1 T z (t)Pξ Bξ BξT Pξ z(t) + ε2ξ f ξT (t, z(t)) f ξ (t, z(t)) ≤ ε2ξ −1 T ≤ ε2ξ z (t)Pξ Bξ BξT Pξ z(t) + ε2ξ ρξ2 z T (t)z(t).

(2.16)

Considering the condition (2.7), one has − 2z T (t)Pξ Bξ (I + Λ)ηξ sgn((Xξ DBξ )T s(t)) = − 2z T (t)Pξ Bξ (I + Λ)ηξ sgn((Xξ DBξ )T D z(t)) = − 2z T (t)Pξ Bξ (I + Λ)ηξ sgn(BξT Pξ z(t)) ≤ − 2ηξ (1 + λ)|Pξ Bξ z(t)| ≤ −2ηξ (1 + λ)||Pξ Bξ z(t)|| < 0,

(2.17)

where ||Pξ Bξ z(t)|| ≤ |Pξ Bξ z(t)|. The transition rate κξρ (l) can be represented by κξρ (l) = θ1 κ ξρ + θ2 κ¯ ξρ , where θ1 + θ2 = 1 and θ1 > 0, θ2 > 0. By tuning θ1 and θ2 , all possible κξρ (l) ∈ [κ ξρ , κ¯ ξρ ] can be obtained. Thus, it holds E {jS1 (z(t), ξ )} ≤ z T (t).ξ z(t),

(2.18)

2.3 Stochastic Stability Analysis

35

where ¯ )BξT Pξ .ξ = PAξ + AξT Pξ + 2Pξ Bξ (I + λI −1 −1 + ε1ξ Pξ Mξ MξT Pξ + ε1ξ Nξ T Nξ + ε2ξ Pξ Bξ BξT Pξ .℘ + ε2ξ ρξ2 I + (θ1 κ ξρ + θ2 κ¯ ξρ )Pρ . ρ=1

Applying Schur complement lemma to (2.8) and (2.9) leads to E {jS1 (z(t), ξ )} < 0,

(2.19)

Therefore, system (2.6) is stochastically stable. Remark 2.2 Different from the sojourn time obeying exponential distribution [1–3, 13–17, 21, 22], we needs to reconstruct the weak infinitesimal operator under semi-Markovian process constraints (see formulas (2.11)–(2.15)). Then, based ¯ ¯ 1−V (l+Δ) V (l)−V (l+Δ) = 0, on Taylor-series formula (2.12) and lim 1−Vξ ξ (l) = 1, lim ξ 1−V ξξ(l) lim

¯ Δ→0

¯ V ξ (l+Δ)−V ξ (l) ¯ (1−V ξ (l)) Δ

¯ Δ→0

¯ Δ→0

= κξ (l), κξρ (l) = λξρ κξ (l), ξ /= ρ, one has the weak infinites-

imal operator.

For the equality constraint (2.7), one has Trace[(Pξ Bξ − D T Xξ DBξ )T × (Pξ Bξ − D T Xξ DBξ )] = 0, which means that (Pξ Bξ − D T Xξ DBξ )T (Pξ Bξ − D T Xξ DBξ ) < γ I , where γ > 0. Applying Schur complement leads to [

] −γ I (Pξ Bξ − D T Xξ DBξ )T < 0. ∗ −I

(2.20)

Therefore, one has following minimization problem γ Pξ , Xξ , ε1ξ , ε2ξ s.t. Inequalities (2.8), (2.9), and (2.20).

min

(2.21)

36

2 Quantized Sliding Mode Control

2.4 Reachability Analysis In this section, we will deal with the reachability of the sliding surface s(t) = 0 under the sliding mode control law (2.5). Theorem 2.2 For system (2.1) and sliding surface (2.4), the finite-time attractiveness of the sliding surface can be achieved under the sliding mode control law (2.5), ¯ ξ ||Xξ DBξ || − ς > 0 with ς > 0. in which ηξ satisfies (1 + λ)η Proof For Lyapunov function S2 (s(t), ξ ) =

1 T s (t)Xξ s(t), 2

(2.22)

one has E {jS2 (s(t), ξ )} .℘ 1 = s T (t)Xξ D z˙ (t) + s T (t) κξρ (l)Xξ s(t) ρ=1 2 = s T (t)Xξ D[(Aξ + ΔAξ (t) + Bξ (I + Λ)BξT Pξ )z(t) + Bξ (−(I + Λ)ηξ sgn((Xξ DBξ )T s(t)) + f ξ (t, z(t)))] .℘ 1 + s T (t) κξρ (l)Xξ s(t) ρ=1 2 T ¯ ≤ ||s(t)||[||Xξ DAξ || + ||Xξ DMξ ||||Nξ || + (1 + λ)||X ξ DBξ Bξ Pξ ||

¯ ξ ||Xξ DBξ || + ρξ ||Xξ DBξ ||]||z(t)|| − ||s(t)||(1 + λ)η .℘ 1 + ||s(t)|||| κ¯ ξρ Xξ D||||z(t)|| ρ=1,ρ/=ξ 2 ¯ ξ ||Xξ DBξ || − ς )] − ς||s(t)||, = ||s(t)||[ςξ ||z(t)|| − ((1 + λ)η where T ¯ ςξ = ||Xξ DAξ || + ||Xξ DMξ ||||Nξ || + (1 + λ)||X ξ DBξ Bξ Pξ || 1 .℘ κ¯ ξρ Xξ D||. + ρξ ||Xξ DBξ || + || ρ=1,ρ/=ξ 2

Defining ¯ ξ ||Xξ DBξ || − ς}, Ω . {z(t) : ςξ ||z(t)|| ≤ (1 + λ)η

(2.23)

2.5 Simulation

37

yields ¯ ξ ||Xξ DBξ || − ς )] ≤ 0. ||s(t)||[ςξ ||z(t)|| − ((1 + λ)η Thus, we can obtain E {jS2 (s(t), ξ )} ≤ −ς||s(t)|| ≤ −

ς. E {S2 (s(t), ξ )}, ζ

(2.24)

/ λ [X ] where ζ = maxξ ∈Θ max2 ξ > 0. Applying Dynkin’s formula to (2.24) from [0, t ' ] leads to . . ς E { S2 (s(t ' ), αt ' )} − E { S2 (s(0), α0 )} ≤ − t ' . 2ζ Furthermore, one has 0≤−

. ς ' t + E { S2 (s(0), α0 )}, 2ζ

which means that . t ' ≤ 2ζ E {S2 (s(0), α0 )}/ς.

(2.25)

√ Then, there exists t ' = 2ζ E {S2 (s(0), α0 )}/ς such that S2 (s(t), αt ) = 0, when t ≥ t ' . Therefore, finite-time attractiveness is realized. Remark 2.3 A mode-independent sliding surface (2.4) is proposed to avoid the potential repetitive jumping effects compared with mode-dependent sliding surface [7–9, 13–16]. Owing to the multimodal characteristic of S-MSSs, a mode-dependent Lyapunov function (2.22) is designed, which can make full use of semi-Markovian switching information. Moreover, the sliding mode control law (2.5) depends on the ¯ When the system trajectories arrive at the sliding region Ω, the quantizer level λ. finite-time reachability of the predefined sliding surface is guaranteed. Meantime, owing to the unavoidable chattering effects, the sliding motion is not always staying on the predefined sliding surface all the time, located in a bounded neighborhood around the predefined sliding surface.

2.5 Simulation Consider the single-link robot arm model expressed by ¨ =− θ(t)

M gL D (t) 1 sin(θ (t)) − u(t), θ˙ (t) + J J J

38

2 Quantized Sliding Mode Control

where θ (t) and u(t) are the angle position of the arm and the control input. J , M , g, L , and D(t) stand for moment of inertia, mass of payload, acceleration of gravity, lengthen of arm, and coefficient of viscous friction, where g = 9.81 m/s2 , L = 0.5 m, and D (t) = D0 = 2. M and J are with three modes: M1 = 0.04, M2 = 0.12, M3 = 0.39, J1 = 5, J2 = 6.67, and J3 = 10. The transformation between different speeds is subject to the semi-Markovian process {αt , t ≥ 0} in Θ = {1, 2, 3} with the transition rate matrices as ⎡

⎡ ⎤ ⎤ −1.5 0.6 0.9 −1.0 0.5 0.5 κ = ⎣ 0.8 −1.2 0.4 ⎦ , κ¯ = ⎣ 1.7 −2.5 0.8 ⎦ . 1.2 0.8 −2.0 0.5 0.6 −1.1 Defining z 1 (t) = θ (t), z 2 (t) = θ˙ (t), one has a linearized model as [

][ ] [ ] [ ] 0 1 0 z˙ 1 (t) z 1 (t) = + 1 u(t). M ξ gL D0 − Jξ − J (t) z˙ 2 (t) z 2 J ξ ξ

Then, we get [

] [ ] [ ] 0 1 0 1 0 1 , A2 = , A3 = , A1 = −0.0392 −0.4 −0.0882 −0.2999 −0.1913 −0.2 [ ] [ ] [ ] 0 0 0 , B2 = , B3 = . B1 = 0.2 0.1499 0.1 [

] [ ] [ ] [ ] 0 0.1 0.2 Consider M1 = , M2 = , M3 = , N1 = 0.1 −0.1 , N2 = 0.3 0 0.1 [ ] [ ] −0.1 0.2 , N3 = 0 0.1 . The other parameters are given as ρξ = 0.5, h ξ = 1/3, ς = 0.1, φ = 0.4, and ξ = 1, 2, 3. Solving the minimization problem (2.23), one has [ ] [ ] 0.2095 −0.0050 0.2098 −0.0074 , P2 = , γ = 2.1120 ∗ 10−5 , P1 = −0.0050 0.0184 −0.0074 0.0207 [ ] 0.2044 −0.0147 , X1 = 0.0227, X2 = 0.0973, X3 = 0.2699. P3 = −0.0147 0.0246 [ ]T For given α0 = 1 and z 0 = −0.4 0.3 , Figs. 2.1 and 2.2 plot the system mode αt and the state response z(t). Figures 2.3 and 2.4 depict the finite-time reachability of the predefined sliding surface and control input u(t). Therefore, the sliding mode control law can ensure stochastic stability and reachability of the sliding region.

2.6 Conclusion Fig. 2.1 System mode αt

39 4 3.5 3 2.5 2 1.5 1 0.5 0

Fig. 2.2 State response z(t)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4

0

1

2

3

4

5

2.6 Conclusion In this chapter, the sliding mode control has been discussed for nonlinear S-MSSs with quantization. First, stochastic stability has been proposed. Then, the sliding mode controller is designed to depend on the quantizer level. At last, the single-link robot arm model is shown to illustrate the proposed method. In order to reduce the occupancy of network bandwidth resources, sliding mode control for event-triggered S-MSSs is significant for future works.

40 Fig. 2.3 Sliding surface s(t)

2 Quantized Sliding Mode Control 0.05 0.04 0.03 0.02 0.01 0 -0.01

Fig. 2.4 Sliding mode control law u(t)

0

1

2

3

4

5

0

1

2

3

4

5

0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35

References 1. Mao, X.R.: Stochastic Differential Equations and Applications, 2nd edn. Horwood, England (2007) 2. Lian, J., Liu, J., Zhuang, Y.: Mean stability of positive Markov jump linear systems with homogeneous and switching transition probabilities. IEEE Trans. Circuits Syst. II: Express Briefs 62(8), 801–805 (2015) 3. Wu, H.N., Cai, K.Y.: Mode-independent robust stabilization for uncertain Markovian jump nonlinear systems via fuzzy control. IEEE Trans. Syst., Man, Cybern. Part B (Cybern.) 36(3), 509–519 (2005) 4. Huang, J., Shi, Y.: Stochastic stability and robust stabilization of semi-Markov jump linear systems. Int. J. Robust Nonlinear Control 23(18), 2028–2043 (2013) 5. Zhang, L.X., Cai, B., Shi, Y.: Stabilization of hidden semi-Markov jump systems: emission probability approach. Automatica 101, 87–95 (2019)

References

41

6. Zong, G.D., Qi, W.H., Karimi, H.R.: L1 control of positive semi-Markov jump systems with state delay. IEEE Trans. Syst., Man, Cybern.: Syst. 51(12), 7569–7578 (2021) 7. Qi, W.H., Zong, G.D., Karimi, H.R.: Finite-time observer-based sliding mode control for quantized semi-Markov switching systems with application. IEEE Trans. Ind. Inform. 16(2), 1259– 1271 (2020) 8. Wei, Y.L., Park, J.H., Qiu, J.B., Wu, L.G., Jung, H.Y.: Sliding mode control for semi-Markovian jump systems via output feedback. Automatica 81, 133–141 (2017) 9. Qi, W.H., Zong, G.D., Karimi, H.R.: Sliding mode control for nonlinear stochastic singular semi-Markov jump systems. IEEE Trans. Autom. Control 65(1), 361–368 (2020) 10. Shen, H., Dai, M., Yan, H., Park, J.H.: Quantized output feedback control for stochastic semiMarkov jump systems with unreliable links. IEEE Trans. Circuits Syst. II: Express Briefs 65(12), 1998–2002 (2018) 11. Emelyanov, S.V.: Variable Structure Control Systems. Nauka, Moscow (1967) 12. Choi, H.H.: Robust stabilization of uncertain fuzzy systems using variable structure system approach. IEEE Trans. Fuzzy Syst. 16(3), 715–724 (2008) 13. Niu, Y.G., Ho, D.W.C., Wang, X.Y.: Sliding mode control for Itô stochastic systems with Markovian switching. Automatica 43(10), 1784–1790 (2007) 14. Li, H.Y., Shi, P., Yao, D.Y., Wu, L.G.: Observer-based adaptive sliding mode control for nonlinear Markovian jump systems. Automatica 64, 133–142 (2016) 15. Zhang, Q.L., Li, L., Yan, X.G., Spurgeon, S.K.: Sliding mode control for singular stochastic Markovian jump systems with uncertainties. Automatica 79, 27–34 (2017) 16. Feng, Z.G., Shi, P.: Sliding mode control of singular stochastic Markov jump systems. IEEE Trans. Autom. Control 62(8), 4266–4273 (2017) 17. Du, C.L., Li, F.B., Yang, C.H.: An improved homogeneous polynomial approach for adaptive sliding-mode control of Markov jump systems with actuator faults. IEEE Trans. Autom. Control 65(3), 955–969 (2020) 18. Delchamps, D.F.: Stabilizing a linear system with quantized state feedback. IEEE Trans. Autom. Control 35(8), 916–924 (1990) 19. Fu, M.Y., Xie, L.H.: The sector bound approach to quantized feedback control. IEEE Trans. Autom. Control 50(11), 1698–1711 (2005) 20. Liu, M., Ho, D.W.C., Niu, Y.G.: Robust filtering design for stochastic system with modedependent output quantization. IEEE Trans. Signal Process. 58(12), 6410–6416 (2010) 21. Xiao, N., Xie, L.H., Fu, M.Y.: Stabilization of Markov jump linear systems using quantized state feedback. Automatica 46(10), 1696–1702 (2010) 22. Tao, J., Lu, R.Q., Su, H.Y., Shi, P., Wu, Z.G.: Asynchronous filtering of nonlinear Markov jump systems with randomly occurred quantization via T-S fuzzy models. IEEE Trans. Fuzzy Syst. 26(4), 1866–1877 (2018)

Chapter 3

Sliding Mode Control Under Stochastic Disturbance

In this chapter, the sliding mode controller is designed for nonlinear stochastic semiMarkovian switching systems (S-MSSs) with time-varying transition rate matrix, where semi-Markovian switching parameters, stochastic disturbance, uncertainty, and nonlinearity are all considered. Thus, the system considered in this chapter is more general. Markovian switching systems with sojourn-time-independent transition rate matrix belongs to a special case. Many practical systems with unpredictable structural variations can be described by nonlinear stochastic S-MSSs, in which the bound of time-varying transition rate matrix is assumed to be known. Then, sojourn-time-dependent sufficient conditions to guarantee the closed-loop sliding mode dynamics stochastically stable are obtained through the stochastic semiMarkovian Lyapunov function. Furthermore, the sliding mode control law is designed to ensure the reachability of the sliding dynamics in a finite time. At last, a space robot manipulator model is given to illustrate the designed sliding mode controller.

3.1 Introduction In the previous chapter, S-MSSs without stochastic disturbance have been investigated. However, many systems may be affected by environmental noise in the actual process, such as aerospace control systems, power systems, and economic systems. Unfortunately, the deterministic system model can not describe these practical systems accurately. Therefore, it is necessary to use differential equations containing random elements to model practical systems. With the development of stochastic differential equations [1], recent works have been largely devoted to stochastic S-MSSs; for details, see [2–7]. In this chapter, we are interested in the design of sliding mode controller for nonlinear stochastic S-MSSs. The main contributions can be highlighted as: (i) The feedback controller is designed to guarantee the stochastic stability of nonlinear © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 W. Qi and G. Zong, Control Synthesis for Semi-Markovian Switching Systems, Studies in Systems, Decision and Control 465, https://doi.org/10.1007/978-981-99-0317-7_3

43

44

3 Sliding Mode Control Under Stochastic Disturbance

stochastic S-MSSs. (ii) Based on stochastic semi-Markovian Lyapunov function, sojourn-time-dependent sufficient conditions for stochastic stability criteria of the closed-loop sliding dynamics are proposed in the form of matrix inequalities. (iii) The sliding mode controller is designed with semi-Markovian switching to guarantee the reachability of the sliding surface in a finite-time interval.

3.2 Problem Statements and Preliminaries Consider the stochastic S-MSSs as d x(t) = [(A (κt ) + ΔA (κt , t))x(t) + B(κt )(u(t) + f (t, κt , x(t)))]dt + D (κt )x(t)dω(t),

(3.1)

where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the input vector, f (t, κt , x(t)) ∈ Rm is the nonlinearity, and ω(t) is a standard Wiener process satisfying E {dω(t)} = 0, E {d 2 ω(t)} = dt. ΔA (κt , t) is represented by ΔA (κt , t) = M (κt )F (κt , t)N (κt ) with F (κt , t) satisfying F T (κt , t)F (κt , t) ≤ I . {κt , t ≥ 0} stands for the semiMarkovian process in Θ = {1, 2, . . . , j} with the probability transition . Pr {κt+Δ¯ = κ|κt = τ } =

¯ + o(Δ), ¯ υτ κ (h)Δ τ /= κ, ¯ ¯ 1 + υτ τ (h)Δ + o(Δ), τ = κ,

where h means the sojourn time, lim (o(h)/ h) = 0, υτ κ (h) ≥ 0 is the transition rate h→0 Σj υτ κ (h) = −υτ τ (h). The transition rate υτ κ (h) from τ to κ for τ /= κ, and κ=1,κ/=τ

is assumed to be υ τ κ ≤ υτ κ (h) ≤ υ¯ τ κ with real constant scalars υ τ κ and υ¯ τ κ . For κt = τ ∈ Θ, A (κt ), ΔA (κt , t), M (κt ), F (κt , t), N (κt ), B(κt ), D (κt ), and f (t, κt , x(t)) are denoted as Aτ , ΔAτ (t), Mτ , Fτ (t), Nτ , Bτ , Dτ , and f τ , respectively. Assumption 3.1 ([8]) f τ satisfies || f τ || ≤ δ||x(t)||,

(3.2)

where δ is the unknown constant with δ > 0. Definition 3.1 ([9]) System{.(3.1) is said to be stochastically stable if for initial } ∞ condition x0 and κ0 ∈ Θ, E 0 ||x(t)||2 dt|(x0 , κ0 ) < ∞ holds. In this section, the practical space robot manipulator model is described as the above-mentioned theoretical dynamic model. To improve the production efficiency, the space robot manipulator has been widely applied to aircraft manufacturing, disaster rescue, nuclear reactor maintenance, etc. The space robot manipulator may

3.2 Problem Statements and Preliminaries Table 3.1 Meanings for Different Parameters Symbol Ω ε Te f f Tde f N υ Jin Jout

45

Parameter Joint angle of inertial axis Joint angle of output axis Effective joint input torque Deformation torque of gearbox Gearbox ratio Damping coefficient Inertia of the input axe Inertia of the output axe

experience some structural and parametrical changes. With the abrupt change of actual operating parameters, the space robot manipulator model can be characterized as S-MSSs, in which the model is described as N 2 Jin Ω¨ + Jout (Ω¨ + ε¨ ) + υ(Ω˙ + ε˙ ) = Te f f , Jout (Ω¨ + ε¨ ) + υ(Ω˙ + ε˙ ) = Tde f ,

(3.3)

where the system parameters are given in Table 3.1. The actuator model of the motor plus the gearbox is Te f f = N Tm , Tm = kt i c and the deformation torque Tde f is described as Tde f = Cε, where Tm and i c denote the motor torque and the motor current, kt and C are the motor torque constant and the spring constant. The stochastic switching law will happen if the space robot manipulator model may encounter failures subject to abrupt changes in the constant motor torque kt and the input inertial axis Jin . The transformation between different speeds obeys the semi-Markovian process {κt , t ≥ 0} in Θ = {1, 2, . . . , j}. [ ]T Define the state vector x = Ω Ω˙ ε ε˙ and the control input u = i c . Considering the stochastic disturbance, the parametrical uncertainty, and the nonlinear factor of control input including friction and air resistance, when κt = τ , the state-space model is presented as d x(t) = [(Aτ + ΔAτ (t))x(t) + Bτ (u(t) + f τ )]dt + Dτ x(t)dω(t),

(3.4)

⎤ 0 1 0 0 C [ ]T ⎢0 0 0 ⎥ N 2 Jin,τ ⎥, Bτ = 0 kt,τ 0 − kt,τ where Aτ = ⎢ , and N Jin,τ N Jin,τ ⎣0 0 0 1 ⎦ 0 − Jυout −( N 2CJin,τ + JCout ) − Jυout ΔAτ (t), f τ , Dτ are the parametrical uncertainty, the nonlinear factor of control input, and the parameter of stochastic disturbance. ⎡

46

3 Sliding Mode Control Under Stochastic Disturbance

3.3 Sliding Mode Control Law Design The sliding surface is proposed as . s(t) =

BτT Xτ x(t)

t

− 0

BτT Xτ (Aτ + Bτ Kτ )x(s)ds,

(3.5)

where Xτ and Kτ are the real matrices to satisfy nonsingularity of BτT Xτ Bτ . Owing to Bτ with full column rank, the nonsingularity of BτT Xτ Bτ is ensured by Xτ > 0, ∀τ ∈ Θ. For system (3.1), one has .

t

x(t) = x(0) + [(Aτ + ΔAτ (s))x(s) + Bτ (u(s) + f τ )]ds 0 . t Dτ x(s)dω(s), +

(3.6)

0

.t where 0 Dτ x(s)dω(s) means the Itˆo’s stochastic integral. If BτT Xτ Dτ = 0, then it follows from (3.5) and (3.6) that . s(t) =

BτT Xτ x(0)

t

+ 0

BτT Xτ [(ΔAτ (s) − Bτ Kτ )x(s)

+ Bτ (u(s) + f τ )]ds.

(3.7)

Then, we have s(t) = 0 and s˙ (t) = 0 when the state trajectories enter the sliding surface. One has the equivalent control law u eq = Kτ x(t) − (BτT Xτ Bτ )−1 BτT Xτ ΔAτ (t)x(t) − f τ .

(3.8)

Based on the controller (3.8), we have d x(t) = [Aτ + Bτ Kτ + ΔAτ (t) − Bτ (BτT Xτ Bτ )−1 BτT Xτ ΔAτ (t)]x(t)dt + Dτ x(t)dω(t).

(3.9)

3.4 Stochastic Stability Analysis Theorem 3.1 If there exist symmetric matrices Pτ > 0 and Xτ > 0, ∀τ ∈ Θ, such that Σ1τ < 0,

(3.10)

BτT Xτ Dτ = 0,

(3.11)

3.4 Stochastic Stability Analysis

47

where Σ1τ = Pτ A˜τ + A˜τT Pτ + DτT Pτ Dτ +

Σj κ=1

υτ κ (h)Pκ , A˜τ = Aτ + Bτ Kτ ,

then system (3.9) under ΔAτ (t) = 0 is stochastically stable. Proof For Lyapunov function V (x(t), κt ) = x T (t)P(κt )x(t),

(3.12)

at time t, κt = τ , one has the weak infinitesimal operator Γ V (x(t), τ ) [ ] 1 ¯ = lim E {V (x(t + Δ), κt+Δ¯ )|κt = τ } − V (x(t), τ ) ¯ ¯ Δ Δ→0 [ 1 Σj ¯ ¯ Pr {κt+Δ¯ = κ|κt = τ }x T (t + Δ)P = lim κ x(t + Δ) ¯ κ=1,κ/=τ ¯ Δ Δ→0 ] T T ¯ ¯ + Pr {κt+Δ¯ = τ |κt = τ }x (t + Δ)Pτ x(t + Δ) − x (t)Pτ x(t) [ Pr {κt+Δ¯ = κ, κt = τ } T 1 Σj ¯ ¯ = lim x (t + Δ)P κ x(t + Δ) ¯ κ=1,κ/ = τ ¯ Pr {κt = τ } Δ Δ→0 ] Pr {κt+Δ¯ = τ, κt = τ } T T ¯ ¯ x (t + Δ)Pτ x(t + Δ) − x (t)Pτ x(t) + Pr {κt = τ } [ ¯ − Gτ (h)) T 1 Σj λτ κ (Gτ (h + Δ) ¯ ¯ = lim x (t + Δ)P κ x(t + Δ) ¯ κ=1,κ/ = τ ¯ 1 − Gτ (h) Δ→0 Δ ] ¯ T 1 − Gτ (h + Δ) ¯ ¯ − x T (t)Pτ x(t) . + (3.13) x (t + Δ)P τ x(t + Δ) 1 − Gτ (h) According to the literature [1], Eq. (3.9) is rewritten as ¯ + Dτ x(t)Δω(t), ¯ = x(t) + A˜τ x(t)Δ x(t + Δ) ¯ − ω(t). where Δω(t) = ω(t + Δ)

(3.14)

48

3 Sliding Mode Control Under Stochastic Disturbance

Then, one has Γ V (x(t), τ ) [ ¯ − Gτ (h)) T 1 Σj λτ κ (Gτ (h + Δ) ¯ [x (t)Pκ x(t) + 2x T (t)A˜τT Pκ x(t)Δ = lim ¯ κ=1,κ/ = τ 1 − Gτ (h) ¯ Δ Δ→0 ¯ + 2x T (t)DτT Pκ x(t)Δω(t) + 2x T (t)A˜τT Pκ Dτ x(t)Δω(t)Δ ¯ G (h) − Gτ (h + Δ) ¯ 2 + x T (t)DτT Pκ Dτ x(t)Δ2 ω(t)] + τ + x T (t)A˜τT Pκ A˜τ x(t)Δ 1 − Gτ (h) ¯ 1 − Gτ (h + Δ) ¯ + 2x T (t)DτT Pτ x(t)Δω(t) [2x T (t)A˜τT Pτ x(t)Δ 1 − Gτ (h) ¯ + x T (t)A˜T Pτ A˜τ x(t)Δ ¯2 + 2x T (t)A˜T Pτ Dτ x(t)Δω(t)Δ

x T (t)Pτ x(t) + τ

τ

+ x T (t)DτT Pτ Dτ x(t)Δ2 ω(t)].

(3.15)

From [10], we have ¯ ¯ 1 − Gτ (h + Δ) Gτ (h) − Gτ (h + Δ) = 1, lim = 0, ¯ ¯ 1 − Gτ (h) 1 − Gτ (h) Δ→0 Δ→0 ¯ − Gτ (h) Gτ (h + Δ) = υτ (h). lim ¯ − Gτ (h)) ¯ Δ(1 Δ→0 lim

(3.16)

For υτ κ (h) = λτ κ υτ (h), τ /= κ, one has ¯ − Gτ (h)) T 1 Σj λτ κ (Gτ (h + Δ) [x (t)Pκ x(t)] ¯ κ=1,κ/=τ ¯ 1 − Gτ (h) Δ Δ→0 Σj Σj = λτ κ υτ (h)[x T (t)Pκ x(t)] = υτ κ (h)x T (t)Pκ x(t). (3.17) lim

κ=1,κ/=τ

κ=1,κ/=τ

Based on Itˆo’s formula [1], we have .

. ¯ ) − Gτ (h)) 1 Σj λτ κ (Gτ (h + Δ [2x T (t)DτT Pκ x(t)Δω(t)] = 0, ¯ κ=1,κ/=τ ¯ 1 − Gτ (h) Δ Δ→0 . . ¯ − Gτ (h)) 1 Σj λτ κ (Gτ (h + Δ) ¯ = 0, E lim [2x T (t)A˜τT Pκ x(t)Δ] ¯ κ=1,κ/=τ ¯ 1 − Gτ (h) Δ Δ→0 . . ¯ − Gτ (h)) 1 Σj λτ κ (Gτ (h + Δ) E lim [2x T (t)DτT Pκ x(t)Δω(t)] = 0, ¯ κ=1,κ/=τ ¯ 1 − Gτ (h) Δ Δ→0 . . ¯ − Gτ (h)) 1 Σj λτ κ (Gτ (h + Δ) ¯ = 0, E lim [2x T (t)A˜τT Pκ Dτ x(t)Δω(t)Δ] ¯ κ=1,κ/=τ ¯ 1 − Gτ (h) Δ Δ→0

E

lim

3.4 Stochastic Stability Analysis

49

.

. ¯ − Gτ (h)) T 1 Σj λτ κ (Gτ (h + Δ) ¯ 2 ] = 0, [x (t)A˜τT Pκ A˜τ x(t)Δ ¯ κ=1,κ/=τ ¯ →0 Δ 1 − Gτ (h) Δ . . ¯ − Gτ (h)) T 1 Σj λτ κ (Gτ (h + Δ) E lim [x (t)DτT Pκ Dτ x(t)Δ2 ω(t)] ¯ κ=1,κ/=τ ¯ →0 Δ 1 − Gτ (h) Δ . . ¯ − Gτ (h)) T 1 Σj λτ κ (Gτ (h + Δ) ¯ ] = 0, = E lim [x (t)DτT Pκ Dτ x(t)Δ ¯ κ=1,κ/=τ ¯ →0 Δ 1 − Gτ (h) Δ . . ¯ T 1 Gτ (h) − Gτ (h + Δ) E lim x (t)Pτ x(t) = −υτ (h)x T (t)Pτ x(t), ¯ ¯ →0 Δ 1 − Gτ (h) Δ . . ¯ 1 1 − Gτ (h + Δ) ¯ = 2x T (t)A˜τT Pτ x(t), E lim 2x T (t)A˜τT Pτ x(t)Δ ¯ ¯ →0 Δ 1 − Gτ (h) Δ . . ¯ 1 1 − Gτ (h + Δ) E lim 2x T (t)DτT Pτ x(t)Δω(t) = 0, ¯ ¯ →0 Δ 1 − Gτ (h) Δ . . ¯ T 1 1 − Gτ (h + Δ) ¯ 2 = 0, E lim x (t)A˜τT Pτ A˜τ x(t)Δ ¯ ¯ →0 Δ 1 − Gτ (h) Δ . . ¯ T 1 1 − Gτ (h + Δ) E lim x (t)DτT Pτ Dτ x(t)Δ2 ω(t) ¯ ¯ →0 Δ 1 − Gτ (h) Δ

E

lim

= x T (t)DτT Pτ Dτ x(t).

(3.18)

Furthermore, Γ V (x(t), τ ) = x T (t)(Pτ A˜τ + A˜τT Pτ + DτT Pτ Dτ Σj + υτ κ (h)Pκ )x(t). κ=1

By the condition (3.10), we get Γ V (x(t), τ ) ≤ − min[λmin (−Σ1τ )]x T (t)x(t). τ ∈Θ

Together with Dynkin’s formula yields [. t ] E [V (x(t), τ )] − E [V (x0 , κ0 )] = E Γ V (x(s), κs )ds 0 [. t ] 2 ≤ − min[λmin (−Σ1τ )]E ||x(s)|| ds|(x0 , κ0 ) , τ ∈Θ

0

which further implies that [.

t

min[λmin (−Σ1τ )]E τ ∈Θ

] ||x(s)||2 ds|(x0 , κ0 )

0

≤E [V (x0 , κ0 )] − E [V (x(t), τ )] ≤ E [V (x0 , κ0 )] .

(3.19)

50

3 Sliding Mode Control Under Stochastic Disturbance

Therefore, [. E

∞

] ||x(s)||2 ds|(x0 , κ0 ) ≤

0

E [V (x0 , κ0 )] < ∞. minτ ∈Θ [λmin (−Σ1τ )]

Therefore, system (3.9) with ΔAτ (t) = 0 is stochastically stable.

.

Remark 3.1 The sliding mode controller design for stochastic MSSs [2–7] is considered with a stochastic Markovian process satisfying exponential distribution. For stochastic semi-Markovian process satisfying non-exponential distribution, there exist time-varying characteristics of transition rate matrix that extend the application range of stochastic switching systems. If the transition rate is fixed and time-invariant, stochastic S-MSSs are transformed into ordinary stochastic MSSs [2–7]. Sojourn-time-dependent sufficient conditions for stochastic stability are based on nominal systems without the uncertain element ΔAτ (t) in Theorem 3.1. Based on Theorem 3.1, robustly stochastic stability is analyzed by replacing A˜τ with A˜τ (t) = Aτ + Bτ Kτ + ΔAτ (t) − Bτ (BτT Xτ Bτ )−1 BτT Xτ ΔAτ (t) in Theorem 3.2. Theorem 3.2 If there exist symmetric matrices Pτ > 0 and Xτ > 0, and positive scalars ε1τ , ε2τ , ∀τ ∈ Θ, such that (3.11) and ] Σ2τ Σ3τ < 0, ∗ Σ4τ ] [ −Pτ Mτ < 0, ∗ −ε2τ I [

(3.20) (3.21)

where Σ2τ = Pτ (Aτ + Bτ Kτ ) + (Aτ + Bτ Kτ )T Pτ + DτT Pτ Dτ + ] [ Σ3τ = ε2τ Pτ Bτ Pτ Mτ ε1τ Nτ T Nτ T ,

Σj κ=1

υτ κ (h)Pκ ,

Σ4τ = − diag{BτT Xτ Bτ , ε1τ I , ε1τ I , ε2τ I }, then system (3.9) is robustly stochastically stable. Proof Replacing A˜τ with A˜τ (t) = Aτ + Bτ Kτ + ΔAτ (t) − Bτ (BτT Xτ Bτ )−1 BτT Xτ ΔAτ (t) in (3.19), the inequality (3.19) is rewritten as E {Γ V (x(t), τ )} = x T (t)[2PτT (Aτ + Bτ Kτ + ΔAτ (t) − Bτ (BτT Xτ Bτ )−1 Σj BτT Xτ ΔAτ (t)) + DτT Pτ Dτ + υτ κ (h)Pκ ]x(t). κ=1

(3.22)

3.4 Stochastic Stability Analysis

51

Then one has 2x T (t)Pτ Δ Aτ (t)x(t) −1 T x (t)Pτ Mτ MτT Pτ x(t) + ε1τ x T (t)Nτ T Nτ x(t), ≤ε1τ

− 2x T (t)Pτ Bτ (BτT Xτ Bτ )−1 BτT Xτ ΔAτ (t)x(t) 2 T ≤ε2τ x (t)Pτ Bτ (BτT Xτ Bτ )−1 BτT Pτ x(t) 1 + 2 x T (t)Nτ T FτT (t)MτT Xτ Mτ Fτ (t)Nτ x(t). ε2τ

(3.23)

Applying Schur complement lemma to (3.22)–(3.23), one has E {Γ V (x(t), τ )} < 0 through the inequality conditions (3.20)–(3.21). . Therefore, system (3.9) is robustly stochastically stable. In Theorem 3.2, sojourn-time-dependent sufficient conditions for robustly stochastic stability are proposed. However, the nonlinear elements PτT Bτ Kτ , BτT KτT Pτ , and υτ κ (h) are not solvable in linear matrix inequalities. Next, strict linear matrix inequalities conditions will be provided in Theorem 3.3. Theorem 3.3 If there exist symmetric matrix Xτ > 0, matrix Yτ , and positive scalars ε1τ , ε2τ , ∀τ ∈ Θ, such that (3.11) and ] Σ6τ < 0, Σ7τ ] Σ˜ 6τ < 0, Σ7τ [ ] −Xτ Xτ Mτ < 0, ∗ −ε2τ I [

Σ5τ ∗ [ Σ˜ 5τ ∗

where Σ5τ = Aτ Xτ + Bτ Yτ + Xτ AτT + YτT BτT + υ τ τ Xτ , √ Σ6τ = [ε2τ Bτ , Mτ , ε1τ Xτ Nτ T , Xτ Nτ T , Xτ DτT , υ τ 1 Xτ , . . . , √ √ √ υ τ τ −1 Xτ , υ τ τ +1 Xτ , . . . , υ τ j Xτ ], Σ7τ = − diag{BτT Xτ Bτ , ε1τ I , ε1τ I , ε2τ I , Xτ , x1 , . . . , xτ −1 , xτ +1 , . . . , xj }, Σ˜ 5τ = Aτ Xτ + Bτ Yτ + Xτ AτT + YτT BτT + υ¯ τ τ Xτ , . Σ˜ 6τ = [ε2τ Bτ , Mτ , ε1τ Xτ Nτ T , Xτ Nτ T , Xτ DτT , υ¯ τ 1 Xτ , . . . , . . . υ¯ τ τ −1 Xτ , υ¯ τ τ +1 Xτ , . . . , υ¯ τ j Xτ ],

(3.24) (3.25) (3.26)

52

3 Sliding Mode Control Under Stochastic Disturbance

then system (3.9) is robustly stochastically stable. Moreover, the controller gain is given as Kτ = Yτ Xτ−1 . Proof Let Xτ = Pτ−1 , Yτ = Kτ Xτ .

(3.27)

Pre- and post-multiply (3.20) by diag{Xτ , I , I , I , I }, one has ] Σ8τ Σ9τ < 0, ∗ Σ10τ

[

(3.28)

where Σ8τ = Aτ Xτ + Bτ Yτ + Xτ AτT + YτT BτT + Xτ DτT Xτ−1 Dτ Xτ + υτ τ (h)Xτ Σj + υτ κ (h)XτT Xκ−1 Xτ , κ=1,κ/=τ

Σ9τ = [ε2τ Bτ , Mτ , ε1τ XτT Nτ T , XτT Nτ T ], Σ10τ = − diag{BτT Xτ Bτ , ε1τ I , ε1τ I , ε2τ I }. For given h, it is got that υτ κ (h) = θ1 υ τ κ + θ2 υ¯ τ κ , where θ1 + θ2 = 1 and θ1 > 0, θ2 > 0. Multiplying (3.24) by θ1 and (3.25) by θ2 and applying Schur complement lemma yield ] Σ11τ Σ9τ < 0, ∗ Σ10τ

[

(3.29)

where Σ11τ = Aτ Xτ + Bτ Yτ + Xτ AτT + YτT BτT + Xτ DτT Xτ−1 Dτ Xτ + (θ1 υ τ τ + θ2 υ¯ τ τ )Xτ +

Σj κ=1,κ/=τ

(θ1 υ τ κ + θ2 υ¯ τ κ )Xτ Xκ−1 Xτ .

By tuning θ1 and θ2 , all possible υτ κ (h) ∈ [υ τ κ , υ¯ τ κ ] can be obtained, which means that (3.20) holds. Performing a congruence transformation to (3.21) by diag{Xτ , I }, (3.26) is . equivalent to (3.21). Remark 3.2 Owing to the equality constraint of the matrix Xτ , it is difficult to solve Eq. (3.11). To research possible solutions, one possible choice is to replace (3.11) by (BτT Xτ Dτ )T × (BτT Xτ Dτ ) < oI ,

(3.30)

3.5 Reachability Analysis

53

where o is a sufficiently small positive scalar. According to Schur complement, (3.30) is equivalent to [

] −oI (BτT Xτ Dτ )T < 0. ∗ −I

(3.31)

Therefore, one has the minimization problem as min

o Xτ , Yτ , ε1τ , ε2τ s.t. Inequalities (3.24) − (3.26) and (3.31),

(3.32)

with the controller gain computed as Kτ = Yτ Xτ−1 . Remark 3.3 By solving this optimization problem, we can obtain the controller gain Kτ . Sufficient conditions are accepted in both theoretically and practical aspects. Considering complex factors such as parametrical uncertainty, stochastic disturbance, and nonlinear factor of control input, one joint of space robot manipulator model in the simulation part is described as nonlinear stochastic S-MSSs to demonstrate the main results.

3.5 Reachability Analysis An adaptive sliding mode control law will be designed to obtain the upper bound δ of unknown nonlinearity f τ . Then, the reachability of the sliding surface will be realized in a finite-time interval. Theorem 3.4 Consider the sliding surface (3.5) and the feasible solutions in optimization (3.32). The states of nonlinear S-MSSs (3.1) will be driven onto the specified sliding surface in finite time by the sliding mode control law u(t) = Kτ x(t) − (||(BτT Xτ Bτ )−1 BτT Xτ Mτ ||||Nτ ||||x(t)|| 1 Σj ˆ υ¯ τ κ (BκT Xκ Bκ )−1 ||||s(t)|| + δ||x(t)|| + ητ )sgn(s(t)), + || κ=1,κ/=τ 2

(3.33)

where ητ is the positive constant. The updated law for estimated parameter δˆ is ˙ˆ = ϑ||x(t)||||s(t)||, δ(0) ˆ > 0 with ϑ > 0. designed as δ(t) Proof Consider the Lyapunov function as ˜ ˜ V (s(t), δ(t), κt ) = V1 (s(t), κt ) + V2 (δ(t)),

(3.34)

54

3 Sliding Mode Control Under Stochastic Disturbance

where 1 T s (t)(B T (κt )X (κt )B(κt ))−1 s(t), 2 1 ˜2 ˜ ˜ = δ − δ(t). ˆ V2 (δ(t)) δ (t), δ(t) = 2ϑ

V1 (s(t), κt ) =

At time t, κt = τ , one has Γ V1 (s(t), τ ) 1 ¯ κt+Δ¯ )|κt = τ } − V1 (s(t), τ )] [E {V1 (s(t + Δ), ¯ ¯ Δ Δ→0 [ j 1 1 Σ T −1 ¯ ¯ Pr {κt+Δ¯ = κ|κt = τ }s T (t + Δ)(B = lim κ Xκ Bκ ) s(t + Δ) ¯ 2 ¯ Δ Δ→0 κ=1,κ/=τ = lim

1 T −1 ¯ ¯ Pr {κt+Δ¯ = τ |κt = τ }s T (t + Δ)(B τ Xτ Bτ ) s(t + Δ) 2 ] 1 − s T (t)(BτT Xτ Bτ )−1 s(t) 2 [ Pr {κt+Δ¯ = κ, κt = τ } T 1 1 Σj ¯ = lim s (t + Δ) ¯ 2 κ=1,κ/=τ ¯ Pr {κt = τ } Δ Δ→0 +

¯ + (BκT Xκ Bκ )−1 s(t + Δ)

1 Pr {κt+Δ¯ = τ, κt = τ } Pr {κt = τ } 2

]

1 T T −1 T −1 ¯ ¯ s (t + Δ)(B τ Xτ Bτ ) s(t + Δ) − s (t)(Bτ Xτ Bτ ) s(t) 2 [ j ¯ − Gτ (h)) T 1 1 Σ λτ κ (Gτ (h + Δ) ¯ = lim s (t + Δ) ¯ 2 ¯ 1 − G (h) Δ Δ→0 τ κ=1,κ/=τ T

¯ T 1 1 − Gτ (h + Δ) ¯ s (t + Δ) 2 1 − Gτ (h) ] 1 T T −1 T −1 ¯ (Bτ Xτ Bτ ) s(t + Δ) − s (t)(Bτ Xτ Bτ ) s(t) . 2 ¯ + (BκT Xκ Bκ )−1 s(t + Δ)

(3.35)

Based on the Taylor formula expansion, one has ¯ + o(Δ), ¯ ¯ = s(t) + s˙ (t)Δ s(t + Δ) ¯ → 0. where Δ

(3.36)

3.5 Reachability Analysis

55

According to (3.16), υτ κ (h) = λτ κ υτ (h), τ /= κ, and (3.35), (3.36), we have Γ V1 (s(t), τ )

Σj 1 = s T (t)(BτT Xτ Bτ )−1 s˙ (t) + s T (t) υτ κ (h)(BκT Xκ Bκ )−1 s(t) κ=1 2 = s T (t)(BτT Xτ Bτ )−1 BτT Xτ [(ΔAτ (t) − Bτ Kτ )x(t) + Bτ (Kτ x(t) 1 Σj υ¯ τ κ (BκT Xκ Bκ )−1 || − (||(BτT Xτ Bτ )−1 BτT Xτ Mτ ||||Nτ ||||x(t)|| + || κ=1,κ/=τ 2 Σj 1 ||s(t)|| + δ||x(t)|| + ητ )sgn(s(t))) + Gτ )] + s T (t) υτ κ (h) κ=1 2 (BκT Xκ Bκ )−1 s(t).

(3.37)

Similarly, it is got that ˙˜ = −(δ − δ(t))||x(t)||||s(t)||. ˜ ˜ δ(t) ˆ = δ(t) Γ V2 (δ(t))

(3.38)

Furthermore, Γ V (s(t), τ ) ≤||s(t)||||(BτT Xτ Bτ )−1 BτT Xτ Mτ ||||Nτ ||||x(t)|| − s T (t)||(BτT Xτ Bτ )−1 BτT Xτ Mτ ||||Nτ ||||x(t)||sgn(s(t)) Σj 1 − s T (t)|| υ¯ τ κ (BκT Xκ Bκ )−1 ||||s(t)||sgn(s(t)) κ=1,κ/=τ 2 Σj 1 υ¯ τ κ (BκT Xκ Bκ )−1 ||||s(t)|| + ||s(t)|||| κ=1,κ/=τ 2 ˆ T (t)||x(t)||sgn(s(t)) + δ||s(t)||||x(t)|| − δs ˆ − (δ − δ(t))||x(t)||||s(t)|| − ητ s T (t)sgn(s(t)) ≤ − ητ ||s(t)||.

(3.39)

Therefore, the finite-time reachability of the predefined sliding surface is . realized.

56

3 Sliding Mode Control Under Stochastic Disturbance

3.6 Simulation Consider one joint of space robot manipulator model (3.4). N , υ, Jout , and C are given by N = −260.6, υ = 0.4 kg m2 /s, Jout = 400 kg m2 , and C = 130000 N m. The parameters Jin,τ and kt,τ are with three different modes as Jin,1 = 0.0011 kg m2 , Jin,2 = 0.0008 kg m2 , Jin,3 = 0.0006 kg m2 , kt,1 = 0.6 N m/A, kt,2 = 0.13 N m/A, and kt,3 = 0.06 N m/A. The transformation between different speeds obeys the semiMarkovian process {κt , t ≥ 0} in Θ = {1, 2, 3}. Other system parameters are given [ ]T [ ]T [ ]T as Dτ = [ 0.1 ∗ I4], M1 = [0 0.1 0 0 ], M2 = [0.1 0 0 0 ] , M3 = 0.1 0.1 0 0 , N1 = 0 0.1 0 0 , N2 = 0.1 0.1 0 0 , N3 = 0 0.1 0 0 , Fτ (t) = sin(t), f τ = 0.3sin(t)x1 (t), ε1τ = 0.1, τ = 1, 2, 3. Consider the transition rate matrices ⎡ ⎡ ⎤ ⎤ −1.0 0.7 0.3 −1.8 0.9 0.9 υ = ⎣ 0.8 −1.2 0.4 ⎦ , υ¯ = ⎣ 1.2 −2.0 0.8 ⎦ . 0.5 0.6 −1.1 0.7 0.9 −1.6 Solving the optimization problem (3.32) results in o = 0.4214 × 10−4 , and ⎡

⎤ 7.7115 −0.1221 0.0052 −0.0405 ⎢−0.1221 0.1045 0.0042 −0.0116⎥ ⎥ X1 = ⎢ ⎣ 0.0052 0.0042 0.0011 −0.0032⎦ , −0.0405 −0.0116 −0.0032 0.0336 ⎡ ⎤ 7.7388 −0.2421 0.0030 −0.0271 ⎢−0.2421 0.7479 −0.0272 0.1257 ⎥ ⎥ X2 = ⎢ ⎣ 0.0030 −0.0272 0.0018 −0.0109⎦ , −0.0271 0.1257 −0.0109 0.1243 ⎡ ⎤ 7.8153 −0.2938 0.0021 −0.0178 ⎢−0.2938 1.3539 −0.0335 0.1207 ⎥ ⎥ X3 = ⎢ ⎣ 0.0021 −0.0335 0.0014 −0.0074⎦ , −0.0178 0.1207 −0.0074 0.0895 [ ] K1 = 104 ∗ 0.0018 0.2952 −3.9982 −1.0536 , [ ] K2 = 104 ∗ −0.0036 −0.0993 −7.7005 −0.9929 , [ ] K3 = 104 ∗ −0.0023 −0.0580 −4.4457 −0.5688 . Then, BτT Xτ Bτ is nonsingular. For given parameters δ1 = δ2 = δ3 = 0.3, η1 = ˆ 0.1, η2 = 0.2, η2 = 0.3, δ(0) = 0.1, and ϑ = 0.5, one has the sliding mode control law (3.33). [ ]T For κ0 = 2 and x0 = −0.25 0.20 −0.20 0.25 , Fig. 3.1 plots the system mode subject to semi-Markovian switching rule. Figure 3.2 describes the state trajectory

3.6 Simulation

57

Fig. 3.1 System mode κt

4 3.5 3 2.5 2 1.5 1 0.5 0

Fig. 3.2 State trajectory x(t) of the open-loop system

0

1

2

3

4

5

6

7

8

9

10

8

9

10

104

8 6 4 2 0 -2 -4 -6 -8 -10

0

1

2

3

4

5

6

7

x(t) of the open-loop system, from which the unstable state trajectory x(t) tends to infinity. Figure 3.3 means the unstable switching function s(t) of the open-loop system. Under the control law (3.33), the state response x(t) of the closed-loop system is depicted in Fig. 3.4, in which the state response of the closed-loop systems converges to zero. The switching function s(t) is shown in Fig. 3.5. Figure 3.6 plots the input u(t).

58 Fig. 3.3 Sliding surface s(t) of the open-loop system

3 Sliding Mode Control Under Stochastic Disturbance 104

5

0

-5

Fig. 3.4 State response x(t) of the closed-loop system

0

2

4

6

8

10

2 1.5 1 0.5 0 -0.5 -1 -1.5

0

1

2

3

4

5

6

7

8

9

10

3.7 Conclusion In this chapter, we have studied the sliding mode control for nonlinear uncertain S-MSSs with the sojourn time subject to non-exponential distribution. An integral sliding surface has been designed to realize finite-time reachability of the predefined sliding surface. Then, applying the weak infinitesimal operator, stochastic stability criteria have been proposed for the underlying system. In light of stochastic stability, the sliding mode control law has been designed related to the semi-Markovian switching law and the controller gain. In the future, finite-time control and fault detection will be investigated for stochastic S-MSSs.

References Fig. 3.5 Sliding surface s(t) of the closed-loop system

59 0.2 0.15 0.1 0.05 0 -0.05 -0.1

Fig. 3.6 Control input u(t)

0

2

4

6

8

10

6

8

10

10 5 0 -5 -10 -15 -20

0

2

4

References 1. Du, H.B., Wen, G.H., Yu, X.H., Li, S.H., Chen, M.Z.Q.: Finite-time consensus of multiple nonholonomic chained-form systems based on recursive distributed observer. Automatica 62, 236–242 (2015) 2. Zhang, Q.L., Li, L., Yan, X.G., Spurgeon, S.K.: Sliding mode control for singular stochastic Markovian jump systems with uncertainties. Automatica 79, 27–34 (2017) 3. Feng, Z.G., Shi, P.: Sliding mode control of singular stochastic Markov jump systems. IEEE Trans. Autom. Control 62(8), 4266–4273 (2017) 4. Zhang, Q.L., Zhang, J.Y., Wang, Y.Y.: Sliding-mode control for singular Markovian jump systems with Brownian motion based on stochastic sliding mode surface. IEEE Trans. Syst. Man Cybern.: Syst. 49(3), 494–505 (2019)

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5. Niu, Y.G., Ho, D.W.C., Wang, X.Y.: Sliding mode control for Itoˆ stochastic systems with Markovian switching. Automatica 43, 1784–1790 (2007) 6. Chen, B., Niu, Y.G., Zou, Y.Y.: Adaptive sliding mode control for stochastic Markovian jumping systems with actuator degradation. Automatica 49, 1748–1754 (2013) 7. Sun, H.B., Li, Y.K., Zong, G.D., Hou, L.L.: Disturbance attenuation and rejection for stochastic Markovian jump system with partially known transition probabilities. Automatica 89, 349–357 (2018) 8. Li, H.Y., Shi, P., Yao, D.Y., Wu, L.G.: Observer-based adaptive sliding mode control of nonlinear Markovian jump systems. Automatica 64, 133–142 (2016) 9. Boukas, E.K.: Stochastic Switching Systems Analysis and Design. Birkhauser, ¨ Boston (2006) 10. Huang, J., Shi, Y.: Stochastic stability and robust stabilization of semi-Markov jump linear systems. Int. J. Robust Nonlinear Control 23(18), 2028–2043 (2013)

Chapter 4

Sliding Mode Control Under Stochastic Disturbance and Singularity

This chapter addresses the issue of sliding mode control design for nonlinear stochastic singular semi-Markovian switching systems (S-MSSs). Stochastic disturbance is given based on stochastic semi-Markovian process related to Weibull distribution at the first time. Specific information including nonlinear boundary is known for control design. Sliding mode control law is constructed to suppress the nonlinearity and the uncertainty. First, sufficient conditions are proposed to ensure stochastic admissibility of the closed-loop sliding mode dynamics by using the Lyapunov function. Second, the sliding mode control law is designed to guarantee the reachability within a finite-time interval. Finally, the effectiveness of the results are verified through the DC motor model.

4.1 Introduction In the previous chapter, the sliding mode control design for nonlinear stochastic S-MSSs has been addressed. The application of singular system in modeling field has become more and more extensive [1], such as biological systems, mechanical engineering systems, network systems, economic systems, and so on. With the gradual deepening of control theory, many significant results for singular MSSs and singular S-MSSs have been proposed; for details, see [2–6]. In this chapter, the sliding mode control for nonlinear singular S-MSSs with stochastic disturbance will be discussed. The main objectives can be summarized as follows: (i) The sliding mode control law is constructed such that the closedloop sliding mode dynamics can dispel the adverse effect of the nonlinearity and the parameter uncertainty; (ii) Stochastic admissibility criteria for the closed-loop sliding mode dynamics are derived by using Lyapunov function; (iii) The sliding mode control law is designed to guarantee the state responses arriving at the sliding switching surface in finite time. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 W. Qi and G. Zong, Control Synthesis for Semi-Markovian Switching Systems, Studies in Systems, Decision and Control 465, https://doi.org/10.1007/978-981-99-0317-7_4

61

62

4 Sliding Mode Control Under Stochastic Disturbance and Singularity

4.2 Problem Statements and Preliminaries Consider the stochastic singular S-MSSs Ed x(t) =[(A (φt ) + ΔA (φt , t))x(t) + B(φt )(u(t) + f (t, φt , x(t)))]dt + W (φt )x(t)dω(t),

(4.1)

where x(t) ∈ Rn , u(t) ∈ Rm , f (t, φt , x(t)) ∈ Rm , and ω(t) are, respectively, the state vector, the control input, the nonlinearity, and the standard Wiener process satisfying E {dω(t)} = 0, E {d 2 ω(t)} = dt. E satisfies Rank{E} = r ≤ n. The matrix ΔA (φt , t) is ΔA (φt , t) = M (φt )F (φt , t)N (φt ) with F (φt , t) satisfying F T (φt , t)F (φt , t) ≤ I . A (φt ), M (φt ), N (φt ), B(φt ), and W (φt ) are known real matrices. {φt , t ≥ 0} means the semi-Markovian process in S = {1, 2, . . . , N } with the probability transition given as . P{φt+Δ¯ = ρ|φt = τ } .

¯ + o(Δ), ¯ δτρ (h)Δ τ /= ρ, ¯ ¯ 1 + δτ τ (h)Δ + o(Δ), τ = ρ,

¯ Δ)) ¯ = 0, and δτρ (h) ≥ 0 denotes where h ≥ 0 denotes the sojourn time, lim (o(Δ/ ¯ Δ→0 .N δτρ (h) = the transition rate from mode τ to mode ρ for τ /= ρ, and ρ=1,ρ/=τ

−δτ τ (h). In practice, the transition rate δτρ (h) is generally bounded as δ τρ ≤ δτρ (h) ≤ δ¯ τρ with real constant scalars δ τρ and δ¯ τρ . For φt = τ ∈ S , A (φt ), ΔA (φt , t), M (φt ), F (φt , t), N (φt ), B(φt ), W (φt ), and f (t, φt , x(t)) are denoted as Aτ , ΔAτ (t), Mτ , Fτ (t), Nτ , Bτ , Wτ , and f τ , respectively. Assumption 4.1 ([4]) The nonlinearity f τ is assumed to be with || f τ || ≤ ατ ||x(t)||,

(4.2)

where ατ is the positive constant. Assumption 4.2 ([5]) For τ ∈ S , Rank{E} = Rank {[E Wτ ]}. Assumption 4.3 ([5]) For τ ∈ S , Bτ is with full column rank. Definition 4.1 ([5]) Consider the nominal singular S-MSSs as Ed x(t) =Aτ x(t)dt + Wτ x(t)dω(t).

(4.3)

System (4.3) is said to be: (i) regular if det(s E − Aτ ) is not identically zero; (ii) impulse-free if deg{det(s E −{. Aτ )} = Rank(E); (iii)}stochastically stable if for initial ∞ condition x0 and φ0 ∈ S , E 0 ||x(t)||2 dt|x0 , φ0 < ∞ holds; (iv) stochastically admissible, if it is regular, impulse-free, and stochastically stable.

4.3 Sliding Mode Control Law Design

63

4.3 Sliding Mode Control Law Design The integral-type sliding mode surface is described as BτT Gˆτ E x(t)

s(t) .

.

t

− 0

BτT Gˆτ (Aτ + Bτ Kτ )x(s)ds,

(4.4)

where Gˆτ and Kτ are the real matrices to be designed with BτT Gˆτ Bτ satisfying nonsingularity. It should be noted that the nonsingularity of BτT Gˆτ Bτ can be guaranteed for Bτ with full column rank by Gˆτ > 0, ∀τ ∈ S . According to (4.1), one has .

t

E x(t) =E x(0) +

[(Aτ + ΔAτ (t))x(t) + Bτ (u(t) + f τ )]dt

0

.

t

+

Wτ x(t)dω(t),

(4.5)

0

.t where 0 Wτ x(t)dω(t) denotes the Itˆo’s stochastic integral. If BτT Gˆτ Wτ = 0, then it follows from (4.4) and (4.5) that s(t)

=BτT Gˆτ E x(0)

. + 0

t

BτT Gˆτ [(ΔAτ (t) − Bτ Kτ )x(t)

+ Bτ (u(t) + f τ )]dt.

(4.6)

When the system states arrive at the integral-type sliding mode surface, we have s(t) = 0 and s˙ (t) = 0. Then, the equivalent controller is shown as u eq =Kτ x(t) − (BτT Gˆτ Bτ )−1 BτT Gˆτ ΔAτ (t)x(t) − f τ .

(4.7)

With the equivalent controller (4.7), the closed-loop sliding mode dynamics are given as Ed x(t) =[Aτ + Bτ Kτ + ΔAτ (t) − Bτ (BτT Gˆτ Bτ )−1 BτT Gˆτ ΔAτ (t)]x(t)dt + Wτ x(t)dω(t).

(4.8)

64

4 Sliding Mode Control Under Stochastic Disturbance and Singularity

4.4 Stochastic Admissibility Analysis Theorem 4.1 If there exist matrix G˜τ > 0, symmetric matrix Gˆτ > 0, nonsingular matrices Q˜τ and Qˆτ , ∀τ ∈ S , such that ∑1τ < 0,

(4.9)

BτT Gˆτ Wτ = 0,

(4.10)

where ∑1τ = GτT A˜τ (t) + A˜τT (t)Gτ + WτT G˜τ Wτ +

.N ρ=1

δτρ (h)E T Gρ ,

A˜τ (t) = Aτ + Bτ Kτ + ΔAτ (t) − Bτ (BτT Gˆτ Bτ )−1 BτT Gˆτ ΔAτ (t), Gτ = G˜τ E + U T Q˜τ V T , Gτ−1 = Gˆτ E T + V Qˆτ U , and U and V are any matrices with full rank satisfying U E = 0 and EV = 0, then system (4.8) realizes stochastic admissibility. Proof First, consider the regularity and absence of impulse. Based on Rank(E) = r ≤ n, M¯ and N¯ satisfy [ [ ] ] I 0 A˜1τ A˜2τ M¯ E N¯ = r , , M¯A˜τ (t)N¯ = 0 0 A˜3τ A˜4τ [ ] −T G1τ G2τ ¯ ¯ M Gτ N = , det (M¯) /= 0, det (N¯) /= 0. G3τ G4τ

(4.11)

From (4.9), one has GτT A˜τ (t) + A˜τT (t)Gτ +

.N ρ=1

δτρ (h)E T Gρ < 0,

(4.12)

which means that G2τ = 0. Pre- and post-multiplying (4.12) by N¯ and N¯ yields T

[

] ∗ ∗ < 0, ∗ A˜4τT G4τ + G4τT A˜4τ

(4.13)

where ∗ will not be used. Then, one has A˜4τT G4τ + G4τT A˜4τ < 0 from the above inequality (4.13), resulting in nonsingularity of A˜4τ . Furthermore, system (4.8) is regular and impulse-free. Next, construct Lyapunov function V (x(t), φt ) . x T (t)E T G˜φt E x(t).

(4.14)

4.4 Stochastic Admissibility Analysis

65

For φt = τ , one has Γ V (x(t), τ ) 1 ¯ φt+Δ¯ )|φt = τ } − V (x(t), τ )] [E {V (x(t + Δ), ¯ Δ [ 1 .N ¯ T G˜ρ E x(t + Δ) ¯ P{φt+Δ¯ = ρ|φt = τ }x T (t + Δ)E = lim ¯ ρ=1,ρ/=τ ¯ Δ Δ→0 = lim

¯ Δ→0

¯ T G˜τ E x(t + Δ) ¯ − x T (t)E T G˜τ E x(t)] + P{φt+Δ¯ = τ |φt = τ }x T (t + Δ)E [ P{φt+Δ¯ = ρ, φt = τ } T 1 .N ¯ T G˜ρ E x(t + Δ) ¯ x (t + Δ)E = lim ¯ ρ=1,ρ/=τ ¯ P{φt = τ } Δ Δ→0 ] P{φt+Δ¯ = τ, φt = τ } T T ˜ T T ˜ ¯ ¯ x (t + Δ)E Gτ E x(t + Δ) − x (t)E Gτ E x(t) + P{φt = τ } [. ¯ − Fτ (h)) T N 1 λτρ (Fτ (h + Δ) ¯ T G˜ρ E x(t + Δ) ¯ x (t + Δ)E = lim ¯ ρ=1,ρ/ = τ ¯ 1 − Fτ (h) Δ→0 Δ ] ¯ T 1 − Fτ (h + Δ) ¯ T G˜τ E x(t + Δ) ¯ − x T (t)E T G˜τ E x(t) . (4.15) x (t + Δ)E + 1 − Fτ (h) Based on Euler-Maruyama [7], Eq. (4.8) is rewritten as ¯ + Wτ x(t)Δω(t), ¯ =E x(t) + A˜τ (t)x(t)Δ E x(t + Δ)

(4.16)

¯ − ω(t). where Δω(t) = ω(t + Δ) Furthermore, Γ V (x(t), τ ) [ ¯ − Fτ (h)) T λτρ (Fτ (h + Δ) 1 .N = lim [x (t)E T G˜ρ E x(t) ¯ ρ=1,ρ/ = τ ¯ F 1 − Δ→0 Δ τ (h) ¯ + 2x T (t)WτT G˜ρ E x(t)Δω(t) + 2x T (t)A˜τT (t)G˜ρ E x(t)Δ ¯ + x T (t)A˜τT (t)G˜ρ A˜τ (t)x(t)Δ ¯2 + 2x T (t)A˜τT (t)G˜ρ Wτ x(t)Δω(t)Δ + x T (t)WτT G˜ρ Wτ x(t)Δ2 ω(t)] +

¯ T ¯ Fτ (h) − Fτ (h + Δ) 1 − Fτ (h + Δ) x (t)E T G˜τ E x(t) + 1 − Fτ (h) 1 − Fτ (h)

¯ + 2x T (t)WτT G˜τ E x(t)Δω(t) [2x T (t)A˜τT (t)G˜τ E x(t)Δ ¯ + x T (t)A˜τT (t)G˜τ A˜τ (t)x(t)Δ ¯2 + 2x T (t)A˜τT (t)G˜τ Wτ x(t)Δω(t)Δ ] + x T (t)WτT G˜τ Wτ x(t)Δ2 ω(t) .

(4.17)

66

4 Sliding Mode Control Under Stochastic Disturbance and Singularity

From [8], we obtain ¯ ¯ 1 − Fτ (h + Δ) Fτ (h) − Fτ (h + Δ) = 1, lim = 0, ¯ ¯ 1 − Fτ (h) 1 − Fτ (h) Δ→0 Δ→0 ¯ − Fτ (h) Fτ (h + Δ) = δτ (h). lim ¯ ¯ Δ(1 − Fτ (h)) Δ→0 lim

(4.18)

For δτρ (h) = λτρ δρ (h), τ /= ρ, one has ¯ − Fτ (h)) T 1 .N λτρ (Fτ (h + Δ) [x (t)E T G˜ρ E x(t)] ¯ ρ=1,ρ/ = τ ¯ 1 − Fτ (h) Δ Δ→0 .N λτρ δτ (h)[x T (t)E T G˜ρ E x(t)] = ρ=1,ρ/=τ .N δτρ (h)x T (t)E T G˜ρ E x(t). = lim

ρ=1,ρ/=τ

(4.19)

By Itˆo’s formula [7], it is got that .

. ¯ − Fτ (h)) λτρ (Fτ (h + Δ) 1 .N T T ˜ [2x (t)Wτ Gρ E x(t)Δω(t)] = 0, E lim ¯ ρ=1,ρ/=τ 1 − Fτ (h) ¯ Δ Δ→0 . . ¯ − Fτ (h)) λτρ (Fτ (h + Δ) 1 .N T T 2 ˜ [x (t)Wτ Gρ Wτ x(t)Δ ω(t)] E lim ¯ ρ=1,ρ/=τ 1 − Fτ (h) ¯ Δ Δ→0 . . ¯ − Fτ (h)) λτρ (Fτ (h + Δ) 1 .N ¯ [x T (t)WτT G˜ρ Wτ x(t)Δ] = 0. = E lim ¯ ρ=1,ρ/=τ 1 − Fτ (h) ¯ Δ Δ→0

(4.20) The rest part of (4.17) can be similarly calculated. Based on U E = 0, one has 0 =2d x T (t)E T U T Q˜τ V T x(t) =2[A˜τ (t)dt + Wτ x(t)dω(t)]T U T Q˜τ V T x(t) .N 0= δτρ (h)x T (t)E T U T Q˜ρ V T x(t) ρ=1

(4.21)

According to (4.15)–(4.21), we have E {Γ V (x(t), τ )} =x T (t)(GτT A˜τ (t) + A˜τT (t)Gτ + WτT G˜τ Wτ .N + δτρ (h)E T Gρ )x(t). ρ=1

(4.22)

4.4 Stochastic Admissibility Analysis

67

From the condition (4.9), we can obtain E {Γ V (x(t), τ )} < 0. Therefore, system . (4.8) realizes stochastic admissibility. Remark 4.1 Sliding mode control for stochastic singular MSSs [5] mainly considers stochastic Markovian process with exponential distribution. Semi-Markovian process is applied to describe more complex practical plants than Markovian process. The system switching in this chapter is subject to a stochastic semi-Markovian process with Weibull distribution, resulting in the time-varying characteristic of the transition rate matrix. When the transition rate is fixed to be time-invariant, stochastic singular S-MSSs are transformed into ordinary stochastic singular MSSs [5]. The stochastic admissibility theorem of stochastic singular S-MSSs covers the stochastic admissibility theorem of stochastic singular MSSs [5]. Remark 4.2 Different from S-MSSs [2, 8–14], the stochastic disturbance is firstly considered in S-MSSs whose sojourn time obeys Weibull distribution, bringing some difficulties to the analysis and synthesis for S-MSSs. In Theorem 4.1, sufficient conditions for stochastic stability for stochastic S-MSSs are proved in more details. Remark 4.3 Different from nonsingular S-MSSs [2, 8, 9], there are some difficulties in investigating singularity. Owing to the existence of singularity, it is necessary to consider the regularity and impulse free of the closed-loop system. Next, there is no related literature about stochastic S-MSSs with singularity. Moreover, we need to derive the weak infinitesimal operator. Sufficient conditions are proposed to ensure stochastic admissibility of system (4.8) in Theorem 4.1. However, the uncertain elements 2x T (t)GτT ΔAτ (t)x(t) and −2x T (t)GτT Bτ (BτT Gˆτ Bτ )−1 BτT Gˆτ ΔAτ (t)x(t) exist in Theorem 4.1. Next, sufficient conditions are given for robustly stochastic admissibility of system (4.8) in Theorem 4.2. Theorem 4.2 If there exist matrix G˜τ > 0, symmetric matrix Gˆτ > 0, nonsingular matrices Q˜τ , Qˆτ , and positive scalars ε1τ , ε2τ , ∀τ ∈ S , such that (4.10) and [

] ∑2τ ∑3τ < 0, ∗ ∑4τ [ ] −Gˆτ−1 Mτ < 0, ∗ −ε2τ I

(4.23) (4.24)

where ∑2τ =GτT (Aτ + Bτ Kτ ) + (Aτ + Bτ Kτ )T Gτ + WτT G˜τ Wτ + ] [ ∑3τ = Bτ GτT Mτ ε1τ Nτ T Nτ T , ∑4τ = − diag{BτT Gˆτ Bτ , ε1τ I, ε1τ I, ε2τ I }, then system (4.8) realizes robust stochastic admissibility.

.N ρ=1

δτρ (h)E T Gρ ,

68

4 Sliding Mode Control Under Stochastic Disturbance and Singularity

Proof The inequality (4.22) is rewritten as E {Γ V (x(t), τ )} =x T (t)[2GτT (Aτ + Bτ Kτ + ΔAτ (t) − Bτ (BτT Gˆτ Bτ )−1 BτT Gˆτ ΔAτ (t)) .N + WτT G˜τ Wτ + δτρ (h)E T Gρ ]x(t). (4.25) ρ=1

Based on [12, Lemma 1], one has 2x T (t)GτT Δ Aτ (t)x(t) −1 T x (t)GτT Mτ MτT Gτ x(t) + ε1τ x T (t)Nτ T Nτ x(t), ≤ε1τ

− 2x T (t)GτT Bτ (BτT Gˆτ Bτ )−1 BτT Gˆτ ΔAτ (t) 2 T x (t)GτT Bτ (BτT Gˆτ Bτ )−1 BτT Gτ ≤ε2τ 1 + 2 Nτ T FτT (t)MτT Gˆτ Mτ Fτ (t)Nτ . ε2τ

(4.26)

Then, one has E {Γ V (x(t), τ )} < 0 by using Schur complement lemma to (4.25) and (4.26). Therefore, system (4.8) is robustly stochastically admissible. . Sufficient conditions are proposed for robustly stochastic admissibility of system (4.8) in Theorem 4.2. However, the nonlinear elements GτT Bτ Kτ , BτT KτT Gτ , Gˆτ−1 , and δτρ (h) are not solvable in strict linear matrix inequality. Next, strict linear matrix inequality conditions will be given for the matrix gain solution. Theorem 4.3 If there exist matrix Gˆτ > 0, nonsingular matrix Qˆτ , matrix Yτ , and positive scalars ε1τ , ε2τ , ∀τ ∈ S , such that (4.10) and [

] ∑5τ ∑6τ < 0, ∗ ∑7τ [ ] ˜ 5τ ∑ ˜ 6τ ∑ < 0, ∗ ∑7τ ] [ −Gˆτ Gˆτ Mτ < 0, ∗ −ε2τ I

(4.27) (4.28) (4.29)

where ∑5τ = Aτ Xτ + Bτ Yτ + XτT AτT + YτT BτT + δ τ τ (EXτ + XτT E T − E Gˆτ E T ), . ∑6τ = [XτT Bτ , Mτ , ε1τ XτT Nτ T , XτT Nτ T , XτT WτTd , δ τ 1 XτT E R , . . . . . . , δ τ τ −1 XτT E R , δ τ τ +1 XτT E R , . . . , δ τ N XτT E R ], ∑7τ = − diag{BτT Gˆτ Bτ , ε1τ I, ε1τ I, ε2τ I, E RT Gˆτ E R , E RT Gˆ1 E R , . . . , E RT Gˆτ −1 E R , E RT Gˆτ +1 E R , . . . , E RT GˆN E R },

4.4 Stochastic Admissibility Analysis

69

˜ 5τ = Aτ Xτ + Bτ Yτ + XτT AτT + YτT BτT + δ¯ τ τ (EXτ + XτT E T − E Gˆτ E T ), ∑ / T T T T T T T ˜ ∑6τ = [Xτ Bτ , Mτ , ε1τ Xτ Nτ , Xτ Nτ , Xτ Wτ d , δ¯ τ 1 XτT E R , / / / . . . , δ¯ τ τ −1 XτT E R , δ¯ τ τ +1 XτT E R , . . . , δ¯ τ N XτT E R ], ˜ 7τ = − diag{BτT Gˆτ Bτ , ε1τ I, ε1τ I, ε2τ I, E RT Gˆτ E R , E RT Gˆ1 E R , ∑ . . . , E RT Gˆτ −1 E R , E RT Gˆτ +1 E R , . . . , E RT GˆN E R }, Xτ = Gˆτ E T + V Qˆτ U , and U , V are any matrices with full rank satisfying U E = 0 and EV = 0, then system (4.8) realizes stochastic admissibility. Moreover, the controller gain is obtained by Kτ = Yτ Xτ−1 . Proof Let Gτ = G˜τ E + U T Q˜τ V T , Xτ = Gτ−1 = Gˆτ E T + V Qˆτ U , Yτ = Kτ Xτ , (4.30) where G˜τ > 0, Q˜τ , and Qˆτ are nonsingular matrices, Gˆτ = GˆτT . It is obtained that E LT G˜τ E L = (E RT Gˆτ E R )−1 with E = E L E RT . E L ∈ Rn×r and E R ∈ Rn×r are of full column rank. Since Rank{E} = Rank{[E Wτ ]}, there exists a matrix Wτ d , such that Wτ = E L Wτ d . Then, WτT G˜τ Wτ is represented as WτTd E LT G˜τ E L Wτ d . Performing a congruence transformation to (4.23) by diag{Xτ , I, I, I, I } yields [

] ∑8τ ∑9τ < 0, ∗ ∑10τ

(4.31)

where ∑8τ = Aτ Xτ + Bτ Yτ + XτT AτT + YτT BτT + XτT WτTd (E RT Gˆτ E R )−1 Wτ d Xτ + δτ τ (h)XτT E R (E RT Gˆτ E R )−1 E RT Xτ .N + δτρ (h)XτT E R (E RT Gˆρ E R )−1 E RT Xτ , ρ=1,ρ/=τ

∑9τ = [XτT Bτ , Mτ , ε1τ XτT Nτ T , XτT Nτ T ], ∑10τ = − diag{BτT Gˆτ Bτ , ε1τ I, ε1τ I, ε2τ I }. On the other side, δτ τ (h)XτT E R (E RT Gˆτ E R )−1 E RT Xτ ≤ δτ τ (h)(EXτ + XτT E T − E Gˆτ E T ). (4.32)

70

4 Sliding Mode Control Under Stochastic Disturbance and Singularity

For a specific h, δτρ (h) can be written as δτρ (h) = θ1 δ τρ + θ2 δ¯ τρ , where θ1 + θ2 = 1 and θ1 > 0, θ2 > 0. Multiplying (4.27) by θ1 and (4.28) by θ2 and using Schur complement lemma lead to ] ∑11τ ∑9τ < 0, ∗ ∑10τ

[

(4.33)

where ∑11τ = Aτ Xτ + Bτ Yτ + XτT AτT + YτT BτT + XτT WτTd (E RT Gˆτ E R )−1 Wτ d Xτ + (θ1 δ τ τ + θ2 δ¯ τ τ )(EXτ + XτT E T − E Gˆτ E T ) .N + (θ1 δ τρ + θ2 δ¯ τρ )XτT E R (E RT Gˆρ E R )−1 E RT Xτ . ρ=1,ρ/=τ

By adjusting θ1 and θ2 , all possible δτρ (h) ∈ [δ τρ , δ¯ τρ ] can be obtained, which means that (4.23) holds. Performing a congruence transformation to (4.24) by diag{Gˆτ , I }, we can get that Eq. (4.29) is equivalent to Eq. (4.24). . The condition (4.10) can be rewritten as tr [(BτT Gˆτ Wτ )T × (BτT Gˆτ Wτ )] = 0.

(4.34)

Next, we use the condition (BτT Gˆτ Wτ )T × (BτT Gˆτ Wτ ) < oI,

(4.35)

where o > 0 is the designed parameter. Furthermore, the inequality (4.35) is equivalent to ] [ −oI (BτT Gˆτ Wτ )T < 0. (4.36) ∗ −I Therefore, the optimization problem is given as min

o Gˆτ , Qˆτ , Yτ , ε1τ , ε2τ s.t. Inequalities (4.27) − (4.29) and (4.36),

(4.37)

with the controller gain given as Kτ = Yτ Xτ−1 . Remark 4.4 Without stochastic disturbance, stochastic S-MSSs are simplified to ordinary S-MSSs [2, 8–14]. Without nonlinear term and stochastic disturbance, system (4.1) is reduced to linear S-MSSs [8, 12]. Furthermore, without singularity and

4.5 Reachability Analysis

71

stochastic disturbance, system (4.1) is reduced to nonsingular S-MSSs [2, 8, 9, 11]. These show that our results are more general.

4.5 Reachability Analysis Theorem 4.4 Consider the integral sliding surface (4.4). Then, the system states are driven onto the integral sliding surface (4.4) in a finite time, and the sliding mode control law is designed as u(t) = Kτ x(t) − (||(BτT Gˆτ Bτ )−1 BτT Gˆτ Mτ ||||Nτ ||||x(t)|| 1 .N + || δ¯ τρ (BρT Gˆρ Bρ )−1 ||||s(t)|| + ατ ||x(t)|| + ητ )sgn(s(t)), ρ=1,ρ/=τ 2 (4.38) with positive constant ητ . Proof Choose Lyapunov function as V (s(t), τ ) .

1 T s (t)(BτT Gˆτ Bτ )−1 s(t). 2

(4.39)

Then, we have Γ V (s(t), τ ) 1 ¯ φt+Δ ¯ )|φt = τ } − V (s(t), τ )] [E {V (s(t + Δ ), = lim ¯ Δ Δ →0 ¯ [ 1 1 .N ¯ BρT Gˆρ Bρ )−1 s(t + Δ ) ¯ P{φt+Δ ¯ = ρ|φt = τ }s T (t + Δ )( = lim ¯ 2 ρ=1,ρ/ =τ ¯ Δ Δ →0 ] 1 1 ¯ BτT Gˆτ Bτ )−1 s(t + Δ ) ¯ − s T (t)(BτT Gˆτ Bτ )−1 s(t) + P{φt+Δ ¯ = τ |φt = τ }s T (t + Δ )( 2 2 [ P{φt+Δ ¯ = ρ, φt = τ } T 1 1 .N T ¯ Bρ Gˆρ Bρ )−1 s(t + Δ ) ¯ = lim s (t + Δ )( ¯ 2 ρ=1,ρ/ =τ ¯ P{φt = τ } Δ Δ →0 ] 1 P{φt+Δ ¯ = τ, φt = τ } T 1 T T ˆ −1 T ˆ −1 ¯ ¯ + s (t + Δ )(Bτ Gτ Bτ ) s(t + Δ ) − s (t)(Bτ Gτ Bτ ) s(t) P{φt = τ } 2 2 [ . ¯ − Fτ (h)) T N λτρ (Fτ (h + Δ ) 1 1 ¯ BρT Gˆρ Bρ )−1 s(t + Δ ) ¯ = lim s (t + Δ )( ¯ 2 ρ=1,ρ/ =τ ¯ 1 − Fτ (h) Δ Δ →0 ¯ T 1 1 − Fτ (h + Δ ) ¯ BτT Gˆτ Bτ )−1 s(t + Δ ) ¯ + s (t + Δ )( 2 1 − Fτ (h) ] 1 − s T (t)(BτT Gˆτ Bτ )−1 s(t) . (4.40) 2

72

4 Sliding Mode Control Under Stochastic Disturbance and Singularity

According to the relevant knowledge of calculus [8], the first order approximation ¯ is of s(t + Δ ) ¯ = s(t) + s˙ (t)Δ ¯ + o(Δ ), ¯ s(t + Δ )

(4.41)

¯ → 0. where Δ Then, it follows from (4.18), δτρ (h) = λτρ δρ (h), τ /= ρ, and (4.40), (4.41) that ΓV (s(t), τ ) .N 1 = s T (t)(BτT Gˆτ Bτ )−1 s˙ (t) + s T (t) δτρ (h)(BρT Gˆρ Bρ )−1 s(t) ρ=1 2 = s T (t)(BτT Gˆτ Bτ )−1 BτT Gˆτ [(Δ Aτ (t) − Bτ Kτ )x(t) + Bτ (Kτ x(t) − (||(BτT Gˆτ Bτ )−1 BτT Gˆτ Mτ ||||Nτ ||||x(t)|| 1 .N + || δ¯ τρ (BρT Gˆρ Bρ )−1 ||||s(t)|| + ατ ||x(t)|| + ητ )sgn(s(t)) + f τ )] ρ=1,ρ/=τ 2 .N 1 δτρ (h)(BρT Gˆρ Bρ )−1 s(t) + s T (t) ρ=1 2 ≤||s(t)||||(BτT Gˆτ Bτ )−1 BτT Gˆτ Mτ ||||Nτ ||||x(t)|| − s T (t)||(BτT Gˆτ Bτ )−1 BτT Gˆτ Mτ ||||Nτ ||||x(t)||sgn(s(t)) .N 1 − s T (t)|| δ¯ τρ (BρT Gˆρ Bρ )−1 ||||s(t)||sgn(s(t)) ρ=1,ρ/=τ 2 .N 1 + ||s(t)|||| δ¯ τρ (BρT Gˆρ Bρ )−1 ||||s(t)|| − ατ s T (t)||x(t)||sgn(s(t)) ρ=1,ρ/=τ 2 + ατ ||s(t)||||x(t)|| − ητ s T (t)sgn(s(t)) ≤ −ητ ||s(t)||.

(4.42)

Therefore, the system states can be driven onto the sliding surface in finite time. . Remark 4.5 It is well known that the main disadvantage of the sliding mode control is to trigger the discontinuous behavior of the sliding surface, which makes chattering and other phenomena appear in the practical systems and involves high frequency dynamics. To avoid the chattering, the sliding mode surface is designed in integral form (4.4). Next, the sliding mode control law (4.38) is constructed by using the reaching law approach. Furthermore, the discontinuous function sgn(s(t)) is replaced s(t) through the continuous treatment for the relay by the continuous function ||s(t)||+ε characteristics, where ε > 0 is a small scalar.

4.6 Simulation

73

4.6 Simulation Consider the DC motor model taken from [3], described by bτ Kt x1 (t) + x2 (t)dt, Jτ Jτ u(t) = K v x1 (t) + Rx2 (t),

d x1 (t) = −

where x1 (t) = v(t) and x2 (t) = i (t) denote the speed of shaft and the electric current respectively; K v , K t , and R denote the electromotive force, the torque constant, and the electric resistor, respectively. Furthermore, define Jτ = Jm + njcτ2 and bτ = bm + bcτ . Moreover, the mode switching of DC motor model obeys the semi-Markovian n2 process {φt , t ≥ 0} in S = {1, 2} with the transition rate matrices [ δ=

] [ ] −0.7 0.7 −1.2 1.2 , δ¯ = . 0.8 −0.8 1.5 −1.5

Let Jm = 0.5 kg m, jc1 = 50 kg m, jc2 = 150 kg m, bc1 = 100, bc2 = 240, R = 1 Ω, bm = 1, K t = 3 N m/A, K v = 1 V s/rad, and n = 10. Then, one has [

] [ ] [ ] [ ] [ ] 10 −2 3 −1.7 1.5 0 1 E= , A1 = , A2 = , B1 = B2 = , EL = , 00 1 1 1.0 1.0 1 0 [ ] [ ] [ ] 1 0 ,V = ,U = 0 1 . ER = 0 1 It is assumed that there exist stochastic disturbance ω(t), parameter uncertainty ΔAτ (t) and nonlinearity f τ in the DC motor model given as [ ] [ ] [ ] [ ] [ ] 0.1 0.2 0.2 0.3 0.2 W1 = , W2 = , W1d = 0.1 0.2 , W2d = 0.2 0.3 , M1 = , 0 0 0 0 0.1 [ ] [ ] [ ] 0.3 , N1 = 0.2 0.1 , N2 = 0.1 0.1 , F1 (t) = F2 (t) = sin(t), M2 = 0.1 f 1 = f 2 = 0.3sin(t)x1 (t), ε11 = ε12 = 0.1.

Solving the optimization problem (4.37) results in o = 1.5 × 10−6 and [ ] [ ] 12.3074 0 7.7018 0 , Gˆ2 = , o = 1.5 × 10−6 , Gˆ1 = 0 39.1947 0 23.6678 [ ] [ ] K1 = −47.3478 −15.2873 , K2 = −29.4379 −20.6759 .

74

4 Sliding Mode Control Under Stochastic Disturbance and Singularity

Fig. 4.1 System mode

2

1

0

Fig. 4.2 x1 (t) of the closed-loop system

0.5

1

1.5

2

2.5

3

2.5

3

5

4

3

2

1

0

-1

0

0.5

1

1.5

2

Then, it is got that BτT Gˆτ Bτ is nonsingular. Next, one has the integral-type sliding surface as . t [ ] [ ] s(t) = 10−5 −0.1006 0 x(t) − 103 −1.8166 −0.5600 x(s)ds, φt = 1, 0 . t [ ] [ ] −673.0626 −465.6853 x(s)ds, φt = 2. s(t) = 10−5 0.3330 0 x(t) − 0

For α1 = α2 = 0.3, η1 = 0.1, and η2 = 0.2, the sliding mode control law (4.38) is computed as

4.6 Simulation Fig. 4.3 x 2 (t) of the closed-loop system

75 20 0 -20 -40 -60 -80 -100 -120

Fig. 4.4 Integral-type sliding surface

0

0.5

1

1.5

2

2.5

3

1.5

2

2.5

3

70 60 50 40 30 20 10 0 -10

0

0.5

1

[ ] u(t) = −47.3478 −15.2873 x(t) − (5.0224||x(t)|| + 0.5||s(t)|| + 0.1)sgn(s(t)), φt = 1, [ ] u(t) = −29.4379 −20.6759 x(t) − (10.0141||x(t)|| + 1.5||s(t)|| + 0.2)sgn(s(t)), φt = 2.

Figure 4.1 describes the system mode. To prevent the control signal from chatters(t) . Figures 4.2 and 4.3 plot the state response ing, sgn(s(t)) is replaced with ||s(t)||+0.001 x(t) satisfying the stochastic admissibility. Figure 4.4 shows the integral-type sliding mode surface s(t). Figure 4.5 plots the control input u(t). These figures demonstrate that the sliding mode control can guarantee stochastic admissibility for nonlinear singular S-MSSs with stochastic disturbance.

76 Fig. 4.5 Control input

4 Sliding Mode Control Under Stochastic Disturbance and Singularity 120 100 80 60 40 20 0 -20

0

0.5

1

1.5

2

2.5

3

4.7 Conclusion In this chapter, we have studied the sliding mode control for nonlinear uncertain stochastic singular S-MSSs. At the beginning, sufficient conditions for stochastic admissibility of nonlinear uncertain stochastic singular S-MSSs are given based on the weak infinitesimal operator. Furthermore, the sliding mode control law is related to the upper bound of the transition rate affected by the semi-Markov switching rule and the controller gain. In future work, the results can be extended to fault detection for nonlinear singular S-MSSs with stochastic disturbance.

References 1. Liu, P., Zhang, Q., Yang, X., Yang, L.: Passivity and optimal control of descriptor biological complex systems. IEEE Trans. Autom. Control 53, 122–125 (2008) 2. Qi, W.H., Park, J.H., Cheng, J., Kao, Y.G.: Robust stabilization for nonlinear time-delay semiMarkovian jump systems via sliding mode control. IET Control Theory Appl. 11(10), 1504– 1513 (2017) 3. Sakthivel, R., Joby, M., Mathiyalagan, K., Santra, S.: Mixed H∞ and passive control for singular Markovian jump systems with time delays. J. Franklin Inst. 352(10), 4446–4466 (2015) 4. Wu, L.G., Su, X.J., Shi, P.: Sliding mode control with bounded L2 gain performance of Markovian jump singular time-delay systems. Automatica 48(8), 1929–1933 (2012) 5. Zhang, Q.L., Li, L., Yan, X.G., Spurgeon, S.K.: Sliding mode control for singular stochastic Markovian jump systems with uncertainties. Automatica 79, 27–34 (2017) 6. Feng, Z.G., Shi, P.: Sliding mode control of singular stochastic Markov jump systems. IEEE Trans. Autom. Control 62(8), 4266–4273 (2017) 7. Mao, X.R.: Stochastic Differential Equations and Applications, 2nd edn. England, Horwood (2007)

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8. Huang, J., Shi, Y.: Stochastic stability and robust stabilization of semi-Markov jump linear systems. Int. J. Robust Nonlinear Control 23(18), 2028–2043 (2013) 9. Wei, Y.L., Park, J.H., Qiu, J.B., Wu, L.G., Jung, H.Y.: Sliding mode control for semi-Markovian jump systems via output feedback. Automatica 81, 133–141 (2017) 10. Jiang, B.P., Kao, Y.G., Karimi, H.R., Gao, C.C.: A novel robust fuzzy integral sliding mode control for nonlinear semi-Markovian jump T-S fuzzy systems. IEEE Trans. Fuzzy Syst. 26(6), 3594–3604 (2018) 11. Zhang, L.X., Yang, T., Liu, M., Shi, P.: Stability and stabilization of a class of discrete-time fuzzy systems with semi-Markov stochastic uncertainties. IEEE Trans. Syst. Man Cybern.: Syst. 46(12), 1642–1653 (2016) 12. Jiang, B.P., Kao, Y.G., Karimi, H.R., Gao, C.C.: Stability and stabilization for singular switching semi-Markovian jump systems with generally uncertain transition rates. IEEE Trans. Autom. Control 63(11), 3919–3926 (2018) 13. Zhang, L.X., Yang, T., Colaneri, P.: Stability and stabilization of semi-Markov jump linear systems with exponentially modulated periodic distributions of sojourn time. IEEE Trans. Autom. Control 62(6), 2870–2885 (2017) 14. Zhang, L.X., Leng, Y.S., Colaneri, P.: Stability and stabilization of discrete-time semi-Markov jump linear systems via semi-Markov kernel approach. IEEE Trans. Autom. Control 61(2), 503–508 (2016)

Chapter 5

Finite-Time Sliding Mode Control Under Quantization

This chapter addresses the sliding mode control for semi-Markovian switching systems (S-MSSs) with quantized measurement under the framework of finite-time interval. The transition between each subsystems obeys a stochastic semi-Markovian process. Additionally, due to the sensor information constraints, the state vectors are not always measurable in practice. Moreover, compared with the existing results, the output quantization is considered for finite-time sliding mode control problem via a logarithmic quantizer at the first time. Our aim is to design an appropriate finite-time sliding mode control law to reduce the impact of parameter uncertainty and external disturbance on the overall performance of the system. Firstly, By using stochastic semi-Markov Lyapunov function and key points of observer design theory, the desired sliding mode control law is constructed to guarantee that the trajectory of the system reaches the specified sliding mode surface in the specified finite-time interval. Then, sufficient conditions, including reaching phase and sliding motion phase, are given to satisfy the requirement of finite-time boundedness. Finally, the applicability of the results is verified by a single link manipulator model.

5.1 Introduction In the previous chapter, the sliding mode control problem for nonlinear stochastic singular S-MSSs has been addressed. In many applications, such as robot positioning systems, communication network systems, flight systems, etc., the main research is related to the transient performance in specific finite-time interval [1, 2]. Finite-time stability means that for given initial condition, the expected state and output variable should not exceed a given physical threshold within a fixed period of time. In recent years, the research for finite-time control has received a rapid development [3–8]. In fact, finite-time theory is totally different from traditional Lyapunov theory: the former describes that the system trajectory is based on the actual demand for preset © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 W. Qi and G. Zong, Control Synthesis for Semi-Markovian Switching Systems, Studies in Systems, Decision and Control 465, https://doi.org/10.1007/978-981-99-0317-7_5

79

80

5 Finite-Time Sliding Mode Control Under Quantization

boundaries within a finite-time level; the latter mainly describes that the system trajectory reaches the equilibrium point in an infinite-time level. It is generally known that the data transmission process in the classical feedback control systems is based on zero transmission delay and high accuracy to some extent. However, due to the constraint of network bandwidth in the practical digital network communication systems, the observational values of system states can only be transmitted to the controller through a finite rate network after being quantized and encoded. The influence of quantization and coding on the system performance cannot be ignored, and system stability should be reanalyzed [9–14]. If the quantization constraint is ignored, the designed controller may not achieve the expected performance or even guarantee the stability of the system. In order to reduce the amount of information transmission in the control channel, save system resources, and achieve good performance, it is necessary to consider the influence of signal quantization. Recently, many scholars have also made some progress in the research of quantized control (see e.g., [15–21]). In this chapter, the finite-time sliding mode control is analyzed for a class of quantized S-MSSs based on the observer design method. The main contributions can be summarized as follows: (i) A class of general stochastic switching systems, namely S-MSSs, are investigated to extend the application range; (ii) An observer-based finite-time sliding mode control law is designed to guarantee that the reachability of the specified sliding surface s(t) = 0 within the given time domain [0, T ∗ ] with T ∗ ≤ T ; (iii) By utilizing Lyapunov function and logarithmic quantizer, sojourntime-dependent sufficient conditions of finite-time boundedness are given for the overall closed-loop system over both reaching phase and sliding motion phase.

5.2 Problem Statements and Preliminaries Consider a class of S-MSSs with quantized output as x(t) ˙ = (A (φt ) + ΔA (φt , t))x(t) + B(φt )u(t) + V (φt )v(t), z(t) = C (φt )x(t), z Θ (t) = Θ(z(t)),

(5.1)

where x(t) ∈ R n , u(t) ∈ R m , v(t) ∈ R r , z(t) ∈ R l , and z Θ (t) ∈ R l are the state, input, disturbance input, output, and quantized output. ΔA (φt , t) is represented by ΔA (φt , t) = M (φt )F (φt , t)N (φt ), where F (φt , t) satisfies F T (φt , t)F (φt , t) ≤ I . {φt , t ≥ 0} denotes the semi-Markovian process in k = {1, 2, . . . , ℘} with the probability transition given as { P{φt+Δ¯ = λ|φt = κ} =

¯ + o(Δ), ¯ δκλ (h)Δ κ /= λ, ¯ ¯ 1 + δκκ (h)Δ + o(Δ), κ = λ,

(5.2)

5.2 Problem Statements and Preliminaries

81

where Σh℘is the sojourn time, δκλ (h) ≥ 0 is the transition rate from κ to λ for κ /= λ, δκλ (h) = −δκκ (h). For real constant scalars δ κλ and δ¯ κλ , the transition and λ=1,λ/=κ rate is assumed to be given as δ κλ ≤ δκλ (h) ≤ δ¯ κλ . For φt = κ ∈ k, A (φt ), ΔA (φt ), B(φt ), C (φt ), and V (φt ) are, respectively, denoted as Aκ , ΔAκ (t), Bκ , Cκ , and Vκ . The disturbance input v(t) ∈ R r satisfies {

T

v T (t)v(t)dt ≤ d, d > 0,

(5.3)

0

where d is a known[ constant. ] Define Θ(·) = Θ1 (·) Θ2 (·) · · · Θl (·) as the logarithmic quantizer, where Θ(·) satisfies Θσ σ (−z γ σ (t)) = −Θσ (z σ (t)), 1 ≤ σ ≤ l.

(5.4)

The set of quantized levels of Θσ (·) takes { } Φσ = ±ϑσ(μ) , |ϑσ(μ) = (φσ )μ ϑσ(0) , μ = ±1, ±2, . . . || { } || {0} , 0 < φσ < 1, ϑσ(0) > 0, ±ϑσ(0)

(5.5)

where φσ and ϑσ(0) denote the quantizer density and the initial quantization values of the sub-quantizer Θσ (·), respectively. Then, the quantizer Θσ (·) is given as ⎧ (μ) ϑσ (μ) ⎪ if 1+λ < z σ (t) < ⎪ ⎨ ϑσ , σ Θσ (z σ (t)) = 0, if z σ (t) = 0, ⎪ ⎪ ⎩ −Θσ (−z σ (t)), if z σ (t) < 0,

(μ)

ϑσ 1−λσ

, (5.6)

σ , 1 ≤ σ ≤ l, μ = ±1, ±2, . . ., which means that 0 < λσ < 1. with λσ = 1−φ 1+φσ Define Υ = diag{λ1 , λ2 , . . . , λl }, and 0 < Υ < I . Based on the logarithmic quantizer (5.6), we have

(1 − λσ )z σ2 (t) ≤ Θσ (z σ (t))z σ (t) ≤ (1 + λσ )z σ2 (t),

(5.7)

which implies that [Θ(z(t)) − (I − Υ )z(t)]T [Θ(z(t)) − (I + Υ )z(t)] ≤ 0.

(5.8)

Then, Θ(·) can be described as Θ(z(t)) = (I − Υ )z(t) + Θs (z(t)),

(5.9)

82

5 Finite-Time Sliding Mode Control Under Quantization

where Θs (·) : R l → R l is the piecewise function satisfying ΘsT (z(t))[Θs (z(t)) − 2Υ z(t)] ≤ 0, Θs (0) = 0.

(5.10)

Substituting (5.9) into (5.1) gives rise to z Θ (t) = (I − Υ )Cκ x(t) + Θs (z(t)).

(5.11)

Therefore, the quantized S-MSSs (5.1) can be described as x(t) ˙ = (Aκ + ΔAκ (t))x(t) + Bκ u(t) + Vκ v(t), z(t) = Cκ x(t), z Θ (t) = (I − Υ )Cκ x(t) + Θs (z(t)).

(5.12)

Consider the state observer as ˙ˆ = Aκ x(t) x(t) ˆ + Bκ u(t) + Lκ (z Θ (t) − yˆ (t)), ˆ yˆ (t) = Cκ x(t),

(5.13)

where x(t) ˆ ∈ R n is the observer state and Lκ is the observer gain. Define the error signal as e(t) = x(t) − x(t) ˆ and W = I − Υ . Then, combining (5.12) and (5.13) leads to the error dynamics as ˆ e(t) ˙ = [Aκ − Lκ W Cκ + ΔAκ (t)]e(t) + [Lκ Cκ − Lκ W Cκ + ΔAκ (t)]x(t) + Vκ v(t) − Lκ Θs (z(t)). (5.14) Furthermore, substituting (5.11) into (5.13) yields ˙ˆ = [Aκ − Lκ Cκ + Lκ W Cκ ]x(t) x(t) ˆ + Bκ u(t) + Lκ W Cκ e(t) + Lκ Θs (z(t)).

(5.15)

Therefore, together with (5.14) and (5.15), one can obtain the augmented system as ξ˙ (t) = A¯κ (t)ξ(t) + B¯κ u(t) + G¯κ v(t) + L¯κ Θs (z(t)), where [ ]T ξ(t) = xˆ T (t) e T (t) , ] [ Aκ − Lκ Cκ + Lκ W Cκ Lκ W Cκ , A¯κ (t) = Lκ Cκ − Lκ W Cκ + ΔAκ (t) Aκ − Lκ W Cκ + ΔAκ (t) [ [ [ ]T ]T ]T B¯κ = BκT 0 , G¯κ = 0 VκT , L¯κ = LκT −LκT .

(5.16)

5.3 Sliding Mode Control Law Design

83

Definition 5.1 ([8]) For given scalars c1 > 0 and c2 > 0 with c1 < c2 , weighting matrix R > 0, and time interval [0, T ], system (5.16) is said to be finite-time bounded w.r.t. (c1 , c2 , [0, T ], R, d), if for t ∈ [0, T ], there holds ξ T (0)Rξ T (0) ≤ c1 ⇒ E {ξ T (t)Rξ T (t)} < c2 .

(5.17)

5.3 Sliding Mode Control Law Design The integral sliding variable is represented as {

t

ˆ − s(t) = Dκ x(t)

Dκ (Aκ + Bκ Kκ )x(s)ds, ˆ

(5.18)

0

where Dκ ∈ R m×n is chosen such that Dκ Bκ is nonsingular and Kκ ∈ R m×n will be designed in Theorem 5.4. Furthermore, we will design an appropriate sliding mode control law (5.19) to guarantee that the system trajectory can reach the specified sliding surface s(t) = 0 at the time T ∗ when T ∗ ≤ T . Theorem 5.1 The state trajectory of system (5.15) will reach the specified sliding surface s(t) = 0 at the finite-time level T ∗ when T ∗ ≤ T by u(t) = Kκ x(t) ˆ + (Dκ Bκ )−1 Dκ Lκ Cκ x(t) ˆ − (Dκ Bκ )−1 Dκ Lκ z Θ (t) − ηκ sgn(s(t)),

(5.19)

where positive constant ηκ satisfies ηκ ≥

ˆ ||Dκ x(0)|| . T

(5.20)

Proof Construct Lyapunov function as Λ(t) =

1 T s (t)s(t). 2

(5.21)

Based on the weak infinitesimal operator, we get Γ Λ(t) = s T (t)˙s (t) = s T (t)[(−Dκ Bκ Kκ − Dκ Lκ Cκ )x(t) ˆ + Dκ Bκ u(t) + Dκ Lκ W Cκ x(t) + Dκ Lκ Θs (z(t))].

(5.22)

84

5 Finite-Time Sliding Mode Control Under Quantization

Recalling Θs (z(t)) = z Θ (t) − W Cκ x(t), one has Γ Λ(t) = s T (t)[(−Dκ Bκ Kκ − Dκ Lκ Cκ )x(t) ˆ + Dκ Bκ u(t) + Dκ Lκ z Θ (t)].

(5.23)

Substituting the sliding mode control law (5.19) into the above equality (5.23) yields √ Γ Λ(t) ≤ −ηκ ||s(t)|| = −ηκ 2Λ(t),

(5.24)

√ Γ Λ(t) dt ≤ − 2ηκ dt. which means that √ Λ(t) Applying Dynkin’s formula to the above inequality from [0, T ∗ ] gives rise to √ √ √ 2ηκ ∗ ∗ E { Λ(T )} − E { Λ(0)} ≤ − T . 2 Then, we have √ √ √ 2ηκ ∗ ∗ E { Λ(T )} ≤ − T + E { Λ(0)}. 2 Furthermore, one has √ 0≤−

√ 2ηκ ∗ T + E { Λ(0)}. 2

From the above inequality, there exists √ E { 2Λ(0)} T ≤ . ηκ ∗

(5.25)

From the sliding function s(0) = Dκ x(0) ˆ and the Lyapunov function Λ(0) = we have

1 T s (0)s(0), 2

T∗ ≤

ˆ Dκ x(0) . ηκ

(5.26)

Considering the condition (5.20), it follows from (5.26) that T ∗ ≤ T,

(5.27)

which shows that the specified sliding surface s(t) = 0 will be reached before the time T . ∎

5.4 Finite-Time Boundedness Analysis over Reaching Phase Within [0, T ∗ ]

85

Remark 5.1 Different from traditional sliding mode control approach in an infinitetime level, the sliding mode control law (5.19) depends on the finite-time level T and the initial condition x(0) ˆ through the constant selection (5.20). Moreover, the connection among specified sliding surface is considered in sliding mode control law (5.19) via the controller gain Kκ . Therefore, under the effect of finite-time rule, the sliding mode control law (5.19) guarantees that the system trajectory can reach the specified sliding surface s(t) = 0 at the time T ∗ when T ∗ ≤ T . Remark 5.2 As is well known, the chattering effect is the main unavoidable drawback of the sliding mode control approach derived from the switching signals, which will cause high-frequency chattering and high-frequency dynamics. In order to avoid the chattering, the discontinuous function sgn(s(t)) is replaced by the continuous s(t) through the continuous treatment for the relay characteristics of the function ||s(t)||+∈ controller, where ∈ > 0 denotes a small scalar.

5.4 Finite-Time Boundedness Analysis over Reaching Phase Within [0, T ∗ ] Substituting the sliding mode control law (5.19) into system (5.16), one has ˆ ξ˙ (t) = A¯¯κ (t)ξ(t) + G¯¯κ v(t) + L¯¯κ Θs (z(t)) − B¯¯κ η(t),

(5.28)

where [ ]T ξ(t) = xˆ T (t) e T (t) , ] [ Aκ + Bκ Kκ Lκ W Cκ ¯ ¯ A κ (t) = , Lκ Cκ − Lκ W Cκ + ΔAκ (t) Aκ − Lκ W Cκ + ΔAκ (t) ]T ]T [ [ [ ]T G¯¯ = 0 V T , L¯¯ = 0 −L T , B¯¯ = B T 0 , η(t) ˆ = η sgn(s(t)). κ

κ

κ

κ

κ

κ

κ

(5.29)

Theorem 5.2 If there exist positive-definite symmetric matrix Pκ , real matrices Kκ , Lκ , and positive scalars γ , c∗ , ε1κ , ε2κ , ∀κ ∈ k, such that the following inequalities hold Π1κ < 0, c1 < c∗ < c2 , λ1 c1 + γ d + λ2 where

(5.30) (5.31) γ ηκ2 T ∗

∗

< e−γ T c∗ ,

(5.32)

86

5 Finite-Time Sliding Mode Control Under Quantization

⎡

Π1κ

11 Π1κ ⎢ ∗ ⎢ =⎢ ⎢ ∗ ⎣ ∗ ∗

12 Π1κ 22 Π1κ ∗ ∗ ∗

0 23 Π1κ 33 Π1κ ∗ ∗

14 Π1κ 0 0 44 Π1κ ∗

⎤ 15 Π1κ 25 ⎥ Π1κ ⎥ 0 ⎥ ⎥, 0 ⎦ 55 Π1κ

11 Π1κ = H e{P1κ (Aκ + Bκ Kκ )} − γ P1κ + ε1κ NκT Nκ +

Σ℘ λ=1

δκλ (h)P1λ ,

12 14 15 = P1κ Lκ W Cκ + (Lκ Cκ − Lκ W Cκ )T P1κ , Π1κ = −P1κ Bκ , Π1κ = CκT Υ, Π1κ −1 22 = H e{P1κ (Aκ − Lκ W Cκ )} − γ P1κ + ε1κ P1κ Mκ MκT P1κ Π1κ Σ℘ −1 + ε2κ P1κ Mκ MκT P1κ + ε2κ NκT Nκ + δκλ (h)P1λ , λ=1

23 Π1κ

=

25 P1κ Vκ , Π1κ

=

33 CκT Υ, Π1κ

44 55 = −γ I, Π1κ = −γ I, Π1κ = −I,

1 1 P¯ κ = R − 2 Pκ R − 2 , λ1 = λmax {P¯ κ }, λ2 = λmin {P¯ κ }, Pκ = diag{P1κ , P1κ },

then system (5.28) realizes finite-time boundedness w.r.t. (c1 , c∗ , [0, T ∗ ], R, d). Proof Construct Lyapunov function as Λ(ξ(t), κ) = ξ T (t)Pκ ξ(t),

(5.33)

where Pκ = diag{P1κ , P1κ }. Along with system (5.28), we have Γ Λ(ξ(t), κ) 1 ¯ φt+Δ¯ )|φt = κ} − Λ(ξ(t), κ)] = lim [E {Λ(ξ(t + Δ), ¯ ¯ Δ Δ→0 [ 1 Σ℘ ¯ ¯ P{φt+Δ¯ = λ|φt = κ}ξ T (t + Δ)P = lim λ ξ(t + Δ) ¯ λ=1,λ/=κ ¯ Δ Δ→0 ] T T ¯ ¯ + P{φt+Δ¯ = κ|φt = κ}ξ (t + Δ)Pκ ξ(t + Δ) − ξ (t)Pκ ξ(t) [ P{φt+Δ¯ = λ, φt = κ} T 1 Σ℘ ¯ ¯ ξ (t + Δ)P = lim λ ξ(t + Δ) ¯ λ=1,λ/ = κ ¯ P{φt = κ} Δ→0 Δ ] P{φt+Δ¯ = κ, φt = κ} T T ¯ ¯ ξ (t + Δ)Pκ ξ(t + Δ) − ξ (t)Pκ ξ(t) + P{φt = κ} [ ¯ − Fκ (h)) T 1 Σ℘ λκλ (Fκ (h + Δ) ¯ ¯ = lim ξ (t + Δ)P λ ξ(t + Δ) ¯ λ=1,λ/=κ ¯ 1 − Fκ (h) Δ Δ→0 ] ¯ T 1 − Fκ (h + Δ) T ¯ ¯ + ξ(t + Δ) − ξ (t)P ξ(t) . (5.34) ξ (t + Δ)P κ κ 1 − Fκ (h)

5.4 Finite-Time Boundedness Analysis over Reaching Phase Within [0, T ∗ ]

87

Considering the Taylor series expansion leads to ¯ = ξ(t) + f κ (t)Δ ¯ + o(Δ), ¯ ξ(t + Δ)

(5.35)

¯ → 0. where f κ (t) = A¯¯κ (t)ξ(t) + G¯¯κ v(t) + L¯¯κ Θs (z(t)) − B¯¯κ η(t) ˆ and Δ Then, one has Γ Λ(ξ(t), κ) ¯ − Fκ (h)) 1 Σ℘ λκλ (Fκ (h + Δ) ¯ + o(Δ)] ¯ T [ [ξ(t) + f κ (t)ξ(t)Δ ¯ λ=1,λ/ = κ ¯ 1 − Fκ (h) Δ→0 Δ

= lim

¯ + o(Δ)] ¯ + Pλ [ξ(t) + f κ (t)Δ

¯ 1 − Fκ (h + Δ) ¯ + o(Δ)] ¯ T [ξ(t) + f κ (t)Δ 1 − Fκ (h)

¯ + o(Δ)] ¯ − ξ T (t)Pκ ξ(t)]. Pκ [ξ(t) + f κ (t)Δ

(5.36)

According to Ref. [22], we have ¯ ¯ 1 − Fκ (h + Δ) Fκ (h) − Fκ (h + Δ) = 1, lim = 0, ¯ ¯ 1 − Fκ (h) 1 − Fκ (h) Δ→0 Δ→0 lim

¯ − Fκ (h) Fκ (h + Δ) = δκ (h). ¯ − Fκ (h)) ¯ Δ(1 Δ→0 lim

(5.37)

Based on δκλ (h) = λκλ δλ (h), κ /= λ, we can obtain ¯ − Fκ (h)) T 1 Σ℘ λκλ (Fκ (h + Δ) [ξ (t)Pλ ξ(t)] ¯ λ=1,λ/=κ ¯ 1 − Fκ (h) Δ Δ→0 Σ℘ λκλ δκ (h)[ξ T (t)Pλ ξ(t)] = lim

λ=1,λ/=κ

=

Σ℘ λ=1,λ/=κ

δκλ (h)ξ T (t)Pλ ξ(t).

It follows from (5.34)–(5.38) that

(5.38)

88

5 Finite-Time Sliding Mode Control Under Quantization

Γ Λ(ξ(t), κ) ˆ = 2ξ T (t)Pκ [A¯¯κ (t)ξ(t) + G¯¯κ v(t) + L¯¯κ Θs (z(t)) − B¯¯κ η(t)] Σ℘ δκλ (h)ξ T (t)Pλ ξ(t) + λ=1

ˆ + Lκ W Cκ e(t) − Bκ η(t)] ˆ = 2 xˆ (t)P1κ [(Aκ + Bκ Kκ )x(t) Σ℘ δκλ (h)xˆ T (t)P1λ x(t) ˆ + 2e T (t)P1κ [(Lκ Cκ − Lκ W Cκ + T

λ=1

+ ΔAκ (t))x(t) ˆ + (Aκ − Lκ W Cκ + ΔAκ (t))e(t) + Vκ v(t) − Lκ Θs (z(t))] Σ℘ + δκλ (h)e T (t)P1λ e(t). (5.39) λ=1

For the uncertainties in (5.39), one has 2e T (t)P1κ ΔAκ (t)x(t) ˆ −1 T e (t)P1κ Mκ MκT P1κ e(t) + ε1κ xˆ T (t)NκT Nκ x(t), ˆ ≤ε1κ

2e T (t)P1κ ΔAκ (t)e(t) −1 T e (t)P1κ Mκ MκT P1κ e(t) + ε2κ e T (t)NκT Nκ e(t). ≤ε2κ

(5.40)

It is noted that the inequality (5.10) is equivalent to −ΘsT (z(t))Θs (z(t)) + 2ΘsT (z(t))Υ Cκ (x(t) ˆ + e(t)) ≥ 0.

(5.41)

Based on (5.39)–(5.41), we obtain Γ Λ(ξ(t), κ) ≤ xˆ T (t)[H e{P1κ (Aκ + Bκ Kκ )} + ε1κ NκT Nκ +

Σ℘

δκλ (h)P1λ ]x(t) ˆ

λ=1 −1 Cκ )} + ε1κ P1κ Mκ MκT P1κ

−1 + ε2κ P1κ Mκ MκT P1κ + e (t)[H e{P1κ (Aκ − Lκ W Σ℘ + ε2κ NκT Nκ + δκλ (h)P1λ ]e(t) + 2 xˆ T (t)P1κ Lκ W Cκ e(t) T

λ=1

− 2 xˆ T (t)P1κ Bκ η(t) ˆ + 2e T (t)P1κ (Lκ Cκ − Lκ W Cκ )x(t) ˆ + 2e T (t)P1κ Vκ v(t) ˆ − 2e T (t)P1κ Lκ Θs (z(t)) − ΘsT (z(t))Θs (z(t)) + 2ΘsT (z(t))Υ Cκ x(t) + 2ΘsT (z(t))Υ Cκ e(t).

(5.42)

Define an auxiliary function as J1 (t) ≙Γ Λ(ξ(t), κ) − γ Λ(ξ(t), κ) − γ v T (t)v(t) − γ ηˆ T (t)η(t). ˆ

(5.43)

5.5 Finite-Time Boundedness Analysis over Sliding Motion Phase Within [T ∗ , T ]

89

By using Schur complement, the inequality (5.30) implies [ ] J1 (t) ≤ xˆ T (t) e T (t) v T (t) ηˆ T (t) ΘsT (z(t)) Π1κ [ T ]T xˆ (t) e T (t) v T (t) ηˆ T (t) ΘsT (z(t)) < 0,

(5.44)

Γ Λ(ξ(t), κ) < γ Λ(ξ(t), κ) + γ v T (t)v(t) + γ ηˆ T (t)η(t). ˆ

(5.45)

i.e.,

Multiplying (5.45) by e−γ t and applying Dynkin’s formula yield { t E {e−γ t Λ(ξ(t), κ)} < Λ(ξ(0), φ0 ) + γ e−γ s v T (s)v(s)ds 0 { t +γ e−γ s ηˆ T (s)η(s)ds ˆ ≤ λ1 ξ T (0)Rξ T (0) + γ d + γ ηκ2 T ∗ 0

≤ λ1 c1 + γ d + γ ηκ2 T ∗ , t ∈ [0, T ∗ ].

(5.46)

On the other hand, it follows from (5.33) that ∗

e−γ t E {Λ(ξ(t), κ)} ≥ e−γ T λ2 E {ξ T (t)Rξ T (t)}, t ∈ [0, T ∗ ],

(5.47)

which means that E {ξ T (t)Rξ T (t)} ≤

λ1 c1 + γ d + γ ηκ2 T ∗ . e−γ T ∗ λ2

(5.48)

Considering the conditions (5.31) and (5.32), one has E {ξ T (t)Rξ T (t)} < c∗ , for t ∈ [0, T ∗ ]. Therefore, system (5.28) realizes finite-time boundedness w.r.t. (c1 , c∗ , ∎ T ∗ , R, d).

5.5 Finite-Time Boundedness Analysis over Sliding Motion Phase Within [T ∗ , T ] If the system trajectory reaches the specified sliding surface s(t) = 0, one has s˙ (t) = 0. Therefore, we get an equivalent control law as ˆ + (Dκ Bκ )−1 [(Dκ Lκ Cκ − Dκ Lκ W Cκ )x(t) ˆ u eq (t) = Kκ x(t) − Dκ Lκ Θs (z(t)) − Dκ Lκ W Cκ e(t)].

(5.49)

90

5 Finite-Time Sliding Mode Control Under Quantization

Substituting (5.49) into the argument system (5.16) yields ξ˙ (t) = A˜κ (t)ξ(t) + G˜κ v(t) + L˜κ Θs (z(t)),

(5.50)

where [ ξ(t) =

] ] [ 11 ˆ x(t) A˜κ (t) A˜κ12 (t) , , A˜κ (t) = e(t) A˜κ21 (t) A˜κ22 (t)

A˜κ11 (t) = Aκ + Bκ Kκ + Bκ (Dκ Bκ )−1 (Dκ Lκ Cκ − Dκ Lκ W Cκ ) − Lκ Cκ + Lκ W Cκ , A˜κ12 (t) = Lκ W Cκ − Bκ (Dκ Bκ )−1 Dκ Lκ W Cκ , A˜κ21 (t) = Lκ Cκ − Lκ W Cκ + ΔAκ (t), A˜κ22 (t) = Aκ − Lκ W Cκ + ΔAκ (t), [ ] [ ] 0 Lκ − Bκ (Dκ Bκ )−1 Dκ Lκ G˜κ = , L˜κ = . −Lκ Vκ Theorem 5.3 If the sliding mode parameter matrix Dκ is chosen as Dκ = BκT P1κ and there exist positive-definite symmetric matrix Pκ , real matrices Kκ , Lκ , and positive scalars γ , c∗ , ε1κ , ε2κ , ∀κ ∈ k, such that (5.31) and Π2κ < 0, λ1 c ∗ + γ d < e−γ T c2 , λ2

(5.51) (5.52)

where ⎡ 11 12 14 ⎤ Π2κ Π2κ 0 Π2κ ⎢ 22 Π 23 Π 24 ⎥ ⎥ ⎢ ∗ Π2κ 2κ 2κ ⎥ ⎢ Π2κ = ⎢ ⎥, 33 ⎢ ∗ ∗ Π2κ 0 ⎥ ⎦ ⎣ 44 ∗ ∗ ∗ Π2κ 11 = H e{P (A + B K )} − H e{P L C } − γ P + H e{P L W C } Π2κ κ κ κ κ 1κ 1κ κ κ 1κ 1κ κ

+ 4P1κ Bκ (BκT P1κ Bκ )−1 BκT P1κ + ε1κ NκT Nκ + CκT W LκT P1κ Lκ W Cκ Σ℘ + CκT LκT P1κ Lκ Cκ + δκλ (h)P1λ , λ=1

12 = P L W C + (L C − L W C )T P , Π 14 = P L + C T Υ, Π2κ κ κ κ κ κ 1κ κ 1κ 1κ κ κ 2κ 22 = H e{P (A − L W C )} − γ P + C T W L T P L W C Π2κ κ κ κ κ 1κ 1κ 1κ κ κ κ Σ℘ −1 −1 T T T + ε1κ P1κ Mκ Mκ P1κ + ε2κ P1κ Mκ Mκ P1κ + ε2κ Nκ Nκ +

δ (h)P1λ , λ=1 κλ

23 = P V , Π 24 = C T Υ − P L , Π 33 = −γ I, Π 44 = L T P L − I, Π2κ 1κ κ 1κ κ 1κ κ κ κ 2κ 1κ 2κ 1

1

P¯ κ = R − 2 Pκ R − 2 , λ1 = λmax {P¯ κ }, λ2 = λmin {P¯ κ }, Pκ = diag{P1κ , P1κ },

then system (5.50) realizes finite-time boundedness w.r.t. (c∗ , c2 , [T ∗ , T ], R, d).

5.5 Finite-Time Boundedness Analysis over Sliding Motion Phase Within [T ∗ , T ]

91

Proof Construct Lyapunov function as Λ(ξ(t), κ) = ξ T (t)Pκ ξ(t),

(5.53)

where Pκ = diag{P1κ , P1κ }. Along with system (5.50), we have the weak infinitesimal operator as Γ Λ(ξ(t), κ) = 2 xˆ T (t)P1κ [(Aκ + Bκ Kκ + Bκ (BκT P1κ Bκ )−1 (BκT P1κ Lκ Cκ − BκT P1κ Lκ W Cκ ) − Lκ Cκ + Lκ W Cκ )x(t) ˆ + (Lκ W Cκ − Bκ (BκT P1κ Bκ )−1 BκT P1κ Lκ W Cκ )e(t) + (Lκ − Bκ (BκT P1κ Bκ )−1 BκT P1κ Lκ )Θs (z(t))] Σ℘ + δκλ (h)xˆ T (t)P1λ x(t) ˆ + 2e T (t)P1κ [(Lκ Cκ − Lκ W Cκ + ΔAκ (t))x(t) ˆ λ=1

+ (Aκ − Lκ W Cκ + ΔAκ (t))e(t) + Vκ v(t) − Lκ Θs (z(t))] Σ℘ + δκλ (h)e T (t)P1λ e(t). λ=1

(5.54)

For the uncertainties in (5.54), we have − 2 xˆ T (t)P1κ Bκ (BκT P1κ Bκ )−1 BκT P1κ Lκ W Cκ x(t) ˆ −1 ≤[xˆ T (t)P1κ Bκ (BκT P1κ Bκ )−1 BκT P1κ ]P1κ

[P1κ Bκ (BκT P1κ Bκ )−1 BκT P1κ x(t)] ˆ + [xˆ T (t)CκT W LκT ]P1κ [Lκ W Cκ x(t)] ˆ = xˆ T (t)P1κ Bκ (BκT P1κ Bκ )−1 BκT P1κ x(t) ˆ + [xˆ T (t)CκT W LκT ]P1κ [Lκ W Cκ x(t)]. ˆ

(5.55)

Similar to the above method, it is got that 2 xˆ T (t)P1κ Bκ (BκT P1κ Bκ )−1 BκT P1κ Lκ Cκ x(t) ˆ ≤xˆ T (t)P1κ Bκ (BκT P1κ Bκ )−1 BκT P1κ x(t) ˆ + [xˆ T (t)CκT LκT ]P1κ [Lκ Cκ x(t)], ˆ − 2 xˆ T (t)P1κ Bκ (BκT P1κ Bκ )−1 BκT P1κ Lκ W Cκ e(t) ≤xˆ T (t)P1κ Bκ (BκT P1κ Bκ )−1 BκT P1κ x(t) ˆ + [e T (t)CκT W LκT ]P1κ [Lκ W Cκ e(t)], − 2 xˆ T (t)P1κ Bκ (BκT P1κ Bκ )−1 BκT P1κ Lκ Θs (z(t)) ≤xˆ T (t)P1κ Bκ (BκT P1κ Bκ )−1 BκT P1κ x(t) ˆ + ΘsT (z(t))LκT P1κ Lκ Θs (z(t)).

(5.56)

92

5 Finite-Time Sliding Mode Control Under Quantization

Recalling (5.40) and (5.41), it follows from (5.54)–(5.56) that Γ Λ(ξ(t), κ) ≤ xˆ T (t)[H e{P1κ (Aκ + Bκ Kκ )} − H e{P1κ Lκ Cκ } + H e{P1κ Lκ W Cκ } + 4P1κ Bκ (BκT P1κ Bκ )−1 BκT P1κ + ε1κ NκT Nκ + CκT W LκT P1κ Lκ W Cκ Σ℘ + CκT LκT P1κ Lκ Cκ + δκλ (h)P1λ ]x(t) ˆ + e T (t)[H e{P1κ (Aκ − Lκ W Cκ )} λ=1

−1 −1 + CκT W LκT P1κ Lκ W Cκ + ε1κ P1κ Mκ MκT P1κ + ε2κ P1κ Mκ MκT P1κ Σ℘ + ε2κ NκT Nκ + δκλ (h)P1λ ]e(t) + 2 xˆ T (t)P1κ Lκ W Cκ e(t) λ=1

ˆ + 2 xˆ T (t)P1κ Lκ Θs (z(t)) + 2e T (t)P1κ (Lκ Cκ − Lκ W Cκ )x(t) + 2e T (t)P1κ Vκ v(t) − 2e T (t)P1κ Lκ Θs (z(t)) + ΘsT (z(t))(LκT P1κ Lκ − I )Θs (z(t)) + 2ΘsT (z(t))Υ Cκ x(t) ˆ + 2ΘsT (z(t))Υ Cκ e(t).

(5.57)

Define an auxiliary function as J2 (t) ≙Γ Λ(ξ(t), κ) − γ Λ(ξ(t), κ) − γ v T (t)v(t).

(5.58)

By Schur complement, the inequality (5.51) implies ] [ J2 (t) ≤ xˆ T (t) e T (t) v T (t) ΘsT (z(t)) Π2κ ]T [ T xˆ (t) e T (t) v T (t) ΘsT (z(t)) < 0,

(5.59)

Γ Λ(ξ(t), κ) < γ Λ(ξ(t), κ) + γ v T (t)v(t).

(5.60)

i.e.,

From Theorem 5.2, the resulting closed-loop system realizes finite-time boundedness w.r.t. (c1 , c∗ , [0, T ∗ ], R, d) within the reaching phase, which means that the initial condition of the sliding motion phase is E {ξ T (t)Rξ T (t)} < c∗ at the instant t = T ∗. Multiplying (5.60) by e−γ t and applying Dynkin’s formula yield e

−γ t

∗

E {Λ(ξ(t), κ)} < Λ(ξ(T ), φT ∗ ) + γ

{

t T∗

e−γ s v T (s)v(s)ds

≤λ1 E {ξ T (T ∗ )Rξ T (T ∗ )} + γ d, ≤ λ1 c∗ + γ d, t ∈ [T ∗ , T ].

(5.61)

It should be noted that e−γ t E {Λ(ξ(t), κ)} ≥ e−γ T E {Λ(ξ(t), κ)} ≥ λ2 e−γ T E {ξ T (t)Rξ T (t)}.

(5.62)

5.6 Gain Matrix Design

93

Together with (5.61) and (5.62), we have E {ξ T (t)Rξ T (t)} ≤

λ1 c∗ + γ d . λ2 e−γ T

(5.63)

Furthermore, from the conditions (5.31) and (5.52), we can get a scalar c∗ with ∎ c1 < c∗ < c2 such that E {ξ T (t)Rξ T (t)} < c2 . Remark 5.3 Different from traditional sliding mode control approach in an infinitetime level, we need to consider two key points when studying sliding mode control during a finite-time level. First, for given finite-time level [0, T ], an appropriate sliding mode control law should be designed to guarantee that the system trajectory can reach the specified sliding surface s(t) = 0 at a time T ∗ when T ∗ ≤ T . Second, for given constraint condition of input variable and finite-time level [0, T ], finitetime boundedness criteria over both reaching phase within [0, T ∗ ] and sliding motion phase within [T ∗ , T ] should be designed for the whole resulting closed-loop system based on sliding mode control law. When the specified sliding surface s(t) = 0 is reached at a time T ∗ , the system trajectory will move to the next phase named as sliding motion phase within [T ∗ , T ]. In such case, finite-time boundedness criteria over sliding motion phase within [T ∗ , T ] are designed in Theorem 5.3. It is well known that sliding mode control strategy has better properties, such as external disturbance suppression and strong robustness to uncertainties.

5.6 Gain Matrix Design For the scheme design in this chapter, the controller gain Kκ and observer gain Lκ should be designed to satisfy the conditions in Theorems 5.2 and 5.3 simultaneously. Next, To deal with some nonlinearities in Theorems 5.2 and 5.3, sufficient conditions that depend on strict linear matrix inequalities will be given in Theorem 5.4. Theorem 5.4 System (5.16) realizes finite-time boundedness w.r.t. (c1 , c2 , [0, T ], R, d), if there exist positive-definite symmetric matrix Pκ , real matrix Hκ , and positive scalars γ , c∗ , ε1κ , ε2κ , ψ1 , ψ2 , ∀κ ∈ k, such that (5.31) and ¯ 3κ < 0, Π 3κ < 0, Π ¯ 4κ < 0, Π 4κ < 0, Π

(5.64)

ψ2 R1 ≤ Pκ ≤ ψ1 R1 ,

(5.66)

ψ1 c1 + γ d ∗

ψ1 c + γ d

+ γ ηκ2 T < ψ2 e−γ T c∗ , < ψ2 e−γ T c2 ,

(5.65) (5.67) (5.68)

94

5 Finite-Time Sliding Mode Control Under Quantization

where

¯ 3κ Π

¯ 4κ Π

Π 4κ

⎡ 11 12 ¯ 3κ Π ¯ 3κ Π ⎢ ⎢ ∗ Π ¯ 22 ⎢ 3κ ⎢ ⎢ ∗ ∗ =⎢ ⎢ ⎢ ∗ ∗ ⎢ ⎢ ∗ ∗ ⎣ ∗ ⎡ 11 ¯ Π ⎢ 4κ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ =⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎣ ∗ ⎡ Π 11 ⎢ 4κ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ =⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎣ ∗

¯ 11 Π 3κ = Π5κ +

∗ ¯ 12 Π 4κ ¯ 22 Π 4κ ∗ ∗ ∗ ∗ ∗ ¯ 12 Π 4κ Π 22 4κ ∗ ∗ ∗ ∗ ∗

14 15 0 Π1κ Π1κ 0

⎤

⎡

⎢ ⎥ 23 25 ¯ 26 ⎥ ⎢ Π1κ 0 Π1κ Π 3κ ⎥ ⎢ ⎢ ⎥ 33 ⎢ ⎥ Π1κ 0 0 0 ⎥ ⎢ = , Π 3κ ⎢ ⎥ 44 0 0 ⎥ ∗ Π1κ ⎢ ⎢ ⎥ 55 ⎢ ⎥ ∗ ∗ Π1κ 0 ⎦ ⎣ 66 ¯ 3κ ∗ ∗ ∗ Π ⎤ 14 15 ¯ 4κ Π ¯ 4κ 0 0 Π 0 ⎥ ⎥ 23 ¯ 24 ¯ 26 Π2κ Π 4κ 0 Π 0 ⎥ 4κ ⎥ 33 0 0 0 ⎥ Π2κ 0 ⎥ ⎥ 44 T ⎥, ¯ ∗ Π 4κ 0 0 Hκ ⎥ ⎥ ¯ 55 ∗ ∗ Π 0 ⎥ ⎥ 4κ 0 ⎥ 66 ¯ ∗ ∗ ∗ Π 4κ 0 ⎥ ⎦ ∗ ∗ ∗ ∗ −P1κ ⎤ ¯ 14 ¯ 15 0 Π 0 4κ Π 4κ 0 ⎥ ⎥ 23 ¯ 24 ¯ 26 Π2κ Π 4κ 0 Π 0 ⎥ 4κ ⎥ 33 0 0 0 ⎥ Π2κ 0 ⎥ ⎥ 44 T ⎥, ¯ ∗ Π 4κ 0 0 Hκ ⎥ ⎥ ¯ 55 ∗ ∗ Π 0 ⎥ ⎥ 4κ 0 ⎥ 66 ¯ ∗ ∗ ∗ Π 4κ 0 ⎥ ⎦ ∗ ∗ ∗ ∗ −P1κ

Σ℘ λ=1

δ¯ κλ P1λ , Π 11 3κ = Π5κ +

14 15 ¯ 12 0 Π 11 3κ Π 3κ 0 Π1κ Π1κ

⎥ 23 25 ¯ 26 ⎥ ∗ Π 22 3κ Π1κ 0 Π1κ Π 3κ ⎥ ⎥ 33 ∗ ∗ Π1κ 0 0 0 ⎥ ⎥, ⎥ 44 0 0 ⎥ ∗ ∗ ∗ Π1κ ⎥ 55 ∗ ∗ ∗ ∗ Π1κ 0 ⎥ ⎦ 66 ¯ 3κ ∗ ∗ ∗ ∗ ∗ Π

Σ℘

δ P1λ , λ=1 κλ

T T T T ¯ 12 Π 3κ = Hκ W Cκ + Cκ Hκ − Cκ W Hκ , Σ℘ Σ℘ ¯ 22 Π δ¯ κλ P1λ , Π 22 = Π6κ + δ P1λ , 3κ 3κ = Π6κ + λ=1 λ=1 κλ ] [ ¯ 26 ¯ 66 Π 3κ = P1κ Mκ P1κ Mκ , Π 3κ = −diag{ε1κ , ε2κ }, Σ Σ℘ ℘ ¯ 11 Π δ¯ κλ P1λ , Π 11 δ κλ P1λ 4κ = Π7κ + 4κ = Π7κ + λ=1

⎤

λ=1

T T T T ¯ 14 T ¯ 12 Π 4κ = Hκ W Cκ + Cκ Hκ − Cκ W Hκ , Π 4κ = Hκ + Cκ Υ, Σ Σ ℘ ℘ ¯ 22 Π δ¯ κλ P1λ , Π 22 δ P1λ , 4κ = Π8κ + 4κ = Π8κ + λ=1 λ=1 κλ ] [ T T T T T ¯ 24 ¯ 44 ¯ 15 , Π 4κ = Cκ Υ − Hκ , Π 4κ = −I, Π 4κ = 2P1κ Bκ Cκ W Hκ Cκ Hκ

5.6 Gain Matrix Design

95

] [ T T T ¯ 55 ¯ 26 ,Π Π 4κ = +P1κ Mκ P1κ Mκ Cκ W Hκ 4κ = −diag{Bκ P1κ Bκ , P1κ , P1κ }, ¯ 66 Π 4κ = −diag{ε1κ , ε2κ , P1κ }, Π5κ = H e{P1κ (Aκ + Bκ Kκ )} − γ P1κ + ε1κ NκT Nκ , Π6κ = H e{P1κ Aκ − Hκ W Cκ } − γ P1κ + ε2κ NκT Nκ , Π7κ = H e{P1κ (Aκ + Bκ Kκ )} − H e{Hκ Cκ } + H e{Hκ W Cκ } + ε1κ NκT Nκ , Π8κ = H e{P1κ Aκ − Hκ W Cκ } − γ P1κ + ε2κ NκT Nκ , Pκ = diag{P1κ , P1κ }, R = diag{R1 , R1 }, 14 15 23 25 33 44 55 23 33 , Π1κ , Π1κ , Π1κ , Π1κ , Π1κ , Π1κ , Π2κ , and Π2κ described in Theorems 5.2 with Π1κ and 5.3, in which Kκ is selected such that Aκ + Bκ Kκ is Hurwitz. Furthermore, −1 Hκ . the observer gain is given as Lκ = P1κ

Proof Let Hκ = P1κ Lκ . For given h, we have δκλ (h) = θ1 δ κλ + θ2 δ¯ κλ , where θ1 + θ2 = 1 and θ1 > 0, θ2 > 0. Multiplying the first inequality by θ1 and the second inequality by θ2 in (5.64), we can get the equivalent relation between (5.64) and (5.30). Similarly, we can also get the equivalent relation between (5.65) and (5.51). Considering the following inequality λ1 c1 + γ d + γ ηκ2 T < e−γ T c∗ , λ2

(5.69)

it is easily noted that the inequality (5.32) is implied by the above inequality (5.69). By the condition (5.66), one has λ1 ≤ ψ1 , λ2 ≥ ψ2 .

(5.70)

Then, the inequalities (5.70) and (5.52) are equivalent to the inequalities (5.67) and (5.68). ∎ Remark 5.4 During the design process of the sliding mode control law (5.19), there are many parameters related to the finite-time stability. First, the initial condition x(0), ˆ the finite-time interval T , and the appropriate matrix Dκ are chosen based on the inequality (5.20). Second, the positive constants c1 , c2 , d, γ , and positive-definite matrix R are known a priori and Kκ is selected such that Aκ + Bκ Kκ is Hurwitz before designing the observer gain Lκ . Finally, we can obtain the observer gain Lκ by finding feasible solution of positive-definite symmetric matrix Pκ and real matrix Hκ . Although there are many constraints in the design of finite-time sliding mode control law, we still obtain a feasible solution. The validity of the main research results is verified by a single link manipulator.

96

5 Finite-Time Sliding Mode Control Under Quantization

Remark 5.5 As is well known, there exist two classical quantization strategies named as static quantization and dynamic quantization. In this chapter, a static logarithmic quantizer is used, in which the parameters of the quantizer are kept constant during the quantization. Although this quantizer is simple in structure and easy to implement, it always produces large quantization error [14, 17]. Since the static quantizer has some limitations, the dynamic quantizer is introduced to solve the above problems, that is, the structure exhibits complex characteristics, and the quantization parameters will change with the system state. In this way, we can get a better quantitative effect. How to apply the dynamic quantization strategy to study the sliding mode control of S-MSSs is a worthwhile topic.

5.7 Simulation Consider the dynamic equation of the single-link robot arm model from [23] given as w ¨ (t) = −

W 1 M (φt )gL sin(w (t)) − w ˙ (t) + u(t), J (φt ) J (φt ) J (φt )

where w (t) and u(t) stand for the angle position of arm and the control input, respectively. The parameters M (φt ), J (φt ), g, L , and W denote mass of payload, moment of inertia, acceleration of gravity, length of arm, coefficient of viscous friction. If the payload changes when the robot works under different environmental conditions, a random switching rule will appear. The transition follows the semi-Markovian ˙ (t). process {φt , t ≥ 0} in k = {1, 2, . . . , ℘}. Define x1 (t) = w (t) and x2 (t) = w Then, considering parametrical uncertainty and external disturbance, when φt = κ, the linearized dynamic model of the single-link manipulator model can be described as x(t) ˙ = (Aκ + ΔAκ (t))x(t) + Bκ u(t) + Vκ v(t), z(t) = Cκ x(t), [ where Aκ =

0

κ gL − MJ κ

1 W −J κ

]

[ , Bκ =

0 1 Jκ

(5.71)

]

[ ]T [ ] , Vκ = 0.1 0.1 , Cκ = 1 0 , and

ΔAκ (t) means the parametrical uncertainty. The parameters g, L , and W are given by g = 9.81 m/s2 , L = 0.5 m, and W = 2. The parameters Mκ and Jκ are with three different modes given as: when κ = 1, M1 = 1 kg, J1 = 1 N.m; when κ = 2, M2 = 1.5 kg, J2 = 2 N.m; when κ = 3, M3 = 2 kg, J3 = 2.5 N.m. The transition follows the semi-Markovian process {φt , t ≥ 0} in k = {1, 2, 3} with the parameters given as

5.7 Simulation

97

Fig. 5.1 System mode 3

2

1

0

5

10

15

[

] [ ] [ ] 0 1 0 1 0 1 , A2 = , A3 = , −4.905 −2 −3.6788 −1 −3.9240 −0.8 [ ] [ ] [ ] 0 0 0 , B2 = , B3 = . B1 = 1 0.5 0.4 A1 =

[ ]T [ ] Other system parameters are given as Mκ = 0 0.1 , Nκ = 0.1 0.1 , Fκ (t) = sin(t), ηκ = 0.15, c1 = 1, c∗ = 2.5, c2 = 4, T = 15, R = I4 , d = 0.1, γ = 0.2, κ = 1, 2, 3, φ1 = 0.5, ϑ1(0) = 0.6. In this model, we choose the transition rate matrices as ⎡

⎡ ⎤ ⎤ −0.8 0.5 0.3 −1.2 0.6 0.6 δ = ⎣ 0.6 −1.0 0.4 ⎦ , δ¯ = ⎣ 0.7 −1.4 0.7 ⎦ . 0.2 0.3 −0.5 0.4 0.6 −1.0 [ ] gains [ are chosen as ] K1 = 3.5875 1.0625 , K2 = [ The controller ] 4.7250 0.1250 , and K3 = 6.5188 −0.3437 , such that Aκ + Bκ Kκ , κ = 1, 2, 3, is Hurwitz. By solving Theorem 5.4, we can obtain [

] [ ] [ ] 0.4130 0.0907 0.3819 0.0800 0.3761 0.0540 , P2 = , P3 = , P1 = 0.0907 0.2130 0.0800 0.1679 0.0540 0.1435 [ ] [ ] [ ] 0.2690 0.0666 0.0657 , L2 = , L3 = . L1 = −0.9755 −1.1167 −1.0143 It is noted that BκT P1κ Bκ is nonsingular, when κ = 1, 2, 3. For the initial condi[ ]T [ ]T ˆ tions φ0 = 3, x0 = −0.45 0.65 , xˆ0 = −0.55 0.85 , ηκ satisfies ηκ ≥ ||D κTx(0)|| , T where Dκ = Bκ P1κ , κ = 1, 2, 3. Then, the sliding mode control law is designed as (5.19). Figure 5.1 plots the system mode. In order to reduce the flutter effect of sliding mode control, the discontinuous function sgn(s(t)) is replaced by the contin-

98 Fig. 5.2 State estimation

5 Finite-Time Sliding Mode Control Under Quantization 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6

Fig. 5.3 Error state

0

5

10

15

10

15

0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2

0

5

s(t) . Figures 5.2 and 5.3 describe the state estimation x(t) ˆ and uous function ||s(t)||+0.001 the error state e(t) of the closed-loop system. Figure 5.4 shows the weighted system state ξ T (t)Rξ T (t) of the closed-loop system. It shows that the weighted system state ξ T (t)Rξ T (t) satisfies the requirement of finite-time boundedness. Figure 5.5 plots the system output z(t) and the quantized output Θ(z(t)). All those figures show the superiority of the proposed sliding mode stabilization law.

5.8 Conclusion Fig. 5.4 Evolution

99 1.2

1

0.8

0.6

0.4

0.2

0

Fig. 5.5 Output and quantized output

0

5

10

15

5

10

15

0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5

0

5.8 Conclusion The finite-time boundedness for uncertain S-MSSs in the presence of quantized output has been discussed via sliding mode control approach, in which the state variable is maintained in a certain range of given physical threshold. An observerbased sliding mode control law is designed to ensure that the system trajectory can reach the specified sliding mode surface s(t) = 0 at a time T ∗ when T ∗ ≤ T . Then, the transient performance of the closed-loop system at the finite-time level is analyzed by the finite-time bounded method. The feasibility of the sliding mode control law is verified by a single link robot arm model. Dynamic quantization strategy will show better performance in future research.

100

5 Finite-Time Sliding Mode Control Under Quantization

References 1. Dorato, P.: Short time stability in linear time-varying systems. In: Proceedings of the IRE International Convention Record, pp. 83–87 (1961) 2. Ren, H.L., Zong, G.D., Li, T.S.: Event-triggered finite-time control for networked switched linear systems with asynchronous switching. IEEE Trans. Syst. Man Cybernet.: Syst. 48(11), 1874–1884 (2018) 3. Amato, F., Ariola, M., Dorato, P.: Finite-time control of linear systems subject to parametric uncertainties and disturbances. Automatica 37(9), 1459–1463 (2001) 4. Zong, G.D., Ren, H.L.: Guaranteed cost finite-time control for semi-Markov jump systems with event-triggered scheme and quantization input. Int. J. Robust Nonlinear Control 29(15), 5251–5273 (2019) 5. Ren, H.L., Zong, G.D., Karimi, H.R.: Asynchronous finite-time filtering of networked switched systems and its application: An event-driven method. IEEE Trans. Circuits Syst. I Regul. Pap. 66(1), 391–402 (2019) 6. Song, J., Niu, Y.G., Zou, Y.Y.: Finite-time stabilization via sliding mode control. IEEE Trans. Autom. Control 62(3), 1478–1483 (2017) 7. Song, J., Niu, Y.G., Zou, Y.Y.: A parameter-dependent sliding mode approach for finite-time bounded control of uncertain stochastic systems with randomly varying actuator faults and its application to a parallel active suspension system. IEEE Trans. Ind. Electron. 65(10), 8124– 8132 (2018) 8. Song, J., Niu, Y.G., Zou, Y.Y.: Finite-time sliding mode control synthesis under explicit output constraint. Automatica 650, 111–114 (2016) 9. Elia, N., Mitter, S.K.: Stabilization of linear systems with limited information. IEEE Trans. Autom. Control 46(9), 1384–1400 (2001) 10. Fu, M.Y., Xie, L.H.: The sector bound approach to quantized feedback control. IEEE Trans. Autom. Control 50(11), 1698–1711 (2005) 11. Chang, X.H., Wang, Y.M.: Peak-to-peak filtering for networked nonlinear DC motor systems with quantization. IEEE Trans. Ind. Inf. 14(12), 5378–5388 (2018) 12. Zhu, H., Fujimoto, H.: Suppression of current quantization effects for precise current control of SPMSM using dithering techniques and Kalman filter. IEEE Trans. Ind. Inf. 10(2), 1361–1371 (2014) 13. Chang, X.H., Xiong, J., Li, Z.M., Park, J.H.: Quantized static output feedback control for discrete-time systems. IEEE Trans. Ind. Inf. 14(8), 3426–3435 (2018) 14. Chang, X.H., Huang, R., Wang, H.Q., Liu, L.: Robust design strategy of quantized feedback control. IEEE Trans. Circuits Syst. II Express Briefs 67(4), 730–734 (2020) 15. Shi, P., Liu, M., Zhang, L.X.: Fault-tolerant sliding-mode-observer synthesis of Markovian jump systems using quantized measurements. IEEE Trans. Ind. Electron. 62(9), 5910–5918 (2015) 16. Li, F.B., Shi, P., Wu, L.G., Basin, M.V., Lim, C.C.: Quantized control design for cognitive radio networks modeled as nonlinear semi-Markovian jump systems. IEEE Trans. Ind. Electron. 62(4), 2330–2340 (2015) 17. Chang, X.H., Huang, R., Park, J.H.: Robust guaranteed cost control under digital communication channels. IEEE Trans. Ind. Inf. 16(1), 319–327 (2020) 18. Xiao, N., Xie, L.H., Fu, M.Y.: Stabilization of Markov jump linear systems using quantized state feedback. Automatica 46(10), 1696–1702 (2010) 19. Liu, M., Ho, D.W.C., Niu, Y.G.: Robust filtering design for stochastic system with modedependent output quantization. IEEE Trans. Signal Process. 58(12), 6410–6416 (2010) 20. Tao, J., Lu, R.Q., Su, H.Y., Shi, P., Wu, Z.G.: Asynchronous filtering of nonlinear Markov jump systems with randomly occcurred quantization via T-S fuzzy models. IEEE Trans. Fuzzy Syst. 26(4), 1866–1877 (2018) 21. Lu, R.Q., Tao, J., Shi, P., Su, H.Y., Wu, Z.G., Xu, Y.: Dissipativity-based resilient filtering of periodic Markovian jump neural networks with quantized measurements. IEEE Trans. Neural Netw. Learn. Syst. 29(5), 1888–1899 (2018)

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22. Huang, J., Shi, Y.: Stochastic stability and robust stabilization of semi-Markov jump linear systems. Int. J. Robust Nonlinear Control 23(18), 2028–2043 (2013) 23. Wu, H.N., Cai, K.Y.: Mode-independent robust stabilization for uncertain Markovian jump nonlinear systems via fuzzy control. IEEE Trans. Syst. Man Cybernet. Part B (Cybernetics) 36(3), 509–519 (2006)

Chapter 6

Adaptive Event-Triggered Sliding Mode Control

In this chapter, the problems of adaptive event-triggered sliding mode control are investigated for semi-Markovian switching systems (S-MSSs). Network-induced communication constraint and uncertain parameter are considered in the designing process of sliding mode controller. Owing to the limitation of the measurement transducers, the states of S-MSSs are not always measurable. Different from the traditional static event-triggered scheme, the adaptive event-triggered scheme is adopted to be efficient in reducing the number of triggering. Due to the network communication, the network-induced delays between the event trigger and the zero-order holder (ZOH) are unavoidable. This chapter aims to design an appropriate sliding mode control law under an adaptive event-triggered mechanism. Based on the stochastic semi-Markovian Lyapunov functional, sojourn-time-dependent sufficient conditions are proposed to realize stochastic stability for S-MSSs. An appropriate sliding mode control law is constructed such that the reachability of the specified sliding surface is guaranteed in a finite-time region. Finally, a boost converter circuit model is presented to demonstrate the theoretical findings.

6.1 Introduction In the previous chapter, the finite-time sliding mode control for S-MSSs with quantized measurement has been addressed. In recent years, the rapid development of computer, modern communication and signal processing technology has brought about great changes in the composition and structure of control systems. The relationship between actuator, sensor, and controller is connected by communication network, which has become one of the most important technologies in control field [1–3]. However, due to the limited network bandwidth, some adverse networkinduced phenomena will occur inevitably in data transfer, such as network delay, network congestion, packet loss, etc., which bring many challenges to system anal© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 W. Qi and G. Zong, Control Synthesis for Semi-Markovian Switching Systems, Studies in Systems, Decision and Control 465, https://doi.org/10.1007/978-981-99-0317-7_6

103

104

6 Adaptive Event-Triggered Sliding Mode Control

ysis and synthesis. Under the limited communication bandwidth, it is necessary and urgent to design an efficient and reasonable control strategy to relieve network congestion and avoid packet loss. The traditional time-triggered mechanism is to update the control periodically. For an ideal working condition, a large amount of redundant data transferred will lead to waste of communication resources and unnecessary energy consumption. Since the late 1990s, an event-triggered mechanism [4, 5] based on the idea of on-demand transmission has been proposed, the core of which is to allow only data transmission that meets the triggered conditions. Up to now, eventtriggered mechanism has received a large number of researchers’ attention (see, e.g., [6–14]). In this chapter, the issue of adaptive event-triggered sliding mode control design is investigated for nonlinear continuous-time S-MSSs. The key contributions are summarized as: (i) Two unrealistic assumptions about the sojourn time in stochastic switching systems following the exponential distribution and the system states being measurable are removed under the framework of complex working environment and sensor information constraints. An adaptive event-triggered mechanism is adopted to save network resources. (ii) To stabilize the addressed S-MSSs, sojourn-timedependent sufficient conditions are developed for the corresponding closed-loop system. (iii) For the induced-delay network environment, an adaptive event-triggered sliding mode control law is constructed, which depends on the transition rate matrix.

6.2 Problem Statements and Preliminaries Consider a class of nonlinear S-MSSs z˙ (t) = (A (ηt ) + ΔA (ηt , t))z(t) + B(ηt )u(t), y(t) = C (ηt )z(t),

(6.1)

where z(t), u(t), and y(t) are the state vector, input vector, and output vector. ΔA (ηt , t) = M (ηt )H (ηt , t)N (ηt ) with H (ηt , t) satisfying H T (ηt , t) H (ηt , t) ≤ I . {ηt , t ≥ 0} signifies the semi-Markovian process and takes value in ℵ = {1, 2, . . . , ℘} with the probability transition given as { Pr {ηt+Δ¯ = μ|ηt = ν} =

¯ + o(Δ), ¯ ρνμ (ι)Δ ν /= μ, ¯ ¯ ν = μ, 1 + ρνν (ι)Δ + o(Δ),

(6.2)

where ι is the sojourn time, ρνμ (ι) ≥ 0 is the transition rate from ν to μ for ν /= μ. Σ ℘ ρνμ (ι) = −ρνν (ι), and ρ νμ ≤ ρνμ (h) ≤ ρ¯ νμ with real constant scalars μ=1,μ/=ν ρ νμ and ρ¯ νμ . For notational brevity, when ηt = ν ∈ ℵ, A (ηt ), ΔA (ηt ), B(ηt ), C (ηt ), M (ηt ), H (ηt , t), and N (ηt ) are denoted as Aν , ΔAν (t), Bν , Cν , Mν , Hν (t), and Nν , respectively.

6.2 Problem Statements and Preliminaries

105

Fig. 6.1 Adaptive event-triggered control systems over network

Figure 6.1 plots the framework of adaptive event-triggered control systems, in which SMC means sliding mode control. Then, the adaptive event-triggered mechanism is designed as lr +1 h = lr h + min j∈N { j h|e T ((lr + j )h)Πν e((lr + j)h) > υ(lr h)y T (lr h)Πν y(lr h)},

(6.3)

where h is the sampled period, y(lh) (l ∈ N) is the current sampled signal, y(lr h) (lr ∈ N, r = 0, 1, 2, . . . , ∞, l0 = 0) is the latest transmitted signal, r is the triggered number, and e((lr + j)h) = y((lr + j)h) − y(lr h). Πν > 0, ν ∈ ℵ, is the weighting matrix. {(lr + j)h}, j ∈ N, denotes the set of sampling instants over [lr h, lr +1 h). υ(lr h) is the adaptive triggered parameter. Remark 6.1 Based on a fixed sampling, an adaptive event-triggered scheme (6.3) is proposed. When the sampled signal satisfies the event-triggered mechanism (6.3), it will be transmitted. Due to the fixed sampling, the minimum event-trigger interval is subject to fixed period h such that it will not trigger countless times during a finite-time interval, resulting in Zeno-free phenomenon. The network-induced transmission delay is always unavoidable denoted as h lr . The ZOH receives the signals at instant lr h + h lr . According to [9], it is assumed that h lr satisfies h m ≤ h lr ≤ h M ≤ h with h m and h M being the minimum and maximum values. The ZOH keeps the previous signal y(lr h) until a new one is updated, that is, y¯ (t) = y(lr h), t ∈ [lr h + h lr , lr +1 h + h lr +1 ), in which [lr h + h lr , lr+1 h + h lr +1 ) is divided into

(6.4)

106

6 Adaptive Event-Triggered Sliding Mode Control

[lr h + h lr , lr+1 h + h lr +1 ) =

kr .

wr, j ,

(6.5)

j=0

where wr, j = [qr, j h + h qr, j , qr, j+1 h + h qr, j+1 ), qr, j = lr + j are the sampling instants during [lr h, lr +1 h), and kr = lr+1 − lr − 1. The delay h qr, j satisfies h m ≤ h qr, j ≤ h M ≤ h and qr, j h + h qr, j < qr, j+1 h + h qr, j+1 . Define the piecewise function w (t) = t − qr, j h, t ∈ wr, j , where w1 ≤ w (t) ≤ w2 , w1 = h m , and w2 = h M + h. Then, we have y¯ (t) = y(lr h) = y(qr, j h) − e(qr, j h) = y(t − w (t)) − e(qr, j h), t ∈ wr, j .

(6.6)

The adaptive threshold υ(t) is updated by 1 υ(t) ˙ = υ(t)

(

) 1 − υ0 e T (qr, j h)Πν e(qr, j h), υ(t)

(6.7)

where υ0 > 0. Remark 6.2 Compared with the traditional static event-triggered mechanism, adaptive event-triggered mechanism has two major advantages: (i) The threshold of adaptive event-triggered mechanism is adjusted on-line through an adaptive law; (ii) The redundant data transmission can be avoided effectively and the triggered number is greatly reduced. Therefore, the adaptive event-triggered mechanism is considered to investigate sliding mode control problem for S-MSSs. Under the framework of adaptive event-triggered mechanism, the observer is designed as z˙ˇ (t) = Aν zˇ (t) + Bν u(t) + Lν ( y¯ (t) − Cν zˇ (t − w (t))), t ∈ wr, j ,

(6.8)

where zˇ (t) ∈ Rn and Lν are the estimated state and the observer gain, respectively. Thus, it is got that z˙ˇ (t) = Aν zˇ (t) + Bν u(t) + Lν (y(t − w (t)) − Cν zˇ (t − w (t)) − e(qr, j h)), t ∈ wr, j .

(6.9)

Define the error e(t) = z(t) − zˇ (t). From (6.1) and (6.9), we have e(t) ˙ = (Aν + ΔAν (t))e(t) + ΔAν (t)ˇz (t) − Lν Cν e(t − w (t)) + Lν e(qr, j h).

(6.10)

6.2 Problem Statements and Preliminaries

107

Therefore, it follows from (6.9) and (6.10) that ξ˙ (t) = (A¯ν + ΔA¯ν (t))ξ(t) + A¯hν ξ(t − w (t)) + B¯ν u(t) + L¯ν e(qr, j h), (6.11) ξ(t0 + θ ) = ϕ(θ ), θ ∈ [−w2 , 0], where [ ] [ ]T ξ(t) = zˇ T (t) e T (t) , A¯ν = Aν 0 , ] [ [ ] 0 Aν 0 0 0 Lν Cν = M¯ν Hν (t)N¯ν , A¯hν = , ΔA¯ν (t) = 0 −Lν Cν ΔAν (t) ΔAν (t) ]T ]T ] [ [ [ [ ]T B¯ν = BνT 0 , L¯ν = −LνT LνT , M¯ν = 0 MνT , N¯ν = 0 Nν . Lemma 6.1 ([15]) For given matrix X > 0 and all continuous function ϑ(·) in [α, β] → R2n , the following inequality holds { − (β − α)

β

α

w ˙ T (ι)X w ˙ (ι)dι

≤ − (w (β) − w (α))T X (w (β) − w (α)) − 3Π T X Π, where Π = w (β) + w (α) −

2 β−α

{β α

(6.12)

w (ι)dι.

Lemma 6.2 ([16]) For continuous function 0 < w1 ≤ w (t) ≤ w2 , positive-definite matrix P] ∈ R2n×2n , a vector z˙ : [−w2 , 0] → R2n , and matrix Z ∈ R4n×4n satisfying [ P˜ Z ≥ 0, there holds Z T P˜ { − (w2 − w1 )

t−w1 t−w2

˜ T − ψ T Pψ ˜ T, x˙ T (ι)P x(ι)dι ˙ ≤ 2ψ1T Z ψ2T − ψ1T Pψ 1 2 2 (6.13)

where ] x(t − w (t)) − x(t − w2 ) , x(t − w (t)) + x(t − w2 ) − 2ρ2 (t) ] [ x(t − w1 ) − x(t − w (t)) , x(t − w1 ) + x(t − w (t)) − 2ρ1 (t) { t−w1 1 diag{P, 3P}, ρ1 (t) = x(ι)dι, w (t) − w1 t−w (t) { t−w (t) 1 x(ι)dι. w2 − w (t) t−w2 [

ψ1 = ψ2 = P˜ = ρ2 (t) =

108

6 Adaptive Event-Triggered Sliding Mode Control

Fig. 6.2 Boost converter circuit

The practical boost converter circuit model can be modeled as the abovementioned theoretical dynamic model. The circuit model from [17] is presented in Fig. 6.2. From Fig. 6.2, when the transfer switch s is a disconnected state, we have L

d Vc Vc di L = u(t) − Vc , C = iL − , dt dt R

(6.14)

where i L is the inductance current, Vc is the capacitance voltage, and u(t) is the control input. When the transfer switch s is a connected state, we have L

d Vc di L = u(t), RC = −Vc . dt dt

(6.15)

Considering abrupt changes in system structures and parameters, S-MSSs are adopted to describe the boost converter circuit model. Different transfer switches obey the semi-Markovian process {ηt , t ≥ 0} in ℵ = {1, 2}. Defining the state vari]T ]T [ [ ables z(t) = z 1 (t) z 2 (t) = i L Vc , one has the state-space model represented by z˙ (t) = (Aν + ΔAν (t))z(t) + Bν u(t), y(t) = Cν z(t),

(6.16)

where [ A1 =

0 − L1 1 C

1 − RC

]

[

[ ] ] 1 0 0 [ ] , A2 = 0 − 1 , Bν = L , Cν = 0 1 , ν = 1, 2, RC 0

and ΔAν (t), ν = 1, 2, denote the parametric uncertainty.

6.3 Stochastic Stability Analysis

109

6.3 Stochastic Stability Analysis The sliding surface is constructed as {

t

s(t) = Dν zˇ (t) −

Dν [(Aν + Bν Fν )ˇz (s) − Lν Cν zˇ (s − w (s))]ds,

(6.17)

0

where Dν ∈ Rm×n is chosen to ensure the nonsingularity of Dν Bν and Fν ∈ Rm×n is the parameter to be designed later. When the state signals arrive the sliding surface s(t) = 0, it follows from s˙ (t) = 0 that u eq (t) = Fν zˇ (t) − (Dν Bν )−1 Dν Lν (y(t − w (t)) − e(qr, j h)).

(6.18)

Substituting (6.18) into (6.11) gives rise to ξ˙ (t) = (A¯¯ν + ΔA¯¯ν (t))ξ(t) + A¯¯hν ξ(t − w (t)) + L¯¯ν e(qr, j h),

(6.19)

where [ ] Aν + Bν Fν 0 ]T ¯ ¯ ξ(t) = zˇ (t) e (t) , A ν = , 0 Aν [ [ ] ] 0 0 −Lν Cν 0 ¯ ¯ ¯ ¯ , = M¯ν Hν (t)N¯ν , A¯hν = ΔA¯ν (t) = ΔAν (t) ΔAν (t) 0 −Lν Cν [

T

T

]T ]T ] [ [ [ L¯¯ν = 0 LνT , M¯¯ν = 0 MνT , N¯¯ν = 0 Nν . Theorem 6.1 If we find symmetric matrices Πν > 0, Pν > 0, Q1ν > 0, Q2ν > 0, R1 > 0, R2 > 0, Z1 > 0, Z2 > 0, matrices Xν , Y , and scalars ε1ν > 0, ε2ν > 0, ∀ν ∈ ℵ, such that ] Z˜2 Y ≥ 0, Y T Z˜2

[

(6.20)

w1 < 0, Σ ℘ Σ ℘ ρνμ (ι)Q1μ ≤ R1 , μ=1

where

μ=1

(6.21) ρνμ (ι)Q2μ ≤ R2 ,

(6.22)

110

6 Adaptive Event-Triggered Sliding Mode Control

Σ s T w1 = a1T (Pν A ¯¯ν + A ¯¯ν Pν +

μ=1

ρνμ (ι)Pμ + Q1ν + Q2ν + w1 R1 + w2 R2

T T T −1 ¯¯ −1 ¯¯ + ε1ν Pν M¯¯ν M¯¯ν Pν + ε1ν N ν N ¯¯ν + ε2ν N ν N ¯¯ν )a1 + a1T Pν A ¯¯hν a2 T T + a2T A ¯¯hν Pν a1 + a1T Xν A ¯¯ν a8 + a8T A ¯¯ν Xν a1 + a2T [I I ]T CνT Πν Cν [I I ]a2

+ a2T Xν A ¯¯hν a8 + a8T A ¯¯hν Xν a2 − a2T [I I ]T CνT Πν a9 − a9T Πν Cν [I I ]a2 T

T − a3T Q1ν a3 − a4T Q2ν a4 + a8T (w12 Z1 + (w2 − w1 )2 Z2 − 2Xν + ε2ν Xν M¯¯ν M¯¯ν Xν )a8 T + a8T Xν L¯¯ν a9 + a9T L¯¯ν Xν a8 + a9T (Πν − κ0 Πν )a9 − b1T Z˜1 b1 + 2b2T Y b3

[ ]T − b2T Z˜2 b2 − b3T Z˜2 b3 , b1 = a1T − a3T a1T + a3T − a7T , Z˜1 = diag{Z1 , 3Z1 }, [ ]T b2 = a2T − a4T a2T + a4T − a5T , Z˜2 = diag{Z2 , 3Z2 }, ] [ [ ]T Y1 Y2 b3 = a3T − a2T a3T + a2T − a6T , Y = , Y3 Y4

where as1 ∈ R2n×(16n+l) (s1 = 1, 2, . . . , 8), and a9 ∈ Rl×(16n+l) stands for the block entry matrix, e.g., a2 ζ (t) = ξ(t − w (t)), then system (6.19) is stochastically stable. Proof Consider Lyapunov functional as W (ξ(t), ηt , t) = W1 (ξ(t), ηt , t) + W2 (ξ(t), ηt , t) + W3 (ξ(t), t) + W4 (ξ(t), t) + W5 (t), (6.23) where W1 (ξ(t), ηt , t) = ξ T (t)P(ηt )ξ(t), { t { T ξ (s)Q1 (ηt )ξ(s)ds + W2 (ξ(t), ηt , t) = { W3 (ξ(t), t) =

t−w1

{

0

−w1

{ + (w2 − w1 ) W5 (t) =

ξ T (s)R1 ξ(s)dsdθ +

t+θ

{

W4 (ξ(t), t) = w1

{

t

{

0

−w1

−w1 −w2

{

t

ξ˙ T (s)Z1 ξ˙ (s)dsdθ

t+θ t

ξ˙ T (s)Z2 ξ˙ (s)dsdθ,

t+θ

1 2 υ (t). 2

At time t, ηt = ν, for ν ∈ ℵ, one has

t

ξ T (s)Q2 (ηt )ξ(s)ds,

t−w2 0

−w2

{

t t+θ

ξ T (s)R2 ξ(s)dsdθ,

6.3 Stochastic Stability Analysis

111

Γ W1 (ξ(t), ηt , t) 1 = lim [E {W1 (ξ(t + Δ), ηt+Δ , t + Δ)|ηt = ν} − W1 (ξ(t), ηt , t)] Δ→0 Δ [ 1 Σ s Pr {ηt+Δ = μ|ηt = ν}ξ T (t + Δ)Pμ ξ(t + Δ) = lim μ=1,μ/=ν Δ→0 Δ + Pr {ηt+Δ = ν|ηt = ν}ξ T (t + Δ)Pν ξ(t + Δ) − ξ T (t)Pν ξ(t)] [ 1 Σ s Pr {ηt+Δ = μ, ηt = ν} T ξ (t + Δ)Pμ ξ(t + Δ) = lim μ=1,μ/ = ν Δ→0 Δ Pr {ηt = ν} ] Pr {ηt+Δ = ν, ηt = ν} T T ξ (t + Δ)Pν ξ(t + Δ) − ξ (t)Pν ξ(t) + Pr {ηt = ν} [ Σ s 1 qνμ (Vν (ι + Δ) − Vν (ι)) T = lim ξ (t + Δ)Pμ ξ(t + Δ) μ=1,μ/ = ν Δ→0 Δ 1 − Vν (ι) ] 1 − Vν (ι + Δ) T T (6.24) ξ (t + Δ)Pν ξ(t + Δ) − ξ (t)Pν ξ(t) . + 1 − Vν (ι) Based on Taylor formula, it is got that ξ(t + Δ) = ξ(t) + ξ˙ (t)Δ + o(Δ) = ξ(t) + [(A¯¯ν + ΔA¯¯ν (t))ξ(t) + A¯¯hν ξ(t − w (t)) + L¯¯ e(q h)]Δ + o(Δ), ν

r, j

when Δ → 0. Based on lim

= 0, lim

and ρνμ (ι)

substituting (6.25) into (6.24) gives rise to

V ν (ι+Δ)−V ν (ι) 1−V ν (ι) Δ→0 = qνμ ρν (ι), ν /= μ,

Δ→0

1−V ν (ι+Δ) 1−V ν (ι)

= 1, lim

Δ→0

V ν (ι+Δ)−V ν (ι) Δ(1−V ν (ι))

(6.25) = ρν (ι),

Γ W1 (ξ(t), ηt , t) = 2ξ T (t)Pν [(A¯¯ν + ΔA¯¯ν (t))ξ(t) + A¯¯hν ξ(t − w (t)) + L¯¯ν e(qr, j h)] Σ s + ξ T (t) ρνμ (ι)Pμ ξ(t). (6.26) μ=1

Furthermore, one has Γ W2 (ξ(t), ηt , t) ≤ ξ T (t)(Q1ν + Q2ν )ξ(t) − ξ T (t − w1 )Q1ν ξ(t − w1 ) − ξ T (t − w2 )Q2ν ξ(t − w2 ) { t Σ s + ρνμ (ι) ξ T (s)Q1μ ξ(s)ds μ=1

+

Σ s μ=1

{ ρνμ (ι)

Γ W3 (ξ(t), t)

t−w1 t t−w2

ξ T (s)Q2μ ξ(s)ds,

(6.27)

112

6 Adaptive Event-Triggered Sliding Mode Control

{ = ξ T (t)(w1 R1 + w2 R2 )ξ(t) − { −

t

ξ T (s)R1 ξ(s)ds

t−w1 t

ξ T (s)R2 ξ(s)ds,

(6.28)

t−w2

Γ W4 (ξ(t), t) = ξ˙ T (t)(w12 Z1 + (w2 − w1 )2 Z2 )ξ˙ (t) − w1 {

t−w1

− (w2 − w1 )

{

t

ξ˙ T (s)Z1 ξ˙ (s)ds

t−w1

ξ˙ T (s)Z2 ξ˙ (s)ds.

(6.29)

t−w2

Combining the event-triggered condition (6.3) and the adaptive law (6.7), we have ( Γ W5 (t) =

) 1 − υ0 e T (qr, j h)Πν e(qr, j h) υ(t)

≤ y T (lr h)Πν y(lr h) − υ0 e T (qr, j h)Πν e(qr, j h) = [y(t − w (t)) − e(qr, j h)]T Πν [y(t − w (t)) − e(qr, j h)] − υ0 e T (qr, j h)Πν e(qr, j h).

(6.30)

For 2ξ T (t)Pν ΔA¯¯ν (t)ξ(t), one has 2ξ T (t)Pν ΔA¯¯ν (t)ξ(t) T T −1 T ξ (t)N¯¯ν N¯¯ν ξ(t), ≤ ε1ν ξ T (t)Pν M¯¯ν M¯¯ν Pν ξ(t) + ε1ν

(6.31)

where ε1ν > 0. Applying Lemma 6.1 yields −w1

{ t t−w1

] ]T [ ξ(t) − ξ(t − w1 ) Z1 0 {t ξ(s)ds ξ(t) + ξ(t − w1 ) − w21 t−w 0 3Z1 1 [ ] ξ(t) − ξ(t − w1 ) (6.32) . {t ξ(s)ds ξ(t) + ξ(t − w1 ) − w21 t−w 1 [

ξ˙ T (s)Z1 ξ˙ (s)ds ≤ −

It follows from Lemma 6.2 and (6.20) that { − (w2 − w1 )

t−w1 t−w2

where

ξ˙ T (s)Z2 ξ˙ (s)ds ≤ 2κ1T Y κ2 − κ1T Z˜2 κ1 − κ2T Z˜2 κ2 , (6.33)

6.3 Stochastic Stability Analysis

113

] ξ(t − w (t)) − ξ(t − w2 ) { t−w (t) , κ1 = 2 ξ(s)ds ξ(t − w (t)) + ξ(t − w2 ) − w2 −w (t) t−w2 [ ] ξ(t − w1 ) − ξ(t − w (t)) { κ2 = . t−w 1 2 ξ(t − w1 ) + ξ(t − w (t)) − w (t)−w t−w (t) ξ(s)ds 1 [

For any appropriate matrices Xν , there holds 2ξ˙ T (t)Xν [−ξ˙ (t) + (A¯¯ν + ΔA¯¯ν (t))ξ(t) + A¯¯hν ξ(t − w (t)) + L¯¯ e(q h)] = 0. ν

r, j

(6.34)

For the uncertainty in (6.34), one has 2ξ˙ T (t)Xν ΔA¯¯ν (t)ξ(t) T T −1 T ≤ ε2ν ξ˙ T (t)Xν M¯¯ν M¯¯ν Xν ξ˙ (t) + ε2ν ξ (t)N¯¯ν N¯¯ν ξ(t),

(6.35)

where ε2ν > 0. Define ζ (t) = [ξ T (t) ξ T (t − w (t)) ξ T (t − w1 ) ξ T (t − w2 )

{ t−w (t) 2 ξ T (s)ds w2 − w (t) t−w2

{ t−w1 { t 2 2 ξ T (s)ds ξ T (s)ds ξ˙ T (t) e T (qr, j h)]T . w (t) − w1 t−w (t) w1 t−w1

(6.36)

According to (6.26)–(6.36), we have Γ W (ξ(t), ηt , t) ≤ ζ T (t)w1 ζ (t),

(6.37)

where as1 ∈ R2n×(16n+l) (s1 = 1, 2, . . . , 8) and a9 ∈ Rl×(16n+l) are given in Theorem 6.1. From (6.21), we have Γ W (ξ(t), ηt , t) < 0. Therefore, system (6.19) is stochastically stable.

(6.38) ∎

Sufficient conditions are developed for stochastic stability in Theorem 6.1. The nonlinear elements in (6.21) and (6.22) make Theorem 6.1 not solvable in linear matrix framework. Next, strict linear matrix framework is provided to find solvability conditions for designing controller gain and observer gain.

114

6 Adaptive Event-Triggered Sliding Mode Control

Theorem 6.2 Consider Dν in (6.17) given as Dν = BνT P1ν . If we find symmetric matrices Πν > 0, Pν > 0, Q1ν > 0, Q2ν > 0, R1 > 0, R2 > 0, Z1 > 0, Z2 > 0, matrices Y , L˜ , and scalars ε1ν > 0, ε2ν > 0, εν > 0, ∀ν ∈ ℵ, such that (6.20) and ⎤ w2 a1T ε1ν Pν M¯¯ν a8T ε2ν εν Pν M¯¯ν ⎦ < 0, ⎣ ∗ −ε1ν I 0 ∗ ∗ −ε2ν I ⎤ ⎡ ¯ T T w3 a1 ε1ν Pν M¯ν a8 ε2ν εν Pν M¯¯ν ⎦ < 0, ⎣ ∗ −ε1ν I 0 ∗ ∗ −ε2ν I Σ ℘ Σ ℘ ρ¯ νμ Q1μ ≤ R1 , ρ¯ νμ Q2μ ≤ R2 , μ=1 μ=1 Σ ℘ Σ ℘ ρ νμ Q1μ ≤ R1 , ρ νμ Q2μ ≤ R2 , ⎡

μ=1

(6.39)

(6.40) (6.41) (6.42)

μ=1

where w2 = a1T

Σ s

ρ¯ Pμ a1 + w4 , w3 = a1T μ=1 νμ

Σ s

ρ

μ=1 νμ

Pμ a1 + w4 ,

−1 ¯¯ w4 = a1T (Pν A ¯¯ν + A ¯¯ν Pν + Q1ν + Q2ν + w1 R1 + w2 R2 + ε1ν N ν N ¯¯ν T

T

−1 ¯¯ + ε2ν N ν N ¯¯ν )a1 + a1T diag{−L˜ν Cν , −L˜ν Cν }a2 + a2T diag{−L˜ν Cν , −L˜ν Cν }T T

T a1 + εν a1T Pν A ¯¯ν a8 + εν a8T A ¯¯ν Pν a1 + a2T [I I ]T CνT Πν Cν [I I ]a2

+ εν a2T diag{−L˜ν Cν , −L˜ν Cν }a8 + εν a8T diag{−L˜ν Cν , −L˜ν Cν }T a2 − a2T [I I ]T CνT Πν a9 − a9T Πν Cν [I I ]a2 − a3T Q1ν a3 − a4T Q2ν a4 [ ]T [ ] + a8T (w12 Z1 + (w2 − w1 )2 Z2 − 2εν Pν )a8 + εν a8T 0 L˜νT a9 + εν a9T 0 L˜νT a8 + a9T (Πν − κ0 Πν )a9 − b1T Z˜1 b1 + 2b2T Y b3 − b2T Z˜2 b2 − b3T Z˜2 b3 , Pν = diag{P1ν , P1ν },

then system (6.19) is stochastically stable. The observer gain is designed as Lν = −1 ˜ Lν and Fν is chosen to make sure that Aν + Bν Fν is Hurwitz. P1ν Proof Define Pν = diag{P1ν , P1ν }, L˜ν = P1ν Lν , Xν = εν Pν .

(6.43)

Then, transition rate ρνμ (ι) is denoted by ρνμ (ι) = s1 ρ νμ + s2 ρ¯ νμ , where s1 + s2 = 1 and s1 ≥ 0, s2 ≥ 0. Applying Schur complement lemma and multiplying (6.39) by s1 and (6.40) by s2 yield

6.4 Reachability Analysis

115

⎤ w5 a1T ε1ν Pν M¯¯ν a8T ε2ν εν Pν M¯¯ν ⎦ < 0, ⎣ ∗ −ε1ν I 0 ∗ ∗ −ε2ν I ⎡

(6.44)

Σ ℘ where w5 = a1T (s1 ρ νμ + s2 ρ¯ νμ )Pμ a1 + w4 and w4 is described in μ=1 Theorem 6.2. By tuning s1 and s2 , all possible ρνμ (ι) ∈ [ρ νμ , ρ¯ νμ ] can be obtained. Then, (6.21) holds. Similar to (6.41) and (6.42), the inequality (6.22) also holds. ∎ Remark 6.3 For the sojourn time subject to probability distribution function, the output signal is only affected by a random semi-Markovian signal. The probability distribution function only satisfies non-exponential distribution. For the solution of the observer gain in Theorem 6.2, the transition probability ρνμ (ι) is time-varying. The upper and lower bounds are considered to deal with the time-varying transition probability which is more general [6–8]. Based on this, a standard linear matrix inequality framework is developed in Theorem 6.2.

6.4 Reachability Analysis An appropriate sliding mode control law (6.45) is designed such that the system trajectories can be driven onto the sliding switching surface s(t) = 0 in a finite time. Theorem 6.3 For system (6.11), the finite-time attractiveness of the sliding surface is achieved by ( 1 Σ ℘ ρ¯ νμ (Dμ Bμ )−1 || u(t) = Fν zˇ (t) − ||(Dν Bν )−1 Dν Lν |||| y¯ (t)|| + || μ=1,μ/=ν 2 ) ||s(t)|| + ω)sgn(s(t) , (6.45) where ω > 0. Proof Consider Lyapunov function: W (s(t), ηt ) = For ηt = ν, one has

1 T s (t)(D (ηt )B(ηt ))−1 s(t). 2

(6.46)

116

6 Adaptive Event-Triggered Sliding Mode Control

Γ W (s(t), ηt )

Σ ℘ 1 = s T (t)(Dν Bν )−1 s˙ (t) + s T (t) ρνμ (ι)(Dμ Bμ )−1 s(t) μ=1 2 Σ ℘ 1 ρνμ (ι)(Dμ Bμ )−1 s(t) = s T (t)[u(t) + (Dν Bν )−1 Dν Lν y¯ (t) + Fν zˇ (t)] + s T (t) μ=1 2 1 Σ ℘ ρνμ (Dμ Bμ )−1 ||||s(t)|| = s T (t)[−(||(Dν Bν )−1 Dν Lν |||| y¯ (t)|| + || μ=1,μ/=ν 2 Σ ℘ 1 ρνμ (ι)(Dμ Bμ )−1 s(t) + ω]sgn(s(t)) + (Dν Bν )−1 Dν Lν y¯ (t) + s T (t) μ=1 2 1 Σ ℘ ρνμ (Dμ Bμ )−1 ||||s(t)|| ≤ s T (t)[−(||(Dν Bν )−1 Dν Lν |||| y¯ (t)|| + || μ=1,μ/=ν 2 + ω)]sgn(s(t)) + ||s(t)||||(Dν Bν )−1 Dν Lν |||| y¯ (t)|| Σ ℘ 1 + ||s(t)|||| ρνμ (Dμ Bμ )−1 ||||s(t)|| μ=1,μ/=ν 2 1 Σ ℘ ρνμ (Dμ Bμ )−1 ||||s(t)|| + ω)] = |s(t)|[−(||(Dν Bν )−1 Dν Lν |||| y¯ (t)|| + || μ=1,μ/=ν 2 Σ ℘ 1 ρνμ (Dμ Bμ )−1 ||||s(t)||. + ||s(t)||||(Dν Bν )−1 Dν Lν |||| y¯ (t)|| + ||s(t)|||| μ=1,μ/=ν 2

For ||s T (t)|| ≤ |s T (t)|, there holds Γ W (s(t), ηt ) ≤ − ω||s(t)||. Therefore, the finite-time reachability is satisfied.

(6.47) ∎

Remark 6.4 The tuning scalar w plays an important part in sliding mode control law (6.45) and directly determines when the sliding switching surface will be reached. For given initial condition, the bigger the value of the tuning scalar w is, the faster the sliding switching surface will be reached. Also, the tuning scalar w is positively associated with the convergence rate of the state variables.

6.5 Simulation The boost converter circuit model is presented in (6.16). The system parameters are chosen as [ ] [ ]T [ ] R = 200 Ω, L = 2 H, C = 200 mF, Cν = 0 1 , Mν = 0 0.3 , Nν = 0.1 −0.1 , Hν (t) = sin(t), ν = 1, 2.

6.5 Simulation Fig. 6.3 System mode

117 3

2.5

2

1.5

1

0.5

0

Fig. 6.4 Release instants with adaptive event-triggered scheme

0

1

2

3

4

5

6

7

8

9

10

1.5

1

0.5

0

0

2

4

6

8

10

The transition rate matrices are set as [ ] [ ] −0.8 0.8 −1.0 1.0 , ρ¯ = . ρ= 0.6 −0.6 1.2 −1.2 [ ] [ ] Choose F1 = −1.9789 −9.1881 and F2 = 1 0 . Then, Aν + Bν Fν , ν = 1, 2, is Hurwitz. Let h m = 0.01, h M = 0.1, h = 0.02, υ0 = 5, ε1ν = 0.1, ε2ν = 0.1, εν = 1.2, ν = 1, 2. Solving the linear matrix inequalities in Theorem 6.2, we have

118

6 Adaptive Event-Triggered Sliding Mode Control

Fig. 6.5 Threshold υ(t)

3 2.5 2 1.5 1 0.5 0

Fig. 6.6 Sliding mode control law u(t)

0

2

4

6

8

10

0

2

4

6

8

10

20 10 0 -10 -20 -30 -40 -50 -60

[

] 1.1553 0.0371 Π1 = 13.4158, Π2 = 37.5691, P11 = , 0.0371 0.9757 [ ] [ ] [ ] 14.0254 −1.2880 −0.8853 0.0013 , L1 = , L2 = . P12 = −1.2880 9.5680 0.9388 0.7201 For w = 0.01, the adaptive sliding mode control law is obtained by (6.45). [ ]T [ ]T For η0 = 2, z 0 = −0.5 0.8 , and zˇ 0 = −0.4 0.7 , we present the simulation results in Figs. 6.3, 6.4, 6.5, 6.6, 6.7, 6.8 and 6.9. Figure 6.3 shows the system mode. Figure 6.4 plots the release instants and intervals with the adaptive eventtriggered scheme that effectively saves communication resources. In Fig. 6.5, the threshold υ(t) converges to a fixed value finally, which means that the traditional

6.5 Simulation Fig. 6.7 Output signal y(t)

119 0.8 0.6 0.4 0.2 0 -0.2 -0.4

Fig. 6.8 System state z(t)

0

2

4

6

8

10

0

2

4

6

8

10

1 0.5 0 -0.5 -1 -1.5

static event-triggered scheme [8] is a special case of the adaptive event-triggered scheme. Figure 6.6 shows the sliding mode control law u(t). Figure 6.7 displays the output signal y(t). Figure 6.8 exhibits the state responses z(t). Figure 6.9 depicts the sliding surface s(t). Thus, the results show the availability of the proposed adaptive event-triggered sliding mode control method.

120 Fig. 6.9 Sliding switching surface s(t)

6 Adaptive Event-Triggered Sliding Mode Control 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6

0

2

4

6

8

10

6.6 Conclusion In this chapter, the sliding mode control problem for S-MSSs has been investigated via the adaptive event-triggered mechanism. In order to save network resources, an adaptive event-triggered mechanism have been constructed. By constructing the stochastic semi-Markovian Lyapunov functional, sojourn-time-dependent sufficient conditions have been developed to realize stochastic stability. The proposed eventtriggered sliding mode control law is related to the controller gain and is more general. Moreover, the adaptive event-triggered mechanism under quantization will be significant in future study.

References 1. Shi, Y., Yu, B.: Output feedback stabilization of networked control systems with random delays modeled by Markov chains. IEEE Trans. Autom. Control 54(7), 1668–1674 (2009) 2. Qiu, J.B., Gao, H.J., Ding, S.X.: Recent advances on fuzzy-model-based nonlinear networked control systems: a survey. IEEE Trans. Ind. Electron. 63(2), 1207–1217 (2016) 3. Wu, Y.Q., Karimi, H.R., Lu, R.Q.: Sampled-data control of network systems in industrial manufacturing. IEEE Trans. Ind. Electron. 65(11), 9016–9024 (2018) 4. Arzen, K.E.: A simple event-based PID controller. IFAC Proc. Vol. 18, 423–428 (1999) 5. Astrom, K.J., Bernhardsson, B.M.: Comparison of periodic and event based sampling for first order stochastic systems. IFAC Proc. Vol. 11, 301–306 (1999) 6. Jiang, B.P., Karimi, H.R., Kao, Y.G., Gao, C.C.: Takagi-sugeno model based event-triggered fuzzy sliding mode control of networked control systems with semi-Markovian switchings. IEEE Trans. Fuzzy Syst. 28(4), 673–683 (2020) 7. Zhang, L.C., Liang, H.J., Sun, Y.H., Ahn, C.K.: Adaptive event-triggered fault detection scheme for semi-Markovian jump systems with output quantization. IEEE Trans. Syst. Man Cybern.: Syst. 51(4), 2370–2381 (2021)

References

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8. Shen, H., Chen, M.S., Wu, Z.G., Cao, J.D., Park, J.H.: Reliable event-triggered asynchronous extended passive control for semi-Markov jump fuzzy systems and its application. IEEE Trans. Fuzzy Syst. 28(8), 1708–1722 (2020) 9. Yue, D., Tian, E.G., Han, Q.L.: A delay system method for designing event-triggered controllers of networked control systems. IEEE Trans. Autom. Control 58(2), 475–481 (2013) 10. Peng, C., Yue, D., Han, Q.L.: Communication and Control for Networked Complex Systems. Springer, Berlin (2015) 11. Zong, G.D., Ren, H.L., Karimi, H.R.: Event-triggered communication and annular finite-time H∞ filtering for networked switched systems. IEEE Trans. Cybern. 51(1), 309–317 (2021) 12. Wu, L.G., Gao, Y.B., Liu, J.X., Li, H.Y.: Event-triggered sliding mode control of stochastic systems via output feedback. Automatica 82, 79–92 (2017) 13. Cheng, J., Park, J.H., Zhang, L.X., Zhu, Y.Z.: An asynchronous operation approach to eventtriggered control for fuzzy Markovian jump systems with general switching policies. IEEE Trans. Fuzzy Syst. 26(1), 6–18 (2018) 14. Song, J., Niu, Y.G., Xu, J.: An event-triggered approach to sliding mode control of Markovian jump Lur’e systems under hidden mode detections. IEEE Trans. Syst. Man Cybern.: Syst. 50(4), 1514–1525 (2020) 15. Seuret, A., Gouaisbaut, F.: Wirtinger-based integral inequality: Application to time-delay systems. Automatica 49(9), 2860–2866 (2013) 16. Zhang, X.M., Han, Q.L.: Global asymptotic stability analysis for delayed neural networks using a matrix-based quadratic convex approach. Neural Netw. 54, 57–69 (2014) 17. Oucheriah, S.: Robust adaptive output feedback controller for a quadratic boost converter. ICIC Express Lett. Part B: Appl. 10(9), 789–795 (2019)

Chapter 7

Finite-Time Synchronization

In this chapter, the finite-time synchronization issue is addressed for quantized semiMarkovian switching neural networks (S-MSNNs) with time delay, in which a logarithmic quantizer is adopted. Semi-Markov process is considered to model practical systems that suffer from random changes in structure and parameters, in which the transition rate is time-varying to depend on the sojourn time. How to design a feedback controller to guarantee the finite-time synchronization between the master system and the slave system is a key issue. This chapter focuses on the finite-time synchronization for the resulting error system over a finite-time interval. By using the weak infinitesimal operator and matrix technique, the solvability conditions for the desired finite-time controller are developed in a linear matrix inequality framework. Finally, the quadruple-tank process model validates the theoretical findings.

7.1 Introduction Owing to the excellent advantages of neural networks in simulating the human nervous systems, neural networks have been favored in pattern recognition [1], signal processing [2], robot manipulator [3], power systems [4], and other aspects [5–11]. However, neural networks face an information latching problem, that is, it is difficult to maintain long-term dependencies for the past input information. In this case, it is a good attempt to model neural networks as a set of finite patterns [12]. The development of Markovian switching systems (MSSs) has prompted the corresponding Markovian switching neural networks (MSNNSs) and become an important model to characterize the system response when the system structure and parameters suffer from change suddenly (see e.g., [13–16]). It is noted that the analysis and synthesis of semi-Markovian switching systems (S-MSSs) and S-MSNNs has gained most of the researchers’ attention [17–28]. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 W. Qi and G. Zong, Control Synthesis for Semi-Markovian Switching Systems, Studies in Systems, Decision and Control 465, https://doi.org/10.1007/978-981-99-0317-7_7

123

124

7 Finite-Time Synchronization

In this chapter, the finite-time synchronization for delayed S-MSNNs with quantized constraint is investigated. The main contributions are concluded as: (i) Sufficient conditions are developed such that the resulting error system is finite-time synchronized, in which stochastic semi-Markovian Lyapunov functional, finite-time stability, and logarithmic quantizer are considered simultaneously. (ii) A feedback controller is designed to realize finite-time synchronization relationship for the master-slave system in a finite-time interval; (iii) Modeling the quadruple-tank process model as delayed S-MSNNs validates the proposed finite-time algorithm.

7.2 Problem Statements and Preliminaries Consider the delayed S-MSNNs as follows: z˙ (t) = − A (δ(t))z(t) + C (δ(t))ψ(z(t)) + Cd (δ(t))ψ(z(t − ν(t))) + V (t), z(t0 + s) = ϕ1 (s), ∀s ∈ [−ν2 , 0],

(7.1)

where z(t) = [z 1 (t), z 2 (t), . . . , z n (t)]T is the state vector with n neurons, ψ(z(t)) = [ψ1 (z(t)), ψ2 (z(t)), . . . , ψn (z(t))]T is the neuron activation function. The activation function ψl (·) is continuous and bounded with Hl − ≤

ψl (ρ1 ) − ψl (ρ2 ) ≤ Hl + , l = 1, 2, . . . , n, ρ1 − ρ2

(7.2)

where ψl (0) = 0, ρ1 , ρ2 ∈ R, ρ1 /= ρ2 , and Hl − and Hl + are known real scalars; V (t) is the external input vector; ν(t) is given as 0 < ν1 ≤ ν(t) ≤ ν2 , ν˙ (t) ≤ h. {δt , t ≥ 0} represents the semi-Markovian process in s = {1, 2, . . . , S }. The transition probability is given as ¯ ) = β|δ(t) = α} = Pr {δ(t + Δ

{

¯ + o(Δ ¯ ), α /= β, ηαβ (d)Δ ¯ + o(Δ ¯ ), α = β, 1 + ηαα (d)Δ

¯ )/Δ ¯ ) = 0 and ηαβ (d) ≥ 0 is the transition where d ≥ 0 is the sojourn time, lim (o(Δ ¯ →0 Δ Σ S ηαβ (d) = −ηαα (d). Generally, assume rate from α to β for α /= β, and β=1,β/=α

the transition rate ηαβ (d) subject to ηαβ ≤ ηαβ (d) ≤ η¯ αβ with real constant scalars ηαβ and η¯ αβ . Remark 7.1 It is noted that the sojourn time of semi-Markovian process is a random variable subject to continuous probability distribution V . When V is Weibull distribution, the cumulative distribution function is

7.2 Problem Statements and Preliminaries

{ V (d) =

125

1 − exp[−( da )b ], d ≥ 0, 0, d < 0,

and the probability distribution function is { v(d) =

b b−1 d exp[−( da )b ], ab

0,

d ≥ 0, d < 0,

v(d) = abb d b−1 . For b = 1, which means that the transition rate function is μ(d) = 1−V (d) the semi-Markovian process is transformed into a standard Markovian process, in which the sojourn time at each mode follows the exponential distribution. Then, it is η eηαβ d

v (d)

αβ got that ηαβ (d) = 1−Vαβαβ (d) = 1−(1−e ηαβ d = ηαβ , that is, the semi-Markovian process ) means a class of more general stochastic processes than Markovian process.

Next, take S-MSNNs (7.1) as the master system. Then, the slave system is given as z˙¯ (t) = −A (δ(t))z¯ (t) + C (δ(t))ψ(z¯ (t)) + Cd (δ(t))ψ(z¯ (t − ν(t))) + V (t) + u(t), (7.3) z¯ (t0 + s) = ϕ2 (s), ∀s ∈ [−ν2 , 0], where z¯ (t) = [z¯ 1 (t), z¯ 2 (t), . . . , z¯ n (t)]T is the state vector, A (δ(t)), C (δ(t)), Cd (δ(t)), and V (t) are described in (7.1), and u(t) ∈ Rn is the appropriate control input. For e(t) = z¯ (t) − z(t), one has the error system e(t) ˙ = − A (δ(t))e(t) + C (δ(t))ψ(e(t)) + Cd (δ(t))ψ(e(t − ν(t))) + u(t), (7.4) where ψ(e(t)) = ψ(z¯ (t)) − ψ(z(t)). According to (7.2), ψ(e(t)) satisfies Hl − ≤

ψl (ρ) ≤ Hl + , l = 1, 2, . . . , n, ρ

(7.5)

where ρ ∈ R, ρ /= 0. Before entering the plant, a logarithmic quantizer is introduced to describe the control signal u(t) as [ ] Q(·) = Q1 (·) Q2 (·) · · · Qn (·) ,

(7.6)

where Q(·) is assumed to be symmetric with Q p (−u p (t)) = −Q p (u p (t)), 1 ≤ p ≤ n.

(7.7)

126

7 Finite-Time Synchronization

The set of quantized levels of Q p (·) takes { } || { (0) } || {0} , Φ p = ±θ p(q) , |θ p(β) = (φ p )q θ p(0) , q = ±1, ±2, . . . ±θ p 0 < φ p < 1, θ p(0) > 0,

(7.8)

where φ p is the quantizer density and θ p(0) is the initial quantization values of the sub-quantizer Q p (·). Then, the quantizer Q p (·) is shown as ⎧ (q) θp ⎪ ⎨ θ p(β) , if 1+λ < u p (t) < p Q p (u p (t)) = 0, if u (t) = 0, p ⎪ ⎩ −Q p (−u p (t)), if u p (t) < 0,

(q)

θp , 1−λ p

(7.9)

1−φ

where λ p = 1+φ pp , 1 ≤ p ≤ n, q = ±1, ±2, . . . with 0 < λ p < 1. Furthermore, Q(u(t)) = (I + Λ)u(t),

(7.10)

where Λ = diag{Λ1 , Λ2 , . . . , Λn } and Λ p ∈ [−λ p , λ p ], 1 ≤ p ≤ n. Furthermore, denote λ = max p=1,2,...,n {λ p }. Letting u(t)=(I + Λ)u(t) in system (7.4) results in e(t) ˙ = − A (δ(t))e(t) + C (δ(t))ψ(e(t)) + Cd (δ(t))ψ(e(t − ν(t))) + (I + Λ)u(t).

(7.11)

Based on the control input u(t) = K (δ(t))e(t), the error system (7.11) is rewritten as e(t) ˙ = (−A (δ(t)) + (I + Λ)K (δ(t)))e(t) + C (δ(t))ψ(e(t)) + Cd (δ(t))ψ(e(t − ν(t))).

(7.12)

Definition 7.1 ([29]) For time constant T , scalars ε1 , ε2 , and matrix R, system (7.12) is finite-time synchronized about (ε1 , ε2 , R, T ), if ∀t ∈ [0, T ], { } { } E sup−ν2 ≤θ ≤0 [e T (θ )Re(θ ), e˙ T (ρ)R e(ρ)] ˙ ≤ ε1 ⇒ E e T (t)Re(t) < ε2 , where 0 ≤ ε1 < ε2 and R > 0. When computing the infinitesimal operator, the derivative integral terms increase the difficulty of calculation. In this case, Lemmas 7.1 and 7.2 are adopted for Wirtinger-based integral inequality and matrix-based quadratic convex approach. Lemma 7.1 (Wirtinger-based integral inequality [30]) For given matrix P > 0 and all continuously θ (·) in [a, b] → Rn , one has

7.3 Finite-Time Synchronization

127

{ − (b − a)

b

θ˙ T (s)P θ˙ (s)ds

a

≤ − (θ (b) − θ (a))T P(θ (b) − θ (a)) − 3Π T PΠ, where Π = θ (b) + θ (a) −

2 b−a

{b a

(7.13)

θ (s)ds.

Lemma 7.2 (Matrix-based quadratic convex approach [31]) For a continuous funcn×n tion as 0 < d1 ≤ d(t) ≤ d2 , positive-definite real matrix [ X ∈]R , vector z˙ : X˜ Y ≥ 0, one has [−d2 , 0] → Xn , and real matrix Y ∈ R2n×2n satisfying Y T X˜ { − (d2 − d1 )

t−d1 t−d2

z˙ T (s)X z˙ (s)ds ≤ 2ψ1T Y ψ2T − ψ1T X˜ ψ1T − ψ2T X˜ ψ2T , (7.14)

where ] ] [ z(t − d(t)) − z(t − d2 ) z(t − d1 ) − z(t − d(t)) , ψ2 = , z(t − d(t)) + z(t − d2 ) − 2χ2 (t) z(t − d1 ) + z(t − d(t)) − 2χ1 (t) { t−d1 1 X˜ = diag{X , 3X }, χ1 (t) = z(s)ds, d(t) − d1 t−d(t) { t−d(t) 1 z(s)ds. χ2 (t) = d2 − d(t) t−d2 [

ψ1 =

7.3 Finite-Time Synchronization By virtue of stochastic semi-Markovian Lyapunov functional and integral inequality, sufficient conditions are developed for finite-time synchronization of the error system (7.12) in Theorem 7.1. Theorem 7.1 For given scalars T > 0, 0 < ε1 < ε2 , ν1 > 0, ν2 > 0, γ > 0, and matrix R > 0, if we find symmetric matrices Pα > 0, Q1α > 0, Q2α > 0, Q3α > 0, R1 > 0, R2 > 0, Z1 > 0, Z2 > 0, matrices Xα and Y , diagonal matrices Γ1α > 0, Γ2α > 0, and scalars ε1α > 0, ε2α > 0, ∀α ∈ ℘, such that [

] Z˜2 Y ≥ 0, Y T Z˜2

(7.15)

Π1 < 0, Σ S qαβ (d)(Q1β + Q3β ) ≤ R2 , β=1

Σ S

q (d)Q2β ≤ R1 , β=1 αβ

Σ S

q (d)Q3β ≤ R2 , β=1 αβ

(7.16) (7.17) (7.18)

128

7 Finite-Time Synchronization

eγ T ε1 Λ < ε2 minα∈s {λmin (P˜ α )},

(7.19)

where Π1 = a1T (−Pα Aα − AαT Pα + Pα Kα + KαT Pα − γ Pα +

Σ S

q (d)Pβ β=1 αβ −1 −1 + Q1α + Q2α + Q3α + ν1 R1 + ν2 R2 + ε1α λ2 Pα Kα KαT Pα + ε1α I + ε2α I )a1 T T T T T T − (1 − h)a2 Q1α a2 − a3 Q2α a3 − a4 Q3α a4 + a1 Pα Cα a5 + a5 Cα Pα a1

T T + a1T Pα Cdα a6 + a6T Cdα Pα a1 + a10 (ν12 Z1 + (ν2 − ν1 )2 Z2 − 2Xα T T + ε2α λ2 Xα Kα KαT Xα )a10 − a10 Xα Aα a1 − a1T AαT Xα a10 + a10 Xα Kα a1 T T + a1T KαT Xα a10 + a10 Xα Cα a5 + a5T CαT Xα a10 + a10 Xα Cdα a6 T ¯ 1 b2 − b T Z˜1 b3 + 2b T Y b5 − b T Z˜2 b4 + a6T Cdα Xα a10 + b1T H 1 b1 + b2T H 3 4 4 ] ] ]T [ [ [ T T ˜ T T T T T T T T − b5 Z2 b5 , b1 = a1 a5 , b2 = a2 a6 , b3 = a1 − a3 a1 + a3T − a9T , [ [ ]T ]T b4 = a2T − a4T a2T + a4T − a7T , b5 = a3T − a2T a3T + a2T − a8T , [ [ ] ] ¯ 1 = −H1 Γ1α H2 Γ1α , H = −H1 Γ2α H2 Γ2α , Z˜1 = diag{Z1 , 3Z1 }, H 1 ∗ −Γ1α ∗ −Γ2α [ ] Y1 Y2 Z˜2 = diag{Z2 , 3Z2 }, Y = , P˜ α = R −1/2 Pα R −1/2 , Y3 Y4 Q˜1α = R −1/2 Q1α R −1/2 , Q˜2α = R −1/2 Q2α R −1/2 , Q˜3α = R −1/2 Q3α R −1/2 ,

R˜ 1 = R −1/2 R1 R −1/2 , R˜ 2 = R −1/2 R2 R −1/2 , Z˜1 = R −1/2 Z1 R −1/2 , Z˜2 = R −1/2 Z2 R −1/2 , Λ = maxα∈s {λmax (P˜ α )} + ν2 maxα∈s {λmax (Q˜1α )} + ν1 maxα∈s {λmax (Q˜2α )} 1 1 1 + ν2 maxα∈s {λmax (Q˜3α )} + ν12 λmax (R˜ 1 ) + ν22 λmax (R˜ 2 ) + ν13 λmax (Z˜1 ) 2 2 2 1 2 + (ν2 − ν1 ) (ν2 + ν1 )λmax (Z˜2 ), 2 ζ (t) = [e T (t) e T (t − ν(t)) e T (t − ν1 ) e T (t − ν2 ) ψ T (e(t)) ψ T (e(t − ν(t))) { t−ν(t) { t−ν1 { 2 2 t T 2 e T (s)ds e T (s)ds e (s)ds e˙ T (t)]T , ν(t) − ν1 t−ν(t) ν1 t−ν1 ν2 − ν(t) t−ν2 where as ∈ R10n×n (s = 1, 2, . . . , 10) means the block entry matrix, e.g. a2 ζ (t) = e T (t − ν(t)), then system (7.12) realizes finite-time synchronization about (ε1 , ε2 , R, T ).

7.3 Finite-Time Synchronization

129

Proof Choose the Lyapunov functional W (e(t), δ(t), t) = W1 (e(t), δ(t), t) + W2 (e(t), δ(t), t) + W3 (e(t), t) + W4 (e(t), t),

(7.20)

where W1 (e(t), δ(t), t) = e T (t)P(δ(t))e(t), { t { e T (s)Q1 (δ(t))e(s)ds + W2 (e(t), δ(t), t) = { +

t−ν(t) t

t

e T (s)Q2 (δ(t))e(s)ds

t−ν1

e T (s)Q3 (δ(t))e(s)ds,

t−ν2

{ 0 { t

e T (s)R1 e(s)dsdρ +

{ 0 { t

e T (s)R2 e(s)dsdρ, −ν1 t+ρ −ν2 t+ρ { 0 { t { −ν1 { t e˙ T (s)Z1 e(s)dsdρ ˙ + (ν2 − ν1 ) e˙ T (s)Z2 e(s)dsdρ. ˙ W4 (e(t), t) = ν1 −ν1 t+ρ −ν2 t+ρ W3 (e(t), t) =

At time t, δ(t) = α, for α ∈ s, the infinitesimal operator is calculated as ℘W1 (e(t), δ(t), t) 1 = lim [E {W1 (e(t + Δ), δ(t + Δ), t + Δ)|δ(t) = α} − W1 (e(t), α, t)] Δ→0 Δ [ 1 Σ S Pr {δ(t + Δ) = β|δ(t) = α}e T (t + Δ)Pβ e(t + Δ) = lim β=1,β/=α Δ→0 Δ ] T T + Pr {δ(t + Δ) = α|δ(t) = α}e (t + Δ)Pα e(t + Δ) − e (t)Pα e(t) [ Pr {δ(t + Δ) = β, δ(t) = α} T 1 Σ S = lim e (t + Δ)Pβ e(t + Δ) β=1,β/=α Δ→0 Δ Pr {δ(t) = α} ] Pr {δ(t + Δ) = α, δ(t) = α} T T e (t + Δ)Pα e(t + Δ) − e (t)Pα e(t) + Pr {δ(t) = α} [ Σ S 1 qαβ (Vα (d + Δ) − Vα (d)) T = lim e (t + Δ)Pβ e(t + Δ) β=1,β/=α Δ→0 Δ 1 − Vα (d) ] 1 − Vα (d + Δ) T T e (t + Δ)Pα e(t + Δ) − e (t)Pα e(t) . (7.21) + 1 − Vα (d) According to Taylor formula, we have

130

7 Finite-Time Synchronization

e(t + Δ) = e(t) + e(t)Δ ˙ + o(Δ) = e(t) + [(−Aα + (I + Λ)Kα )e(t) + Cα ψ(e(t)) + Cdα ψ(e(t − ν(t)))]Δ + o(Δ), when Δ → 0. Based on lim

Δ→0

V α (d+Δ)−V α (d) 1−V α (d)

= 0, lim

Δ→0

1−V α (d+Δ) 1−V α (d)

= 1, lim

Δ→0

(7.22)

V α (d+Δ)−V α (d) Δ(1−V α (d))

=

μα (d), and qαβ (d) = qαβ μα (d), α /= β, substituting (7.22) into (7.21), one has ℘W1 (e(t), δ(t), t) = 2e T (t)Pα [(−Aα + (I + Λ)Kα )e(t) + Cα ψ(e(t)) Σ S qαβ (d)Pβ e(t). + Cdα ψ(e(t − ν(t)))] + e T (t) β=1

(7.23)

Similarly, there holds ℘W2 (e(t), δ(t), t) ≤e T (t)(Q1α + Q2α + Q3α )e(t) − (1 − h)e T (t − ν(t))Q1α e(t − ν(t)) − e T (t − ν1 )Q2α e(t − ν1 ) − e T (t − ν2 )Q3α e(t − ν2 ) { t { Σ S Σ S + qαβ (d) e T (s)Q1β e(s)ds + qαβ (d) β=1

+ +

Σ S β=1

Σ S β=1

qαβ (d)

t−ν(t) { t−ν(t)

{ qαβ (d)

t−ν2 t

β=1

e T (s)Q3β e(s)ds,

(7.24)

t−ν(t)

{

= e (t)(ν1 R1 + ν2 R2 )e(t) −

t

T

{

{ e (s)R1 e(s)ds −

t

T

t−ν1 t−ν(t)

e T (s)Q2β e(s)ds

t−ν1

e T (s)Q3β e(s)ds

℘W3 (e(t), t)

−

t

e T (s)R2 e(s)ds

t−ν(t)

e T (s)R2 e(s)ds,

(7.25)

t−ν2

℘W4 (e(t), t)

{

= e˙ T (t)(ν12 Z1 + (ν2 − ν1 )2 Z2 )e(t) ˙ − ν1 { − (ν2 − ν1 )

t−ν1

t

e˙ T (s)Z1 e(s)ds ˙

t−ν1

e˙ T (s)Z2 e(s)ds. ˙

t−ν2

For 2e T (t)Pα ΛKα e(t), there exist positive scalar ε1α , such that

(7.26)

7.3 Finite-Time Synchronization

131

2e T (t)Pα ΛKα e(t) −1 T e (t)e(t). ≤ε1α λ2 e T (t)Pα Kα KαT Pα e(t) + ε1α

(7.27)

Considering the condition (7.5), for l = 1, 2, . . . , n, one has (ψl (el (t)) − Hl − el (t))(ψl (el (t)) − Hl + el (t)) ≤ 0,

(7.28)

that is, [

][ ]T [ ] H − +H + e(t) e(t) Hl − Hl + wl wlT − l 2 l wl wlT ≤ 0, ψ(e(t)) ψ(e(t)) ∗ wl wlT

(7.29)

where wl is the unit column vector with one element on its lth row and zeros elsewhere. For any diagonal matrix Γ1α , one has [

][ ] ]T [ e(t) −H1 Γ1α H2 Γ1α e(t) ≥ 0, ∗ −Γ1α ψ(e(t)) ψ(e(t))

(7.30)

with H1 = diag{H1− H1+ , H2− H2+ , . . . , Hn− Hn+ }, { − } H1 + H1+ H2− + H2+ H − + Hn+ , ,..., n . H2 = diag 2 2 2 Similarly, for any diagonal matrix Γ2α , there holds [

]T [ ][ ] −H1 Γ2α H2 Γ2α e(t − ν(t)) e(t − ν(t)) ≥ 0. ∗ −Γ2α ψ(e(t) − ν(t)) ψ(e(t − ν(t)))

(7.31)

Applying Lemma 7.1 leads to {

[

e(t) − e(t − {ν1 ) e˙ (s)Z1 e(s)ds ˙ ≤− − ν1 t e(t) + e(t − ν1 ) − ν21 t−ν1 e(s)ds t−ν1 [ ] e(t) − e(t − {ν1 ) . t e(t) + e(t − ν1 ) − ν21 t−ν1 e(s)ds t

]T [

T

Z1 0 0 3Z1

]

(7.32)

Based on Lemma 7.2 and (7.15), we have { t−ν1 e˙ T (s)Z2 e(s)ds ˙ − (ν2 − ν1 ) t−ν2 [ ] [ ] [ ] Y1 Y2 Z2 0 Z2 0 ≤2κ1T κ2 − κ1T κ1 − κ2T κ , Y3 Y4 0 3Z2 0 3Z2 2

(7.33)

132

7 Finite-Time Synchronization

where ] e(t − ν(t)) − e(t − ν2 ) { , κ1 = t−ν(t) 2 e(t − ν(t)) + e(t − ν2 ) − ν2 −ν(t) t−ν2 e(s)ds [ ] e(t − ν1 ) − e(t − ν(t)) { κ2 = . t−ν1 2 e(t − ν1 ) + e(t − ν(t)) − ν(t)−ν t−ν(t) e(s)ds 1 [

(7.34)

For matrix Xα with appropriate dimension, there exists 2e˙ T (t)Xα [−e(t) ˙ + (−Aα + (I + Λ)Kα )e(t) + Cα ψ(e(t)) + Cdα ψ(e(t − ν(t)))] = 0.

(7.35)

For 2e˙ T (t)Xα ΛKα e(t) in (7.34), there exist positive scalar ε2α , such that −1 T 2e˙ T (t)Xα ΛKα e(t) ≤ ε2α λ2 e˙ T (t)Xα Kα KαT Xα e(t) ˙ + ε2α e (t)e(t).

(7.36)

Based on the conditions (7.16) and (7.17), there holds ℘W (e(t), δ(t), t) [ Σ S T ≤ζ (t) a1T (−Pα Aα − AαT Pα + Pα Kα + KαT Pα + qαβ (d)Pβ + Q1α β=1

+ +

−1 Q2α + Q3α + ν1 R1 + ν2 R2 + ε1α λ Pα Kα KαT Pα + ε1α I −1 T T T T ε2α I )a1 − (1 − h)a2 Q1α a2 − a3 Q2α a3 − a4 Q3α a4 + a1 Pα Cα a5 2

T T + a5T CαT Pα a1 + a1T Pα Cdα a6 + a6T Cdα Pα a1 + a10 (ν12 Z1 + (ν2 − ν1 )2 Z2 T − 2Xα + ε2α λ2 Xα Kα KαT Xα )a10 − a10 Xα Aα a1 − a1T AαT Xα a10 T T + a10 Xα Kα a1 + a1T KαT Xα a10 + a10 Xα Cα a5 + a5T CαT Xα a10 T T + a10 Xα Cdα a6 + a6T Cdα Xα a10 [ ]T [ ] [ ] [ ]T [ ][ ] a1 −H1 Γ1α H2 Γ1α a1 a2 −H1 Γ2α H2 Γ2α a2 + + a5 ∗ −Γ1α a5 a6 ∗ −Γ2α a6 [ ]T [ ][ ] [ ]T a1 − a3 Z1 0 a1 − a3 a2 − a4 − + a1 + a3 − a9 a2 + a4 − a7 0 3Z1 a1 + a3 − a9 [ ][ ] [ ]T [ ][ ] 2Y1 2Y2 a3 − a2 a2 − a4 Z2 0 a2 − a4 − 2Y3 2Y4 a3 + a2 − a8 a2 + a4 − a7 0 3Z2 a2 + a4 − a7 [ ]T [ ][ ]] a3 − a2 Z2 0 a3 − a2 − (7.37) ζ (t), a3 + a2 − a8 0 3Z2 a3 + a2 − a8

with as ∈ R10n×n (s = 1, 2, . . . , 10) given in Theorem 7.1. From the condition (7.16), one has

7.3 Finite-Time Synchronization

133

℘W (e(t), δ(t), t) ≤ γ W1 (e(t), δ(t), t) ≤ γ W (e(t), δ(t), t).

(7.38)

Apply Dynkin’s formula to (7.38) leads to e−γ t E {W (e(t), δ(t), t)} − E {W (e(t0 ), δ(t0 ), t0 )} < 0,

(7.39)

which means E {W (e(t), δ(t), t)} < eγ t E {W (e(t0 ), δ(t0 ), t0 )} { } ≤eγ t ΛE sup−ν2 ≤ϑ≤0 [e T (ϑ)Re(ϑ), e˙ T (ϑ)R e(ϑ)] ˙ ≤ eγ t Λε1 ,

(7.40)

with Λ given in Theorem 7.1. According to E {W (e(t), δ(t), t)} ≥ E {e T (t)P(δ(t))e(t)} ≥minα∈s {λmin (P˜ α )}E {e T (t)Re(t)},

(7.41)

we have E {e T (t)Re(t)} ≤

eγ t Λε1 minα∈s {λmin (P˜ α )}

< ε2 .

(7.42)

Therefore, system (7.12) is finite-time synchronization subject to (ε1 , ε2 , R, T ). ∎ Remark 7.2 Wirtinger-based integral inequality and matrix-based quadratic convex are introduced for theoretical analysis. Compared with Jensen’s inequality, Wirtingerbased integral inequality is with a tighter upper bound. Also, matrix-based quadratic convex considers fewer slack variables to estimate the upper bound of delay than free-weighting matrix. Sojourn-time-dependent sufficient conditions are developed such that system (7.12) is finite-time synchronized in Theorem 7.1. Next, based on the aforementioned conditions, the feedback controller will be designed. Theorem 7.2 For given scalars T > 0, 0 < ε1 < ε2 , ν1 > 0, ν2 > 0, γ > 0, and matrix R > 0, if there exist symmetric matrices Pα > 0, Q1α > 0, Q2α > 0, Q3α > 0, R1 > 0, R2 > 0, Z1 > 0, Z2 > 0, matrices K˜α and Y , diagonal matrices Γ1α > 0, Γ2α > 0, and scalars ε1α > 0, ε2α > 0, δα > 0, φs2 , s2 = 1, 2, . . . , 9, ∀α ∈ ℘, such that (7.15) and

134

7 Finite-Time Synchronization

⎤ ⎡ T Π2 ε1α λa1T K˜α ε2α δα λa10 K˜α ⎦ < 0, ⎣ ∗ −ε1α I 0 ∗ ∗ −ε2α I ⎡ ⎤ T ˜ T Π3 ε1α λa1 Kα ε2α δα λa10 K˜α ⎣ ∗ −ε1α I ⎦ < 0, 0 ∗ ∗ −ε2α I Σ S q¯ (Q1β + Q3β ) ≤ R2 , β=1 αβ Σ S q (Q1β + Q3β ) ≤ R2 , β=1 αβ Σ S Σ S q¯ αβ Q2β ≤ R1 , q¯ Q3β ≤ R2 , β=1 β=1 αβ Σ S Σ S qαβ Q2β ≤ R1 , qαβ Q3β ≤ R2 , β=1

(7.43)

(7.44) (7.45) (7.46) (7.47) (7.48)

β=1

0 < φ1 R < Pα < φ2 R, Q1α < φ3 R, Q2α < φ4 R, Q3α < φ5 R, R1 < φ6 R, (7.49) R2 < φ7 R, Z1 < φ8 R, Z2 < φ9 R, 1 2 1 2 1 3 ε1 (φ2 + ν2 φ3 + ν1 φ4 + ν2 φ5 + ν1 φ6 + ν2 φ7 + ν1 φ8 2 2 2 1 2 −γ T ε2 φ1 , (7.50) + (ν2 − ν1 ) (ν2 + ν1 )φ9 ) < e 2 where Π2 = a1T

Σ S

q¯ P a + Π˜ 2 , Π3 = a1T β=1 αβ β 1

Σ S

q

β=1 αβ

Pβ a1 + Π˜ 2 ,

Π˜ 2 = a1T (−Pα Aα − AαT Pα + K˜α + K˜αT − γ Pα + Q1α + Q2α + Q3α −1 −1 + ν1 R1 + ν2 R2 + ε1α I + ε2α I )a1 − (1 − h)a2T Q1α a2 − a3T Q2α a3 T P a − a4T Q3α a4 + a1T Pα Cα a5 + a5T CαT Pα a1 + a1T Pα Cdα a6 + a6T Cdα α 1 T (ν 2 Z + (ν − ν )2 Z − 2δ P )a − δ a T P A a − δ a T A T P a + a10 α α 10 α 10 α α 1 α 1 α α 10 2 1 2 1 1 T K˜ a + δ a T K˜ T a + δ a T P C a + δ a T C T P a + δα a10 α 1 α 1 α 10 α 10 α α 5 α 5 α α 10 T P C a + δ a T C T P a + bT H b + bT H ¯ 1 b2 − b T Z˜1 b3 + 2b T Y b5 + δα a10 α dα 6 α 6 dα α 10 1 1 1 2 3 4

− b4T Z˜2 b4 − b5T Z˜2 b5 ,

¯ 1 , H , Z˜1 , and Z˜2 described in Theorem 7.1, then system with b1 , b2 , b3 , b4 , b5 , H 1 (7.12) is finite-time synchronized subject to (ε1 , ε2 , R, T ) with the controller gain as Kα = Pα−1 K˜α .

7.3 Finite-Time Synchronization

135

Proof Denote K˜α = Pα Kα , Xα = δα Pα .

(7.51)

The transition rate qαβ (d) is described by qαβ (d) = s1 qαβ + s2 q¯ αβ , where s1 + s2 = 1 and s1 ≥ 0, s2 ≥ 0. Based on Schur complement lemma, multiplying (7.43) by s1 and (7.44) by s2 leads to ⎡

⎤ T K˜α Π4 ε1α λa1T K˜α ε2α λa10 ⎣ ∗ −ε1α I ⎦ < 0, 0 ∗ ∗ −ε2α I

(7.52)

Σ S (s1 qαβ + s2 q¯ αβ )Pβ a1 + Π2 , with Π2 described in where Π4 = a1T β=1 Theorem 7.1. By tuning s1 and s2 , all possible qαβ (d) ∈ [qαβ , q¯ αβ ] can be got. Then, (7.16) holds. Similar to (7.45)–(7.48), (7.17) and (7.18) also hold. Pre- and post-multiplying (7.49) by R −1/2 , it is got that φ1 I < R −1/2 Pα R −1/2 < φ2 I , R −1/2 Q1α R −1/2 < φ3 I , R −1/2 Q2α R −1/2 < φ4 I , R −1/2 Q3α R −1/2 < φ5 I , R −1/2 R1 R −1/2 < φ6 I , R −1/2 R2 R −1/2 < φ7 I , R −1/2 Z1 R −1/2 < φ8 I , R −1/2 Z2 R −1/2 < φ9 I ,

(7.53)

which means φ1 < minα∈s {λmin (P˜ α )}, maxα∈s {λmax (P˜ α )} < φ2 , maxα∈s {λmax (Q˜1α )} < φ3 , maxα∈s {λmax (Q˜2α )} < φ4 , maxα∈s {λmax (Q˜3α )} < φ5 , λmax (R˜ 1 ) < φ6 , λmax (R˜ 2 ) < φ7 , λmax (Z˜1 ) < φ8 , λmax (Z˜2 ) < φ9 . Together with (7.50), (7.19) holds.

∎

Remark 7.3 The steady state performance in a finite-time level has been studied in Refs. [7–11], in which the controlled object converges to the equilibrium point in a fixed time region. This finite-time synchronization is essentially based on Lyapunov stability. However, the transient performance during a finite-time level is of great significance in practical systems. To name a few, the trajectory of an aircraft is translated from one point to another in a fixed time; the temperature and pressure does not exceed a certain range in a fixed time. If the transient performance over finitetime interval is ignored in the practical process, it will lead to excessive overshoot and

136

7 Finite-Time Synchronization

damage of the production equipment. In fact, finite-time synchronization is realized based on finite-time stability, in which the expected error signal does not exceed a given physical threshold [29].

7.4 Simulation The quadruple-tank process model (Fig. 7.1) is composed of four interconnected water tanks and two pumps [32], in which the inputs are the voltages of the two pumps and the outputs are the water level of tanks 1 and 2. Considering the abrupt change of actual operating parameters and complicated transition rate, the quadrupletank process model is described by S-MSNNs with ˜ − d2 ) + C˜1 u(t ˜ − d3 ), z˙˜ (t) = A˜0 z˜ (t) + A˜1 z˜ (t − d1 ) + C˜0 u(t

(7.54)

where A˜0 = diag{−0.0021, −0.0021, −0.0424, −0.0424}, ⎡ ⎤ 0 0 0.0424 0 ⎢0 0 0 0.0424⎥ ⎥, A˜1 = ⎢ ⎣0 0 0 0 ⎦ 00 0 0 ]T ]T [ [ 0 00 0.1113m 1 00 0 0.1113(1 − m 1 ) , C˜1 = , C˜0 = 0 0 0.1042m 2 0 0 0 0 0.1042(1 − m 2 )

Fig. 7.1 Schematic representation of the quadruple-tank process model

7.4 Simulation

137

[ ] −0.1603 −0.1765 −0.0795 −0.2073 u(t) ˜ = K˜ z˜ (t), K˜ = , −0.1977 −0.1579 −0.2288 −0.0772 where m 1 means the ratio of water diverted to tank 1 rather than to tank 3, and m 2 means the corresponding ratio diverted from tank 2 to tank 4. m 1 and m 2 are with two different modes with m 11 = 0.333, m 12 = 0.450, and m 21 = 0.307, m 22 = 0.380. The transformation between different ratios obeys the semi-Markovian process {δ(t), t ≥ 0} and takes values in s = {1, 2}. C˜0 and C˜1 is related to δ(t) given as C˜0 (δ(t)) and C˜1 (δ(t)). The transition rate qαβ (d) depends on the sojourn time d, given by ⎧ ¯ d < d, ¯ ⎨ d, ¯ qαβ (d) = d, d ≤ d ≤ d, ⎩ d, d < d,

(7.55)

where d = 0.1 and d¯ = 4.6. Then, one has [ q=

] [ ] −0.1 0.1 −4.6 4.6 , q¯ = . 0.1 −0.1 4.6 −4.6

Set d1 = 0, d2 = 0, and d3 = ν(t). Let ν(t) = 0.4(1.5 − sin(3t)), then ν1 = 0.2, ˜ represents the amount of water supplied by pumps, expressed ν2 = 1.0, h = 1.2. u(t) as ˜ z (t)), u(t) ˜ = K˜ ψ(˜ ˜ z (t)) = [ψ˜ 1 (˜z 1 (t)), ψ˜ 2 (˜z 2 (t)), ψ˜ 3 (˜z 3 (t)), ψ˜ 4 (˜z 4 (t))], ψ(˜ ψ˜ l (˜zl (t)) = 0.1(|˜zl (t) + 1| − |˜zl (t) − 1|), l = 1, 2, . . . , 4. The quadruple-tank process model is described as z˙ (t) = − A (δ(t))z(t) + C (δ(t))ψ(z(t)) + Cd (δ(t))ψ(z(t − ν(t))) + V (t),

(7.56)

where A1 = A2 = A3 = −A˜0 − A˜1 , C1 = C˜01 K˜ , C2 = C˜02 K˜ , C3 = C˜03 K˜ , ˜ z (t)), V (t) = [0 0 0 0], Cd1 = C˜11 K˜ , Cd2 = C˜12 K˜ , Cd3 = C˜13 K˜ , ψ(z(t)) = ψ(˜ and Hl − = 0, Hl + = 0.1, l = 1, 2, . . . , 4. Choose the quantizer density as φ p = 13 . Then, λ p = 21 , p = 1, 2, 3, 4, and λ = 21 . For given T = 5s, ε1 = 1, ε2 = 12.5, γ = 0.5, R = I , ε1α = 0.1, ε2α = 0.1, δα = 1, α = 1, 2, solving Theorem 7.2, one has

138

7 Finite-Time Synchronization

⎡

⎤ 0.0338 0.0000 −0.0092 −0.0001 ⎢ 0.0000 0.0339 −0.0001 −0.0092⎥ ⎥ K1 = ⎢ ⎣−0.0077 −0.0002 0.0482 −0.0010⎦ , −0.0002 −0.0077 −0.0010 0.0483 ⎡ ⎤ 0.0336 −0.0000 −0.0092 −0.0000 ⎢−0.0000 0.0337 −0.0001 −0.0092⎥ ⎥ K2 = ⎢ ⎣−0.0077 −0.0002 0.0482 −0.0008⎦ . −0.0001 −0.0077 −0.0008 0.0485

Fig. 7.2 System mode

2

1

0

Fig. 7.3 System state z(t)

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

1

System State

0.5

0

-0.5

-1

0

1

2

3 Time(s)

4

5

7.4 Simulation

139

Fig. 7.4 System state z¯ (t)

0.8 0.6 System State

0.4 0.2 0 -0.2 -0.4 -0.6

0

1

2

3

4

5

3

4

5

Time(s) Fig. 7.5 Error state e(t)

0.3

Error State

0.2 0.1 0 -0.1 -0.2 -0.3

0

1

2 Time(s)

[ ]T [ ]T For given δ0 =2, z(0)= −0.6 0.8 0.6 −0.8 , and z¯ (0)= −0.4 0.6 0.5 −0.5 , Figures 7.2, 7.3 and 7.4 plot the system mode, the state responses z(t) and z¯ (t). Figure 7.5 shows the error signals e(t). Figure 7.6 plots the input signal. Figure 7.7 depicts the evolution e T (t)Re(t) in a finite-time level. From Fig. 7.7, the evolution e T (t)Re(t) < ε2 satisfies the requirement for finite-time synchronization.

140

7 Finite-Time Synchronization

Fig. 7.6 System input u(t)

System Input

0.05

0

-0.05

0

1

2

3

4

5

3

4

5

Time(s) Fig. 7.7 Evolution e T (t)Re(t)

0.3 0.25

Evolution

0.2 0.15 0.1 0.05 0

0

1

2 Time(s)

7.5 Conclusion In this chapter, the finite-time synchronization for delayed S-MSNNs under quantized mechanism has been addressed. By designing an appropriate stochastic semiMarkovian Lyapunov functional, sufficient conditions have been developed via the corresponding integral inequality. The solvable conditions of quantized controller gains are shown in a standard linear matrix inequality framework. In the future, we will study fuzzy synchronization for delayed S-MSNNs over a finite-time level.

References

141

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Chapter 8

Fuzzy Sliding Mode Control

This chapter is concerned with the issue of robust stabilization for delayed semiMarkovian switching T-S fuzzy systems via sliding mode control. In order to reduce some conservativeness, an integral sliding mode surface is designed without assumption that the input matrices are plat-rule independent with full column rank, which is one of the critical factors for system performance. Based on Lyapunov functional theory, sufficient condition is provided for stochastic stability of the closed-loop system, and then extended to the case where the input matrices are independent of plant-rule. In addition, a fuzzy sliding mode controller is established to guarantee the finite time reachability of the predetermined fuzzy manifold. Finally, the superiorities of the proposed method are validated by a single-link robot arm model.

8.1 Introduction In the previous chapter, the condition on finite-time synchronization is proposed for delayed semi-Markovian switching neural networks under quantized mechanism. It is well known that fuzzy logic theory has provided an attractive and effective method for the synthesis of complex nonlinear systems. Particularly, T-S fuzzy control has become a heated issue, which can approximate nonlinear systems with arbitrary accuracy [1–5]. By the help of T-S fuzzy method, the nonlinear model can be expressed as a set of local linear subsystems with membership functions. Up to now, a growing number of works have been devoted to this topic (see, e.g. [6–10]). This chapter will address the problem of sliding mode control for delayed semiMarkovian switching T-S fuzzy systems. The main contributions can be given as: (i) Based on the T-S fuzzy method, a fuzzy integral switching function is proposed for nonlinear semi-Markovian switching T-S fuzzy systems; (ii) By the use of Lyapunov © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 W. Qi and G. Zong, Control Synthesis for Semi-Markovian Switching Systems, Studies in Systems, Decision and Control 465, https://doi.org/10.1007/978-981-99-0317-7_8

143

144

8 Fuzzy Sliding Mode Control

functional, sufficient conditions are provided to ensure the sliding mode dynamics stochastically stable; (iii) By constructing a fuzzy sliding mode control law, the state trajectory can be driven onto the pre-designed switching surface in finite time level.

8.2 Problem Statements and Preliminaries Consider a class of nonlinear semi-Markovian switching T-S fuzzy systems with delay as: Plant Rule p: IF χ1 (t) is η p1 , χ2 (t) is η p2 and . . . and χm (t) is η pn , THEN ⎧ ⎪ ⎨ z˙ (t) = A p (vt , t)z(t) + Adp (vt , t)z(t − ı(t)) +B p (vt )(u(t) + f (z(t), z(t − ı(t)), vt , t)), ⎪ ⎩ z(t) = ϕ(t), t ∈ [−ı, 0],

(8.1)

where η pq ( p = 1, 2, . . . , m; q = 1, 2, . . . , n) are the fuzzy sets; χ1 (t), χ2 (t), . . . , χm (t) denote the premise variables; u(t) ∈ R m and z(t) ∈ R n stand for the input vector and the state vector; ϕ(t) is the initial condition during [−ı, 0]; {vt , h}t≥0 := {vl , hl }l∈N≥1 represents continuous-time and discrete-state homogeneous semi-Markovian process with right continuous trajectories in a finite set N = {1, 2, . . . , s}, where {vl }l∈N≥1 is the system mode index at the lth transition in N , and {hl }l∈N≥1 is the sojourn time of mode vl−1 between the (l − 1)th transition and lth transition. The transition rate matrix Λ(h) = {πi j (h)} is given by ⎧ Pr {vl+1 = j, hl+1 ≤ h + Δ|vl = i, hl+1 > h} ⎪ ⎪ ⎪ ⎪ ⎨ = πi j (h)Δ + o(Δ), i /= j, πi j (h) = ⎪ Pr {vl+1 = j, hl+1 > h + Δ|vl = i, hl+1 > h} ⎪ ⎪ ⎪ ⎩ = 1 + πii (h)Δ + o(Δ), i = j, where Δ ≥ 0 and limΔ→0 (o(Δ)/Δ) = 0, πi j (h) Σ ≥ 0 represents transition rate from mode i to mode j for i /= j, and πii (h) = − sj=1, j/=i πi j (h) < 0. The time delay ı(t) is known and satisfies 0 ≤ ı(t) ≤ ı,

ı˙(t) ≤ μ < 1,

(8.2)

where ı and μ are positive scalars. For vt ∈ S , A p (vt , t) =A p (vt ) + ΔA p (vt , t), Adp (vt , t) = Adp (vt ) + ΔAdp (vt , t),

(8.3)

8.2 Problem Statements and Preliminaries

145

where A p (vt ) and Adp (vt ) are known constant matrices, the parameter uncertainty matrices Δ A p (vt , t) and ΔAdp (vt , t) are assumed to be norm-bounded and satisfy ,

,

,

ΔA p (vt , t) =E p (vt )F p (vt , t)H p (vt ), ΔAdp (vt , t) = E p (vt )F p (vt , t)H p (vt ), (8.4) ,

,

where E p (vt ), E p (vt ), and H p (vt ), H p (vt ) are known constant matrices, F p (vt , t) and ,

,

T

,

F p (vt , t) satisfies F pT (vt , t)F p (vt , t) ≤ I and F p (vt , t)F p (vt , t) ≤ I . The unknown nonlinear perturbation f (z(t), z(t − ı(t)), vt , t) is bounded and satisfies || f (z(t), z(t − ı(t)), vt , t)|| ≤ κ1 p ||z(t)|| + κ2 p ||z(t − ı(t))|| ≤ κ1 ||z(t)|| + κ2 ||z(t − ı(t))||, where κ1 p , κ2 p are known nonnegative real scalars, and κ1 ≙ max{κ1 p }, κ2 ≙ max{κ2 p }. By using the fuzzy inference technique, we can get the defuzzified mode of system (8.1) as z˙ (t) =

Σ m p=1

h p (χ (t))[A p (vt , t)z(t) + Adp (vt , t)z(t − ı(t))

+ B p (vt )(u(t) + f (z(t), z(t − ı(t)), vt , t))],

(8.5)

where m Σ m μ pq (χq (t)) Πq=1 h p (χ (t)) = Σ m h p (χ (t)) = 1, ≥ 0, n p=1 p=1 Πq=1 μ pq (χq (t))

and μ pq (χq (t)) denotes the grade of membership of χq (t) in μ pq . , When vt = i, i ∈ N , A p (vt ), Δ A p (vt , t), Adp (vt ), ΔAdp (vt , t), E p (vt ), E p (vt ), , , H p (vt ), H p (vt ), F p (vt , t), F p (vt , t), B p (vt ) and f (z(t), z(t − ı(t)), vt , t) are, respec, , , tively, denoted as A pi , ΔA pi (t), Adpi , ΔAdpi (t), E pi , E pi , H pi , H pi , F pi (t), F pi (t), B pi and f i (z(t), z(t − ı(t)), t). Lemma 8.1 ([11]) If F T F ≤ I , then there exist constant matrices E and H , and scalar ε > 0, such that E F H + H T F T E T ≤ ε−1 E E T + ε H T H. Definition 8.1 ([12]) System (8.1) with zero input signal u(t) is said to be stochastically stable, if for any initial condition z 0 and v0 , {{

t

lim E

t→+∞

} ||z(ω)|| dω|(z 0 , v0 ) < +∞. 2

0

(8.6)

146

8 Fuzzy Sliding Mode Control

8.3 Sliding Surface Design Unlike most of the existing results, it doesn’t need to assume that all B pi have full column rank. Therefore, the following transformation based on [10] is adopted in Σ this study. We will assume that Ci = m1 mp=1 B pi , where Ci satisfies full column rank. ¯ = Let Z i = 21 [Ci − B1i , Ci − B2i , ..., Ci − Bmi ], V = [I, I, ..., I ]T , W (h) diag{1 − 2h 1 (χ (t)), 1 − 2h 2 (χ (t)), ..., 1 − 2h m (χ (t))}, where h¯ = [χ (t)), h 2 (χ (t)), ..., h m (χ (t))]. Then, ¯ Ci + Z i W (h)V 1 = Ci + [(Ci − B1i )(1 − 2h 1 (χ (t))) + · · · + (Ci − Bmi )(1 − 2h m (χ (t)))] 2 1 = Ci + Ci [(1 − 2h 1 (χ (t))) + · · · + (1 − 2h m (χ (t)))] 2 1 − [B1i (1 − 2h 1 (χ (t))) + · · · + Bmi (1 − 2h m (χ (t)))] 2 ) 1 Σ m Σ m Σ m 1 ( h p (χ (t)) − B pi + h p (χ (t))B pi = Ci + Ci m − 2 p=1 p=1 p=1 2 2 Σ m = h p (χ (t))B pi . p=1

Therefore, the fuzzy system (8.5) can be rewritten as: z˙ (t) =

Σ m p=1

h p (χ (t))[(A pi + ΔA pi (t))z(t) + (Adpi + ΔAdpi (t))z(t − ı(t))

+ (Ci + ΔCi )(u(t) + f i (z(t), z(t − ı(t)), t))],

(8.7)

¯ ¯ (h) ¯ ≤ I. with W T (h)W where ΔCi = Z i W (h)V As the first step of design procedure, the following integral sliding mode surface is chosen: { t Σ m h p (χ (ω))G i (A pi + Ci K pi )z(ω)dω s(t) =G i z(t) − 0

−

{ t Σ m

p=1

0

{ −

0

t

p=1

h p (χ (ω))G i Adpi z(ω − ı(ω))dω

G i ΔCi (u(ω) + ϑ(ω))dω,

(8.8)

8.4 Stochastic Stability Analysis

147

where G i ∈ R m×n satisfies G i = CiT such that G i Ci is nonsingular, K pi ∈ R m×n is the real matrix to be designed later on, and ϑ(t) is the compensator. By using sliding mode control theory, the equivalent controller is established from s˙ (t) = 0 that u eq (t) =

Σ m p=1

h p (χ (t)){[K pi − (G i Ci )−1 G i ΔA pi (t)]z(t)

− (G i Ci )−1 G i ΔAdpi (t)z(t − ı(t))} − f i (z(t), z(t − ı(t)), t) − (G i Ci )−1 G i ΔCi ( f i (z(t), z(t − ı(t)), t) − ϑ(t)).

(8.9)

Thus, substituting (8.9) into (8.7) yields z˙ (t) =

Σ m p=1

h p (χ (t))[(A pi + (Ci + ΔCi )K pi + Ii ΔA pi (t))z(t)

+ (Adpi + Ii ΔAdpi (t))z(t − ı(t)) − ΔCi ( f i (z(t), z(t − ı(t)), t) − ϑ(t))], (8.10) where Ii = I − (Ci + ΔCi )(G i Ci )−1 G i , ΔCi = (Ci + ΔCi )(G i Ci )−1 G i ΔCi . The compensator parameter is designed as ϑ(t) = −(αi ||z(t)|| + ρ)sign(g(t)),

(8.11)

where g(t) = ΔCiT Pi z(t), ρ > 0 is a small constant and Pi will be defined in Theorem 8.1. The purpose of the compensator ϑ(t) is to eliminate the effect of ΔCi f i (z(t), z(t − ı(t)), t), which could ensure good property of the sliding motion and realize the anti-disturbance performance of integral sliding mode surface.

8.4 Stochastic Stability Analysis In this section, we analyze the stability of the sliding motion described by (8.10), and derive some sufficient conditions for the robustly stochastic stability of the sliding dynamics via linear matrix inequalities method. The following theorem shows the result. Theorem 8.1 If there exist positive-definite symmetric matrices Pi , D1i , D1 , D2i , D2 ∈ R n×n , and positive scalars ε1i , ε2i , ε3i , for all i ∈ N , such that

148

8 Fuzzy Sliding Mode Control

⎛

⎞ Pi Pi Ψ pi Pi Adpi Pi Z i ⎜ ∗ ⎟ Φi 0 0 0 ⎜ ⎟ ⎜ ∗ ⎟ < 0, 0 0 0 −ε1i I ⎜ ⎟ −1 ⎝ ∗ ⎠ 0 0 −ε2i λ I 1 I 0 −1 ∗ 0 0 0 −ε3i λ I 2 I Σ s Σ s π¯ i j D1 j ≤ D1 , π¯ i j D2 j ≤ D2 , j=1

(8.12)

(8.13)

j=1

where T V T V K + ε HT H + Ψ pi = H e{Pi ( A pi + Ci K pi )} + ε1i K pi pi 2i pi pi

Σ s j=1 ,

π¯ i j P j ,

+ D1i + D2i + ı D1 + ı D2 , λ I 1 = λmax {Ii E pi E Tpi IiT }, λ I 2 = λmax {Ii E pi (E pi )T IiT }, { ∞ , , πi j (h)gi (h)dh, Φi = −(1 − μ)D1i + ε3i (H pi )T H pi , π¯ i j = E{πi j (h)} = 0

then system (8.10) realizes robustly stochastic stability. Proof Consider Lyapunov functional ℵ(z(t), i, t) =

Σ 3 ς=1

ℵς (z(t), i, t),

(8.14)

where ℵ1 (z(t), i, t) = z T (t)Pi z(t), { t { z T (ω)D1i z(ω)dω + ℵ2 (z(t), i, t) = {

t

{ z T (ω)D2i z(ω)dω +

t−ı

{

−ı

t−ı(t)

ℵ3 (z(t), i, t) =

0

0

−ı

t

z T (ω)D1 z(ω)dωdθ,

t+θ

{

t

z T (ω)D2 z(ω)dωdθ,

t+θ

where Pi , D1i , D1 , D2i , D2 > 0, and Σ s j=1

π¯ i j D1 j ≤ D1 ,

Σ s j=1

π¯ i j D2 j ≤ D2 .

(8.15)

Then, we have the weak infinitesimal operator as Υ ℵ(z(t), i, t) = lim+ Δ→0

E{ℵ(z(t + Δ), vt+Δ , t + Δ)|vt = i} − ℵ(z(t), i, t) , (8.16) Δ

where Δ is a small positive number.

8.4 Stochastic Stability Analysis

149

Therefore, for each vt = i ∈ N , one has Υ ℵ1 (z(t), i, t) [ Σ s 1 = lim+ Pr {vl+1 = j, hl+1 ≤ h + Δ|vl = i, hl+1 > h} E{ j=1, j/=i Δ→0 Δ T T P j z Δ + Pr {vl+1 = i, hl+1 > h + Δ|vl = i, hl+1 > h}z Δ Pi z Δ } − z T (t)Pi z(t)] × zΔ

= lim+ Δ→0

×

Σ s 1 Pr {vl+1 = j, vl = i} [E{ j=1, j/ = i Δ Pr {vl = i}

Pr {h < hl+1 ≤ h + Δ|vl+1 = j, vl = i} T zΔ Pj zΔ Pr {hl+1 > h|vl = i}

Pr {hl+1 > h + Δ|vl = i} T z Δ Pi z Δ } − z T (t)Pi z(t)] Pr {hl+1 > h|vl = i} Σ s qi j (℘i (h + Δ) − ℘i (h)) T 1 = lim+ [E{ zΔ Pj zΔ j=1, j/=i Δ→0 Δ 1 − ℘i (h) ] 1 − ℘i (h + Δ) T + z Δ Pi z Δ } − z T (t)Pi z(t) , 1 − ℘i (h) +

l+1 = j,vl =i} = Pr {vl+1 = j|vl = i} denotes the probwhere z Δ ≙ z(t + Δ), qi j := Pr{vPr{v l =i} ability intensity of the system switching from i to j. For a small constant Δ, one has

z Δ = z(t) + z˙ (t)Δ + o(Δ).

(8.17)

According to the technique [13], we have lim

1 − ℘i (h + Δ) ℘i (h + Δ) − ℘i (h) = 1, lim+ = 0, Δ→0 1 − ℘i (h) 1 − ℘i (h)

lim+

℘i (h) − ℘i (h + Δ) = πi (h), Δ(1 − ℘i (h))

Δ→0+

Δ→0

where πi (h) means the transition rate of the system switching from i. Define πi j (h) = πi (h)qi j , j /= i. Then, one has

(8.18)

150

8 Fuzzy Sliding Mode Control

Υ ℵ1 (z(t), i, t) Σ m = h p (χ (t))z T (t)Pi [( A pi + (Ci + ΔCi )K pi + Ii ΔA pi (t))z(t) p=1

+ (Adpi + Ii ΔAdpi (t))z(t − ı(t)) − ΔCi ( f i (z(t), z(t − ı(t)), t) − ϑ(t))] Σ m + h p (χ (t))[(A pi + (Ci + ΔCi )K pi + Ii ΔA pi (t))z(t) + (Adpi p=1

+ Ii Δ Adpi (t))z(t − ı(t)) − ΔCi ( f i (z(t), z(t − ı(t)), t) − ϑ(t))]T Pi z(t) Σ s + z T (t) π¯ i j P j z(t), (8.19) j=1

where π¯ i j ≙ E{πi j (h)} = Similarly, we have

{∞ 0

πi j (h)gi (h)dh. {

t

Υ ℵ2 (z(t), i, t) = z (t)D1i z(t) − (1 − ı˙(t))z (t − ı(t))D1i z(t − ı(t)) + T

z T (ω)

(Σ s

T

{

)

z T (ω)D1 z(ω)dω, (8.20) { t z T (ω) Υ ℵ3 (z(t), i, t) = z T (t)D2i z(t) − z T (t − ı)D2i z(t − ı) + t−ı { t ) (Σ s π¯ i j D2 j z(ω)dω + ı z T (t)D2 z(t) − z T (ω)D2 z(ω)dω. (8.21) j=1

π¯ i j D1 j z(ω)dω + ı z T (t)D1 z(t) −

t−ı(t) t

t−ı

j=1

t−ı

By combining (8.19)–(8.21), we have Υ ℵ(z(t), i, t) Σ m = h p (χ (t))H e{z T (t)Pi [( A pi + (Ci + ΔCi )K pi + Ii ΔA pi (t))z(t) p=1

+(Adpi + Ii ΔAdpi (t))z(t − ı(t)) − ΔCi ( f i (z(t), z(t − ı(t)), t) − ϑ(t))]} { t [Σ s ] π¯ i j P j + ı D1 + ı D2 + D1i + D2i z(t) + z T (ω) +z T (t) j=1

(Σ s { −

)

{

t

(Σ s

)

t−ı(t)

π¯ i j D1 j z(ω)dω + z T (ω) π¯ i j D2 j z(ω)dω j=1 t−ı { t z T (ω)D1 z(ω)dω − z T (ω)D2 z(ω)dω − z T (t − ı)D2i z(t − ı) j=1

t t−ı

t−ı

−z T (t − ı(t))(1 − ı˙(t))D1i z(t − ı(t)). From the condition (8.2), it is obviously that

(8.22)

8.4 Stochastic Stability Analysis

151

− z T (t − ı(t))(1 − ı˙(t))D1i z(t − ı(t)) ≤ −z T (t − ı(t))(1 − μ)D1i z(t − ı(t)), { t { t T − z (ω)D1 z(ω)dω ≤ − z T (ω)D1 z(ω)dω, t−ı t

{ −

t−ı(t) t

{ z T (ω)D2 z(ω)dω ≤ −

t−ı

z T (ω)D2 z(ω)dω.

(8.23)

t−ı(t)

By Lemma 8.1, we obtain H e{z T (t)Pi ΔCi K pi z(t)} −1 T T z (t)Pi Z i Z iT Pi z(t) + ε1i z T (t)K pi V T V K pi z(t), ≤ ε1i

(8.24)

−1 H e{z T (t)Pi Ii ΔA pi (t)z(t)} ≤ ε2i λ I 1 z T (t)Pi Pi z(t) + ε2i z T (t)H piT H pi z(t), (8.25)

H e{z T (t)Pi Ii ΔAdpi (t)z(t − ı(t))} ,

,

−1 λ I 2 z T (t)Pi Pi z(t) + ε3i z T (t − ı(t))(H pi )T H pi z(t − ı(t)). ≤ ε3i

(8.26)

From the designed parameter ϑ(t) in (8.11), there holds −H e{z T (t)Pi ΔCi ( f i (z(t), z(t − ı(t)), t) − ϑ(t))} ≤ −2ρ||g(t)|| ≤ 0.

(8.27)

Furthermore, Υ ℵ(z(t), i, t) Σ m Σ m ≤ h p (χ (t))H e{z T (t)Pi (A pi + Ci K pi )z(t)} +

h p (χ (t))z p=1 ε2i H piT H pi

p=1 −1 [ε1i Pi Z i Z iT

T

(t)

−1 T Pi + ε1i K pi V T V K pi + ε2i λ I 1 Pi Pi + Σ s + (ε3i )−1 λ I 2 Pi Pi + π¯ i j P j + D1i + D2i + ı D1 + ı D2 ]z(t) j=1 Σ m , , + z T (t − ı(t))[ h p (χ (t))ε3i (H pi )T H pi − (1 − μ)D1i ]z(t − ı(t)) p=1 Σ m + H e{z T (t)Pi h p (χ (t))Adpi z(t − ı(t))} − z T (t − ı)D2i z(t − ı).

×

p=1

(8.28) Defining ζ (t) = [z T (t) z T (t − ı(t)) z T (t − ı)]T , one has Υ ℵ(z(t), i, t) ≤ where

Σ m p=1

h p (χ (t))ζ T (t)Π ζ (t),

(8.29)

152

8 Fuzzy Sliding Mode Control

⎛

⎞ Ξ Pi Adpi 0 0 ⎠, Π = ⎝ ∗ Φi ∗ 0 −D2i −1 −1 T Ξ = H e{Pi (A pi + Ci K pi )} + ε1i Pi Z i Z iT Pi + ε1i K pi V T V K pi + ε2i λ I 1 Pi Pi Σ s −1 +ε2i H piT H pi + ε3i λ I 2 Pi Pi + D1i + D2i + ı D1 + ı D2 + π¯ i j P j , j=1

,

,

Φi = −(1 − μ)D1i + ε3i (H pi ) H pi . T

By applying Schur complement, we known that the inequality (8.12) is equivalent to Π < 0. Considering λ ≙ min {λmin (−Π )} > 0 gives rise to i∈N

Υ ℵ(z(t), i, t) ≤ −λ||ζ (t)||2 ≤ −λ||z(t)||2 .

(8.30)

By Dynkin’s formula, we obtain {{

t

E{ℵ(z(t), i, t)} − E{ℵ(z 0 , v0 , t0 )} ≤ −λE

} ||z(ω)|| dω|(z 0 , v0 )

(8.31)

} 1 ||z(ω)||2 dω|(z 0 , v0 ) ≤ E{V (z 0 , v0 , t0 )} < +∞. λ

(8.32)

2

0

which yields {{ lim E

t→+∞

0

t

Therefore, system (8.10) realizes robustly stochastic stability.

∎

Remark 8.1 For nonlinear semi-Markovian switching systems with time delay, the Lyapunov functional is usually chosen as ℵ(z(t), i, t) = z T (t)Pi z(t) + {t {0 {t T T t−ı(t) z (ω)D1 z(ω)dω + −ı t+θ z (ω)D2 z(ω)dωdθ whose parameters in integral term are mode-independent, which may lead to some conservativeness. Here, an appropriate mode-dependent Lyapunov functional (8.14) is constructed, and the parameters in integral term ℵ(z(t), i, t) are mode-dependent, which may reduce some conservativeness. Remark 8.2 Although the Lyapunov functional (8.14) is complex and the computational complexity is increased, sufficient conditions have been proposed to reduce the conservativeness of Theorem 8.1. Sufficient conditions are given for robustly stochastic stability of T-S fuzzy system (8.10). However, the parameter matrices K pi in (8.12) cannot be solvable in standard linear matrix inequalities. Next, we will design the controller gain matrices K pi in standard linear matrix inequalities.

8.4 Stochastic Stability Analysis

153

Theorem 8.2 If there exist positive-definite symmetric matrices X i , U1i , U2i , U1i j , U2i j , T1i , T2i ∈ R n×n , matrices Y pi ∈ R m×n , and positive scalars ε1i , ε2i , ε3i , such that ⎞ ⎛ Π1i Adpi X i Π2i Π3i Π4i ⎜ ∗ Π5i 0 0 Π6i ⎟ ⎟ ⎜ ⎜ ∗ 0 Π 0 ⎟ (8.33) 7i 0 ⎟ < 0, ⎜ ⎝ ∗ 0 0 Π8i 0 ⎠ ∗ ∗ 0 0 Π9i Σ s j=1

π¯ i j U1i j ≤ T1i ,

Σ s j=1

π¯ i j U2i j ≤ T2i ,

(8.34)

where Π1i =H e(A pi X i + Ci Y pi ) + π¯ ii X i + U1i + U2i + ı T1i + ı T2i , Π2i = [Z i I I ], √ √ √ √ T Π3i =[ π¯ i1 X i , ..., π¯ ii−1 X i , π¯ ii+1 X i , ..., π¯ is X i ], Π4i = [V Y pi X i H pi 0], ,

−1 Π5i =(μ − 1)U1i , Π6i = [0 0 X i (H pi )T ], Π7i = −diag{ε1i I, ε2i λ−1 I 1 I, ε3i λ I 2 I }, −1 −1 −1 I, ε2i I, ε3i I }, Π8i = − diag{X 1 , ..., X i−1 , X i+1 , ..., X s }, Π9i = −diag{ε1i

then system (8.10) realizes robustly stochastic stability with the controller gains computed by K pi = Y pi X i−1 . Proof Let X i = Pi−1 , Y pi = K pi X i , U1i = X i D1i X i , U2i = X i D2i X i , U1i j = X i D1 j X i , U2i j = X i D2 j X i , T1i = X i D1 X i , T2i = X i D2 X i . Pre-multiplying and post-multiplying (8.12) and (8.13) by diag{X i , X i , I, I, I }, we have ⎞ I I Ψˆ pi Adpi X i Vi ⎟ ⎜ ∗ 0 0 0 Φˆ i ⎟ ⎜ ⎟ < 0, ⎜ ∗ 0 0 0 −ε1i I ⎟ ⎜ −1 ⎠ ⎝ ∗ 0 0 −ε2i λ I I 0 −1 ∗ 0 0 0 −ε3i λ I I ⎛

Σ s j=1

π¯ i j U1i j ≤ T1i ,

Σ s j=1

π¯ i j U2i j ≤ T2i ,

where Ψˆ pi = H e(A pi X i + Ci Yi ) + ε1i Y piT V T V Y pi + ε2i X i H piT H pi X i Σ s +U1i + U2i + ı T1i + ı T2i + X i ( π¯ i j X −1 j )X i , j=1

, , Φˆ i = (μ − 1)U1i + ε3i (H pi X i )T H pi X i .

(8.35)

(8.36)

154

8 Fuzzy Sliding Mode Control

Let Π3i and Π8i be √ √ √ √ Π3i =[ π¯ i1 X i , ..., π¯ ii−1 X i , π¯ ii+1 X i , ..., π¯ is X i ], Π8i = − diag{X 1 , ..., X i−1 , X i+1 , ..., X s }. Σ s

Then, the term X i ( Xi

j=1

π¯ i j X −1 j )X i can be rewritten as

(Σ s j=1

) π¯ i j X −1 X i = π¯ ii X i − Π3i Π8i−1 Π3iT . j

By Schur complement, (8.35) and (8.36) is equivalent to (8.33) and (8.34).

∎

As discussed in most literature, when B pi ≡ Bqi ( p, q = 1, 2, ..., m) is satisfied, there is ΔCi = 0. Then, Corollary 8.1 is valid. Corollary 8.1 If there exist positive-definite symmetric matrices X i , U1i , U2i , U1i j , U2i j , T1i , T2i ∈ R n×n , matrices Y pi ∈ R m×n , and positive scalars ε1i , ε2i , such that ⎛

Π1i Adpi X i Π˜ 2i Π3i

⎜ ⎜ ∗ ⎜ ⎜ ⎜ ∗ ⎜ ⎜ ∗ ⎝ 0 Σ s

j=1

Π5i 0 0 ∗

0

⎞

⎟ , 0 X i (H pi )T ⎟ ⎟ ⎟ ˜ ⎟ < 0, Π4i 0 0 ⎟ ⎟ 0 0 Π8i ⎠ −1 0 0 −ε2i I 0

π¯ i j U1i j ≤ T1i ,

Σ s j=1

(8.37)

π¯ i j U2i j ≤ T2i ,

(8.38)

where , , −1 I }, Π˜ 2i = [I1i I2i X i H piT ], Π˜ 4i = −diag{ε1i I, ε2i I, ε1i ,

,

,

I1i = I − Ci (G i Ci )−1 G i E pi , I2i = I − Ci (G i Ci )−1 G i E pi , then system (8.10) realizes robustly stochastic stability with the controller gains computed by K pi = Y pi X i−1 .

8.5 Reachability Analysis As the last step of design procedure, a fuzzy integral sliding mode controller will be synthesized to drive the trajectories of system (8.7) onto the specified fuzzy sliding manifold.

8.5 Reachability Analysis

155

Theorem 8.3 Suppose the fuzzy integral sliding mode surface is given by (8.8) and K pi can be solved by Theorem 8.2. Then, the state trajectories can be driven onto the specified fuzzy sliding manifold s(t) = 0 in a finite time by the fuzzy sliding mode control law as u(t) =

Σ m p=1

h p (χ(t))K pi z(t) + (G i Ci )−1 G i ΔCi ϑ(t) − (G i Ci )−1 (ρ pi (t) + γ )sign(s(t)),

(8.39) where ρ pi (t) =

Σ m

h p (χ (t))||G i E pi ||||H pi z(t)|| + κ1 ||G i (Ci + ΔCi )||||z(t)|| p=1 Σ m , , + h p (χ (t))||G i E pi ||||H pi z(t − ı(t))|| p=1

+κ2 ||G i (Ci + ΔCi )||||z(t − ı(t))||,

(8.40)

and γ > 0 is a small constant. Proof Choose Lyapunov function as 1 T s (t)s(t). 2

U (t) =

(8.41)

According to (8.8), we can obtain s˙ (t) =

Σ m p=1

h p (χ (t))G i [Δ A pi z(t) + ΔAdpi (t)z(t − ı(t))

+ Ci (u(t) + f i (z(t), z(t − ı(t)), t)) − Ci K pi z(t) + ΔCi ( f i (z(t), z(t − ı(t)), t) − ϑ(t))].

(8.42)

Thus, Σ m

Υ U (t) = s T (t)˙s (t) ≤ −s T (t)G i (

p=1

Σ m

h p (χ (t))Ci K pi z(t) + ΔCi ϑ(t))

+ s T (t)G i Ci u(t) + |s(t)| h p (χ (t))||G i E pi ||||H pi z(t)|| p=1 Σ m , , + |s(t)| h p (χ (t))||G i E pi ||||H pi z(t − ı(t))|| p=1

+ |s(t)|||G i (Ci + ΔCi )||(κ1 ||z(t)|| + κ2 ||z(t − ı(t))||).

(8.43)

Substituting (8.39) into (8.42) yields √ 1 Υ U (t) ≤ −γ ||s(t)|| ≤ − 2γ U 2 (t) < 0,

f or s(t) /= 0,

(8.44)

156

8 Fuzzy Sliding Mode Control

which means that the reachability of the predefined switching surface can be guar∎ anteed in a finite time. Remark 8.3 As is well known, the sliding mode control is an effective robust control strategy because of its insensitivity to parameter variations, complete rejection of external disturbances, and order reduction. However, due to the complexity of nonlinear systems, it is difficult to study the sliding mode control for nonlinear systems. On the other hand, using T-S fuzzy method, nonlinear model can be expressed as local linear subsystem models. The T-S fuzzy model can approximate continuous nonlinear systems with arbitrary accuracy under sufficient fuzzy rules, and ensure the continuity of its output surface. Therefore, the T-S fuzzy model has received extensive attention in the modeling of nonlinear systems. The robot arm is complex nonlinear system, and there exist complex uncertainties in the parameters and loads, which makes it difficult to achieve ideal control performance using traditional control methods. In this article, sliding mode control and T-S fuzzy control are applied to the robot system, and the better control performance can be obtained. Similarly, when the input matrices are independent of plant-rule, Corollary 8.2 is valid.

Corollary 8.2 Suppose the integral sliding mode surface function is given by (8.8) and K pi can be solved by Corollary 8.1. Then, the state trajectories can be driven onto the specified fuzzy sliding manifold s(t) = 0 in a finite time by the fuzzy sliding mode control law as Σ m u(t) = h p (χ (t))K pi z(t) − (G i Ci )−1 (ρ pi (t) + γ )sign(s(t)), (8.45) p=1

where ρ pi (t) =

Σ m p=1

+

h p (χ (t))||G i E pi ||||H pi z(t)|| + κ1 ||G i Ci ||||z(t)||

Σ m p=1

,

,

h p (χ (t))||G i E pi ||||H pi z(t − ı(t))|| + κ2 ||G i Ci ||||z(t − ı(t))||,

and γ > 0 is a small constant.

8.6 Simulation Consider a single-link robot arm in [14], in which the parameters are given in (8.46), and z(t) = [z 1 (t) z 2 (t)]T indicates the state variables with the angle position of the armz 1 (t) and the angle position of the arm change rate z 2 (t). Now, the dynamic equation is given by

8.6 Simulation

157

Table 8.1 Parameters τ and δ Mode i 1 2 3

χ¨ (t) = −

Parameter τ

Parameter δ

1 5 10

1 5 10

δgβ φ(t) 1 sin(χ (t)) − χ(t) ˙ + u(t), τ τ τ

(8.46)

where u(t) is the control input, χ (t) denotes the angle position, and φ(t) is viscous friction. g, τ , β, δ represent the acceleration of gravity, the moment of inertia, the length of the arm, and the mass of the payload, respectively. Let φ(t) = φ0 = 2, β = 0.5, g = 9.81. The parameters τ and δ are listed in Table 8.1. The transition rate matrix is characterized by ⎤ ⎤ ⎡ −2h h h π11 (h) π12 (h) π13 (h) Λ(h) = ⎣π21 (h) π22 (h) π23 (h)⎦ = ⎣0.5h −h 0.5h ⎦ . 2 π31 (h) π32 (h) π33 (h) h 29 h − 49 h 9 ⎡

By considering the characters of Weibull distribution, the transition rate function as above can be called as an approximation that the sojourn time obeys the Weibull with the probability density function gi (h) = νιι hι−1 exp[−( hν )ι ], h ≥ 0. above analysis and Table 8.2, we can compute E{π12 (h)} = { ∞Based on the{ ∞ 2 −h2 hg (h)dh = dh = 0.8862s. Similarly, we have 1 0 0 2h exp ⎡

⎤ −1.7722 0.8862 0.8862 E{Λ(h)} = ⎣ 1.7726 −3.5452 1.7726 ⎦ . 2.6587 2.6587 −5.3174 Defining z 1 (t) = χ (t) and z 2 (t) = χ˙ (t), sin(z 1 (t)) is expresses as sin(z 1 (t)) = h 1 (z 1 (t))z 1 (t) + w h 2 (z 1 (t))z 1 (t), with w = 0.01/π, and h 1 (z 1 (t)) + h 2 (z 1 (t)) = 1 for h 1 (z 1 (t)), h 2 (z 1 (t)) ∈ [0, 1]. Therefore, by solving the above equations, we can get the following membership functions:

Table 8.2 Expression of gi (h) with the parameters ν and ι in mode i Mode i Parameter ν Parameter ι PDF gi (h) 1 2 3

1 2 3

2 2 2

g1 (h) = 2h exp(−h2 ) g2 (h) = 0.5h exp(−0.25h2 ) g3 (h) = 29 h exp(− 19 h2 )

158

8 Fuzzy Sliding Mode Control

h 1 (z 1 (t)) = h 2 (z 1 (t)) =

{ sin(z1 (t))−w z1 (t) z 1 (t)(1−w )

1,

{ z1 (t)−sin(z1 (t)) z 1 (t)(1−w )

1,

, z 1 (t) /= 0, z 1 (t) = 0,

, z 1 (t) /= 0, z 1 (t) = 0.

According to the membership functions, when z 1 (t) is about 0 rad, then h 1 (z 1 (t)) = 1, h 2 (z 1 (t)) = 0, and when z 1 (t) is about π rad or −π rad, then h 1 (z 1 (t)) = 0, h 2 (z 1 (t)) = 1. Thus, we have Plant Rule 1: IF z 1 (t) is about 0 rad, THEN z˙ (t) = (A1i + ΔA1i )z(t) + (Ad1i + ΔAd1i )z(t − ı(t)) +B1i (u(t) + f i (z(t), z(t − ı(t)), t)); Plant Rule 2: IF z 1 (t) is about π rad or −π rad, THEN z˙ (t) = (A2i + ΔA2i )z(t) + (Ad2i + ΔAd2i )z(t − ı(t)) +B2i (u(t) + f i (z(t), z(t − ı(t)), t)), where z(t) = [z 1T (t) z 2T (t)]T , [

[ ] [ ] ] 0 1 0.1 0.2 0 , B11 = B21 = , , Ad11 = −gβ −φ0 0.3 0.1 0.5 [ [ ] [ ] ] 0 1 0.1 0.2 0 = , B12 = B22 = , , Ad12 = −0.75gβ −0.5φ0 0.4 0.2 0.3 [ [ ] [ ] ] 0 1 0.1 0.3 0 = , B13 = B23 = , , Ad13 = −0.8gβ −0.4φ0 0.3 0.2 0.4 [ [ ] [ ] ] , 0 1 0.2 0.2 0 = , E p1 = E p1 = , , Ad21 = −w gβ −φ0 0.4 0.1 0.3 [ [ ] [ ] ] , 0 1 0.3 0.2 0.1 = , E p2 = E p2 = , , Ad22 = −0.75w gβ −0.5φ0 0.4 0.1 0 [ [ ] [ ] ] , 0 1 0.1 0.3 0.2 = , E p3 = E p3 = , , Ad23 = −0.8w gβ −0.4φ0 0.4 0.3 0.1 [ ] [ ] [ ] , , =H p1 = 0.1 0.1 , H p2 = H p2 = −0.1 0.2 , H p3 = H p3 = 0 0.1 ,

A11 = A12 A13 A21 A22 A23 ,

H p1

p ∈{1, 2}. Let f i (z(t), z(t − ı(t)), t) = 0.05 sin(z 1 (t)), i = 1, 2, 3, and ı(t) = 1.2(1 − sin(t)), then κ1 = 1.23, κ2 = 1.23, ı = 2.4, μ = 1.2. G i = CiT such that G i Ci is nonsingular. By solving (8.37) and (8.38), we can obtain

8.6 Simulation Fig. 8.1 System mode

159 3.5 3 2.5 2 1.5 1 0.5

0

5

10

15

10

15

Time(s) Fig. 8.2 State trajectory

0.5

0

-0.5

-1

0

5

[ ] [ ] K 11 = 0.0752 −0.0619 , K 21 = −0.0630 −0.0584 , [ ] [ ] K 12 = 0.0368 −0.0523 , K 22 = −0.0404 −0.0508 , [ ] [ ] K 13 = 0.0326 −0.0630 , K 23 = −0.0501 −0.0749 . According to the above feasible solutions for control gains, we can obtain the switching surface function (8.8) and the fuzzy sliding mode control law (8.45). The results of practical example simulation are shown in Figs. 8.1, 8.2, 8.3, 8.4 and 8.5. Figure 8.1 depicts the switching signal vt . Figure 8.2 demonstrates the state trajectory z(t). Figure 8.3 describes the response of switching surface s(t). Figures 8.4 and 8.5 depict the control input and the membership functions, respectively. Thus, the results show the availability of the proposed integral sliding mode control method.

160 Fig. 8.3 Switching surface

8 Fuzzy Sliding Mode Control 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5

Fig. 8.4 Control input

0

5

10

15

10

15

3 2 1 0 -1 -2

0

5

8.7 Conclusion This paper has proposed a fuzzy integral sliding mode control scheme for nonlinear semi-Markovian switching T-S fuzzy systems with delay. First, a fuzzy integral sliding mode surface function is constructed without assuming that the input matrices are independent of plant-rule, and sufficient conditions are given for robustly stochastic stability. Then, a fuzzy integral sliding mode control law is designed to drive the system trajectory onto the specified fuzzy sliding manifold in finite time. At last, an example has verified the effectiveness and superiorities of this methodology.

References Fig. 8.5 Membership functions

161 1.2 1 0.8 0.6 0.4 0.2 0 -0.2

0

5

10

15

One future work is to extend the results to sliding mode control for uncertain singular semi-Markovian switching T-S fuzzy systems with time delay or event-triggered mechanism.

References 1. T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst. Man Cybernet. SMC-15(1), 116–132 (1985) 2. Chang, X.H., Yang, G.H.: Nonfragile H∞ filtering of continuous-time fuzzy systems. IEEE Trans. Signal Process. 59(4), 1528–1538 (2011) 3. Chang, X.H., Yang, G.H.: Nonfragile H∞ filter design for T-S fuzzy systems in standard form. IEEE Trans. Ind. Electron. 61(7), 3448–3458 (2014) 4. Chang, X.H.: Robust nonfragile H∞ filtering of fuzzy systems with linear fractional parametric uncertainties. IEEE Trans. Fuzzy Syst. 20(6), 1001–1011 (2012) 5. Ge, C., Shi, Y.P., Park, J.H., Hua, C.C.: Robust H∞ stabilization for T-S fuzzy systems with time-varying delays and memory sampled-data control. Appl. Math. Comput. 346, 500–512 (2019) 6. Zhang, H., Wang, J.M.: State estimation of discrete-time Takagi-Sugeno fuzzy systems in a network environment. IEEE Trans. Cybernet. 45(8), 1525–1536 (2015) 7. Xie, X.P., Yue, D., Zhang, H.G., Xue, Y.S.: Control synthesis of discrete-time T-S fuzzy systems via a multi-instant homogenous polynomial approach. IEEE Trans. Cybernet. 46(3), 630–640 (2016) 8. Li, F.B., Shi, P., Lim, C.C., Wu, L.G.: Fault detection filtering for nonhomogeneous Markovian jump systems via a fuzzy approach. IEEE Trans. Fuzzy Syst. 26(1), 131–141 (2018) 9. Syed Ali, M., Vadivel, R., Saravanakumar, R.: Design of robust reliable control for T-S fuzzy Markovian jumping delayed neutral type neural networks with probabilistic actuator faults and leakage delays: an event-triggered communication scheme. ISA Trans. 77, 30–48 (2018) 10. Choi, H.H.: Robust stabilization of uncertain fuzzy systems using variable structure system approach. IEEE Trans. Fuzzy Syst. 16(3), 715–724 (2008) 11. Wang, Y., Xie, L., Souz, C.E.D.: Robust control of a class of uncertain nonlinear systems. Syst. Control Lett. 19(2), 139–149 (1992)

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12. Boukas, E.K.: Stochastic Switching Systems: Analysis and Design. Birkhauser, ¨ Boston (2005) 13. Huang, J., Shi, Y.: Stochastic stability and robust stabilization of semi-Markov jump linear systems. Int. J. Robust Nonlinear Control 23(18), 2028–2043 (2013) 14. Wu, H.N., Cai, K.Y.: Mode-independent robust stabilization for uncertain Markovian jump nonlinear systems via fuzzy control. IEEE Trans. Syst. Man Cybernet.: Syst. 36(3), 509–519 (2005)

Chapter 9

Fuzzy H∞ Sliding Mode Control Under Phase-Type Distribution

The issue of sliding mode control for phase-type stochastic nonlinear semi-Markovian switching systems (S-MSSs) via the T-S fuzzy strategy is investigated. One unrealistic assumption, that is, the sojourn time in stochastic switching systems follows an exponential distribution, is removed by S-MSSs model, which is one of the main features compared to the existing literature. First, by using the plant transformation method and the supplementary variable technique, phase-type S-MSSs are equivalent to associated Markovian switching systems (MSSs). Then, an integral sliding surface function is established to cope with the influence of the switching phenomenon in the plant. By using Lyapunov functions and inequality optimization problems, sufficient conditions are provided for stochastic stability of the system with a prescribed H∞ performance index. In addition, a fuzzy sliding mode control law is synthesized to guarantee that the associated T-S fuzzy MSSs fulfill the reaching condition in bounded time. Finally, the validity of the conclusions is verified by the single-link robot arm model.

9.1 Introduction In the previous chapter, it is investigated based on the traditional semi-Markovian process. In the late 1970s, the phase-type distribution was established for its matrix geometric analytic system [1]. In 1986, the phase-type distribution was applied to communication systems [2]. So far, the phase-type distribution has been used in various performance analysis papers. In fact, for the phase-type distribution based on the finite state Markovian process, a unified mathematical expression is introduced by using the matrix representation method, which makes the model analysis more general. In addition, the matrix expression of phase-type distribution is easy © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 W. Qi and G. Zong, Control Synthesis for Semi-Markovian Switching Systems, Studies in Systems, Decision and Control 465, https://doi.org/10.1007/978-981-99-0317-7_9

163

164

9 Fuzzy H∞ Sliding Mode Control Under Phase-Type Distribution

for numerical analysis and calculation. The typical phase-type distribution includes Erlang distribution, hyper-exponential distribution, and so on. In this chapter, the fuzzy H∞ sliding mode control for phase-type stochastic nonlinear S-MSSs will be addressed. The main contributions are listed as follows: (i) Compared with traditional S-MSSs [3–13] subject to the Weibull distribution, how to deal with S-MSSs subject to phase-type? By using the plant transformation method and the supplementary variable technique, the finite phase-type semi-Markovian process is transformed into its correlative Markovian process. Then, the phase-type S-MSSs are equivalently transformed into their associated MSSs. (ii) How to avoid the defects of linear sliding surface [14, 15]? On the basis of the T-S fuzzy theory, an integral sliding surface function is constructed to cope with the influence of the switching phenomenon in the plant. The integral sliding mode surface can improve the robustness of the control system and avoid the shortcomings of the traditional linear sliding mode surface. (iii) In comparison with MSSs, how to design a fuzzy sliding mode control law to ensure the finite time attractiveness of the sliding surface? Combining with the fuzzy rules, an appropriate sliding mode control law is designed to be dependent of the fuzzy rules. In addition, the connections among integral sliding mode surfaces are reflected in sliding mode control law via the controller gain. Hence, under the effect of external disturbance, the fuzzy sliding mode control law can guarantee the finite-time attractiveness and improve the system performance.

9.2 Problem Statements and Preliminaries Consider a kind of T-S fuzzy S-MSSs: Plant Rule m: IF θ1 (t) is μm1 , θ2 (t) is μm2 , and . . . and θ p (t) is μmq , THEN ˆ t )(u(t) + g(z(t), λt , t)) + Dˆ m (λt )v(t)]dt dz(t) = [( Aˆ m (λt ) + Δ Aˆ m (λt , t))z(t) + B(λ + Cˆ m (λt )z(t)dω(t), y(t) = Mˆ m (λt )z(t) + Nˆ m (λt )v(t), z(0) = ϕ(0),

(9.1)

where μmn (m = 1, 2, . . . , p; n = 1, 2, . . . , q) are the fuzzy sets; z(t), u(t), v(t), and y(t) denote the state vector, the input vector, the external disturbance, and the output vector; ω(t) is the Brownian motion; g(z(t), λt , t) is the unknown nonlinear ˆ t ), Cˆ m (λt ), Dˆ m (λt ), Mˆ m (λt ), and Nˆ m (λt ) are the system perturbation; Aˆ m (λt ), B(λ ˆ matrices; Δ Am (λt , t) is the parameter uncertainty matrix. Consider the stochastic process {λt , t ≥ 0} on the state space {1, 2, . . . , l + 1}, where 1,(2, . . . , l are ) transient, and l + 1 is absorbing. The infinitesimal generator Γ Γ0 , where Γ = (Γi j )l×l satisfies Γii < 0, Γi j ≥ 0, i /= j and the is M = 01×l 0 vector Γ 0 = [Γ10 , Γ20 , . . . , Γm0 ] has non-negative entries such that Γ e + Γ 0 = 0,

9.2 Problem Statements and Preliminaries

165

where the vector e has all components equal to one. The vector of initial probabilities is (b, bl+1 ), where b = (b1 , b2 , . . . , bl ) satisfies be + bl+1 = 1. Lemma 9.1 ([16]) The probability distribution J (·) of the time until absorption in the state l + 1, corresponding to the initial probability vector (b, bl+1 ), is given by J (t) = 1 − bexp(Γ t), for t > 0. Definition 9.1 ([16]) If the probability distribution function is a distribution function of the arrival absorbing state on the state space {1, 2, . . . , l + 1}, it is called the phase-type probability distribution on [0, ∞] represented by (b, Γ ). Remark 9.1 The phase-type distribution is described by (b, Γ ). In other words, as long as the combination of (b, Γ ) is given, the phase-type distribution is uniquely determined. Therefore, (b, Γ ) is called the characteristic matrix representation of phase-type-distribution. According to the fuzzy blending, the defuzzified model of the system (9.1) can be written as dz(t) =

∑p m=1

ˆ t )(u(t) + g(z(t), λt , t)) h m (θ (t)){[( Aˆ m (λt ) + Δ Aˆ m (λt , t))z(t) + B(λ

+ Dˆ m (λt )v(t)]dt + Cˆ m (λt )z(t)dω(t)}, ∑p y(t) = h m (θ (t))[ Mˆ m (λt )z(t) + Nˆ m (λt )v(t)], m=1

z(0) = ϕ(0),

(9.2)

q ∑p Π μmn (θn (t)) where h m (θ (t)) = ∑ p n=1 ≥ 0, q m=1 h m (θ (t)) = 1, and μmn (θn (t)) m=1 Πn=1 μmn (θn (t)) denotes the grade of membership of θn (t) in μmn .

Definition 9.2 ([16]) Let H be a finite set, if the following conditions hold: (1) The sample paths of λt which have left-hand limits with probability one are right-continuous step functions; (2) ı s denotes the sth jump point of the process λt , where 0 = ı 0 < ı 1 < ı 2 < · · · < ı s < · · · , all ı s (s = 1, 2, 3, . . .) are Markovian of the process λt ; (3) Ji j (t) . P(ı s+1 − ı s ≤ t|λıs = i, λıs+1 = j ) = Ji (t), i, j ∈ H , t ≥ 0 are independent of j and s; (4) Ji (t), i ∈ H is a phase-type distribution, then λt is called a phase-type semiMarkovian process on state space H . Let (b(i) , Γ (i ) ), i ∈ H represent the l(i) order representation of Ji (t), and H (i) be the whole transient states set, where b(i ) . (b1(i ) , b2(i) , . . . , bl(i)(i ) ), Γ (i) . (Γ jv(i) , j, v ∈ H (i) ).

166

9 Fuzzy H∞ Sliding Mode Control Under Phase-Type Distribution

Also, let pi j . Pr (λs+1 = j|λs = i ), i, j ∈ H , P . ( pi j ), i, j ∈ H , (b, Γ ) . (b(i) , Γ (i) ), i ∈ H . Obviously, the probability distribution of λt is determined only by {P, (b, Γ )}. For every s(s = 1, 2, . . .), ı s ≤ t ≤ ı s+1 , define E(t) . the phase of Jςˆ (t) (·) at time t − ı s . In addition, for any i ∈ H , define Γ j(i,0) . −

∑li

Γ jv(i) , j = 1, 2, . . . , l(i) ,

(9.3)

Q . {(i, v(i) )|i ∈ H, v(i) = 1, 2, . . . , l(i) }.

(9.4)

v=1

Lemma 9.2 ([16]) Z (t) = (λt , E(t)) is the Markovian chain of state-space Q. M = (m μν , μ, ν ∈ Q) is the infinitesimal generator of Z (t) which is decided only by the pair of (λt , E(t)) given by {P, (b, Γ )} as ⎧ (i ) ⎪ (i, v(i) ) ∈ Q, ⎨ m (i,v(i) )(i,v(i) ) = Γv(i ) vi , (i ) m (i,v(i) )(i,v¯ (i) ) = Γv(i ) v¯ i , v(i) /= v¯ (i) , (i, v(i) ) ∈ Q, (i, v¯ (i) ) ∈ Q, ⎪ (i,0) ( j ) ⎩m (i) ¯ ( j) ) ∈ Q. (i,v (i) )( j,v¯ ( j) ) = pi j Tv (i ) bv( j ) , i / = j, (i, v ) ∈ Q, ( j, v ∑ As shown in (9.4), there are N = i∈H l(i) elements in the state space Q of Z (t). ∑i−1 (r ) Define Φ(i, v) = r =1 l + v, i ∈ H , 1 ≤ v ≤ l(i) and ςt = Φ(Z (t)), λΦ(i,v)Φ(i ' ,v' ) = m Φ(i,v)Φ(i ' ,v' ) . Hence, ςt is an associated Markovian process of λt with the state space N = {1, 2, . . . , N }. The infinitesimal generator of ςt is Λ = (λi j ), 1 ≤ i, j ≤ N , so that Pr {ςt+γ = j|ςt = i} = Pr {Φ(Z (t) + γ ) = j|Φ(Z (t)) = i} { πi j γ + o(γ ), i /= j, = 1 + πi j γ + o(γ ), i = j, where πi j (i /= j) denotes the transition rate from i to j when i /= j; πii = ∑ − Nj=1, j/=i πi j , γ > 0 and limγ →0 (o(γ )/γ ) = 0. According to Lemma 9.1, every finite phase semi-Markovian process can be transformed into its associated Markovian process. Then, the system (9.2) is equivalent to the associated MSSs dz(t) =

∑p m=1

h m (θ (t)){[(Am (ςt ) + ΔAm (ςt , t))z(t) + B(ςt )(u(t) + g(z(t), ςt , t))

+ Dm (ςt )v(t)]dt + Cm (ςt )z(t)dω(t)}, ∑p y(t) = h m (θ (t))[Mm (ςt )z(t) + Nm (ςt )v(t)], m=1

z(0) = ϕ(0),

(9.5)

where ςt is the associated Markovian process of λt . ΔAm (ςt , t) is norm-bounded and satisfies Δ Am (ςt , t) = E m (ςt )Fm (ςt , t) Hm (ςt ), where E m (ςt ) and Hm (ςt ) are

9.3 Sliding Mode Control Law Design

167

known constant matrices, Fm (ςt , t) satisfies FmT (ςt , t)Fm (ςt , t) ≤ I . g(z(t), ςt , t) is unknown nonlinear perturbation and satisfies ∥g(z(t), ςt , t)∥ ≤ κ∥z(t)∥,

(9.6)

where κ is known nonnegative real scalar. The external disturbance v(t) satisfies ℵ . {v(·) : ∥v(t)∥ ≤ h¯ , t ∈ [0, ∞)},

(9.7)

with known constant h¯ > 0. When ςt = i, i ∈ N , Am (ςt ), ΔAm (ςt , t), E m (ςt ), Hm (ςt ), Fm (ςt , t), B(ςt ), Cm (ςt ), Dm (ςt ), Mm (ςt ), Nm (ςt ), and g(z(t), ςt , t) are denoted as Ami , ΔAmi (t), E mi , Hmi , Fmi (t), Bi , Cmi , Dmi , Mmi , Nmi , and gi (z(t), t), respectively. Definition 9.3 ([8]) If for arbitrary initial condition z 0 and ς0 , there holds ∮ t lim E{ ∥z(ω)∥2 dω|(z 0 , ς0 )} < +∞,

t→+∞

0

then system (9.5) (v(t) = 0) is said to be robustly stochastically stable. Definition 9.4 ([8]) For γ > 0, system (9.5) is said to be robustly stochastically stable with a prescribed H∞ performance index, if system (9.5) is robustly stochastically stable} when v(t) {∮ ∞= 0, and for} zero-initial condition, there holds {∮ ∞ E 0 y T (t)y(t)dt ≤ γ 2 E 0 v T (t)v(t)dt , when v(t) /= 0.

9.3 Sliding Mode Control Law Design In [18], Chern and Wu proposed integral sliding mode control that could significantly enhance the robustness against the external disturbance and the parameter uncertainty. The integral sliding mode surface can make the initial state of the system on the sliding mode surface at the beginning by reasonably setting the initial state of the integrator, so as to eliminate the arrival stage and avoid the shortcomings of the approach stage of the traditional linear sliding mode surface. The sliding surface function is constructed s(t) = G i z(t) −

∮ t∑ p 0

m=1

∑p n=1

h m (θ (s))h n (θ (s))G i (Ami + Bi K ni )z(s)ds, (9.8)

where Bi satisfies full column rank, G i satisfies that G i Bi is nonsingular and G i Cm = 0, and K ni is the real matrix which will be constructed subsequently.

168

9 Fuzzy H∞ Sliding Mode Control Under Phase-Type Distribution

Fig. 9.1 Feedback system structure

According to Eq. (9.5), one has s(t) = G i ϕ(0) +

∮ t∑ p

∑p

h m (θ (s))h n (θ (s))G i [(ΔAmi (s) − Bi K ni )z(s) ∮ t∑ ∑p p + Bi (u(s) + gi (z(s), s)) + Dmi v(s)]ds + h m (θ (s))h n (θ (s)) m=1

0

n=1

0

m=1

n=1

G i Cmi z(s)dω(s).

(9.9)

For G i Cmi = 0, the sliding surface function (9.9) can be reduced to s(t) = G i ϕ(0) +

∮ t∑ p m=1

0

∑p n=1

h m (θ (s))h n (θ (s))G i [(ΔAmi (s) − Bi K ni )z(s)

+ Bi (u(s) + gi (z(s), s)) + Dmi v(s)]ds.

(9.10)

Consider the fuzzy integral sliding surface (9.8), where G i = BiT Pi . Next, we will construct an appropriate sliding mode control law to drive the state trajectory of the system (9.5) onto the specific sliding surface in a limited time. Fig. 9.1 represents the feedback system structure. Theorem 9.1 For system (9.5), the sliding mode control law u(t) can be designed as ∑p ∑p h m (θ (t))h n (θ (t))[K ni z(t) − (μ + ρmi (t))sgn(BiT G iT s(t))], u(t) = m=1

n=1

(9.11) where μ > 0 is a small constant and ρmi (t) = h¯ ∥(G i Bi )−1 G i Dmi ∥ + ∥(G i Bi )−1 G i E mi ∥∥Hmi ∥∥z(t)∥ + κ∥z(t)∥.

9.3 Sliding Mode Control Law Design

169

Then, the state trajectories of the closed-loop system can be driven onto the prespecified fuzzy sliding surface s(t) = 0 in a limited time. Proof The Lyapunov function is considered as U (t) =

1 T s (t)s(t). 2

(9.12)

From (9.10), we obtain s˙ (t) =

∑p m=1

∑p n=1

h m (θ (t))h n (θ (t))G i [(ΔAmi − Bi K ni )z(t)

+ Bi (u(t) + gi (z(t), t)) + Dmi v(t)].

(9.13)

Thus, L U (t) = s T (t)˙s (t) ∑p ∑p = h m (θ (t))h n (θ (t))s T (t)G i Bi [(G i Bi )−1 G i ΔAmi z(t) m=1

n=1

− (μ + ρmi (t))sgn(BiT G iT s(t)) + gi (z(t), t) + (G i Bi )−1 G i Dmi v(t)] ∑p ∑p ≤ h m (θ (t))h n (θ (t))∥BiT G iT s(t)∥[∥(G i Bi )−1 G i E mi ∥∥Hmi ∥∥z(t)∥ m=1

n=1

− (μ + ρmi (t))sgn(BiT G iT s(t)) + κ1 ∥z(t)∥ + h¯ ∥(G i Bi )−1 G i Dmi ∥] 1

¯ U 2 (t) < 0, ≤ − μ∥BiT G iT s(t)∥ ≤ −μ √ √ where μ ¯ = 2 mini∈N { λmin (G i Bi BiT G iT )} > 0, which means that the controller ∎ (11) can ensure the reachability of the sliding surface. Remark 9.2 For the sliding mode control law design (9.11), several parameters are required to follow the tuning guideline. According to the constraint condition (9.6), the parameter κ is known as nonnegative real scalar. In addition, the small parameter μ only satisfies μ > 0. Remark 9.3 The chattering problem in sliding mode control is caused by discontinuous high-frequency switching control. The chattering phenomenon will affect the control accuracy of sliding mode control and destroy the system performance. In order to suppress the chattering phenomenon in sliding mode control, sgn{BiT G iT s(t)} is B T G T s(t)

i replaced by continuous ∥B T G Ti s(t)∥+0.001 . Besides that, the common approaches to i i suppress chattering include high-order sliding mode control, filter design, boundary layer design, etc.

170

9 Fuzzy H∞ Sliding Mode Control Under Phase-Type Distribution

9.4 Stochastic Stability with H∞ Performance Index On the basis of the sliding mode control theory, the equivalent controller is constructed from s˙ (t) = 0 that u eq (t) = − (G i Bi )−1 G i

∑p

∑p

m=1

n=1

h m (θ (t))h n (θ (t))[(ΔAmi (t) − Bi K ni )z(t)

+ Dmi v(t)] − gi (z(t), t).

(9.14)

Thus, substituting (9.14) into (9.5) yields dz(t) =

∑p m=1

∑p n=1 −1

h m (θ (t))h n (θ (t)){[(Ami + Bi K ni + ΔAmi (t)

¯ mi v(t)]dt + Cmi z(t)dω(t)}, − Bi (G i Bi ) G i ΔAmi (t))z(t) + D ∑p y(t) = h m (θ (t))[Mmi z(t) + Nmi v(t)], m=1

z(0) = ϕ(0),

(9.15)

¯ mi = Dmi − Bi (G i Bi )−1 G i Dmi . where D Next, by analyzing system (9.15) with v(t) = 0, we give sufficient conditions for robustly stochastic stability of the closed-loop system (9.15). Theorem 9.2 If there exist positive-definite symmetric matrix Pi , and positive scalars ε1i , ε2i for all i ∈ N , m, n = 1, 2, . . . , p, such that Σmmi < 0, Σmni + Σnmi < 0, m < n,

(9.16) (9.17)

BiT Pi Cmi = 0,

(9.18)

where ⎛

Σmni

⎞ Π1mni Π2 0 = ⎝ ∗ Π 3 Π4 ⎠ , ∗ ∗ −ε2i I

T T Π1mni = H e{Pi (Ami + Bi K ni )} + ε1i Hmi Hmi + ε2i Hmi Hmi +

∑N

T Π2 = [0, Cmi Pi , Pi Bi , Pi E mi ],

Π3 = diag{−Pi , −Pi , −BiT Pi Bi , −ε1i I }, Π4 = Pi E mi , then system (9.15) with v(t) = 0 realizes robustly stochastic stability.

j=1

πi j P j ,

9.4 Stochastic Stability with H∞ Performance Index

171

Proof Choose Lyapunov function V (z(t), i ) = z T (t)Pi z(t).

(9.19)

For system (9.15) with (v(t) = 0), one has L V (z(t), i ) ∑p ∑p =

h m (θ (t))h n (θ (t))z T (t){H e[Pi (Ami + ΔAmi (t)) + Bi K ni ∑N T − Bi (G i Bi )−1 G i ΔAmi (t)] + πi j P j + Cmi Pi Cmi }z(t). (9.20) m=1

n=1

j=1

For the uncertainties in (9.20), we have H e{z T (t)Pi ΔAmi (t)z(t)} −1 T T T z (t)Pi E mi E mi Pi z(t) + ε1i z T (t)Hmi Hmi z(t), ≤ε1i

− z T (t)H e{Pi Bi (G i Bi )−1 G i ΔAmi (t)}z(t) T ≤z T (t)Pi Bi (BiT Pi Bi )−1 BiT Pi z(t) + z T (t)Δ Ami (t)Pi ΔAmi (t)z(t).

(9.21)

Furthermore,

≤

L V (z(t), i ) ∑p ∑p m=1

−1 T h m (θ (t))h n (θ (t))z T (t){H e[Pi (Ami + Bi K ni )] + ε1i Pi E mi E mi Pi ∑N T + Pi Bi (BiT Pi Bi )−1 BiT Pi + Δ Ami (t)Pi Δ Ami (t) + πi j P j

n=1

T + ε1i Hmi Hmi

j=1

T + Cmi Pi Cmi }z(t).

(9.22)

By applying Schur complement, one has L V (z(t), i ) ≤

∑p m=1

∑p n=1

h m (θ (t))h n (θ (t))z T (t)Π z(t),

(9.23)

where ⎛

⎞ T T Ξ1 Δ Ami (t)Pi Cmi Pi Pi Bi Pi E mi ⎜ ∗ −Pi 0 0 0 ⎟ ⎜ ⎟ ⎜ 0 0 ⎟ ∗ −Pi Π =⎜ ∗ ⎟, ⎝ ∗ ∗ ∗ −BiT Pi Bi 0 ⎠ ∗ ∗ ∗ ∗ −ε1i I ∑N T Ξ1 = H e{Pi ( Ami + Bi K ni )} + ε1i Hmi Hmi + πi j P j . j=1

Equations (9.16)–(9.18) means that Π < 0. Then, Π < 0 can be rewritten as

172

⎛

Ξ1 ⎜ ∗ ⎜ ⎜ ∗ ⎜ ⎝ ∗ ∗

9 Fuzzy H∞ Sliding Mode Control Under Phase-Type Distribution

⎞ T 0 Cmi Pi Pi Bi Pi E mi −Pi 0 0 0 ⎟ ⎟ T T T 0 0 ⎟ ∗ −Pi ⎟ + Ω Fmi (t)Σ + Σ Fmi (t)Ω < 0, (9.24) T ⎠ ∗ ∗ −Bi Pi Bi 0 ∗ ∗ ∗ −ε1i I

T where Ω = [Hmi , 0, 0, 0, 0]T , Σ = [0, E mi Pi , 0, 0, 0]. By using Schur’s complement, (9.24) can be converted into the following inequality for ε2i > 0,

⎛

⎞ T Pi Pi Bi Pi E mi 0 Π1mni 0 Cmi ⎜ ∗ −Pi 0 0 0 Pi E mi ⎟ ⎜ ⎟ ⎜ ∗ ∗ −Pi 0 0 0 ⎟ ⎜ ⎟ < 0, ⎜ ∗ 0 ⎟ ∗ ∗ −BiT Pi Bi 0 ⎜ ⎟ ⎝ ∗ ∗ ∗ ∗ −ε1i I 0 ⎠ ∗ ∗ ∗ ∗ ∗ −ε2i I

(9.25)

∑ T T Hmi + ε2i Hmi Hmi + Nj=1 πi j P j . where Π1mni = H e{Pi (Ami + Bi K ni )} + ε1i Hmi Equation (9.25) is implied according to (9.16)–(9.17). In addition, (9.23) can be converted into L V (z(t), i ) ∑n−1 ∑ p

≤

+

m=1 ∑p

n=m+1

m=1

h m (θ (t))h n (θ (t))z T (t)(Σmni + Σmni )z(t)

h 2m (θ (t))z T (t)Σmmi z(t).

(9.26)

Define λ . min {λmin (−Π )} > 0. Then, one has i∈N

L V (z(t), i ) ≤ −λ∥z(t)∥2 .

(9.27)

By Dynkin’s formula, one has ∮ t E{V (z(t), i )} − E{V (z 0 , ς0 )} ≤ −λE{ ∥z(s)∥2 ds|(z 0 , ς0 )}, 0

which means that ∮ lim E{

t→+∞

0

t

∥z(s)∥2 ds|(z 0 , ς0 )} ≤

1 E{V (z 0 , ς0 )} < +∞. λ

Therefore, system (9.15) with (v(t) = 0) is robustly stochastically stable.

(9.28) ∎

9.4 Stochastic Stability with H∞ Performance Index

173

Remark 9.4 In this chapter, fuzzy-independent Lyapunov function is adopted to study the robustly stochastic stability for the corresponding sliding dynamics (9.15), which is convenient to calculate the derivative of Lyapunov function. How to design fuzzy-dependent Lyapunov function to reduce some conservatism is an important topic for future work. In Theorem 9.1, we choose G i = BiT Pi , and combine with (9.10). Then, the sliding function s(t) depends upon a group of given matrix Pi , i ∈ N . This means that the semi-Markovian which switches from one mode to another can be reflected in s(t) by Pi , i ∈ N . Next, by analyzing system (9.15), we give sufficient conditions for robustly stochastic stability of closed-loop system (9.15) with a prescribed H∞ performance index. Theorem 9.3 If there exist positive-definite symmetric matrix Pi , and positive scalars γ , ε1i , ε2i for all i ∈ N , m, n = 1, 2, . . . , p, such that (9.18) and Σ1mmi < 0,

(9.29)

Σ1mni + Σ1nmi < 0, m < n,

(9.30)

where ⎛

Σ1mni

Π1mni

⎞ Π1mni Π2 0 = ⎝ ∗ Π 3 Π4 ⎠ , ∗ ∗ −ε2i I ⎞ ⎛ T ¯ Pi Bi 0 Π 1mni Pi Dmi Mmi T ⎜ ∗ −γ 2 I N T 0 Dmi Pi ⎟ mi ⎟ ⎜ ⎜ =⎜ ∗ ∗ −I 0 0 ⎟ ⎟, ⎝ ∗ ∗ ∗ −BiT Pi Bi 0 ⎠ ∗ ∗ ∗ ∗ −Pi

T T ¯ 1mni = H e{Pi (Ami + Bi K ni )} + ε1i Hmi Π Hmi + ε2i Hmi Hmi +

∑N j=1

πi j P j ,

T Π2 = [0, Cmi Pi , Pi Bi , Pi E mi ],

Π3 = diag{−Pi , −Pi , −BiT Pi Bi , −ε1i I }, Π4 = Pi E mi , then system (9.15) realizes robustly stochastic stability with a prescribed H∞ performance index. Proof From (9.29) and (9.30), we can get that (9.16) and (9.17) hold, which means that system (9.15) with v(t) = 0 realizes robustly stochastic stability. Next, when v(t) /= 0, one has

174

9 Fuzzy H∞ Sliding Mode Control Under Phase-Type Distribution

L V (z(t), i ) + y T (t)y(t) − γ 2 v T (t)v(t) ∑p ∑p −1 T ≤ h m (θ (t))h n (θ (t)){z T (t){H e[Pi (Ami + Bi K ni )] + ε1i Pi E mi E mi Pi m=1 n=1 ∑N T T + ε1i Hmi Hmi + Pi Bi (BiT Pi Bi )−1 BiT Pi + ΔAmi (t)Pi Δ Ami (t) + πi j P j j=1

T T T ¯ mi )v(t) + z T (t)H e(Mmi + Cmi Pi Cmi + Mmi Mmi }z(t) + z T (t)H e(Pi D Nmi )v(t) T + v T (t)Nmi Nmi v(t) − γ 2 v T (t)v(t)}.

(9.31)

For the term −z T (t)H e(Pi Bi (G i Bi )−1 G i Dmi )v(t), we have − z T (t)H e(Pi Bi (G i Bi )−1 G i Dmi )v(t) T ≤z T (t)Pi Bi (BiT Pi Bi )−1 BiT Pi z(t) + v T (t)Dmi Pi Dmi v(t),

(9.32)

where G i = BiT Pi . Similar to the proof of Theorem 9.1, it follows from (9.29) and (9.30) that L V (z(t), i ) + y T (t)y(t) − γ 2 v T (t)v(t) < 0,

(9.33)

which means that ∮ E[ 0

∞

∮ z T (t)z(t)dt] ≤γ 2 E[

∞

v T (t)v(t)dt].

0

∎ Theorem 9.3 gives sufficient conditions for robustly stochastic stability of the closed-loop system (9.15) with a prescribed H∞ performance index. However, the controller gain K ni cannot be solved in standard linear matrix inequality. Next, we will design parameters K ni in Theorem 9.4. Theorem 9.4 System (9.15) is robustly stochastically stable with a prescribed H∞ performance index, if there exist positive-definite symmetric matrices Pi , X i , X¯ i , matrix Yni , and positive scalars γ , ε1i , ε2i , β1 for ∀i ∈ N , m, n = 1, 2, . . . , p, such that ⎛ ⎞ Π5 Π6 0 ⎝ ∗ Π7 Π8 ⎠ < 0, (9.34) 0 ∗ −ε2i I ⎛ ⎞ 0 Π9 Π10 ⎝ ∗ Π11 Π12 ⎠ < 0, m < n, (9.35) 0 ∗ −2ε2i I

9.4 Stochastic Stability with H∞ Performance Index

(

−β1 I X¯ i − I ∗ −I

175

) ≤ 0,

BiT Pi Cmi = 0,

(9.36) (9.37)

where ⎛

¯ 5 Dmi X i M T Bi 0 Π mi T ⎜ ∗ −γ 2 I N T 0 Dmi mi ⎜ Π5 = ⎜ ∗ ∗ −I 0 0 ⎜ ⎝ ∗ ∗ ∗ −BiT Pi Bi 0 ∗ ∗ ∗ ∗ −X i

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

T T T Π¯5 =Ami X i + Bi Ymi + X i Ami + Ymi Bi + πii X i , T T T , Bi , E mi , X i Hmi , X i Hmi , Π˜ 1 ], Π6 = [0, X i Cmi −1 −1 Π7 = diag{−X i , −X i , −BiT Pi Bi , −ε1i I, −ε1i I, −ε2i I, Π˜ 2 }, Π8 = E mi , ⎞ ⎛ ¯ 9 Π 12 Π913 2Bi 0 Π 9 ⎜ ∗ −2γ 2 I N T + N T 0 Π915 ⎟ mi ni ⎟ ⎜ Π9 = ⎜ ∗ −2I 0 0 ⎟ ⎟, ⎜ ∗ ⎝ ∗ ∗ ∗ −2BiT Pi Bi 0 ⎠ ∗ ∗ ∗ ∗ −2X i T T T T ¯ 9 = Ami X i + Bi Yni + X i Ami Π + YniT BiT + Ani X i + Bi Ymi + X i Ani + Ymi Bi + 2πii X i , T T + X i Mni , Π912 = Dmi + Dni , Π913 = X i Mmi T T Π915 = Dmi + Dni , T T T T + X i Cni , 2Bi , E mi + E ni , X i Hmi + X i HniT , X i Hmi + X i HniT , Π˜ 1 ], Π10 = [0, X i Cmi 1 −1 −1 I, −2ε2i I, Π˜ 2 }, Π11 = diag{−2X i , −2X i , −2BiT Pi Bi , −2ε1i I, −2ε1i 2 Π12 = E mi + E ni , √ √ √ √ Π˜ 1 = [ πi1 X i , . . . , πii−1 X i , πii+1 X i , . . . , πi N X i ],

Π˜ 2 = − diag{X 1 , . . . , X i−1 , X i+1 , . . . , X N }.

Moreover, the controller gain is given as K ni = Yni X i−1 . Proof Let X i = Pi−1 , Yni = K ni X i , X¯ i = X i Pi . By using diag{X i , I, I, I, X i , X i , X i , I, I, I, I } to pre-multiply and post-multiply (9.29), we can obtain ⎛

⎞ T Bi E mi 0 Ξ1 0 X i Cmi ⎜ 0 −X i 0 0 0 E mi ⎟ ⎜ ⎟ ⎜ ∗ 0 −X i 0 0 0 ⎟ ⎜ ⎟ < 0, ⎜ ∗ 0 0 ⎟ 0 −BiT Pi Bi 0 ⎜ ⎟ ⎝ ∗ 0 0 0 −ε1i I 0 ⎠ 0 ∗ 0 0 0 −ε2i I where

(9.38)

176

9 Fuzzy H∞ Sliding Mode Control Under Phase-Type Distribution

⎞ T Bi 0 Ξ2 Dmi X i Mmi T ⎟ ⎜ ∗ −γ 2 I N T 0 Dmi mi ⎟ ⎜ ⎜ ∗ −I 0 0 ⎟ Ξ1 = ⎜ ∗ ⎟, ⎝ ∗ ∗ ∗ −BiT Pi Bi 0 ⎠ ∗ ∗ ∗ ∗ −X i ⎛

T T T T Ξ2 = Ami X i + Bi Ymi + X i Ami + Ymi BiT + ε1i X i Hmi Hmi X i + ε2i X i Hmi Hmi X i (∑ ) N + Xi πi j X −1 Xi . j j=1

∑N

Then, X i (

j=1

πi j X −1 j )X i can be rewritten as

Xi

(∑

s j=1

) π¯ i j X −1 X i = πii X i − Π˜ 1 Π˜ 2−1 Π˜ 1T . j

(9.39)

Substituting (9.39) into (9.38), the inequality (9.34) can be obtained by the Schur complement. Similarly, one also has (9.35). Because X¯ i = X i Pi = I , the following inequality holds (X¯ i − I )T (X¯ i − I ) ≤ β1 I , where β1 > 0. By applying the Schur complement, (9.36) holds. ∎ Equation (9.37) can be rewritten as (BiT Pi Cmi )T (BiT Pi Cmi ) ≤ β2 I , where β2 > 0. By applying the Schur complement, the above formula is equivalent to (

T Pi Bi −β2 I Cmi T Bi Pi Cmi −I

) ≤ 0, ∀i ∈ N .

(9.40)

For a prescribed H∞ performance index, we can take γ as the optimized variable with the optimization problem as γ Pi , X i , X¯ i , Yni , ε1i , ε2i , β1 , β2 s.t. (34) − (36), and (40).

min

(9.41)

Remark 9.5 By using the linear matrix inequality toolbox, we can solve the above optimization problem involving matrix inequality constraints and linear objective. Therefore, the problem of solving inequalities (9.18), (9.29), and (9.30) is transformed into the problem of solving inequalities (9.34)-(9.36), and (9.40) to solve the parameter K ni .

9.5 Simulation

177

9.5 Simulation In this section, the validity of the conclusions is verified by the single-link robot arm model. First, considering the random variation of work environment, the single-link robot arm model is described by phase-type S-MSSs. Second, by the plant transformation method and the supplementary variable technique, phase-type S-MSSs are equivalent to associated MSSs. Third, by the use of T-S fuzzy rule, phase-type stochastic nonlinear S-MSSs are converted into phase-type stochastic linear S-MSSs through the membership functions. Fourth, for given system parameters, controller gains are solved under the framework of feasp function from MATLAB/LMI toolbox to obtain the sliding mode control law. Finally, some simulation diagrams are adopted to verify the validity of the conclusions by the help of MATLAB/Simulink. Consider a single-link robot arm from [19] given as ¨ =− ϑ(t)

φ(t) J gL 1 ˙ sin(ϑ(t)) − ϑ(t) + u(t), w w w

where ϑ(t) denotes the angle position, u(t) is the control input. The parameters w, L, φ(t), and J stand for moment of inertia, length of the arm, viscous friction, and mass of the payload. First, considering the random variation of work environment, the single-link robot arm model is described by phase-type stochastic S-MSSs as ¨ =− ϑ(t)

φ(t) 1 J gL ˙ sin(ϑ(t)) − u(t), ϑ(t) + w(λt ) w(λt ) w(λt )

where λt is adopted to describe the phase-type semi-Markovian process in {1, 2}. Second, the phase-type S-MSSs are converted into the associated MSSs. For the phase-type semi-Markovian process in {1, 2}, the sojourn time obeys negative exponential distribution with parameter λ1 under the first state. For the second state, the sojourn time obeys a 2-stage Erlang distribution split into two sections. The sojourn time in the first (or second) section obeys negative exponential distribution with parameter λ2 (or λ3 ). Assume that p12 = p21 = 1. Obvi(1) ) = (−λ1 ),Γ (2) = ously, b(1) = (b(1) ) = 1, b(2) = (b1(2) , b2(2) ) = (1, 0),Γ (1) = (Γ11 ] [ (2) (2) ] [ −λ2 λ2 Γ11 Γ12 . (2) (2) = 0 −λ3 Γ21 Γ22 The state space of Z (t) = (λt , E(t)) is Q = ((1, 1), (2, 1), (2, 2)). The elements in Q are enumerated as ϕ((1, 1)) = 1,⎡ϕ((2, 1)) = 2, and ⎤ ϕ((2, 2)) = 3. Hence, the −λ1 λ1 0 infinitesimal generator of Φ(Z (t)) is ⎣ 0 −λ2 λ2 ⎦, where λ1 = 0.5, λ2 = 0.2, λ3 0 −λ3 and λ3 = 0.7. Then, let ςt = ϕ(Z (t)). ςt is the associated Markovian process of λt in {1, 2, 3} and the infinitesimal generator of ςt is given before. Therefore, phase-type S-MSSs can be expressed by the associated MSSs as

178

9 Fuzzy H∞ Sliding Mode Control Under Phase-Type Distribution

¨ ϑ(t) =−

J (ςt )gL φ(t) 1 ˙ sin(ϑ(t)) − u(t). ϑ(t) + w(ςt ) w(ςt ) w(ςt )

For ςt = i, J (ςt ) and w(ςt ) are, respectively, denoted as Ji and wi . Third, the T-S fuzzy model is adopted to describe the system. ˙ sin(z 1 (t)) = h 1 Let z(t) = [z 1 (t) z 2 (t)]T , z 1 (t) = ϑ(t) and z 2 (t) = ϑ(t), (z 1 (t))z 1 (t) + .h 2 (z 1 (t))z 1 (t) with . = 0.01/π, and h 1 (z 1 (t)) + h 2 (z 2 (t)) = 1 for h 1 (z 1 (t)), h 2 (z 1 (t)) ∈ [0, 1]. Therefore, we can get the membership functions as h 1 (z 1 (t)) =

{ sin(z1 (t))−.z1 (t)

h 2 (z 1 (t)) =

z 1 (t)(1−.)

1, { z1 (t)−sin(z1 (t)) z 1 (t)(1−.)

1,

, z 1 (t) /= 0, z 1 (t) = 0, , z 1 (t) /= 0, z 1 (t) = 0.

Based on the membership functions, h 1 (z 1 (t)) = 0, h 2 (z 1 (t)) = 1 when z 1 (t) is about π rad or −π rad, and h 1 (z 1 (t)) = 1, h 2 (z 1 (t)) = 0 when z 1 (t) is about 0 rad. For two modes, the parameters w and J are given as: when i = 1, w1 = 1, J1 = 1; when i = 2, w1 = 5, J1 = 5. Moreover, we assume that the Itô stochastic process, nonlinearity, and external disturbance exist in the system. Thus, we have Plant Rule 1: IF z 1 (t) is about 0 rad, THEN dz(t) = [(A1i + ΔA1i )z(t) + Bi (u(t) + gi (z(t), t)) + D1i v(t)]dt + C1i z(t)dω(t), y(t) = M1i z(t) + N1i v(t); Plant Rule 2: IF z 1 (t) is about π rad or −π rad, THEN dz(t) = (A2i + ΔA2i )z(t) + Bi (u(t) + gi (z(t), t)) + D2i v(t)]dt + C2i z(t)dω(t), y(t) = M2i z(t) + N2i v(t); where z(t) = [z 1T (t) z 2T (t)]T , [

[ ] [ ] [ ] ] 0 1 0.1 0.2 0 0 1 A11 = , B1 = , A12 = , C11 = , −gι −φ0 −0.3 0.1 0.5 −0.75gι −0.5φ0 [ ] [ [ ] [ ] ] 0.1 0.2 0 1 0.2 0.2 0 , B2 , , A21 = , C21 = C12 = −0.4 0.2 −.gι −φ0 −0.4 0.1 0.4 [ [ ] ] [ ]T 0 1 0.3 0.2 , D11 = D21 = 0.1 0 , , C22 = A22 = −0.75.gι −0.5φ0 −0.4 0.1 [ ]T [ ] [ ] D12 = D22 = 0 0.1 , M11 = M21 = 1 0 , M12 = M22 = 0.1 0.5 , [ ]T N11 = N21 = 0.01, N12 = N22 = 0.02, E 11 = E 21 = −0.1 0.1 , [ ]T [ ] [ ] E 12 = E 22 = −0.02 −0.01 , H11 = H21 = 0.1 0 , H12 = H22 = 0 0.1 , ε1i = 0.3, ε2i = 0.1, φ(t) = φ0 = 2, ι = 0.5, g = 9.81, gi (z(t), t) = 0.5 sin(z 1 (t)), v(t) = e−0.5t sin(t), κ = 0.5, h¯ = 0.085, i ∈ {1, 2}.

9.5 Simulation

179 2.5

Fig. 9.2 Switching signal

Switching signal

2

1.5

1

0.5

1

2

3

4

5

6

7

8

9

10

Time(s)

Then, we have the whole fuzzy MSSs as dz(t) =

∑2 m=1

h m (θ (t)){[(Ami + ΔAmi (t))z(t) + Bi (u(t) + gi (z(t), t)) + Dmi v(t)]dt

+ Cmi z(t)dω(t)}, y(t) = Mmi z(t) + Nmi v(t), z(0) = ϕ(0).

(9.42)

Fourth, for given β1 = 0.03 and β2 = 0.01, solving the optimization problem (9.41) results in H∞ optimal performance index γmin = 3.0271 and [ ] [ ] K 11 = −5.6906 −1.7900 , K 12 = −1.9463 −7.3886 , [ ] [ ] K 21 = −13.5107 −2.3505 , K 22 = −10.8218 −6.7894 . For given μ = 1.2, the sliding mode control law is designed as u(t) =

∑2 m=1

∑2 n=1

h m (θ (t))h n (θ (t))[K ni z(t)

− (μ + ρmi (t))sgn(BiT G iT s(t))],

(9.43)

where ρmi (t) = h¯ ∥(G i Bi )−1 G i Dmi ∥ + ∥(G i Bi )−1 G i E mi ∥∥Hmi ∥∥z(t)∥ + κ∥z(t)∥. Fifth, substituting the above sliding mode control law (9.43) into system (9.42), one has the whole closed-loop system. For given initial condition z(0) = [0.2 0.6]T , the results of the example simulation are show in Figs. 9.2, 9.3, 9.4 and 9.5 by the help of MATLAB/Simulink. As we all know, sliding mode control belongs to a class of switching control and frequently switches during the sliding surface. Then, it pro-

180 Fig. 9.3 State response

9 Fuzzy H∞ Sliding Mode Control Under Phase-Type Distribution 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4

Fig. 9.4 Sliding mode control law

0

5

10

15

10

15

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

0

5

duces chattering with a great impact on the system property. In order to avoid chatterB T G iT s(t) . Figure 9.2 displays the switching ing, sgn{BiT G iT s(t)} is updated by ∥B T GiT s(t)∥+0.01 i i signal. Figure 9.3 represents the state response of the closed-loop system with controller. Figures 9.4 and 9.5 represent the control input and the sliding surface function, respectively. These show the superiority of the proposed sliding mode control method.

References Fig. 9.5 Sliding surface

181 0.4 0.3 0.2 0.1 0 -0.1 -0.2

0

5

10

15

9.6 Conclusion In this chapter, we have studied the problem of sliding mode control for Itô stochastic T-S fuzzy systems with semi-Markovian switching parameters. Sufficient condition has been proposed for the robustly stochastic stability of the system with a prescribed H∞ performance index under the action of semi-Markovian switching. Then, a sliding mode control law is designed to guarantee the reachability of system state in a finite time. Finally, the superiority of the conclusion is verified by simulation. In the future, we will extend sliding mode control for phase-type stochastic nonlinear semi-Markovian switching systems to discrete-time case.

References 1. Neuts, M.F.: Probability Distributions of Phase Type. University of Louvain, Belgium (1975) 2. Heffes, H., Lucantoni, D.: A Markov modulated characterization of packetized voice and data traffic and related statistical multiplexer performance. IEEE J. Sel. Areas Commun. 4(6), 856– 868 (1986) 3. Li, M., Chen, Y., Xu, L.Y., Chen, Z.Y.: Asynchronous control strategy for semi-Markov switched system and its application. Inf. Sci. 532, 125–138 (2020) 4. Ning, Z.P., Zhang, L.X., Colaneri, P.: Semi-Markov jump linear systems with incomplete sojourn and transition information: Analysis and synthesis. IEEE Trans. Autom. Control 65(1), 159–174 (2020) 5. Tian, Y.X., Yan, H.C., Zhang, H., Zhan, X.S., Peng, Y.: Dynamic output-feedback control of linear semi-Markov jump systems with incomplete semi-Markov kernel. Automatica 117, Article ID 108997 (2020) 6. Wang, B., Zhu, Q.X.: Stability analysis of semi-Markov switched stochastic systems. Automatica 94(94), 72–80 (2018)

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7. Zhang, L.X., Cai, B., Shi, Y.: Stabilization of hidden semi-Markov jump systems: Emission probability approach. Automatica 101, 87–95 (2019) 8. Jiang, B.P., Karimi, H.R., Kao, Y.G., Gao, C.C.: Takagi-Sugeno model based event-triggered fuzzy sliding mode control of networked control systems with semi-Markovian switchings. IEEE Trans. Fuzzy Syst. 28(4), 673–683 (2020) 9. Qi, W.H., Zong, G.D., Karimi, H.R.: SMC for nonlinear stochastic switching systems with quantization. IEEE Trans. Circuits Syst. II Express Briefs 68(6), 2032–2036 (2021) 10. Qi, W.H., Zong, G.D., Zheng, W.X.: Adaptive event-triggered SMC for stochastic switching systems with semi-Markov process and application to boost converter circuit model. IEEE Trans. Circuits Syst. I Regul. Pap. 68(2), 786–796 (2021) 11. Wang, J., Chen, M.S., Shen, H.: Event-triggered dissipative filtering for networked semiMarkov jump systems and its applications in a mass-spring system model. Nonlinear Dyn. 87(4), 2741–2753 (2017) 12. Shen, H., Men, Y.Z., Wu, Z.G., Cao, J.D., Lu, G.P.: Network-based quantized control for fuzzy singularly perturbed semi-Markov jump systems and its application. IEEE Trans. Circuits Syst. I Regul. Pap. 66(3), 1130–1140 (2019) 13. Dong, S.L., Chen, G.R., Liu, M.Q., Wu, Z.G.: Robust adaptive H∞ control for networked uncertain semi-Markov jump nonlinear systems with input quantization. Science China Inf. Sci. 65, Article ID 189201 (2022) 14. Shi, P., Xia, Y.Q., Liu, G.P., Rees, D.: On designing of sliding-mode control for stochastic jump systems. IEEE Trans. Autom. Control 51(1), 97–103 (2006) 15. Zhu, Q., Yu, X.H., Song, A.G., Fei, S.M., Cao, Z.Q., Yang, Y.Q.: On sliding mode control of single input Markovian jump systems. Automatica 50(11), 2897–2904 (2014) 16. Hou, Z.T., Luo, J.W., Shi, P., Nguang, S.K.: Stochastic stability of Itoˆ differential equations with semi-Markovian jump parameters. IEEE Trans. Autom. Control 51(8), 1383–1387 (2006) 17. Boukas, E.K.: Stochastic Switching Systems: Analysis and Design. Birkhauser, ¨ Boston (2006) 18. Chern, T.L., Wu, Y.C.: Design of integral variable structure controller and application to electrohydraulic velocity servosystems. IEE Proc. D (Control Theory Appl.) 138(5), 439–444 (1991) 19. Wu, H.N., Cai, K.Y.: Mode-independent robust stabilization for uncertain Markovian jump nonlinear systems via fuzzy control. IEEE Trans. Syst. Man Cybern. Part B (Cybern.) 36(3), 509–519 (2005)

Chapter 10

Sliding Mode Control Under Denial-of-Service Attacks

This chapter focuses on the discrete-time sliding mode control of uncertain networked semi-Markovian switching systems (S-MSSs) subject to stochastic denial-of-service attacks. The semi-Markovian kernel approach is exploited to achieve that the switching among different modes is mutually governed by the transition probability and the sojourn-time distribution function. Firstly, by considering randomly occurring denial-of-service attacks, a sliding mode function with respect to the attack probability is well constructed to characterize the effect of such malicious attacks. Then, sufficient conditions under the equivalent discrete-time sliding mode control law are proposed under the framework of σ -error mean square stability criterion. In what follows, the discrete-time sliding mode control law, which ensures that the resulting closed-loop system dynamics can be driven onto the pre-specified sliding region, is synthesised. At last, an electronic throttle control model illustrates the proposed control scheme.

10.1 Introduction Due to the rapid growth of network communication and advanced control techniques, the traditional point-to-point systems no longer meet the practical industrial demand [1–3]. Networked control systems are expected to fulfill some complex and remote control tasks because of their outstanding ability in long-distance task executions through a shared communication network [4–8]. However, the data packets are easy jammed and tampered in an open network environment owing to the industrial control networks subject to many malicious attacks. Currently, many researchers have devoted themselves to mitigate or even eliminate the detrimental effects of cyber attacks, and a great number of fruitful achievements have been made in the field of security issue for networked control systems, see e.g., [9–15].

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 W. Qi and G. Zong, Control Synthesis for Semi-Markovian Switching Systems, Studies in Systems, Decision and Control 465, https://doi.org/10.1007/978-981-99-0317-7_10

183

184

10 Sliding Mode Control Under Denial-of-Service Attacks

The classical cyber-attacks mainly include deception attacks, replay attacks, and denial-of-service attacks. It is worth noting that denial-of-service attacks, as one of the popular attack strategies which launch their attacks through blocking communication channels, is almost inevitable. An obvious result of denial-of-service attacks is that a large scale packet dropouts in the networked control systems. Then, the controller will lose its effectiveness if not appropriately coped with, leading to the degradation of control performance. Up to now, some achievements have attempted to investigate denial-of-service attacks [16–22]. For example, the secure consensus control protocol has been discussed for multi-agent systems subject to the periodic denial-of-service attacks [19]. A novel attack model has been proposed to reflect the randomly occurring behaviors of denial-of-service attacks and the corresponding event-based security control scheme has been proposed [20]. This chapter will study the sliding mode control for discrete-time networked SMSSs under randomly occurring denial-of-service attacks. The main contributions are shown as: (i) A comprehensive networked S-MSSs model is presented in line with semi-Markovian kernel, in which the hypothesis of a single probability density function is removed. (ii) A novel sliding surface, which is dependent of both the current switching instant and the probability of random denial-of-service attacks, is well constructed. (iii) The proposed discrete-time sliding mode control strategy can guarantee that the underlying S-MSSs are mean square stable in the sense of σ -error and the reachability of the specified sliding region is realized.

10.2 Problem Statements and Preliminaries Consider the following discrete-time networked S-MSSs x(k + 1) = (A (ρk ) + ΔA (ρk ))x(k) + B(ρk )u(k),

(10.1)

where x(k) ∈ Rn and u(k) ∈ Rm are the state and the input vectors. For ρk = τ , Aτ and Bτ are known real matrices. Suppose that the parameter uncertainty ΔAτ is represented by ΔAτ = Mτ Fτ Wτ with known matrices Mτ , Wτ , and unknown matrix Fτ satisfying FτT Fτ ≤ I . Moreover, {ρk } is governed by semi-Markovian chain in Γ = {1, 2, . . . , N }. Related concepts and definitions are presented below. The time instant at the nth switch is given as kn with k0 = 0, the system mode index of the nth switch is given as Pn , and the sojourn time of the system mode between the nth switch and (n + 1)th switch is given as Sn+1 with S0 = 0. Please see Fig. 10.1. Definition 10.1 ([23]) {(Pn , kn )}n∈N+ is a Markovian renewal chain, if ∀ω ∈ Γ and ν ∈ N+ , πτ ω (ν) = Pr (Pn+1 = ω, Sn+1 = ν|P0 , . . . , Pn ; k0 , . . . , kn ) = Pr (Pn+1 = ω, Sn+1 = ν|Pn = τ ),

10.2 Problem Statements and Preliminaries

185

Fig. 10.1 A case of stochastic processes (N = 3)

where Π (ν) = [πτ ω (ν)]τ,ω∈Γ is the discrete-time semi-Markovian kernel with ∞ Σν=0 Σω∈Γ πτ ω (ν) = 1 and πτ ω (0) = 0. Furthermore, {Pn } ∈ P is the embedded Markovian chain of Markovian renewal chain {(Pn , kn )}n∈N+ , and the transition probability matrix is Θ = [θτ ω ]τ,ω∈Γ , where θτ ω = Pr (Pn+1 = ω|Pn = τ ) and θτ τ = 0. Definition 10.2 ([23]) {ρk } is a semi-Markovian chain about Markovian renewal chain {(Pn , kn )}n∈N+ , if ∀k ∈ N+ , ρk = PN (k) , where N (k) = max{n ∈ N|kn ≤ k}. Remark 10.1 The main difference between the semi-Markovian chain {ρk } and embedded Markovian chain {Pn } is that the random variable of {ρk } is related to the sampling instant k, while the random variable of the latter varies with the switching instant kn . In this chapter, the probability density function pertaining to both the current and next system mode is given as ρτ ω (ν) = Pr (Sn+1 = ν|Pn = τ, Pn+1 = ω), ∀τ, ω ∈ Γ . Hence, one has πτ ω (ν) = θτ ω ρτ ω (ν). Definition 10.3 ([23]) Consider the discrete-time S-MSSs x(k + 1) = f (x(k), ρk ). τ of sojourn time, if For given initial condition and upper bound Tmax τ |P =τ = 0, lim E {x T (k)x(k)}|z0 ,ρ0 ,Sn+1 ≤Tmax n

k→∞

(10.2)

then system is σ -error mean square stable with τ σ = Στ ∈Γ | ln(Ψτ (Tmax ))|,

(10.3)

where Ψτ (ν) = Pr (Sn+1 ≤ ν|Pn = τ ) = Στν=0 Σω∈Γ πτ ω (τ ) is the cumulative density function of the sojourn time for the τ th switching mode. Without loss of generality, πτ ω (0) = Ψτ (0) = 0 is assumed. τ , Remark 10.2 For Definition 10.3, it is noted that σ is inversely proportional to Tmax τ namely, σ decreases if all Tmax increase. Based on this observation, σ is virtually

186

10 Sliding Mode Control Under Denial-of-Service Attacks

Fig. 10.2 The structure diagram of networked S-MSSs

able to capture the approximation error from σ -error mean square stability to mean τ τ )) → 1 when Tmax → ∞ for ∀τ ∈ Γ and then square stability. Moreover, ln(Ψ (Tmax σ → 0 can be obtained from (10.3). The interested structure of the discrete-time networked S-MSSs is shown in Fig. 10.2, where DoS means denial-of-service. In this chapter, the considered networked S-MSSs suffer from denial-of-service attacks in sensor to controller transmission channel. Therefore, the attacker deteriorates the system performance by preventing the state x(k) through launching denial-of-service attacks. Before preceding further, the attack mechanism of denial-of-service attacks is illustrated briefly. In fact, the denial-of-service attacks will exhaust communication bandwidth to prevent the controller from receiving useful data packets from sensors. Likewise, the networked S-MSSs suffer from denial-of-service attacks from controller to actuators transmission channel. The scenario that the denial-of-service attacks occur both in measurement transmission channel and control transmission channel simultaneously will be regarded as one of the future research directions. In particular, only stochastic denial-of-service attacks, which follow a specific probability distribution rule, are considered in this chapter. In detail, the Bernoulli distribution sequence is adopted to indicate the stochastic denial-of-service attacks. Thus, the received state via communication channel is given as: xc (k) = λ(k)x(k) + (1 − λ(k))xc (k − 1),

(10.4)

where xc (k) is the received state by controller at latest sampling instant k. Here, λ(k) is subject to the Bernoulli distribution with {

Pr (λ(k) = 1) = E {λ(k)} = λ¯ , Pr (λ(k) = 0) = 1 − λ¯ ,

where λ¯ ∈ [0, 1] is a known constant.

(10.5)

10.4 σ -Error Mean Square Stability Analysis

187

Remark 10.3 In fact, Eq. (10.4) can be seen as a compensation strategy of packet loss to handle random denial-of-service attacks tactfully. In another word, the latest received data from the controller will be utilized if denial-of-service attacks occur during the transmission of state x(k), i.e. λ(k) = 0, xc (k) = xc (k − 1). On the contrary, if denial-of-service attacks are sleep, one has λ(k) = 1 and xc (k) = x(k), which implies that the data packet is successfully received by the controller. Lemma 10.1 ([24]) Consider the nonlinear S-MSSs x(t + 1) = f (x(t), ρt ), where x(t) and ρt denote the system state and switching mode index, respectively. The switching instants are defined as t0 , t1 , . . . , ts , . . . with t0 = 0. The system is mean square stable, if there exist a group of C 1 functions Ω(x(t), ρt ) : Rn → R and three class K∞ functions α1 , α2 , α3 , such that α1 (∥x(t)∥) ≤ Ω(x(t), ρt ) ≤ α2 (∥x(t)∥), Ω(x(t), ρts ) ≤ h τ Ω(x(ts ), ρts ), t ∈ N(ts ,ts+1 ] , E {Ω(x(ts+1 ), ρts+1 )}|x(0),ρ0 − Ω(x(ts ), ρts ) ≤ −α3 (∥x(ts )∥).

(10.6) (10.7) (10.8)

for any initial conditions t (0) ∈ Rn , ρ0 ∈ Γ , and a given finite h τ > 0, ∀ρts = τ ∈ Γ .

10.3 Sliding Surface Design Under the stochastic denial-of-service attacks, choose the following mode-dependent sliding surface as s(k) = λ¯ Gτ x(k) + (1 − λ¯ )Gτ Aτ xc (k − 2),

(10.9)

where Gτ ∈ Rm×n is chosen to satisfy nonsingularity of Gτ Bτ with Gτ = BτT Qτ . Here, positive-definite matrix Qτ ∈ Rn×n will be designed in Theorem 10.2, and Bτ is with full column rank. Combing system (10.1) and sliding function (10.9), one has s(k + 1) = λ¯ Gτ [(Aτ + ΔAτ )x(k) + Bτ u(k)] + (1 − λ¯ )Gτ Aτ xc (k − 1). (10.10)

10.4 σ -Error Mean Square Stability Analysis In the sliding phase, one has s(k + 1) = s(k) = 0.

(10.11)

Then, the equivalent discrete-time sliding mode control law is obtained that

188

10 Sliding Mode Control Under Denial-of-Service Attacks

u eq (k) = − (λ¯ Gτ Bτ )−1 [λ¯ Gτ (Aτ + ΔAτ )x(k) + (1 − λ¯ )Gτ Aτ xc (k − 1)]. (10.12) Combing (10.1) and (10.12), we obtain that x(k + 1) = (I − Bτ (Gτ Bτ )−1 Gτ )(Aτ + ΔAτ )x(k) −1

− (λ¯

− 1)Bτ (Gτ Bτ )−1 Gτ Aτ xc (k − 1).

(10.13)

By defining x¯ (k) = [x T (k), xcT (k − 1)]T and combining with (10.4) and (10.13), one has x¯ (k + 1) = A¯τ x¯ (k),

(10.14)

where [ ] A¯1τ A¯2τ A¯τ = , λ(k) 1 − λ(k) A¯1τ = (I − Bτ (Gτ Bτ )−1 Gτ )(Aτ + ΔAτ ), −1 A¯2τ = −(λ¯ − 1)Bτ (Gτ Bτ )−1 Gτ Aτ .

Theorem 10.1 System (10.14) is σ -error mean square stable, if for ∀τ, ω ∈ Γ , there τ ∈ N≥1 } with symmetric matrix Pτ > 0 and scalar h τ > 0 such that exist {Tmax E {(A¯τ )t P¯τ (A¯τ )t } − h τ P¯τ < 0,

(10.15)

τ T Tmax ¯ τ (ν)(A¯τ )ν } E {Σν=1 (A¯τ )ν P

(10.16)

T

− P¯τ < 0,

where ¯ τ (ν) = Σω∈Γ {πτ ω (ν)P¯ω /ητ }, P¯τ = diag{Pτ , Pτ }, P Tτ

max τ ]. Σω∈Γ πτ ω (ν), Gτ = BτT Pτ , ∀t ∈ N[1,Tmax ητ = Σν=1

Proof For system (10.14), choose the following Lyapunov function V (x¯ (k), τ ) = x¯ T (k)P¯τ x¯ (k)| Pn =τ , ∀τ ∈ Γ,

(10.17)

where P¯τ = diag{Pτ , Pτ }. Then, it is got that V (x¯ (k), τ ) ≥ inf {λmin (P¯τ )}∥x¯ (k)∥2 , τ ∈Γ

V (x¯ (k), τ ) ≤ sup{λmax (P¯τ )}∥x¯ (k)∥2 , τ ∈Γ

(10.18)

10.4 σ -Error Mean Square Stability Analysis

189

where λmin (P¯τ ) and λmax (P¯τ ) are of the minimum and maximum eigenvalues of P¯τ individually, which means that (10.6) holds. τ ] }, there is no mode switching at sampling time k, For k ∈ {kn + t, ∀t ∈ N[1,Tmax i.e Pk = τ . Applying (10.15) results in E {V (x¯ (k), ρkn )} − h τ V (x¯ (kn ), ρkn ) = E {x¯ T (k)P¯τ x¯ (k)} − h τ x¯ T (kn )P¯τ x¯ (kn ) = E {x¯ T (kn + t)P¯τ x¯ (kn + t)} − h τ x¯ T (kn )P¯τ x¯ (kn ) T = x¯ T (kn )E {(A¯τ )t P¯τ (A¯τ )t − h τ P¯τ }x¯ (kn ) < 0,

(10.19)

which guarantees (10.7) in Lemma 10.1. Let Pkn = τ, Pkn+1 = ω, ∀τ /= ω ∈ Γ and the sojourn time be expressed by ν. From (10.16), one has τ − V (x¯ (kn ), ρkn ) E {V (x¯ (kn+1 ), ρkn+1 )}|x¯ (0),ρ0 ,Sn+1 ≤Tmax

Tτ

max Σω∈Γ πτ ω (ν)(A¯τ )ν P¯ω (A¯τ )ν /ητ − P¯τ }x¯ (kn ) = x¯ T (kn )E {Σν=1

T

τ

T Tmax ¯ τ (ν)(A¯τ )ν } + P¯τ }∥x¯ (kn )∥2 (A¯τ )ν P ≤ − λmin {−E {Σν=1

≤ − ψ∥x¯ (kn )∥2 ,

(10.20) τ

T Tmax ¯ τ (ν)(A¯τ )ν } − P¯τ )}. where ψ = inf τ ∈Γ {−λmin (E {Σν=1 (A¯τ )ν P τ Furthermore, system (10.14) is mean square stable with the upper bound Tmax of sojourn time. In addition, the σ -error mean square stability in Definition 10.3 is ensured for all possible uncertainties. ∎

Remark 10.4 In this section, if the sojourn time between two adjacent switching τ → ∞), sufficient conditions in Theorem 10.1 will be instants is boundless (i.e. Tmax degraded to mean square stability. Although σ -error mean square stability criteria are constructed for uncertain networked S-MSSs under random denial-of-service attacks, sufficient conditions (10.15) and (10.16) are still challenge issues on how to obtain feasible solutions along with a higher power index of A¯τ . In the following Theorem 10.2, the corresponding issue will be addressed. Theorem 10.2 System (10.14) is said to be σ -error mean square stable, if for τ ∈ N≥1 } and symmetric matrices {Qτ (t, m) > 0}, ∀t ∈ ∀τ, ω ∈ Γ , we can find {Tmax τ ] , ∀m ∈ N[0,t] with Qτ = Qτ (t, t), and {Qτ (ν, n) > 0}, ∀ν ∈ N[1,T τ ] , ∀n ∈ N[1,Tmax max N[0,ν−1] , and scalars ετ > 0, h τ > 0, λ¯ > 0, such that

190

10 Sliding Mode Control Under Denial-of-Service Attacks

[

Ξτ Ξτ1 ∗ Ξτ2

] < 0,

(10.21)

Qτ (t, 0) − h τ Qτ < 0, τ Tmax Σν=n+1 Π˜ τ (ν) < 0, τ Tmax Σν=1 Qτ (ν, 0) − Qτ

(10.23) < 0,

where ⎡

⎤

WτT

0

⎢ ⎥ ⎢ 0 0 ⎥ ⎢ √ ⎥ ⎢ ⎥ 3ετ Qτ (t, m + 1)Mτ 0 ⎥, Ξτ1 = ⎢ ⎢√ ⎥ ⎢ 3ε B T Q (t, m + 1)M 0 ⎥ τ τ τ τ ⎣ ⎦ 0 0 [ ] [ ] Πτ (ν) Πτ1 (ν) Π Π ˜ 11τ 12τ = , , Π Πτ (ν) = 1τ ∗ Πτ2 [ Πτ (ν) =

Π1τ (ν) Π2τ (ν) ∗

⎡

]

Π3τ (ν)

[ ] T , Π2τ = Π21τ Π22τ ,

0

⎢ 0 ⎢ ⎢ √ ⎢ 3ετ Qτ (ν, n + 1)Mτ Πτ1 (ν) = ⎢ ⎢√ ⎢ 3ετ BτT Qτ (ν, n + 1)Mτ ⎣ 0 [ Π11τ (ν) =

λ¯ Qτ (ν, n + 1) − Qτ (ν, n)

⎤

0 ⎥ ⎥ ⎥ 0 ⎥ ⎥, ⎥ 0 ⎥ ⎦ 0

]

0 [

Π12τ (ν) =

WτT

0

, ]

, (1 − λ¯ )Qτ (ν, n + 1) − Qτ (ν, n)

⎡ √ ⎤ 3Qτ (ν, n + 1)Aτ ⎢√ T ⎥ 3Bτ Qτ (ν, n + 1)Aτ ⎥ , Π21τ (ν) = ⎢ ⎣ ⎦ 0

(10.22)

(10.24)

10.4 σ -Error Mean Square Stability Analysis

⎡ ⎢ Π22τ (ν) = ⎢ ⎣

⎤

0 √

191

0 3γ BτT Qτ (ν, n + 1)Aτ

⎥ ⎥, ⎦

Π3τ (ν) = diag{−Qτ (ν, n + 1), −BτT Qτ (ν, n + 1)Bτ , −BτT Qτ (ν, n + 1)Bτ }, Ξτ2 = Πτ2 = diag{−ετ I, −ετ I }, Qτ (ν, ν) = Σω∈Γ [πτ ω (ν)Qω /ητ ], Gτ = BτT Qτ , τ

−1 Tmax Σω∈Γ πτ ω (ν), γ = λ¯ − 1. ητ = Σν=1 τ ] , ∀m ∈ N[0,t−1] , ∀n ∈ N[0,T τ −1] . for ∀t ∈ N[1,Tmax max

Proof From (10.5), (10.13) and Gτ (t, m + 1) = BτT Qτ (t, m + 1), we find T T t−1 T Σm=0 x¯ (kn )E {(A¯τ )m [A¯τ Q¯τ (t, m + 1)A¯τ − Q¯τ (t, m)](A¯τ )m }x¯ (kn ) t−1 T x¯ (kn )E {(A¯τ )m Θτ (A¯τ )m }x¯ (kn ), = Σm=0 T

(10.25)

where ⎡ Q¯τ = diag{Qτ , Qτ }, Θτ = ⎣

Θ1τ Θ2τ ∗ Θ3τ

⎤ ⎦,

Θ1τ = (Aτ + ΔAτ )T [I − Bτ (Gτ (t, m + 1)Bτ )−1 Gτ (t, m + 1)]T Qτ (t, m + 1) [I − Bτ (Gτ (t, m + 1)Bτ )−1 Gτ (t, m + 1)](Aτ + ΔAτ ) + λ¯ Qτ (t, m + 1) − Qτ (t, m), Θ2τ = −γ (Aτ + ΔAτ )T [I − Bτ (Gτ (t, m + 1)Bτ )−1 Gτ (t, m + 1)]T Qτ (t, m + 1)Bτ (Gτ (t, m + 1)Bτ )−1 Gτ (t, m + 1)Aτ , Θ3τ = γ 2 AτT GτT (t, m + 1)(BτT GτT (t, m + 1))−1 BτT Qτ (t, m + 1)Bτ (Gτ (t, m + 1)Bτ )−1 BτT Gτ (t, m + 1)Aτ + (1 − λ¯ )Qτ (t, m + 1) − Qτ (t, m).

Based on [26], we can arrive at

192

10 Sliding Mode Control Under Denial-of-Service Attacks (Aτ + ΔAτ )T (I − Bτ (Gτ (t, m + 1)Bτ )−1 Gτ (t, m + 1))T Qτ (t, m + 1) (I − Bτ (Gτ (t, m + 1)Bτ )−1 Gτ (t, m + 1))(Aτ + ΔAτ )

≤ 2(Aτ + ΔAτ )T Qτ (t, m + 1)Bτ (BτT Qτ (t, m + 1)Bτ )−1 BτT Qτ (t, m + 1)(Aτ + ΔAτ ) + 2(Aτ + ΔAτ )T Qτ (t, m + 1)(Aτ + ΔAτ ), − 2γ x T (k)(Aτ + ΔAτ )T (I − Bτ (Gτ (t, m + 1)Bτ )−1 Gτ (t, m + 1))T Qτ (t, m + 1)Bτ (Gτ (t, m + 1)Bτ )

−1

(10.26)

Gτ (t, m + 1)Aτ x c (k − 1)

≤ x T (k)(Aτ + ΔAτ )T Qτ (t, m + 1)(Aτ + ΔAτ )x(k) + x T (k)(Aτ + ΔAτ )T Qτ (t, m + 1)Bτ (BτT Qτ (t, m + 1)Bτ )−1 BτT Qτ (t, m + 1)(Aτ + ΔAτ )x(k)

+ 2γ 2 xcT (k − 1)AτT Qτ (t, m + 1)Bτ (BτT Qτ (t, m + 1)Bτ )−1 BτT Qτ (t, m + 1)Aτ x c (k − 1), AτT GτT (t, m

(10.27)

+ 1)(BτT GτT (t, m

+ 1))

−1

BτT Qτ (t, m

+ 1)Bτ

(Gτ (t, m + 1)Bτ )−1 Gτ (t, m + 1)Aτ = AτT Qτ (t, m + 1)Bτ (BτT Qτ (t, m + 1)Bτ )−1 BτT Qτ (t, m + 1)Aτ .

(10.28)

Together with ΔAτ = Mτ Fτ Wτ and (10.26)–(10.28), one has ¯ τ = Ξτ + W¯ τT FτT M¯τT + M¯τ Fτ W¯ τ , Θτ ≤ Θ where [ [ ] ] T Ξ1τ = Ξ11τ Ξ12τ , Ξ2τ = Ξ21τ Ξ22τ , [ Ξ11τ =

λ¯ Qτ (t, m + 1) − Qτ (t, m) 0

⎡ √

3Qτ (t, m + 1)Aτ

] ,

⎤

⎢√ T ⎥ 3Bτ Qτ (t, m + 1)Aτ ⎥ , Ξ21τ = ⎢ ⎣ ⎦ 0 [ Ξ12τ =

Ξ22τ

0

(1 − λ¯ )Qτ (t, m + 1) − Qτ (t, m) ⎤ ⎡ 0 ⎥ ⎢ 0 ⎥, =⎢ ⎦ ⎣√ T 3γ Bτ Qτ (t, m + 1)Aτ

] ,

(10.29)

10.4 σ -Error Mean Square Stability Analysis

193

Ξ3τ = diag{−Qτ (t, m + 1), −BτT Qτ (t, m + 1)Bτ , −BτT Qτ (t, m + 1)Bτ }, [ [ ] ] W¯ τ = Wτ 0 0 0 0 , Ξτ = Ξ1τ Ξ2τ , ⎡ ∗ ⎤Ξ3τ 0 ⎥ ⎢ 0 ⎥ ⎢ √ ⎥ ⎢ Q (t, m + 1)M 3 τ τ ¯ Mτ = ⎢√ ⎥. ⎥ ⎢ T ⎣ 3Bτ Qτ (t, m + 1)Mτ ⎦ 0 ¯ τ < 0 is equivalent to (10.21). Recalling (10.21), (10.26)–(10.28) and Thus, Θ using Schur complement, we have x¯ T (k)Θτ x¯ (k) = x T (k)Θ1τ x(k) + 2x T (k)Θ2τ xc (k − 1) + xcT (k − 1)Θ3τ xc (k − 1) < 0,

(10.30)

which means Θτ < 0. Furthermore, one has (10.25) < 0 and T E {(A¯τ )t Q¯τ (t, t)(A¯τ )t } − Q¯τ (t, 0) < 0.

(10.31)

Define Qτ = Qτ (t, t), Q˜τ = Q˜τ (t, t). According to (10.22) and (10.31), we can obtain that E {(A¯τ )t Q¯τ (A¯τ )t } − h τ Q¯τ (t, t) < 0. T

(10.32)

Let Qτ = Pτ . Then E {(A¯τ )t P¯τ (A¯τ )t } − h τ P¯τ < 0, T

(10.33)

which means that (10.15) holds. It is worth noting that T τ −1

max Σn=0

T τ −1

max = Σn=0

τ

T T Tmax ¯ τ (ν, n + 1)A¯τ − Q ¯ τ (ν, n)](A¯τ )n } E {(A¯τ )n [Σν=n+1 A¯τ Q τ

T Tmax E {(A¯τ )n [Σν=n+1 Λτ (ν)](A¯τ )n },

¯ τ = diag{Qτ , Qτ }, and Λτ (ν) by replacing Θτ with the corresponding variwhere Q able. Here, Qτ (t, m + 1) and Qτ (t, m) are replaced by Qτ (ν, n + 1)) and Qτ (ν, n), respectively.

194

10 Sliding Mode Control Under Denial-of-Service Attacks

Similarly, it follows from Eq. (10.23) that T τ −1

max Σn=0

τ

T T Tmax ¯ τ (ν, n + 1)A¯τ − Q ¯ τ (ν, n)](A¯τ )n } < 0, (10.34) E {(A¯τ )n [Σν=n+1 A¯τ Q

which is the same as τ

T T Tmax ν−1 ¯ τ (ν, n + 1)A¯τ − Q ¯ τ (ν, n)](A¯τ )n } < 0, Σν=1 Σn=0 E {(A¯τ )n [A¯τ Q

indicating that Tτ

max ¯ τ (ν, ν)(A¯τ )ν − Q ¯ τ (ν, 0)} < 0. (A¯τ )ν Q E {Σν=1

T

(10.35)

Combing (10.24) and (10.35), we have τ

T Tmax ¯ τ (ν, ν)(A¯τ )ν } − Q¯τ (t, t) < 0. E {(A¯τ )ν Q Σν=1 τ

0.

¯ τ (ν, ν) = P ¯ τ (ν) gives rise to Σ Tmax E {(A¯T )ν P ¯ τ (ν)(A¯τ )ν } − Q¯τ (t, t) < Then, Q τ ν=1

Based on Qτ = Qτ (t, t) = Pτ , (10.16) is achieved. Therefore, system (10.14) is σ -error mean square stable for all possible uncertainties. ∎ Remark 10.5 The probability density function ρτ ω (ν) is related to the current mode τ and the next mode ω. In fact, if the probability density function only depends on the current mode τ , then it will degenerate into the traditional λτ (ν) = Pr (Sn+1 = ν|Pn = τ ) = Σω∈Γ πτ ω (ν) = Σω∈Γ θτ ω ρτ ω (ν). Obviously, different ρτ ω (ν) may lead to the same λτ (ν) under appropriate θτ ω . Thus, the traditional probability density function λτ (ν) can be only one type. Therefore, it is more reasonable to apply the semi-Markovian kernel method for the σ -error mean square stability criteria. τ ], Remark 10.6 The slack matrices are adopted, i.e. {Qτ (t, m) > 0}, ∀t ∈ N[1,Tmax τ ] , ∀n ∈ N[0,ν−1] for the gain Gτ solu∀m ∈ N[0,t] and {Qτ (ν, n) > 0}, ∀ν ∈ N[1,Tmax tion. Since the matrix inequalities (10.21)–(10.24) can be calculated off-line, it is τ for a reasonable calculation cost. important to select an appropriate value Tmax

10.5 Reachability Analysis In this section, a mode-dependent discrete-time sliding mode control law is designed to ensure that the states arrive at the sliding surface. Theorem 10.3 Consider the sliding function (10.9) and the gain Gτ , ∀τ ∈ Γ in Theorem 10.2. The state trajectories can be driven onto the sliding manifold region by the following discrete-time sliding mode control law:

10.5 Reachability Analysis

195

u(k) = − (Gτ Bτ )−1 [Gτ Aτ x(k) + (∥Gτ ∥∥Mτ ∥∥Wτ ∥∥x(k)∥ −1 + λ¯ F)sgn(s(k))],

(10.36)

where F = (1 − λ¯ )(∥Gτ Aτ ∥∥xc (k − 1)∥ + . ), . > 0. Proof Choose the Lyapunov functional as: Vs (k, τ ) =

1 T s (k)s(k). 2

(10.37)

It follows from (10.10) that Δs(k, τ ) = s(k + 1) − s(k) = λ¯ Gτ [(Aτ + ΔAτ )x(k) + Bτ u(k)] + (1 − λ¯ )Gτ Aτ xc (k − 1) − s(k). Based on ∥ΔAτ ∥ ≤ ∥Mτ ∥∥Wτ ∥, and (10.36)–(10.37), one has the increment of Vs (k, τ ) as 1 ΔVs (k, τ ) = s T (k)Δs(k, τ ) + Δs T (k, τ )Δs(k, τ ) 2 ≤λ¯ s T (k)Gτ Aτ x(k) + λ¯ s T (k)Gτ Bτ u(k) + λ¯ ∥s(k)∥∥Gτ ∥∥ΔAτ ∥∥x(k)∥ 1 + (1 − λ¯ )s T (k)Gτ Aτ xc (k − 1) − s T (k)s(k) + Δs T (k, τ )Δs(k, τ ) 2 1 ≤ s T (k)[(1 − λ¯ )Gτ Aτ xc (k − 1) − Fsgn(s(k))] − s T (k)s(k) + Δs T (k, τ )Δs(k, τ ) 2 ≤(1 − λ¯ )∥s(k)∥∥Gτ Aτ ∥∥xc (k − 1)∥ − Fs T (k)sgn(s(k)) 1 − s T (k)s(k) + Δs T (k, τ )Δs(k, τ ) 2 1 T ≤ − . ∥s(k)∥ + Δs (k, τ )Δs(k, τ ) − s T (k)s(k). (10.38) 2

Note that ΔVs (k, τ ) < 0, with an appropriate positive parameter . , indicating that Δs(k, τ ) is bounded and the states can be driven onto the sliding region. ∎ Remark 10.7 It is a common fact that the trajectories of sliding mode control systems in continuous-time domain can reach the predetermined sliding surface in a limited time and then keep moving on it for any initial states. However, for discretetime domain, the states will cross over the sliding surface to form a sawtooth motion

196

10 Sliding Mode Control Under Denial-of-Service Attacks

due to the existence of sampling. During the sliding phase, the sliding mode dynamics can remain in the designated area near the equilibrium point, i.e. the so-called quasi sliding mode motion (10.13) is generated.

10.6 Simulation In this section, we consider a uncertain discrete-time electronic throttle control system from [25]. The state vector is considered as x(k) = [ψ(k), ω(k), i (k)]T , where ψ(k), ω(k), and i (k) stand for angular position of valve, angular velocity of valve, and electrical current, respectively. Set the sampling time as 0.5 s. The electronic throttle control system is modeled as S-MSSs with three modes: normal case (τ = 1), soft failure case (τ = 2), and hard failure case (τ = 3). Then, the system parameters are described as ⎡

(τ ) 0 1 a12

⎤

⎡

0

⎤

⎥ ⎢ (τ ) (τ ) (τ ) ⎥ ⎢ ⎢ 0 ⎥ , Fτ = sin(k), a a a ⎥ Aτ = ⎢ ⎦ ⎣ 21 22 23 ⎦ , Bτ = ⎣ (τ ) (τ ) b3(τ ) 0 a32 a33 Mτ = [0.01 0.01 0.02]T , Wτ = [0.1 0.1 0.2], where the relevant parameters are shown in Table 10.1. Thus, the discrete-time semi-Markovian kernel is given by πτ ω (ν) = θτ ω ρτ ω (ν), ∀τ, ω ∈ {1, 2, 3}, with θτ ω = [0 0.7 0.3; 0.4 0 0.6; 0.5 0.5 0] and ρ12 (ν) =

0.2ν · 0.810−ν · 10! 0.4ν · 0.610−ν · 10! , ρ13 (ν) = , (10 − ν)! · ν! (10 − ν)! · ν!

ρ21 (ν) =

0.3ν · 0.710−ν · 10! 0.8 0.8 , ρ23 (ν) = 0.2(ν−1) − 0.2ν , (10 − ν)! · ν!

ρ31 (ν) = 0.4(ν−1) − 0.4ν , ρ32 (ν) = 0.3(ν−1) − 0.3ν . 1.3

1.3

0.8

0.8

τ = 3(τ = 1, 2, 3). Let ετ = 0.1, h 1 = 1, h 2 = 2, h 3 = 5, λ¯ = 0.75, and Tmax

Table 10.1 Parameters with different modes (τ ) (τ ) (τ ) a21 a22 Parameter a12 τ =1 τ =2 τ =3

0.0109 0.0152 0.0111

–0.1165 –0.0298 –0.0229

–0.8072 0.2737 0.7779

(τ )

(τ )

(τ )

(τ )

a23

a32

a33

b3

–1.5061 –1.1921 –0.1899

–0.2285 –0.3584 –0.6315

0.7967 0.3835 0.4178

–0.0948 –0.0735 –0.2441

10.6 Simulation

197

By solving Theorem 10.2, we have ⎡

⎤ 25.2321 0.2406 0.1205 Q1 = ⎣ 0.2406 24.1876 2.5903 ⎦ , 0.1205 2.5903 38.1031 ⎡ ⎤ 22.5795 0.1258 0.0116 Q2 = ⎣ 0.1258 22.2028 0.9010 ⎦ , 0.0116 0.9010 32.7607 ⎡ ⎤ 21.7157 0.1253 0.1277 Q3 = ⎣ 0.1253 21.9135 0.7961 ⎦ . 0.1277 0.7961 30.3058 Furthermore, one has Gτ = BτT Qτ given as [ ] G1 = −0.0114 −0.2456 −3.6122 , [ ] G2 = −0.0009 −0.0662 −2.4079 , [ ] G3 = −0.0312 −0.1943 −7.3976 . The sliding mode control law is proposed as −1 u(k) = −(Gτ Bτ )−1 [Gτ Aτ x(k) + (∥Gτ ∥∥Mτ ∥∥Wτ ∥∥x(k)∥ + λ¯ F)sgn(s(k))],

where F = (1 − λ¯ )(∥Gτ Aτ ∥∥xc (k − 1)∥ + . ), . > 0. For given x0 = [0.2 − 0.8 0.5]T , under 100 realizations, the states in the openloop case can not achieve σ -error mean square stability for h τ > 0 in Fig. 10.3. The state x(k) in the closed-loop case is given in Figs. 10.4, 10.5 and 10.6. Also, the chattering is necessarily and cannot be completely eliminated due to the existence of inertia and lagging. In order to alleviate the chattering effect of sliding mode control, sgn(s(k)) is sgn(s(k)) . Figures 10.7 and 10.8 describe the sliding mode function replaced by sign(s(k))+0.001 and the control input. Obviously, the state xc (k) received by the controller is plotted in Fig. 10.9, which indicates the non-chattering phenomena of the state information via the network measurement transmission channel. From Figs. 10.4, 10.5, 10.6, 10.7, 10.8 and 10.9, the σ -error mean square stability is realized. τ , and λ¯ . The details are Next, we will discuss the relationship between h τ , Tmax shown in Tables 10.2 and 10.3. It is observed that the maximum value of λ¯ is increasing along with increasing h τ . This means that the tolerance to random denial-ofservice attacks is related to the parameter h τ . Thus, the parameter λ¯ max is obtained by choosing appropriate parameter h τ . From Tables 10.2 and 10.3, the maximum value of sojourn time is increasing along with h τ under the framework of σ -error mean square stability.

198

10 Sliding Mode Control Under Denial-of-Service Attacks 104

3

2

1

0

-1

-2

-3

0

2

4

6

8

10

12

14

16

18

14

16

18

Fig. 10.3 The states in the open-loop system (100 realizations) 5 4 3 2 1 0 -1 -2

0

2

4

6

8

10

Fig. 10.4 Responses of the state x1 (k) (100 realizations)

12

10.6 Simulation

199

1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4

0

2

4

6

8

10

12

14

16

18

12

14

16

18

Fig. 10.5 Responses of the state x2 (k) (100 realizations) 1

0.5

0

-0.5

-1

-1.5

-2

0

2

4

6

8

10

Fig. 10.6 Responses of the state x3 (k) (100 realizations)

200

10 Sliding Mode Control Under Denial-of-Service Attacks 15

10

5

0

-5

-10

0

2

4

6

8

10

12

14

16

18

10

12

14

16

18

Fig. 10.7 Sliding mode function (100 realizations) 4 3 2 1 0 -1 -2 -3 -4

0

2

4

6

Fig. 10.8 Control input (100 realizations)

8

10.7 Conclusion

201

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

0

2

4

6

8

10

12

14

16

18

Fig. 10.9 Responses of the state xc (k) (100 realizations) 1 2 3 = 2, Tmax = Tmax = 3(σ = 1.007) for different Table 10.2 Maximum value of λ¯ WITH Tmax h τ , τ = 1, 2, 3 λ¯ max h1 = h2 = 1 h 1 = 1, h 2 = 1.2 h 1 = h 2 = 1.2

h3 = 1

0.9542

0.9544

0.9553

τ = 3(σ = 0.629) for different h τ , τ = 1, 2, 3 Table 10.3 Maximum value of λ¯ with Tmax ¯λmax h1 = h2 = 1 h 1 = 1, h 2 = 2 h1 = h2 = 2

h3 = 5

0.9485

0.9488

0.9528

1 2 3 Under the same values (i.e. h 1 = 1, h 2 = 2, h 3 = 5, Tmax = Tmax = Tmax = 3) between this chapter and Ref. [27], the discrete-time sliding mode control can make the system states with fast convergence about 10s, while the traditional state feedback control [27] realizes the states convergence about 35 s in Fig. 10.10.

10.7 Conclusion In this chapter, the discrete-time sliding mode control has been investigated for networked uncertain S-MSSs in presence of random denial-of-service attacks. By exploiting the discrete-time semi-Markovian kernel, a new mode-dependent sliding

202

10 Sliding Mode Control Under Denial-of-Service Attacks 40 30 20 10 0 -10 -20 -30

0

5

10

15

20

25

30

35

40

45

50

Fig. 10.10 State responses under the traditional state feedback control [27]

mode function has been designed, which makes full use of the statistics of denialof-service attacks. Sufficient conditions have been achieved to ensure the σ -error mean square stability. The specified discrete-time sliding mode control law has been synthesized to realize the finite-time reachability of the sliding region. In the future, the discrete-time sliding mode control law will be extended to networked S-MSSs subject to channel fading.

References 1. Phat, V.N., Ratchagit, K.: Stability and stabilization of switched linear discrete-time systems with interval time-varying delay. Nonlinear Anal. Hybrid Syst 5(4), 605–612 (2011) 2. Rajchakit, G., Rojsiraphisal, T., Rajchakit, M.: Robust stability and stabilization of uncertain switched discrete-time systems. Adv. Differ. Equ. 2012, Article ID 134 (2012) 3. Ratchagit, K., Phat, V.N.: Stability criterion for discrete-time systems. J. Inequal. Appl. 2010, Article ID 201459 (2010) 4. Park, J.H., Shen, H., Chang, X.H., Lee, T.H.: Recent Advances in Control and Filtering of Dynamic Systems with Constrained Signals. Springer, Switzerland (2018) 5. Muthukumar, P., Arunagirinathan, S., Lakshmanan, S.: Nonfragile sampled-data control for uncertain networked control systems with additive time-varying delays. IEEE Trans. Cybern. 49(4), 1512–1523 (2019) 6. Gao, L.S., Fu, J.Q., Li, F.Q.: Output-based security control of NCSs under resilient eventtriggered mechanism and DoS attacks. Int. J. Control Autom. Syst. 19(4), 1519–1527 (2021) 7. Zhang, J., Peng, C.: Guaranteed cost control of uncertain networked control systems with a hybrid communication scheme. IEEE Trans. Syst. Man Cybern.: Syst. 50(9), 3126–3135 (2020)

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8. Hu, J.W., Zhan, X.S., Wu, J., Yan, H.C.: Optimal tracking performance of NCSs with time-delay and encoding-decoding constraints. Int. J. Control Autom. Syst. 18(4), 1012–1022 (2020) 9. Yan, H.C., Wang, J.N., Zhang, H., Shen, H., Zhan, X.S.: Event-based security control for stochastic networked systems subject to attacks. IEEE Trans. Syst. Man Cybern.: Syst. 50(11), 4643–4654 (2020) 10. Liu, J.L., Wu, Z.G., Yue, D., Park, J.H.: Stabilization of networked control systems with hybriddriven mechanism and probabilistic cyber attacks. IEEE Trans. Syst. Man Cybern.: Syst. 51(2), 943–953 (2021) 11. Shen, B., Zhang, Z.D., Wang, D., Li, Q.: State-saturated recursive filter design for stochastic time-varying nonlinear complex networks under deception attacks. IEEE Trans. Neural Netw. Learn. Syst. 31(10), 3788–3800 (2020) 12. Liu, L., Ma, L.F., Zhang, J., Bo, Y.M.: Distributed non-fragile set-membership filtering for nonlinear systems under fading channels and bias injection attacks. Int. J. Syst. Sci. 52(6), 1192–1205 (2021) 13. Guo, H.B., Pang, Z.H., Sun, J., Li, J.: An output-coding-based detection scheme against replay attacks in cyber-physical systems. IEEE Trans. Circuits Syst. II Express Briefs 68(10), 3306– 3310 (2021) 14. Hu, S.L., Yue, D., Han, Q.L., Xie, X.P., Chen, X.L., Dou, C.X.: Observer-based event-triggered control for networked linear systems subject to denial-of-service attacks. IEEE Trans. Cybern. 50(5), 1952–1964 (2020) 15. Liu, D., Ye, D.: Pinning-observer-based secure synchronization control for complex dynamical networks subject to DoS attacks. IEEE Trans. Circuits Syst. I Regul. Pap. 67(12), 5394–5404 (2020) 16. Ding, L., Han, Q.L., Ning, B.D., Yue, D.: Distributed resilient finite-time secondary control for heterogeneous battery energy storage systems under denial-of-service attacks. IEEE Trans. Ind. Inf. 16(7), 4909–4919 (2020) 17. Li, X.Y., Wang, B.J., Zhang, L., Ma, X.H.: H∞ control with multiple packets compensation scheme for T-S fuzzy systems subject to cyber attacks. Int. J. Control Autom. Syst. 19(1), 230–240 (2021) 18. Gu, Z., Ahn, C.K., Yue, D., Xie, X.P.: Event-triggered H∞ filtering for T-S fuzzy-model-based nonlinear networked systems with multisensors against DoS attacks. IEEE Trans. Cybern. 52(6), 5311–5321 (2022) 19. Zha, L.J., Liu, J.L., Cao, J.D.: Resilient event-triggered consensus control for nonlinear mutiagent systems with DoS attacks. J. Franklin Inst. 356(13), 7071–1090 (2019) 20. Ding, D.R., Wang, Z.D., Wei, G.L., Alsaadi, F.E.: Event-based security control for discrete-time stochastic systems. IET Control Theory Appl. 10(15), 1808–1815 (2016) 21. Liu, J.L., Yin, T.T., Shen, M.Q., Xie, X.P., Cao, J.: State estimation for cyber-physical systems with limited communication resources, sensor saturation and denial-of-service attacks. ISA Trans. 104, 101–114 (2020) 22. Li, X., Wei, G.L., Wang, L.C.: Distributed set-membership filtering for discrete-time systems subject to denial-of-service attacks and fading measurements: A zonotopic approach. Inf. Sci. 547, 49–67 (2021) 23. Barbu, V., Limnios, N.: Empirical estimation for discrete-time semi-Markov processes with applications in reliability. J. Nonparametric Stat. 18(7–8), 483–498 (2006) 24. Zhang, L.X., Yang, T., Colaneri, P.: Stability and stabilization of semi-Markov jump linear systems with exponentially modulated periodic distributions of sojourn time. IEEE Trans. Autom. Control 62(6), 2870–2885 (2017) 25. Ning, Z.P., Zhang, L.X., Mesbah, A.: Stability analysis and stabilization of discrete-time nonhomogeneous semi-Markov jump linear systems: a polytopic approach. Automatica 120, Article ID 109080 (2020) 26. Niu, Y.G., Ho, D.W.C.: Design of sliding mode control subject to packet losses. IEEE Trans. Autom. Control 55(11), 2623–2628 (2010) 27. Zhang, L.X., Leng, Y.S., Colaneri, P.: Stability and stabilization of discrete-time semi-Markov jump linear systems via semi-Markov kernel approach. IEEE Trans. Autom. Control 61(2), 503–508 (2016)

Chapter 11

Sliding Mode Control Under Deception Attacks

In this chapter, the issue of observer-based sliding mode control is addressed for fuzzy stochastic semi-Markovian switching systems (S-MSSs) under cyber attacks. The underlying control signal is vulnerable to the deception attacks, in which the opponents inject fake data into the signal in a probabilistic way. Inspired by the unmeasurable states, an integral sliding surface dependent on a fuzzy observer is designed. Then, in consideration of the deception attacks, the conditions are established to achieve the stochastic stability for fuzzy sliding mode dynamics. Moreover, the proposed observer-based sliding mode control law guarantees the reachability of the sliding region. In the end, a single-link robot arm model is given to demonstrate the availability of the given results.

11.1 Introduction In the previous chapter, it is investigated based on the denial-of-service attacks. Different from denial-of-service attacks, deception attacks belong to a classical type of network attacks, which modify the data integrity of packets transmitted between different parts of the network. Compared with denial-of-services attacks, the security of data resource is more vulnerable to corruption under deception attacks. Due to the complex and hidden characteristics of deception attacks, it is difficult to be detected by the general attack detection mechanism. In recent years, the research for deception attacks has become a hot topic [1–5]. This chapter will solve the sliding mode control for fuzzy S-MSSs under deception attacks. The main contributions of this work are given as follows: (i) Compared with mode-dependent sliding surface [1, 6–14], a common sliding surface is constructed to reduce the repetitive jumps. (ii) Different from S-MSSs under the ideal © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 W. Qi and G. Zong, Control Synthesis for Semi-Markovian Switching Systems, Studies in Systems, Decision and Control 465, https://doi.org/10.1007/978-981-99-0317-7_11

205

206

11 Sliding Mode Control Under Deception Attacks

network transmission [6–12, 15–25], stochastic stability conditions are obtained for the corresponding system under deception attacks. (iii) Based on a fuzzy observer, the sliding mode control law is proposed such that the state trajectories can be driven onto the sliding region within a finite-time interval.

11.2 Problem Statements and Preliminaries Consider the T-S fuzzy S-MSSs described as Plant Rule ι: IF l1 (t) is Qι1 , l2 (t) is Qι2 , . . ., and lb (t) is Qιb , THEN z˙ (t) = (Aι (rt ) + ΔAι (rt , t))z(t) + B(rt )u e (t), y(t) = C(rt )z(t),

(11.1)

where z(t) = [z 1 (t), z 2 (t), . . . , z b (t)]T ∈ Rb and u e (t) ∈ Rm denote the state vector and the control input; l1 (t), l2 (t), . . . , lb (t) are selected as the premise variables; Qιj (ι = 1, 2, . . . , a; j = 1, 2, . . . , b) are the fuzzy sets. Aι (rt ), B(rt ) and C(rt ) are the appropriately dimensional matrices. ΔAι (rt , t) is given as ΔAι (rt , t) = Dι (rt )Hι (rt , t)Nι (rt ), where Hι (rt , t) satisfies HιT (rt , t)Hι (rt , t) ≤ I . {rt , t ≥ 0} denotes the semi-Markovian process in S = {1, 2, . . . , s} with transition probability: ⎧ Pr {rt+δ = θ |rt = τ } =

λτ θ (ρ)h + o(δ), τ /= θ, 1 + λτ θ (ρ)h + o(δ), τ = θ,

where ρ means the sojourn time, o(δ) is defined as limρ→0 o(δ)/δ ∑ = 0, λτ θ (ρ) ≥ 0 denotes the transition rate from τ to θ for τ /= θ , and λτ τ (ρ) = − sθ =1,τ /=θ λτ θ (ρ) < 0 for each τ ∈ S. For simplicity of subsequent expression, define Aι (rt ) . Aι,τ , B(rt ) . Bτ , and C(rt ) . Cτ , for rt = τ . By using fuzzy blending method, the global model of the system (11.1) is described as ∑ z˙ (t) = aι=1 .ι (l(t))[(Aι,τ + ΔAι,τ (t))z(t) + Bτ u e (t)], y(t) = Cτ z(t), where .ι (l(t)) is the membership function as .ι (l(t)) =

(11.2)

.b

j=1 Qιj (l j (t)) .b ι=1 j=1 Qιj (l j (t))

∑a

and

Qιj (l j (t)) is the grade of membership of l j (t) in Qιj . For all t > 0, it is satisfied ∑ that 0 ≤ .ι (l(t)) ≤ 1 and aι=1 .ι (l(t)) = 1.

11.2 Problem Statements and Preliminaries

207

Fig. 11.1 Block diagram for S-MSSs under deception attacks

Under the deception attacks, the control input received by the actuator is given as u e (t) = u(t) + ψ(z(t), t, τ ),

(11.3)

where u(t) is the designed input and ψ(z(t), t, τ ) = ζ (t)G τ (t)Φ(z(t), t, τ ) means the stochastic deception attacks [1]. The attack matrix G τ (t) is unknown and satisfies ||G τ (t)|| ≤ ξτ , with known constant ξτ . φ(z(t), t, τ ) satisfies ||Φ(z(t), t, τ )|| ≤ φ(z(t), t, τ ), with known function φ(z(t), t, τ ). ζ (t) obeys the Bernoulli distribution with Pr {ζ (t) = 1} = E {ζ (t)} = ζ¯ , and Pr {ζ (t) = 0} = 1 − E {ζ (t)} = 1 − ζ¯ , in which ζ (t) is known with ζ¯ ∈ [0, 1]. Figure 11.1 describes the block diagram for S-MSSs under deception attacks. Since the system states are considered unavailable, a state observer in the following form is proposed as ˆ ι1 , lˆ2 (t) is Q ˆ ι2 , . . ., and lˆb (t) is Q ˆ ιb , THEN Observer Rule ι: IF lˆ1 (t) is Q z˙ˆ (t) = Aι,τ zˆ (t) + Bτ u e (t) + L ι,τ (y(t) − Cτ zˆ (t)), yˆ (t) = Cτ zˆ (t),

(11.4)

where zˆ (t) ∈ Rb denotes the state estimation and L ι,τ ∈ Rb×s represents the observer gain.

208

11 Sliding Mode Control Under Deception Attacks

Thus, the fuzzy observer (11.4) is inferred as ∑ ˆ z˙ˆ (t) = aι=1 . ˆ ι (l(t))[A ι,τ zˆ (t) + Bτ u e (t) + L ι,τ (y(t) − C τ zˆ (t))], yˆ (t) = Cτ zˆ (t).

(11.5)

Define ε(t) . z(t) − zˆ (t). Combining (11.2) and (11.5), the error dynamics can be got as ε˙ (t) =

∑a

ˆ . ˆ ι (l(t))[ A˜ ι,τ ε(t) + ΔAι,τ (t)ˆz (t)] + q(t),

ι=1

(11.6)

∑ ˆ ˆ ι (l(t)))[(A where A˜ ι,τ = Aι,τ − L ι,τ Cτ and q(t) = aι=1 (.ι (l(t)) − . ι,τ + ΔAι,τ (t))z(t) + Bι,τ u e (t)]. q(t) is seen as the disturbance to satisfy ||q(t)|| ≤ m ||ε(t)||, where m is a known scalar [26]. Remark 11.1 It is noted that since the variables are defined in a compact set and go to zero if lˆ → l, q(t) is a difference between the real state and the estimated state, called the time-varying difference. In fact, q(t) acts like an unstructured vanishing perturbation that is supposed to be bounded as ||q(t)|| ≤ m ||ε(t)|| for m > 0. As long as the membership function is smooth and the variables are defined on a compact set, there exists a scalar m > 0 to satisfy this assumption ||q(t)|| ≤ m ||ε(t)||. Therefore, the boundary constant m can be obtained by solving the optimization problem as ˆ ι,τ +ΔAι,τ (t))z(t)+Bτ u e (t)] ||. m = maxz,ˆz ,u e || ∂(.ι (l(t))−.ˆ ι (l(t)))[( A∂ε(t) Definition 11.1 ([27]) System (11.2) is said to be stochastically stable if there exists a positive constant T (z(0), r0 ) such that ⎧. lim E

t→∞

t

⎫ ||z(s)||2 ds|(z(0), r0 ) ≤ T (z(0), r0 ).

0

11.3 Stochastic Stability Analysis Choose the common sliding surface as s(t) = W zˆ (t),

(11.7)

where W =

∑s τ =1

γ¯ τ BτT , 0 ≤ γ¯ τ ≤ 1.

Construct the following sliding mode control law u(t) = BτT Pτ zˆ (t) − (ζ¯ ξτ φ(z(t), t, τ ) + ρτ )sgn((Fτ W Bτ )T s(t)),

(11.8)

11.3 Stochastic Stability Analysis

209

where the matrices Pτ > 0, Fτ > 0, and the scalar ρτ > 0 will be determined later. Therefore, the closed-loop system can be obtained as: { ∑ ˆ z˙ˆ (t) = aι=1 . ˆ ι (l(t)) Aι,τ zˆ (t) + Bτ [BτT Pτ zˆ (t) − (ζ¯ ξτ φ(z(t), t, τ ) + ρτ ) sgn((Fτ W Bτ )T s(t)) + ζ (t)G τ (t)Φ(z(t), t, τ )] } + L ι,τ (y(t) − Cτ zˆ (t)) .

(11.9)

Theorem 11.1 If there exist positive-definite symmetric matrices Pτ and Fτ , appropriate matrix Yι,τ , and constants ητ > 0, κτ > 0 , for ∀τ ∈ S, such that Pτ Bτ = W T Fτ W Bτ , ⎡ 1 ⎤ 0 0 0 Ξι,τ 0 Pτ ⎢ ∗ Ξ 2 0 C T Y T Pτ Pτ Dι,τ ⎥ ι,τ τ ι,τ ⎢ ⎥ ⎢ ∗ ∗ −Pτ 0 0 0 ⎥ ⎢ ⎥ < 0, Ξι,τ = ⎢ 0 0 ⎥ ∗ −Pτ ⎢ ∗ ∗ ⎥ ⎣ ∗ ∗ 0 ⎦ ∗ ∗ −ητ I ∗ ∗ ∗ ∗ ∗ −κτ I

(11.10)

(11.11)

where 1 = Ξι,τ 2 Ξι,τ =

∑s θ =1 ∑s θ =1

T λ¯ τ θ Pθ + H e(Pτ Aι,τ + Pτ Bτ BτT Pτ ) + κτ Nι,τ Nι,τ ,

λ¯ τ θ Pθ + H e(Pτ A˜ ι,τ ) + ητ m 2 I,

then the closed-loop system (11.9) with (11.6) is stochastically stable. Besides, the observer gain is computed as L ι,τ = Pτ−1 Yι,τ . Proof Select the Lyapunov function: V1 (ˆz (t), ε(t), τ ) = V2 (ˆz (t), τ ) + V3 (ε(t), τ ), where V2 (ˆz (t), τ ) = zˆ T (t)Pτ zˆ (t) and V3 (ε(t), τ ) = ε T (t)Pτ ε(t). For a smaller ε˜ , it is got that zˆ (t + ε˜ ) = zˆ (t) + z˙ˆ (t)˜ε + o(˜ε ). Then, one has the weak infinitesimal operator as E[℘V2 (ˆz (t), τ )] ⎫ ⎧ 1 E[V2 (ˆz (t + ε˜ ), rt+˜ε )|ˆz (t), rt = τ ] − V2 (ˆz (t), τ ) = lim+ ε˜ →0 ε ˜

(11.12)

210

11 Sliding Mode Control Under Deception Attacks

⎧ [∑ s 1 Pr {rt+˜ε = θ|rt = τ }V2 (ˆz (t + ε˜ ), rt+˜ε ) E θ =1,τ /=θ ε˜ →0 ε ˜ ] ⎫ + Pr {rt+˜ε = τ |rt = τ }V2 (ˆz (t + ε˜ ), rt+˜ε ) − V2 (ˆz (t), τ ) ⎧ [∑ s 1 vτ θ (Vτ (ρ + ε˜ ) − Vτ (ρ)) T = lim+ zˆ (t + ε˜ )Pθ zˆ (t + ε˜ ) E θ =1,τ /=θ ε˜ →0 ε 1 − Vτ (ρ) ˜ ] ⎫ 1 − Vτ (ρ + ε˜ ) T zˆ (t + ε˜ )Pτ zˆ (t + ε˜ ) − zˆ T (t)Pτ zˆ (t) + 1 − Vτ (ρ) ∑s T = 2ˆz (t)Pτ z˙ˆ (t) + λ¯ τ θ zˆ T (t)Pθ zˆ (t) θ =1 { ∑ ˆ z T (t)Pτ Aι,τ zˆ (t) + Bτ [BτT Pτ zˆ (t) − (ζ¯ ξτ φ(z(t), t, τ ) ˆ ι (l(t))ˆ = 2 aι=1 .

= lim+

+ ρτ )sgn((Fτ W Bτ )T s(t)) + ζ (t)G τ (t)Φ(z(t), t, τ )] } ∑s + L ι,τ (y(t) − Cτ zˆ (t)) + λ¯ τ θ zˆ T (t)Pθ zˆ (t),

(11.13)

θ =1

where

λ¯ τ θ = E [λτ θ (ρ)] = 1−Vτ (ρ+˜ε ) 1−Vτ (ρ)

.∞ 0

λτ θ (ρ)vτ (ρ)dρ,

limε˜ →0+

Vτ (ρ+˜ε )−Vτ (ρ) ε˜ (1−Vτ (ρ))

= λτ (ρ),

= 1, vτ (ρ) is the probability distribution function, and Vτ (ρ) limε˜ →0+ is the cumulative distribution function. Similarly, the weak infinitesimal operator of ℘V3 (ε(t), τ ) is given as ∑s E[℘V3 (ε(t), τ )] = 2ε T (t)Pτ ε˙ (t) + λ¯ τ θ ε T (t)Pθ ε(t) θ =1 ∑ T ˆ = 2 aι=1 . ˆ ι (l(t))ε (t)Pτ [ A˜ ι,τ ε(t) + ΔAι,τ (t)ˆz (t)] + 2ε T (t)Pτ q(t) ∑s λ¯ τ θ ε T (t)Pθ ε(t). (11.14) + θ =1

Due to − 2ˆz T (t)Pτ Bτ ζ¯ ξτ φ(z(t), t, τ )sgn((Fτ W Bτ )T s(t)) + 2ˆz T (t)Pτ Bτ ζ (t)G τ (t)Φ(z(t), t, τ ) ≤ − 2ζ¯ ξτ φ(z(t), t, τ )|(Fτ W Bτ )T s(t)| + 2ζ¯ ξτ φ(z(t), t, τ )||(Fτ W Bτ )T s(t)|| < 0, − 2ˆz T (t)Pτ Bτ ρτ ζ¯ ξτ φ(z(t), t, τ )sgn((Fτ W Bτ )T s(t)) = − 2ρτ ζ¯ ξτ φ(z(t), t, τ )|(Fτ W Bτ )T s(t)| < 0.

(11.15)

For κτ > 0 and ητ > 0, there exist T T 2ε T (t)Pτ Δ Aι,τ (t)ˆz (t) ≤ κτ−1 ε T (t)Pτ Dι,τ Dι,τ PτT ε(t) + κτ zˆ T (t)Nι,τ Nι,τ zˆ (t),

2ε T (t)Pτ q(t) ≤ ητ−1 ε T (t)Pτ PτT ε(t) + ητ m 2 ε T (t)ε(t), T 2ˆz T (t)Pτ L ι,τ Cτ ε(t) ≤ zˆ T (t)Pτ Pτ−1 PτT zˆ (t) + ε T (t)CτT L ι,τ Pτ L ι,τ Cτ ε(t).

11.3 Stochastic Stability Analysis

211

Thus, one has E[℘V1 (ˆz (t), ε(t), τ )] ∑a [ T ˆ ≤ . ˆ ι (l(t)) zˆ (t)H e(Pτ Aι,τ + Pτ Bτ BτT Pτ )ˆz (t) + zˆ T (t)Pτ Pτ−1 PτT zˆ (t) ι=1 ∑s T + ε T (t)CτT L ι,τ Pτ L ι,τ Cτ ε(t) + λ¯ τ θ zˆ T (t)Pθ zˆ (t) + ε T (t)H e(Pτ A˜ ι,τ )ε(t) +

κτ−1 ε T (t)Pτ Dι,τ

θ =1 T T Pτ ε(t) + κτ zˆ T (t)Nι,τ Nι,τ zˆ (t) ∑s ] T λ¯ τ θ ε (t)Pθ ε(t) . θ =1

T Dι,τ

+ ητ m 2 ε T (t)ε(t) +

+ ητ−1 ε T (t)Pτ PτT ε(t) (11.16)

Define Pτ L ι,τ = Yι,τ . Applying Schur complement to (11.11), there holds E[℘V1 (ˆz (t), ε(t), τ )] ≤

∑a ι=1

T ˆ . ˆ ι (l(t))ω (t)Ξι,τ ω(t) < 0,

[ ]T where ω(t) = zˆ T (t) ε T (t) . Furthermore, one has E[℘V1 (ˆz (t), ε(t), τ )] ≤ −σ ||ω(t)||2 < 0, [ ] where σ = minτ =1,2,...,s,ι=1,2,...,a λmin −Ξι,τ > 0. According to the mentioned, we have ⎧.

t

lim E

t→∞

0

⎫ 1 ||ω(i )||2 di ≤ V1 (ˆz (0), ε(0), τ ), σ

which implies that the closed-loop system (11.9) with (11.6) is stochastically . stable. Remark 11.2 By constructing the Lyapunov function candidate related to the error dynamics and the fuzzy observer, the stochastic stability analysis is given for the corresponding sliding mode dynamics in the above Theorem 11.1. Then, the closedloop system (11.9) with (11.6) is stochastically stable, which provides the foundation of sliding mode control law design. For the equality constraint (11.10), one has T race[(Pτ Bτ − W T Fτ W Bτ )T (Pτ Bτ − W T Fτ W Bτ )] = 0,

(11.17)

which means that (Pτ Bτ − W T Fτ W Bτ )T (Pτ Bτ − W T Fτ W Bτ ) < ωI, where ω > 0.

(11.18)

212

11 Sliding Mode Control Under Deception Attacks

Applying Schur complement lemma yields ] −ωI (Pτ Bτ − W T Fτ W Bτ )T < 0. ∗ −I

[

(11.19)

Therefore, the feasible problem of Theorem 11.1 is changed into the following optimization problem ω Pτ , Fτ , Yι,τ , ητ , κτ s.t. Inequalities (11.11) and (11.19).

min

(11.20)

11.4 Reachability Analysis In this section, under semi-Markovian switching and deception attacks, the sliding mode control mechanism is constructed to achieve the reachability of sliding region, which means that the sliding mode control law can force the states onto the sliding region within the finite-time interval. Theorem 11.2 Under the sliding mode control law (11.8), the state trajectory of system (11.1) is driven onto the sliding region in a finite time, where ρτ satisfies ρτ ||Fτ W Bτ || − β − d > 0,

(11.21)

where β = maxi∈S {||L ι,τ Cτ ||||ε(t)||} and d > 0. Proof Select the Lyapunov function: V4 (s(t), τ ) =

1 T s (t)Fτ s(t). 2

Then, one has the weak infinitesimal operator as E[℘V4 (s(t), τ )] 1 = lim+ {E[V4 (s(t + ε˜ ), rt+˜ε )|s(t), rt = τ ] − V4 (s(t), τ )} ε˜ →0 ε ˜ ⎧ [∑ s 1 = lim+ Pr {rt+˜ε = θ |rt = τ }V4 (s(t + ε˜ ), rt+˜ε ) E θ =1,τ /=θ ε˜ →0 ε ˜ ] ⎫ + Pr {rt+˜ε = τ |rt = τ }V4 (s(t + ε˜ ), rt+˜ε ) − V4 (s(t), τ )

(11.22)

11.5 Simulation

213

⎧ [∑ s vτ θ (Vτ (ρ + ε˜ ) − Vτ (ρ)) 1 T 1 s (t + ε˜ )Fθ s(t + ε˜ ) E θ =1,τ /=θ ε˜ →0 ε ˜ 1 − Vτ (ρ) 2 ] ⎫ 1 1 − Vτ (ρ + ε˜ ) 1 T s (t + ε˜ )Fτ s(t + ε˜ ) − s T (t)Fτ s(t) + 1 − Vτ (ρ) 2 2 ∑s 1 λ¯ τ θ s T (t)Fθ s(t) = s T (t)Fτ W z˙ˆ (t) + θ =1 2 { ∑ ˆ ˆ ι (l(t)) Aι,τ zˆ (t) + Bτ [BτT Pτ zˆ (t) − (ζ¯ ξτ φ(z(t), t, τ ) = s T (t)Fτ W aι=1 . } + ρτ )sgn((Fτ W Bτ )T s(t)) + ζ (t)G τ (t)Φ(z(t), t, τ )] + L ι,τ Cτ ε(t) 1 ∑s λ¯ τ θ s T (t)Fθ s(t) + θ =1 2 ≤ ||s(t)||[||Fτ W Aι,τ || + ||Fτ W Bτ BτT Pτ ||]||ˆz (t)|| + ||L ι,τ Cτ ||||ε(t)|| ∑s 1 ||λ¯ τ θ ||||Fθ W ||||ˆz (t)|| − ρτ ||s(t)||||Fτ W Bτ || + ||s(t)|| θ =1 2 ≤ ||s(t)||[αι,τ ||ˆz (t)|| − (ρτ ||Fτ W Bτ || − β − d)] − d||s(t)||. (11.23)

= lim+

∑ where αι,τ = ||Fτ W Aι,τ || + ||Fτ W Bτ BτT Pτ || + 21 ||s(t)|| sθ =1 ||λ¯ τ θ ||||Fθ W ||. Define the sliding region Ω = {ˆz (t) : αι,τ ||ˆz (t)|| ≤ ρτ ||Fτ W Bτ || − β − d}. Thus, one has E[℘V4 (s(t), τ )] ≤ −d||s(t)|| ≤ −

d. V4 (s(t), τ ), η

(11.24)

/ −1 τ) ] > 0. where η = λmax [(F 2 From the inequality (11.24), there exists an instant T ∗ satisfying T∗≤

2η . V4 (s(0), r0 ). d

(11.25)

Therefore, the reachability of the sliding region in stochastic sense can be achieved when T ≥ T ∗ . Remark 11.3 Due to the introduction of cyber-attacks, there exist some difficulties in deriving the corresponding Theorem 11.1 and designing the sliding mode control law in Theorem 11.2. In this chapter, the deception attacks are adopted to describe the cyber attacks in network channels, which satisfy the constraint condition (11.3).

11.5 Simulation Consider the single-link robot arm model form [1], where the dynamic equation D0 t )gL sin(ϕ(t)) − J(r ϕ(t) ˙ + J(r1t ) u e (t), where u e (t) is input and ϕ(t) ¨ = − M(r is ϕ(t) J(rt ) t) is angle position of arm. J(rt ) and M(rt ) represent moment of inertia and mass of

214

11 Sliding Mode Control Under Deception Attacks

payload, D0 and L are uncertain coefficient of viscous friction and length of the arm. And D0 and L are given by D0 = 10 and L = 2.5. g is acceleration of gravity with g = 9.81. M(rt ) and J(rt ) are with three different modes that obey the SemiMarkovian process {rt , t ≥ 0} in S = {1, 2, 3} with the transition rate matrix as ⎡ ⎤ ⎡ ⎤ −2ρ ρ ρ λ11 (ρ) λ12 (ρ) λ13 (ρ) Λ(ρ) = ⎣λ21 (ρ) λ22 (ρ) λ23 (ρ)⎦ = ⎣ 21 ρ −ρ 21 ρ ⎦ . 2 λ31 (ρ) λ32 (ρ) λ33 (ρ) ρ 29 ρ − 49 ρ 9 The sojourn time follows the Weibull distribution with PDF as vτ (ρ) = αγτγττ ρ γτ −1 ex p(−( αρτ )γτ ). Consider α1 = 1, γ1 = 2 for τ = 1; α2 = 2, γ2 = 2 for τ = 2; α3 = .∞ 2 3, γ3 = 3 for τ = 3. Then, we can compute E{λ12 (ρ)} = 0 4ρ 2 e−ρ dρ = 1.7722. Similarly, we have ⎡

⎤ −1.7722 0.8861 0.8861 E[Λ(ρ)] = ⎣ 1.7726 −3.5452 1.7726 ⎦ . 2.6587 2.6587 −5.3174 ˙ sin(z 1 (t)) is described by sin(z 1 (t)) = .1 Let z 1 (t) = ϕ(t) and z 2 (t) = ϕ(t). (z 1 (t))z 1 (t) + μ. ˜ 2 (z 1 (t))z 1 (t) with μ˜ = 0.01/π , where .1 (z 1 (t)), .2 (z 1 (t)) ∈ [0, 1] are membership functions that satisfy the condition .1 (z 1 (t)) + .2 (z 1 (t)) = 1. Then, one has .1 (z 1 (t)) = .2 (z 1 (t)) =

⎧ sin(z1 (t))−μz ˜ 1 (t) z 1 (t)(1−μ) ˜

1,

⎧ z1 (t)−sin(z1 (t)) ˜ z 1 (t)(1−μ)

1,

, z 1 (t) /= 0, z 1 (t) = 0,

, z 1 (t) /= 0, z 1 (t) = 0.

It is clear that .1 (z 1 (t)) = 1, .2 (z 1 (t)) = 0 if z 1 (t) is about 0 rad; .1 (z 1 (t)) = 0, .2 (z 1 (t)) = 1 if z 1 (t) is about π rad or −π rad. Thus, the single-link robot arm model can be expressed by the two-rule T-S fuzzy systems: Plant Rule 1: IF z 1 (t) is “about 0 rad”, THEN ⎧

z˙ (t) = (A1,τ + ΔA1,τ (t))z(t) + B1,τ u e (t), y(t) = Cτ z(t),

Plant Rule 2: IF z 1 (t) is “about π rad or −π rad”, THEN ⎧

where

z˙ (t) = (A2,τ + ΔA2,τ (t))z(t) + B2,τ u e (t), y(t) = Cτ z(t),

11.5 Simulation

215

] ] [ [ [ ]T 0 1 0 1 , A1,2 = , z(t) = z 1T (t) z 2T (t) , A1,1 = −gL −D0 −0.75gL −0.5D0 ] ] ] [ [ [ 0 1 0 1 0 1 , A2,1 = , A2,2 = , A1,3 = −0.8gL −0.4D0 −μgL ˜ −D0 −0.75μgL ˜ −0.5D0 ] [ [ ] [ ] [ ] 0 1 0 0 0 , B1 = A2,3 = , B2 = , B3 = , −0.8μgL ˜ −0.4D0 1 0.5 0.4 [ ] [ ] [ ] C1 = 0.1 0.3 , C2 = −0.3 0.4 , C3 = 0.2 −0.1 , [ ] [ ] [ ] 0 0.1 0.2 , Dι,2 = , Dι,3 = , Dι,1 = 0.3 0 0.1 [ ] [ ] [ ] Nι,1 = 0.1 −0.1 , Nι,2 = −0.1 0.2 , Nι,3 = 0 0.1 , ι ∈ {1, 2}.

It is assumed that the attack parameters are given as ζ = 0.75, G 1 (t) = 0.5 cos(t), G 2 (t) = 0.5 sin(t), G 3 (t) = 0.5, Φ(z(t), t, 1) = 0.5z 12 cos(t), Φ(z(t), t, 2) = 0.5z 12 sin(t), Φ(z(t), t, 3) = 0.5z 12 . Select ρ1 = ρ2 = ρ3 = 0.5, γ¯ 1 = γ¯ 2 = γ¯ 3 = 13 , d = 0.1. According to the optimization problem (11.20), one has ω = 7.8633 × 10−4 , and F1 = 0.0917, F2 = 0.1778, F3 = 0.2267, [ ] [ ] [ ] −8.4414 −4.8593 7.6154 , L 1,2 = , L 1,3 = , L 1,1 = 60.6630 31.3625 −64.5598 [ ] [ ] [ ] 4.8949 −4.8906 7.5725 , L 2,2 = , L 2,3 = . L 2,1 = −27.6366 31.0483 −65.8042

Fig. 11.2 System mode rt

4 3.5 3 2.5 2 1.5 1 0.5 0

0

0.5

1

1.5

2

2.5

3

216 Fig. 11.3 State responses of z(t) and zˆ (t)

11 Sliding Mode Control Under Deception Attacks 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3

Fig. 11.4 Control input u(t)

0

0.5

1

1.5

2

2.5

3

0.5

1

1.5

2

2.5

3

0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5

0

[ ]T [ ]T Choose z(0) = −0.2 0.3 and zˆ (0) = −0.3 0.2 , Fig. 11.2 represents the system mode. Figure 11.3 describes the state responses of z(t) and zˆ (t). From sgn(s(t)) Fig. 11.3, the state signals can converge to zero about 2s. The function sgn(s(t))+0.001 takes the place of the function sgn(s(t)) to reduce the chattering of sliding mode control law. Figure 11.4 depicts the input signal u(t). Figure 11.5 plots the sliding surface s(t) that shows the reachability of the sliding region.

References Fig. 11.5 Sliding surface s(t)

217 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1

0

0.5

1

1.5

2

2.5

3

11.6 Conclusion In this chapter, the sliding mode control issue for fuzzy S-MSSs under deception attacks has been studied based on fuzzy observer approach. By designing an appropriate sliding mode control law, the states are driven onto the sliding region despite semi-Markovian switching. And the closed-loop switching systems under deception attacks are stochastically stable. In the end, a practical example demonstrates the validity of the findings. In the future, the sliding mode control law will be discussed for hidden fuzzy S-MSSs.

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6. Qi, W.H., Hou, Y.K., Park, J.H., Zong, G.D., Shi, Y.: SMC for uncertain discrete-time semiMarkov switching systems. IEEE Trans. Circuits Syst. II: Express Briefs 69(3), 1452–1456 (2022) 7. Fu, L., Ma, Y.C., Wang, C.J.: Memory sliding mode control for semi-Markov jump system with quantization via singular system strategy. Int. J. Robust Nonlinear Control 29(18), 6555–6571 (2019) 8. Li, Y.B., Kao, B.H., Park, J.H., Kao, Y.G., Meng, B.: Observer-based mode-independent integral sliding mode controller design for phase-type semi-Markov jump singular systems. Int. J. Robust Nonlinear Control 29(15), 5213–5226 (2019) 9. Qi, W.H., Zong, G.D., Ahn, C.K.: Input-output finite-time asynchronous SMC for nonlinear semi-Markov switching systems with application. IEEE Trans. Syst. Man Cybern.: Syst. 52(8), 5344–5353 (2022) 10. Li, F.B., Wu, L.G., Shi, P., Lim, C.C.: State estimation and sliding mode control for semiMarkovian jump systems with mismatched uncertainties. Automatica 51, 385–393 (2015) 11. Qi, W.H., Zong, G.D., Zheng, W.X.: Adaptive event-triggered SMC for stochastic switching systems with semi-Markov process and application to boost converter circuit model. IEEE Trans. Circuits Syst. I: Regul. Papers 68(2), 786–796 (2021) 12. Qi, W.H., Gao, X.W., Ahn, C.K., Cao, J.D., Cheng, J.: Fuzzy integral sliding mode control for nonlinear semi-Markovian switching systems with application. IEEE Trans. Syst. Man Cybern.: Syst. 52(3), 1674–1683 (2022) 13. Dong, S.L., Liu, M.Q., Wu, Z.G., Shi, K.B.: Observer-based sliding mode control for Markov jump systems with actuator failures and asynchronous modes. IEEE Trans. Circuits Syst. II: Express Briefs 68(6), 1967–1971 (2021) 14. Gao, M., Qi, W.H., Cao, J.D., Cheng, J., Shi, K.B., Gao, Y.T.: SMC for phase-type stochastic nonlinear semi-Markov jump systems. Nonlinear Dyn. 108(1), 279–292 (2022) 15. Ning, Z.P., Zhang, L.X., Mesbah, A., Colaneri, P.: Stability analysis and stabilization of discretetime non-homogeneous semi-Markov jump linear systems: a polytopic approach. Automatica 120, Article ID 109080 (2020) 16. Tian, Y.X., Yan, H.C., Dai, W., Chen, S.M., Zhan, X.S.: Observed-based asynchronous control of linear semi-Markov jump systems with time-varying mode emission probabilities. IEEE Trans. Circuits Syst. II: Express Briefs 67(12), 3147–3151 (2020) 17. Shen, H., Xing, M.P., Yan, H.C., Cao, J.D.: Observer-based l2 -l∞ control for singularly perturbed semi-Markov jump systems with improved weighted TOD protocol. Sci. China Inf. Sci. 65(9), Article ID 199204 (2022) 18. Yan, H.C., Tian, Y.X., Li, H.Y., Zhang, H., Li, Z.C.: Input-output finite-time mean square stabilization of nonlinear semi-Markovian jump systems. Automatica 104, 82–89 (2019) 19. Zong, G.D., Qi, W.H., Shi, Y.: Advances on modeling and control of semi-Markovian switching systems: a survey. J. Frankl. Inst. https://doi.org/10.1016/j.jfranklin.2021.07.056 20. Shen, H., Dai, M.C., Yan, H.C., Park, J.H.: Quantized output feedback control for stochastic semi-Markov jump systems with unreliable links. IEEE Trans. Circuits Syst. II: Express Briefs 65(12), 1998–2002 (2018) 21. Qi, W.H., Yang, X., Park, J.H., Cao, J.D., Cheng, J.: Fuzzy SMC for quantized nonlinear stochastic switching systems with semi-Markovian process and application. IEEE Trans. Cybern. 52(9), 9316–9325 (2022) 22. Qi, W.H., Zong, G.D., Karimi, H.R.: SMC for nonlinear stochastic switching systems with quantization. IEEE Trans. Circuits Syst. II: Express Briefs 68(6), 2032–2036 (2021) 23. Qi, W.H., Zong, G.D., Su, S.F.: Fault detection for semi-Markov switching systems in the presence of positivity constraints. IEEE Trans. Cybern. http://dx.doi.org/10.1109/TCYB. 20213096948 24. Jin, Y.J., Qi, W.H., Zong, G.D.: Finite-time synchronization of delayed semi-Markov neural networks with dynamic event-triggered scheme. Int. J. Control Autom. Syst. 19(6), 2297–2308 (2021)

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Chapter 12

Conclusion and Future Research Direction

In this chapter, conclusion on the thesis is presented and some potential research issues related to the work done in this book are introduced.

12.1 Conclusion In this book, some control synthesis techniques for semi-Markovian switching systems are proposed. By using the weak infinitesimal operator theory, an appropriate sliding mode control law for a class of S-MSSs with signal quantization is designed in Chap. 2. By the use of stochastic semi-Markovian Lyapunov function, stochastic stability and finite-time reachability for stochastic S-MSSs and stochastic singular SMSSs are respectively studied in Chaps. 3 and 4. In consideration of observer design, finite-time sliding mode control law for S-MSSs is established to ensure the required boundedness performance including both reaching phase and sliding motion phase in Chap. 5. In Chap. 6, the adaptive event-triggered mechanism is adopted to effectively reduce the number of triggering in sliding mode control law design for S-MSSs. In Chap. 7, the issue of finite-time synchronization for delayed semi-Markovian switching neural networks with quantized measurement is addressed. T-S fuzzy approach provides a powerful tool in analyzing nonlinear systems. For Chaps. 8 and 9, T-S fuzzy sliding mode control strategy is developed for nonlinear S-MSSs. Under the discrete-time semi-Markovian kernel, the specified discrete-time sliding mode control is synthesized in Chap. 10. The sliding mode control is studied for networked S-MSSs with deception attacks in Chap. 11. All the conditions for the existences of analysis and design are derived in terms of linear matrix inequalities. The effectiveness of the proposed methodologies has been verified by some practical engineering systems.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 W. Qi and G. Zong, Control Synthesis for Semi-Markovian Switching Systems, Studies in Systems, Decision and Control 465, https://doi.org/10.1007/978-981-99-0317-7_12

221

222

12 Conclusion and Future Research Direction

12.2 Future Research Direction Although a great deal of fruitful research results have been reported about semiMarkovian switching systems, there are still many open problems to be further investigated. Finally, some future research directions are provided to be helpful to guide interested readers in studying control synthesis for semi-Markovian switching systems as follows: (i) Most of the results concerning control synthesis for S-MSSs are based on the linear subsystem framework and have some conservatism. As we know, backstepping and adaptive control are effective approaches for general nonlinear systems. Based on these nonlinear control strategies, future research could study the control synthesis for general nonlinear S-MSSs, including event-triggered control, finite-time control, sliding mode control, etc. (ii) Control techniques for discrete-time S-MSSs are still at an early stage and mainly focus on the basic stability and stabilization. For event-triggered control, finite-time control, and sliding mode control in discrete-time S-MSSs, the theory study and design practice have not reached these areas by now. (iii) Now, the networks are ubiquitous, ranging from nature and biological system to society. Security problem of networked S-MSSs is very important. One of the ongoing challenging research issue is to investigate networked S-MSSs subject to mixed attacks. Some advanced techniques used in analyzing and designing for mixed attacks under non-switching systems can be extended to deal with S-MSSs with mixed attacks. (iv) Adaptive event-triggered mechanism has been proposed for S-MSSs recently. Compared with static event-triggered mechanism, it effectively enlarges the minimum inter-event time and has become a hot research topic in the recent years. How to construct an appropriate adaptive event-triggered mechanism to networked S-MSSs subject to positive constraint is significant in both theory and practice.