Analysis and Design of Markov Jump Discrete Systems (Studies in Systems, Decision and Control, 499) [1st ed. 2023] 9819957478, 9789819957477

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Studies in Systems, Decision and Control 499

Yonggui Kao Panpan Zhang Changhong Wang Hongwei Xia

Analysis and Design of Markov Jump Discrete Systems

Studies in Systems, Decision and Control Volume 499

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

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

Yonggui Kao · Panpan Zhang · Changhong Wang · Hongwei Xia

Analysis and Design of Markov Jump Discrete Systems

Yonggui Kao Harbin Institute of Technology Weihai, Shandong, China

Panpan Zhang Harbin Institute of Technology Weihai, Shandong, China

Changhong Wang Space Control and Inertial Technology Research Center Harbin Institute of Technology Harbin, Heilongjiang, China

Hongwei Xia Space Control and Inertial Technology Research Center Harbin Institute of Technology Harbin, Heilongjiang, China

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-981-99-5747-7 ISBN 978-981-99-5748-4 (eBook) https://doi.org/10.1007/978-981-99-5748-4 © 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 Paper in this product is recyclable.

Preface

With their applications in the fields of financial technology, weather forecasting, speech recognition, etc., Markov jump systems have attracted the attention of more and more scholars. Due to the complexity of the system structure, the good properties of many subsystems, such as stability and observability, are not necessarily inherited in the whole Markov jump system, and there is not enough research in this area. Therefore, it is particularly necessary to study the control performance of Markov jump systems theoretically. On the other hand, the network embedding control system has become an increasingly obvious trend, and the resulting network induction phenomenon cannot be ignored. As a widely used Markov jump system, if the existing traditional algorithms are used to guide the practice at this time, the performance of the system will definitely be lost. Therefore, it is very urgent to study the stability and boundedness of Markov jump systems under the circumstances of random communication time delays, network packet loss, and deception attacks from the perspective of a practical application. Sliding mode control technology is an effective nonlinear robust control technology, especially when the system dynamics remain in the sliding mode surface, it has good robustness to external disturbances and uncertainties. The Markov jump system also faces the impact of external interference and uncertainty, especially when embedded in the network, it will also be affected by network inducing factors. So it is very important to discuss how to use the sliding mode control technology to solve the problems such as time delay, packet loss, and attack encountered in the network process of Markov jump system. Moreover, in a complex network environment, the above-mentioned networked phenomena may even occur simultaneously, so it is very necessary to explore the joint analysis and design of Markov jump networked systems. With the popularity of digital sampling, real industrial systems often contain offline dynamics, so this paper mainly considers discrete Markov jump systems. Different from the continuous-time sliding mode control, the discrete-time sliding mode control often cannot realize the ideal sliding mode state, and the discrete-time Markov jump system sliding mode control is no exception. However, noticing the good robustness of sliding mode control technology to external disturbances, it is v

vi

Preface

urgent to study the sliding mode control and quasi-sliding mode problems of discrete Markov jump networked systems. The monograph aims to provide recent research developments and insights on the analysis and design of Markov jump systems in the discrete-time setting. Different from general systems, discrete Markov jump networked systems are more complex because the systems are not only influenced by sampling, but also by Markov chain as well as networks. Owing to the particularity of discrete Markov jump networked systems, many existing methods for general systems cannot be extended to discrete Markov jump networked systems, which makes analysis and synthesis of discrete Markov jump networked systems full of challenges. By using the common sliding surface method and the Lyapunov approach, a basic theoretical framework is formed towards the issues of analysis and design for discrete Markov jump networked systems. The book can be used for researchers to carry out studies on Markov jump systems and is suitable for graduate students of control theory and engineering. It may also be a valuable reference for the control design of switched systems by engineers. The contents of the book are divided into 13 chapters which contain several independent yet related topics, and they are organized as follows. Chapter 1 introduces some basic background knowledge on Markov jump networked systems, and also describes the main work of the book. Chapters 2 and 3 consider the stochastic stability and finite-time boundedness analysis for MJSs with packet losses. By the use of sliding surface related to the initial condition, Chap. 4 studies the finite-time boundedness and passive performance for uncertain MJSs. In consideration of time-delay factor, Chap. 5 addresses the sliding mode control problem for uncertain MJSs under deception attacks based on the hidden Markov model. Noting the theory of analysis and control for normal systems, Chap. 6 discusses the actuator fault problem for MJSs via a sliding mode control. Viewing the transfer disturbance, Chap. 7 presents a bumpless transfer scheme for the Markov jump discrete systems with state feedback and output feedback skills. Chapter 8 considers the mixed H∞ and passivity of semi-Markov jump discrete systems. Chapter 9 involves the problem of adaptive sliding mode fault-tolerant control. All the conditions for the existence of analysis and design are derived in terms of linear programming. Lastly, Chap. 10 concludes some potential study directions. Weihai, China Weihai, China Harbin, China Harbin, China May 2022

Yonggui Kao Panpan Zhang Changhong Wang Hongwei Xia

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. Jun Hu, Harbin University of Science and Technology and Dr. Chengcheng Zhang, Harbin Institute of Technology (Weihai). Then 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 by the Natural Science Foundation of Shandong (ZR2020ZD27) and in part by the National Natural Science Foundation of China (62373119). Weihai, China Weihai, China Harbin, China Harbin, China

Yonggui Kao Panpan Zhang Changhong Wang Hongwei Xia

vii

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Markov Jump Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Several Types of Networked-Induced Phenomena and Corresponding Observer/Output-Feedback Strategy . . . . . . . 1.4 Actuator/Sensor Fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I 2

3

1 1 3 6 11 13 14

Markov Jump Discrete Systems Under Two Kinds of Network-Induced Phenomena

Sliding Mode H∞ Control for Markov Jump Discrete Systems Under Packet Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 2.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Design of Sliding Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Stochastic Stability with H∞ Performance . . . . . . . . . . . . . 2.4 Design of Robust Sliding Mode Controller . . . . . . . . . . . . . . . . . . . 2.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 23 25 25 35 39 40 43 44

Finite-Time Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 3.3 Design of Sliding Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 49 50 60 61 65 66 ix

x

4

Contents

Passivity and Control Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 4.3 Design of Sliding Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Synthesis of Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part II 5

69 69 71 72 83 83 84 87 87

Markov Jump System Under Deception Attacks

Hidden Markov Model-Based Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 5.3 Design of Common Sliding Surface . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 New Reaching Condition and Its Analysis . . . . . . . . . . . . . . . . . . . 5.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93 93 95 96 102 103 106 106

Part III Markov Jump Systems with Actuator Fault 6

Sliding Mode Fault-Tolerant Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 6.3 Design of Sliding Surface and Reachability Analysis . . . . . . . . . . 6.4 Stochastic Finite-Time Boundness of Closed-Loop System . . . . . 6.5 Computational Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 113 115 117 124 124 127 128

Part IV Markov Jump Systems with Bumpless Transfer Constraint 7

Bumpless Transfer Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 State Feedback Based SMBTC of Deterministic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 A Modified SMBTC Scheme of Deterministic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Design of SMBTC for Uncertain Systems via Output Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133 133 135 136 138 140

Contents

8

9

xi

7.2.4 Design of Output-Based Sliding Mode Controller . . . . . . . 7.2.5 Output Feedback-Based Sliding Mode BT Control Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 A Comprehensive Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

142

Semi-Markov Jump Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 8.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Common SMS Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Admissibility Analysis of Closed-Loop System with Mixed H∞ and Passivity . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Adaptive SMC Law Design . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 153 155 157 157

Adaptive Fault-Tolerant Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 9.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Synthesis of Sliding-Mode Control Laws . . . . . . . . . . . . . . 9.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 177 178 179 179 188 190 193 193

Part V

145 147 147 150 150

158 164 166 175 175

Summary

10 Conclusion and Future Research Direction . . . . . . . . . . . . . . . . . . . . . . 197 10.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 10.2 Future Research Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

Symbols

Rn l2 E{•} P{•} |•| • min{•} P > 0(P ≥ 0) Δ * 0 I ∈ AT A−1 sym(A)

n-dimensional Euclidean space Square-summable space Mathematical expectation Mathematic probability Absolute value of element 2-norm Minimun value of set {•} Positive definite (positive semi-definite) matrix Forward difference operator Term induced by the symmetry property Zero matrix Identity matrix Belong to Transpose of matrix A Inverse of matrix A AT + A

xiii

Chapter 1

Introduction

1.1 Background As we all know, the development of network technology has brought more and more convenience to human production and life. Especially during the epidemic, network technology has provided important technical support in terms of personnel flow backup, close contact personnel screening, and vaccination status record review. Then, can the communication network be combined with the control system in order to bring more convenience to the control and analysis of the control system? In order to realize this idea, the concept of networked control system came into being in the 1980s from the 20th century. The communication network is embedded in the traditional control system, and the unified whole formed by the control system, communication network, controller, actuator, and sensor is called a networked control system (NCS) [1]. In this way, the communication network participates in the information exchange of the traditional control system. The networked control system can establish the connection between physical space and cyberspace through the form of network embedded control system, and then realize the remote operation of physical space and other behaviors. Not only that, compared with traditional industrial systems, the embedded network also brings advantages to the control system such as easy reconfiguration, flexibility, strong sharing, reduced wiring, easy operation, low cost, simple installation and maintenance, etc. [2, 3]. Therefore, the networked control system has attracted much attention once it appeared, and it is increasingly popular in many fields such as space and land exploration, entering dangerous areas and related operations, factory automation, and remote diagnosis. At the same time, it is noted that networked control systems often suffer from network-induced phenomena such as actuator component aging, information transmission time delay, random uncertainty, and packet loss, which usually weaken the overall performance of the system [4, 5]. In order to give full play to the advantages of networked control systems and reduce or even avoid unfavorable network-induced phenomena, it is necessary to consider the impact of the above phenomena. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Kao et al., Analysis and Design of Markov Jump Discrete Systems, Studies in Systems, Decision and Control 499, https://doi.org/10.1007/978-981-99-5748-4_1

1

2

1 Introduction

This paper mainly considers a class of common networked control systems: discrete-time Markov jump networked systems, and discusses the sliding mode control problem of such systems under the above-mentioned network-induced phenomena. It is generally believed that the sliding mode variable structure control originated in the former Soviet Union is a widely studied robust nonlinear control technique [6, 7]. Its concept was proposed by the scholar Emelyanov in the 1950s in the 20th century, and after research by scholars such as Utkin and Itkin, the research object was extended from the initial second-order system to the high-level system in the 1960s order system. From the 20th century to the 1970s, the sliding mode variable structure control developed from the general phase plane method to the state space method. Later, this technology attracted the attention of Chinese scholars, among whom Weibing Gao, Yueming Hu, Jinkun Liu and other experts and scholars made significant contributions to it, and published relevant theoretical results or simulation works respectively. In fact, the nonlinear system sliding mode control process does not need to involve linearization processing, so it can effectively simplify the system control design. Moreover, when the system is moving on a sliding surface, it has good robustness against nonlinear perturbations and external disturbances. Therefore, this paper attempts to use sliding mode control technology to solve the design problems in the network process of Markov jump system. It is worth noting that the concept of quasi-sliding mode and the emergence of networked systems have enabled the development of sliding mode control for discrete systems. Especially since the 1980s in the 20th century, the stability analysis and design of networked time-delay systems have attracted the attention of scholars [8–10]. However, much research on networked systems is based on continuous time. The results about continuous systems, especially the sliding mode control theory and results of continuous systems are difficult to be directly applied to modern digital industrial systems. Therefore, related research on discrete networked systems deserves consideration in order to properly complement the theory. At the same time, it is noticed that the industry 4.0 continues to advance worldwide and China puts forward the smart manufacturing 2025 outline, in which development goals such as intelligence, high-end, and green put forward new requirements for industrial production processes and industrial systems. In addition, the development of digital sampling technology continues to follow up. This means that in the context of intelligent manufacturing, modern industrial systems can not only use the information of discrete sampling points to implement control operations, but also pay more and more attention to the stability and performance of the discrete system after sampling. The research on sliding mode control of discrete networked systems can not only supplement the theoretical deficiencies, but also make full use of existing technologies in this context to solve the difficulties behind new requirements, so this research has profound practical significance. In view of the importance of discrete Markov jump networked systems and the technical advantages of sliding mode variable structure control, this paper will focus on the discussion of discrete time Markov jump networked system sliding mode control design. This research aims to apply sliding mode control technology to discrete networked control systems, so as to use sliding mode control technology and

1.2 Markov Jump Systems

3

its advantages to reveal the details of the influence of network-induced factors on the overall performance of Markov jump systems, and further provide a basis for networked control. A series of problems such as performance degradation or even instability caused by network-induced phenomena such as actuator component aging, information transmission time delay, random uncertainty, and data packet loss in the system provide a feasible sliding mode control scheme, and also provide a feasible solution for the network It provides necessary reference for the in-depth application of computerized control systems in fault diagnosis, remote operation, smart home, smart medical care, space survey, industrial automation, etc. Based on the above discussion, this paper will focus on discrete-time Markov jump networked systems, and carry out specific research on the following topics: the sliding mode control problem under the occurrence of two types of networkinduced phenomena (data packet loss and random communication delay), random The sliding mode control strategy based on the hidden Markov model, the sliding mode fault-tolerant control algorithm based on the partition strategy, and the design of the sliding mode bumpless switching control for the discrete Markov jumping system in the case of spoofing attacks.

1.2 Markov Jump Systems Due to the characteristics of complexity and randomness, the Markov jump system model plays an important role in describing the process of mixing phenomena in production and life, especially for systems that are easily affected by random factors, it is more suitable to use the Markov jump model for modeling , so it is widely used in financial technology, weather forecast, speech recognition, etc. In addition, from a theoretical perspective, discrete systems with state time-delays can usually be transformed into Markov jump models to deal with [11]. This makes the research on Markov jump systems very necessary both in practical application and in theory. A typical case of economic neighborhood application is given below. Costa et al. [12] discussed the application of the Markov jump model in the economic neighborhood, and described the discrete Markov jump linear system model for modeling the state of the country’s economy, as shown in Fig. 1.1. Observe the transition probability matrix Pd , and find that its elements are all known non-negative real numbers. This is because the usual Markov jump system model, although the specific moment when each mode jumps to the system is random, but the known probability that the system stays in the mode at the current moment and transfers to the next mode is determined. However, when it comes to practical problems, the mode transition probability of the Markov jump system model is not necessarily completely known to researchers. Therefore, Yang et al. [13] considered some problems in the case of unknown transition probabilities, and used adaptive control technology to consider the realization of H∞ performance of singular Markov jump systems. Kao and Yao etc. [14, 15] generalized the above results, and obtained the sliding mode control scheme of the Markov jumping system in the case of unknown transition probability,

4

1 Introduction

Fig. 1.1 The transition between national economic states and the transition probability matrix

where Yao et al. [15] discusses the adaptive sliding mode control method under the condition that the actuator failure and transition probability information are simultaneously inaccurate, and establishes a sufficient condition for the stochastic stability of the system. It is worth noting that in the process of production practice, the unknown is not necessarily only the transition probability of the Markov jump model, and the jump mode itself may not be easy to capture. In view of this fact, Song et al. [16] utilized the mode detection method to solve the problem of asynchronous control of the hopping system when the mode is unknown, in which part of the mode detection probability is allowed to be uncertain. Shu et al. [17] discussed discrete Markov jump systems by adopting the static output feedback control method, which reduces the dependence on the system state in the design process, and at the same time uses the system augmentation technique to treat the control input as an augmented system Part of the variable and separate the gain K i , which solves the problem that the controller design parameters are inconvenient to solve. Dong etc. [18] associates quantization with Markov chains. For uncertain jumping systems with matching conditions, Huang and Jing [19] discussed the sliding mode state feedback control design based on the exponential reaching law, and proposed a realizable controller to drive the system asymptotically close to the region near the sliding surface. Qi et al. [20] gave a design scheme of sliding mode control for a class of uncertain discrete semi-Markov switching systems. Based on the mode-dependent Lyapunov function, a set of feasible criteria were established to ensure σ—error mean square stability. Obviously different from the sliding mode control of the continuous system, the sliding mode control of the discrete system does not necessarily reach the ideal sliding mode dynamics on the sliding surface. This is because forming the ideal sliding mode dynamics requires the system to perform an infinite number of switching behaviors. For a discrete system with a finite sampling rate, although the trajectory of the closedloop system formed by its sliding mode control is not guaranteed to be a sliding mode dynamic, it is usually driven to the neighborhood near the sliding mode surface

1.2 Markov Jump Systems

5

Fig. 1.2 Discrete time quasi-sliding mode

[21, 22], The so-called quasi-sliding mode is formed, and its principle is shown in Fig. 1.2. Considering this situation, Hu and Wang et al. [23] researched the discrete Markov jump system, and gave a sliding mode controller design scheme based on the arrival condition, which ensures that the system is in Mean square stochastic stability in the case of Markov jump parameters and mixed delays. On the other hand, it is noted that Malloci and Zhao et al. [24–26] have conducted in-depth research on bumpless handover control of switched systems, and proposed a bumpless handover scheme. However, as a special switching system, the Markov jump system also faces the problem of a large jump of the control signal at the switching point, that is, switching disturbance. For this kind of practical Markov switching systems, there are few studies on the bumpless switching control. Among them, an important reason is that the switching rules of this type of switching system are restricted by a specific Markov chain, and the goal of bumpless switching cannot be achieved by designing switching rules like in the bumpless switching scheme of general switching systems. Especially discrete Markov jump systems face system jumps at every moment, which undoubtedly increases the complexity of bumpless switching research. Therefore, this paper will discuss the problem of bumpless switching of discrete Markov jump systems, in order to control the generated switching disturbance within the allowable range through the advantages of sliding mode control technology.

6

1 Introduction

1.3 Several Types of Networked-Induced Phenomena and Corresponding Observer/Output-Feedback Strategy With the development of network technology, it has become a trend to embed the network into the traditional control system. Therefore, the advantages of the network make the traditional system more convenient to control, and at the same time, the wellknown incomplete network channel may pose a potential threat to the stability of the system. For example, data packet loss is one of the manifestations of an incomplete network channel, which usually occurs when the buffer overflows or the network traffic is heavy. However, we should mention that there is also a certain possibility that data loss is thought to be autonomously initiated. For example, in order to reduce the communication load and improve the efficiency of information usage, we may discard obsolete packets that take a long time to reach the receiving end, because the long transmission will cause the rearrangement of packets to some extent [27]. Whether it is objective data packet loss or artificial data packet loss, data packet loss is an unavoidable issue for us. This is why some existing works consider many interesting topics around packet loss, such as the problem of maximizing the sampling period when determining the probability of packet loss and the problem of minimizing the probability of packet loss when determining the sampling period. Specifically, Xue et al. [28] uses a single-exponential smoothing method to compensate for lost packets. In this case, the optimal algorithm with a smoothing parameter of α can still be used to ensure the closed-loop dynamic optimization in the case of packet loss Dissipation index γ ∗ . Wang et al. [29] considered the asynchronous state estimation problem when discrete Markov jumping system occurs packet loss, and proposed a non-static Markov chain correlation The asynchronous filter of , gives the condition of system stochastic stability and performance index of H∞ interference suppression. Hu and Zhang et al. [30] proposed a packet loss compensation strategy based on the current data for the situation where the probability of data packet loss is uncertain, and gave a robust mean square asymptotic guarantee for the Markov jump system stable program. There are channel bit rate limitations and network congestion in the network environment, so there is actually a delay between the transmission of the data packet and the actual use, which is the so-called time lag [27, 31, 32]. Depending on how it is defined, delays can be divided into the following main categories: network access delays, transmission delays, constant and time-varying delays, stochastic and probabilistic delays. In the SMC design, the transient performance of the system [33] can be improved if memory-based sliding surfaces are considered. Grifa and Hien etc. [34, 35] analyzes the stability of discrete control systems with time-varying and constrained delays. In the above research work, the time lag is usually clearly defined in a specific interval, which satisfies the time-varying and bounded condition. However, in fact, the occurrence of time lag is not so simple, and there may be certain statistical laws for its occurrence. That is to say, the time lag τk can also become a random variable, for example, if there is a positive integer τ , satisfying

1.3 Several Types of Networked-Induced Phenomena …

7

P{τk ∈ {τm , τm + 1, τ }} = θ, P{τk ∈ {τ + 1, τm + 1, τ M }} = 1 − θ τk is called the probabilistic lag. In particular, probabilistic lags generally have different probabilities of taking values on different intervals. Sakthivel and Sriraman etc. [36, 37] for a class of fuzzy Markov switching systems and neural network systems with probabilistic time-varying delays, solved H∞ and passive Comprehensive issues of sexual control and stability issues. Zhang et al. [38] discussed a sliding mode fault-tolerant control algorithm for a class of uncertain systems with probabilistic time-delays. The nonlinear part of the designed variable structure controller reflects the details of the probabilistic time-delays in order to offset its influence. Gao et al. [11] such as Gao and Meng directly regard the time lag dk as a kind of random process with a specific distribution, namely P{dk = τ j } = α j , j = 1, 2, . . . , q This fully explains the difference in the probability of network delay values. On this basis, Gao and Meng et al. [11] introduced indicator functions to analyze and design the system under the influence of this kind of random probability delay. Hu and Zhu [39] analyzed the stability of networked systems with long time-delays. Using the smoothness property of conditional expectations, they designed a system that guarantees that the system υk = 0 time exponential mean A stable control law. Liu et al. [40] used the frequency response method to study the robust control problem of uncertain time-delay sampling systems. The time-delay in this research is not necessarily an integer multiple of the sampling period. Note that the generation of time lag is not always detrimental to the system, and may also have a positive auxiliary effect on the stability of the system [3, 35, 41]. For a class of T-S fuzzy systems that cannot be stabilized by a non-time-delay output feedback controller, Zhang and Han et al. [42] developed a time-delay-related system stabilization scheme by introducing a communication network into the system. References [43, 44] pointed out that proper introduction of time delay in the control link can improve the performance of the closed-loop system, and artificially introduced time delay in the design process of the sliding surface/robust controller based on output measurement lag. In fact, not only the introduction of time-delay in the framework of output feedback control sliding mode surface design can improve the performance of the system, but also the introduction of time-delay in the design process of the sliding mode surface based on state information can also greatly improve the transient performance of the system, such as Integral use of sliding surfaces. Zhao et al. [45] used the sliding mode control strategy to discuss the singular finite-time boundedness of the discrete time-varying time-delay singular system. When [0, N ∗ ] and [N ∗ , N ] in the sliding mode stage. Liu et al. [46] provided a stable scheme for stochastic memory neural networks with probabilistic time-delays and leakage phenomena simultaneously. In the study, appropriate Lyapunov-Krasovskill functionals and discrete-time Jensen inequality techniques were used to fully It reveals the law of the influence of the probability timedelay phenomenon on the neural network when the leakage phenomenon occurs. It is worth noting that this type of time delay could also be used by attackers to adjust

8

1 Introduction

attack patterns. Undoubtedly, the phenomenon of spoofing attacks accompanied by probabilistic time lag inevitably poses a greater threat to the security and stability of the system. As we all know, in the network environment, there are not only problems such as packet loss, time delay, and communication constraints, but also the threat of network attacks [27, 47, 48], and the Markov jump networked stochastic system is no exception. Once the network attack phenomenon is not properly dealt with, it will not only affect the performance of the system, but even involve security issues such as privacy leakage in severe cases. Aiming at the multi-channel denial-ofserver attack phenomenon, Liu et al. [49] described it through a set of uncorrelated Bernoulli sequences, where the moment of attack is allowed to be random to match the vulnerable network. Regarding network attacks, attackers usually take advantage of the flaws in the authentication mechanism to disguise themselves as trusted parties by disguising their identities in order to deceive trust and pass the authentication. Their purpose is to communicate with the victim host for information or further attacks. Chen et al. [50] discussed the security of a class of continuous Markov jumping system spoofing attacks under incomplete transfer rates, and described the scenarios of false data injection attacks and denial of service attacks in the same block diagram, where Online Adaptive Estimation is targeted at spoofing attack patterns. Under the spoofing attack, the control signal reaching the actuator is not the original control signal, but the attacker injects false data [51–54] according to the system state or measurement output information. In general, an injection attack with an upper bound consists of a norm-bounded injection pattern matrix/weighting matrix (for example, ζ(t) in the document [52]) and a time-varying matrix related to system information (For example, Φ(y(t), t) in the document [53]), see the document [52–55] for details. For the convenience of analysis, it is usually assumed that the upper bound of the injection attack is directly related to the norm of the system state or its related variables [56, 57]. In order to reduce conservatism, He et al. [58] introduced a nonlinear function f (x(t)) ˜ related to the state of the attacked system when considering the actuator attack. In addition, for the unknown Chen and He et al. [55, 58] respectively proposed an adaptive sliding mode control scheme for Markov transition networked systems and an adaptive anti-attack control scheme based on hidden Markov models plan. Note that the above research is on the phenomenon of spoofing attacks with upper bounds. In fact, due to the complexity of the network environment, attacks initiated by opponents may or are more likely to occur in a random manner, such as random injection attacks. Therefore, in the process of system analysis and synthesis, the phenomenon of random injection attacks has gradually attracted the attention of scholars. [25, 59, 60]. Liu and Cheng et al. used Bernoulli distributed random variables to describe the situation where measurement output information is randomly injected into false data [61, 62]. To reduce conservatism, the embedded signal emitted by the enemy is also assumed to be norm-bounded [59, 63–66]. It is worth mentioning that Wu and Gao etc. [60, 63] introduces a Bernoulli distribution random variable that depends on the system modality to model network attacks. At this time, different subsystems of the Markov switching system are allowed to have different attack

1.3 Several Types of Networked-Induced Phenomena …

9

probabilities. In the face of limited information scheduling and random network injection attacks, Xing and Wang et al. [67, 68] established event-based asynchronous elastic filtering schemes for singular Markov jump linear variable parameter (LPV) system and Markov jump singular perturbation system respectively. In particular, Zhao et al. [25] noticed the impact of denial-of-service attacks and proposed a compensation method for the damaged measurement output signal of the switching system. It is worth mentioning that even if all data can be transmitted safely and in a timely manner, that is, there is no data packet loss, time lag, or network attack, it does not mean that the state information of the system itself is easy to be captured and used, such as black boxes with unpredictable states and certain large-scale systems. However, from the perspective of system analysis and design, it is very necessary to know all or part of the state information of the system, for example, for the design scheme based on system state feedback. There are many schemes we can use to deal with this problem, of which the observer method is a particular one. Because it can provide possible estimates of the system state, rather than aiming to capture it directly. Wang and Zhang et al. [69, 70] discussed the observer-based discrete system design with non-matching uncertainties or external disturbances, assuming non-matching conditions rank([B, Bd ]) = rank(B) for follow-up research [70], using the singular value decomposition technique to decompose the above uncertainty or disturbance into corresponding matching parts and non-matching distributions, based on this model, a state observer and disturbance observer are designed to assist the controller The design to minimize the impact of non-matching interference ensures the stability of the system. In fact, Zhang et al. [70] have introduced the disturbance estimation error into the so-called modified Gaussian reaching law, and used the following conditions to test the accessibility of the discrete background sliding mode control scheme.  (si (k + 1) − si (k))sgn(si (k)) < 0 (si (k + 1) + si (k))sgn(si (k)) > 0 Zhang and Liu etc. [33, 71] generalized the above results, and proposed an observerbased sliding mode control algorithm for continuous time control systems, in which a disturbance observer is designed to provide the possibility of disturbance Estimation, and there are specific bounds on the estimation error, and the integral sliding mode surface is constructed by using the estimation error. It should be mentioned that Zhang et al. [33] also discusses a so-called memory-based sliding surface, which introduces −K τ x(t − τ ) at the technical level to provide historical information of the system. According to the PID theory, since the term −K τ x(t − τ ) can induce a similar differential control realization, the transient performance of the system is expected to be further improved, namely: ) ˆ − K d d(t), K1 = K + τ Kτ , K2 = τ Kτ u 2 (t) = −K 1 x(t) + K 2 x(t)−x(t−τ τ x(t)−x(t−τ ) τ

≈ x(t) ˙

10

1 Introduction

In addition, in order to study the anti-interference scheme more conveniently, Zhang et al. [72] proposed an extended sliding mode observer for Markov jump linear systems. This extended framework simultaneously gives the system state and external interference probably estimated. The output feedback method is another important method to weaken the dependence of the designed scheme on the state. Zhang et al. [73] discussed the design of sliding mode output feedback control for discrete-time systems, and chose The linear sliding mode surface is based on the measured output, and the sliding mode dynamics are further expressed in the form of a singular system. It should be pointed out that due to the special structure of the output matrix, there are certain difficulties in the design process of the controller. Because only part of the system state information can be deduced under the output feedback format, all the system state information cannot be induced through the output information, and the part of the state information that cannot be obtained is difficult to avoid not being used in the controller design process. In order to solve this problem, Zhang et al. [73] carried out a coordinate transformation, estimated part of the system state that cannot be obtained, and directly applied it to the construction of the controller. On the other hand, Zhang and Zheng [74] discuss the design of adaptive sliding mode output feedback control for continuous-time systems, which ensures that the closed-loop system is uniformly ultimately bounded. Compared with the results in the discretetime context, a memory-based sliding mode surface linear in y(t) was chosen to improve the system performance [74], based on which, according to the output from The state information is easy to obtain in the feedback, and the sliding mode output feedback controller is designed under the adaptive framework. It is worth noting that Zhang et al. [75] proposed a dynamic sliding mode surface according to the output information. At this time, the problem of solving the sliding mode control is transformed into the design problem of the dynamic output feedback controller, which can be solved to a certain extent Avoid the difficulty of solving static output controller parameters [43]. Driven by the above work, Xu et al. [76] generalized the abovementioned sliding mode output feedback control problem, and proposed a design method for a class of discrete-time Markov transition systems with asynchronous form modal-dependent sliding surface. In this way, the solution matrix of the static output feedback problem can be solved by an iterative algorithm to finally determine the parameter expression of the sliding mode surface. In addition, the research topic on output also has application scenarios in object tracking. For example, Abidi and Xu [77] established an output tracking scheme in the case of state estimation, and designed a discrete integral sliding surface in this scheme. In summary, regarding several types of network-induced phenomena and observer/output feedback strategies, it is found that there are still two aspects of work that have not been fully studied. On the one hand, it is about the sliding mode control problem of discrete Markov transition networked systems when data packet loss and time delay occur. On the other hand, it is about the stability of networked systems based on hidden Markov models when new random attacks occur. Some chapters of this paper will carry out in-depth research around these two aspects of work. Regarding the sliding mode control problem of discrete Markov

1.4 Actuator/Sensor Fault

11

transition networked systems, the observer method is used to compensate the insufficient information caused by unfavorable conditions (data packet loss and time delay phenomenon) in the system, and a feasible sliding mode control scheme is given. In addition, by analyzing the actual scene of network attack, this paper will also model the new random attack as time-delay-dependent, and give the sliding mode control algorithm based on the hidden Markov model when the attack occurs in the system.

1.4 Actuator/Sensor Fault In production practice, there are systems that operate normally and systems that fail. Under certain conditions, systems that function normally can fail. or, let’s say All systems, whether they are important military systems or commercial/civilian systems, will face failure problems, whether they are major failures or minor failures, whether there are current failures or potential future failures. On the one hand, this is because any system has a limited service life. Once it exceeds a certain number of years or is used irregularly, it will often cause damage to the original components and cause system failures, such as actuator/sensor failures; on the other hand, the impact of external random phenomena It is also an important factor causing the failure of the control system. Therefore, issues such as actuator/sensor failure are fundamental issues in system research, and are also safety issues to ensure the normal operation of faulty systems. Zhang et al. [38] combined adaptive control technology and sliding mode control technology to estimate the failure factor of uncertain continuous system. Under measurable/observable conditions, Li and Huang et al. [78, 79] used the observer method to estimate the faults of linear time-invariant systems and Markov jump linear systems. Li and Guo etc. [80, 81] proposed a sliding-mode fault-tolerant control problem for a class of disturbance switching systems. Li and Chen [80] analyzed actuator faults and sensor faults under the same observer framework. Due to the complexity, the partial actuator degradation problem of a special class of switching systems (continuous Markov jumping systems) has attracted attention [79, 82]. Chen et al. [82] adopted the SMC strategy to ensure the asymptotic stability of the system when the actuator degradation phenomenon with upper and lower bounds occurs. Afterwards, considering the limitations of Lyapunov stability, Cao et al. [83] proposed an acceptable finite-time bounded scheme for continuous Markov jump systems, which is stable under actuator failure and external disturbance In the case of simultaneous existence, a division strategy is adopted to ensure the expected performance of the system arrival phase and sliding mode phase respectively. Zhao and Li et al. [84, 85] combined the SMC method with a segmentation strategy to solve the problem of actuator failure/sensor failure in jumping systems. Zhao and Niu [84] analyzed the impact of a class of randomly occurring actuator failures on the system. Li et al. [85] adopted the average dwell time method to analyze jumping systems with additional switching chains in the case of both actuator and sensor failures, combining the SMC technique, the average dwell time method, and the

12

1 Introduction

segmentation strategy, establishes the discriminant condition that the system is finitetime bounded. Lian and Zhai [86] transform the actuator failure problem of a robot manipulation system into an interference problem through system deformation. Similarly, Li et al. [87] have transformed sensor and actuator faults into a special kind of input perturbation by applying an augmentation method to the initial fault system. It should be pointed out that system fault models can generally be divided into two categories: additive faults and multiplicative faults [88]. Fault detection and estimation, fault isolation, fault-tolerant control, etc. are difficult topics for faulty systems. For example, Liu and Ho [88] simultaneously estimate the effectiveness loss and nonlinear bounds for actuator faults under the same adaptive fault-tolerance framework. In addition, Chen and Li etc. [89, 90] used the projection operator to estimate the effectiveness loss of the adaptive method for a class of stochastic continuous MJS. Regarding the discrete system for, Li and Chen [91] proposed an adaptive fault-tolerant control scheme combining the augmentation method and the sliding mode observer. However, the above results cannot be used to analyze Markov jump systems for which transition probabilities are not fully available. Based on the above observations, Du et al. [92] used the so-called improved homogeneous polynomial method to consider the fault-tolerant synthesis of Markov jump systems when part of the transition probabilities are unknown. Chen and Gao [93] considered the randomness of faults through a hierarchical probability partition method, where the sensor fault value greater than itself is allowed to occur. This study can be seen as an extension of common failures. In view of the simultaneous existence of actuator and sensor faults, Li and Chen et al. [94, 95] adopted the method of system augmentation and adaptive observer for the random jump system. Specifically, Li et al. [94] considered the estimation and control of finite energy faults for a class of MJS with uncertain transition rates, introduced an intermediate variable to estimate the augmented system, and designed a method based on The controller of the observed value, thus ensuring the stochastic stability of the system and having H∞ performance when the system fails. Chen et al. [95] realized the co-design of the sliding sliding surface and the controller for actuator and sensor faults. Yang et al. [96] established a sliding mode prediction model by proposing a two-step scheme, which gave the estimation of the system state and multiple faults. recent, Kaneba et al. [97] designed a hybrid event trigger mechanism and fault-tolerant control for continuous systems in order to reduce redundant packets in the channel, and proposed a hybrid event-triggered fault-tolerant control scheme of time-driven sensors and event-driven controllers. It is guaranteed that the system still has the passive performance level γ when the actuator fails. Although the above literature discusses the analysis and design of faulty systems, it is far from enough for faulty discrete Markov transition networked systems. In particular, there are not many achievements in the study of faults in networked systems resistant to discrete Markov transitions using sliding mode control technology. When using the sliding mode control technology to study the fault problem, the current research focuses more on solving the problems caused by the system fault itself. In fact, when studying the sliding mode fault-tolerant control under the discrete

1.5 Organization of the Book

13

background, some practical issues need to be considered more carefully, such as whether the scheme is more suitable for the non-ideal sliding mode dynamics of the discrete sliding mode control, etc. For this reason, this paper will study the problem of sliding mode fault-tolerant control of faulty discrete Markov transition networked systems, in order to supplement the existing discrete sliding mode fault-tolerant schemes and ensure the normal operation of the system in case of faults. Once the fault problem of the discrete Markov transition networked system is solved, the system can be regarded as a fault-free system and other aspects can be designed, which can also significantly improve the overall performance of the Markov transition system.

1.5 Organization of the Book This book studies the analysis and design for Markov jump 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 proposes a sliding mode H∞ control scheme based on the observer, a finite-time sliding mode control scheme based on the average dwell time method for the discrete Markov transition system with data packet loss phenomenon/ communication time delay phenomenon, according to the sliding mode control technology and the Lyapunov method. Then, a passive sliding mode control scheme is presented. By using Bernoulli distributed random variables to describe the above two network-induced phenomena, the new scheme reveals the influence of data packet loss/communication delay on system performance, and verifies the effectiveness of the scheme through numerical simulation. Chapter 3 focuses on Markov jump discrete systems under random spoofing attacks. Based on stochastic analysis, robust sliding mode control technology and stability theory, an asynchronous sliding mode observer based on hidden Markov model is designed to ensure that the attack situation Stochastic stability of the system. In this study, the random attack mode is not only related to the mode of the Markov transition system, but also related to the network delay state, which makes the attack model more in line with the real situation of network attacks. Chapter 4 studies the design problem of sliding mode fault-tolerant control for Markov jump delayed systems with the actuator fault. Combining the segmentation strategy, the linear matrix inequality technique and the discrete sliding mode control technique, a criterion to ensure the finite-time boundedness of the entire interval of the dynamic system is established. A simulation example is used to test the effectiveness of using the discrete partition strategy to study the sliding mode fault-tolerant control of jumping systems. Chapter 5 considers the problem of bumpless switching for discrete Markov transition uncertain systems. By analyzing the limitations of the existing bumpless switching schemes, the relationship between the bumpless switching and the static distri-

14

1 Introduction

bution of the system Markov chain is established, and applied to the development of the sliding mode bumpless control scheme. In addition, an output-feedback sliding mode controller is designed to reduce the dependence on state information during the implementation of the scheme. The simulation results show that the new bumpless switching scheme proposed in this study can effectively suppress the switching disturbance generated during the system hopping process.

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Part I

Markov Jump Discrete Systems Under Two Kinds of Network-Induced Phenomena

Chapter 2

Sliding Mode H∞ Control for Markov Jump Discrete Systems Under Packet Losses

This paper studies the robust observer-based sliding mode H∞ control issue for discrete-time stochastic Markovian jumping systems (MJSs) subject to packet losses (PLs). Here, a state variable observer of initial system is designed to estimate the state and a series of Bernoulli random variables is utilized to model the PLs, whose occurrence probability is allowed to be imprecise. Then, a linear form of switching function is constructed based on the estimated state space. The aim of our paper is to design an observer-based sliding mode H∞ control strategy such that, the sufficiently stochastic stability conditions with prescribed H∞ performance is established for augmented MJSs under the PLs with imprecise occurrence probability, which is composed of the sliding motion and the error system. The linear form of sliding surface can be determined by solving some linear matrix inequalities (LMIs). In addition, by proposing an improved reaching condition, the desired robust observerbased H∞ controller is synthesized in discrete-time setting to drive the system state from any initial one onto the determined sliding surface. Finally, the usefulness of observer-based H∞ control method is demonstrated via some numerical simulations.

2.1 Introduction Over the last several decades, the Markovian jumping systems (MJSs) as one class of stochastic systems with multiple modes, have attracted increasing attention. As such, some exploratory works on stability analysis and synthesis of MJSs have been developed in [1–3]. The filtering problems have been studied for MJSs in [4, 5], the synchronization control problems in [6–8], the asynchronous state estimation problem for descriptor MJSs with packet losses in [9], the control scheme design problems in [10, 11]. Specially speaking, a resilient guaranteed cost controller has been constructed in [12] for uncertain fuzzy systems with Markov jump parameters and time-varying delays. In [13], the asynchronous output feedback control © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Kao et al., Analysis and Design of Markov Jump Discrete Systems, Studies in Systems, Decision and Control 499, https://doi.org/10.1007/978-981-99-5748-4_2

21

22

2 Sliding Mode H∞ Control for Markov Jump Discrete Systems …

method is investigated for fuzzy MJSs via sliding mode. The result on sliding mode control (SMC) has been reported in [14] for MJSs with different delays via the delayfractioning method. It is noteworthy that an observer-based control scheme has been designed in [15] for singular nonhomogeneous MJSs, where the state-feedback controller based on the estimation has been constructed. The robust SMC method has been widely discussed and investigated due to its effective control for nonlinearity. Moreover, the combinations of robust SMC method and other techniques have been active research topics during the past few years [16– 18]. For instance, the sliding mode dissipative control scheme have been given in [19] for stochastic switched systems under event-triggering condition, the event-triggered SMC methods of discrete-time MJSs in [20]. Recently, since the internal state information of realistic system are generally not easy to be obtained or its expenses are unaffordable, many observer-based nonlinear control methods have been proposed. In [21, 22], the disturbance observer-based SMC approaches have been proposed for a class of fuzzy systems, where the unknown mismatched disturbances are estimated by disturbance observer and the desired controller based on disturbance observer is presented. The H∞ observer-based SMC problems have been addressed in [23, 24] respectively for nonlinear networked systems via a delay-fractioning approach and systems with quantization. However, we notice that few results on H∞ sliding mode observer design have been provided for stochastic MJSs in the discrete-time setting. On the other hand, it is recognized that unknown disturbances and bounded timevarying delays often disturb the most basic stability of the system, even a very small time-delay will cause abnormal fluctuation. For the system to function properly, a great number of methods have been developed for delayed control systems under external disturbances, such as [25–27]. In addition, due to the networks unreliability or various challenges under network environment, the phenomenon of packet losses (PLs) is often encountered in the process of signal transmission of networked control systems (NCSs). As a result, it is necessary to discuss new approaches for NCSs when the data packet is lost and the updated signal is unavailable. For example, the new SMC methods of stochastic systems have been proposed in [28, 29] against the PLs obeying Bernoulli process. Recently, great attention has been paid to the stochastic MJSs with PLs and disturbances [30], where the Markov jumping parameters, PLs phenomenon and unmeasurable state variables have been considered in a same frame. Accordingly, the observer-based control scheme has been provided in [30] to ensure that the corresponding closed-loop systems is finite-time bounded with a given disturbance attenuation level. To the authers’ knowledge, the problem of H∞ sliding mode observer design have not been solved for stochastic MJSs with PLs. Motivated by the above discussions, this paper considers the robust observerbased sliding mode H∞ control problem for a class of stochastic MJSs subject to PLs, time-delay and disturbance. First, the state variable observer of the original system is constructed under the PLs with imprecise occurrence probability. The error system, which is formed by the estimated state variable and original system variable, is generated. Next, based on the estimated state information, the linear sliding surface is chosen and the sliding motion equation is derived. By combining the

2.2 Problem Statements and Preliminaries

23

fore-mentioned error system with sliding motion equation, an augmented system is obtained. Then, the stochastic stability of augmented system with H∞ performance index is investigated via Lyapunov method. Sufficient stochastic stability with H∞ performance criteria are derived with the help of linear matrix inequalities (LMIs) technique and free weighting matrix skills. Recalling the improved reaching condition, the desired H∞ observer-based controller is designed in state-estimation space. At last, a numerical example is provided to verify the effective of obtained robust observer-based sliding mode H∞ control method. The novelties of this paper are summarized briefly as follows. (i) The observer-based SMC problem is discussed for stochastic MJSs with PLs phenomenon and time-varying delay, where the occurrence probability of PLs phenomenon is allowed to be uncertain. (ii) Subject to the PLs under imprecise occurrence probability, new robust sliding mode controller is designed in state-estimation space for addressed stochastic MJSs, which extends the result obtained in [24]. (iii) For the convenience purpose, the common sliding surface is determined, rather than usual mode-dependent one. Besides, the chattering phenomenon can be properly attenuated by adding the constraint to sliding mode gain.

2.2 Problem Statements and Preliminaries Let rk ∈ N  {1, 2, . . . , N } be a discrete homogeneous Markovian chain, where its transition probability matrix Π  [πi j ]i, j∈N is πi j  Pr(rk+1 = j|rk = i) ≥ 0, ∀i, j ∈ N , k ∈ Z +  with Nj=1 πi j = 1 for each i ∈ N . In this paper, we consider the following class of delayed stochastic MJSs with PLs and disturbance:  xk+1 = A(rk )xk + Ad (rk )xk−dk + Bu k + E(rk )ωk , yk = γk C(rk )xk , xk = φk , k = −d M , −d M + 1, . . . , 0,

(2.1)

where xk ∈ Rn is the state vector of system, u k ∈ Rs is the control input, yk ∈ Rq denotes the controlled output, dk ∈ [dm , d M ] is the time-varying delay. Ai  A(rk ), Adi  Ad (rk ), Bi  B(rk ), Ci  C(rk ), Di  D(rk ) and E i  E(rk ) are known matrices with Bi (i ∈ N ) being full column rank. ωk ∈ l2 ([0, +∞); R p ) is the exogenous disturbance. The random variable γk , which takes values from 0 and 1, is employed in order to characterize the packet losses:

24

2 Sliding Mode H∞ Control for Markov Jump Discrete Systems …

Prob{γk = 1} = γ¯ + Δγ , Prob{γk = 0} = 1 − (γ¯ + Δγ ),

(2.2)

where γ¯ + Δγ ∈ [0, 1], γ¯ > 0 is a known scalar, and Δγ satisfies |Δγ | ≤ ε with ε being constant. Obviously, one has 0 ≤ ε ≤ min{γ¯ , 1 − γ¯ }. In this paper, we design an observer for system (2.1) of the following form: 

xˆk+1 = Ai xˆk + Adi xˆk−dk + Bu k + L i (yk − yˆk ), yˆk = γ¯ Ci xˆk ,

(2.3)

where L i (i ∈ N ) are the gain matrices of observer to be designed later. Letting the error state be ek = xk − xˆk , then the following error system can be generated by (2.1) and (2.3) ek+1 = Ai ek + Adi ek−dk + E i ωk − L i (yk − yˆk ).

(2.4)

Our objective is to design an observer-based H∞ controller such that, for Markovian jumping parameters, PLs and time-varying delay, the following requirements can be satisfied. (Q1) the state variables of (2.3) can be driven onto the sliding surface (2.5) in finite time, and then remained thereafter. Moreover, the augmented Markovian jumping system (2.9) are stochastically stable under the ωk = 0 case. (Q2) when φk = 0 and ωk = 0, the following condition is satisfied: ∞ ∞     E ηk 2 ≤ γ 2 ωk 2 k=0

k=0

with γ > 0 being a prescribed scalar. To proceed, the following useful lemmas are recalled to facilitate subsequent developments. Lemma 2.1 [24] For positive-definite matrix P and vectors a, b with compatible dimensions, it holds a T b + b T a ≤ a T Pa + b T P −1 b. Lemma 2.2 (Schur complement) Given constant matrices Q1 , Q2 , Q3 , where Q1 = Q1T and 0 < Q2 = Q2T , then Q1 + Q3T Q2−1 Q3 < 0 if and only if 

Q1 Q3T ∗ −Q2



 < 0 or

−Q2 Q3 ∗ Q1

 < 0.

2.3 Main Results

25

2.3 Main Results This section will discuss the observer-based robust control problem for addressed MJSs subject to time-varying delay and PLs case. First, a common switching surface is given with aid of the state observer. Further, an augmented system, which is formed by the sliding mode dynamics and error system, is presented. Then, the stochastic stability of the sliding mode dynamics with H∞ performance is analyzed and the corresponding delay-dependent sufficient criteria are derived. In what follows, the improved reachability condition is checked by designing a desired robust observerbased sliding mode H∞ controller.

2.3.1 Design of Sliding Surface Firstly, the following sliding surface is first designed for the system (2.1) sk = G xˆk ,

(2.5)

where matrices G are matrices to be designed later such that matrices G B are nonsingular. In this paper, we choose G = B T P1 to ensure the above non-singularities with P1 > 0. It is worthy of noting that the ideal quasi-sliding mode satisfies the following condition sk+1 = sk = 0.

(2.6)

Then, it follows from (2.3), (2.5) and (2.6) that u eq = −(G B)−1 G[(Ai + Δγ L i Ci )xˆk + Adi xˆk−dk +(γ¯ + Δγ )L i Ci ek ].

(2.7)

Substituting (2.7) into (2.3) obtains the following sliding motion in the estimation space:

xˆk+1 = G˜ Ai xˆk + (γk − γ¯ )I − Δγ B(G B)−1 G L i Ci xˆk

+G˜ Adi xˆk−dk + γk I − (γ¯ + Δγ )B(G B)−1 G ×L i Ci ek ,

(2.8)

26

2 Sliding Mode H∞ Control for Markov Jump Discrete Systems …

where G˜ = I − B(G B)−1 G. The augmented system can be given by (2.4) and (2.8) as follows: ηk+1 = A¯ i ηk + A¯ di ηk−dk + E¯ i ωk ,

(2.9)

where    a¯ i,12 xˆk a¯ i,11 ¯ , Ai = , ηk = ek (γ¯ − γk )L i Ci Ai − γk L i Ci     0 G˜ Adi 0 , E¯ i = , A¯ di = Ei 0 Adi a¯ i,11 = G˜ Ai + [(γk − γ¯ )I − Δγ B(G B)−1 G]L i Ci , a¯ i,12 = [γk I − (γ¯ + Δγ )B(G B)−1 G]L i Ci . 

Definition 2.1 [20] The system (2.9) is said to be stochastically stable if for any initial conditions φk ∈ Rn , initial mode r0 ∈ N , the following condition is true   ∞ E  x(k) 2 |φk ,r0 < ∞. k=0

Now, the stochastic stability with ωk = 0 will be discussed for (2.9) based on the Lyapunov stability theorem, LMIs method and free weighting matrix skills. Theorem 2.1 Given scalar λ > 0, the augmented sta (2.9)is stochastically   system Q1 0 P1 0 ,Q= , symmetric ble if there exist positive-definite matrices P = 0 P2 0 Q2     X 11 0 X 21 0 matrices X 1 = , X2 = with proper dimensions and real matri0 X 12 0 X 22 ces L i (i ∈ N ) satisfying  ∗  Σ Σ1 < 0, ∗ −Λ P ≤ λI,

Ξ =

(2.10) (2.11)

where ⎡ ∗ σ11 ⎢ ∗ ⎢ ⎢ ∗ Σ∗ = ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

⎤ 0 AiT P1 Adi 0 −(X 21 )T − X 11 0 ∗ 0 (AiT P2 + X 12 )Adi 0 AiT (X 22 )T − (X 22 )T − X 12 ⎥ σ22 ⎥ ⎥ ∗ σ33 0 0 0 ⎥, ⎥ ∗ ∗ σ44 0 0 ⎥ ∗ ⎦ ∗ ∗ ∗ σ55 0 ∗ ∗ ∗ ∗ ∗ σ66

2.3 Main Results

27

√ ⎤ ⎡√ γ2 X 11 γ5 λCiT L iT 0 0 0 0 √ √ T T ⎥ ⎢ γ2 − 2X 12 γ4 λCi L i 0 0 0 0 ⎥ ⎢ ⎥ ⎢ 0 0 0 0 0 0 ⎥, ⎢ Σ1 = ⎢ ⎥ 0 0 0 0 0 0 ⎥ ⎢ √ ⎦ ⎣ 0 0 0 0 3 + γ¯ + ε X 21 0 √ 0 0 0 0 0 2 + γ¯ + ε X 22 ∗ = −P + Q + 6A T P A − X − (X )T , Λ = diag{P1 , I, P1 , I, P1 , P2 }, σ11 1 1 11 11 i 1 i ∗ = −P + Q + 4 A T P A + sym(X A − X ), σ = (5 + γ¯ + ε)A T P A − Q , σ22 2 2 12 i 12 33 1 i 2 i di 1 di T P A − Q , σ ∗ = −X − (X )T , σ ∗ = −X − (X )T , σ44 = (3 + γ¯ + ε)Adi 2 di 2 55 21 21 22 22 66

 = d M − dm + 1, γ1 = 4(γ¯ − γ¯ 2 ) + 4ε | 1 − 2γ¯ |, γ2 = ε2 + ε + 2 + γ¯ , γ3 = 12(γ¯ + ε)2 + 8(γ¯ + ε), γ4 = γ3 + 6(γ¯ + ε), γ5 = 2 + 15ε2 + 2γ1 . Proof Firstly, choose the following Lyapunov-Krasovskii functional: Vk = V1k + V2k ,

(2.12)

where V1k = ηkT Pηk , V2k =

−d m +1 

k−1 

ηsT Qηs

j=−d M +1 s=k−1+ j

 with P =

   Q1 0 P1 0 > 0, Q = > 0 being matrices to be determined. Then, 0 P2 0 Q2

let’s calculate the difference of Vk along (2.9) ΔVk = E{ΔV1k + ΔV2k },

(2.13)

where E{ΔV1k } T = E{ηk+1 Pηk+1 − ηkT Pηk }   = E ( A¯ i ηk + A¯ di ηk−dk )T P( A¯ i ηk + A¯ di ηk−dk ) − ηkT Pηk  = E ηkT ( A¯ iT P A¯ i − P)ηk + 2ηkT A¯ iT P A¯ di ηk−dk  T T +ηk−d P A¯ di ηk−dk , A¯ di k

E{ΔV2k }  −d k m +1   =E ηsT Qηs − j=−d M +1

s=k+ j

k−1  s=k−1+ j

 ηsT Qηs

(2.14)

28

2 Sliding Mode H∞ Control for Markov Jump Discrete Systems …

 =E

−d m +1 

  T ηkT Qηk − ηk−1+ Qη k−1+ j j

j=−d M +1

  k−d m   T T = E d M − dm + 1 ηk Qηk − η j Qη j .

(2.15)

j=k−d M

Thus, it follows from (2.14) and (2.15) that  T T ΔVk = E ηkT ( A¯ iT P A¯ i − P)ηk + 2ηkT A¯ iT P A¯ di ηk−dk + ηk−d P A¯ di ηk−dk A¯ di k  k−d m   + d M − dm + 1 ηkT Qηk − η Tj Qη j j=k−d M

   ≤ E ηkT A¯ iT P A¯ i − P + (d M − dm + 1)Q ηk + 2ηkT A¯ iT P A¯ di ηk−dk  T T ¯ ¯ (2.16) +ηk−dk ( Adi P Adi − Q)ηk−dk .    X 11 0 X 21 0 For any real matrices X 1 = , X2 = with proper dimen0 X 12 0 X 22 sions, the following equality is true 

  E 2(ηkT X 1 + ΔηkT X 2 )[( A¯ i − I )ηk − Δηk + A¯ di ηk−dk ] = 0.

(2.17)

Combine (2.16) and (2.17), one has    ΔVk ≤ E ηkT A¯ iT P A¯ i − P + (d M − dm + 1)Q + X 1 ( A¯ i − I ) + ( A¯ i − I )T X 1T ηk T T +ηk−d ( A¯ di P A¯ di − Q)ηk−dk − 2ΔηkT X 2 Δηk + 2ηkT ( A¯ iT P A¯ di + X 1 A¯ di ) k  T T T ¯ di (2.18) ×ηk−dk + 2ηkT [( A¯ i − I )T X 2T − X 1 ]Δηk + 2ηk−d X Δη A k 2 k

with A¯ i and A¯ di being defined below (2.9). Then, rewrite the matrix A¯ i by A¯ i = A¯ 1i + A¯ 2i , where A¯ 2i = A¯ 2i(1) + A¯ 2i(2) + A¯ 2i(3) ,      0 L i Ci L i Ci 0 G˜ Ai 0 ¯ ¯ , A2i(1) = (γk − γ¯ ) , , A2i(2) = γk 0 −L i Ci −L i Ci 0 0 Ai   −Δγ¯ B(G B)−1 G L i Ci −(γ¯ + Δγ¯ )B(G B)−1 G L i Ci = . 0 0

A¯ 1i = A¯ 2i(3)



2.3 Main Results

29

Using the Lemma 2.1 and noting (2.18), we know    T T P A¯ 1i ηk + 4ηkT A¯ 2i(1) P A¯ 2i(1) ηk E ηkT A¯ iT P A¯ i ηk ≤ E 4ηkT A¯ 1i

 T T +4ηkT A¯ 2i(2) P A¯ 2i(2) ηk + 4ηkT A¯ 2i(3) P A¯ 2i(3) ηk , (2.19)

   T P A¯ η T E 4ηkT A¯ 1i 1i k ≤ E 4ηk



  AiT P1 Ai − AiT G T (G B)−1 G Ai 0 ηk , (2.20) T 0 Ai P2 Ai      T E 4ηkT A¯ 2i(1) P A¯ 2i(1) ηk ≤ E 4 (γ¯ − γ¯ 2 )+ | 1 − 2γ¯ | ε ηkT    T T C L P L C + CiT L iT P2 L i Ci 0 η , × i i 1 i i (2.21) 0 0 k  ≤ E 4ηkT



  AiT G˜ T P1 G˜ Ai 0 ηk T 0 Ai P2 Ai

  T E 4ηkT A¯ 2i(2) P A¯ 2i(2) ηk     0 0 η , ≤ E 4(γ¯ + ε)ηkT T T T T 0 Ci L i P1 L i Ci + Ci L i P2 L i Ci k   T E 4ηkT A¯ 2i(3) P A¯ 2i(3) ηk

(2.22)

    Δγ 2 CiT L iT G T (G B)−1 G L i Ci (γ¯ + Δγ )Δγ CiT L iT G T (G B)−1 G L i Ci ≤ E 4ηkT ηk ∗ (γ¯ + Δγ )2 CiT L iT G T (G B)−1 G L i Ci     3ε 2 CiT L iT G T (G B)−1 G L i Ci 0 ≤ E 4ηkT ηk . (2.23) 0 3(γ¯ + ε)2 CiT L iT G T (G B)−1 G L i Ci

Thus,     Ω  11 0 η , E ηkT A¯ iT P A¯ i ηk ≤ E ηkT 0 Ω22 k

(2.24)

where Ω11 = 4 AiT P1 Ai − 4 AiT G T (G B)−1 G Ai + γ1 (CiT L iT P1 L i Ci + CiT L iT P2 L i Ci ) Ω22

+12ε2 CiT L iT G T (G B)−1 G L i Ci , = 4 AiT P2 Ai + 12(γ¯ + ε)2 CiT L iT G T (G B)−1 G L i Ci

+4(γ¯ + ε)(CiT L iT P1 L i Ci + CiT L iT P2 L i Ci ), γ1 = 4(γ¯ − γ¯ 2 ) + 4ε | 1 − 2γ¯ | .

30

2 Sliding Mode H∞ Control for Markov Jump Discrete Systems …

Due to

    ˜ G Ai + Δγ G˜ L i Ci (γ¯ + Δγ )G˜ L i Ci ¯ , E Ai = −Δγ L i Ci Ai − (γ¯ + Δγ )L i Ci

we have

    E ηkT X 1 ( A¯ i − I ) + ( A¯ i − I )T X 1T ηk     (γ¯ + Δγ )X 11 G˜ L i Ci X 11 G˜ Ai + Δγ X 11 G˜ L i Ci − X 11 ηk . = 2E ηkT −Δγ X 12 L i Ci X 12 Ai − (γ¯ + Δγ )X 12 L i Ci − X 12

(2.25)

Moreover, it is obvious to see G˜ T P1 G˜ = P1 − G T (G B)−1 G, G˜ P1−1 G˜ T = P1−1 − B(G B)−1 B T . Then,   xˆkT X 11 G˜ Ai + Δγ X 11 G˜ L i Ci − X 11 xˆk 1 T 1 1 xˆ X P −1 (X 11 )T xˆk + xˆkT AiT G˜ T P1 G˜ Ai xˆk − xˆkT X 11 xˆk + ε 2 xˆkT X 11 G˜ P1−1 G˜ T (X 11 )T xˆk 2 k 11 1 2 2 1 + xˆkT CiT L iT P1 L i Ci xˆk 2 1 1 1 ≤ xˆkT X 11 P1−1 (X 11 )T xˆk + ε 2 xˆkT X 11 P1−1 (X 11 )T xˆk + xˆkT CiT L iT P1 L i Ci xˆk 2 2 2 1 + xˆkT AiT P1 Ai xˆk − xˆkT X 11 xˆk 2 1 1 1 ≤ (1 + ε 2 )xˆkT X 11 P1−1 (X 11 )T xˆk − xˆkT X 11 xˆk + xˆkT AiT P1 Ai xˆk + xˆkT CiT L iT P1 L i Ci xˆk , (2.26) 2 2 2 T ˜ (γ¯ + Δγ )xˆ X 11 G L i Ci ek ≤

k

1 1 ≤ (γ¯ + ε)xˆkT X 11 G˜ P1−1 G˜ T (X 11 )T xˆk + (γ¯ + ε)ekT CiT L iT P1 L i Ci ek 2 2 1 1 ≤ (γ¯ + ε)xˆkT X 11 P1−1 (X 11 )T xˆk + (γ¯ + ε)ekT CiT L iT P1 L i Ci ek , 2 2

(2.27)

(−Δγ )ekT X 12 L i Ci xˆk 1 1 (−Δγ )2 ekT X 12 P1−1 (X 12 )T ek + xˆkT CiT L iT P1 L i Ci xˆk 2 2 1 1 ≤ ε 2 ekT X 12 P1−1 (X 12 )T ek + xˆkT CiT L iT P1 L i Ci xˆk , 2 2   ekT X 12 Ai − (γ¯ + Δγ )X 12 L i Ci − X 12 ek

(2.28)

1 1 ≤ ekT (X 12 Ai − X 12 )ek + (γ¯ + ε)ekT X 12 P1−1 (X 12 )T ek + (γ¯ + ε)ekT CiT L iT P1 L i Ci ek . 2 2

(2.29)

≤

Therefore, we can get from (2.25)–(2.29)     E ηkT X 1 ( A¯ i − I ) + ( A¯ i − I )T X 1T ηk  ≤ E (ε 2 + ε + 1 + γ¯ )xˆkT X 11 P1−1 (X 11 )T xˆk − 2 xˆkT X 11 xˆk + xˆkT AiT P1 Ai xˆk +2 xˆkT CiT L iT P1 L i Ci xˆk + (ε 2 + ε + γ¯ )ekT X 12 P1−1 (X 12 )T ek + 2ekT (X 12 Ai − X 12 )ek  +2(γ¯ + ε)ekT CiT L iT P1 L i Ci ek . (2.30)

2.3 Main Results

31

From (2.18), T T ( A¯ di P A¯ di − Q)ηk−dk ηk−d k     T T ˜T T T ˜ A x ˆ A ek−dk = xˆk−d P − Q + e P A − Q G G A 1 di 1 k−d 2 di 2 k di k−dk di k     T T T T Adi Adi ≤ xˆk−d P1 Adi − Q 1 xˆk−dk + ek−d P2 Adi − Q 2 ek−dk , (2.31) k k

  E 2ηkT ( A¯ iT P A¯ di + X 1 A¯ di )ηk−dk     a1 −Δγ CiT L iT P2 Adi T ηk−dk (2.32) = E 2ηk (γ¯ + Δγ )CiT L iT G˜ T P1 G˜ Adi a2 with a1 = AiT G˜ T P1 G˜ Adi + Δγ CiT L iT G˜ T P1 G˜ Adi + X 11 G˜ Adi , a2 = AiT P2 Adi − (γ¯ + Δγ )CiT L iT P2 Adi + X 12 Adi . Similarly, it can be seen that  2 xˆkT

AiT G˜ T

 T T ˜T ˜ ˜ ˜ P1 G Adi + Δγ Ci L i G P1 G Adi + X 11 G Adi xˆk−dk

T T ≤ 2 xˆkT AiT P1 Adi xˆk−dk + xˆkT AiT G T (G B)−1 G Ai xˆk + xˆk−d Adi P1 Adi k   2 T T T T T T ×xˆk−dk + ε xˆk Ci L i P1 L i Ci xˆk + xˆk−dk Adi P1 − G (G B)−1 G   T T ×Adi xˆk−dk + xˆkT X 11 P1−1 (X 11 )T xˆk + xˆk−d P1 − G T (G B)−1 G Adi k

×Adi xˆk−dk ≤ xˆkT X 11 P1−1 (X 11 )T xˆk + ε2 xˆkT CiT L iT P1 L i Ci xˆk + xˆkT AiT G T (G B)−1 G Ai  T T T T P1 ×xˆk + 2 xˆkT AiT P1 Adi xˆk−dk + xˆk−d Adi P1 Adi xˆk−dk + 2 xˆk−d Adi k k  T −1 −G (G B) G Adi xˆk−dk , (2.33) 2(−Δγ )xˆkT CiT L iT P2 Adi ek−dk T T ≤ ε2 xˆkT CiT L iT P2 L i Ci xˆk + ek−d Adi P2 Adi ek−dk , k ≤

2(γ¯ + Δγ )ekT CiT L iT G˜ T P1 G˜ Adi xˆk−dk T (γ¯ + ε)ekT CiT L iT P1 L i Ci ek + (γ¯ + ε)xˆk−d k

(2.34)

  T P1 − G T (G B)−1 G Adi

×Adi xˆk−dk ,

2ekT AiT P2 Adi − (γ¯ + Δγ )CiT L iT P2 Adi + X 12 Adi ek−dk

(2.35)

T T ≤ (γ¯ + ε)ekT CiT L iT P2 L i Ci ek + (γ¯ + ε)ek−d Adi P2 Adi ek−dk k

+2ekT AiT P2 Adi ek−dk + 2ekT X 12 Adi ek−dk ,

(2.36)

32

2 Sliding Mode H∞ Control for Markov Jump Discrete Systems …

and   E 2ηkT ( A¯ iT P A¯ di + X 1 A¯ di )ηk−dk ≤ xˆkT X 11 P1−1 (X 11 )T xˆk + ε2 xˆkT CiT L iT P1 L i Ci xˆk + ε2 xˆkT CiT L iT P2 L i Ci xˆk T +xˆkT AiT G T (G B)−1 G Ai xˆk + 2 xˆkT AiT P1 Adi xˆk−dk + (2 + γ¯ + ε)xˆk−d k   T T T P1 − G T (G B)−1 G Adi xˆk−dk + xˆk−d ×Adi A P A x ˆ + ( γ ¯ + ε) 1 di k−d k di k ×ekT CiT L iT P1 L i Ci ek + (γ¯ + ε)ekT CiT L iT P2 L i Ci ek + 2ekT AiT P2 Adi ek−dk T T +(1 + γ¯ + ε)ek−d Adi P2 Adi ek−dk + 2ekT X 12 Adi ek−dk . k

(2.37)

It holds   E 2ηkT [( A¯ i − I )T X 2T − X 1 ]Δηk   T  T X 21 0 G˜ Ai + Δγ G˜ L i Ci (γ¯ + Δγ )G˜ L i Ci = E 2ηkT 0 X 22 −Δγ L i Ci Ai − (γ¯ + Δγ )L i Ci T     0 X X 11 0 Δηk − 21 − 0 X 22 0 X 12   T T  Ai G˜ + Δγ CiT L iT G˜ T −Δγ CiT L iT = E 2ηkT (γ¯ + Δγ )CiT L iT G˜ T AiT − (γ¯ + Δγ )CiT L iT        0 0 (X 21 )T X 11 0 (X 21 )T − − Δηk × T T 0 X 12 0 (X 22 ) 0 (X 22 )     T a˘ 1 −Δγ Ci L iT (X 22 )T = E 2ηkT Δη k (γ¯ + Δγ )CiT L iT G˜ T (X 21 )T a˘ 2

(2.38)

with a˘ 1 = AiT G˜ T (X 21 )T + Δγ CiT L iT G˜ T (X 21 )T − (X 21 )T − X 11 , a˘ 2 = AiT (X 22 )T − (γ¯ + Δγ )CiT L iT (X 22 )T − (X 22 )T − X 12 . Further,   2 xˆkT AiT G˜ T (X 21 )T + Δγ CiT L iT G˜ T (X 21 )T − (X 21 )T − X 11 Δxˆk = 2 xˆkT AiT G˜ T (X 21 )T Δxˆk + 2Δγ xˆkT CiT L iT G˜ T (X 21 )T Δxˆk −2 xˆkT (X 21 )T Δxˆk − 2 xˆkT (X 11 )Δxˆk ≤ xˆkT AiT G˜ T P1 G˜ Ai xˆk + ΔxˆkT (X 21 )P1−1 (X 21 )T Δxˆk +(Δγ )2 xˆkT CiT L iT G˜ T P1 G˜ L i Ci xˆk + ΔxˆkT (X 21 )P1−1 ×(X 21 )T Δxˆk − 2 xˆkT (X 21 )T Δxˆk − 2 xˆkT (X 11 )Δxˆk     ≤ xˆkT AiT P1 − G T (G B)−1 G Ai xˆk + ε2 xˆkT CiT L iT P1 − G T (G B)−1 G L i Ci xˆk   +2ΔxˆkT (X 21 )P1−1 (X 21 )T Δxˆk − 2 xˆkT (X 21 )T + X 11 Δxˆk , (2.39)

2.3 Main Results

33

2 xˆkT (−Δγ )CiT L iT (X 22 )T Δek

≤ ε2 xˆkT CiT L iT P2 L i Ci xˆk + ΔekT X 22 P2−1 (X 22 )T Δek , 2(γ¯ + Δγ )ekT CiT L iT G˜ T (X 21 )T Δxˆk

(2.40)

≤ (γ¯ + Δγ )ekT CiT L iT G˜ T P1 G˜ L i Ci ek + (γ¯ + Δγ )ΔxˆkT X 21 P1−1 (X 21 )T Δxˆk   ≤ (γ¯ + ε)ekT CiT L iT P1 − G T (G B)−1 G L i Ci ek +(γ¯ + ε)ΔxˆkT X 21 P1−1 (X 21 )T Δxˆk ,   2ekT AiT (X 22 )T − (γ¯ + Δγ )CiT L iT (X 22 )T − (X 22 )T − X 12 Δek

(2.41)

= −2(γ¯ + Δγ )ekT CiT L iT (X 22 )T Δek + 2ekT AiT (X 22 )T Δek −2ekT (X 22 )T Δek − 2ekT X 12 Δek

≤ (γ¯ + Δγ )ekT CiT L iT P2 L i Ci ek + (γ¯ + Δγ )ΔekT X 22 P2−1 ×(X 22 )T Δek + 2ekT AiT (X 22 )T Δek − 2ekT (X 22 )T Δek − 2ekT X 12 Δek ≤ (γ¯ + ε)ekT CiT L iT P2 L i Ci ek + (γ¯ + ε)ΔekT X 22 P2−1 (X 22 )T ×Δek + 2ekT AiT (X 22 )T Δek − 2ekT (X 22 )T Δek − 2ekT X 12 Δek ,

(2.42)

and   E 2ηkT [( A¯ i − I )T X 2T − X 1 ]Δηk      ≤ E xˆkT AiT P1 − G T (G B)−1 G Ai xˆk + ε2 xˆkT CiT L iT P1 − G T (G B)−1 G L i Ci xˆk   +ε2 xˆkT CiT L iT P2 L i Ci xˆk − 2 xˆkT (X 21 )T + X 11 Δxˆk + (2 + γ¯ + ε)ΔxˆkT (X 21 )P1−1   ×(X 21 )T Δxˆk + (γ¯ + ε)ekT CiT L iT P1 − G T (G B)−1 G L i Ci ek + (γ¯ + ε)ekT CiT ×L iT P2 L i Ci ek + 2ekT AiT (X 22 )T Δek − 2ekT (X 22 )T Δek − 2ekT X 12 Δek  +(1 + γ¯ + ε)ΔekT X 22 P2−1 (X 22 )T Δek .

(2.43)

Additionally, we derive   T T X 2T Δηk A¯ di E 2ηk−d k  T T    Adi G˜ P1 G˜ Adi 0 T T −1 T η ≤ ηk−dk + Δηk X 2 P X 2 Δηk T P2 Adi k−dk 0 Adi    T T T T P1 − G T (G B)−1 G Adi xˆk−dk + ek−d Adi Adi P2 Adi ek−dk ≤ xˆk−d k k  (2.44) +ΔηkT X 2 P −1 X 2T Δηk . Recalling the condition (2.11) and formulas (2.19)–(2.44), the inequality (2.18) can be further expressed by  ΔVk ≤ E xˆkT (−P1 + Q 1 )xˆk + 6xˆkT AiT P1 Ai xˆk − 2 xˆkT X 11 xˆk + γ2 xˆkT X 11 P1−1 (X 11 )T xˆk

34

2 Sliding Mode H∞ Control for Markov Jump Discrete Systems …   +(2 + 15ε 2 + 2γ1 )λxˆkT CiT L iT L i Ci xˆk + 2 xˆkT AiT P1 Adi xˆk−dk − 2 xˆkT (X 21 )T + X 11 ×Δxˆk + ekT (−P2 + Q 2 )ek + 4ekT AiT P2 Ai ek + γ4 λekT CiT L iT L i Ci ek + 2ekT (X 12 Ai −X 12 )ek + (γ2 − 2)ekT X 12 P1−1 (X 12 )T ek + 2ekT (AiT P2 + X 12 )Adi ek−dk



T T P A − Q xˆ +2ekT AiT (X 22 )T − (X 22 )T − X 12 Δek + xˆk−d (5 + γ¯ + ε)Adi 1 di 1 k−dk k

−1 T T P A −Q e T T +ek−d (3 + γ¯ + ε)Adi 2 di 2 k−dk + (3 + γ¯ + ε)Δ xˆk (X 21 )P1 (X 21 ) Δ xˆk k

+ΔxˆkT − X 21 − (X 21 )T Δxˆk + (2 + γ¯ + ε)ΔekT X 22 P2−1 (X 22 )T Δek 

+ΔekT − X 22 − (X 22 )T Δek   (2.45) = E ξkT Σξk ,

where ξk =



T ηkT ηk−d ΔηkT

⎡ ⎢ ⎢ ⎢ Σ =⎢ ⎢ ⎢ ⎣

k

σ11 ∗ ∗ ∗ ∗ ∗



,

⎤ 0 AiT P1 Adi 0 −(X 21 )T − X 11 0 T T T T σ22 0 (Ai P2 + X 12 )Adi 0 Ai (X 22 ) − (X 22 ) − X 12 ⎥ ⎥ ⎥ ∗ σ33 0 0 0 ⎥, ⎥ ∗ ∗ σ44 0 0 ⎥ ⎦ ∗ ∗ ∗ σ55 0 ∗ ∗ ∗ ∗ σ66

σ11 = −P1 + Q 1 + 6AiT P1 Ai − X 11 − (X 11 )T + γ2 X 11 P1−1 (X 11 )T + γ5 λCiT L iT L i Ci , σ22 = −P2 + Q 2 + 4 AiT P2 Ai + sym(X 12 Ai − X 12 ) + γ4 λCiT L iT L i Ci + (γ2 − 2)X 12 P1−1 (X 12 )T , T T σ33 = (5 + γ¯ + ε)Adi P1 Adi − Q 1 , σ44 = (3 + γ¯ + ε)Adi P2 Adi − Q 2 ,

σ55 = −X 21 − (X 21 )T + (3 + γ¯ + ε)(X 21 )P1−1 (X 21 )T , σ66 = −X 22 − (X 22 )T + (2 + γ¯ + ε)X 22 P2−1 (X 22 )T , γ1 = 4(γ¯ − γ¯ 2 ) + 4ε | 1 − 2γ¯ |, γ2 = ε2 + ε + 2 + γ¯ , γ3 = 12(γ¯ + ε)2 + 8(γ¯ + ε), γ4 = γ3 + 6(γ¯ + ε), γ5 = 2 + 15ε2 + 2γ1 ,  = d M − dm + 1.

Then, Σ < 0 can be rewritten by Σ ∗ + Σ1 Λ−1 (Σ1 )T < 0

(2.46)

with ⎡ ∗ σ11 ⎢ ∗ ⎢ ⎢ ∗ Σ∗ = ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

⎤ 0 AiT P1 Adi 0 −(X 21 )T − X 11 0 ∗ 0 (AiT P2 + X 12 )Adi 0 AiT (X 22 )T − (X 22 )T − X 12 ⎥ σ22 ⎥ ⎥ ∗ σ33 0 0 0 ⎥, ⎥ ∗ ∗ σ44 0 0 ⎥ ∗ ⎦ ∗ ∗ ∗ σ55 0 ∗ ∗ ∗ ∗ ∗ σ66

2.3 Main Results ⎡

1

35 1

(γ2 ) 2 X 11 (γ5 λ) 2 CiT L iT 0 0 1 1 ⎢ 0 0 (γ2 − 2) 2 X 12 (γ4 λ) 2 CiT L iT ⎢ ⎢ 0 0 0 0 ⎢ Σ1 = ⎢ 0 0 0 0 ⎢ ⎢ ⎣ 0 0 0 0 (3 + γ¯ 0 0 0 0

0 0 0 0 1 + ε) 2 X 21 0 (2 + γ¯

⎤ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥, 0 ⎥ ⎥ ⎦ 0 1 + ε) 2 X 22

∗ Λ = diag{P1 , I, P1 , I, P1 , P2 }, σ11 = −P1 + Q 1 + 6AiT P1 Ai − X 11 − (X 11 )T , ∗ ∗ ∗ σ22 = −P2 + Q 2 + 4 AiT P2 Ai + sym(X 12 Ai − X 12 ), σ55 = −X 21 − (X 21 )T , σ66 = −X 22 − (X 22 )T .

By the Lemma 2.2, the above inequality is equivalent to   ∗ Σ Σ1 < 0. Ξ := ∗ −Λ

(2.47)

Thus, the condition (2.10) in Theorem 2.1 can ensure ΔVk < 0. Based on the stability theorem, the augmented system (2.9) without disturbance is stochastically stable. Remark 2.1 In the derivation of Theorem 2.1, we have made great effort to obtain the sufficient stochastic stability criteria. For this purpose, the condition P ≤ λI has been introduced and the feasibility of new approach has been improved.

2.3.2 Stochastic Stability with H∞ Performance Subsequently, the stochastic stability with H∞ performance will be discussed for the system (2.9) under the conditions of φk = 0 and ωk = 0. Theorem 2.2 For given scalar λ > 0, the augmented system (2.9) is stochastically stable with disturbance attenuation level ϑ if there exist positive-definite matrices         Q1 0 P 0 Y 0 Y 0 ,Q = , symmetric matrices Y1 = 11 , Y2 = 21 P= 1 0 P2 0 Q2 0 Y12 0 Y22 with proper dimensions and real matrices L i (i ∈ N ) satisfying  ∗  Σ¯ Σ¯ 1 Ξ¯ = < 0, ∗ −Λ P ≤ λI,

(2.48) (2.49)

where ⎡ ∗ σ¯ 11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ Σ¯ ∗ = ⎢ ⎢ ∗ ⎢ ∗ ⎢ ⎣ ∗ ∗

0 AiT P1 Adi 0 −(Y21 )T − Y11 ∗ 0 (AiT P2 + Y12 )Adi 0 σ¯ 22 ∗ σ¯ 33 0 0 0 ∗ ∗ σ¯ 44 ∗ ∗ ∗ ∗ σ¯ 55 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 σ¯ 26 AiT P2 E i + Y12 E i 0 0 T P E 0 Adi 2 i 0 0 ∗ Y22 E i σ¯ 66 ∗ σ¯ 77

⎤ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦

36

2 Sliding Mode H∞ Control for Markov Jump Discrete Systems … ⎡

⎢ ⎢ ⎢ ⎢ ⎢ Σ¯ 1 = ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

√

γ2 Y11 0 0 0 0 0 0

 ⎤ (γ5 + ε 2 )λCiT L iT 0 0 0 0 ⎥ √ √ ⎥ 0 γ2 − 2Y12 γ6 λCiT L iT 0 0 ⎥ ⎥ 0 0 0 0 0 ⎥ ⎥, 0 0 0 0 0 ⎥ √ ⎥ ⎥ 3 + γ¯ + εY21 0 0 0 0 ⎥ √ 2 + γ¯ + εY22 ⎦ 0 0 0 0 0 0 0 0 0

∗ = −P + Q + I + 6A T P A − Y − (Y )T , σ¯ ∗ = −P + Q + I + 4 A T P A + sym(Y A − Y ), σ¯ 11 1 1 11 11 2 2 12 i 12 22 i 1 i i 2 i T P A − Q , σ¯ = (3 + γ¯ + ε)A T P A − Q , σ¯ 26 = AiT (Y22 )T − (Y22 )T − Y12 , σ¯ 33 = (5 + γ¯ + ε) Adi 1 di 1 44 2 di 2 di ∗ = −Y − (Y )T , σ¯ ∗ = −Y − (Y )T , σ¯ = (2 + γ¯ + ε)E T P E − ϑ 2 I, γ = γ + γ¯ + ε, σ¯ 55 21 21 22 22 77 6 4 66 i 2 i

and γ j ( j = 1, 2, 3, 4, 5) are defined below (2.11). Proof Choosing the same Lyapunov functional as in Theorem 2.1 and calculating its difference along (2.9), a similar inequality can be derived as follows:    ΔVk ≤ E ηkT A¯ iT P A¯ i − P + (d M − dm + 1)Q ηk T T +2ηkT A¯ iT P A¯ di ηk−dk + ηk−d ( A¯ di P A¯ di − Q)ηk−dk k

 +2ωkT E¯ iT P A¯ i ηk + 2ωkT E¯ iT P A¯ di ηk−dk + ωkT E¯ iT P E¯ i ωk .

(2.50)

   Y11 0 Y21 0 , Y2 = with proper dimensions, the For any real matrices Y1 = 0 Y12 0 Y22 following equality is true 

  E 2(ηkT Y1 + ΔηkT Y2 )[( A¯ i − I )ηk − Δηk + A¯ di ηk−dk + E¯ i ωk ] = 0. (2.51) Combining (2.50) and (2.51), we know    ΔVk ≤ E ηkT A¯ iT P A¯ i − P + (d M − dm + 1)Q + Y1 ( A¯ i − I ) + ( A¯ i − I )T Y1T ηk T T P A¯ − Q)η T T T +ηk−d ( A¯ di di k−dk − 2Δηk Y2 Δηk + 2ηk ( A¯ i P A¯ di + Y1 A¯ di ) k T ×ηk−dk + 2ηkT [( A¯ i − I )T Y2T − Y1 ]Δηk + 2ηk−d A¯ T Y T Δηk + 2ωkT E¯ iT k di 2

×P A¯ i ηk + 2ωkT E iT P2 Adi ek−dk + ωkT E iT P2 E i ωk + 2ekT Y12 E i ωk  +2ΔekT Y22 E i ωk .

(2.52)

Similarly, it can be obtained that 2ωkT E¯ iT P A¯ i ηk ≤ 2ωkT E iT P2 Ai ek + λε2 xˆkT CiT L iT L i Ci xˆk + λ(γ¯ + ε)ekT CiT L iT L i Ci ek + (1 + γ¯ + ε) ×ωkT E iT P2 E i ωk ,

and

(2.53)

2.3 Main Results

37

ΔVk  ≤ E xˆkT (−P1 + Q 1 )xˆk + 6xˆkT AiT P1 Ai xˆk − 2 xˆkT Y11 xˆk + γ2 xˆkT Y11 P1−1 (Y11 )T xˆk   +(2 + 16ε2 + 2γ1 )λxˆkT CiT L iT L i Ci xˆk + 2 xˆkT AiT P1 Adi xˆk−dk − 2 xˆkT (Y21 )T + Y11 ×Δxˆk + ekT (−P2 + Q 2 )ek + 4ekT AiT P2 Ai ek + λ(γ4 + γ¯ + ε)ekT CiT L iT L i Ci ek +2ekT (Y12 Ai − Y12 )ek + (γ2 − 2)ekT Y12 P1−1 (Y12 )T ek + 2ekT (AiT P2 + Y12 )Adi ek−dk

+2ekT AiT (Y22 )T − (Y22 )T − Y12 Δek + 2ekT (AiT P2 E i + Y12 E i )ωk



T T T T (5 + γ¯ + ε)Adi (3 + γ¯ + ε)Adi +xˆk−d P1 Adi − Q 1 xˆk−dk + ek−d P2 Adi − Q 2 k k

T ×ek−dk + 2ek−d A T P E ω + ΔxˆkT − Y21 − (Y21 )T Δxˆk + (3 + γ¯ + ε)ΔxˆkT (Y21 ) k di 2 i k

×P1−1 (Y21 )T Δxˆk + ΔekT − Y22 − (Y22 )T Δek + (2 + γ¯ + ε)ΔekT Y22 P2−1 (Y22 )T Δek  +2ΔekT Y22 E i ωk + (2 + γ¯ + ε)ωkT E iT P2 E i ωk .

In order to deal with the H∞ performance, the following index is given by   n [ηkT ηk − γ 2 ωkT ωk ] . J (n) = E

(2.54)

(2.55)

k=0

Obviously, it is necessary to verify J (n) < 0 (n −→ ∞). Notice that n   [ηkT ηk − ϑ 2 ωkT ωk + ΔVk ] − Vn+1 J (n) = E k=0 n    E ΔVk + xˆkT xˆk + ekT ek − ϑ 2 ωkT ωk . ≤

(2.56)

k=0

We have   E ΔVk + xˆkT xˆk + ekT ek − ϑ 2 ωkT ωk  ≤ E xˆkT (−P1 + Q 1 + I )xˆk + 6xˆkT AiT P1 Ai xˆk − 2 xˆkT Y11 xˆk + γ2 xˆkT Y11 P1−1 (Y11 )T xˆk   +(2 + 16ε2 + 2γ1 )λxˆkT CiT L iT L i Ci xˆk + 2 xˆkT AiT P1 Adi xˆk−dk − 2 xˆkT (Y21 )T + Y11 ×Δxˆk + ekT (−P2 + Q 2 + I )ek + 4ekT AiT P2 Ai ek + λ(γ4 + γ¯ + ε)ekT CiT L iT L i Ci ek +2ekT (Y12 Ai − Y12 )ek + (γ2 − 2)ekT Y12 P1−1 (Y12 )T ek + 2ekT (AiT P2 + Y12 )Adi ek−dk

+2ekT AiT (Y22 )T − (Y22 )T − Y12 Δek + 2ekT (AiT P2 E i + Y12 E i )ωk



T T T T (5 + γ¯ + ε)Adi (3 + γ¯ + ε)Adi +xˆk−d P1 Adi − Q 1 xˆk−dk + ek−d P2 Adi − Q 2 k k

T ×ek−dk + 2ek−d A T P E ω + ΔxˆkT − Y21 − (Y21 )T Δxˆk + (3 + γ¯ + ε)ΔxˆkT Y21 k di 2 i k

×P1−1 (Y21 )T Δxˆk + ΔekT − Y22 − (Y22 )T Δek + (2 + γ¯ + ε)ΔekT Y22 P2−1 (Y22 )T Δek

 +2ΔekT Y22 E i ωk + ωkT (2 + γ¯ + ε)E iT P2 E i − ϑ 2 I ωk   = E ξ¯kT Σ¯ ξ¯k ,

(2.57)

38

2 Sliding Mode H∞ Control for Markov Jump Discrete Systems …

where  T ξ¯k = ξkT ωkT , ⎡ 0 −(Y21 )T − Y11 σ¯ 11 0 AiT P1 Adi T P + Y )A ⎢ ∗ σ¯ 22 0 (A 0 12 di i 2 ⎢ ⎢ ∗ ∗ σ¯ 33 0 0 ⎢ Σ¯ = ⎢ ∗ σ¯ 44 0 ⎢ ∗ ∗ ⎢ ∗ ∗ ∗ ∗ σ¯ 55 ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 σ¯ 26 AiT P2 E i + Y12 E i 0 0 T P E 0 Adi 2 i 0 0 σ¯ 66 Y22 E i ∗ σ¯ 77

⎤ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦

σ¯ 11 = −P1 + Q 1 + I + 6AiT P1 Ai − Y11 − (Y11 )T + γ2 Y11 P1−1 (Y11 )T + (γ5 + ε2 )λCiT L iT L i Ci , σ¯ 22 = −P2 + Q 2 + I + 4 AiT P2 Ai + sym(Y12 Ai − Y12 ) + (γ2 − 2)Y12 P1−1 (Y12 )T + λγ6 CiT L iT L i Ci , T σ¯ 26 = AiT (Y22 )T − (Y22 )T − Y12 , σ¯ 33 = (5 + γ¯ + ε)Adi P1 Adi − Q 1 , T σ¯ 44 = (3 + γ¯ + ε)Adi P2 Adi − Q 2 , σ¯ 55 = −Y21 − (Y21 )T + (3 + γ¯ + ε)(Y21 )P1−1 (Y21 )T ,

σ¯ 66 = −Y22 − (Y22 )T + (2 + γ¯ + ε)Y22 P2−1 (Y22 )T , σ¯ 77 = (2 + γ¯ + ε)E iT P2 E i − ϑ 2 I, γ6 = γ4 + γ¯ + ε.

Then, the inequality Σ¯ < 0 can be presented by Σ¯ ∗ + Σ¯ 1 Λ−1 (Σ¯ 1 )T < 0

(2.58)

with ⎡

⎤ ∗ 0 AT P A σ¯ 11 0 −(Y21 )T − Y11 0 0 i 1 di ∗ T T ⎢ ∗ σ¯ 22 0 (Ai P2 + Y12 )Adi 0 σ¯ 26 Ai P2 E i + Y12 E i ⎥ ⎢ ⎥ ⎢ ∗ ∗ ⎥ σ¯ 33 0 0 0 0 ⎢ ⎥ ∗ T ⎢ ⎥, ¯ Σ =⎢ ∗ ∗ ∗ σ¯ 44 0 0 Adi P2 E i ⎥ ∗ ⎢ ∗ ∗ ⎥ ∗ ∗ σ¯ 55 0 0 ⎢ ⎥ ∗ ⎣ ∗ ∗ ⎦ ∗ ∗ ∗ σ¯ 66 Y22 E i ∗ ∗ ∗ ∗ ∗ ∗ σ¯ 77  ⎡√ ⎤ γ2 Y11 (γ5 + ε2 )λCiT L iT 0 0 0 0 √ √ T T ⎢ ⎥ 0 0 γ2 − 2Y12 γ6 λCi L i 0 0 ⎢ ⎥ ⎢ ⎥ 0 0 0 0 0 0 ⎢ ⎥ ⎥, Σ¯ 1 = ⎢ 0 0 0 0 0 0 ⎢ ⎥ √ ⎢ ⎥ 0 0 0 0 3 + γ¯ + εY21 0 ⎢ ⎥ √ ⎣ 0 0 0 0 0 2 + γ¯ + εY22 ⎦ 0 0 00 0 0 ∗ ∗ ∗ σ¯ 11 = −P1 + Q 1 + I + 6AiT P1 Ai − Y11 − (Y11 )T , σ¯ 55 = −Y21 − (Y21 )T , σ¯ 66 = −Y22 − (Y22 )T , ∗ σ¯ 22 = −P2 + Q 2 + I + 4 AiT P2 Ai + sym(Y12 Ai − Y12 ).

By the Lemma 2.2, it holds   ∗ Σ¯ Σ¯ 1 < 0. Ξ¯ := ∗ −Λ

(2.59)

2.4 Design of Robust Sliding Mode Controller

39

Therefore, the conditions in Theorem 2.2 can ensure Σ¯ < 0 and J (n) ≤ 0. According to the stability theorem, the augmented system (2.9) is stochastically stable with disturbance attenuation level ϑ. Remark 2.2 Now, the observer-based stability problem with H∞ performance has been solved in Theorem 2.2 for the addressed system. Accordingly, the modedependent sufficient criteria, which are in terms of LMIs, have been proposed. It is worthwhile to note that, the free weighting matrix skills and LMIs method have been adopted in order to deal with the conservatism issue of new results. Remark 2.3 In the process of derivation, the Lyapunov function method and LMIs technique are fully developed. For example, by adding the condition P ≤ λI , the feasibility of new approach is further improved and the chattering phenomenon is properly attenuated.

2.4 Design of Robust Sliding Mode Controller In this subsection, the reachability of the sliding surface will be discussed and the observer-based sliding mode H∞ control law will be designed. As in [31], the improved reaching condition, which needs to be tested in order to ensure the desired requirements, is introduced as follows:  Δs = s −μk sgn(sk ) − αsk , sk > 0 k k+1 − sk ≤ −βe Δsk = sk+1 − sk ≥ −βe−μk sgn(sk ) − αsk , sk < 0

(2.60)

with 0 < α < 1, β > 0 and μ ≥ 0. Then, the design of the desired robust observerbased H∞ controller is provided in the subsequent theorem for the reachability analysis. Theorem 2.3 For the sliding surface (2.5) with G = B T P1 , if the SMC problem in Theorem 2.2 is solvable with P1 being the solutions, then the observer-based sliding mode H∞ controller given by  u k = −(G B)−1 G(Ai − I )xˆk + G Adi xˆk−dk + βe−μk sgn(sk )  +αsk + ψsgn(sk )

(2.61)

can ensure the discrete reaching condition, where ψ = G L i yk  +γ¯  G L i Ci xˆk . Proof Recalling the sliding surface (2.5) and the observer (2.3), we derive Δsk = −βe−μk sgn(sk ) − αsk + G L i (yk − γ¯ Ci xˆk ) − ψsgn(sk ). Let’s discuss the sign of Δsk based on sk . Specifically, when sk > 0, one has

(2.62)

40

2 Sliding Mode H∞ Control for Markov Jump Discrete Systems …

G L i (yk − γ¯ Ci xˆk ) − ψsgn(sk ) ⎡  G L i yk  +γ¯  G L i Ci xˆk ⎢ G L i yk  +γ¯  G L i Ci xˆk ⎢ = G L i (yk − γ¯ Ci xˆk ) − ⎢ .. ⎣ .  G L i yk  +γ¯  G L i Ci xˆk ≤ 0.

⎤  ⎥ ⎥ ⎥ ⎦  (2.63)

Combine (2.62) and (2.63), we have Δsk ≤ −βe−μk sgn(sk ) − αsk .

(2.64)

Similarly, when sk < 0, it can be derived that Δsk ≥ −βe−μk sgn(sk ) − αsk .

(2.65)

According to the reaching condition (2.60), the reachability of sliding surface (2.5) in discrete-time setting can be guaranteed by newly proposed the observer-based sliding mode H∞ controller (2.61).

2.5 Simulation In this section, the numerical simulation is proposed such that the usefulness of obtained H∞ observer-based robust SMC approach is demonstrated. The parameter matrices of stochastic MJSs (2.1) are given by ⎡

A1 C1 A2

B

⎤ ⎡ ⎤ ⎡ ⎤ 0.15 −0.25 0 −0.08 −0.02 0.06 0.04 = ⎣ 0 0.13 0.01 ⎦ , E 1 = ⎣ 0.011 ⎦ , Ad1 = ⎣ 0.04 0.01 −0.03⎦ , 0.03 0 −0.05 −0.019 0.01 0.02 0.05 = diag{0.04, 0.04, 0.06}, C2 = diag{−0.08, −0.01, 0.04}, ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0.15 −0.25 0 0.081 0.03 0 0.01 = ⎣ 0 0.13 0.01 ⎦ , E 2 = ⎣0.042⎦ , Ad2 = ⎣0.02 0.03 0 ⎦ , 0.03 0 −0.05 0.023 0.04 0.05 −0.01 ⎡ ⎤ 0.08 −0.017 = ⎣ 0.2 0.3 ⎦ . 0.13 0.26

Choose π11 = 0.4, π12 = 0.6, π21 = 0.8, π22 = 0.2, γ¯ = 0.6 and ε = 0.1. Besides, the bounds of time-delay dk are set as dm = 1 and d M = 4. Then, for prescribed scalars λ = 3 and ϑ = 0.73, solving H∞ SMC problem in Theorem 2.2 yields

2.5 Simulation

41

⎡

⎤ ⎡ 2.0707 −0.1542 −0.2033 1.1689 0.2569 P1 = ⎣−0.1542 2.9158 −0.0151⎦ , L1 = ⎣0.2540 0.9343 −0.2033 −0.0151 2.7707 0.0074 −0.0461 ⎡ ⎤ ⎡ 2.6120 −0.0405 −0.0488 0.6004 0.1582 P2 = ⎣−0.0405 2.8819 0.0080 ⎦ , L2 = ⎣ 0.1560 3.6368 −0.0488 0.0080 2.7824 −0.0090 0.0745

⎤ 0.0077 −0.0462⎦ , 0.9922 ⎤ −0.0093 0.0746 ⎦ . 1.4549

In the simulation, by setting α = 0.1 and β = 1, we can apply newly designed H∞ sliding mode controller with all known parameters. Figures 2.1, 2.2, 2.3, 2.4, 2.5 and 2.6 show the simulation results. Among them, Fig. 2.1 gives one possible realizations of rk . It is shown from Figs. 2.2, 2.3 and 2.4 that despite the PLs condition, the state

Fig. 2.1 Random mode rk

3

rk = 1 rk = 2

2.5

2

1.5

1

0.5

0 0

10

20

30

40

50

60

No. of samples, k

Fig. 2.2 The trajectory of state xk1 and its estimation xˆk1

2

x1k x ˆ1k

1.5 1 0.5 0 −0.5 1

−1

0 −1

−1.5

3

4

5

6

7

8

−2 10

20

30

No. of samples, k

40

50

60

42

2 Sliding Mode H∞ Control for Markov Jump Discrete Systems …

Fig. 2.3 The trajectory of state xk2 and its estimation xˆk2

1

x2k x ˆ2k

0.5

0 0.8 0.6

−0.5

0.4 0.2

−1 0

−1.5

10

7

8

20

9

10

11

30

12

40

50

60

No.of samples, k

Fig. 2.4 The trajectory of state xk3 and its estimation xˆk3

3

x3k x ˆ3k

2

1

0 0.4

−1

0.3 0.2

−2

0.1 0

−3

10

10

20

15

30

20

40

50

60

No.of samples, k

trajectory xk converges to a small neighborhood quickly. Moreover, it can also be seen that the estimation xˆk can track the system trajectories well by letting observer receive and update measurement signals timely. The sliding mode function sk is displayed in Fig. 2.5. In addition, the control signal u k is plotted in Fig. 2.6. Then, it

2.6 Conclusion Fig. 2.5 The trajectory of sliding variable sk

43 3

s1k s2k

2

1

0

−1

−2

−3

20

10

40

30

60

50

No. of samples, k

Fig. 2.6 The control signal uk

10

u1,k u2,k

8 6 4 2 0 −2 −4 −6 −8 −10

10

20

30

40

50

60

No. of samples, k

can be obtained from Fig. 2.6 that the energy of u k decays as the effect of ωk onto the system becomes small. Besides, from the simulation results, we can conclude that the robust observer-based sliding mode H∞ control scheme has a desiring control performance.

2.6 Conclusion In this paper, the robust observer-based sliding mode H∞ control problem has been addressed for delayed stochastic MJSs with PLs and disturbance. First, the PLs phenomenon is characterized by a series of Bernoulli random variables with

44

2 Sliding Mode H∞ Control for Markov Jump Discrete Systems …

imprecise occurrence probability and the sliding mode observer is designed to estimate the state variables. In the sequel, a linear form of switching function is constructed based on the estimated space. Then, the sufficient stochastic stability criteria with H∞ performance has been proposed for the augmented system by using Lyapunov approach, LMIs technique and free weighting matrix skills. In addition, a desired robust observer-based H∞ controller has been constructed in discretetime setting by proposing an improved reaching condition. As a result, the abovementioned sliding surface and the observer-based H∞ controller can be fully determined by solving related parameter matrices in the strict LMIs obtained. The key feature of our results is to propose one way to eliminate the chattering by avoiding the application of larger sliding mode gain.

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Chapter 3

Finite-Time Boundedness

In this chapter, the finite time observer-based sliding mode control (SMC) problem is considered for stochastic Markovian jump systems (MJSs) with deterministic switching chain (DSC) subject to time-varying delay and packet losses (PLs). Firstly, the stochastic MJSs with DSC is appropriately modeled and the PLs case is characterized by using some Bernoulli random variables. Then, a non-fragile finite-time bounded sliding mode observer is designed. Our objective is to propose a finite time observerbased SMC approach such that, for above addressed system, the finite-time boundedness in certain time interval can be guaranteed by giving sufficient criteria via stochastic analysis skills and average dell time (ADT) method. Moreover, a new robust finite-time sliding mode controller can be designed to ensure reachability of common sliding surface in the estimation space. Finally, the theoretical findings are illustrated by a numerical example.

3.1 Introduction Over the last decades, the switched control systems have attracted certain attention due to its good characterization of physical systems [1–3]. Markovian jumping systems (MJSs) are actually a special kind of switched system with certain switching rules. The main feature of MJSs is that the system switches back and forth among several sub-modes according to given Markov chain and thus not traditionally single systems. Some exploratory works on stability analysis and synthesis of MJSs have been developed in [4–7]. The problems of sampled-data synchronization and asynchronous dissipative control have been investigated in [8, 9] for Markovian jump neural networks and fuzzy MJSs respectively, the problems of stochastic admissibility in [10, 11] for descriptor MJSs with time-varying delays and in [12] for descriptor MJSs with switching chains. Indeed, such type of MJSs with switching chains has certain practical application background. For the actual control system, the action time of controller can not be infinite. On the contrary, it is often necessary to achieve the required control goal in a certain time interval. In [13, 14], the finite-time © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Kao et al., Analysis and Design of Markov Jump Discrete Systems, Studies in Systems, Decision and Control 499, https://doi.org/10.1007/978-981-99-5748-4_3

47

48

3 Finite-Time Boundedness

stability problem and synchronization problem have been proposed for memristorbased fractional-order neural networks and networks respectively. The finite-time asynchronous control issue has been investigated in [15] for continuous-time MJSs based on the average dwell time (ADT) approach. However, the existing results on MJSs with switching chains can not meet complex realistic needs obviously. Thus, it seems more significant to study finite-time control problem for stochastic MJSs with switching chain. The robust SMC method has been widely discussed and investigated due to its robustness against disturbance. It is very natural to study the analysis and synthesis for MJSs with switching chain by SMC method. It is worthwhile to note that, many observer-based control methods have been proposed since the internal information of realistic systems are generally not easy to be obtained or its expenses are unaffordable [16–18]. Specially, an observer-based control scheme has been provided in [19] for singular nonhomogeneous MJSs, where the state-feedback controller based on the estimation has been constructed. In order to estimate the unmeasured state, the non-fragile sliding mode observers have been designed in [4, 20] respectively for continuous Markovian neutral-type stochastic systems and uncertain switched systems with state unavailable. It should be mentioned that, the finite-time SMC method has been provided in [21] for uncertain stochastic single systems. However, the existing results on sliding mode observer-based finite time control problem of multiple systems are not much. Thus, it is challenging to explore sliding mode observer-based finite time control scheme for stochastic discrete MJSs subject to deterministic switching chain. On the other hand, time-varying delays and external disturbances often disturb the most basic stability of the system, even a very small time-delay will cause abnormal fluctuation. Accordingly, a great number of methods have been developed for delayed control systems under external disturbances, such as [22–24]. Subject to stochastic communication delays and bounded time-varying delays, the delay-dependent stability conditions have been established in [25, 26] for stochastic networked control systems by delay-fractioning approach, where delay related information can be fully utilized as partition of delay distribution interval becomes more detailed. The finite time stabilization algorithm has been obtained in [12] for descriptor MJSs with switching chains subject to noise signal using ADT method. Moreover, the observerbased finite-time control problem has been investigated in [27] for delayed singular MJSs with actuator saturation, where the stochastic finite-time boundedness of singular system has been guaranteed by designing observer-based feedback. To the best of our knowledge, the finite-time observer-based SMC algorithm has not been proposed for delayed stochastic MJSs with switching chain via ADT method, not to mention packet losses (PLs) case. Motivated by the above discussions, this paper considers the finite time observerbased SMC problem for a class of stochastic MJSs with two switching chain under PLs, time-varying delay and disturbance. A non-fragile observer of original system is first constructed. Then, the error system is obtained by the difference operation of original system and non-fragile observer. Based on the estimated state information, a linear common sliding mode surface is chosen. Further, the equations of

3.2 Problem Statements and Preliminaries

49

sliding mode dynamics and related augmented system are given respectively. By stochastic Lyapunov theorem and ADT approach, the finite time boundedness of augmented system with two switching chains is investigated. Some mode-dependent sufficient criteria are derived via linear matrix inequalities (LMIs) technique and free weighting matrix skills. Recalling the improved reaching condition, the desired observer-based controller is designed in state estimation space. Finally, numerical example is provided to verify the effectiveness of obtained control method. The novelties of this paper are summarized briefly as follows: (i) we make an attempt to study the finite-time sliding mode observer design problem for stochastic MJSs with two switching chains; (ii) the common sliding mode surface is proposed rather than traditional mode-dependent one; (iii) the effects of time-varying delay onto system under consideration are properly attenuated by using ADT method.

3.2 Problem Statements and Preliminaries In this paper, we consider the following class of Markovian jump systems with switching chain: ⎧ ⎪ ⎨ xk+1 = A(rk , σk )xk + Ad (rk , σk )xk−dk + Bu k + D(rk , σk )ωk yk = θk C(rk , σk )xk ⎪ ⎩ xk = φk , k = −d M , −d M + 1, . . . , 0 where xk ∈ Rn is the state vector of system, u k ∈ Rs is the control input, yk ∈ Rq denotes the controlled output, dk ∈ [dτ , d M ] is the time-varying delay. rk (k ∈ Z + ) is a discrete Markov chain and σk is a finite-state switching signal, which take values from N1 = {1, 2, . . . , N1 } and N2 = {1, 2, . . . , N2 } respectively. The sequence {ki , i = 1, 2, . . . } represents the switching instants of σk . Ai,m  A(rk , σk ), Adi,m  Ad (rk , σk ), Bi,m  B(rk , σk ), Ci,m  C(rk , σk ) and Di,m  D(rk , σk ) are known real matrices, among which the matrix B is of column full rank.  The time-varying signal T ωkT ωk ≤ γ 2 . ωk ∈ l2 ([0, T ); R p ) is external disturbance, which satisfies k=0 The above system can be simply expressed as ⎧ ⎪ ⎨ xk+1 = Ai,m xk + Adi,m xk−dk + Bu k + Di,m ωk yk = θk Ci,m xk (3.1) ⎪ ⎩ xk = φk , k = −d M , −d M + 1, . . . , 0 Next, we denote that transition probability matrix of rk can be given by ⎡ m m m ⎤ π11 π12 . . . π1N 1 m m m ⎥ ⎢ π21 π22 . . . π2N 1 ⎥ ⎢ m Π =⎢ . .. .. .. ⎥ . ⎣ .. . . . ⎦ π Nm1 1 π Nm1 2 . . . π Nm1 N1

(3.2)

50

3 Finite-Time Boundedness

In addition, the random variable θk , which takes values from 0 and 1, is employed in order to characterize the packet losses: Prob{θk = 1} = θ¯ ,

Prob{θk = 0} = 1 − θ¯ with θ¯ ∈ [0, 1] is a known scalar. Then, we design a non-fragile observer for system (3.1):

xˆk+1 = Ai,m xˆk + Adi,m xˆk−dk + Bu k + (L i,m + ΔL i,m )(yk − yˆk ) (3.3) ¯ i,m xˆk yˆk = θC where L i (i ∈ N ) are the gain matrices of observer to be designed later, ΔL i,m = T (k)Ji,m (k) ≤ I . Letting ek = xk − xˆk , the error system genHi,m Ji,m (k)E i,m with Ji,m erated by system (3.1) and (3.3) is as follows:   ek+1 = Ai,m − θk (L i,m + ΔL i,m )Ci,m ek + Adi,m ek−dk + Di,m ωk −(θk − θ¯ )(L i,m + ΔL i,m )Ci,m xˆk

(3.4)

Lemma 3.1 (Schur complement) Given constant matrices Q1 , Q2 , Q3 , where Q1 = Q1T and 0 < Q2 = Q2T , then Q1 + Q3T Q2−1 Q3 < 0 if and only if     Q1 Q3T −Q2 Q3 < 0 or < 0. ∗ −Q2 ∗ Q1 Lemma 3.2 Set Q = Q T , M and N be real matrices of compatible dimensions, and uncertain matrix F satisfies the inequality F T F ≤ I . Then Q + M F N + N T F T M T < 0 holds, if and only if there exists ε > 0 such that Q + εM M T + ε−1 N T N < 0 or, ⎡ ⎤ Q εM N T ⎣ εM T −ε I 0 ⎦ < 0. N 0 −ε I

3.3 Design of Sliding Surface In this section, the problem of finite-time boundedness is analysed for Markovian jump systems with switching chain. Some sufficient criteria are given in Theorem 3.1 by traditional SMC technique and ADT approach. Then, the sufficient finitetime bounded conditions are given in Theorem 3.2 by means of strict linear matrix inequalities according to elementary matrix inequality technique. First, we propose a common sliding surface for system (3.1):

3.3 Design of Sliding Surface

51

sk = G xˆk

(3.5)

with G = (B T B)−1 B T . By combining (3.3), (3.5) and condition sk+1 = sk = 0, it can be obtained that u eq = −[G Ai,m xˆk + G Adi,m xˆk−dk + θ¯ G(L i,m + ΔL i,m )Ci,m ek ] Then, substituting u eq into Eq. (3.3), we can get sliding mode dynamics as follows:   xˆk+1 = (I − BG)Ai,m + (θk − θ¯ )(L i,m + ΔL i,m )Ci,m xˆk + (I − BG)Adi,m xˆk−dk +(θk I − θ¯ BG)(L i,m + ΔL i,m )Ci,m ek (3.6) T  Letting ηk = xˆkT ekT , the augmented system formed by (3.4) and (3.6) is ηk+1 = A¯ i,m ηk + A¯ di,m ηk−dk + D¯ i,m ωk

G¯ Ai,m + (θk − θ¯ )(L i,m + ΔL i,m )Ci,m (θk I − θ¯ BG)(L i,m + ΔL i,m )Ci,m −(θk − θ¯ )(L i,m + ΔL i,m )Ci,m Ai,m − θk (L i,m + ΔL i,m )Ci,m     ¯ 0 G Adi,m 0 = , D¯ i,m = , G¯ = I − BG Di,m 0 Adi,m

A¯ i,m = A¯ di,m



(3.7) 

Definition 3.1 [28] The augmented system (3.7) is said to be finite-time bounded if it holds for (c1 , c2 , S, T, γ )   maxk0 −d M ≤k≤k0 E ηkT Sηk ≤ c12   ⇒ E ηkT Sηk ≤ c22 , k ∈ {1, 2, . . . , T }. Theorem 3.1 Given α > 1, β > 0, ν > 1, if there exist positive-definite matrices Pi,m , Q m and R, real matrices X i,m , Yi,m and L i,m satisfying following conditions ⎡

Ξ11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ Ξ := ⎢ ⎢ ∗ ⎢ ∗ ⎢ ⎣ ∗ ∗

Ξ12 Ξ22 ∗ ∗ ∗ ∗ ∗

Ξ13 0 0 Ξ16 Ξ23 Ξ24 Ξ25 Ξ26 1 T T Ξ33 0 0 A G¯ T Y1i,m di,m 2 T 0 ∗ Ξ44 Adi,m P2i,m Di,m ∗ ∗ Ξ55 0 ∗ ∗ ∗ Y1i,m ∗ ∗ ∗ ∗ Pi,m ≤ ν Pi,n , Q m ≤ ν Q n , R ≤ λ1 I λ2 S ≤ Pi,m ≤ λ3 S, λ2 S ≤ Q m ≤ λ3 S ρ1 < ρ2

⎤ 0 ⎥ Ξ27 ⎥ ⎥ 0 ⎥ 1 T T ⎥ A Y

T lnν := τa∗ , lnρ2 − lnρ1

T ∈ Z+. τa

Here, 3 = 1 2 , 1 = d M − dτ + 1, 2 = (d M + dτ )/2, ρ1 = (1 + 3 )(ν − 1)λ3 c12 + α −1 βγ 2 λ1 ν 1 ρ2 = α −T (ν − 1)λ2 c22 + α −1 βγ 2 λ1 , Ξ11 = a˘ 1 − α P1i,m + 1 Q 1,m + sym[X 1i,m (G¯ Ai,m − I )] 2 θ¯ ¯ i,m + ΔL i,m )Ci,m , Ξ13 = A T G¯ T P1i,m G¯ Adi,m + 1 X 1i,m G¯ Adi,m Ξ12 = a˘ 2 + X 1i,m G(L i,m 2 2 1 1 ¯ T T T T T ¯ ¯ Ξ16 = X 1i,m + (G Ai,m − I ) Y1i,m , Ξ23 = θCi,m (L i,m + ΔL i,m ) G P1i,m G¯ Adi,m 2 2 1 θ¯ Ξ22 = a˘ 3 − α P2i,m + 1 Q 2,m + sym[X 2i,m (Ai,m − I )] − sym[X 2i,m (L i,m + ΔL i,m )Ci,m ] 2 2 1 ¯ T (L i,m + ΔL i,m )T P2i,m Adi,m + X 2i,m Adi,m Ξ24 = Ai,m P2i,m Adi,m − θC i,m 2 1 ¯ T (L i,m + ΔL i,m )T P2i,m Di,m + X 2i,m Di,m Ξ25 = Ai,m P2i,m Di,m − θC i,m 2  1 T T ¯T ¯ ¯ T (L i,m + ΔL i,m )T Y T Ξ27 = + (Ai,m − I )T Y2i,m − θC X i,m 2i,m , Ξ33 = Adi,m G P1i,m G Adi,m − α Q 1,m 2 2i,m ¯ θ T T T P T ¯T YT Ξ44 = Adi,m P2i,m Adi,m − α Q 2,m , Ξ55 = Di,m 2i,m Di,m − β R, Ξ26 = Ci,m (L i,m + ΔL i,m ) G 1i,m 2 a˘ 1 = A T G¯ T P1i,m G¯ Ai,m + (θ¯ − θ¯ 2 )C T (L i,m + ΔL i,m )T P1i,m (L i,m + ΔL i,m )Ci,m + (θ¯ − θ¯ 2 )C T i,m

i,m

i,m

×(L i,m + ΔL i,m )T P2i,m (L i,m + ΔL i,m )Ci,m T G ¯ T P1i,m G(L ¯ i,m + ΔL i,m )Ci,m + (θ − θ¯ 2 )C T (L i,m + ΔL i,m )T P1i,m (L i,m + ΔL i,m )Ci,m a˘ 2 = θ¯ Ai,m i,m T (L T +(θ − θ¯ 2 )Ci,m i,m + ΔL i,m ) P2i,m (L i,m + ΔL i,m )Ci,m T (L T 2 2 2 a˘ 3 = Ci,m i,m + ΔL i,m ) (θ¯ P1i,m + θ¯ BGP1i,m BG − θ¯ BGP1i,m − θ¯ P1i,m BG)(L i,m + ΔL i,m )Ci,m T P T ¯ T (L i,m + ΔL i,m )T P2i,m Ai,m +Ai,m 2i,m Ai,m − θ¯ Ai,m P2i,m (L i,m + ΔL i,m )Ci,m − θC i,m

¯ T (L i,m + ΔL i,m )T P2i,m (L i,m + ΔL i,m )Ci,m +θC i,m

Proof To start with, we choose the Lyapunov functional as V (ηk , i, m) = E{V1 (ηk , i, m) + V2 (ηk , i, m)},

(3.8)

where V1 (ηk , i, m) =

ηkT

Pi,m ηk , V2 (ηk , i, m) =

−d τ +1 

k−1 

ηsT Q m ηs

t=−d M +1 s=k−1+t





P1i,m 0 and Q m = with Pi,m = 0 P2i,m V (ηk+1 , rk+1 , m) along (3.7).



 Q 1,m 0 . Then, let’s calculate 0 Q 2,m

3.3 Design of Sliding Surface

53

E{V (ηk+1 , rk+1 , m)|ηk , i, m} = E{V1 (ηk+1 , rk+1 , m)|ηk , i, m} + E{V2 (ηk+1 , rk+1 , m)|ηk , i, m}  T P T T T ¯T P T ¯T ¯ ¯ = E ηkT A¯ i,m i,m A¯ i,m ηk + ηk−dk A¯ di,m Pi,m A¯ di,m ηk−dk + ωk D i,m i,m Di,m ωk + 2ηk Ai,m Pi,m Adi,m T P ¯ i,m ωk + 2η T ¯T ¯ ×ηk−dk + 2ηkT A¯ i,m i,m D k−dk Adi,m Pi,m Di,m ωk +

 N1

−d τ +1

k 

ηsT Q m ηs



t=−d M +1 s=k+t

with Pi,m = j=1 πimj P j,m = diag{P1i,m , P2i,m }, P1i,m =  N1 m j=1 πi j P2 j,m . Further, it can be derived that

 N1

j=1

πimj P1 j,m , P2i,m =

E{V (ηk+1 , rk+1 , m)|ηk , i, m} − αV (ηk , i, m) − βωkT Rωk   T   T   T T A¯ di,m Pi,m A¯ di,m − α Q m ηk−dk + ωkT D¯ i,m ≤ E ηkT A¯ i,m Pi,m A¯ i,m − α Pi,m + 1 Q m ηk + ηk−d Pi,m D¯ i,m k   T P T T ¯ i,m ωk + 2η T ¯T ¯ −β R ωk + 2ηkT A¯ i,m i,m A¯ di,m ηk−dk + 2ηk A¯ i,m Pi,m D k−d Adi,m Pi,m Di,m ωk k

with 1 = d M − dτ + 1. Besides, note that T P A¯ i,m i,m A¯ i,m     ¯ i,m + ΔL i,m )Ci,m (θk I − θ¯ BG)(L i,m + ΔL i,m )Ci,m T P1i,m 0 G¯ Ai,m + (θk − θ)(L = ¯ i,m + ΔL i,m )Ci,m 0 P2i,m −(θk − θ)(L Ai,m − θk (L i,m + ΔL i,m )Ci,m   ¯ i,m + ΔL i,m )Ci,m (θk I − θ¯ BG)(L i,m + ΔL i,m )Ci,m G¯ Ai,m + (θk − θ)(L × ¯ i,m + ΔL i,m )Ci,m −(θk − θ)(L Ai,m − θk (L i,m + ΔL i,m )Ci,m   a1 a2 = (3.9) ∗ a3

where T G ¯ T P1i,m G¯ Ai,m + (θk − θ)A ¯ T G¯ T P1i,m (L i,m + ΔL i,m )Ci,m + (θk − θ)C ¯ T (L i,m + ΔL i,m )T a1 = Ai,m i,m i,m

¯ 2 C T (L i,m + ΔL i,m )T P1i,m (L i,m + ΔL i,m )Ci,m + (θk − θ¯ )2 C T (L i,m + ΔL i,m )T ×P1i,m G¯ Ai,m + (θk − θ) i,m i,m ×P2i,m (L i,m + ΔL i,m )Ci,m T G ¯ T P1i,m (θk I − θ¯ BG)(L i,m + ΔL i,m )Ci,m + (θk − θ)C ¯ T (L i,m + ΔL i,m )T P1i,m (θk I − θ¯ BG) a2 = Ai,m i,m

¯ T (L i,m + ΔL i,m )T P2i,m Ai,m + θk (θk − θ¯ )C T (L i,m + ΔL i,m )T ×(L i,m + ΔL i,m )Ci,m − (θk − θ)C i,m i,m ×P2i,m (L i,m + ΔL i,m )Ci,m T (L T T T T a3 = Ci,m i,m + ΔL i,m ) (θk I − θ¯ BG) P1i,m (θk I − θ¯ BG)(L i,m + ΔL i,m )Ci,m + Ai,m P2i,m Ai,m − θk Ai,m T (L T T T ×P2i,m (L i,m + ΔL i,m )Ci,m + θk2 Ci,m i,m + ΔL i,m ) P2i,m (L i,m + ΔL i,m ) − θk Ci,m (L i,m + ΔL i,m ) ×P2i,m Ai,m Ci,m

Then, considering (3.9), it leads to     T P E ηkT A¯ i,m i,m A¯ i,m − α Pi,m + 1 Q m ηk   a˘ 2 a˘ − α P1i,m + 1 Q 1,m η = ηkT 1 ∗ a˘ 3 − α P2i,m + 1 Q 2,m k = xˆkT (a˘ 1 − α P1i,m + 1 Q 1,m )xˆk + 2 xˆkT a˘ 2 ek + ekT (a˘ 3 − α P2i,m + 1 Q 2,m )ek

(3.10)

54

3 Finite-Time Boundedness

Similarly,    T  T A¯ di,m Pi,m A¯ di,m − α Q m ηk−dk E ηk−d k  T   T  T T Adi,m G¯ T P1i,m G¯ Adi,m − α Q 1,m xˆk−dk + ek−d Adi,m P2i,m Adi,m − α Q 2,m ek−dk = xˆk−d k k     T P ¯ i,m − β R ωk E ωkT D¯ i,m i,m D   T P = ωkT Di,m 2i,m Di,m − β R ωk   T T ¯ E 2ηk Ai,m Pi,m A¯ di,m ηk−dk T (L T ¯TP ¯ Adi,m xˆk−d + 2 xˆ T A T G¯ T P1i,m G¯ Adi,m xˆk−d = 2θ¯ ekT Ci,m i,m + ΔL i,m ) G 1i,m G k i,m k k   T T T TP ¯ +2ek Ai,m P2i,m Adi,m − θC (L + ΔL ) A e i,m 2i,m di,m k−dk i,m i,m   T P ¯ D E 2ηkT A¯ i,m ω i,m i,m k

 = 2ηkT

(3.11) (3.12)

(3.13)

T    ¯ i,m + ΔL i,m )Ci,m G¯ Ai,m P1i,m 0 0 θ¯ G(L ω ¯ 0 Ai,m − θ(L i,m + ΔL i,m )Ci,m 0 P2i,m Di,m k

T P T T T = 2ekT Ai,m 2i,m Di,m ωk − 2θ¯ ek Ci,m (L i,m + ΔL i,m ) P2i,m Di,m ωk   T T A¯ di,m E 2ηk−d Pi,m D¯ i,m ωk

(3.14)

T AT P D ω = 2ek−d k di,m 2i,m i,m k

(3.15)

k

Substituting (3.10)–(3.15) into inequality (3.16), we have E{V (ηk+1 , rk+1 , m)|ηk , i, m} − αV (ηk , i, m) − βωkT Rωk  T G ¯ T P1i,m G¯ Adi,m xˆk−d ≤ E xˆkT (a˘ 1 − α P1i,m + 1 Q 1,m )xˆk + 2 xˆkT a˘ 2 ek + 2 xˆkT Ai,m k ¯ T C T (L i,m + ΔL i,m )T G¯ T P1i,m G¯ Adi,m xˆk−d +ekT (a˘ 3 − α P2i,m + 1 Q 2,m )ek + 2θe k i,m k   T T T T T T ¯ +2ek Ai,m P2i,m Adi,m − θCi,m (L i,m + ΔL i,m ) P2i,m Adi,m ek−dk + 2ek Ai,m P2i,m Di,m ωk  T  ¯T ¯ ¯ T C T (L i,m + ΔL i,m )T P2i,m Di,m ωk + xˆ T −2θe k i,m k−dk Adi,m G P1i,m G Adi,m − α Q 1,m xˆk−dk  T  T T Adi,m P2i,m Adi,m − α Q 2,m ek−dk + 2ek−d +ek−d AT P D ω k k di,m 2i,m i,m k    T P +ωkT Di,m (3.16) 2i,m Di,m − β R ωk

   X 1i,m 0 Y1i,m 0 , Yi,m = with = 0 X 2i,m 0 Y2i,m 

Besides, it holds for matrices X i,m proper dimensions that

  E (ηkT X i,m + ΔηkT Yi,m )[( A¯ i,m − I )ηk − Δηk + A¯ di,m ηk−dk + D¯ i,m ωk ] = 0,

(3.17)

T  where Δηk = ηk+1 − ηk = ΔxˆkT ΔekT . By combining (3.16) and (3.17), we can conclude that   E{V (ηk+1 , rk+1 , m)|ηk , i, m} − αV (ηk , i, m) − βωkT Rωk ≤ E ξkT Ξ ξk   T T ek−d ωkT ΔxˆkT ΔekT , where ξkT = xˆkT ekT xˆk−d k k

3.3 Design of Sliding Surface ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ Ξ = ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Ξ11 Ξ12 Ξ13 0 ∗ Ξ22 Ξ23 Ξ24 ∗ ∗ Ξ33 0 ∗

∗

∗

∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

55

Ξ16 0 Ξ26 Ξ27 1 T ¯T T 0 2 Adi,m G Y1i,m 1 T T T P2i,m Di,m 0 Ξ44 Adi,m 2 Adi,m Y2i,m 1 DT Y T ∗ Ξ55 0 2 i,m 2i,m ∗ ∗ Y1i,m 0 ∗ ∗ ∗ Y2i,m 0 Ξ25 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

1 1 1 T sym[X 1i,m (G¯ Ai,m − I )], Ξ16 = X 1i,m + (G¯ Ai,m − I )T Y1i,m 2 2 2 θ¯ ¯ i,m + ΔL i,m )Ci,m , Ξ13 = A T G¯ T P1i,m G¯ Adi,m + 1 X 1i,m G¯ Adi,m a˘ 2 + X 1i,m G(L i,m 2 2 1 θ¯ a˘ 3 − α P2i,m + 1 Q 2,m + sym[X 2i,m (Ai,m − I )] − sym[X 2i,m (L i,m + ΔL i,m )Ci,m ] 2 2 ¯ ¯ T (L i,m + ΔL i,m )T G¯ T P1i,m G¯ Adi,m , Ξ26 = θ C T (L i,m + ΔL i,m )T G¯ T Y T θC i,m 1i,m 2 i,m 1 T T T P ¯ Ai,m P2i,m Adi,m − θCi,m (L i,m + ΔL i,m ) P2i,m Adi,m + X 2i,m Adi,m , Ξ55 = Di,m 2i,m Di,m − β R 2 1 ¯ T (L i,m + ΔL i,m )T P2i,m Di,m + X 2i,m Di,m , Ξ44 = A T P2i,m Adi,m − α Q 2,m Ai,m P2i,m Di,m − θC i,m di,m 2   1 T T T T T T ¯ ¯ (Ai,m − I ) Y2i,m − θCi,m (L i,m + ΔL i,m ) Y2i,m + X 2i,m , Ξ33 = Adi,m G T P1i,m G¯ Adi,m − α Q 1,m 2

Ξ11 = a˘ 1 − α P1i,m + 1 Q 1,m + Ξ12 = Ξ22 = Ξ23 = Ξ24 = Ξ25 = Ξ27 =

Therefore, Ξ < 0 in Theorem 3.1 can ensure E{V (ηk+1 , rk+1 , m)|ηk , i, m} ≤ αV (ηk , i, m) + βωkT Rωk .

(3.18)

In the sequel, we will analyse finite-time boundedness of system (3.7) based on conditions in Theorem 3.1. For k ∈ [kl−1 , kl ), we have σk = σk−1 = · · · = σkl−1 , E{V (ηk , rk , σk )|ηk−1 , rk−1 , σk } T ≤ αV (ηk−1 , rk−1 , σk ) + βωk−1 Rωk−1 T T ≤ α 2 V (ηk−2 , rk−2 , σk ) + αβωk−2 Rωk−2 + βωk−1 Rωk−1 T T T ≤ α 3 V (ηk−3 , rk−3 , σk ) + α 2 βωk−3 Rωk−3 + αβωk−2 Rωk−2 + βωk−1 Rωk−1 ... k−k

T ≤ α k−kl−1 V (ηkl−1 , rkl−1 , σk ) + βΣ j=1 l−1 α j−1 ωk− j Rωk− j

namely E{V (ηk , rk , σkl−1 )|ηk−1 , rk−1 , σkl−1 } k−k

T ≤ α k−kl−1 V (ηkl−1 , rkl−1 , σkl−1 ) + βΣ j=1 l−1 α j−1 ωk− j Rωk− j

By similar calculations, we have E{V (ηk1 , rk1 , σk0 )|ηk0 , rk0 , σk0 } 1 −k0 ≤ α k1 −k0 V (ηk0 , rk0 , σk0 ) + βΣ kj=1 α j−1 ωkT1 − j Rωk1 − j ,

(3.19)

56

3 Finite-Time Boundedness

and E{V (ηk , rk , σk )|ηk0 , rk0 , σk0 } ≤ ν Nσ (k0 ,k) α k−k0 V (ηk0 , rk0 , σk0 ) + α k−k0 −1 βγ 2 λ1 Nσ (k0 ,k) t ×Σt=0 ν.

(3.20)

Besides, it can be seen that V (ηk0 , rk0 , σk0 ) −dτ +1 k0 −1 Σs=k η T Q σk 0 η s = ηkT0 Prk0 ,σk0 ηk0 + Σ−d M +1 0 −1+t s −dτ +1 k0 −1 ≤ λmax ( Q˘ m )Σ−d Σs=k η T Sηs + λmax ( P˘i,m )ηkT0 Sηk0 M +1 0 −1+t s

≤ (1 + 3 )λ3 c12

(3.21)

with λmax ( P˘i,m ) = max{λ( P˘i,m )|i ∈ N1 , m ∈ N2 }, λmax ( Q˘ m ) = max{λ( Q˘ m )|m ∈ 1 1 1 1 N2 }, P˘i,m = S − 2 Pi,m S − 2 , Q˘ m = S − 2 Q m S − 2 , 3 = 1 2 , 1 = d M − dτ + 1, 2 = d M +dτ . Thus, 2 E{V (ηk , rk , σk )|ηk0 , rk0 , σk0 } Nσ (k0 ,k) t ν + α k−k0 (1 + 3 )ν Nσ (k0 ,k) ≤ α k−k0 −1 βγ 2 λ1 Σt=0

×λ3 c12 T  T ν τa +1 − 1  T 2 −1 2 τ a . ≤ α (1 + 3 )ν λ3 c1 + α βγ λ1 ν−1

(3.22)

On the other hand,     T λ2 E ηk Sηk ≤ E V (ηk , rk , σk ) .

(3.23)

From (3.22) and (3.23), one has   E ηkT Sηk T ν τa +1 − 1  αT  . (1 + 3 )ν τa λ3 c12 + α −1 βγ 2 λ1 λ2 ν−1 T

≤

(3.24)

  According to Theorem 3.1, we can derive lnρ2 − lnρ1 > 0 and E ηkT Sηk ≤ c22 . Recalling Definition 3.1, the system (3.7) is finite-time bounded with regard to (c1 , c2 , S, T, γ ). Remark 3.1 In Theorem 3.1, we have given the conditions to ensure that system (3.7) is bounded in finite time-interval. However, it is not difficult to find that these conditions contain imprecise elements. This leads to certain inconvenience.

3.3 Design of Sliding Surface

57

Subsequently, we will give the form to solve feasible solutions of finite-time boundedness issue easily. Theorem 3.2 Given α > 1, β > 0, ν > 1, if there exist positive-definite matrices Pi,m , Q m , R, W1 and W2 , real matrices X i,m , Yi,m and L i,m satisfying following LMIs ⎡ ⎤ Ξ¯ 11 ε H˜ E˜ T Ξ¯ := ⎣ ∗ −ε I 0 ⎦ < 0, (3.25) ∗ ∗ −ε I (3.26) Pi,m ≤ ν Pi,n , Q m ≤ ν Q n , R ≤ λ1 I, λ2 S ≤ Pi,m ≤ λ3 S, λ2 S ≤ Q m ≤ λ3 S, (3.27) ρ1 < ρ2 ,

(3.28)

then system (3.7) is finite-time bounded with regard to (c1 , c2 , S, T, γ ). Moreover, the average dwell time τa satisfies τa >

T lnν := τa∗ , lnρ2 − lnρ1

T ∈ Z+. τa

Here, ⎡

H˜ T

E˜ T

M¯T

Ξ¯ 1

Υ

M¯1 M¯2

⎤ T (S + S ) 0 . . . 0 Hi,m 0 0 0 0 0 11 22 T ⎢0 ... 0 0 Hi,m (λ6 S11 + λ7 S22 ) 0 0 0 0⎥ ⎥ =⎢ T G ⎣0 ... 0 ¯ T S11 0 0 Hi,m 0 0 0⎦ T 0 ... 0 0 0 0 Hi,m BG S11 0 0 ⎡ ⎤ T ET λ5 Ci,m 0 0 0 i,m T E T λ C T E T 8λ θC T T ⎥ ⎢ 0 Ci,m 3 ¯ i,m E i,m ⎥ i,m 4 i,m i,m ⎢   ⎢ ⎥ ¯ ¯ 0 0 0 0 ⎥ , Ξ¯ 11 = Ξ1 M =⎢ ⎢ ⎥ ∗ −Υ . . . . ⎢ ⎥ . . . . ⎣ ⎦ . . . . 0 0 0 0   = M¯1T M¯2T 0 0 0 M 6T M 7T ⎡ ⎤ Λ¯ 1 0 Ξ13 0 0 Ξ16 0 ⎢ ∗ Λ¯ 3 ⎥ 0 Λ5 Λ7 0 Λ9 ⎢ ⎥ 1 T ⎢ ∗ ∗ Ξ +Ω ⎥ TYT ¯ 0 0 A 0 G 33 3 ⎢ ⎥ 1i,m 2 di,m ⎢ ⎥ 1 T T T 0 A Y ∗ Ξ44 + Ω4 Adi,m P2i,m Di,m =⎢ ∗ ∗ 2 di,m 2i,m ⎥ ⎢ ⎥ 1 T T ⎢ ∗ ∗ ⎥ ∗ ∗ Ξ55 + Ω5 0 D Y 2 i,m 2i,m ⎥ ⎢ ⎣ ∗ ∗ ⎦ ∗ ∗ ∗ Y1i,m 0 ∗ ∗ ∗ ∗ ∗ ∗ Y2i,m  = diag λ5 (S11 + S22 ), λ5 (S11 + S22 ), 2θ¯ S11 , λ6 S11 + λ7 S22 , λ6 S11 + λ7 S22 , 2W1 , W2 , θ¯ S22 ,  λ4 S11 , λ4 S11 , 8λ3 S11 , 8λ3 S11 , 2θ¯ S11 , 2θ¯ S22     T L T (S + S ) 0 θ¯ X ¯ 0 = λ5 Ci,m 22 1i,m 0 . . . 0 , M 6 = 0 . . . 0 θ¯ Y1i,m G i,m 11  T L T (λ S + λ S ) 0 X T T ¯TS = 0 0 0 Ci,m 7 22 2i,m X 2i,m θ¯ X 2i,m λ4 Ci,m L i,m G 11 0 i,m 6 11    ¯ T L T BG S11 0 0 0 , M 7 = 0 . . . 0 θ¯ Y2i,m 8λ3 θC i,m i,m

1 T ¯T ¯ i,m Λ¯ 1 = (1 + θ)A G P1i,m G¯ Ai,m − α P1i,m + 1 Q 1,m + sym[X 1i,m (G¯ Ai,m − I )] 2

58

3 Finite-Time Boundedness 1 sym[X 2i,m (Ai,m − I )], λ4 = (4λ3 + 1)θ¯ 2 1 1 T T T Λ5 = Ai,m P2i,m Adi,m + X 2i,m Adi,m , Λ7 = Ai,m P2i,m Di,m + X 2i,m Di,m , Ω5 = θ¯ Di,m P2i,m Di,m 2 2 1 1 T T T Ω3 = θ¯ Adi,m P2i,m Adi,m , Λ9 = X 2i,m + (Ai,m − I )T Y2i,m G¯ T P1i,m G¯ Adi,m , Ω4 = θ¯ Adi,m 2 2 λ5 = 6λ3 (θ¯ − θ¯ 2 ), λ6 = 2λ3 θ¯ 2 + 6λ3 θ¯ + θ¯ , λ7 = 2λ3 (9θ¯ − θ¯ 2 + 1) + 3θ¯ T ¯ i,m P2i,m Ai,m − α P2i,m + 1 Q 2,m + Λ¯ 3 = (1 + 2θ)A

with parameters Ξ13 , Ξ16 , Ξ33 , Ξ44 , Ξ55 , ρ1 , ρ2 and  j ( j = 1, 2, 3) being defined in Theorem 3.1. Proof From proof of Theorem 3.1, we obtain E{V (ηk+1 , rk+1 , m)|ηk , i, m} − αV (ηk , i, m) − βωkT Rωk   ≤ E ξkT Ξ1 ξk + ξkT Ξ2 ξk ,

(3.29)

where ⎡

Λ1 0 Ξ13 0 0 Ξ16 ⎢ ∗ Λ3 0 Λ 5 Λ7 0 ⎢ 1 AT ⎢ ∗ ∗ Ξ33 0 ¯T T 0 ⎢ 2 di,m G Y1i,m ⎢ T 0 Ξ1 = ⎢ ∗ ∗ ∗ Ξ44 Adi,m P2i,m Di,m ⎢ ⎢ ∗ ∗ ∗ ∗ Ξ 0 55 ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ Y1i,m ∗ ∗ ∗ ∗ ∗ ∗ ⎤ ⎡ Λ2 Ξ12 0 0 0 0 0 ⎢ ∗ Λ4 Ξ23 Λ6 Λ8 Ξ26 Λ10 ⎥ ⎥ ⎢ ⎢∗ ∗ 0 0 0 0 0 ⎥ ⎥ ⎢ ⎢ Ξ2 = ⎢ ∗ ∗ ∗ 0 0 0 0 ⎥ ⎥ ⎢∗ ∗ ∗ ∗ 0 0 0 ⎥ ⎥ ⎢ ⎣∗ ∗ ∗ ∗ ∗ 0 0 ⎦ ∗

∗

∗

∗

∗

∗

0 Λ9 0 1 AT Y T 2 di,m 2i,m 1 T T 2 Di,m Y2i,m 0 Y2i,m

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0

T G ¯ T P1i,m G¯ Ai,m − α P1i,m + 1 Q 1,m + 1 sym[X 1i,m (G¯ Ai,m − I )] Λ1 = Ai,m 2 T (L TP T (L T Λ2 = (θ¯ − θ¯ 2 )Ci,m + ΔL ) (L + ΔL i,m )Ci,m + (θ¯ − θ¯ 2 )Ci,m i,m i,m 1i,m i,m i,m + ΔL i,m )

×P2i,m (L i,m + ΔL i,m )Ci,m 1 T P Λ3 = Ai,m 2i,m Ai,m − α P2i,m + 1 Q 2,m + sym[X 2i,m (Ai,m − I )] 2 1 T P T Λ4 = a˘ 3 − Ai,m 2i,m Ai,m − θ¯ X 2i,m (L i,m + ΔL i,m )Ci,m , Λ5 = Ai,m P2i,m Adi,m + X 2i,m Adi,m 2 1 ¯ T (L i,m + ΔL i,m )T P2i,m Adi,m , Λ7 = A T P2i,m Di,m + X 2i,m Di,m Λ6 = −θC i,m i,m 2 1 1 ¯ T (L i,m + ΔL i,m )T P2i,m Di,m , Λ9 = X 2i,m + (Ai,m − I )T Y T Λ8 = −θC i,m 2i,m 2 2 θ¯ T T Λ10 = − Ci,m (L i,m + ΔL i,m )T Y2i,m 2

3.3 Design of Sliding Surface

59

Note that     E ξkT Ξ2 ξk ≤ E ξkT Ωξk

(3.30)

with Ω = diag{Ω1 , Ω2 , Ω3 , Ω4 , Ω5 , Ω6 , Ω7 } θ¯ −1 T T ¯T T T T T L i,m S11 L i,m Ci,m + X 1i,m S11 X 1i,m + λ5 Ci,m L i,m S22 L i,m Ci,m G P1i,m G¯ Ai,m + λ5 Ci,m Ω1 = θ¯ Ai,m 2 T T T T +λ5 Ci,m ΔL i,m S11 ΔL i,m Ci,m + λ5 Ci,m ΔL i,m S22 ΔL i,m Ci,m 1 T T T + λ7 Ci,m L i,m S22 L i,m Ci,m X 2i,m W1−1 X 2i,m 2 T T T T ¯T + λ7 Ci,m ΔL i,m S22 ΔL i,m Ci,m + λ4 Ci,m L i,m G S11 G¯ L i,m Ci,m

T T T Ω2 = 2θ¯ Ai,m P2i,m Ai,m + λ6 Ci,m L i,m S11 L i,m Ci,m + T T +λ6 Ci,m ΔL i,m S11 ΔL i,m Ci,m

−1 T T T ¯T 2 T T ¯ +λ4 Ci,m ΔL i,m G S11 GΔL i,m Ci,m + 8λ3 θ¯ Ci,m L i,m BG S11 BG L i,m Ci,m + X 2i,m W2 X 2i,m −1 T T T +θ¯ X 2i,m S22 X 2i,m + 8λ3 θ¯ 2 Ci,m ΔL i,m BG S11 BGΔL i,m Ci,m T T T Ω3 = θ¯ Adi,m P2i,m Adi,m , Ω5 = θ¯ Di,m P2i,m Di,m G¯ T P1i,m G¯ Adi,m , Ω4 = θ¯ Adi,m

Ω6 =

θ¯ θ¯ −1 ¯ T T −1 T Y2i,m G Y1i,m , Ω7 = Y2i,m S22 Y1i,m G¯ S11 2 2

Then, the inequality (3.29) can be strengthened by E{V (ηk+1 , rk+1 , m)|ηk , i, m} − αV (ηk , i, m) − βωkT Rωk   ≤ E ξkT (Ξ1 + Ω)ξk .

(3.31)

In addition, we rewrite Ξ1 + Ω < 0 as follows: Ξ¯ 1 + M Υ −1 M T < 0,

(3.32)

where ⎡ ¯ Λ1 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ¯ Ξ1 = ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

0 Λ¯ 3 ∗ ∗ ∗ ∗ ∗

Ξ13 0 0 Ξ16 0 Λ5 Λ7 0 1 AT ¯T T Ξ33 + Ω3 0 0 2 di,m G Y1i,m T ∗ Ξ44 + Ω4 Adi,m P2i,m Di,m 0 ∗ ∗ Ξ55 + Ω5 0 ∗ ∗ ∗ Y1i,m ∗ ∗ ∗ ∗

0 Λ9 0 1 AT Y T 2 di,m 2i,m 1 T T 2 Di,m Y2i,m 0 Y2i,m

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

T G ¯ T P1i,m G¯ Ai,m , Λ¯ 3 = Λ3 + 2θ¯ A T P2i,m Ai,m Λ¯ 1 = Λ1 + θ¯ Ai,m i,m   T T T T T M = M1 M2 0 0 0 M6 M7   T L T (S + S ) λ C T ΔL T (S + S ) θ¯ X M1 = λ5 Ci,m 22 22 1i,m 0 . . . 0 5 i,m i,m 11 i,m 11  T L T (λ S + λ S ) C T ΔL T (λ S + λ S ) X M2 = 0 0 0 Ci,m 7 22 7 22 2i,m X 2i,m θ¯ X 2i,m i,m 6 11 i,m i,m 6 11 T LT G ¯ T S11 λ4 C T ΔL T G¯ T S11 8λ3 θC ¯ T L T BG S11 8λ3 θC ¯ T ΔL T BG S11 λ4 Ci,m i,m i,m i,m i,m i,m i,m i,m

60

3 Finite-Time Boundedness      0 0 , M6 = 0 . . . 0 θ¯ Y1i,m G¯ 0 , M7 = 0 . . . 0 θ¯ Y2i,m  Υ = diag λ5 (S11 + S22 ), λ5 (S11 + S22 ), 2θ¯ S11 , λ6 S11 + λ7 S22 , λ6 S11 + λ7 S22 , 2W1 , W2 , θ¯ S22 ,  λ4 S11 , λ4 S11 , 8λ3 S11 , 8λ3 S11 , 2θ¯ S11 , 2θ¯ S22

Further, by elementary matrix inequality techniques, it can be derived that ⎡

⎤ Ξ¯ 11 ε H˜ E˜ T Ξ¯ := ⎣ ∗ −ε I 0 ⎦ < 0, ∗ ∗ −ε I

(3.33)

where ⎤ T (S + S ) 0 0 0 00 0 . . . 0 Hi,m 11 22 ⎢0 ... 0 T 0 Hi,m (λ6 S11 + λ7 S22 ) 0 0 0 0⎥ ⎥ ⎢ ⎥ ⎢ T T 0 0 0⎦ 0 0 Hi,m G¯ S11 ⎣0 ... 0 T 0 ... 0 0 0 0 Hi,m BG S11 0 0   ¯ ¯   Ξ1 M , M¯T = M¯1T M¯2T 0 0 0 M6T M7T ∗ −Υ   T L T (S + S ) 0 θ¯ X λ5 Ci,m 22 1i,m 0 . . . 0 i,m 11  T L T (λ S + λ S ) 0 X T T ¯TS 0 0 0 0 Ci,m 7 22 11 2i,m X 2i,m θ¯ X 2i,m λ4 Ci,m L i,m G i,m 6 11  ¯ T L T BG S11 0 0 0 8λ3 θC i,m i,m ⎤ ⎡ T T λ5 Ci,m E i,m 0 0 0 ⎢ T E T λ C T E T 8λ θC T T ⎥ 0 Ci,m 3 ¯ i,m E i,m ⎥ ⎢ i,m 4 i,m i,m ⎥ ⎢ 0 0 0 0 ⎥ ⎢ ⎥ ⎢ . . . . ⎥ ⎢ . . . . ⎦ ⎣ . . . . ⎡ H˜ T =

Ξ¯ 11 = M¯1 = M¯2 =

E˜ T =

0

0

0

0

Remark 3.2 Some strict LMIs have been obtained in Theorem 3.2. Then, we can check this scheme directly by solving these LMIs. Now, the finite-time boundedness problem of closed-loop system has been addressed.

3.4 Controller Design In this subsection, the reachability of sliding surface (3.5) is discussed and then finite time observer-based SMC law is constructed. As in [32, 33], the improved reaching condition, which needs to be tested in order to ensure the desired requirements, is introduced as follows:  Δs = s −μk sgn(sk ) − ς sk , sk > 0 k k+1 − sk ≤ −εe Δsk = sk+1 − sk ≥ −εe−μk sgn(sk ) − ς sk , sk < 0

(3.34)

3.5 Simulation

61

with ε > 0, μ ≥ 0 and 0 < ς < 1. Then, the design of the desired robust observerbased controller is provided in the subsequent theorem for reachability analysis. Theorem 3.3 For the sliding surface (3.5) with G = (B T B)−1 B T , if the SMC problem in Theorem 3.2 is solvable, then the observer-based sliding mode controller given by   u k = −G (Ai,m − I )xˆk + Adi,m xˆk−dk + L i,m (yk − yˆk ) −ψsgn(sk ) − εe−μk sgn(sk ) − ς sk

(3.35)

can ensure above reaching condition, where ψ = G Hi,m  E i,m (yk − yˆk ) . Proof Recalling sliding surface (3.5) and observer (3.3), we derive Δsk = G(Ai,m − I )xˆk + G Adi,m xˆk−dk +G(L i,m + ΔL i,m )(yk − yˆk ) + u k .

(3.36)

Let’s discuss the sign of Δsk based on sk . Specifically, when sk > 0, one has Δsk = GΔL i,m (yk − yˆk ) − ψsgn(sk ) − ς sk − εe−μk sgn(sk ) ≤ −εe−μk sgn(sk ) − ς sk .

(3.37)

Similarly, when sk < 0, it can be derived that Δsk ≥ −εe−μk sgn(sk ) − ς sk .

(3.38)

According to the reaching condition (3.34), the reachability of sliding surface (3.5) in discrete-time setting can be guaranteed by newly proposed the observer-based sliding mode controller (3.35).

3.5 Simulation In this section, a numerical simulation is provided to verify the usefulness of proposed finite-time observer-based SMC approach. The parameters of MJSs with switching chain are given by ⎤ ⎡ 0.15 −0.25 0 0.18 A1,3 = ⎣ 0 0.13 0.01 ⎦ , A1,4 = ⎣−0.21 0.03 0 −0.05 0 ⎤ ⎡ ⎡ −0.08 −0.34 0.16 0.15 A2,3 = ⎣ 0.2 0.13 −0.37⎦ , A2,4 = ⎣0.26 0 0.4 −0.5 0.28 ⎡

⎤ ⎤ ⎡ −0.12 0 −0.10 0.16 0 ⎦ , D1,3 = ⎣ 0.011 ⎦ 0 0.05 −0.019 ⎤ ⎤ ⎡ −0.064 0 −0.04 ⎦ ⎣ −0.11 0.01 , D2,3 = 0.02 ⎦ 0.07 0 −0.16

62

3 Finite-Time Boundedness

⎤ ⎤ ⎡ ⎤ ⎤ ⎡ ⎡ 0.08 −0.017 0.06 0.03 −0.08 −0.03 0.3 ⎦ , D1,4 = ⎣ 0.01 ⎦ , D2,4 = ⎣ 0.03 ⎦ 0.01 −0.03⎦ , B = ⎣ 0.2 0.13 0.26 0.02 0.05 −0.023 −0.012 ⎤ ⎤ ⎤ ⎡ ⎡ 0.06 0 0.17 0.06 0.04 −0.02 0.1 0.04 0.01 −0.03⎦ , Ad2,4 = ⎣0.04 0.01 −0.03⎦ , Ad2,3 = ⎣ 0.04 0.12 −0.03⎦ 0.02 0.05 0.01 0.24 0.05 0.01 0.02 0.05   C1,3 = C2,3 = C1,4 = C2,4 = diag{0.04, 0.04, 0.06}, E 1,3 = E 2,3 = E 1,4 = E 2,4 = 0.2 0.1 0  T H1,3 = H2,3 = H1,4 = H2,4 = 0.14 0.2 0.17 ⎡

−0.22 Ad1,3 = ⎣ 0.04 0.01 ⎡ −0.13 Ad1,4 = ⎣ 0.19 0.01

We set time-delay dk as dτ = 2 and d M = 5. Besides, the transition probability matrix of rk in (3.2) are chosen by Π3 =

    0.4 0.6 0.3 0.7 , Π4 = . 0.8 0.2 0.4 0.6

Then, for finite time T = 100 and bounds c1 = 0.3, c2 = 5, solving finite-time observer-based SMC issue in Theorem 3.2 yields ⎡

P11,3

P22,4

2.6445 = ⎣−0.0061 −0.0432 ⎡ 5.0831 = ⎣ 0.8034 −0.4139

⎤ −0.0061 −0.0432 5.1740 0.1723 ⎦ , 0.1723 7.4767 ⎤ 0.8034 −0.4139 3.2765 −0.8923⎦ . −0.8923 2.0779

In the simulation, by choosing ε = 1 and ς = 0.1, the designed finite-time observer-based controller in (3.35) can be applied. The effects of proposed controller and related results are shown in Figs. 3.1, 3.2, 3.3, 3.4, 3.5 and 3.6. Specifically, one Fig. 3.1 Switching chain σk

6

σk

5

4

3

2

1

0 0

20

40

60

No. of samples, k

80

100

3.5 Simulation Fig. 3.2 Random mode rk

63 3

rk = 1 rk = 2

2.5

2

1.5

1

0.5

0 100

80

60

40

20

0

No. of samples, k

Fig. 3.3 The trajectories of state xk1 and its estimation xˆk1 (μ = 1)

2

x1k x ˆ1k

1.5 1 0.5 0 −0.5 −1 −1.5 −2

10

20

30

40

50

60

70

80

90

100

No. of samples, k

possible realizations of piecewise-constant switching chain σk is given in Fig. 3.1 as well as Markov chain rk in Fig. 3.2. It is shown from Figs. 3.3, 3.4 and 3.5 that despite influences of switching chain, Markov jumping parameter and factors induced by networks, the system trajectory xk can still converge to a small neighborhood based on our proposed scheme. Moreover, it is clear to see that the estimation xˆk

64 Fig. 3.4 The trajectories of state xk2 and its estimation xˆk2 (μ = 1)

3 Finite-Time Boundedness 3

x2k x ˆ2k

2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2

10

30

20

40

50

60

80

70

100

90

No.of samples, k

Fig. 3.5 The trajectories of state xk3 and its estimation xˆk3 (μ = 1)

2

x3k x ˆ3k

1.5 1 0.5 0 −0.5 −1

0.6

−1.5

0.4

−2

0.2

−2.5 −3

0

10

20

20

15

10

30

40

50

60

70

80

90

100

No.of samples, k

becomes more and more precise with finite-time nonfragile observer receiving new information by Fig. 3.5. In addition, the finite-time observer-based sliding mode controllers are illustrated in Figs. 3.6 and 3.7. It can conclude from simulation figures that our proposed finite time observer-based SMC approach has a desiring control performance.

3.6 Conclusion Fig. 3.6 The control signal u k (μ = 1)

65 3

u1,k u2,k

2

1

0

−1

−2

−3

10

20

30

40

50

60

70

80

90

100

No. of samples, k

Fig. 3.7 The control signal u k (μ = 0)

3

u1,k u2,k

2

1

0

−1

−2

−3

10

20

30

40

50

60

70

80

90

100

No. of samples, k

3.6 Conclusion The finite time observer-based SMC problem has been addressed for delayed stochastic Markovian jump systems (MJSs) with deterministic switching chain (DSC) under packet losses (PLs). Using Bernoulli random variables, we have been characterized so-called PLs phenomenon first and then designed a non-fragile finite-time bounded sliding mode observer. The achievements of our work are that some sufficient finitetime bounded criteria have been given for related control system. Moreover, the

66

3 Finite-Time Boundedness

desired observer-based sliding mode controller has been designed to ensure reachability. The key feature lies in that a common sliding surface has been proposed for addressed multiple systems with Markovian jumping parameter and deterministic switching chain. To some extent, our proposed common sliding surface approach would help to avoid disadvantages of mode-dependent one. Finally, a numerical example is provided to illustrate our theoretical results.

References 1. L. X. Zhang, S. L. Zhuang, R. D. Braatz, Switched model predictive control of switched linear systems: feasibility, stability and robustness, Automatica, 67 (2016) 8–21. 2. L. X. Zhang, Y. Z. Zhu, W. X. Zheng, State estimation of discrete-time switched neural networks with multiple communication channels, IEEE Transactions on Cybernetics, 47(4) (2017) 1028– 1040. 3. Z. P. Ning, L. X. Zhang, J. Lam, Stability and stabilization of a class of stochastic switching systems with lower bound of sojourn time, Automatica, 92 (2018) 18–28. 4. Y. G. Kao, J. Xie, C. H. Wang, H. R. Karimi, A sliding mode approach to H∞ non-fragile observer-based control design for uncertain Markovian neutral-type stochastic systems, Automatica, 52 (2015) 218–226. 5. Y. G. Kao, J. Xie, L. X. Zhang, H. R. Karimi, A sliding mode approach to robust stabilisation of Markovian jump linear time-delay systems with generally incomplete transition rates, Nonlinear Analysis: Hybrid Systems, 17 (2015) 70–80. 6. J. Xie, Y. G. Kao, J. H. Park, H∞ performance for neutral-type Markovian switching systems with general uncertain transition rates via sliding mode control method, Nonlinear Analysis: Hybrid Systems, 27 (2018) 416–436. 7. Y. Shen, Z.-G. Wu, P. Shi, C. K. Ahn, Model reduction of Markovian jump systems with uncertain probabilities, IEEE Transactions on Automatic Control, 65(1) (2020) 382–388. 8. Z.-G. Wu, P. Shi, H. Y. Su, J. Chu, Stochastic synchronization of Markovian jump neural networks with time-varying delay using sampled data, IEEE Transactions on Cybernetics, 43(6) (2013) 1796–1806. 9. Z.-G. Wu, S. L. Dong, H. Y. Su, C. D. Li, Asynchronous dissipative control for fuzzy Markov jump systems, IEEE Transactions on Cybernetics, 48(8) (2018) 2426–2436. 10. R. F. Tao, Y. C. Ma, C. J. Wang, Stochastic admissibility of singular Markov jump systems with time-delay via sliding mode approach, Applied Mathematics and Computation, 380 (2020) 125282, https://doi.org/10.1016/j.amc.2020.125282. 11. X. W. Gao, H. F. He, W. H. Qi, Admissibility analysis for discrete-time singular Markov jump systems with asynchronous switching, Applied Mathematics and Computation, 313 (2017) 431–441. 12. J. M. Wang, S. P. Ma, C. H. Zhang, Finite-time stabilization for nonlinear discrete-time singular Markov jump systems with piecewise-constant transition probabilities subject to average dwell time, Journal of the Franklin Institute, 354(5) (2017) 2102–2124. 13. C. Y. Chen, S. Zhu, Y. C. Wei, C. Y. Yang, Finite-time stability of delayed memristor-based fractional-order neural networks, IEEE Transactions on Cybernetics, 50(4) (2020) 1607–1616. 14. C. Xu et al., Finite-time synchronization of networks via quantized intermittent pinning control, IEEE Transactions on Cybernetics, 48(10) (2018) 3021–3027. 15. F. B. Li, C. L. Du, C. H. Yang, L. G. Wu, W. H. Gui, Finite-time asynchronous sliding mode control for Markovian jump systems, Automatica, 109 (2019) 108503, https://doi.org/10.1016/ j.automatica.2019.108503. 16. B. Cai, L. X. Zhang, Y. Shi, Control synthesis of hidden semi-Markov uncertain fuzzy systems via observations of hidden modes, IEEE Transactions on Cybernetics, 50 (8) (2020) 3709–3718, https://doi.org/10.1109/TCYB.2019.2921811.

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17. B. P. Jiang, H. R. Karimi, Y. G. Kao, C. C. Gao, Adaptive control of nonlinear semi-Markovian jump T-S fuzzy systems with immeasurable premise variables via sliding mode observer, IEEE Transactions on Cybernetics, 50(2) (2020) 810–820. 18. P. Shi, M. Liu, L. X. Zhang, Fault-tolerant sliding-mode-observer synthesis of Markovian jump systems using quantized measurements, IEEE Transactions on Industrial Electronics, 62(9) (2015) 5910–5918. 19. J. M. Wang, S. P. Ma, C. H. Zhang, M. Y. Fu, Observer-based control for singular nonhomogeneous Markov jump systems with packet losses, Journal of the Franklin Institute, 355(14) (2018) 6617–6637. 20. Y. H. Liu, Y. G. Niu, Y. Y. Zou, Non-fragile observer-based sliding mode control for a class of uncertain switched systems, Journal of the Franklin Institute, 351(2) (2014) 952–963. 21. L. H. Zhang, et al., Non-fragile observer-based H∞ finite-time sliding mode control, Applied Mathematics and Computation, 375 (2020) 125069, https://doi.org/10.1016/j.amc. 2020.125069. 22. Y. Q. Han, Y. G. Kao, C. C. Gao, Robust sliding mode control for uncertain discrete singular systems with time-varying delays and external disturbances, Automatica, 75 (2017) 210–216. 23. H. C. Yan, Y. X. Tian, H. Y. Li, H. Zhang, Z. C. Li, Input-output finite-time mean square stabilisation of nonlinear semi-Markovian jump systems with time-varying delay, Automatica, 104 (2019) 82–89. 24. Q. Li, B. Shen, Z. D. Wang, T. W. Huang, J. Luo, Synchronization control for a class of discrete time-delay complex dynamical networks: a dynamic event-triggered approach, IEEE Transactions on Cybernetics, 49(5) (2019) 1979-1986. 25. P. P. Zhang, J. Hu, H. J. Liu, C. L. Zhang, Sliding mode control for networked systems with randomly varying nonlinearities and stochastic communication delays under uncertain occurrence probabilities, Neurocomputing, 320 (2018) 1–11. 26. P. P. Zhang, J. Hu, H. X. Zhang, D. Y. Chen, H∞ sliding mode control for Markovian jump systems with randomly occurring uncertainties and repeated scalar nonlinearities via delayfractioning method, ISA Transactions., 101(2020) 10–22. 27. Y. C. Ma, X. R. Jia, Q. L. Zhang, Robust observer-based finite-time H∞ control for discretetime singular Markovian jumping system with time delay and actuator saturation, Nonlinear Analysis: Hybrid Systems, 28 (2018) 1–22. 28. H. Y. Song, L. Yu, D. Zhang, W.-A. Zhang, Finite-time H∞ control for a class of discrete-time switched time-delay systems with quantized feedback, Communications in Nonlinear Science and Numerical Simulation, 17 (12) (2012) 4802–4814. 29. Y. Q. Han, Y. G. Kao, J. H. Park, Robust H∞ nonfragile observer-based control of switched discrete singular systems with time-varying delays: A sliding mode control design, International Journal of Robust and Nonlinear Control, 29(5) (2019) 1462–1483. 30. J. Song, Y. G. Niu, Y. Y. Zou, Asynchronous sliding mode control of Markovian jump systems with time-varying delays and partly accessible mode detection probabilities, Automatica, 93 (2018) 33–41. 31. G. W. Yang, B. H. Kao, J. H. Park, Y. G. Kao, H∞ performance for delayed singular nonlinear Markovian jump systems with unknown transition rates via adaptive control method, Nonlinear Analysis: Hybrid Systems, 33 (2019) 33–51. 32. Y. Q. Han, Y. G. Kao, C. C. Gao, J. J. Zhao, C. H. Wang, H∞ sliding mode control of discrete switched systems with time-varying delays, ISA Transactions, 89 (2019) 12–19. 33. Y. Q. Han, Y. G. Kao, C. C. Gao, Robust observer-based H∞ control for uncertain discrete singular systems with time-varying delays via sliding mode approach, ISA Transactions, 80 (2018) 81–88.

Chapter 4

Passivity and Control Synthesis

This paper focuses on the robust sliding mode passive control (SMPC) problem for uncertain Markovian jump discrete systems under the finite time setting. For the considered model, the phenomenon of stochastic communication delays (SCDs) is depicted via some Bernoulli distributed stochastic variables, which involves inaccurate occurrence probability information. In view of initial condition, a common form of sliding surface is chosen. Further, the sliding motion equation can be given for the original system. Our aim of this paper is to propose some potential sufficient criteria for finite-time boundedness and passive requirement of the sliding motion. With the same objective, a comprehensive reaching condition is also utilized to construct the robust sliding mode passive controller such that the controlled system can be kept remain in a region of the sliding surface thereafter. Finally, an example is given to show the usefulness of our proposed scheme here.

4.1 Introduction Over the past years, the sliding mode control (SMC) method has been discussed popularly for the nonlinear dynamic systems. In particular, the significant feature is that the dynamics on the sliding surface is strongly robust against uncertainty or disturbance [1–3]. Accordingly, in light of different requirements, a considerable amount of literature has been published on improvement of the performance for uncertain systems with exogenous disturbance. For instance, the sliding mode tracking control issue has been addressed in [4] for the quadrotor system in the presence of uncertainties and disturbances from a fractional-order perspective. The robust integral SMC approach has been provided in [5] for the controlled system with time-varying parameter uncertainties, which simultaneously exist in the system matrices and the input matrices. It should be noted that the upper bounds of perturbations have been estimated in [6] for uncertain chaotic system via a fractional-order adaptive terminal © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Kao et al., Analysis and Design of Markov Jump Discrete Systems, Studies in Systems, Decision and Control 499, https://doi.org/10.1007/978-981-99-5748-4_4

69

70

4 Passivity and Control Synthesis

SMC. Additionally, subject to exogenous bounded disturbance, the discrete SMC strategy have been proposed respectively in [7, 8] for the system with piece-wise constant switching chain and buck converters. It is worth mentioning that above superior performance of sliding mode dynamics can be achieved only when the system is kept remain the sliding surface. However, it usually takes certain time for the system to approach the sliding surface, which means that the reaching phase is generally inevitable. It is a fact that most literature on SMC results is directly relevant to the reaching phase and the reaching law is generally introduced to force the investigated system onto the sliding surface. Since various practical systems can be characterized via Markovian jump models, the increasingly rapid advances of Markovian jump systems (MJSs) have been seen in the field of control science [9, 10]. It is shown that under a prescribed Markov chain, the changeable structure of system can switch back and forth between different modes. This provides MJSs with the possibility to give an approximation for actual complex systems. However, interestingly, the information of switching probability between multiple modes may not be available exactly in most cases, such as the work in [11] for the MJSs under partly available transition probabilities. The stochastic stability and stabilization have been investigated in [12] for Markovian jump dynamic systems according to imprecise transition probability, where sufficient conditions have been proposed in continuous and discrete-time settings to ensure the considered system to be stochastically stable. As mentioned above, the SMC technique is employed to deal with the uncertainty or exogenous disturbance for the controlled system. It can be noted that the complexity of MJSs not only lies in so-called switching property but the predictable difficulty in studying the uncertain MJSs subject to exogenous disturbance via SMC idea. We can know that much more SMC scheme should become available for uncertain MJSs subject to exogenous disturbance. Then, this present study is motivated by the need to take into account the in-depth analysis of MJSs with imprecise information in a discrete-time setting. Due to the information jam and the limited network scheduling, it frequently occurs time-delay problem for the networked dynamic controlled systems [13–16]. In the light of the non-timeliness of networks, the stochastic communication delays (SCDs) have been taken into account during the investigation. It can be shown that such SCDs model can not only depict the objective network environment, but can be changed into traditional time-varying delay and constant time-delay by setting the relevant factors. In recent years, a large and growing body of literature has been investigated on the finite-time problems of stability and synthesis [17–19]. For example, the stability analysis have been discussed in [20] for linear systems by means of bounded feedback control idea. Based on an average dell time strategy, the asynchronous controller has been designed in [21] for stochastic hybrid system with faulty signals under the requirement of finite-time boundedness. The finite-time adaptive control framework has been given in [22] for the controlled systems with unknown bounded nonlinearity via the aid of functional observer. It can be found that aforesaid results in finite-time mainly serve for the continuous system and are not utilized to the discrete dynamic system directly. Moreover, much more digital information has become available for analysis and design of control system. As a result, the discrete

4.2 Problem Statements and Preliminaries

71

finite-time scheme remains to be further explored. To close this theoretical gap, we will also investigate the effects from SCDs onto the Markovian jump discrete systems in finite-time. Inspired by above discussions, we will take into account the problem of robust SMPC design for addressed MJSs with stochastic time-varying communication delays. For depicting the randomness of time-varying communication delays, we will firstly characterize the SCDs by introducing Bernoulli distributed stochastic variables. Then, an initial condition-dependent sliding surface is chosen. In view of above switching surface, the sliding motion equation can be expressed according to the original system. Our aim of this paper is to propose sufficient criteria such that the aforesaid derived system is finite-time bounded with passive requirement according to Lyapunov method. The key features of our work are as follows: (i) considering the passive performance, the SCDs phenomenon under imprecise probability and robust control issue in same discrete framework; (ii) designing a sliding mode passive scheme for the MJSs against above imperfect information, where the controlled system can be kept on the prescribed sliding surface from the beginning; and (iii) utilizing a comprehensive reaching condition in order to keep the controlled system remain in a region nearby the sliding surface thereafter. Lastly, an example is provided to show the correctness of the sliding mode passive approach obtained.

4.2 Problem Statements and Preliminaries Letting the homogeneous Markovian chain rk ∈ N  {1, 2, . . . , N }, the transition probability matrix can be denoted by   [πi j ]i, j∈N , where πi j  Pr(rk+1 = j|rk =  i) ≥ 0 and Nj=1 πi j = 1 for any i, j ∈ N . Consider the following Markovian jump discrete systems subject to SCDs in this paper:  xk+1 = [A(rk ) + ΔA(rk )]xk + Ad (rk )x˜k + Bu k + E(rk )ωk , z k = C(r k )x k + E ω (r k )ωk , q x˜k = l=1 σk,l xk−τk,l , xk = φk , k ∈ [−τ M , 0],

(4.1)

where xk ∈ Rn is the system state, u k ∈ Rs stands for the control action signal, z k ∈ Rt is the controlled output, τk,l ∈ [τm , τ M ] (l = 1, 2, . . . , q) are stochastic communication delays with τ M , τm being known common upper and lower bounds, respectively. Ai  A(rk ), Adi  Ad (rk ), Ci  C(rk ), E i  E(rk ), E ωi  E ω (rk ) and B with column rank are known real matrices. ωk ∈ R1 is external disturbance T full T satisfying k=0 ωk ωk < ρ with ρ being a known scalar. The norm-bounded parameter uncertainty ΔAi  ΔA(rk ) have the following form: ΔAi = Mi Fi (k)Ni ,

(4.2)

72

4 Passivity and Control Synthesis

where matrices Mi , Ni are known and matrices Fi (k) (i ∈ N ) are time-varying with FiT (k)Fi (k) ≤ I . The Bernoulli distributed random variables σk,l (l ∈ {1, 2, . . . , q}) are employed in order to characterize the stochastic communication time-varying delays and Prob{σk,l = 1} = σ¯ l + Δσl and Prob{σk,l = 0} = 1 − (σ¯ l + Δσl ), where 0 ≤ σ¯ l + Δσl ≤ 1, |Δσl | ≤  with σ¯ l > 0 and  being scalar. Further, it leads to 0 ≤  ≤ min{σ¯ l , 1 − σ¯ l }. Lemma 4.1 [23] For xi ∈ Rn , 0 ≤ M ∈ Rn×n , bi ≥ 0 (i = 1, 2, . . . ), if the series involved are convergent, then  +∞ i=1

T bi xi

    +∞ +∞ +∞ M bi xi ≤ bi bi xiT Mxi . i=1

i=1

i=1

Lemma 4.2 [24] For constant matrix X > 0 and time-varying matrix F(k) satisfying F T (k)F(k) ≤ I , the following inequality is true if αI − E T X E > 0 can be ensured with α > 0. (A + E F(k)H )T X (A + E F(k)H ) ≤ A T X A + A T X E(αI − E T X E)−1 E T X A + αH T H,

where A, E and H are constant matrices.

4.3 Design of Sliding Surface The following sliding surface related to initial condition is firstly designed in this section:  sk = Gxk − G Hk x0 , (4.3) Hk = diag{e−θ1 k , e−θ2 k , . . . , e−θn k }, where G is sliding-mode gain matrix to be determined such that matrix G B is invertible, and Hk is a known time-varying matrix with θ j ( j = 1, 2, . . . , n) being so-called sliding-mode moving parameters. Based on the form of sliding surface, we can easily obtain the equality s0 = 0. This means that the system (4.1) is on the sliding mode surface from the beginning. Furthermore, it follows that limk→∞ sk = Gxk . In this paper, we determine G = (B T B)−1 B T in order to let G B = I . From (4.1), (4.3) and condition sk+1 = sk = 0, we get q    u eq = −(G B)−1 G (Ai + ΔAi )xk + Adi (σ¯ l + Δσl )xk−τk,l + E i ωk − Hk+1 x0 . (4.4) l=1

Combining (4.4) and (4.1), the sliding motion equation can be presented by

4.3 Design of Sliding Surface

73

ˆ i + ΔAi )xk + Adi xk+1 = G(A

q 

σk,l xk−τk,l − B(G B)−1 G Adi

l=1

q 

(σ¯ l + Δσl )xk−τk,l

l=1

+Gˆ E i ωk + B(G B)−1 G Hk+1 x0 ,

(4.5)

where Gˆ = I − BG. Definition 4.1 [25] The closed-loop system (4.5) is taken to be finite-time bounded if for (c1 , c2 , S, T , ρ), we have

sup E xkT Sxk ≤ c1

−τ M ≤k≤0

⇒ E xkT Sxk ≤ c2 , k ∈ {1, 2, . . . , T }. Definition 4.2 [26] The closed-loop system (4.5) is said to be passive if there exist γ > 0 satisfying E{2

Tf 

z Tj ω j }

≥ −γE{

j=0

Tf 

ω Tj ω j }

j=0

under the zero initial condition (i.e., φk = 0, k = −τ M , −τ M + 1, . . . , 0). Definition 4.3 [27] For given real matrices Ψ1 ≤ 0, Ψ2 , Ψ3 > 0, Ψ4 ≥ 0 with ( Ψ1  +  Ψ2 )  Ψ4 = 0, the closed-loop system (4.5) is said to be extended stochastically dissipative, if it holds Tf

 Jk ≥ sup E{z kT Ψ4 z k }, ∀ T f > 0, E k=0

0≤k≤T f

for any ωk ∈ l2 [0, ∞), where J (Ψ1 , Ψ2 , Ψ3 , k) = z kT Ψ1 z k + 2z kT Ψ2 ωk + ωkT Ψ3 ωk . Remark 4.1 From the above definition, we can easily see that the extended stochastic dissipative performance can imply the passive performance by setting Ψ1 = 0, Ψ2 = I , Ψ3 = γ I and Ψ4 = 0. This indicates the relationship between the extended stochastic dissipativeness and passiveness. However, in this paper, only passive performance is explored for the system (4.1) to highlight the idea of our proposed scheme, which can be extended to study the extended stochastic dissipative performance in a similar way. Theorem 4.1 Consider the MJSs (4.1) and the common sliding surface (4.3). For given υ ≥ 1, the corresponding sliding motion (4.5) is finite-time bounded in regard to (c1 , c2 , S, T, ρ) if there exist matrices Pi > 0 (i ∈ N ), Q j > 0 ( j = 1, 2, 3) and W > 0, scalars ζ, λ j > 0 and λˆ 1 > 0 satisfying the following criteria:

74

4 Passivity and Control Synthesis ˆ i + ΔAi ), −υ τm Q 1 , −υ τ M Q 2 , Ω1 , Ω2 , . . . , Ωq , Ωω } < 0, Ω = diag{Ω ∗ + 5(Ai + ΔAi )T Gˆ T P˜i G(A

(4.6) (4.7)

∀i ∈ N , j = 1, 2, 3, G T B T P˜i BG ≤ ζ I, λˆ 1 S ≤ Pi ≤ λ1 S, 0 < Q j ≤ λ2 S, W ≤ λ3 I, υ υ T (λ1 c1 + λ2 c1 ρ) + λ3 υ T −1 ρ + 5

T −1 ˆ e−2θ λ−1 min (S)ζc1 < λ1 c2 , υ−1

(4.8)

where special matrices and specific symbols are presented as Ω ∗ = −υ Pi + Q 1 + Q 2 + qQ 3 ,  = τ M − τm + 1, T ˜ T Ωl = −υ τm Q 3 + (1 + 4σ)( ˜ σ¯ l + )Adi ˜ σ¯ l + )Adi G T B T P˜i BG Adi , l = 1, 2, . . . , q, Pi Adi + 5σ( T ˆT ˜ ˆ Ωω = 5E i G Pi G E i − W, θ = min{θ1 , θ2 , . . . , θn }, σ˜ =

q 

(τ M − τm )(τ M + τm − 1) τ M −2 ]. υ 2

(σ¯ l + ), ρ = (τm υ τm −1 + τ M υ τ M −1 ) + q[τ M υ τ M −1 +

l=1

Proof First of all, we construct the Lyapunov-Krasovskii functional with the following form: Vk = Vk,1 + Vk,2 + Vk,3 ,

(4.9)

where Vk,1 = xkT Pi xk , Vk,2 =

k−1  j=k−τm

 q

Vk,3 =

k−1 

k−1 

υ k−1− j x Tj Q 1 x j +

υ k−1− j x Tj Q 2 x j ,

j=k−τ M

 q

υ k−1− j x Tj Q 3 x j +

−τm 

k−1 

υ k−1− j x Tj Q 3 x j .

l=1 t=−τ M +1 j=k+t

l=1 j=k−τk,l

The difference of Vk along (4.5) can be calculated by ΔVk = E{ΔVk,1 + ΔVk,2 + ΔVk,3 },

(4.10)

where (4.11)

E{ΔVk,1 } = E{Vk+1,1 } − υVk,1 + (υ − 1)Vk,1 ,

ˆ i + ΔAi )xk + E{Vk+1,1 } = E xkT (Ai + ΔAi )T Gˆ T P˜i G(A

q 

σk,l xk−τk,l

T

T ˜ Adi Pi Adi

l=1

q 

σk,l xk−τk,l



l=1

q q T    T (σ¯ l + Δσl )xk−τk,l Adi G T B T P˜i BG Adi (σ¯ l + Δσl )xk−τk,l + l=1

l=1

T +ωkT E iT Gˆ T P˜i Gˆ E i ωk + x0T Hk+1 G T B T P˜i BG Hk+1 x0

+2xkT (Ai + ΔAi )T Gˆ T P˜i Adi

q 

q   σk,l xk−τk,l − 2xkT (Ai + ΔAi )T Gˆ T P˜i BG Adi (σ¯ l + Δσl )xk−τk,l

l=1

l=1

+2xkT (Ai + ΔAi )T Gˆ T P˜i Gˆ E i ωk + 2xkT (Ai + ΔAi )T Gˆ T P˜i BG Hk+1 x0 −2

q 

σk,l xk−τk,l

T

T ˜ Adi Pi BG Adi

l=1

+2

q  l=1

−2

q q    T T ˜ ˆ (σ¯ l + Δσl )xk−τk,l + 2 σk,l xk−τk,l Adi Pi G E i ωk l=1

σk,l xk−τk,l

T

T Adi

P˜i BG Hk+1 x0 − 2

l=1 q 

(σ¯ l + Δσl )xk−τk,l

T

T Adi G T B T P˜i Gˆ E i ωk

l=1

q T

 T (σ¯ l + Δσl )xk−τk,l Adi G T B T P˜i BG Hk+1 x0 + 2ωkT E iT Gˆ T P˜i BG Hk+1 x0 l=1

4.3 Design of Sliding Surface

75

ˆ i + ΔAi )xk + ≤ xkT (Ai + ΔAi )T Gˆ T P˜i G(A

q 

T T ˜ (σ¯ l + )xk−τ Adi Pi Adi xk−τk,l k,l

l=1 q q T    T + (σ¯ l + Δσl )xk−τk,l Adi G T B T P˜i BG Adi (σ¯ l + Δσl )xk−τk,l + ωkT E iT Gˆ T P˜i Gˆ E i ωk l=1

+x0T

l=1

T Hk+1 GT

B P˜i BG Hk+1 x0 + 2xkT (Ai + ΔAi )T Gˆ T P˜i Adi T

q  (σ¯ l + Δσl )xk−τk,l l=1

q   (σ¯ l + Δσl )xk−τk,l + 2x kT (Ai + ΔAi )T Gˆ T P˜i Gˆ E i ωk

−2xkT (Ai

+ ΔAi ) Gˆ T P˜i BG Adi

+2xkT (Ai

+ ΔAi ) G P˜i BG Hk+1 x0 − 2

T

l=1 T

ˆT

q q T    T ˜ (σ¯ l + Δσl )xk−τk,l Adi (σ¯ l + Δσl )xk−τk,l Pi BG Adi l=1

l=1

q q  T T  T ˜ ˆ T ˜ (σ¯ l + Δσl )xk−τk,l Adi (σ¯ l + Δσl )xk−τk,l Adi +2 Pi G E i ωk + 2 Pi BG Hk+1 x0 l=1

l=1

q q   T T T T −2 (σ¯ l + Δσl )xk−τk,l Adi G T B T P˜i Gˆ E i ωk − 2 (σ¯ l + Δσl )xk−τk,l Adi G T B T P˜i BG Hk+1 x0 l=1

l=1

+2ωkT E iT Gˆ T P˜i BG Hk+1 x0 .

Further, using the elementary matrix skills, we get ˆ i + ΔAi )xk + E{Vk+1,1 } ≤ 5xkT (Ai + ΔAi )T Gˆ T P˜i G(A

q 

T (σ¯ l + )xk−τ A T P˜ A x k,l di i di k−τk,l

l=1 q q T    T ˜ Pi Adi +4 (σ¯ l + Δσl )xk−τk,l Adi (σ¯ l + Δσl )xk−τk,l l=1

l=1

q q T    T (σ¯ l + Δσl )xk−τk,l Adi G T B T P˜i BG Adi (σ¯ l + Δσl )xk−τk,l +5 l=1 T T ˆT +5ωk E i G

l=1 T P˜i Gˆ E i ωk + 5x0T Hk+1 G T B T P˜i BG Hk+1 x0 .

(4.12)

Recalling (4.11) and (4.12), it holds E{ΔVk,1 } = E{Vk+1,1 } − υVk,1 + (υ − 1)Vk,1 ˆ i + ΔAi )xk + ≤ −υxkT Pi xk + 5xkT (Ai + ΔAi )T Gˆ T P˜i G(A

q 

T T ˜ (σ¯ l + )xk−τ Adi Pi Adi xk−τk,l k,l

l=1

+4

q  l=1

(σ¯ l + Δσl )xk−τk,l

T

T ˜ Adi Pi Adi

q 

 (σ¯ l + Δσl )xk−τk,l + 5ωkT E iT Gˆ T P˜i Gˆ E i ωk

l=1

q q T    T +5 (σ¯ l + Δσl )xk−τk,l Adi G T B T P˜i BG Adi (σ¯ l + Δσl )xk−τk,l l=1

l=1

T G T B T P˜i BG Hk+1 x0 + (υ − 1)Vk,1 . +5x0T Hk+1

(4.13)

In addition, using the Lemma 4.1 and noting the criteria as in Theorem 4.1, we have q q  T   T ˜ (σ¯ l + Δσl )xk−τk,l Adi Pi Adi (σ¯ l + Δσl )xk−τk,l l=1

l=1

76

4 Passivity and Control Synthesis

≤

q q 

 T T ˜ σ¯ l +  (σ¯ l + )xk−τ Adi Pi Adi xk−τk,l . k,l l=1 q

(4.14)

l=1

q T    T T T ˜ (σ¯ l + Δσl )xk−τk,l Adi G B Pi BG Adi (σ¯ l + Δσl )xk−τk,l l=1 q

l=1

q   T T ≤ σ¯ l +  (σ¯ l + )xk−τ Adi G T B T P˜i BG Adi xk−τk,l , k,l l=1

(4.15)

l=1

T T G T B T P˜i BG Hk+1 x0 } ≤ E{5ζ x0T S 2 (S − 2 Hk+1 Hk+1 S − 2 )S 2 x0 } E{5x0T Hk+1 1

1

1

1

≤ E{5ζ x0T S 2 [S − 2 (e−2θ I )S − 2 ]S 2 x0 } 1

1

1

1

≤ 5e−2θ λ−1 min (S)ζc1 ,

(4.16)

where θ = min{θ1 , θ2 , . . . , θn }. Thus, we can conclude that E{ΔVk,1 } = E{Vk+1,1 } − υVk,1 + (υ − 1)Vk,1 q q  

  ˆ i + ΔAi )xk + 1 + 4 σ¯ l +  ≤ −υxkT Pi xk + 5xkT (Ai + ΔAi )T Gˆ T P˜i G(A (σ¯ l + ) l=1 T ×xk−τ k,l

T Adi

P˜i Adi xk−τk,l + 5

q 

l=1

 T T (σ¯ l + )xk−τ Adi G T B T P˜i BG Adi xk−τk,l σ¯ l +  k,l q

l=1

l=1

+5ωkT E iT Gˆ T P˜i Gˆ E i ωk + 5e−2θ λ−1 min (S)ζc1 + (υ − 1)Vk,1 ,

(4.17)

E{ΔVk,2 } = E{Vk+1,2 } − υVk,2 + (υ − 1)Vk,2 k 

=

j=k+1−τm

−υ

k 

υ k− j x Tj Q 1 x j +

k−1  

=

k 

j=k−τ M

υ k− j x Tj Q 1 x j −

j=k+1−τm

+



k 

υ k−1− j x Tj Q 2 x j + (υ − 1)Vk,2

k−1 

υ k−1− j x Tj Q 1 x j +

j=k−τm



υ k− j x Tj Q 2 x j

j=k+1−τ M

k−1 

υ k− j x Tj Q 1 x j



j=k−τm

j=k+1−τ M

υ k− j x Tj Q 2 x j + (υ − 1)Vk,2

k−1 

υ k− j x Tj Q 2 x j −

j=k−τ M

T T = xkT (Q 1 + Q 2 )xk − υ τm xk−τ Q 1 xk−τm − υ τ M xk−τ Q 2 xk−τ M + (υ − 1)Vk,2 , m M

(4.18)

E{ΔVk,3 } = E{Vk+1,3 } − υVk,3 + (υ − 1)Vk,3 =

q 

k 

υ k− j x Tj Q 3 x j +

q k−1  

υ k−1− j x Tj Q 3 x j +

=

k 

υ k− j x Tj Q 3 x j −

l=1 j=k+1−τk+1,l

+

q 

−τ m 

k 

υ k− j x Tj Q 3 x j

q 

−τ m 

k−1 

υ k−1− j x Tj Q 3 x j + (υ − 1)Vk,3

l=1 t=−τ M +1 j=k+t

l=1 j=k−τk,l q 

−τ m 

l=1 t=−τ M +1 j=k+1+t

l=1 j=k+1−τk+1,l

−υ

q 

q k−1  

υ k− j x Tj Q 3 x j



l=1 j=k−τk,l k 

l=1 t=−τ M +1 j=k+1+t

υ k− j x Tj Q 3 x j −

q 

−τ m 

k−1 

l=1 t=−τ M +1 j=k+t

υ k− j x Tj Q 3 x j + (υ − 1)Vk,3

4.3 Design of Sliding Surface =

q   l=1

+

k 

77

j=k+1−τk+1,l

q 

−τ m 

k−1 

υ k− j x Tj Q 3 x j − 

υ k− j x Tj Q 3 x j



j=k−τk,l

T xkT Q 3 xk − υ −t xk+t Q 3 xk+t + (υ − 1)Vk,3

l=1 t=−τ M +1

=

q   l=1

+

k−1 

xkT Q 3 xk +

T υ k− j x Tj Q 3 x j − υ τk,l xk−τ Q 3 xk−τk,l − k,l

j=k+1−τk+1,l

q 

−τ m 



k−1 

υ k− j x Tj Q 3 x j



j=k−τk,l +1



T xkT Q 3 xk − υ −t xk+t Q 3 xk+t + (υ − 1)Vk,3

l=1 t=−τ M +1

=

q  

T xkT Q 3 xk − υ τk,l xk−τ Q 3 xk−τk,l + k,l

l=1

k−1 

+q(τ M − τm )xkT Q 3 xk −

−τ m 

υ k− j x Tj Q 3 x j



j=k−τk,l +1

j=k+1−τk+1,l q 

k−1 

υ k− j x Tj Q 3 x j −

T υ −t xk+t Q 3 xk+t + (υ − 1)Vk,3

l=1 t=−τ M +1

≤

q  

T xkT Q 3 xk − υ τm xk−τ Q 3 xk−τk,l + k,l

l=1

k−τ m

υ k− j x Tj Q 3 x j



j=k+1−τ M

+q(τ M − τm )xkT Q 3 xk −

q 

k−τ m

υ k− j x Tj Q 3 x j + (υ − 1)Vk,3

l=1 j=k+1−τ M

= qxkT Q 3 xk − υ τm

q 

T xk−τ Q 3 xk−τk,l + (υ − 1)Vk,3 , k,l

(4.19)

l=1

where  is indicated below the Theorem 4.1. Combining with (4.10) and (4.17)– (4.19), it leads to E{ΔVk } − (υ − 1)Vk − ωkT W ωk

 ˆ i + ΔAi )xk ≤ xkT − υ Pi + Q 1 + Q 2 + qQ 3 xk + 5xkT (Ai + ΔAi )T Gˆ T P˜i G(A T T −υ τm xk−τ Q 1 xk−τm − υ τ M xk−τ Q 2 xk−τ M − υ τm m M

q 

T xk−τ Q 3 xk−τk,l k,l

l=1

+5σ˜

q 

T T (σ¯ l + )xk−τ Adi G T B T P˜i BG Adi xk−τk,l + 5ωkT E iT Gˆ T P˜i Gˆ E i ωk k,l

l=1

 T T ˜ + 1 + 4σ˜ (σ¯ l + )xk−τ Adi Pi Adi xk−τk,l + 5e−2θ λ−1 min (S)ζc1 k,l q

=

l=1 T −ωk W ωk ξkT Ωξk + 5e−2θ λ−1 min (S)ζc1 ,

(4.20)

T T T T , xk−τ , xk−τ , . . . , xk−τ , ωkT ] and matrix Ω being defined where ξkT = [xkT , xk−τ m M k,1 k,q in (4.6). As in Theorem 4.1, if the condition Ω < 0 holds for any i ∈ N , we can get that

E{Vk+1 } ≤ υE{Vk } + ωkT W ωk + 5e−2θ λ−1 min (S)ζc1 .

(4.21)

78

4 Passivity and Control Synthesis

Substituting E{Vk } from above inequality by following T W ωk−1 + 5e−2θ λ−1 E{Vk } ≤ υE{Vk−1 } + ωk−1 min (S)ζc1 ,

(4.22)

it leads to T Wω T −2θ λ−1 (S)ζc . E{Vk+1 } ≤ υ 2 E{Vk−1 } + υωk−1 1 k−1 + ωk W ωk + 5(υ + 1)e min

(4.23) By applying Mathematical Induction (MI), E{Vk } ≤ υ k E{V0 } +

k−1 

T υ j ωk−1− j W ωk−1− j + 5(

k−1 

j=0

υ j )e−2θ λ−1 min (S)ζc1 . (4.24)

j=0

Noting the expression in (4.9), we have E{V0 } = E{V0,1 } + E{V0,2 } + E{V0,3 },

(4.25)

where E{V0,1 } = E{x0T Pi x0 } ≤ λ1 E{x0T Sx0 } ≤ λ1 c1 , −1 −1   E{V0,2 } = E{ υ −1− j x Tj Q 1 x j + υ −1− j x Tj Q 2 x j } j=−τm

≤ E{υ τm −1

j=−τ M −1 

1

j=−τm −1 

≤ λ2 υ τm −1 E{

−1 

1

x Tj S 2 Qˆ 1 S 2 x j + υ τ M −1

1

1

x Tj S 2 Qˆ 2 S 2 x j }

j=−τ M −1 

x Tj Sx j } + λ2 υ τ M −1 E{

j=−τm

x Tj Sx j }

j=−τ M

≤ λ2 c1 (τm υ τm −1 + τ M υ τ M −1 ), E{V0,3 } = E{

q 

−1 

υ −1− j x Tj Q 3 x j +

l=1 j=−τ0,l

≤ E{υ τ M −1

q 

−1 

q 

x Tj Q 3 x j + υ τ M −2 q 

−1 

−1 

l=1 t=−τ M +1 j=t

l=1 j=−τ0,l

≤ E{λmax ( Qˆ 3 )υ τ M −1

−τ m 

q 

υ −1− j x Tj Q 3 x j }

−τ m 

−1 

l=1 t=−τ M +1 j=t

x Tj Sx j + λmax ( Qˆ 3 )υ τ M −2

l=1 j=−τ0,l

≤ qλ2 c1 [τ M υ τ M −1 +

x Tj Q 3 x j }

(τ M − τm )(τ M + τm − 1) τ M −2 υ ]. 2

1 1 with Qˆ j = S − 2 Q j S − 2 ( j = 1, 2, 3). Then,

q 

−τ m 

−1 

l=1 t=−τ M +1 j=t

x Tj Sx j }

4.3 Design of Sliding Surface

79

E{V0 }

(τ M − τm )(τ M + τm − 1) τ M −2 υ ] + λ1 c1 2 +λ2 c1 (τm υ τm −1 + τ M υ τ M −1 ).

≤ qλ2 c1 [τ M υ τ M −1 +

(4.26)

Recalling W ≤ λ3 I in Theorem 4.1, one has k−1 

T T −1 υ j ωk−1− λmax (W ) j W ωk−1− j ≤ υ

k−1 

j=0

T T −1 ωk−1− ρ, (4.27) j ωk−1− j ≤ λ3 υ

j=0

and E{Vk } ≤ υ k E{V0 } +

k−1 

T υ j ωk−1− j W ωk−1− j + 5(

j=0

k−1 

υ j )e−2θ λ−1 min (S)ζc1

j=0

(τ M − τm )(τ M + τm − 1) τ M −2 υ ≤ υ T qλ2 c1 [τ M υ τ M −1 + ] + λ1 c1 2

+λ2 c1 (τm υ τm −1 + τ M υ τ M −1 ) + λ3 υ T −1 ρ +5

υ T − 1 −2θ −1 e λmin (S)ζc1 υ−1

= υ T (λ1 c1 + λ2 c1 ρ) + λ3 υ T −1 ρ + 5

υ T − 1 −2θ −1 e λmin (S)ζc1 υ−1 (4.28)

with ρ = (τm υ τm −1 + τ M υ τ M −1 ) + q[τ M υ τ M −1 + other hand,

(τ M −τm )(τ M +τm −1) τ M −2 υ ]. 2

E{Vk } ≥ E{xkT Pi xk } ≥ λˆ 1 E{xkT Sxk }.

On the (4.29)

Combining (4.8) in Theorem 4.1 and above (4.28), (4.29) yields E{xkT Sxk }

≤

υ T (λ1 c1 + λ2 c1 ρ) + λ3 υ T

−1 ρ + 5 υ T −1 e−2θ λ−1 (S)ζc 1 min υ−1

λˆ 1

< c2 .

(4.30)

Corollary 4.1 Consider the MJSs (4.1) and the common sliding surface (4.3). For given υ ≥ 1, the corresponding sliding mode dynamics (4.5) is finite-time bounded in regard to (c1 , c2 , S, T , ρ) if there exist matrices Pi > 0 (i ∈ N ), Q j > 0 ( j = 1, 2, 3) and W > 0, scalars ζ > 0, νi > 0, λˆ 1 > 0, λ1 > 0 and λ2 > 0 satisfying the following LMIs-based conditions: ⎡

⎤ Ωˆ 0 5AiT P˜i Mi ⎣ ∗ Ω˜ ⎦ < 0, 0 ∗ ∗ 5(MiT P˜i Mi − νi I )

(4.31)

80

4 Passivity and Control Synthesis ∀i ∈ N , j = 1, 2, 3, G T B T P˜i BG ≤ ζ I, λˆ 1 S ≤ Pi ≤ λ1 S, 0 < Q j ≤ λ2 S, W ≤ λ3 I, υ T (λ1 c1 + λ2 c1 ρ) + λ3 υ T

−1

ρ+5

υ T − 1 −2θ −1 e λmin (S)ζc1 < λˆ 1 c2 , υ−1 νi I − MiT P˜i Mi > 0,

(4.32) (4.33) (4.34) (4.35)

where Ωˆ = −υ Pi + Q 1 + Q 2 + qQ 3 + 5(AiT P˜i Ai + νi NiT Ni ), Ai = Gˆ Ai , Mi = Gˆ Mi , Ω˜ = diag{−υ τm Q 1 , −υ τ M Q 2 , Ω1 , Ω2 , . . . , Ωq , Ωω }.

Proof From the Lemma 4.2, it can be concluded that for any νi > 0 satisfying νi I − MiT P˜i Mi > 0, ˆ i + ΔAi ) 5(Ai + ΔAi )T Gˆ T P˜i G(A  T  T ≤ 5 Ai P˜i Ai + Ai P˜i Mi (νi I − MiT P˜i Mi )−1 MiT P˜i Ai + νi NiT Ni with Ai = Gˆ Ai and Mi = Gˆ Mi . Then, by the Schur complement, we can know that 

 Ωˆ + 5Ai T P˜i Mi (νi I − MiT P˜i Mi )−1 MiT P˜i Ai 0 0 (i ∈ N ) and Q j > 0 ( j = 1, 2, 3), scalars νi > 0, λˆ 1 > 0, λ > 0, γ > 0 and εi > 0 satisfying the following criteria: ⎤ ⎡¯ Ξ11 + εi NiT Ni 0 0 0 0 0 0 Ξ1,q+4 3A iT P˜i M i 0 ⎥ ⎢ ∗ −Q 1 0 0 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ −Q 2 0 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ Λ1 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ ∗ Λ 0 0 0 0 0 2 ⎥ ⎢ ⎥ < 0, ⎢ .. ⎥ ⎢ ⎥ ⎢ . 0 ∗ ∗ ∗ ∗ ∗ 0 0 0 ⎥ ⎢ ⎥ ⎢ 0 0 ∗ ∗ ∗ ∗ ∗ ∗ Λq ⎥ ⎢ ⎢ TG T P˜ M ⎥ ˆ 0 E ∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξ q+4,q+4 i i⎥ ⎢ i ⎦ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξq+5,q+5 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −εi I

(4.36)

4.3 Design of Sliding Surface

81 νi I − M iT P˜i M i > 0,  1 − 2T λˆ 1 S Ci E ωi

λˆ 1 S − Pi < 0,  T Ci T + (γ − λ)I < 0, + E ωi λρ −

1ˆ λ1 c2 < 0, 2

(4.37) (4.38) (4.39) (4.40)

where special matrices and specific symbols are presented as Ξ¯ 11 = 3A iT P˜i A i − Pi + Q 1 + Q 2 + qQ 3 + 3νi NiT Ni , T ˜ T Λ j = (1 + 3σ)( ˜ σ¯ j + )Adi ˜ σ¯ j + )Adi G T B T P˜i BG Adi − Q 3 , ( j = 1, 2, . . . , q), Pi Adi + 4σ( σ˜ =

q  (σ¯ l + ), Ξ1,q+4 = AiT Gˆ T P˜i Gˆ E i − CiT , l=1

T + E ωi ), Ξq+5,q+5 = −3(νi I − M iT P˜i M i ). Ξq+4,q+4 = 3E iT Gˆ T P˜i Gˆ E i − (γ I + E ωi

Proof To start with, letting − 2z kT ωk − γωkT ωk + E{ΔVk } ≤ 0,

(4.41)

k−1 k−1   {2z sT ωs + γωsT ωs } ≥ E{ΔVs } = E{Vk } ≥ λˆ 1 E{xkT Sxk },

(4.42)

we have

s=0

s=0

i.e., k−1  {2z sT ωs + γωsT ωs } ≥ λˆ 1 E{xkT Sxk }.

(4.43)

s=0

On the other hand, we can analyze by (4.39) that k−1  s=0

{2z sT ωs + γωsT ωs } ≤

k−1  s=0

 T T x s ωs

   1 1 xs λˆ S 0 2T 1 < λˆ 1 c2 + λρ. (4.44) ωs 0 λI 2

Taking (4.40), (4.43) and (4.44) into account, it leads to E{xkT Sxk } ≤ c2 , which completes the proof of finite-time boundedness for the sliding motion (4.5). In the sequel, we will discuss the passive performance under the zero initial conditions. Recalling (4.42), we have k−1   E (2z sT ωs + γωsT ωs ) ≥ 0. s=0

According to the Definition 4.2, we can know that condition (4.41) can ensure the passive performance for the sliding motion (4.5). Therefore, it can be concluded that

82

4 Passivity and Control Synthesis

the sliding motion (4.5) is finite-time bounded in regard to (c1 , c2 , S, T , ρ) with passive performance. Subsequently, we are in a position to propose some criteria to ensure (4.41). In a similar way, choosing the same Lyapunov-Krasovskii functional as in Theorem with υ = 1, it holds for the zero initial condition −2z kT ωk − γωkT ωk + E{ΔVk }   ≤ xkT 3Ai T P˜i Ai + 3Ai T P˜i Mi (νi I − MiT P˜i Mi )−1 MiT P˜i Ai + 3νi NiT Ni xk T T +xkT (−Pi + Q 1 + Q 2 + qQ 3 )xk − xk−τ Q 1 xk−τm − xk−τ Q 2 xk−τ M m M q q q      T T T ˜ σ¯ l +  − xk−τ Q 3 xk−τk,l + 1 + 3 (σ¯ l + )xk−τ Adi Pi Adi k,l k,l l=1

l=1

×xk−τk,l +4

q

l=1

+2xkT (Ai

l=1

  T T σ¯ l + (σ¯ l +)xk−τ Adi G T B T P˜i BG Adi xk−τk,l −2xkT CiT ωk k,l q

l=1

  T + ΔAi ) G P˜i Gˆ E i ωk + ωkT 3E iT Gˆ T P˜i Gˆ E i − (γ I + E ωi + E ωi ) ωk T

ˆT

= ξkT Ξ ξk , where ⎡

⎤ Ξ11 0 0 0 0 0 0 (Ai + ΔAi )T Gˆ T P˜i Gˆ E i − CiT ⎢ ∗ −Q 1 ⎥ 0 0 0 0 0 ⎢ ⎥ ⎢ ∗ ⎥ 0 ∗ −Q 2 0 0 0 0 ⎢ ⎥ ⎢ ∗ ⎥ 0 ∗ ∗ Λ1 0 0 0 ⎢ ⎥ Ξ =⎢ ∗ ⎥, 0 ∗ ∗ ∗ Λ2 0 0 ⎢ ⎥ ⎢ ⎥ . .. 0 ⎢ ∗ ⎥ ∗ ∗ ∗ ∗ 0 ⎢ ⎥ ⎣ ∗ ⎦ 0 ∗ ∗ ∗ ∗ ∗ Λq T ˆT ˜ ˆ T ∗ ∗ ∗ ∗ ∗ ∗ ∗ 3E i G Pi G E i − (γ I + E ωi + E ωi ) T ˜ Ξ11 = 3Ai Pi Ai +3Ai T P˜i Mi (νi I −MiT P˜i Mi )−1 MiT P˜i Ai +3νi NiT Ni − Pi +Q 1 +Q 2 + qQ 3 , νi I − MiT P˜i Mi > 0, Ai and Mi (i ∈ N ) can be found in Corollary 4.1, ξk and Λ j ( j = 1, 2, . . . , q) are defined below (4.20) and (4.40) respectively. Using the Schur complement, it yields ⎤ ⎡ Ξ¯ 11 0 0 0 0 0 0 (Ai + ΔAi )T Gˆ T P˜i Gˆ E i − CiT 3A iT P˜i M i ⎥ ⎢ ∗ −Q 0 0 0 0 0 0 1 ⎥ ⎢ ⎥ ⎢ ∗ ∗ −Q 2 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ Λ 0 0 0 0 0 ⎥ ⎢ 1 ⎥ ⎢ ∗ ∗ ∗ Λ2 0 0 0 0 ⎥ ⎢ ∗ ⎥ < 0, ⎢ ⎥ ⎢ . ⎥ ⎢ .. ∗ ∗ ∗ ∗ 0 0 0 ⎥ ⎢ ∗ ⎥ ⎢ ⎥ ⎢ ∗ 0 0 ∗ ∗ ∗ ∗ ∗ Λq ⎥ ⎢ T T T ⎦ ⎣ ∗ 0 ∗ ∗ ∗ ∗ ∗ ∗ 3E i Gˆ P˜i Gˆ E i − (γ I + E ωi + E ωi ) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −3(νi I − M iT P˜i M i )

4.5 Controller Design

83

where Ξ¯ 11 is defined below (4.40). Besides, using the Lemma 2 in work [7] for the above inequality, (4.36) holds.

4.4 Synthesis of Controller Theorem 4.2 If there exists αi ∈ R+n , ∀i ∈ S , such that (Ai + Bi Ki )T αi +

N j=1

λi j (h)α j + CiT 1s 0. According to above proposed reaching condition (4.49), our purpose is to explore a controller design scheme for the considered Markovian jump discrete systems under the SCDs. Theorem 4.3 For the investigated MJSs (4.1) and the prescribed common sliding surface (4.3), the robust controller given by u k = Gxk + G(Hk+1 − Hk )x0 − ϕsgn(sk ) −αsk − βtanh(xk )sgn(sk )

(4.50)

can keep the system (4.1) remain in a region of the sliding surface (4.3), where q  √ xk−τk,l  + ρG E i . ϕ = G Ai xk  + G Mi Ni xk  + G Adi  l=1

Proof By combining (4.1) and (4.3), it can be applied that Δsk = G(Ai + ΔAi )xk + G Adi x˜k + u k + G E i ωk − Gxk −G(Hk+1 − Hk )x0 .

(4.51)

Substitute the controller in (4.50) into equation in (4.51), i.e., Δsk = G(Ai + ΔAi )xk + G Adi x˜k + G E i ωk − ϕsgn(sk ) −αsk − βtanh( xk )sgn(sk ).

(4.52)

Further, it is clear that the reaching condition (4.49) can be easily ensured by discussing the positive and negative of Δsk . In conclusion, despite the parameter uncertainty and external disturbance, the investigated MJSs (4.1) can be kept remain in a region nearby the sliding surface (4.3) by the controller given in (4.50). This completes the proof of Theorem 4.3.

4.6 Simulation Firstly, take into account the Markovian jump discrete systems (4.1) with following parameters: for i = 1,

4.6 Simulation

85

⎛⎡

⎤ ⎡ ⎤ ⎞ −0.412 0.1946 0 0.04   0.1318 0.0125 ⎦ + ⎣ −0.013 ⎦ sin(0.2k) −0.03 0.2 0 ⎠ xk xk+1 = ⎝⎣ 0 0 0 −0.15 0.02 ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ −0.02 0.06 0.04 0.08 −0.017 0.015 0 −0.01 + ⎣ 0.04 0.01 −0.03 ⎦ x˜k + ⎣ 0.1 0.2 ⎦ u k +⎣ 0.01 0.015 0 ⎦ ωk , 0.01 0.02 0.05 0.13 0.26 0.02 0.025 −0.01 ⎡ ⎤ 0.04 0 0 z k = ⎣ 0 0.04 0 ⎦ xk + diag{−0.02 0.3 − 0.12}ωk ; for i = 2, 0 0 0.06 ⎛⎡

⎤ ⎡ ⎤ ⎞ 0.0866 0.583 0 0.04   xk+1 = ⎝⎣ −0.3187 0.4914 0 ⎦ + ⎣ −0.013 ⎦ sin(0.2k) 0.01 0 −0.1 ⎠ xk 0 0 −0.15 0.02 ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ 0.03 0 0.01 0.08 −0.017 0.015 0 −0.01 + ⎣ 0.02 0.03 0 ⎦ x˜k + ⎣ 0.1 0.2 ⎦ u k + ⎣ 0.01 0.015 0 ⎦ ωk , 0.04 0.05 −0.01 0.13 0.26 0.02 0.025 −0.01 ⎡ ⎤ −0.08 0 0 z k = ⎣ 0 −0.01 0 ⎦ xk + diag{0.17 − 0.02 0.16}ωk . 0 0 0.04

The time-varying communication delays are set as 2 ≤ τk,l ≤ 5 (l = 1, 2). Besides, T  choosing S = I , ωk = 0 0 e−0.5k and θ j = 10−2 ( j = 1, 2, 3), we can get feasible solutions as follows: ⎡

⎤ ⎡ ⎤ 12.2745 0.7819 −1.3200 14.3420 −2.4493 −2.0952 P1 = ⎣ 0.7819 13.2453 1.8123 ⎦ , P2 = ⎣ −2.4493 17.8987 2.5638 ⎦ , γ = 3.3125. −1.3200 1.8123 11.4276 −2.0952 2.5638 11.6764

By letting α = 0.1 and β = 1, the simulation results can be displayed as Figs. 4.1, 4.2, 4.3 and 4.4, which can help theoretical analysis for system (4.1) with SCDs. Fig. 4.1 One passible evolution of system mode rk

3

rk

2

1

0

5

10

15

time (k)

20

25

30

86

4 Passivity and Control Synthesis

Fig. 4.2 Trajectories of xk,1 , xk,2 and xk,3 in (4.1)

5

x1,k x2,k x3,k

4 3 2 1 0 −1 −2 −3 −4 −5

5

10

15

20

25

30

time (k)

Fig. 4.3 Switching function sk and control signal u k

10

s1,k s2,k

5 0

0.5 0

−5 −10

−0.5

1

3

5

7

9

2

11

13

4

15

17

6

19

21

23

25

27

29

time (k) 10

u1,k u2,k

5 0 −5 −10

5

10

15

20

25

30

time (k)

Specifically, one passible evolution of system mode rk is presented in Fig. 4.1. Figure 4.2 shows the trajectories of system state xk,l (l = 1, 2, 3), where xk,l is l-th action signal u k are described element of xk . The switching function sk and control  in Fig. 4.3. In addition, the time-varying rate k

This indicates

(−2) k

j=1

j=1

z jωj

ωjωj

(−2) k

k j=1

z jωj

j=1 ω j ω j

is presented in Fig. 4.4.

γ under the zero initial condition. From the above

discussions, we conclude that the system (4.1) with SCDs is finite-time bounded with passive performance via our comprehensive scheme.

References Fig.  4.4 The actual rate (−2) k

k j=1

j=1

z jωj

ωjωj

87 0.5 rate :=

0.45

(−2)

k j=1 zj ωj k j=1 ωj ωj

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

5

10

15

20

25

30

time (k)

4.7 Conclusion The finite-time robust SMC method has been proposed for uncertain Markovian jump discrete systems with SCDs under imprecise occurrence probability. For the considered model, the phenomenon of SCDs has been firstly described via some Bernoulli distributed stochastic variables, including deterministic occurrence probability information as particular cases. Inspired by considerations about the effective utilization of the imprecise information, the finite-time robust SMPC strategy has been investigated in this paper. The initial condition-dependent sliding surface has been involved. By the Lyapunov method and SMC technique, some potential sufficient criteria have been established for sliding mode dynamics to ensure the required finite-time boundedness and passive performance. According to a comprehensive reaching condition, the robust sliding mode passive controller has been synthesized to keep the system remain in a region of the switching surface. Interestingly, it can be shown that the sliding mode phase can be reached for considered system from the beginning. Therefore, this type of initial condition-dependent sliding surface can help save time by omitting so-called reaching phase of sliding mode. Lastly, the usefulness of our theoretic results can be verified via a given numerical analysis.

References 1. M. Boukattaya, H. Gassara, T. Damak, A global time-varying sliding-mode control for the tracking problem of uncertain dynamical systems, ISA Transactions, 97 (2020) 155–170. 2. X. Wu, K. Xu, M. Lei, X. He, Disturbance-compensation-based continuous sliding mode control for overhead cranes with disturbances, IEEE Transactions on Automation Science and Engineering, 17(4) (2020) 2182–2189.

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3. L. Dong, K. Liu, Adaptive sliding mode control for uncertain nonlinear multi-agent tracking systems subject to node failures, Journal of the Franklin Institute 359 (2) (2022) 1385–1402, https://doi.org/10.1016/j.jfranklin.2021.11.039. 4. M. Labbadi, Y. Boukal, M. Cherkaoui, M. Djemai, Fractional-order global sliding mode controller for an uncertain quadrotor UAVs subjected to external disturbances, Journal of the Franklin Institute, 358(9) (2021) 4822–4847. 5. X. Zhang, Robust integral sliding mode control for uncertain switched systems under arbitrary switching rules, Nonlinear Analysis: Hybrid Systems, 37 (2020) 100900, https://doi.org/10. 1016/j.nahs.2020.100900. 6. A. Modiri, S. Mobayen, Adaptive terminal sliding mode control scheme for synchronization of fractional-order uncertain chaotic systems, ISA Transactions, 105 (2020) 33–50. 7. P. Zhang, Y. Kao, J. Hu, et al, Finite-time observer-based sliding-mode control for Markovian jump systems with switching chain: average dwell-time method, IEEE Transactions on Cybernetics, 53 (1) (2023) 248–261, https://doi.org/10.1109/TCYB.2021.3093162. 8. Y. Cheng, G. Wen, H. Du, Design of robust discretized sliding mode controller: analysis and application to Buck converters, IEEE Transactions on Industrial Electronics, 67 (12) (2020) 10672–10681. 9. K. Ding, Q. Zhu, Extended dissipative anti-disturbance control for delayed switched singular semi-Markovian jump systems with multi-disturbance via disturbance observer, Automatica, https://doi.org/10.1016/j.automatica.2021.109556. 10. P. Zhang, Y. Kao, J. Hu, B. Niu, Robust observer-based sliding mode H∞ control for stochastic Markovian jump systems subject to packet losses. Automatica, https://doi.org/10.1016/j. automatica.2021.109665. 11. M. Shen, H∞ filtering of continuous Markov jump linear system with partly known Markov modes and transition probabilities. Journal of the Franklin Institute, 350(10) (2013) 3384–3399. 12. X. Li, W. Zhang, D. Lu, Stability and stabilization analysis of Markovian jump systems with generally bounded transition probabilities. Journal of the Franklin Institute, 357 (13) (2020) 8416–8434. 13. Y. Cui, H. Feng, W. Zhang, et al, Positivity and stability analysis of T-S fuzzy descriptor systems with bounded and unbounded time-varying delays, IEEE Transactions on Cybernetics, https:// doi.org/10.1109/TCYB.2021.3072392. 14. B. Du, Q. Han, S. Xu, F. Yang, Z. Shu, On joint design of intentionally introduced delay and controller gain for stabilization of second-order oscillatory systems, Automatica, https://doi. org/10.1016/j.automatica.2020.108915. 15. G, Ling, X. Liu, M. Ge, Y. Wu, Delay-dependent cluster synchronization of time-varying complex dynamical networks with noise via delayed pinning impulsive control. Journal of the Franklin Institute, 358 (6) (2021) 3193–3214. 16. J. Wang, M. Chen, L. Zhang, Observer-based discrete-time sliding mode control for systems with unmatched uncertainties. Journal of the Franklin Institute, 358 (16) (2021) 8470–8484. 17. K. Sen, X. Guan, D. Dong, Finite-time stabilization control of quantum systems, Automatica, https://doi.org/10.1016/j.automatica.2020.109327. 18. J. Zhang, S. Tong, Y. Li, Adaptive fuzzy finite-time output-feedback fault-tolerant control of nonstrict-feedback systems against actuator faults, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 52 (2) (2022)1276–1287, https://doi.org/10.1109/TSMC.2020. 3011702. 19. B. Chen, C. Lin, Finite-Time stabilization-based adaptive fuzzy control design, IEEE Transactions on Fuzzy Systems, 29 (8) (2021) 2438–2443. 20. B. Zhou, Finite-time stability analysis and stabilization by bounded linear time-varying feedback, Automatica, https://doi.org/10.1016/j.automatica.2020.109191. 21. F. Li, C. Du, C. Yang, L. Wu, W. Gui, Finite-time asynchronous sliding mode control for Markovian jump systems, Automatica, https://doi.org/10.1016/j.automatica.2019.108503. 22. Y. Wang, B. Zhu, H. Zhang, W. Zheng, Functional observer-based finite-time adaptive ISMC for continuous systems with unknown nonlinear function, Automatica, https://doi.org/10.1016/ j.automatica.2020.109468.

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23. P. Zhang, J. Hu, H. Liu, C. Zhang, Sliding mode control for networked systems with randomly varying nonlinearities and stochastic communication delays under uncertain occurrence probabilities, Neurocomputing, 320 (2018) 1–11. 24. B. Chen, J. Huang, Y. Niu, Sliding mode control for Markovian jumping systems with actuator nonlinearities. International Journal of Systems Science, 43(4) (2012) 656–664. 25. H. Song, L .Yu, D. Zhang, W. Zhang, Finite-time H∞ control for a class of discrete-time switched time-delay systems with quantized feedback, Communications in Nonlinear Science and Numerical Simulation, 17 (12) (2012) 4802–4814. 26. Y. Shan, K. She, S. Zhong, J. Cheng, et al, Event-triggered passive control for Markovian jump discrete-time systems with incomplete transition probability and unreliable channels. Journal of the Franklin Institute, 356 (15) (2019) 8093–8117. 27. Q. Zhu, S. Kumar, R. Raja, F. Rihan, Extended dissipative analysis for aircraft flight control systems with random nonlinear actuator fault via non-fragile sampled-data control, Journal of the Franklin Institute, 356 (15) (2019) 8610–8624. 28. C. Gao, Y. Liu, Y. Li, A reaching-law method for uncertain discrete variable-structure control systems, Control Theory & Applications, 26 (7) (2009) 781–785.

Part II

Markov Jump System Under Deception Attacks

Chapter 5

Hidden Markov Model-Based Control

This paper is concerned with the hidden Markov model (HMM)-based sliding mode control (SMC) problem for uncertain Markovian jump discrete systems (MJDSs) under random deception attacks (DAs). The phenomenon of random DAs is characterized by virtue of the probabilistic delay, where the attack patterns can be adjusted according to the network status. We aim to design the HMM-based SMC strategy via a sliding mode observer such that the stochastic stability can be ensured despite the random DAs and uncertainties. In order to drive the system to approach the sliding surface, a novel reaching condition associated with the attack is proposed. Furthermore, the desired controller is devised with help of the reaching condition above. In the end, a numerical example is given to show the effectiveness of theoretical achievements.

5.1 Introduction Cyber-attack is a kind of data security incident, in which the attacker exploits flaws of configurations, protocols, and programs in the network to launch an attack on the system [1–3]. For example, with the aim of destroying raw data, the fake data (FD) injection attack is a typical representation of deception attacks (DAs), which poses a threat to the original system via data tampering. In order to attenuate DAs, a great number of approaches have been given such as the resilient tracking control and synchronization scheme [4, 5], the attack detection method, and aperiodic sampleddata control [6, 7]. In particular, the multiple cyber-attack issues have been discussed in the same framework against complicated hybrid attack cases [5, 8, 9]. Specifically, the new co-design strategy has been proposed in [10] for input reconstruction and sensor scheduling, where the input signals of the system have got hit by the injection of bad disturbances. Note that in engineering practice, it will have to take a certain amount of time from the transmission of false data to its real impact on the system. Moreover, the time-delay in most existing work is often required to be time-varying and bounded throughout the paper. However, one possibility in the current network © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Kao et al., Analysis and Design of Markov Jump Discrete Systems, Studies in Systems, Decision and Control 499, https://doi.org/10.1007/978-981-99-5748-4_5

93

94

5 Hidden Markov Model-Based Control

environment is that large time-delays are less likely to occur, whereas small timedelays are more likely to occur (i.e., probabilistic delay). Due to the above features, this type of time-delays can be also used by adversary in the course of the cyberattack. The sliding mode control (SMC) has special advantages in attenuating parameter uncertainty as well as external disturbances [11–14]. In fact, the SMC strategy has continued to play a significant role in attenuation of the impact of attacks [15, 16]. For instance, the phenomenon of cyber-attack has been considered in [17] via a sliding mode observer, where the observer-based attack detection scheme has been given under the consensus objective. Besides, the resilient defensive strategies against FD injection attacks have been discussed in [17, 18], where the unknown information of attacks can be assessed in [18] for the power system by an adaptive technique. However, the SMC issue with DAs associated with probabilistic delay is still an open topic. On the other hand, we find that reaching law method has been employed for many complicated SMC problems [19–21], which is driven by the aim to fit system needs by reconfiguring reaching circumstances or altering relative variables. As far as the author knows, the reaching law method has not been utilized against the DAs along with probabilistic delay, not to mention the unavailable real-time dynamic mode of the system. As a result, another motivation for our work is to explore a hidden Markov model (HMM)-based new reaching law that can deal with the attacks under investigation. Inspired by above discussions, we focus on the HMM-based SMC problem for uncertain MJDSs under the DAs with probabilistic delay in this paper. First, the probabilistic delay is depicted via a series of Bernoulli processes, and then the DAs with probabilistic delay is modeled. Second, an asynchronous observer is constructed without knowing the real-time system mode. In view of the observed state, a modedependent common sliding surface is selected and further an augmented system is formed via the augmentation method. Based upon the Lyapunov method, the stochastic stability can be ensured for the augmented system by means of some strict LMIsbased criteria. By proposing a new reaching condition, a desiring attack-resistant controller is synthesized on the basis of HMM such that the original system from any initial state can be driven to approach the sliding surface. Finally, a simulation example is utilized to evaluate the usefulness of our proposal. The challenges in this investigation lie in the following aspects. (i) How to properly characterize the DAs from an attacker’s perspective when occurring probabilistic delay? (ii) How to check the effects of this kind of attacks especially when the realtime mode of the system is not available? (iii) What method should be adopted to design an effective controller, which is attack-resistant and can be adjustable according to the attack situation? Surrounding the challenging issues listed above, we highlight the contributions of this paper as follows: (i) proposing a so-called common sliding surface with respect to the main mode for the first time; (ii) deriving sufficient stochastic stability criteria by adding extra attack factors to the traditional LyapunovKrasovskii (LK) functional; and (iii) constructing a desirable attack-resistant asynchronous controller based on a newly modified reaching condition, where the sampling period is expected to be shorter once the original system suffers an attack.

5.2 Problem Statements and Preliminaries

95

5.2 Problem Statements and Preliminaries In this paper, the variable δk ∈ S = {1, 2, . . . , N } denotes a homogeneous Markov chain whose transition probability matrix Π  [δi j ]i, j∈N is described as follows: δi j  Pr(δk+1 = j|δk = i) ≥ 0, ∀i, j ∈ S , k ∈ Z + ,  and Nj=1 δi j = 1 for each i ∈ S . Consider the following MJDS with uncertainties: 

x(k + 1) = (A(δk ) + ΔA(δk ))x(k) + B u˜ σk (x(k), k), yk = C(δk )x(k),

(5.1)

where x(k) ∈ Rn is the state vector; u˜ σk (x(k), k) ∈ Rm stands for the control input. For simplicity, let Ai  A(δk ), ΔAi  ΔA(δk ) = Hi Fk Ni (δk = i ∈ S ), where the matrices Ai , Hi , Ni and B are known real matrices with B possessing full column rank, and the matrix Fk fulfills FkT Fk ≤ I . Then, the variable σk is represented as detection mode with Ψ  (ψil )i∈S ,l∈H , ψil = E{σk = l|δk = i} and H = {1, 2, . . . , H }. As in [22], the element (δk , σk , Π, Ψ ) constitutes the HMM, which is used to describe the asynchronous case. By virtue of the detection mode σk , one stochastic deception attacks model is introduced by the expression below u˜ σk (x(k), k) = u σk (x(k), k) + γσk ,k μσk (x(k − τk ), k),

(5.2)

where the random variable γσk ,k is a stochastic DA factor with P{γσk ,k = 1} = γ¯σk and P{γσk ,k = 0} = 1 − γ¯σk . In our work, the Bernoulli distributed random variable ϑσk ,k is also utilized for modulating attack patterns according to the probabilistic delay:  ϑσk ,k =

1, τk = τk,1 ∈ Γ1  {τm , τm + 1, τm + 2, . . . , τ } 0, τk = τk,2 ∈ Γ2  {τ + 1, τ + 2, . . . , τ M − 1, τ M }

with the conditional probabilities under γσk ,k = 1 being P(τk = τk,1 | γσk ,k = 1) = P(ϑσk ,k = 1 | γσk ,k = 1) = ϑ¯ σk ,

P(τk = τk,2 | γσk ,k = 1) = P(ϑσk ,k = 0 | γσk ,k = 1) = 1 − ϑ¯ σk .

Hence, it can be seen that μσk (x(k − τk ), k) = ϑσk ,k g1 (x(k − τk,1 )) + (1 − ϑσk ,k ) ×g2 (x(k − τk,2 )),

(5.3)

96

5 Hidden Markov Model-Based Control

where g Tj (x(k − τk, j ))g j (x(k − τk, j )) ≤ x T (k − τk, j )M Tj M j x(k − τk, j ) with M j ( j = 1, 2) being known energy limiting matrices with respect to the attacks. Lemma 5.1 For real matrices Q = QT , M, N and uncertain matrix F with FT F ≤ I , we have Q + MFN + NT FT MT < 0 holds, if and only if there exists ε > 0 such that Q + εMMT + ε−1 NT N < 0 or, ⎡ ⎤ Q εM NT ⎣ εMT −ε I 0 ⎦ < 0. N 0 −ε I Definition 5.1 [23] The uncertain networked discrete system (5.1) under consideration is said to be stochastically stable if it holds for initial condition x(0) and any δ(0) ∈ S E{

∞ 

 x(k) 2 | x(0), δ(0)} < ∞.

k=0

The aim of our work is to design a novel SMC law on the basis of a modified reaching condition such that the stochastic stability of the corresponding system can be ensured despite the simultaneous existence of the Markovian jumping parameter, FD injection attacks and parameter uncertainties.

5.3 Design of Common Sliding Surface In this section, we design the following HMM-based observer to detect the system state:  xˆk+1 = Aχ xˆk + Bu χ + L χ (yk − yˆk ) (5.4) yˆk = Cχ xˆk with the matrix L χ (χ ∈ H ) being resolved one. Next, choose a common switching function below: sk = W xˆk ,

(5.5)



T where sk = sk,1 sk,2 . . . sk,m and the matrix W is a time-invariant matrix to be T P(i 0 )E 1,0 confirmed later. In our work, the matrix W is determined as W = B T E 1,0  I to ensure the validity of (W B)−1 with E 1,0 = n×n . It is necessary to note that 0n×n i 0 is defined as main mode here, which can be determined by virtue of comparing  δ˘ j  i∈S δi j ( j = 1, 2, . . . , N ).

5.3 Design of Common Sliding Surface

97

Remark 5.1 In the near future, we may research sliding surface design with stochastic DAs based on the compensation as xk,c = γσk−1 ,k−1 x˘k + (1 − γσk−1 ,k−1 )xk ,  ( j) where x˘k = xk − 2j=1 ασk−1 ,k−1 g˘ j (xk−1−τk−1, j ). The idea of compensation is that only original uncorrupted data is picked up for the design and analysis of the system (5.1). From the above observation, it can be seen that the compensation scheme is closely related to the fact that the attack occurs. As a matter of fact, this is still an open challenging topic in view of the recent high frequency of cyber-attack. Then, taking the Eqs. (5.1)–(5.5) and ideal sliding-mode condition sk+1 = sk = eq 0 into account, we can get the equivalent controller u χ,k = −(W B)−1 W [Aχ xˆk + eq L χ (yk − yˆk )]. Let’s substitute u χ,k into the system (5.4). So, for σk = χ , the resultant sliding motion can be represented in the estimation space: xˆk+1 = W˘ [Aχ + L χ (Ci − Cχ )]xˆk + W˘ L χ Ci ek ,

(5.6)

where W˘ = I − B(W B)−1 W and ek = xk − xˆk , g˘ j (xk−τk, j ) = Bg j (xk−τk, j ) ( j = (1) (2) 1, 2), αχ,k = ϑχ,k and αχ,k = 1 − ϑχ,k . Recalling (5.1) and (5.4), we can get the error dynamics ek+1 = [(Ai + ΔAi ) − Aχ − L χ (Ci − Cχ )]xˆk + (Ai + ΔAi −L χ Ci )ek + γχ,k

2 

( j)

αχ,k g˘ j (xk−τk, j ).

(5.7)

j=1

By letting υk = [xˆkT ekT ]T , we augment the systems (5.6) and (5.7) and have υk+1 = Ai,χ υk + γχ,k E 0,1

2 

( j)

T αχ,k g˘ j (E 1,1 υk−τk, j ),

(5.8)

j=1

where W˘ L χ Ci W˘ [Aχ + L χ (Ci − Cχ )] , = Ai + ΔAi − Aχ − L χ (Ci − Cχ ) Ai + ΔAi − L χ Ci   0 I = n×n , E 1,1 = n×n . In×n In×n 

Ai,χ E 0,1

Now, we are in a position to analyze the augmented system (5.8) via the Lyapunov stability theory. Theorem 5.1 Consider the MJDS (5.1) with the switching function (5.5) under the DAs (5.2). The augmented system (5.8) is stochastically stable if there exist matrices

98

5 Hidden Markov Model-Based Control

Pi > 0, Q > 0, R > 0 and scalars λi (i ∈ S ) satisfying T ˘ Pi E 0,1 B ≤ λi I, B T E 0,1 Ωiχ < 0,

(5.9) (5.10)

where 

 (1) (2) (3) , P˘i = , Ωiχ , Ωiχ δil Pl , Ωiχ = diag Ωiχ l∈S (1) Ωiχ (2) Ωiχ (3) Ωiχ

= (1 + =

T ˘ γ¯χ )Ai,χ Pi Ai,χ

2γ¯χ ϑ¯ χ λi E 1,1 M1T

+ ϑd1 Q + (1 − ϑ)d2 R − Pi ,

T M1 E 1,1 − ϑ Q,

T = 2γ¯χ (1 − ϑ¯ χ )λi E 1,1 M2T M2 E 1,1 − (1 − ϑ)R, 1  1  E{ϑχ,k }, 1 − θ = E{1 − ϑχ,k }, θ = H H χ∈H

χ∈H

d j = τ M j − τm j + 1, ( j = 1, 2). Proof To start with, we analyze the stochastic stability of the system (5.8). For convenience, set τm1 = τm , τ M1 = τ , τm2 = τ + 1 and τ M2 = τ M . The LK functional is as follows: Vk =

3 j=1

Vk, j ,

(5.11)

where Vk,1 = υkT P(δk )υk ,   k−1 Vk,2 = ϑ υlT Qυl + k−1  

υlT Rυl +

P(δk ) =

υsT

 Qυs ,

Pδ(2) } k

−τ m2 

k−1 

 υsT Rυs ,

l=−τ M2 +1 s=k+l

l=k−τk,2

diag{Pδ(1) , k

k−1 

l=−τ M1 +1 s=k+l

l=k−τk,1

Vk,3 = (1 − ϑ)

−τ m1 

 Pδk .

Calculate the forward difference of Vk,1 , Vk,2 along the system (5.8): E{ΔVk,1 } T = E{υk+1 P(δk+1 )υk+1 − υkT P(δk )υk }

 T ˘ ≤ E ψiχ [(1 + γ¯χ )υkT Ai,χ Pi Ai,χ υk + 2γ¯χ ϑ¯ χ χ∈H

5.3 Design of Common Sliding Surface

99

T T ˘ T ×g1T (E 1,1 υk−τk,1 )B T E 0,1 υk−τk,1 ) Pi E 0,1 Bg1 (E 1,1 T T T T +2γ¯χ (1 − ϑ¯ χ )g2 (E 1,1 υk−τk,2 )B E 0,1 P˘i E 0,1 B  T υk−τk,2 )] − υkT Pi υk , ×g2 (E 1,1

E{ΔVk,2 }  k  = ϑE l=k+1−τk+1,1

+

−τ m1 

k−1 

υlT Qυl − k 

υlT Qυl

l=k−τk,1

υsT

Qυs −

l=−τ M1 +1 s=k+1+l

(5.12)

−τ m1 

k−1 

 υsT

Qυs

l=−τ M1 +1 s=k+l

  T T ≤ ϑE d1 υk Qυk − υk−τk,1 Qυk−τk,1 .

(5.13)

Similarly, one has T E{ΔVk,3 } ≤ (1 − ϑ)E{d2 υkT Rυk − υk−τ Rυk−τk,2 }, k,2

(5.14)

where d j = τ M j − τm j + 1 ( j = 1, 2). From the inequalities (5.12)–(5.14) and the T ˘ conditions B T E 0,1 Pi E 0,1 B ≤ λi I (i ∈ S ), it can be derived that 

 ψiχ ξkT Ωiχ ξk , E{ΔVk } ≤ E

(5.15)

χ∈H



T T T υk−τ where ξk = υkT υk−τ and Ωiχ (i ∈ S , χ ∈ H ) have been defined in k,1 k,2 Theorem 5.1. Letting Ωiχ < 0, we have E{ΔVk } < 0 for ξk = 0. Next, we will prove the stochastic stability of the system (5.8). It follows from (5.15) that 



E{ΔVk } ≤ −λE ξkT ξk ≤ −λE υkT υk ,

(5.16)

where λ = mini∈S ,χ∈H {λmin (−Ωiχ )}. Then, it holds T υk−1 }, E{Vk − Vk−1 } ≤ −λE{υk−1 T E{Vk−1 − Vk−2 } ≤ −λE{υk−2 υk−2 },

... E{V1 − V0 } ≤ −λE{υ0T υ0 }. Summing up on both sides above, it yields   k−1 T E{Vk − V0 } ≤ −λE υl υl , l=0

(5.17)

100

5 Hidden Markov Model-Based Control

i.e.,   k−1 E{V0 } − E{Vk } E{V0 } ≤ < ∞. υlT υl ≤ E λ λ l=0

(5.18)

Thus, letting k → ∞, naturally one has   ∞ 2 E  υl  | υ0 , δ0 < ∞.

(5.19)

l=0

According to the Definition 5.1, the augmented system (5.8) is stochastically stable. The proof of Theorem is complete. Note that there exist uncertainty and nonlinearity in Theorem 5.1. Hence, we give a couple of sufficient criteria below for stochastic stability of the system (5.8) in terms of strict linear matrix inequalities (S-LMIs). Theorem 5.2 Consider the MJDS (5.1) with the switching function (5.5) under the DAs (5.2). The augmented system (5.8) is stochastically stable if for given εi > 0 there exist matrices Pi > 0, Q > 0, R > 0 and scalars λi , i , κiχ > 0 (i ∈ S , χ ∈ H ) satisfying the S-LMIs criteria: ⎤ (1) Ω¯ iχ ∗ ∗ ∗ ∗ ⎢ (2) 0 Ωiχ ∗ ∗ ∗ ⎥ ⎥ ⎢ ⎥ ⎢ (3) ⎢ 0 0 Ωiχ ∗ ∗ ⎥ < 0, ⎥ ⎢ ⎣ 1 + γ¯χ εi A˘iχ 0 0  −εi I ∗ ⎦ 0 0 0 1 + γ¯χ εi Hi T −κiχ I ⎡

⎡

⎤ 2 P˘i(1) − i I ∗ ∗ ⎣ ⎦ < 0, ∗ 0 P˘i(2) − i I 0 −2εi W B 2εi W T T ˘ B E 0,1 Pi E 0,1 B ≤ λi I, P˘i(1)

≤

εi Pi(1) , 0

where (1) Ω¯ iχ = ϑd1 Q + (1 − ϑ)d2 R − Pi + κiχ Ni T Ni ,  P˘i = δil Pl = diag{ P˘i(1) , P˘i(2) }, l∈S

A˘iχ =



L χ Ci Aχ + L χ (Ci − Cχ ) , Ai − Aχ − L χ (Ci − Cχ ) Ai − L χ Ci

(5.20) (5.21) (5.22) (5.23)

5.3 Design of Common Sliding Surface

 Hi =

101

 I 0 0 0 , Ni = , 0 Hi Ni Ni

other matrices or parameters can be checked in Theorem 5.1. Proof To facilitate subsequent simulations, we need to handle the items (1 + (1) T ˘ γ¯χ )Ai,χ (i ∈ S , χ ∈ H ). Given that Ai,χ can be rewritten as Pi Ai,χ in Ωiχ 

Ai,χ =

W˘ 0 0 I



L χ Ci Aχ + L χ (Ci − Cχ ) , Ai + ΔAi − Aχ − L χ (Ci − Cχ ) Ai + ΔAi − L χ Ci

it leads to T ˘ Pi Ai,χ Ai,χ

where A˜ iχ =



T = A˜ iχ



W˘ 0 0 I

T

P˘i



W˘ 0 ˜ Aiχ , 0 I

L χ Ci Aχ + L χ (Ci − Cχ ) Ai + ΔAi − Aχ − L χ (Ci − Cχ ) Ai + ΔAi − L χ Ci

(5.24) .

From the formulas (5.21), (5.23) and (5.24), together with the Schur complement, it is clear that Ωiχ < 0 (i ∈ S , χ ∈ H ) hold if the following criteria are true: ⎡ ⎢ ⎢ ⎢ ⎣

∗ ∗ ∗ ϑd1 Q + (1 − ϑ)d2 R − Pi (2) ∗ ∗ 0 Ωiχ (3) 0 0 Ωiχ ∗  1 + γ¯χ εi (A˘iχ + Hi Fk Ni ) 0 0 −εi I

⎤ ⎥ ⎥ ⎥ 0, l2 > 0 being weighting factors about the stochastic DAs case. Besides, we set l1 > l2 in this work. Proof To start with, a newly modified reaching condition is given by: sk+1 = εk,1 sk − εk,2 T˘k,s sgn(sk ), where εk,1 = 1 − q T˘k,s and εk,2 → 0 (k → ∞). It can be pointed out that the stochastic DAs case is taken into account by introducing βk when designing a newly modified reaching condition. Especially, the time-varying sampling period T˘k,s is constructed, whose value relies upon the attack. Referring to [24], we can easily know that the switching function sk, j ( j = 1, 2, . . . , m) will eventually enter the region {sk, j | |sk, j | < ε2,k T˘k,s }. Letting k → ∞, further, we have sk+1 → 0. Thus, a reaching condition-based sliding mode controller (5.27) can drive the system from any initial condition to approach the sliding surface sk = 0. Remark 5.3 In the proposed scheme, it is clear that the sampling period is not 1 constant. The time-varying T˘k,s = 1+β Ts below (5.27) adapts to the following ideas: k (i) the faster sampling is expected to be reached for the security and stability of the

5.5 Simulation

103

system (5.1) once the DAs occurs; and (ii) T˘k,s will degenerate to a traditional timeinvariant sampling in order to save network resources when there is no attack. Remark 5.4 For a few reasons (i.e., finite sampling and uncertainty), the closedloop systems under discrete SMC are not identical to those under continuous-time setting. Because they do not actually exist an ideal sliding mode phase, but rather a alleged quasi-sliding mode phase [21, 24]. In other words, the closed-loop system under the designed controller (5.27) will come into a vicinity of the sliding surface, which can be checked from the proof of reachability below (5.27). Additionally, it should be noted that a compromise needs to be made on the basis of actual precision and required time during sk → 0.

5.5 Simulation This section will provide an example to prove the feasibility of our proposal. To start with, choose the following parameters for considered networked system (5.1) under random DAs with probabilistic delay: for rk = 1, ⎛⎡ xk+1

= ⎝⎣

⎤ ⎡ ⎤ −1.4120 0.1946 0 0.14 ⎦ ⎣ 0 0.1318 1.0125 + 0.2 ⎦ sin(0.2k) 0 0 −0.15 0.17 ⎡ ⎤ 0.08 −0.017

 × 0.2 0.1 0 xk + ⎣ 0.2 0.3 ⎦ 0.13 0.26   × u χ (x(k), k) + γχ ,k μχ (x(k − τk ), k) ;

for rk = 2, ⎛⎡

xk+1

⎤ ⎡ ⎤ 0.0866 0.5830 0 −0.04 = ⎝⎣ −1.3187 0.4914 0 ⎦ + ⎣ 0.13 ⎦ sin(0.2k) 0 0 −0.15 0.10 ⎡ ⎤ 0.08 −0.017

 ⎣ × 0.13 0 0.21 xk + 0.2 0.3 ⎦ 0.13 0.26   × u χ (x(k), k) + γχ ,k μχ (x(k − τk ), k) .

 Moreover, by setting Π =

 0.7 0.3 0.6 0.4 and Ψ = , together with 0.6 0.4 0.3 0.7

g1 (xk−τk,1 )= tanh(−0.5x2 (k − τk,1 )) tanh(0.8x1 (k − τk,1 )) ,

g2 (xk−τk,2 )= tanh(0.1x3 (k − τk,2 )) tanh(0.2x1 (k − τk,2 )) ,

104 Fig. 5.1 Markovian jump parameter δk

5 Hidden Markov Model-Based Control 2

δk

1.9 1.8 1.7

3

1.6

2

1.5

1

1.4

0 30

40

35

1.3 1.2 1.1 1

0

10

20

30

40

50

60

70

80

90

100

time (k)

Fig. 5.2 Trajectories of state xk in case of u k = 0

50

x1,k x2,k x3,k

40 30 20 10 0 −10 −20 −30 −40 −50

5

10

15

20

25

30

35

40

45

50

time (k)

we can solve the attack-resistant SMC issue in this paper and determine the sliding surface below:  3.8551 10.1572 6.7059 sk = xˆ = 0. −0.6032 15.2473 13.2611 k In the simulation, letting initial value x0 = [0.1 − 0.3 2]T and γ¯1 = 0.8 and γ¯2 = 0.3, the HMM-based sliding mode controller (5.27) with all known parameters can be implemented for the simulation. First of all, observing Figs. 5.1 and 5.2, we would see the evolution of Markovian jump parameter δk as well as the instability of the system (5.1) under condition u k = 0. This means that the original system is divergent once the DAs occurs before the attack-resistant controller can be implemented. Then,

5.5 Simulation Fig. 5.3 Trajectories of state xk when using the controller (5.27)

105 6

x1,k x2,k x3,k

4

2

0

−2

−4

−6

20

10

100

90

80

70

60

50

40

30

time (k)

Fig. 5.4 The proposed sliding surface sk = 0 in 3-D space

20

10

0

0

−20

−10

10 −10

Fig. 5.5 Evolutions of the system state xk in 3-D space and switching function sk when using the controller (5.27)

10

10

0

0

0

−10

−10

−10

10

0

20

s1k s2k

5 15 4

0

10 2

5

−5 2

0

2 0

0 −2 −2

10

5

15

20

0 20

40

60

80

100

time (k)

applying the controller (5.27), the trajectory of the system (5.1) can be seen from Fig. 5.3, which indicates the achievement of required performance via our proposed scheme. Next the sliding surface sk = 0 is drawn in Fig. 5.4 with the switching function sk being plotted in Fig. 5.5. From the above simulations, it can be concluded that the new asynchronous SMC scheme works well for the system suffering random DAs with probabilistic delay.

106

5 Hidden Markov Model-Based Control

5.6 Conclusion The HMM-based SMC problem has been considered for a class of MJDSs under random DAs with probabilistic delay. A common and mode-dependent sliding surface has been proposed in the estimation space. Based on the SMC and Lyapunov method, a few strict LMIs-based criteria have been proposed to ensure the stochastic stability of the augmented system. In addition, a desirable asynchronous controller has also been provided to overcome the uncertainties and the attacks of the system on the strength of a new reaching condition. One of the prospective research topics in future work is to extend this work to cases of limited communication. At last, a numerical example has been given to evaluate the effectiveness of the proposed strategy.

References 1. M. V. Alves, R. J. Barcelos, L. K. Carvalho, et., Robust decentralized diagnosability of networked discrete event systems against DoS and deception attacks, Nonlinear Analysis: Hybrid Systems, https://doi.org/10.1016/j.nahs.2022.101162. 2. R. Meira-Góes, S. Lafortune, H. Marchand, Synthesis of supervisors robust against sensor deception attacks, IEEE Transactions on Automatic Control, 66(10) (2021) 4990–4997. 3. R. Meira-Góes, E. Kang, R. H. Kwong, et., Synthesis of sensor deception attacks at the supervisory layer of cyber-physical systems, Automatica, https://doi.org/10.1016/j.automatica.2020. 109172. 4. E. Mousavinejad, X. Ge, Q. L. Han, et., Resilient tracking control of networked control systems under cyber attacks, IEEE Transactions on Cybernetics, 51(4) (2019) 2107–2119. 5. A. Kazemy, R. Saravanakumar, J. Lam, Master-slave synchronization of neural networks subject to mixed-type communication attacks, Information Sciences, 560 (2021) 20–34. 6. N. Babadi, A. Doustmohammadi, A moving target defence approach for detecting deception attacks on cyber-physical systems, Computers and Electrical Engineering, https://doi.org/10. 1016/j.compeleceng.2022.107931. 7. K. Bansal, P. Mukhija, Aperiodic sampled-data control of distributed networked control systems under stochastic cyber-attacks, IEEE/CAA Journal of Automatica Sinica, 7(4) (2020) 1064– 1073. 8. A. H. Tahoun, M. Arafa, Secure control design for nonlinear cyber-physical systems under DoS, replay, and deception cyber-attacks with multiple transmission channels, ISA Transactions, 128 (2022) 294–308. 9. M. M. Hamdan, M. S. Mahmoud, U. A. Baroudi, Event-triggering control scheme for discrete time cyberphysical systems in the presence of simultaneous hybrid stochastic attacks, ISA Transactions, 122 (2022) 1–12. 10. M. A. Sid, S. Chitraganti, K. Chabir, Medium access scheduling for input reconstruction under deception attacks, Journal of the Franklin Institute, 354 (9) (2017) 3678–3689. 11. L. Zhang, B. Cai, Y. Shi, Stabilization of hidden semi-Markov jump systems: emission probability approach, Automatica, 101 (2019) 87–95. 12. M. Das, C. Mahanta, Optimal second order sliding mode control for nonlinear uncertain systems, ISA Transactions, 53 (4) (2014) 1191–1198. 13. P. Zhang, Y. Kao, J. Hu, B. Niu, Robust observer-based sliding mode H∞ control for stochastic Markovian jump systems subject to packet losses, Automatica, https://doi.org/10.1016/j. automatica.2021.109665.

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14. J. Hu, P. Zhang, Y. Kao, et., Sliding mode control for Markovian jump repeated scalar nonlinear systems with packet dropouts: The uncertain occurrence probabilities case, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124574. 15. W. Qi, C. Lv, J. H. Park, et., SMC for semi-Markov jump cyber-physical systems subject to randomly occurring deception attacks, IEEE Transactions on Circuits and Systems II: Express Briefs, 69 (1) (2022) 159–163. 16. J. Cao, J. Zhou, J. Chen, et., Sliding mode control for discrete-time systems with randomly occurring uncertainties and nonlinearities under hybrid cyber attacks, Circuits, Systems, and Signal Processing, 40 (12) (2021) 5864–5885. 17. Y. Barzegari, J. Zarei, R. Razavi-Far, et., Resilient consensus control design for DC microgrids against false data injection attacks using a distributed bank of sliding mode observers, Sensors, 22 (7) (2022) 1–16. 18. J. Li, D. F. Yang, Y. C. Gao, et., An adaptive sliding-mode resilient control strategy in smart grid under mixed attacks, IET Control Theory & Applications, 15 (15) (2021) 1971–1986. 19. P. Zhang, Y. Kao, J. Hu, et., Finite-time observer-based sliding-mode control for Markovian jump systems with switching chain: average dwell-time method, IEEE Transactions on Cybernetics, 53(1) (2023) 248–261. 20. Q. Ren, Y. Kao, C. Wang, et., New results on the generalized discrete reaching law with positive or negative decay factors, IEEE Transactions on Automatic Control, 67 (2) (2022) 1046–1052. 21. A. Bartoszewicz, Discrete-time quasi-sliding-mode control strategies, IEEE Transactions on Industrial Electronics, 45 (4) (1998) 633–637. 22. Z. G. Wu, P. Shi, Z. Shu, et., Passivity-based asynchronous control for Markov jump systems, IEEE Transactions on Automatic Control, 62 (4) (2017) 2020–2025. 23. M. Xu, L. Ma, H. Ma, et., Sliding mode output feedback controller design of discrete-time Markov jump system based on hidden Markov model approach, IEEE Access, 9 (2021) 101089– 101096. 24. J. Zhang, P. Shi, Y. Xia, et., Discrete-time sliding mode control with disturbance rejection, IEEE Transactions on Industrial Electronics, 66 (10) (2019) 7967–7975.

Part III

Markov Jump Systems with Actuator Fault

Chapter 6

Sliding Mode Fault-Tolerant Control

In this paper, the finite-time sliding mode control (SMC) issue is discussed for stochastic Markovian jump discrete systems (MJDSs) subject to actuator failure, probabilistic delay, and external disturbance. The phenomenon of actuator failure is characterized by utilizing an effectiveness factor and a stuck fault function. Our aim is to propose a new finite-time sliding mode fault-tolerant control (SMFTC) strategy for the corresponding closed-loop system. Additionally, based upon the discrete partitioning strategy, the desired sliding mode controller is provided to drive the system from an initial state into the neighborhood of the prescribed sliding surface. Finally, the simulation examples are given to illustrate the effectiveness of the control scheme obtained.

6.1 Introduction The Markovian jump systems (MJSs) have received growing concerns due to their effective characterization of practical systems over the past years [1–3]. The key feature of MJSs lies in the switching between multiple modes via a given Markov chain, which is much different from single dynamic systems. Accordingly, many interesting methods have been proposed for MJSs according to specific requirements, such as a restricted controller-based stabilization scheme in [4]. To conserve network resources, an event-driven rule has been introduced into the design process of the control scheme, and two kinds of observer-based/state-feedback event-triggered schemes have already been designed in [5] for the purpose of transferring the system data intermittently. It is worthwhile to mention that the information on switching probability between multiple modes may not be available exactly in most cases. For instance, the H∞ filtering problem has been discussed in [6] for Markov jump linear systems (MJLSs) with general unknown transition probabilities, where the transition probabilities of the jumping are assumed to be partly available. The stochastic stability and stabilization have been investigated in [7, 8] for a class of dynamic systems with Markovian jumping parameters based on imprecise transition probability, © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Kao et al., Analysis and Design of Markov Jump Discrete Systems, Studies in Systems, Decision and Control 499, https://doi.org/10.1007/978-981-99-5748-4_6

111

112

6 Sliding Mode Fault-Tolerant Control

where sufficient stability conditions have been proposed in [7] under continuous and discrete-time settings, and a control law against nonlinearity and actuator fault has been given in [8] via the siding mode strategy. As is known to all, the approach of sliding mode control (SMC) is usually adopted to handle the nonlinearity for controlled systems [9–11]. There have existed some works of SMC method under the condition of actuator faults [8, 12–14]. Among them, an adaptive SMC design has been discussed in [14] for stochastic MJSs, where some potential issues about actuator and sensor (e.g., degradation or fault) have been simultaneously taken into account. In fact, the actuator failures are quite common in practical projects, such as AC transmission lines, distribution networks and power swing processes [15–17]. Accordingly, some important algorithms on fault-tolerant aim have been proposed to improve the effectiveness of practical dynamics. For example, the method of detection and isolation with actuator fault has been researched in [18] for helicopter unmanned arial vehicle. In the case of actuator gain loss fault or stuck fault, a novel fault-based observer has been designed in [19] for hypersonic vehicles. Besides, a fault-tolerant observer has also been proposed in order to cope with a sensor fault. In [20], a robust adaptive SMC scheme with actuator fault has been provided for stochastic MJSs of the Itoˆ form, where the sliding function has been designed based on the solution to the original equation. Moreover, the unknown actuator degradation parameter ρ has been estimated later by proposing an adaptive law. To our best knowledge, few achievements have been reported on the robust SMC approach for discrete-time MJSs with actuator fault because of certain difficulty, regardless of the extra probabilistic delay. On the other hand, the discrete robust faulttolerant control algorithm has been given in [21] for uncertain discrete MJLSs when it occurs actuator failures in the more general form. This motivates us to investigate the discrete SMC problem subject to probabilistic delay and actuator failure. Recently, the problems of finite-time stability and finite-time synthesis have attracted great attention due to their certain advantages [22–24]. For example, stability analysis and finite-time stabilization have been discussed in [25] for linear systems by a bounded feedback control idea. Based on the average dwell time strategy, the asynchronous controller has been designed in [26] for stochastic hybrid systems with faulty signals to achieve the goal of finite-time boundedness during the reaching phase and sliding motion phase. The finite-time adaptive control framework has been given in [27] for the continuous-time systems with unknown bounded nonlinearity via the aid of a functional observer. The aforesaid results in finite-time mainly serve for the continuous system and are not utilized to the digital discrete dynamic system directly. However, the discrete finite-time algorithm remains to be further explored. This means that the finite-time SMC problem with actuator failure is still open for stochastic MJDSs in the presence of probabilistic delay. Motivated by the above discussion, the sliding mode fault-tolerant control (SMFTC) design strategy is explored for the addressed MJDSs with probabilistic delay in this paper. Firstly, the probabilistic delay is depicted via a random variable of Bernoulli distribution. The phenomenon of actuator failure is also characterized by an effectiveness factor and a stuck fault function. By proposing a fault-tolerant controller, we can derive the corresponding closed-loop system according to the orig-

6.2 Problem Statements and Preliminaries

113

inal system. Moreover, it can be shown that the aforesaid derived system is stochastic finite-time bounded by sufficient criteria obtained based on the Lyapunov method. The challenges during this investigation mainly lie in the following aspects. (i) How to combine a partitioning strategy primarily used in continuous-time and the quasisliding mode? (ii) Which kind of sliding mode fault-tolerant controller is better to select according to a discrete partitioning strategy? (iii) How to determine the relative factors about the selected controller to ensure the reachability of quasi-sliding mode before a prescribed time T f . In the end, the numerical simulation is conducted to show the usefulness of our sliding mode fault-tolerant scheme. The main contributions of our work are summarized as follows: (i) attempting to incorporate a sliding-surface neighborhood idea into a partitioning strategy; (ii) making a great effort to design a novel fault-tolerant scheme for the reachability of quasi-sliding mode region by a specified time; and (iii) examining the effectiveness of designed scheme against actuator failure and probabilistic delay and applying it to the automotive throttle system. Comparing with the latest results [28, 29], it can be shown that, using a partitioning strategy, the H∞ performance/input-output stability of continuous-time dynamic systems has been explored in a given interval. Similarly, the relative contribution on discrete-time dynamic systems has been given in [30]. However, it is well known that the quasi-sliding mode tends to be reached in a discrete case instead of the ideal one. This makes it impossible to directly use the above methods to deal with the stability of discrete systems. In this paper, a sliding-surface neighborhood idea is really attempted to solve this bottleneck with help of a discrete partitioning strategy. Especially, by taking T ∗ from the partitioning strategy into the SMC design, the operational phases of the considered dynamics (i.e., reaching phase and quasi-sliding mode phase) can thus be well defined in a finite-time interval.

6.2 Problem Statements and Preliminaries The discrete-time Markovian process is defined as rk and its value range is denoted as S = {1, 2, . . . , N }. Besides, it is noted that the probability transition matrix can be expressed by the following form: ⎡

π11 π12 ⎢ π21 π22 ⎢ =⎢ . .. ⎣ .. . πN 1 πN 2

⎤ . . . π1N . . . π2N ⎥ ⎥ .. .. ⎥ , . . ⎦ . . . πN N

 where Nj=1 πi j = 1 (i ∈ S ) and πi j = Prob(rk+1 = j | rk = i). Consider the following class of discrete delayed MJSs subject to actuator failure and external disturbance:

114

6 Sliding Mode Fault-Tolerant Control

xk+1 = A(rk )xk + Ad (rk )xk−dk + B(ρu k + f a,k ) + E ω (rk )ωk , where xk = [xk,1 , xk,2 , . . . , xk,n ]T ∈ Rn is the state vector of system; u k = [u k,1 , u k,2 , . . . , u k,m ]T ∈ Rm is the control input; ωk ∈ Rq is the exogenous disturbance. ρ = diag{ρ1 , ρ2 , . . . , ρm } stands for the effectiveness factor, ρi (i ∈ {1, 2, . . . , m}) may be unknown but bounded, i.e., 0 < ρ i ≤ ρi ≤ ρ i ≤ 1 with scalars ρ i and ρ i being known bounds. f a,k ∈ Rm is the stuck actuator fault and it satisfies  f a,k ≤ ς . For simplicity, Ai  A(rk ), Adi  Ad (rk ) and E ωi  E ω (rk ) (rk = i ∈ S ) are known real matrices. The stochastic variable θk obeying Bernoulli distribution is utilized to model the probabilistic delays: θk =

1, 0,

dk,1  dk ∈ {dm , dm + 1, dm + 2, . . . , d} dk,2  dk ∈ {d + 1, d + 2, . . . , d M − 1, d M }

where E(θk = 1) = θ and E(θk = 0) = 1 − θ . Besides, the time-delay values dm , d M and d are assumed to be known. As a result, the considered system can be briefly presented with the following form:

 xk+1 = Ai xk + Adi θk xk−dk,1 + (1 − θk )xk−dk,2 + B(ρu k + f a,k ) + E ωi ωk . (6.1) Furthermore, the actuator faults are taken into account in this paper, i.e., effectiveness loss faults and actuator stuck faults. The relationship between two types of faults can be illustrated as follows: when f a,k = 0, the stuck fault of actuator occurs; when f a,k = 0, the no stuck fault occurs but other faults may exist right now (i.e., ρi = 1, the ith actuator has partial even complete effectiveness loss faults; ρi = 1, the ith actuator is normal). Please see the following table for details. Here, the signal  denotes the partial effectiveness loss of the ith actuator (i ∈ {1, 2, . . . , m}) (Table 6.1). Assumption 6.1 The exogenous disturbance ωk ∈ Rq belongs to space l2 [0, T f ], i.e., W[0,T f ],γ  {ωkT ωk < γ 2 , ∀k ∈ [0, T f ]}. Remark 6.1 It is noted that the usual assumption (i.e., the input matrix B has full column rank) is not made in our paper. This means rank{B} = l ≤ m is allowed in our work. To a certain extent, this can reduce the conservatism of the method. In order to get the weak-conservative approach, the full rank decomposition of B is carried out as

Table 6.1 Parameters-based failure analysis Parameter f a,k  =0 0 Parameter ρi Fault Situations

– Stuck fault

0 Outage

0

0

(0,1) 

1 Normal

6.3 Design of Sliding Surface and Reachability Analysis

115

Fig. 6.1 The block diagram of SMFTC scheme

in [13], i.e., B = (B1 )n×l (B2 )l×m with rank{B1 } = rank{B2 } = l.On the other hand, this paper assumes that the jump system has a common control channel. Because whether the system input matrix is mode-dependent or not is no longer a major technical issue during the design of the SMC in this article. In fact, in virtue of [31], the mode-dependent input matrices can be transformed into a mode-independent matrix for the scheme design. Our aim of paper is to propose a new fault-tolerant SMC approach such that, for simultaneous presence of Markov jumping parameters, probabilistic delays and actuator failure, the controlled system under consideration is stochastically finitetime bounded on a specified limited interval (Fig. 6.1). Definition 6.1 [32] For a given length (of time) T f , scalars c1 and c2 (0 < c1 < c2 ), a positive-definite weighted matrix R > 0. The system (6.1) with u k is said to be stochastically finite-time bounded with regard to (wrt) (c1 , c2 , [0, T f ], R, W[0,T f ],γ ) if it holds supd M ≤κ≤0 E{xκT Rxκ } ≤ c1 ⇒ xkT Rxk ≤ c2 , ∀k ∈ [0, T f ].

(6.2)

Definition 6.2 [30] (A Discrete-time Partitioning Strategy) For the considered system (6.1), it is taken to be stochastically finite-time bounded (SFTB) wrt (c1 , c2 , [0, T f ], R, W[0,T f ],γ ) if it is SFTB wrt (c1 , c∗ , [0, T ∗ ], R, W[0,T f ],γ ) within the reaching phase and SFTB wrt (c∗ , c2 , [T ∗ , T f ], R, W[0,T f ],γ ) within the sliding motion phase, where c∗ ∈ (c1 , c2 ) is an auxiliary intermediate variable.

6.3 Design of Sliding Surface and Reachability Analysis To begin with, the common state-dependent sliding surface is constructed for (6.1) with the following form: sk = Gxk ,

(6.3)

116

6 Sliding Mode Fault-Tolerant Control

where matrix G = B1T P is a matrix to be determined such that G B1 > 0. As in [13], the controller in this paper is designed as uk , u k = −(B2 ρ B2T )−1 [σk + δsgn(sk )], u k = B2T

(6.4)

where λm ((B1T P B1 )−1 ) s0 2 , T f δ2 = ς B2  + γ (B1T P B1 )−1 G E ωi , x˜k = Ai xk + Adi [θ¯ xk−dk,1 + (1 − θ¯ )xk−dk,2 ],

σk = (B1T P B1 )−1 G x˜k , δ = δ1 + δ2 , δ1 >

From (6.1), (6.3), (6.4), it leads to sk+1 = G{Ai xk + Adi [θk xk−dk,1 + (1 − θk )xk−dk,2 ]} + G Bρu k + G B f a,k + G E ωi ωk (6.5) In the sequel, we are in a position to verify the reachability of prescribed sliding surface (6.3) for the system (6.1). Theorem 6.1 Considering the MJS (6.1) with the prescribed sliding surface (6.3), the above designed sliding mode fault-tolerant controller (6.4) can drive the system into a neighborhood of (6.3) and afterwards keep it there, where P > 0 is a feasible solution to following specified LMIs-based problem. Proof Choose the Lyapunov function as Vks =

1 T T s (B P B1 )−1 sk . 2 k 1

(6.6)

Then, taking the difference of Vks , we have s E{Vks } = E{Vk+1 − Vks }

= E{k + skT (B1T P B1 )−1 sk }

 = E{k + skT (B1T P B1 )−1 [G Ai xk + G Adi θk xk−dk,1 + (1 − θk )xk−dk,2 + G Bρu k +G B f a,k + G E ωi ωk − sk ]}

= E{k + skT (B1T P B1 )−1 [G x˜k + G B(ρu k + f a,k ) + G E ωi ωk − sk ]} = E{k + skT (B1T P B1 )−1 (G B f a,k + G E ωi ωk ) − δsk 1 − skT (B1T P B1 )−1 sk } ≤ k − δ1 sk ,

(6.7)

with scalar δ1 being defined below (6.4) and k = skT (B1T P B1 )−1 sk . By setting sufficiently large δ1 , it leads to E{Vks } ≤ k − δ1 sk  < 0 for sk = 0. As in [30], the reachability of sliding surface (6.3) is guaranteed in spite of the double effects from actuator failure and external disturbance onto the original controlled system (6.1). Next, we solve the instant T ∗ ∈ (0, T f ) satisfying sk  >  (small enough ) for k ∈ {0, 1, 2, . . . , T ∗ − 1} and sk  ≤  for k ∈ {T ∗ , T ∗ + 1, . . . , T f }.

6.4 Stochastic Finite-Time Boundness of Closed-Loop System

117

From (6.7), we can get ∗ T −1

k=0

δ1 ≤

∗ T −1

k=0

∗

T −1 E{Vks } E{skT (B1T P B1 )−1 (sk − sk+1 )} k − = . (6.8) sk  sk  sk  k=0

On the other hand, it can be shown from skT (B1T P B1 )−1 sk ≥ 0 that ∗ T −1

k=0

δ1 ≤

∗ T −1

k=0

T E{sk+1 (B1T P B1 )−1 (sk − sk+1 )} sk 

(6.9)

Adding the two ends of inequality (6.8) and inequality (6.9) respectively, one has 2T ∗ δ1 ≤

∗ T −1

k=0

T E{skT (B1T P B1 )−1 sk − sk+1 (B1T P B1 )−1 sk+1 } 2V0s ≤ . (6.10) sk  

Recalling the definition of δ1 below (6.4), we can conclude that T ∗ < T f . Remark 6.2 Observing the controller (4), it can be easily found that δ1 is associated with the initial value x0 and time bound T f . To ensure the effectiveness of this parameter-related controller, the parameter δ1 need to be appropriately selected, λ ((B T P B )−1 ) which is at least greater than m 1T f 1 s0 2 . Further analysis shows that a larger δ1 needs to be set in the case where x0 is large and T f is small. This is also reasonable from a practical point of view, namely, a larger gain/control force should be matched to balance a larger initial value to meet the performance requirements in a relatively short time. Conversely, then δ1 is not strictly defined.

6.4 Stochastic Finite-Time Boundness of Closed-Loop System To proceed, we will investigate the stochastically finite-time boundedness for the system (6.1) by implementing designed controller (6.4). Firstly, by substituting u k in (6.4) into the system (6.1), we can acquire the following closed-loop system: xk+1 = [I − B1 (B1T P B1 )−1 G]Ai xk + [θk I − θ¯ B1 (B1T P B1 )−1 G]Adi xk−dk,1 +[(1 − θk )I − (1 − θ¯ )B1 (B1T P B1 )−1 G]Adi xk−dk,2 +(B f a,k + E ωi ωk ) − δ B1 sgn(sk ).

(6.11)

Sequentially, the conditions on stochastically finite-time boundedness will be discussed for (6.11) respectively wrt (c1 , c∗ , [0, T ∗ ], R, W[0,T f ],γ ) within the reaching phase and wrt (c∗ , c2 , [T ∗ , T f ], R, W[0,T f ],γ ) within the sliding motion phase.

118

6 Sliding Mode Fault-Tolerant Control

Theorem 6.2 For the MJS (6.1) subject to the designed fault-tolerant controller (6.4), the resultant closed-loop system (6.11) is stochastically finite-time bounded wrt to (c1 , c∗ , [0, T ∗ − 1], R, W[0,T f ],γ ) if for given μ > 1, ι < 1 and λ > 1, there exist positive definite matrices P > 0, Q j > 0, R j > 0, X > 0, Y > 0, real matrices S , T , positive scalars c∗ > 0 and αδ > 0 satisfying the following conditions: ⎡

˜ ∗ +  ⎢ ∗ ⎢ ⎣ ∗ ∗ ∗

μT

∗

−1

3 c1 +

⎤ R T dS d M T −P 0 0 ⎥ ⎥ < 0, 0 ⎦ ∗ −d R1 ∗ ∗ −d M R2

(6.12)

P < λP, B1T P B1 = I,

(6.13)

h < λmin (R − 2 P R − 2 )c∗ ,

(6.14)

c1 < c∗ < c2 ,

(6.15)

T −2

μ(1 − μ 1−μ

)

1

1

where ⎡ ∗ 1 ⎢∗ ⎢ ⎢∗ ∗ = ⎢ ⎢∗ ⎢ ⎣∗ ∗

0 ∗2 ∗ ∗ ∗ ∗

⎤ 0 (d R1 + d M R2 )B1 −(d R1 + d M R2 )B −(d R1 + d M R2 )E ωi ⎥ 0 0 0 0 ⎥ ⎥ ∗3 0 0 0 ⎥, T T T ⎥ −B1 P B −B1 P E ωi ∗ 3B1 P B1 − αδ I ⎥ T T ⎦ ∗ ∗ 3B P B − Y B P E ωi T ∗ ∗ ∗ 3E ωi P E ωi − X

∗1 = 5λAiT [I − ιP]Ai + 1 Q 1 + 2 Q 2 + (d R1 + d M R2 ) − P, P = P + d R1 + d M R2 , T [2θλI ¯ − 2θ¯ 2 λιP]Adi + 4θλA ¯ T [I − ιP]Adi − Q 1 , h = αδ lδ 2 + λm (X )γ 2 + λm (Y )ς 2 , ∗2 = Adi di T [2(1 − θ)λI T ¯ ¯ 2 λιP]Adi + 4(1 − θ)λA ¯ ∗3 = Adi − 2(1 − θ) di [I − ιP]Adi − Q 2 ,

 R = d R1 + d M R2 0 0 0 0 0 , 1 = d − dm + 1, 2 = d M − d,

˜ = T (I − ιP) + S 1 + T S T + T 2 + T T T ,  1 2





 ¯ di 0 0 0 , 1 = I −I 0 0 0 0 , 2 = I 0 −I 0 0 0 ,  = Ai θ¯ Adi (1 − θ)A  1 1 1 1  dm + d − 1 3 = λm (R − 2 P R − 2 ) + λm (R − 2 Q 1 R − 2 ) d + (d − dm ) 2  1 1  d + dM (d M − d − 1) +λm (R − 2 Q 2 R − 2 ) d M + 2  1 1 1 1  +2 d(d + 1)λm (R − 2 R1 R − 2 ) + d M (d M + 1)λm (R − 2 R2 R − 2 ) .

Proof Firstly, we choose a simple Lyapunov-Krasovskii functional as follows: Vk = Vk,1 + Vk,2 + Vk,3 + Vk,4 ,

(6.16)

6.4 Stochastic Finite-Time Boundness of Closed-Loop System

119

where k−1 

Vk,1 = xkT P xk , Vk,2 =

x Tj Q 1 x j +

j=k−dk,1 k−1 

Vk,3 =

x Tj Q 2 x j +

Vk,4 =

k−1 

xlT Q 1 xl ,

j=−d+1 l=k+ j k−1 

xlT Q 2 xl ,

j=−d M +1 l=k+ j

j=k−dk,2 −1 

−d−1 

−dm 

k−1 

ηlT

R1 ηl +

j=−d l=k+ j

−1 k−1  

ηlT R2 ηl

j=−d M l=k+ j

with ηl = xl+1 − xl . By taking the difference of Vk along (6.11) yields E{Vk,1 |χk } = E{Vk+1,1 |χk } − Vk,1 T = xkT AiT G˜ T P G˜ Ai xk + xk−d

k,1

T ¯ Adi [θ I − θ¯ 2 G T (G B1 )−1 G]Adi xk−dk,1 + δ 2 sgn T (sk )G B1 sgn(sk )

T T ¯ 2 G T (G B1 )−1 G]Adi xk−dk,2 + f a,k −xkT P xk + xk−d A T [(1 − θ¯ )I − (1 − θ) B T P B f a,k k,2 di T ¯ kT AiT G˜ T P G˜ Adi xk−dk,2 +ωkT E ωi P E ωi ωk + 2θ¯ xkT AiT G˜ T P G˜ Adi xk−dk,1 + 2(1 − θ)x T +2xkT AiT G˜ T P B f a,k + 2xkT AiT G˜ T P E ωi ωk − 2δxkT AiT G˜ T P B1 sgn(sk ) + 2 f a,k B T P E ωi ωk T T T ¯ − θ)x ¯ k−d −2θ(1 A T G T (G B1 )−1 G Adi xk−dk,2 + 2θ¯ xk−d A T G˜ T P B f a,k − 2δωkT E ωi P B1 sgn(sk ) k,1 di k,1 di T T T ¯ k−d +2θ¯ xk−d A T G˜ T P E ωi ωk − 2δ θ¯ xk−d A T G˜ T P B1 sgn(sk ) + 2(1 − θ)x A T G˜ T P B f a,k k,1 di k,1 di k,2 di T T T ¯ k−d +2(1 − θ)x A T G˜ T P E ωi ωk − 2δ(1 − θ¯ )xk−d A T G˜ T P B1 sgn(sk ) − 2δ f a,k B T P B1 sgn(sk ) k,2 di k,2 di

(6.17) with χk = {xk−dk , xk−dk +1 , . . . , xk−1 , xk } and G˜ = I − B1 (G B1 )−1 G. E{Vk,2 |χk }  = E

k 

k−1 

x Tj Q 1 x j −

j=k+1−dk+1,1

 x Tj Q 1 x j +

j=k−dk,1



j=−d+1

k−d m

T Q 1 xk−dk,1 + ≤ xkT Q 1 xk − xk−d k,1

−d m

k 

k−1 

xlT Q 1 xl −

l=k+1+ j

l=k+ j

x Tj Q 1 x j + (d − dm )xkT Q 1 xk −

j=k+1−d

   xlT Q 1 xl χk

k−d m

x Tj Q 1 x j

j=k+1−d

T = 1 xkT Q 1 xk − xk−d Q 1 xk−dk,1 , k,1

(6.18)

E{Vk,3 |χk }  = E

k  j=k+1−dk+1,2

x Tj Q 2 x j −

k−1  j=k−dk,2

T Q 2 xk−dk,2 , ≤ 2 xkT Q 2 xk − xk−d k,2

E{Vk,4 |χk }

 x Tj Q 2 x j +

−d−1  j=−d M +1



k  l=k+1+ j

xlT Q 2 xl −

k−1 

   xlT Q 2 xl χk

l=k+ j

(6.19)

120

6 Sliding Mode Fault-Tolerant Control

−1    = E j=−d

k 

ηlT R1 ηl −

l=k+1+ j

k−1 

−1    ηlT R1 ηl +

l=k+ j

j=−d M

k 

ηlT R2 ηl −

l=k+1+ j

k−1 

   ηlT R2 ηl χk

l=k+ j

−1  −1       T T ηkT R1 ηk − ηk+ ηkT R2 ηk − ηk+ = E j R1 ηk+ j + j R2 ηk+ j j=−d

j=−d M k−1 

= dηkT R1 ηk + d M ηkT R2 ηk −

k−1 

ηlT R1 ηl −

l=k−d

ηlT R2 ηl

(6.20)

l=k−d M

with 1 and 2 being presented below (6.15). Combining (6.16)–(6.20) and free weight matrix method, it leads to E{Vk |χk } T A T [θ¯ I − θ¯ 2 G T (G B1 )−1 G]Adi xk−dk,1 − xkT P xk ≤ xkT AiT G˜ T P G˜ Ai xk + xk−d k,1 di T ¯ 2 G T (G B1 )−1 G] +δ 2 sgn T (sk )G B1 sgn(sk ) + xk−d A T [(1 − θ¯ )I − (1 − θ) k,2 di T T B T P B f a,k + ωkT E ωi P E ωi ωk + 2θ¯ xkT AiT G˜ T P G˜ Adi xk−dk,1 ×Adi xk−dk,2 + f a,k

¯ kT AiT G˜ T P G˜ Adi xk−dk,2 + 2xkT AiT G˜ T P B f a,k + 2xkT AiT G˜ T P E ωi ωk +2(1 − θ)x T ¯ − θ)x ¯ k−d −2δxkT AiT G˜ T P B1 sgn(sk ) − 2θ(1 A T G T (G B1 )−1 G Adi xk−dk,2 k,1 di T T T +2θ¯ xk−d A T G˜ T P B f a,k + 2θ¯ xk−d A T G˜ T P E ωi ωk − 2δ θ¯ xk−d A T G˜ T P k,1 di k,1 di k,1 di T T ¯ k−d ¯ k−d A T G˜ T P B f a,k + 2(1 − θ)x A T G˜ T P E ωi ωk ×B1 sgn(sk ) + 2(1 − θ)x k,2 di k,2 di T T T ¯ k−d −2δ(1 − θ)x A T G˜ T P B1 sgn(sk ) + 2 f a,k B T P E ωi ωk − 2δ f a,k B T P B1 sgn(sk ) k,2 di T T T P B1 sgn(sk ) + xkT (1 Q 1 + 2 Q 2 )xk − xk−d Q 1 xk−dk,1 − xk−d Q 2 xk−dk,2 −2δωkT E ωi k,1 k,2

 +ηkT (d R1 + d M R2 )ηk + 2ζkT S xk − xk−dk,1 −

k−1   ηl − ηlT R1 ηl

k−1  l=k−dk,1



+2ζkT T xk − xk−dk,2 −

k−1  l=k−dk,2



ηl −

k−1 

l=k−d

ηlT R2 ηl ,

l=k−d M

(6.21) T T T where ζkT = [xkT , xk−d , xk−d , δ˜kT , f a,k , ωkT ] with δ˜k = δsgn(sk ), S and T are k,1 k,2 weight matrices to be determined. From the Jensen inequality, it yields

−

2ζkT S

k−1 

ηl ≤

dζkT S

R1−1 S T ζk

+d

−1

k−1  

l=k−dk,1

l=k−dk,1

≤ dζkT S R1−1 S T ζk +

k−1  l=k−dk,1

and

ηl

ηlT R1 ηl

T R1

k−1  

ηl



l=k−dk,1

(6.22)

6.4 Stochastic Finite-Time Boundness of Closed-Loop System

− 2ζkT T

k−1 

121 k−1 

ηl ≤ d M ζkT T R2−1 T T ζk +

l=k−dk,2

ηlT R2 ηl

(6.23)

l=k−dk,2

Considering (6.21)–(6.23), the condition G B1 = I and inequality skills, it leads to T Y f a,k E{Vk |χk } − αδ δ˜kT δ˜k − ωkT X ωk − f a,k −1 −1 T T ˜ + dS R1 S + d M T R2 T T )ζk ≤ ζk ( +  ˜ k, = ζkT ζ

(6.24)

˜ =+ ˜ + dS R1−1 S T + d M T R2−1 T T and where ζkT is defined below (6.21),  ˜ ∗ , R being clarified in Theorem 6.2.  = ∗ + R T P −1 R with , It can be shown from Schur complement lemma that condition (6.12) in Theorem ˜ < 0. Thus, 6.2 can guarantee  T Y f a,k , E{Vk+1 − μVk |χk } ≤ αδ δ˜kT δ˜k + ωkT X ωk + f a,k E{Vk+1 |χk } ≤ μVk + h

(6.25)

with μ > 1 and h being defined in Theorem 6.2. By recursive method, we have E{Vk |χk−1 } ≤ μk E{V0 |χ−1 } +

μ(1 − μk−1 ) h. 1−μ

(6.26)

Besides, note that the following inequalities hold: E{Vk |χk−1 } ≥ E{xkT P xk } ≥ λmin (R − 2 P R − 2 )E{xkT Rxk }. 1

1

(6.27)

Combining (6.26) and (6.27), one has E{xkT

Rxk } ≤

μk E{V0 |χ−1 } +

μ(1−μk−1 ) h 1−μ

λmin (R − 2 P R − 2 ) 1

1

.

(6.28)

By virtue of (6.16), it follows that E{V0 |χ−1 } =

4  j=1

where

E{V0, j |χ−1 },

(6.29)

122

6 Sliding Mode Fault-Tolerant Control

E{V0,1 |χ−1 } = E{x0T P x0 |χ−1 } ≤ λm (R − 2 P R − 2 )c1 , 1

−1 

E{V0,2 |χ−1 } =

E{x Tj

Q 1 x j |χ−1 } +

j=−d0,1

1

−dm  −1 

E{xlT Q 1 xl |χ−1 }

j=−d+1 l= j

  dm + d − 1 (d − dm ) c1 , ≤ λm (R Q 1 R ) d + 2 −1 −d−1 −1    E{V0,3 |χ−1 } = E{x Tj Q 2 x j |χ−1 } + E{xlT Q 2 xl |χ−1 } − 21

− 21

j=−d M +1 l= j

j=−d0,2

  d + dM (d M − d − 1) c1 , ≤ λm (R Q 2 R ) d M + 2 −1 −1  −1  −1   E{V0,4 |χ−1 } = E{ηlT R1 ηl |χ−1 } + E{ηlT R2 ηl |χ−1 } − 21

− 21

j=−d l= j

j=−d M l= j

  1 1 1 1 ≤ 2 d(d + 1)λm (R − 2 R1 R − 2 ) + d M (d M + 1)λm (R − 2 R2 R − 2 ) c1 . Then,   dm + d − 1 1 1 1 1 (d − dm ) c1 E{V0 |χ−1 } ≤ λm (R − 2 P R − 2 )c1 + λm (R − 2 Q 1 R − 2 ) d + 2   d + d 1 1 M (d M − d − 1) c1 +λm (R − 2 Q 2 R − 2 ) d M + 2   1 1 1 1 +2 d(d + 1)λm (R − 2 R1 R − 2 ) + d M (d M + 1)λm (R − 2 R2 R − 2 ) c1 = 3 c1 ,

(6.30)

where 3 is defined in Theorem 6.2. From (6.14) and (6.28), we have E{xkT Rxk } < c∗ , k ∈ (0, T ∗ − 1]. Theorem 6.3 For the MJS (6.1) subject to the designed fault-tolerant controller (6.4), the resultant closed-loop system (6.11) is stochastically finite-time bounded wrt (c∗ , c2 , [T ∗ , T f ], R, W[0,T f ],γ ) if for given μ > 1, ι < 1 and λ > 1, there exist positive definite matrices P > 0, Q j > 0, R j > 0, X > 0, Y > 0, real matrices S , T , positive scalars αδ > 0 satisfying the conditions (6.12), (6.13), (6.15) and the following conditions: ∗

μT f −T

∗

+1

3 c ∗ +

μ(1 − μT f −T ) 1 1 h < λmin (R − 2 P R − 2 )c2 . 1−μ

(6.31)

Proof Choosing the same Lyapunov-Krasovskii functional and making a similar derivation as in Theorem 6.2, it can be derived from (6.25) that

6.4 Stochastic Finite-Time Boundness of Closed-Loop System

123 ∗

∗ μ(1 − μk−T ) h, k ∈ [T ∗ , T f ], E{Vk |χk−1 } ≤ μk−T +1 E{VT ∗ −1 |χT ∗ −2 } + (6.32) 1−μ   1 1 1 1 dm + d − 1 (d − dm ) c∗ E{VT ∗ −1 |χT ∗ −2 } ≤ λm (R − 2 P R − 2 )c∗ + λm (R − 2 Q 1 R − 2 ) d + 2  1 1  d + dM +λm (R − 2 Q 2 R − 2 ) d M + (d M − d − 1) c∗ 2  1 1 1 1  +2 d(d + 1)λm (R − 2 R1 R − 2 ) + d M (d M + 1)λm (R − 2 R2 R − 2 ) c∗

= 3 c∗ .

(6.33)

Thus, combining (6.32)–(6.33) and recalling (6.27), it holds E{xkT Rxk } ≤

μT f −T

∗

+1

3 c ∗ +

∗

μ(1−μT f −T ) h 1−μ

λmin (R − 2 P R − 2 ) 1

1

, k ∈ [T ∗ , T f ].

(6.34)

In addition, let’s take (6.31) into account and then we have E{xkT Rxk } < c2 for k ∈ [T ∗ , T f ]. Remark 6.3 Observing the right end of inequality (6.14) in Theorem 6.2, we can know that c∗ is a positive scalar to be solved according to the work in [33]. Also, it should be noted that the fault-tolerant issue in Theorems 6.2 and 6.3 can’t be readily 1 1 solved due to some factors, such as λmin (R − 2 P R − 2 ). Thus, in light of Definition 6.2, we will give the following stochastic finite-time boundedness criteria wrt (c1 , c2 , [0, T f ], R, W[0,T f ],γ ) for the system (6.11). Theorem 6.4 For the MJS (6.1) subject to the designed fault-tolerant controller (6.4), the resultant closed-loop system (6.11) is stochastically finite-time bounded wrt (c1 , c2 , [0, T f ], R, W[0,T f ],γ ) if for given μ > 1, ι < 1, λ > 1 and τ 1 , τ¯ p ( p = 1, 2, 3), there exist positive definite matrices P > 0, Q j > 0 and R j > 0 ( j = 1, 2), X > 0, Y > 0, real matrices S , T , positive scalars c∗ > 0, τx , τ y > 0, αδ > 0 satisfying the conditions (6.12), (6.15) and the following: μT

∗

−1

μT f −T

∗



3 c1 +

+1

∗ −2

μ(1−μT 1−μ

)



h < τ 1 c∗ ,

T f −T ∗

) 3 c∗ + μ(1−μ h < τ 1 c2 , 1−μ τ 1 R < P < τ¯1 R, 



0 < Q j < τ¯2 R, 0 < R j < τ¯3 R, j = 1, 2, X < τx I, Y < τ y I, P < λP, B1T P B1 = I, where

(6.35) (6.36) (6.37) (6.38) (6.39) (6.40)

124

6 Sliding Mode Fault-Tolerant Control

    dm + d − 1 d + dM  (d − dm ) + τ¯2 d M + (d M − d − 1) 3 = τ¯1 + τ¯2 d + 2 2   +2 d(d + 1) + d M (d M + 1) τ¯3 , 

h = αδ lδ 2 + τx γ 2 + τ y ς 2 , related matrices and other parameters can be found below (6.15).

6.5 Computational Algorithm By applying the method as in [36], we can know that the condition B1T P B1 = I in Theorem 6.4 can be represented by tr[(B1T P B1 − I )T (B1T P B1 − I )] = 0. To obtain a convex optimization issue, letting introduce (B1T P B1 − I )T (B1T P B1 − I ) ≤ β I with β > 0 and applying the Schur complement, it yields 

−β I B1T P B1 − I ∗ −I

 ≤ 0.

(6.41)

Then, the stochastic finite-time boundedness performance of closed-loop system (6.11) under actuator fault can be guaranteed if the following optimization issue is fully minimized to 0. minβ subject to (6.12), (6.15), (8.32) − (8.36) and (7.39).

(6.42)

Remark 6.4 The fault-tolerant problem subject to probabilistic delay is studied in this paper for the considered systems via adopting a sliding mode technique and a discrete-time partitioning strategy. More precisely, we have described the probabilistic delay using a Bernoulli-type random variable first and then we have dedicated to proposing some delay-dependent criteria to reveal the effects of probabilistic delay onto the stochastic boundedness of the system. Moreover, a matching optimization algorithm has immediately been built to resolve the above problem. Additionally, the effect of the probabilistic delay is reflected mainly by the probability of occurrence for the fault-tolerant controller.

6.6 Simulation This section will provide an example to illustrate the feasibility of SMFTC algorithm obtained. To start with, choose the following parameters for considered MJDSs with probabilistic delay (6.1): for rk = 1,

6.6 Simulation

125

⎤ ⎤ ⎡ −1.4120 0.1946 0 −0.02 0.06 0.04 0 0.1318 1.0125 ⎦ xk + ⎣ 0.04 0.01 −0.03 ⎦ [θk xk−dk,1 + (1 − θk )xk−dk,2 ] xk+1 = ⎣ 0 0 −0.15 0.01 0.02 0.05 ⎤ ⎤ ⎡ ⎡   −0.08 0.08 −0.017   0.7 0 u + 0.1 sin(10(k − 10)) + ⎣ 0.011 ⎦ ωk ; 0.2 ⎦ + ⎣ 0.1 0 0.6 k −0.019 0.13 0.26 ⎡

for rk = 2, ⎤ ⎤ ⎡ 0.0866 0.5830 0 0.03 0 0.01 xk+1 = ⎣ −1.3187 0.4914 0 ⎦ xk + ⎣ 0.02 0.03 0 ⎦ [θk xk−dk,1 + (1 − θk )xk−dk,2 ] 0 0 −0.15 0.04 0.05 −0.01 ⎤ ⎤ ⎡ ⎡   −0.08 0.08 −0.017   0.7 0 ⎦ ⎣ ⎣ 0.011 ⎦ ωk . 0.2 u + 0.1 sin(10(k − 10)) + + 0.1 0 0.6 k −0.019 0.13 0.26 ⎡

√

Moreover, by further setting ωk = 52 cos(kπ ) and π11 = 0.6, π12 = 0.4, π21 = 0.8, π22 = 0.2, solving the SMFTC problem (6.42) yields the feasible solutions ⎤ ⎡ ⎤ 4.9031 0.0167 0.0074 0.4201 0.0047 −0.0004 P = ⎣ 0.0167 4.8869 −0.0293 ⎦ , Q 1 = ⎣ 0.0047 0.4593 −0.0011 ⎦ , 0.0074 −0.0293 4.2745 −0.0004 −0.0011 0.4123 ⎡ ⎤ ⎡ ⎤ 0.3163 0.0071 −0.0005 0.2727 −0.0040 −0.0001 Q 2 = ⎣ 0.0071 0.3514 0.0001 ⎦ , R1 = ⎣ −0.0040 0.2651 −0.0009 ⎦ , −0.0005 0.0001 0.3151 −0.0001 −0.0009 0.2357 ⎡ ⎤ 0.1196 −0.0016 −0.0000 R2 = ⎣ −0.0016 0.1169 −0.0005 ⎦ . −0.0000 −0.0005 0.1040 ⎡

In view of (6.3), the switching function is given with the following form: 

 0.3949 0.4862 0.5533 sk = x . −0.0781 0.9695 1.1054 k In the simulation, letting initial value x0 = [0 0.03 − 1.02]T and the actuator fault factor ς = 0.1, the sliding mode fault-tolerant controller (6.4) with known parameters can be implemented to carry out the simulation. In detail, the discrete-time Markovian process rk is presented in Fig. 6.2. Then, the trajectory of system (6.1) as well as dynamics of signal xkT Rxk can be obtained from Fig. 6.3, which indicates the

126 Fig. 6.2 Random mode rk (ς = 0.1)

6 Sliding Mode Fault-Tolerant Control 3

rk

2

1

0

10

20

70

60

50

40

30

80

100

90

Fig. 6.3 Trajectories of state xk and signal xkT Rxk (ς = 0.1)

Trajectory of system

time (k) 2 xk,1

1

xk,2 xk,3

0 −1 −2

10

20

30

40

50

60

70

80

90

100

time (k)

Signal xTk Rxk

4 xTk Rxk

3 2 1 0 10

20

30

40

50

60

70

80

90

100

time (k)

achievement of stochastic finite-time boundedness performance via our proposed approach. The switching function sk and control signal u k are plotted in Fig. 6.4. Additionally, Fig. 6.5 plots the probabilistic delay dk . By combining Figs. 6.3 and 6.6, we can know that the transient performance of system (6.1) deteriorates but still shows the stochastic finite-time boundedness for severer case of stuck fault (ς = 0.4). Finally, it can be concluded that the proposed SMFTC scheme has a satisfactory stochastic finite-time boundedness performance.

6.7 Conclusion Fig. 6.4 Sliding mode function sk and control input u k (ς = 0.1)

127 1.5

s1k s2k ||sk ||

1 0.5 0 −0.5 −1 −1.5

40

30

20

10

50

time (k)

60

70

80

100

90

5

u1,k u2,k

0

−5

Fig. 6.5 Probabilistic delay dk (ς = 0.1)

10

20

30

40

50

60

time (k)

70

80

90

100

6 P robabilistic delay dk

5.5 5 4.5 4 3.5 3 2.5 2

10

20

30

40

50

60

70

80

90

100

time (k)

6.7 Conclusion The SMFTC problem has been solved for Markovian jump discrete systems with time-varying probabilistic delay. Here, the stochastic finite-time boundedness scheme has been developed for the closed-loop system against actuator failure and probabilistic delay in terms of mode-dependent criteria. Furthermore, according to the discrete partitioning strategy, a sliding mode fault-tolerant controller has also been synthesized to drive the system from an initial state into the neighborhood of the prescribed sliding surface. Finally, we have done numerical simulations to examine the usefulness of the control method obtained.

Fig. 6.6 Trajectories of state xk and signal xkT Rxk (ς = 0.4)

6 Sliding Mode Fault-Tolerant Control Trajectory of system

128 2

xk,1

1

x x

0

k,2 k,3

−1 −2

10

20

30

40

50

60

70

80

90

100

time (k)

Signal xTk Rxk

4 xTk Rxk

3 2 1 0 10

20

30

40

50

60

70

80

90

100

time (k)

References 1. K. Ding, Q. Zhu, Extended dissipative anti-disturbance control for delayed switched singular semi-Markovian jump systems with multi-disturbance via disturbance observer, Automatica, https://doi.org/10.1016/j.automatica.2021.109556. 2. M. Parvizian, K. Khandani, A diffusive representation approach toward H∞ sliding mode control design for fractional-order Markovian jump systems, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 235(7) (2021) 1154–1163. 3. P. Zhang, Y. Kao, J. Hu, B. Niu, Robust observer-based sliding mode H∞ control for stochastic Markovian jump systems subject to packet losses, Automatica, https://doi.org/10.1016/j. automatica.2021.109665. 4. G. Wang, Y. Chen, X. Li, Restricted stabilization of Markovian jump systems based on a period and random switching controller, IEEE Access, 8(2020) 103655–103664. 5. D. Yao, B. Zhang, P. Li, H. Li, Event-triggered sliding mode control of discrete-time Markov jump systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 49(10) (2019) 2016–2025. 6. M. Shen, H∞ filtering of continuous Markov jump linear system with partly known Markov modes and transition probabilities, Journal of the Franklin Institute, 350(10) (2013) 3384–3399. 7. X. Li, W. Zhang, D. Lu, Stability and stabilization analysis of Markovian jump systems with generally bounded transition probabilities, Journal of the Franklin Institute, 357(13) (2020) 8416–8434. 8. D. Yao, M. Liu, R. Lu, Y. Xu, Q. Zhou, Adaptive sliding mode controller design of Markov jump systems with time-varying actuator faults and partly unknown transition probabilities, Nonlinear Analysis: Hybrid Systems, 28(2018) 105–122. 9. M. Rubagotti, G. Incremona, D. Raimondo, A. Ferrara, Constrained nonlinear discrete-time sliding mode control based on a receding horizon approach, IEEE Transactions on Automatic Control, 66(8) (2021) 3802–3809. 10. J. Chang, Passivity-based sliding mode controller/observer for second-order nonlinear systems, International Journal of Robust and Nonlinear Control, 29(6) (2019) 1976–1989. 11. M. Hou, F. Tan, F. Han, G. Duan, Adaptive sliding mode control of uncertain nonlinear systems with preassigned settling time and its applications, International Journal of Robust and Nonlinear Control, 29(18) (2019) 6438–6462.

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Part IV

Markov Jump Systems with Bumpless Transfer Constraint

Chapter 7

Bumpless Transfer Control

This paper considers the sliding mode bumpless transfer control (SMBTC) problem for a class of Markovian jump discrete systems (MJDSs) with external disturbance. Here, the prescribed Markov chain is assumed to be ergodic. Accordingly, based on a unique stationary distribution in relation to ergodicity, a new BT criterion is established to help the characterization of bumpless transfer. This paper aims to construct a desiring expression of the controller such that the resultant closed-loop dynamics is stochastic stable with a prescribed bumpless transfer (BT) attenuation level. By combining the Lyapunov method and stochastic analysis theory, a couple of sufficient conditions are eventually proposed to ensure the reachability of the sliding surface and at the same time achieve the given BT performance. In the end, the usefulness of the scheme can be examined by solving the SMBTC problem and carrying out simulations.

7.1 Introduction The Markovian jump systems (MJSs) are a special type of switching systems (SS), which are switched back and forth in accordance with a particular random process (Markov chain) [1–3]. This characteristic has been viewed and widely used in engineering practice [4, 5]. Given a few problems with SS, for example, one of the more obvious problems is bump transfer (BT). Specifically, in [6, 7], the problems of BT control have been considered for a class of switched systems to reduce the transient behavior, respectively. From a perspective of resource efficiency, a couple of sufficient criteria for output consensus have been set up in [8], achieving the co-design of the event-triggered rules, switching rules and BT control. Accordingly, the problem of BT control for MJSs has recently come into focus. For instance, a generalized form of BT constraint has been given in [9] for continuous Markovian LPV systems, which can be used to solve the H∞ anti-disturbance problem in the presence of both deterministic and stochastic characteristics. It should be noted that in most existing BT schemes with MJSs, the controller is typically mode-dependent and will © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Kao et al., Analysis and Design of Markov Jump Discrete Systems, Studies in Systems, Decision and Control 499, https://doi.org/10.1007/978-981-99-5748-4_7

133

134

7 Bumpless Transfer Control

definitely jump with the system such that the closed-loop system can be effectively controlled in each jumping mode. It is worth mentioning that, in [10], a finite-time BT control algorithm has been presented without the need for synchronization between controller mode and system mode. Moreover, there is a situation where we need to design controllers with very different structures to achieve the desired control performance. This could cause a severe bump transfer. Apart from that, combined with the fact that the system jumping occurs randomly in a discrete-time setting, we know that extending existing BT results to Markovian jump discrete systems (MJDSs) directly is not an easy task. Recently, there are lots of results reported on BT control with imperfect controllers. For example, the problem of BT control has been taken into account in [15], where there exists uncertainty in the controller. The H∞ reliable BT control scheme has been set up in [16] for a class of switched systems with saturation limits, in which the BT performance and H∞ reliable property of the system can be realized under the same framework. It is no doubt that neither the jumping systems nor the single structure system with imperfect controllers will intensely bump theoretically if the BT condition can be reached [11–14]. That seems to be the outcome we are looking for. It should be noted, however, that the existing BT condition is toughly extended to the sliding mode control (SMC) strategy. The key to the difficulty here is the lack of a description of the variable structural item in the BT condition. Moreover, a few new achievements have not been reported to address this difficulty. This encourages us to think about how to build a new BT characterization. On the other hand, there are still few works on sliding mode bumpless transfer control (SMBTC) for the MJDSs. One reason lies in that MJDSs are governed by a certain Markov chain rather than a traditional switching signal, which can participate in the design of the BT control scheme. Obviously, the Markov chain seems not fully exploited if we directly employ the BT condition to the system with Markov jumping parameter. This is another motivation for present work. Inspired by the above discussions, the sliding mode BT control is considered for a class of MJDSs with external disturbance in this paper. First, we take linear Markov jump discrete systems (LMJDSs) without any disturbance into account. Then, a common integral-type sliding surface is chosen, and a corresponding sliding motion is derived for subsequent analysis. To proceed, we study the stochastic stability with prescribed BT performance for resulting sliding motion, where the adopted existing BT constraint involves a virtual controller. Next, viewing the limitation of the existing BT constraint, we propose a new BT criterion via a stationary distribution of the prescribed Markov chain in order to balance the BT performance and the feasibility of the scheme. Accordingly, the stochastic stability of the resulting LMJDSs is ensured by establishing a set of new criteria in the case of BT attenuation. In addition, extensive work is carried out for the sliding mode BT control of the MJDSs with external disturbance via the output feedback. At last, the simulation example is provided to verify the efficiency of the proposed scheme. It should be noted that the challenges in this investigation lie in the following aspects: (i) how to consider the BT control problem of the system with the Markov jumping parameter in the SMC framework? (ii) how to put forward a new BT mechanism taking advantage

7.2 Problem Statements and Preliminaries

135

of jumping parameter information? (iii) how to balance the relationship between the feasibility of the new scheme and the BT performance level. Compared with previous works, the novelties of this paper can be found in the following: (i) co-design of SMC and BT performance control is considered for the MJSs for the first time; (ii) modification of existing BT constraint is undertaken, accordingly, the new BT performance criteria based on state/output-feedback are proposed; (iii) sensitiveness of the newly modified BT criteria to different Markov chains with ergodic nature, which changes the indistinguishable role of different modes in the existing BT performance criterion.

7.2 Problem Statements and Preliminaries In this paper, the variable δk ∈ S = {1, 2, . . . , N } is a discrete time-homogeneous Markov chain whose transition probability matrix Π  [δi j ]i, j∈N is described as follows: δi j  Pr(δk+1 = j|δk = i) ≥ 0, ∀i, j ∈ S , k ∈ Z + ,  and Nj=1 δi j = 1 for each i ∈ S . Consider the following uncertain networked discrete systems: 

x(k + 1) = A(δk )x(k) + B(u(k) + f (k)), y(k) = C(δk )x(k),

(7.1)

where x(k) ∈ Rn , and u(k) ∈ Rm , y(k) ∈ Rq are the state vector, the control input, and the controlled output, respectively. For simplicity, let Ai  A(δk ) (δk = i ∈ S ) and B be known mode-independent matrix. Here, we assume that the pair (Ai , B) is controllable with the matrix B having full column rank, and the prescribed Markov chain has ergodicity. The function f (k) is the external disturbance. The aim of the paper is to develop a novel SMC law based on state/output feedback, which can ensure stochastic stability with a prescribed BT level for consequent closed-loop dynamics with a discrete Markov chain. To proceed, a few lemmas and definition are reviewed for subsequent development. Lemma 7.1 For any vectors x ∈ Rn and y ∈ Rn , it holds for any symmetric matrix Z ∈ R n×n > 0 x T y + y T x ≤ x T Z x + y T Z −1 y. Lemma 7.2 [1] The inequality H + A B −1 C + C T (B −1 )T A T < 0 can be ensured if there exist a scalar ε satisfying

136

7 Bumpless Transfer Control



H ∗ A T + εC −εB − εB T

 < 0.

Definition 7.1 [2] The considered system (7.1) is said to be stochastically stable if it holds for initial condition x(0) and any δ(0) ∈ S ∞ 

E{ x(k) 2 | x(0), δ(0)} < ∞.

k=0

7.2.1 State Feedback Based SMBTC of Deterministic Systems To start with, consider the networked discrete systems without disturbance: x(k + 1) = A(δk )x(k) + Bu(k),

(7.2)

where each pair (A(δk ), B) (δk ∈ S ) is supposed to be controllable. Then, choose a common integral-type sliding surface: 

s(k) = W x(k) − W x(0) +  (k) = 0,  (k) =  (k − 1) + E x(k − 1),

(7.3)

where the variables s(k) ∈ Rm ,  (k) ∈ Rm and  (0) = 0, the matrices E ∈ Rm×n and W = B T P ∈ Rm×n will be designed later with the matrix W satisfying W B > 0. Applying the condition s(k + 1) = s(k) = 0 and the Eqs. (7.2)–(7.3), we can derive the equivalent controller u eq (k) = −(W B)−1 [W (A(δk ) − I ) + E]x(k)  K (δk )x(k). Let’s substitute u eq (k) into the system (7.2). For the purpose of simplicity, denoting Ai  A(δk ) and K i  K (δk ) for each i = δk ∈ S , then the resultant sliding motion can be presented by x(k + 1) = Ai x(k),

(7.4)

where Ai = Ai − B(W B)−1 [W (Ai − I ) + E] = Ai + B K i . Theorem 7.1 For a given BT level ρ, consider the deterministic system (7.2) with the prescribed integral-type sliding surface (7.3). The resultant sliding motion (7.4) is stochastically stable with BT performance level ρ if there exist matrices P > 0, Pi > 0 and K ∗ satisfying the following conditions:  

−Pi ∗ P Ai + W T K i −2P + Pi



< 0,  −ρ I ∗ < 0, K ∗ + (W B)−1 [W (Ai − I ) + E] −ρ I

(7.5) (7.6)

7.2 Problem Statements and Preliminaries

where Pi =

 j∈S

137

δi j P j . Specially, the proposed BT controller u i (k)  u eq (k) with  u i (k) − K ∗ x(k) ≤ ρ  x(k) 

(7.7)

Proof For the system (7.4), choose a Lyapunov functional as follows: V (k) = x(k)T Pi x(k) with each matrix Pi  P(δk ) > 0 (i ∈ S ) to be designed. Calculating the forward difference ΔV (k) along the system (7.4), we can derive ΔV (k) = E{V (k + 1)|(x(k), δk = i)} − V (k) = x(k)T (Ai Pi Ai − Pi )x(k). It can be obtained from the condition (7.5) in Theorem 7.1 and formula −PPi−1 P ≤ −2P + Pi that   Multiplying the matrix 

−Pi ∗ T P Ai + W K i −PPi−1 P

 < 0.

 ∗ −Pi with diag[I P −1 ], it follows that P Ai + W T K i −PPi−1 P

∗ −Pi Ai − B(W B)−1 [W (Ai − I ) + E] −Pi−1

 < 0.

By the Schur complement, we conclude Ai Pi Ai − Pi < 0. Further, it holds ΔV (k) ≤ −λx(k)T x(k) with λ = λmin [−(Ai Pi Ai − Pi )] and mathematical relationships as E{V (k + 1)} − E{V (k)} ≤ −λE{x(k)T x(k)}, E{V (k)} − E{V (k − 1)} ≤ −λE{x(k − 1)T x(k − 1)}, ... E{V (1)} − E{V (0)} ≤ −λE{x(0)T x(0)}. Summing up the relationships yields E{V (k + 1)} − E{V (0)} ≤ −λ

k  l=0

namely,

E{x(l)T x(l)},

138

7 Bumpless Transfer Control k 

E{x(l)T x(l)} ≤

l=0

1 E{V (0)} < ∞. λ

In the light of the Monotone convergence theorem, we conclude ∞ 

E{x(k)T x(k)} < ∞.

k=0

According to the Definition 7.1, the resultant sliding motion (7.4) is stochastically stable. Additionally, it follows from the condition (7.6) in Theorem 7.1 that −ρ I +

1 ∗ (K − K i )T (K ∗ − K i ) < 0, ρ

which derives (7.7) for any state vector x(k). Therefore, the stochastic stability with BT performance level ρ can be guaranteed. Remark 7.1 It is important to note that the sliding mode BT control problem has been solved in terms of the existing BT constraint in (7.7). Through the observation, we find the following details: (i) there will be without significant difference in the structure between the control forces u i (k) (i ∈ S ) satisfying (7.7) if a common upper bound factor ρ exists; (ii) when the state is approaching 0, the error between designed control forces and virtual control force will also be approaching 0. This is how (7.7) can be adopted in previous works. However, we also note that the virtual controller u(k) = K ∗ x(k) may not exist in the course of problem-solving. As a result, we try to propose a new formula of the BT condition to solve this problem. Remark 7.2 We mention that there have existed lots of impressive works on BT for a class of switched systems. In most of them, the aim of BT performance has been achieved by designing the switching signal. Let’s take into account the MJS as a special switched system. We find that few results on BT have been proposed for the MJS since the switching signal at this moment is governed by a Markov chain, which can no longer be designed. Also, it can be found that discrete-time MJLS (7.2) will be jumping between its subsystems, which leads to difficulty in applying a compensation method over a continuous interval for the achievement of BT performance. At this point, we attempt to solve the above problem via stationary distribution of a Markov process.

7.2.2 A Modified SMBTC Scheme of Deterministic Systems + In this paper, we assume that  prescribed Markov chain δk (k ∈ Z ) has a unique stationary distribution ν = ν1 ν2 . . . ν N (ν j ≥ 0, j ∈ S ). Due to the properties of the stationary distribution, it leads to

7.2 Problem Statements and Preliminaries

νΠ = ν,

139



ν j = 1.

j∈S

By solving above two equations, the specific value for each ν j ( j ∈ S ) can be further determined. Based on l  arg max j∈S {ν j }, we will introduce a new BT scheme for the MJSs under consideration, which is expected to properly attenuate the bump induced by the switching between controllers without need of extra reference matrix K ∗ . To start with, according to the previous BT condition, we propose  u δk (k) − u l (k) ≤ bl  x(k) 

(7.8)

δk =l

for the state-feedback controller u δk (k) = K δk x(k) (δk ∈ S \{l }) with u l (k) = K l x(k). Remark 7.3 A new BT condition has been proposed in (7.8) for the closed-loop system with a Markovian jumping parameter under a linear state-feedback controller. In comparison with the previous one, this constraint is only about specified mode l and other modes. Specifically, the extra common virtual controller is no longer involved here. Instead of this, the corresponding control force u l (k) is selected in view of the stationary distribution, which has to do with the prescribed Markov chain. This is quite different from the existing BT condition as in [9]. Referring to extensive one in (7.38) with the output-feedback, it can be shown that the y(k) and sgn(s(k)) are both involved, meaning that the error between u i (k) (i = l ) and u l (k) won’t converge to zero when letting y(k) → 0. That is why the extensive BT condition is the actually weaker one. Owing to formal consistency, we replace K ∗ in Theorem 7.1 with K l when using the above SMBTC scheme with state feedback. Then, the following theorem can be presented. Theorem 7.2 For a given BT level bl > 0, consider the deterministic system (7.2) with the prescribed integral-type sliding surface (7.3). The resultant sliding motion (7.4) is stochastically stable with BT performance level bl if there exist matrices P > 0 and Pi > 0 satisfying the following conditions: 

 ∗ −Pi < 0, i ∈ S , P Ai + W T K i −2P + Pi   −bl I ∗ < 0, j ∈ S \{l }, K l − K j −bl I

(7.9) (7.10)

where K l = −(W B)−1 [W (Al − I ) + E]. Specially, the proposed BT controller u i (k) satisfying (7.8). Proof Based on Theorem 7.1, it is not hard to prove this one. Thus, the proof is omitted here.

140

7 Bumpless Transfer Control

Next, we are ready to give strict LMIs-based conditions using Lemma 7.2 to (7.9) and (7.10) in Theorem 7.2. Theorem 7.3 For a given BT level bl > 0, scalars φ1 ∈ R and φ2 ∈ R, consider the deterministic system (7.2) with the prescribed integral-type sliding surface (7.3). The resultant sliding motion (7.4) is stochastically stable and the proposed controller u i (k) has BT performance level bl if there exist matrices P > 0 and Pi > 0 satisfying the following LMIs: ⎡

⎤ −Pi ∗ ∗ ⎣ ⎦ < 0, i ∈ S , −2P + Pi ∗ P Ai T φ1 W (Ai − I ) + φ1 E −B P −2φ1 W B ⎤ ⎡ ∗ ∗ −bl I ⎦ < 0, j ∈ S \{l }. ⎣ ∗ 0 −bl I

φ2 W (Al − A j ) −I −2φ2 W B

(7.11)

(7.12)

7.2.3 Design of SMBTC for Uncertain Systems via Output Feedback This section aims to present a couple of criteria for the stability of uncertain MJDSs using above new BT mechanism (7.8). Besides, a desiring sliding mode outputfeedback controller is represented for the system to ensure both stochastic stability and BT performance. Let’s take the system with the form: 

x(k + 1) = A¯ i x(k) + B(u(k) + f (k)), y(k) = C¯ i x(k).

(7.13)

Recalling the matrix B with full column rank, we can know there exist a non-singular   0 n×n with B2 ∈ Rm×m being a invertible matrix. satisfying T B = matrix T ∈ R B2 As a result, letting z(k) = T x(k), the system (7.13) can be transformed into the following:  0 (u(k) + f (k)) z(k + 1) = Ai z(k) + B2 

with Ai = T A¯ i T −1 . Partitioning Ai =



 Ai,11 Ai,12 , we have Ai,21 Ai,22

7.2 Problem Statements and Preliminaries

141

 z 1 (k + 1) = Ai,11 z 1 (k) + Ai,12 z 2 (k), z 2 (k + 1) = Ai,21 z 1 (k) + Ai,22 z 2 (k) + B2 (u(k) + f (k)), y(k) = Ci z(k),

(7.14)

where Ai,11 ∈ R(n−m)×(n−m) , Ai,12 ∈ R(n−m)×m , Ai,21 ∈ Rm×(n−m) , Ai,22 ∈ Rm×m , z 1 (k) ∈ Rn−m and z 2 (k) ∈ Rm . Then, choose a common sliding surface s(k) = Gy(k) = 0,

(7.15)

m×q later for the non-singular property where the gain matrix  G ∈ R  will be designed of GCi,2 with Ci = Ci,1 Ci,2 , Ci,1 ∈ Rq×(n−m) and Ci,2 ∈ Rq×m . For convenience, we choose GCi,2 = B2T Mi in this paper with 0 < Mi ∈ Rm×m to be determined. Then, it follows from (7.15) that

z 2 (k) = −(B2T Mi )−1 GCi,1 z 1 (k).

(7.16)

Substituting (7.16) into the first part of (7.14) yields the sliding motion: z 1 (k + 1) = [Ai,11 − Ai,12 (B2T Mi )−1 GCi,1 ]z 1 (k).

(7.17)

Based on previous results, a set of sufficient criteria for stochastic stability is established below for the system (7.17). Theorem 7.4 Consider the indeterministic system (7.13) with the prescribed sliding surface (7.15). The resultant sliding motion (7.17) is stochastically stable if there exist matrices 0 < Mi ∈ Rm×m , 0 < X i ∈ R(n−m)×(n−m) (i ∈ S ) and real matrix G ∈ Rm×q satisfying the following conditions: 

where Xi =

 j∈S

∗ −X i Ai,11 − Ai,12 (B2T Mi )−1 GCi,1 −Xi−1

 < 0,

(7.18)

δi j X j .

Proof For the sake of brevity, the proof is also omitted here. To proceed, we provide a version of strict LMIs-based conditions below to ensure the stochastic stability of the system (7.17). Theorem 7.5 Consider the indeterministic system (7.13) with the prescribed sliding surface (7.15). The resultant sliding motion (7.17) is stochastically stable if for given a scalar φ3 ∈ R, there exist matrices 0 < Mi ∈ Rm×m , 0 < X i ∈ R(n−m)×(n−m) (i ∈ S ) and real matrix G ∈ Rm×q satisfying the following conditions: ⎡

⎤ −X i ∗ ∗ ⎣ Ai,11 −2I + Xi ⎦ < 0. ∗ T T φ3 GCi,1 −Ai,12 −φ3 B2 Mi − φ3 Mi B2

(7.19)

142

7 Bumpless Transfer Control

Proof Applying the Lemma 7.2 to (7.18), we can readily get conditions in Theorem 7.5.

7.2.4 Design of Output-Based Sliding Mode Controller In this section, we focus on designing a robust output-based sliding mode controller and discussing the reachability. Recalling s(k) in (7.15), it holds s(k + 1) = GC(δk+1 )z(k + 1)



= GC(δk+1 )Ai z(k) + GC(δk+1 )

 0 (u(k) + f (k)). B2

(7.20)

Further, using s(k + 1) = 0, it yields GCi Ai z(k) + GCi,2 B2 (u(k) + f (k)) = 0,

(7.21)

  where Ci = j∈S δi j C j and Ci,2 = j∈S δi j C j,2 . Noting GCi,2 = B2T Mi , then   we have GCi,2 = j∈S δi j B2T M j and M˜ i = GCi,2 B2 = j∈S δi j B2T M j B2 > 0. Thus, the virtual controller is expressed as u eq (k) = − M˜ i−1 GCi Ai z(k) − f (k).

(7.22)

Remark 7.4 It can be seen from (7.22) that the virtual controller is linear with respect to z(k) and can be realized despite matched disturbance f (k). Moreover, if outputbased, that would be exactly what we need for the system (7.14). Unfortunately, the controller (7.22) has nothing to do with the measurement output y(k). That is why we give the following design procedure. For the discrete MJS (7.14) with the sliding surface (7.15), the output-based sliding mode controller can be devised as follows: u i (k) = − M˜ i−1 [K i y(k) + Fv + F p sgn(s(k))], where T  F (k)  M˜ i f (k) = F1 (k) F2 (k) . . . Fm (k) , F l ≤ Fl (k) ≤ F l , ∀l ∈ {1, 2, . . . , m}, T

, Fv  F 1 +F 1 F 2 +F 2 . . . F m +F m 2

Fp 

2

2

F −F F −F F −F diag{ 1 2 1 , 2 2 2 , . . . , m 2 m },

(7.23)

7.2 Problem Statements and Preliminaries

143

which can drive the system (7.14) into the succeeding area nearby the prescribed sliding surface (7.15) in mean square: √  1 2 1 + 2τ (maxi∈S  M˜ i−1 ) 2  F p  . B  z : s(k) ≤ ϑ  1 (1 − λ) 2 

(7.24)

Subsequently, implementing the controller (7.23), the reachability of the prescribed sliding surface (7.15) will be analyzed via the Lyapunov method. Theorem 7.6 Consider the discrete-time MJS (7.14) with prescribed sliding surface (7.15). The system (7.14) can be driven into the neighborhood B in mean square if for given scalars ε ∈ R, τ > 0, there exist positive-definite matrices X i > 0, Mi > 0 (i ∈ S ), real matrices G, K i and scalars λi > 0, λ ∈ (0, 1) satisfying ⎡

⎤ AiT (Xi + λi I )Ai − X i ∗ ∗ ∗ ∗ ⎢ ˜T ⎥ ⎢ − B2 (Xi + λi I )Ai + εK i Ci −2ε M˜ i ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ 0 0 −λI ∗ ∗ ⎥ < 0, ⎢ ⎥ ⎣ 2τ K i Ci 0 0 −2τ M˜ i ∗ ⎦ B˜ 2T (Xi + λi I )Ai 0 0 0 − M˜ i CiT G T GCi < λi I, B˜ 2T (Xi + λi I ) B˜ 2 ≤ τ M˜ i , GCi,2 = where Xi = defined

 j∈S

δi j X j , λi =

 j∈S

B2T

Mi ,

(7.25)

(7.26) (7.27) (7.28)

 T δi j λ j and B˜ 2 = 0 B2T , the matrix M˜ i is

below (7.21). Proof For the system (7.14), choose a Lyapunov candidate as follows: V (k) = z T (k)X i z(k) + s T (k)s(k)

(7.29)

with each X i > 0 (i ∈ S ) to be designed. Substituting the composite controller (7.23) into the system (7.14), we have the closed-loop system: z(k + 1) = Ai z(k) − B˜ 2 M˜ i−1 K i y(k) + B˜ 2 M˜ i−1 [F (k) − Fv − F p sgn(s(k))].

(7.30)

Let’s calculate the forward difference ΔV (k) along the above closed-loop system: E{ΔV (k)} = E{z(k + 1)T Xi z(k + 1)} − z(k)T X i z(k) + E{s(k + 1)T s(k + 1)} − s(k)T s(k).

Note the condition (7.26) in Theorem 7.3 and the following E{s T (k + 1)s(k + 1)} =

 j∈S

δi j z T (k + 1)C Tj G T GC j z(k + 1).

(7.31)

144

7 Bumpless Transfer Control

One has E{s(k + 1)T s(k + 1)} ≤ λi z(k + 1)T z(k + 1),

(7.32)

with λi being defined below (7.28). Then, E{ΔV (k)} ≤ z(k + 1)T (Xi + λi I )z(k + 1) − z(k)T X i z(k) − s(k)T s(k) = z(k)T AiT (Xi + λi I )Ai z(k) + z(k)T CiT K iT M˜ −1 B˜ 2T (Xi + λi I ) B˜ 2 M˜ −1 K i Ci z(k) i

i

+ [F (k) − F v − F p sgn(s(k))]T M˜ i−1 B˜ 2T (Xi + λi I ) B˜ 2 M˜ i−1 [F (k) − F v − F p sgn(s(k))] − 2z(k)T AiT (Xi + λi I ) B˜ 2 M˜ i−1 K i Ci z(k) + 2z(k)T AiT (Xi + λi I ) B˜ 2 M˜ i−1 [F (k) − F v − F p sgn(s(k))] − 2z(k)T CiT K iT M˜ i−1 B˜ 2T (Xi + λi I ) B˜ 2 M˜ i−1 [F (k) − F v − F p sgn(s(k))] − z(k)T X i z(k) − s(k)T s(k).

(7.33)

By the Lemma 7.1, it leads to 2z(k)T AiT (Xi + λi I ) B˜ 2 M˜ i−1 [F (k) − Fv − F p sgn(s(k))] ≤ z(k)T AiT (Xi + λi I ) B˜ 2 M˜ i−1 B˜ 2T (Xi + λi I )Ai z(k) + [F (k) − Fv − F p sgn(s(k))]T M˜ i−1 [F (k) − Fv − F p sgn(s(k))],

(7.34)

−2z(k)T CiT K iT M˜ i−1 B˜ 2T (Xi + λi I ) B˜ 2 M˜ i−1 [F (k) − Fv − F p sgn(s(k))] ≤ z(k)T CiT K iT M˜ i−1 B˜ 2T (Xi + λi I ) B˜ 2 M˜ i−1 K i Ci z(k) + [F (k) − Fv − F p sgn(s(k))]T M˜ i−1 B˜ 2T (Xi + λi I ) B˜ 2 M˜ i−1 × [F (k) − Fv − F p sgn(s(k))].

(7.35)

Besides, combining (7.34) and (7.35) together with (7.27), we have E{ΔV (k)} ≤ z(k)T AiT (Xi + λi I )Ai z(k) + 2τ z(k)T CiT K iT M˜ i−1 K i Ci z(k) −2z(k)T AiT (Xi + λi I ) B˜ 2 M˜ i−1 K i Ci z(k) +z(k)T AiT (Xi + λi I ) B˜ 2 M˜ i−1 B˜ 2T (Xi + λi I )Ai z(k) − z(k)T X i z(k) +(1 + 2τ )[F (k) − Fv − F p sgn(s(k))]T M˜ i−1 [F (k) − Fv − F p sgn(s(k))] −s T (k)s(k).

(7.36)

In view of  F (k) − Fv − F p sgn(s(k)) ≤ 2  F p , then (7.36) can be further expressed as E{ΔV (k)} ≤ ξkT Λξk for z ∈ / B, where

(7.37)

7.2 Problem Statements and Preliminaries

145

  ξkT = z(k)T s(k)T , Λ = diag{Λ11 , −λI }, Λ∗11 = AiT (Xi + λi I )Ai − X i + 2τ CiT K iT M˜ i−1 K i Ci + AiT (Xi + λi I ) B˜ 2 M˜ i−1 B˜ 2T (Xi + λi I )Ai , Λ11 = Λ∗11 − AiT (Xi + λi I ) B˜ 2 M˜ i−1 K i Ci − [AiT (Xi + λi I ) B˜ 2 M˜ i−1 K i Ci ]T .

By the Lemma 7.2, we can conclude that Λ < 0 is ensured if ⎡

⎤ Λ∗11 ∗ ∗ ⎣ − B˜ T (Xi + λi I )Ai + εK i Ci −2ε M˜ i ∗ ⎦ < 0. 2 0 0 −λI It can be derived from Schur complement that (7.25) in Theorem 7.6 ensures E{ΔV (k)} < 0 (ξk = 0). Therefore, using the devised controller (7.23), the considered system (7.14) can be driven into the neighborhood B of the sliding surface (7.15) in mean square. The proof of Theorem 7.6 is complete. Remark 7.5 Now, we have designed a new output-feedback sliding mode controller (7.23) for the system (7.14). It should be noticed that the required performance of several switched systems cannot be met just by implementing a single-architecture controller. However, a mode-dependent one tends to cause a few problems, like bump transfer. Next, the new BT mechanism is proposed for a class of output-feedback controllers with a Markovian jumping parameter to attenuate bump transfer.

7.2.5 Output Feedback-Based Sliding Mode BT Control Scheme Based on the BT constraint proposed in (7.8) with state-feedback, we give a suitable version for output feedback based controller u i (k) = K δ(k) y(k):  u i (k) − u l (k) ≤ bl  y(k)  i =l

with u l (k) = K y(k). However, noting the structure of the controller (7.23), we are in a position to extend the above condition as the following weak BT one:  u i (k) − u l (k) ≤ bl  y(k)  +cl  sgn(s(k))  +dl , i =l

(7.38)

where bl > 0, cl > 0 and dl > 0 are BT levels, and u l (k) = −M −1 [K y(k) + Fv + F p sgn(s(k))] with M ∈ Rm×m , K ∈ Rm×q being real matrices to be solved later. In the meantime, we are ready to display some possible conditions that can help satisfy the reachability of the sliding surface (7.15) with prescribed weak BT performance (7.38).

146

7 Bumpless Transfer Control

Theorem 7.7 Consider the discrete-time MJS (7.14) with prescribed sliding surface (7.15). The system (7.14) can be driven in mean square into the neighborhood B by the controller (7.23) and at the same time the BT performance (7.38) can be met if for given BT levels bl > 0, cl > 0, dl > 0 and scalars τ > 0, εl (l = 1, 2), ε, ρ ∈ R, there exist positive-definite matrices X i > 0, Mi > 0 (i ∈ S ), a non-singular matrix M = M T , real matrices G, K i , K and scalars λi > 0, λ ∈ (0, 1) satisfying (7.25)– (7.28) in Theorem 7.6 as well as following LMIs: ⎤ ∗ ∗ −bl2 I ⎦ < 0, i = l , ⎣ K ρ 2 I − 2ρ M ∗ ε1 K i −M −2ε1 M˜ i ⎡ ⎤ ♣ ∗ ∗ ⎣ Fv ρ 2 I − 2ρ M ⎦ < 0, i = l , ∗ ε2 Fv −M −2ε2 M˜ i ⎡

(7.39)

(7.40)

where M = M˜ l , K = K l , ♣ ∈ {−cl2 I, −dl2 I } and other parameters can be found by referring to Theorem 7.6 and previous results. Proof Recalling Theorem 7.6, we know that (7.25)–(7.28) can ensure the reachability of prescribed sliding surface (7.15). Therefore, all we need to do is actually to examine the BT performance in (7.38) via the conditions given in Theorem 7.7. It can be implied from the lemma 7.2 that (7.39) ensures 

∗ −bl2 I K ρ 2 I − 2ρ M



 +

0 −M





  M˜ i−1 K i 0 +

K iT 0



  M˜ i−1 0 −M T < 0,

(7.41)

which is equivalent to 

−bl2 I ∗ −1 2 ˜ K − M Mi K i ρ I − 2ρ M

 < 0.

(7.42)

Applying −M 2 ≤ ρ 2 I − 2ρ M to (7.42), we have 

−bl2 I ∗ K − M M˜ i−1 K i −M 2

 < 0.

(7.43)

Deforming (7.43) by pre-multiplying diag{I M −1 } and pro-multiplying diag{I M −T }, one has   ∗ −bl2 I < 0. (7.44) M −1 K − M˜ i−1 K i −I Similarly, it follows from (7.40) that

7.4 Simulation

147



♣ ∗ (M −1 − M˜ i−1 )Fv −I

 < 0.

(7.45)

According to (7.44) and (7.45), we conclude that the BT performance in (7.38) can be satisfied. Remark 7.6 The SMC issues with a modified BT constraint (7.38) have so far been addressed for the system (7.1) respectively from the perspective of state feedback and output feedback. When applying the devised controllers u i (k) (i ∈ S ) in (7.23), the BT performance (7.38) can be met by the conditions in Theorem 7.7. It is obvious that the phenomenon of bump transfer can theoretically be attenuated when it occurs switching between controllers. Next, a comprehensive algorithm will be provided to render the problem easy to solve via commonly used software.

7.3 A Comprehensive Algorithm Now, we are in a position to handle (7.28) and aim to establish a computational algorithm. As in [17], the equality constraint (7.28) can be changed into the following sub-optimal form: 

−ιI ∗ GCi,2 − B2T Mi −I

 < 0,

(7.46)

where ι > 0. As a result, we can provide a comprehensive algorithm below for our proposed SMBTC scheme. The SMBTC problem with the new BT criterion (7.38) can be addressed for system (7.14) if optimal solution of the following problem min ι subject to (7.19), (7.25)–(7.27), (7.39), (7.40) and (7.46).

(7.47)

is equal to zero.

7.4 Simulation In this section, the effectiveness will be proved for our proposed new SMC scheme with a weak BT performance by utilizing one example. At first, choose the considered MJLSs (7.2) with the following parameters: for rk = 1, 

   −0.4 1.2 0.5 x(k + 1) = x(k) + u(k); −0.3 0.8 −1.8

148

7 Bumpless Transfer Control

for rk = 2,  x(k + 1) =

   −0.5 1.2 0.5 x(k) + u(k); −0.2 −0.4 −1.8

for rk = 3, 

   −1.2 1.1 0.5 x(k + 1) = x(k) + u(k). −0.8 0 −1.8 Moreover, the TPM with above MJLSs is expressed by ⎡

⎤ 0.2 0.6 0.2 Π = ⎣ 0.3 0 0.7 ⎦ . 0.5 0 0.5  By solving νΠ = ν and j∈S ν j = 1, we obtain a formula for the stationary distribution of prescribed Markov chain: ν=

 25

15 31 71 71 71



.

Thus, l  arg max j∈S {ν j } = 3. Besides, setting φ1 = 3.2 and φ1 = 0.4 in Theorem 7.3, we can acquire the feasible solutions to the state-feedback-based SMBTC problem:    0.3268 −0.7169 1.5186 −0.9330 P= , , P1 = −0.7169 2.2419 −0.9330 1.6380     0.1974 −0.5523 1.1045 −0.6321 , P3 = . P2 = −0.5523 2.7871 −0.6321 1.2373 

with the weak BT performance level bl = 0.3394. In view of (7.3), the sliding surface can be presented by     s(k) = 2.4387 −3.4150 x(k) −  2.4387 −3.4150  x(0) +  (k) = 0,  (k) =  (k − 1) + 2.6367 −5.5566 x(k − 1). For conducting the simulation, let initial value x0 = [0.2 − 1]T and then the sliding mode BT controller u(k) = −(W B)−1 [W (A(δk ) − I ) + E]x(k) can be implemented with W = B T P and P being the above feasible solution. Based on the settings of the system parameter, the simulation results can be presented in Figs. 7.1, 7.2 and 7.3. To start with, a possible mode evolution of the system with the prescribed Markov chain is displayed in Fig. 7.1. Observing Fig. 7.2, we can clearly see that the system under the control of u(k) is approaching equilibrium. Furthermore, by drawing the BT controller and the regular sliding mode controller, the trajectories

7.4 Simulation Fig. 7.1 The evolution of the system mode δk

149 4

δk system mode

3

2

1

0

10

20

30

40

50

60

70

80

90

100

time (k)

Fig. 7.2 The trajectories of components of the system state x(k)

2

xk,1 the first component of the system state xk,2 the second component of the system state

1.5 1 0.5 0 −0.5 −1 −1.5 −2 10

20

30

40

50

60

70

80

90

100

time (k)

of these controllers can be seen from Fig. 7.3, which shows the realization of BT performance through our scheme. From the above simulations, it can be obtained that the proposed new SMBTC scheme is effective for the switched linear system subject to the prescribed Markov jumping parameter δk with a BT performance level bl .

150 Fig. 7.3 The control signal u i (k) with/without a new BT constraint

7 Bumpless Transfer Control 0.1

ui (k) without a new BT constaint ui (k) with a new BT constaint

0.08 0.06 0.04 0.02 0 0.01

−0.02

0.005

−0.04

0

−0.06

−0.005

−0.08 −0.1

−0.01 10

10

20

30

15

40

50

20

60

70

80

90

100

time (k)

7.5 Conclusion This paper has addressed the SMBTC problem for a class of the MJDSs with external disturbance. The stationary distribution method has been proposed to attenuate the bump transfer case, which weakens the impact of the transfer on the controller. Based on the state-feedback strategy and newly proposed BT criterion, the desired linear sliding mode controller has been devised, which ensures that the closed-loop system is stochastically stable. According to the mode-dependent Lyapunov functional approach, the parameters of the above controller can be solved by strict LMIs-based sufficient criteria. Also, an extension to output-feedback SMBTC has been provided, which can be solved with the aid of an optimal algorithm with strict LMI constraints. In addition, the effectiveness of our method has been shown via a simulation example. It should be pointed out that the bump of each controller itself will definitely affect the whole behavior of the system. Therefore, it is of great importance to study the future topic about how to attenuate the bump induced by each controller itself and bump transfer between the controllers in the same framework.

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Chapter 8

Semi-Markov Jump Systems

This note is devoted to the analysis of adaptive sliding mode control of H∞ /passivity of semi-Markov jump discrete-time uncertain singular systems. We consider the constraints of parameter uncertainties, unknown nonlinear functions, and external disturbances throughout the model. Firstly, a novel mode-independent sliding mode surface is constructed, then the sliding mode dynamic equation under the equivalent controller is obtained. Secondly, by introducing virtual points, several sufficiency criteria are derived for the sliding mode dynamic ensuring H∞ /passive performance and stochastic admissibility with linear matrix inequality form. Thirdly, the state trajectory can be driven into a switching band containing the origin by adaptive sliding mode control techniques. Finally, the effectiveness of our findings is demonstrated by comparing the existing strategy with three simulation examples.

8.1 Introduction Singular systems (SSs) can describe practical problems more accurately and are widely used with intensive research of modern control theory. During the recent two decades, numerous scholars have carried out thorough research on SS, for example, the theory of continuous-time SS has been systematically explored with attention to robust control problems in [1], and discrete-time (DT) SS with delay and external disturbance is interested in [2], meanwhile, researchers have also devoted their attention to stability study of singular Markov jump systems [3, 4]. Markov process is a highly regarded stochastic process and discussions about it can be found in a large number of studies [3–7]. A simple hypothesis of the Markov process is that its future is only relevant to its present. But this hypothesis is restrictive on the distribution of sojourn-time (ST) in each mode, i.e., the ST obeys geometric and exponential distributions in the discrete and continuous situations, respectively, which is a big drawback of applying the Markov process. Naturally, the semi-Markov process that allows the ST to obey an arbitrary probability distribution in any mode and still satisfies the Markov hypothesis more flexibly has attracted interest [8–13]. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Kao et al., Analysis and Design of Markov Jump Discrete Systems, Studies in Systems, Decision and Control 499, https://doi.org/10.1007/978-981-99-5748-4_8

153

154

8 Semi-Markov Jump Systems

The semi-Markov process has gained attention in continuous-time SS, please see [14, 15] and the references therein. In contrast, there is still more space for DT singular semi-Markov jump systems (SSMJSs). How to be able to fully utilize the distribution information of ST and how to obtain easily detectable stability analysis criterion and controller synthesis are the main technical difficulties in studying DT SSMJSs. As researchers continue to research and innovate, more and more advanced control strategies are applied to improve the system performance, such as event-triggered control [3], fault-tolerant control [7, 9], H∞ control [10, 11], sliding mode control (SMC) [16–18] and so on. Among the above mentioned methods, SMC is already extensively implemented in diverse systems such as SSs [5], fuzzy systems [4], Markov jump systems [6, 7], semi-Markov jump systems [12, 13], reaction-diffusion impulsive systems [17], switched hyperbolic systems [18] and singularly perturbed systems [19] because of its insensitivity to uncertainties and disturbances and its ease of implementation. It is worth noting that the sliding mode surfaces (SMSs) designed for Markov jump systems and semi-Markov jump systems in [6, 7, 12, 13] are mode-dependent. This can cause the system to shift from one mode to another, and the state trajectories may not arrive SMS at all, thus failing to generate the sliding mode and rendering the SMC ineffective. Therefore, the design of a common SMS for DT SSMJSs and the synthesis of sliding mode controllers are also problems that need to be further investigated. Establishing an accurate mathematical model for an actual system is an insurmountable problem subject to variables that cannot be measured precisely and the existence of time-varying factors. Adaptive control approach is a forceful technique to cope with unknown perturbations, disturbances, and inadequate precise information. Therefore, compared to References [20, 21], the time-varying parameter uncertainty that varies with the mode is considered in the system and adaptive control techniques are taken into account. In addition, H∞ and passivity theories are crucial in analyzing the stability of dynamic systems. The theoretical and practical aspects of H∞ and passivity analysis as well as the construction of adaptive sliding mode controller have been significantly developed, however, there are few investigations combining and improving the two approaches in DT semi-Markovian jump systems. Based on the discussion mentioned above, we are inspired to study the mixed H∞ /passive performance and stochastic admissibility analysis and adaptive sliding mode controllers synthesis for DT SSMJS. The major contributions of this note are listed as follows: • A mode-independent common SMS is designed for DT SSMJS, and unknown nonlinear functions, external disturbances, and parameter uncertainties present in the system are estimated using an adaptive controller. The state trajectories can reach inside the switching band containing the origin under the action of the adaptive SMC law. • The resulting criteria are transformed into a convex optimization problem with linear matrix inequality form, and then the inequalities are solved to obtain controller gain matrix while ensuring the realizability of stochastic admissibility with H∞ /passivity.

8.2 Problem Statements and Preliminaries

155

• Virtual points are introduced to obtain the relationship between the jump moment and its previous moment so that the relationship can be established for any two adjacent moments (whether in the same mode or not), which overcomes the effect of semi-Markov jump ST variables. The remainder of this work is summarized below. Section 8.2 introduces the structure of DT SSMJS studied together with some preliminary knowledge. Section 8.3 presents our findings. Section 8.4 is dedicated to demonstrating the validity of results with three simulation examples. Section 8.5 concludes this work.

8.2 Problem Statements and Preliminaries Consider a class of DT SSMJSs defined in the complete probability space (Ω, F , Pr) 

    E xk+1 = Ark + ΔArk (k) xk + B1,rk u k + frk (xk , k) + B2,rk ωk , z k = Crk xk + Drk ωk ,

(8.1)

where rank(E) = r < n, xk ∈ Rn , u k ∈ Rm , and z k ∈ Rz represent system state, control input, and controlled output, respectively, and ωk ∈ Rz is an extraneous disturbance belonging to l2 [0, +∞). ΔArk (k) represents the parameter uncertainty given as ΔArk (k) = Mrk Frk (k)Nrk , where Mrk and Nrk are known real matrices and Frk (k) satisfying FrTk (k)Frk (k) ≤ I stands for unknown time-varying matrix function. The stochastic process {rk }k∈N determines the switching between H modes and is a semiMarkov chain with rk ∈ H = {1, 2, . . . , H }. The unknown nonlinear perturbation function frk (xk , k) ∈ Rm satisfies  frk (xk , k) − frk (x¯k , k) ≤ νrk xk − x¯k  ≤ νxk − x¯k ,

(8.2)

where ν = maxrk ∈H {νrk }, νrk > 0. Matrices Ark , B1,rk , B2,rk , Crk , Drk , Mrk , Nrk are known and have suitable dimensions, and for the convenience of notation, they are abbreviated with frk (xk , k) and νrk as Ai , B1i , B2i , Ci , Di , Mi , Ni , f i (xk , k) and νi when the current system mode rk = i. Remark 8.1 SSs are a more extensive class of systems than ordinary systems, and there are few studies about the stability and stabilization problems of DT SSMJSs, therefore, compared to [22–24], this paper is concerned with DT SSMJSs. Moreover, we also focus on external disturbances, time-varying parameter uncertainties and unknown nonlinear functions associated with the modes, which are more pervasive in practice and more difficult to handle compared to using common uncertainties and nonlinear functions for all modes. To better understand semi-Markov chains, we next describe three stochastic processes and some related matrices, which can be referred to Reference [8] for a more detailed description of them.

156

8 Semi-Markov Jump Systems

(i) Stochastic process {kn }n∈N : kn ∈ N is monotonically increasing and represents the moment corresponding to the nth jump with k0 = 0. (ii) Stochastic process {Rn }n∈N : Rn ∈ H stands for the corresponding mode when nth jump occurs. (iii) Stochastic process {Sn }n∈N : Sn = kn+1 − kn ∈ N+ represents the ST in the mode Rn between nth and (n + 1)th jumps. (iv) DT Semi-Markov kernel Υ (ε) = [υi j (ε)]i, j∈H

,ε∈N+

: υi j (ε)  Pr (Rn+1 = j, Sn = ε|Rn = i) ,

  0 ≤ υi j (ε) ≤ 1, +∞ ε=1 j=i, j∈H υi j (ε) = 1. (v) Transition probability matrix Λ = [λi j ]i, j∈H : λi j  Pr (Rn+1 = j|Rn = i) and λii = 0. (vi) Matrix [i j (ε)]i, j∈H ,i= j,ε∈N+ : i j (ε)  Pr (Sn = ε|Rn = i, Rn+1 = j) represents the probability density function of ST of mode i. Combining above descriptions (iv)-(vi), it is apparent that υi j (ε) = λi j i j (ε). SSs have impulsive behavior and incompatibility with the initial conditions that ordinary systems do not have. Before addressing these issues, we will discuss the unforced form: (8.3) E xk+1 = Ai xk . Definition 8.1 The DT SSMJS (8.3) is defined to be: (i) regular if det(λE − Ai ) is not always equal to 0 for any i ∈ H ; )) = rank(E) for any i ∈ H ;  (ii) causal if deg(det(λE − Ai +∞ 2 i | < +∞ for any ini(iii) stochastically stable if E Rn =i k=0 x(k) |(x0 ,r0 ),S n 0 such that inequality E

 +∞ 

γ

−1

σ z kT z k

− 2(1 −

σ )z kT ωk

k=0



< γE

 +∞ 

ωkT ωk

(8.4)

k=0

holds for ωk = 0 under zero initial condition. Lemma 8.1 ([25]) If real matrices X 1 , X 2 and any G(k) satisfying G T (k)G(k) ≤ I , then X 1 G(k)X 2 + X 2T G T (k)X 1T ≤ ε X 1 X 1T + ε−1 X 2T X 2 holds for any ε > 0.

8.3 Main Results

157

Lemma 8.2 ([26]) If matrices Y1 , Y2 and J > 0 have the appropriate dimensions, then inequality Y1 Y2 + Y2T Y1T ≤ Y1 J Y1T + Y2T J −1 Y2 holds.

8.3 Main Results The core of this part is to realize stochastic admissibility with mixed H∞ /passivity of SSMJS (8.1) subject to matching nonlinear functions, parameter perturbations, and external disturbances with adaptive SMC. To achieve the main objective of this note, the following study steps will be addressed: (a) design of the mode-independent common SMS; (b) admissibility analysis of the closed-loop system; (c) synthesis of an adaptive SMC law for accessibility of (a).

8.3.1 Common SMS Design We design the mode-independent common sliding function sk = G E xk − G K xk−1 ,

(8.5)

where matrix G ∈ Rm×n enables G B1i nonsingular for any i ∈ H , and K ∈ Rn×n is a gain matrix determined subsequently. Remark 8.2 The system is constantly switching between H modes under the domination of a semi-Markov chain. If the designed SMS is mode-dependent as References [6, 7, 12, 13], it will have bad consequences for both the arrival phase and the sliding phase of SMC. For the arrival phase, the sliding function frequently switches from one mode to another randomly, which makes it highly likely that the reachability of the SMS is not guaranteed. Then for the sliding phase, the stability will be unattainable because the state trajectories jump repeatedly on the SMSs determined by different modes. The state trajectory arrives at SMS and proceeds with ideal quasi-sliding mode motion, and the sliding function satisfies sk+1 = sk = 0. eq

The equivalent control u k is calculated by solving above equation u k = (G B1i )−1 G K xk − (G B1i )−1 G B2i ωk − (G B1i )−1 G (Ai + ΔAi (k)) xk − f i (xk , k). eq

(8.6)

158

8 Semi-Markov Jump Systems eq

Substituting equivalent control u k into original system (8.1), hence, the sliding mode dynamics equation is therefore succinctly expressed as 

i (k)xk + Bˆ 2i ωk , E xk+1 = A z k = Ci xk + Di ωk ,

(8.7)

i (k) = Aˆ i + Δ Aˆ i (k), Bˆ 2i =

where A B1i B2i , Aˆ i =

B1i Ai + B¯ 1i K , Δ Aˆ i (k) =

B1i = I − B¯ 1i , B¯ 1i = B1i (G B1i )−1 G. B1i ΔAi (k),

8.3.2 Admissibility Analysis of Closed-Loop System with Mixed H∞ and Passivity This part is devoted to how to make the closed-loop system (8.7) have the desired performance once the state trajectories reach the pre-determined SMS. Theorem 8.1 Given constants γ > 0, 0 ≤ σ ≤ 1, the system (8.7) is stochastically admissible with the mixed H∞ /passive performance index γ , if for any i, j ∈ H and i ∈ N+ , there exists scalars β > 0, εi > 0, a set of symmetric matrices finite ST Tmax i −1] , matrices Si and any matrix R that satisfies R E = 0 {Pi (ε) > 0}, ε ∈ N[0,Tmax and rank(R) = n − r , such that the following inequalities hold ⎡

Φi11 ⎢ ∗ ⎢ Ωi = ⎢ ⎢ ∗ ⎣ ∗ ∗ i Tmax 

ε=1

Φi12 Φi22 ∗ ∗ ∗

Φi13 0 Φi33 ∗ ∗ 

j=i, j∈H υi j (ε) i Tmax  ε=1 j=i, j∈H

Φi14 Φi24 0 Φ44 ∗

⎤ Φi15 Φi25 ⎥ ⎥ Φi35 ⎥ ⎥ < 0, 0 ⎦ Φi55

P j (0) − Pi (ε) υi j (ε)

(8.8)

 < 0,

(8.9)

where Φi11 = −E T Pi (ε)E + SiT R Aˆ i + Aˆ iT R T Si + εi NiT Ni + β I , Φi12 = −(1− √ σ )CiT + Aˆ iT Pi (ε + 1) Bˆ 2i + SiT R Bˆ 2i , Φi13 = Aˆ iT Pi (ε + 1), Φi14 = σ CiT , Φi15 =   √ B1i Mi , Φi22 = −γ I − (1 − σ ) DiT + Di + Bˆ 2iT Pi (ε + 1) Bˆ 2i , Φi24 = σ DiT , SiT R

B1i Mi , Φi33 = −Pi (ε + 1), Φi35 = Pi (ε + 1)

B1i Mi , Φ44 = Φi25 = Bˆ 2iT Pi (ε + 1)

−γ I , Φi55 = −εi I . Proof We consider as the first part of the proof the verification of stochastic admissibility and mixed H∞ /passivity for nominal system, i.e., ΔAi (k) = 0. The regularity and causality are demonstrated here. The nonsingular matrices J1 and J2 are computed with the existing matrices to obtain

8.3 Main Results Eˆ = J1 E J2 =

159 

 Ir 0 , 0 0

 A¯¯ A¯¯ i = J1 Aˆ i J2 = ¯ i11 A¯ i21

 Sˆ Sˆi = J1−T Si J2 = ˆi11 Si21

 Sˆi12 , Sˆi22

   ˆ ˆ A¯¯ i12 ˆ = J1 R J −1 = R11 R12 , , R 1 Rˆ21 Rˆ22 A¯¯ i22

 Pˆ (ε) Pˆi (ε) = J1−T Pi (ε)J1−1 = ˆi11 Pi21 (ε)

 Pˆi12 (ε) . Pˆi22 (ε)

From R E = 0, we have Rˆ Eˆ = 0 and thus Rˆ 11 = Rˆ 21 = 0. It is clear from (8.8) that −E T Pi (ε)E + SiT R Aˆ i + Aˆ iT R T Si < 0 and thus    J2T −E T Pi (ε)E + SiT R Aˆ i + Aˆ iT R T Si J2 = 





¯ ¯¯ T Rˆ A T sym Sˆi12 12 ¯ i22 + Sˆi22 Rˆ22 A i22

  < 0,

(8.10)  where “ ” will not play a role in the subsequent certification process. If A¯¯ i22 is a singular matrix, then there exists ζ = 0 that makes A¯¯ i22 ζ = 0 and thus   T ˆ T ˆ R12 A¯¯ i22 + Sˆi22 R22 A¯¯ i22 ζ = 0, ζ T sym Sˆi12 which contradicts the above inequality (8.10). The calculation shows that     det λE − Aˆ i = det(J1−1 ) det λ Eˆ − A¯¯ i det(J2−1 ). The determinant in the middle of the right side of above equation is not always equal in the to zero for different λ due to the nonsingularity of A¯¯ i22 . Thus,the determinant  left-hand side is also not constant zero, and further deg det λE − Aˆ i = r . To facilitate the next proof, the following Lyapunov candidate functions associated with current mode and the residence time ηk on it are selected V (xk , rk , ηk ) = xkT E T Prk (ηk )E xk ,

(8.11)

where ηk = k − kΣk , Σk = max{n ∈ N|kn ≤ k}. The stochastic stability will be discussed next with ωk = 0. Assuming that rk = rk+1 = i when k ∈ N[kn ,kn+1 −2] , we have   E V (xk+1 , rk+1 , ηk+1 ) |xk ,rk ,ηk − V (xk , rk , ηk )   T E T Pi (ηk+1 )E xk+1 |xk ,rk ,ηk − xkT E T Pi (ηk )E xk =E xk+1 =xkT Aˆ iT Pi

(8.12)

(ηk+1 ) Aˆ i xk − xkT E T Pi (ηk )E xk .

According to the restriction R E = 0, it is straightforward to obtain 2xkT SiT R E xk+1 = 2xkT SiT R Aˆ i xk = 0.

(8.13)

160

8 Semi-Markov Jump Systems

Substituting formulas (8.13) into (8.12) yields   E V (xk+1 , rk+1 , ηk+1 ) |xk ,rk ,ηk − V (xk , rk , ηk ) ≤ xkT Ψi xk ,

(8.14)

where Ψi = Aˆ iT Pi (ηk+1 ) Aˆ i − E T Pi (ηk )E + SiT R Aˆ i + Aˆ iT R T Si . From inequality (8.8), we can obtain   Ψˆ i11 Φi13 < 0, ∗ Φi33 where Ψˆ i11 = −E T Pi (ε)E + SiT R Aˆ i + Aˆ iT R T Si + β I . Implementing Schur complement to above inequality yields Ψˆ i11 + Aˆ iT Pi (ε + 1) Aˆ i < 0. Since ηk = k − kn ∈ i −1] we can subsequently know that N[0,S n −2] ⊂ N[0,Tmax   E V (xk+1 , rk+1 , ηk+1 ) |xk ,rk ,ηk − V (xk , rk , ηk ) ≤ −βxkT xk < 0.

(8.15)

i −1] , from Eq. Meanwhile, when k + 1 = kn+1 , rk+1 = rk = i, ηk = Sn − 1 ∈ N[0,Tmax (8.8) we also get

   E V xkn+1 , rkn+1 −1 , Sn xk

 ,r ,η n+1 −1 kn+1 −1 kn+1 −1

  − V xkn+1 −1 , rkn+1 −1 , ηkn+1 −1 ≤ −βxkTn+1 −1 xkn+1 −1 < 0.

Furthermore, we can deduce from (8.9)       xkn+1 , rkn+1 , ηkn+1 xk ,rk − E V xkn+1 , rkn , Sn xk ,rk n n n n        xkn+1 , rkn+1 , 0 xk ,rk − E V xkn+1 , rkn , Sn xk ,rk n n n n i   Tmax  ε=1 j=i, j∈H υi j (ε) P j (0) − Pi (ε) E xkn+1 ≤ 0. =xkTn+1 E T i Tmax  ε=1 j=i, j∈H υi j (ε)  E V  =E V



(8.16)

Combining (8.15)–(8.16), we can immediately obtain the relationship between two adjacent jump moments    E V xkn+1 , rkn+1 , ηkn+1 xk    ≤E V xkn+1 , rkn , Sn xk    =E V xkn+1 , rkn , Sn xk

 n ,rkn

 n ,rkn

 n ,rkn

  − V x k n , r k n , ηk n

  − V x kn , r kn , 0    − E V xkn+1 −1 , rkn , ηkn+1 −1 xk

(8.17)  n ,rkn

8.3 Main Results

+

161

kn+1 −2 



  E V (xk+1 , rk+1 , ηk+1 ) xk

 n ,rkn

  − E V (xk , rk , ηk ) xk

 n ,rkn

k=kn

= − βE

⎧ −1 ⎨kn+1  ⎩

  T  xk xk 

k=kn

⎫ ⎬ xkn ,rkn ⎭

.

Summing over n from 0 to +∞, we can further obtain +∞      E V xkn+1 , rkn+1 , ηkn+1 xk n=0

 ,r 0 k0

   − E V x k n , r k n , ηk n  x k

 ,r 0 k0

    =E V (+∞)|xk0 ,rk0 − V xk0 , rk0 , 0 ⎫ ⎧  −1 +∞ kn+1 ⎬ ⎨   xkT xk  , ≤ − βE ⎭ ⎩

(8.18)

xk0 ,rk0

n=0 k=kn

which implicitly means that E

 +∞  k=0



 

xkT xk 

xk0 ,rk0

≤

 1  V xk0 , rk0 , 0 < +∞. β

(8.19)

Combined with the above proof and the requirements of Definition 8.1, the nominal system is capable of achieving stochastic admissibility. We next focus our proof on H∞ and passivity by introducing a performance index J +∞  # $ E γ −1 σ z kT z k − 2(1 − σ )z kT ωk − γ ωkT ωk . J = k=0

When k ∈ N[kn ,kn+1 −2] and rkn = i,     E V xk+1 , rk+1 , ηk+1 |xk ,rk ,ηk − V (xk , rk , ηk ) + γ −1 σ z kT z k − 2(1 − σ )z kT ωk − γ ωkT ωk   T E T P (η T T −1 σ z T z − 2(1 − σ )z T ω =E xk+1 i k+1 )E x k+1 |xk ,rk ,ηk − x k E Pi (ηk )E x k + γ k k k k − γ ωkT ωk T     = Aˆ i xk + Bˆ 2i ωk Pi ηk+1 Aˆ i xk + Bˆ 2i ωk − xkT E T Pi (ηk ) E¯ xk # $T # $ # $T + γ −1 σ Ci xk + Di ωk Ci xk + Di ωk − 2(1 − σ ) Ci xk + Di ωk ωk − γ ωkT ωk .

(8.20)

Similar to Eq. (8.13), again according to R E = 0 we know that 2xkT SiT R E xk+1 = 2xkT SiT R( Aˆ i xk + Bˆ 2i ωk ) = 0,

(8.21)

162

8 Semi-Markov Jump Systems

and let ηk = ε and ηk+1 = ε + 1, by inequality (8.8) we know that     E V xk+1 , rk+1 , ηk+1 |xk ,rk ,ηk − V (xk , rk , ηk ) + γ −1 σ z kT z k − 2(1 − σ )z kT ωk − γ ωkT ωk    # $ Ψ¯ i11 Φi12 + γ −1 σ C T Di xk i ≤ −βxkT xk < 0, ≤ xkT ωkT T −1 ∗ Φi22 + γ σ Di Di ωk

(8.22) where Ψ¯ i11 = Aˆ iT Pi (ε + 1) Aˆ i − E T Pi (ε)E + SiT R Aˆ i + Aˆ iT R T Si + γ −1 σ CiT Ci . Inequality (8.23) also holds when k + 1 = kn+1 , rk+1 = i, ηk+1 = Sn by inequality (8.8). Thus combining (8.16) and (8.22) yields       E V xkn+1 , rkn+1 , ηkn+1 xk ,rk − V xkn , rkn , ηkn n n ⎫ ⎧ −1 ⎬ ⎨kn+1  # $ γ −1 σ z kT z k − 2(1 − σ )z kT ωk − γ ωkT ωk +E ⎭ ⎩ k=kn

≤

kn+1 −1 

   E V xk+1 , rkn , ηk+1 xk



 n ,rkn

   − E V x k , r k n , ηk  x k



(8.23)

n ,rkn

k=kn

⎫ ⎧ −1 ⎬ ⎨kn+1  # −1 T $ γ σ z k z k − 2(1 − σ )z kT ωk − γ ωkT ωk < 0. +E ⎭ ⎩ k=kn

Thus ⎧ ⎫ −1 T kn+1 ⎨ ⎬  # −1 T $ E γ σ z k z k − 2(1 − σ )z kT ωk − γ ωkT ωk ⎩ ⎭ n=0 k=kn

+

T      E V xkn+1 , rkn+1 , ηkn+1 xk n=0

 0

,rk0

   − E V x k n , r k n , ηk n  x k

 0

,rk0

< 0, (8.24)

which implicitly means that ⎧ ⎫ & %  T kn+1 ⎨  ⎬ −1  −1 T T T E γ σ z k z k − 2(1 − σ )z k ωk − γ ωk ωk < −E V xk T +1 , rk T +1 , 0 x ,r k0 k0 ⎩ ⎭ n=0 k=kn

(8.25) under zero initial conditions. As n tends to +∞, we can see that J < 0 as well as  +∞  +∞    −1 T T T E γ σ z k z k − 2(1 − σ )z k ωk < γ E ωk ωk . k=0

k=0

(8.26)

8.3 Main Results

163

Combining the above analysis, according to Definitions 8.1 and 8.2, the nominal system can achieve stochastical admissibility with mixed H∞ /passivity when the conditions in Theorem 8.1 are satisfied. Next assume that ΔAi (k) = 0. Implementing the Schur complement to inequality (8.8) yields Ω¯ i + εi−1 Γ1i Γ1iT + εi Γ2i Γ2iT < 0, (8.27) and then according to Lemma 8.1, we can get from the above inequality Ω¯ i + Γ1i Fi (k)Γ2iT + Γ2i FiT (k)Γ1iT < 0,

(8.28)

where ⎡

Ψˆ i11 ⎢ ∗ Ω¯ i = ⎢ ⎣ ∗ ∗

Φi12 Φi22 ∗ ∗

Φi13 0 Φi33 ∗

⎤ ⎤ ⎡ ⎡ T⎤ Φi15 Ni Φi14 ⎥ ⎢ ⎢ ⎥ Φi24 ⎥ ⎥ , Γ1i = ⎢Φi25 ⎥ , Γ2i = ⎢ 0 ⎥ . ⎣Φi35 ⎦ ⎣ 0 ⎦ 0 ⎦ 0 0 Φ44

i (k). Applying Schur complement to In this case, Aˆ i in (8.22) is replaced by A (8.28) that inequality (8.22) holds. And the next analytical steps are similar to (8.24)– (8.27) above, thus system (8.7) can achieve stochastic admissibility with the mixed H∞ /passivity. The sufficient criterion for the mixed H∞ /passivity and stochastic admissibility of system (8.7) is provided in Theorem 8.1, but the actual form of controller gain matrix K is not yet given. The following Theorem 8.2 can guarantee the excellent performance of the system (8.7) and calculate K . Theorem 8.2 Given constants γ > 0, 0 ≤ σ ≤ 1, the system (8.7) is stochastically admissible with the mixed H∞ and passive performance index γ , if for any i, j ∈ H i ∈ N+ , there exists scalars β > 0, εi > 0, matrices Si , R, W , and finite ST Tmax i −1] , U > 0 such that the following symmetric matrix set {Pi (ε) > 0}, ε ∈ N[0,Tmax inequality holds ⎡

i11 Φ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

i12 Φ Φi22 ∗ ∗ ∗ ∗ ∗

i Tmax 

ε=1

i13 Φ 0 Φi33 ∗ ∗ ∗ ∗

Φi14 Φi24 0 Φ44 ∗ ∗ ∗

Φi15 Φi25 Φi35 0 Φi55 ∗ ∗ 

j=i, j∈H υi j (ε) i Tmax  ε=1 j=i, j∈H

i16 Φ

i26 Φ

i36 Φ 0 0 −U ∗

⎤ WT 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ < 0, 0 ⎥ ⎥ 0 ⎦ −U

P j (0) − Pi (ε) υi j (ε)

(8.29)

 < 0,

(8.30)

164

8 Semi-Markov Jump Systems

i11 = −E T Pi (ε)E + SiT R

i12 = where Φ B1i Ai + AiT

B1iT R T Si + εi NiT Ni + β I , Φ T T T T T T ˆ ˆ

i16 = −(1 − σ )Ci + Ai B1i Pi (ε + 1) B2i + Si R B2i , Φi13 = Ai B1i Pi (ε + 1), Φ T T ¯ ˆ ¯ ¯



Si R B1i , Φi26 = B2i Pi (ε + 1) B1i , Φi36 = Pi (ε + 1) B1i . R is an arbitrary matrix that makes R E = 0 and rank(R) = n − r . And controller gain matrix K = U −1 W . Proof Decompose Aˆ i in Ωi of Theorem 8.1, we get

i + Γ3i Γ4iT + Γ4i Γ3iT , Ωi = Ω where ⎡

i11 Φ ⎢ ∗ ⎢

i = ⎢ ∗ Ω ⎢ ⎣ ∗ ∗

i12 Φ Φi22 ∗ ∗ ∗

i13 Φ 0 Φi33 ∗ ∗

Φi14 Φi24 0 Φ44 ∗

⎡ ⎡ T⎤ ⎤ ⎤

i16 Φ K Φi15 ⎢Φ ⎢ 0 ⎥

⎥ Φi25 ⎥ ⎢ i26 ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ Φi35 ⎥ ⎥ , Γ3i = ⎢Φi36 ⎥ , Γ4i = ⎢ 0 ⎥ . ⎣ 0 ⎦ ⎣ 0 ⎦ 0 ⎦ 0 Φi55 0

Furthermore, in accordance with Lemma 8.2, we can directly know that there exist matrices U > 0 to derive

i + Γ3i U −1 Γ3iT + Γ4i U Γ4iT . Ωi ≤ Ω The implementation of Schur complement to the known condition (8.30) soon reveals

i + Γ3i U −1 Γ3iT + Γ4i U Γ4iT < 0, which means that inequality (8.8) holds, and that Ω combined with condition (8.31), the requirements of Theorem 8.1 can be fully satisfied and the system (8.7) can achieve stochastic admissibility with H∞ /passive performance index γ .

8.3.3 Adaptive SMC Law Design The assignment of this part is to synthesize an adaptive SMC law to ensure the accessibility of the designed SMS. Due to the parameter uncertainties ΔAi (k), unknown nonlinear functions f i (xk , k), and external disturbances ωk in system (8.1), the adaptive method is applied to estimate their upper bounds. Although ΔAi (k), f i (xk , k), and ωk are unknown, we can reasonably introduce the unknown parameters ρ1 > 0 and ρ2 > 0 based on the known conditions to obtain the following estimate χk = G Ai  xk  + G Mi  Ni  xk  + G B1i   f i (xk , k) + G B2i  ωk  ≤ρ1 xk  + ρ2 . (8.31) Theorem 8.3 The SMS (8.5) has been designed, where K is derived from Theorem 8.2. In order to enable the state trajectory to reach the neighborhood containing the equilibrium point, the following adaptive controller is devised

8.3 Main Results

165

# $ u k = (G B1i )−1 G K xk − Ξ sk + sk − χˆ k sat(sk ) ,

(8.32)

where matrix Ξ > 0, ⎧ ⎨

sk , sk  ≤ α, α sat(sk ) = ⎩ sign(s ), s  > α, k k χˆ k = ρˆ1 (k)xk  + ρˆ2 (k), ρˆi (k) are the tracking parameters of ρi , i = 1, 2, the error terms are represented by ρ

i (k) = ρi − ρˆi (k), and the adaptation law are devised as Δρˆ1 (k) =ρˆ1 (k + 1) − ρˆ1 (k) = q1 (−ι1 ρˆ1 (k) + xk sk ), Δρˆ2 (k) = ρˆ2 (k + 1) − ρˆ2 (k) = q2 (−ι2 ρˆ2 (k) + sk ), with qi > 0, ιi > 0, i = 1, 2. Proof The proof starts with the selection of candidate Lyapunov functions Vs (k) =

1 T 1 2 1 2 s sk + ρ

(k) + ρ

(k). 2 k 2q1 1 2q2 2

(8.33)

Substituting the controller (8.32) into the system (8.1), and then according to the SMS (8.5) will generate sk+1 = G (Ai + ΔAi (k)) xk + G B1i f i (xk , k) + G B2i ωk − Ξ sk + sk − χˆ k sat(sk ). (8.34) ρ2 (k) = −Δρˆ2 (k), The conditions of Theorem 8.3 lead to Δ

ρ1 (k) = −Δρˆ1 (k) and Δ

which is further combined with (8.31) and (8.34) to yield E {ΔVs (k)} = E {Vs (k + 1)} − Vs (k) 1 1 1 = skT Δsk + ΔskT Δsk + ρ

1 (k)Δ

ρ1 (k) + Δ

ρ1 (k)Δ

ρ1 (k) 2 q1 2q1 1 1 + ρ

2 (k)Δ

ρ2 (k) + Δ

ρ2 (k)Δ

ρ2 (k) q2 2q2 = skT [G (Ai + ΔAi (k)) xk + G B1i f i (xk , k) + G B2i ωk ] − skT Ξ sk 1 1 − skT χˆ k sat(sk ) − ρ

1 (k)Δρˆ1 (k) − ρ

2 (k)Δρˆ2 (k) + Π0 q1 q2 ≤ sk  (ρ1 xk  + ρ2 ) − sk  (

ρ1 (k)xk  + ρ

2 (k)) + ι1 ρ

1 (k)ρˆ1 (k) + ι2 ρ

2 (k)ρˆ2 (k) − skT Ξ sk − χˆ k skT sat(sk ) + Π0 ≤ sk χˆ k +ι1 ρ

1 (k)ρˆ1 (k) + ι2 ρ

2 (k)ρˆ2 (k) − skT Ξ sk − χˆ k skT sat(sk ) + Π0 , (8.35) where Π0 = 21 ΔskT Δsk + If sk  > α, we get

1 Δρˆ12 (k) 2q1

+

1 Δρˆ22 (k). 2q2

166

8 Semi-Markov Jump Systems

  ρ1 2 ρ2 2 E {ΔVs (k)} ≤ − skT Ξ sk − ι1 ρˆ1 (k) − − ι2 ρˆ2 (k) − + Π1 , 2 2

(8.36)

where Π1 = Π0 + 41 ι1 ρ12 + 41 ι2 ρ22 . If sk  ≤ α, we get   ρ1 2 ρ2 2 χˆ k  α 2 E {ΔVs (k)} ≤ − skT Ξ sk − ι1 ρˆ1 (k) − − ι2 ρˆ2 (k) − − + Π2 , sk  − 2 2 α 2

(8.37) where Π2 = Π1 + 41 α χˆ k . Combined with the above analysis, when sk is outside a bounded region containing the equilibrium point, picking an appropriate matrix Ξ so that E {ΔVs (k)} < 0, which means that the state trajectories will remain in the switching region containing the origin under the action of the controller (8.32). Remark 8.3 Adaptive control has the ability of real-time recognition and adjustment, which is important for suppressing and eliminating the limitation of unmeasurable control system variables and enhancing the quality of SMC. Meanwhile, this paper uses saturation function instead of sign function to weaken the chattering problem existing in the traditional SMC. In the next simulation Examples 2 and 3, the adaptive SMC strategy used in this note is compared with the control strategy in [11] to further highlight its effectiveness.

8.4 Simulations Three simulation examples will be performed in this section to verify the stability criterion in Theorem 8.2 that can be numerically tested and the feasibility of adaptive SMC in Theorem 8.3, and the effectiveness of the technique in this work will be further demonstrated by comparing it with the control strategy in [11]. Example 8.1. This simulation example is for DT SSMJS (8.1) with H = 3 and each mode has the under-listed parameters: Mode 1: ⎤ ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ ⎡ 0.31 0.1 0.11 0.09 −0.1 0.18 A1 = ⎣0.09 −0.93 −0.97⎦ , B11 = ⎣0.64⎦ , B21 = ⎣−0.52⎦ , M1 = ⎣−0.55⎦ , 0.25 0.62 0.12 0.38 0.7 −0.17 # $ # $ C1 = 0.17 0.68 −0.03 , D1 = 0.35, N1 = 0.29 −0.12 −0.05 .

8.4 Simulations

167

Mode 2: ⎤ ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ 0.62 −0.56 −0.23 0.49 −0.26 −0.84 A2 = ⎣ 0.04 −0.11 0.08 ⎦ , B12 = ⎣ 0.3 ⎦ , B22 = ⎣ 0.04 ⎦ , M2 = ⎣−0.47⎦ , −0.16 0 0.96 −0.77 −0.18 1.2 ⎡

# $ # $ C2 = −0.41 0.16 −0.36 , D2 = 0.97, N2 = 0.28 −0.8 −0.99 .

Mode 3: ⎤ ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ 0.95 0.17 −0.97 −0.72 0.1 0.21 A3 = ⎣ 0.08 0.69 −0.62⎦ , B13 = ⎣ −0.2 ⎦ , B23 = ⎣ 0.26 ⎦ , M3 = ⎣ 0.04 ⎦ , −0.74 0 −0.18 0.63 −0.38 −0.91 ⎡

# $ # $ C3 = −0.19 −0.56 −1.11 , D3 = 0.42, N3 = 0.23 −0.77 −1.24 .

The other parameters are selected as ⎡

⎤   102 0 0 0.45 E = ⎣0 0 1⎦ , R = . 0 0 1.31 000 Since deg(det(λE − A2 )) = deg(−0.0512λ + 0.0328) = 1 < 2 = rank(E) and deg(det(λE − A3 )) = deg(0.7712λ + 0.5328) = 1 < 2, which indicates that system (8.1) is non-causal and thereby the uncontrolled system is not stochastically admissible. The switching between modes above is controlled by a semi-Markov chain in which the semi-Markov kernel is computed by transition probability matrix ⎡

⎤ 0 0.55 0.45 Λ = ⎣0.64 0 0.36⎦ 0.3 0.7 0 and probability density function ⎡ 1.4 1.4 0 0.5(ε−1) − 0.5ε # $ ⎢ 2 2 i j (ε) = ⎣ 0.6(ε−1) − 0.6ε 0 1.3 1.3 1.2 1.2 0.4(ε−1) − 0.4ε 0.7(ε−1) − 0.7ε

⎤

10!0.2ε 0.810−ε ε!(10−ε)! ⎥ 10!0.510 . ε!(10−ε)! ⎦

0

The evolution of system modes and the residence time of each jump are illustrated in Fig. 8.1.

168

8 Semi-Markov Jump Systems

Fig. 8.1 The evolution of system modes in Example 8.1

4 3.5

system mode

3 2.5 2 1.5 1 0.5 0 0

10

20

30

40

50

time (sec)

The external disturbance ωk = 0.1esin(k) , nonlinear functions f i (xk , k) = 0.2e−k sin(xik# ) and Fi (k) = 0.5 Set $ cos(k), i = 1, 2, 3, are used in the simulation process. 1 G = 0.24 −0.57 0.25 such that G B1i are nonsingular. Choosing σ = 0.49, Tmax = 2 3 = 4 and Tmax = 2, Theorem 8.2 generates a feasible solution γ = 0.73 for 4, Tmax H∞ and passivity, and the following gain matrix ⎡ ⎤ 0.633 −0.028 −0.179 K = ⎣4.015 −0.130 −0.335⎦ . 3.254 −0.128 −0.669 Therefore SMS (8.5) are fully defined and system (8.7) is stochastically admissible. The adjustable parameters are chosen as ι1 = 0.15, ι2 = 0.12, q1 = q2 = 0.001, Ξ = 0.05, α = 0.02, then the controller (8.32) can be constructed. Combining the above results we obtain the following simulation images, which are displayed in # $T Figs. 8.2, 8.3, 8.4 and 8.5 with initial condition x0 = 2.6 2.8 −2.4 . These figures show that system state can eventually be stabilized, indicating that the adaptive SMC tactics designed in this manuscript are effective. Example 8.2. As a verification of the feasibility and effectiveness of the adaptive SMC method, it is applied to a practical DC motor operation in the following. The DC motor model is characterized as a continuous-time Markov jump SS under the assumption of sudden and random load changes [27]. Considering that the system often suffers from measurement noise, external disturbances, and input unmodeled dynamics during the actual drive, in this example, the time-varying uncertainties, nonlinear functions, and external disturbances of interest in the model are taken into account. By setting a suitable sampling interval, such as T = 0.1s, we can discretize the continuous-time model into the system (8.1) with three modes, and the state can be

8.4 Simulations

169

Fig. 8.2 The simulation for state xk in Example 8.1

3

2

state vector

1

0

-1

-2

-3 0

10

20

30

40

50

40

50

time (sec)

Fig. 8.3 The simulation for sliding surface sk in Example 8.1 sliding mode function

1

0.5

0

-0.5 0

10

20

30

time (sec)

 ik , where i k is the current and wk is the shaft speed, and characterized as xk = wk the other parameters are 

 E=

       R Kw 00 −1 0.04 , B1i = , Ai = K t T , B21 = , bT 01 0 0.19 J 1− J i

B22 =

i

    # $ 0.66 0.04 , B23 = , Ci = 0 1 , D1 = −0.23, D2 = 0.4, D3 = −0.04, 0.05 −0.03

R = 3.14, K w = 2.05, K t = 5.45, b = 0.33, J1 = 5.29, J2 = 3.39, J3 = 1.43.

170

8 Semi-Markov Jump Systems

Fig. 8.4 The simulation for controller u k in Example 8.1 adaptive sliding mode controller

2

1

0

-1

-2

-3

-4 0

10

20

30

40

50

40

50

time (sec)

Fig. 8.5 The evolution of adaptive variables ρˆk in Example 8.1

0.03

adaptive variables

0.025

0.02

0.015

0.01

0.005

0 0

10

20

30

time (sec)

    −0.49 0.54 Suppose the system uncertainty is expressed by M1 = , M2 = , 0.26 0.03   # $ # $ # $ 0.4 M3 = , N1 = 0.13 −0.19 , N2 = −0.18 −0.02 , N3 = 0.28 0.36 , −0.06 Fi (k) = 0.5 sin(k), the nonlinearity is f i (xk , k) = 0.1e−k sin(k)x1 (k), and the external disturbance ω(k) = 0.05etan(sin(k)) , i = 1, 2, 3. The jump between modes is determined by the computation of

8.4 Simulations

171

Fig. 8.6 The evolution of the system modes in Example 8.2

4 3.5

system mode

3 2.5 2 1.5 1 0.5 0 0

5

10

15

20

25

30

time (sec)

⎡

⎤ 0 0.51 0.49 Λ = ⎣0.54 0 0.46⎦ , 0.15 0.85 0 ⎤ ⎡ 10!0.7ε 0.310−ε 10!0.4ε 0.610−ε 0 ε!(10−ε)! ε!(10−ε)! $ ⎢ # 1.5 1.5 ⎥ i j (ε) = ⎣0.5(ε−1)1.3 − 0.5ε1.3 0 0.6(ε−1) − 0.6ε ⎦ . 1.3 1.3 1.5 1.5 0.3(ε−1) − 0.3ε 0.6(ε−1) − 0.6ε 0 The evolution of system modes and the residence time of each jump are illustrated in Fig. 8.6. # $ Set G = −0.12 −0.81 such that G B1i are nonsingular. Choosing σ = 0.6, γ = 1 2 3 = 2, Tmax = 3 and Tmax = 5, Theorem 8.2 generates a feasible solution 1.78, Tmax for H∞ and passivity, and the following gain matrices  K =

 −0.5 −9.33 . −0.14 −3.35

Therefore SMS (8.5) are fully defined and system (8.7) is stochastically admissible. The adjustable parameters are chosen as ι1 = 0.21, ι2 = 0.11, q1 = q2 = 0.001, Ξ = 0.1, α = 0.03, then the controller (8.33) can be constructed. Combining all the above results, we show the simulation image in Fig. 8.7 with an # $T initial value x0 = 2.4 −2.6 . To further illustrate the superiority of the approach in this work, the time-varying mode-dependent controller u k = K i (ηk )xk in [11] is applied to the DT SSMJS (8.1), corresponding to the same parameters and initial values to obtain the simulation image in Fig. 8.8. Comparing the two simulation results clearly shows that although the control strategy in [11] is also capable of achieving stabilization of the actual model in this example, the control method in this paper enables the model to be stabilized more expeditiously.

172

8 Semi-Markov Jump Systems

Fig. 8.7 The state xk with controller (8.32) in Example 8.2

3

2

state vector

1

0

-1

-2

-3 0

5

10

15

20

25

30

20

25

30

time (sec)

Fig. 8.8 The state xk with u k = K i (ηk )xk [11] in Example 8.2

3

2

state vector

1

0

-1

-2

-3 0

5

10

15

time (sec)

Example 8.3. In the simulation of this subsection, we apply the controller designed in this paper and the control strategy proposed in [11] to an experimental model of disc rolling [7], which can be described as a DT SSMJS (8.1) with three modal $T # random jumps about the mass of the disc. The system state xk = x1k x2k x3k x4k contains four components, which represent the position and translation velocity of the disc center, the angular velocity of the disc, and the contact force between the disc and the plane, respectively. The relevant parameters correspond to the following:

8.4 Simulations ⎡

1 ⎢0 E =⎢ ⎣0 0

0 1 0 0

0 0 0 0

173

⎡ ⎤ 1 T 0 ⎢− T K s 1 − T b ⎥ 0⎥ mi ⎢ m , Ai = ⎢ 0 i 1 0⎦ ⎣ 0 − mb − mK i i

0 0 −r  2 0 rJ

⎤

⎡ ⎤ ⎡ ⎤ 0 0 ⎥ ⎢ ⎥ ⎢0 ⎥ 0 ⎥ ⎥ ⎢ ⎥ ⎥ , B1i = ⎢ ⎣ 0 ⎦ , B2i = ⎣0⎦ , 0 ⎦ r 1 −J 1 +m 0

T mi

i

# $ # $ # $ C1 = 0 −0.7 0.4 0 , C2 = 0 −0.7 0 −0.1 , C3 = −0.2 0.2 0.7 0.2 , ⎤ ⎤ ⎡ ⎡ ⎤ −0.6 −1 0.5 ⎢ 0.7 ⎥ ⎢−0.3⎥ ⎢0⎥ ⎥ ⎥ ⎢ ⎢ ⎢ , M2 = ⎣ , M3 = ⎣ ⎥ , D1 = 0.07, D2 = −0.25, D3 = −0.03, M1 = ⎣ −0.1⎦ −0.2⎦ 1⎦ 0 0.4 0 ⎡

# $ # $ # $ N1 = 0.8 −0.1 0 0.6 , N2 = 0 0.6 0.4 0.2 , N3 = −0.4 −0.7 0 −0.6 , m 1 = 22, m 2 = 11, m 3 = 15, r = 16.5, K s = 96, b = 30, J = 1.4. Suppose the system suffers from Fi (k) = 0.3 + 0.1 sin(xik ), f i (xk , k) = 0.15 sin(x1k )x2k + 0.1i, ωk = 0.1 sin(5k)e−0.1k , i = 1, 2, 3, and the modes follow ⎡ 1.5 1.5 ⎤ 0 0.6(ε−1) − 0.6ε 0 0.54 0.46 # $ ⎢ 1.6 1.6 (ε−1) ε Λ = ⎣0.68 0 0.32⎦ , i j (ε) = ⎣0.5 − 0.5 0 1.5 1.5 1.4 1.4 0.15 0.85 0 0.4(ε−1) − 0.4ε 0.7(ε−1) − 0.7ε

⎤

⎡

10!0.4ε 0.610−ε ε!(10−ε)! 10!0.3ε 0.710−ε ⎥ . ε!(10−ε)! ⎦

0

The evolution of system modes and the residence time of each jump are illustrated in Fig. 8.9. The other parameters are selected as   # $ 0 0 −0.2 0.2 R= , G = −0.75 0.97 1.21 −0.05 . 00 1 0

Fig. 8.9 The evolution of system modes in Example 8.3

4 3.5

system mode

3 2.5 2 1.5 1 0.5 0 0

50

100 time (sec)

150

200

174

8 Semi-Markov Jump Systems

1 2 3 By solving (8.30) and (8.31) in Theorem 8.2 with Tmax = 4, Tmax = 3, Tmax = 3, σ = 0.8, γ = 0.95, a feasible solution with

⎡

13.31 ⎢−11.98 K =⎢ ⎣ 18.61 −0.59

−2.52 0.72 3.1 −8.02

−0.61 4.54 −6.97 −4.4

⎤ −3.61 −3.27 ⎥ ⎥ −20.64⎦ 1.59

is generated. The other parameters in the controller (8.32) are selected as in Example 8.2, which can be further applied to the original system (8.1) with initial condition # $T x0 = 1.5 −2.7 −1.8 2.2 to obtain the simulation result as shown in Fig. 8.10. To further illustrate the advantage of the presented approach, we will compare it with the approach presented in [11]. As shown in Fig. 8.11, although the approach

Fig. 8.10 State trajectory xk under the action of controller (8.32)

3

2

state vector

1

0

-1

-2

-3 0

50

100

150

200

150

200

time (sec)

Fig. 8.11 State trajectory xk under the action of controller u k = K i (ηk )xk [11]

10 5 1 0.8 0.6

state vector

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0

50

100

time (sec)

References

175

in [11] provides a very effective solution for uncertain semi-Markov jump systems with external disturbances, it probably cannot handle the complex case of SSs with unknown nonlinear functions. Therefore, the adaptive SMC law can be useful to achieve system stability with satisfactory performance.

8.5 Conclusion This paper has addressed the synthesis of the adaptive SMC law for DT SSMJS and stochastic admissibility analysis. A common SMS has been constructed to obtain sufficient criteria for mixed H∞ /passive performance and stochastic admissibility. Linear matrix inequality techniques have been applied to ensure excellent performance of the system while solving for the gain matrix. The adaptive SMC law has been devised for the purpose of ensuring the accessibility of the pre-defined SMS. Finally, the excellence of the obtained results has been fully illustrated by three simulation tests, which are compared with the approach in [11]. The research technique proposed in this work can be tried to be extended to study the analysis and synthesis of DT SSMJSs with time delay, which we will investigate in the future.

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12. L. Fu, Y. Ma, C. Wang, Memory sliding mode control for semi-Markov jump system with quantization via singular system strategy, International Journal of Robust and Nonlinear Control, 29(18) (2019) 6555–6571. 13. H. Wang, G. Liu, Sliding-mode-based admissible consensus of nonlinear singular stochastic uncertain semi-Markov multi-agent systems with time-varying delay, The Journal of Engineering, 2022(7)(2022) 690–700. 14. S. Li, J. Lian, L. Gong, Hidden Markov model based H∞ filtering for singular semi-Markov jump systems, International Journal of Robust and Nonlinear Control, 32(1) (2022) 164–180. 15. Y. Li, B. Kao, J.H. Park, Y. Kao, B. Meng, Observer-based mode-independent integral sliding mode controller design for phase-type semi-Markov jump singular systems, International Journal of Robust and Nonlinear Control, 29(15) (2019) 5213–5226. 16. Q. Ren, Y. Kao, C. Wang, H. Xia, X. Wang, New results on the generalized discrete reaching law with positive or negative decay factors, IEEE Transactions on Automatic Control, 67(2) (2022) 1046–1052. 17. Y. Kao, S. Ma, H. Xia, C. Wang, Y. Liu, Integral sliding mode control for a kind of impulsive uncertain reaction-diffusion systems, IEEE Transactions on Automatic Control, 68(2) (2022) 1154–1160. 18. Y. Kao, X. Liu, M. Song, L. Zhao, Q. Zhang, Non-fragile-observer-based integral sliding mode control for a class of uncertain switched hyperbolic systems, IEEE Transactions on Automatic Control, https://doi.org/10.1109/TAC.2022.3217103. 19. Z. Che, H. Yu, C. Yang, L. Zhou, Passivity analysis and disturbance observer-based adaptive integral sliding mode control for uncertain singularly perturbed systems with input nonlinearity, IET Control Theory & Applications, 13(18) (2019) 3174–3183. 20. P. Zhang, Y. Kao, J. Hu, B. Niu, Robust observer-based sliding mode H∞ control for stochastic Markovian jump systems subject to packet losses, Automatica, 130(2021) 109665. 21. P. Zhang, Y. Kao, G. Ran, C. Wang, Design of sliding mode fault-tolerant control for Markovian jump systems with probabilistic delay: A discrete partitioning strategy, International Journal of Robust and Nonlinear Control, 33(3) (2023) 2060–2077. 22. W. Qi, G. Zong, Y. Hou, M. Chadli, SMC for discrete-time nonlinear semi-Markovian switching systems with partly unknown semi-Markov kernel, IEEE Transactions on Automatic Control, 68(3) (2023) 1855–1861. 23. W. Qi, X. Yang, J. H. Park, J. Cao, J. Cheng, Fuzzy SMC for quantized nonlinear stochastic switching systems with semi-Markovian process and application, IEEE Transactions on Cybernetics, 52(9) (2022) 9316–9325. 24. W. Qi, C. Zhang, G. Zong, S. Su, M. Chadli, Finite-time event-triggered stabilization for discrete-time fuzzy Markov jump singularly perturbed systems, IEEE Transactions on Cybernetics, https://doi.org/10.1109/TCYB.2022.3207430. 25. Y. Wang, L. Xie, C. E. De Souza, Robust control of a class of uncertain nonlinear-systems, Systems & control letters, 19(2) (1992) 139–149. 26. K. Zhou, P. Khargonekar, Robust stabilization of linear systems with norm-bounded timevarying uncertainty, Systems & control letters, 10(1) (1988) 17–20. 27. E.K. Boukas, Control of singular systems with random abrupt changes, Springer Science & Business Media, 2008.

Chapter 9

Adaptive Fault-Tolerant Control

This note is committed to the investigation of the adaptive sliding mode control issue for nonlinear Markovian jumping discrete-time systems subjected to uncertain and actuator-fault signals. In order to improve practicality in practical systems, a linear mode-independent switched surface is constructed. Moreover, a designed adaptive sliding mode controller is employed to guarantee the reachability of the sliding manifold. Sufficient conditions to ensure the stochastic stability of the sliding dynamics are derived in forms of LMIs. In the end, a numerical example is provided to show the effectiveness of our theoretical findings.

9.1 Introduction Markovian jumping discrete-time systems (MJDSs) have received increasing attentions [1–6]. The filtering problems have been studied for MJDSs in [7–9]. The asynchronous control problems is developed in [10]. H∞ control of MJDSs is developed in [11]. In [12], the authors discussed event-triggered fault detection for MJDSs in finite frequency domain. The sliding mode control (SMC) is a well-known robust strategy for its advantages, such as invariance to parametric uncertainties, simple design technique and effective control for nonlinearity. During the past few years, the combinations of SMC method and other control techniques have become hot topics [13–15]. Hu et al. studied the SMC of discrete-time uncertain Markov jumping systems in [16]. Zhang and Kao et al. studied the sliding mode H∞ control for stochastic Markovian jump systems subject to packet losses based on observer method [17]. By singular system method, Yao et al. discussed the sliding mode output-feedback control of Markov jumping discrete-time systems in [18]. In [19], Yao et al. probed the event-triggered SMC for Markov jumping discrete-time systems. Two kinds of event-triggered SMC, which are observer-based event-triggered SMC and event-triggered state-feedback SMC, are employed to handle the proposed problem. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Kao et al., Analysis and Design of Markov Jump Discrete Systems, Studies in Systems, Decision and Control 499, https://doi.org/10.1007/978-981-99-5748-4_9

177

178

9 Adaptive Fault-Tolerant Control

In the physical control plants, actuator fault may occur due to changes in the working environment, aging of components and so on. Actuator fault will not only deteriorate the performance of the control system, but also seriously destabilize it. Therefore, scholars have paid attention to the reliable control for the system subject to actuator faults [20]. Yang et al. designed a fault tolerant adaptive controller for a class of nonlinearity-unknown systems subject to multiple actuators [21]. Wang et al. studied the event-triggered asynchronous H∞ fault-tolerant control for discrete-time Markov jump system subject to actuator faults in [22]. Kchaou et al. investigated the adaptive SMC against the fault in nonlinearity [23]. However, the system in [23] did not take the uncertainty and time delay into consideration and the sliding surface is mode-dependent, which may result in the failure of the SMC in practical systems. Therefore, we devote to solving the adaptive sliding mode fault-tolerant control problem for a class of MJDSs with time-varying delays. The problem formulation is presented in Sect. 9.2. The sliding surface is built and the sufficient conditions for the stability are provided in Sect. 9.3. The adaptive sliding mode fault-tolerant controller which can ensure the reachability of the specified sliding surface is given in Sect. 9.4. An example is presented in Sect. 9.5.

9.2 Problem Statements and Preliminaries Let rk be a homogeneous Markovian chain taking values in discrete space N = {1, 2, . . . , m}. The transition probability matrix Π = [πi j ]i, j∈N , where πi j = Pr (rk+1 = j|rk = i) ≥ 0, ∀i, j ∈ N , k ∈ Z + with mj=1 πi j = 1 for each i ∈N. Consider a class of delayed nonlinear MJDSs as follows xk+1 =(Ark + A¯ rk )xk + (Adrk + A¯ drk )xk−dk + frk (k, xk , xk−dk ) + Brk u kF ,

(9.1)

where xk ∈ R n is the state vector and frk (k, xk , xk−dk ) ∈ R n is a nonlinear function. dk is the time-varying delay with d M ≥ dk ≥ dm , where d M and dm are known scalars. Ark ∈ R n×n , Adrk ∈ R n×n and Brk ∈ R n×s are known matrices and Brk is full column rank. A¯ rk and A¯ drk satisfying [ A¯ rk , A¯ drk ] = Mrk F(k)[Nrk , Ndrk ],

(9.2)

where F(k)T F(k) ≤ I , Mrk , Nrk and Ndrk are known matrices. The actuator fault model is represented as (9.3) u kF = αk u k , where u k ∈ R s is the control inputs and u kF denotes the signals sent from the actuator. αk = diag{α1k , α2k , . . ., αsk } represents the actuator fault matrix, and αik (1 ≤ i ≤ s)

9.3 Main Results

179

is the deterioration level of the ith actuator at instant k, satisfying 0 < αik ≤ αik ≤ αik ≤ 1, where αik and αik are lower and upper bounds of αik respectively. Denotes αk = diag(α1k , α2k , . . . , αsk ).

(9.4)

(9.5)

For simplicity, we use index i to replace rk in corresponding matrices and vectors for each rk = i ∈ N . Then, system (9.1) is rewritten as xk+1 =(Ai + A¯ i )xk + (Adi + A¯ di )xk−dk + f i (k, xk , xk−dk ) + Bi u kF ,

(9.6)

Assumption 9.1 Ai , Adi , Bi are controllable and observable. Assumption 9.2 The non-linear function f i (k, xk , xk−dk ) satisfies T NdiT Ndi xk−dk . f iT f i ≤ h 20 xkT NiT Ni xk + h 21 xk−d k

(9.7)

where h 0 and h 1 are known positive scalars.

9.3 Main Results The sliding function for system (9.6) is designed as follows sk = Gxk , s0 ∈ R s ,

(9.8)

to ensure the where s0 is the initial value of sliding surface. Matrix G mis designed βi BiT , and the scalars nonsingularity of G Bi . In this article, we choose G = i=1 βi (i ∈ N ) should be chosen such that G Bi is not singular for any Bi (i ∈ N ).

9.3.1 Stability Analysis The ideal sliding quasi-surface mode satisfies sk+1 = sk = 0, k > ks ,

(9.9)

where ks is the instant that ideal quasi-sliding mode starts. Then, it follows that

180

9 Adaptive Fault-Tolerant Control

u eq = − αk−1 (G Bi )−1 G[(Ai + A¯ i − I )xk + (Adi + A¯ di )xk−dk + f i (k, xk , xk−dk )].

(9.10)

Substituting (9.10) into (9.6) obtains the following sliding motion xk+1 = [(I − Bi (G Bi )−1 G)(Ai + A¯ i ) + Bi (G Bi )−1 G]xk + (I − Bi (G Bi )−1 G)(Adi + A¯ di )xk−dk + (I − Bi (G Bi )−1 G) f i (k, xk , xk−dk ).

(9.11)

Theorem 9.1 The system (9.11) is stochastically stable if there exist positive-definite matrices Pi (i ∈ N ), Q and positive scalars ε j ( j = 1, 2, 3, 4, 5, 6) satisfying the following LMIs: ⎡

Π1 ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

0 Π2 ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 Π3 ∗ ∗ ∗ ∗ ∗ ∗

⎤ √ √ T 7Ai P Mi 7AiT P˜ Mi √ 0 0 0 0 √ ⎥ T P˜ M T PM 0 0 7Adi 7Adi i i √0 √0 ⎥ ⎥ ˜ 0 0 0 0 7P 7 P ⎥ ⎥ −ε1 0 0 0 0 0 ⎥ ⎥ 0 0 0 0 ⎥ < 0, (9.12) ∗ −ε2 ⎥ 0 0 0 ⎥ ∗ ∗ −ε3 ⎥ ∗ ∗ ∗ −ε4 0 0 ⎥ ⎥ ∗ ∗ ∗ ∗ −P 0 ⎦ ∗ ∗ ∗ ∗ ∗ − P˜



− P˜ P˜ Mi MiT P˜ T −ε5 I −P P Mi MiT P T −ε6 I

< 0,

(9.13)

< 0,

(9.14)



where P=

m 

πi j P j , P˜ = G T (G Bi )−T BiT

j=1

Π1 = 7AiT P Ai

m 

πi j P j Bi (G Bi )−1 G, Π3 = −I,

j=1

+

7AiT

P˜ Ai + 7ε1 NiT Ni + 7ε2 NiT Ni + 7ε5 NiT Ni + 7ε6 NiT Ni

+ 7 P˜ + (d M − dm + 1)Q − Pi + h 20 NiT Ni , T T ˜ Π2 = 7Adi P Adi + 7Adi P Adi + 7ε3 NdiT Ndi + 7ε4 NdiT Ndi + 7ε5 NdiT Ndi + 7ε6 NdiT Ndi − Q + h 21 NdiT Ndi . Proof First, choose the Lyapunov-Krasovskii functional:

9.3 Main Results

181

V (k) = xkT Pi xk +

−d m +1 

k−1 

xlT Qxl

j=−d M +1 l=k−1+ j

= V1 (k) + V2 (k).

(9.15)

Then, along the trajectories of system (9.11), we have E{ΔV (k)} = E{ΔV1 (k)} + E{ΔV2 (k)},

(9.16)

where T Pi xk+1 − xkT Pi xk ) E{ΔV1 (k)} = E(xk+1

= {xkT [(I − Bi (G Bi )−1 G)(Ai + A¯ i ) + Bi (G Bi )−1 G]T T + xk−d [(I − Bi (G Bi )−1 G)(Adi + A¯ di )]T k

+ f iT (k, xk , xk−dk )(I − Bi (G Bi )−1 G)T }P {[(I − Bi (G Bi )−1 G)(Ai + A¯ i ) + Bi (G Bi )−1 G]xk + [(I − Bi (G Bi )−1 G)(Adi + A¯ di )]xk−dk + (I − Bi (G Bi )−1 G) f i (k, xk , xk−dk )} − xkT Pi xk .

(9.17)

Denote E(ΔV1 (k)) as follows E{ΔV1 (k)} = E{ΔV11 (k)} + E{ΔV12 (k)} + E{ΔV13 (k)} + E{V14 (k)} + E{V15 (k)},

(9.18)

where E{ΔV11 (k)} = xkT [(I − Bi (G Bi )−1 G)(Ai + A¯ i ) + Bi (G Bi )−1 G]T P [(I − Bi (G Bi )−1 G)(Ai + A¯ i ) + Bi (G Bi )−1 G]xk − xkT Pi xk = xkT (Ai + A¯ i )T [−G T (G Bi )−T BiT P − P Bi (G Bi )−1 G + P + G T (G Bi )−T BiT P Bi (G Bi )−1 G](Ai + A¯ i )xk + xkT (Ai + A¯ i )T [P Bi (G Bi )−1 G − G T (G Bi )−T BiT P Bi (G Bi )−1 G]xk + xkT [G T (G Bi )−T BiT P − G T (G Bi )−T BiT P Bi (G Bi )−1 G](Ai + A¯ i )xk + xkT G T (G Bi )−T BiT P Bi (G Bi )−1 Gxk − xkT Pi xk . Obviously,

(9.19)

182

9 Adaptive Fault-Tolerant Control

xkT (Ai + A¯ i )T [P Bi (G Bi )−1 G − G T (G Bi )−T BiT P Bi (G Bi )−1 G]xk + xkT [G T (G Bi )−T BiT P − G T (G Bi )−T BiT P Bi (G Bi )−1 G](Ai + A¯ i )xk ≤ xkT (Ai + A¯ i )T [P + G T (G Bi )−T BiT P Bi (G Bi )−1 G](Ai + A¯ i )xk + xkT [G T (G Bi )−T BiT P Bi (G Bi )−1 G + G T (G Bi )−T BiT P Bi (G Bi )−1 G]xk . (9.20) Obviously, xkT (Ai + A¯ i )T [−G T (G Bi )−T BiT P − P Bi (G Bi )−1 G](Ai + A¯ i )xk ≤ xkT (Ai + A¯ i )T [P + G T (G Bi )−T BiT P Bi (G Bi )−1 G](Ai + A¯ i )xk .

(9.21)

In the light of (9.20) and (9.21), we have T ˜ ˜ ¯ E{ΔV11 (k)} = xkT (Ai + A¯ i )T (3P + 3 P)(A i + Ai )x k + 3x k P x k

− xkT Pi xk .

(9.22)

Similarly, we derive E{ΔV12 (k)} = xkT (Ai + A¯ i )T [P − G T (G Bi )−T BiT P − P Bi (G Bi )−1 G + G T (G Bi )−T BiT P Bi (G Bi )−1 G](Adi + A¯ di )xk−dk T + xk−d (Adi + A¯ di )T [P − G T (G Bi )−T BiT P k

− P Bi (G Bi )−1 G + G T (G Bi )−T BiT P Bi (G Bi )−1 G] (Ai + A¯ i )xk + xkT [G T (G Bi )−T BiT P − G T (G Bi )−T BiT P Bi (G Bi )−1 G](Adi + A¯ di )xk−dk T + xk−d (Adi + A¯ di )T [P Bi (G Bi )−1 G k

− G T (G Bi )−T BiT P Bi (G Bi )−1 G]xk .

(9.23)

Obviously, we get xkT [G T (G Bi )−T BiT P − G T (G Bi )−T BiT P Bi (G Bi )−1 G] T (Adi + A¯ di )xk−dk + xk−d (Adi + A¯ di )T k

[P Bi (G Bi )−1 G − G T (G Bi )−T BiT P Bi (G Bi )−1 G]xk T T ˜ ˜ ¯ ˜ ≤ xk−d (Adi + A¯ di )T (P + P)(A di + Adi )x k−dk + x k ( P + P)x k . k Similarly, we obtain

(9.24)

9.3 Main Results

183

xkT (Ai + A¯ i )T [P − G T (G Bi )−T BiT P − P Bi (G Bi )−1 G + G T (G Bi )−T BiT P Bi (G Bi )−1 G](Adi + A¯ di )xk−dk T + xk−d (Adi + A¯ di )T [P − G T (G Bi )−T BiT P k

− P Bi (G Bi )−1 G + G T (G Bi )−T BiT P Bi (G Bi )−1 G] (Ai + A¯ i )xk ˜ ¯ ≤ xkT (Ai + A¯ i )T (2P + 2 P)(A i + Ai )x k T ˜ ¯ + xk−d (Adi + A¯ di )T (2P + 2 P)(A di + Adi )x k−dk . k

(9.25)

Thus, substituting (9.24) and (9.25) into (9.23) obtains T ˜ ¯ E{ΔV12 (k)} ≤ xkT (Ai + A¯ i )T (2P + 2 P)(A i + Ai )x k + x k−dk (Adi T ˜ ˜ ¯ + A¯ di )T (3P + 3 P)(A di + Adi )x k−dk + 2x k P x k .

(9.26)

Similarly, T ˜ ¯ (Ai + A¯ i )T (2P + 2 P)(A E{ΔV13 (k)} ≤ xk−d i + Ai )x k−dk , k

(9.27)

˜ ¯ E{ΔV14 (k)} ≤ xkT (Ai + A¯ i )T (2P + 2 P)(A i + Ai )x k T ˜ ¯ + xk−d (Adi + A¯ di )T (2P + 2 P)(A di + Adi )x k−dk k ˜ f i (k, xk , xk−dk ) + 2xkT P˜ xk , (9.28) + f iT (k, xk , xk−dk )(5P + 5 P) E{ΔV15 (k)} = f iT (k, xk , xk−dk )(I − Bi (G Bi )−1 G)T P (I − Bi (G Bi )−1 G) f i (k, xk , xk−dk ) = f iT (k, xk , xk−dk )[−G T (G Bi )−T BiT P − P Bi (G Bi )−1 G + G T (G Bi )−T BiT P Bi (G Bi )−1 G + P] f i (k, xk , xk−dk ) ˜ f i (k, xk , xk−dk ). ≤ f iT (k, xk , xk−dk )(2P + 2 P) (9.29) From (9.22), (9.26), (9.27), (9.28) and (9.29), we have ˜ ¯ E{ΔV1 (k)} ≤ xkT (Ai + A¯ i )T (7P + 7 P)(A i + Ai )x k T ˜ ¯ + xk−d (Adi + A¯ di )T (7P + 7 P)(A di + Adi )x k−dk k ˜ f i (k, xk , xk−dk ) + f iT (k, xk , xk−dk )(7P + 7 P) + 7xkT P˜ xk − xkT Pi xk . Obviously,

(9.30)

184

9 Adaptive Fault-Tolerant Control

T T T xkT (Ai + A¯ i )T P(Ai + A¯ i )xk ≤ xkT [AiT P Ai + ε−1 1 A i P M i M i P A i + ε1 N i N i + A¯ iT P A¯ i ]xk . (9.31)

Above all, we obtain T T T E{ΔV1 (k)} ≤ 7xkT [AiT P Ai + ε−1 1 A i P M i M i P A i + ε1 N i N i T ˜ T ˜ + A¯ iT P A¯ i ]xk + 7xkT [AiT P˜ Ai + ε−1 2 A i P Mi Mi P A i T T + ε2 NiT Ni + A¯ iT P˜ A¯ i ]xk + 7xk−d [Adi P Adi k T T T + ε−1 3 Adi P Mi Mi P Adi + ε3 Ndi Ndi T T T ˜ + A¯ di P A¯ di ]xk−dk + 7xk−d [Adi P Adi + ε4 NdiT Ndi k T ˜ T ˜ ¯T ˜ ¯ + ε−1 4 Adi P Mi Mi P Adi + Adi P Adi ]x k−dk ˜ f i (k, xk , xk−dk ) + 7 f iT (k, xk , xk−dk )(P + P)

+ 7xkT P˜ xk − xkT Pi xk .

(9.32)

Then, We obtain E{ΔV2 (k)} =

⎛

−d m +1 

⎝

j=−d M +1

=

−d m +1 

k 

l=k+ j

xlT Qxl −

k−1 

⎞ xlT Qxl ⎠

l=k−1+ j

T (xkT Qxk − xk−1+ j Qx k−1+ j )

j=−d M +1 T ≤ (d M − dm + 1)xkT Qxk − xk−d Qxk−dk . k

(9.33)

Thus, from (9.32) and (9.33), we derive T T T E{ΔV (k)} ≤ 7xkT [AiT P Ai + ε−1 1 A i P M i M i P A i + ε1 N i N i T ˜ T ˜ + A¯ iT P A¯ i ]xk + 7xkT [AiT P˜ Ai + ε−1 1 A i P Mi Mi P A i T T + ε1 NiT Ni + A¯ iT P˜ A¯ i ]xk + 7xk−d [Adi P Adi k

T T T + ε−1 1 Adi P Mi Mi P Adi + ε1 Ndi Ndi T T T ˜ + A¯ di P A¯ di ]xk−dk + 7xk−d [Adi P Adi + ε1 NdiT Ndi k T ˜ T ˜ ¯T ˜ ¯ + ε−1 1 Adi P Mi Mi P Adi + Adi P Adi ]x k−dk ˜ f i (k, xk , xk−dk ) + 7 f iT (k, xk , xk−dk )(P + P)

+ 7xkT P˜ xk − xkT Pi xk + (d M − dm + 1)xkT Qxk T − xk−d Qxk−dk . k

Let E{ΔV (k)} = Z kT M Z k , where

(9.34)

9.3 Main Results

185

⎡ Zk = ⎣

xk

⎤

⎦, xk−dk f i (k, xk , xk−dk )

⎤ Π4 0 0 M = ⎣ 0 Π5 0 ⎦ , 0 0 Π6

(9.35)

⎡

(9.36)

T T T ¯T Π4 = 7[AiT P Ai + ε−1 1 A i P M i M i P A i + ε1 N i N i + A i T ˜ T ˜ T P A¯ i ] + 7[AiT P˜ Ai + ε−1 2 A i P M i M i P A i + ε2 N i N i

+ A¯ iT P˜ A¯ i ] + 7 P˜ + (d M − dm + 1)Q − Pi , T T T T P Adi + ε−1 Π5 = 7[Adi 3 Adi P Mi Mi P Adi + ε3 Ndi Ndi T T ˜ + A¯ di P A¯ di ] + 7[Adi P Adi + ε4 NdiT Ndi T ˜ T ˜ ¯T ˜ ¯ + ε−1 4 Adi P Mi Mi P Adi + Adi P Adi ] − Q, ˜ Π6 = 7(P + P).

Then E{ΔV (k)} < 0 equals to M < 0. According to (9.7), it can be obtained that Z T (k) × diag(−h 20 NiT Ni , −h 21 NdiT Ndi , I ) × Z (k) ≤ 0.

(9.37)

Then, (9.38) can guarantee that E{ΔV (k)} < 0. ⎤ Π7 0 0 M = ⎣ 0 Π8 0 ⎦ < 0, 0 0 Π9 ⎡

(9.38)

where Π7 = Π4 + h 20 NiT Ni , Π8 = Π5 + h 21 NdiT Ndi , Π9 = Π6 − I . According to schur complement theorem, M < 0 equals to ⎡

S1 ⎢∗ ⎢ ⎣∗ ∗

0 S2 ∗ ∗

0 0 S3 ∗

⎤ X1 X2 ⎥ ⎥ < 0, X3 ⎦ X4

where S1 = 7AiT P Ai + 7AiT P˜ Ai + 7ε1 NiT Ni + 7ε2 NiT Ni + 7 P˜ + (d M − dm + 1)Q − Pi + h 20 NiT Ni , T T ˜ S2 = 7Adi P Adi + 7Adi P Adi + 7ε3 NdiT Ndi + 7ε4 NdiT Ndi − Q + h 21 NdiT Ndi ,

(9.39)

186

9 Adaptive Fault-Tolerant Control

S3 = − I, √ T  √ √ √ X1 = 7Ai P Mi 7AiT P˜ Mi 7 A¯ iT P 0 7 A¯ iT P˜ 0 0 0 0 0 ,   √ T √ T √ T √ T X 2 = 0 0 0 7 A¯ di P 0 7 A¯ di P Mi 0 0 , P˜ 7Adi P˜ Mi 7Adi  √ √  X 3 = 0 0 0 0 0 0 0 0 7P 7 P˜ ,   X 4 = −ε1 −ε2 −P −P − P˜ − P˜ −ε3 −ε4 −P − P˜ . Formula (9.39) equals to the following formula ⎡

S1 ⎢∗ ⎢ ⎣∗ ∗

0 S2 ∗ ∗

0 0 S3 ∗

⎤ X5 T



X6 ⎥ ⎥ + W T F 0T Y T + Y1 F 0 W1 < 0, 1 1 X3 ⎦ 0 F 0 F X4

(9.40)

where X5 = X6 = X3 = X4 = W1 = Y1 = Obviously,

√ T  √ √ 7Ai P Mi 7AiT P˜ Mi 7 A¯ iT P 0 0 0 0 0 0 0 ,   √ T √ T √ T 0 0 0 7 A¯ di P 0 0 7Adi P Mi 0 0 , P˜ Mi 7Adi  √ √  0 0 0 0 0 0 0 0 7P 7 P˜ ,   −ε1 −ε2 −P −P − P˜ − P˜ −ε3 −ε4 −P − P˜ ,

√ 7Ni √ 0 0 0 0 0 0 0 0 0 , 0 7Ndi 0 0 0 0 0 0 0 0

0 0 0 0 0 0 MiT P˜ 0 0 0 . 0 0 0 0 0 0 0 MiT P˜ 0 0 T T W1T F T Y1T + Y1 F W1 ≤ ε−1 5 Y1 Y1 + ε5 W1 W1 ,

(9.41)

Formula (9.42) and (9.43) can guarantee (9.40) ⎡

S5 ⎢∗ ⎢ ⎣∗ ∗

0 S6 ∗ ∗

0 0 S3 ∗

⎤ X7 X8 ⎥ ⎥ < 0, X9 ⎦ X 10

T ˜ ˜ − P˜ + ε−1 5 P Mi Mi P < 0,

where

(9.42)

(9.43)

9.3 Main Results

187

S5 = 7AiT P Ai + 7AiT P˜ Ai + 7ε1 NiT Ni + 7ε2 NiT Ni + 7 P˜ + 7ε5 NiT Ni + (d M − dm + 1)Q − Pi + h 20 NiT Ni , T T ˜ S6 = 7Adi P Adi + 7Adi P Adi + 7ε3 NdiT Ndi + 7ε4 NdiT Ndi X7 = X8 = X9 = X 10 =

+ 7ε5 NdiT Ndi − Q + h 21 NdiT Ndi , √ T  √ √ 7Ai P Mi 7AiT P˜ Mi 7 A¯ iT P 0 0 0 0 0 ,   √ T √ T √ T 0 0 0 7 A¯ di P 7Adi P Mi 0 0 , P˜ Mi 7Adi  √ √  0 0 0 0 0 0 7P 7 P˜ ,   −ε1 −ε2 −P −P −ε3 −ε4 −P − P˜ .

Formula (9.42) equals to the following formula ⎡

S5 ⎢∗ ⎢ ⎣∗ ∗

0 S6 ∗ ∗

0 0 S3 ∗

⎤ Ω1 T



Ω2 ⎥ ⎥ + W T F 0T Y T + Y2 F 0 W2 < 0, 2 2 X9 ⎦ 0 F 0 F X 10

(9.44)

where √ T  √ 7Ai P Mi 7AiT P˜ Mi 0 0 0 0 0 0 ,   √ T √ T Ω2 = 0 0 0 0 7Adi P Mi 0 0 , P˜ Mi 7Adi

√ 7Ni √ 0 0 0 0 0 0 0 , W2 = 0 7Ndi 0 0 0 0 0 0

0 0 0 0 MiT P 0 0 0 . Y2 = 0 0 0 0 0 MiT P 0 0 Ω1 =

Obviously,

T T W2T F T Y2T + Y2 F W2 ≤ ε−1 6 Y2 Y2 + ε6 W2 W2 .

(9.45)

Formula (9.46), (9.47) and (9.48) can guarantee (9.43) and (9.44) ⎡

Π1 ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

0 Π2 ∗ ∗ ∗ ∗ ∗ ∗ ∗

0 0 Π3 ∗ ∗ ∗ ∗ ∗ ∗

√

7AiT P Mi 0 0 −ε1 ∗ ∗ ∗ ∗ ∗

⎤ √ T 0 0 0 7Ai P˜ Mi √ 0 √ ⎥ T P˜ M T PM 0 7Adi 7Adi i i √0 √0 ⎥ ⎥ ˜ 0 0 0 7P 7 P ⎥ ⎥ 0 0 0 0 0 ⎥ ⎥ 0 0 0 0 ⎥ < 0, (9.46) −ε2 ⎥ ∗ −ε3 0 0 0 ⎥ ⎥ 0 0 ⎥ ∗ ∗ −ε4 ⎥ 0 ⎦ ∗ ∗ ∗ −P ∗ ∗ ∗ ∗ − P˜ T ˜ ˜ − P˜ + ε−1 5 P Mi Mi P < 0,

(9.47)

188

9 Adaptive Fault-Tolerant Control T − P + ε−1 6 P Mi Mi P < 0.

(9.48)

According to schur complement theorem, (9.12), (9.13) and (9.14) can guarantee (9.46), (9.47) and (9.48) respectively. The proof is completed.

9.3.2 Synthesis of Sliding-Mode Control Laws In this section, an adaptive SMC controller is employed to guarantee the reachability of our designed sliding surface. While f i (k, xk , xk−dk ) is known, the additive actuator fault f a (k) may affect the system. Under the assumption that f i (k, xk , xk−dk ) + f a (k) ≤ q1 xk + q2 xk−dk with unknown parameters q1 and q2 , the following theorem can be obtained. Theorem 9.2 Consider the uncertain non-linear system (9.6). If with proper chosen parameters θi (i ∈ N ), there exist matrices Di > 0 such that − sym(θi Di ) − Di +

m 

πi j D j < 0.

(9.49)

j=1

then, the following adaptive SMC can keep a sliding motion in a region contains the equilibrium point: u k = − (G Bi αk )−1 ( G ( Ai + Ni Mi − 1) xk + G ( Ai + Ni Mi ) xk−dk )sign(D T sk ) 1 − (G Bi αk )−1 [(D −1 Σ + D −1 sym(θi Di ))sk 2 + D −1 (qˆ1 (k) xk + qˆ2 (k) xk−dk ) D G sat(sk )], where D =

m j=1

(9.50)

πi j D j and  sat(sk ) =

if sk ≤ σ, sk /σ, if sk > σ. sign(sk ) ,

(9.51)

The adaptive laws are designed as Δqˆ1 (k) = qˆ1 (k + 1) − qˆ1 (k) = c(−ε1 qˆ1 (k) + xk D G sk ),

(9.52)

Δqˆ2 (k) = qˆ2 (k + 1) − qˆ2 (k) = d(−ε2 qˆ2 (k) + xk−dk D G sk ),

(9.53)

9.3 Main Results

189

where Σ is a positive matrix, ε1 , ε2 , σ, c and d are positive scalars, qˆ1 (k) and qˆ2 (k) are the estimations of q1 and q2 respectively. Proof Choose the following Lyapunov function Vs (k) =

1 T 1 1 2 s Di sk + q˜1 2 + q˜2 , 2 k 2c 2d

(9.54)

where q˜1 = q1 − qˆ1 (k), q˜2 = q2 − qˆ2 (k). Then, there exists E{ΔVs (k)} = E{Vs (k + 1) − Vs (k)} 1 1 1 1 = skT DΔsk + skT Dsk − skT Di sk − q˜1 Δqˆ1 (k) − q˜2 Δqˆ2 (k) + Ψ0 2 2 c d 1 = skT D G[(Ai + A¯ i − 1)xk + (Adi + A¯ di )xk−dk + Bi αk u k − q˜2 Δqˆ2 (k) d 1 1 1 + f i (k, xk , xk−dk ) + f a (k)] + skT Dsk − skT Di sk − q˜1 Δqˆ1 (k) 2 2 c + Ψ0 ≤ − skT Σsk + skT D G(Ai + A¯ i )xk − skT D 1 G ( Ai + Ni Mi ) xk − skT DGxk − skT D 1 G xk + skT D G(Adi + A¯ di )xk−dk − skT D 1 G ( Adi 1 + Ndi Mi ) xk−dk + skT (−sym(θi Di ) − Di 2 + D)sk + q1 xk D G sk − q˜1 (−ε1 qˆ1 (k) + xk D G sk ) − qˆ1 (k) xk D G sat(sk ) + q2 xk−dk D G sk − q˜2 (−ε1 qˆ2 (k) + xk−dk D G sk ) − qˆ2 (k) xk−dk D G sat(sk ) + Ψ0 ≤ − skT Σsk + ε1 q˜1 qˆ1 (k) + ε2 q˜2 qˆ2 (k) + qˆ1 (k) xk D G sk − qˆ1 (k) xk D G skT sat(sk ) + qˆ2 (k) xk−dk D G sk − qˆ2 (k) xk−dk D G skT sat(sk ) + Ψ0 .

(9.55)

If sk > σ, it follows (9.55) that E{ΔVs (k)} ≤ − skT Σsk + ε1 q˜1 qˆ1 (k) + ε2 q˜2 qˆ2 (k) + Ψ0 1 1 ≤ − skT Σsk − ε1 (qˆ1 (k) − q1 )2 + ε2 (qˆ2 (k) − q2 )2 + Ψ1 . 2 2 If sk ≤ σ, it follows (9.55) that

(9.56)

190

9 Adaptive Fault-Tolerant Control

E{ΔVs (k)} ≤ − skT Σsk + ε1 q˜1 qˆ1 (k) + ε2 q˜2 qˆ2 (k) qˆ1 (k) xk D G (σ sk − sk 2 ) σ qˆ2 (k) xk−dk D G − (σ sk − sk 2 ) + Ψ0 σ 1 1 ≤ − skT Σsk − ε1 (qˆ1 (k) − q1 )2 − ε2 (qˆ2 (k) − q2 )2 2 2 qˆ1 (k) xk D G 1 2 ( sk − σ) − σ 2 qˆ2 (k) xk−dk D G 1 − ( sk − σ)2 + Ψ2 , σ 2 −

(9.57)

where Ψ0 = 21 ΔskT DΔsk + 21 Δq˜1 2 + 21 Δq˜2 2 , Ψ1 = 41 ε1 q12 + 14 ε2 q22 + Ψ0 , Ψ2 = 1 2 σ + Ψ1 . E{ΔVs (k)} < 0 can be ensured by selecting appropriate Σ and thus the 2 reachability of the sliding surface (9.8) can be guaranteed.

9.4 Simulation Consider the MJDS with two modes and the following parameters: ⎡ A1 =

Ad1 =

B1 =

M1 =

N1 =

Nd1 =

⎤ ⎡ ⎤ 0.2 0.24 0.12 0.1 0.08 0.1 ⎣ 0.18 0.11 0.11 ⎦ , Ad2 = ⎣ 0 0.05 −0.06 ⎦ , 0.5 0.2 −0.3 −0.03 0.04 −0.05 ⎡ ⎤ ⎡ ⎤ −0.02 0.06 0.04 0.1 −0.1 0 ⎣ 0.04 0.01 −0.03 ⎦ , A2 = ⎣ 0.25 0.35 0.2 ⎦ , 0.01 0.02 0.05 0.25 0.35 0.45 ⎡ ⎤ ⎡ ⎤ −0.15 0.03 ⎣ 0.033 ⎦ , B2 = ⎣ 0.03 ⎦ , 0.15 0.158 ⎡ ⎤ ⎡ ⎤ 0.01 0 0 0.02 0 0 ⎣ 0 0.01 0 ⎦ , M2 = ⎣ 0 0.02 0 ⎦ , 0 0 0.02 0 0 0.01 ⎡ ⎤ ⎡ ⎤ −0.1 0 0 −0.05 0 0 ⎣ 0 0 0.1 ⎦ , N2 = ⎣ 0 0 0.05 ⎦ , 0 0.2 0 0 0.1 0 ⎡ ⎤ ⎡ ⎤ −0.1 0 0 −0.05 0 0 ⎣ 0 0 0.1 ⎦ , Nd2 = ⎣ 0 0 0.05 ⎦ , 0 0.2 0 0 0.1 0

9.4 Simulation

191

⎡

⎤ sin(k) 0 0   ⎦ , G = −0.12 0.066 0.308 . 0 F(k) = ⎣ 0 cos(k) 0 0 sin(k) ∗ cos(k)

0.7 0.3 The transition probability matrix Π is chosen as Π = . Taking θ1 = 0.07 0.4 0.6 and θ2 = 0.1, then we obtain D1 = 1.8242 and D2 = 1.9661. Additionally, the adjustable parameters are selected as Σ = 0.17, σ = 0.01, c = 0.0005, d = 0.0005, ε1 = 0.55, ε2 = 0.55, β1 = 1 and β2 = 1. The adaptive controller and the adaptive laws can be obtained referring to Eqs. (9.50)–(9.53). The bounds of time-varying delay dk are set as dm = 3 and d M = 5. For simulation purpose, the nonlinear functions are supposed to be T  f 1 (k, xk , xk−dk ) = 0.1sin(x1k )sin(x2k ) 0.25sin(x2k ) 0.16sin(x3k−dk ) , T  f 2 (k, xk , xk−dk ) = 0.08sin(x1k ) 0.14sin(x2k ) 0.13sin(x3k−dk ) ,

where xik and xik−dk are the ith components of xk and xk−dk respectively. The actuator fault parameters are given as α1k = 0.8 + 0.1 ∗ sin(20 ∗ k), 3 ≤ k ≤ 40, α2k = 0.7 + 0.2 ∗ rand(1), 3 ≤ k ≤ 40. By (9.50)–(9.53), Figs. 9.1, 9.2, 9.3, 9.4 and 9.5 show the simulation results with initial condition x(0) = [0.1 0.2 − 0.23]T . The mode jumps are shown in Fig. 9.1. Figures 9.2 and 9.3 show the trajectories of the system states and the sliding surface, respectively. Figure 9.4 shows the adaptive values, from which we can notice that the estimations qˆ1 and qˆ2 are both bounded. The control input is provided in Fig. 9.5. We can conclude that the system is stabilized in spite of the matched uncertainties, delayed terms and actuator failures.

Fig. 9.1 System mode

System mode

2

1

0 0

10

20

30

40

50

k

60

70

80

90

100

192

9 Adaptive Fault-Tolerant Control

Fig. 9.2 System states

0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 0

10

20

30

40

50

60

70

80

90

100

k

Fig. 9.3 Sliding surface

0.04 s

0.02

s

0

-0.02

-0.04

-0.06

-0.08 0

10

20

30

40

50

60

70

80

90

100

70

80

90

100

k

Fig. 9.4 Estimations qˆ1 and qˆ2

10-5 3

2.5

2

1.5

1

0.5

0 0

10

20

30

40

50

k

60

References

193

Fig. 9.5 Control input

0.4 u

0.3 0.2 0.1

u

0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 0

10

20

30

40

50

60

70

80

90

100

k

9.5 Conclusion In this note, the adaptive SMC problem has been addressed for time-varying delayed MJDSs with uncertainty and nonlinearity against actuator faults. First, a linear modeindependent sliding surface is designed. Then, sufficient conditions for the stability of the sliding mode dynamics are derived. Then, an adaptive fault-tolerant sliding mode controller is designed to guarantee that the mode-independent sliding surface is reachable. In the end, numerical simulations are given to verify the usefulness of our proposed controller.

References 1. Z. Cao, Y. Niu, J. Song, Finite-time sliding mode control of Markvian jump cyber-physical systems against randomly occurring injection attacks, IEEE Transactions on Automatic Control, 65(3) (2020) 1264–1271. 2. N. K. Kwon, I. S. Park, P. Park, C. Park, Dynamic output-feedback control for singular Markovian jump system: Lmi approach, IEEE Transactions on Automatic Control, 62(10) (2017) 5396–5400. 3. P. Zhang, Y. Kao, J. Hu, B. Niu, H. Xia, C. Wang, Finite-time observer-based sliding-mode control for Markovian jump systems with switching chain: Average dwell-time method, IEEE Transactions on Cybernetics, 53(1) (2021) 248–261. 4. Z. Feng, P. Shi, Sliding mode control of singular stochastic markov jump systems, IEEE Transactions on Automatic Control, 62(8) (2017) 4266–4273. 5. J. Zhu, X. Yu, T. Zhang, Z. Cao, Y. Yang, Y. Yi, The mean-square stability probability of H∞ control of continuous markovian jump systems, IEEE Transactions on Automatic Control, 61(7) (2016) 1918–1924. 6. J. Yang, Z. Ning, Y. Zhu, L. Zhang, H.K. Lam, Semi-Markov jump linear systems with biboundary sojourn time: Anti-modal-asynchrony control, Automatica, 140 (2022) 110270.

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7. H. Wang, D. Zhang, R. Lu, Event-triggered H∞ filter design for Markovian jump systems with quantization, Nonlinear Analysis: Hybrid Systems, 28 (2018) 23–41. 8. Y. Zhu, L. Zhang, W. X. Zheng, Distributed H∞ filtering for a class of discrete-time Markov jump lur’e systems with redundant channels, IEEE Transactions on Industrial Electronics, 63(3) (2015) 1876–1885. 9. A. R. Fioravanti, A. P. Gonçalves, J. C. Geromel, H2 and H∞ filtering of discrete-time Markov jump linear systems through linear matrix inequalities, IFAC Proceedings Volumes, 41(2) (2008) 2681–2686. 10. M. Zhang, P. Shi, Z. Liu, J. Cai, H. Su, Dissipativity-based asynchronous control of discrete-time Markov jump systems with mixed time delays, International Journal of Robust and Nonlinear Control, 28(6) (2018) 2161–2171. 11. E.-K. Boukas, Z. Liu, Robust H∞ control of discrete-time markovian jump linear systems with mode-dependent time-delays, IEEE Transactions on Automatic Control, 46 (12) (2001) 1918–1924. 12. P. Cheng, C.-X Cai, Event-triggered fault detection for discrete-time Markovian jump systems in finite frequency domain, International Journal of Robust and Nonlinear Control, 32 (6) (2022) 141–163. 13. Y. Han, Y. Kao, C. Gao, Robust sliding mode control for uncertain discrete singular systems with time-varying delays and external disturbances, Automatica, 75 (2017) 210–216. 14. Q. Ren, Y. Kao, C. Wang, H. Xia, X. Wang, New results on the generalized discrete reaching law with positive or negative decay factors, IEEE Transactions on Automatic Control, 67 (2) (2022) 1046–1052. 15. X. Su, X. Liu, P. Shi, R. Yang, Sliding Mode Control of Discrete-Time Switched Systems with Repeated Scalar Nonlinearities, IEEE Transactions on Automatic Control, 62(9) (2017) 4604–4610. 16. J. Hu, Z. Wang, Y. Niu, H. Gao, Sliding mode control for uncertain discrete-time systems with Markovian jumping parameters and mixed delays, Journal of the Franklin Institute, 351(4) (2014) 2185–2202. 17. P. Zhang, Y. Kao, J. Hu, B. Niu, Robust observer-based sliding mode H∞ control for stochastic Markovian jump systems subject to packet losses, Automatica, 130 (2021) 109665. 18. D. Yao, H. Ren, P. Li, Q. Zhou, Sliding mode output-feedback control of discrete-time Markov jump systems using singular system method, Journal of the Franklin Institute, 355(13) (2018) 5576–5591. 19. D. Yao, B. Zhang, P. Li, H. Li, Event-triggered sliding mode control of discrete-time Markov jump systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 49(10) (2018) 2016–2025. 20. Y. Bai, H. Sun, J. Wen, Y. Wu, Fault-tolerant control for the linearized spacecraft attitude control system with Markovian switching, Journal of the Franklin Institute, 359(17) (2022) 9814–9835. 21. Q. Yang, S. S. Ge, Y. Sun, Adaptive actuator fault tolerant control for uncertain nonlinear systems with multiple actuators, Automatica, 60 (2015) 92–99. 22. A.-M. Wang, J.-N. Li, Event-triggered asynchronous H∞ fault-tolerant control for discretetime Markov jump system with actuator faults, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 235(6) (2021) 781–794. 23. H. Jerbi, R. Abassi, J. VijiPriya, F. Hmidi, A. Kouzou, Passivity-based asynchronous faulttolerant control for nonlinear discrete-time singular Markovian jump systems: A sliding-mode approach, European Journal of Control, 60 (2021) 95–113.

Part V

Summary

Chapter 10

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.

10.1 Conclusion In this book, some analysis and design techniques for Markov jump systems are proposed. Based on Lyapunov function and sliding mode control technique, the stochastic stability and finite-time boundedness analysis for MJSs with packet losses are summarized in Chaps. 2 and 3. By the use of sliding surface related to initial condition, the finite-time boundedness and passive performance for uncertain MJSs are studied in Chap. 4. In consideration of time-delay factor, the hidden Markov modelbased method is adopted to address sliding mode control issue for uncertain MJSs under the attacks in Chap. 5, which is convenient to handle new form of deception attacks with time-delay. Noting the theory of analysis and control for normal systems, Chap. 6 discusses the actuator-fault problem for MJSs via a sliding mode control. Viewing the transfer disturbance, Chap. 7 presents a bumpless transfer scheme for the Markov jump discrete systems with state feedback and output feedback skills. The mixed H∞ and passivity of semi-Markov jump discrete systems and adaptive sliding mode fault-tolerant control are taken into account in Chaps. 8 and 9, respectively. All the conditions for the existences of analysis and design are derived in terms of linear programming. The effectiveness of the proposed methodologies have been verified by some numerical simulations.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Kao et al., Analysis and Design of Markov Jump Discrete Systems, Studies in Systems, Decision and Control 499, https://doi.org/10.1007/978-981-99-5748-4_10

197

198

10 Conclusion and Future Research Direction

10.2 Future Research Direction Although a great deal of fruitful research results have been reported about stochastic jump systems with a prescribed Markov chain, 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 analysis and design for Markov jump systems as follows: (i) Presently, some stability and robust control results for Markov jump systems are based on the Lyapunov method and thus the obtained conditions tend to be sufficient and have some conservatism. How to apply the latest approach like genetic algorithm to study Markov jump discrete systems is still a worth studying topic. (ii) For Markov jump systems, multitudinous works mainly focus on the analysis and design under the framework of continuous-time case. It is of great significance to study discrete-time sliding mode control for Markov jump systems, especially the quasi-sliding mode band problem. (iii) With the rapid development of network communication technology, the amount of data transmission show a substantial increase. As a novel data transmission mechanism, event-triggered mechanism has an obvious advantages in reducing the sampling or transmission of redundant data, saving energy consumption and network bandwidth resources, which has become a hot research topic in the recent years. How to construct an appropriate event-triggered mechanism for networked MJSs under deception attacks is significant in both theory and practice.