Fault-Tolerant Control for Time-Varying Delayed T-S Fuzzy Systems 9789819913565, 9789819913572

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Intelligent Control and Learning Systems 9

Shaoxin Sun Huaguang Zhang Xiaojie Su Jinyu Zhu

Fault-Tolerant Control for Time-Varying Delayed T-S Fuzzy Systems

Intelligent Control and Learning Systems Volume 9

Series Editor Dong Shen , School of Mathematics, Renmin University of China, Beijing, Beijing, China

The Springer book series Intelligent Control and Learning Systems addresses the emerging advances in intelligent control and learning systems from both mathematical theory and engineering application perspectives. It is a series of monographs and contributed volumes focusing on the in-depth exploration of learning theory in control such as iterative learning, machine learning, deep learning, and others sharing the learning concept, and their corresponding intelligent system frameworks in engineering applications. This series is featured by the comprehensive understanding and practical application of learning mechanisms. This book series involves applications in industrial engineering, control engineering, and material engineering, etc. The Intelligent Control and Learning System book series promotes the exchange of emerging theory and technology of intelligent control and learning systems between academia and industry. It aims to provide a timely reflection of the advances in intelligent control and learning systems. This book series is distinguished by the combination of the system theory and emerging topics such as machine learning, artificial intelligence, and big data. As a collection, this book series provides valuable resources to a wide audience in academia, the engineering research community, industry and anyone else looking to expand their knowledge in intelligent control and learning systems.

Shaoxin Sun · Huaguang Zhang · Xiaojie Su · Jinyu Zhu

Fault-Tolerant Control for Time-Varying Delayed T-S Fuzzy Systems

Shaoxin Sun College of Automation Chongqing University Chongqing, China Xiaojie Su College of Automation Chongqing University Chongqing, China

Huaguang Zhang College of Information Science and Engineering Northeastern University Shenyang, China Jinyu Zhu College of Automation Chongqing University Chongqing, China

ISSN 2662-5458 ISSN 2662-5466 (electronic) Intelligent Control and Learning Systems ISBN 978-981-99-1356-5 ISBN 978-981-99-1357-2 (eBook) https://doi.org/10.1007/978-981-99-1357-2 © 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

To My Family Shaoxin Sun To My Family Huaguang Zhang To My Family Xiaojie Su To My Family Jinyu Zhu

Preface

The requirements of system stability, response speed, control accuracy, and adaptability for control systems are increasingly high in different fields along with this prompt development of science technology, production, and life. It is hard to build a precise mathematical model because the plants encountered in practice become increasingly complex and changeable. Takagi-Sugeno (T-S) fuzzy models can approximate nonlinear plants using arbitrary accuracy and can convert original nonlinear plants into weighted sums of multiple subsystems. Therefore, it is necessary to study building T-S fuzzy models so that nonlinear system issues can be overcome through these methods of solving linear systems. In addition, establishing T-S fuzzy systems has recently been one of the hot points in this control field because of the above reasons, and has been widely utilized in different practical production and life. Time delay occurs during signal transmission or system measurement, or because of some physical properties of the system equipment. If these above issues are not analyzed or processed, there will be serious results. Moreover, faults denote an event in which the function of some system components cannot work and this whole system function deteriorates. In general, faults are inevitable in the actual system, because of equipment wear produced by long operation time, abnormal operation, too complex models, overload, unreasonable design, poor maintenance, illegal change of system functions, and so on. There will also be serious resultants without analysis or disposal. Fault estimation and tolerant control can be adopted to resolve these above issues. According to the above discussions, this book researches fault estimation, active fault-tolerant control and passive fault-tolerant control for a class of nonlinear time delayed systems. Firstly, multiple-integral observers are devised to address the issues of fault estimation and active fault-tolerant control for time delayed fuzzy models with the cases of only actuator faults and both actuator and sensor faults. Secondly, the problems of intermittent actuator fault estimation, sensor fault estimation, and active fault-tolerant control are solved for time delayed switched fuzzy systems by exploring the sliding mode observer. The H ∞ guaranteed cost control is researched respectively for time delayed switched fuzzy systems and time delayed switched fuzzy stochastic systems subject to both intermittent actuator and sensor faults. Finally, multiple delay-dependent finite-time fault-tolerant control vii

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is researched for time delayed switched fuzzy systems subject to intermittent faults. The main contributions of this book are listed as follows: 1. In this investigation, the problems of observer-based fault estimation and active fault-tolerant control are studied for a class of T-S fuzzy systems subject to a time-varying delay. In light of the attained (k−1)th information, a new k-step TS fuzzy fast adaptive fault estimation approach is explored to design the observer. The observer improves this fault estimation performance and weakens the input disturbance influence of actuator fault derivatives in the error dynamic. Better estimation results can be attained with faster speed than existing ones, especially for the case in which the actuator fault sizes are constantly changing. In addition, a full-order controller is researched to stabilize this closed-loop system through compensating for this fault influence. Compared with existing research results, the proposed controller can be applied with a wider scope. 2. Fault estimation and tolerant control are studied for T-S fuzzy plants subject to multiple time-varying delays, local nonlinear models, and external disturbances. In light of the (k−1)th induction fault estimation message, a new observer is proposed to make the kth error system stable. The k-step induction fault estimation observer weakens the input disturbance influence of the actuator fault derivatives and improves the fault estimation performance subject to sensor and actuator faults as well as multiple time delays simultaneously. In contrast to existing work, the studied observer realistically better presents the sizes and shapes of actuator and sensor faults. Moreover, based on the online k-step fault estimation message, an active dynamic output feedback fault-tolerant controller is explored to make the closed-loop model asymptotically stable. In addition, delay-dependent sufficient conditions are obtained by establishing a fuzzy Lyapunov-Krasovskii functional and involving slack matrices in the form of linear matrix inequalities. Compared with the existing observers and controllers, the conservatism acquired by our design is less. 3. The problems of observer-based fault estimation and active dynamic output feedback control are developed for switched T-S fuzzy Itô stochastic systems subject to multiple time-varying delays external disturbances, intermittent actuator faults, and sensor faults. First of all, a new descriptor sliding mode observer is researched to construct the error function. In contrast to existing work, the addressed observer can be applied more widely. According to this online fault estimation message, the fault-tolerant controller is investigated to make this closed-loop model meansquare exponentially stable. Besides, a piecewise fuzzy Lyapunov function is described. Then the delay-dependent sufficient conditions are attained through linear matrix inequalities. The devised sliding mode observer is less conservative than the existing ones. The reachability of this sliding mode surface in this estimation error space is guaranteed with the proposed approach. Furthermore, the finite-time boundedness problem is developed. 4. The issue of delay-dependent reliable H ∞ guaranteed cost control is developed for uncertain switched T-S fuzzy systems against multiple interval timevarying delays. This is one of the few attempts to explore fault-tolerant control

Preface

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for nonlinear systems affected by external disturbances, nonlinear dynamics, measurement noise, and intermittent faults in sensors and actuators. The exponential stability of the system is guaranteed by designing the passive dynamic full-order output feedback controller. In contrast to existing work, the restrictions on multiple time-varying delays are relaxed in the derivation process, Therefore, the proposed controller can be used for systems with more time delay cases. Furthermore, with the introduction of piecewise fuzzy Lyapunov function and some slack matrices, the delay-dependent sufficient conditions are obtained in the form of linear matrix inequalities. Compared with the existing fault-tolerant controllers, the conservatism acquired by our design is less. 5. The problems of delay-dependent passive fault-tolerant control as well as optimal guaranteed cost control are discussed for multiple time-varying delayed switched T-S fuzzy stochastic systems with model uncertainties, external disturbances, measurement noise, and intermittent actuator and sensor faults. Few attempts are made to realize H ∞ guaranteed cost control for switched T-S fuzzy stochastic systems subject to intermittent actuator and sensor faults which means that more kinds of faults are allowed. The dynamic output feedback controller is devised. Then an augmented closed-loop system is established. A piecewise fuzzy Lyapunov function is given to obtain stability results of reliable H ∞ guaranteed cost control in the form of linear matrix inequalities. This method does not use slack matrices. Compared with existing fault-tolerant controllers, the conservatism acquired by our design is less. 6. For uncertain multiple time-varying time delayed switched T-S fuzzy systems, the problem of multiple delay-dependent finite-time fault-tolerant control under intermittent process faults, intermittent sensor faults, model uncertainties, external disturbances, and measurement noise is investigated. Few attempts are made to design a nonlinear dynamic output feedback controller for the switched T-S fuzzy systems. Besides, a closed-loop system is established. A piecewise fuzzy Lyapunov function is studied to realize robust finite-time H ∞ control. And finite-time boundedness and input-output finite-time stability are investigated. This publication is a research reference whose intended audience includes researchers and postgraduate and graduate students. Chongqing, China Shenyang, China Chongqing, China Chongqing, China July 2022

Shaoxin Sun Huaguang Zhang Xiaojie Su Jinyu Zhu

Acknowledgements

There are numerous individuals without whose help this book will not have been completed. Special thanks go to Prof. Guanghong Yang from Northeastern University, Prof. Yingchun Wang from Northeastern University, Prof. Jinhai Liu from Northeastern University, and Prof. Jiangshuai Huang from Chongqing University. The content of this book is completed under the careful guidance and strict requirements of Prof. Guanghong Yang. The ideas and methods in the book benefit from Prof. Yang’s inspiration to us. Professor Yang’s extensive and profound knowledge accumulation, diligent scientific research morale, conscientious and responsible scientific research attitude, and frank personality charm are all examples for us to learn from! Professor Yang’s help has benefited us all our life! Gratitude cannot be expressed in words, but can only be deeply engraved in the heart. Here, we sincerely wish Prof. Yang good health, a smooth career, peace and happiness, and the world is full of peach and plum! Professor Yingchun Wang has made a lot of efforts. From the way of thinking to the choice of words and sentences, he made many valuable suggestions. Here, we would like to express our heartfelt thanks to Prof. Wang! Professor Jinhai Liu not only gave us the most patient guidance and selfless help in scientific research, but also provided me with many valuable suggestions in life, which is an important guide in life. Here, we would like to express our heartfelt thanks to Prof. Liu. We would like to thank Prof. Jiangshuai Huang. In our book, Prof. Huang gave many valuable suggestions on the format, structure, and layout of the book. We would like to thank Zhanshan Wang, Jian Feng, Gang Wang, Dongsheng Yang, Qiuye Sun, Zhiliang Wang, Yanhong Luo, Zhenwei Liu, Xinrui Liu, Guotao Hui, Bonan Huang, Bowen Zhou, Zhenning Wu, Senxiang Lu, and other people for their help and support in our research. Finally, we would like to express our sincere thanks to all those who have helped us. We would like to thank all teachers and students for their help in writing this book! We wish all teachers good health and a happy family, and we wish all students who have helped us a bright future.

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The writing of this book was supported in part by the National Natural Science Foundation of China under Grant 62103066, the China Postdoctoral Science Foundation under Grants 2021TQ0392 and 2021M700592, the Chongqing Postdoctoral Innovative Talents Support Program under Grant CQBX2021005, and the Chongqing Postdoctoral Science Foundation under Grant cstc2021jcyj-bshX0178.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Problem Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Overview of the Stability of Fuzzy Systems with Time Delays . . . . 1.3 Overview of Fault Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Overview of Fault-Tolerant Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Active Fault-Tolerant Control . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Passive Fault-Tolerant Control . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Preparatory Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Symbol Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Several Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Publication Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 7 8 9 10 10 11 14 16

2 Fault Estimation and Tolerant Control for Time-Varying Delayed Fuzzy Systems with Actuator Faults . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 System Definition and Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 A New k-Step Induction Actuator Fault Estimation Method . . . . . . . 2.4 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 22 25 32 40 48 48

3 Fault Estimation and Tolerant Control for Multiple Time Delayed Fuzzy Systems with Sensor and Actuator Faults . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 System Definition and Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 State Augmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Observer Design Based on a k-Step Induction Fault Estimation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 52 54 54 56

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3.4 Design of Nonlinear Dynamic Output Feedback Fault-Tolerant Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66 74 81 81

4 Fault Estimation and Tolerant Control for Multiple Time Delayed Switched Fuzzy Stochastic Systems with Sensor Faults and Intermittent Actuator Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 System Definition and Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Design of Sliding Mode Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . j 4.3.1 Design of the Derivative Gain L¯ d . . . . . . . . . . . . . . . . . . . . . . . j 4.3.2 Design of the Proportional Gain L¯ pi . . . . . . . . . . . . . . . . . . . . 4.4 System Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Sliding Motion Reachability Control . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Finite-Time Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Reachability of T ∗ ≤ T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Finite-Time Boundedness Over Reaching Phase Within [0, T ∗ ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Finite-Time Boundedness Over Sliding Motion Phase Within [T ∗ , T ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112 115 120 120

5 Fault-Tolerant Control for Multiple Interval Time Delayed Switched Fuzzy Systems with Intermittent Faults . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 System Definition and Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 123 124 126 128 143 155 155

6 Fault-Tolerant Control for Multiple-Delayed Switched Fuzzy Stochastic Systems with Intermittent Faults . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 System Definition and Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Design of Dynamic Output Feedback Controller . . . . . . . . . . . . . . . . 6.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159 159 160 161 163 178 190 191

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Contents

7 Finite-Time Fault-Tolerant Control for Multiple-Delayed Switched Fuzzy Systems with Intermittent Faults . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 System Definition and Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 Conclusion and Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Chapter 1

Introduction

1.1 Problem Background Generally speaking, the stability analysis of the system is one of the key research problems of the system, which needs to be solved in both system analysis and control system design. In practical systems, such as the chemical industry and metallurgy, launch vehicles, space shuttles, and power grids, due to aging components, environmental changes, insensitive measurement or signal transmission, the system is often affected by time delays, parameter uncertainties, external interference, random noise, and other factors. In the presence of these unstable factors, its stability analysis becomes particularly critical. In the past 100 years, the stability analysis of various systems, especially nonlinear time delayed systems, has received extensive attention and in-depth discussion from scholars. At 4:40 p.m. on April 26, 1982, CAAC flight 3303 226 from Guangzhou to Guilin crashed over Gongcheng, 45 km away from Guilin. A total of 112 passengers and 8 crew members were killed. The reason was that the aircraft rolled due to an autopilot failure. When the roll reached 45 degrees, the pilot noticed the instrument, but at this time, the instrument display was misinterpreted. The pilot continued to press the bar to increase the roll, and finally rolled to 180 degrees, and the nose hit the mountain downward. On March 23, 2005, the isomerization unit of British Petroleum Company in Texas refinery exploded, which was caused by the malfunction of the liquid level gauge at the bottom of the tower. In August 2012, the Russian satellite launch mission failed because the “Proton-M” carrier rocket carrying two communication satellites into the sky malfunctioned during its orbit. On January 22, 2020, the bearing at the fixed end of Pingliang Hailuo 1327 fan suddenly vibrated greatly, resulting in tripping and economic losses. The reason is that the radial load borne by the bearing is too large, resulting in bearing damage. On April 22, 2020, a traffic accident with four deaths and one injury occurred in Shenzhen Cen Expressway (Jiangmen section). It is a life safety responsibility accident, which is caused by improper measures taken by the driver after the vehicle breaks down. These accidents always remind us that © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Sun et al., Fault-Tolerant Control for Time-Varying Delayed T-S Fuzzy Systems, Intelligent Control and Learning Systems 9, https://doi.org/10.1007/978-981-99-1357-2_1

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1 Introduction

the stable operation of modern system equipment is very important, and when faults occur, fault estimation and tolerant control are also very important.

1.2 Overview of the Stability of Fuzzy Systems with Time Delays In actual production and life, as the production process becomes increasingly complex and the degree of industrial modernization increases, nonlinear systems are increasingly common (Wang et al. 2022). It is worth pointing out that modeling is a key problem in the study of nonlinear practical systems. As one of the modeling methods to depict or approach complex nonlinear plants, establishing the Takagi-Sugeno (T-S) fuzzy model is an important and very effective modeling method. T-S fuzzy control takes fuzzy language variables and logical reasoning as tools and uses human experience and knowledge to integrate intuitive reasoning into decision-making. It provides local linear expressions of the underlying nonlinear system through a class of IF-THEN rules (Wang et al. 2018a, b; Chadli and Karimi 2013; Sakthivel et al. 2020, 2016). The T-S fuzzy model can approach nonlinear systems with arbitrary accuracy and is appropriate for describing the dynamic characteristics of complex nonlinear plants (Liu et al. 2019; Li et al. 2019; Zhou et al. 2021; Hu et al. 2012). Therefore, under the framework of T-S fuzzy system, fuzzy “mixing” of submodels of the T-S fuzzy model can deal with many nonlinear models well (Wang et al. 2020a; Zhang et al. 2015a, b; Baranyi 2014; Mirzajani et al. 2019). In 1973, Zadeh put forward the idea of transforming the linguistic description of logic rules into relative control laws (Zadeh 1973), so that it could deal with complex systems in a practical way according to the human thinking mode, which laid a theoretical foundation for the production of early fuzzy controllers. In 1985, Takagi and Sugeno proposed the concept of the T-S fuzzy model. To date, the studies on T-S fuzzy systems have been hot issues. An increasing number of problems of robust control, fault estimation, fault-tolerant control, stability, and stabilization analysis of nonlinear systems have been described and solved under the framework of T-S fuzzy models (Chadli and Karimi 2013; Wang et al. 2020a; Zhang et al. 2015a, b; Baranyi 2014; Mirzajani et al. 2019; Xue et al. 2018; Zhang and Zhao 2020; Wai and Yang 2008; Zhang et al. 2019, 2017; Blanke et al. 2006; Gao and Ding 2007a; Yang and Tong 2016). And great fruits have been made, especially through linear matrix inequalities (Huang and Yang 2014; Han et al. 2018; Qiu et al. 2012; Han et al. 2015). In Wang et al. (2020a), by transforming a nonlinear system into a T-S fuzzy system, the problem of periodic tracking control was studied, and the periodic tracking control of Chua’s circuit dynamic equation with an input term was realized. Zhang et al. (2015a) studied the relative degrees of a cluster of single-input and single-output T-S fuzzy systems and the controller design according to feedback linearization. In Baranyi (2014), the generalized tensor product model transformation was implemented for T-S fuzzy model manipulation and generalized stability verification. Zhang et al. (2015b) studied the

1.2 Overview of the Stability of Fuzzy Systems with Time Delays

3

problem of network-based output tracking control for a series of T-S fuzzy systems that cannot be stabilized by using time delay free output feedback controllers. Mirzajani et al. (2019) discussed the design of adaptive T-S fuzzy control for a group of fractional order systems under the parameter uncertainty and input constraints. In Wai and Yang (2008), according to the T-S fuzzy model, an adaptive fuzzy neural network controller was designed for a manipulator with actuator dynamics. Zhang et al. (2019) studied the sliding mode control of T-S fuzzy multi-agent models. The issue of fault estimation and tolerant control of electromechanical models has been implemented in Han et al. (2018). Qiu et al. (2012) converted a car motion model into a series of T-S fuzzy models and studied the fault estimation and fault-tolerant control of car motion. However, it should be noted that although the related problems of T-S fuzzy systems have received certain attention and study, there are still some problems to be further studied (Cao and Frank 2001; Moodi and Farrokhi 2014). In this book, T-S fuzzy systems are studied. In the following chapters, various nonlinear systems are modeled into different T-S fuzzy models with fuzzy rules for discussion and analysis. In many practical systems, due to a variety of propagation and transmission conditions, time delays are common when establishing a variety of data dynamic models (Wang et al. 2010; Jun 2007; Jiang and Han 2007; Li et al. 2019; Sun et al. 2019; Wu et al. 2011a). For example, in economic systems, a time delay is a widespread and unavoidable phenomenon in nature. In the economic system, time delays exist in some economic fields by using some time intervals in a natural form, such as the evolution of product market investment policies (Niculescu 2001). A large number of practical results have proven that this behavior of the system at the current time caused by a time delay is related to its state at a certain time: the previous input or state value, and the real-time delay will affect the state or input of the nonlinear system, resulting in a series of complex dynamic behaviors, such as vibration, instability, chaos, and so on. Many widely studied T-S fuzzy models have time delays. At present, the stability of nonlinear time delayed systems, especially time delayed T-S fuzzy systems, has attracted extensive attention (Zhang et al. 2009a; Zhao et al. 2009; Cao and Sun 1998; Chen et al. 2008; Wu et al. 2011b; Wu and Li 2007; Yan et al. 2019). Lin et al. (2008) proposed the design of an H∞ filter for a time-varying delayed nonlinear system based on a T-S fuzzy system. Wu et al. (2011a) mainly studied the H∞ model approximation of discrete-time T-S fuzzy time delayed systems. Chen et al. (2008) studied the observer-based stabilization of T-S fuzzy models with an input delay. Wu et al. (2011b) studied the stability analysis and stabilization of discrete-time T-S fuzzy time-varying delayed systems. Wu and Li (2007) studied the modified truck trailer model against a time delay, which was modeled as a T-S model. Yan et al. (2019) studied the event-triggered control of a T-S fuzzy network system with the transmission delay and applied it to practical wireless networks based on the ZigBee protocol. Therefore, the stability analysis of a time delayed T-S fuzzy system is an important characteristic of its system and a necessary condition for the normal operation of the system, which has important theoretical and practical significance. Some stability analysis methods are introduced below.

4

1 Introduction

In classical control theory, the logarithm criterion, Routh criterion, root locus criterion, and Nyquist criterion are often used to analyze the stability of linear steadystate systems, but they are not suitable for the stability analysis of nonlinear time delayed systems. When analyzing the nonlinear system, the linear part of the system is required to have better harmonic filtering performance if the description function law is used; If the phase plane method is used, the nonlinear system is required to be first-order or second-order. In 1892, Russian scholar Lyapunov proposed a more general stability theory to determine the stability of the system. The theory uses state vector description, which can be used not only for the stability analysis of univariate, linear, and constant systems, but also for the stability analysis of multivariable, nonlinear, and time-varying systems. Lyapunov theory can be used to solve the stability analysis of nonlinear time delayed systems. In the process of system stability analysis, Lyapunov theory produces two sets of methods: one is to judge the stability through the solution of the linear system differential equation, which is called Lyapunov’s indirect method; the other is to first build Lyapunov function with experience and skills, and then use the Lyapunov function to judge the stability. It is called Lyapunov’s direct method. Lyapunov’s indirect method needs to solve the differential equation of a linear system in the process of use, which judges the stability by using the characteristics of the solution of the state equation. It should be pointed out that it is difficult to solve the differential equations of linear systems, so this method is limited in application, and can only be used in linear time-varying systems, linear constant systems, linearizable nonlinear functions, and other occasions. Based on the above analysis, Lyapunov’s indirect method is not used in the stability analysis of nonlinear time delayed systems in the book. Lyapunov’s direct method does not need to solve the differential equations of linear systems in the process of use, which makes the stability analysis much simpler, so this method has a wider range of applications. All aspects in modern control theory, such as nonlinear system control, optimal control, time-varying system control, and adaptive control, have been widely used and developed. It can be seen that Lyapunov’s direct method is more suitable for the stability analysis of nonlinear time delayed systems.

1.3 Overview of Fault Estimation The functional failure of some components in the plant will lead to this deterioration of the whole system function. If the system is under normal conditions, at least one parameter or characteristic has unexpected deviation, and then all the unexpected changes that can reduce the overall performance of the system can be called faults. In other words, a fault is the state in which the system cannot perform the specified functions, and the fault may make the system unstable. Faults are common in practice and have many characteristics (Wang et al. 2020b; Shen et al. 2019; Wu et al. 2019, 2020). When the system fails, it is often necessary to conduct fault estimation to determine the location and size of the fault, so as to take the next steps, such as whether to carry out maintenance, or life assessment (Wang et al. 2018c). According

1.3 Overview of Fault Estimation

5

to different classification standards, the fault classification is also different. According to the fault correlation, faults can be divided into related faults and non-related faults. According to the injection form, faults can be divided into additive faults and multiplicative faults. According to the occurrence and development process of faults, faults can be divided into sudden faults and gradual faults. According to the fault location, faults can be divided into actuator faults, sensor faults, and process faults. According to the fault nature, faults can be divided into constant faults, slow time-varying faults (derivative is approximately equal to 0), and time-varying faults. According to the fault cause, they can be divided into external cause faults and internal cause faults. According to the fault severity, they can be divided into destructive faults and non-destructive faults. According to the fault duration, they can be divided into permanent faults, transient faults, and intermittent faults (Shen et al. 2012; Syed et al. 2016; Sakthivel et al. 2017). The permanent fault will not disappear until corrective measures are taken. A transient fault is a kind of fault that only temporarily affects the dielectric performance of electrical equipment and can recover itself in a short time. The intermittent fault refers to the intermittent occurrence of the fault. That is, if no measures are taken, intermittent faults will occur or disappear randomly. According to the faulty components, faults can be divided into hardware faults and software faults. In addition, an equipment fault generally has five basic characteristics: hierarchy, transmissibility, radioactivity, delay, and uncertainty. Generally speaking, the current fault estimation methods are simpler, mainly designing different types of waveguides and observers (Wang et al. 2020c; Liu et al. 2016). In addition, because the sensors that need to be measured in the controller design or system monitoring are expensive or impossible to measure, the design of waves or observers for nonlinear time delayed systems is very necessary. At present, many methods have been used to design observers or waveguides for nonlinear time delayed systems (Wang et al. 2018b; Xie et al. 2018; Xu et al. 2018). This book mainly studies the design of observers. The main purpose of the observer is to estimate the unmeasurable state of the system by measuring the output and input of the system. It is essentially a mathematical replica of the system, injecting the system input into the observer and comparing its output with the system output. Then the difference between the system output and the observer output is called the output estimation error, and it is inverted as a correction term, so that the observer state receives the system state. Some observers commonly used in nonlinear time delayed systems are introduced below. Augmented state observer When there is a fault term in the system, if the traditional state observer is used, the fault value will affect the observer gain matrix, so a more ideal fault estimation value cannot be obtained. In order to overcome this phenomenon, an extended state observer, also known as a generalized observer (Gao and Wang 2006), is proposed. In essence, this kind of observer is a proportional differential observer. For example, in

6

1 Introduction

reference (Han et al. 2018), the augmented state observer estimated the system state and sensor fault at the same time. In Han et al. (2019), the augmented state observer estimated the system state, the process fault, and the sensor fault at the same time. Lomberg observer Luenberger, Kalman, and Bucy et al. proposed the concept of the Lomberg observer (Luenberger 1971). The Lomborg observer is one of the earliest observers. Its essence is to linearly feedback the output estimation error to the observer. It is a common method to solve the problem of the dynamic system control rate. The steps are simply summarized as follows: first, the estimated value of the state vector is obtained, and then the estimated value is substituted into the ideal control law. The Lomborg observer method can be used in dynamic systems where the measurement results are not seriously polluted by noise and the order of the observer is lower than that of the observed system. However, in the presence of unknown signals, the Lomborg observer usually cannot guarantee that the output estimation error is zero, so the observer state will not converge to the system state. Sliding mode observer The sliding mode observer feeds back the output estimation error through the nonlinear switching term (Edwards and Spurgeon 1994; Utkin 1992), which provides an attractive solution to the problem of fault estimation. When the bound of disturbance is known, the sliding mode observer can make the output estimation error converge to zero in finite-time and the observer state asymptotically converges to the system state, which is unlike the linear observer that can only converge asymptotically. In the process of sliding motion, the equivalent output error injection (equivalent control simulation) contains the information of the unknown signal. By properly adjusting the equivalent output error injection, an accurate estimation of the unknown signal can be obtained. The first sliding mode observer appeared in Utkin (1992). Walcott and Zak improved the current existing observer by adding a linear feedback term (Walcott and Zak 1987), so that the sliding surface could be expanded. Edwards and Spurgeon modified the sliding surface of the Walcott-ZaK observer and proposed a numerical design method of the system (Edwards and Spurgeon 1994). In addition, Edwards and Spurgeon (1994) determined the necessary and sufficient conditions for the existence of the observer based on the original system matrices, and thus determined a class of systems in which the observer was feasible. By modeling the fault as an unknown signal, the sliding mode observer can be used to reconstruct the fault, so as to achieve the purpose of fault detection and fault isolation. Edwards et al. reported the early work on fault reconstruction using a sliding mode observer (Edwards et al. 2000). However, in the presence of other disturbances, such as unmodeled dynamics, parameter uncertainties, or external disturbances, the reconstruction method described in references Edwards et al. (2000), Edwards and Spurgeon (2000) will no longer be applicable. Therefore, it is very important to ensure that the fault

1.4 Overview of Fault-Tolerant Control

7

reconstruction is robust to these disturbances, so as to achieve robust fault reconstruction. Tan and Edwards proposed a design method of observer parameters to minimize the L 2 gain of disturbance in the process of fault reconstruction (Tan and Edwards 2003). Proportional integral observer The proportional integral observer is regarded as an extension of the traditional Lomborg observer. Generally speaking, the Lomborg observer can be regarded as a proportional term that introduces the output estimation error into the observer state equation, so the Lomborg observer is essentially a proportional observer. In essence, the proportional integral observer introduces both the proportional and integral terms of the output estimation error into the observer state equation, so it is called the proportional integral observer. The observer reconstructs the fault by outputting the integral term of the estimation error. Recently, proportional integral observers have been used on an increasing number of occasions for fault estimation. Robust fault estimation of nonlinear systems can be achieved by using proportional integral observers and robust control (Zhang et al. 2010; Jia et al. 2015). The advantage of the proportional integral observer is that it has no additional restrictions on the system structure, and is very suitable for constant faults and slow time-varying faults. If this disturbance effect is not considered, the proportional integral observer can obtain this asymptotic estimation of the constant fault. This disadvantage of proportional integral observer is that for time-varying faults, the change rate of faults can affect the estimation results. Multiple-integral observer In order to overcome the shortcomings of proportional integral observers, a variety of proportional integral observers have been studied, among which the relatively common is the multiple-integral observer. There are many integral links in the observer. For example, in Koenig (2005), a proportional multiple-integral observer is used to obtain a more ideal asymptotic estimate of the fault when the higher-order derivative is zero. In Huang and Yang (2014), a k-step fault estimation multiple-integral observer is adopted, which weakens the input disturbance from actuator faults and can obtain better actuator fault estimation. It should be noted that in the face of increasingly complex nonlinear time delayed systems, sometimes a variety of observers need to be combined and applied at the same time, and sometimes the observer needs to be further improved to obtain better estimation results. In this book, the observer will be further studied and applied according to different system characteristics.

1.4 Overview of Fault-Tolerant Control Modern control theory has been increasingly successfully applied to practical industrial systems (Shi et al. 2023; Song et al. 2022), and fault-tolerant control has also attracted widespread attention (Liu et al. 2013, 2018; Tao et al. 2015). When sensors,

8

1 Introduction

actuators, or components fail, fault-tolerant control ensures that the failed system can still remain stable, meet certain satisfactory performance index, and display the tolerance of the system to faults. In 1971, Niederlinski proposed the concept of integrity control, which is considered to be the beginning of fault-tolerant control (Niederlinski 1971). In 1985, Eterno proposed that fault-tolerant control can be divided into two forms: active fault-tolerant control and passive fault-tolerant control (Eterno et al. 1985), which has now been widely recognized. With the maturity of faulttolerant control mechanisms and the continuous improvement of computer control systems, the research on fault-tolerant control has increasingly important theoretical significance and application value. The following is a brief introduction about active fault-tolerant control and passive fault-tolerant control.

1.4.1 Active Fault-Tolerant Control Active fault-tolerant control can also be called self-healing, rescheduling, reconfigurable or self-designed control. In active fault-tolerant control, by actively responding to the component failure of the system, the stability of the system can be maintained through controller reconfiguration, and the acceptable performance can be obtained. In the research of active fault-tolerant control, fault estimation is one of the key parts. More fault information can be obtained through fault estimation (Gao and Ding 2007b). Combining fault diagnosis with fault estimation can solve the problem of fault diagnosis composed of fault detection, fault isolation, and fault identification (Blanke et al. 2006; Zhang et al. 2004). Active fault-tolerant control is very dependent on the fault estimation information provided by observers or filters, and there is relatively little research on fault estimation as a whole. The reason is that there are many constraints in fault estimation and quantitative analysis is difficult. Therefore, it can be seen that the main research goal of active fault-tolerant control is to design a suitable controller in line with the system structure, which can have stability and satisfactory performance under normal and fault conditions of the system. Specifically, when there are no faults, the overall quality and performance of the system should be considered; when a fault occurs, the key consideration is how to avoid the decline of system performance, even if it cannot reach the system performance index without faults. According to the previous content, in recent years, fault estimation has attracted the attention of many scholars (Wang et al. 2018b; Xie et al. 2018; Gao and Wang 2006; Han et al. 2016; Gao et al. 2008; Yoneyama et al. 2000), which has further stimulated people’s enthusiasm for the research on active fault-tolerant control. Han et al. (2018) studied the fault estimation and tolerant control of switched fuzzy stochastic plants against actuator and sensor faults and applied the proposed method to the mechanical system dynamics of a single-link direct-drive manipulator driven through a permanent magnet brush DC motor and the electrical system dynamics of a DC motor. In Gao (2015), a new discrete-time estimator was proposed and based on the estimated fault signal and system state, a discrete-time fault-tolerant design method was proposed. In Ren et al. (2018), when

1.4 Overview of Fault-Tolerant Control

9

a fault happened, a redesigned setpoint fault-tolerant method was used to adapt to the fault, rather than reconstructing the controller. In Ren et al. (2015), the controller was designed for the health subsystem, which could adapt to faults and guarantee that the whole system still had good operation performance. In Zhang et al. (2009b), fast active fault-tolerant control was realized by using an adaptive fault diagnosis observer, and this issue of fault estimation and fault-tolerant control of a single-link flexible joint manipulator system, which was very common and necessary in practical applications, was explored. Liu et al. (2020) discussed the distributed fault estimation of multi-agent systems with nonlinear dynamics and applied fault estimation and tolerant control to the network of four single-link flexible joint manipulator systems. In Van et al. (2017), for uncertain manipulators against actuator faults, a new finite-time fault-tolerant control was proposed, and an online fault estimation method based on time delay estimation was studied to approach actuator faults. Chen and Cao (2013) gave fault estimation and fault-tolerant control of linearized longitudinal dynamics of vertical take-off and landing (VTOL) aircraft. With the efforts of many scholars at home and abroad, active fault-tolerant control has achieved fruitful results and been widely used (Liu et al. 2019; Li and Zhang 2020), but it is still a very attractive research field.

1.4.2 Passive Fault-Tolerant Control Passive fault-tolerant control makes the whole closed-loop system insensitive to the generated faults by means of robust control or H∞ control without changing the parameters and structure of the controller, so as to ensure that the system can still operate stably and achieve the original performance even if there is a fault (Tao et al. 2016). It should be noted that this passive fault-tolerant controller is fixed, which can be robust to some specific faults without fault diagnosis. Therefore, passive faulttolerant control does not need any online fault information, only needs to design the controller according to the fault situation, and has good rapidity and real-time performance for faults. It is insensitive to faults, so it can maintain good performance of the system even in the event of faults (Tao et al. 2015). Although the design of this controller reduces the overall performance of the system, it does not depend on whether the fault occurs, nor does it change with the change of fault type and size. In other words, the parameters of the passive fault-tolerant controller are determined in advance, and a trade-off needs to be made between fault tolerance and performance. If the performance requirements are not high, passive fault-tolerant control is a better choice. Based on the above analysis, it can be seen that active fault-tolerant control and passive fault-tolerant control have their own advantages and disadvantages. Active fault-tolerant control needs a fault diagnosis unit, while passive fault-tolerant control does not need any online fault information. However, active fault-tolerant control does not need to know the fault situation in advance, and passive fault-tolerant control can only deal with the faults considered in advance. The starting point of active fault-

10

1 Introduction

tolerant control is to ensure the system performance when there are no faults, while the starting point of passive fault-tolerant control is that the system can still operate stably under the worst fault condition. However, the fault is an abnormal behavior, and there are no faults in most cases. Thus the conservatism of active fault-tolerant control is relatively low, and the conservatism of passive fault-tolerant control is relatively high. This book will discuss and apply it according to different actual needs.

1.5 Preparatory Knowledge This section lists the mathematical symbols and some basic lemmas often involved in this book.

1.5.1 Symbol Description R n and R n×m represent n × 1 and n × m dimensional Euclidean spaces respectively. I represents the identity matrix with appropriate dimension. In stands for the identity matrix of n dimension. For the matrix K˜ : K˜ T represents the transposed matrix of K˜ , K˜ − represents the left inverse matrix of K˜ , that is, K˜ − K˜ = I , K˜ + represents the Moore-Penrose inverse matrix of K˜ , and K˜ (1) represents a class of generalized inverse matrix of K˜ , which is called the {1}- inverse matrix of K˜ . If K˜¯ only meets K˜ K˜¯ K˜ = K˜ , then K˜¯ is expressed as K˜ (1) . K˜¯ exists and it is not unique. For the left inverse matrix, Moore-Penrose inverse matrix and {1}-inverse matrix of matrix K˜ , see Zhang (2004) for details. K˜ > 0 means that K˜ is a positive definite matrix, K˜ ≥ 0 indicates that K˜ is a semi positive definite matrix and K˜ < 0 means that the matrix K˜ is a negative definite matrix. tr( K˜ ) means the trace of K˜ . sym( K˜ ) = K˜ + K˜ T . λmax ( K˜ ) and λmin ( K˜ ) represent the maximum characteristic value and the minimum characteristic value of K˜ , respectively. P{ K˜ } represents the probability of event K˜ ; E{ K˜ } represents the expected value of event K˜ . For the ˜ t˜)) and min (m( ˜ t˜)) represent the maximum and minimum variables m( ˜ t˜), max (m( t˜∈[t1 ,t2 ]

t˜∈[t1 ,t2 ]

values of m(t) ˜ in the time interval [t1 , t2 ], respectively. L 2 [0, ∞) represents the square sum vector space. diag{ K˜ 1 , K˜ 2 , . . . , K˜ n } represents a block diagonal matrix composed of a square matrix K˜ 1 , K˜ 2 , . . . , K˜ n . The symbol * in the matrix represents the transposed element of the symmetric position.  ·  means the second norm. If hˆ ˆ means taking the real part of h. ˆ C+ reprerepresents a complex number, then Re(h) sents positive and negative numbers. D(β, ) represents the linear matrix inequality ˆ ∈ D(β, ), then −β < aˆ < 0 and |aˆ + Bi| ˆ < . In field. Particularly, if aˆ + bi addition, unless otherwise specified, it is assumed that both matrices or vectors have appropriate dimensions.

1.5 Preparatory Knowledge

11

1.5.2 Several Lemmas Before giving the main results of this book, the following introduces some lemmas that are useful for subsequent proofs. In the following theorems, suppose bi is the membership function and r is the number of IF-THEN rules. Lemma 1.1 (Iwasaki and Hara (2005)) Define ω0 (x), ω1 (x), . . . , ωk (x) as follows: ωi (x) = x T i x, i = 0, 1, . . . , k, where iT = i , x ∈ R m . At the same time, if scalar δ1 ≥ 0, . . . , δk ≥ 0 makes ω1 (x) ≤ 0, . . . , ωk (x) ≤ 0, 0 −

k 

δi i < 0,

i=1

this shows that ω0 (x) < 0. Lemma 1.2 (Guan and Chen (2004)) Assuming that for matrices K¯ i and K¯ i j with suitable dimensions, where i, j, h, l = 1, . . . , r and P¯ > 0, the following inequality holds: r 

bi

r 

i=1

j=1

r 

r 

i=1

bi

j=1

b j K¯ iT P¯ K¯ j ≤

r 

bi K¯ iT P¯ K¯ i ,

i=1

bj

r 

bh

h=1

r 

bl K¯ iTj P¯ K¯ hl ≤

l=l

r  i=1

bi

r 

b j K¯ iTj P¯ K¯ i j .

j=1

Lemma 1.3 (Wang and Lam (2018)) For a matrix Pb =

r 

bi Pi that depends on

i=1

membership function, of which Pi > 0, (i = 1, . . . , r ), is a free variable, satisfying the following formula: P˙b =

r  i=1

b˙i Pi =

r  h=1,h=s

b˙h (Ph − Ps ) ≤ 0.

If h = 1, 2, . . . , R, h = s, the following switching rules are obtained: ⎧ ˙ if bh ≤ 0, then ⎪ ⎪ ⎪ ⎨ Ph − Ps ≥ 0 ⎪ if b˙h > 0, then ⎪ ⎪ ⎩ Ph − Ps < 0

(1.1)

12

1 Introduction

It can also be further expressed as if Hl , then Cl ,

(1.2)

where l = 1, 2, . . . , 2r −1 , Hl means b˙h set of possible combinations, and Cl means the set of constraints of Pi . b˙ H is the time derivative of this membership function, which is positive or negative over time. For example, through Wang and Lam (2018), when r = 3 and s = 3, we have P˙b = b˙1 (P1 − P3 ) + b˙2 (P2 − P3 ) Lemma 1.3 can be written as  if H1 : b˙1 ≤ 0, b˙2 ≤ 0, then C1 :  if H2 : b˙1 ≤ 0, b˙2 > 0, then C2 :  if H3 : b˙1 > 0, b˙2 ≤ 0, then C3 :  if H4 : b˙1 > 0, b˙2 > 0, then C4 :

P1 ≥ P3 P2 ≥ P3 P1 ≥ P3 P2 < P3 P1 < P3 P2 ≥ P3 P1 < P3 P2 < P3

Therefore, there are 22 possible combinations to guarantee P˙b ≤ 0. Lemma 1.4 (Schur complement Boyd et al. (1994), Nguang and Shi (2003), Chen K 11 K 12 (1999)) For a given matrix K = , of which K 11 and K 22 represent sym∗ K 22 metric matrices, and the following three conditions are equivalent: (1) K < 0; (2) −1 T −1 T K 12 < 0; (3) K 11 < 0, K 22 − K 12 K 11 K 12 < 0. K 22 < 0, K 11 − K 12 K 22 ¯ Lemma 1.5 (Chilali and Gahinet (1996)) For D(β, ), the eigenvalue of matrix X is D(β, ), if and only if there exists a matrix P > 0 satisfying conditions as follows: ¯ sym(XP) + 2βP < 0,

¯ −P XP < 0. ∗ −P Lemma 1.6 (Chen (1999)) Given the matrices Aˇ ∈ R n×n , Pˇ ∈ R n×n , Cˇ ∈ R n y ×n , ˇ C), ˇ two equivalent conditions can be obtained: (1) Aˇ is stable; (2) If ( A, ˇ C) ˇ for ( A, T ˇ T ˇ ˇ ˇ ˇ ˇ is detectable, then Lyapunov equation A P + P A = −C C has a unique solution.

1.5 Preparatory Knowledge

13

Lemma 1.7 (Han et al. (2018)) Suppose x(t) ∈ R m represents an m-dimensional Itô process, which obeys the following stochastic differential dx(t) = f 1 (t)dt + f 2 (t)dW (t), where f 1 ∈ R m , f 2 ∈ R m×n . Let V (x, t) ∈ C2,1 (R m × R + ; R + ), and then V (x, t) is an Itô process, which obeys the following stochastic differential d V (x, t) =L V (x, t)dt + Vx (x, t) f 2 (t)dW (t), 1  L V (x, t) =Vt (x, t) + Vx (x, t) f 1 (t) + tr f 2T (t)Vxx (x, t) f 2 (t) , 6 2 where 

2 ∂V (x, t) ∂ V (x, t) , , Vxx (x, t) = ∂t ∂xi x j m×m

∂V (x, t) ∂V (x, t) ∂V (x, t) . Vx (x, t) = , ,..., ∂x1 ∂x2 ∂xm

Vt (x, t) =

Lemma 1.8 (Gu et al. (2003), Zhang et al. (2008)) For any matrix P > 0, scalars χ1 > 0 and χ2 > 0, and vector function ξ¯ : [χ1 − χ2 , χ1 ] → R n that can make the following inequality definition hold, we can obtain χ1 χ2

⎛ ¯ ≥⎝ ξ (θ)Pξ(θ)dθ ¯T

χ1 −χ2

⎞T

χ1

¯ ⎠ P ξ(θ)dθ

χ1 −χ2

χ1

¯ ξ(θ)dθ.

χ1 −χ2

Lemma 1.9 (Jiang and Han (2007)) There exists a symmetric matrix B¯ satisfying

B1 + B¯ L1 ∗ T1



> 0,

B2 − B¯ L2 ∗ T2

> 0,

when and only when ⎡

⎤ B1 + B2 L1 L2 ⎣ ∗ T1 0 ⎦ > 0, ∗ ∗ T2 where B1 , B2 , L1 , L2 , T1 and T2 are matrices with appropriate dimensions.

14

1 Introduction

1.6 Publication Outline This book mainly studies the H∞ fault-tolerant control of time-varying delayed T-S fuzzy systems. Firstly, the problem of fault estimation and fault-tolerant control for a class of time delayed T-S fuzzy systems with actuator faults is studied by designing a multiple-integral observer and an active dynamic output controller. Secondly, by designing a multiple-integral observer and an active dynamic output controller, the problem of fault estimation and fault-tolerant control for a class of time delayed T-S fuzzy systems with actuator faults and sensor faults are studied. Then, for a class of switched T-S fuzzy Itô stochastic systems with multiple time-varying delays, intermittent actuator faults, and sensor faults, the generalized sliding mode observer and sliding mode controller are designed to estimate the system state and the fault at the same time and make the system operate stably. Then, the problem of delaydependent reliable H∞ guaranteed cost control for uncertain switched T-S fuzzy systems with multiple interval time-varying delays, intermittent actuator faults, and intermittent sensor faults is studied. Then, the reliable H∞ guaranteed cost control of uncertain switched fuzzy stochastic systems against multiple time-varying delays, intermittent actuator faults, and intermittent sensor faults is studied. Finally, the problem of delay-dependent finite-time fault-tolerant control for uncertain switched T-S fuzzy systems with multiple time-varying delays, intermittent process faults, and intermittent sensor faults is studied. The main contents of this book are arranged as follows: This chapter mainly expounds on the research background and significance of this book and the development history and research status of fault estimation, active faulttolerant control, and passive fault-tolerant control and gives the relevant mathematical symbol definitions and lemmas in this book. Finally, the main content of this book is briefly introduced. In Chap. 2, based on the T-S fuzzy model with a time delay, observer-based fault estimation and active fault-tolerant control are studied. Using the (k − 1)th information obtained, a new k-step T-S fuzzy fast adaptive fault estimation method is proposed to establish the observer. A full-order controller is proposed to stabilize the closed-loop system through compensating for the fault influence. Compared with the existing research work, the proposed controller has a wider range of applications. The stability of the closed-loop system is analyzed through utilizing a fuzzy Lyapunov functional method. Finally, the advantages of this method are illustrated by two simulation examples. In Chap. 3, observer-based fault estimation and fault-tolerant control are studied for T-S fuzzy models against multiple time-varying delays. Inspired by (k − 1)th induction fault estimation information, a new observer is proposed to establish the kth error dynamic. In the case of sensor and actuator faults and multiple time delays, the k-step induction fault estimation observer reduces this influence of input disturbance affected by actuator fault derivatives and obtains better fault estimation performance. In contrast to the existing results, the observer can more truly display the size and shape of actuator faults and sensor faults. Besides, in light of the online message of

1.6 Publication Outline

15

k-step induction fault estimation, an active dynamic output feedback fault-tolerant controller is designed to make the closed-loop fuzzy system asymptotically stable. Compared with these existing observers and fault-tolerant controllers, the results obtained in this chapter are less conservative. Finally, two simulation examples are used to verify the advantages and effectiveness of the method in this chapter. In Chap. 4, the design of a fault estimation observer and an active dynamic output feedback controller is studied for switched T-S fuzzy Itô stochastic systems with multiple time-varying delays, intermittent actuator faults, and sensor faults. Firstly, a new generalized sliding mode observer is studied to establish this error dynamic. In contrast to existing work, the suggested observer is applied to more cases and has a wider range of applications. Secondly, according to the online fault estimation message, a fault-tolerant controller is proposed to make the closed-loop system meansquare exponentially stable. On this basis, a new piecewise fuzzy Lyapunov function is proposed, and these delay-dependent sufficient conditions are obtained using linear matrix inequalities. The designed sliding mode observer is less conservative than the existing sliding mode observer. This method ensures this reachability of the sliding surface in the estimation error space. And this chapter discusses the finitetime bounded problem. Finally, the novelty and effectiveness of this method are verified by a simulation example. In Chap. 5, the problem of delay-dependent reliable H∞ guaranteed cost control for uncertain switched T-S fuzzy systems against multiple interval time-varying delays is studied. There are few tries to study fault-tolerant control of nonlinear systems affected by external interference, nonlinear functions, measurement noise, intermittent actuator faults, and intermittent sensor faults. In order to achieve exponential stability of the system, a passive dynamic full-order output feedback controller is proposed. Compared with the existing results, this chapter relaxed the limitation of multiple time-varying delays in the derivation process. The controller has wide application prospects. In addition, a piecewise fuzzy Lyapunov function is used, slack matrices are introduced, and sufficient conditions related to time delays are given by a set of linear matrix inequality methods. Compared with the existing fault-tolerant controllers, less conservatism is obtained. Finally, simulation examples are given and comparisons are carried out to demonstrate the merits and availability of the approach in this chapter. In Chap. 6, the problem of delay-dependent passive fault-tolerant control and optimal guaranteed cost control for switched T-S fuzzy stochastic systems under multiple time-varying delays, model uncertainties, intermittent actuator faults, intermittent sensor faults, external disturbances, and measurement noise is studied. There are few attempts to realize H∞ guaranteed cost control for switched T-S fuzzy stochastic systems against intermittent actuator faults and intermittent sensor faults that can allow more kinds of time delays. Therefore, this method has a wider range of applications. In this chapter, a dynamic output feedback controller is constructed. Furthermore, a closed-loop system is formed. Then the stability results of reliable H∞ guaranteed cost control are provided by using piecewise fuzzy Lyapunov function in the form of linear matrix inequalities. The proposed method does not need slack matrices at all,

16

1 Introduction

and its conservatism is relatively low. Finally, an example is provided to illustrate the availability of this method. In Chap. 7, the problem of delay-dependent finite-time fault-tolerant control for uncertain switched T-S fuzzy systems against multiple time-varying delays is studied. There are few tries to study the finite-time fault-tolerant control of nonlinear systems affected by external interference, measurement noise, intermittent process faults, and intermittent sensor faults. In order to achieve finite-time boundedness and input-output finite-time stability of the system, a passive dynamic full-order output feedback controller is proposed and has wide application prospects. In addition, a piecewise fuzzy Lyapunov function is used, slack matrices are introduced, and sufficient conditions related to time delays are given by a set of linear matrix inequality methods. Compared with the existing fault-tolerant controllers, less conservatism is obtained. Finally, a simulation example is given, and a comparison is carried out to demonstrate the merits and availability of the approach in this chapter. Chapter 8 summarizes the main work of this book and gives some problems and possible development directions of this research in theoretical research and practical application. Finally, future work is discussed.

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Liu L, Liu Y, Tong S (2019) Fuzzy-based multierror constraint control for switched nonlinear systems and its applications. IEEE Trans Fuzzy Syst 27(8):1519–1531 Liu C, Jiang B, Patton RJ et al (2019) Hierarchical-structure-based fault estimation and fault-tolerant control for multiagent systems. IEEE Trans Control Netw Syst 6(2):586–597 Liu X, Gao X, Han J (2020) Distributed fault estimation for a class of nonlinear multiagent systems. IEEE Trans Syst Man Cybern: Syst 50(9):3382–3390 Luenberger D (1971) An introduction to observers. IEEE Trans Autom Control 16(6):596–602 Mirzajani S, Aghababa MP, Heydari A (2019) Adaptive T-S fuzzy control design for fractional-order systems with parametric uncertainty and input constraint. Fuzzy Sets Syst 365:22–39 Moodi H, Farrokhi M (2014) On observer-based controller design for Sugeno systems with unmeasurable premise variables. ISA Trans 53(2):305–316 Nguang SK, Shi P (2003) H∞ fuzzy output feedback control design for nonlinear systems: An LMI approach. IEEE Trans Fuzzy Syst 11(3):331–340 Niculescu SI (2001) Delay effects on stability: a robust control approach. Springer, Berlin Niederlinski A (1971) A heuristic approach to the design of linear multivariable interacting control systems. Automatica 7(6):691–701 Qiu J, Feng G, Gao H (2012) Observer-based piecewise affine output feedback controller synthesis of continuous-time T-S fuzzy affine dynamic systems using quantized measurements. IEEE Trans Fuzzy Syst 20(6):1046–1062 Ren Y, Wang A, Wang H (2015) Fault diagnosis and tolerant control for discrete stochastic distribution collaborative control systems. IEEE Trans Syst Man Cybern: Syst 45(3):462–471 Ren Y, Fang Y, Wang A et al (2018) Collaborative operational fault tolerant control for stochastic distribution control system. Automatica 98:141–149 Sakthivel R, Kaviarasan B, Ma Y et al (2016) Sampled-data reliable stabilization of T-S fuzzy systems and its application. Complexity 21(52):518–529 Sakthivel R, Saravanakumar T, Kaviarasan B et al (2017) Finite-time dissipative based fault-tolerant control of Takagi-Sugeno fuzzy systems in a network environment. J Frankl Inst 354(8):3430– 3454 Sakthivel R, Selvaraj P, Kaviarasan B (2020) Modified repetitive control design for nonlinear systems with time delay based on T-S fuzzy model. IEEE Trans Syst Man Cybern: Syst 50(2):646–655 Shen Q, Jiang B, Cocquempot V (2012) Fault-tolerant control for T-S fuzzy systems with application to near-space hypersonic vehicle with actuator faults. IEEE Trans Fuzzy Syst 20(4):652–665 Shen H, Wang Y, Xia J et al (2019) Fault-tolerant leader-following consensus for multi-agent systems subject to semi-Markov switching topologies: an event-triggered control scheme. Nonlinear Anal: Hybrid Syst 34:92–107 Shi PC, Wang XQ, Meng X et al (2023) Adaptive fault-tolerant control for open-circuit faults in dual three-phase PMSM drives. IEEE Trans Power Electron 38(3):3676–3688 Song G, Shi P, Lim CP (2022) Distributed fault-tolerant cooperative output regulation for multiagent networks via fixed-time observer and adaptive control. IEEE Trans Control Netw Syst 9(2):845– 855 Sun K, Zhu S, Wei Y et al (2019) Finite-time synchronization of memristor-based complex-valued neural networks with time delays. Phys Lett A 103:135–140 Syed WA, Perinpanayagam S, Samie M et al (2016) A novel intermittent fault detection algorithm and health monitoring for electronic interconnections. IEEE Trans Compon Packag Manuf Technol 6(3):400–406 Tan CP, Edwards C (2003) Sliding mode observers for robust detection and reconstruction of actuator and sensor faults. Int J Robust Nonlinear Control 13(5):443–463 Tao Y, Shen D, Wang Y et al (2015) Reliable H∞ control for uncertain nonlinear discrete-time systems subject to multiple intermittent faults in sensors and/or actuators. J Frankl Inst 352(11):4721– 4740 Tao Y, Shen D, Fang M et al (2016) Reliable H∞ control of discrete-time systems against random intermittent faults. Int J Syst Sci 47(10):2290–2301 Utkin V (1992) Sliding modes in control optimization. Springer, Berlin

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Chapter 2

Fault Estimation and Tolerant Control for Time-Varying Delayed Fuzzy Systems with Actuator Faults

2.1 Introduction With the prompt increase in the scale and complexity of modern industrial systems, faults may cause disastrous consequences and huge economic losses. Therefore, the security and reliability of the system become increasingly important. The issues of fault diagnosis and tolerant control have become important research topics (Wang et al. 2018; Li et al. 2019; Liang et al. 2020, 2021; Liu et al. 2019). In addition, as an important branch of fault diagnosis, fault estimation has attracted extensive attention from researchers because it can obtain some accurate information, such as the shape and size of faults (Hu et al. 2019; Jiang and Chowdhury 2005). Fault-tolerant control based on fault estimation compensates for faults online and ensures better control performance, which is of practical significance (Moodi and Farrokhi 2014; Mia et al. 2011). In the past two decades, this problem of fault estimation and tolerant control and its application in a wide range of technical and industrial processes has aroused the research interest of a large number of scholars and has always been an in-depth research topic (Blanke et al. 2006; Corless and Tu 1998; Han et al. 2018a). In recent years, many studies have been performed on fault estimation and tolerant control (Gao and Ding 2007; Han et al. 2015, 2018b; Liu et al. 2013; Tao et al. 2015; Liu et al. 2020a; Hassan and Zribi 2014), and many methods have been proposed (Han et al. 2018b; Gao 2015; Van et al. 2017; Mia et al. 2011; Selvaraj et al. 2017). Nevertheless, a lot of issues remain to be further studied, which has stimulated research interest. In the early stage, the ordinary quadratic Lyapunov function was used to analyze the stability of the system (Zhao et al. 2009, 2013; Wang and Lam 2018a). However, it should be noted that the stability obtained by the ordinary quadratic Lyapunov function is more conservative, which has significant limitations. Scholars have proposed a large number of methods to solve this problem (Wang and Lam 2018b; Mozelli et al. 2011; Zhang et al. 2017a; Gao et al. 2009). In Liu et al. (2020b), Zhang et al. (2017a), the barrier Lyapunov function and piecewise Lyapunov function were studied for the system stability, respectively. In Gao et al. (2009), conservatism was significantly © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Sun et al., Fault-Tolerant Control for Time-Varying Delayed T-S Fuzzy Systems, Intelligent Control and Learning Systems 9, https://doi.org/10.1007/978-981-99-1357-2_2

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reduced by using the fuzzy Lyapunov-Krasovskii functional. In addition, the fuzzy Lyapunov-Krasovskii functional uses this time derivative of the membership function and switching behavior, which is a new and meaningful study approach (Wang and Lam 2018b). The stability problem for time-varying delayed fuzzy systems with actuator faults exists widely in practical engineering, but it has not been deeply studied. The contents have aroused people’s challenge and interest and also inspired us to conduct further research.

2.2 System Definition and Description In this chapter, a T-S fuzzy model with a time delay is given by the IF-THEN rule. Plant Rule i: IF θ˜1 (t) is πi1 , and, . . ., and θ˜s¯ (t) is πi s¯ , THEN ⎧ ¯ − t f¯ ) f a (t) x(t) ˙ = Ai x(t) + Aτ i x(t − τ (t)) + Bi u(t) + E f i β(t ⎪ ⎪ ⎪ ⎨ + Bgi g(x(t)) + E wi w(t), ⎪ y(t) = Ci x(t) + E vi v(t), ⎪ ⎪ ⎩ x(t) = ψ(t), t ∈ [−τ˜ , 0], i = 1, 2, . . . , p, ¯

(2.1)

where x(t) ∈ R m represents the system state, u(t) ∈ R n represents the input, f a (t) ∈ R p1 represents the actuator fault, w(t) ∈ R q1 represents the external interference, w(t) ∈ L 2 [0, ∞), y(t) ∈ R n 1 represents this system measurement output, v(t) ∈ R q2 represents the measurement noise and v(t) ∈ L 2 [0, ∞). Ai , Aτ i , Bi , E f i , Bgi , E wi , Ci , and E vi represent the known constant matrices. τ (t) represents this time-varying delay, following 0 < τ (t) ≤ τ˜ , τ˙ (t) ≤ τ M , where τ˜ and τ M represent constants, τ˜ is known, and τ M can be known or unknown. ψ(t) represents the initial function of the vector value, where t belongs to the interval [−τ˜ , 0]. Suppose f˙a (t) is bounded with ¯ − t f¯ ) = diag{β¯1 (t − t f¯ ), . . . , β¯ p1 (t − t f¯ )} ∈ R p1 × p1 respect to the time norm. β(t represents the time profile of the actuator fault, and t f¯ represents the time when ¯ j = 1, . . . , s¯ ) represent premise the actuator fault occurs. θ˜ j (t), πi j , (i = 1, . . . , p; variables and fuzzy sets represented by membership functions, respectively. p¯ and s¯ stand for the IF-THEN rule number and premise variable number, respectively. In addition, the nonlinear dynamic g(x(t)) ∈ R p2 satisfies the Lipschitz condition and the following sector-bounded condition (Zhang et al. 2017b): [g(x(t)) − R1 x(t)]T [g(x(t)) − R2 x(t)] ≤ 0, ∀x(t) ∈ R m ,

(2.2)

{[g(x(t1 )) − g(x(t2 ))] − R1 [x(t1 ) − x(t2 )]}T {[g(x(t1 )) − g(x(t2 ))] −R2 [x(t1 ) − x(t2 )]} ≤ 0, ∀x(t1 ), x(t2 ) ∈ R m ,

(2.3)

where R1 , R2 ∈ R p2 ×m represent the known constant matrices.

2.2 System Definition and Description

23

In short, after fuzzy mixing of every local model, this whole system is described in the following: ⎧ p¯  ⎪ ⎪ ˜ ⎪ x(t) ˙ = bi (θ(t))[A i x(t) + Aτ i x(t − τ (t)) + Bi u(t) ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎨ ¯ − t f¯ ) f a (t) + Bgi g(x(t)) + E wi w(t)], + E f i β(t p¯ ⎪  ⎪ ⎪ ⎪ ˜ ⎪ bi (θ(t))[C y(t) = i x(t) + E vi v(t)], ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎩ x(t) =ψ(t), t ∈ [−τ˜ , 0],

(2.4)

 p¯ ˜ = [θ˜1 (t), . . . , θ˜s¯ (t)], bi (θ(t)) ˜ ˜ ˜ ˜ where θ(t) = o˜ i (θ(t))/ ˜ i (θ(t)), o˜ i (θ(t)) = i=1 o s¯ ˜ ˜ ˜ π ( θ (t)), and here π ( θ (t)) means the membership degree of θ (t) belongj ij j j j=1 i j ˜ ing to fuzzy set πi j , (i = 1, . . . , p, ¯ j = 1, . . . , s¯ ). It is assumed that for each θ(t),  p¯  p¯ ˜ ˜ ˜ ˜ o˜ i (θ(t)) ≥ 0, o˜ i (θ(t)) > 0. Therefore, for each θ(t), i=1 bi (θ(t)) follows  pi=1 ¯ ˜ ˜ bi (θ(t)) ≥ 0, i=1 bi (θ(t)) = 1. ˜ and a set In order to simplify the expression, bi is utilized to denote bi (θ(t)),  for p¯ ¯ they can be expressed as K¯ b = i=1 bi K¯ i of matrices K¯ i and K¯ i j (i, j = 1, . . . , p),  p¯  p¯ and K¯ bb = i=1 bi j=1 b j K¯ i j , respectively. Remark 2.1 Establishing a T-S fuzzy model is an effective method for solving practical problems (Wang et al. 2020). The involvement of g(x(t)) can improve the modeling precision, reduce the computational burden, and reduce the number of fuzzy rules (Sun et al. 2019). The state of the system may be disturbed by a time delay. There are actuator faults and interferences in the plant sometimes. Interferences reduce the control performance of the system, due to the external environment or inaccurate system modeling, and are inevitable in practical applications. Faults are usually caused by one or some system components, making the system unable to work normally at its working point. The parameter matrices of disturbances and faults in this chapter are various and known, indicating that disturbances and faults are injected into these plants with various influence characteristics through different channels. Actuator faults, time delay, and disturbances exist in many practical special physical systems, for instance, balancing and swing-up of the inverted pendulum on the cart, the truck trailer system, this trolley motion model, these linearized longitudinal dynamics of a VTOL aircraft as well as these mechanical system dynamics of the single-link direct-drive manipulator driven via the permanent magnet brush DC motor and these electrical system dynamics of a DC motor (Huang and Yang 2014; Han et al. 2018b; Qiu et al. 2012; Chen and Cao 2013). These prompt us to consider model (2.1). Assumption 2.1 In the fuzzy system (2.4), (Ai , Bi ) is controllable and (Ai , Ci ) is observable. In addition, Bi represents the column full rank matrix, and Ci represents the row full rank matrix. rank(Bi , E f i ) = rank(Bi ).

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According to Zhang et al. (2009), under condition rank(Bi , E f i ) = rank(Bi ), there exists a matrix Bi∗ ∈ R n×m that satisfies (I − Bi Bi∗ )E f i = 0,

(2.5)

where Bi∗ belongs to Bi(1) Assumption 2.2 The measurement noise v(t) can be generated by the exogenous model as follows: v(t) ˙ = E d v(t) + v1 (t),

(2.6)

where v1 (t) ∈ L 2 [0, ∞) represents the additional random signal and E d ∈ R q2 ×q2 represents a known constant matrix. Assumption 2.3 Zhang et al. (2009) Assume that each β¯i (t − t f¯ ) can be defined as β¯i (t − t f¯ ) =



0,

if t < t f¯ , 1 − exp −σi (t − t f¯ ) , if t ≥ t f¯ ,

where σi > 0 represents a ratio and is an unknown constant. In particular, if it is an initial fault, σi will decrease, and if it is a sudden fault, σi will increase. At the same time, if σi → ∞, β¯i (t − t f¯ ) will become a step function, satisfying β¯i (t − t f¯ ) =



0, if t < t f¯ , σi → ∞, 1, if t ≥ t f¯ , σi → ∞.

Assumption 2.4 In fact, the output y(t) can be measured by the sensor. Assume C1 = · · · = Cr = C and E v1 = · · · = E vr = E v . Therefore, this T-S system is described as follows: ⎧ ¯ − t f¯ ) f a (t) x(t) ˙ =Ab x(t) + Aτ b x(t − τ (t)) + Bb u(t) + E f b β(t ⎪ ⎪ ⎪ ⎨ + Bgb g(x(t)) + E wb w(t), ⎪ y(t) =C x(t) + E v v(t), ⎪ ⎪ ⎩ x(t) =ψ(t), t ∈ [−τ˜ , 0].

(2.7)

¯ − t f¯ ) ¯ − t f¯ ) has no effect on the following study. Therefore, for simplicity, β(t β(t is omitted before the simulation results. From formula (2.2), we can see Tao et al. (2015):

x(t) g(x(t))

T

T1 T2 ∗ I



x(t) ≤ 0, g(x(t))

2.3 A New k-Step Induction Actuator Fault Estimation Method

25

where T1 = sym(R1T R2 )/2, T2 = −(R1T + R2T )/2. The same conclusion can be obtained from formula (2.3), so the proof is omitted.

2.3 A New k-Step Induction Actuator Fault Estimation Method This section presents the principal results for T-S fuzzy systems with a state delay through researching a new k-step induction actuator fault estimation method to estimate actuator faults f a (t). (1) 1-step induction actuator fault estimation: The 1-step induction actuator fault estimation observer for system (2.7) can be described as follows: ⎧ x˙ˆ1 (t) =Ab xˆ1 (t) + Aτ b xˆ1 (t − τ (t)) + Bb u(t) + E f b fˆa1 (t) ⎪ ⎪ ⎪ ⎪ ⎨ + Bgb g(xˆ1 (t)) − L 1b e y1 (t), yˆ1 (t) =C xˆ1 (t), ⎪ ⎪ ⎪ ⎪ ⎩ ˙ˆ f a1 (t) = − L 2b (e y1 (t) + e˙ y1 (t)),

(2.8)

where xˆ1 (t) ∈ R m , yˆ1 (t) ∈ R n 1 and fˆa1 (t) ∈ R p1 are the initial estimates of x(t), y(t) and f a (t), separately.  ∈ R p1 × p1 represents the positive definite matrix, which is a learning rate. e y1 (t) = yˆ1 (t) − y(t). L 1b and L 2b are the observer matrices. Let x(t) ¯ = [x T (t), f aT (t)]T , xˆ¯1 (t) = [xˆ1T (t), fˆaT1 (t)]T , ex1 (t) = xˆ1 (t) − x(t), e f1 (t) = fˆa1 (t) − f a (t), e1 (t) = xˆ¯1 (t) − x(t), ¯ g˜1 (x(t)) = g(xˆ1 (t)) − g(x(t)), ρ1 (t) = [w T (t), v T (t), v1T (t), f˙aT (t)]T . By formulas (2.7) and (2.8), the estimated error dynamic is as follows: ⎧ e˙1 (t) =(L 3b + I )(A1b − L b C1 )e1 (t) + (L 3b + I )A1τ b e1 (t − τ (t)) ⎪ ⎪ ⎪ ⎨ + (L + I )B g˜ (T x(t)) ¯ 3b

⎪ ⎪ ⎪ ⎩

1g b 1

0

+ [(L 3b + I )(L b E 1 − F1b ) + G 1b ]ρ1 (t),

(2.9)

e y1 (t) =C1 e1 (t) − E 1 ρ1 (t),

where

  0 0 Ab E f b L 1b , C1 = C 0 , , A1b = , Lb = 0 0 L 2b −L 2b C 0



    Aτ b 0 Bgb , T0 = I 0 , E 1 = 0 E v 0 0 , , B1g b = A1τ b = 0 0 0

E wb 0 0 0 0 0 0 0 F1b = . , G 1b = 0 000 0 L 2b E v E d L 2b E v −I

L 3b =

26

2 Fault Estimation and Tolerant Control for Time-Varying Delayed …

(2) 2-step induction actuator fault estimation: To decrease the disturbance of input perturbation of f˙a (t) and enhance the fault estimation performance, this 2-step induction actuator fault estimation observer of system (2.7) is based on the fˆa1 (t) given by the 1-step induction actuator fault estimation observer, and is designed as follows: ⎧ x˙ˆ2 (t) =Ab xˆ2 (t) + Aτ b xˆ2 (t − τ (t)) + Bb u(t) + E f b fˆa2 (t) ⎪ ⎪ ⎪ ⎪ ⎨ + Bgb g(xˆ2 (t)) − L 1b e y2 (t), yˆ2 (t) =C xˆ2 (t), ⎪ ⎪ ⎪ ⎪ ⎩ ˙ˆ f a2 (t) = − L 2b (e y2 (t) + e˙ y2 (t)) + f˙ˆa1 (t),

(2.10)

where xˆ2 (t) ∈ R m , yˆ2 (t) ∈ R n 1 and fˆa2 (t) ∈ R p1 represent the second estimated value of x(t), y(t) and f a (t), respectively. e y2 (t) = yˆ2 (t) − y(t). Set xˆ¯2 (t) = [xˆ2T (t), fˆaT2 (t)]T , ex2 (t) = xˆ2 (t) − x(t), e f2 (t) = fˆa2 (t) − f a (t), e2 (t) = x¯ˆ2 (t) − x(t), ¯ g˜2 (x(t)) = g(xˆ2 (t)) − g(x(t)), ρ2 (t) = [w T (t), v T (t), v1T (t), f˙T (t) − fˆ˙T (t)]T . From formulas (2.7), (2.8) and (2.10), the estimated error dynamic a

a1

can be acquired in the following: ⎧ e˙2 (t) =(L 3b + I )(A1b − L b C1 )e2 (t) + (L 3b + I )A1τ b e2 (t − τ (t)) ⎪ ⎪ ⎪ ⎨ + (L 3b + I )B1g b g˜2 (T0 x(t)) ¯ (2.11) ⎪ + [(L 3b + I )(L b E 1 − F1b ) + G 1b ]ρ2 (t), ⎪ ⎪ ⎩ e y2 (t) =C1 e2 (t) − E 1 ρ2 (t), where L 3b , A1b , L b , C1 , A1τ b , B1g b , T0 , E 1 , F1b and G 1b are the same as those of this 1-step fault estimation observer. (3) k-step induction actuator fault estimation: Similar to 2-step induction actuator fault estimation, to further decrease the influence of input disturbance from f˙a (t), and further improve the fault estimation performance, the k-step induction actuator fault estimation observer for system (2.7) is based on fˆak−1 (t), that is, the estimated value of the (k − 1)th f a (t), where k ∈ {2, 3, . . .} is designed as follows: ⎧ x˙ˆk (t) =Ab xˆk (t) + Aτ b xˆk (t − τ (t)) + Bb u(t) + E f b fˆak (t) ⎪ ⎪ ⎪ ⎪ ⎨ + Bgb g(xˆk (t)) − L 1b e yk (t), yˆk (t) =C xˆk (t), ⎪ ⎪ ⎪ ⎪ ⎩ ˙ˆ f ak (t) = − L 2b (e yk (t) + e˙ yk (t)) + f˙ˆak−1 (t),

(2.12)

where xˆk (t) ∈ R m , yˆk (t) ∈ R n 1 and fˆak (t) ∈ R p1 represent the kth estimations of x(t), y(t) and f a (t), respectively. e yk (t) = yˆk (t) − y(t).

2.3 A New k-Step Induction Actuator Fault Estimation Method

27

Set xˆ¯k (t) = [xˆ2T (t), fˆaTk (t)]T , exk (t) = xˆk (t) − x(t), e fk (t) = fˆak (t) − f a (t), ek (t) ¯ g˜k (x(t)) = g(xˆk (t)) − g(x(t)), ρk (t) = [w T (t), v T (t), v1T (t), f˙aT (t) = xˆ¯k (t) − x(t), ˙ − fˆaTk−1 (t)]T . From equation (2.7) and the k − 1-step induction actuator fault estimation, the estimated error dynamic can be obtained as ⎧ e˙k (t) =(L 3b + I )(A1b − L b C1 )ek (t) + (L 3b + I )A1τ b ek (t − τ (t)) ⎪ ⎪ ⎪ ⎨ + (L 3b + I )B1g b g˜k (T0 x(t)) ¯ (2.13) ⎪ + [(L 3b + I )(L b E 1 − F1b ) + G 1b ]ρk (t), ⎪ ⎪ ⎩ e yk (t) =C1 ek (t) − E 1 ρk (t), where L 3b , A1b , L b , C1 , A1τ b , B1g b , T0 , E 1 , F1b and G 1b have the same definitions as the 1-step induction actuator fault estimation observer. To sum up, the k-step induction actuator fault estimation observer can reduce the impact of input perturbations generated by f˙a (t) and can enhance this fault estimation performance. These observer matrices under k-step induction actuator fault estimation are designed as follows: 1. Under ρk (t) = 0, the k-step error dynamic (2.13) (k ∈ {1, 2, . . .}) is asymptotically stable; 2. Under ρk (t) = 0 as well as zero initial conditions, the k-step error dynamic (2.13) follows J¯

J¯

e fk (t) dt < 2

0

γk2

ρk (t) 2 dt,

(2.14)

0

where γk > 0 is the H∞ norm bounded constant, representing this H∞ performance index. J¯ > 0, ρk (t) ∈ L 2 [0, ∞). Theorem 2.1 For these specified non-negative scalars 1 , 2 , ε, and positive scalars , ι, γk , the k-step error dynamic (2.13) will be asymptotically stable under the H∞ performance index γk , (k ∈ {1, 2, . . .}) in formula (2.14), if there are matrices L 1i , L 2i , Pi > 0, Q 1i > 0, Q 2i > 0, Ri > 0 and slack matrices Mi , Ni , (i ∈ {1, 2, . . . , p}) ¯ obeying the inequalities as follows: P˙b ≤ 0, Q˙ 1b ≤ 0, Q˙ 2b ≤ 0, R˙ b ≤ 0,

˜ ετ˜ Mb ¯ θ) ϒ( < 0, ∗ −ετ˜ Rb

˜ ετ˜ Nb ¯ θ) ϒ( < 0, ∗ −ετ˜ Rb

(2.15) (2.16) (2.17)

28

2 Fault Estimation and Tolerant Control for Time-Varying Delayed …

where ⎡

⎤ ˜ ˜ ˜ 4 (θ) ˜ 1 (θ) 2 (θ) 0 3 (θ) ⎢ ∗ − 1 (1 − τ M )Q 1b 0 0 0 ⎥ ⎢ ⎥ ⎢ ˜ ¯ ϒ(θ) = ⎢ ∗ 0 ⎥ ∗ − 2 Q 2b 0 ⎥ ⎣ ∗ ∗ ∗ −ιI 0 ⎦ ∗ ∗ ∗ ∗ −γk2 I ˜ b (2 Pb − 2 Rb )−1 Pb ( ˜ ¯ T (θ)P ¯ θ) +ετ˜  ¯1 −W ¯ 2 )) − sym(εNb (W ¯2 −W ¯ 3 )), −sym(εMb (W

and ˜ = sym(Pb (L 3b + I )(A1b − L b C1 )) + 1 Q 1b + 2 Q 2b + I¯T I¯ − ιT T T1 T0 , 1 (θ) 0 T ˜ ˜ 2 (θ) = Pb (L 3b + I )A1τ b , 3 (θ) = Pb (L 3b + I )B1g b − ιT T2 , 0

˜ = Pb [(L 3b + I )(L b E 1 − F1b ) + G 1b ], 4 (θ) ˜ = [(L 3b + I )(A1b − L b C1 ), (L 3b + I )A1τ b , 0, (L 3b + I )B1g b , ¯ T (θ)  (L 3b + I )(L b E 1 − F1b ) + G 1b ], ¯ 1 = [1, 0, 0, 0, 0], W ¯ 2 = [0, 1, 0, 0, 0], W ¯ 3 = [0, 0, 1, 0, 0], I¯ = [0 I ]. W Proof For k ∈ {1, 2, . . .}, construct the fuzzy Lyapunov-Krasovskii functional in the following: Vk (t) = V1k (t) + V2k (t) + V3k (t), where V1k (t) = ekT (t)Pb ek (t), t t T ek (s)Q 1b ek (s)ds + 2 ekT (s)Q 2b ek (s)ds, V2k (t) = 1 t−τ (t)

0

t−τ˜

t

V3k (t) = ε

e˙kT (s)Rb e˙k (s)dsdθ.

−τ˜ t+θ

Based on this trajectory of this error function (2.13), V˙k (t) follows

(2.18)

2.3 A New k-Step Induction Actuator Fault Estimation Method

29

V˙1k (t) = ekT (t)[sym(Pb (L 3b + I )(A1b − L b C1 ))]ek (t) + 2ekT (t)Pb (L 3b + I ) A1τ b ek (t − τ (t)) + 2ekT (t)Pb (L 3b + I )B1g b g˜k (T0 x(t)) ¯ + 2ekT (t)Pb [(L 3b + I )(L b E 1 − F1b ) + G 1b ]ρk (t) + ekT P˙b ek (t), (2.19) V˙2k (t) = ekT (t)( 1 Q 1b + 2 Q 2b )ek (t)− 1 (1 − τ˙ (t))ekT (t − τ (t))Q 1b ek (t − τ (t)) t T − 2 ek (t − τ˜ )Q 2b ek (t − τ˜ ) + 1 ekT (s) Q˙ 1b ek (s)ds t−τ (t)

t + 2

ekT (s) Q˙ 2b ek (s)ds,

(2.20)

t−τ˜

V˙3k (t) = ετ˜ e˙kT (t)Rb e˙k (t) − ε

t e˙kT (s)Rb e˙k (s)ds

t−τ˜

0

t

+ε

e˙kT (s) R˙ b e˙k dsdθ.

(2.21)

−τ˜ t+θ

According to Lemmas 1.1–1.3, we can obtain V˙k (t) + e Tfk (t)e fk (t) − γk2 ρkT (t)ρk (t) = V˙k (t) + ekT (t) I¯T I¯ek (t) − γk2 ρkT (t)ρk (t)

t e˙k (s)ds − 2ες T (t)Mb ek (t) − ek (t − τ (t)) − t−τ (t)

⎡

− 2ες T (t)Nb ⎣ek (t − τ (t)) − ek (t − τ˜ ) −

t−τ  (t)

⎤

e˙k (s)ds ⎦

t−τ˜

≤

1 τ˜

t t−τ (t)

1 + τ˜

ς(t) e˙k (s)

t−τ  (t)

t−τ˜

T

ς(t) e˙k (s)

˜ ετ˜ Mb ¯ θ) ϒ( ∗ −ετ˜ Rb

T



˜ ετ˜ Nb ¯ θ) ϒ( ∗ −ετ˜ Rb

ς(t) ds e˙k (s)



ς(t) ds, e˙k (s)

(2.22)

where ς T (t) = [ekT (t), ekT (t − τ (t)), ekT (t − τ˜ ), g˜kT (T0 x(t)), ¯ ρkT (t)]. At the same time, if formulas (2.16)–(2.17) hold, we can obtain V˙k (t) + e Tfk (t)e fk (t) − γk2 ρkT (t)ρk (t) < 0.

(2.23)

30

2 Fault Estimation and Tolerant Control for Time-Varying Delayed …

In particular, under ρk (t) = 0, formula (2.23) can be described as V˙k (t) + e Tfk (t)e fk (t) < 0, which means V˙k (t) < −e Tfk (t)e fk (t)  0. According to the Lyapunov theory, it can be known that lim ek (t) = 0, and then this k-step error dynamic (2.13) is t→∞

asymptotically stable. At the same time, under ρk (t) = 0 and the zero-valued initial conditions, one has that ek (t) = 0 and e˙k (t) = 0. Therefore, when ek (t) = 0, it fol J¯ lows that Vk (t)|t=0 = 0. In addition, given Vk (t)|t= J¯ ≥ 0, one has 0 ( e fk (t) 2 − γk2 ρk (t) )dt + V (t)|t= J¯ − V (t)|t=0 ≤ 0, which means that formula (2.14) holds. Therefore, the error dynamic (2.13) is asymptotically stable to follow the H∞ performance index γk . Theorem 2.2 For these specified non-negative scalars 1 , 2 , ε, and positive scalars , ι, γk , the k-step error dynamic (2.13) will be asymptotically stable under the H∞ performance index γk , (k ∈ {1, 2, . . .}) in formula (2.14), if there are matrices Yi , L 2i , P = diag{P11 , P22 } > 0, where P22 = −1 , Q 1i > 0, Q 2i > 0, Ri > 0 and ¯ satisfying these following inequalities: slack matrices Mi , Ni , (i ∈ {1, 2, . . . , p}), Q˙ 1b ≤ 0, Q˙ 2b ≤ 0, R˙ b ≤ 0, ii h < 0, i, h = 1, 2, . . . , p, ¯

(2.24) (2.25)

i j h +  ji h < 0, 1 ≤ i < j ≤ p, ¯ h = 1, 2, . . . , p, ¯ ¯ ii h < 0, i, h = 1, 2, . . . , p,

(2.26) (2.27)

i j h +  ji h < 0, 1 ≤ i < j ≤ p, ¯ h = 1, 2, . . . , p, ¯ T E f i P11 = L 2i C, i = 1, 2, . . . , p, ¯

(2.28) (2.29)

where √ √ ⎤ ⎤ ⎡ ϒi j ετ˜ Mi ϒi j ετ˜ Ni ετ˜ iTj ετ˜ iTj ⎦, i j h = ⎣ ∗ −ετ˜ Rh ⎦, = ⎣ ∗ −ετ˜ Rh 0 0 2 2 ∗ ∗ −2 P + Rh ∗ ∗ −2 P + Rh ⎡

i j h

and Mi = [M1iT , M2iT , M3iT , U1iT , V1iT ]T , Ni = [N1iT , N2iT , N3iT , U2iT , V2iT ]T , i j = [11i j , 2i j , 0, 31i j , 4i j ], ¯1 −W ¯ 2 )) − sym(εNi (W ¯2 −W ¯ 3 )), ϒi j = ϒ1i j − sym(εMi (W ⎡ ⎤ 1i j 2i j 0 3i j 4i j ⎢ ∗ − 1 (1 − τ M )Q 1i 0 0 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ∗ − 2 Q 2i 0 ϒ1i j = ⎢ ∗ ⎥, ⎣ ∗ ∗ ∗ −ιI 0 ⎦ ∗ ∗ ∗ ∗ −γk2 I 1i j = sym(11i j ) + 1 Q 1i + 2 Q 2i + I¯T I¯ − ιT0T T1 T0 ,

2.3 A New k-Step Induction Actuator Fault Estimation Method

31

0 P11 Ai − Yi C , I¯ = [0 I ], 11i j = −E Tf j P11 Ai + E Tf j Yi C −E Tf j P11 E f i

0 P11 Aτ i P11 Bgi T 2i j = , , 3i j = 31i j − ιT0 T2 , 31i j = −E Tf j P11 Bgi −E Tf j P11 Aτ i 0

0 0 −P11 E wi Yi E v 4i j = , E Tf j P11 E wi  E Tf j P11 C T (CC T )−1 E v −−1  = −E Tf j Yi E v + E Tf j P11 C T (CC T )−1 E v + E Tf j P11 C T (CC T )−1 E v E d , ¯ 1 = [1, 0, 0, 0, 0], W ¯ 2 = [0, 1, 0, 0, 0], W ¯ 3 = [0, 0, 1, 0, 0]. W −1 Yi . At the same time, the gain matrix L 1i is obtained by L 1i = P11 −1 Proof According to L 1i = P11 Yi , the following equations can be obtained:



P11 Ai − P11 L 1i C 0 11i j = , −E Tf j P11 Ai + E Tf j P11 L 1i C −E Tf j P11 E f i

0 0 −P11 E wi P11 L 1i E v , 4i j =  E Tf j P11 C T (CC T )−1 E v −−1 E Tf j P11 E wi Δ = −E Tf j P11 L 1i E v + E Tf j P11 C T (CC T )−1 E v + E Tf j P11 C T (CC T )−1 E v E d . According to Lemma 1.4, if formulas (2.24)–(2.29) hold, the following inequalities can be obtained: ¯ ii h < 0, i, h = 1, 2, . . . , p,  ¯ ¯ ¯ i j h +  ji h < 0, 1 ≤ i < j ≤ p, ¯ h = 1, 2, . . . , p, ¯ ¯ ii h < 0, i, h = 1, 2, . . . , p, ¯ ¯ i jh +  ¯ ji h < 0, 1 ≤ i < j ≤ p,  ¯ h = 1, 2, . . . , p, ¯

(2.30) (2.31) (2.32) (2.33)

where

˜ ˜ ¯ i j h = ϒi j h ετ˜ Ni , ¯ i j h = ϒi j h ετ˜ Mi ,   ∗ −ετ˜ Rh ∗ −ετ˜ Rh T 2 −1 ϒ˜ i j h =ϒi j +ετ˜ i j [2 P − Rh ] i j .  p¯  p¯  p¯ Therefore, if formulas (2.30)–(2.33) hold, we have i=1 bi j=1 b j h=1  p¯  p¯  p¯ ¯ i j h < 0, i=1 ¯ i j h < 0, which indicates that formulas bh  bi j=1 b j h=1 bh  (2.16)–(2.17) hold. Inspired by Theorem 2.1, it is clear that the k-step error dynamic is asymptotically stable with the H∞ performance index γk in formula (2.14), (k ∈ {1, 2, . . .}). So, we draw the conclusions.

32

2 Fault Estimation and Tolerant Control for Time-Varying Delayed …

2.4 Controller Design The dynamic full-order output feedback controller for the considered system (2.7) is given as follows: ⎧ ⎪ ⎨ x˙c (t) = Acbb xc (t) + Aτ cbb xc (t − τ (t)) + Bcbb y(t), u(t) = Ccb xc (t) + Cτ cb xc (t − τ (t)) + Dc y(t) − Bb∗ E f b fˆak (t), ⎪ ⎩ xc (t) = ψ(t), ∀ ∈ [−τ˜ , 0],

(2.34)

where xc (t) ∈ R m represents this controller state, and Acbb , Aτ cbb , Bcbb , Ccb , Cτ cb and Dc represent the gain matrices of this controller with a form similar to formula (2.7). Bb∗ can be obtained from formula (2.5). ψ(t) is the same as that in formula (2.7). ¯ = [w T (t), v T (t), e Tf (t)]T , where e f (t) = Define η(t) = [x T (t), xcT (t)]T and μ(t) fˆak (t) − f a (t). Combining formulas (2.7) and (2.33), it follows that ⎧ ˜ θ)η(t) ˜ ˜ θ)η(t ˜ ˜ ˙ = A(θ, + Aτ (θ, − τ (t)) + Bg (θ)g(T ⎪ 0 η(t)) ⎨ η(t) ˜ θ) ˜ μ(t), + B(θ, ¯ ⎪ ⎩ y(t) = Cη(t) + Eμ(t), ¯

(2.35)

where

Ab + Bb Dc C Bb Ccb ˜ θ) ˜ = Aτ b Bb Cτ cb , Bg (θ) ˜ = Bgb , , Aτ (θ, Bcbb C Acbb 0 Aτ cbb 0

      ˜ θ) ˜ = E wb Bb Dc E v −E f b , C = C 0 , E = 0 E v 0 , T0 = I 0 . B(θ, 0 Bcbb E v 0 ˜ θ) ˜ = A(θ,



The control aim of this closed-loop fuzzy system (2.35) is to construct a full-order controller with the form (2.33) that satisfies the following conditions: 1. Under μ(t) ¯ = 0, the closed-loop system (2.35) is asymptotically stable; 2. Under μ(t) ¯ = 0, η(t) = 0 and ψ(t) = 0, t ∈ [−τ˜ , 0], output y(t) follows H∞ performance as follows: J˜

J˜

y(t) dt < γ 2

0

2

μ(t)

¯ dt,

2

(2.36)

0

where γ > 0 is a given constant, representing the H∞ performance index, J˜ > 0 and μ(t) ¯ ∈ L 2 [0, ∞).

2.4 Controller Design

33

Theorem 2.3 For the non-negative scalars 1 , 2 , ε and the positive scalars , ι, γ, this closed-loop system (2.35) is asymptotically stable under this H∞ performance index γ of formula (2.36), if there are matrices Aci j , Aτ ci j , Bci j , Cci , Cτ ci , Dc , Pi > 0, Q 1i > 0, Q 2i > 0, Ri > 0 and slack matrices Mi , Ni , (i, j ∈ {1, 2, . . . , p}), ¯ obeying these following inequalities: P˙b ≤ 0, Q˙ 1b ≤ 0, Q˙ 2b ≤ 0, R˙ b ≤ 0,

˜ θ) ˜ ετ˜ Mb ˜ θ,

( < 0, ∗ −ετ˜ Rb

˜ θ) ˜ ετ˜ Nb ˜ θ,

( < 0, ∗ −ετ˜ Rb

(2.37) (2.38) (2.39)

where ⎡

⎤ ˜ θ) ˜ ˜ θ) ˜ ˜ ϑ4 (θ, ˜ θ) ˜ ϑ1 (θ, ϑ2 (θ, 0 ϑ3 (θ) ⎢ ⎥ 0 0 0 ∗ − 1 (1 − τ M )Q 1b ⎢ ⎥ ⎢ ⎥ ˜ ˜ ˜

(θ, θ) = ⎢ 0 ∗ ∗ − 2 Q 2b 0 ⎥ ⎣ ⎦ ∗ ∗ ∗ −ιI 0 2 ∗ ∗ ∗ −γ I T ˜ ˜ 2 −1 ˜ ˜ + ετ˜ $ (θ, θ)Pb (2 Pb − Rb ) Pb $(θ, θ) + T  ¯1 −W ¯ 2 )) − sym(εNb (W ¯2 −W ¯ 3 )), − sym(εMb (W and ˜ θ) ˜ = sym(Pb A(θ, ˜ θ)) ˜ + 1 Q 1b + 2 Q 2b − ιTT T1 T0 , ϑ1 (θ, 0 ˜ θ) ˜ = Pb Aτ (θ, ˜ θ), ˜ ϑ3 (θ) ˜ = Pb Bg (θ) ˜ − ιTT T2 , ϑ4 (θ, ˜ θ) ˜ = Pb B(θ, ˜ θ), ˜ ϑ2 (θ, 0 ˜ θ), ˜ Aτ (θ, ˜ θ), ˜ 0, Bg (θ), ˜ B(θ, ˜ θ)], ˜  = [C, 0, 0, 0, E], $(˜,˜) = [A(θ, ¯ 1 = [1, 0, 0, 0, 0], W ¯ 2 = [0, 1, 0, 0, 0], W ¯ 3 = [0, 0, 1, 0, 0]. W Proof Establish this fuzzy Lyapunov-Krasovskii functional as follows: V (t) = V1 (t) + V2 (t) + V3 (t), where V1 (t) = η T (t)Pb η(t), t t T η (s)Q 1b η(s)ds + 2 η T (s)Q 2b η(s)ds, V2 (t) = 1 t−τ (t)

t−τ˜

0  t V3 (t) = ε −τ˜ t+θ

η˙ T (s)Rb η(s)dsdθ, ˙

(2.40)

34

2 Fault Estimation and Tolerant Control for Time-Varying Delayed …

According to trajectory (2.35) of this closed-loop system, V˙ (t) satisfies: V˙ (t) = V˙1 (t) + V˙2 (t) + V˙3 (t),

(2.41)

where ˜ θ))]η(t) ˜ ˜ θ)η(t ˜ + 2η T (t)Pb Aτ (θ, − τ (t)) + 2η T (t) V˙1 (t) = η T (t)[sym(Pb A(θ, T ˜ ˜ ˜ ¯ + η T (t) P˙b η(t), Pb Bg (θ)g(T 0 η(t)) + 2η (t)Pb B(θ, θ)μ(t)

V˙2 (t) = η T (t)( 1 Q 1b + 2 Q 2b )η(t) − 1 (1 − τ˙ (t))η T (t − τ (t))Q 1b η(t − τ (t)) t T η T (s) Q˙ 1b η(s)ds − 2 η (t − τ˜ )Q 2b η(t − τ˜ ) + 1  + 2

t−τ (t) t t−τ˜

η T (s) Q˙ 2b η(s)ds,

V˙3 (t) = ετ˜ η˙ T (t)Rb η(t)−ε ˙

t

0  t

η˙ T (s) R˙ b η(s)dsdθ. ˙

η˙ (s)Rb η(s)ds ˙ +ε T

t−τ˜

−τ˜ t+θ

According to Lemmas 1.1–1.3, we can obtain V˙ (t) + y T (t)y(t) − γ 2 μ¯ T (t)μ(t) ¯ T 2 T =V˙ (t) + y (t)y(t) − γ μ¯ (t)μ(t) ¯ ⎡

⎤

t

⎢ − 2εξ T (t)Mb ⎣η(t) − η(t − τ (t)) −

⎥ η(s)ds ˙ ⎦

t−τ (t)

⎡

− 2εξ T (t)Nb ⎣η(t − τ (t)) − η(t − τ˜ ) −

t−τ  (t)

⎤

⎦ η(s)ds ˙

t−τ˜

≤

1 τ˜

t t−τ (t)

1 + τ˜

ξ(t) η(s) ˙

t−τ  (t)

t−τ˜

T

ξ(t) η(s) ˙

˜ θ) ˜ ετ˜ Mb ˜ θ,

( ∗ −ετ˜ Rb

T



˜ θ) ˜ ετ˜ Nb ˜ θ,

( ∗ −ετ˜ Rb

ξ(t) ds η(s) ˙



ξ(t) ds, η(s) ˙

(2.42)

where ξ T (t) = [η T (t), η T (t − τ (t)), η T (t − τ˜ ), g T (T0 η(t)), μ¯ T (t)]. At the same time, if formulas (2.37)–(2.39) hold, the inequality can satisfy the following relation: ¯ < 0. V˙ (t) + y T (t)y(t) − γ 2 μ¯ T (t)μ(t)

(2.43)

2.4 Controller Design

35

Particularly, if μ(t) ¯ = 0, formula (2.43) can be described as V˙ (t) + y T (t)y(t) < 0, which indicates that V˙ (t) < −y T (t)y(t)  0. According to Lyapunov theory, it can be concluded that lim η(t) = 0, and then the closed-loop system (2.35) is t→∞

asymptotically stable. At the same time, if μ(t) ¯ = 0, this value of the initial conditions is zero and it can be obtained that xc (t) = 0 and η(t) = 0. Therefore, when η(t) = 0, we can find V (t)|t=0 = 0 and V (t)|t= J˜ ≥ 0. According to V (t)|t= J˜ ≥ 0,  J˜ 2 it follows 0 ( y(t) 2 − γ 2 μ(t)

¯ )dt + V (t)|t= J˜ − V (t)|t=0 ≤ 0. This means that (2.36) holds. Therefore, this closed-loop model (2.35) is asymptotically stable and follows the H∞ performance index γ. Theorem 2.4 For these specified non-negative scalars 1 , 2 , ε, the positive scalars , ι, γ and the matrix X˜ > 0, the closed-loop fuzzy model (2.35) is asymptotically stable under this H∞ performance index γ in formula (2.36), if there are matrices Aˆ ci j , Aˆ τ ci j , Bˆ ci j , Cˆ ci , Cˆ τ ci , Dˆ c , Y˜ > 0, Qˆ 1i > 0, Qˆ 2i > 0, Rˆ i > 0 and slack matrices ¯ satisfying these inequalities as follows: Mˆ i , Nˆ i , (i, j ∈ {1, 2, . . . , p}), Q˙ˆ 1b ≤ 0, Q˙ˆ 2b ≤ 0, R˙ˆ b ≤ 0,

(2.44)

ii h < 0, i, h = 1, 2, . . . , p, ¯ i j h + ji h < 0, 1 ≤ i < j ≤ p, ¯ h = 1, 2, . . . , p, ¯

ii h < 0, i, h = 1, 2, . . . , p, ¯

(2.45) (2.46) (2.47)

i j h + ji h < 0, 1 ≤ i < j ≤ p, ¯ h = 1, 2, . . . , p, ¯

(2.48)

where √ ⎤ ˆT 

i j ετ˜ Mˆ i ετ˜ $ˆ iTj ⎢ ∗ −ετ˜ Rˆ 0 0 ⎥ h ⎥, i j h = ⎢ ⎣ ∗ ∗ −2 ϕ + 2 Rˆ h 0 ⎦ ∗ ∗ ∗ −I √ ⎡ ⎤ T ˆ ˆ ˆT 

i j ετ˜ Ni ετ˜ $i j ⎢ ∗ −ετ˜ Rˆ 0 0 ⎥ h ⎥,

i j h = ⎢ ⎣ ∗ 2 ˆ ∗ −2 ϕ + Rh 0 ⎦ ∗ ∗ ∗ −I ¯ ¯ ¯2 −W ¯ 3 )), ˆ

i j = 1i j − sym(ε Mi (W1 − W2 )) − sym(ε Nˆ i (W ⎡ ⎤ 1i j 2i j 0 3i 4i j ⎢ ∗ − 1 (1 − τ M ) Qˆ 1i 0 0 0 ⎥ ⎢ ⎥ ˆ

1i j = ⎢ ∗ ∗ −

0 0 ⎥ Q 2 2i ⎢ ⎥, ⎣ ∗ ∗ ∗ −ιI 0 ⎦ ⎡

∗

∗

∗

∗ −γ 2 I

1i j = sym(11i j ) + 1 Qˆ 1i + 2 Qˆ 2i − ιTˆ0T T1 Tˆ0 , Tˆ0 = [ X˜ , I ],

Ai X˜ + Bi Cˆ cj Ai + Bi Dˆ c C Aτ i X˜ + Bi Cˆ τ cj Aτ i 11i j = , ,  = 2i j Aˆ ci j Y˜ Ai + Bˆ ci j C Aˆ τ ci j Y˜ Aτ i

36

2 Fault Estimation and Tolerant Control for Time-Varying Delayed …

3i = 31i

− ιTˆ0T T2 , 31i =





Bgi E wi Bi Dˆ c E v −E f i , 4i j = ˜ , Y˜ Bgi Y E wi Bˆ ci j E v −Y˜ E f i

X˜ I ˆ = [5 , 0, 0, 0, 6 ], > 0, $ˆ i j = [11i j , 2i j , 0, 31i , 4i j ],  I Y˜     5 = C X˜ C , 6 = 0 E v 0 , Mˆ i = [ Mˆ 1iT , Mˆ 2iT , Mˆ 3iT , Uˆ 1iT , Vˆ1iT ]T , ¯ 1 = [1, 0, 0, 0, 0], W ¯ 2 = [0, 1, 0, 0, 0], Nˆ i = [ Nˆ 1iT , Nˆ 2iT , Nˆ 3iT , Uˆ 2iT , Vˆ2iT ]T , W ϕ=

¯ 3 = [0, 0, 1, 0, 0]. W

In addition, the controller gain matrices can be obtained: ⎧ Dc = Dˆ c ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Cci =(Cˆ ci − Dˆ c C X˜ ) S˜ −T ⎪ ⎪ ⎪ ⎪ ⎨ Cτ ci =Cˆ τ ci S˜ −T ⎪ Bci j =U˜ −1 ( Bˆ ci j − Y˜ Bi Dˆ c ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Aci j =U˜ −1 [ Aˆ ci j − Y˜ Ai X˜ − Bˆ ci j C X˜ − Y˜ Bi (Cˆ cj − Dˆ c C X˜ )] S˜ −T ⎪ ⎪ ⎪ ⎩ Aτ ci j =U˜ −1 ( Aˆ τ ci j − Y˜ Aτ i X˜ − Y˜ Bi Cˆ τ cj ) S˜ −T

(2.49)

where U˜ and S˜ satisfy S˜ U˜ T = I − X˜ Y˜ . Proof Based on Lemma 1.4, if formulas (2.44)–(2.48) hold, the following inequalities hold ¯ ii h < 0, i, h = 1, 2, . . . , p, ¯ ¯ i j h + ¯ ji h < 0, 1 ≤ i < j ≤ p, ¯ h = 1, 2, . . . , p, ¯ ¯

ii h < 0, i, h = 1, 2, . . . , p, ¯ ¯ ¯

i j h + ji h < 0, 1 ≤ i < j ≤ p, ¯ h = 1, 2, . . . , p, ¯ where

¯ i j h τ˜ Mˆ i ¯ i j h τ˜ Nˆ i



¯ ¯ , i j h = , i j h = ∗ −ετ˜ Rˆ h ∗ −ετ˜ Rˆ h ¯ i j h = i j + ετ˜ $ˆ iTj (2 ϕ − 2 Rˆ h )−1 $ˆ i j +  ˆ ˆ T .

Therefore, if formulas (2.50)–(2.52) hold, we have

(2.50) (2.51) (2.52) (2.53)

2.4 Controller Design

37 p¯ 

bi

p¯ 

bj

p¯ 

i=1

j=1

h=1

p¯ 

p¯ 

p¯ 

bi

i=1

j=1

bj

bh ¯ i j h < 0,

(2.54)

¯ i j h < 0. bh

(2.55)

h=1

Using an approach similar to that in Tao et al. (2015), the partition-symmetric matrices P and P −1 are expressed as

X˜ S˜ Y˜ U˜ −1 , P = . P= ∗ Z2 ∗ Z1

(2.56)

where Z 1 and Z 2 represent matrices and satisfy P P −1 = I . Since P P −1 = I , it follows

X˜ P ˜T S



I I Y˜ X˜ I = . = , P ˜T 0 0 U˜ T S 0

(2.57)

X˜ I I Y˜ ˜ . , D2 = S˜ T 0 0 U˜ T

(2.58)

Define D˜ 1 =



According to formulas (2.56)–(2.58), we can find that P D˜ 1 = D˜ 2 . Applying the contract transformation diag{ D˜ 1−T , D˜ 1−T , D˜ 1−T , I, I, D˜ 1−T } to equations (2.54) and (2.56), inequalities (2.37)–(2.39) are acquired. In conclusion, this closed-loop system (2.35) can be asymptotically stable and follows H∞ performance index γ. Remark 2.2 The proposed observer and controller realize fast adaptive fault estimation and tolerant control for actuator faults, and the design is more flexible for T-S fuzzy systems with time-varying time delay, nonlinear functions, external disturbances, and measurement noises. The application scope is larger and more cases are applied. In contrast to existing integral observers such as Zhang et al. (2010), the new k-step induction actuator fault estimation observer improves this fault estimation performance and decreases this influence of these actuator fault derivatives in this error function. Compared with the existing multiple-integral observers such as Huang and Yang (2014), the proposed observer considers both the output and the derivative of the output, resulting in better estimation results under a higher speed, particularly in this case that actuator fault sizes continue to change. Unlike ordinary Lyapunov functions, such as Yan et al. (2019), the application of this time delaydependent fuzzy Lyapunov-Krasovskii functional decreases this conservation of the maximum delay bounds. Besides, by transforming these integral inequalities into matrix inequalities, using fuzzy Lyapunov-Krasovskii functionals and slack matri-

38

2 Fault Estimation and Tolerant Control for Time-Varying Delayed …

ces, this influence of each integral term is considered, which further reduces the conservatism. In the study, the nonlinear dynamics are attenuated through tolerant control instead of this traditional H∞ performance index method. Different from this static output controller Du and Cocquempot (2017), a dynamic output feedback controller is explored to improve this model stability in light of this estimation of state and actuator faults under k-step induction actuator fault estimation, where an actuator fault compensator is used to decrease the fault impact. Remark 2.3 In contrast to Zhong et al. (2017), this study on parameters 1 , 2 and ε means that this Lyapunov function designed in this chapter is tunable, and is suitable for three cases of no delays, a constant delay and a time-varying delay. Under the condition of no time delays, 1 = 0, 2 = 0 and ε = 0. Under the condition of a constant delay, then 1 = 1, 2 = 0 and ε can be 0 or 1, and τ M = 0. Under the condition of a time-varying delay, if τ M is known and τ M < 1, then 1 = 1, 2 = 1 and ε can be 0 or 1; if τ M ≥ 1 or τ M is unknown or τ (t) is non-differentiable, then

1 = 0, 2 = 1 and ε = 1. Remark 2.4 In this chapter, based on the linear matrix inequalities in Theorems 2.2 and 2.4, through the LMI (the abbreviation of linear matrix inequality) toolbox of MATLAB, the observer matrices and the controller matrices are obtained, respectively. The linear matrix inequality complexity is proportional to Nd3 Nl , where Nd represents this number of scalar decision variables and Nl represents the linear matrix inequality total row size (Jee and Lee 2016). In Theorem 2.1, Nl = (5m + 6 p1 + 2q2 + p2 + q1 )( p¯ 3 + p¯ 2 ) + 3(m + p1 ) p¯ and Nd = 1.5(m + p1 )(m + p1 + 1) p¯ + (m + p1 )(6m + 8 p1 + 4q2 + 2 p2 + 2q1 + n 1 ) p¯ + 0.5m(m + 1). In Theorem 2.4, Nl = (12m + p2 + q1 + q2 + p1 )( p¯ 3 + p¯ 2 ) + 6m p¯ and Nd = m(2m + n 1 ) p¯ 2 + m(6m + 2n + 3) p¯ + 0.5m(5m + 1) + 2m 2 + nn 1 . This calculation complexity of the observer and controller mainly depends on the number of fuzzy rules, the model dimension, and this positive definite matrix number on which the membership functions in the fuzzy Lyapunov-Krasovskii functional depend. When involving new decision variables, the conservativeness of the results can be reduced, but at the same time, the computational complexity can be increased. Therefore, there should be a trade-off between conservatism and complexity. Since the observer matrices and controller matrices can be obtained offline through the LMI toolbox, the online computational difficulty is very small. It is worth pointing out that in Theorem 2.2, ˆ i = 1, 2, . . . , p¯ to we set Ri = R, i = 1, 2, . . . , p¯ and in Theorem 2.4, set Rˆ i = R, decrease the calculation load. Further reducing the calculation difficulty will be a future research topic. Remark 2.5 The minimum H∞ attenuation values of observer (2.13) and controller (2.33), namely, γk and γ, are obtained by the linear matrix inequalities and equations as follows:

2.4 Controller Design

39

Fig. 2.1 The k-step induction actuator fault estimation observer flow chart

1- Step induction actuator fault estimation

fˆa1 ( t ) 2-step induction actuator fault estimation

fˆa 2 ( t )

k-step induction actuator fault estimation

fˆa k ( t )

min γk s.t. (2.24) − (2.29), min γ s.t. (2.44) − (2.48). The research process of this proposed adaptive fault estimation observer and faulttolerant controller strategy is summarized as follows. Research program 1. Design the k-step induction actuator fault estimation observers (2.8), (2.10) and (2.13) in turn. 2. Deal with inequalities (2.24)–(2.28) and equation (2.29) in Theorem 2.2 and obtain feasible solutions of gain matrices L 1i and L 2i . Bring these values of the observer gain matrices L 1i and L 2i into the k-step induction actuator fault estimation observers (2.8), (2.10) and (2.13) in turn and then acquire this value of fˆak . Figure 2.1 represents this k-step induction actuator fault estimation observer flow chart. 3. Design this fault-tolerant controller (2.33). 4. Deal with inequalities (2.44)–(2.48) in Theorem 2.4 and obtain feasible solutions of controller gain matrices Aci j , Aτ ci j , Bci j , Cci , Cτ ci and Dc . The controller gain matrices Aci j , Aτ ci j , Bci j , Cci , Cτ ci and Dc are substituted into the fault-tolerant controller (2.33). At last, this fault-tolerant controller is implemented. The design strategy of the k-step fault estimation observer and fault-tolerant controller is shown in Fig. 2.2, where r¯ (t) represents a given quantity.

40

2 Fault Estimation and Tolerant Control for Time-Varying Delayed …

r (t )

u (t )

f a (t )

w(t )

v(t )

y (t )

fˆa (t ) k

Fig. 2.2 The design strategy of the k-step fault estimation observer and fault-tolerant controller

2.5 Simulation Results Two simulation examples are presented to illustrate the feasibility of this method described in the chapter. Example 2.1 The nonlinear continuous-time system is modeled as a system (2.7) containing two fuzzy rules, considering parameter matrices as follows: τ (t) = 0.8 + 0.6 sin(t), ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0.4 0.2 0.3 0.2 0.1 0 0.1 0.2 −0.1 A1 = ⎣ 0.3 −0.2 0.2 ⎦ , A2 = ⎣ 0 −0.1 0.2 ⎦ , Aτ 1 = ⎣ 0.3 −0.1 0 ⎦ , −0.1 0.3 0.2 0.1 −0.2 0.1 0.2 0 0.1 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 0.1 −0.3 0.01 −0.01 −0.01 −0.01 Aτ 2 = ⎣ 0.2 −0.1 0.1 ⎦ , B1 = ⎣ 0 0.01 ⎦ , B2 = ⎣ −0.01 0 ⎦ , 0.1 0 0.2 0.01 0 0 −0.01 ⎡ ⎤ ⎡ ⎤

0.01 0 −0.02 −0.01 −0.01 0 100 0 0.01 ⎦ , Bg2 = ⎣ −0.01 0 0.01 ⎦ , C = Bg1 = ⎣ 0 , 001 0 0.01 0 0.02 0 −0.01 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 0.1 0.01 0 E f 1 = ⎣ 0.2 ⎦ , E f 2 = ⎣ 0 ⎦ , E w1 = ⎣ 0 ⎦ , E w2 = ⎣ 0.01 ⎦ , 0.2 0 0.01 0

0.01 0 −0.2 0.2 Ev = , Ed = . −0.01 0.01 0.1 −0.1 The nonlinear dynamic is as follows ⎤ 0.04x1 (t) sin(x1 (t)) cos(x2 (t)) − 0.02(x1 (t) − x2 (t) + x3 (t)) ⎦, −0.02(x1 (t) − x3 (t)) g(x(t)) = ⎣ −0.02(x2 (t) + x3 (t)) ⎡

which is bounded by the following matrices:

2.5 Simulation Results

41

⎡

⎤ ⎡ ⎤ 0.02 0.02 −0.02 −0.02 0.02 −0.02 R1 = ⎣ −0.02 0.04 −0.08 ⎦ , R2 = ⎣ −0.06 −0.04 −0.04 ⎦ . −0.04 0.02 0.04 0.04 −0.06 −0.08 The fuzzy rules can be set as follows: ⎧ 1 ⎨ b1 = , 1 + 0.8 exp(−x1 (t)) ⎩ b2 =1 − b1 . Suppose β¯ f a (t) can be generated as β¯ f a (t) =

⎧ ⎪ ⎨ 0,

0 ≤ t < 3,



 −t + 3 ⎪ 5 sin(t − 3) 1 − exp( ) , ⎩ 4

3 ≤ t < 40.

Obviously, τ˜ = 1.4, τ M = 0.6. Choose = 10,  = 1, 1 = 1, 2 = 1, ε = 1, ψ(t) = [0.1, 0.1, 0.1]T . By computing Theorems 2.2 and 2.4, the observer and controller matrices corresponding to H1 and H2 in Lemma 1.3 are acquired, respectively. Denote k as this k-step induction actuator fault estimation and q as the descriptive signal dimension. To clearly discriminate between the k-step induction actuator fault estimation and the qth dimensional signal, the expression (q, t) represents the qth dimensional signal at time t. For example, x(1, t) is x1 (t); ex2 (1, t) stands for this error between the estimated value of x1 (t) under 2-step induction actuator fault estimation and x1 (t). The fault estimation can be abbreviated as FEs (abbreviation for fault estimations) in the figures of this book. Figures 2.3, 2.4, 2.5, and 2.6 present the simulation results. In Fig. 2.3, the actuator fault f a (t) and its estimated values fˆa1 (t), fˆa2 (t) and fˆa3 (t) are given. This shows that the estimated values of the actuator fault f a (t) under 2-step and 3-step induction actuator fault estimations are closer to the actuator fault f a (t) than the one under 1-step induction actuator fault estimation. Evidently, it is unreasonable to say that the

10

Actuator FEs

Fig. 2.3 The actuator fault f a (t) and its estimations fˆa1 (t), fˆa2 (t) and fˆa3 (t)

fa (t) fˆa1 (t) fˆa2 (t) fˆa (t)

5

3

0

-5 0

5

10

15

20

Times

25

30

35

40

2 Fault Estimation and Tolerant Control for Time-Varying Delayed …

ex2 (1, t)

ex3 (1, t)

0.05 0 -0.05 0

State errors ex (3, t) Output errors ey (1, t)

Fig. 2.5 The output errors

ex1 (1, t)

0.1

State errors ex (2, t)

Fig. 2.4 The state errors

State errors ex (1, t)

42

5

10

15

20 Times

25

ex1 (2, t)

0.2

30 ex2 (2, t)

35

40

ex3 (2, t)

0.1 0 -0.1 0

5

10

15

0.3 0.2 0.1 0 -0.1

20 Times

25

ex1 (3, t)

0

5

10

15

20 Times

ex2 (3, t)

25

ey1 (1, t)

0.1

30

30

ey2 (1, t)

35

40

ex3 (3, t)

35

40

ey3 (1, t)

0.05 0 -0.05

Output errors ey (2, t)

0

5

10

15

0.3

20 Times

25

ey1 (2, t)

30

ey2 (2, t)

35

40

ey3 (2, t)

0.2 0.1 0 -0.1 0

5

10

15

20 Times

25

30

35

40

larger the value of the step number k is, the closer to the value of f a (t) the estimated value of the actuator fault is. This is because as the step number k increases, this error dynamic overshoot increases, and the settling time becomes longer. Generally, fault estimations under 2-step and 3-step induction actuator fault estimations work best. In Figs. 2.4 and 2.5, the state error and output error trajectories under 1-step, 2-step, and 3-step induction actuator fault estimations are given, respectively. This shows that the state errors and output errors are closer to zero under 2-step and 3-step induction actuator fault estimation. In Fig. 2.6, the input disturbances caused by the actuator fault are given. f˙a (t) − fˆa1 (t) and f˙a (t) − fˆa2 (t) under 2-step and 3-step induction

2.5 Simulation Results 0.25

f˙a (t) ˙ f˙a (t) − fˆa1 (t) ˙ f˙a (t) − fˆa2 (t) ˙ f˙a (t) − fˆ (t)

0.2 Input disturbances

Fig. 2.6 The input disturbances f˙a (t) and f˙a (t) − f˙ˆak (t) for k = 1, 2, 3

43

0.15

a3

0.1 0.05 0 -0.05 0

Fig. 2.7 The comparison between e f1 (t) and e f1n (t) Actuator fault errors

2

5

10

15

20 Times

25

30

35

40

35

40

ef1 (t) ef1n (t)

1

0

-1 0

5

10

15

20 Times

25

30

actuator fault estimations are nearly zero, which further shows that this approach suggested in this study truly attenuates the effect of input disturbance f˙a (t). To verify this influence of the fuzzy Lyapunov-Krasovskii functional, the same numerical example is carried out for the comparative experiments by using the ordinary Lyapunov function (Yan et al. 2019) instead of the fuzzy Lyapunov-Krasovskii functional under a 1-step induction actuator fault estimation. Define fˆa1n (t) as 1step induction actuator fault estimate not considering fuzzy Lyapunov-Krasovskii functionals, and e f1n (t) = fˆa1n (t) − f a (t). Let Q 1b = Q 1 , Q 2b = Q 2 , Rb = R, calculate linear matrix inequalities in Theorems 2.2 and 2.4, and then the corresponding observer matrices and controller matrices are obtained. Figure 2.7 shows the comparison between e f1 (t) and e f1n (t), which proves that this estimation of f a (t) under the fuzzy Lyapunov-Krasovskii functional is closer to f a (t) than that with this ordinary Lyapunov function. Example 2.2 In the following, the research on this truck trailer system (Cao and Frank 2001) with a time delay is shown:

44

2 Fault Estimation and Tolerant Control for Time-Varying Delayed …

⎧ v t¯ v t¯ v t¯ ⎪ ⎪ x˙1 (t) = −a x1 (t) − (1 − a) x1 (t − τ˜ ) + u(t), ⎪ ⎪ ⎪ Lt0 Lt0 lt0 ⎪ ⎪ ⎨ v t¯ v t¯ x1 (t) + (1 − a) x1 (t − τ˜ ), x˙ (t) = a ⎪ 2 Lt0 Lt0 ⎪ ⎪ ⎪ ⎪ v t¯ v t¯ v t¯ ⎪ ⎪ ⎩ x˙3 (t) = x1 (t) + (1 − a) x1 (t − τ˜ )], sin[x2 (t) + a t0 2L 2L where a = 0.7, l = 2.8, L = 5.5, v = −1.0, t¯ = 2.0, t0 = 0.5, τ˜ = 0.5. Suppose the fuzzy model is used to obtain these following fuzzy rules: Rule 1: ˜ = x2 (t) + a vt¯ x1 (t) + (1 − a) vt¯ x1 (t − τ˜ ) is approximately 0, THEN IF θ(t) 2L 2L 

x(t) ˙ = A1 x(t) + Aτ 1 x(t − τ˜ ) + B1 u(t), y(t) = C x(t),

Rule 2: ˜ = x2 (t) + a vt¯ x1 (t) + (1 − a) vt¯ x1 (t − τ˜ ) is approximately π or −π, THEN IF θ(t) 2L 2L 

x(t) ˙ = A2 x(t) + Aτ 2 x(t − τ˜ ) + B2 u(t), y(t) = C x(t),

where ⎤ ⎤ ⎤ ⎡ ⎡ v t¯ v t¯ v t¯ 0 0 0 0 00 −a Lt −a Lt −(1 − a) Lt 0 0 0 ⎥ ⎥ ⎥ ⎢ vt¯ ⎢ vt¯ ⎢ v t¯ 0 0 ⎦ , A2 = ⎣ a Lt 0 0 ⎦ , Aτ 1 = ⎣ (1 − a) Lt 0 0⎦, A1 = ⎣ a Lt 0 0 0 2 ¯2 2 2 2 2 v t v t¯ v t¯ t¯ hv t¯ a 2Lt a ev (1 − a) 2Lt 0 0 00 t0 2Lt0 t0 0 0 ⎤ ⎡ ⎡ vt¯ ⎤ ⎡ vt¯ ⎤ v t¯ −(1 − a) Lt0 0 0

lt0 lt0 100 ⎥ ⎢ v t¯ ⎣ ⎣ ⎦ ⎦ Aτ 2 = ⎣ (1 − a) Lt0 0 0 ⎦ , B1 = 0 , B2 = 0 , C = , 001 2 ¯2 t 0 0 (1 − a) ev 0 0 2Lt0 ⎡

and e =

10t0 . π

The fuzzy rules can be set as follows:    ⎧ 1 1 ⎪ ⎨ b1 = 1 − , ˜ − 0.5π)) ˜ + 0.5π)) 1 + exp(−3(θ(t) 1 + exp(−3(θ(t) ⎪ ⎩ b2 = 1 − b1 . Suppose β¯ f a (t) can be obtained from

2.5 Simulation Results

45

Fig. 2.8 The actuator fault f a (t) and its estimations Actuator FEs

3

fa (t) fˆa (t) got by this chapter fˆa (t) got by the contrast observer

2

1

0

State errors ex (1, t)

0

State errors ex (3, t)

State errors ex (2, t)

Fig. 2.9 The state errors

 β¯ f a (t) =

5

10 Times

15

20

ex (1, t) got by this chapter ex (1, t) got by the contrast observer

0.4 0.2 0 -0.2 0

5

10 Times

15

20

ex (2, t) got by this chapter ex (2, t) got by the contrast observer

0.04 0.02 0 -0.02 -0.04 0

5

0.06

10 Times

15

20

ex (3, t) got by this chapter ex (3, t) got by the contrast observer

0.04 0.02 0 -0.02 0

5

10 Times

15

20

0, 0 ≤ t < 2, 1.2 + 0.4 sin(0.5t), 2 ≤ t < 20.

Suppose E f 1 = B1 and E f 2 = B2 . Obviously, τ M = 0. Set = 10,  = 1,

1 = 1, 2 = 1, ε = 1, ψ(t) = [0.25π, 0.1π, −0.1]T . By computing the linear matrix inequalities in Theorems 2.2 and 2.4 and choosing this appropriate step k to make the influence best, observer and controller matrices corresponding to H1 and H2 , respectively, in Lemma 1.3 are acquired. Fault-tolerant control can be abbreviated as FTC in the figures of this book. Figures 2.8, 2.9, 2.10, and 2.11 present these simulation results. Figure 2.8 shows the actuator fault f a (t) and the corresponding estimation. This means that if t > 10s, then fˆa (t) and f a (t) are very close. Figures 2.9 and 2.10 depict the state error and output error. It can be seen that the state error and the output error are very close

2 Fault Estimation and Tolerant Control for Time-Varying Delayed …

Fig. 2.10 The output errors

Output errors ey (1, t)

46

ey (1, t) got by this chapter ey (1, t) got by the contrast observer

0.4 0.2 0 -0.2

Output errors ey (2, t)

0

5

15

0.02 0 -0.02 5

10 Times

15

State

0.8

20

x(1, t) η(1, t)

0.4 0 0

5

10 Times

15

20 x(2, t) η(2, t)

0.4 State

20

ey (2, t) got by this chapter ey (2, t) got by the contrast observer

0.04

0

Fig. 2.11 The system state x(t) and controller state η(t) under fault-tolerant control in the chapter

10 Times

0.2 0 0

5

10 Times

15

x(3, t) η(3, t)

10 State

20

0 -10 0

5

10 Times

15

20

to zero. The results show that the method proposed in this chapter can effectively weaken the influence of input perturbation f˙a (t). Besides, this system state x(t) and the controller state η(t) are presented in Fig. 2.11, which shows that with fault-tolerant control, the system is indeed asymptotically stable. A comparative test is performed below. This structure of a few traditional observers is similar to that in Zhang et al. (2010). Figures 2.8, 2.9 and 2.10 describe these simulation results of this contrast observer obtained in Zhang et al. (2010). Figure 2.8 shows this actuator fault f a (t) and the corresponding estimated value fˆa (t) obtained by the method proposed by Zhang et al. (2010). This obviously shows that the fˆa (t)

2.5 Simulation Results

47

Fig. 2.12 The actuator fault and its estimations in different levels of noise

Fig. 2.13 The states in different levels of noise

obtained by the method in this chapter is closer to f a (t). The state error and output error obtained by the method proposed by Zhang et al. (2010) are shown in Figs. 2.9 and 2.10, indicating that the state error and the output error obtained via the method in Zhang et al. (2010) are larger than those obtained by the method in this chapter. The results describe that this approach addressed in the chapter can effectively weaken the influence of input interference brought by f˙a (t). Therefore, the merits and availability of the method proposed in this chapter are proved more powerfully. To illustrate the performance of this suggested method under various noise levels, different noise levels are pumped into the truck trailer system, respectively. Four cases of no noise, small noise, general noise, and large noise are considered. fˆa0 (t), fˆa1 (t), fˆa2 (t) and fˆa3 (t) represent the estimations of f a (t) under the cases of no noise, small noise, general noise, and large noise, respectively. x 0 (t), x 1 (t), x 2 (t), and x 3 (t) represent the states under the cases of no noise, small noise, general noise, and large noise, respectively. The simulation results are shown in Figs. 2.12 and 2.13. From Figs. 2.12 and 2.13, we can find that when the noise is not very large, the noise has

48

2 Fault Estimation and Tolerant Control for Time-Varying Delayed …

little influence on fault estimation and tolerant control. When the noise is large, the curve of the simulation results will become unsmooth, but the performance of fault estimation and tolerant control can still be guaranteed.

2.6 Chapter Summary In this chapter, observer-based fault estimation and active fault-tolerant control problems are investigated according to a cluster of T-S fuzzy models with time-varying delay. Using (k − 1)-step information, the new k-step fuzzy fast adaptive fault estimation approach is studied to obtain this observer. The controller is established to make this closed-loop system asymptotically stable through compensating for this fault influence. Compared with this existing research work, the application range of this controller is wider. This closed-loop model stability is obtained via fuzzy Lyapunov-Krasovskii functional method. These sufficient stability conditions are discussed with the form of linear matrix inequalities solved in the MATLAB toolbox. Finally, the superiority of the method is verified by two simulation examples.

References Blanke M, Kinnaert M, Lunze J et al (2006) Diagnosis and fault-tolerant control. Springer, Berlin Cao YY, Frank P (2001) Stability analysis and synthesis of nonlinear time-delay systems via linear Takagi-Sugeno fuzzy models. Fuzzy Sets Syst 124(2):213–229 Chen J, Cao YY (2013) A stable fault detection observer design in finite frequency domain. Int J Control 86(2):290–298 Corless M, Tu J (1998) State and input estimation for a class of uncertain systems. Automatica 34(6):757–764 Du D, Cocquempot V (2017) Fault diagnosis and fault tolerant control for discrete-time linear systems with sensor fault. IFAC Pap 50(1):15754–15759 Gao Z (2015) Fault estimation and fault-tolerant control for discrete-time dynamic systems. IEEE Trans Ind Electron 62(6):3874–3884 Gao Z, Ding SX (2007) Actuator fault robust estimation and fault-tolerant control for a class of nonlinear descriptor systems. Automatica 43(5):912–920 Gao H, Liu X, Lam J (2009) Stability analysis and stabilization for discrete-time fuzzy systems with time-varying delay. IEEE Trans Cybern 39(2):306–317 Han J, Zhang H, Wang Y et al (2015) Disturbance observer based fault estimation and dynamic output feedback fault tolerant control for fuzzy systems with local nonlinear models. ISA Trans 59:114–124 Han J, Zhang H, Wang Y et al (2018a) Robust fault detection for switched fuzzy systems with unknown input. IEEE Trans Cybern 48(11):3056–3066 Han J, Zhang H, Wang Y et al (2018b) Fault estimation and fault-tolerant control for switched fuzzy stochastic systems. IEEE Trans Fuzzy Syst 26(5):2993–3003 Han J, Liu X, Wei X et al (2019) Reduced-order observer based fault estimation and fault-tolerant control for switched stochastic systems with actuator and sensor faults. ISA Trans 88:91–101 Hassan MF, Zribi M (2014) An observer-based controller for nonlinear systems: a gain scheduling approach. Appl Math Comput 237:695–711

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Huang S, Yang G (2014) Fault tolerant controller design for T-S fuzzy systems with time-varying delay and actuator faults: a k-step fault-estimation approach. IEEE Trans Fuzzy Syst 22(6):1526– 1540 Jee SC, Lee HJ (2016) H _/H∞ fault detection and isolation for nonlinear systems with state delay in T-S form. J Frankl Inst 353(9):2030–2056 Jiang B, Chowdhury FN (2005) Fault estimation and accommodation for linear MIMO discrete-time systems. IEEE Trans Control Syst Technol 13(3):493–499 Li J, Pan K, Zhang D et al (2019) Robust fault detection and estimation observer design for switched systems. Nonlinear Anal: Hybrid Syst 34:30–42 Liang H, Zhang Y, Huang T et al (2020) Prescribed performance cooperative control for multiagent systems with input quantization. IEEE Trans Cybern 50(5):1810–1819 Liang H, Zhang L, Sun Y et al (2021) Containment control of semi-Markovian multiagent systems with switching topologies. IEEE Trans Syst Man Cybern: Syst 51(6):3889–3899 Liu M, Cao X, Shi P (2013) Fault estimation and tolerant control for fuzzy stochastic systems. IEEE Trans Fuzzy Syst 21(2):221–229 Liu L, Liu Y, Tong S (2019) Neural networks-based adaptive finite-time fault-tolerant control for a class of strict-feedback switched nonlinear systems. IEEE Trans Cybern 49(7):2536–2545 Liu X, Gao X, Han J (2020a) Distributed fault estimation for a class of nonlinear multiagent systems. IEEE Trans Syst Man Cybern: Syst 50(9):3382–3390 Liu L, Liu Y, Li D et al (2020b) Barrier Lyapunov function-based adaptive fuzzy FTC for switched systems and its applications to resistance-inductance-capacitance circuit system. IEEE Trans Cybern 50(8):3491–3502 Mao Z, Jiang B, Shi P (2011) Observer-based fault-tolerant control for a class of networked control systems with transfer delays. J Frankl Inst 348(4):763–776 Moodi H, Farrokhi M (2014) On observer-based controller design for Sugeno systems with unmeasurable premise variables. ISA Trans 53(2):305–316 Mozelli LA, Souza FO, Palhares RM (2011) A new discretized Lyapunov-Krasovskii functional for stability analysis and control design of time-delayed TS fuzzy systems. Int J Robust Nonlinear Control 21(1):93–105 Qiu J, Feng G, Gao H (2012) Observer-based piecewise affine output feedback controller synthesis of continuous-time T-S fuzzy affine dynamic systems using quantized measurements. IEEE Trans Fuzzy Syst 20(6):1046–1062 Selvaraj P, Kaviarasan B, Sakthivel R et al (2017) Fault-tolerant SMC for Takagi-Sugeno fuzzy systems with time-varying delay and actuator saturation. IET Control Theory Appl 11(8):1112– 1123 Sun S, Zhang H, Han J et al (2019) A novel double-level observer-based fault estimation for TakagiSugeno fuzzy systems with unknown nonlinear dynamics. Trans Inst Meas Control 41(12):3372– 3384 Tao Y, Shen D, Wang Y et al (2015) Reliable H∞ control for uncertain nonlinear discrete-time systems subject to multiple intermittent faults in sensors and/or actuators. J Frankl Inst 352(11):4721– 4740 Van M, Ge SS, Ren H (2017) Finite time fault tolerant control for robot manipulators using time delay estimation and continuous nonsingular fast terminal sliding mode control. IEEE Trans Cybern 47(7):1681–1693 Wang L, Lam H (2018a) Local stabilization for continuous-time Takagi-Sugeno fuzzy systems with time delay. IEEE Trans Fuzzy Syst 26(1):379–385 Wang L, Lam H (2018b) A new approach to stability and stabilization analysis for continuous-time Takagi-Sugeno fuzzy systems with time delay. IEEE Trans Fuzzy Syst 26(4):2460–2465 Wang Y, Zheng L, Zhang H et al (2018) Event-triggered fault detection filter design for nonlinear networked systems via fuzzy Lyapunov functions. J Frankl Inst 355(17):8392–8411 Wang Z, Sun J, Zhang H (2020) Stability analysis of T-S fuzzy control system with sampleddropouts based on time-varying Lyapunov function method. IEEE Trans Syst Man Cybern: Syst 50(7):2566–2577

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Yan S, Shen M, Nguang SK et al (2019) A distributed delay method for event-triggered control of T-S fuzzy networked systems with transmission delay. IEEE Trans Fuzzy Syst 27(10):1963–1973 Zhang K, Jiang B, Shi P (2009) A new approach to observer-based fault-tolerant controller design for Takagi-Sugeno fuzzy systems with state delay. Circuits Syst Signal Process 28(5):679–697 Zhang K, Jiang B, Staroswiecki M (2010) Dynamic output feedback fault tolerant controller design for Takagi-Sugeno fuzzy systems with actuator faults. IEEE Trans Fuzzy Syst 18(1):194–201 Zhang C, Hu J, Qiu J et al (2017a) Event-triggered nonsynchronized H∞ filtering for discrete-time T-S fuzzy systems based on piecewise Lyapunov functions. IEEE Trans Syst Man Cybern: Syst 47(8):2330–2341 Zhang H, Han J, Luo C et al (2017b) Fault-tolerant control of a nonlinear system based on generalized fuzzy hyperbolic model and adaptive disturbance observer. IEEE Trans Syst Man Cybern: Syst 47(8):2289–2300 Zhao Y, Gao H, Lam J et al (2009) Stability and stabilization of delayed T-S fuzzy systems: a delay partitioning approach. IEEE Trans Fuzzy Syst 17(4):750–762 Zhao L, Gao H, Karimi HR (2013) Robust stability and stabilization of uncertain T-S fuzzy systems with time-varying delay: An input-output approach. IEEE Trans Fuzzy Syst 21(5):883–897 Zhong Z, Wai R, Shao Z et al (2017) Reachable set estimation and decentralized controller design for large-scale nonlinear systems with time-varying delay and input constraint. IEEE Trans Fuzzy Syst 25(6):1629–1643

Chapter 3

Fault Estimation and Tolerant Control for Multiple Time Delayed Fuzzy Systems with Sensor and Actuator Faults

3.1 Introduction The second chapter studies the fault estimation and tolerant control for time delayed T-S fuzzy systems. In general, many factors, such as disturbances, unknown nonlinear dynamics, actuator faults, and sensor faults, may need to be considered in fault estimation and tolerant control for T-S fuzzy systems. In addition, a time-varying delay or even multiple time-varying delays are unavoidable and may lead to system instability, so it is difficult to use traditional controllers (Jun 2007; Cao and Sun 1998; Wu and Li 2007). In Huang and Yang (2014), Cao and Frank (2001), Chen et al. (2006), balancing and swing-up of the inverted pendulum on a cart and the truck trailer system were transformed into T-S fuzzy models with time delay, and the fault estimation and tolerant control were studied. In order to solve the problems of various unfavorable factors in nonlinear systems, robust control, guaranteed cost control, and other stability analysis and stabilization problems for T-S fuzzy systems with time-varying delays have been extensively studied (Huang and Yang 2014; Zhao et al. 2009, 2013; Wang and Lam 2018b; Mozelli et al. 2011; Angulo et al. 2017). It should be noted that all study results of T-S fuzzy systems are divided into two categories: delay-independent results (Cao and Sun 1998) and delay-dependent results (Zhang et al. 2008; Gao et al. 2009). A delay message is not required for the former, but is required for the latter. So the latter can make the results less conservative, particularly when these delay sizes are not large. It is worth pointing out that for T-S fuzzy models against multiple time-varying state delays and sensor and actuator faults, the researches on delay-dependent H∞ control are not sufficient in previous literature. The above discussion motivates us to conduct this study.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Sun et al., Fault-Tolerant Control for Time-Varying Delayed T-S Fuzzy Systems, Intelligent Control and Learning Systems 9, https://doi.org/10.1007/978-981-99-1357-2_3

51

52

3 Fault Estimation and Tolerant Control for Multiple Time Delayed Fuzzy …

3.2 System Definition and Description A T-S fuzzy system against multiple time-varying delays can be described in the following: ⎧  r N  ⎪ ⎪ ⎪ x(t) ˙ = δi (ϑ(t)) Ail x(t − τl (t)) + Bi u(t) + Bai f a (t) ⎪ ⎪ ⎪ ⎪ i=1 l=0 ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ + Bhi h(x(t)) + Bwi w(t) , ⎪ ⎪ ⎪ ⎪ ⎨   r N  ⎪ δi (ϑ(t)) Cil x(t − τl (t)) + Fs f s (t) , y(t) = ⎪ ⎪ ⎪ ⎪ i=1 l=0 ⎪ ⎪ ⎪ r ⎪  ⎪ ⎪ ⎪ y (t) = δi (ϑ(t))C zi x(t), ⎪ z ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎩ x(t) =φ(t), t ∈ [−τ D , 0],

(3.1)

where x(t) ∈ R m denotes this state, u(t) ∈ R n represents this system input, f a (t) ∈ R q1 represents the actuator fault, f˙a (t) ∈ L 2 [0, ∞), w(t) ∈ R n 1 denotes this external disturbance, w(t) ∈ L 2 [0, ∞), y(t) ∈ R g1 represents this measurement output, f s (t) ∈ R q2 represents the sensor fault and yz (t) ∈ R g2 represents this control output. Ail , Bi , Bai , Bhi , Bwi , Cil , Fs and C zi , (i = 1, . . . , r, l = 0, 1, . . . , N ), denote known constant matrices of suitable dimensions. τ0 (t) = 0. τl (t), (l = 1, . . . , N ) represents this time-varying state delay, satisfying τl (t) ≤ τl , and τ˙l (t) ≤ τ Ml , where τl denotes a known scalar, τ Ml denotes a known or unknown scalar. At the same time, τ D is defined as this maximum value of τl , namely, τ D = max{τ1 , . . . , τ N }. φ(t) represents the vector-valued initial function, where t ∈ [−τ D , 0]. r and p represent the numbers of IF-THEN rules and premise variables, respectively. δi (ϑ(t)) can be obtained from ϑ j (t) and oi j , where ϑ j (t) ( j = 1, . . . , p) stands for this via premise variable, oi j (i = 1, . . . , r, j = 1, . . . , p) is a fuzzy set represented r (t), . . . ,ϑ (t)], δ (ϑ(t))=σ (ϑ(t))/ σ (θ(t)), a membership function. ϑ(t)=[ϑ 1 p i i i=1 i

p σi (ϑ(t)) = j=1 oi j (ϑ j (t)) and here oi j (·) represents this order of the membership function of oi j . Suppose that σi (ϑ(t)) ≥ 0, i = 1, . . . , r,

r 

σi (ϑ(t)) > 0,

i=1

for every ϑ(t). Therefore, for every ϑ(t),

r

i=1 δi (ϑ(t))

δi (ϑ(t)) ≥ 0, i = 1, . . . , r,

r  i=1

satisfies

δi (ϑ(t)) = 1.

3.2 System Definition and Description

53

At the same time, h(x(t)) ∈ R m 1 is Lipschitz, which is a nonlinear function and can decrease this fuzzy rule number and this calculation burden and increase this model precision (Moodi and Farrokhi 2015), following the following sector-bounded condition (Zhang et al. 2017): [h(x(t)) − U1 x(t)]T [h(x(t)) − U2 x(t)] ≤ 0, ∀x(t) ∈ R m , T ˜ ˜ [h(x(t)) − U1 x(t)] ˜ [h(x(t)) − U2 x(t)] ˜ ≤ 0, ∀x(t) ∈ R m ,

(3.2) (3.3)

˜ represents the where U1 , U2 ∈ R m 1 ×m represent the known constant matrices. x(t) ˜ difference between two vectors of x(t). h(x(t)) represents the difference between two vectors of h(x(t)). According to Tao et al. (2015), through formula (3.2), it follows 

x(t) h(x(t))

T 

D1 D2 ∗ I



 x(t) ≤ 0, h(x(t))

where D1 = sym(U1T U2 /2), D2 = −(U1T + U2T )/2. Equation (3.3) is similar, so omitted. stand for δi (ϑ(t)). In addition, for matrices Aˆ i and To keep it simple, δi is used to r r ˆ ˆ ˆ ˆ Ai j , (i, j = 1, . . . , r ), set Aδ = i=1 δi Ai and Aδδ = i=1 δi rj=1 δ j Aˆ i j . Assumption 3.1 For some practical models, the output y(t) can be measured by sensors. Thereby suppose that C1l = · · · = Crl = Cl , where l = 1, 2, . . . , N , and then thefollowing T-S model (3.1) is obtained: ⎧ N  ⎪ ⎪ ⎪ x(t) ˙ = Aδl x(t − τl (t)) + Bδ u(t) + Baδ f a (t) + Bhδ h(x(t)) + Bwδ w(t), ⎪ ⎪ ⎪ ⎪ l=0 ⎪ ⎪ ⎪ ⎨ N  (3.4) y(t) = Cl x(t − τl (t)) + Fs f s (t), ⎪ ⎪ ⎪ l=0 ⎪ ⎪ ⎪ ⎪ ⎪ yz (t) = C zδ x(t), ⎪ ⎪ ⎩ x(t) = φ(t), t ∈ [−τ D , 0], Assumption 3.2 In model (3.4), (Ai0 , Bi ) is controllable, and (Ai0 , C0 ) is observable. Moreover, the row rank of matrix C0 is full. Assumption 3.3 Zhang et al. (2009) rank(Bi , Bai ) = rank(Bi ). Afterward, there exists a matrix Bi∗ ∈ R n×m satisfying (I − Bi Bi∗ )Bai = 0, where Bi∗ is part of Bi(1) .

(3.5)

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3 Fault Estimation and Tolerant Control for Multiple Time Delayed Fuzzy …

Remark 3.1 Assumption 3.1 is realistic for many systems, for instance, balancing and swing-up of an inverted pendulum on the cart (Huang and Yang 2014), a cart motion model (Qiu et al. 2012), a truck trailer model (Huang and Yang 2014), a DC motor model (Choi 2007), an electromechanical model (Han et al. 2018) and a chaotic model (Kim et al. 2007). Assumption 3.2 can be adopted to assist stability analysis and is required for stability control. This controllable assumption of (Ai0 , Bi ) ensures the stability of the system; (Ai0 , C0 ) is the observable assumption to ensure the designability of the observer. Without loss of generality, C0 is usually assumed to be a full rank matrix, which theoretically and practically ensures that all output channels are used for state observation. Assumption 3.3 suggests that the fault-tolerant controller only compensates for this influence of actuator faults in this control input channel, with the special condition Bai = Bi . When matrix Bi has full rank, meaning that this control input channel number is equal to or greater than this state channel number, (n ≥ m), then any Bai satisfies Assumption 3.3 (Noura et al. 2000; Theilliol et al. 2014). However, the cases can be hard to encounter in practice (Zadeh 1973). Besides, if this control input channel number is less than this state channel number (n < m), then Assumption 3.3 is only encountered for the existence of Bai such that Bai = Bi B¯ ai , where B¯ ai ∈ R n×q1 is a nonzero matrix (Chen and Saif 2006; Jiang et al. 2006).

3.3 Observer Design This section mainly studies T-S fuzzy systems against multiple time-varying delays. Then, we propose a k-step induction fault estimation approach to estimate the sensor fault f s (t) and the actuator fault f a (t).

3.3.1 State Augmentation Denote x(t) ¯ = [x T (t), f sT (t)]T , and suppose that G ∈ R g1 ×g1 is a non-singular matrix or G is a scalar. Model (3.4) is rewritten as

3.3 Observer Design

55

⎧ N  ⎪ ⎪ ⎪ ˙ ⎪ H x(t) ¯ = A x(t) ¯ + A1δl x(t ¯ − τl (t)) + B1δ u(t) + Ba1δ f a (t) 1δ0 ⎪ ⎪ ⎪ ⎪ l=1 ⎪ ⎪ ⎪ ¯ 10 x(t), ⎪ ˙¯ ¯ + Bw1δ w(t) + E 1 y(t) + GC + Bh1δ h(D0 x(t)) ⎪ ⎨ N  ⎪ ⎪ x(t) ¯ + C1l x(t ¯ − τl (t)), y(t) = C 10 ⎪ ⎪ ⎪ ⎪ l=1 ⎪ ⎪ ⎪ ⎪ ¯ yz (t) = C z1δ x(t), ⎪ ⎪ ⎪ ⎩ x(t) = φ(t), t ∈ [−τ D , 0],

(3.6)

where        Aδ0 0 Aδl 0 Bδ Im 0 , A1δ0 = , A1δl = , B1δ = , H= 0 GC0 G Fs −C0 −Fs −Cl 0          Baδ Bhδ Bwδ 0 , Bh1δ = , D0 = Im 0 , Bw1δ = , E1 = , Ba1δ = 0 0 0 Ig      0 G¯ = , C10 = C0 Fs , C1l = Cl 0 , C z1δ = C zδ 0 . G 

 0 Im . Obviously, H − is one of the generalized inverse −Fs− C0 Fs− G −1 matrices of H . Besides, the T-S fuzzy model (3.6) is redescribed as 

Set H − =

⎧ N  ⎪ ⎪ ⎪ ˙¯ = A2δ0 x(t) ⎪ x(t) ¯ + A2δl x(t ¯ − τl (t)) + B2δ u(t) + Ba2δ f a (t) ⎪ ⎪ ⎪ ⎪ l=1 ⎪ ⎪ ⎪ ⎪ ¯ + Bw2δ w(t) + E 2 y(t) + A¯ 20 ( y˙ (t) + Bh2δ h(D0 x(t)) ⎪ ⎪ ⎪ ⎪ ⎪ N ⎪  ⎪ ⎨ ˙¯ − τ (t))), − C x(t 1l

l

l=1 ⎪ ⎪ ⎪ ⎪ N ⎪  ⎪ ⎪ ⎪ y(t) = C10 x(t) ¯ + C1l x(t ¯ − τl (t)), ⎪ ⎪ ⎪ ⎪ l=1 ⎪ ⎪ ⎪ ⎪ yz (t) = C z1δ x(t), ⎪ ⎪ ⎪ ⎩ x(t) = φ(t), t ∈ [−τ D , 0],

(3.7)

where A2δ0 = H − A1δ0 , A2δl = H − A1δl , B2δ = H − B1δ , Ba2δ = H − Ba1δ , Bh2δ = ¯ H − Bh1δ , Bw2δ = H − Bw1δ , E 2 = H − E 1 , A¯ 20 = H − G.

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3 Fault Estimation and Tolerant Control for Multiple Time Delayed Fuzzy …

3.3.2 Observer Design Based on a k-Step Induction Fault Estimation Method (1) 1-step induction fault estimation: The 1-step induction fault estimation observer for the T-S fuzzy model (3.7) is depicted below: ⎧ N  ⎪ ⎪ ⎪ z ˙ (t) = A z (t) + (A2δl − A2δ0 A¯ 20 C1l )xˆ¯1 (t − τl (t)) + B2δ u(t) 1 2δ0 1 ⎪ ⎪ ⎪ ⎪ l=1 ⎪ ⎪ ⎪ ⎪ ˆ ⎪ + Ba2δ f a1 (t) + Bh2δ h(D0 xˆ¯1 (t)) + (E 2 + A2δ0 A¯ 20 )y(t) ⎪ ⎪ ⎪ ⎪ ⎪ + L 1δ (y(t) − yˆ1 (t)), ⎪ ⎪ ⎨ N  (3.8) ¯ 20 (y(t) − ˆ ⎪ (t) = z (t) + A C1l xˆ¯1 (t − τl (t))), x ¯ 1 1 ⎪ ⎪ ⎪ ⎪ l=1 ⎪ ⎪ ⎪ N ⎪  ⎪ ⎪ ⎪ yˆ1 (t) = C10 xˆ¯1 (t) + C1l xˆ¯1 (t − τl (t)), ⎪ ⎪ ⎪ ⎪ l=1 ⎪ ⎪ ⎪ ⎩ ˙ ˆ f a1 (t) = T L 2δ (y(t) − yˆ1 (t)), where z 1 (t) ∈ R m+q2 represents this initial observer state and xˆ¯1 (t) ∈ R m+q2 , yˆ1 (t) ∈ ¯ y(t) and f a (t). T ∈ R q1 ×q1 R g1 and fˆa1 (t) ∈ R q1 stand for initial estimations of x(t), represents this learning rate and T > 0. The involvement of this learning rate T adjusts this convergence rate and estimation error. L 1δ and L 2δ represent observer gain matrices that will be determined later. ˇ = [x¯ T (t), f aT1 (t)]T , xˆˇ1 (t) = [xˆ¯1T (t), fˆaT1 (t)]T , ex¯1 Set e y1 (t) = yˆ1 (t) − y(t). x(t) (t) = xˆ¯1 (t) − x(t), ¯ e fa1 (t) = fˆa1 (t) − f a (t), e1 (t) = xˆˇ1 (t) − x(t), ˇ h˜ 1 ( D˜ 0 x(t)) ˇ = T T T h(D0 x¯ˆ1 (t)) − h(D0 x(t)), ¯ w¯ 1 (t) = [w (t), e˙x¯ (t − τ1 (t)), . . . , e˙x¯ (t − τ N (t)), f˙aT (t)]T . 1 1 In light of equations (3.7) and (3.8), the estimation error dynamic can be written as ⎧ N  ⎪ ⎪ ⎪ ⎪ e ˙ (t) = (A − L C )e (t) + (A3δl − L δ C2l )e1 (t − τl (t)) 1 3δ0 δ 20 1 ⎪ ⎪ ⎪ ⎪ l=1 ⎨ (3.9) + Bh3δ h˜ 1 ( D˜ 0 x(t)) ˇ + Bw3δ w¯ 1 (t), ⎪ ⎪ ⎪ N ⎪  ⎪ ⎪ ⎪ e (t) = C e (t) + C2l e1 (t − τl (t)), ⎪ y 20 1 ⎩ 1 l=1

where  A3δ0 =

       A2δ0 Ba2δ A2δl 0 L 1δ Bh2δ , Bh3δ = , A3δl = , , Lδ = 0 0 T L 2δ 0 0 0

3.3 Observer Design

57

     B¯ Bw2δ A¯ 20 C11 · · · A¯ 20 C1N 0  ¯w21δ , D˜ 0 = D0 0 , Bw3δ = − 0 0 ··· 0 I Bw22   C20 = C10 0 , C2l = C1l 0 . (2) 2-step induction fault estimation: To decrease this influence of input disturbance caused by f˙a (t) and improve this performance of fault estimation, using fˆa1 (t) obtained from the 1-step induction fault estimation observer, the 2-step induction fault estimation observer for system (3.7) is designed as follows: ⎧ N  ⎪ ⎪ ⎪ z ˙ (t) = A z (t) + (A2δl − A2δ0 A¯ 20 C1l )xˆ¯2 (t − τl (t)) + B2δ u(t) 2 2δ0 2 ⎪ ⎪ ⎪ ⎪ l=1 ⎪ ⎪ ⎪ ⎪ ⎪ + Ba2δ fˆa2 (t) + Bh2δ h(D0 xˆ¯2 (t)) + (E 2 + A2δ0 A¯ 20 )y(t) ⎪ ⎪ ⎪ ⎪ ⎪ + L 1δ (y(t) − yˆ2 (t)), ⎪ ⎪ ⎨ N  (3.10) ¯ ˆ ⎪ x ¯ (t) = z (t) + A (y(t) − C1l xˆ¯2 (t − τl (t))), 2 2 20 ⎪ ⎪ ⎪ ⎪ l=1 ⎪ ⎪ ⎪ N ⎪  ⎪ ⎪ ⎪ yˆ2 (t) = C10 x¯ˆ2 (t) + C1l x¯ˆ2 (t − τl (t)), ⎪ ⎪ ⎪ ⎪ l=1 ⎪ ⎪ ⎪ ⎩ ˙ fˆa2 (t) =T L 2δ (y(t) − yˆ2 (t)) + f˙ˆa1 (t), where z 2 (t) ∈ R m+q2 is the second observer state and xˆ¯2 (t) ∈ R m+q2 , yˆ2 (t) ∈ R g1 and fˆa2 (t) ∈ R q1 denote these second estimations of x(t), ¯ y(t) and f a (t), separately. Other parameter definitions are the same as those in (3.8). Given equations (3.7), (3.8) and (3.10), the estimation error dynamic can be described as ⎧ N  ⎪ ⎪ ⎪ e˙2 (t) = (A3δ0 − L δ C20 )e2 (t) + ⎪ (A3δl − L δ C2l )e2 (t − τl (t)) ⎪ ⎪ ⎪ ⎪ l=1 ⎨ (3.11) + Bh3δ h˜ 2 ( D˜ 0 x(t)) ˇ + Bw3δ w¯ 2 (t), ⎪ ⎪ ⎪ N ⎪  ⎪ ⎪ ⎪ e (t) = C e (t) + C2l e2 (t − τl (t)), ⎪ y 20 2 2 ⎩ l=1

¯ where e y2 (t) = yˆ2 (t) − y(t), xˆˇ2 (t) = [xˆ¯2T (t), fˆaT2 (t)]T , ex¯2 (t) = xˆ¯2 (t) − x(t), ˇ h˜ 2 ( D˜ 0 x(t)) ˇ = h(D0 xˆ¯2 (t)) − h(D0 x(t)), ¯ e fa2 (t) = fˆa2 (t) − f a (t), e2 (t) = xˆˇ2 (t) − x(t), ˙ w¯ 2 (t) = [w T (t), e˙xT¯2 (t − τ1 (t)), . . . , e˙xT¯2 (t − τ N (t)), f˙aT (t) − fˆaT1 (t)]T . A3δ0 , L δ , C20 , A3δl , C2l , Bh3δ , D˜ 0 and Bw3δ (l = 1, 2, . . . , N ) are the same as the ones defined in the 1-step induction fault estimation observer.

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3 Fault Estimation and Tolerant Control for Multiple Time Delayed Fuzzy …

(3) k-step induction fault estimation: To further decrease this influence of input disturbance from f˙a (t) and enhance this fault estimation performance, similar to this 2-step induction fault estimation approach, this k-step fault estimation observer for model (3.7) with the (k − 1)th estimation of f a (t), fˆak−1 (t), obtained from the (k − 1)-step induction fault estimation observer, k = 2, 3, . . ., is described by ⎧ N  ⎪ ⎪ ⎪ z˙ k (t) = A2δ0 z k (t) + (A2δl − A2δ0 A¯ 20 C1l )xˆ¯k (t − τl (t)) + B2δ u(t) ⎪ ⎪ ⎪ ⎪ l=1 ⎪ ⎪ ⎪ ⎪ ⎪ + Ba2δ fˆak (t) + Bh2δ h(D0 xˆ¯k (t)) + (E 2 + A2δ0 A¯ 20 )y(t) ⎪ ⎪ ⎪ ⎪ ⎪ + L 1δ (y(t) − yˆk (t)), ⎪ ⎪ ⎨ N  (3.12) ¯ ˆ ⎪ x ¯ (t) = z (t) + A (y(t) − C1l xˆ¯k (t − τl (t))), k k 20 ⎪ ⎪ ⎪ ⎪ l=1 ⎪ ⎪ ⎪ N ⎪  ⎪ ⎪ ⎪ yˆk (t) = C10 x¯ˆk (t) + C1l x¯ˆk (t − τl (t)), ⎪ ⎪ ⎪ ⎪ l=1 ⎪ ⎪ ⎪ ⎩ ˙ fˆ (t) = T L (y(t) − yˆ (t)) + f˙ˆ (t), ak

2δ

k

ak−1

where z k (t) ∈ R m+q2 represents this kth observer state and xˆk (t) ∈ R m+q2 , yˆk (t) ∈ R g1 and fˆak (t) ∈ R q1 denote these kth estimations of x(t), y(t) and f a (t), separately. Set e yk (t) = yˆk (t) − y(t). xˆˇk (t) = [xˆ¯kT (t), fˆaTk (t)]T , ex¯k (t) = xˆ¯k (t) − x(t), ¯ e f ak ˆ ˆ ˜ ˜ ˆ (t) = f ak (t) − f a (t), ek (t) = xˇk (t) − x(t), ˇ h k ( D0 x(t)) ˇ = h(D0 x¯k (t)) − h(D0 x(t)), ¯ ˙ T T T T T T ˙ ˆ w¯ (t) = [w (t), e˙ (t − τ (t)), . . . , e˙ (t − τ (t)), f (t) − f (t)] . Through k

x¯k

x¯k

1

N

a

ak−1

formulas (3.7), (3.12) and the (k −1)-step induction fault estimation, this estimation error dynamic can be acquired through the following formula: ⎧ N  ⎪ ⎪ ⎪ ⎪ e ˙ (t) = (A − L C )e (t) + (A3δl − L δ C2l )ek (t − τl (t)) k 3δ0 δ 20 k ⎪ ⎪ ⎪ ⎪ l=1 ⎨ + Bh3δ h˜ k ( D˜ 0 x(t)) ˇ + Bw3δ w¯ k (t), ⎪ ⎪ ⎪ N ⎪  ⎪ ⎪ ⎪ C2l ek (t − τl (t)), ⎪ ⎩ e yk (t) = C20 ek (t) +

(3.13)

l=1

where A3δ0 , L δ , C20 , A3δl , C2l , Bh3δ , D˜ 0 and Bw3δ (l = 1, 2, . . . , N ) are the same as those defined in this 1-step induction fault estimation observer. Therefore, this use of a k-step induction fault estimation observer reduces this influence of the input disturbance caused by f˙a (t) on the system and improves this fault estimation performance. And, these gain matrices of this k-step induction fault estimation observer are described below:

3.3 Observer Design

59

1. When w¯ k (t) = 0, (k = 1, 2, . . .), this k-step error dynamic (3.13) is asymptotically stable; 2. When w¯ k (t) = 0 and this initial condition value is zero, this k-step error dynamic (3.13) satisfies the following inequality:

L

L ek (t) dt < 2

0

γk2

w¯ k (t) 2 dt

(3.14)

0

where γk > 0 represents this specified scalar with the H∞ -norm bound, representing this H∞ performance index. L > 0, w¯ k (t) ∈ L 2 [0, ∞). Theorem 3.1 For assigned positive constants τl , τ D , θ, κ, γk , a non-negative scalar υl , a scalar τ Ml , (l = 1, . . . , N ), this k-step error dynamic (3.13) is asymptotically stable against this H∞ performance index γk in formula (3.14), (k = 1, 2, . . .) and these eigenvalues of A3i0 − L i C20 are part of this linear matrix inequality region D(β1 , 1 ), if there exist positive definite matrices Pi , Q 1i , Q 2i , Ri , matrices L 1i , L 2i , and slack matrices Mi , Ni , (i = 1, 2, . . . , r ), making the following inequalities hold: P˙δ ≤ 0, Q˙ 1δ ≤ 0, Q˙ 2δ ≤ 0, R˙ δ ≤ 0,   (ϑ) N τ D Mδ < 0, ∗ −N τ D Rδ   (ϑ) N τ D Nδ < 0, ∗ −N τ D Rδ sym($ˆ¯ 11δ (ϑ)Pδ−1 ) + 2β1 Pδ−1 < 0,   − Pδ−1 $ˆ¯ 11δ (ϑ)Pδ−1 < 0, ∗ − Pδ−1

where ⎤ ⎡ $1 (ϑ)$21 (ϑ)· · ·$2N (ϑ) 0 0 0 $3 (ϑ)$4 (ϑ) ⎢ ∗ $51 (ϑ) 0 0 0 0 0 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ ∗ ∗ 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ $ (ϑ) 0 0 0 0 0 5N ⎥ ⎢ ⎢ ∗ ∗ ∗ −Q 2δ 0 0 0 0 ⎥ (ϑ) = ⎢ ∗ ⎥ ⎥ ⎢ .. ⎥ ⎢ . 0 ⎢ ∗ ∗ ∗ ∗ ∗ 0 0 ⎥ ⎥ ⎢ ⎢ ∗ ∗ ∗ ∗ ∗ ∗ −Q 2δ 0 0 ⎥ ⎥ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −κI 0 ⎦ 2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −γk I +

N 

τl T (ϑ)Pδ (2θ Pδ − θ2 Rδ )−1 Pδ (ϑ)

l=1

− sym(Mδ (E1 − E2 )) − sym(Nδ (E2 − E3 )),

(3.15) (3.16) (3.17) (3.18) (3.19)

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3 Fault Estimation and Tolerant Control for Multiple Time Delayed Fuzzy …

and $1 (ϑ) = sym(Pδ (A3δ0 − L δ C20 )) +

N 

υl Q 1δ + N Q 2δ + I − κ D˜ 0T D1 D˜ 0 ,

l=1

I ∈ R m+q1 +q2 , $2l (ϑ) = Pδ (A3δl − L δ C2l ), $3 (ϑ) = Pδ Bh3δ − κ D˜ 0T D2 , $4 (ϑ) = Pδ Bw3δ , $5l (ϑ) = −υl (1 − τ Ml )Q 1δ , T (ϑ) =[A3δ0 − L δ C20 , A3δ1 − L δ C21 , . . . , A3δ N − L δ C2N , 0,· · ·, 0, Bh3δ , Bw3δ ], E1 = [N , 0, . . . , 0, 0, . . . , 0, 0, 0], E2 = [0, 1, . . . , 1, 0, . . . , 0, 0, 0], E3 = [0, 0, . . . , 0, 1, . . . , 1, 0, 0], $ˆ¯ 11δ (ϑ) = A3δ0 − L δ C20 . Proof In the following, a fuzzy Lyapunov-Krasovskii functional is designed for k = 1, 2, . . .: Vk (t) = V1k (t) + V2k (t) + V3k (t),

(3.20)

where V1k (t) = ekT (t)Pδ ek (t),

t N N t   T υl ek (s)Q 1δ ek (s)ds + ekT (s)Q 2δ ek (s)ds, V2k (t) = l=1

V3k (t) =

t−τl (t)

N 0 t 

l=1 t−τ

l

e˙kT (s)Rδ e˙k (s)dsd .

l=1 −τ t+ l

According to the error dynamic (3.13), V˙k (t) follows: V˙1k (t) = ekT (t)[sym(Pδ (A3δ0 − L δ C20 ))]ek (t) +

N 

2ekT (t)Pδ (A3δl − L δ C2l )

l=1

ek (t − τl (t)) + 2ekT (t)Pδ Bh3δ h˜ k ( D˜ 0 x(t)) ˇ + 2ekT (t)Pδ Bw3δ w¯ k (t) + ekT (t) P˙δ ek (t), V˙2k (t) =

N 

[ekT (t)(υl Q 1δ + Q 2δ )ek (t) − υl (1 − τ˙l (t))ekT (t − τl (t))Q 1δ

l=1

t ek (t − τl (t)) − ekT (t − τl )Q 2δ ek (t − τl ) + υl t−τl (t)

ekT (s) Q˙ 1δ ek (s)ds

3.3 Observer Design

t +

61

ekT (s) Q˙ 2δ ek (s)ds],

t−τl

V˙3k (t) =

N 

t [τl e˙kT (t)Rδ e˙k (t)

l=1

−

e˙kT (s)Rδ e˙k (s)ds

t−τl

0 t +

e˙kT (s) R˙ δ e˙k (t)dsd ].

−τl t+

According to the Newton-Leibniz formula, Lemmas 1.1–1.3 and Iwasaki and Hara (2005), Ahmadizadeh et al. (2014), a straightforward calculation method is obtained below V˙k (t) + ekT (t)ek (t) − γk2 w¯ kT (t)w¯ k (t) = V˙k (t) + ekT (t)ek (t) − γk2 w¯ kT (t)w¯ k (t) − 2η (t)Mδ T

N  

ek (t) − ek (t − τl (t)) −

l=1

− 2η T (t)Nδ

N 

e˙k (s)ds

t−τl (t)

⎡

⎣ek (t − τl (t)) − ek (t − τl ) −

l=1

≤

N  1 N τl l=1

t



t−τl (t)



t

t−τ

l (t)

⎤

e˙k (s)ds ⎦

t−τl

η(t) e˙k (s)

T 

(ϑ) N τ D Mδ ∗ −N τ D Rδ



 η(t) ds e˙k (s)

t−τ  T  

l (t) N  1 η(t) (ϑ) N τ D Nδ η(t) + ds, ∗ −N τ D Rδ e˙k (s) e˙k (s) N τl l=1

(3.21)

t−τl

where η T (t) = [ekT (t), ekT (t − τ1 (t)), . . . , ekT (t − τ N (t)), ekT (t − τ1 ), . . . , ekT (t − τ N ), h˜ k (D0 x(t)), ¯ w¯ kT (t)]. In addition, according to formulas (3.15)–(3.17), we have V˙k (t) + ekT (t)ek (t) − γk2 w¯ kT (t)w¯ k (t) < 0.

(3.22)

If φ(t) = 0, namely, ∀t ∈ [−τ D , 0], x(t) = 0, we have ek (t) = 0 and e˙k (t) = 0. So if ek (0) = 0 and e˙k (0) = 0, we have Vk (t)|t=0 = 0. According to Vk (t)|t=L ≥ 0 and forL mula (3.22), we can obtain 0 ( ek (t) 2 − γk2 w¯ k (t) )dt + Vk (t)|t=L − Vk (t)|t=0 ≤

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3 Fault Estimation and Tolerant Control for Multiple Time Delayed Fuzzy …

0, which indicates that formula (3.14) is satisfied. Therefore, this error dynamic (3.13) is asymptotically stable against this H∞ performance index γk . ¯ P in Lemma Next we discuss this regional pole constraint issue. Choose X, ˆ −1 ¯ 1.5 as $11δ , Pδ , separately. When expressions (3.18) and (3.19) hold, then these eigenvalues of $¯ˆ are part of this linear matrix inequality region D(β , ). This 11δ

1

1

completes the proof. Theorem 3.2 For assigned positive scalars τl , τ D , θ, κ, γk , a non-negative scalar υl , a scalar τ Ml , (l = 1, . . . , N ), this k-step error dynamic (3.13) is asymptotically stable against this H∞ performance index γk , (k = 1, 2, . . .) in formula (3.14), and these eigenvalues of A3i0 − L i C20 are part of this linear matrix inequality region D(β1 , 1 ), if there exist positive definite matrices P, Q 1i , Q 2i , Ri , a matrix Yi , and slack matrices Mi , Ni , (i = 1, 2, . . . , r ), making these following inequalities feasible: P˙δ ≤ 0, Q˙ 1δ ≤ 0, Q˙ 2δ ≤ 0, R˙ δ ≤ 0, ˜ ii < 0, i = 1, 2, . . . , r,

(3.23) (3.24)

˜ i j + ˜ ji < 0, 1 ≤ i < j ≤ r, ˜ ii < 0, i = 1, 2, . . . , r, 

(3.25)

˜ ji < 0, 1 ≤ i < j ≤ r, ˜ ij +  

(3.27)

(3.26)

sym($˜ 11,i ) + 2β1 P < 0, i = 1, 2, . . . , r,   − P $˜ 11,i < 0, i = 1, 2, . . . , r, ∗ − P

(3.28) (3.29)

where ⎡



N

⎤

⎡



N

⎤

˜ N τ D Mi ˜ N τ D Ni ⎢ ⎢ ⎥ ⎥ ⎢ i ⎢ i ⎥ ˜ ⎥ l=1 l=1 ˜ i j = ⎢ ⎥, i j = ⎢ ⎥, ⎣ ∗ −N τ D Ri ⎣ ∗ −N τ D Ri ⎦ ⎦ 0 0 ∗ ∗ −2θ P + θ2 R j ∗ ∗ −2θ P + θ2 R j ⎡ ⎤ N $˜ 1,i $˜ 12,i . . . $˜ 2,i 0 0 0 $˜ 3,i $˜ 4,i ⎢ ⎥ 0 0 ⎥ ⎢ ∗ $˜ 15,i 0 0 0 0 0 ⎢ ⎥ . ⎢ ⎥ ⎢ ∗ ∗ .. 0 0 0 0 0 0 ⎥ ⎢ ⎥ N ⎢ ∗ ∗ ∗ $˜ 5,i 0 0 0 0 0 ⎥ ⎢ ⎥ ˜ i = ⎢ ∗ ∗ ∗ ∗ −Q 0 0  0 0 ⎥ 2i ⎢ ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ ∗ ∗ ... 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ ∗ ∗ ∗ −Q 2i 0 0 ⎥ ⎢ ⎥ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −κI 0 ⎦ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −γk2 I ˜ iT τl 

˜ iT τl 

3.3 Observer Design

63

− sym(Mi (E1 − E2 )) − sym(Ni (E2 − E3 )), $˜ 1,i = sym($˜ 11,i ) +

N 

υl Q 1i + N Q 2i + I − κ D˜ 0T D1 D˜ 0 ,

l=1

$˜ 11,i = P A3i0 − Yi C20 , I ∈ R m+q1 +q2 , $˜ l2,i = P A3il − Yi C2l , $˜ 3,i = $˜ 31,i − κ D˜ 0T D2 , $˜ 31,i = P Bh3i , $˜ 4,i = P Bw3i , ˜ i = [$˜ 11,i , $˜ 1 , . . . , $˜ N , 0, . . . , 0, $˜ 31,i , $˜ 4,i ], $˜ l = −υl (1 − τ Ml )Q 1i ,  5,i

2,i

2,i

T T , . . . , M2iT N , M3i1 , . . . , M3iT N , M4iT , M5iT ]T , Mi = [M1iT , M2i1 T T Ni = [N1iT , N2i1 , . . . , N2iT N , N3i1 , . . . , N3iT N , N4iT , N5iT ]T ,

E1 = [N , 0, . . . , 0, 0, . . . , 0, 0, 0], E2 = [0, 1, . . . , 1, 0, . . . , 0, 0, 0], E3 = [0, 0, . . . , 0, 1, . . . , 1, 0, 0]. Then, this gain matrix L i is obtained by L i = P −1 Yi . Proof Given L i = P −1 Yi , we know: $˜ 11,i = P(A3i0 − L i C20 ), $˜ l2,i = P(A3il − L i C2l ). According to Lemma 1.4 and formulas (3.23)–(3.27), we have ˜¯ ii < 0, i = 1, 2, . . . , r, ˜¯ i j + ˜¯ ji < 0, 1 ≤ i < j ≤ r, ˜¯ < 0, i = 1, 2, . . . , r, 

(3.30) (3.31) (3.32)

ii

˜¯ +  ˜¯ < 0, 1 ≤ i < j ≤ r,  ij ji

(3.33)

where ˜¯ i j =



˜¯  i j N τ D Mi ∗ −N τ D Ri

˜¯ =  ˜i +  ij

N 



˜¯ = , ij



˜¯  i j N τ D Ni ∗ −N τ D Ri

 ,

˜ iT [2θ P − θ2 R j ]−1  ˜ i. τl 

l=1

Therefore, if formulas (3.30)–(3.33) are satisfied, we have ri=1 δi rj=1 δ j ˜¯ i j < ˜¯ < 0. This means that formulas (3.15)–(3.17) are satisfied. In 0, ri=1 δi rj=1 δ j  ij light of Theorem 3.1, the k-step error dynamic (3.13) is asymptotically stable under this H∞ performance index γk , (k = 1, 2, . . .) in formula (3.14). Next we discuss this regional pole constraint issue. If these linear matrix inequalities (3.28) and (3.29) hold, it can be written as

64

3 Fault Estimation and Tolerant Control for Multiple Time Delayed Fuzzy …

sym(P $˜¯ 11,i ) + 2β1 P < 0,   − P P $˜¯ 11,i < 0, ∗ − P

(3.34) (3.35)

where $˜¯ 11,i = A3i0 − L i C20 , (i = 1, 2, . . . , r ). Pre- and post-multiplying (3.34) by P −1 and its transpose, respectively, and (3.35) by diag{P −1 , P −1 } and its transpose, respectively, it can be obtained that: $˜¯ 11,i P −1 + P −1 $˜¯ 11,i + 2β1 P −1 < 0,   − P −1 $˜¯ 11,i P −1 < 0. ∗ − P −1 T

(3.36) (3.37)

According to (3.18)–(3.19), if (3.36) and (3.37) are satisfied, that is, if formulas (3.28) and (3.29) are satisfied, these eigenvalues of $˜¯ 11,i are part of this linear matrix inequality region D(β1 , 1 ). So this completes the proof. Choosing P as P = diag{P1 , T −1 } can not only decrease the variable number, but also greatly attenuate the calculation burden. Corollary 3.1 For assigned positive scalars τl , τ D , θ, κ, γk , a non-negative scalar υl , a scalar τ Ml , (l = 1, . . . , N ), this kth error dynamic (3.13) is asymptotically stable against this H∞ performance index γk , (k = 1, 2, . . .), in formula (3.14), and the eigenvalues of A3i0 − L i C20 belong to the linear matrix inequality region D(β1 , 1 ) , if there exist matrices Yi , L 2i , positive definite matrices P = diag{P1 , T −1 }, Q 1i , Q 2i , Ri , and slack matrices Mi , Ni (i = 1, 2, . . . , r ) such that these following inequalities hold: Q˙ 1δ ≤ 0, Q˙ 2δ ≤ 0, R˙ δ ≤ 0, ii < 0, i = 1, 2, . . . , r,

(3.38) (3.39)

i j +  ji < 0, 1 ≤ i < j ≤ r, ii < 0, i = 1, 2, . . . , r, i j +  ji < 0, 1 ≤ i < j ≤ r,

(3.40) (3.41) (3.42)

sym($¯ 11,i ) + 2β1 P < 0, i = 1, 2, . . . , r,   − P $¯ 11,i < 0, i = 1, 2, . . . , r, ∗ − P T Ba2i P1 = L 2i C10 , i = 1, 2, . . . , r,

(3.43) (3.44) (3.45)

3.3 Observer Design

65

where ⎡



N

⎤

⎡



N

⎤

¯ N τ D Mi ¯ N τ D Ni ⎢ ⎢ ⎥ ⎥ ⎢ i ⎢ i ⎥ ⎥ l=1 l=1 i j = ⎢ ⎥ , i j = ⎢ ⎥, ⎣ ∗ −N τ D Ri ⎣ ∗ −N τ D Ri ⎦ ⎦ 0 0 ∗ ∗ −2θ P + θ2 R j ∗ ∗ −2θ P + θ2 R j ¯ iT τl 

¯ iT τl 

and ⎡ ⎤ N $¯ 1,i $¯ 12,i . . . $¯ 2,i 0 0 0 $¯ 3,i $¯ 4,i ⎢ ⎥ 0 0 ⎥ ⎢ ∗ $¯ 15,i 0 0 0 0 0 ⎢ ⎥ . ⎢ ⎥ ⎢ ∗ ∗ .. 0 0 0 0 0 0 ⎥ ⎢ ⎥ N ⎢ ∗ ∗ ∗ $¯ 5,i 0 0 0 0 0 ⎥ ⎢ ⎥ ¯ i =⎢ ∗ ∗ ∗ ∗ −Q 0 0  0 0 ⎥ 2i ⎢ ⎥ ⎢ ⎥ . ⎢ ∗ ∗ ∗ ∗ ∗ .. 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ ∗ ∗ ∗ −Q 2i 0 0 ⎥ ⎢ ⎥ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −κI 0 ⎦ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −γk2 I − sym(Mi (E1 − E2 ))− sym(Ni (E2 − E3 )), $¯ 1,i = sym($¯ 11,i ) +

N 

υl Q 1i + N Q 2i + I − κ D˜ 0T D1 D˜ 0 ,

l=1



 P1 A2i0 − Yi C10 0 $¯ 11,i = , I ∈ R m+q1 +q2 , 0 0   P1 A2il − P1 A¯ 20 C1l − Yi C1l 0 ¯ $¯ l2,i = , $3,i = $¯ 31,i − κ D˜ 0T D2 , 0 0     P1 Bh2i P1 B¯ w2i1 $¯ 31,i = , $¯ l5,i = −υl (1 − τ Ml )Q 1i , , $¯ 4,i = 0 T −1 B¯ w22 ¯ i = [$¯ 11,i , $¯ 1 , . . . , $¯ N , 0, . . . , 0, $¯ 31,i , $¯ 4,i ],  2,i

2,i

T T , . . . , M2iT N , M3i1 , . . . , M3iT N , M4iT , M5iT ]T , Mi = [M1iT , M2i1 T T Ni = [N1iT , N2i1 , . . . , N2iT N , N3i1 , . . . , N3iT N , N4iT , N5iT ]T ,

E1 = [N , 0, . . . , 0, 0, . . . , 0, 0, 0], E2 = [0, 1, . . . , 1, 0, . . . , 0, 0, 0], E3 = [0, 0, . . . , 0, 1, . . . , 1, 0, 0]. And, this gain matrix L 1i is obtained by L 1i = P1−1 Yi . Proof According to L 1i = P1−1 Yi , we have     ¯ ¯$11,i = P1 A2i0 − P1 L 1i C10 0 , $¯ l = P1 A2il − P1 A20 C1l − P1 L 1i C1l 0 . 2,i 0 0 0 0

66

3 Fault Estimation and Tolerant Control for Multiple Time Delayed Fuzzy …

Using formula (3.45), formulas (3.23)–(3.29) can be converted to formulas (3.38)– (3.44). Therefore, if formulas (3.38)–(3.44) hold, the k-step error dynamic (3.13) is asymptotically stable under this H∞ performance index γk , (k = 1, 2, . . .) in formula (3.14), and these eigenvalues of A3i0 − L i C20 are part of the linear matrix inequality region D(β1 , 1 ). The proof is completed.

3.4 Design of Nonlinear Dynamic Output Feedback Fault-Tolerant Controller This dynamic output feedback controller for system (3.4) is designed in the following: ⎧ N ⎪  ⎪ ⎪ ⎪ x ˙ (t) = Alcδδ xc (t − τl (t)) + Bcδ (y(t) − Fs fˆsk (t)), c ⎪ ⎪ ⎪ ⎪ l=0 ⎨ N (3.46)  l ∗ ⎪ ˆ ˆ ⎪ u(t) = C x (t − τ (t)) + G (y(t) − F (t)) − B B (t), f f c l c s s aδ a k k ⎪ cδ δ ⎪ ⎪ ⎪ l=0 ⎪ ⎪ ⎩ x (t) =φ(t), ∀ ∈ [−τ , 0], c D where xc (t) ∈ R m represents this controller state and fˆsk (t) is this estimation of l and G c denote controller f s (t) via k-step induction fault estimation. Alcδδ , Bcδ , Ccδ matrices, whose forms are similar to those in (3.4), (l = 0, 1, . . . , N ). φ(t) is the same as the definition in formula (3.4). Bδ∗ can be obtained from formula (3.5). Define ξ(t) = [x T (t), xcT (t)]T and ρ(t) = [w T (t), e Tfs (t), e Tfa (t)]T , where e fsk (t) = k k fˆsk (t) − f s (t) , e fa (t) = fˆak (t) − f a (t). Given formulas (3.4) and (3.46), we have k

⎧ N ⎪  ⎪ ⎨ ξ(t) ˙ = Al (ϑ, ϑ)ξ(t − τl (t)) + Bh (ϑ)h(D0 ξ(t)) + E(ϑ)ρ(t), l=0 ⎪ ⎪ ⎩ yz (t) = C(ϑ)ξ(t),

(3.47)

where    l  Aδl + Bδ G c Cl Bδ Ccδ Bhδ , D0 = Im 0 , , Bh (ϑ) = A (ϑ, ϑ) = 0 Bcδ Cl Alcδδ    Bwδ −Bδ G c Fs −Baδ E(ϑ) = , C(ϑ) = C zδ 0 , l = 0, 1, . . . , N . 0 −Bcδ Fs 0 

l

This control objective of the chapter for this closed-loop fuzzy model (3.47) is to construct the dynamic output feedback controller (3.46) such that

3.4 Design of Nonlinear Dynamic Output Feedback Fault-Tolerant Controller

67

1. Under ρ(t) = 0, this closed-loop model (3.47) can be asymptotically stable; 2. Under ρ(t) = 0, ξ(0) = 0 as well as φ(t) = 0, ∀t ∈ [−τ D , 0], this controlled output obeys the following H∞ performance:

L

L yz (t) dt < γ 2

0

ρ(t) 2 dt,

2

(3.48)

0

where L > 0. γ > 0 represents a specific constant, representing the H∞ performance index. ρ(t) ∈ L 2 [0, ∞). Theorem 3.3 For assigned positive scalars τl , τ D , θ, κ, γ, a non-negative scalar υl , a scalar τ Ml , (l = 1, . . . , N ), this closed-loop model (3.47) can be asymptotically stable under this H∞ performance index γ in formula (3.48), and these eigenvalues of   0 Ai0 + Bi G c C0 Bi Ccj Bci C0 A0ci j are part of this linear matrix inequality region D(β2 , 2 ), if there exist positive definite matrices Pi , Q 1i , Q 2i , Ri , matrices Alci j , Bci , Ccil , G c , and slack matrices Mi , Ni , (l = 0, 1, . . . , N , i, j = 1, 2, . . . , r ), so that the following inequalities are feasible: P˙δ ≤ 0, Q˙ 1δ ≤ 0, Q˙ 2δ ≤ 0, R˙ δ ≤ 0,   (ϑ, ϑ) N τ D Mδ < 0, ∗ −N τ D Rδ   (ϑ, ϑ) N τ D Nδ < 0, ∗ −N τ D Rδ

(3.49) (3.50) (3.51)

sym(A0 (ϑ, ϑ)Pδ−1 ) + 2β2 Pδ−1 < 0,   − 2 Pδ−1 A0 (ϑ, ϑ)Pδ−1 < 0, −1 ∗ − 2 Pδ

(3.52) (3.53)

where ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ (ϑ, ϑ) = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎤ ϒ3 (ϑ)ϒ4 (ϑ) 0 0 ⎥ ⎥ ⎥ ⎥ 0 0 0 0 0 0 ⎥ ⎥ 0 0 0 0 0 ⎥ ϒ5N (ϑ) ⎥ ⎥ 0 0 ⎥ ∗ −Q 2δ 0 0 ⎥ ⎥ .. ⎥ . 0 ∗ ∗ 0 0 ⎥ ⎥ 0 ⎥ ∗ ∗ ∗ −Q 2δ 0 ⎥ ∗ ∗ ∗ ∗ −κI 0 ⎦ 2 ∗ ∗ ∗ ∗ ∗ −γ I

ϒ1 (ϑ, ϑ)ϒ21 (ϑ, ϑ). . .ϒ2N (ϑ, ϑ) ∗ ϒ51 (ϑ) 0 0

+

∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

N 

..

. ∗ ∗

∗ ∗ ∗ ∗

0 0

0 0

0 0

τl  T (ϑ, ϑ)Pδ (2θ Pδ − θ2 Rδ )−1 Pδ (ϑ, ϑ) + T (ϑ) (ϑ)

l=1

− sym(Mδ (E1 − E2 )) − sym(Nδ (E2 − E3 )),

68

3 Fault Estimation and Tolerant Control for Multiple Time Delayed Fuzzy …

and ϒ1 (ϑ, ϑ) = sym(Pδ A0 (ϑ, ϑ)) +

N 

υl Q 1δ + N Q 2δ − κD0T D1 D0 ,

l=1

ϒ2l (ϑ, ϑ) = Pδ Al (ϑ, ϑ), ϒ3 (ϑ) = Pδ Bh (ϑ) − κD0T D2 , ϒ4 (ϑ) = Pδ E(ϑ), ϒ5l (ϑ) = −υl (1 − τ Ml )Q 1δ , (ϑ, ϑ) = [A0 (ϑ, ϑ), A1 (ϑ, ϑ), . . . , A N (ϑ, ϑ), 0, . . . , 0, Bh (ϑ), E(ϑ)], (ϑ) = [C(ϑ), 0, . . . , 0, 0, . . . , 0, 0, 0], E1 = [N , 0, . . . , 0, 0, . . . , 0, 0, 0], E2 = [0, 1, . . . , 1, 0, . . . , 0, 0, 0], E3 = [0, 0, . . . , 0, 1, . . . , 1, 0, 0]. Proof A fuzzy Lyapunov-Krasovskii functional is constructed as follows: V (t) = V1 (t) + V2 (t) + V3 (t),

(3.54)

where V1 (t) =ξ T (t)Pδ ξ(t), 

t

t N   T T υl ξ (s)Q 1δ ξ(s)ds + ξ (s)Q 2δ ξ(s)ds , V2 (t) = l=1

V3 (t) =

t−τl (t)

N  0 t  l=1

t−τl

 ˙ ξ˙T (s)Rδ ξ(s)dsd .

−τl t+

According to this trajectory of this closed-loop model (3.47), V˙ (t) follows: V˙1 (t) = ξ T (t)[sym(Pδ A0 (ϑ, ϑ))]ξ(t) +

N 

2ξ T (t)Pδ Al (ϑ, ϑ)ξ(t − τl (t))

l=1 T

+ 2ξ (t)Pδ Bh (ϑ)h(D0 ξ(t)) + 2ξ (t)Pδ E(ϑ)ρ(t) + ξ T (t) P˙δ ξ(t), T

V˙2 (t) =

N  [ξ T (t)(υl Q 1δ + Q 2δ )ξ(t) − υl (1 − τ˙l (t))ξ T (t − τl (t))Q 1δ ξ(t − τl (t)) l=1

t − ξ (t − τl )Q 2δ ξ(t − τl ) + υl T

t−τl (t)

t + t−τl

ξ T (s) Q˙ 2δ ξ(s)ds],

ξ T (s) Q˙ 1δ ξ(s)ds

3.4 Design of Nonlinear Dynamic Output Feedback Fault-Tolerant Controller

69

t

0 t N  T T ˙ ˙ ˙ ˙ ˙ [τl ξ (t)Rδ ξ(t)− ξ (s)Rδ ξ(s)ds + ξ˙T (s) R˙ δ ξ(s)dsd ]. V˙3 (t) = l=1

t−τl

−τl t+

According to the Newton-Leibniz formula, Lemmas 1.1–1.3 and Iwasaki and Hara (2005), Ahmadizadeh et al. (2014), a straightforward calculation method can be obtained as follows: V˙ (t) + yzT (t)yz (t) − γ 2 ρT (t)ρ(t) =V˙ (t) + yzT (t)yz (t)−γ 2 ρT (t)ρ(t) ⎡ ⎤ t

N ⎢ ⎥ ˙ −2ιT (t)Mδ ξ(s)ds ⎣ξ(t)−ξ(t − τl (t))− ⎦ l=1

− 2ιT (t)Nδ

N 

t−τl (t)

⎡

⎣ξ(t − τl (t)) − ξ(t − τl ) −

l=1

≤

N  l=1

1 N τl

t



t−τl (t)

t−τ

l (t)

⎤

˙ ⎦ ξ(s)ds

t−τl

ι(t) ˙ ξ(s)

T 

(ϑ, ϑ) N τ D Mδ ∗ −N τ D Rδ



 ι(t) ds ˙ ξ(s)

t−τ  T  

l (t) N  1 ι(t) ι(t) (ϑ, ϑ) N τ D Nδ + ds, ˙ ˙ ∗ −N τ D Rδ ξ(s) ξ(s) N τl l=1

(3.55)

t−τl

where ιT (t) =[ξ T (t), ξ T (t − τ1 (t)), . . . , ξ T (t − τ N (t)), ξ T (t − τ1 ), . . . , ξ T (t − τ N ), h T (D0 ξ(t)), ρT (t)]. In addition, according to formulas (3.49)–(3.51), the following inequality can be acquired: V˙ (t) + yzT (t)yz (t) − γ 2 ρT (t)ρ(t) < 0.

(3.56)

If φ(t) = 0, namely, x(t) = 0, we have xc (t) = 0 and ξ(t) = 0, ∀t ∈ [−τ D , 0]. So, if ξ(0) = 0, you can find that V (t)|t=0 = 0 and V (t)|t=L ≥ 0. In light of V (t)|t=L ≥ 0 as well as formula (3.56), it yields that

L ( yz (t) 2 − γ 2 ρ(t) 2 )dt + V (t)|t=L − V (t)|t=0 ≤ 0, 0

70

3 Fault Estimation and Tolerant Control for Multiple Time Delayed Fuzzy …

which suggests that formula (3.48) is satisfied. Therefore, this closed-loop model (3.47) can be asymptotically stable under the H∞ performance index γ. ¯ and P in Lemma 1.5 Next we discuss this regional pole constraint. Choose X as A0 (ϑ, ϑ) and Pδ−1 , separately. When formulas (3.52) and (3.53) hold, then these eigenvalues of A0 (ϑ, ϑ) are part of this linear matrix inequality region D(β2 , 2 ). This completes the proof. Theorem 3.4 For given positive scalars τl , τ D , θ, κ, γ, a non-negative scalar υl , and a scalar τ Ml , (l = 1, . . . , N ), the closed-loop fuzzy system (3.47) is asymptotically stable under this H∞ performance index γ in formula (3.48), and the eigenvalues of 

0 Ai0 + Bi G c C0 Bi Ccj 0 Bci C0 Aci j



belong to the linear matrix inequality region D(β2 , 2 ), if there are positive definite matrices Y , Q¯ 1i , Q¯ 2i , R¯ i , matrices S, T , A¯ lci j , B¯ ci , C¯ cil , G¯ c , slack matrices M¯ i , N¯ i , (l = 0, 1, . . . , N , i, j = 1, 2, . . . , r ), so that these following inequalities are feasible: Q˙¯ 1δ ≤ 0, Q˙¯ 2δ ≤ 0, R˙¯ δ ≤ 0,

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

(3.57) (3.58)

i jt + jit < 0, 1 ≤ i < j ≤ r, t = 1, 2, . . . , r, iit < 0, i, t = 1, 2, . . . , r,

(3.59) (3.60)

i jt +  jit < 0, 1 ≤ i < j ≤ r, t = 1, 2, . . . , r, 1ii < 0, i = 1, 2, . . . , r, 1i j + 1 ji < 0, 1 ≤ i < j ≤ r,

(3.61) (3.62) (3.63)

2ii < 0, i = 1, 2, . . . , r, 2i j + 2 ji < 0, 1 ≤ i < j ≤ r,

(3.64) (3.65)

where 

⎡

i jt

i jt

¯ ij ⎢ ⎢ ⎢ =⎢ ∗ ⎢ ⎣ ∗ ∗ ⎡ ¯ ij ⎢ ⎢ ⎢ =⎢ ∗ ⎢ ⎣ ∗ ∗

N τ D M¯ i −N τ D R¯ i ∗ ∗ N τ D N¯ i −N τ D R¯ i ∗ ∗

⎤ T ¯ i ⎥ ⎥ l=1 ⎥ , 0 0 ⎥ ⎥ 2 ¯ −2θψ + θ Rt 0 ⎦ ∗ −I  ⎤ r ¯ iTj ¯ iT ⎥ τl  ⎥ l=1 ⎥ , 0 0 ⎥ ⎥ 2 ¯ ⎦ −2θψ + θ Rt 0 ∗ −I r

¯ iTj τl 

3.4 Design of Nonlinear Dynamic Output Feedback Fault-Tolerant Controller

71

 − 2 ψ ϒ¯ 11,i j , ∗ − 2 ψ ⎡¯ ⎤ 1 ¯N ϒ1,i j ϒ¯ 2,i 0 0 0 ϒ¯ 3,i ϒ¯ 4,i j · · · ϒ2,i j 1 ⎢ ∗ ϒ¯ 5,i 0 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ . ⎢ ∗ ∗ .. 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ ϒ¯ N 0 0 0 0 0 ⎥ 5,i ⎢ ⎥ ¯ i j = ⎢ ∗ ∗ ∗ ∗ − Q¯ 2i 0 0  0 0 ⎥ ⎢ ⎥ ⎢ ⎥ .. ⎢ ∗ ∗ ∗ ∗ . ∗ 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ ∗ 0 ⎥ ∗ ∗ − Q¯ 2i 0 ⎢ ⎥ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −κI 0 ⎦

1i j = sym(ϒ¯ 11,i j ) + 2β2 ψ, 2i j =

∗

∗

∗

∗

∗

∗



∗

∗ −γ 2 I

− sym( M¯ i (E1 − E2 )) − sym( N¯ i (E2 − E3 )), ϒ¯ 1,i j = sym(ϒ¯ 11,i j ) +

N 

υl Q¯ 1i + N Q¯ 2i − κ D¯ 0T D1 D¯ 0 ,

l=1

 0 Ai0 + Bi G¯ c C0 Ai0 X + Bi C¯ cj , D¯ 0 = [X, I ], ϒ¯ 11,i j = Y Ai0 + B¯ ci C0 A¯ 0ci j   l Ail + Bi G¯ c Cl Ail X + Bi C¯ cj l ¯ ϒ2,i j = , ϒ¯ 3,i = ϒ¯ 31,i − κ D¯ 0T D2 , Y Ail + B¯ ci Cl A¯ lci j     Bhi Bwi −Bi G¯ c Fs −Bai ϒ¯ 31,i = , ϒ¯ 4,i = , Y Bhi Y Bwi − B¯ ci Fs −Y Bai   X I l ϒ¯ 5,i = −υl (1 − τ Ml ) Q¯ 1i , ψ = > 0, ∗ Y 1 ¯N ¯ ¯ ¯ i j = [ϒ¯ 11,i j , ϒ¯ 2,i  j , . . . , ϒ2,i j , 0, . . . , 0, ϒ31,i , ϒ4,i ], 

¯ i = [ϒ¯ 6,i , 0, . . . , 0, 0, . . . , 0, 0, 0], ϒ¯ 6,i = [C zi X C zi ], T T , . . . , M¯ 2iT N , M¯ 3i1 , . . . , M¯ 3iT N , M¯ 4iT , M¯ 5iT ]T , M¯ i = [ M¯ 1iT , M¯ 2i1 T T , . . . , N¯ 2iT N , N3i1 , . . . , N¯ 3iT N , N¯ 4iT , N¯ 5iT ]T , N¯ i = [ N¯ 1iT , N¯ 2i1 E1 = [N , 0, . . . , 0, 0, . . . , 0, 0, 0], E2 = [0, 1, . . . , 1, 0, . . . , 0, 0, 0], E3 = [0, 0, . . . , 0, 1, . . . , 1, 0, 0].

In addition, the following controller gain matrices are obtained: ⎧ G c =G¯ c , ⎪ ⎪ ⎪ ⎪ ⎨ C l =(C¯ l − G¯ c Cl X )V −T , ci ci (3.66) −1 ¯ ⎪ Bci =U ( Bci − Y Bi G¯ c ), ⎪ ⎪ ⎪ ⎩ l l Aci j =U −1 [ A¯ lci j − Y Ail X − B¯ ci Cl X − Y Bi (C¯ cj − G¯ c Cl X )]V −T , where U and V satisfy V U T = I − X Y , (l = 0, 1, . . . , N ).

72

3 Fault Estimation and Tolerant Control for Multiple Time Delayed Fuzzy …

Proof Given Lemma 1.4 and formulas (3.57)–(3.61), we have ¯ iit < 0, i, t = 1, 2, . . . , r,

¯ i jt +

¯ jit < 0, 1 ≤ i < j ≤ r, t = 1, 2, . . . , r,

¯ iit < 0, i, t = 1, 2, . . . , r, 

(3.68)

¯ i jt +  ¯ jit < 0, 1 ≤ i < j ≤ r, t = 1, 2, . . . , r, 

(3.70)

(3.67) (3.69)

where  ¯ i jt =

ˆ¯ ¯  i jt N τ D Mi ∗ −N τ D R¯ i

ˆ¯ =  ¯ ij +  i jt

N 



 ¯ i jt = , 

ˆ¯ ¯  i jt N τ D Ni ∗ −N τ D R¯ i

 ,

¯ iTj (2θψ − θ2 R¯ t )−1  ¯ ij + ¯ i. ¯ iT τl 

l=1

Thus, if formulas (3.67)–(3.70) hold, the inequalities are feasible as follows: r 

δi

r 

δj

r 

¯ i jt < 0, δt 

i=1

j=1

t=1

r 

r 

r 

δi

i=1

j=1

δj

¯ i jt < 0. δt 

(3.71)

(3.72)

t=1

Using a method similar to that in Tao et al. (2015), the partitioned symmetric matrices P and P −1 are expressed as  P=

   X V Y U , P −1 = . ∗ W2 ∗ W1

Given P P −1 = I , one has         I X I Y X I = . P = , P 0 VT 0 UT VT 0

(3.73)

(3.74)

Define  J1 =

   X I I Y . , J = 2 VT 0 0 UT

(3.75)

Given formulas (3.73)–(3.75), we know that P J1 = J2 . Apply the congruence transformation diag{J1−T , J1−T , . . . , J1−T , J1−T , . . . , J1−T , I, I, J1−T } to formulas (3.71) and (3.72), and then inequalities (3.49)–(3.51) can be obtained.

3.4 Design of Nonlinear Dynamic Output Feedback Fault-Tolerant Controller

73

In short, this closed-loop model (3.47) can be asymptotically stable and obeys the H∞ performance index γ. Next we discuss the regional pole constraint issue. Pre and post multiplying (3.62)–(3.63) by J1−T and its transpose, respectively, and (3.64)–(3.65) by diag{J1−T , J1−T } and its transpose, respectively, the following equations can be obtained: sym(ϒˆ¯ 11,i j P −1 ) + 2β2 P −1 < 0,   − 2 P −1 ϒˆ¯ 11,i j P −1 < 0, ∗ − 2 P −1

(3.76) (3.77)

where ϒˆ¯ 11,i j =



 0 Ai0 + Bi G c C0 Bi Ccj . Bci C0 A0ci j

Given equations (3.52)–(3.53), if equations (3.76) and (3.77) hold, that is, if equations (3.62)–(3.65) are feasible, these eigenvalues of ϒˆ¯ 11,i j are part of this linear matrix inequality region D(β2 , 2 ). Thus, the proof is completed. Remark 3.2 The setting of υl (l = 1, . . . , N ) means that Theorems 3.1–3.4 and Corollary 3.1 are applicable for two cases where τl (t) is constant or time-varying. Under the case of constant delays, υl = 1 and τ Ml = 0. Under the case of time-varying delays, if τ Ml is known and τ Ml < 1 then υl = 1; if τ Ml is unknown or τ Ml ≥ 1 or τl (t) is non-differentiable, then υl = 0. Remark 3.3 A fault estimation observer and a fault-tolerant controller including the H∞ performance index and regional pole constraint are explored simultaneously for the multiple-delayed T-S fuzzy model with sensor and actuator faults, which can perform fast adaptive fault estimation and fault-tolerant control to sensor and actuator faults simultaneously, be suitable for more practical situations and have a wider application scope. In contrast to these existing integral observers such as Han et al. (2016), the k-step induction fault estimation observer can reduce the influence of the input disturbance caused by the actuator fault derivative and obtain better estimation results. Different from the existing multiple-integral observers in the k-step case such as Huang and Yang (2014), the observer proposed in this chapter performs fault estimation under sensor faults, actuator faults, and multiple time delays simultaneously. Furthermore, in order to stabilize the multiple-delayed T-S fuzzy system, an observer-based dynamic output feedback controller containing the sensor fault compensator and the actuator fault compensator is designed. The switching ideology is used to make the derivative of the Lyapunov-Krasovskii functional negative. Unlike ordinary Lyapunov functions such as Huang and Yang (2014), the integrand of the Lyapunov-Krasovskii functional is not only relevant to the integral variables, but also related to the membership function. Using the fuzzy Lyapunov-Krasovskii functional, we can easily obtain delay-dependent results, and this conservatism of

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3 Fault Estimation and Tolerant Control for Multiple Time Delayed Fuzzy …

the maximum delay bounds is decreased. For the disturbance effect affected via this bounded known nonlinear function, the traditional control method must be attenuated through the H∞ performance index, but this chapter uses the fault-tolerant control method to address it. Remark 3.4 In Theorems 3.1–3.4 and Corollary 3.1, by introducing slack matrices, an enhanced integral inequality approach without neglecting every integral term is proposed, which converts the integral inequalities into matrix inequalities and decreases this conservatism theoretically (Huang and Yang 2014). The matrix inequalities in Theorems 3.1 and 3.3 are bilinear instead of linear. Transforming bilinear matrix inequalities into linear matrix inequalities is a hard problem. Using Schur complement and congruence transformation matrices in Theorems 3.2 and 3.4 and Corollary 3.1, this form of linear matrix inequalities is obtained, and gain matrices of this observer and this controller are acquired. In addition, the application of the fuzzy Lyapunov-Krasovskii functional also greatly decreases this conservatism theoretically (Wang and Lam 2018a). Remark 3.5 It can be denoted that Ri = R, (i = 1, 2, . . . , r ) in Theorems 3.1–3.4 and Corollary 3.1 to reduce the number of linear matrix inequalities.

3.5 Simulation Results In this part, the effectiveness of the method is verified by two simulation examples. Example 3.1 The continuous-time system (3.4) in the case of r = 2, N = 2 has the following data: ⎡

⎤ ⎡ ⎤ ⎡ ⎤ 0.6 0 0.1 0.7 0 0.01 0.12 0 0.01 A10 = ⎣ −0.6 0 0 ⎦ , A20 = ⎣ −0.7 0 0 ⎦ , A11 = ⎣ −0.1 0 0 ⎦ , 0.6 −5 0 0.8 −6 0 0.25 0 0 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0.08 0 0.01 0.11 0 0 0.07 0 0.01 A21 = ⎣ −0.1 0 0 ⎦ , A12 = ⎣ −0.13 0 0 ⎦ , A22 = ⎣ −0.12 0 0 ⎦ , 0.35 0 0 0.28 0 0 0.37 0 0 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −1.5 −1.8 −2 −3 B1 = ⎣ 0 ⎦ , B2 = ⎣ 0 ⎦ , Ba1 = ⎣ 0 ⎦ , Ba2 = ⎣ 0 ⎦ , 0 0 0 0 ⎡ ⎤ ⎡ ⎤ 0.01 0 0.01 0.05 0.01 0  Bh1 = ⎣ 0.01 0.01 0 ⎦ , Bh2 = ⎣ 0.03 0.01 0.02 ⎦ , Bw1 = 0.01 0 0 , 0.01 0.04 0.02 0 −0.01 0.02      0.7 0 0 0.3 0 0 Bw2 = 0 0.01 0.01 , C0 = , C1 = , 0 0.1 0.8 0 0 0.2       0.1 0 0 0.5 , Fs = , C z1 = 1 1 0 , C z2 = 0 0 2 . C2 = 0 0 0.2 0

3.5 Simulation Results

75

Suppose this sensor fault is a constant function (Sun et al. 2018) and this actuator fault is a trigonometric function (Zadeh 1973; Huang and Yang 2014) below:  f s (t) =  f a (t) =

0, 9,

0s ≤ t < 3s, 3s ≤ t < 20s,

0,

0s ≤ t < 2s,

6 + 0.2 cos(0.6t), 2s ≤ t < 20s.

This nonlinear dynamic h(x(t)) can be expressed as ⎡

⎤ 0.022x1 (t) sin(x1 (t)) cos(x2 (t)) − 0.011(x1 (t) − x2 (t) + x3 (t)) ⎦, 0.011(x3 (t) − x1 (t)) h(x(t)) = ⎣ −0.011(x2 (t) + x3 (t)) which is bounded by the following matrices: ⎡

⎤ ⎡ ⎤ 0.011 0.011 −0.011 −0.011 0.011 −0.011 R1 = ⎣ 0.011 0.022 0.044 ⎦ , R2 = ⎣ −0.033 −0.022 −0.022 ⎦ . −0.022 0.011 0.022 0.022 −0.033 −0.044 The assumption rule assumes that: ⎧ 1 ⎨ δ1 = , 1 + exp(−x2 (t)) ⎩ δ2 =1 − δ1 . Assuming τ1 (t) = 0.5 + 0.2 cos(t), τ2 (t) = 0.2, we can know that τ1 = 0.7, τ2 = 0.2, τ M1 = 0.2, τ M2 = 0, τ D = 0.7. Set G = 3I2 , θ = 64, T = 1, υ1 = υ2 = 1. Select the step number k = 1, 2, calculate the matrices inequalities in Theorems 3.2 and 3.4, respectively, and these gain matrices of this observer and this controller corresponding to H1 and H2 , respectively, can be obtained by using Lemma 1.3. In order to differentiate between this p-dimensional signal and this k-step estimation easily, where p represents this signal dimension, k = 1, 2, 3, . . . represents this k-step induction fault estimation. This symbol ( p, t) means this p-dimension signal at time t in these simulations. For example, ex2 (1, t) represents the error between the estimation of x1 (t) and x1 (t) acquired via the 2-step induction fault estimation method. A comparative experiment is carried out under the 1-step induction fault estimation observer and the 2-step induction fault estimation observer to show these merits of this k-step induction fault estimation in this chapter. These simulation results are presented in Figs. 3.1, 3.2, 3.3 and 3.4. Figure 3.1 gives this sensor fault f s (t) and the corresponding estimated values fˆs1 (t) and fˆs2 (t), which shows that the estimation of f s (t) obtained by the 2-step induction fault estimation method is closer to f s (t).

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3 Fault Estimation and Tolerant Control for Multiple Time Delayed Fuzzy …

Fig. 3.1 The sensor fault f s (t) and its estimations

15

Sensor FEs

10 5 9 0

8.9 18.5 19

-5 -10 -15 0

Fig. 3.2 The actuator fault f a (t) and its estimations

5

10 Times

15

fs (t) fˆs1 (t) fˆs2 (t) 20

Actuator FEs

10 5 9 0

8.9 18.5 19

fa (t) fˆa1 (t) fˆa2 (t)

-5 0

5

10 Times

15

20

This actuator fault f a (t) and the corresponding estimations fˆa1 (t) and fˆa2 (t) are given in Fig. 3.2, which shows that the estimated value of f a (t) acquired via the 2-step induction fault estimation method is closer to f a (t). Figures 3.3 and 3.4 show the state errors and output errors obtained by the 1-step induction fault estimation method and the 2-step induction fault estimation method, which shows that this state error and this output error acquired via the two-step induction fault estimation method are closer to zero. Example 3.2 Research a truck trailer system (Cao and Frank, 2001) in the following: ⎧ v t¯ v t¯ ⎪ ⎪ x˙1 (t) = − x1 (t) + u(t), ⎪ ⎪ ⎪ Lt0 lt0 ⎪ ⎪ ⎨ v t¯ x1 (t), x˙ (t) = ⎪ 2 Lt0 ⎪ ⎪ ⎪ ⎪ v t¯ v t¯ ⎪ ⎪ ⎩ x˙3 (t) = x1 (t)], sin[x2 (t) + t0 2L where l represents the length of the truck, l = 2.8[m]. L indicates this length of the trailer, L = 5.5[m]. v represents the constant speed backing up, v = −1.0[m/s]. t¯ represents this sampling time, t¯ = 2.0[s], and t0 = 0.5. x1 (t) represents the angle difference between the truck and the trailer and the unit is [rad]. x2 (t) represents the trailer angle and the unit is [rad]. x3 (t) represents the vertical position of the rear

3.5 Simulation Results

77

State errors

Fig. 3.3 The state errors

0.04 0.02 0 18.519

0

ex1 (1, t) ex2 (1, t)

-10

State errors

0 1 0

5

0.5 0 -0.5 -1

10 15 ×10-3 Times 1 0 -1 18.519

20

ex1 (3, t) ex2 (3, t) 5

Fig. 3.4 The output errors Output errors

20

ex1 (2, t) ex2 (2, t)

0

10 Times

15

20

×10-3 0 -1 -2 -3 18.5 19

0.5 0

ey1 (1, t) ey2 (1, t)

-0.5 0

5

0.5 Output errors

10 -4 15 Times 4 ×10 0 -4 18.519

-1 0

State errors

5

0

10 15 Times -3 ×10 1 0.5 0 18.5 19

20

ey1 (2, t) ey2 (2, t)

-0.5 0

5

10 Times

15

20

end of the trailer and the unit is [m]. u(t) represents the steering angle and the unit is [rad]. Suppose this system state x1 (t) is perturbed by multiple time delays. At the same time, there are nonlinear dynamics, actuator faults, sensor faults, as well as external disturbances in the model. Assuming that a fuzzy model is used with fuzzy rules, the parameter matrices can be obtained as follows:

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3 Fault Estimation and Tolerant Control for Multiple Time Delayed Fuzzy …

⎤ ⎤ ⎤ ⎡ ⎡ v t¯ v t¯ v t¯ −a0 Lt −a0 Lt −a1 Lt 0 0 0 0 00 0 0 0 ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ v t¯ v t¯ v t¯ 0 0 ⎦ , A20 = ⎣ a0 Lt 0 0 ⎦ , A11 = ⎣ a1 Lt 0 0⎦, A10 = ⎣ a0 Lt 0 0 0 2 ¯2 2 2 2 2 v t v t¯ v t¯ t¯ ev t¯ a0 2Lt a0 ev a1 2Lt 0 0 00 t0 2Lt0 t0 0 0 ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ ¯ ¯ ¯ vt vt vt −a1 Lt −a −a 0 0 0 0 0 0 2 2 Lt Lt 0 0 0 ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ v t¯ v t¯ v t¯ 0 0 ⎦ , A12 = ⎣ a2 Lt 0 0 ⎦ , A22 = ⎣ a2 Lt 0 0⎦, A21 = ⎣ a1 Lt 0 0 0 2 ¯2 2 ¯2 2 ¯2 v t t t a1 ev a2 2Lt a2 ev 00 00 00 2Lt0 2Lt0 0 ⎡ vt¯ ⎤ ⎡ ⎤ 0.01 −0.01 0.03 lt0 B1 = B2 = ⎣ 0 ⎦ , Ba1 = B1 , Ba2 = B2 , Bh1 = ⎣ 0 0.02 0.01 ⎦ , 0.01 0 0.03 0 ⎡ ⎤ 0.03 0.02 0   Bh2 = ⎣ 0.01 0.04 0.01 ⎦ , Bw1 = 0 0.001 0 , Bw2 = 0.001 0 0 , 0 −0.02 0.03         0.8 0 0 0.1 0 0 0.1 0 0 1 C0 = , C1 = , C2 = , Fs = , 0 0 0.8 0 0 0.1 0 0 0.1 0   10t0 C z1 = 1 0 1 , C z2 = 0 1 1 , and a0 = 0.8, a1 = 0.1, a2 = 0.1, e = . π ⎡

This resource of this message about sensor faults and actuator faults could be found in references Huang andYang (2014), Sun et al. (2018), Schulte and Pöschke (2016). Sensor faults may be trigonometric functions (Zhang et al. 2017; Han et al. 2016b), and actuator faults may also be trigonometric functions (Zadeh 1973; Huang and Yang 2014). Suppose sensor and actuator faults in the example are:  f s (t) =  f a (t) =

0, 1 + cos t,

0s ≤ t < 5s, 5s ≤ t < 20s,

0, 0s ≤ t < 4s, 4 + 0.5 sin(0.3t), 4s ≤ t < 20s.

This nonlinear function h(x(t)) can be shown ⎡

⎤ 0.01x1 (t) sin2 (x1 (t)) − 0.005(x1 (t) − x2 (t) + x3 (t)) ⎦, −0.005(x1 (t) − x3 (t)) h(x(t)) = ⎣ −0.005(x2 (t) + x3 (t)) which is bounded by the following matrices: ⎡

⎤ ⎡ ⎤ −0.005 0.005 −0.005 0.005 0.005 −0.005 R1 = ⎣ −0.015 −0.01 −0.01 ⎦ , R2 = ⎣ 0.005 0.01 0.01 ⎦ . 0.01 −0.015 −0.02 −0.01 0.005 0.01 Suppose these fuzzy rules are shown:

3.5 Simulation Results

79

Fig. 3.5 The sensor fault f s (t) and its estimations

2 1.95

Sensor FEs

4

1.9 18.5

2 0

fs (t) fˆs2 (t) got by this chapter fˆs (t) got by the contrast observer

-2 0

Fig. 3.6 The actuator fault f a (t) and its estimations

19

5

10 Times

15

20

Actuator FEs

6 4 2

fa (t) fˆa2 (t) got by this chapter fˆa (t) got by the contrast observer

0 0

5

10 Times

15

20

⎧    1 1 ⎪ ⎨δ = 1− , 1 1 + exp(−3(ϑ(t) − 0.5π)) 1 + exp(−3(ϑ(t) + 0.5π)) ⎪ ⎩ δ =1 − δ . 2 1 Assuming τ1 (t) = 0.2 + 0.1 sin(t), τ2 (t) = 0.4 + 0.1 sin(t), we can know that τ1 = 0.3, τ2 = 0.5, τ M1 = 0.1, τ M2 = 0.1, τ D = 0.5. Set G = 2I2 , θ = 64, T = 1, υ1 = υ2 = 1. Choose the step number k = 2, calculate matrix inequalities in Theorems 3.2 and 3.4, respectively, and then these gain matrices of the observer and controller corresponding to H1 and H2 can be obtained separately by Lemma 1.3. Figures 3.5, 3.6, 3.7 and 3.8 show these simulation results. Figure 3.5 shows this sensor fault f s (t) and the corresponding estimated value fˆs2 (t), indicating that when t > 7s, the 2-step induction fault estimated value of f s (t) is very close to f s (t). Figure 3.6 gives this actuator fault f a (t) and the corresponding estimated value fˆa2 (t), showing that when t > 8s, the estimated value of f a (t) acquired via the 2-step induction fault estimation method is very close to the value of f a (t). Figures 3.7 and 3.8 show this state error and this output error acquired via the 2-step induction fault estimation method. It can be seen that the state error and output error are close to zero.

80

3 Fault Estimation and Tolerant Control for Multiple Time Delayed Fuzzy …

State errors

Fig. 3.7 The state errors

0.04 0.02 0 18.5 19

0

ex2 (1, t) got by this chapter ex (1, t) got by the contrast observer

-2

State errors

0 0.1

10 ×10-3 15 Times 2 0 -2

ex2 (2, t) got by this chapter ex (2, t) got by the contrast observer

-0.2 5

0

10 ×10-3 15 Times 2 0 -2 18.5 19

5

10 Times

Fig. 3.8 The output errors Output errors

20

ex2 (3, t) got by this chapter ex (3, t) got by the contrast observer

-0.1 0

15

20

×10-4 4 2 0 18.5 19

0.4 0

ey2 (1, t) got by this chapter ey (1, t) got by the contrast observer

-0.4 0

Output errors

20

18.519

0 -0.1

0

State errors

5

5

10 Times

0.05 0 -0.05

15

20

×10-3 2 1 0 18.5 19

ey2 (2, t) got by this chapter ey (2, t) got by the contrast observer

-0.1 0

5

10 Times

15

20

A comparative test is carried out below. The contrast observer can be obtained from Han et al. (2016), which is one of the traditional observers. By using the contrast observer, the estimated values of the sensor fault, the actuator fault, the state error, and the output error are represented as fˆs (t), fˆa (t), ex (t) and e y (t), respectively. The simulation results of the comparative observer are shown in Figs. 3.5, 3.6, 3.7, and 3.8. Figure 3.5 depicts this sensor fault f s (t) and the corresponding estimated value fˆs (t) obtained by Han et al. (2016), which obviously means that the estimated value of f s (t) in the chapter is closer to f s (t). Figure 3.6 depicts this actuator fault f a (t) and the corresponding estimated value fˆa (t) obtained from Han et al. (2016),

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81

which evidently means that this estimated value of f a (t) in the chapter is closer to f a (t). Figures 3.7 and 3.8 describe this state error ex (t) and this output error e y (t) obtained from Han et al. (2016). It can be seen that the state error and the output error obtained from this chapter are smaller than those obtained from Han et al. (2016). To sum up, the comparison shows that this method proposed in the chapter weakens this influence of input disturbance affected by f˙a (t). Thus, this feasibility and effectiveness of this approach in the chapter are demonstrated more strongly.

3.6 Chapter Summary In the chapter, fault estimation and tolerant control of T-S fuzzy systems against multiple time-varying delays and sensor and actuator faults are studied. There are nonlinear functions and external disturbances in this studied system. Based on this message of this (k − 1)-step induction fault estimation, the new observer is proposed to construct this k-step error dynamic. The k-step induction fault estimation observer can attenuate this influence of the input disturbance affected from this actuator fault derivative and can obtain better fault estimation performance in the presence of sensor and actuator faults and multiple time delays. In contrast to existing achievements, this observer can more authentically show the size and shape of sensor faults and actuator faults. Besides, in light of this online message of the k-step induction fault estimation, the active dynamic output feedback fault-tolerant controller is designed to stabilize the closed-loop fuzzy model. Furthermore, by using the fuzzy Lyapunov-Krasovskii functional and introducing a number of slack matrices, these delay-dependent sufficient conditions for the existence of observers and fault-tolerant controllers are given in the form of a set of linear matrix inequalities with less conservatism. Finally, this superiority and effectiveness of the method in this chapter are verified by two simulation examples.

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Chapter 4

Fault Estimation and Tolerant Control for Multiple Time Delayed Switched Fuzzy Stochastic Systems with Sensor Faults and Intermittent Actuator Faults

4.1 Introduction In Chaps. 2 and 3, we consider the design of multiple-integral observers and faulttolerant controllers for time delayed T-S fuzzy systems with only actuator faults and both actuator and sensor faults, respectively. However, it is worth pointing out that the previous two chapters did not consider the case of intermittent actuator fault. Constant value faults and slow time-varying faults are the commonest and simplest faults, and time-varying faults are also usually seen. Currently, there are many related studies (Ren et al. 2018; Guo and Chen 2019; Li and Yang 2016). In addition, minor faults and intermittent faults are unavoidable. However, the correlative research is relatively insufficient and deserves further research (Sun et al. 2018). Many scholars have put a lot of energy into the study of permanent faults and have achieved fruitful results (Han et al. 2018b; Chibani et al. 2017). Different from the widely studied fault estimation and fault-tolerant control of permanent faults (Gao 2015; Mao et al. 2011), the related research on intermittent faults is insufficient, and the fault estimation and fault-tolerant control of intermittent faults urgently need to be studied. It is worth noting that this sliding mode observer is one of the most popular observers that can be used for fault estimation. It is very robust to uncertainties and nonlinearities in this model and is more robust than the Lomborg observer. The observer is more stable and efficient (Zhang et al. 2019, 2020). It is particularly worth pointing out that the sliding mode observer can perform fault estimation, fault reconstruction, and fault-tolerant control very well and quickly. In addition, the sliding mode observer is very suitable for the estimation of intermittent faults due to its randomness, intermittence, and repeatability. This chapter takes fault estimation and fault-tolerant control as the research object, especially the fault estimation and fault-tolerant control of intermittent faults, and conducts an in-depth study of a sliding mode observer. At the same time, this chapter removes the restrictive assumption that these local input matrices are exactly the same, which is needed in many studies, such as Zhang et al. (2010), Han et al. (2012). © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Sun et al., Fault-Tolerant Control for Time-Varying Delayed T-S Fuzzy Systems, Intelligent Control and Learning Systems 9, https://doi.org/10.1007/978-981-99-1357-2_4

85

86

4 Fault Estimation and Tolerant Control for Multiple Time Delayed Switched …

Through the previous three chapters, we know that the establishment of the T-S fuzzy model is a valid method to handle various complex nonlinear model problems (Zhang et al. 2015; Vatankhah and Asemani 2017; Shen et al. 2020a, b; Sakthivel et al. 2018; Shen et al. 2018). For example, in Huang and Yang (2014), robust fault analysis and robust control of time-varying delayed T-S fuzzy systems involving actuator faults were considered. Nevertheless, sensor faults are not considered. Han et al. (2018b) studied robust fault estimation and tolerant control of a series of switched TS fuzzy stochastic systems with actuator faults and sensor faults. However, the case of intermittent faults is not considered. To sum up, the T-S fuzzy model deserves further study, so the chapter studies the T-S fuzzy system with intermittent faults. On the other hand, switched systems are also common when modeling a large number of practical processes due to frequent mutations in the environment (Gan et al. 2019; Li et al. 2019; Tong et al. 2016; Xing et al. 2019a, b). As far as we know, the switched T-S fuzzy system is a very common and widely applied system that solves many practical problems (Han et al. 2018a; Ren et al. 2019). Switched T-S fuzzy systems are not sufficiently studied, so related research has received increasing attention. Ren et al. (2019) focused on the exponential H∞ synchronization of switched fuzzy models against time-varying delays and impulses. In Li et al. (2019), this problem of adaptive output feedback fuzzy stabilization of sampled data for nonlinear switched systems with asynchronous switching was studied. Tong et al. (2016) aimed at the dead zone problem of switched uncertain nonlinear large-scale systems, and an observer-based adaptive fuzzy decentralized tracking control was studied. Han et al. (2018a) studied this issue of fault detection for switched T-S fuzzy models against unknown inputs, but ignored the random phenomena that are prevalent in real industrial systems. A stochastic system with Brownian motion, also known as a Wiener process, has a control strategy related to its Itô differential equation (Liu and Wu 2018; Wang et al. 2017, 2018). In recent years, stochastic systems have received increasing attention from scholars. Using Itô stochastic differential equations, this approach is extended from deterministic models to stochastic models (Shi et al. 2016; Li and Chen 2018; Liu et al. 2019). In Li and Chen (2018), the controller was researched for a stochastic nonlinear system against matching cases. Liu et al. (2019) proposed the H _ performance index for continuous-time stochastic systems with Markov jump and multiplicative noise. Liu et al. (2013) studied the fault estimation and tolerant control of T-S fuzzy stochastic systems with sensor faults but did not consider the finite-time stability problem. At the same time, in the production process, a time delay and even multiple time delays often occur (Wang et al. 2010; Sakthivel et al. 2017). Zhang et al. (2008) mainly studied the guaranteed cost control of uncertain T-S fuzzy stochastic models against multiple delays. Nevertheless, all states need to be known in this method. Switching behavior, Brownian motion, and time increments all may lead to model instability. To make the system stable, various Lyapunov functions are accordingly proposed (Wu and Li 2007; Zhang et al. 2017; Gao et al. 2009; Angulo et al. 2017; Guo and Zhao 2013; Vu et al. 2018; Chen et al. 2005). Additionally, it is worth pointing out that the piecewise fuzzy Lyapunov function is less conservative than the ordinary Lyapunov function, piecewise Lyapunov function, or fuzzy Lyapunov function (Li et al. 2016). In addition, the research

4.2 System Definition and Description

87

achievements of fuzzy systems are rarely concentrated on switched stochastic models with multiple time-varying delays. In addition, there are few researches on piecewise fuzzy Lyapunov function. This above discussion motivates us to conduct the research.

4.2 System Definition and Description A switched T-S fuzzy stochastic model against multiple time-varying delays can be shown as follows: ⎧   p r S   ⎪ ⎪ j j j ⎪ ⎪ d x(t) = σ (t)  (μ(t)) Aki x(t − τk (t)) + Bui u(t) j ⎪ i ⎪ ⎪ ⎪ j=1 i=1 k=0 ⎪ ⎪ ⎪

⎪ S ⎪  ⎪ j j ⎪ j ⎪ + B α (t) f (t) + B w(t) dt + B x(t − τ (t))dW (t) , a a k ⎪ a wi ki ⎪ ⎪ ⎪ k=0 ⎨ p  (4.1) j ⎪ y(t) = σ (t)[C x(t) + F f (t)], ⎪ j s s 0 ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ p r ⎪   ⎪ ⎪ j j ⎪ yz (t) = σ j (t) i (μ(t))C zi x(t), ⎪ ⎪ ⎪ ⎪ j=1 i=1 ⎪ ⎪ ⎪ ⎩ x(t) =ψ(t), t ∈ [−τ¯ , 0], where x(t) ∈ R n represents this state, u(t) ∈ R m represents this system input, αa (t) f a (t) ∈ R na represents this intermittent actuator fault, w(t) ∈ R n w is the external disturbance, y(t) ∈ R n y represents the system measurement output, f s (t) ∈ R n s j j j is the sensor fault, and yz (t) ∈ R p1 is the system controlled output. Aki , Bui , Ba , j j j j Bwi , Bki , C0 , Fs and C zi stand for known constant matrices of suitable dimensions, (i = 1, . . . , r, k = 0, . . . , S). τ0 (t) = 0; τk (t) represents the time-varying state delay and obeys 0 < τk (t) ≤ τk and τ˙k (t) ≤ τ Mk , where τk and τ Mk represent known scalars. τ¯  max{τ1 , . . . , τ S }. ψ(t) represents a vector-valued initial function, where t is part of the region [−τ¯ , 0]. Suppose f˙a (t) ∈ L 2 [0, ∞). r is the IFTHEN rule number, l is the premise variable number, and S is the of time delay number. W (t) represents the scalar Brownian motion on the complete probabilj ity space. At the same time, E{dW (t)} = 0; E{dW 2 (t)} = dt. i (μ(t)) consists of j premise variable μh (t) and fuzzy set described by membership function πi h (i = r j j j 1, . . . , r, h = 1, . . . , l). μ(t)=[μ1 (t), . . . , μl (t)], i (μ(t))=¯i (μ(t))/ i=1¯i (μ(t)), j j j ¯i (μ(t))= lh=1 πi h (μh (t)), and πi h (·) denotes the order of the membership function r j j j of πi h . Assume that ¯i (μ(t)) ≥ 0, i = 1, . . . , r, > 0, for any μ(t). i=1 ¯i (μ(t)) r r j j j Therefore, for any μ(t), i=1 i (μ(t)) obeys i (μ(t)) ≥ 0, i=1 i (μ(t)) = 1.

88

4 Fault Estimation and Tolerant Control for Multiple Time Delayed Switched …

Besides, αa (t) f a (t) denotes the multiplicative fault. In this intermittent fault αa (t) f a (t), f a (t) denotes this actuator fault under this case that the fault probability is 1 and the form is the same as the permanent fault. This stochastic variable αa (t)  diag{αa1 (t), . . . , αana (t)} indicates that this occurrence of intermittent actuator fault satisfies the Bernoulli distribution and its value is either 0 or 1. The following equations hold:

P{αai (t) = 1} = α¯ ai , P{αai (t) = 0} = 1 − α¯ ai ,

where α¯ ai (i = 1, . . . , n a ) ranges from 0 to 1 and is a known constant. Define E{αa (t)} = α¯ a = diag{α¯ a1 , . . . , α¯ ana }. σ j (t) : [0, ∞) → {0, 1} ( j = 1, 2, . . . , p) represents this switching signal, meaning that the subsystem is activated at this switching instant. That is, when σ j (t) = 1, t ∈ [t1 , t2 ), it means that when t ∈ [t1 , t2 ), the jth switched  psubsystem is activated. p refers to the number of switched subsystems. Obviously j=1 σ j (t) = 1. According to Wang and Tong (2017) and Su et al. (2016), at every assigned time t, assume that this value of σ j (t) is unknown, but its instantaneous value is available in real time. Before obtaining these fault estimation and tolerant control achievements, a number of helpful assumptions and definitions are given. j

j

j

j

Assumption 4.1 In the fuzzy system (4.1), (A0i , C0 ) is detectable, and (A0i , Bui ) j j is stabilizable, (i = 1, . . . , r ; j = 1, . . . , p). In addition, Bui , Ba represent matrices j with column full rank, and C0 represents a matrix with row full rank. And n a + ns ≤ n y . Assumption 4.2 Suppose there exists a positive constant  satisfying the following rank condition:   j j In + A0i Ba α¯ a rank = n + na . j C0 0 Assumption 4.3  f a (t) ≤ κ1 ,  f˙a (t) ≤ κ2 ,  f s (t) ≤ κ3 , where κ1 , κ2 and κ3 denote known positive scalars. Assumption 4.4 w(t) in the chapter obeys the inequality for the time interval [t¯, t˜] as follows:

  t˜ T ¯ [t¯,t˜],  w(t) ∈ L 2 [t¯, t˜] : W w (θ)w(θ)dθ ≤  , t¯

where the scalars t¯, t˜,  > 0.

4.3 Design of Sliding Mode Observer

89

Definition 4.1 (Han et al. (2018a), Zhang et al. (2018)) In system (4.1), if the following inequality is satisfied: N (t1 , t2 ) ≤ N0 +

t2 − t1 , Ta

(4.2)

then Ta is called the average dwell time. Ta > 0 and N0 ≥ 0. N (t1 , t2 ) is the switching number in the time interval [t1 , t2 ). In light of Zhang et al. (2018), it is assumed that N0 = 0 in this chapter. Definition 4.2 (Su et al. (2016)) With σ j (t), this switched fuzzy stochastic model (4.1) against w(t) = 0, f a (t) = 0 and u(t) = 0 can be mean-square exponentially stable, if E{x(t)2 } ≤ δx(t0 )2 e−η(t−t0 ) ,

(4.3)

where δ and η are positive constants, t0 represents this initial time, and x(t0 ) represents the initial value. Definition 4.3 (Su et al. (2016)) For assigned positive constants γ and α, this switched stochastic model (4.1) against f a (t) = 0 and u(t) = 0 will be mean-square exponentially stable under this weighted performance index (γ, α), if 1. When w(t) = 0, this model will be mean-square exponentially stable; 2. When w(t) = 0 and zero initial conditions, the following inequality obeys:  ∞

 ∞

−αt T 2 T E e yz (t)yz (t)dt < γ E w (t)w(t)dt . 0

(4.4)

0

According to Definition 4.3, it can be obtained that if α is close to zero, this weighted H∞ performance index is close to this classical H∞ performance index. Besides, the smaller α we choose, the better interference attenuation performance we can achieve.

4.3 Design of Sliding Mode Observer In the following, these main results of switched T-S fuzzy stochastic models under multiple time-varying delays are presented, and the sliding mode observer approach is proposed to estimate the actuator fault f a (t) and the sensor fault  f s (t).  f a (t) + f˙a (t) T T T T . ¯ = [x (t), f a (t), (Fs f s (t)) ] and f (t) = For simplicity, set x(t) f s (t) Model (4.1) is redescribed by

90

4 Fault Estimation and Tolerant Control for Multiple Time Delayed Switched …

⎧ r S   ⎪  j ⎪ j j j ⎪ ¯ j d x(t) ¯ G ¯ =  (μ(t)) [ A x(t) ¯ + ¯ − τk (t)) + B¯ ui u(t) A¯ ki x(t ⎪ i 0i ⎪ ⎪ ⎪ i=1 k=1 ⎪ ⎪ ⎪ ⎪ j j ⎪ ¯ ¯ + B f f (t) + Bwi w(t) + B¯ aj α˜ a (t) f a (t)]dt ⎪ ⎪ ⎨ S   j ⎪ ¯ − τk (t))dW (t) , B¯ ki x(t + ⎪ ⎪ ⎪ ⎪ k=0 ⎪ ⎪ ⎪ p ⎪ ⎪  ⎪ j ⎪ ⎪ y(t) = σ j (t)C¯ 0 x(t), ¯ ⎪ ⎩

(4.5)

j=1

where ⎡

⎡ j ⎤ ⎤ j In −1 Ba α¯ a 0 A0i 0 0 j G¯ j = ⎣ 0 0 ⎦ , A¯ 0i = ⎣ 0 −Ina 0 ⎦ , In a 0 0 0n y ×n y 0 0 −In y ⎡ j ⎡ j ⎤ ⎡ ⎤ ⎤ j Aki 0 0 Bui −1 Ba α¯ a 0n×n s j j j A¯ ki = ⎣ 0 0 0 ⎦ , B¯ ui = ⎣ 0 ⎦ , B¯ f = ⎣ Ina 0na ×n s ⎦ , 0n y ×na Fs 0 0 00 ⎡ ⎡ ⎤ ⎤ j j Bwi Ba j B¯ wi = ⎣ 0na ×n w ⎦ , B¯ aj = ⎣ 0na ×na ⎦ , α˜ a (t) = αa (t) − α¯ a , 0n y ×n w 0n y ×na ⎤ ⎤ ⎡ j ⎡ j B0i 0 Bki 0 0 0 j j B¯ 0i = ⎣ 0 0na ×na 0 ⎦ , B¯ ki = ⎣ 0 0na ×na 0 ⎦ , 0 0 0n y ×n y 0 0 0n y ×n y   j C¯ 0 = C0j 0 In y , i = 1, . . . , r, j = 1, . . . , p, k = 1, . . . , S. j j Remark 4.1 The involvement of  can make ( A¯ 0i , B¯ ui ) stabilizable, which provides the possibility for fault estimation and tolerant control.

The fuzzy observer designed for system (4.5) has the following form: ⎧ r S   ⎪  ⎪ j j j ¯j j ⎪ ¯ j d x(t) ¯ ¯ ¯ ˆ¯ = S  (μ(t)) [( A − L ) x ¯ (t) − W y(t) + C A¯ ki ⎪ c i 0i pi 0 ⎪ ⎪ ⎪ i=1 k=1 ⎪ ⎨  j j j ¯ ¯ ¯ ˆ × x(t ¯ − τk (t)) + Bui u(t) + L s u s (t)]dt + L d dy(t) , ⎪ ⎪ p ⎪ ⎪  ⎪ j ⎪ ˆ¯ =x¯c (t) + ⎪ σ j (t)( S¯ j )−1 L¯ d y(t), x(t) ⎪ ⎩ j=1

(4.6)

4.3 Design of Sliding Mode Observer

91

⎤ 0n×n y ˆ¯ = [xˆ T (t), fˆaT (t), where W¯ = ⎣ 0na ×n y ⎦. x¯c (t) ∈ R n is the intermediate vector. x(t) In y ×n y (Fs fˆs (t))T ]T means the estimated value of x(t). ¯ We can find that fˆs (t), the actual ˆ¯ estimated value of f s (t), can be obtained by fˆs (t) = (FsT Fs )+ FsT W¯ T x(t). Define j j n×n y ¯ j n×n y n×n y ¯ ¯ , Ls ∈ R and L d ∈ R represent the proporn = n + n y + n a . L pi ∈ R tional gain, the sliding mode gain, and the derivative gain, separately. S¯ j indicates j that the non-singular matrix is obtained by selecting the appropriate gain, L d . ⎡

j 4.3.1 Design of the Derivative Gain L¯ d

Because  rank

j

G¯ j C¯ 0

⎤ j In −1 Ba α¯ a 0 ⎢ 0 0 ⎥ In a ⎥ = n, = rank ⎢ ⎣ 0 0 0n y ⎦ j C0 0 In y ×n y ⎡

j j j there is a gain L d such that S¯ j  G¯ j + L d C¯ 0 is non-singular.   T j Choose L¯ d = 0n y ×n 0n y ×na (H j )T , where H j ∈ R n y ×n y is a diagonal matrix, j j and then the matrix S¯ j = G¯ j + L d C¯ 0 is non-singular. We can find that ( S¯ j )−1 = ⎤ ⎡ j In −−1 Ba α¯ a 0 ⎣ 0 0 ⎦. Given ( S¯ j )−1 , we have C¯ 0j ( S¯ j )−1 L¯ dj = In y , A¯ 0ij ( S¯ j )−1 Ia j −1 j j −C0  C0 Ba α¯ a (H j )−1 j L¯ d = −W¯ .

j 4.3.2 Design of the Proportional Gain L¯ pi j Based on Assumption 4.1, there exists a matrix L¯ pi satisfying that this matrix j j j j S¯ j ( A¯ 0i − L¯ pi C¯ 0 ) is Hurwitz, where i = 1, . . . , r ; j = 1, . . . , p. L¯ pi is considered to obey these following conditions. j It should be noted that there is a constant βi > 0 satisfying j j Re[λh (( S¯ j )−1 A¯ 0i )] ≥ −βi (i = 1, . . . , r ; j = 1, . . . , p; h = 1, . . . , n), namely, j j Re[λh (−(βi In + ( S¯ j )−1 A¯ 0i ))] ≤ 0. In addition, ∀ε ∈ C+ , we have

92

4 Fault Estimation and Tolerant Control for Multiple Time Delayed Switched …



  j j j ε(G¯ j + L¯ d C¯ 0 ) − A¯ 0i ( S¯ j )−1 0 = rank , (4.7) j 0 In y C¯ 0 ⎡ ⎤ j j εIn − A0i ε−1 Ba α¯ a 0 j j j ⎦, 0 0 (ε + )Ina rank(ε(G¯ j + L¯ d C¯ 0 ) − A¯ 0i ) = ⎣ (4.8) j j j εH C0 0 εH + In y ⎡ ⎤ j j εIn − A0i ε−1 Ba α¯ a 0 j rank(εG¯ j − A¯ 0i ) = ⎣ (4.9) 0 (ε + )Ina 0 ⎦ . 0 0 In y

j εIn − ( S¯ j )−1 A¯ 0i rank j C¯ 0





Given ∀ε ∈ C+ , rank(εH j + In y ) = n y , we have j j j j rank(ε(G¯ j + L¯ d C¯ 0 ) − A¯ 0i ) = rank(εG¯ j − A¯ 0i ).

(4.10)

According to formulas (4.7)–(4.10), it yields that 

 j εIn − ( S¯ j )−1 A¯ 0i rank j C¯ 0   j εG ¯ j − A¯ 0i = rank j C¯ 0 ⎤ ⎡ j j εIn − A0i ε−1 Ba α¯ a 0 ⎢ 0 (ε + )Ina 0 ⎥ ⎥ = rank ⎢ ⎣ 0 0 In y ⎦ j C0 0 In y ⎡ ⎤ j j εIn − A0i ε−1 Ba α¯ a ⎢ 0 (ε + )Ina ⎥ ⎥ + ny. = rank ⎢ ⎣ ⎦ 0 0 j C0 0 j

(4.11)

j

If ε = , given that (A0i , C0 ) is detectable, one has 

εIn − ( S¯ j )−1 A¯ 0i rank j C¯ 0 j





j

εIn − A0i = rank j C0



= n + n a + n y = n. If ε = , according to Assumption 4.2, we can know

+ na + n y (4.12)

4.4 System Stability Analysis

93



j εIn − ( S¯ j )−1 A¯ 0i rank j C¯ 0





j

j

−In − A0i −Ba α¯ a = rank j C0 0

 + ny

= n + n a + n y = n,

(4.13)

j j j j this means (( S¯ j )−1 A¯ 0i , C¯ 0 ) is detectable, so (−( S¯ j )−1 A¯ 0i , −C¯ 0 ) is detectable. j j j j j j −1 Then we have L p1i such that −( S¯ ) A¯ 0i − L p1i C¯ 0 is stable, and then −βi In − j j j j j j ( S¯ j )−1 A¯ 0i − L p1i C¯ 0 is stable, which means (−βi In − ( S¯ j )−1 A¯ 0i , C¯ 0 ) is detectable. j Thereby, based on Lemma 1.6, there exists a positive definite matrix P¯i so that j j j j j j j j −(βi In + ( S¯ j )−1 A¯ 0i )T P¯i − P¯i (βi In + ( S¯ j )−1 A¯ 0i ) = −(C¯ 0 )T C¯ 0 .

(4.14)

j j j j ¯j T Assuming L¯ pi = S¯ j ( P¯i )−1 (C¯ 0 )T , we know (βij In + ( S¯ j )−1 ( A¯ 0ij − L¯ pi C0 ))

j j j j j j j j P¯i + P¯i (βi In + ( S¯ j )−1 ( A¯ 0i − L¯ pi C¯ 0 )) = −(C¯ 0 )T C¯ 0 . This means ∀h ∈ {1, . . . , n},

j j j j j j j Re[λh (( S¯ j )−1 ( A¯ 0i − L pi C¯ 0 ))] < −βi . Hence, when L¯ pi = S¯ j ( P¯i )−1 (C¯ 0 )T , matrix j j j ( S¯ j )−1 ( A¯ 0i − L pi C¯ 0 ) is Hurwitz.

4.4 System Stability Analysis According to formula (4.6), we can obtain ˆ¯ S¯ j d x(t) j = S¯ j d x¯c (t) + L¯ d dy(t)  r  j j j j ˆ j j j j j i (μ(t)) ( A¯ 0i − L¯ pi C¯ 0 )x(t) ¯ − A¯ 0i ( S¯ j )−1 L¯ d y(t) + L¯ pi C¯ 0 ( S¯ j )−1 L¯ d y(t) = i=1 S 

− W¯ y(t) +



j ˆ j j j ¯ ¯ ¯ ¯ Aki x(t ¯ − τk (t)) + Bui u(t) + L s u s (t) dt + L d dy(t) . (4.15)

k=1 j j j j It should be noted that C¯ 0 ( S¯ j )−1 L¯ d = In y , A¯ 0i ( S¯ j )−1 L¯ d = −W¯ . Thus formula (4.15) can be written as

ˆ¯ = S¯ j d x(t)

r 

 j

i (μ(t))

i=1

+

S 

ˆ¯ + W¯ y(t) + L¯ pi y(t) − W¯ y(t) ( A¯ 0i − L¯ pi C¯ 0 )x(t) j

j

j

j



j ˆ j j A¯ ki x(t ¯ − τk (t)) + B¯ ui u(t) + L¯ sj u s (t) dt + L¯ d dy(t) . (4.16)

s=1

This further illustrates

94

4 Fault Estimation and Tolerant Control for Multiple Time Delayed Switched …

ˆ¯ = S¯ j d x(t)

r 

j

i (μ(t))

 S  j j j ˆ j j j ˆ ( A¯ 0i − L¯ pi C¯ 0 )x(t) ¯ + L¯ pi C¯ 0 x(t) ¯ + A¯ ki x(t ¯ − τk (t))

i=1

k=1



j j j ¯ . + B¯ ui u(t) + L¯ sj u s (t) dt + L¯ d C¯ 0 d x(t)

(4.17)

According to formula (4.5), we can obtain ¯ = S¯ j d x(t)

r 

j

i (μ(t))

 S  j j j j j j ( A¯ 0i − L¯ pi C¯ 0 )x(t) ¯ + L¯ pi C¯ 0 x(t) ¯ + ¯ − τk (t)) A¯ ki x(t

i=1

k=1

j j j + B¯ ui u(t) + B¯ f f (t) + B¯ wi w(t) + B¯ aj α˜ a (t) f a (t) dt +

S 

j j j ¯ ¯ − τk (t))dW (t) + L¯ d C¯ 0 d x(t). ¯ Bki x(t

(4.18)

k=0

ˆ¯ − x(t). Define the error dynamic as e(t) = x(t) ¯ Formula (4.17) minus formula (4.18) yields de(t) =

p 

σ j (t)

j=1

r 

j i (μ(t))( S¯ j )−1

 S  j j j j ( A¯ 0i − L¯ pi C¯ 0 )e(t) + A¯ ki e(t − τk (t))

i=1

j j + L¯ sj u s (t) − B¯ f f (t) − B¯ wi w(t) − B¯ aj α˜ a (t) f a (t) dt −

S 

k=1

j Bˆ ki x(t − τk (t))dW (t) ,

(4.19)

k=0

where ⎤ ⎡ j ⎤ j B0i Bki j j Bˆ 0i = ⎣ 0na ×n ⎦ , Bˆ ki = ⎣ 0na ×n ⎦ , i = 1, . . . , r, j = 1, . . . , p, k = 1, . . . , p. 0n y ×n 0n y ×n ⎡

Design u(t) as follows: u(t) =

p 

σ j (t)

j=1

 r i=1

r 

j jˆ i (μ(t)) K¯ i x(t) ¯ −

i=1 j

i (μ(t))Bui

p  j=1

r  i=1

σ j (t)

r 

j

i (μ(t))BuiT

i=1

j i (μ(t))BuiT )+ Baj α¯ a fˆa (t)

4.4 System Stability Analysis

=

p 

σ j (t)

j=1

BuiT

 r

r 

95

i (μ(t)) K¯ i (e(t) + x(t)) ¯ − j

j

i=1 j

i (μ(t))Bui

i=1

r 

+ j

i (μ(t))BuiT

p 

r 

σ j (t)

j=1

 j

i (μ(t)

i=1

Baj α¯ a fˆa (t),

(4.20)

i=1

  j j where K¯ i = K ij 0m×na 0m×n y and K i ∈ R m×n represents the state feedback gain. The sliding surface s(t) ∈ R na +n s is expressed as follows: s(t) =

p 

σ j (t)( B¯ f )T ( S¯ j )−T P2 e(t), j

j

(4.21)

j=1

 j j j j where P2 = ri=1 i (μ(t))P2i . There exists a positive definite matrix P2i ∈ R n×n j j j j j such that ( B¯ f )T ( S¯ j )−T P2i = Ci C¯ 0 , where Ci ∈ R (na +n s )×n y is a cluster of matrices to be determined later. As discontinuous input, this sliding mode controller u s (t) is u s (t) = −(κ1 + κ2 + κ3 + κ4 )sgn(s(t)),

(4.22)

where κ4 > 0 denotes a scalar designed later. Substituting formula (4.20) into formula (4.1), we can obtain d x(t) =

p 

σ j (t)

j=1

r 

j

i (μ(t))

i=1

 S

j

j

Aki x(t − τk (t)) + Bui

k=0

r 

h (μ(t)) K¯ h (x(t) ¯ j

j

h=1

S  j j j + e(t)) + Bwi w(t) + Ba α˜ a (t) f a (t) dt + Bki x(t − τk (t))dW (t) −

p 

σ j (t)

j=1 r 

 r

k=0 j

i (μ(t))Bui

i=1

+

j

i (μ(t))BuiT

r 

j

i (μ(t))BuiT

 r

i=1

Baj α¯ a fˆa (t) − Baj α¯ a f a (t) dt.

j

i (μ(t))Bui

i=1

(4.23)

i=1

    Assume rank Buij Baj = rank Buij . Therefore, it can be further obtained d x(t) =

p  j=1

σ j (t)

r  i=1

j i (μ(t))

  S k=0

j

j

Aki x(t − τk (t)) + Bui

r  h=1

j

j

h (μ(t))(K h x(t)

96

4 Fault Estimation and Tolerant Control for Multiple Time Delayed Switched …

j j j j ¯ ¯ + K h e(t)) − V e(t) + Bwi w(t) + Ba α˜ a (t) f a (t) dt +

S 

j Bki x(t

− τk (t))dW (t)

k=0

=

p 

σ j (t)

j=1

r  i=1

j i (μ(t))

r 

j h (μ(t))

  S

h=1

j

j

j

Aki x(t − τk (t)) + Bui (K h x(t)

k=0

j j + K¯ h e(t)) − V¯ j e(t) + Bwi w(t) + Baj α˜ a (t) f a (t) dt +

S 

j Bki x(t − τk (t))dW (t) ,

(4.24)

k=0

  where V¯ j = 0n×n Baj α¯ a 0n×n y . Finally, the entire closed-loop system can be described as ⎧ p r r    ⎪ ⎪ j j j j j ⎪ d x(t) = σ (t)  (μ(t)) h (μ(t)){[(A0i + Bui K h )x(t) ⎪ j i ⎪ ⎪ ⎪ j=1 i=1 h=1 ⎪ ⎪ ⎪ ⎪ S ⎪  j ⎪ ⎪ j j j ⎪ ⎪ + Aki x(t − τk (t)) + (Bui K¯ h − V¯ j )e(t) + Bwi w(t) ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎪ S ⎪  ⎪ ⎪ j j ⎪ + B α ˜ (t) f (t)]dt + Bki x(t − τk (t))dW (t)}, ⎪ a a a ⎪ ⎨ k=0  p r ⎪   j ⎪ j j j ⎪ j −1 ¯ ⎪ de(t) = [( A¯ 0i − L¯ pi C¯ 0 )e(t) σ j (t) i (μ(t))( S ) ⎪ ⎪ ⎪ ⎪ j=1 i=1 ⎪ ⎪ ⎪ ⎪ S ⎪  j ⎪ ⎪ j j ⎪ + A¯ ki e(t − τk (t)) + L¯ sj u s (t) − B¯ f f (t) − B¯ wi w(t) ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎪

⎪ S ⎪  ⎪ j ⎪ j ¯ ˆ ⎪ − Ba α˜ a (t) f a (t)]dt − Bki x(t − τk (t))dW (t) . ⎪ ⎩ k=0

Choose ξ(t) = [x T (t), e T (t)]T , and formula (4.25) is further described

(4.25)

4.4 System Stability Analysis

97

⎧  p r r S     ⎪ ⎪ j j j ¯¯ j ξ(t) + ⎪ dξ(t) = σ (t)  (μ(t))  (μ(t)) A A¯¯ ki ξ(t − τk (t)) ⎪ j i h 0i h ⎪ ⎪ ⎪ j=1 i=1 h=1 k=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ j j ⎪ ⎪ + B¯¯ wi w(t) + B¯¯ aj α˜ a (t) f a (t) + L¯¯ s u s (t) − B¯¯ f f (t) dt ⎪ ⎨ (4.26)

S  ⎪ ⎪ j ¯ ⎪ ¯ ⎪ + Bki ξ(t − τk (t))dW (t) , ⎪ ⎪ ⎪ ⎪ k=0 ⎪ ⎪ ⎪ p r ⎪   ⎪ j j ⎪ ⎪ σ j (t) i (μ(t))C¯ zi ξ(t). ⎪ ⎩ yz (t) = j=1

i=1

where 

 j j j j j A0i + Bi K h Bui K¯ h − V¯ j j j j , A¯¯ ki = diag{Aki , ( S¯ j )−1 A¯ ki }, j j j 0i h = 0 ( S¯ j )−1 ( A¯ 0i − L¯ pi C¯ 0 )      j j 0 Bwi B¯ a j ¯¯ j = ¯¯ = B L B¯¯ wi = , , s a j , j j ( S¯ j )−1 L¯ s −( S¯ j )−1 B¯ wi −( S¯ j )−1 B¯ a      j j 0 Bki 0 j j ¯ ¯ ¯ ¯ ¯ j = C zi . = Bf = , B , C j j j −1 ki zi ( S¯ ) B¯ f 0 ( S¯ j )−1 Bˆ ki 0 j A¯¯

j

Theorem 4.1 Substitute u s (t) (4.22) into model (4.26) and choose K h to follow j j j that A0i + Bui K h is Hurwitz. For scalars τk > 0, γ > 0, α > 0, ϑ > 1, τ Mk , and switching signals whose average dwell time obeys Ta > lnαϑ , this closed-loop model (4.26) can be mean-square exponentially stable under this weighted H∞ performance j j j index (γ, α) in formula (4.4), and all eigenvalues of ( S¯ j )−1 ( A¯ 0i − L¯ pi C¯ 0 ) belong ¯¯ ), ¯¯ if there exists a constant δ > 0, positive definite matrices to this region D(ν,  j j j j j j j j Pi  diag{P1i , P2i }, Q ki , Rki , Z ki and a matrix Cj = ri=1 i (μ(t))Ci such that for every i, h, υ = 1, . . . , r , j, q = 1, . . . , p and k, k1 , k2 = 1, . . . , S, the following optimization problem has feasible solutions: min δ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ s.t.

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

j j j j P˙ ≤ 0, Q˙ k ≤ 0, R˙ k ≤ 0, Z˙ k ≤ 0,

(4.27)

j ϒiυh

(4.28)

j + ϒυi h < 0, i ≤ υ,  j j −δ In (( B¯ f )T ( S¯ j )−T P2i

∗

j

−In a +n s

q

j

q

j j − Ci C¯ 0 )T j

q

 < 0, j

(4.29) q

Pi < ϑPi , Q ki < ϑQ ki , Rki < ϑRki , Z ki < ϑZ ki ,

(4.30)

j 1,iυ j 2,iυ

< 0, i ≤ υ,

(4.31)

< 0, i ≤ υ,

(4.32)

j + 1,υi j + 2,υi

98

4 Fault Estimation and Tolerant Control for Multiple Time Delayed Switched …

where r 

P˙j =

j j ˙i (μ(t))Pi ,

j Q˙ k =

i=1

j Z˙ k =

j j ˙i (μ(t))Q ki ,

j R˙ k =

i=1

r 

j

j

j

˙i (μ(t))Z ki , ϒiυh

 =

j j ϒ¯ 11,iυh ϒ¯ 12,iυh j ∗ ϒ¯ 22,iυ



r 

j

j

˙i (μ(t))Rki ,

i=1

⎡

i=1

j ϒ1,iυh

r 

⎤ j j j j ϒ1,iυh ϒ2,iυ ϒ4,υ ϒ7,iυ j j ⎢ ∗ ϒ 0 ⎥ 3,iυ ϒ5,υ ⎥, =⎢ j ⎣ ∗ ∗ ϒ6,υ 0 ⎦ ∗ ∗ ∗ −γ 2 I

+

S 

j

Q kυ +

k=1

S 

j

τk Rkυ ,

k=1

j j j j j j j j ϒ¯ 11,iυh = sym[P1υ (A0i + Bui K h )] + (B0i )T P1υ B0i j j j j j j + ( Bˆ 0i )T ( S¯ j )−T P2υ ( S¯ j )−1 Bˆ 0i + (C zi )T C zi + αP1υ ,   j ϒ¯ 12,iυh = P1υj Buij K hj −P1υj Baj α¯ a 0n×n y ,   j j j j j j j j j ϒ¯ 22,iυ = sym[P2υ ( S¯ j )−1 ( A¯ 0i − L¯ pi C¯ 0 )] + αP2υ, ϒ2,iυ = ϒ2,1,iυ , . . . , ϒ2,S,iυ ,

ϒ2,k,iυ = diag{P1υ Aki + (B0i )T P1υ Bki + ( Bˆ 0i )T ( S¯ j )−T P2υ ( S¯ j )−1 Bˆ ki , j

j

j

j

j

j

j

j

j

P2υ ( S¯ j )−1 A¯ ki }, k = 1, . . . , S, ⎤ ⎡ j j ϒ3,11,iυ · · · ϒ3,1S,iυ ⎥ ⎢ .. .. = ⎣ ... ⎦, . . j

ϒ3,iυ

j

j

j

∗

· · · ϒ3,SS,iυ

ϒ3,k1 k2 ,iυ = diag{(Bk1 i )T P1υ Bk2 i + ( Bˆ k1 i )T ( S¯ j )−T P2υ ( S¯ j )−1 Bˆ k2 i , 0n×n }, j

j

j

j

j

j

j

k1 < k2 ∈ {1, . . . , S}, j ϒ3,kk,iυ

j j j j j j = diag{(Bki )T P1υ Bki + ( Bˆ ki )T ( S¯ j )−T P2υ ( S¯ j )−1 Bˆ ki , 0n×n }

− e−ατk (1 − τ Mk )Q kυ ,   j j j j j = Z 1υ · · · Z Sυ , ϒ5,υ = diag{−(1 − τ M1 )Z 1υ , . . . , −(1 − τ M S )Z Sυ }, j

j

ϒ4,υ

ϒ6,υ = diag{αZ 1υ − τ1−1 e−ατ1 R1υ , . . . , αZ Sυ − τ S−1 e−ατS R Sυ },   j j P1υ Bwi j j j j j ϒ7,iυ = , 1,iυ = sym[P2υ ( S¯ j )−1 ( A¯ 0i − L¯ pi C¯ 0 )] + 2ν¯¯ P2υ , j j −P2υ ( S¯ j )−1 B¯ wi  j ¯j ¯¯ P2υ P2υ ( S¯ j )−1 ( A¯ 0ij − L¯ pi − C0 ) j . 2,iυ = ¯¯ P2υ ∗ − j

j

j

j

j

4.4 System Stability Analysis

99

j At the same time, the sliding mode observer L¯ s can be designed as j j L¯ sj = S¯ j (P2 )−1 (C¯ 0 )T (Cj )T .

(4.33)

Proof A Lyapunov function is constructed as follows: V (t) = V1 (t) + V2 (t) + V3 (t) + V4 (t),

(4.34)

where V1 (t) =

p 

σ j (t)ξ T (t)Pj ξ(t),

j=1

V2 (t) =

V3 (t) =

V4 (t) =

p 

σ j (t)

t S 

e−α(t−θ) ξ T (θ)Q k ξ(θ)dθ, j

j=1

k=1t−τ (t) k

p 

0  t S 

σ j (t)

j=1

k=1−τ (t) t+ς k

p 

S  t 

σ j (t)

j=1

k=1

e−α(t−θ) ξ T (θ)Rk ξ(θ)dθdς, j

T ξ(θ)dθ

t−τk (t)

t j Z k

ξ(θ)dθ.

t−τk (t)

According to the Itô formula, namely, Lemma 1.7, along the trajectory of system (4.26), L V (t) obeys L V1 (t) =

p 

σ j (t)

j=1

r 

j

i (μ(t))

i=1

x(t) +

S 

r 

j

h (μ(t))

r 

 j j j j υj (μ(t)) 2x T (t)P1υ [(A0i + Bi K h )

υ=1

h=1

Aki x(t − τk (t)) + (Bui K¯ h − V¯ j )e(t) + Bwi w(t) + Baj α˜ a (t) f a (t)] j

j

j

j

k=1

+

 S

j

j

Bki x(t − τk (t))T P1υ

k=0

S 

Bki x(t − τk (t)) + 2e T (t)P2υ ( S¯ j )−1 j

j

k=0

 S  j j j j j j ( A¯ 0i − L¯ pi C¯ 0 )e(t) + A¯ ki e(t − τk (t)) + L¯ sj u s (t) − B¯ f f (t) − B¯ wi w(t) k=1

− B¯ aj α˜ a (t) f a (t)] +

 S k=0

j Bˆ ki x(t − τk (t))

T

j ( S¯ j )−T P2υ ( S¯ j )−1

S  k=0

j Bˆ ki

100

4 Fault Estimation and Tolerant Control for Multiple Time Delayed Switched …

 p x(t − τk (t)) + σ j (t)ξ T (t) P˙j ξ(t),

(4.35)

j=1

L V2 (t) p 

=

σ j (t)

r 

υj (μ(t))

υ=1

j=1

S  j [ξ T (t)Q kυ ξ(t) − e−ατk (t) (1 − τ˙k (t) k=1

t ξ (t − T

j τk (t))Q kυ ξ(t

e−α(t−θ) ξ T (θ)Q k ξ(θ)dθ] j

− τk (t)) − α t−τk (t)

 p

+

σ j (t)

j=1

t

S 

e−α(t−θ) ξ T (θ) Q˙ k ξ(θ)dθ, j

(4.36)

k=1t−τ (t) k

L V3 (t) ≤

p 

σ j (t)

r 

υj (μ(t))

υ=1

j=1

t S  j T [τk ξ (t)Rkυ ξ(t) − e−α(t−θ) k=1

t−τk (t)

0  t j

e−α(t−θ) ξ T (θ)Rk ξ(θ)dθdς] j

ξ T (θ)Rkυ ξ(θ)dθ − α −τk (t) t+ς

+

p 

σ j (t)

j=1

0  t S 

e−α(t−θ) ξ T (θ) R˙ k ξ(θ)dθdς, j

(4.37)

k=1−τ (t) t+ς k

L V4 (t) =

p 

σ j (t)

j=1

r 

υj (μ(t))

υ=1

S  j [2(ξ(t) − (1 − τ˙k (t))ξ(t − τk (t)))T Z kυ k=1

t ξ(θ)dθ] +

p 

t t S  T ˙j σ j (t) ( ξ(θ)dθ) Z k ξ(θ)dθ.

j=1

t−τk (t)

k=1 t−τ (t) k

t−τk (t)

Inspired by Lemma 1.8, one has t −

e−α(t−θ) ξ T (θ)Rkυ ξ(θ)dθ j

t−τk (t) −1 −ατk (t)

 t

≤ − (τk (t)) e

T ξ(θ)dθ

t−τk (t)

t j Rkυ

ξ(θ)dθ t−τk (t)

(4.38)

4.4 System Stability Analysis

≤−

101

τk−1 e−ατk

 t

T ξ(θ)dθ

t−τk (t)

t j Rkυ

ξ(θ)dθ.

(4.39)

t−τk (t)

Given Assumption 4.3, we have  f (t) ≤  f a (t) +  f˙a (t) +  f s (t) ≤ κ1 + κ2 + κ3 .

(4.40)

j j j According to ( B¯ f )T ( S¯ j )−T P2 = Cj C¯ 0 , we can obtain j j j j L¯ sj = S¯ j (P2 )−1 P2 ( S¯ j )−1 B¯ f = B¯ f .

(4.41)

It further follows that 2e T (t)P2 ( S¯ j )−1 [ L¯ sj u s (t) − B¯ f f (t)] = 2e T (t)P2 ( S¯ j )−1 [ B¯ f u s (t) − B¯ f f (t)] j

j

j

j

≤ − 2(κ1 + κ2 + κ3 + κ4 )s T (t)sgn(s(t)) + 2s(t) f (t). Considering that s(t) ∈ R na +n s , s T (t)sgn(s(t)) = |s(t)| and based on formula (4.22), we can know

na +n s i=1

j

(4.42)

|si (t)| = |s(t)|, s(t) ≤

j j 2e T (t)P2 ( S¯ j )−1 [ L¯ sj u s (t) − B¯ f f (t)]

≤ − 2(κ1 + κ2 + κ3 + κ4 )|s(t)| + 2(κ1 + κ2 + κ3 )s(t) ≤ − 2κ4 |s(t)|.

(4.43)

It should be noted that this establishment of inequality (4.23) can be obtained j j j from ( B¯ f )T ( S¯ j )−T P2 = Cj C¯ 0 , which is an equation that cannot be implemented j j j with the LMI toolbox. ( B¯ f )T ( S¯ j )−T P2 = Cj C¯ 0 can be rephrased as tr[(( B¯ f )T ( S¯ j )−T P2 − Cj C¯ 0 )T (( B¯ f )T ( S¯ j )−T P2 − Cj C¯ 0 )] = 0. j

j

j

j

j

j

(4.44)

A minimization problem can be obtained as follows: min δ s.t.

j (( B¯ f )T ( S¯ j )−T

j P2

j j j j − Cj C¯ 0 )T (( B¯ f )T ( S¯ j )−T P2 − Cj C¯ 0 ) < δ In .

(4.45)

Through Lemma 1.4, formula (4.29) can be obtained. Given Lemmas 1.1–1.3, E{α(t)} ˜ = 0, and formulas (4.27), (4.34)–(4.39) and (4.43), we have E{L V (t)} + αV (t) + yzT (t)yz (t) − γ 2 w T (t)w(t) = E{L V1 (t)} + E{L V2 (t)} + E{L V3 (t)} + E{L V4 (t)} + α(V1 (t) + V2 (t)

102

4 Fault Estimation and Tolerant Control for Multiple Time Delayed Switched …

+ V3 (t) + V4 (t)) + yzT (t)yz (t) − γ 2 w T (t)w(t) ≤

p 

σ j (t)

j=1

r 

j

i (μ(t))

r 

υj (μ(t))

υ=1

i=1

r 

j

j

h (μ(t))ζ T (t)ϒiυh ζ(t),

(4.46)

h=1

where ⎡

⎛

⎢ ⎜ ζ(t) = ⎣ξ T (t) ξ T (t − τ1 (t)) · · · ξ T (t − τ S (t)) ⎝ ⎛ ⎜ ⎝

t

⎞T

t

⎞T ⎟ x(θ)dθ⎠ · · ·

t−τ1 (t)

⎤T

⎟ ⎥ x(θ)dθ⎠ w T (t)⎦ , i, h, υ = 1, . . . , r ; j = 1, . . . , p.

t−τ S (t)

First, discuss the condition of w(t) = 0. According to formula (4.28), it can be seen that E{L V (t)} < −αV (t). Then one has E{L (eαt V (t))} < 0. Suppose tk represents the kth switching time, t ∈ [tk , tk+1 ). Then, E{V (t)} < e−α(t−tk ) E{V (tk )}.

(4.47)

Given formulas (4.30) and (4.34), we have E{V (t)} < ϑE{V (t − )},

(4.48)

Formulas (4.47) and (4.48) mean E{V (t)} < ϑ N (0,t) e−αt V (0).

(4.49)

According to Definition 4.1, formula (4.49) can be redescribed as ln ϑ

E{V (t)} < e−(α− Ta )t V (0).

(4.50)

Given formula (4.34), we can obtain E{V (t)} ≥aE{ξ(t)2 },

(4.51)

V (0) ≤bξ(0) ,

(4.52)

2

where j

a = min { min [λmin (Pi )]}, j=1,..., p i=1,...,r

4.4 System Stability Analysis

103



 S S   τk j j j τk λmax (Q ki ) + λmax (Pi ) + λmax (Rki ) j=1,..., p i=1,...,r α k=1 k=1

S  j + τk2 λmax (Z ki ) .

b = max

max

k=1 ln ϑ

The result shows E{ξ(t)2 } < ba e−(α− Ta )t ξ(0)2 . According to Ta > lnαϑ , it follows that α − lnTaϑ > 0. By Definition 4.2, the whole closed-loop model (4.26) can be mean-square exponentially stable. Next, the condition of w(t) = 0 is considered. According to (4.28) and (4.46), it follows that E{L V (t)} + αV (t) < −(yzT (t)yz (t) − γ 2 w T (t)w(t)).

(4.53)

E{d(eαt V (t))} < −eαt E{(yzT (t)yz (t) − γ 2 w T (t)w(t))}.

(4.54)

Therefore,

It yields that αt

E{e V (t)} < E{e

αtk

t V (tk ) −

eαθ (yzT (θ)yz (θ) − γ 2 w T (θ)w(θ))dθ}.

(4.55)

tk

Under zero initial conditions, from formulas (4.49) and (4.55), one has

 t e−α(t−θ)+N (θ,t) ln ϑ (yzT (θ)yz (θ) − γ 2 w T (θ)w(θ))dθ . (4.56) E{V (t)} < −E 0

Noting that V (t) > 0, the following inequality holds:  t

E e−α(t−θ)+N (s,t) ln ϑ (yzT (θ)yz (θ) − γ 2 w T (θ)w(θ))dθ < 0.

(4.57)

0

According to Definition 4.1, one has N (θ, t) = N (0, t) − N (0, θ). We can obtain  t

E e−α(t−θ)−N (0,θ) ln ϑ yzT (θ)yz (θ)dθ 0

104

4 Fault Estimation and Tolerant Control for Multiple Time Delayed Switched …

t 1 and N (0, θ) ≤

θ−0 Ta


1, and switching signals whose average dwell time satisfies Ta > lnαϑ , this closed-loop model (4.26) against constant time delays can be mean-square exponentially stable under this weighted H∞ performance index (γ, α) in formula (4.4), and all eigenvalues of j j j ¯¯ ), ¯¯ if there exists a constant δ > 0, ( S¯ j )−1 ( A¯ 0i − L¯ pi C¯ 0 ) belong to this region D(ν, j

j

j

j

j

j

positive definite matrices Pi  diag{P1i , P2i }, Q ki , Rki , Z ki and a matrix Cj = r j j i=1 i (μ(t))C i , for i, h, υ = 1, . . . , r , j, q = 1, . . . , p and k, k1 , k2 = 1, . . . , S, making these following optimization problem have feasible solutions: min δ  s.t.

(4.27), (4.29)–(4.32), j j ϒ˜ iυh + ϒ˜ υi h < 0, i ≤ υ,

(4.65)

where ⎡

j ϒ˜ iυh

⎤ j j j j ⎡ j ϒ1,iυh ϒ2,iυ ϒ4,υ ϒ7,iυ ϒ˜ 3,11,iυ ⎢ ∗ ϒ˜ j ϒ˜ j ⎥ 0 ⎢ j .. 3,iυ 5,υ ⎥ , ϒ˜ =⎢ 3,iυ = ⎣ j . ⎣ ∗ ∗ ϒ6,υ 0 ⎦ ∗ ∗ ∗ ∗ −γ 2 I

⎤ j · · · ϒ3,1S,iυ ⎥ .. .. ⎦, . . j · · · ϒ˜ 3,SS,iυ

j j j j j j j ϒ˜ 3,kk,iυ = diag{(Bki )T P1υ Bki + ( Bˆ ki )T ( S¯ j )−T P2υ ( S¯ j )−1 Bˆ ki , 0n×n }

− e−ατk Q kυ , j

j j j ϒ˜ 5,υ = diag{−Z 1υ , . . . , −Z Sυ }. j At the same time, the sliding mode observer gain L¯ s can be designed as formula (4.33).

Proof This proof is similar to Theorem 4.1. Thus the proof is omitted for simplicity.

4.5 Sliding Motion Reachability Control This section discusses the reachability of this sliding mode surface s(t). This solution e(t) of formula (4.19) is descried as

106

4 Fault Estimation and Tolerant Control for Multiple Time Delayed Switched …

e(t) =

t  p

σ j (θ)

j=1

0

+

S 

r 

j i (μ(θ))( S¯ j )−1

 j j j ( A¯ 0i − L¯ pi C¯ 0 )e(θ)

i=1 j j j A¯ ki e(θ − τk (θ)) + L¯ sj u s (θ) − B¯ f f (θ) − B¯ wi w(θ)

k=1



S  j − B¯ aj α˜ a (θ) f a (θ) dθ − ¯ − τk (θ))dW (θ) . B¯ ki x(θ

(4.66)

k=0

Substitute formula (4.66) into formula (4.21), and we can obtain s(t) =

p 

σ j (θ)

j=1

+

S 

r 

 t j i (μ(θ))

i=1

j j j j j ( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 [( A¯ 0i − L¯ pi C¯ 0 )e(θ)

0

j j j A¯ ki e(θ − τk (θ)) + L¯ sj u s (θ) − B¯ f f (θ) − B¯ wi w(θ)

k=1

− B¯ aj α˜ a (θ) f a (θ)]dθ −

t  S 0

j j Cj C¯ 0 ( S¯ j )−1 B¯ ki x(θ ¯

− τk (θ))dW (θ) .

k=0

(4.67) j j C¯ 0 ( S¯ j )−1 B¯ ki can be calculated as ⎡ ⎤⎡ j ⎤ j In −−1 Ba α¯ a 0 Bki 0 0   j j 0 ⎦ ⎣ 0 0 0 ⎦ = 0. Ia C¯ 0 S¯ j B¯ ki = C0j 0 In y ⎣ 0 j −1 j j 0 00 −C0  C0 Ba α¯ a (H j )−1

(4.68)

Therefore, formula (4.67) can be written as s(t) =

t  p 0

+

σ j (θ)

j=1 S 

r 

 j j j j j j i (μ(θ)) ( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 [( A¯ 0i − L¯ pi C¯ 0 )e(θ)

i=1 j j j A¯ ki e(θ − τk (θ)) + L¯ sj u s (θ) − B¯ f f (θ) − B¯ wi w(θ)

k=1

− B¯ aj α˜ a (θ) f a (θ)]dθ . Obviously, the following equation holds: s˙ (t) =

p  j=1

σ j (t)

r  i=1

j j j j j j i (μ(t))( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 [( A¯ 0i − L¯ pi C¯ 0 )e(t)

(4.69)

4.5 Sliding Motion Reachability Control

+

S 

107

j j j A¯ ki e(t − τk (t)) + L¯ sj u s (t) − B¯ f f (t) − B¯ wi w(t) − B¯ aj α˜ a (t) f a (t)].

k=1

(4.70) j j j Theorem 4.2 For the given previously designed observer gains L¯ d , L¯ pi and L¯ s , j

assume that there exists a positive constant δ, positive definite matrices Pi  j j j j j j diag{P1i , P2i }, Q ki , Rki , Z ki and a matrix Ci , for i, h, υ = 1, . . . , r , j, q = 1, . . . , p and k, k1 , k2 = 1, . . . , S, making the optimization problem (4.27)–(4.32) hold. Then, in light of this control law u s (t) (4.22) and u(t) (4.20), this error dynamic e(t) of this closed-loop model (4.26) can be globally driven onto the sliding mode surface s(t) under probability 1 in finite-time. Proof Design a Lyapunov function as follows: V5 (t) =

p 

j j j σ j (t)0.5s T (t)(( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 B¯ f )−1 s(t).

(4.71)

j=1

Given formula (4.70), we have V˙5 (t) =

p 

σ j (t)s T (t)(( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 B¯ f )−1 s˙ (t) j

j

j

j=1

=

p  j=1

σ j (t)

r 

i (μ(t))s T (t)(( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 B¯ f )−1 ( B¯ f )T j

j

j

j

j

i=1

j j j j ( S¯ j )−T P2 ( S¯ j )−1 [( A¯ 0i − L¯ pi C¯ 0 )e(t) +

S 

j A¯ ki e(t − τk (t))

k=1

+ L¯ sj u s (t) − B¯ f f (t) − B¯ wi w(t) − B¯ aj α˜ a (t) f a (t)]. j

j

(4.72)

Similar to formulas (4.42)–(4.43), we have s T (t)(( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 B¯ f )−1 ( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 ( L¯ sj u s (t) j

j

j

j

j

j − B¯ f f (t))

= s T (t)(u s (t) − f (t)) ≤ − (κ1 + κ2 + κ3 + κ4 )s T (t)sgn(s(t)) +  f (t)s(t) ≤ − (κ1 + κ2 + κ3 + κ4 )|s(t)| + (κ1 + κ2 + κ3 )s(t) ≤ −κ4 s(t). (4.73) Then, formula (4.72) follows:

108

4 Fault Estimation and Tolerant Control for Multiple Time Delayed Switched …

V˙5 (t) ≤ − κ4 s(t) +

p  j=1

σ j (t)

r 

i (μ(t))s T (t)(( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 B¯ f )−1 j

j

j

i=1



( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 ( A¯ 0i − L¯ pi C¯ 0 )e(t) + j

j

j

j

− B¯ wi w(t) − B¯ aj α˜ a (t) f a (t)

j

j

S 

j A¯ ki e(t − τk (t))

k=1



j

≤ − κ4 s(t) + s(t) ×

p 

σ j (t)

j=1

¯ j −1

(S ) ×

p 

j j B¯ f )−1 ( B¯ f )T ( S¯ j )−T

σ j (t)

j=1

r 

j

i (μ(t))

i=1

r 

i (μ(t))(( B¯ f )T ( S¯ j )−T P2 j

j

j

i=1

j j P2 ( S¯ j )−1 ( A¯ 0i

S 

− L¯ pi C¯ 0 ) × e(t) + s(t) j

j

j j j (( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 B¯ f )−1

k=1

j j j ( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 A¯ ki  × e(t − τk (t)) + s(t) ×

p 

σ j (t)

j=1 r 

j j j j j j i (μ(t))(( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 B¯ f )−1 ( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1

i=1 j B¯ wi  × w(t) + s(t) ×

p 

σ j (t)(( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 B¯ f )−1 j

j

j

j=1 j ( B¯ f )T ( S¯ j )−T

j P2 ( S¯ j )−1 B¯ aj 

× α˜ a (t) f a (t).

(4.74)

Next, define ı 0 (t), ı 1k (t), ı 2 (t) and ı 3 (t) as ı 0 (t) =

p 

σ j (t)

j=1

r 



j

j

j

j

i=1

j ( B¯ f )T ( S¯ j )−T p r

ı 1k (t) =

i (μ(t))(( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 B¯ f )−1

σ j (t)

j=1



P2 ( S¯ j )−1 ( A¯ 0i − L¯ pi C¯ 0 ), j

j

j

j

j j j j i (μ(t))(( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 B¯ f )−1

i=1

j j j ( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 A¯ ki ,

ı 2 (t) =

p  j=1

σ j (t)

r 

 j=1

j

j

j

j

i=1

j ( B¯ f )T ( S¯ j )−T p

ı 3 (t) =

i (μ(t)(( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 B¯ f )−1 P2 ( S¯ j )−1 B¯ wi , j

j

j j j j j σ j (t)(( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 B¯ f )−1 ( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 B¯ aj .

4.6 Finite-Time Boundedness

109

Additionally, suppose ı 0 (t) ≤ ı¯0 , ı 1k (t) ≤ ı¯1k , ı 2 (t) ≤ ı¯2 and ı 3 (t) ≤ ı¯3 . Therefore, based on formula (4.74), we have ¯ V˙5 (t) ≤ −s(t), where ¯  κ4 − ı¯0 e(t) −

S 

(4.75)

ı¯1k e(t − τk (t)) − ı¯2 w(t) − ı¯3 α˜ a (t) f a (t).

k=1

This domain is defined as  = {¯ > 0}. Therefore, this dynamic error e(t) reaches  in finite-time and stays here. This completes this proof.

4.6 Finite-Time Boundedness Definition 4.4 For the time interval [t¯, t˜], assuming there exist constants c2 > c1 > 0 and a positive weighting matrix R, model (4.26) is finite-time bounded about ¯ [t¯,t˜], ), if (c1 , c2 , [t¯, t˜], R, W ξ T (t¯)Rξ(t¯) ≤ c1 ⇒ ξ T (t)Rξ(t) ≤ c2 , ∀t ∈ [t¯, t˜] ¯ [t¯,t˜], is defined in Assumption 4.4. where w(t) ∈ W Recalling this partitioning strategy in Song et al. (2017), this closed-loop model ¯ [0,T ], ) in Definition (4.26) is finite-time bounded with respect to (c1 , c2 , [0, T ], R, W 4.4, if and only if there exists an auxiliary scalar c∗ that satisfies c∗ ∈ (c1 , c2 ) such that ¯ [0,T ], ) during the system (4.26) is finite-time bounded about (c1 , c∗ , [0, T ∗ ], R, W ¯ [0,T ], ) during reaching phase, and finite-time bounded about (c∗ , c2 , [T ∗ , T ], R, W the sliding motion phase.

4.6.1 Reachability of T ∗ ≤ T Theorem 4.3 This reachability of this specified sliding surface s(t) in (4.21) through this sliding mode controller u s (t) in (4.22) can be ensured in finite-time T ∗ , where T ∗ < T , if there exists a positive constant , for i = 1, . . . , r, j = 1, . . . , p following  ≤ ¯ and ≥

1 j j j max { max {λmax [(( B¯ f )T ( S¯ j )−T P2i ( S¯ j )−1 B¯ f )−1 ]}} T j=1,..., p i=1,...,r j j × max { max ( B¯ f )T ( S¯ j )−T P2i e(0)}}. j=1,..., p i=1,...,r

(4.76)

Proof Given formula (4.75) and s(t) ≤ |s(t)|, there exists a positive scalar  ≤ ¯ following

110

4 Fault Estimation and Tolerant Control for Multiple Time Delayed Switched …

 V˙5 (t) ≤ −s(t) ≤ − ℘

&

V5 (t),

(4.77)

' j j j where ℘ = 0.5 max { max {λmax [(( B¯ f )T ( S¯ j )−T P2 ( S¯ j )−1 bar B f )−1 ]}}. j=1,..., p i=1,...,r

There exists a moment T ∗ obeying T∗ ≤

2℘ & V5 (0), 

(4.78)

making V5 (t) = 0, correspondingly, t ≥ T ∗ and s(t) = 0. Additionally, considering Rayleigh’s inequality, we can obtain j j j V5 (0) ≤ 0.5 max { max {λmax [(( B¯ f )T ( S¯ j )−T P2i ( S¯ j )−1 B¯ f )−1 ]}}s(0)2 . j=1,..., p i=1,...,r

(4.79) Substituting formula (4.79) into formula (4.78), we have T∗ ≤

1 j j j max { max {λmax [(( B¯ f )T ( S¯ j )−T P2i ( S¯ j )−1 B¯ f )−1 ]}}  j=1,..., p i=1,...,r p  j j × σ j (0)( B¯ f )T ( S¯ j )−T P2 e(0), j=1

1 j j j ≤ max { max {λmax [(( B¯ f )T ( S¯ j )−T P2i ( S¯ j )−1 B¯ f )−1 ]}} T j=1,..., p i=1,...,r j j × max { max ( B¯ f )T ( S¯ j )−T P2i e(0)}}. j=1,..., p i=1,...,r

(4.80)

According to formula (4.76) for , we have T ∗ ≤ T , meaning that for every finitetime T , this closed-loop trajectory (4.26) is driven onto this specified sliding mode surface s(t) = 0 in finite-time T ∗ with T ∗ ≤ T via this sliding mode control law (4.22). Hence this completes the proof.

4.6.2 Finite-Time Boundedness Over Reaching Phase Within [0, T ∗ ] Theorem 4.4 Substitute this sliding mode control law u s (t) in (4.22) into model j j j j (4.26) and choose K h so that it follows A0i + Bui K h is Hurwitz. For constants τk > 0, γ > 0, α > 0,  > 0, ϑ > 1, τ Mk , and switching signals whose average dwell time follows Ta > lnαϑ , this closed-loop model (4.26) can be mean-square exponentially stable under this weighted H∞ index (γ, α) in formula (4.4) and is ¯ [0,T ], ) during reaching phase index, finite-time bounded about (c1 , c∗ , [0, T ∗ ], R, W

4.6 Finite-Time Boundedness

111

¯¯ ), ¯¯ if there and the eigenvalues of ( S¯ j )−1 ( A¯ 0i − L¯ pi C¯ 0 ) belong to this region D(ν, j

j

j

exist constants δ > 0, c∗ > 0, positive definite matrices Pi  diag{P1i , P2i }, Q ki ,  j j j j Rki , Z ki and a matrix Cj = ri=1 i (μ(t))Ci such that for any i, h, υ = 1, . . . , r , j, q = 1, . . . , p and k, k1 , k2 = 1, . . . , S, the following optimization problem has feasible solutions: j

j

j

j

min δ ⎧ ⎪ ⎨ s.t.

(4.27)–(4.32), c1 < c∗ < c2 ,

⎪ ⎩

¯ 1 +γ 2  bc a¯

0, α > 0,  > 0, ϑ > 1, τ Mk , and switching signals of which the average dwell time follows Ta > lnαϑ , the closed-loop system (4.26) when de(t) = 0 can be mean-square exponentially stable under this weighted H∞ performance index (γ, α) in (4.4), and can be finite-time bounded about ¯ [0,T ], ) during the sliding motion phase, and the eigenval(c∗ , c2 , [T ∗ , T ], R, W j j j −1 ¯ j ¯ ¯¯ ), ¯¯ if there exist constants ues of ( S ) ( A0i − L¯ pi C¯ 0 ) belong to this region D(ν, δ > 0, c∗ > 0, positive definite matrices Pi  diag{P1i , P2i }, Q ki , Rki , Z ki and a  j j matrix Cj = ri=1 i (μ(t))Ci such that for i, h, υ = 1, . . . , r ; j, q = 1, . . . , p and k, k1 , k2 = 1, . . . , S, the following optimization problem has feasible solutions: j

j

j

j

j

j

4.6 Finite-Time Boundedness

113

min δ  s.t.

(4.27)–(4.32), (4.81), ¯ ∗ +γ 2  bc a¯

< e−T c2 .

(4.89)

j At the same time, the sliding mode observer gain L¯ s can be designed as formula (4.33).

Proof Formulas (4.27)–(4.32) guarantee that this closed-loop trajectory (4.26) under de(t) = 0 can be mean-square exponentially stable against this weighted H∞ index (γ, α) in (4.4). Thus obviously, it is only necessary to prove that the system is finite¯ [0,T ], ) during the sliding motion phase. time bounded about (c∗ , c2 , [T ∗ , T ], R, W Multiply both sides of formula (4.84) by e−t , and integrate the resulting inequality from T ∗ to t, where t ∈ [T ∗ , T ], and then we can obtain

e

−t

∗

E{V (t)} 0, γ > 0, α > 0,  > 0, g1 > 0, ϑ > 1, τ Mk , and switching signals whose average dwell time follows Ta > lnαϑ , this closed-loop trajectory (4.26) can be meansquare exponentially stable under this weighted H∞ performance index (γ, α) in ¯ [0,T ], ), and the formula (4.4), and is finite-time bounded about (c1 , c2 , [0, T ], R, W j ¯j j −1 ¯ j ¯ ¯ ¯ ¯ eigenvalues of ( S ) ( A0i − L pi C0 ) belong to this region D(ν, ¯ ), ¯ if there exist constants δ > 0, c∗ > 0, g2 > 0, positive definite matrices Pi  diag{P1i , P2i }, Q ki ,  j j j j Rki , Z ki and a matrix Cj = ri=1 i (μ(t))Ci such that for i, h, υ = 1, . . . , r , j, q = 1, . . . , p and k, k1 , k2 = 1, . . . , S, the following optimal problem has feasible solutions: j

j

j

j

114

4 Fault Estimation and Tolerant Control for Multiple Time Delayed Switched …

min δ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ s.t.

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(4.27)–(4.32), (4.81), 1 1 j R − 2 Pi R − 2 > g11 I, R − 2 Pi R − 2 + j

1

S 

1

S  k=1

(4.92)

τk (R − 2 Q ki R − 2 ) + 1

j

1

S  k=1

τk2 (R − 2 Z ki R − 2 ) < −o2 g2 I + 2oI, + k=1  −T ∗ √  −e c + γ 2  c1 g1 < 0, ∗ −g2 −e

−T

c2 g2 g1

1

j

1

+ c∗ + γ 2 g2 < 0.

1 τk (R − 2 α

Rki R − 2 ) j

1

(4.93) (4.94) (4.95)

j At the same time, the sliding mode observer gain L¯ s can be designed as formula (4.33).

Proof By formula (4.92), a¯ > g11 can be obtained. Through formula (4.93), it can be known that b¯ < g12 . It can be calculated that formulas (4.92)–(4.95) can guarantee that formulas (4.82) and (4.89) hold. Remark 4.2 Fault estimation and tolerant control are presented simultaneously for switched T-S fuzzy stochastic systems against multiple time delays and intermittent actuator faults. In the sliding mode observer, the introduction of proportional gain, derivative gain and sliding mode gain makes this observer design flexible and feasible. This effect of Brownian motion can be removed from this proposed sliding mode function through a tricky matrix parameter. In addition, the reachability of this sliding mode surface can be strictly guaranteed. Unlike Sun et al. (2018), this chapter studies fault estimation and fault-tolerant control, which can greatly improve the performance of control systems. Compared with Liu et al. (2013), the fault estimation and the fault-tolerant control of intermittent actuator faults and sensor faults are realized by the sliding mode observer which has broad application prospects. Multiple time delays are discussed that are an extension of a time delay (Gao et al. 2014). Compared with Liu and Shi (2013), this chapter studies finite-time boundedness problems. A new delay-dependent piecewise fuzzy Lyapunov function is proposed, and is less conservative than the ordinary Lyapunov function (Zhang j j j j et al. 2008), where Pi = P, Q ki = Q, Rki = R, Z ki = Z . The state message is not needed in this chapter compared to Zhang et al. (2008). Unlike the fuzzy Lyapunov function (Wang and Lam 2018) which only considers this influence of membership functions on model stability, or the piecewise Lyapunov function (Han et al. 2018a) that only considers the switching effect on system stability, the piecewise fuzzy Lyapunov function considers this influence of both membership functions and switching behavior on model stability. Using the piecewise fuzzy Lyapunov function, delaydependent achievements can be easily acquired, which reduces this conservatism

4.7 Simulation Results

115

of these finding maximum delay bounds, and this exponential stability is guaranteed, whose convergence speed is faster than that of asymptotic stability. Different from Huang and Yang (2014), these stability conditions of this stochastic model are obtained in this case of no slack matrices, which relatively decreases the computational complexity of linear matrix inequalities. Remark 4.3 This switched T-S fuzzy stochastic system against multiple timevarying delays studied in the chapter simultaneously contains multiple intermittent actuator faults, sensor faults, and external disturbances. The sliding mode observer is used to realize fault estimation and fault-tolerant control, which ensures that the switched T-S fuzzy stochastic system is insensitive to faults and can maintain good performance even if faults occur. A delay-dependent piecewise fuzzy Lyapunov function is established to prove the conservativeness issue which is less conservative than the ordinary Lyapunov function. When constructing the Lyapunov function, all factors are considered to reduce the conservatism. Especially, the sliding mode observer estimates the sensor faults and the intermittent actuator faults that have big danger and widely exist in practice, which enormously decreases the conservativeness of j j this studied fault estimation and fault-tolerant control method. By using Q ki , Rki and j Z ki to design the Lyapunov function, it fully considers the effect of multiple delays, which are common and easy to make the model unstable, on system stability, which enormously decreases this stability result in conservativeness. In practical applications, external disturbances are unavoidable and can be attenuated through this H∞ performance index. The conservativeness of the results is reduced by choosing the smallest value of the H∞ performance index γ. Remark 4.4 To prevent this chattering phenomenon in this gain design of the sliding mode observer, replace sgn(s(t)) with s(t)/(s(t) + ), where this value of this constant  > 0 can be very small, and this value of  is generally 0.01. Remark 4.5 If the membership functions are affected by the Wiener process, then j j j the piecewise Lyapunov function used in this chapter, that is, Pi  diag{P1i , P2i }, j j j j j j j j Q ki , Rki , Z ki is replaced with P j  diag{P1 , P2 }, Q k , Rk , Z k .

4.7 Simulation Results Example 4.1 The effectiveness of this method is illustrated by an example. This mechanical system dynamics of the single-link direct-drive manipulator driven through the permanent magnet brush DC motor and this electrical system dynamics of this DC motor are explored simultaneously (Han et al. 2018b), which are considered as a part of an electromechanical system and have parameters as follows:

116

4 Fault Estimation and Tolerant Control for Multiple Time Delayed Switched …

τ1 (t) = 0.2 + 0.1 sin(t), τ2 (t) = 0.4 + 0.2 cos(t), ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ 0 a0 0 0 a0 0 0 a0 0 A101 = ⎣ θ11 b1 a0 θ21 b11 ⎦ , A102 = ⎣ θ11 b2 a0 θ21 b11 ⎦ , A201 = ⎣ θ12 b1 a0 θ22 b12 ⎦ , 0 a 0 θ3 θ4 0 a 0 θ3 θ4 0 a 0 θ3 θ4 ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ 0 a0 0 0 a1 0 0 a1 0 A202 = ⎣ θ12 b2 a0 θ22 b12 ⎦ , A111 = ⎣ 0 a1 θ21 0 ⎦ , A112 = ⎣ 0 a1 θ21 0 ⎦ , 0 a 0 θ3 θ4 0 a 1 θ3 0 0 a 1 θ3 0 ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ 0 a1 0 0 a1 0 0 a2 0 A211 = ⎣ 0 a1 θ22 0 ⎦ , A212 = ⎣ 0 a1 θ22 0 ⎦ , A121 = ⎣ 0 a2 θ21 0 ⎦ , 0 a 1 θ3 0 0 a 1 θ3 0 0 a 2 θ3 0 ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ 0 a2 0 0 a2 0 0 a2 0 A122 = ⎣ 0 a2 θ21 0 ⎦ , A221 = ⎣ 0 a2 θ22 0 ⎦ , A222 = ⎣ 0 a2 θ22 0 ⎦ , 0 a 2 θ3 0 0 a 2 θ3 0 0 a 2 θ3 0 ⎡ ⎤ 0 2 2 1 1 1 Bu2 = Bu1 = Bu2 = Bu1 = ⎣ 0 ⎦ , Ba2 = Ba1 = Bu1 , b0 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 0.01 0.01 0.001 1 1 2 2 Bw1 = ⎣ 0.01 ⎦ , Bw2 = ⎣ 0 ⎦ , Bw1 = ⎣ 0 ⎦ , Bw2 = ⎣ 0.002 ⎦ , 0 0 0.005 0.006 ⎡ ⎤ ⎡ ⎤ 0 0 0 0 0 0 1 1 B01 = ⎣ −0.001 0 0.001 ⎦ , B02 = ⎣ −0.01 0 0.01 ⎦ , 0.001 0 0.002 0.001 0 0.001 ⎡ ⎤ ⎡ ⎤ 0 0 0 0 0 0 2 2 B01 = ⎣ −0.001 0 0.001 ⎦ , B02 = ⎣ −0.001 0 0.001 ⎦ , 0.001 0 0.001 0.001 0.001 0 ⎡ ⎤ ⎡ ⎤ 0 0 0 0 0 0 1 1 ⎣ B11 = ⎣ 0.002 0 0 ⎦ , B12 = 0.001 0.001 0 ⎦ , 0.003 0 0.001 0 0 0.001 ⎡ ⎤ ⎡ ⎤ 0 0 0 0 0 0 2 2 B11 = ⎣ 0.002 0 0 ⎦ , B12 = ⎣ 0.001 0.002 0 ⎦ , 0 0 0.001 0 0 0.001 ⎡ ⎤ ⎡ ⎤ 0 0 0 0 0 0 1 1 B21 = ⎣ 0.001 −0.001 0.001 ⎦ , B22 = ⎣ −0.001 0.001 0.002 ⎦ , 0 0.001 0.001 0 0 −0.001 ⎡ ⎤ ⎡ ⎤ 0 0 0 0 0 0 2 2 ⎣ B21 = ⎣ 0.001 0.002 0.001 ⎦ , B22 = 0.002 0.001 0 ⎦ , 0 0.001 −0.001 0.001 0 0.002 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 100 10 0 −0.5 C01 = ⎣ 1 0 0 ⎦ , C02 = ⎣ 1 0 0 ⎦ , Fs = ⎣ 0 ⎦ , 100 1 0 −0.1 0.4

4.7 Simulation Results

117

    2     1 1 2 C z1 = 0.1 0 0 , C z2 = 0.08 0 0 , C z1 = 0.1 0 0.01 , C z2 = 0.05 0 0.1 . ¯¯ = 18, c1 = 1.3, c2 = 10, Select  = 1.9, γ = 0.0137, α = 0.1, ϑ = 1.5, ν¯¯ = 1,  T = 5, R = I7 ,  = 0.01 and g1 = 2. This occurrence probability of the actuator fault can be α¯ a = 0.95. This average dwell time is Ta > lnαϑ = 4.0547s, and the switching times are 9.0547s, 18.1093s and 27.1640s. The membership functions are 21 (μ(t)) = 11 (μ(t)) =

sin(x1 (t)) + x1 (t) 2 , 2 (μ(t)) = 12 (μ(t)) = 1 − 11 (μ(t)). 2x1 (t)

Suppose sensor and actuator faults as

f s (t) =

0, t < 4, f a (t) = 0.8 + 3 cos(0.4t), t ≥ 4,

0, t < 3, 0.4 + 2 sin(0.5t), t ≥ 3.

It follows that (A101 , C01 ), (A102 , C01 ), (A201 , C01 ), (A202 , C01 ) are detectable. It can be calculated that τ1 = 0.3, τ2 = 0.6, τ M1 = 0.1, and τ M2 = 0.2. Set κ1 = 2.4, κ2 = 1, and κ3 = 3.8. Choose this state feedback gain as follows:   K 21 = K 11 = −66.1927 −23.2055 −223.9805 ,   K 22 = K 12 = −96.0947 −36.3752 −227.5279 . Firstly, choose H 2 = H 1 = diag{1, 0.4, 0.7}, indicating that ⎡

0 ⎢0 ⎢ ⎢0 ⎢ L¯ 2d = L¯ 1d = ⎢ ⎢0 ⎢1 ⎢ ⎣0 0

0 0 0 0 0 0.4 0

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥. 0 ⎥ ⎥ 0 ⎦ 0.7

Next, select β11 = −λmin (( S¯ 1 )−1 A¯ 101 ) + 0.001 = 2.501, β21 = −λmin (( S¯ 1 )−1 A¯ 102 ) + 0.001 = 2.501, β12 = −λmin (( S¯ 2 )−1 A¯ 201 ) + 0.001 = 2.501, β22 = −λmin (( S¯ 2 )−1 A¯ 202 ) + 0.001 = 2.501. j According to formula (4.14), we can obtain L¯ pi , i = 1, 2, j = 1, 2. Based on Theorem 4.6, we have c∗ = 4.8483 as well as g2 = 0.45.

118

4 Fault Estimation and Tolerant Control for Multiple Time Delayed Switched …

Fig. 4.1 The sensor fault f s (t) and its estimations

fs (t) fˆs (t) fˇs (t)

4

Sensor FEs

3

3.8 3.7

2

3.6 15 15.5 16 16.5

1

0.3

0

0.2

-1 0.1 11.2 11.3 11.4

-2 0

Fig. 4.2 The actuator fault f a (t) and its estimations

5

10

15 Times

20

25

30

Actuator FEs

2 1 0 -1.5 -1

-1.6

-2

-1.7 21 0

Fig. 4.3 The state x(t)

5

10

22

15 Times

fa (t) fˆa (t) fˇa (t)

23 20

25

x1 (t) x2 (t) x3 (t)

4 2 State

30

0 -2 -4 -6 0

5

10

15 Times

20

25

30

Figures 4.1, 4.2, 4.3, 4.4, 4.5 and 4.6 show the simulation results. This sensor fault f s (t) and the corresponding estimation fˆs (t) are described in Fig. 4.1, meaning that this sensor fault f s (t) is well estimated. This actuator fault f a (t) and the corresponding estimation fˆa (t) are shown in Fig. 4.2. This means that this actuator fault f a (t) is well estimated. This state x(t) and this error system e(t) are shown in Figs. 4.3 and 4.4, showing that this closed-loop system can be mean-square exponentially sta¯ [0,5],0.01 ), ble under (0.0137, 0.1), it is finite-time bounded about (1.3, 10, [0, 5], I7 , W j j j and all eigenvalues of ( S¯ j )−1 ( A¯ 0i − L¯ pi C¯ 0 ), (i = 1, 2, j = 1, 2) belong to this region D(1, 18). In light of (4.75), ¯ is obtained, which is described in Fig. 4.5. The results show that the dynamic of the error model enters domain  within finite-time T ∗ ≤ T and remains here. Figure 4.6 shows this sliding mode surface s(t). It shows

4.7 Simulation Results

119

Fig. 4.4 The error dynamic e(t)

1 Error dynamic

0 e1 (t) e2 (t) e3 (t) e4 (t) e5 (t) e6 (t) e7 (t)

-1 -2 -3 -4 0

5

10

15 Times

20

25

Fig. 4.5 The trajectory of ¯

30

¯ 5 0 -5 -10 -15 0

10

15 Times

20

25

30

0 Sliding mode surface

Fig. 4.6 The sliding mode surface s(t)

5

-0.5 -1 -1.5 s1 (t) s2 (t)

-2 0

5

10

15 Times

20

25

30

this existence of the sliding mode and proves this reachability of the sliding mode surface s(t) = 0 within finite-time T ∗ ≤ T . To illustrate the advantages of this approach, a comparison with the similar research can be performed. Ordinary Lyapunov functions are widely used (Zhang et al. 2008). fˇs (t) and fˇa (t) represent these estimations of f s (t) and f a (t), separately, as shown in Figs. 4.1 and 4.2, which means that these estimations of f s (t) and f a (t) using the piecewise fuzzy Lyapunov function are closer to f s (t) and f a (t), and smoother. Thus, this effectiveness of the method is strongly proven.

120

4 Fault Estimation and Tolerant Control for Multiple Time Delayed Switched …

4.8 Chapter Summary In this chapter, observer-based fault estimation and fault-tolerant control are investigated for switched T-S fuzzy Itô stochastic models with sensor faults, external disturbances, intermittent actuator faults, and multiple time-varying delays. Firstly, we investigate the descriptor sliding mode observer to establish this error function. Next, according to this online message of fault estimation, a fault-tolerant controller is proposed to stabilize this closed-loop model. Moreover, this chapter presents a piecewise fuzzy Lyapunov function and obtains delay-dependent sufficient conditions using linear matrix inequalities. The designed sliding mode observer is less conservative than the existing sliding mode observers. This method guarantees this reachability of this sliding mode surface in this estimation error space. At the same time, this finite-time boundedness problem is also studied. Finally, the superiority and effectiveness of the method in this chapter are verified by a simulation example.

References Angulo S, Vazquez D, Márquez R, et al (2017) Stability of Takagi-Sugeno fuzzy systems with time-delay: A non-quadratic functional approach. In: 2017 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Naples, Italy, July 2017 Chen CL, Feng G, Guan XP (2005) Delay-dependent stability analysis and controller synthesis for discrete-time T-S fuzzy systems with time delays. IEEE Trans Fuzzy Syst 13(5):630–643 Chibani A, Chadli M, Shi P et al (2017) Fuzzy fault detection filter design for T-S fuzzy systems in the finite-frequency domain. IEEE Trans Fuzzy Syst 25(5):1051–1061 Gan M, Zhang C, Zhao J (2019) Data-driven optimal switching of switched systems. J Frankl Inst 356(10):5193–5221 Gao Z (2015) Fault estimation and fault-tolerant control for discrete-time dynamic systems. IEEE Trans Ind Electron 62(6):3874–3884 Gao H, Liu X, Lam J (2009) Stability analysis and stabilization for discrete-time fuzzy systems with time-varying delay. IEEE Trans Cybern 39(2):306–317 Gao Q, Feng G, Xi Z et al (2014) A new design of robust H∞ sliding mode control for uncertain stochastic T-S fuzzy time-delay systems. IEEE Trans Cybern 44(9):1556–1566 Guo B, Chen Y (2019) Adaptive fault tolerant control for time-varying delay system with actuator fault and mismatched disturbance. ISA Trans 89:122–130 Guo G, Zhao SP (2013) New approach to delay-dependent stability analysis and stabilization for fuzzy systems with time-varying delay. Appl Mech Mater 389:471–476 Han C, Zhang G, Wu L et al (2012) Sliding mode control of T-S fuzzy descriptor systems with time-delay. J Frankl Inst 349(4):1430–1444 Han J, Zhang H, Wang Y et al (2018a) Robust fault detection for switched fuzzy systems with unknown input. IEEE Trans Cybern 48(11):3056–3066 Han J, Zhang H, Wang Y et al (2018b) Fault estimation and fault-tolerant control for switched fuzzy stochastic systems. IEEE Trans Fuzzy Syst 26(5):2993–3003 Huang S, Yang G (2014) Fault tolerant controller design for T-S fuzzy systems with time-varying delay and actuator faults: A k-step fault-estimation approach. IEEE Trans Fuzzy Syst 22(6):1526– 1540 Li G, Chen Y (2018) Controller design for stochastic nonlinear systems with matched conditions. Syst Eng Electron Technol (English) 29(1):160–165

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Li Y, Yang G (2016) Fuzzy adaptive output feedback fault-tolerant tracking control of a class of uncertain nonlinear systems with nonaffine nonlinear faults. IEEE Trans Fuzzy Syst 24(1):223– 234 Li L, Ding SX, Qiu J et al (2016) Weighted fuzzy observer-based fault detection approach for discrete-time nonlinear systems via piecewise-fuzzy Lyapunov functions. IEEE Trans Fuzzy Syst 24(6):1320–1333 Li S, Ahn CK, Xiang Z (2019) Sampled-data adaptive output feedback fuzzy stabilization for switched nonlinear systems with asynchronous switching. IEEE Trans Fuzzy Syst 27(1):200– 205 Liu M, Shi P (2013) Sensor fault estimation and tolerant control for Itô stochastic systems with a descriptor sliding mode approach. Automatica 49(5):1242–1250 Liu M, Wu H (2018) Stochastic finite-time synchronization for discontinuous semi-Markovian switching neural networks with time delays and noise disturbance. Neurocomputing 310:246– 264 Liu M, Cao X, Shi P (2013) Fault estimation and tolerant control for fuzzy stochastic systems. IEEE Trans Fuzzy Syst 21(2):221–229 Liu X, Zhang W, Li Y (2019) H _ index for continuous-time stochastic systems with Markov jump and multiplicative noise. Automatica 105:167–178 Mao Z, Jiang B, Shi P (2011) Observer-based fault-tolerant control for a class of networked control systems with transfer delays. J Frankl Inst 348(4):763–776 Ren Y, Fang Y, Wang A et al (2018) Collaborative operational fault tolerant control for stochastic distribution control system. Automatica 98:141–149 Ren J, Liu X, Zhu H et al (2019) Exponential H∞ synchronization of switching fuzzy systems with time-varying delay and impulses. Fuzzy Sets Syst 365:116–139 Sakthivel R, Sakthivel R, Selvaraj P et al (2017) Delay-dependent fault-tolerant controller for time-delay systems with randomly occurring uncertainties. Int J Robust Nonlinear Control 27(18):5044–5060 Sakthivel R, Kavikumar R, Ma Y et al (2018) Observer-based H∞ repetitive control for fractionalorder interval type-2 TS fuzzy systems. IEEE Access 6:49828–49837 Shen H, Li F, Yan H et al (2018) Finite-time event-triggered H∞ control for T-S fuzzy Markov jump systems. IEEE Trans Fuzzy Syst 26(5):3122–3135 Shen H, Chen M, Wu Z et al (2020a) Reliable event-triggered asynchronous extended passive control for semi-Markov jump fuzzy systems and its application. IEEE Trans Fuzzy Syst 28(8):1708– 1722 Shen H, Xing M, Wu Z et al (2020b) Multiobjective fault-tolerant control for fuzzy switched systems with persistent dwell-time and its application in electric circuits. IEEE Trans Fuzzy Syst 28(10):2335–2347 Shi P, Su X, Li F (2016) Dissipativity-based filtering for fuzzy switched systems with stochastic perturbation. IEEE Trans Autom Control 61(6):1694–1699 Song J, Niu Y, Zou Y (2017) Finite-time stabilization via sliding mode control. IEEE Trans Autom Control 62(3):1478–1483 Su X, Shi P, Wu L et al (2016) Fault detection filtering for nonlinear switched stochastic systems. IEEE Trans Autom Control 61(5):1310–1315 Sun S, Zhang H, Wang Y et al (2018) Dynamic output feedback-based fault-tolerant control design for T-S fuzzy systems with model uncertainties. ISA Trans 81:32–45 Tong S, Zhang L, Li Y (2016) Observed-based adaptive fuzzy decentralized tracking control for switched uncertain nonlinear large-scale systems with dead zones. IEEE Trans Syst Man Cybern: Syst 46(1):37–47 Vatankhah R, Asemani MH (2017) Output feedback control of piezoelectrically actuated nonclassical micro-beams using T-S fuzzy model. J Frankl Inst 354(2):1042–1065 Vu V, Wang W, Zurada JM et al (2018) Unknown input method based observer synthesis for a discrete time uncertain T-S fuzzy system. IEEE Trans Fuzzy Syst 26(2):761–770

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Wang L, Lam H (2018) A new approach to stability and stabilization analysis for continuous-time Takagi-Sugeno fuzzy systems with time delay. IEEE Trans Fuzzy Syst 26(4):2460–2465 Wang T, Tong S (2017) Observer-based output-feedback asynchronous control for switched fuzzy systems. IEEE Trans Cybern 47(9):2579–2591 Wang Y, Zhang H, Wang X et al (2010) Networked synchronization control of coupled dynamic networks with time-varying delay. IEEE Trans Cybern 40(6):1468–1479 Wang Y, Zheng WX, Zhang H (2017) Dynamic event-based control of nonlinear stochastic systems. IEEE Trans Autom Control 62(12):6544–6551 Wang Z, Shen L, Xia J et al (2018) Finite-time non-fragile l2 − l∞ control for jumping stochastic systems subject to input constraints via an event-triggered mechanism. J Frankl Inst 355(14):6371– 6389 Wu H, Li H (2007) New approach to delay-dependent stability analysis and stabilization for continuous-time fuzzy systems with time-varying delay. IEEE Trans Fuzzy Syst 15(3):482–493 Xing M, Xia J, Huang X et al (2019a) On dissipativity-based filtering for discrete-time switched singular systems with sensor failures: a persistent dwell-time scheme. IET Control Theory Appl 13(12):1814–1822 Xing M, Xia J, Wang J et al (2019b) Asynchronous H∞ filtering for nonlinear persistent dwell-time switched singular systems with measurement quantization. Appl Math Comput 362:124578 Zhang H, Wang Y, Liu D (2008) Delay-dependent guaranteed cost control for uncertain stochastic fuzzy systems with multiple time delays. IEEE Trans Cybern 38(1):126–140 Zhang J, Shi P, Xia Y (2010) Robust adaptive sliding-mode control for fuzzy systems with mismatched uncertainties. IEEE Trans Fuzzy Syst 18(4):700–711 Zhang D, Han QL, Jia X (2015) Network-based output tracking control for T-S fuzzy systems using an event-triggered communication scheme. Fuzzy Sets Syst 273:26–48 Zhang C, Hu J, Qiu J et al (2017) Event-triggered nonsynchronized H∞ filtering for discrete-time T-S fuzzy systems based on piecewise Lyapunov functions. IEEE Trans Syst Man Cybern: Syst 47(8):2330–2341 Zhang H, Han J, Wang Y et al (2018) Sensor fault estimation of switched fuzzy systems with unknown input. IEEE Trans Fuzzy Syst 26(3):1114–1124 Zhang H, Wang Y, Zhang J et al (2019) An SOS-based sliding mode controller design for a class of polynomial fuzzy systems. IEEE Trans Fuzzy Syst 27(4):749–759 Zhang H, Wang Y, Wang Y et al (2020) A novel sliding mode control for a class of stochastic polynomial fuzzy systems based on SOS method. IEEE Trans Cybern 50(3):1037–1046

Chapter 5

Fault-Tolerant Control for Multiple Interval Time Delayed Switched Fuzzy Systems with Intermittent Faults

5.1 Introduction Chapters 2–4 all study the problems of the active fault-tolerant controller. Assertive fault-tolerant control greatly improves the performance of the control system, which has attracted extensive attention from scholars (Gao and Ding 2007). However, active fault-tolerant control requires adjusting the controller parameters after a fault occurs, and may also change this controller structure. Therefore, more control algorithms need to be designed. In addition, the characteristic of passive fault-tolerant control is that an invariant controller is adopted to ensure the robustness of the closed-loop system to specific faults regardless of whether the fault occurs or not. Therefore, a fault diagnosis unit is not required, that is, it does not need any real-time fault message. At the same time, the research on passive fault-tolerant control is not deep enough (Cao and Wang 2018), which is instructive for us to carry out this work. Meanwhile, passive fault-tolerant control of intermittent faults deserves further development. In addition, model uncertainties occur due to many factors, such as additive unknown internal or external noise, environmental influence, nonlinearity, and poor knowledge of objects, which are ubiquitous in real life and industry, and are inevitable (Cao and Sun 1998; Corless and Tu 1998; Cui and Xu 2019; Hashimoto and Amemiya 2011; Chen et al. 2019; Lu et al. 2019), but Chaps. 2–4 are not covered. Cui and Xu (2019) studied the robust H∞ persistent dwell time control problem of switched discrete-time T-S fuzzy models against uncertainty and time-varying delay. Chen et al. (2019) solved the analytical design problem of active disturbance rejection control for nonlinear uncertain time delayed models. Lu et al. (2019) discussed the problem of adaptive tracking control for uncertain Euler-Lagrangian models against external disturbances. It is worth pointing out that the study on uncertain switching T-S fuzzy systems is valuable, but it has not been received enough attention, which arouses our research enthusiasm. On the premise of guaranteeing the system stability, this use of reliable control can tolerate faults and maintain the stability performance of the closed-loop system © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Sun et al., Fault-Tolerant Control for Time-Varying Delayed T-S Fuzzy Systems, Intelligent Control and Learning Systems 9, https://doi.org/10.1007/978-981-99-1357-2_5

123

124

5 Fault-Tolerant Control for Multiple Interval Time Delayed Switched …

(Tao et al. 2015; Qiu et al. 2016). Reliable control is realized through many ways, for instance, H _ control, H2 control and H∞ control (Abdollahi and Khorasani 2011; Li et al. 2017). Abdollahi and Khorasani (2011) designed a decentralized Markovian jump H∞ control routing strategy for mobile multi-agent network systems. Li et al. (2017) realized the control of a grid-forming inverter based on the sliding mode and mixed H2 /H∞ control. Therefore, this use of H∞ control can well ensure the stability of the system. Nevertheless, the related study on switched T-S fuzzy systems against parameter uncertainties and multiple time delays is still insufficient, which inspires our research enthusiasm. In the design of the controller, in order to achieve a certain performance index, many factors need to be considered. One of these approaches is called guaranteed cost control (Liu et al. 2014). In Wang et al. (2010a), an improved predictive control method was used to realize the guaranteed cost control of the network control plant. In Wu et al. (2020), the guaranteed cost control problem of hybrid-triggered network models against stochastic cyberattacks was studied. However, guaranteed cost control for switched fuzzy systems against multiple time-varying delays has rarely been proposed. In many traditional stability analyses, it must be satisfied that the time-varying delay derivative is less than 1 (Huang and Yang 2014; Han 2002). However, the derivative of a time-varying delay is difficult to obtain in a number of cases, or is greater than 1, or even is impossible to differentiate at all. Therefore, when only this time-varying delay bound is known and belongs to [0, ∞), passive fault-tolerant control against time-varying delay in a specific interval is necessary, but the research is weak, which is very important and promotes the research for this chapter.

5.2 System Definition and Description A switched T-S fuzzy model with model uncertainties and multiple time-varying delays can be described below. ⎧  N¯ N˜ r ⎪    j ⎪ j j ⎪ x(t) ˙ = σ (t)  (ν(t)) (Aιi + Aιi )x(t − τι (t)) ⎪ j ⎪ i ⎪ ⎪ ι=0 j=1 i=1 ⎪ ⎪ ⎪ j j j j ⎪ +(Aui + Aui )u(t) + θa (t)(Aai + Aai ) f a (t) ⎪ ⎪  ⎪ ⎪ ⎪ j j j j ⎪ ⎪ + A )g(x(t)) + (A + A )w(t) , +(A ⎪ gi gi wi wi ⎪ ⎪ ⎪ ⎪  ⎨ r N˜ N¯    j j j y(t) = σ j (t) i (ν(t)) (Cιi + Cιi )x(t − τι (t)) ⎪ ⎪ ι=0 j=1 i=1 ⎪  ⎪ ⎪ ⎪ j j j j ⎪ ⎪ +θ (t)(F + F ) f (t) + (A + A )v(t) , s ⎪ si si s vi vi ⎪ ⎪ ⎪ ⎪ ⎪ r N˜ ⎪   ⎪ j j j ⎪ ⎪ y (t) = σ j (t) i (ν(t))(C zi + C zi )x(t), L ⎪ ⎪ ⎪ j=1 i=1 ⎪ ⎪ ⎩ x(t) = φ(t), t ∈ [−τ , 0], i = 1, 2, . . . , r,

(5.1)

5.2 System Definition and Description

125

where x(t) ∈ R n x , u(t) ∈ R n u , f a (t) ∈ R n fa and w(t) ∈ R n w represent the state, system input, actuator fault under this case that the fault probability is 1 and the fault form is the same as the permanent fault, (abbreviated as actuator fault) and external disturbance, separately. y(t) ∈ R n y , f s (t) ∈ R n fs and v(t) ∈ R n v represent the system measurement output, sensor fault under this case in which the fault probability is 1 and the fault form is the same as the permanent fault, (abbreviated as sensor fault) and measurement noise, separately. y L (t) ∈ R n yL represents the desired conj j j j j j j j j trolled output. w(t), v(t) ∈ L 2 [0, ∞). Aιi , Aui , Aai , A gi , Awi , Cιi , Fsi , Avi and C zi (i = 1, . . . , r, j = 1, . . . , N˜ , ι = 1, . . . , N¯ ) denote known constant matrices of suitable dimensions. τ0 (t) = 0. τι (t) represents the time-varying state delay following 0 < τdι ≤ τι (t) ≤ τ Dι , where τdι and τ Dι denote known constants, representing the lower and upper bounds, respectively. In addition, define τ as the maximum value of τ Dι , that is, τ = max{τ D1 , . . . , τ D N¯ }. φ(t) is a vector-valued initial function, where j j j j j j j j j j t ∈ [−τ , 0]. i  {Aιi , Aui , Aai , A gi , Awi , Cιi , Fsi , Avi , C zi } represent unknown matrices with appropriate dimensions, representing the model uncertainties in the system (5.1), and have the following form:



j j j j j j j j j j j j Aιi Aui Aai A gi Awi = E xi G xi Fx1ιi Fx2i Fx3i Fx4i Fx5i ,



j j j j j j j j j j j j Cιi Fsi Avi = E yi G yi Fy1ιi Fy2i Fy3i , C zi = E zi G zi Fzi , j

j

j

j

j

j

j

j

j

j

j

j

where E xi , Fx1ιi , Fx2i , Fx3i , Fx4i , Fx5i , E yi , Fy1ιi , Fy2i , Fy3i , E zi and Fzi all reprej

j

j

sent known constant matrices. G xi , G yi and G zi represent unknown matrix functions j

j

j

j

whose elements are Lebesgue measurable and satisfy (G xi )T G xi ≤ I, (G yi )T G yi ≤ j j I, (G zi )T G zi ≤ I . σ j (t) ( j = 1, . . . , N˜ ) [0, ∞) → {0, 1} is the switching signal, j

representing the activated subsystem at the switching time. i (ν(t)) consists of ¯ ip premise variable ν p (t) and fuzzy set represented by membership function  (i = 1, . . . , r ; p = 1, . . . , l). r is the number of IF-THEN rules, l is the number of premise variables, N¯ denotes this number of delays and N˜ denotes this number of switching subsystems. Besides, g(x(t)) ∈ R n g is Lipschitz, representing a nonlinear function, subject to the following sector bound condition (Zhang et al. 2017a): [g(x(t)) − V¯1 x(t)]T [g(x(t)) − V¯2 x(t)] ≤ 0, ∀x(t) ∈ R n x , where V¯1 , V¯2 ∈ R n g ×n x represent known constant matrices. And it could be converted into this form as follows: 

x(t) g(x(t))

T 

V1 V2 ∗ I



 x(t) ≤ 0, g(x(t))

where V1 = sym(V¯1T V¯2 )/2, V2 = −(V¯1T + V¯2T )/2. For intermittent sensor fault θs (t) f s (t) as well as intermittent actuator fault θa (t) f a (t), θs (t) and θa (t) cause this random occurrence of this fault phenomenon. It is assumed that intermittent faults satisfy the Bernoulli distribution, whose value is either 0 or 1. This occurrence of fault follows:

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5 Fault-Tolerant Control for Multiple Interval Time Delayed Switched …



P{θs (t) = 1} = θ¯s , P{θs (t) = 0} = 1 − θ¯s ,



P{θa (t) = 1} = θ¯a , P{θa (t) = 0} = 1 − θ¯a ,

where θ¯s , θ¯a ∈ [0, 1] are known scalars. Therefore, the expectations of θs (t) and θa (t) are E(θs (t)) = θ¯s and E(θa (t)) = θ¯a , separately. For simplicity, use σ j to represent j j σ j (t) and i to represent i (ν(t)).

5.3 Controller Design This passive dynamic output feedback controller of the multiple time delayed switched T-S fuzzy system against model uncertainties is designed below ⎧ r r N˜ ⎪ ⎪ ⎪ j j j ⎪ x ˙ (t) = σ  κj [Aciκ xc (t) + Bci y(t)], ⎪ c j i ⎪ ⎪ ⎪ κ=1 j=1 i=1 ⎪ ⎨ ˜ r N j j ⎪ ⎪ σj i E ci xc (t), u(t) = ⎪ ⎪ ⎪ ⎪ j=1 i=1 ⎪ ⎪ ⎪ ⎩ xc (t) = 0, t ∈ [−τ , 0], j

j

(5.2)

j

where xc (t) ∈ R n x means this controller state. Aciκ , Bci , E ci stand for controller matrices. T T

ˇ = w T (t), v T (t), f aT (t), f sT (t) , and Define x(t) ˇ = x T (t), xcT (t) , and w(t) this closed-loop system can be written as ⎧ r r N˜ N¯ ⎪ ⎪ ⎪ j j j j ˙ ⎪ x(t) ˇ = σ   [ ˇ − τι (t)) + Aˇ gi g(x(t)) Aˇ ιiκ x(t ⎪ j κ i ⎪ ⎪ ⎪ κ=1 ι=0 j=1 i=1 ⎪ ⎪ ⎪ ⎪ j ⎨ ˇ + Aˇ (t)w(t)], wiκ

⎪ N˜ r ⎪ ⎪ j j ⎪ ⎪ y (t) = σ i Cˇ zi x(t), ˇ L j ⎪ ⎪ ⎪ ⎪ j=1 i=1 ⎪ ⎪ ⎪ ⎩ ˇ x(t) ˇ =φ(t), t ∈ [−τ , 0],

(5.3)

where 

   j j j j j j j A0i + A0i Aιi + Aιi (Aui + Aui )E cκ 0 j ˇ , , Aιiκ = j j j j j Bci (Cικj + Cικj ) 0 Bci (C0κ + C0κ ) Aciκ  j

j  A gi + A gi j j ˇ , Cˇ zi = C zij + C zij 0 , A gi = 0 j Aˇ 0iκ =

5.3 Controller Design

127

(t) = diag{In w , In v , θa (t)In fa , θs (t)In fs },   j j j j A + A 0 A + A 0 j wi wi ai ai Aˇ wiκ = , j j j j j j 0 Bci (Avκ + Avκ ) 0 Bci (Fsκ + Fsκ ) 

ˇ = φT (t) 0T T , t ∈ [−τ , 0], ι = 1, . . . , N¯ . φ(t) Define τaι = 21 (τ Dι + τdι ), ρι = 21 (τ Dι − τdι ). It is obvious that t−τ  aι

x(t ˇ − τaι ) − x(t ˇ − τι (t)) −

˙ˇ x(ϑ)dϑ = 0.

(5.4)

t−τι (t)

This closed-loop model (5.3) is redescribed ⎧ N˜ N¯ r r ⎪ ⎪ j j ⎪ j ˇj ˙ ⎪ x(t) ˇ = σ   [ A x(t) ˇ + ˇ − τaι ) Aˇ ιiκ x(t j ⎪ κ i 0iκ ⎪ ⎪ ⎪ κ=1 ι=1 j=1 i=1 ⎪ ⎪ ⎪ ⎪ t−τ ⎪  aι N¯ ⎨ j j j ˙ˇ ˇ − ˇ Aιiκ x(ϑ)dϑ + Aˇ gi g(x(t)) + Aˇ wiκ (t)w(t)], ⎪ ⎪ ι=1 ⎪ t−τι (t) ⎪ ⎪ ⎪ ⎪ ˜ ⎪ r N ⎪ j j ⎪ ⎪ ⎪ σj i Cˇ zi x(t). ˇ ⎪ y L (t) = ⎩ j=1

(5.5)

i=1

For matrices 1 > 0 and 2 > 0, a constant α > 0, the cost function of this whole closed-loop system is shown by ∞ J=

e−αt [x T (t)1 x(t) + u T (t)2 u(t)]dt.

(5.6)

0

Definition 5.1 (Su et al. (2016)) With σ j , this switched fuzzy model (5.5) when w(t) = 0, v(t) = 0, f a (t) = 0, f s (t) = 0 as well as u(t) = 0 can be called meansquare exponentially stable, if 2 2 −πt ¯ ˇ x(t) ˇ ≤ εφ(0) , a˜ e

(5.7)

2 2 2 ˙ˇ ˇ ˇ , x(t) }. where the constants ε, π¯ > 0, φ(0) a˜  max { x(t) t∈[−τ ,0]

Definition 5.2 (Su et al. (2016)) For the assigned constants γ > 0 and α > 0, this switched model (5.5) when u(t) = 0 is mean-square exponentially stable under this weighted H∞ performance index (γ, α), and follows the guaranteed cost control, if with (t)w(t) ˇ = 0, the system will be mean-square exponentially stable, and with (t)w(t) ˇ = 0 as well as zero initial conditions, one has

128

5 Fault-Tolerant Control for Multiple Interval Time Delayed Switched …

 ∞   ∞  −αt 2 2 2 E e y L (t) dt < γ E (t)w(t) ˇ dt . 0

(5.8)

0

The performance index satisfies J ≤ J ∗ , where J ∗ can be called the guaranteed cost. According to Definition 5.2, if the scalar α is close to 0, this weighted H∞ performance index is close to the classical H∞ performance index. Besides, the smaller α is, the better the interference attenuation performance is.

5.4 Stability Analysis Theorem 5.1 Given these positive constants τdι , τ Dι , α, γ, π¯ > 1 and any switching signal under the average dwell time satisfying Ta > lnαπ¯ , this closed-loop model (5.5) will be mean-square exponentially stable under this H∞ performance index (γ, α) in formula (5.8), and satisfy guaranteed cost control, if there exists a positive constant j j j j j j j Aciκ , Bci , E ci , and slack o, positive definite

matrices Pi , Q ιi , Rιi , Z ιi , matrices j

j

j j j j j j j matrices W1  W10j W111 · · · W11 N¯ W12 , W2i  W20i W211i · · · W21 N¯ i W22i , where i, κ = 1, . . . , r , j, jˆ = 1, . . . , N˜ , ι, ι1 , ι2 , ιˆ = 1, . . . , N¯ , ι1 = ι2 , so that the following inequalities work: j j j ≤ 0, R˙ ι ≤ 0, Z˙ ι ≤ 0, P˙j ≤ 0, Q˙ ι j + ιˆκi < 0, i ≤ κ, ˆ j jˆ ¯ j , Q ι < μQ ¯ ι , Pj < μP

(5.9)

j ιˆiκ

(5.10) j Rι
0, (ι = 1, . . . , N¯ ). According to formula (5.20), for 1 ≤ i ≤ κ ≤ r , we can find that if

134

5 Fault-Tolerant Control for Multiple Interval Time Delayed Switched …



j j j j ϒˇ¯ ιˆiκ + ϒ¯ˇ ιˆκi < 0, ϒιˆiκ + ϒιˆκi < 0, when τιˆ (t) = τaˆι , j j ϒˇ ιˆiκ + ϒˇ ιˆκi < 0,

when τιˆ (t) = τaˆι ,

(5.21)

then 2 ˇ } < 0. E{V˙ (t) + αV (t) + yL (t)2 − γ 2 (t)w(t)

(5.22)

j

In light of |τι (t)| ≤ ρι , through ϒιˆiκ , we can obtain j j ϒιˆiκ ≤ ϒ¯ ιˆiκ ,

(5.23)

where 

 j j −Miκ ϒ¯ 5ˆιiκ = , j ∗ ϒ¯ 55ˆιi

T j j T T ¯j ¯ j ¯ )T , (ϒ¯ j )T , 0T , 0T . ϒ¯ 5ˆιiκ = (ϒ¯ 05ˆ ) , ( ϒ ) , . . . , ( ϒ ιiκ 151ˆιiκ 25ˆιiκ 15 N ιˆiκ j ϒ¯ ιˆiκ

So ϒ¯ ιˆiκ < 0 means ϒιˆiκ < 0. Given Lemma 1.9 and formula (5.23), if (5.10) holds, then formula (5.21) can be obtained, and then (5.22) holds. First, the mean-square exponential stability of system (5.5) is discussed under this condition that (t)w(t) ˇ = 0. Let N (t1 , t2 ) be this switching number over this time interval [t1 , t2 ). According to (5.22), we have V˙ (t) < −αV (t). Then (eαt V (t)) < 0. Assuming t ∈ [tk , tk+1 ), we can obtain V (t) < e−α(t−tk ) V (tk ). According to formulas (5.11) and (5.13), one has V (t) < μV ¯ (t − ). So j

j

V (t) < μ¯ N (0,t) e−αt V (0).

(5.24)

According to Definition 4.1, formula (5.24) can be converted into μ¯

V (t) < e−(α− Ta )t V (0).

(5.25)

According to formula (5.13), we have V (t) ≥

N˜

σ j xˇ T (t)Pj x(t) ˇ ≥

N˜

j=1

σj

r

j=1 j

j

j

i λmin (Pi )xˇ T (t)x(t) ˇ

i=1

ˇ , ≥ min λmin (Pi )x(t) 2

(5.26)

i=1,...,r, j=1,..., N˜

V1 (0) =

N˜ j=1

σ j xˇ

T

(0)Pj x(0) ˇ

≤

N˜ j=1

σj

r i=1

j

j

i λmax (Pi )xˇ T (0)x(0) ˇ

5.4 Stability Analysis

135 j

2 ≤ max λmax (Pi )x(0) ˇ ,

(5.27)

i=1,...,r, j=1,..., N˜

V2 (0) =

N˜

N¯ 0

σj

ι=1 −τ

j=1

≤

N˜

aι

r

σj

j=1

≤

N¯ ι=1

V3 (0) =

N˜

≤

V4 (0) =

N˜

j

σj

ι=1 −τ

aι

r

σj

N˜

ϑ N¯ τaι

α

j ˙ˇ λmax (Rιi ) max x˙ˇ T (t)x(t) t∈[−τ ,0]

2 ˙ˇ max λmax (Rιi ) max x(t) , j

α

−τaι +ρι0 N¯

σj

(5.29)

t∈[−τ ,0]

i=1,...,r, j=1,..., N˜

j ˙ eα x˙ˇ T ()Z ι x()ddϑ ˇ

ι=1 −τ −ρ ϑ aι ι r

σj

i=1

N¯ 2ρι ι=1

j ˙ eα x˙ˇ T ()Rι x()ddϑ ˇ

ι=1

i=1

j=1

≤

i

(5.28)

t∈[−τ ,0]

N¯ 0 0

j=1

≤

t∈[−τ ,0]

i=1,...,r, j=1,..., N˜

N¯ τaι ι=1

ι=1

j

τaι λmax (Q ιi ) max xˇ T (t)x(t) ˇ

2 τaι max λmax (Q ιi ) max x(t) ˇ ,

j=1

≤

N¯

j

i

i=1

j=1 N˜

j eαϑ xˇ T (ϑ)Q ι x(ϑ)dϑ ˇ

α

i

N¯ 2ρι ι=1

α

j ˙ˇ λmax (Z ιi ) max x˙ˇ T (t)x(t) t∈[−τ ,0]

2 ˙ˇ max λmax (Z ιi ) max x(t) . j

(5.30)

t∈[−τ ,0]

i=1,...,r, j=1,..., N˜

Therefore 2 ˇ , V (t) ≥ δ1 x(t)

(5.31)

2 ˇ δ2 φ(0) a˜ ,

(5.32)

V (0) ≤ where j

j

δ1 = min λmin (Pi ), δ2 = max λmax (Pi ) + i=1,...,r, j=1,..., N˜

i=1,...,r, j=1,..., N˜

N¯  τaι max λmax (Q ιi ) ι=1

i=1,...,r, j=1,..., N˜

136

5 Fault-Tolerant Control for Multiple Interval Time Delayed Switched …

 τaι 2ρι j j max λmax (Rιi ) + max λmax (Z ιi ) . + α i=1,...,r, α i=1,...,r, j=1,..., N˜

j=1,..., N˜

2 ≤ Combining formulas (5.25), (5.31) and (5.32), it follows that x(t) ˇ −(α− Tμ¯a

δ2 ˇ φ(0)a2˜ δ1

)t e . Given Definition 5.1, this closed-loop T-S fuzzy model (5.5) can be meansquare exponentially stable. Then, (t)w(t) ˇ = 0 is considered. Formula (5.22) can be transformed to 2 ˇ }. E{V˙ (t) + αV (t)} < −E{yL (t)2 − γ 2 (t)w(t)

(5.33)

2 ˇ }. E{(eαt V (t)) } < −e−αt E{y L (t)2 − γ 2 (t)w(t)

(5.34)

Then

For t ∈ [tk , tk+1 ), it yields   t αtk α T 2 T ¯ E{e V (t)} 1, this matrix X j , as well as any switching signal whose average dwell time satisfies Ta > lnαμ¯ , the closed-loop model (5.5) without model uncerj tainties, i.e., i =0 is mean-square exponentially stable under H∞ performance index (γ, α) in formula (5.8) and follows optimal guaranteed cost control, if there exists a positive scalar o, a positive scalar or apositive definite matrix   0 , j ¯j j j ¯ ¯ ¯ P1i P2i ¯ j Q 1ιi Q 2ιi ¯ j j 1ι , 2ι , 3ι , positive definite matrices P¯i  , Rιi  j , Q ιi  j ∗ P¯3i ∗ Q¯ 3ιi     j j j j Z¯ 1ιi Z¯ 2ιi R¯ 1ιi R¯ 2ιi ¯ j j j j , Z ιi  , matrices Y j , K¯ j , Aˆ ciκ , Bˆ ci , Eˆ ci , and slack matrij j ∗ R¯ 3ιi ∗ Z¯ 3ιi

j j j j j ces W¯ 2i  W¯ 20i W¯ 211i · · · W¯ 21 N¯ i W¯ 22i , where i, κ = 1, . . . , r , j, jˆ = 1, . . . , N˜ , ι, ι1 , ι2 , ιˆ = 1, . . . , N¯ , ι1 = ι2 , such that the following optimal problem is feasible: min{tr{0 } +

N¯ ι=1

[tr{1ι } + tr{2ι } + tr{3ι }]}

138

5 Fault-Tolerant Control for Multiple Interval Time Delayed Switched …

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ s.t.

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

j j j P˙¯j ≤ 0, Q˙¯ ι ≤ 0, R˙¯ ι ≤ 0, Z˙¯ ι ≤ 0, j j ¯ + ¯ < 0, i ≤ κ, ιˆiκ ιˆκi ˆ j j j jˆ jˆ jˆ < μ¯ Q¯ ι , R¯ ι < μ¯ R¯ ι , Z¯ ι < μ¯ Z¯ ι , P¯j < μ¯ P¯j , Q¯ ι   T ˆ −0 0 < 0, j j j ∗ ( 0i )2 P¯3i − 2 0i I   T ˆ 1ι −1ι  < 0, j j j ∗ ( 1ιi )2 Q¯ 3ιi − 2 1ιi I   T ˆ 2ι −2ι  < 0, j j j ∗ ( 2ιi )2 R¯ 3ιi − 2 2ιi I   T ˆ 3ι −3ι  < 0, j j j ∗ ( 3ιi )2 Z¯ 3ιi − 2 3ιi I

(5.41) (5.42) (5.43) (5.44) (5.45) (5.46) (5.47)

where   j j ¯ 12iκ ¯ j 11ˆ ι iκ ¯ ιˆiκ = ¯ 22 , ∗ ⎤ ⎡ j j j j j j j j ϒ¯¯ 00iκ ϒ¯¯ 011iκ · · · ϒ¯¯ 01 N¯ iκ ϒ¯¯ 02iκ ϒˇˇ 03i ϒˇˇ 04iκ ϒˇˇ 05ˆιi ϒ¯¯ 05ˆιiκ ⎥ ⎢ ˇˇ j ϒˇˇ j ˇˇ j ¯¯ j ¯¯ j ¯¯ j ⎥ ⎢ ∗ ϒ¯¯ j ϒ ϒ ϒ ϒ · · · ϒ ⎢ 111iκ 121iκ 131i 141iκ 151ˆιi 151ˆιiκ ⎥ 11 N¯ iκ ⎥ ⎢ .. .. .. .. .. .. .. ⎥ ⎢ ∗ . ∗ . . . . . . ⎥ ⎢ ⎥ ⎢ j j j j j j ˇ ˇ ˇ ¯ ¯ ¯ ⎥ ⎢ ∗ ˇ ˇ ˇ ¯ ¯ ¯ ϒ ϒ ϒ ϒ ϒ ∗ ∗ ϒ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ⎢ N 1 N iκ 12 N iκ 13 N i 14 N i 15 N ιˆi 15 N ιˆiκ ⎥ ¯j = ⎥, ⎢ j j j j j 11ˆιiκ ⎢ ∗ ∗ ∗ ∗ ϒ¯¯ 22i ϒˇ 23i ϒˇ 24iκ ϒˇˇ 25ˆιi ϒ¯¯ 25ˆιiκ ⎥ ⎥ ⎢ ⎢ ∗ ∗ ∗ ∗ ∗ ϒˇ 33 0 0 0 ⎥ ⎥ ⎢ ⎢ ∗ 0 0 ⎥ ∗ ∗ ∗ ∗ ∗ ϒˇ 44 ⎥ ⎢ ⎥ ⎢ j ⎣ ∗ 0 ⎦ ∗ ∗ ∗ ∗ ∗ ∗ ϒˇˇ 55ˆιi j ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ϒ¯¯ 55ˆιi ⎡¯ j ⎤ j j ϒ¯ 06i ( X¯ 1 )T ( E¯ cκ )T ⎢ 0 0 0 ⎥ ⎢ ⎥ ⎢ 0 0 0 ⎥ ⎢ ⎥ ⎢ . .. .. ⎥ ⎢ .. . . ⎥ ⎢ ⎥ ¯ j −1 ¯ 12iκ , = diag{−I, −−1 =⎢ 0 1 , −2 }, 0 0 ⎥ ⎢ ⎥ 22 ⎢ 0 0 0 ⎥ ⎢ ⎥ ⎢ 0 0 0 ⎥ ⎢ ⎥ ⎣ 0 0 0 ⎦ 0 0 0 and

5.4 Stability Analysis N¯

j ϒ¯¯ 00iκ =

ι=1

139

j j j j j j ¯ j, Q¯ ιi + sym[ Aˇ¯ 0iκ + N¯ (W¯ 20i )T ] − o(V¯0 )T V1 V¯0 + α P¯i + 



 j j j j 

A0i X j + Aui Eˆ cκ A0i j , V¯0 = X j I , j j j j j T (Y ) A0i + Bˆ ci C0κ Aˆ ciκ   j T j j j j j T ¯ j = 0 (X ) 1 , ϒ¯¯ 01ιiκ  = Aˇ¯ ιiκ + 1ι Aˇ¯ 0iκ − (W¯ 20i )T + N¯ W¯ 21ιi , ∗ 1   j j j A X A j ιi ιi Aˇ¯ ιiκ = , j j j j (Y j )T Aιi X j + Bˆ ci Cικj X j (Y j )T Aιi + Bˆ ci Cικj j Aˇ¯ 0iκ =

j j j j j j j ϒ¯¯ ι1 1ι2 iκ = 1ι1 Aˇ¯ ι2 iκ − (W¯ 21ι1 i )T + 1ι2 ( A¯ˇ ι1 iκ )T − W¯ 21ι2 i , j j j j j ϒ¯¯ ι1ιiκ = −e−ατaι Q¯ ιi + sym[1ι A¯ˇ ιiκ − (W¯ 21ιi )T ],

 j T j (X ) K¯ j j j j j j j , ϒ¯¯ 02iκ = P¯i − (W¯ 10 )T + 2 ( Aˇ¯ 0iκ )T + N¯ W¯ 22i , W¯ 10 = I Yj

j j j j j j ϒ¯¯ 12ιiκ = −1ι (W¯ 10 )T + 2 ( Aˇ¯ ιiκ )T − W¯ 22i , j ϒ¯¯ 22i =

N¯ j j j j j j j (τaι R¯ ιi + 2ρι Z¯ ιi ) − sym(2 W¯ 10 ), ϒˇˇ 03i = Aˇ¯ gi − o(V¯0 )T V2 , ι=1

j Aˇ¯ gi =



j

A gi j T j (Y ) A gi



j j j j j j , ϒˇˇ 13ιi = 1ι Aˇ¯ gi , ϒˇˇ 23i = 2 Aˇ¯ gi , ϒˇ 33 = −oI,

j j j ϒˇˇ 04iκ = Aˇ¯ wiκ , Aˇ¯ wiκ =



 j j Awi 0 θ¯a Aai 0 , j j j j j j (Y j )T Awi Bˆ ci Avκ θ¯a (Y j )T Aai θ¯s Bˆ ci Fsκ

j j j j j j j T ¯ ϒˇˇ j = − N¯ τaˆι (W¯ 20i ) , ϒˇˇ 14ιiκ = 1ι Aˇ¯ wiκ , ϒˇˇ 24iκ = 2 Aˇ¯ wiκ , ϒˇ 44 = −γ 2 , 05ˆιi

j j j j j j ϒˇˇ 15ιˆιi = − N¯ τaˆι (W¯ 21ιi )T , ϒˇˇ 25ˆιi = − N¯ τaˆι (W¯ 22i )T , ϒˇˇ 55ˆιi = − N¯ e−ατaˆι τaˆι R¯ ιˆi ,

j j j j j j j j ϒ¯¯ 05ˆιiκ = N¯ ριˆ Aˇ¯ ιˆiκ , ϒ¯¯ 15ιˆιiκ = N¯ ριˆ 1ι Aˇ¯ ιˆiκ , ϒ¯¯ 25ˆιiκ = N¯ ριˆ 2 Aˇ¯ ιˆiκ ,

j j j j j ϒ¯¯ 55ˆιi = − N¯ e−ατ Dˆι ριˆ Z¯ ιˆi , ϒ¯¯ 06i = (Cˇ¯ zi )T , Cˇ¯ zi = C zij X j C zij ,



j j j X¯ 1 = X j 0 , E¯ cκ = Eˆ cκ 0 .

Furthermore, the following controller matrices can be obtained: ⎧ j ⎪ E j = Eˆ cκ (U j )−1 , ⎪ ⎨ cκ j j Bci =(S j )−T Bˆ ci , ⎪ ⎪ ⎩ j j j j j j j Aciκ =(S j )−T ( Aˆ ciκ − Bˆ ci C0κ X j − (Y j )T Aui Eˆ cκ − (Y j )T A0i X j )(U j )−1 , (5.48)

140

5 Fault-Tolerant Control for Multiple Interval Time Delayed Switched …

where matrices U j and S j satisfy (U j )T S j = K¯ j − (X j )T Y j . Additionally, this minimum guaranteed cost bound (5.12) can be guaranteed, where ˆ 0T = φ(0)φT (0), ˆ 0  T ˆ 1ι  ˆ 1ι  =

0

eαϑ φ(ϑ)φT (ϑ)dϑ,

−τaι T ˆ 2ι ˆ 2ι   =

0 0

˙ φ˙ T ()ddϑ, eα φ()

−τaι ϑ T ˆ 3ι ˆ 3ι  

−τaι +ρι0

=

˙ φ˙ T ()ddϑ. eα φ()

−τaι −ρι ϑ −1 Proof Denote the  j this matrix W10  and  corresponding inverse matrix (W10 ) as j Y ♦ X ♦ j j W10 = , (W10 )−1 = , where ♦ represents any matrix as long as Sj ♦ Uj ♦ j j −1 W10 (W10 ) = I holds. j j j j j j j j Let W1ι = 1ι W10 , W12= 2 W10 .Moreover,  there are matrices $1 and $2 defined j j X I I Y j j j j , $2 = . Obviously, it follows that W10 $1 = as follows: $1 = 0 Sj Uj 0 j j j j j $2 . Apply this congruence transformation diag{($1 )T , ($1 )T , . . . , ($1 )T , I, I, ($1 )T , j ($1 )T } to formula (5.10), and then formula (5.42) is acquired.  j

j

j

It is worth mentioning that if we define Pi  

j

j

P1i P2i ¯j j , then it follows Pi = ∗ P3i

 j j j j j (X j )T P1i X j + sym[(X j )T P2i U j ] + (U j )T P3i U j (X j )T P1i + (U j )T (P2i )T , j ∗ P1i       j j j j j j Q 1ιi Q 2ιi R1ιi R2ιi Z 1ιi Z 2ιi j j j j j , , Rιi  , Z ιi  that is, P1i = P¯3i . Similarly, define Q ιi  j j j ∗ Q 3ιi ∗ R3ιi ∗ Z 3ιi j

j j

($1 )T Pi $1 =

j j j j j j then we have Q 1ιi = Q¯ 3ιi , R1ιi = R¯ 3ιi , Z 1ιi = Z¯ 3ιi . ˆ 0 < 0 , ˆ 0T P¯3ij  Using Lemma 1.4, formulas (5.44)–(5.47) can be converted into  j j j T T T ˆ 2 R¯ 3ιi  ˆ 3 Z¯ 3ιi  ˆ 1 < 1 ,  ˆ 2 < 2 and  ˆ 3 < 3 , respectively. At the same ˆ 1 Q¯ 3ιi   time, one has     N˜ N˜ N˜ j σ j xˇ T (0)Pj x(0) ˇ = tr σ j xˇ T (0)Pj x(0) ˇ = tr σ j P1 φ(0)φT (0) j=1

=tr

j=1

 N˜ j=1



j σ j P¯3 φ(0)φT (0) = tr

j=1

 N˜ j=1

j ˆ ˆT σ j P¯3  0 0

 = tr

 N˜ j=1

j ˆ ˆ 0T P¯3 0 σj



5.4 Stability Analysis

141

1, the matrix X j , and any switching signal whose average dwell time satisfies Ta > lnαμ¯ , this closed-loop model (5.5) with model uncertainties is mean-square exponentially stable under this H∞ performance index (γ, α) in formula (5.8), and follows optimal guaranteed cost control, if there exists a positive scalar o, positive scalars or positive definite matrices 0 , 1ι , 2ι , 3ι , j j j j j j j positive definite matrices P¯i , Q¯ ιi , R¯ ιi , Z¯ ιi , matrices Y j , K¯ j , Aˆ ciκ , Bˆ ci , Eˆ ci , and j a slack matrix W¯ 2i , whose forms are the same as those in Theorem 5.2, where i, κ = 1, . . . , r , j, jˆ = 1, . . . , N˜ ,ι, ι1 , ι2 , ιˆ = 1, . . . , N¯ , ι1 = ι2 , making the following optimal problem have feasible solutions   N¯ min tr{0 } + [tr{1ι } + tr{2ι } + tr{3ι }] ι=1

⎧ ⎪ ⎨ s.t.

⎪ ⎩

(5.41), (5.43), (5.44)−(5.47),     ¯ j j ¯ j j ιˆiκ 1ˆιiκ + ιˆκi 1ˆικi < 0. j j ∗ 2iκ ∗ 2κi

(5.53)

where ⎡

⎤ j j j j j j η1iκ E¯ iκ ( F¯10iκ )T η2iκ N¯ E¯ iκ 0 ( F¯4i )T 0 ⎢ η j  j E¯ j ( F¯ j )T η j  j N¯ E¯ j 0 0 0 ⎥ ⎢ 1iκ 11 iκ ⎥ 11iκ 2iκ 11 iκ ⎢ .. .. .. .. .. .. ⎥ ⎢ ⎥ . . . . . . ⎥ ⎢ ⎢ j j ¯j ¯j ⎥ j j j ⎢η1iκ  ¯ E iκ ( F ¯ )T η2iκ  ¯ N¯ E¯ iκ 0 0 0 ⎥ 1 N iκ 1N ⎢ j 1 Nj j ⎥ j j j ⎢ η  E¯ 0 η2iκ 2 N¯ E¯ iκ 0 0 0 ⎥ ⎢ 1iκ 2 iκ ⎥ j ⎢ j 0 ( F¯2i )T 0 0 0 0 ⎥

1ˆιiκ = ⎢ ⎥, ⎢ ⎥ j 0 ( F¯3iκ )T 0 0 0 0 ⎥ ⎢ ⎢ ⎥ 0 0 0 0 0 0 ⎥ ⎢ ⎢ ⎥ j ⎢ 0 0 ⎥ 0 0 0 ριˆ ( F¯1ˆιiκ )T ⎢ ⎥ j j ⎢ 0 0 0 0 0 η3i E zi ⎥ ⎢ ⎥ ⎣ 0 0 0 0 0 0 ⎦ 0 0 0 0 0 0     j j ˆ j j j F X + F F E E 0 j j cκ x10i x2i x10i xi E¯ iκ = , F¯10iκ = , j j j j j Fy10κ X j Fy10κ (Y j )T E xi Bˆ ci E yκ

5.5 Simulation Results

143

a

r (t )

(t ) f a (t )

u(t)

Controller

w(t)

Object(switched fuzzy system)

Actuator

y (t )

yL (t)

Sensor v(t )

s

(t ) f s (t )

Fig. 5.1 The overall diagram of the entire system



j F¯1ˆιiκ j F¯3iκ j

  j  j j Fx1ˆιi X j Fx1ˆιi Fx4i j ¯ = = , F , j j 2i 0 Fy1ˆικ X j Fy1ˆικ   j j

Fx5i 0 θ¯a Fx3i 0 ¯4ij = F j X j F j , = , F j j zi zi 0 Fy3κ 0 θ¯s Fy2κ j

j

j

j

j

j

2iκ = diag{−η1iκ I, −η1iκ I, −η2iκ I, −η2iκ I, −η3i I, −η3i I }. In addition, the controller matrix is determined as formula (5.48). The minimum guaranteed cost bound (5.12) can be guaranteed. Proof By Lemma 1.4, the conclusion can be drawn. So the proof is omitted. In this chapter, the overall diagram of the entire system is shown in Fig. 5.1, where r¯ (t) represents a given quantity. Similar to Ren et al. (2018, 2015), the overall schematic diagram of the algorithm designed in this chapter is shown in Fig. 5.2. Firstly, a model is depicted. The dynamic output feedback controller is designed. This model and this controller are then rewritten as the augmented closed-loop model under the interval time-varying delay form. Set appropriate prescribed parameters for later solving the linear matrix inequalities. Solve optimization problems using linear matrix inequalities. Controller matrices and optimal guaranteed cost bound of the system are acquired. Finally, the design of the controller is finished.

5.5 Simulation Results The availability of this method proposed in this chapter is verified by two simulation examples. Example 5.1 A nonlinear system can be described as a switched fuzzy system (5.1) with the following data:

144

5 Fault-Tolerant Control for Multiple Interval Time Delayed Switched …

Fig. 5.2 The overall schematic diagram for the algorithm designed in this chapter

Start Design dynamic output feedback controller

Construct augmented closed-loop system with interval delays Set parameters for solving LMIs Solve the optimal problem

Solutions?

No

Yes Obtain gain matrices of controller and the optimal guaranteed cost bound End

τ1 (t) = 0.2 + 0.1 sin(t), τ2 (t) = 0.3 + 0.1 cos(t),         −2.1 1 −2.8 0 −3.2 0.8 −2 1.2 , A102 = , A201 = , A202 = , A101 = 0.9 0 1.2 −1 0.9 −0.8 1.1 0       0.06 0.1 0.07 0.01 0.08 0 A111 = , A112 = , A211 = , 0 −0.28 0.15 −0.35 0.15 −0.33       0.06 0.12 0.04 0.08 0.02 0 , A121 = , A122 = , A212 = 0 −0.3 0 −0.2 0.05 −0.1         0.02 0 0.03 0.07 1 0 1.1 0 , A222 = , A1u1 = , A1u2 = , A221 = 0.04 −0.15 0 −0.21 0 0.6 0 0.5         1.2 0.01 0.9 0 0.2 0 0.22 0.02 1 1 = = , A2u2 = , Aa1 , Aa2 , A2u1 = 0 0.4 0.01 0.5 0.02 0.16 0 0.18       0.16 0.06 0.24 0 0 −0.03 2 2 = = , Aa2 , A1g1 = , Aa1 0 0.06 0 0.12 −0.01 0.12       −0.01 0.03 0.01 0.02 0.02 0 , A2g1 = , A2g2 = , A1g2 = 0.1 0.02 0.1 0.03 0.01 0.01         0.001 0 0.003 0 1 1 2 2 , Aw2 = , Aw1 = , Aw2 = , Aw1 = 0 0.001 0 0.002

5.5 Simulation Results

145

        0.9 0 0.7 0 0.6 0.01 0.5 0 1 1 2 2 C01 = , C02 = , C01 = , C02 = , 0.01 0.3 0 0.2 0 0.12 0.01 0.3         0.02 0 0.01 0 0.02 0 0.01 0 1 1 2 2 = = = = C11 , C12 , C11 , C12 , 0 0.01 0.01 0.02 0 0.12 0 0         0.01 0 0.01 0.01 0 0 0.06 0 1 1 2 2 = = = = , C22 , C21 , C22 , C21 0 0.01 0 0.01 0 0.01 0.01 0.02         0.5 0.7 0.4 1 1 1 2 1 = = = = , Fs2 , Fs1 , Fs2 , Fs1 0 0 0 0         0.02 0.01 0.01 0.02 , A1v2 = , A2v1 = , A2v2 = , A1v1 = 0 0.01 0 0.01

 1

 2

 2

 1 = 0.02 0.01 , C z2 = 0.01 0 , C z1 = 0.03 0.01 , C z2 = 0 0.02 . C z1 The model uncertainties are as follows:         0.005 0 0.001 0 1 1 2 2 , E x2 = , E x1 = , E x2 = , E x1 = 0 0.002 0 0.004





 1 1 2 Fx101 = 0 0.001 , Fx102 = 0.001 0 , Fx101 = 0 0.0012 ,





 2 1 1 Fx102 = 0.002 0 , Fx111 = 0.0022 0 , Fx112 = 0 0.0014 ,





 2 2 1 Fx111 = 0 0.0028 , Fx112 = 0 0.0011 , Fx121 = 0 0.0019 ,





 1 2 2 Fx122 = 0 0.0037 , Fx121 = 0.0015 0 , Fx122 = 0.003 0 ,





 1 1 2 Fx21 = 0 0.003 , Fx22 = 0.0013 0 , Fx21 = 0.0032 0 ,





 2 1 1 Fx22 = 0.0025 0 , Fx31 = 0.0018 0 , Fx32 = 0 0.0024 ,





 2 2 1 Fx31 = 0 0.0016 , Fx32 = 0.0027 0.001 , Fx41 = 0.0014 0 ,





 1 2 2 Fx42 = 0.0023 0 , Fx41 = 0.001 0.001 , Fx42 = 0.0017 0.001 , 1 1 2 2 Fx51 = 0.001, Fx52 = 0.003, Fx51 = 0.005, Fx52 = 0.0015,         0.003 0 0.002 0.004 1 1 2 2 = = = = E y1 , E y2 , E y1 , E y2 , 0 0.001 0 0





 1 1 2 = 0.0021 0 , Fy102 = 0 0.0017 , Fy101 = 0 0.0031 , Fy101





 2 1 1 = 0 0.0039 , Fy111 = 0 0.001 , Fy112 = 0.001 0 , Fy102





 2 2 1 Fy111 = 0.0033 0 , Fy112 = 0.0011 0.002 , Fy121 = 0.003 0 ,





 1 2 2 Fy122 = 0 0.0015 , Fy121 = 0.003 0.0011 , Fy122 = 0 0.0041 , 1 1 2 2 = 0.001, Fy22 = 0.0012, Fy21 = 0.004, Fy22 = 0.002, Fy21 1 1 2 2 Fy31 = 0.003, Fy32 = 0.0022, Fy31 = 0.005, Fy32 = 0.007, 1 1 2 2 = 0.001, E z2 = 0.002, E z1 = 0.003, E z2 = 0.0011, E z1







 1 1 2 2 Fz1 = 0.001 0 , Fz2 = 0 0.001 , Fz1 = 0.002 0 , Fz2 = 0 0.0013 .

146

5 Fault-Tolerant Control for Multiple Interval Time Delayed Switched …

The nonlinear function is as follows:  g(x(t)) =

 + 0.001x1 (t) + 0.001x2 (t) , 0.003x1 (t) + 0.003x2 (t)

0.003x1 (t)+0.003x2 (t) 1+x12 (t)+x22 (t)

    −0.002 0 0.004 0.002 ¯ ¯ which is bounded by V1 = , V2 = . −0.002 0.002 0 0.004 2 Choose fuzzy rules as 12 = 11 = e−x1 (t) , 22 = 21 = 1 − 11 . j j j Assume G xi = G yi = G zi = sin(t). One has τ D1 = 0.3, τ D2 = 0.4, τd1 = 0.1 and     1.1885 −0.1918 1.2627 −0.1458 j τd2 = 0.2. Set X 1 = , X2 = , 1ι = 0.0001, 0.0696 1.2992 0.0506 1.3791 j j j j 2 = 0.52, η1iκ = 0.12, η3i = η2iκ = 0.14, (i, κ = 1, . . . , r, j = 1, . . . , N˜ , ι = 1, . . . , N¯ ). The H∞ performance index can be selected as (0.8, 1) and μ¯ = 2. Suppose this occurrence probability of faults is θ¯a = 0.8, θ¯s = 0.7, and faults occur as follows: ⎧  0 ⎪ ⎪ , 0 ≤ t < 2, ⎪ ⎪ 0 ⎪ ⎪ ⎪   ⎨ 0.3 + 0.2 cos(0.5t) , 2 ≤ t ≤ 15, f a (t)= 0.4 ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ 0 ⎪ ⎩ , 15 < t ≤ 20, 0 ⎧ 0 ≤ t < 5, ⎪ ⎨0, 5 ≤ t ≤ 15, f s (t)= 0.8 + 0.1 sin(0.1t), ⎪ ⎩ 0, 15 < t ≤ 20.

Fig. 5.3 The intermittent actuator faults

Intermittent actuator faults

The faults are shown in Figs. 5.3 and 5.4. By calculating the matrix inequalities in Theorem 5.3, these corresponding controller matrices are obtained. This guaranteed cost bound is J ∗ = 19.4731.

θa (t)fa1 (t) θa (t)fa2 (t)

0.5 0.4 0.3 0.2 0.1 0 0

5

10 Time (sec)

15

20

Fig. 5.4 The intermittent sensor fault

147 Intermittent sensor fault

5.5 Simulation Results

θs (t)fs (t)

1 0.8 0.6 0.4 0.2 0 0

Fig. 5.5 The states obtained by fault-tolerant control

5

10 Time (sec)

15

States

2

20

x1 (t) x ¯1 (t) x ˆ1 (t)

1

0 0

5

10 Time (sec)

15

States

2

20

x2 (t) x ¯2 (t) x ˆ2 (t)

1

0 0

5

10 Time (sec)

15

20

Figures 5.5, 5.6, 5.7 and 5.8 show the simulation results, where the switching times are 5.6931s, 11.3863s and 17.0794s. Figure 5.5 describes these states x1 (t) and x2 (t), meaning that the states are mean-square exponentially stable via fault-tolerant control. Figure 5.6 shows these output vectors y1 (t) and y2 (t), which means that the output vectors are fast to zero. Figures 5.7 and 5.8 describe the controller states xc1 (t), xc2 (t) and control inputs u 1 (t), u 2 (t), respectively. Obviously, the controller maintains this closed-loop model in a stable state and makes it approach zero quickly after faults occur. In order to demonstrate these advantages of this approach in the chapter, two sets of comparative experiments are designed. A comparison is achieved by comparing the piecewise fuzzy Lyapunov function with this ordinary Lyapunov function (Han et al. 2019). This system state, this output, this controller state, and this control input of this comparison model obtained by this ordinary Lyapunov function are expressed ¯ separately. The other comparison is achieved by comas X¯ (t), y¯ (t), X¯ c (t) and u(t), paring dynamic output feedback controllers with static output feedback controllers (Du et al. 2014). This system state, this output and this control input of this compar-

148

5 Fault-Tolerant Control for Multiple Interval Time Delayed Switched …

Fig. 5.6 The outputs obtained by fault-tolerant control

y1 (t) y¯1 (t) yˆ1 (t)

Outputs

1.5 1 0.5 0 0

5

10 Time (sec)

15

y2 (t) y¯2 (t) yˆ2 (t)

0.4 Outputs

20

0.2 0

Fig. 5.7 Curves of controller states

Controller states

0

5

10 Time (sec)

15

20

xc1 (t) x ¯c1 (t)

1

0.5

0

Controller states

0

5

10 Time (sec)

15

20

xc2 (t) x ¯c2 (t)

2 1.5 1 0.5 0 0

5

10 Time (sec)

15

20

ison model obtained by this static output feedback controller are expressed as x(t), ˆ yˆ (t) and u(t), ˆ separately. These comparison results are shown in Figs. 5.5, 5.6, 5.7 and 5.8. Figure 5.5 shows these system states of these comparison models. Obviously, all three methods can make the system states mean-square exponentially stable. Nevertheless, the curves of these system states x1 (t) and x2 (t) in the chapter are the smoothest. If there are intermittent faults, the curve fluctuations obtained by this approach in the chapter

5.5 Simulation Results 1 Control inputs

Fig. 5.8 Curves of control inputs

149

0 -1

u1 (t) u ¯1 (t) u ˆ1 (t)

-2 -3

Control inputs

0

5

10 Time (sec)

15

20

0 -5 -10

u2 (t) u ¯2 (t) u ˆ2 (t)

-15 -20 0

5

10 Time (sec)

15

20

are the smallest, showing that this method proposed in the chapter has the best faulttolerant control performance. Figure 5.6 gives the output vectors of the comparison systems, showing that in the absence of intermittent faults, all three approaches make the output vectors tend to zero. However, the output curves y1 (t) and y2 (t) of this chapter are the smoothest. If there exist intermittent faults, this curve fluctuations obtained by the method in this chapter are the smallest, indicating that the outputs obtained by this approach in the chapter are the most stable. These controller states and these control inputs of these comparison systems are shown in Figs. 5.7 and 5.8. We can find that these controllers by these three methods keep this closed-loop model in a stable state and make it tend to zero after faults occur. Nevertheless, the controller state curves and control input curves in the chapter are the smoothest. If there exist intermittent faults, this curve fluctuations obtained by this approach in the chapter are the smallest, indicating that the controller designed in this chapter is the best. Thereby, these advantages of this method in the chapter are strongly demonstrated. Furthermore, parametric uncertainties are caused by a number of factors, for example, environmental influences, additive unknown internal or external noise, nonlinearities, and lack of system knowledge. It should be noted that parameter uncertainties are the main factors leading to the degradation of system stability and affect the dynamic response of this studied control mechanism. If parameter uncertainties are neglected and left untreated, the dynamic response will be adversely affected. However, if parameter uncertainties are treated as Theorem 5.3, this dynamic response of this studied control mechanism will only have small fluctuations. Taking this dynamic response of this state and this output, for example, this state and this output without considering parameter uncertainties are represented as x(t) ˜ and y˜ (t), separately. These dynamic response comparisons are depicted in Figs. 5.9 and 5.10. We can find that this dynamic response is smoother and closer to zero in the absence of inter-

150

5 Fault-Tolerant Control for Multiple Interval Time Delayed Switched …

Fig. 5.9 Dynamic response comparisons of states States

1 0.8

x1 (t) 0.038 x ˜1 (t) 0.037

0.6

0.036

0.4

0.035 0.034

0.2

0.033 0 0

10 Time (sec)

2

x2 (t) x ˜2 (t)

1.5 States

20

13

×10-4 2.3

2.1 2

0

1.9 0

10 Time (sec)

20

18.65

18.7

18.75

×10-5

0.8 y1 (t) y˜1 (t)

0.6 Outputs

12.5

2.2 1 0.5

Fig. 5.10 Dynamic response comparisons of outputs

12

1.8 1.6

0.4

1.4

0.2

1.2 1

0 0

10 Time (sec)

20

19

19.2

19.4

×10-5 0.5 Outputs

1.4

y2 (t) y˜2 (t)

0.4

1.3

0.3

1.2

0.2

1.1

0.1

1

0 0

10 Time (sec)

20

0.9 19.7

19.8

19.9

20

5.5 Simulation Results

151

mittent faults by considering this parameter uncertainty influence on this dynamic response of this suggested control mechanism. Example 5.2 This example considers a mass-spring damping system (Yang and Tong 2016). Suppose x1 (t) is disturbed via multiple time-varying delays. The model data are presented below. τ1 (t) = 0.25 + 0.04 sin(t), τ2 (t) = 0.1 + 0.01 cos(t),       0 1 0 1 0 1 1 1 2 , A02 = , A01 = , A01 = −0.02a¯ 0 0 −0.02a¯ 0 −0.225 −1.5275a¯ 0 0       0 1 0 0 0 0 , A111 = , A112 = , A202 = −1.5275a¯ 0 −0.225 −0.02a¯ 1 0 −0.02a¯ 1 0       0 0 0 0 0 0 , A212 = , A121 = , A211 = −1.5275a¯ 1 0 −1.5275a¯ 1 0 −0.02a¯ 2 0       0 0 0 0 0 0 , A221 = , A222 = , A122 = −0.02a¯ 2 0 −1.5275a¯ 2 0 −1.5275a¯ 2 0         0 0 0 0 , A1u2 = , A2u1 = , A2u2 = , A1u1 = 1 1 1 1    



1 1 2 2 C01 = a¯ 0 2 , C02 = a¯ 0 2 , C01 = 0.2a¯ 0 1 , C02 = 0.2a¯ 0 1 ,  1  2  2 



1 C11 = a¯ 1 0 , C12 = a¯ 1 0 , C11 = 0.2a¯ 1 0 , C12 = 0.2a¯ 1 0 ,  1  2  2 



1 C21 = a¯ 2 0 , C22 = a¯ 2 0 , C21 = 0.2a¯ 2 0 , C22 = 0.2a¯ 2 0 . where, a¯ 0 = 0.9, a¯ 1 = 0.04 and a¯ 2 = 0.06. It is assumed that there exists a nonlinear function, parameter uncertainties, external disturbances, measurement noise, intermittent sensor faults, and intermittent actuator faults in the model. The data are as follows: 

         0 0 0 0 0.01 0 1 = 2 = 2 = , Aa2 , Aa1 , Aa2 , A1g1 = , 0.11 0.23 0.01 0.05 0 0.09         0 0.01 0.015 0 0 0.05 0 , A2g1 = , A2g2 = , A1w1 = , A1g2 = 0 0.07 0 0 0.02 0.03 0.01       0.009 0 0.001 1 = 0.8, F 1 = 0.6, , A2w1 = , A2w2 = , Fs1 A1w2 = s2 0 0.008 0.006 1 = Aa1

2 = 0.9, F 1 = 0.3, A1 = 0.009, A1 = 0.012, A2 = 0.014, A2 = 0.008, Fs1 s2 v1 v2 v1 v2 1 = 0.011 0  , C 1 = 0 0.014  , C 2 = 0 0.025  , C 2 = 0.012 0  . C z1 z2 z1 z2

The model uncertainties are as follows:         0 0.001 0 0.002 1 1 2 2 = = = = E x1 , E x2 , E x1 , E x2 , 0.001 0 0.003 0





 1 1 2 = 0.0012 0 , Fx102 = 0 0.0021 , Fx101 = 0.0014 0 , Fx101

152

5 Fault-Tolerant Control for Multiple Interval Time Delayed Switched …





 2 1 1 Fx102 = 0 0.0026 , Fx111 = 0 0.0025 , Fx112 = 0.0016 0 ,





 2 2 1 Fx111 = 0 0.0019 , Fx112 = 0.0017 0 , Fx121 = 0.0018 0.001 ,





 1 2 2 Fx122 = 0.0029 0 , Fx121 = 0 0.0036 , Fx122 = 0 0.0023 , 1 1 2 2 Fx21 = 0.001, Fx22 = 0.0024, Fx21 = 0.0027, Fx22 = 0.0018, 1 1 2 2 Fx31 = 0.0016, Fx32 = 0.0027, Fx31 = 0.0033, Fx32 = 0.002,





 1 1 2 Fx41 = 0 0.0035 , Fx42 = 0 0.0029 , Fx41 = 0 0.0031 ,

 1 2 1 2 Fx42 = 0 0.0028 , Fx51 = 0.003, Fx52 = 0.004, Fx51 = 0.002, 2 1 1 2 2 Fx52 = 0.0014, E y1 = 0.0041, E y2 = 0.0025, E y1 = 0.001, E y2 = 0.003,





 1 1 2 Fy101 = 0 0.0019 , Fy102 = 0.0018 0 , Fy101 = 0.0037 0 ,





 2 1 1 Fy102 = 0.0022 0 , Fy111 = 0.0012 0 , Fy112 = 0 0.002 ,





 2 2 1 Fy111 = 0 0.0026 , Fy112 = 0 0.0031 , Fy121 = 0 0.0041 ,





 1 2 2 = 0.0012 0.0011 , Fy121 = 0 0.0024 , Fy122 = 0 0.005 , Fy122 1 1 2 2 Fy21 = 0.003, Fy22 = 0.0045, Fy21 = 0.0031, Fy22 = 0.0018, 1 1 2 2 Fy31 = 0.0024, Fy32 = 0.0043, Fy31 = 0.001, Fy32 = 0.0064, 1 1 2 2 E z1 = 0.0013, E z2 = 0.0017, E z1 = 0.0014, E z2 = 0.0015,







 1 1 2 2 Fz1 = 0 0.003 , Fz2 = 0.002 0 , Fz1 = 0 0.0014 , Fz2 = 0.0011 0 .

The nonlinear function is as follows:  g(x(t)) =

 + 0.0015x1 (t) + 0.0015x2 (t) , 0.0045x1 (t) + 0.0045x2 (t)

0.0045x1 (t)+0.0045x2 (t) 1+x12 (t)+x22 (t)

    0.006 0.003 −0.003 0 ¯ ¯ which is bounded by V1 = , V2 = . 0 0.006 −0.003 0.003 2 2 2 (t) (t) (t) , 21 = x2.25 , 12 = 1 − x˙2.25 , 22 = The fuzzy rules are chosen as 11 = 1 − x2.25 2 x˙ (t) . 2.25 j j j Assume G xi = G yi = G zi = cos(t). One has τ D1 = 0.29, τ D2 = 0.11, τd1 = 0.21 and τd2 = 0.09. Choose the H∞ performance index as (0.01, 8) and μ¯ = 1.5. Assuming that this occurrence probability of faults is θ¯a = 0.1, θ¯s = 0.5, the fault can be generated by the following formulas: ⎧ 0 ≤ t < 3, ⎪ ⎨ 0, f a (t) = 0.4 + 0.1 sin(0.3t), 3 ≤ t ≤ 8, ⎪ ⎩ 0, 8 < t ≤ 20, ⎧ 0, 0 ≤ t < 3, ⎪ ⎨ f s (t) = 0.3, 3 ≤ t ≤ 8, ⎪ ⎩ 0, 8 < t ≤ 20.

Fig. 5.11 The intermittent actuator fault

153 Intermittent actuator fault

5.5 Simulation Results

θa (t)fa (t)

0.5 0.4 0.3 0.2 0.1 0

Fig. 5.12 The intermittent sensor fault

Intermittent sensor fault

0

10 Time (sec)

15

20

θs (t)fs (t)

0.3 0.2 0.1 0 0

5

10 Time (sec)

15

20

x1 (t) x ¯1 (t) x ˆ1 (t)

1 States

Fig. 5.13 The states obtained by fault-tolerant control

5

0.5 0 0

5

10 Time (sec)

15

20

States

0.5 0 x2 (t) x ¯2 (t) x ˆ2 (t)

-0.5 0

5

10 Time (sec)

15

20

The faults are shown in Figs. 5.11 and 5.12. By calculating the linear matrix inequalities in Theorem 5.3, these corresponding controller matrices can be obtained. This guaranteed cost bound is J ∗ = 0.0134. The simulation results are shown in Figs. 5.13, 5.14, 5.15 and 5.16, where the switching times are 7.0507s and 14.1014s. Figure 5.13 shows these states x1 (t) and x2 (t), which can achieve mean-square exponential stability through fault-tolerant

154

5 Fault-Tolerant Control for Multiple Interval Time Delayed Switched …

Fig. 5.14 The outputs obtained by fault-tolerant control

y(t) y¯(t) yˆ(t)

2 Outputs

1.5 1 0.5 0 -0.5

Fig. 5.15 Curves of controller states

Controller states

0

5

Controller states

15

20

xc1 (t) x ¯c1 (t)

1 0.5 0 0

5

10 Time (sec)

15

20

xc2 (t) x ¯c2 (t)

0.5 0 -0.5 0

Fig. 5.16 Curves of control inputs

10 Time (sec)

5

10 Time (sec)

15

20

Control inputs

1 0 -1 0.1 -2

0

-3

-0.1 10

-4 0

5

10 Time (sec)

15 15

u(t) u ¯(t) u ˆ(t) 20

control. Figure 5.14 shows the output vector y(t), which means that the output vector could quickly approach zero. Figures 5.15 and 5.16 describe the controller states xc1 (t), xc2 (t) and this control input u(t), respectively. Obviously, the controller keeps this closed-loop model in a stable state and quickly zeroes it out after faults occur. Figures 5.13, 5.14, 5.15 and 5.16 exhibit these comparison results. These system states of this comparison system are shown in Fig. 5.13. Obviously, all three methods

References

155

make the system states mean-square exponentially stable. Nevertheless, the curves of these system states x1 (t) and x2 (t) in the chapter are the smoothest. If there exist intermittent faults, the curve fluctuations obtained by this approach in the chapter are the smallest, showing that this method proposed in this chapter has the best faulttolerant control performance. Figure 5.14 shows the output vectors of the comparison system, showing that in the absence of intermittent faults, all three approaches can make the output vector tend to zero. However, the output curve y(t) in this chapter is the smoothest. If there exist intermittent faults, these curve fluctuations obtained by this approach in the chapter are the smallest, indicating that the output obtained by the method in this chapter is the most stable. These controller states and these control inputs of these comparison systems are shown in Figs. 5.15 and 5.16. We can find that the controllers from these three methods can keep this closed-loop model in a stable state and make it zero after faults occur. Nevertheless, the controller state curves and the control input curve in this chapter are the smoothest. If there exist intermittent faults, these curve fluctuations obtained by this approach in the chapter are the smallest, indicating that this controller studied in the chapter is the best. Therefore, these advantages of this method in this chapter are strongly illustrated.

5.6 Chapter Summary This chapter studies the delay-dependent H∞ guaranteed cost control for uncertain switched T-S fuzzy systems with multiple interval time-varying delays. In nonlinear systems, there exist nonlinear functions, external disturbances, measurement noise, intermittent sensor faults, and intermittent actuator faults. In order to make the system mean-square exponentially stable, the passive dynamic full-order output feedback controller is proposed. Compared with existing results, it is not limited by multiple time-varying delay derivatives, so it can handle fast time-varying functions. This controller has broad application prospects. Moreover, by introducing slack matrices and using a piecewise fuzzy Lyapunov function, delay-dependent sufficient conditions are given with linear matrix inequalities, which are less conservative than those from existing approaches. At last, the feasibility and effectiveness of the method in the chapter are illustrated by two simulation examples.

References Abdollahi F, Khorasani K (2011) A decentralized Markovian jump H∞ control routing strategy for mobile multi-agent networked systems. IEEE Trans Control Syst Technol 19(2):269–283 Cao YY, Sun YX (1998) Robust stabilization of uncertain systems with time-varying multistate delay. IEEE Trans Autom Control 43(10):1484–1488 Cao L, Wang Y (2018) Fault-tolerant control for nonlinear systems with multiple intermittent faults and time-varying delays. Int J Control Autom Syst 16(2):609–621

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Chen S, Xue W, Huang Y (2019) Analytical design of active disturbance rejection control for nonlinear uncertain systems with delay. Control Eng Pract 84(MAR):323–336 Corless M, Tu J (1998) State and input estimation for a class of uncertain systems. Automatica 34(6):757–764 Cui Y, Xu L (2019) Robust H∞ persistent dwell time control for switched discrete-time T-S fuzzy systems with uncertainty and time-varying delay. J Frankl Inst 356(7):3965–3990 Du D, Jiang B, Shi P (2014) Sensor fault estimation and accommodation for discrete-time switched linear systems. IET Control Theory Appl 8(11):960–967 Gao Z, Ding SX (2007) Actuator fault robust estimation and fault-tolerant control for a class of nonlinear descriptor systems. Automatica 43(5):912–920 Guan XP, Chen CL (2004) Delay-dependent guaranteed cost control for T-S fuzzy systems with time delays. IEEE Trans Fuzzy Syst 12(2):236–249 Han QL (2002) New results for delay-dependent stability of linear systems with time-varying delay. Int J Syst Sci 33(3):213–228 Han J, Zhang H, Liu X et al (2019) Dissipativity-based fault detection for uncertain switched fuzzy systems with unmeasurable premise variables. IEEE Trans Fuzzy Syst 27(12):2421–2432 Hashimoto T, Amemiya T (2011) Controllability and observability of linear time-invariant uncertain systems irrespective of bounds of uncertain parameters. IEEE Trans Autom Control 56(8):1807– 1817 Huang S, Yang G (2014) Fault tolerant controller design for T-S fuzzy systems with time-varying delay and actuator faults: a k-step fault-estimation approach. IEEE Trans Fuzzy Syst 22(6):1526– 1540 Li Z, Zang C, Zeng P et al (2017) Control of a grid-forming inverter based on sliding-mode and mixed H2 /H∞ control. IEEE Trans Ind Electron 64(5):3862–3872 Liu D, Wang D, Wang F et al (2014) Neural-network-based online HJB solution for optimal robust guaranteed cost control of continuous-time uncertain nonlinear systems. IEEE Trans Cybern 44(12):2834–2847 Lu M, Liu L, Feng G (2019) Adaptive tracking control of uncertain Euler-Lagrange systems subject to external disturbances. Automatica 104:207–219 Qiu J, Ding SX, Gao H et al (2016) Fuzzy-model-based reliable static output feedback H∞ control of nonlinear hyperbolic PDE systems. IEEE Trans Fuzzy Syst 24(2):388–400 Ren Y, Wang A, Wang H (2015) Fault diagnosis and tolerant control for discrete stochastic distribution collaborative control systems. IEEE Trans Syst Man Cybern: Syst 45(3):462–471 Ren Y, Fang Y, Wang A et al (2018) Collaborative operational fault tolerant control for stochastic distribution control system. Automatica 98:141–149 Su X, Shi P, Wu L et al (2016) Fault detection filtering for nonlinear switched stochastic systems. IEEE Trans Autom Control 61(5):1310–1315 Sun S, Zhang H, Wang Y et al (2018) Dynamic output feedback-based fault-tolerant control design for T-S fuzzy systems with model uncertainties. ISA Trans 81:32–45 Tao Y, Shen D, Wang Y et al (2015) Reliable H∞ control for uncertain nonlinear discrete-time systems subject to multiple intermittent faults in sensors and/or actuators. J Frankl Inst 352(11):4721– 4740 Wang L, Lam H (2018) A new approach to stability and stabilization analysis for continuous-time Takagi-Sugeno fuzzy systems with time delay. IEEE Trans Fuzzy Syst 26(4):2460–2465 Wang R, Liu G, Wang W et al (2010a) Guaranteed cost control for networked control systems based on an improved predictive control method. IEEE Trans Control Syst Technol 18(5):1226–1232 Wang Y, Zhang H, Wang X et al (2010b) Networked synchronization control of coupled dynamic networks with time-varying delay. IEEE Trans Cybern 40(6):1468–1479 Wu J, Peng C, Zhang J et al (2020) Guaranteed cost control of hybrid-triggered networked systems with stochastic cyber-attacks. ISA Trans 104:84–92 Yang W, Tong S (2016) Adaptive output feedback fault-tolerant control of switched fuzzy systems. Inf Sci 329:478–490

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Chapter 6

Fault-Tolerant Control for Multiple-Delayed Switched Fuzzy Stochastic Systems with Intermittent Faults

6.1 Introduction In the last chapter, we discuss the delay-dependent reliable H∞ guaranteed cost control for multiple interval time-varying delayed uncertain switched T-S fuzzy models. However, stochastic phenomena that can be seen everywhere in real industrial systems are not considered. This undoubtedly limits the application of the controller in the previous chapter to some occasions. To date, there have been quite a few methods to solve the fault-tolerant control problems of nonlinear models. There exist a kind of controllers, where the static state feedback controller is the simplest controller (Zhang et al. 2008; Lien et al. 2008). Zhang et al. (2008) designed a static state feedback controller to realize delay-dependent guaranteed cost control performance for stochastic fuzzy plants with parameter uncertainties as well as multiple time delays. Nevertheless, these above controllers are all state-dependent, but this system state is not available sometimes. In this case, an observer-based output feedback controller is required. In Gao et al. (2009), the stability analysis and stabilization for discretetime fuzzy systems with time-varying delay are studied through an observer-based output feedback controller. Static output feedback controllers are utilized in some situations. In Du and Cocquempot (2017), fault diagnosis and fault-tolerant control were achieved for a remotely operated underwater vehicle with sensor faults through a static output feedback controller. Dynamic output feedback controllers are very valuable in the field of fault-tolerant control (Zhang et al. 2010; Sun et al. 2018). In Sun et al. (2018), a fault-tolerant control of a nonlinear truck trailer model is implemented using a dynamic output feedback controller. The chapter studies this design of a full-order passive dynamic output feedback controller for uncertain switched T-S fuzzy stochastic models with time-varying delays.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Sun et al., Fault-Tolerant Control for Time-Varying Delayed T-S Fuzzy Systems, Intelligent Control and Learning Systems 9, https://doi.org/10.1007/978-981-99-1357-2_6

159

160

6 Fault-Tolerant Control for Multiple-Delayed Switched Fuzzy …

6.2 System Definition and Description An uncertain switched T-S fuzzy stochastic model with multiple time-varying delays, intermittent actuator faults, and intermittent sensor faults is shown as follows: ⎧   r M¯ H¯ ⎪   ⎪ j j,h j,h ⎪ ⎪ d x(t) = σ j (t) νi (χ(t)) (Bi + Bi (t))x(t − τh (t)) ⎪ ⎪ ⎪ ⎪ j=1 i=1 h=0 ⎪ ⎪ ⎪ ⎪ j j j j j ⎪ + (Bui + Bui (t))u(t) + ϑa (t)(Bai + Bai (t)) f a (t) + (Bωi ⎪ ⎪ ⎪ ⎪ ⎪

H¯ ⎪  ⎪ ⎪ j j,h j,h ⎪ + Bωi (t))ω(t) dt + (Di + Di (t))x(t − τh (t))dW (t) , ⎪ ⎪ ⎪ ⎪ h=0 ⎪ ⎪ ⎪ ⎨  r M¯ H¯   j j,h j,h (6.1) y(t) = σ j (t) νi (χ(t)) (Ci + Ci (t))x(t − τh (t)) ⎪ ⎪ ⎪ ⎪ j=1 i=1 h=0 ⎪ ⎪ ⎪ ⎪ ⎪ j j j j ⎪ ⎪ (t)(B + B (t)) f (t) + (B + B (t))v(t) , + ϑ ⎪ s s si si vi vi ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ r M¯ ⎪   ⎪ ⎪ j j j ⎪ ⎪ yz (t) = σ j (t) νi (χ(t))(C zi + C zi (t))x(t), ⎪ ⎪ ⎪ ⎪ j=1 i=1 ⎪ ⎪ ⎩ x(t) =ψ(t), t ∈ [−τ M , 0]. where x(t) ∈ R m represents this state vector, u(t) ∈ R n represents the system input vector, f a (t) ∈ R m a represents the actuator fault vector, ω(t) ∈ R m ω represents the exogenous disturbance vector, ω(t) ∈ L 2 [0, ∞), y(t) ∈ R m 1 represents the measurement output vector, f s (t) ∈ R m s is the sensor fault vector, v(t) ∈ R m v is the measurement noise vector, and yz (t) ∈ R m 2 represents this desired controlled output. W (t) represents the scalar Brownian motion represented on the complete probability space, E{dW (t)} = 0 and E{dW 2 (t)} = dt. τ0 (t) = 0. τh (t) represents the time-varying state delay and follows 0 < τh (t) ≤ τh , where τh represents a known scalar, which is the upper bound of τh (t). Besides, τ M is the maximum value of all τh . τ˙h (t) ≤ τ Dh , where τ Dh represents a known scalar, which is the upper bound of τ˙h (t). In a plant, especially one with a time delay, it is required to denote the initial values. This initial value range in a time delayed system is from the negative value of the maximum delay to zero Zhang et al. (2008). ψ(t), t ∈ [−τ M , 0] denotes the vector-valued initial function. Define r  1, . . . , r , ι  1, . . . , ι,  M¯  1, . . . , M¯ and  H¯  1, . . . , H¯ , ¯ H¯ represent the numbers of IF-THEN rules, premise variables, where r , ι, M, j,h j j j j,h j,h switched subsystems and time delays, separately. Bi , Bui , Bai , Bωi , Di , Ci , j j j Bsi , Bvi and C zi (i ∈ r , j ∈  M¯ , h ∈  H¯ ) represent the known constant matrices. j,h j j j j,h j,h The unknown matrices Bi (t), Bui (t), Bai (t), Bωi (t), Di (t), Ci (t), j j j Bsi (t), Bvi (t) and C zi (t) represent the time-varying parameter uncertainties of model (6.1), and its form is as follows:

6.3 Design of Dynamic Output Feedback Controller

161

j,h j j j j,h Bi (t) Bui (t) Bai (t) Bωi (t) Di (t)  j j j,h j j j j,h = Mxi Hxi (t) N x1i N x2i N x3i N x4i N x5i ,   j j j,h j j j,h j j Ci (t) Bsi (t) Bvi (t) = M yi Hyi (t) N y1i N y2i N y3i , 

j

j

j

j

C zi (t) = Mzi Hzi (t)Nzi , j

j,h

j

j

j

j,h

j

j,h

j

j

j

j

where Mxi , N x1i , N x2i , N x3i , N x4i , N x5i , M yi , N y1i , N y2i , N y3i , Mzi and Nzi are j

j

j

known constant matrices. Hxi (t), Hyi (t) and Hzi (t) denote unknown time-varying j matrices, obeying (Hxi (t))T

j Hxi (t)

j I, (Hyi (t))T

j

j

j

≤ Hyi (t) ≤ I, (Hzi (t))T Hzi (t) ≤ I . σ j (t) represents this switching signal, mapping from [0, ∞) to {0, 1}. When σ j (t) = j 1, t ∈ [t1 , t2 ), this jth switched subsystem is active during time t ∈ [t1 , t2 ). νi (χ(t)) j j is obtained by χl (t) and ν¯il , where χl (t), (l ∈ ι ), represents the premise variable, ν¯il , (i ∈ r , l ∈ ι ), is a fuzzy set described by the membership function. ϑa (t) f a (t) and ϑs (t) f s (t) are intermittent actuator faults and intermittent sensor faults, respectively. ϑa (t) and ϑs (t) cause the random occurrence of the fault phenomena. Suppose that ϑa (t) and ϑs (t) obey a Bernoulli distribution, and the set of values is {0, 1}. Faults satisfy 

P{ϑa (t) = 1} = ϑ¯ a , P{ϑa (t) = 0} = 1 − ϑ¯ a ,



P{ϑs (t) = 1} = ϑ¯ s , P{ϑs (t) = 0} = 1 − ϑ¯ s ,

where ϑ¯ a , ϑ¯ s ∈ [0, 1] represent known constants. That is, E(ϑa (t)) = ϑ¯ a and E(ϑs (t)) = ϑ¯ s , respectively. j j For simplicity, replace σ j (t) and νi (χ(t)) with σ j and νi , respectively, and replace j,h j j j j,h j,h j j Bi (t), Bui (t), Bai (t), Bωi (t), Di (t), Ci (t), Bsi (t), Bvi (t), j j,h j j j j,h j,h j j j C zi (t) with Bi , Bui , Bai , Bωi , Di , Ci , Bsi , Bvi , C zi , respectively. Remark 6.1 In this chapter, uncertain switched fuzzy stochastic systems with multiple time-varying delays and intermittent faults are studied for one of the few tries. Compared to the study of permanent faults (Gao 2015), this chapter explores intermittent actuator and sensor faults and deals with more fault scenarios. Compared with the case where a delay is studied (Yang et al. 2018), multiple time-varying delays are explored, which can solve the problems of more comprehensive delays. Therefore, the investigated system is novel.

6.3 Design of Dynamic Output Feedback Controller The section studies an augmented closed-loop model. This controller can be designed below

162

6 Fault-Tolerant Control for Multiple-Delayed Switched Fuzzy …

⎧ r r M¯    ⎪ ⎪ j j j j ⎪ η(t) ˙ = σ ν νl [Ail η(t) + Bi y(t)], ⎪ j i ⎪ ⎪ ⎪ j=1 i=1 l=1 ⎪ ⎨ ¯ r M   j j ⎪ ⎪ u(t) = σj νi Ci η(t), ⎪ ⎪ ⎪ ⎪ j=1 i=1 ⎪ ⎪ ⎩ η(t) = 0, t ∈ [−τ M , 0],

(6.2)

j

j

j

where η(t) ∈ R m represents the state vector of the controller, and Ail , Bi , Ci are the controller matrices T T

to be determined. ¯ = f aT (t) f sT (t) ω T (t) v T (t) . AccordDesign η(t) ˇ = x T (t) η T (t) . ω(t) ing to formulas (6.1) and (6.2), this augmented closed-loop system is written by ⎧   r r M¯ H¯ ⎪    ⎪ j j j,h j ⎪ ˇ ˇ ⎪ d η(t) ˇ = σ ν ν η(t ˇ − τ (t)) + B ϑ(t) ω(t) ¯ dt A j h ⎪ i l il il ⎪ ⎪ ⎪ j=1 i=1 l=1 h=0 ⎪ ⎪ ⎪ ⎪

⎪ H¯ ⎪  ⎪ j,h ⎨ ˇ + Di x(t − τh (t))dW (t) , h=0 ⎪ ⎪ ⎪ ¯ ⎪ r M ⎪   ⎪ j j ⎪ ⎪ y (t) = σ νi Cˇ i η(t), ˇ ⎪ z j ⎪ ⎪ ⎪ j=1 i=1 ⎪ ⎪ ⎪ ⎩ ˇ η(t) ˇ =ψ(t), t ∈ [−τ M , 0],

(6.3)

where 

 j,0 j,0 j j j Bi + Bi (Bui + Bui )Cl = , j j,0 j,0 j Bi (Cl + Cl ) Ail   j,h j,h Bi + Bi 0 j,h ˇ Ail = , ϑ(t) = diag{ϑa (t)Im a , ϑs (t)Im s , Im ω , Im v }, j j,h j,h Bi (Cl + Cl ) 0   j j j j B + B 0 B + B 0 j ai ai ωi ωi Bˇ il = , j j j j j j 0 Bi (Bsl + Bsl ) 0 Bi (Bvl + Bvl )  j,0  j,h j,0 j,h Di + Di 0 Di + Di 0 j,0 j,h , Dˇ i = , Dˇ i = 0 0 0 0  j ˇ = [ψ T (t), 0T ]T , t ∈ [−τ M , 0], h ∈  H¯ . Cˇ i = C zij + C zij 0 , ψ(t) j,0 Aˇ il

For matrices 1 , 2 > 0, and a constant α¯ > 0, the following cost function is denoted for model (6.3) as follows:

6.4 Stability Analysis

163

 ∞

−αt ¯ T T J=E e [x (t)1 x(t) + u (t)2 u(t)]dt .

(6.4)

0

Formula (6.4) can be converted into ⎫ ⎧∞ M¯ r r ⎬ ⎨    j j ¯ ˇ ilj η(t)dt e−αt σj νi νl ηˇ T (t) ˇ , J=E ⎭ ⎩ j=1

0

i=1

(6.5)

l=1

  ˇ ilj = diag 1 , (Ci j )T 2 Cl j . where  Definition 6.1 (Su et al. (2016)) Under the jth switching signal, this switched T-S fuzzy stochastic model (6.3) when ϑ(t)ω(t) ¯ = 0 and u(t) = 0 can be mean-square exponentially stable, if   2 2 −κ2 t ˇ E η(t) ˇ ≤ κ1 ψ(0) , aˇ e 2 ˇ where the constants κ1 , κ2 > 0, ψ(0) aˇ  max

t∈[−τ M ,0]



(6.6)

 2 ˇ ψ(t) .

Definition 6.2 (Su et al. (2016)) For the positive scalars γ, α¯ and L, this switched T-S fuzzy stochastic model (6.3) with u(t) = 0 can be mean-square exponentially ¯ and obeys guaranteed cost stable under this weighted H∞ performance index (γ, α) control, if under ϑ(t)ω(t) ¯ = 0, this model can be mean-square exponentially stable, and under ϑ(t)ω(t) ¯ = 0 with zero initial conditions, this inequality is satisfied as follows:

 L

 L ¯ 2 e−αt yz (t)2 dt < γ 2 E ϑ(t)ω(t) ¯ dt . (6.7) E 0

0

And this performance index follows J ≤ J∗ , where J∗ > 0 stands for guaranteed cost control.

6.4 Stability Analysis Theorem 6.1 Given these specified positive constants τh , τ Dh , γ, α, ¯ μˇ > 1 and switching signals with the average dwell time following Ta > lnα¯μˇ , this closed-loop model (6.3) can be mean-square exponentially stable under this weighted H∞ performance index (γ, α) ¯ defined in (6.7) and satisfies guaranteed cost control, if there j j,h j,h j,h j,h are positive definite matrices Pi , Q 1i , Q 2i , Ri and Z i , where i, l, k¯ ∈ r , j, j˜ ∈  M¯ , h, h 1 , h 2 ∈  H¯ , h 1 < h 2 , such that the following inequalities hold:

164

6 Fault-Tolerant Control for Multiple-Delayed Switched Fuzzy …

j,h j,h P˙νj ≤ 0, Q˙ 1ν ≤ 0, Q˙ 2ν ≤ 0, R˙ νj,h ≤ 0, Z˙ νj,h ≤ 0, j kii ¯ j kil ¯

(6.8)

< 0, +

(6.9)

j kli ¯

< 0, i < l,

˜

j,h

ˇ νj , Q 1ν < Pνj < μP

(6.10)

˜ j,h μQ ˇ 1ν ,

j,h

Q 2ν
1, the matrix X 1 > 0, and switching signals whose average dwell time satisfies Ta > lnα¯μˇ , this closed-loop model (6.3) against model uncertainties can be ¯ mean-square exponentially stable under this weighted H∞ performance index (γ, α) in formula (6.7), and obeys the optimal guaranteed cost control, if there are positive j,h j,h j,h j,h j j j definite matrices Y1 , Qˇ 1i , Qˇ 2i , Rˇ i , Zˇ i , matrices Aˇil , Bˇ i , Cˇi , which are the same as those defined in Theorem 6.2, positive scalars or positive definite matrices 0 , 1h , 2h , 3h , 4h , where i, l ∈ r , j, j˜ ∈  M¯ , h ∈  H¯ , so that the following optimal problem has feasible solutions: ⎫ H¯ ⎬  [tr(1h ) + tr(2h ) + tr(3h ) + tr(4h )] min tr(0 ) + ⎭ ⎩ ⎧ ⎨

h=1

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ s.t.

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(6.41), (6.44)−(6.49),   j ˇ iij 1ii  < 0, j ∗ 2ii     j j ˇ ilj 1il ˇ lij 1li   + < 0, i < l, j j ∗ 2il ∗ 2li

(6.57) (6.58)

176

6 Fault-Tolerant Control for Multiple-Delayed Switched Fuzzy …

where ⎡

⎤ j j j,0 j,0 j π¯ 1il Mˇ 1il ( Nˇ 1il )T ( Nˇ 3i )T 0 ( Nˇ zi )T 0 ⎢ ⎥ j,1 j,1 ( Nˇ 1il )T ( Nˇ 3i )T 0 0 0 ⎥ ⎢ 0 ⎢ . .. .. .. .. .. ⎥ ⎢ . ⎥ . . . . . ⎥ ⎢ . ⎢ ⎥ j, H¯ j, H¯ ⎢ 0 0 0 0 ⎥ ( Nˇ 1il )T ( Nˇ 3i )T ⎢ ⎥ ⎢ 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ . .. .. .. .. .. ⎥ ⎢ .. . . . . . ⎥ ⎢ ⎥ ⎢ 0 0 0 0 0 0 ⎥ ⎢ ⎥ j , 1il = ⎢ 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ . ⎥ .. .. .. .. .. ⎥ ⎢ .. . . . . . ⎥ ⎢ ⎢ 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ j 0 0 0 0 ⎥ ( Nˇ 2il )T ⎢ 0 ⎢ ⎥ j ˇ j ⎢ 0 0 0 ⎥ 0 0 π¯ 2i M2i ⎢ j j⎥ ⎢ 0 0 0 0 0 π¯ 3i Mzi ⎥ ⎢ ⎥ ⎣ 0 0 0 0 0 0 ⎦ 0 0 0 0 0 0     j j,0 j ˇ j N j,0 M 0 N C X + N j j,0 1 xi x1i x2i l x1i Mˇ 1il = , Nˇ 1il = j j j j,0 j,0 , Y1 Mxi Bˇ i M yl N y1l X 1 N y1l     j,h j,h j ¯a N j X N 0 N 0 N θ j,h j 1 x1i x1i x3i x4i ˇ Nˇ 1il = , j,h j,h , N2il = j j 0 θ¯s N y2l 0 N y3l N y1l X 1 N y1l   j,0 j,h j,0 j,0 j,h j,h Nˇ 3i = N x5i X 1 N x5i , Nˇ 3i = N x5i X 1 N x5i ,   j  M j j xi Mˇ 2i = , Nˇ zi = Nzij X 1 Nzij , j Y1 Mxi j

j

j

j

j

j

j

2il = diag{−π¯ 1il I, −π¯ 1il I, −π¯ 2i I, −π¯ 2i I, −π¯ 3i I, −π¯ 3i I }. Besides, these controller matrices can be designed by (6.50). At the same time, the minimum guaranteed cost bound (6.12) can be obtained. Proof The results can be obtained using Lemma 1.4. Therefore, the proof process is omitted. Remark 6.2 Instead of using a state feedback controller (Zhang et al. 2008) that requires all states to be known, the dynamic output feedback controller with more adjustable gain matrices is studied, and it is not required to know the states. The controller can be utilized on more practical occasions. Compared with the case (Tao et al. 2015) that only implements fault-tolerant control, this chapter designs an optimal guaranteed cost controller, which means that both stability and a suitable performance index can be satisfied. Therefore, the system structure is novel.

6.4 Stability Analysis Fig. 6.1 The flowchart of the research algorithm

177 Start Design controller structure

Select appropriate parameters

Solve the optimal problem

Solutions?

No

Yes Obtain gain matrices of controller and the optimal guaranteed cost bound End

Remark 6.3 In contrast to the stochastic Lyapunov function (Zhang et al. 2008), fuzzy Lyapunov function (Wang and Lam 2018) or piecewise Lyapunov function (Han et al. 2019a), this chapter uses a piecewise fuzzy Lyapunov function, which considers the influences of time delays, switching signals and membership functions on the model stability. The piecewise fuzzy Lyapunov function makes the obtained results have a larger admissible maximal delay bound, and the obtained results are less conservative, which improves the stability performance. Compared with using R and Q (Zhang et al. 2008), this chapter adopts the multiple delay-dependent Lyapunov j,h j,h j,h j,h function, which uses Q 1i , Q 2i , Ri and Z i , so the obtained results are less conservative. Compared with the method using slack matrices (Huang et al. 2011), this chapter does not need slack matrices, which weakens the freedom of this solution space for this described stability criteria, reduces this stability analysis complexity, and further reduces the computational complexity of this chapter. In contrast to the methods (Xu et al. 2007) where scalars are constants without considering this j,h j,h relationship among subsystems, these scalars in the chapter, for instance, 1i , 2i , j,h j,h j j j 3i , 4i , π¯ 1il , π¯ 2i , π¯ 3i , consider the relationship between multiple time-varying delayed switched fuzzy models, so this adjustable scope of the input parameters is larger, and the optimal guaranteed cost controller is designed with a better effect. This optimal guaranteed cost function is directly handled by the linear matrix inequalities, and the gain matrices of the optimal controller can be acquired simultaneously. Therefore, the method proposed in this chapter is novel. This simulation code composing software adopts MATLAB. The flowchart of the research algorithm is shown in Fig. 6.1.

178

6 Fault-Tolerant Control for Multiple-Delayed Switched Fuzzy …

6.5 Simulation Results In the chapter, the feasibility of this method is demonstrated through numerical examples. Example 6.1 Discuss a switched fuzzy stochastic model (6.1) when r = 2, M¯ = 2, H¯ = 2. Similar to Jiang and Han (2007), these fuzzy rules can be selected as ν12 = ν11 = sin2 (x2 (t)), ν22 = ν21 = cos2 (x2 (t)). This nonlinear time delay model is represented through these switched fuzzy stochastic model parameters as follows: τ1 (t) = 0.8 + 0.2 cos(t), τ2 (t) = 0.5 + 0.3 sin(2t),    −7.2 0.01 −6.9 0 −7.8 0.02 , B21,0 = , B12,0 = , B11,0 = 0.2 −9.1 0.04 −8.7 0.35 −9    −7.5 8 0.4 0.05 0.29 −0.08 B22,0 = , B11,1 = , B21,1 = , 0.41 −7.8 0 0.5 0.18 0.55    0.4 −0.07 0.36 0.47 0.35 0.03 , B22,1 = , B11,2 = , B12,1 = 0.12 0.52 −0.38 0.52 0.04 0.51    0.24 −0.03 0.37 −0.06 0.33 0.29 , B12,2 = , B22,2 = , B21,2 = 0.15 0.55 0.13 0.5 −0.18 0.5     0.9 0 1.2 0 1 0 1.1 0.01 1 1 2 2 = = = = Bu1 , Bu2 , Bu1 , Bu2 , 0 0.5 0 0.4 0.01 0.7 0 0.8     0.2 0 0.3 0.01 0.15 0.05 0.34 0.01 1 1 2 2 = = = = , Ba2 , Ba1 , Ba2 , Ba1 0 0.1 0 0.2 0 0.05 0.02 0.1     0.001 0.002 0.0012 0.0015 1 1 2 2 = = = = , Bω2 , Bω1 , Bω2 , Bω1 0.001 0 0.001 0     0.01 0 0.008 0 0 0.005 0 0.001 , D21,0 = , D12,0 = , D22,0 = , D11,0 = 0 0 0 0 0 0 0 0     0.061 0 0.015 0 0 −0.034 0 0.021 , D21,1 = , D12,1 = , D22,1 = , D11,1 = 0 0 0 0 0 0 0 0     0.002 0 0.001 0 0 0.015 0 0.017 D11,2 = , D21,2 = , D12,2 = , D22,2 = , 0 0 0 0 0 0 0 0    0.21 0.01 0.15 0 0.17 0 , C21,0 = , C12,0 = , C11,0 = 0 0.3 0.01 0.2 0.02 0.3    0.2 0 0.506 0 0.502 0.003 , C11,1 = , C21,1 = , C22,0 = 0 0.25 0.001 0.036 −0.002 0.02    0.5 0.002 0.502 −0.009 0.508 0 , C22,1 = , C11,2 = , C12,1 = −0.001 0.043 0.009 0.014 0.001 0.027    0.504 0.002 0.505 0 0.502 −0.008 , C12,2 = , C22,2 = , C21,2 = −0.002 0.017 0 0.032 0.009 0.009

6.5 Simulation Results

179



   0.5 0.7 0.4 1 1 2 2 , Bs2 = , Bs1 = , Bs2 = , 0 0 0 0     0.01 0 0.03 0 1 1 2 2 = = = = Bv1 , Bv2 , Bv1 , Bv2 , 0 0.02 0 0.04

 1





 1 2 2 = 0.03 0.02 , C z2 = 0 0.02 , C z1 = 0 0.03 , C z2 = 0.01 0.01 . C z1 1 Bs1 =

The model uncertainties are as follows:     0.002 0.001 0.003 0.002 1 1 2 2 Mx1 = , Mx2 = , Mx1 = , Mx2 = , 0 0 0 0





 1,0 1,0 2,0 = 0.03 0 , N x12 = 0.021 0 , N x11 = 0 0.022 , N x11





 2,0 1,1 1,1 N x12 = 0.013 0 , N x11 = 0.04 0.01 , N x12 = 0.02 0 ,





 2,1 2,1 1,2 N x11 = 0.013 0 , N x12 = 0.02 0.03 , N x11 = 0.02 0.01 ,





 1,2 2,2 2,2 N x12 = 0.05 0 , N x11 = 0 0.021 , N x12 = 0 0.018 ,





 1 1 2 N x21 = 0.02 0.07 , N x22 = 0.05 0.02 , N x21 = 0 0.013 ,





 2 1 1 N x22 = 0 0.021 , N x31 = 0.025 0 , N x32 = 0.023 0 ,



 2 2 N x31 = 0.017 0 , N x32 = 0 0.035 , 1 1 2 2 N x41 = 0.02, N x42 = 0.012, N x41 = 0.03, N x42 = 0.01,





 1,0 1,0 2,0 N x51 = 0.011 0 , N x52 = 0.02 0 , N x51 = 0 0.017 ,





 2,0 1,1 1,1 N x52 = 0.024 0 , N x51 = 0 0.03 , N x52 = 0.023 0 ,





 2,1 2,1 1,2 N x51 = 0.031 0 , N x52 = 0.014 0.01 , N x51 = 0 0.026 ,





 1,2 2,2 2,2 N x52 = 0.001 0.015 , N x51 = 0 0.022 , N x52 = 0 0.034 ,     0 0.0012 0 0.0032 1 1 2 2 , M y2 = , M y1 = , M y2 = , M y1 = 0.0024 0 0.0013 0







 1,0 1,0 2,0 2,0 N y11 = 0 0.014 , N y12 = 0.012 0 , N y11 = 0.026 0 , N y12 = 0 0.03 ,







 1,1 1,1 2,1 2,1 N y11 = 0.014 0 , N y12 = 0.019 0 , N y11 = 0.027 0 , N y12 = 0 0.023 ,







 1,2 1,2 2,2 2,2 N y11 = 0.016 0 , N y12 = 0.013 0 , N y11 = 0 0.031 , N y12 = 0.028 0 , 1 1 2 2 N y21 = 0.02, N y22 = 0.024, N y21 = 0.032, N y22 = 0.017, 1 1 2 2 = 0.015, N y32 = 0.013, N y31 = 0.04, N y32 = 0.029, N y31 1 1 2 2 Mz1 = 0.0021, Mz2 = 0.0015, Mz1 = 0.0025, Mz2 = 0.0014,







 1 1 2 2 Nz1 = 0.024 0 , Nz2 = 0.031 0 , Nz1 = 0 0.014 , Nz2 = 0.021 0 . j

j

j

Assuming 1 = diag{0.02, 0.01}, 2 = diag{0.06, 0.1}, Hxi = Hyi = Hzi = cos(t), (i ∈ r , j ∈  M¯ ). The switching times are 5.0068s, 10.0137s, and 15.0205s. In a real system, the fault can be an exponential function or a trigonometric function (Huang and Yang 2014; Han et al. 2019b; Zhang et al. 2017; Han et al.

180

6 Fault-Tolerant Control for Multiple-Delayed Switched Fuzzy …

2016). The exponential function represents a slowly varying fault over time, and the trigonometric function represents a periodically varying fault. Assume that the occurrence probability of actuator and sensor faults is ϑ¯ a = 0.79, ϑ¯ s = 0.81, and this fault form is as follows: ⎧ 0 ⎪ ⎪ , 0 ≤ t < 3, ⎪ ⎪ 0 ⎪ ⎪ ⎪ 1.99−t ⎨ 2(1 − e 5 ) , 3 ≤ t ≤ 16, f a (t)= ⎪ 0.5 + 0.4 sin(0.1t) ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎩ 0 , 16 < t ≤ 20, 0 ⎧ 0 ≤ t < 6, ⎪ ⎨ 0, 0.7 + 0.3 cos(0.2t), 6 ≤ t ≤ 16, f s (t)= ⎪ ⎩ 0, 16 < t ≤ 20.

Fig. 6.2 The intermittent actuator faults

Intermittent actuator faults

The faults are shown in Figs. 6.2 and 6.3. It can be calculated that τ1 = 1, τ D1 = 0.2, τ2 = 0.8 and τ D2 = 0.6. In addition, this weighted H∞ performance index is chosen as (0.2, 0.8) and μˇ = 1.5. Choose j j j j,h j,h j,h j,h 0 = 1i = 2i = 3i = 4i = 1, π¯ 1il = 0.31, π¯ 2i = 0.33, π¯ 3i = 0.3, (i ∈ r , j ∈

T  M¯ ),ψ(0) = 1.5 1.8 .

a (t)fa1 (t) a (t)fa2 (t)

1.5 1 0.5 0

Fig. 6.3 The intermittent sensor fault

Intermittent sensor fault

0

5

10 Time (sec)

15

20

s (t)fs (t)

0.8 0.6 0.4 0.2 0 0

5

10 Time (sec)

15

20

6.5 Simulation Results

181

Fig. 6.4 The resulting states

x1 (t) x2 (t)

States

1.5 1 0.5 0 0

5

10

Time (sec)

15

Fig. 6.5 The outputs

y1 (t) y2 (t)

1.5

Outputs

20

1 0.5 0 0

5

10

15

20

Time (sec)

Controller states

Fig. 6.6 The controller states

η1 (t) η2 (t)

1.5 1 0.5 0 0

5

10 Time (sec)

15

20

In the absence of model uncertainties, these optimal guaranteed cost controller parameters are acquired using Theorem 6.2. Figures 6.4, 6.5, 6.6 and 6.7 are the simulation results. Figure 6.4 shows the states x1 (t) and x2 (t), which means that the states are mean-square exponentially stable. Figure 6.5 depicts the output vectors y1 (t) and y2 (t), showing that the output vectors can quickly converge to zero when faults disappear. Figures 6.6 and 6.7 show controller state vectors η1 (t), η2 (t) and control input vectors u 1 (t), u 2 (t). It is further concluded that the controller can quickly realize mean-square exponential stability. In the case of model uncertainties, these optimal guaranteed cost controller parameters are acquired using Theorem 6.3. This optimal guaranteed cost bound is J∗ = 27.9595. The simulation results are shown in Figs. 6.8, 6.9, 6.10, 6.11. Figure 6.8 describes these states x1 (t) and x2 (t), which means that the states are

182

6 Fault-Tolerant Control for Multiple-Delayed Switched Fuzzy …

Fig. 6.7 The control inputs

u1 (t) u2 (t)

Control inputs

0.4 0.2 0 -0.2 -0.4 0

5

10 Time (sec)

15

Fig. 6.8 The resulting states

x1 (t) x ˆ1 (t)

6 States

20

4 2 0 0

5

10 Time (sec)

15

x2 (t) x ˆ2 (t)

2 1.5 States

20

1 0.5 0 0

5

10 Time (sec)

15

20

mean-square exponentially stable. Figure 6.9 depicts the output vectors y1 (t) and y2 (t), showing that once the faults disappear, the output vectors quickly approach zero. Figures 6.10 and 6.11 show controller state vectors η1 (t), η2 (t) and control input vectors u 1 (t), u 2 (t). It is further concluded that the controller can quickly realize mean-square exponential stability. Next, conduct a comparison experiment between the piecewise fuzzy Lyapunov function and the stochastic Lyapunov function (Zhang et al. 2008). This system state, output, controller state, and control input of this comparison model obtained by the stochastic Lyapunov function are expressed as x(t), ˆ yˆ (t), η(t), ˆ u(t), ˆ respectively. These system state vectors, output vectors, controller state vectors, and control input vectors of the comparison model are presented in Figs. 6.8, 6.9, 6.10 and 6.11, respectively. Both methods can make the system mean-square exponentially stable. However, this overshoot of the curves obtained by this approach in the chapter is small and this convergence speed of the curves obtained by this approach in the

6.5 Simulation Results

183

Fig. 6.9 The outputs

y1 (t) yˆ1 (t)

Outputs

6 4 2 0 0

5

10 Time (sec)

15

0.8

y2 (t) yˆ2 (t)

0.6 Outputs

20

0.4 0.2 0

Fig. 6.10 The controller states

Controller states

0

5

10 Time (sec)

15

20

η1 (t) ηˆ1 (t)

30 20 10 0 0

5

10 Time (sec)

15

Controller states

8

20

η2 (t) ηˆ2 (t)

6 0.4 0.2 0

4

1

2

2

0 0

5

10 Time (sec)

15

20

chapter is faster, which shows that the controller designed in this chapter has better fault-tolerant control performance, which effectively demonstrates the effectiveness of the method in this chapter. Model uncertainties with this form in the chapter are common in practical systems (Jiang and Han 2007; Zhang et al. 2008; Xu et al. 2007). Model uncertainties are

184

6 Fault-Tolerant Control for Multiple-Delayed Switched Fuzzy …

Fig. 6.11 The control inputs Control inputs

200

u1 (t) u ˆ1 (t)

150 100 50 0 0

5

10 Time (sec)

15

20

Control inputs

0 -20 -40 u2 (t) u ˆ2 (t)

-60 0

5

10 Time (sec)

15

20

the main factor that leads to the degradation of system stability and affects this dynamic response of the proposed control system. If uncertainties are neglected and no action is taken, this dynamic response of this proposed control system will be adversely affected. However, if parameter uncertainties are treated according to Theorem 6.3, there will be only small fluctuations in the dynamic response. Taking this dynamic response of state vectors and output vectors as an example, this state and this output without considering parameter uncertainties are defined as x(t) ˇ and yˇ (t), separately. These dynamic response comparisons are presented in Figs. 6.12 and 6.13. This means that these dynamic response curves of this suggested control system are smoother and closer to zero by considering this influence of parameter uncertainties. Delays are common in real systems. Time delays are often an instability source and occur in various practical plants. There exist types of time delays. This delay type of trigonometric functions can be encountered in practical systems (Jeung et al. 1998; Wei et al. 2018). Next, the problem of the admissible maximal delay bounds is considered. Set τ1 (t) = τ1 − τ D1 + τ D1 cos(t), τ2 (t) = τ2 − 0.5τ D2 + 0.5τ D2 sin(2t). Table 6.1 lists the admissible maximal delay bounds of τ1 for different τ2 under τ D1 = 0 and τ D2 = 0.6 calculated according to various criteria. Table 6.2 lists the admissible maximal delay bounds of τ2 for different τ D2 under τ1 = 1 and τ D1 = 0.2 calculated according to various criteria. It is further concluded that the results acquired by this approach described in the chapter are less conservative than those in Zhang et al. (2008).

6.5 Simulation Results

185

Fig. 6.12 The dynamic response of states

x1 (t) x ˇ1 (t)

States

1.5

×10 -4 -9

1 -10 0.5 -11 0 0

10 Time (sec)

x2 (t) x ˇ2 (t)

1.5 States

20

17.4 1.3

17.8

×10 -5

1.2 1.1

1

1

0.5

0.9

0 0

10 Time (sec)

Fig. 6.13 The dynamic response of outputs

20

y1 (t) yˇ1 (t)

1.5 Outputs

17.6

19.4

19.8

×10 -3 10

1

8

0.5

6

0

19.6

4 0

10

17.2517.317.3517.417.45

20

Time (sec) 0.8

y2 (t) yˇ2 (t)

Outputs

0.6

×10 -4 -2 -2.2

0.4 -2.4 0.2 -2.6 0 0

10 Time (sec)

20

17.5

17.6

17.7

17.8

186

6 Fault-Tolerant Control for Multiple-Delayed Switched Fuzzy …

Table 6.1 The admissible maximal delay bounds of τ1 for different τ2 with τ D1 = 0 and τ D2 = 0.6 Methods

τ2 0.7

0.9

1.5

2

2.5

τ1 obtained in this chapter τ1 obtained from Zhang et al. (2008)

19.4 18.5

19.4 18.4

19.4 18.0

19.4 17.6

19.4 17.0

Table 6.2 The admissible maximal delay bounds of τ2 for different τ D2 with τ1 = 1 and τ D1 = 0.2 Methods

τ D2 0

0.2

0.5

0.7

0.9

τ2 obtained in this chapter τ2 obtained from Zhang et al. (2008)

19.4 18.0

19.5 18.1

19.5 18.3

19.6 18.3

19.7 18.3

Example 6.2 The example studies a tunnel diode circuit system (Shen et al. 2020). Suppose x1 (t) is disturbed via multiple time-varying delays. These fuzzy rules can be selected by

ν12

=

ν11

=

⎧ ⎨ ⎩

1− 0,

x12 (t) , −3 ≤ x1 (t) ≤ 3 , 9 otherwise

ν22 = ν21 = 1 − ν11 .

A nonlinear system can be represented by the following data of model parameters: τ1 (t) = 1 + 0.1 sin(t), τ2 (t) = 0.5 + 0.4 cos(t),    2b¯0 10 2.9b¯0 10 2b¯0 10 1,0 1,0 2,0 , B2 = , B1 = , B1 = −b¯0 −1 −b¯0 −1 −b¯0 −2    2.9b¯0 10 2b¯1 0 2.9b¯1 0 2,0 2,1 1,1 2,1 1,1 B2 = , B1 = B1 = , B2 = B2 = , −b¯0 −2 −b¯1 0 −b¯1 0   2b¯2 0 2.9b¯2 0 B12,2 = B11,2 = , B22,2 = B21,2 = , −b¯2 0 −b¯2 0 

 0 2 2 1 1 Bu2 = Bu1 = Bu2 = Bu1 = , C22,0 = C12,0 = C21,0 = C11,0 = b¯0 1 , 1

  C22,1 = C12,1 = C21,1 = C11,1 = b¯1 0 , C22,2 = C12,2 = C21,2 = C11,2 = b¯2 0 , with b¯0 = 0.9, b¯1 = 0.02, and b¯2 = 0.08. Assuming that there exist stochastic systems, intermittent actuator faults and intermittent sensor faults, model uncertainties, external disturbances, and measurement noise in this model, the other data are listed in the following:

6.5 Simulation Results

187



   0.01 0 0 0.04 1 2 2 , Ba2 = , Ba1 = , Ba2 = , 0 0.03 0.02 0     0.002 0 0.001 0.003 1 1 2 2 = = = = Bω1 , Bω2 , Bω1 , Bω2 , 0 0.001 0 0     0 0 0 0 0 0 0 0 , D21,0 = , D12,0 = , D22,0 = , D11,0 = 0 0.001 0.001 0 0.002 0 0.004 0     0 0 0 0 0 0 0 0 , D21,1 = , D12,1 = , D22,1 = , D11,1 = 0 0.002 0.003 0 0 −0.001 −0.002 0     0 0 0 0 0 0 0 0 , D21,2 = , D12,2 = , D22,2 = , D11,2 = 0.0012 0 0 0.0013 0 0.0016 0.0023 0 1 Ba1 =

1 1 2 2 = 0.02, Bs2 = 0.01, Bs1 = 0.07, Bs2 = 0.04, Bs1 1 1 2 2 Bv1 = 0.002, Bv2 = 0.003, Bv1 = 0.001, Bv2 = 0.005,







 1 1 2 2 C z1 = 0.01 0 , C z2 = 0 0.01 , C z1 = 0 0.02 , C z2 = 0.03 0 .

The model uncertainties are as follows:     0 0 0 0 1 1 2 2 , Mx2 = , Mx1 = , Mx2 = , Mx1 = 0.0014 0.0012 0.0013 0.0016





 1,0 1,0 2,0 N x11 = 0 0.0011 , N x12 = 0.0013 0 , N x11 = 0 0.0012 ,





 2,0 1,1 1,1 N x12 = 0.0016 0 , N x11 = 0.0012 0 , N x12 = 0 0.0013 ,





 2,1 2,1 1,2 N x11 = 0 0.0009 , N x12 = 0.0014 0 , N x11 = 0.0021 0 ,





 1,2 2,2 2,2 N x12 = 0 0.0019 , N x11 = 0.0018 0 , N x12 = 0 0.0015 , 1 1 2 2 N x21 = 0.0021, N x22 = 0.0013, N x21 = 0.0016, N x22 = 0.0019, 1 1 2 2 N x31 = 0.0014, N x32 = 0.0018, N x31 = 0.0022, N x32 = 0.0013, 1 1 2 2 N x41 = 0.0016, N x42 = 0.0013, N x41 = 0.0017, N x42 = 0.0011,





 1,0 1,0 2,0 N x51 = 0 0.0013 , N x52 = 0.0015 0 , N x51 = 0.0012 0 ,





 2,0 1,1 1,1 N x52 = 0.0013 0 , N x51 = 0 0.0015 , N x52 = 0 0.0011 ,





 2,1 2,1 1,2 N x51 = 0.0032 0 , N x52 = 0.0018 0 , N x51 = 0 0.0019 ,





 1,2 2,2 2,2 N x52 = 0.0011 0 , N x51 = 0 0.0016 , N x52 = 0 0.0017 , 1 1 2 2 M y1 = 0.0017, M y2 = 0.0011, M y1 = 0.0014, M y2 = 0.0016,





 1,0 1,0 2,0 N y11 = 0.0011 0 , N y12 = 0 0.0013 , N y11 = 0 0.0016 ,





 2,0 1,1 1,1 N y12 = 0.0021 0 , N y11 = 0 0.0018 , N y12 = 0.0024 0 ,





 2,1 2,1 1,2 N y11 = 0.0013 0 , N y12 = 0.0015 0 , N y11 = 0.0018 0 ,





 1,2 2,2 2,2 N y12 = 0 0.0018 , N y11 = 0.0011 0.0013 , N y12 = 0.001 0 , 1 1 2 2 N y21 = 0.0014, N y22 = 0.0021, N y21 = 0.0011, N y22 = 0.0013,

188

6 Fault-Tolerant Control for Multiple-Delayed Switched Fuzzy …

1 1 2 2 N y31 = 0.0018, N y32 = 0.0022, N y31 = 0.0025, N y32 = 0.0031, 1 1 2 2 Mz1 = 0.0023, Mz2 = 0.0012, Mz1 = 0.0016, Mz2 = 0.0011,







 1 1 2 2 Nz1 = 0 0.001 , Nz2 = 0.001 0 , Nz1 = 0 0.002 , Nz2 = 0.003 0 . j

j

j

Assume 1 = diag{0.01, 0.01}, 2 = 0.01, Hxi = Hyi = Hzi = sin(t), (i ∈ r , j ∈  M¯ ). Assuming that this occurrence probability of intermittent actuator faults and intermittent sensor faults is ϑ¯ a = 0.1, ϑ¯ s = 0.5, this fault form is as follows: ⎧ 0, 0 ≤ t < 2, ⎪ ⎨ 0.89−t f a (t) = 3(1 − e 4 ), 2 ≤ t ≤ 14, ⎪ ⎩ 0, 14 < t ≤ 20, ⎧ 0 ≤ t < 3, ⎪ ⎨ 0, f s (t) = 0.5 + 0.2 cos(0.1t), 3 ≤ t ≤ 13, ⎪ ⎩ 0, 13 < t ≤ 20.

Fig. 6.14 The intermittent actuator fault

Intermittent actuator fault

The faults are shown in Figs. 6.14 and 6.15. It is calculated that τ1 = 1.1, τ D1 = 0.1, τ2 = 0.9 and τ D2 = 0.4. In addition, this weighted H∞ performance index is set as (8, 0.1) and μˇ = 1.4. Choose 0 =

3

a (t)fa (t)

2

1

0 0

5

10

15

20

Fig. 6.15 The intermittent sensor fault

Intermittent sensor fault

Time (sec) s (t)fs (t)

0.6 0.4 0.2 0 0

5

10 Time (sec)

15

20

6.5 Simulation Results

189

Fig. 6.16 The resulting states

x1 (t) x2 (t)

States

0.5

0

-0.5 0

5

10

15

20

Time (sec)

Fig. 6.17 The output

0.2

y(t)

Output

0 -0.2 -0.4 0

5

10

15

20

Time (sec)

Fig. 6.18 The controller states

η1 (t) η2 (t)

Controller states

5 0 -5 -10 0

j,h

j,h

j,h

j,h

j

5

j

j

10 Time (sec)

15

20

1i = 2i = 3i = 4i = 1, π¯ 3i = π¯ 2i = π¯ 1il = 0.5, (i ∈ r , j ∈  M¯ ), ψ(0) =

T 0.74 −0.68 . Through Theorem 6.3, this optimal guaranteed cost controller is acquired. This optimal guaranteed cost bound is J∗ = 5.4302. Figures 6.16, 6.17, 6.18 and 6.19 present these simulation results, where the switching times are 8.3647s and 16.7294s. Figure 6.16 shows the states x1 (t) and x2 (t), which means that the states are mean-square exponentially stable. Figure 6.17 describes this output vector y(t), indicating that this output vector can quickly approach zero after faults disappear. Figures 6.18 and 6.19 present these controller state vectors η1 (t) and η2 (t) and the control input vector u(t). We can find that this controller can quickly reach mean-square exponential stability, and the controller is successfully designed.

190

6 Fault-Tolerant Control for Multiple-Delayed Switched Fuzzy …

Fig. 6.19 The control input

u(t)

Control input

3 2 1 0 -1 0

5

10 Time (sec)

15

20

Table 6.3 The admissible maximal delay bounds of τ1 for different τ2 with τ D1 = 0.1 and τ D2 = 0.4 τ2 0.9 1.2 1.5 1.8 2.3 τ1 by this chapter

17.5

16.0

13.6

13.5

13.3

In addition, the admissible maximal delay bound problem is discussed. Set τ1 (t) = τ1 − τ D1 + τ D1 sin(t), τ2 (t) = τ2 − τ D2 + τ D2 cos(t). Table 6.3 lists the admissible maximal delay bounds of τ1 for different τ2 under τ D1 = 0.1 and τ D2 = 0.4 calculated according to various criteria. It can be seen that with the increase of τ2 , the admissible maximal delay bounds of τ1 will become smaller.

6.6 Chapter Summary In this chapter, the delay-dependent passive fault-tolerant control problem for a set of multiple time-varying delayed uncertain switched T-S fuzzy stochastic systems with intermittent actuator faults, intermittent sensor faults external disturbances, and measurement noise is studied. This is one of the few attempts to achieve reliable H∞ guaranteed cost control for a switched T-S fuzzy stochastic model against intermittent actuator and sensor faults, which not only maintains the stability of the system, but also ensures a certain performance index. Therefore, this approach has a wider scope of applications. The dynamic feedback controller is researched. Furthermore, the closed-loop model is established. Using a piecewise fuzzy Lyapunov function, stability results through linear matrix inequalities are given without slack matrices. This chapter has relatively low conservatism. Finally, the effectiveness of the method is demonstrated by simulation examples.

References

191

References Du D, Cocquempot V (2017) Fault diagnosis and fault tolerant control for discrete-time linear systems with sensor fault. IFAC Pap 50(1):15754–15759 Gao Z (2015) Fault estimation and fault-tolerant control for discrete-time dynamic systems. IEEE Trans Ind Electron 62(6):3874–3884 Gao H, Liu X, Lam J (2009) Stability analysis and stabilization for discrete-time fuzzy systems with time-varying delay. IEEE Trans Cybern 39(2):306–317 Han J, Zhang H, Wang Y et al (2016) Robust state/fault estimation and fault tolerant control for T-S fuzzy systems with sensor and actuator faults. J Frankl Inst 353(2):615–641 Han J, Liu X, Wei X et al (2019a) Reduced-order observer based fault estimation and fault-tolerant control for switched stochastic systems with actuator and sensor faults. ISA Trans 88:91–101 Han J, Liu X, Wei X et al (2019b) Dissipativity-based fault estimation for switched non-linear systems with process and sensor faults. IET Control Theory Appl 13(18):2983–2993 Huang S, Yang G (2014) Fault tolerant controller design for T-S fuzzy systems with time-varying delay and actuator faults: A k-step fault-estimation approach. IEEE Trans Fuzzy Syst 22(6):1526– 1540 Huang S, He X, Zhang N (2011) New results on H∞ filter design for nonlinear systems with time delay via T-S fuzzy models. IEEE Trans Fuzzy Syst 19(1):193–199 Jeung ET, Kim JH, Park HB (1998) H∞ -output feedback controller design for linear systems with time-varying delayed state. IEEE Trans Autom Control 43(7):971–974 Jiang X, Han Q (2007) Robust H∞ control for uncertain Takagi-Sugeno fuzzy systems with interval time-varying delay. IEEE Trans Fuzzy Syst 15(2):321–331 Lien CH, Yu KW, Lin YF et al (2008) Robust reliable H∞ control for uncertain nonlinear systems via LMI approach. Appl Math Comput 198(1):453–462 Shen H, Xing M, Wu Z et al (2020) Multiobjective fault-tolerant control for fuzzy switched systems with persistent dwell-time and its application in electric circuits. IEEE Trans Fuzzy Syst 28(10):2335–2347 Su X, Shi P, Wu L et al (2016) Fault detection filtering for nonlinear switched stochastic systems. IEEE Trans Autom Control 61(5):1310–1315 Sun S, Zhang H, Wang Y et al (2018) Dynamic output feedback-based fault-tolerant control design for T-S fuzzy systems with model uncertainties. ISA Trans 81:32–45 Tao Y, Shen D, Wang Y et al (2015) Reliable H∞ control for uncertain nonlinear discrete-time systems subject to multiple intermittent faults in sensors and/or actuators. J Frankl Inst 352(11):4721– 4740 Wang L, Lam H (2018) A new approach to stability and stabilization analysis for continuous-time Takagi-Sugeno fuzzy systems with time delay. IEEE Trans Fuzzy Syst 26(4):2460–2465 Wei Y, Qiu J, Karimi HR et al (2018) A novel memory filtering design for semi-Markovian jump time-delay systems. IEEE Trans Syst Man Cybern: Syst 48(12):2229–2241 Xu S, Lam J, Mao X (2007) Delay-dependent H∞ control and filtering for uncertain Markovian jump systems with time-varying delays. IEEE Trans Circuits Syst I: Regul Pap 54(9):2070–2077 Yang H, Jiang Y, Yin S (2018) Fault-tolerant control of time-delay Markov jump systems with Itô stochastic process and output disturbance based on sliding mode observer. IEEE Trans Ind Inform 14(12):5299–5307 Zhang H, Wang Y, Liu D (2008) Delay-dependent guaranteed cost control for uncertain stochastic fuzzy systems with multiple time delays. IEEE Trans Cybern 38(1):126–140 Zhang K, Jiang B, Staroswiecki M (2010) Dynamic output feedback fault tolerant controller design for Takagi-Sugeno fuzzy systems with actuator faults. IEEE Trans Fuzzy Syst 18(1):194–201 Zhang H, Han J, Luo C et al (2017) Fault-tolerant control of a nonlinear system based on generalized fuzzy hyperbolic model and adaptive disturbance observer. IEEE Trans Syst Man Cybern: Syst 47(8):2289–2300

Chapter 7

Finite-Time Fault-Tolerant Control for Multiple-Delayed Switched Fuzzy Systems with Intermittent Faults

7.1 Introduction In the last chapters, we discuss fault-tolerant control for multiple delayed T-S fuzzy systems, which is concerned with asymptotical stability in infinite-time (Wang et al. 2020a). Nevertheless, finite-time boundedness or finite-time stability is needed in some practical systems (Gu et al. 2020; Du and Lu 2020). In Chen et al. (2020), the issue of finite-time control was researched for Markovian jump systems randomly occurring quantization. In Kang et al. (2020), finite-time stability for neutral fractional order time delayed models against Lipschitz nonlinearities was explored. Besides, it should be noted that much attention has been given to the issues of input-output finite-time stability (Amato and Tommasi 2010). In Wang et al. (2019), this input-output finite-time stability was investigated for network-induced delayed networked control models, which was not sufficiently explored. Therefore, in the chapter, finite-time boundedness and input-output finite-time stability are further investigated.

7.2 System Definition and Description A switched T-S fuzzy system is depicted below. x(t) ˙ =

q 

σj (t)

j =1

h 

δıj (ρ(t))

 m

Bıj,g (t)x(t − τg (t)) + Bjuı (t)u(t)

g=0

ı=1

 + A p (t)Bjpı (t) f p (t) +Bjwı (t)w(t) , y(t) =

q  j =1

σj (t)

h  ı=1

δıj (ρ(t))

 m

(7.1)

Cıj,g (t)x(t − τg (t)) + As (t)Cjsı (t) f s (t)

g=0

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Sun et al., Fault-Tolerant Control for Time-Varying Delayed T-S Fuzzy Systems, Intelligent Control and Learning Systems 9, https://doi.org/10.1007/978-981-99-1357-2_7

193

194

7 Finite-Time Fault-Tolerant Control for Multiple-Delayed Switched …

+ z(t) =

Cjvı (t)v(t)

q 

σj (t)

j =1

 ,

h 

(7.2)

δıj (ρ(t))Dıj x(t),

(7.3)

ı=1

x(t) =ψ(t), t ∈ [−τ¯M , 0],

(7.4)

where x(t) ∈ R a¯ x , A p (t) f p (t) ∈ R a¯ p and w(t) ∈ R a¯ w denote this system state, this intermittent process fault, and this exogenous disturbance, respectively, where A p (t) leads to this random occurrence of this process fault phenomena, complying with the Bernoulli distribution under this range {0, 1}. E{A p (t)} = A¯ p , where the constant A¯ p is known. y(t) ∈ R a¯ y , As (t) f s (t) ∈ R a¯ s and v(t) ∈ R a¯ v denote this system measurement output, this intermittent sensor fault and this measurement noise, respectively, where As (t) results in this random occurrence of this sensor fault phenomena, complying the Bernoulli distribution under this range {0, 1}. E{As (t)} = A¯ s , where the constant A¯ s is known. z(t) ∈ R a¯ z stands for this desired controlled output. τ0 (t) = 0. τg (t) means the time-varying state delay, complying 0 ≤ τdg ≤ τg (t) ≤ τg , where τdg means the minimum of τg (t) and the known scalar τg means the maximum of τg (t). τg ≤ τ¯M , where the known constant τ¯M denotes the maximum of τg . When τg (t) is differentiable, τ˙g (t) ≤ τ˜g , where τ˜g denotes the maximum of τ˙g (t). Besides, ψ(t), t ∈ [−τ¯M , 0], means the vector-valued initial function. Denote ι  {1, . . . , ι}, h  {1, . . . , h}, q  {1, . . . , q} as well as m  {1, . . . , m}, where ι, h, q and m stand for these numbers of premise variables, IF-THEN rules, switched subsystems j j,g and time delays, separately. This constant matrix Dı is known. Matrices Bı (t), j j j j,g j j Buı (t), B pı (t), Bwı (t), Cı (t), Csı and Cvı (t), ı ∈ h , j ∈ q , g ∈ {0, q }, follow Bıj,g (t) = Bıj,g + Bıj,g (t), Bjuı (t) = Bjuı + Bjuı (t), Bjpı (t) = Bjpı + Bjpı (t), Bjwı (t) = Bjwı + Bjwı (t), Cıj,g (t) = Cıj,g + Cıj,g (t),

Cjsı (t) = Cjsı + Cjsı (t), Cjvı (t) = Cjvı + Cjvı (t), j,g

j

j

j

j,g

j

j

where matrices Bı , Buı , B pı , Bwı , Cı , Csı and Cvı are known, and unknown j,g j j j j,g j j matrices Bı (t), Buı (t), B pı (t), Bwı (t), Cı (t), Csı (t), and Cvı (t) are parameter uncertainties, acquired by   j,g j  j,g j j j j j  Bı (t) Buı (t) B pı (t) Bwı (t) = Ajxı Ejxı (t) Fxı Fxuı Fx pı Fxwı ,     j,g j j,g j j j Cı (t) Csı (t) Cvı (t) = Ajyı Ejyı (t), F yı F ysı F yvı , j

j,g

j

j

j

j

j,g

j

j

where constant matrices Axı , Fxı , Fxuı , Fx pı , Fxwı , A yı , F yı , F ysı , and F yvı are j j known. Matrices Exı (t) and E yı (t) are uncertain and time-varying, following (Ejxı (t))T Ejxı (t) ≤ I, (Ejyı (t))T Ejyı (t) ≤ I.

7.3 Controller Design

195

 j  Define ρι¯ (t), ι¯ ∈ ι as this premise variable. ρ(t) = ρ1 (t) · · · ρι (t) . δı (ρ(t)) = h j  j j j j δˆı (ρ(t)), δˆı (ρ(t)) = ιι¯=1 δ¯ı ι¯ (ρ(t)), where δ¯ı ι¯ (·), ı ∈ h , ι¯ ∈ ι , δˆı (ρ(t))/ ı=1 j j j ∈ q stands for this grade of this membership function of δ¯ı ι¯ , and δ¯ı ι¯ denotes this fuzzy set symbolized through this membership function. Denote σj (t), ([0, ∞) ⇒ {0, 1}) as this switching signal. When σj (t) = 1, t ∈ [t¯, t˜), this j th switched subj model is activated under time interval [t¯, t˜). To simplify the expression δı is used to j denote δı (ρ(t)), and σj is used to denote σj (t).

7.3 Controller Design This switched fuzzy controller with zero initial value is devised by η(t) ˙ =

q 

σj

j =1

u(t) =

q  j =1

h 

δıj

h 

j δκj [Bıκ η(t) + Cıj y(t)],

(7.5)

κ=1

ı=1

σj

h 

δıj Dıj η(t),

(7.6)

ı=1 j

j

j

where η(t) ∈ R a¯ x stands for this controller state, and Bıκ , Cı , Dı , ı, κ ∈ h , j ∈ q , mean the controller gain matrices. ¯ = [ f pT (t) w T (t) f sT (t) v T (t)]T . Based on (7.1)– Set η(t) ¯ = [x T (t) η T (t)]T , w(t) (7.6), we have η˙¯ (t) =

q 

σj

h 

δıj

h 

j,0 δκj [Bıκ (t)η(t) ¯ +

j =1

+

ı=1 κ=1 j Bwıκ ¯ ¯ (t)A(t)w(t)],

 q

z(t) =

σj

j =1

h 

m 

j,g Bıκ (t)η(t ¯ − τg (t))

g=1

δıj Dıj η(t), ¯

(7.8)

ı=1

¯ η(t) ¯ =ψ(t), t ∈ [−τ¯M , 0], where    j,g j,0 j Bı (t) 0 Bı (t) Buı (t)Djκ j,g , , B (t) = j j j ıκ Cı Cj,g Cı Cj,0 Bıκ κ (t) 0 κ (t)   j j 0 0 B pı (t) Bwı (t) j Bwıκ , (t) = j j j j ¯ 0 0 Cı Csκ (t) Cı Cvκ (t)  j  A(t) = diag{A p (t), I, As (t), I }, Dıj = Dı 0 ,   ¯ = ψ T (t) 0T T , t ∈ [−τ¯M , 0]. ψ(t) j,0 Bıκ (t) =

(7.7)



(7.9)

196

7 Finite-Time Fault-Tolerant Control for Multiple-Delayed Switched …

Assumption 7.1 Suppose that 2 } < ηˇ¯ 2 , E{u(t)2 } < uˇ 2 , E{η(t) ¯

where known constants ηˇ¯ > 0 and uˇ > 0. Assumption 7.2 Set that T > 0 is a time scalar. During time interval [0, T], suppose that  T  T H(T, M1 , χ1 )  w¯ (t) ∈ L 2 [0, T] : E w¯ T (t)M1 w(t)dt ¯ ≤ χ1 , 0

where M1 is a positive definite matrix, and χ1 is a positive scalar. Definition 7.1 Ren et al. (2018). It is assumed that there are constants c1 > 0, c2 > 0, T > 0, and a matrix R > 0. Under t ∈ [0, T], the closed-loop model (7.7)–(7.9) can be finite-time boundedness subject to (c1 , c2 , R, T), if it follows ˙ ≤ c1 , ⇒ E{x T (t)Rx(t)} < c2 . max {x T (i)Rx(i), x˙ T (i)R x(i)}

i∈[−τ¯ M ,0]

Definition 7.2 Amato et al. (2012). Presume that there exist signals H(T, M1 , χ1 ) defined in Assumption 7.2, a positive definite matrix M2 , and a positive constant χ2 . Under t ∈ [0, T], the closed-loop model (7.7)–(7.9) is called input-output finite-time stability subject to (H(T, M1 , χ1 ), M2 , χ2 ), if it follows w(t) ¯ ∈ H(T, M1 , χ1 ) ⇒ E{z T (t)M2 z(t)} < χ2 . Definition 7.3 Xiang et al. (2012). Presume there are signals (c1 , c2 , R, T) and ¯ γ. This closed-loop model (7.7)–(7.9) (H(T, M1 , χ1 ), M2 , χ2 ), positive constants μ, can be robust finite-time boundedness and input-output finite-time stability with H∞ performance index γ if the following conditions hold (1) The closed-loop model (7.7)–(7.9) is called finite-time boundedness subject to (c1 , c2 , R, T)and input-output finite-time stability subject to (H(T, M1 , χ1 ), M 2 , χ2 ) (2) With w(t) ¯ in Assumption 7.2, one has  T  T −μt ¯ T 2 e z (t)z(t)dt < γ E w¯ T (t)w(t)dt ¯ , +V (η¯0 , 0) E 0

0

where this constant μ¯ > 0 and V (·) means a function under V (0, 0) = 0.

7.4 Stability Analysis

197

7.4 Stability Analysis Theorem 7.1 For input signals (c1 , c2 , R, T) and (H(T, M1 , χ1 ), M2 , χ2 ), positive constants , μ, χ¯ 1 ≥ χ1 , χ¯ 2 ≤ χ2 , χ3 , γ, > 1, and non-negative constants α1g , α2g , α3g , this closed-loop model (7.7)–(7.9) is called robust finite-time boundedness as well as input-output finite-time stability with H∞ performance index γ, if there j j,g j,g j,g j j j exist positive definite matrices Pı , Q 1ı , Q 2ı , Z ı , matrices Bıκ , Cı , Dı and slack j j matrices W1ı , W2ı , , ı, κ ∈ h , j, j¯ ∈ q , g ∈ m , such that inequalities hold as follows: j j,g j,g j,g P˙δ ≤ 0, Q˙ 1δ ≤ 0, Q˙ 2δ ≤ 0, Z˙ δ ≤ 0,

(7.10)

j,g j¯,g j,g j¯,g Pıj < Pıj¯ , Q 1ı < Q 1ı , Q 2ı < Q 2ı , ¯ κı (t) < 0, ı ≤ κ ∈ h , ¯ jıκ (t) +   j ˜ κı (t) < 0, ı ≤ κ ∈ h , ˜ ıκ (t) +   j ¯ ıκ (t) +  ¯ κı (t) < 0, ı ≤ κ ∈ h ,  ˜ jıκ (t) +  ˜ κı (t) < 0, ı ≤ κ ∈ h ,  e(1+ )μT (φ1 c1 + χ3 ) < c2 , φ2

j,g Z 1ı


ln μ and (7.24)–(7.25), integrating (7.23) from 0 to t, it follows that  E V (η¯t , t) μt

E e z (θ)z(θ)dθ 0

(7.38)

0

 t  t −μθ−N (0,θ) ln T T e w¯ (θ)w(θ)dθ ¯ 1, and non-negative constants α1g , α2g , α3g , this closed-loop model (7.7)–(7.9) is called robust finite-time boundedness and input-output finite-time stability with H∞ performance index γ, if there exists a j ¯ j, D ¯ j , positive definite matrices X j , Y j , Q¯ j,g positive constant φ¯ 2 , matrices B¯ ıκ , C 1ı ,  jı T ı T   j T T j,g j,g j j j j ˜ ¯ ˜ ¯ Q 2ı , Z ı and matrices W1ı = (w¯ 1ı ) 0 , W2ı = (w¯ 2ı ) 0 , where w¯ 1ı , w¯ 2ı j j denote slack matrices and have the same dimensions with W1ı , W2ı in Theorem 7.1, , ı, κ ∈ h , j, j¯ ∈ q , g ∈ m , such that inequalities are feasible as follows: j,g j,g j,g Q˙¯ 1δ ≤ 0, Q˙¯ 2δ ≤ 0, Z˙¯ δ ≤ 0,

(7.41)

j,g j¯,g j,g j¯,g j,g j¯,g P¯ j < P¯ j¯ , Q¯ 1ı < Q¯ 1ı , Q¯ 2ı < Q¯ 2ı , Z¯ 1ı < Z¯ 1ı ,

(7.42)

j

¯ ıκ + ¯ κı < 0, ı ≤ κ ∈ h ,

(7.43)

j

˜ ıκ + ˜ κı < 0, ı ≤ κ ∈ h ,

(7.44)

¯ κı < 0, ı ≤ κ ∈ h , ¯j +  ıκ

(7.45)

˜ κı < 0, ı ≤ κ ∈ h , ˜j +  ıκ

(7.46)

 m 1  j j,g j,g (1 − eμτg )(α1g Q¯ 1ı + α2g Q¯ 2ı ) P¯ı − μ g=1  1 1 1 ¯j j,g j + [τg + (1 − eμτg )]α3g Z¯ ı U < 0, − U + μ φ¯ 1 1 φ¯ 1 2

(7.47)

206

7 Finite-Time Fault-Tolerant Control for Multiple-Delayed Switched … 

 j − P¯ j (S1 )T < 0, ∗ −φ¯ 2 R −1 c1 + χ3 ) < c2 , φ¯ 2 e(1+ )μT ( φ¯ 1 ⎡ ⎤ j χ¯ P¯ j (D¯ ı )T − (1+ )μT 2c1 e ( +χ ) 3 ⎣ ⎦ < 0, φ¯ 1 ∗ −M2−1 ⎤ ⎡ j P¯ j (S1 )T − (1+ )μT1 c1 ( ¯ +χ3 ) ⎦ < 0, ⎣ e φ1 ∗ −ηˇ¯ 2 I ⎡ j j ⎤ P¯ı D¯ ı − (1+ )μT1 c1 ( ¯ +χ3 ) ⎦ < 0. ⎣ e φ1 ∗ −uˇ 2 I

(7.48) (7.49) (7.50) (7.51) (7.52)

where j,g Q¯ 1δ =

h  ı=1

j,g j,g δıj Q¯ 1ı , Q¯ 2δ =

h  ı=1

j,g j,g δıj Q¯ 2ı , Z¯ δ =

h 

δıj Z¯ ıj,g ,

ı=1

⎡ j j j j,1 j,m ⎤ ıκ ıκ ıκ α31 τ1 W˜ 1ı · · · α3m τm W˜ 1ı j ⎥ ⎢ ∗ ¯ ıκ 0 0 ··· 0 ⎥ ⎢ j ⎥ ⎢ j ¯ 0 · · · 0 ∗ ∗

ıκ

¯ ıκ = ⎢ ⎥, ⎥ ⎢ . ⎦ ⎣ ∗ ∗ ∗ −α31 τ1 Z¯ j,1 . . 0  j,m ∗ ∗ ∗ ∗ ∗ −α3m τm Z¯  ⎡ j j j j,1 j,m ⎤ ıκ ıκ ıκ α31 τ1 W˜ 2ı · · · α3m τm W˜ 2ı j ⎥ ⎢ ∗ ¯ ıκ 0 0 ··· 0 ⎥ ⎢ j ⎥ ⎢ j ¯ 0 ··· 0

˜ ıκ = ⎢ ∗ ∗ ıκ ⎥, ⎥ ⎢ . ⎦ ⎣ ∗ ∗ ∗ −α31 τ1 Z¯ j,1 . . 0  j,m ∗ ∗ ∗ ∗ ∗ −α3m τm Z¯  ⎤ ⎡ j j j j ˇ ıκ (ϒı )T ıκ ıκ α31 τ1 W˜ 1ıj,1 · · · α3m τm W˜ 1ıj,m  ⎥ ⎢ ∗ −I 0 0 0 ··· 0 ⎥ ⎢ j ⎥ ⎢ ∗ ¯ 0 ··· 0 ∗ ıκ 0 ⎥ ⎢ j ¯ j ıκ = ⎢ ∗ ⎥, ¯ 0 ··· 0 ∗ ∗ ıκ ⎥ ⎢ ⎥ ⎢ j,1 . ⎦ ⎣ ∗ 0 ∗ ∗ ∗ −α31 τ1 Z¯  . . j,m ∗ ∗ ∗ ∗ ∗ ∗ −α3m τm Z¯  ⎡ j j j j,1 j,m ⎤ ˇ ıκ (ϒıj )T ıκ 

ıκ α31 τ1 W˜ 2ı · · · α3m τm W˜ 2ı ⎥ ⎢ ∗ −I 0 0 0 ··· 0 ⎥ ⎢ j ⎥ ⎢ ∗ ¯ 0 ··· 0 ∗ ıκ 0 ⎥ ⎢ j ˜ ıκ = ⎢ j  ⎥, ¯ 0 ··· 0 ∗ ∗ ıκ ⎥ ⎢ ∗ ⎥ ⎢ j,1 . ⎦ ⎣ ∗ 0 ∗ ∗ ∗ −α31 τ1 Z¯  . . j,m ¯ ∗ ∗ ∗ ∗ ∗ ∗ −α3m τm Z 

7.4 Stability Analysis

207

 sym(K j R2 X j ) + K j R3 (K j )T X j R1 + K j R2 , ∗ R1  ı    U2 0 Xj I j j j U¯ 2ı = , , U2ı = (k1 )2 R1−1 − 2k1 X j , S1 = ∗ 0 (K j )T 0  j  j  j ¯ı 0 , D¯ ıj = Dı X j Dı , D¯ ıj = D 

U1ı =

⎡

j j,1 j,m j 1ıκ B¯ıκ · · · B¯ıκ 0 0 0 B¯wıκ ¯ ⎢ ∗  j,1 0 0 0 0 0 0 ⎢ 4ı ⎢ . ⎢ ∗ ∗ .. 0 0 0 0 0 ⎢ ⎢ j,m 0 0 0 0 ∗ ∗ 4ı ⎢ ∗ ⎢ j j,1 ıκ = ⎢ ∗ ∗ ∗ ∗ 5ı 0 0 0 ⎢ ⎢ .. ⎢ ∗ . 0 ∗ ∗ ∗ ∗ 0 ⎢ j,m ⎢ ∗ ∗ ∗ ∗ ∗ ∗  0 5ı ⎢ χ3 ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ − χ¯ 1 M1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎤ ⎡ j,1 j,m 0ıκ · · · 0ıκ 0 ··· 0 ⎢ 0 · · · 0 ( j,1 )T · · · 0 ⎥ ⎥ ⎢ 1ıκ ⎥ ⎢ . . .. .. ⎥ ⎢ . . . . . 0 0 ⎥ ⎢ . ⎢ j,1 T ⎥ ⎢ 0 ··· 0 0 · · · ( 1ıκ ) ⎥ ⎥ ⎢ j ıκ = ⎢ 0 ··· 0 0 ··· 0 ⎥, ⎥ ⎢ . ⎥ ⎢ .. . . .. ⎥ ⎢ . . .. . 0 0 ⎥ ⎢ ⎥ ⎢ 0 ··· 0 0 · · · 0 ⎥ ⎢ ⎣ 0 ··· 0 0 ··· 0 ⎦ j,1 j,m 0 ··· 0 0ıκ · · · 0ıκ j j,1 j,m ¯ = I2 ⊗ diag{−kıκ I, . . . , −kıκ I }, ⎡ ˜j j ⎤ j kıκ 0ıκ ( 1ıκ )T j,1 ⎢ 0 ( 2ıκ )T ⎥ ⎢ ⎥ ⎢ . ⎥ .. ⎢ .. ⎥ . ⎢ ⎥ j,m T ⎥ ⎢ 0 ( 2ıκ ) ⎥ ⎢ j ˜j ¯j

ıκ =⎢ 0 ⎥ ⎢ 0 ⎥ , ıκ = I2 ⊗ (−kıκ I ), ⎢ . ⎥ .. ⎢ .. ⎥ . ⎢ ⎥ ⎢ 0 0 ⎥ ⎢ ⎥ j ⎣ 0 ( 3ıκ )T ,⎦ j j k˜ıκ 0ıκ 0

⎤ j,0 α( ¯ B¯ıκ )T j,1 α( ¯ B¯ıκ )T ⎥ ⎥ ⎥ .. ⎥ . ⎥ ⎥ j,m α( ¯ B¯ıκ )T ⎥ ⎥ j , ⎥ + ıκ 0 ⎥ .. ⎥ ⎥ . ⎥ ⎥ 0 ⎥ j T⎦ ¯ α( ¯ Bwıκ ¯ ) j 4

208

7 Finite-Time Fault-Tolerant Control for Multiple-Delayed Switched …

⎡

j j,1 j,m 1ıκ B¯ıκ · · · B¯ıκ 0 ⎢ ∗  j,1 0 0 0 ⎢ 4ı ⎢ . ⎢ ∗ .. 0 0 ⎢ ∗ ⎢ j,m ⎢ ∗ 0 ∗ ∗ 4ı ⎢ j j,1 ˇ ıκ  =⎢ ∗ ∗ ∗ ∗ 5ı ⎢ ⎢ ⎢ ∗ ∗ ∗ ∗ ∗ ⎢ ⎢ ∗ ∗ ∗ ∗ ∗ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗   ϒıj = D¯ ıj 0 · · · 0 0 · · · 0 0 0 ,

j j,0 1ıκ = sym(B¯ıκ )+



m 

0 0

0 0

0 0 0 .. .

0 0 0

0 j,m ∗ 5ı ∗ ∗ ∗ ∗

⎤ j j,0 B¯wıκ α( ¯ B¯ıκ )T ¯ j,1 0 α( ¯ B¯ıκ )T ⎥ ⎥ ⎥ .. ⎥ 0 . ⎥ ⎥ j,m T 0 α( ¯ B¯ıκ ) ⎥ ⎥ j , ⎥ + ıκ 0 0 ⎥ .. ⎥ ⎥ 0 . ⎥ ⎥ 0 0 ⎥ j T⎦ ¯ B¯wıκ ) −γ 2 I α( ¯ j ∗ 4

j,g j,g (α1g Q¯ 1ı + α2g Q¯ 2ı ) − μ P¯ j ,

g=1

 j,0 + B ı j,0 B¯ıκ = , j j,0 ¯ ıκ ¯ ıj Cj,0 B Y j Bı + C κ    j  j,g j j,g Bı Bı X X I j,g ¯ , B , = P¯ j = j,g ıκ ¯ ıj Cj,g ∗ Yj 0 Y j Bı + C κ   j j Bwı 0 0 A¯ p B pı j ¯ Bwıκ = ¯ j j j j ¯ ¯j j ¯j j , ¯ A p Y B pı Y Bwı As Cı Csκ Cı Cvκ j,g j,g j,g j,g 4ı = −α1g eμτdg (1 − τ˜g ) Q¯ 1ı ,  = −α2g eμτg Q¯ 2ı , j,0 Bı X j

j ¯j Buı D ı

5ı

α¯ = 1 with

m 

α3g > 0, α¯ = 0 with

g=1 j

m 

α3g = 0,

g=1

j

j

4 = −2s P¯ j + (s )2

m  g=1

j,g

α3g τg Z¯  ,

  m  j,g j,1 j,m j α3g w¯ 1ı α31 w¯ 1ı · · · α3m w¯ 1ı 0 · · · 0 0 0 ıκ = sym −  + sym 00 j,g 0ıκ j

0ıκ

g=1

j,1

j,1

j,1

j,m

0 −α31 w¯ 2ı · · · −α31 w¯ 2ı α31 w¯ 2ı · · · α3m w¯ 2ı

 ,

  j  0 j,g j,g X 0 , , 1ıκ = kıκ = j,g ¯ ıj Cj,g Y j Bı + C κ     j j,0 j ¯j j,0 0 Axı Fxı X j + Fxuı D F j xı κ , 1ıκ = = j ¯j j j,0 j,0 , Y j Axı C F yκ X j F yκ ı A yκ 

7.4 Stability Analysis j,g



2ıκ =

209

  j,g j,g  j j 0 0 Fxı X j Fxı A¯ p F pı Fwı j,g ,

. = j,g j,g j j 3ıκ F yκ X j F yκ 0 0 A¯ s F ysκ F yvκ

In addition, switching signals meet this average dwell time Ta > matrices can be obtained by

ln . The controller μ

j j j j j ¯j j j,0 j j −T ¯ ıj Cj,0 = (J j )−1 (B¯ ıκ −C , Bıκ κ X − Y Buı Dκ − Y Bı X )(K ) j j −1 ¯ j Cı = (J ) Cı ,

Dıj

=

¯ ıj (K j )−T , D

(7.53) (7.54) (7.55)

where this matrix J j follows J j = (I − Y j X j )(K j )−T . j

Proof Matrices P j = Pı and (P j )−1 are defined by Pj = j



 j j  Yj Jj X K , (P j )−1 = . ∗ 1 ∗ 2

j

Define matrices S1 and S2 as follows j



S1 =

   Xj I I Yj j = , S , 2 0 (J j )T (K j )T 0

where 1 and 2 are arbitrary matrices as long as P j (P j )−1 = (P j )−1 P j = I . We can find that j j j j S2 = P j S1 , (S1 )T P j S1 = P¯ j . j j j j j j Denote W1 = diag{(S1 )T , Im ⊗ (S1 )T , Im ⊗ (S1 )T , I }, P j = (S1 )−T P¯ j (S1 )−1 , j,g j −T ¯ j,g j −1 j,g j −T ¯ j,g j −1 j,g j −T ¯ j,g j −1 Q 1ı = (S1 ) Q 1ı (S1 ) , Q 2ı = (S1 ) Q 2ı (S1 ) , Z ı = (S1 ) Z ı (S1 ) , j,g j j,g j j,g j j,g j W1ı = (W1 )−T w¯ 1ı (W1 )−1 , W2ı = (W1 )−T w¯ 2ı (W1 )−1 . Pre- and postj −T multiplying (7.41) and (7.42) by (S1 ) and its transpose, separately, yields (7.10) and (7.11). By using Schur complement, it follows from (7.43)–(7.46) that j

j

j

j

j

¯ ıκ (t)W2 + (W2 )T  ¯ κı (t)W2 < 0, ı ≤ κ ∈ h , (W2 )T  j T ˜j j j T ˜ j (W2 ) ıκ (t)W2 + (W2 ) κı (t)W2 < 0, ı ≤ κ ∈ h , j ¯ jıκ (t)Wj3 + (Wj3 )T  ¯ κı (t)Wj3 < 0, ı ≤ κ ∈ h , (W3 )T  j ˜ jıκ (t)Wj3 + (Wj3 )T  ˜ κı (t)Wj3 < 0, ı ≤ κ ∈ h , (W3 )T  j

j

j

j

j

j

(7.56) (7.57) (7.58) (7.59)

where W2 = diag{W1 , Im ⊗ (S1 )T }T , W3 = diag{W1 , I, Im ⊗ (S1 )T }T . Using the j congruence transformation (W2 )−T to (7.56)–(7.57), separately, yields (7.12)– j (7.13). Using the congruence transformation (W3 )−T to (7.58)–(7.59), separately, yields (7.14)–(7.15).

210

7 Finite-Time Fault-Tolerant Control for Multiple-Delayed Switched …

Table 7.1 The values of non-negative constants α1g , α2g and α3g Delay situations

α1g

α2g

α3g

Without time delay With invariant delay (τ˜g = 0) With time-varying delay(τ˜g < 1 is known) With time-varying delay (τ˜g ≥ 1 Or is unknown or τ (t) is not differentiable)

0 1

0 0

0 0 or 1

1

1

0 or 1

0

1

1

j

Employing the congruence transformation (S1 )−T to (7.47), we have φ1
φ¯1 . (7.49) means that (7.16) holds. Pre- and 2 j post-multiplying (7.50)–(7.51) by diag{(S1 )−T , I } and its transpose, respectively,

and then using Schur complement yields (7.17)–(7.18). Pre- and post-multiplying j (7.52) by diag{(S1 )−T , I } and its transpose, separately, ones has (7.19). j,g

Remark 7.1 Define Z¯ ı ber can be reduced.

= Z¯ j,g , ı ∈ h and then the linear matrix inequality num-

Remark 7.2 State constraints and input constraints are rarely discussed simultaneously for lots of switched fuzzy models, for instance (Shi et al. 2019). There are few tries to investigate controllers for uncertain switched fuzzy systems with state and input constraints. Compared with the ordinary Lyapunov function, piecewise Lyapunov function or fuzzy Lyapunov function (Han et al. 2018; Wang et al. 2020b), the piecewise fuzzy Lyapunov function is researched, thinking this influence of membership functions and switching behavior on model stability. By utilizing a piecewise fuzzy Lyapunov function, multiple delay-dependent results with less conservatism are obtained. Compared with only considering finite-time boundedness or input-output finite-time stability (Sun et al. 2022; Hua et al. 2020), finite-time boundedness and input-output finite-time stability are studied at the same time, so the stability issue is solved more comprehensively. Remark 7.3 Compared with (Sun et al. 2021), this involvement of constants α1g , α2g and α3g implies that the Lyapunov function devised in this chapter is regulable, and fit for four situations. How to select these non-negative constants α1g , α2g and α3g for these four situations refers to Table 7.1.

7.5 Simulation Results

211

7.5 Simulation Results The feasibility of the approach proposed in this chapter is illustrated via an example. Example 7.1 The mass-spring-damping system is discussed (Yang and Tong 2016). Presume that x1 (t) is disturbed by time delays. The system data are listed as follows τ1 (t) = 1.05 + 0.1 sin(t), τ2 (t) = 0.7 + 0.35 cos(t),       0 1 0 1 0 1 1,0 1,0 2,0 B1 = , B2 = , B1 = , −0.02b¯0 0 −0.02b¯0 −0.225 −1.5275b¯0 0     0 1 0 0 1,1 1,1 B2,0 2 = −1.5275b ¯0 −0.225 , B2 = B1 = −0.02b¯1 0 ,     0 0 0 0 2,1 1,2 1,2 B2,1 , B , = B = = B = 2 1 2 1 −1.5275b¯1 0 −0.02b¯2 0   0 0 2,2 B2,2 , b¯0 = 0.89, b¯1 = 0.02, b¯2 = 0.09, = B = 2 1 −1.5275b¯2 0   0 B2u2 = B2u1 = B1u2 = B1u1 = , 1     2,0  1,1  1,0 2,0 1,1 ¯ ¯ ¯ C1,0 2 = C1 = b0 2 , C2 = C1 = 0.2b0 1 , C2 = C1 = b1 0 ,     1,2  2,2  2,1 1,2 2,2 ¯ ¯ ¯ C2,1 2 = C1 = 0.2b1 0 , C2 = C1 = b2 0 , C2 = C1 = 0.2b2 0 . Assume that there are intermittent faults, model uncertainties, external disturbances and measurement noise in the system. The relevant data are as follows 

       0.02 0 0.01 0.03 , B1p2 = , B2p1 = , B2p2 = , 0 0.01 0 0         0.001 0 0.002 0 = , B1w2 = , B2w1 = , B2w2 = , 0 0.003 0 0.001

B1p1 = B1w1

C1s1 = 0.015, C1s2 = 0.018, C2s1 = 0.011, C2s2 = 0.024, C1v1 = 0.001, C1v2 = 0.002, C2v1 = 0.004, C2v2 = 0.006,         D11 = 0.012 0 , D12 = 0 0.023 , D21 = 0 0.015 , D12 = 0.017 0 ,         0 0 0.004 0.001 1 1 2 2 , Ax2 = , Ax1 = , Ax2 = , Ax1 = 0.0009 0.0007 0 0 E2x2 (t) = E2x1 (t) = E1x2 (t) = E1x1 (t) = cos(0.5t),   1,0   2,0   2,0   F1,0 x1 = 0.001 0 , Fx2 = 0 0.014 , Fx1 = 0.0015 0 , Fx2 = 0 0.013 ,   1,1   2,1   2,1   F1,1 x1 = 0 0.004 , Fx2 = 0.002 0 , Fx1 = 0.0008 0 , Fx2 = 0 0.0023 ,   1,2   2,2   2,2   F1,2 x1 = 0 0.0031 , Fx2 = 0.0017 0 , Fx1 = 0 0.0013 , Fx2 = 0.0014 0 , F1xu1 = 0.0035, F1xu2 = 0.0021, F2xu1 = 0.0017, F2xu2 = 0.0014,

212

7 Finite-Time Fault-Tolerant Control for Multiple-Delayed Switched …

Fig. 7.1 The response of the states x(t) and x(t) ˆ

1

x1 (t) x ˆ1 (t) x2 (t) x ˆ2 (t)

States

0.5 0 -0.5 -1 0

5

10 Time (sec)

15

20

F1x p1 = 0.0011, F1x p2 = 0.0013, F2x p1 = 0.0015, F2x p2 = 0.0007, F1xw1 = 0.0016, F1xw2 = 0.0015, F2xw1 = 0.0025, F2xw2 = 0.0034, A1y1 = 0.0014, A1y2 = 0.0027, A2y1 = 0.0023, A2y2 = 0.0015,   1,0   2,0   2,0   F1,0 y1 = 0.0021 0 , F y2 = 0 0.0017 , F y1 = 0.0018 0 , F y2 = 0 0.0019 ,   1,1   2,1   2,1   F1,1 y1 = 0.0016 0 , F y2 = 0 0.0033 , F y1 = 0.0013 0 , F y2 = 0 0.0028 ,   1,2   2,2   F1,2 y1 = 0 0.0024 , F y2 = 0 0.0011 , F y1 = 0.0016 0.0021 ,   1 1 2 2 F2,2 y2 = 0.0019 0 , F ys1 = 0.001, F ys2 = 0.003, F ys1 = 0.0024, F ys2 = 0.0015, F1yv1 = 0.0011, F1yv2 = 0.0014, F2yv1 = 0.0017, F2yv2 = 0.0026. Select fuzzy rules in the following δ11 = 1 −

x 2 (t) 1 x˙ 2 (t) 2 , δ2 = 1 − δ11 , δ12 = 1 − , δ = 1 − δ12 . 2.25 2.25 2

The intermittent faults can be described by

A¯ p = 0.2, f p (t) =

⎧ ⎪ ⎨

0,

0 ≤ t < 12

(0.8−t)/4

2(1 − e , 12 ≤ t < 15 ⎪ ⎩ 0, 15 ≤ t < 20 ⎧ 0, 0 ≤ t < 12 ⎪ ⎨ ¯ As = 0.6, f s (t) = 0.1 + 0.1 cos(0.5t), 12 ≤ t < 15 ⎪ ⎩ 0, 15 ≤ t < 20

It can be calculated that τd1 = 0.95, τd2 = 0.35, τ1 = 1.15, τ2 = 1.05, τ˜1 = 0.1, τ˜2 = 0.35. Set ηˇ¯ = 1.5, uˇ = 0.6, c1 = 1, c2 = 10, R = I , T = 10, M1 = I , M2 = I χ1 = 0.1, χ2 = 0.01.

7.5 Simulation Results

213

Fig. 7.2 The response of the outputs y(t) and yˆ (t)

y(t) yˆ(t)

Outputs

0.2 0 -0.2 -0.4 -0.6 0

Fig. 7.3 The response of the controller states η(t) and ηˆ (t)

5

10 Time (sec)

15

Controller states

1

20

η1 (t) ηˆ1 (t) η2 (t) ηˆ2 (t)

0.5 0 -0.5 -1 0

Fig. 7.4 The response of the control inputs u(t) and u(t) ˆ

5

10 Time (sec)

15

Control inputs

0.6

20

u(t) u ˆ(t)

0.4 0.2 0 0

Fig. 7.5 The spectral norms of the control inputs u(t) and u(t) ˆ

5

10 Time (sec)

15

20

u(t) u ˆ(t)

0.5 0.4 0.3 0.2 0.1 0 0

5

10

Time (sec)

15

20

214

7 Finite-Time Fault-Tolerant Control for Multiple-Delayed Switched …

Fig. 7.6 The spectral norms of the closed-loop system states η¯ (t) and ηˆ¯ (t)

η¯(t) ηˆ¯(t) 1

0.5

0 0

5

10

15

20

Time (sec)

Fig. 7.7 The response of z T (t)M2 z(t) and zˆ T (t)M2 zˆ (t)

×10 -3

z T (t)M2 z(t) zˆT (t)M2 zˆ(t)

8 6 4 2 0 0

5

10

15

20

Time (sec)

The simulation results are shown in Figs. 7.1, 7.2, 7.3, 7.4, 7.5, 7.6 and 7.7. This response of this state x(t), this output y(t), this controller state η(t), and this control input u(t) is listed in Figs. 7.1, 7.2, 7.3 and 7.4, showing that finite-time boundedness and fault-tolerant control can be realized. In Figs. 7.5 and 7.6, these spectral norms of this control input u(t) and this closed-loop model state η(t) ¯ can be described, which indicates that the state and input constraints hold. In Fig. 7.7, this response of z T (t)M2 z(t) can be shown, which implies that input-output finite-time stability can be satisfied. To present the merits of these methods studied in the chapter, a contrast test is made by comparing the piecewise fuzzy Lyapunov function with the ordinary Lyapunov function (Han et al. 2019). That is, when constructing the ordinary Lyapunov j j,g j,g j,g function, select Pδ = P, Q 1δ = Q 1 , Q 2δ = Q 2 , Z δ = Z . The state, output, controller state, control input, closed-loop system state, and desired controlled output in this comparison system under the ordinary Lyapunov function are denoted as x(t), ˆ yˆ (t), η(t), ˆ u(t), ˆ ηˆ¯ (t) and zˆ (t), separately. The comparison results are shown in Figs. 7.1, 7.2, 7.3, 7.4, 7.5, 7.6 and 7.7. This response of this system state x(t), ˆ this output yˆ (t), this controller state η(t), ˆ and this control input u(t) ˆ of this comparison system is presented in Figs. 7.1, 7.2, 7.3 and 7.4, which implies that finite-time fault-tolerant control is realized via this comparison approach, but this response of this state x(t), this output y(t), this controller state η(t) and this control input u(t) via this approach in the chapter is smoother and enters

References

215

into this stable boundedness more quickly. So this finite-time fault-tolerant control performance is better via the approach in this chapter. In Figs. 7.5 and 7.6, these spectral norms of this control input u(t) ˆ and this closed-loop model state ηˆ¯ (t) are shown, which indicates that state and input constraints hold via this comparison approach but this response of the spectral norms of the control input u(t) and this closed-loop model state η(t) ¯ is smoother and enters into the stable boundedness more quickly. So this controller devised in the chapter has better control performance under the same cases of state and input constraints. In Fig. 7.7, the response of zˆ T (t)M2 zˆ (t) is obtained, showing that input-output finite-time stability is achieved via this comparison approach, but this response of z T (t)M2 z(t) via this approach in the chapter is smoother and enters into the stable boundedness more quickly. So this input-output finite-time stability performance is better via the approach in this chapter.

7.6 Chapter Summary This issue of multiple delay-dependent finite-time fault-tolerant control is investigated for switched fuzzy models with intermittent faults, model uncertainties, exogenous disturbances, measurement noise, and state and input constraints. There are few tries to research this issue. The nonlinear dynamic output feedback controller is researched for uncertain switched fuzzy models. Besides, a closed-loop model is established to achieve finite-time fault-tolerant control. In addition, sufficient conditions of robust finite-time H∞ control are obtained via a piecewise fuzzy Lyapunov function. Finite-time boundedness and input-output finite-time stability are also realized. The availability and advantages of the method are demonstrated via a simulation example.

References Amato F, Tommasi GD (2010) Input-output finite-time stabilization for a class of hybrid systems. IFAC Proc Vol 43(21):336–341 Amato F, Carannante G, DeTommasi G et al (2012) Input-output finite-time stability of linear systems: necessary and sufficient conditions. IEEE Trans Autom Control 57(12):3051–3063 Chen X, Zhao L, Yu J (2020) Adaptive neural finite-time bipartite consensus tracking of nonstrict feedback nonlinear coopetition multi-agent systems with input saturation. Neurocomputing 397:168–178 Du F, Lu JG (2020) Finite-time stability of neutral fractional order time delay systems with Lipschitz nonlinearities. Appl Math Comput 375:125079 Gu Y, Shen M, Ren Y et al (2020) H∞ finite-time control of unknown uncertain systems with actuator failure. Appl Math Comput 383:125375 Han J, Zhang H, Wang Y et al (2018) Robust fault detection for switched fuzzy systems with unknown input. IEEE Trans Cybern 48(11):3056–3066

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Han J, Zhang H, Liu X et al (2019) Dissipativity-based fault detection for uncertain switched fuzzy systems with unmeasurable premise variables. IEEE Trans Fuzzy Syst 27(12):2421–2432 Hua S, Wang X, Zhu Y (2020) Sliding-mode control for a rolling-missile with input constraints. J Syst Eng Electron 31(5):1041–1050 Kang W, Gao Q, Cao M et al (2020) Finite-time control for Markovian jump systems subject to randomly occurring quantization. Appl Math Comput 385:125402 Ren H, Zong G, Li T (2018) Event-triggered finite-time control for networked switched linear systems with asynchronous switching. IEEE Trans Syst Man Cybern: Syst 48(11):1874–1884 Shi S, Fei Z, Wang T et al (2019) Filtering for switched T-S fuzzy systems with persistent dwell time. IEEE Trans Cybern 49(5):1923–1931 Sun S, Zhang H, Sun J et al (2021) Reliable H∞ guaranteed cost control for uncertain switched fuzzy stochastic systems with multiple time-varying delays and intermittent actuator and sensor faults. Neural Comput Appl 33:1343–1365 Sun S, Zhang H, Zhang J et al (2022) Multiple delay-dependent robust H∞ finite-time filtering for uncertain Itô stochastic Takagi-Sugeno fuzzy semi-Markovian jump systems with state constraints. IEEE Trans Fuzzy Syst 30(2):321–331 Wang K, Tian E, Shen S et al (2019) Input-output finite-time stability for networked control systems with memory event-triggered scheme. J Frankl Inst 356(15):8507–8520 Wang X, Fei Z, Wang T et al (2020a) Dynamic event-triggered actuator fault estimation and accommodation for dynamical systems. Inf Sci 525:119–133 Wang Z, Sun J, Zhang H (2020b) Stability analysis of T-S fuzzy control system with sampleddropouts based on time-varying Lyapunov function method. IEEE Trans Syst Man Cybern: Syst 50(7):2566–2577 Xiang Z, Sun YN, Mahmoud MS (2012) Robust finite-time H∞ control for a class of uncertain switched neutral systems. Commun Nonlinear Sci Numer Simul 17(4):1766–1778 Yang W, Tong S (2016) Adaptive output feedback fault-tolerant control of switched fuzzy systems. Inf Sci 329:478–490

Chapter 8

Conclusion and Prospects

This paper mainly discusses the research of H∞ fault-tolerant control methods for time-varying delay T-S fuzzy systems. Fault-tolerant controllers are divided into active fault-tolerant controllers and passive fault-tolerant controllers according to different design methods. When designing an active fault-tolerant controller, an observer based on fault estimation can be designed to determine the size and shape of the fault, and then a full-order active controller can be designed to make the closedloop system stable by compensating for this fault effect. When designing a passive fault-tolerant controller, a dynamic output feedback fault-tolerant control method can be used to make this closed-loop model stable without any real-time fault information. The effectiveness, feasibility, and superiority of this method are verified by some simulation experiments. To be specific, the following content can be obtained: 1. For time delayed T-S fuzzy systems against only actuator faults and both actuator and sensor faults, a k-step induction actuator fault estimation observer and a kstep induction fault estimation observer are designed, respectively, and the goal of fault estimation can be both achieved. The designed observer reduces this input disturbance influence of the actuator fault derivative in this error function and obtains a better estimation effect. Meanwhile, in order to make the time delayed T-S fuzzy system stable, an observer-based dynamic output feedback controller containing fault compensators is designed. The designed observer and active fault-tolerant controller are suitable for more practical situations and have a wider range of applications. 2. For the switched T-S fuzzy stochastic system with external disturbances, multiple time-varying delays, sensor faults, and intermittent actuator faults, a sliding mode observer and a sliding mode controller are designed. Fault estimation and tolerant control can be carried out for intermittent actuator faults and sensor faults and have broad application prospects. And in the sliding mode observer, the involvement of proportional gain, derivative gain, and sliding mode gain makes © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Sun et al., Fault-Tolerant Control for Time-Varying Delayed T-S Fuzzy Systems, Intelligent Control and Learning Systems 9, https://doi.org/10.1007/978-981-99-1357-2_8

217

218

8 Conclusion and Prospects

the design of the observer more flexible and feasible. This effect of Brownian motion can be removed from the proposed sliding mode function through a complex matrix parameter. Furthermore, the finite-time reachability of the sliding surface is strictly guaranteed. 3. For switched T-S fuzzy systems and switched T-S fuzzy stochastic systems with multiple time delays, model uncertainties, external disturbances, intermittent sensor faults, and intermittent actuator faults, full-order passive output feedback controllers are designed, respectively. Besides, the optimal guaranteed cost control and fault-tolerant control are realized, so that a certain performance index can be guaranteed under the condition that the system is stable. In addition, finite-time fault-tolerant control for switched fuzzy systems with multiple time delays, model uncertainties, external disturbances, intermittent sensor faults, and intermittent process faults is designed. All can realize the stable operation of the system regardless of whether a fault occurs or not without the need for a fault diagnosis unit. The effect of better fault-tolerant control performance and lower conservatism can be obtained. In this book, some researches and explorations on fault estimation, active faulttolerant control, and passive fault-tolerant control are carried out for a series of nonlinear systems such as time delayed T-S fuzzy systems, time delayed switched T-S fuzzy systems, and time delayed switched T-S fuzzy stochastic systems. In the process of writing the book, the authors notice that in the future study work, further research can be focused on the following aspects: 1. With the development of industry, there will be jumping or switching between multiple subsystems. Stochastic phenomena can be seen everywhere. And in a number of nonlinear systems, these premise variables of the system are unmeasurable, or these premise variables of the controller variables are different from these premise variables of the system, so in the future, the design issues of fault estimation observers, active controllers, and passive controllers will be researched for Markovian jump fuzzy systems, switched fuzzy models, and fuzzy stochastic models against unmeasurable premise variables and time-varying delays. 2. For the system dynamics considered in this book, asymptotic tracking can be achieved by an approach similar to the fault estimation approach suggested in the book. Asymptotic tracking is also attractive and widely required in practical applications (Lv et al. 2021). Therefore, asymptotic tracking will also be further developed in future research. 3. In order to reduce communication, the event-triggered mechanism method will be used to study related system stability issues in the future. 4. To overcome safety requirements and/or performance limit situations and physical input saturation phenomena in practical systems, the design issues of fault estimation, and active fault-tolerant control for time-varying delayed fuzzy systems against state and input constraints will be studied. On the whole, although the research on H∞ fault-tolerant control methods for nonlinear time delay systems has been ongoing for decades at home and abroad, and

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some breakthroughs have been achieved, there are still many problems worthy of further research and exploration. At the same time, various problems encountered in the application of theoretical research to practice need to be explored and solved. It is hoped that further in-depth exploration and research can be made in the future, new breakthroughs can be made in theory, and theoretical researches can be applied to a wider range of practical scenarios.

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