Control and Filtering of Fuzzy Systems Under Communication Channels 9819943450, 9789819943456

Table of contents :
Contents
List of Figures
1 Introduction and Preview
1.1 Introduction
1.1.1 T–S Fuzzy Systems
1.1.2 Networked Systems
1.1.3 Fuzzy-Model-Based Nonlinear Networked Systems
1.2 Preliminaries
1.3 Outline of the Monograph
References
2 Fuzzy Robust mathcalHinfty Control with Dynamic Quantization
2.1 Robust Control with Input Quantization
2.1.1 Problem Formulation
2.1.2 Main Results
2.1.3 Simulation Example
2.2 Non-fragile Control with State Quantization
2.2.1 Problem Formulation
2.2.2 Main Results
2.2.3 Simulation Example
2.3 Resilient Control with Input and Output Quantization
2.3.1 Problem Formulation
2.3.2 Resilient mathcalHinfty Controller Design with Quantization
2.3.3 Simulation Example
2.4 Conclusion
References
3 Fuzzy Filtering with Multiple Signal Transmissions
3.1 mathcall2–mathcallinfty Filtering with Static Quantization
3.1.1 Problem Formulation
3.1.2 Main Results
3.1.3 Simulation Example
3.2 Induced mathcallinfty Filtering with Data Packet Dropouts
3.2.1 Problem Formulation
3.2.2 Main Results
3.2.3 Simulation Example
3.3 mathcalHinfty Filtering with Dynamic Quantization
3.3.1 Problem Formulation
3.3.2 Main Results
3.3.3 Simulation Example
3.4 Conclusion
References
4 Fuzzy Output Feedback Control with Communication Constraints
4.1 Feedback Control with Quantization
4.1.1 Problem Formulation
4.1.2 Main Results
4.1.3 Simulation Example
4.2 Feedback Control with Data Packet Dropouts and Time Delays
4.2.1 Problem Formulation
4.2.2 Main Results
4.2.3 Simulation Example
4.3 Feedback Control with Stochastic Communication Protocol
4.3.1 Problem Formulation
4.3.2 Main Results
4.3.3 Simulation Example
4.4 Conclusion
References
5 Event-Triggered Fuzzy Control and Filtering with Communication Constraints
5.1 Tracking Control with Static Quantization
5.1.1 Problem Formulation
5.1.2 Main Results
5.1.3 Simulation Example
5.2 mathcalL2–mathcalLinfty Filtering with Time Delays and DoS Attacks
5.2.1 Problem Formulation
5.2.2 Main Results
5.2.3 Simulation Example
5.3 mathcalHinfty Filtering with Dynamic Quantization and Deception Attacks
5.3.1 Problem Formulation
5.3.2 Main Results
5.3.3 Simulation Example
5.4 Conclusion
References
6 Fuzzy Fault Detection and Fault-Tolerant Control with Quantization
6.1 Fault Detection with Output Quantization
6.1.1 Problem Formulation
6.1.2 Analysis and Synthesis of Quantized Residual System
6.1.3 Simulation Example
6.2 Fault-Tolerant Control with Input Quantization
6.2.1 Problem Formulation
6.2.2 Main Results
6.2.3 Simulation Example
6.3 Guaranteed Cost Fault-Tolerant Control with Input Quantization
6.3.1 Problem Formulation
6.3.2 Fault-Tolerant Guaranteed Cost Control Without Quantization
6.3.3 Fault-Tolerant Guaranteed Cost Control with Input Quantization
6.3.4 Simulation Example
6.4 Conclusion
References

Citation preview

Xiao-Heng Chang Jun Xiong Zhi-Min Li Bo Wu

Control and Filtering of Fuzzy Systems Under Communication Channels

Control and Filtering of Fuzzy Systems Under Communication Channels

Xiao-Heng Chang · Jun Xiong · Zhi-Min Li · Bo Wu

Control and Filtering of Fuzzy Systems Under Communication Channels

Xiao-Heng Chang School of Information Science and Engineering Wuhan University of Science and Technology Wuhan, Hubei, China

Jun Xiong School of Information Science and Engineering Wuhan University of Science and Technology Wuhan, Hubei, China

Zhi-Min Li School of Electronic and Control Engineering North China Institute of Aerospace Engineering Langfang, Hebei, China

Bo Wu School of Information Science and Engineering Wuhan University of Science and Technology Wuhan, Hubei, China

ISBN 978-981-99-4345-6 ISBN 978-981-99-4346-3 (eBook) https://doi.org/10.1007/978-981-99-4346-3 © 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

Contents

1 Introduction and Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 T–S Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Networked Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Fuzzy-Model-Based Nonlinear Networked Systems . . . . . . . 1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Outline of the Monograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 5 6 8 9 11

2 Fuzzy Robust H∞ Control with Dynamic Quantization . . . . . . . . . . . . 2.1 Robust Control with Input Quantization . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Non-fragile Control with State Quantization . . . . . . . . . . . . . . . . . . . . 2.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Resilient Control with Input and Output Quantization . . . . . . . . . . . . 2.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Resilient H∞ Controller Design with Quantization . . . . . . . 2.3.3 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 18 20 27 31 31 33 38 40 40 42 57 62 62

3 Fuzzy Filtering with Multiple Signal Transmissions . . . . . . . . . . . . . . . . 3.1 l2 –l∞ Filtering with Static Quantization . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Induced l∞ Filtering with Data Packet Dropouts . . . . . . . . . . . . . . . . 3.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 64 67 72 76 76 v

vi

Contents

3.2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.2.3 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.3 H∞ Filtering with Dynamic Quantization . . . . . . . . . . . . . . . . . . . . . . 89 3.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.3.3 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4 Fuzzy Output Feedback Control with Communication Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Feedback Control with Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Feedback Control with Data Packet Dropouts and Time Delays . . . 4.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Feedback Control with Stochastic Communication Protocol . . . . . . 4.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Event-Triggered Fuzzy Control and Filtering with Communication Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Tracking Control with Static Quantization . . . . . . . . . . . . . . . . . . . . . 5.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 L2 –L∞ Filtering with Time Delays and DoS Attacks . . . . . . . . . . . 5.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 H∞ Filtering with Dynamic Quantization and Deception Attacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 103 104 106 118 122 123 125 132 135 135 138 142 145 145 147 147 148 152 161 164 164 169 179 181 182 185 195 197 198

Contents

6 Fuzzy Fault Detection and Fault-Tolerant Control with Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Fault Detection with Output Quantization . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Analysis and Synthesis of Quantized Residual System . . . . . 6.1.3 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Fault-Tolerant Control with Input Quantization . . . . . . . . . . . . . . . . . 6.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Guaranteed Cost Fault-Tolerant Control with Input Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Fault-Tolerant Guaranteed Cost Control Without Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Fault-Tolerant Guaranteed Cost Control with Input Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

201 201 202 206 215 220 220 223 229 231 231 233 237 245 249 249

List of Figures

Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 2.5 Fig. 2.6 Fig. 2.7 Fig. 2.8 Fig. 2.9 Fig. 2.10 Fig. 2.11 Fig. 2.12 Fig. 2.13 Fig. 2.14 Fig. 2.15 Fig. 2.16 Fig. 2.17 Fig. 2.18 Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6 Fig. 3.7 Fig. 3.8 Fig. 3.9 Fig. 3.10 Fig. 3.11

Truck-trailer system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Response of the system state x(k) . . . . . . . . . . . . . . . . . . . . . . . . . Response of the control input u(k) . . . . . . . . . . . . . . . . . . . . . . . . Response of the quantized input signal Q(u(k)) . . . . . . . . . . . . . . Response of the quantization error Q(u(k)) − u(k) . . . . . . . . . . .  k k T T History of ζ =0 z (ζ )z(ζ )/ ζ =0 w (ζ )w(ζ ) . . . . . . . . . . . . . . Dynamic quantizer’s parameter μu (k) . . . . . . . . . . . . . . . . . . . . . . Response of the system state x(k) . . . . . . . . . . . . . . . . . . . . . . . . . Response of the control input u(k) . . . . . . . . . . . . . . . . . . . . . . . .  k k T T History of j=0 z ( j)z( j)/ j=0 w ( j)w( j) . . . . . . . . . . . . . Dynamic quantizer’s parameter μx (k) . . . . . . . . . . . . . . . . . . . . . . Response of the system state x(t) . . . . . . . . . . . . . . . . . . . . . . . . . Response of the measurement output y(t) . . . . . . . . . . . . . . . . . . Response of the controlled output z(t) . . . . . . . . . . . . . . . . . . . . . Response of the control inputu(t) . . . . . . . . . . . . . . . . . . . . . . . . . t T t T History of 0 z (i)z(i)di/ 0 w (i)w(i)di . . . . . . . . . . . . . . . . Dynamic quantizer’s parameter μ y (t) . . . . . . . . . . . . . . . . . . . . . . Dynamic quantizer’s parameter μu (t) . . . . . . . . . . . . . . . . . . . . . . Response of the system state x(k) . . . . . . . . . . . . . . . . . . . . . . . . . Response of the filter state x f (k) . . . . . . . . . . . . . . . . . . . . . . . . . . Filtering error ..................................  e(k) . . .  T T History of e (k)e(k)/ ∞ k=0 w (k)w(k) . . . . . . . . . . . . . . . . . . Mass-spring-damper system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Response of the system state x(k) . . . . . . . . . . . . . . . . . . . . . . . . . Response of the filter state x f (k) . . . . . . . . . . . . . . . . . . . . . . . . . . Responses of z(k) and z f (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filtering error  e(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . History of e T (k)e(k)/w(k)2∞ . . . . . . . . . . . . . . . . . . . . . . . . . Response of the system state x(k) . . . . . . . . . . . . . . . . . . . . . . . . .

27 29 29 30 30 30 31 39 39 39 40 59 60 60 60 61 61 61 74 75 75 75 85 87 88 88 88 89 98 ix

x

Fig. 3.12 Fig. 3.13 Fig. 3.14 Fig. 3.15 Fig. 3.16 Fig. 3.17 Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 4.5 Fig. 4.6 Fig. 4.7 Fig. 4.8

List of Figures

Response of the filter state x f (k) . . . . . . . . . . . . . . . . . . . . . . . . . . Responses of z(k) and z f (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filtering error e(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic quantizer’s parameter μ y . . . . . . . . . . . . . . . . . . . . . . . . Dynamic quantizer’s parameter μz . . . . . . . . . . . . . . . . . . . . . . . .  ∞ T ∞ T History of k=0 e (k)e(k)/ k=0 w (k)w(k) . . . . . . . . . . . . . Response of the system state x(k) with w(k) = 0 . . . . . . . . . . . . Response of the measurement output y(k) with w(k) = 0 . . . . . . Response of the control input u(k) with w(k) = 0 . . . . . . . . . . . . Response of x(k) with the quantized H∞ controller . . . . . . . . . . Response of y(k) with the quantized H∞ controller . . . . . . . . . . Response of u(k) with the quantized H∞ controller . . . . . . . . . . Response ofz(k) with the quantized H∞ controller . . . . . . . . . . k k   History of z T (φ)z(φ)/ w T (φ)w(φ)

99 99 99 100 100

with the quantized H∞ controller . . . . . . . . . . . . . . . . . . . . . . . . . History of w(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Response of x(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Response of z(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Response of u(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t T t T History of 0 z (τ )z(τ )dτ/ 0 w (τ )w(τ )dτ . . . . . . . . . . . . . . Response of the state of open-loop system . . . . . . . . . . . . . . . . . . Response of the control input u(k) . . . . . . . . . . . . . . . . . . . . . . . . Response of the state of closed-loop system . . . . . . . . . . . . . . . . .  k k T T History of τ =0 z (τ )z(τ )/ τ =0 ω (τ )ω(τ ) . . . . . . . . . . . . . Response of the system state x(k) . . . . . . . . . . . . . . . . . . . . . . . . . Response of the control input u(k) . . . . . . . . . . . . . . . . . . . . . . . . Responses of y(k) and y (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Response of ec (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Release instants and release intervals of the communication channel from the plant to the controller . . . . . . . . . . . . . . . . . . . . . Release instants and release intervals of the communication channel from the reference model to the controller . . . . . . . . . . . Responses of x(t) and x f (t) under the DoS attacks . . . . . . . . . . . Release time intervals under the DoS attacks . . . . . . . . . . . . . . . . Response of DoS attacks . . . . . . . . . . . . . . . . . . . . z(t) under the ∞ T History of y (t)y(t)/ 0 w T (s)w(s)ds . . . . . . . . . . . . . . . . . . . Responses of z(k) and z f (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Response of the filtering error e(k) . . . . . . . . . . . . . . . . . . . . . . . . Dynamic quantizer’s parameter μ ys (k) . . . . . . . . . . . . . . . . . . . . . Release instants and release intervals . . . . . . . . . . . . . . . . . . . . . .

122 133 133 134 134

φ=0

Fig. 4.9 Fig. 4.10 Fig. 4.11 Fig. 4.12 Fig. 4.13 Fig. 4.14 Fig. 4.15 Fig. 4.16 Fig. 4.17 Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. 5.4 Fig. 5.5 Fig. 5.6 Fig. 5.7 Fig. 5.8 Fig. 5.9 Fig. 5.10 Fig. 5.11 Fig. 5.12 Fig. 5.13 Fig. 5.14

100 120 120 120 121 121 121 122

φ=0



134 143 144 144 145 162 162 162 163 163 163 180 180 180 181 196 196 196 197

List of Figures

Fig. 6.1 Fig. 6.2 Fig. 6.3 Fig. 6.4 Fig. 6.5 Fig. 6.6 Fig. 6.7 Fig. 6.8 Fig. 6.9 Fig. 6.10 Fig. 6.11 Fig. 6.12 Fig. 6.13

xi

Fault signal f (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residual r (k) by robust fault detection filter with quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residual evaluation function r  R M S by robust fault detection filter with quantization . . . . . . . . . . . . . . . . . . . . . . . . . . Residual r (k) by resilient fault detection filter with quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residual evaluation function r  R M S by resilient fault detection filter with quantization . . . . . . . . . . . . . . . . . . . . . . . . . . Response of the system state x(k) . . . . . . . . . . . . . . . . . . . . . . . . . History of the mode jump . . . . . . . . . . . . . . . . . . . . . . . . .  system ∞ ∞ T T History of E k=0 z (k)z(k)/ k=0 w (k)w(k) . . . . . . . . . . Response of the system state x(k) . . . . . . . . . . . . . . . . . . . . . . . . . Response of the control input u(k) . . . . . . . . . . . . . . . . . . . . . . . . Response of the quantized control input u q (k) . . . . . . . . . . . . . . . Dynamic quantizer’s parameter θ (k) . . . . . . . . . . . . . . . . . . . . . . . k   History of J = x T (i)Q 1 x(i) + u˜ T (i)Q 2 u(i) ˜

216 217 217 218 218 230 230 231 247 247 247 248

i=0

Fig. 6.14

for system (6.160) with quantization . . . . . . . . . . . . . . . . . . . . . . . k   History of J = x T (i)Q 1 x(i) + u˜ T (i)Q 2 u(i) ˜

248

i=0

for system (6.160) without quantization . . . . . . . . . . . . . . . . . . . .

248

Chapter 1

Introduction and Preview

1.1 Introduction In the past two decades, increasing attention has been paid to deal with the analysis and design problems for nonlinear networked systems based on Takagi–Sugeno (T–S) fuzzy model and some interesting results have been published in the open literature. In this monograph, some novel advances on the study of nonlinear networked systems based on T–S fuzzy model will be given. Firstly, the various robust control strategies for nonlinear networked systems with dynamic quantization are proposed. Secondly, the various filtering problems are addressed for nonlinear networked systems subject to multiple communication constraints. Thirdly, the various output feedback controller design methods are presented for nonlinear networked systems with various communication constraints. Fourthly, in the presence of quantization and event-triggered communication scheme, the tracking control and security filtering schemes are developed for nonlinear networked systems, respectively. Finally, the quantized fault detection and fault-tolerant control problems are considered for nonlinear networked systems.

1.1.1 T–S Fuzzy Systems It is generally known that nonlinearities are unavoidable in almost all practical control systems and it is difficult to deal with the analysis and design problems for nonlinear systems in both fields of control theory and control engineering. Therefore, considerable efforts have been made to investigate the nonlinear systems and different approaches have been proposed over the past several decades. Among the existing approaches, fuzzy logic control is regarded as one of the most effective and powerful strategies to investigate nonlinear systems [1]. As a result, the study of fuzzy logic control has received a great deal of attention from both scientists and engineers and the corresponding results have been reported in the existing literature, for © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Chang et al., Control and Filtering of Fuzzy Systems Under Communication Channels, https://doi.org/10.1007/978-981-99-4346-3_1

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instance, T–S fuzzy model based control, adaptive fuzzy control, fuzzy proportionalintegral-derivative (PID) control, and so on. In addition, the fuzzy logic control has been successfully found a great variety of practical applications, e.g., artificial intelligence and robotics, management and intelligent decisionmaking, motor industry and driver assistance system, etc. Particularly, in the existing fuzzy logic control methods, the T–S fuzzy modelbased control approach has been widely utilized to study nonlinear systems [2]. The main reason is that the considered nonlinear systems can be approximated as some local linear time-invariant system models connected by IF–THEN rules. In this way, the conventional linear system theory can be utilized to investigate the considered nonlinear systems. In fact, the study of the T–S fuzzy model can be traced back to the publication of pioneering work developed by Takagi and Sugeno in 1985 [3]. Subsequently, it has been proven that the T–S fuzzy model can approximate any smooth nonlinear function to any specified accuracy within any compact set. In recent years, more and more researchers have shown great interest in investigating nonlinear systems based on T–S fuzzy model and a number of significant results on the stability analysis, filter and controller designs have been reported. The stability analysis and stabilization problems were addressed for T–S fuzzy systems based on the parallel distributed compensation (PDC) strategy in [4]. By representing the interactions among the fuzzy subsystems in a single matrix and solving it by linear matrix inequality (LMI), new approaches to relaxed quadratic stability and stabilizability conditions for T–S fuzzy systems were proposed in [5]. Based on the fuzzy Lyapunov function approach, the stability conditions for open-loop fuzzy systems and stabilization conditions for closed-loop fuzzy systems were developed in [6]. Based on fuzzy Lyapunov functions and non-PDC control laws, new stabilization conditions were represented for continuous-time T–S fuzzy systems in [7]. By using new control laws and new nonquadratic Lyapunov functions, LMI-based relaxed nonquadratic stabilization conditions for discrete-time T–S fuzzy systems were given in [8]. The robust stabilization and H∞ control problems for uncertain discrete-time T–S fuzzy systems were addressed in [9]. In [10, 11], the problem of robust static output feedback H∞ control for discrete-time T–S fuzzy systems with time-varying norm-bounded uncertainties was considered. By considering the properties of system output matrices, an H∞ static output feedback control strategy was developed for discrete-time T–S fuzzy systems in [12]. The observer-based output feedback H∞ controller design problem was studied for continuous-time T–S fuzzy systems in [13]. In [14], the observer-based output feedback H∞ tracking controller was designed for continuous-time T–S fuzzy systems. The problem of designing an output feedback controller with pole placement constraints for singular perturbed T–S fuzzy systems was researched in [15]. In [16], the authors discussed the problem of robust dynamic output feedback H∞ control for uncertain discrete-time T–S fuzzy systems with time delays. Based on the piecewise Lyapunov function approach, the state feedback H∞ control problem for continuous-time T–S fuzzy systems was addressed in [17]. By utilizing the descriptor representation approach, the observerbased output feedback H∞ control problem was investigated for discrete-time T–S fuzzy systems in [18]. In [19], the nonfragile H∞ filtering problem was studied for

1.1 Introduction

3

continuous-time T–S fuzzy systems with additive gain variations. By introducing more slack matrices, less-conservative results on H∞ filtering for discrete-time T–S fuzzy systems were given in [20]. In [21], the H∞ filter was designed for discretetime T–S fuzzy systems in finite-frequency domain. The nonfragile H∞ filter design approach for uncertain continuous-time T–S fuzzy systems with linear fractional parametric gain variations was developed in [22]. The nonfragile H∞ filtering problem for continuous-time T–S fuzzy systems with multiplicative gain variations was considered in [23]. In [24], the problem of H∞ filtering for continuous-time T–S fuzzy systems with D stability constraints was studied. The problem of robust H∞ filtering was investigated for continuous-time T–S fuzzy systems in [25]. In [26], the H∞ filtering problem was addressed for continuous-time T–S fuzzy systems with partly immeasurable premise variables. By using basis-dependent Lyapunov functions, the H∞ filter design problem was solved for discrete-time T–S fuzzy systems in [27]. The resilient energy-to-peak filtering strategy was developed for continuoustime T–S fuzzy systems based on the two-step approach in [28]. In [29], the authors investigated the H− /H∞ fault detection observer design problem for discrete-time T–S fuzzy systems with sensor faults and unknown bounded disturbances. The fault detection problem in the finite frequency domain was studied for continuous-time T– S fuzzy systems with sensor faults in [30]. The problem of robust H∞ fault-detection filter design for uncertain continuous-time T–S fuzzy systems was discussed in [31]. In [32], the problem of state feedback fault-tolerant control for continuous-time T–S fuzzy systems with actuator faults was considered. The problem of robust fault estimation and observer-based output feedback fault tolerant control for continuous-time T–S fuzzy systems was investigated in [33]. In this monograph, we use the following T–S fuzzy model to represent the investigated nonlinear networked systems, in which the ith rule is described as follows: Plant Rule i th : IF ε1 (t) is M1i , ε2 (t) is M2i , and, . . ., and εd (t) is Mdi , THEN

F (x(t)) = Ai x(t) + Bi u(t) + E i w(t), z(t) = Ci x(t) + Di u(t) + Fi w(t), y(t) = L i x(t) + Hi w(t),

(1.1)

where ε(t) = [ε1 (t), ε2 (t), . . . , εd (t)] is used to represent the premise variable, M pi (i = 1, 2, . . . , r, p = 1, 2, . . . , d) is used to denote the fuzzy sets, and r refers to the number of fuzzy rules. F(·) stands for the time-derivative for continuous-time systems and forward operator for discrete-time systems. x(t) ∈ Rn x stands for the state variable, u(t) ∈ Rn u denotes the control input, y(t) ∈ Rn y represents the measured output, w(t) ∈ Rn w is the noise input which belongs to L2 [0, ∞) for continuoustime systems and l2 [0, ∞) for discrete-time systems, and z(t) ∈ Rn z is utilized to denote the controlled output. Ai ∈ Rn x ×n x , Bi ∈ Rn x ×n u , E i ∈ Rn x ×n w , Ci ∈ Rn z ×n x , Di ∈ Rn z ×n u , Fi ∈ Rn z ×n w , L i ∈ Rn y ×n x , and Hi ∈ Rn y ×n w for i = 1, 2, . . . , r are the given system matrices.

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1 Introduction and Preview

Denote m i (ε(t)) =

d 

M pi (ε p (t)), i = 1, 2, . . . , r,

(1.2)

p=1

where M pi (ε p (t)) stands for the grade of membership of ε p (t) in M pi . In this monograph, it is assumed that m i (ε(t)) > 0,

r 

m i (ε(t)) > 0, i = 1, 2, . . . , r.

(1.3)

i=1

By formulating the fuzzy basis functions as m i (ε(t)) , i = 1, 2, . . . , r. j=1 m j (ε(t))

ρi (ε(t)) = r

(1.4)

Then, it can be obtained that ρi (ε(t)) ≥ 0,

r 

ρi (ε(t)) = 1, i = 1, 2, . . . , r.

(1.5)

i=1

Moreover, the T–S fuzzy model in (1.1) can be inferred as follows:

F(x(t)) = A(ρ)x(t) + B(ρ)u(t) + E(ρ)w(t), z(t) = C(ρ)x(t) + D(ρ)u(t) + F(ρ)w(t), y(t) = L(ρ)x(t) + H (ρ)w(t), where A(ρ) = C(ρ) = E(ρ) = L(ρ) =

r  i=1 r  i=1 r  i=1 r  i=1

ρi (ε(t))Ai , ρi (ε(t))Ci , ρi (ε(t))E i , ρi (ε(t))L i ,

B(ρ) = D(ρ) = F(ρ) = H (ρ) =

r  i=1 r  i=1 r  i=1 r  i=1

ρi (ε(t))Bi , ρi (ε(t))Di , ρi (ε(t))Fi , ρi (ε(t))Hi .

(1.6)

1.1 Introduction

5

1.1.2 Networked Systems Networked systems are a class of control systems in which the communication network is utilized to exchange information between different components. In the last two decades, as one of the main research topics in both academic and industrial fields, networked systems have been widely investigated by scholars at home and abroad. One reason for this trend is that networked systems have obvious advantages over well-known point-to-point control systems, such as low cost, reduced weight and power requirements, simple installation and maintenance, and high reliability [34]. Another reason is that networked systems have been successfully applied in a great deal of modern complicated industry processes, such as intelligent manufacturing, industrial automation, unmanned vehicles, and teleoperation systems [35]. In networked systems, the most important feature is that the communication network employed for the exchange of information is unreliable and bandwidth-limited. Therefore, a significant problem in networked systems is how to achieve the reasonable and efficacious utilization of limited communication resources. In addition, how to analyze and design networked systems with network-induced constraints caused by the unreliable and bandwidth-limited communication network is also a difficult problem to be investigated. At last, how to develop the security control and filtering strategies for networked systems against cyber attacks is another challenging problem to be addressed. In the existing results, communication protocol, quantization strategy, and eventtriggered strategy are regarded as the three most effective measures to realize the reasonable and efficacious utilization of the limited communication resources in networked systems. The communication protocol is able to decide which nodes have the privileges to transmit information at the prescribed transmission instant, which implies that the data collisions can be relieved to a large extent. Recently, a great number of achievements have been reported for networked systems with communication protocols. Among these results, there are mainly three types of communication protocols, i.e., weighted try-once-discard protocol [36], round-robin protocol [36, 37], and stochastic communication protocol [38]. For the quantization strategy, the amount of data in the transmission of information can be greatly reduced by introducing the quantizer, which implies that the transmission burden of the communication network can be relieved in a great measure. Up to now, there are mainly two quite different approaches to investigating networked systems with quantization in the literature, i.e., static quantization methodology and dynamic quantization methodology. In the static quantization methodology, the quantizer is considered to be static and timeinvariant. While in dynamic quantization methodology, the quantizer is considered to be dynamic and time-varying. Based on static quantization methodology, the control and filtering problems have been addressed for networked systems in [39–44] and references therein. The study of networked systems based on the dynamic quantization methodology has also attracted considerable attention in the past few decades, see, for instance, [37, 45–55]. For the event-triggered strategy, the main idea is to design a suitable event-triggered condition to determine whether the signal should

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be transmitted to the next node or not. In contrast to the conventional time-triggered strategy where the signal is transmitted in a periodic manner, in the event-triggered strategy, the signal will be transmitted only when the presupposed event-triggered condition is satisfied. Therefore, the event-triggered strategy can effectively decrease the unnecessary waste of communication resources, which illustrates that the rational and efficacious utilization of limited communication resources can be realized. Recent years have witnessed a growing interest in studying networked systems based on event-triggered strategy. To mention a few, the event-triggered control problem was considered in [53–58] and the event-triggered filtering problem was studied in [59–61]. On the other hand, due to introducing unreliable and bandwidth-limited communication networks into feedback control loops, some network-induced constraints, such as channel fading, packet dropouts, and communication delays, are unavoidable in networked systems. Among all the network-induced constraints that emerged, packet dropouts and communication delays are considered to be two of the main causes for the performance deterioration or even the instability of the networked systems. For the study of networked systems subject to packet dropouts, there mainly exist two approaches to model the phenomenon of packet dropouts in the existing literature, i.e., Bernoulli stochastic processes and Markov chains. Over the past few decades, the study of networked systems with packet dropouts has attracted growing interest from numerous scholars and a great number of important results have been reported. To mention a few, the filtering problem has been considered in [44, 62–64] and the control problem has been investigated in [42, 43, 50, 51, 54, 55, 65–67]. In the past few decades, the investigation of networked systems with communication delays has also attracted considerable attention and a great deal of results have been published, for instance, [37, 43, 44, 51, 52, 58, 61, 67–69] and references therein. In addition, the unreliable and bandwidth-limited communication networks used in most networked systems are open, which indicates that networked systems are vulnerable to cyber attacks. In the past several years, there have been increasing research interests in developing security control and filtering strategies for networked systems under cyber attacks, and some interesting results have been reported in the open literature. Among these results, there mainly exist two kinds of cyber attacks. One kind is called denial-of-service attacks and another kind is called deception attacks. For networked systems with denial-of-service attacks, security filtering and security control problems have been addressed in [70–72]. For networked systems with deception attacks, security filtering and security control problems have been investigated in [73–75].

1.1.3 Fuzzy-Model-Based Nonlinear Networked Systems In recent years, the study of nonlinear networked systems based on the T–S fuzzy model has attracted a growing interest and a number of significant results have

1.1 Introduction

7

been proposed. Noticeable works include [76–116]. In the presence of weighted try-once-discard protocol, the sliding mode control problem was considered for T–S fuzzy networked systems and T–S fuzzy networked singularly perturbed systems with static quantization in [76, 77], respectively. For T–S fuzzy networked systems with round-robin communication protocol, the H∞ PID control problem was addressed in [78] and the robust H∞ fault detection problem was discussed in [79]. By considering the effect of stochastic communication protocol, the efficient model-predictive control strategy was given for T–S fuzzy networked systems in [80] and the robust dynamic output feedback control strategy was developed for T–S fuzzy networked systems subject to dynamic quantization in [81]. The peak-to-peak filtering problem was considered for T–S fuzzy networked systems with static quantization in [82] and the H∞ filtering problem was addressed for T–S fuzzy networked systems with static quantization and packet dropouts in [83]. The output feedback control problem was investigated for T–S fuzzy networked systems with static quantization in [84, 85]. In [86], the nonfragile H∞ state feedback control problem was considered for T–S fuzzy networked systems with static quantization. In the presence of input and output static quantization, the output feedback tracking control strategy was proposed in [87] and the event-triggered output feedback tracking control strategy was presented in [88]. For T–S fuzzy networked systems with static quantization and packet dropouts, the controller design problem was discussed in [89]. By taking static quantization, network induced delays, and packet dropouts into simultaneous consideration, the robust H∞ control problem and guaranteed cost control problem were studied for T–S fuzzy networked systems in [90, 91], respectively. The H∞ static output feedback control problem was researched for T–S fuzzy networked systems with input and output dynamic quantization in [92], where a novel adjusting scheme was proposed for the parameters of the dynamic quantizers. In [93], a guaranteed cost static output feedback control strategy was proposed for T–S fuzzy networked systems with input and output dynamic quantization. Both the full- and reduced-order energy-to-peak filtering problems were addressed for T–S fuzzy networked systems with dynamic quantization in [94]. For T–S fuzzy networked systems with input and output dynamic quantization, the observer-based output feedback control problem was studied in [95]. In the presence of input dynamic quantization, the robust state feedback control problem was investigated in [96]. In [97], the event-triggered output feedback tracking control strategy was presented for T–S fuzzy networked systems subject to input and output dynamic quantization. In [98], the problem of robust H∞ filtering was investigated for T–S fuzzy networked systems with dynamic quantization and data dropouts. For T–S fuzzy networked systems with dynamic quantization, communication delays, and impulsive effects, the problem of output feedback control was discussed in [99]. In [100], the H∞ state feedback control problem was solved for T–S fuzzy networked systems with dynamic quantization, communication delays, and packet dropouts. In the presence of dynamic quantization and stochastic cyber attacks, both the full- and reduced-order event-triggered H∞ filters were designed for T–S fuzzy networked systems in [101]. In [102], the problem of event-triggered observer-based output feedback non-parallel distribution compensation control was researched for T–S fuzzy networked systems. An observer-based event-triggered slid-

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ing mode control strategy was proposed for T–S fuzzy networked systems in [103]. The reduced-order event-triggered energy-to-peak filtering problem was investigated for T–S fuzzy networked systems in [104]. For T–S fuzzy networked systems with packet dropouts, the H∞ filter design problem was solved in [105] and the peak-topeak filter design problem was considered in [106]. In the presence of multiple packet dropouts, the asynchronous output feedback H∞ control strategy was developed for T–S fuzzy networked systems in [107]. For T–S fuzzy networked systems with packet dropouts, the state feedback H∞ control problem was discussed in [108] and the static output feedback H∞ control was studied in [109]. In [110], the dynamic output feedback H∞ controller design problem was addressed for T–S fuzzy networked systems subject to multiple probabilistic communication delays and multiple packet dropouts. The robust stabilization problem was investigated for T–S fuzzy networked systems with communication delays in [111]. In [112], the fault detection problem was addressed for T–S fuzzy networked systems with communication delays. For T–S fuzzy networked systems with communication delays, both state feedback and output feedback fuzzy delay compensation control strategies were proposed in [113]. In the presence of stochastic deception attacks, the state feedback stabilization problem was studied for T–S fuzzy networked systems with quantization and hybrid-triggered scheme in [114]. For T–S fuzzy networked systems with denial-of-service attacks, a security event-triggered H∞ filtering strategy was proposed in [115]. In [116], the security finite-time filter design problem was solved for T–S fuzzy networked systems subject to deception attacks, packet dropouts, and static quantization. On the basis of the above results, the authors make a further study on controller and filter design problems for nonlinear networked systems based on the T–S fuzzy model in this monograph, which contains some novel technologies and new results. These include the fuzzy robust H∞ control problem with dynamic quantization, which is addressed in Chap. 2, the fuzzy filtering problem with multiple communication constraints, which is considered in Chap. 3, the fuzzy output feedback control problem with communication constraints, which is discussed in Chap. 4, the eventtriggered fuzzy control and filtering problems with communication constraints, which are investigated in Chap. 5, and the fuzzy fault detection and fault-tolerant control problems with quantization, which are researched in Chap. 6.

1.2 Preliminaries The following lemmas will be needed for deriving the main results in this monograph. Lemma 1.1 [110] (Schur complement) For matrices Λ1 , Λ2 , and Λ3 with Λ1 = Λ1T and Λ2 = Λ2T > 0. Then, it can be obtained that Λ1 + Λ3T Λ−1 2 Λ3 < 0 if and only if   Λ1 ∗ < 0. Λ3 −Λ2

(1.7)

1.3 Outline of the Monograph

9

Lemma 1.2 [82] Let X 1 = X 1T , X 2 , X 3 , and Δ be real matrices with appropriate dimensions and ΔT Δ ≤ I . Then, it can be obtained that X 1 + X 2 ΔX 3 + X 3T ΔT X 2T < 0, if and only if there exists a scalar λ > 0 such that X 1 + λ−1 X 2 X 2T + λX 3T X 3 < 0.

(1.8)

Lemma 1.3 [45] (S-procedure) Given quadratic functions for η ∈ Rm , Π0 (η) = η T Σ0 η, Π1 (η) = η T Σ1 η, . . ., Πθ (η) = η T Σθ η, Σi = ΣiT , i = 1, 2, . . . , θ . Then, it can be obtained that Π0 (η) < 0 with Π1 (η) ≥ 0, Π2 (η) ≥ 0, . . ., Πθ (η) ≥ 0, if there exist scalars ε1 > 0, ε2 > 0, . . ., εθ > 0 satisfying Σ0 + ε1 Σ1 + ε2 Σ2 + · · · + εθ Σθ < 0.

(1.9)

Lemma 1.4 [10] For matrices M1 , M2 , M3 , and M4 with appropriate dimensions. Then, it can be obtained that M1 + M2 M3 + M3T M2T < 0, if there exists a scalar λ satisfying   ∗ M1 < 0. (1.10) λM2T + M4 M3 −λM4 − λM4T Lemma 1.5 [51] In terms of any positive definite matrix W , the matrices (or scalars) R and S with appropriate dimensions, the following inequality holds − 2R T S ≤ R T W −1 R + S T W S.

(1.11)

Lemma 1.6 [48] For matrices X > 0 and Y > 0 with same dimension, the following statements are equivalent: − Y −1 + X < 0,

(1.12)

Y − X −1 < 0.

(1.13)

1.3 Outline of the Monograph This monograph is organized into 6 chapters, as follows. Chapter 2 is devoted to studying the H∞ control problem for T–S fuzzy systems with dynamic quantization. More specifically, a two-step design strategy is proposed to deal with the robust H∞ state feedback control problem for uncertain discretetime T–S fuzzy systems with control input quantization based on the fuzzy Lyapunov functional method in Sect. 2.1, a novel two-step design scheme is developed to investigate the non-fragile H∞ state feedback control problem for discrete-time T–S fuzzy systems with the controller gain perturbation and state quantization in Sect. 2.2, the descriptor representation approach is employed to design the resilient H∞ static output feedback controller for continuous-time T–S fuzzy systems with control input

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1 Introduction and Preview

and measurement output quantization, parameter uncertainties, and controller gain perturbation in Sect. 2.3. Chapter 3 is concerned with the filtering problem for T–S fuzzy systems with multiple communication constraints. More specifically, a novel l2 –l∞ filtering strategy is presented for discrete-time T–S fuzzy systems with the effects of static quantization on the measurement output and the performance output in Sect. 3.1, the induced l∞ filter design problem is considered for discrete-time T–S fuzzy systems with the effects of data packet dropouts on the measurement output and the performance output in Sect. 3.2, the H∞ filtering problem is discussed for discrete-time T–S fuzzy systems with the effects of dynamic quantization on the measurement output and the performance output in Sect. 3.3. Chapter 4 is focused on the output feedback control problem for T–S fuzzy systems with communication constraints. More specifically, the static output feedback stabilization and H∞ control problems are addressed for discrete-time T–S fuzzy systems with input and output dynamic quantization in Sect. 4.1, the static output feedback H∞ controller design approach is developed for continuous-time T–S fuzzy systems with the effects of transmission delay and data dropouts in sensor-to-controller and controller-to-actuator communication channels in Sect. 4.2, and the dynamic output feedback H∞ control problem is solved for discrete-time T–S fuzzy systems with the effect of stochastic communication protocol in the communication channel from the plant to the controller based on the fuzzy Markov jump model approach in Sect. 4.3. Chapter 5 studies the event-triggered output feedback tracking control and eventtriggered filtering problems for T–S fuzzy systems with different communication constraints. More specifically, the event-triggered static output feedback tracking control problem is investigated for discrete-time T–S fuzzy systems with input and output static quantization in Sect. 5.1, the event-triggered L2 –L∞ filtering problem is discussed for continuous-time T–S fuzzy systems with time-delay and DoS attacks based on the piecewise Lyapunov–Krasovskii functional approach in Sect. 5.2, and the event-triggered full- and reduced-order H∞ filters design approach is proposed for discrete-time T–S fuzzy systems with dynamic quantization and stochastic deception attacks in Sect. 5.3. Chapter 6 investigates the problems of fault detection and fault-tolerant control for T–S fuzzy systems with quantization. More specifically, the resilient H∞ fault detection filter design approach is presented for uncertain discrete-time T–S fuzzy systems with dynamic quantization and filter gain perturbations in Sect. 6.1, the H∞ fault-tolerant static output feedback control problem is studied for discrete-time T–S fuzzy Markov jump systems with actuator fault and dynamic quantization in Sect. 6.2, and the non-fragile guaranteed cost fault-tolerant state feedback control problem is considered for discrete-time T–S fuzzy systems with input dynamic quantization and actuator fault based on the two-step design approach in Sect. 6.3.

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Chapter 2

Fuzzy Robust H∞ Control with Dynamic Quantization

Abstract This chapter investigates the quantized H∞ control problem for T–S fuzzy systems, where the dynamic quantizer is adopted for signal quantization before transmission to reduce the communication burden. Firstly, the robust H∞ control is addressed for uncertain discrete-time T–S fuzzy systems with the control input quantization via a two-step design approach and the fuzzy Lyapunov functional method. Secondly, a new two-step strategy is proposed to onlinely update the dynamic quantizer’s parameter and design the non-fragile controller for discrete-time T–S fuzzy systems with the controller gain perturbation and state quantization, so that the quantized closed-loop system is asymptotically stable and maintains the same specified H∞ performance as the one without quantization. Thirdly, by fully considering the effect of system parameter uncertainties, controller gain perturbation, control input and measurement output quantization on continuous-time T–S fuzzy systems, the resilient H∞ control problem is solved based on two descriptor representation approaches, and a novel adjusting rule is given for dynamic quantizers’ parameters. Finally, some examples are provided to show the feasibility and effectiveness of the proposed design methods, respectively. Keywords T–S fuzzy systems · Norm-bounded uncertainties · Dynamic quantization · Two-step design method · Descriptor representation approach

2.1 Robust Control with Input Quantization In this section, a two-step method is given to deal with the state feedback control problem of uncertain T–S fuzzy systems with control input quantization. The fuzzy Lyapunov function is used to judge that the T–S fuzzy closed-loop system is asymptotically stable when there are norm-bounded uncertain parameters, and has the same H∞ performance as the system without control input quantization.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Chang et al., Control and Filtering of Fuzzy Systems Under Communication Channels, https://doi.org/10.1007/978-981-99-4346-3_2

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2 Fuzzy Robust H∞ Control with Dynamic Quantization

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2.1.1 Problem Formulation Consider a discrete-time T–S fuzzy model with norm-bounded uncertainties as follows: Plant Rule i th : IF ε1 (k) is M1i , ε2 (k) is M2i , and, . . ., and εd (k) is Mdi , THEN x(k + 1) = (Ai + Δ Ai )x(k) + (Bi + Δ Bi )Q(u(k)) + (E i + Δ Ei )w(k), z(k) = (Ci + ΔCi )x(k) + (Di + Δ Di )Q(u(k)) + (Fi + Δ Fi )w(k),

(2.1)

where ε1 (k), ε2 (k), …, εd (k) represent the premise variables; M pi with i = 1, 2, …, r , p = 1, 2, …, d represents the fuzzy sets, where r is the number of fuzzy rules; x(k) ∈ Rn x and u(k) ∈ Rn u represent the state variable, control input, respectively; w(k) ∈ Rn w denotes the noise signal in l2 [0, ∞); z(k) ∈ Rn z is the controlled output; Ai ∈ Rn x ×n x , Bi ∈ Rn x ×n u , Ci ∈ Rn z ×n x , Di ∈ Rn z ×n u , E i ∈ Rn x ×n w , and Fi ∈ Rn z ×n w , with i = 1, 2, …, r denote the system matrices; Δ Ai , Δ Bi , ΔCi , Δ Di , Δ Ei , and Δ Fi , i = 1, …, r are the uncertain parameter matrices, which can be described as: 

Δ Ai Δ Bi Δ Ei ΔCi Δ Di Δ Fi



 =

   X1 Δ Y1i Y2i Y3i , X2

(2.2)

where ΔT Δ ≤ I , Δ represents the norm-bounded uncertainty, Y1i , Y2i , Y3i , X 1 and X 2 are known constant matrices. By introducing the following fuzzy basis functions: d

p=1 M pi (ε p (k)) d i=1 p=1 M pi (ε p (k))

ρi (ε(k)) = r one has

r 

≥ 0,

ρi (ε(k)) = 1, i = 1, 2, . . . , r,

(2.3)

(2.4)

i=1

where M pi (ε p (k)) is the grade of membership function of ε p (k) in M pi . The uncertain T–S fuzzy model (2.1) can be deduced as follows: x(k + 1) = (A(ρ) + Δ A (ρ)) x(k) + (B(ρ) + Δ B (ρ)) Q(u(k)) + (E(ρ) + Δ E (ρ)) w(k), z(k) = (C(ρ) + ΔC (ρ)) x(k) + (D(ρ) + Δ D (ρ)) Q(u(k)) + (F(ρ) + Δ F (ρ)) w(k),

(2.5)

2.1 Robust Control with Input Quantization

19

where A(ρ) = C(ρ) = E(ρ) =

r  i=1 r  i=1 r 

ρi (ε(k))Ai , ρi (ε(k))Ci , ρi (ε(k))E i ,

B(ρ) = D(ρ) = F(ρ) =

i=1

Δ A (ρ) = X 1 ΔY1 (ρ) = Δ B (ρ) = X 1 ΔY2 (ρ) = ΔC (ρ) = X 2 ΔY1 (ρ) = Δ D (ρ) = X 2 ΔY2 (ρ) = Δ E (ρ) = X 1 ΔY3 (ρ) = Δ F (ρ) = X 2 ΔY3 (ρ) =

r  i=1 r  i=1 r 

ρi (ε(k))Bi , ρi (ε(k))Di , ρi (ε(k))Fi ,

i=1 r  i=1 r  i=1 r  i=1 r  i=1 r  i=1 r 

ρi (ε(k))X 1 ΔY1i , ρi (ε(k))X 1 ΔY2i , ρi (ε(k))X 2 ΔY1i , ρi (ε(k))X 2 ΔY2i , ρi (ε(k))X 1 ΔY3i , ρi (ε(k))X 2 ΔY3i .

i=1

Considering the control input signal needs to be quantized by the quantizer before being sent to the data channel, the dynamic quantizer which is the same as the one in [1] is adopted to quantize the signal variable s(k) ∈ Rn s . Then we define the Δs > 0 and Ms > 0 to denote the error bound and the range of quantizer, respectively, which satisfy the following condition: q (s(k)) − s(k) ≤ Δs , i f s(k) ≤ Ms , q (s(k)) > Ms − Δs , i f s(k) > Ms .

(2.6a) (2.6b)

Next, we will give the following dynamic quantizer to the networked system: 

Q(s(k)) = μs (k)q

 s(k) , μs (k)

(2.7)

where μs (k) > 0 denotes the dynamic quantizer’s parameter. That is, μs (k)Δs and μs (k)Ms mean that the error bound and the range of quantizer can be updated with

2 Fuzzy Robust H∞ Control with Dynamic Quantization

20

the adjustment of dynamic quantizer’s parameter μs (k). Then, the quantized control input signal can be described as 

Q(u(k)) = μu (k)q

 u(k) , μu (k)

(2.8)

with u(k) = K x(k),

(2.9)

where K is controller gain matrix. Substituting (2.8) and (2.9) into (2.5), the quantized fuzzy closed-loop system can be given as follows:

˜ ˜ ˜ ˜ x(k + 1) = A(ρ) + B(ρ)K x(k) + B(ρ)e u (k) + E(ρ)w(k),

˜ ˜ ˜ ˜ z(k) = C(ρ) + D(ρ)K x(k) + D(ρ)e u (k) + F(ρ)w(k), with

(2.10)

    u(k) u(k) − , eu (k) = Q(u(k)) − u(k) = μu (k) q μu (k) μu (k) ˜ ˜ A(ρ) = A(ρ) + Δ A (ρ), B(ρ) = B(ρ) + Δ B (ρ), ˜ ˜ C(ρ) = C(ρ) + ΔC (ρ), D(ρ) = D(ρ) + Δ D (ρ), ˜ ˜ E(ρ) = E(ρ) + Δ E (ρ), F(ρ) = F(ρ) + Δ F (ρ).

The quantized robust state feedback H∞ control problem considered in this section is to design the robust state feedback controller in (2.9) and the dynamic quantizer in (2.8) such that (1) The quantized fuzzy closed-loop system in (2.10) is asymptotically stable when w(k) = 0. (2) The quantized fuzzy closed-loop system in (2.10) has aprescribed H∞ perT 2 formance γ , i.e., under the initial condition x(0) = 0, ∞ k=0 z (k)z(k) < γ  ∞ T k=0 w (k)w(k) is satisfied for any nonzero w(k) ∈ l2 [0, ∞).

2.1.2 Main Results As mentioned above, this section will give an improved two-step algorithm to design the controller and dynamic quantizer. In Step 1, a robust controller is designed, and in Step 2, the dynamic quantizer’s parameter μu (k) is determined.

2.1 Robust Control with Input Quantization

21

Theorem 2.1 For the given positive scalar γ , selected quantization range Mu and quantization error bound Δu , if there exist scalars σu (ρ) > 0, τu > 0, and matrix P(ρ) > 0, such that the following matrix inequalities hold:  1 − σu (ρ) < 0, Mu

(2.11)

χ1T (ρ)P(ρ + )χ1 (ρ) + χ2T (ρ)χ2 (ρ) − diag{P(ρ), 0, γ 2 I }

+ τu χ3T χ3 − diag{0, I, 0} < 0,

(2.12)

where

˜ ˜ ˜ ˜ χ1 (ρ) = A(ρ) , + B(ρ)K B(ρ) E(ρ)

˜ ˜ ˜ ˜ χ2 (ρ) = C(ρ) + D(ρ)K D(ρ) F(ρ) ,

√ χ3 = 2 σu (ρ)Δu K 0 0 , and the online adjustment strategy of dynamic quantizer’s parameter μu (k) is given as:   σu (ρ) u(k) ≤ μu (k) ≤ 2 σu (ρ) u(k) . (2.13) Then, the quantized fuzzy closed-loop system (2.10) is asymptotically stable under the specified H∞ performance index γ . Proof If the condition (2.11) holds and the online adjustment strategy (2.13) can be guaranteed, then the control input signal does not exceed the quantization range. According to (2.6) and (2.13), the following inequality can be obtained:      u(k) u(k)    eu (k) = μu (k) q − μu (k) μu (k)  ≤ μu (k)Δu  ≤ 2 σu (ρ)Δu u(k)  = 2 σu (ρ)Δu K x(k) ,

(2.14)

euT (k)eu (k) ≤ 4σu (ρ)Δ2u x T (k)K T K x(k).

(2.15)

i.e., √ In order to obtain the scalar variable σu (ρ) related to the dynamic quantizer’s parameter μu (k), the following inequality is obtained from (2.15):



  √ ϕ T (k) H a 2 σu (ρ)Δu K 0 0 − diag { 0, I, 0 } ϕ(k) ≥ 0,

(2.16)

2 Fuzzy Robust H∞ Control with Dynamic Quantization

22

T where ϕ(k) = x T (k) euT (k) w T (k) . Next, the Lyapunov function will be con structed as V (x(k)) = x T (k)P(ρ)x(k) with P(ρ) = ri=1 ρi (ε(k))Pi > 0, then one can get: V (x(k + 1)) − V (x(k)) + z T (k)z(k) − γ 2 w T (k)w(k) = ΔV (x(k)) + z T (k)z(k) − γ 2 w T (k)w(k)  

= ϕ T (k) χ1T (ρ)P(ρ + )χ1 (ρ) + χ2T (ρ)χ2 (ρ) − diag P(ρ), 0, γ 2 I ϕ(k). If the matrix inequality (2.12) is satisfied, it is easy to get that when the variable ϕ(k) = 0, one has:    ϕ T (k) χ1T (ρ)P(ρ + )χ1 (ρ) + χ2T (ρ)χ2 (ρ) − diag P(ρ), 0, γ 2 I

 + τu χ3T χ3 − diag{0, I, 0} ϕ(k) < 0,

(2.17)

then based on τu > 0 and (2.16),

ϕ T (k) χ1T (ρ)P(ρ + )χ1 (ρ) + χ2T (ρ)χ2 (ρ) − diag{P(ρ), 0, γ 2 I } ϕ(k) < 0, can be obtained according to the S-procedure (Lemma 1.3). It shows that the H∞ performance γ can be guaranteed with the quantized fuzzy closed-loop system (2.10). The proof is completed. It is worth noting that Theorem 2.1 gives a useful H∞ performance analysis criterion based on matrix inequality, but matrix inequality (2.12) contains some nonlinearity and uncertainties, which leads to matrix inequality (2.12) being nonconvex (non-strict LMI), it means that matrix inequality conditions cannot be directly used in controller design. The following theorem proposes a new H∞ performance analysis criterion for quantized fuzzy system (2.10) by defining new variables and using matrix inequality transformation technology to solve this nonlinear problem. Theorem 2.2 For the given quantization error bound Δu and quantization range Mu , the quantized fuzzy closed-loop system (2.10) is asymptotically stable with the prescribed H∞ performance index γ > 0, if there exist matrices Q(ρ) > 0, G, and N , scalars τu > 0 and νu (ρ) > 0, such that the following matrix inequalities hold: Mu2 νu (ρ) > τu , ⎡ ⎤ −G − G T + Q(ρ) ∗ ∗ ∗ ∗ ∗ ⎢ ⎥ ∗ ∗ ∗ ∗ 0 −γ 2 I ⎢ ⎥ +) ⎢ A(ρ)G ⎥ ˜ ˜ ˜ ∗ ∗ ∗ + B(ρ)N E(ρ) −Q(ρ ⎢ ⎥ < 0, ¯ Λ(ρ) = ⎢ ⎥ ˜ ˜ ˜ C(ρ)G + D(ρ)N F(ρ) 0 −I ∗ ∗ ⎢ ⎥ ⎣ ⎦ T T ∗ 0 0 B˜ (ρ) D˜ (ρ) −τu I 0 0 0 0 −νu (ρ)I 2νu (ρ)Δu N

(2.18)

(2.19)

2.1 Robust Control with Input Quantization

23

and the online adjusting strategy of dynamic quantizer’s parameter μu (k) is given the same as (2.13) with σu (ρ) = νu (ρ)/τu . Proof By using Schur complement (Lemma 1.1) for the matrix inequality (2.12) in Theorem 2.1, the following inequality can be obtained ⎡

⎤ ∗ ⎥ ∗ ⎥ ⎥ ∗ ⎥ < 0. ⎥ − P −1 (ρ + ) ∗ ⎥ ⎦ 0 ∗ 0 − τu σu (ρ)I (2.20) Defining Q(ρ) = P −1 (ρ), Q(ρ + ) = P −1 (ρ + ) and νu (ρ) = τu σu (ρ), the above inequality (2.20) can be rewritten as: −P(ρ) ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ A(ρ) ˜ ˜ + B(ρ)K ⎢ ⎣C(ρ) ˜ ˜ + D(ρ)K 2τu σu (ρ)Δu K

⎡

−Q −1 (ρ) ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ A(ρ) ˜ ˜ + B(ρ)K ⎢ ⎣C(ρ) ˜ ˜ + D(ρ)K 2νu (ρ)Δu K

∗ − τu I 0 ˜ B(ρ) ˜ D(ρ) 0

∗ − τu I 0 ˜ B(ρ) ˜ D(ρ) 0

∗ ∗ − γ2I ˜ E(ρ) ˜ F(ρ) 0

∗ ∗ − γ2I ˜ E(ρ) ˜ F(ρ) 0

∗ ∗ ∗

∗ ∗ ∗ ∗ −I 0

∗ ∗ ∗ − Q(ρ + ) 0 0

∗ ∗ ∗ ∗ −I 0

⎤ ∗ ⎥ ∗ ⎥ ⎥ ∗ ⎥ < 0. (2.21) ⎥ ∗ ⎥ ⎦ ∗ − νu (ρ)I

By introducing a transformation matrix Υ as ⎡

GT ⎢ 0 ⎢ ⎢ 0 Υ =⎢ ⎢ 0 ⎢ ⎣ 0 0

0 0 0 0 I 0

0 I 0 0 0 0

0 0 I 0 0 0

0 0 0 I 0 0

⎤ 0 0⎥ ⎥ 0⎥ ⎥, 0⎥ ⎥ 0⎦ I

and performing congruence transformations with Υ to (2.21), the inequality −(G − Q(ρ))T Q −1 (ρ)(G − Q(ρ)) ≤ 0 can be rewritten as: −G T Q −1 (ρ)G ≤ −G T − G + Q(ρ),  where the symmetric matrix Q(ρ) = ri=1 ρi (ε(k))Q i > 0. By defining a matrix variable N = K G, the inequality (2.18) and (2.19) in Theorem 2.2 can be guaranteed. The proof is completed. Theorem 2.2 proposes a new H∞ performance analysis criterion based on the quantized fuzzy system (2.10), in which matrix inequalities (2.18) and (2.19) effectively avoid product terms (nonlinear coupling terms). However, it is obvious that matrix inequalities (2.18) and (2.19) are still nonconvex, including uncertainty Δ

2 Fuzzy Robust H∞ Control with Dynamic Quantization

24

and nonlinear coupling term νu (ρ)Δu N , which makes it difficult to design the robust controller and the dynamic quantizer for the quantized fuzzy closed-loop system in (2.10) through the H∞ performance analysis criterion in Theorem 2.2. Next, this chapter presents an improved two-step design method in the following algorithm to solve the above problem. Firstly, in Step 1, a robust controller is designed so that the quantized fuzzy closed-loop system (2.10) with norm-bounded uncertainties has the same performance as the system without control input quantization, and then in Step 2, the scalar variable σ (ρ) related to the dynamic quantizer’s parameter μu (k) is calculated. Algorithm 2.1 For the quantized fuzzy closed-loop system (2.10) with normbounded uncertainties, it is asymptotically stable under the given H∞ performance index γ : Step 1. Solving the following LMIs to obtain the values of matrices N , G, Q i > 0, and the controller gain K = N G −1 : ⎡ −G − G T + Q i ⎢ 0 ⎢ ⎢ Ai G + Bi N ⎢ ⎣ Ci G + Di N Y1i G + Y2i N

∗ − γ2I Ei Fi Y3i

∗ ∗ − Q j + α X 1 X 1T α X 2 X 1T 0

∗ ∗ ∗ − I + α X 2 X 2T 0

⎤ ∗ ∗ ⎥ ⎥ ∗ ⎥ ⎥ < 0, ∗ ⎦ − αI (2.22)

for i, j = 1, 2, . . . , r .

 Step 2. Defining νu (ρ) = ri=1 ρi (ε(k))νui , with νui > 0 and solving the following LMIs with known Q i , G, N , and K obtained by Step 1, then the values of scalar variables τu and νui can be determined: Mu2 νui > τu ,

(2.23)

⎤ ∗ ∗ ∗ ∗ ∗ −G − G T + Q i ∗ 2 ⎢ 0 −γ I ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ ⎢ ⎢ Ai G + Bi N E i −Q j + α X 1 X 1T ∗ ∗ ∗ ∗ ⎥ ⎥ ⎢ Λi j = ⎢ Fi α X 2 X 1T −I + α X 2 X 2T ∗ ∗ ∗ ⎥ ⎥ < 0, ⎢ Ci G + Di N ⎥ ⎢ Y1i G + Y2i N Y 0 0 −α I ∗ ∗ 3i ⎥ ⎢ ⎣ DiT Y2iT −τu I ∗ ⎦ 0 0 BiT 2νui Δu N 0 0 0 0 0 −νui I ⎡

(2.24) for i, j = 1, 2, . . . , r . Then, the scalar variable σu (ρ) = νu (ρ)/τu . Through the online adjustment strategy in Theorem 2.2, the dynamic quantizer’s parameter μu (k) can be adjusted by the scalar variable σu (ρ). Discussion 2.1 (1) According to the membership function given in (2.4), if the inequalities in (2.22) are satisfied, then one can obtain

2.1 Robust Control with Input Quantization

⎡

−G − G T + Q(ρ) ⎢ 0 ⎢ ⎢ A(ρ)G + B(ρ)N ⎢ ⎣ C(ρ)G + D(ρ)N Y1 (ρ)G + Y2 (ρ)N

∗ − γ2I E(ρ) F(ρ) Y3 (ρ)

25

∗ ∗ − Q(ρ + ) + α X 1 X 1T α X 2 X 1T 0

∗ ∗ ∗ − I + α X 2 X 2T 0

⎤ ∗ ∗ ⎥ ⎥ ∗ ⎥ ⎥ < 0. ∗ ⎦ − αI (2.25)

By using the Schur complement (Lemma 1.1), the inequality (2.25) can be rewritten as: ⎤ ⎡ ⎡ ⎤ 0 −G − G T + Q(ρ) ∗ ∗ ∗ 2 ⎥ ⎢ ⎢ 0 ⎥

0 −γ I ∗ ∗ ⎥ ⎢ ⎢ ⎥ 0 0 XT XT + α 1 2 ⎣ A(ρ)G + B(ρ)N ⎣X1⎦ E(ρ) − Q(ρ + ) ∗ ⎦ X2 C(ρ)G + D(ρ)N F(ρ) 0 −I ⎡ ⎤ T (Y1 (ρ)G + Y2 (ρ)N ) ⎥

1⎢ Y3T (ρ) ⎥ Y1 (ρ)G + Y2 (ρ)N Y3 (ρ) 0 0 < 0. + ⎢ ⎦ 0 α⎣ 0

(2.26)

Then, based on Lemma 1.2 for α > 0 and ΔT Δ ≤ I , (2.26) means that ⎡ ⎤ −G − G T + Q(ρ) ∗ ∗ ∗ ⎢ ∗ ∗ ⎥ 0 − γ2I ⎢ ⎥ ⎣ A(ρ)G + B(ρ)N E(ρ) − Q(ρ + ) ∗ ⎦ C(ρ)G + D(ρ)N F(ρ) 0 −I ⎧⎡ ⎤ ⎫ 0 ⎪ ⎪ ⎪ ⎪ ⎨⎢ ⎥

⎬ 0 ⎢ ⎥ < 0. + H e ⎣ ⎦ Δ Y1 (ρ)G + Y2 (ρ)N Y3 (ρ) 0 0 X1 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ X2

(2.27)

(2) Similar to (1), if the condition Λi j < 0 in (2.24) holds, then one can obtain ⎡

−G − G T + Q(ρ) ⎢ 0 ⎢ ⎢ A(ρ)G + B(ρ)N ⎢ Λ(ρ) = ⎢ ⎢ C(ρ)G + D(ρ)N ⎢ Y1 (ρ)G + Y2 (ρ)N ⎢ ⎣ 0 2νu (ρ)Δu N ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I + α X 2 X 2T 0 − αI Y2T (ρ) D T (ρ) 0 0

∗ − γ2I E(ρ) F(ρ) Y3 (ρ) 0 0

∗ ∗ − Q(ρ + ) + α X 1 X 1T α X 2 X 1T 0 B T (ρ) 0 ⎤ ∗ ∗ ⎥ ∗ ∗ ⎥ ⎥ ∗ ∗ ⎥ ⎥ < 0. ∗ ∗ ⎥ ⎥ ∗ ∗ ⎥ ⎦ − τu I ∗ 0 − νu (ρ)I

(2.28)

2 Fuzzy Robust H∞ Control with Dynamic Quantization

26

By using Schur complement (Lemma 1.1) to (2.28), one has: ⎡ ⎤ ∗ ∗ ∗ ∗ ∗ −G − G T + Q(ρ) ⎢ ⎥ 0 − γ2I ∗ ∗ ∗ ∗ ⎢ ⎥ ⎢ A(ρ)G + B(ρ)N E(ρ) − Q(ρ + ) ⎥ ∗ ∗ ∗ ⎥ ˜ Λ(ρ) =⎢ ⎢ C(ρ)G + D(ρ)N F(ρ) ⎥ 0 −I ∗ ∗ ⎢ ⎥ T T ⎣ ⎦ 0 0 B (ρ) D (ρ) − τu I ∗ 2νu (ρ)Δu N 0 0 0 0 − νu (ρ)I ⎡ ⎤ (Y1 (ρ)G + Y2 (ρ)N )T ⎢ ⎥ Y3T (ρ) ⎢ ⎥ ⎥

1⎢ 0 ⎢ ⎥ Y1 (ρ)G + Y2 (ρ)N Y3 (ρ) 0 0 Y2 (ρ) 0 (2.29) + ⎢ ⎥ 0 α⎢ ⎥ ⎣ ⎦ Y2T (ρ) 0 ⎡ ⎤ 0 ⎢0⎥ ⎢ ⎥

⎢X ⎥ + α ⎢ 1 ⎥ 0 0 X 1T X 2T 0 0 < 0. ⎢X2⎥ ⎣0⎦ 0

That is ⎡

⎤ −G − G T + Q(ρ) ∗ ∗ ∗ ∗ ∗ ⎢ ⎥ ∗ ∗ ∗ ∗ 0 − γ2I ⎢ ⎥ ⎢ A(ρ)G + B(ρ)N E(ρ) − Q(ρ + ) ⎥ ∗ ∗ ∗ ⎢ ⎥ ¯ Λ(ρ) = ⎢ ⎥ C(ρ)G + D(ρ)N F(ρ) 0 − I ∗ ∗ ⎢ ⎥ ⎣ ⎦ ∗ 0 0 B T (ρ) D T (ρ) − τu I 0 0 0 0 − νu (ρ)I 2νu (ρ)Δu N ⎧⎡ ⎤ ⎫ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎢ 0 ⎥ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎥ ⎨⎢ ⎥ ⎬

X 1⎥ ⎢ + H e ⎢ ⎥ Δ Y1 (ρ)G + Y2 (ρ)N Y3 (ρ) 0 0 Y2 (ρ) 0 < 0. ⎪ ⎪ ⎢ X 2⎥ ⎪ ⎪ ⎪ ⎪ ⎪⎣ 0 ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 0 (2.30) Remark 2.1 According to Discussion 2.1, Algorithm 2.1 finally guarantees that the ¯ condition Mu2 νu (ρ) > τu holds by the definition of τu and νu (ρ) with Λ(ρ) < 0, that is, (2.18) and (2.19) are established. Therefore, the two-step design strategy proposed in Algorithm 2.1 is effective for the robust control design of the quantized fuzzy system (2.10) with norm-bounded uncertainties. Remark 2.2 The quantized control input signal can be obtained by determining the dynamic quantizer’s parameter μu (k) as a unique value and transforming it into a communication channel. The adjustment rule of the dynamic quantizer’s parameter μu (k) considered in this chapter is as [2]:

2.1 Robust Control with Input Quantization

27

The adjusting rule: μu (k) =

  floor 2ϑu u(k) × 10β × 10−β , 0 ≤ 2ϑu u(k) < 1, floor (2ϑu u(k)) , 1 ≤ 2ϑu u(k) ,

    where β = min β ∈ N+ | 2ϑu u(k) × 10β > 1 and floor(θ ) is the integer function and gives the maximum integer that does not exceed θ . It is worth noting that, in Algorithm 2.1, the μu (k) of the dynamic quantizer (2.7) is updated by using ! ϑu =

νu (ρ) τu

through the above adjusting rule.

2.1.3 Simulation Example This section uses the truck-trailer system shown in Fig. 2.1 as an example to verify the effectiveness and feasibility of the proposed method. Set L = 6.0 m, l = 3.5 m, Vm = −15 m/s, and Tm = 0.1 s, where Vm represents the speed of backing up, and Tm means the sampling time [3]. Then the state equation of the truck trailer system in [4] can be presented as follows:   Vm Tm Vm Tm x1 (k) + u(k), x1 (k + 1) = 1 − L l Vm Tm x1 (k) + x2 (k) + 0.2w(k), x2 (k + 1) = (2.31) L x3 (k + 1) = x3 (k) + Vm Tm sin(δ(k)) + 0.1w(k), Vm Tm u(k), z(k) = l

Fig. 2.1 Truck-trailer system

2 Fuzzy Robust H∞ Control with Dynamic Quantization

28

the nonlinear truck-trailer system can be approximately expressed by two local linear systems as: R 1 : x(k + 1) = (A1 + Δ A1 )x(k) + B1 u(k) + E 1 w(k), z(k) = D1 u(k), R 2 : x(k + 1) = (A2 + Δ A2 )x(k) + B2 u(k) + E 2 w(k), z(k) = D2 u(k),

(2.32)

T where x(k) = x1 (k) x2 (k) x3 (k) , the R 1 denotes a region around x2 (k) + Vm Tm m Tm x1 (k) = 0, and R 2 denotes a region around x2 (k) + V2L x1 (k) = ± 179.997π , 2L 180 0.01 and d = L Tm [4]: ⎡

⎡ ⎤ ⎤ 1 − VmLTm 0 0 0 0 V T 1 0⎦ , A2 = ⎣ mL m 1 0⎦ , =⎣ d Vm2 Tm2 Vm Tm 1 d Vm Tm 1 2L

T

T Vm Tm = E 2 = 0 0.2 0.1 , B1 = B2 = 0 0 , l



T = 0 0 0.0018 , Y11 = Y12 = VmLTm 0 0 , Vm Tm = D2 = . l 1−

A1 E1 X1 D1

Vm Tm L Vm Tm L Vm2 Tm2 2L

By setting the system parameters in this example to be the same as [4], and substituting these parameters into the above known parameter matrices, one can get: ⎡

⎤ ⎡ 1.25 0 0 1.25 0 1 A1 = ⎣− 0.25 1 0⎦ , A2 = ⎣− 0.25 0.1875 − 1.5 1 0.0031 − 0.025

T E 1 = E 2 = 0 0.2 0.1 , B1 = B2 = − 0.4286 0

T

X x = 0 0 0.0018 , Y11 = Y12 = − 0.25 0 0 , D1 = D2 = − 0.4286,

⎤ 0 0⎦ , 1

T 0 ,

and the membership functions can be described as: ρ1 (k) =

sin(δ(k)) , ρ2 (k) = 1 − ρ1 (δ(k)). δ(k)

(2.33)

In this simulation example, set the performance index γ = 1, quantization range Mu = 100 and quantization error bound Δu = 0.1 of dynamic quantizer. According to the two-step design method proposed in Algorithm 2.1, one gets the scalar variables τu = 232.9847, νu1 = 0.0468,

νu2 = 0.0468, and the robust controller gain K = 2.4138 − 1.2604 0.0966 .

2.1 Robust Control with Input Quantization Fig. 2.2 Response of the system state x(k)

29

2 x1 (k) x2 (k) x3 (k)

1 0 -1 -2 -3 -4 0

Fig. 2.3 Response of the control input u(k)

20

40

k

60

80

100

80

100

0.3 0.2 0.1 0 -0.1 -0.2 0

20

40

k

60



T Define the initial condition x(0) = 0 0 0 and the disturbance noise signal w(k) = 0.5e−0.1k cos(0.1k). Then, Fig. 2.2 presents the dynamic response of system state x(k). Figure 2.3 shows the response of the control input signal u(k). Figure 2.7 presents the dynamic quantizer’s parameter μu (k). Through the adjusting rule in Remark 2.2, it shows that the adaptive adjustment of dynamic quantizer’s parameters is related to the control input u(k). Figure 2.4 shows the response of the quantized input signal Q(u(k)). Figure 2.5 presents the response of the quantizau(k). Figure 2.6 presents the real-time values of the ratio of tion !error Q(u(k)) − k k T T ζ =0 z (ζ )z(ζ )/ ζ =0 w (ζ )w(ζ ). Given the disturbance noise signal w(k), this ratio eventually tends to stabilize, and the convergence value is lower than the performance index γ = 1. In conclusion, these simulation results show that the proposed two-step design method is robust and can ensure that the quantized fuzzy closed-loop system (2.10) satisfies the prescribed H∞ performance.

30

2 Fuzzy Robust H∞ Control with Dynamic Quantization

Fig. 2.4 Response of the quantized input signal Q(u(k))

0.3 0.2 0.1 0 -0.1 -0.2 0

Fig. 2.5 Response of the quantization error Q(u(k)) − u(k)

20

40

k

60

80

100

80

100

80

100

0.02 0.015 0.01 0.005 0 -0.005 -0.01 0

Fig. !2.6 History of  k k T T ζ =0 z (ζ )z(ζ )/ ζ =0 w (ζ )w(ζ )

20

40

k

60

0.3 0.25 0.2 0.15 0.1 0.05 0 0

20

40

k

60

2.2 Non-fragile Control with State Quantization Fig. 2.7 Dynamic quantizer’s parameter μu (k)

31

0.4

0.3

0.2

0.1

0 0

20

40

k

60

80

100

2.2 Non-fragile Control with State Quantization In this section, a new two-step strategy is proposed to study the non-fragile H∞ control problem of discrete-time fuzzy systems, which can directly provide a simpler and more intuitive design for the non-fragile controller and dynamic quantizer based on the proposed conditions, and make the closed-loop system meet the same specified H∞ performance index as the one without quantization.

2.2.1 Problem Formulation Consider a class of T–S fuzzy models as follows: Plant Rule i th : IF ε1 (k) is M1i , ε2 (k) is M2i , and, . . ., and εd (k) is Mdi , THEN x(k + 1) = Ai x(k) + Bi u(k) + E i w(k), z(k) = Ci x(k) + Di u(k) + Fi w(k),

(2.34)

where x(k) ∈ Rn x and u(k) ∈ Rn u represent the state variable, control input, respectively; w(k) ∈ Rn w denotes the noise signal in l2 [0, ∞); z(k) ∈ Rn z denotes the controlled output. Matrices Ai ∈ Rn x ×n x , Bi ∈ Rn x ×n u , Ci ∈ Rn z ×n x , Di ∈ Rn z ×n u , E i ∈ Rn x ×n w , and Fi ∈ Rn z ×n w , i = 1, 2, …, r represent the constant matrices of appropriate dimensions; ε1 (k), ε2 (k), …, εd (k) represent the premise variables; M pi , i = 1, 2, …, r , p = 1, 2, …, d represents the fuzzy sets; r is the number of fuzzy rules. As in Sect. 2.1, the T–S fuzzy model (2.34) can be deduced as follows: x(k + 1) = A(ρ)x(k) + B(ρ)u(k) + E(ρ)w(k), z(k) = C(ρ)x(k) + D(ρ)u(k) + F(ρ)w(k),

(2.35)

2 Fuzzy Robust H∞ Control with Dynamic Quantization

32

where A(ρ) = C(ρ) = E(ρ) =

r  i=1 r  i=1 r  i=1

ρi (ε(k))Ai , ρi (ε(k))Ci , ρi (ε(k))E i ,

B(ρ) = D(ρ) = F(ρ) =

r  i=1 r  i=1 r 

ρi (ε(k))Bi , ρi (ε(k))Di , ρi (ε(k))Fi ,

i=1

 in which ρi (ε(k)) with ρi (ε(k)) ≥ 0 and ri=1 ρi (ε(k)) = 1 is the membership function defined in (2.3) and (2.4). In this section, it is assumed that the system state is available and needs to be quantized by the quantizer before being sent to the data channel, then the dynamic quantizer which is the same as the one in (2.7) is adopted to quantize the system state x(k). According to the definition in Sect. 2.1, the quantized state signal can be described as:   x(k) . (2.36) Q(x(k)) = μx (k)q μx (k) Due to the inaccuracy of actuator implementation process, environment change, and other factors, the controller gain may have some uncertainties to a certain extent. Therefore, the non-fragile controller based on state feedback can be described as follows: (2.37) u(k) = (K + Δ K )Q(x(k)), where K represents the controller gain, Δ K = K 1 Δk (k)K 2 denotes gain uncertainty, K 1 and K 2 are known matrices, and ΔkT (k)Δk (k) ≤ I is an uncertain term. Substituting (2.36) and (2.37) into (2.35), the quantized fuzzy closed-loop system can be given as follows:

x(k + 1) = A(ρ) + B(ρ)K + B(ρ)Δ K x(k) + B(ρ)K

+ B(ρ)Δ K ex (k) + E(ρ)w(k),

(2.38) z(k) = C(ρ) + D(ρ)K + D(ρ)Δ K x(k) + D(ρ)K

+ D(ρ)Δ K ex (k) + F(ρ)w(k),



− μx(k) . where ex (k) = μx (k) q μx(k) x (k) x (k) The quantized non-fragile state feedback H∞ control problem considered in this section is to design the non-fragile state feedback controller in (2.37) and the dynamic quantizer in (2.36) such that

2.2 Non-fragile Control with State Quantization

33

(1) The quantized fuzzy closed-loop system in (2.38) is asymptotically stable when w(k) = 0. H∞ perfor(2) The quantized fuzzy closed-loop system in (2.38) has   a prescribed T mance γ , i.e., under the initial condition x(0) = 0, ∞ z (k)z(k) < γ2 ∞ k=0 k=0 w T (k)w(k) is satisfied for any nonzero w(k) ∈ l2 [0, ∞). Remark 2.3 For the quantized closed-loop system without uncertainty, [5–8] proposed some effective two-step methods based on system structure constraints to ensure H∞ performance, that is, the system matrices, F = 0 [5], or D = 0 [6, 7], or D = 0 and F = 0 [8]. Once D = 0 and F = 0, it will generate a product term between ex (k) and w(k) [5–8], that is, the design of the quantizer parameter μx (k) may depend on w(k), which will increase the design difficulty. It means that these results are not applicable to system (2.38) synchronously with the control input signal and the external disturbance in z(k). Therefore, compared with the existing results [5–8], the design strategy proposed in this section does not have these constraints and is more general.

2.2.2 Main Results Theorem 2.3 Based on the quantized fuzzy closed-loop system (2.38), given a scalar γ > 0, for known quantization range bound Mx and quantization error Δx , if there exist matrix P > 0, scalars σx > 0 and τx > 0, such that the following matrix inequality holds:

χ1T Pχ1 + χ2T χ2 + τx χ3T χ3 − diag{0, I, 0} − diag{P, 0, γ 2 I } < 0, (2.39) where

χ1 = A(ρ) + B(ρ)(K + Δ K ) B(ρ)(K + Δ K ) E(ρ) ,

χ2 = C(ρ) + D(ρ)(K + Δ K ) D(ρ)(K + Δ K ) F(ρ) ,

√ χ3 = 2 σ x Δ x 0 0 ,

and the online adjustment strategy of dynamic quantizer’s parameter μx (k) is given as: √ √ (2.40) σx x(k) ≤ μx (k) ≤ 2 σx x(k) . Then, through the online adjustment strategy (2.40), the given quantized feedback controller (2.37) can guarantee that the quantized fuzzy closed-loop system (2.38) is asymptotically stable with the prescribed H∞ performance γ . And the quantizer √ range Mx is set to be large enough, that is, Mx > 1/ σx . Proof Substituting (2.36) into (2.6), we can obtain that:

34

2 Fuzzy Robust H∞ Control with Dynamic Quantization

      q x(k) − x(k)  ≤ Δx ,  μx (k) μx (k) 

(2.41)

where x(k)/μx (k) ≤ Mx . If the dynamic parameter μx (k) of the quantizer (2.36) is designed as (2.40), one has that: 1 1 √ μx (k) ≤ x(k) ≤ √ μx (k) < Mx μx (k). 2 σx σx

(2.42)

Consider the homogeneity property of Euclidean norm and combine the equation



x(k) x(k) ex (k) = μx (k) q μx (k) − μx (k) . Then the following condition can be obtained, √ ex (k) ≤ 2 σx Δx x(k) . (2.43)

T Defining ζ (k) = x T (k) exT (k) w T (k) , the following conclusion can be obtained:

(2.44) ζ T (k) χ3T χ3 − diag{0, I, 0} ζ (k) ≥ 0. If the matrix inequality (2.39) is satisfied, then considering the variable ζ (k) = 0, the following inequality can be guaranteed

ζ T (k) Ξ1 + Ξ2 ζ (k) < 0, with

(2.45)

Ξ1 = χ1T Pχ1 + χ2T χ2 − diag{P, 0, γ 2 I },

Ξ2 = τx χ3T χ3 − diag{0, I, 0} ,

where the scalar τx > 0. Combining (2.44) with scalar σx > 0 can give ζ T (k)Ξ1 ζ (k) < 0. Next, constructing the Lyapunov function as V (x(k)) = x T (k)P x(k), P > 0, one can get V (x(k + 1)) − V (x(k)) + z T (k)z(k) − γ 2 w T (k)w(k) = ζ T (k)Ξ1 ζ (k) < 0, (2.46) which shows that the H∞ performance γ can be guaranteed for the quantized fuzzy closed-loop system (2.38) [9, 10]. The proof is completed. It should be noted that the performance condition (2.39) in Theorem 2.3 adds some additional terms to the standard H∞ performance analysis criterion, such as Ξ2 , B(ρ)K , and D(ρ)K , which reflects the impact of the quantization error. In [1, 11], based on the one-step strategy, the controllers (filters) are designed by considering the quantization error (those additional terms), but compared with the controllers (filters) without quantization error, the system performance may be degraded. In Theorem 2.3, the controller and quantizer parameters are designed by a two-step method, and the H∞ performance analysis conditions in (2.39) are applied to the

2.2 Non-fragile Control with State Quantization

35

quantized feedback H∞ control design. This method can ensure that the quantized closed-loop system (2.38) has the same performance as the one without quantization. Theorem 2.4 Based on the quantized fuzzy closed-loop system (2.38), given a scalar γ > 0, for known quantization range bound Mx and quantization error Δx , if there exist matrices K , N , G, and Q > 0, scalar ηx > 0 such that the following matrix inequalities hold: M x > ηx , (2.47) ⎡

Q − G − GT ∗ ∗ 2 ⎢ I ∗ 0 − γ ⎢ ⎢ A(ρ)G + B(ρ)N + B(ρ)Δ K G E(ρ) −Q Λ(ρ) = ⎢ (2.48) ⎢ C(ρ)G + D(ρ)N + D(ρ)Δ K G F(ρ) 0 ⎢ ⎣ 0 0 (B(ρ)(K + Δ K ))T 2Δx G 0 0 ⎤ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ ∗ ∗ ∗ ⎥ ⎥ < 0, −I ∗ ∗ ⎥ ⎥ ∗ ⎦ (D(ρ)(K + Δ K ))T − ηx I 0 0 − ηx I and the online adjustment strategy of dynamic quantizer’s parameter μx (k) is given as: 1 2 x(k) ≤ μx (k) ≤ x(k) . (2.49) ηx ηx Then, through the online adjustment strategy (2.49), the given quantized feedback controller (2.37) can guarantee that the quantized fuzzy closed-loop system (2.38) is asymptotically stable with the prescribed H∞ performance γ . Proof By using Schur complement (Lemma 1.1) for the matrix inequality (2.39) in Theorem 2.3, the following inequality can be obtained ⎡

⎤ ∗ ∗ ∗ ∗ ⎥ ⎥ ∗ ∗ ⎥ ⎥ < 0. ∗ ∗ ⎥ ⎥ −I ∗ ⎦ 0 − τx1σx I (2.50) and P = Q −1 , the inequality (2.50) can be rewritten

−P ∗ ∗ ∗ ⎢ 0 −τ I ∗ ∗ x ⎢ ⎢ 0 0 −γ 2 I ∗ ⎢ ⎢ A(ρ) + B(ρ)(K + Δ K ) B(ρ)(K + Δ K ) E(ρ) −P −1 ⎢ ⎣C(ρ) + D(ρ)(K + Δ K ) D(ρ)(K + Δ K ) F(ρ) 0 2Δx 0 0 0 By defining ηx = τx = as

√1 σx

2 Fuzzy Robust H∞ Control with Dynamic Quantization

36

⎡

−Q −1 ∗ ∗ ⎢ I ∗ 0 −η x ⎢ ⎢ 0 0 −γ 2 I ⎢ ⎢ A(ρ) + B(ρ)(K + Δ K ) B(ρ)(K + Δ K ) E(ρ) ⎢ ⎣C(ρ) + D(ρ)(K + Δ K ) D(ρ)(K + Δ K ) F(ρ) 0 0 2Δx

∗ ∗ ∗ −Q 0 0

⎤ ∗ ∗ ∗ ∗ ⎥ ⎥ ∗ ∗ ⎥ ⎥ < 0. (2.51) ∗ ∗ ⎥ ⎥ −I ∗ ⎦ 0 −ηx I

Set a nonsingular matrix Υ which is expressed as ⎡

GT ⎢ 0 ⎢ ⎢ 0 Υ =⎢ ⎢ 0 ⎢ ⎣ 0 0

0 0 0 0 I 0

0 I 0 0 0 0

0 0 I 0 0 0

0 0 0 I 0 0

⎤ 0 0⎥ ⎥ 0⎥ ⎥, 0⎥ ⎥ 0⎦ I

and perform the congruence transformation with Υ to (2.51). Further, based on the following fact −(G − Q)T Q −1 (G − Q) ≤ 0, where Q is a symmetric matrix and Q > 0, which means that −G T Q −1 G ≤ −G T − G + Q, the inequality (2.48) is obtained by defining a new matrix variable N = K G. The proof is completed. It should be pointed out that, unlike the standard LMIs that cannot be solved directly, because N = K G, (2.48) of Theorem 2.4 is a bilinear matrix inequality (BLMI) on matrix variables G and K . Next, the following algorithm based on Theorem 2.4 and Lemma 1.2 is given, which proposes a new two-step strategy to realize the design of quantized non-fragile feedback controller (2.37) with adjusting strategy (2.49). Algorithm 2.2 For the quantized fuzzy closed-loop system (2.38), it is asymptotically stable under the given H∞ performance index γ : Step 1. Finding the matrix variables Q > 0, G, and N satisfying Λi < 0 for i = 1, . . . , r , where ⎡ ⎤ Q − G − GT ∗ ∗ ∗ ∗ ⎢ ∗ ∗ ∗ ⎥ 0 −γ 2 I ⎢ ⎥ T T ⎢ ∗ ∗ ⎥ Λi = ⎢ Ai G + Bi N E i −Q + α Bi K 1 K 1 Bi ⎥. ⎣ Ci G + Di N Fi α Di K 1 K 1T BiT −I + α Di K 1 K 1T DiT ∗ ⎦ 0 0 0 −α I K2G (2.52) Step 2. Solving the following LMI (2.53) with Q, G, N obtained by Step 1, γ and K = N G −1 (Λi < 0, with i = 1, . . . , r means that the matrix G is nonsingular), and (2.47) can determine the value of scalar variable ηx :

2.2 Non-fragile Control with State Quantization

37

⎡

⎤ ∗ ∗ ∗ ∗ ∗ Q − G − GT ∗ 2 ⎢ 0 −γ I ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ Ai G + Bi N E i −Q + α Bi K 1 K T B T ∗ ∗ ∗ ∗ ⎥ 1 i ⎢ ⎥ ⎢ Ci G + Di N Fi α Di K 1 K 1T BiT −I + α Di K 1 K 1T DiT ∗ ∗ ∗ ⎥ ⎢ ⎥ < 0. ⎢ K2 G 0 0 0 −α I ∗ ∗ ⎥ ⎢ ⎥ ⎣ (Di K )T K 2T −ηx I ∗ ⎦ 0 0 (Bi K )T 2Δx G 0 0 0 0 0 −ηx I

(2.53) According to the online adjustment strategy in (2.49), the dynamic quantizer’s parameter μx (k) can be adjusted by the scalar variable ηx obtained in Step 2. Then, through the online adjustment strategy (2.49), the given quantized feedback controller (2.37) can guarantee that the quantized fuzzy closed-loop system (2.38) is asymptotically stable with the prescribed H∞ performance γ . Remark 2.4 In Algorithm 2.2, according to the standard H∞ controller design conditions of discrete-time systems, Λi < 0 in Step 1 can ensure the desired system performance, that is, the algorithm can provide a quantized state feedback control design with the same one without quantization. Remark 2.5 In Step 1, the state feedback controller gain K is obtained by solving the standard LMI Λi < 0. In Step 2, matrix inequalities (2.47) and (2.53) determine the quantizer parameter ηx based on standard LMIs by using the matrix variables Q, G, and N obtained in Step 1. Compared with [5–8], the results show that the proposed quantized control design strategy can be implemented in the framework of LMI more simply and intuitively. Remark 2.6 The quantized signal can be obtained by determining the dynamic quantizer’s parameter μx (k) as a unique value and transformed it into a communication channel. In this section, the adjustment rule for updating the parameter μx (k) of the quantizer on both sides of the encoder/decoder is the same as [1]: The adjusting rule: ⎧ 1 v −v ⎪ ⎨floor (2ϑx x(k) × 10 ) × 10 , 0 ≤ ϑx x(k) < 2 , 1 μx (k) = 1, ≤ ϑx x(k) < 1, 2 ⎪ ⎩ floor (2ϑx x(k)) , 1 ≤ ϑx x(k) ,   where v = min v ∈ N+ | (2ϑx x(k) × 10v ) > 1 and function floor(θ ) represents the maximum integer that does not exceed θ . It is worth noting that, in Algorithm 2.2, the method of updating μx (k) is to apply the above adjustment strategy together with ϑx = η1x .

2 Fuzzy Robust H∞ Control with Dynamic Quantization

38

2.2.3 Simulation Example In this section, a numerical example shows the effectiveness of the proposed design method. Consider a discrete-time system in the form of (2.34), then related parameters are given as follows: ⎡

⎤ ⎡ ⎤ 1.1159 − 7.1340 0.0000 1.3254 − 8.8267 0.0000 1.0211 − 0.0368⎦ , A2 = ⎣0.0808 1.1163 − 0.0510⎦ , A1 = ⎣0.0808 0.0222 0.6245 1.0880 0.0002 0.7534 0.9967 ⎡ ⎤ ⎡ ⎤ 0.2415 0.0000 0.0000 0.2415 0.0000 0.0000 B1 = ⎣0.0000 0.3444 0.0000⎦ , B2 = ⎣0.0000 0.3444 0.0000⎦ , 0.0000 0.0000 0.2169 0.0000 0.0000 0.2169





C1 = C2 = 1 0 0 , D1 = D2 = 0.2 0 0 , E 1T = E 2T = 0 0 3 , F1 = F2 = 0.1, and the membership functions are considered as follows: ρ1 (k) =

|sin(k)| , ρ2 (k) = 1 − ρ1 (k). 8

In (2.37), the known parameters are assumed to be ⎡

⎤ 2.0000 K 1 = ⎣1.0000⎦ , 0.1000

⎡

⎤ 0.1000 K 2T = ⎣2.0000⎦ . 0.1000

For this numerical example, choose γ = 3, and set the quantizer’s range Mx = 100 and error bound Δx = 0.01. Then, by applying the feedback controller and quantizer parameters designed in Algorithm 2.2, it can be acquired that the scalar variable ηx = 92.8149 and the quantized feedback controller gain ⎡

− 4.9991 K = ⎣− 0.2334 0.0146

0.1191 − 1.9222 − 4.3863

⎤ 0.0024 0.1293 ⎦ . − 4.6732

In order to further illustrate the effectiveness of the designed quantized feedback controller, the simulation results are carried out by using MATLAB with

T the initial condition x(0) = 0 0 0 , and the disturbance noise signal w(k) = 0.5e−0.2k cos(0.2k). Then, Fig. 2.8 shows the dynamic response of the system state x(k); Fig. 2.9 presents the response ! of the control input signal u(k); Fig. 2.10 shows k k T T the real-time values of the ratio of j=0 z ( j)z( j)/ j=0 w ( j)w( j), it can be observed from Fig. 2.10 that the convergence value of the ratio is 0.1512 which is below the given performance index γ = 3; Fig. 2.11 denotes the dynamic quantizer’s

2.2 Non-fragile Control with State Quantization Fig. 2.8 Response of the system state x(k)

39

1.5 x1 (k) x2 (k) x3 (k)

1

0.5

0

-0.5 0

Fig. 2.9 Response of the control input u(k)

10

20

k

30

40

50

2 u1 (k) u2 (k) u3 (k)

0 -2 -4 -6 -8 0

Fig. !2.10 History of k k T T j=0 z ( j)z( j)/ j=0 w ( j)w( j)

10

20

k

30

40

50

40

50

0.16 0.15 0.14 0.13 0.12 0.11 0.1 0.09 0

10

20

k

30

2 Fuzzy Robust H∞ Control with Dynamic Quantization

40 Fig. 2.11 Dynamic quantizer’s parameter μx (k)

30 25 20 15 10 5 0 0

10

20

k

30

40

50

parameter μx (k), which is obtained by the adjusting rule in Remark 2.6. The simulation results of numerical example show that the two-step design method proposed in this section can ensure that the quantized fuzzy control system meets the specified performance.

2.3 Resilient Control with Input and Output Quantization In this section, the resilient H∞ controller for the quantized closed-loop system will be designed by descriptor representation approach in two cases. The measurement output signal and control input signal will be quantized by dynamic quantizers, and the conditions for the resilient controller and dynamic quantizers’ parameters will be given by LMIs.

2.3.1 Problem Formulation For the investigated quantized output feedback control problem in this section, we consider a class of nonlinear systems appropriated by the following T–S fuzzy dynamic model, in which the ith rule is described as follows: Plant Rule i th : IF ε1 (t) is M1i , ε2 (t) is M2i , and, . . ., and εd (t) is Mdi , THEN x(t) ˙ = (Ai + Δ Ai )x(t) + Bi Q(u(t)) + E i w(t), z(t) = (Ci + ΔCi )x(t) + Di Q(u(t)) + Fi w(t), y(t) = (L i + Δ Li )x(t) + Hi w(t),

(2.54)

where x(t) ∈ Rn x is the state variable; u(t) ∈ Rn u is the control input; z(t) ∈ Rn z is the controlled output; y(t) ∈ Rn y is the measurement output; w(t) ∈ Rn w is the

2.3 Resilient Control with Input and Output Quantization

41

disturbance signal that is assumed to be the arbitrary signal in L2 [ 0, ∞). ε(t) = [ ε1 (t), ε2 (t), . . . , εd (t) ], where ε p (t), p = 1, 2, . . . , d are premise variables and measurable; M pi , i = 1, 2, . . . , r, p = 1, 2, . . . , d are the fuzzy sets; r is the number of fuzzy rules. Ai ∈ Rn x ×n x , Bi ∈ Rn x ×n u , E i ∈ Rn x ×n w , Ci ∈ Rn z ×n x , Di ∈ Rn z ×n u , Fi ∈ Rn z ×n w , L i ∈ Rn y ×n x , and Hi ∈ Rn y ×n w for i = 1, 2, . . . , r are known system matrices. Δ Ai ∈ Rn x ×n x , ΔCi ∈ Rn z ×n x , and Δ Li ∈ Rn y ×n x for i = 1, 2, . . . , r are unknown real norm-bounded matrices, which represent time-varying parameter uncertainties. And the uncertainties can be given as the following form: ⎛

⎞ ⎛ ⎞ Δ Ai X Ai ⎝ΔCi ⎠ = ⎝ X Ci ⎠ Δx (t)Y, Δ Li X Li

(2.55)

where X Ai , X Ci , X Li , and Y are known real constant matrices with appropriate dimensions, Δx (t) is an unknown real matrix satisfying ΔTx (t)Δx (t) ≤ I . Then according to Sect. 1.1, the T–S fuzzy model (2.54) is inferred as follows: ˜ x(t) ˙ = A(ρ)x(t) + B(ρ)Q(u(t)) + E(ρ)w(t), ˜ z(t) = C(ρ)x(t) + D(ρ)Q(u(t)) + F(ρ)w(t),

(2.56)

˜ y(t) = L(ρ)x(t) + H (ρ)w(t), where ˜ ˜ A(ρ) = A(ρ) + Δ A (ρ), C(ρ) = C(ρ) + ΔC (ρ), A(ρ) = C(ρ) = L(ρ) = Δ A (ρ) = ΔC (ρ) = Δ L (ρ) =

r  i=1 r  i=1 r  i=1 r  i=1 r  i=1 r 

ρi (ε(t))Ai , ρi (ε(t))Ci , ρi (ε(t))L i ,

B(ρ) = D(ρ) = H (ρ) =

r  i=1 r  i=1 r 

ρi (ε(t))Bi , ρi (ε(t))Di ,

˜ L(ρ) = L(ρ) + Δ L (ρ), E(ρ) = F(ρ) =

r 

ρi (ε(t))E i ,

i=1 r 

ρi (ε(t))Fi ,

i=1

ρi (ε(t))Hi ,

i=1

ρi (ε(t))X Ai Δx (t)Y = X A (ρ)Δx (t)Y, ρi (ε(t))X Ci Δx (t)Y = X C (ρ)Δx (t)Y, ρi (ε(t))X Li Δx (t)Y = X L (ρ)Δx (t)Y.

i=1

The static output feedback controller considered in this chapter is given as follows u(t) = (K + Δ K )Q(y(t)),

(2.57)

42

2 Fuzzy Robust H∞ Control with Dynamic Quantization

where Q(y(t)) is the input of the controller, K is the controller gain to be determined. Δ K is parameter variation matrix and assumed to be the form Δ K = RΔk (t)W,

(2.58)

where R and W are known real constant matrices with appropriate dimensions, and Δk (t) describes the system uncertainty which is assumed to satisfy ΔkT (t)Δk (t) ≤ I . In (2.56) and (2.57), Q(y(t)) and Q(u(t)) represent quantized outputs of y(t) and u(t) by two dynamic quantizers, respectively, and are expressed as 

Q(y(t)) = μ y (t)q 

Q(u(t)) = μu (t)q

 y(t) , μ y (t)

(2.59)

 u(t) . μu (t)

(2.60)

In this chapter, for the quantized resilient control of the fuzzy system (2.56), the considered dynamic quantizer with general form is defined as the one given in Sect. 2.1, and assumed to be the one-parameter family of quantizer as Q(ω(t)) = , ω = y, u, which comprises a dynamic parameter μω (t) and a static μω (t)q μω(t) ω (t) quantizer q(·) and satisfies the following conditions: Q(ω(t)) − ω(t) ≤ μω (t)Δω , i f ω(t) ≤ μω (t)Mω ,

(2.61)

Q(ω(t)) > μω (t)Mω − μω (t)Δω , i f ω(t) > μω (t)Mω .

(2.62)

When the quantizer is unsaturated, the condition (2.61) provides a bound for the quantization error. The condition (2.62) gives a way for detecting the possibility of saturation. Δω represents the quantization error bound and Mω denotes the range of the quantizer q(·), respectively. In fact, it is not hard to notice that the range and the quantization error of this class of dynamic quantizers are μω (t)Mω and μω (t)Δω , respectively. That is to say, we can adjust the dynamic parameter to realize a suitable range and quantization error, and guarantee the desired system performance.

2.3.2 Resilient H∞ Controller Design with Quantization In this section, the output feedback resilient H∞ control design problem for the fuzzy system (2.56) is addressed with the effect of quantized measurement output Q(y(t)) and quantized control input Q(u(t)). By structuring descriptor representation, this section will propose two different design conditions for the quantized output feedback H∞ control. Case A: Combining (2.56) and (2.57) can give

2.3 Resilient Control with Input and Output Quantization

˜ x(t) ˙ = A(ρ)x(t) + B(ρ) (Q(u(t)) − u(t)) + B(ρ)u(t) + E(ρ)w(t), ˜ z(t) = C(ρ)x(t) + D(ρ) (Q(u(t)) − u(t)) + D(ρ)u(t) + F(ρ)w(t), ˜ y(t) = L(ρ)x(t) + H (ρ)w(t), u(t) = K˜ (Q(y(t)) − y(t)) + K˜ y(t),

43

(2.63)

where K˜ = K + Δ K . Then it can be rewritten into the following form based on the definitions of Q(y(t)) and Q(u(t)) in (2.59) and (2.60), respectively,     u(t) u(t) ˜ − + B(ρ)u(t) + E(ρ)w(t), x(t) ˙ = A(ρ)x(t) + B(ρ)μu (t) q μu (t) μu (t)     u(t) u(t) ˜ − + D(ρ)u(t) + F(ρ)w(t), z(t) = C(ρ)x(t) + D(ρ)μu (t) q μu (t) μu (t) ˜ y(t) = L(ρ)x(t) + H (ρ)w(t),     y(t) y(t) ˜ u(t) = K μ y (t) q − + K˜ y(t). μ y (t) μ y (t) (2.64) Further, it can be organized as ˜ x(t) ˙ = A(ρ)x(t) + B(ρ)eu (t) + B(ρ)u(t) + E(ρ)w(t), ˜ z(t) = C(ρ)x(t) + D(ρ)eu (t) + D(ρ)u(t) + F(ρ)w(t), ˜ y(t) = L(ρ)x(t) + H (ρ)w(t), ˜ u(t) = K e y (t) + K˜ y(t),

(2.65)



ω(t) − , ω = y, u. To cast the system into a descripwhere eω (t) = μω (t) q μω(t) μω (t) ω (t) tor form, the controller u(t) in (2.57) is converted into following

˜ + H (ρ)w(t) − u(t). 0 × u(t) ˙ = K˜ e y (t) + K˜ L(ρ)x(t)

(2.66)

As a result, from (2.65) and (2.66), we have the descriptor representation as follows: S A η(t) ˙ = A(ρ)η(t) + Me y (t) + B(ρ)eu (t) + E(ρ)w(t), z(t) = C(ρ)η(t) + D(ρ)eu (t) + F(ρ)w(t), y(t) = L(ρ)η(t) + H(ρ)w(t),

T where η(t) = x T (t) u T (t) and

(2.67)

2 Fuzzy Robust H∞ Control with Dynamic Quantization

44

&

' & ' & ' ˜ 0 I 0 A(ρ) B(ρ) SA = , A(ρ) = ˜ ˜ , M= ˜ , 0 0 K K L(ρ) −I & ' & '

E(ρ) B(ρ) ˜ B(ρ) = , E(ρ) = ˜ , C(ρ) = C(ρ) D(ρ) , 0 K H (ρ)

˜ D(ρ) = D(ρ), F(ρ) = F(ρ), L(ρ) = L(ρ) 0 , H(ρ) = H (ρ). The quantized static output feedback H∞ control problem considered in this section is to design the output feedback controller gain in (2.57) and the dynamic parameters in (2.59) and (2.60) such that (1) The closed-loop system (2.67) is asymptotically stable when w(t) = 0. (2) The closed-loop system (2.67) has a prescribed ( ∞ attenuation, ( ∞ level γ of H∞ noise i.e., under the initial condition x(0) = 0, 0 z T (t)z(t)dt < γ 2 0 w T (t)w(t)dt is satisfied for any nonzero w(t) ∈ L2 [ 0, ∞). To implement the quantized static output feedback H∞ control design for the closed-loop system (2.67), the following condition is firstly presented to ensure the prescribed H∞ performance. Theorem 2.5 Consider the quantized output feedback closed-loop system (2.67). For given scalar γ > 0, quantizers’ ranges Mω and error bounds Δω , ω = y, u, if there exist matrix P, positive scalars αω and δω , ω = y, u, such that the following matrix inequalities hold: (2.68) PS A = STA P T ≥ 0, Mω >

1 , ω = y, u, δω

⎤ H e {P A(ρ)} ∗ ∗ ⎣ Ξ2 ∗ ⎦ < 0, Ξ1 Ξ4 Ξ5 Ξ3

(2.69)

⎡

(2.70)

where

T   Ξ1 = P M P B(ρ) P E(ρ) , Ξ2 = diag −α y I, − αu I, − γ 2 I , ⎡ ⎡ ⎤ ⎤ 0 D(ρ) C(ρ) F(ρ) Ξ3 = ⎣2α y δ y Δ y L(ρ)⎦ , Ξ4 = ⎣0 0 2α y δ y Δ y H(ρ)⎦ , 2αu δu Δu N 0 0 0  

0 I , Ξ5 = diag −I, −α y I, −αu I , N = with on-line adjusting strategy for parameters μω (t) as: δω ω(t) ≤ μω (t) ≤ 2δω ω(t) , ω = y, u.

(2.71)

2.3 Resilient Control with Input and Output Quantization

45

Then, the controller (2.57) and the quantizers (2.59) and (2.60) with the adjusting strategy (2.71) can ensure that the closed-loop system (2.67) achieves the prescribed H∞ performance γ . Proof If the range M y in (2.61) and dynamic parameter μ y (t) of quantizer (2.59) satisfy (2.69) and (2.71), one has

i.e.,

1 y(t) ≤ μ y (t) ≤ 2δ y y(t) , My

(2.72)

1 μ y (t) ≤ y(t) ≤ M y μ y (t). 2δ y

(2.73)

For the measurement output y(t), based  on the property of the quantizer given in  y(t)  (2.73), it is effortless to obtain that  μ y (t)  ≤ M y from (2.61), then       q y(t) − y(t)  ≤ Δ y .  μ y (t) μ y (t) 

(2.74)

Considering the definition of e y (t) in (2.65) and the homogeneity property of Euclidean norm, and the inequalities (2.72) and (2.74), the following condition can be obtained   * )      e y (t) = μ y (t) q y(t) − y(t)   μ y (t) μ y (t)       y(t) y(t)    = μ y (t) q − (2.75) μ (t) μ (t)  y

y

≤ μ y (t)Δ y ≤ 2δ y Δ y y(t) , i.e., e Ty (t)e y (t) ≤ 4δ 2y Δ2y y T (t)y(t).

(2.76)

T Defining ξ(t) = η T (t) e Ty (t) euT (t) w T (t) , the above inequality can be rewritten as

(2.77) ξ T (t) H a {W2 } − diag { 0, I, 0, 0 } ξ(t) ≥ 0,

where W2 = 2δ y Δ y L(ρ) 0 0 2δ y Δ y H(ρ) . Similarly, combining (2.69) and (2.71), considering the definition of eu (t) in (2.65) and the homogeneity property of Euclidean norm, one gets eu (t) ≤ 2δu Δu u(t) ,

(2.78)

euT (t)eu (t) ≤ 4δu2 Δ2u u T (t)u(t).

(2.79)

i.e.,

2 Fuzzy Robust H∞ Control with Dynamic Quantization

46

Further, the above inequality can be rewritten as

ξ T (t) H a {W3 } − diag { 0, 0, I, 0 } ξ(t) ≥ 0,

(2.80)

where W3 = 2δu Δu N 0 0 0 . Now, choose a Lyapunov function as V (η(t)) = η T (t)PS A η(t), PS A = STA P T ≥ 0,

(2.81)

one gives V˙ (η(t)) + z T (t)z(t) − γ 2 w T (t)w(t) = η T (t)PS A η(t) ˙ + η˙ T (t)STA P T η(t) + z T (t)z(t) − γ 2 w T (t)w(t)

= η T (t)P A(ρ)η(t) + Me y (t) + B(ρ)eu (t) + E(ρ)w(t)

T + A(ρ)η(t) + Me y (t) + B(ρ)eu (t) + E(ρ)w(t) P T η(t) + H a {C(ρ)η(t) + D(ρ)eu (t) + F(ρ)w(t)} − γ 2 w T (t)w(t) ' & H e {P A(ρ)}  ∗  ξ(t) + ξ T (t)H a {W1 } ξ(t), = ξ T (t) Ξ1 diag 0, 0, −γ 2 I

where W1 = C(ρ) 0 D(ρ) F(ρ) . Based on (2.70) and Schur complement (Lemma 1.1), it is easy to obtain +

,

 + α H a {W } − diag { 0, I, 0, 0 } y 2 2 diag 0, 0, −γ I Ξ1 (2.82)

+ αu H a {W3 } − diag { 0, 0, I, 0 } + H a {W1 } < 0.

H e {P A(ρ)}

∗



Then based on inequalities (2.77) and (2.80), scalars α y > 0 and αu > 0, the following inequality )& ξ (t) T

' * H e {P A(ρ)}  ∗  + H a {W1 } ξ(t) < 0, Ξ1 diag 0, 0, −γ 2 I

(2.83)

is obtained, which means that V˙ (η(t)) + z T (t)z(t) − γ 2 w T (t)w(t) < 0,

(2.84)

can be guaranteed by the proposed conditions in Theorem 2.5. Further, considering zero initial condition and V (η(∞)) ≥ 0 with the definition of the Lyapunov function (2.81), one has -∞ -∞ T 2 z (t)z(t)dt < γ w T (t)w(t)dt. (2.85) 0

The proof is completed.

0

2.3 Resilient Control with Input and Output Quantization

47

In the following, the design conditions for static output feedback controller gain and dynamic parameters will be given in the form of LMI based on ' & the performance P1 V P2 , where criterion proposed in Theorem 2.5. Let us define the matrix P = 0 P2 T T n x ×n u P1 > 0 is necessary to make sure PS A = S A P ≥ 0 and V ∈ R is a dimension adjustment matrix. The inequality in (2.70) can be expressed as ⎡

⎤ 1 ∗ ∗ ⎣2 3 ∗ ⎦ < 0, 4 5 6

(2.86)

where   , ˜ ˜ H e P1 A(ρ) + V P2 K˜ L(ρ) ∗ = , ˜ P2 K˜ L(ρ) + (P1 B(ρ) − V P2 )T −P2 − P2T

T

T ⎤ ⎡ P2 K˜ V P2 K˜ ⎥ ⎢ ⎥, =⎢ 0 (P1 B(ρ))T ⎣

T

T ⎦ P2 K˜ H (ρ) P1 E(ρ) + V P2 K˜ H (ρ) ⎤ ⎡ ⎡ ⎤ ˜ 0 D(ρ) F(ρ) C(ρ) D(ρ) ⎦, 5 = ⎣0 0 2α y δ y Δ y H (ρ)⎦, ˜ = ⎣2α y δ y Δ y L(ρ) 0 0 0 0 0 2αu δu Δu I     2 = −diag α y I, αu I, γ I , 6 = −diag I, α y I, αu I . +

1

2

4 3

It should be noted that there are still uncertain terms Δ A (ρ), ΔC (ρ), Δ L (ρ), and Δ K , therefore, in the following, we will eliminate these uncertainties. Firstly, according to (2.56), the above inequality can be rewritten as ⎡

⎤ ˜1 ∗ ∗  ⎣2 3 ∗ ⎦ + H e {Π1 Δx (t)Π2 } < 0, ˜ 4 5 6  where

(2.87)

  , ∗ H e P1 A(ρ) + V P2 K˜ L(ρ) , P2 K˜ L(ρ) + (P1 B(ρ) − V P2 )T −P2 − P2T ⎤ ⎡ C(ρ) D(ρ) ⎦, ˜ 4 = ⎣2α y δ y Δ y L(ρ) 0  0 2αu δu Δu I & 'T

T

T  T T ˜ Π1 = P1 X A (ρ) + V P2 K˜ X L (ρ) P2 K X L (ρ) 0 0 0 X C (ρ) 2α y δ y Δ y X L (ρ) 0 ,

Π2 = Y 0 0 0 0 0 0 0 . +

˜1 = 

2 Fuzzy Robust H∞ Control with Dynamic Quantization

48

Then, according to Lemma 1.2, the uncertain terms Δ A (ρ), ΔC (ρ), and Δ L (ρ) can be eliminated, and we can obtain the following inequality ⎡

˜1  ⎢2 ⎢ ⎣ ˜4 7

∗ 3 5 0

∗ ∗ 6 8

⎤ ∗ ∗⎥ ⎥ < 0, ∗⎦ 9

(2.88)

where

T

T , P2 K˜ X L (ρ) P1 X A (ρ) + V P2 K˜ X L (ρ) 7 = , σ1 Y 0 &  T ' 0 X CT (ρ) 2α y δ y Δ y X L (ρ) 8 = , 9 = −diag { σ1 I, σ1 I } . 0 0 0 +

Next, the uncertain term Δ K will be eliminated. The above inequality can be rewritten as ⎡ˆ 1 ⎢ ˆ ⎢ 2 ⎣ ˜4 ˆ7  where

ˆ7  Π3 Π4

∗ ∗ 6 8

⎤ ∗ ∗⎥ ⎥ + H e {Π3 Δk (t)Π4 } < 0, ∗⎦ 9

(2.89)

' ∗ H e {P1 A(ρ) + V P2 K L(ρ)} , P2 K L(ρ) + (P1 B(ρ) − V P2 )T −P2 − P2T ⎡ ⎤ (V P2 K )T (P2 K )T T ⎦, =⎣ 0 (P1 B(ρ)) T T (P1 E(ρ) + V P2 K H (ρ)) (P2 K H (ρ)) ' & (P1 X A (ρ) + V P2 K X L (ρ))T (P2 K X L (ρ))T , = σ1 Y 0

T = (V P2 R)T (P2 R)T 0 0 0 0 0 0 0 0 ,

= W L(ρ) 0 W 0 W H (ρ) 0 0 0 W X L (ρ) 0 .

ˆ1 =  ˆ2 

∗ 3 5 0

&

Then, according to Lemma 1.2, the uncertain term Δ K is eliminated, and the following inequality can be obtained ⎡ ˆ 1 ∗ ⎢ ˆ ⎢ 2 3 ⎢ ⎢ ˜ 4 5 ⎣ ˆ7 0 10 11

⎤ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ 6 ∗ ∗ ⎥ ⎥ < 0, 8  9 ∗ ⎦ 0 12 13

(2.90)

2.3 Resilient Control with Input and Output Quantization

49

where ' ' & 0 0 0 (V P2 R)T (P2 R)T , , 11 = σ2 W 0 σ2 W H (ρ) σ2 W L(ρ) 0 & ' 0 0 = , 13 = −diag { σ2 I, σ2 I } . σ2 W X L (ρ) 0 &

10 = 12

Defining P2 K = T , υω = αω δω , ω = y, u and based on the property of the membership function, the conditions (2.69) and (2.90) can be guaranteed by the following LMIs in Theorem 2.6, which gives the design conditions of the output feedback controller (2.57) and quantizers (2.59) and (2.60) to ensure the H∞ performance γ for the fuzzy system (2.67). Theorem 2.6 For given scalar γ > 0, quantizers’ ranges Mω and error bounds Δω , ω = y, u, if there exist matrices P1 > 0, P2 , and T , positive scalars αω and υω , ω = y, u, σ1 , σ2 satisfying υω Mω > αω , ω = y, u,

(2.91)

⎤ ⎡ ˆ 1i ∗ ∗ ∗ ∗ ⎢ ˆ ∗ ∗ ⎥ ⎥ ⎢ 2i 3 ∗ ⎢ ∗ ⎥ ⎥ < 0, i = 1, 2, . . . , r, ⎢ ˜ 4i 5i 6 ∗ ⎣ ˆ 7i 0 8i 9 ∗ ⎦ 10i 11i 0 12i 13

(2.92)

where ⎡ ' } {P A + V T L ∗ H e 1 i i ˆ 2i = ⎣ ˆ 1i = ,   T L i + (P1 Bi − V P2 )T −P2 − P2T &

⎤ TT (V T )T T 0 ⎦, (P1 Bi ) (P1 E i + V T Hi )T (T Hi )T ⎤

⎡ ⎡ ⎤ Ci 0 Di Di Fi ⎣ ⎦ ˜ 0 4i = 2υ y Δ y L i , 5i = ⎣0 0 2υ y Δ y Hi ⎦ , 0 2υu Δu I 0 0 0 ' & &  T ' T T T 0 , ˆ 7i = (P1 X Ai + V T X Li ) (T X Li ) , 8i = X Ci 2υ y Δ y X Li  σ1 Y 0 0 0 0 ' ' ' & & & 0 0 0 0 0 (V P2 R)T (P2 R)T . , 12i = , 11i = 10i = σ2 W 0 σ2 W Hi σ2 W X Li 0 σ2 W L i 0

Then, the H∞ performance γ can be guaranteed for the fuzzy system (2.67). Moreover, the quantizers (2.59) and (2.60) are designed with on-line adjusting strategy for parameters μω (t), ω = y, u as (2.71) with δω = υω /αω , ω = y, u, and the controller gain K in (2.57) can be obtained by K = P2−1 T.

(2.93)

2 Fuzzy Robust H∞ Control with Dynamic Quantization

50

Case B: Combining (2.56) and (2.57) yields ˜ x(t) ˙ = A(ρ)x(t) + B(ρ) (Q(u(t)) − u(t)) + B(ρ) K˜ Q(y(t)) + E(ρ)w(t), ˜ z(t) = C(ρ)x(t) + D(ρ) (Q(u(t)) − u(t)) + D(ρ) K˜ Q(y(t)) + F(ρ)w(t), ˜ y(t) = L(ρ)x(t) + H (ρ)w(t). (2.94) To cast the system into a descriptor form, the quantized output in (2.59) is converted into the following form     y(t) y(t) ˙ − + y(t) − Q(y(t)). 0 × Q(y(t)) = μ y (t) q μ y (t) μ y (t)

(2.95)

Then with the same definitions of e y (t) and eu (t) in (2.65), one gets ˜ x(t) ˙ = A(ρ)x(t) + B(ρ) K˜ Q(y(t)) + B(ρ)eu (t) + E(ρ)w(t), ˜ z(t) = C(ρ)x(t) + D(ρ) K˜ Q(y(t)) + D(ρ)eu (t) + F(ρ)w(t), ˜ y(t) = L(ρ)x(t) + H (ρ)w(t), ˙ (y(t)) = e y (t) + L(ρ)x(t) ˜ 0×Q + H (ρ)w(t) − Q(y(t)).

(2.96)

Further, from (2.96), the following descriptor representation can be obtained ˙ = A(ρ)ϕ(t) + Me y (t) + B(ρ)eu (t) + E(ρ)w(t), S B ϕ(t) z(t) = C(ρ)ϕ(t) + D(ρ)eu (t) + F(ρ)w(t),

(2.97)

y(t) = L(ρ)ϕ(t) + H(ρ)w(t),

T where ϕ(t) = x T (t) QT (y(t)) and ' & ' ' & ˜

0 A(ρ) B(ρ) K˜ I 0 ˜ , C(ρ) = C(ρ) , M= , A(ρ) = ˜ D(ρ) K˜ , I 0 0 L(ρ) −I ' ' & & E(ρ) B(ρ) , , E(ρ) = D(ρ) = D(ρ), F(ρ) = F(ρ), B(ρ) = H (ρ) 0

˜ L(ρ) = L(ρ) 0 , H(ρ) = H (ρ). &

SB =

Next, we will analyze the H∞ performance of the closed-loop system (2.97) and derive the design conditions for the feedback controller (2.57) and quantizers (2.59) and (2.60). Theorem 2.7 Consider the quantized output feedback closed-loop system (2.97). For given scalar γ > 0, quantizers’ ranges Mω and error bounds Δω , ω = y, u, if there exist matrix P, positive scalars ζω and λω , ω = y, u, such that the following matrix inequalities hold: (2.98) P S B = STB P T ≥ 0,

2.3 Resilient Control with Input and Output Quantization

Mω >



51

ζω , ω = y, u,

⎡

H e {PA(ρ)} ∗ ⎣ ς1 ς2 ς4 ς3

⎤ ∗ ∗ ⎦ < 0, ς5

(2.99)

(2.100)

where

T

)

ζy ζu − I, − I, −γ 2 I 4λ y 4λu ⎤

ς1 = PM PB(ρ) PE(ρ) , ς2 = diag ⎡ ⎡ ⎤ C(ρ) 0 D(ρ) F(ρ) 0 Δ y H(ρ)⎦ , ς3 = ⎣Δ y L(ρ)⎦ , ς4 = ⎣0 ˜ 0 0 0 Δu K N  

ς5 = diag − I, −λ y I, −λu I , N = 0 I ,

* ,

with on-line adjusting strategy for parameters μω (t), ω = y, u as: 1 2 √ ω(t) ≤ μω (t) ≤ √ ω(t) . ζω ζω

(2.101)

Then, the controller (2.57) and the quantizers (2.59) and (2.60) with the adjusting strategy (2.101) can ensure that the closed-loop system (2.97) achieves the prescribed H∞ performance γ .  Proof With (2.99) and (2.101), it is easy to deduce that 21 ζ y μ y (t) ≤ y(t) ≤ M y μ y (t). Then based on the definition of e y (t) in (2.65) and the homogeneity property of Euclidean norm, from (2.61), one has  )   *     e y (t) = μ y (t) q y(t) − y(t)   μ y (t) μ y (t)       y(t)  y(t)  = μ y (t)  q μ (t) − μ (t)  y y

(2.102)

≤ μ y (t)Δ y 2 ≤  Δ y y(t) , ζy

i.e.,

ζy T e (t)e y (t) ≤ Δ2y y T (t)y(t). (2.103) 4 y

Defining φ T (t) = ϕ T (t) e Ty (t) euT (t) w T (t) and considering (2.97), the above inequality can be rewritten as )  * ζy φ T (t) H a {W5 } − diag 0, I, 0, 0 φ(t) ≥ 0, 4

(2.104)

2 Fuzzy Robust H∞ Control with Dynamic Quantization

52

where W5 = Δ y L(ρ) 0 0 Δ y H(ρ) . Similarly, combining (2.99) and (2.101), considering the definition of eu (t) in (2.65) and the homogeneity property of Euclidean norm, one has   2 2   eu (t) ≤ √ Δu u(t) = √ Δu  K˜ Q(y(t)) , ζu ζu

(2.105)

ζu T e (t)eu (t) ≤ Δ2u QT (y(t)) K˜ T K˜ Q(y(t)). 4 u

(2.106)

i.e.,

Based on (2.97), the above inequality can be rewritten as ) *  ζu I, 0 φ(t) ≥ 0, φ T (t) H a {W6 } − diag 0, 0, 4

(2.107)



where W6 = Δu K˜ N 0 0 0 . Now, consider a Lyapunov function as V (ϕ(t)) = ϕ T (t)PS B ϕ(t),

PS B = STB P T ≥ 0,

(2.108)

Then one gets V˙ (ϕ(t)) + z T (t)z(t) − γ 2 w T (t)w(t) = ϕ T (t)PS B ϕ(t) ˙ + ϕ˙ T (t)STB P T ϕ(t) + z T (t)z(t) − γ 2 w T (t)w(t)

= ϕ T (t)P A(ρ)ϕ(t) + Me y (t) + B(ρ)eu (t) + E(ρ)w(t)

T + A(ρ)ϕ(t) + Me y (t) + B(ρ)eu (t) + E(ρ)w(t) P T ϕ(t) + H a {C(ρ)ϕ(t) + D(ρ)eu (t) + F(ρ)w(t)} − γ 2 w T (t)w(t) ' & H e {PA(ρ)} ∗ T   φ(t) + φ T (t)H a {W4 } φ(t), = φ (t) ς1 diag 0, 0, −γ 2 I

where W4 = C(ρ) 0 D(ρ) F(ρ) . Based on (2.100) and Schur complement (Lemma 1.1), it is easy to obtain '  * ) ζy 1 H e {PA(ρ)}  ∗  + H a {W5 } − diag 0, I, 0, 0 ς1 diag 0, 0, −γ 2 I λy 4  * ) ζu 1 H a {W6 } − diag 0, 0, I, 0 + H a {W4 } < 0. + λu 4 (2.109) Then based on inequalities (2.104) and (2.107), scalars λ y > 0 and λu > 0, the following inequality is obtained, &

2.3 Resilient Control with Input and Output Quantization

53

' H e {PA(ρ)}  ∗  φ(t) + φ T (t)H a {W4 } φ(t) < 0, ς1 diag 0, 0, −γ 2 I (2.110) which means that V˙ (ϕ(t)) + z T (t)z(t) − γ 2 w T (t)w(t) < 0. &

φ T (t)

In the following, the design conditions are presented for static output feedback controller gain and dynamic parameters in the form of LMI based on the performance criterion proposed in Theorem 2.7. Firstly, defining Z = P −T , pre- and postmultiplying the inequality (2.100) by diag{Z T , 0, 0, 0, 0, 0, 0} and its transpose, it can be rewritten as ⎡

⎤ H e {A(ρ)Z } ∗ ∗ ⎣ ς˜1 ς2 ∗ ⎦ < 0, ς4 ς5 ς˜3

(2.111)

⎡

⎡ ⎤ ⎤ C(ρ)Z MT ς˜1 = ⎣BT (ρ)⎦ , ς˜3 = ⎣Δ y L(ρ)Z ⎦ . ET (ρ) Δu K˜ N Z

where

As for the condition PS B&= STB P T ' ≥ 0, it can be guaranteed by S B Z = Z T STB ≥ Z1 0 with G ∈ Rn y ×n x being a dimension adjust0. Next, set the matrix Z = Z2G Z2 ζω ment matrix and β1ω = 4λ , ω = y, u. In order to ensure the establishment of the ω T T condition S B Z = Z S B ≥ 0, Z 1 > 0 is necessary. Then the inequality (2.111) can be expressed as follows: ⎤ ⎡ Ω1 ∗ ∗ ⎣Ω2 Ω3 ∗ ⎦ < 0, (2.112) Ω4 Ω5 Ω6 where   ⎤ ˜ ˜ H e A(ρ)Z ∗ 1 + B(ρ) K Z 2 G ⎦,

T =⎣ ˜ ˜ −Z 2 − Z 2T L(ρ)Z 1 − Z 2 G + B(ρ) K Z 2 ⎡ ⎡ ⎤ ⎤ ˜ ˜ ˜ C(ρ)Z 0 βy I 1 + D(ρ) K Z 2 G D(ρ) K Z 2 ⎦, ˜ 0 ⎦ , Ω4 = ⎣ = ⎣βu B T (ρ) Δ y L(ρ)Z 0 1 T T E (ρ) H (ρ) Δu K˜ Z 2 G Δu K˜ Z 2 ⎡ ⎤ 0 βu D(ρ) F(ρ)   0 Δ y H (ρ)⎦ , Ω3 = −diag β y I, βu I, γ 2 I , = ⎣0 0 0 0   = −diag I, λ y I, λu I . ⎡

Ω1

Ω2

Ω5 Ω6

2 Fuzzy Robust H∞ Control with Dynamic Quantization

54

Similar to Case A, there are uncertain terms Δ A (ρ), ΔC (ρ), Δ L (ρ), and Δ K in above inequality, therefore, in the following, these uncertainties will be eliminated. Firstly, according to (2.56), the above inequality can be rewritten as ⎤ Ω˜ 1 ∗ ∗ ⎣Ω2 Ω3 ∗ ⎦ + H e {Π5 Δx (t)Π6 } < 0, Ω˜ 4 Ω5 Ω6 ⎡

(2.113)

  ⎤ H e A(ρ)Z 1 + B(ρ) K˜ Z 2 G ∗ ⎦,

T =⎣ −Z 2 − Z 2T L(ρ)Z 1 − Z 2 G + B(ρ) K˜ Z 2 ⎡ ⎤ C(ρ)Z 1 + D(ρ) K˜ Z 2 G D(ρ) K˜ Z 2 ⎦, Δ y L(ρ)Z 1 0 =⎣ Δu K˜ Z 2 G Δu K˜ Z 2

T = X TA (ρ) X LT (ρ) 0 0 0 X CT (ρ) (Δ y X L (ρ))T 0 ,

= Y Z1 0 0 0 0 0 0 0 . ⎡

where Ω˜ 1

Ω˜ 4 Π5 Π6

Then, according to Lemma 1.2, the uncertain terms Δ A (ρ), ΔC (ρ), and Δ L (ρ) can be eliminated, and one can get the following inequality, ⎤ Ω˜ 1 ∗ ∗ ∗ ⎢Ω2 Ω3 ∗ ∗ ⎥ ⎥ ⎢ ⎣Ω˜ 4 Ω5 Ω6 ∗ ⎦ < 0, Ω7 0 Ω8 Ω9 ⎡

where

(2.114)

&

& T ' ' X TA (ρ) X LT (ρ) X C (ρ) (Δ y X L (ρ))T 0 Ω7 = , Ω8 = , 0 σ3 Y Z 1 0 0 0 Ω9 = −diag { σ3 I, σ3 I } .

as

Next, the uncertain term Δ K will be eliminated. Above inequality can be rewritten ⎤ ⎡ Ωˆ 1 ∗ ∗ ∗ ⎢Ω2 Ω3 ∗ ∗ ⎥ ⎥ ⎢ (2.115) ⎣Ωˆ 4 Ω5 Ω6 ∗ ⎦ + H e {Π7 Δk (t)Π8 } < 0, Ω7 0 Ω8 Ω9

2.3 Resilient Control with Input and Output Quantization

where

' ∗ H e {A(ρ)Z 1 + B(ρ)K Z 2 G} , L(ρ)Z 1 − Z 2 G + (B(ρ)K Z 2 )T −Z 2 − Z 2T ⎡ ⎤ C(ρ)Z 1 + D(ρ)K Z 2 G D(ρ)K Z 2 ⎦, Δ y L(ρ)Z 1 0 =⎣ Δu K Z 2 Δu K Z 2 G

T T = (B(ρ)R) 0 0 0 0 (D(ρ)R)T 0 (Δu R)T 0 0 ,

= W Z2G W Z2 0 0 0 0 0 0 0 0 .

Ωˆ 1 = Ωˆ 4 Π7 Π8

55

&

Then, according to Lemma 1.2, eliminating the uncertain term Δ K can obtain the following inequality, ⎤ ⎡ Ωˆ 1 ∗ ∗ ∗ ∗ ⎢ Ω2 Ω3 ∗ ∗ ∗ ⎥ ⎥ ⎢ ⎢ Ωˆ 4 Ω5 Ω6 ∗ ∗ ⎥ < 0, (2.116) ⎥ ⎢ ⎣ Ω7 0 Ω8 Ω9 ∗ ⎦ Ω10

0 Ω11

0 Ω12

where Ω10

' ' & & 0 σ4 (B(ρ)R)T σ4 (D(ρ)R)T 0 σ4 (Δu R)T , Ω11 = , = W Z2 0 0 0 W Z2G

Ω12 = −diag { σ4 I, σ4 I } . Defining K Z 2 = U , based on the property of membership function, the conditions (2.99) and (2.100) can be guaranteed by the following LMIs in Theorem 2.8, which gives the design conditions of the output feedback controller (2.57) and quantizers (2.59) and (2.60) to ensure the H∞ performance γ for the fuzzy system (2.97). Theorem 2.8 For given scalar γ > 0, quantizers’ ranges Mω and error bounds Δω , ω = y, u, if there exist matrices Z 1 > 0, Z 2 , and U , positive scalars λω and βω , ω = y, u, σ3 , σ4 satisfying βω Mω2 > 4λω , ω = y, u,

(2.117)

⎤ Ωˆ 1i ∗ ∗ ∗ ∗ ⎢ Ω2i Ω3 ∗ ∗ ∗ ⎥ ⎥ ⎢ ⎢ Ωˆ 4i Ω5i Ω6 ∗ ∗ ⎥ < 0, i = 1, 2, . . . , r, ⎥ ⎢ ⎣ Ω7i 0 Ω8i Ω9 ∗ ⎦ Ω10i 0 Ω11i 0 Ω12

(2.118)

⎡

2 Fuzzy Robust H∞ Control with Dynamic Quantization

56

where ⎡ ⎤ ' 0 βy I {A Z + B U G} ∗ H e i 1 i , Ω2i = ⎣βu BiT 0 ⎦ , Ωˆ 1i = L i Z 1 − Z 2 G + (Bi U )T −Z 2 − Z 2T E iT HiT ⎡ ⎡ ⎤ ⎤ Ci Z 1 + Di U G Di U 0 βu Di Fi 0 ⎦ , Ω5i = ⎣0 0 Δ y Hi ⎦ , Ωˆ 4i = ⎣ Δ y L i Z 1 Δu U 0 0 0 Δu U G & T & ' ' T T T Δ y X Li 0 X Ai X Li X Ci , Ω8i = , Ω7i = σ3 Y Z 1 0 0 0 0 ' ' & & σ4 (Bi R)T σ4 (Di R)T 0 σ4 (Δu R)T 0 , Ω11i = Ω10i = . 0 0 0 W Z2G W Z2 &

Then, the H∞ performance γ can be guaranteed for the fuzzy system (2.67). Moreover, the quantizers (2.59) and (2.60) are designed with online adjusting strategy for parameters μω (t), ω = y, u as (2.71) with ζω = 4λω /βω , ω = y, u, and the controller gain K in (2.57) can be obtained by K = U Z 2−1 .

(2.119)

According to the view in [8], the dynamic parameters μω (t), ω = y, u also need to be adjusted. Referring to [8], for the on-line adjusting strategy of parameters μω (t), ω = y, u in Theorems 2.6 and 2.8, the adjusting rule is given as follows: The adjusting rule: For ω = y, u ⎧ 1 s −s ⎪ ⎨floor (2ϑω ω(t) × 10 ) × 10 , 0 ≤ ϑω ω(t) < 2 , 1 μω (t) = 1, ≤ ϑω ω(t) < 1, 2 ⎪ ⎩ floor (2ϑω ω(t)) , 1 ≤ ϑω ω(t) ,

(2.120)

  where s = min s ∈ N+ | (2ϑω ω(t) × 10s ) > 1 and the function floor(θ ) represents the maximum integer which does not exceed θ . More specifically, the ϑω (t) can be updated by applying the above adjusting strategy with ϑω = δω , ω = y, u for Theorem 2.6 and ϑω = √1ζ , ω = y, u for Theorem 2.8. ω

Remark 2.7 Different from [8], the adjusting rule of this chapter gives the deterministic expression for quantizers’ dynamic parameters ϑω (t), ω = y, u, which can be determined as a unique value corresponding to each signal ω(t), ω = y, u. Remark 2.8 It is worth noting that by applying the descriptor representation approach to formulate the two augmented closed-loop systems (2.67) and (2.97), some nonlinear coupling terms between the system matrices and the controller gain matrix which are difficult to deal with, have been successfully avoided, such that we can easily

2.3 Resilient Control with Input and Output Quantization

57

handle the static output feedback control problem and derive strict LMI design conditions. Moreover, it avoids the appearance of coupling between quantization errors e y (t) and eu (t) and facilitates the design of H∞ controller. The obtained conditions (2.91) and (2.92) in Theorem 2.6 or (2.117) and (2.118) in Theorem 2.8 can be used to deal with the output feedback H∞ controller design problem for continuous-time T–S fuzzy systems with quantized measurement output and quantized control input if the dimension adjustment matrices V ∈ Rn x ×n u and G ∈ Rn y ×n x are known in advance.

T In this section, V and G can be chosen in the form as V = In u ×n u 0n u ×(n x −n u ) and G = [ In y ×n y 0n y ×(n x −n y ) ]. Then Theorems 2.6 and 2.8 can be easily solved with the help of LMI control toolbox in MATLAB. Remark 2.9 Based on the LMI conditions in Theorem 2.6 or Theorem 2.8, the static output feedback H∞ control problem for continuous-time T–S fuzzy systems with quantized input and output signals can be solved. The H∞ performance index γ can be optimized by min ι subject to LMIs (2.91) and (2.92) or (2.117) and (2.118) with ι = γ 2 . In this way, the corresponding optimal value of the H∞ performance can be given √ as γmin = ιmin . Remark 2.10 Note that from the on-line adjusting strategies (2.71) and (2.101), quantizers’ parameters μω (t), ω = y, u only depend on the quantized signal ω(t), ω = y, u and the scalars δω , ω = y, u in Theorem 2.6 or ζω , ω = y, u in Theorem 2.8, which can be obtained directly by the solution of LMIs (2.91) and (2.92) or (2.117) and (2.118), respectively, it implies that the computation of proposed algorithm for quantization design is relatively smaller than the design result in [8].

2.3.3 Simulation Example An example for the inverted pendulum system of motor dynamics is used to illustrate the effectiveness of the proposed methods in this section, and it can be given as the following dynamic equation: ρ˙ p (t) = ω p (t), g N (K m + ΔK m ) ω˙ p (t) = sin(ρ p (t)) + Ia (t) + w(t), l ml 2 Kb N Ra 1 I˙a (t) = − ω p (t) − Ia (t) + U (t), La La La

(2.121)

where m, l, g, N , K m , K b , Ra , and L a stand for the mass of pendulum, length of pendulum, gravity acceleration, gear ratio, motor torque coefficient, back EMF

58

2 Fuzzy Robust H∞ Control with Dynamic Quantization

coefficient, resistance of resistor, and inductance of inductor, respectively. ΔK m means the variation of motor torque coefficient caused by the friction. U (t) represents the voltage input. w(t) represents the disturbance signal. The outputs of this system are selected as z(t) = −ρ p (t) + 0.1U (t), y(t) = ν1 ρ p (t) + ν2 ω p (t) + ν3 Ia (t) + ν4 w(t),

(2.122)

where ν1 , ν2 , ν3 , and ν4 represent the sensor coefficient. Select the state of this system

T as x(t) = ρ p (t) ω p (t) Ia (t) . As for the nonlinear term sin(ρ p (t)), it can be approached by the T–S fuzzy model. Giving the physical parameters of the system as m = 0.5 kg, l = 1 m, g = 9.8 m/s2 , N = 5, K m = 0.1 N · m/A, K b = 0.1 V · s/rad, Ra = 0.8 , L a = 100 mH, ν1 = 3, ν2 = 2, ν3 = 0.1, and ν4 = − 0.2, the T–S fuzzy model of this system can be given as R i : i f ρ p (t) is M1i , then x(t) ˙ = (Ai + Δ Ai )x(t) + Bi Q(u(t)) + E i w(t), z(t) = (Ci + ΔCi )x(t) + Di Q(u(t)) + Fi w(t),

(2.123)

y(t) = (L i + Δ Li )x(t) + Hi w(t), with i = 1, 2, where ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 1 0 0 1 0 0 1 ⎦, A2 = ⎣0 0 1 ⎦, B1 = B2 = ⎣ 0 ⎦, A1 = ⎣9.8 0 0 −5 −8 0 −5 −8 10 ⎡ ⎤ 0 E 1 = E 2 = ⎣1⎦, C1 = C2 = [ − 1 0 0 ], D1 = D2 = 0.1, F1 = F2 = 0, 0 ⎡ ⎤ 0 0 0

L 1 = L 2 = 3 2 0.1 , H1 = H2 = − 0.2, X A1 = X A2 = ⎣0 0 10⎦, 0 0 0 ⎡ ⎤ 0.01 0 0

Y = ⎣ 0 0.01 0 ⎦, X C1 = X C2 = 0.1 0.1 0.1 , 0 0 0.01

X L1 = X L2 = 0.1 0.1 0.1 , R = W = 0.01. Moreover, the uncertain terms are selected as Δx (t) = Δk (t) = 0.5 cos(0.1t)e−2t . For this fuzzy system, the membership function is assumed in the form as ρ1 (ε(t)) = sin2 (x1 (t)), ρ2 (ε(t)) = 1 − ρ1 (ε(t)). And the quantization ranges and quantization error bounds are selected as M y = Mu = 50, Δ y = Δu = 0.01. By solving the LMIs

T (2.91) and (2.92) developed in Theorem 2.6 with V = 1 0 0 , there is no a feasible solution due to the design conservatism, whereas solving the LMIs (2.117) and

2.3 Resilient Control with Input and Output Quantization Fig. 2.12 Response of the system state x(t)

59

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

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

2

4

6

8

Time (sec)

10



(2.118) developed in Theorem 2.8 of Case B with G = 1 0 0 , a feasible solution is given as ⎡

⎤ 0.0704 − 0.1597 0.0116 Z 1 = ⎣− 0.1597 1.2717 − 5.6053⎦, 0.0116 − 5.6053 55.2213

Z 2 = 0.1211.

Combining (2.119), it is calculated the output feedback controller gain as K = − 5.9242. Moreover, it gives the optimized H∞ performance index as γmin = 3.2766. Further, in order to demonstrate the validity of the algorithm more intuitively, the designed controller and quantizers are adopted to test the H∞ performance γ = 3.2766 for the fuzzy system (2.123). Let us assume the initial condition to

T be x(0) = 0 0 0 and the disturbance signal to be w(t) = 10 cos(0.1t)e−2t . Moreover, the dynamic parameters μ y (t) and μu (t) are updated by the adjusting rule described in (2.120) with ϑ y = √1 = 0.02 and ϑu = √1ζ = 0.02. Then ζy

u

the response of system state, measurement output, controlled output, and control input are shown in Figs. 2.12, 2.13, 2.14, and 2.15, respectively. The ratio of ! (t (t T T 0 z (i)z(i)di/ 0 w (i)w(i)di showing the disturbance attenuation capability of the system is presented in Fig. 2.16. From this figure, it is easy to observe that the ratio tends to a constant value 1.0550 below the prescribed value γ = 3.2766, which implies that the asymptotical stability and the prescribed H∞ performance of the fuzzy system (2.123) can be guaranteed by Theorem 2.8, and illustrates the feasibility and effectiveness of the proposed design strategy in this chapter. Besides, as shown in Figs. 2.17 and 2.18, the quantizers’ dynamic parameters μ y (t) and μu (t) are adaptively adjusted depending on the measurement output y(t) and the control input u(t), respectively, which is consistent with the proposed adjusting rule (2.120).

60 Fig. 2.13 Response of the measurement output y(t)

2 Fuzzy Robust H∞ Control with Dynamic Quantization 1.5

1

0.5

0

-0.5

Fig. 2.14 Response of the controlled output z(t)

0

2

0

2

0

2

4

6

8

10

4

6

8

10

4

6

8

10

Time (sec)

0.2 0 -0.2 -0.4 -0.6 -0.8 -1

Fig. 2.15 Response of the control input u(t)

Time (sec)

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

Time (sec)

2.3 Resilient Control with Input and Output Quantization Fig. !( 2.16 History(of t T t T 0 z (i)z(i)di/ 0 w (i)w(i)di

61

1.2 1 0.8 0.6 0.4 0.2 0 0

Fig. 2.17 Dynamic quantizer’s parameter μ y (t)

2

4

6

8

10

4

6

8

10

4

6

8

10

Time (sec)

0.05 0.04 0.03 0.02 0.01 0

Fig. 2.18 Dynamic quantizer’s parameter μu (t)

0

2

0

2

Time (sec)

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Time (sec)

62

2 Fuzzy Robust H∞ Control with Dynamic Quantization

From Figs. 2.12, 2.13, 2.14, 2.15, 2.16, 2.17, and 2.18, it can be concluded that both the quantized stabilization control and quantized H∞ control design approaches developed in this chapter are effective.

2.4 Conclusion In this chapter, the H∞ control problem for T–S fuzzy systems with dynamic quantization has been studied sufficiently. For discrete-time fuzzy systems, via the two-step approach and an improved two-step approach, the robust controller and the nonfragile controller based on state feedback have been separately designed to maintain the same H∞ performance with the one without quantization when the control input and the system state are quantized by dynamic quantizers, and the online adjusting rule for dynamic quantizers’ parameters has also been formulated. For the continuous-time case, the descriptor representation approach has been adopted to design the resilient controller based on static output feedback to attenuate the effects of parameter uncertainties, controller gain perturbation, the control input, and measurement output quantization, so as to preserve the desired H∞ performance. Moreover, some simulation results have been given to verify the effectiveness of these quantized robust H∞ control design strategies.

References 1. Chang XH, Yang C, Xiong J (2019) Quantized fuzzy output feedback H∞ control for nonlinear systems with adjustment of dynamic parameters. IEEE Trans Syst, Man, Cybern: Syst 49:2005– 2015 2. Chang XH, Huang R, Wang H, Liu L (2020) Robust design strategy of quantized feedback control. IEEE Trans Circuits Syst II Exp Briefs 67:730–734 3. Tanaka K, Sano M (1994) A robust stabilization problem of fuzzy control systems and its application to backing up control of a truck–trailer. IEEE Trans Fuzzy Syst 2:119–134 4. Lo JC, Lin ML (2004) Robust H∞ nonlinear control via fuzzy static output feedback. IEEE Trans Circuits Syst I Fundam Theory Appl 50:1494–1502 5. Che WW, Yang GH (2008) State feedback H∞ control for quantized discrete-time systems. Asian J Control 10:718–723 6. Zhai G, Chen N, Gui W (2010) Quantizer design for interconnected feedback control systems. J Control Theory Appl 8:93–98 7. Chen N, Zhai G, Gui W, Yang C, Liu W (2010) Decentralised H∞ quantisers design for uncertain interconnected networked systems. IET Control Theory Appl 4:177–185 8. Niu Y, Ho D W C (2014) Control strategy with adaptive quantizer’s parameters under digital communication channels. Automatica 50:2665–2671 9. Wei Y, Park J H, Qiu J, Jung H (2018) Reliable output feedback control for piecewise affine systems with markov–type sensor failure. IEEE Trans Circuits Syst II Exp Briefs 65:913–917 10. Chang XH, Liu RR, Park J H (2019) A further study on output feedback H∞ control for discrete-time systems. IEEE Trans Circuits Syst II Exp Briefs 67:305–309 11. Chang XH, Xiong J, Li ZM, Park J H (2018) Quantized static output feedback control for discrete-time systems. IEEE Trans Industr Inf 14:3426–3435

Chapter 3

Fuzzy Filtering with Multiple Signal Transmissions

Abstract This chapter investigates the filtering problem for T–S fuzzy systems, where multiple signals are considered to be transmitted through the communication channel. Firstly, the l2 –l∞ filtering problem for T–S fuzzy systems with quantization is proposed, in which the measurement output and the performance output signals of the system are quantized by two static quantizers before being transmitted over the communication channel, respectively. Secondly, the induced l∞ filtering problem for T–S fuzzy systems is proposed, in which the measurement output and the performance output are taken into account the data packet dropout phenomenon modeled by two stochastic variables. Sufficient conditions are given to ensure that the filtering error system is not only stochastically stable but also has a prescribed induced l∞ performance. Thirdly, the H∞ filtering problem for the T–S fuzzy systems is proposed, in which the measurement output and the performance output signals of the system are quantized by two dynamic quantizers. And the conditions are given to ensure the filtering error system is asymptotically stable with the prescribed H∞ performance index. Finally, some examples are given to demonstrate the effectiveness of the proposed filtering methods for T–S fuzzy systems, respectively. Keywords T–S fuzzy system · Filtering · Quantization · Data packet dropout

3.1 l2 –l∞ Filtering with Static Quantization In this section, the l2 –l∞ filtering problem for nonlinear system with quantization is studied. The system with two quantized signals is investigated, namely, the measurement output signal and the performance output signal. And based on the fuzzy Lyapunov function approach, the l2 –l∞ filter is designed for the uncertain T–S fuzzy system in terms of a set of strict LMIs.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Chang et al., Control and Filtering of Fuzzy Systems Under Communication Channels, https://doi.org/10.1007/978-981-99-4346-3_3

63

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3 Fuzzy Filtering with Multiple Signal Transmissions

3.1.1 Problem Formulation Consider the following uncertain discrete-time T–S fuzzy model: Plant Rule i th : IF h 1 (k) is N1i , h 2 (k) is N2i , and, . . . , and h θ (k) is Nθi , THEN x(k + 1) = (Ai + Δ Ai )x(k) + Bi w(k), y(k) = (Ci + ΔCi )x(k) + Di w(k),

(3.1)

z(k) = (L i + Δ Li )x(k) + E i w(k), where Nλi are fuzzy sets with λ = 1, . . . , θ and i = 1, . . . , r , in which r is the number of fuzzy rules; h(k) = [ h 1 (k), . . . , h θ (k) ] is premise variable vector; x(k) ∈ Rn x is the state variable, y(k) ∈ Rn y is the measurement output, z(k) ∈ Rn z is the performance output, and w(k) ∈ Rn w is the noise signal belonging to l2 [0, ∞); Ai ∈ Rn x ×n x , Bi ∈ Rn x ×n w , Ci ∈ Rn y ×n x , Di ∈ Rn y ×n w , L i ∈ Rn z ×n x , and E i ∈ Rn z ×n w denote the system matrices which are known constant matrices; Δ Ai , ΔCi , and Δ Li are parameter uncertain matrices, which are considered as the following form in this section: ⎛ ⎞ ⎛ ⎞ Δ Ai FAi ⎝ ΔCi ⎠ = ⎝ FCi ⎠ Δx (k)N x , (3.2) Δ Li FLi where Δx (k) is an uncertainty satisfying ΔTx (k)Δx (k) ≤ I ; FAi , FCi , FLi , and N x are known constant matrices. Denoting θ  si (h(k)) = Nλi (h λ (k)), i = 1, . . . , r, λ=1

  where Nλi h λ (k) is the grade of membership function of h λ (k) in Nλi . Assuming that si (h(k)) > 0,

r

si (h(k)) > 0, i = 1, . . . , r.

i=1

Let

si (h(k)) , i = 1, . . . , r. σi (k) = r i=1 si (h(k))

(3.3)

Then, we have σi (k) ≥ 0,

r i=1

σi (k) = 1, i = 1, . . . , r.

(3.4)

3.1 l2 –l∞ Filtering with Static Quantization

65

The uncertain T–S fuzzy model (3.1) can be described as follows: x(k + 1) = (Aσk + Δ Aσk )x(k) + Bσk w(k), y(k) = (Cσk + ΔCσk )x(k) + Dσk w(k), z(k) = (L σk + Δ Lσk )x(k) + E σk w(k),

(3.5)

where A σk = D σk = Δ Aσk = Δ Lσk =

r i=1 r i=1 r i=1 r

σi (k)Ai , σi (k)Di ,

Bσk = L σk =

r i=1 r

σi (k)Bi , Cσk = σi (k)L i ,

E σk =

i=1

r i=1 r

σi (k)Ci , σi (k)E i ,

i=1

σi (k)FAi Δx (k)N x , ΔCσk =

r

(3.6)

σi (k)FCi Δx (k)N x ,

i=1

σi (k)FLi Δx (k)N x .

i=1

In order to transmit the measurement output y(k) and the performance output z(k) over the communication channel, we consider the static logarithmic quantizer [1]. The quantization levels have the following form: U = {± u i , u i = ρ i u 0 , i = 0, ± 1, ± 2, . . .} ∪ {± u 0 } ∪ {0},

(3.7)

where 0 < ρ < 1 is quantization density and u 0 > 0. The quantizer function Q(·) is defined as ⎧ 1 1 u < ϑ ≤ 1−ξ um , ⎨ um , 1+ξ m Q(ϑ) = 0, (3.8) ϑ = 0, ⎩ −Q(−ϑ), ϑ < 0, with ξ=

1−ρ . 1+ρ

(3.9)

To this end, the following two quantizers are employed in this section:  T y¯ (k) = Q(y(k)) = Q(y1 (k)) · · · Q(yn y (k)) , T  z¯ (k) = Q(z(k)) = Q(z 1 (k)) · · · Q(z n z (k)) .

(3.10)

For the quantization effects, we resort to the sector bound method proposed in [1] to deal with the quantization errors, the two quantizers described in (3.10) can be

66

3 Fuzzy Filtering with Multiple Signal Transmissions

remodeled as

which implies

y¯ (k) − y(k) = Δ y (k)y(k), z¯ (k) − z(k) = Δz (k)z(k), y¯ (k) = (I + Δ y (k))y(k), z¯ (k) = (I + Δz (k))z(k),   Δ y (k) ≤ Ξ y , Ξ y = diag{ ξ 1 , . . . , ξ yn y }, y

where

Δz (k) ≤ Ξz , Ξz = diag{ ξz1 , . . . , ξzn z }.

(3.11)

(3.12)

(3.13)

Then, the following filter form with quantized input is considered: x f (k + 1) = A f x f (k) + B f y¯ (k), z f (k) = C f x f (k) + D f y¯ (k),

(3.14)

where x f (k) and z f (k) are the state and the output of the filter, respectively; A f , B f , C f and D f are filter gain matrices to be determined. By combining (3.5), (3.12), (3.14) and defining the augmented variable υ(k) = [ x T (k) x Tf (k) ]T and the filtering error e(k) = z¯ (k) − z f (k), the filtering error system is given by υ(k + 1) = Aσk υ(k) + Bσk w(k), (3.15) e(k) = Cσk υ(k) + Dσk w(k), where    0 Aσk + Δ Aσk Bσk , B σk = , = B f (I + Δ y (k))Dσk B f (I + Δ y (k))(Cσk + ΔCσk ) A f 

Aσk

Cσk = [ (I + Δz (k))(L σk + Δ Lσk ) − D f (I + Δ y (k))(Cσk + ΔCσk ) − C f ], Dσk = (I + Δz (k))E σk − D f (I + Δ y (k))Dσk . The purpose of this section is to determine the filter (3.14), such that the filtering error system (3.15) satisfies the following two conditions: (1) When w(k) = 0, the filtering error system (3.15) is asymptotically stable; (2) Under zero initial condition, for any nonzero w(k) ∈ l2 [0, ∞) and given γ > 0, the error output satisfies (3.16) e(k)∞ ≤ γ w(k)2 , where e(k)∞ = supk≥0





 ∞ T e T (k)e(k) and w(k)2 = k=0 w (k)w(k).

3.1 l2 –l∞ Filtering with Static Quantization

67

3.1.2 Main Results In this section, sufficient conditions are provided to design the l2 –l∞ filter which also can ensure that filtering error system (3.15) is asymptotically stable with a given l2 –l∞ performance γ . Theorem 3.1 For a given performance index γ > 0, the system (3.15) is asymptotically stable with the l2 –l∞ performance index γ , if there exist matrices Pσk > 0, Pσk+1 > 0 such that 

where Pσk =

 −Pσk + AσTk Pσk+1 Aσk ∗ < 0, BσTk Pσk+1 Aσk −I + BσTk Pσk+1 Bσk

(3.17)

  −Pσk + γ −2 CσTk Cσk ∗ < 0, −I + γ −2 DσTk Dσk γ −2 DσTk Cσk

(3.18)

r i=1

σi (k)Pi .

Proof By performing congruence transformation to (3.17) with [ υ T (k) w T (k)]T = 0, one obtains (Aσk υ(k) + Bσk w(k))T Pσk+1 (Aσk υ(k) + Bσk w(k)) − υ T (k)Pσk υ(k) < w T (k)w(k).

(3.19)

Construct Lyapunov function as V (k) = υ T (k)Pσk υ(k).

(3.20)

Then from (3.15), one has V (k + 1) = (Aσk υ(k) + Bσk w(k))T Pσk+1 (Aσk υ(k) + Bσk w(k)).

(3.21)

Then, (3.19) can be rewritten as V (k + 1) − V (k) < w T (k)w(k).

(3.22)

For w(k) = 0, we can find that ΔV (k) = V (k + 1) − V (k) < 0,

(3.23)

which implies that the filtering error system (3.15) is asymptotically stable. Then by taking the sum on both sides of the inequality (3.22) from 0 to k − 1, one has k−1  k−1  V (s + 1) − V (s) < w T (s)w(s). (3.24) s=0

s=0

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3 Fuzzy Filtering with Multiple Signal Transmissions

After simply calculating, one can be obtained V (k) − V (0)
0, ηl > 0, β > 0, δ > 0 and matrices P1i > 0, P2i , P3i > 0, G 1 , G 2 , G 3 , A¯ f , B¯ f , C¯ f , D¯ f such that ⎡

⎤ −P1i ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎢−P2i −P3i ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ 0 0 −I ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢Θ i T A¯ Θ i T Θ l ∗ ∗ ∗ ∗ ∗ ⎥ f ⎢ 14 ⎥ 34 44 ⎢ i T ¯ ⎥ i T l T l Θil = ⎢Θ15 Θ45 Θ55 ∗ ∗ ∗ ∗ ⎥ < 0, (3.29) A f Θ35 ⎢ ⎥ ⎢ 0 0 0 Ξ yT B¯ Tf Ξ yT B¯ Tf −εl I ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ εl Ci 0 εl Di 0 0 0 −εl I ∗ ∗ ⎥ ⎢ ⎥ i T i T T ⎣ 0 Θ58 0 εl FCi −ηl I ∗ ⎦ 0 0 Θ48 0 0 0 0 0 0 −ηl I ηl N x 0

3.1 l2 –l∞ Filtering with Static Quantization

69

⎡

⎤ −P1i ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎢−P2i −P3i ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ 0 0 −I ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢Σ i T −C¯ Σ i T −γ 2 I ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ 14 ⎥ f 34 ⎢ 0 −β I ∗ ∗ ∗ ∗ ∗ ⎥ 0 0 ΞzT ⎢ ⎥ Σi = ⎢ ⎥ < 0, 0 0 −Ξ yT D¯ Tf 0 −β I ∗ ∗ ∗ ∗ ⎥ ⎢ 0 ⎢ ⎥ 0 0 0 −β I ∗ ∗ ∗ ⎥ ⎢ β L i 0 β Ei ⎢ ⎥ 0 0 0 0 −β I ∗ ∗ ⎥ ⎢ βCi 0 β Di ⎢ ⎥ i T T T ⎣ 0 0 0 β FLi β FCi −δ I ∗ ⎦ 0 0 Σ49 0 0 0 0 0 0 0 −δ I δ Nx 0 (3.30) for i, l = 1, . . . , r , where i i Θ14 = (G 1 Ai + B¯ f Ci )T , Θ15 = (G 3 Ai + B¯ f Ci )T , i i Θ34 = (G 1 Bi + B¯ f Di )T , Θ35 = (G 3 Bi + B¯ f Di )T , l l Θ44 = −G 1 − G 1T + P1l , Θ45 = −G 2 − G 3T + P2l , i l Θ48 = G 1 FAi + B¯ f FCi , Θ55 = −G 2 − G 2T + P3l ,

(3.31)

i i Θ58 = G 3 FAi + B¯ f FCi , Σ14 = (L i − D¯ f Ci )T , i i Σ34 = (E i − D¯ f Di )T , Σ49 = FLi − D¯ f FCi .

In addition, the l2 –l∞ filter gain matrices in (3.14) can be obtained by −1 ¯ ¯ ¯ ¯ A f = G −1 2 A f , B f = G2 B f , C f = C f , D f = D f .

(3.32)

Proof By applying Schur complement (Lemma 1.1) to the matrix inequality (3.17), one has ⎡ ⎤ −Pσk ∗ ∗ ⎣ 0 −I ∗ ⎦ < 0. (3.33) Aσk Bσk −Pσ−1 k+1 Let us consider a nonsingular matrix G with appropriate dimension and pre- and post-multiply diag{I, I, G} and its transpose to (3.33), one yields ⎡

⎤ −Pσk ∗ ∗ ⎣ 0 ⎦ < 0. −I ∗ T G G Aσk G Bσk −G Pσ−1 k+1

(3.34)

(G − Pσk+1 ) ≤ 0 with Pσk+1 > 0 means From the fact that −(G − Pσk+1 )T Pσ−1 k+1 T T −G Pσ−1 G ≤ −G − G + P , we know that (3.34) can be verified by σ k+1 k+1

70

3 Fuzzy Filtering with Multiple Signal Transmissions

⎡

⎤ −Pσk ∗ ∗ ⎣ 0 −I ∗ ⎦ < 0, G Aσk G Bσk Θ p

(3.35)

with Θ p = −G − G T + Pσk+1 . Then, we need to isolate the uncertainties in the matrix inequality (3.35) before they are eliminated. According to the definitions in (3.9) and (3.13), we affirm that the matrix Ξ y is nonsingular, then (3.35) can be rewritten as follows: ⎡

⎤ ⎡ ⎤ 0 ∗ ∗ ⎥ ⎢   0  ⎥ Δ y (k) [ Cσk + ΔCσk 0 ] Dσk 0 −I ∗ ⎦+⎢ ⎦ Ξy ⎣ 0 G B11σk Θ p Ξy G Bf ⎤ T + ΔCσ k ⎥ ΔTy (k)  0 ⎥ [ 0 0 Ξ yT 0 B Tf G T ] < 0, T ⎦ T D σk Ξy 0 (3.36)   0 Aσk + Δ Aσk , B11σk = [ BσTk DσTk B Tf ]T . A11σk = B f (Cσk + ΔCσk ) A f

−Pσk ⎣ 0 G A11σk ⎡ T C σk ⎢ +⎢ ⎣

where

 T Δ (k) Δ y (k) From (3.13), one has Ξy y ≤ I . By using Lemma 1.2 to (3.36), there Ξy will be a sufficient condition guaranteeing that the above inequality holds ⎡ ⎤ ⎤ 0 −Pσk ∗ ∗ ⎥  1 ⎢ ⎢  0  ⎥ [ 0 0 Ξ T 0 B Tf G T ] ⎣ 0 −I ∗ ⎦+ y ⎣ ⎦ εσk+1 G 0 Ξ G A11σk G B11σk Θ p y Bf   T ⎡ ⎤ T Cσσ + ΔCσ k k εσk+1 ⎢ ⎥ 1 ⎢ 0 ⎥ [ εσ [ Cσ + ΔCσ 0 ] εσ Dσ 0 ] < 0. + k k k+1 k T ⎣ ⎦ k+1 εσk+1 Dσk εσk+1 0 (3.37) Using Schur complement (Lemma 1.1) gives ⎡

⎡

−Pσk ⎢ 0 ⎢ ⎢ G A 11σk ⎢ ⎣ 0  εσk+1 Cσk + ΔCσk 0

∗ ∗ −I ∗ G B11σk Θ p  0 Ξ yT 0 B Tf G T εσk+1 Dσk 0

∗ ∗ ∗

∗ ∗ ∗ ∗

−εσk+1 I 0 −εσk+1 I

⎤ ⎥ ⎥ ⎥ < 0. ⎥ ⎦ (3.38)

3.1 l2 –l∞ Filtering with Static Quantization

71

Notice that the uncertainty Δ y (k) dose not exist in (3.38) by using Lemma 1.2, and only the uncertainty Δx (k) is left. Next, the same method as eliminating Δ y (k) is used to eliminate Δx (k), we gain ⎡

∗ −I G B11σk

−Pσk 0 G A12σk

∗ ∗ ∗ ∗ Θ ∗ p   Ξ yT 0 B Tf G T −εσk+1 I

⎢ ⎢ ⎢ ⎢ ⎢ 0 0 ⎢ ⎢ T ⎢ ε T ⎢ σk+1 Cσk εσk+1 Dσk 0 ⎢ 0 ⎢  T ⎢ ⎢ FAσk ⎢ 0 0 GT ⎢ B f FCσk ⎢  T T ⎣ Nx ησk+1 0 0 0

0 0

∗ ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

∗

∗

∗

−εσk+1 I

∗

∗

T εσk+1 FCσ −ησk+1 I k

0

0

0

∗ −ησk+1 I

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(3.39) 

where A12σk =



A σk 0 . B f C σk A f

In order to eliminate the coupling items in (3.39), constructing two matrix variables as

 G=

     r G1 G2 P P ∗ ∗ σi (k) 1i , Pσk = 1σk = , G3 G2 P2σk P3σk P2i P3i

(3.40)

i=1

and according to the definition A¯ f = G 2 A f , B¯ f = G 2 B f in (3.32). We know that (3.39) can be rewritten as ⎡

⎤ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −P1σk ⎢ −P2σk −P3σk ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ 0 0 −I ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ Θ14 T A¯ f Θ34 T Θ44 ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ T T T ⎢ Θ15 A¯ f Θ35 Θ45 Θ55 ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ < 0, TB ¯ T Ξ yT B¯ T −εσk+1 I ⎢ ⎥ ∗ ∗ ∗ 0 0 0 Ξ y f f ⎢ ⎥ ⎢εσ Cσ ⎥ 0 ε D 0 0 0 −ε I ∗ ∗ σk+1 σk σk+1 ⎢ k+1 k ⎥ T T T ⎣ Θ58 0 εσk+1 FCσk −ησk+1 I ∗ ⎦ 0 0 0 Θ48 0 0 0 0 0 0 0 −ησk+1 I ησk+1 N x

(3.41) where Θ14 = (G 1 Aσk + B¯ f Cσk )T , Θ15 = (G 3 Aσk + B¯ f Cσk )T , Θ34 = (G 1 Bσk + B¯ f Dσk )T , Θ35 = (G 3 Bσk + B¯ f Dσk )T , Θ44 = −G 1 − G 1T + P1σk+1 , Θ45 = −G 2 − G 3T + P2σk+1 , Θ48 = G 1 FAσk + B¯ f FCσk , Θ55 = −G 2 − G 2T + P3σk+1 , Θ58 = G 3 FAσk + B¯ f FCσk .

(3.42)

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3 Fuzzy Filtering with Multiple Signal Transmissions

Analogous to the derivation process for the condition (3.17) and definition (3.32), then the matrix inequality (3.18) holds if the following condition is true: ⎡

−P1σk ∗ ⎢−P2σk −P3σk ⎢ ⎢ 0 0 ⎢ T ⎢ Σ14 −C¯ f ⎢ ⎢ 0 0 ⎢ ⎢ 0 0 ⎢ ⎢ β Lσ 0 k ⎢ ⎢ βCσ 0 k ⎢ ⎣ 0 0 δ Nx 0

⎤ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ −I ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ T Σ34 −γ 2 I ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ −β I ∗ ∗ ∗ ∗ ∗ ⎥ 0 ΞzT ⎥ < 0, (3.43) 0 −Ξ yT D¯ Tf 0 −β I ∗ ∗ ∗ ∗ ⎥ ⎥ β E σk 0 0 0 −β I ∗ ∗ ∗ ⎥ ⎥ β D σk 0 0 0 0 −β I ∗ ∗ ⎥ ⎥ T T T 0 0 β FLσ β FCσ −δ I ∗ ⎦ 0 Σ49 k k 0 0 0 0 0 0 0 −δ I

where Σ14 = (L σk − D¯ f Cσk )T , Σ34 = (E σk − D¯ f Dσk )T , Σ49 = FLσk − D¯ f FCσk . (3.44) From (3.41) and (3.43), we have r r

σi (k)σl (k + 1)Θil < 0,

(3.45)

i=1 l=1

and

r

σi (k)Σi < 0.

(3.46)

i=1

By recalling the property of the membership function described in (3.4), we can know that if (3.29) and (3.30) are established, the inequalities (3.45) and (3.46) are true. The proof is completed.

3.1.3 Simulation Example In this section, we consider a factual situation that there are the parameter perturbation and disturbance signal in the DC motor system [2]. The dynamic equations of this system are given by di(t) = −u R (t) − (K b + Δb (t))ω(t), dt dω(t) = K a i(t) − Bω(t) + w(t), J dt L

(3.47)

3.1 l2 –l∞ Filtering with Static Quantization

73

where i(t) is armature current, ω(t) is rotational speed, w(t) is the torque disturbance, L is armature inductor, u R (t) is the voltage across the armature resistor, Δb (t) is an uncertainty satisfying Δb (t) < K b , J is moment of inertia of the mechanical rotating part, K a is the torque constant, B is viscous friction coefficient of the rotating part, respectively. Assume x1 (t) = i(t), x2 (t) = ω(t), and select the parameters as J = 0.082 kg·m2 , B = 0.3 N · m · s/rad, L = 1000 mH, K a = 0.576 N · m/A, K b = 0.612 V · s/rad and Δb (t) = 0.06b(t) (− 1 ≤ b(t) ≤ 1), respectively. As in [2], the system can be described as x˙1 (t) = − 4x1 (t) − x13 (t) − (0.612 + 0.06b(t))x2 (t), x˙2 (t) = (0.576/0.082)x1 (t) − 0.3/0.082x2 (t) + 1/0.082w(t).

(3.48)

Let us rewrite (3.48) as  x(t) ˙ =

   − 4 − x12 (t) − (0.612 + 0.06b(t)) 0 x(t) + w(t). 0.576/0.082 − 0.3/0.082 1/0.082

(3.49)

For the nonlinear system (3.49), one can assume that − 3 ≤ x1 (t) ≤ 3, that is, − 13 ≤ − 4 − x12 (t) ≤ − 4. By fuzzy modeling for the nonlinear system, (3.49) can be represented by the following T–S model with two fuzzy rules: x(t) ˙ =

2

  σi (t) Aib x(t) + Bib w(t) ,

(3.50)

i=1

where x T (t) = [ x1T (t) x2T (t) ] and 

 −4 − (0.612 + 0.06b(t)) , 0.576/0.082 − 0.3/0.082   − 13 − (0.612 + 0.06b(t)) = , 0.576/0.082 − 0.3/0.082



 0 , 1/0.082   0 = . 1/0.082

A1b =

B1b =

A2b

B2b

The membership function can be given by σ1 (t) = x12 (t)/9, σ2 (t) = 1 − x12 (t)/9.

(3.51)

In the following, we will study the l2 –l∞ filtering problem for the system (3.50). By taking a sampling time T = 0.05 s, we get a discrete-time T–S fuzzy system as (3.1) and the corresponding system parameter matrices are

74

3 Fuzzy Filtering with Multiple Signal Transmissions



0.8 0.3512  0.35 A2 = 0.3512 A1 =

 − 0.0306 , 0.8171  − 0.0306 , 0.8171



 0 , 0.05/0.082   0 B2 = , 0.05/0.082

B1 =

C1 = [ − 3 0.5 ], D1 = 2.5, C2 = [ 1.5 2 ], D2 = − 2.3,

L 1 = [ 0.2 1 ], L 2 = [ 0.3 1 ],

(3.52) E 1 = 1, E 2 = − 0.5.

And the following parameters are also considered in (3.2): FA1 = FA2 = [ − 0.6 0 ]T , FL1 = FL2 = 3.5,

FC1 = FC2 = 0.01,

(3.53)

N x = [ 0 0.1 ].

We adopt the other parameters ρ y = ρz = 0.7, the maximum l2 –l∞ attenuation level can be obtained as 1.6855 by solving LMIs (3.29) and (3.30). In this case, the filter matrices can be calculated as the follows:     1.2160 − 0.2963 −0.2008 , Bf = , Af = 0.6312 0.3836 −0.4673 C f = [ 0.3349 − 0.2740 ], D f = 0.4213. Assume that the initial condition is x(0) = x f (0) = [ 0 0 ]T , the uncertainties are Δx (k) = 0.6 cos(0.8k), Δ y (k) = 0.1 sin(k), Δz (k) = 0.1 sin(0.8k), and the external disturbance is w(k) = e−2k cos(0.1k). The response results of x(k) and x f (k) are displayed in Figs. 3.1 and 3.2, respectively. Figure 3.3 describes the response of filtering It can be perceived that the error curve tends to 0. The ratio  error e(k).

∞ T T of e (k)e(k)/ k=0 w (k)w(k) is shown in Fig. 3.4. It can be known that the maximum value of the ratio is less than 1.6855. From the simulation results of this example, we can get that the design method of the l2 –l∞ filter proposed in this section is effective. Fig. 3.1 Response of the system state x(k)

0.6 0.5 0.4 0.3 0.2 0.1 0 0

10

20

30

40

50

60

3.1 l2 –l∞ Filtering with Static Quantization Fig. 3.2 Response of the filter state x f (k)

75 1.5 1 0.5 0 -0.5 -1

Fig. 3.3 Filtering error e(k)

0

10

20

30

40

50

60

0

10

20

30

40

50

60

10

20

30

40

50

60

0.5 0.4 0.3 0.2 0.1 0 -0.1

Fig. of  3.4 History

T e T (k)e(k)/ ∞ k=0 w (k)w(k)

0.5 0.4 0.3 0.2 0.1 0 0

76

3 Fuzzy Filtering with Multiple Signal Transmissions

3.2 Induced l∞ Filtering with Data Packet Dropouts In this section, the induced l∞ filtering problem for a class of nonlinear systems based on T–S fuzzy model is considered. In addition, both the measurement output and the performance output are considered to be suffered from data packet dropout when they are transmitted through the communication channel. Finally, the conditions to design induced l∞ filter are given in the form of LMIs.

3.2.1 Problem Formulation For the induced l∞ filtering problem in this section, we consider a nonlinear discretetime system which is approximated by a T–S fuzzy model as follows: Plant Rule i th : IF h 1 (k) is N1i , h 2 (k) is N2i , and, . . . , and h θ (k) is Nθi , THEN x(k + 1) = Ai x(k) + Bi w(k), y(k) = Ci x(k) + Di w(k),

(3.54)

z(k) = L i x(k) + E i w(k), where x(k) ∈ Rn x , y(k) ∈ Rn y , z(k) ∈ Rn z , and w(k) ∈ Rn w are the state variable, the measurement output, the performance output, and the noise signal in l∞ [0, ∞), respectively; h(k) = [ h 1 (k), . . . , h θ (k) ] is premise variable vector; Nλi with λ = 1, . . . , θ and i = 1, . . . , r are the fuzzy sets with r being the number of fuzzy rules; Ai ∈ Rn x ×n x , Bi ∈ Rn x ×n w , Ci ∈ Rn y ×n x , Di ∈ Rn y ×n w , L i ∈ Rn z ×n x , and E i ∈ Rn z ×n w are known constant matrices with appropriate dimensions. As in Sect. 3.1, the T–S fuzzy model (3.54) can be deduced as follows: x(k + 1) = A(υ(k))x(k) + B(υ(k))w(k), y(k) = C(υ(k))x(k) + D(υ(k))w(k), z(k) = L(υ(k))x(k) + E(υ(k))w(k), where A(υ(k)) = C(υ(k)) = L(υ(k)) =

r i=1 r i=1 r i=1

σi (k)Ai , σi (k)Ci , σi (k)L i ,

B(υ(k)) = D(υ(k)) = E(υ(k)) =

r i=1 r i=1 r i=1

σi (k)Bi , σi (k)Di , σi (k)E i .

(3.55)

3.2 Induced l∞ Filtering with Data Packet Dropouts

77

In this section, the following fuzzy filter is provided to estimate the performance output z(k) in (3.55): x f (k + 1) = A f (υ(k))x f (k) + B f (υ(k)) y¯ (k), z f (k) = C f (υ(k))x f (k) + D f (υ(k)) y¯ (k),

(3.56)

in which x f (k) and z f (k) denote the state and the output of the fuzzy filter, respectively; A f (υ(k)), B f (υ(k)), C f (υ(k)), and D f (υ(k)) are fuzzy filter matrices which can be described as A f (υ(k)) = C f (υ(k)) =

r d=1 r

σd (k)A f d , σd (k)C f d ,

B f (υ(k)) = D f (υ(k)) =

d=1

r d=1 r

σd (k)B f d , σd (k)D f d .

d=1

In this section, suppose that the data packet may be lost, which means that the measurement output y(k) and the performance output z(k) in system (3.55) are transmitted by unreliable communication channel. Then, by using a stochastic method, the data packet dropout phenomenon can be modeled: y¯ (k) = δ y (k)y(k), z¯ (k) = δz (k)z(k),

(3.57)

where {δ y (k)} and {δz (k)} obey Bernoulli process. If y(k) or z(k) fails to be transmitted (i.e., data packet dropout), then δ y (k) = 0 or δz (k) = 0. And δ y (k) = 1 or δz (k) = 1 means the successful transmission of related signals. A nature assumption on {δ y (k)} and {δz (k)} can be obtained: Pr ob{δ y (k) = 0} = 1 − δ¯ y , Pr ob{δ y (k) = 1} = Γ {δ y (k)} = δ¯ y , Pr ob{δz (k) = 0} = 1 − δ¯z , Pr ob{δz (k) = 1} = Γ {δz (k)} = δ¯z .

(3.58)

In the light of (3.54), (3.56), and (3.57), we have x f (k + 1) = A f (υ(k))x f (k) + B f (υ(k))δ y (k)y(k), z f (k) = C f (υ(k))x f (k) + D f (υ(k))δ y (k)y(k), z¯ (k) = δz (k)L(υ(k))x(k) + δz (k)E(υ(k))w(k).

(3.59)

Combining (3.55) and (3.59), the following filtering error system is acquired: r (k + 1) = (A1 + δ˜ y (k)A2 )r (k) + (B1 + δ˜ y (k)B2 )w(k), e(k) = (C1 + δ˜z (k)C2 + δ˜ y (k)C3 )r (k) + (D1 + δ˜z (k)D2 + δ˜ y (k)D3 )w(k), (3.60)

78

3 Fuzzy Filtering with Multiple Signal Transmissions

where r (k) = [ x T (k) x Tf (k) ]T , e(k) = z¯ (k) − z f (k),     A(υ(k)) 0 0 0 A1 = ¯ , , A2 = B f (υ(k))C(υ(k)) 0 δ y B f (υ(k))C(υ(k)) A f (υ(k))     B(υ(k)) 0 , , B2 = B1 = ¯ B f (υ(k))D(υ(k)) δ y B f (υ(k))D(υ(k)) C1 = [ δ¯z L(υ(k)) − δ¯ y D f (υ(k))C(υ(k)) − C f (υ(k)) ],

C2 = [ L(υ(k)) 0 ], C3 = [ −D f (υ(k))C(υ(k)) 0 ], D1 = δ¯z E(υ(k)) − δ¯ y D f (υ(k))D(υ(k)), D2 = E(υ(k)), D3 = −D f (υ(k))D(υ(k)), δ˜ y (k) = δ y (k) − δ¯ y , δ˜z (k) = δz (k) − δ¯z . It is clear that Γ {δ˜ y (k)} = 0, Γ {δ˜z (k)} = 0, Γ {δ˜2y (k)} = δ¯ y (1 − δ¯ y ), and Γ {δ˜z2 (k)} = δ¯z (1 − δ¯z ). Before starting the next section, let us explicitly state the issue of the induced l∞ filtering to be investigated: (1) In the data packet dropout situation, determine a filter (3.56) guaranteeing filtering error system (3.60) to be stochastically stable in the mean square when w(k) = 0, that is, if there exists a finite W > 0 such that ! Γ

∞

# " r (k) "r (0) < r T (0)W r (0). 2"

(3.61)

k=0

(2) For any nonzero w(k) ∈ l∞ [0, ∞), under zero initial condition, the filtering error system meets the following a predefined induced l∞ disturbance attenuation level γ > 0: (3.62) e(k)∞ ≤ γ w(k)∞ .

3.2.2 Main Results In this section, sufficient conditions are given in form of linear matrix inequalities to ensure that filtering error system (3.60) is stochastically stable with a given induced l∞ performance γ [3, 4]. Theorem 3.3 For a given scalar γ > 0, if there exist matrix P > 0 and scalars α ∈ (0, 1), η ∈ (0, 1) such that the following two matrix inequalities hold: 

Ω1 B1T P A1 + μ2y B2T P A2

 ∗ < 0, Ω2

(3.63)

3.2 Induced l∞ Filtering with Data Packet Dropouts

79



Ω3 γ −2 (D1T C1 + μ2z D2T C2 + μ2y D3T C3 )

 ∗ < 0, Ω4

(3.64)

where Ω1 = −P + α P + A1T P A1 + μ2y A2T P A2 , Ω2 = −ηI + B1T P B1 + μ2y B2T P B2 ,  Ω3 = −α P + γ −2 (C1T C1 + μ2z C2T C2 + μ2y C3T C3 ), μ y = δ¯ y (1 − δ¯ y ),  Ω4 = −(1 − η)I + γ −2 (D1T D1 + μ2z D2T D2 + μ2y D3T D3 ), μz = δ¯z (1 − δ¯z ),

the filtering error system in (3.60) is stochastically stable with the induced l∞ performance index γ . Proof Choose a Lyapunov function as V (r (k)) = r T (k)Pr (k). Then, along the trajectories of filtering error system in (3.60) with w(k) = 0, one has Γ {ΔV (r (k))} = Γ {V (r (k + 1))|r (k)} − V (r (k)) = Γ {r T (k)(A1 + δ˜ y (k)A2 )T P(A1 + δ˜ y (k)A2 )r (k)|r (k)} − r T (k)Pr (k) = r T (k)(A1T P A1 + μ2y A2T P A2 − P)r (k).

(3.65) By pre- and post-multiplying [ r T (k) 0 ] and its transpose to (3.63) with r (k) = 0, one obtains r T (k)(A1T P A1 + μ2y A2T P A2 − P)r (k) + αr T (k)Pr (k) < 0. Since 0 < α < 1, the above inequality can be expressed as r T (k)(A1T P A1 + μ2y A2T P A2 − P)r (k) < 0, which implies Γ {ΔV (r (k))} < 0. Define P¯ = A1T P A1 + μ2y A2T P A2 − P. Then one can get ¯ {r (k)2 }. Γ {V (r (k + 1))|r (k)} − V (r (k)) ≤ −λmin (− P)Γ

(3.66)

For any m ≥ 1, by taking mathematical expectation of (3.66) on both sides and summing up the inequality on both sides from 0 to m, it is obtained that ¯ Γ {V (r (m + 1))} − V (r (0)) ≤ −λmin (− P)

m   Γ {r (k)2 } , k=0

80

3 Fuzzy Filtering with Multiple Signal Transmissions

which means that m   ¯ −1 (V (r (0)) − Γ {V (r (m + 1))}). Γ {r (k)2 } ≤ (λmin (− P))

(3.67)

k=0

For all k ≥ 0, in the light of (3.67) and Γ {V (r (k))} ≥ 0, one can derive that Γ

!∞

# " ¯ −1r T (0)λmax (P)r (0) r (k) r (0) ≤ (λmin (− P)) 2"

k=0

¯ −1 λmax (P)r (0) = r T (0)(λmin (− P))

(3.68)

= r T (0)W r (0), ¯ −1 λmax (P). Therefore, the where r (0) is the initial condition and W = (λmin (− P)) filtering error system is stochastically stable in the mean square. In what follows, the induced l∞ performance of the filtering error system in (3.60) with w(k) = 0 will be established. Define ξk = [ r T (k) w T (k) ]T and Λ(k) = Γ {V (r (k))}. Then by pre- and postmultiplying ξkT and its transpose to (3.63), and based on the fact that Λ(k + 1) − Λ(k) = Γ {V (r (k + 1))} − Γ {V (r (k))}, one can deduce that Λ(k + 1) − Λ(k) + αΛ(k) < ηw T (k)w(k).

(3.69)

Set τ = 1 − α, which implies 0 < τ < 1. And multiply (3.69) by τ −k , one gets τ −k Λ(k + 1) − τ −k+1 Λ(k) < τ −k ηw T (k)w(k).

(3.70)

By taking the sum on both sides of the matrix inequality (3.70), from 0 to k − 1, it can be derived that k−1  k−1    τ − j Λ( j + 1) − τ − j+1 Λ( j) < τ − j ηw T ( j)w( j) j=0

j=0

< ηsupw(k)

2

k≥0

k−1

(3.71) τ

−j

.

j=0

After summing up for (3.71), it can be obtained that τ −k+1 Λ(k) − τ Λ(0) < ηsupw(k)2 k≥0

1 − τ −k . 1 − τ −1

(3.72)

Under zero initial condition and 0 < τ < 1, one can conclude from (3.72) that Λ(k)
0, δ¯ y , and δ¯z , if there exist scalars η ∈ (0, 1), β > 0, and σ > 0, matrices M1 , M2 , M3 , M4 , G 1 , G 2 , G 3 , A¯ f d , B¯ f d , C¯ f d , and D¯ f d satisfying the following LMIs: σ − β < 0, (3.76) Υ¯id < 0, i, d = 1, . . . , r,

(3.77)

Ψ¯ id < 0, i, d = 1, . . . , r,

(3.78)

where     ∗ M ∗ Υ Υ¯id = 11id , Υ , Υ3 = diag{ −β I, −σ I, −σ I }, = 11id T Eid G Υ12T Υ3

M = diag{ −M − M T + σ I, −ηI }, G = diag{ −G − G T , −G − G T },      T  T A1id μ y A2id G1 G2 G 1 Ai + δ¯ y B¯ f d Ci A¯ f d , Eid = , , G= A = 1id T T G3 G2 B1id μ y B2id G 3 Ai + δ¯ y B¯ f d Ci A¯ f d       B¯ f d Ci 0 G 1 Bi + δ¯ y B¯ f d Di B¯ f d Di , B2id = ¯ , , B1id = A2id = ¯ B f d Ci 0 G 3 Bi + δ¯ y B¯ f d Di B f d Di    T   M0 M1 M2 Ψ1id ∗ ∗ , Ψ¯ id = , Υ12 = , M= T M3 M4 Ψ3 Ψ2id 0 diag{ M, M }   Ψ˜ ∗ , Ψ˜ 11 = diag{ −M − M T + β I, (η − 1)I }, Ψ1id = ˜ 11 Ψ12id −γ 2 I

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3 Fuzzy Filtering with Multiple Signal Transmissions

1 1 Ψ˜ 12id = [ Ψ˜ 12id δ¯z E i − δ¯ y D¯ f d Di ], Ψ˜ 12id = [ δ¯z L i − δ¯ y D¯ f d Ci − C¯ f d ], ⎡ ⎤ T T [ μz L i 0 ] [ −μ y D¯ f d Ci 0 ] −μ y DiT D¯ Tf d ⎦ , Ψ3 = diag{ −γ 2 I, −γ 2 I }, Ψ2id = ⎣ μz E iT 0 0

the filtering error system in (3.60) is stochastically stable with the induced l∞ performance index γ , and the induced l∞ filter gain matrices in (3.56) can be obtained by the following equations: −1 ¯ ¯ ¯ ¯ A f d = G −1 2 A f d , B f d = G2 B f d , C f d = C f d , D f d = D f d .

(3.79)

Proof By Schur complement (Lemma 1.1), the matrix inequality (3.63) can be rewritten as   F ∗ < 0, (3.80) E T G1 where F = diag{ −P + α P, −ηI }, E =

  T A1 μ y A2T , G1 = diag{ −P −1 , −P −1 }. B1T μ y B2T

Define G I = diag{I, I, G, G} with G being a nonsingular matrix. Then, by performing congruence transformations with G I to (3.80), one has 

 F ∗ < 0, EGT G2

(3.81)

where  A1T G T μ y A2T G T , G2 = −diag{ G P −1 G T , G P −1 G T }. EG = B1T G T μ y B2T G T 

Based on the fact that −(G − P)P −1 (G − P)T ≤ 0 with P > 0, which implies −G P −1 G T ≤ −G − G T + P, (3.81) can be verified by 

 F ∗ < 0, EGT G3

where

(3.82)

G3 = diag{ −G − G T + P, − G − G T + P }.

Next, to eliminate the coupling items in (3.82), such as, α P, A1T G T , μ y A2T G T , and μ y B2T G T , assuming that the Lyapunov matrix has the following form

B1T G T ,

3.2 Induced l∞ Filtering with Data Packet Dropouts

P=

1 T M M, σ

83

(3.83)

where M is a nonsingular matrix and σ > 0. Substituting (3.83) into (3.82) and according to − σ1 M T M ≤ −M − M T + σ I , (3.82) can be rewritten as 

   ˘ ∗ M1 ∗ M + < 0, 0 M4 EGT G

(3.84)

where   $ % ˘ = diag −M − M T + σ I, −ηI , M1 = diag α M T M, 0 , M σ & ' 1 T 1 T M M, M M . M4 = diag σ σ Then based on Schur complement (Lemma 1.1), (3.84) can be rewritten as  Υ˘1 ∗ < 0, Υ˘2T Υ˘3



(3.85)

where  T   ˘ ∗ M0 ∗ M ˘ ˘ Υ1 = , Υ2 = , Υ˘22 = diag{ M, M }, 0 Υ˘22 EGT G   σ Υ˘3 = diag − I, −σ I, −σ I . α In order to further eliminate the coupling items in (3.85), constructing the following matrices:   G1 G2 , A¯ f (υ(k)) = G 2 A f (υ(k)), G= G3 G2   M1 M2 , B¯ f (υ(k)) = G 2 B f (υ(k)), M= M3 M4 where G 2 is nonsingular without loss of generality. Then (3.85) can be ensured by  Υ˜1 ∗ < 0, Υ˘2T Υ˘3



(3.86)

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3 Fuzzy Filtering with Multiple Signal Transmissions

where      T  T ˘ ∗ A1G μ y A2G B¯ f (υ(k))C(υ(k)) 0 M , , Υ˜ G = , Υ˜1 = ˜ T A = 2G T T B1G μ y B2G B¯ f (υ(k))C(υ(k)) 0 ΥG G   G 1 A(υ(k)) + δ¯ y B¯ f (υ(k))C(υ(k)) A¯ f (υ(k)) , A1G = G 3 A(υ(k)) + δ¯ y B¯ f (υ(k))C(υ(k)) A¯ f (υ(k))     G 1 B(υ(k)) + δ¯ y B¯ f (υ(k))D(υ(k)) B¯ f (υ(k))D(υ(k)) , B2G = ¯ , B1G = G 3 B(υ(k)) + δ¯ y B¯ f (υ(k))D(υ(k)) B f (υ(k))D(υ(k)) which means that (3.86) can guarantee (3.63). Along the same line as the proof in the condition (3.63), by setting C¯ f (υ(k)) = C f (υ(k)) and D¯ f (υ(k)) = D f (υ(k)), it can be concluded that the matrix inequality (3.64) holds if the following condition is satisfied:  Ψ˘ 1 ∗ < 0, Ψ˘ 2T Ψ3



(3.87)

with     σ Ψ˘ ∗ ˘ 11 = diag −M − M T + I, (η − 1)I , Ψ˘ 1 = ˘ 11 , Ψ Ψ12 −γ 2 I α 1 ¯ Ψ˘ 12 = [Ψ˘ 12 δ¯z E(υ(k)) − δ¯ y D f (υ(k))D(υ(k)) ], 1 Ψ˘ 12 = [ δ¯z L(υ(k)) − δ¯ y D¯ f (υ(k))C(υ(k)) − C¯ f (υ(k)) ], T  μz E(υ(k)) 0 [ μz L(υ(k)) 0 ] Ψ˘ 2 = . [ −μ y D¯ f (υ(k))C(υ(k)) 0 ] −μ y D¯ f (υ(k))D(υ(k)) 0 Define β = σα . Then, in the light of condition (3.76) with σ > 0 and β > 0, it follows from (3.86) and (3.87) that r r

σi (k)σd (k)Υ¯id < 0,

(3.88)

σi (k)σd (k)Ψ¯ id < 0.

(3.89)

i=1 d=1

and

r r i=1 d=1

Therefore, the inequalities (3.88) and (3.89) can be guaranteed by the conditions (3.77) and (3.78), respectively. The proof is completed.

3.2 Induced l∞ Filtering with Data Packet Dropouts

85

Fig. 3.5 Mass-springdamper system

Table 3.1 The related parameters of the mass-spring-damper system m Mass of car m = 1.5 kg k Stiffness coefficient of spring k = 341.3 N/m f Viscous friction coefficient of damper f = 18.67 N · s/m

3.2.3 Simulation Example In this section, an example of the mass-spring-damper mechanical system [5] is given to demonstrate the effectiveness of the designed method. As shown in Fig. 3.5, we consider a car m vibrating on a horizontal surface within a bounded region and attached to a vertical surface through a spring and a damper. The mass is subject to an external force u = Fu (t). Assume that friction force does not exist due to static and Coulomb friction. Then Newton’s law of motion given as m y¨ (t) = Fu (t) − Fs (t) − Fd (t) + w(t),

(3.90)

where y(t) is the displacement from a reference position, Fu (t) is the restoring force of the spring, Fd (t) is a resistive force of the damper, and w(t) is the external disturbance. Supposing that Fs (t) = ky(t) + y 3 (t) with k being a real constant known as the stiffness coefficient, which signifies that the spring is nonlinear one. Fd (t) = f y˙ (t) is the dynamic property of the viscous damper with f being the viscous friction coefficient. Table 3.1 gives some related parameters of the mechanical system. Notice that when the external force Fu (t) = 0, the mass-spring-damper mechanical system in (3.90) is stable. Thus, we can choose Fu (t) = 0 for studying the filtering problem. Set the displacement y(t) and its derivative y˙ (t) as the state variables of system (3.90), that is x1 (t) = y(t), x2 (t) = y˙ (t).

(3.91)

From (3.91), system (3.90) can be controlled x˙1 (t) = x2 (t), m x˙2 (t) = −kx1 (t) − x13 (t) − f x2 (t) + w(t). Based on parameters in Table 3.1, (3.92) can be rewritten

(3.92)

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3 Fuzzy Filtering with Multiple Signal Transmissions

x˙1 (t) = x2 (t), x˙2 (t) = −(341.3/1.5)x1 (t) − (1/1.5)x13 (t) − (18.67/1.5)x2 (t) + (1/1.5)w(t). (3.93) Suppose that − 1 ≤ x1 (t) ≤ 1, i.e., 0 ≤ x12 (t) ≤ 1. Then a continuous-time T–S system can be obtained to approximate the mass-spring-damper system as follows: x(t) ˙ =

2

  σi (t) Aib x(t) + Bib w(t) ,

(3.94)

i=1



where

 0 1 , − 341.3/1.5 − 18.67/1.5   0 1 = , − 342.3/1.5 − 18.67/1.5



 0 , 1/1.5   0 = . 1/1.5

A1b =

B1b =

A2b

B2b

Define the premise variable h 1 (t) = x12 (t), which implies that the maximum and minimum values of h 1 (t) are 1 and 0, respectively. Then, h 1 (t) can be rewritten as h 1 (t) = σ1 (t) × 1 + σ2 (t) × 0, σ1 (t) + σ2 (t) = 1. And, the membership functions can be calculated as σ1 (t) = x12 (t), σ2 (t) = 1 − σ1 (t).

(3.95)

Based on (3.95), we will address the induced l∞ filtering problem of system (3.93). Set sampling time T = 0.5 s. Then discrete-time system (3.55) is obtained and the corresponding system parameters are given as 

   0.0482 0.0018 0.0028 A1 = , B1 = , − 0.4084 0.0259 0.0012     − 0.0231 0.0017 0.0020 , B2 = , A2 = − 0.5752 − 0.0440 0.0011 C1 = [ 1 0 ], C2 = [ 1 0 ], L 1 = [ 1 0.6 ],

D1 = 0.5, D2 = 0.5,

E 1 = 0.1, E 2 = 0.1,

L 2 = [ 1 0.6 ].

Choose Γ {δ y (k)} = δ¯ y = 0.9 and Γ {δz (k)} = δ¯z = 0.85. Then, by solving LMIs (3.76), (3.77), and (3.78) in Theorem 3.4, we can obtain γmin = 0.0466 and the following related matrix variables:

3.2 Induced l∞ Filtering with Data Packet Dropouts

87

    79.8213 410.8237 − 24.4117 , B¯ f 1 = , A¯ f 1 = 60.0165 38.6745 − 8.3932     79.8213 410.8237 − 24.4117 A¯ f 2 = , B¯ f 2 = , 60.0165 38.6745 − 8.3932 C¯ f 1 = [ − 0.8511 −1.0968 ], D¯ f 1 = 0.1715, C¯ f 2 = [ − 0.8511 −1.0968 ],   3886.1 1202.1 . G2 = 644.2 1904.5

D¯ f 2 = 0.1715,

Substituting the above results into (3.79), the corresponding gains can be obtained 

   0.0121 0.1111 − 0.0055 , Bf1 = , 0.0274 − 0.0173 − 0.0025     0.0121 0.1111 − 0.0055 = , Bf2 = , 0.0274 − 0.0173 − 0.0025 = C¯ f 1 , D f 1 = D¯ f 1 , C f 2 = C¯ f 2 , D f 2 = D¯ f 2 .

Af1 = Af2 Cf1

To show the performance of the designed filter, set the initial conditions x(0) = x f (0) = [ 0 0 ]T and the disturbance w(k) = 12e−2k cos(0.1k). Then, the response of x(k) and its estimate x f (k) are displayed in Figs. 3.6 and 3.7, respectively. The simulation results of signals z(k) and z f (k) are shown in Fig. 3.8. Figure 3.9 displays the response of error signale(k), which can be perceived that the error curve tends to 0. The simulation ratio of e T (k)e(k)/w(k)2∞ is shown in Fig. 3.10, from which we can see that the maximum value of the ratio is 0.0142. It is less than γmin = 0.0466.

Fig. 3.6 Response of the system state x(k)

0.025 0.02 0.015 0.01 0.005 0 -0.005 -0.01 0

5

10

15

20

88 Fig. 3.7 Response of the filter state x f (k)

3 Fuzzy Filtering with Multiple Signal Transmissions 0.01 0 -0.01 -0.02 -0.03 -0.04 0

Fig. 3.8 Responses of z(k) and z f (k)

5

10

15

20

5

10

15

20

5

10

15

20

1.2 1 0.8 0.6 0.4 0.2 0 -0.2 0

Fig. 3.9 Filtering error e(k)

0.2

0.15

0.1

0.05

0 0

3.3 H∞ Filtering with Dynamic Quantization Fig.  3.10 History of e T (k)e(k)/w(k)2∞

89 0.015

0.01

0.005

0 0

5

10

15

20

3.3 H∞ Filtering with Dynamic Quantization In this section, the H∞ filtering problem for T–S fuzzy system with multi-signal dynamic quantization is considered, i.e., the measurement output and the performance output are quantized by dynamic quantizers in this section. And the conditions to design H∞ filter with dynamic quantization are given in the form of a set of LMIs.

3.3.1 Problem Formulation For the investigated H∞ filtering problem in this section, we consider the following T–S fuzzy model: Plant Rule i th : IF h 1 (k) is N1i , h 2 (k) is N2i , and, . . . , and h θ (k) is Nθi , THEN x(k + 1) = Ai x(k) + Bi w(k), y(k) = Ci x(k), z(k) = L i x(k),

(3.96)

where h(k) = [h 1 (k), h 2 (k), . . . , h θ (k)] is the vector of premise variables. N ji is the fuzzy sets with j = 1, 2, . . . , θ, i = 1, 2, . . . , r . x(k) ∈ Rn x stands for the state variable; y(k) ∈ Rn y stands for the measurement output of the system; z(k) ∈ Rn z stands for the performance output of the system; w(k) ∈ Rn w refers to the noise input which belongs to l2 [0, ∞); Ai , Bi , Ci and L i are given system matrices with appropriate dimensions.

90

3 Fuzzy Filtering with Multiple Signal Transmissions

As in Sect. 3.1, the T–S fuzzy model can be deduced as follows: x(k + 1) = A(ρ)x(k) + B(ρ)w(k), y(k) = C(ρ)x(k),

(3.97)

z(k) = L(ρ)x(k), where A(ρ) = C(ρ) =

r i=1 r i=1

σi (k)Ai , σi (k)Ci ,

B(ρ) = L(ρ) =

r i=1 r

σi (k)Bi , σi (k)L i .

i=1

Unlike static quantizer that require an infinite number of quantization levels, we construct the dynamic quantizer which will be defined as the one given in [6], for quantized signal τ ∈ Rl , the relationship between range and error bound of dynamic quantizer is given as !

 f (τ ) − τ  ≤ Δ, i f τ  ≤ M,  f (τ ) > M − Δ, i f τ  > M,

(3.98)

where M and Δ are used to represent the quantizer’s range and error bound, respectively. In the control strategy to be developed below, we consider the one-parameter family of quantizers as ( ) τ f μ (τ ) = μf , μ > 0, (3.99) μ where μ is dynamic parameter of the quantizer. Note that if μ is zero, this will mean that the output of the quantizer is set to zero. Due to the limitation of bandwidth and other factors of the communication channel, the output of the system may be subject to some unpredictable situations. In order to reduce the possibility of uncertainties, it is necessary to quantify the measurement and performance outputs. In this section, we have the following signal quantization expression: ) ( y(k) , f μ (y(k)) = μ y f μy (3.100) ) ( z(k) . f μ (z(k)) = μz f μz Therefore, a filter with measurement output quantization is designed to estimate the state of the system, which is given as follows: x f (k + 1) = A f x f (k) + B f y˜ (k), z f (k) = C f x f (k) + D f y˜ (k),

(3.101)

3.3 H∞ Filtering with Dynamic Quantization

91

where x f (k) ∈ Rn x and z f (k) ∈ Rn z stand for the state variable and the output of filter, respectively; y˜ (k) ∈ Rn y is the quantized measurement output with the form of (3.100); A f , B f , C f and D f are the matrices of the filter system with the appropriate dimensions. Further, the following equation can be obtained: ) y(k) , x f (k + 1) = A f x f (k) + B f μ y f μy ( ) y(k) z f (k) = C f x f (k) + D f μ y f . μy (

(3.102)

Combining (3.99), (3.100) and (3.102), the following equation can be obtained: ( ( ) ) y(k) y(k) f − , μy μy ( ( ) ) y(k) y(k) − . z f (k) = C f x f (k) + D f y(k) + D f μ y f μy μy

x f (k + 1) = A f x f (k) + B f y(k) + B f μ y

(3.103)

By defining e(k) = z˜ (k) − z f (k) where z˜ (k) is the quantized performance output with the form of (3.100), the filtering error system can be obtained as follows: Θ(k + 1) = A1 (ρ)Θ(k) + B(ρ)w(k) + E1 V y (k), e(k) = C L1 (ρ)Θ(k) + D L1 V y (k) + Vz (k), y(k) = C y (ρ)Θ(k), z(k) = L z (ρ)Θ(k),

(3.104)

where  T Θ(k) = x T (k) x Tf (k) ,       A(ρ) 0 B(ρ) 0 , B(ρ) = , A1 (ρ) = , E1 = B f C(ρ) A f 0 Bf  C L1 (ρ) = L(ρ) − D f C(ρ) −C f , D L1 = −D f , ( ( ) )  y(k) y(k) − , C y (ρ) = C(ρ) 0 , V y (k) = μ y f μy μy ( ( ) )  z(k) z(k) − , L z (ρ) = L(ρ) 0 . Vz (k) = μz f μz μz The purpose of this section is to design filter given in (3.102) with multi-signal quantization, such that the filtering error system (3.104) satisfies the following conditions:

92

3 Fuzzy Filtering with Multiple Signal Transmissions

(1) The filtering error system in (3.104) is said to be asymptotically stable when w(k) = 0; (2) Under zero initial condition, for any w(k) = 0 and a given γ > 0, the filtering error e(k) satisfies ∞ k=0

e T (k)e(k) < γ 2

∞

w T (k)w(k).

k=0

3.3.2 Main Results Theorem 3.5 For given quantizer’s ranges Mδ and quantization errors Δδ , δ = y, z, the filtering error system (3.104) is asymptotically stable with the prescribed H∞ performance index γ > 0, if there exists matrix P > 0, positive scalars ρδ and δ , δ = y, z such that the following matrix inequalities hold: M y > 1/ρ y , Mz > 1/ρz , ⎡

−P ∗ ∗ ∗ ∗ 2 ⎢ 0 −γ I ∗ ∗ ∗ ⎢ ⎢ ∗ 0 0 − y I ∗ ⎢ ⎢ I ∗ 0 0 0 − z ⎢ −1 ⎢ A (ρ) B (ρ) E 0 −P 1 1 ⎢ ⎢ 0 D L1 I 0 C L1 (ρ) ⎢ ⎣2 y ρ y Δ y C y (ρ) 0 0 0 0 0 0 0 2z ρz Δz L z (ρ) 0

⎤ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ ∗ ∗ ∗ ⎥ ⎥ ∗ ∗ ∗ ⎥ ⎥ < 0, ∗ ∗ ∗ ⎥ ⎥ −I ∗ ∗ ⎥ ⎥ 0 − y I ∗ ⎦ 0 0 −z I

(3.105)

(3.106)

where the online adjusting strategy for quantizer’s parameters μδ as ρδ δ(k) ≤ μδ ≤ 2ρδ δ(k) , δ = y, z.

(3.107)

Proof Firstly, we will confirm the truth that the filtering error system in (3.104) is asymptotically stable with w(k) = 0 if the conditions in (3.105) and (3.106) can be guaranteed. Combining (3.105) and (3.107), the following inequality can be obtained: 1 μδ ≤ δ(k) ≤ Mδ μδ . 2ρδ It is clear that the definition of (3.98) is satisfied, so    δ(k)     μ  ≤ Mδ , δ = y, z, δ

(3.108)

(3.109)

3.3 H∞ Filtering with Dynamic Quantization

  ( )    f δ(k) − δ(k)  ≤ Δδ .  μδ μδ 

then

93

(3.110)

Based on the homogeneity property of Euclidean norm and the definition of Vδ (k) in (3.104), one gives  ( ( ) )  δ(k) δ(k)    Vδ (k) = μδ f − μδ μδ   (  ) (3.111)  δ(k) δ(k)    = μδ  f ≤ μδ Δδ . − μδ μδ  Obviously, combining (3.107), (3.111) can be rewritten as Vδ (k) ≤ 2ρδ Δδ δ(k) , i.e.,

VTy (k)V y (k) ≤ 4ρ y2 Δ2y (C y (ρ)Θ(k))T C y (ρ)Θ(k), VzT (k)Vz (k) ≤ 4ρz2 Δ2z (L z (ρ)Θ(k))T L z (ρ)Θ(k).

(3.112)

(3.113)

 Then let’s define Ω T (k) = Θ T (k) w T (k) VTy (k) VzT (k) , according to (3.104), it yields Ω T (k)Υ1 Ω(k) ≥ 0, Ω T (k)Υ2 Ω(k) ≥ 0, where

(3.114)

$ % 2ρ y Δ y C y (ρ) 0 0 0 − diag{0, 0, I, 0}, $ % Υ2 = H a 2ρz Δz L z (ρ) 0 0 0 − diag{0, 0, 0, I }.

Υ1 = H a

Then, by using Schur complement (Lemma 1.1), the condition (3.106) can be rewritten as (3.115) Υ0 + Υ3 +  y Υ1 + z Υ2 < 0, where   T Υ0 = A1 (ρ) B(ρ) E1 0 P A1 (ρ) B(ρ) E1 0 − diag{P, 0, 0, 0},  Υ3 = H a{ C L1 (ρ) 0 D L1 I } − diag{0, γ 2 I, 0, 0}. Thus, from Lemma 1.3 and (3.114), we have Ω T (k)(Υ0 + Υ3 )Ω(k) < 0. Then, the Lyapunov function is adopted as

(3.116)

94

3 Fuzzy Filtering with Multiple Signal Transmissions

V (Θ(k)) = Θ T (k)PΘ(k), P > 0,

(3.117)

and we can obtain V (Θ(k + 1)) − V (Θ(k)) T  = A1 (ρ)Θ(k) + B(ρ)w(k) + E1 V y (k) P   × A1 (ρ)Θ(k) + B(ρ)w(k) + E1 V y (k) − Θ T (k)PΘ(k)

(3.118)

= Ω T (k)Υ0 Ω(k). Combining (3.116) and (3.118) can obtain that V (Θ(k + 1)) − V (Θ(k)) + e T (k)e(k) − γ 2 w T (k)w(k) < 0.

(3.119)

Under the condition of external disturbance w(k) = 0, we have ΔV = V (Θ(k + 1)) − V (Θ(k)) < V (Θ(k + 1)) − V (Θ(k)) + e T (k)e(k) < 0, (3.120) which means the filtering error system (3.104) is asymptotically stable. Next, in order to analyze the H∞ performance, taking the summation of the both sides of (3.119) from 0 to ∞, one can be obtained that V (Θ(∞)) − V (Θ(0)) +

∞

e T (k)e(k) −

k=0

∞

γ 2 w T (k)w(k) < 0.

(3.121)

k=0

With zero condition V (Θ(0)) = 0 and V (Θ(∞)) ≥ 0, we obtain ∞ k=0

e T (k)e(k)
0, if there exist matrices P1 > 0, P2 , P3 > 0, G 1 , G 2 , G 3 and positive scalars η, σδ , δ , δ = y, z such that the following linear matrix inequalities hold: σ y M y >  y , σz M z > z ,

(3.123)

⎡

−P1 ∗ ∗ ∗ ∗ ⎢ −P3 ∗ ∗ ∗ −P2 ⎢ 2I ⎢ 0 0 −γ ∗ ∗ ⎢ ⎢ 0 0 0 − y I ∗ ⎢ ⎢ 0 0 0 0 −z I ⎢ ⎢ G 1 Ai + B¯ f Ci A¯ f G 1 Bi B¯ f 0 ⎢ ⎢ G A + B¯ C A¯ ¯ G B 0 B 2 i f i f f ⎢ 2 i ⎢ L − D¯ C ¯ ¯ − C 0 − D I i f i f f ⎢ ⎣ 2σ y Δ y C 0 0 0 0 i 0 0 0 0 2σz Δz L i

⎤ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ < 0, i = 1, . . . , r. T ∗ ∗ ∗ ∗ ⎥ −G 1 − G 1 + P1 ⎥ ∗ ∗ ⎥ −G 2 − G 3T + P2 −G 3 − G 3T + P3 ∗ ⎥ 0 0 −I ∗ ∗ ⎥ ⎥ 0 0 0 − y I ∗ ⎦ 0 0 0 0 −z I

(3.124)

Moreover, the online adjusting strategy for quantizer’s parameters μδ are given in (3.107) and the filtering matrices are given as −1 ¯ ¯ ¯ ¯ A f = G −1 3 A f , B f = G3 B f , C f = C f , D f = D f .

(3.125)

Proof First, by defining ρδ = σδ /δ , δ = y, z, (3.105) can be rewritten as M y >  y /σ y ,

Mz > z /σz .

(3.126)

Therefore, (3.123) can be obtained from (3.126) by inequality transformation. $ Then we perform congruence transformation to (3.106) by G= diag I, I, I, I, G T , I, I, I } and its transpose, respectively, one has

96

3 Fuzzy Filtering with Multiple Signal Transmissions

⎤ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ ∗ ∗ ∗ ⎥ ⎥ ∗ ∗ ∗ ⎥ ⎥ < 0. ∗ ∗ ∗ ⎥ ⎥ −I ∗ ∗ ⎥ ⎥ 0 − y I ∗ ⎦ 0 0 −z I (3.127) By using −G T P −1 G < −G − G T + P, the following inequality can be obtained: ⎡

−P ∗ ∗ ∗ ∗ 2 ⎢ 0 −γ I ∗ ∗ ∗ ⎢ ⎢ ∗ 0 0 − y I ∗ ⎢ ⎢ I ∗ 0 0 0 − z ⎢ ⎢ G T A1 (ρ) G T B(ρ) G T E1 0 −G T P −1 G ⎢ ⎢ 0 D L1 I 0 C L1 (ρ) ⎢ ⎣2 y ρ y Δ y C y (ρ) 0 0 0 0 2z ρz Δz L z (ρ) 0 0 0 0

⎡

⎤ −P ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎢ 0 −γ 2 I ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ ⎥ I ∗ ∗ ∗ ∗ ∗ 0 0 − y ⎢ ⎥ ⎢ ⎥ I ∗ ∗ ∗ ∗ 0 0 0 − z ⎢ ⎥ < 0. ⎢ G T A1 (ρ) G T B(ρ) G T E1 0 −G − G T + P ∗ ⎥ ∗ ∗ ⎢ ⎥ ⎢ ⎥ (ρ) 0 D I 0 −I ∗ ∗ C L1 L1 ⎢ ⎥ ⎣2 y ρ y Δ y C y (ρ) 0 0 0 0 0 − y I ∗ ⎦ 0 0 0 0 0 0 −z I 2z ρz Δz L z (ρ) (3.128)     P1 ∗ G1 G3 T Furthermore, let’s assume that P = ,G = , one has G2 G3 P2 P3 ⎡

−P1 ⎢ −P2 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢Γ1 (ρ) ⎢ ⎢Γ2 (ρ) ⎢ ⎢Γ3 (ρ) ⎣Γ (ρ) 4 Γ5 (ρ)

⎤ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −P3 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ 0 −γ 2 I ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ 0 0 − y I ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ 0 0 0 −z I ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ T G 3 A f G 1 B(ρ) G 3 B f 0 −G 1 − G 1 + P1 ∗ ∗ ∗ ∗ ⎥ ⎥ T T G 3 A f G 2 B(ρ) G 3 B f 0 −G 2 − G 3 + P2 −G 3 − G 3 + P3 ∗ ∗ ∗ ⎥ ⎥ −C f 0 −D f I 0 0 −I ∗ ∗ ⎥ 0 0 0 0 0 0 0 − y I ∗ ⎦ 0 0 0 0 0 0 0 0 −z I < 0,

(3.129) where Γ1 (ρ) = G 1 A(ρ) + G 3 B f C(ρ), Γ2 (ρ) = G 2 A(ρ) + G 3 B f C(ρ), Γ3 (ρ) = L(ρ) − D f C(ρ), Γ4 (ρ) = 2σ y Δ y C(ρ), Γ5 (ρ) = 2σz Δz L(ρ).

3.3 H∞ Filtering with Dynamic Quantization

97

By defining A¯ f = G 3 A f , B¯ f = G 3 B f , C¯ f = C f and D¯ f = D f , then the inequality (3.129) can be rewritten as ⎡

−P1 ∗ ⎢ −P −P 2 3 ⎢ ⎢ 0 0 ⎢ ⎢ 0 0 ⎢ ⎢ 0 0 ⎢ ⎢ G 1 A(ρ) + B¯ f C(ρ) A¯ f ⎢ ⎢ G 2 A(ρ) + B¯ f C(ρ) A¯ f ⎢ ⎢ L(ρ) − D¯ f C(ρ) −C¯ f ⎢ ⎣ 2σ y Δ y C(ρ) 0 0 2σz Δz L(ρ) ∗ ∗ −γ 2 I 0 0 G 1 B(ρ) G 2 B(ρ) 0 0 0

⎤ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ ∗ ∗ ∗ ∗ ∗ ⎥ − y I ∗ ⎥ ∗ ∗ ∗ ∗ ∗ ⎥ 0 −z I ⎥ < 0. B¯ f 0 −G 1 − G 1T + P1 ∗ ∗ ∗ ∗ ⎥ ⎥ B¯ f 0 −G 2 − G 3T + P2 −G 3 − G 3T + P3 ∗ ∗ ∗ ⎥ ⎥ − D¯ f I 0 0 −I ∗ ∗ ⎥ ⎥ 0 0 0 0 0 − y I ∗ ⎦ 0 0 0 0 0 0 −z I (3.130)

Therefore, one can obtain that (3.124) can be guaranteed if the condition in (3.130) is satisfied. The proof is completed. Remark 3.1 In fact, according to the work studied in [7], the dynamic quantizer parameters μδ should also be determined as a unique value by a specific tuning strategy. Such a dynamic adjustment strategy may yield smaller quantization errors compared to the general static adjustment strategy [8–10]. From the online adjusting strategy in (3.107), we know that the dynamic parameters μδ are related to quantized signals δ(k) and the scalars ρδ , δ = y, z. Noted that ρδ = σδ /δ , δ = y, z which can be obtained by solving the conditions in Theorem 3.6. Remark 3.2 In this section, the proposed conditions in Theorem 3.6 are strict LMIs. Therefore, the filter gain matrices can be obtained by solving (3.123) and (3.124) in Matlab LMI toolbox. Then, the following optimization problem is utilized: min γ subject to LMIs (3.123) and (3.124).

(3.131)

98

3 Fuzzy Filtering with Multiple Signal Transmissions

3.3.3 Simulation Example A numerical example will be shown to demonstrate that the designed H∞ filter strategy is effective. And parameter matrices of the system (3.97) will be given as follows:     0.7 − 0.05 0.1 , B1 = , A1 = 0.1 0.8 0.1     0.94 0.3 0.3 , B2 = , A2 = − 0.05 0.6 0.1   C1 = 0.2 0.1 , L 1 = 1 0.1 ,   C2 = 1.2 0 , L 2 = 2 0.5 . In addition, member function is set to σ1 (k) = sin2 (0.05x1 (k)) , σ2 (k) = 1 − σ1 (k).

(3.132)

Furthermore, we adopt the quantization errors Δ y = Δz = 0.01, quantization ranges M y = Mz = 50, the other parameters  y = 5, z = 5 and select the H∞ performance index γ = 3. According to the form of the LMIs of Theorem 3.6, the unknown parameters in inequalities (3.123) and (3.124) can be obtained by using the LMI toolbox, then we have     0.5175 0.2063 − 0.3696 , Bf = , Af = 0.0778 0.5582 0.0512 (3.133)  C f = − 0.9188 − 0.0438 , D f = 0.8616. Moreover, we assume that x(0) = [ 0 0 ]T and w(k) = sin(k)e−k , the simulation results are given in Figs. 3.11, 3.12, 3.13, 3.14, 3.15, 3.16 and 3.17. Figure 3.11 plots the response of system state x(k). Figure 3.12 plots the response of filter state x f (k). Figure 3.13 plots the responses of z(k) and z f (k). Figure 3.14 plots the filtering error e(k). Figure 3.15 plots the response of quantizer’s parameter μ y . Fig. 3.11 Response of the system state x(k)

0.15 x1 (k) x2 (k) 0.1

0.05

0

-0.05 0

20

40

60

k

80

100

3.3 H∞ Filtering with Dynamic Quantization Fig. 3.12 Response of the filter state x f (k)

99 0.02 xf 1 (k) xf 2 (k)

0 -0.02 -0.04 -0.06 -0.08 -0.1 -0.12 0

20

40

60

80

100

k

Fig. 3.13 Responses of z(k) and z f (k)

0.3 z(k) zf (k)

0.25 0.2 0.15 0.1 0.05 0 0

20

40

60

80

100

60

80

100

k

Fig. 3.14 Filtering error e(k)

0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 0

20

40

k

Figure of quantizer’s parameter μz . Figure 3.17 plots the ratio  3.16 plots the response

∞ ∞ T T of k=0 e (k)e(k)/ k=0 w (k)w(k). It can be seen that the convergence value of the trajectory is 0.5079 < 3 from Fig. 3.17.

100 Fig. 3.15 Dynamic quantizer’s parameter μ y

3 Fuzzy Filtering with Multiple Signal Transmissions 0.3 0.25 0.2 0.15 0.1 0.05 0 0

20

40

60

80

100

60

80

100

60

80

100

k

Fig. 3.16 Dynamic quantizer’s parameter μz

0.6 0.5 0.4 0.3 0.2 0.1 0 0

20

40

k

Fig.  3.17 History of

∞ ∞ T T k=0 e (k)e(k)/ k=0 w (k)w(k)

0.6 0.5 0.4 0.3 0.2 0.1 0 0

20

40

k

References

101

3.4 Conclusion In this chapter, the problem of filtering for T–S fuzzy systems with the various induced phenomena, which occur during the measurement output signal and the performance output signal are transmitted through the communication channel, has been studied. The conditions have been given to make the filtering error systems satisfy the l2 –l∞ , induced l∞ and H∞ performance, respectively, while the filtering error systems are stable. First of all, the l2 –l∞ filter has been designed, when the measurement output signal and the performance output signal have been quantized by static quantizers. In the second place, the induced l∞ filter has been designed, when the measurement output signal and the performance output signal have been considered to occur data packet dropout. In addition, the H∞ filter has been designed, when the measurement output signal and the performance output signal have been quantized by dynamic quantizers. At last, some simulation results have been provided to verify the effectiveness of these l2 –l∞ , induced l∞ and H∞ filtering strategies for T–S fuzzy systems.

References 1. Fu M, Xie L (2005) The sector bound approach to quantized feedback control. IEEE Trans Autom Control 50:1698–1711 2. Chang XH, Wang YM (2018) Peak-to-peak filtering for networked nonlinear DC motor systems with quantization. IEEE Trans Industr Inf 14:5378–5388 3. Ahn CK, Shi P, Basin MV (2018) Two-dimensional peak-to-peak filtering for stochastic Fornasini-Marchesini systems. IEEE Trans Autom Control 63:1472–1479 4. Fan C, Lam J, Xie X (2018) Peak-to-peak filtering for periodic piecewise linear polytopic systems. Int J Syst Sci 49:1997–2011 5. Burchett B (2005) Parametric time domain system identification of a mass spring damper system. In 2005 Annual Conference 6. Liberzon D (2003) Hybrid feedback stabilization of systems with quantized signals. Automatica 39:1543–1554 7. Chang XH, Yang C, Xiong J (2019) Quantized fuzzy output feedback H∞ control for nonlinear systems with adjustment of dynamic parameters. IEEE Trans Syst, Man, Cybern: Syst 40:2005– 2015 8. Gao H, Chen T (2008) A new approach to quantized feedback control systems. Automatica 44:534–542 9. Zhang C, Feng G, Gao H, Qiu J (2011) H∞ filtering for nonlinear discrete-time systems subject to quantization and packet dropouts. IEEE Trans Fuzzy Syst 19:353–365 10. Li ZM, Chang XH, Mathiyalagan K, Xiong J (2017) Robust energy-to-peak filtering for discrete-time nonlinear systems with measurement quantization. Signal Process 139:102–109

Chapter 4

Fuzzy Output Feedback Control with Communication Constraints

Abstract In this chapter, the problem of output feedback control is studied for T–S fuzzy systems with communication constraints. Firstly, considering the effect of the quantization error caused from multi-signal quantization, a novel quantization output feedback control scheme is proposed to ensure that the closed-loop fuzzy system is asymptotically stable and satisfies H∞ performance. Secondly, by applying input delay method to describe the control input signal affected by data dropouts and time delays, a static output feedback control strategy is provided to guarantee the asymptotic stability and H∞ performance of closed-loop fuzzy system. Thirdly, by using a Markov chain to describe the communication protocol scheduling behavior over the communication channel, a dynamic output feedback controller is designed to guarantee the stochastic stability and H∞ performance of the closed-loop system. Finally, some simulation examples are given to prove the feasibility and the effectiveness of the developed design methods. Keywords Multi-signal quantization · H∞ output feedback controller · Time delays · Data dropouts · Stochastic communication protocol

4.1 Feedback Control with Quantization In this section, the output feedback control problem is considered for T–S fuzzy system with the effects of multi-signal quantization. By using the concept of PDC, a new design method of output feedback controller under the quantized output signals and control signals is proposed to ensure that the closed-loop fuzzy control system is asymptotically stable and meets H∞ performance.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Chang et al., Control and Filtering of Fuzzy Systems Under Communication Channels, https://doi.org/10.1007/978-981-99-4346-3_4

103

104

4 Fuzzy Output Feedback Control with Communication Constraints

4.1.1 Problem Formulation Consider the following discrete-time T–S fuzzy model: Plant Rule i th : IF ζ1 (k) is J1i and …ζo (k) is Joi , THEN x(k + 1) = Ai x(k) + Bi u(k), ¯

(4.1)

y(k) = Ci x(k),

ny where x(k) ∈ Rn x , u(k) ¯ ∈ Rn u , and y(k) ∈ R  control input vec are the state vector, tor, and output vector, respectively; ζ (k) = ζ1 (k) . . . ζo (k) , ζd (k), d = 1, . . . , o are measurable premise variables; Jdi , i = 1, . . . , r , represent the fuzzy sets and the parameter r means the number of fuzzy rules; The matrices Ai ∈ Rn x ×n x , Bi ∈ Rn x ×n u and Ci ∈ Rn y ×n x are known system matrices. It is assumed that fuzzy basis function is given by

o Jdi (ζd (k))  o , h i (ζ (k)) = r d=1 i=1 d=1 Jdi (ζd (k))

(4.2)

where Jdi (ζd (k)) is the grade of membership function of ζd (k) in Jdi . Considering the properties of membership function, one has h i (ζ (k)) ≥ 0,

r 

h i (ζ (k)) = 1, i = 1, . . . , r.

(4.3)

i=1

Further, a more compact presentation of T–S fuzzy model (4.1) can be derived as follows: x(k + 1) = A(h)x(k) + B(h)u(k), ¯ (4.4) y(k) = C(h)x(k), where A(h) =

r  i=1

h i (ζ (k))Ai , B(h) =

r 

h i (ζ (k))Bi , C(h) =

i=1

r 

h i (ζ (k))Ci .

i=1

Next, assume that the measurement output y(k) and control signal u(k) should be quantized before the signals feedback to controller and system. Then, a dynamic quantizer is given as the following form:  ι¯(k) = qμι (ι(k)) = μι (k)q

 ι(k) , μι (k) > 0, μι (k)

(4.5)

where μι (k) is the quantization parameters of corresponding signals with ι = y, u.

4.1 Feedback Control with Quantization

105

Set μι (k)Mι being the range of the quantizer and μι (k)Δι being the quantization error bound of the quantizer. Then the quantizer qμι (·) satisfies the following conditions: qμι (ι(k)) − ι(k) ≤ μι (k)Δι , if ι(k) ≤ μι (k)Mι , qμι (ι(k)) > μι (k)Mι − μι (k)Δι , if ι(k) > μι (k)Mι .

(4.6)

Considering the effects of the quantization and data packet loss, the following fuzzy controller can be obtained by using the same membership function and premise variables Plant Rule i th : IF ζ1 (k) is J1i and …ζo (k) is Joi , THEN u(k) = K i y¯ (k),

(4.7)

where K i ∈ Rn u ×n y , i = 1, . . . , r , is controller gain matrix to be determined, u(k) represents the control signal output by the controller, and y¯ (k) denotes the signal passed into the controller after quantization of the system measurement output signal. The following terse form is given to represent the fuzzy controller u(k) = K (h) y¯ (k),

(4.8)

where K (h) =

r 

h i (ζ (k))K i .

i=1

Considering the quantization effect of dynamic quantizer on the control signal input to fuzzy system (4.4), it yields

x(k + 1) = A(h)x(k) + B(h)u(k) + B(h) qμu (u(k)) − u(k) , y(k) = C(h)x(k).

(4.9)

Combining the above formula with the static fuzzy controller (4.8), one can obtain

x(k + 1) = A(h)x(k) + B(h)K (h) y¯ (k) + B(h) qμu (u(k)) − u(k) , y(k) = C(h)x(k).

(4.10)

The system output signal will also be quantized when entering the static controller, so the above equation can be rearranged as

x(k + 1) = A(h)x(k) + B(h)K (h)y(k) + B(h) qμu (u(k)) − u(k)

+B(h)K (h) qμ y (y(k)) − y(k) , y(k) = C(h)x(k). (4.11)

106

4 Fuzzy Output Feedback Control with Communication Constraints

Substituting the equation (4.5) into the above system can be reduced to     u(k) u(k) − x(k + 1) = A(h)x(k) + B(h)K (h)y(k) + B(h)μu (k) q μu (k) μu (k)     y(k) y(k) − , +B(h)K (h)μ y (k) q μ y (k) μ y (k) y(k) = C(h)x(k). (4.12)    Set ϑu (k) = μu (k) q μu(k) − u (k)

 u(k) μu (k)

   and ϑ y (k) = μ y (k) q μy(k) − y (k)

y(k) μ y (k)

 .

Then the discrete-time output feedback closed-loop control system is obtained x(k + 1) = A(h)x(k) + B(h)ϑ y (k) + C(h)ϑu (k), where

A(h) = A(h) + B(h)K (h)C(h), B(h) = B(h)K (h), C(h) = B(h).

(4.13)

(4.14)

Next, the quantized static output feedback control problem of fuzzy discrete-time systems can be formulated as: Quantized static output feedback control problem. For the nonlinear discrete-time system, taking full account of the multiple quantization of the output signal y(k) of the system and the output signal u(k) of the controller, the goal of this section is to design the PDC controllers and the appropriate dynamic quantizers such that the closed-loop control system is stable and meets certain control performance.

4.1.2 Main Results 4.1.2.1

Output Feedback Stabilization Control Design

Based on the above construction of the discrete-time output feedback closed-loop control system (4.13) with multi-signal quantization, the problem of the stability analysis for the closed-loop system can be given by the following theorem. Theorem 4.1 For the T–S fuzzy system (4.13). Given quantization ranges M y , Mu and quantization errors Δ y , Δu . If there exist matrix P > 0, scalars σ y , σu , ρ y , and ρu satisfying the following matrix inequalities My >

1 1 , Mu > , ρy ρu

(4.15)

4.1 Feedback Control with Quantization

⎤ −P ∗ ∗ ∗ ⎢ 0 −σ y I ∗ ∗ ⎥ ⎢ ⎥ ⎣ 0 ∗ ⎦ 0 −σu I A(h) B(h) C(h) −P −1

107

⎡

(4.16)

+ σ y (H a{Υ1 (h)}) + σu (H a{Υ2 (h)}) < 0, where     Υ1 (h) = 2ρ y Δ y C(h) 0 0 , Υ2 (h) = Υ21 (h) Υ22 (h) 0 , Υ21 (h) = 2ρu Δu K (h)C(h), Υ22 (h) = 2ρu Δu K (h), and the online adjusting strategy for the parameters μ y (k) and μu (k) are given as: ρ y y(k) ≤ μ y (k) ≤ 2ρ y y(k), ρu u(k) ≤ μu (k) ≤ 2ρu u(k).

(4.17)

Then the closed-loop control system is asymptotically stable.     Proof From the inequality (4.6), it is clear that  μy(k)  ≤ M y and one has y (k)       q y(k) − y(k)  ≤ Δ y .  μ y (k) μ y (k) 

(4.18)

Consider thehomogeneity  property  of Euclidean norm and combine the equation  y(k) y(k) ϑ y (k) = μ y (k) q μ y (k) − μ y (k) . Then one gets with the inequalities (4.15) and (4.17) 1 1 μ y (k) ≤ y(k) ≤ μ y (k) ≤ M y μ y (k). (4.19) 2ρ y ρy Further, one can obtain ϑ y (k) ≤ 2ρ y Δ y y(k).

(4.20)

Square both sides of above inequality gives ϑ yT (k)ϑ y (k) ≤ 4ρ y2 Δ2y y T (k)y(k).

(4.21)

T  Defining δ(k) = x T (k) ϑ yT (k) ϑuT (k) , one has δ T (k)Ω1 (h)δ(k) ≥ 0, where Ω1 (h) = H a

  2ρ y Δ y C(h) 0 0 − diag{ 0, I, 0 }.

(4.22)

108

4 Fuzzy Output Feedback Control with Communication Constraints

    Similarly, when the control signal is quantized there have  μu(k)  ≤ Mu and (k) u       q u(k) − u(k)  ≤ Δu .  μu (k) μu (k) 

(4.23)

Consider thehomogeneity  property  of Euclidean norm and combine the equation  u(k) u(k) ϑu (k) = μu (k) q μu (k) − μu (k) . Then adjusting strategy 2ρ1u μu (k) ≤ u(k) ≤ 1 μ (k) ρu u

≤ Mu μu (k) with inequality (4.15) yields ϑu (k) ≤ 2ρu Δu u(k).

(4.24)

Since the input signal u(k) of the controller is the measurement output signal of the system after quantization, i.e., u(k) = K (h)C(h)x(k) + K (h)ϑ y (k), one has δ T (k)Ω2 (h)δ(k) ≥ 0,

(4.25)

  where Ω2 (h) = H a 2ρu Δu K (h)C(h) 2ρu Δu K (h) 0 − diag{ 0, 0, I }. Next, a Lyapunov function is selected to analyze the stability of closed-loop control system V (k, x(k)) = x T (k)P x(k), P = P T > 0, (4.26) and combine system (4.13), it can be obtained that ΔV (k, (x(k))) =V (k + 1, x(k + 1)) − V (k, x(k))

T = A(h)x(k) + B(h)ϑ y (k) + C(h)ϑu (k) P

× A(h)x(k) + B(h)ϑ y (k) + C(h)ϑu (k) − x T (k)P x(k)  T   =δ T (k) A(h) B(h) C(h) P A(h) B(h) C(h)  − diag{ P, 0, 0 } δ(k) =δ T (k)Ω0 (h)δ(k), (4.27) T    where Ω0 (h) = A(h) B(h) C(h) P A(h) B(h) C(h) −diag{ P, 0, 0 }. Using Lemma 1.3, the following inequality can be divided Ω0 (h) + σ y Ω1 (h) + σu Ω2 (h) < 0.

(4.28)

Finally, the matrix inequality (4.16) can be obtained. In Theorem 4.1, the stability analysis conditions of fuzzy system with quantization are given. However, because of the nonlinearity of the system and the controller and the non-convex terms in the matrix inequality, the inequality conditions

4.1 Feedback Control with Quantization

109

in Theorem 4.1 cannot be directly used in controller design. Next, the main purpose is to decouple the nonlinearity terms so as to obtain controller design conditions in terms of the LMI form. Theorem 4.2 For the T–S fuzzy system (4.13). Given quantization ranges M y , Mu and quantization errors Δ y , Δu . If there exist matrices P > 0, N , N1 , N2 , and Ul , l = 1, . . . , r , scalars η y , and ηu satisfying the following matrix inequalities M y > η y , M u > ηu ,

(4.29)

Γii < 0, i = 1, . . . , r,

(4.30)

Γil + Γli < 0, i < l, i, l = 1, . . . , r,

(4.31)

where ⎡

⎤ −P ∗ ∗ ∗ ∗ ∗ ⎢ ∗ ∗ ∗ ∗ ⎥ 0 −η y I ⎢ ⎥ ⎢ ∗ ∗ ∗ ⎥ 0 0 −ηu I ⎥, Γil = ⎢ ⎢ A ˜l ˆ ˜ il B C −N − NT + P ∗ ∗ ⎥ ⎢ ⎥ ⎣2Δ y Ci Y −1 0 0 0 −η y I ∗ ⎦ i Υ¯21il Yi−1 Υ¯22l 0 0 0 −ηu I       I U N N1 , Ul = l , E¯ = , N= 0 0 N2 0 ˜ l = Ul , C ˆ = N E, ˜ il = NYi Ai Yi−1 + Ul Ci Yi−1 , B ¯ A Υ¯21il = 2Δu K l Ci , Υ¯22l = 2Δu K l . Then the closed-loop control system (4.13) is asymptotically stable. The corresponding dynamic quantization parameters μ y (k) and μu (k) adjustment strategy in Theorem 4.1 can be rewritten as 1 y(k) ≤ μ y (k) ≤ ηy 1 u(k) ≤ μu (k) ≤ ηu

2 y(k), ηy 2 u(k). ηu

(4.32)

In addition, the controller gain matrix can be determined K l = N −1 Ul , l = 1, . . . , r. Proof Defining scalars ηι = σι = be obtained that

1 ,ι ρι

(4.33)

= y, u and using Lemma 1.1 to (4.16), it can

110

4 Fuzzy Output Feedback Control with Communication Constraints

⎡

⎤ −P ∗ ∗ ∗ ∗ ∗ ⎢ 0 −η y I ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ ⎥ 0 0 −ηu I ⎢ ⎥ < 0. ⎢ A(h) B(h) C(h) −P −1 ∗ ∗ ⎥ ⎢ ⎥ ⎣ 2Δ y C(h) 0 0 0 −η y I ∗ ⎦ 0 0 0 −ηu I 2Δu K (h)C(h) 2Δu K (h)

(4.34)

Choose a nonsingular state coordinate transformation matrix Y (h), which satisfies P = Y T (h)PY (h) with P > 0. Then, the inverse matrix of a matrix P is P −1 = Y −1 (h)P−1 Y −T (h). Substituting matrices P and P −1 into the above inequality, and pre- and post-multiplying it by diag{ Y −T (h), I, I, Y (h), I, I } and its transpose matrix, respectively, the following inequality can be obtained ⎡

⎤ −P ∗ ∗ ∗ ∗ ∗ ⎢ ∗ ∗ ∗ ∗ ⎥ 0 −η y I ⎢ ⎥ ⎢ I ∗ ∗ ∗ ⎥ 0 0 −η u ⎢ ⎥ < 0, ⎢Y (h)A(h)Y −1 (h) Y (h)B(h) Y (h)C(h) −P−1 ∗ ∗ ⎥ ⎢ ⎥ ⎣ 2Δ y C(h)Y −1 (h) 0 0 0 −η y I ∗ ⎦ Υ¯21 (h)Y −1 (h) Υ¯22 (h) 0 0 0 −ηu I (4.35) where Υ¯21 (h) = 2Δu K (h)C(h), Υ¯22 (h) = 2Δu K (h). Define E¯ = Y (h)B(h). Then the matrix Y (h) is used as the state coordinate transformation matrix. It yields ⎡

⎤ −P ∗ ∗ ∗ ∗ ∗ ⎢ ∗ ∗ ∗ ∗ ⎥ 0 −η y I ⎢ ⎥ ⎢ ⎥ I ∗ ∗ ∗ 0 0 −η u ⎢ ⎥ < 0, −1 ⎢ ⎥ ¯ (h) ¯ ¯ (h) A B C − P ∗ ∗ ⎢ ⎥ ⎣2Δ y C(h)Y −1 (h) 0 0 0 −η y I ∗ ⎦ 0 0 0 −ηu I Υ¯21 (h)Y −1 (h) Υ¯22 (h)

(4.36)

where ¯ (h) = E¯ K (h), C ¯ = E. ¯ (h) = Y (h)A(h)Y −1 (h) + E¯ K (h)C(h)Y −1 (h), B ¯ A Pre- and post-multiplying (4.36) by diag{ I, I, I, N, I, I, I, I } and its transpose, respectively, one has ⎡

⎤ −P ∗ ∗ ∗ ∗ ∗ ⎢ ∗ ∗ ∗ ∗ ⎥ 0 −η y I ⎢ ⎥ ⎢ 0 0 −ηu I ∗ ∗ ∗ ⎥ ⎢ ⎥ < 0, (4.37) ⎢ ˆ (h) ˆ ˆ (h) A B C −NP−1 NT ∗ ∗ ⎥ ⎢ ⎥ ⎣2Δ y C(h)Y −1 (h) 0 0 0 −η y I ∗ ⎦ Υ¯21 (h)Y −1 (h) Υ¯22 (h) 0 0 0 −ηu I

4.1 Feedback Control with Quantization

where

111

ˆ (h) = NY (h)A(h)Y −1 (h) + N E¯ K (h)C(h)Y −1 (h), A ˆ (h) = N E¯ K (h), C ˆ = N E. ¯ B

Based on the constructed matrices N, U(h), and the new definition matrix U (h) = N K (h), it’s easy to get

N E¯ K (h) =



N N1 0 N2

      I N K (h) U (h) K (h) = = = U(h). 0 0 0

(4.38)

So the inequality (4.37) can be rearranged as ⎡

⎤ −P ∗ ∗ ∗ ∗ ∗ ⎢ ∗ ∗ ∗ ∗ ⎥ 0 −η y I ⎢ ⎥ ⎢ ⎥ I ∗ ∗ ∗ 0 0 −η u ⎢ ⎥ < 0, (4.39) −1 T ⎢ ˜ ˆ ˜ A(h) B(h) C −NP N ∗ ∗ ⎥ ⎢ ⎥ ⎣2Δ y C(h)Y −1 (h) 0 0 0 −η y I ∗ ⎦ Υ¯21 (h)Y −1 (h) Υ¯22 (h) 0 0 0 −ηu I where ˜ (h) = U(h). ˜ (h) = NY (h)A(h)Y −1 (h) + U(h)C(h)Y −1 (h), B A Considering the fact that −NP−1 NT ≤ −N − NT + P, the above inequality can be reorganized as ⎡

⎤ −P ∗ ∗ ∗ ∗ ∗ ⎢ ∗ ∗ ∗ ∗ ⎥ 0 −η y I ⎢ ⎥ ⎢ ∗ ∗ ∗ ⎥ 0 0 −ηu I ⎢ ⎥ < 0. ⎢ ˜ (h) ˆ ˜ (h) A B C −N − NT + P ∗ ∗ ⎥ ⎢ ⎥ ⎣2Δ y C(h)Y −1 (h) 0 0 0 −η y I ∗ ⎦ Υ¯21 (h)Y −1 (h) Υ¯22 (h) 0 0 0 −ηu I (4.40) According to the T–S fuzzy model and membership function, the above inequality can be rewritten as r  r 

=

h i (ζ (k))h l (ζ (k))Γil i=1 l=1 r r  r   h i2 (ζ (k))Γii + h i (ζ (k))h l (ζ (k))(Γil i=1 i=1 i 1}, and the function f loor (ε) means the maximum integer which does not exceed ε.

4.1.2.2

Output Feedback H∞ Control Design

The above content is the analysis and proof of the stability of discrete-time output feedback control problem with quantization. When considering the H∞ performance of the system, based on the same fuzzy rules, the fuzzy control system can be described as x(k + 1) = A(h)x(k) + B(h)u(k) ¯ + E(h)w(k), z(k) = L(h)x(k) + D(h)u(k) ¯ + H (h)w(k), (4.42) y(k) = C(h)x(k) + F(h)w(k), where w(k) ∈ Rn w is the external disturbance signal of the system in l2 [0, ∞); z(k) ∈ Rn z is the controlled output variable; E(h) ∈ Rn x ×n w , L(h) ∈ Rn z ×n x , D(h) ∈ Rn z ×n u , H (h) ∈ Rn z ×n w and F(h) ∈ Rn y ×n w are all known system matrices with appropriate dimensions. Using the same steps, the corresponding closed-loop control system can be obtained as x(k + 1) = A(h)x(k) + B(h)w(k) + C(h)ϑ y (k) + D(h)ϑu (k), z(k) = J(h)x(k) + F(h)w(k) + G(h)ϑ y (k) + H(h)ϑu (k),

(4.43)

where A(h) = A(h) + B(h)K (h)C(h), B(h) = E(h) + B(h)K (h)F(h), C(h) = B(h)K (h), D(h) = B(h), J(h) = L(h) + D(h)K (h)C(h), F(h) = H (h) + D(h)K (h)F(h), G(h) = D(h)K (h), H(h) = D(h).

(4.44)

4.1 Feedback Control with Quantization

113

Similarly, we first analyze the stability and H∞ performance of the closed-loop control system to facilitate the subsequent design conditions. Theorem 4.3 For the T–S fuzzy system (4.43). Given the performance index γ > 0, quantization ranges M y , Mu and quantization errors Δ y , Δu . If there exist matrix P > 0, scalars σ y , σu , ρ y , and ρu satisfying the following matrix inequalities My >

1 1 , Mu > , ρy ρu

⎤ −P ∗ ∗ ∗ ∗ ∗ ⎢ 0 −γ 2 I ∗ ∗ ∗ ∗⎥ ⎥ ⎢ ⎢ 0 ∗ ∗ ∗⎥ 0 −σ y I ⎥ ⎢ ⎢ 0 ∗ ∗⎥ 0 0 −σu I ⎥ ⎢ ⎣A(h) B(h) C(h) D(h) −P −1 ∗ ⎦ J(h) F(h) G(h) H(h) 0 −I

(4.45)

⎡

(4.46)

+ σ y (H a{Ψ1 (h)}) + σu (H a{Ψ2 (h)}) < 0, where   Ψ1 (h) = 2ρ y Δ y C(h) 2ρ y Δ y F(h) 0 0 ,   Ψ2 (h) = Ψ21 (h) Ψ22 (h) Ψ23 (h) 0 , Ψ21 (h) = 2ρu Δu K (h)C(h), Ψ22 (h) = 2ρu Δu K (h)F(h), Ψ23 (h) = 2ρu Δu K (h), and use the same online adjusting strategy for the parameters μ y (k) and μu (k) in Theorem 4.1. Then the closed-loop control system is asymptotically stable and satisfies H∞ control performance. Proof Similar to the derivation of the quantization process in Theorem 4.1, by redefinition the vector φ(k) = [ x T (k) w T (k) ϑ yT (k) ϑuT (k) ]T , the conditions (4.22) and (4.25) can be adjusted as the following form φ T (k)Λ1 (h)φ(k) ≥ 0,

(4.47)

φ T (k)Λ2 (h)φ(k) ≥ 0,

(4.48)

where   2ρ y Δ y C(h) 2ρ y Δ y F(h) 0 0 − diag{ 0, 0, I, 0 },   Λ2 (h) = H a 2ρu Δu K (h)C(h) 2ρu Δu K (h)F(h) 2ρu Δu K (h) 0 − diag{ 0, 0, 0, I }. Λ1 (h) = H a

Taking the same Lyapunov function V (k, (x(k))) and the resulting equation (4.27), one has

114

4 Fuzzy Output Feedback Control with Communication Constraints ΔV (k, (x(k))) + z T (k)z(k) − γ 2 w T (k)w(k) =V (k + 1, x(k + 1)) − V (k, x(k)) + z T (k)z(k) − γ 2 w T (k)w(k) T  = A(h)x(k) + B(h)w(k) + C(h)ϑ y (k) + D(h)ϑu (k) P   × A(h)x(k) + B(h)w(k) + C(h)ϑ y (k) + D(h)ϑu (k) T  + J(h)x(k) + F(h)w(k) + G(h)ϑ y (k) + H(h)ϑu (k)   × J(h)x(k) + F(h)w(k) + G(h)ϑ y (k) + H(h)ϑu (k)

(4.49)

− x T (k)P x(k) − γ 2 w T (k)w(k) =φ T (k)Λ0 (h)φ(k),

where  T   Λ0 (h) = A(h) B(h) C(h) D(h) P A(h) B(h) C(h) D(h)  T   J(h) F(h) G(h) H(h) + J(h) F(h) G(h) H(h) − diag{ P, γ 2 I, 0, 0 }. Using Lemma 1.3 with (4.47) and (4.48), we get the following inequality Λ0 (h) + σ y Λ1 (h) + σu Λ2 (h) < 0.

(4.50)

From the result that Λ0 (h) < 0 and taking the sum from 0 to ∞ of (4.49), the following expression can be get

V (∞, (x(∞))) − V (0, (x(0))) +

∞ 

z T (k)z(k) − γ 2

k=0

∞ 

w T (k)w(k) < 0.

k=0

(4.51) Due to the fact that V (k, (x(k))) ≥ 0 and V (0, x(0)) = 0, the following inequality is obtained under zero initial conditions ∞  k=0

z T (k)z(k) ≤ γ 2

∞ 

w T (k)w(k),

(4.52)

k=0

which demonstrates that the designed performance conditions can ensure that the closed-loop control system is asymptotically stable and satisfies H∞ control performance. The proof is completed. Theorem 4.4 For the T–S fuzzy system (4.43). Given the performance index γ > 0, quantization ranges M y , Mu and quantization errors Δ y , Δu . If there exist matrix P > 0, Q, Wl , αi , and R, scalars β, η y , and ηu satisfying the following matrix inequalities (4.53) M y > η y , M u > ηu ,

4.1 Feedback Control with Quantization

115

Θii < 0, i = 1, . . . , r, Θil + Θli < 0, i < l, i, l = 1, . . . , r,

(4.54) (4.55)

where ⎤ −P ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ −γ 2 I ⎥ ⎢ 0 −η y I ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ 0 ⎥ ⎢ 0 0 −ηu I ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ 0 ⎥ ⎢ Θ52il αi Wl Q Bi Θ55 ∗ ∗ ∗ ∗ Θil = ⎢ Θ51il ⎥, ⎥ ⎢ Θ62il Di Wl Di 0 −I ∗ ∗ ∗ ⎥ ⎢ Θ61il ⎥ ⎢2Δ C 2Δ F 0 0 0 0 −η y I ∗ ∗ ⎥ ⎢ y i y i ⎦ ⎣ Ψ¯ ¯ ¯ Ψ Ψ 0 0 0 0 −η I ∗ 21il 22il 23l u Wl Ci Wl Fi Wl 0 Θ95i Θ96i 0 0 −H e{β R} ⎡

Θ51il = Q Ai + αi Wl Ci , Θ52il = Q E i + αi Wl Fi , Θ55 = −Q − Q T + P, Θ61il = L i + Di Wl Ci , Θ62il = Hi + Di Wl Fi , Ψ¯ 21il = 2Δu K l Ci , Ψ¯ 22il = 2Δu K l Fi , Ψ¯ 23l = 2Δu K l , Θ95i = β(Q Bi − αi R)T , Θ96i = β(Di − Di R)T ,

and use the same online adjusting strategy for the parameters μ y (k) and μu (k) in Theorem 4.2. Then the closed-loop control system is asymptotically stable and satisfies H∞ control performance. In addition, the controller gain matrix can be determined K l = R −1 Wl , l = 1, . . . , r. Proof Similarly, define scalars ηι = σι = (4.46), it can be obtained that ⎡

1 ,ι ρι

(4.56)

= y, u and adopt Lemma 1.1 to

⎤ −P ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎢ 0 −γ 2 I ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ ∗ ∗ ⎥ 0 0 −η y I ⎢ ⎥ ⎢ ∗ ∗ ∗ ∗ ⎥ 0 0 0 −ηu I ⎢ ⎥ < 0, ⎢ A(h) ∗ ∗ ⎥ B(h) C(h) D(h) −P −1 ∗ ⎢ ⎥ ⎢ J(h) F(h) G(h) H(h) 0 −I ∗ ∗ ⎥ ⎢ ⎥ ⎣2Δ y C(h) 2Δ y F(h) 0 0 0 0 −η y I ∗ ⎦ Ψ¯ 21 (h) Ψ¯ 22 (h) Ψ¯ 23 (h) 0 0 0 0 −ηu I (4.57) where Ψ¯ 21 (h)=2Δu K (h)C(h), Ψ¯ 22 (h) = 2Δu K (h)F(h), and Ψ¯ 23 (h) = 2Δu K (h). Performing congruence transformation with diag{ I, I, I, I, Q, I, I, I } to above inequality has

116

4 Fuzzy Output Feedback Control with Communication Constraints ⎡

−P ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ QA(h) ⎢ ⎢ J(h) ⎣2Δ C(h) y Ψ¯ 21 (h)

∗ −γ 2 I 0 0 QB(h) F(h) 2Δ y F(h) Ψ¯ 22 (h)

∗ ∗ ∗ ∗ ∗ ∗ −η y I ∗ ∗ 0 −ηu I ∗ QC(h) QD(h) −Q P −1 Q T G(h) H(h) 0 0 0 0 Ψ¯ 23 (h) 0 0

⎤ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ ∗ ∗ ∗ ⎥ ⎥ ∗ ∗ ∗ ⎥ ⎥ < 0. ∗ ∗ ∗ ⎥ ⎥ −I ∗ ∗ ⎥ 0 −η y I ∗ ⎦ 0 0 −ηu I

(4.58)

Considering the fact that −Q P −1 Q T ≤ −Q − Q T + P, one can obtain ⎡

⎤ −P ∗ ∗ ∗ ∗ ∗ ∗ ∗ 2 ∗ ∗ ∗ ∗ ∗ ∗ ⎥ 0 −γ I ⎢ ⎢ ⎥ 0 0 −η y I ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ ⎢ ⎥ 0 0 0 −ηu I ∗ ∗ ∗ ∗ ⎥ ⎢ ⎢ ⎥ < 0. T ∗ ∗ ⎥ QB(h) QC(h) QD(h) −Q − Q + P ∗ ⎢ QA(h) ⎢ ⎥ F(h) G(h) H(h) 0 −I ∗ ∗ ⎥ ⎢ J(h) ⎣2Δ C(h) 2Δ F(h) 0 0 0 0 −η y I ∗ ⎦ y y Ψ¯ 21 (h) Ψ¯ 22 (h) Ψ¯ 23 (h) 0 0 0 0 −ηu I

(4.59) The above inequality can be rewritten as follows ⎡

⎤ −P ∗ ∗ ∗ ∗ ∗ ∗ ∗ 2 ∗ ∗ ∗ ∗ ∗ ∗ ⎥ 0 −γ I ⎢ ⎢ ⎥ 0 0 −η y I ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ ⎢ ⎥ 0 0 0 −ηu I ∗ ∗ ∗ ∗ ⎥ ⎢ ⎢ ⎥ T ∗ ∗ ⎥ Q E(h) 0 Q B(h) −Q − Q + P ∗ ⎢ Q A(h) ⎢ ⎥ H (h) 0 D(h) 0 −I ∗ ∗ ⎥ ⎢ L(h) ⎣2Δ C(h) 2Δ F(h) 0 0 0 0 −η y I ∗ ⎦ y y Ψ¯ 22 (h) Ψ¯ 23 (h) 0 0 0 0 −ηu I Ψ¯ 21 (h) ⎧⎡ ⎫ ⎤ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎢ 0 ⎥ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ 0 ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎨⎢ ⎥ ⎬   ⎢ 0 ⎥ + He ⎢ ⎥ K (h) C(h) F(h) I 0 0 0 0 0 < 0. ⎪ ⎢ Q B(h)⎥ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪⎢ D(h) ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ ⎦ ⎪ ⎪ 0 ⎪ ⎪ ⎩ ⎭ 0

(4.60)

Define a nonsingular matrix R, which satisfies K (h) = R −1 W (h). Then (4.60) can be rewritten as

4.1 Feedback Control with Quantization ⎤ −P ∗ ∗ ∗ ∗ ∗ ∗ ∗ 2 ∗ ∗ ∗ ∗ ∗ ∗ ⎥ 0 −γ I ⎢ ⎥ ⎢ 0 0 −η y I ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ 0 0 0 −ηu I ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ Θ (h) Θ (h) α(h)W (h) Q B(h) Θ ∗ ∗ ∗ ⎥ ⎢ 51 52 52 ⎢ Θ (h) Θ62 (h) D(h)W (h) D(h) 0 −I ∗ ∗ ⎥ ⎥ ⎢ 61 ⎣2Δ C(h) 2Δ F(h) 0 0 0 0 −η y I ∗ ⎦ y y Ψ¯ 22 (h) Ψ¯ 23 (h) 0 0 0 0 −ηu I Ψ¯ 21 (h) ⎧⎡ ⎫ ⎤ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎥ ⎪ ⎢ ⎪ ⎪ ⎪ ⎥ ⎪ ⎢ ⎪ ⎪ ⎪ 0 ⎥ ⎪⎢ ⎪ ⎪ ⎪ ⎥ ⎨⎢  ⎬ 0 ⎥ −1 ⎢ + He ⎢ ⎥ R W (h) C(h) F(h) I 0 0 0 0 0 < 0, ⎪⎢ Q B(h) − α(h)R ⎥ ⎪ ⎪ ⎪ ⎪ ⎢ D(h) − D(h)R ⎥ ⎪ ⎪ ⎪ ⎥ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎦ ⎣ ⎪ ⎪ 0 ⎪ ⎪ ⎩ ⎭ 0

117

⎡

(4.61)

where Θ51 (h) = Q A(h) + α(h)W (h)C(h), Θ52 = Q E(h) + α(h)W (h)F(h), Θ61 (h) = L(h) + D(h)W (h)C(h), Θ62 = H (h) + D(h)W (h)F(h). By Lemma 1.4, set M4 = R, λ = β, and the following matrices ⎡

⎤ −P ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎢ 0 −γ 2 I ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ ⎥ I ∗ ∗ ∗ ∗ ∗ 0 0 −η y ⎢ ⎥ ⎢ ⎥ I ∗ ∗ ∗ ∗ 0 0 0 −η u ⎢ ⎥, M1 = ⎢ ⎥ (h) Θ (h) α(h)W (h) Q B(h) Θ ∗ ∗ ∗ Θ 52 52 ⎢ 51 ⎥ ⎢ Θ61 (h) ⎥ Θ (h) D(h)W (h) D(h) 0 −I ∗ ∗ 62 ⎢ ⎥ ⎣2Δ y C(h) 2Δ y F(h) 0 0 0 0 −η y I ∗ ⎦ ¯ Ψ¯ 22 (h) Ψ¯ 23 (h) 0 0 0 0 −ηu I ⎤ ⎡ Ψ21 (h) 0 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢   0 ⎥ , M3 = R −1 W (h) C(h) F(h) I 0 0 0 0 0 . ⎢ M2 = ⎢ ⎥ Q B(h) − α(h)R ⎥ ⎢ ⎢ D(h) − D(h)R ⎥ ⎥ ⎢ ⎦ ⎣ 0 0 Then according to the T–S fuzzy model and membership function, inequality (4.61) can be rewritten as

118

4 Fuzzy Output Feedback Control with Communication Constraints r  r 

h i (ζ (k))h l (ζ (k))Θil i=1 l=1 r r r    = h i2 (ζ (k))Θii + h i (ζ (k))h l (ζ (k))(Θil i=1 i=1 i n u .

4.1.3 Simulation Example In order to verify the feasibility and effectiveness of the algorithm in the stability design of closed-loop control system with quantization and the controller design with quantization satisfying H∞ performance, consider the following second-order discrete-time chaotic system [5, 6] x1 (k + 1) = ax1 (k) − x13 (k) + x2 (k) + u(k) ¯ + e1 w(k), x2 (k + 1) = bx1 (k) + e2 w(k), z(k) = L x(k) + D u(k) ¯ + H w(k), y(k) = C x(k) + Fw(k),

(4.63)

where x(k) = [ x1T (k) x2T (k)]T ; the parameters a and b are uncertain satisfying a0 − δa ≤ a ≤ a0 + δa and b0 − δb ≤ b ≤ b0 + δb , in which the a0 , b0 , δa ≥ 0, and δb ≥ 0 are the nominal values of a and b; u(k) ¯ is the quantized signal value. Suppose that x1 (k) ∈ [−d, d] and d > 0 so that 0 ≤ x12 (k) ≤ d 2 . Then using the sector nonlinearity approach [7] in this example, two T–S fuzzy rules are selected to approximate the state-space equation and corresponding fuzzy system parameter matrices are       a0 1 a0 − d 2 1 1 , A2 = , B1 = B2 = , A1 = 0 b0 0 b0 0       0.2 −0.7 0.2 −0.2 E1 = E2 = , L1 = L2 = , D1 = D2 = , (4.64) 0.1 0.1 1 −0.1     −0.3 , F1 = F2 = 0.2, C1 = C2 = −4 0.1 . H1 = H2 = −0.3

4.1 Feedback Control with Quantization

119

Considering the range of the variable x1 (k) and the fuzzy basis functions h 1 (ζ1 (k)) and h 2 (ζ1 (k)) corresponding to the fuzzy rules, and defining the premise variable ζ1 (k) = x1 (k), the fuzzy basis functions need to satisfy x12 (k) = h 1 (x1 (k)) × 0 + h 2 (x1 (k)) × d 2 , h 1 (x1 (k)) + h 2 (x1 (k)) = 1. After equality transformation, the mathematical expression of fuzzy basis functions can be obtained x 2 (k) h 1 (x1 (k)) = 1 − 1 2 , (4.65) d h 2 (x1 (k)) = 1 − h 1 (x1 (k)). In this section, it is assumed that some parameters in the system are a0 = 1, b0 = 0.3, δa = 0.5, δb = 0.1, and d = 1, respectively. Afterwards, select the quantization errors bounds Δ y = Δu = 0.01, the quantizers ranges M y = Mu = 50, and initial condition x(0) = [ 0.5 0 ]T . Then, LMI toolbox of MATLAB is used to solve the formulas (4.29), (4.30), and (4.31). When there is no external disturbance, i.e., w(k) = 0, the feasible matrices satisfying asymptotic stability can be obtained N = 6.4448, U1 = 1.5358, U2 = −0.0111.

(4.66)

According to (4.33), the controller gain matrices gives K 1 = 0.2383, K 2 = −0.0017.

(4.67)

In the design of H∞ controller for the closed-loop control system, adopting the same quantization ranges and quantization errors and selecting the initial condition x(0) = [ 0 0 ]T and the external interference signal w(k) = 2 cos(2k)e−0.25k with the performance index γ = 1, the LMIs (4.53), (4.54), and (4.55) in Theorem 4.4 can be substituted into the feasibility matrices satisfying the stability and H∞ performance of the system are R = 3.5217, W1 = 0.8178, W2 = 0.0832.

(4.68)

According to (4.56), the controller gain matrices are K 1 = 0.2322, K 2 = 0.0236.

(4.69)

In the discrete-time output feedback closed-loop control system (4.13) with multisignal quantization, the corresponding simulation results are shown in the Figs. 4.1, 4.2, and 4.3. When the external interference is zero, but is not in a stable state, i.e.,

120 Fig. 4.1 Response of the system state x(k) with w(k) = 0

4 Fuzzy Output Feedback Control with Communication Constraints 0.5 0.4 0.3 0.2 0.1 0 0

Fig. 4.2 Response of the measurement output y(k) with w(k) = 0

10

20

30

40

50

10

20

30

40

50

20

30

40

50

0.5 0 -0.5 -1 -1.5 -2 0

Fig. 4.3 Response of the control input u(k) with w(k) = 0

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

10

x(0) = [ 0.5 0 ]T , under the control of the controller Fig. 4.3, the system can quickly return to the stable state (Fig. 4.1). For the closed-loop control system (4.43), when there has external interference, the corresponding simulation results are shown in the Figs. 4.4, 4.5, 4.6, 4.7, and 4.8. In the zero initial state, the system state, output and measurement output can return to the stable state (Figs. 4.4, 4.5, 4.6, and 4.7) and obtain the ratio of H∞ performance

4.1 Feedback Control with Quantization Fig. 4.4 Response of x(k) with the quantized H∞ controller

121 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2

Fig. 4.5 Response of y(k) with the quantized H∞ controller

0

10

20

30

40

50

0

10

20

30

40

50

0

10

20

30

40

50

1 0.5 0 -0.5 -1 -1.5 -2

Fig. 4.6 Response of u(k) with the quantized H∞ controller

0.1

0

-0.1

-0.2

-0.3

122

4 Fuzzy Output Feedback Control with Communication Constraints

Fig. 4.7 Response of z(k) with the quantized H∞ controller

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0

Fig.  4.8 History of

20

30

40

50

10

20

30

40

50

0.47

k k   ! z T (φ)z(φ)/ w T (φ)w(φ) φ=0

10

φ=0

0.46

with the quantized H∞ controller

0.45 0.44 0.43 0.42 0

under the action of the controller when the system is affected by external interference. " k k   The convergence value of z T (φ)z(φ)/ w T (φ)w(φ) is 0.4606 < 1 (Fig. 4.8), φ=0

φ=0

so that the design algorithm of PDC controller satisfying H∞ performance is feasible. From the above simulation results, it can be shown that both the quantized stabilization control and quantized H∞ control design approaches developed are effective.

4.2 Feedback Control with Data Packet Dropouts and Time Delays This section addresses the output feedback control problem for the T–S fuzzy system subject to time delays and data dropouts. An input delay method is proposed to describe the control input signal affected by packet losses and time delays, and the output feedback controller design strategy is derived in the form of LMIs to ensure that the closed-loop system is asymptotically stable and meets H∞ performance.

4.2 Feedback Control with Data Packet Dropouts and Time Delays

123

4.2.1 Problem Formulation The following fuzzy system is considered: Plant Rule i th : IF f 1 (t) is M1i and . . . f o (t) is Moi , THEN x(t) ˙ = Ai x(t) + Bi u(t) + E i w(t), y(t) = C1 x(t),

(4.70)

z(t) = C2 x(t), where M ji , j = 1, 2, ..., o, i = 1, . . . , r , are the fuzzy sets; f (t) = [ f 1 (t), f 2 (t), . . . , f o (t)] is the vector of premise variables; x(t) ∈ Rn x and u(t) ∈ Rn u stand for the state variable and the control input, respectively; w(t) ∈ Rn w refers to the noise input, which belongs to L2 [ 0, ∞); y(t) ∈ Rn y and z(t) ∈ Rn z represent the measurement output and the controlled output; Ai ∈ Rn x ×n x , Bi ∈ Rn x ×n u , E i ∈ Rn x ×n w , C1 ∈ Rn y ×n x , and C2 ∈ Rn z ×n x are given system matrices. It is assumed that fuzzy basis function is given by o 



M ji f j (t)

j=1

i ( f (t)) =

r  o 



, M ji f j (t)

(4.71)

i=1 j=1



where M ji f j (t) is the grade of membership function of f j (t) in M ji . Due to the nature of the member function, it needs to satisfy i ( f (t)) ≥ 0,

r 

i ( f (t)) = 1, i = 1, . . . , r.

(4.72)

i=1

Then, the following fuzzy system can be expressed as: x(t) ˙ = A( )x(t) + B( )u(t) + E( )w(t), y(t) = C1 x(t), z(t) = C2 x(t),

(4.73)

where A( ) =

r  i=1

i ( f (t))Ai , B( ) =

r  i=1

i ( f (t))Bi , E( ) =

r 

i ( f (t))E i .

i=1

In the fuzzy system (4.73), the system stability is improved by considering a static output feedback controller. Hence, the considered controller can be represented by the following equation:

124

4 Fuzzy Output Feedback Control with Communication Constraints

Plant Rule i th : IF f 1 (t) is M1i and . . . f o (t) is Moi , THEN u(t) = K i yˆ (tk ), i = 1, . . . , r.

(4.74)

The following terse form is given to represent the fuzzy control u(t) = K ( ) yˆ (tk ), where K ( ) =

r 

(4.75)

i ( f (t))K i .

i=1

In practical applications, signals transmitted through data channels will be affected by a series of practical factors, such as unreliable communication lines and bandwidth delays, which may lead to data dropouts and transmission delays between sensors, controllers, and actuators. In order to fully consider and reduce the impact of this phenomenon, we assume that tk is the sampling instant and h¯ is the sampling time. The transmission delays in communication channel at each sampling period is denoted as dk with dk = dsc + dca , where dsc is the transmission delays between measurement output and controller and dca denotes the transmission delays between controller and actuator, respectively. Now, the packets transmitted to the controller are represented as: ¯ (4.76) yˆ (tk ) = y(tk − pck h), where pck = p1k + p2k represents the number of data dropouts at time tk . Moreover, p1k and p2k denote the data dropouts in sensor-to-controller and controller-toactuator, respectively. In this way, the output signal including transmission delays and data dropouts is given as: yˆ (tk ) = y(tk − pck h¯ − dk ).

(4.77)

Based on the input delay method, define a new variable d(t) = t − (tk − pck h¯ − dk ). Then the controller can be described as: u(t) = K ( )y(t − d(t)),

(4.78)

where d(t) satisfies 0 < d(t) < dmax and dmax is the upper bound of the delay. Combining (4.73) and the control strategy (4.78), the closed-loop system is described as follows:

4.2 Feedback Control with Data Packet Dropouts and Time Delays

x(t) ˙ =

r  r 

125

# $ i ( f (t))l ( f (t)) Ai x(t) + Bi K l C1 x(t − d(t)) + E i w(t)

i=1 l=1

= A( )x(t) + B( )K ( )C1 x(t − d(t)) + E( )w(t), y(t) = C1 x(t), z(t) = C2 x(t). (4.79) Next, we can formulate the H∞ feedback controller design problem: H∞ feedback controller design problem. The purpose of this section is to design a static output feedback controller given in (4.77) with transmission delays and data dropouts such that the closed-loop system (4.79) satisfies the following conditions: 1) When w(t) = 0, the closed-loop system (4.79) is asymptotically stable; 2) Under zero initial conditions, for w(t) = 0 and a given γ > 0, the controlled output z(t) satisfies %

∞ 0

% z (t)z(t)dt < γ T

∞

2

w T (t)w(t)dt.

(4.80)

0

4.2.2 Main Results In this subsection, the output feedback control design problem for the T–S fuzzy system (4.79) with time delays and data packet dropouts will be solved. To begin, we will provide a meaningful stability analysis criterion to ensure that the closed-loop system (4.79) is asymptotically stable with H∞ performance level. It is presented by the following theorem. Theorem 4.5 For given scalar γ > 0, the closed-loop system (4.79) is asymptotically stable with H∞ performance level γ , if there exist positive definite matrices P, Q, and R such that the following matrix inequality holds:   Ω11 ( ) ∗ + dmax Φ T ( )RΦ( ) < 0, (4.81) Ω21 ( ) Ω22 where

Ω11 ( ) = H e{P(A( ) + B( )K ( )C1 )} + Q + C2T C2 , T  Ω21 ( ) = P B( )K ( )C1 0 P E( ) ,   Φ( ) = A( ) B( )K ( )C1 0 E( ) , −1 Ω22 = diag{−dmax R, −Q, −γ 2 I }.

126

4 Fuzzy Output Feedback Control with Communication Constraints

Proof Select the following Lyapunov function as %t V (x(t)) = x (t)P x(t) + T

%0 % t x (s)Qx(s)ds +

x˙ T (s)R x(s)dsdθ. ˙

T

−dmax t+θ

t−dmax

(4.82) Then, one can obtain ˙ + dmax x˙ T (t)R x(t) ˙ V˙ (x(t)) = x˙ T (t)P x(t) + x T (t)P x(t) T T −x (t − dmax )Qx(t − dmax ) + x (t)Qx(t) %t − x˙ T (s)R x(s)ds. ˙

(4.83)

t−dmax

Consider the fact that %t x(t) − x(t − d(t)) −

x(s)ds ˙ = 0.

(4.84)

t−d(t)

Then according to the closed-loop system (4.79), one has x(t) ˙ = (A( ) + B( )K ( )C1 )x(t)

%t

+E( )w(t) − B( )K ( )C1

x(s)ds. ˙

(4.85)

t−d(t)

The following equation can be obtained by calculation x˙ T (t)P x(t) + x T (t)P x(t) ˙ =



%t

A( ) + B( )K ( )C1 x(t) + E( )w(t) − B( )K ( )C1

x(s)ds ˙

T P x(t)

t−d(t)

+x T (t)P



A( ) + B( )K ( )C1 x(t) + E( )w(t) − B( )K ( )C1

%t x(s)ds ˙



t−d(t)





T  = x (t) P A( ) + B( )K ( )C1 + A( ) + B( )K ( )C1 P x(t) + x T (t)P T



×E( )w(t) + x (t)P E( )w(t) T

T

%t − 2x (t)P B( )K ( )C1 T

t−d(t)

x(s)ds. ˙

(4.86)

4.2 Feedback Control with Data Packet Dropouts and Time Delays

127

By using Lemma 1.5, one can obtain %t − 2x (t)P B( )K ( )C1 T



T x(s)ds ˙ ≤ dmax x T (t)P B( )K ( )C1 R −1 P B( )K ( )C1 x(t)

t−d(t)

%t +

x˙ T (s)R x(s)ds. ˙

(4.87)

t−dmax

According to (4.86) and (4.87), (4.83) can be rewritten as ˙ V˙ (x(t)) ≤ x T (t)φ( )x(t) − x T (t − dmax )Qx(t − dmax ) + dmax x˙ T (t)R x(t) T

T T +x (t)P E( )w(t) + x (t)P E( )w(t) , (4.88)



T where φ( ) = P A( ) + B( )K ( )C1 + P A( ) + B( )K ( )C1 +   T

Q + dmax P B( )K ( )C1 R −1 P B( )K ( )C1 . Define   ζ1T (t) = x T (t) x T (t − d(t)) x T (t − dmax ) . Then one has T  ˙ = dmax ζ1T (t) A( ) B( )K ( )C1 0 R dmax x˙ T (t)R x(t)   × A( ) B( )K ( )C1 0 ζ1 (t).

(4.89)

According to (4.81), it’s easy to know that V˙ (x(t)) ≤ 0,

(4.90)

which means that the system (4.79) is asymptotically stable for w(t) = 0. When w(t) = 0, defining ζ T (t) = [x T (t) x T (t − d(t)) x T (t − dmax ) w T (t)], one has T  ˙ = dmax ζ T (t) A( ) B( )K ( )C1 0 E( ) R dmax x˙ T (t)R x(t)   × A( ) B( )K ( )C1 0 E( ) ζ (t). (4.91) Once the condition in (4.81) can be guaranteed, based on (4.83), (4.91), Lemma 1.1, and congruence transformation, the following inequality can be easily obtained V˙ (x(t)) + z T (t)z(t) − γ 2 w T (t)w(t) < 0, which implies that

(4.92)

128

4 Fuzzy Output Feedback Control with Communication Constraints

%∞ %∞ T 2 V (∞) − V (0) + z (t)z(t)dt − γ w T (t)w(t)dt < 0. 0

(4.93)

0

Since zero initial condition x(0) = 0 and V (∞) > 0, the following inequality can be obtained %∞ %∞ T 2 z (t)z(t)dt < γ w T (t)w(t)dt, (4.94) 0

0

which means that the system (4.79) is asymptotically stable with w(t) = 0 and satisfies the prescribed H∞ performance index γ with w(t) = 0. The proof is completed. Theorem 4.6 For given scalars , γ > 0, the T–S fuzzy system (4.79) is asymptotically stable with H∞ performance level γ , if there exist positive definite matrices P, Q, R, matrices U, G l such that the following matrix inequalities hold: Ψii < 0, i = 1, 2, · · · , r,

(4.95)

Ψil + Ψli < 0, i < l = 1, 2, · · · , r,

(4.96)

where ⎤ Υ11 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎢(Bi G l C1 )T −d −1 R ∗ ∗ ∗ ∗ ∗ ∗ ⎥ max ⎥ ⎢ ⎢ 0 0 −Q ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ ⎢ ⎢ (P E i )T 0 0 −γ 2 I ∗ ∗ ∗ ∗ ⎥ ⎥, ⎢ Ψil = ⎢ −1 Bi G l C1 0 P E i −dmax (2P − R) ∗ ∗ ∗ ⎥ ⎥ ⎢ P Ai ⎢ 0 0 0 0 −I ∗ ∗ ⎥ C2 ⎥ ⎢ ⎣ G l C1 0 0 0 0 Υ77 ∗ ⎦ Υ71 0 0 Υ85 0 0 Υ88 0 G l C1 T  Υ11 = P Ai + Bi G l C1 + P Ai + Bi G l C1 + Q, ⎡

Υ71 Υ77 Υ85 Υ88

=  BiT P T − U T BiT + G l C1 , = −U − U T , =  BiT P T − U T BiT , = −U − U T .

Moreover, the controller gain can be obtained by K l = U −1 G l . Proof It should be emphasized that the condition in Theorem 4.5 cannot be employed directly for controller design due to many couple terms. So, in the following, these terms from Theorem 4.5 will be removed, and the condition to design controller will be given in the form of LMIs. One can use Schur complement (Lemma 1.1) to Theorem 4.5, which is equivalent to the following inequality

4.2 Feedback Control with Data Packet Dropouts and Time Delays

⎡

129

⎤ ∗ ∗ ∗ ∗ ⎥ −1 −dmax R ∗ ∗ ∗ ∗⎥ ⎥ 0 −Q ∗ ∗ ∗⎥ ⎥ < 0, 0 0 −γ 2 I ∗ ∗⎥ ⎥ −1 R −1 ∗ ⎦ B( )K ( )C1 0 E( ) −dmax 0 0 0 0 −I (4.97) 

 T where Ωˆ 11 ( ) = P(A( ) + B( )K ( )C1 ) + P A( ) + B( )K ( )C1 . Perform congruence transformation with P = diag{I, I, I, I, P, I } to −1 −1 P R −1 P ≤ −dmax (2P − R), one obtains (4.97). Then, due to the fact that −dmax Ωˆ 11 + Q

T ⎢ ⎢ P B( )K ( )C1 ⎢ 0 ⎢ ⎢

T ⎢ P E( ) ⎢ ⎣ A( ) C2

⎡

Ωˆ 11 + Q ⎢ P B( )K ( )C1 T ⎢ ⎢ 0 ⎢

T ⎢ P E( ) ⎢ ⎣ P A( ) C2

∗

∗

−1 R −dmax 0 0 P B( )K ( )C1 0

⎤ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗⎥ ⎥ −Q ∗ ∗ ∗⎥ ⎥ < 0. ⎥ 0 −γ 2 I ∗ ∗⎥ −1 0 P E( ) −dmax (2P − R) ∗ ⎦ 0 0 0 −I

(4.98) It’s worth noting that there is a coupling term P B( )K ( )C1 in (4.98), which r  l ( f (t))G l . needs to be solved. Now, define K ( ) = U −1 G( ) with G( ) = l=1

Then (4.98) can be rewritten as follows ⎡

Ω˜ 11 ( ) ∗ ∗

T ⎢ −1 −dmax R ∗ ⎢ P B( )U −1 G( )C1 ⎢ 0 0 −Q ⎢

T ⎢ ⎢ 0 0 P E( ) ⎢ ⎣ P A( ) P B( )U −1 G( )C1 0 0 0 C2 ⎤ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ ∗ ∗ ∗ ⎥ ⎥ < 0, ∗ ∗ ⎥ −γ 2 I ⎥ −1 (2P − R) ∗ ⎦ P E( ) −dmax 0 0 −I where Ω˜ 11 ( ) = P(A( ) + B( )U −1 G( )C1 ) 

T + Q. + P A( ) + B( )U −1 G( )C1

(4.99)

130

4 Fuzzy Output Feedback Control with Communication Constraints

The following inequality can be obtained by matrix decomposition on (4.99) ⎡

⎤ Ω¯ 11 ( ) ∗ ∗ ∗ ∗ ∗ −1 ⎢(B( )G( )C1 )T −dmax R ∗ ∗ ∗ ∗⎥ ⎢ ⎥ ⎢ 0 0 −Q ∗ ∗ ∗⎥ ⎢ ⎥ ⎢ (P E( ))T 0 0 −γ 2 I ∗ ∗⎥ ⎢ ⎥ −1 ⎣ (2P − R) ∗ ⎦ P A( ) B( )G( )C1 0 P E( ) −dmax C2 0 0 0 0 −I ⎡ T ⎤ C1 0 ⎢C T C T ⎥ 1⎥ ⎢ 1  −1 T ⎢ 0 U G( ) 0 ⎥ 0 ⎢ ⎥ +⎢ × 0 ⎥ 0 U −1 G( ) ⎢ 0 ⎥ ⎣ 0 0 ⎦ 0 0 ⎤T ⎡ P B( ) − B( )U 0 ⎥ ⎢ 0 0 ⎥ ⎢ ⎥ ⎢ 0 0 ⎥ ⎢ ×⎢ ⎥ 0 0 ⎥ ⎢ ⎣ 0 P B( ) − B( )U ⎦ 0 0 ⎤ ⎡ P B( ) − B( )U 0 ⎥ ⎢ 0 0 ⎥ ⎢ ⎥ ⎢ 0 0 ⎥ ⎢ +⎢ ⎥ 0 0 ⎥ ⎢ ⎣ 0 P B( ) − B( )U ⎦ 0 0 ⎡ T ⎤T C1 0 T⎥ ⎢ T  −1  ⎢C1 C1 ⎥ ⎢ U G( ) 0 0 0 ⎥ ⎥ < 0, × ×⎢ −1 ⎢ 0 U G( ) 0 ⎥ ⎢ 0 ⎥ ⎣ 0 0 ⎦ 0 0 T  where Ω¯ 11 ( ) = P A( ) + B( )G( )C1 + P A( ) + B( )G( )C1 + Q. To make it easier to deal with the previous inequality, let’s define the following equation

4.2 Feedback Control with Data Packet Dropouts and Time Delays

⎤ P B( ) − B( )U 0 ⎥ ⎢ 0 0 ⎥ ⎢   ⎥ ⎢ 0 0 ⎥, U = U 0 , Z=⎢ ⎥ ⎢ 0 U 0 0 ⎥ ⎢ ⎣ 0 P B( ) − B( )U ⎦ 0 0  −1   U G( ) 0 C1 C1 0 0 0 0 Y= . 0 U −1 G( ) 0 C1 0 0 0 0

131

⎡

(4.100)

Then by using Lemma 1.4, it can be obtained that r  r 

i ( f (t))l ( f (t))Ξil < 0,

(4.101)

i=1 l=1

where ⎤ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Ξ11 −1 R ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎢(Bi G l C1 )T −dmax ⎥ ⎢ 0 0 −Q ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ 0 0 −γ 2 I ∗ ∗ ∗ ∗ ⎥ ⎢ (P E i )T Ξil = ⎢ ⎥, −1 Bi G l C1 0 P E i −dmax (2P − R) ∗ ∗ ∗ ⎥ ⎢ P Ai ⎥ ⎢ C2 0 0 0 0 −I ∗ ∗ ⎥ ⎢ ⎦ ⎣ Ξ71 G l C1 0 0 0 0 Ξ77 ∗ 0 G l C1 0 0 Ξ85 0 0 Ξ88

T Ξ11 = P Ai + Bi G l C1 + P Ai + Bi G l C1 + Q, T T T T Ξ71 =  Bi P − U Bi + G l C1 , Ξ77 = −U − U T , Ξ85 =  BiT P T − U T BiT , Ξ88 = −U − U T . ⎡

Further, the following inequality can be obtained r 

i2 ( f (t))Ξii +

i=1

+

r  r 

i ( f (t))l ( f (t))Ξil

(4.102)

i=1 i 0 and G sl satisfying Σsii < 0,

(4.116)

Σsil + Σsli < 0, i < l, where ⎤ ∗ ∗ ∗ −Psi ⎢ 0 ∗ ∗⎥ −γ 2 I ⎥ Σsil = ⎢ ⎣G slT A˜ sil G slT E˜ sil Q sl ∗ ⎦ , D˜ si 0 0 −I  Q sl = −H e{G sl } + πst Pt , ⎡

t∈N

then the stochastic stability and H∞ performance γ > 0 for the closed-loop system (4.113) can be guaranteed by the controller (4.112) under the SCP scheduling governed by the Markov chain σk whose transition probabilities are defined in (4.108). Proof Construct a mode-dependent fuzzy Lyapunov function as follows: V (x(k)) ˜ = x˜ T (k)Psh x(k), ˜ where Psh =

r  i=1

h i (ζ (k))Psi . Then, we assume that Pt hˆ =

(4.117) r  =1

hˆ  Pt with t = σ (k +

1) for any σ (k) = s, where hˆ  denotes the normalized membership function at time k + 1. Along the trajectories of the closed-loop system (4.113), the difference of the Lyapunov function V (x(k)) ˜ can be calculated as follows:     ˜ + 1) − x˜ T (k)Psh x(k) ˜ E ΔV (x(k)) ˜ = E x˜ T (k + 1)Pt hˆ x(k + ny ) , *  πst Pt hˆ x(k ˜ = E x˜ T (k + 1) ˜ + 1) − x˜ T (k)Psh x(k) = η (k) T

#

t=1 T

A˜ sh E˜ sh

$   P¯s hˆ A˜ sh E˜ sh − diag{Psh , 0} η(k)

= η T (k)Ψsh η(k), (4.118)

4.3 Feedback Control with Stochastic Communication Protocol

139



 ny  x(k) ˜ and P¯s hˆ = πst Pt hˆ . ω(k) t=1 On the other hand, it can be inferred from condition (4.116) that

where η(k) =

Σsh =

r  r r  

hˆ  h i (ζ (k))h l (ζ (k))Σsil

=1 i=1 l=1

=

r  =1

hˆ 

+ r 

h i2 (ζ (k))Ξsii

i=1

+

r r −1  

,

(4.119)

h i (ζ (k))h l (ζ (k))(Σsil + Σsli ) < 0,

i=1 l=i+1

where ⎤ ∗ ∗ ∗ −Psh ⎢ 0 ∗ ∗⎥ −γ 2 I ⎥, =⎢ T ˜ T ˜ ⎣G sh Ash G sh E sh Q sh ∗ ⎦ D˜ sh 0 0 −I ⎡

Σsh

G sh =

r 

h l (ζ (k))G sl , Q sh =

r  r 

hˆ  h l (ζ (k))Q sl .

=1 l=1

l=1

According to that the transition probabilities πst > 0 and the Lyapunov matrices ( P¯s hˆ − Psi > 0, we obtain P¯s hˆ > 0. And it is easy to know that ( P¯s hˆ − G sh )T P¯s−1 hˆ G sh ) ≥ 0, which implies that T ¯ −1 Ps hˆ G sh . − H e{G sh } + P¯s hˆ ≥ −G sh

(4.120)

Based on (4.119) and (4.120), one has ⎡

⎤ ∗ ∗ ∗ −Psh ⎢ 0 ∗ ∗⎥ −γ 2 I ⎢ T ⎥ < 0. T ˜ T ¯ −1 ⎣G sh A˜ sh G sh E sh −G sh Ps hˆ G sh ∗ ⎦ 0 0 −I D˜ sh

(4.121)

By pre- and post-multiplying diag{I, I, G −T sh , I } and its transpose to (4.121), one can obtain that ⎤ ⎡ −Psh ∗ ∗ ∗ 2 ⎢ 0 ∗ ∗⎥ −γ I ⎥ < 0. ⎢ (4.122) −1 ¯ ˜ ⎣ A˜ sh ∗⎦ E sh − Ps hˆ 0 0 −I D˜ sh By using Lemma 1.5 to (4.122), one gets diag{−Psh , −γ 2 I } + Γ1T P¯s hˆ Γ1 + Γ2T Γ2 < 0,     where Γ1 = A˜ sh E˜ sh and Γ2 = D˜ sh 0 .

(4.123)

140

4 Fuzzy Output Feedback Control with Communication Constraints

By pre-multiplying η T (k) and post-multiplying η(k) to (4.123), it can be inferred from (4.118) that η T (k)Ψsh η(k) + z T (k)z(k) − γ 2 ω T (k)ω(k) < 0.

(4.124)

For (4.124), when ω(k) = 0, one has Ψsh < 0, which implies that   E ΔV (x(k)) ˜ < −λmin (−Ψsh )η T (k)η(k) < 0.

(4.125)

Summing both sides of (4.125) from 0 to ∞ and taking mathematical expectation, it is easily obtained that ) E

∞ 

* η(k) |η(0), σ (0) < λ−1 ˜ < ∞. min (−Ψsh )V ( x(0)) 2

(4.126)

k=0

# $ # $ ∞ ∞ 2 Obviously, E η(k)2 |η(0), σ (0) ≥ E x(k) ˜ |x(0), ˜ σ (0) . Accordk=0

k=0

ing to Definition 4.1, it can be concluded that the closed-loop system (4.113) is stochastically stable. Next, for any non-zero ω(k), according to (4.118) and (4.125), one has E {ΔV (x(k))} ˜ + z T (k)z(k) − γ 2 ω T (k)ω(k) < 0.

(4.127)

Under zero initial condition V (x(0)) ˜ = 0, integrating both sides of (4.127) from 0 to ∞ and taking mathematical expectation, one can obtain E

)∞ 

* (z(k)2 − γ 2 ω(k)2 ) < −V (x(∞)) ˜ < 0.

(4.128)

k=0

From the above analysis and Definition 4.2, the closed-loop system (4.113) is stochastically stable with H∞ performance γ > 0. The proof is completed. Next, we will give design conditions to determine the gain matrices. Theorem 4.8 The T–S fuzzy system (4.113) is stochastically stable with H∞ performance γ > 0, given a scalar λ,  probabilities  ∈ M and s, t ∈ N,  transition  Gπ1slst , i,G l, ∗ 1si 2sl > 0, G = if there exist matrices Psi = PP2si sl P3si λG 3sl G 3sl , Rsl , and scalar  such that the following inequalities hold: Λsii < 0, Λsil + Λsli < 0, i < l,

(4.129)

4.3 Feedback Control with Stochastic Communication Protocol

where

Λsil

141

⎡

⎤ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −P1si ⎢ −P2si −P3si ⎥ ∗ ∗ ∗ ∗ ∗ ∗ ⎢ ⎥ 2I ⎢ 0 ⎥ 0 −γ ∗ ∗ ∗ ∗ ∗ ⎢ ⎥ ⎢ ψ1sil ⎥ ψ ψ ψ ∗ ∗ ∗ ∗ 2sl 3sil 4sl ⎥, =⎢ ⎢ ψ5sil ⎥ ψ6sl ψ7sil ψ8sl ψ9sl ∗ ∗ ∗ ⎢ ⎥ ⎢ D¯ si ⎥ 0 0 0 0 −I ∗ ∗ ⎢ ⎥ T T ¯ sl ¯ sl C¯ si C ⎣D ⎦ 0 ψ10sil ψ11sil 0 − H e{Rsl } ∗ T T ¯ sl Φs Fi  B¯ G 1sl  B¯ G 2sl 0 D 0 − H e{R } 0 0 sl si si

T ¯ ¯ sl C¯ si + U D ¯ sl C¯ si , ψ2sl = λA ¯ sl , ¯ sl + U C Asi + λB ψ1sil = G 1sl  T ¯ ¯ sl Φs Fi , ψ4sl = −H e{G 1sl } + E si + λB ψ3sil = G 1sl πst P1t , t∈N T ¯ ¯ sl C¯ si + U D ¯ sl C¯ si , ψ6sl = A ¯ sl , ¯ sl + U C Asi + B ψ5sil = G 2sl  T ¯ ¯ sl Φs Fi , ψ8sl = −λG 3sl − G T + E si + B ψ7sil = G 2sl πst P2t , 2sl

ψ9sl = −H e{G 3sl } +



t∈N

πst P3t , ψ10sil =

T ¯ Bsi G 1sl

T ¯ Bsi − U Rsl , − U Rsl , ψ11sil = G 2sl

t∈N

  and U is a dimension regulation matrix and defined as I , In xˆ ×n xˆ 0n xˆ ×(n u −n xˆ ) , and  T In u ×n u 0n u ×(n xˆ −n u ) when n xˆ = n u , n xˆ < n u , and n xˆ > n u , respectively. Then, the dynamic output feedback controller is given as T −1 ¯ T −1 ¯ Asl = (G 3sl ) Asl , Bsl = (G 3sl ) Bsl , −1 ¯ −1 ¯ Csl = Rsl Csl , Dsl = Rsl Dsl .

Proof Firstly, by defining  Psi =

P1si P2si

   ∗ G 1sl G 2sl , G sl = , P3si λG 3sl G 3sl

(4.130)

T ¯ sl = G T Bsl , C ¯ sl = Rsl Csl , and D¯sl = Rsl Dsl , the matrix ¯ sl = G 3sl and A Asl , B 3sl Σsil in Theorem 4.7 can be rewritten as follows:

  Σsil = Λ1sil + H e Λ2sil R¯ sl−1 Λ3sil , where

(4.131)

142

4 Fuzzy Output Feedback Control with Communication Constraints

⎡

−P1si ∗ ∗ ⎢−P2si −P3si ∗ ⎢ 2I ⎢ 0 0 −γ Λ1sil = ⎢ ⎢ ψ1sil ψ ψ 2sl 3sil ⎢ ⎣ ψ5sil ψ6sl ψ7sil 0 0 D¯ si  ¯ ¯ ¯ Dsl Csi Csl 0 Λ3sil = ¯ sl Φs Fi 0 0 D

∗ ∗ ∗

ψ4sl ψ8sl 0 0 0 0 0

⎤ ⎤ ⎡ 0 0 ∗ ⎢ 0 0 ⎥ ∗⎥ ⎥ ⎥ ⎢ ⎢ 0 0 ⎥ ∗⎥ ⎥ , Λ2sil = ⎢ ⎥ ⎢ψ10sil G T B¯ si ⎥ , ∗⎥ ⎥ ⎥ ⎢ 1sl T ⎦ ⎣ψ ψ9sl ∗ ⎦ 11sil G 2sl B¯ si 0 −I 0 0  0 , R¯ sl = diag{Rsl , Rsl }. 0 ∗ ∗ ∗ ∗

Further, (4.116) in Theorem 4.7 can be rewritten as follows:   Λ1sii + H e Λ2sii R¯ sl−1 Λ3sii < 0,   (Λ1sil + Λ1sli ) + H e Λ2sil R¯ sl−1 Λ3sil   +H e Λ2sli R¯ sl−1 Λ3sli < 0, i < l.

(4.132)

It is easy to know that (4.132) can be guaranteed if the condition (4.129) is satisfied by applying Lemma 1.4. The proof is completed.

4.3.3 Simulation Example Let us consider the discrete-time T–S fuzzy model in the form of (4.106) with x(k) = T  T x1 (k) x2T (k) . Then the rules are given as follows: Plant Rule 1: IF x1 (k) is J11 , THEN x(k + 1) = A1 x(k) + B1 u(k) + E 1 ω(k), y(k) = C1 x(k) + F1 ω(k), z(k) = D1 x(k), Plant Rule 2: IF x1 (k) is J12 , THEN x(k + 1) = A2 x(k) + B2 u(k) + E 2 ω(k), y(k) = C2 x(k) + F2 ω(k), z(k) = D2 x(k),

4.3 Feedback Control with Stochastic Communication Protocol

143

where 

   0.7238 0.4972 0.1176 , B1 = , E1 0.5148 0.0877 0.5759     0.9652 0.0969 0.1470 , B2 = , E2 A2 = 0.4813 0.0027 0.6748     0.8282 0.4565 −0.0169 , F1 = , C1 = −0.5649 0.1428 0.0842     0.1408 0.5053 −0.3267 C2 = , F2 = , −0.6831 0.2858 −0.0144 A1 =

  −0.1250 , 0.0148   −0.0530 = , −0.2478   D1 = 0.1426 −0.4029 ,

=

  D2 = −0.4803 −0.3073 .

The disturbance signal is given as ω(k) = 10e−0.4k cos(0.1k). In addition, the membership functions are set to be h 1 (x1 (k)) = sin2 (x1 (k)) and h 2 (x1 (k)) = 1 −  T h 1 (x1 (k)). Then, under the initial condition x(0) = 0 0 and u(k) = 0, the state response of the open-loop system is depicted in Fig. 4.14, which shows that this example is instable without considering the controller. Next, the effectiveness of the controller designed in this paper is demonstrated. Due to the SCP scheduling, we assume the transition probability matrix

P=

  0.45 0.55 . 0.65 0.35

By solving LMI-based inequalities conditions in Theorem 4.8 with H∞ index γ = 1, we obtain the designed controller gain matrices as follows: ⎡

⎡ ⎤ ⎤ 0.0010 0.0013 0.0002 −0.0001 −0.0773 0.0097 ⎢−0.0010 −0.0010 −0.0003 0.0001 ⎥ ⎢ 0.0147 ⎥ ⎥ , B11 = ⎢ 0.0193 ⎥, A11 = ⎢ ⎣ 0.0006 ⎣ ⎦ 0.0008 0.0001 −0.0001 −0.0929 −0.0042⎦ −0.0002 −0.0003 −0.0001 0 0.0194 −0.0510

Fig. 4.14 Response of the state of open-loop system

0

-0.5

-1

-1.5

-2 0

10

20

30

40

50

144

4 Fuzzy Output Feedback Control with Communication Constraints

⎡

A12

A21

A22 C11 C12 C21 C22

⎡ ⎤ ⎤ 0.0003 −0.0026 −0.0026 0.0011 −0.0351 0.0354 ⎢−0.0025 −0.0044 −0.0110 0.0041 ⎥ ⎢ 0.0133 −0.0266⎥ ⎢ ⎥ ⎥ =⎢ ⎣ 0.0006 −0.0000 0.0012 −0.0004⎦ , B12 = ⎣−0.0762 0.0111 ⎦ , −0.0002 0.0002 −0.0001 0 0.0089 −0.0637 ⎡ ⎡ ⎤ ⎤ 0.0057 0.0056 −0.0010 0.0014 −0.0215 0.0845 ⎢−0.0067 −0.0061 −0.0004 0.0002⎥ ⎢ ⎥ ⎥ , B21 = ⎢−0.0226 −0.0855⎥ , =⎢ ⎣ 0.0038 ⎣−0.0746 0.0241 ⎦ 0.0037 −0.0003 0.0005⎦ 0.0004 0.0004 0 0.0001 0.0198 −0.0676 ⎡ ⎡ ⎤ ⎤ 0.0026 0.0019 0.0013 −0.0006 −0.0223 0.0673 ⎢−0.0017 −0.0037 0.0016 ⎢ ⎥ 0.0003 ⎥ ⎥ , B22 = ⎢−0.0205 −0.0656⎥ , =⎢ ⎣ ⎣−0.0904 0.0164 ⎦ ⎦ 0 0 0 0 −0.0008 −0.0009 −0.0001 0.0002 −0.0143 −0.1102     = −0.0147 −0.0188 −0.0111 0.0042 , D11 = −0.9218 −0.0810 ,     = 0.0267 0.0514 0.1456 −0.0555 , D12 = −1.1533 0.4405 ,     = 0.1169 0.1106 −0.0029 0.0084 , D21 = 0.0467 1.8981 ,     = 0.0387 0.0517 −0.0048 −0.0076 , D22 = −0.0097 1.2577 .

Fig. 4.15 Response of the control input u(k)

1.2 1 0.8 0.6 0.4 0.2 0 -0.2

Fig. 4.16 Response of the state of closed-loop system

0

10

20

30

40

50

0

10

20

30

40

50

0.5 0 -0.5 -1 -1.5 -2

References Fig. &4.17 History of k k T T τ =0 z (τ )z(τ )/ τ =0 ω (τ )ω(τ )

145 0.12 0.1 0.08 0.06 0.04 0.02 0 0

10

20

30

40

50

We apply the above controller to the simulation example. The control input is shown in Fig. 4.15. The state response of the closed-loop system is shown in Fig. 4.16. It is obvious that the& system is stabilized by the controller designed above. Figure 4.17 k k T T depicts the ratio of τ =0 z (τ )z(τ )/ τ =0 ω (τ )ω(τ ), which is less than γ = 1.

4.4 Conclusion In this chapter, the output feedback control problem of T–S fuzzy system with several typical communication constraints has been addressed. Firstly, the quantization effects from measurement output of the system and the output of the controller have been fully considered. By using the concept of PDC, the novel design strategy for both the output feedback controller and the quantizers’ dynamic parameters has been proposed. Secondly, based on sampling data, the input delay approach has been introduced to the control synthesis for the fuzzy system with time delays and data dropouts. By using the Lyapunov function approach, the desired output feedback controller gain has been obtained. Thirdly, the Markov chain has been adopted to describe the SCP scheduling behavior. By using the fuzzy Markov jump model method, the dynamic output feedback control scheme has been provided. Finally, some examples have been given to demonstrate the effectiveness of the proposed methods.

References 1. Tanaka K, Ikeda T, Wang HO (1998) Fuzzy regulator and fuzzy observer: Relaxed stability conditions and LMI-based designs. IEEE Trans Fuzzy Syst 6:250-265 2. Chang XH, Huang R, Wang H, Liu L (2020) Robust design strategy of quantized feedback control. IEEE Trans Circuits Syst II Exp Briefs 67:730–734 3. Chang XH, Xiong J, Li ZM, Park JH (2018) Quantized static output feedback control for discrete-time systems, IEEE Trans Industr Inf 14:3426–3435 4. Chang XH, Park JH, Zhou J (2015) Robust static output feedback H∞ control design for linear systems with polytopic uncertainties. Syst Control Lett 85:23–32

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4 Fuzzy Output Feedback Control with Communication Constraints

5. Ushio T (1995) Chaotic synchronization and controlling chaos based on contraction mappings. Phys Lett A 198:14–22 6. Wu HN (2007) Robust H2 fuzzy output feedback control for discrete-time nonlinear systems with parametric uncertainties. Int J Approx Reason 46:151–165 7. Collins EG (2003) Fuzzy control systems design and analysis: a linear matrix inequality approach. Automatica 39:2011–2013 8. Zhao T, Dian SY (2017) Fuzzy dynamic output feedback H∞ control for continuous-time T-S fuzzy systems under imperfect premise matching. ISA Trans 70:248–259 9. Zou L, Wang Z, Gao H (2016) Observer-based H∞ control of networked systems with stochastic communication protocol: The finite-horizon case. Automatica 63:366–373

Chapter 5

Event-Triggered Fuzzy Control and Filtering with Communication Constraints

Abstract This chapter investigates the problem of control and filtering under the event-triggered mechanism for T–S fuzzy systems with different communication constraints. Firstly, the event-triggered output feedback tracking control is studied for discrete-time T–S fuzzy systems with static quantization, and sufficient conditions to design tracking controller are given by LMIs. Secondly, a piecewise Lyapunov– Krasovskii functional method is applied to the event-triggered L2 –L∞ filtering issue for T–S fuzzy systems with time-delay and external interference under the limited communication resources and DoS attacks, so that the filtering error system is exponentially stable and maintains the prescribed L2 –L∞ performance. Thirdly, the problem of event-triggered filtering for discrete-time T–S fuzzy systems with dynamic quantization and stochastic deception attacks is investigated. The sufficient design conditions for the full- and reduced-order H∞ filters are presented as two strict LMIs. Finally, some examples are provided to show the feasibility and effectiveness of the proposed design methods, respectively. Keywords T–S fuzzy systems · Event-triggered mechanism · Quantization · Denial-of-service attacks · Stochastic deception attacks

5.1 Tracking Control with Static Quantization In this section, an event-triggered static output feedback tracking controller is designed. In the presence of input and output static quantization, both the static logarithmic quantizers and the asynchronous event-triggered communication scheme are considered such that the closed-loop system is asymptotically stable with the prescribed H∞ tracking performance.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Chang et al., Control and Filtering of Fuzzy Systems Under Communication Channels, https://doi.org/10.1007/978-981-99-4346-3_5

147

148

5 Event-Triggered Fuzzy Control and Filtering …

5.1.1 Problem Formulation Consider a nonlinear system represented by T–S fuzzy dynamic model, in which the ith rule is described as follows: Plant Rule i th : IF 1 (k) is M1i , 2 (k) is M2i , and, · · · , and d (k) is Mdi , THEN x(k + 1) = Ai x(k) + Bi u g (k) + Di w(k), y(k) = Ci x(k),

(5.1)

where x(k) ∈ Rn x stands for the state variable, u g (k) ∈ Rn u represents the control input, y(k) ∈ Rn y means the measured output, and w(k) ∈ Rn w denotes the noise signal in l2 [0, ∞). The fuzzy sets are described by M pi , i = 1, 2, …, r , p = 1, 2, …, d, where r is the number of fuzzy rules. The premise variables are denoted by 1 (k), 2 (k), …, d (k). The matrices Ai ∈ Rn x ×n x , Bi ∈ Rn x ×n u , Ci ∈ Rn y ×n x , and Di ∈ Rn x ×n w with i = 1, 2, …, r stand for given system matrices. By introducing the following fuzzy basis functions: d

p=1 M pi ( p (k)) , d i=1 p=1 M pi ( p (k))

ρi ((k)) = r

(5.2)

where M pi ( p (k)) is the grade of membership function of  p (k) in M pi . According to the basis functions defined in (5.2), it can be obtained that ρi ((k)) ≥ 0,

r 

ρi ((k)) = 1, i = 1, 2, . . . , r.

(5.3)

i=1

Besides, the discrete-time T–S fuzzy model in (5.1) can be expressed as x(k + 1) = A(ρ)x(k) + B(ρ)u g (k) + D(ρ)w(k), y(k) = C(ρ)x(k), with A(ρ) = C(ρ) =

r  i=1 r  i=1

ρi ((k))Ai , B(ρ) = ρi ((k))Ci , D(ρ) =

r  i=1 r  i=1

ρi ((k))Bi , ρi ((k))Di .

(5.4)

5.1 Tracking Control with Static Quantization

149

In this section, the reference model is presented as follows:  x (k + 1) = Aι x (k) + Bι ι(k),  y(k) = Cι x (k),

(5.5)

where  x (k) ∈ Rnx , ι ∈ Rn ι , and  y(k) ∈ Rn y are used to denote the state, the energy bounded reference input, and the measurement output of the reference model, respectively. Aι ∈ Rnx ×nx is a known Hurwitz matrix, Bι ∈ Rnx ×n ι and Cι ∈ Rn y ×nx stand for known matrices. In order to reduce the amount of data in the communication of network and realize the effective utilization of the limited network communication resources, the following improved event-triggered mechanism will be employed. The output of the fuzzy model (5.4), i.e., y(k) will be transmitted to the quantizer g y () only if ξ yT (k)J y ξ y (k) − b y (y(k) − ξ y (k))T J y (y(k) − ξ y (k)) ≥ 0

(5.6)

is satisfied. Here, ξ y (k) = y(k) − y(ks ), y(k) is the present data for output, and y(ks ) represents the last triggering instant. b y ∈ [0, 1) is a predetermined scalar, J y > 0 stands for a weighting matrix to be determined. The output of the reference model (5.5), i.e.,  y(k) will be transmitted to the quantizer gy () only when y(k) − ξy (k))T Jy ( y(k) − ξy (k)) ≥ 0 ξyT (k)Jy ξy (k) − by (

(5.7)

y(k) −  y(kr ), y(k) represents the present data for reference is satisfied. Here, ξy (k) =  output, and  y(kr ) is the last triggering instant. by ∈ [0, 1) means a predetermined scalar, Jy > 0 is a weighting matrix to be determined. Remark 5.1 It is worth noting that the event-triggered mechanism used in this section is asynchronous, i.e., different event-triggered conditions (5.6) and (5.7) are predefined to independently determine whether the plant’s output y(k) and the reference model’s output  y(k) should be released to the tracking controller or not. In contrast with the synchronous event-triggered mechanism used in [1–3], i.e., a specific event-triggered condition is predefined to judge whether the augmented variable of the states for the controlled system and the reference model should be released to the tracking controller at the same certain time-point, the asynchronous eventtriggered mechanism used in this section is more general. This is mainly because it is difficult for various parts of the system to be located at the same place in networked based tracking control and the asynchronous event-triggered mechanism can effectively reduce the network communication burdens in both communication channels from the plant to the controller and from the reference model to the controller.

150

5 Event-Triggered Fuzzy Control and Filtering …

To further reduce the amount of data in the communication of network and realize the effective utilization of the limited network communication resources, three static y(kr ), and u(k) before logarithmic quantizers will be utilized to quantize y(ks ),  they are transmitted to the tracking controller and the plant, respectively. The static logarithmic quantizers considered in this section are given as T  n yg (ks ) = g y (y(ks )) = g 1y (y 1 (ks )) g 2y (y 2 (ks )) . . . g y y (y n y (ks )) , j

j

j

j

g yy (−y jy (ks )) = −g yy (y jy (ks )), j y = 1, 2, . . . , n y , T  n y 1 (kr )) g2y ( y 2 (kr )) . . . gy y ( y n y (kr )) , y(kr )) = g1y (  yg (kr ) = gy ( gyy (− y jy (kr )) = −gyy ( y jy (kr )), jy = 1, 2, . . . , n y , T  u g (k) = gu (u(k)) = gu1 (u 1 (k)) gu2 (u 2 (k)) . . . gun u (u n u (k)) , guju (−u ju (k)) = −guju (u ju (k)), ju = 1, 2, . . . , n u . (5.8) j As in [4–6], the set of quantization levels for quantizer guu (u ju (k)) is assumed to be

j

j

j

j

U ju = {±ηi u , ηi u = (θuju )i η0u , i = ±1, ±2, · · · } ∪ {±η0u } ∪ {0}, j

j

(5.9) j

where θu u ∈ (0, 1) stands for the quantization density of guu (u ju (k)) , η0u > 0. The j definition of guu (u ju (k)) is formulated as

guju (u ju (k)) =

⎧ ⎪ ⎪ηiju , ⎨

j

ηi u j 1+βuu ju

if

< u ju (k) ≤

0, if u (k) = 0, ⎪ ⎪ ⎩−g ju (−u ju (k)), if u ju (k) < 0, u

where

j

ηi u j 1−βuu

, (5.10)

j

βuju =

1 − θu u j

1 + θu u

j

.

(5.11)

j

Besides, assuming that θ y y ∈ (0, 1) and θyy ∈ (0, 1) are quantization densities j

j

for g yy (y jy (ks )) and gyy ( y jy (kr )), respectively. Then, there are j

j

j

βyy =

1 − θy y

j

, βyy = jy

1 + θy

1 − θyy j

1 + θyy

.

(5.12)

Based on the important conclusions developed in [4], it can be obtained that yg (ks ) − y(ks ) = Ξ y (k)y(ks ), Ξ y (k) ∈ [−β y , β y ], y(kr ) = Ξy (k) y(kr ), Ξy (k) ∈ [−βy , βy ],  yg (kr ) −  u g (k) − u(k) = Ξu (k)u(k), Ξu (k) ∈ [−βu , βu ],

(5.13)

5.1 Tracking Control with Static Quantization

where

151

n

Ξ y (k) = diag{Ξ y1 (k), Ξ y2 (k), . . . , Ξ y y (k)}, n

Ξy (k) = diag{Ξy1 (k), Ξy2 (k), . . . , Ξy y (k)}, Ξu (k) = diag{Ξu1 (k), Ξu2 (k), . . . , Ξun u (k)}, n

β y = diag{β y1 , β y2 , . . . , β y y }, n

βy = diag{βy1 , βy2 , . . . , βy y }, βu = diag{βu1 , βu2 , . . . , βun u }. Remark 5.2 In this section, three static logarithmic quantizers are used to reduce the number of data transmission in the communication channels from the plant and the reference model to the controller and from the controller to the plant, which is a more general assumption in practical situations. It is a significant contribution relative to the existing results on quantized tracking control where only quantized measurement output (see [7] for more details) or quantized control input (see [8] for more details) was considered. The output feedback tracking controller discussed in this section is presented as yg (kr ), u(k) = G 1 yg (ks ) + G 2

(5.14)

where G 1 and G 2 denote the parameters for the tracking controller. By considering the equation in (5.13), u(k) defined in (5.14) can be expressed as y(kr ). u(k) = G 1 (I + Ξ y (k))y(ks ) + G 2 (I + Ξy (k))

(5.15)

y(kr ) =  y(k) − ξy (k) into conMoreover, by taking y(ks ) = y(k) − ξ y (k) and  sideration, one can be obtained that u(k) = G 1 (I + Ξ y (k))(C(ρ)x(k) − ξ y (k)) + G 2 (I + Ξy (k))(Cι x (k) − ξy (k)) = G 1 (I + Ξ y (k))C(ρ)x(k) − G 1 (I + Ξ y (k))ξ y (k)) + G 2 (I + Ξy (k))Cι x (k) − G 2 (I + Ξy (k))ξy (k)).

(5.16)

Based on the discussions above, the closed-loop system can be established as δ(k + 1) = Ac (ρ)σ (k) + Bc (ρ)wι (k), ec (k) = Cc (ρ)σ (k),

(5.17)

152

5 Event-Triggered Fuzzy Control and Filtering …

where     x T (k) ξ yT (k) ξyT (k) , x T (k) , σ T (k) = x T (k)  δ T (k) = x T (k)    wιT (k) = w T (k) ιT (k) , ec (k) = y(k) −  y(k),



Δ1 (ρ) Δ2 (ρ) Δ3 (ρ) Δ4 (ρ) D(ρ) 0 , , Bc (ρ) = Ac (ρ) = 0 0 0 Aι 0 Bι   Cc (ρ) = C(ρ) − Cι 0 0 , Δ1 (ρ) = A(ρ) + B(ρ)G 1 (I + Ξ y (k))C(ρ) + B(ρ)Ξu (k)G 1 (I + Ξ y (k))C(ρ), Δ2 (ρ) = B(ρ)G 2 (I + Ξy (k))Cι + B(ρ)Ξu (k)G 2 (I + Ξy (k))Cι , Δ3 (ρ) = −B(ρ)G 1 (I + Ξ y (k)) − B(ρ)Ξu (k)G 1 (I + Ξ y (k)), Δ4 (ρ) = −B(ρ)G 2 (I + Ξy (k)) − B(ρ)Ξu (k)G 2 (I + Ξy (k)). The event-triggered static output feedback tracking control problem considered in this section is to design an event-based output feedback tracking controller in (5.14) such that (1) The closed-loop system (5.17) is asymptotically stable when wι (k) = 0. (2) The closed-loop system (5.17) has a prescribed H∞ tracking performance γ > 0, i.e., under the zero-initial condition, ∞ 

ecT (k)ec (k) < γ 2

k=0

∞ 

wι T (k)wι (k)

k=0

is satisfied for any wι (k) = 0.

5.1.2 Main Results In this subsection, firstly, the sufficient conditions will be given, such that the closedloop system (5.17) is asymptotically stable with the specified H∞ tracking performance γ > 0 in the presence of event-triggered communication scheme and quantization. Based on the Lyapunov theory, a useful H∞ tracking performance analysis criterion will be given in the following theorem, which can ensure the closed-loop system (5.17) is asymptotically stable and has the specified H∞ tracking performance. Theorem 5.1 Consider the fuzzy system (5.1), the reference model (5.5), and the output feedback tracking controller (5.14). Suppose that the scalars 0 ≤ b y < 1 and 0 ≤ by < 1 are known. The closed-loop system (5.17) is asymptotically stable with a specified H∞ tracking performance γ > 0, if there exist matrices P > 0, J y > 0, and Jy > 0 satisfying

5.1 Tracking Control with Static Quantization

153

⎤  + b y Σ yT J y Σ y + by ΣyT Jy Σy − Jy −  Jy − P ∗ ∗ ∗ ⎢ 0 − γ2I ∗ ∗ ⎥ ⎥ < 0, ⎢ −1 ⎣ Bc (ρ) − P ∗ ⎦ Ac (ρ) 0 0 −I Cc (ρ) (5.18) where    J y = diag 0, 0, J y , 0 ,    Jy = diag 0, 0, 0, Jy , ⎡

 = diag {P, 0, 0} , P   Σ y = C(ρ) 0 − I 0 ,   Σy = 0 Cι 0 − I . Proof Firstly, we will show that if the condition (5.18) is satisfied, then, for wι (k) = 0, the closed-loop system (5.17) is asymptotically stable. To this end, a Lyapunov function is constructed as V (δ(k)) = δ T (k)Pδ(k), P > 0.

(5.19)

According to the event-triggered conditions given in (5.6) and (5.7), one can be obtained that b y (y(k) − ξ y (k))T J y (y(k) − ξ y (k)) − ξ yT (k)J y ξ y (k) ≥ 0, by ( y(k) − ξy (k))T Jy ( y(k) − ξy (k)) − ξyT (k)Jy ξy (k) ≥ 0. Then, it is easily to have that V (δ(k + 1)) − V (δ(k)) ≤ δ T (k + 1)Pδ(k + 1) − δ T (k)Pδ(k) − ξ yT (k)J y ξ y (k) + b y (y(k) − ξ y (k))T J y (y(k) − ξ y (k)) − ξyT (k)Jy ξy (k) + by ( y(k) − ξy (k))T Jy ( y(k) − ξy (k)) = σ T (k)AcT (ρ)P Ac (ρ)σ (k) − δ T (k)Pδ(k) − ξyT (k)Jy ξy (k) + by (Cι x (k) − ξy (k))T Jy (Cι x (k) − ξy (k)) − ξ yT (k)J y ξ y (k) + b y (C(ρ)x(k) − ξ y (k))T J y (C(ρ)x(k) − ξ y (k))   − = σ T (k) AcT (ρ)P Ac (ρ) − P Jy −  Jy + b y Σ yT J y Σ y + by ΣyT Jy Σy σ (k). (5.20) When wι (k) = 0, the inequality in (5.18) reduces to

 + b y Σ yT J y Σ y + by ΣyT Jy Σy − Jy −  Jy − P ∗ < 0. − P −1 Ac (ρ) By using Lemma 1.1 to (5.21), it can be obtained that

(5.21)

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5 Event-Triggered Fuzzy Control and Filtering …

 + b y Σ yT J y Σ y + by ΣyT Jy Σy < 0. AcT (ρ)P Ac (ρ) −  Jy −  Jy − P

(5.22)

Then, for σ (k) = 0, it can be obtained that V (δ(k + 1)) − V (δ(k)) < 0.

(5.23)

Thus, it can be concluded that the closed-loop system (5.17) is asymptotically stable for wι (k) = 0 if the condition (5.18) is satisfied. Next, for wι (k) = 0, the H∞ tracking performance for the closed-loop system (5.17) under zero-initial conditions will be discussed. When wι (k) = 0, the following conclusion can be drawn V (δ(k + 1)) − V (δ(k)) + ecT (k)ec (k) − γ 2 wιT (k)wι (k) ≤ δ T (k + 1)Pδ(k + 1) − δ T (k)Pδ(k) + ecT (k)ec (k) − γ 2 wιT (k)wι (k) + b y (y(k) − ξ y (k))T J y (y(k) − ξ y (k)) − ξ yT (k)J y ξ y (k) + by ( y(k) − ξy (k))T Jy ( y(k) − ξy (k)) − ξyT (k)Jy ξy (k) = (Ac (ρ)σ (k) + Bc (ρ)wι (k))T P (Ac (ρ)σ (k) + Bc (ρ)wι (k)) − −

(5.24)

δ (k)Pδ(k) + σ (k)CcT (ρ)Cc (ρ)σ (k) − γ 2 wιT (k)wι (k) ξyT (k)Jy ξy (k) + by (Cι x (k) − ξy (k))T Jy (Cι x (k) − ξy (k)) T

T

− ξ yT (k)J y ξ y (k) + b y (C(ρ)x(k) − ξ y (k))T J y (C(ρ)x(k) − ξ y (k)) = σ T (k)Z(ρ) σ (k),   where  σ T (k) = σ T (k) wιT (k) and  T    T   Cc (ρ) 0 Z(ρ) = Ac (ρ) Bc (ρ) P Ac (ρ) Bc (ρ) + Cc (ρ) 0      γ 2 I } + b y Σ y 0 T J y Σ y 0 − diag{ J y , 0} − diag{ P, T    Jy , 0}. + by Σy 0 Jy Σy 0 − diag{ By applying Lemma 1.1 to (5.18), it can be obtained that Z(ρ) < 0.

(5.25)

Then, for  σ (k) = 0, there is V (δ(k + 1)) − V (δ(k)) + ecT (k)ec (k) − γ 2 wιT (k)wι (k) < 0.

(5.26)

Moreover, by summing up (5.26) from k = 0 to k = ∞, it can be obtained that V (δ(∞)) − V (δ(0)) +

∞  k=0

ecT (k)ec (k) − γ 2

∞  k=0

wιT (k)wι (k) < 0.

(5.27)

5.1 Tracking Control with Static Quantization

155

According to zero-initial conditions, one can obtain V (δ(0)) = 0 and V (δ(∞)) ≥ 0, which implies that ∞  k=0

ecT (k)ec (k) − γ 2

∞ 

wιT (k)wι (k) < 0.

(5.28)

k=0

Thus, one can conclude that the closed-loop system (5.17) has a specified H∞ tracking performance γ > 0 if the condition (5.18) is satisfied. The proof is completed. Then, according to the analysis results proposed in Theorem 5.1, the design problem of the event-triggered output feedback tracking controller will be addressed. More specifically, by using two-step uncertainty elimination method combined with the extended matrix inequality decoupling strategy, the sufficient design conditions for the event-triggered quantized output feedback tracking controller defined in (5.14) will be proposed in terms of LMIs. Theorem 5.2 Consider the fuzzy system (5.4), the reference model (5.5), and the output feedback tracking controller (5.14). Suppose that the scalars 0 ≤ b y < 1, 0 ≤ by < 1, and λ are known. The closed-loop system (5.17) is asymptotically stable with a specified H∞ tracking performance γ > 0, if there exist matrices P1 , P2 , P3 , J y > 0, Jy > 0, S1 , S2 , N1 , N2 , R1 , and R2 , scalars ϑu > 0, ϑ y > 0, and ϑy > 0 satisfying

P1 ∗ > 0, (5.29) P2 P3 ⎡ 11i ∗ ⎢ 0 − γ2I ⎢ ⎢31i 32i ⎢ ⎢41i 0 ⎢ ⎢51i 0 ⎢ ⎢61i 0 ⎢ ⎣ 71 0 0 81i

⎤ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ 33 ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ 0 −I ∗ ∗ ∗ ∗ ⎥ ⎥ < 0, i = 1, 2, . . . , r, 53i 0 − ϑu I ∗ ∗ ∗ ⎥ ⎥ 63 0 65 − ϑ y I ∗ ∗ ⎥ ⎥ 73 0 75 0 − ϑy I ∗ ⎦ 83i 0 85 86 87 88 (5.30)

where ⎤ ∗ ∗ ∗ −P1 + b y CiT J y Ci ⎥ ⎢ − P3 + by CιT Jy Cι ∗ ∗ −P2 ⎥, =⎢ ⎦ ⎣ 0 (b y − 1)J y ∗ −b y J y Ci 0 (by − 1)Jy 0 − by Jy Cι



S1 Ai + Θ1 N1 Ci Θ2 N2 Cι − Θ1 N1 − Θ2 N2 S1 Di 0 , = , 32i = 0 S2 Aι 0 0 0 S2 Bι

  ∗ P − S1 − S1T , 41i = Ci − Cι 0 0 , = 1 P2 P3 − S2 − S2T ⎡

11i

31i 33

156

5 Event-Triggered Fuzzy Control and Filtering …

0 0 0 0 (S1 Bi )T 0 , , 53i = 0 0 N1 C i N2 C ι − N1 − N2

0 0 0 0 (Θ1 N1 )T 0 0 N1T = , 63 = , , 65 = 0 0 0 0 ϑ y β y Ci 0 − ϑ y β y 0



0 0 0 0 (Θ2 N2 )T 0 0 N2T , 73 = , = , 75 = 0 0 0 0 0 ϑy βy Cι 0 − ϑy βy



− N1 0 N1 C i 0 λ(S1 Bi − Θ1 R1 )T 0 , 83i = = , 0 − N2 0 N2 C ι λ(S1 Bi − Θ2 R2 )T 0



0 λ(ϑu βu − R1 )T N1 0 ,  = = , 86 0 λ(ϑu βu − R2 )T 0 0

∗ 0 0 −λ(R1 + R1T ) , 88 = . = 0 − λ(R2 + R2T ) N2 0

51i = 61i 71 81i 85 87

Moreover, the parameters for the tracking controller defined in (5.14) can be obtained as G 1 = R1−1 N1 , G 2 = R2−1 N2 . (5.31) Proof First of all, Lemma 1.2 will be used to eliminate the quantization error terms method will be Ξu (k), Ξ y (k), and Ξy (k), and the two-step uncertainty elimination   used to deal with the crossing terms between Ξu (k) and Ξ y (k) Ξy (k) . Specifically, the quantization error term Ξu (k) will be separated and eliminated in the first step and the quantization error terms Ξ y (k) and Ξy (k) will be separated and eliminated in the second step. The condition in (5.18) can be expressed as ⎡ ⎤  + b y Σ yT J y Σ y + by ΣyT Jy Σy Jy − P ∗ ∗ ∗ − Jy −  ⎢ 0 − γ2I ∗ ∗ ⎥ ⎢ ⎥ −1 ⎣ c (ρ) Bc (ρ) − P ∗ ⎦ A 0 0 −I Cc (ρ)   T Ξu (k)    Q 2 (ρ) 0 0 0 < 0, + H e 0 0 Q 1T (ρ) 0 βu where



2 (ρ) Δ 3 (ρ) Δ 4 (ρ) 1 (ρ) Δ B(ρ) Δ  , Q 1 (ρ) = Ac (ρ) = , 0 0 0 0 Aι   Q 2 (ρ) = M1 (ρ) M2 M3 M4 , 1 (ρ) = A(ρ) + B(ρ)G 1 (I + Ξ y (k))C(ρ), Δ 2 (ρ) = B(ρ)G 2 (I + Ξy (k))Cι , Δ

(5.32)

5.1 Tracking Control with Static Quantization

157

3 (ρ) = −B(ρ)G 1 (I + Ξ y (k)), Δ 4 (ρ) = −B(ρ)G 2 (I + Ξy (k)), Δ M1 (ρ) = βu G 1 (I + Ξ y (k))C(ρ), M2 = βu G 2 (I + Ξy (k))Cι , M3 = −βu G 1 (I + Ξ y (k)), M4 = −βu G 2 (I + Ξy (k)). According to Lemma 1.2, it can be obtained that the inequality in (5.32) is true if and only if there exists a scalar ϑu > 0 such that the following inequality ⎡

⎤  + b y Σ yT J y Σ y + by ΣyT Jy Σy − Jy −  Jy − P ∗ ∗ ∗ ∗ ⎢ 0 − γ2I ∗ ∗ ∗ ⎥ ⎢ ⎥ −1 ⎢ c (ρ) B (ρ) − P ∗ ∗ ⎥ A c ⎢ ⎥ 0 and ϑy > 0 such that the following inequality ⎤ Σp ∗ ∗ ∗ ∗ ∗ ∗ ⎢ 0 ∗ ∗ ∗ ∗ ∗ ⎥ − γ2I ⎥ ⎢ ⎢ Ac (ρ) Bc (ρ) − P −1 ∗ ∗ ∗ ∗ ⎥ ⎥ ⎢ ⎢Cc (ρ) 0 0 −I ∗ ∗ ∗ ⎥ ⎥ 0, it can be obtained that −S T P −1 S ≤ −S T − S + P. Then, the inequality (5.36) can be guaranteed by ⎤ Σp ∗ ∗ ∗ ∗ ∗ ∗ ⎢ ∗ ∗ ∗ ∗ ∗ ⎥ − γ2I ⎥ ⎢ T 0 ⎢ S Ac (ρ) S T Bc (ρ) − S T − S + P ∗ ∗ ∗ ∗ ⎥ ⎥ ⎢ ⎢ Cc (ρ) 0 0 −I ∗ ∗ ∗ ⎥ ⎥ < 0. ⎢ ⎥ ⎢ W 51 0 W S 0 − ϑ I ∗ ∗ 53 u ⎥ ⎢ ⎣ W61 0 W63 S 0 W65 − ϑ y I ∗ ⎦ 0 W73 S 0 W75 0 − ϑy I W71 (5.37)

P1 ∗ S 0 1 By choosing P = and S T = , the condition in (5.37) can be P2 P3 0 S2 expressed as ⎡

⎡

F11 ∗ ⎢ 0 − γ2I ⎢ ⎢ F31 F32 ⎢ ⎢ F41 0 ⎢ ⎢W 51 0 ⎢ ⎣ F61 0 0 71

⎤ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ 33 ∗ ∗ ∗ ∗ ⎥ ⎥ 0 −I ∗ ∗ ∗ ⎥ ⎥ < 0, F53 0 − ϑu I ∗ ∗ ⎥ ⎥ F63 0 F65 − ϑy I ∗ ⎦ F73 0 F75 0 − ϑy I

(5.38)

  where 33 and 71 have been defined in (5.30) and F41 = C(ρ) −Cι 0 0 , ⎤ ∗ ∗ ∗ −P1 + b y C T (ρ)J y C(ρ) ⎥ ⎢ − P3 + by CιT Jy Cι ∗ ∗ −P2 ⎥, =⎢ ⎦ ⎣ 0 (b y − 1)J y ∗ −b y J y C(ρ) 0 (by − 1)Jy 0 − by Jy Cι

S1 A(ρ) + S1 B(ρ)G 1 C(ρ) S1 B(ρ)G 2 Cι − S1 B(ρ)G 1 − S1 B(ρ)G 2 , = 0 S2 Aι 0 0

  S D(ρ) 0 (S1 B(ρ))T 0 , , F41 = C(ρ) − Cι 0 0 , F53 = = 1 0 S2 Bι 0 0

0 0 0 0 (S1 B(ρ)G 1 )T 0 , F63 = , = 0 0 ϑ y β y C(ρ) 0 − ϑ y β y 0



0 ϑu (βu G 1 )T (S1 B(ρ)G 2 )T 0 0 ϑu (βu G 2 )T , F73 = . = , F75 = 0 0 0 0 0 0 ⎡

F11

F31 F32 F61 F65

However, it is worth noting that the condition in (5.38) is still non-convex with respect to the parameters of the event-triggered output feedback tracking controller defined in (5.14). Therefore, the matrix inequality decoupling strategy developed in [9, 10] will be extended to transform the condition in (5.38) into convex linear matrix inequality form.

160

5 Event-Triggered Fuzzy Control and Filtering …

To this end, suppose that G 1 = R1−1 N1 and G 2 = R2−1 N2 . Then, the condition of (5.38) can be reorganized as ⎡

⎤ F11 ∗ ∗ ∗ ∗ ∗ ∗ ⎢ 0 − γ2I ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢F ⎥ 31 F32 33 ∗ ∗ ∗ ∗ ⎢ ⎥ ⎢ F41 ⎥ 0 0 − I ∗ ∗ ∗ ⎢ ⎥ ⎢F  0 F53 0 − ϑu I ∗ ∗ ⎥ ⎢ 51 ⎥ ⎣ F61 0 63 0 65 − ϑ y I ∗ ⎦ 0 73 0 75 0 − ϑy I 71 





 T 0 0 1T 0 3T 0 0 R1−1 ∗ 5 0 0 0 0 6 0 +H e < 0, 0 0 2T 0 4T 0 0 0 R2−1 7 0 0 0 0 0 8 (5.39) where 33 , 63 , 65 , 71 , 73 , and 75 have been defined in (5.30) and

S1 A(ρ) + Θ1 N1 C(ρ) Θ2 N2 Cι − Θ1 N1 − Θ2 N2  , F31 = 0 S2 Aι 0 0

0 0 0 0 S1 B(ρ) − Θ1 R1  , 1 = , F51 = 0 N1 C(ρ) N2 Cι − N1 − N2

S1 B(ρ) − Θ2 R2 0 0 , 4 = , , 3 = 2 = 0 ϑu βu − R1 ϑu βu − R2     5 = N1 C(ρ) 0 − N1 0 , 6 = N1 0 ,     7 = 0 N2 Cι 0 − N2 , 8 = N2 0 . By applying Lemma 1.4, it can be concluded that the condition of (5.39) is true if and only if there exists a scalar λ such that the following inequality ⎡

F11 ∗ ⎢ 0 − γ2I ⎢ ⎢F  ⎢ 31 F32 ⎢ F41 0 ⎢ ⎢F 51 0 ⎢ ⎢ F61 0 ⎢ ⎣71 0 0 F81

⎤ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ 33 ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ 0 −I ∗ ∗ ∗ ∗ ⎥ ⎥ n u .

5.1.3 Simulation Example Let us consider a discrete-time T–S fuzzy model in the form of (5.1), where





0.6985 − 0.2942 0.3665 0.1576 , B1 = , D1 = , A1 = 0.2547 0.9533 0.0582 −0.0832

0.6985 − 0.2942 0.1430 −0.0128 A2 = , B2 = , D2 = , 0.2547 0.9533 0.0227 0.1324     C1 = 1 0 , C2 = 1 0 . Then, the reference model is assumed to be  x (k + 1) = 0.2 x (k) − 0.2ι(k),  y(k) = 0.2 x (k).

162 Fig. 5.1 Response of the system state x(k)

5 Event-Triggered Fuzzy Control and Filtering … 0.4

0.2

0

-0.2

-0.4 0

Fig. 5.2 Response of the control input u(k)

50

100

150

200

250

300

150

200

250

300

150

200

250

300

0.06 0.04 0.02 0 -0.02 -0.04 -0.06 0

Fig. 5.3 Responses of y(k) and  y(k)

50

100

0.3 0.2 0.1 0 -0.1 -0.2 -0.3 0

50

100

5.1 Tracking Control with Static Quantization Fig. 5.4 Response of ec (k)

163

0.3 0.25 0.2 0.15 0.1 0.05 0 -0.05 0

Fig. 5.5 Release instants and release intervals of the communication channel from the plant to the controller

50

100

150

200

250

300

5 4 3 2 1 0

Fig. 5.6 Release instants and release intervals of the communication channel from the reference model to the controller

0

50

100

150

200

250

300

0

50

100

150

200

250

300

5 4 3 2 1 0

164

5 Event-Triggered Fuzzy Control and Filtering …

By applying Theorem 5.2 with γ = 1.7, b y = 0.25, by = 0.25, λ = 0.5, β y =  T  T 0.25, βy = 0.25, βu = 0.25, Θ1 = 1 0 , and Θ2 = 1 0 , it can be obtained that G 1 = −0.2858, G 2 = 0.0042, J y = 7.2937, Jy = 52.1520.  T x (0) = 0, w(k) = − sin(0.5k)e−0.02k , Moreover, assume that x(0) = 0 0 ,  Ξu (k) = 0.2 sin(0.2k), Ξ y (k) = 0.2 sin(0.5k), Ξy (k) = 0.2 sin(0.1k), ι(k) = 5 cos (0.5k − 0.7)e−0.02k , ρ1 ((k)) = sin2 (x1 (k)), ρ2 ((k)) = 1 − ρ1 ((k)). The simulation results of the closed-loop system (5.17) are given in Figs. 5.1, 5.2, 5.3, 5.4, 5.5 and 5.6. The trajectories of x1 (k) and x2 (k) are shown in Fig. 5.1. The trajectory of u(k) is shown in Fig. 5.2. The trajectories of y(k) and  y(k) are shown in Fig. 5.3. The trajectory of ec (k) is shown in Fig. 5.4. The simulation result on release instants and release intervals of the communication channel from the plant to the controller is depicted in Fig. 5.5. The simulation result on release instants and release intervals of the communication channel from the reference model to the controller is depicted in Fig. 5.6. Based on the simulation results given in Figs. 5.1, 5.2, 5.3, 5.4, 5.5, and 5.6, it can be concluded that the event-triggered output feedback tracking controller design approach developed in this section is effective.

5.2 L2 –L∞ Filtering with Time Delays and DoS Attacks In this section, the event-triggered L2 –L∞ filtering issue for T–S fuzzy systems with time-delay and DoS attacks is studied. A piecewise Lyapunov–Krasovskii functional method is applied to judge the stability of the T–S fuzzy closed-loop system and ensure the T–S fuzzy closed-loop system having a prescribed L2 –L∞ performance.

5.2.1 Problem Formulation Consider a T–S fuzzy model with time-delay as follows: Plant Rule i th : IF q1 (t) is R1i , q2 (t) is R2i , and, · · · , and q p (t) is R pi , THEN x(t) ˙ = Ai x(t) + Bi x(t − ρ) ¯ + E i w(t), y˜ (t) = Ci x(t),

(5.42)

where x(t) ∈ Rn x , y˜ (t) ∈ Rn y , and w(t) ∈ Rn w are the state vector, the output vector, and the arbitrary disturbance belonging to L2 [0, ∞), respectively. q(t) = [ q1 (t), q2 (t), . . . , q p (t) ], where qd (t), d = 1, 2, . . . , p are measurable premise variables; Rdi , i = 1, 2, . . . , r, d = 1, 2, . . . , p are the fuzzy sets; r is the number

5.2 L2 –L∞ Filtering with Time Delays and DoS Attacks

165

of fuzzy rules. Ai ∈ Rn x ×n x , Bi ∈ Rn x ×n x , E i ∈ Rn x ×n w and Ci ∈ Rn y ×n x are known constant matrices for i = 1, 2, . . . , r ; ρ¯ is time delay. The fuzzy basic function is provided as follows: p Rdi (qd (t)) p , θi (q(t)) = r d=1 i=1 d=1 Rdi (qd (t)) where Rdi (qd (t)) is the grade of the membership of qd (t) in Rdi . Thus, it is easily r  θi (q(t)) = 1. get that θi (q(t)) ≥ 0 and i=1

Then system (5.42) can be denoted as: x(t) ˙ = A(θ )x(t) + B(θ )x(t − ρ) ¯ + E(θ )w(t), y˜ (t) = C(θ )x(t), where A(θ ) = E(θ ) =

r  i=1 r 

θi (q(t))Ai , B(θ ) = θi (q(t))E i , C(θ ) =

i=1

r  i=1 r 

(5.43)

θi (q(t))Bi , θi (q(t))Ci .

i=1

According to system (5.43), the following filter is proposed x˙ f (t) = A(θ )x f (t) + B(θ )x f (t − ρ) ¯ + L( y˜ (t) − y f (t)), y f (t) = C(θ )x f (t),

(5.44)

where x f (t) ∈ Rn x , y f (t) ∈ Rn y , and L ∈ Rn x ×n y are the state vector of the filter, the output vector of the filter, and the gain matrix of the filter. Define the filtering error z(t) = x(t) − x f (t) and the output error y(t) = y˜ (t) − y f (t), then the filtering error system is represented as follows: z˙ (t) = A(θ )z(t) + B(θ )z(t − ρ) ¯ + E(θ )w(t) − L y(t), y(t) = C(θ )z(t).

(5.45)

Consider system (5.42) with prescribed DoS attacks in this section. The DoS model is given by [12]:  F DoS (t) =

1, t ∈ [hT, hT + To f f ), 0, t ∈ [hT + To f f , (h + 1)T ),

(5.46)

where h ∈ N is the periodic count; {hT }h∈N is the time instants which means that there is no DoS attack. The sets ∪h∈N [hT, hT + To f f ) are DoS-free interval, in which DoS

166

5 Event-Triggered Fuzzy Control and Filtering …

attacks are stopping and signals are sent, while the sets ∪h∈N [hT + To f f , (h + 1)T ) denote the set of DoS interval, where DoS attacks are active and no signal is sent. To f f ∈ R>0 (To f f < T ) signifies the period where the DoS attacks are asleeping. max Assuming that To f f is time-varying and exists real scalars Tomin f f ∈ R>0 and To f f max ∈ R>0 such that Tomin f f ≤ To f f < To f f < T < ∞ holds. Consider y(t) is sampled at time nτ (n = 1, 2, . . .) with τ > 0. Then we predefined an event-triggering condition [13], only signals that meet the specified conditions are sent. Firstly, when there are no DoS attacks, the delivery of data is decided by the following triggering condition: [y(tιn τ ) − y(tι τ )]T H [y(tιn τ ) − y(tι τ )] > σ y T (tιn τ )H y(tιn τ ), n ∈ N,

(5.47)

where σ ∈ (0, 1) is a threshold and H > 0 is a weighting matrix. tι τ is the last eventtriggering instant and tιn τ = tι τ + nτ (ι, n ∈ N) is the sampling instant. However, considering the impact of DoS attacks, the event-triggered mechanism in (5.47) is not able to be used directly. Thus, to offset the DoS attacks in (5.46), the eventtriggered mechanism in (5.47) needs to be revised. Similar to [14], for ∀ h ∈ N, set To f f ≡ Tomin f f , then the event triggering instant    tι,h+1 τ = tιn τ satisfying (5.47) tιn τ ∈ [hT, hT + Tomin f f ) ∪ {(h + 1)T } , (5.48) in which h, tιn , ιn , and n ∈ N. ι represents the amount of triggering times arising in (h + 1)-th period. In the effect of DoS attacks, the y(t) is successfully sent to the filter, i.e., y¯ (t) = y(tι,h+1 τ ), else the y(t) is zero, i.e., y¯ (t) = 0. Thereby, the following results are obtained  y(tι,h+1 τ ), t ∈ [tι,h+1 τ, tι+1,h+1 τ ) ∩ [hT, hT + Tomin f f ), y¯ (t) = (5.49) , hT + T ), 0, t ∈ [hT + Tomin ff where {tι,h τ } represents the sequence of successful transmission instants ((t0,h+1 τ ) = hT ), which is generated by the event-triggered mechanism in (5.48); ι ∈ l(h)    min {0, 1, 2, ..., ι(h)} with h ∈ N and ι(h) = sup{ι ∈ N hT + To f f ≥ tι,h+1 τ }. In what follows, for the sake of brevity, define

Iι,h = [tι,h+1 τ, tι+1,h+1 τ ), ι ∈ l(h)(h ∈ N), 1,h = [hT, hT + Tomin f f ), 2,h = [hT + Tomin f f , (h + 1)T ),  κι,h  sup{k ∈ N tι,h+1 τ + kτ < tι+1, h+1 τ },

5.2 L2 –L∞ Filtering with Time Delays and DoS Attacks



Kkι,h = [tι,h+1 τ + (k − 1)τ, tι,h+1 τ + kτ ], k ∈ {1, . . . , κι,h }, κ +1

Kι,hι,h

= [tι,h+1 τ + κι,h τ, tι+1,h+1 τ ].

167

(5.50)

Then Iι,h is able to express as κ

κ +1

ι,h Iι,h = ∪k=1 Kkι,h ∪ Kι,hι,h .

Noting that

(5.51)

ι(h) 1,h = ∪ι(h) ι=0 {Iι,h ∩ 1,h } ⊆ ∪ι=0 Iι,h ,

and combining (5.50) and (5.51), 1,h is reorganized as κ +1

ι,h k 1,h = ∪ι(h) ι=0 ∪k=1 {Kι,h ∩ 1,h }.

Set

¯ kι,h = Kkι,h ∩ 1,h . 

Then

κι,h +1 ¯ k 1,h = ∪ι(h) ι=0 ∪k=1 ι,h .

For ι ∈ l(h)(h ∈ N), giving the following two piecewise functions:

and

⎧ ¯ 1ι,h , t − tι,h+1 τ, t ∈ ⎪ ⎪ ⎪ ⎪ ⎨t − tι,h+1 τ − τ, ¯ 2ι,h , t ∈ φι,h (t) = . .. ⎪ ⎪ ⎪ ⎪ ⎩ ¯ κι,hι,h +1 , t − tι,h+1 τ − κι,h τ, t ∈ 

(5.52)

⎧ ¯ 1ι,h , t ∈ ⎪ ⎪0, ⎪ ⎪ ⎨z(tι,h+1 τ ) − z(tι,h+1 τ + τ ), ¯ 2ι,h , t ∈ z¯ ι,h (t) = .. ⎪ ⎪ . ⎪ ⎪ ⎩ ¯ κι,hι,h +1 . z(tι,h+1 τ ) − z(tι,h+1 τ + κι,h τ ), t ∈ 

(5.53)

Then, according to the above two functions, it can be observed that φι,h (t) ∈ [0, τ ), t ∈ Iι,h ∩ 1,h . The event-triggered sampling signal can be represented as z(tι,h+1 τ ) = z(t − φι,h (t)) + z¯ ι,h (t), t ∈ Iι,h ∩ 1,h , and z¯ ι,h (t) meets T (t)H z¯ ι,h (t) ≤ σ z T (t − φι,h (t))H z(t − φι,h (t)). z¯ ι,h

(5.54)

168

5 Event-Triggered Fuzzy Control and Filtering …

Define L c (θ ) = LC(θ ). Then filtering error system (5.45) can be rewritten as ⎧ ⎪ ¯ + E(θ)w(t) − L c (θ)z(t − φι,h (t)) ⎨ A(θ)z(t) + B(θ)z(t − ρ) z˙ (t) = −L c (θ)¯z ι,h (t), t ∈ Iι,h ∩ 1,h , ⎪ ⎩ A(θ)z(t) + B(θ)z(t − ρ) ¯ + E(θ)w(t), t ∈ 2,h , y(t) = C(θ)z(t), z(t) = κ(t), t ∈ [−τ, 0],

(5.55) where z¯ ι,h (t) satisfies (5.54), κ(t) is an initial value. Based on the above analysis, the piecewise Lyapunov–Krasovskii functional for filtering error system (5.55) is chosen as following : t Vα(t) (t) = z (t)Pα(t) z(t) +

z T (s)q(·)Sα(t) z(s)ds

T

t−ρ¯

t z T (s)q(·)Q α(t) z(s)ds

+ t−τ

0  t +

z˙ T (s)q(·)Uα(t) z˙ (s)dsd −τ t+

0  t +

z˙ T (s)q(·)Z α(t) z˙ (s)dsd, −τ t+ α(t)

where Pα(t) , Sα(t) , Q α(t) , Uα(t) , and Z α(t) ∈ Rn+ ; q(·)  e2(−1) gα(t) (t−s) , in which α (t) = 1 for t ∈ [−τ, 0] ∪ (∪h∈N 1,h ), and α(t) = 2 for t ∈ ∪h∈N 2,h . It implies that − ) = 3 − m. α(tm,h ) = m and α(tm,h The event-triggered L2 –L∞ filtering problem considered in this section is to design the L2 –L∞ filter in the form of (5.44) and a triggering matrix H > 0 such that (1) The filtering error system in (5.55) is exponentially stable when w(t) = 0, i.e. there exist scalars b and μ, such that z(t) ≤ be−μt κ(0) for ∀t ≥ 0. (2) The filtering error system in (5.55) has a prescribed L2 –L∞ performance γ , i.e., under the zero-initial condition, ∞ sup{y (t)y(t)} ≤ γ T

w T (t)w(t)dt

2

t≥0 0

is satisfied for any nonzero w(t) ∈ L2 [0, ∞).

(5.56)

5.2 L2 –L∞ Filtering with Time Delays and DoS Attacks

169

5.2.2 Main Results In this subsection, a criterion of the exponential stability and an L2 –L∞ performance analysis result will be given for filtering error system (5.55) subject to DoS attacks (5.46). Then, a co-design method for the needed L2 –L∞ filtering gain and eventtriggering parameter will be provided. Firstly, the following Lemma is given to derive the main results. Lemma 5.1 For given DoS signals (5.46), the filter gain L , and several scalars σ ∈ (0, 1), gm ∈ (0, ∞), and τ > 0. If there exist matrices Pm , Sm , Q m , Um , Z m , and H ∈ Rn+ , and matrices Υm , Υ¯m , Υ˜m , m ∈ {1, 2} such that the following matrix inequalities hold: (5.57) Γm (θ ) < 0, where

⎡ ¯ Γm (θ ) + H e{Γ¯m } √ T ⎢ τ Υ¯ ⎢ √ mT ⎢ τ Υ˜ ⎢ Γm (θ ) = ⎢ √ mT ⎢ √ τ Υm ⎣ τ U A (θ ) √ m m τ Z m Am (θ )

∗ m − e2(−1) gm βm τ Um 0 0 0 0

⎤ ∗ ∗ ⎥ ⎥ m −e2(−1) gm βm τ Um ∗ ⎥ ⎥, m 0 − e2(−1) gm βm τ Z m ∗ ⎥ ⎥ 0 0 ∗ ⎦ 0 0 − Zm   β1  1, β2  0, A1 (θ ) = A(θ ) − L c (θ ) 0 − L c (θ ) B(θ ) ,   A2 (θ ) = A(θ ) 0 0 B(θ ) , ⎤ ⎡ Γ˜1 (θ ) ∗ ∗ ∗ ∗ T T ⎥ ⎢−L c (θ )P σ H ∗ ∗ ∗ 1 ⎥ ⎢ ⎥, Γ¯1 (θ ) = ⎢ 0 0 − e−2g1 τ Q 1 ∗ ∗ ⎥ ⎢ T T ⎦ ⎣−L (θ )P 0 0 −H ∗ ∗ ∗

c

∗ ∗ ∗

∗ ∗ ∗ ∗ − Um 0

1

B T (θ )P1T

0

0

− e−2g1 ρ¯ S1

0

Γ˜1 (θ ) = H e{P1 A(θ )} + S1 + Q 1 + 2g1 P1 , ⎡ ⎤ H e{P2 A(θ )} + S2 + Q 2 − 2g2 P2 ∗ ∗ ∗ ⎢ ⎥ 0 0 ∗ ∗ ⎥, Γ¯2 (θ ) = ⎢ ⎣ ⎦ 0 0 − e2g2 τ Q 2 ∗ 0 0 − e2g2 ρ¯ S2 B T (θ )P2T   Γ¯1 = Υ1 + Υ¯1 − Υ¯1 + Υ˜1 − Υ1 − Υ˜1 0 0 ,   Γ¯2 = Υ2 + Υ¯2 − Υ¯2 + Υ˜2 − Υ2 − Υ˜2 0 , !T   T T ΥT ΥT ΥT ΥT T ΥT ΥT T , , Υ2 = Υ21 Υ1 = Υ11 Υ22 12 13 14 23 24 15 !T T  , Υ¯2 = Υ¯ T Υ¯ T Υ¯ T Υ¯ T , Υ¯1 = Υ¯ T Υ¯ T Υ¯ T Υ¯ T Υ¯ T 11

12

13

14

15

T Υ˜ T Υ˜ T Υ˜ T Υ˜ T Υ˜1 = Υ˜11 12 13 14 15

!T

21

22

23

24

  T T Υ˜ T Υ˜ T T , , Υ˜2 = Υ˜21 Υ˜22 23 24

170

5 Event-Triggered Fuzzy Control and Filtering …

then along the trajectories of filtering error system (5.55) with w(t) = 0, for h ∈ N, one can obtain that m (5.58) Vm (t) ≤ e2(−1) gm (t−tm,h ) Vm (tm,h ), in which

t ∈ [tm,h , t3−m,h+m−1 ], m = {1, 2},  hT, m = 1, tm,h = min hT + To f f , m = 2.

Proof Define  T T (t) z ρT¯ (t) , ζ (t) = z T (t) z T (t − φι,h (t)) z τT (t) z¯ ι,h z τ (t) = z(t − τ ), z ρ (t) = z(t − ρ). ¯ Then, when t ∈ [t1,h , t2,h ], V1 (t) along the trajectories of filtering error system (5.55), one can get V˙1 (t) = 2z T (t)P1 z˙ (t) + z T (t)(S1 + Q 1 )z(t) − z ρT¯ (t)e−2g1 ρ¯ S1 z ρ¯ (t) − z τT (t)e−2g1 τ Q 1 z τ (t) + τ z˙ T (t)(U1 + Z 1 )˙z (t) t t −2g1 (t−s) z(s)e S1 z(s)ds − 2g1 z(s)e−2g1 (t−s) Q 1 z(s)ds − 2g1 t−ρ¯

0 z˙ (t + )e

−

T

t−τ 2g1 

0  t U1 z˙ (t + )d − 2g1

−τ

z˙ T (s)e−2g1 (t−s) U1 z˙ (s)dsd

−τ t+

0 −

z˙ T (t + )e2g1  Z 1 z˙ (t + )d − 2g1

−τ

0  t

z˙ T (s)e−2g1 (t−s) Z 1 z˙ (s)dsd,

−τ t+

by −τ <  < 0, one can be obtained 0 −

z˙ (t + )e T

2g1 

t U1 z˙ (t + )d ≤ −

−τ

t−τ

0 −

z˙ T (τ )e−2g1 τ U1 z˙ (τ )dτ,

z˙ (t + )e T

2g1 

t Z 1 z˙ (t + )d ≤ −

−τ

Then, V˙1 (t) can be reorganized as follows

t−τ

z˙ T (τ )e−2g1 τ Z 1 z˙ (τ )dτ.

5.2 L2 –L∞ Filtering with Time Delays and DoS Attacks

171

V˙1 (t) ≤ 2z T (t)P1 z˙ (t) + z T (t)(S1 + Q 1 )z(t) − z ρT¯ (t)e−2g1 ρ¯ S1 z ρ¯ (t) − z τT (t)e−2g1 τ Q 1 z τ (t) + τ z˙ T (t)(U1 + Z 1 )˙z (t) t t −2g1 (t−s) z(s)e S1 z(s)ds − 2g1 z(s)e−2g1 (t−s) Q 1 z(s)ds − 2g1 t−ρ¯

t

t−τ

z˙ T (τ )e−2g1 τ U1 z˙ (τ )dτ −

− t−τ

t

z˙ T (τ )e−2g1 τ Z 1 z˙ (τ )dτ

t−τ

0

t

− 2g1

z˙ T (s)e−2g1 (t−s) U1 z˙ (s)dsd

−τ t+

0  t − 2g1

z˙ T (s)e−2g1 (t−s) Z 1 z˙ (s)dsd

−τ t+

≤ −2g1 V1 (t) + z T (t)2g1 P1 z(t) + z˙ T (t)2P1 z(t) + z T (t)(S1 + Q 1 )z(t) − z ρT¯ (t)e−2g1 ρ¯ S1 z ρ¯ (t) + τ z˙ T (t)(U1 + Z 1 )˙z (t) − z τT (t)e−2g1 τ Q 1 z τ (t) − 1 − 2 − 3 + 2ζ T (t)(Υ1 N1 + Υ¯1 N2 + Υ˜1 N3 ), where t 1 =

z˙ (s)e T

−2g1 τ

t Z 1 z˙ (s)ds, 2 = t−φι,h (t)

t−τ t−φ  ι,h (t)

3 =

z˙ T (s)e−2g1 τ U1 z˙ (s)ds,

z˙ (s)e T

−2g1 τ

t U1 z˙ (s)ds, N1 = z(t) − z τ (t) −

t−τ

t−τ

t N2 = z(t) − z(t − φι,h (t)) −

z˙ (s)ds  0,

t−φι,h (t) t−φ  ι,h (t)

N3 = z(t − φι,h (t)) − z τ (t) −

z˙ (s)ds  0. t−τ

According to [15], one can be obtained

z˙ (s)ds  0,

172

5 Event-Triggered Fuzzy Control and Filtering …

t − 2ζ (t)Υ1

z˙ (s)ds

T

t−τ

t

≤τ ζ T (t)Υ1 e2g1 τ Z 1−1 Υ1T ζ (t) +

(5.59) z˙ T (s)e−2g1 τ Z 1 z˙ (s)ds,

t−τ

− 2ζ (t)Υ¯1

t z˙ (s)ds

T

t−φι,h (t)

≤τ ζ T (t)Υ¯1 e2g1 τ U1−1 Υ¯1T ζ (t) +

t

(5.60) z˙ T (s)e−2g1 τ U1 z˙ (s)ds,

t−φι,h (t)

− 2ζ T (t)Υ˜1

t−φ  ι,h (t)

z˙ (s)ds t−τ

≤τ ζ T (t)Υ˜1 e2g1 τ U1−1 Υ˜1T ζ (t) +

t−φ  ι,h (t)

(5.61)

z˙ T (s)e−2g1 τ U1 z˙ (s)ds,

t−τ

by (5.54), (5.59), (5.60), and (5.61), one has 1 V˙1 (t) ≤ − 2g1 V1 (t) + ζ T (t)[Γ11 (θ ) + τ Υ1 e2g1 τ Z 1−1 Υ1T + τ Υ¯1 e2g1 τ U1−1 Υ¯1T + τ Υ˜1 e2g1 τ U1−1 Υ˜1T + τ A1T (θ )(U1 + Z 1 )A1 (θ )]ζ (t).

Using Lemma 1.1, Γ1 < 0 in (5.57) is equivalent to 1 (θ ) + τ Υ1 e2g1 τ Z 1−1 Υ1T + τ Υ¯1 e2g1 τ U1−1 Υ¯1T Γ11 + τ Υ˜1 e2g1 τ U1−1 Υ˜1T + τ A1T (θ )(U1 + Z 1 )A1 (θ ) < 0,

which implies

V˙1 (t) + 2g1 V1 (t) ≤ 0.

When t ∈ [t2,h , t1,h+1 ), h ∈ N. Along the same line as the proof in V1 (t), one obtains 2 (θ ) + τ Υ2 Z 2−1 Υ2T + τ Υ¯2 U2−1 Υ¯2T + τ Υ˜2 U2−1 Υ˜2T V˙2 (t) ≤2g2 V2 (t) + ζ T (t)[Γ11

+ τ A2T (U2 + Z 2 )A2 ]ζ (t). Using Lemma 1.1, Γ2 < 0 in (5.57) is equivalent to

5.2 L2 –L∞ Filtering with Time Delays and DoS Attacks

173

2 Γ11 (θ ) + τ Υ2 Z 2−1 Υ2T + τ Υ¯2 U2−1 Υ¯2T + τ Υ˜2T U2−1 Υ˜2T

+ τ A2T (θ )(U2 + Z 2 )A2 (θ ) < 0, which implies

V˙2 (t) − 2g2 V2 (t) ≤ 0.

Therefore, in the light of the above analysis, (5.57) can guarantee (5.58). This completes the proof. Theorem 5.3 For given DoS signals (5.46), the filtering gain L , and several scalars σ ∈ (0, 1), gm ∈ (0, ∞), πm > 1, and τ > 0. If there exist matrices Pm , Sm , Q m , Um , Z m , and H ∈ Rn+ , and matrices Υm , Υ¯m , Υ˜m , m ∈ {1, 2} such that (5.57) and the following conditions hold: ⎧ P1 ≤ π2 P2 , ⎪ ⎪ ⎪ ⎪ ⎪ P2 ≤ e2(g1 +g2 )τ π1 P1 , ⎪ ⎪ ⎪ ⎨Q ≤π m 3−m Q 3−m , ⎪ Sm ≤ π3−m S3−m , ⎪ ⎪ ⎪ ⎪ ⎪ U m ≤ π3−m U3−m , ⎪ ⎪ ⎩ Z m ≤ π3−m Z 3−m , min 0 < ψ = 2g1 Tomin f f − 2g2 (T − To f f ) − 2(g1 + g2 )τ − ln(π1 π2 ),

(5.62)

(5.63)

then filtering error system (5.55) is exponentially stable with w(t) = 0 and the decay ψ under DoS attacks. rate ν = 2T Proof According to (5.58) in Lemma 5.1, for any t ≥ 0 one gets  e−2g1 (t−t1,h ) V1 (t1,h ), t ∈ [t1,h , t2,h ), V (t) ≤ 2g2 (t−t2,h ) V2 (t2,h ), t ∈ [t2,h , t1,h+1 ). e

(5.64)

From (5.62), one has 

− ), V1 (t1,h ) ≤ π2 V2 (t1,h − V2 (t2,h ) ≤ π1 e2(g1 +g2 )τ V1 (t2,h ).

(5.65)

It can find a h ∈ N to ensure t ∈ [t1,h , t2,h ) or t ∈ [t2,h , t1,h+1 ) for ∀t ≥ 0. Thereupon, the following two cases will be considered: Case 1: For t ∈ [t1,h , t2,h ), it can be obtained from (5.64) and (5.65)

174

5 Event-Triggered Fuzzy Control and Filtering …

V (t) ≤e−2g1 (t−t1,h ) V1 (t1,h ) − ) ≤π2 e−2g1 (t−t1,h ) V2 (t1,h

≤π2 e−2g1 (t−t1,h ) e2g2 (t1,h −t2,h−1 ) V2 (t2,h−1 ) ≤π1 π2 e2(g1 +g2 )τ e−2g1 (t−t1,h ) e2g2 (t1,h −t2,h−1 ) e−2g1 (t2,h−1 −t1,h−1 ) V1 (t1,h−1 ) .. . ≤(π1 π2 )h e2h(g1 +g2 )τ e2hg2 (T −To f f ) e−2hg1 To f f e−2g1 (t−t1,h ) V1 (0) min

≤eh[ln(π1 π2 )+2(g1 +g2 )τ +2g2 (T −To f f )−2g1 To f f ] e−2g1 (t−t1,h ) V1 (0) min

min

≤e−ψh V1 (0). min Notice that t < t2,h = hT + Tomin f f , i.e., h > [(t − To f f )/T ] , it follows that

V (t) ≤ V1 (0)e

ψ Tomin ff T

ψ

e− T t .

(5.66)

Case 2: For t ∈ [t2,h , t1,h+1 ), one has V (t) ≤e2g2 (t−t2,h ) V2 (t2,h ) − ) ≤π1 e2(g1 +g2 )τ e2g2 (t−t2,h ) V1 (t2,h

≤π1 e2(g1 +g2 )τ e2g2 (t−t2,h ) e−2g1 (t2,h −t1,h ) V1 (t1,h ) .. . (π1 π2 )h+1 2(h+1)(g1 +g2 )τ 2hg2 (T −Tomin min f f ) e −2(h+1)g1 To f f e 2g2 (t−t2,h ) V (0) e e 1 π2 ! min 1 (h+1) ln(π1 π2 )+2(g1 +g2 )τ +2g2 (T −Tomin f f )−2g1 To f f ≤ e V1 (0) π2 1 ≤ e−ψ(h+1) V1 (0). π2

≤

Same as Case 1, one can get that V (t) ≤

V1 (0) − ψ t e T . π2

(5.67)

  ψ T min of f Define λ0 = max e T , π12 , λ1 = min{λmin (Pm )}, λ2 = max{λmax (Pm )}, λ3 = λ2 + ρλ ¯ max (S1 ) + τ λmax (Q 1 ) + τ2 λmax (U1 + Z 1 ). Then it can be obtained from (5.66) and (5.67) that ψ V (t) ≤ λ0 e− T t V1 (0). (5.68) 2

According to definition of V (t) in (5.58), it is easy to get

5.2 L2 –L∞ Filtering with Time Delays and DoS Attacks

175

V (t) ≥ λ1 z(t) 2 , V1 (0) ≤ λ3 κ(0) 2τ .

(5.69)

Combining (5.63), (5.68), and (5.69), one can be deduced that "

z(t) ≤

λ0 λ3 −νt e κ(0) τ , ∀t ≥ 0. λ1

Thus, the filtering error system is exponentially stable with w(t) = 0 and the decay rate ν under DoS attacks. This completes the proof. Based on Theorem 5.3, this subsection provides sufficient conditions to ensure the L2 –L∞ performance γ¯ for filtering error system (5.55) under DoS attacks. Theorem 5.4 For given DoS signals (5.46), the filter gain L , and several scalars σ ∈ (0, 1), gm ∈ (0, ∞), πm > 1, γ > 0, and τ > 0. If there exist matrices Pm , Sm , Q m , Um , Z m , and H ∈ Rn+ , and matrices Υm , Υ¯m , Υ˜m , m ∈ {1, 2} such that (5.62), (5.63), and the following matrix inequalities hold:

where

Σm (θ ) < 0,

(5.70)

C T (θ )C(θ ) − P1 < 0,

(5.71)

⎡ ¯ Σm (θ ) + H e{Σ¯ m } √ T ⎢ τ Υ¯ ⎢ √ mT ⎢ τ Υ˜ ⎢ Σm (θ ) = ⎢ √ mT ⎢ √ τ Υm ⎣ τ U A (θ ) √ m m τ Z m Am (θ )

∗ m − e2(−1) gm βm τ Um 0 0 0 0

⎤ ∗ ∗ ⎥ ⎥ m −e2(−1) gm βm τ Um ∗ ⎥ ⎥, m 0 − e2(−1) gm βm τ Z m ∗ ⎥ ⎥ 0 0 ∗ ⎦ 0 0 − Zm   β1 1, β2  0, A1 (θ ) = A(θ ) − L c (θ ) 0 − L c (θ ) E(θ ) B(θ ) ,   A2 (θ ) = A(θ ) 0 0 E(θ ) B(θ ) , ⎡ ⎤ Σ˜ 1 (θ ) ∗ ∗ ∗ ∗ ∗ ⎢−L T (θ )P T σ H ⎥ ∗ ∗ ∗ ∗ ⎢ c ⎥ 1 ⎢ ⎥ −2g τ 1 0 0 −e Q1 ∗ ∗ ∗ ⎢ ⎥ ¯ Σ1 (θ ) = ⎢ T ⎥, ⎢−L c (θ )P1T 0 ⎥ 0 −H ∗ ∗ ⎢ T ⎥ T 2 ⎣ E (θ )P1 ⎦ 0 0 0 −γ I ∗ B T (θ )P1T 0 0 0 0 − e−2g1 ρ¯ S1 ∗ ∗

∗ ∗ ∗

Σ˜ 1 (θ ) =H e{P1 A(θ )} + S1 + Q 1 + 2g1 P1 ,

∗ ∗ ∗ ∗ − Um 0

176

5 Event-Triggered Fuzzy Control and Filtering …

⎡

H e{P2 A(θ )} + S2 + Q 2 − 2g2 P2 ∗ ∗ ∗ ⎢ 0 0 ∗ ∗ ⎢ ∗ 0 0 − e2g2 τ Q 2 Σ¯ 2 (θ ) = ⎢ ⎢ T T ⎣ 0 0 − γ2I E (θ )P2 0 0 0 B T (θ )P2T   ¯ ¯ ¯ ˜ ˜ Σ1 = Υ1 + Υ1 − Υ1 + Υ1 − Υ1 − Υ1 0 0 0 ,   Σ¯ 2 = Υ2 + Υ¯2 − Υ¯2 + Υ˜2 − Υ2 − Υ˜2 0 0 ,   T Υ1 = Υ11T Υ12T Υ13T Υ14T Υ15T Υ16T , Υ2 = Υ21T Υ22T Υ23T Υ24T T   Υ¯1 = Υ¯11T Υ¯12T Υ¯13T Υ¯14T Υ¯15T Υ¯16T , Υ¯2 = Υ¯21T Υ¯22T Υ¯23T Υ¯24T T   Υ˜1 = Υ˜11T Υ˜12T Υ˜13T Υ˜14T Υ˜15T Υ˜16T , Υ˜2 = Υ˜21T Υ˜22T Υ˜23T Υ˜24T

⎤

∗ ∗ ∗ ∗

⎥ ⎥ ⎥, ⎥ ⎦

− e2g2 ρ¯ S2

Υ25T

T

T Υ¯25T T Υ˜25T

, , ,

then filtering error system (5.55) is exponentially stable with the prescribed L2 –L∞ # 2g T max min 1 e 1 of f performance γ¯ = ηηmax γ with η = { , 1}, η = { , e2g2 (T −To f f ) }. min max π π2 min 2 Proof For any w(t) ∈ L2 [0, ∞), using a same proof way as Lemma 5.1, it follows from (5.70) that 

V˙1 (t) + 2g1 V1 (t) − γ 2 w T (t)w(t) ≤ 0, t ∈ [t1,h , t2,h ), V˙2 (t) − 2g2 V2 (t) − γ 2 w T (t)w(t) ≤ 0, t ∈ [t2,h , t1,h+1 ).

(5.72)

For any t ∈ [0, (h + 1)T ), h ∈ N, one can be get from (5.72) t1, t2, j −2g1 (t1, j −t)  j+1 h  e 2 T ( γ w (t)w(t)dt + e2g2 (t1, j+1 −t) γ 2 w T (t)w(t)dt) π2 j=0 t1, j

≥

h 

t2, j (

j=0 t1, j

t2, j

e−2g1 (t1, j −t) ˙ (V1 (t) + 2g1 V1 (t))dt + π2 ⎡

V1 (t1,h+1 ) − V1 (0)  + V1 (t2, j ) ⎣ ≥ π2 h

j=0

2g T min e 1 of f

π2

t1,  j+1

e2g2 (t1, j+1 −t) (V˙2 (t) − 2g2 V2 (t))dt)

t2, j

− π1

2g1 Tomin ff

⎤ 2(g +g )τ +2g2 (T −Tomin f f )⎦ . e 1 2

From (5.63), one can be obtained that e π2 − π1 e2(g1 +g2 )τ +2g2 (T −To f f ) > 0. Note that V1 (t1,h+1 ) > 0 and V1 (t2, j ) > 0, j ∈ {0, 1, . . . , h}, h ∈ N, then, under zero initial condition, i.e., V1 (0) = 0, it can be rewritten from the above inequality that min

5.2 L2 –L∞ Filtering with Time Delays and DoS Attacks

177

t 1, j+1 t2, j −2g1 (t1, j −t) h  e 2 T ( γ w (t)w(t)dt + e2g2 (t1, j+1 −t) γ 2 w T (t)w(t)dt) π 2 j=0 t1, j

t2, j

V1 (t1,h+1 ) ≥ . π2 For ∀t ∈ [t1, j , t2, j ), j ∈ {0, 1, . . . , h}, h ∈ N, there is 1 ≤ e−2g1 (t1, j −t) ≤ e2g1 To f f . max

For ∀t ∈ [t2, j , t1, j+1 ), j ∈ {0, 1, . . . , h}, h ∈ N, there is 1 ≤ e2g2 (t1, j+1 −t) ≤ e2g2 (T −To f f ) . min

Set ηmin = { π12 , 1} and ηmax = { e

2g1 Tomax ff

π2

, e2g2 (T −To f f ) }. Then min

t1, j+1 h   ηmax γ 2 w T (t)w(t)dt ≥ ηmin V1 (t1,h+1 ). j=0 t 1, j

By (5.71), it is easy to get t 1, j+1

γ2 0

ηmax T w (t)w(t)dt ≥V1 (t1,h+1 ) ηmin ≥z T (t)P1 z(t) ≥z T (t)C T (θ )C(θ )z(t) ≥y T (t)y(t).

When h → ∞, i.e., t1, j+1 → ∞, it can be derived that ∞ sup{y (t)y(t)} ≤

γ¯ 2 w T (t)w(t)dt,

T

t≥0 0

which means system (5.55) possesses an L2 –L∞ performance index γ¯ in the light of (5.56). The proof is completed.

178

5 Event-Triggered Fuzzy Control and Filtering …

Based on the L2 –L∞ performance analysis result in Theorem 5.4, a joint method of the L2 –L∞ filtering gain and weighting matrix H in event-triggered mechanism (5.48) is presented. Theorem 5.5 For given DoS signals (5.46), several scalars σ ∈ (0, 1), gm ∈ (0, ∞), πm > 1, δm > 0, γ > 0, and τ > 0. If there exist matrices Pm , Sm , Q m and H ∈ Rn+ , and matrices M, Υm , Υ¯m , Υ˜m , m ∈ {1, 2} such that (5.63) and the following matrix inequalities hold: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

−P1 ∗ < 0, Cr −I

(5.73)

P1 ≤ π2 P2 , P2 ≤ e2(g1 +g2 )τ π1 P1 , Q m ≤ π3−m Q 3−m , Sm ≤ π3−m S3−m ,

(5.74)

δm Pm ≤ π3−m δ3−m P3−m , Ψim < 0,

(5.75)

for i = 1, · · · , r , where ⎡

Ψim

⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣

Ψ¯ mi + H e{Σ¯ m } √ T τ Υ¯ √ mT ˜ √τ ΥmT τΥ √ m δm τ A¯ mi √ δm τ A¯ mi

m

−e2(−1)

∗ ∗ ∗

gm βm τ

0 0

m

− e2(−1)

δm Pm

∗

gm βm τ

0 0 0 0 ∗ ∗ ∗ ∗ − δm Pm 0

δm Pm

m

− e2(−1)

⎤ ∗ ⎥ ∗ ⎥ ⎥ ∗ ⎥, ⎥ ∗ ⎥ ⎦ ∗ − δm Pm

∗ ∗

gm βm τ

0 0 0

δm Pm

5.2 L2 –L∞ Filtering with Time Delays and DoS Attacks

179

  A¯ 1i = P1 Ai −MCi 0 −MCi P1 E i P1 Bi ,   A¯ 2i = P2 Ai 0 0 P2 E i P2 Bi , ς1 = 1, ς2 = 0, ⎤ ⎡ H e{P1 Ai } + S1 + Q 1 + 2g1 P1 ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ σH ∗ ∗ ∗ ∗ −C T (θ )M T ⎥ ⎢ −2g1 τ ⎥ ⎢ Q1 ∗ ∗ ∗ 0 0 −e ⎥, ⎢ ¯ Ψ1i = ⎢ T T T ⎥ −Ci M 0 −H ∗ ∗ C˜ i ⎥ ⎢ T T 2 ⎦ ⎣ 0 0 0 −γ I ∗ E i P1 0 0 0 0 −e−2g1 ρ¯ S1 BiT P1T ⎡ ⎤ H e{P2 Ai } + S2 + Q 2 − 2g2 P2 ∗ ∗ ∗ ∗ ⎢ ⎥ 0 0 ∗ ∗ ∗ ⎢ ⎥ 2g τ 2 ⎢ ⎥, Q2 ∗ ∗ 0 0 −e Ψ¯ 2i = ⎢ ⎥ T T 2 ⎣ ⎦ 0 0 −γ I ∗ E i P2 0 0 0 − e2g1 ρ¯ S2 BiT P2T and the other notations are the same as those in Theorem 5.4. Then filtering error system (5.55) is exponentially stable with the prescribed L2 –L∞ performance γ¯ and the needed gain matrix can be chosen as L = P1−1 M. Proof Define Um = Z m = δm Pm . Then by M = P1 L , Σ1 in (5.70) with m = 1 can be equivalently expressed as Ψ1 in (5.74), which ensures that Σ1 < 0. The proof is completed.

5.2.3 Simulation Example In this subsection, a numerical example is given to demonstrate the effectiveness of the proposed event-triggered L2 –L∞ filter design method for the error system in (5.55 ) under DoS attacks. The parameters of nonlinear system (5.55) are chosen as follows:

−2.5 0.1 −0.1 0 , B1 = , 0.2 − 1.5 0 − 0.1

−2.1 0.2 −0.2 0 , B2 = , A2 = 0.5 − 2 0 − 0.2



0.3 0.2 1 0 , C1 = , E1 = 0 0.4 0 1

0.1 0 0.5 0 E2 = , C2 = , 0.2 0.4 0 0.5 A1 =

and the membership functions are given as θ1 (q(t))= sin2 (t), θ2 (q(t)) = 1 − θ1 (q(t)). Set

180 Fig. 5.7 Responses of x(t) and x f (t) under the DoS attacks

5 Event-Triggered Fuzzy Control and Filtering … 0.04 0.02 0 0

5

10

15

20

25

30

0

5

10

15

20

25

30

15

20

25

30

20

25

30

0.1 0.05 0

Fig. 5.8 Release time intervals under the DoS attacks

2

1.5

1

0.5

0 0

Fig. 5.9 Response of z(t) under the DoS attacks

5

10

0.06 0.05 0.04 0.03 0.02 0.01 0 -0.01 0

5

10

15

5.3 H∞ Filtering with Dynamic Quantization and Deception Attacks Fig. of # 5.10 History $∞ y T (t)y(t)/ 0 w T (s)w(s)ds

181

0.06 0.05 0.04 0.03 0.02 0.01 0

0

5

10

15

20

25

30

g1 = 0.05, g2 = 0.5, τ = 0.01, ρ¯ = 2, T = 2, Tomin f f = 1.85, Tomax f f = 1.90, γ = 0.28, π1 = π2 = 1.01, σ = 0.1. Then, by solving LMIs in Theorem 5.5, the corresponding gain and trigger matrices can be obtained as follows:



0.6344 0.9557 2.4941 0.3786 L= , H= . 0.6299 1.9466 0.3786 3.2784  T  T The initial conditions are chosen as x(0) = 0 0 , x f (0)= 0 0 , and w(t) = e−0.4t cos(0.1t). Then the trajectories of state x(t) and its estimate x f (t) are displayed in Fig. 5.7. The event-triggered instant and event time of error system (5.55) is depicted in Fig. 5.8. The trajectory of state z(t) is given in Fig. 5.9. # $∞ T The plot of y (t)y(t)/ 0 w T (s)w(s)ds versus time with the zero-initial condition is given in Fig. 5.10, from which one can be observed that the maximum of # $∞ y T (t)y(t)/ 0 w T (s)w(s)ds is 0.0584, which is less than the prescribed L2 –L∞ performance index γ¯ = 0.3079.

5.3 H∞ Filtering with Dynamic Quantization and Deception Attacks In this section, the problem of event-triggered filtering for discrete-time T–S fuzzy systems is investigated. In the presence of dynamic quantization and stochastic deception attacks, the full- and reduced-order event-triggered filters are designed such that the filtering error system is stochastically stable with the prescribed H∞ filtering performance.

182

5 Event-Triggered Fuzzy Control and Filtering …

5.3.1 Problem Formulation Consider a nonlinear system represented by T–S fuzzy model, in which the ith rule is described as follows: Plant Rule i th : IF 1 (k) is M1i , 2 (k) is M2i , and, · · · , and d (k) is Mdi , THEN x(k + 1) = Ai x(k) + Bi w(k), y(k) = Ci x(k) + Di w(k), z(k) = E i x(k) + Fi w(k),

(5.76)

where x(k) ∈ Rn x stands for the state variable, w(k) ∈ Rn w is the noise input that is assumed to be the arbitrary signal in l2 [0, ∞), y(k) ∈ Rn y means the measurement output, and z(k) ∈ Rn z is the performance output. M pi , i = 1, 2, …, r , p = 1, 2, …, d is the fuzzy sets, where r is the number of fuzzy rules. The premise variables are denoted by 1 (k), 2 (k), …, d (k). The matrices Ai ∈ Rn x ×n x , Bi ∈ Rn x ×n w , Ci ∈ Rn y ×n x , Di ∈ Rn y ×n w , E i ∈ Rn z ×n x , and Fi ∈ Rn z ×n w with i = 1, 2, …, r indicate given system matrices. As in Sect. 5.1, the T–S fuzzy model (5.76) can be deduced as follows: x(k + 1) = A(ρ)x(k) + B(ρ)w(k), y(k) = C(ρ)x(k) + D(ρ)w(k),

(5.77)

z(k) = E(ρ)x(k) + F(ρ)w(k), with A(ρ) = C(ρ) = E(ρ) =

r  i=1 r  i=1 r  i=1

ρi ((k))Ai , B(ρ) = ρi ((k))Ci , D(ρ) = ρi ((k))E i , F(ρ) =

r  i=1 r  i=1 r 

ρi ((k))Bi , ρi ((k))Di , ρi ((k))Fi .

i=1

The output of the fuzzy model (5.77), i.e., y(k) will be transmitted to the dynamic quantizer q() only if ξ yT (k)Jξ y (k) − b(y(k) − ξ y (k))T J(y(k) − ξ y (k)) ≥ 0

(5.78)

is satisfied. Here, ξ y (k) = y(k) − y(ks ), y(k) is the present data, and y(ks ) represents the data at the last triggering instant. b ∈ [0, 1) is a predetermined scalar, J > 0 stands for a weighting matrix to be determined.

5.3 H∞ Filtering with Dynamic Quantization and Deception Attacks

183

In this section, to further reduce the amount of data in the communication of network and realize the effective use of the limited network communication resources, y(ks ) is quantized by using the same form of dynamic quantizer in (2.7). According to the definition in Sect. 2.1, the quantized y(ks ) can be described as: 

Q(y(ks )) = μ ys (k)q

 y(ks ) . μ ys (k)

(5.79)

Assume that the quantized output Q(y(ks )) is transmitted to the filter via imperfect communication network where the deception attacks will be encountered unavoidably and randomly. According to [23, 24], in this section, the quantized output with stochastic deception attacks can be modeled as y(k) = h(k)C(ρ)α(x(k)) + (1 − h(k))Q(y(ks )).

(5.80)

In Eq. (5.80), α(x(k)) is used to stand for a nonlinear function about x(k), which satisfies (5.81) x T (k)Q T Qx(k) − α T (x(k))α(x(k)) ≥ 0, here, Q stands for a known matrix parameter for α(x(k)). In addition, h(k) is a Bernoulli process, which satisfies Pr ob{h(k) = 1} = E{h(k)} = h, Pr ob{h(k) = 0} = 1 − h,

(5.82)

here, 0 ≤ h ≤ 1 refers to a given scalar for h(k). More specifically, h(k) = 1 implies that the signal received by the filter is the deception signal, i.e., y(k) = C(ρ)α(x(k)). h(k) = 0 implies that the signal received by the filter is the quantized output, i.e., y(k) = Q(y(ks )). Moreover, the signal received by the filter can be further rewritten as y(k) = h(k)C(ρ)α(x(k)) + (1 − h(k))Q(y(ks ))       y(ks ) y(ks ) − + y(ks ) = h(k)C(ρ)α(x(k)) + (1 − h(k)) μ ys (k) q μ ys (k) μ ys (k)

(5.83)

= h(k)C(ρ)α(x(k)) + (1 − h(k))(β(k) + y(ks )) = h(k)C(ρ)α(x(k)) + (1 − h(k))(β(k) + y(k) − ξ y (k)),

& & % % y(ks ) s) where β(k) = μ ys (k) q μy(k − . μ ys (k) ys (k) The filter discussed in this section is presented as x f (k + 1) = A f x f (k) + B f y(k), z f (k) = E f x f (k),

(5.84)

184

5 Event-Triggered Fuzzy Control and Filtering …

where x f (k) ∈ Rn x f stands for the state of the filter, z f (k) ∈ Rn z denotes the output of the filter. The matrices A f , B f , and E f represent the parameters for the filter which need to be designed. Moreover, for n x f = n x , the filter model in (5.84) stands for a full-order filter; for 1 ≤ n x f < n x , the filter model in (5.84) refers to a reduced-order filter. By defining  x T (k) = [ x T (k) x Tf (k) ], e(k) = z(k) − z f (k), the filtering error system can be obtained as 1 +  2 )  x (k + 1) = ( A h(k) A x (k) + (  B1 +  h(k)  B2 )w(k) h(k)S2 )α(x(k)) + (S3 +  h(k)S4 ) + (S1 +  h(k)S4 )ξ y (k), × β(k) − (S3 +  x (k) + Fw(k),  e(k) = E where

(5.85)

A(ρ) 0 0 0  , A2 = , −B f C(ρ) 0 (1 − h)B f C(ρ) A f



B(ρ) 0  ,  B2 = B1 = , −B f D(ρ) (1 − h)B f D(ρ)

0 0 , S2 = , S1 = B f C(ρ) h B f C(ρ)

0 0 , S4 = , S3 = −B f (1 − h)B f    = E(ρ) − E f , F  = F(ρ),  E h(k) = h(k) − h.

1 = A



The following important definition on stochastic stability will be used in the analysis of event-triggered filtering. Definition 1 [18, 24] For any initial condition  x (0), when w(k) = 0, it can be concluded that the filtering error system (5.85) is stochastically stable, if there exists a matrix W > 0 such that the following inequality  E

∞ 

   x (0) <  x T (0)W

 x (k) 2  x (0)

(5.86)

k=0

is satisfied. Then the event-triggered filtering problem considered in this section is to design the event-triggered filter in (5.84) such that (1) The filtering error system (5.85) is stochastically stable when w(k) = 0 in the sense of Definition 1. (2) The filtering error system (5.85) has a prescribed H∞ filtering performance γ > 0, i.e., under the zero-initial condition,

5.3 H∞ Filtering with Dynamic Quantization and Deception Attacks

 E

∞ 

 e (k)e(k) < γ 2 T

k=0

∞ 

185

w T (k)w(k)

k=0

is satisfied for any nonzero w(k) ∈ l2 [0, ∞).

5.3.2 Main Results In this subsection, we shall to investigate under what conditions, such that, in the presence of event-triggered communication scheme, dynamic quantization, and stochastic cyber attacks, the filtering error system (5.85) is stochastically stable with the prescribed H∞ filtering performance γ > 0 to give filter gain matrices. Based on the Lyapunov theory, a useful H∞ filtering performance analysis criterion will be formulated in the following theorem, which can ensure the stochastic stability and the H∞ filtering performance of the filtering error system (5.85). Theorem 5.6 Consider the fuzzy system (5.77) and the filter (5.84). Suppose that the scalars 0 ≤ b < 1, h, quantization range bound M ys , and quantization error Δ are known. The filtering error system (5.85) is stochastically stable with the prescribed H∞ filtering performance γ > 0, if there exist matrices P > 0, J > 0, and M, scalars ξ > 0, f 1 > 0, and f 2 > 0 satisfying M ys −

1 > 0, ξ

(5.87)

⎡

⎤ 11 ∗ ∗ ∗ ∗ ⎢21 P − M − M T ∗ ∗ ∗ ⎥ ⎢ ⎥ T ⎢31 0 P−M−M ∗ ∗ ⎥ ⎢ ⎥ < 0, ⎣41 0 0 −I ∗ ⎦ 51 0 0 0 − f1 I

(5.88)

where ⎤ ⎡ T JC  + f2 Q T Q  −P + bC ∗ ∗ ∗ ∗ ⎢  bD T (ρ)JC −γ 2 I + bD T (ρ)JD(ρ) ∗ ∗ ∗ ⎥ ⎥ ⎢ ⎢ 11 = ⎢ ∗ ∗ ⎥ 0 0 − f2 I ⎥, ⎣ ∗ ⎦ 0 0 0 − f1 I  −bJC − bJD(ρ) 0 0 (b − 1)J  T  T T T T 1 M  21 = M A B1 M S1 M S3 − M S3 ,   T 2  31 =  hM A h M T S4 , hMT  B2  h M T S2  h M T S4 −       F  0 0 0 , 51 = 2 f 1 ξ ΔC  2 f 1 ξ ΔD(ρ) 0 0 − 2 f 1 ξ ΔI , 41 = E 1/2       = C(ρ) 0 , Q = Q 0 ,  , C h = h(1 − h)

186

5 Event-Triggered Fuzzy Control and Filtering …

and the parameter of the dynamic quantizer defined in (5.79), i.e., μ ys (k) obeys the following online adjusting strategy: ξ y(ks ) ≤ μ ys (k) ≤ 2ξ y(ks ) .

(5.89)

Proof Firstly, we will prove that when the conditions (5.87) and (5.88) are satisfied, for w(k) = 0, the filtering error system (5.85) is stochastically stable with the online adjusting strategy for the dynamic quantizer’s parameter μ ys (k) given in (5.89). When w(k) = 0, the inequality in (5.88) can be rewritten as ⎡ ⎤ 11 ∗ ∗ ∗  T ⎢  ∗ ∗ ⎥ ⎢ 21 P − M − M ⎥ < 0, T ⎣ 31 0 P−M−M ∗ ⎦ 51  0 0 − f1 I

(5.90)

where ⎡

11  21  31  51 

⎤  + f2 Q T Q  ∗ T JC ∗ ∗ −P + bC ⎢ ∗ ∗ ⎥ 0 − f2 I ⎥, =⎢ ⎣ ∗ ⎦ 0 0 − f1 I  −bJC 0 0 (b − 1)J   T T T T 1 M S1 M S3 − M S3 , = M A   2  =  hMT A h M T S4 , h M T S2  h M T S4 −     0 0 − 2 f 1 ξ ΔI . = 2 f 1 ξ ΔC

Based on the inequality (P − M)T P −1 (P − M) ≥ 0 and P > 0, it can be obtained that P − M − M T ≥ −M T P −1 M. Then, the inequality of (5.90) implies that ⎤ ⎡ 11 ∗ ∗ ∗  T −1 ⎢  ∗ ∗ ⎥ ⎥ < 0. ⎢ 21 − M P M (5.91) ⎣ 31 0 − M T P −1 M ∗ ⎦ 51  0 0 − f1 I Performing congruence transformation to (5.91) by diag{I, M −T , M −T , f 1−1 I }, the following conclusions can be drawn ⎡ 11 ∗ ∗  ⎢21 − P −1 ∗ ⎢ ⎣31 0 − P −1 51 0 0 − where

⎤ ∗ ∗ ⎥ ⎥ < 0, ∗ ⎦ f 1−1 I

(5.92)

5.3 H∞ Filtering with Dynamic Quantization and Deception Attacks

187

  1 S1 S3 − S3 , 21 = A   2  31 =  hA h S4 , h S2  h S4 −     0 0 − 2ξ ΔI . 51 = 2ξ ΔC By using Lemma 1.1 to (5.92), it can be obtained that Υ0 + Υ1 + f 1 Υ2 + f 2 Υ3 < 0,

(5.93)

where       2 S2 S4 − S4 T 1 S1 S3 − S3 T P A 1 S1 S3 − S3 +  Υ0 = A h2 A   2 S2 S4 − S4 − diag{P, 0, 0, 0}, ×P A      0 0 −I TJ C  0 0 − I − diag{0, 0, 0, J}, Υ1 = b C      0 0 − 2ξ ΔI − diag{0, 0, I, 0},  0 0 − 2ξ ΔI T 2ξ ΔC Υ2 = 2ξ ΔC      0 0 0 − diag{0, I, 0, 0}.  0 0 0 T Q Υ3 = Q Now, the Lyapunov function is constructed as x (k), P > 0. V ( x (k)) =  x T (k)P

(5.94)

 2  h 2 , the following result Then, by considering the fact E  h (k) = h(1 − h) =  can be haven x (k)) E {V ( x (k + 1))} − V (  T  =E  x (k + 1)P x (k + 1) −  x T (k)P x (k) '%% & % & 1 +  2  =E A x (k) + S1 +  h(k) A h(k)S2 α(x(k)) & % & % &T + S3 +  h(k)S4 β(k) − S3 +  h(k)S4 ξ y (k) P & % & %% 1 +  2  x (k) + S1 +  × A h(k) A h(k)S2 α(x(k)) % & % & &( + S3 +  h(k)S4 β(k) − S3 +  h(k)S4 ξ y (k) −  x T (k)P x (k) '    2 S2 S4 − S4 T 1 S1 S3 − S3 +  =) x T (k)E h(k) A A     2 S2 S4 − S4 ) 1 S1 S3 − S3 +  h(k) A x (k) − ) x T (k)P1) ×P A x (k) =) x T (k)Υ0) x (k), (5.95)   T x (k) α T (x(k)) β T (k) ξ yT (k) , P1 = diag{P, 0, 0, 0}. where ) x T (k) =  According to the event-triggered condition given in (5.78), one can be obtained that b(y(k) − ξ y (k))T J(y(k) − ξ y (k)) − ξ yT (k)Jξ y (k) ≥ 0, which is equivalent to

188

5 Event-Triggered Fuzzy Control and Filtering …

) x T (k)Υ1) x (k) ≥ 0.

(5.96)

The condition in (5.89) can be expressed as 1 1 μ y (k) ≤ y(ks ) ≤ μ ys (k). 2ξ s ξ

(5.97)

By considering the condition in (5.87), the condition in (5.97) can be rewritten as 1 μ y (k) ≤ y(ks ) ≤ M ys μ ys (k). (5.98) 2ξ s * * * s) * Based on the condition in (5.98), * μy(k * ≤ M ys can be gained, which implies ys (k) that * *   * * *q y(ks ) − y(ks ) * ≤ Δ. (5.99) * μ ys (k) μ ys (k) * In addition, according to the homogeneity property of Euclidean norm, it can be obtained that *  *   * y(ks ) * y(ks ) * *

β(k) = *μ ys (k) q − μ ys (k) μ ys (k) * * *   * y(ks ) y(ks ) * * * = μ ys (k) *q − (5.100) μ ys (k) μ ys (k) * ≤ μ ys (k)Δ ≤ 2ξ Δ y(ks ) , i.e., β T (k)β(k) ≤ 4ξ 2 Δ2 (y(k) − ξ y (k))T (y(k) − ξ y (k)).

(5.101)

The inequality in (5.101) can be further rewritten as x (k) ≥ 0. ) x T (k)Υ2)

(5.102)

Moreover, the inequality in (5.81) can be further expressed as x (k) ≥ 0. ) x T (k)Υ3)

(5.103)

By using Lemma 1.3 with inequalities (5.95), (5.96), (5.102), (5.103), and scalars f 1 > 0, f 2 > 0, it’s easy to conclude that if the inequality in (5.93) is satisfied, then, ) x T (k)Υ0) x (k) < 0, i.e., x (k)) < 0, E {V ( x (k + 1))} − V ( which implies that

(5.104)

5.3 H∞ Filtering with Dynamic Quantization and Deception Attacks

189

x (k)) ≤ −λmin (−Υ0 )) E {V ( x (k + 1))} − V ( x T (k)) x (k).

(5.105)

Then, taking mathematical expectation for both sides of the inequality (5.105), for any c ≥ 1, summing up the inequality (5.105) on both sides from k = 0 to k = c, the following conclusion can be given  c     T 2 T

) x (k) , x (c + 1) −  x (0)P x (0) ≤ −λmin (−Υ0 )E E  x (c + 1)P k=0

(5.106) where  x (0) is the initial condition. The inequality (5.106) can be expressed as E

 c 



) x (k) 2

 T  T  x (0)P x (0) − E  x (c + 1)P x (c + 1) . ≤ (λmin (−Υ0 ))−1 

k=0

(5.107)  T   ∞ 2

) When c=1, . . . , ∞, considering E  x (∞)P x (k) ≥ x (∞) ≥0 and E k=0  ∞ 2

 x (k) , E k=0  E

∞ 



 x (k) 2

x T (0)P x (0) ≤ (λmin (−Υ0 ))−1 

k=0

= x T (0) (λmin (−Υ0 ))−1 P x (0) = x T (0)W x (0), where W = (λmin (−Υ0 ))−1 P. x (k) < 0, Υ0 < 0 can be known, λmin (−Υ0 ) > 0. Thus, W = Based on ) x T (k)Υ0) −1 (λmin (−Υ0 )) P > 0. Then, according to Definition 1, it can be concluded that when w(k) = 0, the filtering error system (5.85) is stochastically stable with the online adjusting strategy for the dynamic quantizer’s parameter μ ys (k) given in (5.89) if the conditions given in (5.87) and (5.88) are satisfied. Next, for w(k) = 0, the H∞ filtering performance constraints will be considered of the filtering error system (5.85) with zero-initial condition. Similar to the above proof process of stochastic stability, the inequality in (5.88) implies that ⎡ ⎤ 11 ∗ ∗ ∗ ∗ ⎢Λ21 − P −1 ∗ ∗ ∗ ⎥ ⎢ ⎥ −1 ⎢Λ31 ⎥ < 0, 0 − P ∗ ∗ (5.108) ⎢ ⎥ ⎣41 0 0 −I ∗ ⎦ 0 0 0 − f 1−1 I Λ51 where

  1  Λ21 = A B1 S1 S3 − S3 ,   2  Λ31 =  hA h S4 , h B2  h S2  h S4 −     2ξ ΔD(ρ) 0 0 − 2ξ ΔI . Λ51 = 2ξ ΔC

190

5 Event-Triggered Fuzzy Control and Filtering …

According to Lemma 1.1, the inequality in (5.108) can be indicated as 0 + Υ 2 + f 2 Υ 3 < 0, 1 + f 1 Υ Υ

(5.109)

where  T   1  1  0 = A Υ B1 S1 S3 − S3 P A B1 S1 S3 − S3  T   2  2  + h2 A B2 S2 S4 − S4 P A B2 S2 S4 − S4      F  0 0 0 T E  F  0 0 0 − diag{P, γ 2 I, 0, 0, 0}, + E      D(ρ) 0 0 − I T J C  D(ρ) 0 0 − I − diag{0, 0, 0, 0, J}, 1 = b C Υ      2ξ ΔD(ρ) 0 0 − 2ξ ΔI  2ξ ΔD(ρ) 0 0 − 2ξ ΔI T 2ξ ΔC 2 = 2ξ ΔC Υ − diag{0, 0, 0, I, 0},      0 0 0 0 − diag{0, 0, I, 0, 0}.  0 0 0 0 T Q 3 = Q Υ Based on the filtering error system (5.85) and the Lyapunov function defined in (5.94), it can be obtained that x (k)) + e T (k)e(k) − γ 2 w T (k)w(k) E {V ( x (k + 1))} − V ( '%% & % & 1 +  2  =E A x (k) +  B1 +  h(k) A h(k)  B2 w(k) % & % & + S1 +  h(k)S2 α(x(k)) + S3 +  h(k)S4 β(k) & & &T %% % 1 +  2  x (k) h(k)S4 ξ y (k) P A h(k) A − S3 +  % & % & +  B1 +  h(k)  B2 w(k) + S1 +  h(k)S2 α(x(k)) % & % & &( + S3 +  h(k)S4 β(k) − S3 +  h(k)S4 ξ y (k)  T   x (k) + Fw(k)  x (k) + Fw(k)  + E E − x T (k)P x (k) − γ 2 w T (k)w(k) %    &T  → 2  1  h(k) A =− x T (k)E A B1 S1 S3 − S3 +  B2 S2 S4 − S4 %    &( − → 2  1  h(k) A x (k) ×P A B1 S1 S3 − S3 +  B2 S2 S4 − S4     → →  F  0 0 0 −  F  0 0 0 T E +− x T (k) E x (k) − → − → T − x (k)P x (k) 2

→ → 0 − =− x T (k)Υ x (k),

(5.110) where

  T − → x (k) w T (k) α T (x(k)) β T (k) ξ yT (k) , x T (k) =  P2 = diag{P, γ 2 I, 0, 0, 0}.

5.3 H∞ Filtering with Dynamic Quantization and Deception Attacks

191

For w(k) = 0, the inequality in (5.96) can be rewritten as → − → 1 − x (k) ≥ 0, x T (k)Υ

(5.111)

the inequality in (5.102) can be modified as → − → 2 − x (k) ≥ 0, x T (k)Υ

(5.112)

and the inequality in (5.103) can be expressed as → − → 3 − x (k) ≥ 0. x T (k)Υ

(5.113)

Then, by using Lemma 1.3 with inequalities (5.110), (5.111), (5.112), and (5.113) and scalars f 1 > 0 and f 2 > 0, it can be concluded that if the inequality in (5.109) → → 0 − x (k) < 0, i.e., is satisfied, then, − x T (k)Υ x (k)) + e T (k)e(k) − γ 2 w T (k)w(k) < 0. E {V ( x (k + 1))} − V (

(5.114)

Moreover, summing up (5.114) for k = 0, 1, 2, . . . , ∞, then x (0)) + E {V ( x (∞))} − V (

∞ 

e T (k)e(k) − γ 2

k=0

∞ 

w T (k)w(k) < 0.

(5.115)

k=0

By considering the fact that V ( x (0)) = 0 and E {V ( x (∞))} ≥ 0, it can be concluded that ∞ ∞   e T (k)e(k) < γ 2 w T (k)w(k), (5.116) k=0

k=0

which implies that if the conditions given in (5.87) and (5.88) are satisfied, the specified H∞ filtering performance for the filtering error system (5.85) can be ensured by using the online adjustment strategy for the dynamic quantizer’s parameter μ ys (k) given in (5.89). The proof is completed. According to the analysis results developed in Theorem 5.6, the design problem of event-triggered full-order H∞ filter will be addressed in this subsection. More specifically, the sufficient design conditions for the full-order filter defined in (5.84) and the additional parameter of the online adjusting strategy for the quantizer’s dynamic parameter defined in (5.89) will be proposed in terms of LMIs. Theorem 5.7 Consider the fuzzy system (5.77) and the filter (5.84). Suppose that the scalars 0 ≤ b < 1, h, quantization range bound M ys , quantization error Δ and matrix Θ = In x ×n x are known. The filtering error system (5.85) is stochastically stable with a prescribed H∞ filtering performance γ > 0, if there exist matrices f ,  f , scalars ϕ > 0, f 1 > 0, B f , and E P1 > 0, P2 , P3 > 0, M1 , M2 , M3 , J > 0, A and f 2 > 0 satisfying

192

5 Event-Triggered Fuzzy Control and Filtering …

ϕ M ys − f 1 > 0, ⎡

K 11i ∗ ∗ ⎢ K 21i K 22 ∗ ⎢ ⎢ K 31i 0 K 22 ⎢ ⎣ K 41i 0 0 0 K 51i 0

⎤ ∗ ∗ ∗ ∗ ⎥ ⎥ ∗ ∗ ⎥ ⎥ < 0, i = 1, 2, . . . , r, −I ∗ ⎦ 0 − f1 I

(5.117)

(5.118)

where ⎡

⎤ T11i ∗ ∗ ∗ ∗ ⎢T21i − γ 2 I + bD T JDi ∗ ∗ ∗ ⎥ i ⎢ ⎥ I ∗ ∗ ⎥ 0 0 − f K 11i = ⎢ 2 ⎢ ⎥, ⎣ 0 ∗ ⎦ 0 0 − f1 I − bJDi 0 0 (b − 1)J T51i   K 21i = T61i T62i T63i T64 − T64 ,

P1 − M1 − M1T ∗ K 22 = , P2 − M2 − M3T Θ T P3 − M3 − M3T     K 31i = T71i T72i T73i T74 − T74 , K 41i = T81i Fi 0 0 0 ,   K 51i = T91i 2ϕΔDi 0 0 − 2ϕΔI ,

−P1 + bCiT JCi + f 2 Q T Q ∗ , T11i = − P3 −P2    T  T21i = bDi JCi 0 , T51i = −bJCi 0 ,

f B f Ci Θ A M1 Ai + (1 − h)Θ  T61i = f , M2 Ai + (1 − h)  B f Ci A

B f Di hΘ  B f Ci M1 Bi + (1 − h)Θ  T62i = , T , = 63i M2 Bi + (1 − h)  B f Di h B f Ci

Bf (1 − h)Θ  − hΘ  B f Ci 0 , T64 = , T71i = − h B f Ci 0 (1 − h)  Bf

 − hΘ  B f Di hΘ  B C T72i = , T73i =  f i , − h B f Di h B f Ci

    − hΘ  Bf f , T91i = 2ϕΔCi 0 , , T81i = E i − E T74 = − h Bf and the parameter of the dynamic quantizer defined in (5.79), i.e., μ ys (k) obeys the online adjusting strategy presented in (5.89) with ξ = ϕ/ f 1 . Moreover, the parameters for the full-order filter defined in (5.84) can be obtained as f . f , B f = M3−1  (5.119) Bf , E f = E A f = M3−1 A

5.3 H∞ Filtering with Dynamic Quantization and Deception Attacks

193

Proof Assume that the related matrices M T and P have the form as



P1 ∗ M1 Θ M3 M = , P= , M2 M3 P2 P3

T

(5.120)

 f = M3 A f ,  f = where M3 is nonsingular. Moreover, by defining A B f = M3 B f , E E f , and ϕ = f 1 ξ , then, the inequality in (5.117) can be obtained based on (5.87) and the inequality in (5.88) can be rewritten as ⎡

⎤ K 11i ∗ ∗ ∗ ∗ ⎢ K 21i K 22 ∗ r ∗ ∗ ⎥  ⎢ ⎥ 0 K ∗ ∗ ⎥ K ρi ((k)) ⎢ 22 ⎢ 31i ⎥ < 0, ⎣ K 41i 0 0 −I ∗ ⎦ i=1 0 0 − f1 I K 51i 0

(5.121)

it is easy to know that if the inequality of (5.118) is satisfied, then the inequality of (5.121) is true. The proof is completed. According to the analysis results developed in Theorem 5.6, the design problem of event-triggered reduced-order H∞ filter will be addressed in this subsection. More specifically, the sufficient design conditions for the reduced-order filter defined in (5.84) and the additional parameter of the online adjusting strategy for the quantizer’s dynamic parameter defined in (5.89) will be proposed in terms of LMIs. Theorem 5.8 Consider the fuzzy system (5.77) and the filter (5.84). Suppose that the scalars 0 ≤ b < 1, h, quantization range bound M ys , quantization error Δ and ! & T % matrix Θ = In x f ×n x f 0n × n −n are known. The filtering error system (5.85) xf

x

xf

is stochastically stable with a prescribed H∞ filtering performance γ > 0, if there f ,  f , scalars B f , and E exist matrices P1 > 0, P2 , P3 > 0, M1 , M2 , M3 , J > 0, A ϕ > 0, f 1 > 0, and f 2 > 0 satisfying the conditions in (5.117) and (5.118) with the online adjusting strategy for the parameter of the dynamic quantizer given in Theorem 5.7. Moreover, the parameters of the reduced-order filter defined in (5.84) can be obtained according to (5.119). Proof The proof of Theorem 5.8 is similar to Theorem 5.7. Thus, the proof is omitted. For the event-triggered quantized filter design strategy proposed in this section, some remarks are given as follows: Remark 5.4 If the scalars b, h, quantization error Δ, quantization range bound M ys and the dimension adjustment matrix Θ are given, then, the design conditions for the full- and reduced-order event-triggered H∞ filters presented in Theorems 5.7 and 5.8 are strictly LMIs which are able to be solved easily by the LMI toolbox in Matlab. In general, the scalars b, h, quantization error Δ, and quantization range bound M ys are given for the event-triggered condition, the occurring probability of

194

5 Event-Triggered Fuzzy Control and Filtering …

stochastic deception attacks, and the dynamic quantizer. The co-design approach given in [25] can be used to deal with the case that the scalar b is unknown. However, how to deal with the filter design problems with the unknown parameters for dynamic quantizer and unknown occurring probability for stochastic deception attacks are still challenging problems, which deserve further study. Remark 5.5 By introducing a dimension adjustment matrix Θ, both the full- and reduced-order event-triggered H∞ filtering problems have been addressed for discrete-time nonlinear networked systems with dynamic quantization and stochastic deception attacks in a unified framework. Moreover, the H∞ filtering performance γ can be optimized by min μ subject to LMIs (5.117) and (5.118) with μ = γ 2 . Then, we have that γmin =

√

μmin .

Remark 5.6 In [20–22], the filtering problem has been considered for networked control systems with dynamic quantization. However, it should be noted that the transmission problem of the dynamic quantizer’s parameter has not been considered in the above results, i.e., the same dynamic quantizer’s parameter can be computed on both sides of the network communication channel according to the received signal, independently (see [20–22] for more details). In fact, according to [19, 26], the parameter of the dynamic quantizer also should be transmitted via the unreliable network communication channel for practical applications. Therefore, as in [26], a feasible adjusting rule for the dynamic quantizer’s parameter considered in this section, i.e., μ ys (k) is given as follows: ⎧ 1 s −s ⎪ ⎨floor (2ξ |y(ks )| × 10 ) × 10 , 0 ≤ ξ |y(ks )| < 2 , 1 μ ys (k) = 1, ≤ ξ |y(ks )| < 1, 2 ⎪ ⎩ 1 ≤ ξ |y(ks )|, floor (2ξ |y(ks )|) ,    where s = min s ∈ N+  (2ξ |y(ks )| × 10s ) > 1 and the scalar ξ = ϕ/ f 1 can be obtained by solving the linear matrix inequalities in (5.117) and (5.118). Here, the function floor( ) denotes the maximum integer that is not bigger than  . Remark 5.7 In contrast with the existing results on quantized filtering in [16–18, 20– 22] or event-triggered filtering in [25, 27], and [28], the considered filtering problem in this section is more general for two reasons. On the one hand, in this section, both the dynamic quantizer and the event-triggered communication scheme are employed to reduce the number of data transmission in the communication channel from the plant and the filter. On the other hand, in the presence of stochastic deception attacks, both the full- and reduced-order filtering problem have been solved in a unified framework.

5.3 H∞ Filtering with Dynamic Quantization and Deception Attacks

195

5.3.3 Simulation Example Let us consider a discrete-time T–S fuzzy model in the form of (5.76), where ⎡

A1

A2 C1 C2

0.7819 1.1899 = ⎣−0.0238 0.7231 −0.2380 − 0.1769 ⎡ 0.6749 1.1093 = ⎣−0.0222 0.7239 −0.2219 − 0.1688   = 1 0 0 , D1 = 1, E 1   = 1 0 0 , D2 = 1, E 2

⎤ 1.1899 − 0.0177⎦ , 0.5640 ⎤ 1.1093 − 0.0169⎦ , 0.5720   = 1 0 0 ,   = 1 0 0 ,

⎡

⎤ 0.0196 B1 = ⎣ 0.0257 ⎦ , −0.0019 ⎡ ⎤ 0.0187 B2 = ⎣ 0.0257 ⎦ , −0.0018 F1 = 0.5, F2 = 0.5.

By applying Theorem 5.7 with M ys = 200, Δ = 0.5, b = 0.25, h = 0.5, and Q = diag{0.1, 0.2, 0.1}, it can be obtained that γ = 0.5747, J = 0.0490, ξ = 0.0050, and ⎡ ⎤ ⎡ ⎤ 0.6702 1.1430 1.1433 −0.0595 A f = ⎣−0.0611 0.7202 − 0.0205⎦ , B f = ⎣−0.0384⎦ , −0.1951 − 0.1644 0.5761 0.0322   E f = −1.0831 − 0.1568 − 0.1502 .  T  T Moreover, initial conditions are assumed as x(0)= 0 0 0 , x f (0)= 0 0 0 , w(k) = 10 cos(0.5k)e−0.5k , ρ1 ((k)) = 1 − ρ2 ((k)), ρ2 ((k)) = x1 (k) ≤ 3, and ⎡ ⎤ −tanh(0.2x2 (k)) α(x(k)) = ⎣−tanh(0.1x1 (k))⎦ . −tanh(0.1x3 (k))

x12 (k) , where −3 9

≤

The simulation results of the filtering error system (5.85) are given in Figs. 5.11, 5.12, 5.13, and 5.14. The trajectories of z(k) and z f (k) are shown in Fig. 5.11. The trajectory of filtering error e(k) is shown in Fig. 5.12. The trajectory the dynamic quantizer’s parameter μ ys (k) is shown in Fig. 5.13. The simulation result on release instants and release intervals is depicted in Fig. 5.14. Based on the simulation results given in Figs. 5.11, 5.12, 5.13, and 5.14, it can be concluded that the event-triggered full-order H∞ filter design approach developed in this section is effective. In the following, the event-triggered reduced-order H∞ filter design problem will be studied for this example. By applying Theorem 5.8 with n x f = 2, the following variables are obtained

196

5 Event-Triggered Fuzzy Control and Filtering …

Fig. 5.11 Responses of z(k) and z f (k)

3 2.5 2 1.5 1 0.5 0 -0.5

Fig. 5.12 Response of the filtering error e(k)

0

10

20

30

40

50

0

10

20

30

40

50

10

20

30

40

50

3 2.5 2 1.5 1 0.5 0 -0.5

Fig. 5.13 Dynamic quantizer’s parameter μ ys (k)

0.05 0.04 0.03 0.02 0.01 0 0

5.4 Conclusion

197

Fig. 5.14 Release instants and release intervals

5 4 3 2 1 0 0

10

20

30

40

50





0.3613 − 0.4773 −0.1231 , Bf = , −0.0318 0.7510 −0.0307   E f = −0.4246 0.2102 , γ = 0.6077. Af =

Moreover, by applying Theorem 5.8 with n x f = 1, then γ = 0.6098, A f = 0.1171, B f = −0.1288, and E f = −0.4404 can be obtained.

5.4 Conclusion In this chapter, the event-triggered output feedback tracking control problem and filtering problem for T–S fuzzy systems have been addressed. For discrete-time fuzzy systems, the event-triggered H∞ tracking controller has been designed based on the tracking controller output quantization and the plant output quantization to maintain the prescribed H∞ performance. For the continuous-time fuzzy systems, the event-triggered L2 –L∞ filtering has been studied with time-delay and DoS attacks. A piecewise Lyapunov–Krasovskii functional method has been proposed to ensure the closed-loop system is exponentially stable and has a prescribed L2 –L∞ performance. For discrete-time case, the event-triggered H∞ filtering has been investigated with dynamic quantization and stochastic deception attacks. In order to ensure the filtering error system is stochastically stable with the prescribed H∞ performance, the full- and reduced-order filters have been designed by LMIs, respectively. Moreover, some simulation results have been given to verify the effectiveness of these control and filtering design strategies.

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5 Event-Triggered Fuzzy Control and Filtering …

References 1. 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 2. Pan Y, Yang GH (2019) Event-based output tracking control for fuzzy networked control systems with network-induced delays. Appl Math Comput 346:513–530 3. Gu Z, Yue D, Liu J, Ding Z (2017) H∞ tracking control of nonlinear networked systems with a novel adaptive event-triggered communication scheme. J Franklin Inst 354:3540–3553 4. Fu M, Xie L (2005) The sector bound approach to quantized feedback control. IEEE Trans Autom Control 50:1698–1711 5. Gao H, Chen T (2008) A new approach to quantized feedback control systems. Automatica 44:534–542 6. Li ZM, Chang XH, Park JH (2021) Quantized static output feedback fuzzy tracking control for discrete-time nonlinear networked systems with asynchronous event-triggered constraints. IEEE Trans Syst, Man, Cybern: Syst 51:3820–3831 7. Liu S, Wei G, Song Y, Liu Y (2016) Error-constrained reliable tracking control for discrete time-varying systems subject to quantization effects. Neurocomputing 174:897–905 8. Liu Z, Wang F, Zhang Y, Chen CLP (2016) Fuzzy adaptive quantized control for a class of stochastic nonlinear uncertain systems. IEEE Trans Cybern 46:524–534 9. Chang XH (2014) Robust output feedback H-infinity control and filtering for uncertain linear systems. Berlin Heidelberg, Springer 10. Chang XH, Yang GH (2014) New results on output feedback H∞ control for linear discretetime systems. IEEE Trans Autom Control 59:1355–1359 11. Yue D, Tian E, Han QL (2013) A delay system method for designing event-triggered controllers of networked control systems. IEEE Trans Autom Control 58:475–481 12. Hu S, Yue D, Xie X, Chen X, Yin X (2018) Resilient event-triggered controller synthesis of networked control systems under periodic DoS jamming attacks. IEEE Trans Cybern 49:4271– 4281 13. Yue D, Tian E, Han Q (2012) A delay system method for designing event-triggered controllers of networked control systems. IEEE Trans Autom Control 58:475–481 14. Shisheh FH, Martínez S (2016) On triggering control of single-input linear systems under pulse-width modulated DoS signals. SIAM J Control Optim 54:3084–3105 15. Zhang X, Han Q (2013) Novel delay-derivative-dependent stability criteria using new bounding techniques. Int J Robust Nonlinear Control 23:1419–1432 16. Chang XH, Wang YM (2018) Peak-to-peak filtering for networked nonlinear DC motor systems with quantization. IEEE Trans Industr Inf 14:5378–5388 17. Dong S, Su H, Shi P, Lu R, Wu ZG (2017) Filtering for discrete-time switched fuzzy systems with quantization. IEEE Trans Fuzzy Syst 25:1616–1628 18. Zhang C, Feng G, Gao H, Qiu J (2010) H∞ filtering for nonlinear discrete-time systems subject to quantization and packet dropouts. IEEE Trans Fuzzy Syst 19:353–365 19. Niu Y, Ho DWC (2014) Control strategy with adaptive quantizer’s parameters under digital communication channels. Automatica 50:2665–2671 20. Che WW, Yang GH (2013) H∞ filter design for continuous-time systems with quantised signals. Int J Syst Sci 44:265–274 21. Che WW, Yang G H (2009) Quantised H∞ filter design for discrete-time systems. Int J Control 82:195–206 22. Chang XH, Li ZM, Park JH (2018) Fuzzy generalized H2 filtering for nonlinear discrete-time systems with measurement quantization. IEEE Trans Syst, Man, Cybern: Syst 48:2419–2430 23. Liu J, Wei L, Xie X, Tian E, Fei S (2018) Quantized stabilization for T–S fuzzy systems with hybrid-triggered mechanism and stochastic cyber-attacks. IEEE Trans Fuzzy Syst 26:3820– 3834 24. Li ZM, Xiong J (2022) Event-triggered fuzzy filtering for nonlinear networked systems with dynamic quantization and stochastic cyber attacks. ISA Trans 121:53–62

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Chapter 6

Fuzzy Fault Detection and Fault-Tolerant Control with Quantization

Abstract In this chapter, the quantized fault detection and fault-tolerant control problems are investigated for T–S fuzzy system. The quantized fault detection problem is first investigated for the uncertain T–S fuzzy system. Particularly, we consider two different situations of the filter parameters with and without uncertain terms. And by using Lyapunov function method for the considered system, both robust and resilient fault detection filters are designed respectively such that the residual system is asymptotically stable and guarantees a prescribed H∞ noise attenuation level bound. Next, the problem of quantized fault-tolerant control for T–S fuzzy system with Markov jumps and actuator fault is studied. A time-varying coefficient matrix is introduced to describe the relation of the fault signal and the input signal of the actuator. Then the fault-tolerant controller is designed based on the mode-dependent idea and robust stochastic stability approach, and the existence conditions of such controller are formulated in terms of a set of LMIs. In addition, quantized guaranteed cost fault-tolerant control problem is studied for the T–S fuzzy system, where there has the uncertainty term for the controller parameter. Consider the coupling term caused from the uncertainties of controller and the fault model, a two-step design method is introduced to develop a design scheme which can be solved by standard LMI technique, and for the quantization effect, an adjusting rule for the dynamic quantization parameter is also proposed to ensure the stability and the guaranteed cost performance for the resulting closed-loop system. The effectiveness of the proposed design methods are shown through some simulation examples. Keywords Fault detection · Fault-tolerant control · Resilient control · Guaranteed cost control · Markov jumps

6.1 Fault Detection with Output Quantization Considering the problem of fault detection in T–S fuzzy system with uncertainty and quantization, whether the fault detection filter contains uncertain terms is fully considered, the main object is to design the filters such that the corresponding residual systems are asymptotically stable and satisfy the certain H∞ performance. By © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 X. Chang et al., Control and Filtering of Fuzzy Systems Under Communication Channels, https://doi.org/10.1007/978-981-99-4346-3_6

201

202

6 Fuzzy Fault Detection and Fault-Tolerant …

introducing the corresponding lemmas, the fault detection filter design conditions for the designed filter are obtained in the form of LMIs.

6.1.1 Problem Formulation In this section, to investigate the fault detection filter design problem, we consider the following T–S fuzzy model Plant Rule i th : IF ε1 (k) is M1i , ε2 (k) is M2i , and, . . ., and εd (k) is Mdi , THEN x(k + 1) = (Ai + Δ Ai (k))x(k) + (Bi + Δ Bi (k))u(k) + (E i + Δ Ei (k))v(k) + (E f i + Δ E f i (k)) f (k), y(k) = (Ci + ΔCi (k))x(k) + (Di + Δ Di (k))u(k)

(6.1)

+ (Fi + Δ Fi (k))v(k) + (F f i + Δ F f i (k)) f (k), where x(k) ∈ Rn x is the state variable, u(k) ∈ Rn u is the control input, y(k) ∈ Rn y is the measurement output, v(k) ∈ Rn v is the unknown input vector (including external disturbance, uninterested fault as well as some norm-bounded unstructured model uncertainty), f (k) ∈ Rn f is the fault to be detected and isolated. ε1 (k), . . . , εd (k) are measurable premise variables; M pi represent the fuzzy sets with p = 1, . . . , d, i = 1, . . . , r , and the parameter r means the number of fuzzy rules. Ai ∈ Rn x ×n x , Bi ∈ Rn x ×n u , E i ∈ Rn x ×n v , E f i ∈ Rn x ×n f , Ci ∈ Rn y ×n x , Di ∈ Rn y ×n u , Fi ∈ Rn y ×n v , and F f i ∈ Rn y ×n f are known system matrices. Without loss of generality, assume v(k) and f (k) are l2 -norm bounded. The uncertain parameter matrices are given by 

     Δ Ai (k) Δ Bi (k) Δ Ei (k) Δ E f i (k) X x1i Δx (k) Yx1i Yx2i Yx3i Yx4i , = X x2i ΔCi (k) Δ Di (k) Δ Fi (k) Δ F f i (k)

where X x1i , X x2i , Yx1i , Yx2i , Yx3i , and Yx4i are known matrices with appropriate dimensions and Δx (k) is an uncertainty satisfying the condition ΔTx (k)Δx (k) ≤ I . It is assumed that fuzzy basis function is given by d

p=1 M pi (ε p (k)) , d i=1 p=1 M pi (ε p (k))

ρi (ε(k)) = r

(6.2)

where M pi (ε p (k)) is the grade of membership function of ε p (k) in M pi . Considering the properties of membership function, there has ρi (ε(k)) ≥ 0,

r  i=1

ρi (ε(k)) = 1, i = 1, . . . , r.

(6.3)

6.1 Fault Detection with Output Quantization

203

Then a more compact presentation of T–S fuzzy model (6.1) can be derived as follows: x(k + 1) = (A(ρ) + Δ A (ρ))x(k) + (B(ρ) + Δ B (ρ))u(k) + (E(ρ) + Δ E (ρ))v(k) + (E f (ρ) + Δ E f (ρ)) f (k), y(k) = (C(ρ) + ΔC (ρ))x(k) + (D(ρ) + Δ D (ρ))u(k) + (F(ρ) + Δ F (ρ))v(k) + (F f (ρ) + Δ F f (ρ)) f (k), where A(ρ) = E(ρ) = C(ρ) = F(ρ) =

r  i=1 r  i=1 r  i=1 r 

ρi (ε(k))Ai , B(ρ) =

r 

ρi (ε(k))Bi ,

i=1 r 

ρi (ε(k))E i , E f (ρ) = ρi (ε(k))Ci , D(ρ) =

ρi (ε(k))Fi , F f (ρ) =

i=1

Δ A (ρ) = X x1 (ρ)Δx (k)Yx1 (ρ) = Δ B (ρ) = X x1 (ρ)Δx (k)Yx2 (ρ) = Δ E (ρ) = X x1 (ρ)Δx (k)Yx3 (ρ) = Δ E f (ρ) = X x1 (ρ)Δx (k)Yx4 (ρ) = ΔC (ρ) = X x2 (ρ)Δx (k)Yx1 (ρ) = Δ D (ρ) = X x2 (ρ)Δx (k)Yx2 (ρ) = Δ F (ρ) = X x2 (ρ)Δx (k)Yx3 (ρ) = Δ F f (ρ) = X x2 (ρ)Δx (k)Yx4 (ρ) =

ρi (ε(k))E f i ,

i=1 r 

ρi (ε(k))Di ,

i=1 r 

ρi (ε(k))F f i ,

i=1 r  i=1 r  i=1 r  i=1 r  i=1 r  i=1 r  i=1 r  i=1 r  i=1

ρi (ε(k))X x1i Δx (k)Yx1i , ρi (ε(k))X x1i Δx (k)Yx2i , ρi (ε(k))X x1i Δx (k)Yx3i , ρi (ε(k))X x1i Δx (k)Yx4i , ρi (ε(k))X x2i Δx (k)Yx1i , ρi (ε(k))X x2i Δx (k)Yx2i , ρi (ε(k))X x2i Δx (k)Yx3i , ρi (ε(k))X x2i Δx (k)Yx4i .

(6.4)

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The fault detection of the system is generally to select the fault detection filter and fault weight system to construct a residual, so as to determine the fault through the residual evaluation function.

6.1.1.1

Residual Generation

To obtain the residual term, the fault detection filter of the following form is selected as x f (k + 1) = (A f + Δ A f (k))x f (k) + (B f + Δ B f (k))yq (k), (6.5) r (k) = (C f + ΔC f (k))x f (k) + (D f + Δ D f (k))yq (k), where x f (k) ∈ Rn x f and r (k) ∈ Rnr represent the state vector of the fault detection filter and the residual signal. A f ∈ Rn x f ×n x f , B f ∈ Rn x f ×n y , C f ∈ Rnr ×n x f , and D f ∈ Rnr ×n y are the filter gain matrices to be determined. Δ A f (k), Δ B f (k), ΔC f (k), and Δ D f (k) represent the uncertain parameter terms, which are assumed to have the similar form as       X f1 Δ A f (k) Δ B f (k) = Δ f (k) Y f 1 Y f 2 , X f2 ΔC f (k) Δ D f (k) where X f 1 , X f 2 , Y f 1 , and Y f 2 are known matrices with appropriate dimensions and Δ f (k) is an uncertainty satisfying ΔTf (k)Δ f (k) ≤ I . Moreover, yq (k) is the quantized signal of the measurement output y(k). Consider a dynamic quantization scheme [1] which is described as yq (k) = qμ (y(k)) = μ(k)q

y(k) , μ(k)

(6.6)

where μ(k) is the quantization parameter that follows the change of quantization signals and q(·) is a static quantizer. For the quantization range M and quantization error bound Δ of the above quantizer, the following conditions should be satisfied 

 q y(k) −  μ(k) 

 q y(k) −  μ(k)

 y(k)   ≤ Δ, μ(k)   y(k)   > Δ, μ(k) 

   y(k)     μ(k)  ≤ M,    y(k)     μ(k)  > M.

(6.7)

In fault detection, a reference residual model is usually needed to describe the desired behavior of the residual signal r (k). In this section, we consider the following fault weighting system [2], xw (k + 1) = Aw xw (k) + Bw f (k), rw (k) = Cw xw (k) + Dw f (k),

(6.8)

6.1 Fault Detection with Output Quantization

205

where xw (k) is the state vector of the fault weighting system and rw (k) is the so-called reference model with rw (z) = W (z) f (z), in which the priori W (z) is given and the choice of Wz is to impose frequency weighting on the spectrum of the fault signal for detection.

6.1.1.2

Residual Evaluation

The residual generated by the above fault detection filter and fault weight system can be compared with the residual characteristics and threshold values, so as to obtain the judgment condition of whether the system has a fault, and the judgment condition can distinguish the external disturbance and uncertainty. The logic for detecting a potential fault is described as: r  R M S > Jth ⇒ alarm, a fault is detected, r  R M S ≤ Jth ⇒ no alarm, fault-free, where r  R M S



k2

 1 Δ =  r T (k)r (k), K = k2 − k1 + 1 K k=k

(6.9)

(6.10)

1

is the evaluation function which measures the average energy of the residual signal r (k) over a time interval (k1 , k2 ). In order to detect fault as early as possible, the length of time window is set to be limited, so as to obtain the evaluation function and determine the threshold Jth of fault detection. Remark 6.1 In this section, the main attention is paid to the design of fault detection filter, thus to highlight this topic, we will not discuss how to set the threshold and make the decision. Surely, after generating the residual signal by fault detection filter and formulating the evaluation function, this issue can be easy to be handled according to [3].

6.1.1.3

Fault Detection Problem

In order to design fault detection filter with dynamic quantization, a residual system needs to be designed. Therefore, by defining error signal er (k) = rw (k) − r (k) and combining (6.4), (6.5), (6.6), and (6.8), the resulting residual system can be obtained as φ(k + 1) = A(ρ)φ(k) + B(ρ)w(k) + Eυ y (k), er (k) = C(ρ)φ(k) + D(ρ)w(k) + Fυ y (k), y(k) = C y (ρ)φ(k) + D y (ρ)w(k),

(6.11)

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6 Fuzzy Fault Detection and Fault-Tolerant …

where     φ T (k) = xwT (k) x T (k) x Tf (k) , w T (k) = u T (k) v T (k) f T (k) , υ y (k) = yq (k) − y(k), ⎡ ⎡ ⎤ ⎤ Aw 0 0 0 ⎦, E = ⎣ ⎦, 0 0 A(ρ) + Δ A (ρ) A(ρ) = ⎣ 0 0 (B f + Δ B f )(C(ρ) + ΔC (ρ)) A f + Δ A f B f + ΔB f ⎡ ⎤ 0 0 Bw B(ρ) = ⎣ B(ρ) + Δ B (ρ) E(ρ) + Δ E (ρ) E f (ρ) + Δ E f (ρ)⎦ , B1 (ρ) B2 (ρ) B3 (ρ)

B1 (ρ) = (B f + Δ B f )(D(ρ) + Δ D (ρ)), B2 (ρ) = (B f + Δ B f )(F(ρ) + Δ F (ρ)),   B3 (ρ) = (B f + Δ B f )(F f (ρ) + Δ F f (ρ)), D(ρ) = D1 (ρ) D2 (ρ) D3 (ρ) ,   C(ρ) = Cw −(D f + Δ D f )(C(ρ) + ΔC (ρ)) −(C f + ΔC f ) , D1 (ρ) = −(D f + Δ D f )(D(ρ) + Δ D (ρ)), D2 (ρ) = −(D f + Δ D f )(F(ρ) + Δ F (ρ)), D3 (ρ) = Dw − (D f + Δ D f )(F f (ρ) + Δ F f (ρ)), F = −(D f + Δ D f ),   C y (ρ) = 0 C(ρ) + ΔC (ρ) 0 ,   D y (ρ) = D(ρ) + Δ D (ρ) F(ρ) + Δ F (ρ) F f (ρ) + Δ F f (ρ) .

In this section, in order to fully consider the influence of fuzzy system, dynamic quantization and uncertain terms between system and fault detection filter, the design problem of fault detection filter is transformed into an H∞ filtering problem by referring to the method in [3]. In other words, we will find the suitable fault detection filter such that the residual system is asymptotically stable and the following performance index J is made small: er (k)2 . J= sup w(k)∈l2 ,w(k)=0 w(k)2 The fault detection problem in this section is to design the fault detection filter in (6.5) and the dynamic quantizer in (6.7) such that (1) The residual system (6.11) is asymptotically stable; (2) The residual system (6.11) has a prescribed performance γ , i.e., ∞  k=0

erT (k)er (k) < γ 2

∞ 

w T (k)w(k).

(6.12)

k=0

6.1.2 Analysis and Synthesis of Quantized Residual System There are two kinds of uncertainties in the residual system (6.11), which increases the difficulty of fault detection filter design. In this section, a two-step method is used

6.1 Fault Detection with Output Quantization

207

to design the corresponding filter to make the residual system (6.11) asymptotically stable and preserve the desired H∞ performance.

6.1.2.1

Performance Analysis of Residual System

In this subsection, the fault detection filter with no uncertainty is considered as x f (k + 1) = A f x f (k) + B f yq (k), r (k) = C f x f (k) + D f yq (k).

(6.13)

Then the corresponding residual system can be presented as ˇ (ρ)w(k) + E ˇ υ y (k), ˇ (ρ)φ(k) + B φ(k + 1) = A ˇ (ρ)φ(k) + D ˇ (ρ)w(k) + F ˇ υ y (k), er (k) = C y(k) = C y (ρ)φ(k) + D y (ρ)w(k),

(6.14)

where ⎡

⎡ ⎤ ⎤ Aw 0 0 0 ˇ = ⎣ 0 ⎦, ˇ (ρ) = ⎣ 0 0 ⎦, E A(ρ) + Δ A (ρ) A 0 B f (C(ρ) + ΔC (ρ)) A f Bf ⎡ ⎤ 0 0 Bw ˇ (ρ) = ⎣ B(ρ) + Δ B (ρ) E(ρ) + Δ E (ρ) E f (ρ) + Δ E f (ρ) ⎦ , B B f (D(ρ) + Δ D (ρ)) B f (F(ρ) + Δ F (ρ)) B f (F f (ρ) + Δ F f (ρ))   ˇ ˇ = −D f , C(ρ) = Cw −D f (C(ρ) + ΔC (ρ)) −C f , F   ˇ (ρ) = −D f (D(ρ) + Δ D (ρ)) −D f (F(ρ) + Δ F (ρ)) Dw − D f (F f (ρ) + Δ F (ρ)) . D f

In the following theorem, the analysis conditions for the stability of the residual system (6.14) and the H∞ performance can be obtained. Theorem 6.1 Consider the residual system (6.14). For known quantization range M and quantization error bound Δ, if there exist matrix P > 0, scalars θ ≥ 1 and γ such that the following matrix inequality holds: ⎡

−P ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ A ⎢ ˇ (ρ) ⎣ C ˇ (ρ) θ C y (ρ)

∗ ∗ ∗ −γ 2 I ∗ ∗ 2 0 −M I ∗ Δ2 ˇ ˇ B(ρ) E −P −1 ˇ (ρ) ˇ D F 0 θ D y (ρ) 0 0

∗ ∗ ∗ ∗ −I 0

⎤ ∗ ∗⎥ ⎥ ∗⎥ ⎥ < 0, ∗⎥ ⎥ ∗⎦ −I

then the system is asymptotically stable and preserves the H∞ performance.

(6.15)

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6 Fuzzy Fault Detection and Fault-Tolerant …

Proof  When thesignal   y(k) to be quantized is in the quantization range, i.e.,  y(k)  y(k)  y(k)   μ(k)  ≤ M, q μ(k) − μ(k)  ≤ Δ can be obtained according to the quantizer properties. Considering the homogeneity property of Euclidean norm, one has   



      υ y (k) = μ(k) q y(k) − y(k)  = μ(k) q y(k) − y(k)  ≤ μ(k)Δ.    μ(k) μ(k) μ(k) μ(k)  (6.16) By defining μ(k) = Mθ y(k), where θ is a scalar satisfying θ ≥ 1. Substituting μ(k) defined into (6.16), we obtain θ 2 Δ2 T y (k)y(k). (6.17) M2   Defining δ T (k) = φ T (k) w T (k) υ yT (k) , the above inequality can be rewritten as υ yT (k)υ y (k) ≤

δ T (k)Σ1 (ρ)δ(k) ≥ 0,

(6.18)

where  T    C y (ρ) D y (ρ) 0 − diag 0, 0, Σ1 (ρ) = C y (ρ) D y (ρ) 0

M2 I θ 2 Δ2

 .

Construct a Lyapunov function as V (φ(k)) = φ T (k)Pφ(k), P > 0,

(6.19)

then, we have V (φ(k + 1)) − V (φ(k)) + erT (k)er (k) − γ 2 w T (k)w(k) = φ T (k + 1)Pφ(k + 1) − φ T (k)Pφ(k) + erT (k)er (k) − γ 2 w T (k)w(k)  T   ˇ (ρ)w(k) + E ˇ υ y (k) P A ˇ (ρ)w(k) + E ˇ υ y (k) ˇ (ρ)φ(k) + B ˇ (ρ)φ(k) + B = A  T ˇ (ρ)φ(k) + D ˇ (ρ)w(k) + F ˇ (ρ)φ(k) + D ˇ (ρ)w(k) ˇ υ y (k) (C − φ T (k)Pφ(k) + C ˇ υ y (k)) − γ 2 w T (k)w(k) +F = δ T (k)Σ0 (ρ)δ(k),

(6.20) where T    ˇ (ρ) E ˇ P A ˇ (ρ) E ˇ ˇ (ρ) B ˇ (ρ) B Σ0 (ρ) = A T      ˇ (ρ) D ˇ (ρ) D ˇ (ρ) F ˇ (ρ) F ˇ ˇ − diag P, γ 2 I, 0 . + C C

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209

Based on the Lemma 1.3 and defining a positive scalar ε1 = θ 2 such that Σ0 (ρ) + θ 2 Σ1 (ρ) < 0,

(6.21)

which means  T    T   ˇ (ρ) D ˇ (ρ) D ˇ (ρ) E ˇ P A ˇ (ρ) E ˇ + C ˇ (ρ) F ˇ (ρ) F ˇ ˇ ˇ (ρ) B ˇ (ρ) B A C   T    M2 C y (ρ) D y (ρ) 0 < 0. − diag P, γ 2 I, 2 I + θ 2 C y (ρ) D y (ρ) 0 Δ (6.22) Applying the Lemma 1.1, the condition (6.15) for system stability and the desired performance can be obtained. On the other hand, under the condition of Σ0 (δ) < 0, we have ∞  k=0

erT (k)er (k) < γ 2

∞ 

w T (k)w(k),

k=0

with V (φ(∞)) > 0, V (φ(0)) = 0, which proves that the residual system (6.14) is asymptotically stable and guarantees the performance defined in (6.12) under the condition (6.15).

6.1.2.2

Design of Robust Fault Detection Filter with Quantization

Due to the existence of nonlinear terms and the uncertainty terms in the matrix inequalities of Theorem 6.1, the LMIs cannot be used directly for filter design, the following theorem gives the design conditions of robust fault detection filter with quantization. Theorem 6.2 Consider the residual system (6.14). For known quantization range M and quantization error bound Δ, given slack scalars β1 , β2 , and β3 , if there exist matrices P11 , P21 , P22 , P33 , A f , B f , C f , D f , scalars θ ≥ 1, γ , and ε1 such that the following LMIs hold (6.23) Λˇ i < 0, i = 1, ..., r, where

210

6 Fuzzy Fault Detection and Fault-Tolerant …

⎤ ⎡ ⎡ ⎡ ⎤ ⎤ ∗ Ξˇ 11i ∗ ˇ 11i  ˇ 12i  ˇ 13 Λˇ 11i ∗ ∗  ⎥ ˇ ⎢Ξˇ ˇ ⎣ ⎣ ⎦ ˇ ˇ ¯ ¯ ∗ Ξ ˇ ˇ , Λ11i = ⎣ 21i 22i Λi = Λ21i Λ22 ∗ Ci Di −D f ⎦ , ⎦ , Λ21i = 2 ¯ yi θ D ¯ yi 0 θC 0 Λˇ 32i −ε1 I 0 0 − M2 I Δ ⎤ ⎡ ⎡ ⎤ T Y 0 ε1 Yx2i −P11 ∗ ∗ x1i 0 ⎥ ⎢ T Y T ⎦ ˇ = , Ξ Ξˇ 11i = ⎣ −P21 −P22 + ε1 Yx1i ∗ Y 0 ε ⎣ 21i 1 x3i Yx1i 0⎦ , x1i T −β2 P33 −β3 P33 −β1 P33 0 ε1 Yx4i Yx1i 0 ⎤ ⎡ T 2 −γ I + ε1 Yx2i Yx2i ∗ ∗ ⎥ ⎢ T Y T Y Ξˇ 22i = ⎣ −γ 2 I + ε1 Yx3i ∗ ε1 Yx3i ⎦, x2i x3i T Y T Y T Y ε1 Yx4i −γ 2 I + ε1 Yx4i ε1 Yx4i x2i x3i x4i ⎡ ⎡ ⎤ ⎤ T A +β B C β1 B f P11 Aw P21 β1 A f 1 f i i ⎣ ⎣ ⎦ ˇ ˇ 11i = P21 Aw P22 Ai + β2 B f Ci β2 A f , 13 = β2 B f ⎦ , β3 B f β1 P33 Aw β2 P33 Ai + β3 B f Ci β3 A f ⎤ ⎡ T T T E +β B F P21 E i + β1 B f Fi P11 Bw + P21 P21 Bi + β1 B f Di 1 f fi fi ˇ 12i = ⎣ P22 Bi + β2 B f Di  P22 E i + β2 B f Fi P21 Bw + P22 E f i + β2 B f F f i ⎦ , β2 P33 Bi + β3 B f Di β2 P33 E i + β3 B f Fi β1 P33 Bw + β2 P33 E f i + β3 B f F f i   T DT θ X T Λˇ 22 = −diag{P, I, I }, Λˇ 32i = Γˇ13i −X x2i f x2i ,   T  T  T , Γˇ13i = P T X x1i + β1 B f X x2i P22 X x1i + β2 B f X x2i β2 P33 X x1i + β3 B f X x2i 21

    ¯ i = −D f Di −D f Fi Dw − D f F f i , ¯ i = Cw −D f Ci −C f , D C     ¯ yi = Di Fi F f i , ¯ yi = 0 Ci 0 , D C

the residual system (6.14) is asymptotically stable with the H∞ performance γ . Then, the robust fault detection filter with quantization gain matrices in (6.13) can be obtained as −1 −1 A f , B f = P33 Bf, Cf = Cf, Df = Df. A f = P33

(6.24)

Proof Pre- and post-multiplying (6.15) by diag{ I, I, I, P, I, I } and diag{ I, I, I, P T , I, I }, respectively, gives ⎡

−P ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ PA ⎢ ˇ (ρ) ⎣ C ˇ (ρ) θ C y (ρ)

∗ ∗ −γ 2 I ∗ 2 0 −M I Δ2 ˇ (ρ) P E ˇ PB ˇ (ρ) ˇ D F θ D y (ρ) 0

∗ ∗ ∗ −P 0 0

∗ ∗ ∗ ∗ −I 0

⎤ ∗ ∗⎥ ⎥ ∗⎥ ⎥ < 0. ∗⎥ ⎥ ∗⎦ −I

(6.25)

By separating the uncertainty Δx (k) from above inequality, we have Ξ (ρ) + H e {W1 (ρ)Δx (k)V1 (ρ)} < 0,

(6.26)

6.1 Fault Detection with Output Quantization

211

with   T (ρ) , W1T (ρ) = 0 0 0 (PU (ρ))T (−D f X x2 (ρ))T θ X x2     T (ρ) (B X (ρ))T , V (ρ) = V (ρ) V (ρ) 0 0 0 0 , U T (ρ) = 0 X x1 11 12 1 f x2     V11 (ρ) = 0 Yx1 (ρ) 0 , V12 (ρ) = Yx2 (ρ) Yx3 (ρ) Yx4 (ρ) ,     ¯ (ρ) = −D f D(ρ) −D f F(ρ) Dw − D f F f (ρ) , ¯ (ρ) = Cw −D f C(ρ) −C f , D C     ¯ y (ρ) = 0 C(ρ) 0 , D ¯ y (ρ) = D(ρ) F(ρ) F f (ρ) , C ⎡ ⎤ −P ∗ ∗ ∗ ∗ ∗ ⎢ 0 ∗ ∗ ∗ ∗⎥ − γ2I ⎢ ⎥ ⎢ 0 M2 0 − I ∗ ∗ ∗⎥ ⎢ ⎥ 2 Δ Ξ (ρ) = ⎢ , ¯ (ρ) P E ˇ −P ∗ ∗ ⎥ ¯ (ρ) P B ⎢ PA ⎥ ⎢ ⎥ ¯ (ρ) ¯ (ρ) ˇ ⎣ C D F 0 −I ∗ ⎦ ¯ y (ρ) θ D ¯ y (ρ) 0 θC 0 0 −I ⎡ ⎡ ⎤ ⎤ Aw 0 0 Bw 0 0 ¯ (ρ) = ⎣ B(ρ) ¯ (ρ) = ⎣ 0 E(ρ) E f (ρ) ⎦ . A(ρ) 0 ⎦ , B A B f D(ρ) B f F(ρ) B f F f (ρ) 0 B f C(ρ) A f

Apply Lemmas 1.1 and 1.2, for a scalar parameter ε1 , the matrix inequality (6.25) holds if and only if the following matrix condition is satisfied   Ξ (ρ) + ε1 V1T (ρ)V1 (ρ) ∗ < 0. −ε1 I W1T (ρ)

(6.27)

⎡

⎤ P11 ∗ ∗ ∗ ⎦, and defining Choosing the Lyapunov matrix as P = ⎣ P21 P22 β1 P33 β2 P33 β3 P33 r  A f = P33 A f , B f = P33 B f , C f = C f , D f = D f , Λˇ ρ = ρi (ε(k))Λˇ i < 0, the i=1

condition (6.23) can be rearranged as the one of Theorem 6.2.

6.1.2.3

Design of Resilient Fault Detection Filter with Quantization

Similar to the design process of robust fault detection filter, the system stability and the performance of the residual system (6.11) need to be analyzed first for the design of corresponding resilient filter. Theorem 6.3 Consider the residual system (6.11). For known quantization range M and quantization error bound Δ, if there exists matrix P > 0, scalars θ ≥ 1 and γ such that the following matrix inequality holds:

212

6 Fuzzy Fault Detection and Fault-Tolerant …

⎡

−P ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ A(ρ) ⎢ ⎣ C(ρ) θ C y (ρ)

∗ ∗ ∗ −γ 2 I ∗ ∗ 2 0 −M I ∗ Δ2 B(ρ) E −P −1 D(ρ) F 0 θ D y (ρ) 0 0

∗ ∗ ∗ ∗ −I 0

⎤ ∗ ∗⎥ ⎥ ∗⎥ ⎥ < 0, ∗⎥ ⎥ ∗⎦ −I

(6.28)

then the system is asymptotically stable and preserves the H∞ performance. Proof This performance analysis condition for residual system (6.11) is an extension of the Theorem 6.1 to a more general case with T    Σ2 (ρ) = A(ρ) B(ρ) E P A(ρ) B(ρ) E T      C(ρ) D(ρ) F − diag P, γ 2 I, 0 , + C(ρ) D(ρ) F and its proof is straightforward and hence omitted. Due to the existence of nonlinear terms in the matrix inequalities, the quantization error term and coupling terms between two types of uncertainties in Theorem 6.3, some existing results [2] do not work, we choose a two-step approach to give the design conditions of resilient fault detection filter with quantization. Theorem 6.4 Consider the residual system (6.11). For known quantization range M and quantization error bound Δ, given scalars β1 , β2 , and β3 , if there exist matrices P11 , P21 , P22 , P33 , A f , B f , C f , D f , scalars θ ≥ 1, γ , ε1 , and ε2 such that the following LMIs hold (6.29) Λi < 0, i = 1, ..., r, where ⎡

⎤ ⎡ ⎤ ∗ ∗ ∗ Ξ11i ∗ ∗ ⎥ ˇ Λ22 ∗ ∗ ⎥ , Λ11i = ⎣Ξ21i Ξ22i ∗ ⎦ , Λˇ 32i Λ33i ∗ ⎦ Ξ31i Ξ32i Ξ33 Λ42 0 −ε2 I  T Γ12i ε2 X x2i Y fT2 Y f 2 ,   T = −ε1 I + ε2 X x2i Y fT2 Y f 2 X x2i , Λ42 = Γ14 −X Tf 2 0 , ⎡ ⎤ −P11 ∗ ∗ T T T ⎦, ∗ = ⎣ −P21 −P22 + ε1 Yx1i Yx1i + ε2 Ci Y f 2 Y f 2 Ci T T −β1 P33 −β2 P33 + ε2 Y f 1 Y f 2 Ci −β3 P33 + ε2 Y f 1 Y f 1 ⎡ ⎤ T T T T T 0 ε1 Yx2i Yx1i + ε2 Di Y f 2 Y f 2 Ci ε2 Di Y f 2 Y f 1 T Yx1i + ε2 FiT Y fT2 Y f 2 Ci ε2 FiT Y fT2 Y f 1 ⎦ , = ⎣0 ε1 Yx3i T 0 ε1 Yx4i Yx1i + ε2 F fTi Y fT2 Y f 2 Ci ε2 F fTi Y fT2 Y f 1   = −diag γ 2 I, γ 2 I, γ 2 I +

Λ11i ⎢Λˇ 21i Λi = ⎢ ⎣Λ31i 0  Λ31i = Γ11i Λ33i Ξ11i

Ξ21i Ξ22i

6.1 Fault Detection with Output Quantization

213

⎡

Ξ221i

⎤ T ε1 Yx2i Yx2i + ε2 DiT Y fT2 Y f 2 Di ∗ ∗ T T ⎣ ε1 Yx3i Yx2i + ε2 FiT Y fT2 Y f 2 Di ε1 Yx3i Yx3i + ε2 FiT Y fT2 Y f 2 Fi ∗ ⎦, T T T T ε1 Yx4i Yx2i + ε2 F f i Y f 2 Y f 2 Di ε1 Yx4i Yx3i + ε2 F fTi Y fT2 Y f 2 Fi Ξ221i   T = ε1 Yx4i Yx4i + ε2 F fTi Y fT2 Y f 2 F f i , Ξ31i = 0 ε2 Y fT2 Y f 2 Ci ε2 Y fT2 Y f 1 ,

  M2 Ξ32i = ε2 Y fT2 Y f 2 Di ε2 Y fT2 Y f 2 Fi ε2 Y fT2 Y f 2 F f i , Ξ33 = − 2 I + ε2 Y fT2 Y f 2 , Δ    T  T Γ11i = 0 ε2 Y f 2 X x2i Y f 2 Ci ε2 Y f 2 X x2i Y f 1 ,         Γ12i = ε2 Y f 2 X x2i T Y f 2 Di ε2 Y f 2 X x2i T Y f 2 Fi ε2 Y f 2 X x2i T Y f 2 F f i ,        Γ14 = β1 P33 X f 1 T β2 P33 X f 1 T β3 P33 X f 1 T ,

the residual system (6.11) is asymptotically stable with the H∞ performance γ . Then, the resilient fault detection filter with quantization gain matrices in (6.5) can be obtained with (6.24). Proof Pre- and post-multiplying (6.28) by diag{ I, I, I, P, I, I } and its transpose, respectively, gives ⎡

−P ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ P A(ρ) ⎢ ⎣ C(ρ) θ C y (ρ)

∗ ∗ −γ 2 I ∗ 2 0 −M I Δ2 P B(ρ) P E D(ρ) F θ D y (ρ) 0

∗ ∗ ∗ −P 0 0

∗ ∗ ∗ ∗ −I 0

⎤ ∗ ∗⎥ ⎥ ∗⎥ ⎥ < 0. ∗⎥ ⎥ ∗⎦ −I

(6.30)

To eliminate the effects of uncertainties Δx (k) and Δ f (k) on the residual system (6.11), a two-step approach is adopted. In the first step, we first eliminate the uncertainty Δx (k). Above inequality can be arranged as Ξˆ (ρ) + H e{Wˆ 1 (ρ)Δx V1 (ρ)} < 0.

(6.31)

with   T Wˆ 1T (ρ) = 0 0 0 (P Uˆ (ρ))T (−(D f + Δ D f )X x2 (ρ))T θ X x2 (ρ) ,   T (ρ) ((B f + Δ B f )X x2 (ρ))T , Uˆ T (ρ) = 0 X x1   ˆ (ρ) = Cw −(D f + Δ D f )C(ρ) −(C f + ΔC f ) , C   ˆ (ρ) = −(D f + Δ D f )D(ρ) −(D f + Δ D f )F(ρ) Dw − (D f + Δ D f )F f (ρ) , D

214

6 Fuzzy Fault Detection and Fault-Tolerant …

⎡

⎤ −P ∗ ∗ ∗ ∗ ∗ ⎢ 0 −γ 2 I ∗ ∗ ∗ ∗⎥ ⎢ ⎥ M2 ⎢ 0 0 − I ∗ ∗ ∗⎥ 2 Δ ⎢ ⎥, ˆ Ξ (ρ) = ⎢ ˆ ⎥ ˆ ⎢ P A(ρ) P B(ρ) P E −P ∗ ∗ ⎥ ⎣ C ˆ (ρ) D ˆ (ρ) F 0 −I ∗ ⎦ ¯ ¯ 0 0 −I θ C y (ρ) θ D y (ρ) 0 ⎤ ⎡ Aw 0 0 ˆ (ρ) = ⎣ 0 ⎦, A(ρ) 0 A 0 (B f + Δ B f )C(ρ) A f + Δ A f ⎡ ⎤ 0 0 Bw ˆ (ρ) = ⎣ ⎦. B(ρ) E(ρ) E f (ρ) B (B f + Δ B f )D(ρ) (B f + Δ B f )F(ρ) (B f + Δ B f )F f (ρ)

Applying the Lemmas 1.1 and 1.2 yields   Ξˆ (ρ) + ε1 V1T (ρ)V1 (ρ) ∗ < 0. Wˆ 1T (ρ) −ε1 I

(6.32)

Now, after separating the Δx (k) terms we’re going to do the second step to get rid of uncertainty Δ f (k). Separating the uncertainty Δ f (k) from (6.32) gives 

 Ξ (ρ) + ε1 V1T (ρ)V1 (ρ) ∗ + H e{Wˆ 2 (ρ)Δ f (k)Vˆ2 (ρ)} < 0, −ε1 I W1T (ρ)

(6.33)

with     T (ρ) = 0 0 X Tf 1 (ρ) , Wˆ 2T (ρ) = 0 0 0 (P W21 (ρ))T (−X f 2 (ρ))T 0 0 , W21   Vˆ2 (ρ) = Vˆ21 (ρ) Vˆ22 (ρ) Y f 2 0 0 0 Y f 2 X x2 (ρ) ,     Vˆ21 (ρ) = 0 Y f 2 C(ρ) Y f 1 , Vˆ22 (ρ) = Y f 2 D(ρ) Y f 2 F(ρ) Y f 2 F f (ρ) . Once again, by applying Lemmas 1.1 and 1.2, for a scalar ε2 , (6.33) holds if and only if  ⎤ ⎡ Ξ (ρ) + ε1 V1T (ρ)V1 (ρ) ∗ ˆ2T (ρ)Vˆ2 (ρ) ∗ + ε V 2 T ⎦ < 0, ⎣ −ε1 I W1 (ρ) T ˆ W2 (ρ) −ε2 I is met.

(6.34)

6.1 Fault Detection with Output Quantization

215

⎤ P11 ∗ ∗ ∗ ⎦, and Then let us choose the Lyapunov matrix as P = ⎣ P21 P22 β1 P33 β2 P33 β3 P33 define A f = P33 A f , B f = P33 B f , C f = C f , D f = D f , and the matrix Λρ = r  ρi (ε(k))Λi < 0, the condition (6.34) can be rearranged as the one of Theorem ⎡

i=1

6.4. Remark 6.2 By solving the LMIs conditions of Theorems 6.2 and 6.4, we can find a feasible solution for the robust fault detection filter with quantization and resilient fault detection filter with quantization, respectively. To obtain an optimal H∞ performance bound for corresponding residual system, we can solve the optimization problem min μ, subject to (6.23) or (6.29) with μ = γ 2 . Then, the optimal H∞ performance is γmin =

√

μmin .

Remark 6.3 It should be noted that, in Theorems 6.2 and 6.4, the scalars β1 , β2 , β3 are free parameters, which need to be determined in advance. In that case, (6.23) and (6.29) are LMIs in terms of unknown variables. To handle this problem, we can apply a numerical optimization algorithm, such as the program “fminsearch” in the optimization toolbox of Matlab, then a locally convergent solution to the problem is obtained.

6.1.3 Simulation Example In this section, we provide a numerical example to illustrate the efficiency of the design strategies for robust and resilient fault detection filters with quantization. A two-rule polynomial fuzzy system is adopted, which is described in (6.4). The relevant system parameters are given as follows: 

       0.5 −0.2 0.4 −0.2 0.3 0.3 , A2 = , B1 = B2 = , E1 = E2 = , 0.2 0.4 0.2 0.4 0.2 0.1       0.3 0.6 −0.3 −0.1 0.5 −0.1 Ef1 = E f2 = , C1 = , C2 = , 0.4 0.2 0.2 −0.6 0.2 −0.6       −0.3 0.1 −0.7 0.1 , F1 = F2 = , Ff 1 = Ff 2 = , D1 = D2 = 0.2 −0.3 0.1 −0.5      T   0.1 −0.3 X x11 = , X x12 = , X x21 = X x22 = −0.4 0.1 , Yx11 = −0.3 0.1 , 0.3 −0.3     Yx12 = −1 0.2 , Yx21 = Yx22 = −0.5, Yx31 = Yx32 = −0.3, Yx41 = Yx42 = 0.4 0.2 . A1 =

216

6 Fuzzy Fault Detection and Fault-Tolerant …

The reference residual model in the form (6.8) with  Aw =

       −0.4 −1 −0.2 0.6 0.4 0.2 −0.1 0.4 , Bw = , Cw = , Dw = . 0.4 −0.2 −1 0.2 1 −0.7 0.8 0.2

Case I: Design of Robust Fault Detection Filter with Quantization In the following, we first design the robust fault detection filter with quantization in the form of (6.13) for the above system. For given Δ = 0.01 and M = 10, by solving the LMI in Theorem 6.2 with β1 = −0.7, β2 = 0.2, β3 = 0.2, we have θ = 5489.3, and the gain matrices of robust fault detection filter with quantization in (6.13) as 

   −0.2443 −0.6190 0.0013 −0.0027 , Bf = , 0.3744 −0.1994 0.0093 −0.0004     0.0789 0.0153 0.0011 −0.0005 , Df = . Cf = 0.1679 −0.1184 0.0008 −0.0008 Af =

To demonstrate the effectiveness of the design method, an unknown input is assume as v(k) = 3 cos(0.1k)e−2k , and the input u(k) = 0. The fault signal f (k) is simulated as a pulse of unit amplitude occurred from k1 = 30 to k2 = 60 (and is zero otherwise) as shown in Fig. 6.1. Moreover, some additive assumptions are taken as Δx (k) = 0.3 sin(k), the initial condition x(0) = x f (0) = xr (0) = [ 0 0 ]T , and υ y (k) =

Fig. 6.1 Fault signal f (k)

5489.3 × 0.01 θΔ sin (y(k)) = sin(y(k)). M 10

1 0.8 0.6 0.4 0.2 0 0

20

40

60

80

100

6.1 Fault Detection with Output Quantization Fig. 6.2 Residual r (k) by robust fault detection filter with quantization

217 10

1

-3

0.5 0 -0.5 -1 -1.5 0

20

40

60

80

100

80

100

80

100

(a) r1 (k) 10 -3

1.5 1 0.5 0 -0.5 -1 0

20

40

60

(b) r2 (k) Fig. 6.3 Residual evaluation function r  R M S by robust fault detection filter with quantization

0.25 0.2 0.15 0.1 0.05 0 0

20

40

60

218 Fig. 6.4 Residual r (k) by resilient fault detection filter with quantization

6 Fuzzy Fault Detection and Fault-Tolerant … 0.1

0.05

0

-0.05

-0.1 0

20

40

60

80

100

80

100

80

100

(a) r1 (k) 0.05

0

-0.05

-0.1 0

20

40

60

(b) r2 (k) Fig. 6.5 Residual evaluation function r  R M S by resilient fault detection filter with quantization

0.25 0.2 0.15 0.1 0.05 0 0

20

40

60

6.1 Fault Detection with Output Quantization

219

The response of residual r (k) generated by the robust fault detection filter with quantization is presented in Fig. 6.2. Suppose the time window K = 10. Figure 6.3 gives the response of residual evaluation function r  R M S . Compared with the case of no fault, the dashed line of the residual evaluation function has an obvious jump, which means the designed robust fault detection filter with quantization can effectively detect a possible fault when its occurrence.

Case II: Design of resilient Fault Detection Filter with Quantization Now we design the resilient fault detection filter with quantization for the above uncertain discrete-time system. Consider a non-fragile filter in the form of (6.5) with some uncertainties and assume the known parameters are given as  T  T X f 1 = −0.4 0.5 , X f 2 = −0.3 0.2 ,     Y f 1 = −0.7 −0.1 , Y f 2 = −0.2 0.6 . Then for given Δ = 0.01 and M = 10, solving the LMIs in Theorem 6.4 with β1 = −0.7, β2 = 0.2, β3 = 0.2 gives the gain matrices of resilient fault detection filter with quantization in (6.5) as     −0.3162 −0.3517 0.0018 −0.0047 , Bf = , 0.2430 0.0343 0.0138 −0.0008     0.0611 0.0320 0.0018 −0.0006 , Df = , Cf = 0.0871 0.0007 0.0018 −0.0014 Af =

and θ = 5902.9. Following the same simulation conditions, i.e., conduct the simulation with the same d(k), u(k), f (k), Δx (k), and initial conditions x(0), x f (0), xr (0), we will demonstrate the effectiveness of the designed resilient fault detection filter with quantization by assuming Δ f (k) = 0.5 cos(2k) and υ y (k) =

5902.9 × 0.01 θΔ sin (y(k)) = sin(y(k)). M 10

Then the simulation results of residual r (k) generated by resilient fault detection filter with quantization are provided in Fig. 6.4 and the response of residual evaluation function r  R M S is illustrated in Fig. 6.5. It can be seen from these figures that the designed resilient fault detection filter with quantization can also effectively detect the fault.

220

6 Fuzzy Fault Detection and Fault-Tolerant …

6.2 Fault-Tolerant Control with Input Quantization In this section, we investigate quantized fault-tolerant controller design problem for T–S fuzzy Markov jump system with actuator fault. The main purpose is to design a mode-dependent controller to ensure the stochastic H∞ stability of the closed loop system. And the fault-tolerant controller design scheme is derived based on the mode-dependent idea and robust stochastic stability approach in terms of a set of LMIs.

6.2.1 Problem Formulation The T–S fuzzy Markov jump system is described as the following form: Plant Rule i th : IF ε1 (k) is M1i , ε2 (k) is M2i , and, . . ., and εd (k) is Mdi , THEN ¯ + Diτ (k) w(k), x(k + 1) = Aiτ (k) x(k) + Biτ (k) u(k) z(k) = L iτ (k) x(k),

(6.35)

where i = 1, 2, . . . , r with r being the number of fuzzy rules, M pi are fuzzy sets with p = 1, . . . , d, i = 1, . . . , r , and ε1 (k), ε2 (k), . . ., εd (k) are the premise variables; ¯ ∈ Rn u is the control input, z(k) ∈ Rn z is the regulated x(k) ∈ Rn x is the state, u(k) nw output, and w(k) ∈ R is the disturbance vector belonging to l2 [0, ∞), respectively; Aiτ (k) , Biτ (k) , Diτ (k) and L iτ (k) are known constant real matrices with appropriate dimensions; the subscript τ (k) ∈ S  {1, 2, . . . , },

(6.36)

is a Markov chain with the transition probability matrix Ω = [πmn ] given by P{τ (k + 1) = n|τ (k) = m} = πmn , where 0 ≤ πmn ≤ 1, ∀m, n ∈ S and

 

(6.37)

πmn = 1, ∀m ∈ S.

n=1

System (6.35) can be represented by the following compact form: x(k + 1) = Aτ (k) (ρ)x(k) + Bτ (k) (ρ)u(k) ¯ + Dτ (k) (ρ)w(k), z(k) = L τ (k) (ρ)x(k), where

(6.38)

6.2 Fault-Tolerant Control with Input Quantization

Aτ (k) (ρ) = Dτ (k) (ρ) =

r  i=1 r 

221

ρi (ε(k))Aiτ (k) , Bτ (k) (ρ) = ρi (ε(k))Diτ (k) , L τ (k) (ρ) =

i=1

6.2.1.1

r  i=1 r 

ρi (ε(k))Biτ (k) , (6.39) ρi (ε(k))L iτ (k) .

i=1

Actuator Fault

Now, we will develop a fault-tolerant quantization control strategy. Consider that there exist some types of fault during execution of actuator. Then, we can describe the control input by u(k) ¯ = β(k)u(k), ˆ (6.40) where u(k) ˆ represents the transmitted signal over the communication channel, β(k) = diag{β1 (k), β2 (k), . . ., βn u (k)}, 0 < β p ≤ β p (k) ≤ β p < 1, p = 1, 2, . . . , n u , β p , and β p are known and can be monitored in practice.  β +β β +β  β −β β +β n Further, define β1 = diag 1 2 1 , 2 2 2 , . . . , nu 2 u , β p = −diag 1 2 1 , β 2 −β 2 , 2

...,

β n −β n u u

2

. Then we can describe the faulty input as u(k) ¯ = (β1 + β p Δ f )u(k), ˆ

(6.41)

where Δ f is an uncertainty term satisfying ΔTf Δ f ≤ I . 6.2.1.2

Dynamic Quantizer

Next, a dynamic quantizer [1] is introduced to handle the signal to be transmitted, which is described as the following form: u(k) ˆ = μ(k)q

u(k) , μ(k)

(6.42)

where μ(k) is an adjustable dynamic parameter and q (·) is uniform static quantizer with the quantization range M and quantization error bound Δ satisfying q(ı) − ı ≤ Δ, i f ı ≤ M, q(ı) ≥ M − Δ, i f ı ≥ M.

(6.43)

Furthermore, introduce v(k)  u(k) ˆ − u(k). Then it follows that v(k) ≤ Δμ(k).

(6.44)

222

6 Fuzzy Fault Detection and Fault-Tolerant …

Assuming that there exists a parameter θ > 1 to be designed and the parameter μ(k) will be adjusted dynamically based on quantizer input to reach the required task. And the adjusting scheme is given as √ 2 θ θ u(k) ≤ μ(k) ≤ u(k) . M M

√

(6.45)

According to (6.44) and (6.45), we have v(k) ≤

√ 2Δ θ u(k) . M

(6.46)

Based on the definition of v(k), we can easily obtain that u(k) ˆ = u(k) + v(k).

6.2.1.3

(6.47)

Controller

Consider the mode-dependent feedback controller as the following form u(k) = K τ (k) x(k),

(6.48)

where K τ (k) , τ (k) ∈ S, is the controller gain to be designed later.

6.2.1.4

Closed-Loop System

From (6.38), (6.41), (6.47) and (6.48), we can obtain the following closed-loop system form x(k + 1) = Aτ (k) (ρ)x(k) + Bτ (k) (ρ)v(k) + Dτ (k) (ρ)w(k), z(k) = L τ (k) (ρ)x(k),

(6.49)

where Aτ (k) (ρ) = Aτ (k) (ρ) + Bτ (k) (ρ)(β1 + β p Δ f )K τ (k) , Bτ (k) (ρ) = Bτ (k) (ρ)(β1 + β p Δ f ). In the following, Aτ (k) (ρ), Aτ (k) (ρ), Bτ (k) (ρ), Bτ (k) (ρ), Dτ (k) (ρ), L τ (k) (ρ) and K τ (k) will be represented by Am (ρ), Am (ρ), Bm (ρ), Bm (ρ), Dm (ρ), L m (ρ) and K m , respectively. Before beginning the main results, we firstly introduce the following definitions. Definition 6.1 [4] Under the condition that w(k) = 0, if the following inequality is satisfied: ∞    E x T (k)x(k) < ∞, (6.50) k=0

6.2 Fault-Tolerant Control with Input Quantization

223

system (6.49) is stochastically stable. Definition 6.2 [5] For a given scalar γ > 0, under zero initial condition, if system (6.49) is stochastically stable, and the following inequality holds " !∞ ∞    T  E z (k)z(k) < γ 2 w T (k)w(k) , k=0

(6.51)

k=0

the system in (6.49) is said to be stochastically stable with the H∞ performance index γ . The quantized fault-tolerant controller design problem considered in this section is to design the quantized fault-tolerant controller in (6.48) and the dynamic quantizer in (6.42) such that (1) The quantized fuzzy closed-loop system in (6.49) is stochastically stable when w(k) = 0. (2) The quantized fuzzy closed-loop system in (6.49) has a prescribed H∞ performance γ .

6.2.2 Main Results The stability of system (6.49) with H∞ performance is discussed and the analysis conditions are given in the following theorem. Theorem 6.5 Consider system (6.49). For known parameters M and Δ, the system is stochastically stable with the required H∞ performance γ , if there exist scalar θ > 1, matrices Pm > 0 and U > 0, m ∈ S such that ⎤ −U ∗ ∗ ∗ ∗ ∗ ⎢ 0 −I ∗ ∗ ∗ ∗ ⎥ ⎥ ⎢ 2 ⎢ 0 0 −γ I ∗ ∗ ∗ ⎥ ⎥ < 0, ⎢ ⎢ L m (ρ) 0 0 −I ∗ ∗ ⎥ ⎥ ⎢ 2 M ⎣ Km 0 0 0 − 4Δ ∗ ⎦ 2θ I π Am (ρ) π Bm (ρ) π Dm (ρ) 0 0 − P¯

(6.52)

U − Pm < 0,

(6.53)

⎡

where π = [

  √ √ √ πm1 πm2 . . . πm ]T and P¯ = diag P1−1 , P2−1 , . . . , P−1 .

Proof Adopting Lemma 1.1 can obtain that

224

6 Fuzzy Fault Detection and Fault-Tolerant …

⎡

⎤ ∗ ⎣ 0 ∗ ⎦ 0 − γ2I  πmn Pn [ Am (ρ) Bm (ρ) Dm (ρ) ] + [ Am (ρ) Bm (ρ) Dm (ρ) ]T −U +

4Δ2 θ M2

K mT K m

∗ −I 0

(6.54)

n∈S

+ [ L m (ρ) 0 0 0 ]T [ L m (ρ) 0 0 0 ] < 0. T  Performing congruence transformation with x T (k) v T (k) w T (k) = 0 to (6.54), we have  πmn Pn (Am (ρ)x(k) + Bm (ρ)v(k) + Dm (ρ)w(k))T n∈S

× (Am (ρ)x(k) + Bm (ρ)v(k) + Dm (ρ)w(k)) − x T (k)U x(k) + x T (k)L mT (ρ)L m (ρ)x(k) − γ 2 w T (k)w(k) +

(6.55)

4Δ2 θ T x (k)K mT K m x(k) − v T (k)v(k) < 0. M2

Recalling (6.46), we can obtain v T (k)v(k) ≤

4Δ2 θ T 4Δ2 θ T u (k)u(k) = x (k)K mT K m x(k). 2 M M2

(6.56)

It can be derived that 4Δ2 θ T x (k)K mT K m x(k) − v T (k)v(k) ≥ 0. M2

(6.57)

According to the conditions (6.55) and (6.57), we have (Am (ρ)x(k) + Bm (ρ)v(k) + Dm (ρ)w(k))T



πmn Pn

n∈S

× (Am (ρ)x(k) + Bm (ρ)v(k) + Dm (ρ)w(k))

(6.58)

− x T (k)U x(k) + x T (k)L mT (ρ)L m (ρ)x(k) − γ 2 w T (k)w(k) < 0, which means that (Am (ρ)x(k) + Bm (ρ)v(k) + Dm (ρ)w(k))T



πmn Pn

n∈S

× (Am (ρ)x(k) + Bm (ρ)v(k) + Dm (ρ)w(k)) − x T (k)Pm x(k) + z T (k)z(k) − γ 2 w T (k)w(k) < x T (k)(U − Pm )x(k). Construct the Lyapunov function as:

(6.59)

6.2 Fault-Tolerant Control with Input Quantization

225

V (k) = x T (k)Pm x(k).

(6.60)

Further, we calculate the difference ΔV (k) along the trajectory of system (6.49) and take expectation E {ΔV (k)} = x T (k + 1)



πmn Pn x(k + 1) − x T (k)Pm x(k)

n∈S

= (Am (ρ)x(k) + Bm (ρ)v(k) + Dm (ρ)w(k))T



πmn Pn

n∈S

× (Am (ρ)x(k) + Bm (ρ)v(k) + Dm (ρ)w(k)) − x T (k)Pm x(k). (6.61) Combining (6.59) with w(k) = 0, there is E {ΔV (k)} + z T (k)z(k) < x T (k)(U − Pm )x(k),

(6.62)

which means that E {ΔV (k)} < −λmin (−U + Pm ) x T (k)x(k) < − x T (k)x(k) < 0, where

(6.63)

 = inf {λmin (−U + Pm ) , m ∈ S} > 0.

(6.64)

Summing up the inequality of the both sides from 0 to ∞ can obtain that ∞    1 E x T (k)x(k) < E {V (0) − V (∞)}  k=0

(6.65)

1 E {V (0)}  < ∞.
0, V > 0, K¯ m , and G m , m ∈ S such that ⎡

m ∗ ∗ 11 ⎢ 0 −I ∗ ⎢ 2 ⎢ 0 I 0 −γ ⎢ ⎢ L mi G m 0 0 ⎢ ⎢ K¯ m 0 0 ⎢ ⎢ mi π Bmi β1 π Dmi ⎢ 61 ⎣ 0 0 0 K¯ m I 0

⎤ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ −I ∗ ∗ ∗ ∗ ⎥ ⎥ < 0, 2¯ θ 0 −M I ∗ ∗ ∗ ⎥ ⎥ 4Δ2 0 0 −Y¯ ∗ ∗ ⎥ ⎥ 0 0 ηm (π Bmi β p )T −ηm I ∗ ⎦ 0 0 0 0 −ηm I (6.72) V > Ym , (6.73)

6.2 Fault-Tolerant Control with Input Quantization

227

mi T ¯ for i = 1, ..., r , where m 11 = −G m − G m + V , 61 = π Ami G m + π Bmi β1 K m , Y¯ = diag{Y1 , Y2 , . . . , Y } and the controller gain matrix is obtained by

K m = K¯ m G −1 m .

(6.74)

Proof From Theorem 6.5, we know that system (6.49) is stochastically stable and guarantees the prescribed H∞ performance if inequalities (6.52) and (6.53) hold. Pre- and post-multiplying (6.52) by diag{G mT , I, I, I, I, I } and its transpose, we can obtain that ⎤ ⎡ −G mT U G m ∗ ∗ ∗ ∗ ∗ ⎢ 0 −I ∗ ∗ ∗ ∗ ⎥ ⎥ ⎢ 2 ⎢ I ∗ ∗ ∗ ⎥ 0 0 −γ ⎥ < 0. ⎢ (6.75) ⎢ L m (ρ)G m 0 0 −I ∗ ∗ ⎥ ⎥ ⎢ 2 M ⎣ Km Gm 0 0 0 − 4Δ ∗ ⎦ 2θ I π Am (ρ)G m π Bm (ρ) π Dm (ρ) 0 0 − P¯ From the fact that −G mT U G m ≤ −G m − G mT + U −1 with U > 0, we can derive a sufficient condition for the above inequality as ⎤ −G m − G mT + U −1 ∗ ∗ ∗ ∗ ∗ ⎢ 0 −I ∗ ∗ ∗ ∗ ⎥ ⎥ ⎢ 2 ⎢ I ∗ ∗ ∗ ⎥ 0 0 −γ ⎥ ⎢ ⎢ 0 0 −I ∗ ∗ ⎥ L m (ρ)G m ⎥ ⎢ M2 ⎣ Km Gm 0 0 0 − 4Δ ∗ ⎦ 2θ I π (Am (ρ) + Bm (ρ)β1 K m )G m π Bm (ρ)β1 π Dm (ρ) 0 0 − P¯ ⎡ ⎤ 0 ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥   0 ⎢ ⎥ Δ f Km Gm I 0 0 0 0 (6.76) +⎢ ⎥ 0 ⎢ ⎥ ⎣ ⎦ 0 π Bm (ρ)β p ⎡ ⎤ (K m G m )T ⎢ ⎥ I ⎢ ⎥ ⎢ ⎥ T  0 ⎢ ⎥ Δ 0 0 0 0 0 (π Bm (ρ)β p )T < 0. +⎢ ⎥ f 0 ⎢ ⎥ ⎣ ⎦ 0 0 ⎡

Then a condition for (6.76) can be obtained as follows by using Lemma 1.2,

228

6 Fuzzy Fault Detection and Fault-Tolerant …

⎡

⎤ −G m − G mT + U −1 ∗ ∗ ∗ ∗ ∗ ⎢ 0 −I ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ 2 ⎢ ∗ ∗ ∗ ⎥ 0 0 −γ I ⎢ ⎥ ⎢ 0 0 −I ∗ ∗ ⎥ L m (ρ)G m ⎢ ⎥ 2 M ⎣ Km Gm 0 0 0 − 4Δ ∗ ⎦ 2θ I π (Am (ρ) + Bm (ρ)β1 K m )G m π Bm (ρ)β1 π Dm (ρ) 0 0 − P¯ ⎡ ⎤ 0 ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥  0 −1 ⎢ ⎥ 0 0 0 0 0 (ηm π Bm (ρ)β p )T (6.77) + ηm ⎢ ⎥ 0 ⎢ ⎥ ⎣ ⎦ 0 ηm π Bm (ρ)β p ⎡ ⎤ (K m G m )T ⎢ ⎥ I ⎢ ⎥ ⎢ ⎥  0 −1 ⎢ ⎥ K m G m I 0 0 0 0 < 0. + ηm ⎢ ⎥ 0 ⎢ ⎥ ⎣ ⎦ 0 0 Employing Lemma 1.1 can obtain ⎡

¯m  ∗ ∗ 11 ⎢ 0 −I ∗ ⎢ 2I ⎢ 0 0 −γ ⎢ ⎢ L m (ρ)G m 0 0 ⎢ ⎢ ⎢ Km Gm 0 0 ⎢ m ⎢  ¯ (ρ) π Bm (ρ)β1 π Dm (ρ) ⎢ 61 ⎣ 0 0 0 I 0 Km Gm

∗ ∗ ∗ −I

∗ ∗ ∗ ∗

2

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

⎤

∗ ∗ ∗ ∗

0 − M2 I ∗ ∗ ∗ 4Δ θ 0 0 − P¯ ∗ ∗ 0 0 ηm (π Bm (ρ)β p )T −ηm I ∗ 0 0 0 0 −ηm I

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎥ ⎦

(6.78) m T −1 ¯ ¯m = −G − G + U and  (ρ) = π (A (ρ) + B (ρ)β K )G where  m  m m 1 m m. m 11 61 By defining θ¯ = θ1 , K¯ m = K m G m , V = U −1 and Y = P−1 , the following matrix inequality ⎡

m ∗ ∗ 11 ⎢ 0 −I ∗ ⎢ 2I ⎢ 0 0 −γ ⎢ ⎢ L m (ρ)G m 0 0 ⎢ ⎢ ⎢ K¯ m 0 0 ⎢ m ⎢  (ρ) π Bm (ρ)β1 π Dm (ρ) ⎢ 61 ⎣ 0 0 0 I 0 K¯ m

∗ ∗ ∗ −I

∗ ∗ ∗ ∗

2¯

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

0 − M 2θ I ∗ ∗ ∗ 4Δ 0 0 −Y¯ ∗ ∗ 0 0 ηm (π Bm (ρ)β p )T −ηm I ∗ 0 0 0 0 −ηm I

¯ and (6.73) are obtained, where m 61 (ρ) = π Am (ρ)G m + π Bm (ρ)β1 K m .

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎥ ⎦

(6.79)

6.2 Fault-Tolerant Control with Input Quantization

229

From the result in [6] for the fuzzy parameters, condition (6.72) ensures the above inequality holds. And via simple calculation, (6.53) means that (6.73) holds. The proof is completed.

6.2.3 Simulation Example In this section, the developed result will be demonstrated by the following example for system (6.38) with l = 2 and r = 2. Plant Rule 1:     0.2 0.8 0.8 0.6 A11 = , A21 = , 0 −1 0 −1     0.2 0.2 , B21 = , B11 = 0.2 0.4     0.8 0.6 , D21 = , D11 = 0.7 0.1     L 11 = 0.6 0.4 , L 21 = 0.1 0.3 .



Plant Rule 2:

   1 0 1 0 , A22 = , 0.3 0.8 0.9 0.7     0.5 0.1 = , B22 = , 0.2 0.4     0.4 0.6 = , D22 = , 0.9 0.2     = 0.3 0.5 , L 22 = 0.4 0.2 .

A12 = B12 D12 L 12

The fuzzy weighting function is defined as ρ1 (k) = |sin(k)| , ρ2 (k) = 1 − ρ1 (k).

(6.80)

The transition rate matrix of the Markov jump is given by   0.3 0.7 Ω= . 0.9 0.1

(6.81)

We suppose that the other parameters are β1 = 0.9, β p = 0.1, M = 10 and Δ = 0.01. We employ the MATLAB LMI toolbox to cope with the conditions (6.72) and (6.73) with γ = 2 in Theorem 6.6. Then the desired controller gain can be given via (6.74) as the follows

230

6 Fuzzy Fault Detection and Fault-Tolerant …

Fig. 6.6 Response of the system state x(k)

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0

Fig. 6.7 History of the system mode jump

20

40

60

80

100

20

40

60

80

100

3 2.5 2 1.5 1 0.5 0 0

  K 1 = −1.5881 −0.6324 , K 2

  = −2.6675 −0.1496 .

Next, we give the simulation results to verify the validity of Theorem 6.6.  T Adopt x(0) = 0 0 and w(k) = cos(k)e−0.1k . Then the simulation results are given in Figs. 6.6, 6.7 and 6.8. Figure 6.6 describes the state trajectories of x(k). Figure mode jumps. Figure 6.8 plots the history of ratio of # 6.7 shows the system  ∞ T ∞ T E k=0 z (k)z(k)/ k=0 w (k)w(k) . From all the figures, it can be concluded that the fault-tolerant H∞ control design approach developed in this section is effective.

6.3 Guaranteed Cost Fault-Tolerant Control with Input Quantization

231

Fig. 6.8 History of 0.12 #   ∞ ∞ T (k)z(k)/ T (k)w(k) E z w k=0 k=0

0.1

0.08 0.06 0.04 0.02 0 0

20

40

60

80

100

6.3 Guaranteed Cost Fault-Tolerant Control with Input Quantization In this section, the quantized guaranteed cost fault-tolerant control problem is studied for the T–S fuzzy system. On the one hand, using the two-step method to design the feedback controller of the quantized closed-loop system can obtain the same guaranteed cost performance as the non-signal quantization system. On the other hand, the dynamic parameters of the quantizer are designed by introducing some auxiliary scalars. The design conditions of dynamic quantizer and controller in LMI form are obtained by decoupling the nonlinear terms with some lemmas.

6.3.1 Problem Formulation In this section, T–S fuzzy model is introduced as the following form: Plant Rule i th : IF ε1 (k) is M1i , ε2 (k) is M2i , and, . . ., and εd (k) is Mdi , THEN x(k + 1) = Ai x(k) + Bi u(k),

(6.82)

where x(k) ∈ Rn x and u(k) ∈ Rn u , denote the state vector and control input, respectively. Ai ∈ Rn x ×n x and Bi ∈ Rn x ×n u are system matrices; ε1 (k), ε2 (k), . . . , εd (k) are measurable premise variables; M pi , i = 1, 2, . . . , r, p = 1, 2, . . . , d, are the fuzzy sets, r is the number of fuzzy rules. Then the T–S fuzzy model (6.82) is inferred as follows: x(k + 1) = A(ρ)x(k) + B(ρ)u(k), where

(6.83)

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6 Fuzzy Fault Detection and Fault-Tolerant …

A(ρ) =

r 

ρi (ε(k))Ai , B(ρ) =

i=1

r 

ρi (ε(k))Bi .

i=1

The guaranteed cost controller is considered as state feedback controller, and the controller is given as follows: u(k) = (K + ΔK )x(k),

(6.84)

where K ∈ Rn u ×n x is the controller gain to be determined. ΔK is a time-variable matrix and assumed to be of the form ΔK = FΔk (k)E,

(6.85)

where F and E are known real constant matrices with appropriate dimensions, and Δk (k) describes the system uncertainty which is assumed to satisfy ΔkT (k)Δk (k) ≤ I . Furthermore, the actuator fault is considered in the closed-loop system u(k) ¯ = Ru(k),

(6.86)

, R = α−α ,α where R = R + ΔR × R denotes the actuator fault matrix. R = α+α 2 2 and α are known fault factors, 0 < α < α < 1. ΔR is a time-variable matrix and assumed to be of the form (6.87) ΔR = DΔr (k)N , where D and N are known real constant matrices with appropriate dimensions, and Δr (k) describes the system uncertainty which is assumed to satisfy ΔrT (k)Δr (k) ≤ I . When the actuator fault is taken into consideration, the T–S fuzzy system can be referred as x(k + 1) = A(ρ)x(k) + B(ρ)u(k). ¯ (6.88) Then, based on the system (6.88) and the controller (6.84), the closed-loop system can be given in the following form: x(k + 1) = A(ρ)x(k),

(6.89)

with A(ρ) = A(ρ) + B(ρ)R K˜ , K˜ = K + ΔK . The guaranteed cost index is constructed as ∞    T J= ¯ , (6.90) x (k)Q 1 x(k) + u¯ T (k)Q 2 u(k) k=0

where Q 1 and Q 2 are known positive definite matrices.

6.3 Guaranteed Cost Fault-Tolerant Control with Input Quantization

233

6.3.2 Fault-Tolerant Guaranteed Cost Control Without Quantization For the sake of elaborating the guaranteed cost fault-tolerant control problem, it is necessary to introduce the following definition, which will be significant for the subsequent development of our results. Definition 6.3 [7] When a positive scalar J ∗ and a control law (6.84) can ensure that the closed-loop system is asymptotically stable with J ≤ J ∗ for the cost function (6.90), then J ∗ is the upper bound of the guaranteed cost index and u(k) is the guaranteed cost control law for the system (6.83). Associated with the resulting closed-loop system (6.89), our goal is to find the control law (6.84) such that the following two conditions are satisfied. (1) The closed-loop system (6.89) is asymptotically stable. (2) For the closed-loop system (6.89) with the control law (6.84), the cost function (6.90) meets J ≤ J ∗ , where J ∗ is the upper bound of the guaranteed cost index. In the following, we present a guaranteed cost performance analysis criterion for the system (6.89) with the control law (6.84), i.e., conducting the analysis for (6.89) under the assumption that the controller gain has been given. Theorem 6.7 Consider the system (6.88) and the control input (6.84). For the given positive definite matrices Q 1 and Q 2 , if there exist matrices P > 0 and G > 0 such that the following inequality holds:

AT (ρ)P A(ρ) − P + Q 1 + K˜ T R T Q 2 R K˜ < −G,

(6.91)

then the closed-loop system (6.89) is asymptotically stable with the guaranteed cost performance index J satisfying J < x T (0)P x(0).

(6.92)

Proof Firstly, we shall to show that if the condition (6.91) holds, then the closed-loop system (6.89) is asymptotically stable. Let us choose a Lyapunov function as V (x(k)) = x T (k)P x(k), P > 0,

(6.93)

then one has ΔV (x(k)) = V (x(k + 1)) − V (x(k)) and ¯ Λ = ΔV (x(k)) + x T (k)Q 1 x(k) + u¯ T (k)Q 2 u(k) = x T (k)AT (ρ)P A(ρ)x(k) − x T (k)P x(k) + x T (k)Q 1 x(k) + x T (k) K˜ T R T Q 2 R K˜ x(k)   = x T (k) AT (ρ)P A(ρ) − P + Q 1 + K˜ T R T Q 2 R K˜ x(k).

(6.94)

234

6 Fuzzy Fault Detection and Fault-Tolerant …

Obviously, the condition (6.91) guarantees the establishment of the following inequality: (6.95) Λ < x T (k)(−G)x(k). Because of G > 0, then from above inequalities, we can obtain ΔV (x(k)) < −x T (k)Q 1 x(k) − x T (k) K˜ T R T Q 2 R K˜ x(k).

(6.96)

It is should be noted that ΔV (x(k)) < 0 with Q 1 > 0 and Q 2 > 0 for any x(k) = 0. Hence, if the condition (6.91) is satisfied, we can conclude that the closed-loop system (6.89) is asymptotically stable. In the following, we are in a position to establish the guaranteed cost index. It is noted that, for all x(k) = 0, integrating the inequality (6.96) yields ∞  

 ¯ < 0. ΔV (x(k)) + x T (k)Q 1 x(k) + u¯ T (k)Q 2 u(k)

(6.97)

k=0

Based on the definition (6.90), (6.96) is equivalent to J < V (x(0)) − V (x(∞)).

(6.98)

With V (x(∞)) ≥ 0, the above condition implies that J < V (x(0)) = x T (0)P x(0).

(6.99)

Accordingly, the guaranteed cost function (6.90) satisfies (6.92). In view of Definition 6.3, (6.84) is a guaranteed cost control law for system (6.88) and J ∗ = x T (0)P x(0) is an upper bound of guaranteed cost index for system (6.89). The proof is completed. In the following, we study the guaranteed cost controller design problem based on Theorem 6.7, that is, we will give design condition to determine the controller gain K such that the closed-loop system (6.89) is asymptotically stable with the upper bound of the guaranteed cost control index given as (6.92). Theorem 6.8 Consider the system (6.88) and the control input (6.84). For the given positive definite matrices Q 1 and Q 2 , if there exist matrices T , V , and L, scalars ρ1 > 0 and ρ2 > 0 such that the following LMIs hold: L < Q −1 1 ,

(6.100)

⎤ ∗ Ξ1i ∗ ∗ ⎢ Ξ2 Ξ3 ∗ ∗ ⎥ ⎥ ⎢ ⎣Ξ4i Ξ5 Ξ6 ∗ ⎦ < 0, i = 1, 2, . . . , r, Ξ7i Ξ8 Ξ9 Ξ10

(6.101)

⎡

6.3 Guaranteed Cost Fault-Tolerant Control with Input Quantization

235

where 

     −V ∗ RT 0 0 ρ1 (Bi R F)T , Ξ2 = , , Ξ4i = Ai V + Bi RT −V EV 0 V 0         0 0 0 ρ2 (Bi D)T ρ DT 0 ρ (R F)T 0 , Ξ8 = 2 , Ξ9 = , Ξ7i = , Ξ5 = 1 ρ1 N R F 0 N RT 0 0 0 0 0

Ξ1i =

Ξ3 = − diag{Q −1 2 , L}, Ξ6 = −diag{ρ1 I, ρ1 I }, Ξ10 = −diag{ρ2 I, ρ2 I }.

Then the closed-loop system (6.89) is asymptotically stable with an upper bound of the guaranteed cost control index in (6.92) with P = V −1 . The controller gain in (6.84) can be derived as (6.102) K = T V −1 . Proof Applying the Lemma 1.1 to (6.91) gives ⎤ ∗ −P + Q 1 + G ∗ ⎣ A(ρ) −P −1 ∗ ⎦ < 0. ˜ RK 0 −Q −1 2 ⎡

(6.103)

By defining V = P −1 , pre- and post-multiplying the matrix inequality (6.103) by diag{V, I, I } and applying the Lemma 1.1 to it, one produces ⎡

−V ⎢A(ρ)V ⎢ ⎣ R K˜ V V

⎤ ∗ ∗ ∗ ⎥ −V ∗ ∗ ⎥ < 0. −1 ⎦ 0 −Q 2 ∗ −1 0 0 −(Q 1 + G)

(6.104)

. Obviously, there exist uncertain items ΔK and ΔR in condition (6.104). In order to design the feedback controller, one should remove these uncertain factors. Firstly, separating uncertain items ΔK from (6.104), one obtains ⎡

⎤ −V ∗ ∗ ∗ ⎢(A(ρ) + B(ρ)R K )V −V ∗ ⎥ ∗ ⎢ ⎥ + H e{T Δk (k)2 } < 0, 1 ⎣ ⎦ RK V 0 −Q −1 ∗ 2 −1 V 0 0 −(Q 1 + G) (6.105)     where 1 = 0 (B(ρ)R F)T (R F)T 0 , 2 = E V 0 0 0 . By using Lemma 1.2, one can produce

236

6 Fuzzy Fault Detection and Fault-Tolerant …

⎡

⎤ −V ∗ ∗ ∗ ∗ ∗ ⎢(A(ρ) + B(ρ)R K )V −V ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ −1 ⎢ RK V 0 −Q 2 ∗ ∗ ∗ ⎥ ⎢ ⎥ < 0. ⎢ V 0 0 −(Q 1 + G)−1 ∗ ∗ ⎥ ⎢ ⎥ ⎣ 0 ρ1 (B(ρ)R F)T ρ1 (R F)T 0 −ρ1 I ∗ ⎦ EV 0 0 0 0 −ρ1 I

(6.106) Secondly, use the same way to eliminate uncertain items ΔR. By defining new variables T = K V and L = (Q 1 + G)−1 , and according to the definition of fuzzy basis functions, condition (6.101) can be obtained. It should be noted that G = L −1 − Q 1 > 0 can be converted to (6.100) by adopting Lemma 1.6. The proof is completed. Remark 6.4 In Theorem 6.8, the two-step method first eliminates the uncertain term ΔK of the controller (6.84), and then eliminates the time-variable matrix ΔR defined in (6.84), so that the design conditions can be solved by standard LMI technique. Theorem 6.8 has offered a solution for designing the controller (6.84) to ensure that the system (6.89) is asymptotically stable with the upper bound of guaranteed cost control index as (6.92). We can acutely observe that the upper bound of J is related to the initial state. Therefore, we desire to find an upper bound J  , which is independent of the initial state. As for the initial state, it is supposed to be arbitrary, but belongs to the set [8] S = {x(0) ∈ Rn x : x(0) = H υ0 , υ0T υ0 ≤ 1}, where H is a given matrix (or a matrix to be determined). In the light of above assumption, we have x T (0)P x(0) = υ0T H T P H υ0 .

(6.107)

Next, selecting a scalar δ > 0 satisfy H T P H < δ I,

(6.108)

based on the Lemma 1.1, it can be rearranged as

i.e.,

  −δ I ∗ < 0, H −P −1

(6.109)

  −δ I ∗ < 0. H −V

(6.110)

6.3 Guaranteed Cost Fault-Tolerant Control with Input Quantization

237

From the above analysis, it can be concluded that

Define

x T (0)P x(0) < δ.

(6.111)

J  = δ.

(6.112)

Thereby, for (6.92), there is the establishment of J ≤ J  = δ. Then, Theorem 6.8 can be deduced as the following one. Theorem 6.9 Consider the system (6.88) and the control input (6.84). For the given positive definite matrices Q 1 and Q 2 , if there exist matrices V , L, and T , positive scalars δ, ρ1 , ρ2 satisfy (6.100), (6.101), and (6.110), then the closed-loop system (6.89) is asymptotically stable and the guaranteed cost index J has the upper bound J  . The controller gain K can be derived as (6.102). Remark 6.5 For Theorem 6.9, by applying LMI toolbox in MATLAB to solve (6.100), (6.101), and (6.110), the controller (6.84) with the suitable upper bound of the cost performance (6.112) can be derived. From conditions (6.110) and (6.112), it is easy to obtain that the selection of matrix H will have a great influence on the upper bound of guaranteed cost index. It can be given based on the system performance requirement or determined by solving the controller design problem in Theorem 6.9, and in this section, the former one will be adopted. Remark 6.6 Based on Theorem 6.9, we can also optimize the upper bound of guaranteed cost performance to formulate an optimal guaranteed cost control law via minimizing J  . Then, the guaranteed cost control issue is transformed into a convex optimization problem as min J  = δ, subject to LMIs (6.100), (6.101), and (6.110),

(6.113)

such that the closed-loop system (6.89) is asymptotically stable and the guaranteed cost index J has the optimal upper bound J  . The controller gain K can be derived as (6.102).

6.3.3 Fault-Tolerant Guaranteed Cost Control with Input Quantization Based on (6.88), the T–S fuzzy system with input quantization can be inferred as follows: x(k + 1) = A(ρ)x(k) + B(ρ)u(k), ˜ (6.114)

238

6 Fuzzy Fault Detection and Fault-Tolerant …

where u(k) ˜ = Ru q (k), u q (k) ∈ Rn u is the quantized control input which is defined as

u(k) , (6.115) u q (k) = f θ (u(k)) = θ (k) f θ (k) where f θ (·) is a dynamic quantizer with general form which is a one-parameter family of quantizer given in [1], with θ (k) denoting the dynamic quantizer parameter. Assume that there exist scalars M > 0 and Δ > 0 satisfying the following conditions:  f θ (ι) − ι ≤ θ Δ, i f ι ≤ θ M,

(6.116)

 f θ (ι) − ι > θ Δ, i f ι > θ M.

(6.117)

When the quantizer is unsaturated, condition (6.116) provides a bound for the quantization error. Condition (6.117) gives a way for detecting the possibility of saturation. θ Δ and θ M represent the quantization error bound and the quantization range of the quantizer f θ (·), respectively. As a consequence, we can rewrite the system model (6.114) as x(k + 1) = A(ρ)x(k) + B(ρ)Ru(k) + B(ρ)Re(k),

(6.118)

    u(k) − . Further, based on (6.118) and (6.84), the quanwhere e(k) = θ (k) f u(k) θ(k) θ(k) tized closed-loop system can be given as: x(k + 1) = A(ρ)x(k) + B(ρ)Re(k),

(6.119)

where A(ρ) has been defined in (6.89). The guaranteed cost index is introduced for constructing a quantized feedback control law (6.115) as J=

∞  

 x T (k)Q 1 x(k) + u˜ T (k)Q 2 u(k) ˜ .

(6.120)

k=0

Remark 6.7 It should be pointed out that in the closed-loop system (6.119), there exists additional item B(ρ)Re(k) caused by the effect of quantization. Therefore, the effect of quantization should be removed to obtain the same guaranteed cost performance as the one without quantization. It is important to note that the performance function (6.120) includes the quantized control input, which will cause great difficulty to design guaranteed cost controller once the input signal is quantized. Next, we shall propose an effective quantization design strategy so that the same guaranteed cost performance as the one without quantization can be still achieved despite the input signal is quantized.

6.3 Guaranteed Cost Fault-Tolerant Control with Input Quantization

239

Theorem 6.10 Suppose that the LMIs (6.100) and (6.101) in Theorem 6.8 are satisfied, and the range M of quantizer is chosen to be large enough so that: M>

1 , ξ

(6.121)

and the quantized control input (6.115) is designed with on-line adjusting strategy for parameter θ (k) as: 1 u(k) ≤ θ (k) ≤ ξ u(k) , M where ξ=

λmin (G) − h

, # Δr2 β1 + β12 + β2 (λmin (G) − h)

(6.122)

(6.123)

0 < h < λmin (G),

(6.124)

$ 2   β1 = r3 +  Q 2  r12 r2 ,

(6.125)

$ 2   β2 = r4 +  Q 2  r12 ,

(6.126)

with G = L −1 − Q 1 and r1 , r2 , r3 , r4 satisfying R < r1 ,    ˜  K  < r2 ,

(6.128)

    (A(ρ) + B(ρ)R K˜ )T P B(ρ) < r3 ,

(6.129)

  T  B (ρ)P B(ρ) < r4 .

(6.130)

(6.127)

Then, the controller (6.115) with the adjusting strategy (6.122) can ensure that the closed-loop system (6.119) is asymptotically stable with the guaranteed cost performance index J satisfying (6.131) J < x T (0)P x(0).

240

6 Fuzzy Fault Detection and Fault-Tolerant …

Proof According to the same definition in (6.93), we have ˜ Λq =ΔV (x(k)) + x T (k)Q 1 x(k) + u˜ T (k)Q 2 u(k) T    = A(ρ)x(k) + B(ρ)Re(k) P A(ρ)x(k) + B(ρ)Re(k) − x T (k)P x(k) + x T (k)Q 1 x(k) + u˜ T (k)Q 2 u(k) ˜   =x T (k) A(ρ)T P A(ρ) − P + Q 1 + K˜ T R T Q 2 R K˜ x(k)

(6.132)

− u¯ T (k)Q 2 u(k) ¯ + 2x T (k)AT (ρ)P B(ρ)e(k) + e T (k)B T (ρ)P B(ρ)e(k) + u˜ T (k)Q 2 u(k). ˜ Under the condition (6.91), thus, from (6.132), we have Λq 0 which implies that On the other hand, based on (6.122) and (6.123), we know that θ (k) ≤

Δ(β1 +

#

λmin (G) − h β12 + β2 (λmin (G) − h))

x(k) ,

< 0.

(6.145)

which means x(k) −

β1 +

#

β12 + β2 (λmin (G) − h) λmin (G) − h

θ (k)Δ ≥ 0,

(6.146)

then (6.144) yields Λq < −h x(k)2 < 0.

(6.147)

Accordingly, the guaranteed cost function (6.120) satisfies (6.131). In view of Definition 6.3, the quantized control strategy (6.115) is a guaranteed cost control law for the system (6.114) and J ∗ = x T (0)P x(0) is an upper bound of guaranteed cost index for the closed-loop system (6.119). The proof is completed. Remark 6.8 For the system (6.114), the matrices A(ρ), R and K˜ contain uncertain parameters ΔK and ΔR which means that the determination of scalars r1 , r2 , r3 , and r4 in the conditions (6.127), (6.128), (6.129), and (6.130) is hard. Thus, further efforts in the following are needed to transform the determination problem into a feasible computation problem. According to the properties of 2-norm of a matrix A, we know that the condition A T A < μ2 I holds if and only if A < μ. Based on the conclusion, the conditions (6.127), (6.128), (6.129), and (6.130) are equivalent to the following inequalities: (R + ΔR × R)T (R + ΔR × R) < r12 I,

(6.148)

6.3 Guaranteed Cost Fault-Tolerant Control with Input Quantization

243

(K + ΔK )T (K + ΔK ) < r22 I,

(6.149)

 T   AT (ρ)P B(ρ) AT (ρ)P B(ρ) < r32 I,

(6.150)



B T (ρ)P B(ρ)

T 

 B T (ρ)P B(ρ) < r42 I.

(6.151)

Then, applying the Lemmas 1.1 and 1.2 to (6.148), (6.149), (6.150), and (6.151) yields, respectively ⎡

⎤ −r12 I ∗ ∗ ∗ ⎢ R −I ∗ ∗ ⎥ ⎢ ⎥ < 0, ⎣ 0 η1 D T −η1 I ∗ ⎦ 0 0 −η1 I NR

(6.152)

⎡ 2 ⎤ −r2 I ∗ ∗ ∗ ⎢ K −I ∗ ∗ ⎥ ⎢ ⎥ < 0, ⎣ 0 ∗ ⎦ η2 F T −η2 I E 0 0 −η2 I

(6.153)

⎡

⎤ −r32 I ∗ ∗ ∗ ∗ ∗ ⎢(A(ρ) + B(ρ)R K )T P B(ρ) −I ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ T T T 0 −η3 I ∗ ∗ ∗ ⎥ F R B (ρ)P B(ρ) ⎢ ⎥ < 0, ⎢ 0 −η3 I ∗ ∗ ⎥ 0 η3 E ⎢ ⎥ ⎣ 0 0 0 −η4 I ∗ ⎦ D T B T (ρ)P B(ρ) 0 −η4 I 0 η4 N R K η 4 N R F 0 (6.154)   −r42 I ∗ < 0. (6.155) B T (ρ)P B(ρ) −I Define l1 = r12 , l2 = r22 , l3 = r32 , and l4 = r42 , we know that above conditions can be guaranteed by the following LMIs: ⎡

⎤ −l1 I ∗ ∗ ∗ ⎢ R −I ∗ ∗ ⎥ ⎢ ⎥ < 0, ⎣ 0 η1 D T −η1 I ∗ ⎦ 0 0 −η1 I NR

(6.156)

⎡

⎤ ∗ ∗ ∗ −l2 I ⎢ K −I ∗ ∗ ⎥ ⎢ ⎥ < 0, ⎣ 0 ∗ ⎦ η2 F T −η2 I E 0 0 −η2 I

(6.157)

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6 Fuzzy Fault Detection and Fault-Tolerant …

⎡

⎤ −l3 I ∗ ∗ ∗ ∗ ∗ ⎢(A + B R K )T P B −I ∗ ∗ ∗ ∗ ⎥ i j ⎢ i ⎥ ⎢ ⎥ T T T ⎢ F R Bi P B j 0 −η3 I ∗ ∗ ∗ ⎥ ⎢ ⎥ < 0, i, j = 1, 2, . . . , r, ⎢ 0 −η3 I ∗ ∗ ⎥ 0 η3 E ⎢ ⎥ ⎣ 0 0 0 −η4 I ∗ ⎦ D T BiT P B j 0 −η4 I 0 η4 N R K η 4 N R F 0

(6.158) 

 −l4 I ∗ < 0, i, j = 1, 2, . . . , r. BiT P B j −I

(6.159)

Now, based on above analysis and Theorem 6.9, the following result of guaranteed cost control for the system (6.114) subject to quantization is proposed, which ensures the asymptotic stability of system (6.114) with the upper bound of guaranteed cost performance J  . Theorem 6.11 Suppose that the LMI conditions in Theorem 6.9 can be solved, i.e., there exist matrices V , L, and T , scalar δ satisfying the matrix inequalities (6.100), (6.101), and (6.110) to maintain the guaranteed cost performance J  . The range M of quantizer is chosen to be large enough as (6.121), and the on-line adjusting strategy for √ quantizer’s√dynamic parameter θ (k) is designed as (6.122) with √ √ r1 = l1 , r2 = l2 , r3 = l3 , and r4 = l4 satisfying (6.156), (6.157), (6.158), and (6.159), respectively. Then, the quantized feedback controller (6.115) with the adjusting strategy (6.122) can ensure that the quantized feedback closed-loop system (6.119) is asymptotically stable with the same guaranteed cost performance index J  as the one without quantization. To facilitate the presentation of design for quantized feedback controller (6.115) with dynamic parameter θ (k), the following algorithm is given first based on Theorem 6.11. Algorithm 6.1 Step 1. Solve (6.100), (6.101), and (6.110) to obtain the upper bound of guaranteed cost index J  ; Step 2. Compute the values of controller gain K with (6.102); Step 3. Compute the values of l1 , l2 , l3 and l4 with (6.156), (6.157), (6.158), and (6.159); √ √ √ Step 4.√Compute the values of r1 , r2 , r3 , and r4 by r1 = l1 , r2 = l2 , r3 = l3 , and r4 = l4 ; Step 5. Compute the values β1 and β2 with (6.125) and (6.126); Step 6. Select h according to (6.124); Step 7. Compute the value of ξ with (6.123). Then, as in [9], according to the condition (6.122), give the adjusting rule of θ (k) as follows: The adjusting rule: Case a: If ξ u(k) − M1 u(k) ≥ 1, θ (k) = p = floor(ξ u(k)) with p ∈ N+ ;

6.3 Guaranteed Cost Fault-Tolerant Control with Input Quantization

245

Case b-i: If ξ u(k) − M1 u(k) < 1 and M1 u(k) < 1 ≤ ξ u(k), θ (k) = p = 1 with p ∈ N+ ; Case b-ii: If ξ u(k) − M1 u(k)