Fault Tolerant Control for Switched Linear Systems [1 ed.] 978-3-319-15161-8, 978-3-319-15162-5

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Studies in Systems, Decision and Control 21

Dongsheng Du Bin Jiang Peng Shi

Fault Tolerant Control for Switched Linear Systems

Studies in Systems, Decision and Control Volume 21

Series editor Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland e-mail: [email protected]

About this Series The series "Studies in Systems, Decision and Control" (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control- quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. More information about this series at http://www.springer.com/series/13304

Dongsheng Du · Bin Jiang Peng Shi

Fault Tolerant Control for Switched Linear Systems

ABC

Dongsheng Du Faculty of Electronic and Electrical Engineering Huaiyin Institute of Technology Huaian Jiangsu China

Peng Shi School of Electrical and Electronic Eng. School of Engineering and Science The University of Adelaide and Victoria University Adelaide, Melbourne Australia

Bin Jiang College of Automation Engineering Nanjing University of Aeronautics and Astronautics Nanjing China

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-319-15161-8 ISBN 978-3-319-15162-5 (eBook) DOI 10.1007/978-3-319-15162-5 Library of Congress Control Number: 2015930509 Springer Cham Heidelberg New York Dordrecht London c Springer International Publishing Switzerland 2015  This work is subject to copyright. All rights are reserved 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, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com)

Preface

Switched linear system belongs to hybrid system, which includes of several linear subsystems (continuous or discrete) and a rule that orchestrates the switching among them. Many real engineering systems can be modeled as such systems, such as circuit system, network system, and chemical systems etc.. To study the framework of switched linear systems can construct a bridge between the linear systems and the uncertain systems or nonlinear systems. On one hand, switching among linear systems may produce complex system behaviors such as chaos and multiple limit cycles. On the other hand, switched linear systems are relatively easy to handle as many powerful tools from linear and multi-linear analysis are available to cope with these systems. Moreover, the study of switched linear systems provides additional insights into some long-standing and sophisticated problems, such as intelligent control, adaptive control, and robust analysis and control. In the past decades, the control and synthesis for switched linear systems have been intensively investigated and have attracted increasingly more attention. The literature grew exponentially and quite a number of fundamental concepts and powerful tools have been developed from various disciplines. Despite the rapid progress made so far, many fundamental problems are still either unexplored or less well understood. In particular, there still lacks a unified framework that can cope with the issues of fault diagnosis and fault tolerant control for switched linear systems. This motivated us to write the related work. The book presents several fundamental problems of fault diagnosis and fault tolerant control for switched linear systems. By using switched Lyapunov function method, linear matrix inequality and average dwell-time technique, a basic theoretical framework is formed towards the issues of fault diagnosis and fault tolerant control for switched linear system. The book contains eleven chapters which exploit several independent yet related topics in detail. Chapter 1 investigates the problem of fault detection filter design for discretetime switched systems with interval time-varying delays. By constructing a novel switched Lyapunov functional, a new criterion is obtained for the residual system. Based on this, a sufficient condition for the existence of the above filters is established in terms of linear matrix inequalities.

VI

Preface

Chapter 2 deals with the problem of fault detection for discrete-time switched systems with intermittent measurements. The stochastic variable is assumed to be a Bernoulli distributed white sequence appearing in measured output. Attention is focused on designing a fault detection filter such that, for any control input and unknown inputs, the estimation error between the residual and the fault is minimized in the sense of H∞ norm. Chapter 3 considers the problem of fault detection for continuous-time switched systems under asynchronous switching. The filter is assumed to be unmatched with the original system. An efficient condition has been given to construct the filter under the switching signal with average dwell time. In Chapter 4, we address sensor fault estimation and accommodation for discretetime switched linear systems. By using descriptor observer method and switched Lyapunov function technique, efficient conditions to design the fault observer are obtained via a linear matrix inequality formulation. The problems of sensor fault estimation and compensation approaches for timedelay switched systems are investigated in chapter 5. First, a novel time-delay switched descriptor state observer is proposed to estimate both the state and sensor fault. Then, based on the estimation of the sensor fault, an efficient fault-tolerant operation can be realized via sensor fault compensation. Chapter 6 addresses an adaptive fault diagnosis observer is designed for nonlinear switched systems. The proposed method improves rapidity and accuracy on fault estimation. Chapter 7 investigates the problems of actuator fault estimation and accommodation for discrete-time switched systems with state delay. By using reduced-order observer method and switched Lyapunov function technique, a fault estimation algorithm is achieved for the delayed discrete-time switched system with actuator fault. Then based on the obtained online fault estimation information, a switched dynamic output feedback controller is employed to compensate for the effect of faults by stabilizing the closed-loop systems. Chapter 8 focuses on the problem of active fault-tolerant control for switched systems with time delay. By utilizing the fault diagnosis observer, an adaptive fault estimate algorithm is proposed. Based on the fault estimation information, an observer-based fault-tolerant controller is designed to guarantee the stability of the closed-loop system. Chapter 9 considers the problem of fault estimation and accommodation for a class of switched systems with time-varying delay. An adaptive fault estimation algorithm is proposed to estimate the fault. Meanwhile, a delay-dependent criteria is obtained with the purpose of reducing the conservatism of the fault estimation algorithm design. On the basis of fault estimation, an observer-based fault tolerant controller is designed to the stability of the closed-loop system. Chapter 10 deals with the problem of reliable control for discrete time systems with actuator failures. By using an average dwell time method, an observer-based feedback controller is developed in terms of linear matrix inequality such that the resulting closed-loop system is exponentially stable.

Preface

VII

At last, in Chapter 11, the perspectives of FTC for switched systems are concluded and predicated. Huaiyin Institute of Technology, Huaian, Jiangsu, China Dongsheng Du Nanjing University of Aeronautics and Astronautics, Nanjing, China Bin Jiang The University of Adelaide, Adelaide, Australia Peng Shi May, 2014

Acknowledgements

The contents included in this book are an outgrowth of our academic research activities over the past several years. This book was partially supported by National Natural Science Foundation of China (Grants No. 61104116) and Jiangsu Province Natural Science Fundation for Youths (BK20140457).

Contents

1

2

3

Fault Detection for Discrete-Time Switched Systems with Interval Time-Varying Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Fault Detection Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Fault Weighting System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Residual Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Residual System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Fault Detection Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Fault Detection Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Fault Detection Filter Design . . . . . . . . . . . . . . . . . . . . . . . . 1.4 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 3 4 4 4 5 6 10 13 16

Fault Detection for Discrete-Time Switched Systems with Intermittent Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 H∞ Fault Detection Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 A Reluctance Motor Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 18 21 28 31

Fault Detection for Continues-Time Switched Systems under Asynchronous Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Fault Detection Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 H∞ Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Fault Detection Filter Design . . . . . . . . . . . . . . . . . . . . . . . .

33 33 34 37 37 42

XII

Contents

3.4 An Illustrate Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44 45

Sensor Fault Estimation and Accommodation for Discrete-Time Switched Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 4.3 Fault Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Fault Estimate without Unknown Disturbance . . . . . . . . . . 4.3.2 Robust Fault Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Fault Accommodation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 An Illustrate Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 48 50 50 53 55 56 60

Sensor Fault Estimation and Compensation for Switched Systems with State Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Descriptor Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Fault Estimate without Unknown Disturbance . . . . . . . . . . 5.2.2 Robust Fault Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Extension to Nonlinear Case . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Fault Compensation Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 62 63 69 71 75 76 78

6

Fault Estimation for Nonlinear Continuous-Time Switched Systems 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Fault Diagnosis Observer Design . . . . . . . . . . . . . . . . . . . . . 6.3.2 Fault Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 80 82 82 83 86 88

7

Actuator Fault Estimation and Accommodation for Discrete-Time Switched Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Without State Delay Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . 7.2.2 Fault Estimation Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Fault Accommodation Design . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 An Illustrate Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 With State Delay Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Problem Statements and Preliminaries . . . . . . . . . . . . . . . . . 7.3.2 Fault Estimation Design . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89 89 90 90 91 97 102 103 103 104

4

5

Contents

XIII

7.3.3 Fault Accommodation Design . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 An Illustrate Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 115 117

Active Fault Tolerant Control for Switched Systems with Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Fault Detection and Fault Estimation . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Fault Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Observer-Based Fault Estimation . . . . . . . . . . . . . . . . . . . . . 8.4 Fault Accommodation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 119 121 122 122 123 129 131 134

Fault Estimation and Accommodation for Switched Systems with Time-Varying Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Fault Diagnosis Observer Design . . . . . . . . . . . . . . . . . . . . . 9.3.2 Fault Estimation Algorithm Design . . . . . . . . . . . . . . . . . . . 9.3.3 Fault Accommodation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135 135 137 138 138 139 145 147 151

10 Observer-Based Reliable Control for Discrete Time Switched Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Preliminaries and Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 10.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 153 155 157 164 165

11 Conclusions and Future Research Direction . . . . . . . . . . . . . . . . . . . . .

167

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169

8

9

Acronyms

R Rn Rn×m I Is×s O A Am×n A−1 AT det(A) rank(A) λ (A) λmin (A) λmax (A) A≥0 A>0 A≤0 A 1 is the number of subsystems. At an arbitrary discrete time k, σ (k) is dependent on k or xk , or both, or other switching rules. As in [28], we assume that the sequence of subsystems in switching signal σ is unknown a priori, but its instantaneous value is available in real time. A(σ (k)), Ad (σ (k)), Eu (σ (k)), Ed (σ (k)), E f (σ (k)), C(σ (k)), Cd (σ (k)), Fu (σ (k)), Fd (σ (k)), and Ff (σ (k)) are constant matrices with appropriate dimensions for all σ (k) ∈ ψ . We denote the matrices associated with σ (k) = i by A(σ (k)) = Ai , Ad (σ (k)) = Adi , Eu (σ (k)) = Eui , Ed (σ (k)) = Edi , E f (σ (k)) = E f i , C(σ (k)) = Ci , Cd (σ (k)) = Cdi , Fu (σ (k)) = Fui , Fd (σ (k)) = Fdi , and Ff (σ (k)) = Ff i .

1.2.1

Fault Detection Filter

The first step of FD is residual generation, which must be sensitive to the fault. This is often realized by utilizing FD observers or filters. As we know, disturbances often inevitably appear in many systems, the residual must also be capable of distinguish faults from exogenous signals. H∞ filter can not only describe the estimated signal accurately but also suppress the disturbance effectively. Thus, for the system S with disturbance, the following switched FD filter is constructed as a residual generator: xˆk+1 = A f (σ (k))xˆk + B f (σ (k))yk

(1.4)

rk = C f (σ (k))xˆk + D f (σ (k))yk

(1.5)

where xˆk ∈ Rn f is the filter’s state, rk is the so-called residual signal. The matrices A f (σ (k)), B f (σ (k)), C f (σ (k)), and D f (σ (k)) are the filter parameters to be determined. Remark 1.1. In this chapter, the filter’s output is selected as residual signal to detect the fault. On another hand, one can get that the designed filter is model-dependent, which means that every subsystem has its own filter formulation.

4

1.2.2

1 Fault Detection for Discrete-Time Switched Systems

Fault Weighting System

Since the main purpose of this chapter is fault detection, it is not necessary to estimate the fault fk . Sometimes one is more interested in the fault signal of a certain frequency interval, which can be formulated as the weighted fault f¯(z) = W f (z) f (z) with W f (z) being a given stable weighting matrix. One minimal realization of f¯(z) = W f (z) f (z) is supposed to be x¯k+1 = AW x¯k + BW fk f¯k = CW x¯k + DW fk

(1.6) (1.7)

where x¯k ∈ RnW is the state of weight fault, f¯k ∈ Rl is the weight fault. AW , BW , CW , and DW are known constant matrices.

1.2.3

Residual Evaluation

The second step for FD is the residual evaluation stage including an evaluation function and a threshold. After the residual signal being constructed, a residual evaluation value will be computed through a prescribed evaluation function, and it will be compared with a predefined threshold. When the evaluation value is larger than the threshold, an alarm of fault is generated. In this paper, the threshold Jth and residual evaluation function JL (r) are selected as 

k0 +L

JL (r) = rk 2 = Jth =

sup ϑk ∈l2 , f k =0

∑

1 2

rkT rk

k=k0

rk 2

(1.8) (1.9)

where k0 denotes the initial evaluation time instant. L is the evaluation time steps. For the detailed discussion of the threshold Jth , readers are referred to [2] and [23]. Based on this, the occurrence of a fault can be detected by comparing JL (r) and Jth according the following test:

1.2.4

JL (r) > Jth =⇒ a f ault occurs

(1.10)

JL (r) ≤ Jth =⇒ no f aults

(1.11)

Residual System

Denoting ek = rk − f¯k and augmenting the model of system (1.1)-(1.2) to include the state of (1.4)-(1.5) and (1.6)-(1.7), the following augmented system can be obtained:

1.3 Fault Detection Filter Design

5

 σ (k))x˜k + A d (σ (k))x˜k−d + Bω (σ (k))ωk x˜k+1 = A( k  σ (k))x˜k + Cd (σ (k))x˜k−d + D  ω (σ (k))ωk ek = C( k

where

 σ (k)) = A( d (σ (k)) = A Bω (σ (k)) =

 σ (k)) = C( Cd (σ (k)) =  ω (σ (k)) = D

(1.12) (1.13)

T T   x˜k = x¯Tk xTk xˆTk , ωk = uTk ϑkT fkT

⎤ 0 0 AW ⎦, ⎣ 0 A(σ (k)) 0 0 B f (σ (k))C(σ (k)) A f (σ (k)) ⎡ ⎤ 0 0 0 ⎣0 0 ⎦, Ad (σ (k)) 0 B f (σ (k))Cd (σ (k)) 0 ⎡ ⎤ 0 0 BW ⎣ ⎦, Eu (σ (k)) Ed (σ (k)) E f (σ (k)) B f (σ (k))Fu (σ (k)) B f (σ (k))Fd (σ (k)) B f (σ (k))Ff (σ (k))   −CW D f (σ (k))C(σ (k)) C f (σ (k)) ,   0 D f (σ (k))Cd (σ (k)) 0 ,   D f (σ (k))Fu (σ (k)) D f (σ (k))Fd (σ (k)) D f (σ (k))Ff (σ (k)) − DW . ⎡

(1.14)

In order to minimize the effect of the disturbance and improve the sensitivity of the residual to the fault, the FD filter design can be formulated as an H∞ filter problem. i.e., the problem to be addressed in this work is expressed as follows: to develop filter (1.4)-(1.5) for system (1.1)-(1.2) such that the augmented system (1.12)-(1.13) • is asymptotically stable when ωk = 0; and • under zero-initial condition, the minimum of γ is made small in the feasibility of ek 2 < γ, γ > 0 ωk 2 =0 ωk 2 sup

1.3

(1.15)

Fault Detection Filter Design

In this section, the FD filter design problem is solved for the discrete-time switched. The FD analysis problem is first solved, and then, based on this, a full-rank detection filter is designed (i.e. n f = n + nW ).

6

1 Fault Detection for Discrete-Time Switched Systems

1.3.1

Fault Detection Analysis

In this subsection, we assume that the FD filter matrices are known, and the conditions are investigated under which the residual system is asymptotically stable and guarantees the performance defined in (1.15), which are summarized in the following theorem, in terms of LMIs. Theorem 1.1. For a given scalar γ > 0, system (1.12)-(1.13) is asymptotically stable with H∞ performance γ if there exit matrices Pi , Qi , and Ri , such that the following LMIs hold, ⎤ ⎡ i − I)T Ri CT τ Qi T Pj τ¯ (A −Pi 0 0 A i i T ⎢ ∗ −Ql 0 A T Pj τ¯ ATdi Ri Cdi 0 ⎥ ⎥ ⎢ di ⎢ ∗ ∗ −γ 2 I BT P T T  τ¯ Bω i Ri D 0 ⎥ ⎥ ⎢ ωi j ωi ⎢ ∗ ∗ (1.16) 0 0 0 ⎥ ∗ −Pj ⎥ 0, system (1.12)-(1.13) is asymptotically stable with H∞ performance γ if there exit matrices Pi , Qi , Ri , and G, such that the following LMIs hold, ⎡

−Pi ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

T G τ¯ (Ai − I)T G 0 0 A i T  G −Ql 0 A τ¯ ATdi G di T G ∗ −γ 2 I Bω τ¯ BTω i G i T ∗ ∗ Pj − (G + G ) 0 τ¯ [Ri − (G + GT )] ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

where τ = d¯ − d + 1, τ¯ = 21 (d¯ + d)(d¯ − d + 1).

⎤ CiT τG T ⎥ 0 Cdi ⎥ ⎥ T  Dω i 0 ⎥ ⎥

P1i P2i Q1i Q2i R1i R2i > 0, Qi = > 0, > 0, Ri = there exit matrices Pi = T T T P2i P3i R2i R3i Q2i Q3i A f i , B f i , C f i , D f i , U, X, and W , for any i, j, l ∈ ψ , satisfying the following LMIs: ⎡ ⎤ 0 Ξ14i jl Ξ15i jl Ξ16i jl Ξ17i jl Ξ11i jl 0 ⎢ ∗ Ξ22i jl 0 Ξ24i jl Ξ25i jl Ξ26i jl 0 ⎥ ⎥ ⎢ ⎢ ∗ ∗ −γ 2 I Ξ34i jl Ξ35i jl Ξ36i jl 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ Ξ44i jl 0 0 0 ⎥ (1.29) ⎢ ⎥ Jth for L = 17, which means that the fault fk can be successfully detected seventeen time steps after its occurrance.

16

1.5

1 Fault Detection for Discrete-Time Switched Systems

Summary

In this chapter, the problem of FD for discrete-time switched system with time delays is investigated. The time delays are assumed to be time-varying and bounded. A switched FD filter has been designed such that the residual system is stable with H∞ performance. An example is offered to illustrate the application of the results.

Chapter 2

Fault Detection for Discrete-Time Switched Systems with Intermittent Measurements

This chapter deals with the problem of FD for discrete-time switched systems with intermittent measurements. The stochastic variable is assumed to be a Bernoulli distributed white sequence appearing in measured output. Attention is focused on designing a FD filter such that, for any control input and unknown inputs, the estimation error between the residual and the fault is minimized in the sense of H∞ norm. By employing a Lyapunov-like function and ADT technique, a sufficient condition for the existence of such a filter is exploited in terms of certain LMIs. Finally, an example is provided to illustrate the effectiveness of the proposed approach.

2.1

Introduction

In recent years, the process of FDI for dynamic systems has been of considerable interest, and fruitful model-based fault detection results have been obtained in several excellent papers [8, 9, 38, 39, 40, 41, 42] and the books [4, 43, 44]. Among these model-based approaches of FDI, the basic idea is to use state observer or filter to construct a residual signal and, based on this, to determine a residual evaluation function to compare with a predefined threshold. When the residual evaluation function has a value larger than the threshold, an alarm of faults is generated. On another hand, it is well known that control inputs, unavoidable unknown inputs, and faults are coupled in many industrial systems, which are potential sources of false alarm. This means that FDI systems have to be robust to control inputs and unknown inputs, and at the same time enhance the sensitivity to the faults. Therefore, it is of great significance to design a model-based FD system. In reviewing the development of the theories and techniques for different FDI system designs, there are a number of results have been obtained in designing FDI system. For examples, in [45] and [46], an H∞ filter formulation of robust FDI has been considered for uncertain LTI systems and Markovian jump linear systems. The issue of H∞ FD filter design for linear discrete-time systems with time delays is investigated in [41, 47].

c Springer International Publishing Switzerland 2015  D. Du, B. Jiang, and P. Shi, Fault Tolerant Control for Switched Linear Systems, Studies in Systems, Decision and Control 21, DOI: 10.1007/978-3-319-15162-5_2

17

18

2 Fault Detection for Discrete-Time Switched Systems

On another research direction, switched systems have attracted increasing attention in the literature of control problems due to their great significance in theory and practical applications. Switched systems belong to hybrid systems, which consist of a class of subsystems and a switching signal. The switching signal specifies which subsystem will be activated along the trajectory at each instant of time. Presently, a lot of achievements have been achieved on the control of switched system [26, 48, 49]. Especially for recent years, it is a hot topic to study the problem of switched systems with time delay. For examples, [48] studied the stabilization and H∞ control problem for switched systems with time delay. The Kalman filter design issue for switched systems with time delay was considered in [30, 31]. It is noted that the time delay mentioned above are assumed to be deterministic. However, they may occur in a randomly varying way in many practical applications as pointed out in [50]. There has been some attention on the research of dynamics systems with intermittent measurements [32, 51]. Until now, to the best of our knowledge, the problem of FD for discrete-time switched systems with intermittent measurements has not been considered yet. Research is still under way into the development of an effective solutions for this issue, which motivates us to study this interesting and challenging issue. In this chapter, the problem of FD for discrete-time switched systems with intermittent measurements is solved. Firstly, by using Lyapunov-like function and ADT technique, a sufficient condition for the H∞ FD filter is exploited in the formation of LMI. Then, based-on the obtained condition, an desired FD filter is constructed. Finally, to demonstrate the feasibility and effectiveness of the proposed method, a simulation example is included. The rest of this chapter is organized as follows. In Section 2.2, system descriptions and problem formulation are presented. A sufficient condition on the existence of a FD filter for discrete-time switched systems with intermittent measurements is derived in terms of LMIs, and the parameters of the desired filter are constructed by solving the corresponding LMIs in Section 2.3. To demonstrate the validity of the proposed approach, an example is given in Section 2.4 which is followed by a conclusion in Section 2.5.

2.2

Problem Formulation

The FD problem with intermittent measurements is shown in Figure 2.1. The data missing phenomenon occurs intermittently from the switched systems to the FD filter. In the following, we will deal with the fault detection problem mathematically. In this paper, we will consider the following class of discrete-time switched systems: xk+1 = A(σk )xk + Bu (σk )uk + Bd (σk )dk + G f (σk ) fk

(2.1)

2.2 Problem Formulation

19

Fig. 2.1 Fault detection filter with data loss

where xk ∈ Rn is the state, uk ∈ Rm , dk ∈ R p , and fk ∈ Rl are the control input, unknown input, and the fault respectively, which belong to l2 [0, ∞). σk is a piecewise constant function of time, called a switching signal, which takes its values in the finite set ψ = {1, · · · , N}, N > 1 is the number of subsystems. At an arbitrary discrete time k, σk is dependent on time k or state xk , or both, or other switching rules. The measurement with random delays is given by yk = C(σk )xk + Eu(σk )uk + Ed (σk )dk + Ff (σk ) fk yck = θk yk

(2.2) (2.3)

where yk ∈ Rq is the output, yck ∈ Rq is the measured output. The stochastic variable θk is a Bernoulli distributed white sequence take the values in the set of {0, 1}, and satisfies the following mathematical expectation values Prob{θk = 1} = E {θk } = ρ

(2.4)

Prob{θk = 0} = E {1 − θk } = 1 − ρ

(2.5)

where ρ ∈ [0, 1] is a known constant. A(σk ), Bu (σk ), Bd (σk ), G f (σk ), C(σk ), Eu (σk ), Ed (σk ), and Ff (σk ) are constant matrices with appropriate dimensions for all σk = i ∈ ψ . We denote the matrices associated with σk = i ∈ ψ by A(σk ) = Ai , Bu (σk ) = Bui , Bd (σk ) = Bdi , G f (σk ) = G f i , C(σk ) = Ci , Eu (σk ) = Eui , Ed (σk ) = Edi , and Ff (σk ) = Ff i . Remark 2.1. From above description, one can get that the process of missing data considered is assumed to satisfy a Bernoulli distributed process. The probability distribution of the process can be estimated based on experimental measurements of data transmitting from output of the switched system to the input of the fault detection filter, which can be achieved by sending a sequence of indexed data through the communication medium and measuring the data dropout characteristics. The inferred statistics of the Bernoulli process will then be used to design the FD filter. As we know, an FDI system is constituted by a residual generator and evaluation stage including an evaluation function and a threshold. For the purpose of residual generation, the following FD filter is constructed as a residual generator: xˆk+1 = A f (σk )xˆk + B f (σk )yck

(2.6)

rˆk = C f (σk )xˆk + D f (σk )yck

(2.7)

20

2 Fault Detection for Discrete-Time Switched Systems

where xˆk ∈ Rn f is the filter’s state, rˆk ∈ Rl is the so-called residual signal. The matrices A f (σk ), B f (σk ), C f (σk ), and D f (σk ) are the filter parameters to be determined. In order to detect the fault fk more convenient and fast, sometimes one is more interested in the fault signal of a certain frequency interval, which can be formulated as the weighted fault fˆ(z) = W f (z) f (z) with W f (z) being a given stable weighting matrix. One minimal realization of fˆ(z) = W f (z) f (z) is supposed to be x¯k+1 = AW x¯k + BW fk fˆk = CW x¯k + DW fk

(2.8) (2.9)

where x¯k ∈ RnW is the state of weight fault, fˆk ∈ Rl is the weight fault. AW , BW , CW , and DW are known constant matrices. Denoting ek = rˆk − fˆk and augmenting the model of system (2.1)-(2.2) to include the state of (2.6)-(2.7) and (2.8)-(2.9), the following augmented system can be obtained: 1 (σk , ρ )x˜k + Bω (σk , ρ )ωk + θ˜k B1ω (σk , ρ )ωk (2.10)  σk , ρ )x˜k + θ˜k A x˜k+1 = A(  σk , ρ )x˜k + θ˜kC1 (σk , ρ )x˜k + D  1ω (σk , ρ )ωk (2.11)  ω (σk , ρ )ωk + θ˜k D ek = C(

 T where θ˜k = θk − ρ , E {θ˜k } = 0, E {θ˜k θ˜k } = ρ (1 − ρ ), x˜k = x¯Tk xTk xˆTk , ωk =  T T T T uk dk f k .  σk , ρ ) A( Bω (σk , ρ ) B1ω (σk , ρ )  σk , ρ ) C(  ω (σk , ρ ) D

1ω (σk , ρ ) D

⎤ ⎤ ⎡ 0 0 AW 0 0 0 1 (σk , ρ ) = ⎣ 0 A(σk ) 0 ⎦, A 0 0 ⎦, =⎣ 0 0 ρ B f (σk )C(σk ) A f (σk ) 0 B f (σk )C(σk ) 0 ⎤ ⎡ 0 0 BW ⎦, Bd (σk ) G f (σk ) Bu (σk ) =⎣ ρ B f (σk )Eu (σk ) ρ B f (σk )Ed (σk ) ρ B f (σk )Ff (σk ) ⎤ ⎡ 0 0 0 ⎦, 0 0 0 =⎣ B f (σk )Eu (σk ) B f (σk )Ed (σk ) B f (σk )Ff (σk )     = −CW ρ D f (σk )C(σk ) C f (σk ) , C1 (σk , ρ ) = 0 D f (σk )C(σk ) 0 ,   = ρ D f (σk )Eu (σk ) ρ D f (σk )Ed (σk ) ρ D f (σk )Ff (σk ) − DW ,   = D f (σk )Eu (σk ) D f (σk )Ed (σk ) D f (σk )Ff (σk ) . (2.12) ⎡

Remark 2.2. The introduction of W f (z) can limit the frequency ranges of interest, but the system performance would be improved, and the frequency characteristics required to reflect the emphases of different frequency ranges could be captured. Therefore,we are interested in minimizing the error between the residual signal and the weighted fault signal, rather than the error between the residual and the original fault signals.

2.3 H∞ Fault Detection Filter Design

21

Now, the problem of FD filter design can be formulated as an H∞ filter problem: to develop filter (2.6)-(2.7) for system (2.1)-(2.2) such that the augmented system (2.10)-(2.11) stable when ωk = 0 and, under zero-initial condition, the minimum of γ is made small in the feasibility of sup E { ωk 2 =0

ek 2 } < γ, γ > 0  ωk  2

(2.13)

After designing the residual generator, the last step to a successful FD is the residual evaluation stage including an evaluation function and a threshold. In this paper, the threshold Jth and residual evaluation function JL (ˆr ) are selected as 

k0 +L

JL (ˆr) = ˆrk 2 = Jth =

∑

1 2

rˆkT rˆk

k=k0

sup ˆrk 2

(2.14) (2.15)

d∈l2 , f =0

where k0 denote the initial evaluation time instant, and L is the evaluation time window. Based on this, the occurrence of faults can be detected by comparing JL (ˆr ) and Jth according the following test: JL (ˆr ) > Jth =⇒ with f aults =⇒ alarm

(2.16)

JL (ˆr ) ≤ Jth =⇒ no f aults

(2.17)

Remark 2.3. From the relationship in (14) and (15), the fault can be successfully detected. Note that the length of the evaluation window L is limited since it is desired that the faults will be detected as early as possible, while an evaluation of residual signal over the whole time range is not practical. This point has been mentioned in [45, 18]. Remark 2.4. Based on Monte Carlo method to design FD may be an additional approach to get the target [52]. Compared with H∞ performance design, sometimes, much better results may be obtained by using Monte Carlo simulations.

2.3

H∞ Fault Detection Filter Design

In this section, a sufficient condition on the existence of the FD filter will be exploited in the formation of LMI. The following definition and lemmas are given for the convenes of the proof. Definition 2.1. [53] For any switching signal σ (k) and any k2 > k1 > 0, let Nσ (τ , k) denote the number of switchings of σ (k) on an interval (k1 , k2 ). If Nσ (τ , k) ≤ N0 +

k2 − k1 τa

22

2 Fault Detection for Discrete-Time Switched Systems

holds for a given N0 ≥ 0 and τa > 0, then the constant τa is called the average dwelltime and N0 the chattering bound. Remark 2.5. Definition 2.1 means that if there exists a positive number τa such that a switching signal has the ADT property, the average time interval between consecutive switching is at least τa . This is a kind of slowly switching signals, which is less conservative than dwell-time switching signal and arbitrary switching signals. As commonly used in the literature, for convenience, we choose N0 = 0 in this paper.

Fig. 2.2 Lyapunov-like function

Lemma 2.1. [54] Consider the discrete-time switched system xk+1 = fσ (xk ), σ ∈ ψ and let 0 < α < 1, β > −1, and µ > 1 be given constants. Suppose that there exist C 1 functions Vσ (k) : Rn → R, σ ∈ ψ , and two class K∞ functions κ1 and κ2 such that κ1 (|xk |) ≤ Vi (xk ) ≤ κ1 (|xk |)

∆ Vi (xk ) ≤



−α Vi (xk ), ∀k ∈ T↓ (kl , kl+1 ) β Vi (xk ), ∀k ∈ T↑ (kl , kl+1 )

and ∀(σ (kl ) = i, σ (kl−1 ) = j) ∈ N × N, i = j, Vi (xk ) ≤ µ V j (xk ) then, letting β˜ = 1 + β and α˜ = 1 − α , the system is globally uniformly asymptotically stable (GUAS) for any switching signal with ADT

τa > τa∗ = −{Tmax [ln β˜ − ln α˜ ] + ln µ }/ ln α˜

(2.18)

where T↓ (kl , kl+1 ) and T↑ (kl , kl+1 ) represent the unions of the dispersed intervals during which Lyapunov function is increasing and decreasing within the interval [kl , kl+1 ), Tmax = max T↑ (kl+1 − kl ), l ∈ N.

2.3 H∞ Fault Detection Filter Design

23

Lemma 2.2. [55] Consider the discrete-time switched system xk+1 = fσ (xk ), σ ∈ ψ and let 0 < α < 1, β > −1, γi > 0, and µ > 1 be given constants. Suppose that there exist C 1 functions Vσ (k) : Rn → R, σ ∈ ψ , ∀(σ (kl ) = i, σ (kl−1 ) = j) ∈ N × N, i = j, such that Vi (xk ) ≤ µ V j (xk ) and  −α Vi (xk ) − zTk zk + γi2 ωkT ωk , ∀k ∈ T↓ (kl , kl+1 ) ∆ Vi (xk ) ≤ ∀k ∈ T↑ (kl , kl+1 ) β Vi (xk ) − zTk zk + γi2 ωkT ωk , then the system is GUAS for any switching signal with ADT (2.18) and has an l2 gain no greater than γ = max{γi }. Based on the existing results in Lemmas 2.1 and 2.2, bounded real lemma result for the augmented system (2.10)-(2.11) is derived in the following theorem. Theorem 2.1. Given scalars 0 < α < 1, β > −1, µ ≥ 1 and γi > 0, system (2.10)(2.11) under switching signal σk with average dwell time τa is GUAS when ωk = 0 and, under zero-initial conditions, guarantees the performance index (2.13) for all nonzero ωk ∈ l2 [0, ∞), if there exist positive definite matrices Pi and matrix Ω , such that the following LMIs hold: Pi ≤ µ Pj ,

∀i, j ∈ ψ

(2.19)

 Θi Ξ i ϒ1i = 0, β > 0, µ1 ≥ 1, and µ2 ≥ 1, if there exist matrices Pi > 0, Pi j > 0, for i = j, i, j ∈ N, such that Pj ≤ µ1 Pi j ,

Pi j ≤ µ2 Pi ;

(3.11)

⎤ T Pi + Pi A i + α Pi Pi Bi CT A i i ⎣  Ti ⎦ < 0 ∗ −γ 2 I D ∗ ∗ −I ⎡

⎡ T i j − β Pi j Pi j Bi j Ai j Pi j + Pi j A ⎣ ∗ −γ 2 I ∗ ∗ ′

(3.12)

⎤ CiTj Tij ⎦ < 0 D −I

(3.13)

then the system Σ (or Σ ) is asymptotically stable with H∞ performance γ for any switching signal with ADT satisfying

τa > τa∗ =

ln(µ1 µ2 ) , ζ∗

T − (t0 ,t) β + ζ ∗ ≥ , 0 < ζ∗ < α T + (t0 ,t) α − ζ ∗

(3.14)

where T − (t0 ,t) or T + (t0 ,t) denotes the total matched or mismatched period during [t0 ,t]. ′

Proof: We first establish the stability of the the system Σ (or Σ ). To this end, assume that ω (t) = 0. When t ∈ [t0 ,t1 ) ∪ [tk−1 + ∆k−1 ,tk ), k = 2, 3, 4, · · ·, the augmented system can be written as in (3.3). Consider the following Lyapunov function: Vi (t) = x˜T (t)Pi x(t) ˜

(3.15)

38

3 Fault Detection for Continues-Time Switched Systems

Then, along the trajectory of system (3.3), we have T ˙˜ + α x˜T (t)Pi x(t) V˙i (t) + α Vi (t) = x˙˜ (t)Pi x(t) ˜ + x˜T (t)Pi x(t) ˜ T T   = x˜ (t)(Ai Pi + Pi Ai + α Pi )x(t) ˜

(3.16)

then by (3.12), we get

V˙i (t) + α Vi (t) ≤ 0

(3.17)

V˙i (t) ≤ −α Vi (t)

(3.18)

it follows that

then during the matched period, Vi (t) satisfy  t0 ≤ t < t1 V (t )e−α (t−t0 ) , (3.19) Vi (t) ≤ i 0 Vi (tk−1 + ∆k−1 )e−α (t−tk−1 −∆k−1 ) , tk−1 + ∆k−1 ≤ t < tk , k = 2, 3, · · · When t ∈ [tk ,tk + ∆k ), k = 1, 2, 3, · · ·, the augmented system can be written as in (3.4). Consider the following Lyapunov function: Vi j (t) = x˜T (t)Pi j x(t) ˜

(3.20)

Then, along the trajectory of system (3.4), we have T ˙˜ − β x˜T (t)Pi j x(t) V˙i j (t) − β Vi j (t) = x˙˜ (t)Pi j x(t) ˜ + x˜T (t)Pi j x(t) ˜ T T i j Pi j + Pi j A i j − β Pi j )x(t) ˜ = x˜ (t)(A

(3.21)

then by (3.12), we get

V˙i j (t) − β Vi j (t) ≤ 0

(3.22)

V˙i j (t) ≤ β Vi (t)

(3.23)

it follows that

then during the unmatched period, Vi j (t) satisfy Vi j (t) ≤ Vi j (tk )eβ (t−tk ) , tk ≤ t < tk + ∆k , k = 1, 2, 3, · · ·

(3.24)

Let t1 ,t2 , · · · ,tk , · · · denote the switching instant of σ (t) over the interval [t0 ,t]. Consider the following piecewise Lyaounov functional candidate for system Σ in (3.3) and (3.4):  Vi (t) = x˜T (t)Pi x(t), ˜ t ∈ [t0 ,t1 ) ∪ [tk−1 + ∆k−1 ,tk ), k = 2, 3, 4, · · · V (t) = (3.25) Vi j (t) = x˜T (t)Pi j x(t), ˜ t ∈ [tk ,tk + ∆k ), k = 1, 2, 3, · · ·

3.3 Fault Detection Filter Design

39

When t ∈ [tk ,tk + ∆k ), k = 1, 2, 3, · · ·, and with the condition in (3.11), (3.19) and (3.24), we have V (t) = Vσ ′ (tk−1 +∆k−1 )σ (tk ) (t) ≤ Vσ ′ (tk−1 +∆k−1 )σ (tk ) (tk )eβ (t−tk ) ≤ µ2Vσ (tk−1 ) (tk− )eβ (t−tk ) ≤ µ2Vσ (tk−1 ) (tk−1 + ∆k−1 )eβ (t−tk )−α (t−tk−1 −∆k−1 ) ≤ µ2 (µ1Vσ ′ (tk−2 +∆k−2 )σ (tk−1 ) [(tk−1 + ∆k−1 )− ])eβ (t−tk )−α (t−tk−1 −∆k−1 ) N

(tk−1 ,t) Nσ ′ (t) (tk−2 +∆ k−2 ),t) µ1 Vσ ′ (tk−2 +∆k−2 )σ (tk−1 ) (tk−1 ) β (t−tk )−α (t−tk−1 −∆ k−1 )

= µ2 σ (t) ×e

≤ ························ N

≤ µ2 σ (t)

(t0 ,t)

N ′

µ1 σ (t)

(t0 ,t)

Vσ (t0 ) (t0 )e−α (t1 −t0 ) eβ (t−tk +∆k−1 +···+∆1 ) ×

e−α [(tk −tk−1 −∆k−1 )+(tk−1 −tk−2 −∆k−2 )+···+(t2 −t1 −∆1 )] N

= µ2 σ (t)

(t0 ,t)

N

µ1 σ (t)

(t0 ,t)−1

Vσ (t0 ) (t0 )eβ (t−tk +∆k−1+···+∆1 ) ×

e−α [(tk −tk−1 −∆k−1 )+(tk−1 −tk−2 −∆k−2 )+···+(t2 −t1 −∆1 )+(t1 −t0 )]

(3.26)

By (3.14), we have

β T + (t0 ,t) − α T − (t0 ,t) ≤ −ζ ∗ (t − t0 )

(3.27)

then (3.26) and (3.27) imply that V (t) ≤ (µ1 µ2 )Nσ (t) (t0 ,t) µ1−1Vσ (t0 ) (t0 )eβ T ≤ (µ1 µ2 )N0 +

t−t0 τa

µ1−1Vσ (t0 ) (t0 )e−ζ

= µ1−1Vσ (t0 ) (t0 )eN0 ln(µ1 µ2 ) e−(ζ

+ (t ,t)−α T − (t ,t) 0 0

∗ (t−t ) 0

∗ − ln(µ1 µ2 ) )(t−t ) 0 τa

(3.28)

Therefore, if the ADT satisfies (3.14), we conclude V (t) converges to zero as t −→ ′ ∞. Then the stability of system Σ (or Σ ) can be deduced. For any nonzero ω (t) ∈ L2 [0, ∞) and zero initial condition x(0) ˜ = 0. When t ∈ [t0 ,t1 ) ∪ [tk−1 + ∆k−1 ,tk ), k = 2, 3, 4, · · ·, the augmented system can be written as in (3.3). Consider the Lyapunov function as in (3.15) and set Γ (t) = −eT (t)e(t) + γ 2 ω T (t)ω (t), one has V˙i (t) + α Vi (t) − Γ (t) = ξ T (t)Θi ξ (t) where ξ (t) =



x˜T (t)

T ω T (t)

By (3.12), it follows that

(3.29)

 T Pi + PiA i + α Pi + CT Ci Pi Bi + CT D i A i i i , Θi =  Ti D  Ti Ci i . BTi Pi + D −γ 2 I + D

V˙i (t) < −α Vi (t) + Γ (t)

(3.30)

40

3 Fault Detection for Continues-Time Switched Systems

From Lemma 1, one has ! ⎧ −α (t−t0 ) e Vi (t0 ) + tt0 e−α (t−s)Γ T (s)Γ (s)ds, t0 ≤ t < t1 ⎪ ⎪ ⎨ −α (t−tk−1 −∆k−1 ) Vi (tk−1 + ∆k−1 ) e !t Vi (t) ≤ (3.31) −α (t−s) Γ T (s)Γ (s)ds, t ⎪ e + k−1 + ∆ k−1 ≤ t < tk , ⎪ tk−1 +∆ k−1 ⎩ k = 2, 3, · · ·

When t ∈ [tk ,tk + ∆k ), k = 1, 2, 3, · · ·, the augmented system can be written as in (3.4). Consider the Lyapunov function as in (3.20), one has V˙i (t) − β Vi(t) − Γ (t) = ξ T (t)Θi j ξ (t) where

(3.32)



T Pi j + Pi j A i j − β Pi j + CT Ci j Pi j Bi j + CT D  A ij ij ij ij Θi j = T 2 T T  i j  i jCi j i jD −γ I + D Bi j Pi j + D

.



By (3.13), it follows that

V˙i j (t) < β Vi j (t) + Γ (t)

(3.33)

From Lemma 3.1, one has Vi j (t) ≤ eβ (t−tk )Vi j (tk ) +

 t tk

eβ (t−s)Γ T (s)Γ (s)ds, tk ≤ t < tk + ∆k

(3.34)

Consider the piecewise Lyapunov function as in (3.25), when t ∈ [tk ,tk + ∆k ), k = 1, 2, 3, · · ·, it follows from (3.31) and (3.34) that V (t) ≤ Vσ ′ (tk−1 +∆k−1 )σ (tk ) (tk )eβ T ≤ µ2Vσ (tk−1 ) (tk− )eβ T

+ (t ,t)−α T − (t ,t) k k

+ (t ,t)−α T − (t ,t) k k

+

 t

 t

tk−1 +∆ k−1

eβ T

eβ T

t

 t

eβ T

+ (s,t)−α T − (s,t)

eβ T

+ (s,t)−α T − (s,t)

+ (s,t)−α T − (s,t)

Γ T (s)Γ (s)ds

Γ T (s)Γ (s)ds

tk

≤ µ2 µ1Vσ ′ (tk−2 +∆k−2 )σ (tk−1 ) (tk−1 )eβ T + µ2 µ1 + µ2

 tk−1 +∆ k−1

 tk

eβ T

+ (t − k−1 ,t)−α T (tk−1 ,t)

+ (s,t)−α T − (s,t)

Γ T (s)Γ (s)ds

tk−1

tk−1 +∆ k−1

eβ T

Γ T (s)Γ (s)ds

+ (t − k−1 +∆ k−1 ,t)−α T (tk−1 +∆ k−1 ,t)

+ (s,t)−α T − (s,t)

+ (s,t)−α T − (s,t)

Γ T (s)Γ (s)ds

tk

tk

≤ µ2Vσ (tk−1 ) (tk−1 + ∆k−1 )eβ T + µ2

+

+

Γ T (s)Γ (s)ds

3.3 Fault Detection Filter Design  t

eβ T

N

(t0 ,t)

+

+ (s,t)−α T − (s,t)

41

Γ T (s)Γ (s)ds

tk

≤ ··············· ≤ µ2 σ (t) N

+µ2 σ (t) N

+ µ2 σ (t) N

+ µ2 σ (t) N

= µ2 σ (t) +

 t t0

N ′

µ1 σ (t)

(t0 ,t)

N ′

µ1 σ (t)

(tk−1 ,t)

(tk ,t)

(t0 ,t)

(t0 ,t)

N ′

N ′

eβ T

(tk−1 +∆ k−1 ,t)

+ (t ,t)−α T − (t ,t) 0 0

+ (s,t)−α T − (s,t)

(tk−1 +∆ k−1 ,t)

 t

 t

tk−1 +∆ k−1

eβ T

Γ T (s)Γ (s)ds + · · ·

eβ T

+ (s,t)−α T − (s,t)

+ (s,t)−α T − (s,t)

Γ T (s)Γ (s)ds

Γ T (s)Γ (s)ds

tk

(t0 ,t)

µ1 σ (t)

N ′

(t0 ,t)

N

(s,t)

µ1 σ (t)

N ′

µ2 σ (t)

 t1 t0

µ1 σ (t)

µ1 σ (t)

Vσ (t0 ) (t0 )eβ T

Vσ (t0 ) (t0 )eβ T

+ (t ,t)−α T − (t ,t) 0 0

(s,t) β T + (s,t)−α T − (s,t)

e

Γ T (s)Γ (s)ds

(3.35)

Under the initial condition x(t0 ) = 0, we get V (t) ≤ =

 t

t0  t

N

µ2 σ (t) e

(s,t)

N ′

µ1 σ (t)

(s,t) β T + (s,t)−α T − (s,t)

e

Γ T (s)Γ (s)ds

Nσ (t) (s,t) ln µ2 +Nσ ′ (t) (s,t) ln µ1 β T + (s,t)−α T − (s,t)

e

Γ T (s)Γ (s)ds (3.36)

t0

Multiplying both sides of (3.36) by e e

≤

−[Nσ (t) (t0 ,t) ln µ2 +Nσ ′ (t) (t0 ,t) ln µ1 ]

yields

−[Nσ (t) (t0 ,t) ln µ2 +Nσ ′ (t) (t0 ,t) ln µ1 ]

 t

V (t)

e

Nσ (t) (t0 ,s) ln µ2 +Nσ ′ (t) (t0 ,s) ln µ1 β T + (s,t)−α T − (s,t)

e

Γ T (s)Γ (s)ds

(3.37)

t0

When t ∈ [tk ,tk + ∆k ), k = 1, 2, 3, · · ·, by the condition in (3.14), we have −[Nσ (t) (t0 ,t) ln µ2 + Nσ ′ (t) (t0 ,t) ln µ1 ] = −[Nσ (t) (t0 ,t)(ln µ2 + ln µ1 ) − ln µ1 ] t − t0 ≥ −[ ∗ ln(µ1 µ2 ) − ln µ1 ] τa ∗ = −ζ (t − t0 ) + ln µ1 By (3.27), (3.37) and (3.38), we have e− ζ ≤e ≤

∗ (t−t )+ln µ 1 0

V (t)

−[Nσ (t) (t0 ,t) ln µ2 +Nσ ′ (t) (t0 ,t) ln µ1 ]

 t t0

V (t)

eNσ (t) (t0 ,s) ln µ2 +Nσ ′ (t) (t0 ,s) ln µ1 e−ζ

∗ (t−s)

Γ T (s)Γ (s)ds

(3.38)

42

3 Fault Detection for Continues-Time Switched Systems

≤

 t

Γ T (s)Γ (s)ds

(3.39)

t0

which means  t

eT (s)e(s)ds ≤ γ 2

 t

ω T (s)ω (s)ds

(3.40)

t0

t0

when t → ∞, then the proof is completed.

3.3.2

Fault Detection Filter Design

Theorem 3.2. Given constants α > 0, β > 0, µ1 ≥ 1, and µ2 ≥ 1, if there exist positive-definite matrices Xi , Ri , P22i , P11i j , P22i j , and any matrices P12i , P12i j , A f i , B f i , C f i , and D f i , for i = j, i, j ∈ N, such that  

P11i j P12i j Xi P12i >0 (3.41) > 0, T P T P P12i P12i 22i j 22i j

   

X j P12 j P11i j P12i j P11i j P12i j Xi P12i ≤ µ µ ≤ , 1 2 T P T P T P T P P12 P12i P12i P12i 22i j 22 j j 22i j j 22i j ⎡

ϕ11 ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

T ⎤ T Ri B1i Ri D1i CiT D f i + C f i ϕ12 T ⎥ ϕ22 Xi B1i + B f i B2i Xi D1i + B f i D2i CiT D f i ⎥ ⎥ T t0 where t0 is the time of fault occurring. That is, f (t) is zero prior to the failure time t ≤ t0 and is f0 (t) after the failure occurs t > t0 . For the purpose of this note, we give the following assumption: Assumption 1: The matrix D f is full collum rank. In order to estimate the state vector x(t) and the sensor fault vector f (t), an augmented system is employed using the descriptor observer technique. As a result, the switched system with sensor fault is transformed as follows: N

N

i=1

i=1

¯ + ∑ δi (t)B¯ i u(t) + B¯ d d(t) + D¯ f x f (t) E¯ x(t ¯ + 1) = ∑ δi (t)A¯ i x(t)

(4.5)

y(t) = C¯ x(t) ¯ = C˜ x(t) ¯ + x f (t)

(4.6)

where  x(t) ¯ = xT (t)

 I 0 ¯ E= , 00

 Bd B¯ d = , 0

xTf (t)

T

x f (t) = D f (t) f (t), 

 Ai 0 Bi ¯ ¯ , Bi = , Ai = 0 −I 0

     0 ¯ ˜ ¯ C I C 0 . C= , C= , Df = I

,

(4.7)

50

4 Sensor Fault Estimation and Accommodation for Discrete-Time Switched Systems

In this study, we construct the following state observer for the system (4.5): ˆ¯ = z(t) + Ly(t) x(t)

(4.8)

y(t) ˆ = Cx(t) ˆ

(4.9) N

N

Kz(t + 1) = ∑ δi (t)Fi z(t) + ∑ δi (t)B¯ i u(t)

(4.10)

i=1

i=1

ˆ¯ ∈ Rn+p is the state estimation, and z(t) ∈ Rn+p is an auxiliary middle where x(t) state variable. Matrices Fi ∈ R(n+p)×(n+p), K ∈ R(n+p)×(n+p), and L ∈ R(n+p)×p are the observer gains to be determined. Substituting ˆ¯ − Ly(t) z(t) = x(t)

(4.11)

in (4.8) lead to N N   ˆ¯ + 1) − LC¯ x(t K x(t ¯ + 1) = ∑ δi (t)Fi z(t) + ∑ δi (t)B¯ i u(t)

(4.12)

i=1

i=1

The objective of this LMI is to determine the observer gains in order to guarantee asymptotic convergence of the estimated state in (4.8) to the state in (4.5). That is to say, the estimated error ˆ¯ e(t) = x(t) ¯ − x(t)

(4.13)

tends to zero when t → ∞. In the following, we will present a method on how to design the switched descriptor observer parameters.

4.3 4.3.1

Fault Estimation Fault Estimate without Unknown Disturbance

In this subsection, we firstly consider the problem of fault estimate for system (4.5) without external disturbance, i.e., d(t) = 0, that is N

N

¯ f x f (t) ¯ + ∑ δi (t)B¯ i u(t) + D E¯ x(t ¯ + 1) = ∑ δi (t)A¯ i x(t)

(4.14)

y(t) = C¯ x(t) ¯

(4.15)

i=1

i=1

4.3 Fault Estimation

51

where x(t) ¯ and x f (t) are defined in (4.7). As a result, the state estimation of system (4.14) simultaneously gives the state estimation vector of x(t) and the fault estimation vector of f (t). Now, subtracting (4.12) from (4.14), one can get ¯ x(t (E¯ + KLC) ¯ + 1) − K xˆ¯(t + 1) =

N

N

N

i=1

i=1

i=1

ˆ¯ + ∑ δi (t)(Fi L + D¯ f )x f (t) (4.16) ¯ − ∑ δi (t)Fi x(t) ∑ δi (t)(Fi LC˜ + A¯ i)x(t)

If the following conditions are fulfilled ¯ K = E¯ + KLC,

Fi = Fi LC˜ + A¯ i ,

Fi L = −D¯ f .

One can obtain a feasible solution of constraints (4.17) is





 Ai 0 0 I + QC Q Fi = , L= , K= . −C −I I RC R

(4.17)

(4.18)

where matrices P and Q are free parameters to be determined. In order to guarantee the matrix K is non-singular, we assume that the matrix R is non-singular. Under the condition in (4.17), one can obtain the estimation error: N

Ke(t + 1) = ∑ δi (t)Fi e(t).

(4.19)

i=1

As a result, that is N

e(t + 1) = ∑ δi (t)Si e(t)

(4.20)

i=1

where Si = K −1 Fi with K −1 =



 I −QR−1 , −C R−1 + CQR−1

 Ai + QR−1C QR−1 Si = . −CAi − (R−1 + CQR−1)C −R−1 − CQR−1

(4.21)

(4.22)

In what follows, the method on how to design the switched descriptor observer parameters will be presented. Theorem 4.1. For the augmented system (4.14), if there exist positive definite matrices Pi1 and Pi3 , matrices Pi2 , Z1 , and Z2 , non-singular matrices G1 and G2 , such that the following LMI hold for all i, j ∈ {1, 2, · · · , N} 

Γi1 ∗ >0 (4.23) Γi2 Γj3

52

4 Sensor Fault Estimation and Accommodation for Discrete-Time Switched Systems

where

Γi1 =





 Pi1 ∗ I ,E = , Pi2 Pi3 0

(4.24)

 G1 Ai + (Z1 − E Z2 )C + E G2CAi Z1 − E Z2 , −Z2C − Z2CAi −Z2 

G1 + GT1 − Pj1 ∗ . Γj3 = GT2 E T − Pj2 G2 + GT2 − Pj3

Γi2 =

(4.25) (4.26)

∗ denotes the symmetric elements in a symmetric matrix. Then there exists a switched observer in the form of (4.8)-(4.10) with parameters as follows −1 −1 R = (G−1 2 Z2 − CG1 Z1 ) ,

Q = G−1 1 Z1 R.

(4.27)

Proof: Choose the following switched Lyapunov function:   N

V (t, e(t)) = eT (t)

∑ δi (t)Pi

(4.28)

e(t)

i=1

Then along the trajectory of system (4.20) with the particular case αi (k) = 1, αl =i (k) = 0, α j (k + 1) = 1, αl = j (k + 1) = 0, we have

∆ Vt = V (t + 1, e(t + 1)) − V(t, e(t))   N

T

∑ δi (t + 1)Pi

= e (t + 1) T

e(t + 1) − e (t)

∑

δi (t)SiT

N

∑ δi (t)Pi

i=1

i=1

N



N

N

N

∑

∑

∑

δi (t + 1)Pi δi (t)Si − δi (t)Pi i=1 i=1 i=1 i=1   T T e (t) Si Pj Si − Pi e(t)

= e (t) =



T





e(t)

e(t) (4.29)

It follows from (4.27) that

Z1 = QR−1 ,

Z2 = G2 (R−1 + CQR−1).

(4.30)

(4.31)

Define

 G1 E G2 G= , 0 G2

 Pi1 ∗ , Pi = Pi2 Pi3

together with (4.22), (4.30), and (4.31), equation (4.25) becomes

Γi2 = GSi

(4.32)

For the same reason, the equations (4.24) and (4.26) can be written respectively

Γi1 = Pi ,

Γi3 = G + GT − Pj .

(4.33)

4.3 Fault Estimation

53

Utilizing the above expressions, one can get that (4.23) is equivalent to

 Pi ∗ > 0, GSi G + GT − Pj

(4.34)

which is equivalent to ∆ V (t) < 0. Then the proof is concluded. Based on the obtained result in theorem 1, the following algorithm is given to design the descriptor observer: Step 1: Determine the nonsingular matrices G1 and G2 , matrices Z1 and Z2 from the condition (4.23). Step 2: Using the condition (4.27) to compute the matrices R and Q. Step 3: Using the condition (4.18) to compute the observer parameters Fi , L and K. Remark 4.1. Since the matrix D f is assumed to be full column rank, then the sensor FE can be obtained by fˆ(t) = (DTf D f )−1 DTf xˆ f (t)

(4.35)

 T ˆ¯ where xˆ f (t) = 0 I x(t).

Remark 4.2. It can be seen that the designed descriptor observer is also a switched system, which is also governed by switching signal δi (t). This means that the observer is model-dependent, that is, every subsystem has its own state observer.

4.3.2

Robust Fault Estimation

In this subsection, we will design a robust switched descriptor observer in the form of (4.8)-(4.10) such that the error system is robustly asymptotically stable and satisfies a prescribed H∞ performance index γ > 0, i.e. ed (t)2 < γ d(t)2 . Now consider the switched system (4.5) and the observer (4.8)-(4.10) with conditions (4.18), one can get N

ed (t + 1) = ∑ δi (t)Si ed (t) + Td d(t)

(4.36)

i=1

where Si = K −1 Fi is defined in (4.22) and Td = K −1 B¯ d =

 Bd . −CBd

(4.37)

Lemma 4.1. If there exist matrices Pi > 0, non-singular matrix G, constant γ > 0, such that the following conditions hold for ∀i, j ∈ {1, 2, · · · , N}

54

4 Sensor Fault Estimation and Accommodation for Discrete-Time Switched Systems

⎡

G + GT − Pj ⎢ ∗ ⎢ ⎣ ∗ ∗

0 I ∗ ∗

GSi I Pi ∗

⎤ GTd 0 ⎥ ⎥>0 0 ⎦ γ 2I

(4.38)

then the system (4.36) is stable with H∞ performance γ .

Proof: Choose the following switched Lyapunov function:   V (t, e(t)) = eTd (t)

N

∑ δi (t)Pi

ed (t)

(4.39)

∑ eTd (t)ed (t) − γ 2 ∑ d T (t)d(t)

(4.40)

i=1

and define Jk =

k−1

k−1

t=0

t=0

where k is an arbitrary positive integer. For any nonzero d(t) ∈ l2 [0, ∞) and zero initial condition ed (0) = 0, one has Jk =

k−1 

∑

t=0

 eTd (t)ed (t) − d T (t)d(t) + ∆ Vt − V (k, ed (k))

(4.41)

where

∆ Vt = V (t + 1, ed (t + 1)) − V(t, ed (t))

(4.42)

defines the increment of V (t, ed (t)) along the trajectories of system (4.36). Then we have eTd (t)ed (t) − γ 2 d T (t)d(t) + ∆ Vt = η T (t)Θi j η (t)

(4.43)

where

Θi j =

SiT Pj Si − Pi + I SiT Pj Td T ∗ Td Pj Td − γ 2 I



which can be rewritten as

 T −T −1 −1   −Pi 0 G Pj G 0 GSi GTd GSi GTd Θi j = + 0 −γ 2 I I 0 I 0 0 I

(4.44)

(4.45)

where G is a nonsingular matrix. On another hand, knowing that GT Pj−1 G > G + GT − Pj and combining the condition in (4.38) lead to

(4.46)

4.4 Fault Accommodation

55

⎡

GPj−1 GT ⎢ ∗ ⎢ ⎣ ∗ ∗

0 I ∗ ∗

GSi I Pi ∗

⎤ GTd 0 ⎥ ⎥>0 0 ⎦ γ 2I

(4.47)

which guarantees Θi j < 0, i.e. Jk < 0 for any k. This completes the proof. Theorem 4.2. If there exist positive definite matrices Pi1 and Pi3 , matrices Pi2 , Z1 , Z2 , and W , non-singular matrices G1 and G2 , constant scalar γ > 0, such that the following LMI hold for all i, j ∈ {1, 2, · · ·, N} 

Ψj1 ∗ >0 (4.48) Ψi2 Ψi3 where 

 Γj3 0 Γi2 Γi4 , , Ψi2 = Ψj1 = I 0 0 γI

  G1 − E G2 C Γi1 0 Bd . , Γi4 = Ψi3 = −G2C 0 γI

(4.49) (4.50)

where Γi1 , Γi2 , and Γi3 are defined respectively in (4.24)-(4.26). Then the observer (4.8)-(4.10) is robust convergence with H∞ performance γ and completely defined in (4.18) and definitions (4.27). Proof: Based on the result in Lemma 4.1, with Pi and G defined in (4.31). (4.22) and variable changes defined by (4.30), then the expression (4.38) is equivalent to (4.48).

4.4

Fault Accommodation

In this chapter, the considered sensor fault can be in many forms, such as constant, time-varying, even unbounded. Therefore, the system may not function normally, and the basic stability of the system may even be destroyed once a sensor fault occurs. This motivates us to address an FTC scheme. A simple FTC scheme (i.e. sensor fault compensation approach) is proposed in the following. Using the switched descriptor observer technique proposed in this study, the sensor FE fˆ(t) can be obtained in (4.35), subtracting the estimate D f fˆ(t) from the output y(t), yields yc (t) = y(t) − D f fˆ(t) = Cx(t) + D f ( f (t) − fˆ(t))   = Cx(t) + 0 I e(t)

(4.51)

56

4 Sensor Fault Estimation and Accommodation for Discrete-Time Switched Systems

For simplicity of narration, we suppose a pre-existing controller can ensure that system (4.1)-(4.2) functions normally. The pre-designed of this controller is obviously not our concern in this study. For simplicity, we suppose that the controller is a static output-feedback controller described by N

u(t) = − ∑ δi (t)Ki y(t)

(4.52)

i=1

Using the compensated measurement output yc (t) to replace the actual measurement output y(t), the closed-loop systems can be written as follows: N

x(t + 1) = ∑ δi (t)(Ai − Bi KiC)x(t) + Bd d(t)

(4.53)

i=1

  y(t) = Cx(t) + 0 I e(t)

(4.54)

Remark 4.3. Using the compensated senor measurement output yc (t), the predesigned controller can generate correct commands whether or not a fault occurs. Therefore the sensor fault compensation can perform fault-tolerant operation. Remark 4.4. In addition, we would like to point out that the static output feedback controller design is not concerned in this note. But, it is not a complex task. One may much more care on how to develop a dynamic output feedback controller for its less conservativeness. Remark 4.5. Based on descriptor observer approach, this note considered sensor FE and accommodation problems for discrete-time switched systems. The proposed method can also be extended to other discrete-time systems, such as Markovian jump linear systems, time-delay systems etc.. Remark 4.6. Further relaxation conditions and fault diagnosis for time delay case will be focused in the future work. Moreover, combining with the methods in [110, 111], the frameworks of industrial fault-tolerant process control and continuoustime switched linear systems will be considered.

4.5

An Illustrate Example

Consider the discrete-time switched linear system (4.1) consisting of two subsystems described by





    1 1 0 0.3 0 0.3 , C = 1 −1 . , B2 = , B1 = , A2 = A1 = 2 1 −0.2 −0.1 −0.2 0.1 By solving the condition (4.23) in Theorem 1, one can get a set of solution as follows:

4.5 An Illustrate Example

57



 18.3875 0.0303 18.3875 0.0303 , P21 = , 0.0303 19.0146 0.0303 19.0146     P12 = 5.2802 0 , P22 = 5.2802 0 , P13 = 10.5605,

 18.3875 3.9670 , G2 = 10.5605, P23 = 10.5605, G1 = −3.9064 19.0146

 1.9978 Z1 = , Z2 = 0. 0.9314 P11 =

Then, based on the obtained matrices G1 , G2 , Z1 , Z2 , and the condition in (4.24), one can get

 −3.6632 R = −39.0043, Q = . −2.6632 Through the condition in (4.18), the descriptor observer gains are computed as follows: ⎤ ⎡ ⎤ ⎡ 0 0.3 0 0 0.3 0 F1 = ⎣ −0.2 0.1 0 ⎦ , F2 = ⎣ −0.2 −0.1 0 ⎦ , −1 1 −1 −1 1 −1 ⎡ ⎤ ⎡ ⎤ 0 −2.6632 3.6632 −3.6632 L = ⎣ 0 ⎦ , K = ⎣ −2.6632 3.6632 −2.6632 ⎦ . 1 −39.0043 39.0043 −39.0043  T When the system (4.1) is affected by the disturbance d(t), and set Bd = 1 1 , γ = 2. By solving the condition in (4.48), one can get a set of solutions as follows:  

  3.0134 −1.8551 2.7140 −1.4922 , P12 = 0.1339 −0.1042 , , P21 = −1.8551 2.5920 −1.4922 2.1792 

  2.0929 −0.9168 , P22 = 0.1209 −0.0783 , P13 = 0.0655, P23 = 0.0616, G1 = −1.3338 2.0056 

0.5612 , Z2 = −0.0504. G2 = 0.4040, Z1 = −0.5091 P11 =

Then, based on the obtained matrices G1 , G2 , Z1 , Z2 , and the condition in (4.24), one can get

 −0.4893 R = −2.2090, Q = . 0.2353 Through the condition in (4.18), the descriptor observer gains are computed as follows: ⎡ ⎤ ⎡ ⎤ 0 0.3 0 0 0.3 0 F1 = ⎣ −0.2 0.1 0 ⎦ , F2 = ⎣ −0.2 −0.1 0 ⎦ , −1 1 −1 −1 1 −1

58

4 Sensor Fault Estimation and Accommodation for Discrete-Time Switched Systems

⎡ ⎤ 0 L = ⎣ 0 ⎦, 1

⎡

⎤ 0.5107 0.4893 −0.4893 K = ⎣ 0.2353 0.7647 0.2353 ⎦ . −2.2090 2.2090 −2.2090

Figure 4.1 depicts the estimation error for the state x(t). When the sensor fault is a constant as follows:  0, 0 ≤ t ≤ 10 f1 (t) = 1, 10 < t Figure 4.2 depicts the fault f1 (t) and its estimate fˆ1 (t). If the external disturbance is set as a step signal, the FE result is given in Figure 4.3. When the sensor fault is set a time-varying signal as follows:  0, 0 ≤ t ≤ 10 f2 (t) = sint + 2, 10 < t

2.5

2

1.5

1

0.5

0

−0.5

−1

0

5

10

15

20

25 Time in second

30

35

40

45

50

35

40

45

50

Fig. 4.1 State estimation error of x(t)

1.5

1 Sensor fault f1(t) 0.5

0

Estimated f1(t)

−0.5

−1

−1.5

0

5

10

15

20

25 Time in second

Fig. 4.2 Sensor fault f1 (t) and its estimate fˆ1 (t)

30

4.5 An Illustrate Example

59

2.5 2 1.5 1 Sensor fault f1(t) 0.5 0 −0.5

Estimated fault f1(t) with disturbance

−1 −1.5

0

5

10

15

20

25 Time in second

30

35

40

45

50

Fig. 4.3 Sensor fault f1 (t) and its estimate fˆ1 (t) with disturbance d(t)

3.5 Sensor fault f2(t)

3

2.5

2

1.5

1 Estimated f2(t) 0.5

0

−0.5

0

5

10

15

20

25 Time in second

30

35

40

45

50

30

35

40

45

50

Fig. 4.4 Sensor fault f2 (t) and its estimate fˆ2 (t)

3.5

3

2.5

2

1.5

1

0.5

0

−0.5

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5

10

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20

25 Time in second

Fig. 4.5 Sensor fault f2 (t) and its estimate fˆ2 (t) with disturbance d(t)

60

4 Sensor Fault Estimation and Accommodation for Discrete-Time Switched Systems

Figure 4.4 depicts the fault f2 (t) and its estimate fˆ2 (t). If the external disturbance is set as a step signal, the FE result is given in Figure 4.5. It follows from Figure 4.2-4.5 that the proposed descriptor switched observer has good performance to estimate the sensor fault. Therefore, the simulation results exhibit well estimate operation.

4.6

Summary

By utilizing the augmented descriptor observer technique, FE and compensation for discrete-time switched system with sensor faults are studied in this chapter. Both cases of with and without disturbance input are discussed, and efficient fault estimation results are obtained in the form of linear matrix inequality. Through a simple fault compensation, an fault-tolerant operation is realized. Finally, simulation results are presented to verify the effectiveness of the proposed method.

Chapter 5

Sensor Fault Estimation and Compensation for Switched Systems with State Delay

In this chapter, the problems of sensor fault estimation and compensation approaches for time-delay switched systems are investigated based on switched descriptor observer approach. First, a novel time-delay switched descriptor state observer is proposed to estimate both the state and sensor fault. The proposed observer technique is also extended to systems with nonlinearities. Then, based on the estimation of the sensor fault, an efficient fault-tolerant operation can be realized via sensor fault compensation. Finally, an example is given to show the efficiency of the developed techniques.

5.1

Introduction

Due to the demand of reliability and safety for dynamic systems, a great amount of effort has been devoted to this significant issue in recent years. Therefore, fault diagnosis and FTC for various systems have been extensively considered, and a number of achievements have been obtained, which can be found in several excellent papers, see for example [112, 113, 114, 115, 116, 117, 118, 119] and the references therein. As for FTC, which includes two main approaches: passive FTC, which uses feedback control laws such that it is robust with respect to all possible system faults [89, 120]; and active FTC, which utilizes a FDI and accommodation technique [121, 122]. Though FDI can give information whether there exist faults occurring, the magnitude of the fault can not be precisely provided. Therefore, the next step is to search an efficient approach to estimate the magnitude of the fault, which is called fault estimate or fault reconstruction. Finally, using the estimated fault information, a fault-tolerant controller can be designed to compensate the effect of the fault. From the above discussion, it can be seen that FE plays a very important role in active FTC. Therefore, fault estimation has become a hot research topic owing to its importance in active FTC. During the past decade, various effective methods, such as sliding mode observer approach using equivalent output injection signal [94], adaptive technique [112, 123], and learning method based on neural network [103, 124] and so on, have been developed for FE problem. However, most of the proposed c Springer International Publishing Switzerland 2015  D. Du, B. Jiang, and P. Shi, Fault Tolerant Control for Switched Linear Systems, Studies in Systems, Decision and Control 21, DOI: 10.1007/978-3-319-15162-5_5

61

62

5 Sensor Fault Estimation and Compensationfor Switched Systems with State Delay

approaches are not suitable to estimate sensor fault. This is due to the fact that the sensor fault is unavoidable amplified by the observer gain matrix. Some efforts were made to solve this problem, and several efficient methods have been presented [99, 100, 101, 102]. Specially the augmented descriptor observer technique proposed in [99] was proved to be an effective scheme to estimate sensor fault. Moreover, no limitations or previous information of the considered sensor fault is required. This means that the sensor fault can be in different forms, such as constant, time-varying, or even unbounded. On another research direction, switched system has been widely investigated for its extensively practical application [55, 125, 126, 127]. Many real-world process and systems can be modeled as switched systems, including chemical processes, computer controlled systems, switched circuits, and so on. Switched system belongs to the class of hybrid systems, which consists of a finite number of subsystems (described by differential or difference equations) and an associated switching signal governing the switching among them. The switching signals may belong to a certain set and the set may be diverse. On the other hand, time delays are the inherent features of many physical process and the big sources of instability and poor performances. Switched systems with time delay have strong engineering background, such as in network control systems [128] and power systems [129]. To the best of our knowledge, the problems of sensor FE and compensation for time-delay switched systems are still under research, which motivate us to study this meaningful and challenge topic. In this chapter, FE and fault compensation for time-delay switched systems under arbitrary switching signal are considered based on descriptor observer design scheme. Firstly, a time-delay switched descriptor observer is designed for augmented switched system, which can simultaneously estimate state, sensor fault for time-delay switched systems. The proposed method is also extended to nonlinear case for time-delay switched systems. By using estimated fault, sensor fault compensation will be performed to realize a fault-tolerant performance. The rest of this LMI is organized as follows: In Section 5.2, the switched descriptor observer design scheme is presented based on LMI technique. The fault compensation is given in Section 5.3. An example is illustrated in Section 5.4 to show the effectiveness of the proposed approach, and the chapter is concluded in Section 5.5.

5.2

Descriptor Observer Design

Consider the following time-delay switched system:  x(t) ˙ = Aσ (t) x(t) + Ahσ (t) x(t − h) + Buσ (t)u(t) + Bd σ (t)d(t) y(t) = Cσ (t) x(t) + Duσ (t) u(t) + D f σ (t) f (t)

(5.1)

where x(t) ∈ Rn is the state vector, y(t) ∈ R p is the output vector, u(t) ∈ Rm is the control input, d(t) ∈ Rl is the unknown disturbance input or modeling error, f (t) ∈ Rq is the sensor fault, the scalar h is the constant time delay and satisfies

5.2 Descriptor Observer Design

63

¯ where h¯ is a constant scalar. σ (t) : [0, +∞) −→ ψ = {1, · · · , N} is the 0 < h < h, switching signal, N > 1 is the number of subsystems. At an arbitrary continuous time t, the switching signal σ (t), denoted by σ for simplicity, is dependent on time t or state x(t), or both, or other switching rules. As in [28], we assume that the sequence of subsystems in switching signal σ (t) is unknown a priori, but its instantaneous value is available in real time. Aσ (t) , Ahσ (t) , Buσ (k) , Bd σ (k) , Cσ (k) , Duσ (k) , and D f σ (t) are constant matrices with appropriate dimensions for all σ (t) ∈ ψ . When the isubsystem is activated, we denote the matrices associated with σ (t) = i by Aσ (t) = Ai , Ahσ (t) = Ahi , Buσ (k) = Bui , Bd σ (k) = Bdi , Cσ (k) = Ci , Duσ (k) = Dui , and D f σ (t) = D f i . For the purpose of this work, we give the following assumption: Assumption 1. For any i ∈ ψ , the matrices Bui and D f i are assumed to be full collum rank. The failure f (t) = β (t − t0 ) f0 (t) can be thought of as an additional signal, and the function β (t − t0 ) is given by  0, t ≤ t0 β (t − t0) = (5.2) 1, t > t0 where t0 is the time of fault occurring. That is, f (t) is zero prior to the failure time t ≤ t0 and is f0 (t) after the failure occurs t > t0 .

5.2.1

Fault Estimate without Unknown Disturbance

In this subsection, we firstly consider the problem of fault estimate for system (5.1) without external disturbance, i.e., d(t) = 0, that is  x(t) ˙ = Aσ (t) x(t) + Ahσ (t) x(t − h) + Buσ (t)u(t) (5.3) y(t) = Cσ (t) x(t) + Duσ (t) u(t) + D f σ (t) f (t) The traditional observer design for time-delay switched system can be constructed as follows:  ˙ˆ = Aσ (t) x(t) x(t) ˆ + Ahσ (t)x(t ˆ − h) + Buσ (t)u(t) + Lσ (t)(y(t) − y(t)) ˆ (5.4) y(t) ˆ = Cσ (t) x(t) ˆ + Duσ (t) u(t) where x(t) ˆ is the estimate vector of x(t), and Lσ (t) is the observer gain matrix to be determined, which can realize the following time-delay system (5.5) is stable without fault. Let x(t) ˜ = x(t) − x(t), ˆ the error dynamics system can be represented as ˙˜ = (Aσ (t) − Lσ (t)Cσ (t) )x(t) ˜ + Ahσ (t)x(t ˜ − h) + Lσ (t)D f σ (t) f (t) x(t)

(5.5)

64

5 Sensor Fault Estimation and Compensationfor Switched Systems with State Delay

From (5.5), one can see that the sensor fault term f (t) is multiplied by the gain matrix Lσ (t) . There is a well known tradeoff between the rapidity of the observer and the amplification of the fault. It is not desirable to decrease the fault effect by reducing the response speed of the observer. Moreover, the conventional observer is only suitable to estimate state, which can not give the estimation of sensor fault. However, the accurate sensor FE is very important in FD and FTC. Therefore, this encourages us to develop a novel switched observer design approach for switched system (5.1) to obtain the state and sensor fault simultaneous estimation. Motivated by the descriptor observer design scheme in [99], we utilize the descriptor system approach to construct a novel time-delay switched descriptor observer to estimate the system state and sensor fault term simultaneously. Let

 



 Aσ (t) 0 Ahσ (t) 0 x(t) I 0 , A¯ hσ (t) = , x¯ = , E¯ = , A¯ σ (t) = f (t) 00 0 0 0 0  (5.6)   Buσ (t) , C¯σ (t) = Cσ (t) D f σ (t) . B¯ σ (t) = 0 we can construct the following augmented system:  ˙¯ = A¯ σ (t) x(t) E¯ x(t) ¯ + A¯ hσ (t) x(t ¯ − h) + B¯ σ (t)u(t) ¯ y(t) = Cσ (t) x(t) ¯ + Duσ (t)u(t)

(5.7)

It can be seen that the system (5.7) is a time-delay switched descriptor system model. The state vector x(t) ¯ of the augmented switched descriptor system (5.7) is consisted by the state x(t) and the sensor fault f (t). Therefore, if we can design a state observer for the augmented system (5.7), then the state of this observer is just a simultaneous state and fault estimator for the original system (5.1). For this purpose, we construct the following state observer for the augmented system (5.7): ˆ¯ + (A¯ hσ (t) − Lhσ (t)C¯σ (t) )x(t ˆ¯ − h) ξ˙ (t) = (A¯ σ (t) − Ld σ (t)C¯σ (t) )x(t) +(B¯ σ (t) − Ld σ (t) Duσ (t) )u(t) +Ld σ (t) y + Lhσ (t) [y(t − h) − Duσ (t)u(t − h)] ˆ¯ = (E¯ + Lσ (t)C¯σ (t) )−1 [ξ (t) + Lσ (t) y(t) − Lσ (t) Duσ (t) u(t)] x(t)

(5.8) (5.9)

ˆ¯ ∈ Rn+q is the estimate of the augmented descriptor state vector x(t) where x(t) ¯ ∈ n+q R , Lσ (t) , Ld σ (t) , and Lhσ (t) are the observer parameters to be designed. Remark 5.1. From above discussion, one can get that, compared with other FE methods, descriptor observer approach provided an efficient method to realize sensor FE. Once a fault occurs, a fast and exactly estimation can be given by the descriptor observer (5.9). Besides, the fault’s type has no limitation, which can be constant, time-varying, or even no limitation. In the following, we will present a method on how to design the switched descriptor observer parameters.

5.2 Descriptor Observer Design

65

Theorem 5.1. For the augmented system (5.7), there exists a switched observer in the form of (5.9), if there exist positive definite matrices Pi , Qi , and matrices Yi , Zi , Wdi , and Whi , for all i, j, k ∈ ψ , such that the following linear matrix inequality holds ⎡ ⎤ (1, 1) (1, 2) (1, 3) (1, 4) ⎢ ∗ (2, 2) (2, 3) (2, 4) ⎥ ⎢ ⎥ 0, let Nσ (τ , k) denote the number of switchings of σ (k) on an interval (k1 , k2 ). If Nσ (τ , k) ≤ N0 +

k2 − k1 τa

holds for a given N0 ≥ 0 and τa > 0, then the constant τa is called the average dwelltime and N0 the chattering bound. Remark 6.2. Definition 6.1 means that if there exists a positive number τa such that a switching signal has the ADT property, the average time interval between consecutive switching is at least τa . This is a kind of slowly switching signals, which is less conservative than dwell-time switching signal. As commonly used in the literature, for convenience, we choose N0 = 0 in this chapter.

Fig. 6.1 Lyapunov-like function

Lemma 6.1. [54] Consider the continuous-time switched system x(t) ˙ = fσ (x(t)), σ ∈ ψ and let 0 < α < 1, β > −1, and µ > 1 be given constants. Suppose that there exist C 1 functions Vσ (t) : Rn → R, σ ∈ ψ , and two class K∞ functions κ1 and κ2 such that κ1 (|x(t)|) ≤ Vi (x(t)) ≤ κ1 (|x(t)|) V˙i ((t)) ≤



−α Vi (x(t)), ∀t ∈ T↓ (tl ,tl+1 ) β Vi (x(t)), ∀t ∈ T↑ (tl ,tl+1 )

and ∀(σ (tl ) = i, σ (tl− ) = j) ∈ N × N, i = j, Vi (x(t)) ≤ µ V j (x(t))

82

6 Fault Estimation for Nonlinear Continuous-Time Switched Systems

then, the system is globally uniformly asymptotically stable (GUAS) for any switching signal with ADT

τa > τa∗ = {Tmax [α + β ] + ln µ }/α

(6.5)

where T↓ (tl ,tl+1 ) and T↑ (tl ,tl+1 ) represent the unions of the dispersed intervals during which Lyapunov function is increasing and decreasing within the interval [tl ,tl+1 ), Tmax = max T↑ (tl+1 − tl ), l ∈ N. Lemma 6.2. ([145]) Given matrices W , X and Y of appropriate dimensions and with W symmetrical, then W + X F (t)Y + Y T F T (t)X T < 0 holds for all F (t) satisfying F T (t)F (t) ≤ I if and only if for some ε > 0, W + ε X X T + ε −1 Y T Y < 0 Lemma 6.3. ([146]) For any real vectors a, b and matrix G > 0 of compatible dimensions, the following inequality holds: aT b + bT a ≤ aT Ga + bT G−1 b, a, b ∈ Rn

6.3 6.3.1

Main Results Fault Diagnosis Observer Design

The fault diagnosis observer is designed as follows: ˙ˆ = Aσ (t) x(t) x(t) ˆ + gσ (t)(t, x(t)) ˆ + Bσ (t)u(t) +Eσ (t) fˆ(t) − Lσ (t) (y(t) ˆ − y(t)) ˆ y(t) ˆ = Cσ (t) x(t)

(6.6) (6.7)

where x(t) ˆ ∈ Rn is the state vector of the observer, y(t) ˆ ∈ Rm is the output vector of the observer, fˆ(t) is an estimate of f (t), Lσ (t) is the observer gain. Denote e(t) = x(t) ˆ − x(t); r(t) = y(t) ˆ − y(t); ˜f (t) = fˆ(t) − f (t);

(6.8)

then the error dynamics is described by e(t) ˙ = Aσ (t) e(t) + Gσ (t)(t, x(t), ˆ e(t)) +Eσ (t) f˜(t) r(t) = Cσ (t) e(t)

(6.9) (6.10)

6.3 Main Results

83

where Aσ (t) = Aσ (t) − Lσ (t)Cσ (t) , Gσ (t) (t, x(t), ˆ e(t)) = gσ (t) (t, x(t)) ˆ − gσ (t) (t, x(t)). As a result, the purpose of FE is to find a diagnostic algorithm for fˆ(t) such that lim e(t) = 0;

t→∞

6.3.2

lim fˆ(t) = f (t)

t→∞

(6.11)

Fault Identification

In the sequel, a convergent adaptive fault diagnostic algorithm to estimate the fault f (t) is given, which is obtained from the residual r(t). Consequently, f˙(t) = 0 implies that the residual f˜(t) with respect to time is ˙f˜(t) = ˙fˆ(t) − f˙(t)

(6.12)

Theorem 6.1. For given scalars 0 < α < 1, β > −1, µ ≥ 1 and li > 0, if there exist matrices Pi > 0, Xi > 0, Yi , Fi , W , and scalars ε1i > 0, ε2i > 0, ηi > 0, such that Pi ≤ µ Pj ,

 Pi I ≥ 0, I Xi

∀i, j ∈ ψ

Pi Xi = I

(6.13)

(6.14)

 Ξ i Πi δ

(6.25)

Similarly, one can prove that (6.16) implies V˙i (t) − β Vi(t) < 0

(6.26)

According to Lemma 6.1, witch means that ξ (t) converges to a small set according to Lyapunov stability theory. Remark 6.3. From the adaptive FE algorithm, one can get that it contains the derivative of r(t) and r˙(t). It is feasible when r˙(t) can be obtained. But if the signal r˙(t) can not be easily obtained from certain systems, we should resort to other alternative methods. In order to deal with this problem, r˙ f (t) is introduced to be a substitute for r˙(t) [34]. The relationship is defined as follows:

86

6 Fault Estimation for Nonlinear Continuous-Time Switched Systems

1 r˙ f (t) = − (r f (t) − r(t)) ε

(6.27)

From (6.27), one can get that under zero initial condition, using Laplace transform yields r˙ f (t) =

1 r˙(t) εs + 1

(6.28)

Therefore, it it easy to show that the substitute r˙ f (t) can approximate to r˙(t) with any desired accuracy as ε → 0. Meanwhile, when s → 0, that is t → ∞, r˙ f (t) asymptotically converges to r˙(t). Remark 6.4. It can be seen that the condition (6.14) is not a strict LMI formation due to the equation Pi Xi = I, which can not be solved directly by Matlab LMI Control Toolbox. However, we can solve this nonconvex feasibility problem by formulating it into a special sequential optimization problem subject to LMI constraints. In the following, a specific algorithm is given by utilizing the result in [147]. Algorithm 1 (0) (0) (0) (0) (0) (0) (0) Step 1: Find a feasible set {Pi , Xi ,Yi , Fi ,W (0) , ε1i , ε2i , ηi } satisfying (6.13)-(6.16). Set k = 0. Step 2: Solve the following LMI problem   # N " (k) (k) min tr ∑ Pi Xi + Pi Xi i=1

subject to (6.13)-(6.16). Step 3: Substitute the obtained matrix variables {Pi , Xi ,Yi , Fi ,W, ε1i , ε2i , ηi } into (6.13)-(6.16). If the condition (6.13) is satisfied with N

|tr( ∑ Pi Xi ) − (N + 1)n| < δ i=1

for some sufficient small scalar δ > 0, then output the feasible solution {Pi , Xi ,Yi , Fi , W, ε1i , ε2i , ηi }, EXIT. Step 4: If k > N where N is the maximum number of iterations allowed, EXIT. (k) (k) (k) (k) (k) (k) (k) Step 5: Set k = k + 1, {Pi , Xi ,Yi , Fi ,W (k) , ε1i , ε2i , ηi } = {Pi , Xi ,Yi , Fi , W, ε1i , ε2i , ηi }, and go to Step 2.

6.4

An Illustrative Example

Consider the nonlinear switched system consisting of two subsystems described by



 −0.54 1.02 −0.01 0.1 A1 = , A2 = , 0.17 −0.31 0.01 0.04

6.4 An Illustrative Example

87





 0.1 0.2 0.1 , B2 = , E1 = , 0.2 0.4 0.2

     0.2 , C1 = 0.1 0.2 , C2 = 0.2 0.3 , E2 = 0.4

 0.1 sin(x1 (t)) g1 (t, x(t)) = , 0

 0 g2 (t, x(t)) = . 0.2 sin(x2 (t)) B1 =

Supposed that the maximum of asynchronous switching Tmax = 0.5. Given µ = 1.5, α = 0.5, and β = 1.5, one can obtain that τa∗ = 2.81. By utilizing Algorithm 1 to solve the conditions in Theory 6.1, we can get a set of solutions as follows:



 1.0754 0.2633 0.5561 P1 = ,Y1 = , 0.2633 1.2454 1.1191



 1.3522 0.7543 5.5206 P2 = ,Y2 = , 0.7543 1.3600 8.2914



 0.9807 −0.2073 3.1335 X1 = , L1 = , −0.2073 0.8468 8.3238



 1.0709 −0.5940 0.9870 X2 = , L2 = , −0.5940 1.0648 5.5491 F1 = 0.4383, F2 = 0.3171,W = 1.1190. Taking the learning law Γ = 100 and generating a possible switching sequence satisfying τa = 3 > τa∗ . In this example, two cases of faults are considered. When the fault is a constant described as  0, 0≤t ≤5 f1 (t) = 4, 5 < t < 30 In this case, the simulation result is shown in Figure 6.2. When the fault is a timevarying function described as  0, 0≤t ≤5 f2 (t) = sint, 5 < t < 30 The simulation result is shown in Figure 6.3. From the above simulation results, we can conclude that whether the fault is a constant or a time-varying function, the estimate algorithm proposed here can estimate them quickly and exactly.

88

6 Fault Estimation for Nonlinear Continuous-Time Switched Systems

5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0 0

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15 Time in second

20

25

30

Fig. 6.2 Fault f1 (t) (solid line) and its estimate fˆ1 (t) (dotted line

1.5

1

0.5

0

−0.5

−1

−1.5 0

5

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15 Time in second

20

25

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Fig. 6.3 Fault f2 (t) (solid line) and its estimate fˆ2 (t) (dotted line)

6.5

Summary

In this chapter, an adaptive fault diagnosis observer is designed for nonlinear switched systems. The proposed method improves rapidity and accuracy on FE for nonlinear switched systems. Simulation results demonstrate the effectiveness of the proposed techniques.

Chapter 7

Actuator Fault Estimation and Accommodation for Discrete-Time Switched Systems

This chapter investigates the problems of actuator FE and accommodation for discrete-time switched systems with state delay. By using reduced-order observer method and switched Lyapunov function technique, a FE algorithm is achieved for the discrete-time switched system with actuator fault. Then based on the obtained online FE information, a switched dynamic output feedback controller is employed to compensate for the effect of faults by stabilizing the closed-loop systems. Finally, an example is proposed to illustrate the obtained results.

7.1

Introduction

In complex systems, dependability is as important as performances. Once a fault (sensor, actuator, or component failures) occurred in the system, the system behavior may be drastically changed, ranging from performance degradation to instability. FTC technique is an effective method in order to reach the system objectives, or if this turns to be impossible, to achievable new objectives to avoid catastrophic behaviors. Therefore, FTC have been the subjects of intensive investigations over the past two decades, and a number of achievements have been obtained, which can be found in several excellent papers, see for example [65, 148, 149, 150, 151, 152, 153, 154] and the references therein. As for FTC, which includes two main approaches: passive and active. While passive fault tolerance considers systems faults as a special kind of uncertainties [89], active fault tolerance is based on FDI and accommodation technique [90]. Though FDI can give information whether there exist faults occurring [92], the magnitude of the fault can not be precisely provided. Therefore, the next step is to search an efficient approach to estimate the magnitude of the fault, which is called fault estimate or fault reconstruction. During the past decade, various effective methods, such as sliding mode observer approach using equivalent output injection signal [94], adaptive technique [155], and learning method based on neural network [103] and so on, have been developed for FE problem. Finally, using the estimated fault information, a fault-tolerant controller can be designed to compensate the effect of the c Springer International Publishing Switzerland 2015  D. Du, B. Jiang, and P. Shi, Fault Tolerant Control for Switched Linear Systems, Studies in Systems, Decision and Control 21, DOI: 10.1007/978-3-319-15162-5_7

89

90

7 Actuator Fault Estimation and Accommodation

fault. From the above discussion, it can be seen that FE plays a very important role in active FTC. Therefore, the study of FE has become a hot research topic owing to its importance in active FTC. Switched system is a class of hybrid systems, which consists of a finite number of subsystems and an associated switching signal governing the switching among them. Many real-world process and systems can be modeled as switched systems, including chemical processes, computer controlled systems, switched circuits, and so on. Therefore, it has been intensively investigated in the past decades [26, 103, 156, 157, 54, 158, 159]. Compared with fruitful stability and stabilization results for switched systems, fault diagnosis and FTC achievements are relatively few. In [16] and [107], FD for switched system was separately investigated for continuous case and discrete case. [160] investigated sensor FE and accommodation approaches for continuous-time switched systems, and [76, 161] separately studied actuator FE and accommodation approaches for continuous-time switched systems with constant time-delay case and time-varying delay case. However, to the best of our knowledge, the problems of actuator FE and compensation for discrete-time switched systems with time delay are still under research, which motivate us to study this meaningful and challenging topic. Based on the above works, our objective of this chapter is to analyze and develop a general framework of FE and accommodation for delayed discrete-time switched systems with actuator faults. Firstly, a switched reduced-order observer is designed in form of LMI, then a FE algorithm is given. By using the estimated fault signal, actuator fault compensation will be performed to realize a fault-tolerant performance. The rest of this chapter is organized as follows. Section 7.2 presents the system description. In Section 7.3, a switched reduced-order observer design, including an H∞ performance index is proposed to estimate the actuator fault. Furthermore, in Section 7.4, based on the online fault estimate information, a switched dynamic output feedback controller is designed to compensate for the effect of faults. An example is illustrated in Section 7.5 to show the effectiveness of the proposed approach, and the chapter is concluded in Section 7.6.

7.2 7.2.1

Without State Delay Case Problem Statements and Preliminaries

Consider the following discrete-time switched linear system: x(t + 1) = y(t) =

N

N

N

i=1 N

i=1

i=1

∑ δi (t)Ai x(t) + ∑ δi (t)Bi (u(t) + f (t)) + ∑ δi (t)Di ω (t)

(7.1)

∑ δi (t)C1i x(t)

(7.2)

∑ δi (t)C2i x(t)

(7.3)

i=1 N

z(t) =

i=1

7.2 Without State Delay Case

91

where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the control input vector, y(t) ∈ R p is the measurable output vector, f (t) ∈ Rm represents the additive actuator fault, ω (t) ∈ Rr is the disturbance which is assumed to belong to l2 [0, ∞). The vector δi (t) is called switching signal, which specifies which subsystem will be activated at the discrete time t, where

δi (t) : Z + = {0, 1, 2 · · ·} → {0, 1},

N

∑ δi (t) = 1,

∀t ∈ Z + .

i=1

Ai , Bi , C1i , C2i , and Di are constant matrices with appropriate dimensions. For the purpose of this note, we give the following assumption: Assumption 2. The measurable output matrices: C11 = C12 = · · · = C1N = C, which are full collum rank. ⊥ (n−p)×n Since the matrix  C is of full collum rank, there always exist a matrix C ∈ R ⊥ C such that ∈ Rn×n is a nonsingular matrix. C

7.2.2 7.2.2.1

Fault Estimation Design State Transformation

C⊥ We construct nonsingular matrix T = (n−p)×n C p×n

−1

  CT = 0 p×(n−p) I p

, which satisfies: (7.4)

Then define the following state transformation: x(t) = T x(t) ˜

(7.5)

By using the condition in (7.5), the switched systems (7.1) and (7.2) can be transformed as follows: 

 

N x˜1 (t) A11i A12i x˜1 (t + 1) = ∑ δi (t) x˜2 (t) A21i A22i x˜2 (t + 1) i=1  

N N B1i D1i + ∑ δi (t) ω (t) (7.6) [u(t) + f (t)] + ∑ δi (t) B2i D2i i=1 i=1 

 x˜1 (t)  (7.7) y(t) = 0 p×(n−p) I p x˜2 (t)

92

7 Actuator Fault Estimation and Accommodation

where



 x˜1 (t) A11i A12i , T −1 Ai T = , A21i A22i x˜2 (t)

   D1i T −1 Di = , CT = 0 p×(n−p) I p . D2i

x(t) ˜ =

T −1 Bi =

 B1i , B2i

Then, the system (7.6) and (7.7) can be rewritten as N

x˜1 (t + 1) =

N

N

∑ δi (t)A11ix˜1 (t) + ∑ δi (t)A12iy(t) + ∑ δi (t)B1iu(t)

i=1

i=1 N

N

i=1

+ ∑ δi (t)B1i f (t) + ∑ δi (t)D1i ω (t) i=1

i=1 N

N

y(t + 1) =

(7.8) N

∑ δi (t)A21ix˜1 (t) + ∑ δi (t)A22iy(t) + ∑ δi (t)B2iu(t)

i=1

i=1 N

N

i=1

+ ∑ δi (t)B2i f (t) + ∑ δi (t)D2i ω (t) i=1

(7.9)

i=1

By introducing the virtual input η (t) and ρ (t): N

N

η (t) = ∑ δi (t)A12i y(t) + ∑ δi (t)B1i u(t)

(7.10)

i=1

i=1

N

N

i=1

i=1

ρ (t) = y(t + 1) − ∑ δi (t)A22i y(t) − ∑ δi (t)B2i u(t)

(7.11)

one can get N

x˜1 (t + 1) =

N

∑ δi (t)A11ix˜1 (t) + η (t) + ∑ δi (t)B1i f (t)

i=1

i=1

N

+ ∑ δi (t)D1i ω (t)

(7.12)

i=1

N

N

ρ (t) =

i=1

i=1

7.2.2.2

N

∑ δi (t)A21ix˜1 (t) + ∑ δi (t)B2i f (t) + ∑ δi (t)D2i ω (t)

(7.13)

i=1

Reducec-Order Observer Design

For the dynamics (7.12) and (7.13), we construct the following reduced-order switched state observer: N

xˆ˜1 (t + 1) =

N

∑ δi (t)A11i xˆ˜1 (t) + η (t) + ∑ δi (t)B1i fˆ(t)

i=1

i=1

7.2 Without State Delay Case

93 N

− ∑ δi (t)Gi (ρˆ (t) − ρ (t))

(7.14)

i=1

N

N

i=1

i=1

ρˆ (k) =

∑ δi (t)A21i xˆ˜1 (t) + ∑ δi (t)B2i fˆ(t)

(7.15)

N

fˆ(t + 1) = fˆ(t) − ∑ δi (t)Fi (ρˆ (t) − ρ (t))

(7.16)

i=1

ˆ˜ ∈ Rn−p is the reduced-order observer state, ρˆ (t) ∈ R p is the reducedwhere x(t) order observer output, fˆ(t) ∈ Rm is the fault estiation of the fault f (t), Gi and Fi are reduced-order observer gain matirces to be designed. Let e(t) = xˆ˜1 (t) − x˜1(t),

e f (t) = fˆ(t) − f (t)

(7.17)

then the error dynamics are given as follows: N

N

e(t + 1) =

∑ δi (t)(A11i − GiA21i)e(t) + ∑ δi (t)(B1i − GiB2i )e f (t) i=1

i=1

N

+ ∑ δi (t)(Gi D2i − D1i )ω (t)

(7.18)

i=1

e f (t + 1) = fˆ(t + 1) − f (t + 1) N

N

i=1

i=1

= fˆ(t) − ∑ δi (t)Fi A21i e(t) − ∑ δi (t)Fi B2i e f (t) N

+ ∑ δi (t)Fi D2i ω (t) − f (t + 1) i=1

N

N

= fˆ(t) − f (t) − ∑ δi (t)Fi A21i e(t) − ∑ δi (t)Fi B2i e f (t) i=1

i=1

N

+ ∑ δi (t)Fi D2i ω (t) − ( f (t + 1) − f (t)) i=1 N

N

= − ∑ δi (t)Fi A21i e(t) + ∑ δi (t)Fi D2i ω (t) i=1 N

i=1

+ ∑ δi (t)(Im − Fi B2i )e f (t) − ∆ f (t)

(7.19)

i=1

where ∆ f (t) = f (t + 1) − f (t). Combining the dynamics in (7.18) and (7.19), one can get the following augmented system:

94

7 Actuator Fault Estimation and Accommodation

e(t ¯ + 1) =

N

N

i=1

i=1

¯ + ∑ δi (t)(Gi D1i − D2i )ν (t) ∑ δi (t)(A1i − Gi A2i )e(t)

(7.20)

where



     A11i B1i e(t) ω (t) , A2i = A21i B2i , , A1i = , ν (t) = 0m×(n−p) Im e f (t) ∆ f (t)

    D1i 0(n−p)×m Gi , Gi = . (7.21) D1i = D2i 0 p×m , D2i = Fi 0m×d Im e(t) ¯ =

The objective of this section is to determine the observer gains in order to guarantee asymptotic convergence of the estimated state in (7.14) to the state in (7.12). That is to say, the estimated error e(t ¯ + 1) tends to zero when t → ∞. In the following, an reduced-order observer design method under an H∞ performance index is proposed to achieve robust FE for discrete-time switched systems with state delay. Theorem 7.1. Given a constant γ1 > 0, if there exists positive definite matrix Pi , nonsingular matrix Ω , matrix Hi , for any i, j ∈ {1, 2, · · ·, N}, such that ⎤ ⎡ Pj − (Ω + Ω T ) Ω T A1i − Hi A2i Hi D1i − Ω T D2i 0 ⎢ ∗ −Pi 0 I ⎥ ⎥ 0, then output the feasible solution {Pi , Xi , Qi , Fi ,Yi , Zi ,Wi , G, η }, EXIT. Step 4: If k > N where N is the maximum number of iterations allowed, EXIT. (k) (k) (k) (k) (k) (k) (k) Step 5: Set k = k + 1, {Pi , Xi , Qi , Fi ,Yi , Zi ,Wi , G(k) , η (k) } = {Pi , Xi , Qi , Fi ,Yi , Zi , Wi , G, η }, and go to Step 2.

8.4

Fault Accommodation

Since the state x(t) is unavailable, the estimation value x(t) ˆ is substituted for x(t). Therefore, the observer-based normal controller is given

130

8 Active Fault Tolerant Control for Switched Systems with Time Delay

ur (t) = −Kσ (t) x(t) ˆ + d(t)

(8.30)

where Kσ (t) is the feedback gain matrix and d(t) is the reference input. Once a fault occurs, based on the accurate and rapid estimation of the fault , the following observer-based fault-tolerant controller is activated to compensate for the fault u(t) = ur (t) − fˆ(t) Assuming d(t) = 0 and substituting (8.31) into (8.1), one obtains  x(t) ˙ = (Aσ (t) − Bσ (t) Kσ (t) )x(t) + Ahσ (t) x(t − h) + ρ (t) y(t) = Cσ (t) x(t)

(8.31)

(8.32)

where ρ (t) = −Bσ (t) Kσ (t) e(t) − Bσ (t) f˜(t). From the result of Theorem 1, one can get that e(t) → 0 and f˜(t) → 0 when t → ∞. The signal ρ (t) can be treated as a disturbance of the system (8.32). So, if only the feedback gain Ki can ensure that the following system is asymptotically stable.  x(t) ˙ = (Aσ (t) − Bσ (t) Kσ (t) )x(t) + Ahσ (t)x(t − h) (8.33) y(t) = Cσ (t) x(t) Theorem 8.2. The system (8.33) is asymptotically stable for any time delay h satisfying 0 ≤ h ≤ h¯ if there exist positive definite matrices Pi , Qi , matrix Yi , for all i, j, k ∈ ψ , such that ⎡ ⎤ Ai Xi + Xi ATi − BiYi − YiT BTi Ahi R j Xi ⎣ ∗ −R j 0 ⎦ < 0 (8.34) ∗ ∗ −Ri where Yi = Ki Xi .

Proof. Let the Lyapunov function be V5 = xT (t)Pσ (t) x(t) +

 t

t−h

xT (s)Qσ (s) x(s)ds

(8.35)

Then the derivative of V5 along the trajectories of the system in (8.33) is V˙5 = 2xT (t)Pσ (t) (Aσ (t) − Bσ (t) Kσ (t) )x(t) + 2xT (t)Pσ (t) Ahσ (t) x(t − h) +xT (t)Qσ (t) x(t) − xT (t − h)Qσ (t−h) x(t − h)

(8.36)

Under the particular case σ (t) = i and σ (t − h) = j, and let Pi = Xi−1 , Qi = R−1 i , one can get that the LMI in (8.34) means that V˙5 < 0. Therefore, the system (8.33) is asymptotically stable according to standard Lyapunov stability theory.

8.5 An Illustrative Example

131

Remark 8.6. This chapter considers the active FTC for switched systems with constant delay. Combined the existing time-varying delays results for switched system, the result obtained in this wok may be extended to time-varying case.

8.5

An Illustrative Example

Consider the switched system S consisting of two subsystems described by





 −0.54 1.02 −0.01 0.1 0.18 0.36 , A2 = , Ah1 = , A1 = 0.17 −0.31 0.01 0.04 −0.06 −0.12





   0.11 0.18 0.1 0.2 Ah2 = , B1 = , B2 = , C1 = 0.1 0.2 , −0.03 −0.04 0.2 0.4   ¯ C2 = 0.2 0.3 , h = 6.

By utilizing Algorithm 1 to solve the conditions in Theory 8.1, we can get a set of solutions as follows:



 0.0139 0.0460 0.0018 0.0015 P1 = , P2 = , 0.0460 0.1538 0.0015 0.0030

  8.0608 −2.4113 929.2677 −465.9440 3 X1 = 10 × , X2 = , −2.4113 0.7278 −465.9440 562.6869



 −0.0259 −0.0660 0.6589 0.7236 Q1 = 10−3 × , Q2 = 10−4 × , −0.0660 −0.1293 0.7236 0.7870

  −0.0926 −0.1936 −0.1237 −0.2492 −3 −3 Y1 = 10 × , Y2 = 10 × , −0.1936 −0.4894 −0.2488 −0.5011





 0.0064 0.0028 0.0926 0.1936 W1 = , W2 = , Z1 = 10−3 × , 0.0271 0.0029 0.1936 0.4894

 0.1251 0.2506 Z2 = 10−3 × , F1 = 0.0830, F2 = 0.0239, 0.2505 0.5036 G = 8.9436, η = 0.0721. Taking the learning law Γ = 200 and the sampling period T = 0.01, the time delay is chosen as h = 3, and the control input u(t) is a unit step function. In this example, two cases of faults are considered. When the fault is a constant described as ⎧ 0≤t ≤5 ⎨ 0, f1 (t) = 0.2(t − 5), 5 < t ≤ 15 ⎩ 2, 15 < t ≤ 30 In this case, the simulation result is shown in Figure 8.2. When the fault is a timevarying function described as

132

8 Active Fault Tolerant Control for Switched Systems with Time Delay

f2 (t) =



0, 0≤t ≤5 0.3 sin 2t + 0.5, 5 < t < 30

The simulation result is shown in Figure 8.3. From the above simulation results, we can conclude that whether the fault is a constant or a time-varying function, the estimate algorithm proposed here can estimate them quickly and exactly. By solving the LMI in Theorem 8.2, one obtains  

177.1841 −136.2999 80.4373 −69.2977 , X2 = , X1 = −136.2999 99.7892 −69.2977 57.5229 2.5

2

1.5

f1(t) f1hat(t)

1

0.5

0

−0.5 0

5

10

15 time step k

20

25

30

25

30

Fig. 8.2 Fault f1 (t) (solid line) and its estimate fˆ1 (t) (dotted line) 0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

f2(t) f2hat(t)

0

−0.1 0

5

10

15 time step k

20

Fig. 8.3 Fault f2 (t) (solid line) and its estimate fˆ2 (t) (dotted line)

8.5 An Illustrative Example

133



 564.7326 −274.1576 1.5693 −1.1476 , R2 = 103 × , −274.1576 301.1453 −1.1476 1.0134     K1 = −338.6052 −446.6806 , K2 = −452.3254 −530.7370 . R1 =

Take the learning law Γ = 2000 and the sampling period T = 0.01, the time delay  T is assumed as h = 3, and the initial condition is selected as x(0) = 0.3 −0.2 . If there is no fault, the state response of the closed-loop system is given in Figure 8.4. If a fault occurs and is supposed as follows:  0, 0 ≤ t ≤ 20 f3 (t) = 6, 20 < t ≤ 100 0.3

0.2

x1 x2

0.1

0

−0.1

−0.2

0

10

20

30

40

50 time step k

60

70

80

90

100

90

100

Fig. 8.4 Time response of the state viable x1 (t) and x2 (t) with no fault

0.3

0.2 x1(t) x2(t) 0.1

0

−0.1

−0.2

0

10

20

30

40

50 time step k

60

70

80

Fig. 8.5 Time response of the state viable x1 (t) and x2 (t) with fault f3 (t)

134

8 Active Fault Tolerant Control for Switched Systems with Time Delay

the state response of the closed-loop system is given in Figure 8.5. It can be seen from the figure that the closed-loop system is asymptotically stable.

8.6

Summary

In this chapter, the problem of active FTC against actuators failure in switched system with time delay has been addressed. Firstly, an adaptive FE algorithm is proposed, which can exactly and fast estimate the fault. Based on the FE information, observer-based state feedback fault tolerant controller is designed such that the closed-loop system is asymptotically stable. An example is given to illustrate the effectiveness of the proposed method.

Chapter 9

Fault Estimation and Accommodation for Switched Systems with Time-Varying Delay

This chapter considers the problem of FE and accommodation for a class of switched systems with time-varying delay. An adaptive FE algorithm is proposed to estimate the fault, moreover, constant or time-varying fault can be estimated. Meanwhile, a delay-dependent criteria is obtained with the purpose of reducing the conservatism of the fault estimation algorithm design. On the basis of FE, an observer-based fault tolerant controller is designed to guarantee the stability of the closed-loop system. Additionally, simulation results are presented to illustrate the efficiency of the proposed results.

9.1

Introduction

In order to increase the reliability and safety of dynamics systems, fault diagnosis and FTC have been intensively investigated in recent years. Generally speaking, FTC includes two main approaches: passive and active. In passive FTC systems, which use feedback control laws that are robust with respect to some fixed faults [182, 183]. However, it may be failed for other accidental faults. Under this case, we must resort to active FTC technique, which relies on a FD and isolation (FDI) process to identify the fault-induced changes [184]. The control law is reconfigured online in response to the FDI decisions and hence has better fault-tolerance capability. It can be seen that FDI process is the first step in active FTC to monitor the system and determine the location of the fault. Then, FE is utilized to on-line generate magnitude of the fault. Therefore, FD is a very meaningful and challenging topic, which has attracted many researches to devote themselves into this study domain. During the past decades, various effective methods, such as sliding mode observer approach using equivalent output injection signal [184], adaptive technique [185], and learning method based on neural network [186, 103] and so on, have been developed for the FE problem. Among them, adaptive fault diagnosis observer has been proved to be an effective approach. The advantage of the adaptive fault diagnosis observer is that the state vector estimation and actuator FE can be obtained simultaneously. c Springer International Publishing Switzerland 2015  D. Du, B. Jiang, and P. Shi, Fault Tolerant Control for Switched Linear Systems, Studies in Systems, Decision and Control 21, DOI: 10.1007/978-3-319-15162-5_9

135

136

9 Fault Estimation and Accommodation for Switched Systems

On the other hand, relatively few FD work was done for switched system compared with the plentiful FD achievements for general dynamic systems. FD for switched systems was considered in [187, 188], passive and active FTC for nonlinear switched systems with actuator faults were separately investigated in [189] and [190]. Sensor fault case for switched systems was studied in [191]. Switched system belongs to hybrid system, which consists a finite number of subsystems and an associated switching signal governing the switching among them. Many physical or man-made systems can be modeled as switched systems, including chemical processes, computer controlled systems, switched circuits and so on. During the past three decades, lots of stability theoretic results have been reported for switched systems [24, 126, 140, 192, 48]. On the other hand, time delay is the inherent features of many physical process and the big sources of instability and poor performances. Switched systems with time delay have strong engineering background, such as in network control systems [128] and power systems [129]. Existing stability criteria of delay systems can be classified into two types: delay-dependent and delay-independent methods. In recent years, much attention has been drawn to the development of delay-dependent conditions aimed at reducing the conservatism, and many theoretical studies have been conducted for switched systems with time delay [27, 16, 31]. However, to the best of our knowledge, the problem of fault estimation and accommodation for switched systems with time-varying delay has not been considered yet, which motivates us to investigate this meaningful issue. In this work, based on adaptive fault diagnosis observer method and average dwell-time technique, the problem of FE and accommodation for switched systems with time-varying delay is solved. A novel adaptive estimate law is proposed by solving some constrained linear matrix inequalities. An efficient algorithm is developed to solve these constrained LMIs and simulation results prove the effectiveness of the proposed method. The contributions of this chapter can be summarized as the following two aspects: 1:)An adaptive estimate law is developed for switched system with time-varying delay. By utilizing piecewise Lyapunov function and average dwell-time technique, a novel adaptive FE algorithm is designed. Moreover, the obtained result is delay-dependent, which makes further efforts to degrade the conservativeness. On the other hand, many FE law is confined to estimate constant fault. The advantage of such proposed estimation algorithm can not only estimate the fault fast and exactly, but also is adaptive to estimate two type of fault: constant fault case and time-varying fault case. 2:)To review the development of FD for switched systems, the FE and accommodation for switched system with time-varying delay is not considered yet. Additionally, the time-delay is time-varying and constrained in an interval set. This condition is more practical than constant time delay case. This chapter investigates this issue and efficient results are given. Another point should be pointed out that average dwell-time technique is utilized to stabilize the switched system, which is more general and flexible than dwell time switching and arbitrary switching signal.

9.2 Model Description

137

The chapter is organized as follows. Section 9.2 gives the model description. Section 9.3 presents the FE algorithm, and an observer-based fault tolerant controller is designed. An example is illustrated in Section 9.4 to show the usefulness and applicability of the proposed approaches, and the chapter is concluded in Section 9.5.

9.2

Model Description

The switched system is described as follows : x(t) ˙ = Aσ (t) x(t) + Ad σ (t) x(t − d(t)) + Bσ (t)(u(t) + f (t))

(9.1)

y(t) = Cσ (t) x(t)

(9.2)

where x(t) ∈ Rn is the state vector, y(t) ∈ Rm is the output vector, u(t) ∈ R p is the control input, f (t) ∈ Rl is the actuator fault. σ (t) : [0, +∞) → ψ = {1, · · · , N} is the switching signal, N > 1 is the number of subsystems. At an arbitrary continuous time t, σ (t), denoted by σ for simplicity, is dependent on t or x(t), or both, or other switching rules. Aσ (t) , Ad σ (t) , Bσ (t) , and Cσ (t) are constant matrices with appropriate dimensions for all σ (t) ∈ ψ . When the i-subsystem is activated, we denote the matrices associated with σ (t) = i by Aσ (t) = Ai , Ad σ (t) = Adi , Bσ (t) = Bi , and Cσ (t) = Ci . d(t) is the time delay and assumed to satisfy the following condition: C1:d (t) is differentiable and bounded with a constant delay-derivative bound: h1 ≤ d(t) ≤ h2 ,

˙ ≤d t0 where t0 is the time of fault occurring. That is, f (t) is zero prior to the failure time t ≤ t0 and is f0 (t) after the failure occurs t > t0 . It is assumed that the derivation f0 (t) with respect to time is norm bounded, i.e.  f0 (t) ≤ f1 ,  f˙0 (t) ≤ f2 , where 0 ≤ f1 < ∞, 0 ≤ f2 < ∞. The following lemmas are added for the convenes of later proof. Lemma 9.1. ([193]) Given matrices W , X and Y of appropriate dimensions and with W symmetrical, then W + X F (t)Y + Y T F T (t)X T < 0 holds for all F (t) satisfying F T (t)F (t) ≤ I if and only if for some ε > 0, W + ε X X T + ε −1 Y T Y < 0

138

9 Fault Estimation and Accommodation for Switched Systems

Lemma 9.2. ([181]) For any real vectors a, b and matrix G > 0 of compatible dimensions, the following inequality holds: aT b + bT a ≤ aT Ga + bT G−1 b, a, b ∈ Rn Lemma 9.3. [194]Consider the switched system x(t) ˙ = fσ (x(t)), σ ∈ ψ and let α > 0, µ > 1 be given constants. Suppose that there exist functions Vσ (t) : Rn → R, and two class functions β1 and β2 such that β1 (|x(t)|) ≤ Vi (x(t)) ≤ β2 (|x(t)|), V˙i (x(t)) ≤ −α Vi (x(t)), ∀i ∈ ψ , and V˙i (x(t)) ≤ µ V j (x(t)), ∀(i, j) ∈ ψ × ψ then the system is globally uniformly asymptotically stable for any switching signal with ADT

τa > τa∗ = ln µ /α Remark 9.1. As a special class of arbitrary switching signal, ADT switching means that the number of switches in a finite interval is bounded and the average time between consecutive switching is not less than a constant. Rapid progress in the field has shown that ADT switching is more general and flexible than dwell time switching signal [143].

9.3 9.3.1

Main Results Fault Diagnosis Observer Design

To diagnose the actuator fault, the following fault diagnosis observer is designed: ˙ˆ = Aσ (t) x(t) ˆ + Ad σ (t) x(t ˆ − d(t)) + Bσ (t)(u(t) + fˆ(t)) x(t) −Lσ (t) (y(t) ˆ − y(t))

(9.5)

y(t) ˆ = Cσ (t) x(t) ˆ

(9.6)

where x(t) ˆ ∈ Rn is the state vector of the observer, y(t) ˆ ∈ Rm is the output vector of the observer, fˆ(t) is an estimate of f (t), Lσ (t) is the observer gain. Denote e(t) = x(t) ˆ − x(t); r(t) = y(t) ˆ − y(t); f˜(t) = fˆ(t) − f (t);

(9.7)

then the error dynamics is described by e(t) ˙ = Aσ (t) e(t) + Ad σ (t)e(t − d(t)) + Bσ (t) f˜(t)

(9.8)

r(t) = Cσ (t) e(t)

(9.9)

where Aσ (t) = Aσ (t) − Lσ (t)Cσ (t) . As a result, the purpose of FE is to find a diagnostic algorithm for fˆ(t) such that lim e(t) = 0;

t→∞

lim fˆ(t) = f (t)

t→∞

(9.10)

9.3 Main Results

9.3.2

139

Fault Estimation Algorithm Design

In the sequel, a convergent adaptive fault diagnostic algorithm to estimate the fault f (t) is given, which is obtained from the residual r(t). Consequently, f˙(t) = 0 implies that the residual r(t) with respect to time is r˙(t) = ˙fˆ(t) − f˙(t)

(9.11)

Theorem 9.1. For the time delay d(t) satisfying the condition C1 and a given scalar α > 0, if there exist positive definite matrices Pi , Xi , Ri , Yi , Ui , Vi , Qi1 , Qi2 , Qi3 , Zi1 , Zi2 , G, matrices Wi , Fi , Ni1 , Ni2 , Mi1 , Mi2 , Si1 , Si2 , and a scalar τ > 0, such that

 Pi I (9.12) ≥ 0, Pi Xi = I, i = 1, 2, · · · , N. I Xi

 Ri Xi ≥ 0, Yi Ri = I, UiVi = I, i = 1, 2, · · · , N. Xi Vi

 Πi Γi Ψi = 0 is a given weighting matrix. Proof. Construct the following piecewise Lyapunov function candidate as T

Vσ (t) (t) = e (t)Pσ (t) e(t) + 2

+∑

 t

j=1 t−h j

+ +

t−d(t)

eT (s)Qσ (t)3 eα (s−t) e(s)ds

eT (s)Qσ (t) j eα (s−t) e(s)ds

 0  t

−h2 t+θ  −h1  t −h2

t

e˙T (s)Zσ (t)1 eα (s−t) e(s)dsd θ ˙

t+θ

e˙T (s)Zσ (t)2 eα (s−t) e(s)dsd θ + f˜T (t)Γ −1 f˜(t) (9.18) ˙

Set σ (t) = i and taking the derivative of (9.18), we obtain T ˙ V˙i (t) = 2eT (t)Pi e(t) ˙ + eT (t)Qi3 e(t) − (1 − d(t))e (t − d(t))Qi3 e−α d(t) e(t − d(t)) 2

¯ i2 )e(t) + ∑ [eT (t)Qi j e(t) − eT (t − h j )Qi j e−α h j e(t − h j )] + e˙T (t)(h2 Zi1 + hZ ˙ j=1

−

 t

t−h2

˙ − e˙T (s)Zi1 eα (s−t) e(s)ds

 t−h1 t−h2

˙ e˙T (s)Zi2 eα (s−t) e(s)ds

+α eT (t)Pi e(t) − α V (t) + 2 f˜T (t)Γ −1 ˙f˜(t) ≤ 2eT (t)Pi e(t) ˙ + eT (t)Qi3 e(t) − (1 − d)eT (t − d(t))Qi3 e−α h2 e(t − d(t)) 2

¯ i2 )e(t) + ∑ [eT (t)Qi j e(t) − eT (t − h j )Qi j e−α h j e(t − h j )] + e˙T (t)(h2 Zi1 + hZ ˙ j=1

−

 t

t−h2 T

T

e˙ (s)Zi1 e

−α h2

e(s)ds ˙ −

 t−h1 t−h2

˙ e˙T (s)Zi2 e−α h2 e(s)ds

+α e (t)Pi e(t) − α Vi (t) − 2 f˜T (t)FiCi e(t) ˙ − 2 f˜T (t)FiCi e(t) −2 f˜T (t)Γ −1 f˙(t)

(9.19)

9.3 Main Results

141

From the Newton-Leibniz formula, the following equations are true for any matrices Ni1 , Ni2 , Mi1 , Mi2 , Si1 , and Si2 with appropriate dimensions.

 t T T 2[e (t)Ni1 + e (t − d(t))Ni2 ] × e(t) − e(t − d(t)) − e(s)ds ˙ = 0 (9.20) t−d(t) 

 t−d(t) e(s)ds ˙ =0 2[eT (t)Mi1 + eT (t − d(t))Mi2 ] × e(t − d(t)) − e(t − h2 ) − t−h2

 2[eT (t)Si1 + eT (t − d(t))Si2] × e(t − h1) − e(t − d(t)) −

t−h1

t−d(t)

(9.21)  e(s)ds ˙ =0

(9.22)

Moreover, the following equalities hold:  t

t−h2

e˙T (s)Zi1 e−α h2 e(s)ds ˙ =

 t−d(t) t−h2  t

e˙T (s)Zi1 e−α h2 e(s)ds ˙

+

t−d(t)

 t−h1 t−h2

˙ = e˙T (s)Zi1 e−α h2 e(s)ds

 t−d(t)

˙ e˙T (s)Zi1 e−α h2 e(s)ds

(9.23)

e˙T (s)Zi2 e−α h2 e(s)ds ˙

t−h2  t−h1

+

t−d(t)

e˙T (s)Zi2 e−α h2 e(s)ds ˙

(9.24)

Thus, from (9.20) and (9.23), we can obtain that −

 t

t−d(t)  t

˙ − 2ξ T (t)Ni e˙T (s)Zi1 e−α h2 e(s)ds

 t

e(s)ds ˙

t−d(t)

−α h2 −α h2 [ξ T (t)Ni + e(s)Z ˙ ](Zi1 e−α h2 )−1 [ξ T (t)Ni + e(s)Z ˙ ]ds i1 e i1 e t−d(t) (9.25) +h2 ξ T (t)Ni (Zi1 e−α h2 )−1 Ni ξ (t)

≤−

where ⎡

⎤ e(t) ⎢ e(t − h) ⎥ ⎢ ⎥ ⎥ ξ (t) = ⎢ ⎢ e(t − h1 ) ⎥ , ⎣ e(t − h2 ) ⎦ f˜(t)

⎡

⎤ Ni1 ⎢ Ni2 ⎥ ⎢ ⎥ ⎥ Ni = ⎢ ⎢ 0 ⎥. ⎣ 0 ⎦ 0

By using the same methods as (9.25), from (9.21)-(9.24), we obtain −

 t−d(t) t−h2

˙ − e˙T (s)Zi1 e−α h2 e(s)ds

 t−d(t) t−h2

e˙T (s)Zi2 e−α h2 e(s)ds ˙

(9.26)

142

9 Fault Estimation and Accommodation for Switched Systems

−2ξ T (t)Mi ≤−

 t−d(t) t−h2 T

 t−d(t)

e(s)ds ˙

t−h2

−α h2 [ξ T (t)Mi + e(s)(Z ˙ ][(Zi1 + Zi2 )e−α h2 ]−1 i1 + Zi2 )e

−α h2 ]ds ˙ ×[ξ (t)Mi + e(s)(Z i1 + Zi2 )e T −α h2 −1 ¯ ] Mi ξ (t) +hξ (t)Mi [(Zi1 + Zi2 )e

(9.27)

and −

 t−h1

t−d(t)

≤−

 t

t−d(t) ¯ T

 t−h1

e˙T (s)Zi2 e−α h2 e(s)ds ˙ − 2ξ T (t)Si

e(s)ds ˙

t−d(t)

−α h2 −α h2 ](Zi2 e−α h2 )−1 [ξ T (t)Si + e(s)Z ]ds ˙ ˙ [ξ T (t)Si + e(s)Z i2 e i2 e

+hξ (t)Si (Zi2 e−α h2 )−1 Si ξ (t)

(9.28)

where ⎡

⎤ Mi1 ⎢ Mi2 ⎥ ⎢ ⎥ ⎥ Mi = ⎢ ⎢ 0 ⎥, ⎣ 0 ⎦ 0

By Lemma 9.2, one can get that

⎡

⎤ Si1 ⎢ Si2 ⎥ ⎢ ⎥ ⎥ Si = ⎢ ⎢ 0 ⎥. ⎣ 0 ⎦ 0

− 2 f˜T (t)Γ −1 f˙(t) ≤ f˜T (t)G f˜(t) + f˙T (t)Γ −1 G−1Γ −1 f˙(t) ≤ f˜T (t)G f˜(t) + f22 λmax (Γ −1 G−1Γ −1 )

(9.29)

(9.30)

From (9.19), (9.25), (9.27), (9.28), and (9.30), one can get that T ¯ i2 )Ai + h2 Ni (Zi1 e−α h2 )−1 Ni V˙i (t) + α Vi(t) ≤ ξ T (t){Ξi + Ai (h2 Zi1 + hZ ¯ i (Zi2 e−α h2 )−1 Si }ξ (t) ¯ i [(Zi1 + Zi2 )e−α h2 ]−1 Mi ] + hS +hM

−

 t

t−d(t) T

−α h2 [ξ T (t)Ni + e(s)Z ](Zi1 e−α h2 )−1 ˙ i1 e

−α h2 ×[ξ (t)Ni + e(s)Z ˙ ]ds i1 e

−

 t−d(t) t−h2 T

−α h2 [ξ T (t)Mi + e(s)(Z ˙ ][(Zi1 + Zi2 )e−α h2 ]−1 i1 + Zi2 )e

−α h2 ]ds ˙ ×[ξ (t)Mi + e(s)(Z i1 + Zi2 )e

−

 t

t−d(t)

−α h2 [ξ T (t)Si + e(s)Z ](Zi2 e−α h2 )−1 [ξ T (t)Si ˙ i2 e

−α h2 ]ds + f22 λmax (Γ −1 G−1Γ −1 ) +e(s)Z ˙ i2 e Ti (h2 Zi1 + hZ ¯ i2 )A i + h2 Ni (Zi1 e−α h2 )−1 NiT ≤ ξ T (t){Ξi + A

9.3 Main Results

143

¯ i [(Zi1 + Zi2 )e−α h2 ]−1 MiT +hM ¯ i (Zi2 e−α h2 )−1 SiT }ξ (t) + δ +hS

(9.31)

where ⎡

ϕ11 ⎢ ∗ ⎢ Ξi = ⎢ ⎢ ∗ ⎣ ∗ ∗

⎤ ⎡ T ⎤T ϕ12 ϕ¯ 15 Si1 −Mi1 Ai ⎢ AT ⎥ ϕ22 Si2 −Mi2 ATdiCiT FiT ⎥ ⎥ ⎢ di ⎥ ⎥, A  = ⎢ BT ⎥ , 0 0 ∗ −e−α h1 Qi1 ⎥ ⎢ i ⎥ ⎦ ⎣ 0 ⎦ ∗ ∗ −e−α h2 Qi2 0 ∗ ∗ ∗ −2FiCi Bi + G 0 T

δ = f22 λmax (Γ −1 G−1Γ −1 ).

ϕ¯ 15 = Pi Bi − CiT FiT − Ai CiT FiT ,

Set Wi = Pi Li and use Schur complement lemma, one can get that matrix Ξi is equivalent to the following formula ⎡ T T⎤ ⎤ ⎡ Ci Wi 0 ⎢ 0 ⎥ ⎢ 0 ⎥ ⎥ −1  ⎢ ⎥ −1    ⎢ ⎥ ⎥ 0 0 0 0 CiT FiT WiCi 0 0 0 0 + ⎢ (9.32) Πi + ⎢ ⎢ 0 ⎥ Pi ⎢ 0 ⎥ Pi ⎣ 0 ⎦ ⎣ 0 ⎦ 0 FiCi Set τ = min{ε , ε −1 } and use Lemma 9.1, if the matrix Pi satisfies (9.12), one can see that (9.14) implies (9.32) < 0. By using (9.13), (9.14), and (9.31), one obtains that V˙i (t) + α Vi (t) ≤ −ε ξ (t)2 + δ

(9.33)

where ε is the minimum eigenvalue of −Ψi . It follows that V˙i (t) + α Vi(t) < 0 f or ε ξ (t)2 > δ

(9.34)

witch means that ξ (t) converges to a small set according to Lyapunov stability theory. Integrating the inequalities (9.34) gives that Vi (t) ≤ e−α (t−tk )Vi (tk ) for any given t ∈ [tk ,tk+1 ). From (9.16) and (9.18), at the switching instant tk , it follows that V˙σ (tk ) (tk ) ≤ µ Vσ (t − ) (tk− ), k = 0, 1, 2, · · · k

Thus, using (9.35) and Nσ (t0 ,t) ≤

t−t0 τa ,

one can get

Vi (t) ≤ e−α (t−tk ) µ Vσ (t − ) (tk− ) ≤ · · · ≤ e−α (t−t0 ) µ Nσ (t0 ,t)Vσ (t0 ) (t0 ) k

= e−α (t−t0 ) eNσ (t0 ,t) ln µ Vσ (t0 ) (t0 ) ≤ e−α (t−t0 ) e

t−t0 τa

ln µ

Vσ (t0 ) (t0 )

(9.35)

144

9 Fault Estimation and Accommodation for Switched Systems ln µ

= e−(α − τa =e

−λ (t−t0 )

)(t−t0 )

Vσ (t0 ) (t0 )

Vσ (t0 ) (t0 )

(9.36)

where λ = 12 (α − lnτaµ ). Set a = min λmin (Pi ), b = max λmax (Pi ) + h1 max λmax (Qi1 ) + h2 max λmax (Qi2 ) + h2 max λmax (Qi3 ) +

h2 − h21 h22 max λmax (Zi1 ) + 2 max λmax (Zi2 ). 2 2

(9.37)

It follows (9.36) and (9.37) that a2 e(t)2 ≤ b2 e−2λ (t−t0 ) et0 2c

(9.38)

b e(t) ≤ e−λ (t−t0 ) et0 c a

(9.39)

That is

witch means that the state error e(t) → 0 and the fault error f˜(t) → 0. Then the proof is completed. Remark 9.2. From the adaptive FE algorithm (9.17), one can get that it contains the derivative of r(t) and r˙(t). It is feasible when r˙(t) can be obtained. But if the signal r˙(t) can not be easily obtained from certain systems, we should resort to other alternative methods. In order to deal with this problem, r˙ f (t) is introduced to be a substitute for r˙(t). The relationship is defined as follows: 1 r˙ f (t) = − (r f (t) − r(t)) ε

(9.40)

From (9.40), one can get that under zero initial condition, using Laplace transform yields r˙ f (t) =

1 r˙(t) εs + 1

(9.41)

Therefore, it it easy to show that the substitute r˙ f (t) can approximate to r˙(t) with any desired accuracy as ε → 0. Meanwhile, when s → 0, that is t → ∞, r˙ f (t) asymptotically converges to r˙(t). Remark 9.3. In the proving process, we set ˙f˜(t) = ˙fˆ(t) − f˙(t). If the fault is a constant, which means that f˙(t) = 0. Under this case, the fault estimate algorithm will be changed into ˙fˆ(t) = −Γ Fi r˙(t), which is only adaptive to estimate constant faults. From the above discussion, one can get that the estimate algorithm in (9.17) is not only adaptive to estimate constant faults, but also adaptive to estimate time-varying faults.

9.3 Main Results

145

Remark 9.4. It can be seen that the conditions (9.12) and (9.13) are not strict LMI formation due to the equation Pi Xi = I, RiYi = I, and UiVi = I, which can not be solved directly by Matlab LMI Control Toolbox. However, we can solve this nonconvex feasibility problem by formulating it into a special sequential optimization problem subject to LMI constraints. In the following, a specific algorithm is given by utilizing the result in [147]. Algorithm 1 (0) (0) (0) (0) (0) (0) (0) (0) (0) (0) Step 1: Find a feasible set {Pi , Xi , Ri ,Yi ,Ui ,Vi , Qi1 , Qi2 , Qi3 , Zi1 , (0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

Zi2 ,Wi , Fi , G(0) , Ni1 , Ni2 , Mi1 , Mi2 , Si1 , Si2 , τ (0) , } satisfying (9.12), (9.13) and (9.14). Set k = 0. Step 2: Solve the following LMI problem   # N " (k) (k) (k) (k) (k) (k) min tr ∑ Pi Xi + Pi Xi + RiYi + Ri Yi + UiVi + Ui Vi i=1

subject to (9.12), (9.13) and (9.14). Step 3: Substitute the obtained matrix variables {Pi , Xi , Ri ,Yi ,Ui ,Vi , Qi1 , Qi2 , Qi3 , Zi1 , Zi2 ,Wi , Fi , G, Ni1 , Ni2 , Mi1 , Mi2 , Si1 , Si2 , τ , } into (9.12), (9.13) and (9.14). If the condition (9.12) is satisfied with N

|tr( ∑ (Pi Xi + RiYi + UiVi )) − 3(N + 1)n| < ρ i=1

for some sufficient small scalar ρ > 0, then output the feasible solution {Pi , Xi , Ri ,Yi , Ui ,Vi , Qi1 , Qi2 , Qi3 , Zi1 , Zi2 ,Wi , Fi , G, Ni1 , Ni2 , Mi1 , Mi2 , Si1 , Si2 , τ , }, EXIT. Step 4: If k > N where N is the maximum number of iterations allowed, EXIT. (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) Step 5: Set k = k + 1, {Pi , Xi , Ri ,Yi ,Ui ,Vi , Qi1 , Qi2 , Qi3 , Zi1 , Zi2 , (k)

(k)

(k)

(k)

(k)

(k)

(k)

(k)

Wi , Fi , G(k) , Ni1 , Ni2 , Mi1 , Mi2 , Si1 , Si2 , τ (k) , } = {Pi , Xi , Ri ,Yi ,Ui ,Vi , Qi1 , Qi2 , Qi3 , Zi1 , Zi2 ,Wi , Fi , G, Ni1 , Ni21 , Mi1 , Mi21 , Si1 , Si2 , τ , }, and go to Step 2.

9.3.3

Fault Accommodation

Since the state x(t) is unavailable, the estimation value x(t) ˆ is substituted for x(t). Therefore, the observer-based normal controller is given ˆ + d(t) ur (t) = −Kσ (t) x(t)

(9.42)

where Kσ (t) is the feedback gain matrix and d(t) is the reference input. Once a fault occurs, based on the accurate and rapid estimation of the fault , the following observer-based fault-tolerant controller is activated to compensate for the fault. u(t) = ur (t) − fˆ(t)

(9.43)

146

9 Fault Estimation and Accommodation for Switched Systems

Assuming d(t) = 0 and substituting (9.43) into (9.1), one obtains  x(t) ˙ = (Aσ (t) − Bσ (t) Kσ (t) )x(t) + Ad σ (t) x(t − d(t)) + ρ (t) y(t) = Cσ (t) x(t)

(9.44)

where ρ (t) = −Bσ (t) Kσ (t) e(t) − Bσ (t) f˜(t). From the result of Theorem 9.1, one can get that e(t) → 0 and f˜(t) → 0 when t → ∞. The signal ρ (t) can be treated as a disturbance of the system (9.44). So, if only the feedback gain Ki can ensure that the following system is asymptotically stable.  x(t) ˙ = (Aσ (t) − Bσ (t) Kσ (t) )x(t) + Ad σ (t)x(t − d(t)) (9.45) y(t) = Cσ (t) x(t) Construct the corresponding piecewise Lyapunov function as Vσ (t) (t) = xT (t)Pσ (t) x(t) +

t

t−d(t)

xT (s)Qσ (s) eα (s−t) x(s)ds

(9.46)

where Pσ (t) and Qσ (t) are positive definite matrices. Thus, we have the following theorem. Theorem 9.2. For the time delay d(t) satisfying the condition C1 and a given scalar α > 0, if there exist positive definite matrices Xi , Ti , matrix Ji , such that ⎤ ⎡ Ai Xi + XiATi − Bi Ji − JiT BTi + α Xi Ahi Ti Xi ⎣ (9.47) ∗ −(1 − d)e−α h2 Ti 0 ⎦ < 0 ∗ ∗ −Ti where Ji = Ki Xi . Then, for the switching signal σ (t) with ADT satisfying

τa ≥ τa∗ =

ln µ α

(9.48)

where

µ ≥ 1 with Pi ≤ µ Pj , Qi ≤ µ Q j .

(9.49)

such that the system (9.45) is asymptotically stable. Proof. Taking the derivative of Vσ (t) along the trajectories of the system in (9.45) is V˙i (t) ≤ 2xT (t)Pi (Ai − Bi Ki )x(t) + 2xT (t)Pi Adi x(t − d(t)) + xT (t)Qi x(t) −(1 − d)xT (t − d(t))Qi e−α h2 x(t − d(t)) + α xT (t)Pi x(t) − α Vi = η T (t)ϒi η (t) − α Vi

(9.50)

9.4 An Illustrative Example

147

where

T xT (t) η (t) = T , x (t − d(t))

 Pi (Ai − Bi Ki ) + (Ai − Bi Ki )T Pi + α Pi + Qi Pi Adi ϒi = . ∗ −(1 − d)Qie−α h2 Let Pi = Xi−1 , Qi = Ti−1 , one can get that the LMI in (9.47) means that V˙i (t) + α Vi (t) < 0. Integrating the inequalities gives that Vi (t) ≤ e−α (t−tk ) Vi (tk ) for any given t ∈ [tk ,tk+1 ). From (9.47) and (9.49), at the switching instant tk , it follows that V˙σ (tk ) (tk ) ≤ µ Vσ (t − ) (tk− ), k = 0, 1, 2, · · · k

Thus, using (9.51) and Nσ (t0 ,t) ≤

t−t0 τa ,

(9.51)

one can get

Vi (t) ≤ e−α (t−tk ) µ Vσ (t − ) (tk− ) ≤ · · · ≤ e−α (t−t0 ) µ Nσ (t0 ,t) Vσ (t0 ) (t0 ) k

= e−α (t−t0 ) eNσ (t0 ,t) ln µ Vσ (t0 ) (t0 ) ≤ e−α (t−t0 ) e =e

t−t0 τa

ln µ

−(α − lnτaµ )(t−t0 )

Vσ (t0 ) (t0 )

Vσ (t0 ) (t0 )

= e−λ (t−t0 ) Vσ (t0 ) (t0 )

(9.52)

where λ = 12 (α − lnτaµ ). Set a = min λmin (Pi ), b = max λmax (Pi ) + h2 max λmax (Qi ).

(9.53)

It follows (9.52) and (9.53) that a2 x(t)2 ≤ b2 e−2λ (t−t0 ) xt0 2c

(9.54)

b x(t) ≤ e−λ (t−t0 ) xt0 c a

(9.55)

That is

Therefore, the system (9.45) is asymptotically stable according to standard Lyapunov stability theory.

9.4

An Illustrative Example

Consider a switched electrical circuit model borrowed from [195], which is shown in Figure 9.1.

148

9 Fault Estimation and Accommodation for Switched Systems

Fig. 9.1 A switched electrical circuit

This circuit has two modes, that is, N = 2, σ (t) : [0, ∞) → {1, 2}. In mode 1, called the ‘on’ time, Sw1 is closed and Sw2 is open. In mode 2, called the ‘off’ time, Sw1 is open and Sw2 is closed. In this system, Sw1 is often a bipolar transistor and Sw2 is a diode. Vc is used to denote the capacitor voltage equal to the output volt age delivered to the load R1 , and I1 denotes the inductor current. During the ‘on’ time, the inductor current is also equal to the input source current. During the ‘off’ time, the input source current is zero. On the other hand, time delay can enter into this system due to the existence of inductor and/or the transmission channel. Thus, it is reasonable to describe this electrical circuit into a switched time-delay system as system (9.1). The parameter matrices are chosen as follows:

   −0.54 1.02 −0.01 0.1 0.18 0.36 , A2 = , Ad1 = , A1 = 0.17 −0.31 0.01 0.04 −0.06 −0.12





   0.11 0.18 0.1 0.2 Ad2 = , B1 = , B2 = , C1 = 0.1 0.2 , −0.03 −0.04 0.2 0.4   C2 = 0.2 0.3 .

Choose time-varying delay signal as d(t) = 0.5 + 0.2 sin(t), we can get the upper bound of h1 = 0.3, h2 = 0.7, and d = 0.2, respectively. For given α = 1 and µ = 2.5, by utilizing Algorithm 1 to solve the conditions in Theorem 9.1, we can get a set of solutions as follows:



 3.3636 −1.4334 1.4910 2.1748 P1 = , P2 = , −1.4334 1.2879 2.1748 3.1725



 0.0566 0.0629 6.9935 −4.7942 X1 = 10−3 × , X2 = , 0.0629 0.1477 −4.7942 3.2868



 1.9284 0.3046 3.2193 −1.2598 R1 = , R2 = , 0.3046 2.2953 −1.2598 2.3752



 0.5297 −0.0703 0.3920 0.2079 −4 −4 Y1 = 10 × ,Y2 = 10 × , −0.0703 0.4450 0.2079 0.5313



 −0.4054 −2.7083 −2.9822 −6.7106 U1 = ,U2 = , −2.7083 −4.9279 −6.7106 −3.7803

9.4 An Illustrative Example

149



 0.9233 −0.5074 0.1120 −0.1988 V1 = 10−8 × ,V2 = 10−8 × , −0.5074 0.0760 −0.1988 0.0883



 2.7830 1.3331 2.5064 3.7597 , Q21 = , Q11 = 1.3331 4.9380 3.7597 5.6397





 1.4844 0.9763 1.8480 2.7720 1.5443 1.3323 Q12 = , Q22 = , Q13 = , 0.9763 3.1418 2.7720 4.1581 1.3323 3.8726



  1.9675 2.9514 5.0262 0.0356 4.9712 0.2028 Q23 = , Z11 = , Z21 = , 2.9514 4.4273 0.0356 5.1985 0.2028 5.0999



 5.0154 0.0390 4.9273 0.1110 Z12 = , Z22 = , 0.0390 5.0371 0.1110 5.0003



 −1.8934 −0.6159 −1.7811 −2.6698 N11 = , N21 = , −1.2114 −3.8431 −2.6713 −4.0046



 0.9297 0.5412 0.9868 1.4810 N12 = , N22 = , 0.2402 2.3082 1.4801 2.2216



 −0.6748 −0.4105 −0.7274 −1.0911 M11 = , M21 = , −0.3886 −1.3313 −1.0910 −1.6366



 −0.2935 −0.4025 −0.9154 −1.3732 M12 = , M22 = , −0.4007 −1.1154 −1.3731 −2.0596



 2.0351 0.8498 1.5192 2.2788 S11 = , S21 = , 0.7902 3.3010 2.2787 3.4182





 0.8180 0.9604 1.9584 2.9377 3.3111 S12 = , S22 = ,W1 = , 0.9207 2.7914 2.9374 4.4063 2.4778

 2.0500 W2 = , F1 = 1.9600, F2 = 1.8311, G = 9.1297, τ = 0.0721. 3.0674 From (9.15), we have τa∗ = 0.9163. Taking τa = 2 ≥ τa∗ and the learning law Γ = 200, the sampling period is chosen as T = 0.1, and the control input u(t) is a unit step function. In this example, two cases of faults are considered. When the fault is a constant described as  0, 0≤t ≤5 f1 (t) = 4, 5 < t ≤ 30 In this case, the simulation result is shown in Figure 9.2. When the fault is a timevarying function described as  0, 0≤t ≤5 f2 (t) = 0.3 sin(2t) + 0.5, 5 < t < 30 The simulation result is shown in Figure 9.3. From the above simulation results, we can conclude that whether the fault is a constant or a time-varying function, the estimate algorithm proposed here can estimate them quickly and exactly.

150

9 Fault Estimation and Accommodation for Switched Systems

5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0 0

5

10

15 Time in second

20

25

30

Fig. 9.2 Fault f1 (t) (dotted line) and its estimate fˆ1 (t) (solid line) 0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

f2(t) f2hat(t)

0

−0.1 0

5

10

15 time step k

20

25

30

Fig. 9.3 Fault f2 (t) (dotted line) and its estimate fˆ2 (t) (solid line)

By solving the LMI in Theorem 9.2, one obtains



 12.7121 −13.8843 54.4373 −21.2977 , X2 = , X1 = −13.8843 79.2945 −21.2977 17.5469



 26.7326 −14.1576 11.5603 −1.1476 T1 = , T2 = , −14.1576 −21.1453 −1.1476 45.3134     K1 = −38.6732 −26.6396 , K2 = 52.3254 −30.7370 .

Take the learning law Γ = 100, the time delay is assumed as d(t) = 0.5 + 0.2 sin(t),  T and the initial condition is selected as x(0) = 0.3 −0.2 . If there is no fault, the state response of the closed-loop system is given in Figure 9.4. If a fault occurs and is supposed as follows:

9.5 Summary

151

0.3

0.2

x1 x2

0.1

0

−0.1

−0.2

0

10

20

30

40

50 time step k

60

70

80

90

100

90

100

Fig. 9.4 Time response of the state viable x1 (t) and x2 (t) with no fault 0.3

0.2 x1(t) x2(t) 0.1

0

−0.1

−0.2

0

10

20

30

40

50 time step k

60

70

80

Fig. 9.5 Time response of the state viable x1 (t) and x2 (t) with fault f3 (t)

f3 (t) =



0, 6,

0 ≤ t ≤ 20 20 < t ≤ 100

the state response of the closed-loop system is given in Figure 9.5. It can be seen from the figure that the closed-loop system is asymptotically stable.

9.5

Summary

In this chapter, the problem of FE and accommodation against actuators failure in switched system with time-varying delay has been addressed. Firstly, an adaptive FE algorithm is proposed, which can exactly and fast estimate the fault. Based on the FE information, observer-based state feedback fault tolerant controller is designed such that the closed-loop system is asymptotically stable. An example is given to illustrate the effectiveness of the proposed method.

Chapter 10

Observer-Based Reliable Control for Discrete Time Switched Systems

This chapter deals with the problem of reliable control for discrete time systems with actuator failures. The actuator is assumed to fail occasionally and can recover through a time of interval. During the time of suffering failures, the considered closed-loop system is assumed unstable. By using an ADT method and under the condition that the activation time ratio between the system without actuator failures and the system with actuator failures is not less than a specified constant, an observer-based feedback controller is developed in terms of LMI such that the resulting closed-loop system is exponentially stable. An example is included to demonstrate the effectiveness of the proposed approach.

10.1

Introduction

Once a failure occurs in the field of aerospace and industrial process, the consequence is usual disastrous. Therefore, the demand of a system’s reliability and safety is becoming more and more urgent. The aim of reliable control is to design an appropriate controller such that the closed-loop system can tolerant for some specific failures and preserve an overall system stability. The study of reliable control has recently attracted considerable attention, see for example [196, 197, 198, 199, 200, 201, 120, 202] and references therein. parts. The ultimate goal is to preserve the stability and high-priority performances of the plant by a single controller which can tolerate a severe component failure. Models of control component failures can be classified as outages, partial degradations and lock-in-place failures. When a failure modeled as outage occurs, the measured signal or the control input simply becomes zero. The partial failure is represented by a scaling factor with upper and lower bounds to the signal to be measured or to the control input. Moreover, several approaches have been developed to deal with this issue. Such as in [201], a methodology for the design of reliable linear-quadratic state feedback control by means of algebraic Riccati equation approach was introduced; the pole region assignment scheme for reliable control was proposed in [120]; [202] studied the variable structure control technique to design the reliable controllers. However, all above c Springer International Publishing Switzerland 2015  D. Du, B. Jiang, and P. Shi, Fault Tolerant Control for Switched Linear Systems, Studies in Systems, Decision and Control 21, DOI: 10.1007/978-3-319-15162-5_10

153

154

10 Observer-Based Reliable Control for Discrete Time Switched Systems

mentioned design methods are based on the basic assumption that the never failed actuators must stabilize a given system. For the case when the actuator appears failures, and the corresponding closed-loop system is not stable, the design methods of existing reliable control do not work. In recent years, the study of stability for switched systems has attracted much attention [31, 54, 76, 203, 204, 205, 206, 207, 208, 209]. Switched system is a class of hybrid systems, which consists of a finite number of subsystems (described by differential or difference equations) and an associated switching signal governing the switching among them. The switching signals may belong to a certain set and the set may be diverse. Many practical systems can be modeled as switched systems, such as chemical reaction processes, computer controlled systems, flight and air traffic control, switching power converters and so on. Among the stability analysis approaches for switched systems, dwell time approach has been proved to be an effective technique and has been presented in several relative references [70, 144, 210, 211, 212]. In [210], if the dwell time is set sufficiently large, the switched system is exponentially stable under the condition that all subsystems are stable. Subsequently, at the premise that each subsystem is stable, [144] extended the concept of dwell time to ADT. Furthermore, in [211], the attention results are extended to the case where stable and unstable subsystems co-exist. In recent study, there are several papers on dealing with problems of control systems with occasional controller failures by using ADT approach [213, 214, 215]. In this chapter, the actuator of the linear discrete time systems is assumed to fail occasionally and can recover through a time of interval. The modeled failure covers the outage cases and the possibility of partial faults. Then the linear discretetime systems can be modeled as a switched system. By using ADT technique, the problem of exponential stability for linear discrete time system via observer-based feedback control is introduced. Under the condition that activation time ratio between the system without actuator failure and the system with actuator failures is not less than a specified constant, the design of observer-based feedback controller is derived in terms of LMIs to make the resulting closed-loop system is exponential stable for all admissible actuator failures. Finally, an example is presented to illustrate the effectiveness of the proposed approach. Overall, compared with the existing results on reliable control for switched systems, the novelty of this chapter can be summarized as the following two aspects: • Based on the ADT technology, observer-based reliable controller design scheme is proposed for linear discrete time switched systems. Moreover, the controller can be directly obtained by solving some linear matrix inequalities. • A more general failure form is considered. The results investigated in [214] can be regarded as a special case of our results.

10.2 Preliminaries and Problem Formulation

10.2

155

Preliminaries and Problem Formulation

Consider the following linear discrete systems x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k)

(10.1) (10.2)

where x(k) ∈ Rn is the state vector, y(k) ∈ Rm is the measurement output vector, u(k) ∈ R p is the control input. Matrices A , B, and C are known constant matrices with appropriate dimensions. When the actuator experiences failures, we use uF (k) to describe the control signal sent from actuators. Consider the actuator failure model with failure parameters F: uF (k) = Fu(k)

(10.3)

where the matrix F satisfies 0 ≤ F = diag{ f 1 , · · · , f p } ≤ F = diag{ f1 , · · · , f p } ≤ F = diag{ f¯1 , · · · , f¯p } ≤ I, in which the variables fi (i = 1, · · · , p) quantity the failures of the actuators. Let f p + f¯p f + f¯1 F + F¯ = diag{ 1 ,···, } F˘ = diag{ f˘1 , · · · , f˘p } = 2 2 2 f¯p − f p f¯1 − f 1 F¯ − F = diag{ ,···, } F˜ = diag{ f˜1 , · · · , f˜p } = 2 2 2

(10.4) (10.5)

Applying the above definitions, we can rewrite F as follows: F = F˘ + ∆ = F˘ + diag{δ1, · · · , δ p }

(10.6)

where |δi | ≤ f˜i , i = 1, · · · , p. For the purpose of this work, the following assumptions are adopted: Assumption 7. The pair of (A, B,C) is stabilizable and observable. Assumption 8. The matrix C is of full-row rank, i. e. rank(C) = m. For the matrix C of full-row rank, there always exist two orthogonal matrices U ∈ Rm×n and V ∈ R(n−m)×n , such that   C = U Σ 0 VT (10.7) where Σ = diag{ε1 , · · · , εm }, ε1 , · · · , εm are nonzero singular values of C. Actuator failure is assumed to occur sometimes and can be recovered through a time of interval. Then, the states of the system can be dominated by the following piecewise differential equation

156

10 Observer-Based Reliable Control for Discrete Time Switched Systems

x(k + 1) = Ax(k) + Bσ (k) u(k) y(k) = Cx(k)

(10.8) (10.9)

where σ (k) : Z + = {0, 1, 2, · · ·} −→ ψ = {1, 2}, B1 = B, B2 = BF. The dynamic observer-based reliable control scheme for system (10.8)-(10.9) is described by ˆ x(k ˆ + 1) = Ax(k) ˆ + Bu(k) − Lσ (k)(y(k) − y(k)) y(k)) ˆ = Cx(k) ˆ u(k) = Kσ (k) x(k) ˆ

(10.10) (10.11) (10.12)

where x(k) ˆ ∈ Rn is the estimation state of system (10.8), Kσ (k) and Lσ (k) are controller and observer gains to be determined later. Remark 10.1. It is well known that a system’s state is usually unavailable. Therefore, it is difficult to design a state feedback controller directly to stabilize the system. So in this work, we assume that the systems (10.1) is observable. By utilizing the observer’s state to realize state feedback, such that the switched systems (10.8) is stable. In this chapter, the aim is to seek for a class of switching signals σ (k) and observerbased feedback controller Ki (i ∈ ψ ) such that the switched system (10.8)-(10.9) is exponentially stable. For any switching signal, we let T + (k0 , k) (resp. T − (k0 , k)) denote the total activation time of the system with actuator failures (resp. the system without actuator failures), during [k0 , k). Then, for any given constants α1 > 0, α2 > 0, and α ∈ (0, α1 ), we choose a scalar α ∗ ∈ (α , α1 ). Motivated by the idea in [211], we propose the following switching condition T − (k0 , k) α2 + α ∗ ≥ k≥k0 T + (k0 , k) α1 − α ∗ inf

(10.13)

holds for any given initial time k0 , where α1 and α2 are positive numbers to be chosen later. Remark 10.2. The switching condition (10.13) used to constrain the activation time of the system with controller failures T + (k0 , k) is relatively small compared with that of the system without controller failure T − (k0 , k). We give the following definitions and lemmas, which will be used in the proof of the main results. Definition 10.1. [216]The equilibrium x = 0 of system (10.8) is is globally uniformly exponentially stable under certain switching signals σ (k) if, for u(k) = 0 or and initial condition xk0 , there exist constants Γ > 0 and λ > 0, such that the solution of the system (10.8) satisfies x(k) ≤ Γ e−λ (k−k0 ) xk0 c

10.3 Main Results

157

Definition 10.2. For any switching signal σ (k) and any k2 > k1 > 0, let Nσ (τ , k) denote the number of switchings of σ (k) on an interval (k1 , k2 ). If Nσ (τ , k) ≤ N0 +

k2 − k1 τa

holds for a given N0 ≥ 0 and τa > 0, then the constant τa is called the average dwelltime and N0 the chattering bound. Remark 10.3. Definition 10.2 means that if there exists a positive number τa such that a switching signal has the ADT property, the average time interval between consecutive switching is at least τa . This is a kind of slowly switching signals, which is less conservative than arbitrary switching signals. As commonly used in the literature, for convenience, we choose N0 = 0 in this chapter. Lemma 10.1. ([217]) For a given C ∈ R p×n with rank(C) = p, assume that Q ∈ Rn×n is a symmetric matrix, then there exists a matrix Z ∈ R p×p such that CQ = ZC if and only if 

Q1 0 VT Q =V 0 Q2 where Q1 ∈ R p×p , Q2 ∈ R(n−p)×(n−p), the matrix V satisfies the condition in (10.7). Lemma 10.2. ([218]) For any real vectors a, b and matrix G > 0 of compatible dimensions, the following inequality holds: aT b + bT a ≤ aT Ga + bT G−1 b, a, b ∈ Rn

10.3

Main Results

In this section, we deal with the stability for the system (10.1) with actuator failures. Applying observer-based feedback controller described by (10.10)-(10.12) to the system (10.8), and let e(k) = x(k) − x(k), ˆ then the resulting closed-loop system is

η (k + 1) = Aσ (k) η (k),

σ (k) ∈ ψ

(10.14)

T  where η (k) = xT (k) eT (k) , 

 A + BK1 −BK1 A + BFK2 −BFK2 , A2 = A1 = . 0 A − L1C B(F − I)K2 A − L2C − B(F − I)K2 we first consider system (10.1) with controller (10.12) for the case without actuator failure, that is

η (k + 1) = A1 η (k) For the system (10.15), we have the following theorem.

(10.15)

158

10 Observer-Based Reliable Control for Discrete Time Switched Systems

Theorem 10.1. Given a constant α > 0, if there exists matrix P > 0 such that the following matrix inequality ⎡ ⎤ −e−α P ∗ ∗ ∗ ⎢ ∗ ∗ ⎥ 0 −e−α P ⎢ ⎥ (10.16) ⎣ P(A + BK1) −PBK1 −P ∗ ⎦ < 0 0 P(A − L1C) 0 −P holds, then along the trajectory of system (10.15), we have V (k + 1) < e−α V (k) Proof: Consider the Lyapunov function as follows:

 P0 V (k) = η T (k) η (k) = xT (k)Px(k) + eT (k)Pe(k) 0P

(10.17)

(10.18)

Along the trajectory of system (10.15), the Lyapunov function (10.18) satisfies V (k + 1) − e−αV (k) = xT (k + 1)Px(k + 1) + xT (k + 1)Px(k + 1) −e−α xT (k)Px(k) − e−α eT (k)Pe(k) = [(A + BK1 )x(k) − BK1 e(k)]T P[(A + BK1)x(k) − BK1 e(k)] +eT (k)(A − L1C)T P(A − L1C)e(k) − e−α xT (k)Px(k) −e−α eT (k)Pe(k) = η T (k)Λ1 η (k) where

Λ1 =

   (A + BK1 )T P A + BK1 −BK1 T −(BK1 )

−α  −e P 0 + 0 −e−α P + (A − L1C)T P(A − L1C)

By Schur complement lemma, (10.16) implies that Λ1 < 0, i. e., V (k + 1) < e−α V (k). Then the proof is completed. When the actuator occurs failure, system (10.1) becomes the following form

η (k + 1) = A2 η (k)

(10.19)

For the system (10.19), we have the following theorem. Theorem 10.2. Given a constant β > 0, if there exists matrix P > 0 such that the following matrix inequality

10.3 Main Results

159

⎡

−eβ P ∗ βP ⎢ 0 −e ⎢ ⎣ P(A + BFK2 ) −PBFK2 PB(F − I)K2 P[A − L2C − B(F − I)K2 ]

⎤ ∗ ∗ ∗ ∗ ⎥ ⎥ 0 and α2 > 0, if there exist positive definite matrices Pi , i ∈ ψ , such that the following matrix inequalities ⎡ ⎤ ∗ ∗ ∗ −e−α1 P1 ⎢ 0 −e−α1 P1 ∗ ∗ ⎥ ⎢ ⎥ (10.22) ⎣ P1 (A + BK1) −P1 BK1 −P1 ∗ ⎦ < 0 0 P1 (A − L1C) 0 −P1

160

10 Observer-Based Reliable Control for Discrete Time Switched Systems

⎡

∗ −eα2 P2 ⎢ 0 −eα2 P2 ⎢ ⎣ P2 (A + BFK2 ) −P2 BFK2 P2 B(F − I)K2 P2 [A − L2C − B(F − I)K2 ]

∗ ∗ −P2 0

⎤ ∗ ∗ ⎥ ⎥ k0 , we let k1 < k2 < · · · < kq = kNσ (k ,k) denote the switching time 0 instants of σ (k) over the interval [k0 , k). Considering Theorem 10.1 and 10.2 and using (10.22) and (10.23) yield  −α (k−k ) q V (k ), σ (k) = 1; e 1 q V (k) ≤ (10.30) (k−kq ) α 2 V (kq ), σ (k) = 2; e Combining (10.29) and (10.30) lead to V (k) ≤ eα2 T ≤e

+ (k ,k)−α T − (k ,k) q q 1

α2 T + (kq ,k)−α1 T − (kq ,k)

≤ eα 2 T

V (kq )

µ V (kq− )

+ (k − q−1 ,k)−α1 T (kq−1 ,k)

µ 2V (kq−1 )

10.3 Main Results

161

≤ ··· ··· ··· + − ≤ eα2 T (k0 ,k)−α1 T (k0 ,k) µ Nσ (k0 ,k)V (k0 ) = eα 2 T

+ (k ,k)−α T − (k ,k)+N (k ,k) ln µ σ 0 0 1 0

V (k0 )

(10.31)

It follows from (10.13) that

α2 T + (k0 , k) − α1 T − (k0 , k) ≤ α ∗ (k − k0 )

(10.32)

From Definition 10.2 and (10.24), it holds that Nσ (k0 , k) ≤

k − k0 (α ∗ − α )(k − k0 ) ≤ τa ln µ

(10.33)

Applying (10.31)-(10.33) gives V (k) ≤ e−α (k−k0 )Vσ (k0 ) (k0 )

(10.34)

According to (10.27) and (10.28), we have aη (k)2 ≤ V (k) ≤ bxk 2c

(10.35)

Combining (10.34) and (10.35) result in b 1 x(k)2 ≤ V (k) ≤ e−α (k−k0 ) xk0 2c a a

(10.36)

which is exactly (10.26). Then the proof is completed. Theorem 10.3 shows that the exponentially stability of system (10.8) is guaranteed under the condition of actuator failure. However, one can get that inequalities (10.22) and (10.23) are not LMIs for unknown controller gains Ki (i ∈ ψ ) and observer gains Li (i ∈ ψ ). In the following, the controller design problem will be derived by the form of LMI, which can be easily implemented by using the MATLAB LMI toolbox. Theorem 10.4. For given constants α1 > 0 and α2 > 0, if there exist matrices Xi > 0, Mi , and Ni , i ∈ ψ , such that ⎡ ⎤ ∗ ∗ ∗ −e−α1 X1 ⎢ ∗ ⎥ 0 −e−α1 X1 ∗ ⎢ ⎥ (10.37) ⎣ AX1 + BN1 −BN1 −X1 ∗ ⎦ < 0 0 AX1 − M1C 0 −X1 ⎡

−eα2 X2 ∗ α2 X ⎢ 0 −e 2 ⎢ ⎣ AX2 + BFN2 −BFN2 B(F − I)N2 AX2 − M2C − B(F − I)N2

∗ ∗ −X2 0

⎤ ∗ ∗ ⎥ ⎥ 0 and α2 > 0, if there exist matrices Xi > 0, Mi , and Ni , i ∈ ψ , such that ⎡ ⎤ ∗ ∗ ∗ −e−α1 X1 ⎢ 0 −e−α1 X1 ∗ ∗ ⎥ ⎢ ⎥ (10.46) ⎣ AX1 + BN1 −BN1 −X1 ∗ ⎦ < 0 0 AX1 − M1C 0 −X1

10.3 Main Results

163

⎡

⎤ −eα2 X2 ∗ ∗ ∗ ∗ ∗ ⎢ 0 −eα2 X2 ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ AX2 + BFN ˘ 2 ˘ −BFN2 −X2 ∗ ∗ ∗ ⎥ ⎢ ⎥