Analysis and Synthesis of Singular Systems (Emerging Methodologies and Applications in Modelling, Identification and Control) [1 ed.] 0128237392, 9780128237397

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ANALYSIS AND SYNTHESIS OF SINGULAR SYSTEMS

ANALYSIS AND SYNTHESIS OF SINGULAR SYSTEMS

ZHIGUANG FENG JIANGRONG LI PENG SHI HAIPING DU ZHENGYI JIANG Series Editor

QUAN MIN ZHU

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2021 Elsevier Inc. All rights reserved. MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-823739-7 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Mara Conner Acquisitions Editor: Sonnini R. Yura Editorial Project Manager: Rafael G. Trombaco Production Project Manager: Sojan P. Pazhayattil Designer: Victoria Pearson Typeset by VTeX

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Contents Preface Acknowledgments Acronyms and symbols

1. Introduction 1.1. Background 1.2. Research problems 1.3. Literature review 1.3.1. Singular systems 1.3.2. Singular systems with time-delay 1.3.3. Singular Markovian jump systems (SMJSs) 1.3.4. T-S fuzzy singular systems 1.3.5. Type-2 fuzzy singular systems 1.4. Book outline

xi xii xiii

1 1 4 5 5 9 13 15 16 17

2. Dissipative control and filtering of singular systems

21

2.1. Dissipative control of continuous-time singular systems 2.1.1. Problem formulation 2.1.2. Main results 2.1.3. Examples 2.1.4. Conclusion 2.2. Dissipative control of discrete-time singular systems 2.2.1. Problem formulation 2.2.2. Dissipative control 2.2.3. Illustrative example 2.2.4. Conclusion 2.3. Dissipative filtering of singular systems 2.3.1. Reduced-order dissipative filtering 2.3.2. Illustrative example 2.3.3. Conclusion

21 21 24 27 29 30 30 32 37 38 38 38 42 45

3. H∞ control with transients for singular systems 3.1. 3.2. 3.3. 3.4.

Performance measure Controller design Illustrative examples Conclusion

4. Delay-dependent admissibility and H∞ control of discrete singular delay systems 4.1. New admissibility analysis for discrete singular systems with time-varying delay

47 47 55 58 60

61 61 vii

viii

Contents

4.1.1. Problem formulation 4.1.2. Main results 4.1.3. Numerical example 4.1.4. Conclusion 4.2. Delay-dependent robust H∞ controller synthesis for discrete singular delay systems 4.2.1. Problem formulation 4.2.2. Robust stability 4.2.3. Stabilization 4.2.4. Robust H∞ control 4.2.5. Illustrative examples 4.2.6. Conclusion

5. Delay-dependent dissipativity analysis and synthesis of singular delay systems 5.1. Dissipativity analysis for discrete singular systems with time-varying delay 5.1.1. Problem formulation 5.1.2. Main results 5.1.3. Numerical examples 5.1.4. Conclusion 5.2. Dissipativity analysis and dissipative control of singular time-delay systems 5.2.1. Problem formulation 5.2.2. Dissipative analysis 5.2.3. State-feedback dissipative control 5.2.4. Illustrative examples 5.2.5. Conclusion 5.3. Robust reliable dissipative filtering for discrete delay singular systems 5.3.1. Problem statement 5.3.2. Reliable dissipativity analysis 5.3.3. Filter design 5.3.4. Illustrative examples 5.3.5. Conclusion

6. State-feedback control for singular Markovian systems 6.1. Admissibilization and H∞ control for singular Markovian systems 6.1.1. Admissibility of singular Markovian jump systems 6.1.2. H∞ control of singular Markovian jump systems with time delay 6.1.3. Examples 6.1.4. Conclusion 6.2. Reliable dissipative control for singular Markovian systems 6.2.1. Problem statement 6.2.2. Reliable dissipativity analysis 6.2.3. Controller design

62 64 72 73 73 74 76 85 88 92 99

101 101 101 104 109 113 113 114 115 127 130 135 135 136 139 148 152 157

159 159 159 163 167 171 171 171 174 176

Contents

6.2.4. Illustrative example 6.2.5. Conclusion

7. Sliding mode control of singular stochastic Markov jump systems 7.1. 7.2. 7.3. 7.4. 7.5.

Problem formulation Admissibilization of SSMSs Application to SMC Examples Conclusion

ix

180 181

185 185 186 190 197 201

8. Admissibility and admissibilization for fuzzy singular systems

203

8.1. Admissibility analysis for Takagi–Sugeno fuzzy singular systems with time delay 8.1.1. Problem formulation 8.1.2. Main results 8.1.3. Numerical example 8.1.4. Conclusion 8.2. Admissibilization of singular IT2 fuzzy systems 8.2.1. Preliminaries 8.2.2. State feedback control of singular systems 8.2.3. Static output feedback of singular systems 8.2.4. Examples 8.2.5. Conclusion

203 203 206 210 211 212 212 215 219 223 230

References Notations Index

231 243 245

Preface

Singular systems, also called descriptor systems, and generalized state-space systems, frequently appear in various engineering systems, such as vehicle suspension systems, flexible robots, large-scale electric networks, chemical engineering systems, and complex ecosystems. Such systems provide a more natural description of dynamic systems than the standard state-space systems, due to the fact that singular systems can preserve the structure of physical systems more than accurately by including nondynamic constraints and impulsive elements. On the other hand, singular systems are, in essence, differential equations coupled with functional equations, and thus the stability problem for singular systems is much more complicated than that for standard state-space systems, because it requires considering not only stability, but also regularity and absence of impulses (for continuous-time singular systems) and causality (for discrete-time singular systems). For these reasons, singular systems not only have great practical significance, but also are of theoretical interest. Different analysis and synthesis problems, including admissibility and admissibilization, state-feedback control, static output feedback control, bounded real lemma and H∞ control, dissipative control and filtering, reliable control and filtering, and sliding mode control for linear singular systems and nonlinear singular systems are all thoroughly studied. Less conservative techniques, such as the slack matrix method, the Wirtinger-based inequality, the reciprocally convex combination approach and two equivalent sets, combined with the linear matrix inequality (LMI) technique, are applied to singular systems. This book includes eight chapters. The part of Introduction is given in Chapter 1. Dissipative control and filtering for discrete-time linear singular systems are considered in Chapter 2. For nonzero initial conditions, the H∞ control with transients problem is solved in Chapter 3. Considering the time delay, the problems of delaydependent H∞ control and dissipative synthesis for singular delay systems are stated in Chapters 4 and 5, respectively. For singular Markovian systems, by applying equivalent sets technique, some new formulation of dissipativity conditions are obtained in Chapter 6. Chapter 7 carries out sliding mode control (SMC) problem for singular stochastic Markov systems (SSMSs). In Chapter 8, for nonlinear singular systems, by using Takagi–Sugeno (T-S) xi

xii

Preface

fuzzy model to describe, the issues of admissibility analysis and controller design for T-S fuzzy singular systems are investigated. In sum, the book provides a systematic theory about analysis and synthesis of singular systems by introducing recent theoretical findings. The purpose of this book is to propose a base for further theoretical research or guidance of engineering applications; it can serve as a reference for undergraduate and postgraduate students who are interested in singular systems.

Acknowledgments We are deeply indebted to our colleagues Prof. James Lam, and Prof. Wei Xing Zheng for their great contributions regarding the contents of the book. The financial support of the National Natural Science Foundation of China under Grants 61741305, 61763045, Natural Science Foundation of Heilongjiang Province of China under Grant YQ2019F004, the Fundamental Research Funds for the Central Universities under Grant 3072020CFJ0409, the China Postdoctoral Science Foundation under Grant 2018M63034, and Grant 2018T110275, the Fundamental Research Project of Natural Science Foundation of Shaanxi Province under Grant 2020JM552 are gratefully acknowledged. Harbin, China Yan’an, China Adelaide, Australia Wollongong, Australia Wollongong, Australia

Zhiguang Feng Jiangrong Li Peng Shi Haiping Du Zhengyi Jiang May 2020

Acronyms and symbols

ADE BRL DOF ESPR FMB GARI GLE KYPL LMI IT2 ODE PRL SMJS SOF SSMS SVD R R+ Rn Rn×m

l2 L2 [0, ∞)

∈   |x|

f 2 x, yτ G ∞

I AT A−1 sym(A) diag(A1 , . . . , An ) A>B A≥B   A B  C   A B C D

Algebraic and differential equation Bounded real lemma Dynamic output-feedback Extended strictly positive realness Fuzzy-model-based Generalized algebraic Riccati inequality Generalized Lyapunov equation Kalman–Yakbovich–Popov Lemma Linear matrix inequality Interval type-2 Ordinary differential equation Positive real lemma Singular Markovian jump system Static output-feedback Singular stochastic Markov system Singular value decomposition set of real numbers set of nonnegative real numbers n-dimensional Euclidean space set of n × m real matrices space of infinite summable vector sequences space of square integrable functions belong to defined as end of proof absolute value of the numberx, or Euclidean norm of the vector x  ∞ ∞ 2 2 | 0 |f (t)| dt for f ∈ L2 [0, ∞) k=0 f (k)| for f ∈ l2 , or τ

T 0 x(t) y(t)dt for x, y ∈ L2 [0, ∞), or x, y ∈ l2

τ

T k=0 x (k)y(k) for any vector

H∞ norm of the operator G identity matrix transpose of the matrix A inverse of the matrix A A + AT block diagonal matrix with A1 , . . . , An on the diagonal A − B is positive definite A semidefinite  − B is positive  A B BT C 

explicitly partitioned matrix of

A C

B D



xiii

CHAPTER 1

Introduction 1.1 Background The model of the standard state-space system described by ordinary differential equations (ODEs) is widely used in linear control theory, because the state space approach does not only reveal various properties of the system, but also offers us effective system analysis and synthesis methods. In many practical systems, however, some algebraic constrained laws are imposed on the state components, which lead to the singular system description. A singular system essentially composes of a set of algebraic and differential equations (ADEs), containing information of the static and dynamic constraints of a plant. Singular systems, also called descriptor systems, semistate space systems and generalized state-space systems, can provide a more natural description of dynamic systems than the standard state-space systems, due to the fact that singular systems can preserve the structure of physical systems more accurately by including nondynamic constraints and impulsive elements. Moreover, because the standard state-space system is a special case of the singular system, the singular system form can describe more practical systems than the standard state-space form. Singular systems have strong application background and are frequently employed to model circuit systems [25,137], economic systems [120], constrained mechanical systems [34,67], aircraft control systems [159] and chemical processes [82]. In this book, linear singular systems and nonlinear singular systems are studied. Apart from the different form compared with the standard state-space systems, singular systems have fundamental differences as pointed out as follows [34,209]: • A singular system may not have a solution. If a solution exists, there may be more than one solutions. This point is different with standard statespace systems, which have a unique solution for any initial conditions. • For any initial conditions, the response of a singular system may contain impulse terms and the derivatives of these impulses or noncausal behaviors. Standard state-space systems do not have impulsive or noncausal behaviors. Analysis and Synthesis of Singular Systems https://doi.org/10.1016/B978-0-12-823739-7.00008-2

Copyright © 2021 Elsevier Inc. All rights reserved.

1

2

Analysis and Synthesis of Singular Systems

A singular system usually contains three dynamic models: finite dynamic models, infinite models (which lead to the undesired impulsive behaviors), infinite static models. However, a standard state-space system only contains finite dynamic models. • The transfer function of a singular system may contain a polynomial matrix, which is not strictly proper, whereas that of a standard statespace system is strictly proper. Therefore the analysis and the synthesis problems for singular systems are more complicated than those of standard state-space systems, because it is required to consider not only the stability, but also the regularity and nonimpulsiveness (for continuous-time singular systems), or causality (for discrete-time singular systems) characteristics. In sum, investigating singular systems is significant both in practice and theory. The core of studying singular systems basically consists of analysis and synthesis problems such that the singular systems have a desired or satisfactory property, which mainly contains admissibility, performance, and robustness. The analysis problem in the control-theoretic context is to establish conditions under which a system is guaranteed to have these properties. Admissibility of singular systems is a property as important as the stability of standard state-space systems. A singular system is said to be admissible if it is asymptotically stable, regular, and impulse-free (for continuous-time singular systems) or causal (for discrete-time singular systems). The existence and uniqueness of solution to a singular system can be guaranteed by the regularity. Nonimpulsiveness for a continuous-time singular system means there is not impulsive behavior with the consistent initial conditions, whereas causality for a discrete-time singular system means that the states of a system in the past do not depend on the state in the future. Asymptotic stability guarantees that the state of a singular system approaches the equilibrium when time goes to infinity. The performance of a dynamic system is usually characterized by an input-output relationship. Among various performance specifications, bounded real (H∞ performance) and positive real (passivity performance) are two of the most popular choices. The H∞ performance represents the maximum gain of the system, which characterizes the worst-case norm of the regulated outputs over all exogenous inputs with bounded energy. By designing the loop gain of the system to be less than unity, the closed-loop system is guaranteed to be asymptotically stable. Positive real property is widely used in adaptive control, absolute stability, and robust stable analysis. The stability of a closed-loop system can be realized by •

Introduction

3

passivity control such that the phase lag of the system is less than 180 degrees [201]. However, H∞ control based on the small gain theorem, and positive real control based on the positive real lemma, both consider the gain and phase performance separately. This may lead to conservative results when used in applications. The dissipativity theory introduced in [180] not only unifies the H∞ and positive real control theory, but also provides a more flexible and less conservative robust control design as it allows a better trade-off between the gain and phase performances. Based on an input-output energy-related consideration, dissipativity theory has generalized many independent theorems or lemmas, for example, the passivity theorem, bounded real lemma (BRL), Kalman–Yakbovich–Popov lemma (KYPL) and the circle criterion, and provides a unified framework for the analysis and design of control systems [114]. Dissipative systems are very useful for a wide range of fields, such as system, circuit, network, and control theory [12], [147]. It gives strong links amongst physics, systems theory, and control engineering. Since its introduction, the theory of dissipativity has attracted extensive attention in system control, such as for nonlinear systems [242] and for linear systems [130]. Time delays are sources of instability and poor performance of a dynamical system. They always exist in many dynamical systems [89]. Consequently, many stability results and controller design approaches of delay systems have been reported in the literature. Singular time-delay systems are in essence delay differential equations coupled with functional equations, and thus the robust stability problem for singular systems is much more complicated than that for state-space systems, because it requires to consider not only stability robustness, but also regularity and causality (absence of impulses), which may affect the stability of the system. The problems arising from singular time-delay systems are significant both in theory and in practice. A considerable number of studies have been devoted to singular time-delay systems, such as the results on continuous-time systems [156,226], discrete-time cases [88,233], and the references therein. It is worth noticing that various methods were developed to obtain less conservative results. The free-weighting matrices approach [53,231], Jensen inequality method [89], the reciprocally convex combination approach [48], Wirtinger inequality [149], and delay partitioning method [4] are used in many papers. In this paper, the delay partitioning and the reciprocally convex combination approaches have been used to reduce conservative of results for bilinear system with time-varying delay.

4

Analysis and Synthesis of Singular Systems

On the other hand, with respect to the uncertainties, the method of sliding mode control (SMC) has fast response and good robustness as competitive advantages. Sliding mode control, which is based on the theory of variable structure systems, has been widely applied to robust control of nonlinear systems. The sliding mode control employs a discontinuous control law to drive the state trajectory toward a specified sliding surface and maintain its motion along the sliding surface in the state space [232]. The dynamic performance of the sliding mode control system has been confirmed as an effective robust control approach with respect to system uncertainties and unknown disturbance when the system trajectories belong to predetermined sliding surface. The synthesis problem in the control-theoretic context is to design a controller or a filter such that the closed-loop system has a desired or satisfactory behavior. Generally speaking, an effective way to synthesize controllers/filters investigation is often based on some performance-based criteria, under which the controlled/filtered system has the desired properties. Then, a controller or a filter will be designed to guarantee the closed-loop system or the filtering error system to satisfy the criterion. For filter design problem, among various filter design methods, Kalman filtering approach is one of the most popular way, which estimates the state vector by optimizing the covariance of the estimation error [2]. However, to use the Kalman filtering approach, the exact statistical properties of the external noises are required, but it is not always satisfied in practical applications. Therefore an alternative approach called H∞ filtering, which does not need the exact information of the external noises, has received much attention. Since dissipativity theory provides a better trade-off between gain and phase performances, dissipative filtering, which is more general and unifies the H∞ and passive filtering, has also been an effective approach. Designed filters may be classified into full-order filters and reduced-order filters. When the order of the filter equals the order of the plant, it is called full-order filter; when it is lower, it is called a reduced-order filter. When considering some large-scale systems, a full-order filter may be not suitable to apply, and the reduced-order filter is a better choice.

1.2 Research problems This book studies the analysis and synthesis problems of linear singular systems and nonlinear singular systems. Some widely encountered factors are taken into consideration, they are time-delay, disturbance, uncertainties,

Introduction

5

Markovian, and nonlinear characteristics. Studying singular systems will be more difficult than their standard state-space counterparts, because the regularity and nonimpulsiveness (for continuous case) or causality (for discrete case) characteristics need to be considered along with the stability. The approaches employed to study standard state-space systems cannot be applied directly, and improved methods or new approaches will be derived. The detailed problems being dealt with are listed as follows: • How to establish conditions under which the singular systems are admissible or/and dissipative for a singular system with or without timedelay, or uncertainty or Markovian or nonlinear characteristic? • Based on the analysis results, how to design a state-feedback controller or a static output-feedback controller such that the closed-loop singular systems are admissible or/and dissipative? For filter design problem, how to design a full-order or reduced-order filter such that the filtering error system is admissible or/and dissipative?

1.3 Literature review 1.3.1 Singular systems Formally speaking, singular systems are the generalization of the standard state-space systems. A great many fundamental concepts and results of singular systems have been obtained by extending the concepts and methods utilized in standard state-space systems. Originally, two frequency domain methods, including the geometric approach [91] and the polynomial matrix method [139,168], are used to study control problems of singular systems. Many problems are solved by using the two methods, such as solvability, controllability, and observability [24], disturbance decoupling [49], pole assignment [23], and observer design [139]. However, the mathematical tools used in geometric approaches are very abstract, which makes the computation difficult. Polynomial matrix methods require the same stability margin of all the designed controllers. With the development of the state-space method for standard state-space systems, more and more researchers have extended the method to singular systems. Some fundamental definitions and the control system design results of singular systems are provided systematically by using state-space method in [25]. Until now, a great number of results on singular system about various topics have been obtained, such as minimal realization [61,68], linear-quadratic optimal control [83], model reduction [207], H2 control [71]. Among the above topics, the following three aspects for singular systems will be reviewed:

6

Analysis and Synthesis of Singular Systems

Admissibility and admissibilization As mentioned above, admissibility contains regularity, nonimpulsiveness or causality, and stability, and hence is an important concept for a singular system. Admissibilization problem is to design a feedback controller such that the closed-loop system is admissible. The definition of admissibility is proposed in [25], which contains a basic criterion for checking a singular system whether it is admissibility by calculating the degree of the characteristic polynomial and the roots of the characteristic equation of a singular system. However, the criterion is not used directly, especially for a system of high dimensions. It is well known that the Lyapunov equation approach is a powerful tool in studying standard state-space systems. A Lyapunov theorem of singular systems is given in terms of Riccati equation in [90] under the assumption of regularity. With the assumption, the state feedback control problem is solved in [25]. However, the assumption is very restrictive, because singular systems are not always regular in practice and the regularity of a singular system may be destroyed by feedback control. Without the regularity assumption, Takaba et al. in [161] propose a necessary and sufficient condition in terms of a generalized Lyapunov equation (GLE) for characterizing the admissibility of singular systems. The admissibilization method is proposed in [166], which involves decomposing system matrices. Although the Riccati equation approach plays an important role in the early period of the state-space theory development, the computational difficulties of some kinds of Riccati equation have limited its application. On the other hand, the way of decomposing system matrices makes the design procedure indirect and cumbersome. By using linear matrix inequalities (LMIs), which can be solved via efficient Matlab® LMI toolbox, Sedumi, or Yalmip [9], a necessary and sufficient admissibility condition is established in [129]. However, the condition in [129] contains an equality constraint and a nonstrict inequality, which makes the condition difficult to apply. By proving the equivalence of two sets, the admissibility condition in terms of strict LMIs is provided in [71]. The same strict LMIs are obtained in [209] by using different proof procedure and the admissibilization problem is solved, which can be designed by solving LMIs. Moreover, another necessary and sufficient admissibility condition for continuous-time singular systems is proposed with strict LMIs in [14]. For discrete-time singular system, the admissibility conditions are given in terms of GLE in [70]. Moreover, the assumptions of regularity and causality are needed. Also, under the regularity assumption, a necessary and sufficient admissibility condition is provided in terms of non-strict LMIs

Introduction

7

in [144], and the controller designed method needs two steps. Without the regularity assumption, the direct state-feedback controller design method is established in [217]. However, the admissibility condition also contains a nonstrict LMI. To obtain tractable and reliable conditions, an equivalent admissibility condition in terms of strict LMIs is proposed in [208].

H∞ control and dissipative control H∞ optimal control to suppress disturbances is introduced when there are external disturbances with unknown statistical properties [178]. Very recently, H∞ control results for singular systems have been addressed in the literature, which guarantee the closed-loop system to be regular, impulsefree, and delay-dependent robustly stable, and meet an H∞ norm bound constraint on disturbance attenuation. For instance, a generalized algebraic Riccati equation (GARE) solution and LMI approaches are used to the H∞ control, and dissipative control problem are established in [144], [173], respectively. Due to the effectiveness of solving LMIs and wide applicability of the LMI approach, many efforts are devoted to LMIs conditions. Without the jw-axis zeros condition and rank conditions on the plants, a necessary and sufficient condition for H∞ control problem is reduced to solving certain generalized algebraic Riccati inequalities (GARIs) or equivalent LMIs [129]. However, to obtain a dynamic output-feedback controller, some equality conditions and nonstrict LMIs should be solved firstly, which leads to computational complexity. To overcome the problem, strict LMIs conditions for H∞ state-feedback control problem are presented in [165]. For discrete-time singular systems, a BRL in terms of strict LMIs is given in [229], which makes the condition more tractable. However, the method presented in [229] cannot be easily used to solve the controller design problem. A new BRL and H∞ control problem are solved in [15] by using the augmentation system approach, which can also be employed to study static output-feedback control problem. However, the H∞ control problem mentioned above are standard H∞ optimization problem, in which it is always assumed that the initial condition of the system is zero. Actually, the initial states are often uncertain and might be nonzero, which may affect the performance level of disturbance attenuation of the H∞ control considerably. The case of H∞ control with transients for singular systems remains an unsolved, yet important, problem with many technical issues to be addressed. On the other hand, the BRL only uses the gain information, and positive real lemma (PRL) only considers the phase information, which may

8

Analysis and Synthesis of Singular Systems

lead to conservative results in many applications. Dissipativity property provides a unified framework for considering the gain and phase information simultaneously, which have played an important role in control theory and applications [180]. For continuous-time singular systems, without the constraint on the choice of the system realization, a necessary and sufficient dissipativity condition is established and the state-feedback control problem is solved in [127]. The corresponding output-feedback controller design method is provided in [128]. However, the results in [127] and [128] are nonstrict LMIs, which often give rise some computational difficulties. The new KYP lemma for the dissipativity of singular system is characterized in terms of strict LMI in [39]. For discrete-time singular systems, there exists only one necessary and sufficient dissipativity condition reported up to now. The dissipativity condition in terms of nonstrict LMIs is proposed in [29]. For the motivations given earlier, it is natural to pose a research problem as follows: Problem 1.1. How to study the H∞ control problem of singular systems when the initial condition is nonzero? How to establish a necessary and sufficient dissipativity condition in terms of strict LMIs to check the admissibility and dissipativity of discrete-time singular systems effectively? How to develop a SOF controller design method such that the closed-loop system is admissible and dissipative?

Filtering The filtering problem for singular system has received great attention due to its practical significance in the field of signal processing and communication, control applications. A classical method is Kalman filtering, which is developed for singular systems in [138]. However, the requirements of knowing exact information on both the external noises and the model of system are not satisfied in practice. An alternative method called H∞ filtering has received much attention, because the class of the noise signals only needs to be bounded energy. The H∞ filtering problem for singular system is firstly investigated in [206], where a necessary and sufficient condition for solving the H∞ filtering problem is proposed in terms of LMIs. The reduced-order filter is designed in [210] such that the filtering error system is admissible with a prescribed H∞ performance. By using a similar method, the reduced-order energy-to-peak filtering problems for continuous-time and discrete-time singular systems are tackled in [251] and [250], respectively. It should be pointed out that the conditions obtained in [206], [210],

Introduction

9

[251], and [250] involve rank and equality constraints, which introduce numerical difficulties. Moreover, to obtain the desired filter parameters, the complicated matrix structure is needed in them. A reduced-order l2 -l∞ filter design method for discrete-time singular systems is given in terms of strict LMI in [124]. To the best knowledge of the author, no reduced-order dissipative filter design method for discrete-time singular systems has been reported to date. Hence, a natural research problem is given as follows: Problem 1.2. How to design a reduced-order filter in terms of strict LMSs, which can be obtained by solving the LMIs directly, such that the filtering error singular system is admissible and dissipative?

1.3.2 Singular systems with time-delay Time-delay, often indicating a source of instability and oscillation, is unavoidable in many practical systems, such as chemical plants, neural networks, and networked control systems [179]. Increasing attention has been focused on the delay-dependent analysis results for standard time-delay systems, and various new approaches have been proposed to reduce their conservatism. Many effective methods established in standard state-space time-delay systems have been successfully extended to singular time-delay systems. In this subsection, a number of effective methods for reducing conservatism of the results of time-delay systems are firstly introduced. In what follows, the feedback control and filter synthesis of time-delay singular systems are reviewed.

Techniques to reducing conservatism of results It is well known that delay-dependent stability conditions are generally less conservative than delay-independent ones, especially in the case when the time-delay is small. The aim of reducing the conservatism of these delaydependent stability criteria is to establish new stability criteria to provide a maximal allowable delay as large as possible. Among different techniques, the reduced conservatism is mainly obtained from constructing an improved Lyapunov function and employing a better bound on some weighted cross products. For continuous-time systems, by utilizing Park’s and Moon’s inequalities, delay-dependent stability conditions are proposed in [141] and [133] based on Lyapunov theory, respectively. The descriptor model transformation method and the corresponding Lyapunov function are proposed

10

Analysis and Synthesis of Singular Systems

in [51] to improve the criterion in [141]. Less conservative results generated by introducing the free-weighting matrices method are presented in [66] to investigate the system with time-varying delay in an interval. By using a convex combination approach to estimate an upper bound of the derivative of Lyapunov functional accurately, improved stability criteria are proposed in [151]. However, the difference between delay bounds has to be approximated, because of the inversely weighted nature of the coefficients caused by the Jensen inequality. The reciprocally convex approach, which can deal with the inversely weighted convex combination of the quadratic integral terms directly is provided in [140]. The results in [140] are less conservative than those in [151], and have less decision variables. To further reduce the conservatism, a new Lyapunov functional, which involves some triple integral terms, is constructed in [160]. Another effective and popular method is the delay-partitioning method, which can improve the stability criteria significantly. It is first proposed for systems with constant time-delay in [60]. Then the method is extended to the time-varying case in [134]. Recently, some new and effective methods are proposed to study the stability analysis problem for discrete-time systems with timevarying delay. By constructing a novel Lyapunov functional and using the bounding inequalities for certain cross products, less conservative results are obtained in [53] compared with those in [55]. Furthermore, some useful terms ignored in [53] are introduced in the new Lyapunov functional defined in [229]. By combining a new Lyapunov functional with the free weighting matrix technique, the results in [53] are improved by the result in [229]. To reduce the conservatism of the results in [53], the convex combination approach proposed in [151] is extended to the discrete-time delay systems. Two new and less conservative stability criteria are given in [150]. Moreover, a delay-partitioning method is firstly utilized to deal with the stability analysis problem for discrete-time system with time-varying delay in [132], which have significantly reduced the conservatism compared with existing results. The reciprocally convex approach is also extended to discrete-time systems with time-varying delay in [104], which improves the results in [150]. In sum, two classes approaches are established to make the LMIs hold more easily for reducing the conservatism: one is to establish novel Lyapunov functional and the other is to establish new integral inequality. Up to now, the delay-partitioning method is the most effective way to establish the Lyapunov functional, and the reciprocally convex approach is the best integral inequality.

Introduction

11

H∞ control and dissipative control The robust H∞ control problem for uncertain singular time-delay systems is investigated in [72,81,213]. However, the sufficient conditions in these three references are delay-independent, and thus conservative. To reduce the conservatism, an equivalent model transformation approach is provided in [50], and the delay-dependent conditions for H∞ control problem of singular systems with time-delay are obtained in terms of LMIs. The singular system concerned should be transformed to an augmented system to using the results in [50], which makes the analysis and synthesis procedure relatively intricate. Free-weighting matrix methods, which are employed for the standard state-space system are extended to singular systems with time-delay in [196,214] by using neither system transformation nor bounding technique. Moreover, state-feedback controllers are designed in [196,214] such that the closed-loop systems are admissible, while satisfying a prescribed H∞ performance level. By using the delay-partitioning method and introducing a three-integral term to construct a new Lyapunov functional, an improved delay-dependent BRL is presented in terms of LMIs, and corresponding feedback control problem is solved in [135]. The above-mentioned results are concerned with continuous-time singular systems. In the discrete-time cases, the robust H∞ control problem for timedelay discrete-time singular systems with parameter uncertainties is solved in [212] based on the delay-independent matrix inequality condition. The delay-dependent case is considered in [122] by using the restricted system equivalent transformation. Although this condition plays a key role in solving the H∞ control problem, it involves semidefinite and nonlinear problems, and is thus difficult to implement numerically. By applying the delay-partitioning method, strict LMI conditions are proposed to solve the H∞ control problem in [45]. For dissipative control of state-space systems with time-varying delays, some results have been reported. Different from using the Lyapunov method for analyzing the stability of time-delay system as in [54], dissipative theory is employed in [21]. A considerable number of studies have been devoted to dissipative control for singular systems. Dissipativity theory and KYP conditions are generalized to nonlinear and linear continuous singular systems in [74]. When time-delay appears, there are few results for such systems. By using an LMI approach, sufficient delay-dependent conditions for dissipative control are established in [101]. Based on the extended Itô stochastic differential formula, the problem of dissipative control

12

Analysis and Synthesis of Singular Systems

for stochastic singular time-delay systems is tackled in [117]. Robust dissipative control conditions are presented in [28,143] for time-delay singular systems, but the conditions are delay-independent. The delay-dependent conditions in terms of LMIs for singular time-delay systems are shown in [123], which reduces the conservatism of the results obtained in [143]. It should be pointed out that conservatism still remains in these results. Moreover, a parameter which denotes the level of dissipativeness is often not considered when the problem of dissipative control is investigated. Hence, a meaningful research problem is raised naturally as follows: Problem 1.3. How to establish a new admissibility/H∞ analysis/dissipativity condition in terms of LMIs for singular time-delay systems to further reduce the conservatism of existing results?

Filtering Considerable attention has been devoted to the H∞ filtering problem due to its major theoretical significance in the past decade [56,183], and great many applications in the aerospace industry [145], TV tracking systems [205], and data segmentation [186]. Moreover, the H∞ filtering results based on the theory of state-space systems have been successfully extended to singular time-delay systems. For continuous-time singular time-delay systems, the robust H∞ filtering problem is dealt with in [230] such that the filtering error singular system is admissible with a prescribed H∞ level. When both discrete and distributed delay appear, the robust H∞ filtering problem for singular systems with norm bounded uncertainty are solved in [222]. However, the results in [230] and [222] are delay-independent conditions, which have higher conservatism. By system transformation and decomposing system matrices, a delay-dependent condition of H∞ filtering for singular time-delay systems is given in [50]. To overcome the complexity introduced by the system transformation and the decomposition of system matrices, an LMI-based delay-dependent condition for the existence of H∞ filter is proposed in [195]. For discrete-time cases, robust H∞ filter design method for singular systems with polytopic uncertainty is considered in [78] without system matrix decomposition, or requiring additional assumption on the systems. By using the reciprocally convex combination method, the reliable dissipative filtering problem for singular systems is studied in [41]. Although the results in [78] improve the results in [56,241] when recovering the state-space case, there is still room for improvement.

Introduction

13

Recently, passive filtering problem has also received increasing attention. The delay-dependent passive filtering problem for discrete-time singular systems with time-varying delay is investigated in [80], where a finite sum inequality proposed in [241] is employed. The existence condition of a filter is obtained in [115] such that the singular error system with Markovian jump parameters is admissible and satisfies the proposed passivity performance. The mixed H∞ and passivity performance as a special case of dissipativity is first given [130]. Based on this definition, the mixed H∞ and passive filtering problem for continuous-time singular systems with time-delay is considered in [187], where the time-delay is constant and the uncertainty is not considered. In actual implementation, conventional filters for a multi-input-multioutput plant may lead to unsatisfactory performance, due to the temporary failures of sensors, which results in incomplete signal delivered to actuators. Therefore the reliable filter design problem has attracted increasing research attention. By employing adaptive method, adaptive reliable H∞ filters are designed to compensate the sensor failure effects on systems in [219]. Benefiting from the delay-partitioning method, the problems of reliable H∞ filtering for discrete time-delay systems with randomly occurring nonlinearities and Markovian jump systems with partially unknown transition probabilities are solved in [110,111], respectively. However, there are few results on dissipative filtering, which is more general and unifies the H∞ and passive filtering. By using sector-nonlinearity modeling techniques, a (Q, S, R)-dissipative fuzzy filter is designed for a class of nonlinear systems rewritten by a T-S fuzzy model in [112]. A sufficient condition for dissipative filtering problem of linear discrete systems is proposed in terms of LMI in [92]. A nonfragile dissipative filtering problem for a class of nonlinear discrete-time systems with sector bounded nonlinearities is investigated in [220]. However, the important problem of reliable dissipative filter for discrete singular systems with time-varying delay and uncertainties remains to be considered. Motivated by the discussion, a natural research problem is given as follows: Problem 1.4. How to investigate the reliable dissipative filtering problem for singular systems both with time-varying delay and uncertainty?

1.3.3 Singular Markovian jump systems (SMJSs) Practically not all the systems can be appropriately described by the linear time-invariant model, due to the abrupt changes in their structures and

14

Analysis and Synthesis of Singular Systems

parameters caused by some discrete events, such as failures, repairs, and disconnection of some components, or sudden environmental disturbances. As a special class of stochastic hybrid systems, Markovian jump systems have been widely employed to model the systems with sudden variation in their dynamic characterization, see, for example, manufacturing systems in [126], networked control systems in [125], and fault-tolerant control systems in [163]. Therefore a great deal of attention has been devoted to the study of the Markovian jump systems over the past decades, such as stabilization in [37,204,236], H2 or H∞ control in [26,155], passivity control and reliable mixed passivity control in [154] and [153], robust extended dissipative control in [152].

H∞ control and dissipative control When abrupt changes appear in singular systems, it is reasonable to model them with SMJSs and a considerable number of studies have been devoted to the control of SMJSs. For both continuous-time and discrete-time Markovian singular systems, necessary and sufficient conditions guaranteeing the systems to be admissible and stochastically stable are proposed in [209]. Moreover, the corresponding H∞ control problem is also solved in the book. However, the results of H∞ control include an upper bound constraint, which makes the condition highly conservative. By using some inequalities, a more tractable result without the upper bound constraints is obtained in [7]. Unfortunately, the results are obtained in terms of nonstrict LMIs, which make it difficult to solve. Based on the technique proposed in [165], the conditions in terms of strict LMIs of H∞ control problem for singular Markovian system are given in [235]. For SMJSs with time-delay, delay-dependent H∞ control problem is investigated in [193] but the results in [193] are applicable to SMJSs with singular value decomposition (SVD) form. Therefore the following problem will be investigated: Problem 1.5. How to design a H∞ controller such that the closed-loop SMJS is admissible and strictly (Q, S, R)-dissipative?

Reliable control The results of H∞ control problem are obtained under a full reliability assumption that all sensors, control components, and actuators of the systems are in a good working condition. However, in practical engineering systems, it is unavoidable to encounter the failures of actuators or sensors, which may lead to intolerable performance of the closed-loop systems

Introduction

15

[164,225]. Therefore it is necessary to design a reliable controller that can tolerate the actuator failures and guarantee the required performance of the closed-loop system. More and more attention has been paid to the study of reliable control for dynamic systems. By modeling sensor failures and actuator failures as a scaling factor and a disturbance, a reliable H∞ controller is designed for linear systems such that the closed-loop system is asymptotically stable and satisfies the H∞ performance in [224]. For singular system, the problem of reliable H∞ control for uncertain systems with actuator failures and multiple time delays is investigated in [225]. For Markovian jump system, the robust reliable H∞ control problem for discrete-time systems is solved in [19]. For reliable dissipative control problem, the state feedback controllers and impulsive controllers are designed in [234] for Makovian jump systems with actuator failures and impulsive effects, such that the closed-loop system is robustly stable and strictly (Q, S, R)-dissipative. Reliable dissipative control for singular Markovian jump systems has never been tackled. Therefore the following problem will be investigated: Problem 1.6. How to design a reliable state-feedback controller such that the closed-loop SMJS with actuator failures is admissible and strictly (Q, S, R)-dissipative?

1.3.4 T-S fuzzy singular systems Additionally, fuzzy logic plays a significant role in system modeling and data mining with characterising uncertainty [8,95,203]. The Takagi–Sugeno (T-S) fuzzy rule model is quite a efficient method to approximate the complicated nonlinear systems, which combines the mathematical theory and the fuzzy logic theory [93,176,244]. Based on “IF-THEN” rules, by “blending” every local linear system, the T-S fuzzy-model-based approach is able to handle nonlinearities existing in the application systems [98,119]. A lot of studies have been done on the research of T-S fuzzy systems [99,100,218]. On account of the interval T-S fuzzy approach, the issue of fuzzy control for nonlinear networked control systems with parameter uncertainties and packet dropouts is investigated in [97]. By utilizing delay partitioning approach, the dissipativity analysis is addressed for T-S fuzzy singular systems with constant time delay in [197]. For the interval timevarying delay, a variable delay composition approach is used to solve the issue of the stability of Takagi–Sugeno fuzzy systems in [198]. In [248], the issue of delay-dependent dissipative control of T-S fuzzy singular model

16

Analysis and Synthesis of Singular Systems

with uncertainties and time delay is investigated. However, there exists ample room to improve the proposed results. A new integral inequality is devised to reduce the conservatism of linear systems with a discrete distributed delay in [228], which is much tighter than other existing integral inequalities. This greatly motivates us to extend this new integral inequality technique to singular systems and reduce the conservatism of results about T-S fuzzy singular systems. Therefore the following problem will be investigated: Problem 1.7. How to obtain less conservative conditions such that the T-S fuzzy singular systems with time delays are admissible?

1.3.5 Type-2 fuzzy singular systems The aforementioned T-S fuzzy systems are all about type-1 fuzzy systems. When uncertainties are involved in nonlinear plant, type-2 fuzzy sets are presented in [227] to capture the uncertainties effectively, and can provide a better performance than type-1 fuzzy sets [102]. A number of applications have been reported in autonomous mobile robots in [63], and extended Kalman filtering in [77]. Since 1971, many different kinds of type-2 fuzzy sets are proposed, such as interval-valued fuzzy sets [59], interval type-2 (IT2) fuzzy sets [131], and so on. Two excellent references have revealed the history of the development of fuzzy sets and the relationships between them in detail in [10], [11]. However, these mentioned type-2 fuzzy set theory is not mainly for fuzzy-model-based (FMB) control problem, and few results study the type-2 FMB control problems. Recently, IT2 fuzzy systems can provide better performance than that of type-1 fuzzy systems [148]. On the other hand, IT2 fuzzy model has outstanding feature on describing the nonlinear plant subject to parameter uncertainties, which can be captured by the lower and upper membership functions [87]. Therefore there has been a growing interest in studying FMB control of IT2 fuzzy systems. By using the information of footprint of uncertainties and the lower and upper membership functions, a new IT2 controller is proposed in [85] to guarantee the closed-loop IT2 fuzzy system to be stable. The extended dissipative control problem by state feedback and dynamic output feedback for IT2 fuzzy systems is investigated in [96]. By using Lyapunov theory, a type of IT2 filter is designed for IT2 fuzzy systems, such that the filtering error system is D stable and satisfies some required performance in [94]. In [13], the stability of an IT2 fuzzy model is investigated, and the result is applied on a half-vehicle active suspension model. The nonlinear networked control system with parameter uncertainty is modeled by IT2 fuzzy system,

Introduction

17

and the IT2 fuzzy predictive controller is designed to stabilize the plant in [116]. The state-feedback control problem is solved by designing IT2 fuzzy controller sharing the same lower and upper membership functions with the considered IT2 fuzzy systems in [87]. For continuous-time IT2 fuzzy systems, an IT2 fuzzy controller with different low and upper membership functions is synthesized to guarantee the closed-loop systems to be stable in [86]. For discrete-time IT2 fuzzy systems, the IT2 fuzzy SOF controller is designed to study the reliable stability with mixed H2 /H∞ performance in [57]. By constructing a fuzzy Lyapunov function, the IT2 fuzzy static output feedback controller is designed for IT2 fuzzy systems in [243]. However, the equality constraint MK = KB in [243] makes the result difficult to solve. Moreover, it should be pointed out that no results about static output feedback control of singular IT2 fuzzy systems have been reported, which is another motivation of this study. Therefore the following problem will be investigated: Problem 1.8. How to design a state-feedback controller and static outputfeedback controller such that the closed-loop IT2 fuzzy singular systems are admissible?

1.4 Book outline •

•

Chapter 2 investigates the dissipative control and filtering of delay-free singular systems. Firstly, the dissipative control problem is considered for continuous-time singular systems. By using two equivalent sets, a novel dissipativity analysis condition is proposed in terms of strict LMIs. Based on this criterion, the state-feedback controller is designed. Secondly, the system augmentation approach is extended to the dissipative control problem of discrete-time singular systems. By giving an equivalent representation of the solution set, a necessary and sufficient dissipativity condition is proposed in terms of strict LMI, such that the singular system is admissible and dissipative. Then by using the system augmentation method, the state-feedback control and SOF control problems are dealt with. The reduced-order filtering problem is transformed to a SOF control problem, and a reduced-order filter is obtained directly by solving LMIs, such that the filtering error system is admissible and dissipative. The effectiveness and applicability of the results are demonstrated by numerical and simulation examples. Chapter 3 addresses H∞ control with transients for singular systems. For singular system with nonzero initial condition, a new performance

18

•

•

•

Analysis and Synthesis of Singular Systems

measure is defined firstly. Then a necessary and sufficient condition is given to guarantee the singular system is admissible with an H∞ performance. Based on this, the state feedback control problem is solved in terms of LMIs. Moreover, the relationship between the new H∞ performance and standard H∞ performance is revealed. Chapter 4 explores the delay-dependent admissibility and H∞ control for discrete-time singular delay systems. Firstly, a triple-summation term is introduced into the Lyapunov function, and the improved reciprocally convex combination approach is utilized to bound the forward difference of the double summation term. The new admissibility criterion, given in terms of LMIs, ensures the regularity, causality, and stability of the considered system. Then the result is extended to singular systems with polytopic uncertainty and disturbances. Delay partitioning technique has been introduced to derive improved results for robust stability and stabilization problems of linear uncertain discretetime singular systems with state delay, which guarantees the closed-loop system is admissible. Moreover, the proposed new results have been utilized to investigate robust H∞ control problem, which assures the resulting closed-loop system is admissible with an H∞ disturbance attenuation. Less conservative and easily verifiable conditions have been formulated in terms of strict LMIs, involving no decomposition of the system matrices. It is also proved that the conservatism of the results is nonincreasing with the reduction of the partition size. Chapter 5 studies the dissipativity analysis and dissipative synthesis of singular systems with time-delay. Firstly, by utilizing the improved reciprocally convex approach and integral inequality, the α -dissipativity analysis condition for discrete-time singular systems with time-varying delay is proposed. Secondly, by employing delaypartitioning method, the less conservative dissipativity conditions are obtained for continuous-time singular systems with time delay. Based on the criteria, the state-feedback controller design method is provided such that the closed-loop system is admissible and strictly α dissipativity. Finally, the reliable dissipative filter design method is presented for singular systems with time delay and sensor failures such that the filtering error system subject to possible sensor failures is admissible and strictly (Q, S, R)-dissipative. Chapter 6 considers the admissibilization, H∞ control and reliable dissipative control problems for SMJSs by state-feedback control. Firstly, by proposing two equivalent sets, the necessary and sufficient admissibi-

Introduction

•

•

19

lization criterion of SMJSs is given in terms of strict LMIs. Secondly, for time delay SMJSs, a less conservative H∞ control method is presented benefitting from the equivalent sets. Secondly, a sufficient condition in terms of strict LMIs is derived to guarantee that the unforced SMJSs with actuator failure is stochastically admissible and strictly (Q, S, R)dissipative. Based on the proposed dissipativity analysis results, the condition for the existence of the state-feedback controller is such that the closed-loop system is stochastically admissible and strictly (Q, S, R)dissipative. Chapter 7 investigates the SMC problem for SSMSs. Firstly, a novel mean square admissibility condition is given in terms of strict LMIs by using replacement of matrix variables. Based on the criterion, the desired state-feedback controller is designed such that the closed-loop system is mean square admissible. Then, the method is applied to solve SMC problem of SSMSs. Numerical examples are provided to demonstrate the applicability of the theoretic results developed. Chapter 8 studies the admissibility analysis for Takagi–Sugeno (T-S) fuzzy singular system with time delay and admissibilization of IT2 fuzzy singular system, respectively. Firstly, a novel tighter integral inequality is utilized to derive a sufficient delay-dependent criterion such that the considered system is admissible. Secondly, based on Lyapunov stability theory, state feedback control criterion and SOF control method are proposed to guarantee the closed-loop system to be admissible. To obtain less conservative results, the information of mismatched membership functions is employed. Numerical examples are given to illustrate the effectiveness of the proposed techniques.

CHAPTER 2

Dissipative control and filtering of singular systems In this chapter, the problems of dissipative control and filtering of singular systems are investigated. Based on two equivalent sets and parameterizing the solutions of the constraint set, dissipative control for continuous-time singular system and discrete-time singular system have been studied, respectively. Necessary and sufficient conditions are established in terms of strict LMI, which makes the conditions more tractable. By using the system augmentation approach, state feedback controller and static output feedback controller design methods are proposed to guarantee that the closed-loop discrete-time singular systems are admissible and strictly (Q, S, R)-dissipative. Then, the results are applied to tackle the reducedorder filtering problem. The effectiveness of the obtained results in this section are illustrated by numerical examples.

2.1 Dissipative control of continuous-time singular systems Dissipativity property provides a unified framework considering the gain and phase information simultaneously, which has played an important role in control theory and applications [180]. Many basic tools, such as passivity theorem, bounded real lemma, Kalman–Yakubovic–Popov (KYP) lemma and circle criterion are special cases under dissipativity theory that have attracted considerable attention [29], [40], [189]. In this section, the problems of dissipativity analysis and dissipative control of continuous-time singular systems are addressed by employing the two equivalent sets.

2.1.1 Problem formulation Consider a class of linear continuous singular systems described by 

Ex˙ (t) = Ax(t) + Bw w (t), z(t) = Cx(t) + Dw w (t),

x(0) = x0 ,

(2.1)

where x(t) ∈ Rn is the state vector; x0 is the initial condition; w (t) ∈ Rp represents the exogenous input, which includes disturbances to be rejected, Analysis and Synthesis of Singular Systems https://doi.org/10.1016/B978-0-12-823739-7.00009-4

Copyright © 2021 Elsevier Inc. All rights reserved.

21

22

Analysis and Synthesis of Singular Systems

and z(t) ∈ Rq is the controlled output; A, Bw , C, and Dw are constant matrices with appropriate dimensions. In contrast with standard linear systems with E = I, the matrix E ∈ Rn×n has 0 < rank(E) = r < n. First, we give some definitions and lemmas based on unforced system (2.1): Definition 2.1. [25] 1. The pair (E, A) is regular if det(sE − A) is not identically zero. 2. The pair (E, A) is impulse-free if deg {det(sE − A)} = rank(E). In view of this, we introduce the following definition for singular system (2.1): Definition 2.2. 1. The singular system in (2.1) is said to be regular and impulse-free if the pair (E, A) is regular and impulse-free. 2. The singular system in (2.1) is said to be asymptotically stable if all the finite roots of det(sE − A) = 0 have negative real parts. 3. The singular system in (2.1) or the pair (E, A) is said to be admissible if the system is regular, impulse-free and asymptotically stable.  T  

w w S with S ∈ R(p+q)×(p+q) , the defiz z nition of dissipativity is given as follows: For a supply rate s(w , z) =

Definition 2.3. The singular system (2.1) is said to be strictly dissipative with respective to the supply rate s(w , z), under zero initial condition, if for any t1 ≥ 0 and for any w ∈ L2 [0, t1 ], the following inequality holds: 

t1

s(w (t), z(t))dt < 0. 0







T 



S S 0 I 0 I Denote S = 11 12 , M = S as that in [127];  S22 C D C D the following lemma gives a necessary and sufficient dissipativity condition for singular system (2.1): Lemma 2.4. [165,199] For matrix E ∈ Rn×n with rank(E) = r ≤ n, denote EL and ER are full column rank with E = EL ERT , rank(EL ) = rank(ER ) = r, and let P = P T such that ELT PEL > 0 and Q is nonsingular. U with full row rank and Λ with full column rank are the left and right null matrices of matrix E, respectively,

Dissipative control and filtering of singular systems

23

that is, UE = 0 and EΛ = 0. Then, PE + U T QΛT is nonsingular, and its inverse is expressed as ¯ , ¯ T + ΛQU (PE + U T QΛT )−1 = PE

(2.2)

¯ is nonsingular such that where P¯ = P¯ T and Q ¯ = (ΛT Λ)−1 Q−1 (UU T )−1 . ¯ R = (ELT PEL )−1 , Q ERT PE

(2.3)

Lemma 2.5. The following sets are equivalent:   A = X ∈ Rn×n : ET X = X T E ≥ 0, X is nonsingular ,   B = X = PE + U T ΦΛT : P = P T ∈ Rn×n , ELT PEL > 0, Φ ∈ R(n−r )×(n−r ) , B is nonsingular, where EL , ER , U T ∈ Rn×(n−r ) and Λ ∈ Rn×(n−r ) are defined in

Lemma 2.4, respectively. Proof. (Sufficiency) Let X = PE + U T ΦΛT , then we have ET X = X T E = ER ELT PEL ERT ≥ 0, and X is nonsingular based on Lemma2.4.  X11 X12 (Necessity) Without loss of generality, denote X = , E= X21 X22





Ir 0 , where X11 ∈ Rr ×r and X22 ∈ R(n−r )×(n−r ) , then we have EL = ER = 0 0

 

Ir 0



and U T = Λ =

0

In−r



. By using ET X = X T E, we have X12 = 0,

X11 ≥ 0. Due to rank(ET X ) = rank(X11 ) = r, we arrive at X11 > 0. Recalling X is nonsingular,  that X22 is nonsingular. Then, we can find  it yields T X X21 a matrix P = P T = 11 and Φ = X22 such that ELT PEL = X11 > 0 X21 In−r and X = PE + U T ΦΛT . Remark 2.6. It should be noted that Lemma 2.5 is different from that in [42], because the set X1 in [42] and the set A in this section are different. On the other hand, the matrices EL and ER satisfying E = ELT ER are not used in [42]. The system considered in [42] is discrete-time singular system, and the synthesis results in [42] are just sufficient conditions, whereas the system considered in this section is continuous-time singular systems, and the synthesis results in terms of strict LMIs are necessary and sufficient conditions for delay free singular systems.

24

Analysis and Synthesis of Singular Systems





M11 M12 Lemma 2.7. [127] Consider partition of M as M = , M11 ∈  M22 Rn×n , and suppose that M11 ≥ 0. Then, the singular system in (2.1) is admissible and strictly dissipative if and only if there exists a matrix X such that the following matrix equality and inequalities hold: ET X = X T E ≥ 0,



sym(AT X ) + M11 X T Bw + M12 + AT W  sym(W T Bw ) + M22



< 0,

(2.4) (2.5)

where W ∈ Rn×p is a matrix satisfying ET W = 0.

2.1.2 Main results Based on the above, we present a new necessary and sufficient dissipativity condition below. Theorem 2.8. Suppose that M11 ≥ 0. Then, the singular system (2.1) is admissible and strictly dissipative if and only if there exist a symmetric matrix P and a nonsingular matrix Φ such that the following LMIs hold: ELT PEL > 0,



sym(AT X ) + M 

11

X T Bw

+ M12

+ AT W

sym(W T Bw ) + M22



< 0,

(2.6) (2.7)

where T X = PE + U T ΦΛT , M11 = C T S22 C , M12 = C T S12 + C T S22 Dw ,

M22 = S11 + sym(S12 Dw ) + DwT S22 Dw . EL , U, and Λ are defined in Lemma 2.5, and W ∈ Rn×p is a matrix satisfying ET W = 0. Proof. From the inequality in (2.5), we can see that sym(AT X ) + M11 < 0. Then, it yields sym(AT X ) < 0 from M11 ≥ 0, which implies that matrix X is nonsingular. Therefore together with the condition in (2.4), the matrix X in Lemma 2.7 satisfies the set A in Lemma 2.5. By the two equivalent sets in Lemma 2.5, we have the desired result. Remark 2.9. The result in Theorem 2.8 is more general, because it covers serval important performance analysis criteria as special cases. When S =

Dissipative control and filtering of singular systems

25

  −R − S , the result in Theorem 2.8 changes to (Q, S , R)-dissipativity  −Q   −γ 2 I 0 criterion ([130], [162], [201]). When S = , the condition in  I

Theorem 2.8 will be a new bounded real lemma ([129], [173]). A new  0 −I formulation of positive real lemma can be obtained by setting S =  0 ([237], [247]). Remark 2.10. It can be seen that the equality constrain ET X = X T E and nonstrict LMI ET X ≥ 0 in Lemma 2.7 are removed in the dissipativity criterion in Theorem 2.8, which makes the condition easier to check. Next we will study the dissipative control problem by using state feedback. The aim is to design the following controller: u(t) = Kx(t) + Jw (t)

(2.8)

for the open-loop system 

Ex˙ (t) = Ax(t) + Bu(t) + Bw w (t), z(t) = Cx(t) + Du(t) + Dw w (t)

x(0) = x0 ,

(2.9)

with u(t) ∈ Rm , B, and D denoting constant matrices with appropriate dimensions, such that the closed-loop system 

Ex˙ (t) = (A + BK )x(t) + (Bw + BJ )w (t), z(t) = (C + DK )x(t) + (Dw + DJ )w (t)

(2.10)

is admissible and strictly dissipative. A necessary and sufficient dissipative control method is given in the following theorem: Theorem 2.11. Suppose that S22 = N T N ≥ 0. There exists a state feedback controller in form of (2.8) such that the closed-loop system in (2.10) is admissible ¯ a nonsingular and strictly dissipative if and only if there exist a symmetric matrix P, ¯ matrix Φ , matrices H and G, such that the following LMIs hold: ¯ R > 0, ERT PE

⎡

sym(AY T

+ BH ) Ξ12

 

Ξ22 

⎢ ⎣

(YC T

+ H T DT )N T

⎤

⎥ (VC T + DwT + GT DT )N T ⎦ < 0, −I

(2.11) (2.12)

26

Analysis and Synthesis of Singular Systems

where ¯ T + ΛΦ¯ U )T ; = (PE

Y Ξ12

T T = AV T + BG + Bw + YC T S12 + H T DT S12 ;

Ξ22

= sym(S12 CV T + S12 Dw + S12 DG) + S11 .

ER , U, and Λ are defined in Lemma 2.5, and V ∈ Rp×n is a matrix satisfying EV T = 0. Then, a desired controller can be obtained by K = HY −T , J = G − KV T . Proof. Based on Theorem 2.8, the closed-loop system is admissible and strictly dissipative if and only if the following inequalities hold: ELT PEL > 0,  Π=

sym(ATc X ) + CcT S22 Cc

T + S D ) + AT W X T Bwc + CcT (S12 22 wc c T TS D sym(W Bwc ) + S11 + sym(S12 Dwc ) + Dwc 22 wc



(2.13)



< 0,

(2.14) where Ac = A + BK , Bwc = Bw + BJ , Cc = C + DK , Dwc = Dw + DJ ; 

XT WT

Π = sym







0 I











 Ac Bwc 0  + Cc Dwc 0 0 S12 

  0 0 CcT + S22 Cc Dwc . + T Dwc 0 S11

Considering S22 = N T N ≥ 0 and employing Schur complement equivalence, (2.14) is equivalent to  ⎡ XT ⎣sym W T



0 I

Ac 0





 



Bwc 0  + Cc S12 0

Dwc



0 + 0

0 S11







< 0.

 ⎤ CcT T N T ⎦ Dwc −I

(2.15) 



YT VT , Performing the congruence transformation to (2.15) by diag( 0 I I ), with Y = (PE + U T ΦΛT )−T , V = −W T X −T , and noting that X =

Dissipative control and filtering of singular systems

27

PE + U T ΦΛT , YX T = I, VX T + W T = 0, we have  ⎡ Ac ⎣sym S12 Cc

Bwc S12 Dwc



YT 0

VT I





0 + 0

0 S11





Y V



0 I



 ⎤ CcT T T N ⎦ Dwc −I

< 0.

(2.16)

Then setting H = KY T , G = KV T + J in (2.16), we obtain (2.12). Noting ¯ R )−1 , (2.14) is equivalent to (2.11). that ELT PEL = (ERT PE Remark 2.12. It should be remarked that a necessary and sufficient dissipative control criterion is proposed in Theorem 2.11. Compared with the controller design method in [127], the equality and nonstrict LMI constraints EY T = YET ≥ 0 have been moved, which makes the numerical computations more tractable and reliable.

2.1.3 Examples In this section, numerical examples are provided to show the advantages on numerical computations and the state feedback control of the equivalent sets approach. Example 2.1. Consider the following singular system: ⎡

⎤

⎤

⎡

⎡ ⎤

1 0 0 0 1 0 0 ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎣0 1 0⎦ x˙ (t) = ⎣−3 −2 1 ⎦ x(t) + ⎣0⎦ w (t), 0 0 −1 0 0 0 1 z(t) = ⎡





2 5 −2 x(t) + 3w (t).

⎤

⎡ ⎤

1 0 0 ⎥ ⎢ ⎢ ⎥ We choose EL = ER = ⎣0 1⎦, U T = Λ = ⎣0⎦. Based on Theorem 2.8, 0 0 1 the following three performances are discussed respectively: • H∞ case: S11 = −γ 2 I, S12 = 0, S22 = I. By solving the LMIs in (2.6)–(2.7), the minimal value of γ is γmin = 3.5551 and the corresponding decision variables P and Q are computed to be ⎡

⎤

40.8318 0.7772 −6.0000 ⎥ ⎢ P = ⎣ 0.7772 18.2773 −14.9999⎦ , Q = 10.6386. −6.0000 −14.9999 0

28

Analysis and Synthesis of Singular Systems

To verify the effectiveness of the result, the maximal singular values (MSVs) of the considered system are depicted in Fig. 2.1. From the figure, we can see that the value of γmin is very near the supremum of MSVs, which implies that the system satisfies the prescribed H∞ performance.

Figure 2.1 Maximal singular values and γmin .

•

General dissipative case: S11 = −1.6, S12 = −0.5, S22 = 0.4. Similarly, to verify the dissipativity of the considered system, we can verify whether the LMIs in Theorem 2.8 are feasible. By solving the LMIs, feasible matrices P and Q are computed to be ⎡

⎤

10.8800 0.2980 −1.4001 ⎥ ⎢ P = ⎣ 0.2980 4.8022 −3.4971⎦ , Q = 3.3810. −1.4001 −3.4971 0

•

According to the definition of (Q, N , R)-dissipativity in [201], we can see the considered system is (−0.4, 0.5, 1.6)-dissipativity. Positive realness case: S11 = 0, S12 = −I, S22 = 0. By solving the LMIs in Theorem 2.8, the conditions in (2.6) and (2.7) are satisfied with the

Dissipative control and filtering of singular systems

29

following solution: ⎡

⎤

14.5494 1.8136 1.9068 ⎢ ⎥ P = ⎣ 1.8136 4.0456 4.5228⎦ , Q = 3.0400. 1.9068 4.5228 0 Therefore the system is positive real. Example 2.2. Consider a singular system with the following parameters: ⎤

⎡

⎤

⎡

⎡ ⎤

1 0 0 0 1 0 0 ⎥ ⎥ ⎢ ⎢ ⎢ ⎥ E = ⎣0 1 0⎦ , A = ⎣−3 −2 1 ⎦ , Bw = ⎣0⎦ , 0 0 0 1 0 0 −1 



 

 

⎡ ⎤

0 1 3 −1 2 0 ⎢ ⎥ , Dw = , D= , B = ⎣1⎦ . 0 0 0 0 3 0

C =

⎡

⎤

⎡ ⎤

1 0 0 ⎥ ⎢ ⎢ ⎥ Choose EL = ER = ⎣0 1⎦, U T = Λ = ⎣0⎦. To illustrate the effectiveness 0 0 1 of Theorem 2.11, we consider the H∞ case, that is S11 = −γ 2 I, S12 = 0, S22 = I. By solving the LMIs in Theorem 2.11, the minimal value of γ is obtained as γmin = 2.2472, and the corresponding solutions of decision variables are ⎤

⎡

0.0459 −0.0011 0.0425 ⎥ ⎢ P¯ = ⎣−0.0011 0.1079 −0.2776⎦ , Π¯ = 0.8001, 0.0425 −0.2776 0 H =





0.0000 −0.1099 −0.0007 , G = −0.0030. 



Then, the feedback controller is K = −0.0244 −1.0215 −0.0009 , J = −0.0021.

2.1.4 Conclusion In this section, a new equivalent sets approach is proposed to investigate the problems of admissibilization and dissipative control of singular systems. By employing an equivalent parametrization of the constrained sets, a novel necessary and sufficient dissipativity condition of singular systems is presented without the equality constraint. Based on the criterion, a necessary

30

Analysis and Synthesis of Singular Systems

and sufficient condition for the existence of a state feedback controller is established to render the closed-loop system to be admissible and dissipative. Two examples are given to demonstrate the effectiveness of the obtained results.

2.2 Dissipative control of discrete-time singular systems The problem of dissipative control of discrete-time singular systems is investigated in this section. Based on parametrizing the solutions of the constraint set, a necessary and sufficient dissipativity condition is established in terms of strict LMI, which makes the condition more tractable. By using the system augmentation approach, an SOF controller design method is proposed to guarantee that the closed-loop system is admissible and strictly (Q, S, R)-dissipative.

2.2.1 Problem formulation Consider a class of linear discrete-time singular systems described by ⎧ ⎪ ⎨ Ex(k + 1) = Ax(k) + Bu(k) + Bw w (k), x0 = x(0), z(k) = Cx(k) + Du(k) + Dw w (k), ⎪ ⎩ y(k) = Cy x(k) + Dy w (k),

(2.17)

where x(k) ∈ Rn is the state vector; u(k) ∈ Rm is the control input; w (k) ∈ Rl represents a disturbance, which belongs to l2 ; z(k) ∈ Rq is the controlled output; y(k) ∈ Rg is the measurement output; matrices E, A, B, Bw , C, D, Dw , Cy , and Dy are constant matrices with appropriate dimensions and rank(E) = r ≤ n. Before moving on, some definitions and lemmas are given, which will be used in deriving the main results. Definition 2.13. [209] 1. The singular system in (2.17) is said to be regular if det(zE − A) is not identically zero. 2. The singular system in (2.17) is said to be causal if deg {det(zE − A)} = rank(E). 3. The singular system in (2.17) is said to be stable if the moduli of the roots of det(zE − A) = 0 are less than 1. 4. The singular system in (2.17) is said to be admissible if it is regular, causal, and stable.

31

Dissipative control and filtering of singular systems

Definition 2.14. [29] The system in (2.17) (u(k) = 0) is said to be strictly (Q, S, R)-dissipative if there exists a scalar α > 0, and under zero initial state x0 = 0, the following inequality holds: z, Qzτ + 2z, Sw τ + w , Rw τ ≥ αw , w τ , ∀τ ≥ 0.

(2.18) 1

As in [29], Q ≤ 0 is assumed. Consequently, there exists a matrix Q−2 = 1

1

(−Q) 2 ≥ 0 satisfying −Q = (Q−2 )2 .

Lemma 2.15. [29] Let the matrices Q, S, and R be given with Q and R real symmetric. Then the system in (2.17) is admissible (when u(k) = 0, w (k) = 0) and strictly (Q, S, R)-dissipative, if and only if there exists a real symmetric and invertible matrix X such that ET XE ≥ 0, ⎡

AT XA − ET XE

⎢ ⎣

AT XBw − C T S

(2.19) 1

⎤

1

⎥

C T Q−2

BwT XBw − DwT S − ST Dw − R DwT Q−2 ⎦ < 0. (2.20)  −I

 

Lemma 2.16. The following two sets are equivalent: X1

= {X ∈ Rn×n : ET XE ≥ 0, rank(ET XE) = r , X = X T };

X2

= {X = P − E0T UE0 : P > 0, E0 E = 0, E0 E0T > 0, E0 ∈ R(n−r )×n ,

U = U T }. Proof. (Sufficiency) When X ∈ X2 , one has ET XE = ET PE ≥ 0 and rank(ET XE) = r, which imply X ∈ X1 .   Ir 0 and X = (Necessity) Without loss of generality, set E = 0 0 

X1 X2T



X2 , where X1 = X1T ∈ Rr ×r and X3 = X3T ∈ R(n−r )×(n−r ) . Then one X3 



has E0 = 0 In−r , and it yields from ET XE ≥ 0 that X1 ≥ 0. Combining with rank(ET XE) =   rank(X1 ) = r, one has X1 > 0. By constructing P =

X1 X2 > 0 with > 0 and U = U T = X2T X1−1 X2 + I − X3 , X2T X2T X1−1 X2 + I X = P − E0T UE0 is obtained.

32

Analysis and Synthesis of Singular Systems

2.2.2 Dissipative control The aim of this section is to design an SOF controller in the form of u(k) = Ky(k) such that the closed-loop system 

Ex(k + 1) = (A + BKCy )x(k) + (Bw + BKDy )w (k), z(k) = (C + DKCy )x(k) + (Dw + DKDy )w (k)

(2.21)

is admissible and strictly (Q, S, R)-dissipative. Firstly, the new dissipativity analysis conditions of system (2.17) (when u(k) = 0), in terms of strict LMIs, are proposed in following theorem: Theorem 2.17. Let the matrices Q, S, and R be given with Q and R real symmetric and Q ≤ 0. The following statements are equivalent: (i) System (2.17) is admissible and strictly (Q, S, R)-dissipative. (ii) There exist matrices P > 0, and U = U T such that the following LMI holds: ⎡ ⎢ ⎢ ⎣

−ET PE + AT VA

AT VBw − C T S

1

C T Q−2 1

⎤ ⎥ ⎥

BwT VBw − DwT S − ST Dw − R DwT Q−2 ⎦ < 0,  −I (2.22)

 

where V = P − E0T UE0 . (iii) There exist matrices P > 0, U = U T , F , and G such that the following LMI holds:   −E T PE + sym(LT S + FA) −F + AT G T < 0,  V − GT − G

(2.23)

where  E

= 

A =

E 0

A 1 2

Q− C





0 P , P= I 0 Bw 1 2

Q− Dw





 0 , L= C R 

V , V= 0





Dw ,

 0 , S= 0 I

 −S .

Proof. (i)⇐⇒(ii): The equivalence between item (i) and item (ii) are obtained by using Lemma 2.15 and Lemma 2.16.

33

Dissipative control and filtering of singular systems

(iii)⇒(ii): The following LMI can be derived by pre- and post





multiplying (2.23) with I AT and I AT

T

:

  T PE + AT VA T VB − C T S E A − w V¯ =  BwT VBw − DwT S − ST Dw − R   −

C T QC C T QDw < 0,  DwT QDw

which is equivalent to (2.22) by utilizing Schur complement equivalence. (ii)⇒(iii): By employing Schur complement equivalence, condition (2.22) is equivalent to 

−ET PE + AT VA − C T QC 

AT VBw − C T S − C T QDw BwT VBw − DwT S − ST Dw − R − DwT QDw



= −E T PE + sym(LT S ) + AT VA < 0,

where P , A, and V are defined in (2.23). On the other hand, there always exists a matrix G such that V − G T − G < 0 and  −E T PE + sym(LT S ) + AT VA

0



0 V − GT − G



I −A T By pre- and post-multiplying (2.24) by 0 I yields that



< 0. 

I −A T and 0 I

(2.24) T

, it

  −E T PE + sym(LT S ) + AT sym(V − G )A AT (−V + G + G T ) < 0.  V − G − GT

(2.25) By setting F = AT (V − G ), one gets inequality (2.23). Remark 2.18. The advantage of Item (iii) of Theorem 2.17 lies in separating the Lyapunov matrix P and the system matrices A and C, which is utilized widely [142]. However, if one uses Item (iii) of Theorem 2.17 todesign the  1 F1 F2 2 SOF controller, the term F2 Q− (C + DK ) will appear with F = , F3 F4 which makes the condition in (2.23) difficult to solve. Therefore the separation of the controller K and system matrices D will be helpful for solving the controller design problem.

34

Analysis and Synthesis of Singular Systems

In the following, the SOF controller will be designed such that the closed-loop system in (2.21) is admissible and strictly (Q, S, R)-dissipative. To this end, one augments the closed-loop singular system in (2.21), as follows, by considering u(k) as a state component and choosing x¯ (k) = 

xT (k) uT (k)

T

as the new system state: 

where E¯

¯ x¯ (k) + B¯ w w (k), E¯ x¯ (k + 1) = A ¯ (k) + Dw w (k), z(k) = Cx









(2.26)





  E 0 B Bw ¯ = A ¯ = C D . , A , B¯ w = , C KDy 0 0 KCy −I

=

Before giving the main result, the equivalence of admissibility and dissipativity between the systems in (2.21) and (2.26) is proved firstly. The following two equations are true: 





¯ = zE − A −B = I −B zE¯ − A −KCy I 0 I





zE − A − BKCy 0 0 I



I 0 −KCy I

and ¯ )−1 B¯ w + Dw C¯ (zE¯ − A 



= C

D



 (zE − A − BKCy )−1

I 0 KCy I

0



0 I

I B 0 I





Bw + Dw KDy

= (C + DKCy )(zE − A − BKCy )−1 (Bw + BKDy ) + Dw + DKDy , ¯ and zE − A − BKCy are which derive that the determinants of zE¯ − A the same, and the transfer functions of the systems in (2.21) and (2.26) are equal, respectively. By using Definitions 2.13 and 2.14, the admissibility and dissipativity of system in (2.21) are equivalent to those in (2.26). The following corollary proposes SOF controller design method by utilizing the system augmentation approach:

Corollary 2.19. The system in (2.21) is admissible and strictly (Q, S, R)dissipative if and only if there exist matrices P > 0, U = U T , F , and G such that the following inequality holds:   −E¯ T P E¯ + sym(L¯ T S + F A¯ ) −F + A¯ T G T < 0,  V¯ − G T − G

(2.27)

Dissipative control and filtering of singular systems

where



 E¯

35





    E¯ 0 P 0 ¯ D w , S = 0 −S , , P= , L¯ = C 0 I 0 R

=

 A¯ =

¯ A 1



B¯ w 1

¯ Q−2 Dw Q−2 C



V¯ , V¯ = 0



0 , V¯ = P − E¯ 0T U E¯ 0 . I

Proof. By applying the Items (i) and (iii) of Theorem 2.17 to the augmented singular system in (2.26), and then using the equivalence of the admissibility and dissipativity between the system in (2.21) and the system in (2.26), the results follow. Remark 2.20. It can be seen that the inequality in (2.27) is in terms of bilinear matrix inequality, which can be solved by utilizing the existing numerical method [113]. Moreover, the H∞ control problem and the passivity control problem can also be addressed by setting −Q = I, S = 0, R = γ 2 and −Q = 0, S = I, R = 0 in (2.27), respectively. To obtain a more tractable SOF control condition for system (2.21), the following result is given: Theorem 2.21. There exists an SOF controller such that the closed-loop system in  (2.21) isadmissible and strictly (Q, S, R)-dissipative if there exist matrices P = P11 P12 > 0, U = U T , F11 , F12 , F13 , F21 , F22 , F3 , G11 , G13 , G21 , G22 ,  P22 G3 , and M such that the following LMI holds: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ Λ=⎢ ⎢ ⎢ ⎢ ⎣

1

Λ11

Λ12

Λ13

Λ14

Λ15



Λ22

Λ23

Λ24

Λ25

   

   

Λ33   

Λ34 Λ44  

Λ35 Λ45 Λ55 

−F21 + C T Q−2 G3T

⎤

⎥ 1 −F22 + DT Q−2 G3T ⎥ ⎥ ⎥ 1 2 T T −F3 + Dw Q− G3 ⎥ ⎥ < 0, ⎥ −G21 ⎥ ⎥ ⎦ −G22 I − G3 − G3T

(2.28)

where 1

Λ11

= −ET P11 E + sym(F11 A + LMCy + F21 Q−2 C );

Λ12

T T = F11 B − LF12 + F21 Q−2 D + AT F13 + CyT M T + C T Q−2 F22 ;

Λ13

= −C T S + F11 Bw + LMDy + F21 Q−2 Dw + C T Q−2 F3T ;

1

1

1

1

36

Analysis and Synthesis of Singular Systems 1

Λ14

T T = −F11 + AT G11 + CyT M T L T + C T Q−2 G21 ;

Λ15

T T = −LF12 + AT G13 + CyT M T + C T Q−2 G22 ;

Λ22

= sym(F13 B − F12 + F22 Q−2 D);

Λ23

= −DT S + F13 Bw + MDy + F22 Q−2 Dw + DT Q−2 F3T ;

Λ24

T T T T = −F13 + BT G11 − F12 L + DT Q−2 G21 ;

Λ25

T T T = −F12 + BT G13 − F12 + DT Q−2 G22 ;

Λ33

= −R + sym(F3 Q−2 Dw − DwT S);

Λ34

T T = BwT G11 + DyT M T L T + DwT Q−2 G21 ;

Λ35

T T = BwT G13 + DyT M T + DwT Q−2 G22 ;

1

1

1

1

1

1

1

1

1

T T = P11 − E¯ 01 U E¯ 01 − G11 − G11 ; T T ¯ ¯ = P12 − E01 U E02 − LF12 − G13 ;

Λ44 Λ45

T T = P22 − E¯ 02 U E¯ 02 − F12 − F12 ,   and E¯ 0 = E¯ 01 E¯ 02 with E¯ 0 E¯ = 0, E¯ 0 E¯ 0T > 0, E¯ 0 ∈ R(n+m−r )×(n+m) , E¯ 01 ∈   R(n+m−r )×n , E¯ 02 ∈ R(n+m−r )×m , L T = Im 0m×(n−m) . Then the SOF controller

Λ55

−1 M. can be obtained by K = F12

Proof. Let the matrices F and G in (2.27) be of the following forms: 





F1 F2 G1 G2 , G= F= 0 F3 0 G3 with



















F LF12 F21 G11 LF12 G21 , F2 = , G1 = , G2 = . F1 = 11 F13 F12 F22 G13 F12 G22 By setting M = F12 K, the inequality in (2.27) is obtained. Remark 2.22. The nonsingularity of the matrix F12 in Theorem 2.21 is satisfied without loss of generality. If it is not the case, then one can choose a sufficient small scalar θ such that F¯ 12 = F12 + θ I, satisfying the inequality −1 in (2.28). Then the matrix K can be replaced with F¯ 12 M. Remark 2.23. The SOF controller design method is given in terms of strict LMIs in Theorem 2.21, which can be easily solved by standard software. Although some SOF control problem for singular systems have been reported

Dissipative control and filtering of singular systems

37

such as [22], [169], the controller design method contains equality constrain, which makes the computation difficult. The result in Theorem 2.21 can also be used to deal with state-feedback dissipative control problem for singular systems. For the computation complexity, the total number of scalar decision variables of Theorem 2.21 is 0.5(n + m)(n + m + 1) + 0.5(n + m − r )(n + m − r + 1) + 2n2 + m(2n + m + g) + q(2n + 2m + l + q).

2.2.3 Illustrative example In this section, one example is provided to illustrate the effectiveness of the proposed approach. Theorem 2.17, which provides a necessary and sufficient dissipativity condition, will be used to check the applicability of the static output feedback controller design. Example 2.3. Consider a singular system (2.17) with following parameters: 

E = C =

















1 0 1.6 0.8 1.8 0.1 , A= , B= , Bw = , 0 0 0.6 1.2 1 0.4 





0.5 0.6 , D = 0.6, Dw = 0.8, Cy = 1 0 , Dy = −0.5.

Given Q = 1.2, S = 0.8, R = 1.6 and to test the admissibility and dissipativity of this open-loop system, one  chooses E0 = 0 1 . Solving the LMI in (2.22), no feasible solutions can be found, which means this open-loop system is not admissible and strictly (Q, S, R)-dissipative. To make the closed-loop system in (2.21) admissible and strictly (Q, S, R)-dissipative, the SOF controller design method in Theorem 2.21 is employed. By solving the LMI in (2.28), one has M = −0.5634, F12 = 0.5520, −1 and K = F12 M = −1.0207. Then the closed-loop system becomes

     ⎧  ⎪ − 1 0 0 . 2372 0 . 8000 1 . 0186 ⎪ ⎨ x(k + 1) = x(k) + w (k), −0.4207 1.2000 0 0 0.9103   ⎪ ⎪ ⎩ z(k) = −0.1124 0.6000 x(k) + 1.1062w (k).

(2.29)

38

Analysis and Synthesis of Singular Systems

Utilizing Theorem 2.17 to system (2.29), one can find that the LMI in (2.22) is feasible, and the closed-loop system in (2.29) is admissible and strictly (1.2, 0.8, 1.6) dissipative.

2.2.4 Conclusion This subchapter studied the problem of dissipative control for discrete-time singular system. By using the system augmentation approach, a static output feedback controller design method is proposed to guarantee that the closed-loop singular system is admissible and strictly (Q, S, R)-dissipative. An example is given to demonstrate the effectiveness of the results.

2.3 Dissipative filtering of singular systems In this section, the SOF controller designed approach will be applied to design the dissipative reduced-order filtering of singular systems.

2.3.1 Reduced-order dissipative filtering Consider a class of discrete-time singular system: ⎧ ⎪ ⎨ Ex(k + 1) = Ax(k) + Bw w (k), x0 = x(0), z(k) = Cx(k) + Dw w (k), ⎪ ⎩ y(k) = Cy x(k) + Dy w (k),

(2.30)

where x(k) ∈ Rn is the state vector; w (k) ∈ Rl represents a disturbance, which belongs to l2 ; z(k) ∈ Rq is the controlled output; y(k) ∈ Rg is the measurement output; matrices E, A, Bw , C, Dw , Cy , and Dy are constant matrices with appropriate dimensions. To estimate controlled output z(k), the following reduced-order filtering is constructed: 

xˆ (k + 1) = Af xˆ (k) + Bf y(k), xˆ (0) = 0, zˆ (k) = Cf xˆ (k) + Df y(k),

(2.31)

where xˆ (k) ∈ Rm (0 < m ≤ n) is the state vector of the filter; zˆ (k) ∈ Rq is the estimation of z(k); matrices Af , Bf , Cf , and Df are filter parameters to be determined. T  Denote x˘ (k) = xT (k) xˆ T (k) and the estimation error z˘ (k) = z(k) − zˆ (k), then the filtering error singular system derived from the singular sys-

Dissipative control and filtering of singular systems

39

tem in (2.30) and the filter in (2.31) is 

where

˘ x˘ (k) + B˘ w w (k), E˘ x˘ (k + 1) = A ˘ w w (k), z˘ (k) = C˘ x˘ (k) + D 







E 0 A 0 Bw ˘ = , A , B˘ w = , 0 I Bf Cy Af Bf Dy

E˘ = C˘ =





(2.32)





˘ w = Dw − Df Dy . C − D f C y −C f , D

The aim is to design a filter in (2.31) such that the filtering error system in (2.32) is admissible and strictly (Q, S, R)-dissipative. By carrying out simple manipulation, the following equations hold: ˘ = A ˜ y , B˘ w = B˜ + HK D ˜ + HK C ˜ y, A ˜ y, D ˘ = C ˜ + JK C ˜ y, ˘ w = Dw + JK D C

where ˜ = A





 0 B ˜ = C , B˜ = w , C 0 0

A 0

 ˜y = D









0 0 , H= Dy I



 0 , J= 0 0



(2.33) 

0 0 , C˜ y = Cy 



Af −I , K = Cf



I , 0 

Bf . Df

Then the system in (2.32) can be rewritten as ⎧ ˜ x˘ (k) + H u˘ (k) + Bw ˜ (k), ⎪ ⎨ E˘ x˘ (k + 1) = A z˘ (k) = C˜ x˘ (k) + J u˘ (k) + Dw w (k), ⎪ ⎩ ˜ y w (k), y˘ (k) = C˜ y x˘ (k) + D

(2.34)

with u˘ (k) = K y˘ (k). By now, one can see that the filter design problem is transferred to an SOF control problem, that is, to design a controller K guaranteeing the system in (2.32) to be admissible and strictly (Q, S, R)

T

dissipative. Define x¯ (k) = x˘ T (k) u˘ T (k) as a new state variable, and the system in (2.34) is equivalent to the following augmented one: 

¯ x¯ (k) + B¯ w w (k), E¯ x¯ (k + 1) = A ¯ w w (k), z˘ (k) = C¯ x¯ (k) + D

(2.35)

40

Analysis and Synthesis of Singular Systems

where









E¯ = C¯ =









˜ E˘ 0 H B˜ ¯ = A ¯w = , A , B ˜y , 0 0 KD K C˜ y −I

˜ J , D ¯ w = Dw . C

Therefore by substituting the matrices E, A, B, Cy , Bw , Dy , C, and D in ˜ H, C ˜ y , B, ˜ and J in (2.33), respectively, ˘ A, ˜ D ˜ y , C, Theorem 2.21 with E, one can get the following reduced-order dissipative filter design method: Theorem 2.24. There exists a filter in (2.31) such that the filtering error system in  (2.32) isadmissible and strictly (Q, S, R)-dissipative if there exist matrices P = P11 P12 > 0, U = U T , F11 , F12 , F13 , F21 , F22 , F3 , G11 , G13 , G21 , G22 ,  P22 G3 , and M such that the following LMI holds: ⎡ Θ ⎢ 11 ⎢ ⎢  ⎢ ⎢ Θ =⎢  ⎢ ⎢  ⎢ ⎣  

1

Θ12

Θ13

Θ14

Θ15

Θ22

Θ23

Θ24

Θ25

   

Θ33   

Θ34 Θ44  

Θ35 Θ45 Θ55 

˜ T Q−2 GT −F21 + C 3

⎤

⎥ 1 ⎥ −F22 + J T Q−2 G3T ⎥ ⎥ 1 −F3 + DwT Q−2 G3T ⎥ ⎥ < 0, ⎥ −G21 ⎥ ⎥ ⎦ −G22 I − G3 − G3T

(2.36)

where 1

Θ11

˜ + LM C ˜ y + F21 Q−2 C ˜ ); = −E˘ T P11 E˘ + sym(F11 A

Θ12

˜ T FT + C ˜ T MT + C ˜ T Q−2 F T ; = F11 H − LF12 + F21 Q−2 J + A 13 y 22

Θ13

˜ T S + F11 B˜ + LM D ˜ T Q−2 F T ; ˜ y + F21 Q−2 Dw + C = −C 3

Θ14

˜ T GT + C ˜ T M T LT + C ˜ T Q−2 GT ; = −F11 + A 11 y 21

Θ15

˜ T GT + C ˜ T MT + C ˜ T Q−2 GT ; = −LF12 + A 13 y 22

Θ22

= sym(F13 H − F12 + F22 Q−2 J );

Θ23

˜ y + F22 Q−2 Dw + D ˜ T Q−2 F3T ; = −J T S + F13 B˜ + M D

Θ24

T T T T = −F13 + H T G11 − F12 L + J T Q−2 G21 ;

Θ25

T T T = −F12 + H T G13 − F12 + J T Q−2 G22 ;

Θ33

= −R + sym(F3 Q−2 Dw − DwT S);

1

1

1

1

1

1

1

1

1

1

1

1

Dissipative control and filtering of singular systems

41

1

Θ34

T T ˜ yT M T L T + DwT Q−2 G21 = B˜ T G11 +D ;

Θ35

T T ˜ yT M T + DwT Q−2 G22 = B˜ T G13 +D ;

1

Θ45

T T = P11 − E¯ 01 U E¯ 01 − G11 − G11 ; T T ¯ ¯ = P12 − E01 U E02 − LF12 − G13 ;

Θ55

T T = P22 − E¯ 02 U E¯ 02 − F12 − F12 ;

Θ44





and E¯ 0 = E¯ 01 E¯ 02 with E¯ 0 E¯ = 0, E¯ 0 E¯ 0T > 0, E¯ 0 ∈ R(n+m+q−r )×(n+2m+q) , 



E¯ 01 ∈ R(n+m+q−r )×(n+m) , E¯ 02 ∈ R(n+m+q−r )×(m+q) , L T = Im+q 0(m+q)×(n−q) . 

−1

Then the desired filter can be obtained by K = F12 M =

Af Cf

Bf Df



.

Proof. Set 

F1 =















F11 LF12 F21 G11 LF12 G21 , F2 = , G1 = , G2 = , F13 F12 F22 G13 F12 G22

and let the matrices F and G be the following forms: 







F1 F2 G1 G2 , G= . F= 0 F3 0 G3 −1 Then noting that K = F12 M, one can obtain the inequality in (2.27) from the inequality in (2.36) by straightforward manipulation. Therefore the admissibility and dissipativity of system (2.35), which is equivalent to those of system (2.32) are proved.

Remark 2.25. The reduced-order filter design method is given in terms of strict LMIs in Theorem 2.24, and the total number of scalar decision variables of Theorem 2.24 is 0.5(n + 2m + q)(n + 2m + q + 1) + 0.5(n + m + q − r )(n + m + q − r + 1) + 2(m + n)(n + m + q) + (m + q)(2n + 4m + g + 3q) + (q + l)q. The reduced-order filtering problem is also investigated in [79], [118] and [210], respectively. However, to obtain the desired filter parameters, a complicated matrix structure is needed, and the rank of the difference of two decision variables should be less than the order of the filter in [210]. For the method developed in this section, the filter parameters can be obtained directly by solving the LMI in (2.36), which avoids considering the rank constraint, non-strict LMIs in [210], [118], or constructing some complicated matrices in [79].

42

Analysis and Synthesis of Singular Systems

2.3.2 Illustrative example In this subsection, an example is provided to illustrate the effectiveness of the proposed approach and filter design method. Example 2.4. In this example, a first-order filter in the form of (2.31) will be designed for the following discrete-time singular systems: ⎧ ⎡ ⎤ ⎤ ⎤ ⎡ ⎡ ⎪ − − 1 1 0 1 0 . 5 1 0 . 1 ⎪ ⎪ ⎪ ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ ⎪ ⎪ ⎣1 −1 1⎦ x(k + 1) = ⎣−1 −0.3 1⎦ x(k) + ⎣ 0 ⎦ w (k), ⎪ ⎪ ⎪ ⎪ 2 0 1 0.5 0.1 0 1 ⎪ ⎪ ⎪ ⎪ ⎤ ⎤ ⎡ ⎡ ⎪ ⎪ ⎪ −3.2 0 3.2 −0.1 ⎨ ⎥ ⎥ ⎢ ⎢ z(k) = ⎣ 3.2 0 1.6⎦ x(k) + ⎣ 0.5 ⎦ w (k), ⎪ ⎪ ⎪ 0.1 0 0 3.2 ⎪ ⎪ ⎪ ⎪ ⎡ ⎡ ⎤ ⎤ ⎪ ⎪ ⎪ 1 1 0 0.1 ⎪ ⎪ ⎢ ⎢ ⎥ ⎥ ⎪ ⎪ y ( k ) = x ( k ) + 1 1 0 0 ⎦ w (k). ⎪ ⎣ ⎣ ⎦ ⎪ ⎪ ⎩ 0 0 1 0.1

(2.37)

¯ one gets From the value of E, ⎡

1 ⎢0 ⎢ ⎢ E¯ 01 = ⎢0 ⎢ ⎣0 0

1 0 0 0 0

−1

0 0 0 0

⎡

⎤

0 0 ⎢ ⎥ 0⎥ ⎢1 ⎢ ⎥ 0⎥ , E¯ 02 = ⎢0 ⎢ ⎥ ⎣0 0⎦ 0 0

0 0 1 0 0

⎤

0 0 0 1 0

0 0⎥ ⎥ ⎥ 0⎥ . ⎥ 0⎦ 1

By setting ⎡

⎤

⎡

⎤

−0.1 0.6200 −0.8000 0.1600 ⎢ ⎢ ⎥ ⎥ Q = − ⎣−0.8000 1.2500 −0.0500⎦ , S = ⎣ 0.5 ⎦ , R = 1.5, 0.1600 −0.0500 1.0100 0.2

and solving the LMI in (2.36), the matrix ⎡ ⎢ ⎢ ⎣

K =⎢

0.1256 1.3812 −1.7594 −0.7707 −0.0637 0.0899 −0.5044 −0.8806 0.1001 1.8047 −1.3853 −2.4015 −1.0660 −0.8425 1.1057 0.7424

⎤ ⎥ ⎥ ⎥. ⎦

Dissipative control and filtering of singular systems

43

Then one obtains a first-order filter as   ⎧ xˆ (k + 1) = 0.1256xˆ (k) + 1.3812 −1.7594 −0.7707 y(k), xˆ (0) = 0, ⎪ ⎪ ⎪ ⎡ ⎡ ⎤ ⎤ ⎨ −0.0637 0.0899 −0.5044 −0.8806 ⎢ ⎢ ⎥ ⎥ ⎪ zˆ (k) = ⎣ 0.1001 ⎦ xˆ (k) + ⎣ 1.8047 −1.3853 −2.4015⎦ y(k), ⎪ ⎪ ⎩ −1.0660 −0.8425 1.1057 0.7424

and the parameters of the filtering error system in (2.32) are given as follows: ⎡

1 1 ⎢1 −1 ⎢ E˘ = ⎢ ⎣2 0 0 0

0 1 1 0

⎤

0 0⎥ ⎥ ⎥, 0⎦ 1

⎡ ⎤ 1.0000 0 −1.0000 0.5000 ⎢−1.0000 −0.3000 1.0000 0 ⎥ ⎥ ˘ = ⎢ A ⎢ ⎥; 0 1.0000 0 ⎦ ⎣ 0.5000 −0.3782 −0.3782 −0.7707 0.1256

B˘ w = C˘

T  −0.1000 0 0.1000 0.0610 ;

⎤ ⎤ ⎡ ⎡ −2.7855 0.4145 4.0806 0.0637 −0.0209 ⎥ ˘ ⎥ ⎢ ⎢ = ⎣ 2.7806 −0.4194 4.0015 −0.1001⎦ , D w = ⎣ 0.5597 ⎦ . 0.1100 −0.2632 −0.2632 2.4576 1.0660

(2.38) To check whether the obtained filtering error system is admissible and strictly (Q, S, R)-dissipative, Theorem 2.17 is utilized again. By solving the LMI in (2.22), a feasible solution is found, which shows the applicability and effectiveness of the method. By giving the initial condition with 

T

x˘ (0) = −1.6756 −0.2870 0.9170 0

and

w (k) = 0,

the state responses of system (2.32) are given in Fig. 2.2, which illustrates the stability of the system. The characteristic polynomial C (z) = −11750z3 +136008z2 −45229z+3768 shows the regularity and causality of the system 25000 from the first two items of Definition 2.13; the admissibility of the system is obtained. To demonstrate the dissipativity of the system, one chooses

44

Analysis and Synthesis of Singular Systems

Figure 2.2 State responses.

Figure 2.3 Output signal z˘ (k).

w (k) = 0.1e−0.1k sin(k) and zero initial conditions, the output signal z˘ (k), and the performance signal G(z˘ , w , τ ) = z, Qzτ + 2z, Sw τ + w , Rw τ are proposed in Figs. 2.3 and 2.4, respectively. From Fig. 2.4, one can see that G(z˘ , w , τ ) is greater than or equal to zero when τ ≥ 0. Then a suf-

Dissipative control and filtering of singular systems

45

Figure 2.4 Performance signal G(z˘ , w, τ ) = ˘z, Q˘zτ + 2˘z, Swτ + w, Rwτ .

ficiently small scalar α > 0 can always be found such that the inequality in (2.18) holds, which shows the dissipativity of the system in (2.32).

2.3.3 Conclusion In this section, the augmentation system approach is utilized to solve the filtering problem, which aims to guarantee the augmentation singular systems to be admissible and strictly (Q, S, R)-dissipative. The results presented in this section are in terms of strict LMIs, which make the conditions more tractable numerically. A numerical example is given to demonstrate the effectiveness of the proposed method.

CHAPTER 3

H∞ control with transients for singular systems

In this chapter, the problem of a generalized type of H∞ control is investigated for continuous-time singular systems, which treats a mixed attenuation of exogenous inputs and initial conditions. First, a performance measure that is essentially the worst-case norm of the regulated outputs over all exogenous inputs and initial conditions is introduced. Necessary and sufficient conditions are obtained to ensure the singular system to be admissible and the performance measure to be less than a prescribed scalar. Based on the criteria, a sufficient condition for the existence of a state-feedback controller is established in terms of LMIs. Moreover, the relationship between the performance measure and the standard H∞ norm of the system is provided. Two numerical examples are given to demonstrate the properties of the obtained results.

3.1 Performance measure Consider a class of linear continuous singular systems described by 

Ex˙ (t) = Ax(t) + Bw (t), z(t) = Cx(t) + Dw (t),

x(0− ) = x0 ,

(3.1)

where x(t) ∈ Rn is the state vector; x0 is the initial condition; w (t) ∈ Rp represents a set of exogenous inputs, which includes disturbances to be rejected, and z(t) ∈ Rs is the controlled output; A, B, C, and D denote constant matrices with appropriate dimensions. In contrast with standard linear systems with E = I, the matrix E ∈ Rn×n has 0 < rank(E) = r < n. Since system (3.1) can be rewritten as ⎧  E ⎪ ⎪ ⎪ ⎪ ⎨ 0 ⎪ ⎪ ⎪ ⎪ ⎩

0 0



x˙ η˙



 =

z(t) =

A 0 0 −I



Analysis and Synthesis of Singular Systems https://doi.org/10.1016/B978-0-12-823739-7.00010-0

C I

 

x η

x



 +

η 

B D



w,

, Copyright © 2021 Elsevier Inc. All rights reserved.

47

48

Analysis and Synthesis of Singular Systems

we will assume D = 0 from now on without loss of generality. Then, the plant under investigation is as follows: 

Ex˙ (t) = Ax(t) + Bw (t), z(t) = Cx(t).

x(0− ) = x0 ,

(3.2)

First, we give some definitions and lemmas concerning the unforced system of (3.2): Lemma 3.1. [216] Suppose the pair (E, A) is regular and impulse-free, then the solution to system (3.2) is impulse-free and unique on [0, ∞). Remark 3.2. As there exist nonsingular matrices M and N, such that 







¯ ¯ A A B¯ MEN = diag(I , 0), MAN = ¯ 11 ¯ 12 , MB = ¯ 1 , A21 A22 B2 



x¯ (t) , N x(t) = x¯ (t) = 1 x¯ 2 (t) −1

(3.3)

system (3.2) is restricted system equivalent to 











¯ ¯ I 0 ˙ A A B¯ x¯ (t) = ¯ 11 ¯ 12 x¯ (t) + ¯ 1 w (t), x¯ (0− ) = N −1 x0 . A21 A22 B2 0 0

(3.4)

¯ 22 is invertNotice that singular system (3.2) is impulse-free if and only if A ible. Then, under an impulse-free condition, a consistent initial condition of singular system (3.2) is characterized as follows:     x¯ 1 (0+ ) limt→0+ x¯ 1 (t) =N x(0+ ) = lim x(t) = N t→0+ limt→0+ x¯ 2 (t) x¯ 2 (0+ )   x¯ 1 (0− ) . = N −1 ¯ ¯ −A22 (A21 x¯ 1 (0− ) + B¯ 2 w (0+ ))

(3.5)

If x(0− ) = x0 = x(0+ ), there is a finite jump at t = 0. There are some papers dealing with the elimination and minimization of initial jumps, see [107, 108,157]. In view of this, we introduce the following definition for singular system (3.2):

H∞ control with transients for singular systems

49

A general excitation performance measure of asymptotically stable system (3.2) is used in this section, which is defined as the worst-case norm of the controlled output over all admissible exogenous signals and initial states: γg (R) =

sup T w 2 +xT 0 E REx0 =0



z w 2 + xT0 ET REx0

1/2

for any w ∈ L2 [0, ∞) and x0 ∈ Rn , where R > 0 is a given weighting matrix. In the following, the problem of H∞ control with transients (HCT) is stated as follows: Problem HCT. Establish a necessary and sufficient condition such that the following conditions hold: 1. The singular system in (3.2) with w (t) = 0 is admissible. 2. The general excitation performance measure γg (R) of system (3.2) is less than a prescribed scalar γ > 0. Remark 3.3. In this section, both the exogenous input and initial state are taken into account when assessing the performance of the singular system. When Ex0 = 0, the problem considered reduces to the H∞ control problem of singular system, that is, z z sup = G∞ =  1/2 . w =0 w  w =0, Ex0 =0 w 2 + xT E T REx0

γ∞ = sup

0

Therefore the definition domain of γ∞ belongs to that of γg , and we get γ∞ = G∞ ≤ γg . On the other hand, when Ex0 = 0 and w (t) = 0, γg becomes z γ0 (R) = sup  1/2 . Ex0 =0 xT E T REx0 0 Based on the definitions of γg (R), γ∞ , and γ0 (R), some properties about these performance measures are given in following theorem: Theorem 3.4. 1. If ET R1 E ≥ ET R2 E, then γg (R1 ) ≤ γg (R2 ). 2. γg (R) ≥ max{γ∞ , γ0 (R)}. Proof. The proving of Items 1 and 2 can be carried out following the similar line as in the proof of Theorem 2 in [3]. Before giving our main result, we provide the generalized bounded real lemma for the H∞ control problem of singular systems.

50

Analysis and Synthesis of Singular Systems

Lemma 3.5. [3] The performance measure of system (3.1) with E = I satisfying γg (R) < γ if and only if there exists a matrix X > 0 such that the following LMIs hold: ⎡

AT X + XA

⎢ ⎣

 

⎤

CT ⎥ 2 −γ I DT ⎦ < 0, X < γ 2 R.  −I XB

Lemma 3.6. The following sets are equivalent: = ℵ =

 



X ∈ Rn×n : ET X = X T E ≥ 0, rank(ET X ) = r ,



X = PE + E0 Φ : P = P T ∈ Rn×n , ELT PEL > 0, Φ ∈ R(n−r )×n ,

where EL and ER are full column rank with E = EL ERT , and E0 ∈ Rn×(n−r ) with full column rank is the right null matrix of ET , that is, ET E0 = 0. Proof. (Sufficiency) Let X = PE + E0 Φ , we have ET X = X T E = ER ELT PEL ERT ≥ 0, and rank(ET X ) = rank(ER ERT ) = r . 





X11 X12 (Necessity) Without loss of generality, denote X = , E= X21 X22 





Ir 0 0 , where X11 ∈ Rr ×r and X22 ∈ R(n−r )×(n−r ) , then we have E0 = . 0 0 In−r

By using ET X = X T E, we have X12 = 0, X11 ≥ 0. Due to rank(ET X ) = rank(X11 ) = r, we arrive at X 11 > 0. Based on above discussion, we can 

X11 0 > 0 and Φ = X21 X22 such that X = find a matrix P = 0 In−r PE + E0 Φ . Moreover, ELT PEL > 0 holds. Then Problem HCT can be solved in terms of LMIs as follows: Theorem 3.7. Given a scalar γ > 0 and E0 ∈ Rn×(n−r ) with full column rank is the right null matrix of ET , that is, ET E0 = 0. Then the following statements are equivalent: 1. The system in (3.2) is admissible and its general excitation performance measure γg (R) < γ . 2. There exists a matrix X such that the following LMIs hold: ET X = X T E ≥ 0,

(3.6)

H∞ control with transients for singular systems

⎡

51

⎤

AT X + X T A X T B C T ⎢ ⎥  −γ 2 I 0 ⎦ < 0, ⎣   −I

(3.7)

ET X ≤ γ 2 ET RE.

(3.8)

3. There exist matrices P = P T and Φ such that the following LMIs hold: ⎡

⎤

AT (PE + E0 Φ) + (PE + E0 Φ)T A (PE + E0 Φ)T B C T ⎢ ⎥  −γ 2 I 0 ⎦ < 0, ⎣   −I (3.9) ET PE ≤ γ 2 ET RE,

(3.10)

ELT PEL > 0,

(3.11)

where EL and ER are full column rank with E = EL ERT . Proof. 3⇒1 Suppose that there exist matrices P and Φ such that the LMIs in (3.9) and (3.11) holds, then AT (PE + E0 Φ) + (PE + E0 Φ)T A < 0, ET (PE + E0 Φ) = (PE + E0 Φ)T E = ET PE ≥ 0 are obtained which gives the necessary and sufficient condition for the admissibility of the singular system in (3.2) [209]. On the other hand,   x(t) ≡ 0 with t ∈ [0, ∞): from (3.9), the following inequality holds for w (t) 



xT (t) AT (PE + E0 Φ) + (PE + E0 Φ)T A + C T C x(t) +2xT (t)(PE + E0 Φ)T Bw (t) − γ 2 w T (t)w (t) ≤ 0.

(3.12)

We choose a Lyapunov candidate as V (t) = xT (t)ET PEx(t) = xT (t)ET (PE + E0 Φ)x(t). Hence, the derivative of the Lyapunov function along the trajectory of system (3.2) is given by 



V˙ (t) = xT (t) AT (PE + E0 Φ) + (PE + E0 Φ)T A x(t) + 2xT (t)(PE + E0 Φ)T Bw (t),

52

Analysis and Synthesis of Singular Systems

and (3.12) implies that V˙ (t) + |z(t)|2 − γ 2 |w (t)|2 ≤ 0. Integrating the above inequality from zero to infinity and noting that V˙ (t)+ |z(t)|2 − γ 2 |w (t)|2 ≡ 0 with t ∈ [0, ∞), we get z2

< γ 2 w 2 + x(0+ )T ET PEx(0+ ).

By using ET PE ≤ γ 2 ET RE in (3.10) and x¯ 1 (0+ ) = x¯ 1 (0− ) in (3.4), it yields z2

< γ 2 w 2 + γ 2 x(0+ )T ET REx(0+ ) = γ 2 w 2 + γ 2 x¯ (0+ )T N T ET REN x¯ (0+ )    = γ w  + γ x¯ (0+ ) 2

2

2

T

 = γ w  + γ x¯ (0− ) 2

2

2

T



I 0 I 0 M −T RM −1 x¯ (0+ ) 0 0 0 0 





I 0 I 0 M −T RM −1 x¯ (0− ) 0 0 0 0

= γ 2 w 2 + γ 2 xT0 ET REx0 ,

(3.13)

where M and N are defined in (3.3). Hence, γg (R) < γ holds true and the result is proved. 1⇒2 Suppose that the singular system in (3.2) is admissible and γg (R) < γ , then there exist two nonsingular matrices S and T such that 



I SET = 0



0 A1 , SAT = 0 0







0 B1 , SB = , CT = C1 I B2 



C2 ,



x˜ (t) where A1 is stable. We set x(t) = T x˜ (t) = T 1 , and the system in (3.2) x˜ 2 (t) is restricted system equivalent to ⎧  I ⎪ ⎪ ⎨















0 ˙ A 0 B x˜ x˜ (t) = 1 x˜ (t) + 1 w (t), x˜ 0 = T −1 x0 ≡ 10 , B2 x˜ 20 0 0 0 I

⎪ ⎪ ⎩ z(t) = C 1

that is,





C2 x˜ (t),

x˙˜ 1 (t) = A1 x˜ 1 (t) + B1 w (t), x˜ 10 = x˜ 1 (0), z(t) = C1 x˜ 1 (t) − C2 B2 w (t).

(3.14)

H∞ control with transients for singular systems

53

z  1/2 T w 2 + xT0 ET REx0 w 2 +xT 0 E REx0 =0 z = sup  1/2 2 T T ˜ T0 T T ET RET x˜ 0 w 2 +x˜ T ˜ 0 =0 w  + x 0 T E RET x

γg (R) =

= =

sup

sup T T T −T RS−1 SET x w 2 +x˜ T ˜ 0 =0 0T E S S

z   1 /2 T E T ST S−T RS−1 SET x w 2 + x˜ T T ˜0 0

z  1/2  γN , 2 ˜ T10 R1 x˜ 10 w 2 +x˜ T ˜ 10 =0 w  + x 10 R1 x sup

where R1 ∈ Rr ×r satisfying R1 > 0 is the (1, 1) block of matrix S−T RS−1 . It can be seen that γg (R) can be transformed to the performance measure γN of the system in (3.14) with a given weighting matrix R1 > 0. Based on Lemma 3.5, for γN < γ , we obtain that there exists a matrix X1 > 0 such that the following LMIs hold: ⎡

⎤

AT X1 + X1 A1 X1 B1 C1T ⎢ 1 ⎥ 2  −γ I −(C2 B2 )T ⎦ < 0, ⎣   −I

(3.15)

X 1 < γ 2 R1 .

(3.16)

Denoting W = C2T C2 + α I with a sufficiently small scalar α > 0, it yields the following inequality from (3.15): 

AT1 X1 + X1 A1 + C1T C1 X1 B1 − C1T C2 B2  −γ 2 I + B2T WB2

 < 0,

which is equivalent to ⎡

⎤

AT X1 + X1 A1 + C1T C1 X1 B1 − C1T C2 B2 0 ⎢ 1 ⎥  −γ 2 I B2T W ⎦ < 0. ⎣   −W Then there exists ⎡

AT X1 + X1 A1 + C1T C1 X1 B1 − C1T C2 B2 ⎢ 1  −γ 2 I ⎣ 



⎤

0 ⎥ T B2 W ⎦ < 0, −W − α I

54

Analysis and Synthesis of Singular Systems

which implies ⎡ ⎢ Π =⎣

AT1 X1 + X1 A1 + C1T C1

⎤

X1 B1 − C1T C2 B2 ⎥ −W − α I −WB2 ⎦ < 0.  −γ 2 I 0

 

Then following the same lines as in the proof of Theorem 5.1 in [209], we can construct a matrix X given by 

X =S



X1 0 T −1 , −C2T C1 −C2T C2 − α I

T

(3.17)

such that the following inequality holds: 

T T (AT X + X T A + C T C )T 

TT XT B −γ 2 I

 = Π < 0,

which is equivalent to (3.7). Moreover, X in (3.17) also satisfies the equality in (3.6) and the inequality in (3.8), because of that 

0≤E X =X E=T T

T

−T







X1 0 −1 γ 2 R1 0 T ≤ T −T T −1 = γ 2 ET RE. 0 0 0 0

2⇒3 It is easy to establish the proof by replacing X = PE + E0 Φ based on Lemma 3.6. 



Remark 3.8. The condition rank ET (P − γ 2 R)E = rank(E) in (3.9) and (3.10) also can be satisfied without loss of generality. If it is not the case, then we can choose a sufficient small scalar α > 0 and a matrix P˜ > 0 such T 2 ¯ ˜ ¯ rank E (P − γ R)E = rank(E), (3.9) and that P = P − α P > 0 satisfying  (3.10). The condition rank ET (X − γ 2 RE) = rank(E) in (3.6), (3.7), and (3.8) can be satisfied following a similar argument. When w (t) ≡ 0, the performance measure γ0 (R) can be characterized by the following LMIs based on Theorem 3.7: Corollary 3.9. Given a scalar γ > 0, the system in (3.2) is admissible and its performance measure γ0 (R) 0,

(3.20)

where E0 and EL are defined in Lemma 3.6. Note that γg (R) is the infimum of γ , for which the LMIs in (3.9)–(3.11) are feasible; γ0 (R) is the infimum of γ , for which the LMIs in (3.18)–(3.20) are feasible, whereas γ∞ is the infimum of γ , for which the inequality in (3.9) is feasible.

3.2 Controller design Consider the following singular system: 

Ex˙ (t) = Ax(t) + Bw (t) + Fu(t), z(t) = Cx(t) + Gu(t),

x(0− ) = x0 ,

(3.21)

where x(t), w (t), A, B, C are defined in (3.1); u(t) ∈ Rm is the control input, F and G are constant matrices. A state-feedback controller in the form of u(t) = Kx(t), K ∈ Rm×n ,

(3.22)

will be designed for the singular system in (3.21) such that the general excitation performance measure of following closed-loop system is less than a prescribed positive scalar: 

Ex˙ (t) = (A + FK )x(t) + Bw (t), z(t) = (C + GK )x(t).

x(0− ) = x0 ,

(3.23)

If we use item 3 in Theorem 3.7 to design the state feedback controller, matrices A and C should be replaced by A + FK and C + GK, respectively. The inequality in (3.9) becomes ⎡

(A + FK )T Ξ + Ξ T (A + FK ) Ξ T B ⎢  −γ 2 I ⎣  

(C + GK )T

0 −I

⎤ ⎥ ⎦ < 0,

(3.24)

56

Analysis and Synthesis of Singular Systems

where Ξ = PE + E0 Φ . Then performing congruence transformation to inequality (3.24) with diag{Ξ −T , I , I } and its transpose, we have ⎡

Ξ −T (A + FK )T + (A + FK )Ξ −1 ⎢  ⎣ 

B

Ξ −T (C + GK )T

−γ 2 I 

0 −I

⎤ ⎥ ⎦ < 0.

(3.25) If the inequality in (3.10) is not present, the controller can be obtained as K = Y Ξ by setting Y = K Ξ −1 in inequality (3.25). However, the inequality in (3.10) should also be solved. The matrix P appears in (3.10), and (PE + E0 Φ)−1 appears in (3.25). On the other hand, there is no expansion formula for (PE + E0 Φ)−1 as that given in Lemma 2.4 for (PE + U T QΛT )−1 , which makes the matrix inequalities (3.10) and (3.25) nonlinear, and hence difficult to solve. Before giving the controller design method, a sufficient condition guaranteeing the general excitation performance of closed-loop system (3.23) to be less than a prescribed positive scalar can be obtained by applying the LMIs in (3.9) and (3.10) of Theorem 3.7, in which E0 and Φ are replaced by U T and QΛT with P = P T , Q unknown, that is, the following two LMIs hold: ⎡

Θ + ΘT ⎢  ⎣ 

(PE + U T QΛT )T B − γ 2I 

(C + GK )T

0 −I

⎤ ⎥ ⎦ < 0,

(3.26)

ET PE ≤ γ 2 ET RE,

(3.27)

ELT PEL > 0,

(3.28)

where Θ = (A + FK )T (PE + U T QΛT ), U ∈ R(n−r )×n with full row rank, and Λ ∈ Rn×(n−r ) with full column rank are the left and right null matrices of matrix E, respectively, that is, UE = 0 and EΛ = 0. Based on discussion below Eq. (3.23) and Lemma 2.4, we are now in a position to present the state-feedback controller design method for system (3.21). Theorem 3.10. There exists a state-feedback controller such that closed-loop system (3.23) is admissible and its general excitation performance measure γg (R) 0, Q if there exist matrices P¯ = P¯ T satisfying ERT PE

57

H∞ control with transients for singular systems

following LMIs hold: 

¯ + FW ) ¯ T + AΛQU sym(APE  



¯ T ΛT C T + W T GT ¯ T + UT Q EPC 0 −I

B −γ 2 I 

−γ 2 ET RE

ER ¯ R −ERT PE

ERT

 < 0,

(3.29)

 ≤ 0,

(3.30)

where U, Λ, EL , ER are defined in Lemma 2.4. Under the conditions, a desired controller can be obtained by ¯ )−1 . ¯ T + ΛQU K = W (PE

(3.31)

Proof. Performing the congruence transformation to the inequality in (3.26) by diag(H , I , I ), with H = (PE + U T QΛT )−1 , we have ⎡

⎤

H T (C + GK )T ⎢ ⎥  −γ 2 I 0 ⎣ ⎦ 0, Considering ELT PEL = (ERT PE which is the inequality in (3.28). In addition, based on the definition of ¯ )−1 is obtained. Then the desired result follows ¯ T + ΛQU W , K = W (PE immediately. Remark 3.11. Note that an H∞ state-feedback controller can be obtained by ¯ ∞ are the values of ¯ ∞ U )−1 , where W∞ , P¯ ∞ , and Q K∞ = W∞ (P¯ ∞ ET + ΛQ ¯ obtained by solving only inequality (3.29) with the minimal ¯ and Q W , P, value of γ . A γg (R) state-feedback controller can be obtained by Kw (R) = ¯ w are the values of W , P, ¯ w U )−1 , where Ww , P¯ w , and Q ¯ Ww (P¯ w ET + ΛQ ¯ obtained by solving inequality (3.29) and inequality (3.30), with and Q the minimal value of γ . Qualitatively, the inequality in (3.30) is relaxed to be always satisfied for sufficiently large positive definite R. Hence, the

58

Analysis and Synthesis of Singular Systems

performance measure γg (R) will be close to γ∞ for large R, and a statefeedback controller for γg (R) obtained by Theorem (3.10) will approach an H∞ state-feedback controller. Moreover, a γ0 (R) state-feedback controller ¯ 0 U )−1 , where W0 , P¯ 0 , and can be obtained by K0 (R) = W0 (P¯ 0 ET + ΛQ ¯ 0 are the values of W , P, ¯ obtained by solving inequality (3.29) ¯ and Q Q with the second row and column omitted and inequality (3.30), with the minimal value of γ . Remark 3.12. For the static output feedback (SOF) control problem, let y(t) = Cy x(t) and u(t) = Ky(t). Then, the terms FKCy H and GKCy H will appear by using the method in Theorem 3.10. These nonlinear terms cannot be solved by using LMI toolbox directly. Recently, an augmented system approach is proposed to address the SOF problem in [158]. The advantages of the approach can separate the input matrix F or G, and gainout matrix KCy and decouple the Lyapunov matrix P and the controller matrix K, which may help to solve the SOF control problem easily.

3.3 Illustrative examples In this section, we use numerical examples to illustrate the properties of obtained results. The following example is given to illustrate the relation obtained in Theorem 3.4. Example 3.1. Given a singular system with following parameters: 







−1 0 1 1 , A= , B= E= 1 1 0 2



√1











1 0 1 , C= , E0 = , 0 1 −1 0 2

we have γ∞ = 1. Let R = ρ 2 I, then γg (ρ 2 I ) and γ0 (ρ 2 I ) can be computed according to description given in the paragraph above Theorem 3.4. In Fig. 3.1, Curve 1 presents the plot of γg (ρ 2 I ) versus ρ ; Curve 2 gives the plot of γ∞ ; Curve 3 denotes the plot of γ0 (ρ 2 I ). From Fig. 3.1, we can see γg (ρ 2 I ) = γ∞ when ρ becomes larger, while the curve γg (ρ 2 I ) asymptotically approaches γ0 (ρ 2 I ) for small ρ . The following example is given to illustrate the property mentioned in Remark 3.11. Example 3.2. Given a singular system with following parameters: 

E =







 





1 1 0 2 4 1 0 , A= , B= , C= , 1 1 4 0 1 −1 1

H∞ control with transients for singular systems

59

Figure 3.1 Performance measure versus parameter ρ .

Figure 3.2 Performance measure γ = γg (ρ 2 I) of the closed-loop system versus parameter ρ under different state-feedback controllers.



F =











0 1 −0.3 , G= , Λ= , U = −1 1 , −1 0.3 0.5

since deg {det(sE − A)} = rank(E), the singular system is not impulse-free. Based on the LMI in (3.29), we obtain a standard H∞ state-feedback con-

60

Analysis and Synthesis of Singular Systems





troller K∞ = −0.8332 −3.3333 and γ∞ = 2.0000. Similarly, let R = ρ 2 I,



the state-feedback controller Kw (ρ) = −0.0002 −2.5020 and γg (ρ 2 I ) = 2.2714 can be obtained by solving the LMIs in (3.29) and (3.30) with ρ = 0.22. Fig. 3.2 shows three curves: Curve 1 denotes the general excitation performance measure γg (ρ 2 I ) for closed-loop system with K∞ ; Curve 2 is the general excitation performance measure γg (ρ 2 I ) for the closed-loop system with K0 (ρ); Curve 3 refers to the general excitation performance measure γg (ρ 2 I ) for the closed-loop system with Kw (ρ). From this figure, we can see that the general excitation performance measure γg (ρ 2 I ) is close to γ∞ for large ρ , and is close to γ0 (ρ 2 I ) for small ρ .

3.4 Conclusion In this chapter, the problem of H∞ control with transients for continuoustime singular systems has been studied. The necessary and sufficient conditions in terms of LMIs were proposed for performance analysis at first. Based on this, a state-feedback controller has been designed to guarantee the closed-loop system to be admissible and minimize the general excitation performance measure. Moreover, the relationship between the H∞ performance measure with transients and the standard H∞ performance has been revealed, which has been illustrated by numerical examples.

CHAPTER 4

Delay-dependent admissibility and H∞ control of discrete singular delay systems In this chapter, the issues of admissibility and H∞ control for discrete singular systems with time-varying delay are addressed, respectively. Firstly, to reduce the conservatism of existing admissibility conditions, we adopt an improved reciprocally convex combination approach to bound the forward difference of the double summation term, and utilize the reciprocally convex combination approach to bound the forward difference of the triple summation term in the Lyapunov function. Without employing decomposition and equivalent transformation of the considered system, a strict delay-dependent LMI criterion is built to guarantee the considered system to be regular, causal, and stable. Secondly, with the introduction of the delay partitioning technique, strict LMI sufficient criteria are obtained for discrete-time singular systems to be regular, causal, and stable. Based on these criteria, the robust stabilization and robust H∞ control problems are addressed and the desired state-feedback controllers are given. Numerical examples are given to illustrate the reduced conservatism of the developed results.

4.1 New admissibility analysis for discrete singular systems with time-varying delay In this section, the problem of admissibility analysis for discrete-time singular system with time-varying delay is investigated. By employing the reciprocally convex combination approach to bound the forward difference of a triple-summation term, a sufficient criterion is presented in terms of LMIs to guarantee the considered system to be regular, causal, and stable. Finally, a numerical example is exhibited to illustrate the effectiveness and the reduced conservatism of the proposed result. Analysis and Synthesis of Singular Systems https://doi.org/10.1016/B978-0-12-823739-7.00011-2

Copyright © 2021 Elsevier Inc. All rights reserved.

61

62

Analysis and Synthesis of Singular Systems

4.1.1 Problem formulation Consider discrete-time singular systems with time-varying delay described by Ex(k + 1) = Ax(k) + Ad x(k − d(k));

(4.1)

x(k) = φ(k), k ∈ [−d2 , 0], where x(k) ∈ Rn is the state vector and d(k) is a time-varying delay satisfying 1 ≤ d1 ≤ d(k) ≤ d2 , where d1 and d2 are prescribed positive integers representing the lower and upper bounds of the time delay, respectively. φ(k) is the compatible initial condition. The matrix E ∈ Rn×n may be singular, and it is assumed that rank(E) = r ≤ n. A and Ad are known real constant matrices with appropriate dimensions. The following definitions and lemmas will be used in the proof of the main results: Definition 4.1. [25] 1. The pair (E, A) is said to be regular if det(zE − A) is not identically zero. 2. The pair (E, A) is said to be causal if deg(det(zE − A)) = rank(E). Definition 4.2. [191] The singular system in (4.1) is said to be stable if for any scalar ε > 0, there exists a scalar δ(ε) > 0 such that, for any compatible initial conditions φ(k) satisfying sup−d2 ≤k≤0 φ(k) ≤ δ(ε), the solution x(k)   of system (4.1) satisfies x(k) ≤ ε for any k ≥ 0, moreover limk→∞ x(k) = 0. Definition 4.3. [121] 1. The singular system in (4.1) is said to be regular and causal if the pairs (E, A) is regular and causal. 2. The singular system in (4.1) is said to be admissible if it is regular, causal, and stable. Lemma 4.4. [140] Let f1 , f2 , ..., fN : Rm → R have positive values in an open subset D of Rm . Then the reciprocally convex combination of fi over D satisfies 1

min 

{αi |αi >0,

i αi =1}



subject to

i

αi

fi (k) =

 i

gi,j : R → R, gj,i (k) = gi,j (k), m



fi (k) + max gi,j (k)



gi,j (k),

i=j

fi (k) gj,i (k) gi,j (k) fj (k)



(4.2)  ≥0 .

Delay-dependent admissibility and H∞ control of discrete singular delay systems

63

Lemma 4.5. For any matrix M > 0, integers a < b ≤ c, vector function x(i) : [k + a, k + c − 1] → Rn , there hold − (b − a)

b−1 

b−1

T b−1

  x (i)Mx(i) ≤ − x(i) M x(i) , T

i=a

−β

i=a

b−1 k +c −1 

(4.3)

i=a

xT (i)Mx(i) ≤ −ϑ T (k)M ϑ(k),

(4.4)

j=a i=k+j

where β =

(2c +1)(b−a)−(b2 −a2 )

2

, ϑ(k) =

b−1 k+c−1 j=a

i=k+j

x(i).

Remark 4.6. Based on similar arguments as those of Lemma 1 in [84], Lemma 4.5 can be established. Moreover, Lemma 4.5 is more general than Lemma 1 in [84], since the latter can be derived by setting a = −d2 , b = c = −d1 in Lemma 4.5. 



S Lemma 4.7. For any constant matrix > 0, integers a ≤ d(k) ≤  M b ≤ c, vector function w(i) = x(i + 1) − x(i), x(i) : [k + a, k + c ] → Rn , then −β

b−1 k +c −1 

M



w (i)Mw (i) ≤ −η (k) T

T

M 

j=a i=k+j

S M

 η(k),

where β is defined in Lemma 4.5, and  η(k) =

η1 (k) η2 (k)



 =

(d(k) − a)x(k + c ) −

d(k)−1

x(k + j) j=a b−1 (b − d(k))x(k + c ) − j=d(k) x(k + j)

 .

Proof. When a < d(k) < b, by utilizing (4.4), we have −β

b−1 k +c −1 

w T (i)Mw (i)

j=a i=k+j

=−β

d( k)−1 k +c −1 j=a

w T (i)Mw (i) − β

i=k+j

b−1 k +c −1 

w T (i)Mw (i)

j=d(k) i=k+j

β β ≤ − η1T (k)M η1 (k) − η2T (k)M η2 (k)w , β1 β2

where β1 = (2c+1)(d(k)−2a)−(d

2 (k)−a2 )

, β2 =

(2c +1)(b−d(k))−(b2 −d2 (k))

2

(4.5) .

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Analysis and Synthesis of Singular Systems

Noting ββ1 + ββ2 = 1 and that according to Lemma 4.4, it follows from the inequality in (4.5) that  η(k) =

η1 (k) η2 (k)



 =

(d(k) − a)x(k + c ) −

d(k)−1

x(k + j) j=a b−1 (b − d(k))x(k + c ) − j=d(k) x(k + j)

 .

(4.6)



When d(k) = a or d(k) = b, we have (d(k) − a)x(k + c ) − jd=(ka)−1 x(k + j)  = 0 or (b − d(k))x(k + c ) − bj=−d1(k) x(k + j) = 0, and the inequality in (4.6) still holds by using Lemma 4.5. Hence, the lemma is proved. Remark 4.8. In [190], the reciprocally convex combination approach is extended to discrete-time systems, where it is used to bound the single summation term. Here the approach is applied in Lemma 4.7 to bound the double-summation term. Lemma 4.9. [104] For a matrix R > 0, a matrix Γ , and a matrix Ξ , the following statements are equivalent: • Ξ − Γ T RΓ < 0. • There exists a matrix Ψ such that 



Ξ + sym(Γ T Ψ ) Ψ T  −R

< 0.

The purpose of this section is to analyze the admissibility of system (4.1), and a new stability criterion with less conservatism than some existing ones will be proposed.

4.1.2 Main results Theorem 4.10. For given positive integers d1 , d2 , the discrete-time singular time-delay system in (4.1) is admissible for any time-varying delay d(k) satisfying 1 ≤ d1 ≤ d(k) ≤ d2 , if there exist matrices P > 0, Qi > 0, Pi = PiT , i = 1, 2, 3, Rj > 0, Sj > 0, j = 1, 2, P4 , X, W , and Ψ such that the following LMIs hold:  R1 > 0, R2 > 0, ⎡ ⎢ ⎣

Ξ + sym(ΛT1 Ψ ) 

S2 P4  S2

> 0;

(4.7)

S2

⎤  ⎥ P4 ⎦ < 0;

(4.8)



S2

T  Ψ

−



Delay-dependent admissibility and H∞ control of discrete singular delay systems

⎡ ⎢ ⎣

Ξ + sym(ΛT2 Ψ )

⎤

T  Ψ

S2 P4  S2

−



65

 ⎥ ⎦ < 0,

(4.9)

where R ∈ Rn×(n−r ) is any matrix with full column rank and satisfies ET R = 0, and T T Ξ=WP1 PWP1 − WP2 PWP2 + W1T Q1 W1 − W2T Q1 W2 T +W2T Q2 W2 − d1 W2T ET P1 EW2 − W3T Q3 W3 + d12 WR1 R1 WR1 2 T +d12 WR1 R2 WR1 + d1 W1T ET P1 EW1 − W4T Q2 W4 + d12 W2T ET P2 EW2

−d12 W3T ET P2 EW3 + d12 W3T ET P3 EW3 − d12 W4T ET P3 EW4 T T 2 T −WR 1 R1 WR1 − WR2 R2 WR2 + α1 ((A − E )W1 + Ad W3 )

×S1 ((A − E)W1 + Ad W3 ) + α22 ((A − E)W1 + Ad W3 )T ×S2 ((A − E)W1 + Ad W3 ) − (d1 EW1 + (d12 + 1)W1T Q3 W1 +sym(W1T WRT (AW1 + Ad W3 )) − W5 )T S1 (d1 EW1 − W5 ), ⎤ ⎤ ⎡ ⎡ P11P12P13 AW1 + Ad W3 ⎥ ⎥ ⎢ ⎢ P =⎣  P22P23 ⎦ , WP1 = ⎣ EW1 − EW2 + W5 ⎦ ,   P33 EW2 − EW4 + W6 + W7 ⎤ ⎡   EW1 EW1 ⎥ ⎢ , WP2=⎣ W5 ⎦ , WR1 = (A − E)W1 + Ad W3 W6 + W7      

R1 =

R11aR12a R R W5 , R2 = 11b 12b , WR1 = ,  R22a  R22b EW1 − EW2

⎡

⎤

W6   ⎢EW − EW ⎥ −W6 ⎢ 2 3⎥ , WR2 =⎢ ⎥ , Λ1 = W7 d12 EW1 − W7 ⎣ ⎦ EW3 − EW4 











d12 EW1 − W6 X X R R + P1 , Λ2 = , X = 11 12 , R1 = 11a 12a −W7 X21X22  R22a + P1 ⎡

⎤

R11bR12b + P2 X11 X12 ⎢  R +P X X22 ⎥ ⎢ ⎥ 22b 2 21 R2 =⎢ ⎥,  R11bR12b + P3 ⎦ ⎣     R22b + P3 Wi =[0n,(i−1)nIn0n,(7−i)n ], i = 1,2,...,7, d12 = d2 − d1.

66

Analysis and Synthesis of Singular Systems

Proof. According to the given condition, we first prove that system (4.1) is regular and causal. Since rank (E) = r, we choose two nonsingular matrices M and N such that 

MEN =



Ir 0 0 0

(4.10)

.

Set 

MAN =

A1 A2 A3 A4





,N W= T

W1 W2





,M

−T

R=

0 I



F,

(4.11)

where F ∈ R(n−r )×(n−r ) is nonsingular. According to Lemma 4.9, the equivalent representation of condition (4.8) is  Θ1 = Ξ − ΛT1

Expand Ξ as

S2 P4  S2

 Ξ=

Ξ11 •

• •

 Λ1 < 0.

 ,

where • represents the elements of the matrix that are not relevant in our discussion, Ξ11 ∈ Rn×n and Ξ11 = Q1 + (d12 + 1)Q3 + d1 ET P1 E + α12 (A − E)T S1 (A − E) + α22 (A − E)T S2 (A − E) + WRT A + AT RW T − ET P11 E T + AT P11 A + AT P12 E + ET P12 A + ET P22 E + d12 ET R12a (A − E) T 2 T + d12 (A − E)T R12a E + d12 (A − E)T R22a (A − E) + d12 E R12b (A − E) 2 T 2 + d12 (A − E)T R12b E + d12 (A − E)T R22b (A − E) − d12 ET S1 E 2 T + d12 ET R11a E + d12 E R11b E − ET (R22a + P1 )E. 2 E T S E < 0, which implies Due to Θ1 < 0, we have Ξ11 − d12 2

Ω = d1 ET P1 E − d12 ET S1 E + WRT A + AT RW T − ET P11 E + AT P12 E T T + ET P12 A + ET P22 E + d12 ET R12a (A − E) + d12 (A − E)T R12a E 2 T 2 T + d12 E R12b (A − E) + d12 (A − E)T R12b E − ET (R22a + P1 )E 2 T − d12 E S2 E < 0.

Delay-dependent admissibility and H∞ control of discrete singular delay systems

67

Premultiplying and postmultiplying Ω < 0 by N T and N, respectively, substituting (4.10) and (4.11) into the above inequality give 

• • • W2 F T A4 + AT4 FW2T

 < 0.

From the above inequality, it is easy to see that W2 F T A4 + AT4 FW2T < 0, which implies A4 is nonsingular. Thus the pair (E, A) is regular and causal. According to Definition 4.1 and Definition 4.3, system (4.1) is regular and causal. To prove the stability of the system in (4.1), design a Lyapunov function as V (k) = V1 (k) + V2 (k) + V3 (k) + V4 (k), where V1 (k) = εT (k)P ε(k), V2 (k) =

k−1 

xT (i)Q1 x(i) +

i=k−d1

+

−d1 

k− d1 −1

xT (i)Q2 x(i) +

i=k−d2 k−1 

k−1 

xT (i)Q3 x(i)

i=k−d(k)

xT (j)Q3 x(j),

i=−d2 +1 j=k+i

V3 (k) = d1

−1  k−1 

η (j)R1 η(j) + d12 T

i=−d1 j=k+i

V4 (k) = α1

− d1 −1  k−1

ηT (j)R2 η(j),

i=−d2 j=k+i

k−1 −1  −1  

θ T (l)ET S1 Eθ (l) + α2

i=−d1 j=i l=k+j

− d1 −1  k−1 −1 

θ T (l)ET S2 Eθ (l),

i=−d2 j=i l=k+j

with θ (i) = x(i + 1) − x(i) and ⎤   Ex(k)  Ex ( k ) k−1 ⎥ ⎢ , ε(k) = ⎣ i=k−d1 Ex(i) ⎦ , η(k) = Eθ (k) k−d1 −1 Ex ( i ) i=k−d2 ⎡

α1 =

d1 (d1 + 1) d12 (d1 + d2 + 1) , α2 = . 2 2

By denoting the forward difference of V (k) as V (k) = V (k + 1) − V (k) and calculating it along the solution of system (4.1), we have V1 (k) = ε T (k + 1)P ε(k + 1) − ε T (k)P ε(k),

68

Analysis and Synthesis of Singular Systems

where ⎡ ⎢ ε(k + 1) = ⎣

Ex(k + 1)  Ex(k) − Ex(k − d1 ) + ki=−k1−d1 Ex(i) Ex(k − d1 ) − Ex(k − d2 ) +

k−d1 −1 i=k−d2

⎤ ⎥ ⎦,

Ex(i)

so T T V1 (k) = ξ T (k)(WP1 PWP1 − WP2 PWP2 )ξ(k),

(4.12)

where ξ T (k) =[ xT (k) xT (k − d1 ) xT (k − d(k)) xT (k − d2 ) k−d1 −1 k−d(k)−1 k−1 T T T i=k−d1 (Ex(i)) i=k−d(k) (Ex(i)) i=k−d2 (Ex(i)) ].

The estimation of the forward difference of V2 (k) is V2 (k) = xT (k)Q1 x(k) − xT (k − d1 )Q1 x(k − d1 ) + xT (k − d1 )Q2 x(k − d1 ) + xT (k)Q3 x(k) − xT (k − d(k))Q3 x(k − d(k)) + d12 xT (k)Q3 x(k) − xT (k − d2 )Q2 x(k − d2 ) −

k −d1

xT (i)Q3 x(i)

i=k−d2 +1

≤ ξ (k)(W1T Q1 W1 − W2T Q1 W2 + W2T Q2 W2 + (d12 + 1)W1T Q3 W1 − W3T Q3 W3 )ξ(k). T

− W4T Q2 W4

(4.13)

Introducing three equalities, just as those in [84] with any real symmetric matrices P1 , P2 , P3 , according to (4.3) in Lemma 4.5, and considering R1 > 0 in (4.7), we have V3 (k) = d12 ηT (k)R1 η(k) − d1

k−1 

2 T ηT (i)R1 η(i) + d12 η (k)R2 η(k)

i=k−d1

+ d1 x (k)E P1 Ex(k) − d1 xT (k − d1 )ET P1 Ex(k − d1 )   k− d1 −1 k−1  0n P1 T T η(i) − d12 η (i)R2 η(i) − d1 η (i) T

T

i=k−d2

i=k−d1

+ d12 xT (k − d1 )ET P2 Ex(k − d1 ) − d12 xT (k − d(k))ET P2 Ex(k − d(k))   k− d1 −1 0 P n 2 η(i) ηT (i) − d12 i=k−d(k)

P2 P2

P1 P1

69

Delay-dependent admissibility and H∞ control of discrete singular delay systems

+ d12 xT (k − d(k))ET P3 Ex(k − d(k)) k− d(k)−1

− d12 x (k − d2 )E P3 Ex(k − d2 ) − d12 T

T

 T

η (i)

i=k−d2

0n P3 P3 P3

 η(i)

2 T ≤ d12 ηT (k)R1 η(k) + d12 η (k)R2 η(k) + d1 xT (k)ET P1 Ex(k)

− d1 xT (k − d1 )ET P1 Ex(k − d1 ) + d12 xT (k − d1 )ET P2 Ex(k − d1 ) − d12 xT (k − d(k))ET P2 Ex(k − d(k)) + d12 xT (k − d(k))ET P3 Ex(k − d(k)) − d12 xT (k − d2 )ET P3 Ex(k − d2 ) T    k−1 k−1 Ex ( i ) Ex ( i ) i=k−d1 i=k−d1 − R1 Ex(k) − Ex(k − d1 ) Ex(k) − Ex(k − d1 ) − d12

d1 −1 k−

ηT (i)(R2 + P2 )η(i) − d12

d(k)−1 k−

i=k−d(k)

ηT (i)(R2 + P3 )η(i),

i=k−d2

(4.14) where 

0n P2 P2 P2

P2 =



 , P3 =

0n P3 P3 P3

 .

Then by employing Lemmas 4.5 and 4.7, the following inequality holds for any matrix R2 > 0 when d1 < d(k) < d2 : k− d1 −1

− d12

η (i)(R2 + P2 )η(i) − d12 T

i=k−d(k)

≤−

ηT (i)(R3 + P3 )η(i)

i=k−d2 k− d1 −1

2 d12

d(k)−1 k−

d(k) − d1 i=k−d(k)

ηT (i)(R2 + P2 )η(i)

k− d(k)−1 2 d12 ηT (i)(R3 + P3 )η(i) − d2 − d(k) i=k−d 2

d12 d12 ≤− ψ1T (R2 + P2 )ψ1 − ψ T (R2 + P3 )ψ2 d(k) − d1 d2 − d(k) 2 

≤−

ψ1 ψ2

T



R2



ψ1 ψ2

,

(4.15)

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Analysis and Synthesis of Singular Systems

where  ψ1 =



k−d1 −1

i=k−d(k) Ex(i)

Ex(k − d1 ) − Ex(k − d(k))

 , ψ2 =

k−d(k)−1 i=k−d2



Ex(i)

Ex(k − d(k)) − Ex(k − d2 )

.

When d(k) = d1 or d(k) = d2 , the inequality (4.15) still holds based on the notation. From (4.14) and (4.15), V3 can be bounded as V3 (k) ≤ ξ T (k)(d12 W2T ET P2 EW2 − d12 W3T ET P2 EW3 T 2 T + d12 W3T ET P3 EW3 + d12 WR1 R1 WR1 + d12 WR1 R2 WR1

+ d1 W1T ET P1 EW1 − d1 W2T ET P1 EW2 T − d12 W4T ET P3 EW4 − WR 1 R1 WR1 T − WR 2 R2 WR2 )ξ(k).

(4.16)

The forward difference of V4 (k) is calculated as −1  −1  V4 (k) =α1 (θ T (k)ET S1 Eθ (k) − θ T (k + j)ET S1 Eθ (k + j))

i=−d1 j=i

+ α2

− d1 −1  −1

(θ T (k)ET S2 Eθ (k) − θ T (k + j)ET S2 Eθ (k + j))

i=−d2 j=i

=α12 θ T (k)ET S1 Eθ (k) − α1

−1  k−1 

θ T (j)ET S1 Eθ (j)

i=−d1 j=k+i

+ α22 θ T (k)ET S2 Eθ (k) − α2

− d1 −1  k−1

θ T (j)ET S2 Eθ (j).

i=−d2 j=k+i

By using (4.4) in Lemma 4.5, yields − α1

−1  k−1 

θ T (j)ET S1 Eθ (j)

i=−d1 j=k+i

≤ −(d1 Ex(k) −

k−1  i=k−d1

Ex(i)) S1 (d1 Ex(k) − T

k−1  i=k−d1

Ex(i))

Delay-dependent admissibility and H∞ control of discrete singular delay systems

71

Considering Lemma 4.9, we have inequality for d1 <  the following  S2 P4 > 0: d(k) < d2 and a matrix P4 satisfying  S2 − α2

− d1 −1  k−1

 θ T (j)ET S2 Eθ (j) ≤ −γ T (k)

i=−d2 j=k+i

S2 P4  S2

 γ (k),

(4.17)

where  γ (k) =

(d(k) − d1 )Ex(k) −

k−d1 −1

i=k−d(k) Ex(i) k−d(k)−1 (d2 − d(k))Ex(k) − i=k−d2 Ex(i)

 .

When d(k) = d1 or d(k) = d2 , according to the notation, we have (d(k) −   1 d1 )Ex(k)− ik=−kd−1 −d(1k) Ex(i) = 0 or (d2 − d(k))Ex(k)− ik=−kd−(kd)− Ex(i) = 0, and 2 the inequality (4.17) still holds by using inequality (4.4) in Lemma 4.5. One can obtain V4 (k) ≤ξ T (k)[α12 ((A − E)W1 + Ad W3 )T S1 ((A − E)W1 + Ad W3 ) + α22 ((A − E)W1 + Ad W3 )T S2 ((A − E)W1 + Ad W3 ) − (d1 EW1 − W5 )T S1 (d1 EW1 − W5 )   − ΛT (k)

S2 P4  S2

(4.18)

Λ(k)]ξ(k),

where  Λ(k) =

(d(k) − d1 )EW1 − W6 (d2 − d(k))EW1 − W7

 .

On the other hand, it is clear that f (k) = 2xT (k)WRT Ex(k + 1) ≡ 0.

(4.19)

Then we can obtain V (k) = V1 (k) + V2 (k) + V3 (k) + V4 (k) + f (k) ≤ ξ T (k)Θξ(k),

where

 Θ = Ξ − Λ (k) T

S2 P4  S2

 Λ(k).

(4.20)

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Analysis and Synthesis of Singular Systems

Based on Lemma 4.9, the equivalent condition Θ < 0 is that there exists a matrix Ψ such that ⎡ ⎢ Θ(k) = ⎣

Ξ + sym(ΛT (k)Ψ ) −



S2

⎤  ⎥ P4 ⎦ < 0.



S2

T  Ψ

(4.21)

Due to the convexity of Θ(k) with respect to d(k), the conditions (4.7) and (4.8) can guarantee the condition (4.21) holds. Hence, there exists a scalar α > 0 such that V (k) ≤ −α||x(k)||2 . Therefore (4.8) and (4.9) guarantee that the system (4.1) is asymptotically stable for any time-varying delay d(k) that satisfies 0 < d1 ≤ d(k) ≤ d2 . The proof of Theorem 4.10 has been completed. Remark 4.11. The main difference from other papers, such as [188], is that the improved reciprocally convex combination approach is applied to bound the forward difference of the double summation term in V3 (k), and the reciprocally convex combination approach to bound the forward difference of the triple summation term in V4 (k). This technique leads to a less conservative result. On the other hand, without decomposition and equivalent transformation of the considered system, a strict delay-dependent LMI criterion is established, which reduces the computational complexity compared with nonstrict LMI conditions [33]. The number of decision variables needed in Theorem 4.10 is given by 31.5n2 + (7.5 − r )n.

4.1.3 Numerical example The following example illustrates the effectiveness of our method and the advantage over some previous ones: Example 4.1. Consider discrete singular time-delay system (4.1) with 

E=

1 0 0 0





, A=

0.8 0 0.05 0.9





, Ad =

0 −0.1 −0.2 −0.1

 .

Applying the approaches of [33] [36] [188] and the result presented in this section, the admissible maximum values of d2 that guarantees the system (4.1) to be admissible, with various d1 are shown in Table 4.1. From the results, it is clear to see that the criterion in this section has reduced conservatism for admissibility condition of discrete-time singular systems with time-varying delay.

Delay-dependent admissibility and H∞ control of discrete singular delay systems

73

Table 4.1 Comparison of the admissible maximum values of d2 for various d1 . d1 0 3 6 9 12

Theorem 2 [121] Theorem 2 [33] Theorem 2 [36] Theorem 1 [73] Corollary 2 [188] Theorem 4.10

7 12 15 18 18 35

8 13 16 18 19 37

10 14 19 21 21 39

13 15 22 24 24 42

15 17 25 27 27 44

4.1.4 Conclusion In this section, the delay-dependent admissibility problem of discrete-time singular system with time-varying delay is investigated. A triple-summation term is introduced into the Lyapunov function, and the improved reciprocally convex combination approach is utilized to bound the forward difference of the double summation term. The new admissibility criterion, given in terms of LMIs, ensures the regularity, causality, and stability of the considered system. The result reduces the conservatism of existing results significantly. A numerical example is given to demonstrate the effectiveness and advantage of the proposed result. Further research is to find a trade-off between conservatism and computational complexity, and design an H∞ or dissipativity controller with reliability condition.

4.2 Delay-dependent robust H∞ controller synthesis for discrete singular delay systems In this section, the problems of delay-dependent robust stability analysis, robust stabilization, and robust H∞ control are investigated for uncertain discrete-time singular systems with state delay. First, by making use of the delay-partitioning technique, a new delay-dependent criterion is given to ensure the nominal system to be regular, causal, and stable. This new criterion is further extended to singular systems with both delay and parameter uncertainties. Then, without the assumption that the considered systems being regular and causal, robust controllers are designed for discrete-time singular time-delay systems such that the closed-loop systems have the characteristics of regularity, causality, and asymptotic stability. Moreover, the problem of robust H∞ control is solved following a similar line. The obtained results are dependent not only on the delay, but also the partitioning size, and the conservatism are nonincreasing with reducing partitioning size.

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Analysis and Synthesis of Singular Systems

These results are shown, via extensive numerical examples, to be much less conservative than existing results in the literature.

4.2.1 Problem formulation Consider a class of linear discrete-time uncertain singular systems with state delay described by ⎧ ⎪ Ex(k + 1) = ⎪ ⎪ ⎨ + ⎪ z(k) = ⎪ ⎪ ⎩ x(k) =

(A + A(k))x(k) + (Ad + Ad (k))x(k − d) Bw w (k) + (B + B(k))u(k), Cx(k) + Du(k), φ(k), k ∈ [−d¯ , 0],

(4.22)

where x(k) ∈ Rn is the state vector; u(k) ∈ Rq is the control input; w (k) ∈ Rp is the disturbance input, and z(k) ∈ Rs is the controlled output; A, Ad , B, Bw , C, and D are constant matrices with appropriate dimensions; d is a constant positive integer satisfying 0 < d ≤ d¯ (d can always be described by d = mτ , where m and τ are integers), where d¯ is a positive integer representing the upper bound of the delay; matrix E may be singular and rank E = r ≤ n; φ(k) is a compatible vector valued initial function; A, Ad , and B are time-varying uncertain matrices of the form [A Ad B] = MF (k)[N1 N2 N3 ],

where M, N1 , N2 , and N3 are constant matrices, and F (k) ∈ Rl×b is an unknown real matrix satisfying F (k)F (k)T ≤ I. Before moving on, we give some definitions and lemmas concerning the following nominal unforced counterpart of the system in (4.22): 

Ex(k + 1) = Ax(k) + Ad x(k − d), x(k) = φ(k), k ∈ [−d¯ , 0]

(4.23)

Lemma 4.12. [72] Suppose the pair (E, A) is regular and causal, then the solution to system (4.23) is causal and unique on [0, ∞) for any constant time¯ delay d satisfying 0 < d ≤ d. In view of this, we introduce the following definition for singular delay system (4.23): Definition 4.13. 1. The singular delay system in (4.23) is said to be regular and causal if the pair (E, A) is regular and causal.

Delay-dependent admissibility and H∞ control of discrete singular delay systems

75

2. The singular system in (4.23) is said to be asymptotically stable if, for any ε > 0, there exists a scalar δ(ε) > 0, such that for any compatible initial conditions φ(k) satisfying sup−d¯ ≤k≤−1 φ(k) ≤ δ(ε), the solution x(k) of (4.23) satisfies x(k) ≤ ε for k ≥ 0; furthermore, x(k) → 0, when k → ∞. 3. The singular time-delay system in (4.23) is said to be admissible if it is regular, causal, and asymptotically stable. Before giving the next definition, we present the formulation of the closed-loop system with a state feedback controller, u(k) = Kx(k), K ∈ Rq×n , ⎧ ⎪ Ex(k + 1) = ⎪ ⎪ ⎨ + ⎪ z ( k ) = ⎪ ⎪ ⎩ x(k) =

(A + BK + A(k) + B(k)K )x(k) (Ad + Ad (k))x(k − d) + Bw w (k), (C + DK )x(k), φ(k), k ∈ [−d¯ , 0].

(4.24)

(4.25)

Definition 4.14. System (4.25) is said to be robust asymptotically stable with γ disturbance attenuation if the following requirements are satisfied: 1. With w (k) = 0, the closed-loop system in (4.25) is asymptotically stable for all uncertainties. 2. Under zero initial condition, the closed-loop system in (4.25) satisfies ||z||2 < γ ||w ||2 for any nonzero w ∈ l2 [0, ∞), where γ > 0 is a prescribed scalar. Lemma 4.15. [175] The system in (4.23) is asymptotically stable if and only if det(zE − A − z−d Ad ) = 0 for |z| ≥ 1. Lemma 4.16. [209] Given matrices Ω , Γ , and Φ with appropriate dimensions, and with Ω symmetric, then Ω + Γ FΦ + ΦT FT Γ T < 0

for any F satisfying F T F ≤ I, if and only if there exists a scalar ε > 0 such that Ω + εΓ Γ T + ε −1 Φ T Φ < 0.

In this subchapter, three problems for uncertain discrete singular delay system (4.22) are investigated. 1. The robust stability problem: the objective is to establish new robust stability criterion such that the discrete-time singular system in (4.22) with u(k) = 0 and w (k) = 0 is admissible.

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Analysis and Synthesis of Singular Systems

2. The robust stabilization problem: the purpose is to design a state feedback controller such that the resulting closed-loop system is admissible. 3. The robust H∞ control problem: the aim is the design of a state feedback controller such that the resulting closed-loop system is admissible with an H∞ disturbance attenuation.

4.2.2 Robust stability In this subsection, we obtain a solution to the robust stability analysis problem formulated previously by using a strict LMI approach. First, we present the following result for the nominal singular delay systems, which will play a key role in solving the aforementioned problems:

4.2.2.1 Stability analysis: nominal case In this subsection, we will present a new delay-dependent sufficient condition guaranteeing the nominal system in (4.23) is admissible. The main idea is based on the delay-partitioning technique, which constitutes the major difference from most of the existing results. Our new result is given as follows: Theorem 4.17. Given positive integers m, τ , the system in (4.23) is admissible if there exist matrices P1 > 0, Q > 0, Z > 0, S1 , S2 , S3 , P2 , P3 , and P4 , such that Θ < 0,

(4.26)

where R ∈ Rn×(n−r ) is any full-column rank matrix satisfying ET R = 0 and Θ

˜ Q − W T ET ZEWZ = WPT (P1 + τ 2 Z )WP + WQT QW Z

˜ = Q

S = WP1 = WQ = WZ =

T T +sym(WP1 E P1 WP + P T WP2 + SRT WP ), ⎤ ⎡   Q11 · · · Q1m Q 0mn,mn ⎢ .. ⎥ , .. , Q=⎣  . . ⎦ 0mn,mn −Q   Qmm T    , WP = 0n,(m+1)n In , S1T S2T 0n−r ,(m−1)n S3T     In 0n,(m+1)n , WP2 = A − E 0n,(m−1)n Ad −In ,     Imn 0mn,2n , P = P2 P4 0n,(m−1)n P3 ,



0mn,n Imn 0mn,n

In −In 0n,mn



.

Delay-dependent admissibility and H∞ control of discrete singular delay systems

77

Proof. First, we prove the regularity and causality of the system. Let 

E¯ = 

P¯ = 

S¯ =

E 0 0 0





E

¯ = , A 

P1 0 0 0

  ¯ = , R

R 0 0 I

Z 0 0 0

 ,



Q11 0 0 τ 2Z



 ¯ = , Z

A − E −I

¯ = , Q

S1 P2T S3 P3T



I

,

 .

Since rankE¯ = rankE = r ≤ n, there exist nonsingular matrices U and V , such that 

¯ = U EV

Denote

 ¯ U AV

= 

V S¯ = T

A11 A12 A21 A22 S11 S21



Ir 0 0 0

.

 ,



 , U

−T

R¯ =

0 I



H,

¯ Dewhere H ∈ R(2n−r )×(2n−r ) is a nonsingular matrix determined by U −T R. fine ⎡

In 0

0 0

0

0(m−1)n,n In

⎢ ⎢ ⎣ 0(m−1)n,n

L=⎢

0n,(m−1)n 0 0n,(m−1)n In I(m−1)n 0(m−1)n,n 0 0

⎤

⎥ ⎥ ⎥. ⎦

Then performing a congruence transformation to (4.26) by L, we obtain the following inequality: ⎡ ⎢ ⎢ ⎢ ⎣

Θˆ 11  • •

Θˆ 13 Θˆ 33 • •

• • • •

• • • •

⎤ ⎥ ⎥ ⎥ < 0, ⎦

where Θˆ 11

= P2T (A − E) + (A − E)T P2 − ET ZE + Q11 ,

(4.27)

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Analysis and Synthesis of Singular Systems

Θˆ 13 Θˆ 33

= −P2T + (A − E)T P3 + S1 RT + ET P1 , = τ 2 Z + P1 − P3 − P3T + S3 RT + RS3T .

From (4.27), we have ¯ T P¯ A ¯ − E¯ T P¯ E¯ + S¯ R ¯ +A ¯ TR ¯ < 0, ¯ TA ¯ S¯ T − E¯ T Z ¯ E¯ + Q A

which implies that ¯ +A ¯ TR ¯ TA ¯ S¯ T − E¯ T Z ¯ E¯ < 0. − E¯ T P¯ E¯ + S¯ R

(4.28)

Performing a congruence transformation to (4.28) by V T and V , we obtain 

• • T • S21 H T A22 + AT22 HS21

 < 0,

which implies that A22 is nonsingular. Hence, ¯ ) = det(U −1 ) det(zIr − A11 + A12 A−1 A21 ) det(−A22 ) det(V −1 ) det(zE¯ − A 22 ¯ ) = r. This, together with Defiis not identically zero, and deg det(zE¯ − A ¯ ¯ nition 4.13, leads to the pair (E, A) being regular and causal. Noticing the fact that ¯ ), det(zE − A) = det(zE¯ − A ¯ )), deg(det(zE − A)) = deg(det(zE¯ − A

we can easily see that the pair (E, A) is regular and causal. Then according to Lemma 4.12 and Definition 4.13, the system in (4.23) is regular and causal. Then we are in the position to show that system (4.23) is asymptotically stable. To this end, we choose a new Lyapunov functional candidate as V (k) = V1 (k) + V2 (k) + V3 (k), where V1 (k) = xT (k)ET P1 Ex(k), V2 (k) =

k−1  i=k−τ

Υ T (i)QΥ (i),

(4.29)

Delay-dependent admissibility and H∞ control of discrete singular delay systems −1  k−1 

V3 (k) = τ

79

ηT (j)Z η(j),

i=−τ j=k+i

and

⎡ ⎢ ⎢ ⎢ Υ (i) = ⎢ ⎢ ⎢ ⎣

⎤

x(i) x(i − τ ) x(i − 2τ ) .. . x (i − τ m + τ )

⎥ ⎥ ⎥ ⎥ , η(j) = Ex(j + 1) − Ex(j). ⎥ ⎥ ⎦

Taking the forward difference of the functional in (4.29) along the solution of system (4.23), and defining ⎡

⎤ Υ (k) ⎢ ⎥ ξ(k) = ⎣ x(k − mτ ) ⎦ , η(k)

we have V1 (k) = xT (k + 1)ET P1 Ex(k + 1) − xT (k)ET P1 Ex(k) T    = Ex(k + 1) − Ex(k) P1 Ex(k + 1) − Ex(k)   +2x(k)T ET P1 Ex(k + 1) − Ex(k)    T +2 x(k)T P2T + Ex(k + 1) − Ex(k) P3T + x(k − τ )T P4T     × (A − E)x(k) − Ex(k + 1) − Ex(k) + Ad x(k − mτ )  +2 x(k)T S1 RT + x(k − τ )T S2 RT   T  + Ex(k + 1) − Ex(k) S3 RT Ex(k + 1) − Ex(k) T T = ξ T (k)WPT P1 WP ξ(k) + 2ξ T (k)(WP1 E P1 WP

+P T WP2 + SRT WP )ξ(k),

(4.30)

V2 (k) = Υ T (k)QΥ (k) − Υ T (k − τ )QΥ (k − τ ) ˜ Q ξ(k), = ξ T (k)WQT QW V3 (k) = τ 2 ηT (k)Z η(k) − τ

k−1  i=k−τ

(4.31) ηT (i)Z η(i).

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Analysis and Synthesis of Singular Systems

By Lemma 4.5, we have V3 (k) ≤ τ 2 ηT (k)Z η(k) − [x(k) − x(k − τ )]T ET ZE[x(k) − x(k − τ )].

(4.32) By connecting (4.30)–(4.32), we obtain V (k)

≤

T T ξ(k)T WPT P1 WP ξ(k) + 2ξ(k)T (WP1 E P1 WP + P T WP2 ˜ Q ξ(k) + ξ(k)T W T τ 2 ZWP ξ(k) + SRT WP )ξ(k) + ξ(k)T WQT QW P

− ξ(k)T WZT ET ZEWZ ξ(k) = ξ(k)T Θξ(k).

(4.33)

Thus (4.26) implies V (k) < − x(k) 2 , where  is a positive scalar. Then we have −V (0) ≤ V (k + 1) − V (0) =

k 

V (i) ≤ −

i=0

k 

x(i) 2 ≤ 0,

i=0

which implies 0≤

k 

x(i) 2 ≤

i=0

Hence, the series

∞ 

i=0

V (0) 

.

x(i) 2 converge, and we have limi→∞ x(i) = 0. Then

from Definition 4.13, we conclude that the system is asymptotically stable and this completes the proof. Remark 4.18. For state-space systems with state delay, that is, rank E = n, then R = 0 as a result of ET R = 0. Therefore the term SRT WP in (4.26) disappears, and Theorem 4.17 is specialized to the following corollary: Corollary 4.19. Given positive integers m, τ , the system in (4.23) with rank E = n is asymptotically stable if there exist matrices P1 > 0, Q > 0, Z > 0, Y1 , Y2 , T1 , P2 , P3 , and P4 , such that Θ˘ < 0,

where Θ˘

˜ Q − W T ET ZEWZ = WPT (P1 + τ 2 Z )WP + WQT QW Z T T + sym(WP1 E P1 WP + P T WP2 ).

Delay-dependent admissibility and H∞ control of discrete singular delay systems

81

When m = 1, in deriving the result of Theorem 4.17, we give up the delay-partitioning approach. In such a case, we have the following result: Corollary 4.20. The system in (4.23) is admissible if there exist matrices P1 > 0, Q > 0, Z > 0, S1 , S2 , S3 , P2 , P3 , and P4 , such that ⎡

Θˆ 11 ⎢ ⎣  

Θˆ 12 Θˆ 22 

⎤ Θˆ 13 ⎥ Θˆ 23 ⎦ < 0, Θˆ 33

(4.34)

where R, Θˆ 11 , Θˆ 13 , Θˆ 33 are defined in (4.27), and Θˆ 12 Θˆ 22

= P2T Ad + (A − E)T P4 + ET ZE,

Θˆ 23

= ATd P3 − P4T + S2 RT .

= P4T Ad + ATd P4 − ET ZE − Q,

Proof. When m = 1, the matrices in (4.26) become τ

WP1 WQ

T    = d, S = S1T S2T S3T , WP = 0n,2n In ,     = In 0n,2n , WP2 = A − E Ad −In ,     In 0n,2n , P = P2 P4 P3 , =

WZ =

0n,n In 0n,n



In −In 0n,n







˜ = Q 0n,n . , Q11 = Q, Q 0n,n −Q

After some simple manipulations, (4.34) can be obtained and this completes the proof. Remark 4.21. The number of LMI decision variables in Theorem 4.17 is 2n [(m2 + 14)n − 6r + m + 2]. Thus the computational burden, including the number of the variables and the computation time will significantly increase with the partitioning number m becoming bigger. However, the LMI toolbox in MATLAB® can still be used to efficiently solve the LMIs.

4.2.2.2 Stability analysis: uncertain case In this subsection, the robust stability analysis for the uncertain singular system in (4.22) with w (k) = 0 is considered; that is, we consider the following

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Analysis and Synthesis of Singular Systems

uncertain system: ⎧ ⎪ ⎨ Ex(k + 1) = (A + A(k))x(k) + (Ad + Ad (k))x(k − d) +(B + B(k))u(k), ⎪ ⎩ x(k) = φ(k), k ∈ [−d¯ , 0].

(4.35)

For system (4.35) with time-varying structured uncertainties, we have the following theorem: Theorem 4.22. Given positive integers m, τ , the time-delay system in (4.35) with u(k) = 0 is admissible for all parameter uncertainties if there exist matrices P1 > 0, Q > 0, Z > 0, S1 , S2 , S3 , P2 , P3 , P4 , and a scalar ε > 0, such that 

Θ + εΞ T Ξ 

PT M −ε I

 < 0,

(4.36)

where Θ is defined in (4.26), and Ξ=





N1 0b,(m−1)n N2 0b,n

.

Proof. Based on Theorem 4.17, by replacing A and Ad in (4.26) with A + MF (k)N1 and Ad + MF (k)N2 , respectively, the stability criterion for the uncertain system can be rewritten as Θ + sym(P T MF (k)Ξ ) < 0.

(4.37)

Applying Schur complement to (4.36), we obtain Θ + εΞ T Ξ + ε −1 P T MM T P < 0,

which by Lemma 4.16 implies (4.37), and the proof is completed. Now we are in the position to show that the proposed result will demonstrate its superiority in terms of reduced conservatism with m increasing. Proposition 4.23. Suppose that τm and dm are the maximal τ and the maximal delay obtained by Theorem 4.17 for a given number of partitions m. Then for any positive integer β such that mm+β τm is an integer, we have mm+β τm ≤ τm+β , and thus dm ≤ dm+β .

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Delay-dependent admissibility and H∞ control of discrete singular delay systems

Proof. From Theorem 4.17, we know that the following inequality holds for given partitioning number m and the integer τm : V (k)

≤

xT (k + 1)ET P1 Ex(k + 1) − xT (k)ET P1 Ex(k) + τm2 ηT (k)Z η(k) −[x(k) − x(k − τm )]T ET ZE[x(k) − x(k − τm )] ⎡ ⎤T ⎡ x(k) x(k) ⎢ ⎢ ⎥ x(k − τm ) x(k − τm ) ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ x ( k − 2 τ ) x (k − 2τm ) ⎢ ⎢ ⎥ m +⎢ ⎥ Q⎢ .. .. ⎢ ⎢ ⎥ ⎣ ⎣ ⎦ . .     x k − (m − 1)τm x k − (m − 1)τm ⎤T ⎡ ⎤ ⎡ x(k − τm ) x(k − τm ) ⎢ x(k − 2τ ) ⎥ ⎢ x(k − 2τ ) ⎥ ⎢ ⎢ m ⎥ m ⎥ ⎢ ⎥ ⎥ ⎢ x ( k − 3 τ ) x ( k − 3 τ ⎢ ⎥ ⎢ m m) ⎥ Q⎢ −⎢ ⎥ ⎥ .. .. ⎢ ⎥ ⎥ ⎢ ⎣ ⎦ ⎦ ⎣ . .     x k − mτm x k − mτm

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

< 0.

(4.38)

Take τ = mm+β τm < τm . Since τm is the maximal value of τ satisfying (4.38), even if we replace τm in (4.38) by mm+β τm , we still have V˜ (k)

≤

xT (k + 1)ET P1 Ex(k + 1) m −xT (k)ET P1 Ex(k) + ( τm )2 ηT (k)Z η(k) m+β m m τm )]T ET ZE[x(k) − x(k − τm )] −[x(k) − x(k − m+β m+β ⎡

⎢ ⎢ ⎢ ⎢ +⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎢ −⎢ ⎢ ⎢ ⎣

⎤T

x(k) x(k − mm+β τm ) x(k − 2 mm+β τm ) .. .



x k − (m − 1) mm+β τm x(k − mm+β τm ) x(k − 2 mm+β τm ) x(k − 3 mm+β τm ) 

.. .

x k − m mm+β τm

⎤T

⎡

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Q⎢ ⎥ ⎢ ⎥ ⎢ ⎣ ⎦ ⎡

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ Q⎢ ⎢ ⎥ ⎢ ⎥ ⎣  ⎦

x(k) x(k − mm+β τm ) x(k − 2 mm+β τm ) .. .



x k − (m − 1) mm+β τm

x(k − mm+β τm ) x(k − 2 mm+β τm ) x(k − 3 mm+β τm ) 

.. .

x k − m mm+β τm

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥  ⎦

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

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Analysis and Synthesis of Singular Systems

= xT (k + 1)ET P1 Ex(k + 1) − xT (k)ET P1 Ex(k)

m τm )2 ηT (k)Z η(k) m+β m m τm )]T ET ZE[x(k) − x(k − τm )] − [x(k) − x(k − m+β m+β ` Υ1 (K ) − Υ2 (K )T Q ` Υ2 (K ) + Υ1 (K )T Q < 0, + (

where ⎡

⎤

⎡ x(k − mm+β τm ) ⎥ ⎢ x(k − 2 m τ ) ⎥ ⎢ m+β m ⎥ ⎢ m ⎥ x ( k − 3 ⎢ ⎥ m+β τm ) ⎢ ⎥ .. .. ⎢ ⎥ . .   ⎥ , Υ2 (k) = ⎢ ⎢   ⎥ m ⎢ x k − (m − 1) m+β τm ⎥ ⎢ x k − m mm+β τm ⎥ ⎢ ⎥ .. ⎢ .. ⎥ . ⎣ .   ⎦   x k − (m + β − 1) mm+β τm x k − mτm  

x(k)

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Υ1 (k) = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

` = Q

x(k − mm+β τm ) x(k − 2 mm+β τm )

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Q 0mn,β n .  0β n,β n

We can always find a small enough scalar δ > 0 such that V˜ (k)

≤

xT (k + 1)ET P1 Ex(k + 1) − xT (k)ET P1 Ex(k) m τm )2 ηT (k)Z η(k) +( m+β m m τm )]T ET ZE[x(k) − x(k − τm )] −[x(k) − x(k − m+β m+β ´ Υ1 (K ) − Υ2 (K )T Q ´ Υ2 (K ) +Υ1 (K )T Q < 0, (4.39)

where





´ = Q 0mn,β n > 0, Q  δI

which implies that the inequality in (4.26) still holds for given partitioning number m + β and the integer τ . We can obtain τ=

m m+β

τm ≤ τm+β .

(4.40)

Delay-dependent admissibility and H∞ control of discrete singular delay systems

85

Then by multiplying (4.40) by m + β , we have dm ≤ dm+β , and this proposition is proved.

4.2.3 Stabilization In this section, we devote our attention to design a robust state feedback controller for system (4.22) with w (k) = 0, such that the closed-loop system is admissible for all uncertainties. Based on Theorem 4.17, we have the result that follows. Before giving the main result, we give the closed-loop system of nominal singular discrete-time system: Ex(k + 1) = (A + BK )x(k) + Ad x(k − d).

(4.41)

Theorem 4.24. Given scalars λ1 , λ2 , λ3 , and positive integers m, τ , there exists a state-feedback controller in the form of (4.24) such that the closed-loop system in (4.25) is admissible if there exist matrices P1 > 0, Q > 0, Z > 0, S1 , S2 , S3 , J, X, and a scalar ε > 0, such that 

Ψ + εΓ Γ T 

ΦT −ε I

 < 0,

(4.42)

where R ∈ Rn×(n−r ) is any full-column rank matrix satisfying ET R = 0 and ˜ Q − W T EZET WZ Ψ = WPT (P1 + τ 2 Z )WP + WQT QW Z T + sym(WP1 EP1 WP + WET Λ + SRT WP ),   WE = λ1 In λ3 In 0n,(m−1)n λ2 In ,

Λ = [ J T (A − E)T + X T BT 0n,(m−1)n J T ATd −J T ],   λ1 (N1 J + N3 X ) λ3 (N1 J + N3 X ) 0b,(m−1)n λ2 (N1 J + N3 X ) Φ= , λ1 N2 J λ3 N2 J 0b,(m−1)n λ2 N2 J   ΓT =

MT 0l,n

0l,(m−1)n 0l,n 0l,(m−1)n M T

0l,n 0l,n

.

Moreover, if the above condition is feasible, a desired controller gain matrix in the form of (4.24) is given by K = XJ −1 . Proof. It is easy to see that det(zE − (A + BK )) = det(zET − (A + BK )T ),

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Analysis and Synthesis of Singular Systems

deg(det(zE − (A + BK ))) = deg(det(zET − (A + BK )T )),

and that det(zE − (A + BK ) − z−d Ad ) = 0 and det(zET − (A + BK )T − z−d ATd ) = 0 have the same solution set. With respect to the regularity, causality, and stability of a system, we obtain that the system in (4.41) is equivalent to the following system based on Definition 4.1 and Lemma 4.12: ET δ(k + 1) = (A + BK )T δ(k) + ATd δ(k − d).

(4.43)

Substituting E, A, and Ad with ET , (A + BK )T , and ATd in (4.26), respectively, we have the following inequality: Ψ1 < 0,

where ˜ Q − W T EZET WZ = WPT (P1 + τ 2 Z )WP + WQT QW Z  T +sym(WP1 EP1 WP + SRT WP + P2 P4 0n,(m−1)n   × (A − E + BK )T 0n,(m−1)n ATd −In ).

Ψ1

P3

T

Then, denoting P2 = λ1 J, P3 = λ2 J, P4 = λ3 J and X = KJ, we obtain Ψ < 0.

(4.44)

Replacing A, Ad , and B by A + MF (k)N1 , Ad + MF (k)N2 , and B + MF (k)N3 in (4.44), we have Ψ + Ψ2 < 0,

(4.45)

where ⎛⎡ Ψ2

F¯

⎜⎢ ⎜⎢ ⎢ = sym ⎜ ⎜⎢ ⎝⎣

MF (k)(N1 J + N3 X ) 0(m−1)n,n MF (k)N2 J 0n,n

= Γ F¯ Φ + Φ T F¯ T Γ T ,   F (k) 0l,b . = 0l,b F (k)

⎤ ⎥ ⎥ ⎥ γ1 In ⎥ ⎦

⎞ γ3 In

0n,(m−1)n

⎟ ⎟ γ2 In ⎟ ⎟ ⎠

Delay-dependent admissibility and H∞ control of discrete singular delay systems

87

By applying Schur complement, (4.42) is equivalent to Ψ + εΓ Γ T + ε −1 Φ T Φ < 0.

(4.46)

Based on Lemma 4.16, (4.46) implies (4.45) holds. Thus the theorem is proved. Remark 4.25. To establish the stability conditions for the singular systems with constant delay, the considered systems are converted to delay-free systems by the state augmentation approach in [20,212]. However, the order of the transformed systems is high if the delay is large, and the method becomes difficult to apply for unknown delay, or for time-varying delay cases. Remark 4.26. It is the utilization of the delay-partitioning technique that constitutes the major difference when compared with existing results about singular systems in the literature, from which the reduced conservatism can benefit. The reduction of conservatism is more prominent when the partitioning number m increases, which has been proved in [60] for continuoustime systems. In addition, the delay-partitioning technique has also been applied to stability analysis of continuous systems with multiple delay components in [31], neutral delay systems in [32], and delayed complex network in [177]. Remark 4.27. Although this section deals with the constant delay case, it can be readily extended to discrete-time singular systems with time-varying delay d(k) satisfying 1 ≤ dm ≤ d(k) ≤ dM in terms of the following Lyapunov functional candidate: Λ(k) = Λ1 (k) + Λ2 (k) + Λ3 (k) + Λ4 (k),

where Λ1 (k) = xT (k)ET P1 Ex(k), Λ2 (k) =

k−1 

Υ (i)Q1 Υ (i) + T

i=k−τ

Λ3 (k) =

−1  k−1 

Λ4 (k) =

xT (i)Q2 x(i),

i=k−dM

ηT (j)Z1 η(j) +

i=−τ j=k+i − mτ +1

k−1 

− mτ −1  k−1

i=−dM j=k+i k−1 

i=−dM +1 j=k−1+i

xT (j)Rx(j).

ηT (j)Z2 η(j),

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Analysis and Synthesis of Singular Systems

Then, the robust stability and stabilization problems for uncertain discretetime singular systems with time-varying delay can be addressed by following similar lines as developed in this section.

4.2.4 Robust H∞ control In this subsection, we consider the admissibility with an H∞ disturbance attenuation for system (4.22). Based on this condition, we will design a robust H∞ state feedback controller for system (4.22) such that the closedloop system is admissible with an H∞ disturbance attenuation.

4.2.4.1 H∞ performance analysis Before presenting the main results, we establish a new version of delaydependent bounded real lemma for the system in (4.22) with u(k) = 0, that is, we consider the following nominal time-delay system: ⎧ ⎪ ⎨ Ex(k + 1) = Ax(k) + Ad x(k − d) + Bw w (k), z(k) = Cx(k), ⎪ ⎩ ¯ 0]. x(k) = φ(k), k ∈ [−d,

(4.47)

Theorem 4.28. Given positive integers m, τ , and a prescribed scalar γ > 0, the system in (4.47) is admissible with an H∞ disturbance attenuation γ , if there exist matrices P1 > 0, Q > 0, Z > 0, S1 , S2 , S3 , P2 , P3 , and P4 , such that Θ¯ < 0,

(4.48)

where R ∈ Rn×(n−r ) is any full-column rank matrix satisfying ET R = 0 and Θ¯

T ˜W ¯ Q + WCT WC − γ 2 WW ¯ P +W ¯ QT Q ¯ PT (P1 + τ 2 Z )W WW = W

˜ = Q ¯P = W ¯ P2 = W ¯Z = W

T T ¯ TW ¯ Z + sym(W ¯ P + P¯ T W ¯ P2 + SR ¯ P ), ¯ ZT ET ZEW ¯ P1 −W E P1 W   T  Q 0mn,mn , S¯ = S1T S2T 0n−r ,(m−1)n S3T 0n−r ,p , 0mn,mn −Q     ¯ P1 = In 0n,(m+1)n 0n,p , 0n,(m+1)n In 0n,p , W   A − E 0n,(m−1)n Ad −In Bw ,     I 0 mn mn , 2n + p ¯Q= , In −In 0n,mn+p , W

0mn,n Imn 0mn,n+p

89

Delay-dependent admissibility and H∞ control of discrete singular delay systems



=

WW





P¯ =

P2 P4 0n,(m−1)n P3 0n,p 

0q,(m+2)n Ip

, WC =





C 0s,(m+1)n+p

,

.

Proof. First, (4.48) implies (4.26), and thus the system in (4.47) is admissible. Next, we shall establish the H∞ performance of the system in (4.47) under zero initial condition. To this end, we introduce the following index: Jzw =

N 

[zT (k)z(k) − γ 2 w (k)w (k)].

k=0

Applying the same technique as in the proof of Theorem 4.17 and noting zero initial condition, it can be shown that for any nonzero w ∈ l2 [0, ∞), Jzw = ≤ ≤

N  k=0 N  k=0 N 

[zT (k)z(k) − γ 2 w T (k)w (k) + V (k + 1) − V (k)] − V (N + 1) [zT (k)z(k) − γ 2 w T (k)w (k) + V (k)] ξ¯ T (k)Θ¯ ξ¯ (k),

k=0

where ξ¯ (k) =



xT (k) xT (k − τ ) · · · xT (k − mτ ) ηT (k) w T (k)

T .

From (4.48), we have Jzw < 0 for all nonzero w ∈ l2 [0, ∞) such that z 2 < γ w 2 , and this completes the proof. Based on Theorem 4.28, we investigate the problem of delay-dependent robust H∞ performance analysis for discrete-time uncertain singular timedelay system: ⎧ ⎪ ⎨ Ex(k + 1) = (A + A(k))x(k) + (Ad + Ad (k))x(k − d) + Bw w (k), z(k) = Cx(k), ⎪ ⎩ x(k) = φ(k), k ∈ [−d, ¯ 0].

(4.49) Theorem 4.29. Given positive integers m, τ , and a prescribed scalar γ > 0, the system in (4.49) is admissible with an H∞ disturbance attenuation γ , if there exist

90

Analysis and Synthesis of Singular Systems

matrices P1 > 0, Q > 0, Z > 0, S1 , S2 , S3 , P2 , P3 , P4 , and a scalar ε > 0, such that 

Θ¯ + ε Ξ¯ T Ξ¯ 

P¯ T M −ε I



< 0,

(4.50)

where Θ¯ , P¯ are defined in (4.48) and Ξ¯ =





N1 0b,(m−1)n N2 0b,n+q

.

Proof. By replacing A and Ad in (4.48) with A + MF (k)N1 and Ad + MF (k)N2 , respectively, the admissibility criterion for the uncertain system can be rewritten as Θ¯ + sym(P¯ T MF (k)Ξ¯ ) < 0.

(4.51)

Applying Schur complement to (4.50), we obtain Θ¯ + ε Ξ¯ T Ξ¯ + ε −1 P¯ T MM T P¯ < 0,

which, by Lemma 4.16, is equivalent to (4.51), and the proof is completed.

4.2.4.2 H∞ controller design In this subsection, we design a robust H∞ state feedback controller in the form of (4.24) for system (4.22) such that the closed-loop system is admissible with H∞ performance. For notational simplicity, we give the following closed-loop system: ⎧ ⎪ ⎨ Ex(k + 1) = (A + BK )x(k) + Ad x(k − d) + Bw w (k), z(k) = (C + DK )x(k), ⎪ ⎩ x(k) = φ(k), k ∈ [−d, ¯ 0].

(4.52)

Based on Theorem 4.28, we can obtain the following result: Theorem 4.30. Given scalars λ1 , λ2 , λ3 , positive integers m, τ , and a prescribed scalar γ > 0, there exists a state-feedback controller in the form of (4.24) such that the closed-loop system in (4.22) is admissible with an H∞ disturbance attenuation γ if there exist matrices P1 > 0, Q > 0, Z > 0, S1 , S2 , S3 , J, X, and a scalar ε > 0, such that 

Ψ˜ + ε Γ˜ Γ˜ T 

Φ˜ T −ε I



< 0,

(4.53)

Delay-dependent admissibility and H∞ control of discrete singular delay systems

91

where R ∈ Rn×(n−r ) is any full-column rank matrix satisfying ET R = 0 and Ψ˜

T ˜ ˜W ˜ Q+W ˜ C − γ 2W ˜W ˜ P +W ˜ QT Q ˜ CT W ˜ PT (P1 + τ 2 Z )W WW = W

˜E = W ˜ P1 = W Λ˜ = ˜P = W

S˜ = ˜Q = W ˜C = W Φ˜

T ˜ TW ˜ Z + sym(W ˜ P +W ˜ P ), ˜ T EZET W ˜ P1 ˜ ET Λ˜ + SR −W EP1 W   Z λ1 In λ3 In 0n,(m−1)n λ2 In 0n,s ,   In 0n,(m+1)n 0n,s ,  T , S1T S2T 0n−r ,(m−1)n S3T 0n−r ,s   0n,(m+1)n In 0n,s , T  , S1T S2T 0n−r ,(m−1)n S3T 0n−r ,s     Imn 0mn,2n+s ˜ Z = In −In 0n,mn+s , , W

0mn,n Imn 0mn,n+s



BwT

#

= 

Γ˜ T

=



0p,(m+1)n+s

˜W= , W

λ1 (N1 J + N3 X )

λ3 (N1 J + N3 X )

λ1 N2 J

λ3 N2 J

MT 0l,n

0l,(m−1)n 0l,n 0l,(m−1)n M T





0l,n+s 0l,n+s

0s,(m+2)n Is 

0b,(m−1)n 0b,(m−1)n

, λ2 (N1 J + N3 X ) λ2 N2 J

0e,s 0e,s

$ ,

.

Moreover, if the above condition is feasible, a desired controller gain matrix in the form of (4.24) is given by K = XJ −1 . Proof. It is easy to see that det(zE − (A + BK )) = det(zET − (A + BK )T ), deg(det(zE − (A + BK ))) = deg(det(zET − (A + BK )T )),

and there are the same solutions to det(zE − (A + BK ) − z−d Ad ) = 0 as well as det(zET − (A + BK )T − z−d ATd ) = 0. Furthermore, the H∞ norm of the system in (4.52) is given by H (z) ∞ = sup σmax ((C + DK )(ejw E − (A + BK + e−jwd Ad ))−1 Bw ), w∈[0,2π )

which is equivalent to H (z) ∞ = sup σmax (BwT (ejw ET − (A + BK + e−jwd Ad )T )−1 (C + DK )T ). w∈[0,2π )

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Analysis and Synthesis of Singular Systems

With respect to the regularity, causality, stability, and H∞ performance problems of a system, we can obtain that the system in (4.52) is equivalent to the following system based on Definition 4.1, Definition 4.14 and Lemma 4.15: 

ET δ(k + 1) = (A + BK )T δ(k) + ATd δ(k − d) + (C + DK )T ϕ(k), κ(k) = BwT δ(k).

Substituting E, A, Ad , Bw , p and C with ET , (A + BK )T , ATd , (C + DK )T , s and BwT in (4.48), respectively, we have the following inequality Ψ1 < 0,

where Ψ1

T ˜ ˜W ˜ Q+W ˜ C − γ 2W ˜W ˜ PT (P1 + τ Z )W ˜ P +W ˜ QT Q ˜ CT W = W WW T ˜ TW ˜ Z + sym(W ˜ P + SR ˜P ˜ ZT EZET W ˜ P1 −W EP1 W T  + P2 P4 0n,(m−1)n P3 0n,s   × (A − E + BK )T 0n,(m−1)n ATd −In (C + DK )T ).

Then, by denoting P2 = λ1 J, P3 = λ2 J, P4 = λ3 J, and X = KJ, Ψ˜ < 0 is readily obtained. Following a similar line of argument as in the proof of Theorem 4.24, Theorem 4.30 can be established.

4.2.5 Illustrative examples In this subsection, we use numerical examples to illustrate the advantages of the developed results and the applicability of the proposed controller design methods. We first demonstrate the improvement by considering a nominal singular system in Example 4.2. Example 4.2. Consider the singular system 

3.5 0 0 0











−1.3 1.5 a 0 x(k + 1) = 11 x(k) + x(k − d). 0 −3 0 0.5

Our purpose is to determine the allowable time-delay upper bounds d¯ such that the system is admissible. To compare our results with those in [72, 172,191,249], we consider a11 = 2.3, or a11 = 2.4. The maximum delay

Delay-dependent admissibility and H∞ control of discrete singular delay systems

93

¯ Table 4.2 Comparisons of maximum allowed delay d. a11 2.3 2.4 d¯ [72] 10 7 d¯ [172] 11 7 d¯ [191] 11 7 d¯ [249] 11 7 ¯d (Theorem 4.17) 11 (m = 1, τ = 11) 7 (m = 1, τ = 7) d¯ (Theorem 4.17) 12 (m = 3, τ = 4) 9 (m = 3, τ = 3) d¯ (Theorem 4.17) 14 (m = 7, τ = 2) 10 (m = 5, τ = 2) d¯ (Exact solution) 14 10

bounds with a11 = 2.3 such that the above system is admissible are found by using the methods of [72,172,191,249], to be 10, 11, 11, 11, respectively. However, the maximum bound obtained by using Theorem 4.17 is 14 for m = 7, τ = 2. Table 4.2 gives more detailed comparison results on the maximum allowed bounds for d¯ via the methods in [72,172,191,249] and Theorem 4.17 in this section. The results in Table 4.2 clearly show that Theorem 4.17 in this section outperforms those in [72,172,191,249] in terms of conservatism. On the other hand, by using Lemma 4.15, the exact solutions of the maximal delay are listed under the given parameters. From Table 4.2, we can see that the results obtained is the same as the exact solution, so there is no room for further improvement. In this example, we obtain the same maximal delay as that in [191]. However, the number of variables involved in Corollary 4.20 is 2n (15n − 6r + 3), whereas the method in [191] requires 2n (21n − 6r + 3). Thus the presented approach is more favorable computationally. Next, the advantages of our results will be shown by considering an uncertain discrete-time singular delay system in Example 4.3. Example 4.3. Consider the uncertain discrete-time singular system with the following parameters (borrowed from [191]): 

2 0 0 0





x(k + 1) =  +

0.9977 + 0.1α 1.1972 −1.9 0.1001 −1.1972

0



1.5772 0.9757 + 0.1α

x(k) 

x(k − d).

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Analysis and Synthesis of Singular Systems

Table 4.3 Allowable maximum absolute value of α obtained by different methods. d¯ 2 3 4

α¯ [72] α¯ [191] α¯ (Theorem 4.22) α¯ (Theorem 4.22)

1.9464 2.1359 2.1359 (m = 1, τ = 2) 2.6847 (m = 2, τ = 1)

0.8033 1.0325 1.0325 (m = 1, τ = 3) 1.6734 (m = 3, τ = 1)

0.1563 0.2853 0.2853 (m = 1, τ = 4) 0.8972 (m = 4, τ = 1)

Table 4.4 Allowable maximum time delays obtained by different methods. d¯ [72] d¯ [191] d¯ (Corollary 4.19) d¯ (Corollary 4.19) d¯ (Corollary 4.19) m=1 m=5 m=7 41 42 42 (τ = 42) 55 (τ = 11) 56 (τ = 8)

The purpose is to determine the upper bounds for the absolute value of uncertain parameter α , that is, α¯ . To illustrate the benefits of our results, Table 4.3 gives the comparison results on α¯ . These comparison results show that the result in Theorem 4.22 for delay singular systems with uncertainties in this section is less conservative than those in [72,191]. From the above two examples, it can be seen that better results have been obtained in this section for discrete-time singular systems with time delay. Moreover, it is worth pointing out that our result is also effective for standard state-space delay systems, which is shown in Example 4.4. Example 4.4. Consider the following standard state-space system: 







−0.1 0.8 0 0 x(k + 1) = x(k) + x(k − d). 0 0.91 −0.1 −0.1

By solving the feasibility problem of the LMIs in Corollary 4.19, we conclude that this system is stable for any constant time-delay d satisfying d ≤ 56 (with m = 7, τ = 8). However, the methods of [72] and [191] fail to yield feasible solutions when d > 41 and d > 42, see computational results shown in Table 4.4. In Example 4.5, the applicability of the proposed controller design methods will be demonstrated.

Delay-dependent admissibility and H∞ control of discrete singular delay systems

95

Example 4.5. Consider the uncertain singular system in (4.22) with the following parameters: 

E = 

B = N3 =



1 0 0 0 −2

0

 , A=

1.7 2 1 2



3 −2

 , M= 



0.01 0.02

0.2 0.2



 , Ad =

1.5 1 1 0.05

 , N1 = N2 =

 , 



0.2 0.2

,

, F (k) = sin(k).

In this example, we choose λ1 = 0.6, λ2 = 1, λ3 = −0.35, R =



0 1

T

, d = 3,

and obtain the solution as follows by solving the LMIs in (4.42): 

X = 

J =

−396.3955 752.8114 −138.9248 406.9560



181.7492 −314.1157 −314.3584 352.1828

,  .

Therefore, by Theorem 4.24, an admissible state-feedback control law can be obtained as 

u(k) =

−2.7939 −0.3544 −2.2744 −0.8730



x(k).

Fig. 4.1 and Fig. 4.2 give the simulation results of two states with and without the state-feedback control, respectively. From Fig. 4.1 and Fig. 4.2, we can see that the open-loop system is unstable and the closed-loop system is stable. Example 4.6. Consider the following system: 

E = 

Bw =

3 −0.5 0 0 0.3 0.3











0.8 0 −2 1 , A= , Ad = , 0 −1.6 0 1

 , C=





0.5 0.5

.

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Analysis and Synthesis of Singular Systems

Figure 4.1 The state trajectories of the open-loop system.

Figure 4.2 The state trajectories of the closed-loop system.

For a given γ > 0, the maximum allowed delay d¯ satisfying the LMI in Theorem 4.28 can be calculated by solving a quasi-convex optimization problem. Similarly, for a given d > 0, the minimum allowed γ satisfying the LMI in Theorem 4.28 can also be computed by solving a quasiconvex optimization problem. Tables 4.5 and 4.6 give the comparison results on

Delay-dependent admissibility and H∞ control of discrete singular delay systems

Table 4.5 Comparisons of maximum allowed delay d given γ > 0. γ 0.66 0.65 0.64 d¯ [196] 27 13 10 d¯ (Theorem 4.28) 27 (m = 1, τ = 27) 13 (m = 1, τ = 13) 11 (m = 1, d¯ (Theorem 4.28) 38 (m = 2, τ = 19) 18 (m = 2, τ = 9) 14 (m = 2, d¯ (Theorem 4.28) 39 (m = 3, τ = 13) 18 (m = 3, τ = 6) 15 (m = 3, ¯d (Theorem 4.28) 40 (m = 4, τ = 20) 20 (m = 4, τ = 5) 15 (m = 5, Table 4.6 Comparisons of minimum allowed γ given d > 0. d 6 12 γ ∗ [196] 0.5977 0.6457 γ ∗ (Theorem 4.28), m = 1 0.5977 0.6457 (τ = 6) (τ = 12) 0.5493 0.6290 γ ∗ (Theorem 4.28), m = 2 (τ = 3) (τ = 6) 0.5402 0.6251 γ ∗ (Theorem 4.28), m = 3 (τ = 2) (τ = 4)

18 0.6555 0.6555 (τ = 18) 0.6477 (τ = 9) 0.6458 (τ = 6)

97

τ = 11) τ = 7) τ = 5) τ = 3)

24 0.6590 0.6590 (τ = 24) 0.6546 (τ = 12) 0.6534 (τ = 8)

the maximum allowed delay d for given γ > 0, and the minimum allowed γ for given d > 0, respectively, via the methods in [196] and Theorem 4.28 in this section. It can be seen that these comparisons show that our results for delay systems without uncertainties (in this section) is less conservative than those in [196]. Example 4.7. Consider the linear uncertain discrete singular delay system in (4.22) with parameters as follows: 

E = 

B =

N2 λ1

0.36 0.5



 , A=



 , Bw =

1.47 0 0.5 −1.5 0.3 0.3

 , C=



 , Ad =

−1 0.5 0 0.5

 ,





0.5 0.5

,

   −0.02 0.02 0.025 0.01 , N1 = , D = 0.3, = 0.1 0.025 −0.02 −0.02     0.02 0.01 0.3 , N3 = , d = 6, = 0.01 0.02 −0.4 T  = 0.2, λ2 = 0.1, λ3 = −0.1, R = 0 1 . 

M

2 0 0 0

98

Analysis and Synthesis of Singular Systems

The purpose is to design a state-feedback controller in the form of (4.24) such that the system in (4.22) is asymptotically stable with a guaranteed H∞ performance γ . In this example, the minimum H∞ performance γ ∗ = 0.48 is obtained by solving LMI (4.53), and the feasible solutions are 

J = 

X =

2.6517 2.9523 2.1264 7.0467

 , 

−7.1433 −11.0540

.

Therefore, by Theorem 4.30, the corresponding stabilizing state-feedback controller can be obtained as K=



−2.1625 −0.6627



.

Next, to illustrate the disturbance attenuation performance, assume zero initial condition and the external disturbance ⎧ ⎪ ⎨ 3, w (k) = −3, ⎪ ⎩ 0,

0 ≤ k ≤ 10 11 ≤ k ≤ 20 elsewhere.

Fig. 4.3 shows the signals w (k) and z(k). It is found that ||z||2 = 2.3628 and ||w ||2 = 13.4164, which yields γ = 0.1761 (below the minimum γ ∗ = 0.48),

Figure 4.3 Signals w(k) and z(k).

Delay-dependent admissibility and H∞ control of discrete singular delay systems

99

Figure 4.4 Maximum singular values over frequency range [0, π ).

showing the effectiveness of the controller design. For illustration, we take one of the worst-case perturbation such that F (k) = I. The maximum singular value plot of the linear uncertain discrete singular delay system in (4.22) is shown in Fig. 4.4 for the frequency range [0, π). The effectiveness of the guaranteed H∞ disturbance is apparent.

4.2.6 Conclusion In this chapter, improved functionals based on the delay partitioning technique have been introduced to derive improved results for robust stability and stabilization problems of linear uncertain discrete-time singular systems with state delay, which guarantees the closed-loop system is admissible. Moreover, the proposed new results have been utilized to investigate robust H∞ control problem which assures the resulting closed-loop system is admissible with an H∞ disturbance attenuation. Less conservative and easily verifiable conditions have been formulated in terms of strict LMIs involving no decomposition of the system matrices. It is also proved that the conservatism of the results is non-increasing with the reduction of the partition size. Numerical examples have been given to demonstrate the advantages and the merits of the proposed results. Extending the method proposed here to tackle the robust control, guaranteed cost and variable structure control for singular system with time-varying delay will be interesting topics for future research.

CHAPTER 5

Delay-dependent dissipativity analysis and synthesis of singular delay systems The problem of delay-dependent dissipativity analysis and synthesis for singular delay system has been addressed in this chapter. Firstly, based on developed inequality and improved reciprocally convex combination approach, the problem of dissipative control for discrete-time singular delay system is studied. Secondly, using the delay partitioning technique, the state-feedback controller is designed. And the sufficient conditions of dissipativity synthesis for continuous-time singular system with time delay is constructed. Finally, by introducing some variables to decouple the Lyapunov matrices and the filtering error system matrices, we consider the problem of robust reliable dissipative filtering for discrete-time singular systems with polytopic uncertainties, time-varying delays, and sensor failures.

5.1 Dissipativity analysis for discrete singular systems with time-varying delay In this section, the issue of dissipativity analysis for discrete singular systems with time-varying delay is investigated. By using a recently developed inequality, which is less conservative than the Jensen inequality, and the improved reciprocally convex combination approach, sufficient criteria are established to guarantee the admissibility and dissipativity of the considered system. Moreover, H∞ performance characterization and passivity analysis are carried out. Numerical examples are presented to illustrate the effectiveness of the proposed method.

5.1.1 Problem formulation Consider discrete-time singular systems with time-varying delay described by Ex(k + 1) = Ax(k) + Ad x(k − d(k)) + Bω ω(k), z(k) = Lx(k) + Ld x(k − d(k)) + Gω ω(k), x(k) = φ(k), k ∈ [−d2 , 0], Analysis and Synthesis of Singular Systems https://doi.org/10.1016/B978-0-12-823739-7.00012-4

Copyright © 2021 Elsevier Inc. All rights reserved.

(5.1)

101

102

Analysis and Synthesis of Singular Systems

where x(k) ∈ Rn is the state vector; ω(k) ∈ Rl represents a set of exogenous inputs, which includes disturbances to be rejected; z(k) ∈ Rq is the control output; d(k) is a time-varying delay satisfying 0 < d1 ≤ d(k) ≤ d2 , where d1 and d2 are prescribed positive integers representing the lower and upper bounds of the time delay, respectively. φ(k) is the compatible initial condition. The matrix E ∈ Rn×n may be singular, and it is assumed that rank(E) = r ≤ n. A, Ad , Bω , C, Cd , D, L, Ld , and Gω are known real constant matrices with appropriate dimensions. The lemmas and definitions that follow will be used in the proof of the primary results in this section. Denote y(k) = x(k + 1) − x(k) and a new inequality is derived in the following lemma: Lemma 5.1. [136] For a given positive definite matrix R and three given nonnegative integers a, b, k satisfying a ≤ b ≤ k, denote  χ (k, a, b) =

1 b−a

  k−a−1

2

s=k−b



x(s) + x(k − a) + x(k − b) , a < b a = b. 2x(k − a),

Then, we have − (b − a)

k −a−1



yT (s)Ry(s) ≤ −

s=k−b

Θ0 Θ1

T 

R 0 0 3R



 Θ0 Θ1

,

(5.2)

where Θ0 = x(k − a) − x(k − b), Θ1 = x(k − a) + x(k − b) − χ (k, a, b).

Remark 5.2. There is a difference between Lemma 3 in [136] and Lemma 5.1 in this section. After checking, when a < b, it is found that the sign of x(k − b) in χ (k, a, b) should be “+” instead of “−” in [136]. Moreover, it is pointed out that there are minor typographical errors in the Lyapunov function. V3 is written as V3 =τm

−1  k−1 

y (ν)S1 y(ν) + (τa − τm ) T

s=−τm ν=k+s

+ (τM − τa )

τ k−1 m −1 

s=−τa ν=k+s τ a −1

k−1 

s=−τM ν=k+s

yT (ν)S3 y(ν).

yT (ν)S2 y(ν)

103

Delay-dependent dissipativity analysis and synthesis of singular delay systems

However, to get the forward difference of V3 in (25) of [136], it should be written as V3 =τm

−1  k−1 

−τm 

yT (ν)S1 y(ν) + (τa − τm )

s=−τm ν=k+s

yT (ν)S2 y(ν)

s=−τa +1 ν=k+s−1 −τa 

+ (τM − τa )

k−1 

k−1 

yT (ν)S3 y(ν).

s=−τM +1 ν=k+s−1

Lemma 5.3. [140] Let n, m be two positive integers and two matrices R1 in + S+ guarantees that, if n and R2 in Sm . The improved reciprocallyconvex combination  R1 X ≥ 0, then the following there exists a matrix X in Rn×m such that X T R2 inequality holds for any scalar α in the interval (0, 1): 

1

R α 1 0

0 1 1−α R2



 ≥

R1 XT

X R2

 .

(5.3)

The nominal discrete singular system with time-varying delay of system (5.1) can be written as Ex(k + 1) =Ax(k) + Ad x(k − d(k)), x(k) =φ(k), k ∈ [−d2 , 0].

(5.4)

Throughout the section, the following definitions will be adopted: Definition 5.4. [101] Given some scalar α > 0, matrices Q, S and R with Q ≤ 0 and R real symmetric, singular system (5.1) is called strictly (Q, S , R)-α -dissipative if for any τ ≥ 0, under zero initial state, the following condition is satisfied: z, Qzτ + 2 z, S ωτ + ω, Rωτ ≥ α ω, ωτ .

(5.5)

The aim of this section is to study issue of α -dissipativity analysis for singular system (5.1). By utilizing the discrete Wirtinger inequality and the improved reciprocally convex combination approach, we will derive some sufficient criteria in terms of LMIs so that singular system (5.1) is admissible and strictly (Q, S , R)-α -dissipative.

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Analysis and Synthesis of Singular Systems

5.1.2 Main results In this section, the admissibility and strict (Q, S , R)-α -dissipativity of the discrete singular system with time-varying delay are addressed by adopting the discrete Wirtinger inequality and the improved reciprocally convex combination approach. To simplify the matrices and vector notations, the following notations will be used in our development: μ1 (k) = χ (k, 0, d1 ), μ2 (k) = χ (k, d1 , d(k)), μ3 (k) = χ (k, d(k), d2 ),  ξ T (k) = xT (k) xT (k − d1 ) xT (k − d(k)) xT (k − d2 )  μT1 (k)ET μT2 (k)ET μT3 (k)ET ωT (k) ,   ei = 0n×(i−1)n In 0n×(7−i)n 0n×l , i = 1, 2, ..., 7,   e8 = 0l×7n Il , ρ1 ρ2 ρ3 ρ4T

1 d1 e5 − Ee1 − Ee2 , 2 = Ae1 + Ad e3 + Bω e8 , =

= ρ2 − Ee1 ,  = ρ3T (Ee1 − Ee2 )T

(Ee2 − Ee4 )T

 ,

= Le1 + Ld e3 + Gω e8 ,

1 φ1 (k) = (d(k) − d1 )e6 − Ee2 − Ee3 , ρ5

2 1 ((d2 − d(k))e7 − Ee3 − Ee4 ), φ2 (k) = 2 φ3 (k) = φ1 (k) + φ2 (k)

1 (d(k) − d1 )e6 + (d2 − d(k))e7 − Ee2 − 2Ee3 − Ee4 , = 2 φ4T (k) = ΠT

=





(Ee1 )T

ρ1T

(Ee1 − Ee2 )T

φ3T (k)



,

(Ee1 + Ee2 − e5 )T

(Ee2 + Ee3 − e6 )T

(Ee3 − Ee4 )T

(Ee2 − Ee3 )T (Ee3 + Ee4 − e7 )T

 .

Theorem 5.5. For given integers d1 and d2 satisfying 0 < d1 ≤ d(k) ≤ d2 , let scalar α > 0, matrices Q, S and R be given with Q and R real symmetric and Q ≤ 0. Then system (5.1) is admissible and strictly (Q, S , R)-α -dissipative if there exist matrices P > 0, Qi > 0 (i = 1, 2, 3), Rj > 0 (j = 1, 2) and matrices X, W

Delay-dependent dissipativity analysis and synthesis of singular delay systems

105

such that the following set of LMIs hold: Ξ + sym(ρ4T P φ41 ) < 0, Ξ

+ sym(ρ4T P φ42 )

(5.6) (5.7)

< 0,

where R ∈ Rn×(n−r ) is any matrix with full column rank and satisfies ET R = 0, and Ξ Ξ1

= Ξ1 − Ξ2, = ρ4T P ρ4 + e1T (Q1 + ( d + 1)Q3 )e1 + e2T (−Q1 + Q2 )e2 − e3T Q3 e3 −eT Q2 e4 + ρ T (d2 R1 +  d2 R2 )ρ3 − Π T ΦΠ + sym(eT WRT ρ2 ), 4

Ξ2 T φ41 T φ42

3

1

1

= ρ5T Qρ5 + sym(ρ5T S e8 ) + e8T Re8 ,   = de7 − Ee2 − 2Ee3 − Ee4 )T , (Ee1 )T ρ1T 12 (   = de6 − Ee2 − 2Ee3 − Ee4 )T , (Ee1 )T ρ1T 12 ( ⎤

⎡

P11 P12 P13 ⎥ ⎢ P = ⎣  P22 P23 ⎦ ,  d = d2 − d1 ,   P33 ⎡

1 0 R ⎢ 2 Φ = ⎣ 0 R 0 XT

⎤

  0 Ri 0 ⎥  , i = 1,2. X ⎦ , Ri = 0 3Ri 2 R

Proof. Firstly, we will prove that the system (5.1) is regular and causal. Due to rank(E) = r, we choose two nonsingular matrices M and N such that 

MEN = Set



MAN =

A1 A2 A3 A4





,N W= T

I 0

0 0

W1 W2



(5.8)

.





,M

−T

R=

where F ∈ R(n−r )×(n−r ) is nonsingular. Expand Ξ + sym(ρ4T P φ41 ) as  Ξ

+ sym(ρ4T P φ41 ) =

Ξ11 •

• •

 ,

0 I



F,

(5.9)

106

Analysis and Synthesis of Singular Systems

where • means that the elements of the matrix are not relevant in our discussion, and T Ξ11 =(A − E)T P11 (A − E) + (A − E)T P12 E + ET P12 (A − E) + Q1 + ( d + 1)Q3 − 4ET R1 E + (A − E)T (d2 R1 +  d2 R2 )(A − E) 1

+ WR A + A RW + (A − E) P11 E + ET P11 (A − E) − L T QL T

T

1 2

T

T

1 2

T T − (A − E)T P12 E − ET P12 (A − E) + ET P12 E + ET P12 E.

Due to Ξ + sym(ρ4T P φ41 ) < 0, thus Ξ11 < 0, together with P > 0, Q1 > 0, Q3 > 0, Ri > 0, Si > 0 (i = 1, 2), and −Q ≥ 0, we have T Ω =(A − E)T P12 E + ET P12 (A − E) − 4ET R1 E + WRT A + AT RW T

1 2

1 2

T + (A − E)T P11 E + ET P11 (A − E) − (A − E)T P12 E − ET P12 (A − E) T E < 0. + ET P12 E + ET P12

Premultiplying and postmultiplying Ω < 0 by N T and N, respectively, substituting (5.8) and (5.9) into the previous inequalities gives 

• • • W2 F T A4 + AT4 FW2T

 < 0.

From the above inequality, it is easy to see that W2 F T A4 + AT4 FW2T < 0, which implies A4 is nonsingular. Hence, the pair (E, A) is regular and causal. To prove the stability of system (5.1), we design a Lyapunov function as V (k) = V1 (k) + V2 (k) + V3 (k) + V4 (k), where V1 (k) =εT (k)P ε(k), V2 (k) =

k−1 

xT (i)Q1 x(i) +

i=k−d1

+

−d1 

k− d1 −1

xT (i)Q2 x(i) +

i=k−d2 k−1 

i=−d2 +1 j=k+i

xT (j)Q3 x(j),

k−1  i=k−d(k)

xT (i)Q3 x(i)

107

Delay-dependent dissipativity analysis and synthesis of singular delay systems

V3 (k) =d1

0 

k−1 

yT (j)ET R1 Ey(j) +  d

i=−d1 +1 j=k+i

−d1 

k−1 

yT (j)ET R2 Ey(j)

i=−d2 +1 j=k+i

with

⎤

⎡

Ex(k) ⎢ k−1 Ex(i) ⎥ ε(k) = ⎣ ⎦. i=k−d1 k−d1 −1 i=k−d2 Ex(i) By denoting the forward difference of V (k) as V (k)= V (k+1) − V (k) and calculating it along the solution of system (5.1), we have V1 (k) = ε T (k)P ε(k) + 2ε T (k)P ε(k),

where

⎡

⎤ (A − E)x(k) + Ad x(k − d1 ) ⎢ ⎥ ε(k) = ⎣ Ex(k) − Ex(k − d1 ) ⎦, Ex(k − d1 ) − Ex(k − d2 )

so V1 (k) = ξ T (k)(ρ4T P ρ4 + 2ρ4T P φ4 (k))ξ(k)

(5.10)

The estimation of the forward difference of V2 (k) is V2 (k) =xT (k)Q1 x(k) − xT (k − d1 )Q1 x(k − d1 ) + xT (k − d1 )Q2 x(k − d1 ) − xT (k − d2 )Q2 x(k − d2 ) + xT (k)Q3 x(k)+

k−1 

xT (i)Q3 x(i)

i=k+1−d(k+1)

−

k−1 

xT (i)Q3 x(i) − xT (k − d(k))Q3 x(k − d(k))

i=k+1−d(k)

+ dxT (k)Q3 x(k) −

k −d1

xT (i)Q3 x(i)

i=k−d2 +1

d + 1)Q3 )x(k) + xT (k − d1 )(−Q1 + Q2 )x(k − d1 ) =x (k)(Q1 + ( T

−xT (k − d2 )Q2 x(k − d2 ) − xT (k − d(k))Q3 x(k − d(k)) +

k−1  i=k+1−d1

−

k −d1

xT (i)Q3 x(i) +

k−1  i=k+1−d(k)

xT (i)Q3 x(i)

i=k+1−d(k+1)

xT (i)Q3 x(i) −

k −d1 i=k−d2 +1

xT (i)Q3 x(i)

108

Analysis and Synthesis of Singular Systems

≤ξ T (k)(e1T (Q1 + ( d + 1)Q3 )e1 + e2T (−Q1 + Q2 )e2 − e3T Q3 e3 − e4T Q2 e4 )ξ(k).

(5.11)

According to Lemma 5.1 and Lemma 5.3, when d1 < d(k) < d2 , the forward difference of V3 (k) is calculated as k 

V3 (k) =yT (k)(d12 ET R1 E +  d2 ET R2 E)y(k) − d1

yT (i)ET R1 Ey(i)

i=k−d1 +1

− d

k −d1

yT (i)ET R2 Ey(i) −  d

i=k−d(k)+1

≤y (k)E T

T

k −d(k)

(d12 R1 ⎡

yT (i)ET R2 Ey(i)

i=k−d2 +1

+ d2 R2 )Ey(k)

1 R T T⎢ 0 − ξ (k)Π ⎣ 0

0

 d  d(k)−d1 R2

0

0 0  d  d2 −d(k) R2

⎤ ⎥ ⎦ Πξ(k)

  ≤ξ T (k) ρ3T (d12 R1 +  d2 R2 )ρ3 − Π T ΦΠ ξ(k). 

(5.12)

Notice that when d(k) = d1 , we have ξ T (k) E(e2 − e3 ) Ee2 + Ee3 − e6 = 0, so the inequality (5.12) still stands. So does it when d(k) = d2 . In addition, it is clear that 2xT (k)WRT Ex(k + 1) ≡ 0.



(5.13)

According to the inequalities from (5.10) to (5.13), we can obtain V (k) = V1 (k) + V2 (k) + V3 (k) + 2xT (k)WRT Ex(k + 1) ≤ ξ T (k)(Ξ1 + sym(ρ4T P φ4 (k)))ξ(k).

Define J (k) = zT (k)Qz(k) + 2zT (k)S ω(k) + ωT (k)Rω(k). Then V (k) − J (k) ≤ ξ T (k)(Ξ + sym(ρ4T P φ4 (k)))ξ(k).

Due to the convexity of Ξ + sym(ρ4T P φ4 (k)) with respect to d(k), conditions (5.6) and (5.7) can guarantee Ξ + sym(ρ4T P φ4 (k) < 0, so V (k) − J (k) < 0, which implies the strict (Q, S , R)-α -dissipativity.When 2 ω(k) ≡ 0, from V (k) − J (k) ≤ 0 and Q ≤ 0, we have V (k) ≤ −c x(k) for some

109

Delay-dependent dissipativity analysis and synthesis of singular delay systems

c > 0, so the discrete singular system (5.1) is stable. Then, the singular system (5.1) is admissible and strictly (Q, S , R)-α -dissipative in the sense of Definition 4.1, Definition 4.2, and Definition 5.4. Then the proof is completed. To facilitate comparison of our result with the existing ones, considering the stability of the nominal system (5.4) and following the same lines as that in proof of Theorem 5.5, it is easy to acquire the admissibility criterion of the singular system (5.4). Corollary 5.6. For given integers d1 and d2 satisfying 0 < d1 ≤ d(k) ≤ d2 , system (5.4) is admissible if there exist matrices P > 0, Qi > 0 (i = 1, 2, 3), Rj > 0 (j = 1, 2), and matrices X, W such that the following set of LMIs hold: Ξ 1 + sym(ρ T4 P φ 41 ) < 0, Ξ 1 + sym(ρ T4 P φ 42 ) < 0,

where R ∈ Rn×(n−r ) is any matrix with full column rank and satisfies ET R = 0, and Ξ1

= ρ T4 P ρ 4 + eT1 (Q1 + ( d + 1)Q3 )e1 + eT2 (−Q1 + Q2 )e2 − eT3 Q3 e3 −eT Q2 e4 + ρ T (d2 R1 +  d2 R2 )ρ − Π T ΦΠ + sym(eT WRT ρ ) 4

ρ1 ρ T4 φ 41

T

=

T

=

φ 42

3

3

1

2

1

1 = d1 e5 − Ee1 − Ee2 , ρ 2 = Ae1 + Ad e3 , ρ 3 = ρ 2 − Ee1 2   = ρ T3 (Ee1 − Ee2 )T (Ee2 − Ee4 )T

ei =

  

(Ee1 )T

ρ T1

1  2 (de7

(Ee1 )T

ρ T1

1  2 (de6

0n×(i−1)n In

− Ee2 − 2Ee3 − Ee4 )T

− Ee2 − 2Ee3 − Ee4 )T  0n×(7−i)n , i = 1, 2, ..., 7

 

5.1.3 Numerical examples Example 5.1. Considering the singular system (5.4) with the following parameters: 

E=

3.5 0 0 0



 , A=

2.3 0 0 −3



 , Ad =

−1.3 1.5 0 0.5

 .

To guarantee the admissibility of the above system, we calculate the allowable maximum of d(k). Set d(k) as a constant time delay d. By using z-transformation method, one can get the characteristic polynomial as

110

Analysis and Synthesis of Singular Systems

det(zE − A − Ad z−d ) = (3.5z − 2.3 + 1.3z−d )(3 − 0.5z−d ). The characteristic values must satisfy |z| < 1 to ensure the above system admissible, which

yields that the maximum value of d is 15. The allowable maximum values of d2 that guarantees system (5.4) to be admissible, with various d1 by applying Corollary 5.6 and some other methods are shown in Table 5.1. From the results, it is clear to see that the criterion in this section has reduced conservatism significantly for the admissibility condition of Example 5.1. Example 5.2. In this example, the generality of the dissipativity is demonstrated, which unifies H∞ performance and passivity performance. Consider singular system (5.1) with the parameters 

E= 

Bω = •

•

•

1 0 0 0 0.1 0.5



 , A=

 , L=



0.8 0 0.05 0.9

−0.1 0.4





 , Ad =

, Ld =



−0.1 0 −0.2 −0.1

−0.2 −0.1



 ,

, Gω = 0.1.

H∞ performance: Let Q = −Iq , S = 0q×l , R − α Il = γ 2 Il . The dissipativity reduces to standard H∞ performance. By adopting Theorem 5.5, the allowable minimum H∞ disturbance attenuation γ can be acquired for stationary d1 = 1 and varied d2 . The relationship between γ and d2 is illustrated in Table 5.2, which demonstrates that the minimal value of γ increases as the value of d2 grows. Compared with Corollary 1 of paper [252] and Corollary 1 in [27], it is obvious that the method proposed in this paper has a better performance to bear the tolerance according to the lower disturbance attenuation level γ . Passivity: When q = l and Q = 0q , S = Iq , R − α Iq = γ Iq , it becomes passivity performance. Setting d1 = 3, the different values of minimum γ are shown in Table 5.3, according to Theorem 5.5, with various d2 . It is clear to see that the minimum value of γ also becomes larger when d2 increases. Dissipativity: Set Q = −0.4, S = 1.1, R = 3.

If the LMIs (5.6) and (5.7) are feasible, singular system (5.1) is admissible and strictly (Q, S , R)-α -dissipative, where α denotes the level of dissipativity. A higher α means that singular system (5.1) has a better performance to tolerate uncertainties and disturbances. From Table 5.4,

Theorem 2 [33] Theorem 2 [36] Theorem 1 [188] Theorem 1 [46] Corollary 5.6

3 3 3 4 10

3 3 4 4 7

4 4 4 5 5

infeasible 5 5 5 6

infeasible infeasible 6 6 6

infeasible infeasible 7 7 7

7

infeasible infeasible infeasible infeasible 8

Table 5.1 Comparison of the allowable maximum values of d2 for various d1 . d1 1 2 3 4 5 6

8

infeasible infeasible infeasible infeasible 9

Delay-dependent dissipativity analysis and synthesis of singular delay systems

111

112

Analysis and Synthesis of Singular Systems

Table 5.2 Allowable minimum γ for different d2 . d2 3 6 9

Corollary 4.1 [252] Corollary 1 [27] Theorem 5.5

0.1945 0.1945 0.1907

0.2614 0.2379 0.2313

0.4465 0.3044 0.2628

12

15

3.763 0.4335 0.2865

infeasible 0.8182 0.3107

Table 5.3 Different minimal values of γ for different d2 . d2 6 11 16 21

Theorem 5.5

0.3892

0.4304

0.4569

0.4754

26

0.4904

Table 5.4 Allowable maximum dissipativity level α for different d2 . d2 5 10 15 20 25

Theorem 5.5

2.558

2.508

2.476

2.454

2.433

we can see that for fixed d1 = 2, when the upper bound of delay d2 becomes larger, the dissipativity level becomes smaller. Example 5.3. In this example, Corollary 5.6 is applied to check the admissibility of a class of economic system, which can be modeled by singular system [120]. We use the Leontief model given in [4], Ex(k + 1) = (I − U + E)x(k) − Wu(k). The physical meaning of x(k) is the production level in the sectors at time k. Ux(k) represents the amount required as direct input for the current production, and u(k) denotes the amount of production for the current demand, respectively. The element eij of matrix E denotes the amount of stock of commodity i, as a capital good, that sector j must have on hand for each unit of production. Since not every sector produces significant capital goods, it is common for some rows of the matrix E to contain only zero elements such that matrix E is singular. Therefore the Leontief model is a singular system. By using the system parameters as those in [4], the singular system is given as follows: 



 0.6 −0.05 x(k) −0.7695 −1.83   0.1676 0.117 + x(k − d(k)). −0.5112 −0.3569

1 1 x(k + 1) = 0 0



Delay-dependent dissipativity analysis and synthesis of singular delay systems

113

The lower bound and upper bound of time-delay d(k) are chosen as d1 = 1 and d2 = 4, respectively. By solving the LMIs in Corollary 5.6, the condition is feasible, which implies that the singular system is admissible under the variation of delay d(k) between 1 and 4.

5.1.4 Conclusion The problem of admissibility and strict (Q, S , R)-α -dissipativity analysis for discrete singular system with time-varying delay has been studied in this section. Through combining a discrete Wirtinger inequality and the improved reciprocally convex combination approach, a sufficient criterion in terms of strict LMIs has been proposed for the admissibility and dissipativity analysis of the considered systems. Compared with existing results, the results of this section show less conservative performance. The obtained results also address the analysis of H∞ performance, passivity, and strict (Q, S , R)-α -dissipativity of the discrete singular system with timevarying delay in a unified framework. The effectiveness and advantages of the proposed approach have been shown by numerical examples. The dissipative controller design of the considered system will be addressed in our future work.

5.2 Dissipativity analysis and dissipative control of singular time-delay systems In this section, the problem of delay-dependent α -dissipative control is investigated for continuous-time singular systems with time delay. Using the delay-partitioning technique, sufficient conditions are established in terms of LMIs, which guarantees a singular system to be admissible and strictly (Q, S, R)-α -dissipative. Based on the criteria, a design algorithm for a state feedback controller is proposed. In addition to delay dependence, the obtained results are also dependent on the parameter α . Moreover, the results of H∞ control and passive control are also derived from the dissipative control results. The results developed in this section are less conservative than existing ones in the literature, which are illustrated by several examples.

114

Analysis and Synthesis of Singular Systems

5.2.1 Problem formulation Consider a class of linear continuous-time singular systems described by ⎧ ⎪ ⎨ Ex˙ (t) = Ax(t) + Ad x(t − h) + Bu(t) + Bw w (t) z(t) = Cx(t) + Cd x(t − h) + Du(t) + Dw w (t) ⎪ ⎩ x(t) = ϕ(t), ∀t ∈ [−h, 0],

(5.14)

where x(t) ∈ Rn is the state vector; u(t) ∈ Rp is the control input; w (t) ∈ Rl represents a set of exogenous inputs, which includes disturbances to be rejected, and z(k) ∈ Rq is the controlled output; h represents the system delay ¯ A, Ad , B, Bw , C, Cd , D, and Dw denote constant masatisfying 0 < h ≤ h; trices with appropriate dimensions. In contrast with standard linear systems with E = I, the matrix E ∈ Rn×n has rank(E) = p ≤ n. Consider the following memoryless state-feedback controller for system (5.14): u(t) = Kx(t).

(5.15)

By applying controller (5.15) to system (5.14), the closed-loop system can be described by ⎧ ⎪ ⎨ Ex˙ (t) = (A + BK )x(t) + Ad x(t − h) + Bw w (t) z(t) = (C + DK )x(t) + Cd x(t − h) + Dw w (t) ⎪ ⎩ x(t) = ϕ(t), ∀t ∈ [−h, 0].

(5.16)

Before moving on, we give some definitions and lemmas concerning the following nominal unforced counterpart of the system in (5.14) with w (t) = 0: ⎧ ⎪ ⎨ Ex˙ (t) = Ax(t) + Ad x(t − h) z(t) = Cx(t) + Cd x(t − h) ⎪ ⎩ x(t) = ϕ(t), ∀t ∈ [−h, 0].

(5.17)

Lemma 5.7. [216] Suppose the pair (E, A) is regular and impulse free, then the solution to system (5.17) is impulse free and unique on [0, ∞). In view of this, we introduce the following definition for singular delay system (5.17): Definition 5.8. 1) The singular delay system in (5.17) is said to be regular and impulse free if the pair (E, A) is regular and impulse free. 2) The singular system in (5.17) is said to be asymptotically stable if, for any ε > 0, there exists a scalar δ(ε) > 0, such that for any compatible initial

Delay-dependent dissipativity analysis and synthesis of singular delay systems

115

conditions x0 satisfying x0 ≤ δ(ε), the solution x(t) of (5.17) satisfies

x(t) ≤ ε for t ≥ 0; furthermore, x(t) → 0, when t → ∞. 3) The singular system in (5.17) is said to be admissible, or the pair (E, A) is admissible, if the system is regular, impulse free, and asymptotically stable. Definition 5.9. [101] Given some scalar α > 0, matrices Q, R, and S with Q and R real symmetric, systems (5.14) with u(t) = 0 is called strictly (Q, S, R)-α -dissipative if, for any τ ≥ 0, under zero initial state, the following condition is satisfied: z, Qzτ + 2z, Sw τ + w , Rw τ ≥ αw , w τ .

(5.18)

As in [202], we assume that Q ≤ 0. Then we can get T −Q = Q− Q−

for some Q− . Our main objective is to study the problem of α -dissipative control for singular system (5.14). More specially, we are concerned with the following two problems: 1. Establish a sufficient condition in terms of LMIs such that singular system (5.14) is admissible and strictly (Q, S, R)-α -dissipative. 2. Design a state-feedback controller in the form of (5.15) such that the closed-loop systems in (5.16) is admissible and strictly (Q, S, R)-α dissipative.

5.2.2 Dissipative analysis In this section, the improved sufficient condition is derived firstly by employing the delay-partitioning technology, which guarantees that the unforced system of (5.14) is admissible and strictly (Q, S, R)-α -dissipative. The result of strict dissipative analysis for system (5.14) is presented in the following theorem: Theorem 5.10. Let scalar α > 0, matrices Q, S, and R be given with Q and R real symmetric and Q ≤ 0. Then the system in (5.14) with u(t) = 0 is admissible and strictly (Q, S, R)-α -dissipative if there exist matrices P, Y , U, V > 0, and W > 0 such that the following set of LMIs hold: ET P = P T E ≥ 0,

(5.19)

116

Analysis and Synthesis of Singular Systems

Ω < 0,

(5.20)

where Ω

WX

h T T h W Y WV − WXT Y T WE + WdT U T WV m X m h h −WdT U T WE − WST SWQ ) + WDT VWD − WVT VWV m m T ¯ +WW W WW − WQT QWQ − WST (R − α I )WS

= sym(WXT P T WD +





=





WQ =

C 0q,(m−1)n Cd 0q,n Dw , 



WD =

A 0n,(m−1)n Ad 0n,n Bw , 





WS =

0l,(m+2)n Il , WW = 



Wd =

Imn 0mn,2n+l 0mn,n Imn 0mn,n+l 

 , 

0n,n In 0n,mn+l , WV = 0n,(m+1)n In 0n,l ,

 ¯ W





In 0n,(m+1)n+l , WE = E −E 0n,mn+l ,

W

=





0mn,mn . −W

Proof. The proof contains two parts. The admissibility of the singular system is tackled in the first part, and the second part deals with the α -dissipative analysis of the singular system. For the first part, we consider system (5.17) and choose two nonsingular matrices M and N such that 

E=M Denote

I

 

T

M PN

−1

P¯ 1 = ¯ P3





¯ 0 A N, A = M ¯ 1 0 A3 



¯2 A ¯ 4 N. A 

P¯ 2 Y¯ 1 T −1 ¯P4 , M YN = Y¯ 3



Y¯ 2 . Y¯ 4

Furthermore, the following inequality is obtained from (5.20): ⎡

sym(AT P − ET Y ) + W

⎢ ⎢ ⎢ ⎣

  

⎤ • • • • • •⎥ ⎥ ⎥  • •⎦   •

Delay-dependent dissipativity analysis and synthesis of singular delay systems

⎡ +

117

⎤

AT

⎥   h⎢ ⎢0(m−1)×n,n ⎥ V A 0 A B ⎢ ⎥ n,(m−1)×n d w m ⎣ ATd ⎦ Bw

⎡

CT

⎤

⎢0 ⎥  ⎢ (m−1)×n,q ⎥ −⎢ ⎥Q C T ⎣ Cd ⎦



0q,(m−1)×n Cd Dw < 0.

(5.21)

Dw Due to −Q ≥ 0 and V > 0, sym(AT P − ET Y ) + W < 0 is derived from (5.21). Then following a similar analysis method of [214], we can ob¯ 4 is nonsingular. Therefore following from [25] and Definition 5.8, tain A we conclude that the system in (5.17) is regular and impulse-free. For the stability property, let us choose the following Lyapunov functional: 

V (x(t)) = x(t)T P T Ex(t) +  +

where

0

− mh



t

t+θ

t

t− mh

Υ (s)T W Υ (s)ds

(Ex˙ (s))T VEx˙ (s)dsdθ,

⎡

(5.22)

⎤

x(t) ⎢ ⎥ x ( t − mh ) ⎥ ⎢

Υ (t) = ⎢ ⎢ ⎣

⎥. ⎥ ⎦ (m−1)h x(t − m ) .. .

Evaluating the derivative of V (x(t)) along the solutions of system (5.17), we obtain h V˙ (x(t)) = 2x(t)T P T Ex˙ (t) + (Ex˙ (t))T V (Ex˙ (t)) m h h T +Υ (t) W Υ (t) − Υ (t − )T W Υ (t − ) m m  t

(Ex˙ (s))T V (Ex˙ (s))ds  t h +2x(t)T Y T E[ x˙ (s)ds − x(t) + x(t − )] m t− mh  t h h +2x(t − )T U T E[ x˙ (s)ds − x(t) + x(t − )] h m m t− m  m t = ζ1 (t, s)T Ω1 ζ1 (t, s)ds, −

h

t− mh

t− mh

118

Analysis and Synthesis of Singular Systems

where h T T T Y T WE1 W Y WV 1 − WX1 m X1 h T T h T T T + Wd1 U WV 1 − Wd1 U WE1 ) + WD1 VWD1 m m h T ¯ − WVT1 VWV 1 + WW 1 W WW 1 , m     = In 0n,(m+1)n , WE1 = E −E 0n,mn , T T = sym(WX1 P WD1 +

Ω1

WX1

WD1 =

Wd1 =







A 0n,(m−1)n Ad 0n,n , WW 1 = 





Imn 0mn,2n 0mn,n Imn 0mn,n

0n,n In 0n,mn , WV 1 = 0n,(m+1)n In



 ,

⎤ Υ (t) ⎥ ⎢ , ζ1 (t, s) = ⎣x(t − h)⎦ . Ex˙ (s) ⎡

It can be seen that Ω < 0 implies Ω1 < 0. Then V˙ (x(t)) < 0

(5.23)

is derived. On the other hand, from (5.20), we have

Ω2

⎡ Λ11 ⎢  ⎢ = ⎢ ⎣    −

Λ12 Λ22  

0n,(m−2)n 0n,n 0(m−2)n,(m−2)n 



⎤

P T Ad  0n,n ⎥ W ⎥ ⎥+ 0(m−2)n,n ⎦  0n,n

0n,n 0n,mn < 0,  W

0mn,n 0n,n



(5.24)

where Λ11

= P T A + AT P − Y T E − E T Y ,

Λ12

= Y T E − ET U ,

Λ22

= U T E + ET U .

Pre- and postmultiplying (5.24) by [I , . . . , I ] ∈ Rn×mn and its transpose, respectively, the following inequality is obtained P T (A + Ad ) + (A + Ad )T P < 0,

119

Delay-dependent dissipativity analysis and synthesis of singular delay systems

which implies P is nonsingular. Due to the regularity and the absence of ˜ impulses of the pair (E, A), there always exist two nonsingular matrices M and N˜ such that 







˜ ˜ , A=M ˜. ˜ Ip 0p,n−p N ˜ A1 0p,n−p N E=M 0 0n−p,n−p 0 In−p,n−p

(5.25)

Define ˜ ˜ PN P˜ = M T

Y˜













˜ ˜ P˜ 1 P˜ 2 ˜ −1 = U1 U2 , (5.26) ˜ =M ˜ T UN , U = ˜ ˜ ˜ P3 P4 U3 U˜ 4 



˜ ˜ ˜ ˜ ˜d=M ˜ −1 = Y1 Y2 , A ˜ −1 = Ad1 Ad2 , ˜ TYN ˜ −1 Ad N = M ˜ ˜ d4 ˜ ˜ Y3 Y4 Ad3 A ⎡

W

−1

W11 · · · W1m

⎢ = ⎣ ∗ ∗



˜ .. ⎥ , W ˜ −1 = Wjj1 ˜ −T Wjj N T . ⎦ ˜ jj = N ˜ jj2 W

..

. ∗

(5.27)

⎤

Wmm



˜ jj2 W ˜ jj4 , W

(5.28) where the partition is compatible with that of A in (5.25). Then, by (5.19), it can be shown that P˜ 2 = 0 and P˜ 1 > 0. Now, pre- and postmultiplying (5.24) by diag(N˜ −T , . . . , N˜ −T ) ∈ R(m+1)n×(m+1)n and its transpose, respectively, we obtain ⎡• ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣









• • • • .. .

    

    

    

    

··· ··· ··· ··· ···

  

P˜ 4T

•

• • ˜ 114 • + P˜ 4 + W •  • • ˜ 114 ˜ 224 − W   W

• • • • . . . .. . ··· ··· ··· ···

• • • • • • • • . . . . . . . . . . . . • • ˜ (m−1)(m−1)4 ˜ mm4 − W  W    

• • ˜ d4 • P˜ 4T A • • • • . . . . . . . . . . . . • • • • • • ˜ mm4  −W

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎥ ⎦

(5.29) which implies



˜ 114 P˜ 4T + P˜ 4 + W 



˜ d4 P˜ 4T A < 0, ˜ mm4 −W

(5.30)

120

Analysis and Synthesis of Singular Systems

˜ 224 − W ˜ 114 < 0, W ˜ 334 − W ˜ 224 < 0, · · · , W ˜ mm4 − W ˜ (m−1)(m−1)4 < 0. W (5.31)   T ˜ Pre- and postmultiplying (5.30) by −Ad4 I and its transpose, we have ˜TW ˜ ˜ ˜ A d4 114 Ad4 − Wmm4 < 0,

that is, ˜TW ˜ ˜ ˜ ˜ ˜ A d4 114 Ad4 − W114 + (W114 − Wmm4 ) < 0.

(5.32)

˜ 114 − W ˜ mm4 > 0, and we have from (5.32) that Since (5.31) implies W ˜TW ˜ ˜ ˜ A d4 114 Ad4 − W114 < 0,

therefore ˜ d4 ) < 1. ρ(A

Noting this and noting (5.22) and (5.23), and following a similar line to that in the proof of Theorem 1 in [214], we can deduce that the singular time-delay system in (5.17) is stable. Let us prove that the system is strictly dissipative. For this purpose, considering (5.14) with u(t) = 0 and (5.23), we obtain V˙ (x(t)) − z(t)T Qz(t) − 2w (t)T Sz(t) − w (t)T (R − α I )w (t) h = 2x(t)T P T [Ax(t) + Ad x(t − h) + Bw w (t)] + (Ex˙ (t))T V (Ex˙ (t)) m h h +Υ (t)T W Υ (t) − Υ (t − )T W Υ (t − ) m m 

−

t

t− mh

(Ex˙ (s))T V (Ex˙ (s))ds

 +2x(t)T Y T E[ +2x(t −

t t− mh

x˙ (s)ds − x(t) + x(t − 

h T T ) U E[ m

t t− mh

h )] m

x˙ (s)ds − x(t) + x(t −

h )] m

−2w (t) S[Cz x(t) + Cd x(t − h) + Dw w (t)] − w (t)T (R − α I )w (t) T

−[Cz x(t) + Cd x(t − h) + Dw w (t)]T Q[Cz x(t) + Cd x(t − h) + Dw w (t)]  m t = ζ (t, s)Ωζ (t, s)ds, (5.33)

h

t− mh

Delay-dependent dissipativity analysis and synthesis of singular delay systems

where

121

⎡

⎤ Υ (t) ⎢x(t − h)⎥ ⎢ ⎥ ζ (t, s) = ⎢ ⎥. ⎣ Ex˙ (s) ⎦ w (t)

Then, since Ω < 0, we have V˙ (x(t)) − z(t)T Qz(t) − 2w (t)T Sz(t) − w (t)T (R − α I )w (t) < 0.

(5.34)

Integrating (5.34) over the range [0, τ ], we obtain 

V (x(T )) + α w (t)T w (t)
0, the H∞ performance optimizing problem corresponds to Q = −I, S = 0 and R = (α + γ 2 )I. Based on Remark 5.16, the following corollaries give LMIs conditions for strict positive realness and H∞ performance analysis: Corollary 5.17. The system in (5.14) with u(t) = 0 is admissible and strictly positive real if there exist matrices P, Y , U, V > 0, and W > 0 such that the following set of LMIs hold: ET P = P T E ≥ 0,

(5.37)

ΩP < 0,

(5.38)

where ΩP

h T T h WX Y WV − WXT Y T WE + WdT U T WV m m h T h T T T T − Wd U WE − WS WQ ) + WD VWD − WV VWV m m T ¯ + WW W WW . = sym(WXT P T WD +

Corollary 5.18. For a given γ > 0, the system in (5.14) with u(t) = 0 is admissible with disturbance rejection of level γ if there exist matrices P, Y , U, V > 0, and W > 0 such that the following set of LMIs hold: ET P = P T E ≥ 0,

(5.39)

Ω∞ < 0,

(5.40)

where Ω∞

h T T h W Y WV − WXT Y T WE + WdT U T WV m X m h h −WdT U T WE ) + WDT VWD − WVT VWV m m T ¯ +WW W WW + WQT WQ − γ 2 WST WS .

= sym(WXT P T WD +

The result obtained in Corollary 5.18 coincides with the following corollary as a special case when m = 1:

Delay-dependent dissipativity analysis and synthesis of singular delay systems

125

Corollary 5.19. Given scalars γ > 0 and h > 0, then the unforced singular system of (5.14) is admissible with disturbance rejection of level γ if there exist matrices P, Y , U, and matrices V > 0 and W > 0 such that the following set of LMIs hold: ET P = P T E ≥ 0,

⎡



hY T hU T −hV













J1 J2 ⎢ ⎢  J3 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 

P T Bw hAT V 0 hATd V 0 0 −γ 2 I hBdT V  −hV 



(5.41) ⎤

CT ⎥ CdT ⎥ ⎥ 0 ⎥ ⎥ ⎥ < 0, Dw ⎥ ⎥ 0 ⎥ ⎦ −I

(5.42)

where J 1 = P T A + AT P + W − Y T E + E T Y , J2 = P T Ad + Y T E − E T U , J3 = U T E + E T U − W . Remark 5.20. Corollary 5.19 is equivalent to Theorem 1 in [214] when the system considered therein takes the same form as (5.14). Furthermore, it can be easily shown that Corollary 5.17 will be less conservative than Corollary 5.19 when m > 1. To compare our results with that in [62], the admissibility criterion of singular delay system is obtained from Corollary 5.18 as follows: Corollary 5.21. For a given h > 0, the singular delay system Ex˙ (t) = Ax(t) + Ad x(t − h)

(5.43)

is admissible if there exist matrices P, Y , U, V > 0, and W > 0 such that the following set of LMIs hold: ET P = P T E ≥ 0,

(5.44)

ΩA < 0,

(5.45)

where ΩA

T T = sym(WX0 P WD0 +

h T T T Y T WE0 W Y WV 0 − WX0 m X0

126

Analysis and Synthesis of Singular Systems

h m h T ¯ − WVT0 VWV 0 + WW 0 W WW 0 , m

T T T T + Wd0 U WV 0 − Wd0 U WE0 ) +

h T W VWD0 m D0

and WX0 , WD0 , WV 0 , WE0 , Wd0 , WW 0 are obtained from WX , WD , WV , WE , Wd , WW defined in Theorem 5.10 by setting l = 0. Remark 5.22. Corollary 5.21 provides a delay-dependent condition for singular delay system (5.43) to be admissible. Although only discrete delay is considered in system (5.43), the obtained condition can be extended to the case of systems with both discrete and distributed delays. For the system 

Ex˙ (t) = Ax(t) + Ad x(t − h) + Ah

t

t−d

x(s)ds,

we can choose a new Lyapunov functional as follows: 

V (x(t)) = x(t)T ET Px(t) +  +

− mh

 + +

0

t



t

t+s

t− mh



η(θ )T Qη(θ )dθ +

θ

t+λ

0

− dr



t

t+θ

x(s)T Zx(s)dsdθ

x˙ (s)T ET REx˙ (s)dsdλdθ,

where ⎡

λ(θ )T Z1 λ(θ )dθ

x˙ (θ )T ET Z2 Ex˙ (θ )dθ ds

t− dr  0  0 t − dr

t

⎡ θ

⎤

θ − dr

x(s)ds

⎤

x(θ ) ⎢ ⎥ .. ⎢ ⎥ ⎢ ⎥ . ⎢ x(θ − mh ) ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ( r −2 ) d  λ(θ ) = ⎢ , η(θ ) = .. ⎢ ⎥ ⎢ θ − r x(s)ds⎥ ; ⎢ θ − ( r −1 )d ⎥ . ⎣ ⎦ r ⎣ ⎦ x(θ − (m−m1)h )  θ − (r −r1)d x(s)ds θ −d m and r denote the different number of partitions for discrete delay h and distributed delay d, respectively. Remark 5.23. For singular system with multiple delays such as Ex˙ (t) = Ax(t) +

q  i=1

Adi x(t − hi ),

127

Delay-dependent dissipativity analysis and synthesis of singular delay systems

we can choose a new Lyapunov functional as follows: V (x(t)) = x(t) E Px(t) + T

+

T

q   i=1

where

0



h − mi i

q   i=1

t

t+s

⎡

t h

t− mi

λi (θ )T Qi λi (θ )dθ

i

x˙ (θ )T ET Zi Ex˙ (θ )dθ ds, ⎤

x(θ )

⎢ x(θ − hi ) ⎥ ⎢ ⎥ mi ⎥, λi (θ ) = ⎢ .. ⎢ ⎥ . ⎣ ⎦ (mi −1)hi x(θ − mi )

and mi denotes the different number of partitions for different delays hi , i = 1, . . . , q.

5.2.3 State-feedback dissipative control In this section, attention will be devoted to design the state-feedback controller in the form of (5.15) such that closed-loop system (5.16) is admissible and strictly (Q, S, R)-α -dissipative. Based on the results of Theorem 5.10, the dissipative stabilization results for singular system (5.14) is presented in the following theorem: Theorem 5.24. Let scalar α > 0, matrices Q, S, and R be given with Q and R real symmetric and Q ≤ 0. Then the system in (5.14) is admissible and strictly (Q, S, R)-α -dissipative if there exist matrices P , Z , U , Y , V > 0, and W > 0 such that the following set of LMIs hold: EP = P T ET ≥ 0, ⎡ Ω˜ ⎢ ⎣ 

h ˜ T m WD − mh V



(5.46)

⎤

T ˜ QT Q− W ⎥ 0 ⎦ < 0, −I

(5.47)

where Ω˜

h T T h ˜ E + WdT U T WV W Y WV − WXT Y T W m X m h ˜ E − WST SW ˜ Q ) − WVT (P + P T − V )WV −WdT U T W m T ˜ +WW W WW − WST (R − α I )WS ,

˜D+ = sym(WXT W

128

Analysis and Synthesis of Singular Systems





˜Q = W

C P + DZ 0q,(m−1)n Cd P 0q,n Dw ,



˜E = W

ET

−E T



0n,mn+l ,



˜D = W

AP + BZ 0n,(m−1)n Ad P 0n,n Bw



 ˜ = W , W 

0mn,mn −W

 .

When the above conditions are satisfied, an admissible and (Q, S, R)-α -dissipative controller is given by K = ZP −1 .

(5.48)

Proof. Substitute A and C in (5.20) with A + BK and C + DK, respectively. Then applying the Schur complement lemma, we have  Ω¯

h ¯ T m WD V − mh V



 < 0,

(5.49)

where Ω¯

¯D+ = sym(WXT P T W

h T T W Y WV − WXT Y T WE m X

h h T W VWV m m V T ¯ ¯ Q − WST (R − α I )WS ¯ QT QW +WW W WW − W ¯ Q) − + WdT U T WV − WdT U T WE − WST SW

¯D = W ¯Q = W





A + BK 0n,(m−1)n Ad 0n,n Bw , 



C + DK 0q,(m−1)n Cd 0q,n Dw .

Then we introduce the new variable P = P −1 , V = V −1 , and let T1 = diag(P , P , P , . . . , P ) ∈ Rmn×mn , T2 = diag(T1 , P T , Il , V T ) ∈ R((m+3)n+l)×((m+3)n+l) . Pre- and postmultiplying to (5.49) by T2T and T2 , yields  ˜Q ˜ QT QW Ω˘ − W 

h ˜ T m WD − mh V



0, and W > 0 such that the following set of LMIs hold: EP = P T ET ≥ 0,  Ω˜ P 

h ˜ T m WD − mh V

(5.52)

 < 0.

(5.53)

When the above conditions are satisfied, a desired state-feedback controller is given by K = ZP −1 ,

(5.54)

where Ω˜ P

h T T h ˜ E + WdT U T WV WX Y WV − WXT Y T W m m h T T T ˜ T ˜ T −Wd U WE − WS WQ ) − WV (P + P − V )WV m T ˜ +WW W WW .

˜D+ = sym(WXT W

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Analysis and Synthesis of Singular Systems

Corollary 5.26. For a given γ > 0, the system in (5.14) is admissible with disturbance rejection of level γ if there exist matrices P , Z , U , Y , V > 0, and W > 0 such that the following set of LMIs hold: EP = P T ET ≥ 0, ⎡

Ω˜ ∞ ⎢ ⎣  

h ˜ T m WD − mh V



(5.55)

⎤

˜ QT W ⎥ 0 ⎦ < 0. −I

(5.56)

When the above conditions are satisfied, a desired state-feedback controller is given by K = ZP −1 ,

(5.57)

where Ω˜ ∞

h T T ˜E W Y WV − WXT Y T W m X h h ˜ E ) − WVT (P + P T − V )WV + WdT U T WV − WdT U T W m m T ˜ +WW W WW − γ 2 WST WS .

˜D+ = sym(WXT W

Remark 5.27. There are nonstrict inequality constraints in our results, such as ET P = P T E in (5.19) and EP = P T ET in (5.46), which will lead to numerical problems when solving such nonstrict LMIs. For this case, the methods in [192], [216], and [223] can be employed by setting P = Z1 E + Φ1 H1 and P = Z2 ET + Φ2 H2 , where Z1 > 0, Z2 > 0, H1 , H2 ∈ R(n−p)×n , and Φ1 , Φ2 ∈ Rn×(n−p) with rank(Φ1 ) = rank(Φ2 ) = n − p such that ET Φ1 = 0 and EΦ2 = 0.

5.2.4 Illustrative examples In this section, some examples are provided to illustrate the applicability and reduced conservatism of the proposed approach. Example 5.4. Consider a singular time-delay system in (5.14) with following parameters: 



E

=

C =









1 1 −2 0 −1.5 0.5 , A= , Ad = , 0 0 0 −3 0.4 −0.3

  −1 0.4

1

1









1 0 1 1.5 , Cd = 0, Bw = , Dw = . 2.5 0 3 2

Delay-dependent dissipativity analysis and synthesis of singular delay systems

131

Table 5.5 Allowable maximum time-delay h obtained by different methods. Methods [143] [123] m=1 m=2 m = 4 m = 10

h

delay-independent

infeasible

0.8501

0.9816

1.0152

1.0245

Table 5.6 Allowable maximum scalar α obtained by different methods. Methods [123] m=1 m=2 m=4 m = 10 α infeasible 0.3859 0.4462 0.4569 0.4597

The purpose is to find the maximal allowable time-delay h such that the singular system is admissible and strictly (Q, S, R)-α -dissipative. To this end, we choose 











−0.4 0 1.1 3 3 0 , Q= , R= , α = 0.2. S= 0.5 2 0 0 1 −1

Then, the maximal allowable time-delay h satisfying (5.19) and (5.20) can be calculated by using standard software. Table 5.5 presents the comparison, which shows that the result in Theorem 5.10 is less conservative than those in [143] and [123]. Furthermore, the conservatism is reduced as the number of partitions increases. On the other hand, to illustrate the dissipative results are also dependent on α , we give the different maximal allowed α in Table 5.6 with h = 0.7 for different partitioning size. From Table 5.6, we can see that the more we partition the time-delay, the larger α is. Example 5.5. To compare the delay-dependent BRL in Corollary 5.18 with existing results, we consider a singular system described by [214]: 



  −0.3012 0.1257 x(t) 0.2351 −2.5652     0.2102 −0.5124 0.9648 + x(t − h) + w (t) −0.8152 0.1023 0.8197   z(t) = 1.2321 0.3185 x(t).

1 0 x˙ (t) = 0 0

For a given γ > 0, we can calculate the maximum allowable time-delay h satisfying the LMIs in (5.39) and (5.40) by solving a quasiconvex optimization problem. Some comparison results between [6], [194], [196], [214], [246] and Corollary 5.18 in this secttion are presented in Table 5.7, which illustrates the reduced conservatism of our results.

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Analysis and Synthesis of Singular Systems

Table 5.7 Allowable maximum time-delay h obtained by different methods. γ 2 2.5 3 3.5

[6] [194] [196] [214] [246]

infeasible 2.6921 2.5920 2.5920 2.2465

infeasible 2.9586 2.9090 2.9090 2.5736

infeasible 3.1710 3.1522 3.1522 2.8348

infeasible 3.3443 3.3408 3.3408 3.0505

Corollary 5.18 m=1 2.5920 m=2 3.0673 3.1635 m=4

2.9090 3.3954 3.5083

3.1522 3.6582 3.7847

3.3408 3.8737 3.8737

Example 5.6. To illustrate the applicability of our controller design method, we consider the system in (5.14) with parameters as follows: 



E =



Bw =









2 1 0 1 0.5 0 , A= , Ad = , −1 1 2 1 0 0.5 





  −0.5 1 , B= , C = 0.8 0.8 , 0.5 0.5 



Dw = 0.1, D = 0.1, Cd = −0.2 0.5 . The purpose is to design a state feedback controller such that the closedloop singular system in (5.16) is admissible and strictly (Q, S, R)-α dissipative. We select S = 1, Q = −1, R = 10. For given time-delay h = 1, a state feedback controller can be calculated by solving the LMIs in (5.46) and (5.47). To illustrate the reduced conservatism of our result, Table 5.8 is given to compare the value of α for different m. Moreover, Fig. 5.1 and Fig. 5.2 give the simulation results of two states with w (t) = e−t sin t. Fig. 5.1 depicts the state response for open-loop system, whereas  state trajectories for the closed-loop system  Fig. 5.2 give the with K = 1.4248 1.4248 . From Fig. 5.1 and Fig. 5.2, we can see that the open-loop system is unstable and the closed-loop system is stable. Example 5.7. To compare Corollary 5.21 obtained by employing the delay-partitioning approach with Theorem 8 in [62], obtained by using

Delay-dependent dissipativity analysis and synthesis of singular delay systems

Figure 5.1 State trajectories of open-loop system.

Figure 5.2 State trajectories of closed-loop system.

133

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Analysis and Synthesis of Singular Systems

Table 5.8 Allowable maximum scalar α obtained for different m. m α K   1 8.0508 10.8299 5.4150   2 9.6380 6.5408 3.2704   3 9.8477 6.6569 3.3284   4 9.9255 6.9973 3.4987 Table 5.9 Comparison of maximum allowable time-delay h. Corollary 5.21 m=1 m=2 m = 10 m = 24

h

0.3111

0.3154

N =1 0.3166

Theorem 8 in [62]

h

0.3166 N =2 0.3167

0.3167 N =3 0.3167

discretized Lyapunov method, we consider a singular system described by 











−1 − 1 0.5 0 x(t) + x(t − h). 0 −1 1 0.5

1 0 x˙ (t) = 0 0

Let x¯ (t) = x1 (t), y(t) = x(t). The system considered in this example can be rewritten as 



x˙¯ (t) = 0.5x¯ (t) + −1 −1 y(t − h),  

y(t) =





1 0 0 x¯ (t) + y(t − h). 0 1 0.5

We can calculate the maximum allowable time-delay h satisfying the LMIs in (5.44) and (5.45) by solving a quasiconvex optimization problem. Some comparison results between [62] and Corollary 5.21 in this section are presented in Table 5.9, which illustrates that both the maximum allowable time-delay h obtained by the two different methods are very close to the analytical solution hmax = 0.3167. However, there are some disadvantages of the discretized Lyapunov method compared with the delay partitioning method:

Delay-dependent dissipativity analysis and synthesis of singular delay systems

135

1. The discretized Lyapunov method is effective in the analysis of delaydependent stability for time-delay systems, but it is not desirable for investigating synthesis problems due to the cross terms of AT P and AT Qi , (i = 1, . . . , N ) [52], [211]. 2. The results in [62] obtained by discretized Lyapunov method has not been shown to be monotonic with respect to the delay h. For this reason, it is only useful to test the stability of delay systems for a given time delay and is not easily employed to test the stability of delay systems on a given interval of delay. 3. It is difficult to extend the discretized Lyapunov method to singular delay systems. The results obtained in [62] were not developed for checking the admissibility of singular delay systems; the regularity and impulse free have to be checked separately.

5.2.5 Conclusion The problem of α -dissipative control for continuous-time singular systems with time-delay has been studied in this section. The delay-dependent conditions in terms of LMIs have been proposed for guaranteeing singular systems admissible and strictly (Q, S, R)-α -dissipative. Based on this result, a state-feedback controller has been designed to guarantee the closedloop system is admissible and strictly (Q, S, R)-α -dissipative. The results presented in this section are not only dependent on the delay, but also dependent on the dissipative margin α . Moreover, the results on H∞ control and passive control of singular systems with time-delay are unified in the proposed results. Finally, some examples are given to demonstrate the reduced conservatism and the applicability of our methods.

5.3 Robust reliable dissipative filtering for discrete delay singular systems This section is concerned with the problem of robust reliable dissipative filtering for uncertain discrete-time singular system with interval timevarying delay and sensor failures. The uncertainty and the sensor failures considered are polytopic uncertainty and varying in a given interval, respectively. The purpose is to design a filter such that the filtering error singular systems is regular, causal, asymptotically stable, and strictly (Q, S, R)-dissipative. By utilizing reciprocally convex approach, firstly, sufficient reliable dissipativity analysis condition is established in terms of LMIs for singular systems with time-varying delay and sensor failures. Based on

136

Analysis and Synthesis of Singular Systems

this criterion, the result is extended to uncertain singular systems with time-varying delay and sensor failures. Moreover, the reliable dissipative filter is designed in terms of LMIs for uncertain singular systems with timevarying delay and sensor failures. Finally, the effectiveness of the filter design method in this section is illustrated by numerical examples.

5.3.1 Problem statement Consider a class of linear discrete-time singular systems with time-varying delay described by ⎧ ⎪ Ex(k + 1) ⎪ ⎪ ⎨ y(k) ⎪ z (k) ⎪ ⎪ ⎩ x(k)

= = = =

Ax(k) + Ad x(k − d(k)) + Bw (k) Cx(k) + Cd x(k − d(k)) + Dw (k) Lx(k) + Ld x(k − d(k)) + Gw (k) φ(k), k = −d2 , −d2 + 1, . . . , 0,

(5.58)

where x(k) ∈ Rn is the state vector; y(k) ∈ Rm is the measured output; z(k) ∈ Rp represents the signal to be estimated; w (k) ∈ Rl is assumed to be an arbitrary noise belonging to l2 , and φ(k) is a known given initial condition sequence; d(k) is a time-varying delay satisfying 1 ≤d1 ≤ d(k) ≤ d2 < ∞, k = 1, 2, . . .

(5.59)

The system matrices A, Ad , B, C, Cd , D, L, Ld , and G with appropriate dimension belong to a convex polytopic set χ := (A, Ad , B, C , Cd , D, L , Ld , G) ∈ Ω,

where

 Ω := χ (λ) =

q  i=1

λi χ i ,

q 

(5.60)

 λi = 1, λi ≥ 0

i=1

and χi := (Ai , Adi , Bi , Ci , Cdi , Di , Li , Ldi , Gi ) ∈ Ω , i = 1, . . . , q denoting the ith vertex of the polyhedral domain Ω . In contrast with standard linear state-space systems with E = I, the matrix E ∈ Rn×n has rank(E) = r ≤ n. The singular system in (5.58) is assumed to be admissible over the whole domain Ω . Our purpose is to design a full order linear filter with sensor failures for the estimate of z(k): 

xˆ (k + 1) = Af xˆ (k) + Bf yF (k), xˆ (0) = 0 zˆ (k) = Cf xˆ (k) + Df yF (k),

(5.61)

Delay-dependent dissipativity analysis and synthesis of singular delay systems

137

where Af , Bf , Cf , and Df are the filter gains to be determined, and yF (k) = T yF1 (k), . . . , yFm (k) denotes the signal from the sensor that may be faulty. The following failure model from [110] is adopted here: 

yFi (k) = αsi yi (k), i = 1, 2, . . . , m, where 0 ≤ α si ≤ αsi ≤ α¯ si , i = 1, 2, . . . , m with 0 ≤ α si ≤ α¯ si ≤ 1, in which the variables αsi quantify the failures of the sensors. Then we have yF (k) = As y(k), As = diag{αs1 , αs2 , . . . , αsm }. Remark 5.28. In the model mentioned above, when α si = α¯ si = 1, it corresponds to the normal fully operating case, that is, yFi = yi ; when α si = 0, then it covers the outage case in [167]; when αai = 0 and αai = 1, then it corresponds to the case, where the intensity of the feedback signal from sensor may vary. Denote A¯ s

= diag{α¯ s1 , α¯ as2 , . . . , α¯ sm },

As

= diag{α s1 , α s2 , . . . , α sm },

As0 B

s

where αs0i =

α si +α¯ si

2

= diag{αs01 , αs02 , . . . , αs0m }, = diag{β1 , β2 , . . . , βm }, = diag{s1 , s2 , . . . , sm },

and βi =

α¯ si −α si

2

. Then we have

As = As0 + s , |si | ≤ βi .

Let the augmented state vector x˜ (k) = [xT (k) xˆ T (k)]T and z˜ (k) = z(k) − zˆ (k). Then the filtering error singular system is described as ⎧ ˜ ˜ ˜ ˜ ⎪ ⎪ ⎨ Ex˜ (k + 1) = Ax˜ (k) + Ad Φ x˜ (k − d(k)) + Bw (k) ˜ (k) z˜ (k) = L˜ x˜ (k) + L˜ d Φ x˜ (k − d(k)) + Gw ⎪ ⎪ ⎩ T T x˜ (k) = [φ (k) 0] , k = −d2 , −d2 + 1, . . . , 0,

(5.62)

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Analysis and Synthesis of Singular Systems

where Φ = [I 0], and 

E˜

=

L˜ =

















E 0 A 0 Ad B ˜ = ˜d= , A , A , B˜ = , 0 I Bf As C Af Bf As D Bf As Cd 

˜ = G − Df As D . L − Df As C −Cf , L˜ d = Ld − Df As Cd , G

Before moving on, we give some definitions and lemmas concerning the following system: Ex(k + 1) = Ax(k) + Ad x(k − d(k)).

(5.63)

Definition 5.29. [209] 1. The singular delay system in (5.63) is said to be regular and causal if det(sE − A) is not identically zero. 2. The singular delay system in (5.63) is said to be causal if deg {det(sE − A)} = rank E. 3. The singular delay system in (5.63) is said to be asymptotically stable if, for any ε > 0, there exists a scalar δ(ε) > 0 such that for any consistent initial conditions φ(k) satisfying sup−d2 ≤k≤−1 φ(k) ≤ δ(ε), the solution x(k) of (5.63) satisfies x(k) ≤ ε for k ≥ 0; furthermore, x(k) → 0 when k → ∞. 4. The singular delay system in (5.63) is said to be admissible if the system is regular, causal, and asymptotically stable. Lemma 5.30. [208] The discrete-time singular system Ex(k + 1) = Ax(k) is admissible if and only if there exist matrices P > 0 and Q such that AT PA − ET PE + sym(AT SQT ) < 0, where S ∈ Rn×(n−r ) is any matrix with full column and satisfies ET S = 0. Lemma 5.31. [7] For any matrices U and V > 0, the following inequality holds UV −1 U T ≥U + U T − V . The purpose of this section is to design a filter in the form of (5.61) such that the filter error system in (5.62) is robustly admissible and strictly (Q, S, R)-dissipative.

139

Delay-dependent dissipativity analysis and synthesis of singular delay systems

5.3.2 Reliable dissipativity analysis In this subsection, we first give the result of reliable dissipative analysis for system (5.62) with fixed system matrix χ . Theorem 5.32. Let matrices Q, S, and R be given with Q and R real symmetric and Q ≤ 0. Then the system in (5.62) with sensor failure is admissible and strictly ¯ i > 0, i = 1, 2, 3, Sj > 0, (Q, S, R)-dissipative if there exist matrices P > 0, Q j = 1, 2, M, and Z˜ such that the following LMIs hold: 



S2 M ≥ 0,  S2

⎡

˜d ˜ TU ˜ TA 0 Π˜ 11 Φ TETS1E Z TM TE ⎢  ˜ 22 ˜ 23 Π E Π ⎢

⎢ ⎢ ⎢ ⎢ Π˜ = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣





Π˜ 33







    

    

    

Π˜ 34 Π˜ 44     

(5.64)

Π˜ 15

Π˜ 16S1

Π˜ 17S2

0

0

0

1

⎤

˜ TP L˜ TQ−2 A 0 0 ⎥ ⎥

⎥ ˜ TS1 d˜ A ˜ TS2 L˜ TQ 2 A ˜ TP ⎥ −L˜ dTS d1A d d d − d ⎥ 1

0

0

Π˜ 55    

0

0 ⎥

0

⎥ < 0,

˜ TQ 2 B˜ TP ⎥ d1B˜ TS1 d˜B˜ TS2 G − ⎥ −S 1 0 0 0 ⎥ ⎥  −S 2 0 0 ⎥ ⎦   −I 0    −P 1

(5.65) where Π˜ 11

˜ TU ˜Z ˜ ) − Φ T ET S1 EΦ, = −E˜ T P E˜ + Q1 + Q2 + (d˜ + 1)Q3 + sym(A

Π˜ 22

¯ 1 − ET (S1 + S2 )E, Π˜ 23 = ET (S2 − M T )E, = −Q

Π˜ 33

¯ 3 + sym(ET (M − S2 )E), Π˜ 34 = ET (S2 − M T )E, = −Q

Π˜ 44

¯ 2 − ET S2 E, Π˜ 15 = Z ˜ T B˜ − L˜ T S, ˜ TU = −Q

Π˜ 55

˜ − R, Π˜ 16 = d1 Φ T (A − E)T , Π˜ 17 = d˜ Φ T (A − E)T , ˜ T S − ST G = −G

¯ i, Q ¯ i }, i = 1, 2, 3, d˜ = d2 − d1, Qi = diag{Q

and U˜ ∈ R2n×(2n−r ) is any matrix with full column rank satisfying E˜ T U˜ = 0. Proof. First, we prove that the system in (5.62) is regular and causal. From the LMI in (5.66), we have  Π˜ 11 



˜ TP A < 0, −P

which implies the following inequality holds by using Schur complement equivalence:

140

Analysis and Synthesis of Singular Systems



Letting S˜ =

S1 0 0 0

˜ < 0. ˜ T PA Π˜ 11 + A



(5.66)

˜ it follows from and noting Φ T ET S1 EΦ = E˜ T S˜ E,

(5.66) that ˜ T PA ˜ − E˜ T (P + S˜ )E˜ + sym(A ˜ TU ˜Z ˜ T ) < 0. A

Combining Lemma 5.30 and Definition 5.29, the regularity and causality of the system in (5.62) are guaranteed. For the stability property, let us define η( ˜ k) = x˜ (k + 1) − x˜ (k) and choose the following Lyapunov functional: V (k) = V1 (k) + V2 (k) + V3 (k) + V4 (k) + V5 (k),

(5.67)

where V1 (k) = x˜ T (k)E˜ T P E˜ x˜ (k), V2 (k) =

k−1 2  

x˜ T (i)Qi x˜ (i),

j=1 i=k−dj

V3 (k) =

k−1 

x˜ T (i)Q3 x˜ (i) +

i=k−d(k)

V4 (k) =

−1  k−1 

−d1 

k−1 

x˜ T (i)Q3 x˜ (i),

j=−d2 +1 i=k+j

d1 η˜ T (i)E˜ T Φ T S1 Φ E˜ η( ˜ i),

j=−d1 i=k+j

V5 (k) =

− d1 −1  k−1

d˜ η˜ T (i)E˜ T Φ T S2 Φ E˜ η( ˜ i).

j=−d2 i=k+j

Calculate the forward difference of V (k) along the trajectories of filtering error singular system (5.62) with w (k) = 0 yields V1 (k) = x˜ T (k + 1)E˜ T P E˜ x˜ (k + 1) − x˜ T (k)E˜ T P E˜ x˜ (k) ˜ x˜ (k) + A ˜ d Φ x˜ (k − d(k)))TP(A ˜ x˜ (k) + A ˜ d Φ x˜ (k − d(k))) = (A − x˜ T (k)E˜ TPE˜ x˜ (k), V2 (k) =

2 

x˜ T (k)Qj x˜ (k) −

j=1

≤

2  j=1

(5.68) 2 

x˜ (k − dj )Qj x˜ (k − dj )

j=1

x˜ T (k)Qj x˜ (k) −

2  j=1

¯ j Φ x˜ (k − dj ), x˜ (k − dj )Φ T Q

(5.69)

141

Delay-dependent dissipativity analysis and synthesis of singular delay systems

V3 (k) = (d˜ + 1)˜xT (k)Q3 x˜ (k) − x˜ T (k − d(k))Q3 x˜ (k − d(k)) +

k−1 

x˜ T (i)Q3 x˜ (i) −

i=k+1−d(k+1)

−

k −d1

k−1 

x˜ T (i)Q3 x˜ (i)

i=k+1−d(k)

x˜ T (i)Q3 x˜ (i)

i=k−d2 +1

= (d˜ + 1)˜xT (k)Q3 x˜ (k) − x˜ T (k − d(k))Q3 x˜ (k − d(k)) +

k−1 

x˜ (i)Q3 x˜ (i) +

i=k+1−d1

−

k −d1

T

k−1 

x˜ T (i)Q3 x˜ (i)

i=k+1−d(k+1) k −d1

x˜ (i)Q3 x˜ (i) − T

i=k+1−d(k)

x˜ T (i)Q3 x˜ (i)

i=k−d2 +1

≤

(d˜ + 1)˜x (k)Q3 x˜ (k) − x˜ (k − d(k))Q3 x˜ (k − d(k))

≤

¯ 3 Φ x˜ (k − d(k)). (d˜ + 1)˜xT (k)Q3 x˜ (k) − x˜ T (k − d(k))Φ T Q

T

T

(5.70) By using Lemma 4.5, we have ˜ k) − d 1 V4 (k) = d12 η˜ T (k)E˜ T Φ T S1 Φ E˜ η( ≤

k−1 

η˜ T (i)E˜ T Φ T S1 Φ E˜ η( ˜ i)

i=k−d1 2 d1 ((A − E)Φ x˜ (k) + Ad Φ x˜ (k − d(k)))T S1 ((A − E)Φ x˜ (k)

+Ad Φ x˜ (k − d(k))) − (EΦ x˜ (k) − EΦ x˜ (k − d1 ))T S1 (EΦ x˜ (k) −EΦ x˜ (k − d1 )). 

(5.71)



S M ≥ 0, the following inequality holds: Since 2  S2 ⎡

α1 (Ex(k − d(k)) − Ex(k − d2 )) α2

⎤T 

S2 M  S2

⎣  ⎦ − αα12 (Ex(k − d1 ) − Ex(k − d(k)) ⎤ ⎡ α1 ( Ex ( k − d ( k )) − Ex ( k − d )) 2 α ⎦ ≥ 0, × ⎣ 2α − α21 (Ex(k − d1 ) − Ex(k − d(k))



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Analysis and Synthesis of Singular Systems

where α1 = d2 −d˜d(k) , α2 = d(k) < d2 , we have

d(k)−d1 . d˜

Then employing Lemma 4.4, for d1
V (T + 1), and there exists a sufficiently small α > 0 such that J (T ) ≥ α

T 

w T (k)w (k).

k=0

By Definition 2.14, the system in (5.62) is strictly (Q, S, R)-dissipative. This completes the proof. Remark 5.33. The result in Theorem 5.32 is obtained using the reciprocally convex approach proposed in [140]. The method has led to less conservative stability criterion for continuous-time system with time-varying delay than that in [151]. It is extended to solve the filtering problem for discrete-time singular system with time-varying delay in this section. Remark 5.34. The reduction of conservatism of the approach used in this section, due to utilizing the reciprocally convex approach, which bounds  the integral term − ik=−kd−1 −d21 (d2 − d1 )ηT (i)Sη(i) with d1 ≤ d(k) ≤ d2 , η(i) =

145

Delay-dependent dissipativity analysis and synthesis of singular delay systems

x(i + 1) − x(i) by the term 

x(k − d(k)) − x(k − d1 ) − x(k − d(k)) − x(k − d2 ) 

T 

ET SE ET ZE  ET SE





x(k − d(k)) − x(k − d1 ) x(k − d(t)) − x(k − d2 )



S Z ≥ 0. When Z = 0, it reduces to the Jensen inequality method with  S used in [36], [239]. Moreover, as explained in [140], the reciprocally convex approach directly deals with the inversely weighted convex combination of quadratic terms of integral quantities, rather than approximating the difference between delay bounds using a convex combination approach in [150]. Based on Theorem 5.32, we give the following by-product condition for the admissibility of the delay singular system in (5.63): Corollary 5.35. Given integers 1 ≤d1 ≤ d2 , system (5.63) with time-varying delay d(k) satisfying (5.59) is admissible if there exist matrices P > 0, Qi > 0, i = 1, 2, 3, Sj > 0, j = 1, 2, M, and Z such that the following LMIs hold: 



S2 M ≥ 0,  S2

⎡ Π11 ⎢  ⎢ ⎢  ⎢ ⎢ Π =⎢  ⎢ ⎢  ⎢ ⎣  

ET S1 E Z T U T Ad Π22     

Π23 Π33    

0 ET M T E Π34 Π44   

(5.75) Π16 S1

Π17 S2

0 0 T ˜ d1 Ad S1 dATd S2 0 0 −S1 0  −S2 



⎤

AT P 0 ⎥ ⎥ ATd P ⎥ ⎥ ⎥ 0 ⎥ < 0, ⎥ 0 ⎥ ⎥ 0 ⎦ −P (5.76)

where Π11

= −ET PE + Q1 + Q2 + (d˜ + 1)Q3 + sym(AT UZ ) − ET S1 E,

Π22

= −ET S2 E − Q1 − ET S1 E, Π23 = ET S2 E − ET M T E,

Π33

= −Q3 − 2ET S2 E + sym(ET ME), Π34 = ET S2 E − ET M T E, = −Q2 − ET S2 E, Π˜ 16 = d1 (A − E)T , Π˜ 17 = d˜ (A − E)T ,

Π44

and U ∈ Rn×(n−r ) is any matrix with full column rank satisfying ET U = 0. Proof. Choose a Lyapunov functional candidate as follows: V˜ (k) = V˜ 1 (k) + V˜ 2 (k) + V˜ 3 (k) + V˜ 4 (k) + V˜ 5 (k),

146

Analysis and Synthesis of Singular Systems

where V˜ 1 (k) = x(k)ET PEx(k), V˜ 2 (k) =

k−1 2  

xT (i)Qi x(i),

j=1 i=k−dj

V˜ 3 (k) =

k−1 

x (i)Q3 x(i) + T

i=k−d(k)

V˜ 4 (k) =

−d1 

k−1 

xT (i)Q3 x(i),

j=−d2 +1 i=k+j

−1  k−1 

d1 ηT (i)ET S1 Eη(i),

j=−d1 i=k+j

V˜ 5 (k) =

− d1 −1  k−1

d˜ ηT (i)ET S2 Eη(i),

j=−d2 i=k+j

η(k) = x(k + 1) − x(k).

Then following a similar line as in the proof of Theorem 5.32 yields Corollary 5.35. Similar to Theorem 2 in [56], a less conservative condition of LMI (5.65) is obtained by introducing three slack matrices H1 , H2 , and T, which is presented in the following theorem: Theorem 5.36. Let matrices Q, S, and R be given with Q and R real symmetric and Q ≤ 0. Then the system in (5.62) with sensor failure is admissible and strictly ¯ i > 0, i = 1, 2, 3, Sj > 0, (Q, S, R)-dissipative if there exist matrices P > 0, Q ˜ Hj , j = 1, 2, T, M, and Z such that the following LMIs hold:  ⎡

˜d ˜ TU ˜ TA 0 Π˜ 11 Φ TE TS1E Z Π˜ 15 Π˜ 22 ET M T E 0 Π˜ 23 ⎢ 

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣



S2 M ≥ 0,  S2

 

 

Π˜ 33 

Π˜ 34 Π˜ 44

    

    

    

    

−L˜ dT S

0

Π˜ 16H1T

(5.77) Π˜ 17H2T

1

˜ TT T A 0

1

˜ T TT A d 0

0

0

L˜ TQ−2 0

˜ T HT d1A d 1 0

˜ T HT d˜ A d 2 0

L˜ dT Q−2 0 1

˜ TQ2 Π˜ 55 d1B˜ T H1T d˜ B˜ T H2T G B˜ T T T −  S1 − H1T − H1 0 0 0   S2 − H2T − H2 0 0    −I 0     P − TT − T

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎦

(5.78) where Π˜ ii , i = 1, . . . , 5, Π˜ 15 , Π˜ 16 , Π˜ 17 , Π˜ 23 , Π˜ 34 , and U˜ are defined in (5.66).

147

Delay-dependent dissipativity analysis and synthesis of singular delay systems

Proof. If (5.65) holds, then there exist Hj = HjT = Sj , j = 1, 2, and T = T T = P such that (5.78) holds. On the other hand if (5.78) holds, we have the following inequality based on Lemma 5.31: ⎡

˜d ˜ TU ˜ TA Π˜ 11 Φ TE TS1E Z Π˜ 15 Π˜ 16 H1T 0 ⎢  T T ˜ ˜ Π22 0 Π23 E M E 0 ⎢ ⎢ ˜ T HT ⎢   Π˜ 33 Π˜ 34 −L˜ dT S d1 A d 1 ⎢   Π˜ 44 0 0 ⎢ 

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

    

    

    

    

⎤

1

˜ T TT L˜ T Q−2 A ⎥ 0 0 ⎥

Π˜ 17 H2T

0

⎥

1

˜ T H T L˜ T Q 2 A ˜ T TT ⎥ d˜ A d 2 d − d ⎥ 0 0 0 ⎥

⎥

˜ T Q 2 B˜ T T T ⎥ < 0. Π˜ 55 d1 B˜ T H1T d˜ B˜ T H2T G − ⎥ ⎥  −H1 S1−1 H1T 0 0 0 ⎥ ⎥   −H2 S2−1 H2T 0 0 ⎦    −I 0 − 1 T     −TP T 1

(5.79) In addition, matrices Hj , j = 1, 2, and T are nonsingular due to Sj − HjT − Hj < 0, j = 1, 2, and P − T T − T < 0. Then, pre- and postmultiplying (5.79) by diag{I , I , I , I , I , S1 H1−1 , S2 H2−1 , PT −1 } and its transpose yield (5.65). Therefore the equivalence between (5.78) and (5.65) is proved. Then a sufficient condition for robust admissibility and strict (Q, S, R)dissipativity of uncertain singular system (5.62) can be derived by using a parameter-dependent Lyapunov functional. Theorem 5.37. Let matrices Q, S, and R be given with Q and R real symmetric and Q ≤ 0. Then the system in (5.62) with sensor failure and polytopic uncertainty (5.60) is robustly admissible and strictly (Q, S, R)-dissipative if there exist matrices ¯ li > 0, l = 1, 2, 3, Sji > 0, Hj , j = 1, 2, T, Mi , and Z ˜ such that the Pi > 0, Q following set of LMIs hold for i = 1, . . . , q: 



S2i Mi ≥ 0,  S2i

⎡

0 Π˜ 11i Π˜ 12i Π˜ 13i ⎢  Π˜ 22i Π˜ 23i ET M T E ⎢ i

⎢ ⎢ ⎢ ⎢ Ω˜ i = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

      

      

Π˜ 33i      

Π˜ 34i Π˜ 44i     

Π˜ 15i

Π˜ 16i H1T

(5.80) Π˜ 17i H2T

0

0

0

TS −L˜ di

˜ T HT d1 A di 1

˜ T HT d˜ A di

2

0

0

0

Π˜ 55i    

d1 B˜ iT H1T Π˜ 66i

d˜ B˜ T H T

  

i

2

0

Π˜ 77i  

⎤

1

˜ T TT L˜ iT Q−2 A i 0 0 ⎥ ⎥

⎥

1

˜ T TT ⎥ L˜ diT Q−2 A di ⎥ 0 0 ⎥

⎥ < 0,

˜ T Q 2 B˜ T T T ⎥ G − i i ⎥ 0 0 ⎥ ⎥ 0 0 ⎥ −I 0 ⎦ 1



Π˜ 99i

(5.81)

148

Analysis and Synthesis of Singular Systems

where ˜ TU ˜Z ˜ ) − Φ T ET S1i EΦ, Π˜ 11i = −E˜ T Pi E˜ + Q1i + Q2i + (d˜ + 1)Q3i + sym(A i ˜ di , ˜ TA ˜ TU Π˜ 12i = Φ T ET S1i E, Π˜ 13i = Z ¯ 1i − ET S1i E, Π˜ 23i = ET S2i E − ET M T E, Π˜ 22i = −ET S2i E − Q i

¯ 3i − 2ET S2i E + sym(ET Mi E), Π˜ 34 = ET S2i E − ET M T E, Π˜ 33i = −Q i T T ˜T˜ T ¯ ˜ ˜ ˜ ˜ Π44i = −Q2i − E S2i E, Π15i = Z U Bi − Li S, ˜ i − R, Π˜ 16i = d1 Φ T (Ai − E)T , Π˜ 17i = d˜ Φ T (Ai − E)T , ˜ T S − ST G Π˜ 55i = −G i Π˜ 66i = S1i − H1T − H1 , Π˜ 77i = S2i − H2T − H2 , Π˜ 99i = Pi − T T − T , ¯ li , Q ¯ li }, l = 1, 2, 3, d˜ = d2 − d1 , Qli = diag{Q

and U˜ ∈ R2n×(2n−r ) is any matrix with full column rank satisfying E˜ T U˜ = 0. ¯ l , l = 1, 2, 3, Sj , j = 1, 2, and M be expressed as Proof. Let P, Q #

P

¯l Q

$

Sj M =

q 

# λi Pi

¯ li Q

$

Sji Mi ,

i=1

where λi is defined in (5.60). By multiplying conditions (5.80), (5.81) with λi and summing up from 1 to q, we obtain (5.77) and (5.78), respectively.

5.3.3 Filter design Now, our attention will be devoted to design a filter in the form of (5.61) such that filtering error system (5.62) subject to possible sensor failures is admissible and strictly (Q, S, R)-dissipative. Based on the result of Theorem 5.36, the reliable dissipative filter design method for singular system (5.58) is presented in the following theorem: Theorem 5.38. Let matrices Q, S, and R be given with Q and R real symmetric and Q ≤ 0. Then the system in (5.62) with sensor failure and polytopic uncertainty (5.60) is robustly admissible and strictly (Q, S, R)-dissipative if there exist matrices   P1i P2i ˜ li > 0, Q ¯ li > 0, l = 1, 2, 3, Sji > 0, Hj , Fj , j = 1, 2, diagonal > 0, Q  P3i matrix N > 0, T1 , Mi , and Z such that the following set of LMIs hold for i = 1, . . . , q: 



S2i Mi ≥ 0,  S2i

(5.82)

149

Delay-dependent dissipativity analysis and synthesis of singular delay systems

 Γi < 0, Λi

 Ξi Ωi = 

(5.83)

where ⎡

Ξ11i −ET P2i ET S1i E Z T U T Adi + CiT NCdi

⎢ ⎢ ⎢ Ξi = ⎢ ⎢ ⎣ ⎡



0

Ξ22i

   

   

Γ11i

0

Γ12i

Γ13i

Ξ16i C¯ T S⎥

⎥

f

ET MiT E 0 ⎥ ⎥, T T E (S2i − Mi )E Ξ46i ⎥ ⎦ Ξ55i 0

− MiT )E Ξ44i  

ET (S2i

Ξ33i   

⎤

0 0



Ξ66i

ATi T1T + CiT As0 B¯ fT ATi F1T + CiT As0 B¯ fT

1 ⎢ ¯T ¯T C¯ fT Q−2 A A 0 ⎢ 0 f f ⎢ ⎢ 0 0 0 0 0 Γi = ⎢ T T ˜ T HT ⎢d1 Adi H1 dA Γ43i ATdi T1T + CdiT As0 B¯ fT ATdi F1T + CdiT As0 B¯ fT di 2 ⎢ ⎣ 0 0 0 0 0

⎡

˜ T HT d1 BiT H1T dB i 2

S1i − H1 − H1T

⎢ ⎢ ⎢ Λi = ⎢ ⎢ ⎢ ⎣



Γ63i

BiT T1T + DiT As0 B¯ fT BiT F1T + DiT As0 B¯ fT

0 S2i − H2 − H2T

0 0





−I







 

 

 

0 0

0 0

0 0 P1i − T1 − T1T P2i − F2 − F1T  P3i − F2 − F2T 



0

⎤ ⎥

0 ⎥ ⎥ 0 ⎥ ⎥, 0 ⎥ ⎥ 0 ⎦ ¯fB ST D 0 0

⎤

⎥ ⎥ ¯ f B⎥ −Q− D ⎥, B¯ f B ⎥ ⎥ B¯ B ⎦ 1 2

f

−N

¯ 1i + Q ¯ 2i + (d˜ + 1)Q ¯ 3i + sym(AT UZ )+ C T NCi , Ξ11i = −ET (P1i + S1i )E + Q i i ˜ 1i + Q ˜ 2i + (d˜ + 1)Q ˜ 3i , Ξ22i = −P3i + Q ¯ 1i − ET S1i E, Ξ33i = −ET S2i E − Q ¯ 3i + sym(ET (Mi − S2i )E ) + C T NCdi , Ξ44i = −Q di 1

¯ 2i − ET S2i E, Γ63i = (Gi − D ¯ f As0 Di )T Q−2 , Ξ55i = −Q ¯ f As0 Di ) + DiT NDi , Ξ66i = sym(−GiT S − R + ST D ¯ f S + CiT NDi , Ξ16i = Z T U T Bi − LiT S + CiT As0 D ¯ fT S + CdiT NDi , Ξ46i = −LdiT S + CdiT As0 D Γ11i = d1 (Ai − E)T H1T , Γ12i = d˜ (Ai − E)T H2T , 1

1

¯ f As0 Ci )T Q−2 , Γ43i = (Ldi − D ¯ f As0 Cdi )T Q−2 , Γ13i = (Li − D

150

Analysis and Synthesis of Singular Systems

where U ∈ Rn×(n−r ) is any matrix with full column rank satisfying ET U = 0. Moreover, a suitable robustly reliable dissipative filter is given by ¯ f F −1 , Bf = B¯ f , Cf = C ¯ f F −1 , Df = D ¯f. Af = A 2 2

(5.84)

Proof. By using Schur complement equivalence, (5.83) is equivalent to  Ξ¯ i

 Γ¯i + Ψ1iT N Ψ1i + Ψ2T B N −1 B Ψ2 < 0, Λ¯ i



(5.85)

where Ξ¯ i is Ξi in (5.83) with the terms, including N disappear; Γ¯i is Γi in (5.83) without the last column; Λ¯ i is Λi in (5.83) without the last row and last column, and Ψ1i

=

Ψ2i

=





Ci

0

0

Cdi

0



0

0

0

0

Di

¯ TS D f

0

0

0

0

0

0 1

0

0

¯ T Q−2 −D f

,

B¯ fT

B¯ fT

 .

Then, using the elementary inequality xT y + yT x ≤ εxT x + ε−1 yT y, we have Ω¯ i

 Ξ¯ i = 

 T

Γ¯i + sym  Ψ Ψ < 0. s 2 1i Λ¯ i

(5.86)

Introduce four matrices T1 , T2 , T3 , and T4 with T4 invertible and define 

J1 =

I 0



0 , F1 = T2 T4−1 T3 , F2 = T2 T4−T T2T , T2 T4−1

˜ i = T2 T −1 Q ¯ i T −T T2 , J = sym{J1 , I , I , I , I , I , I , I , J1 }, Q 4 4  

U˜ =

 U ˜ = Z , Z 0



T 

¯f A C¯ f

B¯ f ¯f D

=



T1 T3

 =

T2 0

 0 ,



P1i 



P2i = J1 Pi J1T , P3i



T2 , T4 0 I



Af Cf

Bf Df



T4−T T2T 0



0 . I

(5.87)

From (5.83), we have F2 + F2T = T2 T4−T T2T + T2 T4−1 T2T > 0, which implies that T2 is nonsingular. Hence, J is nonsingular. Noting that, Ω¯ i = J Ω˜ i J T ,

Delay-dependent dissipativity analysis and synthesis of singular delay systems

151

we have Ω˜ i < 0, and the filtering error system in (5.62) is robustly admissible and strictly (Q, S, R)-dissipative from (5.81). Because T2 and T4 cannot be obtained from (5.83), we cannot determine the filters from (5.87). From (5.87), we have Af Bf Cf Df

¯ f F −1 T2 , = T2−1 A 2 −1 ¯ = T2 Bf , ¯ f F −1 T −1 , = C 2 2 ¯ = Df .

¯ f F −1 , B¯ f , C ¯ f F −1 , D ¯ f ) are algeHence, the systems (Af , Bf , Cf , Df ) and (A 2 2 braically equivalent, and the desired filter can be obtained from (5.84). This completes the proof.

Remark 5.39. The number of LMI decision variables in Theorem 5.38 is 7(q + 1)n2 + (5q + p + m − r )n + (p + 1)m, where n, m, q, p, and r are defined in (5.58). If the delay-partitioning method is employed, the number of decision variables are dependent not only on system dimensions, but also the partitioning number l. The computational burden will be significantly increased with the growth of l. Consequently, a longer central processing unit time will be consumed when testing the criterion. Therefore the reciprocal convex approach reduces the conservatism with less variables, whereas the delay-partitioning method improves the result by costing more computer processing unit time. The reciprocal convex approach can be seen as a method striking a balance between time-consumption and reduction of conservatism. Based on Theorem 5.36, the following by-product condition can be easily obtained for the dissipative filtering of the delay singular system in (5.63) without sensor failures. Corollary 5.40. Let matrices Q, S, and R be given with Q and R real symmetric and Q ≤ 0. Then the system in (5.62) with As = I and polytopic uncertainty (5.60) is robustly admissible and strictly (Q, S, R)-dissipative if there exist matrices   P1i P2i ˜ li > 0, Q ¯ li > 0, l = 1, 2, 3, Sji > 0, Hj , Fj , j = 1, 2, T1 , Mi , > 0, Q  P3i and Z such that the following set of LMIs hold for i = 1, . . . , q: 

S2i 



Mi ≥ 0, S2i

(5.88)

152

Analysis and Synthesis of Singular Systems

 Ξ¯ i Ωi = 

 Γ¯i < 0, Λ¯ i

(5.89)

where Ξ¯ i , Γ¯i , and Λ¯ i are defined in (5.85). Remark 5.41. The reliably dissipative filtering result in Theorem 5.38 includes reliable H∞ filtering and passivity filtering as special cases: 1. When Q = −I, S = 0, R = γ 2 I, it reduces to the reliable H∞ filtering case, such as [111], [110]. 2. When Q = 0, S = I, R = 0, and As = 1, it reduces to the passivity filtering case without considering reliability, such as [1], [35], [65].

5.3.4 Illustrative examples In this section, three examples are provided to illustrate the effectiveness of the proposed approach. The examples chosen here have been widely used in existing references [78], [150], [241], and the reduced conservatism can be easily illustrated and compared. Example 5.8. To illustrate the reduced conservatism of the robust H∞ filtering result and the applicability of the filter design method, consider a delay singular system in (5.58) with parameters borrowed from [78]: 

1 0

E =





0 0.9 , A= 0 0

 

 0 , C= 1 1

B = Ld = •



0.5



0 0.7 + 1 

1 , Cd = 0.2



 −0.1 , Ad = −0.1 



0.5 , L = 1

 2 , − 0.1 

2 ,



0.6 , D = 1, G = −0.5, | 1 |≤ 0.2, | 2 |≤ 0.1.

H∞ case: Q = −I, S = 0, R = γ 2 I, As = I. To compare with the H∞ result in [78], the detailed comparisons on the minima of γ for given d1 and d2 are listed in Table 5.10. From Table 5.10, we can see that the minimum values of the H∞ performance index obtained from [78] and Corollary 5.40 are 4.9515 and 4.7221, with d1 = 1 and d2 = 5, Table 5.10 Comparisons of minimum allowed γ for d2 = 5. Methods \ d1 1 2 3 4 5 [78] 4.9515 4.5463 4.2144 3.9073 3.5960 Corollary 5.40 4.7221 4.3631 4.0815 3.8359 3.5960

Delay-dependent dissipativity analysis and synthesis of singular delay systems

153

respectively. When d1 = 3, d2 = 5, and γ = 4.0815, a feasible solution can be found by applying Corollary 5.40, whereas it fails to do that by using the results in [78]. It is clearly seen that the minimum γ value obtained in this section is less than that in [78] when the time delays d1 and d2 are the same. Moreover, when d1 = 1, d2 = 5, the corresponding robust filter is 

•

Af

=

Cf

=

0.2470 6.4106



0.3723

   − 0.0006 −0.0218 , Bf = × 10−3 0.6068 − 0.0158  0.0002 , Df = 1.5823.

General dissipative case: Q = −0.25, S = −2, R = 4, As = 1, d1 = 2, d2 = 6. By solving the LMIs in Corollary 5.40, the corresponding robust filter is 

Af

=

Cf

=

0.5884 0.0861



0.1701

   − 0.0001 0.0156 , Bf = , − 0.0002 −0.0000  − 0.0002 , Df = 0.9548.

(5.90)

The estimate error z˜ of all vertices of the polytopic system is given in Fig. 5.3.

Figure 5.3 The estimate error z˜ of all vertices of the polytopic system.

154

•

Analysis and Synthesis of Singular Systems

Reliably dissipative case: Q = −0.25, S = −2, R = 4, As = 0.4, A¯ s = 0.6, d1 = 2, d2 = 6. By solving the LMIs in Theorem 5.38, the corresponding robust reliability filter is Af

=

Cf

=













0.5060 0.0009 0.0066 , Bf = , 0.4604 0.0011 0.0386

0.3256 0.0008 , Df = 1.9694.

(5.91)

The estimate error z˜ of all vertices of the polytopic system with the filter in (5.91) for different sensor failure cases is given in Figs. 5.4–5.6, respectively. Example 5.9. To show the generality of the dissipative filtering, some system matrices in Example 5.9 are changed as follows: D = −1, G = 1. •

(5.92)

H∞ case: Q = −γ −1 I, S = 0, R = γ I, As = 1, d1 = 2, and d2 = 6. By solving the LMIs in Corollary 5.40 with γ being 5 and the correspond-

Figure 5.4 As = 0.4.

Delay-dependent dissipativity analysis and synthesis of singular delay systems

155

Figure 5.5 As = 0.5.

Figure 5.6 As = 0.6.

ing robust filter is 

Af

=

Cf

=







0.5203 0.6648

0.0011 0.0053 , Bf = , 0.0013 0.0132

2.4475

0.0020 , Df = 1.3850.





(5.93)

156

•

Analysis and Synthesis of Singular Systems

Passive case: Q = 0, S = I, R = 0, As = 1, d1 = 2, and d2 = 6. By solving the LMIs in Corollary 5.40, the corresponding robust filter is 

•

Af

=

Cf

=



0.3823 −0.1285

0.4744

   −0.1832 − 0.0013 , Bf = , − 0.0007 −0.2188  − 0.0017 , Df = 21.8321.

(5.94)

Dissipative case: Q = −γ −1 θ I, S = (1 − θ )I, R = γ θ I, θ ∈ (0, 1), As = 1, d1 = 2, and d2 = 6. It can be seen that it is the trade-off between H∞ and passivity performance with θ ∈ (0, 1). By solving the LMIs in Corollary 5.40 with θ = 0.5 and γ = 3.8565, the corresponding robust dissipative filter is 

Af

=

Cf

=









0.5115 0.1282

0.0007 0.0051 , Bf = , 0.0005 0.0074

3.3265

0.0001 , Df = 1.8731.



(5.95)

Example 5.10. To show the reduction of conservatism for the stability criterion, we consider the state-space system borrowed from [53]: 

0.8 x(k + 1) = 0.05





0 −0.1 x(k) + 0.9 −0.2



0 x(k − d(k)). − 0.1

To compare with the results in [150], which proposed latest results on stability condition for discrete-time with time-varying delay, Corollary 5.35 is used with E = I and U = 0. When the lower bounds of d(k) are 7 and 10, the upper bounds such that this system is asymptotically stable are 16 and 18, respectively, by using Proposition 2 in [150]. However, by employing the result in Corollary 5.35, the upper bounds can reach up to 18 and 20, respectively. Moreover, for different value of d1 , the admissible upper bound d2 is obtained from different methods in Table 5.11. It can be seen that Corollary 5.35 can tolerate larger delay upper bound with the same delay lower bound than Proposition 2 in [150], and it can get the same delay upper bound with Proposition 1 in [150]. However, the decision variables in Proposition 1 is larger than that in Corollary 5.35 due to directly dealing with the inversely weighted convex combination of quadratic terms of summable quantities by utilizing the reciprocally convex approach, which illustrates the reduced conservatism of our method.

Delay-dependent dissipativity analysis and synthesis of singular delay systems

157

Table 5.11 Comparisons of maximum allowed d2 for given d1 . Methods \ d1 7 10 15 20 25 Variables

[150] Proposition 2 [150] Proposition 1 Corollary 5.35

16 18 18

18 20 20

21 23 23

25 27 27

30 31 31

18 38 22

5.3.5 Conclusion In this subchapter, the problem of robust reliable dissipative control for uncertain discrete-time singular systems with time-varying delay and sensor failures has been studied. The sufficient conditions in terms of LMIs have been proposed for rendering considered filtering error singular systems admissible and strictly (Q, S, R)-dissipative. Based on these results, a desired filter is designed to guarantee the filtering error singular system to be admissible and strictly (Q, S, R)-dissipative. The results presented in this section are in terms of strict LMIs, which make the conditions more tractable. Moreover, the results benefiting from the reciprocally convex approach proposed in [140] have improved the existing results, and H∞ filtering and passive filtering of singular systems are unified in the proposed results. Finally, numerical examples are given to demonstrate the effectiveness of our methods.

CHAPTER 6

State-feedback control for singular Markovian systems In this chapter, we consider the problem of state-feedback control for continuous singular Markovian systems. Firstly, for delay-free singular Markovian systems, necessary and sufficient conditions are proposed for the system to be admissible and for the state-feedback control design by applying equivalent sets technique. For time-delay singular Markovian systems, a new bounded real lemma is proposed and the corresponding H∞ control problem is studied. Secondly, we consider the problem of reliable dissipative control for continuous-time singular Markovian systems with actuator failure. Our attention is focused on the state-feedback controller design method such that the closed-loop system is regular, impulse free, stochastically stable, and strictly (Q, S, R)-dissipative. Numerical examples are given to illustrate the effectiveness of the theoretic results developed.

6.1 Admissibilization and H∞ control for singular Markovian systems In this section, the issues of admissibilization and H∞ control for continuous singular Markovian systems and singular delay Markovian systems are addressed, respectively, by applying equivalent sets technique.

6.1.1 Admissibility of singular Markovian jump systems Consider the following singular Markovian jump systems: Ex˙ (t) = A(rt )x(t) + B(rt )u(t),

(6.1)

where x(t) ∈ Rn is the state vector; u(t) ∈ Rm is the control vector. The matrices A(·) and B(·), which are functions of rt , are known real matrices with appropriate dimensions. {rt , t ≥ 0} is a continuous-time Markov process, which takes values in a finite set N = {1, 2, . . . , N } and describes the evolution of the mode at time t. For notational simplicity, for each rt = i, i ∈ N , matrix A(rt ) will be denoted by Ai . The Markov process describes Analysis and Synthesis of Singular Systems https://doi.org/10.1016/B978-0-12-823739-7.00013-6

Copyright © 2021 Elsevier Inc. All rights reserved.

159

160

Analysis and Synthesis of Singular Systems

the switching between different modes, and its evolution is governed by the following probability transitions: 

Pr{rt+δ = j|rt = i} =

i = j πij δ + o(δ) 1 + πii δ + o(δ) i = j,

(6.2)

where δ > 0 and limδ→0 o(δ) = 0, and πij ≥ 0, for i = j, is the transition δ rate from mode i at time t to mode j at time t + δ , which satisfies πii =  − N j=1,i=j πij . We recall the following lemma: Lemma 6.1. [209] System (6.1) with u(t) = 0 is stochastically admissible if and only if there exist matrices Xi , i ∈ N , such that the following coupled LMIs holds for each i ∈ N : ET Xi = Xi T E ≥ 0, Xi T Ai + ATi Xi +

N 

πij ET Xj < 0.

(6.3) (6.4)

j=1

Remark 6.2. The matrices Xi , i ∈ N , in Lemma 6.1 are nonsingular. If they are singular, then there exist nonzero vectors ξi , i ∈ N , such that Xi ξi = 0. Then for the nonzero vector ξi , we have the following inequality from (6.4): N 

πij ξiT ET Xj ξi + πii ξiT ET Xi ξi + ξiT Xi T Ai ξi + ξiT ATi Xi ξi

j=1,j=i

=

N 

πij ξiT ET Xj ξi < 0.

(6.5)

j=1,j=i

As ET Xj ≥ 0 and πij ≥ 0, i = j, the inequality in (6.5) cannot hold, which implies the inequality in (6.4) does not hold. Based on the discussions, matrices Xi , i ∈ N , are nonsingular. Then combining Lemma 2.5 with Remark 6.2, a new necessary and sufficient admissibility condition of singular Markovian jump systems can be obtained from Lemma 6.1, which is given in the following theorem: Theorem 6.3. System (6.1) with u(t) = 0 is stochastically admissible if and only if there exist symmetric matrices Pi , i ∈ N , and nonsingular matrices Φi , i ∈ N , such that the following coupled LMIs hold for each i ∈ N :

State-feedback control for singular Markovian systems

sym(ATi (Pi E + U T Φi ΛT )) +

N 

161

ELT Pi EL > 0,

(6.6)

πij ET Pj E < 0,

(6.7)

j=1

where EL , U, and Λ are defined in Lemma 2.5. Remark 6.4. It should be mentioned that the obtained admissibility condition in Theorem 6.3 does not require positive definite matrices Pi , i ∈ N . From this point of view, the result in Theorem 6.3 is less conservative. Now we consider the state feedback control problem for system (6.1) with u(t) = K (rt )x(t) such that the closed-loop system ¯ (rt )x(t) = (A(rt ) + B(rt )K (rt ))x(t) Ex˙ (t) = A

(6.8)

is stochastically admissible. A necessary and sufficient state feedback controller design condition is proposed in the following result, based on Theorem 6.3: Theorem 6.5. There exists a state feedback controller such that the closed-loop system in (6.8) is stochastically admissible if and only if there exist symmetric matrices P¯ i , i ∈ N ; nonsingular matrices Φ¯ i , i ∈ N , and matrices Hi , i ∈ N , such that the following LMIs hold for each i ∈ N : 

ERT P¯ i ER > 0, sym(Ai Yi + Bi Hi ) + πii EP¯ i ET

YiT Ωi −Ψi



(6.9)



< 0,

(6.10)

where Yi = P¯ i ET + ΛΦ¯ i U , Ωi =

√

πi1 ER

√

πi2 ER

···

√

πi(i−1) ER

√

πi(i+1) ER

···

√

 πiN ER ,

Ψi = diag{ERT P¯ 1 ER , ERT P¯ 2 ER , . . . , ERT P¯ i−1 ER , ERT P¯ i+1 ER , . . . , ERT P¯ N ER }.

ER , U, and Λ are defined in Lemma 2.5. Then, the desired controller can be obtained by Ki = Hi Yi−1 . Proof. Based on Theorem 6.3, the closed-loop system (6.8) is stochastically admissible if and only if there exist symmetric matrices Pi , i ∈ N and nonsingular matrices Φi , i ∈ N , such that the following coupled LMIs hold for

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each i ∈ N :

sym((Ai + Bi Ki )T (Pi E + U T Φi ΛT )) +

N 

ELT Pi EL > 0,

(6.11)

πij ET Pj E < 0.

(6.12)

j=1

Considering πij > 0, i = j, ELT Pi EL > 0, and using Schur complement equivalence, (6.12) is equivalent to 

sym((Ai + Bi Ki )T (Pi E + U T Φi ΛT )) + πii ET Pi E 

 =

 Ωi −Ψi

sym((Ai + Bi Ki )T (Pi E + U T Φi ΛT )) + πii ET (Pi E + U T Φi ΛT ) 

 Ωi −Ψi

0, the singular Markovian jump system in (6.13) with u(t) = 0 is stochastically admissible with an H∞ performance γ if there exist nonsingular matrices Xi , i ∈ N ; matrices Q > 0, R > 0 and Wi , i ∈ N , such that the following LMIs hold for each i ∈ N : ⎡ Ψ11i ⎢ ⎢  ⎢ ⎢  ⎢ ⎢ ⎢  ⎢ ⎢ ⎣  

ET Xi = XiT E ≥ 0, ⎤ T

(6.14)

XiT Adi − Wi E XiT Bwi dWi E dATi R Ci ⎥ −Q 0 0 dATdi R CdiT ⎥ 

−γ 2 I





0 −dR













⎥

T R DT ⎥ dBwi wi ⎥ ⎥ < 0, 0 0 ⎥ ⎥ −dR 0 ⎥ ⎦  −I

where Ψ11i = XiT Ai + ATi Xi + Q + Wi E + ET WiT +

N

j=1 πij E

TX

(6.15)

j.

Proof. The result can be obtained by choosing the following Lyapunov function and free-weighting equation: 

V (xt ) = x (t)E Xi x(t) + T

 +

t

T

0

−d



t t+θ

t−d

x(τ )T Qx(τ )dτ

x˙ (τ )T ET REx˙ (τ )dτ dθ,

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Analysis and Synthesis of Singular Systems





0 = 2x(t)T Wi E x(t) − x(t − d) −

t

t−d



x˙ (τ )dτ .

To prove the invertibility of matrices Xi , i ∈ N , the following inequality will be used  Ψ11i 



XiT Adi − Wi E < 0, −Q

(6.16)

which can be derived from (6.15). Then pre- and postmultiplying (6.16) by [I , I ] and its transpose, respectively, we obtain sym(XiT (Ai + Adi )) +

N 

πij ET Xj < 0,

j=1

which implies the matrices Xi , i ∈ N , are nonsingular based on discussions in Remark 6.2. Noting that the invertibility of matrices Xi , i ∈ N , in Lemma 6.7, a new bounded real lemma for singular system (6.13) is presented below in terms of strict LMIs based on Lemma 2.5. Theorem 6.8. Given a scalar γ > 0, the singular Markovian jump system in (6.13) is stochastically admissible with an H∞ performance γ if there exist symmetric matrices Pi , i ∈ N , and nonsingular matrices Φi , i ∈ N , such that the following LMIs hold for each i ∈ N : ELT Pi EL > 0, ⎡¯ Ψ11i ⎢ ⎢  ⎢ ⎢  ⎢ ⎢ ⎢  ⎢ ⎣  

XiT Adi − Wi E XiT Bwi dWi E dATi R CiT −Q

0

0



−γ 2 I

  

  

0 −dR  

⎤

⎥ ⎥ T⎥ Dwi ⎥ < 0, ⎥ 0 ⎥ ⎥ 0 ⎦

dATdi R CdiT ⎥ TR dBwi 0 −dR



−I

where Ψ¯ 11i = XiT Ai + ATi Xi + Qi + Wi E + ET WiT +

N  j=1

Xi = Pi E + U T Φi ΛT . EL , U, and Λ are defined in Lemma 2.5.

(6.17)

πij ET Pj E,

(6.18)

State-feedback control for singular Markovian systems

165

Remark 6.9. A bounded real lemma for singular Markovian systems with time delay is proposed   in Lemma 1 of [184]. However, the matrix E is I 0 restricted to be r , and the free-weighting matrix Wi needs to satisfy 0 0 Wi E = Wi in the process of deriving the bounded real lemma. In Theorem 6.8, these constraints are both removed. From this point of view, the result in Theorem 6.8 is more general and less conservative. Based on Theorem 6.8, the state feedback controller u(t) = Ki x(t) is designed in the following theorem: Theorem 6.10. Given a scalar γ > 0, the singular system in (6.13) is stochastically admissible with an H∞ performance γ if there exist symmetric matrices P¯ i , ¯ > 0, R ¯ > 0, W ¯ i and Hi , i ∈ N ; nonsingular matrices Φ¯ i , i ∈ N ; matrices Q i ∈ N , such that the following LMIs hold for each i ∈ N : ERT P¯ i ER > 0,

(6.19)

⎡˜

⎤

¯ i E T Bwi dW ¯ i E T d(Ai Yi + Bi Hi )T (Ci Yi + Di Hi )T Y T Y T Ωi Ψ11i Adi Yi − W i i ⎢  Ψ˜ 22i 0 0 d (Adi Yi )T (Cdi Yi )T 0 0 ⎥ ⎢ ⎥ T T 2 ⎢   −γ I 0 dBwi Dwi 0 0 ⎥ ⎢ ⎥ ⎢    Ψ˜ 44i 0 0 0 0 ⎥ ⎢ ⎥ ⎢  ¯    −dR 0 0 0 ⎥ ⎢ ⎥ ⎢      −I 0 0 ⎥ ⎢ ⎥ ⎣  ¯      −Q 0 ⎦        −Ψi

< 0,

(6.20)

where ¯, ¯ i ET ) + πii EP¯ i ET , Ψ˜ 22i = −YiT − Yi + Q Ψ˜ 11i = sym(Ai Yi + Bi Hi + W ¯ , Yi = P¯ i ET + ΛΦ¯ i U . Ψ˜ 44i = −dYiT − dYi + dR

ER , U, and Λ are defined in Lemma 2.5. Moreover, if the above LMIs are feasible, then the state feedback controller Ki can be given by Ki = Hi Yi−1 . Proof. Based on Theorem 6.8 and using Schur complement equivalence for closed-loop system, it is stochastically admissible with an H∞ performance if the following inequalities hold:

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Analysis and Synthesis of Singular Systems

ELT Pi EL > 0, (6.21)

⎡

¯ TR C ¯T Ψˆ 11i XiT Adi − Wi E XiT Bwi dWi E dA i i T ⎢  −Q 0 0 dAdi R CdiT ⎢

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

     

−γ 2 I     

     

I 0 T R DT 0 dBwi 0 wi −dR 0 0 0  −dR 0 0   −I 0    −Q−1 







⎤

Ωi

0 0 0 0 0 0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, (6.22) ⎥ ⎥ ⎥ ⎦

−Ψi

¯ i + Wi E)+πii ET Pi E, A ¯ i = Ai + Bi Ki , C ¯ i = C i + D i Ki , where Ψˆ 11i = sym(XiT A Ωi , and Ψi are defined in Theorem 6.5. ¯ = Q−1 , R ¯ = R−1 , Hi = Ki Yi , and W ¯ i = YiT Wi YiT , Defining Yi = Xi−1 , Q pre- and postmultiplying the inequality in (6.22) by diag{YiT , YiT , I , YiT , R−1 , I , YiT , I } and its transpose, and factoring in EYi = YiT ET = EP¯ i ET , we have ⎡˜

⎤

¯ i E T d(Ai Yi + Bi Hi )T (Ci Yi + Di Hi )T Y T Y T Ωi dW i i 0 0 dYiT ATdi YiT CdiT 0 0 ⎥ ⎥ T T −γ 2 I 0 dBwi Dwi 0 0 ⎥  −dYiT RYi 0 0 0 0 ⎥ ⎥ < 0. ¯   −dR 0 0 0 ⎥    −I 0 0 ⎥ ⎦ ¯     −Q 0

¯ i E T Bwi Xi Ψ11i Adi Yi − W

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣



−YiT QYi

     

     











−Ψi

(6.23) ¯ and −Y T RYi ≤ −Y T − Yi + R, ¯ Considering −YiT QYi ≤ −YiT − Yi + Q, i i the inequality in (6.20) implies that the inequality in (6.23) holds. On the other hand, the inequality in (6.21) is equivalent to the inequality in (6.19). Therefore the proof is completed. Remark 6.11. Note that the H∞ control problem is also investigated in [184], where the invertibility of matrix Pj E + σ N with N = 

0r ×r

0r ×(n−r )



is required, that is, (Pj E + σ N )−1 = Pj−1 E + σ N . The I(n−r )×(n−r ) above constraints have the following two disadvantages: one is that the inPj E + σ N may not exist when matrix E is not in the form verse  of matrix  I 0 of r ; on the other hand, applying the parameter σ to enlarge matrix 0 0 Pj E, that is Pj E ≤ Pj E + σ N , will derive conservatism. In Theorem 6.10, the invertibility of matrix Xi = Pi E + U T Φi ΛT is needed. From Lemma 2.5, we can see the invertibility of matrix Xi can be guaranteed, regardless of the form of matrix E and without any enlargement. 

167

State-feedback control for singular Markovian systems

Remark 6.12. In [174], based on the equivalence of the H∞ norms of transfer functions of original system 

Ex˙ (t) = (Ai + Bi Ki )x(t) + Adi x(t − d) + Bwi ω(t) z(t) = (Ci x(t) + Di Ki )x(t)

(6.24)

and its dual system 

ET x˙ (t) = (Ai + Bi Ki )T x(t) + ATdi x(t − d) + (Ci + Di Ki )T ω(t) T x(t ), z(t) = Bwi

(6.25)

an H∞ control method is given in [174]. However, the method cannot be used when the delay is time-varying, because the transfer function of a system with time-varying delay cannot be expressed explicitly. The two equivalent sets method without using transfer function in this section can be extended to systems with time-varying delay.

6.1.3 Examples In this subsection, numerical examples are provided to show the advantages on numerical computations and the state feedback control of the equivalent sets approach. Example 6.1. Consider the singular Markovian jump system in (6.1) with two operating modes, that is, N = 2 and the following parameters: ⎡

1 ⎢ E = ⎣0 0

0 1 0

⎡

0.7 ⎢ A2 = ⎣1.1 0.5

0.9 1.4 0.3

⎡

0

0.8 0.8 0.2 ⎡

⎤

⎤

⎡

1.0 1.5 ⎥ ⎢ 0.9 ⎦ , B1 = ⎣0.9 1.1 − 0.7 ⎤

⎡

⎤ − 1.5 ⎥ 0.6 ⎦ ,

5 ⎤

0.3 2 0.9 1 0 ⎢ ⎢ ⎥ ⎥ ⎥ − 0.7⎦ , B2 = ⎣−2 0.6⎦ , EL = ER = ⎣0 1⎦ , 1.6 0.7 1 0 0 ⎡

⎤

 0 π11 ⎢ ⎥ − 0.7 , Λ = ⎣ 0 ⎦ , π21 0.8 



U= 0

⎤

0 1.3 ⎥ ⎢ 0⎦ , A1 = ⎣0.7 0.4 0

  π12 −0.9 = π22 0.5



0.9 . − 0.5

By solving the characteristic equation sE − A1 = 0, two finite characteristic roots are 2.6825 and 0.24604, respectively. Therefore the open-loop system is not admissible. The controller design method in Theorem 6 and Theorem 7 in [199] cannot be used. The purpose is to design a state feedback

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Analysis and Synthesis of Singular Systems

controller to guarantee the closed-loop system to be stochastically admissible. By solving the LMIs in Theorem 6.5, we have 

34.8763 K1 = 37.9459

28.5998 28.4043

 −102.5126 K2 = −490.4724



0.6234 , 0.0044  − 0.5429 . − 1.2175

− 18.2368 − 90.5122

Example 6.2. A direct current (DC) motor has been modeled as a singular Markovian jump system in [146], [170], [171], [238]. The singular Markovian model is given as follows: Ex˙ (t) = A(rt )x(t) + B(rt )u(t).

(6.26)

The matrix parameters are borrowed from Example 2 in [170]: ⎡

1 ⎢ E = ⎣0 0

⎤

0 1 0

⎡

0 ⎢ A2 = ⎣ 9.8 −20 ⎡

⎡

0 0 ⎥ ⎢ 0⎦ , A1 = ⎣ 9.8 0 −20 ⎤

1 0 ⎤

1 0 −3 ⎡

⎤

0 ⎥ 1 ⎦, −1 ⎤

0 0.4 ⎥ ⎥ ⎢ 1 ⎦ , B1 = ⎣−0.2⎦ , 0.5 − 0.5

−3

 −1.2 π11 ⎥ ⎢ B2 = ⎣ 0.5 ⎦ , π21 −0.2 ⎡ ⎤

   π12 −0.6 0.6 = , 0.2 −0.2 π22 ⎡

⎤

1 0 0   ⎢ ⎥ ⎢ ⎥ EL = ER = ⎣0 1⎦ , U = 0 0 −0.7 , Λ = ⎣ 0 ⎦ . 0 0 0.8 By solving the condition in Theorem 6.5, the state-feedback controllers are obtained: 



K1 = 186.3514 

K2 = 188.1730

38.5301 2.0267 , 

21.4539 −2.3464 .

Then the closed-loop system is ¯ (rt )x(t) Ex˙ (t) = A

State-feedback control for singular Markovian systems

169

with ⎡

74.5406 ⎢ A1 = ⎣−27.4703 73.1757

⎤

16.4120 − 7.7060 16.2650

⎡

−225.8076 ⎢ A2 = ⎣ 103.8865 −57.6346

0.8107 ⎥ 0.5947⎦ , 0.0133

− 24.7447 10.7270 − 7.2908

⎤

2.8157 ⎥ − 0.1732⎦ . − 0.0307

To testify the effectiveness of the state-feedback control method, Theorem 2.2 in [171], Theorem 6.5 in this section, and Theorem 4 in [199] are utilized. By solving the LMIs in these theorems, we note that all the conditions are feasible, which implies the state-feedback controller is applicable. Example 6.3. Consider the following singular Markovian jump timedelay system with two operating modes (N = 2) and the following parameters: ⎡

0.5023 ⎢ A1 = ⎣ 0.3025 −0.1002

2.0125 0.4004 0.3002

⎤

0.0150 ⎥ − 4.0020⎦ , − 3.5001

⎤ − 0.1002 ⎥ 0.3003 ⎦ , − 2.0045 ⎤ ⎡ −0.1669 0.0802 1.6820 ⎥ ⎢ Ad1 = ⎣−0.8162 − 0.9373 0.5936⎦ , 2.0941 0.6357 0.7902 ⎤ ⎡ 0.1053 − 0.1948 − 0.6855 ⎥ ⎢ Ad2 = ⎣0.1586 0.0755 − 0.2684⎦ , 0.7709 − 0.5266 − 1.1883 ⎡

0.5005 ⎢ A2 = ⎣0.1256 0.1033

⎡

1 ⎢ E = ⎣0 0 ⎡

0 1 0 ⎤

⎤

0.5052 − 0.0552 1.0015

⎤

⎡

⎡

⎤

0 0.9 1.5 ⎥ ⎢ ⎥ ⎢ ⎥ , , B = B = 0⎦ 1 ⎣1.8⎦ 2 ⎣0.9⎦ , 1.4 1.1 0 ⎡

⎤

−0.6 0.1 ⎥ ⎢ ⎥ ⎢ Bw1 = ⎣0.2⎦ , Bw2 = ⎣ 0.5 ⎦ , 0.4 0.8

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Analysis and Synthesis of Singular Systems





C1 = 0.8

0.3





0.9 , C2 = −0.5

0.2

 − 0.0410 ,   Cd2 = −2.2476 − 0.5108 0.2492 ,   Dw1 = 0.2, Dw2 = 0.5, U = 0 0 − 0.7 , ⎤ ⎡ 1 0   ⎥ ⎢ Λ = 0 0 0.8 , ER = EL = ⎣0 1⎦ , 

Cd1 = 0.2486

 π11 π21

 π12 π22

0.3 , D1 = D2 = 0,

0.1025

0

0

  −0.3 0.3 = , d = 0.2. 0.5 −0.5

The purpose is to design a state feedback controller such that the closedloop system is stochastically admissible with an H∞ performance γ . By solving LMIs in Theorem 6.10, the minimal value of γ is obtained as γmin = 2.43. When γ = 2.43, the state feedback controllers are obtained as follows: 







K1 = −4.4763 −2.2181 −0.1445 , K2 = −5.3581 −0.4309 −1.1303 . By using the method in Theorem 1 of [184], the minimal value of γ is solved as γ = 2.68, which illustrates the reduced conservatism of out result in Theorem 6.10. On the other hand, when some system parameters change to ⎡

2 ⎢√ E=⎣ 2 0

√

⎤

⎡

⎤

2 0 1 √1 ⎥ ⎥ ⎢ 2⎦ , D1 = 0.1, D2 = −0.5, 2 0⎦ , ER = EL = ⎣0 0 0 0 0

the condition in Theorem 1 of [184] cannot be applied. By solving the conditions with γ = 5.60 in Theorem 6.10 in this note, the corresponding state feedback controller is given as follows: 



K1 = −4.7783 −0.8063 −0.9794 , 



K2 = −4.4531 0.3942 −4.0966 .

State-feedback control for singular Markovian systems

171

6.1.4 Conclusion In this section, the application of the equivalent sets is extended to admissibility and corresponding state feedback control of singular Markovian jump systems, which gives a novel admissibility condition and an effective state feedback control method. H∞ control of time-delay singular Markovian jump systems is also addressed. A new bounded real lemma and improved H∞ control result are given. The effectiveness and improvement of the presented method have been illustrated by numerical examples. Compared with existing results, the main contributions of this section are not only presenting the new two equivalent sets, but also developing some improved results as follows: • New necessary and sufficient admissibilization condition is obtained for singular Markovian jump systems. • For singular Markovian jump systems with time-delay, a new bounded real lemma is given in terms of strict LMIs without any constraint on system matrix E.

6.2 Reliable dissipative control for singular Markovian systems In this section, the problem of reliable dissipative control is investigated for continuous-time singular Markovian system with actuator failure. Firstly, a sufficient condition is established in terms of LMIs, which guarantees a singular Markovian system to be stochastically admissible and strictly (Q, S, R)-dissipative. Based on the criterion, a state feedback controller design method is given in terms of strict LMIs. Moreover, the dissipative control results include the results of H∞ control and passive control as special cases. The effectiveness of the controller designed in this section is illustrated by a numerical example.

6.2.1 Problem statement Consider a class of linear continuous-time singular Markovian systems described by ⎧ ⎪ ⎨ Ex˙ (t) = A(rt )x(t) + B(rt )u(t) + Bw (rt )w (t) z(t) = C (rt )x(t) + D(rt )u(t) + Dw (rt )w (t) ⎪ ⎩ x(t ) = x , 0 0

(6.27)

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Analysis and Synthesis of Singular Systems

where x(t) ∈ Rn is the state vector; u(t) ∈ Rm is the control input; w (t) ∈ Rl represents a set of exogenous inputs (which belongs to L2 [0, ∞)), and z(t) ∈ Rq is the controlled output; {rt , t ≥ 0} is a continuous-time Markov process taking values in a finite space S  {1, 2, . . . , N } and describing the evolution of the mode at time t; A(rt ), B(rt ), Bw (rt ), C (rt ), D(rt ), and Dw (rt ) denote constant matrices with appropriate dimensions. In contrast with standard linear state-space systems with E = I, the matrix E ∈ Rn×n has rank(E) = p ≤ n. The transition probability rate matrix of the Markov process {rt , t ≥ 0} is Π  {πij }, and we have 

Pr{rt+δ = j|rt = i} =

i = j πij δ + o(δ) 1 + πii δ + o(δ) i = j,

(6.28)

where δ > 0 and limδ→0 o(δ) = 0, and πij ≥ 0, for i = j, is the transition δ rate from mode i at time t to mode j at time t + δ , which satisfies πii =  − N j=1,i=j πij . For notational simplicity, a matrix W (rt ) will be denoted by Wi for each possible rt = i, i ∈ S; for example, A(rt ) is denoted by Ai and so on. For the control input ui , i = 1, 2, . . . , m, let uFi denote the signal from the actuator that may be faulty. The following failure model is adopted here: uFi = αai ui , i = 1, 2, . . . , m, where 0 ≤ α ai ≤ αai ≤ α¯ ai , i = 1, 2, . . . , m with α ai ≤ 1 ≤ α¯ ai . Then we have uF = Aa u, Aa = diag{αa1 , αa2 , . . ., αam }. Remark 6.13. In the model mentioned above, when α ai = α¯ ai = 1, it corresponds to the normal fully operating case, that is, uFi = ui ; when α ai = 0, then it covers the outage case in [167]; when αai = 0 and αai = 1, then it corresponds to the case that the intensity of the feedback signal from actuator changes. Denote A¯ a = diag{α¯ a1 , α¯ a2 , . . ., α¯ am }, Aa = diag{α a1 , α a2 , . . ., α am }, Ba0 = diag{βa01 , βa02 , . . ., βa0m },

State-feedback control for singular Markovian systems

173

B = diag{β1 , β2 , . . ., βm },

a = diag{a1 , a2 , . . ., am }, ai where βa0i = α¯ ai +α 2 , βi =

α¯ ai −α ai α¯ ai +α ai

and ai =

αai −βa0i . βa0i

Then we have

Aa = (I + a )Ba0 ,

diag{| a1 |, | a2 |, . . . , | am |} ≤ B ≤ I .

(6.29)

Consider the following memoryless state-feedback controller for system (6.27): u(t) = K (rt )x(t).

(6.30)

By applying controller (6.30) to system (6.27), the closed-loop system can be described by ⎧ ⎪ ⎨ Ex˙ (t) = (A(rt ) + B(rt )Aa K (rt ))x(t) + Bw (rt )w (t) z(t) = (C (rt ) + D(rt )Aa K (rt ))x(t) + Dw (rt )w (t) ⎪ ⎩ x(t ) = x . 0 0

(6.31)

Before moving on, we give some definitions and lemmas concerning the following nominal unforced counterpart of the system in (6.27) with w (t) = 0: Ex˙ (t) = A(rt )x(t).

(6.32)

Then, we introduce the following definition for singular Markovian system (6.32): Definition 6.14. [209] 1. The singular Markovian system in (6.32) is said to be regular and impulse free if det(sE − Ai ) is not identically zero ∀i ∈ S. 2. The singular Markovian system in (6.32) is said to be impulse free if deg {det(sE − Ai )} = rank E, ∀i ∈ S. 3. The singular Markovian system in (6.32) is said to be stochastically stable, if, for any x0 ∈ Rn and r0 ∈ S, there exists a scalar δ(x0 , r0 ) > 0 such that  lim E

t→∞

t



x (s, x0 , r0 )x(s, x0 , r0 )ds | x0 , r0 ≤ δ(x0 , r0 ), T

0

where E is the mathematical expectation, and x(t, x0 , r0 ) denotes the solution to system (6.32) at time t under the initial conditions x0 and r0 .

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4. The singular Markovian system in (6.32) is said to be stochastically admissible if the system is regular, impulse free, and stochastically stable. Lemma 6.15. [200] For matrices Y > 0, M, and N with appropriate dimensions, the following inequality holds: M T N + N T M ≤ N T YN + M T Y −1 M . Lemma 6.16. [7] For any matrices U and V > 0, the following inequality holds: UV −1 U T ≥U + U T − V . Definition 6.17. [234] Given matrices Q, R, and S with Q and R real symmetric, systems (6.27) with u(t) = 0 is called strictly (Q, S, R)dissipative if, for any τ ≥ 0, under zero initial state, the following condition is satisfied for some scalar α > 0: E { z, Qzτ } + 2E { z, Sw τ } + E { w , Rw τ } ≥ α E { w , w τ } .

(6.33)

As in [202], we assume that Q ≤ 0. Then we can get 1

−Q = (Q−2 )2 1

for some Q−2 ≥ 0. Our main objective is to design a state-feedback controller in the form of (6.30) such that the closed-loop systems in (6.31) is stochastically admissible and strictly (Q, S, R)-dissipative. In this section, a sufficient condition is derived to guarantee that the unforced system of (6.27) is stochastically admissible and strictly (Q, S, R)dissipative. Based on this result, a state-feedback controller is designed to render the closed-loop system stochastically admissible and strictly (Q, S, R)-dissipative.

6.2.2 Reliable dissipativity analysis The result of reliable dissipativity analysis for system (6.27) is presented in the following theorem: Theorem 6.18. Let matrices Q, S, and R be given with Q and R real symmetric and Q ≤ 0. Then the system in (6.27) is stochastically admissible and strictly

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State-feedback control for singular Markovian systems

(Q, S, R)-dissipative if there exist matrices Pi , i = 1, 2, . . . , N, such that the following set of LMIs hold for i = 1, 2, . . . , N:

ET Pi = PiT E ≥ 0,

(6.34)

Ωi < 0,

(6.35)

where ⎡

PiT Ai + ATi Pi +

⎢ Ωi = ⎢ ⎣

N

j=1 πij E

TP

1

PiT Bwi − CiT S

j



T S − ST D − R −Dwi wi





CiT Q−2 1

⎤ ⎥ ⎥

T Q2 . Dwi −⎦ −I

Proof. From the LMI in (6.35), we have ATi Pi + PiT Ai +

N 

πij ET Pj < 0.

j=1

Combining with the condition in (6.34), the stochastic admissibility of the system in (6.32) is guaranteed, based on Lemma 6.1. Next, we show the dissipativity of system (6.27). To this end, we choose the stochastic Lyapunov function as V (x(t), rt = i) = xT (t)ET Pi x(t). Let L be the weak infinitesimal generator of the stochastic process {x(t), rt }. Then, for each rt = i, i ∈ S, we have LV (x(t), rt = i) = xT (t)(ATi Pi + PiT Ai +

N 

πij ET Pj )x(t)

j=1

+x

T

T (t)PiT Bwi w (t) + w T (t)Bwi Pi x(t).

(6.36)

Denote 

J (τ ) = E

τ







zT (t)Qz(t) + 2zT (t)Sw (t) + w T (t)Rw (t) dt .

0

Then under the zero initial condition, that is, x0 = 0, it can be shown that for any nonzero w ∈ L2 [0, ∞),

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E {V (x(τ ), rτ )} − J (τ )  τ    LV (x(t), rt ) − zT (t)Qz(t) − 2zT (t)Sw (t) − w T (t)Rw (t) dt =E 0 τ    T =E ξ (t)Ω1i ξ(t) dt , 0

where  Ω1i =

PiT Ai + ATi Pi + 

 − 

N

j=1 πij E



TP

j

PiT Bwi − CiT S T S − ST D − R −Dwi wi



  CiT Q Ci Dwi , T Dwi 

x(t) ξ(t) = . w (t) By using Schur complement equivalence, the inequality in (6.35) is equivalent to Ω1i < 0. Then we have J (τ ) > 0, and there exists a sufficiently small α > 0 such that 

J (τ ) ≥ α E

τ



w T (t)Rw (t)dt . 0

By the definition, the system in (6.27) is strictly (Q, S, R)-dissipative. This completes the proof. Remark 6.19. When A¯ a = Aa = diag{1, 1, . . . , 1}, Q = −I, S = 0 and R = γ 2 I, the result in Theorem 6.18 reduces to the H∞ result Lemma 1 in [235], which implies the more generality of our result.

6.2.3 Controller design Now, our attention will be devoted to design a state-feedback controller in the form of (6.30), such that closed-loop system (6.31) subject to possible actuator failures is stochastically admissible and strictly (Q, S, R)-dissipative. There are equality constraints EPi = PiT ET in Theorem 6.18, which will lead to numerical problems when solving such nonstrict LMIs. However, the method to deal with the constraints EPi = PiT ET in [7] is difficult to make the equality exactly hold. Moreover, by employing the state feedback controller design approach in [235], a strict LMI condition guaranteeing the stochastic admissibility and strict dissipativity of the closed-loop system in (6.31) is proposed in the next theorem.

State-feedback control for singular Markovian systems

177

Theorem 6.20. Let matrices Q, S, and R be given with Q and R real symmetric and Q ≤ 0. Then the system in (6.27) is stochastically admissible and strictly ¯ i > 0, (Q, S, R)-dissipative if there exist matrices Ni , nonsingular matrix Φ¯ i , Z i = 1, 2, . . . , N, and diagonal matrix G > 0 such that the following set of LMIs hold for i = 1, 2, . . . , N: ⎡

Ξ11i ⎢  ⎢ ⎢ Ξi = ⎢  ⎢ ⎣  

Ξ12i Ξ22i   

Ξ13i Ξ23i Ξ33i  

NiT Ba0 B 0 0 −G

Ξ15i



Ξ55i

0 0 0

⎤ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎦

(6.37)

where Ξ11i = sym(Ai Mi + Bi Ba0 Ni ) + πii MiT ET + Bi GBiT , Ξ12i = Bwi − (MiT CiT + NiT Ba0 DiT )S − Bi GDiT S, 1

1

Ξ13i = (MiT CiT + NiT Ba0 DiT )Q−2 + Bi GDiT Q−2 ,  √ √ √ √ Ξ15i = πii−1 MiT ER πii+1 MiT ER · · · πi1 MiT ER · · · πiN MiT ER , T S − ST Dwi − R + ST Di GDiT S, Ξ22i = −Dwi 1

1

T Q−2 − ST Di GDiT Q−2 , Ξ23i = Dwi 1

1

Ξ33i = −I + Q−2 Di GDiT Q−2 , ¯ 1 ER , . . . , ERT Z ¯ i−1 ER , ERT Z ¯ i+1 ER , . . . , ERT Z ¯ N ER ), Ξ55i = −diag(ERT Z

Mi = Z¯ i ET + V Φ¯ i U T , and U, V are defined in Lemma 2.4. When the above conditions are satisfied, an admissible and (Q, S, R)-dissipative controller is given by Ki = Ni Mi −1 . Proof. Employing Schur complement equivalence, (6.37) is equivalent to the following inequality: ⎤

⎡

Bi  ⎢ −S T D ⎥ ˜ Ψi + ⎣ i ⎦ G BT i

1

⎡

Q−2 Di

− DiT S

⎤

NiT Ba0 B  ⎢ ⎥ +⎣ 0 ⎦ G−1 BBa0 Ni 0

1



DiT Q−2 

0

0 < 0,

(6.38)

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Analysis and Synthesis of Singular Systems

where ⎡

Ψ˜ 11i

⎢ Ψ˜ i = ⎢ ⎣ 

1

Bwi − (Ci Mi + Di Ba0 Ni )T S (Ci Mi + Di Ba0 Ni )T Q−2 1

T S − ST D − R −Dwi wi



T Q2 Dwi − −I



⎤ ⎥ ⎥, ⎦

¯ i ET Ψ˜ 11i = sym(Ai Mi + Bi Ba0 Ni + πii EMi ) − πii EZ +

N 

¯ j ER )−1 ERT Mi . πij MiT ER (ERT Z

j=1,j=i

By using the inequality πii MiT ER (ERT Z¯ i ER )−1 ERT Mi ≤ πii (MiT ET + EMi − EZ¯ i ET ), based on Lemma 6.15, (6.38) implies the following inequality holds: ⎤

⎡

Bi  ⎢ −S T D ⎥ ˘ Ψi + ⎣ i ⎦ G BT i

1

⎡

Q−2 Di

−DiT S

1



DiT Q−2

⎤

NiT Ba0 B   ⎢ ⎥ +⎣ 0 ⎦ G−1 BBa0 Ni 0 0 < 0, 0

(6.39)

where ⎡

Ψ˘ 11i

⎢ Ψ˘ i = ⎣  

1

Bwi − (Ci Mi + Di Ba0 Ni )T S (Ci Mi + Di Ba0 Ni )T Q−2 T S − ST D − R −Dwi wi 

1

T Q2 Dwi − −I

⎤ ⎥ ⎦,

¯ i ER )−1 ERT Mi Ψ˘ 11i = sym(Ai Mi + Bi Ba0 Ni ) + πii MiT ER (ERT Z +

N 

¯ j ER )−1 ERT Mi πij MiT ER (ERT Z

j=1,j=i

= sym(Ai Mi + Bi Ba0 Ni ) +

N 

¯ j ER )−1 ERT Mi . πij MiT ER (ERT Z

j=1

Then, according to Lemma 2.4, we have Mi−1 = Zi E + U Φi V T , and let T2i = diag(Mi−1 , Il , Iq ) ∈ R(n+l+q)×(n+l+q), Ni = Ki Mi .

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State-feedback control for singular Markovian systems

Pre- and postmultiplying to (6.39) by T2iT and T2i , and using Lemma 6.1 yields ⎤ ⎡ (Zi E + U Φi V T )T Bi  ⎥ ⎢ −S T D i Ψ¯ i + ⎣ ⎦ G BiT (Zi E + U Φi V T ) −DiT S 1 ⎡ ⎢ +⎣

⎤

1



DiT Q−2

Q−2 Di

KiT Ba0  ⎥ 0 ⎦ BG−1 B Ba0 Ki 0



0

0 < 0,

(6.40)

where ⎡

Ψ¯ 11i

⎢ Ψ¯ i = ⎣ 

1

(Zi E + U Φi V T )T Bwi − (Ci + Di Ba0 Ki )T S

(Ci + Di Ba0 Ki )T Q−2

T S − ST D − R −Dwi wi

T Q2 Dwi − −I



1



Ψ¯ 11i = sym((Zi E + U Φi V T )T (Ai + Bi Ba0 Ki )) +

N 

⎤ ⎥ ⎦,

πij ET Zj E.

j=1

From (6.29) and the elementary inequality xT y + yT x ≤ εxT x + ε−1 yT y, (6.40) implies ⎛⎡

⎤ (Zi E + U Φi V T )T Bi  ⎜⎢ ⎥ −S T D i Ψ¯ i + sym ⎝⎣ ⎦ a Ba0 Ki 1 2

⎡



¯ TS (Zi E + U Φi V T )T Bwi − C i



T S − ST D − R −Dwi wi 

⎢ =⎣ 

⎟

0 0 ⎠

Q− Di Θi

⎞

1

⎤

1

⎥

C¯ iT Q−2

T Q 2 ⎦ < 0, Dwi − −I

(6.41)

where ¯ i) + Θi = sym((Zi E + U Φi V T )T A

N 

πij ET Zj E,

j=1

¯ i = Ai + Bi Aa Ki , C ¯ i = C i + D i A a Ki . A ¯ i, C ¯ i , and Zi E + Replacing Ai , Ci , and Pi in (6.34) and (6.35) with A T U Φi V , respectively, the conditions in (6.34) and (6.35) hold from (6.41), which guarantees the stochastic admissibility and the strict dissipativity of the closed-loop system in (6.31).

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Remark 6.21. When A¯ a = Aa = diag{1, 1, . . . , 1}, Q = −I, S = 0, and R = γ 2 I, the result in Theorem 6.5 reduces to Theorem 2 in [235]. Therefore our result is more general.

6.2.4 Illustrative example Example 6.4. Consider a two-modes singular system in (6.27) with following parameters: • mode 1 ⎤ ⎤ ⎡ −0.7 −0.3 0 1 −0.2 ⎥ ⎥ ⎢ ⎢ A1 = ⎣ 0.7 −0.7 −0.5⎦ , B1 = ⎣1 −3.5⎦ , 0.1 0 −1 0 1.2 ⎡ ⎤   1 0.3 −1 0 1 ⎢ ⎥ , Bw1 = ⎣0.2 2 ⎦ , C1 = 0 1 1 0 −0.5     2.7 0.1 0.7 0 , Dw1 = ; D1 = 0 0 0 −1 ⎡

•

mode 2 ⎤ ⎡ −0.5 1.8 1.3 ⎥ ⎢ A2 = ⎣−0.2 − 2.1 − 0.1⎦ , 1.2 2.5 −1 ⎤ ⎡  0.1 − 1 −2 ⎥ ⎢ Bw2 = ⎣0.5 1 ⎦ , C2 = −1 0.1 0    0.1 0 −3 , Dw2 = D2 = 0 2 −0.8

⎡

1 ⎢ B2 = ⎣−1.6 −1 1 0



0 . 0.2

The matrix E is ⎡

1 ⎢ E = ⎣0 0

0.5 1 0



1 , 0.3

⎤

0 ⎥ 0⎦ , 0

⎤

2 ⎥ 1 ⎦, − 0.7

State-feedback control for singular Markovian systems

181

and the switching between the two modes is described by the probability rate matrix   −1.4 1.4 . Π= 1.1 −1.1

It can be verified by using Lemma 6.1 that the system considered in this example is not stochastically admissible. The purpose is to design state feedback controller such that the singular system in (6.31) is stochastically admissible and strictly (Q, S, R)-dissipative. To this end, we choose 

0.5 S= 0.1





−1 0.2 , Q= 0.6 0





0 16 , R= −1 0



0 . 16

To use Theorem 6.20, we choose 

U =V = 0

0

1

T

⎡

1 ⎢ , ER = ⎣0 0

⎤

0 ⎥ 1⎦. 0

Then, the state feedback controller in the form of (6.30) can be calculated by using standard software. Table 6.1 presents the obtained controllers by using Theorem 6.20 for different actuator failure cases, respectively. To illustrate the effectiveness of our obtained controllers, we give the simulation results for the first case, that is, Aa = diag{1, 1} and the fourth case, that is, A¯ a = diag{1.5, 1.5}, Aa = diag{0.5, 0.5}. One of the possible realizations of the jumping mode r (t) is presented in Fig. 6.1. Figs. 6.2 and 6.3 illustrate the state trajectories of closed-loop system without actuator failure and with Aa = diag{1.1, 0.8}, respectively, which show the stochastic stability of the closed-loop systems are guaranteed.

6.2.5 Conclusion In this section, the problem of reliable dissipative control for continuoustime singular Markovian systems with actuator failure has been studied. A sufficient condition in terms of LMIs has been proposed for guaranteeing singular Markovian systems stochastically admissible and strictly (Q, S, R)-dissipative. Based on the result, a state-feedback controller characterization has been given to guarantee the stochastically admissibility and strictly (Q, S, R)-dissipativity of the closed-loop system. The results presented in this section are in terms of strict LMIs, which make the conditions

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Table 6.1 State feedback controllers by using Theorem 6.20 with different actuator failure cases. Actuator cases Theorem 6.20   −0.4860 −0.7418 −0.2308 ¯ Aa = diag{1, 1} K1 = 15.0454 13.2740 −4.2807 Aa = diag{1, 1}   33.5587 −2.9683 −11.6581 K2 = −2.0174 −2.1034 −0.1496 A¯ a = diag{1, 1} Aa = diag{0.5, 0.3}

A¯ a = diag{1, 1} Aa = diag{0.05, 0.1}

A¯ a = diag{1.5, 1.5} Aa = diag{0.5, 0.5}

  −0.5375 −0.7746 −0.4725 K1 = 4.1089 −0.9051 3.6674   −31.4550 9.5591 12.7751 K2 = −6.8894 −3.8126 0.8952   −4.0080 −4.2692 −0.5972 9.7568 13.9286 −0.8025   −0.5840 5.0246 1.2211 K2 = −15.7071 −18.3959 0.3777

K1 =

 −0.3771 K1 = 1.9874  −7.3258 K2 = −3.9245

Figure 6.1 Random jumping mode r(t).

−0.5744 3.1286

2.1828 −3.1026

 −0.3424 −0.7061  2.8738 0.3638

State-feedback control for singular Markovian systems

183

Figure 6.2 State trajectories of closed-loop system without actuator failure.

Figure 6.3 State trajectories of closed-loop system with Aa = diag{1.1, 0.8}.

more tractable. Moreover, the results on H∞ control and passive control of singular Markovian systems are unified in the proposed results. Finally, a numerical example is given to demonstrate the effectiveness of our methods.

CHAPTER 7

Sliding mode control of singular stochastic Markov jump systems An investigation has been made into the SMC problem for SSMSs in this chapter. Firstly, by using replacement of matrix variables, a new mean square admissibility criterion of SSMSs is proposed. Based on this admissibility criterion, the desired state-feedback controller is designed to guarantee the closed-loop system to be mean square admissible. Then, the obtained results, given in the form of strict LMIs, are applied to solve SMC problem of SSMSs. To illustrate the workability and applicability of the theoretic results developed, numerical examples are provided.

7.1 Problem formulation Let SSMSs be described as follows: 

Edx(t) = (A(rt )x(t) + B(rt )u(t))dt + C (rt )x(t)dw (t) y(t) = D(rt )x(t),

(7.1)

where x(t) ∈ Rn and u(t) ∈ Rm are the system state vector and the control input, respectively; w (t) is one-dimensional Brownian motion defined on the filtered probability space (Ω, F , Ft , P ) with a filtering {Ft }{t≥0} . Matrix E ∈ Rn×n ; we assume that rank(E) = r ≤ n. The matrices A(·), B(·), C (·), and D(·), which are functions of rt are real matrices with appropriate dimensions. {rt , t ≥ 0} be a right-continuous Markov chain that takes values in a finite state set N = {1, 2, . . . , N } and illustrates the jumping mode at time t. For notational simplicity, we write A(rt = i)  Ai . The jumping feature between different modes is illustrated by Markov process with generator π = [πij ], (i, j ∈ N ) given by 

Pr{rt+σ = j|rt = i} =

πij σ + o(σ )

i = j 1 + πii σ + o(σ ) i = j,

(7.2)

where σ > 0, limσ →0 o(σσ ) = 0, and πij ≥ 0 for i = j stands for the transition  rate from mode i to mode j, which satisfies πii = − N j=1,i=j πij . Analysis and Synthesis of Singular Systems https://doi.org/10.1016/B978-0-12-823739-7.00014-8

Copyright © 2021 Elsevier Inc. All rights reserved.

185

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Analysis and Synthesis of Singular Systems

As that in [245], the assumption rank(E, Ci ) = rank(E) for every i ∈ N is needed. Considering this assumption with other conditions, the SSMS in (7.1) is guaranteed to have a solution, which is impulse free. The conservatism of this assumption is less than in previous work, such as [69], and the details can be referred to in Remark 7 in [245].

7.2 Admissibilization of SSMSs The matrix variables replacement method is employed to deal with the admissibility of SSMSs in this section. Based on the admissibility criterion, the admissibilization problem is solved. Firstly, an existing admissibility criterion of system (7.1) is provided. Lemma 7.1. [245] For u(t) = 0, system (7.1) is mean square admissible if there exists a matrix Xi for each i ∈ N such that the following coupled matrix inequalities are satisfied ∀i ∈ N :

sym(XiT Ai ) +

N 

ET Xi = XiT E ≥ 0,

(7.3)

πij ET Xj + CiT E+ ET Xi E+ Ci < 0.

(7.4)

T

j=1

Remark 7.2. Notice that the nonstrict LMI and equality constraint exist in Lemma 7.1, which will cause numerical difficulties in checking the nonstrict inequality condition, because of round-off errors in numerical calculation and fragility and nonperfect satisfaction of equality constraints. Now a new admissibility condition of system (7.1) is given in the following theorem: Theorem 7.3. For each i ∈ N , there exist a matrix Pi = PiT , an invertible matrix Φi , such that the following LMIs hold: ELT Pi EL > 0, sym((Pi E + U T Φi ΛT )T Ai ) +

N 

(7.5)

πij ET Pj E

j=1

+CiT E

+T

ET Pi EE+ Ci < 0,

(7.6)

where EL , U, and Λ are defined in Lemma 2.4, then system (7.1) is mean square admissible.

Sliding mode control of singular stochastic Markov jump systems

187

Proof. We first prove that the matrix Xi , for each i ∈ N in Lemma 7.1, is nonsingular. Due to ET Xi ≥ 0, we have sym(XiT Ai ) +

N 

πij ET Xj < 0.

(7.7)

j=1

If matrix Xi is noninvertible, there is a nonzero vector ξi for each i ∈ N leading to Xi ξi = 0. Then for the nonzero vector ξi , we have the following inequality from (7.7): N 

πij ξiT ET Xj ξi + πii ξiT ET Xi ξi + ξiT XiT Ai ξi + ξiT ATi Xi ξi

j=1,j=i

=

N 

πij ξiT ET Xj ξi < 0.

(7.8)

j=1,j=i

Due to ET Xj ≥ 0 and πij ≥ 0, i = j, the inequality in (7.8) cannot hold, which implies the inequality in (7.7) does not hold. Based on the discussions, matrix Xi , for each i ∈ N , is nonsingular. Then combining Lemma 2.5, equivalent conditions of inequalities (7.3) and (7.4) in Lemma 7.1 are obtained in (7.5) and (7.6), which guarantee the mean square admissibility of system (7.1). Remark 7.4. The sufficient condition in Theorem 7.3 is given in the form of strict LMIs, which are reliable and tractable in numerical computation. Although the strict LMIs are also proposed in Proposition 2 of [181] and in Theorem 3 of [245] for admissibility of SSMSs, the condition in Theorem 7.3 does not require positive definite matrix Pi for each i ∈ N , which is needed in Proposition 2 of [181] and in Theorem 3 of [245]. From this point of view, the result in Theorem 7.3 is less conservative. Additionally, the condition in Theorem 7.3 is easily applied to the problem of admissibilization, which can be discussed in the theorem that follow. Now we are ready to tackle with the admissibilization problem for system (7.1) with u(t) = K (rt )x(t) such that the resultant closed-loop system Edx(t) = (A(rt ) + B(rt )K (rt ))x(t)dt + C (rt )x(t)dw (t) is mean square admissible.

(7.9)

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Analysis and Synthesis of Singular Systems

Theorem 7.5. If there is a matrix P¯ i = P¯ iT , an invertible matrix Φ¯ i , and a matrix Hi such that the following LMIs hold for each i ∈ N : ⎡ Υi ⎢ ⎣

ERT P¯ i ER > 0, YiT CiT E+ T ER

YiT Ωi −Ψi

(7.10)

⎤ ⎥

0 ⎦ < 0, T ¯ −ER Pi ER

(7.11)

where Υi =sym(Ai Yi + Bi Hi ) + πii EP¯ i ET , Yi = P¯ i ET + ΛΦ¯ i U ,  √ √ √ √ Ωi = πi1 ER πi2 ER · · · πi(i−1) ER πi(i+1) ER ···

√



πiN ER

,

Ψi =diag ERT P¯ 1 ER , ERT P¯ 2 ER , . . . , ERT P¯ i−1 ER , ERT P¯ i+1 ER ,  . . . , ERT P¯ N ER .

ER , U, and Λ are given in Lemma 2.5, then the resultant closed-loop system (7.9) is mean square admissible, and the desired state-feedback controller is obtained by Ki = Hi Yi−1 . Proof. Based on Theorem 7.3, system (7.9) is mean square admissible if there is a matrix Pi = PiT , a nonsingular matrix Φi , guaranteeing that the following coupled LMIs hold for each i ∈ N :

+

N 

ELT Pi EL > 0, sym((Ai + Bi Ki )T (Pi E + U T Φi V T ))

(7.12)

πij ET Pj E + CiT E+ ET Pi EE+ Ci < 0.

(7.13)

T

j=1

Considering πij > 0, i = j, ELT Pi EL > 0, and applying Schur complement equivalence, inequality (7.13) is equivalent to the following inequality: ⎡ Γ1i ⎢ ⎣

Ωi −Ψi

⎤

⎡

⎤

Γ2i Ωi CiT E+ T ER CiT E+ T ER ⎥ ⎢ ⎥ 0 0 ⎦ = ⎣ −Ψi ⎦ < 0, T T ¯ ¯ −ER Pi ER −ER Pi ER

where Γ1i =sym((Ai + Bi Ki )T (Pi E + U T Φi V T )) + πii ET Pi E,

Sliding mode control of singular stochastic Markov jump systems

189

Γ2i =sym((Ai + Bi Ki )T (Pi E + U T Φi V T )) + πii ET (Pi E + U T Φi V T ).

Then performing congruence transformation to above inequality with matrix ⎡

⎤

Yi 0 0 ⎢ ⎥ ⎣ 0 I 0⎦ , 0 0 I in which Yi = P¯ i ET + ΛΦ¯ i U = (Pi E + U T Φi V T )−1 , and setting Hi = Ki Yi , the inequality in (7.11) is obtained. Because of ERT P¯ i ER = (ELT Pi EL )−1 , the inequalities in (7.10) and (7.12) are equivalent. Remark 7.6. For SSMSs, a sufficient admissibilization condition by statefeedback control is proposed in Theorem 7.5, whereas only the analysis result is provided in Theorem 3 in [245]. Although strict LMIs are given in Theorem 3 in [245], it is difficult to apply on state-feedback control problem, because matrices Ki (Pi E + FQi )−1 , Pi−1 , and (Pi E + FQi )−1 will appear simultaneously when the state-feedback controller is designed. When there is no the Brownian motion w (t), t > 0, then the resultant system in (7.9) will become the following SMS: Ex˙ (t) = (Ai + Bi Ki )x(t).

(7.14)

Combining the admissibility condition in Theorem 10.1 in [209], and by employing similar procedure in Theorem 7.5, a necessary and sufficient admissibility condition of the system in (7.14) will be obtained: Corollary 7.7. [47] The system in (7.14) is stochastically admissible if and only if there exist a matrix P¯ i = P¯ iT , an invertible matrix Φ¯ i , and a matrix Hi , resulting in the following LMIs holding for each i ∈ N : 

ERT P¯ i ER > 0, sym(Ai Yi + Bi Hi ) + πii EP¯ i ET

YiT Ωi −Ψi



< 0,

(7.15) (7.16)

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Analysis and Synthesis of Singular Systems

where Yi , Ωi , Ψi are defined in Theorem 7.5. Moreover, ER , U, and Λ are the same as that in Lemma 2.5. Then, we can design the state-feedback controller by Ki = Hi Yi−1 .

7.3 Application to SMC The observer-based SMC problem is solved firstly when the system involves unmeasured states in this section. When the states are available, the SMC law is also developed to assure the admissibility of resultant closed-loop system. The obtained results in Theorem 7.3 and Corollary 7.7 will be used to solve the observer-based SMC problem for SSMSs. For this aim, the following observer is designed to estimate state of system (7.1): 

Ex˙˘ (t) = Ai x˘ (t) + Bi u(t) + Li (y(t) − y˘ (t)) y˘ (t) = Di x˘ (t),

(7.17)

where x˘ (t) is the estimation of state x(t), and we will design the observer Li later. By denoting the error e(t) = x(t) − x˘ (t), we write the estimation error dynamics from systems (7.1) and (7.17) in the following formula: Ede(t) = (Ai − Li Di )e(t)dt + Ci x(t)dw (t).

(7.18)

For every Markov mode i ∈ N , the integral sliding surface function will be chosen as the following form: 

s(t) = Gi Ex˘ (t) −

t

Gi (Ai + Bi Ki )˘x(δ)dδ,

(7.19)

0

where controller gain Ki is designed to guarantee Ex˙˘ (t) = (Ai + Bi Ki )˘x(t)

(7.20)

to be stochastically admissible. Moreover, the matrix Gi is chosen to satisfy that Gi Bi is nonsingular. Remark 7.8. It can be seen from systems (7.14) and (7.20) that the controller gain Ki can be designed by using Corollary 7.7. Although the work in [43], [182] provides a controller design approach, the conditions given there are sufficient only, which may have some conservatism. Corollary 7.7 provides

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Sliding mode control of singular stochastic Markov jump systems

a necessary and sufficient admissibilization result, which shows the novelty of the result. Following SMC theory, when the system states reach onto the sliding surface, we have s(t) = 0 and ˙s(t) = 0, which implies the equivalent control input can be chosen as ueq (t) = Ki x˘ (t) − (Gi Bi )−1 Gi Li Di e(t).

(7.21)

Replacing u(t) in (7.17) with (7.21), the resultant SMDs in the state space of x˘ (t) are obtained: Ex˙˘ (t) = (Ai + Bi Ki )˘x(t) + (I − Bi (Gi Bi )−1 Gi )Li Di e(t).

(7.22)

Considering the system in (7.18), the SMDs both in x˘ (t) and e(t) are 





Edx˘ (t) = (Ai + Bi Ki )˘x(t) + (I − Bi (Gi Bi )−1 Gi )Li Di e(t) dt Ede(t) = (Ai − Li Di )e(t)dt + Ci x(t)dw (t). 

(7.23)



x˘ (t) , the SMDs are rewritThen by denoting augmented state as η(t) = e(t) ten as ¯ i η(t)dt + C ¯ i η(t)dw (t), E dη(t) = A

(7.24)

where 





−1 E 0 ¯ i = Ai + Bi Ki (I − Bi (Gi Bi ) Gi )Li Di , A E= 0 Ai − Li D i 0 E







0 0 . C¯ i = Ci Ci ¯ i ) = rank(E ) = 2r, where r = rank(E), the rank constraint Because rank(E , C above Section 7.2 is still satisfied. Based on Theorem 7.3, the observer gain Li design method is given in the following theorem to guarantee the admissibility of SMDs in (7.24):

Theorem 7.9. Given scalars μ1 > 0, μ2 > 0, μ3 > 0, if there exist matrices Pi > 0, Zi = ZiT , L¯ i and an invertible matrix Y¯ i such that following LMIs are satisfied ∀i ∈ N :

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Analysis and Synthesis of Singular Systems

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Π11

Π12 Π22

Π13

Π14

Π15

0 T −Bi Pi Bi

0 0 −P i

0 0 0 −P i







ELT Zi EL > 0, ⎤

(7.25)

⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎦

(7.26)

0

Π26

0 0 0 T −Bi Pi Bi

where

  μ1 L¯ i Di μ1 sym (Pi (Ai + Bi Ki )) Π11 = μ2 sym(Pi Ai − L¯ i Di ) +

N 

¯ i, ¯ T (E + )T E T Zi EE + C πij E T Zj E + C i

j=1

  (Ai + Bi Ki )T Pi − μ1 Pi 0 , + μ3 (ATi PiT − DiT L¯ iT ) − μ2 Pi DiT L¯ iT       √ √ Pi Bi 0 0 Π13 = μ1 , Π14 = μ1 , Π15 = , DiT L¯ iT DiT L¯ iT 0     0 Pi Bi −2Pi ¯ T Y¯ i Λ¯ T . Π22 = , Π26 = , Qi = Zi E + U −2μ3 Pi 0 0 Π12 =QiT

Similar as in Lemma 2.4, E = EL ERT meets rank(EL ) = rank(ER ) = 2r and ¯ ∈ R(2n−2r )×2n with rank(U ¯ ) = 2n − 2r and Λ¯ ∈ R2n×(2n−2r ) EL , ER ∈ R2n×2r . U ¯ = 2n − 2r satisfy U ¯ E = 0 and E Λ¯ = 0, respectively. Then system with rank(Λ) (7.24) with Gi = BiT Pi is mean square admissible. Then the observer gain Li can be obtained by Li = Pi−1 L¯ i . Proof. Let Gi = BiT Pi . Due to matrix Pi > 0 and matrix Bi with full column rank, the matrix Gi Bi is nonsingular. By using the Schur complement equivalence and noting that L¯ i = Pi Li , the inequality in (7.26) is equivalent to  Π¯ 11

where Π¯ 11 =Π11 +

 Π12 < 0, Π¯ 22

 μ1 Pi Bi (BiT Pi Bi )−1 BiT Pi

0

0 (μ1 + 1)DiT LiT Pi Li Di

(7.27) 

Sliding mode control of singular stochastic Markov jump systems

193





Pi Bi (BiT Pi Bi )−1 BiT Pi 0 . Π¯ 22 =Π22 + 0 0 On the other hand, due to 



Pi I ≥0 I Pi−1

by premultiplying and postmultiplying the above inequality by 

Pi Bi (BiT Pi Bi )−1 BiT 0

0 T Di LiT Pi



and its transpose, the following inequality is obtained: 



Pi Bi (BiT Pi Bi )−1 BiT Pi Pi Bi (BiT Pi Bi )−1 BiT Pi Li Di ≥ 0. DiT LiT Pi Li Di

(7.28)

Similarly, the following inequality holds: 



DiT LiT Pi Li Di DiT LiT Pi Bi (BiT Pi Bi )−1 BiT Pi ≥ 0. Pi Bi (BiT Pi Bi )−1 BiT Pi

(7.29)

Combining inequalities (7.27) to (7.29), we have  Π˜ 11 Π˜ =

 Π˜ 12 < 0, Π22

(7.30)

where 







0 −μ1 Pi Bi (BiT Pi Bi )−1 BiT Pi Li Di Π˜ 11 =Π11 + 0

0 0 . Π˜ 12 =Π12 + −DiT LiT Pi Bi (BiT Pi Bi )−T BiT Pi 0 Noting that Gi = BiT Pi , the inequality in (7.30) is rewritten as  Π˜ =

Πˆ i



¯ T F¯ T − G ¯i Qi + A i i < 0, T −F¯ i − F¯ i

(7.31)

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Analysis and Synthesis of Singular Systems

where N 

¯ i) + ¯ iA Πˆ i =sym(G ¯i= G

¯ i, ¯ T (E + )T E T Qi E + C πij E T Qj + C i

j=1



 μ1 Pi





0 Pi 0 ¯ T Y¯ i Λ¯ T . , F¯ i = , Qi = Zi E + U 0 μ3 Pi μ2 Pi

0

Then carrying  out the pre- and postmultiplications on inequality (7.31) ¯ T and its transpose, we have with matrix I A i ¯ T Qi ) + sym(A

N 

i

¯ i < 0. ¯ T (E + )T E T Qi E + C πij E T Qj + C i

j=1

Combining with inequality (7.25), we can see that there exists a matrix Zi = ZiT , an invertible matrix Y¯ i , assuring the following LMIs to be satisfied ∀i ∈ N : ELT Zi EL > 0,

¯ i) sym((Zi E + U¯ T Y¯ i Λ¯ T )T A +

N 

¯ i < 0. ¯ T (E + )T E T Zi EE + C πij E T Zj E + C i

j=1

Based on Theorem 7.3, the mean square admissibility of system (7.24) is guaranteed. Remark 7.10. An invertible matrix Y¯ i can be obtained by solving the LMI in (7.26). If the obtained matrix Y¯ i is singular, we can tune one of the elements of matrix Y¯ i slightly such that the new Y¯ i is invertible, which does not affect the holding of inequality (7.26). Remark 7.11. Although some observer-based SMC conditions have been published for different systems in literature references, such as Markov neutral-type stochastic systems [75], stochastic time-delay systems [106], no results about observer-based SMC for SSMSs have been reported. The result in Theorem 7.9 provides an effective method, and the condition is given in the form of strict LMIs, which is one of contributions in this case. Remark 7.12. The system considered in this section is without uncertainty. Consider an uncertain system: Edx(t) = (Ai + Ai (t))x(t) + Bi u(t))dt + Ci x(t)dw (t),

Sliding mode control of singular stochastic Markov jump systems

195

¯ i Fi (t)N ¯ i . Here where Ai (t) is parameter uncertainty satisfying Ai (t) = M ¯ ¯ Mi and Ni are known constant matrices, and Fi (t) is an unknown matrix function satisfying FiT (t)Fi (t) ≤ I. The main technique in this manuscript is to use the equivalent sets in Lemma 2.5 to solve the admissibilization and SMC problems. The appearing of uncertainty does not affect the utilization of this technique. For this uncertain system, the basic admissibility criterion will be obtained by replacing matrix Ai in (7.4) with Ai + Ai (t), that is,

sym(XiT Ai (t)) + Θi < 0,

(7.32)

where Θi = sym(XiT Ai ) +

N 

πij ET Xj + CiT E+ ET Xi E+ Ci . T

j=1

By using Lemma 1 in [105], the inequality in (7.32) can be guaranteed by the following inequality:  ¯i ¯ iT N Θi + ε N



¯i XiT M < 0, −ε I

(7.33)

where ε is a positive scalar. Combining inequality (7.33) and the inequality in (7.3), the admissibility condition of SSMSs with uncertainty is obtained. Then using the two equivalent sets and following similar line as that in the section, the admissibilization and SMC problems of uncertain SSMSs will be solved. Remark 7.13. The reason of choosing the special structures is to make the inequality in (7.26) easier to be solved, because it is LMI when scalars μi , i = 1, 2, 3 are given. The obtaining LMI condition introduce some con¯ i and F¯ i are not completely free. Different servatism, since the matrices G ¯ i and F¯ i will create different conservatism, and thus structures of matrices G no inclusion property can be found. We now design the SMC law: u(t) = Ki x˘ (t)+

N 

πij (Gj Bj )−1 s(t) − (λ + ε(t))sign(s(t)),

j=1

where ε(t) = max( (Gi Bi )−1 Gi Li y(t) + (Gi Bi )−1 Gi Li Di x˘ (t) ), i ∈N

(7.34)

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Analysis and Synthesis of Singular Systems

Gi =BiT Pi , λ > 0. In the following result, the trajectory of the observer in (7.17) being driven on to the designed sliding surface s(t) = 0 in a finite time will be proved. Theorem 7.14. By utilizing SMC law (7.34) and integral sliding surface (7.19), the state trajectories of (7.17) will be driven onto the sliding surface s(t) = 0 in a finite time. Proof. Constructing Lyapunov function: 1 V (t) = sT (t)(Gi Bi )−1 s(t). 2 From (7.19), we have ˙s(t) = − Gi Bi (λ + ε(t))sign(s(t)) + Gi Li (y(t) − y˘ (t)) + Gi Bi

N 

πij (Gj Bj )−1 s(t).

j=1

Let L be the weak infinitesimal generator. Then considering |s(t)| ≥ s(t) , we have LV (t)

=sT (t)(Gi Bi )−1 ˙s(t)+sT (t)

N 

πij (Gj Bj )−1 s(t)

j=1

= − s (t)(λ + ε(t))sign(s(t)) + sT (t)(Gi Bi )−1 Gi Li (y(t) − y˘ (t)) T

≤ − (λ + ε(t)) s(t) + (Gi Bi )−1 Gi Li y(t) s(t) + (Gi Bi )−1 Gi Li Di x˘ (t) s(t) ≤ − λ s(t) ,

which implies that system trajectories of (7.17) converge to the sliding surface in a finite time due to λ > 0. When the states are measurable, the observer is not needed, and the SMC can be solved following the similar lines as that in [181]. Therefore the process will not be given in detail here. When all the states of system (7.1) are available, the sliding surface function can be obtained directly without using the observer to estimate its state. Following the similar lines

Sliding mode control of singular stochastic Markov jump systems

197

as that in [181] to solve the SMC problem, SMDs can be obtained as in (7.9) there. To design the controller gain, Theorem 7.5 in this section can be utilized directly. For details, according to Subsection 3.1 of [181], the integral sliding surface function will be chosen as 

s(t) = Gi Ex(t) −

t

Gi (Ai + Bi Ki )x(δ)dδ,

0

where Gi Bi is nonsingular, and Gi Ci = 0. The SMC law is given as u(t) = Ki x(t) +

N 

πij (Gj Bj )−1 s(t) − λsign(s(t)),

j=1

where λ > 0.

7.4 Examples Two examples are provided to show the workability of admissibilization results by state feedback control and the observer-based SMC of SSMSs in this section, respectively. Example 7.1. Let us consider the modeling of a direct current (DC) motor controlling an inverted pendulum (depicted in Fig. 7.1), which has been modeled by the following SMS in [170]: Ex˙ (t) = Ai x(t) + Bi u(t),

Figure 7.1 Block diagram of a DC motor controller inverted pendulum.

(7.35)

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Analysis and Synthesis of Singular Systems

where

⎤

⎡

⎤

⎡

⎡

⎤

1 0 0 0 1 0 0.4 ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ E = ⎣0 1 0⎦ , A1 = ⎣ 9.8 0 1 ⎦ , B1 = ⎣−0.2⎦ , 0 0 0 −20 −3 −1 −0.5 ⎤

⎡

⎤

⎡

−1.2 0 1 0 ⎥ ⎥ ⎢ ⎢ A2 = ⎣ 9.8 0 1 ⎦ , B2 = ⎣ 0.5 ⎦ , −20 −3 −0.5 −0.2 ⎡

⎤

⎤

⎡

⎡

⎤

1 0 1 0 1 0 0 ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ + EL = ⎣0 1⎦ , ER = ⎣0 1⎦ , E = ⎣0 1 0⎦ , 0 0 0 0 0 0 0 ⎤ ⎡ ⎤ ⎡   0 0 −0.6 0.6 ⎥ ⎢ ⎥ ⎢ , U = ⎣ 0 ⎦, Λ = ⎣ 0 ⎦. π= 0.2 −0.2 −0.7 0.8

Similarly, as examples in [245], if the system matrix Ai is affected by some random environmental effects, such as Ai becoming Ai + Ci “noise”, the system in (7.35) will be Edx(t) = (Ai x(t) + Bi u(t))dt + Ci x(t)dw (t). Set

⎤

⎡

(7.36) ⎤

⎡

0.2 0 1 1 0 0.1 ⎥ ⎥ ⎢ ⎢ C1 = ⎣0.1 0.2 0⎦ , C2 = ⎣0.2 0.1 0 ⎦ . 0 0 0 0 0 0 For system (7.36) with u(t) = 0, by solving the LMIs in Theorem 7.3, no feasible solutions can be found. By using admissibilization criterion in Theorem 7.5, the LMIs in (7.10) and (7.11) are feasible and the statefeedback controllers are



K1 = −38.9127 −7.1955 −18.8725 , 



K2 = −18.4065 −1.1594 −1.6126 , which implies the closed-loop system is mean square admissible. To check the effectiveness of the feedback control method, on the other hand, for the closed-loop system Edx(t) = (Ai + Bi Ki )x(t)dt + Ci x(t)dw (t)

Sliding mode control of singular stochastic Markov jump systems

199

with the given parameters and the obtained controllers K1 and K2 , the result in Theorem 7.3 will be applied. For this system, by solving the LMIs in (7.5) and (7.6), the following feasible solutions can be obtained: ⎡

⎤

⎡

⎤

0.7597 0.6504 0.2792 ⎢ ⎥ P1 = ⎣0.6504 0.6007 0.2291⎦ , Φ1 = 0.1909, 0.2792 0.2291 0 0.7020 0.5730 0.2904 ⎥ ⎢ P2 = ⎣0.5730 0.5254 0.2131⎦ , Φ2 = 0.1112, 0 0.2904 0.2131 which still shows the closed-loop system is mean square admissible and that the admissibilization method is effective. Remark 7.15. By now, only a few results about SSMSs have been reported, such as [245] and [181]. In [245], only the stability condition is proposed and the state-feedback controller design is not considered. One method of designing controller is presented in Theorem 1 in [181]. However, the controller design condition in [181] is based on the stability criterion Lemma 1 there, which is cited from [5]. The authors in [240] have pointed out that the stability condition in [5] is improper. Example 7.2. In this example, the effectiveness of SMC method is demonstrated. The considered SSMS is given with the following parameters: 











1 1 0.2 0.7 0.1 0.1 , A1 = , A2 = , E= 0 0 0.4 −0.5 0.3 −2 













T



−1 1 0.4 0.2 0.2 0.4 , C2 = , π= , C1 = 0 0 0 0 2 −2 





1 −2.5 −0.6 , B2 = , D1 = B1 = 1 −9 −0.9  

 

T  −0.3 , D2 = , 0.7  

 1 1 1 , ER = , U = 0 1 , Λ= . EL = −1 0 1

An observer in (7.17) will be designed to estimate state of system (7.1) and apply the estimation state to design an SMC law u(t) in (7.34), such that

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overall SMDs are mean square admissible. Firstly, according Remark 7.8, by using Corollary 7.7, the controller Ki can be designed as







K1 = 1.3847 2.6201 , K2 = 5.5530 0.8508 . Then Theorem 7.9 will be sued to obtain the observer gain Li . By solving the LMIs in (7.25) and (7.26) of Theorem 7.9 with the following parameters:    −1 1 0 1 0 0 T ¯ ¯ , Λ = U=

0 0 0 1

⎡

1 0

0

⎡

⎤



0 0 , 0 −1 1

⎤

1 0

⎢0 0 ⎥ ⎢1 0⎥ ⎢ ⎢ ⎥ ⎥ EL = ⎢ ⎥ , ER = ⎢ ⎥ , μ1 = 3, μ2 = 5, μ3 = 1.5, ⎣0 1 ⎦ ⎣0 1⎦

0 0

0 1

the feasible variables are given 







0.1003 0.0308 0.0746 0.1143 P1 = , P2 = , 0.0308 0.0430 0.1143 0.2532     − − 0 . 1476 0 . 4231 , L¯ 2 = , L¯ 1 = −0.1863 −0.4424     −0.1793 −9.7293 −1 ¯ −1 ¯ , L2 = P 2 L2 = , L 1 =P 1 L 1 = −4.2050 2.6459  G1 =B1T P1 = −0.1770 −0.3561 ,  G2 =B2T P2 = −0.0721 −0.0326 .

Based on obtained controller and observer, the system parameters of SMDs in (7.24) are given as ⎡

1 ⎢0 ⎢ E =⎢ ⎣0 0

1 0 0 0

0 0 1 0

⎤

0 0⎥ ⎥ ⎥, 1⎦ 0

Sliding mode control of singular stochastic Markov jump systems

⎡

201

⎤

1.5847 3.3201 0.4106 0.6159 ⎢−12.0623 −24.0809 −0.2041 −0.3062⎥ ⎥ ¯ 1 =⎢ A ⎢ ⎥, 0 0 0.0924 0.5386 ⎦ ⎣ −2.1230 −4.2845 0 0 ⎡

⎤ −13.7825 −2.0270 −0.2064 0.4817 ⎢ 5.8530 −1.1492 0.4563 −1.0647⎥ ⎥ ¯ 2 =⎢ A ⎢ ⎥, −2.8188 6.9105 ⎦ 0 0 ⎣ 0 0 1.0938 −3.8522 ⎡ ⎡ ⎤

⎤

0 0 0 0 0 0 0 0 ⎢ 0 ⎢ ⎥ 0 0 0⎥ ¯ 0 0 0⎥ ⎢ ⎢ 0 ⎥ C¯ 1 = ⎢ ⎥ , C2 = ⎢ ⎥. ⎣0.4 0.2 0.4 0.2⎦ ⎣0.2 0.4 0.2 0.4⎦ 0 0 0 0 0 0 0 0 By solving the conditions in Theorem 7.3 for system (7.24) with above parameters, the feasible variables can be found, which implies that overall SMDs (7.24) are mean square admissible.

7.5 Conclusion The chapter is to study the SMC with or without the observer for SSMSs in this section. Firstly, sufficient admissibility and admissibilization criteria in terms of strict LMIs for SSMSs are proposed by using matrix replacement approach. Based on these criteria, the observer-based SMC problem and SMC problem without using the observer are solved, respectively. The workability of the proposed methods about admissibilization and SMC has been illustrated by numerical examples. On the other hand, the Markovian jump systems with incomplete transition rate have attracted considerable attention in recent years. By designing an SMC law with different robust terms for known and unknown modes, respectively, a good SMC result proposed in [17] for stochastic Markovian jump systems has been extended to stochastic Markovian jump systems with incomplete transition rate in [16]. Combining the technique provided in [16] and the two equivalent sets in this section, the observer-based SMC problem will be studied in our future work for SSMSs with incomplete transition rate. Moreover, similar to some existing results such as [95], [109], [182], the controller gain Ki and observer gain Li are designed separately. For how to design the controller gain Ki and observer gain Li simultaneously, which is also an interesting topic, is addressed in our future work.

CHAPTER 8

Admissibility and admissibilization for fuzzy singular systems The issues of admissibility analysis for T-S fuzzy singular system with time delay and admissibilization for IT2 fuzzy singular system are investigated in this chapter, respectively. By adopting a novel tighter integral inequality, a sufficient delay-dependent criterion is built on the basis of strict LMIs, which guarantees the admissibility of the T-S fuzzy singular system. And by designing the state feedback controller and static output feedback controller, the problems of admissibility analysis and admissibilization for continuous singular IT2 fuzzy systems have been studied in the second subchapter. Several numerical examples show that the proposed methods are efficient and less conservative.

8.1 Admissibility analysis for Takagi–Sugeno fuzzy singular systems with time delay This section aims to study the problem of admissibility of T-S fuzzy singular system with time delay by adopting a novel integral inequality, which uses a double integral of the system state and includes the Wirtinger-based inequality. Based on this inequality, a delay-dependent sufficient criterion is put forward to guarantee the admissibility of the considered system. A numerical example shows that the proposed method is efficient and less conservative.

8.1.1 Problem formulation Consider a class of nonlinear singular system with time delay that is denoted by T-S fuzzy singular model as follows: Plant Rule i: IF θ1 (t) is μi1 ; θ2 (t) is μi2 ; ... and θp (t) is μip , THEN Ex˙ (t) = Ai x(t) + Adi x(t − d), x(t) = φ(t), t ∈ [−d, 0], i ∈ S, Analysis and Synthesis of Singular Systems https://doi.org/10.1016/B978-0-12-823739-7.00015-X

Copyright © 2021 Elsevier Inc. All rights reserved.

(8.1) 203

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where x(t) ∈ Rn is the state vector; θ (t) = [θ1 (t), θ2 (t), ..., θp (t)] is the presumed variable; μij is the fuzzy set; S = {1, 2, 3, ..., r }, and r denotes the number of IF-THEN rules; E probably is singular, and rank(E) = g ≤ n; d is the known constant time delay satisfying 0 < dmin ≤ d ≤ dmax ; φ(t) means a compatible vector-value initial function, which satisfies certain strict consistent condition and guarantees a unique solution for any sufficiently differentiable input function [34]; Ai , Adi represent known real constant matrices. Afterwards, the whole model of the above systems can be described by the following model: Ex˙ (t) =

r 

hi (θ (t))[Ai x(t) + Adi x(t − d)],

i=1

x(t) = φ(t), t ∈ [−d, 0], i ∈ S,

(8.2)

where p  ωi (θ (t)) hi (t) = r , ωi (θ (t)) = μij (θ (t)), i=1 ωi (θ (t )) j=1

and μij (θ (t)) denotes the grade of membership of θj (t) in μij . Clearly, for all t, one can see hi (θ (t)) ≥ 0,

r 

hi (θ (t)) = 1.

i=1

Definition 8.1. [215]  (i) If det(sE − ri=1 hi (θ (t))Ai ) is not identically zero, the singular system (8.2) is regarded as regular.  (ii) If deg(det(sE − ri=1 hi (θ (t))Ai ))=rank(E), the singular system (8.2) is referred to as impulse free. (iii) If, for any ε > 0, there exists a scalar δ(ε) > 0 such that, for any compatible initial conditions φ(t) satisfying sup−dmax ≤t≤0  φ(t) < δ(ε), the solution x(t) of system (8.2) satisfies  x(t) < ε for t ≥ 0, moreover, limt→+∞ x(t) = 0, the singular system (8.2) is said to be asymptotically stable. (iv) If the singular system (8.2) is regular, impulse-free, and asymptotically stable, it is referred to as admissible.

Admissibility and admissibilization for fuzzy singular systems

205

The following nomenclature is adopted to simplify vector and matrix symbolizations: 

v1 (t) = η1 = ξ(t) =

ei = Γi =

   

t

t−d



x(s)ds, v2 (t) =

xT (t)ET

t

s

t−d t−d

v1T (t)ET

xT (t) xT (t − d)



x(u)duds,

v2T (t)ET

T

1 T T d v1 (t )E



0n×(i−1)n In 0n×(4−i)n



Ai Adi 0n×n 0n×n

, 2 T v (t)ET d2 2

T ,

, i = 1, 2, 3, 4,

, i ∈ S.

Lemma 8.2. Presume x as a differentiable function: [α, β] −→ Rn . For symmetric matrices S ∈ Rn×n > 0, N1 , N2 , N3 ∈ R4n×n , and E ∈ Rn×n (rank(E) = g ≤ n), the following inequality holds:  −

β

x˙ T (s)ET SEx˙ (s)ds ≤ ϑ T Ωϑ,

(8.3)

α

where 1 1 3 5 ET N1 Π1 +  ET N2 Π2 +  ET N3 Π3 ), + sym(

Ω =τ ( ET N1 S−1 N1T  E+  ET N2 S−1 N2T  E+  ET N3 S−1 N3T  E) Π1 =E(e1 − e2 ), Π2 = E(e1 + e2 ) − 2e3 , Π3 = E(e1 − e2 ) − 6e3 + 6e4 ,  E =diag(E, E, I , I ), E = diag( E,  E,  E), T 



ϑ = xT (β) xT (α) τ1 αβ (Ex(s))T ds τ22 αβ αs (Ex(u))T duds , τ = β − α.

Proof. Define f1 (s) =

2s − β − α

, β −α 6s2 − 6(β + α)s + β 2 + 4βα + α 2 f2 (s) = , (β − α)2 T  N = N1T N2T N3T , T  ζ (s) = ϑ T f1 (s)ϑ T f2 (s)ϑ T .

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Analysis and Synthesis of Singular Systems

It is clear that − 2ζ T (s) ET NEx˙ (s) ≤ ζ T (s) ET NS−1 N Eζ (s) + x˙ T (s)ET SEx˙ (s).

(8.4)

Integrating (8.4) from α to β derives − 2ϑ T  ET N1 E(e1 − e2 )ϑ − 2ϑ T  ET N2 (Ee1 + Ee2 − 2e3 )ϑ − 2ϑ T  ET N3 (Ee1 − Ee2 − 6e3 + 6e4 )ϑ ≤(β − α)ϑ T  ET N1 S−1 N1T  Eϑ + +

(β − α)

5

(β − α)

ϑT ET N3 S−1 N3T  Eϑ +

3



β

ϑT ET N2 S−1 N2T  Eϑ

·T

·

x (s)ET SEx(s)d(s).

(8.5)

α

Rearranging (8.5) derives (8.3), so the proof is completed. Remark 8.3. When E = I, Lemma 8.2 becomes into Lemma 1 in [228]. If rank(E) = g ≤ n, Lemma 8.2 can be used in singular systems, which shows that Lemma 8.2 has a wider application. The prime object of this subchapter is to put forward a new admissibility criterion of the considered systems and reduce the conservatism of existing results by utilizing this new inequality technique in Lemma 8.2.

8.1.2 Main results Theorem 8.4. Given a constant time delay d ∈ [dmin , dmax ], the T-S fuzzy singular systems with time delay in (8.2) is admissible, if, for all i ∈ S, there exist P ∈ R3n×3n > 0, Q ∈ Rn×n > 0, S ∈ Rn×n > 0, W ∈ Rn×g , and matrices N1 , N2 , N3 ∈ R4n×n , making the following LMIs hold: ⎡ ⎢ ⎢ ⎢ ⎣

Ωi ∗ ∗ ∗

√

d ET N1 −S ∗ ∗

√

d ET N2 0 −3S ∗

√

ET N3 d 0 0 −5S

⎤ ⎥ ⎥ ⎥ < 0, ⎦

(8.6)

where R ∈ Rn×(n−g) is any matrix that is full column rank and satisfies ET R = 0, and Ωi =Ω1i + Ω2 + Ω3i , Ω1i =sym(Π4T P Π5i ) + e1T Qe1 − e2T Qe2 + dΓiT SΓi , Ω2 =sym( ET N1 Π1 +  ET N2 Π2 +  ET N3 Π3 ),

Admissibility and admissibilization for fuzzy singular systems

207

Ω3i =sym(e1T WRT Γi ), T  2 Π4 = e1T ET de3T d2 e4T , Π5i =

 ⎡

(e1T − e2T )ET d(e3T − e2T ET ) ⎤ ⎤ ⎡

Γi T

T ,

P11 P12 P13 Nja ⎥ ⎥ ⎢ ⎢ P = ⎣ ∗ P22 P23 ⎦ , Nj = ⎣ Njb ⎦ , j = 1, 2, 3, ∗ ∗ P33 Njc and Πi , i = 1, 2, 3,  E are defined in Lemma 8.2. Proof. First of all, system (8.2) will be proved to be regular and impulse-free. Due to rank(E) = g, it is certain that there are two nonsingular matrices F and H making the following equality stand: 

FEH = Set F

 r 





hi (θ (t))Ai H =

i=1

A11 A12 A21 A22



F

−T

R=

Ig 0 0 0



(8.7)

.



 , H W= T

W1 W2

 ,



0

M,

I(n−g)

(8.8)

where M ∈ R(n−g)×(n−g) is nonsingular. According to (8.6), it can be found Ωi < 0 such that 

Θ1i •

• •



< 0,

where “•” represents the elements in matrix that are not related to next discussions, and T Θ1i = ET P11 Ai + ATi P11 E + ET P12 E + ET P12 E + Q + dATi SAi T T + ET N1a E + ET N1a E + ET N2a E + ET N2a E T + ET N3a E + ET N3a E + WRT Ai + ATi RW T .

Because of Q > 0, S > 0, we have T T 1i = ET P11 Ai + ATi P11 E + ET P12 E + ET P12 Θ E + ET N1a E + ET N1a E

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Analysis and Synthesis of Singular Systems

T T + ET N2a E + ET N2a E + ET N3a E + ET N3a E

+ WRT Ai + ATi RW T < 0.

Due to hi (θ (t)) ≥ 0 and  Θ1 =ET P11

r 

r

i=1 hi (θ (t )) = 1,



hi (θ (t))Ai +

 r 

i=1

we have 

T hi (θ (t))ATi P11 E + ET P12 E + ET P12 E

i=1

T T T + ET N1a E + ET N1a E + ET N2a E + ET N2a E + ET N3a E + ET N3a E  r   r    + WRT hi (θ (t))Ai + hi (θ (t))ATi RW T < 0. i=1

i=1

Premultiply and postmultiply Θ1 < 0 by H T and H, respectively. Substituting (8.7) and (8.8) in the above inequality gets 

• • • W2 M T A22 + AT22 MW2T

 < 0.

T A + AT MW T < 0, From the above inequality, it is easy to see that W2 M 22 22 2    which means A22 is nonsingular. Hence, the pair E, ri=1 hi (θ (t))Ai is regular and impulse-free. Based on Definition 8.1, system (8.2) is regular and impulse-free. To prove the stability of system (8.2), choose Lyapunov–Krasovskii functional as follows:

V (xt ) = V1 (xt ) + V2 (xt ) + V3 (xt ), where V1 (xt ) =η1T (t)P η1 (t), 

V2 (xt ) =

t

xT (s)Qx(s)ds,

t−d 0 t



V3 (xt ) =

−d

t+u

x˙ T (s)ET SEx˙ (s)dsdu.

According to the solutions of system (8.2), take the derivative of V (xt ), and one can get V˙ 1 (xt ) =ξ T (t)sym(Π4T P Π5i )ξ(t), V˙ 2 (xt ) =ξ T (t)(e1T Qe1 − e2T Qe2 )ξ(t),

(8.9) (8.10)

209

Admissibility and admissibilization for fuzzy singular systems

V˙ 3 (xt ) =dξ T (t)ΓiT SΓi ξ(t) −



t

t−d

x˙ T (s)ET SEx˙ (s)ds.

(8.11)

Adopting Lemma 8.2, we have  −

t

t−d

x˙ T (s)ET SEx˙ (s)ds

 d T d T ≤ξ T (t) d ET N1 S−1 N1T  E+  E N2 S−1 N2T  E+  E N3 S−1 N3T  E 3 5  +sym( ET N1 Π1 +  ET N2 Π2 +  ET N3 Π3 ) ξ(t) =ξ T (t)(Ω2 + Ω4 )ξ(t),

(8.12)

where Ω4 = d ET N1 S−1 N1T  E + 3d  ET N2 S−1 N2T  E + 5d  ET N3 S−1 N3T  E. In addition, it is clear to see that 2xT (t)WRT Ex˙ (s) ≡ 0.

(8.13)

Considering the inequalities from (8.9) to (8.13), we have V˙ (xt ) ≤ ξ T (t)(Ω1i + Ω2 + Ω3i + Ω4 )ξ(t). According to Schur complement, Ω1i + Ω2 + Ω3i + Ω4 < 0 is equivalent to (8.6), so we have V˙ (xt ) < 0. Therefore system (8.2) is asymptotically stable. Then, we complete the proof of Theorem 8.4. The following corollary can simplify the computation by eliminating the three free matrices N1 , N2 , N3 in Theorem 8.4: Corollary 8.5. Given a constant time-delay d ∈ [dmin , dmax ], the T-S fuzzy singular systems with time delay in (8.2) is admissible, if there exist P ∈ R3n×3n > 0, Q ∈ Rn×n > 0, S ∈ Rn×n > 0, W ∈ Rn×g satisfying (8.6) with T 1 , −S S 0 0 d T 3 N2 = , −S −S 2S 0 d T 5 N3 = . −S S 6S −6S d

N1 =

Remark 8.6. The prime advantage of this section is to adopt a new inequality in Lemma 8.2, which is an improved version of Lemma 1 in [228] by adding the singular matrix E. Through employing three free matrices

210

Analysis and Synthesis of Singular Systems



N1 , N2 , and N3 to cope with the relationships among xt , xt−d , tt−d x(s)ds,



and tt−d ts−d x(u)duds, we obtain a less conservative result, which is much tighter than Jensen inequality. Corollary 8.5 gives a simple version of Theorem 8.4, which greatly reduces the number of decision variables. Although it is not more conservative than Theorem 8.4, it is a good trade-off between computation and conservatism. To compare some existing results, according to the approach in [64], we obtain a proposition relevant to the admissibility of the considered system. Proposition 8.7. Given an integer m ≥ 1, the T-S fuzzy singular system with time delay in (8.2) is admissible if, for all i ∈ S, there exist matrices Q > 0, R > 0,    P11 0 P= , U = U1 0n×(n−g) (with 0 < P11 ∈ Rg×g , and U1 ∈ P21 P22 Rn×g ), making the following LMIs hold: ⎡

Θi

⎢ ⎣  

d T m WRi R − md R

d T m WX U

0

⎤ ⎥ ⎦ < 0,

(8.14)

− md R



where Θi =sym(WXT P T Ai WX + WXT UWX + WXT P T Adi WD − WXT UWD ) + WQT QWQ ,    WX = In 0n,mn , WD = 0n,mn   

Q=

Q 0mn 0mn −Q

, WQ =



In

, WRi = 

Imn 0mn,n 0mn,n Imn





Ai 0n,(m−1)n Adi

,

.

8.1.3 Numerical example A numerical example is given to show the effectiveness of our approach in this section. Example 8.1. Consider the following T-S fuzzy singular system (8.2): Plant Rule i: IF θ1 (t) is μi1 , THEN Ex˙ (t) =Ai x(t) + Adi x(t − d), i = 1, 2,

Admissibility and admissibilization for fuzzy singular systems

where



E= 

A2 =

1 0 0 0



 , A1 =

1.6 0.3 0 −1.2



1.5 0.2 0 −1.1 

, Ad2 =



 , Ad1 =

−2.2 0.1 0.5 0.6



−2 0.2 0.6 0.5

211

 ,

.

Applying the approaches of Theorem 8.4, Corollary 8.5, and Proposition 8.7, the admissible maximum values of time delay dmax of different methods are calculated by utilizing LMI Tool-box. The results of dmax and the corresponding number of decision variables (NDVs) are shown in Table 8.1. Proposition 8.7 adopts the delay partitioning approach. It is clear to see that the results become less conservative, when the delay-partitioning number m gets larger. Nevertheless, the number of variables also increases, which means the computation becomes more complicated. The time-delay result of Theorem 8.4 is a little better than the result of Proposition 8.7 (when m = 6), and there are less variables in Theorem 8.4. Particularly, the time-delay result of Corollary 8.5 is close to the above two methods, but the number of variables decreases sharply. Considering both conservatism and computational burden, Corollary 8.5 is a good trade-off. Table 8.1 Comparison of the number of decision variables and the maximum admissible values dmax of different methods. Methods dmax NDVs Proposition 8.7 (m = 2) 0.1885 3.5n2 + 1.5n + 0.5g(g + 1) Proposition 8.7 (m = 4) 0.3771 9.5n2 + 2.5n + 0.5g(g + 1) Proposition 8.7 (m = 6) 0.5657 19.5n2 + 3.5n + 0.5g(g + 1) Theorem 8.4 0.5660 17.5n2 + 2.5n 0.5049 5.5n2 + 2.5n Corollary 8.5

8.1.4 Conclusion This subchapter concerns the admissibility of Takagi–Sugeno fuzzy timedelay singular system. According to a new integral inequality and Lyapunov method, a sufficient admissible criterion is developed to guarantee the regularity, absence of impulses and stability of the considered system. A numerical example demonstrates the advantages of this method in reducing the conservatism and computational burden. The proposed method is also easy to apply to uncertain systems, so the issue of the robust dissipative control of the considered system will be considered in our future work.

212

Analysis and Synthesis of Singular Systems

8.2 Admissibilization of singular IT2 fuzzy systems In this section, we consider the state feedback and static output feedback control problems of continuous singular IT2 T-S fuzzy systems with mismatched membership functions. Firstly, based on Lyapunov stability theory, a state feedback control criterion is proposed to guarantee the closed-loop system to be admissible. Secondly, the result is extend to static output feedback control problem, and a sufficient condition for synthesis is derived in terms of strict LMIs, which eliminates the disadvantages in some existing results, such as same output matrices and bilinear matrix inequality problem. To obtain less conservative results, the information of mismatched membership functions is employed. Finally, numerical examples are given to illustrate the effectiveness of the proposed techniques.

8.2.1 Preliminaries Consider the following singular IT2 fuzzy system described by: Plant Rule i (i = 1, 2, . . . , r): If f1 (x(t)) is Wi1 and · · · and fp (x(t)) is Wip , THEN 

Ex˙ (t) = Ai x(t) + Bi u(t) y(t) = Ci x(t),

(8.15)

where Wis is an IT2 fuzzy set of rule i corresponding to the function fs (x(t)), i = 1, 2, . . . , r, s = 1, 2, . . . , p, p is the number of premise variables; x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the input vector, and y(t) ∈ Rl is the measured output; Ai ∈ Rn×n , Bi ∈ Rn×m , and Ci ∈ Rl×n denote constant matrices. In contrast with standard linear systems with E = I, the matrix E ∈ Rn×n has 0 < rank(E) = q < n. The firing strength of the ith rule is of the following interval sets: ¯ i (x(t)) = [w i (x(t)), w¯ i (x(t))], i = 1, 2, . . . , r , W

where w i (x(t)) =

p  s=1 p

w¯ i (x(t)) =



μW (fs (x(t))) ≥ 0, is

μ¯ Wis (fs (x(t))) ≥ 0,

s=1

μ¯ Wis (fs (x(t)))≥ μW (fs (x(t))) ≥ 0, is

w¯ i (x(t)) ≥ wi (x(t)) ≥ 0

(8.16)

Admissibility and admissibilization for fuzzy singular systems

213

with μW (fs (x(t))) ∈ [0, 1] and μ¯ Wis (fs (x(t))) ∈ [0, 1] denote the lower grade is of membership and upper grade of membership governed by lower and upper membership functions, respectively. The inferred singular IT2 T-S fuzzy model is defined as follows: 



Ex˙ (t) = ri=1 wi (x(t))(Ai x(t) + Bi u(t)),  y(t) = ri=1 wi (x(t))Ci x(t),

(8.17)

where wi (x(t)) = α i (x(t))w i (x(t)) + α¯ i (x(t))w¯ i (x(t)) ≥ 0 ∀i, r 

wi (x(t)) = 1,

i=1

0 ≤ α i (x(t)) ≤ 1 ∀i, 0 ≤ α¯ i (x(t)) ≤ 1 ∀i, α i (x(t)) + α¯ i (x(t)) = 1 ∀i,

in which α i (x(t)) and α¯ i (x(t)) are nonlinear functions not necessarily known but exist, and wi (x(t)) denotes the grade of membership of the embedded membership functions α i (x(t)) and α¯ i (x(t)). Because the membership degrees are no longer crisp, but characterized by lower and upper membership degrees, the system in (8.17) belongs to type-2 fuzzy system. Remark 8.8. From the expression in (8.17), we can see that the linear combination of w i (x(t)) and w¯ i (x(t)) is used to describe the actual grades of membership wi (x(t)) scaled by the nonlinear functions. The uncertainty of nonlinear plant may lead to the uncertain wi (x(t)), α i (x(t)), and α¯ i (x(t)). The system model presented in (8.17) facilitates the stability and control problems, and is not necessarily implemented. The first purpose of this section is to design a state feedback IT2 fuzzy controller with r rules of the following format to admissibilize the IT2 T-S fuzzy system in (8.17): Controller Rule j: IF g1 (x(t)) is Mj1 and · · · and gp (x(t)) is Mjp , THEN u(t) = Kj x(t),

(8.18)

where Mjs stands for the jth fuzzy set of the function gs (x(t)), j = 1, 2, . . . , r, s = 1, 2, . . . , p; p is the number of premise variable; Kj ∈ Rm×n is the state

214

Analysis and Synthesis of Singular Systems

feedback gain matrix of rule j. The firing interval of the jth rule is as follows: ¯ j (x(t)) = [mj (x(t)), m M ¯ j (x(t))], j = 1, 2, . . . , r ,

(8.19)

where mj (x(t)) =

p  s=1 p

m¯ j (x(t)) =



μM (gs (x(t))) ≥ 0, js

μ¯ Mjs (gs (x(t))) ≥ 0,

s=1

μ¯ Mjs (gs (x(t)))≥ μM (gs (x(t))) ≥ 0, js

m¯ j (x(t)) ≥ mj (x(t)) ≥ 0 with μM (gs (x(t))) and μ¯ Mjs (gs (x(t))) denote the lower grade of memberjs ship, upper grade of membership, respectively. The overall IT2 state feedback control law is given by u(t) =

r 

mj (x(t))Kj x(t),

(8.20)

j=1

where β j (x(t))mj (x(t)) + β¯j (x(t))m ¯ j (x(t))  ≥ 0, mj (x(t)) =   r ¯ β ( x ( t )) m ( x ( t )) + β ( x ( t )) m ¯ ( x ( t )) k k k k=1 k r 

mj (x(t)) = 1,

j=1

0 ≤ β j (x(t)) ≤ 1 ∀j, 0 ≤ β¯j (x(t)) ≤ 1 ∀j, β j (x(t)) + β¯j (x(t)) = 1 ∀j,

in which β j (x(t)) and β¯j (x(t)) are predefined functions, and mj (x(t)) denotes the grades of membership of the embedded membership functions. For simple notation, wi (x(t)) and mj (x(t)) are denoted as wi and mj , respectively, in the analysis that follows. Then the closed-loop singular IT2 fuzzy system formed by the singular IT2 T-S fuzzy model of (8.17), and the IT2 fuzzy

Admissibility and admissibilization for fuzzy singular systems

215

controller of (8.20) can be expressed as Ex˙ (t) = As (w , m)x(t), 

(8.21)



where As (w , m) = A(w ) + B(w )K (m) = ri=1 rj=1 wi mj (Ai + Bi Kj ), A(w ) =   r   wi Ai , B(w ) = ri=1 wi Bi , K (m) = ri=1 mi Ki , ri=1 wi = 1, rj=1 mj = 1, i=1  and ri=1 rj=1 wi mj = 1. The second aim is to design the following form of static output feedback controller: u(t) =

r 

mi (x(t))Si y(t),

(8.22)

i=1

where mi (x(t)) defined in (8.20) and Si ∈ Rm×l , such that the closed-loop system given by Ex˙ (t) = Ao (w , m)x(t)

(8.23)

is admissible, where Ao (w , m) = A(w ) + B(w )S(m)C (w ) =

r  r r  

wi mj wk (Ai + Bi Sj Ck ),

i=1 j=1 k=1

C (w ) =

r  i=1

wi Ci , S(m) =

r 

mi Si ,

i=1

A(w ), and B(w ) are defined in (8.21).

8.2.2 State feedback control of singular systems In this section, the admissibilization by state feedback control of singular IT2 fuzzy systems is addressed by employing the two equivalent sets. Theorem 8.9. The singular IT2 fuzzy system in (8.21) is admissible if the membership functions of the fuzzy model and fuzzy controller satisfy mj − ρj wj ≥ 0 for all j with 0 < ρj < 1, and there exist matrices Q < 0, P, i , Ki , i = 1, 2, . . . , r such that the following matrix inequalities hold: ET P = P T E ≥ 0, sym[(Ai + Bi Kj )T P ] − i < 0, sym[ρi (Ai + Bi Ki )T P ] + (1 − ρi )i ) < Qii , ρj sym[(Ai + Bi Kj )T P ] + (1 − ρj )i + ρi sym[(Aj + Bj Ki )T P ] + (1 − ρi )j < Qij + Qji , i < j,

(8.24) (8.25) (8.26) (8.27)

216

Analysis and Synthesis of Singular Systems

where

⎡

⎤

Q11 Q12 · · · Q1r ⎢ ⎥ ⎢Q21 Q22 · · · Q2r ⎥

Q=⎢ ⎢ ..

.. .

⎣ .

Qr1

Qr2

..

. ...

.. ⎥ ⎥. . ⎦

Qrr

Proof. Firstly, we will prove the stability of the system and choose the following Lyapunov function: V (x(t)) = x(t)T ET Px(t),

(8.28)

where ET P = P T E ≥ 0. Considering the condition in (8.24), the first-order time-derivative of the Lyapunov function along the trajectories of the system in (8.21) is given below: V˙ (x(t)) = 2x(t)T P T Ex˙ (t) =

r r  

wi mj x(t)T sym[(Ai + Bi Kj )T P ]x(t).

(8.29)

i=1 j=1

To further alleviate the conservatism, the following equalities and some free weighting matrices are introduced: r r  

wi (wj − mj )i = 0,

(8.30)

i=1 j=1

where i = Ti is arbitrary matrix. From (8.29) and (8.30), we have V˙ (x(t)) =

r r  

wi (mj + ρj wj − ρj wj )x(t)T sym[(Ai + Bi Kj )T P ]x(t)

i=1 j=1 r r  

+

wi (wj − ρj wj )x(t)T i x(t)

i=1 j=1

−

r r  

wi (mj − ρj wj )x(t)T i x(t)

i=1 j=1

=

r r  

wi wj xT (t)(sym[ρj (Ai + Bi Kj )T P ] + (1 − ρj )i )x(t)

i=1 j=1 r r  

+

i=1 j=1

wi (mj − ρj wj )xT (t)(sym[(Ai + Bi Kj )T P ] − i )x(t).

(8.31)

217

Admissibility and admissibilization for fuzzy singular systems

Considering mj − ρj wj > 0 and sym[(Ai + Bi Kj )P ] − i < 0 for i, j = 1, 2, . . . , r, we have V˙ (x(t)) ≤

r 

wi2 x(t)T (sym[ρi (Ai + Bi Ki )T P ] + (1 − ρj )i )x(t)

i=1

+

r  



wi wj x(t)T ρj sym[(Ai + BiT Kj )P ]

i=1 i 0, ¯ i < 0, sym[(Ai P¯ + Bi Hj )] −  ¯ ii , ¯i 0, sym[(Ai + Bi Kj )T (YE + U T ΦΛT )] − i < 0, sym[ρi (Ai + Bi Ki )T (YE + U T ΦΛT )] + (1 − ρi )i < Qii ,

(8.42) (8.43) (8.44)

ρj sym[(Ai + Bi Kj )T (YE + U T ΦΛT )] + (1 − ρj )i + ρi sym[(Aj + Bj Ki )T P ] + (1 − ρi )j < Qij + Qji , i < j.

(8.45)

Admissibility and admissibilization for fuzzy singular systems

219

Let P¯ = (YE + U T ΦΛT )−1 = Y¯ ET + ΛΦ¯ U. Pre- and postmultiplying in¯ we obtain equality (8.43) with P¯ T and P, sym[P¯ T (Ai + Bi Kj )T ] − P¯ T i P¯ < 0. ¯ ¯ i = P¯ T i P, ¯ the condition in (8.39) is obtained. Denoting Hj = Kj P, Similarly, performing the same congruent transformation to inequalities ¯ ij = P¯ T Qij P, ¯ the conditions (8.44) and (8.45), respectively, and denoting Q T −1 in (8.40) and (8.41) can be derived. Due to (EL YEL ) = ERT Y¯ ER , the inequality in (8.38) is equivalent to (8.42).

Remark 8.11. It can be seen that the equality constraint ET P = P T E and nonstrict LMI ET P ≥ 0 in Theorem 8.9 are removed in the novel admissibility criterion in Theorem 8.10, which makes the condition easy to check by using standard Matlab® toolbox such as LMI or Yalmip.

8.2.3 Static output feedback of singular systems In this section, the equivalent sets are applied to deal with the SOF control problem of singular IT2 fuzzy systems. To design the SOF controller in terms of LMIs, the following equivalent system of (8.23) is given: ¯ (w , m)¯x(t), E¯ x˙¯ (t) = A

(8.46)

where 











A(w ) B(w ) x(t) E 0 ¯ (w , m) = , E¯ = , x¯ (t) = . A S(m)C (w ) −I u(t) 0 0 Note that the admissibility of system (8.23) is equivalent to that of system (8.46) which is because 



¯ (w , m) = sE − A(w ) −B(w ) sE¯ − A −S(m)C (w ) I 

I −B(w ) = 0 I





sE − A(w )S(m)C (w ) 0 0 I



I 0 , S(m)C (w ) I

¯ (w , m)) = det(sE − A(w )S(m)C (w )). Firstly, a new admissiand det(sE¯ − A bility condition of system (8.46) is proposed in the following theorem:

220

Analysis and Synthesis of Singular Systems

Theorem 8.12. The system in (8.46) is admissibility if there exist matrices Z, F, G, and nonsingular matrix Θ satisfying the following matrix inequalities: E¯ LT Z E¯ L > 0,

 ¯ (w , m)T GT ) Γ12 (w , m) sym(A < 0, Γ (w , m) =  −F − F T 

(8.47) (8.48)

¯ (w , m)T F T − G, E¯ L , and E¯ R are where Γ12 (w , m) = (Z E¯ + U¯ T Θ Λ¯ T )T + A T full column rank matrices with E¯ = E¯ L E¯ R . U¯ T ∈ R(n+m)×(n+m−q) and Λ¯ ∈ ¯ R(n+m)×(n+m−q) with full column rank are the right null matrix of E¯ T and E, T T ¯ ¯ ¯ ¯ that is, E U = 0 and EΛ = 0, respectively.

Proof. Based on Lemma 2.5, the conditions in (8.47) and (8.48) are equivalent to that when there exists a matrix X = Z E¯ + U¯ Θ Λ¯ such that the following inequalities hold: E¯ T X = X T E¯ ≥ 0,



¯ (w , m)T GT ) sym(A

XT

−F

 

Then premultiplying (8.50) by I 

I

¯ (w , m)T A

T

¯ (w , m)T F T +A

−G

− FT



< 0.

(8.49) (8.50)



¯ (w , m)T and postmultiplying it by A

, we have ¯ (w , m)T X + X T A ¯ (w , m) < 0. A

(8.51)

Choosing Lyapunov function V (¯x(t)) = x¯ (t)T ET X x¯ (t), considering (8.47) and (8.51), we can obtain V˙ (¯x(t)) < 0, which implies that the singular IT2 fuzzy system in (8.46) is asymptotically stable. Following similar lines as that from (8.34) to (8.37), the regularity and nonimpulsiveness of the system can be proved. Therefore the condition in Theorem 8.12 can guarantee the admissibility of the singular IT2 fuzzy system in (8.46). The following theorem gives an LMI admissibility condition for closedloop singular IT2 fuzzy system (8.23): Theorem 8.13. The system in (8.23) is admissibility if the membership functions of the fuzzy model and fuzzy controller satisfy mj − ρj wj ≥ 0 for all j with 0 < ρj < 1 and there exist matrices Z = Z T , F1 , F2 , G1 , G2 , Ti , i = 1, . . . , r, R < 0, Ψi = ΨiT and nonsingular matrices Θ , G3 satisfying the following LMIs: E¯ LT Z E¯ L > 0,

(8.52)

221

Admissibility and admissibilization for fuzzy singular systems

Γij − Ψi < 0,

(8.53)

ρi Γii + (1 − ρi )Ψi < Rii ,

(8.54)

ρj Γij + (1 − ρj )Ψi + ρi Γji + (1 − ρi )Ψj < Rij + Rji , i < j,

(8.55)

where ⎡ ⎢ Γij = ⎢ ⎣

⎤ ¯ T Θ Λ¯ T )T + Υij (Z E¯ + U  ⎥  F1 + F1T LG3 + F2T ⎥ ⎦, −  G3 + G3T

Γij11 



Γij11



sym(G1 Ai + LTj Ci ) G1 Bi − LG3 + ATi G2T + CiT TjT , =  sym(G2 Bi − G3 ) ⎡

R11 ⎢ ⎢R21

R =⎢ ⎢ .. ⎣ . 

Rr1

⎤

.. .

··· ··· .. .

R1r R2r ⎥ ⎥

Rr2

...

Rrr

R12 R22

.. ⎥ ⎥, . ⎦ 

ATi F1T + CiT TjT L T − G1 ATi F2T + CiT TjT − LG3 . Υij = BiT F1T − G3T L T − G2 B(w )T F2T − G3T − G3 E¯ R , U¯ T ∈ R(n+m)×(n+m−r ) , and Λ¯ ∈ R(n+m)×(n+m−r ) are defined in Theorem 8.12. Then the SOF controller is given as Si = Ti G3−1 . Proof. Set



G1 G= G2







LG3 F1 LG3 , F= , G3 F2 G3

and T (m) = G3 S(m). We have 



¯ (w , m) = G1 A(w ) + LT (m)C (w ) G1 B(w ) − LG3 , GA G2 A(w ) + T (m)C (w ) G2 B(w ) − G3 

 F A(w ) + LT (m)C (w ) ¯ F A(w , m) = 1 F2 A(w ) + T (m)C (w )

F1 B(w ) − LG3 . F2 B(w ) − G3

It yields from (8.48) and (8.56)–(8.57) that Γ (w , m) =

r r   i=1 j=1

(8.56)

wi mj Γij .

(8.57)

222

Analysis and Synthesis of Singular Systems

For any matrices Ψi , noting that

r

i=1

r

j=1 wi (wj

− mj )Ψi = 0, we have

Γ (w , m) = =

r r  

wi mj Γij +

i=1 j=1 r r  

wi (wj − mj )Ψi

i=1 j=1

wi (mj i=1 j=1 r r  

+ =

r r  

+ ρj wj − ρj wj )Γij

wi (wj − ρj wj )Ψi −

i=1 j=1 r r  

wi (mj − ρj wj )Ψi

i=1 j=1

wi wj (ρj Γij i=1 j=1 r r  

+

r r  

+ (1 − ρj )Ψi )

wj (mj − ρj wj )(Γij − Ψi ).

i=1 j=1

Considering mj − ρj wj > 0, and Γij − Ψi < 0 for i, j = 1, 2, . . . , r, we have Γ (w , m) ≤

r 

wi2 (ρi Γii + (1 − ρi )Ψi ) +

i=1

r  

wi wj (ρj Γij

i=1 i 0 (≥ 0) eig(A) sup inf d(a, S) dH (S1 , S2 ) 

matrix transposition n-dimensional Euclidean space set of nonnegative real numbers set of nonnegative integers {k ∈ Z+ | k ≥ s1 } {k ∈ Z+ | s2 ≥ k ≥ s1 } Euclidean vector norm    ∞ T (k)w (k) for w (k) ∈ l [0, ∞) w 2 k=0    supk {eT (k)e(k)} for e(k) ∈ l∞ [0, ∞)





distance of a vector x to set S , x ∈ Rn , S ⊂ Rn , xS := infy∈S x − y space of square-summable infinite sequences space of all essentially bounded functions nearest integer greater than or equal to a an ellipsis for the terms that are introduced by symmetry set of n × n symmetric positive definite matrices {s ∈ Rn |s + s2 ∈ S1 , ∀s2 ∈ S2 } {s1 + s2 ∈ Rn |s1 ∈ S1 , s2 ∈ S2 } convex hull of S {x ∈ Rn : x2 ≤ 1} space of continuously differentiable functions A is equivalent to B block-diagonal matrix identity matrix zero matrix P is a real symmetric and positive definite (semipositive definite) matrix set of eigenvalues of a matrix A supremum or least upper bound infimum or greatest lower bound distance between an element a and a set S Hausdorff distance between sets S1 and S2 equal by definition or is defined by

243

Index

A Actuator failures, 15, 19, 159, 171, 176, 181 Admissibility, 2, 6, 8, 34, 35, 37, 41, 43, 88, 101, 109, 159, 203, 219, 220, 222 analysis, 203, 223 condition, 6, 7, 19, 61, 160–162, 171, 186, 189, 219, 220 criterion, 18, 73, 90, 109, 125, 185, 186, 195, 206, 219 mean square, 194 stochastic, 175, 176, 179, 181 Admissibilization, 6, 18, 29, 159, 201, 203, 212, 215, 223 condition, 162 criterion, 198, 201 method, 6, 199 Admissible robustly, 138, 147, 148, 151 singular systems, 24, 49, 52 stochastically, 19, 160, 161, 163–165, 171, 174, 176, 177, 181, 189, 190 Approach controller design, 190 convex combination, 10, 145 delay partitioning, 211 reciprocally convex, 10, 135, 144, 145, 151, 156, 157 system augmentation, 17, 21, 30, 34, 38, 230

B Bounded real lemma (BRL), 3

C Causal, 30, 61, 62, 66, 67, 73, 74, 78, 105, 106, 138 Causality, 2, 3, 5, 6, 18, 43, 73, 140 Computational burden, 81, 151, 211 Congruence transformation, 26, 57, 77, 78 Conservatism, 9–12, 73, 87, 121, 131, 144, 151, 156, 166, 195, 206, 211

Constant delay, 87 matrices, 22, 25, 30, 38, 47, 55, 74, 114, 172, 195, 212 Continuous-time singular systems, 21, 23, 60 Control dissipative, 7, 11, 12, 14, 17, 21, 29, 30, 32, 38, 101, 113, 157, 171 H∞ , 3, 7, 11, 14, 15, 47, 49, 113, 159, 163 input, 30, 55, 74, 114, 172, 185 output, 102 passive, 113, 135, 171, 183 passivity, 3, 14 reliable, 14 dissipative, 171 state feedback, 6, 18, 19, 27, 159, 161, 162, 167, 197, 212, 215, 223

D Delay, 73, 74, 87, 113, 122, 135, 167 bounds, 145 dependence, 113 discrete, 126 partitioning approach, 211 method, 134 technique, 18, 61, 99, 101 singular system, 94, 145, 151, 152 state, 18, 73, 74, 80, 99 Direct current (DC) motor, 168, 197 Discrete delay, 126 singular systems, 103, 104, 113 Discrete-time singular systems, 23, 30 Dissipative analysis, 115 control, 7, 11, 12, 14, 17, 21, 29, 30, 32, 38, 101, 113, 157, 171 filtering, 4, 13, 38, 101, 135, 151, 154 margin, 124, 135 result, 123, 131 245

246

Index

stabilization, 127 synthesis, 18 Dissipativity, 3, 8, 13, 22, 28, 34, 35, 37, 41, 43, 45, 101, 110 analysis, 15, 18, 21, 101, 113 condition, 8, 17, 18, 22, 24, 29, 30, 37 controller, 73 level, 112 property, 8, 21 Disturbance, 18, 21, 30, 38, 47, 102, 114 attenuation, 7, 18, 75, 76, 88–90, 99, 110 rejection, 124, 125, 130 Double summation term, 18, 61, 73

E Equivalent conditions, 72, 187 model transformation approach, 11 sets, 171 approach, 27, 29, 167 Existence condition, 13 Exogenous inputs, 2, 21, 47, 49, 102, 114, 163, 172

F Filter, 4, 13, 38–41, 135, 138, 148, 151, 154 design, 148 method, 4, 9, 12, 18, 40, 42, 136, 148, 152 error singular system, 143 system, 138 gains, 137 Filtering, 8, 12, 17, 21, 185 dissipative, 4, 13, 38, 101, 135, 151, 154 error system, 4, 5, 8, 9, 12, 16–18, 38–40, 43, 101, 135, 137, 140, 148, 151, 157 H∞ , 4, 8, 12, 13, 157 reliable, 152 Kalman, 4, 8 passive, 157 passivity, 152 reduced-order dissipative, 38 Forward difference, 18, 61, 67, 68, 70, 72, 73, 79, 103, 107, 108, 140

Fuzzy model, 215, 218, 220 sets, 16 type-1, 16 type-2, 16 systems, 230 IT2, 16, 17, 203, 212, 214, 215, 218–220, 223, 229, 230 singular, 230 type-1, 16 type-2, 213

G Generalized algebraic Riccati equation (GARE), 7 Generalized algebraic Riccati inequalities (GARI), 7 Generalized Lyapunov equation (GLE), 6

H H∞ control, 7, 11, 14, 47, 49, 159, 163 with transients, 49, 60

I Impulse free, 114, 115, 135, 159, 173, 174, 186 IT2 fuzzy model, 16 singular system, 19, 203

J Jensen inequality, 10, 101, 210 Jumping mode, 181, 185

K KYP conditions, 11

L Leontief model, 112 Linear matrix inequalities, 162 systems, 3, 15 Linear matrix inequalities (LMIs), 6, 8, 9, 11, 47, 50, 60, 61, 73, 103, 113, 115, 135, 136, 157, 171, 181 Lyapunov function, 9, 18, 51, 61, 67, 73, 102, 106, 121, 163, 216, 220

Index

matrix, 33, 58 method, 11, 211

M Markovian jump parameters, 13 systems, 13–15, 122, 160, 163, 168, 171, 201 singular, 159, 163 stochastic, 201 Maximal singular values (MSVs), 28 Maximum delay, 93, 96, 97 Membership degrees, 213 functions, 16, 17, 213–215, 218, 220, 223–225, 227, 228, 230 grade, 204, 213, 214, 227 Mismatched membership functions, 19, 212

N Neutral delay systems, 87 Nonimpulsiveness, 2, 5, 6, 217, 220 Nonsingular, 23, 24, 66, 67, 78, 105, 106, 117, 119, 147, 150, 160, 164, 187, 190, 192, 197, 207, 208, 218 matrix, 24, 36, 77, 220

O Observer, 190, 196, 199–201 for SSMSs, 201 gain, 191, 192, 200, 201 trajectory, 196 Observer-based SMC problem, 190

P Parallel distributed compensation (PDC) method, 227 Parameter uncertainties, 11, 15, 16, 73, 82, 195, 230 Passivity, 110, 113 analysis, 101 filtering, 152, 157 performance, 2, 13, 110, 156 Performance, 2, 15, 27 analysis, 24, 60 excitation, 56

247

H2 /H∞ , 17 H∞ , 2, 8, 15, 18, 28, 89, 90, 110, 165 analysis, 88, 89, 124 level, 7 Polynomial matrix, 2, 5 Polytopic system, 153, 154 uncertainty, 12, 18, 135, 147, 148, 151 Positive real lemma (PRL), 7

Q Quasiconvex optimization problem, 96, 131, 134

R Reciprocally convex approach, 10, 135, 144, 145, 151, 156, 157 combination, 3, 61, 62, 64, 72 Reduced-order dissipative filtering, 38 Regular, 30, 61, 62, 66, 67, 73, 78, 105, 106 Regularity, 2, 3, 5, 6, 18, 43, 73, 140 Reliable control, 14 Reliable dissipativity analysis, 139, 174 Robust H∞ control, 88 reliable dissipative filtering, 135 stability, 76 Robustness, 2

S Schur complement, 57, 82, 87, 90, 209 equivalence, 26, 33, 139, 144, 150, 162, 165, 176, 177, 188, 192 Semistate space systems, 1 Singular Markovian jump systems (SMJSs), 13, 14, 19 Singular systems, 1, 5, 22, 38, 215 Singular value decomposition (SVD), 14 Slack matrices, 146 Sliding mode control (SMC), 4, 190 Sliding surface, 4, 191, 196 SMDs, 191, 197, 200 Stability, 6, 18, 67, 73, 106, 208, 211, 213, 216, 217 Stabilization, 85

248

Index

condition, 230 Stable, 30, 61, 62, 73, 94, 95, 109, 120, 132 stochastically, 14, 159, 173, 174 Static output feedback (SOF), 58 Stochastic admissibility, 175, 176, 179 hybrid systems, 14 Lyapunov function, 175 Markovian jump systems, 201 Stochastically admissible, 19, 160, 161, 163–165, 171, 174, 176, 177, 181, 189, 190 stable, 14, 159, 173, 174 Symmetric matrix, 24, 25, 160, 161, 164, 165, 205 Synthesis, 212 controller, 73 dissipative, 18, 101 Systems delay, 3, 97 fuzzy, 230 IT2, 16, 17, 203, 212, 214, 215, 218–220, 223, 229, 230 singular, 230

Markovian jump, 13–15, 122, 160, 163, 168, 171, 201 T-S fuzzy, 15, 16 time-delay, 3, 9, 12 type-2 fuzzy singular, 16

T T-S fuzzy systems, 15, 16 singular, 15 Time delay, 62, 94, 163 Transients, 7, 17, 49, 60 Transition rate, 160, 172, 185, 201 Type-2 FMB control problems, 16 Type-2 fuzzy singular systems, 16

U Uncertainty, 5, 12, 13, 135, 195, 227 nonlinear plant, 213 parameter, 11, 15, 16, 73, 82, 195, 230 polytopic, 12, 18, 135, 147, 148, 151

W Weighting matrix, 10, 49, 53, 216 Wirtinger inequality, 103, 104, 113