Min Wu Yong He Jin-Hua She. Stability Analysis and Robust Control of Time-Delay Systems

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2 Min Wu Yong He Jin-Hua She Stability Analysis and Robust Control of Time-Delay Systems

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4 Min Wu Yong He Jin-Hua She Stability Analysis and Robust Control of Time-Delay Systems With 12 figures

5 Authors Min Wu School of Information Science & Engineering Central South University Changsha, Hunan, , China Yong He School of Information Science & Engineering Central South University Changsha, Hunan, , China Jin-Hua She School of Computer Science Tokyo University of Technology ISBN Science Press Beijing ISBN e-isbn Springer Heidelberg Dordrecht London New York Library of Congress Control Number: Science Press Beijing and Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Frido Steinen-Broo, EStudio Calamar, Spain Printed on acid-free paper Springer is a part of Springer Science+Business Media (

6 Preface A system is said to have a delay when the rate of variation in the system state depends on past states. Such a system is called a time-delay system. Delays appear frequently in real-world engineering systems. They are often a source of instability and poor performance, and greatly increase the difficulty of stability analysis and control design. So, many researchers in the field of control theory and engineering study the robust control of time-delay systems. The study of such systems has been very active for the last 20 years; and new developments, such as fixed model transformations based on the Newton- Leibnitz formula and parameterized model transformations, are continually appearing. Although these methods are a great improvement over previous ones, they still have their limitations. We recently devised a method called the free-weighting-matrix (FWM) approach for the stability analysis and control synthesis of various classes of time-delay systems; and we obtained a series of not so conservative delaydependent stability criteria and controller design methods. This book is based primarily on our recent research. It focuses on the stability analysis and robust control of various time-delay systems, and includes such topics as stability analysis, stabilization, control design, and filtering. The main method employed is the FWM approach. The effectiveness of this method and its advantages over other existing ones are proven theoretically and illustrated by means of various examples. The book will give readers an overview of the latest advances in this active research area and equip them with a state-ofthe-art method for studying time-delay systems. This book is a useful reference for control theorists and mathematicians working with time-delay systems, engineering designing controllers for plants or systems with delays, and for graduate students interested in robust control theory and/or its application to time-delay systems. We are grateful for the support of the National Natural Science Foundation of China ( ), the National Science Fund for Distinguished Young

7 vi Preface Scholars ( ), the Program for New Century Excellent Talents in University (NCET ), the Specialized Research Fund for the Doctoral Program of Higher Education of China ( and ), and the Hunan Provincial Natural Science Foundation of China (08JJ1010). We are also grateful for the support of scholars both at home and abroad. We would like to thank Prof. Zixing Cai of Central South University, Prof. Qingguo Wang of the National University of Singapore, Profs. Guoping Liu and Peng Shi of the University of Glamorgan, Prof. Tongwen Chen of the University of Alberta, Prof. James Lam of the University of Hong Kong, Prof. Lihua Xie of Nanyang Technological University, Prof. Keqin Gu of Southern Illinois University Edwardsville, Prof. Zidong Wang of Brunel University, Prof. Li Yu of Zhejiang University of Technology, Prof. Xinping Guan of Yanshan University, Prof. Shengyuan Xu of Nanjing University of Science & Technology, Prof. Qinglong Han of Central Queensland University, Prof. Huanshui Zhang of Shandong University, Prof. Huijun Gao of the Harbin Institute of Technology, Prof. Chong Lin of Qingdao University, and Prof. Guilin Wen of Hunan University for their valuable help. Finally, we would like to express our appreciation for the great efforts of Drs. Xianming Zhang, Zhiyong Feng, Fang Liu, Yan Zhang and Chuanke Zhang, and graduate student Lingyun Fu. Min Wu Yong He Jin-Hua She July 2009

8 Contents 1. Introduction Review of Stability Analysis for Time-Delay Systems Introduction to FWMs Outline of This Book...12 References Preliminaries Lyapunov Stability and Basic Theorems Types of Stability Lyapunov Stability Theorems Stability of Time-Delay Systems Stability-Related Topics Lyapunov-Krasovskii Stability Theorem Razumikhin Stability Theorem H Norm Norm H Norm H Control LMI Method Common Specifications of LMIs Standard LMI Problems Lemmas Conclusion...38 References Stability of Systems with Time-Varying Delay Problem Formulation Stability of Nominal System Replacing the Term ẋ(t) Retaining the Term ẋ(t) Equivalence Analysis...50

9 viii Contents 3.3 Stability of Systems with Time-Varying Structured Uncertainties Robust Stability Analysis Numerical Example Stability of Systems with Polytopic-Type Uncertainties Robust Stability Analysis Numerical Example IFWM Approach Retaining Useful Terms Further Investigation Numerical Examples Conclusion...68 References Stability of Systems with Multiple Delays Problem Formulation Two Delays Nominal Systems Equivalence Analysis Systems with Time-Varying Structured Uncertainties Numerical Examples Multiple Delays Conclusion...91 References Stability of Neutral Systems Neutral Systems with Time-Varying Discrete Delay Problem Formulation Nominal Systems Systems with Time-Varying Structured Uncertainties Numerical Example Neutral Systems with Identical Discrete and Neutral Delays FWM Approach FWM Approach in Combination with Parameterized Model Transformation FWM Approach in Combination with Augmented Lyapunov- Krasovskii Functional Numerical Examples...114

10 Contents ix 5.3 Neutral Systems with Different Discrete and Neutral Delays Nominal Systems Equivalence Analysis Systems with Time-Varying Structured Uncertainties Numerical Example Conclusion References Stabilization of Systems with Time-Varying Delay Problem Formulation Iterative Nonlinear Minimization Algorithm Parameter-Tuning Method Completely LMI-Based Design Method Numerical Example Conclusion References Stability and Stabilization of Discrete-Time Systems with Time-Varying Delay Problem Formulation Stability Analysis Controller Design SOF Controller DOF Controller Numerical Examples Conclusion References H Control Design for Systems with Time-Varying Delay Problem Formulation BRL Design of State-Feedback H Controller Numerical Examples Conclusion References H Filter Design for Systems with Time-Varying Delay H Filter Design for Continuous-Time Systems Problem Formulation H Performance Analysis...180

11 x Contents Design of H Filter Numerical Examples H Filter Design for Discrete-Time Systems Problem Formulation H Performance Analysis Design of H Filter Numerical Example Conclusion References Stability of Neural Networks with Time-Varying Delay Stability of Neural Networks with Multiple Delays Problem Formulation Stability Criteria Numerical Examples Stability of Neural Networks with Interval Delay Problem Formulation Stability Criteria Numerical Examples Exponential Stability of Continuous-Time Neural Networks Problem Formulation Stability Criteria Derived by FWM Approach Stability Criteria Derived by IFWM Approach Numerical Examples Exponential Stability of Discrete-Time Recurrent Neural Networks Problem Formulation Stability Criterion Derived by IFWM Approach Numerical Examples Conclusion References Stability of T-S Fuzzy Systems with Time-Varying Delay Problem Formulation Stability Analysis Numerical Examples Conclusion...260

12 Contents xi References Stability and Stabilization of NCSs Modeling of NCSs with Network-Induced Delay Stability Analysis Controller Design Numerical Examples Conclusion References Stability of Stochastic Systems with Time-Varying Delay Robust Stability of Uncertain Stochastic Systems Problem Formulation Robust Stability Analysis Numerical Example Exponential Stability of Stochastic Markovian Jump Systems with Nonlinearities Problem Formulation Exponential-Stability Analysis Numerical Example Conclusion References Stability of Nonlinear Time-Delay Systems Absolute Stability of Nonlinear Systems with Delay and Multiple Nonlinearities Problem Formulation Nominal Systems Systems with Time-Varying Structured Uncertainties Numerical Examples Absolute Stability of Nonlinear Systems with Time-Varying Delay Problem Formulation Nominal Systems Systems with Time-Varying Structured Uncertainties Numerical Example Stability of Systems with Interval Delay and Nonlinear Perturbations Problem Formulation...318

13 xii Contents Stability Results Further Results Obtained with Augmented Lyapunov- Krasovskii Functional Numerical Examples Conclusion References Index...335

14 Abbreviations inf lim max min sup infimum limit maximum minimum supremum BRL CCL DOF FWM ICCL IFWM LFT LMI MADB MATI NCS NFDE NLMI RFDE SOF bounded real lemma cone complementarity linearization dynamic output feedback free weighting matrix improved cone complementarity linearization improved free weighting matrix linear fractional transaction linear matrix inequality maximum allowable delay bound maximum allowable transfer interval networked control system neutral functional differential equation nonlinear matrix inequality retarded functional differential equation static output feedback

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16 Symbols R, R n, R n m setofrealnumbers,setofn-dimensional real vectors, and set of n m real matrices C, C n, C n m set of complex numbers, set of n component complex vectors, and set of n m complex matrices R + Z + Re(s) set of non-negative real numbers set of non-negative integers real part of s C A T A 1 A T I n diag {A 1,,A n } X Y Z A > 0(< 0) A 0( 0) det(a) Tr{A} λ(a) λ max (A) λ min (A) transpose of matrix A inverse of matrix A shorthand for (A 1 ) T n n identity matrix (the subscript is omitted if no confusion will occur) diagonal matrix with A i as its ith diagonal element symmetric matrix X Y Y T Z symmetric positive (negative) definite matrix symmetric positive (negative) semi-definite matrix determinant of matrix A trace of matrix A eigenvalue of matrix A largest eigenvalue of matrix A smallest eigenvalue of matrix A

17 xvi Symbols σ max (A) L 2 [0, + ) largest singular value of matrix A set of square integrable functions on [0, + ) l 2 [0, + ) C ([a, b], R n ) C b F 0 ([a, b], R n ) L 2 F 0 ([a, b], R n ) set of square multipliable functions on [0, + ) family of continuous functions φ from [a, b] tor n family of all bounded F 0 -measurable C([a, b], R n )-valued random variables family of all bounded F 0 -measurable C([a, b], R n )-valued { } random variables ξ = ξ(t) : sup a t b E ξ(t) 2 < absolute value (or modulus) φ c L Euclidean norm of a vector or spectral norm of a matrix induced l -norm continuous norm sup φ(t) for φ C([a, b], R n ) a t b weak infinitesimal of a stochastic process Dx t operator that maps C([ h, 0], R n ) R n ;thatis,dx t = x(t) Cx(t h) E A C B D mathematical expectation shorthand for state space realization C(sI A) 1 B +D for a continuous-time system or C(zI A) 1 B + D for a discrete-time system for all belongs to there exists is a subset of union tends toward or is mapped into (case sensitive) implies := is defined as end of proof

18 1. Introduction In many physical and biological phenomena, the rate of variation in the system state depends on past states. This characteristic is called a delay or a time delay, and a system with a time delay is called a time-delay system. Timedelay phenomena were first discovered in biological systems and were later found in many engineering systems, such as mechanical transmissions, fluid transmissions, metallurgical processes, and networked control systems. They are often a source of instability and poor control performance. Time-delay systems have attracted the attention of many researchers [1 3] because of their importance and widespread occurrence. Basic theories describing such systems were established in the 1950s and 1960s; they covered topics such as the existence and uniqueness of solutions to dynamic equations, stability theory for trivial solutions, etc. That work laid the foundation for the later analysis and design of time-delay systems. The robust control of time-delay systems has been a very active field for the last 20 years and has spawned many branches, for example, stability analysis, stabilization design, H control, passive and dissipative control, reliable control, guaranteed-cost control, H filtering, Kalman filtering, and stochastic control. Regardless of the branch, stability is the foundation. So, important developments in the field of time-delay systems that explore new directions have generally been launched from a consideration of stability as the starting point. This chapter reviews methods of studying the stability of time-delay systems and points out their limitations, and then goes on to describe a new method called the free-weighting-matrix (FWM) approach. 1.1 Review of Stability Analysis for Time-Delay Systems Stability is a very basic issue in control theory and has been extensively discussed in many monographs [4 6]. Research on the stability of time-delay

19 2 1. Introduction systems began in the 1950s, first using frequency-domain methods and later also using time-domain methods. Frequency-domain methods determine the stability of a system from the distribution of the roots of its characteristic equation [7] or from the solutions of a complex Lyapunov matrix function equation [8]. They are suitable only for systems with constant delays. The main time-domain methods are the Lyapunov-Krasovskii functional and Razumikhin function methods [1]. They are the most common approaches to the stability analysis of time-delay systems. Since it was very difficult to construct Lyapunov-Krasovskii functionals and Lyapunov functions until the 1990s, the stability criteria obtained were generally in the form of existence conditions; and it was impossible to derive a general solution. Then, Riccati equations, linear matrix inequalities (LMIs) [9], and Matlab toolboxes came into use; and the solutions they provided were used to construct Lyapunov- Krasovskii functionals and Lyapunov functions. These time-domain methods are now very important in the stability analysis of linear systems. This section reviews methods of examining stability and their limitations. Consider the following linear system with a delay: ẋ(t) =Ax(t)+A d x(t h), (1.1) x(t) =ϕ(t), t [ h, 0], where x(t) R n is the state vector; h>0is a delay in the state of the system, that is, it is a discrete delay; ϕ(t) is the initial condition; and A R n n and A d R n n are the system matrices. The future evolution of this system depends not only on its present state, but also on its history. The main methods of examining its stability can be classified into two types: frequencydomain and time-domain. Frequency-domain methods: Frequency-domain methods provide the most sophisticated approach to analyzing the stability of a system with no delay (h = 0). The necessary and sufficient condition for the stability of such asystemisλ(a+a d ) < 0. When h>0, frequency-domain methods yield the result that system (1.1) is stable if and only if all the roots of its characteristic function, f(λ) =det(λi A A d e hλ )=0, (1.2) have negative real parts. However, this equation is transcendental, which makes it difficult to solve. Moreover, if the system has uncertainties and a

20 1.1 Review of Stability Analysis for Time-Delay Systems 3 time-varying delay, the solution is even more complicated. So the use of a frequency-domain method to study time-delay systems has serious limitations. Time-domain methods: Time-domain methods are based primarily on two famous theorems: the Lyapunov-Krasovskii stability theorem and the Razumikhin theorem. They were established in the 1950s by the Russian mathematicians Krasovskii and Razumikhin, respectively. The main idea is to obtain a sufficient condition for the stability of system (1.1) by constructing an appropriate Lyapunov-Krasovskii functional or an appropriate Lyapunov function. This idea is theoretically very important; but until the 1990s, there was no good way to implement it. Then the Matlab toolboxes appeared and made it easy to construct Lyapunov-Krasovskii functionals and Lyapunov functions, thus greatly promoting the development and application of these methods. Since then, significant results have continued to appear one after another (see [10] and references therein). Among them, two classes of sufficient conditions have received a great deal of attention. One class is independent of the length of the delay, and its members are called delay-independent conditions. The other class makes use of information on the length of the delay, and its members are called delay-dependent conditions. The Lyapunov-Krasovskii functional candidate is generally chosen to be V 1 (x t )=x T (t)px(t)+ x T (s)qx(s)ds, (1.3) t h where P > 0andQ>0areto be determined and are called Lyapunov matrices; and x t denotes the translation operator acting on the trajectory: x t (θ) =x(t+θ) for some (non-zero) interval [ h, 0] (θ [ h, 0]). Calculating the derivative of V 1 (x t ) along the solutions of system (1.1) and restricting it to less than zero yield the delay-independent stability condition of the system: PA+ A T P + Q PA d < 0. (1.4) Q Since this inequality is linear with respect to the matrix variables P and Q, it is called an LMI. If the LMI toolbox of Matlab yields solutions to LMI (1.4) for these variables, then according to the Lyapunov-Krasovskii stability theorem, system (1.1) is asymptotically stable for all h 0; and furthermore, an appropriate Lyapunov-Krasovskii functional is obtained.

21 4 1. Introduction Since delay-independent conditions contain no information on a delay, they are overly conservative, especially when the delay is small. This consideration has given rise to another important class of stability conditions, namely, delay-dependent conditions, which do contain information on the length of a delay. First of all, they assume that system (1.1) is stable when h = 0. Since the solutions of the system are continuous functions of h, there must exist an upper bound, h, on the delay such that system (1.1) is stable for all h [0, h]. Thus, the maximum possible upper bound on the delay is the main criterion for judging the conservativeness of a delay-dependent condition. The hot topics in control theory are delay-dependent problems in stability analysis, robust control, H control, reliable control, guaranteed-cost control, saturation input control, and chaotic-system control. Since the 1990s, the main approach to the study of delay-dependent stability has involved the addition of a quadratic double-integral term to the Lyapunov-Krasovskii functional (1.3): where V (x t )=V 1 (x t )+V 2 (x t ), (1.5) V 2 (x t )= 0 h t+θ x T (s)zx(s)dsdθ. The derivative of V 2 (x t )is V 2 (x t )=hx T (t)zx(t) x T (s)zx(s)ds. (1.6) t h Delay-dependent conditions can be obtained from the Lyapunov-Krasovskii stability theorem. However, how to deal with the integral term on the right side of (1.6) is a problem. So far, three methods of studying delay-dependent problems have been devised: the discretized Lyapunov-Krasovskii functional method, fixed model transformations, and parameterized model transformations. The main use of the discretized Lyapunov-Krasovskii functional method is to study the stability of linear systems and neutral systems with a constant delay. It discretizes the Lyapunov-Krasovskii functional, and the results can be written in the form of LMIs [11 15]. The advantage of doing this is that the estimate of the maximum allowable delay that guarantees the stability of the system is very close to the actual value. The drawbacks are that it is

22 1.1 Review of Stability Analysis for Time-Delay Systems 5 computationally expensive and that it cannot easily handle systems with a time-varying delay. Consequently, this method has not been widely studied or used since it was first proposed by Gu in 1997 [11]. The primary way of dealing with the integral term on the right side of equation (1.6) is by using a fixed model transformation. It transforms a system with a discrete delay into a new system with a distributed delay (the integral term in (1.10)). The following inequalities play an important role in deriving the stability conditions: Basic inequality: a, b R n and R >0, 2a T b a T Ra + b T R 1 b. (1.7) Park s inequality [16]: a, b R n, R >0, and M R n n, T 2a T b a R RM a. (1.8) b (M T R + I)R 1 (RM + I) b Moon et al. s inequality [17]: a R na, b R n b, N R na n b,and for X R na na, Y R na n b,andz R n b n b,if X Y 0, then Z 2a T Nb a b T X Y N Z a. (1.9) b The basic features of the typical model transformations discussed in [18] are described below. Model transformation I ẋ(t) =(A + A d )x(t) A d t h [Ax(s)+A d x(s h)]ds. (1.10) The following Lyapunov-Krasovskii functional is used to determine a delay-dependent stability condition: where V (x t )=V 1 (x t )+V 2 (x t )+V 3 (x t ), (1.11) V 3 (x t )= h 2h t+θ x T (s)z 1 x(s)dsdθ.

23 6 1. Introduction The derivative of V (x t ) along the solutions of system (1.10) is where V (x t )=Ψ + η 1 + η 2 x T (s)zx(s)ds t h h t 2h x T (s)z 1 x(s)ds, Ψ = x T (t)[2p (A + A d )+Q + h(z + Z 1 )]x(t) x T (t h)qx(t h), η 1 = 2 η 2 = 2 t h h t 2h x T (t)pa d Ax(s)ds, x T (t)pa d A d x(s)ds. η 1 and η 2 are called cross terms. Using the basic inequality (1.7) yields η 1 hx T (t)pa d AZ 1 A T A T d Px(t)+ x T (s)zx(s)ds, t h h η 2 hx T (t)pa d A d Z1 1 AT d AT d Px(t)+ x T (s)z 1 x(s)ds. t 2h (1.12) Applying these two inequalities to (1.12) eliminates the quadratic integral terms, and a delay-dependent condition is established. This process has two key points: (1) The purpose of a model transformation is to bring the integral term into the system equation so as to produce both cross terms and quadratic integral terms in the derivative of a Lyapunov-Krasovskii functional along the solutions of the system. (2) The bounding of the cross terms, η 1 and η 2, eliminates the quadratic integral terms in the derivative of the Lyapunov-Krasovskii functional, thereby yielding a delay-dependent condition. Model transformation II d t ] [x(t)+a d x(s)ds =(A + A d )x(t). (1.13) dt t h In 2000 and 2001, Prof. Gu [19, 20] pointed out that, since model transformations I and II introduce additional dynamics into the transformed system, the transformed system is not equivalent to the original one. Thus, these transformations were soon replaced by others.

24 1.1 Review of Stability Analysis for Time-Delay Systems 7 Model transformation III ẋ(t) =(A + A d )x(t) A d t h In this case, the Lyapunov-Krasovskii functional is where ẋ(s)ds. (1.14) V (x t )=V 1 (x t )+V 4 (x t ), (1.15) V 4 (x t )= 0 h t+θ ẋ T (s)zẋ(s)dsdθ. The derivative of V (x t )is V (x t )=Φ + η 3 ẋ T (s)zẋ(s)ds, (1.16) t h where Φ = x T (t)[2p (A + A d )+Q]x(t) x T (t h)qx(t h)+hẋ T (t)zẋ(t), η 3 = 2 t h x T (t)pa d ẋ(s)ds. Just as for model transformation I, the bounding of the cross term, η 3, eliminates the quadratic integral terms in the derivative of Lyapunov-Krasovskii functional (1.16), thereby producing a delay-dependent condition. Model transformation III was presented in [16]. The basic idea is the same as that of model transformation I, with the difference being that, after model transformation III, the transformed system is equivalent to the original one. In addition, after the transformation of system (1.1) into (1.14), when dealing with the term hẋ T (t)rẋ(t) in the derivative of V (x t ), system (1.1) is used as a substitute for system (1.14). That is, to obtain system (1.14), the statedelay term x(t h) in system (1.1) is replaced by using the Newton-Leibnitz formula; but x(t h) is not replaced in the derivative of V (x t ). This inconsistency in the elimination of the integral terms leads to conservativeness. In 2001, Fridman devised the following descriptor model transformation [21], which attracted a great deal of attention in subsequent years. Model transformation IV ẋ(t) =y(t), (1.17) y(t) =(A + A d )x(t) A d y(s)ds. t h

25 8 1. Introduction Fridman employed the following generalized Lyapunov-Krasovskii functional: 0 V (x t )=ξ T (t)epξ(t)+ x T (s)qx(s)ds + y T (s)zy(s)dsdθ, t h h t+θ (1.18) where ξ(t) = x(t), E = I 0, P = P 1 0. y(t) 0 0 P 2 P 3 The derivative of V (x t ) along the solutions of system (1.17) is V (x t )=Σ + η 4 y T (s)zy(s)ds, (1.19) t h where Σ = ξ T (t) 2P T 0 I + Q 0 A + A d I 0 hz ξ(t) xt (t h)qx(t h), η 4 = 2 ξ T (t)p T 0 y(s)ds. t h A d As before, the bounding of the cross term, η 4, eliminates the quadratic integral terms in the derivative of Lyapunov-Krasovskii functional (1.19), thereby producing a delay-dependent condition. There are four important points regarding the development of model transformations. (1) When double-integral terms are introduced into the Lyapunov-Krasovskii functional to produce a delay-dependent stability condition, it results in quadratic integral terms appearing in the derivative of that functional. (2) Model transformations emerged as a way of dealing with those quadratic integral terms. (3) More specifically, the purpose of a model transformation is to bring the integral terms into the system equation so as to produce cross terms and quadratic integral terms in the derivative of the Lyapunov-Krasovskii functional. (4) Then, the bounding of the cross terms eliminates the quadratic integral terms.

26 1.1 Review of Stability Analysis for Time-Delay Systems 9 The basic feature of all model transformations is that they produce cross terms in the derivative of the Lyapunov-Krasovskii functional. However, since no suitable bounding methods have yet been discovered, the bounding of cross terms results in conservativeness; and attempts to reduce the conservativeness have naturally focused on this point. For example, in 1999, Park extended the basic inequality (1.7) to produce Park s inequality [16]. In 2001, Moon et al. explored ideas in the proof of Park s inequality to extend it, resulting in Moon et al. s inequality [17], which has greater generality. The use of Park s or Moon et al. s inequality in combination with model transformation III or IV brought forth a series of delay-dependent conditions with less conservativeness that are very useful in stability analysis and control synthesis. However, model transformations III and IV still have limitations: In a stability or performance analysis, they basically use the Newton-Leibnitz formula to replace delay terms in the derivative of the Lyapunov-Krasovskii functional; but not all the delay terms are necessarily replaced. For example, in [17], the derivative of the Lyapunov-Krasovskii functional is V (x t )=2x T (t)p ẋ(t)+ + hẋ T (t)zẋ(t)+, (1.20) where P > 0andZ>0 are matrices to be determined in the Lyapunov- Krasovskii functional. When dealing with the term x(t h) (which appears when ẋ(t) is replaced with the system equation) in V (x t ), the x(t h) in 2x T (t)p ẋ(t) is replaced, but the x(t h)inhẋ T (t)zẋ(t)isnot.thistreatment is equivalent to adding the following zero-equivalent term to the derivative of the Lyapunov-Krasovskii functional: 2x T (t)pa d [x(t) x(t h) t h ] ẋ(s)ds. (1.21) Fixed weighting matrices are used to express the relationships among the terms of the Newton-Leibnitz formula in (1.21). That is, the weighting matrix of x(t) ispa d and that of x(t h) is zero. Similarly, in [18, 22 24], which employ the descriptor model transformation, the delay term x(t h) in T 2 [ x T (t), ẋ T (t) ] P P 2 P 3 A d x(t h) in the derivative of the Lyapunov-Krasovskii functional is replaced with x(t) t h ẋ(s)ds. This treatment is equivalent to adding the following zeroequivalent term to the derivative of the Lyapunov-Krasovskii functional:

27 10 1. Introduction 2 [ x T (t)p2 T A d +ẋ T (t)p3 T ] [ ] A d x(t) ẋ(s)ds x(t h). (1.22) t h Here, fixed weighting matrices are also used to express the relationships among the terms of Newton-Leibnitz formula. (The weighting matrix of x(t) is P T 2 A d,thatofẋ(t) isp T 3 A d, and that of x(t h) is zero). This substitution method is currently used in model transformations III and IV to obtain a delay-dependent condition. Note that, when weighting matrices are used for the above purpose, optimal weights do exist and the values should not be chosen simply for convenience. However, no effective way of determining the weights has yet been devised. The chief feature of a parameterized model transformation [25 28] is the division of the delay term of system (1.1) into two parts: a delay-independent one and one to which a fixed model transformation is applied. That transforms system (1.1) into ẋ(t) =Ax(t)+(A d C)x(t h)+cx(t h), (1.23) where C is a matrix parameter to be determined. In this way, a parameterized model transformation is combined with a fixed model transformation; so the limitations of the latter remain. On the other hand, although an effective approach to matrix decomposition was presented by Han in [28] (Remark 7 on page 378), three undetermined matrices have to be equal, which leads to unavoidable conservativeness. The stabilization problem is closely related to stability. Stabilization involves finding a feedback controller that stabilizes the closed-loop system, with the main feedback schemes being state and output feedback. Methods of stability analysis include both frequency- and time-domain approaches, but the latter are more commonly used for stabilization problems because the former do not lend themselves readily to solving such problems. For synthesis problems (such as delay-dependent stabilization and control), there is no effective controller synthesis algorithm, even for simple state feedback; solutions are even more difficult for output feedback. The main problem is that, even if model transformation I or II is used to derive an LMI-based controller synthesis algorithm, they both introduce additional eigenvalues into the original system, as mentioned above; so the transformed system is not equivalent to the original one. Moreover, they employ conservative vector inequalities. So, they have been replaced by model transformations III and IV. However, when using either of them to solve a

28 1.2 Introduction to FWMs 11 synthesis problem, the design of the controller depends on one or more nonlinear matrix inequalities (NLMIs). There are two main methods of solving this type of inequality: One is the iterative algorithm of Moon et al. [17], who used it on a robust stabilization problem. [23,24] also used it on an H control problem. This method yields a small controller gain, which is easy to implement; but the solutions are suboptimal [17]. The other is the widely used parameter-tuning method of Fridman et al. [18,22,29 34]. It transforms the NLMI(s) into an LMI(s) by using scalar parameters to set one or more undetermined matrices in the NLMI(s) to specific forms; and then the tuning of those parameters produces a controller. This method also yields a suboptimal solution, and experience is required to properly tune the parameters. 1.2 Introduction to FWMs In Section 1.1, we saw that the method of Moon et al. [17] adds the equation (1.21) to V (x t ); and the descriptor model transformation [18, 22 24, 28 35] adds the term (1.22) to it. The difference is that the weighting matrices of terms such as x(t) andẋ(t) are different, but they are all constant. For example, in Moon et al. [17], the weighting matrix of x(t) ispa d,where A d is a coefficient matrix and P is a Lyapunov matrix. P is closely related to other matrices and cannot be freely chosen. For other terms, also, the weighting matrix is constant (for example, for x(t h) it is zero). Moreover, in the descriptor model transformation, they are also constant. This is where FWMs come in. In equations (1.21) and (1.22), the weighting matrices of x(t), ẋ(t), and x(t h) are replaced by unknown FWMs. From the Newton- Leibnitz formula, the following equation is true for any matrices N 1 and N 2 with appropriate dimensions: 2 [ x T (t)n 1 + x T ] [ ] (t h)n 2 x(t) ẋ(s)ds x(t h) =0. (1.24) t h Now, we add the left side of this equation to the derivative of the Lyapunov- Krasovskii functional. The fact that N 1 and N 2 arefreeandthattheiroptimal values can be obtained by solving LMIs overcomes the conservativeness arising from the use of fixed weighting matrices [36 43]. On the other hand, since the two sides of the system equation are equal, FWMs thus express the relationships among the terms of that equation. That is, from system equation (1.1), the following equation is true for any matrices

29 12 1. Introduction T 1 and T 2 with appropriate dimensions: 2 [ x T (t)t 1 +ẋ T (t)t 2 ] [ẋ(t) Ax(t) Ad x(t h)] = 0. (1.25) And from the Newton-Leibnitz formula, the following equation is true for any matrices N i,i=1, 2, 3 with appropriate dimensions: 2 [ x T (t)n 1 +ẋ T (t)n 2 + x T ] [ ] (t h)n 3 x(t) ẋ(s)ds x(t h) =0. t h (1.26) Reserving the term ẋ(t) in the derivative of the Lyapunov-Krasovskii functional and adding the left sides of these two equations to the derivative produce another type of result; Chapter 3 theoretically proves the equivalence of these two methods. This shows that the descriptor model transformation of Fridman et al. is a special case of the FWM approach. Furthermore, this treatment in combination with a parameter-dependent Lyapunov-Krasovskii functional is easily extended to deal with the delay-dependent stability of systems with polytopic-type uncertainties [44 47]. 1.3 Outline of This Book This book is organized as follows: Chapter 1 reviews research on the stability of time-delay systems and describes the free-weighting-matrix approach. Chapter 2 provides the basic knowledge and concepts on the stability of time-delay systems that are needed in later chapters. Chapter 3 deals with linear systems with a time-varying delay. FWMs are used to express the relationships among the terms in the Newton-Leibnitz formula, and delay-dependent stability conditions are derived. The criteria are then extended to delay-dependent and rate-independent stability conditions without any limitations on the derivative of the delay. Two classes of criteria are obtained for two different treatments of the term ẋ(t) (retaining it or replacing it with the system equation) in the derivative of the Lyapunov- Krasovskii functional; and their equivalence is proved. On this basis, the criteria are extended to systems with time-varying structured uncertainties. Furthermore, since retaining the term ẋ(t) allows the Lyapunov matrices and system matrices to readily be separated, this treatment in combination with a parameter-dependent Lyapunov-Krasovskii functional is easily extended to

30 1.3 Outline of This Book 13 deal with the delay-dependent stability of systems with polytopic-type uncertainties. Finally, systems with a time-varying delay are investigated based on an improved FWM (IFWM) approach that yields less conservative results. Chapter 4 focuses on systems with multiple constant delays. For a system with two delays, delay-dependent criteria are derived by using the FWM approach to take the relationship between the delays into account. When the delays are equal, the criteria are equivalent to those for a system with a single delay. This idea is extended to the derivation of delay-dependent stability criteria for a system with multiple delays. Chapter 5 investigates neutral systems. The FWM approach is used to analyze the discrete-delay-dependent and neutral-delay-independent stability of a neutral system with a time-varying discrete delay. Delay-dependent stability criteria for neutral systems are derived for identical discrete and neutral delays using the FWM approach and using that approach in combination with a parameterized model transformation and an augmented Lyapunov-Krasovskii functional, respectively. Again based on the FWM approach, discrete-delayand neutral-delay-dependent stability criteria are obtained for a neutral system with different discrete and neutral delays. It is shown that these criteria include those for identical discrete and neutral delays as a special case. Chapter 6 deals with the stabilization of linear systems with a timevarying delay. Based on the delay-dependent stability criteria obtained in Chapter 3, a static-state-feedback controller that stabilizes the system is designed by an iterative method that uses the cone complementarity linearization (CCL) algorithm or the improved CCL (ICCL) algorithm that we devised by using a new stop condition, along with a method of adjusting the parameters. In addition, an LMI-based method of controller design is developed from a delay-dependent and rate-independent stability condition. Chapter 7 employs the IFWM approach to investigate the output-feedback control of a linear discrete-time system with a time-varying interval delay. The delay-dependent stability is first analyzed by a new method of estimating the upper bound on the difference of a Lyapunov function that does not ignore any terms; and based on the stability criterion, a design criterion for a static-output-feedback (SOF) controller is derived. Since the conditions thus obtained for the existence of admissible controllers are not expressed strictly in terms of LMIs, the ICCL algorithm is employed to solve the nonconvex feasibility SOF control problem. Furthermore, the problem of designing a dynamic-output-feedback (DOF) controller is formulated as one of designing

31 14 1. Introduction an SOF controller, and a DOF controller is obtained by transforming the design problem into one for an SOF controller. Chapter 8 concerns the design of an H controller for systems with a timevarying interval delay. The IFWM approach is used to devise an improved delay-dependent bounded real lemma (BRL). A method of designing an H controller is given that employs the ICCL algorithm. Chapter 9 focuses on the design of an H filter for both continuous-time and discrete-time systems with a time-varying delay. The IFWM approach is used to carry out a delay-dependent H performance analysis for error systems. The resulting criteria are extended to systems with polytopic-type uncertainties. Based on the results of the analysis, H filters are designed in terms of LMIs. Chapter 10 discusses stability problems for neural networks with timevarying delays. First, the stability of neural networks with multiple timevarying delays is considered; and the FWM approach is used to derive a delay-dependent stability criterion, from which both a delay-independent and rate-dependent criterion, and a delay-dependent and rate-independent criterion are obtained as special cases. Next, the IFWM approach is used to establish stability criteria for neural networks with a time-varying interval delay. Moreover, the FWM and IFWM approaches are used to investigate the exponential stability of neural networks with a time-varying delay. Finally, the IFWM approach is used to deal with the exponential stability of a class of discrete-time recurrent neural networks with a time-varying delay. Chapter 11 shows how the IFWM approach can be used to study the asymptotic stability of a Takagi-Sugeno (T-S) fuzzy system with a timevarying delay. By considering the relationships among the time-varying delay, its upper bound, and their difference, and without ignoring any useful terms in the derivative of the Lyapunov-Krasovskii functional, an improved LMIbased asymptotic-stability criterion is obtained for a T-S fuzzy system with a time-varying delay. Then the criterion is extended to a T-S fuzzy system with time-varying structured uncertainties. Chapter 12 investigates the problem of designing a controller for a networked control system (NCS). The IFWM approach is used to derive an improved stability criterion for a networked closed-loop system. This leads to the establishment of a method of designing a state-feedback controller based on the ICCL algorithm.

32 References 15 Chapter 13 concerns the delay-dependent stability of a stochastic system with a delay. The robust stability of an uncertain stochastic system with a time-varying delay is discussed; and the exponential stability of a stochastic Markovian jump system with nonlinearity and a time-varying delay is investigated. Less conservative results are established using the IFWM approach. Chapter 14 investigates the stability of nonlinear systems with delays. First, for Lur e control systems with multiple nonlinearities and a constant delay, LMI-based necessary and sufficient conditions for the existence of a Lyapunov-Krasovskii functional in the extended Lur e form that ensures the absolute stability of the system are obtained and extended to systems with time-varying structured uncertainties. Then, the FWM approach is used to derive delay-dependent criteria for the absolute stability of a Lur e control system with a time-varying delay. Finally, the IFWM approach is used to discuss the stability of a system with nonlinear perturbations and a timevarying interval delay. Less conservative delay-dependent stability criteria are established because the range of the delay is taken into account and an augmented Lyapunov-Krasovskii functional is used. References 1. J. K. Hale and S. M. Verduyn Lunel. Introduction to Functional Differential Equations. New York: Springer-Verlag, S. I. Niculescu. Delay Effects on Stability: A Robust Control Approach. London: Springer, K. Gu, V. L. Kharitonov, and J. Chen. Stability of Time-Delay Systems. Boston: Birkhäuser, N. P. Bhatia and G. P. Szegö. Stability Theory of Dynamical Systems. New York: Springer-Verlag, J. P. LaSalle. The Stability of Dynamical Systems. Philadelphia: SIAM, X. X. Liao, L. Q. Wang, and P. Yu. Stability of Dynamical Systems. London: Elsevier, T. Mori and H. Kokame. Stability of ẋ(t) =Ax(t)+Bx(t τ). IEEE Transactions on Automatic Control, 34(4): , S. D. Brierley, J. N. Chiasson, E. B. Lee, and S. H. Zak. On stability independent of delay. IEEE Transactions on Automatic Control, 27(1): , S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. Philadelphia: SIAM, P. Richard. Time-delay systems: An overview of some recent advances and open problems. Automatica, 39(10): , 2003.

33 16 1. Introduction 11. K. Gu. Discretized LMI set in the stability problem for linear uncertain timedelay systems. International Journal of Control, 68(4): , K. Gu. A generalized discretization scheme of Lyapunov functional in the stability problem of linear uncertain time-delay systems. International Journal of Robust and Nonlinear Control, 9(1): 1-4, K. Gu. A further refinement of discretized Lyapunov functional method for the stability of time-delay systems. International Journal of Control, 74(10): , Q. L. Han and K. Gu. On robust stability of time-delay systems with normbounded uncertainty. IEEE Transactions on Automatic Control, 46(9): , E. Fridman and U. Shaked. Descriptor discretized Lyapunov functional method: analysis and design. IEEE Transactions on Automatic Control, 51(5): , P. Park. A delay-dependent stability criterion for systems with uncertain timeinvariant delays. IEEE Transactions on Automatic Control, 44(4): , Y. S. Moon, P. Park, W. H. Kwon, and Y. S. Lee. Delay-dependent robust stabilization of uncertain state-delayed systems. International Journal of Control, 74(14): , E. Fridman and U. Shaked. Delay-dependent stability and H control: constant and time-varying delays. International Journal of Control, 76(1): 48-60, K. Gu and S. I. Niculescu. Additional dynamics in transformed time delay systems. IEEE Transactions on Automatic Control, 45(3): , K. Gu and S. I. Niculescu. Further remarks on additional dynamics in various model transformations of linear delay systems. IEEE Transactions on Automatic Control, 46(3): , E. Fridman. New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems. Systems & Control Letters, 43(4): , E. Fridman and U. Shaked. An improved stabilization method for linear timedelay systems. IEEE Transactions on Automatic Control, 47(11): , H. Gao and C. Wang. Comments and further results on A descriptor system approach to H control of linear time-delay systems. IEEE Transactions on Automatic Control, 48(3): , Y. S. Lee, Y. S. Moon, W. H. Kwon, and P. G. Park. Delay-dependent robust H control for uncertain systems with a state-delay. Automatica, 40(1): 65-72, S. I. Niculescu. On delay-dependent stability under model transformations of some neutral linear systems. International Journal of Control, 74(6): , S. I. Niculescu. Optimizing model transformations in delay-dependent analysis of neutral systems: A control-based approach. Nonlinear Analysis, 47(8): , 2001.

34 References Q. L. Han. Robust stability of uncertain delay-differential systems of neutral type. Automatica, 38(4): , Q. L. Han. Stability criteria for a class of linear neutral systems with timevarying discrete and distributed delays. IMA Journal of Mathematical Control and Information, 20(4): , E. Fridman and U. Shaked. A descriptor system approach to H control of linear time-delay systems. IEEE Transactions on Automatic Control, 47(2): , E. Fridman and U. Shaked. H Control of linear state-delay descriptor systems: an LMI approach. Linear Algebra and Its Applications, 3(2): , E. Fridman and U. Shaked. On delay-dependent passivity. IEEE Transactions on Automatic Control, 47(4): , E. Fridman and U. Shaked. Parameter dependent stability and stabilization of uncertain time-delay systems. IEEE Transactions on Automatic Control, 48(5): , E. Fridman, U. Shaked, and L. Xie. Robust H filtering of linear systems with time-varying delay. IEEE Transactions on Automatic Control, 48(1): , E. Fridman and U. Shaked. An improved delay-dependent H filtering of linear neutral systems. IEEE Transactions on Signal Processing, 52(3): , E. Fridman. Stability of linear descriptor systems with delay: a Lyapunov-based approach. Journal of Mathematical Analysis and Applications, 273(1): 24-44, M. Wu, Y. He, J. H. She, and G. P. Liu. Delay-dependent criteria for robust stability of time-varying delay systems. Automatica, 40(8): , M. Wu, S. P. Zhu, and Y. He. Delay-dependent stability criteria for systems with multiple delays. Proceedings of 23rd Chinese Control Conference, Wuxi, China, , Y. He, M. Wu, J. H. She, and G. P. Liu. Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays. Systems & Control Letters, 51(1): 57-65, Y. He and M. Wu. Delay-dependent robust stability for neutral systems with mixed discrete- and neutral-delays. Journal of Control Theorey and Applications, 2(4): , Y. He, M. Wu, and J. H. She. Delay-dependent robust stability criteria for neutral systems with time-varying delay. Proceedings of 23rd Chinese Control Conference, Wuxi, China, , Y. He, M. Wu, J. H. She, and G. P. Liu. Robust stability for delay Lur e control systems with multiple nonlinearities. Journal of Computational and Applied Mathematics, 176(2): , M. Wu, Y. He, and J. H. She. Delay-dependent robust stability and stabilization criteria for uncertain neutral systems. Acta Automatica Sinica, 31(4): , 2005.

35 18 1. Introduction 43. Y. He and M. Wu. Delay-dependent conditions for absolute stability of Lur e control systems with time-varying delay. Acta Automatica Sinica, 31(3): , M. Wu, Y. He, and J. H. She. New delay-dependent stability criteria and stabilizing method for neutral systems. IEEE Transactions on Automatic Control, 49(12): , M. Wu and Y. He. Parameter-dependent Lyapunov functional for systems with multiple time delays. Journal of Control Theory and Applications, 2(3): , Y. He, M. Wu, J. H. She, and G. P. Liu. Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic type uncertainties. IEEE Transactions on Automatic Control, 49(5): , Y. He, M. Wu, and J. H. She. Improved bounded-real-lemma representation and H control for systems with polytopic uncertainties. IEEE Transactions on Circuits and Systems II, 52(7): , 2005.

36 2. Preliminaries This chapter provides basic knowledge and concepts on the stability of timedelay systems, including the concept of stability, H norm, H control, the LMI method, and some useful lemmas. They form the foundation for subsequent chapters. 2.1 Lyapunov Stability and Basic Theorems The stability of a system generally refers to its ability to return to its initial state when an external disturbance ceases. Stability is the primary condition for the normal operation of a control system. The Lyapunov stability theorem defines the stability of a system in terms of energy, the biggest advantage of which is that the stability can be determined without the need to solve the motion equation of the system Types of Stability This subsection defines various types of stability for continuous-time and discrete-time systems. 1) Continuous-Time Systems Consider the following continuous-time system: ẋ(t) =f(t, x(t)), x(t 0 )=x 0, (2.1) where x(t) R n is the state vector; f : R+ R n R n ;andt is a continuous time variable. Apointx e R n is called an equilibrium point of system (2.1) if f(t, x e )= 0 for all t t 0. The system remains at that point as long as there is no external action on it. The question is, if there is an external action, will the

37 20 2. Preliminaries system remain near the equilibrium point or will it move farther and farther away? The problem of stability at an equilibrium point is discussed below. Shifting the origin of the system allows us to move the equilibrium point to x e = 0. If there are multiple equilibrium points, the stability of each must be studied by an appropriate shift of the origin. Various types of stability are defined below for system (2.1) at the equilibrium point, x e =0. Definition [1] (1) If, for any t 0 0 and ε>0, there exists a δ 1 = δ(t 0,ε) > 0 such that x(t 0 ) <δ(t 0,ε)= x(t) <ε, t t 0, (2.2) then the system is stable (in the Lyapunov sense) at the equilibrium point, x e =0. (2) If the system is stable at the equilibrium point x e =0and if there exists δ 2 = δ(t 0 ) > 0 such that x(t 0 ) <δ(t 0 )= lim t x(t) =0, (2.3) then the system is asymptotically stable at the equilibrium point, x e =0. (3) If there exist constants δ 3 > 0, α>0, and β>0 such that x(t 0 ) <δ 3 = x(t) β x(t 0 ) e α(t t0), (2.4) then the system is exponentially stable at the equilibrium point, x e =0. (4) If δ 1 in (1) (or δ 2 in (2)) can be chosen independently of t 0, then the system is uniformly stable (or uniformly asymptotically stable) at the equilibrium point, x e =0. (5) If δ 2 in (2) (or δ 3 in (3)) can be an arbitrarily large, finite number, then the system is globally asymptotically stable (or globally exponentially stable) at the equilibrium point, x e =0. The two figures below may help the reader acquire an intuitive understanding of the concepts of stability and asymptotic stability. Fig. 2.1 illustrates the property of stability in the Lyapunov sense. It shows that the track, x(t), remains in the neighborhood Ω(ε) of the equilibrium point as long as x(t 0 ) is in the neighborhood Ω(δ). Fig. 2.2 illustrates the property of asymptotic stability. It shows that the system is asymptotically stable at

38 2.1 Lyapunov Stability and Basic Theorems 21 Fig Stability in the Lyapunov sense Fig Asymptotic stability the equilibrium point, x e = 0, if it is stable at that point and if all solutions starting near that point approach it as t. 2) Discrete-Time Systems Consider the following discrete-time system: x(k +1)=f(k, x(k)), x(k 0 )=x 0, (2.5) where x(k) R n is the state vector; f : Z+ R n R n ;andf(k, x) is continuous in x. Apointx e in R n is called an equilibrium point of (2.5) if f(k, x e )=x e for all k k 0. In the literature, x e is usually assumed to be the origin and is called the zero solution. We now define various types of stability for system (2.5) at the equilibrium point, x e =0. Definition [2] (1) If, for any k 0 0 and ε>0, there exists a δ 1 = δ(k 0,ε) > 0 such that x(k 0 ) <δ(k 0,ε)= x(k) <ε, k k 0, (2.6)

39 22 2. Preliminaries then the system is stable (in the Lyapunov sense) at the equilibrium point, x e =0. (2) If the system is stable at the equilibrium point, x e =0, and if there exists a δ 2 = δ(k 0 ) > 0 such that x(t 0 ) <δ(k 0 )= lim t x(k) =0, (2.7) then the system is asymptotically stable at the equilibrium point, x e =0. (3) If there exist a δ 3 and constants α>0 and β>0 such that x(k 0 ) <δ 3 = x(k) β x(k 0 ) e α(k k0). (2.8) then the system is exponentially stable at the equilibrium point, x e =0. (4) If δ 1 in (1) (or δ 2 in (2)) can be chosen independently of k 0, then the system is uniformly stable (or uniformly asymptotically stable) at the equilibrium point, x e =0. (5) If δ 2 in (2) (or δ 3 in (3)) can be an arbitrarily large, finite number, then the system is globally asymptotically stable (or globally exponentially stable) at the equilibrium point, x e =0. The figures below illustrate the concepts of stability and asymptotic stability. Regarding stability in the Lyapunov sense, when the movement of a solution starts inside a sphere of radius δ in the phase plane (Fig. 2.3), all states x(k) fork k 0 remain in a disk with a radius of ε. The time trajectory in three-dimensional space (Fig. 2.4) provides another perspective on stability. Fig. 2.5 depicts the asymptotic stability of the zero solution Lyapunov Stability Theorems Lyapunov used classical mechanics to investigate how the distribution of the energy field of a system influences its stability, and then devised a method based on the above definition of stability to determine the stability of a system without explicitly integrating a differential equation. This is called Lyapunov s direct method or the second method of Lyapunov. Classical mechanics tells us that, in a physical system, a mass is less stable when it has a high energy than when it has a low energy. Thus, when a particle moves from an unstable state towards a stable state, its energy must necessarily continuously decrease. If we denote the energy by E, then this situation is described by

40 2.1 Lyapunov Stability and Basic Theorems 23 Fig Stability in the Lyapunov sense in phase plane Fig Stability in the Lyapunov sense in three-dimensional space E>0, de dt < 0. Take a mechanical oscillation, for example. Even though the speed of the oscillator varies, the total energy of the system (which is the sum of the kinetic and potential energies) keeps decreasing and ultimately becomes zero at the equilibrium point. At that point, a passive system is stable; and it is impossible for the total energy of an independent passive system to increase.

41 24 2. Preliminaries Fig Asymptotic stability That is, in the neighborhood of an equilibrium point, no positive change in the total energy of the system can occur. Based on the above principles, Lyapunov constructed an energy function, V (t, x(t)), that is expressed solely in terms of the state energy. If > 0, if x 0, V (t, x(t)) =0, if x =0, and V (t, x(t)) 0, then the stability at the equilibrium point can be proven without using any information on the solutions of the motion equation of the system. V (t, x(t)) is called a Lyapunov function. Theorem [1] (Lyapunov stability theorem for continuous-time system) Consider system (2.1). Let f(t,0) = 0, t, which means that the equilibrium point of the system is x e =0. If (1) there exists a positive definite function V (t, x(t)) and (2) V (t, x(t)) := d V (t, x(t)) is negative semi-definite, dt then the system is stable at the equilibrium point, x e =0. If (1) there exists a positive definite function V (t, x(t)) and (2) V (t, x(t)) := d dt V (t, x(t)) is negative definite, then the system is asymptotically stable at the equilibrium point, x e =0. If (1) the system is asymptotically stable at x e =0and (2) V (t, x(t)) as x,

42 2.2 Stability of Time-Delay Systems 25 then the system is globally asymptotically stable at the equilibrium point, x e =0. Theorem [2] (Lyapunov stability theorem for discrete-time system) Consider system (2.5). Let f(0, k) =0, k, which means that the equilibrium point of the system is x e =0. If (1) there exists a positive definite function V (k, x(k)) and (2) ΔV (k, x(k)) := V (k +1,x(k +1)) V (k, x(k)) 0 k, x 0, then the system is stable at the equilibrium point x e =0. If (1) there exists a positive definite function V (k, x(k)) and (2) ΔV (k, x(k)) := V (k +1,x(k +1)) V (k, x(k)) < 0 k, x 0, then the system is asymptotically stable at the equilibrium point, x e =0. If (1) the system is asymptotically stable at x e =0and (2) V (k, x(k)) as x, then the system is globally asymptotically stable at the equilibrium point, x e = Stability of Time-Delay Systems This section presents some basic definitions and theoretical results in the theory of time-delay systems Stability-Related Topics This subsection presents some basic information on time-delay systems, specifically, fundamental concepts, descriptions, and types of stability. 1) Time-Delay Systems In science and engineering, differential equations are often used as mathematical models of systems. A fundamental assumption about a system that is modeled in this way is that its future evolution depends solely on the current values of the state variables and is independent of their history. For example, consider the following first-order differential equation: ẋ(t) =f(t, x(t)), x(t 0 )=x 0. The future evolution of the state variable x at time t depends only on t and x(t), and does not depend on the values of x before time t.

43 26 2. Preliminaries If the future evolution of the state of a dynamic system depends not only on current values, but also on past ones, then the system is called a timedelay system. Actual systems of this type cannot be satisfactorily modeled by an ordinary differential equation; that is, a differential equation is only an approximate model. One way to describe such systems precisely is to use functional differential equations. 2) Functional Differential Equations In many systems, there may be a maximum delay, h. In this case, we are often interested in the set of continuous functions that map [ h, 0] to R n, which we denote simply by C = C([ h, 0], R n ). For any a>0, any continuous function of time ψ C([t 0 h, t 0 + a], R n ), and t 0 t t 0 + a, letψ t Cbe the segment of ψ given by ψ t (θ) =ψ(t + θ), h θ 0. The general form of a retarded functional differential equation (RFDE) (or functional differential equation of retarded type) is ẋ(t) =f(t, x t ), (2.9) where x(t) R n and f : R C R n. This equation indicates that the derivative of the state variable x at time t depends on t and x(ζ) fort h ζ t. Thus, to determine the future evolution of the state, it is necessary to specify the initial value of the state variable, x(t), in a time interval of length h, say,fromt 0 h to t 0 ;thatis, x t0 = φ, (2.10) where φ C is given. In other words, x(t 0 + θ) =φ(θ), h θ 0. It is important to note that, in an RFDE, the derivative of the state contains no term with a delay. If such a term does appear, then we have a functional differential equation of neutral type. For example, 5ẋ(t)+2ẋ(t h)+x(t) x(t h) =0 is a neutral functional differential equation (NFDE). For an a>0, a function x is said to be a solution of RFDE (2.9) in the interval [t 0 h, t 0 + a) ifx is continuous and satisfies that RFDE in that interval. Here, the time derivative should be interpreted as a one-sided derivative in the forward direction. Of course, a solution also implies that (t, x t ) is within the domain of the definition of f. If the solution also satisfies the initial condition (2.10), we say that it is a solution of the equation with

44 2.2 Stability of Time-Delay Systems 27 the initial condition (2.10), or simply a solution through (t 0,φ). We write it as x(t 0,φ,f) when it is important to specify the particular RFDE and the given initial condition. The value of x(t 0,φ,f)att is denoted by x(t; t 0,φ,f). We omit f and write x(t 0,φ)orx(t; t 0,φ)whenf is clear from the context. A fundamental issue in the study of both ordinary differential equations and functional differential equations is the existence and uniqueness of a solution. We state the following theorem without proof. Theorem [3] (Uniqueness) Suppose that Ω R C is an open set, function f : Ω R n is continuous, and f(t, φ) is Lipschitzian in φ in each compact set in Ω. That is, for a given compact set, Ω 0 Ω, there exists a constant L such that f(t, φ 1 ) f(t, φ 2 ) L φ 1 φ 2 for any (t, φ 1 ) Ω 0 and (t, φ 2 ) Ω 0. If (t 0,φ) Ω, then there exists a unique solution of RFDE (2.9) through (t 0,φ). 3) Concept of Stability Let y(t) be a solution of RFDE (2.9). The stability of the solution depends on the behavior of the system when the system trajectory, x(t), deviates from y(t). Throughout this book, without loss of generality, we assume that RFDE (2.9) admits the solution x(t) = 0, which will be referred to as the trivial solution. If the stability of a nontrivial solution, y(t), needs to be studied, then we can use the variable transformation z(t) =x(t) y(t) to produce the new system ż(t) =f(t, z t + y t ) f(t, y t ), (2.11) which has the trivial solution z(t) =0. For the function φ C([a, b], R n ), define the continuous norm c to be φ c = sup φ(θ). a θ b In this definition, the vector norm represents the 2-norm 2. As we did above for continuous- and discrete-time systems, we now define various types of stability for the trivial solution of time-delay system (2.9). Definition [4] If, for any t 0 R and ɛ>0, there exists a δ = δ(t 0,ɛ) > 0 such that x t0 c <δimplies x(t) <ɛfor t t 0, then the trivial solution of (2.9) is stable.

45 28 2. Preliminaries If the trivial solution of (2.9) is stable, and if, for any t 0 R and any ɛ>0, there exists a δ a = δ a (t 0,ɛ) > 0 such that x t0 c <δ a implies lim x(t) =0, then the trivial solution of (2.9) is asymptotically stable. t If the trivial solution of (2.9) is stable and if δ(t 0,ɛ) can be chosen independently of t 0, then the trivial solution of (2.9) is uniformly stable. If the trivial solution of (2.9) is uniformly stable and if there exists a δ a > 0 such that, for any η>0, there exists a T = T (δ a,η) such that x t0 c <δ a implies x(t) <ηfor t t 0 + T, and t 0 R, then the trivial solution of (2.9) is uniformly asymptotically stable. If the trivial solution of (2.9) is (uniformly) asymptotically stable and if δ a canbeanarbitrarilylarge, finite number, then the trivial solution of (2.9) is globally (uniformly) asymptotically stable. If there exist constants α>0 and β>0 such that x(t) β sup x(θ) e αt, h θ 0 then the trivial solution of (2.9) is globally exponentially stable; and α is called the exponential convergence rate Lyapunov-Krasovskii Stability Theorem Just as for a system without a delay, the Lyapunov method is an effective way of determining the stability of a system with a delay. When there is no delay, this determination requires the construction of a Lyapunov function, V (t, x(t)), which can be viewed as a measure of how much the state, x(t), deviates from the trivial solution, 0. Now, in a delay-free system, we need x(t) to specify the future evolution of the system beyond t. Inatime-delay system, we need the state at time t for that purpose; it is the value of x(t) intheinterval[t h, t] (that is, x t ). So, it is natural to expect that, for a time-delay system, the Lyapunov function is a functional, V (t, x t ), that depends on x t and indicates how much x t deviates from the trivial solution, 0. This type of functional is called a Lyapunov-Krasovskii functional. More specifically, let V (t, φ) : R C R be differentiable; and let x t (τ,φ) be the solution of RFDE (2.9) at time t for the initial condition x τ = φ. Calculating the time derivative of V (t, x t ) and evaluating it at t = τ yield

46 2.2 Stability of Time-Delay Systems 29 V (τ,φ) = d dt V (t, x t) t=τ,xt=φ = lim sup Δt 0 V (τ +Δt, x τ +Δt (τ,φ)) V (τ,φ). Δt If V (t, xt )isnonpositive,thenx t does not grow with t, which means that the system under consideration is stable in the sense of Definition The following theorem states this more precisely. Theorem [4] (Lyapunov-Krasovskii stability theorem) Suppose that f : R C R n in (2.9) maps R (bounded sets in C) into bounded sets in R n, and that u, v, w : R+ R + are continuous nondecreasing functions, where u(τ) and v(τ) are positive for τ>0and u(0) = v(0) = 0. If there exists a continuous differentiable functional V : R C R such that u( φ(0) ) V (t, φ) v( φ c ) and V (t, φ) w( φ(0) ), then the trivial solution of (2.9) is uniformly stable. If the trivial solution of (2.9) is uniformly stable, and w(τ) > 0 for τ>0, then the trivial solution of (2.9) is uniformly asymptotically stable. If the trivial solution of (2.9) is uniformly asymptotically stable and if lim u(τ) =, then the trivial solution of (2.9) is globally uniformly τ asymptotically stable Razumikhin Stability Theorem That the Lyapunov-Krasovskii functional requires the state variable x(t) in the interval [t h, t] necessitates the manipulation of functionals, which makes the Lyapunov-Krasovskii theorem difficult to apply. This difficulty can sometimes be circumvented by using the Razumikhin theorem, an alternative that involves only functions, but no functionals. The key idea behind the Razumikhin theorem is the use of a function, V (x), to represent the size of x(t): V (x t )= max V (x(t + θ)). θ [ h, 0] This function indicates the size of x t.ifv (x(t)) < V (x t ), then V (x t )doesnot grow when V (x(t)) > 0. In fact, for V (x t ) not to grow, it is only necessary that

47 30 2. Preliminaries V (x(t)) not be positive whenever V (x(t)) = V (x t ). The precise statement is given in the next theorem. Theorem [4](Razumikhin theorem) Suppose that f : R C R n in (2.9) maps R (bounded sets of C) into bounded sets of R n and also that u, v, w : R+ R + are continuous nondecreasing functions, u(τ) and v(τ) are positive for τ > 0, u(0) = v(0) = 0, and v is always increasing. If there exists a continuously differentiable function V : R R n R such that u( x ) V (t, x) v( x ), t R, x R n, (2.12) and the derivative of V along the solution, x(t), of system (2.9) satisfies V (t, x(t)) w( x(t) ) whenever V (t+θ, x(t+θ)) V (t, x(t)) (2.13) for θ [ h, 0], then the trivial solution of (2.9) is uniformly stable. If there exists a continuously differentiable function V : R R n R such that u( x ) V (t, x) v( x ), t R, x R n, (2.14) if w(τ) > 0 for τ > 0, and if there exists a continuous nondecreasing function p(τ) >τ for τ>0such that condition (2.13) is strengthened to V (t, x(t)) w( x(t) ) if V (t + θ, x(t + θ)) p(v (t, x(t))) (2.15) for θ [ h, 0], then the trivial solution of (2.9) is uniformly asymptotically stable. If the trivial solution of (2.9) is uniformly asymptotically stable and if lim u(τ) =, then the trivial solution of (2.9) is globally uniformly τ asymptotically stable. 2.3 H Norm This section presents some basic concepts that are used in this book Norm Let X be a vector space over the complex field C. Forx X,letf(x) :x R be a real-valued function. If it has the following properties:

48 2.3 H Norm 31 (1) f(x) 0, (2) f(αx) = α f(x), α R, (3) f(x, y) f(x)+f(y), y X, and (4) f(x) = 0 if and only if x =0, then f(x) is said to be a norm on x. It is denoted by x H Norm H space is a space of matrix functions, F (s), that are analytic on the open right-half plane (that is, Re(s) > 0), take values in C m n, and satisfy F =sup{σ max [F (s)] : Re(s) > 0} < +. (2.16) s This equation defines F,theH norm of the matrix functions F (s) [5]. Consider the following linear time-invariant system: ẋ(t) =Ax(t)+Bw(t), (2.17) z(t) =Cx(t)+Dw(t), where x(t) R n is the state vector and x(0) = 0; w(t) R m is a disturbance input vector; and A, B, C, andd are real matrices with appropriate dimensions. G(s) =C(sI A) 1 B + D is the transfer function matrix of the system. For convenience, we denote G(s) = A B. C D From (2.16) and the Maximum Modular Theorem, the H norm of the proper transfer function matrix, G(s), of a stable linear time-invariant system is defined to be G =supσ max [G(jω)]. (2.18) ω When G(s) is a scalar transfer function, the H norm is defined to be G =sup G(jω). (2.19) ω The H norm can also be defined in the time domain. Let w(t) bea square, integrable input signal; and let z(t) be the output signal. Their energies are defined to be

49 32 2. Preliminaries + w 2 2 = w T (t)w(t)dt, + z 2 2 = z T (t)z(t)dt. So, the H norm of G(s) is z 2 G =sup. (2.20) ω 0 w 2 The H norm reflects the maximum ratio of the output signal energy to the input signal energy, or in other words, the maximum energy amplification ratio of the system. 2.4 H Control Fig. 2.6 shows a block diagram of a standard H control problem. G is the generalized plant, which is given in the problem statement; and K is the controller, which needs to be designed. Here, we assume that the system and controller are finite-dimensional, linear, and time-invariant. The external input, w, the control input, u, the controlled output, z, andthemeasured output, y, are all vector signals. Fig Block diagram of standard H control problem Assume G(s) andk(s) in Fig. 2.6 are both proper real-rational transform function matrices that describe a linear time-invariant system. From Fig. 2.6, we have z = G(s) w. y u

50 2.4 H Control 33 The state space realization of G(s) is ẋ = Ax + B 1 w + B 2 u, z = C 1 x + D 11 w + D 12 u, y = C 2 x + D 21 w + D 22 u. So, A B 1 B 2 G(s) = C 1 D 11 D 12, (2.21) C 2 D 21 D 22 where x R n is the state vector. Also, assume w R m1, u R m2, z R p1, and y R p2. Decomposing G(s) into G(s) = G 11(s) G 12 (s), (2.22) G 21 (s) G 22 (s) and comparing (2.22) and (2.21), we have G ij (s) =C i (si A) 1 B j + D ij, i,j =1, 2. (2.23) The closed-loop transform function matrix from w to z is T zw (s) =G 11 (s)+g 12 (s)k(s)(i G 22 (s)k(s)) 1 G 21 (s) :=F l (G, K), (2.24) where F l (G, K) is called the lower linear fractional transformation (LFT) of G(s) andk(s). The H optimal control problem for the closed-loop control system in Fig. 2.6 involves (1) finding a proper real-rational controller, K(s), that stabilizes the system internally and (2) minimizing the H norm of T zw (s), that is, finding min F l(g, K). (2.25) K stabilizes G The H suboptimal control problem for the closed-loop control system in Fig. 2.6 involves (1) finding all proper real-rational controllers, K(s), that stabilize the closedloop system internally and (2) making the H norm of T zw (s) less than a given constant γ>0: F l (G, K) <γ. (2.26)

51 34 2. Preliminaries 2.5 LMI Method In the past couple of decades, LMIs have become a hot topic in the field of the analysis and design of control systems [6]. This is due to the good properties of LMIs, breakthroughs in mathematical programming, and the discovery of useful algorithms and ways of using them to solve problems. Of particular importance are the development of interior-point algorithms and the launch of the LMI toolbox in Matlab. Previously, Riccati equations and inequalities were used to represent and solve most control problems; but that involved a large number of parameters, and symmetric positive definite matrices needed to be adjusted beforehand. So, even though a solution might exist, it might not necessarily be found. This is a big drawback when dealing with real-world problems. LMIs, on the other hand, do not suffer from this handicap and, furthermore, require no adjustment of parameters Common Specifications of LMIs An LMI is an expression of the form F (x) =F 0 + x 1 F x m F m < 0, (2.27) where x 1,x 2,,x m are real variables, which are called the decision variables of the LMI (2.27); x =(x 1,x 2,,x m ) T R m is a vector consisting of decision variables, which is called the decision vector; and F i = F T i R n n, i =0, 1,,m are given symmetric matrices. In many system and control problems, the variables are matrices. One example is a Lyapunov matrix inequality: F (X) =A T X + XA + Q<0, (2.28) where A R n n and Q = Q T R n n are given constant matrices, and the variable X = X T R n n is an unknown matrix. That is, the variable in this matrix inequality is a matrix. Let E 1,E 2,,E m be a basis in S n = {N : N = N T R n n }. For any symmetric matrix X = X T R n n,there exist x 1,x 2,,x m such that X = m i=1 x ie i. Therefore, ( m ) F (X) =F x i E i = A T( m ) ( m x i E i + x i E i )A + Q i=1 i=1 i=1 = Q + x 1 (A T E 1 + E 1 A)+ + x m (A T E m + E m A) < 0.

52 2.5 LMI Method 35 Thus, (2.27) is the general form of a Lyapunov matrix inequality written in terms of LMIs. Replacing < with in (2.27) produces a non-strict LMI. For arbitrary affine functions F (x) andg(x) : R m S n, F (x) > 0andF (x) < G(x) are also LMIs because they can be written as F (x) < 0, F (x) G(x) < 0. The set of all x satisfying LMI (2.27) is a convex set. This property of LMIs makes it possible to solve some LMI problems by methods commonly used to solve convex optimization problems Standard LMI Problems This section presents three generic LMI problems for which the Matlab LMI toolbox has solvers. Let F, G, andh be symmetric matrix affine functions; and let c be a given constant vector. LMI problem (LMIP): For the LMI F (x) < 0, the problem is to determine whether or not there exists an x such that F (x ) < 0 holds. This is called a feasibility problem. That is, if there exists such an x, then the LMI is feasible; otherwise, it is infeasible. Eigenvalue problem (EVP): The problem is to minimize the maximum eigenvalue of a matrix subject to an LMI constraint (or to prove that the constraint is infeasible). The general form of an EVP is: Minimize λ subject to G(x) <λi, H(x) < 0. EVPs can also appear in the equivalent form of minimizing a linear function subject to an LMI: Minimize c T x subject to F (x) < 0. This is the standard form for the EVP solver in the LMI toolbox. The feasibility problem for the LMI F (x) < 0 can also be written as an EVP:

53 36 2. Preliminaries Minimize λ subject to F (x) λi < 0. Clearly, for any x, ifλ is chosen large enough, (x, λ) is a feasible solution to the above problem. So, the problem certainly has a solution. If the minimum λ, λ,satisfiesλ 0, then the LMI F (x) < 0 is feasible. Generalized eigenvalue problem (GEVP): The problem is to minimize the maximum generalized eigenvalue of a pair of affine matrix functions, subject to an LMI constraint. For two given symmetric matrices G and F of the same order and a scalar λ, if there exists a nonzero vector y such that Gy = λf y, thenλ is called the generalized eigenvalue of matrices G and F. The problem of calculating the maximum generalized eigenvalue of G and F can be transformed into an optimization problem subject to an LMI constraint. Suppose that F is positive definite and that λ is a scalar. If λ is sufficiently large, G λf < 0. As λ decreases, G λf will become singular at some point. So there exists a nonzero vector y such that Gy = λf y. This λ is the generalized eigenvalue of matrices G and F. Using this idea, we can obtain the generalized eigenvalue of G and F by solving the following optimization problem: Minimize λ subject to G λf < 0. If G and F areaffinefunctionsofx, the general form of the problem of minimizing the maximum generalized eigenvalue of the matrix functions G(x) andf (x) subject to an LMI constraint is Minimize λ subject to G(x) <λf(x), F (x) > 0, H(x) < 0. Note that, in this problem, the constraints are not linear in x and λ simultaneously. 2.6 Lemmas In this section, some basic lemmas that are used extensively throughout this book are given without proof.

54 2.6 Lemmas 37 Lemma [6](Schur complement) For a given symmetric matrix S = S T = S 11 S 12, where S 11 R r r, the following conditions are equivalent: S 22 (1) S<0; (2) S 11 < 0, S 22 S T 12 S 1 11 S 12 < 0; and (3) S 22 < 0, S 11 S 12 S 1 22 ST 12 < 0. Lemma dimensions, [7] For given matrices Q = Q T,H,and E with appropriate Q + HF(t)E + E T F T (t)h T < 0 holds for all F (t) satisfying F T (t)f (t) I ifandonlyifthereexistsε>0 such that Q + ε 1 HH T + εe T E<0. Lemma [8] There exists a symmetric matrix X such that P 1 + XQ 1 > 0, P 2 XQ 2 > 0 R 1 R 2 if and only if P 1 + P 2 Q 1 Q 2 R 1 0 > 0. R 2 Lemma [9] (S-procedure) Let T i R n n,i=0, 1,,pbe symmetric matrices. Consider the following condition on T 0,T 1,,T p : ζ T T 0 ζ>0, for all ζ 0 such that ζ T T i ζ 0, i =1, 2,,p. (2.29) Clearly, if there exist τ i 0, p i =0, 1,, p such that T 0 τ i T i > 0, (2.30) i=1 then (2.29) holds. It is a nontrivial fact that, when p =1, the converse also holds (that is, (2.29) and (2.30) are equivalent), provided that there exists a ζ 0 such that ζ T 0 T 1ζ 0 > 0.

55 38 2. Preliminaries Lemma [10] Let A, D, E, F, and P be real matrices with appropriate dimensions, and let F T F I and P > 0. Then, the following propositions are true: (1) For any x, y R n, 2x T y x T P 1 x + y T Py. (2) For any x, y R n and any ε>0, 2x T DFEy ε 1 x T DD T x + εy T E T Ey. (3) For any ε>0 satisfying P εdd T > 0, (A + DFE) T P 1 (A + DFE) ε 1 E T E + A T (P εdd T ) 1 A. 2.7 Conclusion This chapter provides basic knowledge and concepts on the stability of timedelay systems, including the concept of Lyapunov stability and some basic theorems, fundamental concepts related to the stability of time-delay systems, H norm, H control, the LMI method, and some useful lemmas. This knowledge is the foundation for the study of later chapters. References 1. X. X. Liao, L. Q. Wang, and P. Yu. Stability of Dynamical Systems. London: Elsevier, S. Elaydi. An Introduction to Difference Equations. New York: Springer-Verlag, J. K. Hale and S. M. Verduyn Lunel. Introduction to Functional Differential Equations. New York: Springer-Verlag, K. Gu, V. L. Kharitonov, and J. Chen. Stability of Time-Delay Systems. Boston: Birkhäuser, K. Zhou, J. C. Doyle, and K. Glover. Robust and Optimal Control. New Jersey: Prentice Hall, S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. Philadelphia: SIAM, I. R. Petersen and C. V. Hollot. A Riccati equation approach to the stabilization of uncertain linear systems. Automatica, 22(4): , 1986.

56 References K. Gu. A further refinement of discretized Lyapunov functional method for the stability of time-delay systems. International Journal of Control, 74(10): , V. A. Yakubovič. The S-procedure in nonlinear control theory. Vestnik Leningrad University: Mathematics, 4(1): 73-93, Y. Y. Cao, Y. X. Sun, and C. W. Cheng. Delay-dependent robust stabilization of uncertain systems with multiple state delays. IEEE Transactions on Automatic Control, 43(11): , 1998.

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58 3. Stability of Systems with Time-Varying Delay Increasing attention is being paid to delay-dependent stability criteria for linear time-delay systems. Investigations of such criteria for constant delays [1 10] usually involve either some type of frequency-domain method or the Lyapunov-Krasovskii functional method in the time domain. For time-varying delays, studies of such criteria [11 17] generally employ a fixed model transformation because frequency-domain methods and the discretized Lyapunov-Krasovskii functional method are too difficult to use in this case. Of the four types of fixed model transformations, three can handle time-varying delays: Model transformation I [12 14]; Model transformation III [18]; and Model transformation IV, which uses both Park s and Moon et al. s inequalities. Model transformations III and IV are the most effective, and they can also be used to obtain delay-independent stability criteria. However, their use of fixed weighting matrices imposes certain limitations, as pointed out in Chapter 1. On the other hand, there are often uncertainties due to errors in system modeling and changes in operating conditions. One way of describing a system uncertainty is a parameter uncertainty in the state equation, of which there are two types: a time-varying structured uncertainty and a polytopic-type uncertainty. For the former, [19] gave a necessary and sufficient condition under which stability criteria for nominal systems can easily be extended to uncertain systems. For the latter, research has shown that a parameterdependent Lyapunov function or functional can eliminate the conservativeness of quadratic stability (See, e.g., [20 25] regarding linear continuous systems; [9 11, 26, 27] regarding time-delay systems; and [28 30] regarding discrete systems). However, one problem with using a parameter-dependent Lyapunov function or functional is that it is difficult to separate the system matrices from the Lyapunov matrices in the derivative. Many researchers have endeavored to do that. Some have devised a method of separating them that yields only a sufficient condition. Others have devised sufficient and nec-

59 42 3. Stability of Systems with Time-Varying Delay essary criteria, but they have the drawback that they cannot be formulated solely in terms of LMIs and require the manual adjustment of parameters. So, they also impose limitations. This chapter concerns delay-dependent stability problems for systems with a time-varying delay. For nominal systems, two approaches are first used to derive stability conditions for two different treatments of the term ẋ(t). The first approach is to replace the term ẋ(t) in the derivative of the Lyapunov-Krasovskii functional with the system equation and to use FWMs to express the relationships among the terms of the Newton-Leibnitz formula. This technique yields delay-dependent stability criteria. We also show that these criteria include delay-independent ones, and that Moon et al. s criterion [5] is a special case of the criterion in this chapter [31]. The other approach is to retain the term ẋ(t) in the derivative of the Lyapunov-Krasovskii functional and to use FWMs to express the relationships among the terms of the state equation of the system. This technique yields delay-dependent stability criteria for systems with a time-varying delay. Moreover, we show that Fridman et al. s criterion [11], which was obtained by using the descriptor model transformation in combination with Park s and Moon et al. s inequalities, is a special case of the criterion in this chapter [32]. Then, we prove that the criteria obtained by these two different approaches are equivalent. Furthermore, we use Lemma to extend these two categories of criteria to systems with time-varying structured uncertainties. The criteria obtained by using FWMs to express the relationships among the terms of the system state equations separate the system matrices from the Lyapunov matrices in a natural way. That makes it easy to extend these criteria to systems with polytopic-type parameter uncertainties, which are handled using a parameter-dependent Lyapunov-Krasovskii functional. The resulting delaydependent and delay-independent stability criteria are formulated in terms of LMIs [32]. Finally, the stability of systems with a time-varying delay are examined using the IFWM approach, which considers the relationships among the timevarying delay, its upper bound, and their difference, and does not ignore any useful terms in the derivative of the Lyapunov-Krasovskii functional [33 35]. This is in contrast to [26, 31, 32, 36 38], which did ignore some useful terms, thereby leading to considerable conservativeness.

60 3.1 Problem Formulation Problem Formulation Consider the following nominal linear system with a time-varying delay: ẋ(t) =Ax(t)+A d x(t d(t)), t > 0, (3.1) x(t) =φ(t), t [ h, 0], where x(t) R n is the state vector; A and A d are constant matrices with appropriatedimensions; the delay, d(t), is a time-varying continuous function; and the initial condition, φ(t), is a continuously differentiable initial function of t [ h, 0]. In this chapter, the delay is assumed to satisfy one or both of the following conditions: 0 d(t) h, d(t) μ, (3.2) (3.3) where h and μ are constants. A system containing time-varying structured uncertainties is described by ẋ(t) =(A +ΔA(t))x(t)+(A d +ΔA d (t))x(t d(t)), t > 0, (3.4) x(t) =φ(t), t [ h, 0]. The uncertainties are assumed to be of the form [ΔA(t) ΔA d (t)] = DF(t)[E a E ad ], (3.5) where D, E a,ande ad are constant matrices with appropriate dimensions; and F (t) is an unknown, real, and possibly time-varying matrix with Lebesguemeasurable elements satisfying F T (t)f (t) I, t. (3.6) This chapter also discusses another class of uncertainties, namely, polytopictype uncertainties. For this class, matrices A and A d of system (3.1) contain uncertainties and satisfy the real, convex, polytopic-type model p p [A A d ] Ω, Ω = [A(ξ) A d(ξ)]= ξ j [A j A dj ], ξ j =1,ξ j 0, j=1 j=1 (3.7) where A j and A dj,j =1, 2,,p are constant matrices with appropriate dimensions; and ξ j,j=1, 2,,p are time-invariant uncertainties.

61 44 3. Stability of Systems with Time-Varying Delay 3.2 Stability of Nominal System Now, we use the FWM approach to obtain delay-dependent stability conditions for systems with a time-varying delay. Rather than using the Newton- Leibnitz formula to directly replace the delay term, we use FWMs to take into account the relationships among the terms of the Newton-Leibnitz formula in the derivation of delay-dependent stability criteria for nominal system (3.1). This section is divided into three parts. In the first part, the term ẋ(t) is replaced with system equation (3.1) in the conventional way. In the second part, it is retained, and FWMs are used to express the relationships among the terms of the state equation of the system. This method enables the system matrices and Lyapunov matrices to be easily separated, which lays the foundation for a discussion of a parameter-dependent Lyapunov-Krasovskii functional in Section 3.4. In the third part, we prove that these two treatments are equivalent. This section first examines delay- and rate-dependent stability conditions; and based on those conditions, a simple procedure yields delay-dependent and rate-independent ones Replacing the Term ẋ(t) The Newton-Leibnitz formula gives us x(t d(t)) = x(t) t d(t) ẋ(s)ds. For any appropriately dimensioned matrices N 1 and N 2, the following is true: 2 [ x T (t)n 1 + x T ] [ ] t (t d(t))n 2 x(t) ẋ(s)ds x(t d(t)) =0. t d(t) (3.8) In the next theorem, the terms on the left side of this equation are added to the derivative of the Lyapunov-Krasovskii functional. The FWMs, N 1 and N 2, indicate the relationships among the terms of the Newton-Leibnitz formula; and optimal values for them can be obtained by solving LMIs. Now, replacing the term ẋ(t) with system equation (3.1) yields the following theorem. Theorem Consider nominal system (3.1) with a delay, d(t), that satisfies both (3.2) and (3.3). Given scalars h > 0 and μ, the system is asymptotically stable if there exist matrices P > 0, Q 0, Z>0, and

62 3.2 Stability of Nominal System 45 X = X 11 X 12 0, and any appropriately dimensioned matrices N 1 and X 22 N 2 such that the following LMIs hold: Φ 11 Φ 12 ha T Z Φ = Φ 22 ha T d Z < 0, (3.9) hz where X 11 X 12 N 1 Ψ = X 22 N 2 0, (3.10) Z Φ 11 = PA+ A T P + N 1 + N T 1 + Q + hx 11, Φ 12 = PA d N 1 + N T 2 + hx 12, Φ 22 = N 2 N T 2 (1 μ)q + hx 22. Proof. Choose the Lyapunov-Krasovskii functional candidate to be 0 V (x t )=x T (t)px(t)+ x T (s)qx(s)ds+ ẋ T (s)zẋ(s)dsdθ, (3.11) t d(t) h t+θ where P>0, Q 0, and Z>0areto be determined. This type of functional is called a quadratic Lyapunov-Krasovskii functional. For any matrix X = X 11 X 12 0, the following inequality is true: X 22 hη1 T (t)xη 1 (t) η1 T (t)xη 1 (t)ds 0, (3.12) t d(t) where η 1 (t) =[x T (t), x T (t d(t))] T. Calculating the derivative of V (x t ) along the solutions of (3.1) and adding the left sides of (3.8) and (3.12) to it yield V (x t )=x T (t)[pa+ A T P ]x(t)+2x T (t)pa d x(t d(t)) + x T (t)qx(t) [1 d(t)]x T (t d(t))qx(t d(t)) + hẋ T (t)zẋ(t) t h ẋ T (s)zẋ(s)ds

63 46 3. Stability of Systems with Time-Varying Delay where x T (t)[pa+ A T P ]x(t)+2x T (t)pa d x(t d(t)) + x T (t)qx(t) (1 μ)x T (t d(t))qx(t d(t)) + hẋ T (t)zẋ(t) t d(t) ẋ T (s)zẋ(s)ds +2 [ x T (t)n 1 + x T (t d(t))n 2 ] [ x(t) +hη1 T (t)xη 1(t) t d(t) η T 1 (t)xη 1(t)ds = η1 T (t)ξη 1 (t) η2 T (t, s)ψη 2 (t, s)ds, t d(t) η 2 (t, s) =[x T (t), x T (t d(t)), ẋ T (s)] T, Ξ = Φ 11 + ha T ZA Φ 12 + ha T ZA d, Φ 22 + ha T d ZA d t d(t) ẋ(s)ds x(t d(t)) with Φ 11, Φ 12,andΦ 22 being defined in (3.9) and Ψ being defined in (3.10). If Ξ<0andΨ 0, then V (x t ) < ε x(t) 2 holds for any sufficiently small ε>0, which ensures the asymptotic stability of system (3.1). From the Schur complement, Ξ<0isequivalent to (3.9). Thus, if LMIs (3.9) and (3.10) are true, then system (3.1) is asymptotically stable. This completes the proof. Remark For a system with a constant delay, setting X 12 =0,X 22 =0, and N 2 = 0 in Theorem yields Theorem 1 in [5]. That is, the above theorem is an extension of Moon et al. s. Instead of making X 12, X 22,and N 2 fixed matrices, Theorem selects them by solving LMIs. So, it always chooses suitable ones, thus overcoming the conservativeness of Theorem 1 in [5]. Remark If the matrices N 1, N 2,andX in (3.10) are all set to zero, and Z = εi (where ε is a sufficiently small positive scalar), then Theorem is identical to the well-known delay-independent stability criterion in [39] and [40], which is now stated. Corollary Consider nominal system (3.1) with a delay, d(t), that satisfies both (3.2) and (3.3). When μ =0, the system is asymptotically stable if there exist matrices P>0 and Q 0 such that the following LMI holds: ]

64 3.2 Stability of Nominal System 47 PA+ AT P + Q PA d < 0. Q Thus, any system that exhibits delay-independent stability, as determined by Corollary 3.2.1, is, for all practical purposes, asymptotically stable for any delay satisfying 0 d(t) h, whereh is a positive real number. So, Theorem contains the well-known delay-independent stability criterion. On the other hand, Theorem contains a delay-dependent and rateindependent condition. That is, although there is a limit on the upper bound on the delay, there is no limit on the upper bound on the derivative of the delay. In fact, setting Q = 0 in the theorem yields a delay-dependent and rate-independent stability criterion. Corollary Consider nominal system (3.1) with a delay, d(t), that satisfies (3.2) [but not necessary (3.3)]. Given a scalar h > 0, the system is asymptotically stable if there exist matrices P > 0, Z > 0, and X = X 11 X 12 0, and any appropriately dimensioned matrices N 1 and X 22 N 2 such that LMI (3.10) and the following LMI hold: Φ 11 Φ 12 ha T Z Φ = Φ22 ha d Z < 0, (3.13) hz where Φ 11 = PA+ A T P + N 1 + N T 1 + hx 11, Φ 22 = N 2 N T 2 + hx 22, and Φ 12 is defined in (3.9) Retaining the Term ẋ(t) In contrast to the previous subsection, where ẋ(t) in V (x t ) was replaced with Ax(t)+A d x(t d(t)), in this subsection the term ẋ(t) is retained, and FWMs are used to express the relationships among the terms of the system equation. The following equation holds for any matrices T j, j =1, 2 with appropriate dimensions:

65 48 3. Stability of Systems with Time-Varying Delay 2 [ x T (t)t 1 +ẋ T (t)t 2 ] [ẋ(t) Ax(t) Ad x(t d(t))] = 0. (3.14) Just as in Theorem 3.2.1, FWMs express the relationships among the terms of the Newton-Leibnitz formula, and we combine them with equation (3.14) to obtain the following theorem. Theorem Consider nominal system (3.1) with a delay, d(t), that satisfies both (3.2) and (3.3). Given scalars h > 0 and μ, the system is asymptotically stable if there exist matrices P > 0, Q 0, Z>0, and X 11 X 12 X 13 X = X 22 X 23 0, and any appropriately dimensioned matrices X 33 N i,i=1, 2, 3 and T j,j=1, 2 such that the following LMIs hold: where Γ 11 Γ 12 Γ 13 Γ = Γ 22 Γ 23 < 0, (3.15) Γ 33 X 11 X 12 X 13 N 1 X Θ = 22 X 23 N 2 0, (3.16) X 33 N 3 Z Γ 11 = Q + N 1 + N T 1 AT T T 1 T 1A + hx 11, Γ 12 = P + N T 2 + T 1 A T T T 2 + hx 12, Γ 13 = N T 3 N 1 T 1 A d + hx 13, Γ 22 = hz + T 2 + T T 2 + hx 22, Γ 23 = N 2 T 2 A d + hx 23, Γ 33 = (1 μ)q N 3 N T 3 + hx 33. Proof. From the Newton-Leibnitz formula, we know that the following equation holds for any appropriately dimensioned matrices N i,i=1, 2, 3: 2 [ x T (t)n 1 +ẋ T (t)n 2 +x T ] [ ] t (t d(t))n 3 x(t) x(t d(t)) ẋ(s)ds =0; t d(t) (3.17)

66 3.2 Stability of Nominal System 49 and (3.14) holds on the basis of (3.1). On the other hand, for any matrix X 0, the following holds: hζ T 1 (t)xζ 1(t) t d(t) ζ T 1 (t)xζ 1(t)ds 0, (3.18) where ζ 1 (t) =[x T (t), ẋ T (t), x T (t d(t))] T. Calculating the derivative of V (x t ) and using (3.14), (3.17), and (3.18) yield V (x t ) ζ T 1 (t)γζ 1(t) t d(t) ζ T 2 (t, s)θζ 2(t, s)ds, (3.19) where ζ 2 (t, s) = [ζ1 T (t), ẋ T (s)] T ;andγ and Θ are defined in (3.15) and (3.16), respectively. If Γ < 0andΘ 0, then, for any sufficiently small positive scalar ε, V (x t ) < ε x(t) 2, which ensures that system (3.1) is asymptotically stable. This completes the proof. Remark If the matrices N 3, X 13, X 23,andX 33 are all set to zero, then Theorem is equivalent to Lemma 1 in [11] for systems with a single delay. However, in our theorem, the optimal values of these matrices can be obtained by solving LMIs. That is, Lemma 1 in [11] is a special case of Theorem In the above theorem, the system matrices and Lyapunov matrices are separated, which sets the stage for a discussion of a parameter-dependent Lyapunov-Krasovskii functional in Section 3.4. Now, if we set Q = 0, we can use Theorem to obtain a delaydependent and rate-independent stability criterion. Corollary Consider nominal system (3.1) with a delay, d(t), that satisfies (3.2) [but not necessary (3.3)]. Given a scalar h>0, the system is asymptotically stable if there exist matrices P > 0, Z>0, and X = X 11 X 12 X 13 X 22 X 23 0, and any appropriately dimensioned matrices N i,i= X 33 1, 2, 3 and T j,j=1, 2 such that LMI (3.16) and the following LMI hold: ˇΓ 11 Γ 12 Γ 13 Γ 22 Γ 23 < 0, (3.20) ˇΓ33

67 50 3. Stability of Systems with Time-Varying Delay where ˇΓ 11 = N 1 + N T 1 AT T T 1 T 1A + hx 11, ˇΓ 33 = N 3 N T 3 + hx 33; and Γ 12,Γ 13,Γ 22, and Γ 23 are defined in (3.15) Equivalence Analysis In this subsection, we prove that Theorem is equivalent to Theorem Let I A T [ ] 0 J J 1 = 0 I 0, J 2 = A T 0 I d I Now, ˆΓ 11 ˆΓ12 ˆΓ13 ˆΓ = J 1 ΓJ1 T = ˆΓ22 ˆΓ23 < 0, (3.21) ˆΓ33 ˆΘ 11 ˆΘ12 ˆΘ13 N 1 + A T N 2 ˆΘ = J 2 ΘJ2 T X 22 ˆΘ23 N 2 = ˆΘ33 N 3 + A T d N 0, (3.22) 2 Z where Γ and Θ are defined in (3.15) and (3.16), respectively; and ˆΓ 11 = PA+ A T P + Q + N 1 + N T 1 + N T 2 A + A T N 2 + ha T ZA + h ˆΘ 11, ˆΘ 11 = X 11 + X 12 A + A T X T 12 + AT X 22 A, ˆΓ 12 = P + T 1 + ha T Z + A T T 2 + N T 2 + h ˆΘ 12, ˆΘ 12 = X 12 + A T X 22, ˆΓ 13 = PA d + N T 2 A d + N T 3 N 1 A T N 2 + ha T ZA d + h ˆΘ 13, ˆΘ 13 = X 13 + A T X 23 + X 12 A d + A T X 22 A d, ˆΓ 22 = hz + T 2 + T T 2 + hx 22, ˆΓ 23 = N 2 + T T 2 A d + hza d + h ˆΘ 23, ˆΘ 23 = X 23 + X 22 A d, ˆΓ 33 = (1 μ)q N 3 N T 3 A T d N 2 N T 2 A d + ha T d ZA d + h ˆΘ 33, ˆΘ 33 = X 33 + X T 23 A d + A T d X 23 + A T d X 22A d. On the one hand, if LMIs (3.21) and (3.22) are feasible, then setting N 1 = N 1 + A T N 2, N 2 = N 3 + A T d N 2, X 11 = ˆΘ 11, X 12 = ˆΘ 13,andX 22 = ˆΘ 33

68 3.3 Stability of Systems with Time-Varying Structured Uncertainties 51 (where the right sides of these equations are the feasible solutions of LMIs (3.21) and (3.22)) guarantees that LMIs (3.9) and (3.10) hold. On the other hand, if LMIs (3.9) and (3.10) are feasible, then setting T 1 = P, T 2 = hz, N 2 =0,X 12 =0,X 22 =0,X 23 =0,X 13 = X 12, X 33 = X 22,andN 3 = N 2 in LMIs (3.21) and (3.22) (where the right sides of these equations are the feasible solutions of LMIs (3.9) and (3.10)) makes all the elements in the second row and all those in the second column zero, except for ˆΓ 22 = hz. Removing that row and that column converts the equations into LMIs (3.9) and (3.10) (LMI (3.9) is equivalent to Ξ < 0), which guarantees that LMIs (3.21) and (3.22) hold. Therefore, LMIs (3.21) and (3.22) are equivalent to LMIs (3.9) and (3.10), which means that LMIs (3.15) and (3.16) are equivalent to LMIs (3.9) and (3.10). 3.3 Stability of Systems with Time-Varying Structured Uncertainties This section explains how to extend the stability criteria for nominal systems to systems with time-varying structured uncertainties using Lemma Robust Stability Analysis We can use Lemma to extend the stability criteria for nominal system (3.1) to system (3.4), which has time-varying structured uncertainties. First of all, extending Theorem to system (3.4) yields the following theorem. Theorem Consider system (3.4) withadelay, d(t), that satisfies both (3.2) and (3.3). Given scalars h>0 and μ, the system is robustly stable if there exist matrices P>0, Q 0, Z>0, and X = X 11 X 12 0, any X 22 appropriately dimensioned matrices N 1 and N 2, and a scalar λ>0 such that LMI (3.10) and the following LMI hold: Φ 11 + λea T E a Φ 12 + λea T E ad ha T Z PD Φ 22 + λead T E ad ha T d Z 0 < 0, (3.23) hz hzd λi

69 52 3. Stability of Systems with Time-Varying Delay where Φ 11,Φ 12, and Φ 22 are defined in (3.9). Proof. Replacing A and A d in (3.9) with A + DF(t)E a and A d + DF(t)E ad, respectively, makes (3.9) equivalent to the following condition: PD Φ+ 0 F (t)[e a E ad 0]+ hzd E T a E T ad 0 F T (t) [ D T P 0 hd T Z ] < 0. (3.24) From Lemma 2.6.2, we know that a necessary and sufficient condition guaranteeing (3.24) is that there exists a scalar λ>0 such that PD [ Φ+λ 1 0 D T P hzd ] 0 hd T Z +λ E T a E T ad 0 [ ] E a E ad 0 < 0. (3.25) Applying the Schur complement shows that (3.25) is equivalent to (3.23). This completes the proof. Similarly, extending Theorem yields another theorem. Theorem Consider system (3.4) withadelay, d(t), that satisfies both (3.2) and (3.3). Given scalars h>0 and μ, the system is robustly stable if X 11 X 12 X 13 there exist matrices P>0, Q 0, Z>0, and X = X 22 X 23 0, X 33 any appropriately dimensioned matrices N i,i=1, 2, 3 and T j,j=1, 2, and ascalarλ>0 such that LMI (3.16) and the following LMI hold: Γ 11 + λea TE a Γ 12 Γ 13 + λea TE ad T 1 D Γ 22 Γ 23 T 2 D Γ 33 + λead T E < 0, (3.26) ad 0 λi where Γ ij,i=1, 2, 3, i j 3 are defined in (3.15). We can easily derive delay-dependent and rate-independent criteria from Theorems and by setting Q =0.

70 3.3 Stability of Systems with Time-Varying Structured Uncertainties 53 Corollary Consider system (3.4) with a delay, d(t), that satisfies (3.2) [but not necessary (3.3)]. Given a scalar h>0, the system is robustly stable if there exist matrices P>0, Z>0, and X = X 11 X 12 0, any X 22 appropriately dimensioned matrices N 1 and N 2, and a scalar λ>0 such that LMI (3.10) and the following LMI hold: Φ 11 + λea TE a Φ 12 + λea TE ad ha T Z PD Φ22 + λe ad T E ad ha T d Z 0 < 0, (3.27) hz hzd λi where Φ 12 is defined in (3.9), and Φ 11 and Φ 22 are defined in (3.13). Corollary Consider system (3.4) with a delay, d(t), that satisfies (3.2) [but not necessary (3.3)]. Given a scalar h>0, the system is robustly X 11 X 12 X 13 stable if there exist matrices P>0, Z>0, and X = X 22 X 23 0, X 33 any appropriately dimensioned matrices N i,i=1, 2, 3 and T j,j=1, 2, and ascalarλ>0 such that LMI (3.16) and the following LMI hold: ˇΓ 11 + λea T E a Γ 12 Γ 13 + λea T E ad T 1 D Γ 22 Γ 23 T 2 D ˇΓ33 + λe ad T E < 0, (3.28) ad 0 λi where Γ 12,Γ 13,Γ 22, and Γ 23 are defined in (3.15), and ˇΓ 11 and ˇΓ 33 are defined in (3.20) Numerical Example Example Consider the robust stability of system (3.4) with the following parameters: A = , A d = , E a =diag{1.6, 0.05}, E ad =diag{0.1, 0.3}, D = I.

71 54 3. Stability of Systems with Time-Varying Delay Table 3.1. Allowable upper bound, h, for various μ (Example 3.3.1) μ unknown μ [7] [13] < 0.2 < 0.1 [14] [5] [11] Theorems and Corollaries and Table 3.1 shows the upper bounds on the delay for different μ obtained from Theorems and and Corollaries and For comparison, the table also lists the upper bounds obtained from the criteria in [5,7,11,13,14]. Note that the values for [11] were obtained by using Lemma 1 in [11] together with Lemma in Chapter 2. It is clear that the theorems and corollaries in this chapter produce much better results than those in [5,7,13,14], and the same or better results than those in [11]. This example shows that Theorems and produce the same results, as do Corollaries and 3.3.2, which demonstrates the equivalence of the two classes of criteria in Subsection Stability of Systems with Polytopic-Type Uncertainties This section employs a parameter-dependent Lyapunov-Krasovskii functional to examine the stability of systems with polytopic-type uncertainties Robust Stability Analysis Polytopic-type uncertainties (3.7) are an important class of uncertainties because they can be used to represent uncertainties described in terms of interval matrices. Recent research has shown that a parameter-dependent Lyapunov-Krasovskii functional can overcome the conservativeness of a quadratic Lyapunov-Krasovskii functional of the form (3.11). The basic procedure for using one to handle polytopic-type uncertainties has three steps: Step 1: Obtain a stability condition for the nominal system.

72 3.4 Stability of Systems with Polytopic-Type Uncertainties 55 Step 2: Derive either a sufficient condition or a necessary and sufficient condition for the original condition by separating the Lyapunov matrices from the system matrices. Step 3: Extending the condition obtained in Step 2 with a parameterdependent Lyapunov-Krasovskii functional yields a new condition. However, it is difficult to separate the Lyapunov and system matrices. Conditions obtained from the separation are usually only sufficient conditions for the original conditions, which leads to conservativeness. Even if some of them are necessary and sufficient conditions for the original ones, the criteria obtained cannot be expressed in terms of LMIs because of newly introduced parameters. Regarding the delay-dependent conditions of a parameterdependent Lyapunov-Krasovskii functional, it is not difficult to separate the matrices by the method of Fridman et al. [11, 26, 27, 41]; but the problem is that the weighting matrices they employ are fixed, which leads to conservativeness. Theorem separates the Lyapunov and system matrices in a natural way and is equivalent to Theorem So, considering the delay-dependent conditions of a parameter-dependent Lyapunov-Krasovskii functional based on Theorem leads to the following theorem. Theorem Consider system (3.1) with polytopic-type uncertainties (3.7) and a delay, d(t), that satisfies both (3.2) and (3.3). Given scalars h>0 and μ, the system is robustly stable if there exist matrices P j > 0, Q j 0, Z j > 0, and X (j) = X (j) 11 X (j) 12 X (j) 13 X (j) 22 X (j) 23 0, j =1, 2,,p, and appropri- X (j) 33 ately dimensioned matrices N ij,i=1, 2, 3, j =1, 2,,p and T k,k=1, 2 such that LMI (3.29) and the following LMIs hold for j =1, 2,,p: Γ (j) = Ψ (j) = Γ (j) 11 Γ (j) 12 (j) Γ 22 Γ (j) 13 Γ (j) 23 Γ (j) 33 X (j) 11 X (j) 12 X (j) X (j) 22 X (j) X (j) < 0, (3.29) 13 N 1j 23 N 2j 33 N 3j Z j 0, (3.30)

73 56 3. Stability of Systems with Time-Varying Delay where Γ (j) 11 = Q j + N 1j + N T 1j AT j T T 1 T 1A j + hx (j) 11, Γ (j) 12 = P j + N T 2j + T 1 A T j T T 2 + hx(j) 12, Γ (j) 13 = N 3j T N 1j T 1 A dj + hx (j) 13, Γ (j) 22 = hz j + T 2 + T2 T + hx (j) 22, Γ (j) 23 = N 2j T 2 A dj + hx (j) 23, Γ (j) 33 = (1 μ)q j N 3j N3j T + hx(j) 33. Proof. Choose the following parameter-dependent Lyapunov-Krasovskii functional candidate: p p V u (x t )= x T (t)ξ j P j x(t)+ x T (s)ξ j Q j x(s)ds j=1 + p j=1 0 h t+θ j=1 t d(t) ẋ T (s)ξ j Z j ẋ(s)dsdθ, (3.31) where P j > 0, Q j 0, and Z j > 0, j =1, 2,,p are to be determined. Following a line similar to the one in Theorem yields V u (x t ) p p ζ1 T (t)ξ j Γ (j) ζ 1 (t) j=1 j=1 t d(t) ζ T 2 (t, s)ξ j Ψ (j) ζ 2 (t, s)ds, (3.32) where ζ 1 (t) andζ 2 (t, s) are defined in (3.18) and (3.19), respectively; and Γ (j) and Ψ (j),j=1, 2,,p are defined in (3.29) and (3.30), respectively. If Γ (j) < 0and Ψ (j) 0, j =1, 2,,p,then V u (x t ) < ε x(t) 2 for a sufficiently small ε>0, which ensures the robust stability of system (3.1) with polytopic-type uncertainties. This completes the proof. Now, setting Q j =0,j=1, 2,,p, yields the following delay-dependent and rate-independent corollary. Corollary Consider system (3.1) with polytopic-type uncertainties (3.7) and a delay, d(t), that satisfies (3.2) [but not necessary (3.3)]. Given ascalarh>0, the system is robustly stable if there exist matrices P j > 0, Z j > 0, and X (j) = X (j) 11 X (j) 12 X (j) 13 X (j) 22 X (j) 23 0, j =1, 2,,p, and appropri- X (j) 33 ately dimensioned matrices N ij,i=1, 2, 3, j =1, 2,,p and T k,k=1, 2 such that LMI (3.30) and the following LMI hold for j =1, 2,,p:

74 where and Γ (j) Stability of Systems with Polytopic-Type Uncertainties 57 Γ (j) 12 (j) Γ 22 Γ (j) 13 Γ (j) 23 Γ (j) 33 < 0, (3.33) Γ (j) 11 = N 1j + N1j T AT j T 1 T T 1 A j + hx (j) 11 ; Γ (j) 33 = N 3j N3j T + hx(j) 33 ; Γ (j) 12, Γ (j) 13, (j) Γ 22, and Γ (j) 23 are defined in (3.29). On the other hand, Ψ (j) must be positive semi-definite, not positive definite, to prove Theorem Setting all the matrix elements of Ψ (j) (namely, Z j, X (j),andn ij,i=1, 2, 3, j=1, 2,,p) to zero produces the following delay-independent and rate-dependent condition. That is, although a limit is imposed on the upper bound on the derivative of the delay, there is no limit on the upper bound on the delay. Corollary Consider system (3.1) with polytopic-type uncertainties (3.7) and a delay, d(t), that satisfies (3.3) [but not necessary (3.2)]. Given a scalar μ, the system is robustly stable if there exist matrices P j > 0 and Q j 0, j =1, 2,,p, and any appropriately dimensioned matrices T k,k=1, 2 such that the following LMI holds for j =1, 2,,p: where Γ (j) 11 Γ (j) 12 (j) Γ 22 Γ (j) 13 Γ (j) 23 Γ (j) 33 Γ (j) 11 = Q j A T j T T 1 T 1 A j, Γ (j) 12 = P j + T 1 A T j T T 2, Γ (j) 13 = T 1A dj, Γ (j) 22 = T 2 + T T 2, Γ (j) 23 = T 2A dj, Γ (j) 33 = (1 μ)q j Numerical Example < 0, (3.34) The numerical example in this subsection demonstrates the effectiveness of the above method and shows how much of an improvement it is over other methods.

75 58 3. Stability of Systems with Time-Varying Delay Example Consider system (3.1) with polytopic-type uncertainties (3.7) and with A 1 = , A 2 =, A 3 = 1.9 0, A d1 = 0.1 0, A d2 = 01, A d3 = Assume the delay is time-invariant (μ = 0). The upper bound on the delay is in [9], in [10], and in [11, 26, 27]. However, Theorem in this section shows that the system is robustly stable for h = Theorem yields a larger maximum upper bound on the allowable size of the delay than [9 11, 26, 27] do. Moreover, Table 3.2 compares the upper bounds obtained by Fridman & Shaked [11,26,27] and those we obtained with Theorem and Corollary for various μ. Clearly, ours are bigger than theirs. Table 3.2. Allowable upper bound, h, for various μ (Example 3.4.1) μ unknown μ [9] [10] [11, 26, 27] Theorem Corollary IFWM Approach Although the FWM approach produces better results than other methods, there is room for further investigation. An important point is that previous sections in this chapter and some reports by other authors, such as [11,26,27], ignore useful terms in the derivative of a Lyapunov-Krasovskii functional. This is the issue discussed below. Section 3.2 uses the following Lyapunov-Krasovskii functional:

76 3.5 IFWM Approach 59 0 V 1 (x t )=x T (t)px(t)+ x T (s)qx(s)ds + ẋ T (s)z 1 ẋ(s)dsdθ. t d(t) h t+θ (3.35) However, in the derivative of V 1 (x t ), the term t h ẋt (s)z 1 ẋ(s)ds is increased to t d(t) ẋt (s)z 1 ẋ(s)ds. Notethat ẋ T (s)z 1 ẋ(s)ds = ẋ T (s)z 1 ẋ(s)ds t h t d(t) d(t) t h ẋ T (s)z 1 ẋ(s)ds. (3.36) The term d(t) t h ẋ T (s)z 1 ẋ(s)ds was ignored in previous studies, which may lead to considerable conservativeness. We can reduce the conservativeness by using the IFWM approach presented below to examine the stability of systems with a time-varying delay. It retains useful terms in the derivative of the Lyapunov-Krasovskii functional and takes into account the relationships among the delay, its upper bound, and their difference Retaining Useful Terms In this subsection, d(t) t h ẋ T (s)z 1 ẋ(s)ds is retained when estimating the upper bound on the derivative of a Lyapunov-Krasovskii functional. A new class of Lyapunov-Krasovskii functional candidates is used to handle this term: V 2 (x t )=x T (t)px(t)+ x T (s)qx(s)ds + x T (s)rx(s)ds t d(t) t h 0 + ẋ T (s)(z 1 + Z 2 )ẋ(s)dsdθ, (3.37) h t+θ where P>0, Q 0, R 0, and Z i > 0, i=1, 2 are to be determined. Now, we give the following theorem. Theorem Consider system (3.1) withadelay, d(t), that satisfies both (3.2) and (3.3). Given scalars h > 0 and μ, the system is asymptotically stable if there exist matrices P > 0, Q 0, R 0, and Z i > 0, i=1, 2, [ ] T and any appropriately dimensioned matrices N = N1 T N 2 T N 3 T, S = [ ] T [ ] T S1 T ST 2 ST 3, and M = M1 T M 2 T M 3 T such that the following LMI

77 60 3. Stability of Systems with Time-Varying Delay holds: Φ hn hs hm ha T c1(z 1 + Z 2 ) hz hz < 0, (3.38) hz 2 0 h(z 1 + Z 2 ) where Φ = Φ 1 + Φ 2 + Φ T 2, PA+ A T P + Q + R PA d 0 Φ 1 = (1 μ)q 0, R Φ 2 =[N + M N + S M S], A c1 =[AA d 0]. Proof. From the Newton-Leibnitz formula, we know that the following equations are true for any matrices N, S,andM with appropriate dimensions: [ ] 2ζ T 1 (t)n x(t) x(t d(t)) 2ζ T 1 (t)s [x(t d(t)) x(t h) t d(t) ẋ(s)ds d(t) t h ] 2ζ1 [x(t) T (t)m x(t h) ẋ(s)ds t h ẋ(s)ds =0, (3.39) ] =0, (3.40) =0, (3.41) where ζ 1 (t) = [ x T (t), x T (t d(t)), x T (t h) ] T. Calculating the derivative of V 2 (x t ) along the solutions of system (3.1), adding the left sides of (3.39)-(3.41) to it, and using (3.36) yield V 2 (x t )=2x T (t)p ẋ(t)+x T (t)qx(t) (1 d(t))x T (t d(t))qx(t d(t)) +x T (t)rx(t) x T (t h)rx(t h) +hẋ T (t)(z 1 + Z 2 )ẋ(t) t h ẋ T (s)(z 1 + Z 2 )ẋ(s)ds

78 3.5 IFWM Approach 61 2x T (t)p ẋ(t)+x T (t)(q + R)x(t) (1 μ)x T (t d(t))qx(t d(t)) x T (t h)rx(t h) +hẋ T (t)(z 1 + Z 2 )ẋ(t) t d(t) t h ẋ T (s)z 1 ẋ(s)ds ẋ T (s)z 2 ẋ(s)ds d(t) t h +2ζ T 1 (t)n [x(t) x(t d(t)) +2ζ T 1 (t)s [ x(t d(t)) x(t h) ẋ T (s)z 1 ẋ(s)ds ẋ(s)ds t d(t) d(t) t h [ ] +2ζ1 T (t)m x(t) x(t h) ẋ(s)ds t h ζ1 T (t) [ Φ + ha T c1(z 1 + Z 2 )A c1 + hnz1 1 N T t d(t) d(t) t h t h ] ẋ(s)ds +hsz1 1 ST + hmz2 1 M T] ζ 1 (t) [ ζ T 1 (t)n +ẋ T ] [ (s)z 1 Z 1 1 N T ζ 1 (t)+z 1 ẋ(s) ] ds [ ζ T 1 (t)s +ẋ T ] [ (s)z 1 Z 1 1 S T ζ 1 (t)+z 1 ẋ(s) ] ds [ ζ T 1 (t)m +ẋ T ] [ (s)z 2 Z 1 2 M T ζ 1 (t)+z 2 ẋ(s) ] ds. (3.42) Since Z i > 0, i=1, 2, the last three parts of (3.42) are all less than 0. So, if Φ + ha T c1 (Z 1 + Z 2 )A c1 + hnz1 1 N T + hsz1 1 ST + hmz2 1 M T < 0, which is equivalent to (3.38) by the Schur complement, V2 (x t ) < ε x(t) 2 for a sufficiently small ε>0, which means that system (3.1) is asymptotically stable. This completes the proof. Remark If N 3 = 0, S = M = 0, Z 2 = ε 1 I,andR = ε 2 I (where ε i > 0, i = 1, 2 are sufficiently small scalars), Theorem reduces to Theorem So, if we choose suitable values for N 3, S, M, Z 2 and R, Theorem overcomes the conservativeness of Theorem and is an improvement over the criterion in [5]. The alternative version of Theorem below enables us to use a parameter-dependent Lyapunov-Krasovskii functional for the investigation of systems with polytopic-type uncertainties. ]

79 62 3. Stability of Systems with Time-Varying Delay Theorem Consider system (3.1) withadelay, d(t), that satisfies both (3.2) and (3.3). Given scalars h > 0 and μ, the system is asymptotically stable if there exist matrices P > 0, Q 0, R 0, and Z i > 0, i=1, 2, [ ] T and any appropriately dimensioned matrices Ñ = N1 T N 2 T N 4 T, S = [ ] T [ ] T [ ] T S1 T ST 2 ST 4, M = M1 T M 2 T M 4 T, and T = T1 T T 2 T T 4 T such that the following LMI holds: where Θ hñ h S h M hz < 0, (3.43) hz 1 0 hz 2 Θ = Θ 1 + Θ 2 + Θ2 T, Q + R 0 0 P (1 μ)q 0 0 Θ 1 =, R 0 h(z 1 + Z 2 ) Θ 2 = [Ñ + M Ñ + S M S 0] + TA c2 + A T c2 T T, A c2 =[ A A d 0 I]. Proof. Choose the same Lyapunov-Krasovskii functional candidate as in (3.37) and note that 2ζ T 2 (t)t [ẋ(t) Ax(t) A dx(t d(t))] = 0, (3.44) where ζ 2 (t) = [ x T (t), x T (t d(t)), x T (t h), ẋ T (t) ] T. Then, the proof uses equations similar to (3.39)-(3.41) and follows a line similar to the one in Theorem Remark Subsection shows that Theorem is equivalent to Theorem Note that Theorem can be extended to deal with a system with polytopic-type uncertainties by using a parameter-dependent Lyapunov-Krasovskii functional, as in Section 3.4. This is because the LMI condition in Theorem does not involve any product of system matrices and Lyapunov matrices.

80 3.5 IFWM Approach 63 Remark If N 4 = 0, S = M = 0, Z2 = ε 1 I,andR = ε 2 I (where ε i > 0, i = 1, 2 are sufficiently small scalars), Theorem reduces to Theorem and is an improvement over the theorems in [11]. Now, we consider two classes of uncertainties mentioned in Section 3.1. For system (3.4), which has time-varying structured uncertainties, we have a corollary similar to Theorem in Subsection Corollary Consider system (3.4) with a delay, d(t), that satisfies both (3.2) and (3.3). Given scalars h>0 and μ, the system is robustly stable if there exist matrices P>0, Q 0, R 0, and Z i > 0, i=1, 2, any appropriately dimensioned matrices N = N1 T N 2 T N 3 T,S= S1 T ST 2 ST 3, [ ] T [ ] T [ ] T and M = M1 T M 2 T M 3 T, andascalarλ>0 such that the following LMI holds: ˆΦ hn hs hm ha T c1 (Z 1 + Z 2 ) ˆPD hz hz < 0, (3.45) hz h(z 1 + Z 2 ) h(z 1 + Z 2 )D λi where λe a TE a λea TE d 0 P ˆΦ = Φ + λe d TE d 0, ˆP = 0 ; 0 0 and Φ and A c1 are defined in (3.38). The next corollary is derived from Theorem by using a parameterdependent Lyapunov-Krasovskii functional. Corollary Consider system (3.1) with polytopic-type uncertainties (3.7) and a delay, d(t), that satisfies both (3.2) and (3.3). Given scalars h>0 and μ, the system is robustly stable if there exist matrices P j > 0, Q j 0, R j 0, and Z ij > 0, i=1, 2, j=1, 2,,p, and any appropriately dimensioned matrices N [ ] T [ ] T j = N1j T N 2j T N 4j T, Sj = S1j T ST 2j ST 4j,

81 64 3. Stability of Systems with Time-Varying Delay [ ] T [ ] T M j = M1j T M 2j T M 4j T, and T = T1 T T 2 T T4 T,i=1, 2, j = 1, 2,,p such that the following LMI holds for j =1, 2,,p: where Θ (j) h N j h S j h M j hz 1j 0 0 < 0, (3.46) hz 1j 0 hz 2j Θ (j) (j) (j) = Θ 1 + Θ 2 +[ Θ (j) 2 ]T, Q j + R j 0 0 P j Θ (j) (1 μ)q 1 = j 0 0, R j 0 h(z 1j + Z 2j ) Θ (j) 2 = [ Nj + M j N j + S j M j S j 0 ] + T Āj + ĀT j T T, Ā j =[ A j A dj 0 I] Further Investigation Even though in the previous subsection we retained the term d(t) ẋ T (s)z t h 1 ẋ(s)ds in the derivative of the Lyapunov-Krasovskii functional and obtained improved delay-dependent stability criteria for systems with a time-varying delay, there is still room for further investigation. In the last part of formula (3.42), the terms d(t)nz1 1 N T and (h d(t))sz1 1 ST are increased to hnz1 1 N T and hsz1 1 ST, respectively. Since d(t) andh d(t) are closely related because their sum is h, this treatment may lead to conservativeness. We now present a new theorem for nominal system (3.1) and a new corollary for system (3.4) that do not ignore any useful terms and take the relationships among d(t), h d(t), and h into account. Theorem Consider nominal system (3.1) with a delay, d(t), that satisfies both (3.2) and (3.3). Given scalars h>0 and μ, the system is asymptotically stable if there exist matrices P > 0, Q 0, R 0, Z>0, and X 0, and any appropriately dimensioned matrices N and S such that the following LMIs hold:

82 3.5 IFWM Approach 65 where Φ hat c1 Z < 0, hz X N 0, Z X S 0, Z (3.47) (3.48) (3.49) Φ = Φ 1 + Φ 2 + Φ T 2 + hx, PA+ A T P + Q + R PA d 0 Φ 1 = (1 μ)q 0, R [ ] Φ 2 = N N + S S, [ ] A c1 = A A d 0. Proof. Choose the Lyapunov-Krasovskii functional candidate to be V 3 (x t )=x T (t)px(t)+ + 0 h t+θ t d(t) ẋ T (s)zẋ(s)dsdθ, x T (s)qx(s)ds + t h where P>0, Q 0, R 0, and Z>0areto be determined. For any matrix X 0, the following holds: hζ T 1 (t)xζ 1(t) t d(t) ζ T 1 (t)xζ 1(t)ds d(t) t h x T (s)rx(s)ds (3.50) ζ T 1 (t)xζ 1(t)ds =0. (3.51) Calculating the derivative of V 3 (x t ) along the solutions of nominal system (3.1) and using (3.39), (3.40), and (3.51) yield V 3 (x t )=ζ1 T (t) ( Φ + ha T ) t c1za c1 ζ1 (t) ζ2 T (t, s) X N ζ 2 (t, s)ds t d(t) Z d(t) ζ2 T (t, s) X S ζ 2 (t, s)ds, (3.52) t h Z

83 66 3. Stability of Systems with Time-Varying Delay where ζ 2 (t, s) = [ ζ1 T(t), ẋt (s) ] T. The rest of the proof follows a line similar to the one in Theorem Applying Lemma gives us the following corollary for system (3.4). Corollary Consider system (3.4) with a delay, d(t), that satisfies both (3.2) and (3.3). Given scalars h > 0 and μ, the system is robustly stable if there exist matrices P > 0, Q 0, R 0, Z>0, and X 0, any appropriately dimensioned matrices N and S, and a scalar λ>0 such that LMIs (3.48) and (3.49), and the following LMI hold: Φ ha T c1 Z ˆPD λê hz hzd 0 < 0, (3.53) λi 0 λi where P E a T ˆP = 0, Ê = E b T ; 0 0 and Φ and A c1 are defined in (3.47). Remark Retaining the term ẋ(t) enables us to use parameter-dependent Lyapunov-Krasovskii functionals in combination with the IFWM approach to derive improved criteria for systems with polytopic-type uncertainties. Moreover, combining the IFWM approach with the augmented Lyapunov- Krasovskii functional method in Subsection produces even better results. Remark The conditions in this section are all delay- and rate-dependent. However, if we set the Q (or Q j,j=1, 2,,p) in them to zero, they reduce to delay-dependent and rate-independent ones, which can be used when the delay is not differentiable or the derivative of the delay is unknown Numerical Examples Example Consider the robust stability of system (3.4) with the following parameters:

84 3.5 IFWM Approach A =,A d =, E a = E ad =diag{0.2, 0.2}, D = I Table 3.3 lists the upper bounds on the delay obtained from Theorems and 3.3.2, and Corollaries 3.3.1, 3.3.2, 3.5.1, and for various μ. In addition, the results obtained from Lemma 1 in [11] in combination with Lemma are also listed. Clearly, our results are significantly better than those in [11]. Note that the method in [11] fails for μ =0.9, while the upper bound is for Theorems and 3.3.2, for Corollary 3.5.1, and for Corollary Furthermore, the table also shows a gradual improvement as our method progresses. Table 3.3. Allowable upper bound, h, for various μ (Example 3.5.1) μ unknown μ [11] Theorems and Corollaries and Corollary Corollary Example Consider system (3.1) with the following parameters: A = ρ, 0.1 A d = 0.35, ρ and ρ If we let ρ m =0.035 and set A 1 = ρ m, A 2 = ρ m, ρ m ρ m A d1 = A d2 = A d =, then the system is recast as a system with polytopic-type uncertainties that are described by (3.7). Table 3.4 lists the upper bounds on the delay obtained

85 68 3. Stability of Systems with Time-Varying Delay Table 3.4. Allowable upper bound, h, for various μ (Example 3.5.2) μ unknown μ [11, 26, 27] Theorem Corollary Corollary from [11, 26, 27], Theorems 3.4.1, and Corollaries and for various μ. Note that our methods produce larger upper bounds on the delay than [11, 26, 27] do. 3.6 Conclusion This chapter explains how the FWM and IFWM approaches can be used to examine the delay-dependent stability of systems with a delay. First, the FWM approach and two different treatments of the term ẋ(t) (retaining it or replacing it) are used to produce two different forms of delay- and ratedependent stability conditions. They are proven to be equivalent and are easily extended to delay-dependent and rate-independent conditions without any limit on the derivative of the delay. Second, the robust stability of two classes of uncertainties is examined. Lemma is used to extend the conditions for nominal systems obtained by the FWM approach to systems with time-varying structured uncertainties; and retaining the term ẋ(t) and using a parameter-dependent Lyapunov-Krasovskii functional extends the conditions for the nominal system to systems with polytopic-type uncertainties. Finally, the IFWM approach is used to study the delay-dependent stability of systems with a time-varying delay. The resulting criteria are less conservative than those produced by other methods. References 1. K. Gu. Discretized LMI set in the stability problem for linear uncertain timedelay systems. International Journal of Control, 68(4): , K. Gu. A generalized discretization scheme of Lyapunov functional in the stability problem of linear uncertain time-delay systems. International Journal of Robust and Nonlinear Control, 9(1): 1-4, 1999.

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87 70 3. Stability of Systems with Time-Varying Delay 19. I. R. Petersen and C. V. Hollot. A Riccati equation approach to the stabilization of uncertain linear systems. Automatica, 22(4): , J. C. Geromel, M. C. de Oliveira, and L. Hsu. LMI characterization of structural and robust stability. Linear Algebra and its Applications, 285(1-3): 68-80, D. Peaucelle, D. Arzelier, O. Bachelier, and J. Bernussou. A new robust Dstability condition for real convex polytopic uncertainty. Systems & Control Letters, 40(1): 21-30, H. D. Tuan, P. Apkarian, and T. Q. Nguyen. Robust and reduced-order filtering: new characterizations and methods. Proceedings of the American Control Conference, Chicago, USA, , U. Shaked. Improved LMI representations for analysis and design of continuoustime systems with polytopic-type uncertainty. IEEE Transactions on Automatic Control, 46(4): , P.J.deOliveira,R.C.L.F.Oliveira,V.J.S.Oliveira,V.F.Montagner,andP. L. D. Peres. LMI based robust stability conditions for linear uncertain systems: a numerical comparison. Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, USA, , Y. Jia. Alternative proofs for improved LMI representation for the analysis and the design of continuous-time systems with polytopic-type uncertainty: a predictive approach. IEEE Transactions on Automatic Control, 48(8): , E. Fridman and U. Shaked. Delay-dependent stability and H control: constant and time-varying delays. International Journal of Control, 76(1): 48-60, E. Fridman and U. Shaked. Parameter dependent stability and stabilization of uncertain time-delay systems. IEEE Transactions on Automatic Control, 48(5): , O. Bachelier, J. Bernussou, M. C. de Oliveira, and J. C. Geromel. Parameterdependent Lyapunov control design: numerical evaluation. Proceedings of the 38th IEEE Conference on Decision and Control, New York, USA, , M. C. de Oliveira, J. Bernussou, and J. C. Geromel. A new discrete-time robust stability condition. Systems & Control Letters, 37(4): , M. C. de Oliveira, J. C. Geromel, and L. Hsu. LMI characterization of structural and robust stability: the discrete-time case. Linear Algebra and its Applications, 296(1-3): 27-38, M. Wu, Y. He, J. H. She, and G. P. Liu. Delay-dependent criteria for robust stability of time-varying delay systems. Automatica, 40(8): , Y. He, M. Wu, J. H. She, and G. P. Liu. Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic type uncertainties. IEEE Transactions on Automatic Control, 49(5): , Y. He, Q. G. Wang, L. Xie, and C. Lin. Further improvement of free-weighting matrices technique for systems with time-varying delay. IEEE Transactions on Automatic Control, 52(2): , Y. He, Q. G. Wang, C. Lin, and M. Wu. Delay-range-dependent stability for systems with time-varying delay. Automatica, 43(2): , 2007.

88 References Y. He, G. P. Liu, and D. Rees. New delay-dependent stability criteria for neural networks with time-varying delay. IEEE Transactions on Neural Networks, 18(1): , Y. He, M. Wu, and J. H. She. Delay-dependent exponential stability of delayed neural networks with time-varying delay. IEEE Transactions on Circuits and Systems II, 53(7): , Q. L. Han. Robust stability for a class of linear systems with time-varying delay and nonlinear perturbations. Computers & Mathematics with Applications, 47(8-9): , Q. L. Han and D. Yue. Absolute stability of Lur e systems with time-varying delay. IET Proceedings: Control Theory & Applications, 1(3): , K. Gu, V. L. Kharitonov, and J. Chen. Stability of Time-Delay Systems. Boston: Birkhäuser, J. K. Hale and S. M. Verduyn Lunel. Introduction to Functional Differential Equations. New York: Springer-Verlag, E. Fridman and U. Shaked. A descriptor system approach to H control of linear time-delay systems. IEEE Transactions on Automatic Control, 47(2): , 2002.

89

90 4. Stability of Systems with Multiple Delays If a linear system with a single delay, h, is not stable for a delay of some length, but is stable for h = 0, then there must exist a positive number h for which the system is stable for 0 h h. Many researchers have simply extended this idea to a system with multiple delays, but this simple extension may lead to conservativeness. For example, Fridman & Shaked [1, 2] investigated a linear system with two delays: ẋ(t) =A 0 x(t)+a 1 x(t h 1 )+A 2 x(t h 2 ). (4.1) The upper bounds h 1 and h 2 on h 1 and h 2, respectively, are selected so that this system is stable for 0 h 1 h 1 and 0 h 2 h 2. However, the ranges of h 1 and h 2 that guarantee the stability of this system are conservative because they start from zero, even though that may not be necessary. One reason for this is that the relationship between h 1 and h 2 was not taken into account in the procedure for finding the upper bounds. Another point concerns a linear system with a single delay, ẋ(t) =A 0 x(t)+(a 1 + A 2 )x(t h 1 ), (4.2) which is a special case of system (4.1), namely, the case h 1 = h 2. The stability criterion for system (4.2) should be equivalent to that for system (4.1) for h 1 = h 2 ; but this equivalence cannot be demonstrated by the methods in [1,2]. This chapter presents delay-dependent stability criteria for systems with multiple constant delays based on the FWM approach [3, 4]. Criteria are first established for a linear system with two delays. They take into account not only the relationships between x(t h 1 )andx(t) t h 1 ẋ(s)ds, and x(t h 2 )andx(t) t h 2 ẋ(s)ds, but also the one between x(t h 2 )and x(t h 1 ) h 1 t h 2 ẋ(s)ds. Note that the last relationship is between h 1 and h 2. All these relationships are expressed in terms of FWMs, and their parameters are determined based on the solutions of LMIs. In addition, the equivalence

91 74 4. Stability of Systems with Multiple Delays between system (4.2) and system (4.1) for h 1 = h 2 is demonstrated. Numerical examples show that the methods presented in this chapter are effective and are a significant improvement over others. Finally, these ideas are extended from systems with two delays to systems with multiple delays. 4.1 Problem Formulation Consider the following linear system with multiple delays: m ẋ(t) = A i x(t h i ), t > 0, i=0 x(t) =φ(t), t [ h, 0], (4.3) where x(t) R n is the state vector; h 0 =0,h i 0, i =1, 2,,m are constant delays; h =max{h 1,h 2,,h m }; A i R n n,i=0, 1,,m are constant matrices; and the initial condition, φ(t), is a continuously differentiable initial function of t [ h, 0]. If the system contains time-varying structured uncertainties, it can be described by m ẋ(t) = (A i +ΔA i (t))x(t h i ), t > 0, i=0 (4.4) x(t) =φ(t), t [ h, 0]. The uncertainties are assumed to be of the form ΔA i (t) =DF(t)E i, i =0, 1,,m, (4.5) where D and E i,i = 0, 1,,m are constant matrices with appropriate dimensions; and F (t) is an unknown, real, and possibly time-varying matrix with Lebesgue-measurable elements satisfying F T (t)f (t) I, t. (4.6) This chapter also concerns system (4.3) with polytopic-type uncertainties. In this case, the matrices A i,i=0, 1,,mof the system contain uncertainties and satisfy the real convex polytopic model [A 0 A 1 A m ] Ω, p Ω = [A 0(ξ) A 1 (ξ) A m (ξ)]= ξ j [A 0j A 1j A mj ], j=1 p ξ j =1,ξ j 0, j=1

92 4.2 Two Delays 75 (4.7) where A ij,i=0, 1,,m, j =1, 2,,p are constant matrices with appropriate dimensions, and ξ j,j=1, 2,,p are time-invariant uncertainties. 4.2 Two Delays This section considers the stability of systems with two delays, the relationship between which is taken into account for the first time Nominal Systems First, consider the case m = 2. A delay-dependent stability criterion is established for system (4.3) with two delays by taking the relationship between h 1 and h 2 into account and replacing the term ẋ(t) in the derivative of the Lyapunov-Krasovskii functional with the system equation. Theorem Consider nominal system (4.3) with m =2. Given scalars h i 0, i=1, 2, the system is asymptotically stable if there exist matrices P>0, Q i 0, i=1, 2, W j > 0, X jj 0, Y jj 0, and Z jj 0, j=1, 2, 3, and any appropriately dimensioned matrices N i,s i,m i,i=1, 2, 3, X ij,y ij, and Z ij,i=1, 2, 3, i<j 3 such that the following LMIs hold: Φ 11 Φ 12 Φ 13 Φ = Φ 22 Φ 23 < 0, (4.8) Φ 33 X 11 X 12 X 13 N 1 X 22 X 23 N 2 Ψ 1 = 0, (4.9) X 33 N 3 W 1 Y 11 Y 12 Y 13 S 1 Y 22 Y 23 S 2 Ψ 2 = 0, (4.10) Y 33 S 3 W 2

93 76 4. Stability of Systems with Multiple Delays Z 11 Z 12 Z 13 km 1 Z 22 Z 23 km 2 Ψ 3 = 0, (4.11) Z 33 km 3 W 3, where 1, if h 1 h 2, k = 1, ifh 1 <h 2 ; and Φ 11 = PA 0 +A T 0 P +Q 1 +Q 2 +N 1 +N T 1 +S 1 +S T 1 +A T 0 HA 0 +h 1 X 11 +h 2 Y 11 + h 1 h 2 Z 11, Φ 12 = PA 1 N 1 +N T 2 +ST 2 M 1+A T 0 HA 1+h 1 X 12 +h 2 Y 12 + h 1 h 2 Z 12, Φ 13 = PA 2 +N T 3 +S T 3 S 1 +M 1 +A T 0 HA 2 +h 1 X 13 +h 2 Y 13 + h 1 h 2 Z 13, Φ 22 = Q 1 N 2 N T 2 M 2 M T 2 +A T 1 HA 1 +h 1 X 22 +h 2 Y 22 + h 1 h 2 Z 22, Φ 23 = N T 3 S 2 + M 2 M T 3 + AT 1 HA 2 + h 1 X 23 + h 2 Y 23 + h 1 h 2 Z 23, Φ 33 = Q 2 S 3 S T 3 +M 3+M T 3 +AT 2 HA 2+h 1 X 33 +h 2 Y 33 + h 1 h 2 Z 33, H = h 1 W 1 + h 2 W 2 + h 1 h 2 W 3. Proof. First, consider the case h 1 h 2. Choose the following Lyapunov- Krasovskii functional candidate: V 2 (x t )=x T (t)px(t) h 1 t+θ h2 h 1 t+θ t h 1 x T (s)q 1 x(s)ds + ẋ T (s)w 1 ẋ(s)dsdθ + 0 h 2 t h 2 x T (s)q 2 x(s)ds t+θ ẋ T (s)w 2 ẋ(s)dsdθ ẋ T (s)w 3 ẋ(s)dsdθ, (4.12) where P>0, Q i 0, i=1, 2, and W j > 0, j=1, 2, 3 are to be determined. Calculating the derivative of V 2 (x t ) along the solutions of system (4.3) yields V 2 (x t )=2x T (t)p [A 0 x(t)+a 1 x(t h 1 )+A 2 x(t h 2 )] +x T (t)q 1 x(t) x T (t h 1 )Q 1 x(t h 1 ) +x T (t)q 2 x(t) x T (t h 2 )Q 2 x(t h 2 ) +h 1 ẋ T (t)w 1 ẋ(t) t h 1 ẋ T (s)w 1 ẋ(s)ds

94 4.2 Two Delays 77 +h 2 ẋ T (t)w 2 ẋ(t) ẋ T (s)w 2 ẋ(s)ds t h 2 +(h 1 h 2 )ẋ T (t)w 3 ẋ(t) h2 t h 1 ẋ T (s)w 3 ẋ(s)ds. (4.13) From the Newton-Leibnitz formula, the following equations are true for any appropriately dimensioned matrices N i, S i,andm i,i=1, 2, 3: 2 [ x T (t)n 1 + x T (t h 1 )N 2 + x T ] (t h 2 )N 3 [ x(t) x(t h 1 ) t h 1 ] ẋ(s)ds =0, (4.14) 2 [ x T (t)s 1 + x T (t h 1 )S 2 + x T ] (t h 2 )S 3 [ ] x(t) x(t h 2 ) ẋ(s)ds =0, (4.15) t h 2 2 [ x T (t)m 1 + x T (t h 1 )M 2 + x T ] (t h 2 )M 3 [ x(t h 2 ) x(t h 1 ) h2 t h 1 ẋ(s)ds ] =0. (4.16) On the other hand, for any matrices X jj 0, Y jj 0, and Z jj 0, j = 1, 2, 3, and any appropriately dimensioned matrices X ij, Y ij,andz ij,i= 1, 2, 3, i<j 3, the following equation holds: where x(t) x(t h 1 ) x(t h 2 ) Λ 11 Λ 12 Λ 13 x(t) Λ 22 Λ 23 x(t h 1 ) =0, (4.17) Λ 33 x(t h 2 ) T Λ ij = h 1 (X ij X ij )+h 2 (Y ij Y ij )+(h 1 h 2 )(Z ij Z ij ),i=1, 2, 3, i j 3. Adding the left sides of (4.14)-(4.17) to V 2 (x t ) yields V 2 (x t )=η1 T (t)φη 1 (t) η2 T (t, s)ψ 1 η 2 (t, s)ds t h 1 h2 η2 T (t, s)ψ 2 η 2 (t, s)ds η2 T (t, s)ψ 3 η 2 (t, s)ds, (4.18) t h 2 t h 1 where η 1 (t) = [ x T (t), x T (t h 1 ), x T (t h 2 ) ] T, η 2 (t, s) = [ η T 1 (t), ẋt (s) ] T,

95 78 4. Stability of Systems with Multiple Delays and Φ and Ψ i,i=1, 2, 3(wherek =1inΨ 3 ) are defined in (4.8)-(4.11), respectively. If Φ<0andΨ i 0, i=1, 2, 3, then V 2 (x t ) < ε x(t) 2 for a sufficiently small ε>0. So, system (4.3) is asymptotically stable if LMIs (4.8)-(4.11) hold. On the other hand, when h 1 <h 2, one Lyapunov-Krasovskii functional candidate is V 2 (x t )=x T (t)px(t) h 1 t+θ h1 h 2 t+θ t h 1 x T (s)q 1 x(s)ds + ẋ T (s)w 1 ẋ(s)dsdθ + Equation (4.16) can be rewritten as 0 h 2 t h 2 x T (s)q 2 x(s)ds t+θ ẋ T (s)w 2 ẋ(s)dsdθ ẋ T (s)w 3 ẋ(s)dsdθ. (4.19) 2 [ x T (t)m 1 + x T (t h 1 )M 2 + x T ] (t h 2 )M 3 [ x(t h 2 ) x(t h 1 )+ h1 t h 2 ẋ(s)ds ] =0. (4.20) Then,followingtheprocedureforthecaseh 1 h 2 yields a similar result; but note that, in this case, k = 1 in (4.11). This completes the proof. Remark The main modification to the Lyapunov-Krasovskii functional candidate is the addition of the last term, which contains an integral of the state that is a function of the upper bounds on h 1 and h 2.Thisisavery important term: Without it, the stability is guaranteed from 0 to the upper bounds; but with it, the stability range for each delay can begin at a nonzero lower bound. This enlarges the stability range, thereby reducing the conservativeness. Now, retaining the term ẋ(t) in the derivative of V2 (x t ) rather than replacing it with the system equation yields another theorem. Theorem Consider nominal system (4.3) with m =2. Given scalars h i 0, i=1, 2, the system is asymptotically stable if there exist matrices P>0, Q i 0, i=1, 2, W j > 0, j=1, 2, 3, X ll 0, Y ll 0, and Z ll 0, l =1, 2,, 4, and any appropriately dimensioned matrices N i,s i,m i,i= 1, 2,, 4, X ij,y ij, and Z ij,i=1, 2,, 4, i<j 4 such that the following LMIs hold:

96 4.2 Two Delays 79 Ξ 11 Ξ 12 Ξ 13 Ξ 14 Ξ Ξ = 22 Ξ 23 Ξ 24 < 0, (4.21) Ξ 33 Ξ 34 Ξ 44 X 11 X 12 X 13 X 14 N 1 X 22 X 23 X 24 N 2 Θ 1 = X 33 X 34 N 3 0, (4.22) X 44 N 4 W 1 Y 11 Y 12 Y 13 Y 14 S 1 Y 22 Y 23 Y 24 S 2 Θ 2 = Y 33 Y 34 S 3 0, (4.23) Y 44 S 4 W 2 Z 11 Z 12 Z 13 Z 14 km 1 Z 22 Z 23 Z 24 km 2 Θ 3 = Z 33 Z 34 km 3 0, (4.24) Z 44 km 4 W 3 where 1, if h 1 h 2, k = 1, if h 1 <h 2 ; and Ξ 11 = Q 1 + Q 2 T 1 A 0 A T 0 T T 1 + N 1 + N T 1 + S 1 + S T 1 + h 1X 11 + h 2 Y 11 + h 1 h 2 Z 11, Ξ 12 = P + T 1 A T 0 T T 2 + N T 2 + ST 2 + h 1X 12 + h 2 Y 12 + h 1 h 2 Z 12, Ξ 13 = T 1 A 1 + N T 3 N 1 + S T 3 M 1 + h 1 X 13 + h 2 Y 13 + h 1 h 2 Z 13, Ξ 14 = T 1 A 2 + N T 4 + ST 4 S 1 + M 1 + h 1 X 14 + h 2 Y 14 + h 1 h 2 Z 14,

97 80 4. Stability of Systems with Multiple Delays Ξ 22 = h 1 W 1 +h 2 W 2 + h 1 h 2 W 3 +T 2 +T T 2 +h 1 X 22 +h 2 Y 22 + h 1 h 2 Z 22, Ξ 23 = T 2 A 1 N 2 M 2 + h 1 X 23 + h 2 Y 23 + h 1 h 2 Z 23, Ξ 24 = T 2 A 2 S 2 + M 2 + h 1 X 24 + h 2 Y 24 + h 1 h 2 Z 24, Ξ 33 = Q 1 N 3 N T 3 M 3 M T 3 + h 1X 33 + h 2 Y 33 + h 1 h 2 Z 33, Ξ 34 = N T 4 S 3 + M 3 M T 4 + h 1X 34 + h 2 Y 34 + h 1 h 2 Z 34, Ξ 44 = Q 2 S 4 S T 4 + M 4 + M T 4 + h 1X 44 + h 2 Y 44 + h 1 h 2 Z 44. Proof. From the Newton-Leibnitz formula, we know that the following equations hold for any appropriately dimensioned matrices N i,s i,and M i,i=1, 2,, 4: 2 [ x T (t)n 1 +ẋ T (t)n 2 + x T (t h 1 )N 3 + x T ] (t h 2 )N 4 [ x(t) x(t h 1 ) t h 1 ] ẋ(s)ds =0, (4.25) 2 [ x T (t)s 1 +ẋ T (t)s 2 + x T (t h 1 )S 3 + x T ] (t h 2 )S 4 [ ] x(t) x(t h 2 ) ẋ(s)ds =0, (4.26) t h 2 2 [ x T (t)m 1 +ẋ T (t)m 1 + x T (t h 1 )M 3 + x T ] (t h 2 )M 4 [ ] x(t h 2 ) x(t h 1 ) h2 t h 1 ẋ(s)ds =0. (4.27) Moreover, from system equation (4.3), the following is true for any appropriately dimensioned matrices T i,i=1, 2: 2 [ x T (t)t 1 +ẋ T ] (t)t 2 [ẋ(t) A0 x(t) A 1 x(t h 1 ) A 2 x(t h 2 )] = 0. (4.28) On the other hand, the following holds for any matrices X jj 0, Y jj 0, and Z jj 0, j =1, 2,, 4, and any appropriately dimensioned matrices X ij, Y ij,andz ij, i =1, 2,, 4, i<j 4: T x(t) Λ 11 Λ 12 Λ 13 Λ 14 x(t) ẋ(t) Λ 22 Λ 23 Λ 24 ẋ(t) =0, (4.29) x(t h 1 ) Λ 33 Λ 34 x(t h 1 ) x(t h 2 ) Λ 44 x(t h 2 ) where

98 4.2 Two Delays 81 Λ ij =h 1 (X ij X ij )+h 2 (Y ij Y ij )+(h 1 h 2 )(Z ij Z ij ),i=1, 2,, 4, i j 4. We retain the term ẋ(t) in V 2 (x t ) and add the left sides of (4.25)-(4.29) to V 2 (x t ). The proof is completed by following a line similar to the one in Theorem Remark The equivalence of Theorems and can be proven in the same way that equivalence was proven in Subsection Equivalence Analysis Although the criterion for the case h 1 = h 2 should be equivalent to the criterion for a single delay, that cannot be demonstrated using previous methods. In contrast, since Theorem takes the relationship between h 1 and h 2 into account, it is easy to show the equivalence of Theorem for two identical delays and a criterion for a single delay, as explained below. We begin with a criterion for a single delay derived directly from Theorem Corollary Consider nominal system (4.3) with m =1. Given a scalar h 1 0, the system is asymptotically stable if there exist matrices P >0, Q 0, W > 0, and X = X 11 X12 0, and any appropriately dimensioned X22 matrices N 1 and N 2 such that the following LMIs hold: Φ 11 Φ12 < 0, (4.30) Φ22 X 11 X12 N1 X22 N2 0, (4.31) W where Φ 11 = PA 0 + A T 0 P + Q + N 1 + N T 1 + AT 0 HA 0 + h 1 X11, Φ 12 = PA 1 N 1 + N T 2 + A T 0 HA 1 + h 1 X12, Φ 22 = Q N 2 N T 2 + AT 1 HA 1 + h 1 X22, H = h 1 W.

99 82 4. Stability of Systems with Multiple Delays Now, we will show that Corollary is equivalent to Theorem for h 1 = h 2 when A 1 is replaced with A 1 + A 2 in Φ 12 and Φ 22. If the third row and third column of (4.8) are added to the second row and second column, respectively, then (4.8) is equivalent to the LMI Φ 11 Π 12 Φ 13 Π = Π 22 Π 23 < 0, (4.32) Φ 33 where Π 12 = PA 1 + PA 2 + N T 2 + N T 3 N 1 + S T 2 + ST 3 S 1 + A T 0 H(A 1 + A 2 ) +h 1 (X 12 + X 13 )+h 2 (Y 12 + Y 13 )+ h 1 h 2 (Z 12 + Z 13 ), Π 22 = (Q 1 + Q 2 ) N 3 N T 3 S 3 S T 3 N 2 N T 2 S 2 S T 2 +(A 1 + A 2 ) T H(A 1 + A 2 )+h 1 (X 22 + X 23 + X T 23 + X 33) +h 2 (Y 22 + Y 23 + Y T 23 + Y 33 )+ h 1 h 2 (Z 22 + Z 23 + Z T 23 + Z 33 ), Π 23 = Q 2 S 3 S T 3 + M 3 N T 3 S 2 + M 2 +(A 1 + A 2 ) T HA 2 +h 1 (X 23 + X 33 )+h 2 (Y 23 + Y 33 )+ h 1 h 2 (Z 23 + Z 33 ); and Φ 11, Φ 13, Φ 33,andH are defined in (4.8). On the one hand, if LMIs (4.30) and (4.31) in Corollary are feasible (when A 1 is replaced with A 1 + A 2 ), then the solutions can be expressed as appropriate forms of the feasible solutions of LMIs (4.9)-(4.11) and (4.32). In fact, for the feasible solutions of LMIs (4.30) and (4.31) in Corollary 4.2.1, we can make the assignments: P = P, S i =0,i =1, 2, 3, N 1 = N 1, N 2 = N 2, N 3 = 0, 0 < Q 2 < Q, Q 1 = Q Q 2, M 1 = PA 2 A T HA 0 2, M 2 = Q 2 (A 1 + A 2 ) T HA2 M 3 =0,W 1 = W, W 2 =0,X 11 = X 11, X 12 = X 12, X 13 =0,X 22 = X 22, X 23 =0,X 33 =0,andY ij =0,i =1, 2, 3, i j 3. Then, Z ij, i =1, 2, 3, i j 3, and W 3 are the feasible solutions of the LMI Z 11 Z 12 Z 13 M 1 Z 22 Z 23 M 2 0. (4.33) Z 33 0 W 3 The above matrices must be the feasible solutions of LMIs (4.9)-(4.11) and (4.32). Consequently, Theorem for h 1 = h 2 contains Corollary

100 4.2 Two Delays 83 On the other hand, for the feasible solutions of LMIs (4.9)-(4.11) and (4.32), we make the assignments: P = P, Q = Q 1 +Q 2, W = W1 +W 2, N1 = N 1 +S 1, N 2 = N 2 +N 3 +S 2 +S 3, X 11 = X 11 +Y 11, X 12 = X 12 +Y 12 +X 13 +Y 13, and X 22 = X 22 +Y 22 +X 23 +Y 23 +X T 23 +Y T 23 +X 33+Y 33. This yields the feasible solutions of LMIs (4.30) and (4.31) in Corollary That is, Corollary contains Theorem for h 1 = h 2. Thus, Corollary and Theorem are equivalent for the case h 1 = h Systems with Time-Varying Structured Uncertainties Using Lemma to extend Theorem to system (4.4), which has timevarying structured uncertainties, produces a new theorem. Theorem Consider system (4.4) with m = 2. Given scalars h i 0, i=1, 2, the system is robustly stable if there exist matrices P>0, Q i 0, i =1, 2, W j > 0, X jj 0, Y jj 0, and Z jj 0, j=1, 2, 3, any appropriately dimensioned matrices N i,s i,m i,i=1, 2, 3, X ij,y ij, and Z ij,i=1, 2, 3, i< j 3, andascalarλ>0 such that LMIs (4.9)-(4.11) and the following LMI hold: ˆΦ 11 ˆΦ12 ˆΦ13 A T 0 H PD ˆΦ22 ˆΦ23 A T 1 H 0 ˆΦ33 A T 2 H 0 < 0, (4.34) H HD λi where ˆΦ 11 = PA 0 + A T 0 P + Q 1 + Q 2 + N 1 + N T 1 + S 1 + S T 1 + λet 0 E 0 + h 1 X 11 + h 2 Y 11 + h 1 h 2 Z 11, ˆΦ 12 = PA 1 N 1 +N T 2 +S T 2 M 1 +λe T 0 E 1 +h 1 X 12 +h 2 Y 12 + h 1 h 2 Z 12, ˆΦ 13 = PA 2 +N T 3 +ST 3 S 1 +M 1 +λe T 0 E 2 +h 1 X 13 +h 2 Y 13 + h 1 h 2 Z 13, ˆΦ 22 = Q 1 N 2 N T 2 M 2 M T 2 +λe T 1 E 1 +h 1 X 22 +h 2 Y 22 + h 1 h 2 Z 22, ˆΦ 23 = N T 3 S 2 + M 2 M T 3 + λe T 1 E 2 + h 1 X 23 + h 2 Y 23 + h 1 h 2 Z 23, ˆΦ 33 = Q 2 S 3 S T 3 +M 3 +M T 3 +λet 2 E 2 +h 1 X 33 +h 2 Y 33 + h 1 h 2 Z 33, H = h 1 W 1 + h 2 W 2 + h 1 h 2 W 3.

101 84 4. Stability of Systems with Multiple Delays Proof. Applying the Schur complement and Lemma 2.6.2, and following a procedure similar to the one in the proof of Theorem 3.3.1, yield Theorem Theorem can also be extended to system (4.4), although we omit the explanation here for brevity. Furthermore, Theorem can be extended to a system with polytopictype uncertainties, as shown next. Theorem Consider system (4.3) with polytopic-type uncertainties (4.7) and m = 2. Given scalars h i 0, i = 1, 2, the system is robustly stable if there exist matrices P j > 0, Q ij 0, i=1, 2, W ij > 0, i=1, 2, 3, X (j) kk (j) 0, Y kk 0, and Z (j) kk 0, k = 1, 2,, 4, j = 1, 2,,p, and any appropriately dimensioned matrices N ij,s ij,m ij,x (j) ik 1, 2,, 4, i<k 4, j=1, 2,,p, and T i,i=1, 2 such that the following (j),y ik,z(j) ik,i= LMIs hold for j =1, 2,,p: Ξ (j) 11 Ξ (j) 12 Ξ (j) 13 Ξ (j) 14 Ξ (j) Ξ (j) 22 Ξ (j) 23 Ξ (j) 24 = Ξ (j) 33 Ξ (j) < 0, (4.35) 34 Ξ (j) 44 Θ (j) 1 = Θ (j) 2 = X (j) 11 X (j) 12 X (j) 13 X (j) 14 N 1j X (j) 22 X (j) 23 X (j) 24 N 2j X (j) 33 X (j) 34 N 3j 0, (4.36) X (j) 44 N 4j W 1j Y (j) 11 Y (j) 12 Y (j) 13 Y (j) 14 S 1j Y (j) 22 Y (j) 23 Y (j) 24 S 2j Y (j) 33 Y (j) 34 S 3j 0, (4.37) Y (j) 44 S 4j W 2j

102 where Θ (j) 3 = Z (j) 11 Z (j) 12 Z (j) 13 Z (j) Z (j) 22 Z (j) 23 Z (j) Z (j) 33 Z (j) Z (j) 14 M 1j 24 M 2j 34 M 3j 44 M 4j W 3j 4.2 Two Delays 85 0, (4.38) Ξ (j) 11 = Q 1j + Q 2j T 1 A 0j A T 0j T 1 T + N 1j + N1j T + S 1j + S1j T + h 1X (j) 11 + h 2 Y (j) 11 + h 1 h 2 Z (j) 11, Ξ (j) 12 = P j + T 1 A T 0j T T 2 + N T 2j + ST 2j + h 1X (j) 12 + h 2Y (j) 12 + h 1 h 2 Z (j) 12, Ξ (j) 13 = T 1A 1j + N3j T N 1j + S3j T M 1j + h 1 X (j) 13 + h 2Y (j) 13 + h 1 h 2 Z (j) 13, Ξ (j) 14 = T 1A 2j + N4j T + ST 4j S 1j + M 1j + h 1 X (j) 14 + h 2Y (j) 14 + h 1 h 2 Z (j) 14, Ξ (j) 22 = h 1W 1j + h 2 W 2j + h 1 h 2 W 3j + T 2 + T2 T + h 1X (j) 22 + h 2Y (j) 22 + h 1 h 2 Z (j) 22, Ξ (j) 23 = T 2A 1j N 2j M 2j + h 1 X (j) 23 + h 2Y (j) 23 + h 1 h 2 Z (j) 23, Ξ (j) 24 = T 2A 2j S 2j + M 2j + h 1 X (j) 24 + h 2Y (j) 24 + h 1 h 2 Z (j) 24, Ξ (j) 33 = Q 1j N 3j N T 3j M 3j M T 3j + h 1X (j) 33 + h 2Y (j) 33 + h 1 h 2 Z (j) 33, Ξ (j) 34 = N 4j T S 3j + M 3j M4j T + h 1X (j) 34 + h 2Y (j) 34 + h 1 h 2 Z (j) 34, Ξ (j) 44 = Q 2j S 4j S4j T + M 4j + M4j T + h 1X (j) 44 + h 2Y (j) 44 + h 1 h 2 Z (j) Numerical Examples Example Consider the stability of system (4.3) with m = 2and A 0 = 2 0, 1 A 1 = 0.6, A 2 = (4.39) If h 1 = h 2, this system is equivalent to system (4.3) with m =1and A 0 = 2 0, A 1 = 1 0. (4.40) The methods in [1, 2] and Corollary show that system (4.3) with m = 1 and (4.40) is asymptotically stable for 0 h However, [1, 2] show that it is asymptotically stable for 0 h 1 = h This result is conservative for multiple delays primarily because the relationship between h 1 and h 2 was not taken into account. In contrast, Theorem 4.2.1

103 86 4. Stability of Systems with Multiple Delays shows that system (4.3) with m = 2 and (4.39) is asymptotically stable for 0 h 1 = h This upper bound is much larger than the one in [1, 2] and is the same as the one for a single delay. Regarding the calculated range of h 2 that ensures that system (4.3) with m = 2 and (4.39) is asymptotically stable for a given h 1, Table 4.1 compares the results for our method and for the one in [1, 2]; the results are also illustrated in Fig Clearly, our method produces significantly larger stability domains for h 1 and h 2. Since the stable range for a single delay is generally from 0 to an upper bound, we usually just need to find that upper bound. As Fridman & Shaked simply extended the method for a single delay to two delays [1,2], they were able to provide only a stable upper bound for h 2, but not an appropriate (possibly non-zero) lower bound. In the numerical example, their method yielded h 1 < 2.25, and it was impossible to find the stable range of h 2 for h In contrast, our method employs a cross term for h 1 and h 2 (the last term in (4.12)) to construct a new type of Lyapunov-Krasovskii functional. Unlike other methods, this is not a simple extension of the treatment for a single delay; and it takes the relationship between the two delays into account. Consequently, our method yields a stable range for h 2 rather than a simple upper bound. In the numerical example, the stable range of h 2 is much larger than that given by the method in [1, 2] when h 1 < 2.25; and we can even obtain a stable range for h 2 when h Note that in this case, the stable range of h 2 no longer starts from 0. Example Consider system (4.3) with two delays and the following parameters: ρ A 0 =,A 1 =,A 2 =, (4.41) ρ and ρ When h 1 = h 2, the above system is equivalent to the one in Example in Chapter 3 for a constant delay. The methods for a system with a single delay in [2, 5] tell us that the system is stable for 0 h On the other hand, Theorem in Chapter 3 gives 0 h as the stable range. Above we proved that a system with two equal delays is equivalent to the same system with a single delay. Solving LMIs (4.35)-(4.38) in Theorem also demonstrates this point; that is, system (4.3) with two delays and

104 4.2 Two Delays 87 Fig Stability ranges ensuring asymptotic stability of system (4.3) with m = 2 and parameters (4.39) Table 4.1. Range of h 2 ensuring asymptotic stability of system (4.3) with m =2 and parameters (4.39) for a given h 1 (Example 4.2.1) h h 2 (Theorem 4.2.1) [0, + ] [0, 3.36] [0, 3.35] [0, 3.34] h 2 ( [1, 2]) [0, + ] [0, 1.84] [0, 1.81] [0, 1.78] h h 2 (Theorem 4.2.1) [0, 3.33] [0, 3.33] [0, 3.33] [0, 3.36] h 2 ( [1, 2]) [0, 1.71] [0, 1.64] [0, 1.57] [0, 1.42] h h 2 (Theorem 4.2.1) [0, 3.39] [0, 3.43] [0, 3.47] [0, 3.52] h 2 ( [1, 2]) [0, 1.22] [0, 0.88] [0, 0.40] [0, 0.07] h h 2 (Theorem 4.2.1) [0, 3.55] [0.08, 3.57] [0.22, 3.61] [0.35, 3.65] h 2 ( [1, 2]) [0, 0] h h 2 (Theorem 4.2.1) [1.04, 3.77] [1.88, 3.90] [3.59, 4.18] [4.47, 4.47] h 2 ( [1, 2])

105 88 4. Stability of Systems with Multiple Delays (4.41) is robustly stable for 0 h 1 = h However, applying Lemma 1 in [2] and Corollary 3 in [1] or extending Theorem 1 in [5] to the twodelay case shows that system (4.3) with two delays and (4.41) is robustly stable for 0 h 1 = h Clearly, the upper bound on h 1 = h 2 obtained in [1,5,6] is much smaller than the one produced by Theorem Furthermore, Table 4.2 compares the methods in [1,5,6] and our method for the problem of calculating the upper bound on h 2 for a given h 1 ;theresults are illustrated in Fig It is clear that our method produces significantly larger stability domains for h 1 and h 2. Table 4.2. Range of h 2 ensuring asymptotic stability of system (4.3) with two delays and parameters (4.41) for a given h 1 (Example 4.2.2) h h 2 (Theorem 4.2.4) [0, 0.50] [0, 0.54] [0, 0.55] [0, 0.57] h 2 ( [1, 5, 6]) [0, 0.36] [0, 0.26] [0, 0.235] [0, 0.17] h h 2 (Theorem 4.2.4) [0, 0.60] [0.05, 0.64] [0.41, 0.69] [0.61, 0.75] h 2 ( [1, 5, 6]) [0, 0.08] h h 2 (Theorem 4.2.4) [0.77, 0.82] [0.863, 0.863] h 2 ( [1, 5, 6]) Fig Stability ranges ensuring asymptotic stability of system (4.3) with two delays and parameters (4.41)

106 4.3 Multiple Delays Multiple Delays This section extends Theorem to system (4.3) with m>2. For convenience, we assume that 0=h 0 h 1 h 2 h m. (4.42) We have the following theorem. Theorem Consider system (4.3). Given scalars h i 0,i=1, 2,,m satisfying (4.42), the system is asymptotically stable if there exist matrices X (ij) 00 X (ij) 01 X (ij) 0m X (ij) P>0, Q i 0, i=1, 2,,m, X (ij) 11 X (ij) 1m = 0, i=... X mm (ij) 0, 1,,m 1, i<j m, and W (ij) > 0, i=0, 1,,m 1, i<j m, and any appropriately dimensioned matrices N (ij) l,l = 0, 1,,m, i = 0, 1,,m 1, i<j m such that the following LMIs hold: Ξ 00 Ξ 01 Ξ 0m Ξ 11 Ξ 1m Ξ = < 0, (4.43)... Ξ mm Γ (ij) = X (ij) 00 X (ij) 01 X (ij) 0m X (ij) mm N (ij) 0 X (ij) 11 X (ij) 1m N (ij) , i=0, 1,,m 1, i<j m, N m (ij) W (ij) where m m ( Ξ 00 = PA 0 + A T 0 P + Q i + N (0j) 0 +[N (0j) 0 ] T) i=1 m 1 +A T 0 GA 0 + i=0 j=i+1 j=1 m (h j h i )X (ij) 00, (4.44)

107 90 4. Stability of Systems with Multiple Delays k 1 Ξ 0k = PA k N (ik) 0 + i=0 m 1 +A T 0 GA k + i=0 m i=1 i=0 j=i+1 [N (0i) k ] T + k 1 ( Ξ kk = Q k N (ik) k +[N (ik) k ] T) + k 1 Ξ lk = G = m 1 +A T k GA k + i=0 N (ik) l i=0 j=i+1 l 1 i=0 m 1 +A T l GA k + m 1 i=0 j=i+1 i=0 j=i+1 m j=k+1 N (kj) 0 m (h j h i )X (ij), k =1, 2,,m, 0k m j=k+1 ( N (kj) k +[N (kj) k ] T) m (h j h i )X (ij), k =1, 2,,m, [N (il) k ] T + m (h j h i )W (ij). m j=k+1 kk N (kj) l + m j=l+1 [N (lj) k ] T m (h j h i )X (ij),l=1, 2,,m, l <k m lk Proof. Choose the following Lyapunov-Krasovskii functional candidate: m V m (x t )=x T (t)px(t)+ x T (s)q i x(s)ds t h i m 1 + m i=0 j=i+1 i=1 hi h j t+θ ẋ T (s)w (ij) ẋ(s)dsdθ, (4.45) where P>0, Q i 0, i=1, 2,,m,andW (ij) > 0, i=0, 1,,m 1, i< j m are to be determined. According to the Newton-Leibnitz formula, for any appropriately dimensioned matrices N (ij) l, i =0, 1,,m 1, i<j m, l =0, 1,,m,the following equation holds: [ m ][ ] t hi 2 x T (t h l )N (ij) l x(t h i ) x(t h j ) ẋ(s)ds =0. (4.46) l=0 t h j On the other hand, for any matrices X (ij) 0, i=0, 1,,m 1, i<j m, the following holds: m 1 i=0 j=i+1 m (h j h i )ζ1 [X T (t) (ij) X (ij)] ζ 1 (t) =0, (4.47)

108 4.4 Conclusion 91 where ζ 1 (t) = [ x T (t), x T (t h 1 ), x T (t h 2 ),, x T (t h m ) ] T. So, the derivative of V m (x t ) along the solutions of system (4.3) can be written as where V m (x t )=ζ1 T (t) Ξζ m 1 1 (t) ζ 2 (t, s) = [ ζ T 1 (t), ẋ(s)] T ; m i=0 j=i+1 hi t h j ζ T 2 (t, s)γ (ij) ζ 2 (t, s)ds, (4.48) and Ξ and Γ (ij),i=0, 1,,m 1, i<j m are defined in (4.43) and (4.44), respectively. From (4.48), if LMIs (4.43) and (4.44) hold, system (4.3) is asymptotically stable. This completes the proof. Remark If i {1, 2,,m 1} such that h i = h i+1, then the system can be transformed into a system with m 1 delays. From the explanations for Theorem and Corollary 4.2.1, it is easy to see that the delay-dependent condition is equivalent to the one for a system with m 1delays. Remark Following a similar line, Theorem can also be extended to a system with multiple delays and polytopic-type uncertainties using a parameter-dependent Lyapunov-Krasovskii functional, although we do not give the details here for brevity. 4.4 Conclusion This chapter presents new delay-dependent stability criteria for linear systems with multiple constant delays derived by the FWM approach. This method is less conservative than previous ones because it employs neither a system transformation nor an inequality to estimate the upper bound on a cross term, but instead uses FWMs to take the relationships among the delays into account. FWMs that express the reciprocal influences of the terms of the Newton-Leibnitz formula are easy to calculate and are determined by LMIs. In contrast to other methods, the stability domain of a delay provided by our method is a range, rather than just an upper bound.

109 92 4. Stability of Systems with Multiple Delays References 1. E. Fridman and U. Shaked. Delay-dependent stability and H control: constant and time-varying delays. International Journal of Control, 76(1): 48-60, E. Fridman and U. Shaked. An improved stabilization method for linear timedelay systems. IEEE Transactions on Automatic Control, 47(11): , Y. He, M. Wu, and J. H. She. Delay-dependent stability criteria for linear systems with multiple time delays. IEE Proceedings: Control Theory and Applications, 153(4): , M. Wu and Y. He. Parameter-dependent Lyapunov functional for systems with multiple time delays. Journal of Control Theory and Applications, 2(3): , E. Fridman and U. Shaked. Parameter dependent stability and stabilization of uncertain time-delay systems. IEEE Transactions on Automatic Control, 48(5): , E. Fridman and U. Shaked. A descriptor system approach to H control of linear time-delay systems. IEEE Transactions on Automatic Control, 47(2): , 2002.

110 5. Stability of Neutral Systems A neutral system is a system with a delay in both the state and the derivative of the state, with the one in the derivative being called a neutral delay. That makes it more complicated than a system with a delay in only the state. Neutral delays occur not only in physical systems, but also in control systems, where they are sometimes artificially added to boost the performance. For example, repetitive control systems constitute an important class of neutral systems [1]. Stability criteria for neutral systems can be classified into two types: delay-independent [2 4] and delay-dependent [5 24]. Since the delayindependent type does not take the length of a delay into consideration, it is generally conservative. The basic methods for studying delay-dependent criteria for neutral systems are similar to those used to study linear systems, with the main ones being fixed model transformations. As mentioned in Chapter 1, the four types of fixed model transformations impose limitations on possible solutions to delay-dependent stability problems. The delay in the derivative of the state gives a neutral system special features not shared by linear systems. In a neutral system, a neutral delay can be the same as or different from a discrete delay. Neutral systems with identical constant discrete and neutral delays were studied in [6 8,12,14]; and systems with different discrete and neutral delays were studied in [10,11,13, 16 21,23]. The criteria in these reports usually require the neutral delay to be constant, but allow the discrete delay to be either constant [10,11,13,17,18,23] or time-varying [16,19 21]. Almost all these criteria take only the length of a discrete delay into account and ignore the length of a neutral delay. They are thus called discrete-delay-dependent and neutral-delay-independent stability criteria [13]. Discrete-delay- and neutral-delay-dependent criteria are rarely investigated, with two exceptions being [23, 25]. This chapter offers a comprehensive analysis of these various types of criteria based on the FWM approach. First, this approach is used to examine systems with a time-varying discrete delay and a constant neutral

111 94 5. Stability of Neutral Systems delay, and discrete-delay-dependent and neutral-delay-independent stability criteria are obtained. We show that the criterion in [19], which relies on a descriptor model transformation, is a special case of ours. Furthermore, we point out that another reason why criteria derived using Park s inequality in combination with a descriptor model transformation are conservative is that, when the coefficient matrix of a term with a discrete delay is nonsingular, Park s inequality leads to conservativeness. Then, for a neutral system with identical constant discrete and neutral delays, we use the FWM approach to derive delay-dependent stability criteria; and we obtain less conservative results [26, 27] by using the FWM approach in combination with either a parameterized model transformation or an augmented Lyapunov-Krasovskii functional. Finally, for a neutral system with different constant discrete and neutral delays, we use the FWM approach to derive a discrete-delay- and neutral-delay-dependent stability criterion; and we show that, when the two delays are identical, the criterion is equivalent to the one obtained by using the FWM approach to directly handle identical discrete and neutral delays [23, 25]. 5.1 Neutral Systems with Time-Varying Discrete Delay This section uses the FWM approach to examine the stability of neutral systems with a time-varying discrete delay and a constant neutral delay Problem Formulation Consider the following neutral system with a time-varying discrete delay: ẋ(t) Cẋ(t τ) =Ax(t)+A d x(t d(t)), t > 0, (5.1) x(t) =φ(t), t [ r, 0], where x(t) R n is the state vector; A, A d,andc are constant matrices with appropriate dimensions; all the eigenvalues of matrix C are inside the unit circle; the delay, d(t), is a time-varying continuous differentiable function satisfying 0 d(t) h and d(t) μ, (5.2) (5.3)

112 5.1 Neutral Systems with Time-Varying Discrete Delay 95 where h and μ are constants; r is defined to be max{h, τ}; and the initial condition, φ(t), is a continuously differentiable initial function of t [ r, 0]. If the system contains time-varying structured uncertainties, it can be written as ẋ(t) Cẋ(t τ) =(A +ΔA(t))x(t)+(A d +ΔA d (t))x(t d(t)), t > 0, x(t) =φ(t), t [ r, 0]. (5.4) The uncertainties are assumed to be of the form [ΔA(t) ΔA d (t)] = DF(t)[E a E ad ], (5.5) where D, E a,ande ad are constant matrices with appropriate dimensions; and F (t) is an unknown, real, and possibly time-varying matrix with Lebesguemeasurable elements satisfying F T (t)f (t) I, t. (5.6) Nominal Systems Choose the Lyapunov-Krasovskii functional candidate to be V (x t )=x T (t)px(t)+ x T (s)qx(s)ds + ẋ T (s)rẋ(s)ds t d(t) t τ 0 + ẋ T (s)zẋ(s)dsdθ, (5.7) h t+θ where P>0, Q 0, R 0, and Z>0areto be determined. For any appropriately dimensioned matrices N i,i=1, 2, 3, the Newton- Leibnitz formula gives us 2 [ x T (t)n 1 + x T (t d(t))n 2 +ẋ T ] (t τ)n 3 [ ] t x(t) ẋ(s)ds x(t d(t)) =0. (5.8) t d(t) X 11 X 12 X 13 On the other hand, for any matrix X = X 22 X 23 0, the following X 33 inequality holds: hη T 1 (t)xη 1 (t) t d(t) η T 1 (t)xη 1 (t)ds 0, (5.9)

113 96 5. Stability of Neutral Systems where η 1 (t) = [ x T (t), x T (t d(t)), ẋ T (t τ) ] T. Then, calculating the derivative of V (x t ) along the solutions of system (5.1), adding the left sides of (5.8) and (5.9) to it, and replacing the term ẋ(t) in V (x t ) with the system equation yield the following theorem. Theorem Consider nominal system (5.1). Given scalars h>0 and μ, the system is asymptotically stable if there exist matrices P>0, Q 0, X 11 X 12 X 13 R 0, Z>0, and X = X 22 X 23 0, and any appropriately X 33 dimensioned matrices N i,i=1, 2, 3 such that the following LMIs hold: Φ 11 Φ 12 Φ 13 A T H Φ Φ = 22 Φ 23 A T d H Φ 33 C T < 0, (5.10) H H where X 11 X 12 X 13 N 1 X 22 X 23 N 2 0, (5.11) X 33 N 3 Z Φ 11 = PA+ A T P + Q + N 1 + N T 1 + hx 11, Φ 12 = PA d + N T 2 N 1 + hx 12, Φ 13 = PC + N T 3 + hx 13, Φ 22 = (1 μ)q N 2 N T 2 + hx 22, Φ 23 = N T 3 + hx 23, Φ 33 = R + hx 33, H = R + hz. On the other hand, for any appropriately dimensioned matrices N i,i= 1, 2,, 4, the Newton-Leibnitz formula gives us 2 [ x T (t)n 1 +ẋ T (t)n 2 + x T (t d(t))n 3 +ẋ T ] (t τ)n 4 [ ] x(t) t d(t) ẋ(s)ds x(t d(t)) =0. (5.12)

114 5.1 Neutral Systems with Time-Varying Discrete Delay 97 In addition, from system equation (5.1), we know that, for any appropriately dimensioned matrices T j,j=1, 2, 2 [ x T (t)t 1 +ẋ T ] (t)t 2 [ẋ(t) Ax(t) Ad x(t d(t)) Cẋ(t τ)] = 0. (5.13) X 11 X 12 X 13 X 14 X Furthermore, for any matrix X = 22 X 23 X 24 0, the following X 33 X 34 X 44 inequality holds: where hη T 2 (t)xη 2 (t) t d(t) η T 2 (t)xη 2 (t)ds 0, (5.14) η 2 (t) = [ x T (t), ẋ T (t), x T (t d(t)), ẋ T (t τ) ] T. Calculating the derivative of V (x t ) along the solutions of system (5.1) and using (5.12)-(5.14) yield V (x t )=η2 T (t)γη 2 (t) η3 T (t, s)ψη 3 (t, s)ds, (5.15) t d(t) where and η 3 (t, s) = [ η T 2 (t), ẋt (s) ] T, (5.16) Γ 11 Γ 12 Γ 13 Γ 14 Γ Γ = 22 Γ 23 Γ 24 < 0, (5.17) Γ 33 Γ 34 Γ 44 X 11 X 12 X 13 X 14 N 1 X 22 X 23 X 24 N 2 Ψ = X 33 X 34 N 3 0, (5.18) X 44 N 4 Z

115 98 5. Stability of Neutral Systems Γ 11 = Q + N 1 + N T 1 A T T T 1 T 1 A + hx 11, Γ 12 = P + N T 2 + T 1 A T T T 2 + hx 12, Γ 13 = N T 3 N 1 T 1 A d + hx 13, Γ 14 = N T 4 T 1C + hx 14, Γ 22 = R + hz + T 2 + T T 2 + hx 22, Γ 23 = N 2 T 2 A d + hx 23, Γ 24 = T 2 C + hx 24, Γ 33 = (1 μ)q N 3 N T 3 + hx 33, Γ 34 = N T 4 + hx 34, Γ 44 = R + hx 44. Thus, we arrive at the following theorem. Theorem Consider nominal system (5.1). Given scalars h>0 and μ, the system is asymptotically stable if there exist matrices P>0, Q 0, X 11 X 12 X 13 X 14 X R 0, Z>0, and X = 22 X 23 X 24 0, and any appropriately X 33 X 34 X 44 dimensioned matrices N i,i =1, 2,, 4 and T j,j =1, 2 such that LMIs (5.17) and (5.18) hold. If (5.12) is replaced with 2 [ x T (t)n 1 +ẋ T (t)n 2 + x T (t d(t))n 3 +ẋ T ] (t τ)n 4 Ad [ ] x(t) t d(t) ẋ(s)ds x(t d(t)) =0, the Z in Lyapunov-Krasovskii functional (5.7) is replaced with A T d ZA d, X is [ ] T [ ] set to N1 T N2 T N3 T N4 T Z 1 N1 T N2 T N3 T N4 T,andη 3 (t, s) issetto η 4 (t, s) = [ η2 T(t), ] T, ẋt (s)a T d then we get a corollary. Corollary Consider nominal system (5.1). Given scalars h>0 and μ, the system is asymptotically stable if there exist matrices P>0, Q 0, R 0, and Z > 0, and any appropriately dimensioned matrices N i, i = 1, 2,, 4 and T j, j =1, 2 such that the following LMI holds:

116 where 5.1 Neutral Systems with Time-Varying Discrete Delay 99 Π 11 Π 12 Π 13 Π 14 hn 1 Π 22 Π 23 Π 24 hn 2 Π = Π 33 Π 34 hn 3 < 0, (5.19) Π 44 hn 4 hz Π 11 = Q + N 1 A d + A T d N T 1 AT T T 1 T 1A, Π 12 = P + A T d N T 2 + T 1 A T T T 2, Π 13 = A T d N T 3 N 1 A d T 1 A d, Π 14 = A T d N T 4 T 1C, Π 22 = R + ha T d ZA d + T 2 + T T 2, Π 23 = N 2 A d T 2 A d, Π 24 = T 2 C, Π 33 = (1 μ)q N 3 A d A T d N T 3, Π 34 = A T d N T 4, Π 44 = R. Remark Corollary is equivalent to Theorem when A d is nonsingular. However, if A d is singular, N i A d,i=1, 2,, 4 cannot describe all the FWMs, which means that Corollary is more conservative than Theorem Remark The condition in Corollary includes Theorem 1 in [19] for a single delay. In fact, if we set P = P 1, N 1 = W T 11 + P T 2, N 2 = W T 12 + P T 3, N 3 =0,N 4 =0,T 1 = P T 2, T 2 = P T 3, Q = S 1,andZ = R 1 (where the terms on the right are the parameter matrices of Theorem 1 in [19]), then Corollary yields precisely Theorem 1 in [19]. Moreover, N 3 and N 4 are selected by solving LMIs, rather than simply being set to zero, which makes our criterion an improvement over the one in [19]. Remark This section concerns systems with a time-varying discrete delay and a constant neutral delay. Since the criterion obtained depends on the length of the discrete delay but not on that of the neutral delay, it is a discrete-delay-dependent and neutral-delay-independent condition.

117 Stability of Neutral Systems Systems with Time-Varying Structured Uncertainties We can use Lemma to extend Theorems and to systems with time-varying structured uncertainties. Corollary Consider system (5.4). Given scalars h>0and μ, the system is robustly stable if there exist matrices P>0, Q 0, R 0, Z>0, X 11 X 12 X 13 and X = X 22 X 23 0, any appropriately dimensioned matrices X 33 N i,i=1, 2, 3, and a scalar λ>0 such that LMI (5.11) and the following LMI hold: Φ 11 + λea T E a Φ 12 + λea TE ad Φ 13 A T H PD Φ 22 + λead T E ad Φ 23 A T d H 0 Φ 33 C T H 0 < 0, (5.20) H HD λi where Φ ij,i=1, 2, 3, i j 3 and H are defined in (5.10). Corollary Consider system (5.4). Given scalars h>0and μ, the system is robustly stable if there exist matrices P > 0, Q 0, R 0, Z> X 11 X 12 X 13 X 14 X 0, and X = 22 X 23 X 24 0, any appropriately dimensioned X 33 X 34 X 44 matrices N i,i=1, 2,, 4 and T j,j =1, 2, and a scalar λ>0 such that LMI (5.18) and the following LMI hold: Γ 11 + λea T E a Γ 12 Γ 13 + λea TE ad Γ 14 T 1 D Γ 22 Γ 23 Γ 24 T 2 D Γ 33 + λead T E ad Γ 34 0 < 0. (5.21) Γ 44 0 λi

118 5.2 Neutral Systems with Identical Discrete and Neutral Delays 101 where Γ ij,i=1, 2,, 4, i j 4 are defined in (5.17) Numerical Example This subsection uses a numerical example to compare the above method with the one in [19]. Example Consider the stability of system (5.1) with A = 2 1, A d = 0 0, C = Table 5.1 shows the allowable upper bound, h, for various μ. Note that A d is singular. As explained in Remarks and 5.1.3, our results are better than those in [19]. Table 5.1. Allowable upper bound, h, for various μ (Example 5.1.1) μ [19] Theorems and Neutral Systems with Identical Discrete and Neutral Delays If the discrete delay, d(t), has a constant value of h, and if the neutral delay is also equal to h, then system (5.1) becomes a system with identical discrete and neutral delays: ẋ(t) Cẋ(t h) =Ax(t)+A d x(t h), t > 0, (5.22) x(t) =φ(t), t [ h, 0]. Moreover, system (5.4) becomes ẋ(t) Cẋ(t h) =(A +ΔA(t))x(t)+(A d +ΔA d (t))x(t h), t > 0, x(t) =φ(t), t [ h, 0], (5.23)

119 Stability of Neutral Systems where the structured uncertainties are defined in (5.5) and (5.6). The structures of systems (5.22) and (5.23) are different from those of systems (5.1) and (5.4) in that they have only one delay. We can exploit this to overcome the conservativeness arising from the use of a discrete-delaydependent and neutral-delay-independent stability condition. It is known that Dx t must be stable if systems (5.22) and (5.23) are to be stable [28]. Several delay-dependent criteria derived by the FWM approach are given below FWM Approach First, we give a theorem for nominal system (5.22) that is based on the FWM approach. Theorem Consider nominal neutral system (5.22). Given a scalar h > 0, the system is asymptotically stable if the operator Dx t is stable and there exist matrices P > 0, Q 0, R 0, W > 0, and X 11 X 12 X 13 X = X 22 X 23 0, and any appropriately dimensioned matrices X 33 N i,i=1, 2, 3 such that the following LMIs hold: where Φ 11 Φ 12 Φ 13 A T H Φ Φ = 22 Φ 23 A T d H Φ 33 C T < 0, (5.24) H H X 11 X 12 X 13 N 1 X Ψ = 22 X 23 N 2 0, (5.25) X 33 N 3 W Φ 11 = PA+ A T P + Q + N 1 + N T 1 + hx 11, Φ 12 = PA d A T PC N 1 + N T 2 + hx 12, Φ 13 = N T 3 + hx 13,

120 5.2 Neutral Systems with Identical Discrete and Neutral Delays 103 Φ 22 = Q C T PA d A T d PC N 2 N T 2 + hx 22, Φ 23 = N T 3 + hx 23, Φ 33 = R + hx 33, H = R + hw. Proof. Choose the Lyapunov-Krasovskii functional candidate to be V (x t )=(Dx t ) T P (Dx t )+ t h x T (s)qx(s)ds + t h ẋ T (s)rẋ(s)ds 0 + ẋ T (s)w ẋ(s)dsdθ, (5.26) h t+θ where P>0, Q 0, R 0, and W>0 are to be determined. From the Newton-Leibnitz formula, we obtain the following for any appropriately dimensioned matrices N i,i=1, 2, 3: 2 [ x T (t)n 1 +x T (t h)n 2 +ẋ T ] [ ] (t h)n 3 x(t) ẋ(s)ds x(t h) =0. t h (5.27) X 11 X 12 X 13 On the other hand, for any matrix X = X 22 X 23 0, the following X 33 equation holds: where x(t) x(t h) ẋ(t h) Λ 11 Λ 12 Λ 13 x(t) Λ 22 Λ 23 x(t h) =0, (5.28) Λ 33 ẋ(t h) T Λ ij = h(x ij X ij ), i =1, 2, 3, i j 3. Calculating the derivative of V (x t ) along the solutions of system (5.22) and using (5.27) and (5.28) yield V (x t )=2[x(t) Cx(t h)] T P [Ax(t)+A d x(t h)] + x T (t)qx(t) x T (t h)qx(t h)+ẋ T (t)rẋ(t) ẋ T (t h)rẋ(t h) +hẋ T (t)w ẋ(t) t h ẋ T (s)w ẋ(s)ds

121 Stability of Neutral Systems +2 [ x T (t)n 1 +x T (t h)n 2 +ẋ T ] [ (t h)n 3 x(t) +hη1 T (t)xη 1 (t) t h η T 1 (t)xη 1 (t)ds = η1 T (t)ξη 1 (t) η2 T (t, s)ψη 2 (t, s)ds, t h where t h η 1 (t) = [ x T (t), x T (t h), ẋ T (t h) ], η 2 (t, s) = [ x T (t), x T (t h), ẋ T (t h), ẋ T (s) ] T, Φ 11 + A T HA Φ 12 + A T HA d N3 T + AT HC Ξ = Φ 22 + A T d HA d N3 T + AT d HC ; R + C T HC ] ẋ(s)ds x(t h) Φ 11, Φ 12, Φ 22,andH are defined in (5.24); and Ψ is defined in (5.25). If Ξ<0, which is equivalent to Φ<0 from the Schur complement, and if Ψ 0, then V (x t ) ε x(t) 2 for a sufficiently small ε>0. In addition, operator Dx t is stable. So, system (5.22) is asymptotically stable if LMIs (5.24) and (5.25) are feasible. This completes the proof. Remark If ẋ(t) is retained and FWMs are used to express the relationships among the terms of the system equation, we obtain a result equivalent to Theorem that can be extended to systems with polytopic-type uncertainties through the use of a parameter-dependent Lyapunov-Krasovskii functional. If we do not add the left side of (5.28) to the derivative of V (x t )inthe proof of Theorem 5.2.1, we can write the derivative of V (x t )as V (x t )= 1 h t h ζ T 1 (t, s)θζ 1(t, s)ds, where Φ + A T HA Φ 12 + A T HA d N3 T + A T HC hn 1 Φ Θ = 22 + A T d HA d N3 T + AT d HC hn 2 R + C T. HC hn 3 W That leads to the following corollary.

122 5.2 Neutral Systems with Identical Discrete and Neutral Delays 105 Corollary Consider nominal system (5.22). Given a scalar h>0, the system is asymptotically stable if the operator Dx t is stable and there exist matrices P > 0, Q 0, R 0, and W > 0, and any appropriately dimensioned matrices N i,i=1, 2, 3 such that the following LMI holds: Φ + A T HA Φ 12 + A T HA d N3 T + AT HC hn 1 Φ 22 + A T d HA d N3 T + AT d HC hn 2 < 0. (5.29) R + C T HC hn 3 W Extending Theorem to neutral system (5.23), which has time-varying structured uncertainties, gives us the following stability criterion. Theorem Consider neutral system (5.23). Given a scalar h>0, the system is robustly stable if the operator Dx t is stable and there exist matrices X 11 X 12 X 13 P > 0, Q 0, R 0, W >0, and X = X 22 X 23 0, any X 33 appropriately dimensioned matrices N i,i=1, 2, 3, andascalarλ>0 such that LMI (5.25) and the following LMI hold: Φ 11 + λea TE a Φ 12 + λea TE ad Φ 13 A T H PD Φ 22 + λead T E ad Φ 23 A T d H CT PD Φ = Φ 33 C T H 0 < 0, (5.30) H HD λi where Φ ij,i=1, 2, 3, i j 3 and H are defined in (5.24) FWM Approach in Combination with Parameterized Model Transformation Now, we use the FWM approach in combination with a parameterized model transformation to investigate the stability of systems (5.22) and (5.23). First, we have the following theorem.

123 Stability of Neutral Systems Theorem Consider nominal neutral system (5.22). Given a scalar h>0, the system is asymptotically stable if the operator Dx t is stable and there exist matrices P = P 11 P 12 0(with P 11 > 0), Q 0, R 0, Z P 22 0, and W > 0, and any appropriately dimensioned matrices N i,i=1, 2, 3 such that the following LMI holds: Φ 11 Φ 12 Φ 13 Φ 14 hn 1 A T H Φ 22 Φ 23 Φ 24 hn 2 A T d H Φ Φ = 33 0 hn 3 C T H < 0, (5.31) hw 0 0 hz 0 H where Φ 11 = P 11 A + A T P 11 + P 12 + P T 12 + Q + hw + N 1 + N T 1, Φ 12 = P 11 A d A T P 11 C P 12 P T 12 C + N T 2 N 1, Φ 13 = N T 3, Φ 14 = h(a T P 12 + P 22 ), Φ 22 = Q + P T 12C + C T P 12 C T P 11 A d A T d P 11C N 2 N T 2, Φ 23 = N T 3, Φ 24 = h(a T d P 12 P 22 ), Φ 33 = R, H = R + hz. Proof. Choose the Lyapunov-Krasovskii functional candidate to be V (x t )=(Dx t ) T P 11 (Dx t )+2(Dx t ) T P 12 t h x(s)ds [ T + x(s)ds] P 22 x(s)ds t h t h + x T (s)qx(s)ds + ẋ T (s)rẋ(s)ds t h t h + 0 h t+θ ẋ T (s)zẋ(s)dsdθ + 0 h t+θ x T (s)wx(s)dsdθ,

124 5.2 Neutral Systems with Identical Discrete and Neutral Delays 107 where P = P 11 P 12 0(withP 11 > 0), Q 0, R 0, Z 0, and P 22 W>0areto be determined. It is clear that where α 1 Dx t 2 V (x t ) α 2 x t 2 c1, x t c1 = sup h θ 0 { x(t + θ), ẋ(t + θ) }, α 1 = λ min (P ), α 2 = λ max (P )(1+ C + h)+h {λ max (Q)+λ max (R)} h2 {λ max (Z)+λ max (W )}. From the Newton-Leibnitz formula, we have the following for any appropriately dimensioned matrices N i,i=1, 2, 3: 2 [ x T (t)n 1 +x T (t h)n 2 +ẋ T ] [ ] (t h)n 3 x(t) ẋ(s)ds x(t h) =0. t h (5.32) Calculating the derivative of V (x t ) along the solutions of system (5.22) and using (5.32) yield V (x t )=2[x(t) Cx(t h)] T P 11 [Ax(t)+A d x(t h)] where +2 [Ax(t)+A d x(t h)] T P 12 t h x(s)ds +2[x(t) Cx(t h)] T P 12 [x(t) x(t h)] +2[x(t) x(t h)] T P 22 x(s)ds+x T (t)qx(t) x T (t h)qx(t h) = 1 h t h +ẋ T (t)rẋ(t) ẋ T (t h)rẋ(t h)+hẋ T (t)zẋ(t) +hx T (t)wx(t) x T (s)wx(s)ds t h +2 [ x T (t)n 1 +x T (t h)n 2 +ẋ T (t h)n 3 ] [ x(t) t h t h t h ẋ T (s)zẋ(s)ds ] ẋ(s)ds x(t h) ζ T (t, s)ξζ(t, s)ds, (5.33) ζ(t, s) =[x T (t), x T (t h), ẋ T (t h), x T (s), ẋ T (s)] T,

125 Stability of Neutral Systems Φ 11 + A T HA Φ 12 + A T HA d Φ 13 + A T HC Φ 14 hn 1 Φ 22 + A T d HA d Φ 23 + A T d HC Φ 24 hn 2 Ξ = Φ 33 + C T HC 0 hn 3, hw 0 hz and H is defined in (5.31). If Ξ<0, which is equivalent to Ξ<0fromthe Schur complement, then V (x t ) ε x(t) 2 for a sufficiently small ε>0. In addition, operator Dx t is stable. Thus, system (5.22) is asymptotically stable if LMI (5.31) is true. This completes the proof. Remark The matrix P in Theorem can be chosen to be positive semi-definite. Setting P 12 =0,P 22 =0,andW = 0, turns Theorem into Corollary 5.2.1, which was obtained by directly using the FWM approach. So, we can get appropriate values for the elements of matrices P 12,P 22,and W by solving an LMI rather than by setting these matrices to zero. On the other hand, setting Z = 0 and N i following corollary. = 0,i = 1, 2, 3 yields the Corollary Consider nominal system (5.22). Given a scalar h>0, the system is asymptotically stable if the operator Dx t is stable and there exist matrices P = P 11 P 12 0(with P 11 > 0), Q 0, R 0, and W > 0 P 22 such that the following LMI holds: Φ 11 Φ12 0 Φ14 A T R Φ22 0 Φ24 A T d R Φ33 0 C T R < 0, (5.34) hw 0 R where Φ 11 = P 11 A + A T P 11 + P 12 + P T 12 + Q + hw, Φ 12 = P 11 A d A T P 11 C P T 12C P 12, Φ 14 = h(a T P 12 + P 22 ),

126 5.2 Neutral Systems with Identical Discrete and Neutral Delays 109 Φ 22 = Q + P T 12C + C T P 12 C T P 11 A d A T d P 11C, Φ 24 = h(a T d P 12 P 22 ), Φ 33 = R, Remark We can use a parameterized model transformation to derive Corollary by combining parameter matrices, such as P 12 and P 22,with a Lyapunov-Krasovskii functional. That enables the matrices to be obtained by solving an LMI. Theorem can be obtained from this corollary and the explanation in Remark by combining the FWM approach with a parameterized model transformation. Theorem can also be extended to system (5.23), which has timevarying structured uncertainties, as stated in the following theorem. Theorem Consider neutral system (5.23). Given a scalar h>0, the system is robustly stable if the operator Dx t is stable and there exist matrices P = P 11 P 12 0(with P 11 > 0), Q 0, R 0, Z 0, and W>0, any P 22 appropriately dimensioned matrices N i,i=1, 2, 3, andascalarλ>0 such that the following LMI holds: Φ 11 + λea TE a Φ 12 + λea TE ad Φ 13 Φ 14 hn 1 A T H P 11 D Φ 22 + λead T E ad Φ 23 Φ 24 hn 2 A T d H CT P 11 D Φ 33 0 hn 3 C T H 0 hw 0 0 hp12 T D <0, hz 0 0 H HD λi (5.35) where Φ 11,Φ 12,Φ 13,Φ 14,Φ 22,Φ 23,Φ 24,Φ 33, and H are defined in (5.31) FWM Approach in Combination with Augmented Lyapunov-Krasovskii Functional At present, it is difficult to further reduce the conservativeness by using a general type of Lyapunov-Krasovskii functional.

127 Stability of Neutral Systems This subsection describes an augmented Lyapunov-Krasovskii functional that takes the delay into account through augmentation of the terms of the general Lyapunov-Krasovskii functional. This functional in combination with the FWM approach yields an improved delay-dependent stability criterion for neutral system (5.22). It can be extended to systems with time-varying structured uncertainties and polytopic-type uncertainties although we do not do so here for brevity. An augmented Lyapunov-Krasovskii functional can be used for various types of time-delay systems; interested readers are referred to [27, 29, 30]. Theorem Consider nominal system (5.22). Given a scalar h>0, the system is asymptotically stable if the operator Dx t is stable and there exist L 11 L 12 L 13 matrices L = L 22 L 23 0(with L 11 > 0), Q= Q 11 Q 12 Q 22 L 33 0, and Z = Z 11 Z 12 0, and any appropriately dimensioned matrices Z 22 M i,i=1, 2, 3 such that the following LMI holds: Γ 11 Γ 12 Γ 13 Γ 14 hm 1 A T S Γ 22 Γ 23 Γ 24 hm 2 A T d S Γ Γ = 33 Γ 34 hm 3 C T S < 0, (5.36) hz 11 hz 12 0 hz 22 0 S where Γ 11 = GA + A T G T + L 13 + L T 13 + Q 11 + hz 11 + M 1 + M T 1, Γ 12 = GB + A T L 12 + L T 23 L 13 + M T 2 M 1, Γ 13 = GC + L 12 + M T 3, Γ 14 = h(l 33 + A T L 13 ), Γ 22 = A T d L 12 + L T 12 A d L 23 L T 23 Q 11 M 2 M T 2, Γ 23 = L T 12C + L 22 Q 12 M T 3, Γ 24 = h( L 33 + A T d L 13), Γ 33 = Q 22,

128 5.2 Neutral Systems with Identical Discrete and Neutral Delays 111 Γ 34 = h(l 23 + C T L 13 ), S = Q 22 + hz 22, G = L 11 + Q 12 + hz 12. Proof. Choose the Lyapunov-Krasovskii functional candidate to be 0 V (x t )=ζ1 T (t)lζ 1 (t)+ ζ2 T (s)qζ 2 (s)ds + ζ2 T (s)zζ 2 (s)dsdθ, t h h t+θ (5.37) L 11 L 12 L 13 where L = L 22 L 23 0(withL 11 > 0), Q = Q 11 Q 12 0, Q 22 L 33 and Z = Z x(t) 11 Z 12 0 are to be determined; ζ 1 (t) = x(t h) ζ 2 (t) = Z 22 t h x(s)ds ;and [ x T (t), ẋ T (t) ] T. From the Newton-Leibnitz formula, the following equation is true for any appropriately dimensioned matrices M i,i=1, 2, 3: 2 [ ] [ ] x T (t)m 1 + x T (t h)m 2 +ẋ T (t h)m 3 x(t) ẋ(s)ds x(t h) =0. t h (5.38) Calculating the derivative of V (x t ) along the solutions of system (5.22) and using (5.38) yield V (x t )=2ζ T 1 (t)l ζ 1 (t)+ζ T 2 (t)qζ 2 (t) ζ T 2 (t h)qζ 2 (t h) +hζ T 2 (t)zζ 2(t) t h ζ T 2 (s)zζ 2(s)ds [ ] T =2ζ1 T (t)l ẋ T (t) ẋ T (t h) x T (t) x T (t h) + ζ T 2 (t)qζ 2 (t) ζ T 2 (t h)qζ 2 (t h)+hζ T 2 (t)zζ 2 (t) t h +2 [ x T (t)m 1 + x T (t h)m 2 +ẋ T ] (t h)m 3 [ ] x(t) ẋ(s)ds x(t h) = 1 h t h t h η T 1 (t, s) ˆΓη 1 (t, s)ds, ζ T 2 (s)zζ 2 (s)ds

129 Stability of Neutral Systems where η 1 (t, s) =[x T (t), x T (t h), ẋ T (t h), x T (s), ẋ T (s)] T, Γ 11 + A T SA Γ 12 + A T SA d Γ 13 + A T SC Γ 14 hm 1 Γ 22 + A T d SA d Γ 23 + A T d SC Γ 24 hm 2 ˆΓ = Γ 33 + C T SC Γ 34 hm 3, hz 11 hz 12 hz 22 S = Q 22 + hz 22. From the Schur complement, we find that Γ<0isequivalentto ˆΓ <0, which means that V (x t ) ε x(t) 2 for a sufficiently small ε>0. Therefore, nominal system (5.22) is asymptotically stable. This completes the proof. Theorem was established by using the Newton-Leibnitz formula and the FWMs M i,i=1, 2, 3 (see (5.38)). Below, we derive an alternative delaydependent criterion by retaining the term ẋ(t) and employing another set of FWMs to express the relationships among the terms of system equation (5.22). Theorem Consider nominal system (5.22). Given a scalar h>0, the system is asymptotically stable if the operator Dx t is stable and there exist L 11 L 12 L 13 matrices L = L 22 L 23 0(with L 11 > 0), Q= Q 11 Q 12 0, Q 22 L 33 Z = Z 11 Z 12 0, and any appropriately dimensioned matrices U, M i, Z 22 i =1, 2, 3, and T j,j=1, 2,, 6 such that the following LMI holds: Φ 11 Φ 12 Φ 13 Φ 14 Φ 15 Φ 16 Φ 22 Φ 23 Φ 24 Φ 25 Φ 26 Φ Φ = 33 Φ 34 Φ 35 Φ 36 < 0, (5.39) Φ 44 Φ 45 Φ 46 hz 11 hz 12 hz 22

130 5.2 Neutral Systems with Identical Discrete and Neutral Delays 113 where Φ 11 = L 13 + L T 13 + Q 11 + hz 11 + M 1 + M T 1 T 1A A T T T 1, Φ 12 = L 11 + Q 12 + hz 12 + U T + T 1 A T T T 2, Φ 13 = L T 23 L 13 + M T 2 M 1 T 1 A d A T T T 3, Φ 14 = L 12 + M T 3 T 1C A T T T 4, Φ 15 = hl 33 A T T T 5, Φ 16 = hm 1 A T T T 6, Φ 22 = Q 22 + hz 22 + T 2 + T T 2, Φ 23 = L 12 U + T T 3 T 2 A d, Φ 24 = T 2 C + T T 4, Φ 25 = hl 13 + T T 5, Φ 26 = hu + T T 6, Φ 33 = L 23 L T 23 Q 11 M 2 M T 2 T 3A d A T d T T 3, Φ 34 = L 22 Q 12 M T 3 T 3 C A T d T T 4, Φ 35 = hl 33 A T d T T 5, Φ 36 = hm 2 A T d T T 6, Φ 44 = Q 22 T 4 C C T T T 4, Φ 45 = hl 23 C T T T 5, Φ 46 = hm 3 C T T T 6. Proof. Choose the same Lyapunov-Krasovskii functional candidate as in (5.37). From system equation (5.22), we know that 2 η2 T (t, s)t [ẋ(t) Cẋ(t h) Ax(t) A d x(t h)] ds =0, (5.40) t h where T = [ T1 T T2 T T3 T T4 T T5 T ] T, T6 T η 2 (t, s) =[x T (t), ẋ T (t), x T (t h), ẋ T (t h), x T (s), ẋ T (s)] T. On the other hand, ẋ(t) in V (x t ) is retained (Note that, in contrast to the proof of Theorem (5.2.5), ẋ(t) is replaced with system equation (5.22)), and (5.38) is slightly modified to 2 [ x T (t)m 1 +ẋ T (t)u + x T (t h)m 2 +ẋ T ] (t h)m 3 [ ] x(t) ẋ(s)ds x(t h) =0. (5.41) t h

131 Stability of Neutral Systems If we follow a line similar to the proof of Theorem 5.2.5, but add (5.40) and (5.41) to V (x t ), we get the desired result immediately. This completes the proof Numerical Examples The next two examples illustrate the effectiveness and advantages of the methods described above. Example Consider the stability of nominal system (5.22) with 0.9 A = 0.2, A d =, C = The allowable upper bound on the delay that guarantees the stability of the system is 0.3 in[7], in [8], and 0.74 in [12]. In contrast, solving LMIs (5.24) and (5.25) in Theorem yields a maximum upper bound of h =1.7855, which is about 451%, 192%, and 123% larger than the three values just mentioned. Example Consider the robust stability of system (5.23) with A = 2 0, A d = 1 0, C = c 0, 0 c<1, c D = I, E a = E ad = αi. [14] used a parameterized model transformation to solve this problem, but that method requires that coefficient matrix A d be artificially decomposed. (Although Han [16] devised an effective way of decomposing it, that method is still conservative because it requires that three matrices be the same.) In contrast, if we use either the FWM approach or the FWM approach in combination with a parameter model transformation, solving LMIs gives us all the parameter matrices. Table 5.2 shows the maximum delay that ensures the asymptotic stability of nominal system (5.22) for α = 0. The method in this section produces significantly better results than those in Han [14] and Fridman & Shaked [19], especially when c is large. The results also show that a parameterized matrix transformation (Corollary 5.2.2) is almost equivalent to Theorem but is conservative for c = 0; that is, the FWM approach in combination

132 5.2 Neutral Systems with Identical Discrete and Neutral Delays 115 Table 5.2. Allowable upper bound on h for α = 0 (Example 5.2.2) c [19] Corollary [14] Corollary Theorem Theorems and with a parameter model transformation (Theorem 5.2.3) is superior to a simple model transformation. Moreover, they also indicate that the FWM approach in combination with an augmented Lyapunov-Krasovskii functional (Theorems and 5.2.6) yields the best results. Table 5.3 shows the maximum delay that ensures the robust stability of a system with time-varying structured uncertainties for α =0.2 andvariousc. The results obtained with Theorem are much better than those obtained by the method in Han [14]. Table 5.3. Allowable upper bound on h for α =0.2 (Example 5.2.2) c [14] Theorem Table 5.4 shows how the uncertainty bound, α, affects the upper bound on h for c =0.1. Again, Theorem produces better results than the method in Han [14]. Table 5.4. Allowable upper bound on h for c =0.1 andvariousα (Example 5.2.2) α [14] Theorem

133 Stability of Neutral Systems 5.3 Neutral Systems with Different Discrete and Neutral Delays If the discrete delay, d(t), has a constant value of h, then system (5.1) turns into a system with different neutral and discrete delays: ẋ(t) Cẋ(t τ) =Ax(t)+A d x(t h), t > 0, (5.42) x(t) =φ(t), t [ r, 0]; and system (5.4) turns into ẋ(t) Cẋ(t τ) =(A +ΔA(t))x(t)+(A d +ΔA d (t))x(t h), t > 0, x(t) =φ(t), t [ r, 0], (5.43) where r =max{h, τ}. In addition, Dx t must be stable if systems (5.42) and (5.43) are to be stable [28] Nominal Systems First, we give a theorem for the nominal system that is based on the FWM approach. Theorem Consider nominal system (5.42). Given scalars h 0 and τ 0, the system is asymptotically stable if the operator Dx t is stable and there exist matrices P > 0, Q i 0, i=1, 2, R 0, W i > 0, i=1, 2, 3, X jj 0, Y jj 0, and Z jj 0, j=1, 2,, 4, and any appropriately dimensioned matrices N i,s i,m i,x ij,y ij, and Z ij,i=1, 2,, 4, i<j 4 such that the following LMIs hold: Φ 11 Φ 12 Φ 13 Φ 14 A T H Φ 22 Φ 23 Φ 24 A T d H Φ = Φ 33 Φ 34 0 < 0, (5.44) Φ 44 C T H H

134 5.3 Neutral Systems with Different Discrete and Neutral Delays 117 X 11 X 12 X 13 X 14 N 1 X 22 X 23 X 24 N 2 Ψ 1 = X 33 X 34 N 3 0, (5.45) X 44 N 4 W 1 Y 11 Y 12 Y 13 Y 14 S 1 Y 22 Y 23 Y 24 S 2 Ψ 2 = Y 33 Y 34 S 3 0, (5.46) Y 44 S 4 W 2 Z 11 Z 12 Z 13 Z 14 km 1 Z 22 Z 23 Z 24 km 2 Ψ 3 = Z 33 Z 34 km 3 0, (5.47) Z 44 km 4 W 3 where 1, if h τ, k = 1, if h<τ, and Φ 11 = PA+ A T P + Q 1 + Q 2 + N 1 + N T 1 + S 1 + S T 1 + Ξ 11, Φ 12 = PA d N 1 + N T 2 + ST 2 M 1 + Ξ 12, Φ 13 = A T PC + N T 3 + S T 3 S 1 + M 1 + Ξ 13, Φ 14 = N T 4 + ST 4 + Ξ 14, Φ 22 = Q 1 N 2 N T 2 M 2 M T 2 + Ξ 22, Φ 23 = A T d PC N T 3 S 2 + M 2 M T 3 + Ξ 23, Φ 24 = N T 4 M T 4 + Ξ 24, Φ 33 = Q 2 S 3 S T 3 + M 3 + M T 3 + Ξ 33, Φ 34 = S T 4 + M T 4 + Ξ 34, Φ 44 = R + Ξ 44, H = R + hw 1 + τw 2 + τ h W 3,

135 Stability of Neutral Systems Ξ ij = hx ij + τy ij + τ h Z ij, i =1, 2,, 4, i j 4. Proof. First, consider the case h τ. Choose the Lyapunov-Krasovskii functional candidate to be V (x t )=(Dx t ) T P (Dx t )+ x T (s)q 1 x(s)ds + x T (s)q 2 x(s)ds + + t τ 0 τ t h ẋ T (s)rẋ(s)ds + t+θ 0 h ẋ T (s)w 2 ẋ(s)dsdθ + t+θ τ t τ ẋ T (s)w 1 ẋ(s)dsdθ h t+θ ẋ T (s)w 3 ẋ(s)dsdθ, where P > 0, Q i 0, i=1, 2, R 0, and W i > 0, i=1, 2, 3aretobe determined. Calculating the derivative of V (x t ) along the solutions of system (5.42) yields V (x t )=2(Dx t ) T P [Ax(t)+A d x(t h)]+x T (t)q 1 x(t) x T (t h)q 1 x(t h) +x T (t)q 2 x(t) x T (t τ)q 2 x(t τ)+ẋ T (t)rẋ(t) ẋ T (t τ)rẋ(t τ) +hẋ T (t)w 1 ẋ(t) +τẋ T (t)w 2 ẋ(t) t h t τ +(h τ)ẋ T (t)w 3 ẋ(t) ẋ T (s)w 1 ẋ(s)ds ẋ T (s)w 2 ẋ(s)ds τ t h ẋ T (s)w 3 ẋ(s)ds. From the Newton-Leibnitz formula, the following equations hold for any appropriately dimensioned matrices N i, S i,andm i, i =1, 2,, 4: 2 [ x T (t)n 1 + x T (t h)n 2 + x T (t τ)n 3 +ẋ T ] (t τ)n 4 [ x(t) x(t h) t h ] ẋ(s)ds =0, (5.48) 2 [ x T (t)s 1 + x T (t h)s 2 + x T (t τ)s 3 +ẋ T ] (t τ)s 4 [ ] x(t) x(t τ) ẋ(s)ds =0, (5.49) 2 [ x T (t)m 1 + x T (t h)m 2 + x T (t τ)m 3 +ẋ T ] (t τ)m 4 [ τ ] x(t τ) x(t h) ẋ(s)ds =0. (5.50) On the other hand, the following is also true for any matrices X jj 0, Y jj 0, and Z jj 0, j =1, 2,, 4, and any appropriately dimensioned matrices X ij, Y ij,andz ij,i=1, 2,, 4, i<j 4: t τ t h

136 where 5.3 Neutral Systems with Different Discrete and Neutral Delays 119 x(t) x(t h) x(t τ) ẋ(t τ) Λ 11 Λ 12 Λ 13 Λ 14 x(t) Λ 22 Λ 23 Λ 24 x(t h) =0, (5.51) Λ 33 Λ 34 x(t τ) Λ 44 ẋ(t τ) T Λ ij = h(x ij X ij )+τ(y ij Y ij )+(h τ)(z ij Z ij ),i=1, 2,, 4, i j 4. Now, adding the terms on the left sides of equations (5.48)-(5.51) to V (x t ), we get where V (x t )=η T 1 (t)ωη 1 (t) t h η2 T (t, s)ψ 2 η 2 (t, s)ds t τ η T 2 (t, s)ψ 1 η 2 (t, s)ds τ t h η T 2 (t, s)ψ 3 η 2 (t, s)ds, (5.52) η 1 (t) =[x T (t), x T (t h), x T (t τ), ẋ T (t τ)] T, η 2 (t, s) =[η1 T(t), ẋt (s)] T, Φ 11 + A T HA Φ 12 + A T HA d Φ 13 Φ 14 + A T HC Φ Ω = 22 + A T d HA d Φ 23 Φ 24 + A T d HC. Φ 33 Φ 34 Φ 44 + C T HC If Ω<0andΨ i 0, i=1, 2, 3, then V (x t ) < ε x(t) 2 for a sufficiently small scalar ε>0. From the Schur complement, we find that Φ<0 implies Ω<0. Thus, system (5.42) is asymptotically stable if LMIs (5.44)-(5.47) are feasible. Next, consider the case h<τ. In this case, the candidate Lyapunov- Krasovskii functional is chosen to be V (x t )=(Dx t ) T P (Dx t )+ + + t τ 0 τ t h ẋ T (s)rẋ(s)ds + t+θ and (5.50) can be rewritten as x T (s)q 1 x(s)ds + 0 h ẋ T (s)w 2 ẋ(s)dsdθ + t+θ h t τ ẋ T (s)w 1 ẋ(s)dsdθ τ t+θ x T (s)q 2 x(s)ds ẋ T (s)w 3 ẋ(s)dsdθ;

137 Stability of Neutral Systems 2 [ x T (t)m 1 + x T (t h)m 2 + x T (t τ)m 3 +ẋ T ] (t τ)m 4 [ ] x(t τ) x(t h)+ h t τ ẋ(s)ds =0. (5.53) Then, following the same procedure as for h τ yields the same conclusion. Note that, in this case, k in (5.47) is 1. This completes the proof Equivalence Analysis Now we consider the special case of identical delays (τ = h) in system (5.42). This turns system (5.42) into system (5.22), for which we have already used the FWM approach to obtain a delay-dependent stability criterion, namely Theorem Theorem should be equivalent to that theorem for τ = h. This point is discussed below. If the third row and third column of (5.44) are added to the second row and second column, respectively, (5.44) is equivalent to the following LMI: Φ 11 Π 12 Φ 13 Φ 14 A T H Π 22 Π 23 Π 24 A T d H Π = Φ 33 Φ 34 0 < 0, (5.54) Φ 44 C T H H where Π 12 = PA d A T PC + N T 2 + N T 3 N 1 + S T 2 + ST 3 S 1 + Ξ 12 + Ξ 13, Π 22 = C T PA d A T d PC (Q 1 + Q 2 ) N 3 N T 3 S 3 S T 3 N 2 N T 2 S 2 S T 2 + Ξ 22 + Ξ 23 + Ξ T 23 + Ξ 33, Π 23 = Q 2 A T d PC S 3 S T 3 + M 3 N T 3 S 2 + M 2 + Ξ 23 + Ξ 33, Π 24 = N T 4 ST 4 + Ξ 24 + Ξ 34, and Φ 11, Φ 13, Φ 14, Φ 33, Φ 34, Φ 44, Ξ 12, Ξ 13, Ξ 22, Ξ 23, Ξ 33, Ξ 24, Ξ 34,andH are defined in (5.44). First, we show that, if LMIs (5.24) and (5.25) in Theorem are feasible, then the solutions can be written as appropriate forms of the feasible solutions of LMIs (5.44)-(5.47). In fact, for the feasible solutions of LMIs (5.24) and (5.25) in Theorem 5.2.1, if we set P = P, R = R, S i =0,i=1, 2,, 4, N 1 = N 1, N 2 = N 2, N 3 =0,N 4 = N 3,0<Q 2 < Q, Q 1 = Q Q 2, M 1 = A T PC, M2 = A T d PC + Q 2, M 3 = 0, M 4 = 0,

138 5.3 Neutral Systems with Different Discrete and Neutral Delays 121 W 1 = W, W 2 = 0, X 11 = X 11, X 12 = X 12, X 13 = 0, X 14 = X 13, X 22 = X 22, X 23 =0,X 24 = X 23, X 33 =0,X 34 =0,X 44 = X 33,and Y ij =0,i=1, 2,, 4, i j 4; and if we let Z ij,i=1, 2,, 4, i j 4 and W 3 be the feasible solutions of the following LMI for a given M 1 and M 2 : Z 11 Z 12 Z 13 Z 14 M 1 Z 22 Z 23 Z 24 M 2 Z 33 Z , Z 44 0 W 3 then the above matrices must be the feasible solutions of LMIs (5.44)-(5.47). Therefore, Theorem includes Theorem for τ = h. Next, we show that, if LMIs (5.44)-(5.47) are feasible, then the solutions are feasible solutions of LMIs (5.24) and (5.25). That is, for the feasible solutions of LMIs (5.44)-(5.47), setting P = P, R = R, Q = Q1 + Q 2, W = W 1 +W 2, N1 = N 1 +S 1, N2 = N 2 +N 3 +S 2 +S 3, N3 = N 4, X11 = X 11 +Y 11, X 12 = X 12 +Y 12 +X 13 +Y 13, X 13 = X 14 +Y 14, X 22 = X 22 +Y 22 +X 23 +Y 23 + X23 T +Y 23 T+X 33+Y 33, X 23 = X 24 +Y 24 +X 34 +Y 34,and X 33 = X 44 +Y 44 yields the feasible solutions of LMIs (5.24) and (5.25) in Theorem Therefore, Theorem includes Theorem for τ = h. Thus, Theorems and are equivalent for the case τ = h. Remark Better results can be obtained by using the FWM approach in combination with either a parameterized model transformation or an augmented Lyapunov-Krasovskii functional. For brevity, we do not give the details here Systems with Time-Varying Structured Uncertainties The next theorem extends Theorem to a neutral system with timevarying structured uncertainties. Theorem Consider neutral system (5.43). Given scalars τ 0 and h 0, the system is robustly stable if the operator Dx t is stable and there exist matrices P>0, Q i 0, i=1, 2, R 0, W i > 0, i=1, 2, 3, X jj 0, Y jj 0, and Z jj 0, j=1, 2,, 4, any appropriately dimensioned matrices

139 Stability of Neutral Systems N i,s i,m i,i=1, 2,, 4, X ij,y ij, and Z ij,i=1, 2,, 4, i<j 4, and ascalarλ>0 such that LMIs (5.45)-(5.47) and the following LMI hold: Φ 11 + λea T E a Φ 12 + λea T E ad Φ 13 Φ 14 A T H PD Φ 22 + λead T E ad Φ 23 Φ 24 A T d H 0 Φ 33 Φ 34 0 C T PD Φ 44 C T < 0, (5.55) H 0 H HD λi where Φ ij,i=1, 2,, 4, i j 4 and H are defined in (5.44). Since the criteria obtained in this section depend not only on the length of the discrete delay but also on that of the neutral delay, they are discretedelay- and neutral-delay-dependent conditions Numerical Example Example Consider the robust stability of system (5.43) with A = 2 0, A d = 1 0, C = c 0, 0 c<1, c D = I, E a = E ad =0.2I. Table 5.5. Allowable upper bound on h for different c c [14] (τ = h) Theorem Theorem (τ = h) Theorem (τ = 0.1) Table 5.5 compares Theorems and in this chapter with the method in [14]. The values in the table are the upper bound on the delay, h. Note that the values in the third and fourth rows are the same, which reflects the equivalence of Theorems and for τ = h. Theresults are better for τ =0.1 thanforτ = h. That shows that reducing the neutral delay, τ, increases the upper bound on the discrete delay, h, whichmeans that discrete-delay- and neutral-delay-dependent criteria are very important.

140 References Conclusion In this chapter, the FWM approach is first used to derive discrete-delaydependent and neutral-delay-independent stability criteria for neutral systems with a time-varying discrete-delay. Next, delay-dependent stability criteria for neutral systems with identical discrete and neutral delays are derived by the FWM approach and by that approach in combination with either a parameterized model transformation or an augmented Lyapunov-Krasovskii functional. Finally, the FWM approach is used to derive discrete-delay- and neutral-delay-dependent stability criteria for neutral systems with different discrete and neutral delays. Moreover, it is shown that, if we make the two delays the same in this criterion, the result is equivalent to the one obtained by using the FWM approach to directly handle identical discrete and neutral delays. References 1. S. Hara, Y. Yamamoto, T. Omata, and M. Nakano. Repetitive control system: a new type servo system for periodic exogenous signals. IEEE Transactions on Automatic Control, 33(7): , G. D. Hu and G. D. Hu. Some simple stability criteria of neutral delaydifferential systems. Applied Mathematics and Computation, 80(2-3): , G. D. Hui and G. D. Hu. Simple criteria for stability of neutral systems with multiple delays. International Journal of Systems Science, 28(12): , M. S. Mahmoud. Robust H control of linear neutral systems. Automatica, 36(5): , W. H. Chen and W. X. Zheng. Delay-dependent robust stabilization for uncertain neutral systems with distributed delays. Automatica, 43(1): , C. H. Lien. New stability criterion for a class of uncertain nonlinear neutral time-delay systems. International Journal of Systems Science, 32(2): , C. H. Lien, K. W. Yu, and J. G. Hsieh. Stability conditions for a class of neutral systems with multiple delays. Journal of Mathematical Analysis and Applications, 245(1): 20-27, J.D.Chen,C.H.Lien,K.K.Fan,andJ.H.Chou.Criteriaforasymptotic stability of a class of neutral systems via an LMI approach. IEE Proceedings Control Theory and Application, 148(6): , 2001.

141 Stability of Neutral Systems 9. J. H. Park. A new delay-dependent criterion for neutral systems with multiple delays. Journal of Computational and Applied Mathematics, 136(1-2): , S. I. Niculescu. On delay-dependent stability under model transformations of some neutral linear systems. International Journal of Control, 74(6): , S. I. Niculescu. Optimizing model transformations in delay-dependent analysis of neutral systems: A control-based approach. Nonlinear Analysis, 47(8): , E. Fridman. New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems. Systems & Control Letters, 43(4): , Q. L. Han. On delay-dependent stability for neutral delay-differential systems. International Journal of Applied Mathematics and Computer Science, 11(4): , Q. L. Han. Robust stability of uncertain delay-differential systems of neutral type. Automatica, 38(4): , J. H. Park. Stability criterion for neutral differential systems with mixed multiple time-varying delay arguments. Mathematics and Computers in Simulation, 59(5): , Q. L. Han. Stability criteria for a class of linear neutral systems with timevarying discrete and distributed delays. IMA Journal of Mathematical Control and Information, 20(4): , D. Ivănescu, S. I. Niculescu, L. Dugard, J. M. Dionc, and E. I. Verriestd. On delay-dependent stability for linear neutral systems. Automatica, 39(2): , E. Fridman and U. Shaked. A descriptor system approach to H control of linear time-delay systems. IEEE Transactions on Automatic Control, 47(2): , E. Fridman and U. Shaked. Delay-dependent stability and H control: constant and time-varying delays. International Journal of Control, 76(1): 48-60, Q. L. Han. Robust stability for a class of linear systems with time-varying delay and nonlinear perturbations. Computers & Mathematics with Applications, 47(8-9): , Q. L. Han and L. Yu. Robust stability of linear neutral systems with nonlinear parameter perturbations. IEE Proceedings Control Theory & Applications, 151(5): , Q. L. Han. A descriptor system approach to robust stability of uncertain neutral systems with discrete and distributed delays. Automatica, 40(10): , Y. He, M. Wu, J. H. She, and G. P. Liu. Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays. Systems & Control Letters, 51(1): 57-65, 2004.

142 References Y. He, M. Wu, and J. H. She. Delay-dependent robust stability and stabilization of uncertain neutral systems. Asian Journal of Control, 10(3): , Y. He and M. Wu. Delay-dependent robust stability for neutral systems with mixed discrete- and neutral-delays. Journal of Control Theory and Applications, 2(4): , M. Wu, Y. He, and J. H. She. New delay-dependent stability criteria and stabilizing method for neutral systems. IEEE Transactions on Automatic Control, 49(12): , Y. He, Q. G. Wang, C. Lin, and M. Wu. Augmented Lyapunov functional and delay-dependent stability criteria for neutral systems. International Journal of Robust and Nonlinear Control, 15(18): , J. K. Hale and S. M. Verduyn Lunel. Introduction to Functional Differential Equations. New York: Springer-Verlag, Y. He, Q. G. Wang, L. Xie, and C. Lin. Further improvement of free-weighting matrices technique for systems with time-varying delay. IEEE Transactions on Automatic Control, 52(2): , Y. He, G. P. Liu, and D. Rees. Augmented Lyapunov functional for the calculation of stability interval for time-varying delay. IET Proceedings: Control Theory & Applications, 1(1): , 2007.

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144 6. Stabilization of Systems with Time-Varying Delay At present, there is no effective controller synthesis algorithm for solving delay-dependent stabilization problems, even for the simple situation of statefeedback; for output feedback, the problem is even more difficult. It is possible to use model transformations I and II to derive an LMI-based controller synthesis algorithm. However, as mentioned in [1,2], they add eigenvalues to the system, with the result that the transformed system is not equivalent to the original one. Thus, they have been abandoned in favor of model transformations III and IV, for which an NLMI is used to design a controller in synthesis problems. Two methods are available to solve the NLMI. One is an iterative algorithm devised by Moon et al. [3]. It has two steps: The original nonconvex problem is first reduced to an LMI-based nonlinear minimization problem; and then the CCL algorithm is used to obtain a suboptimal solution. [3] used this method to deal with a robust stabilization problem, and [4,5] used it for an H control problem. The controller obtained by this method has a small gain and is easy to implement, but the drawback is that the solution is suboptimal. There is still room for further investigation of the CCL algorithm in [3 5]. For instance, the iteration stop condition is very strict; and the gain matrix and some derived Lyapunov matrices must satisfy one or more matrix inequalities. However, once the gain matrix is derived, the delay-dependent stabilization conditions reduce to LMI ones, which means that the iteration can actually be stopped when the LMIs for that gain matrix are feasible. Moreover, some Lyapunov matrices can be used as decision variables rather than as fixed matrices. The other method of solving the NLMI is a parameter-tuning method oftenusedbyfridmanet al. [6 9]. It transforms the NLMI into an LMI by setting one or more undetermined matrices in the NLMI to a specific form with a scalar parameter, and then tunes the parameter to obtain a controller. This method also produces a suboptimal solution, and the parameter needs

145 Stabilization of Systems with Time-Varying Delay to be continuously tuned based on experience. Although these two methods yield only suboptimal solutions, they are still the most effective methods now available. This chapter first explains how the two methods just mentioned can be used to extend the stability theorems in Chapter 3 to delay-dependent stabilization design. Then, an ICCL algorithm with a better stop condition is presented that gives a suboptimal solution when an iterative algorithm is used. The theorems obtained with model transformations III and IV are special cases of the ones derived by the FWM approach. Thus, a stabilization design method based on FWMs is less conservative than other methods [10,11]. Furthermore, an LMI-based controller synthesis algorithm based on delay-dependent and rate-independent stabilization is derived that has no parameter tuning or iterative processing. 6.1 Problem Formulation Consider the following nominal linear system with a time-varying delay: ẋ(t) =Ax(t)+A d x(t d(t)) + Bu(t), t > 0, (6.1) x(t) =φ(t), t [ h, 0], where x(t) R n is the state vector; u(t) R m is the control input; A, A d, and B are constant matrices with appropriate dimensions; the delay, d(t), is a time-varying continuous function; and the initial condition, φ(t), is a continuously differentiable initial function of t [ h, 0]. The delay is assumed to satisfy one or both of the following conditions: 0 d(t) h, (6.2) d(t) μ, (6.3) where h and μ are constants. Our objective in this chapter is to design a memoryless state-feedback controller with the following form to stabilize system (6.1): u(t) =Kx(t), (6.4) where K R m n is a constant gain matrix. Then, we extend the results for the nominal system to a system with time-varying structured uncertainties:

146 6.2 Iterative Nonlinear Minimization Algorithm 129 ẋ(t) =(A +ΔA(t))x(t)+(A d +ΔA d (t))x(t d(t)) +(B +ΔB(t))u(t), t > 0, x(t) =φ(t), t [ h, 0]. (6.5) The uncertainties are assumed to be of the form [ΔA(t) ΔA d (t) ΔB(t)] = DF(t)[E a E ad E b ], (6.6) where D, E a, E ad,ande b are constant matrices with appropriate dimensions; and F (t) is an unknown, real, and possibly time-varying matrix with Lebesgue measurable elements satisfying F T (t)f (t) I, t. (6.7) 6.2 Iterative Nonlinear Minimization Algorithm This section explains how to use an iterative nonlinear minimization algorithm to obtain the controller gain from NLMIs. This involves reducing the original nonconvex problem to an LMI-based nonlinear minimization problem and using the CCL algorithm to obtain a suboptimal solution. Moreover, an ICCL algorithm with a better iteration stop condition is presented that leads to less conservativeness. First, we give a theorem that follows from Theorem Theorem Consider nominal system (6.1) withadelay,d(t), that satisfies both (6.2) and (6.3). For given scalars h>0 and μ, if there exist matrices L>0, W 0, R>0, and Y = Y 11 Y 12 0, and any appropriately Y 22 dimensioned matrices M 1,M 2, and V such that the following matrix inequalities hold: Π 11 Π 12 h(la T + V T B T ) Π 22 hla T < 0, (6.8) d hr Y 11 Y 12 M 1 Y 22 M 2 0, (6.9) LR 1 L

147 Stabilization of Systems with Time-Varying Delay where Π 11 = LA T + AL + BV + V T B T + M 1 + M T 1 + W + hy 11, Π 12 = A d L M 1 + M T 2 + hy 12, Π 22 = M 2 M T 2 (1 μ)w + hy 22, then the system can be stabilized by control law (6.4), and the controller gain is K = VL 1. Proof. Applying memoryless state-feedback controller (6.4) to closedloop system (6.1) yields ẋ(t) =(A + BK)x(t)+A d x(t d(t)). (6.10) Now, we replace the A in Theorem with A + BK, pre- and postmultiply (3.9) by diag {P 1,P 1,Z 1 }, pre- and post-multiply (3.10) by diag {P 1,P 1,P 1 }, and make the following assignments: L = P 1, W = P 1 QP 1, Y =diag{p 1,P 1 } X diag {P 1,P 1 }, R = Z 1, M 1 = P 1 N 1 P 1, M 2 = P 1 N 2 P 1, V = KP 1. These operations result in (6.8) and (6.9). This completes the proof. Since the conditions in Theorem are no longer LMIs because of the term LR 1 L in (6.9), we cannot use a convex optimization algorithm to find an appropriate controller gain. However, as mentioned in [3], we can use the method in [12], which involves solving a cone complementarity problem. First, we convert the problem into a nonlinear minimization problem. Define a new variable, S, forwhichlr 1 L S; and replace (6.9) with Y 11 Y 12 M 1 Y 22 M 2 0 (6.11) S and LR 1 L S. (6.12) Inequality (6.12) is equivalent to L 1 RL 1 S 1, which the Schur complement allows us to write as S 1 L 1 R 1 0. (6.13)

148 6.2 Iterative Nonlinear Minimization Algorithm 131 We introduce the new variables J, U, andh so that we can write the original condition (6.9) as (6.11) and U J 0, J = L 1, U = S 1, H = R 1. (6.14) H Thus, the problem is converted into the following LMI-based nonlinear minimization problem: Minimize subject to Tr{LJ + SU + RH} (6.8) and Y 11 Y 12 M 1 L>0, S > 0, Y 22 M 2 0, U S L I 0, S I 0, R I 0. J U H J 0, H (6.15) If the solution to this problem is 3n, thatis,iftr{lj + SU + RH} =3n, then from Theorem closed-loop system (6.10) is asymptotically stable. Although it is still impossible to always find the global optimal solution, this nonlinear minimization problem is easier to solve than the original non-convex feasibility problem. Actually, we can easily find a suboptimal maximal delay by using the linearization method in [12] and the following CCL algorithm. Algorithm To maximize h: Step 1: Choose a sufficiently small initial h > 0 such that there exists a feasible solution to (6.8) and (6.15). Set h max = h. Step 2: Find a feasible set (L 0,J 0,W 0,R 0,H 0,Y 0,M 10,M 20,V 0,S 0,U 0 ) satisfying (6.8) and (6.15). Set k =0. Step 3: Solve the following LMI problem for the variables L, J, W, R, H, Y,M 1,M 2,V,S,and U: Minimize Tr{LJ k + L k J + SU k + S k U + RH k + R k H} subject to (6.8) and (6.15). Set L k+1 = L, J k+1 = J, S k+1 = S, U k+1 = U, R k+1 = R, and H k+1 = H.

149 Stabilization of Systems with Time-Varying Delay Step 4: If (6.9) is satisfied, then set h max = h, increase h, and return to Step 2.If it is not satisfied within a specified number of iterations, then exit. Otherwise, set k = k +1 andgotostep3. This algorithm can find a suboptimal maximum h for which the controller u(t) =VL 1 x(t) stabilizes system (6.1). However, the iteration stop condition is very strict. In addition, the gain matrix and some derived Lyapunov matrices must satisfy one or more matrix inequalities, which makes the iteration process very long. So, there is still room for investigation to reduce the number of iterations. In fact, [13] presents an ICCL algorithm with a better stop condition that does just that. To make the description of this algorithm easier, we first give a corollary obtained by replacing the A in Theorem with A + BK. Corollary Consider nominal system (6.1) with a delay, d(t), that satisfies both (6.2) and (6.3). For given scalars h>0 and μ, if there exist matrices P>0, Q 0, Z>0, and X = X 11 X 12 0, and any appropriately X 22 dimensioned matrices N 1 and N 2 such that the following matrix inequalities hold: Φ 11 Φ 12 ha T k Z Φ = Φ 22 ha T d Z < 0, (6.16) hz where X 11 X 12 N 1 Ψ = X 22 N 2 0, (6.17) Z Φ 11 = PA k + A T k P + N 1 + N T 1 + Q + hx 11, Φ 12 = PA d N 1 + N T 2 + hx 12, Φ 22 = N 2 N T 2 (1 μ)q + hx 22, A k = A + BK, then the system can be stabilized by control law (6.4). Clearly, once the controller gain, K, is derived, the conditions in this corollary become LMI conditions; and the iteration stop condition can be

150 6.2 Iterative Nonlinear Minimization Algorithm 133 modified to include a determination of whether or not LMIs (6.16) and (6.17) are feasible for decision variables P, Q, Z, X, N 1,andN 2. Algorithm To maximize h: Step 1: Choose a sufficiently small initial h > 0 such that there exists a feasible solution to (6.8) and (6.15). Set h max = h. Step 2: Find a feasible set (L 0,J 0,W 0,R 0,H 0,Y 0,M 10,M 20,V 0,S 0,U 0 ) satisfying (6.8) and (6.15). Set k =0. Step 3: Solve the following LMI problem for the variables L, J, W, R, H, Y, M 1,M 2,V,S,U,and K: Minimize Tr{LJ k + L k J + SU k + S k U + RH k + R k H} subject to (6.8) and (6.15). Set L k+1 = L, J k+1 = J, S k+1 = S, U k+1 = U, R k+1 = R, and H k+1 = H. Step 4: For the K obtained in Step 3, if LMIs (6.16) and (6.17) are feasible for the variables P, Q, Z, X, N 1, and N 2, then set h max = h, increase h, andreturntostep2. If they are not satisfied within a specified number of iterations, then exit. Otherwise, set k = k +1 and go to Step 3. Remark Note that the stop condition at the beginning of Step 4 is different from the one in Algorithm If we follow the idea in Algorithm 6.2.1, the stop condition could be that matrix inequality (6.9) holds, which means that matrix inequalities (6.8) and (6.9) should be true for given L, W, R, Y, M 1, M 2,andV. However, once the gain matrix, K, is obtained, the stop condition reduces to the question of the feasibility of LMIs (6.16) and (6.17) for the decision variables P, Q, Z, X, N 1,andN 2 rather than the fixed matrices L, W, R, Y, M 1,andM 2. So, in Algorithm 6.2.2, the stop condition is modified to include a determination of whether or not LMIs (6.16) and (6.17) are feasible for the specified K, which provides more freedom in the selection of variables, such as P, Q, Z, X, N 1,andN 2. Now, we present a theorem derived from Theorem Theorem Consider nominal system (6.1) withadelay,d(t), that satisfies both (6.2) and (6.3). For given scalars h>0 and μ, if there exist matrices L > 0, W 0, R>0, and Y = Y Y 11 Y 12 Y Y 23 0, and any Y 33

151 Stabilization of Systems with Time-Varying Delay appropriately dimensioned matrices M i,i=1, 2, 3, S j,j=1, 2, and V such that the following matrix inequalities hold: Λ 11 Λ 12 Λ 13 hs1 T Λ Λ = 22 Λ 23 hs T 2 < 0, (6.18) Λ 33 0 hr where Y 11 Y 12 Y 13 M 1 Y 22 Y 23 M 2 0, (6.19) Y 33 M 3 LR 1 L Λ 11 = W + M 1 + M T 1 + S 1 + S T 1 + hy 11, Λ 12 =(AL + BV ) T S T 1 + S 2 + M T 2 + hy 12, Λ 13 = M T 3 M 1 + hy 13, Λ 22 = S 2 S T 2 + hy 22, Λ 23 = A d L M 2 + hy 23, Λ 33 = (1 μ)w M 3 M T 3 + hy 33, then the system can be stabilized by control law (6.4), and the controller gain is K = VL 1. Proof. Replace the A in (3.15) with A + BK. ThetermΓ 22 in Theorem is negative definite, which implies that T 2 + T2 T is also negative definite. Thus, T 2 is nonsingular. Define H = P 0, H 1 = H = L 0. T1 T T2 T S 1 S 2 Pre- and post-multiply Γ in (3.15) by diag { H T, L} and diag { H, L}, respectively; pre- and post-multiply Θ in (3.16) by diag { H T, L, L} and diag { H, L, L}, respectively; and set Y =diag{ H T,L} X diag { H, L}, W = LQL, R = Z 1, V = KL, [ ] T ] T M1 T M2 T M3 T =diag{,l} [ HT N1 T N2 T N3 T L.

152 6.2 Iterative Nonlinear Minimization Algorithm 135 Then, we can use the Schur complement to write (3.15) and (3.16) as (6.18) and (6.19), respectively. This completes the proof. The next two theorems extend Theorems and to system (6.5), which has time-varying structured uncertainties. Theorem Consider system (6.5) with a delay, d(t), that satisfies both (6.2) and (6.3). For given scalars h>0 and μ, if there exist matrices L>0, W 0, R>0, and Y = Y 11 Y 12 0, any appropriately dimensioned Y 22 matrices M 1,M 2, and V, andascalarλ>0 such that matrix inequality (6.9) and the following matrix inequality hold: Π 11 +λdd T Π 12 h(la T +V T B T +λdd T ) (E a L+E b V ) T Π 22 hla T d LE T ad hr + λh DD T < 0, (6.20) 0 λi where Π 11,Π 12, and Π 22 are defined in (6.8), then the system can be stabilized by control law (6.4), and the controller gain is K = VL 1. Theorem Consider system (6.5) withadelay, d(t), that satisfies both (6.2) and (6.3). For given scalars h>0 and μ, if there exist matrices L>0, Y 11 Y 12 Y 13 W 0, R>0, and Y = Y 22 Y 23 0, any appropriately dimensioned matrices M i,i=1, 2, S j,j=1, 2, 3, and V, andascalarλ>0 such Y 33 that matrix inequality (6.19) and the following matrix inequality hold: Λ 11 Λ 12 Λ 13 hs1 T (E a L + E b V ) T Λ 22 + λdd T Λ 23 hs T 2 0 Λ 33 0 LEad T < 0, (6.21) hr 0 λi where Λ ij,i=1, 2, 3, i j 3 are defined in (6.18), then the system can be stabilized by control law (6.4), and the controller gain is K = VL 1.

153 Stabilization of Systems with Time-Varying Delay The conditions in Theorems are no longer LMI conditions. Just as for Theorem 6.2.1, this problem can be transformed into an LMI-based nonlinear minimization problem; and the CCL or ICCL algorithm can be used to obtain a suboptimal maximum value for the bound h. Chapter 3 showed that the stability theorems derived using Moon et al. s inequalities or a descriptor model transformation are special cases of the ones in Chapter 3. So, we can conclude that Theorems , which are stabilization design methods derived from the stability theorems in Chapter 3, are more effective than other methods. The controller thus obtained has a small gain and is easy to implement; but the iterative algorithm takes a long time to finish. 6.3 Parameter-Tuning Method This section explains a parameter-tuning method that uses an NLMI to obtain the controller gain. If we put some of the matrices in the NLMI into a special form, such as the product of a scalar and a matrix, then we can transform the NLMI in Section 6.2 into an LMI with only one scalar that needs to be tuned. For example, by writing matrix R in Theorem as R = εl, we obtain a new corollary. Corollary Consider nominal system (6.1) with a delay, d(t), that satisfies both (6.2) and (6.3). For given scalars h>0 and μ, if there exist matrices L>0, W 0, R>0, and Y = Y 11 Y 12 0, any appropriately Y 22 dimensioned matrices M 1,M 2, and V, and a scalar ε>0 such that the following matrix inequalities hold: Π 11 Π 12 h(la T + V T B T ) Π 22 hla T < 0, (6.22) d hεl Y 11 Y 12 M 1 Y 22 M 2 0, (6.23) ε 1 L

154 6.3 Parameter-Tuning Method 137 where Π 11,Π 12, and Π 22 are defined in (6.8), then the system can be stabilized by control law (6.4), and the controller gain is K = VL 1. A similar treatment produces three corollaries from Theorems Corollary Consider nominal system (6.1) with a delay, d(t), that satisfies both (6.2) and (6.3). For given scalars h>0 and μ, if there exist matrices L>0, W 0, R>0, and Y = Y Y 11 Y 12 Y Y 23 0, any appropriately Y 33 dimensioned matrices M i,i=1, 2, 3, S j,j=1, 2, and V, andascalarε>0 such that the following matrix inequalities hold: Λ 11 Λ 12 Λ 13 hs1 T Λ 22 Λ 23 hs T 2 < 0, (6.24) Λ 33 0 hεl Y 11 Y 12 Y 13 M 1 Y 22 Y 23 M 2 0, (6.25) Y 33 M 3 ε 1 L where Λ ij,i=1, 2, 3, i j 3 are defined in (6.18), then the system can be stabilized by control law (6.4), and the controller gain is K = VL 1. Corollary Consider system (6.5) with a delay, d(t), that satisfies both (6.2) and (6.3). For given scalars h>0 and μ, if there exist matrices L>0, W 0, R>0, and Y = Y 11 Y 12 0, any appropriately dimensioned Y 22 matrices M 1,M 2, and V, andascalarε>0 such that matrix inequality (6.23) and the following matrix inequality hold: Π 11 +λdd T Π 12 h(la T +V T B T +λdd T ) (E a L+E b V ) T Π 22 hla T d LE T ad hεl + λh DD T < 0, (6.26) 0 λi

155 Stabilization of Systems with Time-Varying Delay where Π 11,Π 12, and Π 22 are defined in (6.8), then the system can be stabilized by control law (6.4), and the controller gain is K = VL 1. Corollary Consider system (6.5) with a delay, d(t), that satisfies both (6.2) and (6.3). For given scalars h>0 and μ, if there exist matrices L>0, Y 11 Y 12 Y 13 W 0, R>0, and Y = Y 22 Y 23 0, any appropriately dimensioned matrices M i,i=1, 2, 3, S j,j=1, 2, and V, and a scalar ε>0 such Y 33 that LMI (6.25) and the following matrix inequality hold: Λ 11 Λ 12 Λ 13 hs1 T (E a L + E b V ) T Λ 22 + λdd T Λ 23 hs T 2 0 Λ 33 0 LEad T < 0, (6.27) hεl 0 λi where Λ ij,i=1, 2, 3, i j 3 are defined in (6.18), then the system can be stabilized by control law (6.4), and the controller gain is K = VL 1. These four corollaries prevent an excessive amount of time from being spent in a nonlinear iterative algorithm. However, the parameter ε needs to be tuned and the matrix R must be put into a special form. So, the solution is still suboptimal. Fridman et al. has often used this method [6 9], and the theorems thus derived are special cases of the ones in this chapter. In other words, our theorems are more general and better than those, as is demonstrated by the numerical example in Section Completely LMI-Based Design Method Sections 6.2 and 6.3 presented the results of the best attempts so far at developing a method of solving delay-dependent robust stabilization problems; they are not general LMI-based methods. Even though it is possible to use an iterative nonlinear minimization algorithm or a parameter-tuning method to obtain a suboptimal solution, the iterations and the parameter tuning both take a long time. This section presents a completely LMI-based design

156 6.4 Completely LMI-Based Design Method 139 method, in which the controller is obtained by directly solving LMI-based stabilization conditions without any parameter tuning or iterative process, for a special case that is based on the delay-dependent and rate-independent stability conditions in Chapter 3. A simple transformation converts Corollary to the following one. Corollary Consider nominal system (6.1) withadelay, d(t), that satisfies (6.2). Given a scalar h>0, the system is asymptotically stable when u(t) =0if there exist matrices P > 0, Z>0, and X = X 11 X 12 0, X 22 and any appropriately dimensioned matrices N 1 and N 2 such that matrix inequality (3.10) and the following matrix inequality hold: Φ 11 Φ12 ha T Φ = Φ22 ha d < 0, (6.28) hz 1 where Φ 11 = PA+ A T P + N 1 + N T 1 + hx 11, Φ 12 = PA d N 1 + N T 2 + hx 12, Φ 22 = N 2 N T 2 + hx 22. From this corollary, we obtain an LMI-based delay-dependent and rateindependent stabilization condition. Theorem Consider nominal system (6.1) withadelay,d(t), that satisfies (6.2). For a given scalar h>0, if there exist matrices L>0, R>0, and Y = Y 11 Y 12 0, and any appropriately dimensioned matrices M 1, Y 22 M 2, and V such that the following LMIs hold: S + S T + hy 11 A d M 2 + L M1 T + hy 12 hs T Π = M 2 M2 T + hy 22 hm2 T A T d < 0, (6.29) hr Θ = Y 11 Y 12 0, (6.30) Y 22 R

157 Stabilization of Systems with Time-Varying Delay where S = AL + A d M 1 + BV, then the system can be stabilized by control law (6.4), and the controller gain is K = VL 1. Proof. Setting H = P 0 [ ] [ ] [ ] T, A K = A + BK A d, Ī = I I, Ā= A T N1 T N2 T K Ī, and replacing the A in (6.28) with A + BK yield Ξ = HT Ā + ĀT H + hx ha T K < 0. (6.31) ha K hz 1 Also, (6.28) implies that N 2 N2 T is negative definite, which means that N 2 is nonsingular. So, we set 1 H 1 = P 0 = L 0. N1 T N2 T M 1 M 2 Then, we pre- and post-multiply Ξ in (6.31) by diag {H T,I} and diag {H 1, I}, respectively, and set Y = H T XH 1, R = Z 1, V = KL. This transforms (6.31) into (6.29). Pre- and post-multiplying Ψ in (3.10) by diag {H T, I} and diag {H 1, I}, respectively, and applying the Schur complement yield (6.30). This completes the proof. Remark For a given scalar h>0, conditions (6.29) and (6.30) are LMI-based; that is, they constitute a completely LMI-based solution to the delay-dependent and rate-independent stabilization problem. Next, we use Lemma to extend Theorem to system (6.5), which has time-varying structured uncertainties. Theorem Consider system (6.5) with a delay, d(t), that satisfies (6.2). For a given scalar h>0, if there exist matrices L>0, R>0, and Y = Y 11 Y 12 0, any appropriately dimensioned matrices M 1,M 2, and Y 22 V, and a scalar λ>0 such that LMI (6.30) and the following LMI hold:

158 6.5 Numerical Example 141 S+S T +λdd T +hy 11 A d M 2 +L M1 T+hY 12 hs T +hλdd T E T M 2 M2 T + hy 22 hm2 T A T d M2 T E T ad hr+λh 2 DD T <0, 0 λi (6.32) where E = E a L + E ad M 1 + E b V and S is defined in (6.29), then the system can be stabilized by control law (6.4), and the controller gain is K = VL 1. Remark There is an error in Theorem 2 of [7], which gives the delaydependent and rate-independent conditions as LMIs for ε i = I, i =1, 2. In (28a) in the theorem, rows 4-7 and columns 4-7 were deleted. That is, both S 1 and S 2 must be zero. However, in (17), L = E 0, L 1 = E 1,andL 2 = E 2 ; and (28a) was derived from (17) by pre- and post-multiplying it by Δ T and Δ (Δ =diag{q, I, S1, S2,I}, respectively, not by diag {Q, I} and S 1 = S1 1, S 2 = S2 1 ). Since S 1 and S 2 must be zero, this treatment is equivalent to making rows 3-4 and columns 3-4 in (17) all zero and then deleting them, which is clearly not correct. On the other hand, if either L 1 or L 2 in (17) is non-zero, the BRL representation of (17) cannot be extended to make it delay-dependent and rate-independent. In fact, it is a mistake that (28) does not contain E 1 or E 2 when there are uncertainties in A 1 and A 2.So,the delay-dependent and rate-independent conditions in [7] are not valid for a delay with an uncertainty. Note that all the equation numbers mentioned here are those in [7]. 6.5 Numerical Example Example Consider a system with time-varying structured uncertainties and the following parameters: A = , A d =, B = 0, D = I, E a =0.2I, E ad = αi, E b =0. If we assume that this system contains a constant delay (μ = 0), then the upper bound, h, for which the system is stabilizable by a state-feedback

159 Stabilization of Systems with Time-Varying Delay Table 6.1. Maximum upper bound, h, forα =0.2 μ Method h Feedback gain, K Iterations or param. 0 [3] 0.45 [ ] 99 0 Theorem 6.2.3, Algorithm [ ] 203 Theorem 6.2.3, Algorithm [ ] Theorem 6.2.3, Algorithm [ ] 243 Theorem 6.2.3, Algorithm [ ] Theorem 6.2.4, Algorithm [ ] 220 Theorem 6.2.4, Algorithm [ ] Theorem 6.2.4, Algorithm [ ] 240 Theorem 6.2.4, Algorithm [ ] Corollary [ ] ε = Corollary [ ] ε = Corollary [ ] ε = Corollary [ ] ε = 1.2 controller is 0.45 in [3] (where α =0.2) and in [7] (where α =0). Tables 6.1 and 6.2 list these values along with results for α =0.2 orα =0 obtained by Theorems and using Algorithm or 6.2.2, and also Corollaries and Clearly, the conditions in this chapter produce the largest upper bounds. In addition, when the same theorem is used, Algorithm produces a smaller controller gain in fewer iterations than Algorithm does. Simulations were also run on a closed-loop system for α =0,ΔA(t) = 0.2I, andμ = 0. Fig. 6.1 shows input and state response curves for the statefeedback controller gain obtained with Theorem and that obtained by

160 6.5 Numerical Example 143 Fridman et al. [7] for h = Both controllers stabilize the system; but the one from Theorem makes the state converge to zero more quickly, although it does produce a larger initial control input. Fig. 6.2 shows results for h = In this case, Fridman et al. s controller [7] cannot stabilize the system at all, while that of Theorem can. Table 6.2. Maximum upper bound, h, forα =0 μ Method h Feedback gain, K Iterations or param. 0 [7] 0.58 [ ] 0 Theorem 6.2.3, Algorithm [ ] 200 Theorem 6.2.3, Algorithm [ ] Theorem 6.2.3, Algorithm [ ] 279 Theorem 6.2.3, Algorithm [ ] Theorem 6.2.4, Algorithm [ ] 245 Theorem 6.2.4, Algorithm [ ] Theorem 6.2.4, Algorithm [ ] 208 Theorem 6.2.4, Algorithm [ ] Corollary [ ] ε = Corollary [ ] ε = Corollary [ ] ε = Corollary [ ] ε = 1.1 In addition, when the derivative of the time-varying delay, d(t), is unknown and a delay-dependent and rate-independent stabilization condition is used to find the upper bound, h, for which the system is stabilizable by a state-feedback controller, then the values are in [6] and in [7] for

161 Stabilization of Systems with Time-Varying Delay Fig Simulation results for h =0.5865: (a) method of Fridman et al. and (b) Theorem Fig Simulation results for h =0.84: (a) method of Fridman et al. and (b) Theorem 6.2.4

162 References 145 α = 0, and for the LMI-based condition in Theorem However, in contrast to LMIs (6.29) and (6.30) in Theorem 6.4.1, for which no parameters need to be tuned, the condition in [7] is not completely LMI-based and requires parameter tuning. A more important point is that, as mentioned in Remark 6.4.2, the method in [7] is ineffective when α =0.2 because of the error in the condition. However, Theorem yields a value of for the upper bound, h, for which the system is stabilizable by a state-feedback controller. 6.6 Conclusion This chapter uses an LMI-based iterative nonlinear minimization algorithm and a parameter-tuning method to establish methods of designing controllers for systems with a time-varying delay. The methods are based on the delaydependent stabilization conditions obtained by the FWM approach. It also presents an ICCL algorithm that requires fewer iterations than the CCL algorithm. Furthermore, it describes an approach for designing an LMI-based delay-dependent and rate-independent stabilizable controller. Finally, a numerical example demonstrates the benefits of our method. References 1. K. Gu and S. I. Niculescu. Additional dynamics in transformed time-delay systems. IEEE Transactions on Automatic Control, 45(3): , K. Gu and S. I. Niculescu. Further remarks on additional dynamics in various model transformations of linear delay systems. IEEE Transactions on Automatic Control, 46(3): , Y. S. Moon, P. Park, W. H. Kwon, and Y. S. Lee. Delay-dependent robust stabilization of uncertain state-delayed systems. International Journal of Control, 74(14): , H. Gao and C. Wang. Comments and further results on A descriptor system approach to H control of linear time-delay systems. IEEE Transactions on Automatic Control, 48(3): , Y. S. Lee, Y. S. Moon, W. H. Kwon, and P. G. Park. Delay-dependent robust H control for uncertain systems with a state-delay. Automatica, 40(1): 65-72, E. Fridman and U. Shaked. Delay-dependent stability and H control: constant and time-varying delays. International Journal of Control, 76(1): 48-60, 2003.

163 Stabilization of Systems with Time-Varying Delay 7. E. Fridman and U. Shaked. An improved stabilization method for linear timedelay systems. IEEE Transactions on Automatic Control, 47(11): , E. Fridman and U. Shaked. A descriptor system approach to H control of linear time-delay systems. IEEE Transactions on Automatic Control, 47(2): , E. Fridman and U. Shaked. Parameter dependent stability and stabilization of uncertain time-delay systems. IEEE Transactions on Automatic Control, 48(5): , M. Wu, Y. He, and J. H. She. Delay-dependent robust stability and stabilization criteria for uncertain neutral systems. Acta Automatica Sinica, 31(4): , M. Wu, Y. He, and J. H. She. New delay-dependent stability criteria and stabilizing method for neutral systems. IEEE Transactions on Automatic Control, 49(12): , E. L. Ghaoui, F. Oustry, and M. AitRami. A cone complementarity linearization algorithms for static output feedback and related problems. IEEE Transactions on Automatic Control, 42(8): , Y. He, G. P. Liu, D. Rees, and M. Wu. Improved stabilization method for networked control systems. IET Control Theory & Applications, 1(6): , 2007.

164 7. Stability and Stabilization of Discrete-Time Systems with Time-Varying Delay Increasing attention is being paid to the delay-dependent stability, stabilization, and H control of linear systems with delays [1 14]. The literature discusses discrete-time systems with two types of time-varying delays: small and non-small. For a small delay, [15 17] presented methods of designing an H state-feedback controller. More recently, a time-varying interval delay, which is a kind of non-small delay, has become a focus of attention for both continuous-time systems [18] and discrete-time systems [9, 19, 20]. [9] solved the robust H control problem using an output-feedback controller; but the limitations of that approach are that matrix inequalities must be solved to obtain the decision matrix variables, and only a range of delays can be dealt with. [21] handled the problem of designing an H filter by using a finite-sum inequality. [20] used Moon et al. s inequality and criteria containing both the range and upper bound of the time-varying delay for the delay-dependent output-feedback stabilization of discrete-time systems with a time-varying state delay. And [19] derived H control criteria using a descriptor model transformation in combination with Moon et al. s inequality for uncertain linear discrete-time systems with a time-varying interval delay; in that approach, the delay was decomposed into a nominal part and an uncertain part. However, as mentioned in Chapter 3, Moon et al. s inequality is more conservative than the FWM approach for continuous-time systems. This is also true for discrete-time systems. In addition, [20, 21] ignored some terms when estimating the upper bound on the difference of a Lyapunov function; but [6, 7, 22] showed that retaining those terms yields less conservative stability results. Another point is that the delay was increased to make it easy to handle. That is, the delay, d(k), where d 1 d(k) d 2, was increased to d 2 in many studies [19 22]; and d 2 d(k) was increased to d 2 d 1 in [22], or in other words, d 2 = d(k)+(d 2 d(k)) was increased to 2d 2 d 1.These methods may lead to considerable conservativeness. On the other hand, the

165 Stability and Stabilization of Discrete-Time Systems... conditions for a delay-dependent stabilizing controller obtained by improved methods cannot be expressed strictly in terms of LMIs, although this type of problem can be solved by using either an iterative nonlinear minimization algorithm or a parameter-tuning method. Recently, [20] employed the CCL algorithm to deal with the problem of the output-feedback stabilization of discrete-time systems with a time-varying delay. However, as mentioned in Chapter 6, there is still room for further investigation of that algorithm. This chapter discusses the output-feedback control of a linear discretetime system with a time-varying delay [23]. First, the IFWM explained in Chapter 3 is used to derive a criterion for delay-dependent stability. Next, this criterion is used to design an SOF controller. Since the conditions for the existence of an admissible controller are not expressed strictly in terms of LMIs, the ICCL algorithm described in Chapter 6 is used to solve the nonconvex feasibility SOF stabilization control problem. Then, the problem of designing a DOF controller is transformed into one of designing an SOF controller, and a DOF controller is obtained by following the design method for an SOF controller. Finally, numerical examples demonstrate the effectiveness of this method and its advantages over other methods. 7.1 Problem Formulation Consider the following linear discrete-time system with a time-varying delay: x(k +1)=Ax(k)+A d x(k d(k)) + Bu(k), y(k) =Cx(k)+C d x(k d(k)), (7.1) x(k) =φ(k), d 2 k 0, where x(k) R n is the state vector; u(k) R m is the control input; A, A d, and B are constant matrices with appropriate dimensions; d(k) isatimevarying interval delay satisfying d 1 d(k) d 2, (7.2) where d 1 and d 2 are known positive integers; and φ(k) is the initial condition. This chapter considers both SOF and DOF controllers. The SOF control law is u(k) =Fy(k), (7.3)

166 7.2 Stability Analysis 149 where F R m p is a constant gain matrix to be determined. The DOF control law is x c (k +1)=A c x c (k)+b c y(k), u(k) =C c x c (k)+d c y(k), (7.4) x c (k) =0, k < 0, where x c (k) R r is the state of the controller; and A c, B c, C c,andd c are appropriately dimensioned matrices to be determined. The aim of this chapter is to find an SOF controller of the form (7.3) and a DOF controller of the form (7.4) such that the closed-loop system is asymptotically stable. 7.2 Stability Analysis This section presents a stability analysis of system (7.1) with u(k) = 0. It is used in subsequent sections to design output-feedback controllers. First, we give a theorem. Theorem Consider system (7.1) with u(k) = 0. Given scalars d 1 and d 2 with d 2 d 1 > 0, the system is asymptotically stable if there exist matrices P>0, Q i 0, i=1, 2, 3, Z j > 0, j=1, 2, X= X 11 X 12 0, X 22 and Y = Y 11 Y 12 0, and any appropriately dimensioned matrices N = Y 22 [ ] N T 1 N2 T T [ ],S= S T 1 S2 T T [ ], and M = M T 1 M2 T T such that the following LMIs hold: Φ Ξ1 TP d 2 Ξ2 TZ 1 d12 Ξ2 TZ 2 P 0 0 < 0, (7.5) Z 1 0 Z 2 Ψ 1 = X N 0, (7.6) Z 1

167 Stability and Stabilization of Discrete-Time Systems... Ψ 2 = Y S 0, (7.7) Z 2 Ψ 3 = X + Y M 0, (7.8) Z 1 + Z 2 where Φ 11 Φ 12 S 1 M 1 Φ Φ = 22 S 2 M 2, Q 1 0 Q 2 Φ 11 = Q 1 + Q 2 +(d 12 +1)Q 3 P + N 1 + N1 T + d 2 X 11 + d 12 Y 11, Φ 12 = N2 T N 1 + M 1 S 1 + d 2 X 12 + d 12 Y 12, Φ 22 = [ Q 3 N 2 ] N2 T + M 2 + M2 T S 2 S2 T + d 2X 22 + d 12 Y 22, Ξ 1 = A A d [ 0 0, ] Ξ 2 = A I A d 0 0, d 12 = d 2 d 1. Proof. Defining η(l) =x(l +1) x(l) (7.9) and using system equation (7.1) with u(k) = 0 yield and where x(k +1)=x(k)+η(k) (7.10) η(k) =x(k +1) x(k) =(A I)x(k)+A d x(k d(k)). (7.11) Choose the Lyapunov function candidate to be V (k) =V 1 (k)+v 2 (k)+v 3 (k)+v 4 (k), (7.12) V 1 (k) =x T (k)px(k), 0 V 2 (k) = θ= d 2+1 k 1 l=k 1+θ η T (l)z 1 η(l)+ d 1 θ= d 2+1 k 1 l=k 1+θ η T (l)z 2 η(l),

168 7.2 Stability Analysis 151 V 3 (k) = V 4 (k) = k 1 l=k d 1 x T (l)q 1 x(l)+ d 1+1 θ= d 2+1 k 1 l=k 1+θ k 1 l=k d 2 x T (l)q 2 x(l), x T (l)q 3 x(l); and P>0, Q i 0, i=1, 2, 3, and Z j > 0, j=1, 2 are to be determined. Defining ΔV (k) =V (k +1) V (k) yields ΔV 1 (k) =x T (k +1)Px(k +1) x T (k)px(k) = ζ T 1 (k)ξt 1 PΞ 1ζ 1 (k) x T (k)px(k), (7.13) ΔV 2 (k) =d 2 η T (k)z 1 η(k) k 1 l=k d 2 η T (l)z 1 η(l) k d d 12 η T (k)z 2 η(k) η T (l)z 2 η(l) l=k d 2 =ζ T 1 (k)ξ T 2 (d 2 Z 1 + d 12 Z 2 )Ξ 2 ζ 1 (k) k d 1 1 l=k d(k) k 1 l=k d(k) η T (l)z 1 η(l) k d(k) 1 η T (l)z 2 η(l) η T (l)(z 1 + Z 2 )η(l), (7.14) l=k d 2 ΔV 3 (k) =x T (k)(q 1 + Q 2 )x(k) x T (k d 1 )Q 1 x(k d 1 ) x T (k d 2 )Q 2 x(k d 2 ), (7.15) where ΔV 4 (k) =(d 2 d 1 +1)x T (k)q 3 x(k) k d 1 l=k d 2 x T (l)q 3 x(l) (d 12 +1)x T (k)q 3 x(k) x T (k d(k))q 3 x(k d(k)), (7.16) ζ 1 (k) = [ x T (k), x T (k d(k)), x T (k d 1 ), x T (k d 2 ) ] T. From (7.9), the following equations hold for any matrices N, M, ands with appropriate dimensions: 0=2ζ2 T (k)n x(k) x(k d(k)) k 1 η(l), (7.17) l=k d(k)

169 Stability and Stabilization of Discrete-Time Systems... where 0=2ζ2 T (k)m x(k d(k)) x(k d 2 ) 0=2ζ2 T (k)s x(k d 1 ) x(k d(k)) k d(k) 1 l=k d 2 k d 1 1 l=k d(k) η(l), (7.18) η(l), (7.19) ζ 2 (k) = [ x T (k), x T (k d(k)) ] T. On the other hand, for any matrices X = X 11 X 12 0 and Y = Y 11 Y 12 0, the following equations are true: X 22 Y 22 k 1 k 1 0= ζ2 T (k)xζ 2(k) ζ2 T (k)xζ 2(k) l=k d 2 l=k d 2 =d 2 ζ T 2 (k)xζ 2 (k) k 1 l=k d(k) k d 1 1 k d 0= ζ2 T (k)yζ 1 1 2(k) ζ2 T (k)yζ 2(k) l=k d 2 l=k d 2 =d 12 ζ T 2 (k)yζ 2 (k) k d 1 1 l=k d(k) k d(k) 1 ζ2 T (k)xζ 2 (k) ζ2 T (k)xζ 2 (k), l=k d 2 k d(k) 1 ζ2 T (k)yζ 2 (k) ζ2 T (k)yζ 2 (k). l=k d 2 (7.20) (7.21) Taking the forward difference of V (k) and adding the terms on the right sides of (7.17)-(7.21) to ΔV (k) allow us to write ΔV (k) as where ΔV (k) ζ T 1 (k) [ Φ + Ξ T 1 PΞ 1 + Ξ T 2 (d 2Z 1 +d 12 Z 2 )Ξ 2 ] ζ1 (k) k 1 l=k d(k) ζ T 3 (k, l)ψ 1 ζ 3 (k, l) k d 1 1 l=k d(k) ζ T 3 (k, l)ψ 2 ζ 3 (k, l) k d(k) 1 ζ3 T (k, l)ψ 3 ζ 3 (k, l), (7.22) l=k d 2

170 7.3 Controller Design 153 ζ 3 (k, l) = [ ζ T 2 (k), η T (l) ] T. Thus, if Ψ i 0, i=1, 2, 3andΦ + Ξ1 TPΞ 1 + Ξ2 T(d 2Z 1 + d 12 Z 2 )Ξ 2 < 0, which is equivalent to (7.5) from the Schur complement, then ΔV (k) < 0. That is, system (7.1) with u(k) = 0 is asymptotically stable. This completes the proof. Remark Lyapunov function (7.12) is different from the one in [20] in two ways: V 4 (k) includes information on d 2 that was not used in previous studies; and only V 4 (k) = d 1+1 k 1 θ= d 2+1 l=k 1+θ xt (l)q 3 x(l) isemployedto handle the time-varying delay. In contrast, V 4 (k) = k 1 d1+1 θ= d 2+2 l=k d(k) xt (l)qx(l)+ k 1 l=k 1+θ xt (l)qx(l) was used in [9,20]. Our treatment considerably simplifies the proof. Remark Let Q 1 = Q 2 = Z 2 = εi (where ε is a sufficiently small positive scalar) and let M =0,S =0,Y =0,X = X 0, andn = [ Y T 0 ] T. 0 Then, Theorem reduces to Theorem 1 in [20]. That is, Theorem provides more freedom in the selection of Q 1, Q 2, Z 2, M, S, X, N, andy. Remark In the proof of Theorem 7.2.1, d 2 is separated into two parts: d 2 = d(k)+(d 2 d(k)). In contrast, in (6) in [22], the inequalities and d 2 ζ T (k)mz 1 1 M T ζ(k) k 1 l=k d(k) ζ T (k)mz 1 1 M T ζ(k) 0 k d(k) 1 d 12 ζ T (k)sz1 1 ST ζ(k) ζ T (k)sz1 1 ST ζ(k) 0 l=k d 2 are employed; and d(k) and(d 2 d(k)) are increased to d 2 and d 12, respectively. So, their theorem is more conservative than Theorem Controller Design The results of the stability analysis in the previous section are now employed to design both SOF and DOF controllers for system (7.1).

171 Stability and Stabilization of Discrete-Time Systems SOF Controller Connecting SOF controller (7.3) to system (7.1) yields the closed-loop system x(k +1)=Âx(k)+Âdx(k d(k)), (7.23) x(k) =φ(k), d 2 k 0, where  = A + BFC,  d = A d + BFC d. We obtain the following theorem from Theorem Theorem Consider system (7.1). For given scalars d 1 and d 2 with d 2 d 1 > 0, if there exist matrices P>0, L>0, Q i 0, i=1, 2, 3, Z j > 0, R j > 0, j =1, 2, X= X 11 X 12 0, and Y = Y 11 Y 12 0, and X 22 Y 22 any appropriately dimensioned matrices N = [ N1 T N ] 2 T T [ ],S= S T 1 S2 T T, M = [ ] M1 T M2 T T, and F such that LMIs (7.6)-(7.8) and the following matrix inequalities hold: Φ ˆΞ 1 T d2 ˆΞT 2 d12 ˆΞT 2 L 0 0 < 0, (7.24) R 1 0 R 2 PL = I, Z j R j = I, j =1, 2, (7.25) where [ ˆΞ 1 =   d 0 0], [ ˆΞ 2 =  I Âd 0 0], and Φ is defined in (7.5), then the system can be stabilized by control law (7.3). Proof. In (7.5), replacing A and A d with  and Âd, respectively, and preand post-multiplying the left and right sides by diag {I, P 1, Z1 1, Z 1 2 } yield

172 7.3 Controller Design 155 Φ ˆΞT 1 d2 ˆΞT 2 d12 ˆΞT 2 P Z Z2 1 < 0. (7.26) Defining L = P 1 and R j = Zj 1,j=1, 2 in (7.26) gives equations (7.24) and (7.25). This completes the proof. Since the conditions in Theorem are no longer LMI conditions owing to (7.25), we cannot use a convex optimization algorithm to find a maximum d 2, d 2max. However, as mentioned in Chapter 6, we can use the idea for solving a cone complementarity problem first proposed in [24]. Defining L = P 1 and R j = Z 1 j,j=1, 2 converts this nonconvex problem into the following LMI-based nonlinear minimization problem: Minimize subject to 2 Tr{PL+ Z j R j } j=1 (7.6)-(7.8), (7.24), and P>0, L > 0, Q i > 0, Z j > 0, R j > 0 P I 0, Z j I 0, i=1, 2, 3, j=1, 2. L R j (7.27) Then, for a given d 1 > 0, a suboptimal maximum d 2 can be found by using either the CCL or the ICCL algorithm. Here, we employ the ICCL algorithm because of its advantages. Algorithm To maximize d 2 : Step 1: Choose a sufficiently small initial d 2 d 1 such that there exists a feasible solution to (7.6)-(7.8), (7.24), and (7.27). Set d 2max = d 2. Step 2: Find a feasible set (P 0,L 0,Q i0,i=1, 2, 3, Z j0,r j0,j=1, 2, M 0, N 0,S 0,X 0,Y 0,F 0 ) satisfying (7.6)-(7.8), (7.24), and (7.27). Setk =0. Step 3: Solve the following LMI problem for the variables P, L, Q i,i=1, 2, 3, Z j,r j,j=1, 2, M,N,S,X,Y,and F : 2 Minimize Tr P kl + PL k + [Z j R jk + Z jk R j ] j=1 subject to (7.6)-(7.8), (7.24), and (7.27).

173 Stability and Stabilization of Discrete-Time Systems... Set P k+1 = P, L k+1 = L, Z j,k+1 = Z j, and R j,k+1 = R j,j=1, 2. Step 4: For the F obtained in Step 3, if LMIs (7.6)-(7.8) and Φ ˆΞ 1 TP d 2 ˆΞT 2 Z 1 d12 ˆΞT 2 Z 2 P 0 0 < 0 (7.28) Z 1 0 Z 2 are feasible for the variables P, Q i,i=1, 2, 3, Z j,j=1, 2, M,N,S, X, and Y, then set d 2max = d 2, increase d 2, andreturntostep2. If LMIs (7.6)-(7.8) and (7.28) are not feasible within a specified number of iterations, then stop. Otherwise, set k = k + 1 and go to Step 3. Remark The iteration stop condition in [20,25] is very strict. The gain matrix, F, and the other matrices (P, Q i,i=1, 2, 3, Z j j =1, 2, M, N, S, X, and Y ) must satisfy matrix inequality (7.26). However, once F is obtained, condition (7.28), which is equivalent to (7.24) and (7.25) (that is, if (7.28) holds, then (7.24) and (7.25) are true), reduces to an LMI for P, Q i,i=1, 2, 3, Z j,j =1, 2, M, N, S, X, andy. So, we modified stop condition for Algorithm to include a determination of whether or not LMIs (7.6)-(7.8) and (7.28) are feasible. This provides more freedom in the selection of the variables P, Q i,i=1, 2, 3, Z j,j=1, 2, M, N, S, X, andy. Remark The choice of the initial matrices for iteration in Step 2 is an important factor determining whether or not a feasible solution is found. However, even if a feasible solution is obtained, sometimes it is not satisfactory. Although randomly choosing the initial matrices may not lead to a feasible solution every time, many simulations have shown that a random choice often produces a good solution DOF Controller Connecting DOF controller (7.4) to system (7.1) yields the closed-loop system ξ(k +1)=Āξ(k)+Ādξ(k d(k)), (7.29) where ξ(k) = [ x T (k), x T c (k)] T,

174 7.3 Controller Design 157 Ā = A + BD cc BC c = Ã + B F C, B c C A c Ā d = A d + BD c C d 0 = Ãd + B F C d, B c C d 0 Ã= A 0, Ã d = A d 0, B = B 0, C = C 0, Cd = C d 0, I 0 I 0 0 and F = D c C c are to be determined. Then, the problem of designing B c A c DOF controller (7.4) for system (7.1) is transformed into the problem of designing an SOF controller, u(k) = F ȳ(k), for the system ξ(k +1)=Ãξ(k)+Ãdξ(k d(k)) + Bu(k), ȳ(k) = Cξ(k)+ C d ξ(k d(k)). (7.30) (7.31) This leads to the following corollary. Corollary Consider system (7.1). For given scalars d 1 and d 2 with d 2 d 1 > 0, if there exist matrices P>0, L>0, Q i 0, i=1, 2, 3, Z j > 0, R j > 0, j =1, 2, X= X 11 X 12 0, and Y = Y 11 Y 12 0, and X 22 Y 22 any appropriately dimensioned matrices N = [ ] N1 T N2 T T [ ],S= S T 1 S2 T T, M = [ M1 T M ] 2 T T, and F such that matrix inequalities (7.6)-(7.8), (7.25) and the following matrix inequality hold: Φ Ξ 1 T d2 ΞT 2 d12 ΞT 2 L 0 0 < 0, (7.32) R 1 0 R 2 where Ξ 1 = [ Ā Ā d 0 0],

175 Stability and Stabilization of Discrete-Time Systems... Ξ 2 = [ Ā I Ād 0 0], and Φ is defined in (7.5) (Note that the dimensions here are different from those in Theorem ), then the system can be stabilized by control law (7.4). Remark Since the problem of designing a DOF controller has been transformed into one of designing an SOF controller, Algorithm can be used to solve the design problem by replacing A, A d, B, C, andc d in the algorithm with Ã, Ã d, B, C, and Cd, respectively. 7.4 Numerical Examples The two examples below demonstrate the effectiveness of our method and its advantage over other methods. Example Consider the stability of system (7.1) with u(k) = 0andthe following parameters: A = 0.8 0, A d = This example was discussed in [20,22]. Table 7.1 lists values of the upper bound on d 2 that guarantee the stability of system (7.1) with u(k) =0that were reported in [20, 22] and those that were obtained with Theorem The method presented in this chapter is significantly better than the others. Table 7.1. Upper bound on d 2 for various d 1 (Example 7.4.1) d [20] [22] Theorem Example Consider system (7.1) with the following parameters: A=,A d = 0.3 0,B= 1,C= 1 1,C d =

176 References 159 [20] stated that system (7.1) could be stabilized by SOF controller (7.3) for 3 d(k) 11 and by DOF controller (7.4) with r =1orr =2for 3 d(k) 100. However, using Algorithm shows that system (7.1) can be stabilized by SOF controller (7.3) with F =[ ] for 3 d(k) 12. On the other hand, even for 3 d(k) 1000, system (7.1) can be stabilized by DOF controller (7.4) with an order of either one [ A c =0.0279, B c = C c = , D c = ] , [ or two A c =, B c =, [ ] [ ] C c = , D c = ], 7.5 Conclusion This chapter discusses the output-feedback stabilization control of a linear discrete-time system with a time-varying delay. First, the IFWM approach is used to carry out a delay-dependent stability analysis. Next, based on that, a design criterion for an SOF controller is derived; and the problem of designing a DOF controller is reduced to the problem of designing an SOF controller. A design criterion for a DOF controller is also established. Then, since the conditions for the existence of an admissible controller are not expressed strictly in terms of LMIs, the ICCL algorithm described in Chapter 6 is used to solve the nonconvex feasibility SOF stabilization control problem. Finally, numerical examples demonstrate the effectiveness of this method and its advantage over other methods. References 1. E. Fridman and S. I. Niculescu. On complete Lyapunov-Krasovskii functional techniques for uncertain systems with fast-varying delays. International Journal of Robust and Nonlinear Control, 18(3): , K. Gu, V. L. Kharitonov, and J. Chen. Stability of Time-Delay Systems. Boston: Birkhäuser, 2003.

177 Stability and Stabilization of Discrete-Time Systems P. Richard. Time-delay systems: An overview of some recent advances and open problems. Automatica, 39(10): , H. Gao and C. Wang. Comments and further results on A descriptor system approach to H control of linear time-delay systems. IEEE Transactions on Automatic Control, 48(3): , M. Wu, Y. He, J. H. She, and G. P. Liu. Delay-dependent criteria for robust stability of time-varying delay systems. Automatica, 40(8): , Y. He, Q. G. Wang, L. Xie, and C. Lin. Further improvement of free-weighting matrices technique for systems with time-varying delay. IEEE Transactions on Automatic Control, 52(2): , Y. He, Q. G. Wang, C. Lin, and M. Wu. Delay-range-dependent stability for systems with time-varying delay. Automatica, 43(2): , C. Lin, Q. G. Wang, and T. H. Lee. A less conservative robust stability test for linear uncertain time-delay systems. IEEE Transactions on Automatic Control, 51(1): 87-91, S. Xu and T. Chen. Robust H control for uncertain discrete-time systems with time-varying delays via exponential output feedback controllers. Systems & Control Letters, 51(3-4): , S. Xu and J. Lam. On equivalence and efficiency of certain stability criteria for time-delay systems. IEEE Transactions on Automatic Control, 52(1): , X. M. Sun, J. Zhao, and D. J. Hill. Stability and L 2-gain analysis for switched delay systems: a delay-dependent method. Automatica, 42(10): , X. M. Sun, G. P. Liu, D. Rees, and W. Wang. Delay-dependent stability for discrete systems with large delay sequence based on switching techniques. Automatica, 44(11): , X. M. Sun, W. Wang, G. P. Liu, and J. Zhao. Stability analysis for linear switched systems with time-varying delay. IEEE Transactions on Systems, Man, and Cybernetics-Part B, 38(2): , M. Wu, Y. He, and J. H. She. Delay-dependent stabilization for systems with multiple unknown time-varying delays. International Journal of Control, Automation, and Systems, 4(6): , S. H. Song and J. K. Kim. H control of discrete-time linear systems with norm-bounded uncertainties and time delay in state. Automatica, 34(1): , S.H.Song,J.K.Kim,C.H.Yim,andH.C.Kim.H control of discrete-time linear systems with time-varying delays in state. Automatica, 35(9): , Z. Wang, B. Huang, and H. Unbehauen. Robust H observer design of linear state delayed systems with parametric uncertainties: the discrete-time case. Automatica, 35(6): , Q. L. Han and K. Gu. Stability of linear systems with time-varying delay: A generalized discretized Lyapunov functional approach. Asian Journal of Control, 3(3): , 2001.

178 References E. Fridman and U. Shaked. Stability and guaranteed cost control of uncertain discrete delay systems. International Journal of Control, 78(4): , H. Gao, J. Lam, C. Wang, and Y. Wang. Delay-dependent output-feedback stabilization of discrete-time systems with time-varying state delay. IEE Proceedings Control Theory & Applications, 151(6): , X. M. Zhang and Q. L. Han. Delay-dependent robust H filtering for uncertain discrete-time systems with time-varying delay based on a finite sum inequality. IEEE Transactions on Circuits and Systems II, 53(12): , H. Gao and T. Chen. New results on stability of discrete-time systems with time-varying state delay. IEEE Transactions on Automatic Control, 52(2): , Y. He, M. Wu, G. P. Liu, and J. H. She. Output feedback stabilization for a discrete-time system with a time-varying delay. IEEE Transactions on Automatic control, 53(10): , E. L. Ghaoui, F. Oustry, and M. AitRami. A cone complementarity linearization algorithms for static output feedback and related problems. IEEE Transactions on Automatic Control, 42(8): , Y. S. Moon, P. Park, W. H. Kwon, and Y. S. Lee. Delay-dependent robust stabilization of uncertain state-delayed systems. International Journal of Control, 74(14): , 2001.

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180 8. H Control Design for Systems with Time-Varying Delay During the last decade, considerable attention has been devoted to the problems of delay-dependent stability, stabilization, and H controller design for time-delay systems [1 6]. However, as pointed out in [7, 8], most studies to date have ignored some useful terms in the derivative of the Lyapunov- Krasovskii functional [1, 5, 9 11]. Although [7, 8] retained these terms and established an improved delay-dependent stability criterion for systems with a time-varying delay, there is room for further investigation. For instance, in [1, 5, 7 12], the delay, d(t), where 0 d(t) h, was often increased to h. And in [7,8], another term, h d(t), was also taken to be equal to h; thatis, h = d(t)+(h d(t)) was increased to 2h, which may lead to conservativeness. Moreover, these methods are not applicable to systems with a time-varying interval delay. On the other hand, design conditions for a delay-dependent H controller with memoryless state feedback obtained by improved methods cannot be expressed strictly in terms of LMIs. So, either an iterative nonlinear minimization algorithm or a parameter-tuning method is needed to solve the problem. Recently, [5] used the FWM approach and the CCL algorithm to solve the problem of designing an H controller. However, as mentioned in Chapter 6, that algorithm can be improved. This chapter shows how the IFWM approach can be used to obtain an improved delay-dependent BRL for systems with a time-varying interval delay. Then, that BRL along with the ICCL algorithm described in Chapter 6 are used to design an H controller. Finally, numerical examples demonstrate the effectiveness and advantages of this method. 8.1 Problem Formulation Consider the following linear system with a time-varying delay:

181 H Control Design for Systems with Time-Varying Delay ẋ(t) =Ax(t)+A d x(t d(t)) + Bu(t)+B ω ω(t), t > 0, Cx(t)+D ω ω(t) z(t) = C d x(t d(t)), Du(t) x(t) =φ(t), t [ h 2, 0], (8.1) where x(t) R n is the state vector; u(t) R m is the control input; ω(t) L 2 [0, ) is an exogenous disturbance signal; z(t) R r is the control output; A, A d, B, B ω, C, D ω, C d,andd are constant matrices with appropriate dimensions; and the delay, d(t), is a time-varying differentiable function that satisfies h 1 d(t) h 2 (8.2) and d(t) μ, (8.3) where h 2 h 1 0andμare constants. Note that h 1 may be non-zero. The initial condition, φ(t), is a continuously differentiable initial function of t [ h 2, 0]. For a given scalar γ>0, the performance of the system is defined to be J(ω) = 0 (z T (t)z(t) γ 2 ω T (t)ω(t))dt. (8.4) The H control problem addressed in this chapter is stated as follows: For a memoryless state-feedback controller, find a value for the gain, K R m n, in the control law u(t) =Kx(t) (8.5) such that, for any delay, d(t), satisfying (8.2) and (8.3), (1) the closed-loop system of (8.1), ẋ(t) =(A + BK)x(t)+A d x(t d(t)) + B ω ω(t), (8.6) is asymptotically stable under the condition ω(t) =0, t 0; and (2) J(ω) < 0 for all non-zero ω(t) L 2 [0, ) andagivenγ>0 under the condition x(t) =0, t [ h 2, 0].

182 8.2 BRL BRL Below, we use the IFWM approach to derive a new delay-dependent BRL. Theorem Consider system (8.1) with u(t) =0. Given scalars h 2 h 1 0, μ, andγ>0, the system is asymptotically stable and satisfies J(ω) < 0 for all non-zero ω(t) L 2 [0, ) under the condition x(t) =0, t [ h 2, 0] if there exist matrices P > 0, Q i 0, i =1, 2, 3, Z j > 0, andx j 0, j =1, 2 and any appropriately dimensioned matrices N i i =1, 2, 3 such that the following LMIs hold: Φ 1 + Φ 2 + Φ T 2 + h 2X 1 + h 12 X 2 h2 Φ T 3 Z 1 h12 Φ T 3 Z 2 Φ T 4 Φ T 5 Z Φ = Z < 0, I 0 I N j (8.7) Ψ j = X j 0, j =1, 2, (8.8) Z j Ψ 3 = X 1 + X 2 N 3 0, (8.9) Z 1 + Z 2 where 3 PA+ A T P + Q i PA d 0 0 PB ω i=1 (1 μ)q Φ 1 =, Q Q 2 0 γ 2 I Φ 2 =[N 1 N 3 N 1 N 2 N 2 N 3 0], Φ 3 =[A A d 0 0 B ω ], Φ 4 =[C D ω ], Φ 5 =[0 C d 0 0 0], h 12 = h 2 h 1.

183 H Control Design for Systems with Time-Varying Delay Proof. Choose the Lyapunov-Krasovskii functional candidate to be 2 V (x t )=x T (t)px(t)+ x T (s)q i x(s)ds + x T (s)q 3 x(s)ds i=1 t h i t d(t) 0 h1 + ẋ T (s)z 1 ẋ(s)dsdθ+ ẋ T (s)z 2 ẋ(s)dsdθ, (8.10) h 2 t+θ h 2 t+θ where P>0, Q i 0, i=1, 2, 3, and Z j > 0, j=1, 2 are to be determined. From the Newton-Leibnitz formula, the following equations are true for any matrices N i i =1, 2, 3 with appropriate dimensions: ] where 0=2ζ T (t)n 1 [x(t) x(t d(t)) t d(t) ẋ(s)ds, (8.11) ] t h1 0=2ζ T (t)n 2 [x(t h 1 ) x(t d(t)) ẋ(s)ds, (8.12) t d(t) ] 0=2ζ T (t)n 3 [x(t d(t)) x(t h 2 ) d(t) t h 2 ẋ(s)ds ζ(t) = [ x T (t), x T (t d(t)), x T (t h 1 ),x T (t h 2 ), ω T (t) ] T., (8.13) On the other hand, for any matrices X j 0, j=1, 2, the following equalities are true: d(t) 0=h 2 ζ T (t)x 1 ζ(t) ζ T (t)x 1 ζ(t)ds ζ T (t)x 1 ζ(t)ds, (8.14) 0=h 12 ζ T (t)x 2 ζ(t) t h 2 d(t) t h 2 ζ T (t)x 2 ζ(t)ds Moreover, the following equations are also true: t d(t) h1 t d(t) ẋ T (s)z 1 ẋ(s)ds = ẋ T (s)z 1 ẋ(s)ds t h 2 t d(t) ζ T (t)x 2 ζ(t)ds. (8.15) d(t) t h 2 ẋ T (s)z 1 ẋ(s)ds, (8.16) h1 ẋ T (s)z 2 ẋ(s)ds = t h 2 d(t) t h 2 h1 ẋ T (s)z 2 ẋ(s)ds ẋ T (s)z 2 ẋ(s)ds. t d(t) (8.17) Calculating the derivative of V (x t ) along the solutions of system (8.1), adding the right sides of (8.11)-(8.15) to it, and using equations (8.16) and (8.17) yield

184 8.2 BRL 167 V (x t )+z T (t)z(t) γ 2 ω T (t)ω(t) ζ T (t) [ Φ 1 + Φ 2 + Φ T 2 + h 2X 1 + h 12 X 2 +Φ T 3 (h 2 Z 1 + h 12 Z 2 )Φ 3 + Φ T 4 Φ 4 + Φ T 5 Φ 5 ] ζ(t) t d(t) d(t) ξ T (t, s)ψ 1 ξ(t, s)ds h1 t d(t) ξ T (t, s)ψ 2 ξ(t, s)ds t h 2 ξ T (t, s)ψ 3 ξ(t, s)ds, (8.18) where ξ(t, s) = [ ζ T (t), ẋ T (s) ] T. Thus, if Ψ i 0, i=1, 2, 3andΦ 1 + Φ 2 + Φ T 2 + h 2X 1 + h 12 X 2 + Φ T 3 (h 2Z 1 + h 12 Z 2 )Φ 3 + Φ T 4 Φ 4 + Φ T 5 Φ 5 < 0, which is equivalent to (8.7) by the Schur complement, then V (x t )+z T (t)z(t) γ 2 ω T (t)ω(t) < 0, which ensures that J(ω) < 0. On the other hand, (8.7)-(8.9) imply that the following LMIs hold: ˆΦ 1 + ˆΦ 2 + ˆΦ T 2 + h 2 ˆX 1 + h 12 ˆX2 h2 ˆΦT 3 Z 1 h12 ˆΦT 3 Z 2 Z 1 0 < 0, (8.19) Z 2 ˆΨ j = ˆX j ˆNj 0, j =1, 2, (8.20) Z j ˆΨ 3 = ˆX 1 + ˆX 2 ˆN3 0, (8.21) Z 1 + Z 2 where 3 PA+ A T P + Q i PA d 0 0 i=1 ˆΦ 1 = (1 μ)q 3 0 0, Q 1 0 Q 2 [ ˆΦ 2 = ˆN1 ˆN3 ˆN 1 ˆN 2 ˆN2 ˆN ] 3, ˆΦ 3 =[A A d 0 0]. Thus, V (xt ) < ε x(t) 2 for a sufficiently small ε>0. Therefore, system (8.1) with u(t) = 0andω(t) = 0 is asymptotically stable. This completes the proof.

185 H Control Design for Systems with Time-Varying Delay Remark Often there is no information on the derivative of the delay. In that case, a delay-dependent and rate-independent BRL for a delay satisfying only (8.2) can be derived by setting Q 3 = 0 in Theorem From the procedure used in the proof of Theorem 8.2.1, we obtain a corollary on the stability of system (8.1) with u(t) = 0 and ω(t) = 0. Corollary Consider system (8.1) with u(t) =0and ω(t) =0. Given scalars h 2 h 1 0 and μ, the system is asymptotically stable if there exist matrices P>0, Q i 0, i=1, 2, 3, Z j > 0, and ˆX j 0, j=1, 2, andany appropriately dimensioned matrices ˆN i,i = 1, 2, 3 such that LMIs (8.19)- (8.21) hold. Remark For h 1 =0,ifZ 1 = Z, Q 3 = Q, ˆN2 =0, ˆN3 =0, ˆN1 = X 11 X [ Y T T T 0 0 ] T X, ˆX2 =0, ˆX1 = , Q i = ε i I, i =1, 2, and Z 2 = ε j I (where ε j > 0, j=1, 2, 3 are sufficiently small scalars), then Corollary reduces to Theorem 2 in [9]. 8.3 Design of State-Feedback H Controller This section extends Theorem to the design of an H controller for system (8.1) under control law (8.5). Theorem Consider closed-loop system (8.6). For given scalars h 2 h 1 0, μ, andγ > 0, if there exist matrices L>0, R i 0, i=1, 2, 3, Y j 0, andw j > 0, j=1, 2, and any appropriately dimensioned matrices M j,j=1, 2, 3, andv such that the following matrix inequalities hold: Ξ 1 + Ξ 2 + Ξ2 T + h 2Y 1 + h 12 Y 2 h2 Ξ3 T h12 Ξ3 T Ξ4 T Ξ5 T Ξ6 T W W I 0 0 <0, I 0 I (8.22)

186 8.3 Design of State-Feedback H Controller 169 Y j M j 0, j =1, 2, (8.23) LW 1 j L Y 1 + Y 2 M 3 0, (8.24) LW1 1 L + LW2 1 L where 3 AL+LA T +BV +V T B T + R i A d L 0 0 B ω i=1 (1 μ)r Ξ 1 =, R R 2 0 γ 2 I Ξ 2 =[M 1 M 3 M 1 M 2 M 2 M 3 0], Ξ 3 =[AL + BV A d L 0 0 B ω ], Ξ 4 =[CL D ω ], Ξ 5 =[0 C d L 0 0 0], Ξ 6 =[DV ], then the system is asymptotically stable and satisfies J(ω) < 0 for all nonzero ω(t) L 2 [0, ) under the condition x(t) =0, t [ h 2, 0], andu(t) = VL 1 x(t) is a stabilizing H controller. Proof. If system (8.1) in Theorem is replaced with closed-loop system (8.6), then (8.7) should be changed to Φ 1 +Φ 2 +Φ T 2 +h 2X 1 +h 12 X 2 h2 ΦT 3 Z 1 h12 ΦT 3 Z 2 Φ T 4 Φ T 5 Φ T 6 Z Z Φ= <0, I 0 0 I 0 I where (8.25)

187 H Control Design for Systems with Time-Varying Delay Φ 3 =[A + BK A d 0 0 B ω ], Φ 6 =[DK ], and Φ 1 is obtained by replacing A in Φ 1 in (8.7) with A + BK; and the other parameters are defined in Theorem Define Π =diag{p 1, P 1,P 1, P 1, I}, Θ =diag{π, Z 1 1,Z 1 2, I, I, I}. Pre- and post-multiply Φ in (8.25) by Θ; pre- and post-multiply Ψ i,i=1, 2 and Ψ 3 in (8.8) and (8.9) by diag {Π, L}; and make the following changes in the variables: L = P 1, V = KL, M i = ΠN i L, R i = LQ i L, i =1, 2, 3, Y j = ΠX j Π, W j = Z 1 j, j =1, 2. Then, (8.22)-(8.24) are derived using the Schur complement. This completes the proof. Note that the conditions in Theorem are no longer LMI conditions due to the terms LW 1 j L, j =1, 2 in (8.23) and (8.24). As mentioned in Chapter 6, we can solve this nonconvex problem by using the idea for solving a cone complementarity problem in [13]. Defining new variables S j,j=1, 2forwhichLWj 1 L S j,j=1, 2and letting P = L 1, T j = S 1 j,andz j = W 1 j, j= 1, 2 convert this nonconvex problem into the following LMI-based nonlinear minimization problem: 2 Minimize Tr LP + (S j T j + W j Z j ) subject to j=1 (8.22) and Y j M j 0, Y 1 + Y 2 M 3 0, T j P 0, S j S 1 + S 2 Z j L I 0, S j I 0, W j I 0, j=1, 2. P T j Z j (8.26) The minimum H performance, γ min, can be found for given h 2 h 1 0 by using either the CCL or ICCL algorithm described in Chapter 6. Below, we use the ICCL algorithm because of its advantages.

188 8.4 Numerical Examples 171 Algorithm To find γ min : Step 1: Choose a sufficiently large initial γ>0 such that there exists a feasible solution to (8.22) and (8.26). Setγ min = γ. Step 2: Find a feasible set (P 0, L 0, V 0, R i0, M i0, Y j0, Z j0, W j0, S j0, T j0,i= 1, 2, 3, j=1, 2) that satisfies (8.22) and (8.26). Setk =0. Step 3: Solve the following LMI problem for the variables P, L, R i, M i,i= 1, 2, 3, Y j, Z j,w j, S j, T j,j=1, 2, andv : 2 Minimize Tr LP k+l k P + (S j T jk +S jk T j +W j Z jk +W jk Z j ) subject to (8.22) and (8.26). j=1 Set P k+1 = P, L k+1 = L, S j,k+1 = S j, T j,k+1 = T j, W j,k+1 = W j,and Z j,k+1 = Z j,j=1, 2. Step 4: For the K obtained in Step 3, if LMIs(8.8), (8.9), and(8.25) are feasible for the variables P, Q i, N i,i=1, 2, 3, X j,andz j,j=1, 2, then set γ min = γ, decreaseγ, and return to Step 2. Iftheyarenotfeasible within a specified number of iterations, then exit. Otherwise, set k = k+1 and go to Step 3. Remark Note that the iteration stop condition at the beginning of Step 4 in [2] and [14] is very strict. The gain matrix, K, and other decision variables, such as L, R i, M i,i = 1, 2, 3, W j,andy j,j = 1, 2thatwere obtained in the previous step must satisfy (8.22)-(8.24), which is the same as saying that the matrices P, Q i, N i,i=1, 2, 3, Z j,andx j,j=1, 2 obtained in the previous step must satisfy (8.8), (8.9), and (8.25) for the specified K. However, once K is obtained, the conditions in Theorem reduce to LMIs for P, Q i, N i,i =1, 2, 3, Z j,andx j,j =1, 2. So, we modified the stop condition in Algorithm to include a determination of whether or not LMIs (8.8), (8.9), and (8.25) are feasible, which may provide more freedom in the selection of variables, such as P, Q i, N i,i=1, 2, 3, Z j,and X j,j=1, Numerical Examples This section presents two numerical examples that demonstrate the benefits of the method described above.

189 H Control Design for Systems with Time-Varying Delay Example Consider the stability of system (8.1) with u(t) =0,ω(t) =0, and A = 0 1, A d = If (h 1 + h 2 )/2 =1andμ =0.8, the lower and upper bounds on d(t) are h 1 = 0.88 and h 2 = 1.12 in [15], and h 1 = 0.46 and h 2 = 1.54 in [8]. However, Corollary yields better values, namely, h 1 = 0.60 and h 2 =1.40. Table 8.1 lists values of the upper bound, h 2, for various μ and h 1 that were obtained with Corollary and those that are given in [4,8,16]. Although our corollary produces the same values as those in [16] for h 1 =0, the criteria in [16] are not applicable when h 1 > 0. Table 8.1. Allowable upper bound, h 2, for various h 1 (Example 8.4.1) μ Method h [4] h unknown μ [8] h [16] h Corollary h [8] h μ =0.3 [16] h Corollary h Example Consider system (8.1) with A = 0 0, A d = 1 1, B = 0, B ω = 1, [ ] [ ] C = 01, D ω =0, C d = 0 0, D =0.1. When d(t) is unknown and (h 1 + h 2 )/2 = 2, the range of delays for which system (8.1) with ω(t) = 0 is stable is [1.78, 2.22] for K = [ , ] in [17]. However, Algorithm produces the range

190 8.5 Conclusion 173 Table 8.2. Controller gain and number of iterations for Algorithm for γ = (Example 8.4.2) h 1 = h 2 Feedback gain Number of iterations from Algorithm Algorithm [3] [5] 1.1 [ , ] [ , ] [ , ] [0.0009, ] 7 [1.70, 2.30] for K =[ , ]. Thus, the method in this chapter produces both a larger range of delays and smaller gains than the method in [17] does. For a constant delay (d(t) =h 2 = h 1 )andγ =0.1287, the range of h 2 for which the system is stable is 0 h in [2], 0 h in [3], and 0 h in [5]. However, Algorithm yields the range 0 h Table 8.2 shows that this algorithm requires far fewer iterations than the ones in [3] or [5] because the stop condition is less strict. Regarding a time-varying delay, [5] only considered the case h 1 = 0; but d(t) can vary within an interval. Table 8.3 lists the minimum H performance, γ min, for the closed-loop system obtained for various h 1 and h 2.Note that our method yields better results than those in [5], even for h 1 =0, because we use both an improved BRL and a new algorithm. Table 8.3. γ min for various h 1 and h 2 (Example 8.4.2) h 2 h 1 μ =0.5 unknown μ [5] Algorithm [5] Algorithm Conclusion This chapter describes a new delay-dependent BRL derived using the IFWM approach of Chapter 3. Based on this BRL, an H controller is designed

191 H Control Design for Systems with Time-Varying Delay using the ICCL algorithm of Chapter 6. Two numerical examples show that this method is less conservative than others. References 1. E. Fridman and U. Shaked. Delay-dependent stability and H control: constant and time-varying delays. International Journal of Control, 76(1): 48-60, H. Gao and C. Wang. Comments and further results on A descriptor system approach to H control of linear time-delay systems. IEEE Transactions on Automatic Control, 48(3): , Y. S. Lee, Y. S. Moon, W. H. Kwon, and P. G. Park. Delay-dependent robust H control for uncertain systems with a state-delay. Automatica, 40(1): 65-72, X. Jiang and Q. L. Han. On H control for linear systems with interval timevarying delay. Automatica, 41(12): , S. Xu, J. Lam, and Y. Zou. New results on delay-dependent robust H control for systems with time-varying delays. Automatica, 42(2): , X. M. Zhang, M. Wu, Q. L. Han, and J. H. She. A new integral inequality approach to delay-dependent robust H control. Asian Journal of Control, 8(2): , Y. He, Q. G. Wang, L. Xie, and C. Lin. Further improvement of free-weighting matrices technique for systems with time-varying delay. IEEE Transactions on Automatic Control, 52(2): , Y. He, Q. G. Wang, C. Lin, and M. Wu. Delay-range-dependent stability for systems with time-varying delay. Automatica, 43(2): , M. Wu, Y. He, J. H. She, and G. P. Liu. Delay-dependent criteria for robust stability of time-varying delay systems. Automatica, 40(8): , Y. He, M. Wu, J. H. She, and G. P. Liu. Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic type uncertainties. IEEE Transactions on Automatic Control, 49(5): , Q. L. Han. On robust stability of neutral systems with time-varying discrete delay and norm-bounded uncertainty. Automatica, 40(6): , J. Lam, H. Gao, and C. Wang. Stability analysis for continuous systems with two additive time-varying delay components. Systems & Control Letters, 56(1): 16-24, E. L. Ghaoui, F. Oustry, and M. AitRami. A cone complementarity linearization algorithms for static output feedback and related problems. IEEE Transactions on Automatic Control, 42(8): , Y. S. Moon, P. Park, W. H. Kwon, and Y. S. Lee. Delay-dependent robust stabilization of uncertain state-delayed systems. International Journal of Control, 74(14): , 2001.

192 References E. Fridman. Stability of systems with uncertain delays: a new complete Lyapunov-Krasovskii functional. IEEE Transactions on Automatic Control, 51(5): , P. G. Park and J. W. Ko. Stability and robust stability for systems with a time-varying delay. Automatica, 43(10): , E. Fridman and U. Shaked. Descriptor discretized Lyapunov functional method: analysis and design. IEEE Transactions on Automatic Control, 51(5): , 2006.

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194 9. H Filter Design for Systems with Time-Varying Delay One problem with estimating the state from corrupted measurements is that, if we assume that the noise source is an arbitrary signal with bounded energy, then the well-known Kalman filtering scheme is no longer applicable. To handle that case, H filtering was proposed in [1]. It provides a guaranteed noise attenuation level [2 5]. H filtering for time-delay systems has been a hot topic in recent years, and design methods for delay-independent H filters have been presented in [6 10]. However, since delay-independent methods tend to be conservative, especially when the delay is small, attention has shifted to delay-dependent H filtering [11 16]. A descriptor model transformation was first employed in H filter design in [11]. Later, [12 16] devised less conservative results using Park s or Moon et al. s inequality. An improved method based on the FWM approach was reported in [17] for systems with a time-varying interval delay. Recently, Zhang & Han employed a new integral inequality, which is equivalent to the FWM approach, to study H filtering for neutral delay systems in [18] and robust H filtering for uncertain linear systems with a time-varying delay in [19]. These methods are less conservative than previous ones. A similar method was used to design H filters for discrete-time systems with a time-varying delay in [20, 21]. However, there is room for further improvement for systems with a time-varying delay. For example, for continuous-time systems with a time-varying delay, d(t), where 0 d(t) h and d(t) μ, [12, 19] ignored the term d(t) t h ẋ T (s)zẋ(s)ds in the derivative of the Lyapunov-Krasovskii functional, which may lead to considerable conservativeness. The best way to deal with this term is to keep it; that is, all the terms in hẋ T (t)zẋ(t) t h ẋt (s)zẋ(s)ds = hẋ T (t)zẋ(t) t d(t) ẋt (s)zẋ(s)ds d(t) ẋ T (s)zẋ(s)ds should be retained. On the other hand, for discrete-time systems with a time-varying t h delay, in the calculation of the difference of the Lyapunov function in [20, 22], the term k 1 l=k h ηt (l)zη(l) was increased to k 1 l=k d(k) ηt (l)zη(l) and

195 H Filter Design for Systems with Time-Varying Delay the term k d(k) 1 η T (l)zη(l) was ignored, which may also lead to considerable conservativeness. l=k h This chapter uses the IFWM approach to analyze the delay-dependent H performance of the error systems of both continuous-time and discretetime systems [23] with a time-varying delay. Useful terms in the derivative of a Lyapunov-Krasovskii functional and those in the difference of a Lyapunov function are retained, and the relationships among a time-varying delay, its upper bound, and their difference are taken into consideration. That treatment yields H filters that are designed in terms of LMIs. The resulting criteria are extended to systems with polytopic-type uncertainties. Numerical examples demonstrate the effectiveness of the method and its advantages. 9.1 H Filter Design for Continuous-Time Systems First, we use the IFWM approach to design a delay-dependent H filter for continuous-time systems with a time-varying delay Problem Formulation Consider the following system with a time-varying delay: ẋ(t) =Ax(t)+A d x(t d(t)) + Bω(t), y(t) =Cx(t)+C d x(t d(t)) + Dω(t), z(t) =Lx(t)+L d x(t d(t)), x(t) =φ(t), t [ h, 0], (9.1) where x(t) R n is the state vector; y(t) R r is the vector of the measured state; ω(t) R m is a noise signal vector belonging to L 2 [0, ); z(t) R p is the signal to be estimated; A, A d, B, C, C d,d, L, andl d are constant matrices with appropriate dimensions; the delay, d(t), is a time-varying differentiable function satisfying 0 d(t) h (9.2) and d(t) μ, (9.3)

196 9.1 H Filter Design for Continuous-Time Systems 179 where h>0andμare constants; and the initial condition, φ(t), is a continuously differentiable initial function of t [ h, 0]. The aim of this section is to design a full-order, linear, time-invariant, asymptotically stable filter for system (9.1). The state-space realization of the filter has the form ˆx(t) =A F ˆx(t)+B F y(t), ẑ(t) =C F ˆx(t)+D F y(t), (9.4) ˆx(0) = 0, where A F R n n, B F R n r, C F R p n,andd F R p r are filter parameters to be determined. Denote ζ(t) = x(t), e(t) =z(t) ẑ(t). (9.5) ˆx(t) Then, the filtering-error dynamics of system (9.1) are ζ(t) =Āζ(t)+ĀdEζ(t d(t)) + Bω(t), e(t) = Cζ(t)+ C d Eζ(t d(t)) + Dω(t), ζ(t) =[φ T (t) 0] T, t [ h, 0], (9.6) where Ā = A 0, Ā d = A d, B = B, B F C A F B F C d B F D [ ] [ ] E = I 0, C = L D F C C F, Cd = L d D F C d, D = DF D. The H filtering problem addressed in this section is stated as follows: Given scalars h>0, μ, and γ>0, find a full-order, linear, time-invariant, asymptotically stable filter with a state-space realization of the form (9.4) for system (9.1) such that, for any delay, d(t), satisfying (9.2) and (9.3), (1) filtering-error system (9.6) with ω(t) = 0 is asymptotically stable; and (2) the H performance e(t) 2 γ ω(t) 2 (9.7) is guaranteed under zero-initial conditions for all nonzero ω(t) L 2 [0, ) and a given γ>0.

197 H Filter Design for Systems with Time-Varying Delay Denote Λ =[A A d B C C d D L L d ]. (9.8) For a system with polytopic-type uncertainties, we consider the set of system matrices Λ Ω, whereω is the real, convex, polytopic domain p p Ω = Λ(λ) = λ j Λ j, λ j =1, λ j 0. (9.9) j=1 j=1 Here, Λ j =[A j A dj B j C j C dj D j L j L dj ],j=1, 2,,p are constant matrices with appropriate dimensions, and λ j,j=1, 2,,pare time-invariant uncertainties H Performance Analysis This subsection considers the case where the set of system matrices, Λ, is fixed. For this case, the following theorem holds. Theorem Consider filtering-error system (9.6). Given scalars h> 0, μ, andγ > 0, the system is asymptotically stable and (9.7) is satisfied under zero-initial conditions for all nonzero ω(t) L 2 [0, ) if there exist matrices P = P 1 P 2 > 0, Q i 0, i=1, 2, Z>0, andx 0, andany P 3 appropriately dimensioned matrices N and M such that the following matrix inequalities hold: Φ 1 + Φ 2 + Φ T 2 + hx hφt 3 Z ΦT 4 Φ(Λ) = hz 0 < 0, (9.10) I Ψ 1 = X N 0, (9.11) Z Ψ 2 = X M 0, (9.12) Z where

198 9.1 H Filter Design for Continuous-Time Systems 181 Φ 11 Φ 12 Φ 13 0 Φ 15 Φ 22 Φ 23 0 Φ 25 Φ 1 = Φ , Q 1 0 γ 2 I Φ 2 =[N 0 M N M 0], Φ 11 = P 1 A + A T P 1 + P 2 B F C + C T BF TP 2 T + Q 1 + Q 2, Φ 12 = P 2 A F + A T P 2 + C T BF TP 3, Φ 13 = P 1 A d + P 2 B F C d, Φ 15 = P 1 B + P 2 B F D, Φ 22 = P 3 A F + A T F P 3, Φ 23 = P2 T A d + P 3 B F C d, Φ 25 = P2 TB + P 3B F D, Φ 33 = (1 μ)q 2, Φ 3 =[A 0 A d 0 B], Φ 4 =[L D F C C F L d D F C d 0 D F D]. Proof. Choose the Lyapunov-Krasovskii functional candidate to be V (t, ζ t )=ζ T (t)pζ(t)+ t h x T (s)q 1 x(s)ds + x T (s)q 2 x(s)ds t d(t) 0 + ẋ T (s)zẋ(s)dsdθ, (9.13) h t+θ where P>0, Q i 0, i=1, 2, and Z>0areto be determined. From the Newton-Leibnitz formula, the following equations are true for any matrices N and M with appropriate dimensions: [ ] where 0=2η T (t)n 0=2η T (t)m [ x(t) x(t d(t)) x(t d(t)) x(t h) t d(t) ẋ(s)ds d(t) t h ẋ(s)ds η(t) = [ x T (t), ˆx T (t), x T (t d(t)), x T (t h), ω T (t) ] T., (9.14) ], (9.15) On the other hand, for any matrix X 0, the following equation is true:

199 H Filter Design for Systems with Time-Varying Delay 0= t h η T (t)xη(t)ds = hη T (t)xη(t) t h d(t) t h η T (t)xη(t)ds η T (t)xη(t)ds In addition, the following equation is also true: t d(t) ẋ T (s)zẋ(s)ds = ẋ T (s)zẋ(s)ds t h t d(t) η T (t)xη(t)ds. (9.16) d(t) t h ẋ T (s)zẋ(s)ds. (9.17) Calculating the derivative of V (t, ζ t ) along the solutions of system (9.6), adding the right sides of (9.14)-(9.16) to it, and using (9.17) yield V (t, ζ t )=2ζ T (t)p ζ(t)+x T (t)q 1 x(t) x T (t h)q 1 x(t h)+x T (t)q 2 x(t) Thus, (1 d(t))x T (t d(t))q 2 x(t d(t)) + hẋ T (t)zẋ(t) t h ẋ T (s)zẋ(s)ds 2ζ T (t)p ζ(t)+x T (t)(q 1 + Q 2 )x(t) x T (t h)q 1 x(t h) (1 μ)x T (t d(t))q 2 x(t d(t)) + hẋ T (t)zẋ(t) t d(t) +2η T (t)n +2η T (t)m d(t) t h ẋ T (s)zẋ(s)ds [ [ d(t) x(t) x(t d(t)) t h x(t d(t)) x(t h) V (t, ζ t )+e T (t)e(t) γ 2 ω T (t)ω(t) ẋ T (s)zẋ(s)ds ẋ(s)ds ] t d(t) d(t) t h ẋ(s)ds ] + hη T (t)xη(t) η T (t)xη(t)ds η T (t)xζ(t)ds. (9.18) t d(t) = η T (t) [ Φ 1 + Φ 2 + Φ T 2 + hx + hφt 3 ZΦ 3 + Φ T 4 Φ 4] η(t) t d(t) ξ T (t, s)ψ 1 ξ(t, s)ds d(t) t h ξ T (t, s)ψ 2 ξ(t, s)ds, (9.19) where ξ(t, s) = [ η T (t), ẋ T (s) ] T. Therefore, if Ψ i 0, i = 1, 2, and Φ 1 + Φ 2 + Φ T 2 + hx + hφt 3 ZΦ 3 + Φ T 4 Φ 4 < 0, which is equivalent to (9.10) by the Schur complement, then

200 9.1 H Filter Design for Continuous-Time Systems 183 V (t, ζ t )+e T (t)e(t) γ 2 ω T (t)ω(t) < 0. Following an argument similar to the one in [15] ensures that (9.7) holds. On the other hand, (9.10)-(9.12) imply that (9.20)-(9.22) hold, which guarantees that V (t, ζ t ) < ε ζ(t) 2 for a sufficiently small ε>0 and thus that error system (9.6) with ω(t) = 0 is asymptotically stable. ˆΦ 1 + ˆΦ 2 + ˆΦ T 2 + h ˆX hˆφ T 3 Z < 0, (9.20) hz ˆΨ 1 = ˆX ˆN 0, (9.21) Z ˆΨ 2 = ˆX ˆM 0, (9.22) Z where Φ 11 Φ 12 Φ 13 0 Φ ˆΦ 1 = 22 Φ 23 0, Φ 33 0 Q 1 [ ˆΦ 2 = ˆN 0 ˆM ˆN ˆM], ˆΦ 3 =[A 0 A d 0], and ˆN, ˆM, and ˆX 0 are decision variables with appropriate dimensions. This completes the proof. Remark Regarding the design of a full-order filter for a single delay in [15], where s =1,F =0,andG = 0, it is easy to see that Theorem in this section reduces to Theorem 1 in [15] for μ = 0 if we make the following assignments: X Y X = , N = 0, M =0, Q 1 = εi, where ε>0 is a sufficiently small scalar.

201 H Filter Design for Systems with Time-Varying Delay Remark Often there is either no information on the derivative of a delay, or the function d(t) is continuous but not differentiable. For these situations, we can carry out a delay-dependent and rate-independent performance analysis for a delay satisfying (9.2), but not necessarily (9.3), by choosing Q 2 = 0 in Theorem Note that the method in [15] cannot handle this case Design of H Filter This subsection uses Theorem to solve the H filter synthesis problem. Theorem Consider filtering-error system (9.6). For given scalars h> 0, μ, andγ>0, ifthereexistmatricesp 1 > 0, V > 0, Q i 0, i =1, 2, Z>0, and X 0, and any appropriately dimensioned matrices N, M, ĀF, B F, C F,and D F such that the following LMIs hold: Φ 1 + Φ 2 + Φ T 2 + h X h Φ T 3 Z Φ T 4 Φ(Λ) = hz 0 < 0, (9.23) I Ψ 1 = X N 0, (9.24) Z Ψ 2 = X M 0, (9.25) Z P 1 V > 0, (9.26) where Φ 11 Φ12 Φ13 0 Φ15 Φ22 Φ23 0 Φ25 Φ 1 = Φ33 0 0, Q 1 0 γ 2 I Φ 2 = [ ] N 0 M N M 0, Φ 11 = P 1 A + A T P 1 + B F C + C T BT F + Q 1 + Q 2, Φ 12 = ĀF + A T V + C T BT F,

202 9.1 H Filter Design for Continuous-Time Systems 185 Φ 13 = P 1 A d + B F C d, Φ 15 = P 1 B + B F D, Φ 22 = ĀF + ĀT F, Φ 23 = VA d + B F C d, Φ 25 = VB+ B F D, Φ 33 = (1 μ)q 2, Φ 3 =[A 0 A d 0 B], Φ 4 = [ L D F C C F L d D F C d 0 D F D ], then the system is asymptotically stable, the bound on the H noise attenuation level is γ, and either of the following is a suitable filter of the form (9.4): or A F = V 1 Ā F, B F = V 1 BF, C F = C F, D F = D F (9.27) A F = ĀF V 1, B F = B F, C F = C F V 1, D F = D F. (9.28) Proof. For P = P 1 P 2 > 0 defined in Theorem 9.1.1, define P 3 J 1 =diag{i, P 2 P3 1, I, I, I}, J 2 =diag{j 1,I,I}, J 3 =diag{j 1,I}. (9.29) Pre- and post-multiply Φ(Γ ) by J 2 and J2 T, respectively; pre- and postmultiply Ψ i,i =1, 2byJ 3 and J3 T, respectively; and define the following new variables: N = J 1 N, M = J1 M, X = J1 XJ1 T, V = P 2 P3 1 P2 T, Ā F = P 2 A F P3 1 P2 T, BF = P 2 B F, CF = C F P3 1 P2 T, DF = D F. (9.30) Then, LMIs (9.10)-(9.12) are equivalent to LMIs (9.23)-(9.25). In addition, by the Schur complement, P = P 1 P 2 > 0isequivalentto P 3 0 <P 1 P 2 P 1 3 P T 2 = P 1 V. (9.31)

203 H Filter Design for Systems with Time-Varying Delay Since V = P 2 P3 1 P2 T > 0, P 2 is nonsingular. Thus, using (9.30), we find that the following holds: A F B F = P ĀF BF P 2 T P 3 0. (9.32) C F D F 0 I C F DF 0 I However, P 2 and P 3 cannot be derived from the solutions of LMIs (9.23)- (9.26). Just as in [15], let the filter transfer function from y(t) toẑ(t) be Tẑy (s) =C F (si A F ) 1 B F + D F. (9.33) Replacing the filter matrices with (9.32) and considering the relationship V = P 2 P 1 3 P T 2 yield Tẑy (s) =C F (si A F ) 1 B F + D F = C F P2 T P 3 (si P2 1 Ā F P2 T P 3 ) 1 P2 1 B F + D F = C F (sv ĀF ) 1 BF + D F = C F (si V 1 Ā F ) 1 V 1 BF + D F = C F V 1 (si ĀF V 1 ) 1 BF + D F. In this way, the state-space realization of filter (9.27) or (9.28) is readily established. This completes the proof. Remark Proposition 11 in [19] can be derived from the above theorem by setting G in that paper to 0, making the following assignments, and using the Schur complement: N = π T 2, M =0, X = π T 2 Z 1 π 2, V = U, Ā F = N 1, B F = N 2, CF = N 3, DF = N 4,andQ 1 = εi (where ε>0 is a sufficiently small scalar). Thus, N and M provide extra freedom in the choice of these variables. For polytopic-type uncertainties, we have the following corollary. Corollary Consider filtering-error system (9.6) with polytopic-type uncertainties (9.9). For given scalars h>0, μ, andγ > 0, ifthereexist matrices P 1 > 0, V > 0, Q i 0, i=1, 2, Z > 0, and X 0, andany appropriately dimensioned matrices N, M, ĀF, BF, C F,and D F such that LMIs (9.24)-(9.26) and the following LMI hold: Φ(Λ j ) < 0, j =1, 2,,p, (9.34)

204 9.1 H Filter Design for Continuous-Time Systems 187 where Φ(Λ j ) is derived by replacing Λ with Λ j in Φ(Λ), which is defined in (9.23), then the system is robustly stable, the bound on the H noise attenuation level is γ, and either (9.27) or (9.28) constitute a suitable filter of the form (9.4) Numerical Examples The next two examples demonstrate the benefits of the method explained above. Example Consider system (9.1) with A = 2 0, A d = 1 0, B = 0, [ ] [ ] [ ] [ ] C = 1 0, C d = 0 0, D =1, L = 1 2, L d = 0 0. As mentioned in [19], the method in [11] fails for this example. Table 9.1 lists the results obtained by the method in this section along with those in [12] and [19] for h = 1 and various μ. Our method yields a smaller H - performance, γ min. Table 9.1. Minimum γ for h = 1 and various μ (Example 9.1.1) μ [12] [19] Theorem Example Consider system (9.1) with A = σ, A d =, B = 0.5, ρ [ ] [ ] [ ] [ ] C = 1 1, C d = 0.5 1, D =1, L = 0.5 2, L d = 0 0, ρ < 0.8, σ < 0.2.

205 H Filter Design for Systems with Time-Varying Delay Table 9.2. Minimum γ for various h and μ (Example 9.1.2) h μ [15] [19] Corollary unknown μ unknown μ The uncertainties can be expressed as polytopic-type ones [24]. Table 9.2 compares the results in [15, 19] with those obtained by Corollary Our results are clearly better. 9.2 H Filter Design for Discrete-Time Systems In this section, we use the IFWM approach to design a delay-dependent H filter for discrete-time systems with a time-varying delay Problem Formulation Consider the following system with a time-varying delay: x(k +1)=Ax(k)+A d x(k d(k)) + Bω(k), y(k) =Cx(k)+C d x(k d(k)) + Dω(k), z(k) =Lx(k)+L d x(k d(k)) + Gω(k), x(k) =φ(k), k = d 2, d 2 +1,, 0, (9.35) where x(k) R n is the state vector; y(k) R m is the vector of the measured state; ω(k) R q is a noise signal vector belonging to l 2 [0, + ); z(k) R p is thesignaltobeestimated;a, A d, B, C, C d, D, L, L d,andg are constant matrices with appropriate dimensions; d(k) is a time-varying delay satisfying d 1 d(k) d 2, (9.36)

206 9.2 H Filter Design for Discrete-Time Systems 189 where d 1 and d 2 are known positive integers and d 12 = d 2 d 1 ;andφ(k), k = d 2, d 2 +1,, 0 is a known given initial condition. The aim of this section is to design a full-order, linear, asymptotically stable filter for system (9.35). The state-space realization of the filter has the form ˆx(k +1)=A F ˆx(k)+B F y(k), ẑ(k) =C F ˆx(k)+D F y(k), (9.37) ˆx(0) = 0, where A F R n n, B F R n m, C F R p n,andd F R p m are filter parameters to be determined. Denote ζ(k) = x(k), e(k) =z(k) ẑ(k). (9.38) ˆx(k) Then, the filtering-error dynamics of system (9.35) are ζ(k +1)=Āζ(k)+ĀdEζ(k d(k)) + Bω(k), e(k) = Cζ(k)+ C d Eζ(k d(k)) + Dω(k), ζ(k) =[φ T (k) 0] T, k = d 2, d 2 +1,, 0, (9.39) where Ā = A 0, Ā d = A d, B = B, B F C A F B F C d B F D [ ] C = L D F C C F, Cd = L d D F C d, [ ] D = G D F D, E = I 0. The H filtering problem addressed in this section is stated as follows: Given integers d 2 d 1 > 0andascalarγ>0, find a full-order, linear, timeinvariant filter for system (9.35) with a state-space realization of the form (9.37) such that, for any delay, d(k), satisfying (9.36), (1) the filtering-error system (9.39) with ω(k) = 0 is asymptotically stable; and (2) the H performance e 2 γ ω 2 (9.40) is guaranteed under zero-initial conditions for all nonzero ω(k) l 2 [0, + ) and a given γ>0.

207 H Filter Design for Systems with Time-Varying Delay Denote Λ =[A A d B C C d D L L d G]. (9.41) For a system with polytopic-type uncertainties, we consider the set of system matrices, Λ Ω, whereω is the real, convex, polytopic domain p p Ω = Λ(λ) = λ j Λ j, λ j =1, λ j 0. (9.42) j=1 j=1 Here, Λ j =[A j A dj B j C j C dj D j L j L dj G j ],j =1, 2,,p are constant matrices with appropriate dimensions and λ j,j = 1, 2,,p are timeinvariant uncertainties H Performance Analysis This subsection considers the case where the set of system matrices, Λ, is fixed. For this case, the following theorem holds. Theorem Consider filtering-error system (9.39). Given integers d 2 d 1 > 0 andascalarγ>0, the system is asymptotically stable and (9.40) is satisfied under zero-initial conditions for all nonzero ω(k) l 2 [0, + ) if there exist matrices P > 0, Q i 0, i = 1, 2, 3, Z j > 0, j = 1, 2, X = X 11 X 12 0, Y = Y 11 Y 12 0, and any appropriately dimensioned X 22 Y 22 [ ] T, [ ] T,andS [ ] T matrices N = N1 T N2 T M = M1 T M2 T = S1 T S2 T such that the following matrix inequalities hold: Φ 1 d2 Φ T 2 Z 1 d12 Φ T 2 Z 2 Φ T 3 Φ T 4 P Z Φ = Z < 0, (9.43) I 0 P Ψ 1 = X N 0, (9.44) Z 1

208 9.2 H Filter Design for Discrete-Time Systems 191 Ψ 2 = Y S 0, (9.45) Z 2 Ψ 3 = X + Y M 0, (9.46) Z 1 + Z 2 where Φ 11 Φ 12 E T S 1 E T M 1 0 Φ 22 S 2 M 2 0 Φ 1 = Q 1 0 0, Q 2 0 γ 2 I Then, where Φ 11 = E T [Q 1 + Q 2 +(d 12 +1)Q 3 + N 1 + N T 1 + d 2X 11 + d 12 Y 11 ]E P, Φ 12 = E T [ N T 2 N 1 + M 1 S 1 + d 2 X 12 + d 12 Y 12 ], Φ 22 = Q 3 N 2 N T 2 + M 2 + M T 2 S 2 S T 2 + d 2X 22 + d 12 Y 22, Φ 2 =[(A I)E A d 0 0 B], Φ 3 = [ C Cd 0 0 D ], Φ 4 = [ Ā Ād 0 0 B ], d 12 = d 2 d 1. Proof. Let η(l) =x(l +1) x(l). (9.47) x(k +1)=x(k)+η(k), (9.48) η(k) =x(k +1) x(k) =(A I)x(k)+A d x(k d(k)) + B 1 ω(k) =(A I)Eζ(k)+A d x(k d(k)) + B 1 ω(k) = Φ 2 ξ 1 (k), (9.49) ξ 1 (k) =[ζ T (k), x T (k d(k)), x T (k d 1 ), x T (k d 2 ), ω T (k)] T. Choose the Lyapunov function candidate to be V (k) =V 1 (k)+v 2 (k)+v 3 (k)+v 4 (k), (9.50)

209 H Filter Design for Systems with Time-Varying Delay where V 1 (k) =ζ T (k)pζ(k), 0 V 2 (k) = V 3 (k) = V 4 (k) = θ= d 2+1 k 1 k 1 l=k 1+θ l=k d 1 x T (l)q 1 x(l)+ d 1+1 θ= d 2+1 k 1 l=k 1+θ η T (l)z 1 η(l)+ k 1 d 1 θ= d 2+1 l=k d 2 x T (l)q 2 x(l), x T (l)q 3 x(l); k 1 l=k 1+θ η T (l)z 2 η(l), and P>0, Q i 0, i=1, 2, 3, and Z j > 0, j=1, 2 are to be determined. Defining ΔV (k) =V (k +1) V (k) yields where ΔV (k) =ΔV 1 (k)+δv 2 (k)+δv 3 (k)+δv 4 (k), ΔV 1 (k) =ζ T (k +1)Pζ(k +1) ζ T (k)pζ(k) = ξ T 1 (k)φ T 4 PΦ 4 ξ 1 (k) ζ T (k)pζ(k), ΔV 2 (k) =d 2 η T (k)z 1 η(k) k 1 l=k d 2 η T (l)z 1 η(l) k d 1 1 +d 12 η T (k)z 2 η(k) η T (l)z 2 η(l), l=k d 2 = ξ T 1 (k)φt 2 (d 2Z 1 + d 12 Z 2 )Φ 2 ξ 1 (k) k d 1 1 l=k d(k) k 1 l=k d(k) η T (l)z 1 η(l) k d(k) 1 η T (l)z 2 η(l) η T (l)(z 1 + Z 2 )η(l), l=k d 2 ΔV 3 (k) =x T (k)(q 1 + Q 2 )x(k) x T (k d 1 )Q 1 x(k d 1 ) x T (k d 2 )Q 2 x(k d 2 ) = ζ T (k)e T (Q 1 + Q 2 )Eζ(k) x T (k d 1 )Q 1 x(k d 1 ) x T (k d 2 )Q 2 x(k d 2 ), ΔV 4 (k) =(d 2 d 1 +1)x T (k)q 3 x(k) k d 1 l=k d 2 x T (l)q 3 x(l) (d 12 +1)ζ T (k)e T Q 3 Eζ(k) x T (k d(k))q 3 x(k d(k)). From (9.47), we have

210 9.2 H Filter Design for Discrete-Time Systems 193 0=x(k) x(k d(k)) = Eζ(k) x(k d(k)) k 1 l=k d(k) k 1 l=k d(k) η(l) η(l), k d(k) 1 0=x(k d(k)) x(k d 2 ) η(l), 0=x(k d 1 ) x(k d(k)) l=k d 2 k d 1 1 l=k d(k) η(l). [ Then, the following equations hold for any matrices N = N1 T N2 T ] T with appropriate dimensions: [ M T 1 M T 2 ] T,andS = [ S T 1 S T 2 0=2 [ x T (k)n 1 + x T ] (k d(k))n 2 Eζ(k) x(k d(k)) l=k d(k) k 1 l=k d(k) ] T, M = η(l) =2 [ ζ T (k)e T N 1 + x T ] (k d(k))n 2 k 1 Eζ(k) x(k d(k)) η(l), (9.51) 0=2 [ ζ T (k)e T M 1 + x T ] (k d(k))m 2 x(k d(k)) x(k d 2 ) 0=2 [ ζ T (k)e T S 1 + x T ] (k d(k))s 2 x(k d 1 ) x(k d(k)) k d(k) 1 l=k d 2 k d 1 1 l=k d(k) η(l), (9.52) η(l). (9.53) On the other hand, for any matrices X = X 11 X 12 0andY = X 22 Y 11 Y 12 0, the following equations are true: Y 22

211 H Filter Design for Systems with Time-Varying Delay where 0= 0= k 1 l=k d 2 ξ T 2 (k)xξ 2(k) = d 2 ξ T 2 (k)xξ 2 (k) k d 1 1 k 1 l=k d(k) l=k d 2 ξ T 2 (k)yξ 2(k) = d 12 ξ T 2 (k)yξ 2 (k) k d 1 1 l=k d(k) k 1 l=k d 2 ξ T 2 (k)xξ 2(k) k d(k) 1 ξ2 T (k)xξ 2 (k) ξ1 T (k)xξ 2 (k), (9.54) l=k d 2 k d 1 1 l=k d 2 ξ T 2 (k)yξ 2(k) k d(k) 1 ξ2 T (k)yξ 2 (k) ξ2 T (k)yξ 2 (k), (9.55) l=k d 2 ξ 2 (k) = [ x T (k), x T (k d(k)) ] T = [ ζ T (k)e T, x T (k d(k)) ] T. Adding the right sides of (9.51)-(9.55) to ΔV (k) yields ΔV (k)+e T (k)e(k) γ 2 ω T (k)ω(k) ξ T 1 (k) [ Φ 1 + Φ T 2 (d 2 Z 1 + d 12 Z 2 )Φ 2 + Φ T 3 Φ 3 + Φ T 4 PΦ 4 ] ξ1 (k) k 1 l=k d(k) ξ T 3 (k, l)ψ 1ξ 3 (k, l) k d 1 1 l=k d(k) ξ T 3 (k, l)ψ 2ξ 3 (k, l) k d(k) 1 ξ3 T (k, l)ψ 3ξ 3 (k, l), (9.56) l=k d 2 where ξ 3 (k, l) = [ ξ2 T (k), η T (l) ] T. Thus, if Ψ i 0, i = 1, 2, 3andΦ 1 + Φ T 2 (d 2Z 1 + d 12 Z 2 )Φ 2 + Φ T 3 Φ 3 + Φ T 4 PΦ 4 < 0, which is equivalent to (9.43) by the Schur complement, then ΔV + e T (k)e(k) γ 2 ω T (k)ω(k) < 0. If we follow an argument similar to the one in [20], this ensures that (9.40) holds under zero-initial conditions for all nonzero ω(k) L 2 [0, + ) andagivenγ>0. On the other hand, (9.43) implies that the following matrix inequality holds: ˆΦ 1 d2 ˆΦT 2 Z 1 d12 ˆΦT 2 Z 2 ˆΦT 4 P Z ˆΦ = < 0, Z 2 0 P where

212 Φ 11 Φ 12 E T S 1 E T M 1 Φ ˆΦ 1 = 22 S 2 M 2, Q 1 0 Q 2 ˆΦ 2 =[(A I)E A d 0 0], ˆΦ 4 = [ Ā Ād 0 0 ], 9.2 H Filter Design for Discrete-Time Systems 195 thereby guaranteeing that ΔV (k) < 0, which means that error system (9.39) with ω(k) = 0 is asymptotically stable This completes the proof. Remark The term k d(k) 1 l=k d 2 η T (l)zη(l), which was ignored in [20, 22], is retained in Theorem to overcome the conservativeness of those methods. Furthermore, d 2, which was increased to 2d 2 d 1 in [25], is separated into two parts (d(k), d 2 d(k)) in the procedure for proving Theorem Just as for Theorem 2 in [16] and Proposition 2 in [20], (9.43) has an equivalent form, which we obtain by introducing the three variables H 1, H 2, and T : Φ 1 d2 Φ T 2 H 1 d12 Φ T 2 H 2 Φ T 3 Φ T 4 T Λ Φ = Λ < 0, I 0 Λ 3 where Λ i = Z i H i Hi T Λ 3 = P T T T., i =1, 2, We employ a parameter-dependent Lyapunov function to deal with error system (9.39) when it has polytopic-type uncertainties (9.42), which gives us the following corollary. Corollary Consider filtering-error system (9.39) with polytopic-type uncertainties (9.42). Given integers d 2 d 1 > 0 and a scalar γ > 0, the system is robustly stable and (9.40) is satisfied under zero-initial conditions for all nonzero ω(k) l 2 [0, + ) if there exist matrices P j > 0, Q ij 0,

213 H Filter Design for Systems with Time-Varying Delay Z 1j > 0, Z 2j = Z2j T > 0, i=1, 2, 3, j=1, 2,,p, X = X 11 X 12 0, X 22 and Y = Y 11 Y 12 0, and any appropriately dimensioned matrices N j = Y 22 [ ] T, [ ] T, [ N1j T N2j T Mj = M1j T M2j T Sj = S1j T ] T, S2j T j =1, 2,,p, H1, H 2,andT such that the following matrix inequalities hold for j =1, 2,,p: Φ (j) ] T 1 d2 [Φ (j) [ ] T [ ] T [ ] T 2 H1 d12 Φ (j) 2 H2 Φ (j) 3 Φ (j) 4 T Λ 1j Λ 2j 0 0 < 0, (9.57) I 0 Λ 3j Ψ 1j = X N j 0, (9.58) Z 1j Ψ 2j = Y M j 0, (9.59) Z 2j Ψ 3j = X + Y S j 0, (9.60) Z 1j + Z 2j where Φ (j) 1 = Φ (j) 11 Φ (j) 12 E T S 1j E T M 1j 0 Φ (j) 22 S 2j M 2j 0 Q 1j 0 0, Q 2j 0 γ 2 I Φ (j) 11 = ET Ξ (j) 11 E P j, Ξ (j) 11 = Q 1j + Q 2j +(d 12 +1)Q 3j + N 1j + N1j T + d 2X 11 + d 12 Y 11, Φ (j) 12 = ET Ξ (j) 12, Ξ (j) 12 = N 2j T N 1j + M 1j S 1j + d 2 X 12 + d 12 Y 12, Φ (j) 22 = Q 3j N 2j N2j T + M 2j + M2j T S 2j S2j T + d 2X 22 + d 12 Y 22,

214 9.2 H Filter Design for Discrete-Time Systems 197 Φ (j) 2 =[(A j I)E A dj 0 0 B j ], Φ (j) 3 = [ Cj Cdj 0 0 D ] j, Φ (j) 4 = [ Ā j Ā dj 0 0 B ] j, Λ ij = Z ij H i Hi T, i =1, 2, Λ 3j = P j T T T ; and Āj, Ādj, B j, C j, C dj,and D j are defined in the same way as Ā, Ād, B, C, C d,and D in (9.39) by replacing the elements in Λ with those in Λ j Design of H Filter This subsection uses Corollary to solve the H filter synthesis problem. Theorem Consider filtering-error system (9.39). For given integers d 2 d 1 > 0 and a scalar γ>0, if there exist matrices P j = P 1j P 2j > 0, P 3j Q ij 0, Z 1j > 0, Z 2j > 0, i=1, 2, 3, j=1, 2,,p, X = X 11 X 12 X 22 0, andy = Y 11 Y 12 0, and any appropriately dimensioned matrices Y 22 [ ] T, [ ] T, [ N j = N1j T N2j T Mj = M1j T M2j T Sj = S1j T ] T, S2j T j =1, 2,,p, H 1, H 2, T 1, V 1, V 2, ĀF, B F, C F,and D F such that LMIs (9.58)-(9.60) and the following LMI hold for j =1, 2,,p: Ξ (j) 1 Ξ (j) 2 Ξ (j) 4 Ξ (j) 3 0 < 0, (9.61) Ξ (j) 5 where Ξ (j) 1 = ˆΞ (j) 11 P 2j Ξ (j) 12 S 1j M 1j 0 P 3j Φ (j) 22 S 2j M 2j 0, Q 1j 0 0 Q 2j 0 γ 2 I

215 H Filter Design for Systems with Time-Varying Delay ˆΞ (j) 11 = Ξ(j) Ξ (j) 2 = Ξ (j) [ 11 P 1j, d2 [ ] T [ H1 d12 Φ (j) 2 Φ (j) 2 ] T H2 [ Φ (j) 3 ] T ], 3 =diag { Z 1j H 1 H1 T, Z 2j H 2 H2 T, I }, A T j T 1 T + C T B j F T AT j V 1 T + C T B j F T Ā T F Ā T F Ξ (j) A 4 = T dj T 1 T + C T B dj F T AT dj V 1 T + C T B dj T F, Bj TT 1 T + DT B j F T BT j V 1 T + DT B j F T Ξ (j) 5 = P 1j T 1 T1 T P 2j V1 T V 2, P 3j V2 T V 2 T and Ξ (j) 11, Ξ(j) 12,andΦ(j) 22 are defined in (9.57), then the system is robustly stable and (9.40) is satisfied under zero-initial conditions for all nonzero ω(k) l 2 [0, + ), and either of the following is a suitable filter of the form (9.37): A F = V2 1 Ā F, B F = V 1 2 B F, C F = C F, D F = D F (9.62) or A F = ĀF V 1 2, B F = B F, C F = C F V 1 2, D F = D F. (9.63) Proof. Let T = T 1 T 2. T 3 T 4 (9.64) Inequality (9.57) shows that T + T T > 0. Thus, T 4 + T4 T > 0, which implies that T 4 is invertible. Define J 1 = I 0, J 2 =diag{j 1,I,I,I,I,I,I,I,J 1 }. (9.65) 0 T 2 T4 1 Pre- and post-multiply (9.57) by J 2 following new variables: and J2 T, respectively; and define the

216 V 1 = T 2 T 1 4 T 3, V 2 = T 2 T 1 4 T T 2, Pj = 9.2 H Filter Design for Discrete-Time Systems 199 P 1j P 2j P 3j = J 1 P j J1 T, Ā F = T 2 A F T 1 4 T T 2, BF = T 2 B F, CF = C F T 1 4 T T 2, DF = D F. (9.66) Thus, (9.57) is equivalent to (9.61). On the other hand, Ξ (j) 5 < 0 implies that V 2 = T 2 T4 1 T2 T > 0, which further implies that T 2 is nonsingular. Thus, (9.66) yields the following: A F B F = T ĀF BF T 2 T T 4 0. (9.67) C F D F 0 I C F DF 0 I However, T 2 and T 4 cannot be derived from the solutions of LMIs (9.58)- (9.61). Let the filter transfer function from y(k) toẑ(k) be Tẑy (z) =C F (zi A F ) 1 B F + D F. (9.68) Replacing the filter matrices with (9.67) and using the fact that V 2 = T 2 T 1 4 T T 2 yield Tẑy (z) =C F (zi A F ) 1 B F + D F = C F T2 T T 4 (zi T2 1 Ā F T2 T T 4 ) 1 T2 1 B F + D F = C F (zv 2 ĀF ) 1 BF + D F = C F (zi V2 1 Ā F ) 1 V2 1 B F + D F = C F V2 1 (zi ĀF V2 1 ) 1 BF + D F. Thus, the state-space realization of filter (9.62) or (9.63) is readily established. This completes the proof Numerical Example The numerical example in this subsection demonstrates the benefits of the method described above. Example Consider system (9.35) with A = 0.9 0, A d = 0.1 σ, B = 0, C = φ [ ] [ ] C d = , D =1, L = 1 2, L d = φ 0.2, σ 0.1. [ ] 1 1, [ ] , G = 0.5,

217 H Filter Design for Systems with Time-Varying Delay It is clear that the uncertainties can be expressed as polytopic-type uncertainties [24]. Table 9.3 compares the results in [20] with those obtained with Theorem Our results are markedly better. Moreover, we can use (9.62) and (9.63) to calculate the filter parameters for d 1, d 2,andγ. For example, when d 1 =1,d 2 =5,andγ =3.5555, the parameters are A F =, B F = , C F = [ ] 10 4, D F = In addition, when d 1 =1andd 2 =2,weobtainaγ min of by setting A F =, B F = , [ ] C F = , D F = Table 9.3. γ min for d 2 = 5 and various d 1 (Example 9.2.1) d [20] Theorem Conclusion This chapter focuses on the design of H filters for both continuous-time and discrete-time systems with a time-varying delay. The IFWM approach is first used to carry out a delay-dependent H performance analysis for error systems. The resulting criteria are extended to systems with polytopic-type uncertainties. Then, based on the results of the performance analysis, H filters are designed in terms of LMIs. Finally, numerical examples demonstrate that this method is less conservative than others. References 1. A. Elsayed and M. J. Grimble. A new approach to H design of optimal digital linear filters. IMA Journal of Mathematical Control and Information, 6(2): , 1989.

218 References C. E. de Souza, L. Xie, and Y. Wang. H filtering for a class of uncertain nonlinear systems. Systems & Control Letters, 20(6): , K. M. Nagpal and P. P. Khargonekar. Filtering and smoothing in an H setting. IEEE Transactions on Automatic Control, 36(2): , L. Xie, C. E. de Souza, and M. Fu. H estimation for discrete-time linear uncertain systems. International Journal of Robust and Nonlinear Control, 1(6): , H. Gao, J. Lam, and C. Wang. Mixed H 2/H filtering for continuous-time polytopic systems: a parameter-dependent approach. Circuits, Systems & Signal Processing, 24(6): , A. Pila, U. Shaked, and C. E. de Souza. H filtering for continuous-time linear systems with delay. IEEE Transactions on Automatic Control, 44(7): , Z. Wang and F. Yang. Robust filtering for uncertain linear systems with delayed states and outputs. IEEE Transactions on Circuits and Systems I, 49(1): , Z. Wang and D. W. C. Ho. Filtering on nonlinear time-delay stochastic systems. Automatica, 39(1): , Z. Wang, D. W. C. Ho, and X. Liu. Robust filtering under randomly varying sensor delay with variance constraints. IEEE Transactions on Circuits and Systems II, 51(6): , Z. Wang, F. Yang, D. W. C. Ho, and X. Liu. Robust H filtering for stochastic time-delay systems with missing measurements. IEEE Transactions on Signal Processing, 54(7): , E. Fridman and U. Shaked. A new H filter design for linear time-delay systems. IEEE Transactions on Signal Processing, 49(11): , E. Fridman, U. Shaked, and L. Xie. Robust H filtering of linear systems with time-varying delay. IEEE Transactions on Automatic Control, 48(1): , E. Fridman and U. Shaked. An improved delay-dependent H filtering of linear neutral systems. IEEE Transactions on Signal Processing, 52(3): , H. Gao and C. Wang. Robust L 2 L filtering for uncertain systems with multiple time-varying state delays. IEEE Transactions on Circuits and Systems I, 50(4): , H. Gao and C. Wang. Delay-dependent robust H and L 2 L filtering for a class of uncertain nonlinear time-delay systems. IEEE Transactions on Automatic Control, 48(9): , H. Gao and C. Wang. A delay-dependent approach to robust H filtering for uncertain discrete-time state-delayed systems. IEEE Transactions on Signal Processing, 52(6): , Y. He, Q. G. Wang, and C. Lin. An improved H filter design for systems with time-varying interval delay. IEEE Transactions on Circuits and Systems II, 53(11): , 2006.

219 H Filter Design for Systems with Time-Varying Delay 18. X. M. Zhang and Q. L. Han. Stability analysis and H filtering for delay differential systems of neutral type. IET Control Theory & Applications, 1(3): , X. M. Zhang and Q. L. Han. Robust H filtering for a class of uncertain linear systems with time-varying delay. Automatica, 44(1): , X. M. Zhang and Q. L. Han. Delay-dependent robust H filtering for uncertain discrete-time systems with time-varying delay based on a finite sum inequality. IEEE Transactions on Circuits and Systems II, 53(12): , X. M. Zhang and Q. L. Han. A new finite sum inequality approach to delaydependent H-infinity control of discrete-time systems with time-varying delay. International Journal of Robust and Nonlinear Control, 18(6): , H. Gao, X. Meng, and T. Chen. A new design of robust H filters for uncertain discrete-time state-delayed systems. Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, USA, , Y.He,G.P.Liu,D.Rees,andM.Wu.H filtering for discrete-time systems with time-varying delay. Signal Processing, 89(3): , Y. He, M. Wu, J. H. She, and G. P. Liu. Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties. IEEE Transactions on Automatic Control, 49(5): , Y. He, M. Wu, Q. L. Han, and J. H. She, Delay-dependent H control of linear discrete-time systems with an interval-like time-varying delay. International Journal of Systems Science, 39(4): , 2008.

220 10. Stability of Neural Networks with Time-Varying Delay Neural networks are useful in signal processing, pattern recognition, static image processing, associative memory, combinatorial optimization, and other areas [1]. Although considerable effort has been expended on analyzing the stability of neural networks without a delay, in the real world such networks often have a delay due, for example, to the finite switching speed of amplifiers in electronic networks and to the finite signal propagation speed in biological networks. So, the stability of different classes of neural networks with a delay has become an important topic [2 20]. The criteria in these papers are based on various types of stability (asymptotic, complete, absolute, exponential, and so on); and they can be classified into two categories according to their dependence on information about the length of a delay: delay-independent [3 10, 13] and delay-dependent [2, 11 20]. Since delay-independent criteria tend to be conservative, especially when the delay is small or varies within an interval, the delay-dependent type receives greater attention. [13] presented stability criteria for the global asymptotic stability of a class of neural networks with multiple delays. They have two main weaknesses. First, the nonlinear parts were handled using inequalities rather than the S-procedure, which is the most effective way of dealing with nonlinearities. Second, no information on nonlinearities was included in the Lyapunov- Krasovskii functional, even though including it would have yielded better results. On the other hand, [16 19] took the range of a time-varying delay in a neural network to be from 0 to an upper bound. In practice, however, timevarying interval delays are often encountered for which the interval does not necessarily start from zero. The stability criteria for neural networks with a time-varying delay in [16 19] are conservative for this case because they do not use information on the lower bound of the delay. To our knowledge, few reports have appeared on the stability of neural networks with a time-varying interval delay.

221 Stability of Neural Networks with Time-Varying Delay As pointed out in [12], the property of exponential stability is particularly important when the exponential convergence rate is used to determine the speed of neural computations. Thus, in general, it is not only of theoretical interest but also of practical importance to determine the exponential stability of, and to estimate the exponential convergence rate of, dynamic neural networks. Accordingly, a great number of sufficient conditions guaranteeing the global exponential stability of both continuous-time [3, 4, 7, 9, 12, 14, 15, 17, 20 22] and discrete-time [23 26] neural networks with constant and/or time-varying delays have been derived. Among them, delaydependent exponential stability criteria have been attracting a great deal of attention [12,17,20,23 26] because they exploit information on the length of a delay, which makes them less conservative than delay-independent ones. However, for continuous-time neural network systems with a time-varying delay, some negative terms in the derivative of the Lyapunov-Krasovskii functional tend to be ignored when delay-dependent stability criteria are derived [12,17,18,20]. For example, the negative term derivative of 0 h t h e2ks ż T (s)zż(s)ds in the t+θ e2ks ż T (s)zż(s)dsdθ was ignored in [20], which may lead to considerable conservativeness. [17] used e 2k(t h) t t d(t) żt (s)zż(s)ds as an estimate of the negative term e 2k(t h) t t h żt (s)zż(s)ds; but they ignored the other term, e 2k(t h) t d(t) ż T (s)zż(s)ds, whichmayalsolead t h to considerable conservativeness. For discrete-time recurrent neural network systems with a time-varying delay, the derivations of delay-dependent criteria generally employ a fixed model transformation of the original system [23, 24, 26], which may lead to conservativeness. Recently, [25] employed the FWM approach to study the delay-dependent stability of discrete-time neural networks with a time-varying delay. However, further research is still possible. For example, the delay, d(k) (whereh 1 d(k) h 2 ), was increased to h 2 ; and h 2 d(k) was increased to h 2 h 1.Inotherwords,h 2 = d(k)+h 2 d(k) was increased to 2h 2 h 1, which may lead to considerable conservativeness. In this chapter, first, the delay-dependent stability of neural networks with multiple time-varying delays is investigated by constructing a class of Lyapunov-Krasovskii functionals that contain information on nonlinearities [16]; and the S-procedure and FWM approach are used to derive a delaydependent stability criterion. This criterion is shown to include the delayindependent and rate-dependent one, and it is extended to a delay-dependent and rate-independent stability criterion for multiple unknown time-varying delays. Second, the IFWM approach is used to establish stability criteria for

222 10.1 Stability of Neural Networks with Multiple Delays 205 neural networks with a time-varying interval delay [27]. Third, the FWM and IFWM approaches are employed to derive delay-dependent exponential stability criteria for neural networks with a time-varying delay [17, 28]. Finally, for discrete-time recurrent neural networks with a time-varying delay, the IFWM approach is used to establish less conservative criteria without ignoring any useful terms in the difference of a Lyapunov function [29]. Numerical examples illustrate the effectiveness of these methods and the improvement over others Stability of Neural Networks with Multiple Delays This section examines the stability of neural networks with multiple timevarying delays by constructing a class of Lyapunov-Krasovskii functionals containing information on nonlinearities Problem Formulation Consider the following neural network with multiple time-varying delays: u(t) = Au(t)+W (0) g(u(t)) + r W (k) g(u(t d k (t))) + J, (10.1) k=1 where u(t) =[u 1 (t), u 2 (t),, u n (t)] T is the neural state [ vector; ] A = diag {a 1,a 2,,a n } is a positive diagonal matrix; W (k) = w (k) ij n n,k= 0, 1,,rare interconnection matrices; g(u) =[g 1 (u 1 ),g 2 (u 2 ),,g n (u n )] T is the vector of neural activation functions, and g(0) = 0; J =[J 1, J 2,, J n ] T is a constant input vector; and the delays, d k (t), k =1, 2,,r,are time-varying differentiable functions. In this section, the delays are assumed to satisfy one or both of the following conditions: 0 d k (t) h k, d k (t) μ k, (10.2) (10.3) where h k,andμ k,k=1, 2,,r are constants. In addition, the activation functions, g j ( ), j=1, 2,,n, of the neurons in system (10.1) are assumed to satisfy the condition

223 Stability of Neural Networks with Time-Varying Delay 0 g j(x) g j (y) x y σ j, x, y R, x y, (10.4) where σ j,j=1, 2,,n are positive constants. We use the transformation x( ) =u( ) u to shift the equilibrium point u =[u 1, u 2,, u n ]T of system (10.1) to the origin, which means that the following holds: r 0= Au (t)+w (0) g (u (t)) + W (k) g (u (t d k (t))) + J. (10.5) k=1 System equation (10.1) minus (10.5) yields r ẋ(t) = Ax(t)+W (0) f(x(t)) + W (k) f(x(t d k (t))), (10.6) k=1 where x =[x 1,x 2,,x n ] T is the state vector of the transformed system; f(x) =[f 1 (x 1 ),f 2 (x 2 ),,f n (x n )] T ;andf j (x j )=g j (x j +u j ) g j(u j ),j= 1, 2,,n. Note that the functions f j ( ) satisfy the condition 0 f j(x j ) x j σ j, x j 0,j=1, 2,,n, (10.7) which is equivalent to f j (x j )[f j (x j ) σ j x j ] 0, j=1, 2,,n. (10.8) Stability Criteria In this subsection, we use a new class of Lyapunov-Krasovskii functionals, the S-procedure, and the FWM approach to establish stability criteria. Theorem Consider neural network (10.6) with time-varying delays, d k (t), k=1, 2,,r, that satisfy both (10.2) and (10.3). Given scalars h k > 0 and μ k, the system is asymptotically stable at the origin if there exist matrices P > 0, Q k 0, R k 0, Z k > 0, Λ=diag{λ 1, λ 2,, λ n } 0, T = diag{t 1, t 2,, t n } 0, ands k =diag{s k1, s k2,, s kn } 0, and any appropriately dimensioned matrices N kj and M kj,k =0, 1,,r, j = 1, 2,,r such that the following LMI holds:

224 where 10.1 Stability of Neural Networks with Multiple Delays 207 Ξ 11 Ξ 12 Ξ 13 + ΣT Ξ 14 h 1 M 01 h r M 0r Ξ 22 Ξ 23 Ξ 24 + Σ s h 1 M 1 h r M r Ξ 33 2T Ξ 34 h 1 N 01 h r N 0r Ξ = Ξ 44 2S h 1 N 1 h r N r < 0, (10.9) h 1 Z h r Z r Ξ 11 = Φ 11 + A T HA + [ r Ξ 12 = M1k T M 01, k=1 r k=1 Ξ 13 = Φ 13 A T HW (0) + Ξ 14 = Φ 14 A T HW 1r + ( M0k + M0k T ), r M2k T M 02,, k=1 r k=1 [ r k=1 N T 0k, N T 1k, ] r Mrk T M 0r, k=1 r N2k, T, k=1 r k=1 N T rk M 11 M11 T M 12 M21 T M 1r Mr1 T M 22 M22 T Ξ 22 = Φ 22 + M 2r M T r2...,... M rr Mrr T Ξ 23 =[ N 01, N 02,, N 0r ] T, N11 T N21 T Nr1 T N22 T Ξ 24 = N T r2...,... Nrr T Σ s =diag{σs 1, ΣS 2,, ΣS r }, Ξ 33 = Φ 33 +[W (0) ] T HW (0), Ξ 34 = Φ 34 +[W (0) ] T HW 1r, Ξ 44 = Φ 44 + W T 1rHW 1r, ],

225 Stability of Neural Networks with Time-Varying Delay and Φ 11 = PA AP + r Q k, k=1 S =diag{s 1, S 2,, S r }, Σ =diag{σ 1, σ 2,, σ r }, W 1r = [ W (1), W (2),, W (r)], r H = h k Z k, k=1 M k = [ M T 1k, MT 2k,, MT rk] T, k =1, 2,,r, N k = [ N T 1k, NT 2k,, NT rk] T, k =1, 2,,r. Φ 13 = PW (0) A T Λ, Φ 14 = PW 1r, Φ 22 =diag{ (1 μ 1 )Q 1, (1 μ 2 )Q 2,, (1 μ r )Q r }, r [ Φ 33 = R k + ΛW (0) + W (0)] T Λ, k=1 Φ 34 = ΛW 1r, Φ 44 =diag{ (1 μ 1 )R 1, (1 μ 2 )R 2,, (1 μ r )R r }. Proof. Choose the Lyapunov-Krasovskii functional to be n xj r V (x t )=x T (t)px(t)+2 λ j f j (s)ds+ + r k=1 t d k (t) j=1 0 k=1 0 h k t+θ ẋ T (s)z k ẋ(s)dsdθ [ x T (s)q k x(s)+f T (x(s))r k f(x(s)) ] ds, (10.10) where P > 0, Q k 0, R k 0, Z k > 0, k = 1, 2,,r, and Λ = diag {λ 1,λ 2,,λ n } 0 are to be determined. Calculating the derivative of V (x t ) along the solutions of system (10.6) yields V (x t )=2x T (t)p ẋ(t)+2 + r k=1 n λ j f j (x j (t))ẋ j (t) j=1 [ x T (t)q k x(t) (1 d ] k (t))x T (t d k (t))q k x(t d k (t))

226 10.1 Stability of Neural Networks with Multiple Delays r [ f T (x(t))r k f(x(t)) k=1 (1 d ] k (t))f T (x(t d k (t)))r k f(x(t d k (t))) [ r ] t h k ẋ T (t)z k ẋ(t) ẋ T (s)z k ẋ(s)ds k=1 t d k (t) 2x T (t)p ẋ(t)+2f T (x(t))λẋ(t) r [ + x T (t)q k x(t) (1 μ k )x T (t d k (t))q k x(t d k (t)) ] + + k=1 r [ f T (x(t))r k f(x(t)) (1 μ k )f T (x(t d k (t)))r k f(x(t d k (t))) ] k=1 [ r h k ẋ T (t)z k ẋ(t) k=1 t d k (t) ẋ T (s)z k ẋ(s)ds ]. (10.11) From the Newton-Leibnitz formula, the following equations hold for any appropriately dimensioned matrices N jk and M jk,k=1, 2,,r, j = 0, 1,,r: r r x T (t)m 0k + x T (t d j (t))m jk +f T (x(t))n 0k + f T (x(t d j (t)))n jk j=1 [ x(t) x(t d k (t)) t d k (t) On the other hand, for any matrices X (k) 11 X (k) 12 X (k) 13 X (k) 14 X (k) X k = 22 X (k) 23 X (k) 24 X (k) 33 X (k) 0, 34 X (k) 44 j=1 ] ẋ(s)ds =0. (10.12) k =1, 2,,r, the following holds: h k ξ T (t)x k ξ(t) ξ T (t)x k ξ(t)ds 0, (10.13) t d k (t) where ξ(t) =[x T (t), x T (t d 1 (t)),,x T (t d r (t)), f T (x(t)), f T (x(t d 1 (t))),, f T (x(t d r (t)))] T.

227 Stability of Neural Networks with Time-Varying Delay Now, adding the terms on the left sides of (10.12) and (10.13) to V (x t ) allows us to write V (x t )as [ ] r r V (x t ) ξ T (t) ˆΞ + h k X k ξ(t) ζ T (t, s)ψ k ζ(t, s)ds, (10.14) where k=1 k=1 t d k (t) ζ(t, s) = [ ξ T (t), ẋ T (s) ] T, Ξ 11 Ξ 12 Ξ 13 Ξ 14 Ξ ˆΞ = 22 Ξ 23 Ξ 24, Ξ 33 Ξ 34 Ξ 44 X (k) 11 X (k) 12 X (k) 13 X (k) 14 M 0k X (k) 22 X (k) 23 X (k) 24 M k Ψ k = X (k) 33 X (k) 34 N 0k, k =1, 2,,r. X (k) 44 N k Z k From (10.8), we have f j (x j (t)) [f j (x j (t)) σ j x j (t)] 0, j =1, 2,,n, (10.15) f j (x j (t d k (t))) [f j (x j (t d k (t))) σ j x j (t d k (t))] 0, j =1, 2,,n, k =1, 2,,r. (10.16) Thus, by applying the S-procedure, we find that system (10.6) is asymptotically stable if there exist T = diag{t 1, t 2,, t n } 0andS k = diag {s k1,s k2,, s kn } 0, k=1, 2,,r such that n V (x t ) 2 t j f j (x j (t)) [f j (x j (t)) σ j x j (t)] 2 j=1 r k=1 j=1 ξ T (t) n {s kj f j (x j (t d k (t))) [f j (x j (t d k (t))) σ j x j (t d k (t))]} [ Ξ + ] r h k X k ξ(t) k=1 r k=1 t d k (t) ζ T (t, s)ψ k ζ(t, s)ds < 0 (10.17)

228 10.1 Stability of Neural Networks with Multiple Delays 211 for ξ(t) 0,where Ξ 11 Ξ 12 Ξ 13 + ΣT Ξ 14 Ξ Ξ = 22 Ξ 23 Ξ 24 + Σ s. Ξ 33 2T Ξ 34 Ξ 44 2S r Inequality (10.17) is satisfied if Ξ + k=1 h kx k < 0andΨ k 0, k = 1, 2,,r.Specifically,wecanchooseX k to be T M 0k M 0k X k = M k N 0k N k Z 1 k M k N 0k N k, which ensures that X k 0, k =1, 2,,r and Ψ k 0, k =1, 2,,r. In this case, according to the Schur complement, Ξ + r k=1 h kx k < 0is equivalent to Ξ<0. This completes the proof. Remark The term 2 n j=1 λ xj j 0 f j (s)ds in the Lyapunov-Krasovskii functional (10.10) contains information on nonlinearities. Furthermore, since the proof uses the S-procedure, all the Lyapunov matrices are full-block matrices, in contrast to Theorem 1 of [13], where some of them were limited to being diagonal matrices. So, our criterion is an improvement over that theorem. Remark If the matrices N kj and M kj,k=0, 1,,r, j =1, 2,,r in (10.9) are all set to zero, and Z k = ε k I, k =1, 2,,r (where ε k,k= 1, 2,,r are sufficiently small positive scalars), then Theorem can be extended to the delay-independent and rate-dependent stability criterion in the following corollary. Corollary Consider neural network (10.6) with time-varying delays, d k (t), k=1, 2,,r, that satisfy (10.3) [but not necessarily (10.2)]. Given scalars μ k,k=1, 2,,r, the system is asymptotically stable at the origin if there exist matrices P > 0, Q k 0, R k 0, Λ =diag{λ 1, λ 2,, λ n } 0, T =diag{t 1,t 2,,t n } 0, ands k =diag{s k1,s k2,,s kn } 0, k=1, 2,,r such that the following LMI holds:

229 Stability of Neural Networks with Time-Varying Delay Φ 11 0 Φ 13 + ΣT Φ 14 Φ Φ = 22 0 Σ s < 0, (10.18) Φ 33 2T Φ 34 Φ 44 2S where Φ 11, Φ 13, Φ 14, Φ 22, Φ 33, Φ 34, Φ 44, Σ s, Σ, ands are defined in (10.9). Setting Q k and R k,k=1, 2,,r to zero results in a delay-dependent and rate-independent criterion for which the derivative of the delay may be unknown. Corollary Consider neural network (10.6) with time-varying delays, d k (t), k=1, 2,,r, that satisfy (10.2) [but not necessarily (10.3)]. Given scalars h k > 0, k =1, 2,,r, the system is asymptotically stable at the origin if there exist matrices P>0, Z k > 0, Λ =diag{λ 1,λ 2,,λ n } 0, T =diag{t 1,t 2,,t n } 0, ands k =diag{s k1,s k2,,s kn } 0, k= 1, 2,,r, and any appropriately dimensioned matrices N kj and M kj,k= 0, 1,,r, j =1, 2,,r such that the following LMI holds: Ξ 11 Ξ 12 Ξ 13 + ΣT Ξ 14 h 1 M 01 h r M 0r Ξ22 Ξ 23 Ξ 24 + Σ s h 1 M 1 h r M r Ξ33 2T Ξ 34 h 1 N 01 h r N 0r Ξ44 2S h 1 N 1 h r N r < 0, (10.19) h 1 Z 1 0 where h r Z r Ξ 11 = PA AP + A T HA + r k=1.. ( M0k + M0k T ), M 11 M11 T M 12 M21 T M 1r Mr1 T M 22 M22 T Ξ 22 = M 2r M T r2,... M rr Mrr T Ξ 33 = ΛW (0) + [ W (0)] T Λ + [ W (0) ] T HW (0),..

230 10.1 Stability of Neural Networks with Multiple Delays 213 Ξ 44 = W T 1rHW 1r, and the other terms are defined in (10.9). Remark Regarding the stability of a neural network with time-varying delays, previously derived criteria restricted the derivatives of the delays to being less than 1. Corollary does not impose this limitation because it is based on the FWM approach. This enables us to obtain a stability criterion for neural networks with unknown time-varying delays Numerical Examples Example Consider system (10.6) with r = 2 and the following parameters: A = , W (0) =, W (1) =, W (2) = Assume μ 1 =0.2 andμ 2 =0.1 for the time-varying delays; and let σ 1 = 5.5 andσ 2 =1.19. Under these conditions, LMI (10.18) in Corollary is feasible, which implies that the system is asymptotically stable regardless of the lengths of the delays. In contrast, LMI (4) in [13] is infeasible; and it is infeasible even for σ 1 =4.8 andσ 2 =1.0, which shows how conservative the method in [13] is. Now, assume h 1 = 2 and h 2 = 1. LMI (10.18) in Corollary is infeasible in any of the following three cases: (1) σ 1 exceeds 5.5 and σ 2 equals 1.19; (2) σ 1 equals 5.5 and σ 2 exceeds 1.19; or (3) σ 1 exceeds 5.5 and σ 2 exceeds But LMI (10.9) in Theorem is feasible for σ 1 =5.6 andσ 2 =1.39, which demonstrates that it is better than the delay-independent criterion in Corollary In fact, if we let g 1 (x) =5.6tanh(x), g 2 (x) =1.39 tanh(x), u 1 (θ) =0.2, u 2 (θ) = 0.5, (θ [ 2, 0]), J 1 = J 2 =1,d 1 (t) = sint, and d 2 (t) = sint, then system (10.1) is asymptotically stable at its unique equilibrium point u =[1.933, 1.520] T. Figs and show the convergence dynamics.

231 Stability of Neural Networks with Time-Varying Delay Fig Time response curve of u 1(t) of system (10.6) (Example ) Fig Time response curve of u 2(t) of system (10.6) (Example ) Finally, suppose that no information on the derivatives of delays is available; that is, μ 1 and μ 2 could take any values. If the upper bounds, h 1 and h 2, on the delays are known to be 2 and 1, respectively, then LMI (10.19) in Corollary is feasible for σ 1 =4.2 andσ 2 =0.92. Example Consider system (10.6) with r = 2 and the following parameters: A = 1 0, W (0) = 1 2 5, W (1) = 2 3 0, W (2) = [21] discusses the case μ 1 = μ 2 =0andσ 1 = σ 2 = 1. Corollary shows that this system is asymptotically stable regardless of the lengths of