Research Article Exponential Estimates and Stabilization of Discrete-Time Singular Time-Delay Systems Subject to Actuator Saturation

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1 Discrete Dynamics in Nature and Society Volume 0, Article ID 44373, 7 pages doi:0.55/0/44373 Research Article Exponential Estimates and Stabilization of Discrete-Time Singular Time-Delay Systems Subject to Actuator Saturation Jinxing Lin, College of Automation, Nanjing University of Posts and Telecommunications, Nanjing 0003, China Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Southeast University, Nanjing 0096, China Correspondence should be addressed to Jinxing Lin, jxlin004@6.com Received 9 February 0; Revised 0 June 0; Accepted 4 July 0 Academic Editor: Recai Kilic Copyright q 0 Jinxing Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper is concerned with exponential estimates and stabilization of a class of discrete-time singular systems with time-varying state delays and saturating actuators. By constructing a decay-rate-dependent Lyapunov-Krasovskii function and utilizing the slow-fast decomposition technique, an exponential admissibility condition, which not only guarantees the regularity, causality, and exponential stability of the unforced system but also gives the corresponding estimates of decay rate and decay coefficient, is derived in terms of linear matrix inequalities LMIs. Under the proposed condition, the exponential stabilization problem of discrete-time singular time-delay systems subject actuator saturation is solved by designing a stabilizing state feedback controller and determining an associated set of safe initial conditions, for which the local exponential stability of the saturated closed-loop system is guaranteed. Two numerical examples are provided to illustrate the effectiveness of the proposed results.. Introduction Singular time-delay systems STDSs arise naturally in many engineering fields such as electric networks, chemical processes, lossless transmission lines, and so forth. ASTDS is a mixture of delay differential equations and delay difference equations; such a complex nature of STDS leads to abundant dynamics, for example, non-strictly proper transcendental equations, irregularity, impulses or non-causality. Therefore, the study of such systems is much more complicated than that for normal state-space time-delay systems. In the past two decades, a great number of stability results on STDSs have been reported in the literature; see, for example, 8 and the references therein.

2 Discrete Dynamics in Nature and Society It is noted that many stability results for STDSs are concerned with asymptotic stability. Practically, however, exponential stability is more important because the transient process of a system can be described more clearly once the decay rate is determined 9. Therefore, in recent years, the study of exponential estimates problem of STDSs has received increasing attention, and a few approaches have been proposed. For example, in 0,, the STDS was decomposed into slow differential and fast algebraic subsystems and the exponential stability of the slow subsystem was proved by using the Lyapunov method. Subsequently, the solutions of the fast subsystem was bounded by an exponential term using a function inequality. However, this approach cannot give an estimate of the convergence rate of the system. To overcome this difficulty, Shu and Lam and Lin et al. 3 adopted the Lyapunov-Krasovskii function method 4, 5 and some improvements have been obtained. In 6, 7, an exponential estimates approach for SSTDs was presented by employing the graph theory to establish an explicit expression of the state variables of fast subsystem in terms of those of slow subsystem and the initial conditions, which allows to prove the exponential stability of the fast subsystem. However, all of the above results are related to continuous-time STDSs. To the best of the authors knowledge, the problem of exponential estimates of discrete-time STDSs has not been investigated yet. One possible reason is the difficulty in obtaining the estimates for solutions of the corresponding fast subsystem. Therefore, the first aim is to develop effective approach to give the exponential estimates of discrete-time STDSs. On the other hand, actuator saturation is also an important phenomenon arising in engineering. Saturation nonlinearity not only deteriorates the performance of the closedloop systems but also is the source of instability. Stabilization of normal state-space systems subject to actuator saturation has therefore attracted much attention from many researchers; see, for example, 8, and the references cited therein. Recently, some results for normal state-space systems have been generalized to singular systems. For example, semiglobal stabilization and output regulation of continuous-time singular system subject to input saturation were addressed in 3 by assuming that the open-loop system is semistable and impulse free which allows a state transformation such that the singular system is transformed into a normal system. Also, an algebraic Riccati equation approach to semiglobal stabilization of discrete-time singular linear systems with input saturation was proposed in 4 without any transformation of the original singular system. The invariant set approach developed for state-space system in 8 was extended to general, not necessarily semistable, continuoustime singular system in 5. This approach was further extended to the analysis of the L gain and L performance for continuous-time singular systems under actuator saturation 6 and the analysis and design of discrete-time singular systems under actuator saturation 7, 8, respectively. In 7, estimation of domain of attraction for continuous-time STDSs with actuator saturation and the design of static output feedback controller that maximize it were proposed. However, so far, few work exists to address the stabilization problem for discrete-time STDSs subject to actuator saturation, which forms the second object of this paper. In this paper, we investigate the problems of exponential estimates and stabilization for a class of discrete-time singular systems with time-varying delays and saturating actuators. The main contributions of the paper are twofold: In terms of linear matrix inequalities LMIs, an exponential admissibility condition, which not only guarantees the regularity, causality and exponential stability of the unforced system but also gives the corresponding estimates of

3 Discrete Dynamics in Nature and Society 3 decay rate and decay coefficient, is derived by constructing a decay-rate-dependent Lyapunov-Krasovskii function and using the slow-fast decomposition. The exponential stabilization problem of STDSs with saturating actuators is solved by designing a stabilizing state feedback controller and determining an associated set of safe initial conditions, for which the local exponential stability of the saturated closed-loop system is guaranteed. The existence criterion of the desired controller is formulated, and an LMI optimization approach is proposed to enlarge the domain of safe initial conditions. The paper is organized as follows. Problem statement and the preliminaries are given in Section. InSection 4, we present the exponential estimates for the STDSs and the solutions to the stabilization problem for the system with saturating actuators. Numerical examples will be given in Section 4 to illustrate the effectiveness of the proposed method. The paper will be concluded in Section 5. Notation. For real symmetric matrices P, P>0 P 0 means that matrix P is positive definite semipositive definite. λ max P λ min P denotes the largest smallest eigenvalue of the positive definite matrix P. R n is the n-dimensional real Euclidean space and R m n is the set of all real m n matrices. Z represents the sets of all non-negative integers. The superscript T represents matrix transposition, and in a matrix is used to represent the term which is induced by symmetry. diag{ }stands for a block-diagonal matrix. Sym{A} is the shorthand notation for A A T. For two integers n and n with n n,weusei n,n to denote the integer set {n,n,...,n }.LetC n,d {φ : I d, 0 R n } denote the Banach space of family continuous vector valued functions mapping the interval I d, 0 to R n with the topology of uniform convergence. Denote x k s x k s, s I d, 0. refers to either the Euclidean vector norm or the induced matrix two-norm. For a function φ C n,d, its norm is defined as φ c sup s I d,0 φ s.. Problem Statement and Preliminaries.. Problem Statement Consider a class of discrete-time singular system subject to time-varying delay and actuator saturation as follows: Ex k Ax k A d x k d k B sat u k, x s φ s, s I d, 0,. where x k R n is the system state, u k R m is the control input, and φ s C n,d is a compatible vector valued initial function. d k is a time-varying delay satisfying 0 < d d k d, where d and d are constant positive scalars representing the minimum and maximum delays, respectively. The matrix E is singular and rank E r < n. A, A d and B are known constant matrices. The function sat : R m R m is the standard saturation function defined as sat u k sat u k sat u k sat u m k T, where sat u i k Sign u i k min{, u i k }. Note that the notation of sat is slightly abused to denote scalar values and vector valued saturation functions. Also note that it is without loss of generality to assume unity saturation level 0.

4 4 Discrete Dynamics in Nature and Society In this paper, we consider the design of a linear state feedback control law of the following form 7 : u k Fx k, F FE,. where F R m n and F R m n. The closed-loop system under this feedback is given by ) Ex k Ax k A d x k d k B sat FEx k..3 Remark.. The state feedback of the form. is used to guarantee the uniqueness of the solution of system.. For that, since rank E r<n, there exist two nonsingular matrices G, H R n n such that GEH Ir 0 A A, GAH Ad A, GA 0 0 A A d H d, A d A d GB [ B x k H ], FH F B F, FG [F F ], x k, x x k k R r, x k R nr..4 If the state feedback is taken as the general form, that is, u k Fx k, byusing.4, then system.3 is restricted system equivalent r.s.e. to the following one: x k A x k A x k A d x k d k A d x k d k B sat F x k F x k, 0 A x k A x k A d x k d k A d x k d k B sat F x k F x k..5 From the second equation of.5, it can be seen that x k is in the function sat. Hence, if B / 0, for given x 0, the solution of x 0 is not unique even A is nonsingular. However, if u k FEx k, then system.3 is r.s.e. to ) x k A x k A x k A d x k d k A d x k d k B sat F x k, ) 0 A x k A x k A d x k d k A d x k d k B sat F x k,.6 which implies that, for given x k, the unique solution of x k can be obtained when A is nonsingular. Nevertheless, it should be pointed that the state feedback. may be conservative due to its special structure.

5 Discrete Dynamics in Nature and Society 5 To describe the main objective of this paper more precisely, we introduce the following definitions. Definition. see 9. SystemEx k Ax k or the pair E, A is said to be regular if det ze A is not identically zero, and if deg det ze A rank E, then it is further said to be causal. Definition.3 see 6. System. with u k 0 is said to be regular and causal, if the pair E, A is regular and causal. Note that regularity and causality of system. with u k 0 ensure that the solution to this system exists and is unique for any given compatible initial value φ s. Definition.4. System. under feedback law. is said to be exponentially stable with decay rate λ λ > if, for any compatible initial conditions x k0 s x k 0 s, s I d, 0, its solution x k, x k0 satisfies x k, x k0 ɛλ kk0 x k0 c for all k k 0, where x k0 c sup s I d,0 x k 0 s, k 0 is the initial time step, and ɛ>0isthe decay coefficient. We are interested in the exponential estimates and design for system.3. For any compatible initial condition x 0 φ C n,d, denote the state trajectory of system. as x k, x 0 ; then the domain of attraction of the origin is S { } φ C n,d : lim x k, x k In general, for a given stabilizing state feedback gain F, it is impossible to determine exactly the domain of attraction of the origin with respect to system.3. Therefore, the purpose of this paper is to design a state feedback gain F and determine a suitable set of initial condition D {φ C n,d : φ c δ} Sfrom which the regularity, causality, and exponential stability of the closed-loop system.3 is ensured. Also, we are interested in maximizing the size of this set, that is, obtaining the maximal value of δ... Preliminary Results Define ζ k [ x T k x T k d k x T k d x T k d ], y k x k x k..8

6 6 Discrete Dynamics in Nature and Society Lemma.5. For any appropriately dimensioned matrices R>0 and N, two positive time-varying integer d k and d k satisfying d k d k d, and a scalar λ>0, the following equality holds kd k l kd k y T l E T λ kl REy l kd k cζ T k NR N T ζ k ζ T k N Ey l kd k l kd k l kd k ) ) ) ζ T k N λ kl y T l E T R λ kl R N T ζ k λ kl REy l,.9 where c λ d k λ d k / λ. Proof. See the Appendix. Lemma.6 see 30. Given a matrix D, let a positive-definite matrix S and a positive scalar η 0, exist such that D T SD η S<0.0 then the matrix D satisfies the bound D i χe λi, i 0,,...,. where χ λ max S /λ min S and λ ln η. Lemma.7. Let 0 <d d k d. Consider the following system: x k Dx k d k f k, k 0,. where D i χe λi, λ>0, i 0,,..., and f k κe βk, k 0. If βd λ<0..3 Then, for any compatible initial function φ C n,d, the solution x k, φ of.0 satisfies ) x k, φ χ ) φ c χκ κ e rk, k 0,.4 eβdλ where r min{λ/d, β}. Proof. See the Appendix.

7 Discrete Dynamics in Nature and Society 7 For a matrix F R m n, denote the jth row of F as f j and define L F { x k R n : fj x k,j I,m }..5 Let P R n n be a positive-definite matrix and E T PE 0, the set Ω E T PE, is defined by ) { } Ω E T PE, x k R n : x T k E T PEx k..6 Also, let V be the set of m m diagonal matrices whose diagonal elements are either or 0. There are m elements in V. Suppose that each element of V is labeled as D i,i I, m and denote D i I D i. Clearly, D i is also an element of V if D i V. Lemma.8 see 8. Let F, H R m n be given. If x k L H,thensat Fx k can be expressed as sat Fx k m i α i k D i F D i H) x k,.7 where α i k for i I, m are some variables satisfying α i k 0 and m i α i k. Lemma.9 see 3. Given matrices X, Y, and Z with appropriate dimensions, and Y is symmetric. Then there exists a scalar ρ>0, such that ρi Y > 0 and Sym {X T Z}Z T YZ X T ρi Y X ρz T Z. 3. Main Results In this section, we will first present a delay-dependent LMI condition which guarantees the regularity, causality and exponential stability of the unforced system. i.e., with u k 0 with a predefined decay rate. Theorem 3.. Given constants 0 <α<and 0 <d< d. If there exist symmetric matrices X>0, Q l > 0, l,, 3, Z v > 0, v, and S, and matrices M M T M T M T 3 M T 4 T, N N T N T N T 3 N T 4 T and T T T T T T T 3 T T 4 T such that the following inequality holds Φ Ψ < 0, 3. Γ

8 8 Discrete Dynamics in Nature and Society where { Φ diag α E T XE d } )Q Q Q 3, α d Q, α d Q, α d Q 3 Sym {MΛ NΛ TΛ 3 } A T XA A T RT SRA A T UA, Ψ [ ϱ M ϱ N ϱ T ], Γ diag { ϱ Z, ϱ Z Z, ϱ Z }, A [ A A d 0 0 ], A [ A E A d 0 0 ], Λ [ E E 0 0 ], Λ [ 0 E 0 E ], Λ 3 [ 0 E 0 E ], d d d, U dz dz, ϱ α d α, ϱ α d α d, α 3. and R R n n is any constant matrix satisfying RE 0 and rank R n r, then the unforced system. is regular, causal, and exponentially stable with λ / α. Proof. The proof is divided into three parts: i To show the regularity and causality; ii to show the exponential stability of the difference subsystem; iii to show the exponential stability of the algebraic subsystem. Part i : Regularity and causality. Since E is regular and rank E r, there exist two nonsingular matrices G and H such that G T XG G EH X X X T X Ir 0, G 0 0 AH, H T M G ] [Ã Ã, Ã Ã M M M M, RG [ R R ]. 3.3 Note that RE 0 and rank R n r, it can be verified that R 0, rank R n r and R R n nr,thatis, ] RG [0 R. 3.4 From 3., itiseasytoobtainφ < 0. In view of X>0, Q l > 0, l,, 3, Z l > 0, Z > 0, d> 0, and d >0, it can be further obtained that A T R T SRA α E T XE Sym{M E} <

9 Discrete Dynamics in Nature and Society 9 Pre- and postmultiplying 3.5 by H T obtained that and H, respectively, and using 3.3 and 3.4, itis à T R T S < 0, 3.6 R à where represents matrices that are not relevant in the following discussion. Thus, à T R T S R à < Now, we assume that the matrix à is singular, then, there exists a vector η R nr and η / 0 such that à η 0. Pre- and postmultiplying 3.7 by η T and η, respectively, result in η T à T R T S R à η 0. Then, it is easy to see that 3.7 is a contradiction since η T à T R T S R à η>0. Thus, à is nonsingular, which implies that the unforced system. is regular and causal by Definition and Theorem in 4. Part ii : Exponential stability of the difference subsystem. From 9, the regularity and causality of the unforced system. imply that there exist two nonsingular matrices G and H such that E G EH Ir 0 A, A G 0 0 AH I nr According to 3.8, define Ad A A d G A d H d, X G T X X A d A XG d X T, X Q l H T Q Ql Q lh l, Z v G T Z vg Zv Z v, Q T l M v Q l M v Z T v N v Z v M v H T M vg Mv M v, N v H T N vg Nv N v, [ T T v H T T vg v T v T v T v ], R RG N v [0 R ], l,, 3, v,. 3.9 By using Schur complement on 3., weget [ Φ ] Φ Φ < 0, 3.0

10 0 Discrete Dynamics in Nature and Society where Φ AT R T SRA α E T PE d ) Q Q Q 3 Sym{M E}, Φ AT R T SRA d M E N E T E E T M T, 3. Φ AT d RT SRA d α d Q Sym{M E N E T E}. Substituting 3.8 and 3.9 into the above inequality, pre- and postmultiplying by diag{h T,HT } and diag{h,h }, respectively, and using Schur complement yields RT SR d ) Q Q Q 3 R T SR A d α d Q A T dr T < 0. SR A d 3. Pre- and postmultiplying this inequality by A T d noting A T d Q Q 3 A d 0, we have I and its transpose, respectively, and A T α d dq A d d Q < Therefore, according to Lemma.6, there exist constants χ λ ln α d/ ln d / such that λ max Q /λ min Q and A i d χe λi, i,, Let ξ k H x k ξt k ξt k T, where ξ k R r and ξ k R nr. Then, the unforced system. is r.s.e. to the following one: ξ k A ξ k A d ξ k d k A d ξ k d k, 0 ξ k A d ξ k d k A d ξ k d k. 3.5 Now, choose the following Lyapunov-Krasovskii function: 4 V ξ k V s ξ k, s 3.6

11 Discrete Dynamics in Nature and Society where V ξ k ξ T k X ξ k ξ T k E T XEξ k, V ξ k V 3 ξ k k l kd k k l kd ξ T l α kl Q ξ l ξ T l α kl Q ξ l d k θ d l k θ k l kd ξ T l α kl Q ξ l, ξ T l α kl Q 3 ξ l, 3.7 V 4 ξ k k θ d l k θ η T l E T d α kl Z Eη l k θ d l k θ η T l E T α kl Z Eη l, with ξ k s ξ k s, s I d, 0 and η k ξ k ξ k. Define 4 4 V ξ k V s ξ k V s ξ k α V s ξ k. s s 3.8 Then, it follows from. that V ξ k ξ T k E T X Eξ k α ξ T k E T X Eξ k ) TX ) Aξ k A d ξ k d k Aξ k A d ξ k d k α ξ T k E T X Eξ k, V ξ k d ) ξ T k Q ξ k ξ T k d k α d k Q ξ k d k, V 3 ξ k ξ T k Q Q 3 )ξ k ξ T k d ) α d Q ξ k d ) ξ T ) ) k d α d Q 3 ξ k d, V 4 ξ k η T k E T d Z d Z )Eη k η T l E T α kl Z Eη l kd l kd η T k E T d Z d Z )Eη k kd k l kd k l kd k l kd k η T l E T α kl Z Z )Eη l η T l E T α kl Z Eη l η T l E T α kl Z Eη l kd l kd k η T l E T α kl Z Eη l. 3.9

12 Discrete Dynamics in Nature and Society Let ψ k [ ξ T k ξ T k d k ξ T k d ξ T k d ] T. 3.0 Using Lemma.5 for the last three terms of V 4 ξ k, respectively, and noting d d k d, we have k l kd k η T l E T α kl Z Eη l ϱ ψ T k M Z M T ψ k ψ T k M k l kd k k l kd k Eη l [ψ T k M α kl η T l E T Z ] α kl Z ) [ M T ] ψ k α kl Z Eη l kd k l kd η T l E T α kl Z Z )Eη l ) N kd k ϱ ψ T T k N Z Z ψ k ψ T k N Eη l kd k l kd l kd [ψ T k N α kl η T l E T Z Z )] α kl Z Z )) [N T ψ k α kl ) ] Z Z Eη l 3. kd l kd k η T l E T α kl Z Eη l ϱ ψ T k T Z T T ψ k ψ T k T kd l kd k kd l kd k Eη l [ψ T k T α kl η T l E T Z ] α kl Z ) [ T T ] ψ k α kl Z Eη l.

13 Discrete Dynamics in Nature and Society 3 Note that η l ξ l ξ l provides k l kd k kd k l kd kd l kd k Eη l Eη l Eη l E E 0 0 ψ k, 0 E 0 E ψ k, 0 E E 0 ψ k Also, it follows from RE 0that 0 ξ T k E T R T SREξ T k ) TR T ) Aξ k A d ξ k d k SR Aξ k A d ξ k d k. 3.5 Then, substituting into 3.8, using 3.5, and noting α>0, Z > 0and Z > 0, we get V ξ k ψ T k [Φ ϱ M Z M T ϱ N Z Z N T ϱ T Z T T] ψ k k l kd k [ψ T k M α kl η T l E T Z ] α kl Z ) [ M T ] ψ k α kl Z Eη l kd k l kd [ψ T k N α kl η T l E T Z Z )][ α kl Z Z )] [N T ψ k α kl ) ] Z Z Eη l 3.6 kd l kd k [ψ T k T α kl η T l E T Z ] α kl Z ) [ T T ] ψ k α kl Z Eη l ψ T k [Φ ϱ M Z M T ϱ N Z Z N T ϱ T Z T T] ψ k, where Φ follows the same definition as Φ defined in 3. with A, A d, X, Q l, Z v, R, M, N,and T instead of A, A d, X, Q l, Z v, R, M, N, andt. Performing a congruence transformation on 3. by diag{h T,HT,HT,HT,HT,HT,HT }, and then using the Schur complement implies

14 4 Discrete Dynamics in Nature and Society Φ ϱ M Z M T ϱ N Z Z N T ϱ T Z T T < 0. Thus, it follows from 3.8 and 3.6 that V ξ k V ξ k α V ξ k 0, which leads to V ξ k α V ξ k. 3.7 By iterative substitutions, inequality 3.7 yields V ξ k α k V ξ On the other hand, it follows from the Lyapunov functional 3.6 that β ξ k V ξ k, V ξ 0 β φ c, 3.9 where β λ min X ), ) β λ max X d d ) ) ) d ))λ max Q dλ max Q dλ max Q 3 d ) λ max Z Z, 3.30 Then, combining 3.8 and 3.9 leads to ξ k β β α k/ φ c β e ln α / k φ β c. 3.3 Therefore, the difference subsystem of 3.5 is exponentially stable with a decay rate which is not less than e ln α /. The remaining task is to show the exponential stability of the algebraic subsystem. Part iii : Exponential stability of the algebraic subsystem. Set f k A d ξ kd k ; then, it follows from 3.3 that f k α d/ β Ad e ln α / k φ c. β 3.3 Using the second equation in 3.5, 3.4 and Lemma.7, onegets ξ k β 3 e ln α / k φ c, 3.33

15 Discrete Dynamics in Nature and Society 5 where ) λ max Q β 3 ) φ c λ min Q λ max Q ) λ min Q ) e / ln d β β α d/ Ad Combining 3.3 and 3.33 yields that x k ξ T k H T H ξ k H ξ k ξ k ) β β β 3 ) e ln α k φ c Thus, we have x k β β α / k 3e ln β φ c β β β 3 α ) k φ c 3.36 The proof is completed. Remark 3.. Theorem 3. is obtained by applying a Lyapunov-Krasovskii function method to both the difference and algebraic subsystems of a discrete-time singular system with timevarying delay. Such a method in dealing with the algebraic subsystem of the discrete-time singular delay system has not been reported in the literature. Remark 3.3. When E I in system., then R 0, and we get the exponential stability condition for the standard delay systems x k Ax k A d x k d k from 3.. Based on the result of Theorem 3., we now present the existence conditions of a stabilizing state feedback controller for system. and the corresponding set of initial condition. Theorem 3.4. Given constants 0 <α<, 0 <d< d, ρ>0, ε > 0, ε > 0, ε 3 > 0, ε 4, and ε 5. If there exist symmetric matrices X>0, Q l > 0, l,, 3, Z v > 0, v,, and S, and matrices

16 6 Discrete Dynamics in Nature and Society M M T M T M T 3 M T 4 T, N N T N T N T 3 N T 4 T, T T T T T T T 3 T T 4 T, F and H such that the following inequalities hold: Φ j Ψ Υ j dπ j dπj Ξ Υ j R T Γ ε I ε X d ε I ε Z ) d ε 3 I ε 3 Z ) < 0, j I, m, 0 0 ρi S 0 ρ I 3.37 hl E T 0, XE l I,m, 3.38 where h l denotes the lth row of H, { Φ j diag α E T XE d } )Q Q Q 3, α d Q, α d Q, α d Q 3 } Sym {ΞRΥ T j MΛ NΛ TΛ 3, Υ j [ A j A d 0 0 ] T, Πj [ A j E A d 0 0 ] T Ξ [ ε 4 I ε 5 I 0 0 ] T, ) Aj A B D j FE D j H Λ, Λ, Λ 3, Ψ, Γ are defined in 3., and R R n n is any constant matrix satisfying RE 0 with rank R n r, then system.3 is regular, causal, and locally exponentially stable with λ / α for any compatible initial condition in the ball B δ {φ C n,d : φ c δ }, 3.40 where δ λ max ET XE ) d d )). d λ max Q dλ max Q dλ max Q 3 d λ max Z Z 3.4 Proof. If 3.38 holds, then the ellipsoid Ω E T XE, is included in L H 7 For more details about the ellipsoids and ellipsoid algorithm, we refer the readers to Suppose that

17 Discrete Dynamics in Nature and Society 7 x k L H, k > 0 will be proved later. Hence, by Lemma.8, sat FEx k can be expressed as ) sat FEx k m j ) α j k D j FE D i H x k 3.4 and it follows that Ex k m j α j k A j x k A d x k d k Choose a Lyapunov function as in 3.6, and then, by Theorem 3., system 3.43 is regular, causal and locally exponentially stable if there exist symmetric matrices X>0, Q l > 0, l,, 3, Z v > 0, v,, and S, and matrices M, N,andT defined in 3. such that Φ Ψ < 0, 3.44 Γ where { Φ diag α E T XE d } )Q Q Q 3, α d Q, α d Q, α d Q 3 A T j X A j A T j RT SRA j A T j U A j Sym{MΛ NΛ TΛ 3 }, 3.45 m m A j α j k A j A d 0 0, A j α j k A j E A d j j Now, provided that inequalities 3.37 hold, then by using Schur complement and Lemma.9, one has Φ j Ξ ρi S ) Ξ T Ψ Υ j dπ j dπj Υ j R T Γ X dz < 0, j I, m dz 0 ρ I

18 8 Discrete Dynamics in Nature and Society This, together with α j k 0and m j α j k, implies that m j Φ j Ξ ρi S ) Ξ T Ψ Υ j dπ j dπj Υ j R T Γ X α j k dz < 0, dz 0 ρ I that is, m j α j k Φ j Ξ ρi S ) Ξ T Ψ m m m α j k Υ j α j k dπ j α j k dπ m j α j k Υ j R T j j j j Γ X < 0. dz 0 0 dz 0 ρ I 3.49 Applying Schur complement to 3.49 leads to Λ j Ψ m m m α j k Υ j α j k dπ j α j k dπ j j j j Γ X 0 0 < 0, 3.50 dz 0 dz where Λ j m α j k Φ j Ξ ρi S ) Ξ T ρ j j j m m α j k Υ j R T α j k RΥ T j. 3.5 Rewrite m α j k Φ j m j j { α j k Sym Υ j R T Ξ T} Φ, 3.5

19 Discrete Dynamics in Nature and Society 9 where { Φ diag α E T XE d } )Q Q Q 3, α d Q, α d Q, α d Q 3 Sym{MΛ NΛ TΛ 3 } From 3.37, it follows that ρi S >0, and then the following inequality holds from Lemma.9 m α j k Υ j R T S m j j α j k RΥ T j m j ρ { α j k Sym Υ j R T Ξ T} Ξ ρi S ) Ξ T m m α j k Υ j R T α j k RΥ T j, j j 3.54 which, together with 3.50 and 3.5, implies that m α j k Υ j R T S m α j k RΥ T j Φ Ψ m m m α j k Υ j α j k dπ j α j k dπ j j j j j j Γ X 0 0 < dz 0 dz Applying Schur complement, it results that inequality 3.55 is equivalent to inequality Therefore, if inequalities 3.37 hold, then inequality 3.44 holds, that is, system 3.43 is regular, causal, and exponentially stable, provided that x k L H, k 0. On the other hand, by the proof in Theorem 3., inequality 3.44 holding implies V ξ k < α V ξ k <V ξ k. Assume the initial condition x s φ s is included in the ball B δ, s I d, 0, and inequalities 3.37 and 3.38 hold. Then, from the definition of V ξ k, we get x T 0 E T XEx 0 ξ T 0 E T XEξ 0 V ξ 0 δ φ c, that is, x 0 Ω E T XE,. Since inequalities 3.38 are satisfied, it follows that x 0 L H. Furthermore, because V ξ k <V ξ k, we can conclude that x T k E T XEx k ξ T k E T XEξ k V ξ k V ξ 0 δ φ c, which means that x k L H, k >0. This completes the proof. Remark 3.5. In practice, we may be interested to obtain a ball B δ of initial condition as large as possible. To maximize the ball of initial condition, the following approximate optimization procedure can be used: min X>0,Q >0,Q >0,Q 3 >0,Z >0,Z >0,F,H r s.t. 3.44, 3.45 and E T XE w, Q w, P Q w 3,Q 3 w 4,Z w 5, Z w 6, where r w d d d w dw 3 dw 4 d w 5 d w 6.

20 0 Discrete Dynamics in Nature and Society In the special case that E I, the following corollary directly from Remark 3.3 and Theorem 3.4. Corollary 3.6. Given constants 0 <α<, 0 <d< d, ρ>0, ε > 0, ε > 0, and ε 3 > 0. If there exist matrices X > 0, Q l > 0, l,, 3, Z v > 0, v,, M M T M T M T 3 M T 4 T, N N T N T N T 3 N T 4 T, T T T T T T T 3 T T 4 T, F and H such that the following hold: Φ j Ψ Υ j dπ j dπj Γ ε I ε X 0 0 d < 0, j I, ε I ε Z ) m, 0 d ε 3 I ε 3 Z ) hl 0, l I,m, X 3.56 where h l denotes the lth row of H, { Φ j diag α X d } )Q Q Q 3, α d Q, α d Q, α d Q 3 Sym {MΛ NΛ TΛ 3 }, Υ j [ A j A d 0 0 ] T, Πj [ A j I A d 0 0 ] T, Λ [ I I 0 0 ], Λ [ 0 I 0 I ], Λ 3 [ 0 I I 0 ] ), A j A B D j F D j H, 3.57 and Ψ and Γ are defined in 3., then, under the state feedback u k Fx k, system.3 with E I is locally exponentially stable with λ / α for any compatible initial condition in the ball B δ {φ C n,d : φ c δ }, 3.58 where δ λ max X d d )). d λ max Q dλ max Q dλ max Q 3 d λ max Z Z 3.59

21 Discrete Dynamics in Nature and Society Table : Maximum allowable decay rates λ for different d with d. d λ Also, as Remark 3.5, the corresponding approximate optimization problem of Corollary 3.6 becomes lim X>0,Q >0,Q >0,Q 3 >0,Z >0,Z >0,F,H r s.t. 3.56, 3.58 and X w,q w, P Q w 3,Q 3 w 4,Z w 5,Z w 6, where r w d d d w dw 3 dw 4 d w 5 d w 6. Remark 3.7. Scalars ρ, ε f, f,,...,5, in Theorem 3.4 are tuning parameters which need to be given first. In fact, 3.37 and 3.38, for fixed α > 0and0 < d < d, are bilinear matrix inequalities BMIs regarding to these tuning parameters. If one can accept more computational burden, the optimal values of these parameters can be obtained by applying some global optimization algorithms 35 to solve the BMIs. 4. Numerical Examples In this section, two examples are given to show the effectiveness of the proposed method. Example 4.. Consider the discrete-time singular time-delay system. without input and with the following parameters: E , A, A d Let R [ 00 0]. For various delay upper bound d and by using Theorem 3. in this paper, the maximum allowable values of decay rate λ, that guarantees the regularity, causality and exponential stability of the system for given delay lower bound d, are listed in Table. It can be seen from Table that the delay upper bound d is related to the decay rate λ, and a smaller λ allows for a larger d. Figure depicts the simulations results of x k and x k as compared to.5 k when d k sin.5πk and initial function is φ k, T, k 3,,...,0. It is clearly observed from Figure that the states x k and x k exponentially converge to zero with a decay rate greater than.5. Example 4.. Consider the discrete-time singular time-delay system. with the following parameters: E.9, A , A.. d 0. 0., B

22 Discrete Dynamics in Nature and Society Table : δ max and F for different values of α. α λ δ max F The time-delay is d k 3. Our purpose is to design the stabilizing controllers for different decay rate λ / α and estimate the domain of attraction for the above system. Let ρ 0.00, ε.6, ε 67, ε 3 7, ε , ε 5 0.0, and R [ 00 05]. For various α, solving P by using the mincx problem in LMI Toolbox 36, the maximal values of δ, denoted δ max, and the corresponding controller gains F are listed in Table. Especially, when α 0.3, the feasible solutions to P are X Q , Q , Z , Q , Z , , H [ ], F [.4 0 ], Furthermore, applying the obtained controller and giving a possible time-varying delay d k sin.5πk, we obtain the state response of the resulting closed-loop system and the corresponding actuator output as shown in Figures and 3 for the given initial function φ k.6, T, k 3,,...,0. For clarity, we give some comments to explain the above simulation results. Firstly, it is easily seen from Table that the smaller α, that is, the smaller decay rate λ, then the larger is the domain B δ for which we guarantee the exponential stability of the saturated system. It shows the tradeoff between the size of the ball of the admissible initial condition and the desired decay rate. Secondly, from the curves in Figures and 3, one can see that despite the partly unknown transition probabilities, either the designed state-feedback or the designed output-feedback controllers are feasible and effective ensuring that the resulting closed-loop systems are stable. It should be pointed out that the results in 7 cannot be applied to this system, because the system considered here contains state delay. 5. Conclusions The problems of exponential estimates and stabilization for a class of discrete-time singular systems with time-varying state delays and saturating actuators have been investigated in this paper. An exponential admissibility condition has been developed for the system. Also, a saturated state feedback controller and a domain of safe initial conditions have been determined by using LMI optimization-based approach. The effectiveness of the results has been illustrated by numerical examples. Note that in the present work, a quadratic Lyapunov- Krasovskii function was used for the saturated closed-loop system modeled by differential

23 Discrete Dynamics in Nature and Society 3.5 State response Time k) x x.5 k.5 k Figure : Simulation results of and as compared.5 k..5 State response Time k) x x Figure : State response of the closed-loop system under saturated control. inclusion, which may lead to conservatism. Applying the saturation-dependent Lyapunov function approach proposed in 0 to capture the real-time information on the severity of saturation and thus leading to less conservative results will be interesting topics for further research.

24 4 Discrete Dynamics in Nature and Society Actuator output response Time k) uk) Figure 3: The actuator output under saturated control. Appendix Proof of Lemma.5. For given ζ k and y k defined in.8, the following equation holds kd k l kd k ) ) ) ζ T k N λ kl y T l E T R λ kl R N T ζ k λ kl REy l kd k ζ T k NR N T ξ k λ lk ζ T k N l kd k kd k l kd k Ey l kd k l kd k y T l E T λ kl REy l. A. Rearranging the above equation yields.9. This completes the proof. Proof of Lemma.7. Case k d k. Clearly, k d k d, 0, and then x k, φ D φ c f k χe λ φ c κe βk χe λ k/d φ c κe βk χ φ c κ e rk. Case k > d k. Inspired by the idea in 6, to model the dependency of x k on past instances, define k 0 k, k i k i d k i, i,,... A. Note that k i <k i then, by iterative substitutions of.0, onegets x k, φ ) D k t x k t ) k k t D i f k i f k 0, k 0, A.3 where k t is a positive integer. i

25 Discrete Dynamics in Nature and Society 5 i There exists a finite positive integer k t such that 0 k k t and d k k t < 0. Then it follows from A. that k k t d, i k i k 0 d ) k j k0 id. j 0 A.4 By A.3, A.4,. and r min{λ/d, β}, the following is true: x k, φ ) D k t φ c D i f ki f k0 k t i χe λk t φ k t c χe λi κe βk i κe βk 0 i χe λ k/d k t φ c χκ e λi e β k 0 i j 0 d kj κe βk 0 i χe λ k/d φ k t c χκ e λi e βid e βk κe βk i χ ) k t φ c χκ e βdλ i κ i e rk χ ) φ c χκ κ e rk. eβdλ A.5 ii k 0 i j 0 d k j 0 for all positive integer i with i. Since d k 0, the sequence is monotonically decreased and has a greatest lower bound ɛ with ɛ 0; {k 0 i j 0 d k j } i that is lim i k 0 i j 0 d k j ɛ. Notingλ>0, we have lim i e λk i x ki 0 8. Then, similar to the process to derive A.5, the results in the lemma holds. Acknowledgment This work was supported by the National Natural Science Foundation of China under Grant , Open Fund of Key Laboratory of Measurement and Control of CSE no. MCCSE0A06, Ministry of Education, China, Southeast University, and the Scientific Research Foundation of Nanjing University of Posts and Telecommunications NY0080. References J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer, New York, NY, USA, 993. S. Xu, P. Van Dooren, R. Ştefan, and J. Lam, Robust stability and stabilization for singular systems with state delay and parameter uncertainty, IEEE Transactions on Automatic Control, vol. 47, no. 7, pp. 8, 00.

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