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

Abstract

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.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Jinxing Lin 1, 2

Recommended by Recai Kilic

1, College of Automation, Nanjing University of Posts and Telecommunications, Nanjing 210003, China
2, Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Southeast University, Nanjing 210096, China

Received 29 February 2012; Revised 20 June 2012; Accepted 24 July 2012

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.

1. Introduction

Singular time-delay systems (STDSs) arise naturally in many engineering fields such as electric networks, chemical processes, lossless transmission lines, and so forth [1]. A STDS 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, [2-8] and the references therein.

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 [10, 11], 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 [12] and Lin et al. [13] adopted the Lyapunov-Krasovskii function method [14, 15] and some improvements have been obtained. In [16, 17], 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 closed-loop 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, [18-22], 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 [23] 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 [24] without any transformation of the original singular system. The invariant set approach developed for state-space system in [18] was extended to general, not necessarily semistable, continuous-time singular system in [25]. This approach was further extended to the analysis of the _{[Lagrangian (script capital L)]2} gain and _{[Lagrangian (script capital L)]∞} performance for continuous-time singular systems under actuator saturation [26] and the analysis and design of discrete-time singular systems under actuator saturation [27, 28], respectively. In [17], 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:

(1) 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 decay rate and decay coefficient, is derived by constructing a decay-rate-dependent Lyapunov-Krasovskii function and using the slow-fast decomposition.

(2) 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 2. In Section 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 1.

For real symmetric matrices P , P>0 (P...5;0 ) means that matrix P is positive definite (semipositive definite). _λmax (P) (_λmin (P) ) denotes the largest (smallest) eigenvalue of the positive definite matrix P . ^Rn is the n -dimensional real Euclidean space and ^Rm×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+^AT . For two integers _n1 and _n2 with _n1 ...4;_n2 , we use I[_n1 ,_n2 ] to denote the integer set {_n1 ,_n1 +1,...,_n2 } . Let _...9E;n,d ={[varphi]:I[-d,0][arrow right]^Rn } denote the Banach space of family continuous vector valued functions mapping the interval I[-d,0] to ^Rn with the topology of uniform convergence. Denote _xk (s)=x(k+s) , ∀s∈I[-d,0] . ||·|| refers to either the Euclidean vector norm or the induced matrix two-norm. For a function [varphi]∈_...9E;n,d , its norm is defined as ||[varphi]_||c =su_ps∈I[-d,0] ||[varphi](s)|| .

2. Problem Statement and Preliminaries

2.1. Problem Statement

Consider a class of discrete-time singular system subject to time-varying delay and actuator saturation as follows: [figure omitted; refer to PDF] where x(k)∈^Rn is the system state, u(k)∈^Rm is the control input, and [varphi](s)∈_...9E;n,d¯ is a compatible vector valued initial function. d(k) is a time-varying delay satisfying 0<d_...4;d(k)...4;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 , _Ad and B are known constant matrices. The function sat :^Rm [arrow right]^Rm is the standard saturation function defined as sat (u(k))=[sat (_u1 (k))sat (_u2 (k))...sat (_um (k))^]T , where sat (_ui (k))=Sign(_ui (k))min {1,|_ui (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 [20].

In this paper, we consider the design of a linear state feedback control law of the following form [27]: [figure omitted; refer to PDF] where F∈^Rm×n and F¯∈^Rm×n . The closed-loop system under this feedback is given by [figure omitted; refer to PDF]

Remark 2.1.

The state feedback of the form (2.2) is used to guarantee the uniqueness of the solution of system (2.1). For that, since rank E=r<n , there exist two nonsingular matrices G,H∈^Rn×n such that [figure omitted; refer to PDF] If the state feedback is taken as the general form, that is, u(k)=Fx(k) , by using (2.4), then system (2.3) is restricted system equivalent (r.s.e.) to the following one: [figure omitted; refer to PDF] From the second equation of (2.5), it can be seen that _x2 (k) is in the function sat (·) . Hence, if _B2 ...0;0 , for given _x1 (0) , the solution of _x2 (0) is not unique even _A22 is nonsingular. However, if u(k)=F¯Ex(k) , then system (2.3) is r.s.e. to [figure omitted; refer to PDF] which implies that, for given _x1 (k) , the unique solution of _x2 (k) can be obtained when _A22 is nonsingular. Nevertheless, it should be pointed that the state feedback (2.2) may be conservative due to its special structure.

To describe the main objective of this paper more precisely, we introduce the following definitions.

Definition 2.2 (see [29]).

System Ex(k+1)=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 2.3 (see [6]).

System (2.1) 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 (2.1) with u(k)=0 ensure that the solution to this system exists and is unique for any given compatible initial value [varphi](s) .

Definition 2.4.

System (2.1) under feedback law (2.2) is said to be exponentially stable with decay rate λ ( λ>1 ) if, for any compatible initial conditions _xk0 (s)=x(_k0 +s) , s∈I[-d¯,0] , its solution x(k,_xk0 ) satisfies ||x(k,_xk0 )||...4;...^{λ-(k-_k0 )} ||_xk0||c for all k...5;_k0 , where ||_xk0||c =su_{ps∈I[-d¯,0]} ||x(_k0 +s)|| , _k0 is the initial time step, and ...>0 is the decay coefficient.

We are interested in the exponential estimates and design for system (2.3). For any compatible initial condition _x0 =[varphi]∈_...9E;n,d¯ , denote the state trajectory of system (2.1) as x(k,_x0 ) ; then the domain of attraction of the origin is [figure omitted; refer to PDF] 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 (2.3). Therefore, the purpose of this paper is to design a state feedback gain F¯ and determine a suitable set of initial condition ...9F;={[varphi]∈_Cn,d¯ :||[varphi]^||c2 ...4;δ}∈...AE; from which the regularity, causality, and exponential stability of the closed-loop system (2.3) is ensured. Also, we are interested in maximizing the size of this set, that is, obtaining the maximal value of δ .

2.2. Preliminary Results

Define [figure omitted; refer to PDF]

Lemma 2.5.

For any appropriately dimensioned matrices R>0 and N , two positive time-varying integer d(_k1 ) and d(_k2 ) satisfying d(_k1 )+1...4;d(_k2 )...4;d¯ , and a scalar λ>0 , the following equality holds [figure omitted; refer to PDF] where c=(^{λ-d(_k2 )} -^{λ-d(_k1 )} )/(1-λ) .

Proof.

See the Appendix.

Lemma 2.6 (see [30]).

Given a matrix D , let a positive-definite matrix S and a positive scalar η∈(0,1) exist such that [figure omitted; refer to PDF] then the matrix D satisfies the bound [figure omitted; refer to PDF] where χ=_λmax (S)/_λmin (S) and λ=-ln (η) .

Lemma 2.7.

Let 0<d_...4;d(k)...4;d¯ . Consider the following system: [figure omitted; refer to PDF] where ||^Di ||...4;χ^e-λi , λ>0 , i=0,1,... , and ||f(k)||...4;κ^e-βk , k...5;0 . If [figure omitted; refer to PDF] Then, for any compatible initial function [varphi]∈_...9E;n,d¯ , the solution x(k,[varphi]) of (2.10) satisfies [figure omitted; refer to PDF] where r=min {λ/d¯,β} .

Proof.

See the Appendix.

For a matrix F∈^Rm×n , denote the jth row of F as _fj and define [figure omitted; refer to PDF] Let P∈^Rn×n be a positive-definite matrix and ^ET PE...5;0 , the set Ω(^ET PE,1) is defined by [figure omitted; refer to PDF] Also, let ...B1; be the set of m×m diagonal matrices whose diagonal elements are either 1 or 0 . There are ^2m elements in ...B1; . Suppose that each element of ...B1; is labeled as _Di ,i∈I[1,^2m ] and denote ^Di- =I-_Di . Clearly, _Di is also an element of ...B1; if _Di ∈...B1; .

Lemma 2.8 (see [18]).

Let F,H∈^Rm×n be given. If x(k)∈[Lagrangian (script capital L)](H) , then sat (Fx(k) can be expressed as [figure omitted; refer to PDF] where _αi (k) for i∈I[1,^2m ] are some variables satisfying _αi (k)...5;0 and ^∑i=12m ..._αi (k)=1 .

Lemma 2.9 (see [31]).

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 {^XT Z}-^ZT YZ...4;^XT (ρI+Y^)-1 X+ρ^ZT 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 (2.1) (i.e., with u(k)=0 ) with a predefined decay rate.

Theorem 3.1.

Given constants 0<α<1 and 0<d_<d¯ . If there exist symmetric matrices X>0 , _Ql >0 , l=1,2,3 , _Zv >0 , v=1,2 and S , and matrices M=^{[M1T M2T M3T M4T ]T} , N=^{[N1T N2T N3T N4T ]T} and T=^{[T1T T2T T3T T4T ]T} such that the following inequality holds [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and R∈^Rn×n is any constant matrix satisfying RE=0 and rank (R)=n-r , then the unforced system (2.1) is regular, causal, and exponentially stable with λ=1/1-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 _G1 and _H1 such that [figure omitted; refer to PDF] Note that RE=0 and rank (R)=n-r , it can be verified that _R~1 =0 , rank (_R~2 )=n-r and _R~2 ∈^Rn×(n-r) , that is, [figure omitted; refer to PDF] From (3.1), it is easy to obtain _Φ11 <0 . In view of X>0 , _Ql >0 , l=1,2,3 , _Zl >0 , _Z2 >0 , d¯>0 , and d~>0 , it can be further obtained that [figure omitted; refer to PDF] Pre- and postmultiplying (3.5) by ^H1T and _H1 , respectively, and using (3.3) and (3.4), it is obtained that [figure omitted; refer to PDF] where * represents matrices that are not relevant in the following discussion. Thus, [figure omitted; refer to PDF] Now, we assume that the matrix _A~22 is singular, then, there exists a vector η∈^Rn-r and η...0;0 such that _A~22 η=0 . Pre- and postmultiplying (3.7) by ^ηT and η , respectively, result in ^ηTA~22TR~2T S_R~2A~22 η=0 . Then, it is easy to see that (3.7) is a contradiction since ^ηTA~22TR~2T S_R~2A~22 η>0 . Thus, _A~22 is nonsingular, which implies that the unforced system (2.1) is regular and causal by Definition 2 and Theorem 1 in [4].

Part (ii): Exponential stability of the difference subsystem. From [29], the regularity and causality of the unforced system (2.1) imply that there exist two nonsingular matrices _G2 and _H2 such that [figure omitted; refer to PDF] According to (3.8), define [figure omitted; refer to PDF] By using Schur complement on (3.1), we get [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Substituting (3.8) and (3.9) into the above inequality, pre- and postmultiplying by diag {^H2T ,^H2T } and diag {_H2 ,_H2 } , respectively, and using Schur complement yields [figure omitted; refer to PDF] Pre- and postmultiplying this inequality by [-^A¯d22T I] and its transpose, respectively, and noting ^A¯d22T (_Q¯222 +_Q¯322 )_A¯d22 ...5;0 , we have [figure omitted; refer to PDF] Therefore, according to Lemma 2.6, there exist constants χ=_λmax (_Q¯122 )/_λmin (_Q¯122 ) and λ=-ln (1-α^)d¯/2 +ln (1+d~^)1/2 such that [figure omitted; refer to PDF]

Let ξ(k)=^H2-1 x(k)=^{[ξ1T (k) ξ2T (k)]T} , where _ξ1 (k)∈^Rr and _ξ2 (k)∈^Rn-r . Then, the unforced system (2.1) is r.s.e. to the following one: [figure omitted; refer to PDF] Now, choose the following Lyapunov-Krasovskii function: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] with _ξk (s)=ξ(k+s) , ∀s∈I[-d¯,0] and η(k)=ξ(k+1)-ξ(k) . Define [figure omitted; refer to PDF] Then, it follows from (2.1) that [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] Using Lemma 2.5 for the last three terms of [white triangle up]_V4 (_ξk ) , respectively, and noting d_...4;d(k)...4;d¯ , we have [figure omitted; refer to PDF] Note that η(l)=ξ(l+1)-ξ(l) provides [figure omitted; refer to PDF] Also, it follows from RE=0 that [figure omitted; refer to PDF] Then, substituting (3.19)-(3.24) into (3.18), using (3.25), and noting 1-α>0 , _Z¯1 >0 and _Z¯2 >0 , we get [figure omitted; refer to PDF] where Φ¯ follows the same definition as Φ defined in (3.1) with A , _Ad , X , _Ql , _Zv , R , M , N , and T instead of A¯ , _A¯d , X¯ , _Q¯l , _Z¯v , R¯ , M¯ , N¯ , and T¯ . Performing a congruence transformation on (3.1) by diag {^H2T ,^H2T ,^H2T ,^H2T ,^H2T ,^H2T ,^H2T } , and then using the Schur complement implies Φ¯+_[varrho]1 M¯ ^Z¯1-1M¯T +_[varrho]2 N¯(_Z1 +_Z2^)-1N¯T +_[varrho]2 T¯ ^Z¯2-1T¯T <0 . Thus, it follows from (3.18) and (3.26) that [white triangle up]V(_ξk )=V(_ξk+1 )-(1-α)V(_ξk )...4;0 , which leads to [figure omitted; refer to PDF] By iterative substitutions, inequality (3.27) yields [figure omitted; refer to PDF] On the other hand, it follows from the Lyapunov functional (3.16) that [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Then, combining (3.28) and (3.29) leads to [figure omitted; refer to PDF] Therefore, the difference subsystem of (3.15) is exponentially stable with a decay rate which is not less than ^{eln (1-α)-1 /2} . 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¯d21ξ1 (k-d(k)) ; then, it follows from (3.31) that [figure omitted; refer to PDF] Using the second equation in (3.15), (3.14) and Lemma 2.7, one gets [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Combining (3.31) and (3.33) yields that [figure omitted; refer to PDF] Thus, we have [figure omitted; refer to PDF] The proof is completed.

Remark 3.2.

Theorem 3.1 is obtained by applying a Lyapunov-Krasovskii function method to both the difference and algebraic subsystems of a discrete-time singular system with time-varying 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 (2.1), then R=0 , and we get the exponential stability condition for the standard delay systems x(k+1)=Ax(k)+_Ad x(k-d(k)) from (3.1).

Based on the result of Theorem 3.1, we now present the existence conditions of a stabilizing state feedback controller for system (2.1) and the corresponding set of initial condition.

Theorem 3.4.

Given constants 0<α<1 , 0<d_<d¯ , ρ>0 , _{[varepsilon]1} >0 , _{[varepsilon]2} >0 , _{[varepsilon]3} >0 , _{[varepsilon]4} , and _{[varepsilon]5} . If there exist symmetric matrices X>0 , _Ql >0 , l=1,2,3 , _Zv >0 , v=1,2 , and S , and matrices M=^{[M1T M2T M3T M4T ]T} , N=^{[N1T N2T N3T N4T ]T} , T=^{[T1T T2T T3T T4T ]T} , F¯ and H such that the following inequalities hold: [figure omitted; refer to PDF] where _hl denotes the lth row of H , [figure omitted; refer to PDF] _Λ1 , _Λ2 , _Λ3 , Ψ , Γ are defined in (3.1), and R∈^Rn×n is any constant matrix satisfying RE=0 with rank (R)=n-r , then system (2.3) is regular, causal, and locally exponentially stable with λ=1/1-α for any compatible initial condition in the ball [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

If (3.38) holds, then the ellipsoid Ω(^ET XE,1) is included in [Lagrangian (script capital L)](H) [27] (For more details about the ellipsoids and ellipsoid algorithm, we refer the readers to [32-34]). Suppose that x(k)∈[Lagrangian (script capital L)](H) , ∀k>0 (will be proved later). Hence, by Lemma 2.8, sat (F¯Ex(k)) can be expressed as [figure omitted; refer to PDF] and it follows that [figure omitted; refer to PDF] Choose a Lyapunov function as in (3.16), and then, by Theorem 3.1, system (3.43) is regular, causal and locally exponentially stable if there exist symmetric matrices X>0 , _Ql >0 , l=1,2,3 , _Zv >0 , v=1,2 , and S , and matrices M , N , and T defined in (3.1) such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Now, provided that inequalities (3.37) hold, then by using Schur complement and Lemma 2.9, one has [figure omitted; refer to PDF] This, together with _αj (k)...5;0 and ^∑j=12m ..._αj (k)=1 , implies that [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF] Applying Schur complement to (3.49) leads to [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Rewrite [figure omitted; refer to PDF] where [figure omitted; refer to PDF] From (3.37), it follows that ρI+S>0 , and then the following inequality holds from Lemma 2.9 [figure omitted; refer to PDF] which, together with (3.50) and (3.52), implies that [figure omitted; refer to PDF] Applying Schur complement, it results that inequality (3.55) is equivalent to inequality (3.44). 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)∈[Lagrangian (script capital L)](H) , ∀k...5;0 .

On the other hand, by the proof in Theorem 3.1, inequality (3.44) holding implies V(_ξk+1 )<(1-α)V(_ξk )<V(_ξk ) . Assume the initial condition x(s)=[varphi](s) is included in the ball [Bernoulli](δ) , ∀s∈I[-d¯,0] , and inequalities (3.37) and (3.38) hold. Then, from the definition of V(_ξk ) , we get ^xT (0)^ET XEx(0)=^ξT (0)^E¯T X¯E¯ξ(0)...4;V(_ξ0 )...4;^δ-1 ||[varphi]^||c2 ...4;1 , that is, x(0)∈Ω(^ET XE,1) . Since inequalities (3.38) are satisfied, it follows that x(0)∈[Lagrangian (script capital L)](H) . Furthermore, because V(_ξk+1 )<V(_ξk ) , we can conclude that ^xT (k)^ET XEx(k)=^ξT (k)^E¯T X¯E¯ξ(k)...4;V(_ξk )...4;V(_ξ0 )...4;^δ-1 ||[varphi]^||c2 ...4;1 , which means that x(k)∈[Lagrangian (script capital L)](H) , ∀k>0 . This completes the proof.

Remark 3.5.

In practice, we may be interested to obtain a ball [Bernoulli](δ) of initial condition as large as possible. To maximize the ball of initial condition, the following approximate optimization procedure can be used: [figure omitted; refer to PDF] where r=_w1 +(d¯+d~(d¯-1))_w2 +d__w3 +d¯_w4 +^d¯2_w5 +^d¯2_w6 .

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<1 , 0<d_<d¯ , ρ>0 , _{[varepsilon]1} >0 , _{[varepsilon]2} >0 , and _{[varepsilon]3} >0 . If there exist matrices X>0 , _Ql >0 , l=1,2,3 , _Zv >0 , v=1,2 , M=^{[M1T M2T M3T M4T ]T} , N=^{[N1T N2T N3T N4T ]T} , T=^{[T1T T2T T3T T4T ]T} , F and H such that the following hold: [figure omitted; refer to PDF] where _hl denotes the l th row of H , [figure omitted; refer to PDF] and Ψ and Γ are defined in (3.1), then, under the state feedback u(k)=Fx(k) , system (2.3) with E=I is locally exponentially stable with λ=1/1-α for any compatible initial condition in the ball [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Also, as Remark 3.5, the corresponding approximate optimization problem of Corollary 3.6 becomes [figure omitted; refer to PDF] where r=_w1 +(d¯+d~(d¯-1))_w2 +d__w3 +d¯_w4 +^d¯2_w5 +^d¯2_w6 .

Remark 3.7.

Scalars ρ , _{[varepsilon]f} , f=1,2,...,5 , in Theorem 3.4 are tuning parameters which need to be given first. In fact, (3.37) and (3.38), for fixed α>0 and 0<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 4.1.

Consider the discrete-time singular time-delay system (2.1) without input and with the following parameters: [figure omitted; refer to PDF] Let R=[0001] . For various delay upper bound d¯ and by using Theorem 3.1 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 1. It can be seen from Table 1 that the delay upper bound d¯ is related to the decay rate λ , and a smaller λ allows for a larger d¯ . Figure 1 depicts the simulations results of _x1 (k) and _x2 (k) as compared to 1.2^5-k when d(k)=2+sin (2.5πk) and initial function is _[varphi]k =[-1,1^]T , k=-3,-2,...,0 . It is clearly observed from Figure 1 that the states _x1 (k) and _x2 (k) exponentially converge to zero with a decay rate greater than 1.25 .

Table 1: Maximum allowable decay rates λ for different d¯ with d =1 .

d¯	3	4	5	6	7	10	11
λ	1.2642	1.1501	1.0936	1.0613	1.0408	1.0101	1.0049

Figure 1: Simulation results of and as compared 1.2^5-k .

[figure omitted; refer to PDF]

Example 4.2.

Consider the discrete-time singular time-delay system (2.1) with the following parameters: [figure omitted; refer to PDF] The time-delay is 1...4;d(k)...4;3 . Our purpose is to design the stabilizing controllers for different decay rate λ=1/1-α and estimate the domain of attraction for the above system.

Let ρ=0.001 , _{[varepsilon]1} =2.6 , _{[varepsilon]2} =67 , _{[varepsilon]3} =71 , _{[varepsilon]4} =-0.08 , _{[varepsilon]5} =0.01 , and R=[0005] . For various α , solving (P1) by using the mincx problem in LMI Toolbox [36], the maximal values of δ , denoted _δmax , and the corresponding controller gains F¯ are listed in Table 2. Especially, when α=0.3 , the feasible solutions to (P1) are [figure omitted; refer to PDF] Furthermore, applying the obtained controller and giving a possible time-varying delay d(k)=2+sin (2.5πk) , we obtain the state response of the resulting closed-loop system and the corresponding actuator output as shown in Figures 2 and 3 for the given initial function _[varphi]k =[-1.6,1^]T , k=-3,-2,...,0 .

Table 2: _δmax and F- for different values of α .

α (λ)	δmax	F-
0.1 (1.0541)	2.1025	[1.3690 0]
0.2 (1.1180)	2.0932	[1.3206 0]
0.3 (1.1952)	2.0318	[1.2124 0]

Figure 2: State response of the closed-loop system under saturated control.

[figure omitted; refer to PDF]

Figure 3: The actuator output under saturated control.

[figure omitted; refer to PDF]

For clarity, we give some comments to explain the above simulation results. Firstly, it is easily seen from Table 2 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 2 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 [27] 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 inclusion, which may lead to conservatism. Applying the saturation-dependent Lyapunov function approach proposed in [20] 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.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant 60904020, Open Fund of Key Laboratory of Measurement and Control of CSE (no. MCCSE2012A06), Ministry of Education, China, Southeast University, and the Scientific Research Foundation of Nanjing University of Posts and Telecommunications (NY210080).

References

[1] J. K. Hale, S. M. Verduyn Lunel Introduction to Functional-Differential Equations , vol. 99, of Applied Mathematical Sciences, pp. x+447, Springer, New York, NY, USA, 1993.

[2] S. Xu, P. Van Dooren, R. Stefan, 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. 1122-1128, 2002.

[3] Z. Wu, H. Su, J. Chu, "Robust stabilization for uncertain discrete singular systems with state delay," International Journal of Robust and Nonlinear Control , vol. 18, no. 16, pp. 1532-1550, 2008.

[4] X. Ji, Z. Yang, H. Su, "Robust stabilization for uncertain discrete singular time-delay systems," Asian Journal of Control , vol. 12, no. 2, pp. 216-222, 2010.

[5] Z. Du, Q. Zhang, G. Chang, "Delay-dependent robust _H∞ control for uncertain descriptor systems with multiple state delays," Optimal Control Applications & Methods , vol. 31, no. 4, pp. 375-387, 2010.

[6] J. H. Kim, "Delay-dependent robust _H∞ control for discrete-time uncertain singular systems with interval time-varying delays in state and control input," Journal of the Franklin Institute , vol. 347, no. 9, pp. 1704-1722, 2010.

[7] S. Zhu, Z. Li, C. Zhang, "Delay decomposition approach to delay-dependent stability for singular time-delay systems," IET Control Theory & Applications , vol. 4, no. 11, pp. 2613-2620, 2010.

[8] F. Li, X. Zhang, "A delay-dependent bounded real lemma for singular LPV systems with time-variant delay," International Journal of Robust and Nonlinear Control , vol. 22, no. 5, pp. 559-574, 2012.

[9] S. Xu, J. Lam, M. Zhong, "New exponential estimates for time-delay systems," IEEE Transactions on Automatic Control , vol. 51, no. 9, pp. 1501-1505, 2006.

[10] D. Yue, Q.-L. Han, "Robust _H∞ filter design of uncertain descriptor systems with discrete and distributed delays," IEEE Transactions on Signal Processing , vol. 52, no. 11, pp. 3200-3212, 2004.

[11] D. Yue, Q. L. Han, "Delay-dependent robust _H∞ controller design for uncertain descriptor systems with time-varying discrete and distributed delays," IEE Proceedings: Control Theory and Applications , vol. 152, no. 6, pp. 628-638, 2005.

[12] Z. Shu, J. Lam, "Exponential estimates and stabilization of uncertain singular systems with discrete and distributed delays," International Journal of Control , vol. 81, no. 6, pp. 865-882, 2008.

[13] J. X. Lin, X. L. Zhao, S. M. Fei, "New delay-range-dependent exponential estimates for singular systems with time-varying delay," International Journal of Control, Automation and Systems , vol. 9, no. 2, pp. 218-227, 2011.

[14] Y. Y. Cao, P. M. Frank, "Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach," IEEE Transactions on Fuzzy Systems , vol. 8, no. 2, pp. 200-211, 2000.

[15] J. J. Rubio, W. Yu, "Stability analysis of nonlinear system identification via delayed neural networks," IEEE Transactions on Circuits and Systems II , vol. 54, no. 2, pp. 161-165, 2007.

[16] A. Haidar, E. K. Boukas, "Exponential stability of singular systems with multiple time-varying delays," Automatica , vol. 45, no. 2, pp. 539-545, 2009.

[17] A. Haidar, E. K. Boukas, S. Xu, J. Lam, "Exponential stability and static output feedback stabilisation of singular time-delay systems with saturating actuators," IET Control Theory & Applications , vol. 3, no. 9, pp. 1293-1305, 2009.

[18] T. Hu, Z. Lin Control Systems with Actuator Saturation: Analysis and Design , Birkhäuser, Basel, Switzerland, 2001.

[19] S. Tarbouriech, J. M. Gomes da Silva, Jr., "Synthesis of controllers for continuous-time delay systems with saturating controls via LMI's," IEEE Transactions on Automatic Control , vol. 45, no. 1, pp. 105-111, 2000.

[20] Y.-Y. Cao, Z. Lin, "Stability analysis of discrete-time systems with actuator saturation by a saturation-dependent Lyapunov function," Automatica , vol. 39, no. 7, pp. 1235-1241, 2003.

[21] L. Zhang, E.-K. Boukas, A. Haidar, "Delay-range-dependent control synthesis for time-delay systems with actuator saturation," Automatica , vol. 44, no. 10, pp. 2691-2695, 2008.

[22] B. Zhou, Z. Lin, G. Duan, "Robust global stabilization of linear systems with input saturation via gain scheduling," International Journal of Robust and Nonlinear Control , vol. 20, no. 4, pp. 424-447, 2010.

[23] W. Lan, J. Huang, "Semiglobal stabilization and output regulation of singular linear systems with input saturation," IEEE Transactions on Automatic Control , vol. 48, no. 7, pp. 1274-1280, 2003.

[24] B. Zhou, J. Lam, G.-R. Duan, "An ARE approach to semi-global stabilization of discrete-time descriptor linear systems with input saturation," Systems & Control Letters , vol. 58, no. 8, pp. 609-616, 2009.

[25] Z. Lin, L. Lv, "Set invariance conditions for singular linear systems subject to actuator saturation," IEEE Transactions on Automatic Control , vol. 52, no. 12, pp. 2351-2355, 2007.

[26] L. Lv, Z. Lin, "Analysis and design of singular linear systems under actuator saturation and _L2 /_L∞ disturbances," Systems & Control Letters , vol. 57, no. 11, pp. 904-912, 2008.

[27] S. Ma, E. K. Boukas, "Stability and _H∞ control for discrete-time singular systems subject to actuator saturation," in Proceedings of the American Control Conference (ACC '09), pp. 1244-1249, Hyatt Regency Riverfront, St Louis, Mo, USA, June 2009.

[28] X. Ji, Y. Sun, H. Su, "Analysis and design for singular discrete linear systems subject to actuator saturation," Asian Journal of Control , vol. 13, no. 2, pp. 350-355, 2011.

[29] L. Dai Singular Control Systems , vol. 118, of Lecture Notes in Control and Information Sciences, pp. x+332, Springer, Berlin, Germany, 1989.

[30] V. Kharitonov, S. Mondié, J. Collado, "Exponential estimates for neutral time-delay systems: an LMI approach," IEEE Transactions on Automatic Control , vol. 50, no. 5, pp. 666-670, 2005.

[31] S. Ma, E.-K. Boukas, Y. Chinniah, "Stability and stabilization of discrete-time singular Markov jump systems with time-varying delay," International Journal of Robust and Nonlinear Control , vol. 20, no. 5, pp. 531-543, 2010.

[32] F. C. Schweppe Uncertain Dynamic Systems , Prentice Hall, Englewood Cliffs, NJ, USA, 1973.

[33] W. Yu, J. J. Rubio, "Recurrent neural networks training with stable bounding ellipsoid algorithm," IEEE Transactions on Neural Networks , vol. 20, no. 6, pp. 983-991, 2009.

[34] J. J. Rubio, W. Yu, A. Ferreyra, "Neural network training with optimal bounded ellipsoid algorithm," Neural Computing and Applications , vol. 18, no. 6, pp. 623-631, 2009.

[35] K. C. Goh, M. G. Safonov, G. P. Papavassilopoulos, "Global optimization approach for the BMI problem," in Proceedings of the 2nd IEEE International Symposium on Requirements Engineering, pp. 2009-2014, Lake Buena Vista, Fla, USA, March 1995.

[36] S. Boyd, L. F. Ghaout, E. Feron, V. Balakrishnan Linear Matrix Inequality in Systems and Control Theory , SIAM, Philadelphia, Pa, USA, 1994.

Appendix

Appendix

Proof of Lemma 2.5.

For given ζ(k) and y(k) defined in (2.8), the following equation holds [figure omitted; refer to PDF] Rearranging the above equation yields (2.9). This completes the proof.

Proof of Lemma 2.7.

Case 1 (k...4;d(k)) . Clearly, k-d(k)∈[-d¯,0] , and then ||x(k,[varphi])||...4;||D||||[varphi]_||c +||f(k)||...4;χ^e-λ ||[varphi]_||c +κ^e-βk ...4;χ^e-λ(k/d¯) ||[varphi]_||c +κ^e-βk ...4;(χ||[varphi]_||c +κ)^e-rk .

Case 2 (k>d(k)) . Inspired by the idea in [16], to model the dependency of x(k) on past instances, define [figure omitted; refer to PDF] Note that _ki <_ki-1 then, by iterative substitutions of (2.10), one gets [figure omitted; refer to PDF] where k(t) is a positive integer.

(i) There exists a finite positive integer k(t) such that 0...4;_kk(t)-1 and -d¯...4;_kk(t) <0 . Then it follows from (A.2) that [figure omitted; refer to PDF] By (A.3), (A.4), (2.11) and r=min {λ/d¯,β} , the following is true: [figure omitted; refer to PDF]

(ii) _k0 -^∑j=0i-1 d(_kj )...5;0 for all positive integer i with i...5;1 . Since d(k)...5;0 , the sequence {_k0 -^∑j=0i-1 d(_kj )^}i=1+∞ is monotonically decreased and has a greatest lower bound ... with ......5;0 ; that is li_{mi[arrow right]+∞k0} -^∑j=0i-1 d(_kj )=... . Noting λ>0 , we have li_{mi[arrow right]+∞}^e-λ_ki ||_xki ||=0 [8]. Then, similar to the process to derive (A.5), the results in the lemma holds.