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

Authors: Jinxing Lin
Journal: Discrete Dynamics in Nature and Society
 (2012)

---

## 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.


---

## Body

## 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 ℒ2 gain and 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 Section2. 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 matricesP, P>0(P≥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≤n2, we use I[n1,n2] to denote the integer set {n1,n1+1,…,n2}. Let 𝒞n,d={ϕ:I[-d,0]→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 ϕ∈𝒞n,d, its norm is defined as ∥ϕ∥c=sups∈I[-d,0]∥ϕ(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:(2.1)Ex(k+1)=Ax(k)+Adx(k-d(k))+Bsat(u(k)),x(s)=ϕ(s),s∈I[-d¯,0],
where x(k)∈Rn is the system state, u(k)∈Rm is the control input, and ϕ(s)∈𝒞n,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 rankE=r<n. A, Ad and B are known constant matrices. The function sat:Rm→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]:
(2.2)u(k)=Fx(k),F=F¯E,
where F∈Rm×n and F¯∈Rm×n. The closed-loop system under this feedback is given by
(2.3)Ex(k+1)=Ax(k)+Adx(k-d(k))+Bsat(F¯Ex(k)).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 rankE=r<n, there exist two nonsingular matrices G,H∈Rn×n such that
(2.4)GEH=[Ir000],GAH=[A11A12A21A22],GAdH=[Ad11Ad12Ad21Ad22],GB=[B1B2],FH=[F1F2],F¯G-1=[F¯1F¯2],x(k)=H[x1(k)x2(k)],x1(k)∈Rr,x2(k)∈Rn-r.
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:
(2.5)x1(k+1)=A11x1(k)+A12x2(k)+Ad11x1(k-d(k))+Ad12x2(k-d(k))+B1sat(F1x1(k)+F2x2(k)),0=A21x1(k)+A22x2(k)+Ad21x1(k-d(k))+Ad22x2(k-d(k))+B2sat(F1x1(k)+F2x2(k)).
From the second equation of (2.5), it can be seen that x2(k) is in the function sat(·). Hence, if B2≠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
(2.6)x1(k+1)=A11x1(k)+A12x2(k)+Ad11x1(k-d(k))+Ad12x2(k-d(k))+B1sat(F¯1x1(k)),0=A21x1(k)+A22x2(k)+Ad21x1(k-d(k))+Ad22x2(k-d(k))+B2sat(F¯1x1(k)),
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]).
SystemEx(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))=rankE, 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 ϕ(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)∥≤ϵλ-(k-k0)∥xk0∥c for all k≥k0, where ∥xk0∥c=sups∈I[-d¯,0]∥x(k0+s)∥, k0 is the initial time step, and ϵ>0 is the decay coefficient.We are interested in the exponential estimates and design for system (2.3). For any compatible initial condition x0=ϕ∈𝒞n,d¯, denote the state trajectory of system (2.1) as x(k,x0); then the domain of attraction of the origin is
(2.7)𝒮={ϕ∈𝒞n,d¯:limk→∞x(k,x0)=0}.
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 𝒟={ϕ∈Cn,d¯:∥ϕ∥c2≤δ}∈𝒮 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(2.8)ζ(k)=[xT(k)xT(k-d(k))xT(k-d_)xT(k-d¯)]2,y(k)=x(k+1)-x(k).Lemma 2.5.
For any appropriately dimensioned matricesR>0 and N, two positive time-varying integer d(k1) and d(k2) satisfying d(k1)+1≤d(k2)≤d¯, and a scalar λ>0, the following equality holds
(2.9)-∑l=k-d(k2)k-d(k1)-1‍yT(l)ETλk-lREy(l)=cζT(k)NR-1NTζ(k)+2ζT(k)N∑l=k-d(k2)k-d(k1)-1‍Ey(l)-∑l=k-d(k2)k-d(k1)-1‍(ζT(k)N+λk-lyT(l)ETR)(λk-lR)-1(NTζk+λk-lREy(l)),
where c=(λ-d(k2)-λ-d(k1))/(1-λ).Proof.
See the Appendix.Lemma 2.6 (see [30]).
Given a matrixD, let a positive-definite matrix S and a positive scalar η∈(0,1) exist such that
(2.10)DTSD-η2S<0
then the matrix D satisfies the bound
(2.11)∥Di∥≤χe-λi,i=0,1,…,
where χ=λmax(S)/λmin(S) and λ=-ln(η).Lemma 2.7.
Let0<d_≤d(k)≤d¯. Consider the following system:
(2.12)x(k)+Dx(k-d(k))=f(k),k≥0,
where ∥Di∥≤χe-λi,  λ>0,  i=0,1,…,  and  ∥f(k)∥≤κe-βk, k≥0. If
(2.13)βd¯-λ<0.
Then, for any compatible initial function ϕ∈𝒞n,d¯, the solution x(k,ϕ) of (2.10) satisfies
(2.14)∥x(k,ϕ)∥≤(χ∥ϕ∥c+χκ11-eβd¯-λ+κ)e-rk,k≥0,
where r=min{λ/d¯,β}.Proof.
See the Appendix.For a matrixF∈Rm×n, denote the jth row of F as fj and define
(2.15)ℒ(F)={x(k)∈Rn:|fjx(k)|≤1,j∈I[1,m]}.
Let P∈Rn×n be a positive-definite matrix and ETPE≥0, the set Ω(ETPE,1) is defined by
(2.16)Ω(ETPE,1)={x(k)∈Rn:xT(k)ETPEx(k)≤1}.
Also, let 𝒱 be the set of m×m diagonal matrices whose diagonal elements are either 1 or 0. There are 2m elements in 𝒱. Suppose that each element of 𝒱 is labeled as Di,i∈I[1,2m] and denote Di-=I-Di. Clearly, Di is also an element of 𝒱 if Di∈𝒱.Lemma 2.8 (see [18]).
LetF,H∈Rm×n be given. If x(k)∈ℒ(H), then sat(Fx(k) can be expressed as
(2.17)sat(Fx(k))=∑i=12m‍αi(k)(DiF+Di-H)x(k),
where αi(k) for i∈I[1,2m] are some variables satisfying αi(k)≥0 and ∑i=12m‍αi(k)=1.Lemma 2.9 (see [31]).
Given matricesX, Y, and Z with appropriate dimensions, and Y is symmetric. Then there exists a scalar ρ>0, such that ρI+Y>0 and -Sym{XTZ}-ZTYZ≤XT(ρI+Y)-1X+ρZTZ.


## 2.1. Problem Statement

Consider a class of discrete-time singular system subject to time-varying delay and actuator saturation as follows:(2.1)Ex(k+1)=Ax(k)+Adx(k-d(k))+Bsat(u(k)),x(s)=ϕ(s),s∈I[-d¯,0],
where x(k)∈Rn is the system state, u(k)∈Rm is the control input, and ϕ(s)∈𝒞n,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 rankE=r<n. A, Ad and B are known constant matrices. The function sat:Rm→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]:
(2.2)u(k)=Fx(k),F=F¯E,
where F∈Rm×n and F¯∈Rm×n. The closed-loop system under this feedback is given by
(2.3)Ex(k+1)=Ax(k)+Adx(k-d(k))+Bsat(F¯Ex(k)).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 rankE=r<n, there exist two nonsingular matrices G,H∈Rn×n such that
(2.4)GEH=[Ir000],GAH=[A11A12A21A22],GAdH=[Ad11Ad12Ad21Ad22],GB=[B1B2],FH=[F1F2],F¯G-1=[F¯1F¯2],x(k)=H[x1(k)x2(k)],x1(k)∈Rr,x2(k)∈Rn-r.
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:
(2.5)x1(k+1)=A11x1(k)+A12x2(k)+Ad11x1(k-d(k))+Ad12x2(k-d(k))+B1sat(F1x1(k)+F2x2(k)),0=A21x1(k)+A22x2(k)+Ad21x1(k-d(k))+Ad22x2(k-d(k))+B2sat(F1x1(k)+F2x2(k)).
From the second equation of (2.5), it can be seen that x2(k) is in the function sat(·). Hence, if B2≠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
(2.6)x1(k+1)=A11x1(k)+A12x2(k)+Ad11x1(k-d(k))+Ad12x2(k-d(k))+B1sat(F¯1x1(k)),0=A21x1(k)+A22x2(k)+Ad21x1(k-d(k))+Ad22x2(k-d(k))+B2sat(F¯1x1(k)),
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]).
SystemEx(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))=rankE, 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 ϕ(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)∥≤ϵλ-(k-k0)∥xk0∥c for all k≥k0, where ∥xk0∥c=sups∈I[-d¯,0]∥x(k0+s)∥, k0 is the initial time step, and ϵ>0 is the decay coefficient.We are interested in the exponential estimates and design for system (2.3). For any compatible initial condition x0=ϕ∈𝒞n,d¯, denote the state trajectory of system (2.1) as x(k,x0); then the domain of attraction of the origin is
(2.7)𝒮={ϕ∈𝒞n,d¯:limk→∞x(k,x0)=0}.
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 𝒟={ϕ∈Cn,d¯:∥ϕ∥c2≤δ}∈𝒮 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(2.8)ζ(k)=[xT(k)xT(k-d(k))xT(k-d_)xT(k-d¯)]2,y(k)=x(k+1)-x(k).Lemma 2.5.
For any appropriately dimensioned matricesR>0 and N, two positive time-varying integer d(k1) and d(k2) satisfying d(k1)+1≤d(k2)≤d¯, and a scalar λ>0, the following equality holds
(2.9)-∑l=k-d(k2)k-d(k1)-1‍yT(l)ETλk-lREy(l)=cζT(k)NR-1NTζ(k)+2ζT(k)N∑l=k-d(k2)k-d(k1)-1‍Ey(l)-∑l=k-d(k2)k-d(k1)-1‍(ζT(k)N+λk-lyT(l)ETR)(λk-lR)-1(NTζk+λk-lREy(l)),
where c=(λ-d(k2)-λ-d(k1))/(1-λ).Proof.
See the Appendix.Lemma 2.6 (see [30]).
Given a matrixD, let a positive-definite matrix S and a positive scalar η∈(0,1) exist such that
(2.10)DTSD-η2S<0
then the matrix D satisfies the bound
(2.11)∥Di∥≤χe-λi,i=0,1,…,
where χ=λmax(S)/λmin(S) and λ=-ln(η).Lemma 2.7.
Let0<d_≤d(k)≤d¯. Consider the following system:
(2.12)x(k)+Dx(k-d(k))=f(k),k≥0,
where ∥Di∥≤χe-λi,  λ>0,  i=0,1,…,  and  ∥f(k)∥≤κe-βk, k≥0. If
(2.13)βd¯-λ<0.
Then, for any compatible initial function ϕ∈𝒞n,d¯, the solution x(k,ϕ) of (2.10) satisfies
(2.14)∥x(k,ϕ)∥≤(χ∥ϕ∥c+χκ11-eβd¯-λ+κ)e-rk,k≥0,
where r=min{λ/d¯,β}.Proof.
See the Appendix.For a matrixF∈Rm×n, denote the jth row of F as fj and define
(2.15)ℒ(F)={x(k)∈Rn:|fjx(k)|≤1,j∈I[1,m]}.
Let P∈Rn×n be a positive-definite matrix and ETPE≥0, the set Ω(ETPE,1) is defined by
(2.16)Ω(ETPE,1)={x(k)∈Rn:xT(k)ETPEx(k)≤1}.
Also, let 𝒱 be the set of m×m diagonal matrices whose diagonal elements are either 1 or 0. There are 2m elements in 𝒱. Suppose that each element of 𝒱 is labeled as Di,i∈I[1,2m] and denote Di-=I-Di. Clearly, Di is also an element of 𝒱 if Di∈𝒱.Lemma 2.8 (see [18]).
LetF,H∈Rm×n be given. If x(k)∈ℒ(H), then sat(Fx(k) can be expressed as
(2.17)sat(Fx(k))=∑i=12m‍αi(k)(DiF+Di-H)x(k),
where αi(k) for i∈I[1,2m] are some variables satisfying αi(k)≥0 and ∑i=12m‍αi(k)=1.Lemma 2.9 (see [31]).
Given matricesX, Y, and Z with appropriate dimensions, and Y is symmetric. Then there exists a scalar ρ>0, such that ρI+Y>0 and -Sym{XTZ}-ZTYZ≤XT(ρI+Y)-1X+ρZTZ.


## 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 constants0<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=[M1TM2TM3TM4T]T, N=[N1TN2TN3TN4T]T and T=[T1TT2TT3TT4T]T such that the following inequality holds
(3.1)[ΦΨ*Γ]<0,
where
(3.2)Φ=diag{-(1-α)ETXE+(1+d~)Q1+Q2+Q3,-(1-α)d¯Q1,-(1-α)d_Q2,-(1-α)d¯Q3}+Sym{MΛ1+NΛ2+TΛ3}+𝒜1TX𝒜1-𝒜1TRTSR𝒜1+𝒜2TU𝒜2,Ψ=[ϱ1Mϱ2Nϱ2T],Γ=diag{-ϱ1Z1,-ϱ2(Z1+Z2),-ϱ2Z2},𝒜1=[AAd00],𝒜2=[A-EAd00],Λ1=[E-E00],Λ2=[0E0-E],Λ3=[0E0-E],d~=d¯-d_,U=d¯Z1+d~Z2,ϱ1=(1-α)-d¯-1α,ϱ2=(1-α)-d¯-(1-α)-d_α,
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. SinceE is regular and rank(E)=r, there exist two nonsingular matrices G1 and H1 such that
(3.3)G1EH1=[Ir000],G1AH1=[A~11A~12A~21A~22],G1-TXG1-1=[X~11X~12X~12TX~22],H1TM1G1-1=[M~11M~12M~21M~22],RG1-1=[R~1R~2].
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,
(3.4)RG1-1=[0R~2].
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
(3.5)-ATRTSRA-(1-α)ETXE+Sym{M1E}<0.
Pre- and postmultiplying (3.5) by H1T and H1, respectively, and using (3.3) and (3.4), it is obtained that
(3.6)[⋆⋆⋆-A~22TR~2TSR~2A~22]<0,
where ⋆ represents matrices that are not relevant in the following discussion. Thus,
(3.7)-A~22TR~2TSR~2A~22<0.
Now, we assume that the matrix A~22 is singular, then, there exists a vector η∈Rn-r and η≠0 such that A~22η=0. Pre- and postmultiplying (3.7) by ηT and η, respectively, result in ηTA~22TR~2TSR~2A~22η=0. Then, it is easy to see that (3.7) is a contradiction since ηTA~22TR~2TSR~2A~22η>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
(3.8)E¯=G2EH2=[Ir000],A¯=G2AH2=[A¯1100In-r].
According to (3.8), define
(3.9)A¯d=G2AdH2=[A¯d11A¯d12A¯d21A¯d22],X¯=G2-TXG2-1=[X¯11X¯12X¯12TX¯22],Q¯l=H2TQlH2=[Q¯l11Q¯l12Q¯l12TQ¯l22],Z¯v=G2-TZvG2-1=[Z¯v11Z¯v12Z¯v12TZ¯v22],M¯v=H2TMvG2-1=[M¯v11M¯v12M¯v21M¯v22],N¯v=H2TNvG2-1=[N¯v11N¯v12N¯v21N¯v22],T¯v=H2TTvG2-1=[T¯v11T¯v12T¯v21T¯v22],R¯=RG2-1=[0R¯2],l=1,2,3,v=1,2.
By using Schur complement on (3.1), we get
(3.10)[Φ11′Φ12′*Φ22′]<0,
where
(3.11)Φ11′=-ATRTSRA-(1-α)ETPE+(1+d~)Q1+Q2+Q3+Sym{M1E},Φ12′=-ATRTSRAd-M1E+N1E-T1E+ETM2T,Φ22′=-AdTRTSRAd-(1-α)d¯Q1+Sym{-M2E+N2E-T2E}.
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
(3.12)[-R¯2TSR¯2+(1+d~)Q¯122+Q¯222+Q¯322-R¯2TSR¯2A¯d22*-(1-α)d¯Q¯122-A¯d22TR¯2TSR¯2A¯d22]<0.
Pre- and postmultiplying this inequality by [-A¯d22TI] and its transpose, respectively, and noting A¯d22T(Q¯222+Q¯322)A¯d22≥0, we have
(3.13)A¯d22TQ¯122A¯d22-(1-α)d¯1+d~Q¯122<0.
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
(3.14)∥A¯d22i∥≤χe-λi,i=1,2,…
Letξ(k)=H2-1x(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:
(3.15)ξ1(k+1)=A¯11ξ1(k)+A¯d11ξ1(k-d(k))+A¯d12ξ2(k-d(k)),0=ξ2(k)+A¯d21ξ1(k-d(k))+A¯d22ξ2(k-d(k)).
Now, choose the following Lyapunov-Krasovskii function:
(3.16)V(ξk)=∑s=14‍Vs(ξk),
where
(3.17)V1(ξk)=ξ1T(k)X¯11ξ1(k)=ξT(k)E¯TX¯E¯ξ(k),V2(ξk)=∑l=k-d(k)k-1‍ξT(l)(1-α)k-1-lQ¯1ξ(l)+∑θ=-d¯+1-d_‍∑l=k+θk-1‍ξT(l)(1-α)k-1-lQ¯1ξ(l),V3(ξk)=∑l=k-d_k-1‍ξT(l)(1-α)k-1-lQ¯2ξ(l)+∑l=k-d¯k-1‍ξT(l)(1-α)k-1-lQ¯3ξ(l),V4(ξk)=∑θ=-d¯-1‍∑l=k+θk-1‍ηT(l)E¯T(1-α)k-1-lZ¯1E¯η(l)+∑θ=-d¯-d_-1‍∑l=k+θk-1‍ηT(l)E¯T(1-α)k-1-lZ¯2E¯η(l),
with ξk(s)=ξ(k+s), ∀s∈I[-d¯,0] and η(k)=ξ(k+1)-ξ(k). Define
(3.18)▵V(ξk)=∑s=14‍[▵Vs(ξk)]=∑s=14‍[Vs(ξk+1)-(1-α)Vs(ξk)].
Then, it follows from (2.1) that
(3.19)▵V1(ξk)=ξT(k+1)E¯TX¯E¯ξ(k+1)-(1-α)ξT(k)E¯TX¯E¯ξ(k)=(A¯ξ(k)+A¯dξ(k-d(k)))TX¯(A¯ξ(k)+A¯dξ(k-d(k)))-(1-α)ξT(k)E¯TX¯E¯ξ(k),▵V2(ξk)≤(1+d~)ξT(k)Q¯1ξ(k)-ξT(k-d(k))(1-α)d(k)Q¯1ξ(k-d(k)),▵V3(ξk)=ξT(k)(Q¯2+Q¯3)ξ(k)-ξT(k-d_)(1-α)d_Q¯2ξ(k-d_)-ξT(k-d¯)(1-α)d¯Q¯3ξ(k-d¯),▵V4(ξk)=ηT(k)E¯T(d¯Z¯1+d~Z¯2)E¯η(k)-∑l=k-d¯k-1‍ηT(l)E¯T(1-α)k-lZ¯1E¯η(l)-∑l=k-d¯k-d_-1‍ηT(l)E¯T(1-α)k-lZ¯2E¯η(l)=ηT(k)E¯T(d¯Z¯1+d~Z¯2)E¯η(k)-∑l=k-d(k)k-1‍ηT(l)E¯T(1-α)k-lZ¯1E¯η(l)-∑l=k-d¯k-d(k)-1‍ηT(l)E¯T(1-α)k-l(Z¯1+Z¯2)E¯η(l)-∑l=k-d(k)k-d_-1‍ηT(l)E¯T(1-α)k-lZ¯2E¯η(l).
Let
(3.20)ψ(k)=[ξT(k)ξT(k-d(k))ξT(k-d_)ξT(k-d¯)]T.
Using Lemma 2.5 for the last three terms of ▵V4(ξk), respectively, and noting d_≤d(k)≤d¯, we have
(3.21)-∑l=k-d(k)k-1‍ηT(l)E¯T(1-α)k-lZ¯1E¯η(l)≤ϱ1ψT(k)M¯Z¯1-1M¯Tψ(k)+2ψT(k)M¯∑l=k-d(k)k-1‍E¯η(l)-∑l=k-d(k)k-1‍[ψT(k)M¯+(1-α)k-lηT(l)E¯TZ¯1]((1-α)k-lZ¯1)-1×[M¯Tψ(k)+(1-α)k-lZ¯1E¯η(l)]-∑l=k-d¯k-d(k)-1‍ηT(l)E¯T(1-α)k-l(Z¯1+Z¯2)E¯η(l)≤ϱ2ψT(k)N¯(Z¯1+Z¯2)-1N¯Tψ(k)+2ψT(k)N¯∑l=k-d¯k-d(k)-1‍E¯η(l)-∑l=k-d¯k-d(k)-1‍[ψT(k)N¯+(1-α)k-lηT(l)E¯T(Z¯1+Z¯2)]((1-α)k-l(Z¯1+Z¯2))-1·[N¯Tψ(k)+(1-α)k-l(Z¯1+Z¯2)E¯η(l)]-∑l=k-d(k)k-d_-1‍ηT(l)E¯T(1-α)k-lZ¯2E¯η(l)≤ϱ2ψT(k)T¯Z¯2-1T¯Tψ(k)+2ψT(k)T¯∑l=k-d(k)k-d_-1‍E¯η(l)-∑l=k-d(k)k-d_-1‍[ψT(k)T¯+(1-α)k-lηT(l)E¯TZ¯2]((1-α)k-lZ¯2)-1×[T¯Tψ(k)+(1-α)k-lZ¯2E¯η(l)].
Note that η(l)=ξ(l+1)-ξ(l) provides
(3.22)∑l=k-d(k)k-1‍E¯η(l)=[E¯-E¯00]ψ(k),(3.23)∑l=k-d¯k-d(k)-1‍E¯η(l)=[0E¯0-E¯]ψ(k),(3.24)∑l=k-d(k)k-d_-1‍E¯η(l)=[0-E¯E¯0]ψ(k).
Also, it follows from RE=0 that
(3.25)0=-ξT(k+1)E¯TR¯TSR¯E¯ξT(k+1)=-(A¯ξ(k)+A¯dξ(k-d(k)))TR¯TSR¯(A¯ξ(k)+A¯dξ(k-d(k))).
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
(3.26)▵V(ξk)≤ψT(k)[Φ¯+ϱ1M¯Z¯1-1M¯T+ϱ2N¯(Z1+Z2)-1N¯T+ϱ2T¯Z¯2-1T¯T]ψ(k)-∑l=k-d(k)k-1‍[ψT(k)M¯+(1-α)k-lηT(l)E¯TZ¯1]((1-α)k-lZ¯1)-1×[M¯Tψ(k)+(1-α)k-lZ¯1E¯η(l)]-∑l=k-d¯k-d(k)-1‍[ψT(k)N¯+(1-α)k-lηT(l)E¯T(Z¯1+Z¯2)][(1-α)k-l(Z¯1+Z¯2)]-1·[N¯Tψ(k)+(1-α)k-l(Z¯1+Z¯2)E¯η(l)]-∑l=k-d(k)k-d_-1‍[ψT(k)T¯+(1-α)k-lηT(l)E¯TZ¯2]((1-α)k-lZ¯2)-1×[T¯Tψ(k)+(1-α)k-lZ¯2E¯η(l)]≤ψT(k)[Φ¯+ϱ1M¯Z¯1-1M¯T+ϱ2N¯(Z1+Z2)-1N¯T+ϱ2T¯Z¯2-1T¯T]ψ(k),
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¯+ϱ1M¯Z¯1-1M¯T+ϱ2N¯(Z1+Z2)-1N¯T+ϱ2T¯Z¯2-1T¯T<0. Thus, it follows from (3.18) and (3.26) that ▵V(ξk)=V(ξk+1)-(1-α)V(ξk)≤0, which leads to
(3.27)V(ξk+1)≤(1-α)V(ξk).
By iterative substitutions, inequality (3.27) yields
(3.28)V(ξk)≤(1-α)kV(ξ0).
On the other hand, it follows from the Lyapunov functional (3.16) that
(3.29)β1∥ξ1(k)∥2≤V(ξk),V(ξ0)≤β2∥ϕ∥c2,
where
(3.30)β1=λmin(X¯11),β2=λmax(X¯11)+(d¯+d~(d¯-1))λmax(Q¯1)+d_λmax(Q¯2)+d¯λmax(Q¯3)+d¯2λmax(Z¯1+Z¯2),
Then, combining (3.28) and (3.29) leads to
(3.31)∥ξ1(k)∥≤β2β1(1-α)k/2∥ϕ∥c=β2β1e-(ln(1-α)-1/2)k∥ϕ∥c.
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. Setf(k)=-A¯d21ξ1(k-d(k)); then, it follows from (3.31) that
(3.32)∥f(k)∥≤(1-α)-d¯/2∥A¯d21∥β2β1e-(ln(1-α)-1/2)k∥ϕ∥c.
Using the second equation in (3.15), (3.14) and Lemma 2.7, one gets
(3.33)∥ξ2(k)∥≤β3e-(ln(1-α)-1/2)k∥ϕ∥c,
where
(3.34)β3=λmax(Q¯122)λmin(Q¯122)∥ϕ∥c+(λmax(Q¯122)λmin(Q¯122)11-e-(1/2)ln(1+d~)+1)β2β1(1-α)-d¯/2∥A¯d21∥.
Combining (3.31) and (3.33) yields that
(3.35)∥x(k)∥2=ξT(k)H2TH2ξ(k)≤∥H2∥2(∥ξ1(k)∥2+∥ξ2(k)∥2)≤(β2β1+β32)e-(ln(1-α)-1)k∥ϕ∥c2.
Thus, we have
(3.36)∥x(k)∥≤β2β1+β32e-(ln(1-α)-1/2)k∥ϕ∥c=β2β1+β32(11-α)-k∥ϕ∥c
The proof is completed.Remark 3.2.
Theorem3.1 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.
WhenE=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)+Adx(k-d(k)) from (3.1).
Based on the result of Theorem3.1, 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 constants0<α<1, 0<d_<d¯, ρ>0, ɛ1>0, ɛ2>0, ɛ3>0, ɛ4, and ɛ5. If there exist symmetric matrices X>0, Ql>0, l=1,2,3, Zv>0, v=1,2, and S, and matrices M=[M1TM2TM3TM4T]T, N=[N1TN2TN3TN4T]T, T=[T1TT2TT3TT4T]T, F¯ and H such that the following inequalities hold:
(3.37)[Φ¯jΨΥjd¯Πjd~ΠjΞΥjRT*Γ00000**-2ɛ1I+ɛ12X0000***-d¯(2ɛ2I-ɛ22Z1)000****-d~(2ɛ3I-ɛ32Z2)00*****-ρI-S0******-ρ-1I]<0,j∈I[1,2m],(3.38)[1hl*ETXE]≥0,l∈I[1,m],
where hl denotes the lth row of H,
(3.39)Φ¯j=diag{-(1-α)ETXE+(1+d~)Q1+Q2+Q3,-(1-α)d¯Q1,-(1-α)d_Q2,-(1-α)d¯Q3}+Sym{ΞRΥjT+MΛ1+NΛ2+TΛ3},Υj=[AjAd00]T,Πj=[Aj-EAd00]TΞ=[ɛ4Iɛ5I00]T,Aj=A+B(DjF¯E+Dj-H).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
(3.40)ℬ(δ)={ϕ∈Cn,d¯:∥ϕ∥c2≤δ},
where
(3.41)δ=1λmax(ETXE)+(d¯+d~(d¯-1))λmax(Q1)+d_λmax(Q2)+d¯λmax(Q3)+d¯2λmax(Z1+Z2).Proof.
If (3.38) holds, then the ellipsoid Ω(ETXE,1) is included in ℒ(H) [27] (For more details about the ellipsoids and ellipsoid algorithm, we refer the readers to [32–34]). Suppose that x(k)∈ℒ(H), ∀k>0 (will be proved later). Hence, by Lemma 2.8, sat(F¯Ex(k)) can be expressed as
(3.42)sat(F¯Ex(k))=∑j=12m‍αj(k)(DjF¯E+Di-H)x(k)
and it follows that
(3.43)Ex(k+1)=∑j=12m‍αj(k)Ajx(k)+Adx(k-d(k)).
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
(3.44)[Φ^Ψ*Γ]<0,
where
(3.45)Φ^=diag{-(1-α)ETXE+(1+d~)Q1+Q2+Q3,-(1-α)d¯Q1,-(1-α)d_Q2,-(1-α)d¯Q3}+𝒜^1jTX𝒜^1j-𝒜^1jTRTSR𝒜^1j+𝒜^2jTU𝒜^2j+Sym{MΛ1+NΛ2+TΛ3},(3.46)𝒜^1j=[∑j=12m‍αj(k)AjAd00],𝒜^2j=[∑j=12m‍αj(k)Aj-EAd00].
Now, provided that inequalities (3.37) hold, then by using Schur complement and Lemma 2.9, one has
(3.47)[Φ¯j+Ξ(ρI+S)-1ΞTΨΥjd¯Πjd~ΠjΥjRT*Γ0000**-X-1000***-d¯Z1-100****-d~Z2-10*****-ρ-1I]<0,j∈I[1,2m].
This, together with αj(k)≥0 and ∑j=12m‍αj(k)=1, implies that
(3.48)∑j=12m‍αj(k)[Φ¯j+Ξ(ρI+S)-1ΞTΨΥjd¯Πjd~ΠjΥjRT*Γ0000**-X-1000***-d¯Z1-100****-d~Z2-10*****-ρ-1I]<0,
that is,
(3.49)[∑j=12m‍αj(k)Φ¯j+Ξ(ρI+S)-1ΞTΨ∑j=12m‍αj(k)Υj∑j=12m‍αj(k)d¯Πj∑j=12m‍αj(k)d~Πj∑j=12m‍αj(k)ΥjRT*Γ0000**-X-1000***-d¯Z1-100****-d~Z2-10*****-ρ-1I]<0.
Applying Schur complement to (3.49) leads to
(3.50)[ΛjΨ∑j=12m‍αj(k)Υj∑j=12m‍αj(k)d¯Πj∑j=12m‍αj(k)d~Πj*Γ000**-X-100***-d¯Z1-10****-d~Z2-1]<0,
where
(3.51)Λj=∑j=12m‍αj(k)Φ¯j+Ξ(ρI+S)-1ΞT+ρ∑j=12m‍αj(k)ΥjRT∑j=12m‍αj(k)RΥjT.
Rewrite
(3.52)∑j=12m‍αj(k)Φ¯j=∑j=12m‍αj(k)Sym{ΥjRTΞT}+Φ~,
where
(3.53)Φ~=diag{-(1-α)ETXE+(1+d~)Q1+Q2+Q3,-(1-α)d¯Q1,-(1-α)d_Q2,-(1-α)d¯Q3}+Sym{MΛ1+NΛ2+TΛ3}.
From (3.37), it follows that ρI+S>0, and then the following inequality holds from Lemma 2.9(3.54)-∑j=12m‍αj(k)ΥjRTS∑j=12m‍αj(k)RΥjT≤∑j=12m‍αj(k)Sym{ΥjRTΞT}+Ξ(ρI+S)-1ΞT+ρ∑j=12m‍αj(k)ΥjRT∑j=12m‍αj(k)RΥjT,
which, together with (3.50) and (3.52), implies that
(3.55)[-∑j=12m‍αj(k)ΥjRTS∑j=12m‍αj(k)RΥjT+Φ~Ψ∑j=12m‍αj(k)Υj∑j=12m‍αj(k)d¯Πj∑j=12m‍αj(k)d~Πj*Γ000**-X-100***-d¯Z1-10****-d~Z2-1]<0.
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)∈ℒ(H), ∀k≥0.
On the other hand, by the proof in Theorem3.1, inequality (3.44) holding implies V(ξk+1)<(1-α)V(ξk)<V(ξk). 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(ξk), we get xT(0)ETXEx(0)=ξT(0)E¯TX¯E¯ξ(0)≤V(ξ0)≤δ-1∥ϕ∥c2≤1, that is, x(0)∈Ω(ETXE,1). Since inequalities (3.38) are satisfied, it follows that x(0)∈ℒ(H). Furthermore, because V(ξk+1)<V(ξk), we can conclude that xT(k)ETXEx(k)=ξT(k)E¯TX¯E¯ξ(k)≤V(ξk)≤V(ξ0)≤δ-1∥ϕ∥c2≤1, 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 ballℬ(δ) of initial condition as large as possible. To maximize the ball of initial condition, the following approximate optimization procedure can be used:
(P1)minX>0,Q1>0,Q2>0,Q3>0,Z1>0,Z2>0,F¯,Hrs.t.(3.44),(3.45)andETXE≤w1,Q1≤w2,s.t.Q2≤w3,Q3≤w4,Z1≤w5,Z2≤w6,
where r=w1+(d¯+d~(d¯-1))w2+d_w3+d¯w4+d¯2w5+d¯2w6.In the special case thatE=I, the following corollary directly from Remark 3.3 and Theorem 3.4.Corollary 3.6.
Given constants0<α<1, 0<d_<d¯, ρ>0, ɛ1>0, ɛ2>0, and ɛ3>0. If there exist matrices X>0, Ql>0, l=1,2,3, Zv>0, v=1,2, M=[M1TM2TM3TM4T]T, N=[N1TN2TN3TN4T]T, T=[T1TT2TT3TT4T]T, F and H such that the following hold:
(3.56)[Φ¯jΨΥjd¯Πjd~Πj*Γ000**-2ɛ1I+ɛ12X00***-d¯(2ɛ2I-ɛ22Z1)0****-d~(2ɛ3I-ɛ32Z2)]<0,j∈I[1,2m],[1hl*X]≥0,l∈I[1,m],
where hl denotes the lth row of H,
(3.57)Φ¯j=diag{-(1-α)X+(1+d~)Q1+Q2+Q3,-(1-α)d¯Q1,-(1-α)d_Q2,-(1-α)d¯Q3}+Sym{MΛ1+NΛ2+TΛ3},Υj=[AjAd00]T,Πj=[Aj-IAd00]T,Λ1=[I-I00],Λ2=[0I0-I],Λ3=[0-II0],Aj=A+B(DjF+Dj-H),
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
(3.58)ℬ(δ)={ϕ∈Cn,d¯:∥ϕ∥c2≤δ},
where
(3.59)δ=1λmax(X)+(d¯+d~(d¯-1))λmax(Q1)+d_λmax(Q2)+d¯λmax(Q3)+d¯2λmax(Z1+Z2).Also, as Remark3.5, the corresponding approximate optimization problem of Corollary 3.6 becomes
(P2)limX>0,Q1>0,Q2>0,Q3>0,Z1>0,Z2>0,F,Hrs.t.(3.56),(3.58)andX≤w1,Q1≤w2,s.t.Q2≤w3,Q3≤w4,Z1≤w5,Z2≤w6,
where r=w1+(d¯+d~(d¯-1))w2+d_w3+d¯w4+d¯2w5+d¯2w6.Remark 3.7.
Scalarsρ, ɛf, 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:
(4.1)E=[1000],A=[0.500.350.35],Ad=[0.1300.20.1].
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 1.25-k when d(k)=2+sin(2.5πk) and initial function is ϕk=[-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 ratesλ 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
1.0049Figure 1
Simulation results of  and  as compared1.25-k.Example 4.2.
Consider the discrete-time singular time-delay system (2.1) with the following parameters:
(4.2)E=[2.9100],A=[1.20.21.22.2],Ad=[0.1-0.10.3-0.2],B=[-0.1-0.4].
The time-delay is 1≤d(k)≤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.
Letρ=0.001, ɛ1=2.6, ɛ2=67, ɛ3=71, ɛ4=-0.08, ɛ5=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
(4.3)X=[0.01510.01180.01180.0288],Q1=[0.01200.00260.00260.0011],Q2=[0.01800.00260.00260.0016],Q3=[0.02100.00150.00150.0016],Z1=[0.00910.00190.00190.0043],Z2=[0.00970.00230.00230.0013],H=[1.62870.5616],F¯=[1.21240],
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 ϕk=[-1.6,1]T, k=-3,-2,…,0.Table 2
δ
max and F- for different values of α.
α
(
λ
)
δ
max
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 3
The actuator output under saturated control.For clarity, we give some comments to explain the above simulation results. Firstly, it is easily seen from Table2 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.



---

*Source: 414373-2012-09-10.xml*