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

Discrete Dynamics in Nature and Society Volume 2012, Article ID 414373,27pages doi:10.1155/2012/414373

Research Article

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

Jinxing Lin

1College of Automation, Nanjing University of Posts and Telecommunications, Nanjing 210003, China

2Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Southeast University, Nanjing 210096, China

Correspondence should be addressed to Jinxing Lin,jxlin2004@126.com Received 29 February 2012; Revised 20 June 2012; Accepted 24 July 2012 Academic Editor: Recai Kilic

Copyrightq2012 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.

1. Introduction

Singular time-delay systems STDSs arise naturally in many engineering fields such as electric networks, chemical processes, lossless transmission lines, and so forth1. 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,2–8and 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 slowdifferentialand fastalgebraicsubsystems 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 Lam12and Lin et al.13adopted the Lyapunov-Krasovskii function method14,15and some improvements have been obtained.

In16,17, 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 in23by 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 in24without any transformation of the original singular system. The invariant set approach developed for state-space system in18was extended to general, not necessarily semistable, continuous- time singular system in25. This approach was further extended to the analysis of theL2

gain and L∞ performance for continuous-time singular systems under actuator saturation 26and the analysis and design of discrete-time singular systems under actuator saturation 27,28, respectively. In17, 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:

1In 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.

2The 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 inSection 2. 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 inSection 4to illustrate the effectiveness of the proposed method. The paper will be concluded inSection 5.

Notation. For real symmetric matricesP,P >0P ≥0means that matrixPis positive definite semipositive definite. λmaxP λminP denotes the largest smallest eigenvalue of the positive definite matrixP. Rⁿis then-dimensional real Euclidean space and R^m×n is the set of all realm×nmatrices. Zrepresents 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 AA^T. For two integersn1 and n2 with n1 ≤ n2, we use In1, n2 to denote the integer set{n1, n11, . . . , n2}. LetCn,d {φ : I−d,0 → Rⁿ}denote the Banach space of family continuous vector valued functions mapping the interval I−d,0to Rⁿ with the topology of uniform convergence. Denotexks xks,∀s∈I−d,0. · refers to either the Euclidean vector norm or the induced matrix two-norm. For a functionφ∈ Cn,d, its norm is defined as φ c sup_s∈I−d,0 φs .

2. Problem Statement and Preliminaries

Consider a class of discrete-time singular system subject to time-varying delay and actuator saturation as follows:

Exk1 Axk Adxk−dk Bsatuk, xs φs, s∈I

−d,0 ,

2.1

where xk ∈ Rⁿ is the system state, uk ∈ R^m is the control input, and φs ∈ C_n,d is a compatible vector valued initial function.dk is a time-varying delay satisfying 0 <

d ≤ dk ≤ d, wheredand dare 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 : R^m → R^m is the standard saturation function defined as satuk satu1ksatu2k· · ·satumk^T, where satuik Signuikmin{1,|uik|}. 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 level20.

In this paper, we consider the design of a linear state feedback control law of the following form27:

uk Fxk, F FE, 2.2

whereF∈R^m×nandF∈R^m×n. The closed-loop system under this feedback is given by Exk1 Axk Adxk−dk Bsat

FExk

. 2.3

Remark 2.1. The state feedback of the form2.2is used to guarantee the uniqueness of the solution of system2.1. For that, since rankE r < n, there exist two nonsingular matrices G, H∈R^n×nsuch that

GEH

Ir 0 0 0

, GAH

A11 A12

A21 A22

, GAdH

Ad11 Ad12

Ad21 Ad22

,

GB B1

B2

, FH

F1 F2 , FG⁻¹ F1 F2

,

xk H x1k

x2k

, x1k∈R^r, x2k∈R^n−r.

2.4

If the state feedback is taken as the general form, that is,uk Fxk, by using2.4, then system2.3is restricted system equivalentr.s.e.to the following one:

x1k1 A11x1k A12x2k Ad11x1k−dk Ad12x2k−dk B1satF1x1k F2x2k,

0 A21x1k A22x2k Ad21x1k−dk Ad22x2k−dk B2satF1x1k F2x2k.

2.5

From the second equation of2.5, it can be seen thatx2kis in the function sat·. Hence, if B2/0, for givenx10, the solution ofx20is not unique evenA22 is nonsingular. However, ifuk FExk, then system2.3is r.s.e. to

x1k1 A11x1k A12x2k Ad11x1k−dk Ad12x2k−dk B1sat

F1x1k , 0 A21x1k A22x2k Ad21x1k−dk Ad22x2k−dk B2sat

F1x1k ,

2.6

which implies that, for given x1k, the unique solution of x2k can be obtained when A22 is nonsingular. Nevertheless, it should be pointed that the state feedback 2.2may be conservative due to its special structure.

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

Definition 2.2see29. SystemExk1 Axk or the pairE, Ais said to be regular if detzE−Ais not identically zero, and if degdetzE−A rankE, then it is further said to be causal.

Definition 2.3see6. System2.1withuk 0 is said to be regular and causal, if the pair E, Ais regular and causal.

Note that regularity and causality of system2.1withuk 0 ensure that the solution to this system exists and is unique for any given compatible initial valueφs.

Definition 2.4. System2.1under feedback law2.2is said to be exponentially stable with decay rateλλ >1if, for any compatible initial conditionsxk0s xk0s,s ∈I−d,0, its solution xk, xk0 satisfies xk, xk0 ≤ λ^−k−k0 xk0 c for all k ≥ k0, where xk0 c

sup_s∈I−d,0 xk0s ,k0is the initial time step, and >0 is the decay coefficient.

We are interested in the exponential estimates and design for system2.3. For any compatible initial condition x0 φ ∈ C_n,d, denote the state trajectory of system 2.1 as xk, x0; then the domain of attraction of the origin is

S

φ∈ C_n,d: lim

k→ ∞xk, x0 0

. 2.7

In general, for a given stabilizing state feedback gainF, it is impossible to determine exactly the domain of attraction of the origin with respect to system2.3. Therefore, the purpose of this paper is to design a state feedback gainFand determine a suitable set of initial condition D {φ∈C_n,d : φ ²_c ≤δ} ∈ Sfrom which the regularity, causality, and exponential stability of the closed-loop system2.3is ensured. Also, we are interested in maximizing the size of this set, that is, obtaining the maximal value ofδ.

2.2. Preliminary Results Define

ζk

x^Tk x^Tk−dk x^Tk−d x^Tk−d₂ , yk xk1−xk.

2.8

Lemma 2.5. For any appropriately dimensioned matricesR > 0 andN, two positive time-varying integerdk1anddk2satisfyingdk1 1≤dk2≤d, and a scalarλ >0, the following equality holds

−^k−dk1−1

l k−dk2

y^TlE^Tλ^k−lREyl

cζ^TkNR⁻¹N^Tζk 2ζ^TkN^k−dk1−1

l k−dk2

Eyl

−^k−dk1−1

l k−dk2

ζ^TkNλ^k−ly^TlE^TR

λ^k−lR₋₁

N^Tζkλ^k−lREyl ,

2.9

wherec λ^−dk2−λ^−dk1/1−λ.

Proof. See the Appendix.

Lemma 2.6see30. Given a matrixD, let a positive-definite matrixSand a positive scalarη ∈ 0,1exist such that

D^TSD−η²S <0 2.10

then the matrixDsatisfies the bound

Dⁱ≤χe^−λi, i 0,1, . . . , 2.11 whereχ

λmaxS/λminSandλ −lnη.

Lemma 2.7. Let 0< d≤dk≤d. Consider the following system:

xk Dxk−dk fk, k≥0, 2.12

where Dⁱ ≤χe^−λi,λ >0,i 0,1, . . ., and fk ≤κe^−βk,k≥0. If

βd−λ <0. 2.13

Then, for any compatible initial functionφ∈ C_n,d, the solutionxk, φof 2.10satisfies x

k, φ≤ χφ

cχκ 1

1−e^βd−λκ

e^−rk, k≥0, 2.14

wherer min{λ/d, β}.

Proof. See the Appendix.

For a matrixF ∈R^m×n, denote thejth row ofFasfjand define

LF

xk∈Rⁿ:fjxk≤1, j ∈I1, m

. 2.15

LetP∈R^n×nbe a positive-definite matrix andE^TP E≥0, the setΩE^TP E,1is defined by

Ω

E^TP E,1

xk∈Rⁿ:x^TkE^TP Exk≤1

. 2.16

Also, letVbe the set ofm×mdiagonal matrices whose diagonal elements are either 1 or 0.

There are 2^melements inV. Suppose that each element ofVis labeled asDi, i∈I1,2^mand denoteD_i⁻ I−Di. Clearly,Diis also an element ofVifDi ∈ V.

Lemma 2.8see18. LetF, H∈R^m×nbe given. Ifxk∈ LH, then satFxkcan be expressed as

satFxk ² m

i 1

αik

DiFD_i⁻H

xk, 2.17

whereαikfori∈I1,2^mare some variables satisfyingαik≥0 and₂^m

i 1αik 1.

Lemma 2.9 see 31. Given matrices X, Y, and Z with appropriate dimensions, and Y is symmetric. Then there exists a scalarρ > 0, such thatρIY > 0 and−Sym{X^TZ} −Z^TY Z ≤ X^TρIY⁻¹XρZ^TZ.

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 system2.1 i.e., withuk 0 with a predefined decay rate.

Theorem 3.1. Given constants 0< α < 1 and 0< d < d. If there exist symmetric matricesX >0, Ql > 0, l 1,2,3,Zv > 0,v 1,2 andS, and matrices M M^T₁ M^T₂ M^T₃ M^T₄^T,N N₁^T N₂^T N₃^T N^T₄^TandT T₁^T T₂^T T₃^T T₄^TTsuch that the following inequality holds

Φ Ψ

∗ Γ

<0, 3.1

where

Φ diag

−1−αE^TXE 1d

Q1Q2Q3,−1−α^dQ1,−1−α^dQ2,−1−α^dQ3

Sym{MΛ1NΛ2TΛ3}A^T₁XA1− A^T₁R^TSRA1A^T₂UA2, Ψ

1M 2N 2T , Γ diag

−1Z1,−2Z1Z2,−2Z2

, A1

A Ad 0 0 , A2

A−E Ad 0 0 , Λ1

E −E 0 0 , Λ2

0 E 0 −E , Λ3

0 E 0 −E , d d−d, U dZ1dZ 2,

1 1−α^−d−1

α , 2 1−α^−d−1−α^−d

α ,

3.2

andR ∈ R^n×n is any constant matrix satisfyingRE 0 and rankR n−r, then the unforced system2.1is regular, causal, and exponentially stable withλ 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;iiito show the exponential stability of the algebraic subsystem.

Parti: Regularity and causality. SinceEis regular and rankE r, there exist two nonsingular matricesG1andH1such that

G1EH1

Ir 0 0 0

, G1AH1

A11 A12

A21 A22

,

G^−T₁ XG⁻¹₁

X11 X12

X₁₂^T X22

, H₁^TM1G⁻¹₁

M11 M12

M21 M22

, RG⁻¹₁ R1 R2

.

3.3

Note thatRE 0 and rankR n−r, it can be verified thatR1 0, rankR2 n−rand R2 ∈R^n×n−r, that is,

RG⁻¹₁ 0 R2

. 3.4

From3.1, it is easy to obtainΦ11 <0. In view ofX >0, Ql>0,l 1,2,3,Zl>0, Z2>0,d >

0, andd > 0, it can be further obtained that

−A^TR^TSRA−1−αE^TXESym{M1E}<0. 3.5

Pre- and postmultiplying3.5byH₁^T and H1, respectively, and using3.3and 3.4, it is obtained that

−A^T₂₂R^T₂SR2A22

<0, 3.6

whererepresents matrices that are not relevant in the following discussion. Thus,

−A^T₂₂R^T₂SR2A22 <0. 3.7

Now, we assume that the matrix A22 is singular, then, there exists a vector η ∈ R^n−r and η /0 such that A22η 0. Pre- and postmultiplying 3.7 by η^T and η, respectively, result in η^TA^T₂₂R^T₂SR2A22η 0. Then, it is easy to see that 3.7 is a contradiction since η^TA^T₂₂R^T₂SR2A22η >0. Thus,A22is nonsingular, which implies that the unforced system2.1 is regular and causal by Definition 2 and Theorem 1 in4.

Partii: Exponential stability of the difference subsystem. From29, the regularity and causality of the unforced system2.1imply that there exist two nonsingular matricesG2

andH2such that

E G2EH2

Ir 0 0 0

, A G2AH2

A11 0

0 In−r

. 3.8

According to3.8, define

Ad G2AdH2

Ad11 Ad12

Ad21 Ad22

, X G^−T₂ XG⁻¹₂

X11 X12

X^T₁₂ X22

,

Q_l H₂^TQlH2

Q_l11 Q_l12 Q^T_l12 Q_l22

, Zv G^−T₂ ZvG⁻¹₂

Zv11 Zv12

Z^T_v12 Zv22

,

Mv H₂^TMvG⁻¹₂

Mv11 Mv12

Mv21 Mv22

, Nv H₂^TNvG⁻¹₂

Nv11 Nv12

Nv21 Nv22

,

Tv H₂^TTvG⁻¹₂

Tv11 Tv12

Tv21 Tv22

, R RG⁻¹₂ 0 R2

, l 1,2,3, v 1,2.

3.9

By using Schur complement on3.1, we get Φ₁₁ Φ₁₂

∗ Φ₂₂

<0, 3.10

where

Φ₁₁ −A^TR^TSRA−1−αE^TP E 1d

Q1Q2Q3Sym{M1E}, Φ₁₂ −A^TR^TSRAd−M1EN1E−T1EE^TM^T₂,

Φ₂₂ −A^T_dR^TSRAd−1−α^dQ1Sym{−M2EN2E−T2E}.

3.11

Substituting 3.8 and 3.9 into the above inequality, pre- and postmultiplying by diag{H₂^T, H₂^T}and diag{H2, H2}, respectively, and using Schur complement yields

⎡

⎣−R^T₂SR2 1d

Q₁₂₂Q₂₂₂Q₃₂₂ −R^T₂SR2Ad22

∗ −1−α^dQ₁₂₂−A^T_d22R^T₂SR2Ad22

⎤

⎦<0. 3.12

Pre- and postmultiplying this inequality by−A^T_d22 Iand its transpose, respectively, and notingA^T_d22Q₂₂₂Q₃₂₂Ad22≥0, we have

A^T_d22Q₁₂₂Ad22−1−α^d

1d Q₁₂₂<0. 3.13

Therefore, according to Lemma 2.6, there exist constantsχ

!

λmaxQ₁₂₂/λminQ₁₂₂and λ −ln1−α^d/2ln1d^1/2such that

Aⁱ_d22≤χe^−λi, i 1,2, . . . 3.14

Letξk H₂⁻¹xk ξ₁^Tk ξ₂^Tk^T, whereξ1k ∈ R^r andξ2k ∈ R^n−r. Then, the unforced system2.1is r.s.e. to the following one:

ξ1k1 A11ξ1k Ad11ξ1k−dk Ad12ξ2k−dk,

0 ξ2k Ad21ξ1k−dk Ad22ξ2k−dk. 3.15

Now, choose the following Lyapunov-Krasovskii function:

Vξk ⁴

s 1

Vsξk, 3.16

where

V1ξk ξ^T₁kX11ξ1k ξ^TkE^TXEξk, V2ξk ^k−1

l k−dk

ξ^Tl1−α^k−1−lQ₁ξl −d θ −d1

k−1 l kθ

ξ^Tl1−α^k−1−lQ₁ξl,

V3ξk ^k−1

l k−d

ξ^Tl1−α^k−1−lQ₂ξl ^k−1

l k−d

ξ^Tl1−α^k−1−lQ₃ξl,

V4ξk ⁻¹

θ −d k−1

l kθ

η^TlE^T1−α^k−1−lZ1Eηl ^−d−1

θ −d k−1

l kθ

η^TlE^T1−α^k−1−lZ2Eηl,

3.17

withξks ξks,∀s∈I−d,0andηk ξk1−ξk. Define

Vξk ⁴

s 1

Vsξk ⁴

s 1

Vsξ_k1−1−αVsξk. 3.18

Then, it follows from2.1that

V1ξk ξ^Tk1E^TX Eξk1−1−αξ^TkE^TX Eξk Aξk Adξk−dk_T

X

Aξk Adξk−dk

−1−αξ^TkE^TX Eξk, V2ξk≤

1d

ξ^TkQ₁ξk−ξ^Tk−dk1−α^dkQ₁ξk−dk, V3ξk ξ^Tk

Q₂Q₃

ξk−ξ^T k−d

1−α^dQ₂ξ k−d

−ξ^T k−d

1−α^dQ₃ξ k−d

, V4ξk η^TkE^T

d Z1d Z 2

Eηk− ^k−1

l k−d

η^TlE^T1−α^k−lZ1Eηl

−^k−d−1

l k−d

η^TlE^T1−α^k−lZ2Eηl

η^TkE^T

d Z1d Z 2

Eηk− ^k−1

l k−dk

η^TlE^T1−α^k−lZ1Eηl

−^k−dk−1

l k−d

η^TlE^T1−α^k−l Z1Z2

Eηl− ^k−d−1

l k−dk

η^TlE^T1−α^k−lZ2Eηl.

3.19

Let

ψk

ξ^Tk ξ^Tk−dk ξ^Tk−d ξ^Tk−d_T

. 3.20

UsingLemma 2.5for the last three terms ofV4ξk, respectively, and notingd ≤dk≤ d, we have

− ^k−1

l k−dk

η^TlE^T1−α^k−lZ1Eηl

≤1ψ^TkM Z⁻¹₁ M^Tψk 2ψ^TkM ^k−1

l k−dk

Eηl

− ^k−1

l k−dk

ψ^TkM 1−α^k−lη^TlE^TZ1

1−α^k−lZ1

₋₁

×

M^Tψk 1−α^k−lZ1Eηl

−^k−dk−1

l k−d

η^TlE^T1−α^k−l Z1Z2

Eηl

≤2ψ^TkN Z1Z2

₋₁

N^Tψk 2ψ^TkN^k−dk−1

l k−d

Eηl

−^k−dk−1

l k−d

ψ^TkN 1−α^k−lη^TlE^T Z1Z2

1−α^k−l Z1Z2

₋₁

·

N^Tψk 1−α^k−l Z1Z2

Eηl

−

k−d−1

l k−dk

η^TlE^T1−α^k−lZ2Eηl

≤2ψ^TkT Z⁻¹₂ T^Tψk 2ψ^TkT

k−d−1

l k−dk

Eηl

− ^k−d−1

l k−dk

ψ^TkT 1−α^k−lη^TlE^TZ2

1−α^k−lZ2

₋₁

×

T^Tψk 1−α^k−lZ2Eηl .

3.21

Note thatηl ξl1−ξlprovides k−1 l k−dk

Eηl

E −E 0 0

ψk, 3.22

k−dk−1

l k−d

Eηl

0 E 0 −E

ψk, 3.23

k−d−1

l k−dk

Eηl

0 −E E 0

ψk. 3.24

Also, it follows fromRE 0 that

0 −ξ^Tk1E^TR^TSREξ^Tk1

−

Aξk Adξk−dk_T R^TSR

Aξk Adξk−dk .

3.25

Then, substituting3.19–3.24into 3.18, using3.25, and noting 1−α > 0,Z1 > 0 and Z2>0, we get

Vξk≤ψ^Tk

Φ 1M Z⁻¹₁ M^T2NZ1Z2⁻¹N^T2T Z⁻¹₂ T^T ψk

− ^k−1

l k−dk

ψ^TkM 1−α^k−lη^TlE^TZ1

1−α^k−lZ1

₋₁

×

M^Tψk 1−α^k−lZ1Eηl

−^k−dk−1

l k−d

ψ^TkN 1−α^k−lη^TlE^T Z1Z2

1−α^k−l Z1Z2

−1

·

N^Tψk 1−α^k−l Z1Z2

Eηl

− ^k−d−1

l k−dk

ψ^TkT 1−α^k−lη^TlE^TZ2

1−α^k−lZ2

₋₁

×

T^Tψk 1−α^k−lZ2Eηl

≤ψ^Tk

Φ 1M Z⁻¹₁ M^T2NZ1Z2⁻¹N^T2T Z⁻¹₂ T^T ψk,

3.26

whereΦfollows the same definition asΦdefined in3.1withA,Ad,X,Ql,Zv,R,M,N, and T instead ofA,Ad,X,Q_l,Zv,R,M,N, andT. Performing a congruence transformation on 3.1by diag{H₂^T, H₂^T, H₂^T, H₂^T, H₂^T, H₂^T, H₂^T}, and then using the Schur complement implies

Φ 1M Z⁻¹₁ M^T2NZ1Z2⁻¹N^T2T Z⁻¹₂ T^T <0. Thus, it follows from3.18and3.26 thatVξk Vξ_k1−1−αVξk≤0, which leads to

Vξk1≤1−αVξk. 3.27

By iterative substitutions, inequality3.27yields

Vξk≤1−α^kVξ0. 3.28

On the other hand, it follows from the Lyapunov functional3.16that β1 ξ1k ²≤Vξk, Vξ0≤β2φ²

c, 3.29

where β1 λmin

X11

, β2 λmax

X11

dd

d−1 λmax

Q₁

dλmax

Q₂

dλmax

Q₃

d²λmax

Z1Z2

, 3.30

Then, combining3.28and3.29leads to

ξ1k ≤

"

β2

β11−α^k/2φ

c

"

β2

β1e^{−ln1−α−1/2k}φ

c.

3.31

Therefore, the difference subsystem of3.15is exponentially stable with a decay rate which is not less than e^{ln1−α−1/2}. The remaining task is to show the exponential stability of the algebraic subsystem.

Partiii: Exponential stability of the algebraic subsystem. Setfk −Ad21ξ1k−dk;

then, it follows from3.31that

fk≤1−α^−d/2Ad21

"

β2

β1e^{−ln1−α−1/2k}φ

c. 3.32

Using the second equation in3.15,3.14andLemma 2.7, one gets ξ2k ≤β3e^{−ln1−α−1/2k}φ

c, 3.33

where

β3

#$

$$

$%λmax

Q₁₂₂ λmin

Q₁₂₂φ

c

⎛

⎜⎜

⎝

#$

$$

$%λmax

Q₁₂₂ λmin

Q₁₂₂ 1

1−e^−1/2ln1d 1

⎞

⎟⎟

⎠

"

β2

β11−α^−d/2Ad21. 3.34

Combining3.31and3.33yields that

xk ² ξ^TkH₂^TH2ξk

≤ H2 2

ξ1k ² ξ2k ²

≤ β2

β1 β²₃

e^{−ln1−α−1k}φ²

c.

3.35

Thus, we have

xk ≤

"

β2

β1 β²₃e^{−ln1−α−1/2k}φ

c

"

β2

β1 β²₃ 1

√1−α _−k

φ

c 3.36

The proof is completed.

Remark 3.2. Theorem 3.1is 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 system2.1, then R 0, and we get the exponential stability condition for the standard delay systemsxk1 Axk Adxk−dkfrom3.1.

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,ε1 > 0,ε2 > 0,ε3 > 0,ε4, andε5. If there exist symmetric matricesX > 0,Ql >0,l 1,2,3,Zv > 0,v 1,2, andS, and matrices

M M^T₁ M₂^T M^T₃ M₄^TT,N N^T₁ N₂^T N₃^T N₄^TT,T T₁^T T₂^T T₃^T T₄^TT,FandHsuch that the following inequalities hold:

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

Φj Ψ Υj dΠj dΠ j Ξ ΥjR^T

∗ Γ 0 0 0 0 0

∗ ∗ −2ε1Iε²₁X 0 0 0 0

∗ ∗ ∗ −d

2ε2I−ε²₂Z1

0 0 0

∗ ∗ ∗ ∗ −d

2ε3I−ε₃²Z2

0 0

∗ ∗ ∗ ∗ ∗ −ρI−S 0

∗ ∗ ∗ ∗ ∗ ∗ −ρ⁻¹I

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦

<0, j∈I1,2^m,

3.37 1 hl

∗ E^TXE

≥0, l∈I1, m, 3.38

wherehldenotes thelth row ofH,

Φj diag

−1−αE^TXE 1d

Q1Q2Q3,−1−α^dQ1,−1−α^dQ2,−1−α^dQ3

Sym

ΞRΥ^T_j MΛ1NΛ2TΛ3

, Υj

Aj Ad 0 0 ^T, Πj

Aj−E Ad 0 0 ^T Ξ

ε4I ε5I 0 0 ^T, Aj AB

DjFED⁻_jH .

3.39

Λ1, Λ2,Λ3,Ψ, Γ are defined in3.1, and R ∈ R^n×n is any constant matrix satisfyingRE 0 with rankR n−r, then system2.3 is regular, causal, and locally exponentially stable with λ 1/√

1−αfor any compatible initial condition in the ball

Bδ

φ∈C_n,d:φ²

c≤δ

, 3.40

where

δ 1

λmax

E^TXE

dd d−1

λmaxQ1 dλmaxQ2 dλmaxQ3 d²λmaxZ1Z2. 3.41

Proof. If3.38holds, then the ellipsoidΩE^TXE,1is included inLH 27 For more details about the ellipsoids and ellipsoid algorithm, we refer the readers to32–34. Suppose that

xk ∈ LH, ∀k > 0 will be proved later. Hence, by Lemma 2.8, satFExk can be expressed as

sat

FExk ^2m

j 1

αjk

DjFED_i⁻H

xk 3.42

and it follows that

Exk1 ² m

j 1

αjkAjxk Adxk−dk. 3.43

Choose a Lyapunov function as in3.16, and then, byTheorem 3.1, system3.43is regular, causal and locally exponentially stable if there exist symmetric matricesX > 0,Ql > 0,l 1,2,3,Zv>0,v 1,2, andS, and matricesM,N, andTdefined in3.1such that

Φ Ψ.

∗ Γ

<0, 3.44

where

Φ . diag

−1−αE^TXE 1d

Q1Q2Q3,−1−α^dQ1,−1−α^dQ2,−1−α^dQ3

A._1j^TXA.1j−A._1j^TR^TSRA.1jA.^T_2jUA.2jSym{MΛ1NΛ2TΛ3},

3.45 A.1j

2^m

j 1αjkAj Ad 0 0

, A.2j

2^m

j 1αjkAj−E Ad 0 0

. 3.46

Now, provided that inequalities 3.37 hold, then by using Schur complement and Lemma 2.9, one has

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎣ Φj Ξ

ρIS₋₁

Ξ^T Ψ Υj dΠj dΠ j ΥjR^T

∗ Γ 0 0 0 0

∗ ∗ −X⁻¹ 0 0 0

∗ ∗ ∗ −dZ⁻¹₁ 0 0

∗ ∗ ∗ ∗ −dZ ₂⁻¹ 0

∗ ∗ ∗ ∗ ∗ −ρ⁻¹I

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎦

<0, j∈I1,2^m. 3.47

This, together withαjk≥0 and₂^m

j 1αjk 1, implies that

2^m

j 1

αjk

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎣ Φj Ξ

ρIS₋₁

Ξ^T Ψ Υj dΠj dΠ j ΥjR^T

∗ Γ 0 0 0 0

∗ ∗ −X⁻¹ 0 0 0

∗ ∗ ∗ −dZ⁻¹₁ 0 0

∗ ∗ ∗ ∗ −dZ ⁻¹₂ 0

∗ ∗ ∗ ∗ ∗ −ρ⁻¹I

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎦

<0, 3.48

that is,

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

2^m

j 1αjkΦj Ξ

ρIS₋₁

Ξ^T Ψ ² m

j 1αjkΥj 2^m

j 1αjkdΠj 2^m

j 1αjkdΠ j 2^m

j 1αjkΥjR^T

∗ Γ 0 0 0 0

∗ ∗ −X⁻¹ 0 0 0

∗ ∗ ∗ −dZ⁻¹₁ 0 0

∗ ∗ ∗ ∗ −dZ ₂⁻¹ 0

∗ ∗ ∗ ∗ ∗ −ρ⁻¹I

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦

<0.

3.49

Applying Schur complement to3.49leads to

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

Λj Ψ ² m

j 1αjkΥj 2^m

j 1αjkdΠj 2^m

j 1αjkdΠ j

∗ Γ 0 0 0

∗ ∗ −X⁻¹ 0 0

∗ ∗ ∗ −dZ₁⁻¹ 0

∗ ∗ ∗ ∗ −dZ ₂⁻¹

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦

<0, 3.50

where

Λj 2^m

j 1

αjkΦj Ξ

ρIS₋₁ Ξ^Tρ

2^m

j 1

αjkΥjR^T

2^m

j 1

αjkRΥ^T_j. 3.51

Rewrite

2^m

j 1

αjkΦj 2^m

j 1

αjkSym

ΥjR^TΞ^T

Φ, 3.52