# Delay-Dependent Stability Criterion of Arbitrary Switched Linear Systems with Time-Varying Delay

Authors: Jun Li; Weigen Wu; Jimin Yuan; Qianrong  Tan; Xing Yin
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

This paper deals with the problem of delay-dependent stability criterion of arbitrary switched linear
systems with time-varying delay. Based on switched quadratic Lyapunov functional approach and free-weighting
matrix approach, some linear matrix inequality criterions are found to guarantee delay-dependent asymptotically
stability of these systems. Simultaneously, arbitrary switched linear system can be expressed as a problem of
uncertain liner system, so some delay-dependent stability criterions are obtained with the result of uncertain liner
system. Two examples illustrate the exactness of the proposed criterions.


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## Body

## 1. Introduction

Recently, switched linear systems have got more and more attention in the research community, which consists of a family of liner subsystems described by liner differential or difference equations and a switching law that orchestrates switching between them; see, for example, [1–4]. Simultaneously, systems with delays abound in the world and time-delay systems frequently appear in vast engineering systems [5–7]. Therefore, many papers consider switched linear systems with time constant delay or time-varying delay [8–24]. Naturally, stability is a fundamental property which has been investigated from the very beginning for this class of systems [25]. For stability analysis under arbitrary switching, even when all subsystems of a switched system are asymptotically stable or exponentially stable, it is still possible to construct a divergent trajectory from any initial state for such a switched system [4]. Thus, this paper aims to study the stability of arbitrary switched linear system with time-varying delay.On one hand, many methods have been developed in the study of arbitrary switched systems such as common quadratic Lyapunov functional approach (CQLF), converse Lyapunov theorem, and switched quadratic Lyapunov functional approach (SQLF) [4, 26–28]. On the other hand, Wu M. and He Y. develop free-weighting matrix approach for stability of liner system and uncertain liner system [29–33]. In this paper, Based on SQLF and free-weighting matrix approach, we consider the linear switched system:(1.1)x(k+1)=Ar(k)x(k)+Adr(k)x(k-d(k))+Br(k)u(k),k∈Z+,r(k)∈Ω,
where x(k)∈Rn is the state, u(k)∈Rn is the control input, and r(k) is a switching rule defined by r(k):N→Ω with Ω={1,2,…,N}. Moreover, r(k)=i means the subsystem (Ai,Adi,Bi) is active. d(k) is nonnegative differential time-varying functions which denote the time delays and satisfy  0≤d1≤d(k)≤d2.At the same time, the uncertain linear system(1.2)x(k+1)=(A-ΔA(k))x(k)+(Ad-ΔAd(k))x(k-d(k))+(B-ΔB(k))u(k),k∈Z+,
where x(k)∈Rn is the state, u(k)∈Rn is the control input, A, Ad, and B are given constant matrices, ΔA(k), ΔAd(k), and ΔB(k) are the parameter uncertainties matrices which are assumed to be of the form(1.3)[ΔA(k)ΔAd(k)ΔB(k)]=DF(k)[EaEadEb],
where Ea, Ead, and Eb are given constant matrices of appropriate dimensions and F(k) is the uncertain matrix such that(1.4)FT(k)F(k)≤I.
From (1.1) and (1.2), we know that when one subsystem switches to another subsystem, there exist matrixes A, Ad, and B such that(1.5)[Ar(k)Adr(k)Br(k)]=[A-ΔA(k)Ad-ΔAd(k)B-ΔB(k)]
so system (1.1) be equivalent to system (1.2). The key ideas of this paper are that SQLF is connected with free-weighting matrix approach and arbitrary switched linear system can be expressed as a problem of uncertain liner system.This paper is organized as follows. In Section2, we give some basic definitions. We analyze the stability of the system (1.1) with the SQLF and free-weighting matrix approach in Section 3. Based on uncertain liner system, we study the stability of the system (1.1) in Section 4. Some examples are given in Section 5. The last section offers the conclusions of this paper.


## 2. Preliminaries

In this section, with the switched quadratic Lyapunov functional approach, we investigate the stability of the origin of an autonomous switched system given by(2.1)x(k+1)=Ar(k)x(k),k∈Z+,r(k)∈Ω.
Define the indicatorfunction(2.2)ξ(k)=[ξ1(k),…,ξi(k),…,ξN(k)]T,with(2.3)ξi(k)={1ifr(k)=i,0,otherwise.Then, the switched system (2.1) can also be written as(2.4)x(k+1)=∑i=1Nξi(k)Aix(k).
This corresponds to the switched Lyapunov function defined as(2.5)V(k,x(k))=xT(k)Pr(k)x(k)=xT(k)(∑i=1Nξi(k)Pi)x(k)
with Pi is symmetric positive definite matrices. If such a positive-definite Lyapunov function exists and(2.6)ΔV(k,x(k))=V(k+1,x(k+1))-V(k,x(k))
is negative definite along the solutions of (2.1), then the origin of the switched system (2.1) is asymptotically stable. In order to represent, we give the following notation.Throughout this paper, the superscriptT stands for the inverse and transpose of a matrix; Rn×m is the set of all n×m real matrices; P>0 means that the matrix P is positive definite; and the symmetric terms in a symmetric matrix are denoted by ⋆, for example,(2.7)[MO⋆N]=[MOOTN].Lemma 2.1 (see [4]).
If there exist positive definite symmetric matricesPi∈Rn×n(Pi=PiT), satisfying
(2.8)[PiAiTPj⋆Pj]<0
for all i,j∈Ω, then the switched linear system (2.1) is asymptotically stable.Lemma 2.2 (see [4]).
If there exist positive definite symmetric matricesPi∈Rn×n(Pi=PiT) and matrices Fi,Gi∈Rn×n(i∈Ω), satisfying
(2.9)[AiFiT+FiAiT-PiAiGi-Fi⋆Pj-Gi-GiT]<0
for all i,j∈Ω, then the switched linear system (2.1) is asymptotically stable.Lemma 2.3 (see [33]).
Letd1 and d2 be positive integers such that 0≤d1≤d2. When u(k)=0, the systems (1.2) is asymptotically stability if there exist symmetric matrices P=PT>0, Q=QT>0, Z=ZT>0, X=[X11X12⋆X22]≥0 and any appropriate dimensional matrices N1, N2 and λ>0 such that the following LMIs hold,
(2.10)[Ψ11+λEaTEaΨ12+λEaTEad(A-I)THPD⋆Ψ22+λEadTEadAdTH0⋆⋆-HHD⋆⋆⋆-λI]<0,[X11X12N1⋆X22N2⋆⋆Z]≥0,
where Ψ11=(d2-d1+1)Q+P(A-I)+(A-I)TP+N1+N1T+d2X11, Ψ12=PAd+NZT-N1+d2X12,Ψ22=-Q-N2T-N2+d2X22, and H=P+d2Z.


## 3. Stability Analysis of System (1.1) with SQLF

In this section, firstly, when we do not consider the control input, the linear switched system (1.1) is rewritten as(3.1)x(k+1)=Ar(k)x(k)+Adr(k)x(k-d(k)),k∈Z+,r(k)∈Ω.With SQLF and free-weighting matrix approach, we have the following theorem.Theorem 3.1.
Letd1 and d2 be positive integers such that 0≤d1≤d2; the systems (3.1) is asymptotically stability,if there exist symmetric matrices Pi=PiT>0, Pj=PjT>0, Q=QT≥0, Z=ZT≥0, Xij=[X11ijX12ij⋆X22ij]≥0 and any appropriate dimensional matrices N1ij and N2ij such that the following LMIs hold,
(3.2)Φij=[Φ11ijΦ12ijd2(Ai-I)TZ⋆Φ22ijd2AdiTZ⋆⋆-d2Z]<0,(3.3)Θij=[X11ijX12ijN1ij⋆X22ijN2ij⋆⋆Z]≥0,i,j∈Ω,
where Φ11ij=(d2-d1+1)Q+AiTPjAi-Pi+N1ij+(N1ij)T+d2X11ij, Φ12ij=AiTPjAdi+(N2ij)T-N1ij+d2X12ij and Φ22ij=AdiTPjAdi-Q-(N2ij)T-N2ij+d2X22ij.Proof.
Suppose thaty(l)=x(l+1)-x(l), then we have x(k+1)=x(k)+y(k) and x(k)=x(k-d(k))+∑i=k-d(k)k-1y(l).
Combined with (2.2), we consider the following SQLF:
(3.4)V(k,x(k))=V1(k,x(k))+V2(k,x(k))+V3(k,x(k)),V1(k,x(k))=xT(k)Pr(k)x(k)=xT(k)(∑i=1Nξi(k)Pi)x(k),V2(k,x(k))=∑θ=-d2+10∑l=k-1+θk-1yT(l)Zr(k)y(l)=∑θ=-d2+10∑l=k-1+θk-1yT(l)(∑i=1Nξi(k)Zi)y(l),V3(k,x(k))=∑θ=-d2+1-d1+1∑l=k-1+θk-1xT(l)Qr(k)x(l)=∑θ=-d2+1-d1+1∑l=k-1+θk-1xT(l)(∑i=1Nξi(k)Qi)x(l),
where Pi=PiT>0,Zi=ZiT≥0,andQi=QiT≥0.
With (2.6), we obtain
(3.5)ΔV1(k,x(k))=xT(k+1)Pr(k+1)x(k+1)-xT(k)Pr(k)x(k)=xT(k)[Ar(k)TPr(k+1)Ar(k)-Pr(k)]x(k)+xT(k-d(k))[Adr(k)TPr(k+1)Adr(k)]x(k-d(k))+xT(k)Ar(k)TPr(k+1)Adr(k)x(k-d(k))+xT(k-d(k))Adr(k)TPr(k+1)Ar(k)x(k),(3.6)ΔV2(k,x(k))=∑θ=-d2+10∑l=k+θkyT(l)Zr(k+1)y(l)-∑θ=-d2+10∑l=k-1+θk-1yT(l)Zr(k)y(l)=d2yT(k)Zr(k+1)y(k)-∑l=k-d2k-1yT(l)Zr(k)y(l)+∑θ=-d2+10∑l=k+θk-1yT(l)[Zr(k+1)-Zr(k)]y(l)
when Zr(k+1)=Zr(k),
(3.7)ΔV2(k,x(k))≤d2yT(k)Zr(k+1)y(k)-∑l=k-d2k-1yT(l)Zr(k)y(l),(3.8)ΔV3(k,x(k))=∑θ=-d2+1-d1+1∑l=k+θkxT(l)Qr(k+1)x(l)-∑θ=-d2+1-d1+1∑l=k-1+θk-1xT(l)Qr(k)x(l)=(d2-d1+1)xT(k)Qr(k+1)x(k)-∑l=k-d2k-d1xT(l)Qr(k)x(l)+∑θ=-d2+1-d1+1∑l=k+θk-1xT(l)[Qr(k+1)-Qr(k)]x(l)
when Qr(k+1)=Qr(k),
(3.9)ΔV3(k,x(k))≤(d2-d1+1)xT(k)Qr(k+1)x(k)-xT(k-d(k))Qr(k)x(k-d(k)).
Suppose thatr(k)=i and r(k+1)=j mean that the subsystem i switches to the subsystem j in the switching system. As this has to be satisfied under arbitrary switching laws, it follows that this has to hold for the special configuration ξi(k)=1, ξl≠i(k)=0, ξj(k+1)=1, and ξl≠j(k+1)=0. And supposing that Zj=Zi=Z and Qj=Qi=Q, we obtain
(3.10)ΔV(k,x(k))≤xT(k)[AiTPjAi-Pi]x(k)+xT(k-d(k))[AdiTPjAdi]x(k-d(k))+d2yT(l)Zy(l)+xT(k)AiTPjAdix(k-d(k))+xT(k-d(k))AdiTPjAix(k)-∑l=k-d2k-1yT(l)Zy(l)+(d2-d1+1)xT(k)Qx(k)-xT(k-d(k))Qx(k-d(k)).
By using the Leibniz-Newton formula, for any appropriately dimensioned matrices N1ij and N2ij, the following equation is true:
(3.11)2[xT(k)N1ij+xT(k-d(k))N2ij]×[x(k)-x(k-d(k))-∑l=k-d(k)k-1y(l)]=0.
In addition, for any semipositive definite matrix Xij=[X11ijX12ij⋆X22ij]≥0, the following equation holds:
(3.12)∑l=k-d2k-1ϕ1T(k)Xijϕ1(k)-∑l=k-d(k)k-1ϕ1T(k)Xijϕ1(k)=d2ϕ1T(k)Xijϕ1(k)-∑l=k-d(k)k-1ϕ1T(k)Xijϕ1(k)≥0,
where ϕ1(k)=[xT(k)xT(k-d(k))]T.
With (3.1), (3.10), (3.11), and (3.12), we have
(3.13)ΔV(k,x(k))≤ϕ1T(k)Γijϕ1(k)-∑l=k-d(k)k-1ϕ2T(k,l)Θijϕ2(k,l)≤0,
where
(3.14)Γij=[Φ11ij+d2(Ai-I)TZ(Ai-I)Φ12ij+d2I(Ai-I)TZAdi⋆Φ22ij+d2AdiZAdi],ϕ2(k,l)=[ϕ1T(k)yT(l)]T.
Andϕ1(k) is defined in (3.12); Φ11ij, Φ12ij, and Φ22ij are defined in (3.2). Therefore, when Γij<0 and Θij≥0, the system (3.1) is asymptotically stability. Applying Schur's complement, Γij<0 is equivalent to Φij<0,i,j∈Ω. This completes the proof of Theorem 3.1.If we havey(k), for any appropriately dimensioned matrices T1ij, T2ij, N1ij, N2ij, N2ij, and
(3.15)Xij=[X11ijX12ijX13ij⋆X22ijX23ij⋆⋆X33ij]≥0,
the following equations are also true:(3.16)2[xT(k)T1ij+yT(k)T2ij]×[y(k)-(Ai-I)x(k)-Adix(k-d(k))]=0,2[xT(k)N1ij+yT(k)N2ij+xT(k-d(k))N3ij]×[x(k)-x(k-d(k))-∑l=k-d(k)k-1y(l)]=0,∑l=k-d2k-1η1T(k)Xijη1(k)-∑l=k-d(k)k-1η1T(k)Xijη1(k)=d2η1T(k)Xijη1(k)-∑l=k-d(k)k-1η1T(k)Xijη1(k)≥0,
where η1(k)=[xT(k)yT(l)xT(k-d(k))]T.Considering (3.16), similar to the proof of Theorem 3.1, we can obtain the following corollary.Corollary 3.2.
Letd1 and d2 be positive integers such that 0≤d1≤d2; the systems (3.1) is asymptotically stability if there exist symmetric matrices Pi=PiT>0, Pj=PjT>0, Q=QT≥0, Z=ZT≥0, and any appropriate dimensional matrices T1ij, T2ij, N1ij, N2ij, N2ij, and
(3.17)Xij=[X11ijX12ijX13ij⋆X22ijX23ij⋆⋆X33ij]≥0
such that the following LMIs hold,
(3.18)Υij=[Υ11ijΥ12ijΥ13ij⋆Υ22ijΥ23ij⋆⋆Υ33ij]<0,[X11ijX12ijX13ijN1ij⋆X22ijX23ijN2ij⋆⋆X33ijN3ij⋆⋆⋆Z]≥0,i,j∈Ω,
where Υ11ij=(d2-d1+1)Q+AiTPjAi-Pi-T1ij(Ai-I)-(Ai-I)T(T1ij)T+(N1ij)T+N1ij+d2X11ij, Υ12ij=T1ij-(Ai-I)T(T2ij)T+(N2ij)T+d2X12ij, Υ13ij=AiTPjAdi-T1ijAdi+(N3ij)T-N1ij+d2X13ij, Υ22ij=d2Z+(T2ij)T+T2ij+d2X22ij, Υ23ij=-T2ijAdi-N2ij+d2X23ij, and Υ33ij=AdiTPjAdi-Q-N3ij+(N3ij)T+d2X33ij.Next, we consider the design of a switched state feedback:(3.19)u(k)=Kr(k)x(k).
Ensuring stability of the closed-loop switched system:(3.20)x(k+1)=(Ar(k)+Br(k)Kr(k))x(k)+Adr(k)x(k-d(k)),k∈Z+,r(k)∈Ω.Based on Theorem3.1, we obtain the following theorem.Theorem 3.3.
Letd1 and d2 be positive integers such that 0≤d1≤d2. Under arbitrary switch, the systems (1.1) is asymptotically stability if there exist symmetric matrices Pi=PiT>0, Pj=PjT>0, Q=QT≥0, Z=ZT≥0, Yij=[Y11ijY12ij⋆Y22ij]≥0, and any appropriate dimensional matrices M1ij and M2ij such that the following LMIs hold,
(3.21)Ξij=[Ξ11ijΞ12ijd2(AiLi+BiVi-Vi)T⋆Ξ22ijd2LiAdiT⋆⋆d2R]≥0,[Y11ijY12ijM1ij⋆Y22ijM2ij⋆⋆LR-1L]≥0,i,j∈Ω,
where Ξ11ij=(d2-d1+1)Wi+(AiLi)TPj(AiLi)+(AiLi)Pj(BiVi)+(BiVi)TPj(AiLi)+(BiVi)TPj(BiVi)-Li+(M1ij)T+M1ij+d2Y11ij, Ξ12ij=(AiLi)TPj(AdiLi)+(BiVi)TPj(AdiLi)+(M2ij)T-M1ij+d2Y12ij, and Ξ22ij=(AdiLi)TPj(AdiLi)-W-M2ij-(M2ij)T+d2Y22ij.Proof.
To the system (3.1), Ai is replaced by Ai+BiKi in (3.2). Simultaneously, two parts of inequality (3.2) multiply the same matrix diag[Pi-1,Pi-1,Z-1] and two parts of inequality (3.3) multiply the same matrix diag[Pi-1,Pi-1,Pi-1]. Suppose that Li=Pi-1, Wi=Pi-1QPi-1, Yij=diag[Pi-1,Pi-1]Xijdiag[Pi-1,Pi-1], R=Z-1, M1ij=Pi-1N1ijPi-1, M2ij=Pi-1N2ijPi-1, and Vi=KiPi-1; then we obtain (3.21).This completes the proof of Theorem 3.3.


## 4. Stability Analysis of System (1.1) with Uncertain Liner System

In this section, results of uncertain liner system are extended to arbitrary switched linear system for arbitrary switched linear system can be expressed as a problem of uncertain liner system. Whenu(k)=0, (1.5) are rewritten as(4.1)[ΔA(k)ΔAd(k)]=[A-Ar(k)Ad-Adr(k)].
Then the system (3.1) is rewritten as(4.2)x(k+1)=(A-(A-Ar(k)))x(k)+(Ad-(Ad-Adr(k)))x(k-d(k)),k∈Z+,r(k)∈Ω.Combined with Lemma2.3, we easily have the following theorem.Theorem 4.1.
Letd1 and d2 be positive integers such that 0≤d1≤d2. Under arbitrary switch, the system (4.2) is asymptotically stability if there exist matrices P=PT>0, Q=QT>0, Z=ZT>0, X=[X11X12⋆X22]≥0 and any appropriate dimensional matrices A,Ad,D, Ea,Ead,Fi, N1, N2, and λ>0 such that the LMIs (2.10) and the following LMIs hold,
(4.3)[A-AiAd-Adi]=DFi[EaEad]FiTFi≤I,i∈Ω.Next, we consider the design of a switched state feedback. With (4.1) and (4.2), the system (3.20) is rewritten as(4.4)x(k+1)=(A-(A-(Ar(k)+Br(k)Kr(k))))x(k)+(Ad-(Ad-Adr(k)))x(k-d(k)),k∈Z+,r(k)∈Ω.Combined with Theorem4.1, we easily have the following theorem.Theorem 4.2.
Letd1 and d2 be positive integers such that 0≤d1≤d2. Under arbitrary switch, the systems (4.4) is asymptotically stability if there exist matrices P=PT>0, Q=QT>0, Z=ZT>0, X=[X11X12⋆X22]≥0, and any appropriate dimensional matrices A,Ad,D, Ea,Ead,Fi, N1, N2 and λ>0 such that the LMIs  (2.10) and (4.3), and the following LMIs hold,
(4.5)[A-(Ai+BiKi)Ad-Adi]=DFi[EaEad],i∈Ω.


## 5. Examples

Example 5.1.
Consider the following switched delay systems with two subsystems(5.1)x(k+1)=Aix(k)+Adix(k-d(k)),k∈Z+,i∈Ω,
where
(5.2)A1=[0.1000.5],Ad1=[-0.20-0.1-0.1],A2=[0.2000.3],Ad2=[-0.100.3-0.2],
and Ω={1,2}.
Whend1=d2 that is, d(k)=d2, d2 is without limit. To time-varying delay d(k), when d1 is given, d2 is a maximum value of the solvability of LMIs (3.2) and (3.3), and some results are in Table 1.
In this example, the switching system has two subsystems, so there are there switches that are between subsystem 1 and subsystem 2, between subsystem 1 and subsystem 1, and between subsystem 2 and subsystem 2. According to Theorem3.1, when d1=1 and d2=3, solving the LMIs (3.2) and (3.3) leads to
(5.3)Q=[29.9101-3.0324-3.032414.4926]Z=[5.34740.03410.03416.8596],P1=[143.7206-9.2029-9.2029102.1555],P2=[144.9590-8.5971-8.597193.0312],N112=[-1.25920.5437-0.4002-2.1476],N212=[1.9723-0.14010.45883.0624],N121=[-1.3023-0.86190.8282-2.1259],N221=[2.20160.3203-0.64053.2331],N111=[-1.25530.6225-0.4427-2.2909],N211=[1.9636-0.19490.51263.0790],N122=[-1.3199-0.70260.7078-2.0469],N222=[2.20490.2493-0.53193.1800],X12=[10.23480.0282-0.87220.30920.02828.28570.3092-0.5114-0.87220.30926.3470-0.87160.3092-0.5114-0.87164.7227],X21=[10.15290.2694-0.4215-0.65880.26948.2610-0.6588-0.6227-0.4215-0.65885.03570.9594-0.6588-0.62270.95943.9515],X11=[10.23690.0251-0.87880.36590.02517.79270.3659-0.4471-0.87880.36596.3347-0.89280.3659-0.4471-0.89284.7126],X22=[10.14670.2491-0.4220-0.54940.24918.4372-0.5494-0.7209-0.4220-0.54945.24770.8376-0.5494-0.72090.83764.0014].
It can be seen from Figure1 that when d1=1 and d2=3, all the state solutions corresponding to the 10 random initial points are convergent asymptotically to the unique equilibrium x*={0,0}.Table 1
Allowable upper bound ofd2 with given d1.
d1012345678950100d21112131415161718192061111Global convergence of statesx1 and x2 in Example 5.1, when d1=1 and d2=3.
(a)(b)Example 5.2.
Consider the following switched delay systems with two subsystems:(5.4)x(k+1)=Aix(k)+Adix(k-d(k)),k∈Z+,i∈Ω,
where
(5.5)A1=[0.38500.00900.01800.5880],Ad1=[-0.41500.0090-0.0820-0.3120],A2=[0.39700.01200.01500.5820],Ad2=[-0.40300.0120-0.0850-0.3180],Ω={1,2}.
When there exit matrixes
(5.6)A=[0.40000.00000.00000.6000],Ad=[-0.40000.0000-0.1000-0.3000],D=[0.03000.00000.00000.0300],Ea=[1.00000.00000.00001.0000],Ead=[1.00000.00000.00001.0000],F1=[-0.50000.30000.6000-0.4000],F2=[-0.10000.40000.5000-0.6000],
which satisfied (4.3), and d1=1 and d2=3, based on Theorem 4.1 solving the LMIs (2.10) leads to
(5.7)P=(1.0e+004)*[1.0888-0.0129-0.01290.2758],Q=(1.0e+003)*[2.51930.03790.03790.4439],Z=(1.0e+003)*[1.49270.08090.08090.4826],N1=[-472.4815-6.3448-53.9979-181.8198],N2=[478.2798-0.008645.6698185.5393],X=[216.7792-41.8304-134.0982-18.7083-41.8304113.2493-18.7083-62.9775-134.0982-18.7083174.9568-10.6693-18.7083-62.9775-10.669384.3589],λ=148.1177.
It can be seen from Figure2 that when d1=1 and d2=3, all the state solutions corresponding to the 10 random initial points are convergent asymptotically to the unique equilibrium x*={0,0}.Global convergence of statesx1 and x2 in Example 5.2, when d1=1 and d2=3.
(a)(b)


## 6. Conclusions

This paper was dedicated to the delay-dependent stability of arbitrary switched linear systems with time-varying delay. We obtain two main results. Firstly, using switched quadratic Lyapunov functional approach and free-weighting matrix approach, less conservative LMI conditions have been proposed. Secondly, based on the result of uncertain liner system, some delay-dependent stability criterions are obtained.



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