# Robust Exponential Stability for LPD Discrete-Time System with Interval Time-Varying Delay

Authors: Kanit Mukdasai
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper investigates the problem of robust exponential stability for uncertain linear-parameter dependent (LPD) discrete-time system with delay. The delay is of an interval type, which means that both lower and upper bounds for the time-varying delay are available. The uncertainty under consideration is norm-bounded uncertainty. Based on combination of the linear matrix inequality (LMI) technique and the use of suitable Lyapunov-Krasovskii functional, new sufficient conditions for the robust exponential stability are obtained in terms of LMI. Numerical examples are given to demonstrate the effectiveness and less conservativeness
of the proposed methods.


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## Body

## 1. Introduction

Over the past decades, the problem of stability analysis of delay discrete-time systems has been widely investigated by many researchers. Because the existence of time delay is frequent, a source of oscillation instability performances degradation of systems. Stability criteria for discrete-time systems with time delay is generally divided into two classes: delay-independent ones and delay-dependent ones. Delay-independent stability criteria tend to be more conservative, especially for small-size delay; such criteria do not give any information on the size of the delay. On the other hand, delay-dependent stability criteria is concerned with the size of the delay and usually provide a maximal delay size. Moreover, robust stability of linear continuous-time and discrete-time systems subject to time-invariant parametric uncertainty has received considerable attention. An important class of linear time-invariant parametric uncertain system is linear parameter-dependent (LPD) system in which the uncertain state matrices are in the polytope consisting of all convex combination of known matrices. To address this problem, several results have been obtained in terms of sufficient (or necessary and sufficient) conditions; see [1–15] and references cited therein. Most of these conditions have been obtained via the Lyapunov theory approaches in which parameter dependent Lyapunov functions have been employed. These conditions are always expressed in terms of linear matrix inequalities (LMIs) which can be solved numerically by using available tools such as LMI toolbox in MATLAB. The results have been obtained for robust stability for LPD systems in which time delay occur in state variable such as [6, 11, 14] present sufficient conditions for robust stability of LPD continuous-time system with delays. However, a few results have been obtained for robust stability for LPD discrete-time systems with delay.In this paper, we deal with the problem of robust exponential stability for uncertain LPD discrete-time system with interval time-varying delay. Combined with the linear matrix inequality technique and the use of suitable Lyapunov-Krasovskii functional, new sufficient conditions for the robust exponential stability are obtained in terms of LMI. Finally, numerical examples have demonstrated the effectiveness of the criteria.


## 2. Problem Formulation and Preliminaries

We introduce some notations and definitions that will be used throughout the paper.ℤ+ denotes the set of non negative integer numbers; ℝn denotes the n-dimensional space with the vector norm ∥·∥; ∥x∥ denotes the Euclidean vector norm of x∈ℝn; that is, ∥x∥2=xTx; Mn×r denotes the space of all matrices of (n×r)-dimensions; AT denotes transpose of the Matrix A; A is symmetric if A=AT; I denotes the identity matrix; λ(A) denotes the set of all eigenvalues of A; λmax(A)=max{Reλ:λ∈λ(A)}; Matrix A is called semi positive definite (A≥0) if xTAx≥0, for all x∈ℝn; A is positive definite (A>0) if xTAx>0 for all x≠0; Matrix B is called semi-negative definite (B≤0) if xTBx≤0, for all x∈ℝn;B is negative definite (B<0) if xTBx<0 for all x≠0; A>B means A-B>0; A≥B means A-B≥0; * represents the elements below the main diagonal of a symmetric matrix.Consider the following uncertain LPD discrete-time system with interval time-varying delay in the state(2.1)x(k+1)=[A(α)+ΔA(k)]x(k)+[B(α)+ΔB(k)]x(k-h(k)),x(s)=ϕ(s),s=-h2,…,-1,0,
where k∈ℤ+, x(k)∈ℝn is the system state and ϕ(s) is a initial value at s. A(α), B(α)∈Mn×n are uncertain matrices belonging to the polytope of the form
(2.2)A(α)=∑i=1N‍αiAi,B(α)=∑i=1N‍αiBi,∑i=1N‍αi=1,αi≥0,Ai,Bi∈Mn×n,i=1,…,N.ΔA(k) and ΔB(k) are unknown matrices representing time-varying parameter uncertainties, we assumed to be of the form
(2.3)ΔA(k)=K(α)Δ(k)A1(α),ΔB(k)=K(α)Δ(k)B1(α),A1(α)=∑i=1N‍αiAi1,B1(α)=∑i=1N‍αiBi1,K(α)=∑i=1N‍αiKi,∑i=1N‍αi=1,αi≥0,Ai1,Bi1∈Mn×n,i=1,…,N.
The class of parametric uncertainties Δ(k), which satisfies
(2.4)Δ(k)=F(k)[I-JF(k)]-1,
is said to be admissible where J is a known matrix satisfying
(2.5)I-JJT>0,
and F(k) is uncertain matrix satisfying
(2.6)F(k)TF(k)≤I.
In addition, we assume that the time-varying delay h(k) is upper and lower bounded. It satisfies the following assumption of the form
(2.7)h1≤h(k)≤h2,
where h1 and h2 are known positive integers.Definition 2.1.
The uncertain LPD discrete-time-delayed system in (2.1) is said to be robustly exponentially stable if there exist constant scalars 0<a<1 and b>0 such that
(2.8)‖x(k)‖2≤baksup⁡-h2≤l≤0‖ϕ(l)‖2,
for all admissible uncertainties.Lemma 2.2 (see [5] (Schur complement lemma)).
Given constant matricesX,Y,Z of appropriate dimensions with Y>0. Then X+ZTY-1Z<0 if and only if
(2.9)(XZTZ-Y)<0or(-YZZTX)<0.Lemma 2.3 (see [2]).
Given constant matricesM1,M2, and M3 of appropriate dimensions with M1=M1T. Then,
(2.10)M1+M2Δ(k)M3+M3TΔ(k)TM2T<0,
where Δ(k)=F(k)[I-JF(k)]-1, F(k)TF(k)≤I,for allk∈ℤ+ if and only if
(2.11)M1+[ϵ-1M3TϵM2][I-J-JTI]-1[ϵ-1M3TϵM2]T<0,
for some scalar ϵ>0.


## 3.  Main Results

In this section, we present our main results on the robust exponential stability criteria for uncertain LPD discrete-time system with interval time-varying delays. We introduce the following notation for later use:(3.1)A^(α)=A(α)+ΔA(k),B^(α)=B(α)+ΔB(k),h^=h2-h1+1.Lemma 3.1.
For anyA^(α),B^(α),h^ in (3.1), P(α) and Q(α) given by
(3.2)P(α)=∑i=1N‍αiPi,Q(α)=∑i=1N‍αiQi,∑i=1N‍αi=1,αi≥0,i=1,…,N,
are parameter-dependent positive definite Lyapunov matrices such that
(3.3)[A^T(α)P(α)A^(α)-P(α)+h^Q(α)A^T(α)P(α)B^(α)B^T(α)P(α)A^(α)B^T(α)P(α)B^(α)-Q(α)]<0,
if and only if
(3.4)[-P(α)+h^Q(α)0A(α)TP(α)ϵ-1A1(α)T0*-Q(α)B(α)TP(α)ϵ-1B1(α)T0**-P(α)0ϵP(α)K(α)***-IJ****-I]<0.Proof.
Consider(3.5)[A^T(α)P(α)A^(α)-P(α)+h^Q(α)A^T(α)P(α)B^(α)B^T(α)P(α)A^(α)B^T(α)P(α)B^(α)-Q(α)]=[-P(α)+h^Q(α)00-Q(α)]+[A^T(α)P(α)A^(α)A^T(α)P(α)B^(α)B^T(α)P(α)A^(α)B^T(α)P(α)B^(α)]=[-P(α)+h^Q(α)00-Q(α)]+[A^T(α)B^T(α)]P(α)[A^(α)B^(α)].
We assume that
(3.6)[-P(α)+h^Q(α)00-Q(α)]+[A-T(α)B-T(α)]P(α)[A-(α)B-(α)]<0.
Using Lemma 2.2, we obtain
(3.7)[-P(α)+h^Q(α)0A(α)T+[K(α)Δ(k)A1(α)]T*-Q(α)B(α)T+[K(α)Δ(k)B1(α)]T**-P(α)-1]<0.
We rewrite the latter inequality as
(3.8)[-P(α)+h^Q(α)0A(α)T*-Q(α)B(α)T**-P(α)-1]+[00K(α)]Δ(k)[A1(α)B1(α)0]+[A1(α)B1(α)0]TΔ(k)T[00K(α)]T<0.
Using Lemma 2.3, inequality (3.8) holds if and only if there exists ϵ>0 such that
(3.9)[-P(α)+h^Q(α)0A(α)T*-Q(α)B(α)T**-P(α)-1]+[ϵ-1A1(α)T0ϵ-1B1(α)T00ϵK(α)][I-J-JI]-1[ϵ-1A1(α)T0ϵ-1B1(α)T00ϵK(α)]T<0.
If we apply to (3.9), then we obtain
(3.10)[-P(α)+h^Q(α)0A(α)Tϵ-1A1(α)T0*-Q(α)B(α)Tϵ-1B1(α)T0**-P(α)-10ϵK(α)***-IJ****-I]<0.
Premultiplying (3.10) by diag{I,I,P(α),I,I} and postmultiplying by diag{I,I,P(α),I,I}, we get that (3.4) and the lemma is proved.Lemma 3.2.
If there exist positive definite symmetric matricesPi, Qi, i=1,2,…,N, and positive real numbers ϵ,ζ such that
(3.11)[-Pi+h^Qi0AiTPiϵ-1Ai1T0*-QiBiTPiϵ-1Bi1T0**-Pi0ϵPiKi***-IJ****-I]<-ζI,i=1,2,…,N,[-Pi+h^Qi0AiTPjϵ-1Ai1T0*-QiBiTPjϵ-1Bi1T0**-Pi0ϵPiKj***-IJ****-I]+[-Pj+h^Qj0AjTPiϵ-1Aj1T0*-QjBjTPiϵ-1Bj1T0**-Pj0ϵPjKi***-IJ****-I]<2ζIN-1,i=1,…,N-1,j=i+1,…,N,
then, for any A(α),A1(α),B(α),B1(α),K(α),h^ in (3.1), P(α) and Q(α) are parameter-dependent positive definite Lyapunov matrices in Lemma 3.1 such that (3.4) holds.Proof.
Consider(3.12)[-P(α)+h^Q(α)0A(α)TP(α)ϵ-1A1(α)T0*-Q(α)B(α)TP(α)ϵ-1B1(α)T0**-P(α)0ϵP(α)K(α)***-IJ****-I]=∑i=1N‍∑j=1N‍αiαj[-Pi+h^Qi0AiTPjϵ-1Ai1T0*-QiBiTPjϵ-1Bi1T0**-Pi0ϵPiKj***-IJ****-I].
Using the fact that ∑i=1N‍αi=1, we obtain the following identities:
(3.13)∑i=1N‍∑j=1N‍αiαjAiBj=∑i=1N‍α2AiBi+∑i=1N-1‍∑j=i+1N‍αiαj[AiBj+AjBi],(N-1)∑i=1N‍αi2ζ-2∑i=1N-1‍∑j=i+1N‍αiαjζ=∑i=1N-1‍∑j=i+1N‍[αi-αj]2ζ≥0.
Then, it follows from (3.11), (3.12), and (3.13) that (3.4) holds. The proof of the lemma is complete.Theorem 3.3.
The system (2.1) is robustly exponentially stable if the LMI conditions (3.11) are feasible.Proof.
Consider the following Lyapunov-Krasovskii function for system (2.1) of the form
(3.14)V(x(k))=V1(x(k))+V2(x(k))+V3(x(k)),
where
(3.15)V1(x(k))=xT(k)P(α)x(k),V2(x(k))=∑i=k-h(k)k-1‍xT(i)Q(α)x(i),V3(x(k))=∑j=-h2+2-h1+1‍∑l=k+j-1k-1‍xT(l)Q(α)x(l).
A Lyapunov-Krasovskii difference for the system (2.1) is defined as
(3.16)ΔV(x(k))=ΔV1(x(k))+ΔV2(x(k))+ΔV3(x(k)).
Taking the difference of V1(x(k)) and V2(x(k)), the increments of V1(x(k)) and V2(x(k)) are
(3.17)ΔV1(x(k))=V1(x(k+1))-V1(x(k))=xT(k+1)P(α)x(k+1)-xT(k)P(α)x(k)=xT(k)A^T(α)P(α)A^(α)x(k)+xT(k-h(k))B^T(α)P(α)A^(α)x(k)+xT(k-h(k))B^T(α)P(α)B^(α)x(k-h(k))+xT(k)A^T(α)P(α)B^(α)x(k-h(k))-xT(k)P(α)x(k),(3.18)ΔV2(x(k))=V2(x(k+1))-V2(x(k))=∑i=k+1-h(k+1)k‍xT(i)Q(α)x(i)-∑i=k-h(k)k-1‍xT(i)Q(α)x(i)=xT(k)Q(α)x(k)-xT(k-h(k))Q(α)x(k-h(k))+∑i=k+1-h(k+1)k-h1‍xT(i)Q(α)x(i)-∑i=k+1-h(k+1)k-1‍xT(i)Q(α)x(i)+∑i=k+1-h1k-1‍xT(i)Q(α)x(i).
Form h(k)≥h1, the two last terms of the right-hand side of the latter equality yield
(3.19)∑i=k+1-h1k-1‍xT(i)Q(α)x(i)-∑i=k+1-h(k+1)k-1‍xT(i)Q(α)x(i)≤0.
Thus, we obtain
(3.20)ΔV2(x(k))≤xT(k)Q(α)x(k)-xT(k-h(k))Q(α)x(k-h(k))+∑i=k+1-h(k+1)k-h1‍xT(i)Q(α)x(i).
The increment of V3(x(k)) is easily computed as
(3.21)ΔV3(x(k))=V3(x(k+1))-V3(x(k))=∑j=-h2+2-h1+1‍[xT(k)Q(α)x(k)+∑l=k+j-1k-1‍xT(l)Q(α)x(l)-∑l=k+j-1k-1‍xT(l)Q(α)x(l)]=(h2-h1)xT(k)Q(α)x(k)-∑i=k+1-h2k-h1‍xT(i)Q(α)x(i).
It is easy to see that
(3.22)ΔV2(x(k))+ΔV3(x(k))≤(h2-h1+1)xT(k)Q(α)x(k)-xk-hTQ(α)xk-h+∑i=k+1-h(k+1)k-h1‍xT(i)Q(α)x(i)-∑i=k+1-h2k-h1‍xT(i)Q(α)x(i),
for simplicity, we let x(k-h(k))=xk-h. Since, h(k)≤h2, we obtain that
(3.23)∑i=k+1-h(k+1)k-h1‍xT(i)Q(α)x(i)-∑i=k+1-h2k-h1‍xT(i)Q(α)x(i)≤0.
Therefore, we conclude that
(3.24)ΔV(x(k))≤xT(k)A^T(α)P(α)A^(α)x(k)+xk-hTB^T(α)P(α)A^(α)x(k)+xT(k)A^T(α)P(α)B^(α)xk-h+xk-hTB^T(α)P(α)B^(α)xk-h-xT(k)P(α)x(k)+(h2-h1+1)xT(k)Q(α)x(k)-xk-hTQ(α)xk-h.
It follows form (3.24) that
(3.25)ΔV(x(k))≤YT[Δ11(α)A^T(α)P(α)B^(α)B^T(α)P(α)A^(α)B^T(α)P(α)B^(α)-Q(α)]Y,
where Δ11(α)=A^T(α)P(α)A^(α)-P(α)+h^Q(α) and YT=[x(k)Tx(k-h(k))T]. By (3.11), (3.25), and Lemma 3.1, and 3.2, we obtain
(3.26)ΔV(x(k))<-ω‖x‖2,
where ω>0. By (3.14), it is easy to see that
(3.27)V(x(k))≤β1‖x‖2+β1h^∑i=k-h2k-1‍‖x(i)‖2,
where
(3.28)β1=max⁡{λmax⁡(Pi),λmax⁡(Qi);i=1,2,…,N}.
It can be shown that there always exists a scalar θ>1 satisfying
(3.29)(θ-1)β1-λθ+h2θh2(θ-1)β1h^=0.
For any scalar θ>1, it follows from (3.26) and (3.27) that
(3.30)θk+1V(x(k+1))-θk+1V(x(k))=θk+1(V(x(k+1))-V(x(k)))+θk(θ-1)V(x(k))<α1(θ)θk‖x(k)‖2+α2(θ)θk∑i=k-h2k-1‍‖x(i)‖2,
where
(3.31)α1(θ)=(θ-1)β1-λθ,α2(θ)=(θ-1)β1h^.
Therefore, for any integer T≥h2+1, summing up both sides of (3.30) from 0 to T-1 gives
(3.32)θTV(x(T))-V(x(0))≤α1(θ)∑i=0T-1‍θi‖x(i)‖2+α2(θ)∑i=0T-1‍∑l=i-h2i-1‍θi‖x(l)‖2.
For h2≥1,
(3.33)∑i=0T-1‍∑l=i-h2i-1‍θi‖x(l)‖2≤∑l=-h2-1‍∑i=0l+h2‍θi‖x(l)‖2+∑l=0T-1-h2‍∑i=l+1l+h2‍θi‖x(l)‖2+∑l=T-h2T-1‍∑i=l+1T-1‍θi‖x(l)‖2≤h2∑l=-h2-1‍θl+h2‖x(l)‖2+h2∑l=0T-1-h2‍θl+h2‖x(l)‖2+h2∑l=T-h2T-1‍θl+h2‖x(l)‖2≤h2(h2+1)θh2sup⁡-h2≤l≤0‖ϕ(l)‖2+h2θh2∑l=1T-1‍θl‖x(l)‖2.
From (3.32) and (3.33), we obtain
(3.34)θTV(x(T))≤V(x(0))+h2(h2+1)θh2α2(θ)sup⁡-h2≤l≤0‖ϕ(l)‖2+[α1(θ)+α2(θ)h2θh2]∑l=0T-1‍θl‖x(l)‖2.
Observe
(3.35)V(x(T))≥γ‖x(T)‖2,V(x(0))≤(β1+β1h^h2)sup⁡-h2≤l≤0‖ϕ(l)‖2,γ=min⁡{λmin⁡(Pi);i=1,2,…,N}.
Then, it follows from (3.29), (3.33), and (3.35) that
(3.36)‖x(T)‖2≤h2[(h2+1)θh2α2(θ)+β1+β1h^h2]γ(1θ)Tsup⁡-h2≤l≤0‖ϕ(l)‖2.
By Definition 2.1, this means that the system (2.1) is robustly exponentially stable. The proof of the theorem is complete.


## 4. Numerical Example

Example 4.1.
Consider the following uncertain LPD discrete-time system with time-varying delays (2.1) where h(k)=2+cos(kπ/2), that is, h1=1,h2=3 and
(4.1)A1=[-0.60.020.02-0.6],A2=[-0.70.030.03-0.7],B1=[-0.60.020.02-0.08],B2=[-0.80.030.03-0.09],A11=[0.0050.00010.00010.005],A21=[0.0060.00020.00020.006],B11=[-0.0070.00050.0005-0.007],B21=[-0.0040.00020.0002-0.004],K1=[0.010.0030.0030.01],K2=[0.020.0010.0010.02],
and J=[0.001000.001]. By using LMI Toolbox in MATLAB, we use condition (3.11) in Theorem 3.3 for this example. The solutions of LMI verify as follows of the form ϵ=1, P1=[31.36351.23651.236529.4763],P2=[37.63540.25430.254341.3745],Q1=[9.43250.55870.558711.4534], and Q2=[10.85641.38561.385611.9781] (see Figure 1).Figure 1
The simulation solution of the statesx1(k) and x2(k) in Example 4.1 for uncertain LPD discrete-time delayed system with initial conditions x1(k)=2 and x2(k)=4, k=-3,-2,-1,0, and α1=α2=1/2 by using the method of Runge-Kutta order 4(h=0.01) with Matlab.Example 4.2.
Consider the following the LPD discrete-time system with time-varying delays (2.1) where, ΔA(k)=ΔB(k)=0 with
(4.2)A1=[0.6000.010.60],A2=[0.8000.050.70],B1=[0.1000.200.10],B2=[-0.100-0.20-0.10].
Table 1 lists the comparison of the upper-bound delay for asymptotic stability of system (2.1) where ΔA(k)=ΔB(k)=0 by different method. We apply Theorem 3.3 and see from Table 1 that our result is superior to those in [7, Theorem 3.2].Table 1
Comparison of the maximum allowed time delayh2.
Methods
h
2 (h1=2)
h
2 (h1=4)
h
2 (h1=5)
h
2 (h1=7)
Liu et al. [7] 2006
2
4
5
7
Our results
4
6
7
9



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