# Robust Exponential Stability Criteria of LPD Systems with Mixed Time-Varying Delays and Nonlinear Perturbations

Authors: Kanit Mukdasai; Akkharaphong Wongphat; Piyapong Niamsup
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper investigates the problem of robust exponential stability for linear parameter-dependent (LPD) systems with discrete and distributed time-varying delays and nonlinear perturbations. Parameter dependent Lyapunov-Krasovskii functional, Leibniz-Newton formula, and linear matrix inequality are proposed to analyze the stability. On the basis of the estimation and by utilizing free-weighting matrices, new delay-dependent exponential stability criteria are established in terms of linear matrix inequalities (LMIs). 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, dynamical systems with state delays have attracted much interest in the literature over the half century, especially in the last decade. Since time delay is frequently a source of instability or poor performances in various systems such as electric, chemical processes, and long transmission line in pneumatic systems [1]. The problems of stability and stabilization for dynamical systems with or without state delays have been intensively studied in the past years by many researchers in mathematics and control communities [2, 3]. Stability criteria for dynamical systems with time delay is generally divided into two classes: delay-independent one and delay-dependent one. Delay-independent stability criteria tends 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 concerned with the size of the delay and usually provide a maximal delay size. Various stability of linear continuous-time and discrete-time systems subject to time-invariant parametric uncertainty have 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. Most of sufficient (or necessary and sufficient) conditions have been obtained via Lyapunov-Krasovskii theory approaches in which parameter-dependent Lyapunov-Krasovskii functional has been employed. These conditions are always expressed in terms of linear matrix inequalities (LMIs). The results have been obtained for robust stability for LPD systems in which time delay occurs in state variable such as [4–6] which present sufficient conditions for robust stability of LPD continuous-time system with delays.Recently, many researchers have studied the problem of stability for time-delay systems with nonlinear perturbations such as [7] which considers the robust stability for a class of linear systems with interval time-varying delay and nonlinear perturbations. In [8], exponential stability of time-delay systems with nonlinear uncertainties is studied. Based on the Lyapunov theory approach and the approaches of decomposing the matrix, a new exponential stability criterion is derived in terms of LMI. In [9], they propose a new delay-dependent stability criterion in terms of linear matrix inequality for dynamic systems with time-varying delays and nonlinear perturbations by using Lyapunov theory. However, many researchers have studied the problem of stability for systems with discrete and distributed delays such as [10] which presented some stability conditions for uncertain neutral systems with discrete and distributed delays. The robust stability of uncertain linear neutral systems with discrete and distributed delays has been studied in [11]. In [12, 13], they studied the problem of stability for linear switching system with discrete and distributed delays. Moreover, a descriptor model transformation and a corresponding Lyapunov-Krasovskii functionals have been introduced for stability analysis of systems with delays in [14, 15].In this paper, we will investigate the problems of robust exponential stability for LPD system with mixed time-varying delays and nonlinear perturbations. Based on the combination of Leibniz-Newton formula and linear matrix inequality, the use of suitable Lyapunov-Krasovskii functional, new delay-dependent exponential stability criteria will be obtained in terms of LMIs. Finally, numerical examples will be given to show the effectiveness of the obtained results.


## 2. Problem Formulation and Preliminaries

We introduce some notations, definition, and lemmas that will be used throughout the paper.R+ denotes the set of all real nonnegative numbers; Rn denotes the n-dimensional space with the vector norm ∥·∥; ∥x∥ denotes the Euclidean vector norm of x∈Rn; Rn×r denotes the set n×r real matrices; AT denotes the 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)}; λmin(A)=min{Reλ:λ∈λ(A)}; λmax(A(α))=max{λmax(Ai):i=1,2,…,N}; λmin(A(α))=min{λmin(Ai):i=1,2,…,N}; matrix A is called a semipositive definite (A≥0) if xTAx≥0, for all x∈Rn; A is a positive definite (A>0) if xTAx>0 for all x≠0; matrix B is called a seminegative definite (B≤0) if xTBx≤0, for all x∈Rn;  B is a negative definite (B<0) if xTBx<0 for all x≠0; A>B means A-B>0; A≥B means A-B≥0; C([-h-,0],Rn) denotes the space of all continuous vector functions mapping [-h-,0] into Rn where h-=max{h,g}; * represents the elements below the main diagonal of a symmetric matrix.Consider the system described by the following state equation of the form(2.1)x˙(t)=A(α)x(t)+B(α)x(t-h(t))+f(t,x(t))+g(t,x(t-h(t)))+C(α)∫t-g(t)t‍x(s)ds,t>0;x(t)=ϕ(t),x˙(t)=ψ(t),t∈[-h-,0],
where x(t)∈Rn is the state variable, A(α), B(α), C(α)∈Rn×n are uncertain matrices belonging to the polytope
(2.2)A(α)=∑i=1N‍αiAi,B(α)=∑i=1N‍αiBi,C(α)=∑i=1N‍αiCi,∑i=1N‍αi=1,αi≥0,Ai,Bi,Ci∈Rn×n,i=1,…,N.h(t) and g(t) are discrete and distributed time-varying delays, respectively, satisfying
(2.3)0≤h(t)≤h,h˙(t)≤hd,0≤g(t)≤g,
where h,hd,g are given positive real constants. Consider the initial functions ϕ(t), ψ(t)∈C([-h-,0],Rn) with the norm ∥ϕ∥=supt∈[-h,0]∥ϕ(t)∥ and ∥ψ∥=supt∈[-h,0]∥ψ(t)∥. The uncertainties f(·),g(·) represent the nonlinear parameter perturbations with respect to the current state x(t) and the delayed state x(t-h(t)), respectively, and are bounded in magnitude of the form
(2.4)fT(t,x(t))f(t,x(t))≤η2xT(t)x(t),gT(t,x(t-h(t)))g(t,x(t-h(t)))≤ρ2xT(t-h(t))x(t-h(t)),
where η,ρ are given real constants.Definition 2.1.
The system (2.1) is robustly exponentially stable, if there exist positive real numbers β and M such that for each ϕ(t),ψ(t)∈C([-h-,0],Rn), the solution x(t,ϕ,ψ) of the system (2.1) satisfies
(2.5)∥x(t,ϕ,ψ)∥≤Mmax{∥ϕ∥,∥ψ∥}e-βt,∀t∈R+.Lemma 2.2 (Schur complement lemma, see [9]).
Given constant symmetric matricesX,Y,Z where Y>0. Then X+ZTY-1Z<0 if and only if
(2.6)(XZTZ-Y)<0or(-YZZTX)<0.Lemma 2.3 (Jensen's inequality, see [1]).
For any constant matrixQ∈Rn×n, Q=QT>0, scalar h>0, vector function x˙:[0,h]→Rn such that the integrations concerned are well defined, then
(2.7)-h∫-h0‍x˙T(s+t)Qx˙(s+t)ds≤-(∫-h0‍x˙(s+t)ds)TQ(∫-h0‍x˙(s+t)ds).
Rearranging the term ∫-h0‍x˙(s+t)ds with x(t)-x(t-h), one can yield the following inequality:
(2.8)-h∫-h0‍x˙T(s+t)Qx˙(s+t)ds≤[x(t)x(t-h)]T[-QQQ-Q][x(t)x(t-h)].Lemma 2.4 (see [16]).
Letx(t)∈Rn be a vector-valued function with first-order continuous-derivative entries. Then, the following integral inequality holds for any matrices X,Mi∈Rn×n, i=1,2,…,5 and a scalar function h:=h(t)≥0:
(2.9)-∫t-ht‍x˙T(s)Xx˙(s)ds≤[x(t)x(t-h)]T[M1T+M1-M1T+M2-M1+M2T-M2T-M2][x(t)x(t-h)]+h[x(t)x(t-h)]T[M3M4M4TM5][x(t)x(t-h)],
where
(2.10)[XM1M2M1TM3M4M2TM4TM5]≥0.


## 3. Main Results

In this section, we first study the robust exponential stability criteria for the system (2.1) by using the combination of linear matrix inequality (LMI) technique and Lyapunov theory method. We introduce the following notations for later use:
(3.1)Pj(α)=∑i=1N‍αiPij,Wj(α)=∑i=1N‍αiWij,Nj(α)=∑i=1N‍αiNij,Qj(α)=∑i=1N‍αiQij,Ml(α)=∑i=1N‍αiMil,Rs(α)=∑i=1N‍αiRis,∑i=1N‍αi=1,αi≥0,Pij,Wij,Nij,Qij,Mil,Ris∈Rn×n,j=1,2,…,6,l=1,2,…,5,s=1,2,3,i=1,2,…,N;(3.2)∏i,j,k=[Σi,j,k11Σi,j,k12Σi,j13Σi,j14Σi,j15Σi16Σi,j,k17*Σi,j,k22Σi,j23Σi,j24Σi,j25Σi26Σi,j,k27**Σi33Σi34Σi35-Ni3TΣi,j37***Σi44Σi45-Ni4TΣi,j47****Σi55Σi56Σi,j57*****Σi660******Σi,j,k77],
where
Σi,j,k11=2βPi1+Pi1Aj+AiTPj1+Pi2+h2AiTPj5Ak-e-2βhPi5+Qi1+Qi1T+Qi4TAj+AiTQj4+Ni1T+Ni1+Wi1TAj+AiTWj1+hMi1T+hMi1+h2Mi3+ϵ1η2I+g2Pi6,Σi,j,k12=Pi1Bj+Qi2-Qi1T+AiTQj5+Qi4TBj+h2AiTPj5Bk+e-2βhPi5+hRi2T-hMi1T-Ni1T+Ni2+Wi1TBj+Wi2TAj+hMi2+h2Mi4,Σi,j13=Ni3+Wi1T+AiTWj3+Pi1+Qi4T+h2AiTPj5,Σi,j14=Ni4+Wi1T+AiTWj4+Pi1+Qi4T+h2AiTPj5,Σi,j15=Ni5-Wi1T+AiTWj5+Qi3-Qi4T+AiTQj6,Σi16=-Qi1T-Ni1T+Ni6,Σi,j,k17=h2AiTPj5Ck+Wi1TCj+AiTWj6+Pi1Cj+Qi4TCj,Σi,j,k22=-Qi2T-Qi2+Qi5TBj+BiTQj5-e-2βhPi2+hdPi2+h2BiTPj5Bk-e-2βhPi5+h2Ri1-Ni2T-Ni2+Wi2TBj+BiTWj2-hRi2T-hRi2-hMi2T-hMi2+h2Mi5+ϵ2ρ2I,Σi,j23=Wi2T-Ni3+BiTWj3+Qi5T+h2BiTPj5,Σi,j24=Wi2T-Ni4+BiTWj4+Qi5T+h2BiTPj5,Σi,j25=-Wi2T-Ni5+BiTWj5+BiTQj6-Qi3-Qi5T,Σi26=-Qi2T-Ni2T-Ni6,Σi,j,k27=h2BiTPj5Ck+Wi2TCj+BiTWj6+Qi5TCj,Σi33=Wi3T+Wi3+h2Pi5-ϵ1I,Σi34=Wi3T+Wi4+h2Pi5,Σi35=-Wi3T+Wi5+Qi6,Σi,j37=h2Pi5Cj+Wi3TCj+Wi6,Σi44=Wi4T+Wi4+h2Pi5-ϵ2I,Σi45=-Wi4T+Wi5+Qi6,Σi,j47=h2Pi5Cj+Wi4TCj+Wi6,Σi55=-Wi5T-Wi5-Qi6T-Qi6+h2Pi3+h2Pi4,Σi56=-Qi3T-Ni5T,Σi,j57=Wi5TCj-Wi6+Qi6TCj,Σi66=-Ni6T-Ni6-e-2βhPi4,Σi,j,k77=-e2βgPi6+h2CiTPj5Ck+CiTWj6+Wi6TCj.Theorem 3.1.
For given positive real constantsh,hd,g,η and ρ, system (2.1) is robustly exponentially stable with a decay rate β, if there exist positive definite symmetric matrices Pis, any appropriate dimensional matrices Wis, Qis, Nis, Mir, Rit, s=1,2,…,6, r=1,2,…,5,t=1, 2, 3, i=1, 2,…,N and positive real constants ϵ1 and ϵ2 satisfying the following LMIs:
(3.4)[Ri1Ri2*Ri3]>0,i=1,2,…,N,(3.5)[e-2βhPi3-Ri3Mi1Mi2*Mi3Mi4**Mi5]≥0,i=1,2,…,N,(3.6)∏i,i,i<-I,i=1,2,…,N,(3.7)∏i,i,j+∏i,j,i+∏j,i,i<1(N-1)2I,i=1,2,…,N,i≠j,j=1,2,…,N,(3.8)∏i,j,k+∏i,k,j+∏j,i,k+∏j,k,i+∏k,i,j+∏k,j,i<6(N-1)2I,i=1,2,…,N-2,j=i+1,…,N-1,k=j+1,…,N.
Moreover, the solution x(t,ϕ,ψ) satisfies the inequality
(3.9)∥x(t,ϕ,ψ)∥≤Nλmin(P1(α))max[∥ϕ∥,∥ψ∥]e-βt,∀t∈R+,
where N=λmax(P1(α))+hλmax(P2(α))+h3λmax(P3(α))+h3λmax(P4(α))+h3λmax(P5(α)) + h3λmax([R1(α)R2(α)R2T(α)R3(α)]).Proof.
Choose a parameter-dependent Lyapunov-Krasovskii functional candidate for the system (2.1) of the form
(3.10)V(t)=∑i=17‍Vi(t),
where
(3.11)V1(t)=xT(t)P1(α)x(t)=[x(t)x(t-h(t))x˙(t)]T[I00000000][P1(α)00Q1(α)Q2(α)Q3(α)Q4(α)Q5(α)Q6(α)][x(t)x(t-h(t))x˙(t)],V2(t)=∫t-h(t)t‍e2β(s-t)xT(s)P2(α)x(s)ds,V3(t)=h∫-h0‍∫t+θt‍e2β(s-t)x˙T(s)P3(α)x˙(s)dsdθ,V4(t)=h∫-h0‍∫t+θt‍e2β(s-t)x˙T(s)P4(α)x˙(s)dsdθ,V5(t)=h∫-h0‍∫t+θt‍e2β(s-t)x˙T(s)P5(α)x˙(s)dsdθ,V6(t)=h∫-ht‍∫θ-h(θ)θ‍e2β(θ-t)[x(θ-h(θ))x˙(s)]T[R1(α)R2(α)R2T(α)R3(α)][x(θ-h(θ))x˙(s)]dsdθ,V7(t)=g∫-g0‍∫t+θt‍e2β(s-t)xT(s)P6(α)x(s)dsdθ.
Calculating the time derivatives of Vi(t), i=1,2,3,…,6, along the trajectory of (2.1) yields
(3.12)V˙1(t)=2[x(t)x(t-h(t))x˙(t)]T[P1(α)Q1T(α)Q4T(α)0Q2T(α)Q5T(α)0Q3T(α)Q6T(α)][x˙(t)00]=2[x(t)x(t-h(t))x˙(t)]T[P1(α)Q1T(α)Q4T(α)0Q2T(α)Q5T(α)0Q3T(α)Q6T(α)][ω11ω21ω31],
where
(3.13)ω11=A(α)x(t)+B(α)x(t-h(t))+f(t,x(t))+g(t,x(t-h(t)))+C(α)∫t-g(t)t‍x(s)ds,ω21=x(t)-x(t-h(t))-∫t-h(t)t‍x˙(s)ds,ω31=A(α)x(t)+B(α)x(t-h(t))+f(t,x(t))+g(t,x(t-h(t)))+C(α)∫t-g(t)t‍x(s)ds-x˙(t).
Taking the time-derivative of V2(t) leads to
(3.14)V˙2(t)=xT(t)P2(α)x(t)-(1-h˙(t))e-2βh(t)xT(t-h(t))P2(α)x(t-h(t))-2βV2(t)≤xT(t)P2(α)x(t)-e-2βhxT(t-h(t))P2(α)x(t-h(t))+hdxT(t-h(t))P2(α)x(t-h(t))-2βV2(t).
Obviously, for any scalar s∈[t-h,t], we get e-2βh≤e-2β(s-t)≤1. Together with Lemma 2.3 (Jensen's inequality), we obtain
(3.15)V˙3(t)=h2x˙T(t)P3(α)x˙(t)-h∫-h0‍e2βsx˙T(t+s)P3(α)x˙(t+s)ds-2βV3(t)≤h2x˙T(t)P3(α)x˙(t)-h∫t-ht‍e2β(s-t)x˙T(s)P3(α)x˙(s)ds-2βV3(t)≤h2x˙T(t)P3(α)x˙(t)-he-2βh∫t-ht‍x˙T(s)P3(α)x˙(s)ds-2βV3(t).
Following the estimation of V˙3(t), we have
(3.16)V˙4(t)≤h2x˙T(t)P4(α)x˙(t)-he-2βh∫t-ht‍x˙T(s)P4(α)x˙(s)ds-2βV4(t)≤h2x˙T(t)P4(α)x˙(t)-e-2βh∫t-ht‍x˙T(s)dsP4(α)∫t-ht‍x˙(s)ds-2βV4(t)≤h2x˙T(t)P4(α)x˙(t)-e-2βh∫t-h(t)t‍x˙T(s)dsP4(α)∫t-h(t)t‍x˙(s)ds-2βV4(t).
From (3.16), it follows that
(3.17)V˙5(t)≤h2x˙T(t)P5(α)x˙(t)-e-2βh∫t-h(t)t‍x˙T(s)dsP5(α)∫t-h(t)t‍x˙(s)ds-2βV5(t)=h2x˙T(t)P5(α)x˙(t)-e-2βh[xT(t)-xT(t-h(t))]P5(α)[x(t)-x(t-h(t))]-2βV5(t)=h2[A(α)x(t)+B(α)x(t-h(t))+f(t,x(t))+g(t,x(t-h(t)))+C(α)∫t-g(t)t‍x(s)ds]T×P5(α)[∫t-g(t)t‍A(α)x(t)+B(α)x(t-h(t))+f(t,x(t))+g(t,x(t-h(t)))+C(α)∫t-g(t)t‍x(s)ds]-e-2βh[xT(t)-xT(t-h(t))]P5(α)[x(t)-x(t-h(t))]-2βV5(t).
Taking the time derivative of V6(t) and V7(t), we obtain
(3.18)V˙6(t)=hh(t)xT(t-h(t))R1(α)x(t-h(t))+2hxT(t-h(t))R2(α)x(t)-2hxT(t-h(t))R2(α)x(t-h(t))+h∫t-ht‍x˙T(s)R3(α)x˙(s)ds-2βV6(t)≤h2xT(t-h(t))R1(α)x(t-h(t))+2hxT(t-h(t))R2(α)x(t)-2hxT(t-h(t))R2(α)x(t-h(t))+h∫t-ht‍x˙T(s)R3(α)x˙(s)ds-2βV6(t);V˙7(t)≤g2xT(t)P6(α)x(t)-e-2βg∫t-g(t)t‍xT(s)dsP6(α)∫t-g(t)t‍x(s)ds-2βV7(t).
From the Leibinz-Newton formula, the following equation is true for any real matrices Ni(α), i=1, 2,…,6 with appropriate dimensions
(3.19)2[∫t-h(t)t‍xT(t)N1T(α)+xT(t-h(t))N2T(α)+fT(t,x(t))N3T(α)+gT(t,x(t-h(t)))N4T(α)+x˙T(t)N5T(α)+∫t-h(t)t‍x˙T(s)dsN6T(α)]×[x(t)-x(t-h(t))-∫t-h(t)t‍x˙(s)ds]=0.
From the utilization of zero equation, the following equation is true for any real matrices Wi, i=1,2,…,5 with appropriate dimensions
(3.20)2[∫t-h(t)t‍xT(t)W1T(α)+xT(t-h(t))W2T(α)+fT(t,x(t))W3T(α)+gT(t,x(t-h(t)))W4T(α)+x˙T(t)W5T(α)+∫t-g(t)t‍xT(s)dsW6T(α)]×[A(α)x(t)+B(α)x(t-h(t))+f(t,x(t))+g(t,x(t-h(t)))+C(α)∫t-g(t)t‍x(s)ds-x˙(t)]=0.
From (2.4), we obtain for any positive real constants ϵ1 and ϵ2,
(3.21)0≤ϵ1η2xT(t)x(t)-ϵ1fT(t,x(t))f(t,x(t)),0≤ϵ2ρ2xT(t-h(t))x(t-h(t))-ϵ2gT(t,x(t-h(t)))g(t,x(t-h(t))).
By (3.5), Lemma 2.4 and the integral term of the right-hand side of V˙3(t) and V˙6(t), we obtain
(3.22)-h∫t-ht‍x˙T(s)[e-2βhP3(α)-R3(α)]x˙(s)ds≤h[x(t)x(t-h(t))]T[M1T(α)+M1(α)-M1T(α)+M2(α)-M1(α)+M2T(α)-M2T(α)-M2(α)][x(t)x(t-h(t))]+h2[x(t)x(t-h(t))]T[M3(α)M4(α)M4T(α)M5(α)][x(t)x(t-h(t))].
According to (3.12)–(3.22), it is straightforward to see that
(3.23)V˙(t)≤ζT(t)∑i=1N‍∑j=1N‍∑k=1N‍αiαjαk∏i,j,kζ(t)-2βV(t),
where ζT(t)=[xT(t), xT(t-h(t)), fT(t,x(t)), gT(t, x(t-h(t))), x˙T(t), ∫t-h(t)t‍x˙T(s)ds, ∫t-g(t)t‍xT(s)ds] and ∏i,j,k‍ is defined in (3.2). The facts that ∑i=1N‍αi=1, we obtain the following identities:
(3.24)∑i=1N‍∑j=1N‍∑k=1N‍αiαjαk∏i,j,k=∑i=1N‍αi3∏i,i,i+∑i=1N‍∑i≠j,j=1N‍αi2αj[∏i,i,j+∏i,j,i+∏j,i,i]+∑i=1N-2‍∑j=i+1N-1‍∑k=j+1N‍αiαjαl[∏i,j,k+∏i,k,j+∏j,i,k+∏j,k,i+∏k,i,j+∏k,j,i].
We define Φ and Λ as
(3.25)Φ≡∑i=1N‍∑j=1N‍αi(αi-αj)2=(N-1)∑i=1N‍αi3-∑i=1N‍∑j≠i;j=1N‍αi2αj≥0,Λ≡∑i=1N‍∑j≠i;j=1N-1‍∑k≠i;k=2N‍αi[αj-αk]2=(N-2)∑i=1N‍∑j≠i;j=1N-1‍αi2αj-6∑i=1N-2‍∑j=i+1N-1‍∑k=j+1N‍αiαjαl≥0.
From (N-1)Φ+Λ≥0, we obtain
(3.26)∑i=1N‍αi3-1(N-1)2∑i=1N‍∑i≠j,j=1N‍αi2αj-6(N-1)2∑i=1N-2‍∑j=i+1N-1‍∑k=j+1N‍αiαjαl≥0.
By (3.23)–(3.26), if the conditions (3.6)–(3.8) are true, then
(3.27)V˙(t)+2βV(t)≤0,∀t∈R+,
which gives
(3.28)V(t)≤V(0)e-2βt,∀t∈R+.
From (3.28), it is easy to see that
(3.29)λmin(P1(α))∥x(t)∥2≤V(t)≤V(0)e-2βt,V(0)=∑i=16‍Vi(0),
where
(3.30)V1(0)=xT(0)P1(α)x(0),V2(0)=∫-h(0)t‍e2βsxT(s)P2(α)x(s)ds,V3(0)=h∫-h0‍∫θ0‍e2βsx˙T(s)P3(α)x˙(s)dsdθ,V4(0)=h∫-h0‍∫θ0‍e2βsx˙T(s)P4(α)x˙(s)dsdθ,V5(0)=h∫-h0‍∫θ0‍e2βsx˙T(s)P5(α)x˙(s)dsdθ,V6(0)=h∫-h0‍∫θ-h(θ)0‍e2βθ[x(θ-h(θ))x˙(s)]T[R1(α)R2(α)R2T(α)R3(α)][x(θ-h(θ))x˙(s)]dsdθ,V7(0)=g∫-g0‍∫θ0‍e2βsxT(s)P6(α)x(s)dsdθ.
Therefore, we get
(3.31)λmin(P1(α))∥x(t)∥2≤V(0)e-2βt≤Nmax[∥ϕ∥,∥φ∥]2e-2βt,
where N=λmax(P1(α))+hλmax(P2(α))+h3λmax(P3(α))+h3λmax(P4(α))+h3λmax(P5(α)) + h3λmax([R1(α)R2(α)R2T(α)R3(α)]). From (3.31), we get
(3.32)∥x(t,ϕ,φ)∥≤Nλmin(P1(α))max[∥ϕ∥,∥φ∥]e-βt,∀t∈R+.
This means that system (2.1) is robustly exponentially stable. The proof of the theorem is complete.IfA(α)=A, B(α)=B and C(α)=0 when A and B are appropriate dimensional constant matrices, then system (2.1) reduces to the following system:
(3.33)x˙(t)=Ax(t)+Bx(t-h(t))+f(t,x(t))+g(t,x(t-h(t))),t>0;x(t)=ϕ(t),x˙(t)=ψ(t),t∈[-h,0].
Take the Lyapunov-Krasovskii functional as
(3.34)V(t)=∑i=16‍Vi(t),
where
(3.35)V1(t)=xT(t)P1x(t)=[x(t)x(t-h(t))x˙(t)]T[I00000000][P100Q1Q2Q3Q4Q5Q6][x(t)x(t-h(t))x˙(t)],V2(t)=∫t-h(t)t‍e2β(s-t)xT(s)P2x(s)ds,V3(t)=h∫-h0‍∫t+θt‍e2β(s-t)x˙T(s)P3x˙(s)dsdθ,V4(t)=h∫-h0‍∫t+θt‍e2β(s-t)x˙T(s)P4x˙(s)dsdθ,V5(t)=h∫-h0‍∫t+θt‍e2β(s-t)x˙T(s)P5x˙(s)dsdθ,V6(t)=h∫-ht‍∫θ-h(θ)θ‍e2β(θ-t)[x(θ-h(θ))x˙(s)]T[R1R2R2TR3][x(θ-h(θ))x˙(s)]dsdθ.
According to Theorem 3.1, we have the following Corollary 3.2 for the delay-dependent exponential stability criteria of system (3.33).Corollary 3.2.
For given positive real constantsh,hd,η and ρ, system (3.33) is exponentially stable with a decay rate β, if there exist positive definite symmetric matrices Pi,i=1,2,…,5, any appropriate dimensional matrices Qi,Ni,i=1,2,…,6,Wi,Mi,i=1,2,…,5,Ri,i=1,2,3 and positive real constants ϵ1 and ϵ2 satisfying the following LMIs:(3.36)[R1R2*R3]>0,[e-2βhP3-R3M1M2*M3M4**M5]≥0,[Σ11Σ12Σ13Σ14Σ15Σ16*Σ22Σ23Σ24Σ25Σ26**Σ33Σ34Σ35-N3T***Σ44Σ45-N4T****Σ55Σ56*****Σ66]<0,
where
(3.37)Σ11=2βP1+P1A+ATP1+P2+h2ATP5A-e-2βhP5+Q1+Q1T+Q4TA+ATQ4+N1T+N1+W1TA+ATW1+hM1T+hM1+h2M3+ϵ1η2I,Σ12=P1B+Q2-Q1T+ATQ5+Q4TB+h2ATP5B+e-2βhP5+hR2T-hM1T-N1T+N2+W1TB+W2TA+hM2+h2M4,Σ13=N3+W1T+ATW3+P1+Q4T+h2ATP5,Σ14=N4+W1T+ATW4+P1+Q4T+h2ATP5,Σ15=N5-W1T+ATW5+Q3-Q4T+ATQ6,Σ16=-Q1T-N1T+N6,Σ22=Q2T-Q2+Q5TB+BTQ5-e-2βhP2+hdP2+h2BTP5B-e-2βhP5+h2R1-N2T-N2+W2TB+BTW2-hR2T-hR2-hM2T-hM2+h2M5+ϵ2ρ2I,Σ23=W2T-N3+BTW3+Q5T+h2BTP5,Σ24=W2T-N4+BTW4+Q5T+h2BTP5,Σ25=-W2T-N5+BTW5+BTQ6-Q3-Q5T,Σ26=-Q2T-N2T-N6,Σ33=W3T+W3+h2P5-ϵ1I,Σ34=W3T+W4+h2P5,Σ35=-W3T+W5+Q6,Σ44=W4T+W4+h2P5-ϵ2I,Σ45=-W4T+W5+Q6,Σ55=-W5T-W5-Q6T-Q6+h2P3+h2P4,Σ56=-Q3T-N5T,Σ66=-N6T-N6-e-2βhP4.
Moreover, the solution x(t,ϕ,ψ) satisfies the inequality
(3.38)∥x(t,ϕ,ψ)∥≤Nλmin(P1)max[∥ϕ∥,∥ψ∥]e-βt,∀t∈R+,
where N=λmax(P1)+hλmax(P2)+h3λmax(P3)+h3λmax(P4)+h3λmax(P5)+h3λmax([R1R2R2TR3]).


## 4. Numerical Examples

In order to show the effectiveness of the approaches presented in Section3, four numerical examples are provided.Example 4.1.
Consider the LPD time-delay system (2.1) with the following parameters (N=3):
(4.1)A1=[-201-3],A2=[-310-4],A3=[-300-2],B1=[101-1],B2=[-110-1],B3=[-100-1],C1=[0.20.10.1-0.3],C2=[-0.30.20.10.2],C3=[-0.40.10.10.5],f(t,x(t))=[0.2sintx1(t)0.2costx2(t)],g(t,x(t-h(t)))=[0.3sintx1(t-h(t))0.3costx2(t-h(t))],h(t)=0.2134sin2(0.3t0.4268),g(t)=0.4cos2(t),ϕ(t)=[-75],t∈[-0.4,0].
It is easy to see that hd=0.3, η=0.2, ρ=0.3, and g=0.4. Find the discrete delay time h to guarantee system (2.1) with the above parameters to be robustly exponentially stable with a decay rate β=0.15.Solution 1.
By using the LMI Toolbox in Matlab (with accuracy 0.01) and conditions (3.4)–(3.8) of Theorem 3.1, this system is robustly exponentially stable for discrete delay satisfying h=0.2134 and
(4.2)P11=[225.4987-143.6565-143.6565300.6876],P21=[316.66330.31220.3122426.8816],P31=[361.5534-0.4974-0.4974306.5360],P12=[153.3987-105.3621-105.3621242.2669],P22=[284.5598-17.5849-17.5849306.8985],P32=[283.8792-0.3896-0.3896254.6003],P13=[484.794531.159831.1598398.0539],P23=[380.09519.98569.9856387.3373],P33=[391.5353-0.0741-0.0741392.5712],P14=[285.1892-5.6549-5.6549230.3520],P24=[229.7296-0.3010-0.3010239.7532],P34=[239.92360.04300.0430251.9889],P15=[285.3879-8.7124-8.7124227.7653],P25=[231.9963-0.7154-0.7154239.6643],P35=[239.9146-0.0007-0.0007251.9987],P16=[195.2662-48.4444-48.4444261.3741],P26=[285.1103-2.8941-2.8941286.4945],P36=[291.38972.44182.4418303.1851],Q11=[153.8322-6.2058-6.2058511.1849],Q21=[1,057.7119.5119.51,113.8],Q31=[-442.42.62.6-1,942.6],Q12=[-1,438.7472.6472.6503.4],Q22=[-3,162.4-435.4-435.4-107.8],Q32=[1,383.2-0.1-0.11,874.5],Q13=[-1,217.0-623.5-623.5-135.3],Q23=[-9.6-1,232.0-1,232.091.0],Q33=[564.8910-1.6044-1.6044-90.0137],Q14=[-61.6-1,470.3-1,470.3-4,166.8],Q24=[-4,191.213.613.6-4,402.9],Q34=[3,768.2-0.5-0.5-89.7],Q15=[214.0-15,185.0-15,185.015,214.0],Q25=[7,593.67,529.07,529.01,325.3],Q35=[3,542.10.00.0-3,023.0],Q16=[-40.43759.93899.938928.8514],Q26=[54.7453-6.5632-6.563245.3682],Q36=[53.4114-0.2701-0.270142.9778],N11=[-123.76829.31179.3117-461.4212],N21=[-1,007.0-119.6-119.6-1,067.4],N31=[483.1-2.9-2.91,962.9],N12=[1,415.9-464.4-464.4-534.9],N22=[3,122.6434.1434.164.8],N32=[-1,429.40.10.1-1,922.0],N13=[38.2260-6.0579-6.0579-13.3891],N23=[-13.44126.75796.75799.3172],N33=[-9.1322-0.0352-0.0352-24.2347],N14=[31.0135-6.3425-6.3425-14.6572],N24=[-14.15766.05736.05737.3776],N34=[-9.65310.02820.0282-23.7517],N15=[1,234.7657.0657.0110.7],N25=[-18.21,241.11,241.1-134.2],N35=[-604.67481.60131.601349.2281],N16=[14.9282-0.7772-0.777233.2457],N26=[36.94360.41950.419533.4742],N36=[33.2443-0.0821-0.082125.6059],M11=[222.2533104.5561104.556154.1789],M21=[2.13140.04760.04763.1486],M31=[2.0463-0.0108-0.01081.4850],M12=[-11.9251-13.1913-13.1913-5.6759],M22=[1.4344-2.6047-2.60473.3445],M32=[1.0670-0.0153-0.01530.0826],M13=[393.282156.334856.3348295.3780],M23=[269.8280-0.1703-0.1703270.2562],M33=[270.0284-0.0192-0.0192268.5208],M14=[-63.2823-34.4679-34.4679-20.2534],M24=[-0.8017-0.1312-0.1312-0.8706],M34=[-0.6584-0.0064-0.0064-1.2408],M15=[208.1212-29.9392-29.9392252.5413],M25=[268.68880.66400.6640268.0517],M35=[268.9342-0.0012-0.0012268.6650],R11=[196.3135-36.7812-36.7812248.7288],R21=[268.68630.66310.6631268.0488],R31=[268.9326-0.0013-0.0013268.6593],R12=[28.51967.48177.48174.9413],R22=[1.4407-2.6031-2.60313.3546],R32=[1.0720-0.0153-0.01530.0894],R13=[158.8158-17.7934-17.7934171.3160],R23=[178.25394.68244.6824181.6405],R33=[183.6198-0.0347-0.0347184.1092],
and ϵ1=414.9151 and ϵ2=381.9944. It is known that the maximum value of h for the stability of this system is h=0.6246. The stability is also assured for h<0.6246. The numerical solution x1(t) and x2(t) of (2.1) with (4.1) are plotted in Figure 1.Figure 1
State trajectoriesx1(t) and x2(t) of LPD time-delay system (2.1) with (4.1), α1=α2=α3=1/3 by using program dde45lin with Matlab.Example 4.2.
Consider the following linear systems, which are considered in [17]:
(4.3)x˙(t)=[-1.20.1-0.1-1]x(t)+[-0.60.7-1-0.8]x(t-h(t))+f(t,x(t))+g(t,x(t-h(t))),
where ∥f(t,x(t))∥≤η∥x(t)∥, ∥g(t,x(t-h(t)))∥≤ρ∥x(t-h(t))∥.By Corollary3.2 to the system (4.3), we can obtain the maximum upper bounds of the time delay under different values of η, ρ, and hd as shown in Table 1. From Table 1, we see that Corollary 3.2 gives larger delay bounds than some of the recent results in literatures.Table 1
Upper bounds of time delays in Example4.2 for various conditions.
η
=
0
,
ρ
=
0.1
η
=
0
,
ρ
=
0.1
η
=
0.1
,
ρ
=
0.1
η
=
0.1
,
ρ
=
0.1
h
d
=
0.5
h
d
≥
1
h
d
=
0.5
h
d
≥
1
Cao and Lam [18] (2000)
0.5467
—
0.4950
—
Zuo and Wang [3] (2006)
1.1424
—
1.0097
—
Chen et al. [17] (2008)
1.1425
0.7355
1.0097
0.7147
Corollary3.2
2.0925
0.8412
1.8235
0.8406Example 4.3.
Consider the following linear systems, which are considered in [19]:
(4.4)x˙(t)=[-200-1]x(t)+[-10-1-1]x(t-h)+f(t,x(t))+g(t,x(t-h)),
where ∥f(t,x(t))∥≤η∥x(t)∥, ∥g(t,x(t-h(t)))∥≤ρ∥x(t-h(t))∥. By using Corollary 3.2 to the system (4.4), we obtain the maximum upper bounds of the time delay for different values of η, ρ, and hd as shown in Table 2. From Table 2, it can be seen that Corollary 3.2 gives larger delay bounds than the recent results in [19].Table 2
Upper bounds of time delays in Example4.3 for various conditions.
η
=
0.05
,
ρ
=
0.1
β
=
0
β
=
0.1
β
=
0.3
β
=
0.5
Kwon and Park [19] (2008)
3.40
1.36
0.76
0.55
Corollary3.2
2.87
1.53
0.97
0.75Example 4.4.
Consider the following linear systems, which is considered in [8]:
(4.5)x˙(t)=[-410-4]x(t)+[0.1040.1]x(t-h)+f(t,x(t))+g(t,x(t-h)),
where ∥f(t,x(t))∥≤0.2∥x(t)∥, ∥g(t,x(t-h))∥≤0.2∥x(t-h)∥. The maximum value of convergence rate is 1.410 by using Corollary 3.2 for system (4.5). From Table 3, we can see that Corollary 3.2 gives larger convergence rate than the results in [8, 20].Table 3
Comparison of convergence rate obtained for Corollary3.2 and from [8, 20] in Example 4.4.
Method
Year
Convergence rateβ
Mondié and Kharitonov [20]
2005
0.470
Nam [8]
2009
1.153
Corollary3.2
2012
1.410


## 5. Conclusions

The problem of robust exponential stability for LPD systems with time-varying delays and nonlinear perturbations was studied. Based on the combination of Leibniz-Newton formula and linear matrix inequality, the use of suitable Lyapunov-Krasovskii functional, new delay-dependent exponential stability criteria are formulated in terms of LMIs. Numerical examples have shown significant improvements over some existing results.



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*Source: 348418-2012-12-30.xml*