# Robust Stability Criteria for Uncertain Neutral Systems with Interval Nondifferentiable Time-Varying Delay and Nonlinear Perturbations

Authors: W. Weera; P. Niamsup
Journal: Journal of Applied Mathematics
 (2011)

---

## Abstract

We study the robust stability criteria for uncertain neutral systems
with interval time-varying delays and time-varying nonlinear perturbations simultaneously. The constraint on the derivative of the time-varying delay is not required, which allows the time-delay to be a fast time-varying function. Based on the Lyapunov-Krasovskii
theory, we derive new delay-dependent stability conditions in terms of linear matrix inequalities (LMIs) which can be solved by various available algorithms. Numerical examples are given to demonstrate that the derived conditions are much less conservative than those given in the literature.


---

## Body

## 1. Introduction

It is well known that the existence of time delay in a system may cause instability and oscillations. Example, of time-delay systems are chemical engineering systems, biological modeling, electrical networks, physical networks, and many others, [7–16]. The stability criteria for system with time delays can be classified into two categories: delay-independent and delay-dependent. Delay-independent criteria do not employ any information on the size of the delay; while delay-dependent criteria make use of such information at different levels. Delay-dependent stability conditions are generally less conservative than delay-independent ones especially when the delay is small.In many practical systems, models of system are described by neutral differential equations, in which the models depend on the delays of state and state derivatives. Heat exchanges, distributed networks containing lossless transmission lines and population ecology are examples of neutral systems because of its wider application. Therefore, several researchers have studied neutral systems and provided sufficient conditions to guarantee the asymptotic stability of neutral time delay systems, see [5, 9, 11–14, 16, 17] and references cited therein.Well-known nonlinearities, as the delays, may cause instability and poor performance of practical systems, which have driven many researchers to study the problem of nonlinear perturbed systems with state delays during the recent years [5, 7, 9, 18]. In [18], the delay-dependent robust stability for linear time-varying systems with nonlinear perturbations is given, by using the Newton-Leibniz formula which has been taken into account instead of applying an integral inequality. In [7], a model transformation technique is used to deal with the stability of system with time varying for delays and nonlinear perturbations. In [9], based on a descriptor model transformation combined with a matrix decomposition approach, the robust stability of uncertain systems with time varying discrete delay is studied by applying an integral inequality. However, these model transformations often introduce additional dynamics which leads to relatively conservative results. In [5], the neutral delay and the discrete delay are all time-varying, while the derivative of discrete delay is less than 1 which limits its bigger application. In most studies the time-varying delays are required to be differentiable [1–5, 7, 9, 11–14, 16, 18]. Therefore their methods have a conservatism which can be improved upon. However, in most cases, these conditions are difficult to satisfy. From these reasons, the conditions are interesting to study, but there are fewer results for removing restriction to the derivative of interval time-varying delays. Therefore, in this paper we will employ some new techniques so that the above conditions can be removed.In this paper, the problem of delay-dependent criterion for asymptotic stability for uncertain neutral system is studied with interval time-varying delay and time-varying nonlinear perturbations simultaneously. The restriction to the derivative of the interval time-varying delays is removed, which means that a fast interval time-varying delay is allowed. Based on the Lyapunov-Krasovskii theory, we derive new delay-dependent stability conditions in terms of linear matrix inequalities (LMIs) which can be solved by various available algorithms. The new stability condition is much less conservative and is more general than some existing results. Numerical examples are given to illustrate the effectiveness of our theoretical results.


## 2. Problem Formulation and Preliminaries

The following notations will be used in this paper:ℝ+ denotes the set of all real nonnegative numbers; ℝn denotes the n-dimensional space and the vector norm ∥·∥; Mn×r denotes the space of all matrices of (n×r)-dimensions. AT denotes the transpose of 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)}.xt∶={x(t+s):s∈[-h,0]}, ∥xt∥=sups∈[-h,0]∥x(t+s)∥; C([0,t],ℝn) denotes the set of all ℝn-valued continuous functions on [0,t]; Matrix A is called semipositive definite (A≥0) if xTAx≥0, for all x∈ℝn; A is positive definite (A>0) if xTAx>0, for all x≠0; A>B means A-B>0. The symmetric term in a matrix is denoted by *.Consider the following neutral system with time-varying delay:(2.1)ẋ(t)-Cẋ(t-d(t))=A(t)x(t)+B(t)x(t-τ(t))+D1(t)f1(t,x(t))+D2(t)f2(t,x(t-τ(t))),x(t0+θ)=ϕ(θ),θ∈[-h,0],
where x(t)∈Rn is the state vector, d(t) is a neutral delay, τ(t) is a time-varying continuous function which satisfies(2.2)0≤τm≤τ(t)≤τM,0≤d(t)≤d,ḋ(t)≤δ,
where τm, τM, d, δ are constants and h=max{d,τM}; the initial condition function ϕ(t) denotes a continuous vector-valued initial function of t∈[-h,0], f1(t,x(t)) and f2(t,x(t-τ(t))) are unknown nonlinear perturbations satisfying f1(t,0)=0, f2(t,0)=0 and(2.3)f1T(t,x(t))f1(t,x(t))≤α2xT(t)x(t),f2T(t,x(t-τ(t)))f2(t,x(t-τ(t)))≤β2xT(t-τ(t))x(t-τ(t)),
where α and β are positive real numbers.The uncertain matricesA(t), B(t), D1(t), and D2(t) satisfy (2.4)A(t)=A+ΔA(t),B(t)=B+ΔB(t),D1(t)=D1+ΔD1(t),D2(t)=D2+ΔD2(t),
where A, B, D1, D2∈ℝn×n are constant matrices with appropriate dimension, and ΔA(t), ΔB(t), ΔD1(t), and ΔD2(t) are unknown real matrices of appropriate dimension representing the systems time-varying parameter uncertainties which satisfy(2.5)ΔA(t)=G1F(t)EA,ΔB(t)=G2F(t)EB,ΔD1(t)=G3F(t)ED1,ΔD2(t)=G4F(t)ED2,
where G1, G2, G3, G4, EA, EB, ED1, and ED2 are known real constant matrices of appropriate dimension. F(t) is unknown time-varying matrix satisfying(2.6)FT(t)F(t)≤I.
For simplicity, we denote f1(t,x(t)), f2(t,x(t-τ(t))), by f1, f2, respectively.Letτe=(1/2)(τM+τm) and ρ=(1/2)(τM-τm). Then τ(t) can be expressed as(2.7)τ(t)=τe+ρξ(t),
where (2.8)ξ(t)={2τ(t)-(τM+τm)τM-τm,τM>τm,0,τM=τm.
Obviously, |ξ(t)|≤1. For this case, τ(t) is a function belonging to the interval [τe-ρ,τe+ρ], where ρ can be taken as the range of variation of the time-varying delay τ(t). Using the fact that (2.9)x(t-τe)-x(t-τ(t))=∫t-τ(t)t-τeẋ(s)ds
system (2.1) can be rewritten as(2.10)ẋ(t)-Cẋ(t-d(t))=A(t)x(t)+B(t)x(t-τe)-B(t)∫t-τ(t)t-τeẋ(s)ds+D1(t)f1+D2(t)f2.Lemma 2.1 (see [17]).
There exists a symmetric matrixX such that
(2.11)[P1-LXLTQ1Q1TR1]<0,[P2+XQ2Q2TR2]<0
if and only if
(2.12)[P1+LP2LTQ1LQ2Q1TR10Q2TLT0R2]<0.Lemma 2.2 (see [3]).
For any constant symmetric matrixM∈Rn×n, M=MT>0, 0≤hm≤h(t)≤hM, t≥0, and any differentiable vector function x(t)∈Rn, we have
(2.13)(a)[∫t-hmtẋ(s)ds]TM[∫t-hmtẋ(s)ds]≤hm∫t-hmtẋT(s)Mẋ(s)ds,(b)[∫t-h(t)t-hmẋ(s)ds]TM[∫t-h(t)t-hmẋ(s)ds]≤(h(t)-hm)∫t-h(t)t-hmẋT(s)Mẋ(s)ds≤(hM-hm)∫t-h(t)t-hmẋT(s)Mẋ(s)ds.Lemma 2.3 (see [19]).
Given matricesQ=QT, H, E, and R=RT>0 with appropriate dimensions. Then
(2.14)Q+HFE+ETFTHT<0
for all F satisfying FTF≤R, if and only if there exists an ϵ>0 such that
(2.15)Q+ϵHHT+ϵ-1ETRE<0.Proposition 2.4 (Cauchy inequality).
For any symmetric positive definite matrixN∈Mn×n and  x,y∈ℝn, we have
(2.16)±2xTy≤xTNx+yTN-1y.


## 3. Main Results

Now we present a new delay-dependent condition for the asymptotic stability of system (2.1).Assumption 3.1.
All the eigenvalues of matrix C are inside the unit circle.
First, we study the problem of stability for nominal system of (2.10) with ΔA(t)=0, ΔB(t)=0, ΔD1(t)=0, and ΔD2(t)=0.Theorem 3.2.
Under  Assumption3.1, nominal system of (2.10) with time-varying delay satisfying (2.2) is asymptotically stable if there exist positive definite matrices  P, Q, Q1, R, S, W, matrices  K1, K2, Li, Mi,i=1,2,…,7 of appropriate dimension and  δ1,δ2>0 such that
(3.1)Σ1=[ϕ11ϕ12ϕ13ϕ14ϕ15ϕ16ϕ17τeL1TρM1*ϕ22ϕ23ϕ24ϕ25ϕ26ϕ27τeL2TρM2**ϕ33ϕ34ϕ35ϕ36ϕ37τeL3TρM3***ϕ44ϕ45ϕ46ϕ47τeL4Tρ(K1TB+M4)****ϕ55ϕ56ϕ57τeL5Tρ(K2TB+M5)*****ϕ660τeL6TρM6******ϕ77τeL7TρM7*******-τeR0********-ρS]<0,
where
(3.2)ϕ11=Q+L1+L1T+ɛ1α2I,ϕ12=M1T+L2,ϕ13=-L1T+L3+M1,ϕ14=P+ATK1+L4,ϕ15=ATK2+L5,ϕ16=L6,ϕ17=L7,ϕ22=M2T+M2+ɛ2β2I-W,ϕ23=-L2T-M2T+M3+W,ϕ24=M4,ϕ25=M5,ϕ26=M6,ϕ27=M7,ϕ33=-Q-L3T-L3-M3T-M3-W,ϕ34=BTK1-L4-M4,ϕ35=BTK2-L5-M5,ϕ36=-L6-M6,ϕ37=-L7-M7,ϕ44=Q1+τeR+ρS-K1T-K1+ρ2W,ϕ45=K1TC-K2,ϕ46=K1TD1,ϕ47=K1TD2,ϕ55=-(1-δ)Q1+K2TC+CTK2,ϕ56=K2TD1,ϕ57=K2TD2,ϕ66=-δ1I,ϕ77=-δ2I.Proof.
We prove that Theorem3.2 is true for three cases, namely, τm≤τ(t)<τe; τ(t)=τe; τe<τ(t)≤τM.
Case 1 (τm≤τ(t)<τe).
Choose a Lyapunov-Krasovskii functional candidate as(3.3)V1(xt)=xT(t)Px(t)+∫t-τetxT(s)Qx(s)ds+∫t-d(t)tẋT(s)Q1ẋ(s)ds∫-τe0∫t+stẋT(θ)Rẋ(θ)dθds+∫-τe-τm∫t+stẋT(θ)Sẋ(θ)dθds+ρ∫-τe-τm∫t+stẋT(θ)Wẋ(θ)dθds,
where P, Q, Q1, R, S, and W are positive definite matrices. Taking the derivative of V1(xt) with respect to t along the trajectory of (2.10) yields
(3.4)V̇1(xt)=2xT(t)Pẋ(t)+xT(t)Qx(t)-xT(t-τe)Qx(t-τe)+ẋT(t)Q1ẋ(t)-(1-ḋ(t))ẋT(t-d(t))Q1ẋT(t-d(t))+ẋT(t)(τeR+ρ2W+ρS)ẋ(t)-∫t-τetẋT(s)Rẋ(s)ds-∫t-τet-τmẋT(s)Sẋ(s)ds-ρ∫t-τet-τmẋT(s)Wẋ(s)ds≤2xT(t)Pẋ(t)+xT(t)Qx(t)-xT(t-τe)Qx(t-τe)+ẋT(t)Q1ẋ(t)-(1-δ)ẋT(t-d(t))Q1ẋT(t-d(t))+ẋT(t)(τeR+ρ2W+ρS)ẋ(t)-∫t-τetẋT(s)Rẋ(s)ds-∫t-τet-τmẋT(s)Sẋ(s)ds-ρ∫t-τet-τmẋT(s)Wẋ(s)ds,
since
(3.5)-∫t-τet-τmẋT(s)Sẋ(s)ds≤-∫t-τet-τ(t)ẋT(s)Sẋ(s)ds,-ρ∫t-τet-τmẋT(s)Wẋ(s)ds≤-ρ∫t-τet-τ(t)ẋT(s)Wẋ(s)ds.
Based on Lemma 2.2, we obtain
(3.6)-ρ∫t-τet-τ(t)ẋT(s)Wẋ(s)ds≤-(τe-τ(t))∫t-τet-τ(t)ẋT(s)Wẋ(s)ds≤-xT(t-τe)Wx(t-τe)+2xT(t-τe)Wx(t-τ(t))-xT(t-τ(t))Wx(t-τ(t)),
and from the following equalities:
(3.7)2[ẋT(t)K1T+ẋT(t-d(t))K2T]×[Ax(t)+Bx(t-τe)-B∫t-τ(t)t-τeẋ(s)ds+D1f1+D2f2+Cẋ(t-d(t))-ẋ(t)B∫t-τ(t)t-τeẋ(s)ds]=0,(3.8)2[xT(t)L1T+xT(t-τ(t))L2T+xT(t-τe)L3T+ẋT(t)L4T+ẋT(t-d(t))L5T+f1TL6T+f2TL7T]×[x(t)-x(t-τe)-∫t-τetẋ(s)ds]=0,(3.9)2[xT(t)M1T+xT(t-τ(t))M2T+xT(t-τe)M3T+ẋT(t)M4T+ẋT(t-d(t))M5T+f1TM6T+f2TM7T]×[x(t-τ(t))-x(t-τe)-∫t-τet-τ(t)ẋ(s)ds]=0,
where K1, K2, and Li, Mi, i=1,2,…,7 are some matrices of appropriate dimension. Next, from (4.5), for any scalars δ1>0 and δ2>0, we obtain
(3.10)δ1[α2xT(t)x(t)-f1Tf1]≥0,δ2[β2xT(t-τ(t))x(t-τ(t))-f2Tf2]≥0.
By adding the terms on left of (3.7)–(3.10) to V̇1(xt), we may express V̇1(xt) as
(3.11)V̇1(xt)≤2xT(t)Pẋ(t)+xT(t)Qx(t)-xT(t-τe)Qx(t-τe)+ẋT(t)Q1ẋ(t)-(1-δ)ẋT(t-d(t))Q1ẋ(t-d(t))+ẋT(t)(τeR+ρS+ρ2W)ẋ(t)-∫t-τetẋT(s)Rẋ(s)ds-∫t-τet-τ(t)ẋT(s)Sẋ(s)ds+2[ẋT(t)K1T+ẋT(t-d(t))K2T]×[Ax(t)+Bx(t-τe)+B∫t-τet-τ(t)ẋ(s)ds+D1f1+D2f2+Cẋ(t-d(t))-ẋ(t)]+2[xT(t)L1T+xT(t-τ(t))L2T+xT(t-τe)L3T+ẋT(t)L4T+ẋT(t-d(t))L5T+f1TL6T+f2TL7T]×[x(t)-x(t-τe)-∫t-τetẋ(s)ds]+2[xT(t)M1T+xT(t-τ(t))M2T+xT(t-τe)M3T+ẋT(t)M4T+ẋT(t-d(t))M5T+f1TM6T+f2TM7T]×[x(t-τ(t))-x(t-τe)-∫t-τet-τ(t)ẋ(s)ds]+δ1[α2xT(t)x(t)-f1Tf1]-xT(t-τe)Wx(t-τe)+2xT(t-τe)Wx(t-τ(t))-xT(t-τ(t))Wx(t-τ(t))+δ2[β2xT(t-τ(t))x(t-τ(t))-f2Tf2]=1τe∫t-τetωT(t,s)ϕ1ω(t,s)ds+1τe-τ(t)∫t-τet-τ(t)ωT(t,s)ϕ2ω(t,s)ds,
where
(3.12)ωT(t,s)=[xT(t)xT(t-τ(t))xT(t-τe)ẋT(t)ẋT(t-d(t))f1Tf2T-ẋT(s)],ϕ1=[ϕ11ϕ12ϕ13ϕ14ϕ15ϕ16ϕ17τeL1T*ϕ22ϕ23ϕ24ϕ25ϕ26ϕ27τeL2T**ϕ33ϕ34ϕ35ϕ36ϕ37τeL3T***ϕ44ϕ45ϕ46ϕ47τeL4T****ϕ55ϕ56ϕ57τeL5T*****ϕ660τeL6T******ϕ77τeL7T*******-τeR]+Z,ϕ2=[-Z11-Z12-Z13-Z14-Z15-Z16-Z17(τe-τ(t))M1*-Z22-Z23-Z24-Z25-Z26-Z27(τe-τ(t))M2**-Z33-Z34-Z35-Z36-Z37(τe-τ(t))M3***-Z44-Z45-Z46-Z47φ1****-Z55-Z56-Z57φ2*****-Z66-Z67(τe-τ(t))M6******-Z77(τe-τ(t))M7*******-|τe-τ(t)|S],Z=[Z11Z12Z13Z14Z15Z16Z170*Z22Z23Z24Z25Z26Z270**Z33Z34Z35Z36Z370***Z44Z45Z46Z470****Z55Z56Z570*****Z66Z670******Z770*******0],φi=(τe-τ(t))(-KiTB+Mi), i=1,2, Zii>0, i=1,2,…,7, Zij, i=1,2,…,7, j=i+1,…,7 are some parameter matrices of appropriate dimensions. From (3.11) if ϕ1<0 and ϕ2<0,  then V̇1(xt)≤-λ1∥x(t)∥2 for some λ1>0. Pre and postmultiplying both sides of ϕ2<0 by diag{I,I,I,I,I,I,I,sgn(τ(t)-τe)}, we get that
(3.13)ϕ2≤[-Z11-Z12-Z13-Z14-Z15-Z16-Z17ρM1*-Z22-Z23-Z24-Z25-Z26-Z27ρM2**-Z33-Z34-Z35-Z36-Z37ρM3***-Z44-Z45-Z46-Z47ρ(-K1TB+M4)****-Z55-Z56-Z57ρ(-K2TB+M5)*****-Z66-Z67ρM6******-Z77ρM7*******-ρS]<0. By Schur complement lemma, this implies ϕ2<0. In light of Lemma 2.1, (3.1) holds if and only if ϕ1<0 and (3.13) simultaneously hold. Then (3.1) holds if and only if there exists a symmetric matrix Z,ϕ1<0 and (3.13) simultaneously hold. Therefore, nominal system of (2.10) is asymptotically stable.
Case 2 (τ(t)=τe).
For this case, we choose a Lyapunov-Krasovskii functional candidate as(3.14)V2(xt)=xT(t)Px(t)+∫t-τetxT(s)Qx(s)ds+∫t-d(t)tẋT(s)Q1ẋ(s)ds+∫-τe0ds∫t+stẋT(θ)Rẋ(θ)dθ,
where P, Q, Q1, and R positive definite matrices are the same as those in V1(xt).
Case 3 (τe<τ(t)<τM).
For this case, we choose the Lyapunov-Krasovskii functional candidate as(3.15)V3(xt)=xT(t)Px(t)+∫t-τetxT(s)Qx(s)ds+∫t-d(t)tẋT(s)Q1ẋ(s)ds∫-τe0∫t+stẋT(θ)Rẋ(θ)dθds+∫-τM-τe∫t+stẋT(θ)Sẋ(θ)dθds+ρ∫-τM-τe∫t+stẋT(θ)Wẋ(θ)dθds,
where P, Q, Q1, R, S, and W are positive definite matrices and are the same as those in V1(xt).
By similar arguments used in proof of Theorem3.2, we conclude that the nominal system of (2.10) is robustly asymptotically stable. The proof is complete.Based on Theorem3.2, we can perform the robust stability analysis for system (2.10) with uncertainties (2.5) and (2.6).Theorem 3.3.
Under Assumption3.1, system (2.10) with time-varying delay satisfying (2.2) and uncertainties (2.5) and (2.6) is asymptotically stable if there exist positive definite matrices P, Q, Q1, R, S, W and K1, K2, Li, Mi, i=1,2,…,7 of appropriate dimension and scalars ϵi>0, i=1,2,…,10 such that
(3.16)M=[M11ϕ12ϕ13ϕ14ϕ15ϕ16ϕ17τeL1TρM1*ϕ22ϕ23ϕ24ϕ25ϕ26ϕ27τeL2TρM2**M33ϕ34ϕ35ϕ36ϕ37τeL3TρM3***M44ϕ45ϕ46ϕ47τeL4Tρ(K1TB+M4)****M55ϕ56ϕ57τeL5Tρ(K2TB+M5)*****M660τeL6TρM6******M77τeL7TρM7*******-τeR0********M99]<0,M1=[-0.1K1T-0.1K1K1TG1K1TG2K1TG2K1TG3K1TG4G1TK1-ϵ1I0000G2TK10-ϵ2I000G2TK100-ϵ3I00G3TK1000-ϵ4I0G4TK10000-ϵ5I]<0,M2=[-0.1Q1+0.1K2TC+0.1CTK2K2TG1K2TG2K2TG2K2TG3K2TG4G1TK2-ϵ6I0000G2TK20-ϵ7I000G2TK200-ϵ8I00G3TK2000-ϵ9I0G4TK20000-ϵ10I]<0,
where
(3.17)M11=ϕ11+ϵ1EATEA+ϵ6EATEA,M33=ϕ33+ϵ2EBTEB+ϵ7EBTEB,M44=Q1+τeR+ρS-0.9K1T-0.9K1+ρ2W,M55=-0.9Q1+δQ1+0.9K2TC+0.9CTK2,M66=ϕ66+ϵ4ED1TED1+ϵ9ρ2EBTEB,M77=ϕ77+ϵ5ED2TED2+ϵ10ρ2EBTEB,M99=-ρS+ϵ3ρ2EBTEB+ϵ8ρ2EBTEB.Proof.
We choose Lyapunov-Krasovskii functional as in Theorem3.2,  we may proof this Theorem by using a similar arguments as in the proof of Theorem 3.2. By replacing A, B, D1, and D2 in (3.11) with A+GF(t)EA, B+GF(t)EB, D1+GF(t)ED1 and D2+GF(t)ED2, respectively. For Case 1(3.18)V̇1(xt)≤2xT(t)Pẋ(t)+xT(t)Qx(t)-xT(t-τe)Qx(t-τe)+ẋT(t)Q1ẋ(t)-(1-δ)ẋT(t-d(t))Q1ẋ(t-d(t))+ẋT(t)(τeR+ρS+ρ2W)ẋ(t)-∫t-τetẋT(s)Rẋ(s)ds-∫t-τet-τ(t)ẋT(s)Sẋ(s)ds+2[ẋT(t)K1T+ẋT(t-d(t))K2T]×[∫t-τet-τ(t)ẋ(s)ds(A+G1F(t)EA)x(t)+(B+G2F(t)EB)x(t-τe)+(B+G2F(t)EB)×∫t-τet-τ(t)ẋ(s)ds+(D1+G3F(t)ED1)f1+(D2+G4F(t)ED2)f2+Cẋ(t-d(t))-ẋ(t)]+2[xT(t)L1T+xT(t-τ(t))L2T+xT(t-τe)L3T+ẋT(t)L4T+ẋT(t-d(t))L5T+f1TL6T+f2TL7T]×[x(t)-x(t-τe)-∫t-τetẋ(s)ds]+2[xT(t)M1T+xT(t-τ(t))M2T+xT(t-τe)M3T+ẋT(t)M4T+ẋT(t-d(t))M5T+f1TM6T+f2TM7T][x(t-τ(t))-x(t-τe)-∫t-τet-τ(t)ẋ(s)ds]+δ1[α2xT(t)x(t)-f1Tf1]-xT(t-τe)Wx(t-τe)+2xT(t-τe)Wx(t-τ(t))-xT(t-τ(t))Wx(t-τ(t))+δ2[β2xT(t-τ(t))x(t-τ(t))-f2Tf2].
Applying Lemmas 2.3. and 2.4., the following estimations hold:
(3.19)2ẋT(t)K1T(A+G1F(t)EA)x(t)≤2ẋT(t)K1TAx(t)+ϵ1-1ẋT(t)K1TG1G1TK1ẋ(t)+ϵ1xT(t)EATEAx(t),(3.20)2ẋT(t)K1T(B+G2F(t)EB)x(t-τe)≤2ẋT(t)K1TBx(t-τe)+ϵ2-1ẋT(t)K1TG2G2TK1ẋ(t)+ϵ2xT(t-τe)EBTEBx(t-τe),(3.21)2ẋT(t)K1T(B+G2F(t)EB)∫t-τet-τ(t)ẋ(s)ds≤2ẋT(t)K1TB∫t-τet-τ(t)ẋ(s)ds+ϵ3-1ẋT(t)K1TG2G2TK4ẋT(t)+ϵ3ρ(∫t-τet-τ(t)ẋ(s)EBTEBẋ(s)ds),(3.22)2ẋT(t)K1T(D1+G3F(t)ED1)f1≤2ẋT(t)K1TD1f1+ϵ4-1ẋT(t)K1TG3G3TK1ẋ(t)+ϵ4f1TED1TED1f1,(3.23)2ẋT(t)K1T(D2+G4F(t)ED2)f2≤2ẋT(t)K1TD2f2+ϵ5-1ẋT(t)K1TG4G4TK1ẋ(t)+ϵ5f2TED2TED2f2,(3.24)2ẋT(t-d(t))K2T(A+G1F(t)EA)x(t)≤2ẋT(t-d(t))K2TAx(t)+ϵ6-1ẋT(t-d(t))K2TG1G1TK2ẋ(t-d(t))+ϵ6xT(t-d(t))EATEAx(t),(3.25)2ẋT(t-d(t))K2T(B+G2F(t)EB)x(t-τe)≤2ẋT(t-d(t))K1TBx(t-τe)+ϵ7-1ẋT(t-d(t))K1TG2G2TK1ẋ(t-d(t))+ϵ7xT(t-τe)EBTEBx(t-τe),(3.26)2ẋT(t-d(t))K2T(B+G2F(t)EB)∫t-τet-τ(t)ẋ(s)ds≤2ẋT(t-d(t))K2TB∫t-τet-τ(t)ẋ(s)ds+ϵ8-1ẋT(t-d(t))K2TG2G2TK2ẋT(t-d(t))+ϵ8ρ(∫t-τet-τ(t)ẋ(s)EBTEBẋ(s)ds),(3.27)2ẋT(t-d(t))K2T(D1+G3F(t)ED1)f1≤2ẋT(t-d(t))K2TD1f1+ϵ9-1ẋT(t-d(t))K2TG3G3TK2ẋ(t-d(t))+ϵ9f1TED1TED1f1,(3.28)2ẋT(t-d(t))K2T(D2+G4F(t)ED2)f2≤2ẋT(t-d(t))K2TD2f2+ϵ10-1ẋT(t-d(t))K2TG4G4TK2ẋ(t-d(t))+ϵ10f2TED2TED2f2.
Therefore, from (3.18)–(3.28), it follows that
(3.29)V̇1(xt)≤ωT(t,s)Mω(t,s)+ẋT(t)Ω1ẋ(t)+ẋT(t-d(t))Ω2ẋ(t-d(t)),
where
(3.30)Ω1=-0.1K1T-0.1K1+ϵ1-1K1TG1G1TK1+ϵ2-1K1TG2G2TK1+ϵ3-1K1TG2G2TK1+ϵ4-1K1TG3G3TK1+ϵ5-1K1TG4G4TK1,Ω2=-0.1Q1T+0.1K2TC+0.1CTK2+ϵ6-1K2TG1G1TK2+ϵ7-1K2TG2G2TK2+ϵ8-1K2TG2G2TK2+ϵ9-1K2TG3G3TK2+ϵ10-1K2TG4G4TK2.
Applying Schur complement lemma, the inequalities Ω1<0 and Ω2<0 are equivalent to ℳ1<0 and ℳ2<0, respectively. Therefore, system (2.10) is robust asymptotically stable if the condition (3.16) holds.
By using arguments similar to the proof of Case1 for Case 2 and Case 3, we may conclude that the close-loop system (2.10) is robust asymptotically stable.Remark 3.4.
In this paper, the restriction that the state delay is differentiable is not required, which allows state delay to be fast time varying. Meanwhile, this restriction is required in some existing results, see [1–5, 7, 9, 11–14, 16, 18].Remark 3.5.
In the proof of Theorem3.3, we need negative definiteness of matrices ℳ, Ω1 and Ω2 simultaneously. In order to do so, we need to have certain diagonal terms of matrices ℳ, Ω1 and Ω2 being negative. This leads to the splitting of the term K1 as (0.1+0.9)K1 which is one possibility to achieve such goal.


## 4. Numerical Examples

In this section, we provide numerical examples to show the effectiveness of our theoretical results.Example 4.1.
Consider the following uncertain neutral system with time-varying delay and nonlinear uncertainties which is studied in [1, 2]:
(4.1)ẋ(t)-Cẋ(t-d)=(A+ΔA(t))x(t)+(B+ΔB(t))x(t-τ(t))+(D1+ΔD1(t))f1(t,x(t))+(D2+ΔD2(t))f2(t,x(t-τ(t))),
where
(4.2)A=[-1.2-0.1-0.1-1],B=[-0.60.7-1-0.8],C=[-0.200.2-0.1],D1=[-0.100-0.1],D2=[-0.100-0.1],ΔA(t)=GF(t)EA,ΔB(t)=GF(t)EB,ΔD1(t)=GF(t)ED1,ΔD2(t)=GF(t)ED2,FT(t)F(t)≤I,G=γI,EA=I,EB=I,ED1=I,ED2=I.
It is assumed that the nonlinear uncertainties satisfy
(4.3)‖f1(t,x(t))‖≤α‖x(t)‖,‖f2(t,x(t-τ(t)))‖≤β‖x(t-τ(t))‖,α>0,β>0.
Applying Theorem 3.3, the maximum allowable value of τM is given in Table 1 when γ=0.1 and in Table 2 for γ=0.5. The results obtained in [1, 2] may not be used for the case when τm≠0. Moreover, the differentiability of the time delay τ(t) is not required in Theorem 3.3. Tables 1 and 2 show that our results significantly improve the results of [1, 2].
Moreover, it should be pointed out that if we letτm=0.1 and τM=0.85, then from Theorem 3.3, the solutions of LMI (3.16) are given as follows:
(4.4)P=[2.45970.35090.35091.9870],Q=[0.99550.19670.19671.0236],Q1=[0.5615-0.0545-0.05450.2292],S=[1.42110.12240.12241.1393],R=[1.81640.27180.27181.7699],W=[0.41310.05200.05200.3768],L1=[-1.8399-0.0430-0.1215-1.9400],L2=[1.53411.0797-1.10611.2226],L3=[-0.1656-0.0202-0.1437-0.3157],L4=[-1.3144-0.45670.1806-1.3580],L5=[-0.1514-0.0592-0.04100.0423],L6=[-0.00470.0064-0.0036-0.0045],L7=[-0.00470.0064-0.0037-0.0046],K1=[1.76520.22070.09281.4950],K2=[-0.28160.02030.0408-0.1346],M1=[-1.4428-0.0867-0.0535-1.3096],M2=[-1.2780-1.02581.1266-0.9849],M3=[0.2312-0.17170.06600.4381],M4=[0.3926-1.06870.91080.5805],M5=[0.36010.1664-0.19280.1095],M6=[0.0032-0.00460.01100.0081],M7=[0.0032-0.00470.01110.0081],δ1=2.0156,δ2=1.9922,ϵ1=0.4383,ϵ2=0.4197,ϵ3=0.5513,ϵ4=0.9140,ϵ5=0.9042,ϵ6=0.3288,ϵ7=0.3143,ϵ8=0.4376,ϵ9=0.9021,ϵ10=0.9003.
Figure1 shows the trajectories of solutions x1(t) and x2(t) of the system (4.1) with time-varying delay τ(t)=0.1+0.75|sin10t|, d=1, ϕ(t)=[sint,cost)], for all t∈[-1,0], f1(t,x(t))=[0.1sin|x1(t)|,0.1cos|x2(t)|]T, f2(t,x(t-τ(t)))=[0.1e-sin2x1(t-τ(t)),0.1e-cos2x2(t-τ(t))]T and F(t)=diag{sin2(t),sin2(t)}. Since the time-delay τ(t) is not differentiable, the stability criterion in [1, 2] cannot be applied to this case because it is only applicable to the system with the differentiable delay.Table 1
Comparison of the maximum valueτM for γ=0.1.
α=0,β=0.1τ̇(t)=0.5No restriction onτ̇(t)Zhang  et al. [1]0.704—Shen and Zhong [2]1.060—Ours—1.0762α=0.1,β=0.1τ̇(t)=0.5No restriction onτ̇(t)Zhang  et al. [1]0.689—Shen and Zhong [2]1.040—Ours—1.0638Table 2
Comparison of the maximum valueτM for γ=0.5.
α=0,β=0.1τ̇(t)=0.5No restriction onτ̇(t)Zhang  et al. [1]0.259—Shen and Zhong  [2]0.450—Ours—0.5333α=0.1,β=0.1τ̇(t)=0.5No restriction onτ̇(t)Zhang  et al. [1]0.240—Shen and Zhong  [2]0.429—Ours—0.5207Figure 1
The trajectories ofx1(t) and x2(t) of the system (4.1) with time-varying delay τ(t)=0.1+0.75|sin10t|.Example 4.2.
Consider the following uncertain neutral system with time-varying delay in [3, 4]:
(4.5)ẋ(t)-Cẋ(t-d(t))=(A+ΔA(t))x(t)+(B+ΔB(t))x(t-τ(t)),
where
(4.6)A=[-200-1],B=[-10-1-1],C=[c00c],ΔA(t)=[γ100γ2],ΔB(t)=[γ300γ4],
where 0≤|c|<1, and γi, i=1,2,…,4 are unknown parameter satisfying |γ1|≤1.6, |γ2|≤0.05, |γ3|<0.1, and |γ4|<0.3.Case 1.
Forc=0.1, δ=0, the maximum values of τM are listed in Table 3 for c=0.1 by applying criteria in [3, 4] and in this paper. We see that the maximum allowable bounds for τM obtained from Theorem 3.3 are much better than that obtained in [3, 4].Table 3
Maximum allowable upper boundsτM of the time-varying delay for different values of the lower bounds τm and c=0.1.
τm=0τm=0.5Yu and Lien [3]0.6580.793Kwon et al.  [4]0.8520.894Ours0.8730.951Case 2.
Forc=0.1, δ=0.1, the maximum value τM obtained form Theorem 3.3 is listed in Table 4. In [3, 4] the neutral delay is constant, then its stability criterion cannot be applied to systems with time-varying neutral delay. Furthermore, the stability criterion in [5, 6] cannot be applied to this case because Theorem 3.3 does not have restriction on the derivative of time-varying delay. It is obvious that the obtained results are significantly better than those in [3–6].Table 4
Maximum allowable upper boundsτM of the time-varying delay for different values of the lower bounds τm, c=0.1 and δ=0.1.
τm=0τ̇(t)=0.5τ̇(t)=0.9τ̇(t)≥1No restriction onτ̇(t)Han and Yu [5]0.570.17——Qiu et al.  [6]0.78900.71990.7216—Ours———0.8676τm=0.5τ̇(t)=0.5τ̇(t)=0.9τ̇(t)≥1No restriction onτ̇(t)Ours———0.9460Figure2 shows the trajectories of solutions x1(t) and x2(t) of the system (4.5) with time-varying delay τ(t)=0.3+0.5|cos10t|, d(t)=0.1sin2t, ϕ(t)=[sint,cost)], for all t∈[-0.8,0].Figure 2
The trajectories ofx1(t) and x2(t) of the system (4.5) with time-varying delay τ(t)=0.3+0.5|cos10t|.


## 5. Conclusions

In this paper, we have investigated the delay-dependent robust stability criteria for uncertain neutral systems with interval time-varying delays and time-varying nonlinear perturbations simultaneously. Based on Lyapunov-krasovskii theory, new delay-dependent sufficient conditions for robust stability have been derived in terms of LMIs. The interval time-varying delay function is not required to be differentiable, which allows time-delay function to be a fast time-varying function. Numerical examples are given to illustrate the effectiveness of the theoretic results which show that our results are much less conservative than some existing results in the literature.



---

*Source: 138912-2011-12-13.xml*