# Robust Stability of a Class of Uncertain Lur'e Systems of Neutral Type

Authors: W. Weera; P. Niamsup
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper deals with the problem of stability for a class of Lur’e systems with interval time-varying delay and sector-bounded nonlinearity. The interval time-varying delay function is not assumed to be differentiable. We analyze the global exponential stability for uncertain neutral and Lur’e dynamical systems with some sector conditions. By constructing a set of improved Lyapunov-Krasovskii functional combined with Leibniz-Newton’s formula, we establish some stability criteria in terms of linear matrix inequalities. Numerical examples are given to illustrate the effectiveness of the results.


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## Body

## 1. Introduction

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 [1–6] and references cited therein.It is well known that nonlinearities may cause instability and poor performance of practical systems, which have driven many researchers to study [3–9]. Many nonlinear control systems can be modeled as a feedback connection of a linear neutral system and a nonlinear element. One of the important classes of nonlinear systems is the Lur’e system whose nonlinear element satisfies certain sector constraints. Lur’e systems with sector bound have been widely interested in the control system such as Kalman-YaKubovih-Popov lemma, Popov criterion, and Circle criterion [10–12]. On the other hand, 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, [3–5].The stability criteria for system with time delays can be classified into two categories: delay-independent and delay-dependent. Delay-independent criteria does not employ any information on the size of the delay; while delay-dependent criteria makes 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 most of the existing results, the range of time-varying delay considered in paper is form 0 to an upper bound. In practice, the range of delay may vary in a range for which the lower bound is not restricted to be 0, that is, interval time-varying delay. A typical example with interval time delay is the networked control system, which has been widely studied in the recent literature (see, e.g., [3, 13, 14]). However there are fewer results for removing restriction to the derivative of interval time-varying delays. Therefore their methods have a conservatism which can be improved upon.It is known that exponential stability is more favorite property than asymptotic stability since it gives a faster convergence rate to the equilibrium point, the decay rates (i.e., convergent rates) are important indices of practical systems, and the exponential stability analysis of time-delay systems has been a popular topic in the past decades; see, for example, [3, 13, 15] and their references. In [13], delay-dependent exponential stability for uncertain linear systems with interval time-varying delays, in [15], global exponential periodicity and global exponential stability of a class of recurrent neural networks with time-varying delays.Recently, there are many research study on the asymptotic stability of a class of neutral and Lur’e dynamical systems with time delay, see, for example, [8, 9, 16, 17]. The problems have been dealt with stability analysis for neutral systems with mixed delays and sector-bounded nonlinearity [8], robust absolute stability criteria for uncertain Lur’e systems of neutral type [16], and robust stability criteria for a class of Lur’e systems with interval time-varying delay [9]. However, it is worth pointing out that, even though these results above were elegant, there still exist some points waiting for the improvement. Firstly, most of the work above the time-varying delays are required to be differentiable. In fact, 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. Secondly, in most studies on the asymptotic stability of Lur’e dynamical systems still need to be improved to the exponential stability.Based on the above discussions, we consider the problem of robust stability for a class of uncertain neutral and Lur’e dynamical systems with sector-bounded nonlinearity. The time delay is a continuous function belonging to a given interval, which means that the lower and upper bounds for the time varying delay are available, but the delay function is not necessary to be differentiable. To the best of the authors knowledge, there were no global stability results for uncertain neutral and Lur’e dynamical systems with some sector conditions [8, 9, 16, 17]. Based on the construction of improved Lyapunov-Krasovskii functionals combined with Liebniz-Newton’s formula and the integral terms, new delay-dependent sufficient conditions for the uncertain neutral and Lur’e dynamical of system are established of LMIs. The new stability condition is much less conservative and more general than some existing results. Numerical examples are given to illustrate the effectiveness of our theoretical results.The rest of this paper is organized as follows. In Section2, we give notations, definition, propositions, and lemma to be used in the proof of the main results. Delay-dependent sufficient conditions for uncertain neutral and Lur’e dynamical systems with sector-bounded nonlinearity are presented in Section 3. Numerical examples illustrated the obtained results and are given in Section 4. The paper ends with conclusions in Section 5 and cited references.


## 2. Problem Formulation and Preliminaries

The following notation 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; diag(c1,c2,…,cm) denotes block diagonal matrix with diagonal elements ci, i=1,2,…,m. The symmetric term in a matrix is denoted by *.Consider the following uncertain Lur’e system of neutral type with interval time-varying delays and sector-bounded nonlinearity:(2.1)x˙(t)-Cx˙(t-η(t))=(A+ΔA(t))x(t)+(A1+ΔA1(t))x(t-h(t))+(B+ΔB(t))f(σ(t)),σ(t)=HTx(t)=[h-1h-2⋯h-m]Tx(t),∀t≥0,x(t+s)=ϕ(t+s),x˙(t+s)=φ(t+s),s∈[-m,0],m=max{h2,η},
where x(t)∈ℝn is the state vector; σ(t)∈ℝn is the output vector; A∈ℝn×n, B∈ℝn×m, C∈ℝn×n, A1∈ℝn×n, H∈ℝn×m are constant known matrices; f(σ(t))∈ℝm is the nonlinear function in the feedback path, which is denoted as f for simplicity in the sequel. Its form is formulated as
(2.2)f(σ(t))=[f1(σ1(t))f2(σ2(t))⋯fm(σm(t))]T,σ(t)=[σ1(t)σ2(t)⋯σm(t)]T=[h1Tx(t)h2Tx(t)⋯hmTx(t)],
wherein, each term fi(σi(t)),i=1,2,…,m satisfies any one of the following sector conditions:
(2.3)fi(σi(t))∈K[0,ki]={fi(σi(t))∣fi(0)=0,0<σi(t)fi(σi(t))≤kiσi(t)2,σi(t)≠0}
or
(2.4)fi(σi(t))∈K[0,∞]={fi(σi(t))∣fi(0)=0,σi(t)fi(σi(t))>0,σi(t)≠0}.ΔA(t), ΔB(t), and  ΔA1(t) are time-varying uncertainties of appropriate dimensions, which are assumed to be of the following form:
(2.5)[ΔA(t)ΔB(t)ΔA1(t)]=DF(t)[E1E2E3],
where D, E1, E2, and E3 are known matrices of appropriate dimensions, and the time-varying matrix F(t) satisfies
(2.6)FT(t)F(t)≤I,∀t≥0.
The delays h(t) and τ(t) are time-varying continuous functions that satisfy
(2.7)0≤h1≤h(t)≤h2,0≤η(t)≤η,η˙(t)≤ηd<1.
We introduce the following technical well-known propositions and definition, which will be used in the proof of our results.Definition 2.1.
If there existγ>0 and ψ(γ)>0 such that
(2.8)∥x(t)∥≤ψ(γ)e-γt,∀t>0,
system (2.1) is said to be exponentially stable at the equilibrium point, where γ is called the degree of exponential stability.Proposition 2.2 (Cauchy inequality).
For any symmetric positive definite matrixN∈Mn×n and x,y∈ℝn we have
(2.9)±2xTy≤xTNx+yTN-1y.Proposition 2.3 (see [2]).
For any symmetric positive definite matrixM>0, scalar γ>0, and vector function ω:[0,γ]→ℝn such that the integrations concerned are well defined, the following inequality holds
(2.10)(∫0γ‍ω(s)ds)TM(∫0γ‍ω(s)ds)≤γ(∫0γ‍ωT(s)Mω(s)ds).Proposition 2.4 (Schur complement lemma, [2]).
Given constant symmetric matricesX, Y, Z with appropriate dimensions satisfying X=XT, Y=YT>0. Then X+ZTY-1Z<0 if and only if
(2.11)(XZTZ-Y)<0or(-YZZTX)<0.


## 3. Main Results

Now we present a new delay-dependent condition for the uncertain system (2.1) satisfying the sector conditions (2.3).Assumption 3.1.
All the eigenvalues of matrix C are inside the unit circle.Theorem 3.2.
Under Assumption3.1, given α>0. The system (2.1) satisfying the sector condition (2.3) is α-exponentially stabilizable if there exist symmetric positive definite matrices P, Q, R, U, F,L; symmetric positive semidefinite matrices Z=diag(z1,z2,…,zm) and J=diag(j1,j2,…,jm); scalars ϵ1>0 and ϵ2>0; matrices Ni; i=1,2,3 of appropriate dimension such that the following LMI holds:
(3.1)ℳ1=ℳ-[0I-I000000]T×e-2αh2U[0-II000000]<0,ℳ2=ℳ-[0-I0I00000]T×e-2αh2U[0I0-I00000]<0,(3.2)ℳ=[ϕ11ϕ12ϕ13ϕ14ϕ15ϕ16ϕ17ϕ18ϕ19*ϕ22ϕ23ϕ24ϕ25ϕ26ϕ27ϕ28ϕ29**ϕ33000000***ϕ4400000****ϕ55ϕ56ϕ57ϕ58ϕ59*****ϕ66ϕ67ϕ680******ϕ7700*******ϕ880********ϕ99],
where
(3.3)ϕ11=PA+ATP+Q+QT-e-2αh1R-e-2αh2R+2αxT(t)Px(t),ϕ12=PA1+ATN1T,ϕ13=e-2αh1R,ϕ14=e-2αh2R,ϕ17=PC,ϕ18=PD,ϕ19=ϵiE1T,ϕ16=ATL+ATN3T,ϕ15=ATHZT+PB+HKJ+ATN2T+2αHZ,ϕ22=-2e-2αh2U+N1A1+A1TN1T,ϕ23=e-2αh2U,ϕ24=e-2αh2U,ϕ25=A1THZT+N1B+A1TN2T,ϕ26=A1TL-N1+A1TN3T,ϕ27=N1C,ϕ28=N1D,ϕ29=ϵiE3T,ϕ33=-e-2αh1Q-e-2αh1R-e-2αh2U,ϕ44=-e-2αh2Q-e-2αh2R-e-2αh2U,ϕ57=ZHTC+N2C,ϕ55=ZHTB+BTHZT-J-JT+N2B+BTN2T,ϕ56=BTL-N2+BTN3T,ϕ59=ϵiE2T,ϕ58=ZHTD+N2D,ϕ88=-ϵiI,ϕ99=-ϵiI,ϕ66=h12R+h22R+F+(h2-h1)2U-L-LT-N3-N3T,ϕ67=LC+N3C,ϕ68=LD+N3D,ϕ77=-e-2αη(1-ηD)F.The solutionx(t) of the system satisfies,
(3.4)∥x(t)∥≤a∥ϕ∥2+b∥M1∥2+c∥M2∥2λm(P),
where a=λM(P)+2h2λM(R)+h2λM(U)+2λM(HZKHT), b=2λM(Q)(1-e-2αh2)/2α+2h2λM(R)((1-e-2αh2)/2α)+h2λM(U)((1-e-2αh2)/2α), and c=λM(F)((1-e-2αη)/2α), ∥M1∥=sup-m≤s≤0∥x(s)∥, ∥M2∥=sup-m≤s≤0∥x˙(s)∥.Proof.
Using (2.5), the uncertain system (2.1) can be represented as
(3.5)x˙(t)=Ax(t)+A1x(t-h(t))+Bf(σ(t))+Cx˙(t-η(t))+Dp(t),p(t)=F(t)(E1x(t)+E2f(σ(t))+E3x(t-h(t))),σ(t)=HTx(t)=[h-1h-2⋯h-m]Tx(t),∀t≥0,x(s)=ϕ(s),s∈[-max(h2,η2),0].
We consider the following Lyapunov-Krasovskii functional
(3.6)V(x(t))=∑i=18‍Vi,
where
(3.7)V1(x(t))=e2αtxT(t)Px(t),V2(x(t))=∫t-h1t‍e2αsxT(s)Qx(s)ds,V3(x(t))=∫t-h2t‍e2αsxT(s)Qx(s)ds,V4(x(t))=h1∫-h10‍∫t+st‍e2αsx˙T(τ)Rx˙(τ)dτds,V5(x(t))=h2∫-h20‍∫t+st‍e2αsx˙T(τ)Rx˙(τ)dτds,V6(x(t))=(h2-h1)∫-h2-h1‍∫t+st‍e2αsx˙T(τ)Ux˙(τ)dτds,V7(x(t))=∫t-η(t)t‍e2αsx˙T(s)Fx˙(s)ds,V8(x(t))=2∑i=1m‍λie2αt∫0σi‍fi(σi)dσi.
Taking the derivative of V(xt) along the solution of system (3.5), we have
(3.8)V˙1(x(t))=2αe2αtxT(t)Px(t)+2e2αtxT(t)Px˙(t),=e2αt[2αxT(t)Px(t)+2xT(t)P(Ax(t)+A1x(t-h(t))+Bf(σ(t))+Cx˙(t-η(t))Dp(t))2αxT],V˙2(x(t))=e2αt[xT(t)Qx(t)-e-2αh1xT(t-h1)Qx(t-h1)],V˙3(x(t))=e2αt[xT(t)Qx(t)-e-2αh2xT(t-h2)Qx(t-h2)],V˙4(x(t))≤e2αt[h12x˙T(t)Rx˙(t)-h1e-2αh1∫t-h1t‍x˙T(s)Rx˙(s)ds],V˙5(x(t))≤e2αt[h22x˙T(t)Rx˙(t)-h2e-2αh2∫t-h2t‍x˙T(s)Rx˙(s)ds],V˙6(x(t))≤e2αt[(h2-h1)2x˙T(t)Ux˙(t)-(h2-h1)e-2αh2∫t-h2t-h1‍x˙T(s)Ux˙(s)ds],V˙7(x(t))≤e2αt[x˙T(t)Fx˙(t)-e-2αη(1-ηd)x˙T(t-η(t))Fx˙(t-η(t))],V˙8(x(t))=2∑i=1m‍λie2αt(2α∫0σi(t)‍fi(σi)dσi+fi(σi(t))σ˙i(t))≤e2αt[4αfT(σ(t))Zσ(t)+2fT(σ(t))Zσ˙(t)]≤e2αt[4αfT(σ(t))ZHx(t)+2fT(σ(t))ZHTx˙(t)]=e2αt[4αfT(σ(t))ZHx(t)+2fT(σ(t))ZHT×(Ax(t)+A1x(t-h(t))+Bf(σ(t))+Cx˙(t-η(t))+Dp(t))2fT(σ(t))].
Applying Proposition 2.3 and the Leibniz-Newton formula, we have
(3.9)-h1∫t-h1t‍x˙T(s)Rx˙(s)ds≤-[∫t-h1t‍x˙(s)]TR[∫t-h1t‍x˙(s)]≤-[x(t)-x(t-h1)]TR[x(t)-x(t-h1)]=-xT(t)Rx(t)+2xT(t)Rx(t-h1)-xT(t-h1)Rx(t-h1),-h2∫t-h2t‍x˙T(s)Rx˙(s)ds≤-[∫t-h2t‍x˙(s)]TR[∫t-h2t‍x˙(s)]≤-[x(t)-x(t-h2)]TR[x(t)-x(t-h2)]=-xT(t)Rx(t)+2xT(t)Rx(t-h2)-xT(t-h2)Rx(t-h2).
Note that
(3.10)-(h2-h1)∫t-h2t-h1x˙T(s)Ux˙(s)ds=-(h2-h1)∫t-h2t-h(t)‍x˙T(s)Ux˙(s)ds-(h2-h1)∫t-h(t)t-h1‍x˙T(s)Ux˙(s)ds=-(h2-h(t))∫t-h2t-h(t)‍x˙T(s)Ux˙(s)ds-(h(t)-h1)∫t-h2t-h(t)‍x˙T(s)Ux˙(s)ds-(h(t)-h1)∫t-h(t)t-h1‍x˙T(s)Ux˙(s)ds-(h2-h(t))∫t-h(t)t-h1‍x˙T(s)Ux˙(s)ds.
Using Proposition 2.3 gives
(3.11)-(h2-h(t))∫t-h2t-h(t)‍x˙T(s)Ux˙(s)ds≤-[∫t-h2t-h(t)‍x˙(s)ds]TU[∫t-h2t-h(t)‍x˙(s)ds]≤-[x(t-h(t))-x(t-h2)]TU[x(t-h(t))-x(t-h2)],-(h(t)-h1)∫t-h(t)t-h1‍x˙T(s)Ux˙(s)ds≤-[∫t-h(t)t-h1‍x˙(s)ds]TU[∫t-h(t)t-h1‍x˙(s)ds]≤-[x(t-h1)-x(t-h(t))]TU[x(t-h1)-x(t-h(t))].
Let β=(h2-h(t))/(h2-h1)≤1, so we have
(3.12)-(h2-h(t))∫t-h(t)t-h1‍x˙T(s)Ux˙(s)ds=-β∫t-h(t)t-h1‍(h2-h1)x˙T(s)Ux˙(s)ds≤-β∫t-h(t)t-h1‍(h(t)-h1)x˙T(s)Ux˙(s)ds≤-β[x(t-h1)-x(t-h(t))]TU[x(t-h1)-x(t-h(t))],-(h(t)-h1)∫t-h2t-h(t)‍x˙T(s)Ux˙(s)ds=-(1-β)∫t-h2t-h(t)‍(h2-h1)x˙T(s)Ux˙(s)ds≤-(1-β)∫t-h2t-h(t)‍(h2-h(t))x˙T(s)Ux˙(s)ds≤-(1-β)[x(t-h(t))-x(t-h2)]T×U[x(t-h(t))-x(t-h2)].
Therefore from (3.10)–(3.12), we obtain
(3.13)-(h2-h1)∫t-h2t-h1‍x˙T(s)Ux˙(s)ds≤-[x(t-h(t))-x(t-h2)]TU[x(t-h(t))-x(t-h2)]-[x(t-h1)-x(t-h(t))]TU[x(t-h1)-x(t-h(t))]-β[x(t-h1)-x(t-h(t))]TU[x(t-h1)-x(t-h(t))]-(1-β)[x(t-h(t))-x(t-h2)]TU[x(t-h(t))-x(t-h2)].
We add the following zero equation:
(3.14)2ξT(t)N-[Ax(t)+A1x(t-h(t))+Bf(σ(t))+Cx˙(t-η(t))+Dp(t)-x˙(t)]=0,
where N-=[N1TN2TN3T]T, ξ(t)=[xT(t-h(t))fT(σ(t))x˙T(t)]T and by using the identity relation
(3.15)-x˙(t)+Ax(t)+A1x(t-h(t))+Bf(σ(t))+Cx˙(t-η(t))+Dp(t)=0,
we have
(3.16)-2x˙T(t)Lx˙(t)+2x˙T(t)LAx(t)+2x˙T(t)LA1x(t-h(t))+2x˙T(t)LBf(σ(t))+2x˙T(t)LCx˙(t-η(t))+2x˙T(t)LDp(t)=0.
For system (2.1) with nonlinearity located in the sectors [0,kj](j=1,2,…,m), if there exists J=diag(j1,j2,…,jm), then we have
(3.17)jifi(σi)[kihiTx(t)-fi(σi)]≥0,i=1,2,…,m,
which is equivalent to
(3.18)xT(t)HKJf(σ(t))-fT(σ(t))Jf(σ(t))≥0.
Similarly, for any ϵ>0, from (3.4), we have
(3.19)-ϵpT(t)p(t)+ϵ(E1x(t)+E2f(σ(t))+E3x(t-h(t)))T×(E1x(t)+E2f(σ(t))+E3x(t-h(t)))≥0.
Hence, according to (3.9), (3.13), and by adding the zero term (3.16) and (3.18)-(3.19), we get
(3.20)V˙(x(t))≤e2αt{[x(t-h1)-x(t-h(t))]T2αxT(t)Px(t)+2xT(t)P(Ax(t)+A1x(t-h(t))+Bf(σ(t))+Cx˙(t-η(t))+Dp(t))+xT(t)Qx(t)-e-2αh1xT(t-h1)Qx(t-h1)-e-2αh2xT(t-h2)Qx(t-h2)+h12x˙T(t)Rx˙(t)+h22x˙T(t)Rx˙(t)-e-2αh1xT(t)Rx(t)+2e-2αh1xT(t)Rx(t-h1)+xT(t)Qx(t)-e-2αh1xT(t-h1)Rx(t-h1)-e-2αh2xT(t-h2)Rx(t-h2)-e-2αh2xT(t)Rx(t)+2e-2αh2xT(t)Rx(t-h2)×x˙T(t)Fx˙(t)-e-2αη(1-ηD)x˙T(t-η(t))Fx˙(t-η(t))2αfT(σ(t))ZHx(t)+2fT(σ(t))ZHT(Ax(t)+A1x(t-h(t))+Bf(σ(t))+Cx˙(t-η(t))+Dp(t))-e-2αh2xT(t)Rx(t)+2e-2αh2xT(t)Rx(t-h2)(h2-h1)2x˙T(t)Ux˙(t)+2xT(t)HKJf(σ(t))-2fT(σ(t))Jf(σ(t))-2x˙T(t)Lx˙(t)+2x˙T(t)LAx(t)+2x˙T(t)LA1x(t-h(t))+2x˙T(t)LBf(σ(t))+2x˙T(t)LCx˙(t-η(t))+2x˙T(t)LDp(t)-ϵpT(t)p(t)+ϵ(E1x(t)+E2f(σ(t))+E3x(t-h(t)))T×(E1x(t)+E2f(σ(t))+E3x(t-h(t)))-e-2αh2[x(t-h(t))-x(t-h2)]T×U[x(t-h(t))-x(t-h2)]-e-2αh2[x(t-h1)-x(t-h(t))]TU[x(t-h1)-x(t-h(t))]}-β[x(t-h1)-x(t-h(t))]TU[x(t-h1)-x(t-h(t))]-(1-β)[x(t-h(t))-x(t-h2)]TU[x(t-h(t))-x(t-h2)]=e2αt{ζT(t)ℳζ(t)-β[x(t-h1)-x(t-h(t))]Te-2αh2U[x(t-h1)-x(t-h(t))]-(1-β)[x(t-h(t))-x(t-h2)]Te-2αh2U[x(t-h(t))-x(t-h2)]}=e2αt{ζT(t)[(1-β)ℳ1+βℳ2]ζ(t)},
where ℳ1 and ℳ2 are defined as in (3.1), respectively, and ζ(t)=[x(t)x(t-h(t))x(t-h1)x(t-h2)f(σ(t))x˙(t)x˙(t-η(t))p(t)]. By (1-β)ℳ1+βℳ2<0 holds if and only if ℳ1<0 and ℳ2<0. For showing the convergence rate, we have V˙(x(t))≤0, and then V(x(t))≤V(x(0)). However,
(3.21)V1(x(0))=e2αtxT(0)Px(0)≤λmax(P)∥ϕ∥2,V2(x(0))=∫-h10‍e2αsxT(s)Qx(s)ds,≤λmax(Q)∫-h10‍e2αsds∥M1∥2=λmax(Q)1-e-2αh22α∥M1∥2,V3(x(0))=∫-h20‍e2αsxT(s)Qx(s)ds,≤λmax(Q)∫-h20‍e2αsds∥M1∥2=λmax(Q)1-e-2αh22α∥M1∥2,V4(x(0))=h1∫-h10‍∫s0‍e2αsx˙T(τ)Rx˙(τ)dτds=h1∫-h10‍e2αs[xT(0)Rx(0)-xT(s)Rx(s)]ds≤h2λM(R)∫-h10‍e2αsds∥ϕ∥2-h2λM(R)∫-h10‍e2αsds∥M1∥2=h2λM(R)1-e-2αh22α∥ϕ∥2-h2λM(R)1-e-2αh22α∥M1∥2,V5(x(0))=h2∫-h20‍∫s0‍e2αsx˙T(τ)Rx˙(τ)dτds≤h2λM(R)1-e-2αh22α∥ϕ∥2-h2λM(R)1-e-2αh22α∥M1∥2,V6(x(0))=h2∫-h2-h1‍∫s0‍e2αsx˙T(τ)Ux˙(τ)dτds≤h2λM(U)1-e-2αh22α∥ϕ∥2-h2λM(U)1-e-2αh22α∥M1∥2,V7(x(0))=∫η(0)0‍e2αsx˙T(s)Fx˙(s)ds,≤λM(F)∫-η0‍e2αsds∥M2∥2≤λM(F)1-e-2αη2α∥M2∥2,V8(x(t))=2∑i=1m‍λie2αt∫0σi‍fi(σi)dσi≤2∑i=1m‍λie2αt∫0σi‍kσi(t)dσi≤2∑i=1m‍e2αtλikσi2(t)≤2∑i=1m‍e2αtλikxT(t)HHTx(t),V8(x(0))≤2λM(HZKHT)∥ϕ∥2,
and V(x(t))≥e2αtxT(t)Px(t)≥e2αtλm(P)∥x(t)∥2. Then, from Definition 2.1, we conclude that the equilibrium point is globally exponentially stable. This completes the proof.Remark 3.3.
For system (2.1) with uncertainty located in the sector conditions (2.4), if there exists J=diag(j1,j2,…,jm), it follows that
(3.22)rifi(σi(t))hiTx(t)≥0,i=1,2,…,m,
which is equivalent to
(3.23)xT(t)HJf(σ(t))≥0.
By replacing (3.18) with (3.23) and letting α=0, we obtain the following robust stability criterion.Corollary 3.4.
Under Assumption3.1, the system (2.1) satisfying the sector condition (2.4) is asymptotically stable if there exist symmetric positive definite matrices P, Q, R, U, F, L; symmetric positive semidefinite matrices Z=diag(z1,z2,…,zm) and J=diag(j1,j2,…,jm); scalars ϵ1>0 and ϵ2>0; matrices Ni; i=1,2,3 of appropriate dimension such that the following LMI holds:
(3.24)ℳ1=ℳ-[0I-I000000]T×e-2αh2U[0-II000000]<0,ℳ2=ℳ-[0-I0I00000]T×e-2αh2U[0I0-I00000]<0,ℳ=[ϕ11ϕ12ϕ13ϕ14ϖ15ϕ16ϕ17ϕ18ϕ19*ϕ22ϕ23ϕ24ϕ25ϕ26ϕ27ϕ28ϕ29**ϕ33000000***ϕ4400000****ϖ55ϕ56ϕ57ϕ58ϕ59*****ϕ66ϕ67ϕ680******ϕ7700*******ϕ880********ϕ99],
where
(3.25)ϖ15=ATHZT+PB+HJ,ϖ55=ZHTB+BTHZT.Remark 3.5.
The following stability criteria are presented for finite and infinite sector conditions, for systems without uncertainties.Corollary 3.6.
Under Assumption3.1, given α>0. The system (2.1) without uncertainties satisfying the sector condition (2.3) is α-exponentially stabilizable if there exist symmetric positive definite matrices P, Q, R, U, F, L; symmetric positive semidefinite matrices Z=diag(z1,z2,…,zm) and J=diag(j1,j2,…,jm); matrices Ni; i=1,2,3 of appropriate dimension such that the following LMI holds:
(3.26)ℳ1=ℳ-[0I-I0000]T×e-2αh2U[0-II0000]<0,ℳ2=ℳ-[0-I0I000]T×e-2αh2U[0I0-I000]<0,ℳ=[ϕ11ϕ12ϕ13ϕ14ϕ15ϕ16ϕ17*ϕ22ϕ23ϕ24ϕ25ϕ26ϕ27**ϕ330000***ϕ44000****ϕ55ϕ56ϕ57*****ϕ66ϕ67******ϕ77].Corollary 3.7.
Under Assumption3.1, the system (2.1) without uncertainties satisfying the sector condition (2.4) is asymptotically stable if there exist symmetric positive definite matrices P, Q, R, U, F, L; symmetric positive semidefinite matrices Z=diag(z1,z2,…,zm) and J=diag(j1,j2,…,jm); matrices Ni; i=1,2,3 of appropriate dimension such that the following LMI holds:
(3.27)ℳ1=ℳ-[0I-I0000]T×e-2αh2U[0-II0000]<0,ℳ2=ℳ-[0-I0I000]T×e-2αh2U[0I0-I000]<0,ℳ=[ϕ11ϕ12ϕ13ϕ14ϖ15ϕ16ϕ17*ϕ22ϕ23ϕ24ϕ25ϕ26ϕ27**ϕ330000***ϕ44000****ϖ55ϕ56ϕ57*****ϕ66ϕ67******ϕ77].Remark 3.8.
In this paper, the restriction that the state delay is differentiable is not required which allows the state delay to be fast time-varying. Meanwhile, this restriction is required in some existing result, see [8, 9, 16, 17].Remark 3.9.
It is worth pointing out that we can extend this method to more complex dynamical network models, such as neutral-type neural networks [18, 19] or BAM neutral-type neural networks [20, 21].


## 4. Numerical Examples

In this section, we provide numerical examples to show the effectiveness of our theoretical results.Example 4.1.
Consider the following nominal Lur’e system with time-varying delays which is studied in [8, 9, 17]:
(4.1)x˙(t)-Cx˙(t-η(t))=Ax(t)+A1x(t-h(t))+Bf(σ(t)),σ(t)=HTx(t)=[h1h2]Tx(t),∀≥0,
where
(4.2)A=[-20.50-1],A1=[10.40.4-1],B=[-0.5-0.75],C=[0.20.10.10.2],H=[0.20.6].
Tables 1 and 2 give comparison of maximum allowable value of h2 for (4.1) obtained in Corollary 3.6 with nonlinearity satisfying (2.3), where k=100 and Corollary 3.7 with nonlinearity satisfying (2.4), respectively. We see that, when h1=0, the maximum allowable bounds for h2 obtained from Corollaries 3.6 and 3.7 are much better than those obtained in [8, 9, 17]. The results obtained in [8, 9] may not be used for the case when h1≠0. Moreover, the differentiability of the time delay h(t) is not required in Corollaries 3.6 and 3.7.Table 1
Upper delay boundh2 with h1=0.
η
D
Method
h
D
0.6
0.8
No restriction onh˙D
0.1
[8]
h
2
0.9888
0.7228
—
Corollary3.4
h
2
—
—
1.0651
0.5
[8]
h
2
0.7793
0.6282
—
Corollary3.4
h
2
—
—
0.9210
0.9
[8]
h
2
0.0983
0.0967
—
Corollary3.4
h
2
—
—
0.1177Table 2
Upper delay boundh2 with h1=0.
η
D
Method
h
D
0.6
0.8
No restriction onh˙D
0.5
[8]
h
2
0.7793
0.6282
—
[17]
h
2
0.7901
0.6321
—
Corollary3.6
h
2
—
—
0.9209
0.9
[8]
h
2
0.0983
0.0967
—
[17]
h
2
0.0994
0.0981
—
[9]
h
2
0.1086
0.1086
—
Corollary3.6
h
2
—
—
0.1181We leth(t)=0.7|cost|, η(t)=1, ϕ(t)=[-0.1cost,cost], for all t∈[-1,0], and f(x(t))=δ|x(t)|, |δ|≤0.5. Figure 1 shows the trajectories of solutions x1(t) and x2(t) of the nominal Lur’e system with time-varying delays (4.1).Figure 1
The trajectories ofx1(t) and x2(t) of the the nominal Lur’e system with time-varying delays (4.1).Example 4.2.
Consider the following uncertain Lur’e system with interval time-varying delay with the following parameters:(4.3)x˙(t)-Cx˙(t-η(t))=(A+ΔA(t))x(t)+(A1+ΔA1(t))x(t-h(t))+(B+ΔB(t))f(σ(t)),
where
(4.4)A=[-200-0.9],A1=[-10-1-1],B=[-0.2-0.3],C=[0.2000.2],H=[0.40.8],D=E1=E3=[0.1000.1],E2=[0.10].Solution.
From the conditions (3.1) of Theorem 3.2, we let α=0.4, h1=0.1, h2=0.4, t=0.8, η=0.1, and K=0.5. By using the LMI Tool-box in MATLAB, we obtain
(4.5)P=[28.6713-0.2237-0.22372.4026],Q=[25.11760.76160.76161.0772],U=[23.67983.00453.00453.1148],F=[7.0939-0.3657-0.36570.8018],N1=[-3.17360.4426-1.46011.2008],L=[16.55540016.5554],N3=[-5.5006-0.4441-0.9892-14.4101],N2=[-1.64100.2843],R=[12.09060.23130.23132.2354],Z=0.9717,J=4.7851,e1=10.2112,e2=18.1870.
Thus, the system (2.1), is 0.4-exponentially stabilizable. Given α>0, we will give the values of the maximum allowable upper bounds of the uncertain Lur’e system with interval time-varying delay (4.3) for difference ηd of the delay for different decay rates 0.1≤α≤0.4. From Theorem 3.2, we obtain the maximum allowable upper bound of the time-varying delay h2, as shown in Table 3.
We leth(t)=0.1+0.65|sint|, η(t)=0.5+0.4|sint|, ϕ(t)=[-cost,cost], for all t∈[-0.9,0], and f(x(t))=δ|x(t)|, |δ|≤0.5. Figure 2 shows the trajectories of solutions x1(t) and x2(t) of the uncertain Lur’e system with interval time-varying delay (4.3).Table 3
Maximum allowable upper boundsh2 of the uncertain Lur'e system with interval time-varying delay (4.3) for different values of the ηd and decay rates.
η
d
=
0.2
η
d
=
0.4
η
d
=
0.6
η
d
=
0.8
α
=
0.1
0.7423
0.7094
0.6541
0.5294
α
=
0.2
0.6798
0.6513
0.6031
0.4923
α
=
0.3
0.6291
0.6037
0.5606
0.4604
α
=
0.4
0.5870
0.5637
0.5245
0.4326Figure 2
The trajectories ofx1(t) and x2(t) of the uncertain Lur’e system with interval time-varying delay (4.3).


## 5. Conclusions

In this paper, we have investigated the delay-dependent robust stability criteria for uncertain neutral and Lur’e dynamical systems with sector-bounded nonlinearity. 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. The global exponential stability for uncertain neutral and Lur’e dynamical systems with some conditions are investigated. 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.



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*Source: 961382-2012-12-23.xml*