This paper is concerned with the problem of delay-dependent robust stability analysis for a class of uncertain neutral type Lur’e systems with mixed time-varying delays. The system has not only time-varying uncertainties and sector-bounded nonlinearity, but also discrete and distributed delays, which has never been discussed in the previous literature. Firstly, by employing one effective mathematical technique, some less conservative delay-dependent stability results are established without employing the bounding technique and the mode transformation approach. Secondly, by constructing an appropriate new type of Lyapunov-Krasovskii functional with triple terms, improved delay-dependent stability criteria in terms of linear matrix inequalities (LMIs) derived in this paper are much brief and valid. Furthermore, both nonlinearities located in finite sector and infinite one have been also fully taken into account. Finally, three numerical examples are presented to illustrate lesser conservatism and the advantage of the proposed main results.
1. Introduction
Time delay arises naturally in connection with system process and information flow for different part of dynamic systems. Dynamical systems with time delays, also called dynamic systems with hereditary systems, after effect or dead time, equations with deviating argument, have been investigated extensively for the past few decades due to their wide applications in many research fields, covering nuclear reactor, manual control, engineering systems, neural networks, and other scientific areas [1–42]. The main reason is that time-delay phenomenon frequently happens in various engineering systems and is also a key source of instability and oscillation. Hence, the delayed systems have been widely considered by a great quantity of research results in the recent years. To mention a few, stability analysis is carried out in [1, 3, 6, 7, 11, 12, 15, 17, 25, 32–34, 36]; H∞ controllers and H∞ estimation are investigated in [31, 38, 41]; Lur’e dynamical systems are addressed in [4–6, 26–29]; neutral systems with discrete and distributed delays are researched in [32–34]; passivity analysis and reliable fuzzy control are considered, respectively, in [39, 42]. For the new results on time-delay systems, see the works [3, 21, 29, 34, 38–42] and references cited therein. Therefore, it is of both theoretical and practical importance to study the problem of stability analysis on delayed systems. So far in the literatures, delayed systems can be classified into two types: retarded type and neutral type [1–12]. The retarded type system only depends on the state delay, while the neutral type system not only defines the derivative term of the current state but also explains the derivative term of the past state. And what is more, the delayed stability criteria are also classified into two categories according to their dependence on the size of the delays, namely, delay-independent stability criteria and delay-dependent stability criteria [13–18]. It is well known that delay-dependent stability criteria are less conservative than delay-independent ones when the sizes of time-delays are small.
As everyone knows, many nonlinear physical systems can be expressed as a feedback connection of a linear dynamical system and nonlinearity. One of the important classes of nonlinear systems is the Lur’e dynamical system whose nonlinear element satisfies certain sector constraints. Since the notion of absolute stability was introduced by Lur’e [19], stability analysis for the Lur’e systems has been extensively studied for several decades [20–29]. Recently, various approaches have been proposed by many researchers to obtain robust stability criteria for time-delay Lur’e control systems. Therefore, many methods have been proposed in these results to reduce the conservatism of the stability criteria, such as model transformation method, the bounding technique, free-weighting matrix method, the method of constructing Lyapunov-Krasovskii functionals, delay decomposition technique, and weighting-matrix decomposition method. By using suitable Lyapunov-Krasovskii functionals and free-weighting matrices, a new delay-dependent robust stability criterion of the class of uncertain mixed neutral and Lur’e systems has been proposed in [21]. Some conditions have been derived by employing Lyapunov-Krasovskii functionals and generalized convex combination in [26]. Sufficient conditions for the robustly asymptotic stability have been achieved by eliminating nonlinearity and removing free-weighting matrices in [29]. In addition, delayed systems often have a spatial extent because of the presence of an amount of parallel pathways of varying axon sizes and lengths. Then, there may exist either a distribution of conduction velocities along these pathways or a distribution of propagation delays over a period of time in some cases, which may cause another type of time delays, namely, distributed time delays in delayed systems. Hence, the stability problem for neutral systems with nonlinearity perturbation and time delay was thus of interest to a great number of researchers. The results in [30] are neutral-delay-independent by a method based on the equivalent equation of zero and the Leibniz-Newton formula in the derivative of the Lyapunov-Krasovskii functionals. In [32], Li and Zhu have obtained the discrete-, neutral-, and distributed-delay-dependent stability conditions for uncertain neutral system with discrete and distributed delays by constructing an augmented Lyapunov-Krasovskii functional and using the free-weighting matrices combined with the bounding technique. In [34], by constructing a proper Lyapunov-Krasovskii functional, Chen et al. have derived some new delay-dependent stability criteria for uncertain neutral system with mixed constant delays by making full use of the free-weighting matrices method and without bounding technique bringing much more conservatism used to deal with these triple-integral terms in [33]; however, the research of uncertain neutral system with mixed constant delays has a lot of limitations because of the delay-dependent robust stability criteria for uncertain neutral systems often connected with time-varying delay. Since time delay and parameter uncertainty exist in Lur’e systems in practice, the robust stability analysis for uncertain Lur’e systems with time delays still requires further consideration. And theoretically, except the traditional free-weighting matrices approach, how to get rid of the rigorous constraint that the time derivatives of time-varying delays must be less than one still needs much more study.
Motivated by the above discussion, in the paper, it is the first attempt to investigate one effective mathematical technique [29] to extend for a class of uncertain neutral type Lur’e systems with mixed time-varying delays which has not only time-varying uncertainties and sector-bounded nonlinearity, but also discrete and distributed delays. The effective mathematical technique is that dividing [hL,hU] into [hL,hL+hLU(1)/s)∪⋯∪[hL+hLU(i-1)/s,hL+hLU(i)/s)∪⋯∪[hL+hLU(s-1)/s,hU] is defined as a delay segmentation technique and supposing hL+hLU(i-1)/s≤h(t)≤hL+hLUi/s (i=1,2,…,s, hLU=hU-hL). Based on the mathematical technique, the derivative of ∫t-hUt(s-(t-hU))x˙T(s)Q2x˙(s)ds was estimated by hUx˙T(t)Q2x˙(t)-(1/hL)∫t-hLtx˙T(s)dsQ1∫t-hLtx˙(s)ds-(s/hLU(s-i+1))∫t-hUt-h(t)x˙T(s)dsQ2∫t-hUt-h(t)x˙(s)ds-(s/hLUi)∫t-h(t)t-hLx˙T(s)dsQ2∫t-h(t)t-hLx˙(s)ds, which plays an important role in the improvement of less conservative results. Secondly, by constructing a new type of Lyapunov-Krasovskii functional with triple terms, improved delay-dependent stability criteria in terms of linear matrix inequalities (LMIs) derived in this paper are brief and effective, which can be easily solved by using MATLAB LIM control toolbox [37]. Furthermore, both nonlinearities located in a finite sector and infinite one have been also fully taken into account. Finally, three numerical examples are given to illustrate the less conservatism of the proposed method.
Notation. Notations used in this paper are fairly standard: Rn denotes the n-dimensional Euclidean space; Rn×m is the set of all n×m dimensional matrices; I denotes the identity matrix of appropriate dimensions; T stands for matrix transposition; the notation X>0 (resp., X≥0), for X∈Rn×n, means that the matrix is real symmetric positive definite (resp., positive semidefinite); diag{r1,r2,…,rn} denotes block diagonal matrix with diagonal elements ri, i=1,2,…,m; the symbol * represents the elements below the main diagonal of a symmetric matrix.
2. Preliminaries
Consider the following class of uncertain neutral type Lur’e systems with mixed time-varying delays:
(1)x˙(t)=(A+ΔA(t))x(t)+(Ad+ΔAd(t))x(t-h(t))+(C+ΔC(t))x˙(t-τ(t))+(D+ΔD(t))∫t-rtx(s)ds+(B+ΔB(t))f(σ(t)),σ(t)=HTx(t)=[h1,h2,…,hm]Tx(t),t≥0,x(t)=ϕ(t),t∈[-max{hU,τU,r},0],
where x(t)∈Rn is the state vector, σ(t)∈Rm is the output vector, A∈Rn×n, Ad∈Rn×n, D∈Rn×n, C∈Rn×n, B∈Rn×m, and H∈Rn×m are known matrices, and 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)f(σ(t))=[f1(σ1(t))f2(σ2(t))⋯fm(σm(t))]Tσ(t)=[σ1(t)σ2(t)⋯σm(t)]T=HTx(t)=[h1h2⋯hm]Tx(t),
wherein each term fi(σi(t)), i=1,2,…,m, satisfies one of the following sector conditions:
(3)fi(σi(t))∈K[0,ki]={fi(σi(t))∣fi(0)=0,0<fi(σi(t))σi(t)≤ki,σi(t)≠0}
or
(4)fi(σi(t))∈K[0,∞]={fi(σi(t))∣fi(0)=0,fi(σi(t))σi(t)>0,σi(t)≠0},
where ΔA(t), ΔAd(t), ΔC(t), ΔD(t), and ΔB(t) are time-varying uncertain matrices of appropriate dimensions, which are assumed to be of the following form:
(5)[ΔA(t)ΔAd(t)ΔC(t)ΔD(t)ΔB(t)]=NF(t)[E1E2E3E4E5],
where N, E1, E2, E3, E4, and E5 are known real constant matrices of appropriate dimensions and F(t) is a time-varying uncertain matrix satisfying
(6)F(t)TF(t)≤I,∀t≥0.
By using (5), the uncertain system (1) can be rewritten as follows:
(7)x˙(t)=(A+ΔA(t))x(t)+(Ad+ΔAd(t))x(t-h(t))+(C+ΔC(t))x˙(t-τ(t))+(D+ΔD)∫t-rtx(s)ds+(B+ΔB(t))f(σ(t))=Ax(t)+Adx(t-h(t))+Cx˙(t-τ)+D∫t-rtx(s)ds+Bf(σ(t))+NP(t),
where(8)ξT(t)=[∫t-rtxT(t)xT(t-hL)xT(t-h(t))xT(t-hU)x˙T(t-τ(t))∫t-rtxT(s)dsfT(σ(t))PT(t)],ξ=[E10E20E3E4E50],P(t)=F(t)ξξ(t).
The delays h(t) and τ(t) are time-delaying continuous functions that satisfy
(9)0≤hL≤h(t)≤hU,h˙(t)≤hD,0≤τ(t)≤τU,τ˙(t)≤τD<1,
where hL, hU, hD, τU, and τD are known scalars.
The following fact and lemmas are introduced, which will be used in the proof of the main results.
Fact 1 (Schur complement).
For a given symmetric matrix S=ST=[S11S12*S22], where S11∈Rn×n, the following conditions are equivalent:
S<0;
S11<0, S22-S12TS11-1S12<0;
S22<0, S11-S12S22-1S12T<0.
Lemma 1 (see [<xref ref-type="bibr" rid="B35">35</xref>]).
For any constant matrix 0<R=RT∈Rn×n, a scalar r>0, a vector function x:[0,r]→Rn such that the integrations concerned are well defined, and then
(10)-r∫0rxT(s)Rx(s)ds≤-(∫0rx(s)ds)TR(∫0rx(s)ds).
Lemma 2 (see [<xref ref-type="bibr" rid="B36">36</xref>]).
For any constant matrix R∈Rn×n, R=RT>0, a scalar function h:=h(t)>0, and a vector-valued function x˙:[-h,0]→Rn such that the following integrations are well defined:
(11)-h∫t-htx˙(s)Rx˙(s)ds≤[x(t)x(t-h)]T[-RRR-R][x(t)x(t-h)],-h22∫-h0∫t+θtx˙T(s)Rx˙(s)dsdθ≤[hx(t)∫t-htx(s)ds]T[-RRR-R][hx(t)∫t-htx(s)ds].
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In this section, we will give sufficient conditions under which the system (1) is robustly asymptotically stable.
Theorem 3.
For a given positive integer s and some scalars hL, hU, hD, τU, τD, and ϵ with hU>hL>0, τD<1, and ϵ>0, the system (1) with a nonlinear function f(σ(t)) satisfying the finite sector condition (3) and time-varying delays h(t) and τ(t) satisfying (9) is robustly asymptotically stable if there exist PiT=Pi>0(i=1,2,…,5), QiT=Qi>0, MiT=Mi>0(i=1,2), diagonal matrices R=diag{r1,r2,…,rm}≥0, and Λ=diag{λ1,λ2,…,λm}≥0 with dimensions such that the following symmetric linear matrix inequality holds:
(12)[Φ+ΠiΓTP5hLΓTQ1ΓTQ2r22ΓTM2*-P5000**-Q100***-1hUQ20****-M2]<0,(i=1,2,…,s),
where
(13)Φ=[Φ11Φ12Φ13Φ14Φ15Φ16Φ17Φ18*Φ22Φ23Φ24Φ25Φ26Φ27Φ28**Φ33Φ34Φ35Φ36Φ37Φ38***Φ44Φ45Φ46Φ47Φ48****Φ55Φ56Φ57Φ58*****Φ66Φ67Φ68******Φ77Φ78*******Φ88],Πi=[00000000*ϕi1ϕi200000**ϕi3ϕi40000***ϕi50000****0000*****000******00*******0],Φ11=P1A+ATP1+P2+P3+P4-Q1-1hLQ2+rM1-r2M2+ϵE1TE1,Φ12=Q1+1hLQ2,Φ13=P1Ad+ϵE1TE2,Φ15=P1C+ϵE1TE3,Φ16=P1D+rM2+ϵE1TE4,Φ17=P1B+ATHΛ+HKR+ϵE1TE5,Φ18=P1N,Φ22=-P2-Q1-1hLQ2,Φ33=-(1-hD)P3+ϵE2TE2,Φ35=ϵE2TE3,Φ36=ϵE2TE4,Φ37=AdTHΛ+ϵE2TE5,Φ44=-P4,Φ55=-(1-τD)P5+ϵE3TE3,Φ56=ϵE3TE4,Φ57=CTHΛ+ϵE3TE3,Φ66=-1rM1-M2+ϵE4TE4,Φ67=DTHΛ+ϵE4TE5,Φ77=ΛHTB+BTHΛ-2R+ϵE5TE5,Φ78=ΛHTN,Φ88=-ϵI,ϕi1=-shLUiQ2,ϕi2=shLUiQ2,ϕi3=-(shLUi+shLU(s-i+1))Q2,ϕi4=shLU(s-i+1)Q2,ϕi5=-shLU(s-i+1)Q2,Γ=[A0Ad0CDBN],hLU=hU-hL,othertermsare0.
Proof.
Consider a novel augmented Lyapunov-Krasovskii functional for the system (1) as follows:
(14)V(xt)=V1(xt)+V2(xt)+V3(xt)+V4(xt)+V5(xt)+V6(xt)+V7(xt)+V8(xt)+V9(xt)+V10(xt),
where
(15)V1(xt)=xT(t)P1x(t),V2(xt)=∫t-hLtxT(s)P2x(s)ds,V3(xt)=∫t-h(t)txT(s)P3x(s)ds,V4(xt)=∫t-hUtxT(s)P4x(s)ds,V5(xt)=∫t-τ(t)tx˙T(s)P5x˙(s)ds,V6(xt)=∫t-hLthL(s-(t-hL))x˙T(s)Q1x˙(s)ds,V7(xt)=∫t-hUt(s-(t-hU))x˙T(s)Q2x˙(s)ds,V8(xt)=∫t-rt(s-(t-r))xT(s)M1x(s)ds,V9(xt)=r22∫-r0∫0θ∫t+λtx˙T(s)M2x(s)dsdλdθ,V10(xt)=2∑i=1Nλi∫0σifi(s)ds.
The derivative of V(xt) with respect to time along the trajectory (1) is
(16)V˙(xt)=V˙1(xt)+V˙2(xt)+V˙3(xt)+V˙4(xt)+V˙5(xt)+V˙6(xt)+V˙7(xt)+V˙8(xt)+V˙9(xt)+V˙10(xt).
By utilizing (7), the time derivative of V1(xt) is obtained as
(17)V˙1(xt)=2xT(t)P1x˙(t)=2xT(t)P1[∫t-rtAx(t)+Adx(t-h(t))+Cx˙(t-τ(t))mmmmmmm+D∫t-rtx(s)ds+Bf(σ(t))+NP(t)]=xT(t)(P1A+ATP1)x(t)+2xT(t)P1Adx(t-h(t))+2xT(t)P1Cx˙(t-τ(t))+2xT(t)P1D∫t-rtx(s)ds+2xT(t)P1Bf(σ(t))+2xT(t)P1NP(t),V˙2(xt)=xT(t)P2x(t)-xT(t-hL)P2x(t-hL),V˙3(xt)=xT(t)P3x(t)-(1-h˙(t))xT(t-h(t))P3x(t-h(t))≤xT(t)P3x(t)-(1-hD)xT(t-h(t))P3x(t-h(t)),V˙4(xt)=xT(t)P4x(t)-xT(t-hU)P4x(t-hU),V˙5(xt)=x˙T(t)P5x˙(t)-(1-τ˙(t))x˙T(t-τ(t))P5x˙(t-τ(t))≤x˙T(t)P5x˙(t)-(1-τD)x˙T(t-τ(t))P5x˙(t-τ(t)).
By utilizing Lemma 2, the time derivative of V6(xt) is obtained as
(18)V˙6(xt)=hL2x˙T(t)Q1x˙(t)-∫t-hLthLx˙T(s)Q1x˙(s)ds≤hL2x˙T(t)Q1x˙(t)-(∫t-hLtx˙(s)ds)TQ1(∫t-hLtx˙(s)ds)=hL2x˙T(t)Q1x˙(t)-[x(t)-x(t-hL)]T×Q1[x(t)-x(t-hL)]=hL2x˙T(t)Q1x˙(t)-xT(t)Q1x(t)+2xT(t)Q1x(t-hL)-xT(t-hL)Q1x(t-hL).
By utilizing Lemma 2, for every i=1,2,…,s, if hL+hLU(i-1)/s≤h(t)≤hL+hLUi/s, hLU=hU-hL, the time derivative of V7(xt) is obtained as
(19)V˙7(xt)=hUx˙T(t)Q2x˙(t)-∫t-hUtx˙T(s)Q2x˙(s)ds=hUx˙T(t)Q2x˙(t)-∫t-hLtx˙T(s)Q1x˙(s)ds-∫t-h(t)t-hLx˙T(s)Q2x˙(s)ds-∫t-hUt-h(t)x˙T(s)Q2x˙(s)ds≤hUx˙T(t)Q2x˙(t)-1hL[x(t)-x(t-hL)]T×Q2[x(t)-x(t-hL)]-shLU(s-i+1)×[x(t-h(t))-x(t-hU)]T×Q2[x(t-h(t))-x(t-hU)]-shLUi[x(t-hL)-x(t-h(t))]T×Q2[x(t-hL)-x(t-h(t))]=hUx˙T(t)Q2x˙(t)-1hLxT(t)Q2x(t)+2hLxT(t)Q2x(t-hL)-xT(t-hL)(1hL+shLUi)Q2x(t-hL)+2shLUixT(t-hL)Q2x(t-h(t))-xT(t-h(t))×(shLU(s-i+1)+shLUi)Q2x(t-h(t))+2shLU(s-i+1)xT(t-h(t))Q2x(t-hU)-shLU(s-i+1)xT(t-hU)Q2x(t-hU),V˙8(xt)=rxT(t)M1x(t)-∫t-rtxT(s)M1x(s)ds≤rxT(t)M1x(t)-1r(∫t-rtx(s)ds)TM1(∫t-rtx(s)ds),V˙9(xt)=r44x˙T(t)M2x˙(t)-r22∫-r0∫t+θtx˙T(s)M2x˙(s)dsdθV˙9(xt)≤r44x˙T(t)M2x˙(t)-r2xT(t)M2x(t)V˙9(xt)=+2rxT(t)M2∫t-rtx(s)dsV˙9(xt)=-(∫t-rtx(s)ds)TM2(∫t-rtx(s)ds),V˙10(xt)=2∑i=1mλiσ˙i(t)fi(σi)=2fT(σ(t))ΛHTx˙(t)=2fT(σ(t))ΛHT×[∫t-rtAx(t)+Adx(t-h(t))+Cx˙(t-τ)mmli+D∫t-rtx(s)ds+Bf(σ(t))+NP(t)]=2xT(t)ATHΛf+2xT(t-h(t))AdTHΛf+2x˙(t-τ(t))CTHΛf+∫t-rtxT(s)dsDTHΛf+fT(ΛHTB+BTHΛ)f+fTΛHTNP(t).
From system (1) with nonlinearity located in the sectors K[0,ki](i=1,2,…,m), if there exists R=diag{r1,r2,…,rm}, it is easy to achieve
(20)rifi(σi)[kihiTx(t)-fi(σi)]≥0,i=1,2,…,m,
which is equivalent to
(21)2[xT(t)HKRf(σ(t))-fT(σ(t))Rf(σ(t))]≥0;
also, for any positive scalar ϵ, the following inequality is satisfied:
(22)ϵξT(t)ξTξξ(t)-ϵPT(t)P(t)≥0.
The combination of (17)–(22) gives that
(23)V˙(xt)≤ξT(t)[Φ+Πi+ΓT(P5+hL2Q1+hUQ2+r44M2)Γ]ξ(t),mmmmmmmmmmmmmmmmmimmmmmiiimli=1,2,…,s.
By Fact 1, (23) is equivalent to (12), which implies V˙(xt)<0, which guarantees the robustly asymptotic stability of the system (1). This completes the proof.
When hL=0, by simplifying the Lyapunov-Krasovskii functional V(xt) used in the proof of Theorem 3 to
(24)V¯(xt)=V1(xt)+V3(xt)+V4(xt)+V5(xt)+V7(xt)+V8(xt)+V9(xt)+V10(xt).
Theorem 4.
For a given positive integer s and some scalars hU, hD, τU, τD, and ϵ with 0<hU, hD<1, τD<1, and ϵ>0, the system (1) with a nonlinear function f(σ(t)) satisfying the finite sector condition (3) and time-varying delays h(t) and τ(t) satisfying (9) is robustly asymptotically stable if there exist PiT=Pi>0(i=1,3,4,5), Q2T=Q2>0, MiT=Mi>0(i=1,2), R=diag{r1,r2,…,rm}≥0, and diagonal matrices Λ=diag{λ1,λ2,…,λm}≥0 with dimensions such that the following symmetric linear matrix inequality holds:
(25)[Ψ+ΛiΣTP5ΣTQ2r22ΣTM2*-P500**-1hUQ20***-M2]<0,(i=1,2,…,s),
where
(26)Ψ=[Ψ11Ψ12Ψ13Ψ14Ψ15Ψ16Ψ17*Ψ22Ψ23Ψ24Ψ25Ψ26Ψ27**Ψ33Ψ34Ψ35Ψ36Ψ37***Ψ44Ψ45Ψ46Ψ47****Ψ55Ψ56Ψ57*****Ψ66Ψ67******Ψ77],Λi=[φi1φi200000*φi3φi40000**00000***0000****000*****00******0],Ψ11=P1A+ATP1+P3+P4+rM1-r2M2+ϵE1TE1,Ψ12=P1Ad+ϵE1TE2,Ψ14=P1C+ϵE1TE3,Ψ15=P1D+rM2+ϵE1TE4,Ψ16=P1B+ATHΛ+HKR+ϵE1TE5,Ψ17=P1N,Ψ22=-(1-hD)P3+ϵE2TE2,Ψ24=ϵE2TE3,Ψ25=ϵE2TE24,Ψ26=AdHTΛ+ϵE2TE5,Ψ33=-P4,Ψ44=-(1-τD)P5+ϵE3TE3,Ψ45=ϵE3TE4,Ψ46=CTHΛ+ϵE3TE5,Ψ55=-1rM1-M2+ϵE4TE4,Ψ56=DTHΛ+ϵE4TE5,Ψ66=ΛHTB+BTHΛ-2R+ϵE5TE5,Ψ67=ΛHTN,Ψ77=-ϵI,Ψi1=-shUiQ2,Ψi2=shUiQ2,Ψi3=-(shUi+shU(s-i+1))Q2,Ψi4=shU(s-i+1)Q2,Σ=[AAd0CDBN],hLU=hU-hL,othertermsare0.
When the nonlinear function f(σ(t)) vanishes, the system (1) reduces to
(27)x˙(t)=(A+ΔA(t))x(t)+(Ad+ΔAd(t))x(t-h(t))+(C+ΔC(t))x˙(t-τ(t))+(D+ΔD)∫t-rtx(s)ds,x(t)=ϕ(t),t∈[-max{hU,r},0].
Similarly, based on Theorem 3, we can obtain the robustly asymptotical stability for system (27) as follows.
Theorem 5.
For a given positive integer s and some scalars hL, hU, hD, τU, τD, and ϵ with hU>hL>0, hD<1, τD<1, and ϵ>0, the system (27) with time-varying delays h(t) and τ(t) satisfying (9) is robustly asymptotically stable if there exist PiT=Pi>0(i=1,2,…,5), QiT=Qi>0, and MiT=Mi>0(i=1,2) with dimensions such that the following symmetric linear matrix inequality holds:
(28)[Ξ+ΔiΥTP5hLΥTQ1ΥTQ2r22ΥTM2*-P5000**-Q100***-1hUQ20****-M2]<0,(i=1,2,…,s),
where
(29)Ξ=[Ξ11Ξ12Ξ13Ξ14Ξ15Ξ16Ξ17*Ξ22Ξ23Ξ24Ξ25Ξ26Ξ27**Ξ33Ξ34Ξ35Ξ36Ξ37***Ξ44Ξ45Ξ46Ξ47****Ξ55Ξ56Ξ57*****Ξ66Ξ67******Ξ77],Δi=[0000000*ψi1ψi20000**ψi3ψi4000***ϕi5000****000*****00******0],Ξ11=P1A+ATP1+P2+P3+P4-Q1-1hLQ2+rM1-r2M2+ϵE1TE1,Ξ12=Q1+1hLQ2,Ξ13=P1Ad+ϵE1TE2,Ξ15=P1C+ϵE1TE3,Ξ16=P1D+rM2+ϵE1TE4,Ξ17=P1N,Ξ22=-P2-Q1-1hLQ2,Ξ33=-(1-hD)P3+ϵE2TE2,Ξ35=ϵE2TE3,Ξ36=ϵE2TE4,Ξ44=-P4,Ξ55=-(1-τD)P5+ϵE3TE3,Ξ56=ϵE3TE4,Ξ66=-1rM1-M2+ϵE4TE4,Ξ77=-ϵI,ψi1=-shLUiQ2,ψi2=shLUiQ2,ψi3=-(shLUi+shLU(s-i+1))Q2,ψi4=shLU(s-i+1)Q2,ψi5=-shLU(s-i+1)Q2,hLU=hU-hL,Υ=[A0Ad0CDN],othertermsare0.
3.2. Stability Analysis in the Infinite Sector <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M171"><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mo mathvariant="bold">[</mml:mo><mml:mn>0</mml:mn><mml:mo mathvariant="bold">,</mml:mo><mml:mi>∞</mml:mi><mml:mo mathvariant="bold">]</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>Theorem 6.
For a given positive integer s and scalars hL, hU, hD, τU, τD, and ϵ with hU>hL>0, hD<1, τD<1, and ϵ>0, the system (1) with a nonlinear function f(σ(t)) satisfying the finite sector condition (4) and time-varying delays h(t) and τ(t) satisfying (9) is robustly asymptotically stable if there exist PiT=Pi>0(i=1,2,…,5), QiT=Qi>0, MiT=Mi>0(i=1,2), a scalar α>0, and diagonal matrix Λ=diag{λ1,λ2,…,λm}≥0 with dimensions such that the following symmetric linear matrix inequality holds:
(30)[Φ¯+ΠiΓTP5hLΓTQ1ΓTQ2r22ΓTM2*-P5000**-Q100***-1hUQ20****-M2],(i=1,2,…,s),
where
(31)Φ¯=[Φ11Φ12Φ13Φ14Φ15Φ16Φ¯17Φ18*Φ22Φ23Φ24Φ25Φ26Φ27Φ28**Φ33Φ34Φ35Φ36Φ37Φ38***Φ44Φ45Φ46Φ47Φ48****Φ55Φ56Φ57Φ58*****Φ66Φ67Φ68******Φ¯77Φ78*******Φ88],Φ¯17=P1B+ATHΛ+αH,Φ¯77=ΛHTB+BTHΛ.
The remaining terms are defined as in Theorem 3.
Proof.
For system (1) with nonlinearity located in the sector K[0,∞], the condition (4) is equivalent to
(32)2αxT(t)Hf(σ(t))>0.
Now, replacing (21) by (32), then combining (17)–(22) and (32) yields
(33)V˙(xt)≤ξT(t)Ψiξ(t),i=1,2,…,s.
Then, similarly to the proof of Theorem 3, the result follows immediately. This completes the proof.
When hL=0, we can get a delay-dependent asymptotical stability criterion for the system (1) using the same method of Theorem 4. This result is shown in Theorem 7.
Theorem 7.
For a given positive integer s and scalars hU, hD, τU, τD, and ϵ with 0<hU, hD<1, τD<1, and ϵ>0, the system (1) with a nonlinear function f(σ(t)) satisfying the finite sector condition (4) and time-varying delays h(t) and τ(t) satisfying (9) is robustly asymptotically stable if there exist PiT=Pi>0(i=1,3,4,5), Q2T=Q2>0, MiT=Mi>0(i=1,2), a scalar α>0, and diagonal matrices Λ=diag{λ1,λ2,…,λm}≥0 with dimensions such that the following symmetric linear matrix inequality holds:
(34)[Ψ~+ΠiΣTP5ΣTQ2r22ΣTM2*-P500**-1hUQ20***-M2]<0,(i=1,2,…,s),
where
(35)Ψ~=[Ψ11Ψ12Ψ13Ψ14Ψ15Ψ~16Ψ17*Ψ22Ψ23Ψ24Ψ25Ψ26Ψ27**Ψ33Ψ34Ψ35Ψ36Ψ37***Ψ44Ψ45Ψ46Ψ47****Ψ55Ψ56Ψ57*****Ψ~66Ψ67******Ψ77],Ψ~16=P1B+ATHΛ+αH,Ψ~66=ΛHTB+BTHΛ.
Remark 8.
Recently, Wang et al. [29] present the effective mathematical technique to improve significantly the stability criterion. Motivated by this technique, it is the first attempt of the integral partitioning method to extend to a class of uncertain neutral type Lur’e systems with discrete and distributed delays, which has never been mentioned in the previous literatures.
Remark 9.
In the augmented Lyapunov-Krasovskii functional V(xt), this term (r2/2)∫-r0∫0θ∫t+λtx˙T(s)M2x˙(s)dsdλdθ plays a key role in reducing the conservatism of our results.
Remark 10.
Compared with those in previous methods, this method proposed in this paper has two differences: (i) the item ∫t-hUtx˙T(s)Q2x˙(s)ds is added into the Lyapunov functional, which will reduce the conservativeness of results; (ii) the Jensen inequality technique is used, which results in LMIs criteria with less variables than free-weighting matrix approach.
4. Numerical Examples
In this section, numerical simulation examples are given to show the effectiveness and correctness of the derived main results.
Example 11.
Consider the following class of uncertain neutral type Lur’e systems with mixed time-varying delays:
(36)x˙(t)=(A+ΔA(t))x(t)+(Ad+ΔAd(t))x(t-h(t))+(C+ΔC(t))x˙(t-τ(t))+(D+ΔD(t))∫t-rtx(s)ds+(B+ΔB(t))f(σ(t)),
where
(37)A=[-200-1],Ad=[-10-1-1],C=[0000],D=[-0.120-0.120.12],B=[-0.3-0.2],H=[0.60.8],N=[1.00.00.01.0],E1=[0.1000.1],E2=[0.1000.1],E3=[0.1000.1],E4=[0.1000.1],E5=[0.10],ΔA(t)=[α100α2],ΔAd(t)=[α300α4],ΔC(t)=[α500α6],ΔC(t)=[α700α8],ΔB(t)=[α90].αi(i=1,2,…,9) denote the uncertainties which satisfy
(38)-1.6<α1<1.6,-0.05<α2<0.05,-1.8<α3<1.8,-0.15<α4<0.15,-1.65<α5<1.65,-0.25<α6<0.25,-1.5<α7<1.5,-0.2<α8<0.2,-1.7<α9<1.7.
Remark 12.
For Example 11, our results of the maximum allowable delay bound (MADB) hU for different values of hD and τD are listed in Tables 1 and 2. It is clear to show that the maximum allowable delay bound (MADB) hU is dependent on different values of hD and τD. The values of hD and τD increase, respectively, as the maximum allowable delay bound (MADB) hU decreases. In order to verify the stability condition, the simulation results are given as hL=0.1, hD=0.1, r=1, and f(σ(t))=[f1(σ(s))=0.1(|s+1|-|s-1|)f2(σ(s))=0.1(|s+1|-|s-1|)]T with initial condition [-0.50.5]T in Figures 1–4. It is clear that the state trajectories approach to zero asymptotically.
Allowable time delay hU for hL=0.1, hD=0.1, and r=1 and different values of τD in Example 11.
τD
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Theorem 6
3.3587
3.1749
3.1258
3.0987
2.9859
2.9525
2.9012
2.8753
2.5537
Theorem 3
3.5201
3.5016
3.4857
3.4512
3.4185
3.3125
3.1517
2.9513
2.8718
Allowable time delay hU for hL=0, τD=0.1, and r=1 and different values of hD in Example 11.
hD
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Theorem 7
4.4132
4.2052
4.1231
4.0532
3.8962
3.7132
3.4891
3.2567
3.1032
Theorem 4
4.5658
4.3562
4.2578
4.1365
4.0596
3.9862
3.7253
3.5329
3.3568
State trajectories of x(t) in the plane (hU=3.5201, hL=0.1, hD=0.1, τD=0.1, and r=1, in Example 11, where the initial value is [-0.50.5]T).
State trajectories of x(t) on the space (hU=3.5201, hL=0.1, hD=0.1, τD=0.1, and r=1, in Example 11, where the initial value is [-0.50.5]T).
State trajectories of x(t) on the space (hU=3.5201, hL=0.1, hD=0.1, τD=0.1, and r=1, in Example 11, where the initial value is [-0.50.5]T).
State trajectories of x(t) in polar coordinates (hU=3.5201, hL=0.1, hD=0.1, τD=0.1, and r=1, in Example 11, where the initial value is [-0.50.5]T).
Remark 13.
Table 3 shows the maximum values of hU which guarantee stability of this system by applying Theorems 3 and 6 in this paper, where A1=K[0,0.5), A2=K[0,10), A3=K[0,1000), and A4=K[0,∞).
Allowable time delay hU for hL=0.1, hd=0.1, and r=1 and different values of τD in Example 11.
τD
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Theorem 3 (f∈A1)
3.5127
3.4125
3.3652
3.2158
3.1569
3.0237
2.9853
2.9356
2.8765
Theorem 3 (f∈A2)
3.4419
3.3589
3.2518
3.1321
3.0128
2.9512
2.8758
2.8133
2.7567
Theorem 3 (f∈A3)
3.2158
3.1675
3.1291
3.0962
2.9866
2.8534
2.7962
2.7531
2.6158
Theorem 6 (f∈A4)
3.2013
3.1326
3.0987
2.9765
2.9215
2.8312
2.7538
2.7312
2.5983
Example 14.
Consider the following uncertain system:
(39)x˙(t)=(A+ΔA(t))x(t)+(Ad+ΔAd(t))x(t-h)+(C+ΔC(t))x˙(t-τ)+(D+ΔD)∫t-rtx(s)ds,
where
(40)A=[-0.90.20.1-0.9],Ad=[-1.1-0.2-0.1-1.1],C=[-0.200.2-0.1],D=[-0.120-0.120.12],E1=[0.1000.1],E2=[0.1000.1],E3=[0.1000.1],E4=[0.1000.1],N=[1.00.00.01.0],ΔA(t)=[β100β2],ΔAd(t)=[β300β4],ΔC(t)=[β500β6],ΔC(t)=[β700β8].βi(i=1,2,…,8) denote the uncertainties which satisfy
(41)-1.6<β1<1.6,-0.05<β2<0.05,-1.5<β3<1.5,-0.06<β4<0.06,-1.6<β5<1.6,-0.2<β6<0.2,-1.4<β7<1.4,-0.1<β8<0.1.
Remark 15.
By applying Theorem 5 and solving the LMI (28) using MATLAB LMI control toolbox, we obtain the maximum allowable upper bounds which are listed in Tables 4 and 5. From Tables 4 and 5, it can be seen that our results show significant improvements over the results obtained in [15, 33, 34]. Further, if the values of s become larger, then the upper bound of the delay will also become larger which is the advantage of the delay partitioning approach used in this paper. Figures 5–8 show clearly that the state vector x(t) stabilizes to zero asymptotically.
Maximum allowable delay bound (MADB) r for τ=0.1 and different values of h in Example 14.
h
0.1
0.5
1
1.5
1.6
1.7
Li and Zhu [32]
6.64
5.55
1.62
—
—
—
Sun et al. [33]
6.67
6.12
2.75
1.31
0.93
0.42
Theorem 3.5inChen et al. [34]
6.60
4.32
0.91
—
—
—
Theorem 3.4 in Chen et al. [34]
6.67
6.02
3.19
1.50
1.25
1.02
Theorem 5 (s=18)
7.32
7.13
5.89
4.56
4.08
3.16
Theorem 5 (s=28)
8.16
7.96
7.68
6.58
5.32
4.65
Maximum allowable delay bound (MADB) h for τ=0.1 and different values of r in Example 14.
r
1
2
3
4
5
6
Li and Zhu [32]
1.12
0.93
0.77
0.65
0.55
0.43
Sun et al. [33]
1.58
1.20
0.95
0.77
0.64
0.51
Theorem 3.5 in Chen et al. [34]
0.97
0.77
0.63
0.52
0.44
0.35
Theorem 3.4 in Chen et al. [34]
1.71
1.71
1.09
0.83
0.68
0.51
Theorem 5 (s=18)
2.12
2.09
1.98
1.87
1.54
1.12
Theorem 5 (s=28)
2.65
2.56
2.35
2.13
1.87
1.52
State trajectories of x(t) in the plane (r=8.16, h=0.1, and τ=0.1 in Example 14, where the initial value is [-33]T).
State trajectories of x(t) on the space (r=8.16, h=0.1, and τ=0.1 in Example 14, where the initial value is [-33]T).
State trajectories of x(t) on the space (r=8.16, h=0.1, and τ=0.1 in Example 14, where the initial value is [-33]T).
State trajectories of x(t) in polar coordinates (r=8.16, h=0.1, and τ=0.1 in Example 14, where the initial value is [-33]T).
Example 16.
Consider the system (1) with the following parameters [7]:
(42)A=[-20.50-1],Ad=[10.40.4-1],C=[0.20.10.10.2],B=[-0.5-0.75],H=[0.20.6],ΔA(t)=ΔAd(t)=ΔB(t)=ΔC(t)=ΔD(t)=D=0.
Remark 17.
When hL=0, the maximum upper bounds hU of the delay h(t) for different hD and τD obtained from [7] and Theorem 3 of this paper are listed in Tables 6, 7, and 8. From the tables, we can see that the proposed result in this paper is less conservative than the one in [7]. Besides, Figures 9–12 show the simulation results for the state trajectories with initial conditions [-33]T, f(σ(t))=[f1(σ(s))=0.1(|s+1|-|s-1|)f2(σ(s))=0.1(|s+1|-|s-1|)]T, and time-delay hU=2.6052. One can see that the state trajectories approach to zero asymptotically.
Maximum allowable delay bound (MADB) hU for different values of hD in Example 16.
hD (hL=0, τD=0.1)
0.2
0.4
0.6
0.8
Gao et al. [7]
2.5405
1.8002
1.3605
0.7314
Theorem 3 (s=18)
2.5617
1.9216
1.5845
1.0123
Theorem 3 (s=28)
2.6052
1.9538
1.6232
1.1659
Maximum allowable delay bound (MADB) hU for different values of hD in Example 16.
hD (hL=0, τD=0.5)
0.2
0.4
0.6
0.8
Gao et al. [7]
2.1066
1.5581
1.0701
0.6329
Theorem 3 (s=18)
2.1258
1.6689
1.3210
0.9836
Theorem 3 (s=28)
2.1532
1.6986
1.3687
1.0066
Maximum allowable delay bound (MADB) hU for different values of hD in Example 16.
hD (hL=0, τD=0.9)
0.2
0.4
0.6
0.8
Gao et al. [7]
0.1086
0.1022
0.0989
0.0972
Theorem 3 (s=18)
0.1219
0.1086
0.1082
0.1082
Theorem 3 (s=28)
0.1221
0.1087
0.1083
0.1083
State trajectories of x(t) in the plane (hU=2.6052, hL=0, hD=0.2, and τ=0.1 in Example 16, where the initial value is [-0.60.8]T).
State trajectories of x(t) on the space (hU=2.6052, hL=0, hD=0.2, and τ=0.1 in Example 16, where the initial value is [-0.60.8]T).
State trajectories of x(t) on the space (hU=2.6052, hL=0, hD=0.2, and τ=0.1 in Example 16, where the initial value is [-0.60.8]T).
State trajectories of x(t) in polar coordinates (hU=2.6052, hL=0, hD=0.2, and τ=0.1 in Example 16, where the initial value is [-0.60.8]T).
Finally, in order to better understand the effectiveness of the paper, we give some explanation about Figures 1–12 presented for Examples 11, 14, and 16.
Figures 1, 5, and 9 are 2D line graph using linear axes (vectors create a single line; matrices create one line per column). Plotted variables are
single variable: plot a vector or each column of a matrix as one line versus its index;
N variable pairs: plot each pair of variables in the selected sequence. For example, the sequence var1, var2, var3, and var4 is plotted as var2 versus var1, var4 versus var3, and so on. Both variables in associated pairs must contain the same number of elements.
Figures 2, 6, and 10 are 3D mesh plot displaying a matrix as a wireframe surface. Plotted variables are as follows:
single variable (z): plot matrix values as heights above the x-y plane and map data values to colormap;
three variables (X,Y,Z): X and Y are vectors or matrices defining the x and y components of a surface.
Figures 3, 7, and 11 are 3D stem graph (display lines extending from the x-y-plane, terminating in circular markers). Plotted variables are as follows:
single variable (z): when z is a row vector, plot all elements at equally spaced x-values against the same y-value. When z is a column vector, plot all elements at equally spaced y-values against the same x-value;
three variables (x,y,z): plot z at values specified by x and y. x, y, and z must be of the same size.
Figures 4, 8, and 12 are line graph in polar coordinates. Plotted variables are as follows:
two variables (theta, rho): polar coordinate plot of angle theta (radians) versus radius rho (data units).
Figures 1–12 show clearly that the state trajectories of the system (1), (27) are asymptotically stable from different angles, respectively.
5. Conclusions
This paper is focused on the problem of some improved delay-dependent robust stability criteria for a class of uncertain neutral type Lur’e systems with discrete and distributed delays. In the first place, by using one effective mathematical technique, some less conservative delay-dependent stability results are established without employing the bounding technique and the mode transformation approach. In the second place, by constructing an appropriate Lyapunov-Krasovskii functional with triple terms playing an important role in reducing the conservatism of our results, improved delay-dependent stability criteria in terms of linear matrix inequalities (LMIs) derived in this paper are much simple and valid. What is more, both nonlinearities located in a finite sector and infinite one have been also fully taken into account. Finally, three numerical examples are given to illustrate the less conservatism of the proposed main results.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by National Basic Research Program of China (2010CB732501) and National Natural Science Foundation of China (61273015).
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