# Delay-Dependent RobustH∞ Performance for Uncertain Neutral Systems with Mixed Time-Varying Delays and Nonlinear Perturbations

Authors: Peerapongpat Singkibud; Luu Thi Hiep; Piyapong Niamsup; Thongchai Botmart; Kanit Mukdasai
Journal: Mathematical Problems in Engineering
 (2018)

---

## Abstract

This paper deals with the problems of delay-dependent stability andH∞ performance for uncertain neutral systems with time-varying delays, and nonlinear perturbations. The time-varying delays are neutral, discrete, and distributed time-varying delays that the upper bounds for the delays are available. The restrictions on the derivatives of the discrete and distributed time-varying delays are removed, which mean that a fast discrete time-varying delay is allowed. The uncertainties under consideration are nonlinear time-varying parameter perturbations and norm-bounded uncertainties, respectively. Firstly, by applying a novel Lyapunov-Krasovskii functional approach, Wirtinger-based integral inequality, Peng-Park’s integral inequality, decomposition technique of constant matrix, descriptor model transformation, Leibniz Newton formula and utilization of zero equation, and improved delay-dependent bounded real lemmas (BRL) for systems are established in terms of linear matrix inequalities (LMIs). Then, based on the obtained BRL, some less conservative delay-dependent stability criteria of uncertain neutral systems with mixed time-varying delays and nonlinear perturbations are obtained and improved H∞ performance criterion with the framework of LMIs is introduced. Finally, some numerical examples are given to illustrate that the presented method is effective.


---

## Body

## 1. Introduction

Time delay is frequently a source of instability and a source of generation of oscillation in many dynamic systems such as hybrid systems (and practical application) [1, 2], networked control systems, biological systems, mechanical systems, and chemical or process control systems [3]. Thus, analysis and synthesis problem for systems with time-varying delay have become an important issue and large varieties of problems have been researched since the nineties by several researchers [4–8], H∞ performance [9–11].In some physical system, the system models can be described by functional differential equation of neutral type, in which the models depend on the state delay but also depend on the state derivatives, are often encountered in various fields, such as population ecology [12], distributed networks containing lossless transmission lines [13], heat exchangers, and robots in contact with rigid environments [14]. For interesting research methods, stability criteria for application neutral stochastic systems and neural networks have been discussed in [15–19]. On the one hand, some LMI criteria on robust stability for uncertain ones have been deeply derived in [20–23]. Very recently, improved H∞ performance analysis and stability for uncertain systems with time-varying delays were proposed in [10, 11, 24]. However, there are rooms for further improvements in the feasible region of criteria for H∞ performance and stability.Stability criteria for time-delay systems are generally divided into two classes: delay-independent one and delay-dependent one. Delay-independent stability criteria tend to be more conservative, especially for small size delay, and such criteria do not give any information on the size of the delay. On the other hand, delay-dependent stability criteria are concerned with the size of the delay and usually provide a maximal delay size. Generally speaking, the latter ones are less conservative than the former ones when the time-delay values are small. Much time and efforts have been put into the development of some techniques and new Lyapunv-Krasovskii functional because how to choose Lyapunov-Krasovskii functional and estimate an upper bound of time-derivative of Lyapunov-Krasovskii functional play key roles to improve the feasible region of stability criteria. Delay-dependent stability criteria for these systems are established in terms of linear matrix inequalities (LMIs).With above motivations, based on Lyapunov stability theory, improvedH∞ performance criteria and stability analysis for uncertain neutral systems with mixed time-varying delays and nonlinear perturbations are derived by the framework of LMIs which will be introduced in Theorem 11. Some numerical examples are given to illustrate that the presented method is effective.


## 2. Problem Formulation

We introduce some notations 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; 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; matrix B is called seminegative definite (B≤0) if xTBx≤0, for all x∈Rn; B is negative definite (B<0) if xTBx<0 for all x≠0; A>B means A-B>0(B-A<0); A≥B means A-B≥0(B-A≤0); C([-h¯,0],Rn) denotes the space of all continuous vector functions mapping [-h¯,0] into Rn when h¯=max{rM,hM}, rM,hM∈R+; xt=x(t+s),s∈[-h¯,0]; ∗ represents the elements below the main diagonal of a symmetric matrix.Consider the following uncertain neutral system with mixed time-varying delays of the form:(1)x˙t=Atxt+Btxt-ht+Ctx˙t-rt+Dt∫t-httxsds+f1t,xt+f2t,xt-ht+f3t,x˙t-rt+D1wt,t≥0;zt=C1xt+C2xt-ht+D2wt,t≥0;xt=ϕt,t∈-h¯,0,where x(t)∈Rn is the state variable, w(t)∈Rp is the disturbance input that belongs to L2[0,∞], z(t)∈Rq is the controlled output, r(t) is the neutral time-varying delay, and h(t) is the discrete and distributed time-varying delays satisfying(2)0≤rt≤rM,0≤r˙t≤rd,(3)0≤ht≤hM,where rM,rd and hM are positive real constants. ϕ(t) is initial condition function and A(t), B(t), C(t), and D(t) are uncertain matrices. The uncertainties f1(.), f2(.), and f3(.) represent the nonlinear parameter perturbations with respect to the current state x(t), the delayed state x(t-h(t)), and the neutral delayed state x˙(t-r(t)), respectively, and are bounded in magnitude(4)f1Tt,xtf1t,xt≤α2xTtxt,(5)f2Tt,xt-htf2t,xt-ht≤β2xTt-htxt-ht,(6)f3Tt,x˙t-rtf3t,x˙t-rt≤η2x˙Tt-rtx˙t-rt,where α,β and η are given positive real constants. We assume that the uncertainties are norm-bounded and can be described as(7)At=A+ΔAt,Bt=B+ΔBt,Ct=C+ΔCt,Dt=D+ΔDt,ΔAtΔBtΔCtΔDt=EΔtG1G2G3G4,where A, B, C, C1, C2, D, D1, D2, E, G1, G2, G3, and G4 are real constant matrices with appropriate dimensions. The uncertain matrix Δ(t) satisfies(8)Δt=FtI-HFt-1and is said to be admissible, where H is known matrix satisfying(9)I-HHT>0.The uncertain matrix F(t) satisfies(10)FtTFt≤I.Definition 1.
Given a scalarγ>0, the system (1) is said to be asymptotically stable with the H∞ performance level γ, if it is asymptotically stable and satisfies the H∞-norm constraint z(t)2≤γw(t)2, for all nonzero w(t)∈L2[0,∞) under zero initial condition.Lemma 2 (see [25]).
Suppose thatΔ(t) is given by (8)-(10). Let M, L, and N be real constant matrices of appropriate dimension with M=MT. Then, the inequality(11)M+LΔtN+NTΔtTLT<0holds if and only if for any positive real constant δ,(12)MLδNT∗-δIδHT∗∗-δI<0.Lemma 3 (Jensen’s inequality).
For any symmetric positive definite matrixQ, positive real number h, and vector function x˙:[-h,0]→Rn such that the following integral is well defined, then(13)-h∫-h0x˙Ts+tQx˙s+tds≤-∫-h0x˙s+tdsTQ∫-h0x˙s+tds.Lemma 4 (Wirtinger-based integral inequality, [26]).
For any matrixZ>0, the following inequality holds for all continuously differentiable functions x˙:[α,β]→Rn:(14)-β-α∫αβx˙TsZx˙sds≤ωTΦω,where (15)ω=xTβ,xTα,1β-α∫αβxTsdsTand(16)Φ=-4Z-2Z6Z∗-4Z6Z∗∗-12Z.Lemma 5 (Peng-Park’s integral inequality, [27, 28]).
For any matrixZS∗Z≥0, positive scalars τ and τ(t) satisfying 0<τ(t)<τ, vector function x˙:[-τ,0]→Rn, such that the concerned integrations are well defined; then(17)-τ∫t-τtx˙TsZx˙sds≤ξTΘξ,where(18)ξ=xTt,xTt-τt,xTt-τTand(19)Θ=-ZZ-SS∗-2Z+S+STZ-S∗∗-Z.Lemma 6 (see [11]).
For a positive matrixM, the following inequality holds:(20)-α-β22∫βα∫sαxTuMxududs≤-∫βα∫sαxududsTM∫βα∫sαxududs.Lemma 7 (see [11]).
For a positive matrixM, the following inequality holds:(21)-α-β36∫βα∫sα∫uαxTλMxλdλduds≤-∫βα∫sα∫uαxλdλdudsTM∫βα∫sα∫uαxλdλduds.Lemma 8 (see [29]).
For any constant symmetric positive definite matrixQ∈Rn×n, h(t) is discrete time-varying delays with (3) and vector function ω:[-hM,0]→Rn such that the integrations concerned are well defined; then(22)-hM∫-hM0ωTsQωsds≤-∫-ht0ωTsdsQ∫-ht0ωsds-∫-hM-htωTsdsQ∫-hM-htωsds.Lemma 9 (see [29]).
For any constant matricesQ1,Q2,Q3∈Rn×n,Q1≥0,Q3>0, Q1Q2∗Q3≥0, h(t) is discrete time-varying delays with (3) and vector function x˙:[-hM,0]→Rn such that the integration is well defined, and then(23)-hM∫t-hMtxsx˙sTQ1Q2∗Q3xsx˙sds≤xtxt-htxt-hM∫t-httxsds∫t-hMt-htxsdsT-Q3Q30-Q2T0∗-Q3-Q3TQ3Q2T-Q2T∗∗-Q30Q2T∗∗∗-Q10∗∗∗∗-Q1xtxt-htxt-hM∫t-httxsds∫t-hMt-htxsds.Lemma 10 (see [29]).
Letx(t)∈Rn be a vector-valued function with first-order continuous-derivative entries. Then, the following integral inequality holds for any constant matrices X,Mi∈Rn×n,i=1,2,…,5, and h(t) is discrete time-varying delays with (3):(24)-∫t-hMtx˙TsXx˙sds≤xtxt-htxt-hMTM1+M1T-M1T+M20∗M1+M1T-M2-M2T-M1T+M2∗∗-M2-M2Txtxt-htxt-hM+hMxtxt-htxt-hMTM3M40∗M3+M5M4∗∗M5xtxt-htxt-hM,where (25)XM1M2∗M3M4∗∗M5≥0.


## 3. Main Results

In this section, a newH∞ performance and stability analysis for the system (1) will be introduced by using the Lyapunov functional method combining with linear matrix inequality technique. Firstly, we will establish a new version of delay-dependent BRL for the nominal system (1). We can rewrite the nominal system (1) as follows:(26)x˙t=Axt+Bxt-ht+Cx˙t-rt+D∫t-httxsds+f1t,xt+f2t,xt-ht+f3t,x˙t-rt+D1wt,t≥0;zt=C1xt+C2xt-ht+D2wt,t≥0;xt=ϕt,t∈-h¯,0.We introduce the following notations for later use:(27)∑​=Σi,j18×18,where Σi,j=Σj,iT,i,j=1,2,3,…,18,(28)Σ1,1=ϵ1α2I+W1+W2+W1T+W2T+P2TA1+P2TB1+A1TP2+B1TP2+P3T+P3+Q1+R1+hM2Q2+M1+M1T+hMM3-4Q4-Q5+hM2R4-R6+hM44Q6-2hM2Q7-hM44Q8+Z3,Σ1,2=P1-P2T+A1TP14+B1TP14+P15+R2+hM2R5,Σ1,3=A1TP11+B1TP11+P12-P4T-2Q4+S,Σ1,4=-W1+P2TA2+P2TB2+A1TP5+B1TP5-P3T+P6+P4T-M1T+M2+hMM4+Q5-S+R6,Σ1,5=-W2+L1,Σ1,6=-W1-P2TB1+P2TA2-P3T+L2,Σ1,7=-W2+L3,Σ1,8=-P4T,Σ1,11=-R5T+P2TD+A1TP8+B1TP8+P9,Σ1,12=6hMQ4,Σ1,13=2hMQ7,Σ1,14=hM2Q8,Σ1,15=P2TC,Σ1,16=Σ1,17=Σ1,18=P2T,Σ2,2=rMQ9-P14T-P14+R3+hMQ3+hM2Q4+hM2Q5+hM2R6+hM44Q7+hM636Q8+hM2Z1+rM2Z2,Σ2,3=-P11-P16T,Σ2,4=-P5+P14TA2+P14TB2-P15T+P16,Σ2,6=-P14TB1+P14TA2-P15T,Σ2,8=-P16T,Σ2,11=P14TD-P8,Σ2,15=P14TC,Σ2,16=Σ2,17=Σ2,18=P14T,Σ3,3=-P13T-P13-Q1-R1-M2-M2T+hMM5-4Q4-Q5-R6,Σ3,4=P11TA2+P11TB2-P12T-P7+P13T-M1+M2T+hMM4T+Q5T-ST+R6,Σ3,6=-P11TB1+P11TA2-P12T,Σ3,8=P13T,Σ3,9=-R2,Σ3,10=R5T,Σ3,11=P11TD-P10,Σ3,12=6hMQ4,Σ3,15=P11TC,Σ3,16=Σ3,17=Σ3,18=P11T,Σ4,4=ϵ2β2I+P5TA2+P5TB2+A2TP5+B2TP5-P6T-P6+P7T+P7+M1+M1T-M2-M2T+ST+hMM3+hMM5-2Q5+S-R6-R6T,Σ4,6=-P5TB1+P5TA2-P6T-L2,Σ4,8=-P7T,Σ4,10=-R5T,Σ4,11=R5T+P5TD+A2TP8+B2TP8-P9+P10,Σ4,15=P5TC,Σ4,16=Σ4,17=Σ4,18=P5T,Σ5,5=-L1T-L1-Z3+rdZ3,Σ5,7=-L1T-L3,Σ6,6=-L2T-L2-Z1,Σ6,8=-Z1,Σ6,11=-B1TP8+A2TP8-P9,Σ7,7=-L3T-L3-Z2,Σ8,8=-Z1,Σ8,11=-P10,Σ9,9=-R3,Σ10,10=-Q2-R4,Σ11,11=-Q2-R4+P8TD+DTP8,Σ11,15=P8TC,Σ11,16=Σ11,17=Σ11,18=P8T,Σ12,12=-12h2Q4,Σ13,13=-Q7,Σ14,14=-Q6-Q8,Σ15,15=ϵ3η2I-rM1-rdQ9,Σ16,16=ϵ1I,Σ17,17=ϵ2I,Σ18,18=ϵ3I,W1=P1J,W2=P2K, and other terms are 0.(29)∑​^=Σ^i,j19×19,where Σ^i,j=Σ^j,iT=Σi,j,i,j=1,2,3,…,19, except(30)Σ^1,1=ϵ1α2I+W1+W2+W1T+W2T+P2TA1+P2TB1+A1TP2+B1TP2+P3T+P3+Q1+R1+hM2Q2+M1+M1T+hMM3-4Q4-Q5+hM2R4-R6+hM44Q6-2hM2Q7-hM44Q8+Z3+C1TC1,Σ^1,4=-W1+P2TA2+P2TB2+A1TP5+B1TP5-P3T+P6+P4T-M1T+M2+hMM4+Q5-S+R6+C1TC2,Σ^1,19=C1TD2+P2TD1,Σ^2,19=P14TD1,Σ^3,19=P11TD1,Σ^4,4=ϵ2β2I+P5TA2+P5TB2+A2TP5+B2TP5-P6T-P6+P7T+P7+M1+M1T-M2-M2T+ST+hMM3+hMM5-2Q5+S-R6-R6T+C2TC2,Σ^4,19=C2TD2+P5TD1,Σ^11,19=P8TD1,Σ19,19=D2TD2-γ2I,and other terms are 0.Theorem 11.
For a prescribed scalarγ>0, given scalars rM>0,rd>0 and hM>0, the system (26) is asymptotically stable with the H∞ performance γ; if C+η<1, there exist positive definite symmetric matrices P1, R4, R6, Qi, and Zj, i=1,2,..,9, j=1,2,3, any appropriate dimensional matrices, J, K, S, Rk, Ml, Pm, and k=1,2,..,6, l=1,2,…,5, m=2,3,…,16, and positive real constants ϵ1, ϵ2, and ϵ3 satisfying the following LMIs:(31)R1R2R2TR3>0,(32)R4R5R5TR6>0,(33)Q5SSTQ5≥0,(34)Q3M1M2M1TM3M4M2TM4TM5≥0,(35)∑​^<0.Proof.
Under the conditions of the theorem, we first show the asymptotic stability of the system (26). To this end, we consider the nominal system (26) with w(t)=0, that is,(36)x˙t=Axt+Bxt-ht+Cx˙t-rt+D∫t-httxsds+f1t,xt+f2t,xt-ht+f3t,x˙t-rt.From model transformation method, we rewrite the system (36) in the following system:(37)x˙t=yt,(38)0=-yt+Axt+Bxt-ht+Cyt-rt+D∫t-httxsds+f1t,xt+f2t,xt-ht+f3t,yt-rt.In order to improve of the discrete delay h(t) in (26), let us decompose constant matrices A and B as(39)A=A1+A2,(40)B=B1+B2,where A1, A2, B1, and B2∈Rn×n are real constant matrices. By utilizing the zero equations, we obtain(41)0=Jxt-Jxt-ht-J∫t-httx˙sds,(42)0=Kxt-Kxt-rt-K∫t-rttx˙sds,where J and K∈Rn×n will be chosen to guarantee the asymptotic stability of the system (26). By (39)-(42), the systems (37)-(38) can be represented by the form(43)x˙t=yt+Jxt-Jxt-ht-J∫t-httysds+Kxt-Kxt-rt-K∫t-rttysds,(44)0=-yt+A1+B1xt+A2+B2txt-ht-B1-A2∫t-httysds+D∫t-httxsds+f1t,xt+f2t,xt-ht+f3t,yt-rt+Cyt-rt.Construct a Lyapunov-Krasovskii functional candidate for the systems (43)-(44) of the form(45)Vt=∑i=19Vit,where (46)V1t=ζTtEPζt,V2t=∫t-hMtxTsQ1xsds+∫t-hMtxsysTR1R2∗R3xsysds,V3t=hM∫-hM0∫t+stxTθQ2xθdθds+∫-hM0∫t+styTθQ3yθdθds,V4t=hM∫-hM0∫t+styTθQ4yθdθds+hM∫-hM0∫t+styTθQ5yθdθds,V5t=hM∫-hM0∫t+stxθyθTR4R5∗R6xθyθdθds,V6t=hM22∫t-hMt∫st∫utxTλQ6xλdλduds+hM2∫t-hMt∫st∫utyTλQ7yλdλduds,V7t=hM36∫t-hMt∫st∫ut∫λtyTθQ8yθdθdλduds,V8t=hM∫-hM0∫t+styTθZ1yθdθds+rM∫-rM0∫t+styTθZ2yθdθds,V9t=∫t-rttxTsZ3xsds+rM∫t-rttyTsQ9ysds,with(47)E=I0000000000000000000,P=P10000P2P5P8P11P14P3P6P9P12P15P4P7P10P13P16,ζt=xtxt-ht∫t-httxsdsxt-hMyt. The time derivative of V(t) along the trajectory of systems (43)-(44) is given by(48)V˙t=∑i=19V˙it.It is noted that ζT(t)EPζ(t) is really xT(t)P1x(t). Then the time derivative of V1(t) is calculated as(49)V˙1t=2xTtP1x˙t=2ζTtPTx˙t000=2xtxt-ht∫t-httxsdsxt-hMytTP1P2TP3TP4T0P5TP6TP7T0P8TP9TP10T0P11TP12TP13T0P14TP15TP16Tx˙t000=2xTtP1×yt+Jxt-Jxt-ht-J∫t-httysds+Kxt-Kxt-rt-K∫t-rttysds+2xTtP2T+xTt-htP5T+∫t-httxsdsTP8T+xTt-hMP11T+yTtP14T×-yt+A1+B1xt+A2+B2txt-ht-B1-A2∫t-httysds+D∫t-httxsds+f1t,xt+f2t,xt-ht+f3t,yt-rt+Cyt-rt+2xTtP3T+xTt-htP6T+∫t-httxsdsTP9T+xTt-hMP12T+yTtP15T×xt-xt-ht-∫t-httysds+2xTtP4T+xTt-htP7T+∫t-httxsdsTP10T+xTt-hMP13T+yTtP16T×xt-ht-xt-hM-∫t-hMt-htysds+2L1TxTt-rt×xt-xt-rt-∫t-rttysds+2L2T∫t-httysds×xt-xt-ht-∫t-httysds+2L3T∫t-rttysds×xt-xt-rt-∫t-rttysds. Differentiating V2(t), we have(50)V˙2t=xTtQ1xt-xTt-hMQ1xt-hM+xtytTR1R2R2TR3xtyt-xt-hMyt-hMTR1R2R2TR3xt-hMyt-hM. Using Lemmas 8 and 10, V3(t) is calculated as(51)V˙3t=hM∫-hM0xTtQ2xt-xTt+sQ2xt+sds+∫-hM0yTtQ3yt-yTt+sQ3yt+sds=hM∫-hM0xTtQ2xtds-hM∫-hM0xTt+sQ2xt+sds+∫-hM0yTtQ3ytds-∫-hM0yTt+sQ3yt+sds=hM2xTtQ2xt-hM∫t-hMtxTsQ2xsds+hMyTtQ3yt-∫t-hMtx˙TsQ3x˙sds≤hM2xTtQ2xt+hMyTtQ3yt-∫t-httxTsdsQ2∫t-httxsds-∫t-hMt-htxTsdsQ2∫t-hMt-htxsds+xtxt-htxt-hMTM1+M1T-M1T+M20-M1+M2TM1+M1T-M2-M2T-M1T+M20-M1+M2T-M2-M2Txtxt-htxt-hM+hMxtxt-htxt-hMTM3M40M4TM3+M5M40M4TM5xtxt-htxt-hM.Using Lemma 4 (Wirtinger-base integral inequality) and Lemma 5 (Peng-Park’s integral inequality), an upper bound of V4(t) can be obtained as(52)V˙4t=hM2yTtQ4yt-hM∫t-hMtx˙TsQ4x˙sds+hM2yTtQ5yt-hM∫t-hMtx˙TsQ5x˙sds≤hM2yTtQ4yt+hM2yTtQ5yt+xTtxTt-hM1hM∫t-hMtxTsds-4Q4-2Q46Q4-2Q4T-4Q46Q46Q4T6Q4T-12Q4×xtxt-hM1hM∫t-hMtxsds+xTtxTt-htxTt-hM-Q5Q5-SSQ5T-ST-2Q5+S+STQ5-SSTQ5T-ST-Q5×xtxt-htxt-hM. It is from Lemma 9 that we have(53)V˙5t=hM∫-hM0xtytTR4R5R5TR6xtyt-xt+syt+sTR4R5R5TR6xt+syt+sds=hM2xtytTR4R5R5TR6xtyt-hM∫t-hMtxsysTR4R5R5TR6xsysds≤hM2xtytTR4R5R5TR6xtyt+xtxt-htxt-hM∫t-httxsds∫t-hMt-htxsdsT-R6R60-R5T0R6T-R6-R6TR6R5T-R5T0R6T-R60R5T-R5R50-R400-R5R50-R4xtxt-htxt-hM∫t-httxsds∫t-hMt-htxsds. By Lemma 6, we can obtain V˙6(t) as follows:(54)V˙6t=hM22hM22xTtQ6xt-∫t-hMt∫utxTλQ6xλdλdu+hM2hM22yTtQ7yt-∫t-hMt∫utx˙TλQ7x˙λdλdu=hM42xTtQ6xt-hM22∫t-hMt∫utxTλQ6xλdλdu+hM42yTtQ7yt-hM2∫t-hMt∫utx˙TλQ7x˙λdλdu≤hM44xTtQ6xt-∫t-hMt∫utxTλdλduQ6∫t-hMt∫utxλdλdu+hM42yTtQ7yt-2∫t-hMt∫utx˙TλdλduQ7∫t-hMt∫utx˙λdλdu=hM44xTtQ6xt+hM42yTtQ7yt-∫t-hMt∫utxTλdλduQ6∫t-hMt∫utxλdλdu-2hMxTt-2∫t-hMtxTuduQ72hMxt-2∫t-hMtxudu. By Lemma 7 and calculating V˙7(t), we have(55)V˙7t=hM36hM36yTtQ8yt-∫t-hMt∫ut∫λtx˙TθQ8x˙θdθdλdu=hM636yTtQ8yt-hM36∫t-hMt∫ut∫λtx˙TθQ8x˙θdθdλdu≤hM636yTtQ8yt-∫t-hMt∫ut∫λtx˙TθdθdλduQ8∫t-hMt∫ut∫λtx˙θdθdλdu=hM636yTtQ8yt-hM22xTt-∫t-hMt∫utxTλdλduQ8hM22xt-∫t-hMt∫utxλdλdu. Using Lemma 3 (Jensen’s Inequality) that we have(56)V˙8t≤hM2yTtZ1yt-∫t-hMtyTsdsZ1∫t-hMtysds+rM2yTtZ2yt-∫t-rMtyTsdsZ2∫t-rMtysds≤hM2yTtZ1yt-∫t-httyTsds+∫t-hMt-htyTsdsZ1∫t-httysds+∫t-hMt-htysds+rM2yTtZ2yt-∫t-rttyTsdsZ2∫t-rttysds. Calculating V˙9(t) leads to(57)V˙9t=xTtZ3xt-1-r˙txTt-rtZ3xt-rt+rMyTtQ9yt-rM1-r˙tyTt-rtQ9yt-rt≤xTtZ3xt-1-rdxTt-rtZ3xt-rt+rMyTtQ9yt-rM1-rdyTt-rtQ9yt-rt. From (4), (5), and (6), we obtain for any positive real constants ϵ1,ϵ2, and ϵ3,(58)ϵ1α2xTtxt-f1Tt,xtf1t,xt≥0,(59)ϵ2β2xTt-htxt-ht-f2Tt,xt-htf2t,xt-ht≥0,(60)ϵ3η2yTt-rtyt-rt-f3Tt,yt-rtf3t,yt-rt≥0.According to (48)-(60), it is straightforward to see that(61)V˙t≤ζTt∑ζt,where ζT(t)=[xT(t),yT(t),xT(t-hM),xT(t-h(t)), xT(t-r(t)),∫t-h(t)t‍yT(s)ds,∫t-r(t)t‍yT(s)ds,∫t-hMt-h(t)‍yT(s)ds, yT(t-hM),∫t-hMt-h(t)‍xT(s)ds,∫t-h(t)t‍xT(s)ds,∫t-hMt‍xT(s)ds, 2∫t-hMt‍xT(u)du,∫t-hMt‍∫ut‍xT(λ)dλdu,y(t-r(t)),f1(t,x(t)),  and f2(t,x(t-h(t))),f3(t,y(t-r(t)))].
If the conditions (31)-(34) and ∑‍<0 hold, then (61) implies that there exists δ>0 such that V˙(t)≤-δx(t)2. Therefore system (26) is asymptotically stable.
Next, we shall establish theH∞ performance of system (26) under zero initial condition. To this end, we introduce(62)Jzwt=∫0tzTszs-γ2wTswsds,t>0.
Noting zero initial condition, it can be shown that for any nonzerow(t)∈L2[0,∞) and t>0. This can be written as(63)Jzwt=∫0tzTszs-γ2wTswsds=∫0tzTszs-γ2wTswsds+∫0tV˙sds-Vt+V0=∫0tzTszs-γ2wTsws+V˙sds-Vt≤∫0tzTszs-γ2wTsws+V˙sds, where V(t) is define in (45). After some algebraic manipulations, we obtain(64)zTtzt-γ2wTtwt+V˙t≤ζ^Tt∑​^ζ^t, where ζ^T(t)=[xT(t),yT(t),xT(t-hM),xT(t-h(t)), xT(t-r(t)),∫t-h(t)t‍yT(s)ds,∫t-r(t)t‍yT(s)ds,∫t-hMt-h(t)‍yT(s)ds, yT(t-hM),∫t-hMt-h(t)‍xT(s)ds,∫t-h(t)t‍xT(s)ds,∫t-hMt‍xT(s)ds, 2∫t-hMt‍xT(u)du,∫t-hMt‍∫ut‍xT(λ)dλdu,y(t-r(t)),f1(t,x(t)),  and f2(t,x(t-h(t))),f3(t,y(t-r(t))),w(t)].
If the condition (35) holds, we have zT(t)z(t)-γ2wT(t)w(t)+V˙(t)≤0. Thus,(65)V˙t≤-zTtzt+γ2wTtwt.Integrating both sides of (65) from t0 to t, we obtain(66)Vt-Vt0≤-∫t0tzTtzt+∫t0tγ2wTtwt.Then, letting t→∞ and under zero initial condition, we have V(t0)=V(0)=0 and V(∞)=0, which leads to(67)∫t0∞zTtzt≤∫t0∞γ2wTtwt.Therefore z(t)2≤γw(t)2 is satisfied for any nonzero w(t)∈L2[0,∞).
The proof of theorem is complete.According to Theorem11, we can obtain delay-dependent robust asymptotic stability criteria with the H∞ performance level γ of system (1). We introduce the following notations for later use(68)LT=ETP2ETP14ETP11ETP5000000ETP800000000,N=G100G2000000G4000G30000.Theorem 12.
For a prescribed scalarγ>0, given scalars rM>0,rd>0 and hM>0, the system (1) is robustly asymptotically stable with the H∞ performance γ; if C(t)+η<1, there exist positive definite symmetric matrices P1, R4, R6, Qi, and Zj, i=1,2,..,9, j=1,2,3, any appropriate dimensional matrices, J, K, S, Rk, Ml, and Pm, k=1,2,..,6, l=1,2,…,5, m=2,3,…,16 and positive real constants δ, ϵ1, ϵ2, and ϵ3 satisfying the following LMIs:(69)R1R2R2TR3>0,(70)R4R5R5TR6>0,(71)Q5SSTQ5≥0,(72)Q3M1M2M1TM3M4M2TM4TM5≥0,(73)∑​^LδNT∗-δIδHT∗∗-δI<0.Proof.
ReplacingA,B,C, and D in (35) with A=A+EΔ(t)G1, B=B+EΔ(t)G2, C=C+EΔ(t)G3, and D=D+EΔ(t)G4, respectively, we find that condition (35) is equivalent to the following condition:(74)∑​^+LΔtN+NTΔtTLT<0.By using Lemma (1), we can find that (74) is equivalent to the LMIs as follows:(75)∑​^LδNT∗-δIδHT∗∗-δI<0, where δ is a positive real constant. From Theorem (26) and conditions (69)-(73), system (1) is robustly asymptotically stable and satisfies z(t)2≤γw(t)2. The proof of theorem is complete.


## 4. Numerical Examples

Example 1.
Consider the following uncertain neutral system with mixed time-varying delays (26). We consider robust asymptotic stability with H∞ performance γ of system (26) by using Theorem 11. The system (26) is specified as follows:(76)A=-200-2,B=00.40.40,C=0.1000.1,C1=10,C2=10,D=000-1,D1=0.51,D2=0.1,I=1001,f1t,xt=0.1cos⁡tx1t0.1cos⁡tx2t,f2t,xt-ht=0.1sin⁡tx1t-ht0.1sin⁡tx2t-ht,f3t,x˙t-rt=0.1cos⁡tx˙1t-rt0.1sin⁡tx˙2t-rt.Decompose the matrices A=A1+A2 and B=B1+B2, where(77)A1=-1.5-0.5-0.1-1.5,A2=-0.50.50.1-0.5,B1=-0.60-0.5-1.1,B2=0.60.40.91.1,ht=cos⁡t,rt=sin2⁡0.6t,ϕt=0.51,t∈-1,0.It is easy to see that α=0.1, β=η=0.05, hM=1, rM=1, rd=0.6, and γ=11. By using LMI Toolbox in MATLAB, we use (31)-(35) in Theorem 11. This example shows that the solutions of LMIs are given as follows:(78)P1=168.3110-44.5433-44.543385.4171,P2=127.4372-29.2853-29.285396.9456,P3=5.822234.893734.893724.4550,P4=-4.38902.15942.1594-0.9552,P5=-9.5746-7.5072-7.5072-1.7680,P6=-0.11850.12990.1299-0.0008,P7=-0.51260.25220.2522-0.2031,P8=5.172511.912011.9120-1.9884,P9=11.157112.719212.719231.2439,P10=-0.15330.11970.11970.1812,P11=27.3684-4.8605-4.86058.2454,P12=15.07974.87374.873716.5943,P13=4.1994-2.0019-2.00191.0377,P14=75.3600-12.6160-12.616049.4866,P15=-6.561639.989639.989629.4163,P16=-23.5965-7.6251-7.62513.1193,Q1=62.0192-23.0061-23.006111.6538,Q2=15.1796-10.2201-10.220121.4165,Q3=6.9057-3.1644-3.16441.7515,Q4=15.58546.98686.986819.8625,Q5=14.1662-5.1961-5.19616.1072,Q6=64.9034-31.5581-31.558121.8466,Q7=10.6144-5.0964-5.09643.0301,Q8=24.1919-11.7046-11.70467.6700,Q9=14.9436-2.3454-2.345412.6511,L1=44.57560.75400.754045.5384,L2=50.62592.50762.507624.1936,L3=46.8341-0.3771-0.377146.2886,R1=62.0196-23.0062-23.006211.6539,R2=-0.03200.01520.0152-0.0079,R3=2.8943-1.3916-1.39160.8241,R4=28.2168-12.4294-12.4294138.1825,R5=0.33520.45580.455812.2682,R6=7.1169-3.1950-3.19502.8227,J=0.19220.31860.58840.2678,K=-0.3073-0.1700-0.1716-0.6207,S=-8.19802.42012.4201-4.6544,M1=-6.75883.09523.0952-1.7091,M2=6.7510-3.0908-3.09081.7066,M3=10.7350-5.0861-5.08612.8906,M4=-7.14683.29473.2947-1.9222,M5=10.9985-5.2411-5.24112.9857,Z1=4.8285-2.2335-2.23351.1915,Z2=-0.4052-0.1913-0.19130.1191,Z3=17.9504-8.9631-8.96315.9259,ϵ1=1.3735×103,ϵ2=1.2228×103,ϵ3=609.7809.Example 2.
Consider system (26) with D=0, ω(t)≡0, f1(.)=f2(.)=f3(.)≡0, and C1=C2=0, which means neutral system with time-varying delay(79)x˙t-Cx˙t-rt=Axt+Bxt-ht,t≥0;xt=ϕt,t∈-h¯,0,with the parameters (80)A=-110-1,B=0.500.5-0.5,C=0.4000.4. Decompose matrix A and B as follows: A=A1+A2 and B=B1+B2, where (81)A1=-1.5-0.5-0.1-0.5,A2=0.51.50.1-0.5,B1=-0.60-0.5-1.1,B2=1.1010.6. Table 1 lists the comparison of the upper bounds delays for asymptotic stability of system (79) by different methods. And the derivative of the discrete time-varying delay is unknown, because we removed the derivative of the discrete time-varying delay. We can see from Table 1 that our results are superior to those in [6, 30–33].Table 1
Maximum allowable upper boundshM with rM=1 and given rd for different μm in Example 2.
Method μm 0.7 0.9 1.1 Unknown  r d=0.5 Kwon et al. (2008) [30] - - - 1.0864 Guo et al. (2010) [6] - - - 1.2223 Liu et al. (2015) [32] 1.655 1.615 - - Liu (2016) [31] 1.864 1.742 - - Liu (2016) [31] - - - 1.8611 Wang et al. (2017) [33] 1.895 1.862 1.820 - This paper - - - 2.0447Example 3.
Consider system (26) with D=0,ω(t)≡0 and C1=C2=0,(82)x˙t-Cx˙t-ht=Axt+Bxt-ht+f1t,xt+f2t,xt-ht+f3t,x˙t-ht,t≥0,xt=ϕt,t∈-h¯,0,with the parameters (83)A=-200-2,B=00.40.40,C=0.1000.1,α=0.1,β=η=0.05. Decompose matrix A and B as follows: A=A1+A2and B=B1+B2, where (84)A1=-1.5-0.5-0.1-1.5,A2=-0.50.50.1-0.5,B1=-0.60-0.5-1.1,B2=0.60.40.91.1. Table 2 lists the comparison of the upper bounds delays for asymptotic stability of system (82) by different methods. And the derivative of the discrete time-varying delay is unknown, because we removed the derivative of the discrete time-varying delay. It is clear that our results are superior to those in [22, 32, 34, 35].Table 2
Maximum allowable upper boundshM with rM=1 and given rd for different μm in Example 3.
Method μm 0.5 0.9 1.1 Unknown  r d=0.6 Lakshmanan et al. (2011) [34] - - - 3.9563 Cheng et al. (2013) [22] - - - 4.6235 Qiu et al. (2015) [35] - - - 4.9423 Ours - - - 8.7375 r d=0 Liu et al. (2015) [32] 8.975 8.820 - - Qiu et al. (2015) [35] 9.646 9.225 - - Cheng et al. (2013) [22] 9.975 9.756 9.685 - This paper - - - 9.7967Example 4.
We consider system (82) with the parameters (85)A=-1.20.1-0.1-1,B=-0.60.7-1-0.8,C=0.1000.1,β=0.1. Decompose matrix A and B as follows: A=A1+A2and B=B1+B2, where (86)A1=-1-0.5-0.1-0.5,A2=-0.20.60-0.5,B1=-0.60.3-1.1-1.1,B2=00.40.10.3. Table 3 shows the comparison of the upper bounds delay allowed obtained for asymptotic stability of system (82) by other methods. And the derivative of the discrete time-varying delay is unknown, because we removed the derivative of the discrete time-varying delay. It can be found from Table 3 that our results are significantly better than those in [22, 34].Table 3
Maximum allowable upper boundshM with μm=0.5, rM=1 and different values of η in Example 4.
Method α = 0    α = 0.1 η 0.1 0.2 0.3  0.1 0.2 0.3  r d=0 Lakshmanan et al. (2011) [34] 1.3386 1.0816 0.8563  1.1950 0.9734 0.7760 Cheng et al. (2013) [22] 1.3721 1.0923 0.8613  1.2466 0.9996 0.7804 This paper (μm is unknown) 1.3496 1.1232 0.9027  1.2479 1.0450 0.8522


## 5. Conclusions

The problem of robust stability andH∞ performance for uncertain neutral systems with time-varying delays was studied. The restriction on the derivative of the discrete time-varying delay is removed. The uncertainties under consideration are nonlinear time-varying parameter perturbations and norm-bounded uncertainties, respectively. By applying a novel Lyapunov-Krasovskii functional approach, Wirtinger-based integral inequality and Peng-Park’s integral inequality, decomposition technique of constant matrix, descriptor model transformation, Leibniz-Newton formula and utilization of zero equation, and improved delay-dependent bounded real lemmas (BRL) for systems are established in terms of linear matrix inequalities (LMIs). Then, based on the obtained BRL, some less conservative delay-dependent stability criteria of uncertain neutral systems with mixed time-varying delays and nonlinear perturbations are obtained and improved H∞ performance criterion with the framework of LMIs is introduced. Numerical examples have shown significant improvements over some existing results.



---

*Source: 5721695-2018-10-16.xml*