# ℋ

Authors: M. J. Park; O. M. Kwon; Ju H. Park; S. M. Lee; E. J. Cha
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper deals with the problems ofℋ∞ performance and stability analysis for linear systems with interval time-varying delays. It is assumed that the parameter uncertainties are of stochastic properties to represent random change of various environments. By constructing a newly augmented Lyapunov-Krasovskii functional, less conservative criteria of the concerned systems are introduced with the framework of linear matrix inequalities (LMIs). Four numerical examples are given to show the improvements over the existing ones and the effectiveness of the proposed methods.


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## Body

## 1. Introduction

The mathematical models representing physical systems are generally not exact due to the various reasons such as noises and parameter changes in electrical elements. For this reason, in some cases, the stability of the mathematical model cannot guarantee the stability of the physical systems. In order to take into account such problem, the parameter uncertainties should be considered in the theoretical stability analysis for various systems. The aforementioned parameter uncertainties are the internal sources of the model, whereas the disturbances can be their external sources. Then, the objective of anH∞ performance analysis is to find a saddle point of objective functional calculus depending on the disturbance. In other words, we find its minimum for the worst-case disturbances. Moreover, from the point of view of stability, it is also needed to pay attention to a delay in the time. It is well known that time delays frequently occur in various systems due to the finite speed limit of information processing and transmission in the implementation of the systems. For this reason, the undesirable dynamic behaviors such as poor performance and instability can be caused by the wake of the delay.In this regard,H∞ performance and/or stability of time-delay systems were dealt with in the literature [1–12]. Above all, in [5], the robust H∞ performance conditions for uncertain networked control systems with time-delay were derived by the use of some slack matrix variable. Jeong et al. [6] introduced the improved conditions of H∞ performance analysis and stability for systems with interval time-varying delays and uncertainties. In [10], the robust H∞ performance analysis and stability problems of linear systems with interval time-varying delays were investigated by constructing some new augmented Lyapunov-Krasovskii functional. Also, in order to obtain tighter lower bounds of integral terms of quadratic form, Wirtinger-based inequality in [11] is the recent remarkable tool in reducing the conservatism of delay-dependent stability criteria for dynamic systems with time delays. Therefore, there are scopes for further improved results in stability analysis with time-delay.Returning to the concept of parameter uncertainties, in this work, it is assumed that the parameter uncertainties occur by stochastic property to represent random change of various environments. This exemplifies why considering the stochastic property includes the fact that when the earthquake happens, although the seismic intensity is the same, at all times, its wavy pattern and effects are different. However, the systems with stochastic parameter uncertainties have not been fully investigated yet. Specially, in this work, two stochastic indexes, the mean and the variance, are utilized. Thus, the concerned problems highlighting the difference between the effects of the mean and the variance on the systems will be dealt in this work.With this motivation mentioned above, in this paper, theH∞ performance and stability problems to get improved sufficient conditions for uncertain systems with interval time-varying delays and stochastic parameter uncertainties are studied. Here, stability of system with interval time-varying delays has been a focused topic of theoretical and practical importance [13]. The interval time-varying delays mean that its lower bounds which guarantee the stability of system are not restricted to be zero and include networked control system as one of typical examples. To achieve this, by construction of a suitable augmented Lyapunov-Krasovskii functional and utilization of Wirtinger-based inequality [11], an H∞ performance condition is derived in Theorem 8 with the framework of LMIs which can be formulated as convex optimization algorithms which are amenable to computer solution [14]. Also, inspired by the works of [4, 12], the reciprocally convex and zero equality approaches are utilized with some decision variables to reduce the conservatism of the concerned condition. Based on the result of Theorem 8, H∞ performance condition with deterministic parameter uncertainties and an improved stability condition for the nominal form without parameter uncertainties and disturbances will be proposed, respectively, in Theorem 11 and Corollary 12. Finally, four numerical examples are included to show the effectiveness of the proposed methods.Notation. The notations used throughout this paper are fairly standard. Rn is the n-dimensional Euclidean space, and Rm×n denotes the set of all m×n real matrices. L2[0,∞) is the space of square integrable vector on [0,∞). For symmetric matrices X and Y, X>Y means that the matrix X-Y is positive definite, whereas X≥Y means that the matrix X-Y is nonnegative definite. In, 0n, and 0n·m denote n×n identity matrix, n×n and n×m zero matrices, respectively. · refers to the Euclidean vector norm or the induced matrix norm. diag{⋯} denotes the block diagonal matrix. For square matrix X, sym{X} means the sum of X and its transposed matrix XT, that is, sym{X}=X+XT. col{x1,x2,…,xn} means [x1T,x2T,…,xnT]T. X[f(t)]∈Rm×n means that the elements of matrix X[f(t)] include the scalar value of f(t), that is, X[f0]=X[f(t)=f0]. E{·} stands for the mathematical expectation operator. Pr⁡{A} means the occurrence probability of the event A.


## 2. Preliminaries and Problem Statement

Consider the uncertain systems with time-varying delays and disturbances given by(1)x˙t=A+ΔAtxt+Ad+ΔAdtxt-ht+B1wt,zt=Cxt+Cdxt-ht+B2wt,
where x(t)∈Rn is the state vector, z(t)∈Rnz is the output vector, and w(t)∈Rnw is the disturbances; A, Ad∈Rn×n, B1∈Rn×nw, B2∈Rnz×nw, and C, Cd∈Rnz×n are the system matrices, and ΔA(t) and ΔAd(t) are the parameter uncertainties of the form
(2)ΔAt,ΔAdt=DFtEa,Ed,
where D∈Rn×nf, Ea∈Rnu×n, and Ed∈Rnu×n are real known constant matrices and F(t)∈Rnf×nu is a real uncertain matrix function with Lebesgue measurable elements satisfying FT(t)F(t)≤Inu.The delayh(t) is a time-varying continuous function satisfying
(3)0≤hm≤ht≤hM,h˙t≤dM,
where hm, hM, and dM are known constant values.For simplicity of system representation, the system can be formulated as follows:(4)x˙t=Axt+Adxt-ht+B1wt+Dpt,pt=Ftqt,qt=Eaxt+Edxt-ht,zt=Cxt+Cdxt-ht+B2wt.
Also, the following definition and lemmas will be used in main results.Assumption 1.
The parameter uncertainties are changed with the stochastic sequencesρ(t), which are a family of time functions depending on the outcome of the set of experimental outcomes. Then, the uncertainty term, q(t), is represented by
(5)qt=ρtEaxt+Edxt-ht,
where ρ(t) satisfies E{ρ(t)}=ρ0 and E{(ρ(t)-ρ0)2}=σ2. Here, ρ0 and σ2 are mean and variance of ρ(t), respectively.Remark 2.
After the introduction of the Bernoulli property to engineering, it has been applied in many situations such as random delays [15] and sensors fault [16]. In very recent times, various forms of randomly occurring concept, for example, randomly occurring uncertainties, randomly occurring nonlinearities, randomly occurring delays, and so on, are represented by the Bernoulli property [17, 18]. Besides, the Markov property, which is a favorite stochastic sequence, is used to describe the unexpected changes of parameters in hybrid systems [19–21]. It should be noted that the existing results utilizing Bernoulli and Markov property have not utilized the information about the variance. However, in this work, the system parameter uncertainties are described by the general stochastic property with its two indexes, mean and variance. By defining ρ(t) in (5), the variance value of ρ(t) will be considered in analyzing the robust H∞ performance of system (4). The necessity of these considerations will be explained in Example 1. Therefore, to analyze this problem mentioned above, in this work, the stochastic parameter uncertainties are dealt by adopting the property of the stochastic sequence, which contains two indexes, mean and variance, instead of studying the problem of previous stochastic analysis method considering Wiener process, that is, the form (Eax(t)+Edx(t-h(t)))dω(t), where ω(t) is Wiener process. Moreover, by utilizing the proposed model, the dynamic behavior of practical problem nearer to the random change of real environment will become accessible.The aim of this paper is to investigate theH∞ performance and stability analysis of system (4) with interval time-varying delays and stochastic parameter uncertainties. Before deriving our main results, the following definition and lemmas are introduced.Definition 3.
H
∞-optimization seeks a state-feedback controller that minimizes the H∞-norm of the system's closed-loop transfer function between the controlled output z(t) and the external disturbance w(t), which belongs to L2[0,∞); that is, Gzw∞=sup⁡∥w(t)∥2≠0z(t)2/w(t)2. Then, an equivalent definition of the H∞-norm is
(6)Gzw∞2=sup⁡w≠0∫0∞zTtztdt∫0∞wTtwtdt,
where it is assumed that x(0)=0. Therefore, Gzw∞ is the maximum possible gain in signal energy. This fact can be used to express constraints on the H∞-norm in terms of LMIs. From the above it follows that Gzw∞<γ is equivalent to
(7)J=∫0∞zTtzt-γ2wTtwtdt<0.Lemma 4 (see [11]).
The following inequality holds for a given matrixM>0 and all continuously differentiable functions x in [a,b]→Rn:
(8)∫abxTsMxsds≥1b-aξ1TMξ1+3b-aξ2TMξ2,
where ξ1=∫ab‍x(s)ds and ξ2=∫ab‍x(s)ds-2/b-a∫ab‍∫as‍xududs.Lemma 5 (see [12]).
For any vectorsx1, x2, constant matrices M, S, and real scalars 0<α<1 satisfying that MSSTM≥0, the following inequality holds:
(9)1αx1TMx1+11-αx2TMx2≥x1x2TMSSTMx1x2.Lemma 6 (see [22]).
Letζ∈Rn, Φ=ΦT∈Rn×n, and B∈Rm×n such that rank(B)<n. Then, the following statements are equivalent:(i)
ζTΦζ<0, Bζ=0, ζ≠0,(ii)
B⊥TΦB⊥<0, where B⊥ is a right orthogonal complement of B.Lemma 7 (see [23]).
For any matricesΩ>0, Ξ, matrix Λ, the following statements are equivalent:(i)
Ξ-ΛTΩΛ<0,(ii)
∃F:Ξ+ΛTF+FTΛFTF-Ω<0.


## 3. Main Results

In this section, some newH∞ performance and stability criteria for the system (4) will be derived. For convenience, the notations of several matrices are defined as follows:
(10)ζt=col∫t-hMt-ht∫t-hMsxududsxt,xt-ht,xt-hm,xt-hM,mmmx˙t,x˙t-hm,mmmx˙t-hM,1hm∫t-hmtxsds,mmm1ht-hm∫t-htt-hmxsds,mmm1hM-ht∫t-hMt-htxsds,mmm1hm∫t-hmt∫t-hmsxududs,mmm1ht-hm∫t-htt-hm∫t-htsxududs,mmm1hM-ht∫t-hMt-ht∫t-hMsxududs,pt,Π1ht=e1,e3,e4,e8,e9,e10·I3n03n03n·nhmIn02n·n04n·nht-hmInhM-htIn,V1ht=symΠ1htRe5,e6,e7,e1-e3,e3-e4T,V2=e1,e5Te3,e6T∑..d∑..de4,e7TT·diagN1,N2-N1,-N2e1,e5Te3,e6T∑..d∑..de4,e7T,V3ht=e3,e0T∑..de2,e3-e2TT·diagG,-1-dMGe3,e0T∑..de2,e3-e2TT+ht-hmsyme9,e3-e9G12∑..dG22e6T,V4,1=e5,e1hm2Q1e5,e1T-e1-e3,hme8Te1+e3-2e8T∑..d∑..dhme8-2e11TT·diagQ1,3Q1e1-e3,hme8Te1+e3-2e8T∑..d∑..dhme8-2e11T,Θz=e3,e2,e4hM-hm·diagZ1,Z2-Z1,-Z2e3,e2,e4T,V4,2=e6,e3hM-hm2Q2e6,e3T+hM-hmΘz,V4=V4,1+V4,2,Q2,1=Q2+0nZ1Z10n,Q2,2=Q2+0nZ2Z20n,Ω=diagQ2,1,3Q2,1SST∑..ddiagQ2,2,3Q2,2,Λht=e3-e2,ht-hme9Te3+e2-2e9,ht-hme9-2e12T∑..de2-e4,hM-hte10T∑..d∑..de2+e4-2e10,hM-ht*e10-2e13T,Δρ0,σ=ϵρ02+σ2Ea,Ede1,e2TTmi·Ea,Ede1,e2T-e14Infe14TEa,Ed,e1,e2TT,Ψ=e1,e2C,CdTC,Cde1,e2Te1,e2C,CdTB2B2TC,Cde1,e2T∑..dB2TB2,Υ=A,Ad,-In,De1,e2,e5,e14T,Vht=V1ht+V2+V3ht+V4,Ξht,ρ0,σ=Υ,B1⊥T·diagVht+Δρ0,σ,0nw+ΨΥ,B1⊥+symϜTdiagΛht,InwΥ,B1⊥,
where ei∈R(13n+nf)×n(i=1,2,…,14) are defined as block entry matrices, for example, e3Tζ(t)=x(t-hm).Then, the following theorem is given by the main result.Theorem 8.
For given scalars0≤hm<hM and dM, the system (4) is stochastically stable with H∞ performance γ, stochastic parameters ρ0 and σ, for 0≤hm≤h(t)≤hM and h˙(t)≤dM, if there exist positive definite matrices R∈R5n×5n, Ni∈R2n×2n(i=1,2), G=[Gij]∈R2n×2n, and Qi∈R2n×2n(i=1,2) and a positive scalar ϵ, any symmetric matrices Zi∈Rn×n(i=1,2), and any matrices S∈R4n×4n and Ϝ∈R(8n+nw)×(12n+nf+nw) satisfying the following LMIs:
(11)Ξiρ0,σϜTϜ∑..d-diagΩ,γ2Inw<0,i=1,2,(12)diagΩ,γ2Inw>0,(13)Q2,i≥0,i=1,2,
where Ξi[ρ0,σ] are the two vertices of Ξ[h(t),ρ0,σ] with the bounds of h(t). That is, h(t)=hM when i=1 and h(t)=hm when i=2.Proof.
Let us consider the following Lyapunov-Krasovskii functional candidate as follows:(14)Vt=V1+V2+V3+V4,
where
(15)V1=ν1TtRν1t,V2=∫t-hmtν2TsN1ν2sds+∫t-hMt-hmν2TsN2ν2xsds,V3=∫t-htt-hmν3Tt,sGν3t,sds,V4=hm∫t-hmt∫stν4TuQ1ν4ududs+hM-hm∫t-hMt-hm∫st-hmν4TuQ2ν4ududs
with
(16)ν1t=col∫t-hMt-hmxsdsxt,xt-ht,xt-hM,mmm∫t-hmtxsds,∫t-hMt-hmxsds,ν2s=colxs,x˙s,ν3t,s=colxs,∫st-hmx˙udu,ν4u=colx˙u,xu.
By infinitesimal operator L in [24], the LVi(i=1,2,3) can be calculated as follows:
(17)LV1=2xtxt-hmxt-hMhmhm∫t-hmtxsdsht-hmht-hm∫t-htt-hmxsds+hM-hthM-ht∫t-hMt-htxsdsT︸=ζTtΠ1ht·Rx˙tx˙t-hmx˙t-hMxt-xt-hnxt-hm-xt-hM=ζTtV1htζt,LV2=ν2TtN1ν2t-ν2Tt-hmN1ν2t-hm+ν2Tt-hmN2ν2xt-hm-ν2Tt-hMN2ν2xt-hM=ζTtV2ζt,LV3=ν3Tt,t-hmGν3t,t-hm-1-h˙tν3Tt,t-htGν3t,t-ht+2∫t-htt-hmν3Tt,sG∂∂tν3t,sds≤xt-hm∫st-hmx˙sdss=t-hmT·Gxt-hm∫st-hmx˙sdss=t-hm-1-dMxt-ht∫st-hmx˙sdss=t-htT·Gxt-ht∫st-hmx˙sdss=t-ht+2∫t-htt-hmν3Tt,sG11G12G12TG220n·1x˙t-hmds=xt-hm0n·1TGxt-hm0n·1-1-dMxt-htxt-hm-xt-htT·Gxt-htxt-hm-xt-ht+2∫t-htt-hmxsds∫t-htt-hm∫st-hmx˙ududsTG12G22x˙t-hm=xt-hm0n·1TGxt-hm0n·1-1-dMxt-htxt-hm-xt-htT·Gxt-htxt-hm-xt-ht+2ht-hm·1ht-hm∫t-htt-hmxsdsxt-hm-1ht-hm∫t-htt-hmxsdsT·G12G22x˙t-hm=ζTtV3htζt.
Prior to obtaining the bound of LV4, the V4 is divided into the following two parts:
(18)V4,1=hm∫t-hmt∫stν4TuQ1ν4ududs.V4,2=hM-hm∫t-hMt-hm∫st-hmν4TuQ2ν4ududs.
Inspired by the work of [4], the following zero equalities with any symmetric matrices Z1 and Z2 are considered as a tool of reducing the conservatism of criterion:
(19)0=xTt-hmZ1xt-hm-xTt-htZ1xt-ht-2∫t-htt-hmxTsZ1x˙sds+xTt-htZ2xt-ht-xTt-hMZ2xt-hM-2∫t-hMt-htxTsZ2x˙sds=ζTtΘzζt-ξ1t,
where
(20)ξ1t=2∫t-htt-hmxTsZ1x˙sds+2∫t-hMt-htxTsZ2x˙sds.
By utilizing Lemma 4, calculating the LV4,1 and LV4,2, and adding (19) into the LV4,2, the following relations can be obtained as follows:
(21)LV4,1=hm2ν4TtQ1ν4t-hm∫t-hmtν4TsQ1ν4sds≤hm2ν4TtQ1ν4t-∫t-hmtν4sdsTQ1∫t-hmtν4sds-3∫t-hmtν4sdsmmm-2hm∫t-hmt∫t-hmsν4ududsTQ1·∫t-hmtν4sds-2hm∫t-hmt∫t-hmsν4ududs=hm2ν4TtQ1ν4t-xt-xt-hm∫t-hmtxsdsTQ1xt-xt-hm∫t-hmtxsds-3xt+xt-hm-2hm∫t-hmtxsds∫t-hmtxsds-2hm∫t-hmt∫t-hmsxududsTQ1·xt+xt-hm-2hm∫t-hmtxsds∫t-hmtxsds-2hm∫t-hmt∫t-hmsxududs=ζTtV4,1ζt,LV4,2+hM-hmζTtΘzζt-ξ1t︸=0=hM-hm2ν4Tt-hmQ2ν4t-hm+hM-hmζTtΘzζt-hM-hm·∫t-htt-hmν4TsQ2ν4sdsmm-2∫t-htt-hmxTsZ1x˙sds-hM-hm∫t-hMt-htν4TsQ2ν4sdsmmmmmmmm-2∫t-hMt-htxTsZ2x˙sds=ζTtV4,2ζt-ξ2t,
where
(22)ξ2t=hM-hm∫t-htt-hmν4TsQ2,1ν4sds+hM-hm∫t-hMt-htν4TsQ2,2ν4sds.
By Lemmas 4 and 5, the integral terms, ξ2(t), of the LV4,2 are bounded as follows:
(23)ξ2t≥hM-hmht-hmϖ1,1TtQ2,1ϖ1,1t+3hM-hmht-hmϖ1,2TtQ2,1ϖ1,2t+hM-hmhM-htϖ2,1TtQ2,2ϖ2,1t+3hM-hmhM-htϖ2,2TtQ2,2ϖ2,2t=1αtϖ1,1tϖ1,2tTdiagQ2,1,3Q2,1·ϖ1,1tϖ1,2t+11-αtϖ2,1tϖ2,2tT·diagQ2,2,3Q2,2ϖ2,1tϖ2,2t≥ϖ1,1tϖ1,2tTdiagQ2,1,3Q2,1·ϖ1,1tϖ1,2t+ϖ1,1tϖ1,2tTSϖ2,1tϖ2,2t+ϖ2,1tϖ2,2tTSTϖ1,1tϖ1,2t+ϖ2,1tϖ2,2tTdiagQ2,2,3Q2,2·ϖ2,1tϖ2,2t=ζTtΛhtTΩΛhtζt,
where 1/α(t)=hM-hm/h(t)-hm,
(24)ϖ1,1t=∫t-htt-hmν4sds=xt-hm-xt-ht∫t-htt-hmxsdsmmmh=e3-e2,ht-hme9Tζt,ϖ1,2t=∫t-htt-hmν4sds-2ht-hm·∫t-htt-hm∫t-htsν4ududs=xt-hm+xt-ht-2ht-hm∫t-htt-hmxsds∫t-htt-hmxsds-2ht-hm∫t-htt-hm∫t-htsxududs=e3+e2-2e9,ht-hme9-2e12ζt,ϖ2,1t=∫t-hMt-htν4sds=xt-ht-xt-hM∫t-hMt-htxsdsmmmh=e2-e4,hM-hte10ζt,ϖ2,2t=∫t-hMt-htν4sds-2hM-ht·∫t-hMt-ht∫t-hMsν4ududs=xt-ht+xt-hM-2hM-ht∫t-hMt-htxsds∫t-htt-hmxsds-2hM-ht∫t-hMt-ht∫t-hMsxududs=e2+e4-2e10,hM-hte10-2e13ζt,
and Λ[h(t)]ζ(t)=col{ϖ1,1(t),ϖ1,2(t),ϖ2,1(t),ϖ2,2(t)}.
Hence,(25)LV4≤ζTtV4ζt-ζTtΛhtTΩΛhtζt.
Here, when Q2,i≥0(i=1,2) hold, the bound of LV4 is valid.
In succession, with the relational expression betweenp(t) and q(t), pT(t)p(t)≤qT(t)q(t), from the system (4), there exists any scalar ϵ>0 satisfying the following inequality
(26)0≤EϵqTtqt-pTtpt=EϵρtEaxt+Edxt-htT︸=φtmmiiim·ρtφtEaxt+Edxt-htT︸=φt-ϵpTtpt=Eρ0+ρt-ρ0φtφT-ϵpTϵρ0+ρt-ρ0φTtm0miii·ρ0+ρt-ρ0φtφT-ϵpTtpt=ζTtΔρ0,σζt,
where ϕ(t)=ρ02+2ρ0(ρ(t)-ρ0)+(ρ(t)-ρ0)2 and its mathematical expectation is as follows:
(27)Eϕt=Eρ02+Eρt-ρ02=ρ02+σ2.
From (17) to (26), the LV(t) has a new upper bound as follows:
(28)LVt≤ζTtΛhtTV1ht+V2+V3ht+V4mmmm-ΛhtTΩΛht+Δρ0,σζt.
From Definition 3, with the zero initial condition, we rewrite J as follows:
(29)J=E∫t=0∞zTtzt-γ2wTtwtmmmmn+zTtzt-γ2wTtwtLVt-LVtdt∫t=0∞zTtzt-γ2wTtwt.
Here, we get
(30)γ2wTtwt=ζtwtTdiag013n+nf,γ2Inw·ζtwt,zTtzt=Cxt+Cdxt-ht+B2wtT·Cxt+Cdxt-ht+B2wt=C,Cd,B2e1,e2TζtwtT·C,Cd,B2e1,e2Tζtwt=e1,e2TζtwtTC,CdTB2T·C,Cd,B2e1,e2Tζtwt=ζtwtT·$1e1,e2C,CdTB2B2TC,Cde1,e2T∑..dB2TB2·ζtwt=ζtwtTΨζtwt,
where $1=e1,e2[C,Cd]TC,Cd[e1,e2]T.
From (29) with (30), considering V(t)|t=0=0 and V(t)|t=∞→0, the J is bounded as follows:
(31)J≤E∫t=0∞zTtzt-γ2wTtwt+LVtdt=∫t=0∞ζtwtTΨ-diag013n+nf,γ2Inwmmmi·ζtwt+ζtwtTmmmi·diagVht-ΛhtTΩΛht+Δρ0,σ,0nwmmmi·ζtwtζtwtTdt=J*.
Thus, the following inequality
(32)ζtwtT·Ψ+diagVht-ΛhtTΩΛht+Δρ0,σ,-γ2Inw·ζtwt<0
is equivalent to the J≤J*<0. Therefore, if the condition (32) then system (4) is asymptotically stable with H∞ performance γ.
Applying Lemmas6 and 7 to (32) with
(33)0=EΥζt+B1wt=Υ,B1ζtwt
leads to
(34)Υ,B1⊥T·Ψ+diagVht-ΛhtTΩΛht+Δρ0,σ,-γ2Inw·Υ,B1⊥=Υ,B1⊥T·Ψ+diagVht+Δρ0,σ,0nwΥ,B1⊥-Υ,B1⊥TdiagΛhtTΩΛht,γ2Inw·Υ,B1⊥<0⟺$2ϜTϜ∑..d-diagΩ,γ2Inw<0
for any matrix Ϝ∈R(8n+nw)×(12n+nf+nw), where $2=[Υ,B1]⊥T(Ψ+diag{V[h(t)]+Δ[ρ0,σ],0nw})[Υ,B1]⊥+sym{ϜTdiag{Λ[h(t)],Inw}[Υ,B1]⊥}.
The above condition is affinely dependent onh(t). Therefore, if LMIs (11) hold, then system (4) is stochastically stable with H∞ performance γ and stochastic indexes ρ0 and σ2 for 0≤hm≤h(t)≤hM and h˙(t)≤dM. It should be noted that the inequality (12) is satisfied if the inequalities (11) hold. This completes our proof.Remark 9.
To achieve the less conservatism of stability condition, Wirtinger-based inequality with the basic Lyapunov-Krasovskii functional was introduced in [11]. However, a newly Lyapunov-Krasovskii functional was not proposed. In view of this, the main contribution in this work is the use of V3 included in a new Lyapunov-Krasovskii functional (14). As a result, some cross terms such as 2ht-hm1/ht-hm∫t-htt-hmxsdsxt-hm-1/ht-hm∫t-htt-hmxsdsTG12G22x˙t-hm and -1-dMxt-htxt-hm-xt-htTGxt-htxt-hm-xt-ht are utilized in estimating the LV.Remark 10.
In deriving lower bounds ofhm∫t-hmt‍νT(s)Q1ν(s)ds and ξ2(t) obtained by calculating the time-derivative values of V4, Lemma 4 which is the remarkable result in reducing the conservatism of delay-dependent stability criteria is utilized. However, unlike the results in [11], the utilized vectors of the two quadratic integral terms hm∫t-hmt‍νT(s)Q1ν(s)ds and ξ2(t) are [x˙T(s),xT(s)]T. As a result, some new integral terms such as 1/hm∫t-hmt‍∫t-hms‍xududs, 1/ht-hm∫t-htt-hm‍∫t-hts‍xududs, and 1/hM-ht∫t-hMt-ht‍∫t-hMs‍xududs are utilized as elements of the augmented vector ζ(t), which is different from the works [11].In case of the deterministic uncertainties, the following theorem can be obtained.Theorem 11.
For given scalars0≤hm<hM, dM, the system (4) is asymptotically stable with H∞ performance γ, for 0≤hm≤h(t)≤hM and h˙(t)≤dM, if there exist positive definite matrices R∈R5n×5n, Ni∈R2n×2n(i=1,2), G=[Gij]∈R2n×2n, and Qi∈R2n×2n(i=1,2), a positive scalar ϵ, any symmetric matrices Zi∈Rn×n(i=1,2), any matrices S∈R4n×4n and Ϝ∈R(8n+nw)×(12n+nf+nw) satisfying the LMIs (13) and
(35)Ξiρ0=1,σ=0ϜTϜ∑..d-diagΩ,γ2Inw<0,i=1,2,(36)diagΩ,γ2Inw>0,
where all notations were defined in (10).Proof.
When the mean,ρ0, and the variance, σ2, of ρ(t) are, respectively, 1 and 0, it means the uncertainties are deterministic. Therefore, by setting ρ0=1 and σ=0 in the result of Theorem 8, LMIs (35) can be easily obtained. So, it is omitted.As a special case of Theorem11, when the system (1) is the nominal form without parameter uncertainties and disturbances given by
(37)x˙t=Axt+Adxt-ht,
then, based on same Lyapunov-Krasovskii functional candidate in (14), the following corollary can be obtained.Corollary 12.
For given scalars0≤hm<hM, dM, the system (37) is asymptotically stable, for 0≤hm≤h(t)≤hM and h˙(t)≤dM, if there exist positive definite matrices R∈R5n×5n, Ni∈R2n×2n(i=1,2), G=[Gij]∈R2n×2n, and Qi∈R2n×2n(i=1,2), any symmetric matrices Zi∈Rn×n(i=1,2), and any matrices S∈R4n×4n and Ϝ^∈R8n×12n satisfying the LMIs (13) and
(38)Ξ^iϜ^TϜ^∑..d-Ω<0,i=1,2,Ω>0,
where Ξ^i is the two vertices of Ξ^[h(t)]=Υ^⊥TVhtΥ^⊥+symϜ^TΛhtΥ⊥ with Υ^=[A,Ad,-In][e1,e2,e5]T and other notations were defined in (10).Proof.
Upper bound of time-derivative of (14) can be calculated as follows:
(39)V˙t≤ζ^TtVhtζ^t,
where V[h(t)] was defined in (10) and
(40)ζ^t=col∫t-hMt-ht∫t-hMsxududsxt,xt-ht,xt-hm,xt-hM,mmmiix˙t,x˙t-hm,x˙t-hM,mmmii1hm∫t-hmtxsds,mmmii1ht-hm∫t-htt-hmxsds,mmmii1hM-ht∫t-hMt-htxsds,mmmii1hm∫t-hmt∫t-hmsxududs,mmmii1ht-hm∫t-htt-hm∫t-htsxududs,mmmii1hM-ht∫t-hMt-ht∫t-hMsxududs
with replacing the block entry matrices to ei∈R13n×n(i=1,…,13), which is very similar to the proofs of Theorems 8 and 11, so it is omitted.Remark 13.
When the information ofh˙(t) is unknown, the corresponding results of Theorems 8, 11 and Corollary 12 can be obtained by choosing G=0, respectively.


## 4. Illustrative Examples

Example 1.
Consider the system (1) with
(41)A=-200-1,Ad=-10-1-1,B1=0.010.05,C=0.1,0.2,CD=01·2,B2=0,D=I2,Ea=diag1.6,0.05,Ed=diag0.1,0.3.
For the above system, the maximum allowable delay bounds (MADBs) with various ρ0 and σ2, fixed hm=0 and γ=1, and unknown dM are listed in Table 1. When the stochastic indexes (the mean ρ0 and the variance σ2) increase, the MADBs become smaller, which means that Theorem 8 becomes more conservative. In order to verify this, the MADBs with ranges ρ0={0,0.2,…,2} and σ2={0,0.2,…,4} are shown in Figure 1. This figure demonstrates that a larger ρ0 or σ2 will lead to a smaller hM. Then, from Table 1 and Figure 1, it can be seen that the mean ρ0 and the variance σ2 can be addressed in the parameter uncertainties since the MADBs for guaranteeing the H∞ performance are influenced by the stochastic indexes. Moreover, Figures 2 and 3 are drawn to show the state trajectories with ρ0 and σ2. At this time, the initial condition x(0)=[0,0]T and the disturbance w(t) is 1 if 3≤t≤5 and 0, otherwise, and the time-delays are 0.67/2sin(2/0.67t)+0.67/2 and 0.43/2sin(2/0.43t)+0.43/2, respectively, in Figures 2 and 3. Also, in order to verify the stochastic indexes, F(t) is set as I2.Table 1
MADBs with fixed unknowndM, fixed hm=0 and γ=1 (Example 1).
ρ
0
σ
2
0.2
0.6
0.8
1.2
1.6
2.0
2.4
2.8
3.2
0.0
1.64
1.30
1.08
0.91
0.77
0.66
0.53
0.31
0
0.2
1.60
1.28
1.06
0.89
0.76
0.65
0.51
0.23
0
0.4
1.48
1.21
1.00
0.85
0.73
0.61
0.47
0
0
0.8
1.33
1.10
0.92
0.79
0.67
0.54
0.35
0
0
1.0
1.16
0.97
0.82
0.70
0.58
0.43
0
0
0
1.2
0.99
0.84
0.71
0.60
0.45
0
0
0
0
1.4
0.82
0.70
0.58
0.43
0
0
0
0
0
1.6
0.67
0.54
0.35
0
0
0
0
0
0
1.8
0.47
0
0
0
0
0
0
0
0
2.0
0
0
0
0
0
0
0
0
0
0: infeasible.Figure 1
Effect ofρ0 and σ2 (Example 1).Figure 2
State trajectories with variousρ0 and fixed σ2=0.2 (Example 1).Figure 3
State trajectories with variousσ2 and fixed ρ0=1.0 (Example 1).These figures give the relations between state trajectories andσ2 for the fixed ρ0=1.0 and the relations between the state trajectories and ρ0 for the fixed σ2=0.2. Also, these figures show that a lager ρ0 or σ2 will lead to the poor performance of system.Here, one of significant points is that the effect of the mean and the variance on system performance is different. From Table1, it can be seen that the growth of stochastic indexes leads to conservatism, whereas, from Figures 2 and 3, one can confirm the following two facts: (i) the mean ρ0 deteriorates the dynamic behavior of systems (see Figure 2) and (ii) the variance σ2 influences the system performance (see Figure 3).Example 2.
Consider the system (1) with (41). For the above system, the results of MADBs with various hm, fixed unknown dM, and γ=1 are listed in Table 2. By applying Theorem 11, it can be guaranteed that the MADBs under the same conditions are larger than the ones in the existing works which supports the fact that the proposed Lyapunov-Krasovskii functional and some utilized techniques effectively reduce the conservatism in H∞ performance.Table 2
MADBs with fixed unknowndM and γ=1 (Example 2).
h
m
0.0
0.2
0.4
0.6
0.8
1.0
Yue et al. [5]
0.6695
0.7343
0.8118
0.8962
0.9852
1.0784
Jeong et al. [6]
1.0400
1.0404
1.0411
1.0426
1.0508
1.0794
Kwon et al. [10]
1.0736
1.0995
1.0955
1.0986
1.1254
1.1740
Theorem11
1.0833
1.1019
1.1042
1.1198
1.1625
1.2296Example 3.
Consider the system (1) with
(42)A=010-0.1,Ad=00-0.375-1.15,B1=0.10.1,C=0,1,CD=-0.375,-1.15,B2=0,D=Ea=Ed=02.
For the above system, the minimized H∞ performance γ with various ordered pair (hm,hM) and unknown dM are listed in Table 3. In this table, the recent results [5, 6, 10] are compared with ones in this works. From Table 3, it is clear that our results for this example give smaller γ than the ones in [5, 6, 10].Table 3
MinimizedH∞ performance γ with various ordered pair (hm, hM) and unknown dM (Example 3).
(
h
m, hM)
(
0,0.8695
)
(
0.5695,0.8695
)
(
0,1
)
(
0.9,1
)
Yue et al. [5]
6.82
1.26
—
3.98
Jeong et al. [6]
0.87
0.81
4.05
2.59
Kwon et al. [10]
0.7810
0.7348
2.6153
1.8788
Theorem11
0.7019
0.6718
1.9819
1.3521Example 4.
Consider the system (1) with
(43)A=01-1-2,Ad=00-11.
In Table 4, the results for different condition of various hm and dM for guaranteeing stability are compared with the results of the existing works. From Table 4, it can be shown that our result for this example gives larger MABD than the ones in [7–10].Table 4
MADBs with varioushm and fixed dM=0.3 (Example 4).
0.3
0.5
0.8
1
Lee and Park [9]
1.07
1.21
1.45
1.61
Sun et al. [7]
2.2634
2.2858
2.3078
2.3167
Liu et al. [8]
2.2887
2.3094
2.3370
2.3516
Kwon et al. [10]
2.4503
2.4756
2.5069
2.5279
Corollary12
2.5198
2.5289
2.5492
2.5704


## 5. Conclusions

TheH∞ performance and stability analysis for linear systems with interval time-varying delays and disturbances were studied in this paper. In Theorem 8, the H∞ performance criterion for interval time-delayed systems with stochastic parameter uncertainties was proposed with the stochastic indexes, the mean ρ0 and the variance σ2. In Theorem 11, based on the result of Theorem 8, the interval time-delayed systems with deterministic parameter uncertainties were dealt. Afterward, in Corollary 12, the improved stability criterion for the nominal form of linear systems without parameter uncertainties and disturbances was derived. Four illustrative examples have been given to show the effectiveness and usefulness of the presented criteria. By utilizing the proposed criteria, future works will focus on solving various problems in [25–30].



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