# Further Results on Stability Analysis for Markovian Jump Systems with Time-Varying Delays

Authors: O. M. Kwon; R. Sakthivel; M. J. Park
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper is concerned with the problem of stability analysis for Markovian jump systems with time-varying delays. By constructing a newly augmented Lyapunov-Krasovskii functional and combining Wirtinger-based integral inequality, an improved delay-dependent stability criterion within the framework of linear matrix inequalities (LMIs) is introduced. Based on the result of delay-dependent stability criterion, when linear systems have fast time-varying delays, a corresponding stability condition is given. Via three numerical examples, the improvements of the proposed criteria are shown by comparing maximum delay bounds provided by our theorems with the recent results.


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## Body

## 1. Introduction

Stability analysis of dynamic systems is a prerequisite and essential job before designing a controller to achieve the prescribed specifications. In particular, a great concern of stability analysis for systems with time-delays has been received due to the fact that time-delay naturally occurs in many practical systems such as networked control system, chemical processing, hot rolling mill, synchronization between chaotic systems, neural networks, and multiagent systems. For instance, see [1, 2] and references therein.The main issue in delay-dependent stability analysis for time-delay systems with the framework of LMIs is how to increase maximum delay bounds for guaranteeing the asymptotic stability of systems. Thus, the choosing of Lyapunov-Krasovskii functional (LKF) and some techniques in estimating an upper bound of time-derivative value of the constructed LKF are the most important factors in enhancing the stability feasible region. In the LKF aspect, quadratic form, single integral, and double integral of quadratic form are the most utilized functionals. Recently, since the triple integral form of LKF was introduced in [3], this form of LKF has been utilized in many works such as [4–6]. Moreover, in [4, 5], it was shown that some augmented LKFs can increase the feasible region of stability criteria. In estimating an upper bound of time-derivative value of LKF, Jensen’s inequality [7], free-weighting matrix technique [8], and reciprocally convex optimization theory [9] make big impacts on the enhancement of delay-dependent stability and stabilization. Seuret and Gouaisbaut [10] proposed the Wirtinger-based integral inequality which provides more tight lower bounds than Jensen’s inequality and showed that the utilization of Wiritinger-based integral inequality can improve maximum delay bounds in many systems such as systems with constant and known delay, systems with a time-varying delay, systems with a constant distributed delay, and sampled-date systems. Cheng and Xiong [11] reduced conservative condition of stabilization criteria for continuous-time systems with time-varying input by introducing a new integral inequality. Recently, in [12, 13], for neural network with time-varying delay, it can be confirmed that the utilization of Wirtinger-based integral inequality in obtaining an upper bound of time-derivative values of some augmented LKFs can provide larger delay bounds than some other literatures. Very recently, in [14], it was shown that the results obtained by [10] can be further improved by choosing some new augmented LKFs. From the statements mentioned above, one can see that the choosing of LKF and some techniques play key roles to reduce the conservatism of stability criteria.On the other hand, increasing attention has been paid to Markovian jumping systems (MJSs) which are a special sort of hybrid systems and driven by Markov chain. MJSs may undergo unexpected changes in their structure and parameters including economic systems, aerospace systems, power systems, and networked control system [15, 16]. Very recently, a survey on recent developments of modeling, analysis, and design of MJSs was reported in Shi and Li [17].In this regard, many researchers put their times and efforts into stability and stabilization of Markovian jumping systems with time-delays. In [18], the problems of robust H∞ control and H∞ filtering for uncertain MJSs with time-varying delays were investigated by utilizing bounded real lemma. In [19], some new results on stabilization of MJSs with time-delays were proposed based on a delay-partitioning approach. Wu et al. [20] investigated the problem of stability and H∞ filtering for singular Markovian jump systems with time-delay via a delay-dependent bounded real lemma. Li et al. [21] utilized an input-output approach to stability and stabilization of MJSs with time-varying delays and showed the reduction of conservatism of the concerned criteria by a precise approximation of time-varying delay. By constructing new LKFs having distinct Lyapunov matrices for different modes, the mean square exponential stability and stabilization problems were studied in [22] for MJSs with constant time-delays. In [23], improved delay-dependent stability and H∞ control for singular Markovian jump systems with time-delay by utilizing delay-partitioning technique with a tuning parameter. Zhu [24] derived some new conditions for ensuring the asymptotic stability of singular nonlinear MJSs with unknown parameters and continuously distributed delays. Recently, some new augmented LKFs and techniques in estimating upper bounds of time-derivative of LKFs were introduced in [25] in studying stability and H∞ performance analysis of MJSs with time-varying delays. Very recently, in [26], an input-output approach to the delay-dependent stability analysis and H∞ control for MJSs with time-varying delays and deficient transition descriptions. The problem of finite-time H∞ estimation for a class of discrete-time Markov jump systems with time-varying transition probabilities subject to average dwell time switching was investigated in [27]. However, as mentioned in [17], the results on stability have still some conservativeness. Thus, there are rooms for further reduction of conservativeness caused by time-delays with the construction of a newly augmented Lyapunov-Krasovskii functional and utilization of a Wirtinger-based integral inequality [10].Motivated by [17] and based on the result of [25], the goal of this paper is to propose a further improved result of delay-dependent stability for MJSs with time-varying delays. In Theorem 5, a new and improved stability criterion will be proposed based on the results of [25]. To derive less conservative results, Wirtinger-based integral inequality is applied to the augmented LKFs and some new techniques are introduced. When an upper bound of time-derivative value of time-varying delay is larger than one or unknown, a corresponding result will be presented in Corollary 6 by constructing some part of LKF utilized in Theorem 5. Comparing with the result of [25], the constructed Lyapunov-Krasovskii functionals in Theorem 5 and Corollary 6 are simple since the triple and quadruple integral form of Lyapunov-Krasovskii functionals will not be utilized. Via three numerical examples, the advantage and effectiveness of the proposed results will be explained by comparing maximum delay bounds with some recent results presented in other literatures.Notation. Throughout this paper, the following notations will be used. X>0(X≥0) means that X is a real symmetric positive definitive matrix (positive semidefinite). The subscript “T” represents the transpose. X⊥ denotes a basis for the null-space of X. Rn denotes the n-dimensional Euclidean space and Rm×n is the set of all m×n real matrix. Cn,h=C([-h,0],Rn) denotes the Banach space of continuous functions mapping the interval [-h,0] into Rn with the topology of uniform convergence. L2[0,∞) means the space of square-integrable vector functions over [0,∞). E{·} denotes the expectation operator with respect to some measure P. In, 0n, and 0m·n denote n×n identity matrix and n×n and m×n zero matrices, respectively. · refers to the induced matrix 2-norm. diag⁡⋯ denotes the block diagonal matrix. X[f(t)]∈Rm×n means that the elements of matrix X[f(t)] include the scalar value of f(t). For any matrix M, Sym{M} means M+MT. col{x1,x2,…,xn} means [x1T,x2T,…,xnT]T.


## 2. Problem Statement and Preliminaries

Consider the Markovian jump system with time-varying delays:(1)x˙t=Artxt+Adrtxt-ht,xt=ϕt,∀t∈-hU,0,where x(t)∈Rn is the state vector, ϕ(t) which belongs to Cn,hU means the initial function, A(r(t)) and Ad(r(t)) are known system matrices with appropriate dimensions, and r(t) denotes a finite state Markovian jump process representing the system mode. That is, r(t) takes values in the finite discrete set S={1,2,…,N} with transition probability matrix Π=[πij].The transition probability is described as(2)Prrt+δ=j∣rt=i=πijδ+oδ,j≠i,1+πijδ+oδ,j=i,where δ>0, limδ→0+(o(δ)/δ)=0, πij≥0 for j≠i and πii=-∑j≠iπij.The delay in states,h(t), is a time-varying and continuous function satisfying (3)0≤ht≤hU,h˙t≤hD,where hU is a known positive scalar and hD is any constant one.For simplicity, a matrixM(r(t)) of ith node is denoted by Mi for each possible r(t)=i,i∈S in the rest of this paper. For example, A(r(t)) and Ad(r(t)) of ith node will be represented as Ai and Adi, respectively. Let xt=x(t+s) for s∈[-hU,0]. From [28], it should be noted that {(xt,r(t))} is a Markov process for t≥0. Then, its weak infinitesimal operator L acting on a functional V(xt,i) is defined by (4)LVxt,i=limδ→0+⁡1δEVxt+δ,rt+δ∣xt,rt=i-Vxt,i.In stability analysis of system (1), the following definition will be utilized.Definition 1 (see [29]).
For any finiteϕ(t)∈Cn,hU, and the initial condition of the mode r0∈S, the system x˙(t)=A(r(t))x(t)+Ad(r(t))x(t-h(t)) is said to (a)
be stochastically stable if there exists a constantT(r0,ϕ(t)) such that (5)E∫0∞xt2∣r0,ϕt≤Tr0,ϕt,(b)
be mean square stable if(6)limt→∞Ext2=0,hold for any initial condition(r0,ϕ(t)),(c)
be mean exponentially stable if there exist constantsα>0 and β>0 such that the following holds for any initial condition (r0,ϕ(t)): (7)Ext2∣r0,ϕt≤αϕte-βt.Based on the results of [25], the objective of this paper is to develop further improved delay-dependent stability criteria of system (1) which will be conducted in next section.The following lemmas will be utilized in deriving main results.Lemma 2.
Consider a given matrixM>0. Then, for all continuous function η in a,b→Rn, the following inequality holds: (8)∫abηTsMηsds≥1b-a∫abηsdsT·M∫abηsds+3b-a∫abηsds-2b-a∫ab∫sbηududsT·M∫abηsds-2b-a∫ab∫sbηududs.Proof.
From the original Wirtinger-based integral inequality [10], since (9)∫abηsds-2b-a∫ab∫asηududs=∫abηsds-2b-a∫ab∫abηudu-∫sbηududs=∫abηsds-2b-a∫ab∫abηududs︸1/b-a∫abηsds+2b-a∫ab∫sbηududs=-∫abηsds+2b-a∫ab∫sbηududs,inequality (8) holds.Lemma 3 (see [30]).
Letζ∈Rn, Φ=ΦT∈Rn×n, and B∈Rm×n such that rank(B)<n. Then, the following two statements are equivalent:(a)
ζTΦζ<0, Bζ=0,ζ≠0,(b)
(B⊥)TΦB⊥<0, where B⊥ is a right orthogonal complement of B.Lemma 4 (see [31]).
For the symmetric appropriately dimensional matricesΩ>0, Ξ, an any matrix Λ, the following two statements are equivalent:(a)
Ξ-ΛTΩΛ<0,(b)
there exists a matrix of appropriate dimensionΨ such that (10)Ξ+ΛTΨ+ΨTΛΨTΨ-Ω<0.


## 3. Main Results

In this section, improved delay-dependent stability criteria for MJSs (1) will be proposed. To express vectors and matrices in simple forms, block entry matrices ei(i=1,…,9)∈R9n×n will be used. For example, e3 means 0n·2n,In,0n·6nT. And some of scalars, vectors, and matrices are defined as(11)ζt=colxt,xt-ht,xt-hU,x˙t,x˙t-hU,∫t-httxsds,∫t-hUt-htxsds,∫t-htt∫stxududs,∫t-hUt-ht∫stxududs,ηt=colxt,xt-hU,∫t-hUtxsds,∫t-hUt∫stxududs,αt,s=colx˙s,xs,∫stx˙udu,βt,s=colxs,∫stx˙udu,Qaug1=Q+0nP10nP10n0n0n0n0n,Qaug2=Q+0nP20nP20n0n0n0n0n,Ξ1i=Syme1,e3,e6+e7,e8+e9·Rie4,e5,e1-e3,hUe1-e6-e7T,+e1,e3,e6+e7,e8+e9∑j=1sπijRje1,e3,e6+e7,e8+e9T,Ξ2i=e4,e1,09n·nNie4,e1,09n·nT-e5,e3,e1-e3·Nie5,e3,e1-e3T+Syme1-e3,e6+e7,hUe1-e6-e7·Ni09n·n,09n·n,e4T,Ξ3iht=e1,09n·nGie1,09n·nT-1-hDe2,e1-e2Gie2,e1-e2T+Syme6,hte1-e6·Gi09n·n,e4T,Ξ4=e4,e1,09n·nhU2Qe4,e1,09n·nT+2hUhUe1-e6-e7,e8+e9,hU22e1-e8-e9Q09n·n,09n·n,e4T,Ξ5=hUe1P1e1T-e2P1e2T+e2P2e2T-e3P2e3T,Λ1ht=e1-e2,e6,hte1-e6,ht-e1-e2+2e6,hte6-2e8,2e8-hte6T,Λ2ht=e2-e3,e7,hU-hte1-e7,hU-ht·-e2-e3+2e7,hU-hte7-2e9,-hU-ht2e6+e7+2e9T,Λ3ht=Λ1htT,Λ2htTT,Θ1i=Qaug1-1hU∑j=1sπijNj-1hU·0n0n·2n02n·n∑j=1sπijGj,Θ2i=Qaug2-1hU∑j=1sπijNj,Ω1i=Θ1i03n03n3Θ1ihU2,Ω2i=Θ2i03n03n3Θ2ihU2,Ω3i=Ω1iSSTΩ2i,Υi=Ai,Adi,0n,-In,0n,0n,0n,0n,0n,Σiht=Ξ1i+Ξ2i+Ξ3iht+Ξ4+Ξ5,Φiht=Υi⊥TΣihtΥi⊥+SymΥi⊥T·Λ3ihtTΨ.Now, we have the following theorem.Theorem 5.
For given scalarshU>0 and hD, system (1) is stochastically stable for 0≤h(t)≤hU and h˙(t)≤hD if there exist positive definite matrices Ri∈R4n×4n, Ni∈R3n×3n, Gi∈R2n×2n, and Q∈R3n×3n, any matrices S∈R6n×6n and Ψ∈R12n×8n, and any symmetric matrices P1∈Rn×n and P2∈Rn×n satisfying the following LMIs for all r(t)=i, i∈S: (12)Φi,kΨTΨ-Ω3i<0,k=1,2,where {Φi,k}k=12 means the two vertices of Φi[h(t)] with the bounds of 0≤h(t)≤hM. That is, Φi,1=Φi[h(t)=0] and Φi,2=Φi[h(t)=hU].Proof.
For eachr(t)=i, i∈S, let us consider the Lyapunov-Krasovskii functional candidate: (13)Vxt,i=∑j=14Vjxt,i,where(14)V1xt,i=ηTtRiηt,V2xt,i=∫t-hUtαTt,sNiαt,sds,V3xt,i=∫t-httβTt,sGiβTt,sds,V4xt,i=∫t-hUt∫stαTt,uQαt,ududs.From the following relationship: (15)ηt=xtxt-hU∫t-httxsds+∫t-hUt-htxsds∫t-htt∫stxududs+∫t-hUt-ht∫stxududs​=e1,e3,e6+e7,e8+e9Tζt,η˙t=x˙tx˙t-hUxt-xt-hUhUxt-∫t-hUtxsds︸∫t-httxsds+∫t-hUt-htxsds=e4,e5,e1-e3,hUe1-e6-e7Tζt,LV1(xt,i) can be represented as (16)LV1xt,i=2ηTtRη˙t+ηTt∑j=1sπijRjηt=ζTtΞ1iζt,where Ξ1i is defined in (11).
Note that(17)∫t-hUtαt,sds=∫t-hUtx˙sxs∫stx˙ududs=∫t-hUtx˙sds∫t-hUtxsds∫t-hUt∫stx˙udu=xt-xt-hU∫t-httxsds+∫t-hUt-htxsdshUxt-∫t-httxsds-∫t-hUt-htxsds=e1-e3,e6+e7,hUe1-e6-e7Tζt.From (17), calculation of LV2(xt,i) leads to (18)LV2xt,i=αTt,tNiαt,t·ddtt-αTt,t-hUNiαt,t-hUT·ddtt-hU+∫t-hUtddtαTt,sNiαt,s︸2αTt,sNidαt,s/dtds+∫t-hUtαTt,s·∑j=1sπijNjαt,sds=ζTte4,e1,09n·n·Nie4,e1,09n·nT-e5,e3,e1-e3Nie5,e3,e1-e3T+Syme1-e3,e6+e7,hUe1-e6-e7·Ni09n·n,09n·n,e4Tζt+∫t-hUtαTt,s·∑j=1sπijNjαt,sds=ζTtΞ2iζt+∫t-hUtαTt,s∑j=1sπijNjαt,sds.An upper bound of LV3(xt,i) can be obtained as (19)LV3xt,i≤xt0n·1TGixt0n·1-1-hD·xt-ht∫t-httx˙sdsTGixt-ht∫t-httx˙sds+2∫t-httβTt,sG∂βt,s∂tds+∫t-httβTt,s·∑j=1sπijGjβt,sds=ζTt·e1,09n·nGie1,09n·nT-1-hDe2,e1-e2Gie2,e1-e2T+Syme6,hte1-e6Gi09n·n,e4Tζt+∫t-httβTt,s∑j=1sπijGjβt,sds=ζTt·Ξ3ihtζt+∫t-httβTt,s∑j=1sπijGjβt,sds.Inspired by the work of [32], for any symmetric matrices Pi∈Rn×n(i=1,2), the following two zero equalities are satisfied:(20)0=hUxTtP1xt-xTt-htP1xt-ht-2∫t-httxTsP1x˙sds,0=hUxTt-htP2xt-ht-xTt-hUP2xt-hU-2∫t-hUt-htxTsP2x˙sds.By summing the two zero equalities in (20), we have (21)0=ζTtΞ5ζt-2hU∫t-httxTsP1x˙sds-2hU∫t-hUt-htxTsP2x˙sds.Let φ(t,s)=∫stαT(t,u)Qα(t,u)du. By using the similar methods presented in (18) to (19), the calculation of LV4(xt,i) can be represented as (22)LV4xt,i=hUφt,ss=t︸0-hUφt,ss=t-hU+hU∫t-hUt∂φt,s∂tds=-hUφt,ss=t-hU+hU∫t-hUtαTt,tQαt,t+2∫stαTt,uQ∂αt,u∂tduds=hU2αTt,t·Qαt,t-hU∫t-hUtαTt,sQαt,sds+2hU∫t-hUt∫stαTt,uQ∂αt,u∂tduds=ζTt·e4,e1,09n·nhU2Qe4,e1,09n·nT+2hUhUe1-e6-e7,e8+e9,hU22e1-e8-e9·Q09n·n,09n·n,e4Tζt-hU∫t-hUtαTt,s·Qαt,sds=ζTtΞ4ζt-hU∫t-hUtαTt,s·Qαt,sds.Here, the following equations are utilized in (22): (23)2hU∫t-hUt∫stαTt,uQ∂αt,u∂tduds=2hU∫t-hUt∫stx˙uxu∫utx˙uduTQ0n·10n·1x˙tduds=2hU∫t-hUt∫stx˙ududs∫t-hUt∫stxududs∫t-hUt∫st∫utx˙vdvdudsTQ0n·10n·1x˙t=2hUhUxt-∫t-httxsds-∫t-hUt-htxsds∫t-htt∫stxududs+∫t-hUt-ht∫stxududshU22xt-∫t-htt∫stxududs-∫t-hUt-ht∫stxududsTQ0n·10n·1x˙t.With the consideration of ∫t-hUtαT(t,s)∑j=1sπijNjα(t,s)ds in (18), ∫t-h(t)tβT(t,s)∑j=1sπijGjβ(t,s)ds in (19), and the two integral terms -2hU∫t-h(t)txT(s)P1x˙(s)ds and -2hU∫t-hUt-h(t)xT(s)P2x˙(s)ds in (21), the last integral term -hU∫t-hUtα2T(t,s)Q1α2(t,s)ds at (22) with the addition of integral terms mentioned above can be estimated by the use of (a) in Lemma 2 and reciprocally convex optimization approach [9] as (24)-hU∫t-hUtαTt,sQαt,sds-2hU∫t-httxTsP1x˙sds-2hU∫t-hUt-htxTsP2x˙sds+∫t-hUtαTt,s∑j=1sπijNjαt,sds+∫t-httβTt,s∑j=1sπijGjβt,sds=-hU∫t-httαTt,s·Qaug1-1hU∑j=1sπijNj-1hU0n0n·2n02n·n∑j=1sπijGj︸Θ1i·αt,sds-hU∫t-hUt-htαTt,sQaug2-1hU∑j=1sπijNj︸Θ2iαt,sds,where (25)Qaug1=Q+0nP10nP10n0n0n0n0n,Qaug2=Q+0nP20nP20n0n0n0n0n,which were defined in (11).
With the use of Lemma2, the integral term -hU∫t-h(t)tαT(t,s)Θ1iα(t,s)ds can be bounded as (26)-hU∫t-httαTt,sΘ1iαt,sds≤-hUht∫t-httαt,sdsT·Θ1i∫t-httαt,sds-3hUht∫t-httαt,sds-2ht∫t-htt∫stαt,ududsT·Θ1i∫t-httαt,sds-2ht∫t-htt∫stαt,ududs=-hUht∫t-httαt,sdsTΘ1i∫t-httαt,sds-3hUhtht∫t-httαt,sds-2∫t-htt∫stαt,ududsT·Θ1ih2t·ht∫t-httαt,sds-2∫t-htt∫stαt,ududs≤-hUht∫t-httαt,sdsTΘ1i∫t-httαt,sds-3hUhtht∫t-httαt,sds-2∫t-htt∫stαt,ududsT·Θ1ihU2·ht∫t-httαt,sds-2∫t-htt∫stαt,ududs=-hUhtxt-xt-ht∫t-httxsdshtxt-∫t-httxsdsT·Θ1ixt-xt-ht∫t-httxsdshtxt-∫t-httxsds-3hUhtht-xt-xt-ht+2∫t-httxsdsht∫t-httxsds-2∫t-htt∫stxududs2∫t-htt∫stxududs-ht∫t-httxsdsT·Θ1ihU2ht-xt-xt-ht+2∫t-httxsdsht∫t-httxsds-2∫t-htt∫stxududs2∫t-htt∫stxududs-ht∫t-httxsds=-hUhtζTtΛ1htTΩ1iΛ1htζt,where (27)Λ1ht=e1-e2,e6,hte1-e6,ht-e1-e2+2e6,hte6-2e8,2e8-hte6T,Ω1i=Θ1i03n03n3Θ1ihU2.The other integral term -hU∫t-hUt-h(t)αT(t,s)Θ1iα(t,s)ds can be estimated as (28)-hU∫t-hUt-htαTt,sΘ2iαt,sds≤-hUhU-ht∫t-hUt-htαt,sdsT·Θ2i∫t-hUt-htαt,sds-3hUhU-ht∫t-hUt-htαt,sds-2hU-ht∫t-hUt-ht∫st-htαt,ududsT·Θ2i∫t-hUt-htαt,sds-2hU-ht∫t-hUt-ht∫st-htαt,ududs=-hUhU-ht∫t-hUt-htαt,sdsT·Θ2i∫t-hUt-htαt,sds-3hUhU-hthU-ht∫t-hUt-htαt,sds-2∫t-hUt-ht∫st-htαt,ududsTΘ2ihU-ht2·hU-ht∫t-hUt-htαt,sds-2∫t-hUt-ht∫st-htαt,ududs≤-hUhU-ht∫t-hUt-htαt,sdsT·Θ2i∫t-hUt-htαt,sds-3hUhU-hthU-ht∫t-hUt-htαt,sds-2∫t-hUt-ht∫st-htαt,ududsTΘ2ihU2·hU-ht∫t-hUt-htαt,sds-2∫t-hUt-ht∫st-htαt,ududs.Since(29)∫t-hUt-htαt,sds=xt-ht-xt-hU∫t-hUt-htxsds∫t-hUt-ht∫stx˙ududs=e2-e3,e7,hU-hte1-e7Tζt,2∫t-hUt-ht∫stht∫utx˙vdvduds=2∫t-hUt-ht∫st-htxt-xududs=2·hU-ht22xt-2∫t-hUt-ht∫st-htxududs-2∫t-hUt-ht∫t-httxududs+2∫t-hUt-ht∫t-httxududs︸=0=hU-ht2xt-2∫t-hUt-ht∫stxududs+2∫t-hUt-ht∫t-httxududs︸=2·hU-ht∫t-httxsds,from (28) to (29), we have (30)-hU∫t-hUt-htαTt,sΘ2iαt,sds≤-hUhU-htxt-ht-xt-hU∫t-hUt-htxsdshU-htxt-∫t-hUt-htxsdsT·Θ2ixt-ht-xt-hU∫t-hUt-htxsdshU-htxt-∫t-hUt-htxsds-3hUhU-hthU-ht-xt-ht-xt-hU+2∫t-hUt-htxsdshU-ht∫t-hUt-htxsds-2∫t-hUt-ht∫stxududs+2hU-ht∫t-httxsds-hU-ht2∫t-httxsds+∫t-hUt-htxsds+2∫t-htt∫stxududsTΘ2ihU2·hU-ht-xt-ht-xt-hU+2∫t-hUt-htxsdshU-ht∫t-hUt-htxsds-2∫t-hUt-ht∫stxududs+2hU-ht∫t-httxsds-hU-ht2∫t-httxsds+∫t-hUt-htxsds+2∫t-htt∫stxududs=-hUhU-htζTt·Λ2htTΩ2iΛ2htζt,where (31)Λ2ht=e2-e3,e7,hU-hte1-e7,hU-ht-e2-e3+2e7,hU-hte7-2e9,-hU-ht2e6+e7+2e9T,Ω2i=Θ2i03n03n3Θ2ihU2.From (22) to (30), by utilizing reciprocally convex optimization approach [9], it can be confirmed that (32)LV4xt,i-2hU∫t-httxTsP1x˙sds-2hU∫t-hUt-htxTsP2x˙sds+∫t-hUtαTt,s∑j=1sπijNjαt,sds+∫t-httβTt,s∑j=1sπijGjβt,sds≤ζTtΞ4-Λ3htTΩ3iΛ3htζt,where Λ3[h(t)]=Λ1[h(t)]T,Λ2[h(t)]TT and Ω3i=Ω1iSSTΩ2i>0.
From (13) to (32), an upper bound of LV(xt,i) with the addition of (21) can be represented as (33)LVxt,i+ζTtΞ5ζt-2hU∫t-httxTsP1x˙sds-2hU∫t-hUt-htxTsP2x˙sds≤ζTt·Ξ1i+Ξ2i+Ξ3iht+Ξ4+Ξ5︸Σiht-Λ3htTΩ3iΛ3htζt.By utilizing Lemma 3, the following inequality (34)ζTtΣiht-Λ3htTΩ3iΛ3htζt<0subject to 0=Υiζ(t) is equivalent to (35)Υi⊥TΣiht-Λ3htTΩ3iΛ3htΥi⊥<0.By Lemma 4, condition (35) can be casted into the following inequality with an appropriate dimension Ψ: (36)Υi⊥TΣihtΥi⊥+SymΥi⊥TΛ3htTΨΨTΨ-Ω3i<0.It should be noted that inequality (36) is affinely dependent on h(t). Therefore, if inequalities (12) hold for k=1,2, then inequality (36) is satisfied for 0≤h(t)≤hU. Furthermore, one can see that Ω3i>0 holds if inequalities (12) are satisfied. Therefore, if condition (12) holds, then there exists a sufficiently small positive scalar ɛ such that LV(xt,i)<-ɛx(t)2. Thus, by using the similar method in [33] and Definition 1, system (1) is stochastically stable. This completes our proof.In many cases, the information about an upper bound ofh˙(t) is unknown. For this case, based on the result of Theorem 5, the corresponding stability condition will be presented in Corollary 6. In Corollary 6, for simplicity of matrix notations, some of vectors and matrices are redefined as (37)Θ~1i=Qaug1-1hU∑j=1sπijNj,Ω~1i=Θ~1i03n03n3Θ~1ihU2,Ω~3i=Ω~1iSSTΩ2i,Σ~iht=Ξ1i+Ξ2i+Ξ4+Ξ5,Φ~iht=Υi⊥TΣ~ihtΥi⊥+SymΥi⊥TΛ3ihtTΨ.Except the above notations, all the notations defined in (11) will be used in Corollary 6. Now, the following result is given by Corollary 6.Corollary 6.
For a given scalarhU>0, system (1) is stochastically stable for 0≤h(t)≤hU if there exist positive definite matrices Ri∈R4n×4n, Ni∈R3n×3n, and Q∈R3n×3n, any matrices S∈R6n×6n and Ψ∈R12n×8n, and any symmetric matrices P1∈Rn×n and P2∈Rn×n satisfying the following LMIs for all r(t)=i, i∈S: (38)Φ~i,kΨTΨ-Ω~3i<0,k=1,2,where {Φ~i,k}k=12 means the two vertices of Φ~i[h(t)] with the bounds of 0≤h(t)≤hM. That is, Φ~i,1=Φ~i[h(t)=0] and Φ~i,2=Φ~i[h(t)=hU].Proof.
Let us choose LKF as(39)Vxt,i=ηTtRiηt+∫t-hUtαTt,sNiαt,sds+∫t-hUt∫stαTt,uQαt,ududs.The proof of Corollary 6 is very similar to the proof of Theorem 5. Thus, it is omitted. This completes our proof.Remark 7.
Theorem5 and Corollary 6 are derived based on the result of [25]. LKFs V5(xt,i), V6(xt,i) of Theorem  1 in [25] are not included in this paper. Instead, an upper bound of LV4(xt,i) is derived by utilizing Wirtinger-based integral inequality.Remark 8.
Unlike the previous results [12–14], the integral terms ∫t-h(t)tx(s)ds, ∫t-hUt-h(t)x(s)ds, ∫t-h(t)t∫stx(u)duds, and ∫t-hUt-h(t)∫stx(u)duds which were utilized as elements of augmented vector ζ(t) are not multiplied by 1/h(t) or 1/(hU-ht). As shown in [10], the appearance of the terms (1/h(t))∫t-h(t)tx(s)ds and (1/(hU-h(t)))∫t-h(t)tx(s)ds is unavoidable in utilizing Wirtinger-based integral inequality. However, with the terms (1/ht)∫t-h(t)tx(s)ds and (1/(hU-ht))∫t-hUt-h(t)x(s)ds as elements of augmented vector, the derivation of L{ηT(t)Riη(t)}+L{∫t-hUtαT(t,s)Niα(t,s)ds} is more difficult than the case of the terms ∫t-h(t)tx(s)ds and ∫t-h(t)tx(s)ds as elements of augmented vector. In this paper, with the process shown in (26), the utilized integral terms in augmented vector are not multiplied by 1/h(t) or 1/(hU-ht).


## 4. Numerical Examples

In this section, three numerical examples are introduced to show the improvements of the proposed methods. In the examples, MATLAB, YALMIP, and SeDuMi 1.3 are used to solve LMI problems.Example 1.
Consider Markovian jump system (1) with the parameters(40)A1=-3.48880.8057-0.6451-3.2684,A2=-2.48980.28951.3396-0.0211,Ad1=-0.8620-1.2919-0.6841-2.0729,Ad2=-2.83060.4978-0.8436-1.0115.In Table 1, when π22=-0.8 and hD=0, the obtained maximum delay bounds by Theorem 5 are compared with some recent results and [25] under some various π11. From Table 1, one can see that Theorem 5 significantly improves the feasible region of stability, which shows the advantages of the proposed Theorem 5.Table 1
Maximum delay boundshU with hD=0 and various π11 (Example 1).
Methods
π
11
=
-
0.1
π
11
=
-
0.5
π
11
=
-
0.8
π
11
=
-
1
[18]
0.6797
0.5794
0.5562
0.5465
[19] (m=5)
0.8232
0.7327
0.7039
0.6934
[22] (m=2)
1.2550
0.8816
0.8065
0.7783
[25]
1.2132
0.9797
0.9345
0.8986
Theorem5
1.3954
1.1138
1.0566
1.0367
∗
m is delay-partitioning number.Example 2.
Consider Markovian jump system (1) where (41)A1=-2.30.81.0-2.9,A2=-1.90.20.6-0.8,Ad1=0.81.20.7-3.5,Ad2=-1.3-2.60.5-1.4.When hD is unknown and π22=-0.8, in Table 2, maximum delay bounds obtained by Corollary 6 are compared with those of [18, 21, 25]. Table 2 shows the less conservatism of Corollary 6.Table 2
Maximum delay boundshU with unknown hD and various π11 (Example 2).
Methods
π
11
=
-
0.1
π
11
=
-
0.5
π
11
=
-
0.8
π
11
=
-
1
[18]
0.271
0.271
0.271
0.271
[21]
0.500
0.496
0.493
0.492
[25]
0.6003
0.5909
0.5862
0.5836
Corollary6
0.6209
0.6166
0.6152
0.6146Example 3.
Consider Markovian jump system (1) with the parameters (42)A1=-200-0.9,A2=-10-1-1,Ad1=-10.50.1-1,Ad2=-100.1-1.In Table 3, the results of maximum delay bounds hU obtained by Theorem 5 with hD=0.9, π22=-0.8, and various π11 are listed and some recent results [18, 25, 26] are also listed. The results in Table 3 also show that Theorem 5 provides larger delay bound than those of very recent results such as [26].Table 3
Maximum delay boundshU with hD=0.9 and various π11 (Example 3).
Methods
π
11
=
-
0.1
π
11
=
-
0.5
π
11
=
-
0.8
π
11
=
-
1
[18]
1.0224
1.0148
1.0141
1.0130
[26]
1.3671
1.3565
1.3541
1.3535
[25]
1.7858
1.7006
1.6803
1.6713
Theorem5
1.8270
1.7320
1.7093
1.6999Example 4.
Consider Markovian jump system (1) with the parameters (43)A1=-3.50.8-0.6-3.3,A2=-2.50.31.4-0.1,Ad1=-0.9-1.3-0.7-2.1,Ad2=-2.80.5-0.8-1.0.In Table 4, when hD=0 and π22=-0.8, maximum delay bounds hU obtained by Theorem 5 are listed and compared with those of [20, 23, 26] for various π11. From the result of Table 4, the superiority of Theorem 5 can be verified.Table 4
Maximum delay boundshU with hD=0 and various π11 (Example 4).
Methods
π
11
=
-
0.4
π
11
=
-
0.55
π
11
=
-
0.7
π
11
=
-
0.85
π
11
=
-
1.00
[20]
0.6708
0.5894
0.5768
0.5675
0.5603
[23]
0.6322
0.6120
0.5981
0.5881
0.5805
[25]
1.0328
0.9933
0.9681
0.9523
0.9429
Theorem5
1.1826
1.1335
1.1016
1.0799
1.0650


## 5. Conclusion

In this paper, further improved results on stability for Markovian jump systems with time-varying delays were proposed in Theorem5 and Corollary 6. With simple LKFs comparing with [25], it was shown that from three numerical examples, all the results obtained by Theorem 5 and Corollary 6 are larger than those of [25] by applying Wirtinger-based integral inequality and some new techniques to LV4(xt,i). With the ideas proposed in this paper, stability and stabilization for various systems such as multiagent systems, complex networks, and neural networks will be conducted in future works.



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*Source: 148565-2015-10-19.xml*