# Reachable Set Estimation for Uncertain Markovian Jump Systems with Time-Varying Delay and Disturbances

Authors: Shouwei Zhou; Jiangliu Gu; Changchun Shen; Min Jiang
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we are concerned with the problem of reachable set estimation for uncertain Markovian jump systems with time-varying delays and disturbances. The main consideration is to find a proper method to obtain the no-ellipsoidal bound of the reachable set of Markovian jump system as small as possible. Based on an augmented Lyapunov–Krasovskii functional, by dividing the time-varying delay into two nonuniform subintervals, more general delay-dependent stability criteria for the existence of a desired ellipsoid are derived. An optimized integral inequality which is based on distinguished Wirtinger integral inequality and reciprocally convex combination inequality is used to deal with the integral terms. Finally, numerical examples are presented to demonstrate the effectiveness of the theoretical results.


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## Body

## 1. Introduction

The reachable set for a dynamic system with disturbances is a set that bounds the state trajectories starting from the origin by inputs with peak value. In practice and engineering applications, many dynamical systems may cause abrupt variations in their structure due to stochastic failures or repairs of the components, changes in the interconnections of subsystems, sudden environment changes, and so on. Markovian jump systems, modeled by a set of subsystems with transitions among the models determined by a Markov chain taking values in a finite set, have appealed to a lot of researchers in the control community. In the past few decades, the Markovian jump systems have been extensively studied (see [1–3] and the references therein).For the bound of reachable sets for linear systems without any time delay, we can find a well-known result which has been formulated in terms of linear matrix inequality (LMI) [4], and it is widely used to design control systems that have saturating actuators [5, 6]. However, time-delay phenomenon is frequently encountered in many practical systems, such as biological systems, chemical systems, hydraulic systems, and electrical networks. In recent years, the problem of reachable set estimation for time-delay systems has attracted much attention. Then, an increasing number of researchers have devoted their efforts to the problem of reachable set estimation [7–11]. In [7], a delay-dependent condition for an ellipsoid bounding the set of reachable states was presented by using the Lyapunov–Razumikhin function and the S-procedure. Five nonconvex scalar parameters have to be treated as tuning parameters to find the smallest possible ellipsoid. Based on a relaxed Lyapunov–Krasovskii function, the delay-dependent and delay-rate-dependent conditions for the existence of a desired ellipsoid are obtained [8]. Chen and Zhong [10] studied the reachable set of neutral systems with perturbations and uncertainties via novel inequality. In [11], the time-varying delay is split into two nonuniform subintervals based on the Lyapunov–Krasovskii functional, and using the well-known Wirtinger integral inequality and reciprocating convex combination inequality, the RSE boundary of the time-delay system is obtained.However, the Markov jump system is different from the general time-delay system. It is a random system with multiple modes. The jump transfer between the various modes of the system is determined by a set of Markov chains. In practical application, the equation of state of the system often has some randomness. On the one hand, the stability and stability of Markov jump system have been widely studied [12–16]. On the other hand, there are few studies on the reachable set estimation problem of the Markov jump system (see [17–20] and the references therein). Therefore, this paper studies the reachable set estimation of uncertain Markov jump system.Besides, in real life, parameter uncertainty is inevitable in the mathematical modeling due to the failure or maintenance of parts, external perturbations, parameter fluctuations, data errors and the change of connection mode of subsystems, etc. Therefore, inspired by the issues discussed above, the problem of reachable set estimation for uncertain Markov jump systems with time-varying delays and disturbance is investigated in this study. The interval is divided by time-varying delay split based on Lyapunov–Krasovskii functional, and the partial integral term of derivative of Lyapunov function is optimized by using Wirtinger-based inequality and reciprocating convex matrix inequality. Finally, numerical examples are given to illustrate the validity of the results.Notations: the notations used throughout the paper are fairly standard. The superscript “T” stands for matrix transposition; ℜn denotes the n-dimensional Euclidean space; the notation P>0 means that P is a positive definite matrix; In and 0n×n represent identity matrix and zero matrix with dimension n, respectively; and symX11=X11+X11T. In symmetric block matrices, we use an asterisk (∗) to represent a term which is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.


## 2. Problem Statement

Consider the following Markov jump systems with uncertainties:(1)x˙t=At,rt+ΔAt,rttxt+Bt,rt+ΔBt,rttxt−ht+Dt,rt+ΔDt,rttωt,xt≡0,∀t∈−h2,0,where xt∈ℜn is the state vector and ωt∈ℜm is the disturbance which satisfies(2)ωTtωt≤ωm2≤1.The discrete time-varying delayht>0 satisfies(3)0≤h1≤ht≤h2,where h1, h2 are constants and hm=γh1+1−γh2, 0≤γ≤1. rt,t≥0 is a Markovian process taking values on the probability space in a finite state ℘=1,2,…,N with generator Λ=λij,i,j∈℘ given by(4)Prrt+Δ=j|rt=i=λijΔ+oΔ,j≠i,1+λijΔ+oΔ,j=i,where Δ>0, limΔ⟶0+oΔ/Δ=0, λij≥0, for j≠i is the transition probability from mode i at time t to mode j at time t+Δ, λii=−∑j=1,j≠iNλij. At,rt, Bt,rt, and Dt,rt are known constant matrices of the Markovian process. For notational simplicity, when t,rt=i,i∈℘, the matrix At,rt will be represented by Ai, and the other symbols are similarly defined.Since the state transition probability of the Markovian jump process considered in this paper is partially known, the transition probability matrix of Markovian jumping processΛ is defined as(5)Λ=λ11?⋯λ1N?λ22⋯λ2N⋮⋮⋱⋮λN1?⋯λNN,where ? represents the unknown transition rate. For notational clarity, ∀i∈℘, and the set Ui=Uki∪Uuki with(6)Uki≜j:λijis known forj∈℘,Uuki≜j:λijis unknown forj∈℘.Moreover, ifUki≠∅, it is further described as Uki=k1i,k2i,…,kmi, where m is a non-negation integer with 1≤m≤N and kji∈Z+, 1≤kji≤N,j=1,2,…,m represent the known element of the ith row and jth column in the state transition probability matrix Λ.Besides,ΔAi, ΔBi, and ΔDi are the parametric uncertainties in system (1), which are assumed to be in the following form:(7)ΔAi,ΔBi,ΔDi=LiKitEia,Eib,Eid,where Kit is an unknown real and possibly time-varying matrix with Lebesgue measurable elements satisfying(8)KiTtKit≤I,and Li, Eia, Eib, and Eid are known real constant matrices which characterize how the uncertainty enters the nominal matrices Ai, Bi, and Di.Before proceeding further, system (1) can be written as(9)x˙t=Aixt+Bixt−ht+Diωt+Liu,z=Eiaxt+Eibxt−ht+Eidωt,with the constraint: u=Kitz.We further have(10)uTu≤Eiaxt+Eibxt−ht+EidωtT⋅Eiaxt+Eibxt−ht+Eidωt.For the sake of brevity,xt is used to represent the solution of the system under initial conditions xt,t0,x0, and xt,t satisfies the initial condition x0,r0. And its weak infinitesimal generator, acting on function V, is defined in [21].(11)LVxt,t,i=limΔ⟶0+1ΔεVxt+Δ,t+Δ,rt+Δ|xt,rt=i−Vxt,t,i.A reachable set that bounds the state of system (1) is defined by(12)ℜx=xt∈ℜn|xt,ωtsatisfy1and2.Based on the ideas proposed in [4], this reachable set estimation problem can be transformed into the problem of finding an ellipsoid to bound the ℜx. We will bound ℜx by an ellipsoid of the form(13)ℑP≜ξt∈ℜn:ξTtPξt≤1;P>0.Before proceeding further, we will state well-known lemmas.Lemma 1 (see [4]).
LetVt,x0=0 and ωTtωt≤ωm2; if(14)LVt,xt+αVt,xt−βωTtωt≤0,α>0,β>0,then we have Vt,xt≤β/αωm2 for ∀t≥0.Lemma 2 (see [22]).
For any positive definite matrixM∈ℝn×n, scalar h2>h1≥0, and vector function ω:h1,h2↦ℜn such that the integrations concerned are well defined,(15)−h2−h1∫t−h2t−h1ωTsMωsds≤−∫t−h2t−h1ωsdsTM∫t−h2t−h1ωsds.Lemma 3 (see [23, 24]).
For given a matrixR>0, the following inequality holds for all continuously differentiable functions ωt in ω:a,b↦ℝn:(16)∫abω˙TsRω˙sds≥1b−aΩ1TRΩ1+3b−aΩ2TRΩ2+5b−aΩ3TRΩ3,where(17)Ω1=ωb−ωa,Ω2=ωb+ωa−2b−a∫abωsds,Ω3=ωb−ωa+6b−a∫abωsds−12b−a2∫ab∫ubωsdsdu.Lemma 4 (reciprocally convex combination inequality [25]).
For all vectorsξ∈ℜn, the function(18)Hα,Q=1αξTW1TQW1ξ+11−αξTW2TQW2ξ,where α∈0,1,W1,W2 and Q are matrices with appropriate dimensions. Then, the following inequality holds:(19)minα∈0,1Hα,Q≥ξTW1W2TQX∗QW1W2ξ,if there exists a matrix X such that QX∗Q>0.Lemma 5 (Schur complement [26]).
Given constant symmetric matrices∑1,∑2,∑3, where ∑1=∑1T and ∑2=∑2T>0, then ∑1+∑3T∑2−1∑3<0 holds if and only if(20)∑1∑3T∑3−∑2<0or−∑2∑3∑3T∑1<0.


## 3. Main Results

In this section, some delay-dependent criteria for the existence of ellipsoidℑP bounding the states of system (1) will be obtained. The notations for some matrices are defined in Appendixes.Theorem 1.
Consider the uncertain Markov jump system (1) with constraints (2) and (3); if there exist real matrices Pi=Pi11Pi12Pi13Pi14Pi15∗Pi22Pi23Pi24Pi25∗∗Pi33Pi34Pi35∗∗∗Pi44Pi45∗∗∗∗Pi55>0, Wi>0i=1,2,…,N, P˜1,i>0, Q1>0, Q2>0, Q3>0 and Ri>0i=1,…,6, R˜5=R50003R50005R5, R˜6=R60003R60005R6, any matrices Mii=1,2,X=X11X12X13X21X22X23X31X32X33, Y=Y11Y12Y13Y21Y22Y23Y31Y32Y33 with appropriate dimension such that R˜5X∗R˜5>0, R˜6Y∗R˜6>0, Pi≥P˜1,i0000∗0000∗∗000∗∗∗00∗∗∗∗0, scalar ε>0 satisfying the following matrix inequality:(21)Ψi+Φ1<0,Ψ^i+Φ2<0,Ψ^i+Φ3<0,(22)Ψi+Π1<0,Ψ^i+Π2<0,(23)Pj−Wi≤0,j∈Uuki,j≠i,(24)Pj−Wi≥0,j∈Uuki,j=i,where Ψi, Ψ^i, Φ1, Φ2, Φ3, Π1, and Π2, are defined in Appendix A. Then, the reachable sets of system (1) with (2) and (3) are bounded by boundaries ∩i=1NℑP˜1,i, which is defined in (12).Proof.
We choose the following Lyapunov–Krasovskii functional candidate as follows:(25)Vt,xt=∑i=15Vit,xt,where(26)V1t,xt=ηTtPi11Pi12Pi13Pi14∗Pi22Pi23Pi24∗∗Pi33Pi34∗∗∗Pi44ηt,with(27)ηTt=xTt∫t−h1txsdsT∫t−hmt−h1xsdsT∫t−h2t−hmxsdsT,V2t,xt=∫t−h1teαs−txTsQ1xsds+∫t−hmt−h1eαs−txTsQ2xsds+∫t−h2t−hmeαs−txTsQ3xsds,V3t,xt=h1∫−h10∫t+θteαs−txTsR1xsdsdθ+hm−h1∫−hm−h1∫t+θteαs−txTsR2xsdsdθ+h2−hm∫−h2−hm∫t+θteαs−txTsR3xsdsdθ,V4t,xt=h1∫−h10∫t+θteαs−tx˙TsR4x˙sdsdθ,V5t,xt=hm−h1∫−hm−h1∫t+θteαs−tx˙TsR5x˙sdsdθ+h2−hm∫−h2−hm∫t+θteαs−tx˙TsR6x˙sdsdθ,Pi=Pi,j5×5,hm=γh1+1−γh2,0≤γ≤1.
Taking derivative ofVt,xt along the trajectories of system (1), we can obtain the following:(28)LV=LV1+LV2+LV3+LV4+LV5,where(29)LV1t,xt=2ηTtPiη˙t+ηTt∑j=1NλijPjηt=2ηTtuTPi11Pi12Pi13Pi14Pi15∗Pi22Pi23Pi24Pi25∗∗Pi33Pi34Pi35∗∗∗Pi44Pi45∗∗∗∗Pi55Aixt+Bixt−ht+Diωt+Liuxt−xt−h1xt−h1−xt−hmxt−hm−xt−h2Aixt−x˙t+Bixt−ht+Diωt+Liu−2Aixt+Bixt−ht+Diωt+Liuxt−xt−h1xt−h1−xt−hmxt−hm−xt−h2TPi15Pi25Pi35Pi45u+ηTt∑j=1NλijPjηt.
Taking into account the situation that the information of transition probabilities are not accessible completely, due to∑j=1Nλij=0, the following equations hold for arbitrary appropriate matrices Wi=WiT are satisfied(30)−ηTt∑j=1NλijWiηt=0.
Hence,(31)LV1t,xt=2ηTtuTPi11Pi12Pi13Pi14Pi15∗Pi22Pi23Pi24Pi25∗∗Pi33Pi34Pi35∗∗∗Pi44Pi45∗∗∗∗Pi55Aixt+Bixt−ht+DiωtLiuxt−xt−h1xt−h1−xt−hmxt−hm−xt−h2Aixt−x˙t+Bixt−ht+Diωt+Liu,−2Aixt+Bixt−ht+Diωt+Liuxt−xt−h1xt−h1−xt−hmxt−hm−xt−h2TPi15Pi25Pi35Pi45u+ηTt∑j=1NλijPj−Wiηt,LV2t,xt=xTtQ1xt+e−αh1xTt−h1Q2−Q1xt−h1+e−αhmxTt−hmQ3−Q2⋅xt−hm−e−αh2xTt−h2Q3xt−h2−αV2,LV3t,xt=xTth12R1+hm−h12R2+h2−hm2R3xt−h1∫t−h1teαs−txTsR1xsds−hm−h1∫t−hmt−h1eαs−txTsR2xsds−h2−hm∫t−h2t−hmeαs−txTsR3xsds−αV3.
Based on Lemma2, LV3t,xt can be rewritten as(32)LV3t,xt≤xTth12R1+hm−h12R2+h2−hm2R3xt−e−αh1∫t−h1txsdsTR1⋅∫t−h1txsds−e−αhm∫t−hmt−h1xsdsTR2∫t−hmt−h1xsds−e−αh2∫t−h2t−hmxsdsTR3∫t−h2t−hmxsds−αV3.
So, according to Lemma3, we have(33)LV4t,xt≤x˙Tth12R4x˙t−e−αh1⋅h1∫t−h1tx˙TsR4x˙sds−αV4≤x˙Tth12R4x˙t−e−αh1xt−xt−h1TR4xt−xt−h1+xt+xt−h1−2h1∫t−h1txsdsT3R4xt+xt−h1−2h1∫t−h1txsds+xt−xt−h1+6h1∫t−h1txsds−12h12∫t−h1t∫utxsdsdu5R4⋅xt−xt−h1+6h1∫t−h1txsds−12h12∫t−h1t∫utxsdsdu−αV4,(34)LV5t,xt≤x˙Tthm−h12R5+h2−hm2R6x˙t−e−αhmhm−h1∫t−hmt−h1x˙TsR5x˙sds−e−αh2h2−hm∫t−h2t−hmx˙TsR6x˙sds−αV5.
Whenht∈h1,hm, based on the Lemmas 3 and 4, we have(35)−hm−h1∫t−hmt−h1x˙TsR5x˙sds=−hm−h1∫t−hmt−htx˙TsR5x˙sds−hm−h1∫t−htt−h1x˙TsR5x˙sds≤−ξ1Tt1hm−ht/hm−h1Γ1TR˜5Γ1+1ht−h1/hm−h1Γ2TR˜5Γ2ξ1t≤−ξ1TtΓ1Γ2TR˜5X∗R˜5Γ1Γ2ξ1t,where(36)ξ1t=xTtx˙TtxTt−h1xTt−htxTt−hmxTt−h2∫t−h1txsdsT∫t−hmt−h1xsdsT∫t−h2t−hmxsdsT1hm−ht∫t−hmt−htxsdsT1ht−h1∫t−htt−h1xsdsT1hm−ht2∫t−hmt−ht∫ut−htxsdsduT1ht−h12∫t−htt−h1∫ut−h1xsdsduT1h12∫t−h1t∫utxsdsduT1h2−hm2∫t−h2t−hm∫ut−hmxsdsduTωTtuTT,χ1=000I−I000000000000,χ2=000II0000−2I0000000,χ3=000I−I00006I0−12I00000,χ4=00I−I0000000000000,χ5=00II000000−2I000000,χ6=00I−I0000006I0−12I0000,Γ1=χ1χ2χ3,Γ2=χ4χ5χ6,R˜5=R50003R50005R5,R˜5X∗R˜5>0.
Using Lemma3, we further have(37)−h2−hm∫t−h2t−hmx˙TsR6x˙sds≤−9xTt−hmR6xt−hm−6xTt−hmR6xt−h2+9xTt−h2R6xt−h2+48h2−hm⋅xTt−hmR6∫t−h2t−hmxsds−120h2−hm2xTt−hmR6∫t−h2t−hm∫ut−hmxsdsdu−72h2−hm⋅xTt−h2R6∫t−h2t−hmxsds+120h2−hm2xTt−h2R6∫t−h2t−hm∫ut−hmxsdsdu+192h2−hm2⋅∫t−h2t−hmxsdsTR6∫t−h2t−hmxsds−720h2−hm3∫t−h2t−hmxsdsTR6∫t−h2t−hm∫ut−hmxsdsdu+720h2−hm4∫t−h2t−hm∫ut−hmxsdsduTR6∫t−h2t−hm∫ut−hmxsdsdu.
Thus,LV5t,xt can be acquired as(38)LV5t,xt≤x˙Tthm−h12R5+h2−hm2R6x˙t−e−αhmξ1TtΓ1Γ2TR˜5X∗R˜5Γ1Γ2ξ1t−e−αh29xTt−hmR6xt−hm−6xTt−hmR6xt−h2+9xTt−h2R6xt−h2+48h2−hmxTt−hmR6∫t−h2t−hmxsds−120h2−hm2xTt−hmR6∫t−h2t−hm∫ut−hmxsdsdu−72h2−hmxTt−h2R6∫t−h2t−hmxsds+120h2−hm2xTt−h2R6∫t−h2t−hm∫ut−hmxsdsdu+192h2−hm2∫t−h2t−hmxsdsTR6∫t−h2t−hmxsds−720h2−hm3∫t−h2t−hmxsdsTR6⋅∫t−h2t−hm∫ut−hmxsdsdu+720h2−hm4∫t−h2t−hm∫ut−hmxsdsduT⋅R6∫t−h2t−hm∫ut−hmxsdsdu−αV5.
Meanwhile, for any matricesM1 and M2 with appropriate dimension, the following equation is true:(39)2xTtM1+x˙TtM2Aixt−x˙t+Bixt−ht+Diωt+Liu=0.
Combining equations (25)–(37), we can obtain(40)LVt,xt+αVt,xt−αωm2ωTtωt≤ξ1TtΨi+Φ1ξ1t+ηTt∑j∈UukiλijPj−Wiηt,where Ψi, Φ1 are the same as defined in Theorem 1. Using the S-procedure in [4], one can see that this condition is implied by the existence of a non-negative scalar ε>0 such that(41)LVt,xt+αVt,xt−αωm2ωTtωt≤ξ1TtΨi+Φ1ξ1t+ηTt∑j∈UukiλijPj−Wiηt+εEiaxt+Eibxt−ht+EidωtTEiaxt+Eibxt−ht+Eidωt−uTu<0,for all ξ1t≠0. By using Lemma 5, the matrix inequalities (21), (23), and (24) imply (41).
Similarly, whenht=h1 or ht=hm, inequality (35) can be reduced to the following two inequalities, respectively. When ht=h1, we obtain the following inequality:(42)−hm−h1∫t−hmt−h1x˙TsR5x˙sds=−hm−ht∫t−hmt−htx˙TsR5x˙sds≤−xt−ht−xt−hmTR5xt−ht−xt−hm+xt−ht+xt−hm−2hm−ht∫t−hmt−htxsdsT3R5xt−ht+xt−hm−2hm−ht∫t−hmt−htxsds+xt−ht−xt−hm+6hm−ht∫t−hmt−htxsds−12hm−ht2∫t−hmt−ht∫ut−htxsdsduT⋅5R5xt−ht−xt−hm+6hm−ht∫t−hmt−htxsds−12hm−ht2∫t−hmt−ht∫ut−htxsdsdu.
Combining Equations (25)–(39) and (42), we can obtain(43)LVt,xt+αVt,xt−αωm2ωTtωt≤ξ1TtΨ^i+Φ2ξ1t+ηTt∑j∈UukiλijPj−Wiηt,where Ψ^i, Φ2 are the same as defined in Theorem 1. Using the S-procedure in [4], one can see that this condition is implied by the existence of a non-negative scalar ε>0 such that(44)LVt,xt+αVt,xt−αωm2ωTtωt≤ξ1TtΨ^i+Φ2ξ1t+ηTt∑j∈UukiλijPj−Wiηt+εEiaxt+Eibxt−ht+EidωtTEiaxt+Eibxt−ht+Eidωt−uTu<0,for all ξ1t≠0. By using Lemma 5, the matrix inequalities (21), (23), and (24) imply (44).
Whenht=hm, we get(45)−hm−h1∫t−hmt−h1x˙TsR5x˙sds=−ht−h1∫t−htt−h1x˙TsR5x˙sds≤−xt−h1−xt−htTR5xt−h1−xt−ht+xt−h1+xt−ht−2ht−h1∫t−htt−h1xsds3R5xt−h1+xt−ht−2ht−h1∫t−htt−h1xsds+xt−h1−xt−ht+6ht−h1∫t−htt−h1xsds−12ht−h12∫t−htt−h1∫ut−h1xsdsduT⋅5R5xt−h1−xt−ht+6ht−h1∫t−htt−h1xsds−12ht−h12∫t−htt−h1∫ut−h1xsdsdu.
Combining Equations (25)–(39) and (45), we can obtain(46)LVt,xt+αVt,xt−αωm2ωTtωt≤ξ1TtΨ^i+Φ3ξ1t+ηTt∑j∈UukiλijPj−Wiηt,where Ψ^i, Φ3 are the same as defined in Theorem 1. Using the S-procedure in [4], one can see that this condition is implied by the existence of a non-negative scalar ε>0 such that(47)LVt,xt+αVt,xt−αωm2ωTtωt≤ξ1TtΨ^i+Φ3ξ1t+ηTt∑j∈UukiλijPj−Wiηt+εEiaxt+Eibxt−ht+EidωtTEiaxt+Eibxt−ht+Eidωt−uTu<0,for all ξ1t≠0. By using Lemma 5, the matrix inequalities (21), (23), and (24) imply (47).
It should be noted thatΨi+Φ1<0, Ψ^i+Φ2<0, Ψ^i+Φ3<0 according to inequalities (21), (23), and (24), which implies that LVt,xt+αVt,xt−α/ωm2ωTtωt<0.
Whenht∈hm,h2, the last two integral terms of LV5t,xt are handled in the following way based on Lemmas 3 and 4:(48)−hm−h1∫t−hmt−h1x˙TsR5x˙sds≤−9xTt−h1R5xt−h1−6xTt−h1R5xt−hm+9xTt−hmR5xt−hm+48hm−h1xTt−h1R6∫t−hmt−h1xsds−120hm−h12xTt−h1R5∫t−hmt−h1∫ut−h1xsdsdu−72hm−h1xTt−hmR5∫t−hmt−h1xsds+120hm−h12xTt−hmR5∫t−hmt−h1∫ut−h1xsdsdu+192hm−h12∫t−hmt−h1xsdsTR5∫t−hmt−h1xsds−720hm−h13∫t−hmt−h1xsdsTR5⋅∫t−hmt−h1∫ut−h1xsdsdu+720hm−h14∫t−hmt−h1∫ut−h1xsdsduTR5∫t−hmt−h1∫ut−h1xsdsdu.
Using the same method to deal with the following integral inequality, we further have(49)−h2−hm∫t−h2t−hmx˙TsR6x˙sds=−h2−hm∫t−h2t−htx˙TsR6x˙sds−h2−hm∫t−htt−hmx˙TsR6x˙sds≤−ξ2Tt1h2−ht/h2−hmΓ3TR˜6Γ3+1ht−hm/h2−hmΓ4TR˜6Γ4ξ2t≤−ξ2TtΓ3Γ4TR˜6Y∗R˜6Γ3Γ4ξ2t,where(50)ξ2t=xTtx˙TtxTt−h1xTt−htxTt−hmxTt−h2∫t−h1txsdsT∫t−hmt−h1xsdsT∫t−h2t−hmxsdsT1h2−ht∫t−h2t−htxsdsT1ht−hm∫t−htt−hmxsdsT1h2−ht2∫t−h2t−ht∫ut−htxsdsduT1ht−hm2∫t−htt−hm∫ut−hmxsdsduT1h12∫t−h1t∫utxsdsduT1hm−h12∫t−hmt−h1∫ut−h1xsdsduTωTtuTT,χ7=000I0−I00000000000,χ8=000I0I000−2I0000000,χ9=000I0−I0006I0−12I00000,χ10=000−II000000000000,χ11=000II00000−2I000000,χ12=000−II000006I0−12I0000,Γ3=χ7χ8χ9,Γ4=χ10χ11χ12,R˜6=R60003R60005R6,R˜6Y∗R˜6>0.
Then,LV5t,xt can be rewritten as(51)LV5t,xt≤x˙Tthm−h12R5+h2−hm2R6x˙t−e−αhm9xTt−h1R5xt−h1−6xTt−h1R5xt−hm+9xTt−hmR5xt−hm+48hm−h1xTt−h1R5⋅∫t−hmt−h1xsds−120hm−h12xTt−h1R5∫t−hmt−h1∫ut−h1xsdsdu−72hm−h1⋅xTt−hmR5∫t−hmt−h1xsds+120hm−h12xTt−hmR5∫t−hmt−h1∫ut−h1xsdsdu+192hm−h12∫t−hmt−h1xsdsTR5∫t−hmt−h1xsds−720hm−h13∫t−hmt−h1xsdsTR5∫t−hmt−h1∫ut−h1xsdsdu+720hm−h14∫t−hmt−h1∫ut−h1xsdsduTR5⋅∫t−hmt−h1∫ut−h1xsdsdu−e−αh2ξ2TtΓ3Γ4TR˜6Y∗R˜6Γ3Γ4ξ2t−αV5.
Based on Equations (25)–(34), (39), and (51), we have(52)LVt,xt+αVt,xt−αωm2ωTtωt≤ξ2TtΨi+Π1ξ2t+ηTt∑j∈UukiλijPj−Wiηt,where Ψi, Π1 are the same as defined in Theorem 1. Using the S-procedure in [4], one can see that this condition is implied by the existence of a non-negative scalar ε>0 such that(53)LVt,xt+αVt,xt−αωm2ωTtωt≤ξ2TtΨi+Π1ξ2t+ηTt∑j∈UukiλijPj−Wiηt+εEiaxt+Eibxt−ht+EidωtTEiaxt+Eibxt−ht+Eidωt−uTu<0,for all ξ2t≠0. By using Lemma 5, the matrix inequalities (22)–(24) imply (53).
Whenht=h2, inequality (49) can simplify the following equation:(54)−h2−hm∫t−h2t−hmx˙TsR6x˙sds=−ht−hm∫t−htt−hmx˙TsR6x˙sds≤−xt−hm−xt−htTR6xt−hm−xt−ht+xt−hm+xt−ht−2ht−hm∫t−htt−hmxsdsT3R6xt−hm+xt−ht−2ht−hm∫t−htt−hmxsds+xt−hm−xt−ht+6ht−hm∫t−htt−hmxsds−12ht−hm2∫t−htt−hm∫ut−hmxsdsT⋅5R6xt−hm−xt−ht+6ht−hm∫t−htt−hmxsds−12ht−hm2∫t−htt−hm∫ut−hmxsdsT.
Based on Equations (25)–(34), (39), and (54), we have(55)LVt,xt+αVt,xt−αωm2ωTtωt≤ξ2TtΨ^i+Π2ξ2t+ηTt∑j∈UukiλijPj−Wiηt,where Ψ^i, Π2 are the same as defined in Theorem 1. Using the S-procedure in [4], one can see that this condition is implied by the existence of a non-negative scalar ε>0 such that(56)LVt,xt+αVt,xt−αωm2ωTtωt≤ξ2TtΨ^i+Π2ξ2t+ηTt∑j∈UukiλijPj−Wiηt+εEiaxt+Eibxt−ht+EidωtTEiaxt+Eibxt−ht+Eidωt−uTu<0,for all ξ2t≠0. By using Lemma 5, the matrix inequalities (22)–(24) imply (56).
It should be noted thatΨi+Π1<0, Ψ^i+Π2<0 according to inequalities (22)–(24), which implies that LVt,xt+αVt,xt−α/ωm2ωTtωt<0.
In conclusion,LVt,xt+αVt,xt−α/ωm2ωTtωt<0 is true on the basis of equations (21)–(24). Since Pi=pi,j5×5, there is a positive matrix P˜1,i∈ℝn×n such that(57)Pi≥P˜1,i000000000000000000000000.
We further have the inequalityxTtP˜1,ixt≤V1t,xt≤Vt,xt≤1 when ht∈h1,h2. Hence, Vt,xt≤1 is true by using Lemma 1. This completes the proof.Remark 1.
We choose an augmented Lyapunov–Krasovskii functionalV1t,xt=ηTtPrtηt to establish more general delay-dependent conditions, where Prt is a fifth-order matrix and ηt is a five-dimensional column vector. Therefore, the reachable set estimation criteria can utilize more information on state variables via using these augmented variables in the Lyapunov–Krasovskii functional.Remark 2.
Parameterγ is used to divide the time-delay interval into two subintervals in this paper. ht is the time-varying delay satisfying 0≤h1≤ht≤h2. Generally, the authors divide the time-delay interval into two or more equal subintervals in previous studies to get less conservative stability criteria. Different from them, the time-delay interval h1,h2 in our study is partitioned into h1,hm and hm,h2 by introducing an adjustable parameter γ, where hm=γh1+1−γh2,0≤γ≤1; when γ=0.5, the two subintervals are equal as those in the literatures. Moreover, the integral interval is decomposed in the same way to estimate the bounds of integral terms more exactly.Remark 3.
An optimized integral inequality is provided to deal with the integral term−hm−h1∫t−hmt−h1x˙TsR5x˙sds and −h2−hm∫t−h2t−hmx˙TsR6x˙sds. Recently, the reciprocally convex combination approach [18, 25, 27] has been widely used to deduce the results. According to the applications shown in these literatures, it is easy to see that the results based on this method are less conservative than the existing ones. Therefore, we adopt the distinguished Wirtinger integral inequality [23] together with the reciprocally convex combination inequality to handle these two integral terms to get more general reachable set estimation criteria.Corollary 1.
Consider the uncertain Markov jump system (1)–(4) with all elements completely known in transition rate matrix (5); if there exist real matrices Pi=Pi11Pi12Pi13Pi14Pi15∗Pi22Pi23Pi24Pi25∗∗Pi33Pi34Pi35∗∗∗Pi44Pi45∗∗∗∗Pi55>0, Wi>0i=1,2,…,N, P˜1,i>0, Q1>0, Q2>0, Q3>0 and Ri>0i=1,…,6, R˜5=R50003R50005R5, R˜6=R60003R60005R6, any matrices Mii=1,2,X=X11X12X13X21X22X23X31X32X33, Y=Y11Y12Y13Y21Y22Y23Y31Y32Y33 with appropriate dimension such that R˜5X∗R˜5>0, R˜6Y∗R˜6>0, Pi≥P˜1,i0000∗0000∗∗000∗∗∗00∗∗∗00 a scalar ε>0 such that the LMIs equations (21)–(22) hold, which all elements Ψi, Ψ^i, Φ1, Φ2, Φ3, Π1, and Π2 are defined in Appendix B.Corollary 2.
Consider the uncertain Markov jump system (1)–(4) with all elements completely unknown in transition rate matrix (5); if there exist real matrices Pi=Pi11Pi12Pi13Pi14Pi15∗Pi22Pi23Pi24Pi25∗∗Pi33Pi34Pi35∗∗∗Pi44Pi45∗∗∗∗Pi55>0, Wi>0i=1,2,…,N, P˜1,i>0, Q1>0, Q2>0, Q3>0, Ri>0i=1,…,6, R˜5=R50003R50005R5, R˜6=R60003R60005R6, any matrices Mii=1,2,X=X11X12X13X21X22X23X31X32X33, Y=Y11Y12Y13Y21Y22Y23Y31Y32Y33 with appropriate dimension such that R˜5X∗R˜5>0, R˜6Y∗R˜6>0, Pi≥P˜1,i0000∗0000∗∗000∗∗∗00∗∗∗00, scalar ε>0 such that the LMIs equations (21) and (22) hold, where Ψi, Ψ^i, Φ1, Φ2, Φ3, Π1, and Π2 are defined in Appendix C.
Next, we consider the following Markov jump system with disturbances(58)x˙t=At,rtxt+Bt,rtxt−ht+Dt,rtωt,xt≡0,∀t∈−h2,0.
Theorem2 is a result for the ellipsoidal bound of a reachable set for a Markov jump time-delayed system (58) having constraints in equations (2) and (3).Theorem 2.
Consider the Markov jump system (56) with all elements partly known in transition rate matrix (5); if there exist real matrices Q1>0, Q2>0, Q3>0, Pi=Pi11Pi12Pi13Pi14∗Pi22Pi23Pi24∗∗Pi33Pi34∗∗∗Pi44>0, Wi>0, P˜1,1>0 and Ri>0i=1,…,6, R˜5=R50003R50005R5, R˜6=R60003R60005R6, any matrices Mii=1,2, X=X11X12X13X21X22X23X31X32X33, Y=Y11Y12Y13Y21Y22Y23Y31Y32Y33 with appropriate dimension such that R˜5X∗R˜5>0, R˜6Y∗R˜6>0, Pi≥P˜1,1000∗000∗∗00∗∗∗0, satisfying the following matrix inequalities:(59)Ψi+Φ1<0,Ψ^i+Φ2<0,Ψ^i+Φ3<0,(60)Ψi+Π1<0,Ψ^i+Π2<0,(61)Pj−Wi≤0,j∈Uuki,j≠i,(62)Pj−Wi≥0,j∈Uuki,j=i,where all the elements are defined in Appendix D.
Then, the reachable sets of system (58) having the constraints (2) and (3) are bounded by a boundary ℑP defined in equation (13).Proof .
of Theorem 2 Following a similar line as in the proof of Theorem1, one can simply obtain this theorem. This completes our proof.Corollary 3.
Consider the Markov jump system (58) with all elements completely known in transition rate matrix (5); if there exist real matrices Pi=Pi11Pi12Pi13Pi14∗Pi22Pi23Pi24∗∗Pi33Pi34∗∗∗Pi44>0, P˜1,1>0, Q1>0, Q2>0, Q3>0 and Ri>0i=1,…,6, R˜5=R50003R50005R5, R˜6=R60003R60005R6, any matrices Mii=1,2, X=X11X12X13X21X22X23X31X32X33, Y=Y11Y12Y13Y21Y22Y23Y31Y32Y33 with appropriate dimension such that R˜5X∗R˜5>0, R˜6Y∗R˜6>0, Pi≥P˜1,1000∗000∗∗00∗∗∗0 , satisfying the following matrix inequalities:(63)Ψi+Φ1<0,Ψ^i+Φ2<0,Ψ^i+Φ3<0,(64)Ψi+Π1<0,Ψ^i+Π2<0,where all elements are defined in Appendix E.Corollary 4.
Consider the Markov jump system (58) with all elements completely unknown in transition rate matrix (5); if there exist real matrices Pi=Pi11Pi12Pi13Pi14∗Pi22Pi23Pi24∗∗Pi33Pi34∗∗∗Pi44>0, P˜1,1>0, Q1>0, Q2>0, Q3>0 and Ri>0i=1,…,6, R˜5=R50003R50005R5, R˜6=R60003R60005R6, any matrices Mii=1,2, X=X11X12X13X21X22X23X31X32X33, Y=Y11Y12Y13Y21Y22Y23Y31Y32Y33 with appropriate dimension such that R˜5X∗R˜5>0, R˜6Y∗R˜6>0, Pi≥P˜1,1000∗000∗∗00∗∗∗0, satisfying the following matrix inequalities:(65)Ψi+Φ1<0,Ψ^i+Φ2<0,Ψ^i+Φ3<0,(66)Ψi+Π1<0,Ψ^i+Π2<0,where all elements are defined in Appendix F.Remark 4.
Since this is an estimation problem, our aim is to find an ellipsoid as small as possible to bound the reachable sets of system (1).We use a simple approximation as that in [7]. That is, maximize δ>0 subject to P˜1,1≥δI, which can be transformed into the following optimization problem:(67)minδ¯=1δs.t.iδ¯IIIP˜1,1≥0,iiEqs.13−16or48−51,


## 4. Numerical Examples

In this section, two examples are used to demonstrate the effectiveness and correctness of the main results derived above.Example 1.
Consider the following uncertain time-delayed system (58) which has been studied in [9, 11]:(68)A1=−200−0.7,B1=−10−1−0.9,D1=−0.51,A2=−200−1.1,B2=−10−1−1.1,D2=D1,ωTtωt≤ωm2=1,h1=0.
The transition rate matrixΛ is considered as in the following three cases.Case 1. The transition rate matrix Λ is completely known, which is considered as Λ=−0.60.60.2−0.2.Case 2. The transition rate matrix Λ is partly known, which is considered as Λ=−0.60.6??.Case 3. The transition rate matrix Λ is completely unknown, which is considered as Λ=????.
By solving the optimization problem (59)–(62), the allowable minimum values of δ¯ obtained by different methods for 0≤ht≤0.7 and 0≤ht≤0.75 are listed in Table 1. According the Table 1, it is inescapably clear that our results decrease the size of the ellipsoid significantly.
Furthermore, for the parameters listed above, leth2=0.75, α=0.7, and we can get the following feasible solutions by Theorem 2 in this paper. In this example, when the transition probability matrix is completely unknown (at this point, the state trajectory is general switching system, not the system studied in this paper), it is a general switching system, such as [9, 11]. But when the Λ is partly unknown or completely known, it is the Markov jump system. And there are two elliptical boundaries in the picture. So, we choose the smaller one as the reachable set boundary. Due to the limitation of the length of this paper, we just show some of them here, and the reachable set is ∩i=12ℑP˜1,i.
By using Theorem2 and solving the problem (59)–(62) in Case 2, we can obtain(69)Pi11=2.4425−0.4772−0.47721.0098,Pi44=24.88670.34330.343324.9046,Q3=175.92610.44670.4467173.8194,R6=0.0493−0.0035−0.00350.0365,X11=−50.20370.08910.0893−50.3448,Y11=−52.98100.2084−0.2832−52.6961,P˜1,1=1.6344−0.3169−0.31690.6826.
By using Corollary3 and solving the problems (63) and (64) in Case 1, we can obtain(70)Pi11=4.3786−0.5368−0.53682.1556,Pi44=72.98590.36830.368372.5227,Q3=474.53810.79490.7949467.5645,P˜1,1=2.9222−0.3632−0.36321.4403.
By using Corollary4 and solving the problems (65) and (66) in Case 3, we can obtain(71)Pi11=4.9321−0.4496−0.44962.5786,Pi44=99.74400.39440.394499.3030,Q3=678.15731.03811.0381672.4074,R3=0.1860−0.0122−0.01220.1374,X11=−195.76360.49190.4548−197.1652,Y11=−203.48270.7165−0.8392−202.9476,P˜1,1=3.2976−0.3031−0.30311.7280.
Figure1 is the plot of the state trajectory of system (56). Figures 2–4 are the plots of the ellipsoidal sets ℑ defined in equation (13), which are obtained in Theorem 2 and Corollaries 3 and 4 when h2=0.75, respectively.Table 1
The minimum sizes of ellipsoidal boundδ¯ obtained by different methods.
Method[28][22][29][30][13][12][11] cor 3.3[11] cor 3.4Corollary40≤ht≤0.72.971.91511.891.791.381.151.10961.13750.92790≤ht≤0.753.342.31992.282.031.271.341.15511.20640.5982Figure 1
The time responses of state variablext of Markov jump system (1) for Case 1.Figure 2
The ellipsoidal boundℑ and state trajectory of system (58) for Case 1.Figure 3
The ellipsoidal boundℑ and state trajectory of system (58) for Case 2.Figure 4
The ellipsoidal boundℑ and state trajectory of system (58) for Case 3.Example 2.
Consider the uncertainMJSs with time-varying delays and disturbances:(72)x˙t=At,rt+ΔAt,rttxt+Bt,rt+ΔBt,rttxt−ht+Dt,rt+ΔDt,rttωt,where(73)A1=−500−6,B1=−1.60−1.8−1.5,D1=0.5−1,L1=0.1000.1,E13=0,A2=−400−5,B2=−20−0.9−1.2,D2=−0.51,L2=000−0.1,E23=0,E11=000−0.1,E12=−0.1000.1,E21=0000.1,E22=0.1000.1.
By using Corollary1 and solving the optimization problem (72) in Case 1, we can get that the minimization of δ¯ is 0.0785 (the short half axis length of the ellipsoid is 0.2802) when α=0.7 and the corresponding feasible matrices are given as P˜1,1=21.3278−0.3828−0.382812.7572,P˜1,2=10.29380.90420.904210.2938. The reachable sets of system (72) in Case 1 are bounded by a intersection of two ellipsoids: ∩i=12ℑP˜1,i, which is depicted in Figure 5.
By using Theorem1 and solving the optimization problem (72) in Case 2, we can get that the minimization of δ¯ is 0.0524 (the short half axis length of the ellipsoid is 0.2288) when α=0.7 and the corresponding feasible matrices are given as P˜1,1=32.80012.01902.019019.3954,P˜1,2=15.87312.55652.556513.5311. The reachable sets of system (72) in Case 1 are bounded by a intersection of two ellipsoids: ∩i=12ℑP˜1,i, which is depicted in Figure 6.
By using Corollary2 and solving the optimization problem (72) in Case 3, we can get that the minimization of δ¯ is 0.0598 (the short half axis length of the ellipsoid is 0.2445) when α=0.7 and the corresponding feasible matrices are given as P˜1,1=32.80284.44964.449617.9617,P˜1,2=21.13765.71355.713514.6146. The reachable sets of system (72) in Case 1 are bounded by a intersection of two ellipsoids: ∩i=12ℑP˜1,i, which is depicted in Figure 7.
Figure8 is the plot of the time responses of state variable for system (72). Figures 5–7 are the plots of the ellipsoidal sets ℑ defined in equation (13), which are obtained in Theorem 1 and Corollaries 1 and 2 when h2=0.75, respectively.Figure 5
The ellipsoidal boundℑ and state trajectory of system(72) for Case 1.Figure 6
The ellipsoidal boundℑ and state trajectory of system (72) for Case 2.Figure 7
The ellipsoidal boundℑ and state trajectory of system (72) for Case 3.Figure 8
The time responses of state variablext of Markov jump system (72) for Case 1.


## 5. Conclusion

The reachable set bounding for uncertain Markov jump systems with time-varying delays and disturbances has been investigated in our study. We partition the time-varying delay into two nonuniform subintervals and consider the delay in these two cases separately. Furthermore, some new reachable set estimation conditions are derived in terms of linear matrix inequalities by constructing a novel augmented Lyapunov–Krasovskii functional, combining with optimized integral inequality which is based on distinguished Wirtinger integral inequality and reciprocally convex combination inequality. Finally, the feasibility and the comparisons with recent results obtained in the latest literatures are shown through numerical examples.



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