# Exponential Passification of Markovian Jump Nonlinear Systems with Partially Known Transition Rates

Authors: Mengzhuo Luo; Shouming Zhong
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The problems of delay-dependent exponential passivity analysis and exponential passification of uncertain Markovian jump systems (MJSs) with partially known transition rates are investigated. In the deterministic model, the time-varying delay is in a given range and the uncertainties are assumed to be norm bounded. With constructing appropriate Lyapunov-Krasovskii functional (LKF) combining with Jensen’s inequality and the free-weighting matrix method, delay-dependent exponential passification conditions are obtained in terms of linear matrix inequalities (LMI). Based on the condition, desired state-feedback controllers are designed, which guarantee that the closed-loop MJS is exponentially passive. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach.


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## Body

## 1. Introduction

In recent years, more and more attention has been devoted to the Markovian jump systems since they are introduced by Krasovskii and Lidskii [1]. It is known that systems with Markovian jump parameters are a set of systems with transition among the models governed by a Markov chain taking values in a finite set. They have the character of stochastic hybrid systems with two components in the state. The first one refers to the mode which is described by a continuous-time finite-state Markov process, and the second one refers to the state which is represented by a system of differential equations. Markovian jump systems have got the virtue of modeling the abrupt phenomena such as random failures and repairs of the components changes in the interconnections of subsystems, sudden environment changes, and so forth, which often takes place in many dynamical systems [2–4]. So due to extensive applications of such systems in manufacturing systems, power systems, communication systems, and network-based control systems, recently, many works have been reported about MJSs, which including filtering problems [5–7], stability analysis problems [8–12], and control problems [13–20], and so forth.However, the aforementioned references almost considered that the transition probabilities are known exactly. In some practical applications, the mode information is transmitted through unreliable networks, it may be lost or observed simultaneously. That means the systems mode is neither totally accessible or inaccessible. So the ideal assumption on the transition probabilities inevitably limits the application of the traditional Markovian jump systems theory. Therefore, whether in theory or in practice, it is necessary to further consider more general systems with partially mode information [21–27].Recently, the passivity problems for a variety of practical systems have been attracting renewing attention [28–31]. The passivity theory was first proposed in the circuit analysis [32] so it has played an efficient role in both electrical network and nonlinear control systems. The main point of passivity theory is that the passive properties of system can keep the system internal stability. Thus, the passivity theory provides a nice tool for analyzing the stability of a nonlinear system, and the passivity analysis has received a lot of attention and has found applications in diverse areas such as signal processing, complexity, chaos control and synchronization, and fuzzy control [33–38]. In [33] authors dealt with global robust passivity analysis for stochastic interval neural networks with interval time varying delays and Markovian jumping parameter; in [34] both delay-independent and delay-dependent stochastic passivity conditions are presented for uncertain neural networks; in [35–37] authors discussed the robust passivity and passification of Markovian jump systems and fuzzy time-delay systems; in [38], the exponential passivity of neural networks with time-varying are studied and the results are extended to two types of uncertainties.In practice, input delays are often encountered in control systems because of the transmission of measurement information. Especially, in networked control systems, sensors controllers, and plants are often connected by a net medium hence it is quite meaningful to study the effect of the input delay in the design of controllers. However, to the best of the authors’ knowledge providing less conservative delay-dependent exponential passification criteria for uncertain MJS with input delays and partially known transition rates to desired performance are still open problems.Motivated by this observation, in this paper, we study the exponential passification problem of nonlinear Markovian jump systems with partially known transition rates, including state and input delays, the aim of this problem is to design a controller such that the resulting closed-loop systems satisfy a certain passivity performance index. Comparing with the large amount of the literature on the analysis of stability of Markovian jump systems, passivity analysis and passification for these systems have many obvious advantages. Thus, research in this area should be of both theoretical and practical importance, which motivates us to carry out the present work. Based on the LKF theory and the free-weighting matrix method, some desired exponentially passification controllers are designed, which guarantee that the closed-loop MJS is exponential passive. Finally, a numerical example is used to illustrate the designed method.Notations.The notations are quite standard. Throughout this letterℝn and ℝn×m denote, resp., the n-dimensioned Euclidean space and the set of all n×m real matrices. The notation X≥Y (resp., X>Y) means that X and Y are symmetric matrices, and that X-Y is positive semidefinitive (resp., positive definite). ∥·∥ is the Euclidean norm in ℝn. I is the identity matrix with compatible dimension. If A is a matrix, λmax(A) (respective λmin(A)) means the largest (respective smallest) eigervalue of A. Moreover, let (Ω,𝔽,(𝔽t)t≥0,ℙ) be a complete probability space with a filteration. (𝔽t)t≥0 satisfies the usual conditions (i.e, the filtration contains all P-null sets and is right continuous).  E{·} stands for the mathematical expectation operator with respect to the given probability measure. Denote by L𝔽02([-τ̅2,0]:ℝn) the family of all 𝔽0 measurable C([-τ̅2,0]:ℝn)-valued random variables φ={φ(s):-τ̅2≤s≤0} such that sup-τ̅2≤s≤0E∥φ(s)∥2<∞. The asterisk * in a matrix is used to denote term that is induced by symmetry. Matrices, if not explicitly specified, are assumed to have appropriate dimensions. Sometimes, the arguments of function will be omitted in the analysis when no confusion can be arised.


## 2. Problem Formulation and Preliminaries

Consider the following uncertain MJS with time-varying delays(2.1)ẋ(t)=A(t,rt)x(t)+Ad(t,rt)x(t-τ(t,rt))+B1(t,rt)u(t)+E1(t,rt)u(t-τ(t,rt))+D0(rt)f(x(t),rt)+D1(rt)ω(t).(2.2)z(t)=C(t,rt)x(t)+Cd(t,rt)x(t-τ(t,rt))+B2(t,rt)u(t)+E2(t,rt)u(t-τ(t,rt))+D2(rt)ω(t),
Here x(t)∈ℝn is the state vector, u(t)∈ℝp is the control input, z(t)∈ℝq is the control output, and ω(t)∈ℝl is the exogenous disturbance input which belongs to 𝕃2[0,∞],{rt,t≥0} is a homogenous finite-state Markov process with right continuous trajectories, which takes value in a finite-state space S={1,2,…,N} with generator Π={πij},i,j∈S and has the mode transition probabilities(2.3)Pr⁡{rt+Δt=j∣rt=i}={πijΔt+o(Δt)i≠j,1+πiiΔt+o(Δt)i=j,
where Δt>0,limΔt→0(o(Δt)/Δt)=0,πij is the transition rete from i to j, and(2.4)πii=-∑j≠iπij,πij≥0,j≠i.For notational simplicity, whichrt=i,i∈S, the matrices A(t,rt),Ad(t,rt),B1(t,rt),E1(t,rt),C(t,rt), Cd(t,rt), B2(t,rt), E2(t,rt),  D0(rt), D1(rt), and D2(rt) will be described by Ai(t),Adi(t),B1i(t),E1i(t), Ci(t),Cdi(t),B2i(t), E2i(t), D0i,D1i, and D2i. We denote that(2.5)Ai(t)=Ai+ΔAi(t),Adi(t)=Adi+ΔAdi(t),B1i(t)=B1i+ΔB1i(t),E1i(t)=E1i+ΔE1i(t),Ci(t)=Ci+ΔCi(t),Cdi(t)=Cdi+ΔCdi(t),B2i(t)=B2i+ΔB2i(t),E2i(t)=E2i+ΔE2i(t),
where Ai,Adi,B1i,E1i,Ci,Cdi,B2i,E2i, and D0i,D1i,D2i are known constant matrices with appropriate dimensions. In this paper, the transition rates of Markov chain are partially known, that is, some elements in matrix Π are unknown. We denote that(2.6)Ikni={j:ifπijisknow}Iuki={j:ifπijisunknow}
moreover, if Ikni≠∅, it is further described as Ikni={k1i,k2i,…,kmi},1≤m≤N-2.Remark 2.1.
kli∈N+,l∈{1,2,…,m} represents the index of the lth known element in the ith row of transition rate matrix. The case m=N-1 is excluded, which means if we have only one unknown element, one can naturally calculate it from the known elements in each row and the transition rate matrix property.Now the mode-dependent state-feedback controller is taken to be as follows:(2.7)u(t)=Kix(t),
then, the closed-loop MJS can be represented as(2.8)ẋ(t)=(Ai(t)+B1i(t)Ki)x(t)+(Adi(t)+E1i(t)Ki)x(t-τi(t))+D0if(x(t),i)+D1iω(t),z(t)=(Ci(t)+B2i(t)Ki)x(t)+(Cdi(t)+E2i(t)Ki)x(t-τi(t))+D2iω(t).Before proceeding further, we will introduce the following assumptions, definition and some lemmas which will be used in the next section.Assumption 1.
The uncertain parameters are assumed to be of the form:(2.9)(ΔAi(t)ΔAdi(t)ΔB1i(t)ΔE1i(t)ΔCi(t)ΔCdi(t)ΔB2i(t)ΔE2i(t))=(T1iT2i)Fi(t)(N1iN2iN3iN4i),
where T1i,T2i, and Nki,k=1,2,3,4,i∈S are known real constant matrices with appropriate dimensions and Fi(t),foralli∈S, are unknown time-varying matrix functions satisfying
(2.10)FiT(t)Fi(t)≤I.Remark 2.2.
It is assumed that all the elementsFi(t),foralli∈S, are Lebesgue measurable. The matrices ΔAi(t),ΔAdi(t),ΔB1i(t),ΔE1i(t),ΔCi(t),ΔCdi(t),ΔB2i(t), and ΔE2i(t) are said to be admissible if and only if both (2.9) and (2.10) hold. The parameter uncertainty structure as in Assumption 1 is an extension of the so-called matching condition, which has been widely used in the problems of control and robust filtering of uncertain linear systems.Assumption 2.
The time-varying delayτi(t) satisfies  0≤τ1i≤τi(t)≤τ2i,τ̇i(t)≤μi, with τ1i,τ2i,  and  μi being real constant scalars for each for all i∈S.Assumption 3.
For a fixed system modert=i∈S, there exists a know real constant mode-dependent matrix Γi=diag(k1i,k2i,…,kni)>0 such that the nonlinear vector function f(·,·) satisfy the following conditions:
(2.11)fT(x(t),i)(f(x(t),i)-Γix(t))≤0.Definition 2.3 (see [39]).
The MJS (2.8) is said to be passive if there exists a constant δ such that
(2.12)2E{∫0TzT(t)ω(t)dt}≥δ
holds for all T≥0.Definition 2.4.
The MJS (2.8) is said to be exponentially passive from input ω(t) to output z(t), if there exists an exponential Lyapunov function (or called the exponential storage funtion)  V defined on ℝn, and positive scalars ρ, γ such that for all ω(t), all initial conditions x(0), all t≥0, the following inequality holds:
(2.13)LV(xt,rt)+ρV(xt,rt)-γωT(t)ω(t)≤2zT(t)ω(t).Remark 2.5.
From Definition2.4, if ρ=0, then the MJS in the form (2.8) is passive, in other words, exponential passivity implies passivity. It follows from (2.13) that
(2.14)2E{∫0TzT(t)ω(t)dt}≥-E{V(x0)}-γE{∫0TωT(t)ω(t)dt}=δ.
Then from Definition 2.3, we can see that MJS (2.8) is passive. But the converse does not necessarily hold, that is, we can not obtain the exponential passive if systems are passive.Lemma 2.6 (see [36]).
LetQ(x)=QT(x),R(x)=RT(x), and S(x) depend affinely on x. Then the following linear matrix inequality:
(2.15)[Q(x)S(x)ST(x)R(x)]>0
holds if and only if one of the following conditions holds:(1)
R(x)>0,Q(x)-S(x)R-1(x)ST(x)>0;(2)
Q(x)>0,R(x)-ST(x)Q-1(x)S(x)>0.Lemma 2.7 (see [40]).
LetA,D,S,F, and P be real matrices of appropriate dimensions with P>0 and F satisfy FT(t)F(t)≤I. Then the following statement holds.(1)
For any scalarε>0(2.16)DFS+(DFS)T≤ε-1DDT+εSTS.(2)
For any vectorsx and y with appropriate dimensions(2.17)2xTADy≤xTAPATx+yTDTP-1Dy.Lemma 2.8 (see [41]).
LetA,X be real matrices with appropriate dimensions. Then there exist a matrix P=PT>0 such that PAT+AP+X<0, if and only if, there exists a scalar ε>0 and Z such that
(2.18)[-Z-ZTZTAT+PZT*-ε-1P+X0**-εP]<0.


## 3. Main Results

### 3.1. Exponential Passivity Analysis

In this section, we assumed the transition rates are partially known and given the state-feedback controller gain matrixKi,i∈S, at first, we will present a sufficient condition, which guarantees the MLS (2.8) is exponential passive.Theorem 3.1.
Given the state-feedback controller gain matrixKi, the uncertain MJS (2.8) is exponentially passive in the sense of expectation if there exists positive definite matrices Pi,Qi,Q̅1,Q̅2,Q̅3,Q*,Z1,Z2,  positive scalars γ,ε1i,ε2i, and for any matrices Gi,Mi,Ri,Ui,Vi,Hi with appropriate dimensions such that the following matrices inequalities hold for all i=1,2,…,N:
(3.1)(Ωi,9×91Λ1Λ2Λ3Λ4kΛ5kΛ6Λ7*-Z200002τ̅2Z2T1iε2i0**-Z200000***-Z20000****-Z2000*****-Z200******-ε2i0*******-ε2i)<0k=1,2.Case 1.
Ifπii∈Ikni(3.2)(πiiQi-Q*Qj*-Qj)∀j∈Iuki<0,(3.3)((1+∑j∈Ikniπij)(πiiQi-Q*)πik1iQk1i⋯πikmiQkmi*-Qk1i00**⋱0***-Qkmi)<0,(3.4)PiAi+AiTPi+Pj<0∀j∈Iuki.Case 2.
Ifπii∈Iuki(3.5)Qj-Q*>0∀j∈Iuki,j=i,(3.6)Qj-Q*<0∀j∈Iuki,j≠i,(3.7)((-1-∑j∈Ikniπij)Q*πik1iQk1i⋯πikmiQkmi*-Qk1i00**⋱0***-Qkmi)<0,(3.8)PiAi+AiTPi+Pj>0∀j∈Iuki,j=i,(3.9)PiAi+AiTPi+Pj<0∀j∈Iuki,j≠i,where(3.10)Ωi,111=Pi((1+∑j∈Ikniπij)Ai+B1iKi)+((1+∑j∈Ikniπij)Ai+B1iKi)TPi+∑j∈IkniπijPj+Qi+τ2iQ*+Q̅1+Q̅2+Q̅3+(τ2i-τ1i)Z1+G1iT+G1i,Ωi,121=-G1i+G2iT+M1i,Ωi,131=R1i+G3iT-M1i,Ωi,141=-R1i+G4iT+U1i+Pi(Adi+E1iKi),Ωi,151=V1i+G5iT-U1i,Ωi,161=-V1i+G6iT+H1i,Ωi,171=G7iT-H1iΩi,181=G8iT+ε1iΓi+PiD0i,Ωi,191=PiD1i-(Ci+B2iKi)T,Ωi,221=-G2iT-G2i+M2iT+M2i-Q̅1,Ωi,231=-G3iT+R2i+M3iT-M2i,Ωi,241=-G4iT-R2i+M4iT+U2i,Ωi,251=-G5iT+V2i+M5iT-U2i,Ωi,261=-G6iT-V2i+M6iT+H2i,Ωi,271=-G7iT+M7iT-H2i,Ωi,281=-G8iT+M8iT,Ωi,291=0,Ωi,331=R3iT+R3i-M3iT-M3i,Ωi,341=R4iT-R3i-M4iT+U3i,Ωi,351=R5iT+V3i-M5iT-U3i,Ωi,361=R6iT-V3i-M6iT+H3i,Ωi,371=R7iT-M7iT-H3i,Ωi,381=R8iT-M8iTΩi,391=0,Ωi,441=-R4iT-R4i+U4iT+U4i-(1-μi)Qi,Ωi,451=-R5iT+V4i+U5iT-U4i,Ωi,461=-R6iT-V4i+U6iT+H4i,Ωi,471=-R7iT+U7iT-H4i,Ωi,481=-R8iT+U8iT,Ωi,491=-(Cdi+E2iKi)T,Ωi,551=V5iT+V5i-U5iT-U5i,Ωi,561=-U6iT-V5i+V6iT+H5i,Ωi,571=V7iT-U7iT-H5i,Ωi,581=V8iT-U8iTΩi,591=0,Ωi,661=-V6iT-V6i+H6iT+H6i-Q̅2,Ωi,671=-V7iT+H7iT-H6i,Ωi,681=-V8iT+H8iT,Ωi,691=0,Ωi,771=-H7iT-H7i-Q̅3,Ωi,781=-H8iT,Ωi,791=0,Ωi,881=-2ε1iI,Ωi,891=0,Ωi,991=-D2i-D2iT-γI,τ̅2=max⁡i∈S{τ2i},τ̲1=min⁡i∈S{τ1i},Λ1=(2τ̅2Z2(Ai+B1iKi),0,0,2τ̅2Z2(Adi+E1iKi),0,0,0,2τ̅2Z2D0i,2τ̅2Z2D1i)T,Λ1(t)=(2τ̅2Z2(Ai(t)+B1i(t)Ki),0,0,2τ̅2Z2(Adi(t)+E1i(t)Ki),0,0,0,2τ̅2Z2D0i,2τ̅2Z2D1i)T,Λ2=τ̅2-τ2iHi,Λ3=τ1iGi,Λ41=τ2i-τ1i2Mi,Λ42=τ2i-τ1i2Ui,Λ51=τ2i-τ1i2Ri,Λ52=τ2i-τ1i2Vi,Λ6=(ε2iT1iTPi,0,0,0,0,0,0,0,-ε2iT2iT)T,Λ7=(N1i+N3iKi,0,0,N2i+N4iKi,0,0,0,0,0)T,Gi=(G1iTG2iTG3iTG4iTG5iTG6iTG7iTG8iT0)TMi=(M1iTM2iTM3iTM4iTM5iTM6iTM7iTM8iT0)T,Ri=(R1iTR2iTR3iTR4iTR5iTR6iTR7iTR8iT0)T,Ui=(U1iTU2iTU3iTU4iTU5iTU6iTU7iTU8iT0)T,Vi=(V1iTV2iTV3iTV4iTV5iTV6iTV7iTV8iT0)T,Hi=(H1iTH2iTH3iTH4iTH5iTH6iTH7iTH8iT0)T,Proof.
First, in order to cast our model involved in the framework of the Markov process, we define a new processxt(s)=x(t+s),s∈[-τ̅2,0], and let L be the weak infinitesimal generator of the random process xt(s),t≥0 and
(3.11)Lv(xt,rt)=lim⁡Δ→0+1Δ{E[v(xt+Δ,rt+Δ)∣xt,rt=i]-v(xt,rt)}.
Now consider the Lyapunov-Krasovskii functional as follows for rt=i,i∈1,2,…,S:
(3.12)v(xt,i)=v1(xt,i)+v2(xt,i)+v3(xt,i)+v4(xt,i)+v5(xt,i),
where
(3.13)v1(xt,i)=xT(t)P(i)x(t),v2(xt,i)=∫t-τi(t)txT(s)Q(i)x(s)ds,v3(xt,i)=∫-τ2i0∫t+θtxT(s)Q*x(s)dsdθ,v4(xt,i)=∫t-τ1itxT(s)Q̅1x(s)ds+∫t-τ2itxT(s)Q̅2x(s)ds+∫t-τ̅2txT(s)Q̅3x(s)d,v5(xt,i)=∫-τ2i-τ1i∫t+θtxT(s)Z1x(s)dsdθ+2∫-τ̅20∫t+θtẋT(s)Z2ẋ(s)dsdθ,
where
(3.14)∑j=1NπijQj≤Q*.
In order to show the exponential passivity of the MJS (2.8) under the given controller gain matrix Ki, we set
(3.15)J*=Lv(xt,i)-γωT(t)ω(t)-2zT(t)ω(t).
Notice that(3.16)Lv1(xt,i)=xT(t)(Pi(Ai(t)+B1i(t)Ki)+(Ai(t)+B1i(t)Ki)TPi)x(t)+xT(t)∑j=1NπijPjx(t)+2xT(t)Pi(Adi(t)+E1i(t)Ki)x(t-τi(t))+2xT(t)PiD0if(x(t),i)+2xT(t)PiD1iω(t),Lv2(xt,i)≤xT(t)Qix(t)-(1-μi)xT(t-τi(t))Qix(t-τi(t))+∫t-τi(t)txT(s)∑j=1NπijQjx(s)ds,Lv3(xt,i)≤τ2ixT(t)Q*x(t)-∫t-τi(t)txT(s)Q*x(s)ds,Lv4(xt,i)=xT(t)(Q̅1+Q̅2+Q̅3)x(t)-xT(t-τ1i)Q̅1x(t-τ1i)-xT(t-τ2i)Q̅2x(t-τ2i)-xT(t-τ̅2)Q̅3x(t-τ̅2),Lv5(xt,i)=(τ2i-τ1i)xT(t)Z1x(t)+2τ̅2ẋT(t)Z2ẋ(t)-∫t-τ2it-τ1ixT(s)Z1x(s)ds-2∫t-τ̅2tẋT(s)Z2ẋ(s)ds=(τ2i-τ1i)xT(t)Z1x(t)+2τ̅2ẋT(t)Z2ẋ(t)-∫t-τ̅2t-τ2iẋT(s)Z2ẋ(s)ds-∫t-τ2it-(τi(t)+τ2i)/2ẋT(s)Z2ẋ(s)ds-∫t-(τi(t)+τ2i)/2t-τi(t)ẋT(s)Z2ẋ(s)ds-∫t-τi(t)t-(τi(t)+τ1i)/2ẋT(s)Z2ẋ(s)ds-∫t-(τi(t)+τ1i)/2t-τ1iẋT(s)Z2ẋ(s)ds-∫t-τ1itẋT(s)Z2ẋ(s)ds-∫t-τ2it-τ1ixT(s)Z1x(s)ds-∫t-τ̅2tẋT(s)Z2ẋ(s)ds.
Then using Newton-Leibniz formula, for any matricesHi,Gi,Mi,Ri,Ui,Vi we have(3.17)2ξT(t)Gi(x(t)-x(t-τ1i)-∫t-τ1itẋ(s)ds)=0,2ξT(t)Mi(x(t-τ1i)-x(t-τ1i+τi(t)2)-∫t-(τ1i+τi(t))/2t-τ1iẋ(s)ds)=0,2ξT(t)Ri(x(t-τ1i+τi(t)2)-x(t-τi(t))-∫t-τi(t)t-(τ1i+τi(t))/2ẋ(s)ds)=0,2ξT(t)Ui(x(t-τi(t))-x(t-τ2i+τi(t)2)-∫t-(τ2i+τi(t))/2t-τi(t)ẋ(s)ds)=0,2ξT(t)Vi(x(t-τ2i+τi(t)2)-x(t-τ2i)-∫t-τ2it-(τ2i+τi(t))/2ẋ(s)ds)=0,2ξT(t)Hi(x(t-τ2i)-x(t-τ̅2)-∫t-τ̅2t-τ2iẋ(s)dt)=0,
where
(3.18)ξT(t)=(xT(t),xT(t-τ1i),xT(t-τi(t)+τ1i2),xT(t-τi(t)),xT(t-τi(t)+τ2i2),xT(t-τ2i)xT(t-τ̅2),fT(x(t),i),ωT(t)(t-τi(t)+τ1i2)).
From the Lemma2.7 (2.2), it is easy to see that(3.19)-2ξT(t)Gi∫t-τ1itẋ(s)ds≤τ1iξT(t)GiZ2-1GiTξ(t)+∫t-τ1itẋT(s)Z2ẋ(s)ds-2ξT(t)Mi∫t-(τ1i+τi(t))/2t-τ1iẋ(s)ds≤τi(t)-τ1i2ξT(t)MiZ2-1MiTξ(t)+∫t-(τ1i+τi(t))/2t-τ1iẋT(s)Z2ẋ(s)ds-2ξT(t)Ri∫t-τi(t)t-(τ1i+τi(t))/2ẋ(s)ds≤τi(t)-τ1i2ξT(t)RiZ2-1RiTξ(t)+∫t-τi(t)t-(τ1i+τi(t))/2ẋT(s)Z2ẋ(s)ds-2ξT(t)Ui∫t-(τ2i+τi(t))/2t-τi(t)ẋ(s)ds≤τ2i-τi(t)2ξT(t)UiZ2-1UiTξ(t)+∫t-(τ2i+τi(t))/2t-τi(t)ẋT(s)Z2ẋ(s)ds-2ξT(t)Vi∫t-τ2it-(τ2i+τi(t))/2ẋ(s)ds≤τ2i-τi(t)2ξT(t)ViZ2-1ViTξ(t)+∫t-τ2it-(τ2i+τi(t))/2ẋT(s)Z2ẋ(s)ds-2ξT(t)Hi∫t-τ̅2t-τ2iẋ(s)ds≤(τ̅2-τ2i)ξT(t)HiZ2-1HiTξ(t)+∫t-τ̅2t-τ2iẋT(s)Z2ẋ(s)ds.
Now by Assumption3, it can be deduced that for any positive scalar ε1i,i∈1,2,…,S,(3.20)2ε1ifT(x(t),i)(Γix(t)-f(x(t),i))≥0.
Then from the above discussion, we can see that
(3.21)J*≤ξT(t)(τi(t)-τ1iτ2i-τ1iΦ1(t)+τ2i-τi(t)τ2i-τ1iΦ2(t))ξ(t)-∫t-τ2it-τ1ixT(s)Z1x(s)ds-∫t-τ̅2tẋT(s)Z2ẋ(s)ds,Φ1(t)=Ωi,9×9(t)+Λ1(t)Z2-1Λ1(t)T+(τ̅2-τ2i)HiZ2-1HiT+τ1iGiZ2-1GiT+τ2i-τ1i2(MiZ2-1MiT+RiZ2-1RiT),Φ2(t)=Ωi,9×9(t)+Λ1(t)Z2-1Λ1(t)T+(τ̅2-τ2i)HiZ2-1HiT+τ1iGiZ2-1GiT+τ2i-τ1i2(UiZ2-1UiT+ViZ2-1ViT),
where
(3.22)Ωi,11(t)=Pi(Ai(t)+B1i(t)Ki)+(Ai(t)+B1i(t)Ki)TPi+∑j=1NπijPj+Qi+τ2iQ*+Q̅1+Q̅2+Q̅3+(τ2i-τ1i)Z1+G1iT+G1i,Ωi,14(t)=Pi(Adi(t)+E1i(t)Ki)-R1i+G4iT+U1iΩi,19(t)=PiD1i-(Ci(t)+B2i(t)Ki)T,Ωi,49(t)=-(Cdi(t)+E2i(t)Ki)T
other terms of Ωi,i×j(t) are similar to Ωi,i×j1. In order to get our results, we will describe that the Φ1(t)<0 and Φ2(t)<0. By the Schur complement, Φ1(t)<0 and Φ2(t)<0 under the restriction of (3.14) if and only if
(3.23)(Ωi,9×9Λ1Λ2Λ3Λ4kΛ5k*-Z20000**-Z2000***-Z200****-Z20*****-Z2)k=1,2+(T1iTPi,0n×7n,-T2iT,2τ̅2T1iTZ2,0n×4n)TFi(k)(N1i+N3iKi,0n×2n,N2i+N4iKi,0n×10n)+(N1i+N2iKi,0n×2n,N2i+N4iKi,0n×10n)TFiT(k)(T1iTPi,0n×7n,-T2iT,2τ̅2T1iTZ2,0n×4n)<0,
where Ωi,9×9 is the nominal matrix of Ωi,9×9(t). Then from the Lemma 2.7 (2.1), above matrix inequality holds, which is equivalent to
(3.24)(Ωi,9×9Λ1Λ2Λ3Λ4kΛ5kΛ6Λ7*-Z200002τ̅2Z2T1iε2i0**-Z200000***-Z20000****-Z2000*****-Z200******-ε2i0*******-ε2i)k=1,2<0.Case 1.
Ifπii∈Ikni
then (3.24) is equivalent to(3.25)(Ωi,9×91Λ1Λ2Λ3Λ4kΛ5kΛ6Λ7*-Z200002τ̅2Z2T1iε2i0**-Z200000***-Z20000****-Z2000*****-Z200******-ε2i0*******-ε2i)k=1,2+diag⁡(∑j∈Iukij≠iπijPj,0,…,0︷15)+diag⁡(∑j∈Iukij≠iπij(PiAi+AiTPi),0,…,0︷15)<0.
Obviously, we can see that if (3.1) and (3.4) hold, then Φ1(t)<0 and Φ2(t)<0 under the restriction of (3.14). Next we will further consider the equivalent form of (3.14).∑j=1NπijQj<Q* is equivalent to(3.26)∑j≠ij∈IkniπijQj+∑j≠ij∈IukiπijQj+πiiQi-Q*+∑j∈Ikniπij(πiiQi-Q*)+∑j∈Iukiπij(πiiQi-Q*)<0.
If we have the following matrix inequalities hold, we can have that (3.14) is satisfied
(3.27)(1+∑j∈Ikniπij)(πiiQi-Q*)+∑j≠ij∈IkniπijQj<0,πiiQi-Q*+Qj<0j∈Iuki.
Obviously, (3.27) is equivalent to (3.2) and (3.3) by the Schur complement. Case 2.
Ifπii∈Iuki
then (3.24) is equivalent to(3.28)(Ωi,9×91Λ1Λ2Λ3Λ4kΛ5kΛ6Λ7*-Z200002τ̅2Z2T1iε2i0**-Z200000***-Z20000****-Z2000*****-Z200******-ε2i0*******-ε2i)k=1,2+diag⁡(∑j∈Iukij≠iπijPj,0,…,0︷15)+diag⁡(∑j∈Iukij≠iπij(PiAi+AiTPi),0,…,0︷15)+πiidiag⁡(PiAi+AiTPi+Pi,0,…,0︷15)<0.
Then if (3.1), (3.8), and (3.9) hold, then Φ1(t)<0 and Φ2(t)<0 under the restriction of (3.14), furthermore, with the similar consideration, we can deduce that if (3.5)–(3.7) are established, then (3.14) is founded. So there exists a positive scalar ρ1, then(3.29)J*≤-ρ1‖x(t)‖2-λmin⁡(Z1)∫t-τ̅2t‖x(s)‖2ds-λmin⁡(Z2)∫t-τ̅2t‖ẋ(s)‖2ds.
On the other hand, it is easy to obtain that
(3.30)v(x(t),i)≤‖P‖‖x(t)‖2+(‖Q‖+‖Q*‖+‖Q̅1‖+‖Q̅2‖+‖Q̅3‖+(τ̅2-τ̲1)‖Z1‖)×∫t-τ̅2t‖x(s)‖2ds+2τ̅2‖Z2‖∫t-τ̅2t‖ẋ(s)‖2ds,
where ∥P∥=maxi∈S{∥Pi∥}, ∥Q∥=maxi∈S{∥Qi∥}.Let ρ>0 be sufficiently small such that
(3.31)ρ‖P‖-ρ1<0,ρ(‖Q‖+‖Q*‖+‖Q̅1‖+‖Q̅2‖+‖Q̅3‖+(τ̅2-τ̲1)‖Z1‖)-λmin⁡(Z1)<02τ̅2ρ‖Z2‖-λmin⁡(Z2)<0.,
So, by Definition 2.4, the MJS (2.8) is exponentially passive. This completes the proof.Remark 3.2.
It is easy to derive that the MJS (2.8) is exponential mean square stability with ω(t)=0 if the MJS (2.8) is exponentially passive. Moreover, the result of Theorem 3.1 makes use of the information of the subsystems upper bounds of the time varying delays, which may bring us less conservativeness, and from the free-weighting matrix and Newton-Leibnitz formula, the upper bounds of μi are not restricted to be less than 1 in this paper. Therefore, our result is more natural and reasonable to the Markovian jump systems.Remark 3.3.
In order to obtain the gain matricesKi for convenience in the next section, (3.1) is not LMI, if we substitute ε2i by ε2i-1 and use the Lemma 2.7 (2.1), we can obtain the equivalent form of LMI.


### 3.2. Exponential Passification

In this section, we will determine the feedback controller gain matricesKi,i∈S in (2.7), which guarantee that the closed-loop MJS (2.8) is exponentially passive with partially known transition rates.Theorem 3.4.
Given a positive constantε, there exists a state-feedback controller in the form (2.7) such that the closed-loop MJS (2.8) is exponentially passive if there exist positive definite matrices P̅i,Q̅i,Q̃1,Q̃2,Q̃3,Q̅*,Z̅1,Z̅2, positive scalar ε1i,ε2i, and for any matrices G̅i,M̅i,R̅i,U̅i,V̅i,H̅i,Zii with appropriate dimensions satisfying the following LMIs under the two cases for all i=1,2,…,N. Case 1.
Ifπii∈Ikni(3.32)(πiiQ̅i-Q̅*Q̅j*-Q̅j)∀j∈Iuki<0,(3.33)((1+∑j∈Ikniπij)(πiiQ̅i-Q̅*)πik1iQ̅k1i⋯πikmiQ̅kmi*-Q̅k1i00**⋱0***-Q̅kmi)<0,(3.34)(Ω̅i,9×91Λ̅1Λ̅2Λ̅3Λ̅4kΛ̅5kΛ̅6Λ̅7Λ̅8*-Z̅200002τ̅2T1iε2i00**Θ000000***Θ00000****Θ0000*****Θ000******-ε2i00*******-ε2i0********Λ̅9)<0k=1,2,(3.35)(-Zii-ZiiTZiiTAiT+P̅iZiiT0*-ε-1P̅i0P̅i**-εP̅i0***-P̅j)∀j∈Iuki<0,Case 2.
Ifπii∈Iuki(3.36)Q̅j-Q̅*>0∀j∈Iuki,j=i,(3.37)Q̅j-Q̅*<0∀j∈Iuki,j≠i,(3.38)((-1-∑j∈Ikniπij)Q̅*πik1iQ̅k1i⋯πikmiQ̅kmi*-Q̅k1i00**⋱0***-Q̅kmi)<0,(3.39)(Ω̅i,9×92Λ̅1Λ̅2Λ̅3Λ̅4kΛ̅5kΛ̅6Λ̅7Λ̅8*-Z̅200002τ̅2T1iε2i00**Θ000000***Θ00000****Θ0000*****Θ000******-ε2i00*******-ε2i0********Λ̅9)<0k=1,2,(3.40)(-Zii-ZiiT-ZiiTAiT+P̅iZiiT*-ε-1P̅i-P̅j0**-εP̅i)∀j∈Iukij=i<0,(3.41)(-Zii-ZiiTZiiTAiT+P̅iZiiT0*-ε-1P̅i0P̅i**-εP̅i0***-P̅j)∀j∈Iukij≠i<0,
where
(3.42)Ω̅i,111=((1+∑j∈Ikniπij)AiP̅i+B1iYi)+((1+∑j∈Ikniπij)AiP̅i+B1iYi)T+πiiP̅i+Q̅i+τ2iQ̅*+Q̃1+Q̃2+Q̃3+(τ2i-τ1i)Z̅1+G̅1iT+G̅1i,Ω̅i,112=((1+∑j∈Ikniπij)AiP̅i+B1iYi)+((1+∑j∈Ikniπij)AiP̅i+B1iYi)T+Q̅i+τ2iQ̅*+Q̃1+Q̃2+Q̃3+(τ2i-τ1i)Z̅1+G̅1iT+G̅1i,Ω̅i,121=Ω̅i,122=-G̅1i+G̅2iT+M̅1i,Ω̅i,131=Ω̅i,132=R̅1i+G̅3iT-M̅1i,Ω̅i,141=Ω̅i,142=-R̅1i+G̅4iT+U̅1i+(AdiP̅i+E1iYi)Ω̅i,151=Ω̅i,152=V̅1i+G̅5iT-U̅1i,Ω̅i,161=Ω̅i,162=-V̅1i+G̅6iT+H̅1i,Ω̅i,171=Ω̅i,172=G̅7iT-H̅1i,Ω̅i,181=Ω̅i,182=G̅8iT+P̅iΓi+D0iε̅1i,Ω̅i,191=Ω̅i,192=D1i-(CiP̅i+B2iYi)T,Ω̅i,221=Ω̅i,222=-G̅2iT-G̅2i+M̅2iT+M̅2i-Q̃1,Ω̅i,231=Ω̅i,232=-G̅3iT+R̅2i+M̅3iT-M̅2i,Ω̅i,241=Ω̅i,242=-G̅4iT-R̅2i+M̅4iT+U̅2i,Ω̅i,251=Ω̅i,252=-G̅5iT+V̅2i+M̅5iT-U̅2i,Ω̅i,261=Ω̅i,262=-G̅6iT-V̅2i+M̅6iT+H̅2i,Ω̅i,271=Ω̅i,272=-G̅7iT+M̅7iT-H̅2i,Ω̅i,281=Ω̅i,282=-G̅8iT+M̅8iT,Ω̅i,291=Ω̅i,292=0,Ω̅i,331=Ω̅i,332=R̅3iT+R̅3i-M̅3iT-M̅3i,Ω̅i,341=Ω̅i,342=R̅4iT-R̅3i-M̅4iT+U̅3i,Ω̅i,351=Ω̅i,352=R̅5iT+V̅3i-M̅5iT-U̅3i,Ω̅i,361=Ω̅i,362=R̅6iT-V̅3i-M̅6iT+H̅3i,Ω̅i,371=Ω̅i,372=R̅7iT-M̅7iT-H̅3i,Ω̅i,381=Ω̅i,382=R̅8iT-M̅8iT,Ω̅i,391=Ω̅i,392=0,Ω̅i,441=Ω̅i,442=-R̅4iT-R̅4i+U̅4iT+U̅4i-(1-μi)Q̅i,Ω̅i,451=Ω̅i,452=-R̅5iT+V̅4i+U̅5iT-U̅4i,Ω̅i,461=Ω̅i,462=-R̅6iT-V̅4i+U̅6iT+H̅4i,Ω̅i,471=Ω̅i,472=-R̅7iT+U̅7iT-H̅4i,Ω̅i,481=Ω̅i,482=-R̅8iT+U̅8iT,Ω̅i,491=Ω̅i,492=-(CdiP̅i+E2iYi)T,Ω̅i,551=Ω̅i,552=V̅5iT+V̅5i-U̅5iT-U̅5i,Ω̅i,561=Ω̅i,562=-U̅6iT-V̅5i+V̅6iT+H̅5i,Ω̅i,571=Ω̅i,572=V̅7iT-U̅7iT-H̅5i,Ω̅i,581=Ω̅i,582=V̅8iT-U̅8iTΩ̅i,591=Ω̅i,592=0,Ω̅i,661=Ω̅i,662=-V̅6iT-V̅6i+H̅6iT+H̅6i-Q̃2,Ω̅i,671=Ω̅i,672=-V̅7iT+H̅7iT-H̅6i,Ω̅i,681=Ω̅i,682=-V̅8iT+H̅8iT,Ω̅i,691=Ω̅i,692=0,Ω̅i,771=Ω̅i,772=-H̅7iT-H̅7i-Q̃3,Ω̅i,781=Ω̅i,782=-H̅8iT,Ω̅i,791=Ω̅i,792=0,Ω̅i,881=Ω̅i,882=-2ε̅1iI,Ω̅i,891=Ω̅i,892=0,Ω̅i,991=Ω̅i,992=-D2i-D2iT-γ,Θ=JTZ̅2J-JTP̅i-P̅i,Λ̅1=(2τ̅2(AiP̅i+B1iYi),0,0,2τ̅2(AdiP̅i+E1iYi),0,0,0,2τ̅2D0iε̅1i,2τ̅2D1i)T,Λ̅2=τ̅2-τ2iH̅i,Λ̅3=τ1iG̅i,Λ̅41=τ2i-τ1i2M̅i,Λ̅42=τ2i-τ1i2U̅i,Λ̅51=τ2i-τ1i2R̅i,Λ̅52=τ2i-τ1i2V̅i,Λ̅6=(ε2iT1iT,0,0,0,0,0,0,0,-ε2iT2iT)T,Λ̅7=(N1iP̅i+N3iYi,0,0,N2iP̅i+N4iYi,0,0,0,0,0)T,Λ̅8=(πik1iP̅iπik2iP̅i⋯πikmiP̅i08n×n08n×n⋯08n×n),Λ̅9=diag⁡(-P̅k1i,-P̅k2i,…,-P̅kmi),Z̅2=Z2-1,Yi=KiP̅i,ε̅1i=ε1i-1,P̅i=Pi-1,Q̅i=Pi-1QiPi-1,Q̃1=Pi-1Q̅1Pi-1,Q̃2=Pi-1Q̅2Pi-1,Q̃3=Pi-1Q̅3Pi-1,Σ=diag⁡(P̅i,…,P̅i︷7),G̅i=(G̅1iT,…,G̅8iT,0)T=(P̅i(G1iT,…,G7iT)Σ,ε̅1iP̅iG8iT,0)T,H̅i=(H̅1iT,…,H̅8iT,0)T=(P̅i(H1iT,…,H7iT)Σ,ε̅1iP̅iH8iT,0)T,M̅i=(M̅1iT,…,M̅8iT,0)T=(P̅i(M1iT,…,M7iT)Σ,ε̅1iP̅iM8iT,0)T,R̅i=(R̅1iT,…,R̅8iT,0)T=(P̅i(R1iT,…,R7iT)Σ,ε̅1iP̅iR8iT,0)T,U̅i=(U̅1iT,…,U̅8iT,0)T=(P̅i(U̅1iT,…,U̅7iT)Σ,ε̅1iP̅iU̅8iT,0)T,V̅i=(V̅1iT,…,V̅8iT,0)T=(P̅i(V1iT,…,V7iT)Σ,ε̅1iP̅iV8iT,0)T,
when the LMIs are feasible, a desired state-feedback controller can be obtained in the form of (2.7) with the controller gains given by Ki=YiPifor alli∈S.Proof.
At first, we list the following fact:(3.43)JTZ̅2J-JTP̅i-P̅iJ+P̅iZ2P̅i=(Z̅2J-P̅i)TZ2(Z̅2J-P̅i)≥0
which implies that
(3.44)-P̅iZ2P̅≤Θ=JTZ̅2J-JTP̅i-P̅iJ.
Now perform a congruence transformation to (3.1) by(3.45)diag⁡(Pi-1,…,Pi-1︷7,ε1i-1,I,Z2-1,Pi-1,…,Pi-1︷4,I,I).
Ifπii∈Ikni, then by the Schur complement and (3.44), we can infer that (3.34) is established. In the same way, if πii∈Iuki, (3.39) is established.
From Lemma2.8, we can see that (3.35) is equivalent to AiP̅i+P̅iAiT+P̅iPjP̅i<0forallj∈Iuki, so (3.4) can be established. Furthermore, using the same method that proposed above, we can deduced that (3.32), (3.33), (3.36)–(3.38), and (3.40) are equivalent to (3.2) (3.3), (3.5)–(3.7), and (3.8), respectively. In conclusion, the gain matrix of desired controller in the form of (2.7) is given by Ki=YiPi. This completes the proof.Remark 3.5.
To reduce the conservatism, when estimatingLv5(xt,i),-∫t-τ̅2tẋT(s)Z2ẋ(s)ds is not simply enlarged as -∫t-τi(t)tẋT(s)Z2ẋ(s)ds, but -∫t-τi(t)t-(τ1i+τi(t))/2ẋT(s)Z2ẋ(s)ds, -∫t-(τ2i+τi(t))/2t-τi(t)ẋT(s)Z2ẋ(s)ds are considered as well, and different free-weighting matrices are introduced. This method above may lead to obtain improved feasible region for delay-dependent exponential passivity criteria.Remark 3.6.
In fact, Theorem3.1 gives a exponential passivity criteria for MJS (2.8) with τ1i≤τi(t)≤τ2i,τ̇i(t)≤μi, where μi is a given constant. In many cases, μi is unknown. Considering this case, a rate-independent criteria for a delay satisfying τ1i≤τi(t)≤τ2i is derived as follows by setting Qi=Q*=0,foralli∈S in the proof of Theorem 3.1.


## 3.1. Exponential Passivity Analysis

In this section, we assumed the transition rates are partially known and given the state-feedback controller gain matrixKi,i∈S, at first, we will present a sufficient condition, which guarantees the MLS (2.8) is exponential passive.Theorem 3.1.
Given the state-feedback controller gain matrixKi, the uncertain MJS (2.8) is exponentially passive in the sense of expectation if there exists positive definite matrices Pi,Qi,Q̅1,Q̅2,Q̅3,Q*,Z1,Z2,  positive scalars γ,ε1i,ε2i, and for any matrices Gi,Mi,Ri,Ui,Vi,Hi with appropriate dimensions such that the following matrices inequalities hold for all i=1,2,…,N:
(3.1)(Ωi,9×91Λ1Λ2Λ3Λ4kΛ5kΛ6Λ7*-Z200002τ̅2Z2T1iε2i0**-Z200000***-Z20000****-Z2000*****-Z200******-ε2i0*******-ε2i)<0k=1,2.Case 1.
Ifπii∈Ikni(3.2)(πiiQi-Q*Qj*-Qj)∀j∈Iuki<0,(3.3)((1+∑j∈Ikniπij)(πiiQi-Q*)πik1iQk1i⋯πikmiQkmi*-Qk1i00**⋱0***-Qkmi)<0,(3.4)PiAi+AiTPi+Pj<0∀j∈Iuki.Case 2.
Ifπii∈Iuki(3.5)Qj-Q*>0∀j∈Iuki,j=i,(3.6)Qj-Q*<0∀j∈Iuki,j≠i,(3.7)((-1-∑j∈Ikniπij)Q*πik1iQk1i⋯πikmiQkmi*-Qk1i00**⋱0***-Qkmi)<0,(3.8)PiAi+AiTPi+Pj>0∀j∈Iuki,j=i,(3.9)PiAi+AiTPi+Pj<0∀j∈Iuki,j≠i,where(3.10)Ωi,111=Pi((1+∑j∈Ikniπij)Ai+B1iKi)+((1+∑j∈Ikniπij)Ai+B1iKi)TPi+∑j∈IkniπijPj+Qi+τ2iQ*+Q̅1+Q̅2+Q̅3+(τ2i-τ1i)Z1+G1iT+G1i,Ωi,121=-G1i+G2iT+M1i,Ωi,131=R1i+G3iT-M1i,Ωi,141=-R1i+G4iT+U1i+Pi(Adi+E1iKi),Ωi,151=V1i+G5iT-U1i,Ωi,161=-V1i+G6iT+H1i,Ωi,171=G7iT-H1iΩi,181=G8iT+ε1iΓi+PiD0i,Ωi,191=PiD1i-(Ci+B2iKi)T,Ωi,221=-G2iT-G2i+M2iT+M2i-Q̅1,Ωi,231=-G3iT+R2i+M3iT-M2i,Ωi,241=-G4iT-R2i+M4iT+U2i,Ωi,251=-G5iT+V2i+M5iT-U2i,Ωi,261=-G6iT-V2i+M6iT+H2i,Ωi,271=-G7iT+M7iT-H2i,Ωi,281=-G8iT+M8iT,Ωi,291=0,Ωi,331=R3iT+R3i-M3iT-M3i,Ωi,341=R4iT-R3i-M4iT+U3i,Ωi,351=R5iT+V3i-M5iT-U3i,Ωi,361=R6iT-V3i-M6iT+H3i,Ωi,371=R7iT-M7iT-H3i,Ωi,381=R8iT-M8iTΩi,391=0,Ωi,441=-R4iT-R4i+U4iT+U4i-(1-μi)Qi,Ωi,451=-R5iT+V4i+U5iT-U4i,Ωi,461=-R6iT-V4i+U6iT+H4i,Ωi,471=-R7iT+U7iT-H4i,Ωi,481=-R8iT+U8iT,Ωi,491=-(Cdi+E2iKi)T,Ωi,551=V5iT+V5i-U5iT-U5i,Ωi,561=-U6iT-V5i+V6iT+H5i,Ωi,571=V7iT-U7iT-H5i,Ωi,581=V8iT-U8iTΩi,591=0,Ωi,661=-V6iT-V6i+H6iT+H6i-Q̅2,Ωi,671=-V7iT+H7iT-H6i,Ωi,681=-V8iT+H8iT,Ωi,691=0,Ωi,771=-H7iT-H7i-Q̅3,Ωi,781=-H8iT,Ωi,791=0,Ωi,881=-2ε1iI,Ωi,891=0,Ωi,991=-D2i-D2iT-γI,τ̅2=max⁡i∈S{τ2i},τ̲1=min⁡i∈S{τ1i},Λ1=(2τ̅2Z2(Ai+B1iKi),0,0,2τ̅2Z2(Adi+E1iKi),0,0,0,2τ̅2Z2D0i,2τ̅2Z2D1i)T,Λ1(t)=(2τ̅2Z2(Ai(t)+B1i(t)Ki),0,0,2τ̅2Z2(Adi(t)+E1i(t)Ki),0,0,0,2τ̅2Z2D0i,2τ̅2Z2D1i)T,Λ2=τ̅2-τ2iHi,Λ3=τ1iGi,Λ41=τ2i-τ1i2Mi,Λ42=τ2i-τ1i2Ui,Λ51=τ2i-τ1i2Ri,Λ52=τ2i-τ1i2Vi,Λ6=(ε2iT1iTPi,0,0,0,0,0,0,0,-ε2iT2iT)T,Λ7=(N1i+N3iKi,0,0,N2i+N4iKi,0,0,0,0,0)T,Gi=(G1iTG2iTG3iTG4iTG5iTG6iTG7iTG8iT0)TMi=(M1iTM2iTM3iTM4iTM5iTM6iTM7iTM8iT0)T,Ri=(R1iTR2iTR3iTR4iTR5iTR6iTR7iTR8iT0)T,Ui=(U1iTU2iTU3iTU4iTU5iTU6iTU7iTU8iT0)T,Vi=(V1iTV2iTV3iTV4iTV5iTV6iTV7iTV8iT0)T,Hi=(H1iTH2iTH3iTH4iTH5iTH6iTH7iTH8iT0)T,Proof.
First, in order to cast our model involved in the framework of the Markov process, we define a new processxt(s)=x(t+s),s∈[-τ̅2,0], and let L be the weak infinitesimal generator of the random process xt(s),t≥0 and
(3.11)Lv(xt,rt)=lim⁡Δ→0+1Δ{E[v(xt+Δ,rt+Δ)∣xt,rt=i]-v(xt,rt)}.
Now consider the Lyapunov-Krasovskii functional as follows for rt=i,i∈1,2,…,S:
(3.12)v(xt,i)=v1(xt,i)+v2(xt,i)+v3(xt,i)+v4(xt,i)+v5(xt,i),
where
(3.13)v1(xt,i)=xT(t)P(i)x(t),v2(xt,i)=∫t-τi(t)txT(s)Q(i)x(s)ds,v3(xt,i)=∫-τ2i0∫t+θtxT(s)Q*x(s)dsdθ,v4(xt,i)=∫t-τ1itxT(s)Q̅1x(s)ds+∫t-τ2itxT(s)Q̅2x(s)ds+∫t-τ̅2txT(s)Q̅3x(s)d,v5(xt,i)=∫-τ2i-τ1i∫t+θtxT(s)Z1x(s)dsdθ+2∫-τ̅20∫t+θtẋT(s)Z2ẋ(s)dsdθ,
where
(3.14)∑j=1NπijQj≤Q*.
In order to show the exponential passivity of the MJS (2.8) under the given controller gain matrix Ki, we set
(3.15)J*=Lv(xt,i)-γωT(t)ω(t)-2zT(t)ω(t).
Notice that(3.16)Lv1(xt,i)=xT(t)(Pi(Ai(t)+B1i(t)Ki)+(Ai(t)+B1i(t)Ki)TPi)x(t)+xT(t)∑j=1NπijPjx(t)+2xT(t)Pi(Adi(t)+E1i(t)Ki)x(t-τi(t))+2xT(t)PiD0if(x(t),i)+2xT(t)PiD1iω(t),Lv2(xt,i)≤xT(t)Qix(t)-(1-μi)xT(t-τi(t))Qix(t-τi(t))+∫t-τi(t)txT(s)∑j=1NπijQjx(s)ds,Lv3(xt,i)≤τ2ixT(t)Q*x(t)-∫t-τi(t)txT(s)Q*x(s)ds,Lv4(xt,i)=xT(t)(Q̅1+Q̅2+Q̅3)x(t)-xT(t-τ1i)Q̅1x(t-τ1i)-xT(t-τ2i)Q̅2x(t-τ2i)-xT(t-τ̅2)Q̅3x(t-τ̅2),Lv5(xt,i)=(τ2i-τ1i)xT(t)Z1x(t)+2τ̅2ẋT(t)Z2ẋ(t)-∫t-τ2it-τ1ixT(s)Z1x(s)ds-2∫t-τ̅2tẋT(s)Z2ẋ(s)ds=(τ2i-τ1i)xT(t)Z1x(t)+2τ̅2ẋT(t)Z2ẋ(t)-∫t-τ̅2t-τ2iẋT(s)Z2ẋ(s)ds-∫t-τ2it-(τi(t)+τ2i)/2ẋT(s)Z2ẋ(s)ds-∫t-(τi(t)+τ2i)/2t-τi(t)ẋT(s)Z2ẋ(s)ds-∫t-τi(t)t-(τi(t)+τ1i)/2ẋT(s)Z2ẋ(s)ds-∫t-(τi(t)+τ1i)/2t-τ1iẋT(s)Z2ẋ(s)ds-∫t-τ1itẋT(s)Z2ẋ(s)ds-∫t-τ2it-τ1ixT(s)Z1x(s)ds-∫t-τ̅2tẋT(s)Z2ẋ(s)ds.
Then using Newton-Leibniz formula, for any matricesHi,Gi,Mi,Ri,Ui,Vi we have(3.17)2ξT(t)Gi(x(t)-x(t-τ1i)-∫t-τ1itẋ(s)ds)=0,2ξT(t)Mi(x(t-τ1i)-x(t-τ1i+τi(t)2)-∫t-(τ1i+τi(t))/2t-τ1iẋ(s)ds)=0,2ξT(t)Ri(x(t-τ1i+τi(t)2)-x(t-τi(t))-∫t-τi(t)t-(τ1i+τi(t))/2ẋ(s)ds)=0,2ξT(t)Ui(x(t-τi(t))-x(t-τ2i+τi(t)2)-∫t-(τ2i+τi(t))/2t-τi(t)ẋ(s)ds)=0,2ξT(t)Vi(x(t-τ2i+τi(t)2)-x(t-τ2i)-∫t-τ2it-(τ2i+τi(t))/2ẋ(s)ds)=0,2ξT(t)Hi(x(t-τ2i)-x(t-τ̅2)-∫t-τ̅2t-τ2iẋ(s)dt)=0,
where
(3.18)ξT(t)=(xT(t),xT(t-τ1i),xT(t-τi(t)+τ1i2),xT(t-τi(t)),xT(t-τi(t)+τ2i2),xT(t-τ2i)xT(t-τ̅2),fT(x(t),i),ωT(t)(t-τi(t)+τ1i2)).
From the Lemma2.7 (2.2), it is easy to see that(3.19)-2ξT(t)Gi∫t-τ1itẋ(s)ds≤τ1iξT(t)GiZ2-1GiTξ(t)+∫t-τ1itẋT(s)Z2ẋ(s)ds-2ξT(t)Mi∫t-(τ1i+τi(t))/2t-τ1iẋ(s)ds≤τi(t)-τ1i2ξT(t)MiZ2-1MiTξ(t)+∫t-(τ1i+τi(t))/2t-τ1iẋT(s)Z2ẋ(s)ds-2ξT(t)Ri∫t-τi(t)t-(τ1i+τi(t))/2ẋ(s)ds≤τi(t)-τ1i2ξT(t)RiZ2-1RiTξ(t)+∫t-τi(t)t-(τ1i+τi(t))/2ẋT(s)Z2ẋ(s)ds-2ξT(t)Ui∫t-(τ2i+τi(t))/2t-τi(t)ẋ(s)ds≤τ2i-τi(t)2ξT(t)UiZ2-1UiTξ(t)+∫t-(τ2i+τi(t))/2t-τi(t)ẋT(s)Z2ẋ(s)ds-2ξT(t)Vi∫t-τ2it-(τ2i+τi(t))/2ẋ(s)ds≤τ2i-τi(t)2ξT(t)ViZ2-1ViTξ(t)+∫t-τ2it-(τ2i+τi(t))/2ẋT(s)Z2ẋ(s)ds-2ξT(t)Hi∫t-τ̅2t-τ2iẋ(s)ds≤(τ̅2-τ2i)ξT(t)HiZ2-1HiTξ(t)+∫t-τ̅2t-τ2iẋT(s)Z2ẋ(s)ds.
Now by Assumption3, it can be deduced that for any positive scalar ε1i,i∈1,2,…,S,(3.20)2ε1ifT(x(t),i)(Γix(t)-f(x(t),i))≥0.
Then from the above discussion, we can see that
(3.21)J*≤ξT(t)(τi(t)-τ1iτ2i-τ1iΦ1(t)+τ2i-τi(t)τ2i-τ1iΦ2(t))ξ(t)-∫t-τ2it-τ1ixT(s)Z1x(s)ds-∫t-τ̅2tẋT(s)Z2ẋ(s)ds,Φ1(t)=Ωi,9×9(t)+Λ1(t)Z2-1Λ1(t)T+(τ̅2-τ2i)HiZ2-1HiT+τ1iGiZ2-1GiT+τ2i-τ1i2(MiZ2-1MiT+RiZ2-1RiT),Φ2(t)=Ωi,9×9(t)+Λ1(t)Z2-1Λ1(t)T+(τ̅2-τ2i)HiZ2-1HiT+τ1iGiZ2-1GiT+τ2i-τ1i2(UiZ2-1UiT+ViZ2-1ViT),
where
(3.22)Ωi,11(t)=Pi(Ai(t)+B1i(t)Ki)+(Ai(t)+B1i(t)Ki)TPi+∑j=1NπijPj+Qi+τ2iQ*+Q̅1+Q̅2+Q̅3+(τ2i-τ1i)Z1+G1iT+G1i,Ωi,14(t)=Pi(Adi(t)+E1i(t)Ki)-R1i+G4iT+U1iΩi,19(t)=PiD1i-(Ci(t)+B2i(t)Ki)T,Ωi,49(t)=-(Cdi(t)+E2i(t)Ki)T
other terms of Ωi,i×j(t) are similar to Ωi,i×j1. In order to get our results, we will describe that the Φ1(t)<0 and Φ2(t)<0. By the Schur complement, Φ1(t)<0 and Φ2(t)<0 under the restriction of (3.14) if and only if
(3.23)(Ωi,9×9Λ1Λ2Λ3Λ4kΛ5k*-Z20000**-Z2000***-Z200****-Z20*****-Z2)k=1,2+(T1iTPi,0n×7n,-T2iT,2τ̅2T1iTZ2,0n×4n)TFi(k)(N1i+N3iKi,0n×2n,N2i+N4iKi,0n×10n)+(N1i+N2iKi,0n×2n,N2i+N4iKi,0n×10n)TFiT(k)(T1iTPi,0n×7n,-T2iT,2τ̅2T1iTZ2,0n×4n)<0,
where Ωi,9×9 is the nominal matrix of Ωi,9×9(t). Then from the Lemma 2.7 (2.1), above matrix inequality holds, which is equivalent to
(3.24)(Ωi,9×9Λ1Λ2Λ3Λ4kΛ5kΛ6Λ7*-Z200002τ̅2Z2T1iε2i0**-Z200000***-Z20000****-Z2000*****-Z200******-ε2i0*******-ε2i)k=1,2<0.Case 1.
Ifπii∈Ikni
then (3.24) is equivalent to(3.25)(Ωi,9×91Λ1Λ2Λ3Λ4kΛ5kΛ6Λ7*-Z200002τ̅2Z2T1iε2i0**-Z200000***-Z20000****-Z2000*****-Z200******-ε2i0*******-ε2i)k=1,2+diag⁡(∑j∈Iukij≠iπijPj,0,…,0︷15)+diag⁡(∑j∈Iukij≠iπij(PiAi+AiTPi),0,…,0︷15)<0.
Obviously, we can see that if (3.1) and (3.4) hold, then Φ1(t)<0 and Φ2(t)<0 under the restriction of (3.14). Next we will further consider the equivalent form of (3.14).∑j=1NπijQj<Q* is equivalent to(3.26)∑j≠ij∈IkniπijQj+∑j≠ij∈IukiπijQj+πiiQi-Q*+∑j∈Ikniπij(πiiQi-Q*)+∑j∈Iukiπij(πiiQi-Q*)<0.
If we have the following matrix inequalities hold, we can have that (3.14) is satisfied
(3.27)(1+∑j∈Ikniπij)(πiiQi-Q*)+∑j≠ij∈IkniπijQj<0,πiiQi-Q*+Qj<0j∈Iuki.
Obviously, (3.27) is equivalent to (3.2) and (3.3) by the Schur complement. Case 2.
Ifπii∈Iuki
then (3.24) is equivalent to(3.28)(Ωi,9×91Λ1Λ2Λ3Λ4kΛ5kΛ6Λ7*-Z200002τ̅2Z2T1iε2i0**-Z200000***-Z20000****-Z2000*****-Z200******-ε2i0*******-ε2i)k=1,2+diag⁡(∑j∈Iukij≠iπijPj,0,…,0︷15)+diag⁡(∑j∈Iukij≠iπij(PiAi+AiTPi),0,…,0︷15)+πiidiag⁡(PiAi+AiTPi+Pi,0,…,0︷15)<0.
Then if (3.1), (3.8), and (3.9) hold, then Φ1(t)<0 and Φ2(t)<0 under the restriction of (3.14), furthermore, with the similar consideration, we can deduce that if (3.5)–(3.7) are established, then (3.14) is founded. So there exists a positive scalar ρ1, then(3.29)J*≤-ρ1‖x(t)‖2-λmin⁡(Z1)∫t-τ̅2t‖x(s)‖2ds-λmin⁡(Z2)∫t-τ̅2t‖ẋ(s)‖2ds.
On the other hand, it is easy to obtain that
(3.30)v(x(t),i)≤‖P‖‖x(t)‖2+(‖Q‖+‖Q*‖+‖Q̅1‖+‖Q̅2‖+‖Q̅3‖+(τ̅2-τ̲1)‖Z1‖)×∫t-τ̅2t‖x(s)‖2ds+2τ̅2‖Z2‖∫t-τ̅2t‖ẋ(s)‖2ds,
where ∥P∥=maxi∈S{∥Pi∥}, ∥Q∥=maxi∈S{∥Qi∥}.Let ρ>0 be sufficiently small such that
(3.31)ρ‖P‖-ρ1<0,ρ(‖Q‖+‖Q*‖+‖Q̅1‖+‖Q̅2‖+‖Q̅3‖+(τ̅2-τ̲1)‖Z1‖)-λmin⁡(Z1)<02τ̅2ρ‖Z2‖-λmin⁡(Z2)<0.,
So, by Definition 2.4, the MJS (2.8) is exponentially passive. This completes the proof.Remark 3.2.
It is easy to derive that the MJS (2.8) is exponential mean square stability with ω(t)=0 if the MJS (2.8) is exponentially passive. Moreover, the result of Theorem 3.1 makes use of the information of the subsystems upper bounds of the time varying delays, which may bring us less conservativeness, and from the free-weighting matrix and Newton-Leibnitz formula, the upper bounds of μi are not restricted to be less than 1 in this paper. Therefore, our result is more natural and reasonable to the Markovian jump systems.Remark 3.3.
In order to obtain the gain matricesKi for convenience in the next section, (3.1) is not LMI, if we substitute ε2i by ε2i-1 and use the Lemma 2.7 (2.1), we can obtain the equivalent form of LMI.


## 3.2. Exponential Passification

In this section, we will determine the feedback controller gain matricesKi,i∈S in (2.7), which guarantee that the closed-loop MJS (2.8) is exponentially passive with partially known transition rates.Theorem 3.4.
Given a positive constantε, there exists a state-feedback controller in the form (2.7) such that the closed-loop MJS (2.8) is exponentially passive if there exist positive definite matrices P̅i,Q̅i,Q̃1,Q̃2,Q̃3,Q̅*,Z̅1,Z̅2, positive scalar ε1i,ε2i, and for any matrices G̅i,M̅i,R̅i,U̅i,V̅i,H̅i,Zii with appropriate dimensions satisfying the following LMIs under the two cases for all i=1,2,…,N. Case 1.
Ifπii∈Ikni(3.32)(πiiQ̅i-Q̅*Q̅j*-Q̅j)∀j∈Iuki<0,(3.33)((1+∑j∈Ikniπij)(πiiQ̅i-Q̅*)πik1iQ̅k1i⋯πikmiQ̅kmi*-Q̅k1i00**⋱0***-Q̅kmi)<0,(3.34)(Ω̅i,9×91Λ̅1Λ̅2Λ̅3Λ̅4kΛ̅5kΛ̅6Λ̅7Λ̅8*-Z̅200002τ̅2T1iε2i00**Θ000000***Θ00000****Θ0000*****Θ000******-ε2i00*******-ε2i0********Λ̅9)<0k=1,2,(3.35)(-Zii-ZiiTZiiTAiT+P̅iZiiT0*-ε-1P̅i0P̅i**-εP̅i0***-P̅j)∀j∈Iuki<0,Case 2.
Ifπii∈Iuki(3.36)Q̅j-Q̅*>0∀j∈Iuki,j=i,(3.37)Q̅j-Q̅*<0∀j∈Iuki,j≠i,(3.38)((-1-∑j∈Ikniπij)Q̅*πik1iQ̅k1i⋯πikmiQ̅kmi*-Q̅k1i00**⋱0***-Q̅kmi)<0,(3.39)(Ω̅i,9×92Λ̅1Λ̅2Λ̅3Λ̅4kΛ̅5kΛ̅6Λ̅7Λ̅8*-Z̅200002τ̅2T1iε2i00**Θ000000***Θ00000****Θ0000*****Θ000******-ε2i00*******-ε2i0********Λ̅9)<0k=1,2,(3.40)(-Zii-ZiiT-ZiiTAiT+P̅iZiiT*-ε-1P̅i-P̅j0**-εP̅i)∀j∈Iukij=i<0,(3.41)(-Zii-ZiiTZiiTAiT+P̅iZiiT0*-ε-1P̅i0P̅i**-εP̅i0***-P̅j)∀j∈Iukij≠i<0,
where
(3.42)Ω̅i,111=((1+∑j∈Ikniπij)AiP̅i+B1iYi)+((1+∑j∈Ikniπij)AiP̅i+B1iYi)T+πiiP̅i+Q̅i+τ2iQ̅*+Q̃1+Q̃2+Q̃3+(τ2i-τ1i)Z̅1+G̅1iT+G̅1i,Ω̅i,112=((1+∑j∈Ikniπij)AiP̅i+B1iYi)+((1+∑j∈Ikniπij)AiP̅i+B1iYi)T+Q̅i+τ2iQ̅*+Q̃1+Q̃2+Q̃3+(τ2i-τ1i)Z̅1+G̅1iT+G̅1i,Ω̅i,121=Ω̅i,122=-G̅1i+G̅2iT+M̅1i,Ω̅i,131=Ω̅i,132=R̅1i+G̅3iT-M̅1i,Ω̅i,141=Ω̅i,142=-R̅1i+G̅4iT+U̅1i+(AdiP̅i+E1iYi)Ω̅i,151=Ω̅i,152=V̅1i+G̅5iT-U̅1i,Ω̅i,161=Ω̅i,162=-V̅1i+G̅6iT+H̅1i,Ω̅i,171=Ω̅i,172=G̅7iT-H̅1i,Ω̅i,181=Ω̅i,182=G̅8iT+P̅iΓi+D0iε̅1i,Ω̅i,191=Ω̅i,192=D1i-(CiP̅i+B2iYi)T,Ω̅i,221=Ω̅i,222=-G̅2iT-G̅2i+M̅2iT+M̅2i-Q̃1,Ω̅i,231=Ω̅i,232=-G̅3iT+R̅2i+M̅3iT-M̅2i,Ω̅i,241=Ω̅i,242=-G̅4iT-R̅2i+M̅4iT+U̅2i,Ω̅i,251=Ω̅i,252=-G̅5iT+V̅2i+M̅5iT-U̅2i,Ω̅i,261=Ω̅i,262=-G̅6iT-V̅2i+M̅6iT+H̅2i,Ω̅i,271=Ω̅i,272=-G̅7iT+M̅7iT-H̅2i,Ω̅i,281=Ω̅i,282=-G̅8iT+M̅8iT,Ω̅i,291=Ω̅i,292=0,Ω̅i,331=Ω̅i,332=R̅3iT+R̅3i-M̅3iT-M̅3i,Ω̅i,341=Ω̅i,342=R̅4iT-R̅3i-M̅4iT+U̅3i,Ω̅i,351=Ω̅i,352=R̅5iT+V̅3i-M̅5iT-U̅3i,Ω̅i,361=Ω̅i,362=R̅6iT-V̅3i-M̅6iT+H̅3i,Ω̅i,371=Ω̅i,372=R̅7iT-M̅7iT-H̅3i,Ω̅i,381=Ω̅i,382=R̅8iT-M̅8iT,Ω̅i,391=Ω̅i,392=0,Ω̅i,441=Ω̅i,442=-R̅4iT-R̅4i+U̅4iT+U̅4i-(1-μi)Q̅i,Ω̅i,451=Ω̅i,452=-R̅5iT+V̅4i+U̅5iT-U̅4i,Ω̅i,461=Ω̅i,462=-R̅6iT-V̅4i+U̅6iT+H̅4i,Ω̅i,471=Ω̅i,472=-R̅7iT+U̅7iT-H̅4i,Ω̅i,481=Ω̅i,482=-R̅8iT+U̅8iT,Ω̅i,491=Ω̅i,492=-(CdiP̅i+E2iYi)T,Ω̅i,551=Ω̅i,552=V̅5iT+V̅5i-U̅5iT-U̅5i,Ω̅i,561=Ω̅i,562=-U̅6iT-V̅5i+V̅6iT+H̅5i,Ω̅i,571=Ω̅i,572=V̅7iT-U̅7iT-H̅5i,Ω̅i,581=Ω̅i,582=V̅8iT-U̅8iTΩ̅i,591=Ω̅i,592=0,Ω̅i,661=Ω̅i,662=-V̅6iT-V̅6i+H̅6iT+H̅6i-Q̃2,Ω̅i,671=Ω̅i,672=-V̅7iT+H̅7iT-H̅6i,Ω̅i,681=Ω̅i,682=-V̅8iT+H̅8iT,Ω̅i,691=Ω̅i,692=0,Ω̅i,771=Ω̅i,772=-H̅7iT-H̅7i-Q̃3,Ω̅i,781=Ω̅i,782=-H̅8iT,Ω̅i,791=Ω̅i,792=0,Ω̅i,881=Ω̅i,882=-2ε̅1iI,Ω̅i,891=Ω̅i,892=0,Ω̅i,991=Ω̅i,992=-D2i-D2iT-γ,Θ=JTZ̅2J-JTP̅i-P̅i,Λ̅1=(2τ̅2(AiP̅i+B1iYi),0,0,2τ̅2(AdiP̅i+E1iYi),0,0,0,2τ̅2D0iε̅1i,2τ̅2D1i)T,Λ̅2=τ̅2-τ2iH̅i,Λ̅3=τ1iG̅i,Λ̅41=τ2i-τ1i2M̅i,Λ̅42=τ2i-τ1i2U̅i,Λ̅51=τ2i-τ1i2R̅i,Λ̅52=τ2i-τ1i2V̅i,Λ̅6=(ε2iT1iT,0,0,0,0,0,0,0,-ε2iT2iT)T,Λ̅7=(N1iP̅i+N3iYi,0,0,N2iP̅i+N4iYi,0,0,0,0,0)T,Λ̅8=(πik1iP̅iπik2iP̅i⋯πikmiP̅i08n×n08n×n⋯08n×n),Λ̅9=diag⁡(-P̅k1i,-P̅k2i,…,-P̅kmi),Z̅2=Z2-1,Yi=KiP̅i,ε̅1i=ε1i-1,P̅i=Pi-1,Q̅i=Pi-1QiPi-1,Q̃1=Pi-1Q̅1Pi-1,Q̃2=Pi-1Q̅2Pi-1,Q̃3=Pi-1Q̅3Pi-1,Σ=diag⁡(P̅i,…,P̅i︷7),G̅i=(G̅1iT,…,G̅8iT,0)T=(P̅i(G1iT,…,G7iT)Σ,ε̅1iP̅iG8iT,0)T,H̅i=(H̅1iT,…,H̅8iT,0)T=(P̅i(H1iT,…,H7iT)Σ,ε̅1iP̅iH8iT,0)T,M̅i=(M̅1iT,…,M̅8iT,0)T=(P̅i(M1iT,…,M7iT)Σ,ε̅1iP̅iM8iT,0)T,R̅i=(R̅1iT,…,R̅8iT,0)T=(P̅i(R1iT,…,R7iT)Σ,ε̅1iP̅iR8iT,0)T,U̅i=(U̅1iT,…,U̅8iT,0)T=(P̅i(U̅1iT,…,U̅7iT)Σ,ε̅1iP̅iU̅8iT,0)T,V̅i=(V̅1iT,…,V̅8iT,0)T=(P̅i(V1iT,…,V7iT)Σ,ε̅1iP̅iV8iT,0)T,
when the LMIs are feasible, a desired state-feedback controller can be obtained in the form of (2.7) with the controller gains given by Ki=YiPifor alli∈S.Proof.
At first, we list the following fact:(3.43)JTZ̅2J-JTP̅i-P̅iJ+P̅iZ2P̅i=(Z̅2J-P̅i)TZ2(Z̅2J-P̅i)≥0
which implies that
(3.44)-P̅iZ2P̅≤Θ=JTZ̅2J-JTP̅i-P̅iJ.
Now perform a congruence transformation to (3.1) by(3.45)diag⁡(Pi-1,…,Pi-1︷7,ε1i-1,I,Z2-1,Pi-1,…,Pi-1︷4,I,I).
Ifπii∈Ikni, then by the Schur complement and (3.44), we can infer that (3.34) is established. In the same way, if πii∈Iuki, (3.39) is established.
From Lemma2.8, we can see that (3.35) is equivalent to AiP̅i+P̅iAiT+P̅iPjP̅i<0forallj∈Iuki, so (3.4) can be established. Furthermore, using the same method that proposed above, we can deduced that (3.32), (3.33), (3.36)–(3.38), and (3.40) are equivalent to (3.2) (3.3), (3.5)–(3.7), and (3.8), respectively. In conclusion, the gain matrix of desired controller in the form of (2.7) is given by Ki=YiPi. This completes the proof.Remark 3.5.
To reduce the conservatism, when estimatingLv5(xt,i),-∫t-τ̅2tẋT(s)Z2ẋ(s)ds is not simply enlarged as -∫t-τi(t)tẋT(s)Z2ẋ(s)ds, but -∫t-τi(t)t-(τ1i+τi(t))/2ẋT(s)Z2ẋ(s)ds, -∫t-(τ2i+τi(t))/2t-τi(t)ẋT(s)Z2ẋ(s)ds are considered as well, and different free-weighting matrices are introduced. This method above may lead to obtain improved feasible region for delay-dependent exponential passivity criteria.Remark 3.6.
In fact, Theorem3.1 gives a exponential passivity criteria for MJS (2.8) with τ1i≤τi(t)≤τ2i,τ̇i(t)≤μi, where μi is a given constant. In many cases, μi is unknown. Considering this case, a rate-independent criteria for a delay satisfying τ1i≤τi(t)≤τ2i is derived as follows by setting Qi=Q*=0,foralli∈S in the proof of Theorem 3.1.


## 4. Examples

In this section, we will consider a interval time-varying delay MJS in the form of (2.8) with three modes, and the parameters of the system are given as follows:(4.1)A1=(-0.05-0.050.5-0.5),A2=(-0.05-0.091.5-0.1),A3=(-0.03-0.0150.05-0.01),Ad1=(0.110.24-0.53-0.37),Ad2=(-0.590.01-0.07-0.61),Ad3=(0.520.240.02-0.45),D01=(00-0.020),D02=(-0.0200-0.02),D03=(0-1.200),B11=(2.01.0),B12=(1.00.5),B13=(1.02.0),E11=(0.51.0),E12=(0.82.0),E13=(1.00.5),D11=(1.00.2),D12=(1.01.0),D13=(0.50.5),C1=(1.00.2),C2=(0.51.0),C3=(0.50.5),Cd1=(-1.00.2),Cd2=(0.1-0.1),Cd3=(0.5-0.5),B21=1.0,B22=-0.5,B23=0.5,D21=1.0,D22=0.5,D23=-0.5,T11=T12=T13=(0.020.01),N11=N12=N13=(0.020.01),N21=N22=N23=(0.010.02),N31=N32=N33=N41=N42=N43=0.01,T21=T22=T23=0.1,μ1=0.2,μ2=0.3,μ3=0.1,Γ1=Γ2=Γ3=(0.06000.06),τ11=0.12,τ12=0.11,τ13=0.13,τ21=0.23,τ22=0.28,τ23=0.25,
The two cases of the transition rates matrices are described as follows:(4.2)Case1:Π=(-0.50.20.30.2-0.60.40.50.3-0.8),Case2:Π=(-0.5??0.2-0.60.40.5??),
where ? means the unknown element. With the choice of ε=0.2 and J=(0.3000.3), we can obtain the feasibility solution of case 1 and case 2 as follows. Case 1:(4.3)P̅1=(3.2525-0.1821-0.18214.2081),P̅2=(2.6050-0.2795-0.27953.6119),P̅3=(3.2386-0.2026-0.20263.8813),Q̅1=(1.8478-0.0727-0.07272.0824),Q̅2=(1.5779-0.5056-0.50561.7197),Q̅3=(1.44620.21090.21091.8895),Q̃1=(0.5074-0.1035-0.10350.7639),Q̃2=(0.3321-0.0760-0.07600.5493),Z̅1=(1.4315-0.2159-0.21591.8214),Z̅2=(3.0280-0.2085-0.20854.1893),Y1=(-0.73760.1326),Y2=(-1.0932-0.3296),Y3=(-0.8734-0.5801).
case 2:(4.4)P̅1=1.0e+004*(0.13040.35860.35862.6532),P̅2=1.0e+004*(0.57680.46640.46643.9812),P̅3=1.0e+004*(0.49620.76080.76084.3848),Q̅1=1.0e+004*(0.08640.23300.23301.7220),Q̅2=1.0e+003*(0.19180.39250.39253.7184),Q̅3=1.0e+004*(0.05370.21330.21331.5653),Q̃1=1.0e+003*(0.06580.32810.32812.4649),Q̃2=1.0e+003*(0.04290.22060.22061.6811),Z̅1=1.0e+003*(0.23120.92730.92737.3269),Z̅2=1.0e+004*(0.67541.42461.42465.4089),Y1=1.0e+003*(-2.2126-1.5022),Y2=1.0e+003*(-2.25302.3481),Y3=1.0e+004*(-0.5783-1.2112).Under the two cases above, Table1 lists the state-feedback controller gains matrix Ki, which can be determined by the method of Theorem 3.4. If the ρ is sufficiently small, we can check that the MJS (2.8) is exponentially passive under the condition of Theorem 3.4. Given the initial condition as x(t)=(2.0-2.0)T and r(t)=2, from Figures 1 and 2, we can easily see that the closed-loop system in (2.8) is mean square exponential stable with ω(t)=0.Table 1
Calculated the controller gains matrix for different cases.
Case 1Case 2K1(-0.2256,0.0218)(-2.4527,0.2749)K2(-0.4330,-0.1248)(-0.4842,0.1157)K3(-0.2800,-0.1641)(-1.0109,-0.1008)Figure 1
State response of case 1 and the switch signal.Figure 2
State response of case 2 and the switch signal.Remark 4.1.
In order to illustrate the effectiveness of the proposed approach, a numerical example is given which included two cases, that is, case 1, the transition rate matrix is completely known; case 2, some elements in the transition rate matrix are inaccessible. By using Matlab Toolbox, we can obtain the gain matrixKi, which guarantees that the Markovian jump systems (2.8) is robust exponential passivity. If we choose the switch signal as Figures 1 and 2, we can know that the closed-loop system (2.8) is exponentially stable in the mean square under the state-feedback controllers obtained above, which have been listed in Table 1.


## 5. Conclusions

In this paper, the problems of exponential passification of uncertain MJS have been investigated. To reflect more realistic dynamical behaviors of the system, both the partially known transition rates, state and input delays have been considered. With utilizing the Lyapunov functional method and free-weighting matrix method, delay-dependent exponential passivity conditions are established. Finally, an illustrative example has been given to demonstrate the effectiveness of the proposed approach.



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*Source: 950590-2012-03-14.xml*