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

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.

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 letter ℝ^𝑛 and ℝ^𝑛×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 ℝ^𝑛. 𝐼 is the identity matrix with compatible dimension. If 𝐴 is a matrix, 𝜆_max(𝐴) (respective 𝜆_min(𝐴)) means the largest (respective smallest) eigervalue of 𝐴. Moreover, let (Ω,𝔽,(𝔽_𝑡)_𝑡≥0,ℙ) be a complete probability space with a filteration. (𝔽_𝑡)_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 𝐿²_𝔽0([−𝜏₂,0]∶ℝ^𝑛) the family of all 𝔽₀ measurable 𝐶([−𝜏₂,0]∶ℝ^𝑛)-valued random variables 𝜑={𝜑(𝑠)∶−𝜏₂≤𝑠≤0} such that sup_{−𝜏2≤𝑠≤0}𝐸‖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 delays𝑥̇(𝑡)=𝐴𝑡,𝑟_𝑡𝑥(𝑡)+𝐴_𝑑𝑡,𝑟_𝑡𝑥𝑡−𝜏𝑡,𝑟_𝑡+𝐵₁𝑡,𝑟_𝑡𝑢(𝑡)+𝐸₁𝑡,𝑟_𝑡𝑢𝑡−𝜏𝑡,𝑟_𝑡+𝐷₀𝑟_𝑡𝑓𝑥(𝑡),𝑟_𝑡+𝐷₁𝑟_𝑡𝜔(𝑡).(2.1)𝑧(𝑡)=𝐶𝑡,𝑟_𝑡𝑥(𝑡)+𝐶_𝑑𝑡,𝑟_𝑡𝑥𝑡−𝜏𝑡,𝑟_𝑡+𝐵₂𝑡,𝑟_𝑡𝑢(𝑡)+𝐸₂𝑡,𝑟_𝑡𝑢𝑡−𝜏𝑡,𝑟_𝑡+𝐷₂𝑟_𝑡𝜔(𝑡),(2.2) Here 𝑥(𝑡)∈ℝ^𝑛 is the state vector, 𝑢(𝑡)∈ℝ^𝑝 is the control input, 𝑧(𝑡)∈ℝ^𝑞 is the control output, and 𝜔(𝑡)∈ℝ^𝑙 is the exogenous disturbance input which belongs to 𝕃₂[0,∞],{𝑟_𝑡,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={𝜋_𝑖𝑗},𝑖,𝑗∈𝑆 and has the mode transition probabilitiesPr𝑟_𝑡+Δ𝑡=𝑗∣𝑟_𝑡=𝑖=𝜋_𝑖𝑗Δ𝑡+𝑜(Δ𝑡)𝑖≠𝑗,1+𝜋_𝑖𝑖Δ𝑡+𝑜(Δ𝑡)𝑖=𝑗,(2.3) where Δ𝑡>0,lim_Δ𝑡→0(𝑜(Δ𝑡)/Δ𝑡)=0,𝜋_𝑖𝑗 is the transition rete from 𝑖 to 𝑗, and𝜋_𝑖𝑖=−_𝑗≠𝑖𝜋_𝑖𝑗,𝜋_𝑖𝑗≥0,𝑗≠𝑖.(2.4)

For notational simplicity, which 𝑟_𝑡=𝑖,𝑖∈𝑆, the matrices 𝐴(𝑡,𝑟_𝑡),𝐴_𝑑(𝑡,𝑟_𝑡),𝐵₁(𝑡,𝑟_𝑡),𝐸₁(𝑡,𝑟_𝑡),𝐶(𝑡,𝑟_𝑡), 𝐶_𝑑(𝑡,𝑟_𝑡), 𝐵₂(𝑡,𝑟_𝑡), 𝐸₂(𝑡,𝑟_𝑡),  𝐷₀(𝑟_𝑡), 𝐷₁(𝑟_𝑡), and 𝐷₂(𝑟_𝑡) will be described by 𝐴_𝑖(𝑡),𝐴_𝑑𝑖(𝑡),𝐵_1𝑖(𝑡),𝐸_1𝑖(𝑡), 𝐶_𝑖(𝑡),𝐶_𝑑𝑖(𝑡),𝐵_2𝑖(𝑡), 𝐸_2𝑖(𝑡), 𝐷_0𝑖,𝐷_1𝑖, and 𝐷_2𝑖. We denote that𝐴_𝑖(𝑡)=𝐴_𝑖+Δ𝐴_𝑖(𝑡),𝐴_𝑑𝑖(𝑡)=𝐴_𝑑𝑖+Δ𝐴_𝑑𝑖(𝑡),𝐵_1𝑖(𝑡)=𝐵_1𝑖+Δ𝐵_1𝑖(𝑡),𝐸_1𝑖(𝑡)=𝐸_1𝑖+Δ𝐸_1𝑖(𝑡),𝐶_𝑖(𝑡)=𝐶_𝑖+Δ𝐶_𝑖(𝑡),𝐶_𝑑𝑖(𝑡)=𝐶_𝑑𝑖+Δ𝐶_𝑑𝑖(𝑡),𝐵_2𝑖(𝑡)=𝐵_2𝑖+Δ𝐵_2𝑖(𝑡),𝐸_2𝑖(𝑡)=𝐸_2𝑖+Δ𝐸_2𝑖(𝑡),(2.5) where 𝐴_𝑖,𝐴_𝑑𝑖,𝐵_1𝑖,𝐸_1𝑖,𝐶_𝑖,𝐶_𝑑𝑖,𝐵_2𝑖,𝐸_2𝑖, and 𝐷_0𝑖,𝐷_1𝑖,𝐷_2𝑖 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 that𝐼^𝑖_kn=𝑗∶if𝜋_𝑖𝑗isknow𝐼^𝑖_uk=𝑗∶if𝜋_𝑖𝑗isunknow(2.6) moreover, if 𝐼^𝑖_kn≠∅, it is further described as 𝐼^𝑖_kn={𝑘^𝑖₁,𝑘^𝑖₂,…,𝑘^𝑖_𝑚},1≤𝑚≤𝑁−2.

Remark 2.1. 𝑘^𝑖_𝑙∈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:𝑢(𝑡)=𝐾_𝑖𝑥(𝑡),(2.7) then, the closed-loop MJS can be represented as𝑥̇(𝑡)=𝐴_𝑖(𝑡)+𝐵_1𝑖(𝑡)𝐾_𝑖𝑥(𝑡)+𝐴_𝑑𝑖(𝑡)+𝐸_1𝑖(𝑡)𝐾_𝑖𝑥𝑡−𝜏_𝑖(𝑡)+𝐷_0𝑖𝑓(𝑥(𝑡),𝑖)+𝐷_1𝑖𝜔(𝑡),𝑧(𝑡)=𝐶_𝑖(𝑡)+𝐵_2𝑖(𝑡)𝐾_𝑖𝑥(𝑡)+𝐶_𝑑𝑖(𝑡)+𝐸_2𝑖(𝑡)𝐾_𝑖𝑥𝑡−𝜏_𝑖(𝑡)+𝐷_2𝑖𝜔(𝑡).(2.8)

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: Δ𝐴_𝑖(𝑡)Δ𝐴_𝑑𝑖(𝑡)Δ𝐵_1𝑖(𝑡)Δ𝐸_1𝑖(𝑡)Δ𝐶_𝑖(𝑡)Δ𝐶_𝑑𝑖(𝑡)Δ𝐵_2𝑖(𝑡)Δ𝐸_2𝑖(𝑡)=𝑇_1𝑖𝑇_2𝑖𝐹_𝑖(𝑡)𝑁_1𝑖𝑁_2𝑖𝑁_3𝑖𝑁_4𝑖,(2.9) where 𝑇_1𝑖,𝑇_2𝑖, and 𝑁_𝑘𝑖,𝑘=1,2,3,4,𝑖∈𝑆 are known real constant matrices with appropriate dimensions and 𝐹_𝑖(𝑡),forall𝑖∈𝑆, are unknown time-varying matrix functions satisfying 𝐹^𝑇_𝑖(𝑡)𝐹_𝑖(𝑡)≤𝐼.(2.10)

Remark 2.2. It is assumed that all the elements 𝐹_𝑖(𝑡),forall𝑖∈𝑆, are Lebesgue measurable. The matrices Δ𝐴_𝑖(𝑡),Δ𝐴_𝑑𝑖(𝑡),Δ𝐵_1𝑖(𝑡),Δ𝐸_1𝑖(𝑡),Δ𝐶_𝑖(𝑡),Δ𝐶_𝑑𝑖(𝑡),Δ𝐵_2𝑖(𝑡), and Δ𝐸_2𝑖(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 delay 𝜏_𝑖(𝑡) satisfies 0≤𝜏_1𝑖≤𝜏_𝑖(𝑡)≤𝜏_2𝑖,.𝜏_𝑖(𝑡)≤𝜇_𝑖, with 𝜏_1𝑖,𝜏_2𝑖, and 𝜇_𝑖 being real constant scalars for each for all 𝑖∈𝑆.

Assumption 3. For a fixed system mode 𝑟_𝑡=𝑖∈𝑆, there exists a know real constant mode-dependent matrix Γ_𝑖=diag(𝑘_1𝑖,𝑘_2𝑖,…,𝑘_𝑛𝑖)>0 such that the nonlinear vector function 𝑓(⋅,⋅) satisfy the following conditions: 𝑓^𝑇(𝑥(𝑡),𝑖)𝑓(𝑥(𝑡),𝑖)−Γ_𝑖𝑥(𝑡)≤0.(2.11)

Definition 2.3 (see [39]). The MJS (2.8) is said to be passive if there exists a constant 𝛿 such that 2𝐸^𝑇₀𝑧^𝑇(𝑡)𝜔(𝑡)𝑑𝑡≥𝛿(2.12) 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 ℝ^𝑛, and positive scalars 𝜌, 𝛾 such that for all 𝜔(𝑡), all initial conditions 𝑥(0), all 𝑡≥0, the following inequality holds: 𝐿𝑉𝑥_𝑡,𝑟_𝑡+𝜌𝑉𝑥_𝑡,𝑟_𝑡−𝛾𝜔^𝑇(𝑡)𝜔(𝑡)≤2𝑧^𝑇(𝑡)𝜔(𝑡).(2.13)

Remark 2.5. From Definition 2.4, 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 2𝐸^𝑇₀𝑧^𝑇(𝑡)𝜔(𝑡)𝑑𝑡≥−𝐸𝑉𝑥₀−𝛾𝐸^𝑇₀𝜔^𝑇(𝑡)𝜔(𝑡)𝑑𝑡=𝛿.(2.14) 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]). Let 𝑄(𝑥)=𝑄^𝑇(𝑥),𝑅(𝑥)=𝑅^𝑇(𝑥), and 𝑆(𝑥) depend affinely on 𝑥. Then the following linear matrix inequality: 𝑄(𝑥)𝑆(𝑥)𝑆^𝑇(𝑥)𝑅(𝑥)>0(2.15) holds if and only if one of the following conditions holds:(1)𝑅(𝑥)>0,𝑄(𝑥)−𝑆(𝑥)𝑅⁻¹(𝑥)𝑆^𝑇(𝑥)>0;(2)𝑄(𝑥)>0,𝑅(𝑥)−𝑆^𝑇(𝑥)𝑄⁻¹(𝑥)𝑆(𝑥)>0.

Lemma 2.7 (see [40]). Let 𝐴,𝐷,𝑆,𝐹, and 𝑃 be real matrices of appropriate dimensions with 𝑃>0 and 𝐹 satisfy 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼. Then the following statement holds.(1)For any scalar 𝜀>0𝐷𝐹𝑆+(𝐷𝐹𝑆)^𝑇≤𝜀⁻¹𝐷𝐷^𝑇+𝜀𝑆^𝑇𝑆.(2.16)(2)For any vectors 𝑥 and 𝑦 with appropriate dimensions2𝑥^𝑇𝐴𝐷𝑦≤𝑥^𝑇𝐴𝑃𝐴^𝑇𝑥+𝑦^𝑇𝐷^𝑇𝑃⁻¹𝐷𝑦.(2.17)

Lemma 2.8 (see [41]). Let 𝐴,𝑋 be real matrices with appropriate dimensions. Then there exist a matrix 𝑃=𝑃^𝑇>0 such that 𝑃𝐴^𝑇+𝐴𝑃+𝑋<0, if and only if, there exists a scalar 𝜀>0 and 𝑍 such that ⎡⎢⎢⎣−𝑍−𝑍^𝑇𝑍^𝑇𝐴^𝑇+𝑃𝑍^𝑇∗−𝜀⁻¹𝑃+𝑋0∗∗−𝜀𝑃⎤⎥⎥⎦<0.(2.18)

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 matrix 𝐾_𝑖,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 matrix 𝐾_𝑖, the uncertain MJS (2.8) is exponentially passive in the sense of expectation if there exists positive definite matrices 𝑃_𝑖,𝑄_𝑖,𝑄₁,𝑄₂,𝑄₃,𝑄^∗,𝑍₁,𝑍₂, positive scalars 𝛾,𝜀_1𝑖,𝜀_2𝑖, and for any matrices 𝐺_𝑖,𝑀_𝑖,𝑅_𝑖,𝑈_𝑖,𝑉_𝑖,𝐻_𝑖 with appropriate dimensions such that the following matrices inequalities hold for all 𝑖=1,2,…,𝑁: ⎛⎜⎜⎜⎜⎜⎜⎜⎝Ω¹_𝑖,9×9Λ₁Λ₂Λ₃Λ_4𝑘Λ_5𝑘Λ₆Λ₇∗−𝑍₂0000√2𝜏₂𝑍₂𝑇_1𝑖𝜀_2𝑖0∗∗−𝑍₂00000∗∗∗−𝑍₂0000∗∗∗∗−𝑍₂000∗∗∗∗∗−𝑍₂00∗∗∗∗∗∗−𝜀_2𝑖0∗∗∗∗∗∗∗−𝜀_2𝑖⎞⎟⎟⎟⎟⎟⎟⎟⎠<0𝑘=1,2.(3.1)Case 1. If 𝜋_𝑖𝑖∈𝐼^𝑖_kn𝜋_𝑖𝑖𝑄_𝑖−𝑄^∗𝑄_𝑗∗−𝑄_𝑗_{∀𝑗∈𝐼𝑖uk}<0,(3.2)⎛⎜⎜⎜⎜⎜⎝⎛⎜⎜⎝1+_{𝑗∈𝐼𝑖kn}𝜋_𝑖𝑗⎞⎟⎟⎠𝜋_𝑖𝑖𝑄_𝑖−𝑄^∗√𝜋_𝑖𝑘^𝑖₁𝑄_𝑘^𝑖₁⋯√𝜋_𝑖𝑘^𝑖_𝑚𝑄_𝑘^𝑖_𝑚∗−𝑄_𝑘^𝑖₁00∗∗⋱0∗∗∗−𝑄_𝑘^𝑖_𝑚⎞⎟⎟⎟⎟⎟⎠<0,(3.3)𝑃_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑗<0∀𝑗∈𝐼^𝑖_uk.(3.4)Case 2. If 𝜋_𝑖𝑖∈𝐼^𝑖_uk𝑄_𝑗−𝑄^∗>0∀𝑗∈𝐼^𝑖_uk,𝑗=𝑖,(3.5)𝑄_𝑗−𝑄^∗<0∀𝑗∈𝐼^𝑖_uk,𝑗≠𝑖,(3.6)⎛⎜⎜⎜⎜⎜⎝⎛⎜⎜⎝−1−_{𝑗∈𝐼𝑖kn}𝜋_𝑖𝑗⎞⎟⎟⎠𝑄^∗√𝜋_𝑖𝑘^𝑖₁𝑄_𝑘^𝑖₁⋯√𝜋_𝑖𝑘^𝑖_𝑚𝑄_𝑘^𝑖_𝑚∗−𝑄_𝑘^𝑖₁00∗∗⋱0∗∗∗−𝑄_𝑘^𝑖_𝑚⎞⎟⎟⎟⎟⎟⎠<0,(3.7)𝑃_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑗>0∀𝑗∈𝐼^𝑖_uk,𝑗=𝑖,(3.8)𝑃_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑗<0∀𝑗∈𝐼^𝑖_uk,𝑗≠𝑖,(3.9)whereΩ¹_𝑖,11=𝑃_𝑖⎛⎜⎜⎝⎛⎜⎜⎝1+_{𝑗∈𝐼𝑖kn}𝜋_𝑖𝑗⎞⎟⎟⎠𝐴_𝑖+𝐵_1𝑖𝐾_𝑖⎞⎟⎟⎠+⎛⎜⎜⎝⎛⎜⎜⎝1+_{𝑗∈𝐼𝑖kn}𝜋_𝑖𝑗⎞⎟⎟⎠𝐴_𝑖+𝐵_1𝑖𝐾_𝑖⎞⎟⎟⎠^𝑇𝑃_𝑖+_{𝑗∈𝐼𝑖kn}𝜋_𝑖𝑗𝑃_𝑗+𝑄_𝑖+𝜏_2𝑖𝑄^∗+𝑄₁+𝑄₂+𝑄₃+𝜏_2𝑖−𝜏_1𝑖𝑍₁+𝐺^𝑇_1𝑖+𝐺_1𝑖,Ω¹_𝑖,12=−𝐺_1𝑖+𝐺^𝑇_2𝑖+𝑀_1𝑖,Ω¹_𝑖,13=𝑅_1𝑖+𝐺^𝑇_3𝑖−𝑀_1𝑖,Ω¹_𝑖,14=−𝑅_1𝑖+𝐺^𝑇_4𝑖+𝑈_1𝑖+𝑃_𝑖𝐴_𝑑𝑖+𝐸_1𝑖𝐾_𝑖,Ω¹_𝑖,15=𝑉_1𝑖+𝐺^𝑇_5𝑖−𝑈_1𝑖,Ω¹_𝑖,16=−𝑉_1𝑖+𝐺^𝑇_6𝑖+𝐻_1𝑖,Ω¹_𝑖,17=𝐺^𝑇_7𝑖−𝐻_1𝑖Ω¹_𝑖,18=𝐺^𝑇_8𝑖+𝜀_1𝑖Γ_𝑖+𝑃_𝑖𝐷_0𝑖,Ω¹_𝑖,19=𝑃_𝑖𝐷_1𝑖−𝐶_𝑖+𝐵_2𝑖𝐾_𝑖^𝑇,Ω¹_𝑖,22=−𝐺^𝑇_2𝑖−𝐺_2𝑖+𝑀^𝑇_2𝑖+𝑀_2𝑖−𝑄₁,Ω¹_𝑖,23=−𝐺^𝑇_3𝑖+𝑅_2𝑖+𝑀^𝑇_3𝑖−𝑀_2𝑖,Ω¹_𝑖,24=−𝐺^𝑇_4𝑖−𝑅_2𝑖+𝑀^𝑇_4𝑖+𝑈_2𝑖,Ω¹_𝑖,25=−𝐺^𝑇_5𝑖+𝑉_2𝑖+𝑀^𝑇_5𝑖−𝑈_2𝑖,Ω¹_𝑖,26=−𝐺^𝑇_6𝑖−𝑉_2𝑖+𝑀^𝑇_6𝑖+𝐻_2𝑖,Ω¹_𝑖,27=−𝐺^𝑇_7𝑖+𝑀^𝑇_7𝑖−𝐻_2𝑖,Ω¹_𝑖,28=−𝐺^𝑇_8𝑖+𝑀^𝑇_8𝑖,Ω¹_𝑖,29=0,Ω¹_𝑖,33=𝑅^𝑇_3𝑖+𝑅_3𝑖−𝑀^𝑇_3𝑖−𝑀_3𝑖,Ω¹_𝑖,34=𝑅^𝑇_4𝑖−𝑅_3𝑖−𝑀^𝑇_4𝑖+𝑈_3𝑖,Ω¹_𝑖,35=𝑅^𝑇_5𝑖+𝑉_3𝑖−𝑀^𝑇_5𝑖−𝑈_3𝑖,Ω¹_𝑖,36=𝑅^𝑇_6𝑖−𝑉_3𝑖−𝑀^𝑇_6𝑖+𝐻_3𝑖,Ω¹_𝑖,37=𝑅^𝑇_7𝑖−𝑀^𝑇_7𝑖−𝐻_3𝑖,Ω¹_𝑖,38=𝑅^𝑇_8𝑖−𝑀^𝑇_8𝑖Ω¹_𝑖,39=0,Ω¹_𝑖,44=−𝑅^𝑇_4𝑖−𝑅_4𝑖+𝑈^𝑇_4𝑖+𝑈_4𝑖−1−𝜇_𝑖𝑄_𝑖,Ω¹_𝑖,45=−𝑅^𝑇_5𝑖+𝑉_4𝑖+𝑈^𝑇_5𝑖−𝑈_4𝑖,Ω¹_𝑖,46=−𝑅^𝑇_6𝑖−𝑉_4𝑖+𝑈^𝑇_6𝑖+𝐻_4𝑖,Ω¹_𝑖,47=−𝑅^𝑇_7𝑖+𝑈^𝑇_7𝑖−𝐻_4𝑖,Ω¹_𝑖,48=−𝑅^𝑇_8𝑖+𝑈^𝑇_8𝑖,Ω¹_𝑖,49=−𝐶_𝑑𝑖+𝐸_2𝑖𝐾_𝑖^𝑇,Ω¹_𝑖,55=𝑉^𝑇_5𝑖+𝑉_5𝑖−𝑈^𝑇_5𝑖−𝑈_5𝑖,Ω¹_𝑖,56=−𝑈^𝑇_6𝑖−𝑉_5𝑖+𝑉^𝑇_6𝑖+𝐻_5𝑖,Ω¹_𝑖,57=𝑉^𝑇_7𝑖−𝑈^𝑇_7𝑖−𝐻_5𝑖,Ω¹_𝑖,58=𝑉^𝑇_8𝑖−𝑈^𝑇_8𝑖Ω¹_𝑖,59=0,Ω¹_𝑖,66=−𝑉^𝑇_6𝑖−𝑉_6𝑖+𝐻^𝑇_6𝑖+𝐻_6𝑖−𝑄₂,Ω¹_𝑖,67=−𝑉^𝑇_7𝑖+𝐻^𝑇_7𝑖−𝐻_6𝑖,Ω¹_𝑖,68=−𝑉^𝑇_8𝑖+𝐻^𝑇_8𝑖,Ω¹_𝑖,69=0,Ω¹_𝑖,77=−𝐻^𝑇_7𝑖−𝐻_7𝑖−𝑄₃,Ω¹_𝑖,78=−𝐻^𝑇_8𝑖,Ω¹_𝑖,79=0,Ω¹_𝑖,88=−2𝜀_1𝑖𝐼,Ω¹_𝑖,89=0,Ω¹_𝑖,99=−𝐷_2𝑖−𝐷^𝑇_2𝑖−𝛾𝐼,𝜏₂=max_𝑖∈𝑆𝜏_2𝑖,𝜏₁=min_𝑖∈𝑆𝜏_1𝑖,Λ₁=2𝜏₂𝑍₂𝐴_𝑖+𝐵_1𝑖𝐾_𝑖,0,0,2𝜏₂𝑍₂𝐴_𝑑𝑖+𝐸_1𝑖𝐾_𝑖,0,0,0,2𝜏₂𝑍₂𝐷_0𝑖,2𝜏₂𝑍₂𝐷_1𝑖^𝑇,Λ₁(𝑡)=2𝜏₂𝑍₂𝐴_𝑖(𝑡)+𝐵_1𝑖(𝑡)𝐾_𝑖,0,0,2𝜏₂𝑍₂𝐴_𝑑𝑖(𝑡)+𝐸_1𝑖(𝑡)𝐾_𝑖,0,0,0,2𝜏₂𝑍₂𝐷_0𝑖,2𝜏₂𝑍₂𝐷_1𝑖^𝑇,Λ₂=𝜏₂−𝜏_2𝑖𝐻_𝑖,Λ₃=√𝜏_1𝑖𝐺_𝑖,Λ₄₁=𝜏_2𝑖−𝜏_1𝑖2𝑀_𝑖,Λ₄₂=𝜏_2𝑖−𝜏_1𝑖2𝑈_𝑖,Λ₅₁=𝜏_2𝑖−𝜏_1𝑖2𝑅_𝑖,Λ₅₂=𝜏_2𝑖−𝜏_1𝑖2𝑉_𝑖,Λ₆=𝜀_2𝑖𝑇^𝑇_1𝑖𝑃_𝑖,0,0,0,0,0,0,0,−𝜀_2𝑖𝑇^𝑇_2𝑖^𝑇,Λ₇=𝑁_1𝑖+𝑁_3𝑖𝐾_𝑖,0,0,𝑁_2𝑖+𝑁_4𝑖𝐾_𝑖,0,0,0,0,0^𝑇,𝐺_𝑖=𝐺^𝑇_1𝑖𝐺^𝑇_2𝑖𝐺^𝑇_3𝑖𝐺^𝑇_4𝑖𝐺^𝑇_5𝑖𝐺^𝑇_6𝑖𝐺^𝑇_7𝑖𝐺^𝑇_8𝑖0^𝑇𝑀_𝑖=𝑀^𝑇_1𝑖𝑀^𝑇_2𝑖𝑀^𝑇_3𝑖𝑀^𝑇_4𝑖𝑀^𝑇_5𝑖𝑀^𝑇_6𝑖𝑀^𝑇_7𝑖𝑀^𝑇_8𝑖0^𝑇,𝑅_𝑖=𝑅^𝑇_1𝑖𝑅^𝑇_2𝑖𝑅^𝑇_3𝑖𝑅^𝑇_4𝑖𝑅^𝑇_5𝑖𝑅^𝑇_6𝑖𝑅^𝑇_7𝑖𝑅^𝑇_8𝑖0^𝑇,𝑈_𝑖=𝑈^𝑇_1𝑖𝑈^𝑇_2𝑖𝑈^𝑇_3𝑖𝑈^𝑇_4𝑖𝑈^𝑇_5𝑖𝑈^𝑇_6𝑖𝑈^𝑇_7𝑖𝑈^𝑇_8𝑖0^𝑇,𝑉_𝑖=𝑉^𝑇_1𝑖𝑉^𝑇_2𝑖𝑉^𝑇_3𝑖𝑉^𝑇_4𝑖𝑉^𝑇_5𝑖𝑉^𝑇_6𝑖𝑉^𝑇_7𝑖𝑉^𝑇_8𝑖0^𝑇,𝐻_𝑖=𝐻^𝑇_1𝑖𝐻^𝑇_2𝑖𝐻^𝑇_3𝑖𝐻^𝑇_4𝑖𝐻^𝑇_5𝑖𝐻^𝑇_6𝑖𝐻^𝑇_7𝑖𝐻^𝑇_8𝑖0^𝑇,(3.10)

Proof. First, in order to cast our model involved in the framework of the Markov process, we define a new process 𝑥_𝑡(𝑠)=𝑥(𝑡+𝑠),𝑠∈[−𝜏₂,0], and let 𝐿 be the weak infinitesimal generator of the random process 𝑥_𝑡(𝑠),𝑡≥0 and 𝐿𝑣𝑥_𝑡,𝑟_𝑡=lim_Δ→0+1Δ𝐸𝑣𝑥_𝑡+Δ,𝑟_𝑡+Δ∣𝑥_𝑡,𝑟_𝑡=𝑖−𝑣𝑥_𝑡,𝑟_𝑡.(3.11) Now consider the Lyapunov-Krasovskii functional as follows for 𝑟_𝑡=𝑖,𝑖∈1,2,…,𝑆: 𝑣𝑥_𝑡,𝑖=𝑣₁𝑥_𝑡,𝑖+𝑣₂𝑥_𝑡,𝑖+𝑣₃𝑥_𝑡,𝑖+𝑣₄𝑥_𝑡,𝑖+𝑣₅𝑥_𝑡,𝑖,(3.12) where 𝑣₁𝑥_𝑡,𝑖=𝑥^𝑇(𝑡)𝑃(𝑖)𝑥(𝑡),𝑣₂𝑥_𝑡,𝑖=^𝑡_{𝑡−𝜏𝑖(𝑡)}𝑥^𝑇(𝑠)𝑄(𝑖)𝑥(𝑠)𝑑𝑠,𝑣₃𝑥_𝑡,𝑖=⁰_−𝜏2𝑖^𝑡_𝑡+𝜃𝑥^𝑇(𝑠)𝑄^∗𝑥(𝑠)𝑑𝑠𝑑𝜃,𝑣₄𝑥_𝑡,𝑖=^𝑡_{𝑡−𝜏1𝑖}𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)𝑑𝑠+^𝑡_{𝑡−𝜏2𝑖}𝑥^𝑇(𝑠)𝑄₂𝑥(𝑠)𝑑𝑠+^𝑡_𝑡−𝜏2𝑥^𝑇(𝑠)𝑄₃𝑥(𝑠)𝑑,𝑣₅𝑥_𝑡,𝑖=^−𝜏1𝑖_−𝜏2𝑖^𝑡_𝑡+𝜃𝑥^𝑇(𝑠)𝑍₁𝑥(𝑠)𝑑𝑠𝑑𝜃+2⁰_−𝜏2^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠𝑑𝜃,(3.13) where ^𝑁_𝑗=1𝜋_𝑖𝑗𝑄_𝑗≤𝑄^∗.(3.14) In order to show the exponential passivity of the MJS (2.8) under the given controller gain matrix 𝐾_𝑖, we set 𝐽^∗=𝐿𝑣𝑥_𝑡,𝑖−𝛾𝜔^𝑇(𝑡)𝜔(𝑡)−2𝑧^𝑇(𝑡)𝜔(𝑡).(3.15)
Notice that𝐿𝑣₁𝑥_𝑡,𝑖=𝑥^𝑇(𝑡)𝑃_𝑖𝐴_𝑖(𝑡)+𝐵_1𝑖(𝑡)𝐾_𝑖+𝐴_𝑖(𝑡)+𝐵_1𝑖(𝑡)𝐾_𝑖^𝑇𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃_𝑖𝐴_𝑑𝑖(𝑡)+𝐸_1𝑖(𝑡)𝐾_𝑖𝑥𝑡−𝜏_𝑖(𝑡)+2𝑥^𝑇(𝑡)𝑃_𝑖𝐷_0𝑖𝑓(𝑥(𝑡),𝑖)+2𝑥^𝑇(𝑡)𝑃_𝑖𝐷_1𝑖𝜔(𝑡),𝐿𝑣₂𝑥_𝑡,𝑖≤𝑥^𝑇(𝑡)𝑄_𝑖𝑥(𝑡)−1−𝜇_𝑖𝑥^𝑇𝑡−𝜏_𝑖(𝑡)𝑄_𝑖𝑥𝑡−𝜏_𝑖(𝑡)+^𝑡_{𝑡−𝜏𝑖(𝑡)}𝑥^𝑇(𝑠)^𝑁_𝑗=1𝜋_𝑖𝑗𝑄_𝑗𝑥(𝑠)𝑑𝑠,𝐿𝑣₃𝑥_𝑡,𝑖≤𝜏_2𝑖𝑥^𝑇(𝑡)𝑄^∗𝑥(𝑡)−∫^𝑡_{𝑡−𝜏𝑖(𝑡)}𝑥^𝑇(𝑠)𝑄^∗𝑥(𝑠)𝑑𝑠,𝐿𝑣₄𝑥_𝑡,𝑖=𝑥^𝑇(𝑡)𝑄₁+𝑄₂+𝑄₃𝑥(𝑡)−𝑥^𝑇𝑡−𝜏_1𝑖𝑄₁𝑥𝑡−𝜏_1𝑖−𝑥^𝑇𝑡−𝜏_2𝑖𝑄₂𝑥𝑡−𝜏_2𝑖−𝑥^𝑇𝑡−𝜏₂𝑄₃𝑥𝑡−𝜏₂,𝐿𝑣₅𝑥_𝑡,𝑖=𝜏_2𝑖−𝜏_1𝑖𝑥^𝑇(𝑡)𝑍₁𝑥(𝑡)+2𝜏₂𝑥̇^𝑇(𝑡)𝑍₂𝑥̇(𝑡)−^{𝑡−𝜏1𝑖}_{𝑡−𝜏2𝑖}𝑥^𝑇(𝑠)𝑍₁𝑥(𝑠)𝑑𝑠−2^𝑡_𝑡−𝜏2𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠=𝜏_2𝑖−𝜏_1𝑖𝑥^𝑇(𝑡)𝑍₁𝑥(𝑡)+2𝜏₂𝑥̇^𝑇(𝑡)𝑍₂𝑥̇(𝑡)−^{𝑡−𝜏2𝑖}_𝑡−𝜏2𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠−^{𝑡−(𝜏𝑖(𝑡)+𝜏2𝑖)/2}_{𝑡−𝜏2𝑖}𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠−^{𝑡−𝜏𝑖(𝑡)}_{𝑡−𝜏𝑖(𝑡)+𝜏2𝑖/2}𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠−^{𝑡−(𝜏𝑖(𝑡)+𝜏1𝑖)/2}_{𝑡−𝜏𝑖(𝑡)}𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠−^{𝑡−𝜏1𝑖}_{𝑡−𝜏𝑖(𝑡)+𝜏1𝑖/2}𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠−^𝑡_{𝑡−𝜏1𝑖}𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠−^{𝑡−𝜏1𝑖}_{𝑡−𝜏2𝑖}𝑥^𝑇(𝑠)𝑍₁𝑥(𝑠)𝑑𝑠−^𝑡_𝑡−𝜏2𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠.(3.16)
Then using Newton-Leibniz formula, for any matrices 𝐻_𝑖,𝐺_𝑖,𝑀_𝑖,𝑅_𝑖,𝑈_𝑖,𝑉_𝑖 we have2𝜉^𝑇(𝑡)𝐺_𝑖𝑥(𝑡)−𝑥𝑡−𝜏_1𝑖−^𝑡_{𝑡−𝜏1𝑖}𝑥̇(𝑠)𝑑𝑠=0,2𝜉^𝑇(𝑡)𝑀_𝑖⎛⎜⎝𝑥𝑡−𝜏_1𝑖−𝑥𝑡−𝜏_1𝑖+𝜏_𝑖(𝑡)2−^{𝑡−𝜏1𝑖}_{𝑡−𝜏1𝑖+𝜏𝑖(𝑡)/2}𝑥̇(𝑠)𝑑𝑠⎞⎟⎠=0,2𝜉^𝑇(𝑡)𝑅_𝑖𝑥𝑡−𝜏_1𝑖+𝜏_𝑖(𝑡)2−𝑥𝑡−𝜏_𝑖(𝑡)−^{𝑡−(𝜏1𝑖+𝜏𝑖(𝑡))/2}_{𝑡−𝜏𝑖(𝑡)}𝑥̇(𝑠)𝑑𝑠=0,2𝜉^𝑇(𝑡)𝑈_𝑖⎛⎜⎝𝑥𝑡−𝜏_𝑖(𝑡)−𝑥𝑡−𝜏_2𝑖+𝜏_𝑖(𝑡)2−^{𝑡−𝜏𝑖(𝑡)}_{𝑡−𝜏2𝑖+𝜏𝑖(𝑡)/2}𝑥̇(𝑠)𝑑𝑠⎞⎟⎠=0,2𝜉^𝑇(𝑡)𝑉_𝑖𝑥𝑡−𝜏_2𝑖+𝜏_𝑖(𝑡)2−𝑥𝑡−𝜏_2𝑖−^{𝑡−(𝜏2𝑖+𝜏𝑖(𝑡))/2}_{𝑡−𝜏2𝑖}𝑥̇(𝑠)𝑑𝑠=0,2𝜉^𝑇(𝑡)𝐻_𝑖𝑥𝑡−𝜏_2𝑖−𝑥𝑡−𝜏₂−^{𝑡−𝜏2𝑖}_𝑡−𝜏2𝑥̇(𝑠)𝑑𝑡=0,(3.17) where 𝜉^𝑇(𝑡)=𝑥^𝑇(𝑡),𝑥^𝑇𝑡−𝜏_1𝑖,𝑥^𝑇𝑡−𝜏_𝑖(𝑡)+𝜏_1𝑖2,𝑥^𝑇𝑡−𝜏_𝑖(𝑡),𝑥^𝑇𝑡−𝜏_𝑖(𝑡)+𝜏_2𝑖2,𝑥^𝑇𝑡−𝜏_2𝑖𝑥^𝑇𝑡−𝜏₂,𝑓^𝑇(𝑥(𝑡),𝑖),𝜔^𝑇(𝑡).(3.18)
From the Lemma 2.7 (2.2), it is easy to see that−2𝜉^𝑇(𝑡)𝐺_𝑖^𝑡_{𝑡−𝜏1𝑖}𝑥̇(𝑠)𝑑𝑠≤𝜏_1𝑖𝜉^𝑇(𝑡)𝐺_𝑖𝑍⁻¹₂𝐺^𝑇_𝑖𝜉(𝑡)+^𝑡_{𝑡−𝜏1𝑖}𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠−2𝜉^𝑇(𝑡)𝑀_𝑖^{𝑡−𝜏1𝑖}_{𝑡−(𝜏1𝑖+𝜏𝑖(𝑡))/2}𝑥̇(𝑠)𝑑𝑠≤𝜏_𝑖(𝑡)−𝜏_1𝑖2𝜉^𝑇(𝑡)𝑀_𝑖𝑍⁻¹₂𝑀^𝑇_𝑖𝜉(𝑡)+^{𝑡−𝜏1𝑖}_{𝑡−(𝜏1𝑖+𝜏𝑖(𝑡))/2}𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠−2𝜉^𝑇(𝑡)𝑅_𝑖^{𝑡−(𝜏1𝑖+𝜏𝑖(𝑡))/2}_{𝑡−𝜏𝑖(𝑡)}𝑥̇(𝑠)𝑑𝑠≤𝜏_𝑖(𝑡)−𝜏_1𝑖2𝜉^𝑇(𝑡)𝑅_𝑖𝑍⁻¹₂𝑅^𝑇_𝑖𝜉(𝑡)+^{𝑡−(𝜏1𝑖+𝜏𝑖(𝑡))/2}_{𝑡−𝜏𝑖(𝑡)}𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠−2𝜉^𝑇(𝑡)𝑈_𝑖^{𝑡−𝜏𝑖(𝑡)}_{𝑡−(𝜏2𝑖+𝜏𝑖(𝑡))/2}𝑥̇(𝑠)𝑑𝑠≤𝜏_2𝑖−𝜏_𝑖(𝑡)2𝜉^𝑇(𝑡)𝑈_𝑖𝑍⁻¹₂𝑈^𝑇_𝑖𝜉(𝑡)+^{𝑡−𝜏𝑖(𝑡)}_{𝑡−(𝜏2𝑖+𝜏𝑖(𝑡))/2}𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠−2𝜉^𝑇(𝑡)𝑉_𝑖^{𝑡−(𝜏2𝑖+𝜏𝑖(𝑡))/2}_{𝑡−𝜏2𝑖}𝑥̇(𝑠)𝑑𝑠≤𝜏_2𝑖−𝜏_𝑖(𝑡)2𝜉^𝑇(𝑡)𝑉_𝑖𝑍⁻¹₂𝑉^𝑇_𝑖𝜉(𝑡)+^{𝑡−(𝜏2𝑖+𝜏𝑖(𝑡))/2}_{𝑡−𝜏2𝑖}𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠−2𝜉^𝑇(𝑡)𝐻_𝑖^{𝑡−𝜏2𝑖}_𝑡−𝜏2𝑥̇(𝑠)𝑑𝑠≤𝜏₂−𝜏_2𝑖𝜉^𝑇(𝑡)𝐻_𝑖𝑍⁻¹₂𝐻^𝑇_𝑖𝜉(𝑡)+^{𝑡−𝜏2𝑖}_𝑡−𝜏2𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠.(3.19)
Now by Assumption 3, it can be deduced that for any positive scalar 𝜀_1𝑖,𝑖∈1,2,…,𝑆,2𝜀_1𝑖𝑓^𝑇(𝑥(𝑡),𝑖)Γ_𝑖𝑥(𝑡)−𝑓(𝑥(𝑡),𝑖)≥0.(3.20) Then from the above discussion, we can see that 𝐽^∗≤𝜉^𝑇(𝑡)𝜏_𝑖(𝑡)−𝜏_1𝑖𝜏_2𝑖−𝜏_1𝑖Φ₁(𝑡)+𝜏_2𝑖−𝜏_𝑖(𝑡)𝜏_2𝑖−𝜏_1𝑖Φ₂(𝑡)𝜉(𝑡)−^{𝑡−𝜏1𝑖}_{𝑡−𝜏2𝑖}𝑥^𝑇(𝑠)𝑍₁𝑥(𝑠)𝑑𝑠−^𝑡_𝑡−𝜏2𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠,Φ₁(𝑡)=Ω_𝑖,9×9(𝑡)+Λ₁(𝑡)𝑍⁻¹₂Λ₁(𝑡)^𝑇+𝜏₂−𝜏_2𝑖𝐻_𝑖𝑍⁻¹₂𝐻^𝑇_𝑖+𝜏_1𝑖𝐺_𝑖𝑍⁻¹₂𝐺^𝑇_𝑖+𝜏_2𝑖−𝜏_1𝑖2𝑀_𝑖𝑍⁻¹₂𝑀^𝑇_𝑖+𝑅_𝑖𝑍⁻¹₂𝑅^𝑇_𝑖,Φ₂(𝑡)=Ω_𝑖,9×9(𝑡)+Λ₁(𝑡)𝑍⁻¹₂Λ₁(𝑡)^𝑇+𝜏₂−𝜏_2𝑖𝐻_𝑖𝑍⁻¹₂𝐻^𝑇_𝑖+𝜏_1𝑖𝐺_𝑖𝑍⁻¹₂𝐺^𝑇_𝑖+𝜏_2𝑖−𝜏_1𝑖2𝑈_𝑖𝑍⁻¹₂𝑈^𝑇_𝑖+𝑉_𝑖𝑍⁻¹₂𝑉^𝑇_𝑖,(3.21) where Ω_𝑖,11(𝑡)=𝑃_𝑖𝐴_𝑖(𝑡)+𝐵_1𝑖(𝑡)𝐾_𝑖+𝐴_𝑖(𝑡)+𝐵_1𝑖(𝑡)𝐾_𝑖^𝑇𝑃_𝑖+^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗+𝑄_𝑖+𝜏_2𝑖𝑄^∗+𝑄₁+𝑄₂+𝑄₃+𝜏_2𝑖−𝜏_1𝑖𝑍₁+𝐺^𝑇_1𝑖+𝐺_1𝑖,Ω_𝑖,14(𝑡)=𝑃_𝑖𝐴_𝑑𝑖(𝑡)+𝐸_1𝑖(𝑡)𝐾_𝑖−𝑅_1𝑖+𝐺^𝑇_4𝑖+𝑈_1𝑖Ω_𝑖,19(𝑡)=𝑃_𝑖𝐷_1𝑖−𝐶_𝑖(𝑡)+𝐵_2𝑖(𝑡)𝐾_𝑖^𝑇,Ω_𝑖,49(𝑡)=−𝐶_𝑑𝑖(𝑡)+𝐸_2𝑖(𝑡)𝐾_𝑖^𝑇(3.22) other terms of Ω_{𝑖,𝑖×𝑗}(𝑡) are similar to Ω¹_{𝑖,𝑖×𝑗}. 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 ⎛⎜⎜⎜⎜⎝Ω_𝑖,9×9Λ₁Λ₂Λ₃Λ_4𝑘Λ_5𝑘∗−𝑍₂0000∗∗−𝑍₂000∗∗∗−𝑍₂00∗∗∗∗−𝑍₂0∗∗∗∗∗−𝑍₂⎞⎟⎟⎟⎟⎠_𝑘=1,2+𝑇^𝑇_1𝑖𝑃_𝑖,0_𝑛×7𝑛,−𝑇^𝑇_2𝑖,2𝜏₂𝑇^𝑇_1𝑖𝑍₂,0_𝑛×4𝑛^𝑇𝐹_𝑖(𝑘)𝑁_1𝑖+𝑁_3𝑖𝐾_𝑖,0_𝑛×2𝑛,𝑁_2𝑖+𝑁_4𝑖𝐾_𝑖,0_𝑛×10𝑛+𝑁_1𝑖+𝑁_2𝑖𝐾_𝑖,0_𝑛×2𝑛,𝑁_2𝑖+𝑁_4𝑖𝐾_𝑖,0_𝑛×10𝑛^𝑇𝐹^𝑇_𝑖(𝑘)𝑇^𝑇_1𝑖𝑃_𝑖,0_𝑛×7𝑛,−𝑇^𝑇_2𝑖,2𝜏₂𝑇^𝑇_1𝑖𝑍₂,0_𝑛×4𝑛<0,(3.23) where Ω_𝑖,9×9 is the nominal matrix of Ω_𝑖,9×9(𝑡). Then from the Lemma 2.7 (2.1), above matrix inequality holds, which is equivalent to ⎛⎜⎜⎜⎜⎜⎜⎜⎝Ω_𝑖,9×9Λ₁Λ₂Λ₃Λ_4𝑘Λ_5𝑘Λ₆Λ₇∗−𝑍₂0000√2𝜏₂𝑍₂𝑇_1𝑖𝜀_2𝑖0∗∗−𝑍₂00000∗∗∗−𝑍₂0000∗∗∗∗−𝑍₂000∗∗∗∗∗−𝑍₂00∗∗∗∗∗∗−𝜀_2𝑖0∗∗∗∗∗∗∗−𝜀_2𝑖⎞⎟⎟⎟⎟⎟⎟⎟⎠_𝑘=1,2<0.(3.24)Case 1. If 𝜋_𝑖𝑖∈𝐼^𝑖_kn
then (3.24) is equivalent to⎛⎜⎜⎜⎜⎜⎜⎜⎝Ω¹_𝑖,9×9Λ₁Λ₂Λ₃Λ_4𝑘Λ_5𝑘Λ₆Λ₇∗−𝑍₂0000√2𝜏₂𝑍₂𝑇_1𝑖𝜀_2𝑖0∗∗−𝑍₂00000∗∗∗−𝑍₂0000∗∗∗∗−𝑍₂000∗∗∗∗∗−𝑍₂00∗∗∗∗∗∗−𝜀_2𝑖0∗∗∗∗∗∗∗−𝜀_2𝑖⎞⎟⎟⎟⎟⎟⎟⎟⎠_𝑘=1,2+diag⎛⎜⎜⎜⎝_𝑗∈_{𝐼𝑖uk𝑗≠𝑖}𝜋_𝑖𝑗𝑃_𝑗,0,…,^{15}0⎞⎟⎟⎟⎠+diag⎛⎜⎜⎜⎝_𝑗∈_{𝐼𝑖uk𝑗≠𝑖}𝜋_𝑖𝑗𝑃_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖,0,…,^{15}0⎞⎟⎟⎟⎠<0.(3.25) 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).∑^𝑁_𝑗=1𝜋_𝑖𝑗𝑄_𝑗<𝑄^∗ is equivalent to_𝑗∈_{𝐼𝑖kn𝑗≠𝑖}𝜋_𝑖𝑗𝑄_𝑗+_{𝑗∈𝐼𝑖uk𝑗≠𝑖}𝜋_𝑖𝑗𝑄_𝑗+𝜋_𝑖𝑖𝑄_𝑖−𝑄^∗+_{𝑗∈𝐼𝑖kn}𝜋_𝑖𝑗𝜋_𝑖𝑖𝑄_𝑖−𝑄^∗+_{𝑗∈𝐼𝑖uk}𝜋_𝑖𝑗𝜋_𝑖𝑖𝑄_𝑖−𝑄^∗<0.(3.26) If we have the following matrix inequalities hold, we can have that (3.14) is satisfied ⎛⎜⎜⎝1+_{𝑗∈𝐼𝑖kn}𝜋_𝑖𝑗⎞⎟⎟⎠𝜋_𝑖𝑖𝑄_𝑖−𝑄^∗+_{𝑗∈𝐼𝑖kn𝑗≠𝑖}𝜋_𝑖𝑗𝑄_𝑗<0,𝜋_𝑖𝑖𝑄_𝑖−𝑄^∗+𝑄_𝑗<0𝑗∈𝐼^𝑖_uk.(3.27) Obviously, (3.27) is equivalent to (3.2) and (3.3) by the Schur complement. Case 2. If 𝜋_𝑖𝑖∈𝐼^𝑖_uk
then (3.24) is equivalent to⎛⎜⎜⎜⎜⎜⎜⎜⎝Ω¹_𝑖,9×9Λ₁Λ₂Λ₃Λ_4𝑘Λ_5𝑘Λ₆Λ₇∗−𝑍₂0000√2𝜏₂𝑍₂𝑇_1𝑖𝜀_2𝑖0∗∗−𝑍₂00000∗∗∗−𝑍₂0000∗∗∗∗−𝑍₂000∗∗∗∗∗−𝑍₂00∗∗∗∗∗∗−𝜀_2𝑖0∗∗∗∗∗∗∗−𝜀_2𝑖⎞⎟⎟⎟⎟⎟⎟⎟⎠_𝑘=1,2+diag⎛⎜⎜⎜⎝_𝑗∈_{𝐼𝑖uk𝑗≠𝑖}𝜋_𝑖𝑗𝑃_𝑗,0,…,^{15}0⎞⎟⎟⎟⎠+diag⎛⎜⎜⎜⎝_𝑗∈_{𝐼𝑖uk𝑗≠𝑖}𝜋_𝑖𝑗𝑃_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖,0,…,^{15}0⎞⎟⎟⎟⎠+𝜋_𝑖𝑖diag⎛⎜⎜⎜⎝𝑃_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖,0,…,^{15}0⎞⎟⎟⎟⎠<0.(3.28) 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, then𝐽^∗≤−𝜌₁‖𝑥(𝑡)‖²−𝜆_min𝑍₁^𝑡_𝑡−𝜏2‖𝑥(𝑠)‖²𝑑𝑠−𝜆_min𝑍₂^𝑡_𝑡−𝜏2‖𝑥̇(𝑠)‖²𝑑𝑠.(3.29) On the other hand, it is easy to obtain that 𝑣(𝑥(𝑡),𝑖)≤‖𝑃‖‖𝑥(𝑡)‖²+‖𝑄‖+‖𝑄^∗‖+‖‖𝑄₁‖‖+‖‖𝑄₂‖‖+‖‖𝑄₃‖‖+𝜏₂−𝜏₁‖‖𝑍₁‖‖×^𝑡_𝑡−𝜏2‖𝑥(𝑠)‖²𝑑𝑠+2𝜏₂‖‖𝑍₂‖‖^𝑡_𝑡−𝜏2‖𝑥̇(𝑠)‖²𝑑𝑠,(3.30) where ‖𝑃‖=max_𝑖∈𝑆{‖𝑃_𝑖‖}, ‖𝑄‖=max_𝑖∈𝑆{‖𝑄_𝑖‖}.Let 𝜌>0 be sufficiently small such that 𝜌‖𝑃‖−𝜌₁<0,𝜌‖𝑄‖+‖𝑄^∗‖+‖‖𝑄₁‖‖+‖‖𝑄₂‖‖+‖‖𝑄₃‖‖+𝜏₂−𝜏₁‖‖𝑍₁‖‖−𝜆_min𝑍₁<02𝜏₂𝜌‖‖𝑍₂‖‖−𝜆_min𝑍₂<0.,(3.31) 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 𝜇_𝑖 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 matrices 𝐾_𝑖 for convenience in the next section, (3.1) is not LMI, if we substitute 𝜀_2𝑖 by 𝜀⁻¹_2𝑖 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 matrices 𝐾_𝑖,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 constant 𝜀, 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 𝑃_𝑖,𝑄_𝑖,𝑄₁,𝑄₂,𝑄₃,𝑄^∗,𝑍₁,𝑍₂, positive scalar 𝜀_1𝑖,𝜀_2𝑖, and for any matrices 𝐺_𝑖,𝑀_𝑖,𝑅_𝑖,𝑈_𝑖,𝑉_𝑖,𝐻_𝑖,𝑍_𝑖𝑖 with appropriate dimensions satisfying the following LMIs under the two cases for all 𝑖=1,2,…,𝑁. Case 1. If 𝜋_𝑖𝑖∈𝐼^𝑖_kn⎛⎜⎝𝜋_𝑖𝑖𝑄_𝑖−𝑄^∗𝑄_𝑗∗−𝑄_𝑗⎞⎟⎠_{∀𝑗∈𝐼}^𝑖_uk<0,(3.32)⎛⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎝1+_{𝑗∈𝐼𝑖kn}𝜋_𝑖𝑗⎞⎟⎟⎠𝜋_𝑖𝑖𝑄_𝑖−𝑄^∗√𝜋_𝑖𝑘^𝑖₁𝑄_𝑘^𝑖₁⋯√𝜋_𝑖𝑘^𝑖_𝑚𝑄_𝑘^𝑖_𝑚∗−𝑄_𝑘^𝑖₁00∗∗⋱0∗∗∗−𝑄_𝑘^𝑖_𝑚⎞⎟⎟⎟⎟⎟⎟⎠<0,(3.33)⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝Ω¹_𝑖,9×9Λ₁Λ₂Λ₃Λ_4𝑘Λ_5𝑘Λ₆Λ₇Λ₈∗−𝑍₂0000√2𝜏₂𝑇_1𝑖𝜀_2𝑖00∗∗Θ000000∗∗∗Θ00000∗∗∗∗Θ0000∗∗∗∗∗Θ000∗∗∗∗∗∗−𝜀_2𝑖00∗∗∗∗∗∗∗−𝜀_2𝑖0∗∗∗∗∗∗∗∗Λ₉⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠<0𝑘=1,2,(3.34)⎛⎜⎜⎜⎝−𝑍_𝑖𝑖−𝑍^𝑇_𝑖𝑖𝑍^𝑇_𝑖𝑖𝐴^𝑇_𝑖+𝑃_𝑖𝑍^𝑇_𝑖𝑖0∗−𝜀⁻¹𝑃_𝑖0𝑃_𝑖∗∗−𝜀𝑃_𝑖0∗∗∗−𝑃_𝑗⎞⎟⎟⎟⎠_{∀𝑗∈𝐼}^𝑖_uk<0,(3.35)Case 2. If 𝜋_𝑖𝑖∈𝐼^𝑖_uk𝑄_𝑗−𝑄^∗>0∀𝑗∈𝐼^𝑖_uk,𝑗=𝑖,(3.36)𝑄_𝑗−𝑄^∗<0∀𝑗∈𝐼^𝑖_uk,𝑗≠𝑖,(3.37)⎛⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎝−1−_{𝑗∈𝐼𝑖kn}𝜋_𝑖𝑗⎞⎟⎟⎠𝑄^∗√𝜋_𝑖𝑘^𝑖₁𝑄_𝑘^𝑖₁⋯√𝜋_𝑖𝑘^𝑖_𝑚𝑄_𝑘^𝑖_𝑚∗−𝑄_𝑘^𝑖₁00∗∗⋱0∗∗∗−𝑄_𝑘^𝑖_𝑚⎞⎟⎟⎟⎟⎟⎟⎠<0,(3.38)⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝Ω²_𝑖,9×9Λ₁Λ₂Λ₃Λ_4𝑘Λ_5𝑘Λ₆Λ₇Λ₈∗−𝑍₂0000√2𝜏₂𝑇_1𝑖𝜀_2𝑖00∗∗Θ000000∗∗∗Θ00000∗∗∗∗Θ0000∗∗∗∗∗Θ000∗∗∗∗∗∗−𝜀_2𝑖00∗∗∗∗∗∗∗−𝜀_2𝑖0∗∗∗∗∗∗∗∗Λ₉⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠<0𝑘=1,2,(3.39)⎛⎜⎜⎝−𝑍_𝑖𝑖−𝑍^𝑇_𝑖𝑖−𝑍^𝑇_𝑖𝑖𝐴^𝑇_𝑖+𝑃_𝑖𝑍^𝑇_𝑖𝑖∗−𝜀⁻¹𝑃_𝑖−𝑃_𝑗0∗∗−𝜀𝑃_𝑖⎞⎟⎟⎠^{∀𝑗∈𝐼𝑖}_uk𝑗=𝑖<0,(3.40)⎛⎜⎜⎜⎝−𝑍_𝑖𝑖−𝑍^𝑇_𝑖𝑖𝑍^𝑇_𝑖𝑖𝐴^𝑇_𝑖+𝑃_𝑖𝑍^𝑇_𝑖𝑖0∗−𝜀⁻¹𝑃_𝑖0𝑃_𝑖∗∗−𝜀𝑃_𝑖0∗∗∗−𝑃_𝑗⎞⎟⎟⎟⎠^{∀𝑗∈𝐼𝑖}_{uk𝑗≠𝑖}<0,(3.41) where Ω¹_𝑖,11=⎛⎜⎜⎝⎛⎜⎜⎝1+_{𝑗∈𝐼𝑖kn}𝜋_𝑖𝑗⎞⎟⎟⎠𝐴_𝑖𝑃_𝑖+𝐵_1𝑖𝑌_𝑖⎞⎟⎟⎠+⎛⎜⎜⎝⎛⎜⎜⎝1+_{𝑗∈𝐼𝑖kn}𝜋_𝑖𝑗⎞⎟⎟⎠𝐴_𝑖𝑃_𝑖+𝐵_1𝑖𝑌_𝑖⎞⎟⎟⎠^𝑇+𝜋_𝑖𝑖𝑃_𝑖+𝑄_𝑖+𝜏_2𝑖𝑄^∗+𝑄₁+𝑄₂+𝑄₃+𝜏_2𝑖−𝜏_1𝑖𝑍₁+𝐺^𝑇_1𝑖+𝐺_1𝑖,Ω²_𝑖,11=⎛⎜⎜⎝⎛⎜⎜⎝1+_{𝑗∈𝐼𝑖kn}𝜋_𝑖𝑗⎞⎟⎟⎠𝐴_𝑖𝑃_𝑖+𝐵_1𝑖𝑌_𝑖⎞⎟⎟⎠+⎛⎜⎜⎝⎛⎜⎜⎝1+_{𝑗∈𝐼𝑖kn}𝜋_𝑖𝑗⎞⎟⎟⎠𝐴_𝑖𝑃_𝑖+𝐵_1𝑖𝑌_𝑖⎞⎟⎟⎠^𝑇+𝑄_𝑖+𝜏_2𝑖𝑄^∗+𝑄₁+𝑄₂+𝑄₃+𝜏_2𝑖−𝜏_1𝑖𝑍₁+𝐺^𝑇_1𝑖+𝐺_1𝑖,Ω¹_𝑖,12=Ω²_𝑖,12=−𝐺_1𝑖+𝐺^𝑇_2𝑖+𝑀_1𝑖,Ω¹_𝑖,13=Ω²_𝑖,13=𝑅_1𝑖+𝐺^𝑇_3𝑖−𝑀_1𝑖,Ω¹_𝑖,14=Ω²_𝑖,14=−𝑅_1𝑖+𝐺^𝑇_4𝑖+𝑈_1𝑖+𝐴_𝑑𝑖𝑃_𝑖+𝐸_1𝑖𝑌_𝑖Ω¹_𝑖,15=Ω²_𝑖,15=𝑉_1𝑖+𝐺^𝑇_5𝑖−𝑈_1𝑖,Ω¹_𝑖,16=Ω²_𝑖,16=−𝑉_1𝑖+𝐺^𝑇_6𝑖+𝐻_1𝑖,Ω¹_𝑖,17=Ω²_𝑖,17=𝐺^𝑇_7𝑖−𝐻_1𝑖,Ω¹_𝑖,18=Ω²_𝑖,18=𝐺^𝑇_8𝑖+𝑃_𝑖Γ_𝑖+𝐷_0𝑖𝜀_1𝑖,Ω¹_𝑖,19=Ω²_𝑖,19=𝐷_1𝑖−𝐶_𝑖𝑃_𝑖+𝐵_2𝑖𝑌_𝑖^𝑇,Ω¹_𝑖,22=Ω²_𝑖,22=−𝐺^𝑇_2𝑖−𝐺_2𝑖+𝑀^𝑇_2𝑖+𝑀_2𝑖−𝑄₁,Ω¹_𝑖,23=Ω²_𝑖,23=−𝐺^𝑇_3𝑖+𝑅_2𝑖+𝑀^𝑇_3𝑖−𝑀_2𝑖,Ω¹_𝑖,24=Ω²_𝑖,24=−𝐺^𝑇_4𝑖−𝑅_2𝑖+𝑀^𝑇_4𝑖+𝑈_2𝑖,Ω¹_𝑖,25=Ω²_𝑖,25=−𝐺^𝑇_5𝑖+𝑉_2𝑖+𝑀^𝑇_5𝑖−𝑈_2𝑖,Ω¹_𝑖,26=Ω²_𝑖,26=−𝐺^𝑇_6𝑖−𝑉_2𝑖+𝑀^𝑇_6𝑖+𝐻_2𝑖,Ω¹_𝑖,27=Ω²_𝑖,27=−𝐺^𝑇_7𝑖+𝑀^𝑇_7𝑖−𝐻_2𝑖,Ω¹_𝑖,28=Ω²_𝑖,28=−𝐺^𝑇_8𝑖+𝑀^𝑇_8𝑖,Ω¹_𝑖,29=Ω²_𝑖,29=0,Ω¹_𝑖,33=Ω²_𝑖,33=𝑅^𝑇_3𝑖+𝑅_3𝑖−𝑀^𝑇_3𝑖−𝑀_3𝑖,Ω¹_𝑖,34=Ω²_𝑖,34=𝑅^𝑇_4𝑖−𝑅_3𝑖−𝑀^𝑇_4𝑖+𝑈_3𝑖,Ω¹_𝑖,35=Ω²_𝑖,35=𝑅^𝑇_5𝑖+𝑉_3𝑖−𝑀^𝑇_5𝑖−𝑈_3𝑖,Ω¹_𝑖,36=Ω²_𝑖,36=𝑅^𝑇_6𝑖−𝑉_3𝑖−𝑀^𝑇_6𝑖+𝐻_3𝑖,Ω¹_𝑖,37=Ω²_𝑖,37=𝑅^𝑇_7𝑖−𝑀^𝑇_7𝑖−𝐻_3𝑖,Ω¹_𝑖,38=Ω²_𝑖,38=𝑅^𝑇_8𝑖−𝑀^𝑇_8𝑖,Ω¹_𝑖,39=Ω²_𝑖,39=0,Ω¹_𝑖,44=Ω²_𝑖,44=−𝑅^𝑇_4𝑖−𝑅_4𝑖+𝑈^𝑇_4𝑖+𝑈_4𝑖−1−𝜇_𝑖𝑄_𝑖,Ω¹_𝑖,45=Ω²_𝑖,45=−𝑅^𝑇_5𝑖+𝑉_4𝑖+𝑈^𝑇_5𝑖−𝑈_4𝑖,Ω¹_𝑖,46=Ω²_𝑖,46=−𝑅^𝑇_6𝑖−𝑉_4𝑖+𝑈^𝑇_6𝑖+𝐻_4𝑖,Ω¹_𝑖,47=Ω²_𝑖,47=−𝑅^𝑇_7𝑖+𝑈^𝑇_7𝑖−𝐻_4𝑖,Ω¹_𝑖,48=Ω²_𝑖,48=−𝑅^𝑇_8𝑖+𝑈^𝑇_8𝑖,Ω¹_𝑖,49=Ω²_𝑖,49=−𝐶_𝑑𝑖𝑃_𝑖+𝐸_2𝑖𝑌_𝑖^𝑇,Ω¹_𝑖,55=Ω²_𝑖,55=𝑉^𝑇_5𝑖+𝑉_5𝑖−𝑈^𝑇_5𝑖−𝑈_5𝑖,Ω¹_𝑖,56=Ω²_𝑖,56=−𝑈^𝑇_6𝑖−𝑉_5𝑖+𝑉^𝑇_6𝑖+𝐻_5𝑖,Ω¹_𝑖,57=Ω²_𝑖,57=𝑉^𝑇_7𝑖−𝑈^𝑇_7𝑖−𝐻_5𝑖,Ω¹_𝑖,58=Ω²_𝑖,58=𝑉^𝑇_8𝑖−𝑈^𝑇_8𝑖Ω¹_𝑖,59=Ω²_𝑖,59=0,Ω¹_𝑖,66=Ω²_𝑖,66=−𝑉^𝑇_6𝑖−𝑉_6𝑖+𝐻^𝑇_6𝑖+𝐻_6𝑖−𝑄₂,Ω¹_𝑖,67=Ω²_𝑖,67=−𝑉^𝑇_7𝑖+𝐻^𝑇_7𝑖−𝐻_6𝑖,Ω¹_𝑖,68=Ω²_𝑖,68=−𝑉^𝑇_8𝑖+𝐻^𝑇_8𝑖,Ω¹_𝑖,69=Ω²_𝑖,69=0,Ω¹_𝑖,77=Ω²_𝑖,77=−𝐻^𝑇_7𝑖−𝐻_7𝑖−𝑄₃,Ω¹_𝑖,78=Ω²_𝑖,78=−𝐻^𝑇_8𝑖,Ω¹_𝑖,79=Ω²_𝑖,79=0,Ω¹_𝑖,88=Ω²_𝑖,88=−2𝜀_1𝑖𝐼,Ω¹_𝑖,89=Ω²_𝑖,89=0,Ω¹_𝑖,99=Ω²_𝑖,99=−𝐷_2𝑖−𝐷^𝑇_2𝑖−𝛾,Θ=𝐽^𝑇𝑍₂𝐽−𝐽^𝑇𝑃_𝑖−𝑃_𝑖,Λ₁=2𝜏₂𝐴_𝑖𝑃_𝑖+𝐵_1𝑖𝑌_𝑖,0,0,2𝜏₂𝐴_𝑑𝑖𝑃_𝑖+𝐸_1𝑖𝑌_𝑖,0,0,0,2𝜏₂𝐷_0𝑖𝜀_1𝑖,2𝜏₂𝐷_1𝑖^𝑇,Λ₂=𝜏₂−𝜏_2𝑖𝐻_𝑖,Λ₃=√𝜏_1𝑖𝐺_𝑖,Λ₄₁=𝜏_2𝑖−𝜏_1𝑖2𝑀_𝑖,Λ₄₂=𝜏_2𝑖−𝜏_1𝑖2𝑈_𝑖,Λ₅₁=𝜏_2𝑖−𝜏_1𝑖2𝑅_𝑖,Λ₅₂=𝜏_2𝑖−𝜏_1𝑖2𝑉_𝑖,Λ₆=𝜀_2𝑖𝑇^𝑇_1𝑖,0,0,0,0,0,0,0,−𝜀_2𝑖𝑇^𝑇_2𝑖^𝑇,Λ₇=𝑁_1𝑖𝑃_𝑖+𝑁_3𝑖𝑌_𝑖,0,0,𝑁_2𝑖𝑃_𝑖+𝑁_4𝑖𝑌_𝑖,0,0,0,0,0^𝑇,Λ₈=√𝜋_𝑖𝑘^𝑖₁𝑃_𝑖√𝜋_𝑖𝑘^𝑖₂𝑃_𝑖⋯√𝜋_𝑖𝑘^𝑖_𝑚𝑃_𝑖0_8𝑛×𝑛0_8𝑛×𝑛⋯0_8𝑛×𝑛,Λ₉=diag−𝑃_𝑘^𝑖₁,−𝑃_𝑘^𝑖₂,…,−𝑃_𝑘^𝑖_𝑚,𝑍₂=𝑍⁻¹₂,𝑌_𝑖=𝐾_𝑖𝑃_𝑖,𝜀_1𝑖=𝜀⁻¹_1𝑖,𝑃_𝑖=𝑃⁻¹_𝑖,𝑄_𝑖=𝑃⁻¹_𝑖𝑄_𝑖𝑃⁻¹_𝑖,𝑄₁=𝑃⁻¹_𝑖𝑄₁𝑃⁻¹_𝑖,𝑄₂=𝑃⁻¹_𝑖𝑄₂𝑃⁻¹_𝑖,𝑄₃=𝑃⁻¹_𝑖𝑄₃𝑃⁻¹_𝑖,Σ=diag⎛⎜⎜⎜⎝^{7}𝑃_𝑖,…,𝑃_𝑖⎞⎟⎟⎟⎠,𝐺_𝑖=𝐺^𝑇_1𝑖,…,𝐺^𝑇_8𝑖,0^𝑇=𝑃_𝑖𝐺^𝑇_1𝑖,…,𝐺^𝑇_7𝑖Σ,𝜀_1𝑖𝑃_𝑖𝐺^𝑇_8𝑖,0^𝑇,𝐻_𝑖=𝐻^𝑇_1𝑖,…,𝐻^𝑇_8𝑖,0^𝑇=𝑃_𝑖𝐻^𝑇_1𝑖,…,𝐻^𝑇_7𝑖Σ,𝜀_1𝑖𝑃_𝑖𝐻^𝑇_8𝑖,0^𝑇,𝑀_𝑖=𝑀^𝑇_1𝑖,…,𝑀^𝑇_8𝑖,0^𝑇=𝑃_𝑖𝑀^𝑇_1𝑖,…,𝑀^𝑇_7𝑖Σ,𝜀_1𝑖𝑃_𝑖𝑀^𝑇_8𝑖,0^𝑇,𝑅_𝑖=𝑅^𝑇_1𝑖,…,𝑅^𝑇_8𝑖,0^𝑇=𝑃_𝑖𝑅^𝑇_1𝑖,…,𝑅^𝑇_7𝑖Σ,𝜀_1𝑖𝑃_𝑖𝑅^𝑇_8𝑖,0^𝑇,𝑈_𝑖=𝑈^𝑇_1𝑖,…,𝑈^𝑇_8𝑖,0^𝑇=𝑃_𝑖𝑈^𝑇_1𝑖,…,𝑈^𝑇_7𝑖Σ,𝜀_1𝑖𝑃_𝑖𝑈^𝑇_8𝑖,0^𝑇,𝑉_𝑖=𝑉^𝑇_1𝑖,…,𝑉^𝑇_8𝑖,0^𝑇=𝑃_𝑖𝑉^𝑇_1𝑖,…,𝑉^𝑇_7𝑖Σ,𝜀_1𝑖𝑃_𝑖𝑉^𝑇_8𝑖,0^𝑇,(3.42) 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 𝐾_𝑖=𝑌_𝑖𝑃_𝑖for all𝑖∈𝑆.

Proof. At first, we list the following fact: 𝐽^𝑇𝑍₂𝐽−𝐽^𝑇𝑃_𝑖−𝑃_𝑖𝐽+𝑃_𝑖𝑍₂𝑃_𝑖=𝑍₂𝐽−𝑃_𝑖^𝑇𝑍₂𝑍₂𝐽−𝑃_𝑖≥0(3.43) which implies that −𝑃_𝑖𝑍₂𝑃≤Θ=𝐽^𝑇𝑍₂𝐽−𝐽^𝑇𝑃_𝑖−𝑃_𝑖𝐽.(3.44)
Now perform a congruence transformation to (3.1) bydiag⎛⎜⎜⎜⎝^{7}𝑃⁻¹_𝑖,…,𝑃⁻¹_𝑖,𝜀⁻¹_1𝑖,𝐼,𝑍⁻¹₂,𝑃⁻¹_𝑖,…,𝑃^{4−1}_𝑖,𝐼,𝐼⎞⎟⎟⎟⎠.(3.45)
If 𝜋_𝑖𝑖∈𝐼^𝑖_kn, then by the Schur complement and (3.44), we can infer that (3.34) is established. In the same way, if 𝜋_𝑖𝑖∈𝐼^𝑖_uk, (3.39) is established.
From Lemma 2.8, we can see that (3.35) is equivalent to 𝐴_𝑖𝑃_𝑖+𝑃_𝑖𝐴^𝑇_𝑖+𝑃_𝑖𝑃_𝑗𝑃_𝑖<0forall𝑗∈𝐼^𝑖_uk, 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 𝐾_𝑖=𝑌_𝑖𝑃_𝑖. This completes the proof.

Remark 3.5. To reduce the conservatism, when estimating 𝐿𝑣₅(𝑥_𝑡,𝑖),−∫^𝑡_𝑡−𝜏2𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠 is not simply enlarged as −∫^𝑡_{𝑡−𝜏𝑖(𝑡)}𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠, but −∫^{𝑡−(𝜏1𝑖+𝜏𝑖(𝑡))/2}_{𝑡−𝜏𝑖(𝑡)}𝑥̇^𝑇(𝑠)𝑍₂𝑥̇(𝑠)𝑑𝑠, −∫^{𝑡−𝜏𝑖(𝑡)}_{𝑡−(𝜏2𝑖+𝜏𝑖(𝑡))/2}𝑥̇^𝑇(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, Theorem 3.1 gives a exponential passivity criteria for MJS (2.8) with 𝜏_1𝑖≤𝜏_𝑖(𝑡)≤𝜏_2𝑖,.𝜏_𝑖(𝑡)≤𝜇_𝑖, where 𝜇_𝑖 is a given constant. In many cases, 𝜇_𝑖 is unknown. Considering this case, a rate-independent criteria for a delay satisfying 𝜏_1𝑖≤𝜏_𝑖(𝑡)≤𝜏_2𝑖 is derived as follows by setting 𝑄_𝑖=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:𝐴₁=−0.05−0.050.5−0.5,𝐴₂=−0.05−0.091.5−0.1,𝐴₃=−0.03−0.0150.05−0.01,𝐴_𝑑1=0.110.24−0.53−0.37,𝐴_𝑑2=−0.590.01−0.07−0.61,𝐴_𝑑3=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,𝐶_𝑑1=−1.00.2,𝐶_𝑑2=0.1−0.1,𝐶_𝑑3=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,(4.1) The two cases of the transition rates matrices are described as follows:Case1∶Π=⎛⎜⎝−0.50.20.30.2−0.60.40.50.3−0.8⎞⎟⎠,Case2∶Π=⎛⎜⎝−0.5??0.2−0.60.40.5??⎞⎟⎠,(4.2) 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:𝑃₁=3.2525−0.1821−0.18214.2081,𝑃₂=2.6050−0.2795−0.27953.6119,𝑃₃=3.2386−0.2026−0.20263.8813,𝑄₁=1.8478−0.0727−0.07272.0824,𝑄₂=1.5779−0.5056−0.50561.7197,𝑄₃=1.44620.21090.21091.8895,𝑄₁=0.5074−0.1035−0.10350.7639,𝑄₂=0.3321−0.0760−0.07600.5493,𝑍₁=1.4315−0.2159−0.21591.8214,𝑍₂=3.0280−0.2085−0.20854.1893,𝑌₁=−0.73760.1326,𝑌₂=−1.0932−0.3296,𝑌₃=−0.8734−0.5801.(4.3) case 2:𝑃₁=1.0𝑒+004∗0.13040.35860.35862.6532,𝑃₂=1.0𝑒+004∗0.57680.46640.46643.9812,𝑃₃=1.0𝑒+004∗0.49620.76080.76084.3848,𝑄₁=1.0𝑒+004∗0.08640.23300.23301.7220,𝑄₂=1.0𝑒+003∗0.19180.39250.39253.7184,𝑄₃=1.0𝑒+004∗0.05370.21330.21331.5653,𝑄₁=1.0𝑒+003∗0.06580.32810.32812.4649,𝑄₂=1.0𝑒+003∗0.04290.22060.22061.6811,𝑍₁=1.0𝑒+003∗0.23120.92730.92737.3269,𝑍₂=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.(4.4)

Under the two cases above, Table 1 lists the state-feedback controller gains matrix 𝐾_𝑖, 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.

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 matrix 𝐾_𝑖, 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.

Acknowledgments

This work was supported in part by the Opening Fund of Geomathematics Key Laboratory of Sichuan Province (scsxdz2011001) and the National Basic Research Program of China (2010CB732501).