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Journal of Applied Mathematics
Volume 2012, Article ID 950590, 24 pages
http://dx.doi.org/10.1155/2012/950590
Research Article

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

1School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China
2Key Laboratory for Neuroinformation of Ministry of Education, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China

Received 24 August 2011; Accepted 28 November 2011

Academic Editor: Ying U. Hu

Copyright © 2012 Mengzhuo Luo and Shouming Zhong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [24]. 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 [57], stability analysis problems [812], and control problems [1320], 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 [2127].

Recently, the passivity problems for a variety of practical systems have been attracting renewing attention [2831]. 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 [3338]. 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 [3537] 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 𝑛 and 𝑛×𝑚 denote, resp., the 𝑛-dimensioned Euclidean space and the set of all 𝑛×𝑚 real matrices. The notation 𝑋𝑌 (resp., 𝑋>𝑌) means that 𝑋 and 𝑌 are symmetric matrices, and that 𝑋𝑌 is positive semidefinitive (resp., positive definite). is the Euclidean norm in 𝑛. 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)𝑡0,) be a complete probability space with a filteration. (𝔽𝑡)𝑡0 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 𝐿2𝔽0([𝜏2,0]𝑛) the family of all 𝔽0 measurable 𝐶([𝜏2,0]𝑛)-valued random variables 𝜑={𝜑(𝑠)𝜏2𝑠0} such that sup𝜏2𝑠0𝐸𝜑(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 delayṡ𝑥(𝑡)=𝐴𝑡,𝑟𝑡𝑥(𝑡)+𝐴𝑑𝑡,𝑟𝑡𝑥𝑡𝜏𝑡,𝑟𝑡+𝐵1𝑡,𝑟𝑡𝑢(𝑡)+𝐸1𝑡,𝑟𝑡𝑢𝑡𝜏𝑡,𝑟𝑡+𝐷0𝑟𝑡𝑓𝑥(𝑡),𝑟𝑡+𝐷1𝑟𝑡𝜔(𝑡).(2.1)𝑧(𝑡)=𝐶𝑡,𝑟𝑡𝑥(𝑡)+𝐶𝑑𝑡,𝑟𝑡𝑥𝑡𝜏𝑡,𝑟𝑡+𝐵2𝑡,𝑟𝑡𝑢(𝑡)+𝐸2𝑡,𝑟𝑡𝑢𝑡𝜏𝑡,𝑟𝑡+𝐷2𝑟𝑡𝜔(𝑡),(2.2) Here 𝑥(𝑡)𝑛 is the state vector, 𝑢(𝑡)𝑝 is the control input, 𝑧(𝑡)𝑞 is the control output, and 𝜔(𝑡)𝑙 is the exogenous disturbance input which belongs to 𝕃2[0,],{𝑟𝑡,𝑡0} 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,𝑗𝑆 and has the mode transition probabilitiesPr𝑟𝑡+Δ𝑡=𝑗𝑟𝑡=𝑖=𝜋𝑖𝑗Δ𝑡+𝑜(Δ𝑡)𝑖𝑗,1+𝜋𝑖𝑖Δ𝑡+𝑜(Δ𝑡)𝑖=𝑗,(2.3) where Δ𝑡>0,limΔ𝑡0(𝑜(Δ𝑡)/Δ𝑡)=0,𝜋𝑖𝑗 is the transition rete from 𝑖 to 𝑗, and𝜋𝑖𝑖=𝑗𝑖𝜋𝑖𝑗,𝜋𝑖𝑗0,𝑗𝑖.(2.4)

For notational simplicity, which 𝑟𝑡=𝑖,𝑖𝑆, 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 that𝐴𝑖(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),(2.5) 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 that𝐼𝑖kn=𝑗if𝜋𝑖𝑗isknow𝐼𝑖uk=𝑗if𝜋𝑖𝑗isunknow(2.6) moreover, if 𝐼𝑖kn, it is further described as 𝐼𝑖kn={𝑘𝑖1,𝑘𝑖2,,𝑘𝑖𝑚},1𝑚𝑁2.

Remark 2.1. 𝑘𝑖𝑙𝑁+,𝑙{1,2,,𝑚} represents the index of the 𝑙th known element in the 𝑖th row of transition rate matrix. The case 𝑚=𝑁1 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)=𝐾𝑖𝑥(𝑡),(2.7) then, the closed-loop MJS can be represented aṡ𝑥(𝑡)=𝐴𝑖(𝑡)+𝐵1𝑖(𝑡)𝐾𝑖𝑥(𝑡)+𝐴𝑑𝑖(𝑡)+𝐸1𝑖(𝑡)𝐾𝑖𝑥𝑡𝜏𝑖(𝑡)+𝐷0𝑖𝑓(𝑥(𝑡),𝑖)+𝐷1𝑖𝜔(𝑡),𝑧(𝑡)=𝐶𝑖(𝑡)+𝐵2𝑖(𝑡)𝐾𝑖𝑥(𝑡)+𝐶𝑑𝑖(𝑡)+𝐸2𝑖(𝑡)𝐾𝑖𝑥𝑡𝜏𝑖(𝑡)+𝐷2𝑖𝜔(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: Δ𝐴𝑖(𝑡)Δ𝐴𝑑𝑖(𝑡)Δ𝐵1𝑖(𝑡)Δ𝐸1𝑖(𝑡)Δ𝐶𝑖(𝑡)Δ𝐶𝑑𝑖(𝑡)Δ𝐵2𝑖(𝑡)Δ𝐸2𝑖(𝑡)=𝑇1𝑖𝑇2𝑖𝐹𝑖(𝑡)𝑁1𝑖𝑁2𝑖𝑁3𝑖𝑁4𝑖,(2.9) where 𝑇1𝑖,𝑇2𝑖, and 𝑁𝑘𝑖,𝑘=1,2,3,4,𝑖𝑆 are known real constant matrices with appropriate dimensions and 𝐹𝑖(𝑡),forall𝑖𝑆, are unknown time-varying matrix functions satisfying 𝐹𝑇𝑖(𝑡)𝐹𝑖(𝑡)𝐼.(2.10)

Remark 2.2. It is assumed that all the elements 𝐹𝑖(𝑡),forall𝑖𝑆, 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 delay 𝜏𝑖(𝑡) satisfies 0𝜏1𝑖𝜏𝑖(𝑡)𝜏2𝑖,.𝜏𝑖(𝑡)𝜇𝑖, with 𝜏1𝑖,𝜏2𝑖, and 𝜇𝑖 being real constant scalars for each for all 𝑖𝑆.

Assumption 3. For a fixed system mode 𝑟𝑡=𝑖𝑆, 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: 𝑓𝑇(𝑥(𝑡),𝑖)𝑓(𝑥(𝑡),𝑖)Γ𝑖𝑥(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 2𝐸𝑇0𝑧𝑇(𝑡)𝜔(𝑡)𝑑𝑡𝛿(2.12) holds for all 𝑇0.

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 𝑛, and positive scalars 𝜌, 𝛾 such that for all 𝜔(𝑡), all initial conditions 𝑥(0), all 𝑡0, the following inequality holds: 𝐿𝑉𝑥𝑡,𝑟𝑡+𝜌𝑉𝑥𝑡,𝑟𝑡𝛾𝜔𝑇(𝑡)𝜔(𝑡)2𝑧𝑇(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 2𝐸𝑇0𝑧𝑇(𝑡)𝜔(𝑡)𝑑𝑡𝐸𝑉𝑥0𝛾𝐸𝑇0𝜔𝑇(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)=𝑄𝑇(𝑥),𝑅(𝑥)=𝑅𝑇(𝑥), and 𝑆(𝑥) depend affinely on 𝑥. Then the following linear matrix inequality: 𝑄(𝑥)𝑆(𝑥)𝑆𝑇(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)𝑅1(𝑥)𝑆𝑇(𝑥)>0;(2)𝑄(𝑥)>0,𝑅(𝑥)𝑆𝑇(𝑥)𝑄1(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 𝐹𝑇(𝑡)𝐹(𝑡)𝐼. Then the following statement holds.(1)For any scalar 𝜀>0𝐷𝐹𝑆+(𝐷𝐹𝑆)𝑇𝜀1𝐷𝐷𝑇+𝜀𝑆𝑇𝑆.(2.16)(2)For any vectors 𝑥 and 𝑦 with appropriate dimensions2𝑥𝑇𝐴𝐷𝑦𝑥𝑇𝐴𝑃𝐴𝑇𝑥+𝑦𝑇𝐷𝑇𝑃1𝐷𝑦.(2.17)

Lemma 2.8 (see [41]). Let 𝐴,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 𝑍𝑍𝑇𝑍𝑇𝐴𝑇+𝑃𝑍𝑇𝜀1𝑃+𝑋0𝜀𝑃<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 𝐾𝑖,𝑖𝑆, 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 𝐾𝑖, the uncertain MJS (2.8) is exponentially passive in the sense of expectation if there exists positive definite matrices 𝑃𝑖,𝑄𝑖,𝑄1,𝑄2,𝑄3,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: Ω1𝑖,9×9Λ1Λ2Λ3Λ4𝑘Λ5𝑘Λ6Λ7𝑍200002𝜏2𝑍2𝑇1𝑖𝜀2𝑖0𝑍200000𝑍20000𝑍2000𝑍200𝜀2𝑖0𝜀2𝑖<0𝑘=1,2.(3.1)Case 1. If 𝜋𝑖𝑖𝐼𝑖kn𝜋𝑖𝑖𝑄𝑖𝑄𝑄𝑗𝑄𝑗𝑗𝐼𝑖uk<0,(3.2)1+𝑗𝐼𝑖kn𝜋𝑖𝑗𝜋𝑖𝑖𝑄𝑖𝑄𝜋𝑖𝑘𝑖1𝑄𝑘𝑖1𝜋𝑖𝑘𝑖𝑚𝑄𝑘𝑖𝑚𝑄𝑘𝑖1000𝑄𝑘𝑖𝑚<0,(3.3)𝑃𝑖𝐴𝑖+𝐴𝑇𝑖𝑃𝑖+𝑃𝑗<0𝑗𝐼𝑖uk.(3.4)Case 2. If 𝜋𝑖𝑖𝐼𝑖uk𝑄𝑗𝑄>0𝑗𝐼𝑖uk,𝑗=𝑖,(3.5)𝑄𝑗𝑄<0𝑗𝐼𝑖uk,𝑗𝑖,(3.6)1𝑗𝐼𝑖kn𝜋𝑖𝑗𝑄𝜋𝑖𝑘𝑖1𝑄𝑘𝑖1𝜋𝑖𝑘𝑖𝑚𝑄𝑘𝑖𝑚𝑄𝑘𝑖1000𝑄𝑘𝑖𝑚<0,(3.7)𝑃𝑖𝐴𝑖+𝐴𝑇𝑖𝑃𝑖+𝑃𝑗>0𝑗𝐼𝑖uk,𝑗=𝑖,(3.8)𝑃𝑖𝐴𝑖+𝐴𝑇𝑖𝑃𝑖+𝑃𝑗<0𝑗𝐼𝑖uk,𝑗𝑖,(3.9)whereΩ1𝑖,11=𝑃𝑖1+𝑗𝐼𝑖kn𝜋𝑖𝑗𝐴𝑖+𝐵1𝑖𝐾𝑖+1+𝑗𝐼𝑖kn𝜋𝑖𝑗𝐴𝑖+𝐵1𝑖𝐾𝑖𝑇𝑃𝑖+𝑗𝐼𝑖kn𝜋𝑖𝑗𝑃𝑗+𝑄𝑖+𝜏2𝑖𝑄+𝑄1+𝑄2+𝑄3+𝜏2𝑖𝜏1𝑖𝑍1+𝐺𝑇1𝑖+𝐺1𝑖,Ω1𝑖,12=𝐺1𝑖+𝐺𝑇2𝑖+𝑀1𝑖,Ω1𝑖,13=𝑅1𝑖+𝐺𝑇3𝑖𝑀1𝑖,Ω1𝑖,14=𝑅1𝑖+𝐺𝑇4𝑖+𝑈1𝑖+𝑃𝑖𝐴𝑑𝑖+𝐸1𝑖𝐾𝑖,Ω1𝑖,15=𝑉1𝑖+𝐺𝑇5𝑖𝑈1𝑖,Ω1𝑖,16=𝑉1𝑖+𝐺𝑇6𝑖+𝐻1𝑖,Ω1𝑖,17=𝐺𝑇7𝑖𝐻1𝑖Ω1𝑖,18=𝐺𝑇8𝑖+𝜀1𝑖Γ𝑖+𝑃𝑖𝐷0𝑖,Ω1𝑖,19=𝑃𝑖𝐷1𝑖𝐶𝑖+𝐵2𝑖𝐾𝑖𝑇,Ω1𝑖,22=𝐺𝑇2𝑖𝐺2𝑖+𝑀𝑇2𝑖+𝑀2𝑖𝑄1,Ω1𝑖,23=𝐺𝑇3𝑖+𝑅2𝑖+𝑀𝑇3𝑖𝑀2𝑖,Ω1𝑖,24=𝐺𝑇4𝑖𝑅2𝑖+𝑀𝑇4𝑖+𝑈2𝑖,Ω1𝑖,25=𝐺𝑇5𝑖+𝑉2𝑖+𝑀𝑇5𝑖𝑈2𝑖,Ω1𝑖,26=𝐺𝑇6𝑖𝑉2𝑖+𝑀𝑇6𝑖+𝐻2𝑖,Ω1𝑖,27=𝐺𝑇7𝑖+𝑀𝑇7𝑖𝐻2𝑖,Ω1𝑖,28=𝐺𝑇8𝑖+𝑀𝑇8𝑖,Ω1𝑖,29=0,Ω1𝑖,33=𝑅𝑇3𝑖+𝑅3𝑖𝑀𝑇3𝑖𝑀3𝑖,Ω1𝑖,34=𝑅𝑇4𝑖𝑅3𝑖𝑀𝑇4𝑖+𝑈3𝑖,Ω1𝑖,35=𝑅𝑇5𝑖+𝑉3𝑖𝑀𝑇5𝑖𝑈3𝑖,Ω1𝑖,36=𝑅𝑇6𝑖𝑉3𝑖𝑀𝑇6𝑖+𝐻3𝑖,Ω1𝑖,37=𝑅𝑇7𝑖𝑀𝑇7𝑖𝐻3𝑖,Ω1𝑖,38=𝑅𝑇8𝑖𝑀𝑇8𝑖Ω1𝑖,39=0,Ω1𝑖,44=𝑅𝑇4𝑖𝑅4𝑖+𝑈𝑇4𝑖+𝑈4𝑖1𝜇𝑖𝑄𝑖,Ω1𝑖,45=𝑅𝑇5𝑖+𝑉4𝑖+𝑈𝑇5𝑖𝑈4𝑖,Ω1𝑖,46=𝑅𝑇6𝑖𝑉4𝑖+𝑈𝑇6𝑖+𝐻4𝑖,Ω1𝑖,47=𝑅𝑇7𝑖+𝑈𝑇7𝑖𝐻4𝑖,Ω1𝑖,48=𝑅𝑇8𝑖+𝑈𝑇8𝑖,Ω1𝑖,49=𝐶𝑑𝑖+𝐸2𝑖𝐾𝑖𝑇,Ω1𝑖,55=𝑉𝑇5𝑖+𝑉5𝑖𝑈𝑇5𝑖𝑈5𝑖,Ω1𝑖,56=𝑈𝑇6𝑖𝑉5𝑖+𝑉𝑇6𝑖+𝐻5𝑖,Ω1𝑖,57=𝑉𝑇7𝑖𝑈𝑇7𝑖𝐻5𝑖,Ω1𝑖,58=𝑉𝑇8𝑖𝑈𝑇8𝑖Ω1𝑖,59=0,Ω1𝑖,66=𝑉𝑇6𝑖𝑉6𝑖+𝐻𝑇6𝑖+𝐻6𝑖𝑄2,Ω1𝑖,67=𝑉𝑇7𝑖+𝐻𝑇7𝑖𝐻6𝑖,Ω1𝑖,68=𝑉𝑇8𝑖+𝐻𝑇8𝑖,Ω1𝑖,69=0,Ω1𝑖,77=𝐻𝑇7𝑖𝐻7𝑖𝑄3,Ω1𝑖,78=𝐻𝑇8𝑖,Ω1𝑖,79=0,Ω1𝑖,88=2𝜀1𝑖𝐼,Ω1𝑖,89=0,Ω1𝑖,99=𝐷2𝑖𝐷𝑇2𝑖𝛾𝐼,𝜏2=max𝑖𝑆𝜏2𝑖,𝜏1=min𝑖𝑆𝜏1𝑖,Λ1=2𝜏2𝑍2𝐴𝑖+𝐵1𝑖𝐾𝑖,0,0,2𝜏2𝑍2𝐴𝑑𝑖+𝐸1𝑖𝐾𝑖,0,0,0,2𝜏2𝑍2𝐷0𝑖,2𝜏2𝑍2𝐷1𝑖𝑇,Λ1(𝑡)=2𝜏2𝑍2𝐴𝑖(𝑡)+𝐵1𝑖(𝑡)𝐾𝑖,0,0,2𝜏2𝑍2𝐴𝑑𝑖(𝑡)+𝐸1𝑖(𝑡)𝐾𝑖,0,0,0,2𝜏2𝑍2𝐷0𝑖,2𝜏2𝑍2𝐷1𝑖𝑇,Λ2=𝜏2𝜏2𝑖𝐻𝑖,Λ3=𝜏1𝑖𝐺𝑖,Λ41=𝜏2𝑖𝜏1𝑖2𝑀𝑖,Λ42=𝜏2𝑖𝜏1𝑖2𝑈𝑖,Λ51=𝜏2𝑖𝜏1𝑖2𝑅𝑖,Λ52=𝜏2𝑖𝜏1𝑖2𝑉𝑖,Λ6=𝜀2𝑖𝑇𝑇1𝑖𝑃𝑖,0,0,0,0,0,0,0,𝜀2𝑖𝑇𝑇2𝑖𝑇,Λ7=𝑁1𝑖+𝑁3𝑖𝐾𝑖,0,0,𝑁2𝑖+𝑁4𝑖𝐾𝑖,0,0,0,0,0𝑇,𝐺𝑖=𝐺𝑇1𝑖𝐺𝑇2𝑖𝐺𝑇3𝑖𝐺𝑇4𝑖𝐺𝑇5𝑖𝐺𝑇6𝑖𝐺𝑇7𝑖𝐺𝑇8𝑖0𝑇𝑀𝑖=𝑀𝑇1𝑖𝑀𝑇2𝑖𝑀𝑇3𝑖𝑀𝑇4𝑖𝑀𝑇5𝑖𝑀𝑇6𝑖𝑀𝑇7𝑖𝑀𝑇8𝑖0𝑇,𝑅𝑖=𝑅𝑇1𝑖𝑅𝑇2𝑖𝑅𝑇3𝑖𝑅𝑇4𝑖𝑅𝑇5𝑖𝑅𝑇6𝑖𝑅𝑇7𝑖𝑅𝑇8𝑖0𝑇,𝑈𝑖=𝑈𝑇1𝑖𝑈𝑇2𝑖𝑈𝑇3𝑖𝑈𝑇4𝑖𝑈𝑇5𝑖𝑈𝑇6𝑖𝑈𝑇7𝑖𝑈𝑇8𝑖0𝑇,𝑉𝑖=𝑉𝑇1𝑖𝑉𝑇2𝑖𝑉𝑇3𝑖𝑉𝑇4𝑖𝑉𝑇5𝑖𝑉𝑇6𝑖𝑉𝑇7𝑖𝑉𝑇8𝑖0𝑇,𝐻𝑖=𝐻𝑇1𝑖𝐻𝑇2𝑖𝐻𝑇3𝑖𝐻𝑇4𝑖𝐻𝑇5𝑖𝐻𝑇6𝑖𝐻𝑇7𝑖𝐻𝑇8𝑖0𝑇,(3.10)

Proof. First, in order to cast our model involved in the framework of the Markov process, we define a new process 𝑥𝑡(𝑠)=𝑥(𝑡+𝑠),𝑠[𝜏2,0], and let 𝐿 be the weak infinitesimal generator of the random process 𝑥𝑡(𝑠),𝑡0 and 𝐿𝑣𝑥𝑡,𝑟𝑡=limΔ0+1Δ𝐸𝑣𝑥𝑡+Δ,𝑟𝑡+Δ𝑥𝑡,𝑟𝑡=𝑖𝑣𝑥𝑡,𝑟𝑡.(3.11) Now consider the Lyapunov-Krasovskii functional as follows for 𝑟𝑡=𝑖,𝑖1,2,,𝑆: 𝑣𝑥𝑡,𝑖=𝑣1𝑥𝑡,𝑖+𝑣2𝑥𝑡,𝑖+𝑣3𝑥𝑡,𝑖+𝑣4𝑥𝑡,𝑖+𝑣5𝑥𝑡,𝑖,(3.12) where 𝑣1𝑥𝑡,𝑖=𝑥𝑇(𝑡)𝑃(𝑖)𝑥(𝑡),𝑣2𝑥𝑡,𝑖=𝑡𝑡𝜏𝑖(𝑡)𝑥𝑇(𝑠)𝑄(𝑖)𝑥(𝑠)𝑑𝑠,𝑣3𝑥𝑡,𝑖=0𝜏2𝑖𝑡𝑡+𝜃𝑥𝑇(𝑠)𝑄𝑥(𝑠)𝑑𝑠𝑑𝜃,𝑣4𝑥𝑡,𝑖=𝑡𝑡𝜏1𝑖𝑥𝑇(𝑠)𝑄1𝑥(𝑠)𝑑𝑠+𝑡𝑡𝜏2𝑖𝑥𝑇(𝑠)𝑄2𝑥(𝑠)𝑑𝑠+𝑡𝑡𝜏2𝑥𝑇(𝑠)𝑄3𝑥(𝑠)𝑑,𝑣5𝑥𝑡,𝑖=𝜏1𝑖𝜏2𝑖𝑡𝑡+𝜃𝑥𝑇(𝑠)𝑍1𝑥(𝑠)𝑑𝑠𝑑𝜃+20𝜏2𝑡𝑡+𝜃̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠𝑑𝜃,(3.13) where 𝑁𝑗=1𝜋𝑖𝑗𝑄𝑗𝑄.(3.14) In order to show the exponential passivity of the MJS (2.8) under the given controller gain matrix 𝐾𝑖, we set 𝐽=𝐿𝑣𝑥𝑡,𝑖𝛾𝜔𝑇(𝑡)𝜔(𝑡)2𝑧𝑇(𝑡)𝜔(𝑡).(3.15)
Notice that𝐿𝑣1𝑥𝑡,𝑖=𝑥𝑇(𝑡)𝑃𝑖𝐴𝑖(𝑡)+𝐵1𝑖(𝑡)𝐾𝑖+𝐴𝑖(𝑡)+𝐵1𝑖(𝑡)𝐾𝑖𝑇𝑃𝑖𝑥(𝑡)+𝑥𝑇(𝑡)𝑁𝑗=1𝜋𝑖𝑗𝑃𝑗𝑥(𝑡)+2𝑥𝑇(𝑡)𝑃𝑖𝐴𝑑𝑖(𝑡)+𝐸1𝑖(𝑡)𝐾𝑖𝑥𝑡𝜏𝑖(𝑡)+2𝑥𝑇(𝑡)𝑃𝑖𝐷0𝑖𝑓(𝑥(𝑡),𝑖)+2𝑥𝑇(𝑡)𝑃𝑖𝐷1𝑖𝜔(𝑡),𝐿𝑣2𝑥𝑡,𝑖𝑥𝑇(𝑡)𝑄𝑖𝑥(𝑡)1𝜇𝑖𝑥𝑇𝑡𝜏𝑖(𝑡)𝑄𝑖𝑥𝑡𝜏𝑖(𝑡)+𝑡𝑡𝜏𝑖(𝑡)𝑥𝑇(𝑠)𝑁𝑗=1𝜋𝑖𝑗𝑄𝑗𝑥(𝑠)𝑑𝑠,𝐿𝑣3𝑥𝑡,𝑖𝜏2𝑖𝑥𝑇(𝑡)𝑄𝑥(𝑡)𝑡𝑡𝜏𝑖(𝑡)𝑥𝑇(𝑠)𝑄𝑥(𝑠)𝑑𝑠,𝐿𝑣4𝑥𝑡,𝑖=𝑥𝑇(𝑡)𝑄1+𝑄2+𝑄3𝑥(𝑡)𝑥𝑇𝑡𝜏1𝑖𝑄1𝑥𝑡𝜏1𝑖𝑥𝑇𝑡𝜏2𝑖𝑄2𝑥𝑡𝜏2𝑖𝑥𝑇𝑡𝜏2𝑄3𝑥𝑡𝜏2,𝐿𝑣5𝑥𝑡,𝑖=𝜏2𝑖𝜏1𝑖𝑥𝑇(𝑡)𝑍1𝑥(𝑡)+2𝜏2̇𝑥𝑇(𝑡)𝑍2̇𝑥(𝑡)𝑡𝜏1𝑖𝑡𝜏2𝑖𝑥𝑇(𝑠)𝑍1𝑥(𝑠)𝑑𝑠2𝑡𝑡𝜏2̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠=𝜏2𝑖𝜏1𝑖𝑥𝑇(𝑡)𝑍1𝑥(𝑡)+2𝜏2̇𝑥𝑇(𝑡)𝑍2̇𝑥(𝑡)𝑡𝜏2𝑖𝑡𝜏2̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠𝑡(𝜏𝑖(𝑡)+𝜏2𝑖)/2𝑡𝜏2𝑖̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠𝑡𝜏𝑖(𝑡)𝑡𝜏𝑖(𝑡)+𝜏2𝑖/2̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠𝑡(𝜏𝑖(𝑡)+𝜏1𝑖)/2𝑡𝜏𝑖(𝑡)̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠𝑡𝜏1𝑖𝑡𝜏𝑖(𝑡)+𝜏1𝑖/2̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠𝑡𝑡𝜏1𝑖̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠𝑡𝜏1𝑖𝑡𝜏2𝑖𝑥𝑇(𝑠)𝑍1𝑥(𝑠)𝑑𝑠𝑡𝑡𝜏2̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠.(3.16)
Then using Newton-Leibniz formula, for any matrices 𝐻𝑖,𝐺𝑖,𝑀𝑖,𝑅𝑖,𝑈𝑖,𝑉𝑖 we have2𝜉𝑇(𝑡)𝐺𝑖𝑥(𝑡)𝑥𝑡𝜏1𝑖𝑡𝑡𝜏1𝑖̇𝑥(𝑠)𝑑𝑠=0,2𝜉𝑇(𝑡)𝑀𝑖𝑥𝑡𝜏1𝑖𝑥𝑡𝜏1𝑖+𝜏𝑖(𝑡)2𝑡𝜏1𝑖𝑡𝜏1𝑖+𝜏𝑖(𝑡)/2̇𝑥(𝑠)𝑑𝑠=0,2𝜉𝑇(𝑡)𝑅𝑖𝑥𝑡𝜏1𝑖+𝜏𝑖(𝑡)2𝑥𝑡𝜏𝑖(𝑡)𝑡(𝜏1𝑖+𝜏𝑖(𝑡))/2𝑡𝜏𝑖(𝑡)̇𝑥(𝑠)𝑑𝑠=0,2𝜉𝑇(𝑡)𝑈𝑖𝑥𝑡𝜏𝑖(𝑡)𝑥𝑡𝜏2𝑖+𝜏𝑖(𝑡)2𝑡𝜏𝑖(𝑡)𝑡𝜏2𝑖+𝜏𝑖(𝑡)/2̇𝑥(𝑠)𝑑𝑠=0,2𝜉𝑇(𝑡)𝑉𝑖𝑥𝑡𝜏2𝑖+𝜏𝑖(𝑡)2𝑥𝑡𝜏2𝑖𝑡(𝜏2𝑖+𝜏𝑖(𝑡))/2𝑡𝜏2𝑖̇𝑥(𝑠)𝑑𝑠=0,2𝜉𝑇(𝑡)𝐻𝑖𝑥𝑡𝜏2𝑖𝑥𝑡𝜏2𝑡𝜏2𝑖𝑡𝜏2̇𝑥(𝑠)𝑑𝑡=0,(3.17) where 𝜉𝑇(𝑡)=𝑥𝑇(𝑡),𝑥𝑇𝑡𝜏1𝑖,𝑥𝑇𝑡𝜏𝑖(𝑡)+𝜏1𝑖2,𝑥𝑇𝑡𝜏𝑖(𝑡),𝑥𝑇𝑡𝜏𝑖(𝑡)+𝜏2𝑖2,𝑥𝑇𝑡𝜏2𝑖𝑥𝑇𝑡𝜏2,𝑓𝑇(𝑥(𝑡),𝑖),𝜔𝑇(𝑡).(3.18)
From the Lemma 2.7 (2.2), it is easy to see that2𝜉𝑇(𝑡)𝐺𝑖𝑡𝑡𝜏1𝑖̇𝑥(𝑠)𝑑𝑠𝜏1𝑖𝜉𝑇(𝑡)𝐺𝑖𝑍12𝐺𝑇𝑖𝜉(𝑡)+𝑡𝑡𝜏1𝑖̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠2𝜉𝑇(𝑡)𝑀𝑖𝑡𝜏1𝑖𝑡(𝜏1𝑖+𝜏𝑖(𝑡))/2̇𝑥(𝑠)𝑑𝑠𝜏𝑖(𝑡)𝜏1𝑖2𝜉𝑇(𝑡)𝑀𝑖𝑍12𝑀𝑇𝑖𝜉(𝑡)+𝑡𝜏1𝑖𝑡(𝜏1𝑖+𝜏𝑖(𝑡))/2̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠2𝜉𝑇(𝑡)𝑅𝑖𝑡(𝜏1𝑖+𝜏𝑖(𝑡))/2𝑡𝜏𝑖(𝑡)̇𝑥(𝑠)𝑑𝑠𝜏𝑖(𝑡)𝜏1𝑖2𝜉𝑇(𝑡)𝑅𝑖𝑍12𝑅𝑇𝑖𝜉(𝑡)+𝑡(𝜏1𝑖+𝜏𝑖(𝑡))/2𝑡𝜏𝑖(𝑡)̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠2𝜉𝑇(𝑡)𝑈𝑖𝑡𝜏𝑖(𝑡)𝑡(𝜏2𝑖+𝜏𝑖(𝑡))/2̇𝑥(𝑠)𝑑𝑠𝜏2𝑖𝜏𝑖(𝑡)2𝜉𝑇(𝑡)𝑈𝑖𝑍12𝑈𝑇𝑖𝜉(𝑡)+𝑡𝜏𝑖(𝑡)𝑡(𝜏2𝑖+𝜏𝑖(𝑡))/2̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠2𝜉𝑇(𝑡)𝑉𝑖𝑡(𝜏2𝑖+𝜏𝑖(𝑡))/2𝑡𝜏2𝑖̇𝑥(𝑠)𝑑𝑠𝜏2𝑖𝜏𝑖(𝑡)2𝜉𝑇(𝑡)𝑉𝑖𝑍12𝑉𝑇𝑖𝜉(𝑡)+𝑡(𝜏2𝑖+𝜏𝑖(𝑡))/2𝑡𝜏2𝑖̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠2𝜉𝑇(𝑡)𝐻𝑖𝑡𝜏2𝑖𝑡𝜏2̇𝑥(𝑠)𝑑𝑠𝜏2𝜏2𝑖𝜉𝑇(𝑡)𝐻𝑖𝑍12𝐻𝑇𝑖𝜉(𝑡)+𝑡𝜏2𝑖𝑡𝜏2̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠.(3.19)
Now by Assumption 3, it can be deduced that for any positive scalar 𝜀1𝑖,𝑖1,2,,𝑆,2𝜀1𝑖𝑓𝑇(𝑥(𝑡),𝑖)Γ𝑖𝑥(𝑡)𝑓(𝑥(𝑡),𝑖)0.(3.20) Then from the above discussion, we can see that 𝐽𝜉𝑇(𝑡)𝜏𝑖(𝑡)𝜏1𝑖𝜏2𝑖𝜏1𝑖Φ1(𝑡)+𝜏2𝑖𝜏𝑖(𝑡)𝜏2𝑖𝜏1𝑖Φ2(𝑡)𝜉(𝑡)𝑡𝜏1𝑖𝑡𝜏2𝑖𝑥𝑇(𝑠)𝑍1𝑥(𝑠)𝑑𝑠𝑡𝑡𝜏2̇𝑥𝑇(𝑠)𝑍2̇𝑥(𝑠)𝑑𝑠,Φ1(𝑡)=Ω𝑖,9×9(𝑡)+Λ1(𝑡)𝑍12Λ1(𝑡)𝑇+𝜏2𝜏2𝑖𝐻𝑖𝑍12𝐻𝑇𝑖+𝜏1𝑖𝐺𝑖𝑍12𝐺𝑇𝑖+𝜏2𝑖𝜏1𝑖2𝑀𝑖𝑍12𝑀𝑇𝑖+𝑅𝑖𝑍12𝑅𝑇𝑖,Φ2(𝑡)=Ω𝑖,9×9(𝑡)+Λ1(𝑡)𝑍12Λ1(𝑡)𝑇+𝜏2𝜏2𝑖𝐻𝑖𝑍12𝐻𝑇𝑖+𝜏1𝑖𝐺𝑖𝑍12𝐺𝑇𝑖+𝜏2𝑖𝜏1𝑖2𝑈𝑖𝑍12𝑈𝑇𝑖+𝑉𝑖𝑍12𝑉𝑇𝑖,(3.21) where Ω𝑖,11(𝑡)=𝑃𝑖𝐴𝑖(𝑡)+𝐵1𝑖(𝑡)𝐾𝑖+𝐴𝑖(𝑡)+𝐵1𝑖(𝑡)𝐾𝑖𝑇𝑃𝑖+𝑁𝑗=1𝜋𝑖𝑗𝑃𝑗+𝑄𝑖+𝜏2𝑖𝑄+𝑄1+𝑄2+𝑄3+𝜏2𝑖𝜏1𝑖𝑍1+𝐺𝑇1𝑖+𝐺1𝑖,Ω𝑖,14(𝑡)=𝑃𝑖𝐴𝑑𝑖(𝑡)+𝐸1𝑖(𝑡)𝐾𝑖𝑅1𝑖+𝐺𝑇4𝑖+𝑈1𝑖Ω𝑖,19(𝑡)=𝑃𝑖𝐷1𝑖𝐶𝑖(𝑡)+𝐵2𝑖(𝑡)𝐾𝑖𝑇,Ω𝑖,49(𝑡)=𝐶𝑑𝑖(𝑡)+𝐸2𝑖(𝑡)𝐾𝑖𝑇(3.22) other terms of Ω𝑖,𝑖×𝑗(𝑡) are similar to Ω1𝑖,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 Ω𝑖,9×9Λ1Λ2Λ3Λ4𝑘Λ5𝑘𝑍20000𝑍2000𝑍200𝑍20𝑍2𝑘=1,2+𝑇𝑇1𝑖𝑃𝑖,0𝑛×7𝑛,𝑇𝑇2𝑖,2𝜏2𝑇𝑇1𝑖𝑍2,0𝑛×4𝑛𝑇𝐹𝑖(𝑘)𝑁1𝑖+𝑁3𝑖𝐾𝑖,0𝑛×2𝑛,𝑁2𝑖+𝑁4𝑖𝐾𝑖,0𝑛×10𝑛+𝑁1𝑖+𝑁2𝑖𝐾𝑖,0𝑛×2𝑛,𝑁2𝑖+𝑁4𝑖𝐾𝑖,0𝑛×10𝑛𝑇𝐹𝑇𝑖(𝑘)𝑇𝑇1𝑖𝑃𝑖,0𝑛×7𝑛,𝑇𝑇2𝑖,2𝜏2𝑇𝑇1𝑖𝑍2,0𝑛×4𝑛<0,(3.23) 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 Ω𝑖,9×9Λ1Λ2Λ3Λ4𝑘Λ5𝑘Λ6Λ7𝑍200002𝜏2𝑍2𝑇1𝑖𝜀2𝑖0𝑍200000𝑍20000𝑍2000𝑍200𝜀2𝑖0𝜀2𝑖𝑘=1,2<0.(3.24)Case 1. If 𝜋𝑖𝑖𝐼𝑖kn
then (3.24) is equivalent toΩ1𝑖,9×9Λ1Λ2Λ3Λ4𝑘Λ5𝑘Λ6Λ7𝑍200002𝜏2𝑍2𝑇1𝑖𝜀2𝑖0𝑍200000𝑍20000𝑍2000𝑍200𝜀2𝑖0𝜀2𝑖𝑘=1,2+diag𝑗𝐼𝑖uk𝑗𝑖𝜋𝑖𝑗𝑃𝑗,150,,0+diag𝑗𝐼𝑖uk𝑗𝑖𝜋𝑖𝑗𝑃𝑖𝐴𝑖+𝐴𝑇𝑖𝑃𝑖,150,,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).𝑁𝑗=1𝜋𝑖𝑗𝑄𝑗<𝑄 is equivalent to
𝑗𝐼𝑖kn𝑗𝑖𝜋𝑖𝑗𝑄𝑗+𝑗𝐼𝑖uk𝑗𝑖𝜋𝑖𝑗𝑄𝑗+𝜋𝑖𝑖𝑄𝑖𝑄+𝑗𝐼𝑖kn𝜋𝑖𝑗𝜋𝑖𝑖𝑄𝑖𝑄+𝑗𝐼𝑖uk𝜋𝑖𝑗𝜋𝑖𝑖𝑄𝑖𝑄<0.(3.26) If we have the following matrix inequalities hold, we can have that (3.14) is satisfied 1+𝑗𝐼𝑖kn𝜋𝑖𝑗𝜋𝑖𝑖𝑄𝑖𝑄+𝑗𝐼𝑖kn𝑗𝑖𝜋𝑖𝑗𝑄𝑗<0,𝜋𝑖𝑖𝑄𝑖𝑄+𝑄𝑗<0𝑗𝐼𝑖uk.(3.27) Obviously, (3.27) is equivalent to (3.2) and (3.3) by the Schur complement. Case 2. If 𝜋𝑖𝑖𝐼𝑖uk
then (3.24) is equivalent toΩ1𝑖,9×9Λ1Λ2Λ3Λ4𝑘Λ5𝑘Λ6Λ7𝑍200002𝜏2𝑍2𝑇1𝑖𝜀2𝑖0𝑍200000𝑍20000𝑍2000𝑍200𝜀2𝑖0𝜀2𝑖𝑘=1,2+diag𝑗𝐼𝑖uk𝑗𝑖𝜋𝑖𝑗𝑃𝑗,150,,0+diag𝑗𝐼𝑖uk𝑗𝑖𝜋𝑖𝑗𝑃𝑖𝐴𝑖+𝐴𝑇𝑖𝑃𝑖,150,,0+𝜋𝑖𝑖diag𝑃𝑖𝐴𝑖+𝐴𝑇𝑖𝑃𝑖+𝑃𝑖,150,,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, then
𝐽𝜌1𝑥(𝑡)2𝜆min𝑍1𝑡𝑡𝜏2𝑥(𝑠)2𝑑𝑠𝜆min𝑍2𝑡𝑡𝜏2̇𝑥(𝑠)2𝑑𝑠.(3.29) On the other hand, it is easy to obtain that 𝑣(𝑥(𝑡),𝑖)𝑃𝑥(𝑡)2+𝑄+𝑄+𝑄1+𝑄2+𝑄3+𝜏2𝜏1𝑍1×𝑡𝑡𝜏2𝑥(𝑠)2𝑑𝑠+2𝜏2𝑍2𝑡𝑡𝜏2̇𝑥(𝑠)2𝑑𝑠,(3.30) where 𝑃=max𝑖𝑆{𝑃𝑖}, 𝑄=max𝑖𝑆{𝑄𝑖}.Let 𝜌>0 be sufficiently small such that 𝜌𝑃𝜌1<0,𝜌𝑄+𝑄+𝑄1+𝑄2+𝑄3+𝜏2𝜏1𝑍1𝜆min𝑍1<02𝜏2𝜌𝑍2𝜆min𝑍2<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 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 matrices 𝐾𝑖 for convenience in the next section, (3.1) is not LMI, if we substitute 𝜀2𝑖 by 𝜀12𝑖 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 𝐾𝑖,𝑖𝑆 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 𝑃𝑖,𝑄𝑖,𝑄1,𝑄2,𝑄3,𝑄,𝑍1,𝑍2, positive scalar 𝜀1𝑖,𝜀2𝑖, and for any matrices 𝐺𝑖,𝑀𝑖,𝑅𝑖,𝑈𝑖,𝑉𝑖,𝐻𝑖,𝑍𝑖𝑖 with appropriate dimensions satisfying the following LMIs under the two cases for all 𝑖=1,2,,𝑁. Case 1. If 𝜋𝑖𝑖𝐼𝑖kn𝜋𝑖𝑖𝑄𝑖𝑄𝑄𝑗𝑄𝑗𝑗𝐼𝑖uk<0,(3.32)1+𝑗𝐼𝑖kn𝜋𝑖𝑗𝜋𝑖𝑖𝑄𝑖𝑄𝜋𝑖𝑘𝑖1𝑄𝑘𝑖1𝜋𝑖𝑘𝑖𝑚𝑄𝑘𝑖𝑚𝑄𝑘𝑖1000𝑄𝑘𝑖𝑚<0,(3.33)Ω1𝑖,9×9Λ1Λ2Λ3Λ4𝑘Λ5𝑘Λ6Λ7Λ8𝑍200002𝜏2𝑇1𝑖𝜀2𝑖00Θ000000Θ00000Θ0000Θ000𝜀2𝑖00𝜀2𝑖0Λ9<0𝑘=1,2,(3.34)𝑍𝑖𝑖𝑍𝑇𝑖𝑖𝑍𝑇𝑖𝑖𝐴𝑇𝑖+𝑃𝑖𝑍𝑇𝑖𝑖0𝜀1𝑃𝑖0𝑃𝑖𝜀𝑃𝑖0𝑃𝑗𝑗𝐼𝑖uk<0,(3.35)Case 2. If 𝜋𝑖𝑖𝐼𝑖uk𝑄𝑗𝑄>0𝑗𝐼𝑖uk,𝑗=𝑖,(3.36)𝑄𝑗𝑄<0𝑗𝐼𝑖uk,𝑗𝑖,(3.37)1𝑗𝐼𝑖kn𝜋𝑖𝑗𝑄𝜋𝑖𝑘𝑖1𝑄𝑘𝑖1𝜋𝑖𝑘𝑖𝑚𝑄𝑘𝑖𝑚𝑄𝑘𝑖1000𝑄𝑘𝑖𝑚<0,(3.38)Ω2𝑖,9×9Λ1Λ2Λ3Λ4𝑘Λ5𝑘Λ6Λ7Λ8𝑍200002𝜏2𝑇1𝑖𝜀2𝑖00Θ000000Θ00000Θ0000Θ000𝜀2𝑖00𝜀2𝑖0Λ9<0𝑘=1,2,(3.39)𝑍𝑖𝑖𝑍𝑇𝑖𝑖𝑍𝑇𝑖𝑖𝐴𝑇𝑖+𝑃𝑖𝑍𝑇𝑖𝑖𝜀1𝑃𝑖𝑃𝑗0𝜀𝑃𝑖𝑗𝐼𝑖uk𝑗=𝑖<0,(3.40)𝑍𝑖𝑖𝑍𝑇𝑖𝑖𝑍𝑇𝑖𝑖𝐴𝑇𝑖+𝑃𝑖𝑍𝑇𝑖𝑖0𝜀1𝑃𝑖0𝑃𝑖𝜀𝑃𝑖0𝑃𝑗𝑗𝐼𝑖uk𝑗𝑖<0,(3.41) where Ω1𝑖,11=1+𝑗𝐼𝑖kn𝜋𝑖𝑗𝐴𝑖𝑃𝑖+𝐵1𝑖𝑌𝑖+