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

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.


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, 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 L 2 F 0 −τ 2 , 0 : R n the family of all F 0 measurable C −τ 2 , 0 : R n -valued random variables ϕ {ϕ s : −τ 2 ≤ s ≤ 0} such that sup −τ 2 ≤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.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 L 2 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 probabilities Pr r t Δt j | r t i π ij Δt o Δt i / j, 1 π ii Δt 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, and

Problem Formulation and Preliminaries
For notational simplicity, which r t i, i ∈ S, the matrices A t, r t , A d t, r t , B 1 t, r t , E 1 t, r t , C t, r t , C d t, r t , B 2 t, r t , E 2 t, r t , D 0 r t , D 1 r t , and D 2 r t will be described by and D 2i .We denote that

2.5
where , 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 that 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: then, the closed-loop MJS can be represented as 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: 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 , for all i ∈ S, are unknown time-varying matrix functions satisfying holds if and only if one of the following conditions holds:

Exponential Passivity Analysis
In this section, we assumed the transition rates are partially known and given the statefeedback 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 , 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: where

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 ∈ −τ 2 , 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 Now consider the Lyapunov-Krasovskii functional as follows for r t i, i ∈ 1, 2, . . ., S: where ẋT s Z 2 ẋ s ds dθ,

3.13
where In order to show the exponential passivity of the MJS 2.8 under the given controller gain matrix K i , we set Journal of Applied Mathematics Notice that ẋT s Z 2 ẋ s ds ẋT s Z 2 ẋ s ds ẋT s Z 2 ẋ s ds.

3.16
Then using Newton-Leibniz formula, for any matrices 3.17 where

3.18
From the Lemma 2.7 2.2 , it is easy to see that ẋT s Z 2 ẋ s ds ẋT s Z 2 ẋ s ds ẋT s Z 2 ẋ s ds.

3.19
Now by Assumption 3, it can be deduced that for any positive scalar ε 1i , i ∈ 1, 2, . . ., S, Then from the above discussion, we can see that where other terms of Ω i,i×j t are similar to Ω 1 i,i×j .In order to get our results, we will describe that the Φ 1 t < 0 and Φ 2 t < 0.
By the Schur complement, Φ 1 t < 0 and Φ 2 t < 0 under the restriction of 3.14 if and only if 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 < 0.

3.24
Case 1.If π ii ∈ I i kn then 3.24 is equivalent to

3.25
Obviously, we can see that if 3.1 and 3.4 hold, then Φ 1 t < 0 and Φ 2 t < 0 under the restriction of 3.14 .Next we will further consider the equivalent form of 3.14 .

3.26
If we have the following matrix inequalities hold, we can have that 3.14 is satisfied

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

3.28
Then if 3.1 , 3.8 , and 3.9 hold, then Φ 1 t < 0 and Φ 2 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 ρ 1 , then ẋ s 2 ds.

3.29
On the other hand, it is easy to obtain that

3.30
where 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.

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 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.

and for any matrices
where

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: A 1 −0.05 −0.05 0.5 −0.

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.
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.

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.

Figure 1 :
Figure 1: State response of case 1 and the switch signal.

Figure 2 :
Figure 2: State response of case 2 and the switch signal.

10
Remark 2.2.It is assumed that all the elements F i t , for all i ∈ 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 3.For a fixed system mode r t i ∈ S, there exists a know real constant modedependent matrix Γ i diag k 1i , k 2i , . . ., k ni > 0 such that the nonlinear vector function f •, • satisfy the following conditions: Assumption 2. The time-varying delay τ i t satisfies 0 ≤ τ 1i ≤ τ i t ≤ τ 2i , τi t ≤ μ i , with τ 1i , τ 2i , and μ i being real constant scalars for each for all i ∈ S.

Table 1 :
Calculated the controller gains matrix for different cases.