Stochastic Stability Analysis for Markovian Jump Neutral Nonlinear Systems

In this paper, the stability problem is studied for a class of Markovian jump neutral nonlinear systems with time-varying delay. By Lyapunov-Krasovskii function approach, a novel mean-square exponential stability criterion is derived for the situations that the systems transition rates are completely accessible, partially accessible and non-accessible, respectively. Moreover, the developed stability criterion is extended to the systems with different bounded sector nonlinear constraints. Finally, some numerical examples are provided to illustrate the effectiveness of the proposed methods.


Introduction
Among stochastic hybrid systems, a widely studied class is the so-called Markovian jump systems (MJSs), which are described by a set of systems with commutation generated by a finite state Markov chain.It has been shown that MJSs have great application potentials, such as biomedicine, fault tolerant systems, networked control, economics, aerospace, and communication networks.Due to the probabilistic description of commutations, MJSs are very suitable and effective to describe dynamical systems with the model changes induced by external causes, such as unexpected events, random faults, uncontrolled configuration changes.A great amount of effort has been devoted to MJSs in the past years, and many useful and important results have been reported in literature.In Ma et al. (2012), a stochastic version of a bounded real lemma was derived for a general class of discrete-time time-varying MJSs with state-and disturbance-multiplicative noise, and the necessary and sufficient conditions for the solvability of the control problem in terms of four coupled discrete-time Riccati equations are presented.In Zhang (2009), the estimation problem was studied for a class of MJSs with time-varying transition probabilities, which was considered to be finite piecewise homogeneous.In Costa and Oliveira (2012), the stochastic optimal control problem was researched for discretetime linear systems subject to Markov jumps and multiplicative noises under two criteria.In Fei et al. (2009), a new delay-dependent stochastic stability criterion for a class of MJSs with time delay was derived based on a novel Lyapunov-Krasovskii functional ap-proach.In Wu et al. (2012), the problem on the sliding mode control of Markov jump singular time-delay systems is concerned through investigation of the bounded gain performance in the analysis of sliding mode dynamics in order to improve the system transient performance.Recently in Karimi (2011), an efficient approach is developed for robust H ∞ control problem of uncertain time-delay systems with Markovian switching parameters and mixed discrete, neutral and distributed delays.For more state-of-the-art results on this topic, readers are referred to (Karimi (2012), Wu et al. (2011), Basin and Shi (2009), Shi and Liu (2011), Wu et al. (2010), Liu et al. (2011b)) and the references therein.
In real systems, time delays exist widely.Hence recently much effort has been included in such area , see Cai et al. (2012), Liu et al. (2011a) and references therein.Meanwhile,the transition rates of MJSs are, in general, not completely accessible.Hence the problems of modeling, analysis and design on MJSs with partially transition rates have become an important and challenging issue, and attracted a lot of attentions, see for example, Shi et al. (2012), Yin et al. (2012), Luan et al. (2010), Zhang et al. (2009), Zhang et al. (2011), Zhang et al. (2009), Wang et al. (2010) and the references therein.However, most theoretical results either have some strict limitations with regard to transition rates matrix, such as main diagonal elements are required, or have some conservatism for not fully utilizing the provided transition rates.In addition, for MJSs with time-varying delay, most stability criteria have the constraints that the derivative of time-varying delay must be less than a constant, such as 1, which limits the applied scope.In this paper, all the available commutation information will be utilized to derive a stochastic stability criterion, independent of time-varying delays derivative, for Markovian jump neutral nonlinear delay systems (MJNNSs), no matter if transition rates are completely accessible, partially accessible and non-accessible.Moreover, the developed stability criterion will be extended to systems with different bounded sector nonlinear constraints.Some numerical examples will be included to demonstrate the effectiveness of the obtained theoretical results.
The remainder of this paper is organized as follows.The MJNNS with time-varying delay and some preliminaries are provided in Section 2. Section 3 presents the results on stochastic stability analysis.Section 4 gives an illustrative example, and the paper is concluded in Section 5. Notations used in this paper are fairly standard.Let R n be the n-dimensional Euclidean space, R n×m represents the set of n × m real matrices, the symbol * denotes the elements below the main diagonal of a symmetric block matrix, A > 0 means that A is a real symmetric positive definitive matrix, I denotes the identity matrix with appropriate dimensions.diag {• • • } denotes the diagonal matrix.E{•} refers to the expectation operator with respect to some probability measure P. • refers to the Euclidean vector norm or the induced matrix 2-norm.The superscript T stands for matrix transposition, L n,h = L([−h, 0], R n ) denotes the Banach space of continuous functions mapping the interval [−h, 0] into R n with the topology of uniform convergence.

Model description and preliminaries
In this paper, (Ω, Υ, P ) denotes the probability space, where Ω is the sample space, Υ is σ -algebra of subset of the sample space, and P is the probability measure defined on Υ.Let S = {1, 2, . . ., N } and the random form process {r(t), t ∈ [0, +∞)} be a homogeneous, finite state Markov process with right continuous trajectories with generator Π = (π ij ) N ×N and transition probability form mode i at time t to mode j at time t + ∆t, i ∈ S, j ∈ S, given by where transition rates π ij ≥ 0, j = I, π ii = − N j=1,j =i π ij , ∆t > 0, and lim ∆t→0 o(∆t)/∆t = 0.
For the convenience of study, the transition rates of the Markov chain are considered to be partially accessible.Namely, some elements in transition rates matrix are time invariant but unknown, such as a system with three operation modes may have the transition rates matrix as follow where − represents the inaccessible element.
Throughout the paper, we denote S = S i kn ∪ S i un for ∀i ∈ S, where S i kn = {j : π ij is known} and is the nonlinear term, and satisfies the following Lipchitz condition Then, model (1) can be represented as Now, to present the main objective of this paper more precisely, we also introduce the following definition for system (1), which is essential for the later development.
Definition 1: System (1) is mean-square exponentially stable if there exist scalars α > 0 and β > 0 such that where Therefore, in this paper, the authors attention will be focused on the development of mean-square exponential stability criterion independent of time-varying delays derivative for the situations that the systems transition rates are completely accessible, partially accessible and non-accessible, respectively.
Before ending the section, we give the following lemmas, which will be used in the proof of our main results.
Lemma 1 (Boyd et al. (1994)): For any constant matrices E, G and F with appropriate dimensions, where c is a positive scalar, x ∈ R n , andy ∈ R n .Lemma 2 (Gu (2000)): For any positive definite matrix Φ ∈ R n×n , scalar γ > 0, vector function w : [0, γ] → R n such that the integrations concerned are well defined, then 3 Main results In this section, based on Lyapunov method and linear matrix inequality techniques, the following stochastic stability criteria are derived.
Theorem 1: When π ii is unknown with given h and k, if there exist P 2 , P 3 , U, V, W , and symmetric positive-definite matrices (5) where then, MJNNS (1) is mean-square exponentially stable.
Proof: Choose the following Lyapunov-Krasovskii functional candidate where Let L be the infinitesimal generator of random process.Then for each r(t) = i, i ∈ S, we have where Hence, we get where From ( 7) and ( 8), we know that By Schur complement, the above inequality can be rewritten as With ineq.(5)-ineq.(8),we can conclude that Hence By Dynkins formula, one can obtain Hence From the definition of V (t, i), we know that where Then, the following inequality is deduced where Finally, with Definition 1, we can conclude that MJNNS (1) is mean-square exponentially stable.The proof of Theorem 1 is thus completed.
Theorem 1 investigated MJNNS (1) in the situation that π ii is unknown.Based on Theorem 1, the following theoretical result can be derived.
Corollary 1: When π ii is known with given h and k, if there exist P 2 , P 3 , U, V, W , and symmetric positivedefinite matrices where then, MJNNS (1) is mean-square exponentially stable.Next, we consider the MJNNS with following nonlinear constraints where where The following inequality can be obtained Then MJNNS (1) can be represented as follows where Ā(r(t) It can be seen that system (1) and system ( 27) have the same structure with different system parameters.Hence based on Theorem 1, the following theoretical result can be derived.
Theorem 2: When π ii is unknown with given h and k, if there exist P 2 , P 3 , U, V, W , and symmetric positive-definite matrices where then, MJNNS ( 27) is mean-square exponentially stable.Proof: The proof is similar to that of Theorem 1, and thus is omitted.
Theorem 2 studied MJNNS ( 27) in the situation that π ii is unknown.Based on Theorem 2, the following theoretical result can be derived.
Corollary 2: When π ii is known with given h and k, if there exist P 2 , P 3 , U, V, W , and symmetric positivedefinite matrices where then, MJNNS ( 27) is mean-square exponentially stable.

Simulation Results
In this section, we will verify the proposed methodology by giving an illustrative example.The MJNNS with following system parameters is included First we consider the given MJNNS with transition rates as follow According to the derived stability criteria, Table 1 can be obtained.
According to the presented theoretical results, Table 2 can be obtained.1 shows the maximum stable delay of the MJNNS changes with different transition rates.This means that transition rates have an effect on the stochastic stability of the MJNNS.Table 2 displays the relationship between the maximum stable delay of the MJNNS and transition rates that are unknown, partially known and fully known, respectively.It can be seen that the biggest maximum stable delay occurs in the situation that transition rates are fully known for the best utilization of all commutation information.In addition, it is illustrated that no matter how many transition rates are accessible, the presented stability criteria are still available.
Next, the numerical simulation is carried out to analyze the dynamics of the MJNNS.The numerical simulation is with the initial state value ψ(θ) = [1, −1.2] T , θ ∈ (−1, 0), r(t 0 ) = 2, the simulation step 0.001 second, the transition rates Π 6 and the switching time 0.1 second.Corresponding simulation results are shown in Figs. 1 and 2.
Remark 3: Fig. 1 depicts the time response of the state variable of the MJNNS.response of the state variable of the MJNNS.It can be seen that the MJNNS is mean-square exponential stable, and spends about 6 seconds converging to the zero state.
Remark 4: For the given example, τ (t) = 1.5 sin(30t), and the maximum upper bound of τ (t) is 1.5, greater than 1.Based on the constructed Lyapunov function, the derived stochastic stability criteria have no constrains on the upper bound of τ (t).In addition, the calculated maximum stable delay of the MJNNS is 1.1524, bigger than the real maximum delays upper bound 0.7, which means that this system should be mean-square exponentially stable, and the simulation results in Figs. 1 and 2 accord with the proposed stability criteria.

Conclusions
In this paper, the stochastic stability analysis on Markovian jump neutral nonlinear systems with timevarying delay is concerned.By constructing a novel Lyapunov function, the mean-square exponential stability criterion, independent of time-varying delays derivative, has been concluded for the circumstance that transition rates are completely accessible, partially accessible and non-accessible.Subsequently, the developed stability criterion has been extended to the systems with a different bounded sector nonlinear constrain.At last, some numerical examples have been included to illustrate the effectiveness of the obtained theoretical results.

FigFigure 1 :
Figure 1: Time response of the state variable x 1 (t) of the MJNNS.

Figure 2 :
Figure 2: Time response of the state variable x 2 (t) of the MJNNS.

Table 1 :
The maximum stable delay of the MJNNS with different transition rates

Table 2 :
The maximum stable delay of the MJNNS with transition rates unknown, partly known and fully known