The No-Ellipsoidal Bound of Reachable Sets for Neutral Markovian Jump Systems with Disturbances

This paper is concerned with the reachable set estimation problem for neutral Markovian jump systems with bounded peak disturbances, which was rarely proposed for neutral Markovian jump systems. The main consideration is to find a proper method to obtain the no-ellipsoidal bound of the reachable set for neutral Markovian jump system as small as possible. By applying Lyapunov functional method, some derived conditions are obtained in the form of matrix inequalities. Finally, numerical examples are presented to demonstrate the effectiveness of the theoretical results.


Introduction
In practice and engineering applications, many dynamical systems may cause abrupt variations in their structure, due to stochastic failures or repairs of the components, changes in the interconnections of subsystems, sudden environment changes, and so on. Markovian jump systems, modeled by a set of subsystems with transitions among the models determined by a Markov chain taking values in a finite set, have appealed to a lot of researchers in the control community. In the past few decades, the Markovian jump systems have been extensively studied, see [1] [2] [3] [4] and the references therein.
The reachable set [5] estimation of dynamic systems is to derive some closed bounded set that bounds the state trajectories starting from the origin by inputs with peak value. Reachable set estimation is not only an important issue in the

Problem Statement
Consider the following neutral Markovian jump systems with disturbances ( ) ( ) ( ) ( known constant matrices of the Markov process.
Since the state transition probability of the Markovian jump process is considered in this paper is partially known, the transition probability matrix of Markovian jumping process Λ is defined as , where ? represents the unknown transition rate. For notational clarity, : is known for , : : is unknown for . , x r . And its weak infinitesimal generator, acting on function V, is defined in [3].
For simplicity, there are the following representations: In this paper, the following Lemma and Assumption are needed:  [17]. For any positive-definite matrix

Main Results
Our aim is to find an ellipsoid set as small as possible to bound the reachable set defined in (3). In this section, based on an appropriate Lyapunov functional and matrix inequality techniques (Lemma (1-3)), following Theorems are derived.
Then, the reachable set of the system (1) having the constraints (2) is bounded by a non-ellipsoid boundary Proof. We choose the following Lyapunov-Krasovskii functional candidate as follows: Therefore, we get Then, for given t r i = ∈℘, 1 and the weak infinitesimal operator L of the stochastic process ( ) x t along the evolution of Taking into account the situation that the information of transition probabilities is not accessible completely, due to   T  T  T  T  T  T  2  2  1  2  3   T  T  T  T  T  T  2  2  2   T  T  T  T  3  3  3   T  T  T  3  3   T  1 , ,   T  T  T   T  T  3 , , Then substituting (13)(14)(15)(16) into (12), we further have T  T  T  2  2  1  1   T  T  T  T  T  1  2  3  2   T  T  T  T  2  2   T  T  3  3 , , ,

t P B x t h t x t P C x t x t P D w t x t h t h Q h S x t h t h x t h t Sx t h x t Rx t h x t h Sx t h
where i Φ is the same as defined in the Theorem 1 and which means, by Lemma 1, that

and this results in
So the reachable set of the system (1) is bounded by ellipsoid ( ) (7), which implies that the reachable sets of the system (1) having the constraints (2) Following a similar line as in proof of Theorem 1, we can obtain the following Theorem.
Theorem 2. Consider the Markov neutral system (1) with all elements unknown in transition rate matrix (4), if there exist symmetric matrices 2 P , 3 P , where C. C. Shen et al. Journal of Applied Mathematics and Physics Then, the reachable sets of the system (1) having the constraints (2) is bounded by a non-ellipsoid boundary ( ) (7).

Numerical Examples
In this section, a numerical example demonstrates the effectiveness of the approaches presented in this paper. Consider the neutral Markov jump systems with three operation modes whose state matrices are listed as follow: where  1  2  3   1  2  3   2  1  3 0  2  0  ,  ,  ,  0  2  0  2  1 1.5   1 0  2 0  2 0  ,  ,  ,  1 2  1 1  0 The transition rate matrix Λ is considered as the following three cases. ? ?
The transition rate matrix Λ is completely unknown, which is considered as ? ? ?
Firstly, by giving the transition probabilities Λ , a possible mode evolution of the neutral Markov jump system (22) is derived as shown in Figure 1. Based on the mode evolution shown in Figure 1, and choosing disturbances ( ) w t as the random signal satisfying ( ) ( ) T 1 w t w t ≤ , all the reachable states of neutral Markov jump system (22) starting from the origin are given in Figure 2.
By using theorem 2 and solving the optimization problem (20)   , which is depicted in Figure 3.
By using theorem 1 and solving the optimization problem (22)


, which is depicted in Figure 4. By using theorem 3 and solving the optimization problem (22)


, which is depicted in Figure 5.

Conclusion
In this paper, the problem of robust stability for a class of uncertain neutral systems with time-varying delays is investigated. Sufficient conditions are given in terms of linear matrix inequalities which can be easily solved by LMI Toolbox in Matlab. Numerical examples are given to indicate significant improvements over some existing results.