Stability Analysis for a Class of Discrete-Time Nonhomogeneous Markov Jump Systems with Multiplicative Noises

This paper is concerned with a class of discrete-time nonhomogeneous Markov jump systems with multiplicative noises and timevarying transition probability matrices which are valued on a convex polytope. The stochastic stability and finite-time stability are considered. Some stability criteria including infinite matrix inequalities are obtained by parameter-dependent Lyapunov function. Furthermore, infinitematrix inequalities are converted into finite linearmatrix inequalities (LMIs) via a set of slackmatrices. Finally, two numerical examples are given to demonstrate the validity of the proposed theoretical methods.


Introduction
There are a lot of dynamic systems which are subjected to abrupt variations in parameters caused by external surroundings or internal structure in practical engineering field.Markov jump system models have attracted more and more attention because they effectively described this kind of systems and have become a hot topic in control theory.Many mature and systematic results have been obtained [1][2][3][4][5].
It should be pointed out that most of the current research is done in the framework of homogeneous Markov process (or Markov chain), which is under the assumption that the transition rate (or transition probability) matrix of the system at any time is the same.In fact, due to the impact of the objective environment, this assumption is difficult to satisfy.For example, in a Markov jump networked control system, the transition probability is time-variant because the packet dropouts and network delays are different at different periods.Another typical example can be found in the faultprone systems, where Markov process is used to describe the failure rate that is influenced by the factors of the age and the usage rate.Obviously, it is not a homogeneous process.Driven by these practical problems, people turn to the nonhomogeneous Markov jump systems [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].
In order to describe the nonhomogeneous property, several assumptions are put forward.In [6], the time-varying transition probabilities of discrete-time Markov jump systems are considered to be finite piecewise homogeneous with two types of variations in the finite set: arbitrary variation and stochastic variation, which implies that the transition probabilities are varying but invariant within an interval.The  ∞ estimation problem is investigated.This assumption is generalized to the continuous-time Markov jump system [7], Markov jump neural networks [8,9], complex networks [10], and singular Markov jump systems [11], and the robust stability, stochastic stability, passivity analysis, and synchronization are studied, respectively.Another way of describing timevarying characteristics is in a polytopic sense.It is proposed for the first time in [12] and a sufficient condition for stochastic stability is derived by using a parameter-dependent stochastic Lyapunov function.The main idea is that the transition probability matrix is assumed to be valued in a polytope convex set with some given vertices when the exact transition probability is not known.The polytopic model 2 Complexity is more general and includes the piecewise homogeneous Markov jump system model with arbitrary switch as a special case.Subsequently, many new results are obtained on this model.In [13][14][15], mode-dependent, variation-dependent, and observer-based  ∞ controllers are designed to satisfy the stochastic stability and prescribed  ∞ performance index.The other control problems, such as  2 - ∞ control [16] and fault detection [17], are also considered.An application to DC motors can be found in [18].The model is used not only in discrete-time systems but also in continuous-time systems; see [19].However, up to now, the model has not been applied to Markov jump systems with multiplicative noises by the authors' knowledge.This kind of systems with multiplicative noises is often used in engineering practice.For example, a practical model with the control input dependent noise in CDMA systems can be found in [22].This note aims to make an attempt to investigate the stability of such systems.
Our purpose is to address the stability analysis and stabilization controller design for discrete-time nonhomogeneous Markov jump systems with multiplicative noises and with polytopic transition probability matrices.It is well known that stability analysis is the basis for the design and synthesis in all control systems and fruitful results have been obtained; see [23][24][25][26][27][28][29][30] and the references therein.In this article, two types of stability, stochastic stability (SS) in mean square sense and finite-time stability (FTS), will be considered.
The organization of this paper is as follows.In Section 2, we provide model formulation and give some definitions.Section 3 is devoted to stochastic stability and stabilization.In Section 4, the finite-time stability and stabilization will be discussed.Some numerical examples are presented in Section 5. Finally, a brief concluding remark is given in Section 6.

Problem Formulation
Consider the following discrete-time stochastic Markov jump system with multiplicative noises: And the corresponding controlled system is as follows: where   ∈ R where Γ  ( = 1, 2, . . ., ) are given matrices and the entries of To be convenient, we denote the coefficient matrices associated with   =  as   = (  ).
In this paper, we mainly formulate some conditions of SS and FTS.They are two different and independent stability concepts: SS describes the asymptotic behavior of the systems in infinite time domain, while FTS reflects transient performance of the systems in finite-time interval.Now, let us recall the definitions.
Definition 2 (see [1]).System (1) is called stochastically stable if for any initial state  0 and initial mode  0 , System ( 2) is called stochastically stabilizable if there exist a sequence of feedback controls   = (  )  such that for any initial state  0 and initial mode  0 , the closed-loop system is stochastically stable.

Stochastic Stability and Stabilization
In this section, some sufficient conditions will be provided for the stochastic stability and stabilization of systems ( 1) and (2).
Remark 5. Theorem 4 develops a sufficient condition for SS of system (1) when transition probability matrix is arbitrarily valued in a polytope with given vertices.For finite Markov jump switching systems, the stochastic stability is equivalent to asymptotic mean square stability (AMSS) [1].So Theorem 4 is also a sufficient condition for AMSS.
Remark 6.A convex parameter space method was used in control design of uncertain linear system without jumping.
State feedback conditions can be easily derived by exploiting the change of matrix variable due to Geromel and coworkers [31].
It is hard to verify the conditions in Theorem 4 because there are infinite matrix inequalities.Hence it has more theoretical significance than its practical significance.To be solvable, we introduce a set of slack matrices and convert the infinite matrix inequalities into finite LMIs in the following theorem.

Finite-Time Stability and Control
In this section, we will consider the finite-time stability.Firstly a theoretical criterion is given.
Next, we convert the infinite matrix inequalities ( 15) and ( 16) into finite matrix inequalities by use of slack matrices and properties of the condition number and design a state feedback controller to make system (2) finite-time stabilizable.
(1) The proof can be divided into two parts.Firstly, we prove that (15) holds if (21) holds.We adopt the method used in Proposition 2 of [12].
Next, we prove that (16) holds if (22) is true.Before proving this, we need to recall two properties of the condition number: When (22) holds and    = (   ) −1 , one has (2) This can be easily proved by the same procedure as (1).
Remark 11.We should point out that unlike the equivalence of ( 7) and ( 13), ( 21) is just a sufficient condition of ( 15) due to the existence of .
Remark 12.In fact, ( 22) can be guaranteed by the following LMIs: So we can obtain a set of feasible feedback controls by solving ( 23) and (32) or by solving (23) and checking if they satisfy (22).

Numerical Examples
The initial state is given as  0 = (−0.5,0.5) T .The vertices of the polytope transition probability matrices are given as follows: By solving LMIs ( 14), it can be concluded that system (2) is stochastically stabilized via feedback gains The system modes and the state trajectories of the openloop and closed-loop systems are, respectively, shown in Figures 1-3.(36) The initial state is given as  0 = (−2, 1) T and the vertices of the polytope transition probability matrices are given as follows:

Conclusions
In this paper, we have investigated the stochastic stability and finite-time stability of a kind of discrete-time nonhomogeneous Markov jump systems with multiplicative noises and polytopic transition probability matrices.Some sufficient conditions for stability are proposed by parameter-dependent Lyapunov function and stabilization controllers are designed by using LMI toolbox.The simulation results show the effectiveness of the developed techniques.This research will motivate us to study the continuous-time version in future.

Figure 4 :
Figure 4: The system modes for Example 2.

Figure 5 :
Figure 5: ( T    ) of the open-loop system for Example 2.

Figure 6 :
Figure 6: ( T    ) of the closed-loop system for Example 2.
In this section, two illustrative examples are presented to show the effectiveness of the proposed main results.