Improved Stability Criteria for Markovian Jump Systems with Time-Varying Delays

and Applied Analysis 3 Ξ 1 = [ [ [ [ R 2 − M 0 X 11 X 12


Introduction
Markovian jump systems are a special class of stochastic hybrid systems.Many dynamical systems subject to random abrupt variations, such as mechanical systems, economics, and systems with human operators, can be modeled by Markovian jump systems [1].Due to their extensive applications in many files, the analysis and synthesis of Markovian jump systems have received much research attention and lots of significant results have been reported; see, for example, [2][3][4][5][6][7][8] and the references therein.
Time delay is an inherent characteristic of many dynamic systems such as networked control systems, industrial systems, and process control systems.The systems with or without time delays are convergent when time delays are close to zero; otherwise, they may be divergent.In other words, time delays can degrade the performance of systems designed without considering the delays and can even destabilize the systems.During the past few decades, considerable attention has been paid to the stability analysis of timedelay systems [9][10][11][12][13][14][15][16].The existing stability criteria for linear systems can be classified into two types: delay-independent ones which are applicable to delays of arbitrary size and delaydependent ones which include information on the size of delays.In general, delay-dependent stability criteria are less conservative than delay-independent ones especially when the size of the delay is small.Thus, considerable attention has been paid to the delay-dependent stability criteria; see [11][12][13][14][15][16][17][18][19][20][21][22][23], for example.As for delay-dependent stability, many methods have been taken for deriving stability criteria, such as free-weighting matrices methods [11], model transformation techniques [12,13], convex combination methods [14][15][16][17], delay decomposition approaches [18], multiple integral approaches [19], and input-output approaches [20].Recently, the bounding techniques of the cross terms and integral terms in the derivatives of the Lyapunov-Krasovskii functional are widely investigated, such as improved Jensen's integral inequality [21], reciprocally convex approach [22,23], and improved Wirtinger's integral inequality [24].Some less conservative stability results have been derived by using the above techniques.
In this paper, we develop some new stability criteria by using an improved Wirtinger's integral inequality and the convex combination method to deal with the cross terms and integral terms in the derivatives of the Lyapunov-Krasovskii functional.In addition, the positive definiteness of some Lyapunov matrix is not required.The obtained results can be applied to both slow and fast time-varying delays.
The numerical examples demonstrate the effectiveness and superiority of the presented results.
Notation.Throughout this paper, (Ω, , ) is a probability space, Ω is the sample space,  is the -algebra of the sample space, and  is the probability measure on .E{⋅} refers to the expectation operator with respect to some probability measure . > 0 (<0) means  is a symmetric positive (negative) definite matrix and  −1 denotes the inverse of matrix .  represents the transpose of .Sym() stands for  +   .The symbol * in LMIs denotes the symmetric term of the matrix.col{, } represents a column vector formed by  and .Identity matrix, of appropriate dimensions, will be denoted by .diag(⋅, ⋅) denotes a diagonal matrix.Ω(, ) means the element in the th row and th column of the block matrix Ω.

Problem Statement
Fix a probability space (Ω, , ) and consider the following Markovian jump systems: where () ∈ R  is the state vector and () is a compatible vector-valued initial function defined on [− The evolution of the Markovian process {  ,  ⩾ 0} is governed by the following transition probability: where Δ > 0 and lim Δ → 0 ((Δ)/Δ) = 0;   ⩾ 0 for  ̸ =  is the transition probability from mode  at time  to mode  at time  + Δ and   = − ∑  =1, ̸ =   .For simplicity, when   = ,  ∈ , the matrices (  ) and   (  ) are denoted by   and   , respectively.
The following definition and lemmas are needed in the proof of our main results.

Improved Stability Criterion
In this section, we will present an improved stochastic stability criterion in terms of LMIs by using Lyapunov-Krasovskii functional method and convex combination technique.
Remark 5.In order to guarantee the Lyapunov-Krasovskii functional (  ,   ) > 0, most authors require the Lyapunov matrix   > 0 in  1 (see, e.g., [4][5][6][7][8]).The Lyapunov-Krasovskii functional employed in this paper is very simple, but a less conservative result is developed by using the LMI (10) instead of inequality P  > 0, which can be seen in Section 4.However, it should be pointed out that the provided result cannot be used when the lower delay bound is considered to be zero because of the existence of 1/ 1 in LMI (10).
Remark 6.It is well known that the convex combination approach is effective in reducing conservatism in stability analysis.In some literature, the integral terms (30) Based on the above method, we are now ready to give an improved asymptotic stability criterion for system (30).
Corollary 7. Given scalars 0 <  1 <  2 and , the timevarying delay system (30) is asymptotically stable, if there exist matrix   =    ∈ R × , (,  = 1, 2, 3); symmetric positive definite matrices where (32) Remark 8. Theorem 4 and Corollary 7 can be applied to both slow and fast time-varying delays.But when  is unknown, the above results cannot be used directly to check the stability.
From the construction of Lyapunov-Krasovskii functional, it can be seen by setting  3 = 0 in Theorem 4 and Corollary 7 that the corresponding conclusions are valid for the case when  is unknown.

Numerical Examples
In this section, three numerical examples will be presented to show the validity of the main results derived above.
This example has been taken from [4].To compare the stochastic stability condition in Theorem 4 with that in [25,26], we choose  22 = 0.8,  = 0.9.Using Theorem 4 of our paper, the admissible upper bound  2 for different  1 and  11 can be found in Table 2.It can be seen from Table 2 that Theorem 4 in our paper is less conservative.
This system is a well-known delay-dependent stable system where the maximum allowable delay  max = 6.1725 [24].For known and unknown , the admissible upper bounds  2 for different  1 , which guarantee the asymptotical stability of the system (30), are listed in Tables 3 and 4, respectively.It can be seen from Tables 3 and 4 that the stability results obtained in this paper are less conservative than those in [14,16,17,22,27,28].

Conclusion
In this paper, the problem of stochastic stability for a class of Markovian jump systems has been investigated.By using the convex combination technique and the improved integral inequality, some less conservative delay-dependent stability criteria are established in terms of linear matrix inequalities.