New stability criteria for semi-Markov jump linear systems with time-varying delays

Abstract: In this paper, the delay-dependent stochastic stability problem of semi-Markov jump linear systems (S-MJLS) with time-varying delays is investigated. By constructing a Lyapunov-Krasovskii functional (LKF) with two delay-product-type terms, a new sufficient condition on stochastic stability of S-MJLSs is derived in terms of linear matrix inequalities (LMIs). Furthermore, the combination use of a slack condition on Lyapunov matrix and the improved Wirtinger’s integral inequality reduces the conservatism of the result. Numerical examples are provided to verify the effectiveness and superiority of the presented results.


Introduction
Markov jump linear systems (MJLSs) are a special class of hybrid stochastic systems. Many different types of systems subjects to random abrupt variations can be modeled by MJLSs, such as target tracking systems, mechanical systems and networked control systems [1,2]. In the past decades, the research of MJLSs has received widespread attention and has obtained many significant results, such as [3][4][5][6]. In the previous literature, the sojourn times of MJLSs are assumed to be a random variable that follows an exponential distribution, which results in the constant transition rates due to the memoryless property of exponential distribution. However, such an assumption cannot be appropriate for many situations, for example, DNA analysis [7] and fault-tolerant control systems [8]. Different from the MJLSs, semi-Markov jump linear systems (S-MJLSs) allow the sojourn time to follow an non-exponential distribution. S-MJLSs are determined by an embedded Markov chain of semi-Markov processes and probability density functions of the sojourn time. Obviously, due to the relaxed conditions on the probability distributions, the S-MJLS has a wider application domain by I; diag(·, ·) denotes a diagonal matrix; Ω(i, j) means the element in the i-th row and j-th column of the block matrix Ω.

Problem statement
In this section, we first introduce some concepts, notation and terminology related to semi-Markov processes.
i) The stochastic process {r k } k∈N + takes values in space S = {1, 2, · · · , N}, where r k represents system mode at the kth transition.
ii) The stochastic process {t k } k∈N + takes values in R + , where t k (t 0 =0) stands for the time at kth transition and monotonically increases with k.
iii) The stochastic process {θ k } k∈N + takes values in R + , where θ k = t k − t k−1 for ∀k ∈ N 1 represents the sojourn-time of mode r k−1 between the (k − 1)th transition and kth transition.
In order to introduce the S-MJLS, we consider the following jumping system: where x(t) ∈ R n is the system state vector, A(r t ) is a matrix function of random jumping process {r t } ∈ S and suppose that the initial condition t 0 = 0 and r(0) is a constant.
The stochastic process r t := r k , t ∈ [t k , t k−1 ) is a semi-Markov process, and system (1) is a continuous-time S-MJLS if for i, j ∈ S and t 0 , t 1 , · · · , t k 0, the following conditions are satisfied: i) It hold that the Pr( ii) The probability Pr(r k+1 = j, θ k+1 ≤ h | r k = i) is independent on k.
In this paper,we consider the following smei-Markov jump time-delay systems in the space (Ω, F, P) as: where x(t) ∈ R n is the system state vector, {r t , t 0} is a continuous-time semi-Markov process taking values in a finite space S = {1, 2, · · · , N}, A(r t ) and A d (r t ) are matrix functions of the random jumping process {r t }; For notation simplicity, when the system operates in the mode, A(r t ) and A d (r t ) are respectively denoted by A i and A di . φ(t) is a continuous vector-valued initial function defined on the interval [−h, 0]. The d(t) is the time-varying delay satisfying: (2. 3) The evolution of the Markov process {r t , t 0} is governed by the following probability transitions: where λ i j (θ) is the transition probability from mode i at the time to j at time t + θ when i j and The following definition and lemmas are needed in the proof of our main results.
2) is said to be stochastically stable, if for any initial state (x 0 , r 0 ), the following relation holds: For a positive definite matrix R ∈ R n×n , the following inequality holds for all differential function ω(s) in [a, b] → R n : [44] For given vectors β 1 and β 2 , scalar α in the interval (0,1), symmetric positive definite matrix R ∈ R n×n and any matrix X ∈ R n×n satisfying R X * R 0, then the following inequality holds: Lemma 3.
[41] For a given symmetric positive definite matrix R > 0, scalars a and b satisfying a < b, and any differential function ω in [a, b] → R n , the following inequality holds:

Improved stability criterion
Before stating the main results, some notations are given. Let In this section, we will present a new stochastic stability criterion in terms of LMIs by using Lyapunov-Krasovskii functional method and Wirtinger-based inequality.
Note that Z is a symmetric positive definite matrix. It follows from Lemma 1 that Further, considering Ξ 1 > 0 and applying Lemma 2 yield Based on the inequalities (3.7) and (3.8), LKF (3.6) can be written as . From Ξ 2 > 0, Q > 0 and R > 0, we conclude that there exists a sufficiently small positive number 1 such that V(x t , i) 1 ||x t || 2 . We next show that L V(x t , i) − 2 ||x(t)|| 2 for a sufficiently small positive number 2 . Let L be the weak infinitesimal generator of the random process {x t , r t }.
Similar to the procedures of calculating the derivative of the Lyapunov function in [8], we have The derivative of V 2 (x t , i) is displayed as follows: Calculating the derivative of other terms in V(x t , i) yields Note that Ψ > 0 holds for any matrix X. Applying Lemma 2 and Lemma 3 to estimate the last term in (3.12) yields (3.13) Combining the inequalities (3.10)-(3.13), we have Thus, Φ < 0 leads to L V(x t , i) < 0, which implies there exists a sufficiently small positive number 2 such that L V(x t , i) − 2 ||x(t)|| 2 for any initial condition (x 0 , r 0 ). By Dynkin's formula, we further have The previous inequality means that lim t→+∞ E{||x(t)|| 2 |(x 0 , r 0 )} = 0 (3.17) Thus, system (2.2) is stochastically stable by Definition 2.
Remark 1. In order to guarantee the Lyapunov-Krasovskii functional V(x t , r t ) > 0, most authors require the Lyapunov matrix P i > 0 in V 1 (x t , r t ) (see, e.g., [8,10] ). In this paper, we derive a relaxed condition for ensuring the positive definite of LKF, i.e., Ξ 2 > 0, Q > 0 and R > 0. The relaxed condition can reduce the conservatism of the result.

Remark 2.
As we known, the appropriate augmented Lyapunov functionals is an effective method to reduce the conservatism of the stability conditions (see, e.g., [29,33] ). An augmented LKF with two delay-product-type terms is constructed in this paper, so that the information about the delay is fully considered, which can further reduce the conservatism of the result.
Remark 3. It is well known that the integral inequalities have played a key role when dealing with the integral terms (see, e.g., [40,41,44]). In this paper, we use Lemma 2 and Lemma 3 to deal with the integral terms in the L V 5 (x t , i) and use Lemma 1 to deal with V 4 (x t , i), the conservatism of stability conditions is further reduced efficiently.
When the sojourn-time following an exponential distribution, the transition rate λ i j (θ) will become to a constant λ i j . In such a case, the S-MJLS (2.2) reduces to an MJLS. The stochastic stability criterion for the MJLS is given as follows: Corollary 1. Given scalars h, µ 1 and µ 2 , the MJLS is stochastically stable, if there exist symmetric matrices P i ∈ R 4n×4n , G ∈ R 3n×3n , M ∈ R 3n×3n , symmetric positive definite matrices Q ∈ R 2n×2n , Z ∈ R n×n , R ∈ R n×n , and any matrices S ∈ R n×n , X ∈ R 2n×2n , such that the following holds: where and Ω ls (l, s = 1, 2, 3, 4),Ω mn (m, n = 1, 2, 3, 4, 5, 6) are defined as in Theorem 1.  The sojourn time is assumed to follow the Weibull distribution. Its probability density function is For different modes, we select the parameters α and β as The embedded Markov chain is assumed to be Based on the definition of transition rate functions, one has In this example, we assume that µ 1 = −µ 2 ≤ 0. Using Theorem 1 of our paper, the admissible upper bounds h for different µ 2 can be found in Table 1    In this example, we assume µ 1 = −µ 2 = −0.5 and λ 22 = −λ 21 = −3. Using Corollary 1 of our paper, the admissible upper bounds h for different λ 11 can be found in Table 2.  [49,50].
Take d(t) = 0.5sin(t) + 0.296, the simulation results are provided in Figure 3 and Figure 4. It can be seen from Figure 4 that the MJLS is stochastically stable under the maximum allowable delay h = 0.796.

Conclusions
In this paper, we have provided a new sufficient condition on stochastic stability of S-MJLSs with time-varying delays by construing an augmented LKF with two delay-product-type terms. In order to reduce the conservatism of the result, a slack condition on Lyapunov matrix has been introduced. In addition, the improved Wirtinger's integral inequality are used to deal with the integral term in the time derivative of the LKF. To compare with the existing results, we also provide a sufficient condition on stochastic stability of MJLSs with time-varying delays. Numerical examples are presented to show the superiority of the proposed method.