Analysis for a Class of Uncertain Markovian Jumping System with Multiple Delays

and Applied Analysis 3 Lemma 6 (see [36]). Consider system (1) with 0 ≤ h 1i ≤ h i (t) ≤ h 2i , i = 1, 2, . . . , m, for any matrices Z i ∈ R n×n and i ∈ R n×n satisfying [ Zi Ui ∗ Z i ] ≥ 0; the following inequality holds


Introduction
It is well known that time delay is usually the main reason for instability and poor performance of many practical control systems [1][2][3][4][5].The stability results for delayed systems can be generally classified into two categories: delay-independent stability criteria and delay-dependent criteria.And the delaydependent results are often less conservative than the delayindependent ones, especially when the time delays are small.Therefore, much more attention has been focused on study of the delay-dependent stability conditions in recent years.For example, the system transformation method in [6], the descriptor system method in [7], parameterdependent Lyapunov-Krasovskii functional method in [8], Jensen inequality method in [9], Free-weighting matrix method in [10,11], integral inequality method in [12], augmented Lyapunov functional method in [13], convex domain method in [14], interval partition method in [15,16], reciprocally convex method in [17], and so forth.And those approaches have been widely used in the stability analysis for lots of delayed systems in recent years [18][19][20].
On the other hand, since Markovian jumping systems can model many types of dynamic systems subject to abrupt changes in their structures, such as failure prone manufacturing systems, power systems, and economics systems [21][22][23][24][25][26][27], a great deal of results related to stability analysis and synthesis for this class of systems with time delays has been reported in recent years.For example, for the delay-independent results, sufficient conditions for mean squares to stochastic stability were obtained in [28], while exponential stability conditions were proposed in [29].The robust  ∞ filtering problem was dealt with in [30].For the delay-dependent ones, the stability and  ∞ control results were presented by resorting to some bounding techniques for some cross terms and using model transformation to the original delay system in [31].The  ∞ control and Filtering problem were taken into account in [32] using the Free-weighting matrix method.The stability and  ∞ analysis was proposed in [33] with the idea of delay partition.Filtering problem with a new index was considered in [34] using the reciprocally convex method.It is worth mentioned that inspite of the deep study for the delayed stochastic in recent years as mentioned above, there are few papers that consider the problem of stability analysis for uncertain stochastic systems with multiple delays, which motivates our study.
In this paper, the robust exponential stability and  ∞ performance analysis for a class of uncertain Markovian system with multiple time-varying delays is investigated.Some new delay-dependent stability conditions are derived.
Throughout this paper, we will use the following Definitions and Lemmas.

Main Results
For simplicity, we define

Robust Exponential Stability Analysis.
The criteria of the robust exponential stability for the systems (1)-( 3) are proposed in the following Theorem.

Robust 𝐻 ∞ Exponential Stability Analysis.
The criteria of the robust exponential stability with  ∞ performance for the systems (1)-( 3) are proposed in the following Theorem.
This implies that for any nonzero Therefore, by Definition 3, the system is robustly exponentially stable with a prescribed  ∞ performance level .This completes the proof.

Numerical Example
In this section, we provide an example to demonstrate the effectiveness of the proposed method.Let  = 2 and  = 2; consider the systems (1)-( 3) with parameters as follows.
If we fix the lower bound of ℎ 1 () and ℎ 2 (), that is, ℎ 11 = 0.2 and ℎ 21 = 0.6, for the different ℎ 12 , we can get the upper bounds of ℎ 22 as in Table 2.

Conclusion
The robust exponential stability and  ∞ performance analysis for uncertain Markovian jumping system with multiple time-varying delays has been investigated based on the reciprocally convex approach.Some new delay-dependent stability conditions are obtained in term of LMIs.Numerical example has been proposed to illustrate the effectiveness of result.

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.