Stochastic Stability for Time-Delay Markovian Jump Systems with Sector-Bounded Nonlinearities and More General Transition Probabilities

This paper is concerned with delay-dependent stochastic stability for time-delay Markovian jump systems (MJSs) with sectorbounded nonlinearities and more general transition probabilities. Different from the previous results where the transition probability matrix is completely known, a more general transition probability matrix is considered which includes completely known elements, boundary known elements, and completely unknown ones. In order to get less conservative criterion, the state and transition probability information is used as much as possible to construct the Lyapunov-Krasovskii functional and deal with stability analysis. The delay-dependent sufficient conditions are derived in terms of linear matrix inequalities to guarantee the stability of systems. Finally, numerical examples are exploited to demonstrate the effectiveness of the proposed method.


Introduction
During the past decades, much attention has been devoted to the stochastic systems since stochastic modeling has an important science and engineering application [1,2].As an important class of stochastic systems, MJSs have attracted a lot of interest, since they can be used to model many practical dynamical systems, such as power systems, manufacturing systems, and economic systems in which they may experience failure or repairs, abrupt environmental disturbances, and abrupt changes in the operating point of a nonlinear plant (see [3,4]).Theoretical results of this class of systems have been applied in many processes, such as target tracking, manufactory processes, solar thermal receivers, and faultdetection and fault-tolerant systems.Some examples can be found in [5][6][7].
For MJSs, the transition probabilities of the jumping process are important, but most of the previous issues on this kind of systems usually assumed that the elements of the transition probability matrix are completely known.Based on this condition, a lot of research results have been worked out in the literature, such as [8][9][10][11][12][13].However, in many practical engineering applications, we cannot get all of the transition probabilities elements, and some elements might be time variable in some cases.Therefore, the study of Markov jump systems with partly known transition probabilities becomes necessary and some well-known results have been published, see [11][12][13].In some cases, the exact value of some elements of the transition probability matrix can not be obtained, but their lower and upper bounds can be determined.The case has been addressed by [8], but its method cannot be used to deal with the case where there is no information available for transition probabilities.In [9], the more general partly known transition probabilities are investigated, which covers the cases that the transition probabilities are exactly known, unknown, and unknown but with known bounds.However, a conservative condition to relax inequality is used to deal with this kind of transition probabilities, and the time delay is not considered.
It is well known that time delay is very common in variously practical systems such as manufacturing systems, chemical processes, network control systems (NCSs), and telecommunication and economic systems.They are often the causes of oscillation, instability, and poor performance in many control systems.In the past years, time-delay system has been widely studied, and many analysis results have been reported (see [14,15]).However, many approaches for timedelay problems only employed partial information of the delay-related terms.The quadratic integral terms of the form ∫  −ℎ() ()() have been the major choices to construct the Lyapunov-Krasovskii functional, ignoring the employment of the delay upper bound ℎ (see [16]).Recently, Markovian systems with time delay have been widely studied (see [17][18][19]).But the time delays in [17,19] are invariant, which is limited in the practical application.Although [18] considers the systems with time-varying delay, some useful terms are ignored to construct the Lyapunov-Krasovskii functional, which may lead to considerable conservativeness.Unfortunately, in the aforementioned papers about time delay, the elements of the considered transition probability matrix are completely known.To the best of the authors' knowledge, up to now, there are few papers concerning both time delay and more general transition probability to deal with stochastic stability for nonlinear Markov jump systems.
Motivated by the previous points, in this paper, we are concerned with the stochastic stability analysis for a class of nonlinear MJSs with time-varying delay and general transition probabilities.The considered nonlinearity is the socalled sector-bounded nonlinearity [20], which is quite more general than the usual Lipschitz conditions.By exploiting full information of the delay-related terms including ℎ() and ℎ and inherent relationship between the elements in transition probabilities, less conservative stability criterions are obtained in the framework of LMIs.Numerical examples are given to demonstrate the superiorities and effectiveness of the proposed method.
Notations.The superscript  stands for matrix transposition, and   denotes the  dimensional Euclidean space.[⋅] stands for the  mathematical expectation; () represents the sum of  and   .In addition, in symmetric block matrices or long matrix expressions, * is used as an ellipsis for the terms that are introduced by symmetry.The notation  > 0 (≥0) means that  is real symmetric positive (semipositive) definite.[⋅] stands for the expectation operator with respect to the given probability measure . 2 Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.
Remark 1.In order to get more general result, this paper considers that L   includes two cases.One is that   is exactly known, and the other is that it is unknown but with upper and lower known bounds, which has been considered in [8].If   is exactly known, it also can be seen as a rate with equal bounds.
The set L comprises the various operation modes of system (1), and for each possible value of () =  ( ∈ L), the matrices and the vector-valued nonlinear functions associated with the th mode will be denoted by In system (1), the vector-valued nonlinear functions ( * ) and   ( * ) are assumed to satisfy the following sectorbounded conditions: where for all ,  ∈   ,  1 ,  2 ,  1 ,  2 ∈  × are known constant matrices.Without loss of generality, the following equations are always assumed: Remark 2. As in [20], the nonlinear functions ( * ),   ( * ) are said to belong to sectors.In other words, the nonlinearities are bounded by sectors.Further, both the filter design problems and control analysis have been investigated; see [21,22].The nonlinear descriptions in (7) are more general than the usual sigmoid functions and the recently commonly used Lipschitz conditions [21], and  1 , it can be obtained that with Proof.Notice the fact that Then, ( 10) can be obtained.Thus, the proof is complete.
For the sake of simplicity, the solution (,  0 ,  0 ) of system (1) with  0 ∈ L is denoted by ().It is known that {(), ()} is a Markov process with an initial state ( 0 ,  0 ), and its weak infinitesimal generator acting on function  is defined as follows: Throughout this paper, the following definition is used.1) is said to be stochastically stable if for finite () ∈   defined on [−ℎ, 0] and (0) ∈ L, the following is satisfied: where (, , (0)) denotes the solution to system at time  under the initial conditions () and (0).

Stochastic Stability Analysis
In this section, new LMI-based sufficient conditions for system (1) will be presented based on a special Lyapunov-Krasovskii functional and Lemma 3.
So, Ω 1 (ℎ()) < 0 is equivalent to Then, the proof is separated into two cases,  ∈ L   and  ∈ L   .
Taking into account ( 17)-( 22), there exists a scalar  > 0 for each  ∈ L such that which implies that system (1) is stochastically stable.Therefore, the proof is completed.
Then, the following corollary is obtained.

Numerical Examples
In this section, numerical examples are given to demonstrate the benefits and effectiveness of the proposed methods.
First, Example 1 in [16] is given to show the superiority of Theorem 5 for MJS (1) with no nonlinear terms and completely known transition probability matrix.
Table 1 presents the comparison results, which shows that the stochastic stability result in Theorem 5 is less conservative than those in [16,18] when no nonlinear terms exist and transition probabilities are completely known.
Next numerical example is for uncertain MJS (43) with more general transition probability.
The more general transition probability matrix is chosen as follows, which includes completely known elements, boundary known elements, and completely unknown elements: Choose −1.5 ≤  ≤ −0.9.According to Corollary 6, Table 2 can be obtained for different values of .Also, the uncertain MJS (43) is stochastically stable.Moreover, as the upper bound of ḣ (), that is, , increases, the maximum allowed upper bound of ℎ() may be reduced.Furthermore, a nonlinear MJS with sector-bounded nonlinearity is given.(50) According to (7), the corresponding matrices, which describe the sector bound, can be chosen as From Table 3, it can be seen that the stochastic stability can be guaranteed for nonlinear MJS (1) when more general transition probabilities (4) are considered.

Conclusion
This paper has presented a new method for solving the stochastic stability analysis problem for MJS with timevarying delay and sector-bounded nonlinearity.A general transition probability matrix is adopted, which includes completely known elements, boundary known elements, and completely unknown elements.Based on a new modedependent stochastic Lyapunov-Krasovskii functional and some new techniques to deal with transition probabilities, sufficient conditions with less conservativeness are derived in terms of LMIs to obtain stochastic stability of systems.
Numerical examples have also demonstrated the effectiveness of the results.

Table 1 :
Comparison of maximum allowed ℎ.