Stability and stabilization of Markov jump systems with generally uncertain transition rates
Introduction
Markov jump systems (MJSs) and semi-MJSs, because of their important roles in many practical systems, have received a significant attention of researchers over the past few decades. The transition rates (TRs) of MJSs (accordingly, the semi-Markov kernel of semi-MJSs) govern the system behavior to a great degree, and by now, a lot of investigations on MJSs assume that exact TRs are available (see, e.g., [1], [2], [3], [4], [5], [6], [7], [8], [9] and the references therein). However, in practical systems, it is often hard or even infeasible to obtain the accurate values of TRs. Thus, it is necessary to consider MJSs with uncertain TRs.
In the literature, two classes of models have been considered to describe the uncertain TRs. The first one is the polytopic model where the TR matrix is assumed to be in a convex hull with known vertices [10], [11], [12], [13], [14]. The other class describes the uncertain TRs in an element-wise way. Bounded uncertain TRs model is one of the element-wise descriptions, in which only the bounds of each TR instead of its accurate value need to be available (see, e.g., [15], [16], [17], [18]). Another element-wise description is partially unknown TR model, in which each TR is either exactly known or totally unknown (see, e.g., [19], [20], [21]). It is noted that although the polytopic approach can describe the bounded uncertain TRs by an equivalent transformation, the total number of vertex matrices of the convex hull increase exponentially as the increase of the number of modes, which leads to a combinatoric complexity explosion [22], [23]. On the other hand, despite the capability that the polytopic approach can also describe the partially unknown TRs when the lower bounds of unknown diagonal elements in TR matrices are assumed to be known, it is powerless when the lower bounds are unknown [20]. Moreover, in reality the element-wise uncertainty description is usually more natural as well as handier than the polytopic description [15]. In spite of these advantages, both element-wise models are still limited in practice. For instance, the bounded uncertain TR model cannot deal with the totally unknown TRs although it can handle the imprecise TRs. As for the partially unknown TR model, its limitation is exact opposite to that of the bounded uncertain model. By considering the limitation of both models, in [24] a new element-wise description is proposed, that is called generally uncertain TR model. This model allows that the bounds of each TR are either known or unknown. It is therefore more general, by which much more practical situation can be modeled. Specifically, the bounded uncertain and partially unknown TR models can be covered under the same framework.
Due to the high applicability for modelling the uncertain TRs, MJSs with generally uncertain TRs have been received more and more attention of the researchers and a range of results have been developed (see e.g. [24], [25], [26], [27], [28], [29], [30] and the references therein). However, as the fundamental problems in system analysis and synthesis, stability and stabilization of such systems have not been fully investigated. In fact, the existing stability analysis does not fully use theproperties of TRs, which leads to the conservatism of the obtained stability criterion. Moreover, the existing stability criteria cannot yield equivalent linear matrix inequality (LMI) conditions but only equivalent nonlinear matrix inequality (NLMI) ones for state-feedback stabilization of MJSs. Unfortunately, there is still no any method to completely solve such NLMIs up to now [31,32]. The reason of resulting in NLMIs is that the derivation of stabilization condition requires congruent transformations. Due to the presence of terms in the exiting stability criteria, the congruent transformations lead to NLMI conditions of state-feedback stabilization for MJSs. In order to obtain stabilization conditions in terms of LMIs, the existing results have to introduce additional constraints, which further aggravates conservatism. Therefore, the stability and stabilization problem of MJSs with generally uncertain TRs needs to be further investigated.
In this paper, we first address the stability analysis problem of Markov jump system with generally uncertain TRs. By fully considering the properties of TRs, we obtain a new criterion on mean-square stability. Its conservatism is shown to be lower than some current ones. Moreover, unlike these current results, the proposed criterion avoids introducing the terms so that we can establish equivalent LMI condition for controller design of MJSs based on the proposed stability criterion. Examples are given to illustrate the validity and advantages of the developed results.
Notation: In this paper, and denote, the -dimensional Euclidean space and the set of all real matrices, respectively. The notation ( means that is a real symmetric and positive definite (semi-positive-definite) matrix. stands for the mathematical expectation. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.
Section snippets
System formulation
Consider the following Markov jump system:where the system state is initial state and the control input . taking values in is a Markov process. Its TR matrix satisfieswhere for denotes the transition rate from mode at time to mode at time and . For the sake of simplicity, when we use to
Stability analysis
Here the mean-square stability for MJSs with generally uncertain TRs is investigated. Theorem 1 System (1) with generally uncertain TRs (2) and is mean-square stable if the LMIs below have a feasible solution and . If if where and
State-feedback stabilization
In this section, we address the problem of stabilizing controller design for system (1) with generally uncertain TRs. Specially, the form of the controller to be constructed iswhere for all are the controller gains to be determined.
Using Eq. (28), system (1) is represented as
The following theorem provides solvable condition to determine the gains of the controller (28) for system (1). Theorem 4 For system (1) with generally uncertain TRs (2),
Examples
This section gives three examples to illustrate the validity and the advantages of the developed approach. Example 1 In this example, the stability problem is addressed for MJSs (1) with and four modes. The system data are given byThe generally uncertain TR matrix is given aswhere and
Conclusion
This paper first addressed the stability analysis problem of Markov jump systems with generally uncertain transition rates. The conservatism of the proposed stability criterion is lower than some current ones, which benefits from the more full use of the relationships among the transition rates. Besides, the proposed criterion can be readily used for deriving equivalent solvable condition for stabilizing controller design. Three examples covering stability analysis and stabilizing controller
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowlgedgments
This work was supported by National Natural Science Foundation of China under Grant No. 61773291.
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