Abstract
This paper first investigates the problem of finite-time boundedness of Markovian jump system with piecewise-constant transition probabilities via dynamic output feedback control, which leads to both stochastic jumps and deterministic switches. Based on stochastic Lyapunov functional, the concept of finite-time boundedness, average dwell time, and the coupling relationship among time delays, several sufficient conditions are established for finite-time boundedness and filtering finite-time boundedness. The system trajectory stays within a prescribed bound. Finally, an example is given to illustrate the efficiency of the proposed method.
1. Introduction
Markovian jump systems were introduced as a class of stochastic switched systems, which can be governed by a Markov chain in a finite mode set of linear dynamics. In recent years, because it is appropriate to model many physical systems with economics, random failures, and networked control systems, more and more people draw their attention to Markovian jump systems [1–4]. As a special class of stochastic systems in the finite operation modes, Markovian jump systems can switch from one to another at different time. Up to now, many important results in the literature are based on the assumption that the complete knowledge of transition probabilities is available in the jump process. However, at mode observation instants, the Markovian jump modes of the systems cannot be accurately obtained, and to get the ideal information on all transition rates is hard or generally expensive in reality, and the obtained results are not accurate. Therefore, it is very important to consider systems based on the assumption that transition probabilities are completely unknown. Recently, the Markovian jump systems subject to partially known transition probabilities have been reported [5–10]. However, the Markov processes are time-invariant in most of aforementioned obtained results.
Nowadays, piecewise-homogeneous (namely, time-varying transition probabilities) Markovian jump systems are developed for practical applications, affecting not only the time-varying transition probabilities but also the state dynamics. The evolution between two operating modes with time-varying transition probabilities was proposed in economy systems [11, 12]. Because of the important issue of the possibility in measuring the variations, up till now, a few people in view of stochastic Markovian jump systems with time-varying transition probabilities except in [13–19]. In [14], there is a bounded real lemma for Markovian jump linear systems with time-varying transition probabilities in discrete-time domain. The Markov switching is employed for sustainability of US external debt in [15]. The linear matrix inequalities are used for control theory in Markov switching [16]. In [19], newly Lyapunov functional is proposed with piecewise-constant transition probabilities. It should be noted that average dwell time switching is very important in dynamic systems [20–23]. In [20], the average dwell time switching and uncertainties are considered. Correspondingly, a dependent average dwell time approach is proposed in [21]. The piecewise-homogeneous is taken into account which makes the considering dynamic of the Markovian jump systems more controllable and optimizes the performance of systems.
Furthermore, in a finite horizon, the practical application problems tend to care about the described systems’ transient characteristics state, especially the transient performances of control systems. It is necessary to consider the state in a fixed region; therefore, the concept of finite-time stability was introduced [22, 23]. Some research results in finite-time case for Markovian jump systems can be found in [24–30]. For example, the finite-time stabilization with output feedback control is introduced in [24]. Finite-time boundedness is considered with state-dependent switching strategy in [26]. In [27], finite-time control is proposed for nonlinear jump systems. In [29], the partially unknown transition rates are introduced for finite-time filtering of stochastic systems. It is noted that, in the engineering area, there are still some problems related to stochastic systems to be solved. In order to make the finite-time behaviour of stochastic Markovian jump systems more reasonable and satisfy the requirements, the finite-time boundedness of Markov jump systems with piecewise-constant transition probabilities via dynamic output feedback control has not been studied. The problem is interesting but also challenging, which motivates us to conduct this study.
The main contribution of this paper is that we present a novel approach for finite-time boundedness of Markovian jump system with piecewise-constant transition probabilities via dynamic output feedback control. We establish a more general model to extend the existing results into more feedback control systems. The deterministic switches and stochastic jumps are taken into account at the same times. The finite-time stability is an independent concept, which is different from Lyapunov stability and can be determined by switching. By selecting the appropriate Lyapunov-Krasovskii functional, under average dwell time constraint on switching signals, the sufficient conditions among average dwell times, transition probabilities, and time-varying delay are derived to guarantee finite-time boundedness of the Markovian jump systems.
2. Preliminaries
In this paper, fixing the probability space , we consider the following Markovian jump system described bywhere is the state vector of the system, is the measured output, and is the exogenous noise signal. , , , , and are constant real matrices with appropriate dimension. represent the constant delay and is the differentiable vector-valued initial function on . Let the random form process , be the Markov stochastic process taking values on a finite set , governing the switching from mode at time to mode at time with the following transition probabilities:with transition rates , , , , and . Here, is now a function of . By , we mean that the transition rates are time-varying. Moreover, is assumed to be piecewise-constant function of time , and transition rates can be defined by
Furthermore, to determine the time-varying property, represents a high-level average dwell time switching signal. is a given initial condition sequence. For simplicity, let represent as a piecewise-constant function of time, which takes values in the finite set . At an arbitrary time , may be dependent on or , or both, or other logic rules. For a switching sequence , is continuous from right everywhere and maybe either autonomous or controlled. When , we say that the th transition probabilities matrix is active and therefore the trajectory of system (1) is trajectory of system (1) with the th transition probabilities matrix.
In this paper, our goal is to design the following dynamic output feedback controller, which can guarantee the system is finite-time boundness:where , , , and are matrices to be determined.
Substituting (4) into (1) and , , , and , we have where
Throughout the paper, suppose that the matrices have full row rank, in other words, . Then we have the singular decomposition of as where is a diagonal positive matrix and and are unitary matrices.
Remark 1. In this paper, matrices are singular decomposition as unitary matrices, which reduce the conservatism.
First of all, we will give definitions and lemmas about system (5), which plays an important role in the derivation of our result.
Definition 2 (see [29]). System (5) is said to be finite-time bounded with respect to , where , is positive define matrix, and is a switching signal. We have where , .
Definition 3 (see [21]). For any , let denote the switching number of during . If holds for and , then and are called chattering bound and average dwell time, respectively. Here we assume for simplicity as commonly used in the literature.
Definition 4 (see [31]). Consider as the stochastic Lyapunov function of the resulting system (4); its weak infinitesimal operator is defined as
Definition 5 (see [32]). The jump rates of the visited modes from a given mode are assumed to satisfy where and are known parameters for a given mode and represent the lower and upper bounds when all the jump rates are known; that is, and . Meanwhile, the number of the visited modes from a given mode is denoted by including the mode itself.
Lemma 6 (Schur complement [14]). Given constant matrices , , and , where and , then if and only if
3. Finite-Time Boundedness Analysis
Theorem 7. System (5) is finite-time stochastic boundedness (FTSB) with respect to if there exist matrices , , and and constants , , and (), such that we have the following linear matrix inequalities: with the average dwell time of the switching signal satisfyingwhere
Proof. We consider the following Lyapunov-Krasovskii functional: Taking the time derivative of along the trajectory of the system (5), one hasMoreover, we have Assuming that condition (12) is satisfied, we obtain Notice that Integrate (23) from to , from which we can get that Noting that , where is the th switching instant and , from condition (15) it yields From condition (24) and (25), we can easily have Thus, from (24)–(26), it yields Note that Thus On the other hand Substituting (29) and (30) into (19), one obtains When , which is the trivial case, from (17), . When , from (17), we have Substituting (32) into (31) yields The proof is completed.
Remark 8. It should be noted that the linear feedback control subject to piecewise constant transition probability is first considered in the paper, and it is classical and effective to stabilize the Markov jump system.
4. Finite-Time Performance Analysis
Theorem 9. For a given constant , , system (5) is robustly finite-time stochastic boundedness with respect to , if there exist positive definite matrices , , , , and , such that the following linear matrix inequalities where with the average dwell time of the switching signal satisfying and the feasible solutions are given as follows: Then the closed-loop systems (5) are finite-time boundedness with respect to .
Proof. Pre- and postmultiply inequality (12) by , it yields that Denote , ; using Schur complement, we can obtain where Define And can guarantee that ; then we can obtain (34).
5. Illustrative Example
Consider the system as follows:
The piecewise-constant transition probabilities matrices are given as
Choosing , , , , and , by solving the matrix equalities in Theorem 9, we have the following filter parameters:
From (38), we have . Moreover, we can obtain the average dwell time
By Theorem 9, through the program in the optimization toolbox of MATLAB, the optimal bound with minimum value of relies on the parameter . We can find feasible solution when . Figure 1 shows the solution trajectory of the system. The state trajectory of the closed-loop system is shown in Figures 2–4, where the initial state . From Figures 2–4, it is easy to see that the system is finite-time boundedness.
6. Conclusions
In this paper, the problems of finite-time boundedness of Markovian jump system with piecewise-constant transition probabilities via dynamic output feedback control is concerned. By allowing new Lyapunov-Krasovskii functional, the switching signal is constraint by average dwell time, and a numerical example is also given to demonstrate the effectiveness of the proposed approach.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Nature Science Foundation of China (Grant no. 61173121), the Postdoctoral Foundation of China (Grant no. 20100471665), the Youth Science Foundation of Sichuan Province, in China (Grant no. 2010JQ0032), and the Science Foundation of Chengdu City in China (Grant no. 60873092).