ZHANG Baoyong， XIA Weifeng， LI Yongmin

(1. School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China；2. School of Science, Huzhou Teachers College, Huzhou 313000, China)

0 Introduction

In a great many of practical systems, such as mechanical systems, electric power systems, flight control systems and networked control systems, there may exist sudden environment changes, random failures and repairs[1-2]. These phenomena make the system structures or parameters changing randomly. Thus, the systems can be modeled as randomly switched systems, where the switching law is described by a finite Markov chain. In references, this class of systems is referred to systems with Markovian/random switching, systems with Markovian jumping parameters (MJPs) and Markovian jump systems (MJSs). Over the past 50 years, the MJSs have been extensively studied and numerous results have been reported. The readers are referred to Refs.[1-8] for fundamental theory and recent developments on MJSs.

On the other hand, time delays are unavoidable when designing control systems. More importantly, time delays are always the cause of instability and poor performance of control systems. For these reasons, time-delay systems have received considerable attention; see, Refs.[9-14] and the references therein. The study of delayed MJSs started in the middle of 1990s and has made a great progress in the past 20 years.

The pioneer research on linear delayed MJSs was carried out in the late five years of 1990s by Benjelloun and Boukas, who investigated a series of problems on stochastic stability, robust stabilization andH∞controller design[15-19]. Almost at the same time, Shaikhet studied the asymptotic mean-square stability for a class of stochastic hereditary systems with MJPs and constant delays[20]. In 1999, Cao and Lam addressed the discrete-time MJSs with delays[21]. In 2000, Mao and his collaborators introduced a class of stochastic differential delay equations (SDDEs) with MJPs[22]. Since then, Mao has devoted his research efforts greatly to the analysis of nonlinear delayed MJSs described by SDDEs[2, 23-28]. In 2003, the neutral stochastic differential delay equations with MJPs were introduced by Kolmanovskii[29], which can be regarded as a general expression of the delayed MJSs. The stability of delayed neural networks with MJPs was primarily studied in Refs.[30-31]. In 2005, the study of delayed normal MJSs was extended to the case of singular systems[32-33].

Up to date, the study of delayed MJSs has become an important and hot research branch in the control and system area. A great number of results and methods have been developed in the literature. As more and more researchers, especially PhD and master students, are choosing the delayed MJSs as their research topics, it is necessary to provide a survey on the analysis and synthesis of delayed MJSs. However, it is impossible to include all the results on delayed MJSs in a single paper. Thus, the emphasis of this paper will be given on the study of stability analysis, feedback stabilization, and control and filtering with disturbance attenuation performances for continuous-time time-delay systems with MJPs. It is worth mentioning that the notations in the mathematical expressions in the context of this paper can be found in every paper related to the delayed MJSs, and thus their explanations are omitted in this paper.

1 Stability

Stability is the prerequisite for designing automatic control systems. Thus, the stability analysis is a fundamental problem in the study of delayed MJSs. In the literature, there are a few of different descriptions of stability concepts for delayed MJSs, among which the stochastic stability[15-18], asymptotic mean-square stability[19-20]and exponential mean-square stability[34-35]have been largely used. It is now known that the Lyapunov-Krasovskii (L-K) functional approach is a powerful tool for analyzing the stability of delayed MJSs. Therefore, in the following we briefly introduce the L-K functional approach.

1.1 Mode-dependent L-K functional approach

In general, the L-K functional approach to stability analysis of delayed MJSs contains two steps. The first step is to construct an appropriate L-K functionalV(xt,rt,t). It should be pointed out that the functional must be dependent on the system modert, otherwise there is no difference between the stability analysis of the delayed MJSs and that of deterministic time-delay systems. Therefore, the mode-dependence is the main characteristic of the L-K functional approach for delayed MJSs. The second step of the approach is to compute the weak infinitesimal generator ofV(xt,rt,t) whenrt=i, which is defined by[2,16,34]

(1)

If we can find conditions that ensure the negative definiteness of the generator defined in (1), then the stability defined in the stochastic and mean-square manner could be guaranteed.

The structure of the mode-dependent L-K functional plays a key role in the conservatism reduction of stability conditions for delayed MJSs. For ease of understanding, it is better for us to start from a simple but important class of retarded-type linear systems, which are described by

(2)

whereτ>0 denotes the time delay that can be either constant or time-varying; the modertis a right-continuous Markov chain on a probability space taking values in a finite state spaceS={1,2,…,s}. The definition of transition probability of the modes can be found in every reference related to delayed MJSs, and thus it is omitted here.

For system (2), the simplest mode-dependent L-K functional is of the following form[15-18, 36]

(3)

whereP(rt) andQare positive-definite matrices. This functional can lead to delay-independent stability conditions, which are quite conservative, especially when the delay is small. In order to reduce the conservatism, it is generally necessary to develop delay-dependent stability conditions. In general, the L-K functional used for developing delay-dependent stability conditions consists of three parts: non-integral terms, single-integral terms and double-integral terms. For system (2), an efficient L-K functional is of the following form[37]

(4)

(5)

In this functional, not only the non-integral term but also the single-integral term are dependent on the system mode. If the last double integral term in (5) is further required to be dependent on the system mode, the following L-K functional is utilizable[39-42]

(6)

Clearly, this functional is strongly dependent on the system mode, and thus it has particular efficiency for reducing the conservatism of delay-dependent stability conditions. The advantage of functional (6) in the stability and performance analysis of linear delayed MJSs has been discussed in Refs.[40-42].

In the above, we provide a general idea to construct strongly mode-dependent L-K functionals for system (2). This idea has not been fully applied in the study of complicated MJSs, and thus the idea is expected to get more applications in the future. Next, in the context of L-K functional approach, we will briefly review the recent studies on the stability analysis problem for different kinds of delayed MJSs.

1.2 Stability analysis of linear time-delay systems with MJPs

The delay-independent stability problem for linear time-delay systems with MJPs was primarily studied in Refs.[15-17] based on the L-K functional in (3). The conditions obtained in Refs.[15-17] are expressed in terms of algebraic matrix equations. The functional (3) was also employed in Refs.[18, 36] to derive LMI-based stability conditions. In Ref.[43], the delay-dependent stability analysis for system (2) was firstly studied based on an L-K functional containing one mode-dependent non-integral term and two mode-independent double integrals. Another version of delay-dependent conditions for system (2) was obtained in Ref.[44], where the Newton-Leibniz formula was applied. In Refs.[37, 45-48], the free-weighting matrix method was applied to derive less conservative stability conditions. Some of these results were further improved in Refs.[38, 49-50] by using the L-K functionals similar to (5). Delay-dependent stability conditions based on the L-K functional (6) were given in Refs.[39-42]. Delay-partitioning techniques were applied in Refs.[51-54] to analyze the stability of system (2).

A system is called to be neutral if the differential of the system state involves delays. The stability problem for neutral systems with MJPs has been also studied. For example, the stochastic stability analysis for linear neutral MJSs with multiple constant delays was tackled in Ref.[55], where delay-independent conditions were presented. Different versions of delay-dependent stability conditions for linear neutral MJSs with time-varying delays were obtained in Refs.[56-59]. When the transition probability is partially unknown, the stability of neutral MJSs has been analyzed in Refs.[60-61].

1.3 Stability analysis of stochastic time-delay systems with MJPs

In many of works on delayed MJSs, the Brownian motions are involved in the system model. Such systems are called stochastic time-delay systems with MJPs, which are described by It-type delay differential equations. The fundamental theory of general stochastic time-delay systems with MJPs can be found in Mao’s book[2] and the recent journal papers[23-28]. The exponential mean-square stability for linear stochastic systems with MJPs, constant delays and interval uncertainties was studied in Ref.[34], where a rigorous discussion on the L-K functional with mode-dependent integrals was provided. The delay-dependent conditions for robust stability of stochastic time-delay systems with MJPs and norm-bounded uncertainties were developed in Refs.[62-63]. In Refs.[64-65], the nonlinear uncertainties were taken into account in the stochastic delayed MJSs and delay-dependent conditions of exponential mean-square stability were obtained. The neutral-type stochastic systems with MJPs were investigated in Refs.[66-68], where delay-dependent stability conditions were obtained by using different techniques.

1.4 Stability analysis of delayed neural networks with MJPs

In recent years, the artificial neural networks with time delays and MJPs have been largely studied. It is known that the artificial neural networks are described in mathematics as nonlinear systems with the nonlinear terms satisfying certain bounding conditions. By making the use of the bounding conditions in the stability analysis procedure, LMI-based conditions can be always obtained. For this reason, most of the techniques in the stability study of linear time-delay systems with or without MJPs have been generalized to delay neural networks with MJPs.

For recurrent neural networks with MJPs and time delays, the delay-independent stability conditions were obtained in Refs.[31, 69], while the delay-dependent stability conditions were presented in Refs.[70-75]. The delay-dependent stability problem for delayed Cohen-Grossberg neural networks with MJPs was investigated in Refs.[76-78]. The stochastic stability problem for delayed BAM neural networks with MJPs was considered in Refs.[79-81]. The Markovian genetic regulatory networks were studied in Refs.[82-84]. Delay-dependent stability results for neutral-type neural networks with MJPs have been also reported; see, for example, Refs.[85-88] and the references therein.

2 Stabilization

In general, the stabilization problem is formulated as designing feedback controllers such that the resulting closed-loop system is stable. When the system states are fully available, the state-feedback controllers are desirable. Otherwise, if the system states are not fully available, then the output-feedback controllers need to be designed by using measured output of the original system. It should be pointed out that, for delayed MJSs, particular attention has been paid to the design of mode-dependent controllers. In the following, we are going to survey some recent studies on the stabilization problem for different delayed MJSs.

The robust stabilization problem for linear MJSs with constant delays was addressed in the pioneer works [15-16], where two kinds of state-feedback controllers, namely linear-type controller and saturation-type controller, were designed. It is noted that the controllers in Refs.[15-16] were designed based on a constructive method. The exponential stabilization using state-feedback controllers for linear MJSs with constant delays was investigated in Refs.[47, 54, 89], where delay-dependent conditions were obtained in terms of LMIs. It is worth mentioning that the conditions obtained in Refs.[47, 54, 89] are dependent not only on the delay size but also on the decay rate of the exponential stability. When the delays depend on the system mode, the delay-independent and delay-dependent conditions for solving the state-feedback stabilization problems have been given in Refs.[50, 90-91], respectively. The memory state-feedback controller using delayed states was designed in Ref.[52] for linear delayed MJSs. When the system states are not fully available, the output-feedback stabilization problem has been considered in Refs.[45, 92-94] for different kinds of delayed MJSs. In Refs. [95-96], the finite-time stabilization problem was considered for MJSs with constant and time-varying delays, respectively. In Refs.[97-99], the stabilization of delayed MJSs was studied by using sliding-mode control approach. When the transition rates are partially unknown, the stabilization problems have been addressed in Refs.[100-102].

3 Control and filtering with disturbance attenuation performances

In this section, we are going to review the control and filtering problems for delayed MJSs with external disturbances. These problems are always formulated based on the disturbance attenuation performances such asH∞performance,L2-L∞performance, passivity and dissipativity. Hence, in the following we first give the descriptions of these performances.

3.1 Disturbance attenuation performances

An MJS is said to have anH∞performance levelγif the following inequality holds[18, 36, 38]

(7)

An MJS is said to have anL2-L∞performance levelγif the following inequality holds[110-111]

(8)

An MJS is said to be passive if the following inequality holds for all terminal timetp≥0[112]

(9)

An MJS is said to be (Q,S,R)-dissipative if the following inequality holds for some scalarα>0 and for all terminal timetp≥0[113]

(10)

whereQ,S,Rare prescribed weighting matrices.

It is easy to find that, when the weighting matricesQ,S,Rare chosen as special values, the dissipative performance defined by (10) covers the passivity andH∞performance as special cases. However, theL2-L∞performance cannot be covered by the dissipativity. Regarding this, the so-called extended dissipativity is introduced in Ref.[42]. Specifically, an MJS is said to be extended dissipative if there exists a scalarρsuch that the following inequality holds for all terminal timetp≥0

(11)

whereΨ1≤0 is a semi-negative definite matrix,Ψ3≥0 andΦ≥0 are semi-positive definite matrices, and these matrices satisfy (‖Ψ1‖+‖Ψ2‖)‖Φ‖=0. It is worth noting that (11) reduces to (10) whenΦ=0,Ψ1=Q,Ψ2=S,Ψ3=R-αIandρ=0. The inequality (11) also reduces to (8) whenΦ=I,Ψ1=0,Ψ2=0,Ψ3=γ2Iandρ=0. Therefore, the extended dissipative performance defined by(11) is quite general since it covers the (Q,S,R)-dissipative performance and theL2-L∞performance. More discussions on the definition and efficiency of the extended dissipative performance can be found in Ref.[42].

3.2 Control with disturbance attenuation performances

For delayed MJSs with energy-bounded external disturbances, it is necessary to design state-feedback and output-feedback controllers ensuring both the stability and the disturbance attenuation performances of the resulting closed-loop systems. It seems that theH∞control problem for delayed linear MJSs was first studied independently in Refs.[18, 36, 114], where state-feedback controllers were designed based on the LMI approach. These results were then improved by using different methods; see, for example, Refs.[37-38, 44, 46, 115]. TheH∞control problem for MJSs with mode-dependent delays was studied in Refs.[116-117]. TheH∞control problem for neutral type MJSs was addressed in Refs.[55-56, 107]. TheH∞control problem for stochastic time-delay systems with MJPs was investigated in Refs.[105, 118-119].

TheL2-L∞control problem for stochastic systems with time delays and MJPs was studied in Ref.[120]. By taking the passivity into account for time-delay systems with MJPs, the state-feedback controller design problem was considered in Refs.[121-122], while the output-feedback controllers were designed in Refs.[123-124]. It is also noted that the dissipative control problems for delayed MJSs have been addressed in Refs.[125-126].

3.3 Filtering with disturbance attenuation performances

TheH∞filtering problem has been extensively studied for delayed MJSs. For example, TheH∞filtering problem for MJSs with time-varying delays was investigated in Refs.[38, 41, 127-129]. For MJSs with mode-dependent delays, theH∞filtering problem was studied in Refs.[35, 130]. In context of stochastic time-delay systems with MJPs, theH∞filtering problems were studied in Refs.[131-136]. It should be noted that reduced-orderH∞filters for delayed MJSs were designed in Refs.[137-138].

TheL2-L∞filtering problems for retarded and neutral MJSs with time-varying delays were investigated in Refs.[111,139], respectively, where the transition probabilities are assumed to be partially unknown. The exponentialL2-L∞filtering for linear MJSs with distributed delays was studied in Ref.[140] by applying the delay partitioning techniques. The decentralizedL2-L∞filtering problem for a class of interconnected MJSs with constant delays was addressed in Ref.[141]. The exponentialL2-L∞filtering problem for stochastic MJSs with mixed mode-dependent delays was considered in Ref.[142]. For delayed MJSs with nonlinear uncertainties, theL2-L∞filtering problem was studied in Ref.[143].

In Ref.[42], the filter design problem for linear MJSs with time-varying delays was studied by considering the extended dissipative performance defined by (11). In that work, both mode-dependent and mode-independent filters were designed, and the delay-dependent conditions were given in terms of LMIs. It is noted that the results obtained in Ref.[42] are valid for designingH∞filters,L2-L∞filers, passivity-based filters and dissipative filters, respectively. Therefore, the Ref.[42] provides a unified framework for designing filters with different disturbance attenuation performances. The method developed in Ref.[42] is not limited to the filtering problem of delayed MJSs. Actually, the method has been applied in a number of works on different kinds of systems; see, for example, Refs.[144-148].

4 Conclusions

This paper has surveyed the studies of time-delay systems with MJPs. Since there are many subjects in the research of delayed MJSs, we cannot cover all of them. Thus, our emphasis has been mainly given on the problems of stability analysis, feedback stabilization, robust control and filtering for continuous-time systems. The study of discrete-time systems is not included in this paper. In addition, for delayed neural networks with MJPs, the stability analysis has been reviewed briefly, but the estimation and synchronization problems have not been mentioned, which have also received lots of attention recently. Therefore, this paper only includes a very small amount of references on the delayed MJSs. The readers are encouraged to pay attention to the follow-up researches based on the references provided in this paper.

Although the delay-dependent stability of delayed MJSs has been extensively studied, the results reported in the literature are still conservative to some extent, because they are only sufficient but not necessary. It is of interest to further reduce the conservatism of the stability results. For this purpose, the relaxed L-K functional approach[149-150]and complicated integral inequality techniques[151-153]may be applicable. It is also important research topics that apply the stability conditions derived by using recently developed techniques to control and filtering synthesis.

Recently, a particular attention has been paid to the semi-Markov jump systems and hidden Markov jump systems; see, for example, Refs.[154-157] and the references therein. In semi-Markov jump systems, the transition rates (or probabilities) are no longer constant, because at each time they involve the past information of elapsed jumping sequences[154-155]. In the design of hidden Markov jump systems, the modes of the original system are not available for controllers. In this case, an estimator (called detector in Refs.[156-157]) needs to be introduced to estimate the system mode. Therefore, the mode estimator and feedback controllers should be designed simultaneously for hidden Markov jump systems. It is obvious that the semi and hidden Markov jump systems generalize the traditional MJSs. However, when time delays are taken into account in the two kinds of generalized systems, the control and filtering problems have not been fully investigated, which is an interesting research topic in the future.

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