马尔可夫跳变系统综述
马尔可夫切跳变系统和切换系统
马尔可夫切跳变系统和切换系统1. 引言随着现代技术的发展，许多复杂的系统需要进行控制和优化。
其中一个问题是如何有效地对系统的状态进行建模和控制。
针对这个问题，学者们提出了许多理论和方法。
本文将介绍其中的两种方法：马尔可夫切跳变系统和切换系统。
2. 马尔可夫切跳变系统马尔可夫切跳变系统是一个随机过程系统，它在离散时间下的状态具有马尔可夫性质。
然而，它的状态并不是固定不变的，而是会随着时间的推移而发生跳变。
这些跳变可以是由内部的系统变化造成的，也可以是由外部环境的变化所引起的。
当状态发生跳变时，传统的控制方法可能变得难以应对，因此马尔可夫切跳变系统需要特殊的控制策略。
2.1 马尔可夫切跳变系统的建模马尔可夫切跳变系统的最重要的特征就是状态的跳变。
为了对这种变化进行建模，我们需要引入一个新的概念——模式。
模式是一种特定的状态序列，通常表示系统在某些时间段内遵循的固定行为规律。
在马尔可夫切跳变系统中，模式的出现取决于外部环境或内部状态的变化，因此模式的选择是随机的。
在建立马尔可夫切跳变系统的模型时，我们需要考虑以下几个因素：1. 状态-状态转移概率矩阵。
2. 模式的出现概率。
3. 状态跳变造成的系统性能损失和成本开销。
4. 确定合适的控制策略。
2.2 马尔可夫切跳变系统的控制马尔可夫切跳变系统的控制需要根据系统当前的状态做出决策。
在系统的状态发生跳变时，传统的控制方法可能失效，因此需要采用更为特殊的控制策略。
常见的控制方法包括随机控制、模糊控制和最优控制等。
3. 切换系统切换系统也是一种随机过程系统，其状态会随着时间变化而发生跳变。
与马尔可夫切跳变系统不同的是，切换系统的状态跳变与外部环境无关，而是由内部系统元件的切换行为所决定。
在切换系统中，系统行为可以看作是根据一组预定义的行为模式坚持一种特定的行为规律。
3.1 切换系统的建模切换系统的建模同样需要考虑与状态跳变相关的因素。
为了对系统进行建模，我们需要确定以下几个要素：1. 系统状态-状态转移概率矩阵。
具有多噪声的马尔科夫跳变随机系统的精确能观性
具有多噪声的马尔科夫跳变随机系统的精确能观性王维;张永华;梁向前【摘要】本文研究了具有多噪声的马尔科夫(Markov)跳变随机系统的精确能观性问题,利用H-表示和谱算子的方法以及伊藤公式,建立了马尔科夫跳变随机系统的系数矩阵和确定性系统的系数矩阵之间的关系,将随机系统的精确能观性转化为确定性系统的完全能观性,从而得到了离散时间马尔科夫跳变随机系统的精确能观性的格拉姆矩阵判据(Gramian matrix criterion).【期刊名称】《山东科技大学学报（自然科学版）》【年(卷),期】2016(035)003【总页数】7页(P99-105)【关键词】精确能观性;H-表示;格拉姆矩阵判据【作者】王维;张永华;梁向前【作者单位】山东科技大学数学与系统科学学院,山东青岛266590;山东科技大学信息科学与工程学院,山东青岛266590;山东科技大学数学与系统科学学院,山东青岛266590【正文语种】中文【中图分类】O231在现代控制理论中，能观性是控制问题中的一个基本而重要的特性。
系统的能观性可以反映系统直接测量输入输出量的量测值以便确定系统状态的可能性。
随着控制理论的发展，能观性对于控制和状态估计问题研究的作用越来越重要。
马尔科夫(Markov)跳变系统有着广泛的实际应用背景，也是近年来控制领域热门的研究方向之一。
近年来，谱技术被成功的运用到线性随机系统的精确能观测问题中，Zhang等[1-2]用广义Lyapunov算子的方法给出了连续时间随机时不变系统的精确能观测的随机Popov-Belevitch-Hautus(PBH)判据，并将确定性系统的能观性问题推广到随机系统上，采用随机谱方法得到了判定精确能观性的PBH判据。
Zhang等[3]研究了线性随机时变系统的精确能观性，给出了判定连续时间和离散时间的精确能观性的格拉姆矩阵判据和秩判据。
由于马尔科夫跳变系统能被应用到自然和工程中，因此该类系统也已经得到了广泛的研究。
时滞马尔科夫跳变系统的分析与综合研究综述
时滞马尔科夫跳变系统的分析与综合研究综述张保勇;夏卫锋;李永民【摘要】The study of Markovian jump systems (MJSs)is an important and hot research branch in the control and system area.Over the past 20 years,the analysis and synthesis problems for delayed MJSs have been extensively studied.A great many of important methods and valuable results have been proposed in the literature.The general idea of Lyapunov-Krasovskii functional approach to stability analysis was briefly introduced.Then, the studies of stability analysis,feedback stabilization,and control and filtering with disturbance attenuation performances for different kinds of delayed MJSs were reviewed. Some future possible research topics were also introduced at the end of this paper.%马尔科夫跳变系统研究是控制与系统领域的重点和热点方向.过去20年来,时滞马尔科夫跳变系统的分析与综合问题得到了广泛研究,提出了许多重要的方法,得到了很多有价值的结论.简要介绍稳定性分析的 Lyapunov-Krasovskii泛函方法的一般思想,然后针对不同的时滞马尔科夫跳变系统,概述稳定性分析、反馈镇定以及基于干扰抑制性能指标的控制和滤波的研究现状,最后给出未来可能的研究方向.【期刊名称】《安徽大学学报（自然科学版）》【年(卷),期】2018(042)002【总页数】16页(P3-17,12)【关键词】马尔科夫跳变系统;时滞系统;Lyapunov-Krasovskii泛函方法;鲁棒控制;鲁棒滤波【作者】张保勇;夏卫锋;李永民【作者单位】南京理工大学自动化学院,江苏南京210094;南京理工大学自动化学院,江苏南京210094;湖州师范学院理学院,浙江湖州313000【正文语种】中文【中图分类】O1510 IntroductionIn 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 controlsystems. 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 stabili ty, robust stabilization and H∞ 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 delayedMJSs 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 StabilityStability 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 approachIn 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 functional V(xt,rt,t). It should be pointed out that the functional must be dependent on the system mode rt, otherwise there is no differencebetween 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 of V(xt,rt,t) when rt=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)where τ>0 denotes the time delay that can be either constant or time-varying; the mode rt is a right-continuous Markov chain on a probability space taking values in a finite state space S={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]V(x,t,rt)=xT(t)P(rt)x(t)+xT(σ)Qx(σ)dσ,(3)where P(rt) and Q are 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)It is obvious that the integral terms in (4) are independent from the system mode. The mode-independence of the integral terms is a main cause of the conservatism in the corresponding stability conditions. Therefore, it is necessary to make the matrices Q and Z depending on the system mode. By observing this fact, the following mode-dependent L-K functional was constructed in Ref.[38] for system (1) with time-varying delayV(x,t,rt)=xT(t)P(rt)x(t)+xT(σ)Q(rt)x(σ)dσ+(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]V(x,t,rt)=xT(t)P(rt)x(t)+xT(σ)Q(rt)x(σ)dσ+(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 MJPsThe 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-dependentconditions 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 MJPsIn 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 MJPsIn 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 stochasticstability 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 StabilizationIn 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 butalso 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].For It-type stochastic systems with time delays and MJPs, there have been a number of results on the stabilization problem. For example, the exponential stabilization problem for a class of stochastic time-delay systems with MJPs was studied in Ref.[103], where both the discrete and distributed mode-dependent delays were considered. The stabilization of stochastic time-delay systems with MJPs and nonlinear uncertainties was investigated in Refs.[104-105]. The robust finite-time stabilization of stochastic MJSs with time-varying delays was addressed in Ref.[106]. The stabilization of neutral-type stochastic systems with MJPs was studied in Refs.[67, 107]. The delayed controllers for stochastic time-delay systems with MJPs were designed in Refs.[108-109].3 Control and filtering with disturbance attenuation performancesIn 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 performancesFor a dela yed MJS, suppose that z(t) denotes the output and ω(t) denotes the external disturbance. The disturbance ω(t) is assumed to be deterministic and square-integrable; that is, ωT(t)ω(t)dt<∞. The disturbances satisfying this assumption are said to be energy-bounded. This is a standard assumption for investigating the disturbance attenuation performances.An MJS is said to have an H∞ performance level γ if the following inequality holds[18, 36, 38](7)An MJS is said to have an L2-L∞ performance level γ if the following inequality holds [110-111](8)An MJS is said to be passive if the following inequality holds for all terminal time tp≥0[112](9)An MJS is said to be (Q,S,R)-dissipative if the following inequality holds for some scalar α>0 and for all terminal time tp≥0[113](10)where Q,S,R are prescribed weighting matrices.It is easy to find that, when the weighting matrices Q,S,R are chosen as special values, the dissipative performance defined by (10) covers the passivity and H∞ performance as special ca ses. However, the L2-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 scalar ρ such that thefol lowing inequality holds for all terminal time tp≥0(11)where Ψ1≤0 is a semi-negative definite matrix, Ψ3≥0 and Φ≥0 are semi-positive definite matrices, and these matrices satisfy (‖Ψ1‖+‖Ψ2‖)‖Φ‖=0. It is worth noting that (11) reduces to (10) when Φ=0, Ψ1=Q, Ψ2=S, Ψ3=R-αI and ρ=0. The inequality (11) also reduces to (8) when Φ=I, Ψ1=0, Ψ2=0, Ψ3=γ2I and ρ=0. Therefore, the extended dissipative performance defined by(11) is quite general since it covers the (Q,S,R)-dissipative performance and the L2-L∞ perfo rmance. More discussions on the definition and efficiency of the extended dissipative performance can be found in Ref.[42].3.2 Control with disturbance attenuation performancesFor 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 the H∞ 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]. The H∞ control problem for MJSs with mode-dependent delays was studied in Refs.[116-117]. The H∞ control problem for neutral type MJSs was addressed in Refs.[55-56, 107]. The H∞ control problem for stochastic time-delay systems with MJPs was investigated in Refs.[105, 118-119].The L2-L∞ control problem f or stochastic systems with time delays and MJPs was studied in Ref.[120]. By taking the passivity into account fortime-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 performancesThe H∞ filtering problem has been extensively stud ied for delayed MJSs. For example, The H∞ filtering problem for MJSs with time-varying delays was investigated in Refs.[38, 41, 127-129]. For MJSs with mode-dependent delays, the H∞ filtering problem was studied in Refs.[35, 130]. In context ofstochastic time-delay systems with MJPs, the H∞ filtering problems were studied in Refs.[131-136]. It should be noted that reduced-order H∞ filters for delayed MJSs were designed in Refs.[137-138].The L2-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 exponential L2-L∞ filtering for linear MJSs with distributed delays was studied in Ref.[140] by applying the delay partitioning techniques. The decentralized L2-L∞ filtering problem for a class of interconnected MJSs with constant delays was addressed in Ref.[141]. The exponential L2-L∞ filtering problem for stochastic MJSs with mixed mode-dependent delays was considered in Ref.[142]. For delayed MJSs with nonlinear uncertainties, the L2-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 designing H∞ 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 ConclusionsThis 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 jumpsystems 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. References:[1] MARITON M. Jump linear systems in automatic control[M]. New York: Marcel Dekker, 1990.[2] MAO X, YUAN C. Stochastic differential equations with Markovian switching[M]. London: Imperial College Press, 2006.[3] BOUKAS E K. Control of singular systems with random abrupt changes[M]. Berlin: Springer, 2008.[4] COSTA O L V, FRAGOSO M D, TODOROV M G. Continuous-time Markovian jump linear systems[M]. Berlin: Springer, 2013.[5] ZHANG L, YANG T, SHI P, et al. Analysis and design of Markov jump systems with complex transition probabilities[M]. Switzerland: Springer,2016.[6] MAHMOUDM S, SHI P. Methodologies for control of jump time-delay systems[M]. Boston: Kluwer Academic Publishers, 2003.[7] HE S, SHEN H. Finite short time control and synthesis for stochastic Marov jump systems[M]. Beijing: Science Press, 2018 (in Chinese).[8] SHI P, LI F. A survey on Markovian jump systems: modeling and design[J]. International Journal of Control, Automation, and Systems, 2015, 13 (1): 1-16.[9] GU K, KHARITONOV V L, CHEN J. Stability of time-delay systems[M]. Boston: Birkhauser, 2003.[10] KHARITONOV V L. Time-delay systems[M]. New York: Springer, 2013.[11] FRIDMAN E. Introduction to time-delay systems: analysis andcontrol[M]. Switzerland: Springer, 2014.[12] WU M, HE Y, SHE J H. Stability analysis and robust control of time-delay systems[M]. New York: Springer, 2010 (in Chinese).[13] RICHARD J P. Time-delay systems: an overview of some recent advances and open problems[J]. Automatica, 2003, 39: 1667-1694. [14] XU S, LAM J. A survey of linear matrix inequality techniques in stability analysis of delay systems[J]. International Journal of Systems Science, 2008, 39 (12): 1095-1113.[15] BENJELLOUN K, BOUKAS E K, YANG H. Robust stabilizability of uncertain linear time delay systems with Markovian jumpingparameters[C]//Proceedings of the American Control Conference, Washington, 1995: 330-334.[16] BENJELLOUN K, BOUKAS E K, YANG H. Robust stabilizability of uncertain linear time-delay systems with Markovian jumping parameters[J]. Journal of Dynamic Systems, Measurement, and Control, 1996, 118: 776-783.[17] BENJELLOUN K, BOUKAS E K. Stochastic stability of linear time-delay system with Markovian jumping parameters[J]. Mathematical Problems in Engineering, 1997, 3: 187-201.[18] BENJELLOUN K, BOUKAS EK, COSTA L V. H∞ control for linear time-delay systems with Markovian jumping parameters[J]. Journal of Optimization Theory and Applications, 2000, 105 (1): 73-95.[19] BENJELLOUN K, BOUKAS E K. Mean square stochastic stability of linear time-delay system with Markovian jumping parameters[J]. IEEE Transactions on Automatic Control, 1998, 43 (10): 1456-1460.[20] SHAIKHET L. Stability of stochastic hereditary systems with Markov switching[J]. Theory of Stochastic Processes, 1996, 2 (18): 180-184. [21] CAO Y Y, LAM J. Stochastic stabilizability and H∞ control for discrete-time jump linear systems with time delay[J]. Journal of the Franklin Institute, 1999, 336: 1263-1281.[22] MAO X, MATASOV A, PIUNOVSKIY A B. Stochastic differential delay equations with Markovian switching[J]. Bernoulli, 2000, 6 (1): 73-90. [23] MAO X. Robustness of stability of stochastic differential delay equations with Markovian switching[J]. Stability and Control: Theory and Applications, 2000, 3 (1): 48-61.[24] MAO X, SHAIKHET L. Delay-dependent stability criteria for stochasticdifferential delay equations with Markovian switching[J]. Stability and Control: Theory and Applications, 2000, 3 (2): 87-101.[25] YUAN C, ZOU J, MAO X. Stability in distribution of stochastic differential delay equations with Markovian switching[J]. Systems & Control Letters, 2003, 50: 195-207.[26] YUAN C, MAO X. Robust stability and controllability of stochastic differential delay equations with Markovian switching[J]. Automatica, 2004, 40: 343-354.[27] HUANG L, MAO X. On input-to-state stability of stochastic retarded systems with Markovian switching[J]. IEEE Transactions on Automatic Control, 2009, 54 (8): 1898-1902.[28] FENG L, LI S, MAO X. Asymptotic stability and boundedness of stochastic functional differential equations with Markovian switching[J]. Journal of the Franklin Institute, 2016, 353 (18): 4924-4949.[29] KOLMANOVSKII V, KOROLEVA N, MAIZENBERG T, et al. Neutral stochastic differential delay equations with Markovian switching[J]. Stochastic Analysis and Applications, 2003, 21 (4): 839-867.[30] XIE L. Stochastic robust stability analysis for discrete-time neural networks with Markovian jumping parameters and timedelays[C]//Proceedings of the 30th Annual Conference of the IEEE Industrial Electronics Society, Korea, 2004: 1743-1748.[31] XIE L. Stochastic robust stability analysis for Markovian jumping neural networks with time delays[C]// Proceedings of 2005 Conference on Networking, Sensing and Control, Tucson, USA, 2005: 923-928.。
马尔可夫切跳变系统和切换系统
马尔可夫切跳变系统和切换系统马尔可夫切跳变系统和切换系统是两种常见的数学模型，用于描述具有随机性质的系统。
这两个模型在不同领域中都有广泛的应用，如物理学、生物学、经济学等。
本文将分别介绍马尔可夫切跳变系统和切换系统的基本概念和特点，并探讨它们的应用。
一、马尔可夫切跳变系统马尔可夫切跳变系统是一种离散时间、离散状态的随机系统。
它的基本思想是将系统的状态看作是在不同状态之间跳跃的，而每次跳跃的概率只依赖于当前的状态，而与之前的状态无关。
这种特性称为马尔可夫性质。
马尔可夫切跳变系统可以用一个状态转移矩阵来描述，矩阵的每个元素表示从一个状态跳转到另一个状态的概率。
根据这个转移矩阵，可以计算系统在不同时间点处于不同状态的概率分布。
通过观察系统的状态序列，可以估计系统的参数，并预测未来的状态。
马尔可夫切跳变系统在物理学中有广泛的应用。
例如，用于描述粒子在空间中的运动，粒子在不同状态之间跳跃的概率可以用马尔可夫切跳变系统来描述。
在生物学中，马尔可夫切跳变系统可以用来描述基因在染色体上的位置变化。
此外，马尔可夫切跳变系统还可以用于金融市场的模型建立，用来预测股票价格的波动。
二、切换系统切换系统是一种动态系统，它的行为在不同的模式之间切换。
每个模式对应着系统的一种状态和一种行为。
在不同的模式之间切换的规律可以用一个切换矩阵来描述，矩阵的每个元素表示从一个模式切换到另一个模式的概率。
切换系统可以用于描述具有多种行为模式的系统。
例如，在机器人控制中，机器人可能会根据不同的任务需求切换工作模式。
在通信系统中，根据信道的状态，系统可能会切换不同的调制方式和编码方式。
切换系统还可以用于描述经济系统中的市场行为，不同的市场状态对应着不同的交易策略。
切换系统是一种复杂的系统模型，它的行为受到多种因素的影响。
在实际应用中，需要通过观察系统的行为来估计模型的参数，并预测未来的行为。
切换系统的建模和分析是一个具有挑战性的问题，需要运用概率论和统计学的方法来解决。
马可夫跳变系统
ISSN1751-8644continuous-884IET Control Theory Appl.,2008,Vol.2,No.10,pp.884–894 &The Institution of Engineering and Technology2008doi:10.1049/iet-cta:20070297IET Control Theory Appl.,2008,Vol.2,No.10,pp.884–894885 doi:10.1049/iet-cta:20070297&The Institution of Engineering and Technology2008886IET Control Theory Appl.,2008,Vol.2,No.10,pp.884–894&The Institution of Engineering and Technology2008doi:10.1049/iet-cta:20070297IET Control Theory Appl.,2008,Vol.2,No.10,pp.884–894887 doi:10.1049/iet-cta:20070297&The Institution of Engineering and Technology2008888IET Control Theory Appl.,2008,Vol.2,No.10,pp.884–894&The Institution of Engineering and Technology2008doi:10.1049/iet-cta:20070297IET Control Theory Appl.,2008,Vol.2,No.10,pp.884–894889 doi:10.1049/iet-cta:20070297&The Institution of Engineering and Technology2008890IET Control Theory Appl.,2008,Vol.2,No.10,pp.884–894&The Institution of Engineering and Technology2008doi:10.1049/iet-cta:20070297Figure1Behaviours of the system states in function of time t Figure2Behaviours of the system states in function of time tIET Control Theory Appl.,2008,Vol.2,No.10,pp.884–894891 doi:10.1049/iet-cta:20070297&The Institution of Engineering and Technology2008system with the computed controller is piecewise regular,impulse-free and stochastically stable.5ConclusionThis paper dealt with a class of continuous-time singular linear systems with Markovian switching.Results on stochastic stability and its robustness,and the stochastic stabilisation and its robustness are developed.The LMI framework is used to establish the different results on stability,stabilisation and their robustness.Full and partial knowledge of the jump rates are considered.The results we developed here can easily be solved using any LMI toolbox like the one of Matlab or the one of Scilab.Figure 4Behaviours of the system states in function of time tFigure 3Behaviours of the system states in function of time t892IET Control Theory Appl.,2008,Vol.2,No.10,pp.884–894&The Institution of Engineering and Technology 2008doi:10.1049/iet-cta:20070297IET Control Theory Appl.,2008,Vol.2,No.10,pp.884–894893 doi:10.1049/iet-cta:20070297&The Institution of Engineering and Technology2008894IET Control Theory Appl.,2008,Vol.2,No.10,pp.884–894&The Institution of Engineering and Technology 2008doi:10.1049/iet-cta:20070297。
不确定离散时间马尔可夫跳变模糊系统鲁棒H∞控制
不确定离散时间马尔可夫跳变模糊系统鲁棒H∞控制
不确定离散时间马尔可夫跳变模糊系统鲁棒H∞控制
讨论不确定离散时间马尔可夫跳变模糊系统(MJFS)的鲁棒H∞控制.首先,本文给出了能够保证系统鲁棒稳定且具有H∞鲁棒度的一个充分条件.然后采用并行分布补偿算法,将系统鲁棒H∞控制控制器的设计转化成为了一组线性矩阵不等式的求解问题,方便使用Matlab求解.最后的仿真结果表明,本文所提出的方法是有效的.
作者：李长滨何熠吴爱国 LI Chang-bin HE Yi WU Ai-guo 作者单位：天津大学,电气与自动化学院,天津,300072 刊名：模糊系统与数学ISTIC PKU 英文刊名：FUZZY SYSTEMS AND MATHEMATICS 年，卷(期)： 2007 21(5) 分类号： O159 关键词：马尔可夫跳变非线性系统马尔可夫跳变模糊系统均方收敛。
随机马尔可夫跳变系统的弹性动态输出反馈控制
随机马尔可夫跳变系统的弹性动态输出反馈控制作者：李艳恺陈谋吴庆宪来源：《南京信息工程大学学报（自然科学版）》2018年第06期摘要本文討论了随机噪声影响下马尔可夫跳变系统的弹性动态输出反馈控制问题.在系统随机干扰和控制输入扰动的情况下，设计的弹性控制器可以确保闭环系统的依概率渐近稳定性.通过运用随机微分方程理论和线性矩阵不等式技术对系统进行稳定性分析，获得了系统依概率渐近稳定的充分条件和控制器增益.最后通过数值算例和直升机系统仿真验证了所提弹性动态输出反馈控制方法的有效性.关键词随机马尔可夫跳变系统;弹性控制;动态输出反馈控制;依概率渐近稳定;线性矩阵不等式中图分类号 TP273文献标志码 A0 引言马尔可夫跳变系统是一类特殊的随机切换系统，它的切换规律依赖于转移概率矩阵.随着对马尔可夫跳变系统的深入研究，很多实际系统控制的问题，例如电子通信、生物医学以及经济分析等，都可以利用马尔可夫跳变系统的控制方法来处理[1-3] .近年来，很多控制领域知名专家在马尔可夫跳变控制系统问题上取得了很多研究成果[4-8] .另一方面，由于随机噪声的存在，实际系统的控制性能往往会受到影响，甚至导致系统的不稳定.随着随机控制系统理论的不断发展，很多关于随机噪声的抑制问题得到解决[9-10] .如文献[11]讨论了关于随机拉格朗日系统的输出反馈控制问题;文献[12]研究了非线性随机系统的状态反馈H ∞控制问题;文献[13]给出了随机非线性系统的稳定性法则.对于马尔可夫跳变系统，随机噪声也是普遍存在的.通常情况下，随机噪声存在于各个独立的子系统中，并且与模态间的随机跳变相互独立.随机噪声的存在使得马尔可夫跳变系统问题变得更加复杂.文献[14]分别采用状态反馈反步控制方法和输出反馈反步控制方法处理了一类随机马尔可夫跳变系统的控制问题;文献[15]解决了奇异随机马尔可夫跳变系统的稳定性问题.这些研究工作很大程度上促进了随机马尔可夫跳变系统的理论发展，也为许多实际工程上的控制问题提供了可行的解决方案.由于在很多实际工程应用中，系统状态往往很难测量，或者测量的成本极高，因此，输出反馈控制方案成为处理这类问题的首选方案.在文献[8，16]中，应用动态输出反馈控制器处理连续时间马尔可夫跳变系统和连续时间奇异马尔可夫跳变系统问题，并取得了良好的控制效果.然而，由于控制器设备的老化、计算器维数限制以及传感器灵敏性过强或过弱等，都会使系统的控制过程中混入一定的扰动.因此，在设计控制器的时候充分考虑到这些扰动对系统的影响是有必要的.针对这一问题，一些学者设计了弹性控制器，提高了闭环系统的鲁棒性.文献[17]研究了带有脉冲异步切换系统的弹性控制器设计问题;文献[18]结合基于干扰观测器控制方法，设计抗干扰弹性控制器，讨论了多干扰下的马尔可夫跳变系统的稳定性问题.本文主要研究了随机马尔可夫跳变系统的弹性动态输出反馈控制器设计问题.首先，在系统混有随机干扰、控制器存在扰动的情况下设计动态输出反馈控制器，保证闭环系统稳定.然后，利用随机控制理论、李雅普诺夫稳定性理论以及线性矩阵不等式技术，分析闭环系统的稳定性，获得可解的充分条件.最后通过数值仿真和直升机控制系统算例验证本文所提控制方案的有效性.4 总结本文研究了随机噪声和输入扰动下随机马尔可夫跳变系统的弹性动态输出控制问题.为了保证闭环系统的依概率渐近稳定性，设计了弹性动态输出反馈控制器，并应用随机控制系统理论、李雅普诺夫稳定性理论以及线性矩阵不等式技术，获取了保证系统具有相应控制性能的可解的充分条件.最后通过一个数值算例和无人直升机系统模型验证了本文所提方法的可行性.参考文献References[ 1 ]Boukas E K.Stochastic switching systems：analysis and 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T—S模糊Markov跳变系统控制策略综述
Ａ ｓｕｒｖｅｙ ｏｆ ｃｏｎｔｒｏｌ ｓｔｒａｔｅｇｙ ｆｏｒ Ｔ—Ｓ ｆｕｚｚｙ Ｍａｒｋｏｖ ｊ ｕｍｐ ｓｙｓｔｅｍｓ
ＳＵ Ｌｅｉ 。Ｙ Ｅ Ｄ ａｎ ， （１．Ｃｏｌｌｅｇｅ ｏｆ Ｉｎｆｏｒｍ ａｔｉｏｎ Ｓｃｉｅｎｃｅ ａｎｄ Ｅｎｇｉｎｅｅｒｉｎｇ， Ｎｏｒｔｈｅａｓｔｅｒｎ Ｕ ｎｉｖｅｒｓｉｔｙ，Ｓｈｅｎｙａｎｇ １１０８１９，Ｃｈｉｎａ
统 状 态 、模 糊 规 则 及 系 统 模 态 组 成 ．系 统 各 模 态 间 的 随 机 跳 变 服 从 Ｍ ａｒｋｏｖ过 程 ，且 用 此 来 描 述 系 统 参 数 的 随
机 变 化 ．首先 ，介 绍 Ｔ—ｓ模 糊 Ｍａｒｋｏｖ跳 变 系 统 的 研 究 背 景 ；其 次 。从 Ｔ—ｓ模 糊 Ｍａｒｋｏｖ跳 变 系 统 的 复 杂 结 构 出
发 ，阐 述 Ｔ—ｓ模 糊 Ｍ ａｒｋｏｖ跳 变 系统 及其 控制 的研 究进 展 ；最 后 ，简 要 介 绍 面临 的挑 战．
关 键 词 ：Ｔ Ｓ模 糊 Ｍ ａｒｋｏｖ跳 变 系 统 ；模 糊 规 则 ；随机 跳 变
中 图 分 类 号 ：ＴＭ７３４
文 献 标 志 码 ：Ａ
文 章 编 号 ：１０００ ２１６２（２Ｏ１８）０２ ００１８ ０５
２．Ｓｔａｔｅ Ｋ ｅｙ Ｌａｂｏｒａｔｏｒｙ ｏｆ Ｓｙｎｔｈｅｔｉｃａｌ Ａ ｕｔｏｍ ａｔｉｏｎ ｏｆ Ｐｒｏｃｅｓｓ Ｉｎｄｕｓｔｒｉｅｓ， Ｎ ｏｒｔｈｅａｓｔｅｒｎ Ｕ ｎｉｖｅｒｓｉｔｙ，Ｓｈｅｎｙａｎｇ １１０８１９，Ｃｈｉｎａ）
Ａｂｓｔｒａｃｔ：Ｔ—Ｓ ｆｕｚｚｙ Ｍ ａｒｋｏｖ ｊ ｕｍｐ ｓｙｓｔｅｍ ｉｓ ａ ｋｉｎｄ ｏｆ ｓｔｏｃｈａｓｔｉｃ ｈｙｂｒｉｄ ｎｏｎｌｉｎｅａｒ ｓｙｓｔｅｍ ｓ ｄｒｉｖｅｎ ｂｙ ｔｉｍ ｅ ａｎｄ ｅｖｅｎｔｓ．Ｉｔｓ ｓｔａｔｅ ｖａｒｉａｂｌｅｓ ｄｅｐｅｎｄ ｏｎ ｓｙｓｔｅｍ ｓｔａｔｅ，ｆｕｚｚｙ ｒｕｌｅｓ ａｎｄ ｓｙｓｔｅｍ ｍｏｄｅ．Ａ Ｍ ａｒｋｏｖ ｐｒｏｃｅｓｓ ｉＳ ｕｓｅｄ ｔｏ ｄｅｓｃｒｉｂｅ ｔｈｅ ｒａｎｄｏｍ ｉｕｍｐｓ ｂｅｔｗｅｅｎ ｖａｒｉｏｕｓ ｍｏｄｅｓ ｏｆ ｔｈｅ ｓｙｓｔｅｍ．Ｔｈｅ ｂａｃｋｇｒｏｕｎｄ ａｎｄ ｓｉｇｎｉｆｉｃａｎｃｅ ｏｆ Ｔ—Ｓ ｆｕｚｚｙ Ｍ ａｒｋｏｖ ｊ ｕｍ ｐ ｓｙｓｔｅｍ ｗｅｒｅ ｉｎｔｒｏｄｕｃｅｄ ｆｉｒｓｔ ｉｎ ｔｈｉｓ ｐａｐｅｒ．Ｔｈｅｎ ｔｈｅ ｒｅｓｅａｒｃｈ ｐｒｏｇｒｅｓｓｅｓ ｏｆ ｃｏｎｔｒｏｌ ｓｔｒａｔｅｇｙ ｆｏｒ Ｔ—Ｓ ｆｕｚｚｙ Ｍ ａｒｋｏｖ ｊ ｕｍｐ ｓｙｓｔｅｍ ｗｅｒｅ ａｎａｌｙｚｅｄ ｉｎ ｔｅｒｍｓ ｏｆ ｉｔｓ ｃｏｍｐｌｅｘ ｓｔｒｕｃｔｕｒｅ．Ｆｉｎａｌｌｙ， ｔｈｅ ｃｏｒｒｅｓｐｏｎｄｉｎｇ
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马尔可夫跳变系统综述
马尔可夫跳变线性系统（MJLS）是一种具有多个模态的随机系统，系统在各个模态之间的跳变转移由一组马尔可夫链来决定。
MJLS模型因其在表示过程中可以产生突变而更能精确的描述实际工程应用中的系统。
近年来，MJLS的最优控制问题成为了研究的热点，动态规划、极大值原理以及线性矩阵不等式等成为了解决此类问题的主流方法。
本文对MJLS最优控制领域的研究现状进行了综述。
分别对一般情况下、带有噪声的情况下、带有时滞的情况下以及某些特定情况下的MLJS最优控制问题的国内外研究现状进行论述。
最后进行了总结并提出MJLS最优控制领域未来值得关注的研究方向。