Stabilization of Markov jump linear systems using quantized state feedback

This paper addresses the stabilization problem for single-input Markov jump linear systems via mode-dependent quantized state feedback. Given a measure of quantization coarseness, a mode-dependent logarithmic quantizer and a mode-dependent linear state feedback law can achieve optimal coarseness for mean square quadratic stabilization of a Markov jump linear system, similar to existing results for linear time-invariant systems. The sector bound approach is shown to be non-conservative in investigating the corresponding quantized state feedback problem, and then a method of optimal quantizer/controller design in terms of linear matrix inequalities is presented. Moreover, when the mode process is not observed by the controller and quantizer, a mode estimation algorithm obtained by maximizing a certain probability criterion is given. Finally, an application to networked control systems further demonstrates the usefulness of the results.     

Robust mean field games for coupled Markov jump linear systems

We consider robust stochastic large population games for coupled Markov jump linear systems (MJLSs). The N agents’ individual MJLSs are governed by different infinitesimal generators, and are affected not only by the control input but also by an individual disturbance (or adversarial) input. The mean field term, representing the average behaviour of N agents, is included in the individual worst-case cost function to capture coupling effects among agents. To circumvent the computational complexity and analyse the worst-case effect of the disturbance, we use robust mean field game theory to design low-complexity robust decentralised controllers and to characterise the associated worst-case disturbance. We show that with the individual robust decentralised controller and the corresponding worst-case disturbance, which constitute a saddle-point solution to a generic stochastic differential game for MJLSs, the actual mean field behaviour can be approximated by a deterministic function which is a fixed-point solution to the constructed mean field system. We further show that the closed-loop system is uniformly stable independent of N, and an approximate optimality can be obtained in the sense of ε-Nash equilibrium, where ε can be taken to be arbitrarily close to zero as N becomes sufficiently large. A numerical example is included to illustrate the results.     

On almost sure stability of continuous-time Markov jump linear systems

In this paper, we study the almost sure stability of continuous-time jump linear systems with a finite-state Markov form process. A sufficient condition for almost sure stability is derived that refers to the statistics of the transition matrix over m switches. It is shown that, if the system is exponentially almost sure stable, there exists a finite m such that the criterion is satisfied. In order to evaluate the expected value appearing in the condition, an efficient Monte Carlo algorithm is worked out.     

Sliding mode control for Markov jump linear uncertain systems with partly unknown transition rates

In this paper, based on sliding mode control approach, the robust stabilisation problem for a class of continuous-time Markovian jump linear uncertain systems with partly unknown transition rates is investigated. The transition rate matrix under consideration covers completely known, boundary known and completely unknown elements. By making use of linear matrix inequalities technique, sufficient conditions are presented to derive the linear switching surface and guarantee the stochastic stability of sliding mode dynamics. Then a sliding mode control law is designed to drive the state trajectory of the closed-loop system to the specified linear switching surface in finite time in spite of the existing uncertainties and unknown transition rates. Finally, an example is given to verify the validity of the theoretical results.     

基于策略迭代算法的连续时间线性Markov跳变系统非零和微分反馈Nash控制

针对一类连续时间线性Markov跳变系统,本文提出了一种新的策略迭代算法用于求解系统的非零和微分反馈Nash控制问题.通过求解耦合的数值迭代解,以获得具有线性动力学特性和无限时域二次成本的双层非零和微分策略的Nash均衡解.在每一个策略层,采用策略迭代算法来计算与每一组给定的反馈控制策略相关联的最小无限时域值函数.然后,通过子系统分解将Markov跳变系统分解为N个并行的子系统,并将该算法应用于跳变系统.本文提出的策略迭代算法可以很容易求解非零和微分策略所对应的耦合代数Riccati方程,且对高维系统有效.最后通过仿真示例证明了本文设计方法的有效性和可行性.     

Discrete‐time output feedback for Markov jump systems with uncertain transition probabilities

This article addresses the output feedback control for discrete‐time Markov jump linear systems. With fully known transition probability, sufficient conditions for an internal model based controller design are obtained. For the case where the transition probabilities are uncertain and belong to a convex polytope with known vertices, we provide a sufficient LMI condition that guarantees the norm of the closed‐loop system is below a prescribed level. That condition can be improved through an iterative procedure. Additionally, we are able to deal with the case of cluster availability of the Markov mode, provided that some system matrices do not vary within a given cluster, an assumption that is suitable to deal with packet dropout models for networked control systems. A numerical example shows the applicability of the design and compares it with previous results. Copyright © 2012 John Wiley & Sons, Ltd.     

State estimation for discrete-time Markov jump linear systems with multiplicative noises and delayed mode measurements

In this paper, the state estimation problem for discrete-time Markov jump linear systems affected by multiplicative noises is considered. The available measurements for the system under consideration have two components: the first is the model measurement and the second is the output measurement, where the model measurement is affected by a fixed amount of delay. Using Bayes' rule and some results obtained in this paper, a novel suboptimal state estimation algorithm is proposed in the sense of minimum mean-square error under a lot of Gaussian hypotheses. The proposed algorithm is recursive and does not increase computational and storage load with time. Computer simulations are carried out to evaluate the performance of the proposed algorithm.     

On the detectability and observability of discrete-time Markov jump linear systems

This paper presents a new detectability concept for discrete-time Markov jump linear systems with finite Markov state, which generalizes the MS-detectability concept found in the literature. The new sense of detectability can similarly assure that the solution of the coupled algebraic Riccati equation associated to the quadratic control problem is a stabilizing solution. In addition, the paper introduces a related observability concept that also generalizes previous concepts. A test for detectability based on a coupled matrix equation is derived from the definition, and a test for observability is presented, which can be performed in a finite number of steps. The results are illustrated by examples, including one that shows that a system may be detectable in the new sense but not in the MS sense.     

Filtering of discrete‐time Markov jump linear systems with uncertain transition probabilities

This article addresses the filtering design problem for discrete‐time Markov jump linear systems (MJLS) under the assumption that the transition probabilities are not completely known. We present the methods to determine ??₂‐ and ??_∞‐norm bounded filters for MJLS whose transition probability matrices have uncertainties in a convex polytope and establish an equivalence with the ones with partly unknown elements. The proposed design, based on linear matrix inequalities, allows different assumptions on Markov mode availability to the filter and on system parameter uncertainties to be taken into account. Under mode‐dependent assumption and internal model knowledge, observer‐based filters can be obtained and it is shown theoretically that our method outperforms some available ones in the literature to date. Numerical examples illustrate this claim. Copyright © 2010 John Wiley & Sons, Ltd.     

具有干扰的马尔科夫跳跃线性系统几乎处处稳定性充分条件

跳跃线性系统是一类具有随机跳变参数的线性系统,其跳变参数根据给定的有限状态马尔科夫链演化,这样的模型可以用来描述出现故障或者在结构卜突然发生变化的系统.本文采用随机李雅谱诺夫第二方法研究了具有干扰的离散时间跳跃线性系统的几乎处处稳定性,得到了一类充分条件.并由此条件进一步得出了更易于检测其几乎处处稳定性的允分条件.     

带有乘性过程噪声和模式观测时滞离散马氏跳线性系统的状态估计

研究受乘性过程噪声干扰的离散马氏跳线性系统状态估计问题. 系统可得到的观测包括两部分: 模式观测和输出观测, 其中模式观测受到固定时滞的影响. 利用贝叶斯定理及所得到的一些结果, 提出一种新颖的最小均方误差意义下次优状态估计算法. 该次优算法是回归的, 并且不随着时间增加而加重计算存储负荷. 通过计算机仿真来评估所提出次优算法的性能, 仿真结果验证了该算法的优越性.

     

随机时滞马尔可夫跳跃系统的比例-积分跟踪控制

针对具有时变时滞和未知非线性的随机马尔可夫跳变系统,基于传统的PI控制策略和线性矩阵不等式算法,提出一种具有随机稳定性能、跟踪性能和鲁棒性能的多目标控制器设计方案.运用Lyapunov稳定性理论并引入L<,1>性能指标,构造出具有PI结构的跟踪控制器,保证了随机马尔可大跳跃系统的稳定性和跟踪性能,实现了系统跟踪性能的优...     

Infinite horizon linear quadratic stochastic Nash differential games of Markov jump linear systems with its application

In this paper, we deal with the problem of stochastic Nash differential games of Markov jump linear systems governed by Itô-type equation. Combining the stochastic stabilizability with the stochastic systems, a necessary and sufficient condition for the existence of the Nash strategy is presented by means of a set of cross-coupled stochastic algebraic Riccati equations. Moreover, the stochastic H₂/H_∞ control for stochastic Markov jump linear systems is discussed as an immediate application and an illustrative example is presented.     

An iterative approach for the discrete‐time dynamic control of Markov jump linear systems with partial information

The , and mixed dynamic output feedback control of Markov jump linear systems in a partial observation context is studied through an iterative approach. By partial information, we mean that neither the state variable x(k) nor the Markov chain θ(k) are available to the controller. Instead, we assume that the controller relies only on an output y(k) and a measured variable coming from a detector that provides the only information of the Markov chain θ(k). To solve the problem, we resort to an iterative method that starts with a state‐feedback controller and solves at each iteration a linear matrix inequality optimization problem. It is shown that this iterative algorithm yields to a nonincreasing sequence of upper bound costs so that it converges to a minimum value. The effectiveness of the iterative procedure is illustrated by means of two examples in which the conservatism between the upper bounds and actual costs is significantly reduced.     

Markov跳变系统的有限时间状态反馈镇定

讨论一类含有限能量未知扰动的线性Markov跳变系统的有限时间镇定问题.针对连续系统和离散系统两种情况,利用构造的Lyapunov-Krasovskii函数,并结合线性矩阵不等式方法,分别证明并给出了跳变系统有限时间镇定控制器有解的充分条件.采用该方法设计的镇定控制器可使连续系统和离散系统对所有满足条件的未知扰动是有限时问有界和有限时间镇定的.最后通过数值示例表明了该设计方法的有效性.     

一类不确定离散Markov 跳变系统滑模状态反馈控制

针对具有不确定的离散Markov跳变系统,研究其滑模状态反馈控制问题.考虑系统的不确定满足匹配条件,以线性矩阵不等式形式给出了离散滑模面存在的充分条件,设计了具有指数趋近律的滑模控制器,保证了系统状态到达滑模面并在滑模带上随机镇定.数值仿真验证了所提出的控制方案的有效性.     

Quantized feedback stabilization of continuous-time Markov jump systems under asynchronous control

This paper addresses a new asynchronous control scheme for continuous-time Markov jump linear systems (MJLSs). Both controlled system and quantizer are asynchronous with the controller due to the process by which the controller can accurately observe and emit the switching signal being stochastic. The random variable satisfying Bernoulli distribution is introduced to describe this observation. On this basis, two methods are proposed to obtain sufficient conditions for exponential almost sure stability and almost surely asymptotically stability, respectively from the perspective of the linear matrix inequality (LMI). The results are independent of the asynchronous time interval. Finally, a numerical example demonstrates the validity and feasibility of developed theoretical results.     

Derandomisation-based multiple frequency control for stochastic Markov jump systems

Under the framework of derandomisation approach, the state feedback control issues for both continuous-time and discrete-time Markov jump linear systems (MJLSs) are investigated to meet multiple performance objectives over multiple frequency ranges. Because of the stochastic jumping among different modes, the generalised Kalman–Yakubovic–Popov lemma-based finite-frequency controller design approach cannot be directly applied to MJLSs. To overcome this limitation, a derandomisation approach is established by transforming the original stochastic multiple modes systems to deterministic ones. Then the multiple frequency controllers for both discrete-time and continuous-time MJLSs are designed to guarantee the multiple performances of the closed-loop systems. To verify the effectiveness of the developed algorithms, examples are presented, where the performance requirements include specifications in low-frequency and high-frequency ranges, respectively.     

转移概率部分未知的随机Markov 跳跃系统的镇定控制

研究一类随机Markov跳跃系统的稳定性与镇定控制问题.此类系统跳跃过程的转移概率部分未知,包括转移概率完全已知和完全未知两种情形,因而更具一般性.首先,给出保证随机Markov跳跃系统均方渐近稳定的充分性判据,并设计了相应的状态反馈镇定控制器;然后,基于矩阵的奇异值分解给出了系统静态输出反馈镇定控制器的设计方法,并将其归结为求解一组线性矩阵不等式（LMIs）的可行性问题;最后,通过数值仿真验证了所得结论的正确性.