Robust H 2 Filtering for Discrete‐Time Markovian Jump Linear Systems

This paper studies the problems of H₂ filtering for discrete‐time Markovian jump linear systems with non‐accessible mode information. The objective is to design a less‐conservative mode‐independent H₂ filter such that the filtering error system is stochastically stable. To this end, sufficient conditions for the existence of an upper bound of H₂ norm are given in terms of linear matrix inequalities. With the introduction of a slack variable, a less‐conservative filter is derived. The proposed robust H₂ filter design method is also applicable to cover the cases where the system matrices are subject to polytopic uncertainty, as well as the case where the transition probability matrix is subject to a given polytopic uncertainty. Application of the proposed method to three examples from the literature demonstrates the favorable performance of the proposed solution to existing approaches.     

Sliding Mode Control for Markovian Jump Systems with Generalized Switching

In this paper, the sliding mode control (SMC) problem for continuous‐time Markovian jump systems (MJSs) is considered, in which the transition rate matrix (TRM) is partially unknown and uncertain. Firstly, the sliding mode surface S(t) = 0 is designed, which is mode‐dependent. Therefore, is used instead of in the SMC algorithm. Via adopting a linear matrix inequality (LMI) approach, sufficient conditions are proposed to ensure that the reduced order system is exponentially stable in mean square. Furthermore, the reduced order system is completely insensitive to the external disturbance. Secondly, SMC law is designed correspondingly which dominated by a Markov process. It could drive the state trajectories onto the specified sliding mode surface in finite time quickly and maintain them on the surface in subsequently time. Thirdly, a new term in will be introduced in the designed SMC and should be handled by a new approach. Finally, a numerical example is provided to show the effectiveness of the proposed method.     

Robust Stochastic Stability and Control for Uncertain Singular Markovian Jump Systems with Multiplicative Noise

This paper focuses on the problems of robust stability and stabilization and robust control for uncertain singular Markovian jump systems with (x,v)‐dependent noise. The parameter uncertainties appearing in state, input, disturbance as well as diffusion terms are assumed to be time‐varying but norm‐bounded. Based on the approach of generalized quadratic stability, the memoryless state feedback controller is designed for the robust stabilization problem, which ensures that the resulting closed‐loop system has an impulse‐free solution and is asymptotically stable in the mean square. Furthermore, the results of robust control problem are derived. The desired state feedback controller is presented, which not only meets the requirement of robust stabilization but also satisfies a prescribed performance level. The obtained results are formulated in terms of strict LMIs. What we have obtained can be viewed as corresponding extensions of existing results on uncertain singular systems. A numerical example is finally given to demonstrate the application of the proposed method.     

A Bounded Real Lemma for Discrete‐Time Markovian Jump Singular Systems

This paper presents a bounded real lemma for discrete‐time Markovian jump singular systems. First, a new necessary and sufficient condition is proposed in terms of a strict linear matrix inequality, which guarantees the stochastic admissibility of the unforced Markovian jump singular systems. Then, a bounded real lemma for discrete‐time Markovian jump singular systems is derived. It is also proven that the bounded real lemma can be described by a strict matrix inequality. Finally, a numerical example is provided to illustrate the effectiveness of the proposed theory. The results are more tractable and reliable in numerical computations than existing conditions.     

Finite‐Time H∞ Filtering for Nonlinear Continuous‐Time Singular Semi‐Markov Jump Systems

The stochastic finite‐time H_∞ filtering issue for a class of nonlinear continuous‐time singular semi‐Markov jump systems is discussed in this paper. Firstly, sufficient conditions on singular stochastic H_∞ finite‐time boundedness for the filtering error system are established. The existence of a unique solution for the corresponding system is also ensured. Secondly, based on the bounds of the time‐varying transition rate, without imposing constraints on slack variables, a novel approach to finite‐time H_∞ filter design is proposed in the forms of strict LMIs, which guarantees the filtering error system is singular stochastic H_∞ finite‐time bounded and of a unique solution. Compared with the existing ones, the presented results reveal less conservativeness. Finally, one numerical example is exploited to testify the advantage of the proposed design technique.     

Stochastic stability of semi‐Markovian jump systems with mode‐dependent delays

Semi‐Markovian jump systems, due to the relaxed conditions on the stochastic process, and its transition rates are time varying, can be used to describe a larger class of dynamical systems than conventional full Markovian jump systems. In this paper, the problem of stochastic stability for a class of semi‐Markovian systems with mode‐dependent time‐variant delays is investigated. By Lyapunov function approach, together with a piecewise analysis method, a sufficient condition is proposed to guarantee the stochastic stability of the underlying systems. As more time‐delay information is used, our results are much less conservative than some existing ones in literature. Finally, two examples are given to show the effectiveness and advantages of the proposed techniques. Copyright © 2013 John Wiley & Sons, Ltd.     

Probability Analysis of Terminal Sliding Mode Control of Second‐Order Markovian Jump Systems

This paper explores the probability problems on terminal sliding mode control of second‐order nonlinear continuous Markovian jump systems. An equivalent control based terminal sliding mode control scheme is proposed that can guarantee the systems' finite time mean‐square stability. By introducing a multi‐step transition conditional probability, the novel reaching and sliding probabilities are derived for the situations where the control force is not strong enough. This indicates that the reaching and sliding probabilities are both monotonically bounded non‐decreasing non‐negative piecewise right‐continuous functions of the control parameter. A numerical example is given to show the feasibility of the theoretical results.     

奇异Markov跳跃系统的鲁棒故障检测

研究受L2有界未知输入影响的一类奇异Markov跳跃系统的鲁棒故障检测问题。采用基于观测器的故障检测滤波器（FDF）作为残差产生器,将故障检测滤波器的设计归结为随机意义下的H∞滤波问题。推导并证明了问题可解的充分条件,并通过求解线性矩阵不等式得到了故障检测滤波器参数矩阵的解。算例验证了所给算法的有效性。     

State and Mode Feedback Control for Discrete-time Markovian Jump Linear Systems With Controllable MTPM

In this note, the state and mode feedback control problems for a class of discrete-time Markovian jump linear systems (MJLSs) with controllable mode transition probability matrix (MTPM) are investigated. In most achievements, controller design of MJLSs pays more attention to state/output feedback control for stability, while the system cost in practice is out of consideration. In this paper, we propose a control mechanism consisting of two parts: finite-path-dependent state feedback controller design with which uniform stability of MJLSs can be ensured, and mode feedback control which aims to decrease system cost. Differing from the traditional state/output feedback controller design, the main novelty is that the proposed control mechanism not only guarantees system stability, but also decreases system cost effectively by adjusting the occurrence probability of system modes. The effectiveness of the proposed mechanism is illustrated via numerical examples.      

Distributed Control Design for Spatially Interconnected Markovian Jump Systems With Time‐Varying Delays

This paper focuses on spatially interconnected Markovian jump systems with time‐varying delays. Firstly, a sufficient condition is given to check the well‐posedness, stochastic stability and contractiveness of spatially interconnected systems with the effect of time‐varying delays and Markovian jumping parameters. Secondly, the distributed controllers which inherit the structure of the plants are designed. A sufficient and necessary condition is proposed to guarantee the existence of the distributed controllers. Finally, an illustrative numerical example is given to show the effectiveness of the results.     

Quantized filtering for Markovian jump LPV systems with intermittent measurements

This paper investigates the problem of quantized filtering for a class of discrete‐time linear parameter‐varying systems with Markovian switching under data missing. The measured output of the plant is quantized by a logarithmic mode‐independent quantizer. The data missing phenomenon is modeled by a stochastic variable. The purpose of the problem addressed is to design a full‐order filter such that the filtering error dynamics is stochastically stable and the prescribed noise attenuation level in the sense can be achieved. Sufficient conditions are derived for the existence of such filters in terms of parameterized linear matrix inequalities. Then the corresponding filter synthesis problem is transformed into a convex optimization problem that can be efficiently solved by using standard software packages. A simulation example is utilized to demonstrate the usefulness of the developed theoretical results. Copyright © 2011 John Wiley & Sons, Ltd.     

Dynamic output‐feedback control for continuous‐time singular Markovian jump systems

This paper considers a dynamic output‐feedback control for continuous‐time singular Markovian jump systems, whereas the existing research studies in literature focused on state‐feedback or static output‐feedback control. While they have only provided the sufficient conditions, this paper successfully obtains the necessary and sufficient condition for the existence of the dynamic output‐feedback control. Furthermore, this condition is expressed with linear matrix inequalities by the so‐called replacement technique. Two numerical examples show the validity of the resulting control.     

State Feedback Control for Discrete‐time Markovian Jump Systems with Random Communication Time Delay

This paper concerns the stabilization problem for a class of networked control system with Markovian parameter using mode dependent state feed‐back controller through a wireless networked control system. The problem that the random network‐induced delays generated both by wireless connection contention and the movement of wireless nodes is taken into consideration. The mode dependent controller can be obtained by solving a set of parameterized bilinear matrix inequalities (BMIs). A numerical example is exhibited to show the effectiveness of the proposed design method.     

Sliding Mode Control for Nonlinear Markovian Jump Systems Under Denial-of-Service Attacks

This paper investigates the sliding mode control (SMC) problem for a class of discrete-time nonlinear networked Markovian jump systems (MJSs) in the presence of probabilistic denial-of-service (DoS) attacks. The communication network via which the data is propagated is unsafe and the malicious adversary can attack the system during state feedback. By considering random Denial-of-Service attacks, a new sliding mode variable is designed, which takes into account the distribution information of the probabilistic attacks. Then, by resorting to Lyapunov theory and stochastic analysis methods, sufficient conditions are established for the existence of the desired sliding mode controller, guaranteeing both reachability of the designed sliding surface and stability of the resulting sliding motion. Finally, a simulation example is given to demonstrate the effectiveness of the proposed sliding mode control algorithm.      

Stochastic Stability and Stabilization of Singular It ‐type Markovian Jump Systems with Uncertain Transition Rates: An LMI Approach

This paper investigates the stochastic stability and stabilization for a class of singular stochastic systems of Itô‐type with Markovian switching, the transition rates (TRs) in the jumping processes are uncertain. The aims are to establish sufficient conditions to ensure the considered system to be stochastically stable in the mean square sense, which is supported by a detailed proof of existence and uniqueness of the system solution, and to propose a controller such that the system can be stabilizable. The controller is first proposed and has advantage over traditional ones, the controller gain matrices are obtained by solving a strict linear matrix inequality (LMI). Finally, a numerical example is provided to illustrate the validity of the obtained methodology.     

Delay‐range‐dependent H∞ control for Markovian jump systems with mode‐dependent time delays

This paper considers mean‐square exponential stability and H_∞ control problems for Markovian jump systems (MJSs) with time delays which are time‐varying in an interval and depend on system mode. By exploiting a novel Lyapunov‐Krasovskii functional which takes into account the range of delay, and by making use of some techniques, new delay‐range‐dependent stability result and bounded real lemma for MJSs are obtained, where the introduction of the lower bound of delay is shown to be advantageous for reducing conservatism. Moreover, a sufficient condition for the solvability of the H_∞ control problem is derived in terms of linear matrix inequalities. Finally, illustrative examples are presented to show the advantage and effectiveness of the proposed approaches. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Robust Stochastic Stability for a Class of Singular Systems with Uncertain Markovian Jump and Time‐Varying Delay

This paper deals with the problem of the robust stochastic stability for a class of singular systems with uncertain Markovian jump and time‐varying delay. Sufficient conditions on the stochastic stability are presented. The obtained results are formulated in terms of strict linear matrix inequalities. A numerical example is provided to show the effectiveness of the proposed approaches.     

广义马氏跳变系统在一般转移速率下的控制器设计

针对转移概率不能精确获得并且部分未知的广义马氏跳变系统, 分别讨论在模态依赖控制器和模态独立控制器条件下的系统镇定问题. 与现有结果相比, 所提研究方法具有较小的保守性, 能够有效地解决实际问题. 首先, 运用自由权矩阵方法, 得到了广义马氏跳变系统在转移速率满足上述一般条件时系统随机容许的充分条件. 在此基础上, 以线性矩阵不等式(Linear matrix inequality, LMI)的形式分别给出了模态依赖和模态独立控制器的求解条件. 最后, 通过数值算例验证设计方法的有效性和优越性.