Delay-dependent stability analysis for discrete-time singular Markovian jump systems with time-varying delay

This article considers the problem of delay-dependent stability analysis for a class of discrete-time singular systems with Markovian jump and time-varying delay. The transition probabilities in Markov chain are assumed to be partially unknown. In terms of linear matrix inequality approach, the delay-dependent criteria are proposed to ensure the underlying system to be regular, causal and stochastically stable. Several numerical examples are given to demonstrate the effectiveness and less conservatism of the obtained results.     

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.     

Stability and passivity analysis for Lur’e singular systems with Markovian switching

This paper is concerned with the stochastic stability and passivity analysis for a class of Lur’e singular systems with time-varying delay and Markovian switching. By using the free-weighting matrices approach, a delay-dependent stability criterion, which guarantees that the system is stochastically stable and robustly passive, is derived in terms of linear matrix inequality (LMI). Two numerical examples are provided to illustrate the effectiveness of the proposed method.     

Asymptotic stability in probability of singular stochastic systems with Markovian switchings

This paper investigates the problem of asymptotic stability in probability for singular stochastic systems with Markovian switchings. A stochastic Lyapunov theorem on asymptotic stability in probability for the considered systems is provided. Also, we show that the original system has the same stability property as its difference‐algebraic form based on singular value decomposition. By utilizing the earlier results, a sufficient condition is obtained in terms of linear matrix inequalities, which is easy to check by using standard software. Copyright © 2017 John Wiley & Sons, Ltd.     

Sliding mode control of singular stochastic hybrid systems

This paper is concerned with the sliding mode control (SMC) of nonlinear singular stochastic systems with Markovian switching. An integral sliding surface function is designed, and the resulting sliding mode dynamics is a full-order Markovian jump singular stochastic system. By introducing some specified matrices, a new sufficient condition is proposed in terms of strict linear matrix inequality (LMI), which guarantees the stochastic stability of the sliding mode dynamics. Then, a SMC law is synthesized for reaching motion. Moreover, when there exists an external disturbance, the ?₂ disturbance attenuation performance is analyzed for the sliding mode dynamics. Some related sufficient conditions are also established.     

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.     

Stabilization of discrete‐time singular Markovian jump repeated vector nonlinear systems

This paper studies the control problem for discrete‐time singular Markovian jump systems with repeated vector nonlinearities. Sufficient conditions for stochastic stability are established, where the uniqueness of solution to the underlying system is also guaranteed. Then, a series of formulations of stabilizing conditions is further developed to design mode‐dependent and mode‐independent controllers by using the linear matrix inequality approach. Based on the proposed results, more special cases for stabilizing controller are considered. Finally, numerical examples are used to demonstrate the effectiveness and superiority of the proposed methods. Copyright © 2015 John Wiley & Sons, Ltd.     

Reliable finite-time H∞ control of uncertain singular nonlinear Markovian jump systems with bounded transition probabilities and time-varying delay

This paper investigates the problem of reliable finite-time H_∞ control for one class of uncertainsingular nonlinear Markovian jump systems with time-varying delay subject to partial information on the transition probabilities. Continuous fault model is more general and practical to serve as the actuator fault. Time delay is a kind of positive time-varying differentiable bounded delays. First, based on a state estimator the resulting closed-loop error system is constructed and sufficient criteria are provided to guarantee that the augmented system is singular stochastic finite-time boundedness and singular stochastic H_∞ finite-time boundedness in both normal and fault cases via constructing a delay-dependent Lyapunov–Krasonskii function. Then, the gain matrices of state-feedback controller and state estimator are fixed by solving a feasibility problem in terms of linear matrix inequalities through decoupling technique, respectively. Finally, numerical examples are given to show the validity of the proposed design approach.     

Delay‐dependent robust control for singular discrete‐time Markovian jump systems with time‐varying delay

The problem of delay‐dependent robust stabilization for uncertain singular discrete‐time systems with Markovian jumping parameters and time‐varying delay is investigated. In terms of free‐weighting‐matrix approach and linear matrix inequalities, a delay‐dependent condition is presented to ensure a singular discrete‐time system to be regular, causal and stochastically stable based on which the stability analysis and robust stabilization problem are studied. An explicit expression for the desired state‐feedback controller is also given. Some numerical examples are provided to demonstrate the effectiveness of the proposed approach. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

Reliable Dissipative Control for a Class of Uncertain Singular Markovian Jump Systems via Hybrid Impulsive Control

In this paper, the problem of robust normalization and reliable dissipative control is investigated for a class of uncertain singular Markovian jump systems with actuator failures. The uncertainties exhibit in both system matrices and transition rate matrix of the Markovian chain. A new impulsive and proportional–derivative control strategy is presented. The gain matrices of the impulsive control part can be obtained together with the design approach. Moreover, the dissipative control results obtained in this paper include the results of H‐infinity and passive control as special cases. Finally, two numerical examples are provided to illustrate the effectiveness and applicability of the proposed methods.     

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.     

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.     

Robust normalization and guaranteed cost control for a class of uncertain singular Markovian jump systems via hybrid impulsive control

This paper investigates the problem of robust normalization and guaranteed cost control for a class of uncertain singular Markovian jump systems. The uncertainties exhibit in both system matrices and transition rate matrix of the Markovian chain. A new impulsive and proportional‐derivative control strategy is presented, where the derivative gain is to make the closed‐loop system of the singular plant to be a normal one, and the impulsive control part is to make the value of the Lyapunov function does not increase at each time instant of the Markovian switching. A linearization approach via congruence transformations is proposed to solve the controller design problem. The cost function is minimized via solving an optimization problem under the designed control scheme. Finally, three examples (two numerical examples and an RC pulse divider circuit example) are provided to illustrate the effectiveness and applicability of the proposed methods. Copyright © 2013 John Wiley & Sons, Ltd.     

Output feedback stabilization for discrete singular systems with random abrupt changes

The problem of static output feedback control is investigated for discrete singular systems with Markovian jump. Two necessary and sufficient conditions for the discrete singular Markovian jump system to be regular, causal and stochastically stable are proposed in terms of linear matrix inequality (LMI) approach. Two kinds of design methods of the desired mode‐independent static output feedback controller are given. The explicit expressions for the desired controller are also given. Numerical examples are proposed to show the validness of the developed results. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust H ∞ filtering for singular time‐delayed systems with uncertain Markovian switching probabilities

This paper is concerned with the robust H _∞ filter design for a class of uncertain singular time‐delayed Markovian jump systems, whose transition rate matrix has elementwise bounded uncertainties. By the LMI approach, a novel bounded real lemma is proposed such that the singular Markovian jump system is robustly exponentially mean‐square admissible with a prescribed H _∞ performance index. Based on this, a sufficient condition for the existence of a robust H _∞ filter is developed in terms of LMIs. Finally, a numerical example is provided to show the effectiveness of the theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Stochastic stability and stabilization of discrete‐time singular Markovian jump systems with partially unknown transition probabilities

This paper considers the stochastic stability and stabilization of discrete‐time singular Markovian jump systems with partially unknown transition probabilities. Firstly, a set of necessary and sufficient conditions for the stochastic stability is proposed in terms of LMIs, then a set of sufficient conditions is proposed for the design of a state feedback controller to guarantee that the corresponding closed‐loop systems are regular, causal, and stochastically stable by employing the LMI technique. Finally, some examples are provided to demonstrate the effectiveness of the proposed approaches. Copyright © 2014 John Wiley & Sons, Ltd.     

Stability analysis and saturation control for nonlinear positive Markovian jump systems with randomly occurring actuator faults

This article investigates the stability analysis and control design of a class of nonlinear positive Markovian jump systems with randomly occurring actuator faults and saturation. It is assumed that the actuator faults of each subsystem are varying and governed by a Markovian process. The nonlinear term is located in a sector. First, sufficient conditions for stochastic stability of the underlying systems are established using a stochastic copositive Lyapunov function. Then, a family of reliable L₁‐gain controller is proposed for nonlinear positive Markovian jump systems with actuator faults and saturation in terms of a matrix decomposition technique. Under the designed controllers, the closed‐loop systems are positive and stochastically stable with an L₁‐gain performance. An optimization method is presented to estimate the maximum domain of attraction. Furthermore, the obtained results are developed for general Markovian jump systems. Finally, numerical examples are given to illustrate the effectiveness of the proposed techniques.     

网络控制系统随机稳定性研究

研究了具有随机网络诱导时延及数据包丢失的网络控制系统随机稳定性问题. 本文用一个具有两个状态的马尔可夫链来描述数据通过网络传输时随机数据包丢失过程, 利用马尔可夫跳变线性系统理论, 将网络控制系统建模为一个具有两种运行模式的马尔可夫跳变线性系统, 给出了在状态反馈控制下网络控制系统随机稳定的线性矩阵不等式形式的充分条件, 最后用一个仿真示例验证了该方法的有效性.     

Robust stability and H∞ control for uncertain discrete Markovian jump singular systems with mode‐dependent time‐delay

The robust stochastic stability, stabilization and H_∞ control for mode‐dependent time‐delay discrete Markovian jump singular systems with parameter uncertainties are discussed. Based on the restricted system equivalent (r.s.e.) transformation and by introducing new state vectors, the singular system is transformed into a standard linear system, and delay‐dependent linear matrix inequalities (LMIs) conditions for the mode‐dependent time‐delay discrete Markovian jump singular systems to be regular, causal and stochastically stable, and stochastically stable with γ‐disturbance attenuation are obtained, respectively. With these conditions, robust stabilization problem and robust H_∞ control problem are solved, and the LMIs sufficient conditions are obtained. A numerical example illustrates the effectiveness of the method given in the paper. Copyright © 2008 John Wiley & Sons, Ltd.