Stochastic stability of Positive Markov Jump Linear Systems

This paper investigates on the stability properties of Positive Markov Jump Linear Systems (PMJLS’s), i.e. Markov Jump Linear Systems with nonnegative state variables. Specific features of these systems are highlighted. In particular, a new notion of stability (Exponential Mean stability) is introduced and is shown to be equivalent to the standard notion of 1-moment stability. Moreover, various sufficient conditions for Exponential Almost-Sure stability are worked out, with different levels of conservatism. The implications among the different stability notions are discussed. It is remarkable that, thanks to the positivity assumption, some conditions can be checked by solving Linear Programming feasibility problems.     

Almost Sure Stabilization of Uncertain Continuous-Time Markov Jump Linear Systems

This work deals with the robust Almost Sure (AS) stabilization problem for continuous-time Markov Jump Linear Systems (MJLS). Norm-bounded uncertainties affecting the system state and input matrices are considered. A deterministically testable sufficient condition for robust AS stability is provided which relies on a bound on the 2-norm of the system transition matrix. Such a condition can be profitably employed also to design a robust feedback stabilization strategy. Such feedback design is based on a formulation of the sufficient condition for robust AS stability in terms of an equivalent LMI problem.      

Active Estimation for Jump Markov Linear Systems

Jump Markov Linear Systems are convenient models for systems that exhibit both continuous dynamics and discrete mode changes. Estimating the hybrid discrete-continuous state of these systems is important for control and fault detection. Existing solutions for hybrid estimation approximate the belief state by maintaining a subset of the possible discrete mode sequences. This approximation can cause the estimator to lose track of the true mode sequence when the effects of discrete mode changes are subtle.      

具有双边随机时延和丢包的网络控制系统稳定性分析

针对具有双边随机网络时延和丢包的网络控制系统,首先基于Markov的随机过程描述系统丢包的特性;然后利用系统增广矩阵的方法建立参数不确定的离散时间跳变系统模型,在考虑Markov链转移概率矩阵中部分元素未知甚至完全未知的条件下,采用Lyapunov稳定理论和随机理论的分析方法,设计依赖于丢包特性且满足系统均方稳定要求的时变控制器;最后通过数值算例仿真表明所提出方法的有效性.     

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.     

Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems

This note deals with robust stability and control of uncertain discrete-time linear systems with Markovian jumping parameters. Systems with polytopic-type parameter uncertainty in either the state-space model matrices, or in the transition probability matrix of the Markov process, are considered. This note develops methods of robust stability analysis and robust stabilization in the mean square sense which are dependent on the system uncertainty. The design of both mode-dependent and mode-independent control laws is addressed. The proposed methods are given in terms of linear matrix inequalities. Numerical examples are provided to demonstrate the effectiveness of the derived results.     

Linear minimum mean square filter for discrete-time linear systems with Markov jumps and multiplicative noises

In this paper we obtain the linear minimum mean square estimator (LMMSE) for discrete-time linear systems subject to state and measurement multiplicative noises and Markov jumps on the parameters. It is assumed that the Markov chain is not available. By using geometric arguments we obtain a Kalman type filter conveniently implementable in a recurrence form. The stationary case is also studied and a proof for the convergence of the error covariance matrix of the LMMSE to a stationary value under the assumption of mean square stability of the system and ergodicity of the associated Markov chain is obtained. It is shown that there exists a unique positive semi-definite solution for the stationary Riccati-like filter equation and, moreover, this solution is the limit of the error covariance matrix of the LMMSE. The advantage of this scheme is that it is very easy to implement and all calculations can be performed offline.     

Analysis of linear asynchronous hybrid stochastic systems and its application to multi-agent systems with Markovian switching topologies

This paper studies the stability and stabilisation of a class of asynchronous hybrid stochastic systems described by linear Markov jump systems with asynchronous parameters induced by hysteretic mode and input delay. By exploring the essential properties of the Markov chain and developing novel Lyapunov function/functional based approaches, we derive sufficient conditions in terms of sizes of the delays ensuring the mean square asymptotical stability of the systems. Interestingly, the Lyapunov functional based approach further leads to the mean square stability criteria that depend only on the drift part and allow an arbitrary large delay for the diffusion part, which is different from the Lyapunov-Razumikhin based results. The theoretical results are applied to solve the consensus problems of multi-agent systems with Markovian switching topologies and input delay. Numerical examples demonstrate the established results.     

Robust Stability of a Class of Uncertain Markov Jump Nonlinear Systems

This note addresses the problem of robust stability analysis for a class of Markov jump nonlinear systems subject to polytopic-type parameter uncertainty. A condition for robust local exponential mean square stability in terms of linear matrix inequalities is developed. An estimate of a robust domain of attraction of the origin is also provided. The approach is based on a stochastic Lyapunov function with polynomial dependence on the system state and uncertain parameters. A numerical example illustrates the proposed result     

State estimation for Markovian Jump Linear Systems with bounded disturbances

In this paper, we investigate the state estimation problem for a class of Markovian Jump Linear Systems (MJLSs) in the presence of bounded polyhedral disturbances. A set-membership estimation algorithm is first proposed to find the smallest consistent set of all possible states, which is shown to be expressed by a union of multiple polytopes. The posterior probabilities of the system jumping modes are then estimated by introducing the Lebesgue measure, based on which the optimal point estimate is further provided. Moreover, a similarity relationship for polytopes is defined and an approximate method is presented to calculate the Minkowski sum of polytopes, which can help reduce the computational complexity of the overall estimation algorithm.     

基于混合随机Petri网的一类混合系统的模型及稳定性

混合动态系统是包含离散事件系统（ＤＥＳ）和连续变量系统（ＣＶＳ）的复杂系统。讨论了一类每个离散状态包含一个连续动态系统,且离散事件具有Ｍａｒｋｏｖ链性质的随机混合系统,在提出利用混合随机Ｐｅｔｒｉ网的建模方法的基础上,给出混合均方稳定的概念、稳定性和可镇定条件,最后以一个简单的例子做子说明和仿真。     

Stochastic stability properties of jump linear systems

Jump linear systems are defined as a family of linear systems with randomly jumping parameters (usually governed by a Markov jump process) and are used to model systems subject to failures or changes in structure. The authors study stochastic stability properties in jump linear systems and the relationship among various moment and sample path stability properties. It is shown that all second moment stability properties are equivalent and are sufficient for almost sure sample path stability, and a testable necessary and sufficient condition for second moment stability is derived. The Lyapunov exponent method for the study of almost sure sample stability is discussed, and a theorem which characterizes the Lyapunov exponents of jump linear systems is presented. Finally, for one-dimensional jump linear system, it is proved that the region for δ-moment stability is monotonically converging to the region for almost sure stability at δ↓0⁺     

多模态It随机系统的均方稳定性与鲁棒镇定

研究由连续时间Markov链所确定的多模态It^o随机系统的均方稳定性与鲁棒镇定 ,得到了一般多模态It^o随机系统的k阶矩指数稳定性定理 ,线性不确定系统的均方稳定性定理 ,给出了线性不确定系统的鲁棒镇定控制器 .     

Stabilization of fuzzy systems with quantization and packet dropout

The discrete‐time and continuous‐time fuzzy systems with quantization and packet dropout are considered in this paper. The quantizer used here is a uniform one with arbitrary quantization regions, and the packet dropout process is modeled as a time‐homogenous Markov process. Because of the properties of packet dropout process, the fuzzy systems considered here are seen as Markov jump fuzzy systems, and the theories of Markov jump system are used to discuss the mean square stability of the closed‐loop systems. On the basis of the zoom strategy and Lyapunov theory, for the given failure rate and recovery rate, sufficient conditions are given for the closed‐loop fuzzy systems to be mean square stable, and the feedback controllers are designed to ensure the stability of fuzzy systems. A single link direct joint driven manipulator model is presented to show the effectiveness of the main results. Copyright © 2013 John Wiley & Sons, Ltd.     

State Estimation for Discrete-Time Neural Networks with Markov-Mode-Dependent Lower and Upper Bounds on the Distributed Delays

This paper is concerned with the state estimation problem for a new class of discrete-time neural networks with Markovian jumping parameters and mixed time-delays. The parameters of the neural networks under consideration switch over time subject to a Markov chain. The networks involve both the discrete-time-varying delay and the mode-dependent distributed time-delay characterized by the upper and lower boundaries dependent on the Markov chain. By constructing novel Lyapunov-Krasovskii functionals, sufficient conditions are firstly established to guarantee the exponential stability in mean square for the addressed discrete-time neural networks with Markovian jumping parameters and mixed time-delays. Then, the state estimation problem is coped with for the same neural network where the goal is to design a desired state estimator such that the estimation error approaches zero exponentially in mean square. The derived conditions for both the stability and the existence of desired estimators are expressed in the form of matrix inequalities that can be solved by the semi-definite programme method. A numerical simulation example is exploited to demonstrate the usefulness of the main results obtained.     

Fault Detection Filtering for a Class of Nonhomogeneous Markov Jump Systems with Random Sensor Saturations

This paper is concerned with the problem of fault detection filtering for a class of nonhomogeneous Markov jump systems with randomly occurring sensor saturations, where it is assumed that sensor saturation is randomly occur satisfying the Bernoulli distribution, meanwhile transition probabilities in nonhomogeneous Markov chain are time-varying. By employing the suitable Lyapunov technique in combination with Linear Matrix Inequalities (LMIs), some sufficient criteria are derived to guarantee that the augmented system is mean square stochastically stable with an H∞ performance level. Finally, a representative example of industrial stirred tank reactor systems is presented to illustrate the theoretical results, and it then turns out that the feasible filtering parameters can be exactly fixed by Matlab Tool Box, and the H∞ performance of the system can be fully guaranteed by the proposed control strategy.     

Consensus in multi-agent systems with random delays governed by a Markov chain

This paper investigates the consensus problem in a multi-agent system with random delays governed by a Markov chain. The communication topology is assumed to be directed and fixed. With first-order dynamics under the sampled-data setting, we first convert the original system into a reduced-order one featuring the error dynamics. Accordingly, the consensus problem is transformed into the stabilization of the error dynamic system. Thereafter, based on the theory in stochastic stability for time-delay systems, a sufficient condition is established in terms of a set of linear matrix inequalities (LMIs). The mean square stability of the error dynamics is shown to guarantee consensus of the multi-agent system. By explicitly incorporating the transition probability of the random delay into consideration, the conservativeness in control design is reduced. A delay-dependent switching control scheme is developed by redesigning the adjacency matrix. Finally, simulation results are provided to verify the effectiveness of the proposed approach.     

切换拓扑结构下多智能体系统的一致性研究

针对一类多智能体系统,研究了系统在切换拓扑结构下的一致性控制问题。假定系统拓扑结构的变化符合马尔可夫过程,设计与切换拓扑结构相关的一致性协议,利用线性矩阵不等式和随机稳定性理论,给出了多智能体系统满足一致性的充分条件,并通过数值仿真验证了所提控制方案的可行性和有效性。     

Consensus of data‐sampled multi‐agent systems with Markovian switching topologies

This paper studies the consensus problem of data‐sampled multi‐agent systems with Markovian switching topologies. Two different consensus algorithms are considered. By a system transformation, the mean‐square consensus problem of multi‐agent systems is converted into the mean‐square stability problem of Markov jump systems. Necessary and sufficient conditions of mean‐square consensus are obtained. The cone complementary linearization algorithm is used to get the allowable control gain. Simulation examples are given to show the effectiveness of the results. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

不确定变时滞随机系统的鲁棒均方指数稳定性

研究了不确定变时滞随机系统的鲁棒均方指数稳定性问题, 不确定性是范数有界的. 通过构造Lyapunov泛函, 得到了基于线性矩阵不等式的鲁棒均方指数稳定的充分条件. 最后给出实例加以验证所提出方法的有效性.