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.     

Distributed delay‐dependent filtering for Markovian jump systems interconnected over an undirected graph with time‐delay

In this paper, the problems of delay‐dependent stochastic stability analysis and distributed filter synthesis are considered for Markovian jump systems interconnected over an undirected graph with state time‐invariant delay. A sufficient condition for the well‐posedness, delay‐dependent stochastic stability and contractiveness of the plant is developed in terms of linear matrix inequalities (LMIs). The distributed filter synthesis aims to design a distributed filter inheriting the structure of the plant such that the filtering error systems is well‐posed, delay‐dependent stochastically stable and contractive. Specifically, a corresponding sufficient condition to guarantee the filtering error system contractive is first presented by a set of nonlinear matrix inequalities. Next, for coupling these nonlinear matrix inequalities, a sufficient condition on the existence of such a distributed filter is proposed via a series of finite‐dimensional LMIs. Finally, a numerical simulation is presented to demonstrate the effectiveness of the proposed approach.     

一类不确定关联时滞大系统的分散H∞ 控制器设计—LMI方法

研究了一类具有N×N个任意未知常时滞和具有范数有界时变不确定的线性连续大系统的分散鲁棒H∞ 状态反馈控制器设计问题 ,基于线性矩阵不等式方法得到了一个使该系统存在无记忆H∞ 状态反馈控制器的充分条件 ,最后通过一个数值例子来说明分散H∞ 状态反馈控制器的设计     

Delay‐dependent stability criterion and H∞ analysis for Markovian jump systems with time‐varying delays

This paper deals with the problems of stochastic stability and H_∞ analysis for Markovian jump linear systems with time‐varying delays. In terms of linear matrix inequalities, a less conservative delay‐dependent stability criterion for Markovian jump systems is proposed by constructing a different Lyapunov‐Krasovskii functional and introducing improved integral‐equalities approach, and a sufficient condition is derived from the H_∞ performance. Numerical examples are provided to demonstrate the efficiency and reduced conservatism of the results in this paper. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Exponential stability of impulsive positive switched systems with discrete and distributed time‐varying delays

This paper studies the exponential stability problems of discrete‐time and continuous‐time impulsive positive switched systems with mixed (discrete and distributed) time‐varying delays, respectively. By constructing novel copositive Lyapunov‐Krasovskii functionals and using the average dwell time technique, delay‐dependent sufficient conditions for the solvability of considered problems are given in terms of fairly simple linear matrix inequalities. Compared with the most existing results, by introducing an extra real vector, restrictive conditions on derivative of the time‐varying delays (less than 1) are relaxed, thus the obtained improved stability criteria can deal with a wider class of continuous‐time positive switched systems with time‐varying delays. Finally, two simple examples are provided to verify the validity of theoretical results.     

Event‐triggered finite‐time reliable control for nonhomogeneous Markovian jump systems

This article investigates the event‐triggered finite‐time reliable control problem for a class of Markovian jump systems with time‐varying transition probabilities, time‐varying actuator faults, and time‐varying delays. First, a Luenberger observer is constructed to estimate the unmeasured system state. Second, by applying an event‐triggered strategy from observer to controller, the frequency of transmission is reduced. Third, based on linear matrix inequality technique and stochastic finite‐time analysis, event‐triggered observer‐based controllers are designed and sufficient conditions are given, which ensure the finite‐time boundedness of the closed‐loop system in an H_∞ sense. Finally, an example is utilized to show the effectiveness of the proposed controller design approach.     

Stabilization of stochastic Markovian switching systems on networks with multilinks based on aperiodically intermittent control: A new differential inequality technique

In this paper, the stabilization problem of stochastic Markovian switching systems on networks with multilinks and time‐varying delays (SMNMT) is investigated via aperiodically intermittent control. At first, a new differential inequality is established for SMNMT, which relaxes the conditions of time‐varying delays compared with existing literature. Different from previous approaches of studying multilinks systems, new differential inequality technique combined with graph theory and Lyapunov method is adopted, based on which two types of sufficient conditions are derived to ensure the stability of SMNMT. The topological structure of multilinks systems on networks, stochastic perturbation, the transition rate of Markov chain, and intermittent control has a great impact on these developed conditions. The theoretical results are applied to stochastic Markovian switching oscillators networks with multilinks (SMONM), and a stabilization criterion of SMONM is derived as well. Finally, a numerical example is shown to illustrate the feasibility of our theoretical results.     

Delay‐dependent state estimation for discrete Markovian jump neural networks with time‐varying delay

The state estimation problem is discussed for discrete Markovian jump neural networks with time‐varying delays in terms of linear matrix inequality (LMI) approach. The considered transition probabilities are assumed to be time‐variant and partially unknown. The aim of the state estimation problem is to design a state estimator to estimate the neuron states and ensure the stochastic stability of the error‐state system. A delay‐dependent sufficient condition for the existence of the desired state estimator is proposed. An explicit expression of the desired estimator is also given. A numerical example is introduced to show the effectiveness of the given result. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

变时滞不确定组合系统基于观测器的鲁棒镇定

研究一类孤立子系统中状态及控制输入均含有时变时滞,且互联项也含有时变时滞的不确定组合系统基于状态观测器的鲁棒控制问题.基于一组线性矩阵不等式（LMIs）解的存在性,并依据Razumikhin-type理论和Lyapunov稳定性理论,给出了保证系统可鲁棒分散镇定的充分条件及相应控制器的设计方法.分散控制器可通过求解一组LMIs得到.最后,利用一个数值例子验证了所给设计方法的有效性.     

Robust H∞ filtering for a class of Markovian jump systems with time‐varying delays based on delta operator approach

In this paper, the robust H_∞ filtering problem for a class of discrete Markovian jump systems with time‐varying delays and linear fractional uncertainties is investigated based on delta operator approach. Based on Lyapunov‐Krasovskii functional in delta domain, new delay‐dependent sufficient conditions for the solvability of this problem are presented in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, an explicit expression of a desired jump H_∞ filter is given. The proposed method can unify some previous related continuous and discrete systems into the delta operator systems framework. Numerical examples are given to illustrate the effectiveness of the developed techniques. © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Robust stability of Markovian jump discrete-time neural networks with partly unknown transition probabilities and mixed mode-dependent delays

This article discusses the robust stability problem for a class of uncertain Markovian jump discrete-time neural networks with partly unknown transition probabilities and mixed mode-dependent time delays. The transition probabilities of the mode jumps are considered to be partly unknown, which relax the traditional assumption in Markovian jump systems that all of them must be completely known a priori. The mixed time delays consist of both discrete and distributed delays that are dependent on the Markovian jump modes. By employing the Lyapunov functional and linear matrix inequality approach, some sufficient criteria are derived for the robust stability of the underlying systems. A numerical example is exploited to illustrate the developed theory.     

Positivity and stability of mixed fractional‐order systems with unbounded delays: Necessary and sufficient conditions

This article provides a comprehensive study on quantitative properties of linear mixed fractional‐order systems with multiple time‐varying delays. The delays can be bounded or unbounded. We first obtain a result on existence and uniqueness of solutions to these systems. Then, we prove a necessary and sufficient condition for their positivity. Finally, we provide a necessary and sufficient criterion to characterize asymptotic stability of positive linear mixed fractional‐order systems with multiple time‐varying delays.     

Stabilization of linear systems with time‐varying delays

This paper is concerned with the stability and stabilization problems for a class of time‐delayed systems, whose time‐varying delays are studied via Markovian approach. By separating the delay interval into several subintervals and by considering the inherent distribution of time‐varying delay, a new model is firstly developed. On the basis of the established model, a novel Lyapunov functional, which makes full use of each subinterval's delay bounds and the randomicity of time‐varying delay, is constructed to drive less conservative stability criteria. Especially sufficient conditions for the existence of stabilizing controllers are obtained as linear matrix inequalities, which are further used to deal with networked control systems. Finally, numerical examples are used to demonstrate the effectiveness of the proposed methods. Copyright © 2012 John Wiley & Sons, Ltd.     

Finite-Time Stability of Stochastic Cohen–Grossberg Neural Networks with Markovian Jumping Parameters and Distributed Time-Varying Delays

In this paper, the finite-time stability problem is considered for a class of stochastic Cohen–Grossberg neural networks (CGNNs) with Markovian jumping parameters and distributed time-varying delays. Based on Lyapunov–Krasovskii functional and stability analysis theory, a linear matrix inequality approach is developed to derive sufficient conditions for guaranteeing the stability of the concerned system. It is shown that the addressed stochastic CGNNs with Markovian jumping and distributed time varying delays are finite-time stable. An illustrative example is provided to show the effectiveness of the developed results.     

Distributed formation‐containment control with Euler‐Lagrange systems subject to input saturation and communication delays

This paper is concerned with the problem of formation‐containment on networked systems, with interconnected systems modeled by the Euler‐Lagrange equation with bounded inputs and time‐varying delays on the communication channels. The main results are the design of control algorithms and sufficient conditions to ensure the convergence of the network. The control algorithms are designed as distributed dynamic controllers, in such a way that the number of neighbors of each agent is decoupled from the bound of the control inputs. That is, in the proposed approach the amplitude of the input signal does not directly increase with the number of neighbors of each agent. The proposed sufficient conditions for the asymptotic convergence follow from the Lyapunov‐Krasovskii theory and are formulated in the linear matrix inequalities framework. The conditions rely only on the upper bound of delays and on a subset of the controller parameters, but they do not depend on the model of each agent, which makes it suitable for networks with agents governed by distinct dynamics. In order to illustrate the effectiveness of the proposed method we present numerical examples and compare with similar approaches existing in the literature.     

Distributed output-feedback controllers design for Markovian jump systems interconnected over an undirected graph with time-varying delay

This paper considers the problems of stochastic stability analysis and distributed output-feedback controllers design for Markovian jump systems interconnected over an undirected graph with state time-varying delay. Specifically, a sufficient condition for the well-posedness, stochastic stability and contractiveness of the resultant closed-loop system is first developed in the form of non-linear matrix inequalities. Then, to avoid the difficulty of solving non-linear matrix inequalities, the sufficient condition is relaxed to an equivalent one in terms of coupling linear matrix inequalities (LMIs), which provides an effective way to confirm the existence of a distributed output-feedback controller inheriting the structure of the given plant. Furthermore, a constructive algorithm is given for the design of such a distributed output-feedback controller to achieve the well-posedness, stochastic stability as well as the contractiveness of the corresponding closed-loop system. Finally, numerical simulations are given to illustrate the validity of the proposed method.     

New results on stability analysis and stabilization of time‐delay continuous Markovian jump systems with partially known rates matrix

In this note, the problems of stability analysis and controller synthesis of Markovian jump systems with time‐varying delay and partially known transition rates are investigated via an input–output approach. First, the system under consideration is transformed into an interconnected system, and new results on stochastic scaled small‐gain condition for stochastic interconnected systems are established, which are crucial for the problems considered in this paper. Based on the system transformation and the stochastic scaled small‐gain theorem, stochastic stability of the original system is examined via the stochastic version of the bounded realness of the transformed forward system. The merit of the proposed approach lies in its reduced conservatism, which is made possible by a precise approximation of the time‐varying delay and the new result on the stochastic scaled small‐gain theorem. The proposed stability condition is demonstrated to be much less conservative than most existing results. Moreover, the problem of stabilization is further solved with an admissible controller designed via convex optimizations, whose effectiveness is also illustrated via numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

具有Markov跳跃参数的一类随机非线性系统逆最优增益设计

研究了一类随机非线性系统的逆最优增益设计问题,系统中除了方差未知的Wiener噪声之外,还含有Markov跳跃参数. 首先,给出此类系统逆最优增益设计问题可解的一个充分条件. 其次,针对一类具有严格反馈形式的随机非线性系统,利用积分反推法,给出了依概率全局渐近稳定和逆最优控制策略的设计方法. 其中,所设计的Lyapunov函数和控制策略与模态显式无关,克服了由于Markov跳跃模态引起的耦合项所带来的设计困难. 最后,通过仿真验证了控制策略的有效性.     

Finite‐time boundedness of large‐scale systems with actuator faults and gain fluctuations

This paper addresses the issue of finite‐time boundedness of large‐scale interconnected systems with the use of a distributed nonfragile fault‐tolerant controller. The objective of this paper is to design a state‐feedback controller consisting of a time‐varying delay such that the resulting closed‐loop system is finite‐time bounded under a prescribed extended passivity performance level even in the presence of all admissible uncertainties and possible actuator faults. More precisely, based on the Lyapunov‐Krasovskii stability theory, a new set of sufficient conditions is obtained in the framework of linear matrix inequality constraints that ensures finite‐time boundedness and satisfies the prescribed extended passivity performance index of the considered system. Finally, two numerical examples, including the interconnected inverted pendulum, are given to show the effectiveness of the proposed controller design technique.     

Sliding mode observer–controller design for uncertain Markovian jump systems with time delays

This paper proposes a sliding mode observer–controller design method for uncertain Markovian jump systems with time delays and uncertain switching probabilities. Both the structures of a sliding mode observer and a sliding mode controller are given. By the mode‐dependent Lyapunov functional approach, a sufficient condition for the stochastic stability of the closed‐loop system is given, which can be converted into a convex optimization problem. The reachability of the sliding surfaces in both the estimation error space and the state estimate space can be ensured by the presented control scheme. Finally, the effectiveness of the proposed design method is demonstrated by a simulation example. Copyright © 2011 John Wiley & Sons, Ltd.