On stabilization for a class of nonlinear stochastic time-delay systems： a matrix inequality approach

This paper treats the feedback stabilization of nonlinear stochastic time-delay systems with state and control-dependent noise. Some locally （globally） robustly stabilizable conditions are given in terms of matrix inequalities that are independent of the delay size. When it is applied to linear stochastic time-delay systems, sufficient conditions for the state-feedback stabilization are presented via linear matrix inequalities. Several previous results are extended to more general systems with both state and control-dependent noise, and easy computation algorithms are also given.     

On stabilization for a class of nonlinear stochastic time-delay systems:a matrix inequality approach

This paper treats the feedback stabilization of nonlinear stochastic time-delay systems with state and control-dependent noise. Some locally (globally) robustly stabilizable conditions are given in terms of matrix inequalities that are independent of the delay size. When it is applied to linear stochastic time-delay systems, sufficient conditions for the state-feedback stabilization are presented via linear matrix inequalities. Several previous results are extended to more general systems with both state and control-dependent noise, and easy computation algorithms are also eiven.     

Stability and stabilization of discrete‐time singular Markov jump systems with time‐varying delay

The stochastic stability and stochastic stabilization of time‐varying delay discrete‐time singular Markov jump systems are discussed. For full and partial knowledge of transition probabilities cases, delay‐dependent linear matrix inequalities (LMIs) conditions for the systems to be regular, causal and stochastically stable are given. Sufficient conditions are proposed for the existence of state feedback controller in terms of LMIs. Finally, two numerical examples to illustrate the effectiveness of the method are given. Copyright © 2009 John Wiley & Sons, Ltd.     

Algorithm to optimize the quantile criterion for the polyhedral loss function and discrete distribution of random parameters

For the vector of random parameters with discrete distribution and finite number of realizations, consideration was given to the problem of stochastic linear programming with a quantile criterion. The sufficient conditions for existence of problem solution were formulated. A method of reduction of the original problem to the mixed linear programming problem of high dimension was proposed. For the resulting problem, a solution algorithm was constructed on the basis of the methods of decomposition of the linear programming problems.     

Optimal estimation of linear discrete-time systems with stochastic parameters

This paper considers optimal linear state estimation in the general case of linear discrete-time systems with stochastic parameters which are statistically independent with respect to time. The estimator is derived by transforming the system to one with deterministic parameters and state dependent additive system and observation noise. It is shown that mean square stability of the system is a sufficient and almost necessary condition for the existence, uniqueness and stability of the time invariant estimator.     

Input delay compensation of nonlinear stochastic systems with both state and input delays by prediction approach

In this study, we present an extension of the prediction scheme (dynamic control) for nonlinear stochastic systems with both input and state delays. The stochastic system includes multiplicative noise and it is modelled as Ito stochastic differential equation. Input delay is considered to be equal or less than state delay and both delays are considered to be constant. At first, a new formula for prediction of the system's state is presented and then by means of this prediction vector, control input is constructed. To calculate the stabilising gain of the predictive controller, some sufficient delay-independent conditions in the form of linear matrix inequality are presented. Finally, simulation examples are given to illustrate the effectiveness of the proposed approach.     

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.     

Parametrization method for calculating exact stability bounds of stochastic linear systems with multiplicative noise

For deterministic time-invariant linear systems, stability results are quite simple. For stochastic systems, however, even for linear ones, they are rather complicated. In this paper, some results on second mean stability (mean square stability) of time-invariant linear systems with multiplicative noise are summarized and the parametrization method of getting an exact bound for pth mean stability (p ≥ 2) via second mean stability is stated. Moreover, relations between pth mean stabilities for various values of p are given. On the basis of these relations, a simpler method for getting sufficient conditions for pth mean stability is also given, though the resulting sufficient bound is, of course, more conservative. Comparative studies of various conditions are made by using examples.     

Mean-square asymptotic stability of linear hereditary systems

A sufficient condition is given for the mean-square asymptotic stability of the trivial solution of time-homogeneous stochastic linear functional differential equations. We use the stochastic Lyapunov function method and the resulting condition is, in some sense, an extension of the condition for linear ordinary differential equations. This condition is especially useful for equations with small noise and small delays and/or small delay terms. Some extensions for the time-inhomogeneous case are also discussed.     

Stability of discrete-time linear systems with Markovian jumpingparameters and constrained control

The note is devoted to the study of linear discrete-time systems with Markovian jumping parameters and constrained control. The constraints used in the note are of symmetrical inequality type. The approach of positively invariant sets is used to obtain new necessary and sufficient conditions for positive invariance, and sufficient conditions for stochastic stability. These conditions become those given for stationary discrete-time systems with one mode (deterministic linear discrete-time systems) as known in the literature     

Positively invariant polyhedral sets of discrete-time linear systems

The problem of the existence of positively invariant polyhedral sets for linear discrete-time dynamical systems is studied. In the first part of the paper, necessary and sufficient conditions for a given polyhedral set to be a positively invariant set of a linear system are obtained. Then, the spectral properties of systems possessing this kind of invariant set are established. Finally the class of systems possessing positively invariant polyhedral cones is studied.     

Finite‐Time Stability and Stabilization of Linear Itô Stochastic Systems with State and Control‐Dependent Noise

In this paper, finite‐time stability and stabilization problems for a class of linear stochastic systems are studied. First, a new concept of finite‐time stochastic stability is defined for linear stochastic systems. Then, based on matrix inequalities, some sufficient conditions under which the stochastic systems are finite‐time stochastically stable are given. Subsequently, the finite‐time stochastic stabilization is studied and some sufficient conditions for the existence of a state feedback controller and a dynamic output feedback controller are presented by using a matrix inequality approach. An algorithm is given for solving the matrix inequalities arising from finite‐time stochastic stability (stabilization). Finally, two examples are employed to illustrate the results.     

Containment control of stochastic multiagent systems with semi‐Markovian switching topologies

The containment control of stochastic multiagent systems with semi‐Markov switching topologies is investigated in this paper. The general case that the distribution function of the sojourn time is dependent on both the current system mode and the target mode is considered. Taking state multiplicative noise into account and using stochastic techniques, sufficient conditions to achieve the containment control in the asymptotic mean square sense are obtained in a form of linear matrix inequalities and the controller design condition is given. Finally, a simulation is given to demonstrate the effectiveness of the obtained theoretical results.     

具有时滞脉冲的线性随机时滞系统的均方指数稳定

本文研究了具有时滞脉冲的线性随机时滞系统的稳定性问题,基于Lyapunov函数和Razumikhin技巧,针对具有镇定型脉冲和反镇定型脉冲的线性随机时滞系统分别建立了系统均方指数稳定的充分条件,最后给出两个数值例子论证结果的有效性.     

非线性随机时滞系统的稳定性与应用

首先考虑了不确定性的一族非线笥随机时滞系统,建立了这种系统的均方指数稳定与几乎必须指数稳定的充分准则,其准则是时滞无关的,然后应用这些充分条件到一类不确定的随机时滞神经网络,得到了这咱神经网络指数稳定的产用判据。本文的结果是最近文献中某些结果的推广,最后一个数值例子说明的所给准则的有效性。     

Invariant and attracting sets of Hopfield neural networks with delay

This paper studies invariant and attracting sets of Hopfield neural networks system with delay. Sufficient criteria are given for the invariant and attracting sets. In particular, we provide an estimate of the existence range of attractors by using invariant and attracting sets. Moreover, when the system has an equilibrium point, we obtain the sufficient conditions of global asymptotic stability of the equilibrium point. Several examples are also worked out to demonstrate the advantages of our results.     

H2/H∞ control for stochastic systems with delay

This paper is concerned with the H₂/H_∞ control problem for stochastic linear systems with delay in state, control and external disturbance-dependent noise. A necessary and sufficient condition for the existence of a unique solution to the control problem is derived. The resulting solution is characterised by a kind of complex generalised forward–backward stochastic differential equations with stochastic delay equations as forward equations and anticipated backward stochastic differential equations as backward equations. Especially, we present the equivalent feedback solution via a new type of Riccati equations. To explain the theoretical results, we apply them to a population control problem.     

Stability of stochastic θ-methods for stochastic delay integro-differential equations

In this paper, we are concerned with the numerical stability of linear stochastic delay integro-differential equations (SDIDEs). A sufficient condition for mean square stability of the exact solution of a linear SDIDE with multiplicative noise is derived. Then the mean square stability of stochastic θ-methods is investigated, and it is shown that the numerical solution can reproduce the mean square stability of the exact solution under appropriate conditions. At last, we present some numerical experiments to support our conclusions.     

Infinite time horizon nonzero-sum linear quadratic stochastic differential games with state and control-dependent noise

This paper discusses the infinite time horizon nonzero-sum linear quadratic （LQ） differential games of stochastic systems governed by Itoe＇s equation with state and control-dependent noise. First, the nonzero-sum LQ differential games are formulated by applying the results of stochastic LQ problems. Second, under the assumption of mean-square stabilizability of stochastic systems, necessary and sufficient conditions for the existence of the Nash strategy are presented by means of four coupled stochastic algebraic Riccati equations. Moreover, in order to demonstrate the usefulness of the obtained results, the stochastic H-two/H-infinity control with state, control and external disturbance-dependent noise is discussed as an immediate application.     

Exponential H∞ filtering for nonlinear discrete‐time switched stochastic systems with mixed time delays and random missing measurements

This paper is concerned with the exponential H_∞ filtering for a class of nonlinear discrete‐time switched stochastic hybrid systems with mixed time delays and random missing measurements. The switched system under study involves stochastic disturbance, time‐varying discrete delay, bounded distributed delay and nonlinearity. Attention is focused on the design of a mode‐dependent filter that guarantees the exponential stability in the mean‐square sense and a prescribed H_∞ noise attenuation level for the filtering error dynamics. By constructing a new Lyapunov functional and using the average dwell time scheme, a new delay‐dependent sufficient condition for the existence of the filter is presented in terms of linear matrix inequalities. A numerical example is finally given to show the effectiveness of the proposed design method. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society