New results on delay‐dependent stability analysis and stabilization for stochastic time‐delay systems

This paper investigates the problems of stability analysis and stabilization for stochastic time‐delay systems. Firstly, this paper uses the martingale theory to investigate expectations of stochastic cross terms containing the Itô integral. On the basis of this, an improved delay‐dependent stability criterion is derived for stochastic delay systems. In the derivation process, the mathematical development avoids bounding stochastic cross terms, and neither model transformation method nor free‐weighting‐matrix method is used. Thus, the method leads to a simple criterion and shows less conservatism. Secondly, on the basis of this stability result, this paper further proposes a state‐feedback controller that exponentially stabilizes the stochastic delay system by a strict LMI. Therefore, unlike previous results, it is not necessary to transform the nonlinear matrix inequalities into LMIs by the cone complementarity linearization method or parameter‐tuning method, which always yield a suboptimal solution. Finally, examples are provided to demonstrate the reduced conservatism of the proposed conditions.Copyright © 2013 John Wiley & Sons, Ltd.     

A new delay-dependent absolute stability criterion for a class of nonlinear neutral systems

This paper deals with absolute stability for a class of nonlinear neutral systems using a discretized Lyapunov functional approach. A delay-dependent absolute stability criterion is obtained and formulated in the form of linear matrix inequalities (LMIs). The criterion is valid not only for systems with small delay, but also for systems with non-small delay. Neither model transformation nor bounding technique for cross terms, nor free weighting matrix method is involved through derivation of the stability criterion. Numerical examples show that for small delay case, the results obtained in this paper significantly improve the estimate of the stability limit over some existing result in the literature; for non-small delay case, the ideal results can also be achieved.     

New delay‐dependent stability analysis and synthesis of T–S fuzzy systems with time‐varying delay

In this paper, a new method is proposed for stability analysis and synthesis of Takagi–Sugeno (T–S) fuzzy systems with time‐varying delay. Based on a new Lyapunov–Krasovskii functional (LKF), some less conservative delay‐dependent stability criteria are established. In the derivation process, some additional useful terms, ignored in previous methods, are considered and new free‐weighting matrices are introduced to estimate the upper bound of the derivative of LKF for T–S fuzzy systems with time‐varying delay. The proposed stability criterion and stabilization condition are represented in terms of linear matrix inequalities. Numerical examples are given to demonstrate the effectiveness and the benefits of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.     

Passivity Analysis and Synthesis for a Class of Discrete‐Time Switched Stochastic Systems with Time‐Varying Delay

This paper deals with the problems of passivity and passification for a class of discrete‐time switched stochastic systems with time‐varying delay. Based on the average dwell time approach, the piecewise Lyapunov function technique, and the free‐weighting matrix method, a new Lyapunov functional is proposed and sufficient conditions for mean‐square exponential stability and stochastic passivity are developed under average dwell time switching. Moreover, an estimate of state decay can be calculated in terms of linear matrix inequalities (LMIs). Then, the solvability condition for passification is established and the corresponding controller is designed. Two numerical examples are given to show the effectiveness of the proposed methods.     

A Delay‐Dependent Approach to Robust Fast Adaptive Fault Estimation Design for Uncertain Neutral Systems with Time‐Varying Interval Delay

This paper studies the problem of robust fault estimation for neutral systems, which are subjected to uncertainties, actuator fault, time‐varying interval delay, and norm‐bounded external disturbance. Based on the fast adaptive fault estimation (FAFE) algorithm, we focus on the design of a fault estimation filter that guarantees stability in the filtering error system with a prescribed H_∞ performance. A novel Lyapunov‐Krasovskii functional is employed, which includes time delay information. A delay‐dependent criterion of robust fault estimation design is obtained by employing the free‐weighting matrices technique, and the proposed result has advantages over some existing results, in that it is less conservative and it enlarges the application scope. An improved sufficient condition for the existence of such a filter is proposed in terms of the linear matrix inequality (LMI) by the Schur complements and the cone complementary linearization algorithm. Finally, illustrative examples are provided to show the effectiveness of the proposed method.     

Improved H ∞ analysis of Markovian jumping stochastic systems with time-varying delays

This article investigates the problems of H _∞ analysis for Markovian jump stochastic systems with both nonlinear disturbance and time-varying delays. By virtue of the delay partition approach, the improved delay-dependent stochastic stability and bounded real lemma (BRL) for Markovian jump stochastic systems are obtained in terms of linear matrix inequalities (LMIs). The proposed approach involves neither free weighting matrices nor any model transformation, and it is shown that the new criteria have the capability of providing less conservative results than the state-of-the-art. Two numerical simulations are conducted to demonstrate the effectiveness of the proposed method in comparison with existing methods.     

Stability analysis and H∞ control design for a class of nonlinear time‐delay systems

This paper investigates the stability and H_∞ control problem for a class of nonlinear time‐delay systems with a nonsingular Jacobian matrix, and provides a number of new results regarding stability analysis and control design. Firstly, an equivalent form is obtained for this class of systems by means of coordinate transformation and/or orthogonal decomposition of vector fields. Then, based on the equivalent form and free‐weighting matrix method, several sufficient conditions, in terms of nonlinear matrix inequalities, are derived for the stability analysis of the time‐delay systems by constructing suitable Lyapunov functionals. Finally, we use the equivalent form and the obtained stability results to investigate the H_∞ control problem, and present a control design procedure for this class of time‐delay systems. A study of illustrative examples shows that the results obtained in this paper have less conservatism, and work very well in the stability analysis and control design of some nonlinear time‐delay systems. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Exponential H ∞ filtering for stochastic Markovian jump systems with time delays

This paper investigates the problem of exponential H _∞ filtering for stochastic systems with time delays and Markovian jumping parameters. On the basis of Lyapunov–Krasovskii functional theory and generalized Finsler lemma, a delay‐dependent bounded real lemma is established without using any model transformations, bounding techniques for cross terms, or additional free matrix variables. The obtained bounded real lemma guarantees that the filtering error system is both mean‐square exponentially stable and almost surely exponentially stable with a prescribed H _∞ noise attenuation level. Then an exponential H _∞ filter is designed for stochastic retarded Markovian jump systems in terms of a set of LMIs. Meanwhile, the mathematical equivalence of the proposed method to one recent method is presented, but our proposed method is more computationally efficient with fewer matrix variables than that recent method. The validity of the method is verified by a numerical example.Copyright © 2012 John Wiley & Sons, Ltd.     

Robust Exponential Stability and Stabilization of a Class of Nonlinear Stochastic Time‐Delay Systems

This paper presents new exponential stability and delayed‐state‐feedback stabilization criteria for a class of nonlinear uncertain stochastic time‐delay systems. By choosing the delay fraction number as two, applying the Jensen inequality to every sub‐interval of the time delay interval and avoiding using any free weighting matrix, the method proposed can reduce the computational complexity and conservativeness of results. Based on Lyapunov stability theory, exponential stability and delayed‐state‐feedback stabilization conditions of nonlinear uncertain stochastic systems with the state delay are obtained. In the sequence, the delayed‐state‐feedback stabilization problem for a nonlinear uncertain stochastic time‐delay system is investigated and some sufficient conditions are given in the form of nonlinear inequalities. In order to solve the nonlinear problem, a cone complementarity linearization algorithm is offered. Mathematical and/or numerical comparisons between the proposed method and existing ones are demonstrated, which show the effectiveness and less conservativeness of the proposed method.     

Delay‐distribution‐dependent robust stability of uncertain systems with time‐varying delay

By employing the information of the probability distribution of the time delay, this paper investigates the problem of robust stability for uncertain systems with time‐varying delay satisfying some probabilistic properties. Different from the common assumptions on the time delay in the existing literatures, it is assumed in this paper that the delay is random and its probability distribution is known a priori. In terms of the probability distribution of the delay, a new type of system model with stochastic parameter matrices is proposed. Based on the new system model, sufficient conditions for the exponential mean square stability of the original system are derived by using the Lyapunov functional method and the linear matrix inequality (LMI) technique. The derived criteria, which are expressed in terms of a set of LMIs, are delay‐distribution‐dependent, that is, the solvability of the criteria depends on not only the variation range of the delay but also the probability distribution of it. Finally, three numerical examples are given to illustrate the feasibility and effectiveness of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.     

Robust reliable control for systems with random actuator fault and probabilistic nonlinearity with new characters

In this study, the reliable control for time‐varying systems with random actuator faults and probabilistic nonlinearities is investigated. The system under consideration has the following main features: (1) nonlinearities with new characters. The probability information of nonlinearities belonging to different varying bounds is used; (2) its multi‐actuators are subject to various possible faults/failures, and failure rates can vary in some measure; and (3) there are uncertainties in the plant model parameters. Covering these features, a comprehensive model is developed for uncertain time‐varying delay systems. By employing the Lyapunov functional method, free‐weighting matrix method, and the linear matrix inequality technique, we can obtain several delay‐distribution‐dependent sufficient conditions to ensure the exponentially mean square stability of the system. Those conditions are characterized in terms of linear matrix inequalities, and the reliable controller feedback gain can be solved by the standard numerical software. A simulation example is presented to show the effectiveness and applicability of the results. Copyright © 2012 John Wiley & Sons, Ltd.     

A new delay‐dependent stability criterion for time‐delay systems

This paper is concerned with the problem of stability of time‐delay systems. A new type of augmented Lyapunov functional is proposed. By introducing some free‐weighting matrices and using the parameterized model transformation method, a new delay‐dependent stability condition is obtained in terms of a linear matrix inequality (LMI). Numerical examples are given to illustrate the effectiveness of the proposed methods. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Delay‐dependent stability and stabilization of neutral time‐delay systems

This paper is concerned with the problem of stability and stabilization of neutral time‐delay systems. A new delay‐dependent stability condition is derived in terms of linear matrix inequality by constructing a new Lyapunov functional and using some integral inequalities without introducing any free‐weighting matrices. On the basis of the obtained stability condition, a stabilizing method is also proposed. Using an iterative algorithm, the state feedback controller can be obtained. Numerical examples illustrate that the proposed methods are effective and lead to less conservative results. Copyright © 2008 John Wiley & Sons, Ltd.     

New less‐conservative stability results for uncertain stochastic neural networks with fewer slack variables

The exponential stability problem is investigated for a class of uncertain stochastic neural networks with discrete and unbounded distributed time delays. Two types of uncertainty are considered: one is time‐varying structured uncertainty, whereas the other is interval uncertainty. With the application of the Jensen integral inequality and constructing appropriate Lyapunov–Krasovskii functional based on delay partitioning, several improved delay‐dependent criteria are developed to achieve the exponential stability in mean square in terms of linear matrix inequalities. It is established theoretically that two special cases of the obtained criteria are less conservative than some existing results but including fewer slack variables. As the present conditions involve fewer free weighting matrices, the computational burden is largely reduced. Three numerical examples are provided to demonstrate the effectiveness of the theoretical results. Copyright © 2012 John Wiley & Sons, Ltd.     

Exponential stability and exponential stabilization of singularly perturbed stochastic systems with time‐varying delay

In this paper, the problems of exponential stability and exponential stabilization for linear singularly perturbed stochastic systems with time‐varying delay are investigated. First, an appropriate Lyapunov functional is introduced to establish an improved delay‐dependent stability criterion. By applying free‐weighting matrix technique and by equivalently eliminating time‐varying delay through the idea of convex combination, a less conservative sufficient condition for exponential stability in mean square is obtained in terms of ε‐dependent linear matrix inequalities (LMIs). It is shown that if this set of LMIs for ε=0 are feasible then the system is exponentially stable in mean square for sufficiently small ε?0. Furthermore, it is shown that if a certain matrix variable in this set of LMIs is chosen to be a special form and the resulting LMIs are feasible for ε=0, then the system is ε‐uniformly exponentially stable for all sufficiently small ε?0. Based on the stability criteria, an ε‐independent state‐feedback controller that stabilizes the system for sufficiently small ε?0 is derived. Finally, numerical examples are presented, which show our results are effective and useful. Copyright © 2010 John Wiley & Sons, Ltd.     

Robust absolute stability for general neutral type Lurie indirect control systems

This note concerns the robust absolute stability analysis for a class of general neutral type Lurie indirect control systems with nonlinearity located in an infinite sector or finite one. By using Lyapunov functional of quadratic form with integral term and introducing some free‐weighting matrices, some delay‐dependent robust absolute stability criteria are presented in terms of strict linear matrix inequalities. Neither model transformation nor bounding technique is required here. The obtained criteria are less conservative than previous ones, which are illustrated by numerical examples. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A NEW INTEGRAL INEQUALITY APPROACH TO DELAY‐DEPENDENT ROBUST H∞ CONTROL

A design method for the robust H_∞ control of an uncertain linear system with a time‐varying state delay is proposed. First, an integral inequality that we recently obtained is employed to establish a new delay‐dependent bounded real lemma for a system with a time‐varying delay. The lemma uses neither a model transformation nor a bounding technique for cross terms. Then, the lemma is used in combination with a matrix decomposition method to derive delay‐dependent conditions for the existence of robust H_∞ control based on linear matrix inequalities. Finally, some numerical examples are given to demonstrate the validity of the method.     

混合变时滞不确定中立型系统鲁棒稳定性分析

研究一类含混合变时滞不确定中立系统时滞相关鲁棒稳定性问题。基于时滞中点值,把时滞区间均分成两部分,通过构造包含时滞中点信息的增广泛函和三重积分项的Lyapunov-Krasovskii (L-K)泛函,利用L-K稳定性定理、积分不等式方法和自由权矩阵技术,建立了一种基于线性矩阵不等式(LMI)的、与离散时滞和中立时滞均相关的鲁棒稳定性判据。数值算例表明,该判据改善了已有文献的结论,具有更低的保守性。     

An analysis of the exponential stability of linear stochastic neutral delay systems

This paper is concerned with the analysis of the mean square exponential stability and the almost sure exponential stability of linear stochastic neutral delay systems. A general stability result on the mean square and almost sure exponential stability of such systems is established. Based on this stability result, the delay partitioning technique is adopted to obtain a delay‐dependent stability condition in terms of linear matrix inequalities (LMIs). In obtaining these LMIs, some basic rules of the Ito calculus are also utilized to introduce slack matrices so as to further reduce conservatism. Some numerical examples borrowed from the literature are used to show that, as the number of the partitioning intervals increases, the allowable delay determined by the proposed LMI condition approaches h_max, the maximal allowable delay for the stability of the considered system, indicating the effectiveness of the proposed stability analysis. Copyright © 2013 John Wiley & Sons, Ltd.