Robust stability of nonlinear time‐delay systems with interval time‐varying delay

This paper deals with the problem of obtaining delay‐dependent stability conditions and L₂‐gain analysis for a class of nonlinear time‐delay systems with norm‐bounded and possibly time‐varying uncertainties. No restrictions on the derivative of the time‐varying delay are imposed, though lower and upper bounds of the delay interval are assumed to be known. A Lyapunov–Krasovskii functional approach is proposed to derive novel delay‐dependent stability conditions which are expressed in terms of linear matrix inequalities (LMIs). To reduce conservatism, the work exploits the idea of splitting the delay interval in multiple regions, so that specific conditions can be imposed to a unique functional in the different regions. This improves the computed bounds for certain delay‐dependent integral terms, providing less conservative LMI conditions. Examples are provided to demonstrate the reduced conservatism with respect to the available results in the literature. Copyright © 2010 John Wiley & Sons, Ltd.     

H ∞ observer design for uncertain nonlinear discrete‐time systems with time‐delay: LMI optimization approach

We present a robust H _∞ observer for a class of nonlinear discrete‐time systems. The class under study includes an unknown time‐varying delay limited by upper and lower bounds, as well as time‐varying parametric uncertainties. We design a nonlinear H _∞ observer, by using the upper and lower bounds of the delay, that guarantees asymptotic stability of the estimation error dynamics and is also robust against time‐varying parametric uncertainties. The described problem is converted to a standard optimization problem, which can be solved in terms of linear matrix inequalities (LMIs). Then, we expand the problem to a multi‐objective optimization problem in which the maximum admissible Lipschitz constant and the minimum disturbance attenuation level are the problem objectives. Finally, the proposed observer is illustrated with two examples. Copyright © 2014 John Wiley & Sons, Ltd.     

Robust delay‐probability‐distribution‐dependent stability of uncertain stochastic genetic regulatory networks with random discrete delays and distributed delays

This study is concerned with the problem of robust delay‐probability‐distribution‐dependent stability of uncertain stochastic genetic regulatory networks with mixed time‐varying delays. The parameter uncertainties are modeled as having a structured linear fractional form. Besides, we consider that the derivatives of the discrete time delays have different upper bounds in various delay intervals. Moreover, less conservative conditions are obtained by choosing an augmented novel Lyapunov–Krasovskii functional and using the lower bound lemma together with the Jensen inequality lemma. Furthermore, the criteria can be applicable to both fast and slow time‐varying delays. Finally, numerical examples are presented to illustrate the effectiveness of the theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

DELAY‐DEPENDENT ROBUST CONTROL FOR UNCERTAIN SINGULAR TIME‐DELAY SYSTEMS

The problem of delay‐dependent robust stabilization for uncertain singular time‐delay systems is investigated in this paper. The parameter uncertainty is assumed to be norm‐bounded and possibly time‐varying, while the time delay considered here is assumed to be constant but unknown. A delay‐dependent condition is presented for a singular time‐delay system to be regular, impulse free, and stable, based on which robust stability analysis and the robust stabilization problem are studied. An explicit expression for the desired state‐feedback control law is also given. The obtained results are formulated in terms of linear matrix inequalities (LMIs), which involve no decomposition of the system matrices. Some numerical examples are given to show the efficiency of the theoretical conditions.     

State feedback control for uncertain switched systems with interval time‐varying delay

Stability criteria and hybrid controllers' design problems for a class of uncertain switched systems with interval time‐varying delay are considered in this paper. Based on the average dwell time method, by choosing a new appropriate Lyapunov‐Krasovskii functional which fully utilizes the information of both the lower and upper bounds of the interval time‐varying delay, new delay‐range‐dependent stability criteria and stabilization conditions are first derived in terms of linear matrix inequalities. Moreover, in order to obtain much less conservative results, a tighter bounding for some terms is estimated and no redundant matrix variable is introduced. Finally, two numerical examples are given to demonstrate the applicability and the effectiveness of the proposed method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

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.     

New absolute stability criteria for time‐delay Lur'e systems with sector‐bounded nonlinearity

This paper is concerned with the problem of absolute stability of time‐delay Lur'e systems with sector‐bounded nonlinearity. Several novel criteria are presented by using a Lur'e–Postnikov function. For a general Lur'e system with known time delay, the absolute stability of it is analyzed by solving a set of linear matrix inequalities (LMIs). The maximum upper bound of the allowable time delay for a general Lur'e system is derived by solving a convex optimization problem. The feasibility of the LMIs implies some frequency‐domain interpretations which are similar to the frequency‐domain inequalities in the circle criterion and the Popov criterion. Copyright © 2009 John Wiley & Sons, Ltd.     

Passive Control for Stochastic Interval Systems with Interval Time‐Varying Delay

This paper is concerned with the problem of delay‐dependent passive analysis and control for stochastic interval systems with interval time‐varying delay. The system matrices are assumed to be uncertain within given intervals, and the time delay is a time‐varying continuous function belonging to a given range. By the transformation of the interval uncertainty into the norm‐bounded uncertainty, partitioning the delay into two segments of equal length, and constructing an appropriate Lyapunov–Krasovskii functional in each segment of the delay interval, delay‐dependent stochastic passive control criteria are proposed without ignoring any useful terms by considering the information of the lower bound and upper bound for the time delay. The main contribution of this paper is that a tighter upper bound of the stochastic differential of Lyapunov–Krasovskii functional is obtained via a newly‐proposed bounding condition. Based on the criteria obtained, a delay‐dependent passive controller is presented. The results are formulated in terms of linear matrix inequalities. Numerical examples are given to demonstrate the effectiveness of the method.     

Robust delay‐dependent ℋ︁∞ control of time‐delay systems with state and input delays

This paper studies the design problem of robust delay‐dependent ??_∞ controller for a class of time‐delay control systems with time‐varying state and input delays, which are assumed to be noncoincident. The system is subject to norm‐bounded uncertainties and ??₂ disturbances. Based on the selection of an augmented form of Lyapunov–Krasovskii (L‐K) functional, first a Bounded Real Lemma (BRL) is obtained in terms of linear matrix inequalities (LMIs) such that the nominal, unforced time‐delay system is guaranteed to be globally asymptotically stable with minimum allowable disturbance attenuation level. Extending BRL, sufficient delay‐dependent criteria are developed for a stabilizing ??_∞ controller synthesis involving a matrix inequality for which a nonlinear optimization algorithm with LMIs is proposed to get feasible solution to the problem. Moreover, for the case of existence of norm‐bounded uncertainties, both the BRL and ??_∞ stabilization criteria are easily extended by employing a well‐known bounding technique. A plenty of numerical examples are given to illustrate the application of the proposed methodology of this note. The achieved numerical results on the maximum allowable delay bound and minimum allowable disturbance attenuation level are exhibited to be less conservative in comparison to those of existing methods in the literature. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

具有区间时变时滞2-D 离散系统的时滞相关稳定与控制

针对具有区间时变时滞2-D离散系统,利用时滞相关方法,研究其稳定性与控制问题.首先选取含有时滞项上、下界的一个新的Lyapunov函数,对其差分时考虑所有项,得到了基于线性矩阵不等式(LMI)的时滞相关稳定性准则;然后给定时变时滞项的下界,再由一个凸优化问题最大化其上界,进而通过状态反馈实现系统的时滞相关控制,且求解LMI可得到增益矩阵;最后,利用数值算例说明了所得结果有效且优于已有成果.     

New Augmented Lyapunov‐Krasovskii Functional for Stability Analysis of Systems with Additive Time‐Varying Delays

This paper is concerned with stability analysis for continuous‐time systems with additive time‐varying delays in the Lyapunov‐Krasovskii(L‐K) framework. Firstly, in view of the relationships between the upper bounds of the two time‐varying delays, a new augmented L‐K functional is constructed by using the information of the two upper bounds. Secondly, the free‐matrix‐based integral inequality is used to estimate the derivative of the constructed L‐K functional. Thirdly, a less conservative criterion is derived to assess stability. Finally, a numerical example is presented to demonstrate the effectiveness of the criterion.     

Improved robust stability criteria and design of robust stabilizing controller for uncertain linear time‐delay systems

In this paper, an improved linear matrix inequality (LMI)‐based robust delay‐dependent stability test is introduced to ensure a larger upper bound for time‐varying delays affecting the state vector of an uncertain continuous‐time system with norm‐bounded‐type uncertainties. A quasi‐full‐size Lyapunov–Krasovskii functional is chosen and free‐weighting matrix approach is employed. Less restrictive sufficient conditions are derived for robust stability of time‐varying delay systems with norm‐bounded‐type uncertainties. Moreover, the investigation of the stabilization problem with memoryless state‐feedback control is presented such that the stabilizability criteria are obtained in terms of matrix inequalities, which can be solved via utilizing a cone complementarity minimization algorithm. Finally, the problem of output feedback stabilization for square systems is also taken into consideration. The output feedback stabilizability criteria are derived in the form of linear matrix inequalities, which are convex and can be easily solved using interior point algorithms. A plenty of numerical examples are presented indicating that the proposed stability and stabilization methods effectively improve the existing results. Copyright © 2006 John Wiley & Sons, Ltd.     

Delay‐dependent dissipativity analysis and synthesis of switched delay systems

In this paper, we study the problem of dissipative analysis for a class of switched systems with time‐varying delays. Sufficient conditions for dissipativity are developed for a class of switching signals with average dwell time. These conditions express delay‐dependent exponential stability and are provided in terms of linear matrix inequalities (LMIs). It is shown that the derived results encompass some available results on ??_∞ approach and arbitrary switching case. Numerical examples are given to illustrate the developed results. Copyright © 2010 John Wiley & Sons, Ltd.     

Novel delay‐derivative‐dependent stability criteria using new bounding techniques

This paper studies the stability of linear systems with interval time‐varying delays. By constructing a new Lyapunov–Krasovskii functional, two delay‐derivative‐dependent stability criteria are formulated by incorporating with two different bounding techniques to estimate some integral terms appearing in the derivative of the Lyapunov–Krasovskii functional. The first stability criterion is derived by using a generalized integral inequality, and the second stability criterion is obtained by employing a reciprocally convex approach. When applying these two stability criteria to check the stability of a linear system with an interval time‐varying delay, it is shown through some numerical examples that the first stability criterion can provide a larger upper bound of the time‐varying delay than the second stability criterion. Copyright © 2012 John Wiley & Sons, Ltd.     

Robust Absolute Stability Analysis of Multiple Time‐Delay Lur'e Systems With Parametric Uncertainties

The problem of robust absolute stability for time‐delay Lur'e systems with parametric uncertainties is investigated in this paper. The nonlinear part of the Lur'e system is assumed to be both time‐invariant and time‐varying. The structure of uncertainty is a general case that includes norm‐bounded uncertainty. Based on the Lyapunov–Krasovskii stability theory, some delay‐dependent sufficient conditions for the robust absolute stability of the Lur'e system will be derived and expressed in the form of linear matrix inequalities (LMIs). These conditions reduce the conservativeness in computing the upper bound of the maximum allowed delay in many cases. Numerical examples are given to show that the proposed stability criteria are less conservative than those reported in the established literatures.     

Interconnected jumping time‐delay systems: Mode‐dependent decentralized stability and stabilization

In this paper, we deal with the problems of mode‐dependent decentralized stability and stabilization with ??_∞ performance for a class of continuous‐time interconnected jumping time‐delay systems. The jumping parameters are governed by a finite state Markov process and the delays are unknown time‐varying and mode‐dependent within interval. The interactions among subsystems satisfy quadratic bounding constraints. To characterize mode‐dependent local stability behavior, we employ an improved Lyapunov–Krasovskii functional at the subsystem level and express the stability conditions in terms of linear matrix inequalities (LMIs). A class of local decentralized state‐feedback controllers is developed to render the closed‐loop interconnected jumping system stochastically stable. Then, we extend the feedback strategy to dynamic observer‐based control and establish the stochastic stabilization via LMIs. It has been established that the developed results encompass several existing results as special cases which are illustrated by simulation of examples. Copyright © 2011 John Wiley & Sons, Ltd.     

On reachable set estimation of two‐dimensional systems described by the Roesser model with time‐varying delays

In this paper, the problem of reachable set estimation of two‐dimensional (2‐D) discrete‐time systems described by the Roesser model with interval time‐varying delays is considered for the first time. New 2‐D weighted summation inequalities, which provide a tighter lower bound than the commonly used Jensen summation inequality, are proposed. Based on the Lyapunov‐Krasovskii functional approach, and by using the 2‐D weighted summation inequalities presented in this paper, new delay‐dependent conditions are derived to ensure the existence of an ellipsoid that bounds the system states in the presence of bounded disturbances. The derived conditions are expressed in terms of linear matrix inequalities, which can be solved by various computational tools to determine a smallest possible ellipsoidal bound. Applications to exponential stability analysis of 2‐D systems with delays are also presented. The effectiveness of the obtained results are illustrated by numerical examples.