New delay range–dependent stability criteria for interval time‐varying delay systems via Wirtinger‐based inequalities

Stability conditions for time‐delay systems using the Lyapunov‐based methodologies are generically expressed in terms of linear matrix inequalities. However, due to assuming restrictive conditions in deriving the linear matrix inequalities, the established stability conditions can be strictly conservative. This paper attempts to relax this problem for linear systems with interval time‐varying delays. A double‐integral inequality is derived inspired by Wirtinger‐based single‐integral inequality. Using the advanced integral inequalities, the reciprocally convex combination techniques and necessary slack variables, together with extracting a condition for the positive definiteness of the Lyapunov functional, novel stability criteria, have been established for the system. The effectiveness of the criteria is evaluated via 2 numerical examples. The results indicate that more complex stability criteria not only improve the stability region but also bring computational expenses.     

New stability criterion for linear switched systems with time‐varying delay

This paper studies the stability problem of a class of linear switched systems with time‐varying delay in the sense of Hurwitz convex combination. By designing a parameter‐dependent switching law and using a new convex combination technique to deal with delay terms, a new stability criterion is established in terms of LMIs, which is dependent on the parameters of Hurwitz convex combination. The advantage of the new criterion lies in its less conservatism and simplicity. Numerical examples are given to illustrate the effectiveness and the less conservatism of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Less conservative stability criteria for linear systems with interval time‐varying delays

The problem of the stability of a linear system with an interval time‐varying delay is investigated. A new Lyapunov–Krasovskii functional that fully uses information about the lower bound of the time‐varying delay is constructed to derive new stability criteria. It is proved that the proposed Lyapunov–Krasovskii functional can lead to less conservative results than some existing ones. Based on the proposed Lyapunov–Krasovskii functional, two stability conditions are developed using two different methods to estimate Lyapunov–Krasovskii functional's derivative. Two numerical examples are given to illustrate that the two stability conditions are complementary and yield a larger maximum upper bound of the time‐varying delay than some existing results. Copyright © 2013 John Wiley & Sons, Ltd.     

Improved robust stability criteria for uncertain systems with time‐varying delay

This paper is concerned with the delay‐dependent stability and robust stability for uncertain systems with time‐varying delay. Through constructing an appropriate type of Lyapunov‐Krasovskii functional and proving its positive definiteness, using slack matrices and a convex combination condition, the delay‐dependent stability criteria, which are less conservative, are derived in terms of linear matrix inequalities. Numerical examples are also given to illustrate the improvement on the conservatism of the delay bound over some existing results. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Linear quadratic regulation for systems with time‐varying delay

In this paper we study the linear quadratic regulation (LQR) problem for discrete‐time systems with time‐varying delay in the control input channel. We assume that the time‐varying delay is of a known upper bound, then the LQR problem is transformed into the optimal control problem for systems with multiple input channels, each of which has single constant delay. The optimal controller is derived by establishing a duality between the LQR and a smoothing estimation for an associated system with a multiple delayed measurement. Copyright © 2009 John Wiley & Sons, Ltd.     

Delay‐range‐dependent robust H∞ filtering for uncertain systems with interval time‐varying delays

This paper is concerned with the problem of delay‐range‐dependent robust H_∞ filtering for systems with time‐varying delays in a range. The aim of this problem is to design a filter such that, for all admissible uncertainties, the filtering error system is robustly asymptotically stable with a prescribed H_∞ level. The desired filter can be constructed by solving a set of linear matrix inequalities (LMIs). An illustrative numerical example is provided to demonstrate the effectiveness of the proposed method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

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 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.     

Two general integral inequalities and their applications to stability analysis for systems with time‐varying delay

Integral inequalities have been widely used in stability analysis for systems with time‐varying delay because they directly produce bounds for integral terms with respect to quadratic functions. This paper presents two general integral inequalities from which almost all of the existing integral inequalities can be obtained, such as Jensen inequality, the Wirtinger‐based inequality, the Bessel–Legendre inequality, the Wirtinger‐based double integral inequality, and the auxiliary function‐based integral inequalities. Based on orthogonal polynomials defined in different inner spaces, various concrete single/multiple integral inequalities are obtained. They can produce more accurate bounds with more orthogonal polynomials considered. To show the effectiveness of the new inequalities, their applications to stability analysis for systems with time‐varying delay are demonstrated with two numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Synchronization analysis for nonlinear bilateral teleoperator with interval time‐varying delay

In this paper, we investigate synchronization problem of nonlinear teleoperator in the presence of variable time delay. Compared with previous works, we relax the restriction of the communication delay range. Although approaches to the interval time delay can be found in many literatures, they are not suitable for nonlinear teleoperator due to the properties of robotic systems. We first investigate the synchronization of teleoperation system with interval time delay and make full use of the information of both the lower and upper bounds of delay. All criteria are presented in the form of LMI, which can be easily calculated by MATLAB (MathWorks, USA). Finally, numerical example and simulation are given to show the effectiveness of the main results. Copyright © 2014 John Wiley & Sons, Ltd.     

Robust stability criteria for uncertain neutral system with interval time varying discrete delay

This paper deals with the problem of rubost stability for the uncertain neutral system with interval time varying discrete delay. By defining an appropriate Lyapunov‐Krasovskii functional and by employing the developed free weight matrices technique, several less conservative sufficient conditions are derived in term of the linear matrix inequalities. Numerical examples are given to demonstrate the effectiveness and the feasibility of the proposed method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

New results of stability analysis for systems with time‐varying delay

This paper studies the problem of stability analysis for continuous‐time systems with time‐varying delay. By developing a delay decomposition approach, the information of the delayed plant states can be taken into full consideration, and new delay‐dependent sufficient stability criteria are obtained in terms of linear matrix inequalities. The merits of the proposed results lie in their less conservatism, which are realized by choosing different Lyapunov matrices in the decomposed integral intervals and estimating the upper bound of some cross term more exactly. Numerical examples are given to illustrate the effectiveness and less conservatism of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.     

Delay‐range‐dependent robust stability and stabilization for uncertain systems with time‐varying delay

This paper concerns delay‐range‐dependent robust stability and stabilization for time‐delay system with linear fractional form uncertainty. The time delay is assumed to be a time‐varying continuous function belonging to a given range. On the basis of a novel Lyapunov–Krasovskii functional, which includes the information of the range, delay‐range‐dependent stability criteria are established in terms of linear matrix inequality. It is shown that the new criteria can provide less conservative results than some existing ones. Moreover, the stability criteria are also used to design the stabilizing state‐feedback controllers. Numerical examples are given to demonstrate the applicability of the proposed approach. Copyright © 2007 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 conditions for finite time stability of linear time‐varying systems with delay

This paper investigates the problem of finite time stability of linear time‐varying system with delay. By constructing an augmented time‐varying Lyapunov functional and using the Wirtinger‐type inequality deductively, delay‐dependent finite time stability conditions are derived and presented in terms of differential linear matrix inequalities (DLMIs). Then, the DLMIs are transformed into a series of recursive linear matrix inequalities (RLMIs) by discretizing the time interval into equally spaced time distances, and an algorithm is given to solve the RLMIs. Examples illustrate the feasibility and effectiveness of the proposed method.     

Exponential stability criteria for discrete time‐delay systems

In this paper, we study the exponential stability of linear discrete time‐delay systems with slowly varying coefficients and nonlinear perturbations. We establish the robustness of the exponential stability in Hilbert spaces, in the sense that the exponential stability for a given linear equation persists under sufficiently small perturbations. As an application of the main results, we discuss the exponential stability of a general nonlinear system. The main novelty of this work is that we always consider the exponential behavior of solutions with respect to an specific ball. Copyright © 2013 John Wiley & Sons, Ltd.     

A delay decomposition approach to delay‐dependent stability for linear systems with time‐varying delays

This paper is concerned with delay‐dependent stability for linear systems with time‐varying delays. By decomposing the delay interval into multiple equidistant subintervals, on which different Lyapunov functionals are chosen, and new Lyapunov‐Krasvskii functionals are then constructed. Employing these new Lyapunov‐Krasvskii functionals, some new delay‐dependent stability criteria are established. The numerical examples show that the obtained results are less conservative than some existing ones in the literature. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

Delay dependent stabilization of linear time‐varying system with time delay

This paper investigates both stability and stabilization dependent on the delay of a class of time‐varying linear systems with a constant point time delay. The matrices, describing the state space dynamics, are parameterized by time‐varying function matrices. A numerical example is given in order to verify the theoretical results. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Delay‐dependent robust control for singular discrete‐time Markovian jump systems with time‐varying delay

The problem of delay‐dependent robust stabilization for uncertain singular discrete‐time systems with Markovian jumping parameters and time‐varying delay is investigated. In terms of free‐weighting‐matrix approach and linear matrix inequalities, a delay‐dependent condition is presented to ensure a singular discrete‐time system to be regular, causal and stochastically stable based on which the stability analysis and robust stabilization problem are studied. An explicit expression for the desired state‐feedback controller is also given. Some numerical examples are provided to demonstrate the effectiveness of the proposed approach. Copyright © 2009 John Wiley & Sons, Ltd.