Parameter-Dependent Lyapunov Functional for Stability of Time-Delay Systems With Polytopic-Type Uncertainties

Abstract

Abstract-This paper concerns the problem of the robust stability of a linear system with a time-varying delay and polytopictype uncertainties. In order to construct a parameter-dependent Lyapunov functional for the system, we first devised a new method of dealing with a time-delay system without uncertainties. In this method, the derivative terms of the state, which is in the derivative of the Lyapunov functional, are retained and some free weighting matrices are used to express the relationships among the system variables, and among the terms in the Leibniz-Newton formula. As a result, the Lyapunov matrices are not involved in any product terms of the system matrices in the derivative of the Lyapunov functional. This method is then easily extended to a system with polytopic-type uncertainties. Numerical examples demonstrate the validity of the proposed criteria.Index Terms-time-varying delay, robust stability, parameterdependent Lyapunov functional, polytopic-type uncertainties, linear matrix inequality (LMI).

Robust Stability for a Class of Uncertain Singular Time-Delays Systems

“…Therefore, the system (9) is regular and impulse free [17]. According to Lemma 2 in [7], we know that the operator D is stable.…”

“…Remark 2 If A 022 is non-singular, then the system (9) is regular and impulse free [17]. In this case, the compatible initial condition is…”

Stabilization of Neutral Systems with Saturating Actuators

“…If t min > 0, the previous values of ε 1 and ε 2 give the largest domain of attraction; otherwise (t min ≤ 0), to improve the solution, set h m = h m + h mstep and repeat from…”

“…Systems with time delays constitute basic mathematical models of many real phenomena in circuits theory, economics, mechanics, and so forth, so they have been extensively studied in the literature: see, for example, [1][2][3][4] and references therein for several theoretical studies on this subject. An aspect that has not been frequently taken into account is the fact that, in many of these systems, the actuators have strict limitations.…”

“…Moreover, to reduce conservatism, some free matrices P i are used (only one of them restricted to be positive definite). This introduction of free matrices can be viewed as an extension of the model description introduced in [6], or similar to the slack variables used in [1,3], albeit in a different context. Finally, a LMI optimization approach is proposed to design the state feedback gain that maximizes the size of the estimated domain of attraction.…”