Improved Results on Robust Stability for Systems with Interval Time-Varying Delays and Nonlinear Perturbations

Abstract

This paper investigated delay-dependent robust stability criteria for systems with interval time-varying delays and nonlinear perturbations. A delay-partitioning approach is used in this paper, the delay-interval is partitioned into multiple equidistant subintervals, a new Lyapunov-Krasovskii (L-K) functional contains some triple-integral terms, and augment terms are introduced on these intervals. Then, by using integral inequalities method together with free-weighting matrix approach, a new less conservative delay-dependent stability criterion is formulated in terms of linear matrix inequalities (LMIs), which can be easily solved by optimization algorithms. Numerical examples are given to show the effectiveness and the benefits of the proposed method.

Improved Stability Criteria on Linear Systems with Distributed Interval Time-Varying Delays and Nonlinear Perturbations

“…β ∈ [−u, u]. By Lemma 9, LMIs (27) and (28) hold if and only if LMIs (29)-(32) hold. This completes the proof.…”

“…By Lemma 2 [10], if LMIs (27) and (28) are true for α = {0, 1}, β = {−u, u}, then φ(α, β) + ∆(α) < 0 holds for all α ∈ (0, 1). β ∈ [−u, u].…”

“…On the other hand, delay-dependent stability criteria are concerned with the size of the delay and usually provide a maximal delay size. The stability criteria of dynamical systems with time-varying delays and nonlinear perturbations have received the attention of many theoreticians and engineers in this field over the last few decades [4,9,14,18,25,28]. For delay-dependent stability criteria, the main concern is to enlarge the feasible region of the criteria to guarantee the asymptotic stability of time-delay systems in a given time-delay interval [2,3,6,7,22,23].…”

“…A descriptor model transformation and a corresponding Lyapunov-Krasovskii functional were introduced for the stability analysis of systems with delays in [19,21]. Moreover, delay-range-dependent stability criteria for dynamical systems with interval time-varying delays have been attracting the attention of several researchers [4,9,14,20,25,28]. Delay-range-dependent stability criteria make use of information on the lower and upper bounds of delay.…”

“…Delay-range-dependent stability criteria make use of information on the lower and upper bounds of delay. In the past few years, there have been various approaches to reduce the conservatism of delay-dependent conditions by using a new Lyapunov-Krasovskii functional [18,26], improved inequalities [7,[11][12][13][16][17][18]27], the free-weighting matrices technique [3,18], the delay-decomposition approach [2], and model transformation [9,28]. However, the results do not take into account the presence of nonlinear perturbations and distributed time-varying delay in the system.…”