An enhanced stability criterion for linear time-delayed systems via new Lyapunov–Krasovskii functionals

Abstract

AbstractThe stability problem of linear systems with time-varying delays is studied by improving a Lyapunov–Krasovskii functional (LKF). Based on the newly developed LKF, a less conservative stability criterion than some previous ones is derived. Firstly, to avoid introducing the terms with $h^{2}(t)$h2(t), which are not convenient to directly use the convexity of linear matrix inequality (LMI), the type of integral terms $\{\int _{s}^{t}\dot{x}(u)\,du, \int _{t-h}^{s}\dot{x}(u)\,du\}${∫stx˙(u)du,∫t−hsx˙(u)du} is used in the LKF instead of $\{\int _{s}^{t}x(u)\,du, \int _{t-h}^{s}x(u)\,du\}${∫stx(u)du,∫t−hsx(u)du}. Secondly, two couples of integral terms $\{\int _{s}^{t}\dot{x}(u)\,du, \int _{t-h(t)}^{s}\dot{x}(u)\,du\}${∫stx˙(u)du,∫t−h(t)sx˙(u)du}, and $\{\int _{s}^{t-h(t)}\dot{x}(u)\,du, \int _{t-h}^{s}\dot{x}(u)\,du\}${∫st−h(t)x˙(u)du,∫t−hsx˙(u)du} are supplemented in the integral functionals $\int _{t-h(t)}^{t}\dot{x}(u)\,du$∫t−h(t)tx˙(u)du and $\int _{t-h}^{t-h(t)}\dot{x}(u)\,du$∫t−ht−h(t)x˙(u)du, respectively, so that the time delay, its derivative, and information between them can be fully utilized. Thirdly, the LKF is further augmented by two delay-dependent non-integral items. Finally, three numerical examples are presented under two different delay sets, to show the effectiveness of the proposed approach.