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

Advances in Difference Equations

article

Duan, Wenyong¹  Li, Yan²  Chen, Jian¹ 

[1] School of Electrical Engineering, Yancheng Institute of Technology;Undergraduate Office, Yancheng Biological Engineering Higher Vocational Technology School

关键词: Delay-dependent stability;    Lyapunov–Krasovskii functional;    Linear matrix inequalities;    Time-delayed system;    Time-varying delay;   

DOI  :  10.1186/s13662-019-2439-z

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

The 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 $h^{2}(t)$, which are not convenient to directly use the convexity of linear matrix inequality (LMI), the type of integral terms $\{\int _{s}^{t}\dot{x}(u)\,du, \int _{t-h}^{s}\dot{x}(u)\,du\}$ is used in the LKF instead of $\{\int _{s}^{t}x(u)\,du, \int _{t-h}^{s}x(u)\,du\}$. Secondly, two couples of integral terms $\{\int _{s}^{t}\dot{x}(u)\,du, \int _{t-h(t)}^{s}\dot{x}(u)\,du\}$, and $\{\int _{s}^{t-h(t)}\dot{x}(u)\,du, \int _{t-h}^{s}\dot{x}(u)\,du\}$ are supplemented in the integral functionals $\int _{t-h(t)}^{t}\dot{x}(u)\,du$ and $\int _{t-h}^{t-h(t)}\dot{x}(u)\,du$, 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.

【 授权许可】

CC BY