$\mathcal{KL}_*$-stability for a class of hybrid dynamical systems. (29th July 2017)

• Record Type: Journal Article
• Authors: Liu, Bin; Dam, Hai Huyen Heidi; Teo, Kok Lay; Hill, David John
• Is Part Of: IMA journal of applied mathematics. Volume 82:Number 5(2017)
• Page Start: 1043
• Page End: 1060
• Publication Date: 2017-07-29
• Subjects: hybrid dynamical system -- stability -- 𝓚𝓛-stability -- dwell-time; Mathematics -- Periodicals; Mathematics; Periodicals; 519
• DOI: 10.1093/imamat/hxx023
• Languages: English
• ISSNs: 0272-4960

Abstract

Abstract: This article studies $\mathcal{KL}_*$ -stability (the stability expressed by $\mathcal{KL}_*$ -class function) for a class of hybrid dynamical systems (HDS). The notions of $\mathcal{KL}_{*}\mathcal{K}_{*}$ -property and $\mathcal{KL}_{*}$ -stability are proposed for HDS with respect to the hybrid-event-time. The $\mathcal{KL}_{*}$ -stability, which is based on $\mathcal{K}$ or $\mathcal{L}$ property of the continuous flow, the discrete jump, and the event in an HDS, extends the $\mathcal{KLL}$ -stability and the event-stability reported in the literature for HDS. The relationships between $\mathcal{KL}_{*}\mathcal{K}_{*}$ -property and $\mathcal{KL}_{*}$ -stability are established via introducing the hybrid dwell-time condition (HDT). The HDT generalizes the average dwell-time condition in the literature. For an HDS with $\mathcal{KL}_{*}\mathcal{K}_{*}$ -property consisting of stabilizing $\mathcal{L}$ -property and destabilizing $\mathcal{K}$ -property, it is shown that there exists a common HDT under which the HDS will achieve $\mathcal{KL}_{*}$ -stability. Thus HDT may help to derive some easily tested conditions for HDS to achieve uniform asymptotic stability. Moreover, a criterion of $\mathcal{KL}_{*}$ -stability is derived by using the multiple Lyapunov-like functions. Examples are given to illustrate the obtained theoretical results.