$\mathcal{KL}_*$-stability for a class of hybrid dynamical systems. (29th July 2017) - Liu, Bin; Dam, Hai Huyen Heidi; Teo, Kok Lay; Hill, David John
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.