Stability of singularly perturbed hybrid systems with restricted systems evolving on boundary layer manifolds

TL;DR

This paper develops a singular perturbation theory for hybrid systems with boundary layer dynamics, establishing practical attractivity and semi-global stability of attractor sets under certain conditions.

Contribution

It introduces a new theoretical framework for analyzing stability in hybrid systems with boundary layers, including jump dynamics and conditions for practical attractivity.

Findings

Practical attractivity of an attractor set for small tuning parameters

Semi-global practical asymptotic stability under mild jump conditions

Stability results for systems with slow states and boundary layer dynamics

Abstract

We present a singular perturbation theory applicable to systems with hybrid boundary layer systems and hybrid reduced systems {with} jumps from the boundary layer manifold. First, we prove practical attractivity of an adequate attractor set for small enough tuning parameters and sufficiently long time between almost all jumps. Second, under mild conditions on the jump mapping, we prove semi-global practical asymptotic stability of a restricted attractor set. Finally, for certain classes of dynamics, we prove semi-global practical asymptotic stability of the restricted attractor set for small enough tuning parameters and sufficiently long period between almost all jumps of the slow states only.