Stability analysis of a general class of singularly perturbed linear hybrid systems

TL;DR

This paper develops a stability analysis framework for a broad class of singularly perturbed linear hybrid systems with mode-dependent variables, switches, and impulses, providing bounds on dwell-time for stability.

Contribution

It introduces a mode-dependent variable reordering technique and derives a stability condition with a dwell-time bound that accounts for changing variable dynamics and state dimension at switches.

Findings

Derived an upper bound on minimum dwell-time for stability.

Reformulated systems to preserve variable nature over time.

Numerical examples validate the theoretical results.

Abstract

Motivated by a real problem in steel production, we introduce and analyze a general class of singularly perturbed linear hybrid systems with both switches and impulses, in which the slow or fast nature of the variables can be mode-dependent. This means that, at switching instants, some of the slow variables can become fast and vice-versa. Firstly, we show that using a mode-dependent variable reordering we can rewrite this class of systems in a form in which the variables preserve their nature over time. Secondly, we establish, through singular perturbation techniques, an upper bound on the minimum dwell-time ensuring the overall system's stability. Remarkably, this bound is the sum of two terms. The first term corresponds to an upper bound on the minimum dwell-time ensuring the stability of the reduced order linear hybrid system describing the slow dynamics. The order of magnitude of the second term is determined by that of the parameter defining the ratio between the two time-scales of the singularly perturbed system. We show that the proposed framework can also take into account the change of dimension of the state vector at switching instants. Numerical illustrations complete our study.