Stability of Planar Switched Systems under Delayed Event Detection

TL;DR

This paper investigates how delays in event detection affect the stability of planar hybrid automata, providing methods to compute maximum allowable delays and linking instability to the existence of closed orbits.

Contribution

It introduces an exact algorithm for computing the maximum stable delay in planar hybrid automata with delays, extending stability analysis under delayed event detection.

Findings

Maximum delay preserving stability is identified.

Instability corresponds to the existence of a closed orbit.

Algorithm for exact delay computation is developed.

Abstract

In this paper, we analyse the impact of delayed event detection on the stability of a 2-mode planar hybrid automata. We consider hybrid automata with a unique equilibrium point for all the modes, and we find the maximum delay that preserves stability of that equilibrium point. We also show for the class of hybrid automata treated that the instability of the equilibrium point for the equivalent hybrid automaton with delay in the transitions is equivalent to the existence of a closed orbit in the hybrid state space, a result that is inspired by the Joint Spectral Radius theorem. This leads to an algorithm for computing the maximum stable delay exactly. Other potential applications of our technique include co-simulation, networked control systems and delayed controlled switching with a state feedback control.