Stability of Bimodal Planar Switched Linear Systems with Both Stable and Unstable Subsystems

TL;DR

This paper investigates the stability conditions of bimodal planar switched linear systems with both stable and unstable subsystems, deriving dwell-flee relations and extending results to symmetric bilinear and multimodal systems.

Contribution

It introduces new dwell-flee stability relations for bimodal systems and extends these to symmetric bilinear and multimodal configurations with complex switching graphs.

Findings

Dwell-flee relations depend on eigenvalues and eigenvectors.

Results apply to symmetric bilinear systems.

Stability conditions derived for multimodal systems with star graph switching.

Abstract

We study dynamics of a bimodal planar linear switched system with a Hurwitz stable and an unstable subsystem. For given \textit{flee time} from the unstable subsystem, the goal is to find corresponding \textit{dwell time} in the Hurwitz stable subsystem so that the switched system is asymptotically stable. The dwell-flee relations obtained are in terms of certain smooth functions of the eigenvalues and (generalized) eigenvectors of the subsystem matrices. The results are extended to a special class of symmetric bilinear systems. The results are also extended to a multimodal planar linear switched system in which the switching is governed by an undirected star graph, where the internal node corresponds to a Hurwitz stable (unstable, respectively) subsystem and all the leaves correspond to unstable (Hurwitz stable, respectively) subsystems. For this multimodal system, dwell-flee relations are obtained as solutions of a minimax optimization problem.