Feedback stabilization of switched systems under arbitrary switching: A convex characterization

TL;DR

This paper presents a convex LMI-based method for stabilizing discrete-time switched linear systems under arbitrary switching, using graph structures to derive necessary and sufficient conditions for feedback stabilization.

Contribution

It introduces a novel convex characterization of feedback stabilization for switched systems that accounts for mode-dependent and mode-independent controllers, with explicit controller design.

Findings

LMI conditions for stabilizability are both necessary and sufficient.

The approach applies to systems with or without mode information.

Numerical examples demonstrate the effectiveness of the proposed method.

Abstract

In this paper, we study stabilizability of discrete-time switched linear systems where the switching signal is considered as an arbitrary disturbance (and not a control variable). We characterize feedback stabilization via necessary and sufficient linear matrix inequalities (LMIs) conditions based on novel graph structures. We analyze both the cases in which the controller has (or has not) access to the current switching mode, the so-called mode-dependent and mode-independent settings, providing specular results. Moreover, our approach provides explicit piecewise-linear and memory-dependent linear controllers, highlighting the connections with existing stabilization approaches. The effectiveness of the proposed technique is finally illustrated with the help of some numerical examples.