Feedback Stabilization of Switched Systems: Memory is not needed

TL;DR

This paper proves that memoryless static feedback controllers are sufficient for stabilizing switched linear systems, eliminating the need for controllers with memory or dynamic features.

Contribution

It rigorously demonstrates that if a stabilizing controller with memory exists, then a memoryless controller can be constructed, confirming a longstanding assumption in the field.

Findings

Memoryless controllers can stabilize systems previously thought to require memory.

Dynamic controllers do not provide additional stabilization capabilities.

The results validate the folklore claim about the sufficiency of static feedback.

Abstract

A long-standing assumption in the literature on switched linear systems is that static, homogeneous of degree one feedbacks form the most general class of controllers necessary and sufficient for stabilization. In this paper, we provide a rigorous justification. More specifically, we prove by construction that if a switched linear system admits a stabilizing full-information controller, with access to the entire history of states and switching signals, then a memoryless and homogeneous of degree one stabilizing controller also exists. Specifically, in the modeindependent setting the controller can be chosen to depend only on the current state, and in the mode-dependent setting only on the current state and active mode. Our results thus show that dynamic controllers offer no additional stabilizing capability for switched linear systems, formally validating this folklore claim.