Switch System with Stable Period Orbit

TL;DR

This paper introduces a 3D switched system that stabilizes unstable modes through fast switching, demonstrating a novel stabilization mechanism and broadening understanding of stable periodic orbits in switched systems.

Contribution

It presents a new class of unstable systems that can be stabilized via switching, with mathematical analysis elucidating the stabilization mechanism.

Findings

Fast switching induces stability in the system.

The system converges to a stable periodic orbit.

A general class of unstable systems can be stabilized through switching.

Abstract

Stability is a key property of dynamical systems. In some cases, we want to change unstable system into stable one to achieve certain goals in engineering. Here, we present an example of a $3$ dimensional switched system that alternates between unstable modes with the same period orbit, but exhibits stability for fast-switching frequencies and converges to the period orbit. Through mathematical analysis, we elucidate the stabilization mechanism and construct a general class of unstable systems that can form stable switched systems. The demonstration of this new system with period orbit provide a new sight in the study of switched system.