Stabilizing Unstable Periodic Orbits with Delayed Feedback Control in Act-and-Wait Fashion

TL;DR

This paper introduces a delayed feedback control method with an act-and-wait approach to stabilize unstable periodic orbits in linear and nonlinear periodic systems, with proven convergence conditions and numerical demonstrations.

Contribution

It presents a novel act-and-wait delayed feedback control framework for stabilizing unstable periodic orbits in linear time-varying systems, extending to nonlinear systems.

Findings

Derived conditions for convergence of the closed-loop system

Validated the approach with numerical examples

Demonstrated stabilization of unstable orbits in nonlinear systems

Abstract

A delayed feedback control framework for stabilizing unstable periodic orbits of linear periodic time-varying systems is proposed. In this framework, act-and-wait approach is utilized for switching a delayed feedback controller on and off alternately at every integer multiples of the period of the system. By analyzing the monodromy matrix of the closed-loop system, we obtain conditions under which the closed-loop system's state converges towards a periodic solution under our proposed control law. We discuss the application of our results in stabilization of unstable periodic orbits of nonlinear systems and present numerical examples to illustrate the efficacy of our approach.