Time-delayed feedback control method and unstable controllers

TL;DR: An idea of using unstable degrees of freedom in a feedback loop to avoid a well known topological limitation of the method is described in details and is extended for the problem of adaptive stabilization of unknown steady states of dynamical systems.

Abstract: Time delayed-feedback control is an efficient method for stabilizing unstable periodic orbits of chaotic systems. The method is based on applying feedback proportional to the deviation of the current state of the system from its state one period in the past so that the control signal vanishes when the stabilization of the desired orbit is attained. A brief review of experimental implementations, applications for theoretical models, most important modifications as well as recent advancements in the theory of the method is presented. An idea of using unstable degrees of freedom in a feedback loop to avoid a well known topological limitation of the method is described in details. This idea is extended for the problem of adaptive stabilization of unknown steady states of dynamical systems.