Generic stabilizability for time-delayed feedback control

TL;DR

This paper demonstrates that time-delayed feedback control can universally stabilize periodic orbits in nonlinear systems under generic conditions, expanding its applicability in experimental dynamical system control.

Contribution

It proves that for any periodic orbit meeting a mild controllability condition, extended time-delayed feedback control can stabilize it, removing previous restrictions.

Findings

Universal stabilizability of periodic orbits under generic conditions

Control gains can be designed for stabilization in nonlinear systems

Techniques are based on linear stability analysis near periodic orbits

Abstract

Time delayed feedback control is one of the most successful methods to discover dynamically unstable features of a dynamical system in an experiment. This approach feeds back only terms that depend on the difference between the current output and the output from a fixed time T ago. Thus, any periodic orbit of period T in the feedback controlled system is also a periodic orbit of the uncontrolled system, independent of any modelling assumptions. It has been an open problem whether this approach can be successful in general, that is, under genericity conditions similar to those in linear control theory (controllability), or if there are fundamental restrictions to time-delayed feedback control. We show that there are no restrictions in principle. This paper proves the following: for every periodic orbit satisfying a genericity condition slightly stronger than classical linear controllability, one can find control gains that stabilise this orbit with extended time-delayed feedback control. While the paper's techniques are based on linear stability analysis, they exploit the specific properties of linearisations near autonomous periodic orbits in nonlinear systems, and are, thus, mostly relevant for the analysis of nonlinear experiments.