On the stability of cycles by delayed feedback control

Abstract

We present a delayed feedback control (DFC) mechanism for stabilizing cycles of one dimensional discrete time systems. In particular, we consider a delayed feedback control for stabilizing $T$-cycles of a differentiable function $f:\mathbb{R}\to \mathbb{R}$ of the form $$x(k+1)=f(x(k))+u(k)$$ where $$u(k)=(a_1-1)f(x(k))+a_{2}f(x(k-T))+...+a_{N}f(x(k-(N-1)T)),$$ with $a_1+...+a_N=1$. Following an approach of Morgül, we construct a map $F:{\mathbb{R}}^{T+1}\to {\mathbb{R}}^{T+1}$ whose fixed points correspond to $T$-cycles of $f$. We then analyze the local stability of the above DFC mechanism by evaluating the stability of the corresponding equilibrum points of $F$. We associate to each periodic orbit of $f$ an explicit polynomial whose Schur stability corresponds to the stability of the DFC on that orbit. An example indicating the efficacy of this method is provided.