Fejer Polynomials and Control of Nonlinear Discrete Systems

TL;DR

This paper explores the use of Fejér polynomials in optimizing delayed feedback control mechanisms to stabilize cycles in nonlinear discrete systems, providing explicit conditions and minimal parameters for guaranteed stability.

Contribution

It introduces a novel approach linking Fejér kernels to the stability analysis of delayed feedback control in nonlinear systems, and determines minimal control parameters for stabilization.

Findings

Existence of control parameters for stabilization of 1- and 2-cycles.

Explicit polynomial criteria for Schur stability of the control system.

Minimal N guaranteeing stabilization for given cycles.

Abstract

We consider optimization problems associated to 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 $f: \mathbb{R}\rightarrow\mathbb{R}$ of the form $$x(k+1) = f(x(k)) + u(k)$$ where $$u(k) = (a_1 - 1)f(x(k)) + a_2 f(x(k-T)) + \cdots + a_N f(x(k-(N-1)T))\;,$$ with $a_1 + \cdots + a_N = 1$. Following an approach of Morg\"ul, we associate to each periodic orbit of $f$, $N \in \mathbb{N}$, and $a_1,\ldots,a_N$ an explicit polynomial whose Schur stability corresponds to the stability of the DFC on that orbit. We prove that, given any 1- or 2-cycle of $f$, there exist $N$ and $a_1,\ldots,a_N$ whose associated polynomial is Schur stable, and we find the minimal $N$ that guarantees this stabilization. The techniques of proof will take advantage of extremal properties of the Fej\'er kernels found in classical harmonic analysis.