Fejer Polynomials and Control of Nonlinear Discrete Systems

2018/04/12 by Dmitriy Dmitrishin, Dmitrishin, Dmitriy, Paul Hagelstein +7
Mathematics · #37N35 #42A05 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:37N35 #msc:42A05

paper · pdf · doi:10.48550/arxiv.1804.04537

arxiv created 2018/04/12 · arxiv updated 2018/04/13

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: ℝ→ℝ of the form x(k+1) = f(x(k)) + u(k) where u(k) = (a₁ - 1)f(x(k)) + a₂ f(x(k-T)) + ⋯ + a_N f(x(k-(N-1)T)) , with a₁ + ⋯ + a_N = 1. Following an approach of Morgül, we associate to each periodic orbit of f, N ∈ ℕ, and a₁,…,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 a₁,…,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 Fejér kernels found in classical harmonic analysis.

Related

• Fejer and Suffridge polynomials in the delayed feedback control theory
• On the stability of cycles by delayed feedback control
• Generalization of nonlinear control for nonlinear discrete systems
• Search for cycles in non-linear autonomous discrete dynamical system
• Optimal stabilization of a cycle in nonlinear discrete systems