Fejér Polynomials and Control of Nonlinear Discrete Systems

D. Dmitrishin, P. Hagelstein, A. Khamitova, A. Korenovskyi, A. Stokolos

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

We consider optimization problems associated with 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))+a2f(x(k-T))+⋯+aNf(x(k-(N-1)T)),with a1+ ⋯ + aN= 1. Following an approach of Morgül, we associate with each periodic orbit of f, N∈ N, and a1,.., aN 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, … , aN 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.

Original languageEnglish
Pages (from-to)383-412
Number of pages30
JournalConstructive Approximation
Volume51
Issue number2
DOIs
StatePublished - Apr 1 2020

Keywords

  • Control theory
  • Optimization
  • Stability

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