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 : R → R of the formwith a 1 + · · · + a N = 1. Following an approach of Morgül, we construct a map F : R T +1 → 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.2010 Mathematics Subject Classification. Primary 93B52, 93B60.

Fejér Polynomials and Control of Nonlinear Discrete Systems

“…, x * T (N −1) ) where for convenience we set x * j = x * j mod T . In the paper [6], we proved that this polynomial is given by…”

Finding, Stabilizing, and Verifying Cycles of Nonlinear Dynamical Systems

“…We suggest the method developed in [4,9] that allows to write the polynomial in very compact and specific form where only the coefficients a j , b j and the multipliers µ j are used. Using this method we have found [4,9] that the characteristic polynomial of T -cycle can be written in the following elegant form λ N T m f (1/λ) where…”

Characteristic polynomials for a cycle of non-linear discrete systems with time delays

“…The goal of the current paper is to introduce a modified approach to the problem of local asymptotic stability of cycles in special non-linear systems and to Khamitova Anna Dmitrievna -postgraduate student; anna_khamitova@georgiasouthern.edu Хамитова Анна Дмитриевна -аспирант; anna_khamitova@georgiasouthern.edu c Санкт-Петербургский государственный университет, 2016 make a comparison. The standard approach to this problem is well outlined in [1,2] which is a further development of the method describe byÖ. Morgül [3].…”

“…Therefore, the matrix (12) is a generalized form of companion matrix. If system is of special case below, then the product is manageable and the characteristic equation was found in scalar case by means of induction in [1].…”