On the stability of cycles by delayed feedback control

2015/01/19 by D. Dmitrishin, P. Hagelstein, Dmitrishin, D. +5
Mathematics · #93B52 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Optimization and Control (math.OC) #math.CA #math.OC #msc:93B52

paper · pdf · doi:10.48550/arxiv.1501.04573

arxiv created 2015/01/19 · arxiv updated 2015/01/20

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 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 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.

Related

• Fejer Polynomials and Control of Nonlinear Discrete Systems
• Optimal stabilization of a cycle in nonlinear discrete systems
• Fejer and Suffridge polynomials in the delayed feedback control theory
• Stabilizing Unstable Periodic Orbits with Delayed Feedback Control in Act-and-Wait Fashion
• Generalization of nonlinear control for nonlinear discrete systems