On the Stability of Cycles by Delayed Feedback Control

We present a delayed feedback control (DFC) mechanism for stabilizing cycles of one-dimensional discrete time systems. In particular, we consider a DFC for stabilizing T -cycles of a differentiable function 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→ℝ^T+1 whose fixed points correspond to T -cycles of f. We then analyse the local stability of the above DFC mechanism by evaluating the stability of the corresponding equilibrium 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. This polynomial is the characteristic polynomial of a Jacobian matrix that lies in a large class of matrices that encompasses the usual ‘companion matrices’ found in linear algebra; the primary purpose of this paper is to show that this polynomial may be expressed in a surprisingly simple form. An example indicating the efficacy of this method is provided.