Stabilizing Unstable Periodic Orbits via Universal Input–Output Delayed-Feedback Control

Abstract

Estimating the approximated linear state variable equation around an unstable periodic orbit (UPO) is difficult in most chaotic systems since the measurable signal is generally only in the form of a scalar output signal. Therefore, conventional state delayed feedback control (DFC) methods may be unfeasible in practice, particularly for high-dimensional systems. Consequently, this study proposes an input–output delayed-feedback control (IODFC) method. Initially, the approximated input–output description around the desired UPO is estimated using the delay coordinates. Subsequently, a (2n−1)-dimensional controllable canonical state equation is reconstructed, in which the state vector consists of the delayed input and the delayed output. All the eigenvalues of the (2n−1)-dimensional system can therefore be assigned to be inside the unit circle using the pole placement technique. The proposed method requires only that the system be both controllable and observable around the UPO.