Delayed feedback control with a minimal-order observer for stabilization of chaotic discrete-time systems

Delayed feedback control (DFC) is a powerful method for stabilizing unstable periodic orbits embedded in chaotic attractors, which uses a small control input fed by the difference between the current state and the delayed state. One drawback of the DFC is known as the odd number limitation; that is, DFC can never stabilize a target unstable fixed point of a chaotic discrete time system, if the Jacobian of its linearized system around the unstable fixed point has an odd number of real eigenvalues greater than unity. To overcome it, in this paper we propose a dynamic DFC method using output measurements of the chaotic systems. The proposed dynamic DFC is realized by using an output feedback controller with a minimal-order observer that has the least order for estimating the state of the chaotic system from the control input and the output measurements. In addition to the design procedure of the controller, we derive a necessary and sufficient condition for the existence of such controllers.