Limitation of delayed feedback control in nonlinear discrete-time systems

Recently, Pyragas [1993] proposed a control method called delayed feedback control to stabilize unstable periodic orbits in nonlinear continuous-time systems. In this letter, we consider delayed state-feedback control in discrete-time systems. We show that a fixed point can not be stabilized by the delayed state-feedback control if the linearized system around the fixed point has an odd number of real eigenvalues greater than one. Moreover, in one- and two-dimensional systems, we show necessary and sufficient conditions for the local stabilization by the delayed state-feedback control.