Feedback stabilization of nonlinear discrete-time systems

It is the merit of Aeyels (Systems Control Lett. 5 (1985) 289-294) to have shown a way in which center manifold theory can be used in a constructive manner to find a smooth feedback control for stabilizing an equilibrium of a continuous-time system described by a nonlinear ordinary differential equation (x) over dot = f(x, tc) In this paper we are going to extend Aeyels' approach to nonlinear discrete-time systems described by equations of the type x(k + 1) = f(x(k)) + Bu(k), K = 0, 1, 2, ..., where we assume that f is sufficiently smooth and satisfies f(0)= 0. In critical cases. i.e. in situations where the linearization of the system in the neighborhood of the equilibrium includes non-controllable modes, we derive conditions under which there exists a stabilizing quadratic feedback control.