WIAS Preprint No. 85, (1994)

Feedback stabilization of nonlinear discrete-time systems


  • Müller, Wolfdietrich
  • Schneider, Klaus R.

2010 Mathematics Subject Classification

  • 93D15 93C55 34H05


  • Discrete-time control system, Smooth feedback stabilization, Center manifold




It is the merit of D. Aeyels [4] 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 eqution ẋ = ƒ(x,u). In this paper we are going to extend Aeyels' approach to nonlinear discrete-time systems described by equations of the type
x(k + 1)=ƒ(x(k),u(k)), k = 0, 1, 2, ... ,
where we assume that ƒ is sufficiently smooth and satisfies ƒ(0,0) = 0. In critical cases, i.e. in situations where the linearization of the system in the neighborhood of the equilibrium includes non-controllable modes, under some non-resonance conditions we derive sufficient conditions for the existence of a smooth nonlinear stabilizing feedback.

Appeared in

  • J. of Difference Equations and Applications, 4 (1998), pp. 579-596

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