Canonical forms for nonlinear discrete time control systems

In this paper we provide a canonical form for discrete-time control systems whose linear approximation around an equilibrium is controllable and prove that two systems are feedback equivalent if and only if their canonical forms coincide. This is a nice generalization of results obtained for continuous time control systems. We also compute the homogeneous invariants under the action of a homogeneous feedback group. Consequently, as for the continuous systems, we deduce that the discrete time systems in consideration do not admit nontrivial symmetries, i.e., a map preserving the dynamics.