Transforming Nonlinear Single-Input Control Systems to Normal Forms via Feedback

We study the problem of bringing a nonlinear single-input system to a normal form. We follow an approach of Kang and Krener based on studying, step by step, the action of homogeneous transformations on homogeneous terms of the system. The first part of the paper deals with systems whose linearization is controllable. We give explicit transformations bringing the system to a normal form proposed by Kang. Then we construct a canonical form. In the second part, we give a normal form for systems whose linearization is not controllable. The normal forms, in the controllable and uncontrollable case, are illustrated by two physical examples.