Revisiting the Normal Form of Input-Output Linearization

This paper revisits the normal form arising in the context of input-output feedback linearization for nonlinear control systems possessing well-defined relative degree. The objective is to investigate the validity of the normal form in a neighbourhood of the zero dynamics manifold, as opposed to a neighbourhood of a point on the manifold. The two main results of the paper are necessary and sufficient conditions under which the normal form exists in some neighbourhood of the zero dynamics manifold, or in a given a priori neighbourhood of the manifold in question. A special case is the existence of a global normal form. These results naturally lead to conditions for either local or regional (global, as a special case) asymptotic stabilization of the zero dynamics manifold. To illustrate these contributions, a normal form is derived for the kinematic unicycle model leading to a novel global circular path following controller.