Structure at infinity, zero dynamics and normal forms of systems undergoing sliding motions

Abstract

In this article we examine the structure at infinity of non-linear closed-loop systems locally undergoing sliding regimes about a smooth surface defined in state space. By using a locally diffeomorphic state coordinate transformation, associated with the relative degree of the system, one obtains a normal form exhibiting the basic internal dynamic structure of the controlled system. It is found that the local existence of sliding motions demands a considerably simple local structure at infinity of the original non-linear system. The ideal sliding dynamics in local sliding surface coordinates is shown to coincide precisely with the zero dynamics. The stability properties of this internal behaviour model are studied. Several illustrative examples are presented.