Tracking through singularities

Abstract

Results on the invertibility of nonlinear systems and their implication in the study of the output tracking problem, i.e., the determination of whether the output of the system can be made to follow a preassigned trajectory over a specific interval of time, are now well known (see Hirschorn [4] and the references therein or the recent paper by Grasse [3]). In these studies, nonlinear systems are either defined by a state-space representation or by a set of input-output differential equations. However, these approaches can break down when singularity is present. Recently Hirschorn and Davis [5] studied output tracking with "singular points": if the singularity is assumed to be at the origin, the authors give necessary and sufficient conditions involving, as in the regular case, only derivatives at the origin of the output function to be tracked, in terms of an integer α called the degree of singularity and depending on the singular point. However, their paper gives little indication of the complexity of the problem and does not, for example, raise the existence issue.

In this paper we give a new formulation of the problem which directly involves existence, uniqueness and regularity issues. An input-output formulation of the singular output tracking problem is proposed, although the details of this new concept will be discussed in a future paper. The main results are then illustrated with some worked examples.