Robust Feedback Stabilization of Nonlinear Systems

Abstract

The problem of feedback stabilization of nonlinear systems has recently attracted a great deal of interest among control theorists (see e.g. [1–2]) both because of its intrinsic appeal and because of its extreme importance as a part of a more general design theory. In [2–3] a general method for feedback stabilization, based on a nonlinear enhancement of classical root-locus methods, has been developed which contains the more special, yet celebrated, method of feedback linearization as a special case. On the other hand, it is common practice in engineering to arrive at system models, in particular nonlinear control systems, by ignoring faster time-scale transients in a system to arrive at a simpler, reduced order model. For example, flexible modes of a system are often assumed to be rigid, thereby suppressing the higher frequencing flexible modes. While the effects of feedback linearization, when possible, of the reduced model on the larger, singularly perturbed model which includes faster transients have been investigated by Khorasani and Kokotovic [8], no comparable study has been made for the case where the reduced model is not linearizable. In this paper we study the possibility of augmenting the stabilizing laws derived in [2–3] for the reduced model to obtain stabilizing laws for the augmented, singularly perturbed system. Our main result is positive, obtaining stabilizing laws for “flexible” systems which have exponentially stable zero dynamics. This is illustrated for a PUMA 560 robot arm controlled by a PD controller.