On construction of coprime factorizations for nonlinear feedback control systems

Abstract

In this paper we describe a family of operators for constructing nontrivial left and right coprime factorizations under rather weak conditions for a large class of nonlinear feedback control systems that has a stabilizable or unstabilizable plant, using a common criterion of input-output stability. The proposed sufficiency and construction method are based on a generalization of the nonlinear Lipschitz operator theory formulated by us for such systems. This class of generalized nonlinear Lipschitz operators constitutes a very large family (an infinite-dimensional Banach space) of bounded nonlinear operators that describe (part of) the underlying systems. One of the main difficulties in constructing coprine factorizations for nonlinear feedback systems has been in taking care of the nonlinear composite and inverse operators that appear in the closed-loop configuration such that the overall feedback system is well defined in the sense of stability, causality, and uniqueness of the internal signals and such that the coprime factorizations can be achieved. In this paper we show how these difficult issues can be handled nicely under our framework of generalized nonlinear Lipschitz operator theory, at least for a very large class of nonlinear control systems. We give a simple illustrative example to show how these coprime factorizations can actually be characterized and constructed to yield explicit closed-form solutions.