Coprime factorization over a class of nonlinear systems

Abstract

In this paper, we consider the problem of generalizing elements of linear coprime factorization theory to a nonlinear context. The idea is to work with a suitably wide class of nonlinear systems to cover many practical situations, yet not cope with so broad a class as to disallow useful generalizations to the linear results. In particular, we work with nonlinear systems characterized in terms of (possibly time-varying) state-dependent matrices A(x), B(x), C(x), D(x) and an initial state x0. (This class clearly does contain the class of finite-dimensional linear (time-varying) systems.) We achieve first right coprime factorizations for idealized situations. To achieve stable left factorizations we specialize to the case where the matrices are output-dependent. Alternatively, we work with systems, perhaps augmented by a direct feedthrough term, where the input is reconstructible from the output. For nonlinear feedback control systems, with plant and controller having stable left factorizations, then under appropriate regularity-conditions earlier results have allowed the generation of the class of stabilizing controllers for a system in terms of an arbitrary stable system (parameter). Plant uncertainties, including unknown initial conditions are modelled by means of a Yula–Kucera-type parametrization approach developed for nonlinear systems. Certain robust stabilization results are also shown, and simulations demonstrate the regulation of nonlinear plants using the techniques developed. All the results are presented in such a way that specialization for the case of linear systems is immediate.