Irreducibility and reduction of nonlinear control systems: Unification and extension via pseudo-linear algebra

The paper applies the pseudo-linear algebra to unify the study of irreducibility and reduction for continuous-and discrete-time nonlinear control systems. The necessary and sufficient condition for irreducibility of nonlinear input-output equation is presented in terms of the common left factor of two polynomials describing the behaviour of the system. Besides unification, the tools of pseudo-linear algebra allow to extend the results for systems defined by difference, q-shift and q-difference operators.

I. Introduction

Pseudo-linear algebra [2], [3], also known as Ore algebra, is the study of common properties of linear ordinary differential, difference, shift and other type of operators, expressed in terms of the skew polynomials. In [1] pseudo-linear algebra has been introduced into the control community and applied to define the transfer function and study its basic properties in a unified manner both for continuous-and discrete-time nonlinear systems as well for less conventional discrete-time system descriptions in terms of difference, -shift and difference operators. Note that the difference operator description provides a smooth transition from sampled-data algorithms to their continuous-time counterparts [4].