Irreducibility and reduction of MIMO 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 problems for continuous- and discrete-time multi-input multi-output nonlinear control systems. The necessary and sufficient condition for irreducibility of the set of nonlinear input-output equations is presented in terms of the greatest common left divisor of two polynomial matrices describing the behaviour of the system. The basic difference is that, unlike the linear case, the elements of the polynomial matrices belong to a non-commutative skew polynomial ring. The suggested condition provides a bases for finding the (transfer) equivalent minimal irreducible representation of the set of input-output equations, which is a suitable starting point for constructing an observable and accessible state space realization. Besides unification, the tools of pseudolinear algebra allow to extend the results for systems defined by difference, q-shift and q-difference operators.

I. Introduction

Pseudo-linear algebra [5], [1], also known as Ore algebra, allows to study the common properties of linear ordinary differential, difference, shift and other type of operators, expressed in terms of the skew polynomials. The concept of Ore algebra has been used earlier in control theory [6]. In [10], the concept of pseudo-linear algebra of (functional) operators has been combined with the concept of (Kähler) differentials in order to develop a unified “polynomial approach” for continuous-and discrete-time nonlinear control systems, or to be more precise, for generic (i.e not local) linearizations of nonlinear systems. The resulting models in terms of differentials can then be easily used for analysis. However, in order to find the control laws, one has to integrate the respective one-forms. Note that the paper focuses only on the algebraic aspects, and not on specifying the solution space.