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.