Irreducibility conditions for nonlinear input-output difference equations

The purpose of the paper is to present a necessary and sufficient condition for irreducibility of a nonlinear input-output (i/o) difference equation which extends directly the corresponding condition for the linear case. The condition is presented in terms of the common left factors of two polynomials describing the behavior of the system; the basic difference is that unlike the linear case the polynomials related to the nonlinear system belong to a non-commutative polynomial ring. This condition provides a basis for finding the minimal (irreducible) equivalent representation of the i/o equation which is a suitable starting point for constructing a minimal state space representation