Generalization of Transfer Equivalence for Discrete-Time Non-Linear Systems: Comparison of Two Definitions

The problem of system reduction is studied for discrete-time non-linear single-input single-output systems described by high-order input–output (i/o) difference equations, that is, given the i/o equation, we can find a minimal representation which is equivalent to the original system with the order being as small as possible. Comparison of two definitions of reducibility is provided. The reducibility concepts addressed are both generalizations of the well-known notion of transfer equivalence to the case of non-linear control systems. Two roles of transfer equivalence are covered by these extensions. The first is that of identity of the outputs for any fixed control sequence under zero initial conditions. The second property is that of pole/zero cancellation that may occur in the transfer function of equivalent systems. The relationship between two reducibility definitions related to two equivalence notions is examined and the reducibility definition which extends pole/zero cancellation property is shown to be stronger. Finally, the computational aspects of system reduction are discussed.