Transfer equivalence and realization of nonlinear higher order input-output difference equations

Two fundamental modelling problems in nonlinear discrete-time control syste ms are studied using the language of differential forms. The discrete-time nonlinear single-input single-output systems to be studied are described by input-output (i/o) difference equations, i.e. a high order difference equa tion relating the input, the output and a finite number of their time shift s. A new definition of equivalence is introduced which generalizes the noti on of transfer equivalence well known for the linear case. Our definition i s based upon the notion of an irreducible differential form of the system a nd was inspired by the analogous definition for continuous-time systems. Th e second problem to be addressed is the realization problem. The i/o differ ence equation is assumed to be in the irreducible form so that one can obta in an accessible and observable realization. Necessary and sufficient condi tions are given for the existence of a (local) state-space realization of t he irreducible i/o difference equation. These conditions are formulated in terms of the integrability of certain subspaces of one-forms, classified ac cording to their relative degree. The sufficiency part of the proof gives a constructive procedure (up to finding the integrating factors and integrat ion of the set of one-forms) for obtaining a locally observable and accessi ble state-space system. If the system is not in the irreducible form, one h as first to apply the reduction procedure to transform the system into the irreducible form. (C) 2001 Elsevier Science Ltd. All rights reserved.