State-space realization of input-output nonlinear difference equations

The realization problem for single-input single-output discrete-time nonlinear systems is studied through the language of differential forms. Necessary and sufficient conditions are given for the existence of a (local) state-space realization of the input-output difference equation, i.e. a higher order difference equation in the input, the output and a finite number of their time-shifts. The sufficiency part of the proof gives a constructive procedure (up to the integration of one-forms) for obtaining a locally observable and accessible state-space system.