State-space realization of nonlinear input-output equations: Unification and extension via pseudo-linear algebra

The paper focuses on unification of the study of continuous- and discrete-time nonlinear control systems using the mathematical tools of pseudo-linear algebra. The realization problem is studied through the language of differential forms. Necessary and sufficient conditions are given for the existence of the state space realization of the nonlinear input-output equation defined in terms of the pseudo-linear operator. The sufficiency part of the proof gives a constructive procedure (up to integration of the one-forms) for obtaining the observable state equations. The special cases for continuous- and discrete-time systems, described either in terms of the shift or difference operator, follow from the general result as the special cases. Moreover, the result accommodates also the systems described via q-shift or q-difference operators.