Input-output parametric models for non-linear systems Part I: deterministic non-linear systems

Recursive input-output models for non-linear multivariate discrete-time systems are derived, and sufficient conditions for their existence are defined. The paper is divided into two parts. The first part introduces and defines concepts such as Nerode realization, multistructural forms and results from differential geometry which are then used to derive a recursive input-output model for multivariable deterministic non-linear systems. The second part introduces several examples, compares the derived model with other representations and extends the results to create prediction error or innovation input-output models for non-linear stochastic systems. These latter models are the generalization of the multivariable ARM AX models for linear systems and are referred to as NARMAX or Non-linear AutoRegressive Moving Average models with exogenous inputs.