On the input-output approximation of nonlinear systems

Discrete-time nonlinear systems represented in the state-space form are considered. Input (drive signal) and output (response signal) are assumed to be measurable. The problem of approximating the external behavior of such a system - in the form of an Input-Output (I-O) model - is addressed. It is already known that a system with fading-memory can be uniformly I-O approximated by a nonlinear MA filter fed with the input signal. In this paper, we prove a more general result: Almost any (in a precise sense) continuous nonlinear system is uniformly I-O approximable by a nonlinear ARMA filter fed with the input and output signals. In other words, an I-O model can be designed that tracks the external behavior of the system. We suggest this result constitutes a mathematical justification for the practice of 'black box' nonlinear system identification.