Non-linear system modelling based on constrained Volterra series estimates

A simple non-linear system modelling algorithm designed to work with limited a priori knowledge and short data records, is examined. It creates an empirical Volterra series-based model of a system using an l(q)-constrained least squares algorithm with q >= 1. If the system m(.) is a continuous and bounded map with a finite memory no longer than some known tau, then (for a D parameter model and for a number of measurements N) the difference between the resulting model of the system and the best possible theoretical one is guaranteed to be of order root N(-1)ln D, even for D >= N. The performance of models obtained for q = 1, 1.5 and 2 is tested on the Wiener-Hammerstein benchmark system. The results suggest that the models obtained for q > 1 are better suited to characterise the nature of the system, while the sparse solutions obtained for q = 1 yield smaller error values in terms of input-output behaviour.