Adaptive prediction in the presence of unmodelled dynamics

Abstract

This paper considers the prediction problem for a discrete-time stochastic system with unmodelled dynamics. The precisely modelled part of the system is described by an ARMAX model, while unmodelled dynamics is represented by a small constant ζ multiplied by a quantity tending to infinity as the past input, output and noise of the system increase. For the estimation of the unknown predictor parameters, the usual stochastic approximation algorithm is used. Under the standard conditions imposed on the modelled system part, it is shown that the mean-square prediction error converges to a finite limit. This limit depends explicitly on the unmodelled dynamics in such a way that when the unmodelled dynamics decays, the prediction error tends to the minimum possible.