On criterion selection and noise model parametrization for prediction-error identification methods

Abstract

Prediction-error methods for systems with approximate noise models and a possibly non-linear dynamics are studied. It is assumed that the influence of the input on the system is parametrized in an appropriate way. The parameter estimates describing the system dynamics will then, under weak conditions, be asymptotically gaussian distributed. Their covariance matrix is given in explicit form. The results are applicable for a general class of prediction-error criteria. It is then proved that optimal accuracy can be obtained if simultaneously the model structure is taken rich enough to cover the true system and the criterion is chosen as the determinant of the sample covariance of the prediction errors. Some examples are included demonstrating that it is not generally possible to suboptimize by choosing either the model structure or the criterion in this way. This result has obvious practical importance and is somewhat contrary to intuition.