New results on the asymptotic theory of system identification for the assessment of the quality of estimated models

In this paper the problem of estimating uncertainty regions for identified models is considered. Usually, one resorts to the asymptotic theory of system identification, by means of which ellipsoidal uncertainty regions can be constructed for the uncertain parameters. We show that these uncertainty regions supplied by the asymptotic theory can be unreliable in certain situations precisely characterized in the paper. Then, we investigate on the conditions of validity of the asymptotic theory, and we prove a new statement of more general applicability. Thanks to this statement, we can identify for which standard classes of models (ARMAX, Box Jenkins, etc.) the asymptotic theory can be safety used to assess the estimation quality. These results are of interest in many applications, including iterative controller design schemes.