SM system identification with approximated models

Set Membership system identification is investigated, when the identification algorithms are restricted to give n dimensional approximating models. In the recent literature two main approaches have been used to evaluate the quality of such approximated models, based on the concepts of n-width and of radius of information. In this paper it is shown that the n-width approach consists of an asymptotic analysis of the noise free case of the problem posed here, based on the concept of conditional radius of information. Upper and lower bounds of the conditional radius of information are derived for the H 2 identification of exponentially stable systems in the presence of power bounded measurements. These bounds are shown to be convergent to the radius for large number of data and model dimension. Moreover, a linear algorithm is presented, which is shown to be almost optimal for finite sample and convergent to an optimal one for large number of data and model dimension.