Model Quality Evaluation in Identification

Model quality evaluation in set-membership identification is investigated. In the recent literature, two main approaches have been used to investigate this problem, based on the concepts of n-width and of radius of information. In this paper it is shown that the n- width is related to the asymptotic value of the conditional radius of information of the identification problem with noise free measurements. Upper and lower bounds of the conditional radius of information are derived for the identification of exponentially stable systems using approximating n-dimensional models linear in the parameters in the presence of power bounded measurement errors. The derived bounds are shown to be convergent to the radius for a large number of data and model dimensions. Moreover, a formula for computing the worst case identification error for any linear algorithm is given. In particular, it is shown that the identification error of the least square algorithm may be increasing with respect to the model dimension ("peaking effect"). An almost-optimal linear algorithm is presented, that is not affected by this peaking effect, and indeed is asymptotically optimal.

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