SUBSPACE MODEL IDENTIFICATION .3. ANALYSIS OF THE ORDINARY OUTPUT-ERROR STATE-SPACE MODEL IDENTIFICATION ALGORITHM

The ordinary MOESP algorithm presented in the first part of this serie s of papers is analysed and extended in this paper. First, an analysis is made which proves that the asymptotic unbiasedness of the estimate d state-space quadruple [A(T), B(T), C(T), D] critically depends on th e unbiased calculation of the column space of the extended observabili ty matrix. Second, it is proved that the latter quantity can be calcul ated asymptotically unbiasedly only when the stochastic additive error s on the output quantity are zero-mean white noise. The extension of t he ordinary MOESP scheme with instrumental variables increases the app licability of this scheme. Two types of instrumental variables are pro posed: (1) based on past input measurements; and (2) based on reconstr ucted state quantities. The first type yields asymptotic unbiased esti mates when the perturbation on the output quantity is an arbitrary zer o-mean stochastic process independent of the error-free input. However , a detailed sensitivity analysis demonstrates that for the finite dat a-length case the calculations can become very sensitive; this occurs when the particular system at hand has dominant modes close to the uni t circle. In the same sensitivity analysis it is shown that far more r obust results can be obtained with the second type of instrumental var iables when the true state quantities are used. A number of guidelines are derived from the given sensitivity analysis to obtain accurate re constructed state quantities. Efficient numerical implementations are presented for both extensions of the ordinary MOESP scheme. The obtain ed insights are verified by means of two realistic simulation studies. The developed extensions and strategy in these studies demonstrate ex cellent performances in the treatment of both identification problems.