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 series of papers is analysed and extended in this paper. First, an analysis is made which proves that the asymptotic unbiasedness of the estimated state-space quadruple [A(T), B(T), C(T), D] critically depends on the unbiased calculation of the column space of the extended observability matrix. Second, it is proved that the latter quantity can be calculated asymptotically unbiasedly only when the stochastic additive errors on the output quantity are zero-mean white noise. The extension of the ordinary MOESP scheme with instrumental variables increases the applicability of this scheme. Two types of instrumental variables are proposed: (1) based on past input measurements; and (2) based on reconstructed state quantities. The first type yields asymptotic unbiased estimates when the perturbation on the output quantity is an arbitrary zero-mean stochastic process independent of the error-free input. However, a detailed sensitivity analysis demonstrates that for the finite data-length case the calculations can become very sensitive; this occurs when the particular system at hand has dominant modes close to the unit circle. In the same sensitivity analysis it is shown that far more robust results can be obtained with the second type of instrumental variables when the true state quantities are used. A number of guidelines are derived from the given sensitivity analysis to obtain accurate reconstructed state quantities. Efficient numerical implementations are presented for both extensions of the ordinary MOESP scheme. The obtained insights are verified by means of two realistic simulation studies. The developed extensions and strategy in these studies demonstrate excellent performances in the treatment of both identification problems.