Asymptotic variances of subspace estimates

We provide new expressions for the asymptotic covariance of the estimated parameters (A, B, C, D) of a state space model obtained by some popular subspace identification method. The expressions, similar but simpler than the asymptotic covariance formulas which have so far been published in the literature, involve the inverses of the conditional covariance matrices /spl Sigma/(xx|u/sup +/), /spl Sigma/(u/sup +/u/sup +/|x) thus providing a direct link of possible ill-conditioning of the estimation problem with the asymptotic variance of the estimates. A study of ill-conditioning of subspace identification has been presented. The formulas can be applied to several subspace methods including N4SID, MOESP, CVA, etc.

[1]  Anders Lindquist,et al.  Canonical correlation analysis, approximate covariance extension, and identification of stationary time series , 1996, Autom..

[2]  Bart De Moor,et al.  Choice of state-space basis in combined deterministic-stochastic subspace identification , 1995, Autom..

[3]  V. Solo.  The Statistical Theory of Linear Systems E. J. Hannan and Manfred Deistler John Wiley & Sons, 1988 , 1992, Econometric Theory.

[4]  Michel Gevers,et al.  On jointly stationary feedback-free stochastic processes , 1982 .

[5]  Alessandro Chiuso,et al.  Error Analysis of Certain Subspace Methods , 2000 .

[6]  B. Wahlberg,et al.  Counterexample to General Consistency of Subspace System Identification Methods , 1997 .

[7]  Giorgio Picci,et al.  Stochastic realization with exogenous inputs and 'subspace-methods' identification , 1996, Signal Process..

[8]  M. Jansson.  Asymptotic Variance Analysis of Subspace Identification Methods , 2000 .

[9]  Alessandro Chiuso,et al.  Some Algorithmic aspects of Subspace Identification with Inputs , 2001 .

[10]  Lennart Ljung,et al.  Some facts about the choice of the weighting matrices in Larimore type of subspace algorithms , 2002, Autom..

[11]  W. Larimore.  System Identification, Reduced-Order Filtering and Modeling via Canonical Variate Analysis , 1983, 1983 American Control Conference.

[12]  Michel Verhaegen,et al.  Identification of the deterministic part of MIMO state space models given in innovations form from input-output data , 1994, Autom..

[13]  Alessandro Chiuso,et al.  Probing inputs for subspace identification , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[14]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

[15]  Bart De Moor,et al.  Subspace algorithms for the stochastic identification problem, , 1993, Autom..

[16]  P. Caines,et al.  Estimation, Identification and Feedback , 1976 .

[17]  E. Hannan,et al.  The statistical theory of linear systems , 1989 .

[18]  E. J. Hannan,et al.  Unit canonical correlations between future and past , 1988 .

[19]  Giorgio Picci,et al.  Realization of stochastic systems with exogenous inputs and subspace identification methods , 1999, Autom..

[20]  D. Bauer.  Some asymptotic theory for the estimation of linear systems using maximum likelihood methods or subspace algorithms , 1998 .

[21]  Dietmar Bauer.  Asymptotic Efficiency of CCA Subspace Methods in the Case of no Exogenous Inputs , 2000 .

[22]  N. Wiener.  Generalized harmonic analysis , 1930 .

[23]  Subspace identification by orthogonal decomposition , 1999 .

[24]  Bart De Moor,et al.  N4SID: Subspace algorithms for the identification of combined deterministic-stochastic systems , 1994, Autom..

[25]  Dietmar Bauer,et al.  Analysis of the asymptotic properties of the MOESP type of subspace algorithms , 2000, Autom..

[26]  Bo Wahlberg,et al.  Analysis of state space system identification methods based on instrumental variables and subspace fitting , 1997, Autom..

[27]  Bo Wahlberg,et al.  A linear regression approach to state-space subspace system identification , 1996, Signal Process..