### Doctoral Thesis Defense, Optimization and Systems Theory

Wednesday, June 6, 2001, 10.00, Kollegiesalen, Administration building,
Valhallavägen 79, KTH

**Anders Dahlén**

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Identification of stochastic systems: Subspace methods and
covariance extension

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Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan
framlägges till offentlig granskning för avläggande av filosofie
doktorsexamen onsdagen den 6:e juni 2001 kl 10.00 i Kollegiesalen,
Administrationsbyggnaden, Kungliga Tekniska Högskolan, Valhallavägen
79.
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This thesis consists of four papers in identification of
linear stochastic systems.

In the first paper it is briefly explained why certain
subspace methods, for identification of time-series,
may fail for theoretical reasons.
Reproducible experiments are described that make it possible
to test algorithms for failures.
Massive failures of some popular subspace methods are verified
through simulations.

In the second paper an alternative identification procedure which
overcomes these difficulties is presented for scalar times-series. It
is based on identification of a high-order maximum entropy model (AR
model) followed by stochastically balanced truncation. The procedure
is described using just linear algebraic operations, and therefore it
inherits the nice properties of subspace methods. A complete analysis
of the statistical convergence properties of the method is
presented. In particular, it is shown that the transfer function of
the estimated system tends in a "worst case" measure to the true
transfer function. Simulations show convergence to the Cramér-Rao
bounds of the obtained variances, as the data length increases.

In the third paper the identification procedure of the second paper is
generalized to multivariate time-series. As in the scalar case, the
procedure is described using just linear algebraic operations. The
essential differences between the CCA subspace method and the proposed
method are described. CCA estimates all covariances in a block Hankel
matrix directly from data, whereas the proposed procedure uses
covariance extension when constructing the Hankel matrix. A
consistency and asymptotic normality proof for the identification
procedure is given.

The fourth paper studies the relation between CCA and the proposed
method (MEST) in more detail. For the sake of comparison, the two
identification procedures are formulated in a uniform framework using
the same truncation scheme, and from these expressions the essential
difference becomes apparent. It is shown that MEST and CCA are
asymptotically equivalent, which implies that they have the same
asymptotic normal distribution. However, simulations indicate that
AR-modeling and stochastically balanced truncation has a better
performance than CCA in practice.

Calendar of seminars

*Last update: May 9, 2001 by
Anders Forsgren,
anders.forsgren@math.kth.se.
*