Sat Jun 19 15:02:00 CEST 2010

Subspace Identification

Subspace Identification[1] refers to system identification of linear
MIMO dynamical models using linear algebra techniques (QR + SVD).  The
basic idea revolves around a two-step procedure:

  1. Given input/output data, construct a _Kalman state sequence_ by
     projection onto a limited subspace.

  2. Obtain state matrices from this using linear least squares.

The fact that 1. is at all possible provides the main leverage.

( The following is generalized from the HSVD method.  I'm not sure if
it completely applies to the stochastic version in [1], but I have the
impression it does.  I need a more intuitive grasp of the Kalman
filter first. )

Such SVD-based methods are good in _practice_ but the characterization
of the approximation is theoretically very far removed from what can
be obtained by maximum likelyhood methods using a more direct

Subspace methods have been used to categorize the "unreasonable
effectiveness" of mathematics.  I'm definitely no expert, but it seems
to me that they fall more into the class of convenient, accedental
hacks.  (The link between the estimation error and an ML approach is
not clear : probably there is none due to radical re-interpretation of
the problem?)

[1] ftp://ftp.esat.kuleuven.ac.be/pub/SISTA/nackaerts/other/alln.ps.gz