Subspace identification is known to be an excellent tool for the
creation of linear models based on available measurements. Although
Subspace algorithms are much more robust than classical predictor
error methods, which tend to get stuck in local minima, subspace
algorithms are not necessarily free of conditioning problems.
One
such problem is the fact that if certain row-spaces of the
Hankel-matrices used in subspace identification are almost parallel,
all subspace implementations that use oblique projections might
result in very bad estimates. We try to obtain some theoretical
insight into how bad things can actually become.
An even
bigger problem is the so-called positive-realness problem. Subspace
algorithms can not guarantee that a positive-real or passive model is
obtained even if the original system was passive. Also in this
category falls the fact that subspace algorithms can not guarantee
stable models, even if the original system is known to be stable. A
paper suggesting a solution to the positive realness problem can be
found here.
All
in all, you would like to have a good guess of the variance on your
estimates once you obtain your subspace results. We are working on
some practical algorithms to obtain them by using matrix function
differentiation of all the QR and SVD steps in a subspace algorithm
to obtain a first order perturbation analysis.
Researcher(s): Ivan Goethals