Researchers: Anders Lindquist, Anders Dahlén and Jorge Marí, in cooperation with Giorgio Picci (University of Padova).
Sponsors: The Swedish Research Council for Engineering Sciences (TFR) and the Göran Gustafsson Foundation.
In this project we analyze a class of identification algorithms based on canonical correlation analysis in the light of recent results on stochastic systems theory. In principle these so called ``subspace methods" can be described as covariance estimation followed by stochastic realization. The method offers the major advantage of converting the nonlinear parameter estimation phase in traditional ARMA models identification into the solution of a Riccati equation but introduces at the same time some nontrivial mathematical problems related to positivity. The reason for this is that an essential part of the problem is equivalent to the well-known rational covariance extension problem. Therefore the usual deterministic arguments based on factorization of a Hankel matrix are not valid, something that is habitually overlooked in the literature.
We have demonstrated that there is no guarantee that several popular subspace identification procedures will not fail to produce a positive extension, unless some rather stringent assumptions are made which, in general, are not explicitly stated. These assumptions are equivalent to the condition that the positive and algebraic degrees of a certain estimated covariance sequence coincide. We have previously described how to generate data with the property that this condition is not satisfied. Using this data we have shown through simulations that several subspace identification algorithms exhibit massive failure.
Therefore we have studied alternative identification strategies. In
[A22] we consider a three-step procedure for identification of
time series, based on covariance extension and model reduction, and we
present a complete analysis of its statistical convergence properties.
A partial covariance sequence is estimated from statistical data. Then
a high-order maximum-entropy model is determined, which is finally
approximated by a lower-order model by stochastically balanced model
truncation. Such procedures have been studied before, in various
combinations, but an overall convergence analysis comprising all three
steps has been lacking. Supposing the data is generated from a true
finite-dimensional system which is minimum phase, it is shown that the
transfer function of the estimated system tends in 
to
the true transfer function as the data length tends to infinity, if
the covariance extension and the model reduction is done properly. The
proposed identification procedure, and some variations of it, are
evaluated by simulations. In particular, these simulations show that
our procedure compares favorably to the subspace methods.
In [R4] we consider the multivariate case and compare it with the CCA subspace method. It is shown that the two methods are asymptotically equivalent in the sense that the difference of the two estimates multiplied with the square root of the sample size converges to zero. This means that the two procedures have the same asymptotic normal distribution.
Research 1999/2000
Some problems in navigation and motion control of nonlinear systems
The rational covariance extension problem, Cepstral geometry and global analysis of shaping filters