Division of Optimization and Systems Theory
Department of Mathematics
The thesis consists of three closely related papers.
In the first paper it is briefly explained why subspace methods for time-series identification may fail for theoretical reasons. These algorithms are implicitly based on the assumptions that the data are really generated by a linear stochastic system, and that one has an upper bound for the order of the model. Assumptions like this are in general never fulfilled for generic data. The main contribution of the paper is a description of how to construct statistical data, such that certain stochastic subspace identification algorithms exhibit massive failure when applying them to this data. The failures occurs when a parameter, closely related to the choice of order of the identified model, is chosen to low by the user of the identification algorithms. This is verified through simulations of some popular subspace methods. Consequently some care has to be exercised when using these stochastic subspace identification methods.
In the second paper an alternative identification procedure which overcomes these difficulties is presented. The procedure is based on identification of a high-order maximum entropy model (AR model) followed by model reduction. It is shown that the transfer function of the estimated system tends in a "worst case" measure to the true transfer function if the data is generated from a finite-dimensional linear system which is minimum phase. The high-order model will inherit the poles of the true system which lie outside a disc in the complex plane containing all of the zeros of the true system. The rest of the poles of the high-order model will cluster in the interior of this disc. Because of this, we can heuristically justify why stochastically balanced truncation will work better than deterministically balanced truncation as a model reduction procedure. Some simulation results are shown verifying this model reduction choice.
In fact the simulations show convergence to the Cramér-Rao bounds of the obtained variances, as the data length increases. Finally, it is shown that the special structure of the AR-model allows to perform stochastically balancing by linear algebraic methods, which means that the overall procedure will just require linear algebra.
The final paper shows more extensive simulation results of the proposed method in the second paper. This part of the thesis will bring some intuition of how to use the procedure. Some different implementation alternatives of the procedure are tested. The conclusion is that Burg estimation and stochastically balanced truncation gives the best result. Finally, it is shown that Burg estimation and stochastically balanced truncation is more efficient than a tested subspace method.