Model Fitting in Frequency Domain Imposing Stability of the Modell

Abstract

A signal processing method works in time or in frequency domain. Most of the methods
can be formulated equivalently in both domains. In the thesis we focus only on frequency
domain algorithms. Parts of two signal processing topics are investigated: approximation
and identifcation problems. Still, there are similarities they have very di?erent tools and
goals. Topics of the thesis are stable approximation and an identifcation method.
Two problems may appear not independently, but consecutive steps in an identi?ca-
tion problem. The ?rst step is estimation of model parameters from noisy measurement
data. Without imposing stability, the result may determine an unstable model. Then the
unstable model must be approximated by a stable one.
In identifcation algorithms in order to increase to numerical stability the parameter
space is usually transformed. If any constraint of the parameter space is imposed then
they must be transformed, too. In the case of so-called total least squares estimator a
theoretical gap is ?lled by introducing how the constraints can be transformed. Variations
of the method are also analysed.
In practical realizations unstable models are usually undesirable. To overcome this
problem the unstable transfer function must be approximated by a stable one. Approxi-
mation theory has rich literature, nevertheless there are lots of open questions. A possible
solution of the stable approximation in the case of linear time invariant systems is adding
delay to the target function. Its main disadvantage is that it can be used only in open-
loop con?gurations. But a strong bene?t is that the resulted parameter vector of the
minimisation algorithm may determine a stable model without imposing any constraint.
This dissertation presents a theorem which guarantees the existence of a delay added
to a target function such that the minimum of the cost function is stable. Moreover, a
practical approach which is capable to ?nd the appropriate stable minimum is proposed
and analysed. The thesis also compares the results with those found in the literature.

Date of Award	2009
Original language	English
Supervisor	Rik Pintelon (Promotor), Patrick Guillaume (Jury), Steve Vanlanduit (Jury), Joannes Schoukens (Jury), Joos Vandewalle (Jury), Barnabás Garay (Jury) & István Kollár (Co-promotor)