# Algorithms for identifying guaranteed stable and passive models from noisy data

Scriptie/masterproef: Doctoral Thesis

### Uittreksel

The purpose of this thesis is to show an approach to estimate constrained models. The
constraints where we look at are the stability and passivity constraints. Here we propose a two
step approach. In the first step a (high) degree model without any constraints will be estimated
that passes the validation tests (analysis cost function, whiteness weighted residues,...). This
step suppresses in an optimal way the noise without introducing systematic errors. In the
second step the constraint will be added: The (high) order model will be approximated by a
constrained one. For this the weighted difference between the unconstrained model and the
constrained model is minimized. The big advantage of the two step procedure is that it
provides models with uncertainty and bias error bounds, which is not the case when the
constraint is imposed during the noise removal step.
Imposing the stability or passivity constraint in the second step with application of a user
defined weighting is the crucial step and is more difficult than it appears. So starting from a
model that passed step one, the following steps have to be carried out: The transfer function
is already estimated with an optimal noise removal criterion, however the model is
unstable or non passive. To constrain the transfer function, we propose an iterative algorithm
that minimizes the following cost function in a user specified frequency band:
with the constrained version of , the transfer function parameters, a user
defined weighting function, and the number of frequencies. The iterative algorithm needs
stable or passive initial values . This can be created by some simple or advanced
techniques. The iterative algorithm will create from these initial values a stable or passive
model that performs at least as good as the initial values. The basic idea is to decrease the cost
function by leaving some freedom to the gain and the positions of the zeros and the poles of
the transfer function.

This method is applicable for single input single output continuous-time and discrete-time
systems and multiple input multiple output continuous-time and discrete-time systems.
This technique is illustrated on several measurement examples and compared with other
stabilizing and passivity enforcement techniques.
Datum Prijs 21 jan 2008 English Rik Pintelon (Promotor), Gert Desmet (Jury), Jean Vereecken (Jury), Patrick Guillaume (Jury), Gerd Vandersteen (Jury), Tom Dhaene (Jury) & M. Olivier (Jury)

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