### Jens Ejnar Parkum

In the present thesis methods for recursive identification of time-varying
systems are considered.
In chapter 2 some of the facts, methods and assumptions forming the background
of the presentation are described. The model structure is formulated and the
connected assumptions are discussed. A review of the recursive least squares
estimation method and its properties is given. The methods which are studied
in the later chapters are obtained by modifying the least squares scheme,
but the basic principles and variable interpretations remain the same. The
chapter also contains a description of the well known adaptive controller
obtained by combining least squares estimation with minimum variance control.
Modified versions of this algorithm are applied and discussed throughout
the thesis.

Chapter 3 gives a survey of the so-called forgetting methods designed for
tracking slowly drifting system parameters. The main result of the chapter
is the formulation of a general forgetting algorithm containing the existing
methods as special cases.

The analytical properties of the general forgetting algorithm are studied in
chapter 4. The attention is focused on an ideal environment without
disturbances and with a time-invariant plant description. It is of fundamental
importance that the estimator has good performance in this simple case. The
analysis leads to guite general results which can readily be applied to any
forgetting method belonging to the general family.

In chapter 5 a method based on a principle of selective forgetting is
formulated. The theoretical properties of the method are studied and its
practical performance is examined via simulation experiments.

The last chapter, chapter 6, contains development and examination of an
adaptive controller for the nitrification process. The controller has to be
based on a strongly simplified model, since the true process dynamics is
complicated and only partly understood. However, the model captures the
basic features of the variation, and the ability of the adaptive controller to
adjust itself to the current conditions compensates for the simplification.
This is verified via simulation studies by application of an extended model
for representation of the true system. The non-linear structure of the model
and the restrictions on the control signal make it impossible directly to
apply the results known from the theory of linear systems. However, closely
related results are established by using slightly modified proving techniques.

## IMSOR Ph.D Thesis 57, 1992