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On determining unknown functions in differential systems, with an applicationto biological reactors.

Published online by Cambridge University Press:  15 September 2003

Éric Busvelle  and
Jean-Paul Gauthier
Éric Busvelle
Affiliation:
Laboratoire d'Analyse Appliquée et Optimisation, Département de Mathématiques, Université de Bourgogne, bâtiment Mirande, BP. 47870, 21078 Dijon Cedex, France; busvelle@u-bourgogne.fr. gauthier@u-bourgogne.fr.
Jean-Paul Gauthier
Affiliation:
Laboratoire d'Analyse Appliquée et Optimisation, Département de Mathématiques, Université de Bourgogne, bâtiment Mirande, BP. 47870, 21078 Dijon Cedex, France; busvelle@u-bourgogne.fr. gauthier@u-bourgogne.fr.
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Abstract

In this paper, we consider general nonlinear systems with observations, containing a (single) unknown function φ. We study the possibility to learn about this unknown function via the observations: if it is possible to determine the [values of the] unknown function from any experiment [on the set of states visited during the experiment], and for any arbitrary input function, on any time interval, we say that the system is “identifiable”. For systems without controls, we give a more or less complete picture of what happens for this identifiability property. This picture is very similar to the picture of the “observation theory” in [7]: Contrarily to the case of the observability property, in order to identify in practice, there is in general no hope to do something better than using “approximate differentiators”, as show very elementary examples. However, a practical methodology is proposed in some cases. It shows very reasonable performances.
As an illustration of what may happen in controlled cases, we consider the equations of a biological reactor, [2,4], in which a population is fed by some substrate. The model heavily depends on a “growth function”, expressing the way the population grows in presence of the substrate. The problem is to identify this “growth function”. We give several identifiability results, and identification methods, adapted to this problem.

Keywords

Nonlinear systemsobservabilityidentifiabilityidentification.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 509 - 551
Copyright
© EDP Sciences, SMAI, 2003

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