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

There is a french presentation of this paper available at PPF-STIC

There is also an english presentation of this paper containing also new results and application to a fluid catalytic cracker available at BanachC. This conference has been presented at Banach Center in June 2003.

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 (1):

If the number of observations is three or more, then, systems are generically identifiable.

If the number of observations is 1 or 2, then the situation is reversed. Identifiability is not at all generic. In that case, we add a more tractable infinitesimal condition, to define the ''infinitesimal identifiability'' property.

This property is so rigid, that we can almost characterize it (we can characterize it by geometric properties, on an open-dense subset of the product of the state space X by the set of values of φ). This, surprisingly, leads to a non trivial classification, and to certain corresponding ''identifiability normal forms''.

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.

As an illustration of what may happen in controlled cases, we consider the equations of a biological reactor, (2), (3), 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''. Our result, in the case where the observed variable is the concentration of the substrate, is as follows: the system is identifiable along a trajectory, if and only if this trajectory visits the same value of the output at least twice. The simulation result shows actual (Haldane type) growth function in black and identified function along the time (red curve). Initial guess was a Monod growth function.

We propose also a practical methodology for identification, which shows very reasonable performances.

(1) J.-P. Gauthier, I. Kupka, Deterministic Observation Theory and Applications, Cambridge University Press, 2001

(2) G. Bastin, D. Dochain, Adaptive Control of Bioreactors, Elsevier, 1990

(3) J.-P. Gauthier, H. Hammouri, S. Othman, A simple observer for nonlinear systems, Application to bioreactors, IEEE Trans. On Aut. Control, 37, p875-880, 1992