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

Éric Busvelle; Jean-Paul Gauthier

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 9, page 509-551
- ISSN: 1292-8119

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topBusvelle, Éric, and Gauthier, Jean-Paul. "On determining unknown functions in differential systems, with an application to biological reactors.." ESAIM: Control, Optimisation and Calculus of Variations 9 (2010): 509-551. <http://eudml.org/doc/90709>.

@article{Busvelle2010,

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.
},

author = {Busvelle, Éric, Gauthier, Jean-Paul},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Nonlinear systems; observability; identifiability; identification.; identification},

language = {eng},

month = {3},

pages = {509-551},

publisher = {EDP Sciences},

title = {On determining unknown functions in differential systems, with an application to biological reactors.},

url = {http://eudml.org/doc/90709},

volume = {9},

year = {2010},

}

TY - JOUR

AU - Busvelle, Éric

AU - Gauthier, Jean-Paul

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 9

SP - 509

EP - 551

AB -
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.

LA - eng

KW - Nonlinear systems; observability; identifiability; identification.; identification

UR - http://eudml.org/doc/90709

ER -

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