Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-03T04:49:28.529Z Has data issue: false hasContentIssue false
  • English
  • Français

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; [email protected]. [email protected].
Jean-Paul Gauthier
Affiliation:
Laboratoire d'Analyse Appliquée et Optimisation, Département de Mathématiques, Université de Bourgogne, bâtiment Mirande, BP. 47870, 21078 Dijon Cedex, France; [email protected]. [email protected].
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R. Abraham and J. Robbin, Transversal mappings and flows. W.A. Benjamin Inc. (1967).
G. Bastin and D. Dochain, Adaptive control of bioreactors. Elsevier (1990).
E. Busvelle and J.-P. Gauthier, High-Gain and Non High-Gain Observers for nonlinear systems, edited by Anzaldo-Meneses, Bonnard, Gauthier and Monroy-Perez. World Scientific, Contemp. Trends Nonlinear Geom. Control Theory (2002) 257-286.
Gauthier, J.-P., Hammouri, H. and Othman, S., A simple observer for nonlinear systems. Applications to Bioreactors. IEEE Trans. Automat. Control 37 (1992) 875-880. CrossRef
Gauthier, J.-P. and Kupka, I., Observability and observers for nonlinear systems. SIAM J. Control 32 (1994) 975-994. CrossRef
Gauthier, J.-P. and Kupka, I., Observability for systems with more outputs than inputs. Math. Z. 223 (1996) 47-78. CrossRef
J.-P. Gauthier and I. Kupka, Deterministic Observation Theory and Applications. Cambridge University Press (2001).
M. Goresky and R. Mc Pherson, Stratified Morse Theory. Springer Verlag (1988).
Hardt, R., Stratification of real analytic mappings and images. Invent. Math. 28 (1975) 193-208. CrossRef
H. Hironaka, Subanalytic Sets, in Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Y. Akizuki. Kinokuniya, Tokyo (1973) 453-493.
M.W. Hirsch, Differential Topology. Springer-Verlag, Grad. Texts in Math. (1976).
Hirschorn, R.M., Invertibility of control systems on Lie Groups. SIAM J. Control Optim. 15 (1977) 1034-1049. CrossRef
Hirschorn, R.M., Invertibility of Nonlinear Control Systems. SIAM J. Control Optim. 17 (1979) 289-297. CrossRef
Jouan, P. and Gauthier, J.-P., Finite singularities of nonlinear systems. Output stabilization, observability and observers. J. Dynam. Control Systems 2 (1996) 255-288. CrossRef
W. Respondek, Right and Left Invertibility of Nonlinear Control Systems, edited by H.J. Sussmann. Marcel Dekker, New-York, Nonlinear Control. Optim. Control (1990) 133-176.
M. Shiota, Geometry of Subanalytic and Semi-Algebraic Sets. Birkhauser, P.M. 150 (1997).
H.J. Sussmann, Some Optimal Control Applications of Real-Analytic Stratifications and Desingularization, in Singularities Symposium Lojiasiewicz 70, Vol. 43. Banach Center Publications (1998).