
WeB2 Invited
Session 
C102

What's up within
the Theory of the Chemostat? 


Chair: Rapaport,
Alain 
INRA 
CoChair: Harmand,
Jérome 
INRA 
Organizer: Harmand,
Jérome 
INRA 

14:3014:50, Paper
WeB2.1 

Extremum Seeking Via Continuation
Techniques for Optimizing Biogas Production in the
Chemostat (I) 
Rapaport, Alain 
INRA 
Sieber, Jan 
Univ. of Exeter 
Rodrigues, Serafim 
Plymouth Univ. 
Desroches, Mathieu 
INRIA ParisRocquencourt Res. Centre 
Keywords:
Biological
and Biomedical Systems, Stabilization,
Parameter
Estimation
Abstract: We consider the chemostat
model with the substrate concentration as the single
measurement. We propose a control strategy that drives
the system at a steady state maximizing the gas
production without the knowledge of the specific
growth rate. Our approach separates the extremum
seeking problem from the feedback control problem such
that each of the two subproblems can be solved with
relatively simple algorithms. We are then free to
choose any numerical optimization algorithm. We give a
demonstration for two choices: one is based on
slowfast dynamics and numerical continuation, the
other is a combination of goldensection and Newton
iteration. The method copes with nonmonotonic growth
functions.


14:5015:10, Paper
WeB2.2 

FedBatch Bioreactor with Mortality Rate
(I) 
Bayen, Terence 
Univ. of Montpellier 2 
Mairet, Francis 
Inria 
Mazade, Marc 
Univ. Montpellier II 
Keywords:
Optimal
Control, Time
Optimal Control, Chemical
Process
Abstract: We address the problem of
finding an optimal feedback control for feeding a
fedbatch bioreactor with one species and one
substrate, from a given initial condition to a given
target value in a minimal amount of time. Mortality
rate for the biomass and nutrient recycling are taken
into account in this work. The optimal synthesis
(optimal feeding strategy) has been obtained by Moreno
in 1999 when both mortality and recycling are
considered negligible, in the case of Monod and
Haldane growth function. Our objective is to study the
effect of mortality and recycling on the optimal
synthesis. We provide an optimal synthesis of the
problem using Pontryagin maximum principle, which
extends the result of Moreno in the impulsive
framework with mortality and recycling effect.


15:1015:30, Paper
WeB2.3 

ConstantYield Control of the Chemostat
(I) 
Savoglidis, Georgios 
Univ. of Patras 
Kravaris, Costas 
Univ. of Patras 
Keywords:
Chemical
Process, Biological
and Biomedical Systems, Lyapunov
Methods
Abstract: The present work addresses
the problem of chemostat stabilization around an
optimal steady state, in the sense of enlargement of
its stability region. The need for stabilization
becomes imperative under conditions where the growth
of biomass is subject to substrate inhibition, and the
primary concern is to prevent washout of the biomass
in the presence of disturbances. Inspired by the
empirical concept of constantyield control, a
nonlinear state feedback control law is derived, and
the stability basin of resulting closedloop system is
estimated using a Lyapunov function approach. Our
analysis extends previous results in the sense that it
accounts for biomass decay and endogenous metabolism
and, moreover, it covers the case where the product is
soluble in the effluent stream.


15:3015:50, Paper
WeB2.4 

Global Stabilization of the Chemostat with
Delayed and Sampled Measurements and Control (I) 
Mazenc, Frederic 
INRIACNRSSupelec, 
Harmand, Jérome 
INRA 
Mounier, Hugues 
Lab. des Signaux et Systèmes, CNRS
SUPELECUniversité Pari 
Keywords:
Stabilization,
Control
of Sampled Data Systems, Biological
and Biomedical Systems
Abstract: The classical model of the
chemostat with one substrate, one species and a
Haldane type growth rate function is considered. The
input substrate concentration is supposed to be
constant and the dilution rate is considered as the
control. The problem of globally asymptotically
stabilizing a positive equilibrium point of this
system in the case where the measured concentrations
are delayed and piecewise constant with a piecewise
constant control is addressed. The result relies on
the introduction of a dynamic extension of a new type.


15:5016:10, Paper
WeB2.5 

Driving Species Competition in a
LightLimited Chemostat (I) 
Mairet, Francis 
Inria 
MuñozTamayo, Rafael 
INRIA 
Bernard, Olivier 
INRIA 
Keywords:
Biological
and Biomedical Systems, Stabilization
Abstract: In this paper, we tackle
the problem of microalgae selection in a continuous
photobioreactor where microalgae growth is limited by
light. We propose a closedloop control for selecting,
for a given range of light intensity, the strain with
the maximum growth rate from the microalgae
population. In particular, we are interested in
strains with high growth rate for high light
intensity, i.e.,strains with high resistance to
photoinhibition. Firstly, we recall the framework of
the lightlimited chemostat. Then, we propose a
nonlinear adaptive control which regulates the light
intensity at the bottom of the photobioreactor in
monoculture. This light is of particular interest as
it defines the winner of the competition in a
multispecies culture operated in openloop mode.
Finally, we show that the proposed controller allows
the selection of a strain of interest in the case of a
culture with n species.


16:1016:30, Paper
WeB2.6 

The Buffered Chemostat with NonMonotonic
Response Functions (I) 
Rapaport, Alain 
INRA 
Haidar, Ihab 
SupElec 
Harmand, Jérome 
INRA 
Keywords:
Biological
and Biomedical Systems, Stability,
Performance
Issues
Abstract: We show how a particular
spatial structure with a buffer globally stabilizes
the chemostat dynamics with nonmonotonic response
function, while this is not possible with single,
serial or parallel chemostats of the same total volume
and input flow. We give a characterization of the set
of such configurations that satisfy this property, as
well as the configuration that ensures the best
nutrient conversion. Furthermore, we characterize the
minimal buffer volume to be added to a single
chemostat for obtaining the global stability. These
results are illustrated with the Haldane kinetic
function.
