Section: New Results

Analysis and supervision of bioprocesses

Models development and identification

Participants : Fabien Campillo, Amine Charfi, Yessmine Daoud, Jérôme Harmand, Sonia Hassam, Guilherme Pimentel, Alain Rapaport.

Membrane bioreactors combine a filtration process (with a membrane) and a suspended growth rate bioreactor. This recent technology present many advantages compared to conventional ones, but is more sophisticated and requires refined control because of the fouling process. We have proposed new modeling approaches of such bioreactors, where a fouling mechanism is explicitly described:

• Membrane bioreactors allow a perfect separation of biomass and treated waters. However, membranes are subject to clogging - also called fouling - by large organic molecules and solids (biomass, suspended solids, etc...). This phenomenon represents the main problem that limits the development of membranes bioreactors. It is well documented but very studies tried to formalize it under the form of dynamical model, even more under anaerobic conditions. This is what we did in the framework of Amine Charfi's PhD [11] : he contributed to a better understanding of membrane fouling dynamics and proposed a number of strategies to avoid irreversible clogging [24] , [25] .

• In the scope of the PhD supervision of Guilherme Pimentel, we have proposed a simple three time scales model in view of the control of the cake formation [13] , [37] . This model has been validated on real data from a pilot plant at Univ. Mons (Belgium).

As already mentioned in 6.2.1 , anaerobic bioreactors are able to produce valuable energy. However, they are subject to destabilization in case of organic overload. It is thus necessary to develop appropriate models dedicated to the synthesis of stabilizing control feedbacks. Two strategies are followed to obtain such models. Either simple models are proposed from the knowledge we have about the process, either reduced models are obtained from more complex ones. Yessmine Daoud just began her PhD thesis following the first strategy: more precisely, we study simple inhibition models of the anaerobic digestion and tries to establish simple equivalence between these models and the well known ADM1 model. Sonia Hassam, in her PhD, works within the framework of the second strategy to propose simple models obtained in reducing complex ones like the ADM1.

As already proposed last year, we have set a methodology to identify from data observed on a chemostat plant a multi-specific model that suits better than a mono-specific one, when data from molecular biology are available. In [35] , we combine molecular fingerprints obtained at some discrete times (such as the ones provides by the DGGE or SSCP techniques) with on-line macroscopic measurements. In a similar spirit, within the framework of a collaboration with the LOMIC at Banyuls, we have analysed molecular data obtained via SSCP technique to monitor the structure of microbial communities. It was shown that aerial transport of bacteria from desert into the sea influenced its bacterial diversity [32] , [31] .

As far as stochastic approaches are concerned, the thesis of Mohsen Chebbi aims at developing stochastic models of membrane bioreactors following the approach proposed in [57] . A mathematical and simulation framework has been established, as well as the bases of vectorial simulation techniques in Matlab . Developments of Monte Carlo techniques for the identification of bioprocesses are investigated in the thesis of Oussama Hadj-Abdelkader, improving classical particle filtering approaches (sequential Monte Carlo) by integrating MCMC (Monte Carlo Markov Chain) procedures. A software approach has been adopted in C++ .

Synthesis of control laws

Participants : Térence Bayen, Walid Bouhafs, Céline Casenave, Amel Ghouali, Jérôme Harmand, Zeyneb Khedim, Claude Lobry, Alain Rapaport, Victor Riqueleme, Matthieu Sebbah.

We investigate two kinds of bioprocesses to be controlled, arising in industrial biotechnology (digesters, waste-water purification...) or in the bioremediation of natural environments (lakes, landfill...).

Industrial biotechnology

As it is often the case in industry, we distinguish two kinds of process operating: continuous processes, for which the volumes of the bio-reactors are constant, and fed-batch processes, for which the filling rate is the control.

We tackle several optimal control problems related to the maximization of productivity of continuous bioprocesses:

• As far as anaerobic digestion is concerned, we have considered an optimal control problem for the chemostat model with substrate inhibition. The originality of this problem relies on the fact that the quantity to be maximized is not simply a state of the model (the substrate or the biomass) but the volume of biogas produced (which is a nonlinear function of the state) within a given time interval. Optimal strategies have been proposed for a class of initial conditions of the system (PhD thesis of Amel Ghouali). Other optimal control problems are studies by Walid Bouhafs to establish the optimality of controls initially proposed by Djalel Mazouni, a former PhD candidate supervised by the team. Using a tricky projection of the problem into another state space, he has shown that the problem of optimizing the degradation of two different substrates by two antagonist bacterial communities could be solved in a very general sense as the minimization of a specific functional for a very large class of nonlinear systems [19] .

• The work [15] arises in the context of selection of species (widely used in agriculture and biotechnology in order to improve productivity). For microorganisms, the selection process can be based on genetic tools. Our methodology in [15] is to drive the competition between species in a chemostat. We consider a two species chemostat model with one limiting substrate, and our aim is to optimize the selection of the species of interest. Thanks to the Pontryagin Maximum Principle, we introduce a singular feeding strategy which allows to reach the target, and we prove that the feedback control provided by this strategy is optimal. The optimal synthesis of the problem in presence of more than two species will be investigated in a future work.

• In [38] , we study the problem of minimal time for a chemostat system with one limiting substrate and one species. Given a target point, the problem consists in finding an optimal feeding strategy steering any initial condition of the system to this target. This is typically of interest whenever the input substrate concentration changes yielding in a new steady state. We consider the case where the growth rate function is of Haldane type implying the existence of a singular arc that is non-necessary admissible everywhere. We provide an optimal synthesis of the problem using tools from optimal control theory.

• The work [16] studies the coupling of a culture of micro-algae limited by light and an anaerobic digester in a two-tank bioreactor (the model combines a periodic day-night light for the culture of micro-algae and a classical chemostat model for the digester). We first prove the existence and attraction of periodic solutions of this problem for a one day period. Then, we study the optimal control problem of optimizing the production of methane in the digester during a certain time frame, the control on the system being the dilution rate (the input flow of micro-algae in the digester). We also investigate the dependence of the optimal cost with respect to the volume ratio of the two tanks.

In collaboration with researchers of the unit SPO (Sciences For Oenology), we have proposed a control law of a multi-stage continuous fermentor (MSCF) designed for the study of the wine fermentation, that has been implemented on the experimental process [41] . We have also finalized the controllability analysis and minimal time feedback synthesis of models of cascade of continuous bioreactors under input constraints [18] , [40] (that is also related to the control of MSCF).

As far as fed-batch processes are concerned, an extension of former results of the team about the minimal time control of fed-batch processes with impulse controls is presented in [34] ,

The paper [17] is devoted to the study of the minimal time problem of a fed-batch reactor, under the presence of a saturation point on the singular locus (this typically occurs whenever the growth rate function is of Haldane type and when typically the maximum input flow rate is not high enough to maintain the substrate concentration constant). This brings non-intuitive issues for the optimal synthesis (existence of switching curve and point of prior saturation).

Bioremediation of natural environments

In the scope of the associated team with Chile and the supervision of the postdoctoral stay of Matthieu Sebbah in Chile, we have addressed a new model of landfill remediation when controlling the leachate recirculation [53] . We have applied the same methodological approach than the one for the work [18] , [40] mentioned in the previous section, which consists in characterizing first the sub-domains for which the target can be optimally reached with a constant extreme control (no recirculation or maximal speed of recirculation), and further the nature of optimal commutations outside these sets. This analysis provides information for the practitioners on the benefit to implement sensors and real-time controllers.

Also in the scope of the associated team with Chile (see 8.3.2.1 ) and the co-supervision of the PhD of Victor Riquelme, we have carried on the study of optimal syntheses for the minimal time treatment of natural water reservoirs (such as lakes) [52] . We have proved that the minimal time strategy consists in a most-rapid approach to homogeneous concentrations, even though the optimal control problem is non convex. Moreover, we have shown that a large diffusion increases the treatment time when the resource is everywhere highly polluted , while it can at the opposite be beneficial when only part of the resource is polluted. This feature should serve the practitioners in the choice of pumps positioning in a originally clean water resource that is suddenly affected by a local pollution. This work is in connection with the INRA/Inria patent [47] that has been deposited jointly with LEMON Team.