Section: New Results

Identification and control

Reconstruction methods of kinetics functions

Participant : Alain Rapaport.

A collaboration with Sisyphe Inria project-team has led to the development of a new identification method of the kinetics function in the chemostat model, without any a priori on the monotonicity of the function (thus allowing the consideration of bio-processes that are unstable in open loop) [29] . An extension of this method, that is based on singular perturbations, has been proposed for the extremum seeking problem with only two times scale (instead of three for the usual extremum seeking techniques [75] ) [50] .

Parameter estimation and particle filtering

Participants : Amine Boutoub, Fabien Campillo, Jérôme Harmand, Marc Joannides.

We consider a stochastic logistic growth model involving both birth and death rates in the drift and diffusion coefficients for which extinction eventually occurs almost surely. We then use the numerical integration of the Fokker-Planck equation presented in Section 6.2.1 to build a likelihood function for the unknown model parameters, when discretely sampled data is available. The existing estimation methods need adaptation in order to deal with the extinction problem. We propose such adaptations, based on the particular form of the Fokker-Planck equation, and we evaluate their performances with numerical simulations [64] .

We develop particle approximation methods for the nonlinear filtering and parameter estimation with the help of the chemostat model [70] .

Functional assignments methods

Participants : Jérôme Harmand, Alain Rapaport.

Following the philosophy of the work that was achieved within the framework of the former PhD thesis of M. Dumont [3] , we have applied part of the proposed methodology for a better understanding of the dynamics of specific species of the anaerobic digestion [30] , with Chilean collaborators (see Sections 8.4.1.1 and 8.4.2 ).

Using a combinatorial approach, we have also developed together with UMR Eco & Sols (Cirad, Inra, IRD, SupAgro – Montpellier) a new method to study the role of the interactions within bacterial species on the performance of an ecosystem. More precisely, based on the specific characteristics of the species of a community and the way they interact between each other, we propose a method to predict the behavior of the ecosystem with respect to its biodiversity [34] , [25] .

Stabilizing strategies for bioprocesses

Participants : Céline Casenave, Jérôme Harmand, Guilherme Pimentel, Alain Rapaport.

We have carrying on developments of stabilizing strategies for bio-processes, with specific characteristics:

• In [48] , it has been shown how the buffered configuration of two chemostat models studied in Section 6.1.3 provides an efficient way to stabilize bioprocesses with inhibition. In the same spirit, its has been shown how the consideration of a “passive” buffer (i.e. without biological reaction) can play the role of a delay and enlarge the attraction basin of stable equilibria [49] .

• If often happens in bio-processes that measurements are delayed. In [46] , a new stabilizing strategy have been proposed to robustly cope with such delays for single chemostats with inhibition.

• For the stabilization of a series of reactors with multiple inputs, a control strategy based on a linearizing control law coupled with a state observer and an anti windup component has been proposed [66] , [67] , in view of its application in wine fermentation processes. The originality and difficulty of this multi-inputs problem are due to the inputs constraint that imposes that the manipulated dilution rate of each tank has to be less or equal than the one of the previous tank.

Optimal syntheses for bioprocesses control

Participants : Térence Bayen, Amel Ghouali, Jérôme Harmand, Claude Lobry, Alain Rapaport, Tewfik Sari.

We have continued our activities related to the development of optimal control laws for the optimization of bio-processes, notably in taking advantage of the presence of T. Bayen in the team in 2013. Three kinds of results, depending on the kind of processes under interest, were obtained.

a. Control of batch processes. Sequencing Batch bioReactors can be used to efficiently treat water containing both carbonaceous and nitrogenous pollutants. In such a case, an efficient control that can be used is the oxygen concentration. In such systems, oxic and anoxic bacterial are in competition for certain substances. For a simplified version of this complex situation, we have investigated the optimal strategies in order to minimize the energy to be introduced into the system under performance constraints. The originality of the approach lies in the fact that the original problem is transformed into a very general form. Thus, the optimal control problem is formulated and solved for a very general class of systems of ecological relevance [16] .

b. Control of fed-batch processes. References [13] , [36] , [60] are devoted to the study of a bioreactor which is operated in fed-batch mode. We aim at finding an optimal control in feedback form (i.e. depending of the state) that steers the system in a minimal amount of time to a target (which typically has several interests in wastewater treatment). Finding an optimal control in feedback form is crucial from a practical point of view. In [13] , previous works on the subject are extended to the case where the growth function depends on an additional product of the reaction. In the references [36] , [60] , we provide an optimal control in feedback form whenever mortality and recycling rates are considered, and in the case where the maximum dilution rate is not large enough to compete the growth of the species (in the latter case, this implies that the singular arc is non-necessary controllable implying difficulties in determining optimal controls). References [58] , [61] are devoted to the study of optimal control problem governed by a chemostat-type model. In [58] , an optimal feedback control law is provided in order to optimize the selection of a species in a chemostat model with one limiting substrate and two species. This brings an interesting issue in order to extend this result to the case where the number of species is larger than 3.

c. Coupled dynamics. References [59] , [60] give the results of the study of an optimal control problem of a system coupling a culture of micro-algae limited by light and an anaerobic digester. The mathematical model for the dynamics of the reactors takes into account a periodic day-night model of the light in the culture of micro-algae and a chemostat model for the digester. Our aim is to optimize the production of methane in the digester during a certain number of days with respect to the dilution rate. In [59] , some preliminary results on this problem are given for an optimal control problem governed by a one-dimensional Kolmogorov equation. In [60] , the full system is analyzed by combining direct methods and indirect methods based on Pontryagin's Principle. In [62] , we provide a complete characterization of optimal controls for a minimal time control problem where the system describes a two tanks gradostat model under a cascade inputs constraint. This model allows to create a gradient of resources that is expected to be more realistic to mimic real environment for studying micro-organisms growth.