Section: New Results
Advanced automatic control for SBRs
Within the framework of former European project EOLI, a SBR pilot has been designed at INRA Narbonne, treating both the organic carbon and nitrogen (PhD thesis of Djalel Mazouni). To do so, two different operating conditions are needed: one aerated period (also called the "aerobic phase") and one without aeration (also called the "anoxic phase"). Depending on the initial concentrations of the different components (biomasses and substrates), the objective is to find the best switching instants (from the aerobic phase to the anoxic one or conversely from the anoxic phase to the aerobic one) such that the total reaction time is minimized. Because several components and biological reactions are simultaneously present in the different reaction phases, the optimal solution is not obvious and has required a rigorous mathematical analysis. A complete attainability study and the determination of the optimal switchings have been achieved, with the help of Maximum Principle. A synthesis of the work realized within the PhD thesis of D. Mazouni is to be published [Oops!] while, in collaboration with our Chilean colleagues at the CMM (cf. Section 8.1.2 ), we try to extend these results in order to take into account for the oxygen, giving to our results much more realism [Oops!] .
Observers for determining the major species
The use of molecular fingerprinting techniques is about to induce major changes in the modeling of bio-processes. Indeed, under the condition that the ecosystem under interest is dominated by a limited number of species (for instance, it is typically the case in the nitrification process), it becomes possible to monitor the relative abundances of the major species. Viewed as real new sensors, the use of these data pose new problems to modelers : they can be used either as new inputs into models or as data useful to design and validate new ecological models. Among interesting problems for dynamical systems theory, one finds the need for new observers that are needed to recover a number of unmeasured variables of these new models of complex systems. These observers are tunable and insensitive to model uncertainty while allowing us to classify - in terms of function - the major species which can be observed through molecular techniques. These molecular tools together with these new observers give insights to ecological problems of interest for wastewater treatment technology.
Physical bases of density-dependence in the chemostat
The flocculation process is of major importance in wastewater treatment plants. On one hand, the presence of flocks limits the access of the biomass to the substrate. On the other hand, flock formation permits the separation of the biomass from the effluent by clarification. We proposed an effective way to include flocculation in existing models [Oops!] , [Oops!] , and showed that under certain conditions, this leads to a density-dependent growth function. This establishes the link between the limited access to the substrate inside the flocks, and the growth characteristics of the biomass on the level of the bioreactor.
Modeling and inferring of ecological and environmental dynamics
As mentioned in Section 4.6 , most of ecological and environmental data could be treated in a batch way. We could therefore make use of iterative inference techniques (non-sequential), as the Markov Chain Monte Carlo (MCMC), as well as non-iterative methods (sequential), like sequential Monte Carlo (SMC). In the last case, the data are scanned only once, while in the first one, they are scanned as many times as needed for convergence. Thus, in the MCMC methods, the issues of convergence and convergence criterion are crucial.
Both MCMC and SMC methods can exploit the hierarchical structure of the distribution law of the underlying dynamic system; this structure is derived from the Markovian modeling approach (see Section 4.6 ).
We have investigated two different possibilities:
The non-sequential approach - We propose to run many interacting MCMC's in parallel. Each chain independently proposes candidates to all others chains: this induces a mixing effect that improves the convergence properties of the global MCMC [Oops!] .
The sequential approach - SMC scans sequentially the data through a prediction/correction updating process. The prediction step can be improved considerably by using a few iterations of MCMC, see [Oops!] for details.
This work is part of an INRIA Cooperative Research Initiative (see Section 8.3 ), of the SARIMA program (see Section 8.1.1 ) and is done in collaboration with CIRAD Montpellier (“Dynamics of Natural Forests” unit) and the University of Fianarantsoa in Madagascar.