The Modelling and Control of Living Resources

by Jean-Luc Gouzé and Olivier Bernard

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The aim of the research team COMORE (Modelling and Control of Renewable Resources) at INRIA Sophia-Antipolis is to apply methods from control theory (feedback control, estimation, identification, optimal control, game theory) and from the theory of dynamical systems to understand the working of the living exploited resources (renewable resources) and to manage them. COMORE is a joint project with the CNRS Villefranche-sur-Mer. The main research themes are the modelling of biological systems, the study of the properties of nonlinear dynamical biological systems, the design of robust observers, and the control of biological systems. We apply and validate our results to various fields: phytoplankton growth, fisheries and forests, wastewater treatment processes, food agro-industry.

COMORE is interested in the mathematical modelling of biological systems, more particularly of ecosystems subject to a human action (the framework is thus that of renewable resources). It is now clear that it is important to understand the working of these complex dynamical systems in order to regulate the exploitation of these resources by man. Our conceptual framework is that of Control Theory: a system, described by state variables, with inputs (action on the system), and outputs (the measurements available on the system). In our case, the system is an ecosystem, modelled by a mathematical model (generally a differential equation). Its variables are, for example, the number or the density of populations. The inputs can be the actions which one exerts on the ecosystem: eg action of the man (fishing effort, introduction of food, etc), or action of an external factor (pollution, light, etc). The outputs will be some product that one can collect from this ecosystem (harvest, capture, production of a biochemical product, etc.), or some measurements (number of individuals, concentrations, etc.).

This approach begins with the mathematical modelling of the system. This stage is fundamental and difficult, because one does not have rigorous laws as in physics. We develop techniques to identify and validate the structure of a model from a set of available noisy measurements. This approach is based on the qualitative analysis of the data (extrema, relative position,…) that we use to build a model able to reproduce the same qualitative pattern. We work also on methods dedicated to the identification of the mathematical functions that link the dynamics of a state variable to other variables. Finally we verify that the model satisfies some biological constraints: for example the concentrations must remain positive. A fundamental problem is that of the validation, or invalidation, of these models: how to accept, with a certain precision, a model by comparing it with experimental noisy data ? The traditional approach, which consists of identifying the parameters of the model by minimizing a criterion of variation between the outputs of the model and the data, is often inefficient. We are developing new methods more pertinent for the biologists.

From a model, that synthesises the behaviour of such a complex nonlinear biological system, we can study its properties and understand the way it works. One seeks to study the qualitative behaviour of the system, the existence of equilibria, their stability, the existence of periodic solutions. These qualitative questions are fundamental because they tell us whether or not the system is viable (the model does not predict the extinction of any species, etc). Specific problems are posed by the biological origin of the models: functions or parameters are uncertain, or unknown; what can we say on the behaviour of the model? Often, the models have a strong structure belonging to a general class of systems, for which one develops adapted techniques: for example the well-known models of Lotka-Volterra in dimension n, describing the interactions between n species.

Once the dynamics of the considered living system has been understood we consider problems of regulation: how to keep a variable at a given level? This is important for example in the framework of waste water treatment where the pollution levels are imposed by laws. The main problem that we have to address is to try to control a complex system when the model is uncertain. We work mainly on a class of biological systems: the bioreactors that have a growing importance in many domains related to the human environment: alimentary (food production), pharmaceuticals (production of medicine), environment (waste water treatment, plankton study), etc. The strong structure of these systems for which the hydraulic flow plays an important role is used in order to derive controllers.

Finally we develop observers that use the model and on-line measurements to estimate asymptotically the variables that are not measured directly. These so called ‘software sensors’ can help the monitoring of some systems but also replace some expensive measurements. We face with the problem of the various uncertainties that are specific to biological modelling: the model is uncertain (parameters, functions), but also the inputs can be uncertain and the outputs highly variable. We have therefore to deal with these uncertainties in the design of the observers. We developed robust observers that assume that some parameter or input belongs to a given interval. The observer estimates then asymptotically an interval for the state variables (collaboration with INRA Montpellier). The methods that we develop are validated and tested on several applications:

Growth of the Marine Plankton

We work in association with the Station Zoologique of the CNRS (Villefranche-sur-Mer, France), which developed a chemostat (open bioreactor where algae or cells grow on a substrate) fully automated and managed by computers; this system is well adapted to the application of the methods resulting from the theory of control. Our current work consists of studying and validating models of growth for the plankton in a variable environment (light, food, etc). The growth of the plankton is the basis of all the production of the organic matter of the oceans (fishes, etc); however, the existing traditional models (Monod, Droop) are often unsatisfactory. We seek to obtain models valid during the transitory stages, away from the equilibrium.

Fisheries and forests

The scale of the problems changes here; data are rare and noisy. We consider (in collaboration with IFREMER Nantes) some important methodological problems: how to model the stock-recruitment relationship of the fish (the relationship between the number of fertile adults and eggs they produce)? How does one optimize the exploitation of fisheries or forests with respect to some criteria?

Waste Water Treatment Processes

In collaboration with the Laboratory of Environmental Biotechnology of INRA (Narbonne, France), we work on activated sludge wastewater treatment plants and on anaerobic digesters. We build dynamical models and we design robust observers that take into account the large uncertainties encountered in this field. As an example, the amount of waste water to treat, which is an important input, is rarely measured. The software sensors are used to monitor the processes and help to detect a failure.

Food Agro-Industry

The control theory methods are helpful to optimise the production of some agricultural bioproduct, or the biomass of micro-organisms used in fermentation. We work in collaboration with the CESAME (Louvain-la-Neuve, Belgium) in order to optimise the production of vanillin by filamentous fungi. In this framework we have to take advantage from the (relative) good quality of the measurements obtained with gaseous flow rates.

Links:

COMORE website: http://www.inria.fr/Equipes/COMORE-eng.html

Please contact:

Jean-Luc Gouzé and Olivier Bernard - INRIA Sophia-Antipolis
Tel: +33 4 9238 7875 / 7785
E-mail: Jean-Luc.Gouze@inria.fr, Olivier.Bernard@inria.fr