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Theses

Observateurs asymptotiques invariants : théorie et exemples

Abstract : This thesis is about the construction of non-linear estimators of the asymptotic observers type. We will first build an observer in order to estimate the concentrations in reactive in a polymerization reactor of TOTAL. Then we will pursue theoretical questions about the use of symmetries for the design of non-linear estimators. The chemical reactor we worked on is a high pressure polymerization reactor which produces plastics that are polymers made of two or three monomers. The estimation of the concentrations in each reactive in the several zones of the reactor is based on a model of the reaction. The model consists of a mass balance, an energy balance, and the use of a chemical kinetics model. Thanks to the equations of the model, and to the measurements of temperature and flows, we give a real-time estimation of the concentrations. It is a nonlinear estimator, and the convergence is based on contraction properties. This estimator was implemented and validated on the industrial unit. The convergence of the estimator is independent of the choice of the units with which the balances are written. We wondered if it always possible, when one builds an observer for the concentrations of the Luenberger or extended Kalman filter type, to write correction terms which do not depend on the units. We thus considered a more academic example : a chemical exothermic reactor, for which the temperature and flows are measured, and the chemical kinetics is of order one. We want to estimate the concentrations, and we want the convergence properties to be independent of the physical units. This study showed that an approach based on symmetries could suggest non-linear correction terms, and some change of variables which help when it comes to studying global convergence of observers of the Luenberger of extended Kalman filter type. Then we developed a general method in order to write correction terms which systematically preserve the symmetries of the system. The principal theoretical contribution of the thesis is to give a precise method to write all the correction terms which preserve the symmetries. The notion of the error between the true state and the estimated state is reexplored via the notion of invariant state-error. The invariant state error dynamics has very interesting properties. In particular it is independent of the trajectory for a left-invariant trajectory on a Lie group. We apply the theory of invariants observers to mainly three examples, a chemical reactor for which we build a non-linear globally convergent observer, an example of a non-holonomic car for which we build an almost globally convergent observer, and an example of inertial navigation assisted by velocity measurements for which we get the local convergence around any trajectory and such that the global behavior of the error does not depend neither on the trajectory nor on the inputs. Although the theory deals with the cases where the dimension of the symmetry group is smaller than the dimension of the state, it seems very natural to use a similar method to deal with the general case. This is the topic of the last part of the thesis where we look at four examples. The synthesis of reduced observer for a class of lagrangian systems such that all the positions are measured : the transformation group consists of all the change of coordinates on the configuration space. The models of the Saint-Venant type on which the Shallow-water model are based, and which are used in oceanography : the dimension of the state space is infinite since the models use partial differential equations. The data fusion in inertial navigation for which the measurement is an image, and thus is dimension of the output is infinite. Finally, the parametric estimation of a two-states quantum system for which the dimension of the group is bigger than the dimension of the state.
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Silvère Bonnabel. Observateurs asymptotiques invariants : théorie et exemples. Mathématiques [math]. École Nationale Supérieure des Mines de Paris, 2007. Français. ⟨NNT : 2007ENMP1590⟩. ⟨pastel-00004868⟩

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