Méthodes seminumériques en algèbre différentielle~; applications à l'étude des propriétés structurelles de systèmes différentiels algébriques en automatique

Abstract : The work presented here is based on differential algebra and on the methods of symbolic computation to solve problems of nonlinear control theory which do not lend themselves to a direct numerical resolution. The problem of local algebraic observability consists in deciding if the model state variables can be deduced from perfectly known inputs and outputs. We present a probabilistic algorithm of arithmetic complexity polynomial in the size of the input which allows to test local algebraic observability by determining the non observable variables. The use of modular arithmetic enables to obtain a polynomial bit complexity for this test. This complexity depends linearly on the probability of success which can be arbitrarily fixed. An implementation of this algorithm is available and makes it possible to deal with problems unreachable until now. Taking as a starting point these methods mixing symbolic computation and numerical computation, we propose a generalization of the concept of differential flatness to certain models described by nonlinear partial differential equations. An ordinary differential system is differentialy flat if its solutions can locally be parameterized by arbitrary functions. To study certain nonlinear systems of partial differential equations, one brings back to a system of ordinary differential equations by discretization ; our approach consists in seeking flat discretizations such that the associated parametrization converge when the step of discretization tends towards zero. This method is illustrated by the study of the motion planning problem for three nonlinear models : the semilinear heat equation, the Burger equation with diffusion and a nonlinear model of flexible rod.
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Engineering Sciences. Ecole Polytechnique X, 2001. French


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Contributor : Alexandre Sedoglavic <>
Submitted on : Monday, July 6, 2009 - 12:09:08 PM
Last modification on : Friday, March 27, 2015 - 4:02:23 PM

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Alexandre Sedoglavic. Méthodes seminumériques en algèbre différentielle~; applications à l'étude des propriétés structurelles de systèmes différentiels algébriques en automatique. Engineering Sciences. Ecole Polytechnique X, 2001. French. <tel-00401888>

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