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Quantified real constraint solving using modal intervals with applications to control

Abstract : A Quantified Real Constraint (QRC) is a mathematical formalism that is used to model many physical problems involving systems of nonlinear equations linking real variables, some of them affected by logical quantifiers. QRCs appear in numerous contexts, such as Control Engineering, Electrical Engineering, Mechanical Engineering, and Biology. QRC solving is an active research domain for which two radically different approaches are proposed: the symbolic quantifier elimination and the approximate methods. However, solving large problems within a reasonable computational time and solving the general case, still remain open problems. With the aim of contributing to the research on QRC solving, this thesis proposes a new approximate methodology based on Modal Interval Analysis (MIA), a mathematical theory developed by researchers from the University of Barcelona and from the University of Girona. This methodology allows solving in an elegant way, problems involving logical quantifiers over real variables. Simultaneously, this work aims to promote the use of MIA for solving complex problems, such as QRCs. The MIA theory is relatively confidential due to its theoretical complexity and due to its nonconventional mathematical notation. This thesis tries to raise this barrier by presenting the theory in a more intuitive way through examples and analogies from the classical Interval Analysis approach. The proposed methodology has been implemented and validated by resolving several problems from the literature, and comparing the obtained results with different state-of-the-art techniques. Thus, it has been shown that the presented approach extends the class of QCRs that can be solved and improves the computation time in some particular cases. All the presented algorithms in this work are based on an algorithm developed in this thesis and called Fstar algorithm. This algorithm allows the computation with Modal Intervals in an easy way, something that helps to the utilization of MIA and facilitates its diffusion. With this purpose, an Internet site has been created to allow the utilization of most of the algorithms presented in this thesis. Finally, two control engineering applications are presented. The first application refers to the problem of fault detection in dynamic systems and has been validated from experiments involving actual processes. The second application consists of the realization of a controller for a sailboat. This last one has been validated using simulation.
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  • HAL Id : tel-00331668, version 1

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Pau Herrero Vinas. Quantified real constraint solving using modal intervals with applications to control. Automatic. Université d'Angers, 2006. English. ⟨tel-00331668⟩

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