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Méthodes géométriques pour la construction des ensembles invariants. Application à la faisabilité des lois de commande prédictive

Abstract : The present work is based on the explicit formulations for the predictive control of constrained linear systems and underlines the piecewise linear structure of the resulting closed loop dynamics. The main contributions of this thesis are concentrated around the construction of invariant set for Piecewise Affine Systems (PWA). Practical algorithmical procedures are detailed for the case of PWA system defined over a polyhedral partition in the state space even if the priciples are applicable in the general framework. Three specific techniques are presented : – expansive construction, – contractive construction, – construction based on transition graph. At the methodological level, all these constructions being based on the forward of backord set dynamics in the state space, they imply a geometrical treatment at least in what it may concern the comparison with the original feasible domain. This treatement can lead to complex calculations due to the curse of dimensionality. An innovative solution was proposed by exploiting the analysis by intervals. It is interesting to observe that the construction of the invariant sets opens the way to the post-treatment of the of predictive control laws with the aim of the maximization of their domain of functioning with guarantee of safety. In this directions comparisons are carried out between the different MPC structures. The performances of the classical formulations are analysed in comparison with schemes which benefit from the invariance properties from the design stage on one hand and with explicit formulations which benefit from the post analysis for the characterization of the viable domains on the other hand. Furthermore, the present work presents extensions of these geometrical methods for the trajectory tracking, for handling paramer uncertainties or dealing with variable delay in the input of the system. An important part of these theoretical developments are illustrated by examples along their introduction. The manuscript contains also the study of a trajectory tracking problem and a feasibility / viability analysis of a certain profile in view of the electricity production characterization for a hydraulic valley.
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Submitted on : Monday, January 11, 2010 - 3:00:36 PM
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Hichem Benlaoukli. Méthodes géométriques pour la construction des ensembles invariants. Application à la faisabilité des lois de commande prédictive. Automatique / Robotique. Université Paris Sud - Paris XI, 2009. Français. ⟨tel-00445869⟩

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