Contribution à l'étude de la commande prédictive sous contraintes par approche géométrique

This thesis is a contribution to the study of the predictive control under constraints essentially using a geometrical approach. The feasible domain resulting from a set of linear constraints is represented by a polyhedron. But, since the dynamics of the system to be controlled intervenes in the structure of the constraints, it results a parameterization of the optimization problem to be solved on line. The structure of the feasible region can then be analyzed through the concept of parameterized polyhedron. This characterization of the feasible domain initially enables to establish necessary and sufficient feasibility conditions for the predictive law, the relationships with the stability of the closed loop system being highlighted using the invariant set theory. Analysing the position of the unconstrained optimum with respect to the polyhedral feasible domain can lead to a partitioning of the parameters space, allowing the construction of an explicit formulation of the predictive law as well in the nominal case as for multiparametric optimizations constructed with robustness improvement purposes. The originality of the approach, related to this geometrical point of view, allows, beside the construction of explicit laws, the analysis of the redundancy phenomenon. It proposes a partition of the parameters space in regions corresponding to subsets of constraints locally nonredundant. All these results lead to off-line design procedures for the predictive laws such that their effective implementation may use techniques spread from on-line optimization to fully explicit piecewise laws evaluated by look-up table positioning mechanisms.