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Sur la commandabilité des systèmes non linéaires à temps discret

Badis Djeridane 1
1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes
Abstract : The work presented in this thesis deals with the problem of optimal control of discrete-time system by means of dynamic programming methods. A particular formulation of functional equation is considered. This problem leads us to consider the problem of the controllability of nonlinear discrete-time systems, also denoted here by input-state inversion problem. The problem has been carried out in two stages. The first stage is concerned with the analysis of the problem of controllability. We present different approaches for study of the controllability of systems: composition of function, differential geometry and differential algebra. We propose hereafter a necessary controllability condition based on a minimum principle application. Finally a necessary and sufficient controllability condition is developed for bilinear discrete-time systems, which follows immediately from the previous result based on minimum principle. Once the ability of the system to be controllable is checked, the second stage will consist to solve the inversion problem. Firstly, we propose an algebraic technique based on the determination of the sequence of controls directly from the matrix of controllability. We also propose an approach based on the reformulation of the controllability problem as a classical optimal control problem. Secondly, we use techniques of feedback linéarisation. The problem of feedback linéarisation is considered in two ways. We present in the first way an algebraic approach. In the second way, we propose a geometric approach in which we define a sequence of distributions by using vector field associated to system as well as a virtual linear output system. The linearizing transformation and the feedback control are constructed from these distributions. Finally, results on accessibility and feedback linearisation are presented on practical applications in the framework of dynamic programming.
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Submitted on : Friday, June 17, 2005 - 10:15:50 AM
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Badis Djeridane. Sur la commandabilité des systèmes non linéaires à temps discret. Automatique / Robotique. Université Paul Sabatier - Toulouse III, 2004. Français. ⟨tel-00009518⟩

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