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Sur l'(A,B)-invariance de polyèdres convexes ; application à la commande sous contraintes et au problème l1

Abstract : This work analyses the property of (A,B)-invariance of convex polyhedra and its application to the control of constrained systems and to the l 1 control problem. Firstly, an explicit characterization of (A,B)-invariance of convex polyhedra for discrete-time systems is proposed. Such a characterization amounts to necessary and sufficient conditions in the form of linear matrix relations, and presents two major advantages compared to the ones found in the literature: it applies to any convex polyhedron and it does not require the computation of vertices. Such advantages are particularly felt in the computation of the supremal (A,B)-invariant domain included in a given polyhedron, for which we propose a numerical method. The problem of computing a control law which makes an (A,B)-invariant polyhedron positively invariant in closed-loop is treated as well. The (A,B)-invariance relations are then generalized to systems subject to linear constraints on the control vector and to systems subject to bounded additive disturbances. The results obtained for discrete-time systems are then extended to continuous-time systems. Next, the problem of attenuation of persistent additive disturbances, known in the literature as the l 1 control problem, is studied. The internally stabilizable (A,B)-invariant domains are firstly characterized. Then, a decomposed approach is proposed for the computation of the supremal internally stabilizable domain included in the polyhedron defined by the l 1 performance constraints. A given performance is achievable if and only if this associated supremal domain is not empty. Such a geometric approach allows to directly determine the solution of the l 1 problem for an important class of systems. Finally, the study of (A,B)-invariance of polyhedra is extended to systems whose model is subject to structured uncertainties.
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Contributor : Emilie Marchand <>
Submitted on : Friday, September 9, 2005 - 10:49:45 AM
Last modification on : Friday, January 10, 2020 - 9:08:09 PM
Long-term archiving on: : Tuesday, September 7, 2010 - 5:32:36 PM


  • HAL Id : tel-00010088, version 1


Carlos Eduardo Trabuco Dórea. Sur l'(A,B)-invariance de polyèdres convexes ; application à la commande sous contraintes et au problème l1. Automatique / Robotique. Université Paul Sabatier - Toulouse III, 1997. Français. ⟨tel-00010088⟩



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