201. Well-Posedness and Output Regulation for Implicit Time-Varying Evolution Variational Inequalities
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Aneel Tanwani, Christophe Prieur, Bernard Brogliato, Équipe Méthodes et Algorithmes en Commande (LAAS-MAC), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), Modélisation, simulation et commande des systèmes dynamiques non lisses (TRIPOP ), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), GIPSA - Systèmes non linéaires et complexité (GIPSA-SYSCO), Département Automatique (GIPSA-DA), Grenoble Images Parole Signal Automatique (GIPSA-lab ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Grenoble Images Parole Signal Automatique (GIPSA-lab ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), ANR-17-CE40-0019,ConVan,Contrôle des Systèmes Interconnectés sous Contraintes en Utilisant l'Analyse Variationelle(2017), Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), and Université Fédérale Toulouse Midi-Pyrénées
- Subjects
0209 industrial biotechnology ,time-varying maximal monotone mappings ,Control and Optimization ,convex optimization ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,Internal model ,Dynamical Systems (math.DS) ,Systems and Control (eess.SY) ,02 engineering and technology ,output regulation ,01 natural sciences ,020901 industrial engineering & automation ,Differential inclusion ,[INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering ,Full state feedback ,FOS: Mathematics ,FOS: Electrical engineering, electronic engineering, information engineering ,[INFO.INFO-SY]Computer Science [cs]/Systems and Control [cs.SY] ,Applied mathematics ,passivity ,Uniqueness ,0101 mathematics ,Mathematics - Dynamical Systems ,Mathematics ,Applied Mathematics ,010102 general mathematics ,AMS subject classifications - 34A36, 34A60, 49K21, 49J52, 49J53, 93D15, 93D20 ,Lipschitz continuity ,System dynamics ,complementarity programming ,Ordinary differential equation ,Variational inequality ,Computer Science - Systems and Control ,viability control ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,Evolution variational inequalities - Abstract
A class of evolution variational inequalities (EVIs), which comprises ordinary differential equations (ODEs) coupled with variational inequalities (VIs) associated with time-varying set-valued mappings, is proposed in this paper. We first study the conditions for existence and uniqueness of solutions. The central idea behind the proof is to rewrite the system dynamics as a differential inclusion which can be decomposed into a single-valued Lipschitz map, and a time-dependent maximal monotone operator. Regularity assumptions on the set-valued mapping determine the regularity of the resulting solutions. Complementarity systems with time-dependence are studied as a particular case. We then use this result to study the problem of designing state feedback control laws for output regulation in systems described by EVIs. The derivation of control laws for output regulation is based on the use of internal model principle, and two cases are treated: First, a static feedback control law is derived when full state feedback is available, In the second case, only the error to be regulated is assumed to be available for measurement and a dynamic compensator is designed. As applications, we demonstrate how control input resulting from the solution of a variational inequality results in regulating the output of the system while maintaining polyhedral state constraints. Another application is seen in designing control inputs for regulation in power converters., Accepted for publication in SICON, SIAM Journal of Control and Optimization, 2018. Some technical arguments in the proofs from earlier version have been corrected
- Published
- 2018
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