Static and dynamic boundary control of coupled PDE-ODE systems

Directeur de thèse :     Christophe PRIEUR

Encadrant :     Christophe PRIEUR

École doctorale : Electronique, electrotechnique, automatique, traitement du signal (EEATS)

Spécialité : Automatique et productique

Structure de rattachement : Grenoble-INP

Établissement d'origine : SUPELEC

Financement(s) : bourse attribuée par un gouvernement étranger

Date d'entrée en thèse : 00/00/0000

Date de soutenance : 10/03/2017

Composition du jury :
Leonardo Torres, Professeur, Federal University of Minas Gerais, Brésil, rapporteur
Sophie Tarbouriech, Directeur de Recherche, LAAS-CNRS, rapporteur
Eugenio Castelan, professeur, Federal University of Santa Catarina, Brésil, examinateur
Edson de Pieri, professeur, Federal University of Santa Catarina, Brésil, examinateur
Ubirajara Moreno, professeur, Federal University of Santa Catarina, Brésil, examinateur

Résumé : This work studies boundary control strategies for stability analysis and stabilization of first-order hyperbolic system coupled with nonlinear dynamic boundary conditions. The modeling of a flow inside a pipe (fluid transport phenomenon) with boundary control strategy applied in a physical experimental setup is considered as a case study to evaluate the proposed strategies. Firstly, in the context of finite dimension systems, classical control tools are applied to deal with first-order hyperbolic systems having boundary conditions given by the coupling of a heating column dynamical model and a ventilator static model. The tracking problem of this complex dynamics is addressed in a simple manner considering linear approximations, finite difference schemes and an integral action leading to an augmented discrete-time linear system with dimension depending on the step size of discretization in space. Hence, for the infinite dimensional counterpart, two strategies are proposed to address the boundary control problem of first-order hyperbolic systems coupled with nonlinear dynamic boundary conditions. The first one approximates the first-order hyperbolic system dynamics by a pure delay. Then, convex stability and stabilization conditions of uncertain input delayed nonlinear quadratic systems are proposed based on the Lyapunov-Krasovskii (L-K) stability theory which are formulated in terms of Linear Matrix Inequality (LMI) constraints with additional slack variables (introduced by the Finsler's lemma). Thus, strictly Lyapunov functions are used to derive an LMI based approach for the robust regional boundary stability and stabilization of first-order hyperbolic systems with a boundary condition defined by means of a nonlinear quadratic dynamic system. The proposed stability and stabilization LMI conditions are evaluated considering several academic examples and also the flow inside a pipe case study.