The L2 gain of a nonlinear time-dependent system measures the maximal gain in the transfer of energy from admissible input signals to the output signals, in which both the input and output signals are measured with the L2 norm. For general nonlinear systems, obtaining a sharp estimate of the L2 gain is challenging both theoretically and numerically. In this thesis, we explore a computationally efficient way to obtain numerical estimations of L2 gains for systems with quadratic nonlinearity. The approach utilizes a recently developed method that solves a class of Hamilton-Jacobi-Bellman equations via a Taylor series-based approximation, which is scalable to high-dimensional problems given the utilization of linear tensor systems. The ideas are demonstrated through a few concrete examples that include a one-dimensional problem with an explicit energy function and several Galerkin approximations of the viscous Burgers equation. / Master of Science / With nonlinear systems that are of the form of input-output models, questions often arise as to how to measure the energy that passes through such systems and determine strategies to look for specific signals that allow the designer freedom to explore certain system behaviors. The energy comes in the form of a signal. For general nonlinear systems, obtaining a sharp estimate of such energy gain is challenging both theoretically and numerically. In this thesis, we explore a computationally efficient way to obtain numerical estimations of these gains for systems with quadratic nonlinearity. The approach combines fundamental theoretical understandings established in the literature with scalable software recently developed in approximating the solution of the underlying partial differential equation, called the Hamilton-Jacobi-Bellman (HJB) equation. In this approach, the energy gain is linked to a single scalar parameter in the HJB equation. Roughly speaking, the energy gain is the lower bound of this scalar parameter above which the HJB equation always admits a non-negative solution. Thus, it boils down to approximating the HJB solution using the software while changing this scalar parameter. We will present the theoretical foundation of the approach and illustrate the foundation through several academic examples ranging from low to relatively high dimensions.

Cette thèse porte sur la conception d’une classe particulière d’estimateurs d'état, les observateurs intervalles. L’objectif est d’estimer de manière garantie, les bornes supérieure et inférieure de l’ensemble admissible de l'état d’un système, à chaque instant de temps. L’approche considérée repose sur la connaissance a priori du domaine d’appartenance, supposé borné, des incertitudes du système (incertitudes de modélisation, perturbations, bruits, etc). Une classe d'observateurs intervalles à entrées inconnues est proposée pour la classe des systèmes Linéaires à Paramètres Variants (LPV). La synthèse des paramètres de l’observateur repose sur la résolution d’un problème d’optimisation sous contraintes de type inégalités matricielles linéaires (LMI) permettant de garantir simultanément les conditions d’existence de l’observateur ainsi qu’un niveau de performance, soit dans un contexte énergie, soit dans un contexte amplitude ou soit dans un contexte mixte énergie/amplitude. Plus particulièrement, la performance de l'observateur repose sur une technique de découplage pour annuler les effets des entrées inconnues et une technique d’optimisation destinée à minimiser, au sens de critères de type gain L2et/ou gain L∞, les effets des perturbations sur la largeur totale de l’enveloppe de l'état du système LPV. La méthodologie de synthèse proposée est illustrée sur un exemple académique. Enfin, la méthodologie est appliquée au cas de la phase d’atterrissage du véhicule spatial HL20, sous des conditions de simulations réalistes. / This thesis addresses the design of a class of estimator, named interval obser-ver, which evaluates in a guaranteed way, a set for the state of the system at each instant of time. The proposed approach is based on a priori knowledge of bounded sets for the system uncertainties (modeling uncertainties, disturbances, noise, etc.). A methodology to design an interval observer is proposed for the class of Linear Parameter Varying (LPV) Systems. The feasibility of the latter is based on the resolution of linear Matrix Inequalities (LMI) constraints allowing to simultaneously get the existence conditions of the intervalobserver and a certain level of a priori given performance for the state estimation of the system. Specifically, the performance of the estimates is based on a decoupling technique to avoid the effects of unknown inputs and an optimization technique to minimize, in the L2 and/or L∞ gain sense, the effects of disturbances on the estimated interval length for the state of the LPV system. The design methodology is illustrated on academic examples.Finally, the methodology is applied on the landing phase of the HL20 shuttle.