Dynamical systems are mathematical models of physical phenomena widely used throughout the world today. When a dynamical system is too large to effectively use, we turn to model reduction to obtain a smaller dynamical system that preserves the behavior of the original. In many cases these models depend on one or more parameters other than time, which leads to the field of parametric model reduction. Constructing a parametric reduced-order model (ROM) is not an easy task, and for very large parametric systems it can be difficult to know how well a ROM models the original system, since this usually involves many computations with the full-order system, which is precisely what we want to avoid. Building off of efficient 𝐻-infinity approximations, we develop a greedy algorithm for efficiently modeling large-scale parametric dynamical systems in an 𝐻₂-sense. We demonstrate the effectiveness of this greedy search on a fluid problem, a mechanics problem, and a thermal problem. We also investigate Bayesian optimization for solving the optimization subproblem, and end with extending this algorithm to work with MIMO systems. / Master of Science / In the past century, mathematical modeling and simulation has become the third pillar of scientific discovery and understanding, alongside theory and experimentation. Mathematical models are used every day, and are essential to modern engineering problems. Some of these mathematical models depend on quantities other than just time, parameters such as the viscosity of a fluid or the strength of a spring. These models can sometimes become so large and complicated that it can take a very long time to run simulations with the models. In such a case, we use parametric model reduction to come up with a much smaller and faster model that behaves like the original model. But when these large models vary highly with the parameters, it can also become very expensive to reduce these models accurately. Algorithms already exist for quickly computing reduced-order models (ROMs) with respect to one measure of how "good" the ROM is. In this thesis we develop an algorithm for quickly computing the ROM with respect to a different measure - one that is more closely tied to how the models are simulated.

Mathematical models of physical systems are pervasive in engineering. These models can be used to analyze properties of the system, to simulate the system, or synthesize controllers. However, many of these models are too complex or too large for standard analysis and synthesis methods to be applicable. Hence, there is a need to reduce the complexity of models. In this thesis, techniques for reducing complexity of large linear time-invariant (lti) state-space models and linear parameter-varying (lpv) models are presented. Additionally, a method for synthesizing controllers is also presented. The methods in this thesis all revolve around a system theoretical measure called the H2-norm, and the minimization of this norm using nonlinear optimization. Since the optimization problems rapidly grow large, significant effort is spent on understanding and exploiting the inherent structures available in the problems to reduce the computational complexity when performing the optimization. The first part of the thesis addresses the classical model-reduction problem of lti state-space models. Various H2 problems are formulated and solved using the proposed structure-exploiting nonlinear optimization technique. The standard problem formulation is extended to incorporate also frequency-weighted problems and norms defined on finite frequency intervals, both for continuous and discrete-time models. Additionally, a regularization-based method to account for uncertainty in data is explored. Several examples reveal that the method is highly competitive with alternative approaches. Techniques for finding lpv models from data, and reducing the complexity of lpv models are presented. The basic ideas introduced in the first part of the thesis are extended to the lpv case, once again covering a range of different setups. lpv models are commonly used for analysis and synthesis of controllers, but the efficiency of these methods depends highly on a particular algebraic structure in the lpv models. A method to account for and derive models suitable for controller synthesis is proposed. Many of the methods are thoroughly tested on a realistic modeling problem arising in the design and flight clearance of an Airbus aircraft model. Finally, output-feedback H2 controller synthesis for lpv models is addressed by generalizing the ideas and methods used for modeling. One of the ideas here is to skip the lpv modeling phase before creating the controller, and instead synthesize the controller directly from the data, which classically would have been used to generate a model to be used in the controller synthesis problem. The method specializes to standard output-feedback H2 controller synthesis in the lti case, and favorable comparisons with alternative state-of-the-art implementations are presented.

The thesis considers the problems of designing a dynamic output feedback controller to discrete time systems with polytopic uncertainty so that the closed-loop systems are DR stable with their transfer matrices having H2 norm and H¡Û norm bounded by a prescribed value ru. The formar part of the thesis provides less conservative LMI conditions for H2 and H¡Û analysis and the output feedback control of discrete system than those appeared in the current research. While the latter part of the thesis extend the current research to DR stable with H2 and H¡Û design. Finally, numerical examples are illustrated to show improvement of the propered result.

Un connecteur électrique automobile est le siège de points de contact électriques soumis à des échauffements supplémentaires par effet Joule provoqués par leurs résistances de contact. La dégradation d’une résistance de contact peut provoquer des échauffements critiques au niveau des points de contact. La présente étude propose deux approches pour détecter les variations anormales des résistances de contact. La première, appliquée à un connecteur de boîtier électronique automobile, repose sur un diagnostic à base de modèle afin de détecter les variations anormales de la résistance de contact par rapport à sa valeur nominale. La seconde approche,appliquée à un connecteur de recharge de véhicule électrique, consiste d’abord à estimer un flux de chaleur lié à l’effet Joule provoqué par la résistance de contact. La valeur de cette dernière est ensuite estimée grâce à la méthode des moindres carrés. Ces deux approches reposent sur des modèles électrothermiques des connecteurs considérés et de leurs environnements. Cette étude présente également le développement de ces modèles, et notamment l’optimisation du maillage d’un fil électrique basée sur la minimisation de la norme H2 de l’erreur entre un modèle maillé d’ordre entier et un modèle analytique de référence d’ordre non entier. / An automotive electrical connector contains electrical contact points subject to additional temperature rises by Joule effect caused by their contact resistances. The deterioration of a contact resistance can cause critical overheating at the contact points. This study proposes two approaches to detect abnormal changes in contact resistance. The first one, applied to an automotive electric case connector, relies on a model-based diagnosis to detect an abnormal variation in the contact resistance with respect to its nominal value. The second one, applied toan electric vehicle charging connector, consists in estimating a heat flow related to the Joule effect caused by the contact resistance. The value of the latter is then estimated using the least squares.These approaches rely on electrothermal models considered connectors and their environments.This study also presents the development of these models, including the optimization of the mesh of an electrical wire based on the minimization of the H2-norm of the error between ameshed integer order model and a reference analytical fractional order model.