Applied statisticians are often faced with the problem of dealing with high dimensional data sets when attempting to describe the variability of a single set of variables, or trying to predict the variation of one set of variables from another. In this study, two data reduction methods are described: Redundancy Analysis and Partial Least Squares. A hybrid approach developed by Bougeard et al., (2007) and called Continuum Redundancy-Partial Least Squares, is described. All three methods are extended to the frequency domain in order to allow the lower dimensional subspace used to describe the variability to change with frequency. To illustrate and compare the three methods, and their frequency dependent generalizations, an idealized coupled atmosphere-ocean model is introduced in state space form. This model provides explicit expressions for the covariance and cross spectral matrices required by the various methods; this allows the strengths and weaknesses of the methods to be identified.

Este trabalho enfoca os procedimentos de modelagem por elementos finitos determinísticos e estocásticos de sistemas estruturais do tipo placas finas sujeitas a não linearidades distribuídas e discretas na presença de incertezas. Neste caso, o método de Newmark não linear combinado com o método de Newton-Raphson foi utilizado para resolução direta das equações do movimento e obtenção dos envelopes das respostas no tempo dos sistemas não lineares. Com o objetivo de reduzir o esforço computacional requerido para resolver os problemas não lineares, especialmente para os casos envolvendo a presença de incertezas, foi proposto o método modal enriquecido e o método das aproximações combinadas melhorado. No que diz respeito à consideração da inserção de incertezas no modelo determinístico optou-se pela construção de um modelo estocásticos do sistema com não linearidades distribuídas, utilizando-se para isto da técnica de discretização de campos aleatórios de Karhunen-Loève para sistemas bi-dimensionais. Já, para a obtenção das respostas dinâmicas aleatórias, foi utilizado o método de simulação Hyper-Cubo-Latino. Através dos vários exemplos de simulações com estruturas do tipo placas finas não amortecidas e amortecidas sujeitas a diferentes níveis de força e condições de contorno, pode-se ilustrar os desenvolvimentos abordados ao longo deste trabalho. Em particular, pode-se concluir sobre a eficiência e necessidade da utilização de métodos de redução para a avaliação dinâmica de sistemas não lineares, principalmente para sistemas não lineares mais complexos ou na presença de incertezas. Por fim, deve-se salientar a importância de se considerar as incertezas na análise e no projeto de sistemas não lineares para lidar com situações mais realísticas de interesse prático / This work is devoted to the deterministic and stochastic finite element modeling of thin flat plates under large displacements and subjected to geometric and discrete non-linearities. In order to solve the resulting non-linear equations of motion in the time-domain, the non-linear Newmark strategy combined with the Newton-Raphson method has been used herein. With the aim of reducing the computational cost required to solve the non-linear problems, especially for the cases in which the uncertainties are considered, the modal method based on the construction of an enriched reduction basis and an improved version of the combined approximations technique have been retained in the present study. With regard to the insertion of uncertainties in the deterministic model opted by the construction of a stochastic model of the system with no distributed linearity, using for this the Karhunen-Loève expansion technique in bi-dimensional form. In order to generate the envelopes of the dynamic responses of the non-linear systems in time-domain, it has been used the so-called Hyper-Cube-Latino. Based on the numerical simulations with plate structures subjected to various levels of excitation and boundary conditions it is possible to illustrate the methodology presented in this work. In particular, it can be concluded about the efficiency and necessity of performing efficient and accurate model reduction methods to deal with non-linear systems. Finally, it is also important to discuss about the interest in considering uncertainties in the analysis and design of non-linear systems in order to deal with more realistic non-linear situations / Tese (Doutorado)

Nowadays, increasing pressure from legislation and customer demands in the automotive industryare forcing manufacturers to produce greener vehicles with lower emissions and fuel consumption.As a result, electrified and hybrid vehicles are a growing popular alternative to traditional internalcombustion engines (ICE). The noise from an electric vehicle comes mainly from contact betweentyres and road, wind resistance and driveline. The noise emitted from the driveline is for the mostpart related to the gearbox. When developing a driveline, it is a factor of importance to estimatethe noise radiating from the gearbox to achieve an acceptable design.Gears are used extensively in the driveline of electric vehicles. As the gears are in mesh, a mainintrusive concern is known as gear whine noise. Gear whine noise is an undesired vibroacousticphenomenon and is likely to originate through the gear contacts and be transferred through themechanical components to the housing where the vibrations are converted into airborne andstructure-borne noise. The gear whine noise originates primarily from the excitation coming fromtransmission error (TE). Transmission error is defined as the difference between the ideal smoothtransfer of motion of a gear and what is in practice due to lack of smoothness.The main objective of this study is to simulate the vibrations generated by the gear whine noise inan electric powertrain line developed by AVL Vicura. The electric transmission used in this studyprovides only a fixed overall gear ratio, i.e. 9.59, under all operation conditions. It is assumed thatthe system is excited only by the transmission error and the mesh stiffness of the gear contacts. Inorder to perform NVH analysis under different operating conditions, a multibody dynamics modelaccording to the AVL Excite program has been developed. The dynamic simulations are thencompared with previous experimental measurements provided by AVL Vicura.Two validation criteria have been used to analyse the dynamic behaviour of the AVL Excite model:signal processing using the FFT method and comparison with the experimental measurements.The results from the AVL Excite model show that the FFT criterion is quite successful and allexcitation frequencies are properly observed in FFT plots. Nevertheless, when it comes to thesecond criterion, as long as not all dynamic parameters of the system such as damping or stiffnesscoefficients are provided with certainty in the model, it is too difficult to investigate the accuracy ofthe AVL Excite model.Another investigation is a numerical design study to analyses how the damping coefficientsinfluence the response. After reducing the damping parameters, the results show that the housingand bearings have the highest influence on the response. If more acceptable results are desired,future studies must be concentrated on these to obtain more acceptable damping values. / För närvarande tvingar ökat tryck från lagstiftning och kundkrav inom bilindustrin tillverkarna attproducera grönare fordon med lägre utsläpp och bränsleförbrukning. Som ett resultat ärelektrifierade och hybridfordon ett växande populärt alternativ till traditionellaförbränningsmotorer (ICE). Bullret från ett elfordon kommer främst från kontakten mellan däckoch väg, vindmotstånd och drivlinan. Bullret från drivlinan är i huvudsak relaterat till växellådan.Vid utveckling av en drivlina är det av betydelse att uppskatta bullret från växellådan för att uppnåen acceptabel design.Utväxlingar används i stor utsträckning i elfordons drivlina. Eftersom kugghjulen är i kontaktuppstår ett huvudproblem som är känt som ett vinande ljud från kugghjulskontakten.Kugghjulsljud är ett oönskat vibro-akustiskt fenomen och uppstår sannolikt på grund avkugghjulkontakterna och överförs via de mekaniska komponenterna till växellådshuset därvibrationerna omvandlas till luftburet och strukturburet ljud. Kugghjulsljudet härstammarhuvudsakligen från exciteringen som kommer från transmissionsfel (TE) i kugghjulskontakten.Överföringsfelet definieras som skillnaden mellan den ideala smidiga rörelseöverföringen hoskugghjulen och rörelsen som sker i verkligheten på grund av ojämnheter.Huvudsyftet med denna studie är att simulera vibrationerna som genereras avkugghjulskontakterna i en elektrisk drivlina utvecklad av AVL Vicura. Den elektriska drivlinan somanvänds i denna studie har endast ett fast utväxlingsförhållande, dvs 9,59, för alladriftsförhållanden. Det antas att systemet är exciterat endast av överföringsfelet och kugghjulensstyvhet i kuggkontakterna. För att kunna utföra NVH-analys under olika driftsförhållanden har enstelkroppsdynamikmodell utvecklats med hjälp av programmet AVL Excite. De dynamiskasimuleringarna jämförs sedan med tidigare experimentella mätningar som tillhandahålls av AVLVicura.Två valideringskriterier har använts för att analysera det dynamiska beteendet hos AVL Excitemodellen:signalbehandling med FFT-metoden och jämförelse med experimentella mätningar.Resultaten från AVL Excite-modellen visar att FFT-kriteriet är ganska framgångsrikt och allaexcitationsfrekvenser observeras korrekt i FFT-diagrammen. Men när det gäller det andra kriteriet,så länge som inte alla dynamiska parametrar i systemet, såsom dämpnings- ellerstyvhetskoefficienter, är tillförlitliga i modellen, är det för svårt att undersöka exaktheten hos AVLExcite-modellen.En annan undersökning som utförts är en numerisk designstudie för att analysera hurdämpningskoefficienterna påverkar responsen. Efter minskning av dämpningsparametrarna visarresultaten att växellådshus och lager har störst inflytande på resultatet. Om mer acceptabla resultatär önskvärda måste framtida studier koncentreras på dessa parametrar för att uppnå mer acceptabladämpningsvärden.

Cette thèse s'inscrit dans le cadre de l'analyse statistique de données en grande dimension. Nous avons en effet aujourd'hui accès à un nombre toujours plus important d'information. L'enjeu majeur repose alors sur notre capacité à explorer de vastes quantités de données et à en inférer notamment les structures de dépendance. L'objet de cette thèse est d'étudier et d'apporter des garanties théoriques à certaines méthodes d'estimation de structures de dépendance de données en grande dimension.La première partie de la thèse est consacrée à l'étude de modèles parcimonieux et aux méthodes de type Lasso. Après avoir présenté les résultats importants sur ce sujet dans le chapitre 1, nous généralisons le cas gaussien à des modèles exponentiels généraux. La contribution majeure à cette partie est présentée dans le chapitre 2 et consiste en l'établissement d'inégalités oracles pour une procédure Group Lasso appliquée aux modèles linéaires généralisés. Ces résultats montrent les bonnes performances de cet estimateur sous certaines conditions sur le modèle et sont illustrés dans le cas du modèle Poissonien. Dans la deuxième partie de la thèse, nous revenons au modèle de régression linéaire, toujours en grande dimension mais l'hypothèse de parcimonie est cette fois remplacée par l'existence d'une structure de faible dimension sous-jacente aux données. Nous nous penchons dans cette partie plus particulièrement sur la méthode PLS qui cherche à trouver une décomposition optimale des prédicteurs étant donné un vecteur réponse. Nous rappelons les fondements de la méthode dans le chapitre 3. La contribution majeure à cette partie consiste en l'établissement pour la PLS d'une expression analytique explicite de la structure de dépendance liant les prédicteurs à la réponse. Les deux chapitres suivants illustrent la puissance de cette formule aux travers de nouveaux résultats théoriques sur la PLS . Dans une troisième et dernière partie, nous nous intéressons à la modélisation de structures au travers de graphes et plus particulièrement à la détection de communautés. Après avoir dressé un état de l'art du sujet, nous portons notre attention sur une méthode en particulier connue sous le nom de spectral clustering et qui permet de partitionner les noeuds d'un graphe en se basant sur une matrice de similarité. Nous proposons dans cette thèse une adaptation de cette méthode basée sur l'utilisation d'une pénalité de type l1. Nous illustrons notre méthode sur des simulations. / This thesis falls within the context of high-dimensional data analysis. Nowadays we have access to an increasing amount of information. The major challenge relies on our ability to explore a huge amount of data and to infer their dependency structures.The purpose of this thesis is to study and provide theoretical guarantees to some specific methods that aim at estimating dependency structures for high-dimensional data. The first part of the thesis is devoted to the study of sparse models through Lasso-type methods. In Chapter 1, we present the main results on this topic and then we generalize the Gaussian case to any distribution from the exponential family. The major contribution to this field is presented in Chapter 2 and consists in oracle inequalities for a Group Lasso procedure applied to generalized linear models. These results show that this estimator achieves good performances under some specific conditions on the model. We illustrate this part by considering the case of the Poisson model. The second part concerns linear regression in high dimension but the sparsity assumptions is replaced by a low dimensional structure underlying the data. We focus in particular on the PLS method that attempts to find an optimal decomposition of the predictors given a response. We recall the main idea in Chapter 3. The major contribution to this part consists in a new explicit analytical expression of the dependency structure that links the predictors to the response. The next two chapters illustrate the power of this formula by emphasising new theoretical results for PLS. The third and last part is dedicated to graphs modelling and especially to community detection. After presenting the main trends on this topic, we draw our attention to Spectral Clustering that allows to cluster nodes of a graph with respect to a similarity matrix. In this thesis, we suggest an alternative to this method by considering a $l_1$ penalty. We illustrate this method through simulations.

Dans ce mémoire nous avons développé des méthodes de réduction des systèmes hamiltoniens à port en dimension finie et infinie qui préservent leur structure. Dans la première partie, nous avons défini une représentation des systèmes hamiltoniens à port avec contraintes sous la forme d'équations différentielles algébriques (DEA) de type de système descripteur. De cette forme nous avons déduit une réalisation équilibrée du système hamiltonien à port exprimée sous forme de système descripteur contenant les mêmes systèmes d'équations de contrainte. Dans la deuxième partie, nous avons défini une classe de problèmes de commande LQG tels que le contrôleur dynamique LQG est passif et admet une réalisation hamiltonien à port. Deux méthodes de synthèse de commande passive LQG sont proposées et une de ces méthodes LQG nous a permis de définir une réalisation équilibrée LQG. Puis nous avons appliqué la méthode de contrainte de l'effort pour réduire le système hamiltonien à port et obtenir une commande LQG passive d'ordre réduit. Ce contrôleur LQG admettant une réalisation hamiltonienne, la structure hamiltonienne est préservée pour le système en boucle fermée par interconnexion de systèmes hamiltoniens à port. Dans la troisième partie, nous avons généralisé les résultats précédents aux systèmes hamiltoniens à ports linéaires de dimension infinie. Pour cela nous avons considéré une classe de systèmes hamiltoniens à ports de dimension infinie dont l'opérateur d'entrée est borné et un problème de commande LQG passif. Sous des conditions de nucléarité de l'opérateur de Hankel lié au problème LQG, nous définissons une réalisation équilibrée LQG passive du système et une approximation en dimension finie. Le contrôleur LQG passif d'ordre réduit obtenu par cette approximation admet une réalisation hamiltonienne à port et par conséquent la structure hamiltonienne et la passivité sont préservées en boucle fermée / In this thesis we have developed different structure preserving reduction methods for finite and infinite dimensional port Hamiltonian systems by using a balanced model reduction approach. In the first part we have defined a descriptor representation of port Hamiltonian systems with constraints. The balanced realization of the descriptor system has been used for reducing the port Hamiltonian descriptor system and conserving explicitly the constraint equations. In the second part, conditions have been derived on the weighting matrices of the LQG control problem such that the dynamical LQG controller is passive and has a port Hamiltonian realization. Two passive LQG control design methods have been suggested and one of them allows us to define a LQG balanced realization. Based on this realization, the effort constraint method has been used to reduce the LQG balanced port Hamiltonian system and obtain a reduced order passive LQG controller. In this way the closed-loop system is derived from the interconnection of 2 port Hamiltonian systems, hence the Hamiltonian structure has been preserved. In the third part, the proceeding results have been extended to a class of infinite dimensional port Hamiltonian system with bounded input operator. A passive LQG control design method for infinite dimensional port Hamiltonian system has been derived as by Control by Interconnection (CbI). Based on the balanced realization associated with this passive LQG control design, a finite dimensional approximation has been achieved and a reduced order passive LQG controller has been derived. As a consequence, the system in closed-loop with this reduced order LQG controller again admits a port Hamiltonian structure and satisfies the passivity

The transition from internal combustion engine to battery electric cars is accompanied by a shift on the NVH issues damaging the passenger comfort. The rolling noise generated by the wheels and tyres is in particular characterized by an increasing attention from OEMs and car manufacturers. Among the causes of the rolling noise are the vibrations generated at the wheel, which are transmitted to the vehicle interior through its structure. To limit these vibrations at their source, a new innovative concept has been proposed at Stellantis. This idea is based on the use of a specific type of vibration absorber known by the acronym MTMD (Multiple Tuned Mass Damper).First, the existing literature about vibrations absorbers and optimisation techniques has been reviewed. Then, initial simulations on a global model comprising the wheel and beam-like resonators have shown how this system can effectively reduce a resonant peak.Optimisation processes have then revealed an important attenuation of a wheel resonance, even when the peak frequency is shifted. This demonstrated how important the distribution of resonators resonance frequencies are and their damping ratios when designing a robust and efficient MTMD.Finally, local resonators have been designed and modelled by considering the constraints for an integration on a wheel. A satisfactory design for resonators able to vibrate at the wanted frequency has been found. The beam-like resonators used in the first global model have then been replaced by these real resonators, confirming the possible gain with this device in a vehicle. / Övergången från förbränningsmotorer till batterielektriska bilar åtföljs av en övergång till NVH-frågor som kan påverka passagerarnas komfort. Det rullningsljud som genereras av hjul och däck får i synnerhet ökad uppmärksamhet från OEM:er och biltillverkare.Bland orsakerna till rullningsljudet finns de vibrationer som genereras vid hjulet och som överförs till fordonets inre genom dess struktur. För att begränsa dessa vibrationer vid källan har ett nytt innovativt koncept föreslagits av Stellantis. Idén bygger på användningen av en särskild typ av vibrationsdämpare som kallas MTMD (Multiple Tuned Mass Damper).Först har den befintliga litteraturen om vibrationsdämpare och optimeringstekniker gåtts igenom. Därefter har inledande simuleringar på en global modell som omfattar hjulet och balkliknande resonatorer visat hur detta system effektivt kan minska en resonanstopp.Optimeringsprocesserna har sedan bekräftat att hjulresonansen är kraftigt dämpad, även när toppfrekvensen är förskjuten. Detta visade hur viktig fördelningen av resonansfrekvenser och dämpningsförhållanden för resonatorer är när man utformar en robust och effektiv MTMD.Slutligen har lokala resonatorer utformats och modellerats genom att beakta begränsningarna för en integrering i ett hjul. En tillfredsställande utformning av resonatorer som kan vibrera vid den önskade frekvensen har hittats. De balkliknande resonatorer som användes i den första globala modellen har sedan ersatts av dessa riktiga resonatorer, vilket bekräftar den potentiella förbättringen med denna anordning i ett fordon.