De nombreux systèmes utilisent des convertisseurs électroniques de puissance fonctionnant en modulation de largeur d'impulsions (MLI) comme organe de réglage ou de commande. On peut en particulier citer les entraînements à vitesse variable, les alimentations à découpage ou les filtres actifs. L'objectif du travail est de contribuer à l'étude en temps discret du fonctionnement de ces systèmes lorsqu'ils sont munis d'une commande numérique dont la période d'échantillonnage est synchronisée sur la période de modulation de largeur d'impulsions. Pour cela nous avons développé une méthode originale qui consiste à écrire les équations différentielles qui décrivent la dynamique du système en remplaçant sur chaque période de modulation les fonctions binaires qui représentent les états successifs (ON ou OFF) des semi-conducteurs de puissance par un développement limité en série de Fourier. En introduisant de manière judicieuse dans les équations différentielles du système, des termes harmoniques des fonctions binaires qui décrivent les états des interrupteurs, cette manière de procéder nous a permis de décomposer l'étude en deux étapes : · dans une première étape en limitant le développement en série de Fourier des fonctions binaires décrivant les états des interrupteurs à leur premier terme (c'est-à-dire à leurs valeurs moyennes) nous avons construit un modèle discret d'ordre zéro qui rend compte de l'effet moyen de la découpe MLI, · dans une deuxième étape nous avons construit un modèle dynamique approché de l'écart existant entre le modèle d'ordre zéro et le modèle détaillé du système. Ce modèle fournit une bonne estimation des ondulations induites par la découpe MLI dans les grandeurs du système. Pour le cas des entraînements par moteurs synchrones à aimants permanents alimentés par onduleurs de tension nous avons ainsi pu mettre en évidence que l'étude de la stabilité des boucles des courants du moteur à partir du modèle d'ordre zéro reste valable en présence des ondulations dues à la MLI pour autant qu'on utilise une MLI symétrique avec rafraîchissement des références une fois par période. / Nowadays, a large number of systems are using Pulse Width Modulation (PWM) Power Electronic Converters as control part. These systems can, for example, be variable-speed drives, switching power supplies or active filters. The goal of this work is to contribute to the study of operation of this type of system when they are equipped with a digital control part of which the sampling period is synchronized on the modulation one. After having shown how the equations of these systems can be written down and indicated how their evolution in closed loop operation can be described by using iterative maps, we have tried to develop an approach which allows simplifying this study. The iterative map method has been used in order to take into account the “sequential “character of these systems, i.e. the fact that the control signals are sequences of events which correspond to the changes of the state of the converter semiconductor switches. Therefore we have developed an original method which consists in replacing on each modulation period, in the differential equations describing the system dynamics, the binary functions representing the successive ON or OFF states of the power semiconductors, by a limited Fourier series development of these functions. This method has allowed splitting the study of these systems into two steps: • in the first step a “zero order” model was built. This first model, which provides the average effect of the PWM pulse pattern, has been obtained by limiting the Fourier series development of the binary functions describing the states of the converter switches, to their first term, the one corresponding to their average values on the PWM period, • in the second step we have introduced an approached dynamic model characterising the error between the zero order model and the exact model of the system. This model was built by considering a well chosen number of harmonic terms of the binary functions describing the ON-OFF states of the switches in the differential equations of the system. By combining the error model and the zero order model we have been able to estimate in which measure the ripples induced by the PWM modulation affect the results of the system stability study made by using the zero order model. For the case of Permanent Magnets Synchronous Motors fed by Voltage Source Inverters we have shown that the study of the stability of the motor currents loops made with the zero order model remains valid in the presence of ripples induced by the PWM, as long as a symmetrical modulation is used, with references which are refreshed one time on each switching period.

L'objectif de cette thèse est de développer des techniques de modélisation et de simulation multi-échelle avancées et efficaces pour étudier les phénomènes d'instabilité dans trois structures d'ingénierie courantes: membrane, film/substrat et structures sandwich, en combinant la technique des coefficients de Fourier lentement variables (TSVFC) et la méthode numérique asymptotique (ANM). À cette fin, basée sur les équations de la plaque de Von Karman, la TSVFC été utilisée pour développer un modèle de Fourier à bidimensionnel (2D) qui a également été implémenté dans ABAQUS via sa sous-routine UEL. Ensuite, un 2D modèle de Fourier est construit pour le film/substrat. En outre, en utilisant leurs caractéristiques de déformation, un 1D modèle de Fourier est développé en utilisant à la fois le TSVFC et le CUF. Par la suite, sur la base d'une cinématique Zig-Zag d'ordre supérieur, un 2D modèle de Fourier est déduit pour une plaque sandwich. Les équations directrices pour les modèles ci-dessus sont discrétisées par la méthode des éléments finis, et les systèmes non linéaires résultants sont résolus par le solveur non linéaire efficace et robuste ANM. Ces modèles sont ensuite adoptés pour étudier les instabilités dans ces structures. Les résultats montrent que les modèles établis peuvent simuler avec précision et efficacité divers phénomènes d'instabilité. En outre, on constate que l'instabilité membranaire est sensible aux conditions aux limites et qu'il existe un paramètre sans dimension presque constant près du point de bifurcation pour différents cas de charge et paramètres géométriques, ce qui peut être utile pour prédire rapidement l'apparition des rides / The main aim of this thesis is to develop advanced and efficient multi-scale modeling and simulation techniques to study instability phenomena in three common engineering structures, i.e., membrane, film/substrate and sandwich structures, by combining the Technique of Slowly Variable Fourier Coefficients (TSVFC) and the Asymptotic Numerical Method (ANM). Towards this end, based on the Von Karman plate equations, the TSVFC has been firstly used to develop a two-dimensional (2D) Fourier double-scale model for membrane, which has also been implemented into ABAQUS via its subroutine UEL. Then a 2D Fourier model is constructed for film/substrate. Further, making use of deformation features of the film/substrate, a 1D Fourier model is developed by using both the TSVFC and the Carrera’s Unified Formulation (CUF). Subsequently, based on high-order kinematics belonging to Zig-Zag theory, a 2D Fourier model is deduced for sandwich plate. The governing equations for the above models are discretized by the Finite Element Method, and the resulting nonlinear systems are solved by the efficient and robust nonlinear solver ANM. These models are then adopted to study instabilities in these structures. Results show that the established models could accurately and efficiently simulate various instability phenomena. Besides, it’s found that the membrane instability is very sensitive to boundary conditions, and there exists a dimensionless parameter that is almost constant near bifurcation point for various loading cases and geometric parameters, which may be helpful for fast predicting the occurrence of wrinkles

Ce travail est constitué de deux parties indépendantes traitant chacune d'un problème issu de la théorie du contrôle des équations aux dérivées partielles. La première partie est consacrée à l'étude d'un feedback explicite et déjà connu, s'appliquant à des systèmes linéaires, réversibles en temps et éventuellement munis d'un opérateur de contrôle non-borné. On justifie le caractère bien posé du problème en boucle fermée via la théorie des semi-groupes puis on étudie le taux de décroissance des solutions du système régulé. La seconde partie concerne un problème d'observation pour la corde vibrante : on détermine comment choisir des instants d'observation pour que la position de la corde à ces instants permette de retrouver les conditions initiales tout en préservant une certaine régularité. La méthode, qui repose sur des résultats d'approximation diophantienne, est ensuite étendue à d'autres systèmes. En utilisant une méthode de dualité on démontre aussi un résultat de contrôlabilité exacte. / This works contains two independent parts, each one dealing with the control of partial differential equations. In the first part, we study an explicit and already known feedback law that applies to linear, time-reversible systems, with a possibly unbounded control operator. We prove the well-posedness of the closed-loop problem in the semi-group framework and we study the decay rate of the solutions. In the second part, we give conditions on the choice of some time instants, such that the positions of a vibrating string (or beam) at these times enable to recover the initial data. The method relies on Diophantine approximation results. Using a duality method, we give a related exact controllability result.

Neste trabalho definimos o operador não local, raiz quadrada do Laplaciano com condição de fronteira de Neumann, através do método de extensão harmônica. O estudo foi feito com o auxílio das séries de Fourier em domínios limitados, como sendo o intervalo, o quadrado e a bola. Posteriormente, aplicamos nosso estudo, à problemas elípticos não lineares envolvendo o operador não local raiz quadrada do Laplaciano com condição de fronteira de Neumann. / In this work we define the non-local operator, square root of the Laplacian with Neumann boundary condition, using the method of harmonic extension. The study was done with the aid of Fourier series in bounded domains, as the interval, the square and the ball. Subsequently, we apply our study, the nonlinear elliptic problems involving non-local operator square root of the Laplacian with Neumann boundary condition.

Made available in DSpace on 2014-07-29T14:52:09Z (GMT). No. of bitstreams: 1 tese jorge antonini.pdf: 527031 bytes, checksum: f95363ba3889fe76020a491c3ec28afe (MD5) Previous issue date: 2009-08-14 / The regional climatic conditions in the State of Goias Brazil are favorable for herbaceous cotton (Gossypium hirsutum L. r. latifolium hutch) cultivation. However, for achieving the best productivities, it is important that both the planting date and the fruiting period be matched with the adequate soil-water availability as well as that the period from open bull to harvesting be coincident with the dry period. Thus, the knowledge of cotton cycle as function of planting location is very important for choosing the optimum planting date. In this context, the air temperature is one of the climatic variables that mostly influence the cotton growth. Nevertheless, the low density of meteorological stations with capability for measuring temperature has restricted the modeling studies for estimating cotton cycle. This work was carried out with the objective of developing and validating mathematical models to estimate average daily air temperature and based on the degreesday theory, the cycle of herbaceous cotton in the State of Goias, considering altogether its variations in space and time. Both models were based on a linear combination of elevation, latitude, longitude, and the daily time variation, represented by an incomplete Fourier series. The parameter models were adjusted to the data from 21 meteorological stations available in the State of Goiás and Federal District of Brazil, using multiple linear regressions with observations varying from eight to twenty four years. In the case of modeling degrees-day, the maximum and minimum temperature data were limited between 15°C and 40°C, which were taken as the lower and upper threshold temperatures, respectively. The air-temperature model was validated against the measured data from three meteorological stations from different elevations: high (1100 m), medium (554 m) and low (431 m). The coefficients of determination obtained from fitting the models for both daily air-temperature and daily degrees-day were 0.82 and 0.84, respectively, resulting in a medium performance for both low and high altitudes and very good for intermediate altitudes. The validation of the degrees-day model was conducted by comparing the period duration running from crop emergence to 90% open bulls observed from cotton cultivars, cropped in commercial fields. The results showed an overall performance index of 0.85, which was considered as very good. The models developed in this study adequately estimated the average daily air temperature and the cycle of herbaceous cotton cultivars in the State of Goias / As condições climáticas regionais do Estado de Goiás são favoráveis ao cultivo do algodoeiro herbáceo (Gossypium hirsutum L. r. latifolium hutch), contudo, para alcançar as melhores produtividades, é imprescindível que a semeadura e o período de frutificação coincidam com o período de maior disponibilidade de água no solo e os períodos de abertura dos capulhos e da colheita coincidam com o período seco. Assim, o conhecimento acerca do ciclo da cultura em função do local de cultivo, é muito importante na definição da melhor época de plantio. Nesse contexto, é a temperatura do ar, uma das variáveis climáticas, que mais influencia o desenvolvimento do algodoeiro. No entanto, a baixa densidade de estações meteorológicas com capacidade de medição da temperatura tem limitado os estudos de modelagem de estimativa do ciclo deste cultivo. Este trabalho foi conduzido com o objetivo de desenvolver e validar modelos matemáticos para estimar a temperatura média diária do ar e, com base na teoria de graus-dia, o ciclo do algodoeiro herbáceo no Estado de Goiás, considerando, simultaneamente, suas variações no espaço e no tempo. Ambos os modelos basearam-se em uma combinação linear da altitude, latitude, longitude e da variação temporal diária, representada pela série trigonométrica incompleta de Fourier. Os parâmetros dos modelos foram ajustados aos dados de 21 estações meteorológicas disponíveis no Estado de Goiás e Distrito Federal, por meio de regressão linear múltipla, com observações variando de 8 a 24 anos. No caso da modelagem de graus-dia, os dados de temperatura máxima e mínima ficaram restritos ao intervalo de 15°C a 40°C, cujos limites foram adotados como os valores de temperatura de base inferior e superior, respectivamente. O modelo de temperatura foi validado, considerando os dados observados de temperatura em estações localizadas em condições de altitudes diferentes: elevada (1100 m), média (554 m) e baixa (431 m). Os coeficientes de determinação resultantes do ajuste dos modelos aos dados de temperatura média diária ou aos de grausdia foram 0,82 e 0,84, respectivamente. O desempenho do modelo foi mediano nas altitudes baixas e elevadas e muito bom nas altitudes médias. A validação do modelo de graus-dia foi feita comparando-se a duração observada do período entre a emergência e 90% de capulhos abertos de cultivares de algodoeiro, plantados em lavouras comerciais, resultando em um índice de desempenho de 0,85, classificado como muito bom. Os modelos desenvolvidos estimaram adequadamente a temperatura média diária do ar e a duração do ciclo dos cultivares de algodoeiro herbáceo no Estado de Goiás

Neste trabalho definimos o operador não local, raiz quadrada do Laplaciano com condição de fronteira de Neumann, através do método de extensão harmônica. O estudo foi feito com o auxílio das séries de Fourier em domínios limitados, como sendo o intervalo, o quadrado e a bola. Posteriormente, aplicamos nosso estudo, à problemas elípticos não lineares envolvendo o operador não local raiz quadrada do Laplaciano com condição de fronteira de Neumann. / In this work we define the non-local operator, square root of the Laplacian with Neumann boundary condition, using the method of harmonic extension. The study was done with the aid of Fourier series in bounded domains, as the interval, the square and the ball. Subsequently, we apply our study, the nonlinear elliptic problems involving non-local operator square root of the Laplacian with Neumann boundary condition.