In this dissertation, a new numerical method for solving the fractional dynamical systems, is presented. We first introduce Riemann-Liouville fractional integral operator for hybrid functions. Then we will show the spectral accuracy of the present method for solving fractional-order differential equations, and we will extend the present method for solving nonlinear fractional integro-differential equations, fractional Bagley-Torvik equation, distributed order fractional differential equations, two-dimensional fractional partial differential equations, and fractional optimal control problems. In all cases, we will show the rate of convergence is more than some existing numerical methods which were used to solve these kind of problems in the literature. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Cardiovascular diseases (CVDs) remain the leading cause of death worldwide. Patients at risk of evolving CVDs are assessed by evaluating a risk factor-based score that incorporates different bio-markers ranging from age and sex to arterial stiffness (AS). AS depicts the rigidity of the arterial vessels and leads to an increase in the arterial pulse pressure, affecting the heart and vascular physiology. These facts have encouraged researchers to propose surrogate markers of cardiovascular risks and develop simple and non-invasive models to better understand cardiovascular system operations. This work thus fundamentally capitalizes on developing a novel class of low-dimensional physics-based fractional-order models of systemic arteries and exploring the feasibility of fractional differentiation order to portray the vascular stiffness. Fractional-order modeling is a successful paradigm to integrate multiscale and interconnected mechanisms of the complex arterial system. However, this type of modeling alone often fails to efficiently integrate altered variabilities in vascular physiology from various sources of large datasets, multi-modalities, and levels. In this regard, combining fractional-order-based approaches with machine learning techniques presents a unique opportunity to develop a powerful prediction framework that reveals the correlation between intertwined vascular events. This work is divided into three parts. The first part contributes to developing the fractional-order lumped parametric model of the arterial system. First, we propose fractional-order representations to model and characterize the complex and frequency-dependent apparent arterial compliance. Second, we propose fractional-order arterial Windkessel modeling the aortic input impedance and hemodynamic. Subsequently, the proposed models have been applied and validated using both human in-silico healthy datasets and real vascular aging and hypertension. The second part addresses the non-zero initial value problem for fractional differential equations (FDEs) and proposes an estimation technique for joint estimation of the input, parameters, and fractional differentiation order of non-commensurate FDEs. The performance of the proposed estimation techniques is illustrated on arterial and neurovascular hemodynamic response models. The third part explores the feasibility of using machine learning algorithms to estimate the gold-standard measurement of AS, carotid-to-femoral pulse wave velocity. Different modalities have been investigated to generate informative input features and reduce the dimensionality of the time series pulse waves.

<div>In this thesis work, a novel approach for the design of a fractional order proportional integral</div><div>derivative (FOPID) controller is proposed. This design introduces a new time-varying FOPID controller</div><div>to mitigate a voltage spike at the controller output whenever a sudden change to the setpoint occurs. The</div><div>voltage spike exists at the output of the proportional integral derivative (PID) and FOPID controllers when a</div><div>derivative control element is involved. Such a voltage spike may cause a serious damage to the plant if it is</div><div>left uncontrolled. The proposed new FOPID controller applies a time function to force the derivative gain to</div><div>take effect gradually, leading to a time-varying derivative FOPID (TVD-FOPID) controller, which maintains</div><div>a fast system response and signi?cantly reduces the voltage spike at the controller output. The time-varying</div><div>FOPID controller is optimally designed using the particle swarm optimization (PSO) or genetic algorithm</div><div>(GA) to ?nd the optimum constants and time-varying parameters. The improved control performance is</div><div>validated through controlling the closed-loop DC motor speed via comparisons between the TVD-FOPID</div><div>controller, traditional FOPID controller, and time-varying FOPID (TV-FOPID) controller which is created</div><div>for comparison with all three PID gain constants replaced by the optimized time functions. The simulation</div><div>results demonstrate that the proposed TVD-FOPID controller not only can achieve 80% reduction of voltage</div><div>spike at the controller output but also is also able to keep approximately the same characteristics of the system</div><div>response in comparison with the regular FOPID controller. The TVD-FOPID controller using a saturation</div><div>block between the controller output and the plant still performs best according to system overshoot, rise time,</div><div>and settling time.</div>

Cette thèse vise à concevoir des estimateurs non-asymptotiques et robustes pour les systèmes linéaires d’ordre fractionnaire dans un environnement bruité. Elle traite une classe des systèmes linéaires d’ordre fractionnaire modélisée par la dite pseudo représentation d’état avec des conditions initiales inconnues. Elle suppose également que les systèmes étudiés ici peuvent être transformés sous la forme canonique de Brunovsky. Pour estimer le pseudo-état, la forme précédente est transformée en une équation différentielle linéaire d’ordre fractionnaire en prenant en compte les valeurs initiales des dérivées fractionnaires séquentielles de la sortie. Ensuite, en utilisant la méthode des fonctions modulatrices, les valeurs initiales précédentes et les dérivées fractionnaires avec des ordres commensurables de la sortie sont données par des formules algébriques avec des intégrales à l’aide d’une méthode récursive. Ainsi, ces formules sont utilisés pour calculer le pseudo-état dans le cas continu sans bruit. En outre, elle fournit un algorithme pour construire les fonctions modulatrices requises à l’accomplissement de l’estimation. Deuxièmement, inspiré par la méthode des fonctions modulatrices développée pour l’estimation de pseudo-état, cette méthode algébrique basée sur un opérateur est introduite pour estimer la dérivée fractionnée avec un ordre arbitraire fractionnaire de la sortie pour les systèmes considérés. Cet opérateur sert à annuler les valeurs initiales non désirées, puis permet d’estimer la dérivée fractionnaire souhaitée par une nouvelle formule algébrique à l’aide d’une méthode récursive. Troisièmement, l’estimateur du pseudo-état et le différenciateur d’ordre fractionnaire obtenus précédemment sont étudiés respectivement dans le cas discret et bruité. Chacun d’entre eux contient une erreur numérique due à la méthode d’intégration numérique utilisée et au bruit. En particulier, elle fournit une analyse pour diminuer la contribution du bruit au moyen d’une d’erreur bornée qui permet de sélectionner les degrés optimaux des fonctions de modulation à chaque instant. Ensuite, des exemples numériques sont donnés pour mettre en évidence la précision, la robustesse et la propriété non-asymptotique des estimateurs proposés. En outre, les comparaisons avec certaines méthodes existantes et avec un nouvel observateur d’ordre fractionnaire de typeH1sont montrées. Enfin, elle donne des conclusions / This thesis aims to design non-asymptotic and robust estimators for a class of fractional order linear systems in noisy environment. It deals with a class of commensurate fractional order linear systems modeled by the so-called pseudo-state space representation with unknown initial conditions. It also assumed that linear systems under study can be transformed into the Brunovsky’s observable canonical form. Firstly, the pseudo-state of the considered systems is estimated. For this purpose, the Brunovsky’s observable canonical form is transformed into a fractional order linear differential equation involving the initial values of the fractional sequential derivatives of the output. Then, using the modulating functions method, the former initial values and the fractional derivatives with commensurate orders of the output are given by algebraic integral formulae in a recursive way. Thereby, they are used to calculate the pseudo-state in the continuous noise-free case. Moreover, to perform this estimation, it provides an algorithm to build the required modulating functions. Secondly, inspired by the modulating functions method developed for pseudo-state estimation, an operator based algebraic method is introduced to estimate the fractional derivative with an arbitrary fractional order of the output. This operator is applied to cancel the former initial values and then enables to estimate the desired fractional derivative by a new algebraic formula using a recursive way. Thirdly, the pseudo-state estimator and the fractional order differentiator are studied in discrete noisy case. Each of them contains a numerical error due to the used numerical integration method, and the noise error contribution due to a class of stochastic processes. In particular, it provides ananalysis to decrease noise contribution by means of an error bound that enables to select the optimal degrees of the modulating functions at each instant. Then, several numerical examples are given to highlight the accuracy, the robustness and the non-asymptotic property of the proposed estimators. Moreover, the comparisons to some existing methods and a new fractional orderH1-like observer are shown. Finally, conclusions are outlined with some perspectives

Infinite-Dimensional Systems (IDSs) which have been made possible by recent advances in mathematical and computational tools can be used to model complex real phenomena. However, due to physical, economic, or stringent non-invasive constraints on real systems, the underlying characteristics for mathematical models in general (and IDSs in particular) are often missing or subject to uncertainty. Therefore, developing efficient estimation techniques to extract missing pieces of information from available measurements is essential. The human brain is an example of IDSs with severe constraints on information collection from controlled experiments and invasive sensors. Investigating the intriguing modeling potential of the brain is, in fact, the main motivation for this work. Here, we will characterize the hemodynamic behavior of the brain using functional magnetic resonance imaging data. In this regard, we propose efficient estimation methods for two classes of IDSs, namely Partial Differential Equations (PDEs) and Fractional Differential Equations (FDEs). This work is divided into two parts. The first part addresses the joint estimation problem of the state, parameters, and input for a coupled second-order hyperbolic PDE and an infinite-dimensional ordinary differential equation using sampled-in-space measurements. Two estimation techniques are proposed: a Kalman-based algorithm that relies on a reduced finite-dimensional model of the IDS, and an infinite-dimensional adaptive estimator whose convergence proof is based on the Lyapunov approach. We study and discuss the identifiability of the unknown variables for both cases. The second part contributes to the development of estimation methods for FDEs where major challenges arise in estimating fractional differentiation orders and non-smooth pointwise inputs. First, we propose a fractional high-order sliding mode observer to jointly estimate the pseudo-state and input of commensurate FDEs. Second, we propose a modulating function-based algorithm for the joint estimation of the parameters and fractional differentiation orders of non-commensurate FDEs. Sufficient conditions ensuring the local convergence of the proposed algorithm are provided. Subsequently, we extend the latter technique to estimate smooth and non-smooth pointwise inputs. The performance of the proposed estimation techniques is illustrated on a neurovascular-hemodynamic response model. However, the formulations are efficiently generic to be applied to a wide set of additional applications.

L'adoption récente et généralisée des systèmes d'automatisation des véhicules, avec l’incorporation de la connectivité entre voitures, a encouragé l’utilisation des techniques comme le Contrôle Croisière Adaptatif Coopératif (CACC) et la conduite en convoi. Ces techniques ont prouvé l’amélioration du flux de trafic et la sécurité de la conduite, tout en réduisant la consommation d’énergie et les émissions CO_2. Néanmoins, la robustesse et la stabilité stricte du convoi, malgré les délais de communication et l’hétérogénéité des convois, restent des sujets de recherche en cours. Cette thèse a pour sujet la conception, l’analyse et validation de systèmes de contrôle pour le car-following automatisé et coopératif, en ciblant l’augmentation de ses avantages et son usage, en se concentrant sur la robustesse et la stabilité du convoi même sur des séries de véhicules hétérogènes avec des retards de communication. Une structure feedforward/feedback est développée, dont sa modularité est fondamentale pour la mise au point des approches avec des objectifs différents mais complémentaires. L’architecture permet non seulement l’adoption d’une stratégie d’espacement pour la range entière de vitesse, mais elle peut aussi être employée dans le cadre d’un CACC basé sur une machine d’état pour la conduite en convoi sur des environnements urbains avec des capacités de freinage d’urgence et de rejoint du convoi. Des différents algorithmes pour la conception de systèmes de contrôle feedback pour la régulation des distances sont présentés, pour quoi le calcul d’ordre fractionnaire démontre fournir des réponses fréquentielles de boucle fermé plus précises et satisfaire des besoins plus exigeantes. La performance est assurée malgré l’hétérogénéité avec la proposition de deux approches feedforward différents. Le premier est basé sur une topologie en considérant que le véhicule précédent dans la boucle, tandis que le deuxième inclut le véhicule leader pour améliorer la performance de suivi. Les algorithmes proposés sont validés avec des études de stabilité dans le domaine du temps et fréquence, ainsi que simulations et expérimentations réelles. / Recent widespread adoption of vehicle automation and introduction of vehicle-to-vehicle connectivity has opened the doors for techniques as Cooperative Adaptive Cruise Control (CACC) and platooning, showing promising results in terms of traffic capacity and safety improvement, while reducing fuel consumption and CO_2 emissions. However, robustness and strict string stability, despite communication delays and string heterogeneity is still an on-going research field. This thesis deals with the design, study and validation of control systems for cooperative automated car-following, with the purpose of extending their benefits and encourage their employment, focusing on robustness and string stability, despite possible V2V communication delays and string heterogeneity. A feedforward/feedback hierarchical control structure is developed, which modularity is fundamental for the proposal of approaches that target different but complementary performance objectives. The architecture not only permits the adoption of a full speed range spacing policy that target multiple criteria, but can also be employed in a state machine-based CACC framework for urban environments with emergency braking and platoon re-joining capabilities in case of pedestrian interaction. Different feedback control design algorithms are presented for the gap-regulation, for which the fractional-order calculus is demonstrated to provide more accurate closed loop frequency responses and satisfy more demanding requirements. Desired performance is ensured in spite of string heterogeneity through the proposal of two feedforward methods : one based on predecessor-only topology, while the second includes the leader vehicle information on feedforward to gain tracking capabilities. Proposed control algorithms are validated through time and frequency-domain stability studies, simulation and real platforms experiments.

A inclusão de fontes alternativas no sistema de energia elétrica possibilita a otimização dos recursos naturais disponíveis para geração. A conexão dessas fontes deve ser efetuada de forma a evitar índices de qualidade de energia incondizentes aos limites estabelecidos por normas nacionais e internacionais. Nesse contexto, sistemas de Geração Distribuída (GD) conectados à rede de distribuição via controle tensão/potência utilizam controladores Proporcional+Ressonantes (P+R) para produzir tensões com mínimas distorções na presença de cargas não lineares. Além disso, esse tipo de solução apresenta resultados satisfatórios na presença de outros tipos de carga (linear, não linear e conexão de máquina de indução), entretanto, para uma carga RLC com frequência de ressonância igual à frequência fundamental, o sistema de GD torna-se instável e as distorções harmônicas ultrapassam os limites mínimos estabelecidos para assegurar a qualidade de energia. Dessa forma, essa dissertação de mestrado propõe a substituição do controlador P+R da frequência fundamental por um controlador de ordem fracionária PI&#955;, projetado através de um algoritmo de otimização heurística. Os resultados obtidos através de simulação demonstram que o controlador proposto apresenta resultados satisfatórios para todas as cargas testadas: Não Linear; Motor de Indução Trifásico e RLC com frequência de ressonância igual à fundamental da rede. Entretanto, os resultados experimentais mostraram uma sensibilidade quanto à mudança de carga e quanto aos parâmetros de discretização do controlador, sugerindo a necessidade de uma análise mais rigorosa na etapa de projeto de modo a se ter robustez quanto à variação de carga. / The insertion of alternative sources in the power system enables the optimization of natural resources available for electrical generation. The connection of these sources to the grid has to be made in order to avoid indices of power quality greater than the limits established by national and international standards. In this context, a Distributed Generation (DG) systems connected to the grid via voltage/power control use a proportional+resonant (P+R) controller to produce not only the fundamental frequency but also to reduce the voltage distortion when nonlinear loads are connected to the DG terminals. Additionally, this type of solution shows satisfactory results for linear, balanced, unbalanced and direct connection of induction machine as well. However, for parallel RLC load with frequency of resonance equals to the fundamental frequency of the grid, the DG system operation becomes unstable with high levels of harmonic distortions and an oscillatory behavior. In this context, we propose the replacement of the PR controller for the fundamental frequency by a fractional order controller PI&#955; designed via a heuristic optimization algorithm. To prove the feasibility of the proposed approach a set of simulations are presented, however, the experimental prototype shown the sensitivity of the controller for different loads and for the controller discretization parameters suggesting the need for more analysis during the design procedures of the PI&#955; controller to achieve the expected robustness.

A inclusão de fontes alternativas no sistema de energia elétrica possibilita a otimização dos recursos naturais disponíveis para geração. A conexão dessas fontes deve ser efetuada de forma a evitar índices de qualidade de energia incondizentes aos limites estabelecidos por normas nacionais e internacionais. Nesse contexto, sistemas de Geração Distribuída (GD) conectados à rede de distribuição via controle tensão/potência utilizam controladores Proporcional+Ressonantes (P+R) para produzir tensões com mínimas distorções na presença de cargas não lineares. Além disso, esse tipo de solução apresenta resultados satisfatórios na presença de outros tipos de carga (linear, não linear e conexão de máquina de indução), entretanto, para uma carga RLC com frequência de ressonância igual à frequência fundamental, o sistema de GD torna-se instável e as distorções harmônicas ultrapassam os limites mínimos estabelecidos para assegurar a qualidade de energia. Dessa forma, essa dissertação de mestrado propõe a substituição do controlador P+R da frequência fundamental por um controlador de ordem fracionária PI&#955;, projetado através de um algoritmo de otimização heurística. Os resultados obtidos através de simulação demonstram que o controlador proposto apresenta resultados satisfatórios para todas as cargas testadas: Não Linear; Motor de Indução Trifásico e RLC com frequência de ressonância igual à fundamental da rede. Entretanto, os resultados experimentais mostraram uma sensibilidade quanto à mudança de carga e quanto aos parâmetros de discretização do controlador, sugerindo a necessidade de uma análise mais rigorosa na etapa de projeto de modo a se ter robustez quanto à variação de carga. / The insertion of alternative sources in the power system enables the optimization of natural resources available for electrical generation. The connection of these sources to the grid has to be made in order to avoid indices of power quality greater than the limits established by national and international standards. In this context, a Distributed Generation (DG) systems connected to the grid via voltage/power control use a proportional+resonant (P+R) controller to produce not only the fundamental frequency but also to reduce the voltage distortion when nonlinear loads are connected to the DG terminals. Additionally, this type of solution shows satisfactory results for linear, balanced, unbalanced and direct connection of induction machine as well. However, for parallel RLC load with frequency of resonance equals to the fundamental frequency of the grid, the DG system operation becomes unstable with high levels of harmonic distortions and an oscillatory behavior. In this context, we propose the replacement of the PR controller for the fundamental frequency by a fractional order controller PI&#955; designed via a heuristic optimization algorithm. To prove the feasibility of the proposed approach a set of simulations are presented, however, the experimental prototype shown the sensitivity of the controller for different loads and for the controller discretization parameters suggesting the need for more analysis during the design procedures of the PI&#955; controller to achieve the expected robustness.

L’objectif de cette thèse est de comprendre le comportement cinématique d’interception latérale manuelle et locomotrice. Dans l’introduction nous montrons que l’étude du mouvement d’interception peut être engagée avec les concepts de la perception directe et du couplage perception-action. De nombreux travaux ont montré qu’une stratégie prospective serait mise en oeuvre pour intercepter un mobile mais un débat existe quant à la double question (i) quelle information est utilisée pour réguler le mouvement ? (ii) comment est utilisée cette information ? Pour répondre nous avons utilisé la réalité virtuelle. Dans le premier chapitre expérimental nous montrons que les variables environnementales qui ont émergées de la littérature ne peuvent expliquer seules les régulations du mouvement d’interception. Dans le deuxième chapitre expérimental nous proposons qu’une variable d’ordre intermédiaire plutôt qu’entier pourrait guider le mouvement. Enfin dans le troisième chapitre expérimental nous apportons des éléments de preuve en faveur du concept d’information d’ordre fractionnaire qui augmenterait au cours de l’action. Nous développons dans ces deux dernières études deux modèles dynamiques qui nous permettent de simuler de façon satisfaisante le comportement cinématique d’interception latérale pour une situation donnée. Plus généralement, nous suggérons, quant à l’ensemble des modèles dynamiques formalisant les interactions perception-action, que cette proposition conceptuelle pourrait permettre de plus larges avancées encore si l’on considérait l’éventuelle sensibilité des différents organismes vivants aux informations d’ordre intermédiaire plutôt que nécessairement entier. / The goal of the present thesis was to understand the operative perception-action relations underlying the kinematic patterns of movement observed in manual and locomotor lateral interception. The introduction sets the stage for this endeavor, calling upon the theoretical concepts of direct perception and information-movement coupling, understood as a particular form of the online control of movement. Whereas a large body of work has demonstrated that interception is based on a prospective control strategy, debate persists as to the double question of (i) what information is that used to regulate the movement and (ii) how this information is used in the regulation of movement. To address these questions our general methodology was based on interception tasks presented in virtual reality settings. In the first experimental chapter we demonstrate that neither of the two variables proposed in the literature could by themselves explain the kinematic patterns observed. Integrating an modeling approach based on attractor dynamics, the second experimental chapter provides first evidence for reliance on a variable of intermediate rather than integer order. In the third experimental chapter we take our modeling approach one step further and demonstrate that reliance on information of a trajectory-dependent fractional order allows capturing the qualitative and quantitative aspects of the interception behavior observed. In the final chapter we discuss the suggested potential of opening up the information space to non-integer orders for the understanding of perception-action relations.

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.