This thesis solves estimation and control problems in computational neuroscience, mathematically dealing with the first-passage times of diffusion stochastic processes. We first derive estimation algorithms for model parameters from first-passage time observations, and then we derive algorithms for the control of first-passage times. Finally, we solve an optimal design problem which combines elements of the first two: we ask how to elicit first-passage times such as to facilitate model estimation based on said first-passage observations. The main mathematical tools used are the Fokker-Planck partial differential equation for evolution of probability densities, the Hamilton-Jacobi-Bellman equation of optimal control and the adjoint optimization principle from optimal control theory. The focus is on developing computational schemes for the solution of the problems. The schemes are implemented and are tested for a wide range of parameters.

A ground vehicle is a dynamic system containing many non-linear components, ranging from the non-linear engine response to the tyre-road interface. In pursuit of developing driver-assist systems for accident avoidance, as well as fully autonomous vehicles, the application of modern mechatronics systems to vehicles are widely investigated. Extensive work has been done in an attempt to model and control the lateral response of the vehicle system utilising a wide variety of conventional control and intelligent systems theory. The majority of driver models are however intended for low speed applications where the vehicle dynamics are fairly linear. This study proposes the use of adaptive control strategies as robust driver models capable of steering the vehicle without explicit knowledge of vehicle parameters. A Model Predictive Controller (MPC), self-tuning regulator and Linear Quadratic Self-Tuning Regulator (LQSTR) updated through the use of an Auto Regression with eXogenous input (ARX) model that describes the relation between the vehicle steering angle and yaw rate are considered as solutions. The strategies are evaluated by performing a double lane change in simulation using a validated full vehicle model in MSC ADAMS and comparing the maximum stable speed and lateral offset from the required path. It is found that all the adaptive controllers are able to successfully steer the vehicle through the manoeuvre with no prior knowledge of the vehicle parameters. An LQSTR proves to be the best adaptive strategy for driver model applications, delivering a stable response well into the non-linear tyre force regime. This controller is implemented on a fully instrumented Land Rover 110 of the Vehicle Dynamics Group at the University of Pretoria fitted with a semi-active spring-damper suspension that can be switched between two discrete setting representing opposite extremes of the desired response namely: ride mode (soft spring and low damping) and handling mode (stiff spring and high damping). The controller yields a stable response through a severe double lane change (DLC) up to the handling limit of the vehicle, safely completing the DLC at a maximum speed of 90 km/h all suspension configurations. The LQSTR also proves to be robust by following the same path for all suspension configurations through the manoeuvre for vehicle speeds up to 75 km/h. Validation is continued by successfully navigating the Gerotek dynamic handling track, as well as by performing a DLC manoeuvre on an off-road terrain. The study successfully developed and validated a driver model that is robust against variations in vehicle parameters and friction coefficients. / Dissertation (MEng)--University of Pretoria, 2015. / Mechanical and Aeronautical Engineering / Unrestricted

En astronautique, une question importante est de contrôler le mouvement d’un satellite soumis à la gravitation des corps célestes de telle sorte que certains indices de performance soient minimisés (ou maximisés). Dans cette thèse, nous nous intéressons à la minimisation de la norme L¹ du contrôle pour le problème circulaire restreint des trois corps. Les conditions nécessaires à l’optimalité sont obtenues en utilisant le principe du maximum de Pontryagin, révélant l’existence de contrôles bang-bang et singuliers. En s’appuyant sur les résultats de Marchal [1] et Zelikin et al. [2], la présence du phénomène de Fuller est mise en évidence par l’analyse des es extrêmales singulières. La contrôlabilité pour le problème à deux corps (un cas dégénéré du problème circulaire restreint des trois corps) avec un contrôle prenant des valeurs dans une boule euclidienne est caractérisée dans le chapitre 2. Le résultat de contrôlabilité est facilement étendu au problème des trois corps puisque le champ de vecteurs correspondant à la dérive est récurrent. En conséquence, si les trajectoires contrôlées admissibles restent dans un compact fixé, l’existence des solutions du problème de minimisation L¹ peut être obtenu par une combinaison du théorème de Filippov (voir [4, chapitre 10]) et une procédure appropriée de convexification (voir [5]). En dimension finie, le problème de minimisation L¹ est bien connu pour générer des solutions où le contrôle s’annule sur certains intervalles de temps. Bien que le principe du maximum de Pontryagin soit un outil puissant pour identifier les solutions candidates pour le problème de minimisation L¹, il ne peut pas garantir que ces candidats sont au moins localement optimaux sauf si certaines conditions d’optimalité suffisantes sont satisfaites. En effet, il est une condition préalable pour établir (et pour être capable de vérifier) les conditions d’optimalité nécessaires et suffisantes pour résoudre le problème de minimisation L¹. Dans cette thèse, l’idée cruciale pour obtenir de telles conditions est de construire une famille paramétrée d’extrémales telle que l’extrémale de référence peut être intégrée dans un champ d’extrémales. Deux conditions de non-pliage pour la projection canonique de la famille paramétrée d’extrémales sont proposées. En ce qui concerne le cas de points terminaux fixés, ces conditions de non-pliage sont suffisantes pour garantir que l’extrémale de référence est localement minimisante tant que chaque point de commutation est régulier (cf. chapitre 3). Si le point terminal n’est pas fixe mais varie sur une sous-variété lisse, une condition suffisante supplémentaire impliquant la géométrie de variété de cible est établie (cf. chapitre 4). Bien que diverses méthodes numériques, y compris celles considérées comme directes [6, 7], indirectes [5, 8], et hybrides [11], dans la littérature sont en mesure de calculer des solutions optimales, nous ne pouvons pas attendre d’un satellite piloté par le contrôle optimal précalculé (ou le contrôle nominal) de se déplacer sur la trajectoire optimale précalculée (ou trajectoire nominale) en raison de perturbations et des erreurs inévitables. Afin d’éviter de recalculer une nouvelle trajectoire optimale une fois que la déviation de la trajectoire nominale s’est produite, le contrôle de rétroaction optimale voisin, qui est probablement l’application pratique la plus importante de la théorie du contrôle optimal [12, Chapitre 5], est obtenu en paramétrant les extrémales voisines autour de la nominale (cf. chapitre 5). Étant donné que la fonction de contrôle optimal est bang-bang, le contrôle optimal voisin comprend non seulement la rétroaction sur la direction de poussée, mais aussi celle sur les instants de commutation. En outre, une analyse géométrique montre qu’il est impossible de construire un contrôle optimal voisin une fois que le point conjugué apparaisse ou bien entre ou bien à des instants de commutation. / In astronautics, an important issue is to control the motion of a satellite subject to the gravitation of celestial bodies in such a way that certain performance indices are minimized (or maximized). In the thesis, we are interested in minimizing the L¹-norm of control for the circular restricted three-body problem. The necessary conditions for optimality are derived by using the Pontryagin maximum principle, revealing the existence of bang-bang and singular controls. Singular extremals are analyzed, and the Fuller phenomenon shows up according to the theories developed by Marchal [1] and Zelikin et al. [2, 3]. The controllability for the controlled two-body problem (a degenerate case of the circular restricted three-body problem) with control taking values in a Euclidean ball is addressed first (cf. Chapter 2). The controllability result is readily extended to the three-body problem since the drift vector field of the three-body problem is recurrent. As a result, if the admissible controlled trajectories remain in a fixed compact set, the existence of the solutions of the L¹-minimizaion problem can be obtained by a combination of Filippov theorem (see [4, Chapter 10], e.g.) and a suitable convexification procedure (see, e.g., [5]). In finite dimensions, the L¹-minimization problem is well-known to generate solutions where the control vanishes on some time intervals. While the Pontryagin maximum principle is a powerful tool to identify candidate solutions for L1-minimization problem, it cannot guarantee that the these candidates are at least locally optimal unless sufficient optimality conditions are satisfied. Indeed, it is a prerequisite to establish (as well as to be able to verify) the necessary and sufficient optimality conditions in order to solve the L¹-minimization problem. In this thesis, the crucial idea for establishing such conditions is to construct a parameterized family of extremals such that the reference extremal can be embedded into a field of extremals. Two no-fold conditions for the canonical projection of the parameterized family of extremals are devised. For the scenario of fixed endpoints, these no-fold conditions are sufficient to guarantee that the reference extremal is locally minimizing provided that each switching point is regular (cf. Chapter 3). If the terminal point is not fixed but varies on a smooth submanifold, an extra sufficient condition involving the geometry of the target manifold is established (cf. Chapter 4). Although various numerical methods, including the ones categorized as direct [6, 7], in- direct [5, 8, 9], and hybrid [10], in the literature are able to compute optimal solutions, one cannot expect a satellite steered by the precomputed optimal control (or nominal control) to move on the precomputed optimal trajectory (or nominal trajectory) due to unavoidable perturbations and errors. In order to avoid recomputing a new optimal trajectory once a deviation from the nominal trajectory occurs, the neighboring optimal feedback control, which is probably the most important practical application of optimal control theory [11, Chapter 5], is derived by parameterizing the neighboring extremals around the nominal one (cf. Chapter 5). Since the optimal control function is bang-bang, the neighboring optimal control consists of not only the feedback on thrust direction but also that on switching times. Moreover, a geometric analysis shows that it is impossible to construct the neighboring optimal control once a conjugate point occurs either between or at switching times.

L'optique adaptative a révolutionné l'imagerie astronomique en permettant de corriger en temps réel les déformations du front d'onde introduites l'atmosphère, et d'atteindre la limite de diffraction des télescopes. Plus récemment, différentes modalités d'optique adaptative grand-champ ont permis de repousser les limites d'utilisation de ces instruments, compensant l'anisoplanétisme de l'atmosphère, et la faible couverture du ciel. L'asservissement de ces systèmes est sans aucun doute un point clé pour améliorer encore les performances de ces systèmes, pour ainsi converger vers les programmes scientifiques des futurs ELT.Le premier objectif de ces travaux de thèse est de définir un outil général d'analyse de performance, permettant de comparer, sous la forme de budgets d'erreur, différents régulateurs linéaires. Ceci permet d'améliorer les instruments actuels, ou de faire des choix dans la conception des futurs instruments des ELT. Pour cela, un formalisme d'analyse fréquentielle est développé dans le cadre de l'optique adaptative classique, et étendu au cas grand-champ. On montre que cet outil permet aussi bien de décomposer les performances calculées en simulation qu'à partir de données télémétriques enregistrées sur le ciel. De nouvelles stratégies de commande, basées sur de nouveaux modèles de perturbation sont proposées, et leur apport en performance discuté au regard de leur budget d'erreur. Ces résultats ont servi à la caractérisation d'une commande LQG tip-tilt avec filtrage de vibration qui doit être intégrée à l'instrument d'optique adaptative multi-conjuguée GeMS, au Chili. / Adaptive Optics (AO) systems have revolutionized ground-based astronomical imagery, allowing for real-time compensation of turbulence-induced deformations of the optical wavefront, and therefore allowing to reach the diffraction limit. More recently, wide-field AO modalities have been proposed to expand the operational range of instruments by compensating anisoplanatism and increase sky coverage. Controlling such systems is certainly a key issue to further improve their performance and to converge towards the goals of the ELTs science programs.The first objective of this thesis work is to define a general-purpose performance analysis tool, enabling to compare different linear controllers through their error budgets, in order to improve existing instruments or make choices in the design of future instruments. To achieve this aim, a frequency-domain formalism is developed for single-conjugated AO and extended to wide-field configurations. It is shown that this tool allows to decompose controller performance using either simulations or on-sky data. New control strategies based on new disturbance models (turbulence, vibrations...) are proposed, and the improvement in performance is discussed based on their error budget. Furthermore, these results contributed to characterize an LQG controller with vibration mitigation that is to be integrated in the tip/tilt loop of the multi-conjugate AO system GeMS, at Gemini South Observatory, in Chile.

京都大学 / 新制・課程博士 / 博士(情報学) / 甲第23324号 / 情博第760号 / 新制||情||130(附属図書館) / 京都大学大学院情報学研究科システム科学専攻 / (主査)教授 大塚 敏之, 教授 石井 信, 教授 太田 快人 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

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The Diploma thesis deals with the problem of energy-efficient train control. It presents the basic survey of mathematical models used in the problem of energy-efficient train control, analysis of optimal driving regimes, determining optimal switching times between optimal driving regimes and timetabling of the train. The mathematical formulation of the problem is done using Newton's second law of motion and other known physical laws. To analyse optimal driving regimes and determine the switching times between optimal driving regimes, we apply tools of optimal control theory, particularly Pontryagin's Maximum Principle. The timetabling of the train is discussed from the numerical solution of the settled non-linear programming problem.

Motorsport has become a multidisciplinary sport in which skilled engineers and strategists play as big a part in the team’s success as the athlete driving the car. In Formula One it is common practice for teams to have dedicated resources on the track that are supported by a second team back at the home base who analyses telemetry data and performs simulations to refine the racing strategy. Optimal control calculations are typically used to optimise vehicle setup parameters (such as ride height and heave spring stiffness) and driver inputs (such as braking and steering) specific to each track. Traditionally this approach has been employed by minimising time over a single lap. Although this is useful in qualifying simulations, there is an unexplored element of optimising a vehicle’s "race pace". Drivers complete qualifying laps using minimal fuel with new tyres to get the best possible lap time but this performance cannot be sustained throughout the whole race. Drivers need to manage their tyres so that they do not wear prematurely and have a detrimental effect on their performance. This work places an emphasis on tyre modelling and in particular how optimal control can be used to optimise a tyre management strategy. A model has been presented that reduces grip as a function of tyre wear. This ensures that the qualifying pace cannot be sustained indefinitely. A thermodynamic model consisting of two states (surface and carcass temperature) is used to calculate tyre wear, which ultimately dictates how much grip can be provided by each tyre. The objective function for the optimal control problem is to minimise time over multiple laps and the absolute tyre wear (in mm tread) is constrained to a predefined limit. This ensures that the consequences of pushing the car to its limits are considered: overheating temperatures and accelerated wear will be detrimental to racing performance. The optimal control solver needs to manage the tyre temperatures carefully over a racing distance. It has been shown that lap times degrade more severely as the tyres reach the end of their life. At some point in the race this drop off in performance will render the car uncompetitive and strategists can use this model to evaluate the performance of different tyre compounds at each track and to strategically plan pit stops during a race. / Dissertation (MEng (Electronic Engineering))--University of Pretoria, 2020. / Electrical, Electronic and Computer Engineering / MEng (Electronic Engineering) / Unrestricted

Tato diplomová práce se zabývá problematikou spojitého optimálního řízení, což je jedna z nejvýznamnějších aplikací teorie diferenciálních rovnic. Cílem této práce bylo jak nastudování matematické teorie optimálního řízení, tak především ukázat užití Pontrjaginova principu maxima a Bellmanova principu optimality při řešení vybraných úloh optimálního řízení. Důraz byl kladen především na problematiku časově a energeticky optimálního řízení elektrického vlaku, při zahrnutí kvadratické odporové funkce.

This thesis deals with optimization of a vehicle’s (racing) drive on a track. The model of a vehicle and a track is built in this thesis. The first chapter is devoted to the fastest pass problem formulation. The problem optimizes (in the least time) the vehicle’s drive from a start line to a finish line. The problem is formulated as an optimal control theory problem. In the second chapter the optimal control theory problem is suitably discretised and transformed into a nonlinear programming problem. The transformation of the fastest pass problem into nonlinear programming problem, its detailed and illustrative derivation and reformulation form the main part of the thesis. Third chapter presents the implementation and solution of the problem using GAMS and MATLAB. This thesis is a part of a specific research project on which the author has participated. The main contribution of the author is an original formulation of the fastest pass problem as a nonlinear programming problem and its implementation and solving using GAMS.