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.

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.

Approximately 2 to 3 million people in the United States suffer from chronic wounds, which are defined as wounds that do not heal in 30 days time; an estimated $25 billion per year is spent on their treatment in the United States. In our work, we focused on treating chronic wounds with bacterial infections using hyperbaric and topical oxygen therapies. We used a mathematical model describing the interaction between bacteria, neutrophils and oxygen. Optimal control theory was then employed to study oxygen treatment strategies with the mathematical model. Existence of a solution was shown for both therapies. Uniqueness was also shown for hyperbaric therapy. We then used a forward-backward sweep method to find numerical solutions for the therapies. We concluded by putting forth ideas for how this problem could progress toward finding applicable treatment strategies.

<div>Optimal control theory (OCT) exists since the 1950s. However, with the advent of modern computers, the design community delegated the task of solving the optimal control problems (OCPs) largely to computationally intensive direct methods instead of methods that use OCT. Some recent work showed that solvers using OCT could leverage parallel computing resources for faster execution. The need for near real-time, high quality solutions for OCPs has therefore renewed interest in OCT in the design community. However, certain challenges still exist that prohibits its use for solving complex practical aerospace problems, such as landing human-class payloads safely on Mars.</div><div><br></div><div>In order to advance OCT, this thesis introduces Epsilon-Trig regularization method to simply and efficiently solve bang-bang and singular control problems. The Epsilon-Trig method resolves the issues pertaining to the traditional smoothing regularization method. Some benchmark problems from the literature including the Van Der Pol oscillator, the boat problem, and the Goddard rocket problem verified and validated the Epsilon-Trig regularization method using GPOPS-II.</div><div><br></div><div>This study also presents and develops the usage of trigonometry for incorporating control bounds and mixed state-control constraints into OCPs and terms it as Trigonometrization. Results from literature and GPOPS-II verified and validated the Trigonometrization technique using certain benchmark OCPs. Unlike traditional OCT, Trigonometrization converts the constrained OCP into a two-point boundary value problem rather than a multi-point boundary value problem, significantly reducing the computational effort required to formulate and solve it. This work uses Trigonometrization to solve some complex aerospace problems including prompt global strike, noise-minimization for general aviation, shuttle re-entry problem, and the g-load constraint problem for an impactor. Future work for this thesis includes the development of the Trigonometrization technique for OCPs with pure state constraints.</div>

Cette thèse est consacrée au développement de techniques de commande avancées pour des classes de systèmes non linéaires en général et pour des applications automobiles en particulier.Pour répondre au besoin du contrôle moteur, la première partie propose des nouveaux résultats théoriques sur la technique de commande non linéaire à base de modèles de type Takagi-Sugeno soumis à la saturation de la commande. La saturation de la commande est traitée en utilisant sa représentation polytopique ou une stratégie anti-windup.La deuxième partie porte sur la commande du système d'air d'un moteur turbocompressé à allumage commandé. Deux approches originales sont proposées. Dans la première, l'outil théorique concernant les modèles Takagi-Sugeno à commutation développé dans la première partie est directement appliqué. La seconde approche est basée sur une commande linéarisante robuste. L'originalité de ces approches multivariables consiste dans sa simplicité de mise en œuvre et son efficacité par rapport à celles qui existent dans la littérature.La dernière partie vise à développer des stratégies pour la gestion énergétique des systèmes électriques d'un véhicule obtenues en se basant sur le Principe du Minimum de Pontryagin. À cet effet, deux approches sont considérées : l'approche hors ligne d'optimisation utilisant les informations du futur concernant les conditions de roulage et l'approche en ligne qui est adaptée de la précédente. Ensuite, ces deux approches sont implémentées et évaluées dans un simulateur avancé. / This thesis addresses the development of some advanced control design tools for a class of nonlinear systems in general and for automotive systems in particular.Motivated by automotive applications, Part I proposes some novel theoretical results on control design for nonlinear systems under Takagi-Sugeno form subject to the control input saturation. The input saturation is dealt with by using its polytopic representation or an anti-windup strategy.Part II deals with our automotive application concerning the control of a turbocharged air system of a spark ignition engine. To this end, two novel control approaches are proposed in this part. For the first one, the theoretical design tool on switching Takagi-Sugeno controller developed in Part I is directly applied. The second one is based on a robust feedback linearization control technique. The originality of these MIMO approaches consist in their simplicity and effectiveness compared to other ones existing in the literature.Part III aims at developing the strategies, which are based on the Pontryagin's Minimum Principle in optimal control theory, for the energy management of the vehicular electric power systems in a hybrid engine configuration. To this end, both offline optimization approach using the future information of driving conditions and online implementable one have been developed and evaluated in an advanced simulator.

In dieser Arbeit wird erstmalig der Einfluss stochastischer Schwankungen auf endoreversible Modelle untersucht. Hierfür wird die Novikov-Maschine mit drei verschieden Wärmetransportgesetzen (Newton, Fourier, asymmetrisch) betrachtet. Während die maximale verrichtete Arbeit und der dazugehörige Wirkungsgrad recht einfach im Falle konstanter Wärmebadtemperaturen hergeleitet werden können, ändern sich dies, falls die Temperaturen stochastisch fluktuieren können. Im letzteren Fall muss die stochastische optimale Kontrolltheorie genutzt werden, um das Maximum der zu erwartenden Arbeit und die dazugehörige Kontrollstrategie zu ermitteln. Im Allgemeinen kann die Lösung derartiger Probleme auf eine nichtlineare, partielle Differentialgleichung, welche an eine Optimierung gekoppelt ist, zurückgeführt werden. Diese Gleichung wird stochastische Hamilton-Jacobi-Bellman-Gleichung genannt. Allerdings können, wie in dieser Arbeit dargestellt, die Berechnungen vereinfacht werden, wenn man annimmt, dass die Fluktuationen unabhängig von der betrachteten Kontrollvariablen sind. In diesem Fall zeigen analytische Betrachtungen, dass die Gleichungen für die verrichtete Arbeit and den Wirkungsgrad ihre ursprüngliche Form behalten, aber manche Terme müssen durch entsprechende Zeitmittel bzw. Erwartungswerte ersetzt werden, jeweils abhängig von der betrachteten Art der Kontrolle. Basierend auf einer Analyse der Leistungsparameter im Falle einer Gleichverteilung der heißen Temperatur der Novikov-Maschine können Schlussfolgerungen auf deren Monotonieverhalten gezogen werden. Der Vergleich verschiedener, zeitunabhängiger, symmetrischer Verteilungen führt zu einer bis dato unbekannten Erweiterung des Curzon-Ahlborn-Wirkungsgrades im Falle kleiner Schwankungen. Weiterhin wird eine Analyse einer Novikov-Maschine mit asymmetrischen Wärmetransport, bei der das Verhalten der heißen Temperatur durch einen Ornstein-Uhlenbeck-Prozess beschrieben wird, durchgeführt. Abschließend wird eine Novikov-Maschine mit Fourierscher Wärmeleitung, bei der die Dynamik der heißen Temperatur von der Kontrollvariable abhängt, betrachtet. Durch das Lösen der Hamilton-Jacobi-Bellman-Gleichung können neuartige Schlussfolgerungen gezogen werden, wie derartige Systeme optimal zu steuern sind. / In this thesis, the influence of stochastic fluctuations on the performance of endoreversible engines is investigated for the first time. For this, a Novikov-engine with three different heat transport laws (Newtonian, Fourier, asymmetric) is considered. While the maximum work output and corresponding efficiency can be deduced easily in the case of constant heat bath temperatures, this changes, if these temperatures are allowed to fluctuate stochastically. In the latter case, stochastic optimal control theory has to be used to find the maximum of the expected work output and the corresponding control policy. In general, solving such problems leads to a non-linear, partial differential equation coupled to an optimization, called the stochastic Hamilton-Jacobi-Bellman equation. However, as presented in this thesis, calculations can be simplified, if one assumes that the fluctuations are independent of the considered control variable. In this case, analytic considerations show that the equations for performance measures like work output and efficiency keep their original form, but terms have to be replaced by appropriate time averages and expectation values, depending on the considered control type. Based on an analysis of the performance measures in the case of a uniform distribution of the hot temperature of the Novikov engine, conclusions on their monotonicity behavior are drawn. The comparison of several, time independent, symmetric distributions reveals a to date unknown extension to the Curzon-Ahlborn efficiency in the case of small fluctuations. Furthermore, an analysis of a Novikov engine with asymmetric heat transport, where the behavior of the hot temperature is described by an Ornstein-Uhlenbeck process, is performed. Finally, a Novikov engine with Fourier heat transport is considered, where the dynamics of the hot temperature depends on the control variable. By solving the corresponding Hamilton-Jacobi-Bellman equation, new conclusions how to optimally control such systems are drawn.

Applications such as urban surveillance, search and rescue, agricultural applications, military applications, etc., require miniature air vehicles (MAVs) to fly for a long time. But they have restricted flight duration due to their dependence on battery life, which necessitates optimal path planning. The generated optimal path should obey the curvature limits prescribed by the minimum turn radius/ maximum turn rate of the MAV. Further, in a dynamically changing environment, the final configuration that the MAV has to achieve may change en route, which demands the path to be replanned by an airborne processor in real-time. As MAVs are small in size and light in weight, wind has a very significant effect on the flight of MAVs and the computation of the minimum-time path in the presence of wind plays an important role. The thesis develops feasible trajectory generation algorithms which are fast, efficient, optimal and implementable in an onboard computer for rectilinear and circular path convergence problems and waypoint following problems both in the absence and in the presence of wind. The first part of the thesis addresses the problem of computation of optimal trajectories when MAVs fly on a two-dimensional (2D) plane maintaining a constant altitude. The shortest path is computed for MAVs from a given initial position and orientation to a given final path with a specified direction as required for a given mission. Unlike the classical Dubins problem where the shortest path was computed between two given configurations (position and orientation), the final point in this case is not specified. However, the final path, which can either be a rectilinear path or a circular path, and the direction to which the MAV should converge, is specified. The time-optimal path of MAVs is developed in the presence of wind mainly using the geometric approach although a few important properties are also obtained using optimal control theory, specifically, Pontryagin’s minimum principle (which provides only the necessary condition for optimality) for control-constrained systems. The complete optima l solution to this problem in all its generality is a major contribution of this thesis as existing methods in the literature that address this problem are either not optimal or do not give a complete solution. Further, the time-optimal path for specified initial and final configurations is generated in reasonably short time without computing all the path lengths of possible candidate paths, which is the method that exists in the literature for similar problems. Simulation results illustrate path generation for various cases, including the presence of steady and time-varying wind. Another problem in MAV path planning in 2D addressed in this thesis computes an extremal path that transitions between two consecutive waypoint segments (obtained by joining two way points in sequence) in a time-optimal fashion. This designed trajectory, named as γ-trajectory, is also used to track the maximum portion of waypoint segments in minimum time and the shortest distance between this trajectory and the associated waypoint can be set to a desired value. Another optimal path, called the loop trajectory, that goes through the way points as well as through the entire waypoint segments, is also proposed. Subsequently, the thesis proposes algorithms to generate trajectories in the presence of steady wind and compares these with the optimal trajectory generated using nonlinear programming based multiple shooting method to show that the generated paths are optimal in most cases. In three-dimensional (3D) space, if the initial and final configurations – in terms of (X,Y,Z) position, heading angle and flight path angle- of the vehicle are specified then shortest path computation is an interesting problem in literature. The proposed method in this thesis is based on 3D geometry and, unlike the existing iterative methods which yield suboptimal paths and are computationally more intensive, this method generates the shortest path in much less time. Due to its simplicity and low computational requirements, this approach can be implemented on a MAV in real-time. But, If the path demands very high pitch angle (as in the case of steep climbs), the generated path may not be flyable for an aerial vehicle with limited range of flight path angles. In such cases numerical methods, such as multiple shooting, coupled with nonlinear programming, are used to obtain the optimal solution. The time-optimal 3D path is also developed in the presence of wind which has a magnitude comparable to the speed of MAVs. The simulation results show path generation for a few sample cases to show the efficacy of the proposed approach as compared to the available approach in the literature. Next, the path convergence problem is studied in 3D for MAVs. The shortest path is generated to converge to a rectilinear path and a circular path starting from a known initial position and orientation. The method is also extended to compute the time-optimal path in the presence of wind. In simulation, optimal paths are generated for a variety of cases to show the efficacy of the algorithm. The other problem discussed in this thesis considers curvature-constrained trajectory generation technique for following a series of way points in 3D space. Extending the idea used in 2D, a γ-trajectory in 3D is generated to track the maximum portion of waypoint segments with a desired shortest distance between the trajectory and the associated waypoint. Considering the flyability issue of the plane a loop-trajectory is generated which is flyable by a MAV with constrained flight path angle. Simulation results are given for illustrative purposes. The path generation algorithms are all based on a kinematic model, considering the vehicle as a point in space. Implementing these results in a real MAV will require the dynamics of the MAV to be considered. So, a 6-DOF SIMULINK model of a MAV is used to demonstrate the tracking of the computed paths both in 2D plane and in 3D space using autopilots consisting of proportional-integral-derivative (PID )controllers .Achieving terminal condition accurately in real-time, if there is noisy measurement of wind data, is also addressed.

In dieser Arbeit wird erstmalig der Einfluss stochastischer Schwankungen auf endoreversible Modelle untersucht. Hierfür wird die Novikov-Maschine mit drei verschieden Wärmetransportgesetzen (Newton, Fourier, asymmetrisch) betrachtet. Während die maximale verrichtete Arbeit und der dazugehörige Wirkungsgrad recht einfach im Falle konstanter Wärmebadtemperaturen hergeleitet werden können, ändern sich dies, falls die Temperaturen stochastisch fluktuieren können. Im letzteren Fall muss die stochastische optimale Kontrolltheorie genutzt werden, um das Maximum der zu erwartenden Arbeit und die dazugehörige Kontrollstrategie zu ermitteln. Im Allgemeinen kann die Lösung derartiger Probleme auf eine nichtlineare, partielle Differentialgleichung, welche an eine Optimierung gekoppelt ist, zurückgeführt werden. Diese Gleichung wird stochastische Hamilton-Jacobi-Bellman-Gleichung genannt. Allerdings können, wie in dieser Arbeit dargestellt, die Berechnungen vereinfacht werden, wenn man annimmt, dass die Fluktuationen unabhängig von der betrachteten Kontrollvariablen sind. In diesem Fall zeigen analytische Betrachtungen, dass die Gleichungen für die verrichtete Arbeit and den Wirkungsgrad ihre ursprüngliche Form behalten, aber manche Terme müssen durch entsprechende Zeitmittel bzw. Erwartungswerte ersetzt werden, jeweils abhängig von der betrachteten Art der Kontrolle. Basierend auf einer Analyse der Leistungsparameter im Falle einer Gleichverteilung der heißen Temperatur der Novikov-Maschine können Schlussfolgerungen auf deren Monotonieverhalten gezogen werden. Der Vergleich verschiedener, zeitunabhängiger, symmetrischer Verteilungen führt zu einer bis dato unbekannten Erweiterung des Curzon-Ahlborn-Wirkungsgrades im Falle kleiner Schwankungen. Weiterhin wird eine Analyse einer Novikov-Maschine mit asymmetrischen Wärmetransport, bei der das Verhalten der heißen Temperatur durch einen Ornstein-Uhlenbeck-Prozess beschrieben wird, durchgeführt. Abschließend wird eine Novikov-Maschine mit Fourierscher Wärmeleitung, bei der die Dynamik der heißen Temperatur von der Kontrollvariable abhängt, betrachtet. Durch das Lösen der Hamilton-Jacobi-Bellman-Gleichung können neuartige Schlussfolgerungen gezogen werden, wie derartige Systeme optimal zu steuern sind. / In this thesis, the influence of stochastic fluctuations on the performance of endoreversible engines is investigated for the first time. For this, a Novikov-engine with three different heat transport laws (Newtonian, Fourier, asymmetric) is considered. While the maximum work output and corresponding efficiency can be deduced easily in the case of constant heat bath temperatures, this changes, if these temperatures are allowed to fluctuate stochastically. In the latter case, stochastic optimal control theory has to be used to find the maximum of the expected work output and the corresponding control policy. In general, solving such problems leads to a non-linear, partial differential equation coupled to an optimization, called the stochastic Hamilton-Jacobi-Bellman equation. However, as presented in this thesis, calculations can be simplified, if one assumes that the fluctuations are independent of the considered control variable. In this case, analytic considerations show that the equations for performance measures like work output and efficiency keep their original form, but terms have to be replaced by appropriate time averages and expectation values, depending on the considered control type. Based on an analysis of the performance measures in the case of a uniform distribution of the hot temperature of the Novikov engine, conclusions on their monotonicity behavior are drawn. The comparison of several, time independent, symmetric distributions reveals a to date unknown extension to the Curzon-Ahlborn efficiency in the case of small fluctuations. Furthermore, an analysis of a Novikov engine with asymmetric heat transport, where the behavior of the hot temperature is described by an Ornstein-Uhlenbeck process, is performed. Finally, a Novikov engine with Fourier heat transport is considered, where the dynamics of the hot temperature depends on the control variable. By solving the corresponding Hamilton-Jacobi-Bellman equation, new conclusions how to optimally control such systems are drawn.

Les Energies Marines Renouvelables (EMR) se développent aujourd’hui très vite tant au niveau de la recherche amont que de la R&D, et même des premiers démonstrateurs à la mer. Parmi ces EMR, l'énergie des vagues présente un potentiel particulièrement intéressant. Avec une ressource annuelle brute moyenne estimée à 40 kW/m au large de la côte atlantique, le littoral français est plutôt bien exposé. Mais l’exploitation à grande échelle de cette énergie renouvelable ne sera réalisable et pertinente qu'à condition d'une bonne intégration au réseau électrique (qualité) ainsi que d'une gestion et d'un dimensionnement optimisé au sens du coût sur cycle de vie. Une première solution de génération tout électrique pour un houlogénérateur a d’abord été évaluée dans le cadre de la thèse de Marie RUELLAN menée sur le site de Bretagne du laboratoire SATIE (ENS de Cachan). Ces travaux ont mis en évidence le potentiel de viabilité économique de cette chaîne de conversion et ont permis de poser la question du dimensionnement de l’ensemble convertisseur-machine et de soulever les problèmes associés à la qualité de l’énergie produite. Puis une seconde thèse a été menée par Judicaël AUBRY dans la même équipe de recherche. Elle a consisté, entre autres, en l’étude d’une première solution de traitement des fluctuations de la puissance basée sur un système de stockage par supercondensateurs. Une méthodologie de dimensionnement de l’ensemble convertisseur-machine et de gestion de l’énergie stockée fut également élaborée, mais en découplant le dimensionnement et la gestion de la production d’énergie et de ceux de son système de stockage. Le doctorant devra donc : 1. S’approprier les travaux antérieurs réalisés dans le domaine de la récupération de l’énergie des vagues ainsi que les modèles hydrodynamiques et mécaniques réalisés par notre partenaire : le LHEEA de l’Ecole Centrale de Nantes - 2. Résoudre le problème du couplage entre dimensionnement/gestion de la chaîne de conversion et dimensionnement/gestion du système de stockage. 3. Participer à la réalisation d’un banc test à échelle réduite de la chaine électrique et valider expérimentalement les modèles énergétiques du stockage et des convertisseurs statiques associés - 4. Proposer une méthodologie de dimensionnement de la chaine électrique intégrant le stockage et les lois de contrôle préalablement élaborées 5. Déterminer les gains en termes de capacités de stockage obtenus grâce à la mutualisation de la production (parc de machines) et évaluer l’intérêt d’un stockage centralisé - 6. Analyser l’impact sur le réseau d’une production houlogénérée selon divers scenarii, modèles et outils développés par tous les partenaires dans le cadre du projet QUALIPHE. L’exemple traité sera celui de l’Ile d’Yeu (en collaboration avec le SyDEV. / The work of this PhD thesis deals with the minimization of the per-kWh cost of direct-drive wave energy converter, crucial to the economic feasibility of this technology. Despite the simplicity of such a chain (that should provide a better reliability compared to indirect chain), the conversion principle uses an oscillating system (a heaving buoy for example) that induces significant power fluctuations on the production. Without precautions, such fluctuations can lead to: a low global efficiency, an accelerated aging of the fragile electrical components and a failure to respect power quality constraints. To solve these issues, we firstly study the optimization of the direct drive wave energy converter control in order to increase the global energy efficiency (from wave to grid), considering conversion losses and the limit s from the sizing of an electrical chain (maximum force and power). The results point out the effect of the prediction horizon or the mechanical energy into the objective function. Production profiles allow the study of the flicker constraint (due to grid voltage fluctuations) linked notably to the grid characteristics at the connection point. Other models have also been developed to quantify the aging of the most fragile and highly stressed components, namely the energy storage system used for power smoothing (with super capacitors or electrochemical batteries Li-ion) and power semiconductors.Finally, these aging models are used to optimize key design parameters using life-cycle analysis. Moreover, the sizing of the storage system is co-optimized with the smoothing management.