Following results over recent years, this thesis enhances the problem of minimizing a cost functional defined on a state trajectory of an autonomous switched dynamical system. The cost functional traditionally used, is augmented with explicit costs on the switching times and the final time is set by a constraint as opposed to being given. An equation for the gradient of the cost functional is derived and an algorithm is proposed for computing local minima. The algorithm is based on existing steepest descent methods including the Armijo procedure and gradient projection. A matlab implementation of the algorithm is developed in order to solve optimal problems that can be modelled with costs on or between the switching times. An existing problem, the motivation for this research, where repairs on a bridge is to be optimized, is provided and solved.

Steuerungsprobleme treten in vielen Anwendungen in Naturwissenschaft und Technik auf. In dieser Arbeit werden Optimalsteuerungsprobleme mit semilinearen elliptischen partiellen Differentialgleichungen als Nebenbedingungen untersucht. Die Kontrolle wird durch Kontrollschranken als Ungleichungsnebenbedingungen eingeschränkt. Dabei ist die Zielfunktion quadratisch in der Kontrolle. Die Lösung des Optimierungsproblems kann dann durch die Projektionsbedingung mit Hilfe des adjungierten Zustandes dargestellt werden. Ein neuer Zugang ist die variationelle Diskretisierung. Bei dieser wird nur der Zustand und der adjungierte Zustand diskretisiert, nicht aber der Raum der Kontrollen. Dieser Zugang erlaubt höhere Konvergenzraten für die Kontrolle für kontrollrestingierte Probleme als bei einer Diskretisierung des Kontrollraumes. Die Projektionsbedingung für das variationell diskretisierte Problem ist dabei auf die gleiche zulässige Menge wie beim nicht diskretisierten Problem. In der vorliegenden Arbeit wird die Methode der variationellen Diskretisierung auf semilineare elliptische Optimalkontrollprobleme angewendet und Fehlerabschätzungen für die Kontrollen bewiesen. Dabei wird hauptsächlich auf die verteilte Steuerung Wert gelegt, aber auch die Neumann-Randsteuerung mitbehandelt. Nach einem Überblick über die Literatur wird die Aufgabenstellung mit den Voraussetzungen aufgeschrieben und die Optimalitätsbedingungen angegeben. Danach wird die Existenz einer Lösung, sowie die Konvergenz der diskreten Lösungen gegen eine kontinuierliche Lösung gezeigt. Außerdem werden Finite-Elemente-Konvergenzordnungen angegeben. Dann werden optimale Fehlerabschätzungen in verschiedenen Normen für die variationelle Kontrolle bewiesen. Insbesondere werden die Fehlerabschätzung in Abhängigkeit vom Finite-Elemente-Fehler des Zustandes und des adjungierten Zustandes angegeben. Dabei wird die nichtlineare Fixpunktgleichung mittels semismooth Newtonverfahrens linearisiert. Das Newtonverfahren wird auch für die numerische Lösung des Problems eingesetzt. Die Voraussetzung für die Konvergenzordnung ist dabei nicht die SSC, die hinreichende Bedingung zweiter Ordnung, welche eine lokale Konvexität in der Zielfunktion impliziert, sondern die Invertierbarkeit des Newtonoperators. Dies ist eine stationäre Bedingung in der optimalen Kontrolle. Dabei wird nur benötigt, dass der Rand der aktiven Menge eine Nullmenge ist und die Invertierbarkeit des Newtonoperators in der Optimallösung. Der Schaudersche Fixpunktsatz wird benutzt, um für die Newtongleichung die Existenz eines Fixpunktes innerhalb der gewünschten Umgebung zu beweisen. Außerdem wird die Eindeutigkeit eines solchen Fixpunktes für eine gegebene Triangulation bei hinreichend feiner Diskretisierung gezeigt. Das Ergebnis ist, dass die Konvergenzrate nur durch die Finite-Elemente-Konvergenzraten von Zustand und adjungiertem Zustand beschränkt wird. Diese Rate wird nicht nur durch die Ansatzfunktionen, sondern auch durch die Glattheit der rechten Seite beschränkt, so dass der Knick am Rand der aktiven Menge hier ein Grenze setzt. Außerdem wird die Implementation des semismooth Newtonverfahrens für den unendlichdimensionalen Kontrollraum für die variationelle Diskretisierung erläutert. Dabei wird besonders auf den zweidimensionalen verteilten Fall eingegangen. Es werden die bewiesenen Konvergenzraten an einigen semilinearen und linearen Beispielen mittels der variationellen Diskretisierung demonstriert. Es entsprechen sich die bei den analytische Beweisen und der numerischen Lösung eingesetzten Verfahren, die Fixpunktiteration sowie das nach Kontrolle oder adjungiertem Zustand aufgelöste Newtonverfahren. Dabei sind einige Besonderheiten bei der Implementation zu beachten, beispielsweise darf die Kontrolle nicht inkrementell mit dem Newtonverfahren oder der Fixpunktiteration aufdatiert werden, sondern muss in jedem Schritt neu berechnet werden.

A class of parametric optimal control problems for semilinear parabolic equations is considered. Using recent regularity results for solutions of such equations, sufficient conditions are derived under which the solutions to optimal control problems are locally Lipschitz continuous functions of the parameter in the L1-norm. It is shown that these conditions are also necessary, provided that the dependence of data on the parameter is sufficiently strong.

An optimal control problem governed by the heat equation with nonlinear boundary conditions is considered. The objective functional consists of a quadratic terminal part and a quadratic regularization term. It is known, that an SQP method converges quadratically to the optimal solution of the problem. To handle the quadratic optimal control subproblems with high precision, very large scale mathematical programming problems have to be treated. The constrained problem is reduced to an unconstrained one by a method due to Bertsekas. A multigrid approach developed by Hackbusch is applied to solve the unconstrained problems. Some numerical examples illustrate the behaviour of the method.

Gaits have become an integral part of the design method of robots heading to complex terrains. But research into optimal ways to transition between different gaits is still lacking, and is the primary motivation behind this research. An essential characteristic of gaits is periodicity, and considering that a novel notion of graceful transition is proposed: a graceful transition is one that has maximally persisting periodicity. This particular notion of persistence in the characteristic behavior can be generalized. Therefore, a comprehensive framework for the general problem of connecting any two trajectories of a dynamical system, with an underlying characteristic behavior, over a finite time interval and in a manner that the behavior persists maximally during the transition, is developed and presented. This transition is called the Gluskabi Raccordation, and the characteristic behavior is defined by a kernel representation. Along with establishing this framework, the kernel representations for some interesting characteristic behaviors are also identified. The problem of finding the Gluskabi Raccordations is then solved for different combinations of characteristic behaviors and dynamical systems, and compact widely applicable results are obtained. Lastly, the problem of finding graceful gait transitions is treated within this newly established broader framework, and these graceful gait transitions are obtained for the case of a two-piece worm model.

Wind is one of the most promising resources in the renewable energy portfolio. Still, the cost of electrical power produced by small wind turbines impedes the use of this technology, which can otherwise provide power to millions of homes in rural regions worldwide. To encourage their use, small wind turbines must convert wind energy more effectively while avoiding increased equipment costs. A variable ratio gearbox (VRG) can provide this capability to the simple low-cost fixed-speed wind turbine through discrete operating speeds. The VRG concept is based upon mature technology taken from the automotive industry and is characterized by low cost and high reliability. A 100 kW model characterizes the benefits of integrating a VRG into a fixed-speed stall-regulated wind turbine system. Simulation results suggest it improves the efficiency of the fixed-speed turbine in the partial-load region and has the ability to limit power in the full-load region where pitch control is often used. To maximize electrical production, mechanical braking is applied during the normal operation of the wind turbine. A strategy is used to select gear ratios that produce torque slightly above the maximum amount the generator can accept while simultaneously applying the mechanical brake, so that full-load production may be realized over greater ranges of the wind speed. Dynamic programming is used to establish the VRG ratios and an optimal control design. This optimization strategy maximizes the energy production while insuring that the brake pads maintain a predetermined service life. In the final step of the research, a decision-making algorithm is developed to find the gears that emulate the ratios found in the optimal control design. The objective is to match the energy level as closely as possible, minimize the mass of the gears, and insure that tooth failure does not occur over the design life of the VRG. Recorded wind data of various wind classes is used to quantify the benefit of using the VRG. The results suggest that an optimized VRG design can increase wind energy production by roughly 10% at all of the sites in the study. / text

Motor noise and redundancy are vexing issues in motor control; yet their understanding provides great insights on underlying control mechanisms that govern movement. They provide glimpses into how the nervous system organizes and regulates movement within the motor control system. Understand of motor control could spur new advances in motor control could lead to better development of rehabilitation process and technology to counteract debilitating affects of neuromuscular disorders and motor readjustment with prostheses. However, before such process and technology could be developed and adapted for clinical use, a deeper understanding of motor control is needed to unravel the neural roadmap that regulates and generates movement. New theory of motor control could precipitate the development of more robust control mechanisms for robotic-human interaction. This work aims at expanding a more rigorous analytical and mathematical framework to understand how these control mechanisms reconcile redundancy and stochastic noise in human motor control. / text

Optimal control (motion planning) of the free interface in classical two-phase Stefan problems is considered. The evolution of the free interface is modeled by a level set function. The first-order optimality system is derived on a formal basis. It provides gradient information based on the adjoint temperature and adjoint level set function. Suitable discretization schemes for the forward and adjoint systems are described. Numerical examples verify the correctness and flexibility of the proposed scheme.

Ο πρωταρχικός στόχος αυτής της εργασίας είναι να μελετηθεί η υποδιαδικασία κατά την παραγωγή χαρτιού, γνώστη και ως πολτοποίηση. Η πολτοποίηση αφορά το στάδιο οπού το πριονίδι, παρουσία νερού περνά μέσα από τον πολτοποιητή ο οποίος αποτελείται από δύο πιάτα που περιστρέφονται με αντίθετη φορά, το πριονίδι διαλύεται και μαζί με το νερό δημιουργούν το μίγμα του χαρτοπολτού. Η εν λόγω διαδικασία χαρακτηρίζεται από υψηλή πολυπλοκότητα λόγω της δυσκολίας που παρουσιάζει στη μαθηματική μοντελοποίηση. Σε αυτή την εργασία, εξετάζονται ήδη υπάρχουσες προσεγγίσεις στο θέμα της μοντελοποίησης. Διαφορετικές προσεγγίσεις πάνω στη μοντελοποίηση εχουν γίνει τα τελευταία χρόνια από τους μηχανικούς, υπό την έννοια οτι χρησιμοποιήθηκαν διαφορετικές μεταβλητές ως έξοδοι και ως είσοδοι και πληθώρα μεθόδων μοντελοποίησης. Τα σημαντικότερα μοντέλα ανοιχτού βρόχου που βρεθήκαν στη βιβλιογραφία εξομοιώνονται και τα χαρακτηριστικά τους αναλύονται. Ο στόχος της εργασίας είναι να καθοριστεί αν οι προαναφερθείσες μοντελοποιήσεις μπορούν να ελεγχθούν με επιτυχία μέσω ενός Model Predictive Controller (MPC). Εκτεταμένες προσομοιώσεις πραγματοποιούνται όπου οι παράμετροι του ελεγκτή εξετάζονται λεπτομέρως ώστε να διαπιστωθεί το αντίκτυπο τους στην όλη διαδικασία. Προκειμένου να εξεταστεί η αποδοτικότητα του ελεγκτή, δεδομένα τα οποία πάρθηκαν από πραγματική βιομηχανία συγκρίνονται με αυτά του MPC τα οποία παράχθηκαν μέσω του MATLAB. / The initial objective of this diploma thesis is to investigate and examine the papermaking’s sub-process also known as pulping. Pulping concerns the stage where wood chip along with water is being passed through a refiner that consists of two counter rotating discs, woodchip is being smashed and with the presence of water the pulp mixture is being created. This process is being characterized as highly complex because of the difficulties that appear in its mathematical modeling. In this thesis, the existing modeling approaches developed for the pulp and paper refining process are also being investigated. Different approaches have been developed towards the modeling of the process during the past years, where engineers utilized different controlled and manipulated variables as well as a variety of modeling methods. The most important pulp and paper refiner open-loop model approximations, extracted from the existing related literature, are being simulated and their characteristics are being examined in detail. This work’s main and final objective is to determine if the aforementioned modeling approaches of the pulp and paper refining process can be successfully controlled via a Model Predictive Control (MPC) based structure. Extensive simulation trials are being carried out, where the MPC parameters are being investigated in detail regarding their effect on the overall control performance of the pulp and paper refining process. In order to further evaluate the efficacy of the proposed control scheme, the MPC related results are being compared to experimental data extracted from a real refining system that utilizes a generic industrial controller.

Modeling and simulation form an integral role in the engineering design process. An accurate mathematical description of a system provides the design engineer the flexibility to perform trade studies quickly and accurately to expedite the design process. Most often, the mathematical model of the system contains components of different engineering disciplines. A modeling methodology that can handle these types of systems might be used in an indirect fashion to extract added information from the model.This research examines the ability of a modeling methodology to provide added insight into system analysis and design. The modeling methodology used is bond graph modeling. An investigation into the creation of a bond graph model using the Lagrangian of the system is provided. Upon creation of the bond graph, system analysis is performed. To aid in the system analysis, an object-oriented approach to bond graph modeling is introduced. A framework is provided to simulate the bond graph directly. Through object-oriented simulation of a bond graph, the information contained within the bond graph can be exploited to create a measurement of system efficiency. A definition of system efficiency is given. This measurement of efficiency is used in the design of different controllers of varying architectures. Optimal control of a missile autopilot is discussed within the framework of the calculated system efficiency.