The H_infinity control problem is studied for linear constant coefficient descriptor systems. Necessary and sufficient optimality conditions as well as controller formulas are derived in terms of deflating subspaces of even matrix pencils for problems of arbitrary index. A structure preserving method for computing these subspaces is introduced. In combination these results allow the derivation of a numerical algorithm with advantages over the classical methods.

The global growth in demand for transportation has been phenomenal, owing to an exponential increase in population, industrialization and urbanization. This has led to a corresponding increase in the number of motor vehicles on the roads globally which has made the transport industry one of the main contributors to environmental pollution and energy insecurity. The profile of alternative fuels has been rising as an important component of the solutions to the challenge of energy sustainability. Compressed natural gas is one of the most successful alternative fuels for motor vehicle applications because of its compatibility with the internal combustion engine, reduced engine maintenance costs, reduced criteria air pollutants, low cost, abundance and the existence of renewable sourced natural gas from biomass. The infrastructure for the delivery of compressed natural gas forms part of the primary energy supply network, which has a significant interdependence with the electricity supply network. The compressed natural gas fuelling station is one of the vital nodes of the gas delivery network, that is also reliant on the electricity supply due to the energy intensive compressors that are required to achieve the right pressure conditions for gas transfer to vehicle tanks. At the same time, the increase in human population, industrialization, urbanization and market volatility have threatened the reliability and stability of electricity supply networks. Traditional reliance on supply upgrading to meet rising demand has proven to be unsustainable due to prohibitively high costs and associated environmental impact. As a result, demand side management solutions, where better use of the existing capacity is emphasized have received increasing attention. Demand side management requires that electricity consumers also play a role in the efficient operation of the electricity grid by minimizing their electricity usage as well as shifting their flexible loads away from peak electricity demand periods, so that grid stability is sustained. In order to participate in demand side management initiatives, operators of compressed natural gas stations need technically and economically sound strategies for the operation of station compressors and system components so that energy costs are minimized and gas transfer performance is enhanced. The compressed natural gas fast-fill station, being the most used configuration for commercial fuelling service is the focus of the work carried out in this thesis, with a description of solutions to minimize energy consumption, minimize energy costs and improve gas transfer performance through reduction of filling time. For this purpose, firstly, an optimal control strategy that minimizes energy cost by shifting the compressor load optimally away from the peak electricity pricing period under a time-of-use electricity tariff, while meeting the gas demand is modelled and evaluated. The controller further minimizes the switching frequency of the compressor thereby avoiding an increase in wear and tear which would lead to higher maintenance costs. The results show the effectiveness of the optimal operation model to achieve a huge reduction in electricity cost for the compressed natural gas station, when compressor-on time is shifted to offpeak and standard electricity pricing times. Further strategies for the minimization of switching frequency are compared and the superior approach identified. Secondly, a hierarchical operation optimization model is designed and evaluated. The strategy achieves minimized electricity cost and optimal vehicle filling time by optimally controlling the gas dispenser and priority panel valve function under an optimised schedule of compressor operation. The results show that the proposed approach is effective in achieving a minimum electricity costs in the upper layer optimisation while meeting vehicle gas demand over the control horizon. Further, a reduction in filling time is achieved through a lower layer model predictive control of the pressure-ratio-dependent fuelling process. Thirdly, an evaluation of compressor optimal sizing is carried out to minimize energy consumption and cascade the benefits of optimal operation of the compressed natural gas compressor under the time-of-use tariff. A comparison of the implication of using a variable speed drive or a fixed speed drive which are optimally sized is carried out. Results show that indeed further reduction in electricity costs for the compressed natural gas station is realized when optimally sized compressor drives are used in combination with optimal operation strategies. Additionally, the four line priority panel is evaluated for gas transfer performance and found to further increase the efficiency of vehicle fuelling which is a performance indicator for consumer convenience. The outcomes of this work demonstrate the effectiveness of the approaches proposed as necessary to integrate compressed natural gas stations, which are vital nodes of the gas delivery network, with the demand side management of the electricity grid while at the same time enhancing the gas transfer performance. This increases the economic efficiency of the compressed natural gas as an alternative fuel and also advances the goals of demand side management in electricity grid reliability and stability. / Thesis (PhD)--University of Pretoria, 2019. / Electrical, Electronic and Computer Engineering / PhD / Unrestricted

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.

Bei der Kristallzüchtung insbesondere von Halbleitern hat die Form des freien Randes (dem Interface zwischen fester und flüssiger Phase) einen starken Einfluss auf die Qualität des Kristalls. Die Dissertation befasst sich mit der Optimalsteuerung der Form und des Verlaufs des freien Randes. Als Vorlage für die in der Arbeit betrachteten Modellkonfigurationen dient das VGF-Verfahren (Vertical Gradient Freeze). Der Erstarrungsprozess wird durch ein Zweiphasen-Stefan-Problem mit durch Konvektion und Lorentzkräfte getriebener Strömung beschrieben. Der freie Rand wird als Graph formuliert. Das Kontrollziel besteht in der Ansteuerung eines gewünschten Verlaufs des freien Randes. Als Kontrollgrößen dient die Temperatur auf der Wand des Schmelztiegels und/oder wandnahe oder verteilte Lorentzkräfte. Das Kontrollziel wird durch Minimierung eines geeigneten Kosten-Funktionals erreicht. Das daraus resultierende Minimierungsproblem wird mit einem Adjungierten-Ansatz gelöst. Anhand numerischer Experimente mit Aluminium und Gallium-Arsenid Schmelzen wird gezeigt, das das vorgestellte Verfahren gut funktioniert.

The operation of groups of heavy-duty vehicles at small inter-vehicular distances, known as platoons, lowers the overall aerodynamic drag and, therefore, reduces fuel consumption and greenhouse gas emissions. Experimental tests conducted on a flat road and without traffic have shown that platooning has the potential to reduce the fuel consumption up to 10%. However, platoons are expected to drive on public highways with varying topography and traffic. Due to the large mass and limited engine power of heavy-duty vehicles, road slopes can have a significant impact on feasible and optimal speed profiles. Therefore, maintaining a short inter-vehicular distance without coordination can result in inefficient or even infeasible speed trajectories. Furthermore, external traffic can interfere by affecting fuel-efficiency and threatening the safety of the platooning vehicles. This thesis addresses the problem of safe and fuel-efficient control for heavy-duty vehicle platooning. We propose a hierarchical control architecture that splits this complex control problem into two layers. The layers are responsible for the fuel-optimal control based on look-ahead information on road topography and the real-time vehicle control, respectively. The top layer, denoted the platoon coordinator, relies on a dynamic programming framework that computes the fuel-optimal speed profile for the entire platoon. The bottom layer, denoted the vehicle control layer, uses a distributed model predictive controller to track the optimal speed profile and the desired inter-vehicular spacing policy. Within this layer, constraints on the vehicles' states guarantee the safety of the platoon. The effectiveness of the proposed controller is analyzed by means of simulations of several realistic scenarios. They suggest a possible fuel saving of up to 12% for the follower vehicles compared to the use of existing platoon controllers. Analysis of the simulation results shows how the majority of the fuel saving comes from a reduced usage of vehicles brakes. A second problem addressed in the thesis is model predictive control for obstacle avoidance and lane keeping for a passenger car. We propose a control framework that allows to control the nonlinear vehicle dynamics with linear model predictive control. The controller decouples the longitudinal and lateral vehicle dynamics into two successive stages. First, plausible braking and throttle profiles are generated. Second, for each profile, linear time-varying models of the lateral dynamics are derived and used to formulate a collection of linear model predictive control problems. Their solution provides the optimal control input for the steering and braking actuators. The performance of the proposed controller has been evaluated by means of simulations and real experiments. / <p>QC 20150911</p>

The electric power industry and society are facing the challenges and opportunities of transforming the present power grid into a smart grid. To meet these challenges, new types of control systems are connected over IT infrastructures. While this is done to meet highly set economical and environmental goals, it also introduces new sources of uncertainty in the control loops. In this thesis, we consider control design taking some of these uncertainties into account. In Part I of the thesis, some economical and environmental concerns in smart grids are taken into account, and a scheduling framework for static loads (e.g., smart appliances in residential areas) and dynamic loads (e.g., energy storage systems) in the distribution level is investigated. A robust formulation is proposed taking the user behavior uncertainty into account, so that the optimal scheduling cost is less sensitive to unpredictable changes in user preferences. In addition, a novel distributed algorithm for the studied scheduling framework is proposed, which aims at minimizing the aggregated electricity cost of a network of apartments sharing an energy storage system. We point out that the proposed scheduling framework is applicable to various uncertainty sources, storage technologies, and programmable electrical loads. In Part II of the thesis, we study smart grid uncertainty resulting from possible security threats. Smart grids are one of the most complex cyber-physical systems considered, and are vulnerable to various cyber and physical attacks. The attack scenarios consider cyber adversaries that may corrupt a few measurements and reference signals, which may degrade the system’s reliability and even destabilize the voltage magnitudes. In addition, a practical attack-resilient framework for networked control systems is proposed. This framework includes security information analytics to detect attacks and a resiliency policy to improve the performance of the system running under the attack. Stability and optimal performance of the networked control system under attack and by applying the proposed framework, is proved here. The framework has been applied to an energy management system and its efficiency is demonstrated on a critical attack scenario. / <p>QC 20160830</p>

Inthis degree project, a solution on a coarse grid is recovered by fitting apartial differential equation to a few known data points. The PDE to consideris the heat equation and the Dupire’s equation with their synthetic data,including synthetic data from the Black-Scholes formula. The approach to fit aPDE is by optimal control to derive discrete approximations to regularized Hamiltoncharacteristic equations to which discrete stepping schemes, and parameters forsmoothness, are examined. By non-parametric numerical implementation thedervied method is tested and then a few suggestions on possible improvementsare given / I detta examensarbete återskapas en lösning på ett glest rutnät genom att anpassa en partiell differentialekvation till några givna datapunkter. De partiella differentialekvationer med deras motsvarande syntetiska data som betraktas är värmeledningsekvationen och Dupires ekvation inklusive syntetiska data från Black-Scholes formel. Tillvägagångssättet att anpassa en PDE är att med hjälp av optimal styrning härleda diskreta approximationer på ett system av regulariserade Hamilton karakteristiska ekvationer till vilka olika diskreta stegmetoder och parametrar för släthet undersöks. Med en icke-parametrisk numerisk implementation prövas den härledda metoden och slutligen föreslås möjliga förbättringar till metoden.

L'objectif de ce manuscrit est de présenter mes travaux concernant la modélisation des ablations thermiques par ultrasons focalisés. La méthode de simulation du faisceau acoustique, fondée sur l'utilisation de la bibliothèque k-Wave, est appliquée à un cas concret de propagation des ultrasons à travers une couche de tissu superficiel. Des mesures à l'hydrophone réalisées dans différentes configurations sur des échantillons biologiques fournissent une validation en régime linéaire. A partir de ces résultats, l'influence des tissus superficiels sur la focalisation est évaluée en fonction de la géométrie du problème grâce à des simulations non linéaires.La modélisation thermique des traitements est ensuite discutée avec la volonté de réaliser des simulations thermiques réellement quantitatives. En particulier, un modèle équivalent de la déposition de chaleur en présence d'ébullition est proposé et validé grâce à l'utilisation de données expérimentales originales, diversifiées et peu coûteuses à acquérir.Finalement, un algorithme d'optimisation fondé sur le principe du maximum de Pontryagin est proposé afin d'optimiser la durée des traitements. L'approche étudiée consiste à optimiser la trajectoire du point focal pour maximiser l'efficacité de la déposition de chaleur. A travers une série d'exemples, les avantages et les limites de l'algorithme proposé sont discutés. / This manuscript aims at discussing the complex issue of modeling high-intensity focused ultrasound thermal ablations. An acoustical simulation method, based on the use of the k-Wave library, is described and applied to the description of the interaction between the acoustic beam and the superficial tissue layers. It is validated in the linear domain based on hydrophone measurements realized in different configurations with biological samples. Nonlinear simulations are subsequently used to evaluate the influence of the tissue geometry on the beam focusing.The thermal modeling of the treatment is then discussed with intent to design a truly quantitative model. An equivalent model of the modified heat deposition pattern in presence of boiling is presented and validated based on the use of original, diverse and unexpensive data.Finally, an algorithm is proposed to optimize the focal spot trajectory in order to maximize the heat deposition efficiency, thus reducing treatment time. The advantages and the limits of the approach are discussed based on different examples.

La locomotion anthropomorphe est un processus complexe qui met en jeu un très grand nombre de degrés de liberté, le corps humain disposant de plus de trois cents articulations contre une trentaine chez les robots humanoïdes. Pris dans leur ensemble, ces degrés de liberté montrent une certaine cohérence rendant possible la mise en mouvement du système anthropomorphe et le maintien de son équilibre, dans le but d'éviter la chute. Cette thèse met en lumière les fondements calculatoires à l'origine de cette orchestration. Elle introduit un cadre mathématique unifié permettant à la fois l'étude de la locomotion humaine, et la génération de trajectoires locomotrices pour les robots humanoïdes. Ce cadre consiste en une réduction de la dynamique corps-complet du système pour ne considérer que sa projection autour du centre de gravité, aussi appelée dynamique centroïdale. Bien que réduite, nous montrons que cette dynamique centroïdale joue un rôle central dans la compréhension et la formation des mouvements locomoteurs. Pour ce faire, nous établissons dans un premier temps les conditions d'observabilité de cette dynamique, c'est-à-dire que nous montrons dans quelle mesure cette donnée peut être appréhendée à partir des capteurs couramment employés en biomécanique et en robotique. Forts de ces conditions d'observabilité, nous proposons un estimateur capable de reconstruire la position non-biaisée du centre de gravité. A partir de cet estimateur et de l'acquisition de mouvements de marche sur divers sujets, nous mettons en évidence la présence d'un motif cycloïdal du centre de gravité dans le plan sagittal lorsque l'humain marche de manière nominale, c'est-à-dire sans y penser. La présence de ce motif suggère l'existence d'une synergie motrice jusqu'alors ignorée, soutenant la théorie d'une coordination générale des mouvements pendant la locomotion. La dernière contribution de cette thèse porte sur la locomotion multi-contacts. Les humains ont une agilité remarquable pour effectuer des mouvements locomoteurs qui nécessitent l'utilisation conjointe des bras et des jambes, comme lors de l'ascension d'une paroi rocheuse. Comment doter les robots humanoïdes de telles capacités ? La difficulté n'est certainement pas technologique, puisque les robots actuels sont capables de développer des puissances mécaniques suffisantes. Leurs performances, évaluées tant en termes de qualité des mouvements que de temps de calcul, restent très limitées. Dans cette thèse, nous abordons le problème de génération de trajectoires multi-contacts sous la forme d'un problème de commande optimale. L'intérêt de cette formulation est de partir du modèle réduit de la dynamique centroïdale tout en répondant aux contraintes d'équilibre. L'idée originale consiste à maximiser la vraisemblance de cette dynamique réduite vis-à-vis de la dynamique corps-complet. Elle repose sur l'apprentissage d'une mesure d'occupation qui reflète les capacités cinématiques et dynamiques du robot. Elle est effective : l'algorithmique qui en découle est compatible avec des applications temps réel. L'approche a été évaluée avec succès sur le robot humanoïde HRP-2, sur plusieurs modes de locomotions, démontrant ainsi sa polyvalence. / Anthropomorphic locomotion is a complex process that involves a very large number of degrees of freedom, the human body having more than three hundred joints against thirty in humanoid robots. Taken as a whole, these degrees of freedom show a certain coherence making it possible to set the anthropomorphic system in motion and maintain its equilibrium, in order to avoid falling. This thesis highlights the computational foundations behind this orchestration. It introduces a unified mathematical framework allowing both the study of human locomotion and the generation of locomotive trajectories for humanoid robots. This framework consists of a reduction of the body-complete dynamics of the system to consider only its projection around the center of gravity, also called centroid dynamics. Although reduced, we show that this centroidal dynamics plays a central role in the understanding and formation of locomotive movements. To do this, we first establish the observability conditions of this dynamic, that is to say that we show to what extent this data can be apprehended from sensors commonly used in biomechanics and robotics. Based on these observability conditions, we propose an estimator able to reconstruct the unbiased position of the center of gravity. From this estimator and the acquisition of walking motions on various subjects, we highlight the presence of a cycloidal pattern of the center of gravity in the sagittal plane when the human is walking nominally, that is, to say without thinking. The presence of this motif suggests the existence of a motor synergy hitherto unknown, supporting the theory of a general coordination of movements during locomotion. The last contribution of this thesis is on multi-contact locomotion. Humans have remarkable agility to perform locomotive movements that require joint use of the arms and legs, such as when climbing a rock wall. How to equip humanoid robots with such capabilities? The difficulty is certainly not technological, since current robots are able to develop sufficient mechanical powers. Their performances, evaluated both in terms of quality of movement and computing time, remain very limited. In this thesis, we address the problem of generating multi-contact trajectories in the form of an optimal control problem. The interest of this formulation is to start from the reduced model of centroid dynamics while responding to equilibrium constraints. The original idea is to maximize the likelihood of this reduced dynamic with respect to body-complete dynamics. It is based on learning a measurement of occupation that reflects the kinematic and dynamic capabilities of the robot. It is effective: the resulting algorithmic is compatible with real-time applications. The approach has been successfully evaluated on the humanoid robot HRP-2, on several modes of locomotion, thus demonstrating its versatility.

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