The following is an exploration into the optimal guidance and control of gun-launched guided projectiles. Unlike their early counterparts, modern-day gun-launched projectiles are capable of considerable accuracy. This ability is enabled through the use of control surfaces, such as fins or wings, which allow the projectile to maneuver towards a target. These aerodynamic features are part of a control system which lets the projectile achieve some effect at the target. With the advent of very high velocity guns, such as the Navy's electromagnetic railgun, these systems are a necessary part of the projectile design. This research focuses on a control scheme that uses the projectile's angle of attack as the single control in the development of an optimal control methodology that maximizes impact velocity, which is directly related to the amount of damage in icted on the target. This novel approach, which utilizes a reference trajectory as a seed for an iterative optimization scheme, results in an optimal control history for a projectile. The investigation is geared towards examining how poor an approximation of the true optimal solution that reference trajectory can be and still lead to the determination of an optimal control history. Several different types of trajectories are examined for their applicability as a reference trajectory. Although the use of aerodynamic control surfaces enables control of the projectile, there is a potential down side. With steady development of guns with longer ranges and higher launch velocities, it becomes increasingly likely that a projectile will y into a region of the atmosphere (and beyond) in which there is not sufficient air ow over the control surfaces to maintain projectile control. This research is extended to include a minimum dynamic pressure constraint in the problem; the imposition of such a constraint is not examined in the literature. Several methods of adding the constraint are discussed and a number of cases with varying dynamic pressure limits are evaluated. As a result of this research, a robust methodology exists to quickly obtain an optimal control history, with or without constraints, based on a rough reference trajectory as input. This methodology finds its applicability not only for gun-launched weapons, but also for missiles and hypersonic vehicles. / Doctor of Philosophy / As the name implies, optimal control problems involve determining a control history for a system that optimizes some aspect of the system's behavior. In aerospace applications, optimal control problems often involve finding a control history that minimizes time of ight, uses the least amount of fuel, maximizes final velocity, or meets some constraint imposed by the designer or user. For very simple problems, this optimal control history can be analytically derived; for more practical problems, such as the ones considered here, numerical methods are required to determine a solution. This research focuses on the optimal control problem of a gun-launched guided projectile. Guided projectiles have the potential to be significantly more accurate than their unguided counterparts; this improvement is achieved through the use of a control mechanism. For this research, the projectile is modeled using a single control approach, namely using the angle of attack as the only control for the projectile. The angle of attack is the angle formed between the direction the projectile is pointing and the direction it is moving (i.e., between the main body axis and the velocity vector of the projectile). An approach is then developed to determine an optimal angle of attack history that maximizes the projectile's final impact velocity. While this problem has been extensively examined by other researchers, the current approach results in the analytical determination of the costate estimates that eliminates the need to iterate on their solutions. Subsequently, a minimum dynamic pressure constraint is added to the problem. While extensive investigation has been conducted in the examination of a maximum dynamic pressure constraint for aerospace applications, the imposition of a minimum represents a novel body of work. For an aerodynamically controlled projectile, (i.e., one controlled with movable surfaces that interact with the air stream), dropping below a minimum dynamic pressure may result in loss of sufficient control. As such, developing a control history that accommodates this constraint and prevents the loss of aerodynamic control is critical to the ongoing development of very long range, gun-launched guided projectiles. This new methodology is applied with the minimum dynamic pressure constraint imposed and the resulting optimal control histories are then examined. In addition, the possibility of implementing other constraints is also discussed.

A new software tool was needed for flight planning of a high altitude, low speed unmanned aerial vehicle which would be flying in winds close to the actual airspeed of the vehicle. An energy modeled NLP formulation was used to obtain results for a variety of missions and wind profiles. The energy constraint derived included terms due to the wind field and the performance index was a weighted combination of the amount of fuel used and the final time. With no emphasis on time and with no winds the vehicle was found to fly at maximum lift to drag velocity, V_md. When flying in tail winds the velocity was less than V_md, while flying in head winds the velocity was higher than Vmd. A family of solutions was found with varying times of flight and varying fuel amounts consumed which will aid the operator in choosing a flight plan depending on a desired landing time. At certain parts of the flight, the turning terms in the energy constraint equation were found to be significant. An analysis of a simpler vertical plane cruise optimal control problem was used to explain some of the characteristics of the vertical plane NLP results. / Master of Science

It is expected that the grid of the future differs from the current system by the increased integration of distributed generation, distributed storage, demand response, power electronics, and communications and sensing technologies. The consequence is that the physical structure of the system becomes significantly more distributed. The existing centralized control structure is not suitable any more to operate such a highly distributed system. This thesis is dedicated to providing a promising solution to a class of energy management problems in power systems with a high penetration of distributed resources. This class includes optimal dispatch problems such as optimal power flow, security constrained optimal dispatch, optimal power flow control and coordinated plug-in electric vehicles charging. Our fully distributed algorithm not only handles the computational complexity of the problem, but also provides a more practical solution for these problems in the emerging smart grid environment. This distributed framework is based on iteratively solving in a distributed fashion the first order optimality conditions associated with the optimization formulations. A multi-agent viewpoint of the power system is adopted, in which at each iteration, every network agent updates a few local variables through simple computations, and exchanges information with neighboring agents. Our proposed distributed solution is based on the consensus+innovations framework, in which the consensus term enforces agreement among agents while the innovations updates ensure that local constraints are satisfied.

This dissertation investigates advanced concepts in terminal missile guidance. The terminal phase of missile guidance usually lasts less than ten seconds and calls for very accurate maneuvering to ensure intercept. Technological advancements have produced increasingly sophisticated threats that greatly reduce the effectiveness of traditional approaches to missile guidance. Because of this, terminal missile guidance is, and will remain, an important and active area of research. The complexity of the problem and the desire for an optimal solution has resulted in researchers focusing on simplistic, usually linear, models. The fruit of these endeavors has resulted in some of the world's most advanced weapons systems. Even so, the resulting guidance schemes cannot possibly counter the evolving threats that will push the system outside the linear envelope for which they were designed. The research done in this dissertation greatly extends previous research in the area of optimal missile guidance. Herein it is shown that optimal missile guidance is fundamentally a pairing of an optimal guidance strategy and an optimal control strategy. The optimal guidance strategy is determined from a missile's information constraints, which are themselves largely determined from the missile's sensors. The optimal control strategy is determined by the missile's control constraints, and works to achieve a specified guidance strategy. This dichotomy of missile guidance is demonstrated by showing that missiles having different control constraints utilize the same guidance strategy so long as the information constraints are the same. This concept has hitherto been unrecognized because of the difficulty in developing an optimal control for the nonlinear set of equations that result from control constraints. Having overcome this difficulty by indirect means, evidence of the guidance strategy paradigm emerged. The guidance strategy paradigm is used to develop two advanced guidance laws. The new guidance laws are compared qualitatively and quantitatively with existing guidance laws.

We study the loss of information (measured in terms of the Kullback- Leibler distance) caused by observing "grouped" data (observing only a discretized version of a continuous random variable). We analyse the asymptotical behaviour of the loss of information as the partition becomes finer. In the case of a univariate observation, we compute the optimal rate of convergence and characterize asymptotically optimal partitions (into intervals). In the multivariate case we derive the asymptotically optimal regular sequences of partitions. Forthermore, we compute the asymptotically optimal transformation of the data, when a sequence of partitions is given. Examples demonstrate the efficiency of the suggested discretizing strategy even for few intervals. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

Uruguay is a pioneer in the use of renewable sources of energy and can usually satisfy its total demand from renewable sources. Control and optimization of the system is complicated by half of the installed power - wind and solar sources - be- ing non-controllable with high uncertainty and variability. In this work we present a novel optimization technique for efficient use of the production facilities. The dy- namical system is stochastic, and we deal with its non-Markovian dynamics through a Lagrangian relaxation. Continuous-time optimal control and value function are found from the solution to a sequence of Hamilton-Jacobi-Bellman partial differential equations associated with the system. We introduce a monotone scheme to avoid spurious oscillations in the numerical solution and apply the technique to a number of examples taken from the Uruguayan grid. We use parallelization and change of variables to reduce the computational times. Finally, we study the usefulness of extra system storage capacity offered by batteries.

The need to ease pressure from the depleting fossil fuel reserves coupled with the rising global energy demand has seen a drastic increase in research and uptake of renewable energy sources in recent decades. Of the commonly exploited renewable energy resources, hydropower is currently the most popular resource accounting for 17% of the world's total energy generation, a portion which translates to 85% of the renewable energy share. However, despite the huge potential, hydropower is dependent on the availability of water resource, which is affected by climate change. During wet seasons, hydropower system operators are faced with a deluge of floods which results in excess power generation and spillage. The situation reverses in dry seasons where system operators are compelled to curtail power generation because of low water levels in the hydro reservoirs. The later situation is more pronounced in drought prone regions such as Southern Africa where some hydropower plants are completely shut down in dry seasons due to water shortage. This dissertation focuses on the application of optimal control to hydropower plants with pumpback retrofits powered by on-site hydrokinetic and wind power systems. The first section of this work develops an optimal operation strategy for a high head hydropower plant retrofitted with hydrokinetic-powered cascaded pumpback system in dry season. The objective of pumpback operation is to recycle a part of the downstream discharged water back to the main dam to maintain a high water level required for optimal power generation. The problem is formulated as a discrete optimisation problem to simultaneously minimise the grid pumping energy demand, minimise the wear and tear associated with the switching frequency of the two pumps in cascade, maximise restoration of the reservoir volume through pumpback operation and maximise the use of on-site generated hydrokinetic power for pumping operation. Simulation results based on a practical case study show the pumping energy saving advantages of the cascaded pumping system as compared to a classical pumped storage (PS) system. The second section of this work develops an optimal control system for assessing the effects of ecological flow constraints to the operation of a hydropower plant with a hydrokinetic-wind powered pumpback retrofit. The aim of the control law in this case is to use the allocated water to optimally meet the contractual obligations of the power plant. The problem is formulated as a discrete optimisation problem to maximise the energy output of the reservoir subject to some defined technical and hydrological constraints. In this system, pumping power is met primarily by the wind power generator output supplemented by the on-site generated hydrokinetic power. The excess hydrokinetic power is exported to the grid to meet the committed demand. Three different optimisation scenarios are developed: The first scenario is the baseline operation of the hydropower plant without any intervention. The second scenario incorporates the hydrokinetic-wind-powered pumpback operation in the optimal control policy. The third scenario includes the downstream flow constraint to the optimal control policy of the second optimisation scenario. Simulation results based on a practical case study show that ecological flow constraints have negative effects to the economic performance of a hydropower plant. / Dissertation (MEng)--University of Pretoria, 2017. / MasterCard Foundation Scholars Program / Centre of New Energy Systems / University of Pretoria / Electrical, Electronic and Computer Engineering / MEng / Unrestricted

My PhD thesis concentrates on the field of stochastic analysis, with focus on stochastic optimization and applications in finance. It is composed of two parts: the first part studies an optimal stopping problem, and the second part studies an optimal control problem. The first topic considers a one-dimensional transient and downwards drifting diffusion process X, and detects the optimal times of a random time(denoted as ρ). In particular, we consider two classes of random times: (1) the last time when the process exits a certain level l; (2) the time when the process reaches its maximum. For each random time, we solve the optimization problem infτ E[λ(τ- ρ)+ +(1-λ)(ρ - τ)+] overall all stopping times. For the last exit time, the process should stop optimally when it runs below some fixed level k the first time, where k is the solution of an explicit defined equation. For the ultimate maximum time, the process should stop optimally when it runs below a boundary which is the maximal positive solution (if exists) of a first-order ordinary differential equation which lies below the line λs for all s > 0 . The second topic solves an optimal consumption and investment problem for a risk-averse investor who is sensitive to declines than to increases of standard living (i.e., the investor is loss averse), and the investment opportunities are constant. We use the tools of stochastic control and duality methods to solve the resulting free-boundary problem in an infinite time horizon. Briefly, the investor consumes constantly when holding a moderate amount of wealth. In bliss time, the investor increases the consumption so that the consumption-wealth ratio reaches some fixed minimum level; in gloom time, the investor decreases the consumption gradually. Moreover, high loss aversion tends to raise the consumption-wealth ratio, but cut the investment-wealth ratio overall.

Cette thèse est consacrée à l'analyse mathématique et numérique pour les problèmes de transport partiel optimal et d'appariement avec contrainte (constrained matching problem). Ces deux problèmes présentent de nouvelles quantités inconnues, appelées parties actives. Pour le transport partiel optimal avec des coûts qui sont donnés par la distance finslerienne, nous présentons des formulations équivalentes caractérisant les parties actives, le potentiel de Kantorovich et le flot optimal. En particulier, l'EDP de condition d'optimalité permet de montrer l'unicité des parties actives. Ensuite, nous étudions en détail des approximations numériques pour lesquelles la convergence de la discrétisation et des simulations numériques sont fournies. Pour les coûts lagrangiens, nous justifions rigoureusement des caractérisations de solution ainsi que des formulations équivalentes. Des exemples numériques sont également donnés. Le reste de la thèse est consacré à l'étude du problème d'appariement optimal avec des contraintes pour le coût de la distance euclidienne. Ce problème a un comportement différent du transport partiel optimal. L'unicité de solution et des formulations équivalentes sont étudiées sous une condition géométrique. La convergence de la discrétisation et des exemples numériques sont aussi établis. Les principaux outils que nous utilisons dans la thèse sont des combinaisons des techniques d'EDP, de la théorie du transport optimal et de la théorie de dualité de Fenchel--Rockafellar. Pour le calcul numérique, nous utilisons des méthodes du lagrangien augmenté. / The manuscript deals with the mathematical and numerical analysis of the optimal partial transport and optimal constrained matching problems. These two problems bring out new unknown quantities, called active submeasures. For the optimal partial transport with Finsler distance costs, we introduce equivalent formulations characterizing active submeasures, Kantorovich potential and optimal flow. In particular, the PDE of optimality condition allows to show the uniqueness of active submeasures. We then study in detail numerical approximations for which the convergence of discretization and numerical simulations are provided. For Lagrangian costs, we derive and justify rigorously characterizations of solution as well as equivalent formulations. Numerical examples are also given. The rest of the thesis presents the study of the optimal constrained matching with the Euclidean distance cost. This problem has a different behaviour compared to the partial transport. The uniqueness of solution and equivalent formulations are studied under geometric condition. The convergence of discretization and numerical examples are also indicated. The main tools which we use in the thesis are some combinations of PDE techniques, optimal transport theory and Fenchel--Rockafellar dual theory. For numerical computation, we make use of augmented Lagrangian methods.

La technique de moyennation est un moyen efficace pour simplifier les transferts optimaux pour un satellite à faible poussée dans un problème à deux corps contrôlé. Cette thèse est une étude analytique et numérique du transferts orbital à poussée faible en temps optimal qui généralise l'application de la moyennation du problème à deux corps à des transferts dans le problème à deux corps perturbés et aux transfert d'une orbite proche de la Terre au point de Lagrange L1, dans le cadre du problème à quatre corps bi-circulaire où l’effet perturbatif de la Lune et du Soleil est modélisé. Dans le transfert à faible poussée à deux corps, nous comparons le cas du temps minimal et de l'énergie. Nous déterminons que le domaine elliptique pour les transferts orbitaux temps-minimal est géodésiquement convexe pour un transfert coplanaire et vers une orbite circulaire, contrairement au cas de l’énergie. Nous examinons ensuite l’effet la perturbation lunaire, nous montrons que dans ce cas le Hamiltonien moyenné se trouve être celui associé à un problème de navigation de Zermelo. Nous étudions numériquement à l’aide du code Hampath, les points conjugués pour caractériser l’optimalité globale des trajectoires. Enfin, nous construisons et réalisons numériquement un transfert d'une orbite terrestre au point de Lagrange L1, qui utilise la moyennation sur un arc (proche de la Terre) pour simplifier les calculs numériques. Dans ce dernier résultat nous voyons qu'un transfert concaténant une trajectoire moyennée avec une trajectoire temps minimal au voisinage du point de Lagrange est en effet proche d’un transfert de temps optimal calculé avec une méthode numérique de tir. / The technique of averaging is an effective way to simplify optimal low-thrust satellite transfers in a controlled two-body Kepler problem. This study takes the form of both an analytical and numerical investigation of low-thrust time-optimal transfers, extending the application of averaging from the two-body problem to transfers in the perturbed low-thrust two body problem and a low-thrust transfer from Earth orbit to the L1 Lagrange point in the bicircular four-body setting. In the low-thrust two-body transfer, we compare the time-minimal case with the energy-minimal case, and determine that the elliptic domain under time-minimal orbital transfers (reduced in some sense) is geodesically convex. We then consider the Lunar perturbation of an energy-minimal low-thrust satellite transfer, finding a representation of the optimal Hamiltonian that relates the problem to a Zermelo navigation problem and making a numerical study of the conjugate points. Finally, we construct and implement numerically a transfer from an Earth orbit to the L1 Lagrange point, using averaging on one (near-Earth) arc in order to simplify analytic and numerical computations. In this last result we see that such a `time-optimal' transfer is indeed comparable to a true time-optimal transfer (without averaging) in these coordinates.