[pt] Esta tese apresenta um estudo na área de controle robusto não paramétrico, mais precisamente relacionado ao Problema H2/Hinf. O Principal objetivo deste trabalho consiste no projeto de controladores por uma abordagem direta sobre o problema mencionado, baseada em método de Gallerkin. Dois métodos foram propostos, sendo denominados Expansão em Base Pré-Estabelecida (EBPE) e Expansão em Base Otimizada (EBO). Para a aplicação destes métodos, os controladores estabilizantes do sistema em estudo devem ser explicitados por intermédio da parametrização de Youla. Em seguida, uma nova parametrização é realizada a fim de colocar o problema resultante em um formato padrão, com a norma Hinf sob a forma do problema de Nehari. Um controlador calculado por EBO ou EBPE possui ordem previamente determinada, estabiliza internamente a planta, minimiza um funcional de desempenho e satisfaz a um nível pré- especificado de robustez em estabilidade. Em EBPE, uma base é escolhida para o espaço solução e o ajuste dos coeficientes de seus vetores é realizado minimizando o critério proposto. A vantagem deste método é resolver um problema de otimização convexo. Por outro lado, a escolha dos vetores que participarão da base truncada já estará definida, levando a uma solução com ordem superior àquela obtida por EBO e um mesmo nível de desempenho. No método EBO, as tarefas de escolha dos vetores que participarão da base e seus respectivos ajustes de coeficientes são realizados simultaneamente. Embora este problema seja não convexo, sua vantagem reside na possibilidade de encontrar soluções com ordens relativamente menores em comparação com aquelas por EBPE para um mesmo nível de desempenho. Adicionalmente, discute-se alguns resultados teóricos, onde a existência e a unicidade da solução do Problema H2/Hinf são demonstradas. Mostra-se também que a sequência de controladores calculados pelos métodos propostos, à medida que a ordem aumenta, converge para a solução ótima do problema original. / [en] This thesis presents a study on the subject of robust and non parametrical control, more precisely related to the Mixed H2/Hinf problem. The main purpose of this work consists in the controllers design by a direct approach over the mentioned problem, using Galerkin`s method. Two numerical methods were proposed, being known as Pre- Established Basis Expansion (EBPE) and Optimized Basis Expansion (EBO). For the application of these methods, the stabilizing controllers of the system under analisys must be explicated by Youla`s parametrization. After this, a new parametrization is performed in order to set the problem in a standard format, with the Hinf-norm in the Nehari problem. A controller computed by EBO or EBPE has a previously specified order, internally stabilizes the plant, minimizes a performance functional and satisfies a pre- established robustness stability margin. In the EBPE method, a basis is chosen from space solution and the adjustment of the vectors` coefficients is performed minimizing the proposed criterion. The advantage is solve a convex optimization problem. By the other side, the choice of vectors in the truncated basis will be defined, leading to a solution with a higher order than the one obtained by the EBO approach, for the same performance level In the EBO method, both the vectors` choice in the truncated basis and their respective coefficient adjustment are performed simultaneously. The Youla`s parameter is approximated by the class of real, rational, proper and stable transfer function and the project variables are the coefficients of the numerator and denominator polynomials. Although this problem is a non- convex one, its main advantage lies in the possibility of finding controllers of relatively lower orders than ones by EBPE at the same level of performance. In addition, some theoretical results are discussed, where the existence and uniqueness of Mixid H2/Hinf problem solution are proved. It is also shown that the sequence of controllers computed by the proposed methods, as order increase, converges to the solution of the original problem.

[pt] O Planejamento econômico, abordado como um problema de controle, tem por objetivo estabelecer trajetórias ótimas (ou sub-ótimas) para as variáveis que estão sujeitas ao controle do Governo. Isto significa dizer que as varáveis de política (controle) não mais serão arbitrariamente determinadas pelos seus planejadores, sendo agora resultantes de um processo de otimização , tendo em vista o cumprimento de metas previamente estabelecidas. Neste artigo aplicamos um controlador adaptativo de certeza equivalente a um modelo macroeconométrico da economia brasileira, considerando erro de medida nas variáveis de estado. A adoção de um controlador adaptativo é justificada tendo em vista as críticas (principalmente a crítica de Lucas) que recaíram sobre os modelos macroeconométricos estacionários. Uma das formas adequadas de se tratar a não estacionariedade de tais modelos é por intermédio de um controlador adaptativo cujo objetivo será controlar e identificar simultaneamente o modelo em questão. Apresentamos uma pequena resenha das aplicações de controle ótimo e controle adaptativo em problema econômicos, ressaltando a aplicação de ambas as técnicas em modelos macroeconométricos com expectativas racionais. Por intermédio de simulações comparamos a política realmente efetivada pelo governo federal e a política ótima obtida via controle ótimo não adaptativo. / [en] Economic planning, when considered as a control problem, has as its objective establishing optimal (or sub-optimal) trajectories for the variables subject to Government Control. This means that the policy variables (control), instead of being arbitrarily determined by the policymakers, will be the result of an optimization process, with the objective of reaching pre-established goals. In this work a Certainly Equivalence Adaptative Control is applied to a macroeconometric model of the Brazilian economy with measurement error. Since the employment of time-invariant models has been widely criticized (Lucas critique) the model used here is time- varying. An adequate way to treat such a case is through an adaptative control scheme, in which control and identification of the model are perfomed simultaneously. By means of simulations the policy obtained with the adaptative controller is compared to the policy adopted by the Brazilian Government.

We revisit the influential economic growth model by Lucas (1988) ["On the mechanics of economic development." Journal of Monetary Economics, 22(1):3-42], assuming that households optimally allocate consumption and education over the life-cycle given an exogenous interest rate and exogenous wages. We show that in such a partial equilibrium setting, the original two-state (physical capital and human capital) optimization problem can be decomposed into two single-state optimal control models. This transformation allows us to rigorously prove the existence of a singular control describing the allocation of education time along a balanced growth path. We derive a constructive condition for a singular control to exist and show that under this condition infinitely many singular controls are optimal in the individual household problem. In contrast to the original general equilibrium framework in which an agent always chooses part-time education and part-time work, in our framework such an agent might find it optimal to allocate her whole available time to education at the beginning of her life and to focus on labor supply only when she is older. (authors' abstract) / Series: Department of Economics Working Paper Series

We discuss a parallel library of efficient algorithms for the solution of linear-quadratic optimal control problems involving largescale systems with state-space dimension up to $O(10^4)$. We survey the numerical algorithms underlying the implementation of the chosen optimal control methods. The approaches considered here are based on invariant and deflating subspace techniques, and avoid the explicit solution of the associated algebraic Riccati equations in case of possible ill-conditioning. Still, our algorithms can also optionally compute the Riccati solution. The major computational task of finding spectral projectors onto the required invariant or deflating subspaces is implemented using iterative schemes for the sign and disk functions. Experimental results report the numerical accuracy and the parallel performance of our approach on a cluster of Intel Itanium-2 processors.

Parce que la robotique collaborative vise à libérer les robots des barrières physiques les séparant des opérateurs humains, de nouveaux défis apparaissent autour de la sécurité de ces derniers. S'il est possible de diminuer la dangerosité des robots en amont de leur conception, les logiciels qui les contrôlent doivent impérativement intégrer des mesures de sécurité, afin d'être compatibles avec des environnements humains dynamiques. Les algorithmes classiques de planification de trajectoire nécessitant de lourds calculs, il est avantageux de modifier la trajectoire en temps réel pour l'adapter à l'environnement dangereux. Dans ce projet de recherche, un algorithme de cinématique inverse, sous forme de problème d'optimisation, est utilisé afin de générer la commande du robot à partir d'une trajectoire définie hors-ligne. L'ajout de contraintes de sécurité à ce problème est particulièrement étudié : dans un premier temps, l'indice de manipulabilité, qui traduit la distance du robot à une configuration singulière, est considéré. Ainsi, il doit être maximisé tout au long de la trajectoire afin d'assurer la meilleure mobilité disponible. Dans un deuxième temps, le facteur humain a été intégré par la prise en compte du confort de celui-ci : afin de réduire le stress éprouvé par l'opérateur face à un robot aux mouvements imprévisibles, on s'assure de minimiser la distance entre l'effecteur et le regard de l'humain pour garantir une plus grande visibilité de la tâche. Dans les deux cas, nous avons présenté une formulation originale de ces critères afin de les intégrer dans le problème d'optimisation. Par ailleurs la contrainte d'évitement d'obstacles a aussi été utilisée, de même que la relaxation de la trajectoire, qui permet au robot de dévier un peu de cette dernière pendant une portion de la durée de la tâche. Enfin des tests en simulation et avec le robot réel Baxter de Rethink Robotics ont permis de valider notre approche et de vérifier les performances en conditions réelles, en utilisant une caméra RGB-D et un logiciel de détection d'humain en temps réel. / Abstract : Because collaborative robots are aimed at working in the vicinity of human workers without physical security fences, they bring new challenges about security. Even if robots can be conceived to be less harmful, their software has to integrate security features in order to be suitable for dynamic human environments. Since classical path planning algorithms require heavy calculations, it is interesting to modify the trajectory in real time to adapt it to the dangerous environment. In this research project, an inverse kinematics solver, in the form of an optimization problem, is used to generate the command of the robot to follow a trajectory defined offline. The addition of security constraints is studied: first, the manipulability index, which reflects the distance of the robot to singular configurations, is considered. Thus, it should be maximized all along the trajectory to ensure the best mobility available. Then the human is integrated by taking into account its comfort: in order to reduce the stress of working near an unpredictable moving robot, the distance between the end-effector and the human gaze is minimized to guarantee a greater visibility of the task. In both cases, we have presented a new formulation of those criteria to integrate them into the optimization problem. Moreover, the collision avoidance constraint is used, as well as the trajectory relaxation, which allows the robot to deviate from its trajectory for a certain amount of time during the task. Finally tests in simulation and with the real Baxter robot from Rethink Robotics validated our approach and the performance has been evaluated in real conditions, using a RGB-D camera and a real time human tracker software.

Portfolio Management with Multiple Benchmarks Bc. Robert Navrátil Abstract: In this thesis, we study a maximal volatility portfolio that treats all assets in a symmetric way and related option contract. To preserve symmetry we need numeraire that treats all assets symmetrically. We choose market index with equal weights. In case of two assets we focus on a variation of a passport option on the portfolio. The optimal strategy for the investor is the mentioned maximal volatility portfolio. We extend the known optimal strategy for the option to a richer class of convex payoff functions. We also show a modification of the optimal strategy for maximizing the probability of ending above or at a desired level. We later extend the symmetric market model to case of three assets, which can be even further extended to an arbitrary number of assets. The three asset model requires more parameters than are observable from the data, however we show indistinguishably of the model on the choice of parameters under very natural conditions. Both numerical simulations and an application on real data is provided. 1

Doctor of Philosophy / Department of Agricultural Economics / Jeffrey M. Peterson / Nathan P. Hendricks / The two studies presented in this dissertation examine incentives for groundwater extraction and their resulting effect on aquifer depletion. Both studies apply dynamic optimization methods in a context of irrigated agriculture in arid and semi-arid regions such as in western Kansas. The first study examines the effects of capital subsidies aimed at increasing irrigation application efficiency. The second study examines the effects of changing incentives posed by changes in climatic patterns and by technical progress in the form of increasing crop water productivity. Both studies have significant policy and groundwater management implications. Subsidies for the adoption of (more) efficient irrigation technologies are commonly proposed and enacted with the goal of achieving water conservation. These subsidies are more politically feasible than water taxes or water use restrictions. The reasoning behind this type of policy is that increased application efficiency makes it possible to sustain a given level of crop production per acre with lower levels of groundwater pumping, all else equal. Previous literature argues that adoption of more efficient irrigation systems may not reduce groundwater extraction. Rewarding the acquisition of more efficient --and capital intensive-- irrigation equipment affects the incentives farmers have to pump groundwater. For instance, the farmer may choose to produce more valuable and water intensive crops or to expand the irrigated acreage after adopting the more efficient irrigation system. Hence, the actual impact of the policy on overall groundwater extraction and related aquifer depletion is unclear. The first chapter examines the effects of such irrigation technology subsidies using a model of inter-temporal common pool groundwater use with substitutable technology and declining well-yields from groundwater stocks, where pumping cost and stock externalities arise from the common property problem. An optimal control analytical model is developed and simulated with parameters from Sheridan County, Kansas-- a representative region overlying the Ogallala aquifer. The study contrasts competitive and optimal allocations and accounts for endogenous and time-varying irrigation capital on water use and groundwater stock. The analysis is the first to account for the labor savings from improved irrigation technologies. The results show that in the absence of policy intervention, the competitive solution yields an early period with underinvestment in efficiency-improving irrigation technology relative to the socially efficient solution, followed by a period of over-investment. This suggests a potential role for irrigation capital subsidies to improve welfare over certain ranges of the state variables. In contrast to previous work, the findings are evidence that significant returns may be achieved from irrigation capital subsidies. Finally, a policy scenario is simulated where an irrigation technology subsidy is implemented to explore whether such a program can capture significant portions of the potential welfare gain. Results indicate that the technology subsidy can improve welfare, but it captures a relatively small portion of the potential gains in welfare. The second chapter presents a dynamic model of groundwater extraction for irrigation where climate change and technical progress are included as exogenous state variables-- in addition to the usual state variable of the stock of groundwater. The key contributions of this study are (i) an intuitive description of the conditions under which groundwater extraction can be non-monotonic, (ii) a numerical demonstration that extraction is non-monotonic in an important region overlying the Ogallala Aquifer, and (iii) the predicted gains from management are substantially larger after accounting for climate and technical change. Intuitively, optimal extraction is increasing in early periods when the marginal benefits of extraction are increasing sufficiently fast due to climate and technical change compared to the increase in the marginal cost of extraction. In contrast, most previous studies include the stock of groundwater as the only state variable and, consequently, recommend a monotonically decreasing extraction path. In this study, the numerical simulations for a region in Kansas overlying the Ogallala Aquifer indicate that optimal groundwater extraction peaks 23 years in the future and the gains from management are large (29.5%). Consistent with previous literature, the predicted gains from management are relatively small (6.1%) when ignoring climate and technical change. The realized gains from management are not substantially impacted by incorrect assumptions of climate and technical change when formulating the optimal plan.

Running, jumping and walking are physical activities that are performed by humans in a simple and efficient way. However, these types of movements are difficult to perform by humanoid robots. Humans perform these activities without difficulty thanks to their ability to absorb the ground impact force. The absorption of the impact force is based on the human ability to vary muscles stiffness. The principal objective of this dissertation is to study vertical jumps in order to reduce the impact force in the landing phase of the jump motion of humanoid robots. Additionally, the impact force reduction is applied to an arm-oriented movement with the objective of preserving the integrity of falling humanoid robot. This dissertation focuses on researching vertical jump motions by designing, implementing and testing variable stiffness control strategies based on Computed-Torque Control while tracking desired trajectories calculated using the Zero Moment Point (ZMP) and the Center of Mass (CoM) conditions. Variable stiffness method is used to reduce the impact force during the landing phase. The variable stiffness approach was previously presented by Pratt et al. in [1], where they proposed that full stiffness is not always required. In this dissertation, the variable stiffness capability is implemented without the integration of any springs or dampers. All the actuators in the robot are DC Motors and the lower stiffness is achieved by the design and implementation of PID gain values in the PID controller for each motor. The current research proposes two different approaches to generate variable stiffness. The first approach is based on an optimal control theory where the linear quadratic regulator is used to calculate the gain values of the PID controller. The second approach is based on Fuzzy logic theory and it calculates the proportional gain (KP) of the PID controller. Both approaches are based on the idea of computing the PID gains to allow for the displacement of the DC motor positions with respect to the target positions during the landing phase. While a DC motor moves from the target position, the robot CoM changes towards a lower position reducing the impact force. The Fuzzy approach uses an estimation of the impact velocity and a specified desired soft landing level at the moment of impact in order to calculate the P gain of the PID controller. The optimal approach uses the mathematical model of the motor and the factor, which affects the Q matrix of the Linear Quadratic Regulator (LQR), in order to calculate the new PID values. A One-legged robot is used to perform the jump motion verification in this research. In addition, repeatability experiments were also successfully performed with both the optimal control and the Fuzzy logic methods. The results are evaluated and compared according to the impact force reduction and the robot balance during the landing phase. The impact force calculation is based on the displacement of the CoM during the landing phase. The impact force reduction is accomplished by both methods; however, the robot balance shows a considerable improvement with the optimal control approach in comparison to the Fuzzy logic method. In addition, the Optimal Variable Stiffness method was successfully implemented and tested in Falling Robots. The robot integrity is accomplished by applying the Optimal Variable Stiffness control method to reduce the impact force on the arm joints, shoulders and elbows.

This thesis deals with problems arising in the study of nonlinear partial differential equations arising from many-body problems. It is divided into two parts: The first part concerns the derivation of a nonlinear diffusion equation from a microscopic stochastic process. We give a new method to show that in the hydrodynamic limit, the particle densities of a one-dimensional zero range process on a periodic lattice converge to the solution of a nonlinear diffusion equation. This method allows for the first time an explicit uniform-in-time bound on the rate of convergence in the hydrodynamic limit. We also discuss how to extend this method to the multi-dimensional case. Furthermore we present an argument, which seems to be new in the context of hydrodynamic limits, how to deduce the convergence of the microscopic entropy and Fisher information towards the corresponding macroscopic quantities from the validity of the hydrodynamic limit and the initial convergence of the entropy. The second part deals with problems arising in the analysis of nonlinear Schrödinger equations of Gross-Pitaevskii type. First, we consider the Cauchy problem for (energy-subcritical) nonlinear Schrödinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superfluid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities without any restriction on the rotation frequency, generalizing earlier results given in the literature. Moreover, we find that the rotation term has a considerable influence in proving finite time blow-up in the focusing case. Finally, a mathematical framework for optimal bilinear control of nonlinear Schrödinger equations arising in the description of Bose-Einstein condensates is presented. The obtained results generalize earlier efforts found in the literature in several aspects. In particular, the cost induced by the physical work load over the control process is taken into account rather then often used L^2- or H^1-norms for the cost of the control action. We prove well-posedness of the problem and existence of an optimal control. In addition, the first order optimality system is rigorously derived. Also a numerical solution method is proposed, which is based on a Newton type iteration, and used to solve several coherent quantum control problems.

With the increase of world population and industrial growth of developing countries, demand for energy, in particular electric power, has gone up at an unprecedented rate over the last decades. To meet the demand, electric power generation by use of fossil fuel has increased enormously thereby producing increased quantity of greenhouse gases. This contributes more and more to atmospheric pollution, which climate scientists believe can adversly affect the global climate, as well as health and the welfare of the world population. In view of these issues, there is global awareness to look for alternate sources of energy such as natural gas, hydropower, wind, solar, geothermal and biomass. It is recognized that this requires replacement of existing infrastructure with new systems, which cannot be achieved overnight. Optimal control theory has been widely used in diverse areas of physical sciences, medicine, engineering and economics. The main motivation of this thesis is to use this theory to find the optimum strategy for integration of all currently available renewable energy sources with the existing electric power generating systems. The ultimate goal is to eliminate fossil fuels. Eight main energy sources namely, Coal, Petroleum, Natural Gas, Conventional Hydro, Wind, Solar, Geothermal and Biomass are considered in a dynamic model. The state of the dynamic model represents the level of energy generation from each of the sources. Different objective functions are proposed in this thesis. These range from meeting the desired target level of power generation from each of the available sources at the end of a given plan period, to reducing the implementation and investment costs; from minimizing the production from polluted energy sources to meeting the electricity demand during a whole plan period. Official released data from the U.S. Energy Information Administration have been used as a case study. Based on real life data and the mathematics of optimal control theory, we present an optimal policy for integration of renewable energy sources to the national power grid.