Este trabalho trata do controle dos movimentos de um Veículo Submersível Autônomo (VSA). Veículos submersíveis são difíceis de controlar devido à alta não linearidade de seus modelos, ao forte acoplamento de movimentos, ao desconhecimento de certas dinâmicas, às incertezas do próprio modelo, devido a distúrbios externos impostos pelo ambiente e devido ao ruído de sensores. A dificuldade de controle pode ser exacerbada quando o veículo é subatuado. Para realização deste trabalho foi escolhido um VSA do tipo torpedo, cujo modelo matemático disponível na literatura foi devidamente modificado para garantir uma melhor descrição de seus movimentos em seis graus de liberdade (6-GL). O modelo foi então validado através de simulações numéricas. Para a síntese dos controladores utilizou-se uma técnica de controle avançada. Mais especificamente, utilizou-se a abordagem do controle H1 para sistemas multivariáveis. Assim foram obtidos controladores centralizados capazes de superar o problema do forte acoplamento de movimentos. Técnicas de controle avançado permitem também considerar as informações disponíveis sobre perturbações, incertezas, ruídos e diferentes tipos de entrada já na fase de síntese, o que permite obter controladores com desempenho adequado numa ampla faixa de operação. Neste trabalho, em particular, a técnica da Sensibilidade Mista foi escolhida para a síntese de controladores robustos. Nesta abordagem, formatam-se algumas funções de malha fechada ligadas a sensibilidade do sistema buscando garantir estabilidade e desempenho robusto para o sistema controlado. Usando a mesma técnica de controle desenvolveu-se ainda um controlador de dois graus de liberdade (2-GL), apropriado para aplicação no problema de guiagem, onde procura-se seguir trajetórias tridimensionais. Os controladores desenvolvidos foram testados em simulações numéricas, produzindo-se uma grande quantidade de resultados. A análise destes resultados revela o poder e flexibilidade das técnicas escolhidas. / This work focuses the motion control of an Autonomous Underwater Vehicle (AUV). Underwater vehicles are difficult to control due to high non-linearities of its model, coupling between dynamics, unknown dynamics, model uncertainties, disturbances and sensor noises. Difficulty is greater, when the system is subactuated. In this work, a mathematical model of a torpedo-like AUV available in the bibliography was chosen and refined, leading to a six degree of freedom (6-DOF) model. The model was further analyzed and validated by a number of numerical simulations. Advanced approaches were used for the synthesis of controllers. Speciffically, a H1 approach for multivariable systems was used. Thus, a centralized controller was developed, able to avoid the problem of high coupling between the variables. This advanced approach is also able to use informations about perturbations, uncertainties, noises and different types of input signals in the synthesis stage, leading to controllers with better performance in a large operation bandwidth. In this work, a Mixed Sensitivity approach was employed. This control approach is based on the shapping of well known closed-loop sensitivity functions, seeking to achieve stability and performance robustness. Using a similar technique, a controller with two degree of freedom (2-DOF) was also synthesised, to tackle the guidance problem tracking of 3-D trajectories was then fully achieved. The controllers developed were tested in a number of numerical simulations. Analyses of results reveals the power and flexibility of the employed techniques.

<div>Previous studies have shown that implementing trajectory optimization can reduce state estimations errors. These navigation based path planning problems are often diffcult to solve being computationally burdensome and exhibiting other numerical issues, so former studies have often used lower-delity methods or lacked explanatory power.</div><div><br></div><div><div>This work utilizes indirect optimization methods, particularly optimal control theory, to obtain high-quality solutions minimizing state estimation errors approximated by a continuous-time extended Kalman lter. Indirect methods are well-suited to this because necessary conditions of optimality are found prior to discretization and numerical computation. They are also highly parallelizable enabling application to increasingly larger problems.</div></div><div><br></div><div><div>A simple one dimensional problem shows some potential obstacles to solving problems of this type including regions of the trajectory where the control is unimportant. Indirect trajectory optimization is applied to a more complex scenario to minimize location estimation errors of a single cart traveling in a 2-D plane to a goal location and measuring range from a xed beacon. This resulted in a 96% reduction of the location error variance when compared to the minimum time solution. The single cart problem also highlights the importance of the matrix that encodes the linearization of the vehicle's measurement with respect to state. It is shown in this case that the vehicle roughly attempts to maximize the magnitude of its elements. Additionally, the cart problem further illustrates problematic regions of a design space where the objective is not signicantly affected by the trajectory.</div></div><div><br></div><div><div>An aircraft descent problem demonstrates the applicability of these methods to aerospace problems. In this case, estimation error variance is reduced 28.6% relative to the maximum terminal energy trajectory. Results are shown from two formulations of this problem, one with control constraints and one with control energy cost, to show the benets and disadvantages of the two methods. Furthermore, the ability to perform trade studies on vehicle and trajectory parameters is shown with this problem by solving for dierent terminal velocities and different initial locations.</div></div>

L’objectif de cette thèse est d’appliquer la théorie du contrôle optimal à la dynamique d’ensembles inhomogènes de spins. La première partie est dévouée au contrôle d’un ensemble de spins couplé à une cavité. La théorie est introduite en détail, et une méthode générale pour contrôler efficacement les spins est présentée. Plusieurs pulses sont déterminés dans les régimes de bonne et de mauvaise cavité. De même, les fonctions non linéaires généralisées sont utilisées afin de déterminer des approximations simples. Dans un second temps, le problème de la maximisation du Signal-sur-Bruit d’un écho de spin est abordé, et des conditions d’optimisations sont établies. Il est montré que les nouveaux pulses sont supérieurs à ceux de l’état de l’art, en termes de fidélité et d’augmentation du Signal-sur-Bruit. Par ailleurs, ils permettent d’explorer de nouvelles situations (e.g. mesure de FID (Free Induction Decay) en CQED avec un taux de perte de cavité plus long que T2∗). La seconde partie est dévouée à des problèmes de RMN/IRM standard. Deux situations de "sélectivité" sont étudiées. La première consiste à déterminer le pulse le plus court qui produit la transformation la plus sélective par rapport aux offsets. Dans le cas ultra-sélectif, la solution optimale est un arc singulier d’amplitude constante. Cependant, si des contraintes de robustesse sont ajoutées, la solution optimale peut-être un arc régulier. La seconde est celle de l’optimisation de base de données pour des expériences de MR-fingerprinting. Dans ce cas, un champ de contrôle est conçu pour générer une base de données "d’empreinte digitale" qui maximise le processus de reconnaissance entre spins de paramètres différents. / The goal of this thesis is to apply optimal control theory to the dynamics ofinhomogeneous spin ensembles. The first part focuses on the control of a spin ensemble coupled to a cavity. The theory is introduced in detail, and a general method to efficiently control spins ispresented. Several pulses are derived in the bad/good cavity regimes using numerical optimal control techniques. Additionally, non-linear generalized functions are used in order to derivesimple approximated solutions. In a second step, the problem of spin echo Signal to Noise Ratio maximization is investigated, and maximization conditions are derived. It is shown that new pulses are superior to state-of-the-art square pulses in terms of fidelity and SNR maximization. Moreover, they allow us to explore new situations (e.g. Free Induction Decay measurementsin cavity-QED with a cavity damping longer than T2∗). The second part focuses on standard NMR/MRI problems. Two distinct situations of selectivity are investigated. The first one consists of determining the time minimum pulse which produces the most offset-selective transformation. In the ultra-selectivity case, the optimal solution is a singular arc of constant amplitude. However,if additional robustness constraints are taken into account, the optimal solution can be a regular arc. The second situation is the optimization of databases for MR-fingerprinting experiments. In this case, a control field is designed so that it generates a fingerprint database which maximizesthe recognition process between several spins with different parameters.

Many systems operate in uncertain, possibly adversarial environments, and their successful operation is contingent upon satisfying specific requirements, optimal performance, and ability to recover from unexpected situations. Examples are prevalent in many engineering disciplines such as transportation, robotics, energy, and biological systems. This thesis studies designing correct, resilient, and optimal controllers for discrete-time complex systems from elaborate, possibly vague, specifications. The first part of the contributions of this thesis is a framework for optimal control of non-deterministic hybrid systems from specifications described by signal temporal logic (STL), which can express a broad spectrum of interesting properties. The method is optimization-based and has several advantages over the existing techniques. When satisfying the specification is impossible, the degree of violation - characterized by STL quantitative semantics - is minimized. The computational limitations are discussed. The focus of second part is on specific types of systems and specifications for which controllers are synthesized efficiently. A class of monotone systems is introduced for which formal synthesis is scalable and almost complete. It is shown that hybrid macroscopic traffic models fall into this class. Novel techniques in modular verification and synthesis are employed for distributed optimal control, and their usefulness is shown for large-scale traffic management. Apart from monotone systems, a method is introduced for robust constrained control of networked linear systems with communication constraints. Case studies on longitudinal control of vehicular platoons are presented. The third part is about learning-based control with formal guarantees. Two approaches are studied. First, a formal perspective on adaptive control is provided in which the model is represented by a parametric transition system, and the specification is captured by an automaton. A correct-by-construction framework is developed such that the controller infers the actual parameters and plans accordingly for all possible future transitions and inferences. The second approach is based on hybrid model identification using input-output data. By assuming some limited knowledge of the range of system behaviors, theoretical performance guarantees are provided on implementing the controller designed for the identified model on the original unknown system.

An existing mathematical model of ordinary differential equations was studied to better understand the interactions between hepatitis C virus (HCV) and the immune system cells in the human body. Three possible qualitative scenarios were explored: dominant CTL response, dominant antibody response, and coexistence. Additionally, a sensitivity analysis was carried out to rank model parameters for each of these scenarios. Therapy was addressed as an optimal control problem. Numerical solutions of optimal controls were computed using a forward-backward sweep scheme for each scenario. Model parameters were estimated using ordinary least squares fitting from longitudinal data (serum HCV RNA measurements) given in reported literature.

Soft robotic platforms are becoming increasingly popular as they are generally safer, lighter, and easier to manufacture than their more rigid, heavy, traditional counterparts. These soft platforms, while inherently safer, come with significant drawbacks. Their compliant components are more difficult to model, and their underdamped nature makes them difficult to control. Additionally, they are so lightweight that a payload of just a few pounds has a significant impact on the manipulator dynamics. This thesis presents novel methods for addressing these issues. In previous research, Model Predictive Control has been demonstrably useful for joint angle control for these soft robots, using a rigid inverted pendulum model for each link. A model describing the dynamics of the entire arm would be more desirable, but with high Degrees of Freedom it is computationally expensive to optimize over such a complex model. This thesis presents a method for simplifying and linearizing the full-arm model (the Coupling-Torque method), and compares control performance when using this method of linearization against control performance for other linearization methods. The comparison shows the Coupling-Torque method yields good control performance for manipulators with seven or more Degrees of Freedom. The decoupled nature of the Coupling-Torque method also makes adaptive control, of the form described in this thesis, easier to implement. The Coupling-Torque method improves performance when the dynamics are known, but when a payload of unknown mass is attached to the end effector it has a significant impact on the dynamics. Adaptive Control is needed at this point to compensate for the model's poor approximation of the system. This thesis presents a method of layering Model Reference Adaptive Control in concert with Model Predictive Control that improves control performance in this scenario. The adaptive controller modifies dynamic parameters, which are then delivered to the optimizer, which then returns inputs for the system that take all of this information into account. This method has been shown to reduce step input tracking error by 50% when implemented on the soft robot.

This thesis considers the coordinated control of autonomous agents. The agents are modeled as double integrators, one for each Cartesian dimension. The goal is to force the agents to converge to a formation specified by their desired relative positions. To this end a pair of one-step-ahead optimization based control laws are developed. The control algorithms produce a communication topology that mirrors the geometric formation topology due to the careful choice of the minimized cost functions. Through this equivalence a natural understanding of the relationship between the geometric formation topology and the communication infrastructure is gained. It is shown that the control laws are stable and guarantee convergence for all viable formation topologies. Additionally, velocity constraints can be added to allow the formation to follow fixed or arbitrary time dependent velocities. Both control algorithms only require local information exchange. As additional agents attach to the formation, only those agents that share position constraints with the joining agents need to adjust their control laws. When redundancy is incorporated into the formation topology, it is possible for the system to survive loss of agents or communication channels. In the event that an agent drops out of the formation, only the agents with position interdependence on the lost agent need to adjust their control laws. Finally, if a communication channel is lost, only the agents that share that communication channel must adjust their control laws. The first control law falls into the category of distributed control, since it requires either the global information exchange to compute the formation size or an a priori knowledge of the largest possible formation. The algorithm uses the network size to penalize the control input for each formation. When using a priori knowledge, it is shown that additional redundancy not only adds robustness to loss of agents or communication channels, but it also decreases the settling times to the desired formation. Conversely, the overall control strategy suffers from sluggish response when the network is small with respect to the largest possible network. If global information exchange is used, scalability suffers. The second control law was developed to address the negative aspects of the first. It is a fully decentralized controller, as it does not require global information exchange or any a priori knowledge.

In this work, we study the application both of optimal control techniques and a numerical method to a system of partial differential equations arising from a problem in wound healing. Optimal control theory is a generalization of calculus of variations, as well as the method of Lagrange Multipliers. Both of these techniques have seen prevalent use in the modern theories of Physics, Economics, as well as in the study of Partial Differential Equations. The numerical method we consider is the method of lines, a prominent method for solving partial differential equations. This method uses finite difference schemes to discretize the spatial variable over an N-point mesh, thereby converting each partial differential equation into N ordinary differential equations. These equations can then be solved using numerical routines defined for ordinary differential equations.

System Lupus Erythematosus (SLE) is a chronic inflammatory autoimmune disorder that affects many parts of the body including skin, joints, kidneys, brains and other organs. Lupus Nephritis (LN) is a disease caused by SLE. Given the complexity of LN, we establish an optimal treatment strategy based on a previously developed mathematical model.For our thesis work, the model variables are: Immune Complexes (I), Pro-inflammatory mediators (P), Damaged tissue (D), and Anti-inflammatory mediators (A). The analysis in this research project focuses on analyzing therapeutic strategies to control damage using both parameter estimation techniques (integration of data to quantify any uncertainties associated with parameters) and optimal control with the goal of minimizing time spent on therapy for treating damaged tissue by LN.

Cette thèse s'intéresse à une étude des EDSR ergodiques, avec pour principal objectif leur application à l'étude du comportement en temps long de certaines EDP. Dans un premier temps, nous démontrons des résultats (qui sont déjà connus dans le cadre où l'EDS sous-jacente est à bruit additif) dans un cadre de bruit sous-jacent multiplicatif. Par la suite, l'introduction d'un nouvel aléa via un processus de Poisson nous permet de nous intéresser non plus au comportement en temps long d'une seule EDP, mais au comportement en temps long d'un système d'EDP couplées. Enfin, lorsque l'EDS sous-jacente est bruitée par un processus de Lévy, le lien est fait avec des équations intégro-différentielles partielles. L'application de ces équations à la résolution de problèmes de contrôle optimal est également présentée. / In this thesis, we are interested in studying ergodic BSDEs, and our main goal is to apply our results to the large time behaviour of some PDEs. First, we prove some results (already known in the case where the underlying SDE has an additive noise) in the case of an underlying multiplicative noise. Then, we introduce a Poisson process and it leads us to the large time behaviour of a system of coupled PDEs. Finally, when the underlying SDE has a Lévy noise, we make a link with partial integro-differential equations. We also apply these equations to solve some optimal control problems.