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This paper considers the problem of the deployment of a set of agents distributed on a disk-shaped grid onto three-dimensional (3-D) profiles, by using a continuum approximation (valid in the limit for a large number of agents) and then a control methodology for partial differential equations (PDEs). The agents’ collective behavior is modeled by a pair of radially-varying diffusion–reaction PDEs in polar coordinates, whose state determines the agents’ position. Having a radially-varying reaction coefficient not only increases the challenge of kernel equations becoming singular in radius, but also brings more potential deployment manifolds. To stabilize and increase the convergence of the deployment, a boundary controller and a boundary observer are designed by combining an infinite-dimensional backstepping approach with a Fourier series decomposition technique, thus driving all agents to the desired profile. A key feature of the presented result is that the desired profile only needs to be known by the leaders, with the followers only needing to follow a simple control strategy which requires only the measurement of its current position and communication with its neighbors as defined by the multi-agent system topology. The method provides closed-loop exponential stability with any prescribed convergence rate in the L2 norm. Simulation tests are shown to prove the effectiveness of the proposed algorithm.

This article deals with one of the types of “Satellite Range Scheduling” problems arising in Earth Observation Satellite operations, Antenna-Satellite Scheduling. Given a set of satellites, a set of available antennas and a time horizon, the problem consists of designing an operational plan that assigns satellites to antennas in an optimal fashion. Extending a previous integer linear programming (ILP) model (shortening model, with only integer variables), we propose a mixed ILP (MILP) (shaving model, which includes both continuous and integer variables), to more efficiently solve this problem. After computing the passes generated by the satellites' windows of visibility from the antennas, the optimal planner is able to cancel a pass, move it to another antenna, or shorten its duration, in order to avoid scheduling conflicts. In contrast to the shortening model, which used intersections between passes to determine the best schedule, the shortening operation is now referred to as shaving, since the shaving model can arbitrarily adjust the duration of a pass in a razor-like fashion, giving the model its name. Computational results obtained in tests over realistic scenarios prove that the shaving model outperforms the shortening model, producing fewer cancellations, smaller shaved times, and a fairer distribution of cancelled passes among satellites, with much shorter preprocessing times.

La importancia estratégica de la Órbita Baja Terrestre (LEO) ha crecido significativamente con el aumento de despliegues de satélites. Esta región se ha vuelto crucial para la economía espacial, con la Agencia Espacial Europea estimando más de 8500 satélites activos en LEO, que representan solo una fracción del total de objetos en esta banda orbital. La última década ha visto un aumento exponencial en la población de satélites, principalmente debido al advenimiento de mega-constelaciones para Internet global y la reducción de los costes de lanzamiento. Esta tesis profundiza en las complejidades de la detección de maniobras en mitad de los desafíos presentados por la congestión orbital y la basura espacial en LEO. Algunos satélites no comunican la realización de maniobras, lo que hace que sus trayectorias sean impredecibles. La detección de maniobras es fundamental para mantener catálogos orbitales y prever posibles colisiones, necesitando pues de métodos de detección robustos y eficientes. Esta tesis presenta el desarrollo de nuevas métricas y algoritmos para la detección de maniobras en órbita baja terrestre. Se centra en un escenario donde solo se dispone de datos de una única estación de radar, con el objetivo final de crear algoritmos de detección de maniobras para el Radar de Vigilancia y Seguimiento Espacial Español. La estructura sigue la evolución cronológica de la investigación, cada capítulo revisando contribuciones específicas de las etapas del doctorado. Inicialmente, la detección de maniobras se aborda desde el punto de vista del análisis de accesibilidad, estudiando la viabilidad de una métrica de Distancia de Mahalanobis (MD) derivada de un attributable en el espacio de medidas, y su inclusión en un Filtro de Detección de Maniobras. Los métodos desarrollados se prueban en datos reales de satélites con éxito variable, mostrando dónde residen los aspectos críticos del problema. La correcta representación de la incertidumbre bajo largos intervalos de pro, The strategic importance of Low Earth Orbit (LEO) has grown significantly, with increased satellite deployments for various applications. This region has become crucial for the space economy, with the European Space Agency (ESA) estimating over 8500 active satellites in LEO, representing only a fraction of the total objects in this orbital band. The last decade has seen an exponential increase in satellite populations, primarily due to the advent of mega-constellations for global internet coverage like Starlink and OneWeb and the reduction in launch costs. This thesis delves into the complexities of maneuver detection amidst the challenges posed by orbital congestion and space debris in LEO. Non-cooperative satellites do not broadcast their maneuver plans, making their trajectories unpredictable and potentially dangerous. Accurate maneuver detection is vital for maintaining orbital catalogs and predicting potential collisions, necessitating robust and efficient detection methods. This thesis presents the development of novel metrics and algorithms for maneuver detection in LEO. The main focus is on a scenario where only data from a single radar station is available, the final aim being to create automatic maneuver detection algorithms for the Spanish Space Surveillance and Tracking Surveillance Radar (S3TSR). The structure follows the chronological evolution of the research effort, each chapter reviewing specific contributions of the PhD stages. At first maneuver detection is approached from the point of view of reachability analysis, studying the feasibility of an attributable-derived Mahalanobis Distance (MD) metric in measurement space, and its inclusion in a Maneuver Detection Filter (MDF). The developed methods are tested on real satellite data with varying degrees of success, showing where the critical aspects of the problem reside. Stemming from the constrained source of radar data, the correct uncertainty representation under the long propagation gaps become

La Vela Solar Eléctrica (E-sail) es una novedosa tecnología propulsiva sin propelente aplicable a vehículos espaciales, que genera el empuje necesario a partir de la interacción eléctrica entre cables (tethers) muy largos cargados eléctricamente y los iones del viento solar. Así pues, el presente trabajo aborda el problema de control de actitud de las E-sails, con el objectivo principal de ampliar el rango de validez de trabajos recientes ([1], [2]), que logran realizar maniobras de cabeceo manteniendo el eje de giro en una alineación próxima con el Sol, y considerando una deformación simétrica de los tethers. Para tal fin, en este proyecto se propone un modelo de deformación no simétrico, que permite a los tethers deformarse de forma independiente según la orientación relativa al Sol y el voltaje aplicado, tal y como sucedería en realidad. Esta propuesta es entonces analizada bajo control LQR para comprobar su rango de validez y estudiar la respuesta al control de la vela solar eléctrica, y, después, bajo control LQG para incluir la influencia de las perturbaciones y el ruido procedente de los sensores en el análisis., The Electric Solar Sail (E-sail) is an innovative propellantless propulsion technology for spacecraft, which generates thrust from interaction of long electrically-charged tethers with solar wind ions. This thesis addresses the attitude control problem for E-sails, with the main objective of widening the validity range of recent work ([1], [2]), that are able to perform pitch change maneuvers near a Sun-facing configuration, assuming a symmetrical deformation of all tethers. For this purpose, a non-symmetrical deformed shape model is proposed to allow the tethers deform independently according to relative orientation from Sun and applied voltage, as it would happen in reality. This proposal is then analyzed under LQR control to check the validity range and study the behaviour of the E-sail’s control response, and, later, under LQG control to include the effects of disturbances and sensors’ noise.

Debido al efecto de las perturbaciones, el cual es difícil de modelar con precisión, siempre aparece un cierto grado de incertidumbre asociado a la órbita. Para modelar y trabajar con dicha incertidumbre se puede utilizar la teoría de la probabilidad. El objetivo de este proyecto es el de estudiar y analizar distintos métodos que permitan la propagación de la incertidumbre a lo largo de la órbita, tales como el método Montecarlo, Linealización, Unscented Transform (UT) o los Elementos Orbitales Equinocciales. Asimismo, las perturbaciones más importantes para los satélites en órbita baja, el J2 y la resistencia aerodinámica, son estudiadas también. Además, el efecto que tienen en los elementos orbitales de la órbita y en la propia evolución de la incertidumbre es analizado de igual forma., Due to perturbation effects, which are difficult to model accurately, there is always a certain degree of uncertainty associated with the orbit. To model and work with such uncertainty, probability theory can be utilized. The objective of this project is to study and analyze different methods that allow us to propagate this uncertainty along the orbit, such as the Monte Carlo method, Linearization, Unscented Transform (UT), or Equinoctial Orbital Elements. Additionally, the most significant perturbations for satellites in low earth orbit, the J2 effect and aerodynamic drag, are also studied. Besides, their impact on the orbital elements and the evolution of uncertainty itself are also analyzed.

Las Velas Solares Eléctricas son una alternativa interesante a los sistemas de propulsión tradicionales, un nuevo tipo de propulsión espacial sin combustible que obtiene la energía de los protones del viento solar, obteniendo unos niveles de aceleración que las hace potencialmente útiles para misiones interplanetarias. Pese a que estos sistemas surgen como una alternativa para alcanzar objetos lejanos sin depender de maniobras asistidas por gravedad, la combinación de ambos puede ser muy prometedora. El primer objetivo de este proyecto es analizar diferentes trayectorias con asistencia gravitatora para Velas Solares Eléctricas. Tras esto, estas trayectorias han de ser optimizadas con el fin de reducir el tiempo de vuelo. Existen múltiples alternativas de optimización disponibles, y de entre estas opciones, se han elegido específicamente los algoritmos genéticos. Los algoritmos genéticos, un algoritmo evolutivo que ha experimentado un crecimiento significativo en el campo de estudio en la última década debido a los avances computacionales, proporcionan excelentes resultados; no obstante, implican un costo computacional sustancial. Distintas variaciones en los parámetros de control y las operaciones son llevadas a cabo, dando lugar a resultados comparables con un tiempo computacional reducido., Electric Solar Wind Sails are an interesting alternative to traditional propulsion systems, a new type of spacecraft propellantless propulsion system that gets its energy from solar wind protons, making them potentially useful for interplanetary missions. Despite the fact that these systems emerge as an alternative to reach far away objects without depending on gravity assist maneuvers, the combination of both can be very promising. The first goal of this project is to analyze different gravity-assist trajectories for Electric Solar Wind Sails. Then, these trajectories have to be optimized in order to reduce the flight time. There are multiple optimization alternatives available, and out of these options, genetic algorithms have been specifically chosen. Genetic algorithms, an evolutionary algorithm that has experienced significant growth in the field of study over the last decade due to computational advancements, provide excellent results; however, they imply a substantial computational cost. Several variations on control parameters and operations are conducted, resulting in comparable results with reduced computational time. V

Recently, the problem of boundary stabilization and estimation for unstable linear constant-coefficient reaction–diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. Some early success has been achieved under simplifying conditions, such as radially-varying reaction coefficients under revolution symmetry, on a disk or a sphere. These particular cases notwithstanding, the problem remains open. The main issue is that the equations become singular in the radius; when applying the backstepping method, the same type of singularity appears in the kernel equations. Traditionally, well-posedness of these equations has been proved by transforming them into integral equations and then applying the method of successive approximations. In this case, with the resulting integral equation becoming singular, successive approximations do not easily apply. This paper takes a different route and directly addresses the kernel equations via a power series approach (in the spirit of the method of Frobenius for ordinary differential equations), finding in the process the required conditions for the radially-varying reaction (namely, analyticity and evenness) and showing the existence and convergence of the series solution. This approach provides a direct numerical method that can be readily applied, despite singularities, to both control and observer boundary design problems.

The problem of state estimation for vehicles when an initial fix is highly uncertain and/or the number of sensors is not sufficient (and changes with time) is very relevant for both aircraft and spacecraft navigation. This work proposes a Locally Linearized Particle Filter based on a quaternion-adapted Unscented Kalman Filter to estimate the state of a vehicle with minimal sensors and uncertain initial conditions, exploiting geometrical symmetries. The algorithm is applied to a ballistic vehicle navigating towards a laser-illuminated target using on-board sensors, including a triad of accelerometers and gyroscopes, a barometric altimeter and a laser receiver. A symmetry around the vertical axis is identified; based on it, the algorithm becomes capable of solving the navigation problem, even with highly uncertain initial conditions and without enough sensor information; this second condition is particularly severe when the laser receiver is not yet obtaining data. The proposed navigation algorithm offers promising results in simulation, rapidly converging to an accurate estimate of the real trajectory when the laser receiver becomes active.

A escala astronómica, el principal motor de muchos de los procesos del Universo es la gravedad. El objetivo de la misión LISA es la detección de ondas gravitacionales de baja frecuencia desde del espacio. Dada la imposibilidad de sustituir o reparar el equipo de las misiones espaciales no tripuladas, uno de los factores fundamentales es la fiabilidad de la instrumentación a bordo. El objetivo de este proyecto es estudiar la fiabilidad de uno de los módulos electrónicos que componen el fasómetro de la misión LISA, así como la optimización de la misma. El análisis de la fiabilidad se basa en la arquitectura interna de las placas electrónicas y en la interacción entre los componentes. El proceso de optimización se centra en la búsqueda de un aumento de la fiabilidad mediante una mejora del rendimiento térmico de los componentes. Otro de los objetivos desarrollados en este documento es establecer una disposición de los componentes en la placa, de forma que se cumplan los requisitos necesarios de cada uno para su correcto funcionamiento. Por último, se proponen modificaciones en la estructura del módulo para reducir el peso total., On an astronomical scale, the main driver of many of the processes in the Universe is gravity. The goal of the LISA mission is to detect low-frequency gravitational waves from space. Given the impossibility of replacing or repairing the equipment of unmanned space missions, one key factor is the reliability of the on-board instrumentation. The objective of this project is to study the reliability of one of the electronic modules that make up the Phasemeter of the LISA mission, as well as its optimisation. The reliability analysis is based on the internal architecture of the electronic boards and the interaction between components. The process focuses on the search for an increase in reliability based on an improvement in the thermal performance of the components. Another of the objectives developed in this document is to establish a layout of the electronic devices on the board so that the necessary requirements of each one are met in order for them to operate correctly. Finally, modifications to the module structure are proposed in order to reduce the overall weight.

En esta memoria se abordará el tema de control de actitud de las velas solares eléctricas, un nuevo mecanismo de generación de empuje que usa los iones procedentes del plasma del viento solar y que permite, por tanto, eliminar la masa de propulsante necesaria para viajar por el Sistema Solar. Se comenzará realizando un estudio del estado actual de las velas solares eléctricas, sus posibles aplicaciones y sus ventajas respecto a sistemas cuyo fin es el mismo. Posteriormente, se introducirán aspectos básicos de la representación y la cinemática de la actitud, para luego, basándonos en la literatura de referencia presentar el modelo dinámico del que se ha hecho uso. Además, se reproducirán los resultados obtenidos aplicando la ley de control expuesta en [2] con el objetivo de comprender mejor el comportamiento de las velas solares eléctricas. Asimismo, se propondrá una nueva ley de control basada en el Control LQR con el objetivo de mejorar los resultados anteriormente comentados. Por ultimo, se analizarán los resultados obtenidos en varios casos en los que se han modificado distintos parámetros del sistema, para ver como se comporta la E-sail con la nueva ley de control propuesta., This report will address the issue of attitude control of electric solar sails, a new oush generation mechanism that uses ions from solar wind plasma and therefore allows to eliminate the mass of propellant necessary to travel through the Solar System. A study of the current state of electric solar sails, their possible applications and their advantages over systems whose purpose is the same is realised in the first place. Subsequently, basic aspects of the representation and kinematics of attitude are introduced. Then, the dynamic model thata has been used and that has benn obtained from the reference literature is presented. In addition, the results obtained by applying the control law set out in [2] will be reproduced in order to better understand the behaviour of electric solar sails. Likewise, a new control law based on LQR Control will be proposed with the aim of improving the results mentioned above. Finally, the results obtained in several cases in which different parameters of the system have been modified will be analysed, to see how the E-sail behaves with the proposed new control law.

Tras analizar el problema de los dos cuerpos y revisar en profundidad las distintas perturbaciones a las que están sometidos los satélites, se explica la herramienta Orekit. Esta permite el cálculo de la órbita de los satélites afectados bajo diferentes tipos de perturbaciones. Mediante Orekit se realiza un extenso análisis abordando las fuerzas más importantes que pueden afectar a satélites en LEO y GEO, para posteriormente pasar a analizar datos reales de satélites proporcionados por la agencia espacial europea en formato OEM. Con estos datos se diseña un propagador que permita predecir con alta precisión la posición y velocidad de los satélites Sentinel-1A/B y Sentinel-2A/B y finalmente se extraen las conclusiones y se proponen futuras mejoras., The objective of this project is to develop an accurate, efficient propagator for Low Earth Orbit satellites designed and launched by the European Space Agency. Predicting where an object will be after several hours since the last time we measured its position is crucial in order to prevent accidents with other satellites during the time lapse we can not track the satellite’s position and direction of movement. This need resulted in this project, in which the analysis of different kind of perturbations is made in order to decide which ones we should take into account and which ones are disposables. Perturbations can be understood as the forces that disturb the classical Keplerian orbit of orbiting objects. The four main forces that must be taken into account are the non-spherical body’s gravity, the third-body perturbations, the solar-radiation pressure and the atmospheric drag, being the last one the most important in LEO. Other kind of perturbations, such as tides or the precession of Earth’s axis, are discussed throughout this project. To perform the computations to predict the position and velocity of the satellites under study, the tool Orekit is used. This low-level space dynamic library written in Java can be integrated into Matlab, and offers a wide scope when studying space dynamics. This tool is the base of this project, and it is used to analyse the effect that different kind of perturbations have in LEO and GEO satellites. Since it was released as an open source in 2008, its popularity grew exponentially due to the precision that this tool allows. In this project its library is presented, as well as the different uses that this tool can have. The data of the satellites is provided by ESA in the form of Orbit Ephemeris Messages, which are used to specify the position and velocity of a single object at one or multiple epochs. The OEMs format is presented in this project, along with several examples of different versions of the OEMs. In order to design a pr

We present a novel methodology for designing output-feedback backstepping bilateral boundary controllers for an unstable 1D diffusion-reaction partial differential equation (PDE) with spatially varying reaction. Using folding transforms the parabolic PDE into a 2 × 2 coupled PDE system with coupling through compatibility conditions. We apply a two-tiered backstepping approach, where the invertibility of the transformations guarantees the statefeedback controllers exponentially stabilize the trivial solution of the PDE system. A state observer is also designed for two collocated measurements at an arbitrary interior point, generating exponentially stable state estimates. The output feedback control law is formulated by composing the independently designed state-feedback controller with the observer, and the resulting dynamic feedback is shown to stabilize the trivial solution. Some numerical analysis on how the selection of these points affect the responses of the controller and observer are discussed, with simulations illustrating various choices of folding points and their effect on the stabilization in different performance indexes.

In this paper, we study the single-input boundary feedback stabilization of 3 × 3 linear hyperbolic partial differential equations (PDEs) with two counterconvecting PDEs and the third one with zero characteristic speed. We design a full-state backstepping controller which exponentially stabilizes the origin in the L2sense. The zero transport velocity makes the previous backstepping designs inapplicable (their application would result in a controller with infinite gains). To employ backstepping in the presence of zero speed, we use an invertible Volterra transformation only for the PDEs with nonzero speeds, leaving the state of the zero-speed PDE unaltered in the target system, but making the target zero-speed PDE input-to-state stable with respect to the decoupled and stable counterconvecting nonzero-speed PDEs. In addition to achieving stabilization, we produce an explicit bound on the rate of convergence of the target system by a method of successive approximations and the use of Laplace transform. Simulation results are presented to illustrate the effectiveness of the proposed control design.

Las Velas Solares Eléctricas son un nuevo tipo de propulsión espacial que no requiere de combustible ya que obtiene la energía de los protones del viento solar, y que es potencialmente útil para misiones interplanetarias. Con un modelo propulsivo apropiado, que puede ser obtenido de la literatura, es posible estudiar órbitas óptimas para vuelos interplanetarios. El primer objetivo de este proyecto es estudiar las propiedades de las órbitas de tiempo mínimo en misiones hacia Marte y Júpiter utilizando un método indirecto aplicando las ecuaciones de Euler-Lagrange y el Principio del Mínimo de Pontryagin, y luego hallar dichas órbitas usando métodos de transcripción directa para convertir el problema en un problema de Programación No Lineal. Tras esto, se añaden las perturbaciones del viento solar, y múltiples estrategias de guiado basadas en el Regulador Cuadrático Lineal y el Control Predictivo por Modelo se desarrollan, simulan y comparan, obteniendo resultados satisfactorios., Electric Solar Wind Sails are a new type of spacecraft propellantless propulsion system that gathers its energy from solar wind protons and is potentially useful for interplanetary missions. With an appropriate thrust model, which can be obtained from the literature, one can study optimal orbits for interplanetary flights. The first goal of this project is to study the properties of time-optimal orbits for missions to Mars and Jupiter using an indirect approach by applying Euler-Lagrange equations and Pontryagin’s Minimum Principle, and then compute them with direct transcription methods, converting the problem into Non Linear Programming form. Then, solar wind perturbations are added, and several guidance strategies based on Linear Quadratic Regulator Control and Model Predictive Control are developed, simulated and compared, yielding satisfactory results.

In this paper, we present a rapid boundary stabilization of a Timoshenko beam with anti-damping and anti-stiffness at the uncontrolled boundary, by using PDE backstepping. We introduce a transformation to map the Timoshenko beam states into a (2+2) × (2+2) hyperbolic PIDE-ODE system. Then backstepping is applied to obtain a control law guaranteeing closed-loop stability of the origin in the H1 sense. Arbitrarily rapid stabilization can be achieved by adjusting control parameters. Finally, a numerical simulation shows that the proposed controller can rapidly stabilize the Timoshenko beam. This result extends a previous work which considered a slender Timoshenko beam with Kelvin-Voigt damping, allowing destabilizing boundary conditions at the uncontrolled boundary and attaining an arbitrarily rapid convergence rate.

Satellite overcrowding in the low Earth’s orbit, along with the arising problem of the space debris results in an increase of the probability of collision between orbiting bodies. Therefore, orbit determination for the relative movement between orbiting spacecraft in close proximity becomes an important task. This Final Degree Project is focused on orbit determination. Considering an initial condition, a set of propagation equations and a series of external measurements, the reality is reconstructed by using filtering methods. Filters such as the Kalman Filter, the Extended Kalman Filter and the Unscented Kalman Filter are tested for this purpose in several different scenarios. The performance of these filter are analysed highlighting their strengths and weaknesses., La superpoblación de satelites en la órbita baja terrestre, junto con el creciente problema de la basura espacial ha incrementado la probabilidad de colisiones entre satélites operativos. Es por ello que resulta imperativo determinar la órbita del movimiento relativo entre vehículos orbitales próximos entre sí. Este Trabajo Fin de Grado se enmarca en el ámbito de la determinación de órbitas. Considerando una condición inicial, un conjunto de ecuaciones para la propagación y una serie de medidas externas, la realidad se reconstruye utilizando filtros tales como el Filtro de Kalman, el Filtro de Kalman Extendido y el Filtro de Kalman Unscented, que se ponen a prueba en diferentes escenarios. Se analiza el rendimiento de estos filtros y se destacan sus fortalezas y debilidades.

En este proyecto se explora el proceso de planificación y guiado óptimos de la trayectoria de un vehículo lanzador multietapa. Para la definición de la trayectoria óptima de referencia se utiliza un método de colocación indirecto, aplicando la Teoría de Control Óptimo, formulando el Principio del Mínimo de Pontryagin y resolviendo numéricamente el problema de contorno resultante. El sistema de guiado se ha basado en el desarrollo y la sintonización de controladores LQR de horizonte finito, con término integral y sistema anti-windup. Por último, se han evaluado las prestaciones del sistema completo sometido a distintas combinaciones de perturbaciones y errores de modelado, obteniendo resultados satisfactorios., In this project the process of development of the optimal trajectory planning and guidance of a multistage rocket is explored. To define the optimal reference trajectory an indirect transcription method is used, by first utilizing the Optimal Control Theory, formulating Pontryagin’s Minimum Principle, and then numerically solving the resulting boundary value problem. The guidance system is based on the development and tuning of finite-horizon LQR controllers, including integral terms and an anti-windup system. Lastly, the performance of the overall system has been assessed, obtaining successful results.

In an increasingly dense space environment, collision avoidance has become an essential task in satellite operations. In order to avoid these possible collisions, it is needed to determine a possible conjunction and its associated uncertainty. Satellite conjunction analysis is the assessment of collision risk during a close encounter between a satellite and another object in orbit. A counterintuitive phenomenon has emerged in the conjunction analysis literature, namely, probability dilution, in which lower quality data paradoxically appear to reduce the risk of collision. This work presents a review of the current methods used to estimate the probability of collision between two bodies in orbit as well as an exhaustive analysis of the factors to take into account in this process. The concept of uncertainty propagation, probability dilution and the effect of perturbations and noise is developed., En un entorno espacial cada vez más denso, la prevención de colisiones se ha convertido en una tarea esencial en las operaciones de los satélites. Para evitar estas posibles colisiones, es necesario determinar una posible conjunción y su incertidumbre asociada. El análisis de conjunción de satélites es la evaluación del riesgo de colisión durante un encuentro cercano entre un satélite y otro objeto en órbita. En la literatura sobre análisis de conjunciones ha surgido un fenómeno contrario a la intuición, la dilución de la probabilidad, en el que, paradójicamente, los datos de menor calidad parecen reducir el riesgo de colisión. Este trabajo presenta una revisión de los métodos actuales utilizados para estimar la probabilidad de colisión entre dos cuerpos en órbita así como un análisis exhaustivo de los factores a tener en cuenta en este proceso. Se desarrolla el concepto de propagación de la incertidumbre, dilución de la probabilidad y el efecto de las perturbaciones y el ruido.

The overarching goal of this thesis is the design of model predictive control algorithms for spacecraft proximity operations. These include, but it is not limited to, spacecraft rendezvous, hovering phases or orbiting in the vicinity of small bodies. The main motivation behind this research is the increasing demand of autonomy, understood as the spacecraft capability to compute its own control plan, in current and future space operations. This push for autonomy is fostered by the recent introduction of disruptive technologies changing the traditional concept of space exploration and exploitation. The development of miniaturized satellite platforms and the drastic cost reduction in orbital access have boosted space activity to record levels. In the near future, it is envisioned that numerous artificial objects will simultaneously operate across the Solar System. In that context, human operators will be overwhelmed in the task of tracking and commanding each spacecraft in real time. As a consequence, developing intelligent and robust autonomous systems has been identified by several space agencies as a cornerstone technology. Inspired by the previous facts, this work presents novel controllers to tackle several scenarios related to spacecraft proximity operations. Mastering proximity operations enables a wide variety of space missions such as active debris removal, astronauts transportation, flight-formation applications, space stations resupply and the in-situ exploration of small bodies. Future applications may also include satellite inspection and servicing. This thesis has focused on four scenarios: six-degrees of freedom spacecraft rendezvous; near-rectilinear halo orbits rendezvous; the hovering phase; orbit-attitude station-keeping in the vicinity of a small body. The first problem aims to demonstrate rendezvous capabilities for a lightweight satellite with few thrusters and a reaction wheels array. For near-rectilinear halo orbits rendezvous, the goal is to ac, El objeto de esta tesis es el dise˜no de algoritmos de control predictivo basado en modelo para operaciones de veh´ıculos espaciales en proximidad. Esto incluye, pero no se limita, a la maniobra de rendezvous, las fases de hovering u orbitar alrededor de cuerpos menores. Esta tesis est´a motivada por la creciente demanda en la autonom´ıa, entendida como la capacidad de un veh´ıculo para calcular su propio plan de control, de las actuales y futuras misiones espaciales. Este inter´es en incrementar la autonom´ıa est´a relacionado con las actuales tecnolog´ıas disruptivas que est´an cambiando el concepto tradicional de exploraci´on y explotaci´on espacial. Estas son el desarrollo de plataformas satelitales miniaturizadas y la dr´astica reducci´on de los costes de puesta en ´orbita. Dichas tecnolog´ıas han impulsado la actividad espacial a niveles de record. En un futuro cercano, se prev´e que un gran n´umero de objetos artificiales operen de manera simult´anea a lo largo del Sistema Solar. Bajo dicho escenario, los operadores terrestres se ver´an desbordados en la tarea de monitorizar y controlar cada sat´elite en tiempo real. Es por ello que el desarrollo de sistemas aut´onomos inteligentes y robustos es considerado una tecnolog´ıa fundamental por diversas agencias espaciales. Debido a lo anterior, este trabajo presenta nuevos resultados en el control de operaciones de veh´ıculos espaciales en proximidad. Dominar dichas operaciones permite llevar a cabo una gran variedad de misiones espaciales como la retirada de basura espacial, transferir astronautas, aplicaciones de vuelo en formaci´on, reabastecer estaciones espaciales y la exploraci ´on de cuerpos menores. Futuras aplicaciones podr´ıan incluir operaciones de inspecci´on y mantenimiento de sat´elites. Esta tesis se centra en cuatro escenarios: rendezvous de sat´elites con seis grados de libertad; rendezvous en ´orbitas halo cuasi-rectil´ıneas; la fase de hovering; el mantenimiento de ´orbita y actitud en las inmen

Uno de los principales problemas del sector espacial actualmente es la gran población de objetos y basura espacial que se encuentra orbitando la Tierra, y que se prevee que continue creciendo en el futuro. A causa de ello, la detección de maniobras se ha convertido en un tema de actualidad en el sector; ya que permite mejorar la conciencia situacional reduciendo el número de objetos no correlacionados. Este trabajo se centra en un algoritmo de detección de maniobras en órbita baja usando datos de radar. La idea principal del mismo es la comparación, mediante una distancia estadística, de los datos esperados según los parámetros iniciales de la órbita y los datos obtenidos por el radar en la siguiente pasada. El objetivo del trabajo es caracterizar la dependencia de los resultados en función de los diferentes parámetros de los que depende el algoritmo, destacando entre ellos, la distancia estadística empleada., Nowadays, one of the main problems in the space sector is the amount of space objects and debris in orbit about Earth, which will increase continually in the future. This is the reason why manoeuvres detection has become a topic of broad and current interest, because it can enable to improve the space situational awareness by reducing the number of uncorrelated targets detected. This project focuses on an algorithm for manoeuvres detection for satellites in Low Earth Orbit from radar data. The main idea of this algorithm is the comparison, using a statistical distance, between the predicted orbit from the initial data and those obtain from the radar in the next track. The aim of the project is to characterize the impact of different parameters of the algorithm on the results, highlighting among them, the statistical distance used.

This work presents an event-triggered controller for spacecraft rendezvous hovering phases. The goal is to maintain the chaser within a bounded region with respect to the target. The main assumption is that the chaser vehicle has impulsive thrusters. These are assumed to be orientable at any direction and are constrained by dead-zone and saturation bounds. The event-based controller relies on trigger rules deciding when a suitable control law is applied. The local control law consists on a single impulse; therefore the trigger rules design is based on the instantaneous reachability to the admissible set. The final outcome is a very efficient algorithm from both computational burden and footprint perspectives. Because the proposed methodology is based on a single impulse control, the controller invariance is local and assessed through impulsive systems theory. Finally, numerical results are shown and discussed

This work introduces the tools used to teach the Kalman Filter (KF) to Aerospace Engineering students in the University of Seville. In particular, an easy-to-set-up application is introduced; based on the Matlab framework on its 2020b (or newer versions), it is able to display the attitude of a smart device in real time through a wireless connection to a computer. This tool is a simple yet powerful educative resource when teaching about the KF, since it showcases its performance while allowing the student to understand how a complex real-world problem can be solved using a relatively simple implementation of the KF; in particular, the Multiplicative Extended KF (MEKF) is chosen, but the framework can be easily adapted to other versions such as the Extended or Unscented KF. In addition, the tool allows the student to be aware of the inner workings of the filter itself, learning about its advantages and limitations compared with other attitude estimation algorithms. The student is also able to understand how the filter needs to be tuned, and to observe the results of the experiments in a visual and straightforward manner.

En la asignatura de Mecánica Orbital y Vehículos Espaciales impartida en el cuarto curso del Grado en Ingeniería Aeroespacial se proporcionan las bases necesarias para afrontar el cálculo de trayectorias espaciales. Sin embargo, para realizar el análisis y optimización de una misión interplanetaria (o en general, de cualquier transferencia entre dos órbitas genéricas) surge la necesidad de resolver el Problema de Lambert. El objetivo de este material es proporcionar al alumno en formato de práctica de la asignatura unas pautas para programar en Python un algoritmo que permita resolver de forma numérica dicho problema. Con esta nueva herramienta, se procederá a obtener ciertos resultados que permiten ahondar un poco más en un campo tan interesante como es el de misiones espaciales., At the Orbital Mechanics and Space Vehicles course imparted in the fourth year of the Bachelor in Aerospace Engineering, the basis necessary for the calculation of space trajectories are given. However, to perform the analysis and optimization of an interplanetary mission (or generally, of any transfer between two generic orbits) the need to solve Lambert’s Problem arises. The goal of this work is to provide the student with the guidelines for coding in Python an algorithm that allows said problem to be solved numerically, in a laboratory class format. With this new tool, certain results will be obtained which allow to delve a bit more into such a fascinating field as interplanetary missions are.

Humanity’s interest in the heavens has led to space exploration, which helps to expand technology, create new industries and foster the links between nations. As an outgrowth of the mid-20th-century Cold War, the Space Race began. It was a series of competitive technology demonstrations between the United States and the Soviet Union, aiming to show superiority in spaceflight. Although the last years of the Space Race were focused on reaching the Moon, it is worthwhile to mention the effort made by the Soviet Union to visit other planets. In fact, in 1965, Venera 3 became the first spacecraft to land on another planet’s surface (Venus). (excerpted from the Introduction)

El problema propuesto en este documento es la realización de una planificación optimizada para la toma de imágenes de una constelación formada por satélites ágiles. Este tipo de satélites presenta una mayor libertad para modificar su actitud con respecto a sus predecesores. Los objetivos de este proyecto fueron principalmente: minimizar el tiempo de observación de un cierto escenario, dada una nube de puntos objetivos en un área, y maximizar la calidad de las imágenes tomadas por la constelación. Para poder realizar simulaciones y obtener resultados se crearon dos modelos de realidad. El primero consiste en un cierto vehículo sobrevolando un plano, que serviría para ensayar los algoritmos creados, y el segundo es un modelo de una constelación de satélites heliosíncronos orbitando la Tierra. Se estudió el movimiento óptimo de giro de la cámara para el apuntamiento entre dos puntos. Se llegó a la conclusión de que este movimiento era una rotación alrededor del eje de Euler y se analizó como resolver este movimiento de forma numérica. Dada la imposibilidad de obtener la secuencia óptima de observación de un escenario de manera exacta, se desarrolló un Algoritmo de Selección de Objetivos (TSA) para buscar una secuencia con un tiempo de observación aceptable. Además un algoritmo de optimización ha sido probado, obteniendo de manera exitosa caminos de observación alternativos para el problema plano. Se han realizado simulaciones en varios escenarios globales y locales. En los escenarios globales se han estudiado las tendencias que presenta el número de puntos objetivo observados durante 24 horas de simulación. Por otra parte, se han simulado la observación de escenarios locales sobre la península ibérica. El objetivo de estas simulaciones era observar las tendencias del tiempo de observación total al variar ciertos parámetros. Además se ha medido el efecto de ciertos acontecimientos aleatorios como la presencia de nubes en el tiempo de observación.

This work presents a closed-loop guidance algorithm for six-degrees of freedom spacecraft rendezvous with a passive target flying in an eccentric orbit. The main assumption is that the chaser vehicle has an attitude control system, based on reaction wheels, providing the necessary torque to change its orientation whereas the number of thrusters is arbitrary. The goal is to design fuel optimal maneuvers while satisfying operational constraints and rejecting disturbances. The proposed method is as follows; first, the coupled translational and angular dynamics are transformed to equivalent algebraic relations using the relative translational states transition matrix and the attitude flatness property. Then, a direct transcription method, based on B-splines parameterization and discretization of time continuous constraints, is developed to obtain a tractable static program. Finally, a Model Predictive Controller, based on linearization around the previously computed solution, is considered to handle disturbances. Numerical results are shown and discussed.

We investigate the state observer design problem for thermoacoustic instabilities in a Rijke tube using an infinite dimensional perspective. The observer, whose design is based on the backstepping methodology with Volterra and full integral terms, consists of a copy of the linearized plant model plus output injection terms and relies only on a single boundary acoustic pressure sensor. The exponential convergence, in the L 2 sense, of the observed error dynamics is proved, and the analytic expression of the observer gains are derived explicitly. The results are tested in an experimental Rijke tube prototype in order to illustrate the effectiveness of the method.

This paper presents an observer design for estimation of thermoacoustic instabilities in a Rijke tube. To study this problem, we consider that the acoustic dynamics is represented by the wave equation with a point source term representing the heat release. In turn, the heat release dynamics is given by a first-order ordinary differential equation (ODEs). The observer, whose design is based on the backstepping methodology, relies on measurements of acoustic pressure and velocity at an arbitrary point of the domain. The design employs a folding transformation (with two folds) around the measurements and the heat release point, allowing to write the system into a form with more states but boundary measurements and ODE couplings. Then, we formulate a well-posed and invertible integral transformation with both triangular and full terms that maps the observer error dynamics into an exponentially stable target system. The theoretical results were tested through numerical simulations in order to show the effectiveness of the design.

Recently, the problem of boundary stabilization and estimation for unstable linear constant-coefficient reaction-diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. Some early success has been achieved under simplifying conditions, such as radially-varying reaction coefficients under revolution symmetry, on a disk or a sphere. These particular cases notwithstanding, the problem remains open. The main issue is that the equations become singular in the radius; when applying the backstepping method, the same type of singularity appears in the kernel equations. Traditionally, well-posedness of these equations has been proved by transforming them into integral equations and then applying the method of successive approximations. In this case, with the resulting integral equation becoming singular, successive approximations do not easily apply. This paper takes a different route and directly addresses the kernel equations via a power series approach, finding in the process the required conditions for the radially-varying coefficients and stating the existence of the series solution. This approach provides a direct numerical method that can be readily applied, despite singularities, to both control and observer boundary design problems.

We tackle the boundary control and estimation problems for a class of viscous Hamilton–Jacobi PDEs, considering bilateral actuation and sensing, i.e., at the two boundaries of a 1-D spatial domain. We first solve the nonlinear trajectory generation problem for this type of PDEs, providing the necessary feedforward actions at both boundaries. We then design an observer-based output-feedback control law, which consists of two main elements—a nonlinear observer that is constructed utilizing measurements from both boundaries and state-feedback laws, which are employed at the two boundary ends. All of our designs are explicit since they are constructed interlacing a feedback linearizing transformation with backstepping. Due to the fact that the linearizing transformation is locally invertible, only a regional stability result is established, combining this transformation with backstepping, suitably formulated to handle the case of bilateral actuation and sensing. We illustrate the developed methodologies via application to traffic flow control and we present consistent simulation results.

This work presents an event-triggered model predictive controller for spacecraft hovering phases. The target is assumed to be inert on its an elliptic orbit whereas the chaser has impulsive thrusters. The main goal is to design a local controller based on event-triggering principle to maintain the spacecraft inside a given hovering region while minimizing fuel consumption and avoiding unnecessary small amplitude firings. Using hybrid impulsive systems theory and reachability analysis, the invariance of the proposed method is studied. Simulation results are shown and discussed.

We consider the stabilization problem for an unstable 1-D diffusion-reaction partial differential equation using a so-called folding transformation. The diffusion-reaction equation is transformed into a 2 ×2 system of coupled parabolic PDEs with exotic boundary conditions. A first backstepping transformation is designed to map the unstable system into a strict-feedback intermediate target system. A second backstepping transformation is designed to stabilize the intermediate target system. Interestingly, the companion gain kernel PDEs contain the folding boundary condition, exhibiting symmetry with the original system. The kernels posses a cascading structure that allows for sequential solution methods. Finally, the controller derived is shown to be exponentially stabilizing in the L2 sense.

"Remote sensing techniques provide the capacity of collecting information of spots from afar. This tool has an important application on Earth’s monitoring from space, a practice which help to gather multiple data from any point on Earth’s surface in short periods of time: vegetation biomass, water quality, surface profile, surface temperature, human infrastructures spread... This knowledge is used for a diversity of purposes of human interest, such as military, agricultural, meteorological or ecological ends. In order to acquire this information by remote sensing practice an entire infrastructure has been set, divided into space and ground segments. The space segment comprises the satellite or satellites, also called Earth Observation Satellites (EOS), which are equipped with sensors for the reception of data, usually in the form of electromagnetic signals. These satellites establish transmissions with the ground segment by means of antennas placed in certain spots, in order to sent the information collected. Then the ground segment is responsible for the processing of the data acquired for its use on a particular application. Earth observation satellites have been used for decades now. Historically, large satellites were required in order to do the observation tasks. These satellites were manually coordinated and their time resolution, linked with the time needed to revisit and acquire data from the exact location, was very low. Currently, the miniaturization tendencies have also arrived to the space sector and smaller satellites are now developmed. The emergence of the CubeSats, included in the nanosatellites group, and the growth of the number of their launches make clear the future trend: Earth observation with small and numerous satellites" (Fragmento extraido de la Introducción del trabajo)

Debido al efecto de las perturbaciones, el cual es difícil de modelar con precisión, siempre aparece un cierto grado de incertidumbre asociado a la órbita. Para modelar y trabajar con dicha incertidumbre se puede utilizar la teoría de la probabilidad. El objetivo de este proyecto es el de estudiar y analizar distintos métodos que permitiesen la propagación de la incertidumbre a lo largo de la órbita, tales como la linearización de la covarianza, caos polinomial, álgebra diferencial, el método de Monte Carlo o el PTM. Además, los modelos de las perturbaciones se desarrollarán y presentarán, analizando también su efecto en los elementos orbitales., Due to perturbation effects that are difficult to model with enough precision, there is always a degree of uncertainty associated with orbits. Probability theory can be used to model this uncertainty. The goal of this project is to explore several methods that propagate this uncertainty along the orbit, such as linearized covariance, polynomial chaos, differential algebra, or the Monte Carlo method. Moreover, perturbation models will be developed and attached and their effects in the orbital elements will be analyzed as well.

En marzo de 2015 la NASA lanzó la misión MMS (Multiescala Magnetosférica) con el fin de investigar el fenómeno de reconexión magnética, un proceso por el que los campos electromagnéticos de la Tierra y el Sol interaccionan intercambiando grandes cantidades de energía que acelera y calienta las partículas con carga que conforman el plasma circundante. La misión MMS revela por primera vez la microfísica de la reconexión con una resolución y precisión sin precedentes, determinando los procesos cinéticos que ocurren a pequeña escala y que son responsables del desarrollo del fenómeno. Desde su concepción, multitud de laboratorios americanos, europeos y japoneses están implicados en la misión MMS, participando activamente en el análisis científico de las medidas y su interpretación. El presente proyecto recoge los resultados del estudio llevado a cabo en el Instituto francés de Investigación en Astrofísica y Planetología (IRAP), cuyo objeto principal ha sido la caracterización de múltiples poblaciones iónicas en un entorno de plasma no colisional a partir de los datos recogidos por los equipos MMS. Trabajos previos han detectado la presencia conjunta de dos poblaciones de iones en la magnetosfera terrestre durante el proceso de reconexión magnética. Estas poblaciones iónicas se diferencian principalmente por su densidad y temperatura, distinguiéndose un grupo de iones más calientes y un grupo de iones fríos procedentes de la ionosfera. Al tratarse de un régimen no colisional, los iones de distinta temperatura no interaccionan de forma directa, siendo posible la coexistencia en una misma región del espacio. El diseño de los equipos de medición MMS no permite distinguir de manera automática los distintos tipos de poblaciones iónicas, siendo la caracterización de los iones fríos particularmente complicada. Sin embargo, se ha demostrado que, en ocasiones, los iones fríos son predominantes en las principales zonas de la magnetosfera donde se produce la reconexión, y su presenc, In March 2015, NASA launched the MMS (Magnetospheric Multiscale) mission in order to study the phenomenon of magnetic reconnection, a process by which the electromagnetic fields of the Earth and the Sun interact exchanging large amounts of energy that accelerates and heats the charged particles that constitute the surrounding plasma. The MMS mission reveals for the first time the microphysics of reconnection with unprecedented resolution and precision, identifying the kinetic processes that occur on a small scale and are responsible for the development of this phenomenon. Since its conception, many American, European and Japanese laboratories are involved in the MMS mission, actively participating in the scientific analysis of the measures and their interpretation. The present project summarizes the results of the study carried out at the French Institute of Research in Astrophysics and Planetology (IRAP), whose main objective has been the characterization of multiple ionic populations in a non-collisional plasma from the data collected by MMS satellites. Previous works have detected the presence of two ion populations in the Earth's magnetosphere during the process of magnetic reconnection. These ionic populations differ mainly in their density and temperature, distinguishing a group of hot ions and a group of colder ions from the ionosphere. A non-collisional regime implies that ions of different temperatures do not interact directly, so it is possible to coexist in the same space region. MMS measuring equipments usually do not automatically characterize different ion populations based on their differing temperature. However, it has been shown that the cold population can be predominant in the main areas of the magnetosphere where reconnection occurs, and its presence affects not only at large scale but also at the microphysics of the process. Therefore, to fully understand the phenomenon of magnetic reconnection, it is first necessary to identify the present spec

Uno de los mayores problemas a la hora de controlar y recuperar los datos obtenidos de un satélite se debe a que el número de oportunidades viene limitado por la localización del centro desde donde se recuperan y de los parámetros de la propia órbita. Por esta razón, las órbitas con traza repetida, las cuales tienen la peculiaridad de que, tras un número de revoluciones, vuelven a sobrevolar los mismos puntos de la Tierra, son muy ventajosas, ya que permiten controlar de forma precisa los puntos de observación y recuperaciónde datos. Además, este tipo de trazas pueden presentar puntos de cruce, es decir, puntos que el satélite sobrevuela dos veces antes de que la traza se repita. Estos puntos son de especial interés para la colocación de estos centros de recuperación de datos y observación de satélites, ya que permiten duplicar el número de ventanas de observación. Este documento se encarga de caracterizar con el mayor grado de detalle posible cómo los distintos elementos orbitales afectan a la disposición y cantidad de estos puntos de cruce en órbitas geocéntricas de excentricidad pequeña. Para ello, primero se desarrollan brevemente algunos conceptos necesarios de Mecánica orbital, además de realizar un pequeño resumen del caso circular ya estudiado en otros documentos para que sirva de introducción. Tras esto, se obtendrán las ecuaciones que permiten obtener la posición de los puntos de corte,se explicará su resolución y se analizará cómo afecta la variación de algunos elementos orbitales, por ejemplo,la excentricidad, el argumento de perigeo o la inclinación. Después, se ilustrará con ejemplos la precisión y las limitaciones del modelo utilizado para resolver el problema. Por último, se incluirán algunas conclusiones generales y se discutirán algunas líneas de trabajo que podrían tomarse en un futuro a partir de este trabajo.

The objective of this work is to present a closed-loop guidance algorithm for landing a probe on an irregular shaped rotating asteroid. The main assumption is that the spacecraft is orbiting close to the asteroid and has a continuous propulsion system enabling it to do a powered descent. The goal is to minimize fuel consumption while avoiding collision with the asteroid during the manoeuvre. This non-convex time-continuous optimal control problem is transformed to a convex static program by relaxing some constraints, discretizing and using an iterative method to handle with the asteroid gravity field non-linearities. Then, a guidance algorithm based on Model Predictive Control is applied with the purpose of coping with unmodelled dynamics and disturbances. Numerical results are showed and discussed.

This work presents a trajectory planning algorithm for spacecraft rendezvous with a passive target. The main assumption is that the chaser vehicle has a single thruster and an attitude control system (e.g. reaction wheels) providing the necessary torque to change its orientation, which may be the situation for small spacecraft or in the case of thruster failure. The goal is to design fuel-optimal manoeuvres while satisfying operational constraints. This time-continuous optimal control problem is addressed using the translational state transition matrix and the attitude flatness property to transform the dynamics into algebraic relations. Then, the problem is transformed to a non-linear programming problem which has to be solved. Simulation results are showed and discussed.

El objetivo de este proyecto es el desarrollo de una práctica experimental sobre apuntamiento de una cámara que permita ilustrar la cinemática de la actitud y las maniobras que se realizan para modificarla. Para ello, se partirá inicialmente de los conceptos básicos de la cinemática de la actitud. Haciendo uso de un controlador PID y de un microcontrolador Arduino UNO se desarrollarán algoritmos de detección de elementos y de trayectorias que actúan modificando las posiciones de los servomotores de la cámara. La práctica se desarrollará con estos algoritmos y con interfaces de usuario creadas en MATLAB. Finalmente, se propondrán mejoras y líneas de trabajo futuras., The main purpose of this Master's Thesis is to develop an experimental practice about a tracking camera device that allows to illustrate the attitude kinematics and manoeuvres. To do so, the thesis starts with basic concepts of attitude kinematics. Then, using a PID controller and an Arduino UNO board, algorithms are developed to detect elements and to create trajectories. These algorithms work by changing the servomotors’ positions. A practical lab for student is developed, which makes use of these algorithms and graphical interfaces created in MATLAB. To conclude the work, some possible improvements and extensions to the thesis are proposed.

This work can serve the reader as an initiation in numerical continuation through the AUTO software, which is specialized in bifurcation analysis and has proved to be extremely versatile and efficient in calculating ordinary differential system solutions that would otherwise be much more difficult to obtain. A brief review is made about the concepts of non-linear dynamics that rest on the basis of the problem under study, the circular restricted three body problem. Then, the bases of the numerical continuation are presented, as well as the concrete strategies that will be used for the calculation of periodic solutions and their manifolds. After this, the reader finds a guide to the basic concepts that would allow him to use the AUTO software, as well as to understand the subsequent study that is carried out with it. Finally, AUTO is used to generate the families of periodic orbits that arise from the Lagrange points in the circular restricted three body problem, in addition to the associated stable and unstable manifolds when possible. This analysis is used to comment on the characteristics of the orbits considered to be the best candidates for the Deep Space Gateway project, which aims to use near rectilinear Halo orbits., Este trabajo puede servir al lector como iniciación en continuación numérica a través del software AUTO, especializado en análisis de bifurcaciones y que demuestra ser extremadamente versátil y eficiente en el cálculo de soluciones a sistemas diferenciales ordinarios que de otra forma serían mucho más difíciles de obtener. Se hace un breve repaso por los conceptos de dinámica no lineal que descansan en la base del tema principal bajo estudio, el problema circular restringido de los tres cuerpos. Después se plantean las bases de la continuación numérica, así como las estrategias concretas que se usarán para el cálculo de soluciones periódicas y sus variedades. Tras esto, el lector se encuentra con una guía de los conceptos básicos que le permitirían utilizar el software AUTO, así como comprender el posterior estudio que se realiza con este. Finalmente, se utiliza AUTO para calcular las familias de órbitas periódicas que nacen de los puntos de Lagrange en el problema circular restringido de los tres cuerpos, además de las variedades estables e inestables asociadas cuando procedan. Este análisis se aprovecha para comentar las características de las órbitas consideradas como mejores candidatas para el proyecto Deep Space Gateway (Portal de espacio profundo), que pretende utilizar órbitas Halo casi rectilíneas.

This paper is concerned with boundary stabilization of thermoacoustic oscillations in the Rijke tube. This system consists of a vertical tube open in both ends and a heater placed in the lower half of the tube. A speaker placed under the tube is used as actuator while a microphone placed near the top of the tube provides the pressure measurement. To study this problem we consider that the mathematical model takes the form of two interconnected compartments: one for the cold zone and other for the hot zone. The control input is applied on the left boundary condition of the cold zone. From this model we derive an irrational transfer function to design a stabilizing boundary control in the frequency domain. In particular, we derive necessary and sufficient conditions for the input-output stability through the use of Nyquist-type test. Experimental results show the effectiveness and real-life applicability of the method.

This work presents a Model Predictive Controller (MPC) that is able to handle Linear Time-Varying (LTV) plants with Pulse-Width Modulated (PWM) control. The MPC is based on a planner that employs a Pulse-Amplitude Modulated (PAM) or impulsive approximation as a hot-start and then uses explicit linearization around successive PWM solutions for rapidly improving the solution by means of quadratic programming. As an example, the problem of rendezvous of spacecraft for eccentric target orbits is considered. The problem is modeled by the LTV Tschauner–Hempel equations, whose state transition matrix is explicit; this is exploited by the algorithm for rapid convergence. The efficacy of the method is shown in a simulation study.