A Generic Solver for Unconstrained Control Problems with Integral Functional Objectives

We present a generic solver for unconstrained control problems (UCPs) whose objectives take the form of an integral functional of the controllers. The solver generalizes and improves upon the algorithm in [1] for the Witsenhausen’s counterexample, which provides the best-known results. In essence, we show that minimizing the objective implies minimizing the marginal cost functions almost everywhere, and we perform the latter task pointwisely by the adaptive minimization technique, which speeds up the computation. We implement single-threaded and parallelized versions of the proposed algorithm. Our implementation runs 30× faster than the algorithm in [1] on the Witsenhausen’s counterexample, and we demonstrate the applicability of the solver and discuss the possible generalization to constrained problems and multidimensional controllers through three more examples

Conflict resolution algorithms for optimal trajectories in presence of uncertainty

Mención Internacional en el título de doctorThe objective of the work presented in this Ph.D. thesis is to develop a novel method to address the aircraft-obstacle avoidance problem in presence of uncertainty, providing optimal trajectories in terms of risk of collision and time of flight. The obstacle avoidance maneuver is the result of a Conflict Detection and Resolution (CD&R) algorithm prepared for a potential conflict between an aircraft and a fixed obstacle which position is uncertain. Due to the growing interest in Unmanned Aerial System (UAS) operations, CD&R topic has been intensively discussed and tackled in literature in the last 10 years. One of the crucial aspects that needs to be addressed for a safe and efficient integration of UAS vehicles in non-segregated airspace is the CD&R activity. The inherent nature of UAS, and the dynamic environment they are intended to work in, put on the table of the challenges the capability of CD&R algorithms to handle with scenarios in presence of uncertainty. Modeling uncertainty sources accurately, and predicting future trajectories taking into account stochastic events, are rocky issues in developing CD&R algorithms for optimal trajectories. Uncertainty about the origin of threats, variable weather hazards, sensing and communication errors, are only some of the possible uncertainty sources that make jeopardize air vehicle operations. In this work, conflict is defined as the violation of the minimum distance between a vehicle and a fixed obstacle, and conflict avoidance maneuvers can be achieved by only varying the aircraft heading angle. The CD&R problem, formulated as Optimal Control Problem (OCP), is solved via indirect optimal control method. Necessary conditions of optimality, namely, the Euler-Lagrange equations, obtained from calculus of variations, are applied to the vehicle dynamics and the obstacle constraint modeled as stochastic variable. The implicit equations of optimality lead to formulate a Multipoint Boundary Value Problem (MPBVP) which solution is in general not trivial. The structure of the optimality trajectory is inferred from the type of path constraint, and the trend of Lagrange multiplier is analyzed along the optimal route. The MPBVP is firstly approximated by Taylor polynomials, and then solved via Differential Algebra (DA) techniques. The solution of the OCP is therefore a set of polynomials approximating the optimal controls in presence of uncertainty, i.e., the optimal heading angles that minimize the time of flight, while taking into account the uncertainty of the obstacle position. Once the obstacle is detected by on-board sensors, this method provide a useful tool that allows the pilot, or remote controller, to choose the best trade-off between optimality and collision risk of the avoidance maneuver. Monte Carlo simulations are run to validate the results and the effectiveness of the method presented. The method is also valid to address CD&R problems in presence of storms, other aircraft, or other types of hazards in the airspace characterized by constant relative velocity with respect to the own aircraft.L’obiettivo del lavoro presentato in questa tesi di dottorato è la ricerca e lo sviluppo di un nuovo metodo di anti collisione velivolo-ostacolo in presenza di incertezza, fornendo traiettorie ottimali in termini di rischio di collisione e tempo di volo. La manovra di anticollisione è il risultato di un algoritmo di detezione e risoluzione dei conflitti, in inglese Conflict Detection and Resolution (CD&R), che risolve un potenziale conflitto tra un velivolo e un ostacolo fisso la cui posizione è incerta. A causa del crescente interesse nelle operazioni che coinvolgono velivoli autonomi, anche definiti Unmanned Aerial System (UAS), negli ultimi 10 anni molte ricerche e sviluppi sono state condotte nel campo degli algoritmi CD&R. Uno degli aspetti cruciali per un’integrazione sicura ed efficiente dei velivoli UAS negli spazi aerei non segregati è l’attività CD&R. La natura intrinseca degli UAS e l’ambiente dinamico in cui sono destinati a lavorare, impongono delle numerose sfide fra cui la capacità degli algoritmi CD&R di gestire scenari in presenza di incertezza. La modellizzazione accurata delle fonti di incertezza e la previsione di traiettorie che tengano conto di eventi stocastici, sono problemi particolarmente difficoltosi nello sviluppo di algoritmi CD&R per traiettorie ottimali. L’incertezza sull’origine delle minacce, zone di condizioni metereologiche avverse al volo, errori nei sensori e nei sistemi di comunicazione per la navigazione aerea, sono solo alcune delle possibili fonti di incertezza che mettono a repentaglio le operazioni degli aeromobili. In questo lavoro, il conflitto è definito come la violazione della distanza minima tra un veicolo e un ostacolo fisso, e le manovre per evitare i conflitti possono essere ottenute solo variando l’angolo di rotta dell’aeromobile, ovvero virando. Il problema CD&R, formulato come un problema di controllo ottimo, o Optimal Control Problem (OCP), viene risolto tramite un metodo indiretto. Le condizioni necessarie di ottimalità, vale a dire le equazioni di Eulero-Lagrange derivanti dal calcolo delle variazioni, sono applicate alla dinamica del velivolo e all’ostacolo modellizato come una variabile stocastica. Le equazioni implicite di ottimalità formano un problema di valori al controno multipunto, Multipoint Boundary Value Problem(MPBVP), la cui soluzione in generale è tutt’altro che banale. La struttura della traiettoria ottimale viene dedotta dal tipo di vincolo, e l’andamento del moltiplicatore di Lagrange viene analizzato lungo il percorso ottimale. Il MPBVP viene prima approssimato con un spazio di polinomi di Taylor e successimvamente risolto tramite tecniche di algebra differenziale, in inglese Differential Algebra (DA). La soluzione del OCP è quindi un insieme di polinomi che approssima il controllo ottimo del problema in presenza di incertezza. In altri termini, il controllo ottimo è l’insieme degli angoli di prua del velivolo che minimizzano il tempo di volo e che tenendo conto dell’incertezza sulla posizione dell’ostacolo. Quando l’ostacolo viene rilevato dai sensori di bordo, questo metodo fornisce un utile strumento al pilota, o al controllore remoto, al fine di scegliere il miglior compromesso tra ottimalità e rischio di collisione con l’ostacolo. Simulazioni Monte Carlo sono eseguite per convalidare i risultati e l’efficacia del metodo presentato. Il metodo è valido anche per affrontare problemi CD&R in presenza di tempeste, altri velivoli, o altri tipi di ostacoli caratterizzati da una velocità relativa costante rispetto al proprio velivolo.El objetivo del trabajo presentado en esta tesis doctoral es la búsqueda y el desarrollo de un método novedoso de anticolisión con osbstáculos en espacios aéreos en presencia de incertidumbre, proporcionando trayectorias óptimas en términos de riesgo de colisión y tiempo de vuelo. La maniobra de anticolisión es el resultado de un algoritmo de detección y resolución de conflictos, en inglés Conflict Detection and Resolution (CD&R), preparado para un conflicto potencial entre una aeronave y un obstáculo fijo cuya posición es incierta. Debido al creciente interés en las operaciones de vehículos autónomos, también definidos como Unmanned Aerial System (UAS), en los últimos 10 años muchas investigaciones se han llevado a cabo en el tema CD&R. Uno de los aspectos cruciales que debe abordarse para una integración segura y eficiente de los vehículos UAS en el espacio aéreo no segregado es la actividad CD&R. La naturaleza intrínseca de UAS, y el entorno dinámico en el que están destinados a trabajar, suponen un reto para la capacidad de los algoritmos de CD&R de trabajar con escenarios en presencia de incertidumbre. La precisa modelización de las fuentes de incertidumbre, y la predicción de trayectorias que tengan en cuenta los eventos estocásticos, son problemas muy difíciles en el desarrollo de algoritmos CD&R para trayectorias óptimas. La incertidumbre sobre el origen de las amenazas, condiciones climáticas adversas, errores en sensores y sistemas de comunicación para la navegación aérea, son solo algunas de las posibles fuentes de incertidumbre que ponen en peligro las operaciones de los vehículos aéreos. En este trabajo, el conflicto se define como la violación de la distancia mínima entre un vehículo y un obstáculo fijo, y las maniobras de anticolisión se pueden lograr variando solo el ángulo de rumbo de la aeronave, es decir virando. El problema CD&R, formulado como problema de control óptimo, o Optimal Control Problem (OCP), se resuelve a través del método de control óptimo indirecto. Las condiciones necesarias de optimalidad, es decir, las ecuaciones de Euler-Lagrange que se obtienen a partir del cálculo de variaciones, son aplicadas a la dinámica de la aeronave y al obstáculo modelizado como variable estocástica. Las ecuaciones implícitas de optimalidad forman un problema de valor de frontera multipunto (MPBVP) cuya solución en general no es trivial. La estructura de la trayectoria de optimalidad se deduce del tipo de vínculo, y la tendencia del multiplicador de Lagrange se analiza a lo largo de la ruta óptima. El MPBVP se aproxima en primer lugar a través de un espacio de polinomios de Taylor, y luego se resuelve por medio de técnicas de álgebra diferencial, en inglés Differential Algebra(DA). La solución del OCP es un conjunto de polinomios que aproximan los controles óptimos en presencia de incertidumbre, es decir, los ángulos de rumbo óptimos que minimizan el tiempo de vuelo teniendo en cuenta la incertidumbre asociada a la posición del obstáculo. Una vez que los sensores a bordo detectan el obstáculo, este método proporciona una herramienta muy útil que permite al piloto, o control remoto, elegir el mejor compromiso entre optimalidad y riesgo de colisión con el obstáculo. Se ejecutan simulaciones de Monte Carlo para validar los resultados y la efectividad del método presentado. El método también es válido para abordar los problemas de CD&R en presencia de tormentas, otras aeronaves u otros tipos de obstáculos caracterizados por una velocidad relativa constante con respecto a la propia aeronave.Programa de Doctorado en Mecánica de Fluidos por la Universidad Carlos III de Madrid; la Universidad de Jaén; la Universidad de Zaragoza; la Universidad Nacional de Educación a Distancia; la Universidad Politécnica de Madrid y la Universidad Rovira i VirgiliPresidente: Carlo Novara.- Secretario: Lucia Pallotino.- Vocales: Manuel Sanjurjo Rivo; Yoshinori Matsuno; Alfonso Valenzuela Romer

Study of lightweighting structural design considering 3D printing constraints

One of the current challenges of the aerospace industry is the exploration of new lightweighting structures to reduce fuel consumption and limiting the environmental impact. The use of numerical methods concerning topology optimization techniques allows the obtaining of such weight reduction, also minimizing both design time and costs, and hence accelerating the design process. Nevertheless, current structural optimization leads to the apparition of complex shapes and volumes with unintuitive holes, thus needing the use of additive manufacturing constraints - minimum length scales and overhanging - to ensure manufacturability. Considering the background exposed above, the aim of this project is to study the feasibility of heuristic designs concerning lightweighting structures, materialized with additive manufacturing and considering 3D printing constraints. The design stage will be developed by means of topology optimization techniques, applied to anisotropic filtering. The methodology employed has considered all details concerning Computational Solid Mechanics (CSM) techniques used in structures optimization, as well as additive manufacturing techniques, different case studies definition and their feasibility study. More specifically, in the context of CSM, the use of Finite Element Methods (FEM) in the classical elastic problem is reviewed, as well as current topology optimization techniques, so as to implement FEM in optimization algorithms. Thus, theoretical basis in additive manufacturing techniques are reviewed, along with the mathematical formulation of length scale and overhang constraints. Lastly, the programming stage is performed by previously defining the working environment, consisting in the use of Object-Oriented Programming within the git Version Control System, and hence establishing the computational domain definition for all cases, the meshing process and the simulation setup. In the end, the present project has accomplished the main objectives, giving a positive answer to the creation of lightweighting structures and fulfillment of 3D printing constraints. Indeed, FEM combined with topology optimization techniques has led to the obtaining of optimized designs, fulfilling an objective function and a set of constraints, considering both design variables approaches, density and level set. Besides, an additional shape functional has been defined as a penalty contribution to the main cost function in order to fulfill 3D printing constraints - the anisotropic perimeter - being the evolution of the standard isotropic one, both applied to total and relative perimeters. This shape functional self-penalizes length scale constraints and keeps control in overhanging phenomena by orienting the topologies with the definition of a virtual anisotropic stiffness matrix. Results obtained show that the apparition of local features with small length scales has been avoided when including either isotropic or anisotropic perimeter as a penalty term. Furthermore, vertical tendency orientation of topologies has been generally obtained with the anisotropic cases, along with penalization of horizontal features. Overall, this project has become clearly relevant for the exploration of new lightweighting structures, achieving weight reduction with topology optimization techniques. Further exploration remains in the course of PhD professionalization, specially when considering phase-field models, high-performance computing and large-scale optimization inside the non-linear regime

Numerical model for material parameter identification of cells

Bacteria have a complex external layer that render them with an increased stiffness and more resistant to external invasion. The works aims to model the squeezing of a bacteria between two walls, and deduce the composition of bacterial external layer from the observed deformations. A FE based model will be developed for inferring the stiffness of baceria, solving an inverse problem from the applied loading and measured displacements. The results will be applied to laboraotry experiments carried out at Institu of Bioengineering of Catalunya (IBEC)

Computational optimal control of the terminal bunt manoeuvre

This work focuses on a study of missile guidance in the form of trajectory shaping of a generic cruise missile attacking a fixed target which must be struck from above. The problem is reinterpreted using optimal control theory resulting in two formulations: I) minimum time-integrated altitude and 2) minimum flight time. Each formulation entails nonlinear, two-dimensional missile flight dynamics, boundary conditions and path constraints. Since the thus obtained optimal control problems do not admit analytical solutions, a recourse to computational optimal control is made. The focus here is on informed use of the tools of computational optimal control, rather than their development. Each of the formulations is solved using a three-stage approach. In stage I, the problem is discretised, effectively transforming it into a nonlinear programming problem, and hence suitable for approximate solution with the FORTRAN packages DIRCOL and NUDOCCCS. The results of this direct approach are used to discern the structure of the optimal solution, i.e. type of constraints active, time of their activation, switching and jump points. This qualitative analysis, employing the results of stage I and optimal control theory, constitutes stage 2. Finally, in stage 3, the insight of stage 2 are made precise by rigorous mathemati cal formulation of the relevant two-point boundary value problems (TPBVPs), using the appropriate theorems of optimal control theory. The TPBVPs obtained from this indirect approach are then solved using the FORTRAN package BNDSCO and the results compared with the appropriate solutions of stage I. For each formulation (minimum altitude and minimum time) the influence of boundary conditions on the structure of the optimal solution and the performance index is investigated. The results are then interpreted from the operational and computational perspectives. Software implementation employing DIRCOL, NUDOCCCS and BNDSCO, which produced the results, is described and documented. Finally, some conclusions are drawn and recommendations made