52 research outputs found
A Primal-Dual Method for Optimal Control and Trajectory Generation in High-Dimensional Systems
Presented is a method for efficient computation of the Hamilton-Jacobi (HJ)
equation for time-optimal control problems using the generalized Hopf formula.
Typically, numerical methods to solve the HJ equation rely on a discrete grid
of the solution space and exhibit exponential scaling with dimension. The
generalized Hopf formula avoids the use of grids and numerical gradients by
formulating an unconstrained convex optimization problem. The solution at each
point is completely independent, and allows a massively parallel implementation
if solutions at multiple points are desired. This work presents a primal-dual
method for efficient numeric solution and presents how the resulting optimal
trajectory can be generated directly from the solution of the Hopf formula,
without further optimization. Examples presented have execution times on the
order of milliseconds and experiments show computation scales approximately
polynomial in dimension with very small high-order coefficients.Comment: Updated references and funding sources. To appear in the proceedings
of the 2018 IEEE Conference on Control Technology and Application
Polynomial approximation of high-dimensional Hamilton–Jacobi–Bellman equations and applications to feedback control of semilinear parabolic PDES
© 2018 Society for Industrial and Applied Mathematics. A procedure for the numerical approximation of high-dimensional Hamilton–Jacobi–Bellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed. Its main ingredients are a pseudospectral collocation approximation of the PDE dynamics and an iterative method for the nonlinear HJB equation associated to the feedback synthesis. The latter is known as the successive Galerkin approximation. It can also be interpreted as Newton iteration for the HJB equation. At every step, the associated linear generalized HJB equation is approximated via a separable polynomial approximation ansatz. Stabilizing feedback controls are obtained from solutions to the HJB equations for systems of dimension up to fourteen
Multiresolution strategies for the numerical solution of optimal control problems
Optimal control problems are often characterized by discontinuities or switchings in the control variables. One way of accurately capturing the irregularities in the solution is to use a high resolution (dense) uniform grid. This requires a large amount of computational resources both in terms of CPU time and memory. Hence, in order to accurately capture any irregularities in the solution using a few computational resources, one can refine the mesh locally in the region close to an irregularity instead of refining the mesh uniformly over the whole domain. Therefore, a novel multiresolution scheme for data compression has been designed which is shown to outperform similar data compression schemes. Specifically, we have shown that the proposed approach results in fewer grid points in the grid compared to a common multiresolution data compression scheme.
The validity of the proposed mesh refinement algorithm has been verified by solving several challenging initial-boundary value problems for evolution equations in 1D. The examples have demonstrated the stability and robustness of the proposed algorithm. Next, a direct multiresolution-based approach for solving trajectory optimization problems is developed. The original optimal control problem is transcribed into a nonlinear programming (NLP) problem that is solved using standard NLP codes. The novelty of the proposed approach hinges on the automatic calculation of a suitable, nonuniform grid over which the NLP problem is solved, which tends to increase numerical efficiency and robustness. Control and/or state constraints are handled with ease, and without any additional computational complexity. The proposed algorithm is based on a simple and intuitive method to balance several conflicting objectives, such as accuracy of the solution, convergence, and speed of the computations. The benefits of the proposed algorithm over uniform grid implementations are demonstrated with the help of several nontrivial examples. Furthermore, two sequential multiresolution trajectory optimization algorithms for solving problems with moving targets and/or dynamically changing environments have been developed.Ph.D.Committee Chair: Tsiotras, Panagiotis; Committee Member: Calise, Anthony J.; Committee Member: Egerstedt, Magnus; Committee Member: Prasad, J. V. R.; Committee Member: Russell, Ryan P.; Committee Member: Zhou, Hao-Mi
Robust feedback control of nonlinear PDEs by numerical approximation of high-dimensional Hamilton-Jacobi-Isaacs equations
Copyright © by SIAM. We propose an approach for the synthesis of robust and optimal feedback controllers for nonlinear PDEs. Our approach considers the approximation of infinite-dimensional control systems by a pseudospectral collocation method, leading to high-dimensional nonlinear dynamics. For the reducedorder model, we construct a robust feedback control based on the H∞ control method, which requires the solution of an associated high-dimensional Hamilton-Jacobi-Isaacs nonlinear PDE. The dimensionality of the Isaacs PDE is tackled by means of a separable representation of the control system, and a polynomial approximation ansatz for the corresponding value function. Our method proves to be effective for the robust stabilization of nonlinear dynamics up to dimension d ≈ 12. We assess the robustness and optimality features of our design over a class of nonlinear parabolic PDEs, including nonlinear advection and reaction terms. The proposed design yields a feedback controller achieving optimal stabilization and disturbance rejection properties, along with providing a modeling framework for the robust control of PDEs under parametric uncertainties
Stabilisation of spectral/hp element methods through spectral vanishing viscosity: Application to fluid mechanics modelling
Stabilisation of spectral/hp element methods through spectral vanishing viscosity: Application to fluid mechanics modellin
Numerics for finite-dimensional optimal control problems
We survey on numerics for finite-dimensional nonlinear optimal control. The chapter is written as a guide to practitioners who wish to get rapidly acquainted with the main numerical methods used to efficiently solve an optimal control problem. We consider throughout two classical examples, quite simple but representative enough to be complexified and generalized to other problems: the Zermelo and the Goddard problems. We provide their solving codes that are available on the web and make the point on the most up-to-date and efficient methods existing nowadays. We range on direct and indirect methods, on Hamilton-Jacobi approaches and we end with optimistic planning. Our examples illustrate the pros and cons of the methods and we also show how those various approaches can be combined in view of augmenting the efficiency of the numerical solving
Cumulative reports and publications through December 31, 1990
This document contains a complete list of ICASE reports. Since ICASE reports are intended to be preprints of articles that will appear in journals or conference proceedings, the published reference is included when it is available
Robust aircraft trajectory optimization under meteorological uncertainty
Mención Internacional en el título de doctorThe Air Traffic Management (ATM) system in the busiest airspaces in the world
is currently being overhauled to deal with multiple capacity, socioeconomic, and environmental
challenges. One major pillar of this process is the shift towards a concept
of operations centered on aircraft trajectories (called Trajectory-Based Operations or
TBO in Europe) instead of rigid airspace structures. However, its successful implementation
(and, thus, the realization of the associated improvements in ATM performance)
rests on appropriate understanding and management of uncertainty. Due to its complex
socio-technical structure, the design and operations of the ATM system are heavily impacted
by uncertainty, proceeding from multiple sources and propagating through the
interconnections between its subsystems.
One major source of ATM uncertainty is weather. Due to its nonlinear and chaotic
nature, a number of meteorological phenomena of interest cannot be forecasted with
complete accuracy at arbitrary lead times, which leads to uncertainty or disruption in
individual air and ground operations that propagates to all ATM processes. Therefore,
in order to achieve the goals of SESAR and similar programs, it is necessary to deal
with meteorological uncertainty at multiple scales, from the trajectory prediction and
planning processes to flow and traffic management operations.
This thesis addresses the problem of single-aircraft flight planning considering two
important sources of meteorological uncertainty: wind prediction error and convective
activity. As the actual wind field deviates from its forecast, the actual trajectory will
diverge in time from the planned trajectory, generating uncertainty in arrival times,
sector entry and exit times, and fuel burn. Convective activity also impacts trajectory
predictability, as it leads pilots to deviate from their planned route, creating challenging
situations for controllers. In this work, we aim to develop algorithms and methods
for aircraft trajectory optimization that are able to integrate information about the
uncertainty in these meteorological phenomena into the flight planning process at both
pre-tactical (before departure) and tactical horizons (while the aircraft is airborne), in
order to generate more efficient and predictable trajectories.
To that end, we frame flight planning as an optimal control problem, modeling the
motion of the aircraft with a point-mass model and the BADA performance model. Optimal
control methods represent a flexible and general approach that has a long history
of success in the aerospace field. As a numerical scheme, we use direct methods, which
can deal with nonlinear systems of moderate and high-dimensional state spaces in a
computationally manageable way. Nevertheless, while this framework is well-developed
in the context of deterministic problems, the techniques for the solution of practical optimal control problems under uncertainty are not as mature, and the methods proposed
in the literature are not applicable to the flight planning problem as it is now
understood.
The first contribution of this thesis addresses this challenge by introducing a framework
for the solution of general nonlinear optimal control problems under parametric
uncertainty. It is based on an ensemble trajectory scheme, where the trajectories of the
system under multiple scenarios are considered simultaneously within the same dynamical
system and the uncertain optimal control problem is turned into a large conventional
optimal control problem that can be then solved by standard, well-studied direct methods
in optimal control. We then employ this approach to solve the robust flight plan
optimization problem at the planning horizon. In order to model uncertainty in the
wind and estimating the probability of convective conditions, we employ Ensemble Prediction
System (EPS) forecasts, which are composed by multiple predictions instead of
a single deterministic one. The resulting method can be used to optimize flight plans for
maximum expected efficiency according to the cost structure of the airline; additionally,
predictability and exposure to convection can be incorporated as additional objectives.
The inherent tradeoffs between these objectives can be assessed with this methodology.
The second part of this thesis presents a solution for the rerouting of aircraft in
uncertain convective weather scenarios at the tactical horizon. The uncertain motion of
convective weather cells is represented with a stochastic model that has been developed
from the output of a deterministic satellite-based nowcast product, Rapidly Developing
Thunderstorms (RDT). A numerical optimal control framework, based on the pointmass
model with the addition of turn dynamics, is employed for optimizing efficiency
and predictability of the proposed trajectories in the presence of uncertainty about
the future evolution of the storm. Finally, the optimization process is initialized by a
randomized heuristic procedure that generates multiple starting points. The combined
framework is able to explore and as exploit the space of solution trajectories in order to
provide the pilot or the air traffic controller with a set of different suggested avoidance
trajectories, as well as information about their expected cost and risk.
The proposed methods are tested on example scenarios based on real data, showing
how different user priorities lead to different flight plans and what tradeoffs are then
present. These examples demonstrate that the solutions described in this thesis are
adequate for the problems that have been formulated. In this way, the flight planning
process can be enhanced to increase the efficiency and predictability of individual aircraft
trajectories, which would lead to higher predictability levels of the ATM system and thus
improvements in multiple performance indicators.El sistema de gestión del tráfico aéreo (Air Traffic Management, ATM) en los espacios
aéreos más congestionados del mundo está siendo reformado para lidiar con múltiples
desafíos socioeconómicos, medioambientales y de capacidad. Un pilar de este proceso es
el gradual reemplazo de las estructuras rígidas de navegación, basadas en aerovías y waypoints,
hacia las operaciones basadas en trayectorias. No obstante, la implementación
exitosa de este concepto y la realización de las ganancias esperadas en rendimiento ATM
requiere entender y gestionar apropiadamente la incertidumbre. Debido a su compleja
estructura socio-técnica, el diseño y operaciones del sistema ATM se encuentran marcadamente
influidos por la incertidumbre, que procede de múltiples fuentes y se propaga
por las interacciones entre subsistemas y operadores humanos.
Uno de los principales focos de incertidumbre en ATM es la meteorología. Debido a su
naturaleza no-linear y caótica, muchos fenómenos de interés no pueden ser pronosticados
con completa precisión en cualquier horizonte temporal, lo que crea disrupción en las
operaciones en aire y tierra que se propaga a otros procesos de ATM. Por lo tanto,
para lograr los objetivos de SESAR e iniciativas análogas, es imprescindible tener en
cuenta la incertidumbre en múltiples escalas espaciotemporales, desde la predicción de
trayectorias hasta la planificación de flujos y tráfico.
Esta tesis aborda el problema de la planificación de vuelo de aeronaves individuales
considerando dos fuentes importantes de incertidumbre meteorológica: el error en la
predicción del viento y la actividad convectiva. Conforme la realización del viento se
desvía de su previsión, la trayectoria real se desviará temporalmente de la planificada, lo
que implica incertidumbre en tiempos de llegada a sectores y aeropuertos y en consumo
de combustible. La actividad convectiva también tiene un impacto en la predictibilidad
de las trayectorias, puesto que obliga a los pilotos a desviarse de sus planes de vuelo
para evitarla, cambiado así la situación de tráfico. En este trabajo, buscamos desarrollar
métodos y algoritmos para la optimización de trayectorias que puedan integrar
información sobre la incertidumbre en estos fenómenos meteorológicos en el proceso de
diseño de planes de vuelo en horizontes de planificación (antes del despegue) y tácticos
(durante el vuelo), con el objetivo de generar trayectorias más eficientes y predecibles.
Con este fin, formulamos la planificación de vuelo como un problema de control
óptimo, modelando la dinámica del avión con un modelo de masa puntual y el modelo
de rendimiento BADA. El control óptimo es un marco flexible y general con un largo
historial de éxito en el campo de la ingeniería aeroespacial. Como método numérico,
empleamos métodos directos, que son capaces de manejar sistemas dinámicos de alta
dimensión con costes computacionales moderados. No obstante, si bien esta metodología es madura en contextos deterministas, la solución de problemas prácticas de control
óptimo bajo incertidumbre en la literatura no está tan desarrollada, y los métodos
propuestos en la literatura no son aplicables al problema de interés.
La primera contribución de esta tesis hace frente a este reto mediante la introducción
de un marco numérico para la resolución de problemas generales de control óptimo
no-lineal bajo incertidumbre paramétrica. El núcleo de este método es un esquema de
conjunto de trayectorias, en el que las trayectorias del sistema dinámico bajo múltiples
escenarios son consideradas de forma simultánea, y el problema de control óptimo bajo
incertidumbre es así transformado en un problema convencional que puede ser tratado
mediante métodos existentes en control óptimo. A continuación, empleamos este método
para resolver el problema de la planificación de vuelo robusta. La incertidumbre en el
viento y la probabilidad de ocurrencia de condiciones convectivas son modeladas mediante
el uso de previsiones de conjunto o ensemble, compuestas por múltiples predicciones
en lugar de una única previsión determinista. Este método puede ser empleado para
maximizar la eficiencia esperada de los planes de vuelo de acuerdo a la estructura de
costes de la aerolínea; además, la predictibilidad de la trayectoria y la exposición a la
convección pueden ser incorporadas como objetivos adicionales. El trade-off entre estos
objetivos puede ser evaluado mediante la metodología propuesta.
La segunda parte de la tesis presenta una solución para reconducir aviones en escenarios
tormentosos en un horizonte táctico. La evolución de las células convectivas es
representada con un modelo estocástico basado en las proyecciones de Rapidly Developing
Thunderstorms (RDT), un sistema determinista basado en imágenes de satélite.
Este modelo es empleado por un método de control óptimo numérico, basado en un
modelo de masa puntual en el que se modela la dinámica de viraje, con el objetivo de
maximizar la eficiencia y predictibilidad de la trayectoria en presencia de incertidumbre
sobre la evolución futura de las tormentas. Finalmente, el proceso de optimizatión es
inicializado por un método heurístico aleatorizado que genera múltiples puntos de inicio
para las iteraciones del optimizador. Esta combinación permite explorar y explotar el
espacio de trayectorias solución para proporcionar al piloto o al controlador un conjunto
de trayectorias propuestas, así como información útil sobre su coste y el riesgo asociado.
Los métodos propuestos son probados en escenarios de ejemplo basados en datos
reales, ilustrando las diferentes opciones disponibles de acuerdo a las prioridades del
planificador y demostrando que las soluciones descritas en esta tesis son adecuadas para
los problemas que se han formulado. De este modo, es posible enriquecer el proceso de
planificación de vuelo para incrementar la eficiencia y predictibilidad de las trayectorias
individuales, lo que contribuiría a mejoras en el rendimiento del sistema ATM.These works have been financially supported by Universidad Carlos III de Madrid
through a PIF scholarship; by Eurocontrol, through the HALA! Research Network grant
10-220210-C2; by the Spanish Ministry of Economy and Competitiveness (MINECO)'s
R&D program, through the OptMet project (TRA2014-58413-C2-2-R); and by the European
Commission's SESAR Horizon 2020 program, through the TBO-Met project
(grant number 699294).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 iPresidente: Damián Rivas Rivas.- Secretario: Xavier Prats Menéndez.- Vocal: Benavar Sridha
ICASE semiannual report, April 1 - September 30, 1989
The Institute conducts unclassified basic research in applied mathematics, numerical analysis, and computer science in order to extend and improve problem-solving capabilities in science and engineering, particularly in aeronautics and space. The major categories of the current Institute for Computer Applications in Science and Engineering (ICASE) research program are: (1) numerical methods, with particular emphasis on the development and analysis of basic numerical algorithms; (2) control and parameter identification problems, with emphasis on effective numerical methods; (3) computational problems in engineering and the physical sciences, particularly fluid dynamics, acoustics, and structural analysis; and (4) computer systems and software, especially vector and parallel computers. ICASE reports are considered to be primarily preprints of manuscripts that have been submitted to appropriate research journals or that are to appear in conference proceedings
- …