Spectral Algorithm for Pseudospectral Methods in Optimal Control

Gong, Qi; Fahroo, Fariba; Ross, I. Michael

doi:10.2514/1.32908

Cited by 165 publications

(102 citation statements)

References 32 publications

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“…In general, using the elementbased Galerkin optimal control formations discussed (both continuous and discontinuous Galerkin) may lead to higher computational efficiencies and the formulation may be retooled to incorporate hp-adaptive techniques, such as the spectral algorithm discussed in [86]. Additionally, the discontinuous Galerkin formulation may be advantageous from a parallel computing standpoint.…”

Section: Dissertation Summarymentioning

confidence: 99%

Galerkin Optimal Control

Boucher

Kang

Gong

2016

J Optim Theory Appl

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Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instruction, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington ABSTRACT (maximum 200 words)A Galerkin-based family of numerical formulations is presented for solving nonlinear optimal control problems. This dissertation introduces a family of direct methods that calculate optimal trajectories by discretizing the system dynamics using Galerkin numerical techniques and approximate the cost function with Gaussian quadrature. In this numerical approach, the analysis is based on L 2 -norms. An important result in the theoretical foundation is that the feasibility and consistency theorems are proved for problems with continuous and/or piecewise continuous controls. Galerkin methods may be formulated in a number of ways that allow for efficiency and/or improved accuracy while solving a wide range of optimal control problems with a variety of state and control constraints. Numerical formulations using Lagrangian and Legendre test functions are derived. One formulation allows for a weak enforcement of boundary conditions, which imposes end conditions only up to the accuracy of the numerical approximation itself. Additionally, the multi-scale formulation can reduce the dimension of multi-scale optimal control problems, those in which the states and controls evolve on different timescales. Finally, numerical examples are shown to demonstrate the versatile nature of Galerkin optimal control. SUBJECT TERMS

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Section: Dissertation Summarymentioning

confidence: 99%

Galerkin Optimal Control

Boucher

Kang

Gong

2016

J Optim Theory Appl

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“…The coefficients a m in Eq. (44) are called the spectral coefficients [17,18,21]. A key principle in a PS approach is that the spectral coefficients are computed indirectly by transforming Eq.…”

Section: A Standard Pseudospectral Optimal Control Theorymentioning

confidence: 99%

Riemann–Stieltjes Optimal Control Problems for Uncertain Dynamic Systems

Ross

Proulx

Karpenko

et al. 2015

Journal of Guidance, Control, and Dynamics

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Motivated by uncertain parameters in nonlinear dynamic systems, we define a nonclassical optimal control problem where the cost functional is given by a Riemann-Stieltjes "functional of a functional." Using the properties of Riemann-Stieltjes sums, a minimum principle is generated from the limit of a semidiscretization. The optimal control minimizes a Riemann-Stieltjes integral of the Pontryagin Hamiltonian. The challenges associated with addressing the noncommutative operations of integration and minimization are addressed via cubature techniques leading to the concept of hyper-pseudospectral points. These ideas are then applied to address the practical uncertainties in control moment gyroscopes that drive an agile spacecraft. Ground test results conducted at Honeywell demonstrate the new principles. The Riemann-Stieltjes optimal control problem is a generalization of the unscented optimal control problem. It can be connected to many independently developed ideas across several disciplines: search theory, viability theory, quantum control and many other applications involving tychastic differential equations.

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“…The main idea behind all these adaptive grid methods is to use a high resolution (dense) grid only in the vicinity of control switches, constraint boundaries etc, and a coarse grid elsewhere. Examples of such adaptive gridding techniques for the solution of optimal control problems are [10,13,70,12,43,35].…”

Section: Optimal Trajectory Generationmentioning

confidence: 99%

Mathematical Models for Aircraft Trajectory Design: A Survey

Delahaye

Puechmorel

Tsiotras

et al. 2014

Lecture Notes in Electrical Engineering

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Air traffic management ensures the safety of flight by optimizing flows and maintaining separation between aircraft. After giving some definitions, some typical feature of aircraft trajectories are presented. Trajectories are objects belonging to spaces with infinite dimensions. The naive way to address such problem is to sample trajectories at some regular points and to create a big vector of positions (and or speeds). In order to manipulate such objects with algorithms, one must reduce the dimension of the search space by using more efficient representations. Some dimension reduction tricks are then presented for which advantages and drawbacks are presented. Then, front propagation approaches are introduced with a focus on Fast Marching Algorithms and Ordered upwind algorithms. An example of application of such algorithm to a real instance of air traffic control problem is also given. When aircraft dynamics have to be included in the model, optimal control approaches are really efficient. We present also some application to aircraft trajectory design. Finally, we introduce some path planning techniques via natural language processing and mathematical programming.

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Spectral Algorithm for Pseudospectral Methods in Optimal Control

Cited by 165 publications

References 32 publications

Galerkin Optimal Control

Galerkin Optimal Control

Riemann–Stieltjes Optimal Control Problems for Uncertain Dynamic Systems

Mathematical Models for Aircraft Trajectory Design: A Survey

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