Convergence rates for direct transcription of optimal control problems using collocation at Radau points

Authors:
Shivakumar Kameswaran;Lorenz T. Biegler
Affiliations:
Dept. of Chemical Engineering, Carnegie Mellon University, Pittsburgh, USA 15213;Dept. of Chemical Engineering, Carnegie Mellon University, Pittsburgh, USA 15213
Venue:
Computational Optimization and Applications
Year:
2008

Citing 10
Cited 7

A new basis implementation for a mixed order boundary value ODE solver

SIAM Journal on Scientific and Statistical Computing
Introduction to numerical analysis: 2nd edition

Introduction to numerical analysis: 2nd edition
Direct and indirect methods for trajectory optimization

Annals of Operations Research - Special issue on nonlinear methods in economic dynamics and optimal control: Gmo¨or-series No. 2
Consistent Approximations for Optimal Control Problems Based on Runge--Kutta Integration

SIAM Journal on Control and Optimization
Matrix computations (3rd ed.)

Matrix computations (3rd ed.)
Adjoint estimation from a direct multiple shooting method

Journal of Optimization Theory and Applications
Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations

Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
Stable computational procedures for dynamic optimization in process engineering

Stable computational procedures for dynamic optimization in process engineering
On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming

Mathematical Programming: Series A and B
Practical Methods for Optimal Control and Estimation Using Nonlinear Programming

Practical Methods for Optimal Control and Estimation Using Nonlinear Programming

Algorithm 902: GPOPS, A MATLAB software for solving multiple-phase optimal control problems using the gauss pseudospectral method

ACM Transactions on Mathematical Software (TOMS)
Brief paper: A unified framework for the numerical solution of optimal control problems using pseudospectral methods

Automatica (Journal of IFAC)
Inexact Restoration for Runge-Kutta Discretization of Optimal Control Problems

SIAM Journal on Numerical Analysis
Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method

Computational Optimization and Applications
Sequential virtual motion camouflage method for nonlinear constrained optimal trajectory control

Automatica (Journal of IFAC)
Convergence of the forward-backward sweep method in optimal control

Computational Optimization and Applications
Fast, accurate, and small-scale direct trajectory optimization using a Gegenbauer transcription method

Journal of Computational and Applied Mathematics

Quantified Score

Hi-index	0.01

Visualization

Abstract

We present convergence rates for the error between the direct transcription solution and the true solution of an unconstrained optimal control problem. The problem is discretized using collocation at Radau points (aka Gauss-Radau or Legendre-Gauss-Radau quadrature). The precision of Radau quadrature is the highest after Gauss (aka Legendre-Gauss) quadrature, and it has the added advantage that the end point is one of the abscissas where the function, to be integrated, is evaluated. We analyze convergence from a Nonlinear Programming (NLP)/matrix algebra perspective. This enables us to predict the norms of various constituents of a matrix that is "close" to the KKT matrix of the discretized problem. We present the convergence rates for the various components, for a sufficiently small discretization size, as functions of the discretization size and the number of collocation points. We illustrate this using several test examples. This also leads to an adjoint estimation procedure, given the Lagrange multipliers for the large scale NLP.