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Fast Mesh Refinement in Pseudospectral Optimal Control

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Listed:
  • N. Koeppen
  • I. M. Ross
  • L. C. Wilcox
  • R. J. Proulx

Abstract

Mesh refinement in pseudospectral (PS) optimal control is embarrassingly easy --- simply increase the order $N$ of the Lagrange interpolating polynomial and the mathematics of convergence automates the distribution of the grid points. Unfortunately, as $N$ increases, the condition number of the resulting linear algebra increases as $N^2$; hence, spectral efficiency and accuracy are lost in practice. In this paper, we advance Birkhoff interpolation concepts over an arbitrary grid to generate well-conditioned PS optimal control discretizations. We show that the condition number increases only as $\sqrt{N}$ in general, but is independent of $N$ for the special case of one of the boundary points being fixed. Hence, spectral accuracy and efficiency are maintained as $N$ increases. The effectiveness of the resulting fast mesh refinement strategy is demonstrated by using \underline{polynomials of over a thousandth order} to solve a low-thrust, long-duration orbit transfer problem.

Suggested Citation

  • N. Koeppen & I. M. Ross & L. C. Wilcox & R. J. Proulx, 2019. "Fast Mesh Refinement in Pseudospectral Optimal Control," Papers 1904.12992, arXiv.org.
    • Handle: RePEc:arx:papers:1904.12992
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    References listed on IDEAS

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    1. Qi Gong & Isaac Michael Ross & Fariba Fahroo, 2016. "Spectral and Pseudospectral Optimal Control Over Arbitrary Grids," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 759-783, June.
    2. Bruce A. Conway, 2012. "A Survey of Methods Available for the Numerical Optimization of Continuous Dynamic Systems," Journal of Optimization Theory and Applications, Springer, vol. 152(2), pages 271-306, February.
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