N. Koeppen, I. M. Ross, L. C. Wilcox, R. J. Proulx
arXiv 29 Apr 2019 · Mathematics — Optimization · publishedJournal of Guidance Control and Dynamics (2018) · 41 citations (OpenAlex)
arXiv:1904.12992 · PDF · DOI · OpenAlex · Extracted main text
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.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | I. M. Ross and M. Karpenko, “A Review of Pseudospectral Optimal Cont… (2012) A Review of Pseudospectral Optimal Control: From Theory to Flight self | 1.000 | 13 | 4 | 100% |
| 2 | I. M. Ross, A Primer on Pontryagin's Principle in Optimal Control, S… (2015) self | 1.000 | 11 | 4 | 100% |
| 3 | L.-L Wang, M. D. Samson and X. Zhao, “A Well-Conditioned Collocation… (2014) A Well-Conditioned Collocation Method Using a Pseudospectral Integration Matrix | 1.000 | 7 | 3 | 100% |
| 4 | F. Fahroo and I. M. Ross, “Advances in Pseudospectral Methods for Op… (2008) Advances in Pseudospectral Methods for Optimal Control self | 1.000 | 6 | 3 | 100% |
| 5 | Q. Gong, I. M. Ross and F. Fahroo, “Spectral and Pseudospectral Opti… (2016) Spectral and Pseudospectral Optimal Control Over Arbitrary Grids self | 0.874 | 7 | 2 | 100% |
| 6 | Q. Gong and I. M. Ross, “Autonomous Pseudospectral Knotting Methods… (2006) Autonomous Pseudospectral Knotting Methods for Space Mission Optimization self | 0.874 | 7 | 2 | 100% |
| 7 | Q. Gong, F. Fahroo and I. M. Ross, “Spectral Algorithm for Pseudospe… (2008) Spectral Algorithm for Pseudospectral Methods in Optimal Control self | 0.874 | 7 | 2 | 100% |
| 8 | J. Boyd, Chebyshev and Fourier Spectral Methods, Dover Publications,… (2001) | 0.874 | 6 | 2 | 100% |
| 9 | I. M. Ross and F. Fahroo, “Pseudospectral Knotting Methods for Solvi… (2004) Pseudospectral Knotting Methods for Solving Optimal Control Problems self | 0.874 | 5 | 2 | 100% |
| 10 | I. M. Ross, Q. Gong, M. Karpenko and R. J. Proulx, “Scaling and Bala… Scaling and Balancing for High-Performance Computation of Optimal Controls self | 0.843 | 3 | 3 | 100% |
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