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An adaptive mesh refinement method for numerically solving optimal control problems is developed using Legendre-Gauss-Radau direct collocation. In regions of the solution where the desired accuracy tolerance has not been met, the mesh is refined by either increasing the degree of the approximating polynomial in a mesh interval or dividing a mesh interval into subintervals. In regions of the solution where the desired accuracy tolerance has been met, the mesh size may be reduced by either merging adjacent mesh intervals or decreasing the degree of the approximating polynomial in a mesh interval. Coupled with the mesh refinement method described in this paper is a newly developed relative error estimate that is based on the differences between solutions obtained from the collocation method and those obtained by solving initial-value and terminal-value problems in each mesh interval using an interpolated control obtained from the collocation method. Because the error estimate is based on explicit simulation, the solution obtained via collocation is in close agreement with the solution obtained via explicit simulation using the control on the final mesh, which ensures that the control is an accurate approximation of the true optimal control. The method is demonstrated on three examples from the open literature, and the results obtained show an improvement in final mesh size when compared against previously developed mesh refinement methods.
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Mesh refinement method for solving optimal control problems with nonsmooth solutions using jump function approximations
Abstract A mesh refinement method is described for solving optimal control problems using Legendre‐Gauss‐Radau collocation. The method detects discontinuities in the control solution by employing an edge detection scheme based on jump function approximations. When discontinuities are identified, the mesh is refined with a targetedh‐refinement approach whereby the discontinuity locations are bracketed with mesh points. The remaining smooth portions of the mesh are refined using previously developed techniques. The method is demonstrated on two examples, and results indicate that the method solves optimal control problems with discontinuous control solutions using fewer mesh refinement iterations and less computation time when compared with previously developed methods.
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- PAR ID:
- 10451092
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Optimal Control Applications and Methods
- Volume:
- 42
- Issue:
- 4
- ISSN:
- 0143-2087
- Page Range / eLocation ID:
- p. 1119-1140
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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