Adaptive mesh refinement method for optimal control using nonsmoothness detection and mesh size reduction
Adaptive mesh refinement method for optimal control using nonsmoothness detection and mesh size reduction
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DOI：
10.1016/j.jfranklin.2015.05.028
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发表时间：
2015-10
期刊：
影响因子：
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通讯作者：
Fengjin Liu;W. Hager;Anil V. Rao
中科院分区：
文献类型：
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作者：
Fengjin Liu;W. Hager;Anil V. Rao
提出了一种求解最优控制问题的自适应网格细化方法。该方法在legende - gauss - radau点采用正交搭配，并在细化过程中调整网格大小和逼近多项式的程度。使用先前导出的收敛率来指导改进过程。该方法通过检查状态的高阶导数是否有较大的增加，提高了解的精度。在不连续点之间的区域，解是光滑的，通过增加近似多项式的次数来减小近似中的误差。在满足误差容忍度的网格间隔上，可以通过合并相邻网格间隔或降低近似多项式的程度来降低网格密度。最后，通过公开文献中的两个实例对该方法进行了验证，并将其性能与先前开发的自适应方法进行了比较。
An adaptive mesh refinement method for solving optimal control problems is developed. The method employs orthogonal collocation at Legendre–Gauss–Radau points, and adjusts both the mesh size and the degree of the approximating polynomials in the refinement process. A previously derived convergence rate is used to guide the refinement process. The method brackets discontinuities and improves solution accuracy by checking for large increases in higher-order derivatives of the state. In regions between discontinuities, where the solution is smooth, the error in the approximation is reduced by increasing the degree of the approximating polynomial. On mesh intervals where the error tolerance has been met, mesh density may be reduced either by merging adjacent mesh intervals or lowering the degree of the approximating polynomial. Finally, the method is demonstrated on two examples from the open literature and its performance is compared against a previously developed adaptive method.