Approximate Solutions to the Hamilton-Jacobi Equations for Generating Functions
Approximate Solutions to the Hamilton-Jacobi Equations for Generating Functions
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DOI：
10.1007/s11424-019-8334-6
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发表时间：
2020-01
影响因子：
2.1
通讯作者：
Z. Hao;K. Fujimoto;Qiuhua Zhang
中科院分区：
文献类型：
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作者：
Z. Hao;K. Fujimoto;Qiuhua Zhang
针对一类非线性有限时间最优控制问题，提出了一种求解生成函数的Hamilton-Jacobi方程的系统数值算法。该算法允许通过递归求解一阶常微分方程序列来获得生成函数的泰勒级数展开式，直到任意指定阶。此外，在一定的技术条件下，还可以精确地计算生成函数的泰勒级数展开式的系数。一旦一个生成函数被找到，它可以被用来生成一个家庭的最优控制不同的边界条件。由于母函数是离线计算的，因此与传统方法相比，不同边界条件下的按需计算工作量减少了很多。该方法对在线最优轨迹生成问题有一定的参考价值。数值例子说明了所提出的算法的有效性。
For a nonlinear finite time optimal control problem, a systematic numerical algorithm to solve the Hamilton-Jacobi equation for a generating function is proposed in this paper. This algorithm allows one to obtain the Taylor series expansion of the generating function up to any prescribed order by solving a sequence of first order ordinary differential equations recursively. Furthermore, the coefficients of the Taylor series expansion of the generating function can be computed exactly under a certain technical condition. Once a generating function is found, it can be used to generate a family of optimal control for different boundary conditions. Since the generating function is computed off-line, the on-demand computational effort for different boundary conditions decreases a lot compared with the conventional method. It is useful to online optimal trajectory generation problems. Numerical examples illustrate the effectiveness of the proposed algorithm.