基于密度函数的伪谱网格细化算法及应用

Pseudospectral Mesh Refinement Algorithm Based on Density Function and Its Application

摘要: 本文针对伪谱方法在求解不连续或非光滑最优控制问题上的缺点，提出了一种基于密度函数的伪谱网格细化算法，即利用Radau伪谱方法将原连续时间最优控制问题转化为非线性规划问题，取相邻配点的中点作为采样点，将动态约束在采样点上的残差作为近似解的误差评估准则；对于不满足求解精度要求的区间，利用轨迹曲率密度函数及其累积分布函数的性质将该区间进行细化。该算法能够捕捉到状态变量和控制变量的任意不连续性和非平滑性，通过有效的计算方式改进解的精度。仿真算例验证了算法的有效性。

Abstract: For the defects of pseudospectral method in solving discontinuous and non-smooth optimal control problems, a pesudospectral mesh refinement algorithm based on density function was proposed. The continuous-time optimal control problem was converted into nonlinear programming problems by using Radau pesudospectral method. The midpoints of adjacent collocation points were used as sample points, and the residuals of the dynamics constraints at these points were used as the assessment of approximation solution. The intervals where solution needs to be improved were divided into new subintervals by using the properties of curvature density function and corresponding cumulative distribution function. The algorithm can capture any discontinuities and smoothness in state and control variables, and improve the accuracy of the solution in a computational efficient manner. Simulation examples demonstrated the validity of the algorithm.