﻿ 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

Dynamical Systems and Control
Vol.06 No.02(2017), Article ID:20311,8 pages
10.12677/DSC.2017.62005

Pseudospectral Mesh Refinement Algorithm Based on Density Function and Its Application

Liying Wang1, Jie Huang2, Gang Zhang3

1Institute of Systems Science and Mathematics, Naval Aeronautical and Astronautical University, Yantai Shandong

2Naval Aeronautical and Astronautical University, Qingdao Branch, Qingdao Shandong

Received: Apr. 4th, 2017; accepted: Apr. 25th, 2017; published: Apr. 28th, 2017

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.

Keywords:Optimal Control Problems, Radau Pseudospectral Method, Mesh Refinement, Density Function

1海军航空工程学院系统科学与数学研究所，山东 烟台

2海军航空工程学院青岛分院，山东 青岛

3海军航空兵学院，辽宁 葫芦岛

1. 引言

2. 多区间连续时间最优控制问题的一般性描述

(1)

(2)

(3)

(4)

(5)

(6)

3. 伪谱网格细化算法

3.1. 解的误差判定准则

(Lagrange多项式的维数)表示区间的配点数，取相邻配点的中点作为采样点。将动态约束方程在采样点上的残差作为解的误差评估准则，如式(7)所示：

(7)

3.2. 基于曲率密度函数的细化策略

3.2.1. 密度函数的定义

(8)

(9)

3.2.2. 新增区间位置的确定

(10)

3.3. 算法流程

4. 验证算例

Figure 1. Algorithm flow

Figure 2. Nodes distribution pattern of different pseudospectral methods

Figure 3. Control and state curves of different pseudospectral methods with times

Table 1. Comparision of optimization result of difference accuracy

5. 结论

Pseudospectral Mesh Refinement Algorithm Based on Density Function and Its Application[J]. 动力系统与控制, 2017, 06(02): 35-42. http://dx.doi.org/10.12677/DSC.2017.62005

1. 1. Gong, Q., Kang, W. and Ross, I.M. (2006) A Pseudospectral Method for the Optimal Control of Constrained Feedback Linearizable Systems. IEEE Transactions on Automatic Control, 51, 1115-1129. https://doi.org/10.1109/TAC.2006.878570

2. 2. Kang, W. (2010) Rate of Convergence for the Legendre Pseudo-spectral Optimal Control of Feedback Linearizable Systems. Journal of Control Theory and Applications, 8, 391-405. https://doi.org/10.1007/s11768-010-9104-0

3. 3. 雍恩米, 唐国金, 陈磊. 基于Gauss伪谱方法的高超声速飞行器再入轨迹快速优化[J]. 宇航学报, 2008, 29(6): 1766-1772.

4. 4. Guo, X. and Zhu, M. (2013) Direct Trajectory Optimization Based on a Mapped Chebyshev Pseudospectral Method. Chinese Journal of Aeronautics, 26, 401-402. https://doi.org/10.1016/j.cja.2013.02.018

5. 5. 水尊师, 周军, 葛致磊. 基于高斯伪谱方法的再入飞行器预测校正制导方法研究[J]. 宇航学报, 2011, 32(6): 1249-1255.

6. 6. Ross, I.M. and Fahroo, F. (2004) Pseudospectral Knotting Methods for Solving Optimal Control Problems. Journal of Guidance, Control, and Dynamics, 27, 397-405. https://doi.org/10.2514/1.3426

7. 7. Chowdhury, S. and Mehmani, A. (2016) Adaptive Model Refinement in Surrogate-Based Multiobjective Optimization. 57th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Mate-rials Conference, San Diego, 4-8 January 2016, AIAA 2016-0417. https://doi.org/10.2514/6.2016-0417

8. 8. Sa-gliano, M., Mooij, E. and Theil, S. (2016) Onboard Trajectory Generation for Entry Vehicles via Adaptive Multivariate Pseudospectral Interpolation. AIAA Guidance, Navigation, and Control Conference, San Diego, 4 January 2016, AIAA 2016-2115. https://doi.org/10.2514/6.2016-2115

9. 9. Zhao Y. and Tsiotras P. (2009) Mesh Refinement Using Density-Function for Solving Optimal Control Problems. Infotech and Aerospace Conference, Seattle, 6-9 April 2009, AIAA Paper 2009-2019. https://doi.org/10.2514/6.2009-2019

10. 10. Zhao Y. and Tsiotras P. (2011) Density Functions for Mesh Refinement in Numerical Optimal Control. Journal of Guidance, Control, and Dynamics, 34, 271-277. https://doi.org/10.2514/1.45852

11. 11. Betts, J.T. (1998) Survey of Numerical Methods for Trajectory Optimization. Journal of Guidance, Control, and Dynamics, 21, 193-207. https://doi.org/10.2514/2.4231