求解最优控制问题的Chebyshev-Gauss伪谱法

唐小军; 尉建利; 陈凯

doi:10.16383/j.aas.2015.e130297

求解最优控制问题的Chebyshev-Gauss伪谱法

doi: 10.16383/j.aas.2015.e130297

1.
西北工业大学航空学院西安 710072
2.
西北工业大学航天学院西安 710072

基金项目:

Supported by Natural Science Basic Research Plan in Shaanxi Province of China (2014JQ8366), Fundamental Research Foun- dation of Northwestern Polytechnical University (JC20120210, JC20110238), and Aeronautical Science Foundation of China (20120853007)

计量
- 文章访问数: 2749
- HTML全文浏览量: 178
- PDF下载量: 1326
- 被引次数: 0
出版历程
- 收稿日期: 2013-12-18
- 修回日期: 2014-05-28
- 刊出日期: 2015-10-20

A Chebyshev-Gauss Pseudospectral Method for Solving Optimal Control Problems

1.
School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, China;
2.
School of Astronautics, Northwestern Polytechnical University, Xi'an 710072, China

Funds:

摘要

摘要: 提出了一种求解最优控制问题的Chebyshev-Gauss伪谱法, 配点选择为Chebyshev-Gauss点. 通过比较非线性规划问题的Kaursh-Kuhn-Tucker条件和伪谱离散化的最优性条件, 导出了协态和Lagrange乘子的估计公式. 在状态逼近中, 采用了重心Lagrange插值公式, 并提出了一种简单有效的计算状态伪谱微分矩阵的方法. 该法的独特优势是具有良好的数值稳定性和计算效率. 仿真结果表明, 该法能够高精度地求解带有约束的复杂最优控制问题.
- 最优控制 /
- 伪谱法 /
- 协态估计 /
- Chebyshev-Gauss点
Abstract: A pseudospectral method is presented for direct trajectory optimization of optimal control problems using collocation at Chebyshev-Gauss points, and therefore, it is called Chebyshev-Gauss pseudospectral method. The costate and constraint multiplier estimates for the proposed method are rigorously derived by comparing the discretized optimality conditions of an optimal control problem with the Karush-Kuhn-Tucker conditions of the resulting nonlinear programming problem from collocation. The distinctive advantages of the proposed method over other pseudopsectral methods are the good numerical stability and computational efficiency. In order to achieve this goal, the barycentric Lagrange interpolation is substituted for the classic Lagrange interpolation in the state approximation. Furthermore, a simple yet efficient method is presented to alleviate the numerical errors of state differential matrix using the trigonometric identity especially when the number of Chebyshev-Gauss points is large. The method presented in this paper has been taken to two optimal control problems from the open literature, and the results have indicated its ability to obtain accurate solutions to complex constrained optimal control problems.
- Optimal control /
- pseudospectral methods /
- costate estimation /
- Chebyshev-Gauss points

HTML全文

参考文献(21)

[1]	Benson D A. A Gauss Pseudospectral Transcription for Optimal Control [Ph.D. dissertation], Massachusetts Institute of Technology, USA, 2004
[2]	[2] Benson D A, Huntington G T, Thorvaldsen T P, Rao A V. Direct trajectory optimization and costate estimation via an orthogonal collocation method. Journal of Guidance, Control, and Dynamics, 2006, 29(6): 1435-1440
[3]	[3] Huntington G T. Advancement and Analysis of a Gauss Pseudospectral Transcription for Optimal Control Problems [Ph.D. dissertation], Massachusetts Institute of Technology, USA, 2007
[4]	[4] Garg D, Patterson M A, Francolin C, Darby C L, Huntington G T, Hager W W, Rao A V. Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method. Computational Optimization and Applications, 2011, 49(2): 335-358
[5]	[5] Garg D, Hager W W, Rao A V. Pseudospectral methods for solving infinite-horizon optimal control problems. Automatica, 2011, 47(4): 829-837
[6]	[6] Garg D. Advances in Global Pseudospectral Methods for Optimal Control [Ph.D. dissertation], Massachusetts Institute of Technology, USA, 2011
[7]	[7] Elnagar G, Kazemi M A, Rzaazghi M. The pseudospectral Legendre method for discretizing optimal control problems. IEEE Transactions on Automatic Control, 1995, 40(10): 1793-1796
[8]	[8] Elnagar G N, Rzaazghi M. A collocation-type method for linear quadratic optimal control problems. Optimal Control Applications and Methods, 1997, 18(3): 227-235
[9]	[9] Fahroo F, Ross I M. Costate estimation by a Legendre pseudospectral method. Journal of Guidance, Control, and Dynamics, 2001, 24(2): 270-277
[10]	Elnagar G N, Kazemi M A. Pseudospectral Chebyshev optimal control of constrained nonlinear dynamical systems. Computational Optimization and Applications, 1998, 11(2): 195-217
[11]	Fahroo F, Ross I M. Direct trajectory optimization by a Chebyshev pseudospectral method. Journal of Guidance, Control, and Dynamics, 2002, 25(1): 160-166
[12]	Gong Q, Ross I M, Fahroo F. Costate computation by a Chebyshev pseudospectral method. Journal of Guidance, Control, and Dynamics, 2010, 33(2): 623-628
[13]	Fornberg B. A Practical Guide to Pseudospectral Methods. New York: Cambridge University Press, 1998
[14]	Weideman J A C, Trefethen L N. The kink phenomenon in Fejr and Clenshaw-Curtis quadrature. Numerische Mathematik, 2007, 107(4): 707-727
[15]	Berrut J P, Trefethen L N. Barycentric Lagrange interpolation. SIAM Review, 2004, 46(3): 501-517
[16]	Gill P E, Murray W, Saunders M A. SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Review, 2005, 47(1): 99-131
[17]	Biegler L T, Zavala V M. Large-scale nonlinear programming using IPOPT: an integrating framework for enterprise-wide dynamic optimization. Computers and Chemical Engineering, 2009, 33(3): 575-582
[18]	Waldvogel J. Fast construction of the Fejr and Clenshaw-Curtis quadrature rules. BIT Numerical Mathematics, 2006, 46(1): 195-202
[19]	Costa B, Don W S. On the computation of high order pseudospectral derivatives. Applied Numerical Mathematics, 2000, 33(1-4): 151-159
[20]	Gong Q, Kang W, Ross I M. A pseudospectral method for the optimal control of constrained feedback linearizable systems. IEEE Transactions on Automatic Control, 2006, 51(7): 1115-1129
[21]	Betts J T. Practical Methods for Optimal Control and Estimation Using Nonlinear Programming (Second edition). Philadelphia: Society for Industrial and Applied Mathematics, 2010