勒壤得拟谱法

最优控制中的勒壤得拟谱法(Legendre pseudospectral method)是以勒让德多项式为基础的方式。是拟谱最佳控制中的一部分,后者是由I. Michael Ross英语I. Michael Ross所命名的理论[1]。勒壤得拟谱法的基本版本最早是由Elnagar等人在1995年提出[2]。之后,I. Michael Ross、Fariba Fahroo等人[3][4][5][6][7]延伸扩展此方法,应用到更大范围的问题中[8]。其中一个受到广泛宣传的应用[9][10]是用此方法来产生国际空间站的实时轨迹。

基础

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勒壤得拟谱法可以分为三种[1]

以高斯-洛巴度取样点(Gauss-Lobatto points)为基础的勒壤得拟谱法,最早是由Elnagar等人提出[2],之后被Fahroo和Ross所扩展[4],包括了伴随向量映射原理。这种勒壤得拟谱法是求解一般非线性有限时域滚动(finite-horizon)最佳控制问题的基础[1][11][12],像DIDO英语DIDO (software)、OTIS、PSOPT软件中都有此方法。

以高斯-蓝道取样点(Gauss-Radau points)为基础的勒壤得拟谱法,最早是由Fahroo和Ross提出[13],之后也扩展,包括了伴随向量映射原理.[5]求解有一个自由终端点的非线性有限时域滚动最佳控制问题的基础[1][12]

以高斯取样点(Gauss points)为基础的勒壤得拟谱法,最早是由Reddien所提出[14],是求解有多个自由终端点的非线性有限时域滚动最佳控制问题的基础[11][12],在GPOPS-II英语GPOPS-IIPROPT英语PROPT软件中都有整合此方法。

软件

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第一个实现勒壤得拟谱法的软件是2001年的DIDO英语DIDO (software)[12][15],之后也整合到NASA的OTIS程式中[16],几年后,像PSOPT、PROPT及GPOPS等软件也有此机能。

太空船航行上的实现

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美国太空总署已在许多太空船的航行上,用DIDO软件使用高斯-洛巴度取样点(Gauss-Lobatto points)为基础的勒壤得拟谱法已实现在[1]。第一次在太空船的航行上的实现是在2006年11月5日,由DIDO来操作国际空间站,达到零燃料机动英语Zero Propellant Maneuver[17]。 零燃料机动是Nazareth Bedrossian用DIDO所发现的方法。

相关条目

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参考资料

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  1. ^ 1.0 1.1 1.2 1.3 1.4 Ross, I. M.; Karpenko, M. A Review of Pseudospectral Optimal Control: From Theory to Flight. Annual Reviews in Control. 2012, 36 (2): 182–197 [2019-02-01]. doi:10.1016/j.arcontrol.2012.09.002. (原始内容存档于2015-09-24). 
  2. ^ 2.0 2.1 G. Elnagar, M. A. Kazemi, and M. Razzaghi, "The Pseudospectral Legendre Method for Discretizing Optimal Control Problems," IEEE Transactions on Automatic Control, 40:1793–1796, 1995.
  3. ^ Ross, I. M. and Fahroo, F., “Legendre Pseudospectral Approximations of Optimal Control Problems,” Lecture Notes in Control and Information Sciences, Vol. 295, Springer-Verlag, New York, 2003, pp 327-342
  4. ^ 4.0 4.1 Fahroo, F. and Ross, I. M., “Costate Estimation by a Legendre Pseudospectral Method,” Journal of Guidance, Control and Dynamics, Vol.24, No.2, March–April 2001, pp.270-277.
  5. ^ 5.0 5.1 Fahroo, F. and Ross, I. M., “Pseudospectral Methods for Infinite-Horizon Optimal Control Problems,” Journal of Guidance, Control and Dynamics, Vol. 31, No. 4, pp. 927-936, 2008.
  6. ^ Kang, W.; Gong, Q.; Ross, I. M.; Fahroo, F. On the Convergence of Nonlinear Optimal Control Using Pseudospectral Methods for Feedback Linearizable Systems. International Journal of Robust and Nonlinear Control: 2007. 
  7. ^ Ross, I. M.; Fahroo, F. Pseudospectral Knotting Methods for Solving Nonsmooth Optimal Control Problems. Journal of Guidance Control and Dynamics: 2004. 
  8. ^ Q. Gong, W. Kang, N. Bedrossian, F. Fahroo, P. Sekhavat and K. Bollino, "Pseudospectral Optimal Control for Military and Industrial Applications," 46th IEEE Conference on Decision and Control, New Orleans, LA, pp. 4128–4142, Dec. 2007.
  9. ^ Kang, W.; Bedrossian, N. Pseudospectral Optimal Control Theory Makes Debut Flight, Saves NASA $1M in Under Three Hours. SIAM News: 2007. 
  10. ^ Bedrossian, N. S., Bhatt, S., Kang, W. and Ross, I. M., “Zero-Propellant Maneuver Guidance,” IEEE Control Systems Magazine, Vol.29, No.5, October 2009, pp 53-73; Cover Story.
  11. ^ 11.0 11.1 Fahroo F., and Ross, I. M., "Advances in Pseudospectral Methods for Optimal Control," AIAA Guidance, Navigation, and Control Conference, AIAA Paper 2008-7309, Honolulu, Hawaii, August 2008.
  12. ^ 12.0 12.1 12.2 12.3 Ross, Isaac. A Primer on Pontryagin's Principle in Optimal Control. San Francisco: Collegiate Publishers. 2015. 
  13. ^ Fahroo, F. and Ross, I. M., “Pseudospectral Methods for Infinite Horizon Nonlinear Optimal Control Problems,” AIAA Guidance, Navigation and Control Conference, August 15–18, 2005, San Francisco, CA
  14. ^ Reddien, G.W., "Collocation at Gauss Points as a Discretization in Optimal Control," SIAM Journal on Control and Optimization, Vol. 17, No. 2, March 1979.
  15. ^ J. R. Rea, A Legendre Pseudospectral Method for Rapid Optimization of Launch Vehicle Trajectories, S.M. Thesis, Dept. of Aeronautics and Astronautics, Massachusetts Institute of Technology, 2001. http://dspace.mit.edu/handle/1721.1/8608页面存档备份,存于互联网档案馆
  16. ^ [ OTIS ] Optimal Trajectories by Implicit Simulation. otis.grc.nasa.gov. [2016-12-08]. (原始内容存档于2016-11-18). 
  17. ^ Zero Propellant Maneuver. [2008-10-05]. (原始内容存档于2008-10-05).