Ross–Fahroo擬譜法

Ross–Fahroo擬譜法（Ross–Fahroo pseudospectral methods）是由I. Michael Ross（英语：I. Michael Ross）和Fariba Fahroo導入的方法，屬於擬譜最佳控制中的一部份^{[1][2][3][4][5][6][7][8][9]}。Ross–Fahroo擬譜法的例子有擬譜knotting法、平坦擬譜法、Legendre-Gauss-Radau擬譜法^[10][11]以及無限時域滾動最佳控制的擬譜法^{[12] [13]}。

簡介

Ross–Fahroo擬譜法是以位移過的高斯擬譜節點為基礎，位移是靠線性變換或是非線性變換，高斯擬譜點是由Gauss-Lobatto或Gauss-Radau分布，從勒让德多项式或切比雪夫多项式而來。Gauss-Lobatto擬譜點可以用在有限時域滾動的最优控制問題，而Gauss-Radau擬譜點可以用在無限時域滾動的最优控制問題^[14]。

數學應用

Ross–Fahroo擬譜法可以由Ross–Fahroo引理求得，可以應用在統御方程是微分方程、微分幾何方程（英语：differential algebraic equation）、微分包含式的系統，及微分flat系統的。在經過簡單的定義域變換後，也可以應用在無限時域滾動的最优控制問題^{[12] [13]}。Ross–Fahroo擬譜法也是貝爾曼擬譜法的基礎。

飛航應用及獎項

TRoss–Fahroo擬譜法已用在全世界的許多實驗室及實務應用中。NASA在2006年時用Ross–Fahroo擬譜法實現了国际空间站的零燃料機動（zero propellant maneuver）降落^[15]。為了表彰這些進步的成果，AIAA將2010年飛行力學和控制獎（2010 Mechanics and Control of Flight Award）頒給Ross及Fahroo，原因是「改變飛行力學的現狀」。Ross也獲選為美國太空學會（英语：American Astronautical Society）（AAS）的Fellow，原因是「在擬譜最佳控制中開創性的貢獻。」

特點

Ross–Fahroo擬譜法有一個重大特點，和以往強調「直接法」或「間接法」的其他方式不同。透過 Ross 及 Fahroo結合了相關定理^{[5][6][8][16]}，證明了可以設計在「直接法」及「間接法」上都等效的擬譜最佳控制法。因此設計者可以直接用他們設計的方法為「直接法」，同時自動產生一個準確的對偶問題，為「間接法」。這種革命性的作法讓Ross–Fahroo擬譜法廣為使用^[17]。

軟體應用

Ross–Fahroo擬譜法已實現在MATLAB的最佳控制求解器DIDO（英语：DIDO (optimal control)）。

相關條目

• 貝爾曼擬譜法
• DIDO（英语：DIDO (optimal control)），得名自迦太基女王狄多（Dido）
• 羅斯π引理
• Ross–Fahroo引理

參考資料

1. N. Bedrossian, M. Karpenko, and S. Bhatt, "Overclock My Satellite: Sophisticated Algorithms Boost Satellite Performance on the Cheap", IEEE综览, November 2012.
2. Jr-; Li, S; Ruths, J.; Yu, T-Y; Arthanari, H.; Wagner, G. Optimal Pulse Design in Quantum Control: A Unified Computational Method. Proceedings of the National Academy of Sciences. 2011, 108 (5): 1879–1884. PMC 3033291 . PMID 21245345. doi:10.1073/pnas.1009797108. 
3. Kang, W. Rate of Convergence for the Legendre Pseudospectral Optimal Control of Feedback Linearizable Systems. Journal of Control Theory and Application. 2010, 8 (4): 391–405. doi:10.1007/s11768-010-9104-0. 
4. Conway, B. A. A Survey of Methods Available for the Numerical Optimization of Continuous Dynamic Systems. Journal of Optimization Theory Applications. 2012, 152 (2): 271–306. doi:10.1007/s10957-011-9918-z. 
5. I. M. Ross and F. Fahroo, A Pseudospectral Transformation of the Covectors of Optimal Control Systems, Proceedings of the First IFAC Symposium on System Structure and Control, Prague, Czech Republic, 29–31 August 2001.
6. I. M. Ross and F. Fahroo, Legendre Pseudospectral Approximations of Optimal Control Problems, Lecture Notes in Control and Information Sciences, Vol. 295, Springer-Verlag, 2003.
7. Ross, I. M.; Fahroo, F. Pseudospectral Knotting Methods for Solving Optimal Control Problems. Journal of Guidance, Control and Dynamics. 2004, 27 (3): 397–405. doi:10.2514/1.3426. 
8. I. M. Ross and F. Fahroo, Discrete Verification of Necessary Conditions for Switched Nonlinear Optimal Control Systems, Proceedings of the American Control Conference, Invited Paper, June 2004, Boston, MA.
9. Ross, I. M.; Fahroo, F. Pseudospectral Methods for the Optimal Motion Planning of Differentially Flat Systems. IEEE Transactions on Automatic Control. 2004, 49 (8): 1410–1413. doi:10.1109/tac.2004.832972. hdl:10945/29675. 
10. F. Fahroo and I. M. Ross, "Advances in Pseudospectral Methods for Optimal Control," Proceedings of the AIAA Guidance, Navigation and Control Conference, AIAA 2008-7309. [1] （页面存档备份，存于互联网档案馆）
11. Wen, H.; Jin, D.; Hu, H. Infinite-Horizon Control for Retrieving a Tethered Subsatellite via an Elastic Tether. Journal of Guidance, Control and Dynamics. 2008, 31 (4): 889–906. doi:10.2514/1.33224. 
12. F. Fahroo and I. M. Ross, Pseudospectral Methods for Infinite Horizon Nonlinear Optimal Control Problems, AIAA Guidance, Navigation and Control Conference, August 15–18, 2005, San Francisco, CA.
13. Fahroo, F.; Ross, I. M. Pseudospectral Methods for Infinite-Horizon Optimal Control Problems. Journal of Guidance, Control and Dynamics. 2008, 31 (4): 927–936. doi:10.2514/1.33117. 
14. Ross, I. M.; Karpenko, M. A Review of Pseudospectral Optimal Control: From Theory to Flight. Annual Reviews in Control. 2012, 36 (2): 182–197 [2019-02-13]. doi:10.1016/j.arcontrol.2012.09.002. （原始内容存档于2015-09-24）. 
15. N. S. Bedrossian, S. Bhatt, W. Kang, and I. M. Ross, Zero-Propellant Maneuver Guidance, IEEE Control Systems Magazine, October 2009 (Feature Article), pp 53–73.
16. F. Fahroo and I. M. Ross, Trajectory Optimization by Indirect Spectral Collocation Methods, Proceedings of the AIAA/AAS Astrodynamics Conference, August 2000, Denver, CO. AIAA Paper 2000–4028
17. 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.