維基百科

Ross–Fahroo擬譜法

Ross–Fahroo擬譜法（Ross–Fahroo pseudospectral methods）是由I. Michael Ross（英語：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，從勒讓德多項式或切比雪夫多項式而來。Gauss-Lobatto擬譜點可以用在有限時域滾動的最優控制問題，而Gauss-Radau擬譜點可以用在無限時域滾動的最優控制問題[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（英語：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優控制問題[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擬譜法實現了國際空間站的零燃料機動（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（英語：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（英語：DIDO (optimal control)）。

相關條目

參考資料

  1. ^ N. Bedrossian, M. Karpenko, and S. Bhatt, "Overclock My Satellite: Sophisticated Algorithms Boost Satellite Performance on the Cheap", IEEE綜覽, 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. ^ 5.0 5.1 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. ^ 6.0 6.1 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. ^ 8.0 8.1 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] （頁面存檔備份，存於網際網路檔案館）
  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. ^ 12.0 12.1 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. ^ 13.0 13.1 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. （原始內容存檔於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.
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