Ross–Fahroo引理
Ross–Fahroo引理（Ross–Fahroo lemma）得名自以撒·麦克·罗斯和Fariba Fahroo，是最优控制理论中的引理[1][2][3][4]。 该引理提到一般而言，对偶化和离散化不能交换。若配合伴随向量映射原理，才能交换这二个运算[5]。
引理的叙述 编辑
连续时间的最佳控制问题有丰富的资讯。针对特定问题，应用庞特里亚金最大化原理或哈密顿-雅可比-贝尔曼方程可以找到计多有趣的性质。这些定理其有用到其变化量相对时间的连续性[6]。若最佳控制问题离散化时，Ross–Fahroo引理指出在本质上就少了一些资讯。减少的资料可能是在边界点控制量的值[7][8]，也有可能是对偶变数在汉米尔顿量中的值。为了解决资讯减少问题，Ross和Fahroo引进了“闭合条件”（closure condition）的概念，让已知的减少资讯可以再加回去。这是透过伴随向量映射原理达到的[5]。
在拟谱最佳控制中的应用 编辑
当拟谱法用在离散最佳控制时，其中隐含的Ross–Fahroo引理在离散的伴随向量中，看起来似乎是将微分矩阵的转置加以离散化[1][2][3]。若应用了伴随向量映射原理，即为伴随矩阵的转换。此转换的应用产生了Ross–Fahroo拟谱法[9][10]。
相关条目 编辑
参考资料 编辑
- ^ 1.0 1.1 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.
- ^ 2.0 2.1 Ross, I. M.; Fahroo, F. Legendre Pseudospectral Approximations of Optimal Control Problems. Lecture Notes in Control and Information Sciences. 2003, 295.
- ^ 3.0 3.1 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.
- ^ N. Bedrossian, M. Karpenko, and S. Bhatt, "Overclock My Satellite: Sophisticated Algorithms Boost Satellite Performance on the Cheap", IEEE Spectrum, November 2012.
- ^ 5.0 5.1 Ross, I. M.; Karpenko, M. A Review of Pseudospectral Optimal Control: From Theory to Flight. Annual Reviews in Control. 2012, 36: 182–197 [2018-10-19]. doi:10.1016/j.arcontrol.2012.09.002. （原始内容存档于2015-09-24）.
- ^ B. S. Mordukhovich, Variational Analysis and Generalized Differentiation: Basic Theory, Vol.330 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Series, Springer, Berlin, 2005.
- ^ 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.
- ^ 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.
- ^ A. M. Hawkins, Constrained Trajectory Optimization of a Soft Lunar Landing From a Parking Orbit, S.M. Thesis, Dept. of Aeronautics and Astronautics, Massachusetts Institute of Technology, 2005. http://dspace.mit.edu/handle/1721.1/32431 （页面存档备份，存于互联网档案馆）
- ^ 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）