查看“︁擬譜最佳控制”︁的源代码

{{Technical|date=2023年9月}}
'''擬譜最佳控制'''（Pseudospectral optimal control）是一種求解[[最优控制]]問題的方式<ref name="ReviewPOC">{{cite journal |doi=10.1016/j.arcontrol.2012.09.002 |title=A review of pseudospectral optimal control: From theory to flight |journal=Annual Reviews in Control |volume=36 |issue=2 |pages=182–97 |year=2012 |last1=Ross |first1=I. Michael |last2=Karpenko |first2=Mark }}</ref><ref name="RossAcademy">{{cite journal |doi=10.1196/annals.1370.015 |pmid=16510411 |title=A Roadmap for Optimal Control: The Right Way to Commute |journal=Annals of the New York Academy of Sciences |volume=1065 |pages=210–31 |year=2005 |last1=Ross |first1=I M. |bibcode=2005NYASA1065..210R }}</ref><ref name="advances">{{cite book |doi=10.2514/6.2008-7309 |chapter=Advances in Pseudospectral Methods for Optimal Control |title=AIAA Guidance, Navigation and Control Conference and Exhibit |year=2008 |last1=Fahroo |first1=Fariba |last2=Ross |first2=I. Michael |isbn=978-1-60086-999-0 |pages=18–21 }}</ref><ref name="unified-ross">{{cite book |doi=10.1109/CDC.2003.1272946 |chapter=A unified computational framework for real-time optimal control |title=42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475) |volume=3 |pages=2210–5 |year=2003 |last1=Ross |first1=I.M. |last2=Fahroo |first2=F. |isbn=0-7803-7924-1 }}</ref>，結合了[[擬譜法]]的數值方法以及[[最优控制]]的理論。擬譜最佳控制已用在軍事及工業應用的飛行系統中<ref name="ReviewPOC"/><ref name="GKBFSB">{{cite book |doi=10.1109/CDC.2007.4435052 |chapter=Pseudospectral Optimal Control for Military and Industrial Applications |title=2007 46th IEEE Conference on Decision and Control |pages=4128–42 |year=2007 |last1=Qi Gong |last2=Wei Kang |last3=Bedrossian |first3=Nazareth S. |last4=Fahroo |first4=Fariba |last5=Pooya Sekhavat |last6=Bollino |first6=Kevin |isbn=978-1-4244-1497-0 }}</ref>，此技術也廣泛的用在飛彈導引、機械手臂控制、振動阻尼等問題中<ref name="GKBFSB" /><ref name ="unified-li">{{cite journal |doi=10.1073/pnas.1009797108 |pmid=21245345 |pmc=3033291 |jstor=41001785 |title=Optimal pulse design in quantum control: A unified computational method |journal=Proceedings of the National Academy of Sciences |volume=108 |issue=5 |pages=1879–84 |year=2011 |last1=Li |first1=Jr-Shin |last2=Ruths |first2=Justin |last3=Yu |first3=Tsyr-Yan |last4=Arthanari |first4=Haribabu |last5=Wagner |first5=Gerhard |bibcode=2011PNAS..108.1879L }}</ref>。

==簡介==
最佳控制的求解法中，有許多的方法都屬於擬譜最佳控制的範圍，例如[[勒壤得擬譜法]]、[[切比雪夫擬譜法]]、{{link-en|高斯擬譜法|Gauss pseudospectral method}}、[[Ross–Fahroo擬譜法]]、[[貝爾曼擬譜法]]及[[平坦擬譜法]]等<ref name="ReviewPOC"/><ref name="advances"/>。

要求解最佳控制問題，需要將三種數學工具進行近似：成本函數的積分、控制系統的微分方程、以及狀態控制的限制條件<ref name="advances"/>，理想的近似要在上述三種數學工具上可以有效率的近似。有些工具可以適用於其中的一種（例如高效的ODE求解器），但無法適用於其他二種數學工具上，而擬譜法可以適用於這三種數學工具的近似，適合應用在最佳控制問題上<ref name="GKS"/><ref name="HGG">{{cite book |first1=J. S. |last1=Hesthaven |first2=S. |last2=Gottlieb |first3=D. |last3=Gottlieb |title=Spectral methods for time-dependent problems |publisher=Cambridge University Press |year=2007 |isbn=978-0-521-79211-0 }}{{page needed|date=January 2017}}</ref><ref name="GRKF">{{cite journal |doi=10.1007/s10589-007-9102-4 |title=Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control |journal=Computational Optimization and Applications |volume=41 |issue=3 |pages=307–35 |year=2007 |last1=Gong |first1=Qi |last2=Ross |first2=I. Michael |last3=Kang |first3=Wei |last4=Fahroo |first4=Fariba }}</ref>。使用擬譜法時，連續函數會用適當選擇的[[高斯求积|分割格點]]來近似。分割格點會由近似用的對應正交多項式基底函數來決定。在擬譜最佳控制中，常用[[勒让德多项式]]及[[切比雪夫多项式]]。在數學上，利用分割格點可以只用幾個點達到高精度。例如在Legendre–Gauss–Lobatto格點下，針對光滑函數(C<sup><math>\infty</math></sup>)的[[拉格朗日插值法]]可以以譜率（spectral rate）為L<sup>2</sup>的方式收斂，比任何多項式的收斂速率都快<ref name="HGG"/>。

==細節==
最佳控制的基本擬譜法是以[[伴随向量映射原理]]為基礎<ref name="RossAcademy" />，其他的技巧，例如[[貝爾曼擬譜法]]，是用初始時間的網格密集（node-clustering）來進行最佳控制。網格密集會出現在所有的高斯點上<ref name="GKS">{{cite journal |doi=10.1109/TAC.2006.878570 |title=A Pseudospectral Method for the Optimal Control of Constrained Feedback Linearizable Systems |journal=IEEE Transactions on Automatic Control |volume=51 |issue=7 |pages=1115–29 |year=2006 |last1=Gong |first1=Q. |last2=Kang |first2=W. |last3=Ross |first3=I.M. }}</ref><ref name="Elnagar1">{{cite journal |doi=10.1109/9.467672 |title=The pseudospectral Legendre method for discretizing optimal control problems |journal=IEEE Transactions on Automatic Control |volume=40 |issue=10 |pages=1793–6 |year=1995 |last1=Elnagar |first1=G. |last2=Kazemi |first2=M.A. |last3=Razzaghi |first3=M. }}</ref><ref>{{cite journal |doi=10.2514/2.4709 |title=Costate Estimation by a Legendre Pseudospectral Method |journal=Journal of Guidance, Control, and Dynamics |volume=24 |issue=2 |pages=270–7 |year=2001 |last1=Fahroo |first1=Fariba |last2=Ross |first2=I. Michael |bibcode=2001JGCD...24..270F }}</ref>。

而且，擬譜最佳控制的結構會考慮使運算高效進行的方式，例如ad-hoc縮放<ref name="Sagliano2">{{cite journal |doi=10.1016/j.orl.2014.03.003 |title=Performance analysis of linear and nonlinear techniques for automatic scaling of discretized control problems |journal=Operations Research Letters |volume=42 |issue=3 |pages=213–6 |year=2014 |last1=Sagliano |first1=Marco }}</ref>及雅可比計算法，例如已有研究者將[[二元数]]理論<ref name="Sagliano3">{{cite book |doi=10.2514/6.2016-0867 |chapter=Exact Hybrid Jacobian Computation for Optimal Trajectories via Dual Number Theory |title=AIAA Guidance, Navigation, and Control Conference |year=2016 |last1=d'Onofrio |first1=Vincenzo |last2=Sagliano |first2=Marco |last3=Arslantas |first3=Yunus E. |isbn=978-1-62410-389-6 }}</ref>用在擬譜最佳控制上<ref name="Sagliano">{{cite book |doi=10.2514/6.2013-4554 |chapter=Hybrid Jacobian Computation for Fast Optimal Trajectories Generation |title=AIAA Guidance, Navigation, and Control (GNC) Conference |year=2013 |last1=Sagliano |first1=Marco |last2=Theil |first2=Stephan |isbn=978-1-62410-224-0 }}</ref>。

在擬譜最佳控制中，積分會用分割的方式來近似，設法得到最理想的[[數值積分]]結果。例如，若只有N個格點，Legendre-Gauss分割積分在<math>2N-1</math>次或以下的多項式，都可以有完全精確的結果。在最佳控制問題中，用擬譜法離散微分方程時，會用簡單而高精度的微分矩陣來計算導數。因為擬譜法強迫系統在選定的格點上計算結果，狀態控制的限制條件也會直接離散化，這些數學上的優點使擬譜法成為求解連續最佳控制問題的直接離散化工具。

==相關條目==
*[[貝爾曼擬譜法]]
*[[切比雪夫擬譜法]]
*[[伴隨向量映射原理]]
*[[平坦擬譜法]]
*{{link-en|高斯擬譜法|Gauss pseudospectral method}}
*[[勒壤得擬譜法]]
*[[擬譜knotting法]]
*[[Ross–Fahroo引理]]
*[[Ross–Fahroo擬譜法]]
*[[羅斯π引理]]

==參考資料==
{{Reflist}}

==外部連結==
* [http://computer.howstuffworks.com/dido.htm How Stuff Works] {{Wayback|url=http://computer.howstuffworks.com/dido.htm |date=20120901001122 }}
* [https://www.youtube.com/watch?v=faQeCI1IgoQ Pseudospectral optimal control: Part 1] {{Wayback|url=https://www.youtube.com/watch?v=faQeCI1IgoQ |date=20200505093536 }}
* [https://www.youtube.com/watch?v=jRmJwQI_JZw Pseudospectral optimal control: Part 2] {{Wayback|url=https://www.youtube.com/watch?v=jRmJwQI_JZw |date=20200503025858 }}

==軟體==
* {{link-en|DIDO (最佳控制)|DIDO (optimal control)|Dido}}：[http://www.mathworks.com/products/connections/product_detail/product_61633.html DIDO - MATLAB tool for optimal control] {{Wayback|url=http://www.mathworks.com/products/connections/product_detail/product_61633.html |date=20170330083720 }}，得名自Carthage皇后[[狄多]]
* {{link-en|GPOPS-II|GPOPS-II}}：[http://www.gpops2.com General Purpose Optimal Control Software] {{Wayback|url=http://www.gpops2.com/ |date=20200224151605 }}
* [https://web.archive.org/web/20101031195713/http://www.astos.de/products/gesop GESOP – Graphical Environment for Simulation and OPtimization]
* [https://openocl.org/ OpenOCL – Open Optimal Control Library] {{Wayback|url=https://openocl.org/ |date=20190420200752 }}
* [http://tomdyn.com/ PROPT – MATLAB Optimal Control Software] {{Wayback|url=http://tomdyn.com/ |date=20190727174021 }}
* [https://sites.google.com/a/psopt.org/psopt/ PSOPT – Open Source Pseudospectral Optimal Control Solver in C++] {{Wayback|url=https://sites.google.com/a/psopt.org/psopt/ |date=20160412133604 }}
* {{link-en|SPARTAN|SPARTAN}}：[http://elib.dlr.de/112107 Simple Pseudospectral Algorithm for Rapid Trajectory ANalysis ] {{Wayback|url=http://elib.dlr.de/112107 |date=20190722223459 }}
* [https://github.com/istellartech/OpenGoddard OpenGoddard - Python Open Source Pseudospectral Optimal Control Software] {{Wayback|url=https://github.com/istellartech/OpenGoddard |date=20181222060934 }}

{{DEFAULTSORT:Pseudospectral Optimal Control}}
[[Category:最佳控制]]