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词条 Pseudospectral optimal control
释义

  1. Overview

  2. Details

  3. See also

  4. References

  5. External links

  6. Software

{{multiple issues|{{confusing|date=July 2013}}{{overly detailed|date=July 2013}}{{Technical|date=February 2011}}
}}{{Use dmy dates|date=May 2013}}Pseudospectral optimal control is a joint theoretical-computational method for solving optimal control problems.[1][2][3][4] It combines pseudospectral (PS) theory with optimal control theory to produce PS optimal control theory. PS optimal control theory has been used in ground and flight systems[1] in military and industrial applications.[5] The techniques have been extensively used to solve a wide range of problems such as those arising in UAV trajectory generation, missile guidance, control of robotic arms, vibration damping, lunar guidance, magnetic control, swing-up and stabilization of an inverted pendulum, orbit transfers, tether libration control, ascent guidance and quantum control.[5][6]

Overview

There are a very large number of ideas that fall under the general banner of pseudospectral optimal control. Examples of these are the Legendre pseudospectral method, the Chebyshev pseudospectral method, the Gauss pseudospectral method, the Ross-Fahroo pseudospectral method, the Bellman pseudospectral method, the flat pseudospectral method and many others.[1][3] Solving an optimal control problem requires the approximation of three types of mathematical objects: the integration in the cost function, the differential equation of the control system, and the state-control constraints.[3] An ideal approximation method should be efficient for all three approximation tasks. A method that is efficient for one of them, for instance an efficient ODE solver, may not be an efficient method for the other two objects. These requirements make PS methods ideal because they are efficient for the approximation of all three mathematical objects.[12][7][8] In a pseudospectral method, the continuous functions are approximated at a set of carefully selected quadrature nodes. The quadrature nodes are determined by the corresponding orthogonal polynomial basis used for the approximation. In PS optimal control, Legendre and Chebyshev polynomials are commonly used. Mathematically, quadrature nodes are able to achieve high accuracy with a small number of points. For instance, the interpolating polynomial of any smooth function (C) at Legendre–Gauss–Lobatto nodes converges in L2 sense at the so-called spectral rate, faster than any polynomial rate.[7]

Details

A basic pseudospectral method for optimal control is based on the covector mapping principle.[2] Other pseudospectral optimal control techniques, such as the Bellman pseudospectral method, rely on node-clustering at the initial time to produce optimal controls. The node clusterings occur at all Gaussian points.[9][10][11][12][13][14][15][16][17][18][19]

Moreover, their structure can be highly exploited to make them more computationally efficient, as ad-hoc scaling[20] and Jacobian computation methods, involving dual number theory[21] have been developed.[18]

In pseudospectral methods, integration is approximated by quadrature rules, which provide the best numerical integration result. For example, with just N nodes, a Legendre-Gauss quadrature integration achieves zero error for any polynomial integrand of degree less than or equal to . In the PS discretization of the ODE involved in optimal control problems, a simple but highly accurate differentiation matrix is used for the derivatives. Because a PS method enforces the system at the selected nodes, the state-control constraints can be discretized straightforwardly. All these mathematical advantages make pseudospectral methods a straightforward discretization tool for continuous optimal control problems.{{citation needed|date=January 2017}}

See also

  • Bellman pseudospectral method
  • Chebyshev pseudospectral method
  • Covector mapping principle
  • Flat pseudospectral methods
  • Gauss pseudospectral method
  • Legendre pseudospectral method
  • Pseudospectral knotting method
  • Ross–Fahroo lemma
  • Ross–Fahroo pseudospectral methods
  • Ross' π lemma

References

1. ^1 2 {{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 }}
2. ^1 {{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 }}
3. ^1 2 {{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 }}
4. ^{{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 }}
5. ^1 {{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 }}
6. ^{{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 }}
7. ^1 {{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}}
8. ^{{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 }}
9. ^1 {{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. }}
10. ^{{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. }}
11. ^{{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 }}
12. ^{{cite journal |doi=10.2514/1.32908 |title=Spectral Algorithm for Pseudospectral Methods in Optimal Control |journal=Journal of Guidance, Control, and Dynamics |volume=31 |issue=3 |pages=460–71 |year=2008 |last1=Gong |first1=Qi |last2=Fahroo |first2=Fariba |last3=Ross |first3=I. Michael |bibcode=2008JGCD...31..460G }}
13. ^{{cite journal |doi=10.1023/A:1018694111831 |title=Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems |year=1998 |last1=Elnagar |first1=Gamal N. |journal=Computational Optimization and Applications |volume=11 |issue=2 |pages=195–217 |last2=Kazemi |first2=Mohammad A. }}
14. ^{{cite journal |doi=10.2514/2.4862 |title=Direct Trajectory Optimization by a Chebyshev Pseudospectral Method |journal=Journal of Guidance, Control, and Dynamics |volume=25 |pages=160–6 |year=2002 |last1=Fahroo |first1=Fariba |last2=Ross |first2=I. Michael |bibcode=2002JGCD...25..160F }}
15. ^{{cite journal |doi=10.2514/1.20478 |title=Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method |journal=Journal of Guidance, Control, and Dynamics |volume=29 |issue=6 |pages=1435–40 |year=2006 |last1=Benson |first1=David A. |last2=Huntington |first2=Geoffrey T. |last3=Thorvaldsen |first3=Tom P. |last4=Rao |first4=Anil V. |bibcode=2006JGCD...29.1435B }}
16. ^{{cite journal |doi=10.1145/1731022.1731032 |title=Algorithm 902: GPOPS, A MATLAB software for solving multiple-phase optimal control problems using the gauss pseudospectral method |journal=ACM Transactions on Mathematical Software |volume=37 |issue=2 |year=2010 |last1=Rao |first1=Anil V. |last2=Benson |first2=David A. |last3=Darby |first3=Christopher |last4=Patterson |first4=Michael A. |last5=Francolin |first5=Camila |last6=Sanders |first6=Ilyssa |last7=Huntington |first7=Geoffrey T. }}
17. ^{{cite journal |doi=10.1007/s10589-009-9291-0 |title=Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method |journal=Computational Optimization and Applications |volume=49 |issue=2 |pages=335–58 |year=2009 |last1=Garg |first1=Divya |last2=Patterson |first2=Michael A. |last3=Francolin |first3=Camila |last4=Darby |first4=Christopher L. |last5=Huntington |first5=Geoffrey T. |last6=Hager |first6=William W. |last7=Rao |first7=Anil V. }}
18. ^1 {{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 }}
19. ^{{cite conference |last1=Huneker |first1=Laurens |last2=Sagliano |first2=Marco |last3=Arslantas |first3=Yunus |title=SPARTAN: An Improved Global Pseudospectral Algorithm for High-fidelity Entry-Descent-Landing Guidance Analysis |conference=The 30th International Symposium on Space Science and Technology |location=Kobe, Japan |year=2015 |url=http://elib.dlr.de/97510/1/SPARTAN_An%20Improved%20Global%20Pseudospectral%20Algorithm_for_High_EDL_Guidance_Analysis.pdf }}
20. ^{{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 }}
21. ^{{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 }}

External links

  • How Stuff Works
  • [https://www.youtube.com/watch?v=faQeCI1IgoQ Pseudospectral optimal control: Part 1]
  • [https://www.youtube.com/watch?v=jRmJwQI_JZw Pseudospectral optimal control: Part 2]

Software

  • DIDO - MATLAB tool for optimal control named after Dido, the first queen of Carthage.
  • GPOPS-II: General Purpose Optimal Control Software
  • GESOP – Graphical Environment for Simulation and OPtimization
{{Use dmy dates|date=September 2010}}
  • PROPT – MATLAB Optimal Control Software
  • [https://sites.google.com/a/psopt.org/psopt/ PSOPT – Open Source Pseudospectral Optimal Control Solver in C++]
  • SPARTAN: Simple Pseudospectral Algorithm for Rapid Trajectory ANalysis
  • [https://github.com/istellartech/OpenGoddard OpenGoddard - Python Open Source Pseudospectral Optimal Control Software]
{{DEFAULTSORT:Pseudospectral Optimal Control}}

1 : Optimal control
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