Legendre pseudospectral method

There are three basic types of Legendre pseudospectral methods:

1. One based on Gauss-Lobatto points
2. First proposed by Elnagar et al and subsequently extended by Fahroo and RossFahroo, F. and Ross, I. M., “Costate Estimation by a Legendre Pseudospectral Method,” Journal of Guidance, Control, and Dynamics, Vol.24, No.2, March–April 2001, pp.270-277. to incorporate the covector mapping theorem.
3. Forms the basis for solving general nonlinear finite-horizon optimal control problems.{{Cite book|title=A Primer on Pontryagin's Principle in Optimal Control|last=Ross|first=Isaac|publisher=Collegiate Publishers|year=2015|location=San Francisco}}
4. Incorporated in several software products
5. * DIDO, [https://web.archive.org/web/20161118023825/https://otis.grc.nasa.gov/ OTIS], [http://www.psopt.org/ PSOPT]
6. One based on Gauss-Radau points
7. First proposed by Fahroo and RossFahroo, F. and Ross, I. M., “Pseudospectral Methods for Infinite Horizon Nonlinear Optimal Control Problems,” AIAA Guidance, Navigation and Control Conference, August 15–18, 2005, San Francisco, CA and subsequently extended (by Fahroo and Ross) to incorporate a covector mapping theorem.Fahroo, F. and Ross, I. M., “Pseudospectral Methods for Infinite-Horizon Optimal Control Problems,” Journal of Guidance, Control, and Dynamics, Vol. 31, No. 4, pp. 927-936, 2008.
8. Forms the basis for solving general nonlinear infinite-horizon optimal control problems.
9. Forms the basis for solving general nonlinear finite-horizon problems with one free endpoint.
10. One based on Gauss points
11. First proposed by ReddienReddien, G.W., "Collocation at Gauss Points as a Discretization in Optimal Control," SIAM Journal on Control and Optimization, Vol. 17, No. 2, March 1979.
12. Forms the basis for solving finite-horizon problems with free endpointsFahroo F., and Ross, I. M., "Advances in Pseudospectral Methods for Optimal Control," AIAA Guidance, Navigation, and Control Conference, AIAA Paper 2008-7309, Honolulu, Hawaii, August 2008.
13. Incorporated in several software products
14. * GPOPS, PROPT

The first software to implement the Legendre pseudospectral method was DIDO in 2001.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 Subsequently, the method was incorporated in the NASA code OTIS.{{Cite web|url=https://otis.grc.nasa.gov/|title=[ OTIS ] Optimal Trajectories by Implicit Simulation|website=otis.grc.nasa.gov|access-date=2016-12-08|archive-url=https://web.archive.org/web/20161118023825/https://otis.grc.nasa.gov/|archive-date=2016-11-18|url-status=dead}} Years later, many other software products emerged at an increasing pace, such as PSOPT, PROPT and GPOPS.

The Legendre pseudospectral method (based on Gauss-Lobatto points) has been implemented in flight by NASA on several spacecraft through the use of the software, DIDO. The first flight implementation was on November 5, 2006, when NASA used DIDO to maneuver the International Space Station to perform the [https://web.archive.org/web/20081005205928/http://www.nasa.gov/mission_pages/station/science/experiments/ZPM.html Zero Propellant Maneuver]. The Zero Propellant Maneuver was discovered by Nazareth Bedrossian using DIDO. [https://www.youtube.com/watch?v=MIp27Ea9_2I Watch a video] of this historic maneuver.

• DIDO
• Chebyshev pseudospectral method
• Ross–Fahroo pseudospectral methods
• Ross–Fahroo lemma
• Covector mapping principle

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Category:Optimal control

Category:Numerical analysis

Category:Control theory