Legendre pseudospectral method

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The Legendre pseudospectral method for optimal control problems is based on Legendre polynomials. It is part of the larger theory of pseudospectral optimal control, a term coined by Ross. [1] A basic version of the Legendre pseudospectral was originally proposed by Elnagar and his coworkers in 1995. [2] Since then, Ross, Fahroo and their coworkers [3] [4] [5] [6] [7] have extended, generalized and applied the method for a large range of problems. [8] An application that has received wide publicity [9] [10] is the use of their method for generating real time trajectories for the International Space Station.

Contents

Fundamentals

There are three basic types of Legendre pseudospectral methods: [1]

  1. One based on Gauss-Lobatto points
    1. First proposed by Elnagar et al [2] and subsequently extended by Fahroo and Ross [4] to incorporate the covector mapping theorem.
    2. Forms the basis for solving general nonlinear finite-horizon optimal control problems. [1] [11] [12]
    3. Incorporated in several software products
  2. One based on Gauss-Radau points
    1. First proposed by Fahroo and Ross [13] and subsequently extended (by Fahroo and Ross) to incorporate a covector mapping theorem. [5]
    2. Forms the basis for solving general nonlinear infinite-horizon optimal control problems. [1] [12]
    3. Forms the basis for solving general nonlinear finite-horizon problems with one free endpoint. [1] [11] [12]
  3. One based on Gauss points
    1. First proposed by Reddien [14]
    2. Forms the basis for solving finite-horizon problems with free endpoints [11] [12]
    3. Incorporated in several software products

Software

The first software to implement the Legendre pseudospectral method was DIDO in 2001. [12] [15] Subsequently, the method was incorporated in the NASA code OTIS. [16] Years later, many other software products emerged at an increasing pace, such as PSOPT, PROPT and GPOPS.

Flight implementations

The Legendre pseudospectral method (based on Gauss-Lobatto points) has been implemented in flight [1] 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 Zero Propellant Maneuver. The Zero Propellant Maneuver was discovered by Nazareth Bedrossian using DIDO. Watch a video of this historic maneuver.

See also

Related Research Articles

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DIDO is a software product for solving general-purpose optimal control problems. It is widely used in academia, industry, and NASA. Hailed as a breakthrough software, DIDO is based on the pseudospectral optimal control theory of Ross and Fahroo. The latest enhancements to DIDO are described in Ross.

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Zero-propellant maneuver

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Isaac Michael Ross is a Distinguished Professor and Program Director of Control and Optimization at the Naval Postgraduate School in Monterey, CA. He has published papers in pseudospectral optimal control theory, energy-sink theory, the optimization and deflection of near-Earth asteroids and comets, robotics, attitude dynamics and control, real-time optimal control unscented optimal control and a textbook on optimal control. The Kang-Ross-Gong theorem, Ross' π lemma, Ross' time constant, the Ross–Fahroo lemma, and the Ross–Fahroo pseudospectral method are all named after him.

Fariba Fahroo is an American Persian mathematician, a program manager at the Air Force Office of Scientific Research, and a former program manager at the Defense Sciences Office. Along with I. M. Ross, she has published papers in pseudospectral optimal control theory. The Ross–Fahroo lemma and the Ross–Fahroo pseudospectral method are named after her. In 2010, she received, the AIAA Mechanics and Control of Flight Award for fundamental contributions to flight mechanics.

The flat pseudospectral method is part of the family of the Ross–Fahroo pseudospectral methods introduced by Ross and Fahroo. The method combines the concept of differential flatness with pseudospectral optimal control to generate outputs in the so-called flat space.

A Carathéodory-π solution is a generalized solution to an ordinary differential equation. The concept is due to I. Michael Ross and named in honor of Constantin Carathéodory. Its practicality was demonstrated in 2008 by Ross et al. in a laboratory implementation of the concept. The concept is most useful for implementing feedback controls, particularly those generated by an application of Ross' pseudospectral optimal control theory.

GPOPS-II is a general-purpose MATLAB software for solving continuous optimal control problems using hp-adaptive Gaussian quadrature collocation and sparse nonlinear programming. The acronym GPOPS stands for "General Purpose OPtimal Control Software", and the Roman numeral "II" refers to the fact that GPOPS-II is the second software of its type.

References

  1. 1 2 3 4 5 6 Ross, I. M.; Karpenko, M. (2012). "A Review of Pseudospectral Optimal Control: From Theory to Flight". Annual Reviews in Control. 36 (2): 182–197. doi:10.1016/j.arcontrol.2012.09.002.
  2. 1 2 G. Elnagar, M. A. Kazemi, and M. Razzaghi, "The Pseudospectral Legendre Method for Discretizing Optimal Control Problems," IEEE Transactions on Automatic Control, 40:1793–1796, 1995.
  3. ↑ Ross, I. M. and Fahroo, F., “Legendre Pseudospectral Approximations of Optimal Control Problems,” Lecture Notes in Control and Information Sciences, Vol. 295, Springer-Verlag, New York, 2003, pp 327-342
  4. 1 2 Fahroo, 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.
  5. 1 2 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.
  6. ↑ Kang, W.; Gong, Q.; Ross, I. M.; Fahroo, F. "On the Convergence of Nonlinear Optimal Control Using Pseudospectral Methods for Feedback Linearizable Systems". International Journal of Robust and Nonlinear Control. 17 (1251–1277): 2007.
  7. ↑ Ross, I. M.; Fahroo, F. (2004). "Pseudospectral Knotting Methods for Solving Nonsmooth Optimal Control Problems". Journal of Guidance Control and Dynamics. 27 (397–405): 2004. Bibcode:2004JGCD...27..397R. doi:10.2514/1.3426.
  8. ↑ 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.
  9. ↑ Kang, W.; Bedrossian, N. "Pseudospectral Optimal Control Theory Makes Debut Flight, Saves NASA $1M in Under Three Hours". SIAM News. 40: 2007.
  10. ↑ Bedrossian, N. S., Bhatt, S., Kang, W. and Ross, I. M., “Zero-Propellant Maneuver Guidance,” IEEE Control Systems Magazine, Vol.29, No.5, October 2009, pp 53-73; Cover Story.
  11. 1 2 3 Fahroo 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.
  12. 1 2 3 4 5 Ross, Isaac (2015). A Primer on Pontryagin's Principle in Optimal Control. San Francisco: Collegiate Publishers.
  13. ↑ Fahroo, 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
  14. ↑ Reddien, G.W., "Collocation at Gauss Points as a Discretization in Optimal Control," SIAM Journal on Control and Optimization, Vol. 17, No. 2, March 1979.
  15. ↑ 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
  16. ↑ "[ OTIS ] Optimal Trajectories by Implicit Simulation". otis.grc.nasa.gov. Archived from the original on 2016-11-18. Retrieved 2016-12-08.
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