Gauss pseudospectral method

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The Gauss Pseudospectral Method (abbreviated "GPM") is one of the variations of pseudospectral methods that was initially introduced by [1] and structured by [2] for solving optimal control problems.

[edit] Description

The method is based on the theory of orthogonal collocation where the collocation points (i.e., the points at which the optimal control problem is discretized) are the Legendre-Gauss (LG) points. The approach used in the GPM is to use a Lagrange polynomial approximation for the state that includes coefficients for the initial state plus the values of the state at the N LG points. In a somewhat opposite manner, the approximation for the costate (adjoint) is performed using a basis of Lagrange polynomials that includes the final value of the costate plus the costate at the N LG points. These two approximations together lead to the ability to map the KKT multipliers of the nonlinear program (NLP) to the costates of the optimal control problem at the N LG points PLUS the boundary points. The costate mapping theorem that arises from the GPM has been described in several references including two MIT PhD theses[3][4] and journal articles that include the theory along with applications[5][6]

[edit] References and notes

  1. ^ Elnagar, J., Kazemi, M. A. and Razzaghi, M., The Pseudospectral Legendre Method for Discretizing Optimal Control Problems, IEEE Transactions on Automatic Control, Vol. 40, No. 10, 1995, pp. 1793-1796
  2. ^ 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
  3. ^ Benson, D.A., A Gauss Pseudospectral Transcription for Optimal Control, Ph.D. Thesis, Dept. of Aeronautics and Astronautics, MIT, November 2004,
  4. ^ Huntington, G.T., Advancement and Analysis of a Gauss Pseudospectral Transcription for Optimal Control, Ph.D. Thesis, Dept. of Aeronautics and Astronautics, MIT, May 2007
  5. ^ Benson, D.A., Huntington, G.T., Thorvaldsen, T.P., and Rao, A.V., "Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method", Journal of Guidance, Control, and Dynamics. Vol. 29, No. 6, November-December 2006, pp. 1435-1440.,
  6. ^ Huntington, G.T., Benson, D.A., and Rao, A.V., "Optimal Configuration of Tetrahedral Spacecraft Formations", The Journal of The Astronautical Sciences. Vol. 55, No. 2, March-April 2007, pp. 141-169.