Gauss pseudospectral method
The Gauss Pseudospectral Method (abbreviated "GPM") is a pseudospectral method for solving optimal control problems.
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 main idea behind the GPM is is based on the work of Elnagar et al[1] and Fahroo et al.[2][3][4]
The approach used in the GPM 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 covector) 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. Fahroo et al call this the costate mapping theorm. The costate mapping theorem that arises from the GPM has been described in several references including two MIT PhD theses [5][6] and journal articles that include the theory along with applications[7][8]
References and notes
- ^ 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
- ^ 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.
- ^ Fahroo, F. and Ross, I. M., “A Second Look at Approximating Differential Inclusions,” Journal of Guidance, Control and Dynamics, Vol.24, No.1, January-February 2001.
- ^ 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.
- ^ Benson, D.A., A Gauss Pseudospectral Transcription for Optimal Control, Ph.D. Thesis, Dept. of Aeronautics and Astronautics, MIT, November 2004,
- ^ 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
- ^ 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.,
- ^ 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.