Direct trajectory optimization by a Chebyshev pseudospectral method

A Chebyshev pseudospectral method is presented in this paper for directly solving a generic optimal control problem with state and control constraints. This method employs Nth degree Lagrange polynomial approximations for the state and control variables with the values of these variables at the Chebyshev-Gauss-Lobatto (CGL) points as the expansion coefficients. This process yields a nonlinear programming problem (NLP) with the state and control values at the CGL points as unknown NLP parameters. Numerical examples demonstrate this method yields more accurate results than those obtained from the traditional collocation methods.

[1]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[2]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

[3]  Greg Knowles,et al.  An Introduction to Applied Optimal Control , 1982 .

[4]  S. Orszag,et al.  Theory and applications of spectral methods , 1984 .

[5]  C. Hargraves,et al.  DIRECT TRAJECTORY OPTIMIZATION USING NONLINEAR PROGRAMMING AND COLLOCATION , 1987 .

[6]  Jacques Vlassenbroeck,et al.  A chebyshev polynomial method for optimal control with state constraints , 1988, Autom..

[7]  R. V. Dooren,et al.  A Chebyshev technique for solving nonlinear optimal control problems , 1988 .

[8]  C. Canuto.  Spectral methods in fluid dynamics , 1991 .

[9]  Alex Solomonoff,et al.  A fast algorithm for spectral differentiation , 1992 .

[10]  O. V. Stryk,et al.  Numerical Solution of Optimal Control Problems by Direct Collocation , 1993 .

[11]  Gamal N. Elnagar,et al.  The pseudospectral Legendre method for discretizing optimal control problems , 1995, IEEE Trans. Autom. Control..

[12]  A. L. Herman,et al.  Direct optimization using collocation based on high-order Gauss-Lobatto quadrature rules , 1996 .

[13]  D. Hull.  Conversion of optimal control problems into parameter optimization problems , 1996 .

[14]  B. Conway,et al.  Collocation Versus Differential Inclusion in Direct Optimization , 1998 .

[15]  Gamal N. Elnagar,et al.  Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems , 1998, Comput. Optim. Appl..

[16]  I. Michael Ross,et al.  Costate Estimation by a Legendre Pseudospectral Method , 1998 .

[17]  J. Betts.  Survey of Numerical Methods for Trajectory Optimization , 1998 .

[18]  I. Michael Ross,et al.  A Spectral Patching Method for Direct Trajectory Optimization , 2000 .

[19]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .