Legendre Approximation for Solving a Class of Nonlinear Optimal Control Problems

ABSTRACT

This paper introduces a numerical technique for solving a class of optimal control problems containing nonlinear dynamical system and functional of state variables. This numerical method consists of two major parts. In the first part, using linear combination property of intervals, we convert the nonlinear dynamical system into an equivalent linear system. And in the second part, which we are dealing with a linear dynamical system, using Legendre expansions for approximating both the state and associated control together with discretizing the constraints over the Chebyshev-Gauss-Lobatto points, the optimal control problem is transformed into a corresponding NLP problem which is diretly solved. The proposed idea is illustrated by several numerical examples.

This paper introduces a numerical technique for solving a class of optimal control problems containing nonlinear dynamical system and functional of state variables. This numerical method consists of two major parts. In the first part, using linear combination property of intervals, we convert the nonlinear dynamical system into an equivalent linear system. And in the second part, which we are dealing with a linear dynamical system, using Legendre expansions for approximating both the state and associated control together with discretizing the constraints over the Chebyshev-Gauss-Lobatto points, the optimal control problem is transformed into a corresponding NLP problem which is diretly solved. The proposed idea is illustrated by several numerical examples.

KEYWORDS

Optimal Control, Legendre Polynomials, Linear Combination Property of Intervals, Chebyshev-Gauss-Lobatto Points, Nonlinear Programming

Optimal Control, Legendre Polynomials, Linear Combination Property of Intervals, Chebyshev-Gauss-Lobatto Points, Nonlinear Programming

Cite this paper

nullE. Tohidi, O. Samadi and M. Farahi, "Legendre Approximation for Solving a Class of Nonlinear Optimal Control Problems,"*Journal of Mathematical Finance*, Vol. 1 No. 1, 2011, pp. 8-13. doi: 10.4236/jmf.2011.11002.

nullE. Tohidi, O. Samadi and M. Farahi, "Legendre Approximation for Solving a Class of Nonlinear Optimal Control Problems,"

References

[1] R. R. Bless, D. H. Hoges and H. Seywald, “Finite Element Method for the Solution of State-Constrained Optimal Control Problems,” Journal of Guidance, Control, and Dynamics, Vol. 18, No. 5, 1995, pp. 1036-1043. doi:10.2514/3.21502

[2] R. Bulrisch and D. Kraft, “Compu-tational Optimal Control,” Birkhauser, Boston, 1994.

[3] G. Elnagar, M. A. Kazemi and M. Razzaghi, “The Pseudospectral Legendre Method for Discretizing Optimal Control Problems,” IEEE Transactions on Automatic Con- trol, Vol. 40, No. 10, 1995, pp. 1793-1796. doi:10.1109/9.467672

[4] G. N. Elnagar and M. A. Kazemi, “Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems,” Computational Optimization and Applications, Vol. 11, No. 2, 1998, pp. 195-217. doi:10.1109/9.467672

[5] W. W. Hager, “Multiplier Methods for Nonlinear Optimal Control Problems,” SIAM Journal on Numerical Anal, Vol. 27, No. 4, 1990, pp. 1061-1080. doi:10.1137/0727063

[6] A. L. Herman and B. A. Conway, “Direct Trajectory Optimization Using Collocation Based on High-Order Gauss-Lobatoo Quadrature Rules,” Journal of Guidance, Control, and Dynamics, and Dynamics, Vol. 19, No. 3, 1996, pp. 592-599. doi:10.2514/3.21662

[7] M. Ross and F. Fahroo, “Legendre Pseudospectral Approximations of Optimal Control Problems,” Lecture Notes in Control and Information Sciences, Vol. 295, No. 1, 2003, pp. 327-342.

[8] J. Vlassen-broeck and R. V. Dooren, “A Chebyshev Technique for Solving Nonlinear Optimal Control Problems,” IEEE Transactions on Automatic Control, Vol. 33, No. 4, 1988, pp. 333-340. 10.1109/9.192187

[9] H. Seywald and R. R. Kumar, “Finite Difference Scheme for Automatic Costate Calculation,” Journal of Guidance, Control, and Dynamics, and Dynamics, Vol. 19, No. 1, 1996, pp. 231-239. doi:10.2514/3.21603

[10] K. Schittkowskki, “NLPQL: A Fortran Subroutine for Solving Constrained Nonlinear Programming Problems,” Operations Research Annals, Vol. 5, No. 2, 1985, pp. 385-400.

[11] M. Razzaghi and G. Elnagar, “A Legendre Technique For Solving Time-Varing Linear Quadratic Optimal Control Problems,” Journal of the Franklin Institute, Vol. 330, No. 3, 1993, pp. 453-463. doi:10.1016/0016-0032(93)90092-9

[12] O. V. Stryk, “Nu-merical Solution of Optimal Control Problems by Direct Col-location,” In: R. Bulrisch, A. Miele, J. Stoer and K. H. Well, Eds., Optional Control of Variations, Optimal Control Theory and Numerical Methods, International Series of Numerical Mathematics, Birkh?user Verlag, Basel, 1993, pp. 129-143.

[1] R. R. Bless, D. H. Hoges and H. Seywald, “Finite Element Method for the Solution of State-Constrained Optimal Control Problems,” Journal of Guidance, Control, and Dynamics, Vol. 18, No. 5, 1995, pp. 1036-1043. doi:10.2514/3.21502

[2] R. Bulrisch and D. Kraft, “Compu-tational Optimal Control,” Birkhauser, Boston, 1994.

[3] G. Elnagar, M. A. Kazemi and M. Razzaghi, “The Pseudospectral Legendre Method for Discretizing Optimal Control Problems,” IEEE Transactions on Automatic Con- trol, Vol. 40, No. 10, 1995, pp. 1793-1796. doi:10.1109/9.467672

[4] G. N. Elnagar and M. A. Kazemi, “Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems,” Computational Optimization and Applications, Vol. 11, No. 2, 1998, pp. 195-217. doi:10.1109/9.467672

[5] W. W. Hager, “Multiplier Methods for Nonlinear Optimal Control Problems,” SIAM Journal on Numerical Anal, Vol. 27, No. 4, 1990, pp. 1061-1080. doi:10.1137/0727063

[6] A. L. Herman and B. A. Conway, “Direct Trajectory Optimization Using Collocation Based on High-Order Gauss-Lobatoo Quadrature Rules,” Journal of Guidance, Control, and Dynamics, and Dynamics, Vol. 19, No. 3, 1996, pp. 592-599. doi:10.2514/3.21662

[7] M. Ross and F. Fahroo, “Legendre Pseudospectral Approximations of Optimal Control Problems,” Lecture Notes in Control and Information Sciences, Vol. 295, No. 1, 2003, pp. 327-342.

[8] J. Vlassen-broeck and R. V. Dooren, “A Chebyshev Technique for Solving Nonlinear Optimal Control Problems,” IEEE Transactions on Automatic Control, Vol. 33, No. 4, 1988, pp. 333-340. 10.1109/9.192187

[9] H. Seywald and R. R. Kumar, “Finite Difference Scheme for Automatic Costate Calculation,” Journal of Guidance, Control, and Dynamics, and Dynamics, Vol. 19, No. 1, 1996, pp. 231-239. doi:10.2514/3.21603

[10] K. Schittkowskki, “NLPQL: A Fortran Subroutine for Solving Constrained Nonlinear Programming Problems,” Operations Research Annals, Vol. 5, No. 2, 1985, pp. 385-400.

[11] M. Razzaghi and G. Elnagar, “A Legendre Technique For Solving Time-Varing Linear Quadratic Optimal Control Problems,” Journal of the Franklin Institute, Vol. 330, No. 3, 1993, pp. 453-463. doi:10.1016/0016-0032(93)90092-9

[12] O. V. Stryk, “Nu-merical Solution of Optimal Control Problems by Direct Col-location,” In: R. Bulrisch, A. Miele, J. Stoer and K. H. Well, Eds., Optional Control of Variations, Optimal Control Theory and Numerical Methods, International Series of Numerical Mathematics, Birkh?user Verlag, Basel, 1993, pp. 129-143.