# Pseudospectral optimal control

According to Ross et al.,[1][2][3][4] pseudospectral optimal control is a joint theoretical-computational method for solving optimal control problems. It combines pseudospectral (PS) theory with optimal control theory to produce PS optimal control theory. PS optimal control theory has been used in ground and flight systems[1] in military and industrial applications.[5] The techniques have been extensively used to solve a wide range of problems such as those arising in UAV trajectory generation, missile guidance, control of robotic arms, vibration damping, lunar guidance, magnetic control, swing-up and stabilization of an inverted pendulum, orbit transfers, tether libration control, ascent guidance and quantum control.[5][6]

## Overview

There are a very large number of ideas that fall under the general banner of pseudospectral optimal control. Examples of these are the Legendre pseudospectral method, the Chebyshev pseudospectral method, the Gauss pseudospectral method, the Ross-Fahroo pseudospectral method, the flat pseudospectral method and many others.[1][3] Solving an optimal control problem requires the approximation of three types of mathematical objects: the integration in the cost function, the differential equation of the control system, and the state-control constraints.[3] An ideal approximation method should be efficient for all three approximation tasks. A method that is efficient for one of them, for instance an efficient ODE solver, may not be an efficient method for the other two objects. These requirements make PS methods ideal because they are efficient for the approximation of all three mathematical objects.[7][8][9] In a pseudospectral method, the continuous functions are approximated at a set of carefully selected quadrature nodes. The quadrature nodes are determined by the corresponding orthogonal polynomial basis used for the approximation. In PS optimal control, Legendre and Chebyshev polynomials are commonly used. Mathematically, quadrature nodes are able to achieve high accuracy with a small number of points. For instance, the interpolating polynomial of any smooth function (C$\infty$) at Legendre–Gauss–Lobatto nodes converges in L2 sense at the so-called spectral rate, faster than any polynomial rate.[8]

## Details

A basic pseudospectral method for optimal control is based on the covector mapping principle.[2] Other pseudospectral optimal control techniques, such as the Bellman pseudospectral method, rely on node-clustering at the initial time to produce optimal controls. The node clusterings occur at all Gaussian points.[7][10][11][12][13][14][15][16][17]

In pseudospectral methods, integration is approximated by quadrature rules, which provide the best numerical integration result. For example, with just N nodes, a Legendre-Gauss quadrature integration achieves zero error for any polynomial integrand of degree less than or equal to $2N-1$. In the PS discretization of the ODE involved in optimal control problems, a simple but highly accurate differentiation matrix is used for the derivatives. Because a PS method enforces the system at the selected nodes, the state-control constraints can be discretized straightforwardly. All these mathematical advantages make pseudospectral methods a straightforward discretization tool for continuous optimal control problems.

## References

1. ^ a b c I. M. Ross and M. Karpenko, "A Review of Pseudospectral Optimal Control: From Theory to Flight," Annual Reviews in Control, Vol. 36, pp. 182-197, 2012. [1]
2. ^ a b I. M. Ross, "A Roadmap for Optimal Control: The Right Way to Commute," Annals of the New York Academy of Sciences, Vol. 1065, pp. 210–231, January 2006.
3. ^ a b c F. Fahroo and I. M. Ross, "Advances in Pseudospectral Methods for Optimal Control," Proceedings of the AIAA Guidance, Navigation and Control Conference, AIAA 2008-7309. [2]
4. ^ I. M. Ross and F. Fahroo, "A Unified Computational Framework for Real-Time Optimal Control," Proceedings of the 42nd IEEE Conference on Decision and Control, Vol.3, 2003, pp.2210-2215.[3]
5. ^ a b 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.
6. ^ Jr-S Li, J. Ruths, T-Y Yu, H. Arthanari and G. Wagner, "Optimal Pulse Design in Quantum Control: A Unified Computational Method," Proceedings of the National Academy of Sciences, Vol.108, No.5, Feb 2011, pp.1879-1884. http://www.pnas.org/content/108/5/1879.full
7. ^ a b Q. Gong, W. Kang and I. M. Ross, A Pseudospectral Method for The Optimal Control of Constrained Feedback Linearizable Systems, IEEE Trans. Auto. Cont., Vol.~51, No.~7, July 2006, pp.~1115–1129.
8. ^ a b J. S. Hesthaven, S. Gottlieb and D. Gottlieb, Spectral methods for time-dependent problems, Cambridge University Press, 2007. ISBN 978-0-521-79211-0
9. ^ Q. Gong, I. M. Ross, W. Kang and Fahroo, F., Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control, Computational Optimization and Applications, Springer Netherlands, published online: 31 October 2007, to appear in Journal, 2008.
10. ^ 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
11. ^ F. Fahroo and I. M. Ross, Costate Estimation by a Legendre Pseudospectral Method, Journal of Guidance, Control and Dynamics, Vol.24, No.2, March–April 2001, pp.270–277.
12. ^ Gong, Q., Fahroo, F., and Ross, I.M., "Spectral Algorithm for Pseudospectral Methods in Optimal Control," "Journal of Guidance, Control, and Dynamics," Vol. 31, No. 3, 2008.
13. ^ Elnagar, G. and Kazemi, "Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems," Computational Optimization and Applications, Vol. 17, No. 2, pp. 195–217
14. ^ Fahroo, F., and Ross, I. M., "Direct Trajectory Optimization via a Chebyshev Pseudospectral Method," Journal of Guidance, Control, and Dynamics, Vol. 25, pp. 160–166
15. ^ 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
16. ^ Rao, A. V., Benson, D. A., Darby, C. L., Patterson, M. A., Francolin, C., Sanders, I., and Huntington, G. T., "Algorithm 902: GPOPS, A Matlab Software for Solving Multiple-Phase Optimal Control Problems Using the Gauss Pseudospectral Method", ACM Transactions on Mathematical Software, Vol. 37, No. 2, April–June, 2010, Article 22, pages 22:1-22:39.
17. ^ Garg, D. A., Patterson, M. A., Darby, C. L., Francolin, C., Huntington, G. T., Hager, W. W., and Rao, A. V., "Direct Trajectory Optimization and Costate Estimation of Finite-Horizon and Infinite-Horizon Optimal Control Problems Using a Radau Pseudospectral Method," Computational Optimization and Applications, Vol., 49, No. 2, June 2011, pp. 335-358.