# Pseudospectral optimal control

Pseudospectral optimal control is a joint theoretical-computational method for solving optimal control problems.[1][2][3][4] 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 Bellman 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${\displaystyle \infty }$) 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][18][19]

Moreover, their structure can be highly exploited to make them more computationally efficient, as ad-hoc scaling[20] and Jacobian computation methods, involving dual number theory[21] have been developed.[18]

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 ${\displaystyle 2N-1}$. 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.[citation needed]

## References

1. ^ a b c Ross, I. Michael; Karpenko, Mark (2012). "A review of pseudospectral optimal control: From theory to flight". Annual Reviews in Control. 36 (2): 182–97. doi:10.1016/j.arcontrol.2012.09.002.
2. ^ a b Ross, I M. (2005). "A Roadmap for Optimal Control: The Right Way to Commute". Annals of the New York Academy of Sciences. 1065: 210–31. Bibcode:2005NYASA1065..210R. doi:10.1196/annals.1370.015. PMID 16510411.
3. ^ a b c Fahroo, Fariba; Ross, I. Michael (2008). "Advances in Pseudospectral Methods for Optimal Control". AIAA Guidance, Navigation and Control Conference and Exhibit. pp. 18–21. doi:10.2514/6.2008-7309. ISBN 978-1-60086-999-0.
4. ^ Ross, I.M.; Fahroo, F. (2003). "A unified computational framework for real-time optimal control". 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475). 3. pp. 2210–5. doi:10.1109/CDC.2003.1272946. ISBN 0-7803-7924-1.
5. ^ a b Qi Gong; Wei Kang; Bedrossian, Nazareth S.; Fahroo, Fariba; Pooya Sekhavat; Bollino, Kevin (2007). "Pseudospectral Optimal Control for Military and Industrial Applications". 2007 46th IEEE Conference on Decision and Control. pp. 4128–42. doi:10.1109/CDC.2007.4435052. ISBN 978-1-4244-1497-0.
6. ^ Li, Jr-Shin; Ruths, Justin; Yu, Tsyr-Yan; Arthanari, Haribabu; Wagner, Gerhard (2011). "Optimal pulse design in quantum control: A unified computational method". Proceedings of the National Academy of Sciences. 108 (5): 1879–84. Bibcode:2011PNAS..108.1879L. doi:10.1073/pnas.1009797108. JSTOR 41001785. PMC . PMID 21245345.
7. ^ a b Gong, Q.; Kang, W.; Ross, I.M. (2006). "A Pseudospectral Method for the Optimal Control of Constrained Feedback Linearizable Systems". IEEE Transactions on Automatic Control. 51 (7): 1115–29. doi:10.1109/TAC.2006.878570.
8. ^ a b Hesthaven, J. S.; Gottlieb, S.; Gottlieb, D. (2007). Spectral methods for time-dependent problems. Cambridge University Press. ISBN 978-0-521-79211-0.[page needed]
9. ^ Gong, Qi; Ross, I. Michael; Kang, Wei; Fahroo, Fariba (2007). "Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control". Computational Optimization and Applications. 41 (3): 307–35. doi:10.1007/s10589-007-9102-4.
10. ^ Elnagar, G.; Kazemi, M.A.; Razzaghi, M. (1995). "The pseudospectral Legendre method for discretizing optimal control problems". IEEE Transactions on Automatic Control. 40 (10): 1793–6. doi:10.1109/9.467672.
11. ^ Fahroo, Fariba; Ross, I. Michael (2001). "Costate Estimation by a Legendre Pseudospectral Method". Journal of Guidance, Control, and Dynamics. 24 (2): 270–7. Bibcode:2001JGCD...24..270F. doi:10.2514/2.4709.
12. ^ Gong, Qi; Fahroo, Fariba; Ross, I. Michael (2008). "Spectral Algorithm for Pseudospectral Methods in Optimal Control". Journal of Guidance, Control, and Dynamics. 31 (3): 460–71. Bibcode:2008JGCD...31..460G. doi:10.2514/1.32908.
13. ^ Elnagar, Gamal N.; Kazemi, Mohammad A. (1998). "Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems". Computational Optimization and Applications. 11 (2): 195–217. doi:10.1023/A:1018694111831.
14. ^ Fahroo, Fariba; Ross, I. Michael (2002). "Direct Trajectory Optimization by a Chebyshev Pseudospectral Method". Journal of Guidance, Control, and Dynamics. 25: 160–6. Bibcode:2002JGCD...25..160F. doi:10.2514/2.4862.
15. ^ Benson, David A.; Huntington, Geoffrey T.; Thorvaldsen, Tom P.; Rao, Anil V. (2006). "Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method". Journal of Guidance, Control, and Dynamics. 29 (6): 1435–40. Bibcode:2006JGCD...29.1435B. doi:10.2514/1.20478.
16. ^ Rao, Anil V.; Benson, David A.; Darby, Christopher; Patterson, Michael A.; Francolin, Camila; Sanders, Ilyssa; Huntington, Geoffrey T. (2010). "Algorithm 902: GPOPS, A MATLAB software for solving multiple-phase optimal control problems using the gauss pseudospectral method". ACM Transactions on Mathematical Software. 37 (2). doi:10.1145/1731022.1731032.
17. ^ Garg, Divya; Patterson, Michael A.; Francolin, Camila; Darby, Christopher L.; Huntington, Geoffrey T.; Hager, William W.; Rao, Anil V. (2009). "Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method". Computational Optimization and Applications. 49 (2): 335–58. doi:10.1007/s10589-009-9291-0.
18. ^ a b Sagliano, Marco; Theil, Stephan (2013). "Hybrid Jacobian Computation for Fast Optimal Trajectories Generation". AIAA Guidance, Navigation, and Control (GNC) Conference. doi:10.2514/6.2013-4554. ISBN 978-1-62410-224-0.
19. ^ Huneker, Laurens; Sagliano, Marco; Arslantas, Yunus (2015). SPARTAN: An Improved Global Pseudospectral Algorithm for High-fidelity Entry-Descent-Landing Guidance Analysis (PDF). The 30th International Symposium on Space Science and Technology. Kobe, Japan.
20. ^ Sagliano, Marco (2014). "Performance analysis of linear and nonlinear techniques for automatic scaling of discretized control problems". Operations Research Letters. 42 (3): 213–6. doi:10.1016/j.orl.2014.03.003.
21. ^ d'Onofrio, Vincenzo; Sagliano, Marco; Arslantas, Yunus E. (2016). "Exact Hybrid Jacobian Computation for Optimal Trajectories via Dual Number Theory". AIAA Guidance, Navigation, and Control Conference. doi:10.2514/6.2016-0867. ISBN 978-1-62410-389-6.