Pseudospectral optimal control

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Pseudospectral optimal control is a numerical technique for solving optimal control problems. [1] These problems involve finding the best way to control a dynamic system, for example, calculating the most fuel-efficient trajectory for a spacecraft or determining the fastest way for a robot arm to move. [2] The pseudospectral method transforms the original, continuous problem—which is often too complex to be solved directly—into a simpler set of algebraic equations that can be solved efficiently by a computer. [3]

Contents

The method combines pseudospectral (PS) theory with optimal control theory and is notable for its high accuracy with a relatively small number of calculations. [4] [5] It has been used to solve a wide range of problems in military and industrial applications, such as those arising in UAV trajectory generation, missile guidance, control of robotic arms, vibration damping, and lunar guidance. [2] [6]

Overview

There are a very large number of ideas that fall under the general banner of pseudospectral optimal control. [7] 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. [8] [9] 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. [9] 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. [10] [11] [12] 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) at Legendre–Gauss–Lobatto nodes converges in L2 sense at the so-called spectral rate, faster than any polynomial rate. [11]

Details

A basic pseudospectral method for optimal control is based on the covector mapping principle. [13] 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. [10] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

Moreover, their structure can be highly exploited to make them more computationally efficient, as ad-hoc scaling [24] and Jacobian computation methods, involving dual number theory [25] have been developed. [22]

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 . 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 ]

See also

References

  1. ↑ 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. 1 2 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. hdl:10945/29677. ISBN   978-1-4244-1497-0. S2CID   2935682.
  3. ↑ 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. S2CID   17819443.
  4. ↑ 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. S2CID   7625851.
  5. ↑ 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). Vol. 3. pp. 2210–5. doi:10.1109/CDC.2003.1272946. ISBN   0-7803-7924-1. S2CID   122755607.
  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   3033291 . PMID   21245345.
  7. ↑ Ross, I. M.; Proulx, R. J. (September 2019). "Further Results on Fast Birkhoff Pseudospectral Optimal Control Programming" (PDF). Journal of Guidance, Control, and Dynamics. 42 (9): 2086–2092. Bibcode:2019JGCD...42.2086R. doi:10.2514/1.g004297. ISSN   1533-3884. S2CID   191166808.
  8. ↑ 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.
  9. 1 2 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. S2CID   17819443.
  10. 1 2 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. hdl: 10945/29674 . S2CID   16048034.
  11. 1 2 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 ]
  12. ↑ 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. hdl: 10945/48182 . S2CID   38196250.
  13. ↑ 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. S2CID   7625851.
  14. ↑ 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.
  15. ↑ 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. hdl: 10945/29649 . S2CID   122759455.
  16. ↑ 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. CiteSeerX   10.1.1.301.3354 . doi:10.2514/1.32908. hdl:10945/56995. S2CID   18145691.
  17. ↑ 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. S2CID   30241469.
  18. ↑ Fahroo, Fariba; Ross, I. Michael (2002). "Direct Trajectory Optimization by a Chebyshev Pseudospectral Method". Journal of Guidance, Control, and Dynamics. 25 (1): 160–6. Bibcode:2002JGCD...25..160F. doi:10.2514/2.4862.
  19. ↑ 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. CiteSeerX   10.1.1.658.9510 . doi:10.2514/1.20478.
  20. ↑ 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 . S2CID   15375549.
  21. ↑ 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. CiteSeerX   10.1.1.663.4215 . doi:10.1007/s10589-009-9291-0. S2CID   8817072.
  22. 1 2 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.
  23. ↑ 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.
  24. ↑ Sagliano, Marco (2014). "Performance analysis of linear and nonlinear techniques for automatic scaling of discretized control problems" (PDF). Operations Research Letters. 42 (3): 213–6. doi:10.1016/j.orl.2014.03.003.
  25. ↑ d'Onofrio, Vincenzo; Sagliano, Marco; Arslantas, Yunus E. (2016). "Exact Hybrid Jacobian Computation for Optimal Trajectories via Dual Number Theory" (PDF). AIAA Guidance, Navigation, and Control Conference. doi:10.2514/6.2016-0867. ISBN   978-1-62410-389-6.

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