# Ross–Fahroo pseudospectral method

Short description: Optimal control technique

Introduced by I. Michael Ross and F. Fahroo, the Ross–Fahroo pseudospectral methods are a broad collection of pseudospectral methods for optimal control.[1][2][3][4][5][6][7][8][9] Examples of the Ross–Fahroo pseudospectral methods are the pseudospectral knotting method, the flat pseudospectral method, the Legendre-Gauss-Radau pseudospectral method[10][11] and pseudospectral methods for infinite-horizon optimal control.[12][13]

## Overview

The Ross–Fahroo methods are based on shifted Gaussian pseudospectral node points. The shifts are obtained by means of a linear or nonlinear transformation while the Gaussian pseudospectral points are chosen from a collection of Gauss-Lobatto or Gauss-Radau distribution arising from Legendre or Chebyshev polynomials. The Gauss-Lobatto pseudospectral points are used for finite-horizon optimal control problems while the Gauss-Radau pseudospectral points are used for infinite-horizon optimal control problems.[14]

## Mathematical applications

The Ross–Fahroo methods are founded on the Ross–Fahroo lemma; they can be applied to optimal control problems governed by differential equations, differential-algebraic equations, differential inclusions, and differentially-flat systems. They can also be applied to infinite-horizon optimal control problems by a simple domain transformation technique.[12] [13] The Ross–Fahroo pseudospectral methods also form the foundations for the Bellman pseudospectral method.

## Flight applications and awards

The Ross–Fahroo methods have been implemented in many practical applications and laboratories around the world. In 2006, NASA used the Ross–Fahroo method to implement the "zero propellant maneuver" on board the International Space Station.[15] In recognition of all these advances, the AIAA presented Ross and Fahroo, the 2010 Mechanics and Control of Flight Award, for "... changing the landscape of flight mechanics." Ross was also elected AAS Fellow for "his pioneering contributions to pseudospectral optimal control."

## Distinctive features

A remarkable feature of the Ross–Fahroo methods is that it does away with the prior notions of "direct" and "indirect" methods. That is, through a collection of theorems put forth by Ross and Fahroo,[5][6][8] [16] they showed that it was possible to design pseudospectral methods for optimal control that were equivalent in both the direct and indirect forms. This implied that one could use their methods as simply as a "direct" method while automatically generating accurate duals as in "indirect" methods. This revolutionized solving optimal control problems leading to widespread use of the Ross–Fahroo techniques.[17]

## Software implementation

The Ross–Fahroo methods are implemented in the MATLAB optimal control solver, DIDO.

## References

1. N. Bedrossian, M. Karpenko, and S. Bhatt, "Overclock My Satellite: Sophisticated Algorithms Boost Satellite Performance on the Cheap", IEEE Spectrum, November 2012.
2. Jr-; Li, S; Ruths, J.; Yu, T-Y; Arthanari, H.; Wagner, G. (2011). "Optimal Pulse Design in Quantum Control: A Unified Computational Method". Proceedings of the National Academy of Sciences 108 (5): 1879–1884. doi:10.1073/pnas.1009797108. PMID 21245345. Bibcode2011PNAS..108.1879L.
3. Kang, W. (2010). "Rate of Convergence for the Legendre Pseudospectral Optimal Control of Feedback Linearizable Systems". Journal of Control Theory and Application 8 (4): 391–405. doi:10.1007/s11768-010-9104-0.
4. Conway, B. A. (2012). "A Survey of Methods Available for the Numerical Optimization of Continuous Dynamic Systems". Journal of Optimization Theory Applications 152 (2): 271–306. doi:10.1007/s10957-011-9918-z.
5. I. M. Ross and F. Fahroo, A Pseudospectral Transformation of the Covectors of Optimal Control Systems, Proceedings of the First IFAC Symposium on System Structure and Control, Prague, Czech Republic, 29–31 August 2001.
6. I. M. Ross and F. Fahroo, Legendre Pseudospectral Approximations of Optimal Control Problems, Lecture Notes in Control and Information Sciences, Vol. 295, Springer-Verlag, 2003.
7. Ross, I. M.; Fahroo, F. (2004). "Pseudospectral Knotting Methods for Solving Optimal Control Problems". Journal of Guidance, Control and Dynamics 27 (3): 397–405. doi:10.2514/1.3426.
8. I. M. Ross and F. Fahroo, Discrete Verification of Necessary Conditions for Switched Nonlinear Optimal Control Systems, Proceedings of the American Control Conference, Invited Paper, June 2004, Boston, MA.
9. Ross, I. M.; Fahroo, F. (2004). "Pseudospectral Methods for the Optimal Motion Planning of Differentially Flat Systems". IEEE Transactions on Automatic Control 49 (8): 1410–1413. doi:10.1109/tac.2004.832972.
10. 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. [1]
11. Wen, H.; Jin, D.; Hu, H. (2008). "Infinite-Horizon Control for Retrieving a Tethered Subsatellite via an Elastic Tether". Journal of Guidance, Control and Dynamics 31 (4): 889–906. doi:10.2514/1.33224. Bibcode2008JGCD...31..899W.
12. F. Fahroo and I. M. Ross, Pseudospectral Methods for Infinite Horizon Nonlinear Optimal Control Problems, AIAA Guidance, Navigation and Control Conference, August 15–18, 2005, San Francisco, CA.
13. Fahroo, F.; Ross, I. M. (2008). "Pseudospectral Methods for Infinite-Horizon Optimal Control Problems". Journal of Guidance, Control and Dynamics 31 (4): 927–936. doi:10.2514/1.33117.
14. Ross, I. M.; Karpenko, M. (2012). "A Review of Pseudospectral Optimal Control: From Theory to Flight". Annual Reviews in Control 36 (2): 182–197. doi:10.1016/j.arcontrol.2012.09.002.
15. N. S. Bedrossian, S. Bhatt, W. Kang, and I. M. Ross, Zero-Propellant Maneuver Guidance, IEEE Control Systems Magazine, October 2009 (Feature Article), pp 53–73.
16. F. Fahroo and I. M. Ross, Trajectory Optimization by Indirect Spectral Collocation Methods, Proceedings of the AIAA/AAS Astrodynamics Conference, August 2000, Denver, CO. AIAA Paper 2000–4028
17. 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.