In mathematics, unscented optimal control combines the notion of the unscented transform with deterministic optimal control to address a class of uncertain optimal control problems.[1][2][3] It is a specific application of Riemmann-Stieltjes optimal control theory,[4][5] a concept introduced by Ross and his coworkers.

Mathematical description

Suppose that the initial state of a dynamical system,

is an uncertain quantity. Let be the sigma points. Then sigma-copies of the dynamical system are given by,

Applying standard deterministic optimal control principles to this ensemble generates an unscented optimal control.[6][7][8] Unscented optimal control is a special case of tychastic optimal control theory.[1][9][10] According to Aubin[10] and Ross,[1] tychastic processes differ from stochastic processes in that a tychastic process is conditionally deterministic.

Applications

Unscented optimal control theory has been applied to UAV guidance,[8][11] spacecraft attitude control,[12] air-traffic control[13] and low-thrust trajectory optimization[2][6]

References

  1. 1 2 3 Ross, Isaac (2015). A primer on Pontryagin's principle in optimal control. San Francisco: Collegiate Publishers. pp. 75–82. ISBN 978-0-9843571-1-6.
  2. 1 2 Unscented Optimal Control for Orbital and Proximity Operations in an Uncertain Environment: A New Zermelo Problem I. Michael Ross, Ronald Proulx, Mark Karpenko August 2014, American Institute of Aeronautics and Astronautics (AIAA) doi:10.2514/6.2014-4423
  3. ↑ Ross et al, Unscented Control for Uncertain Dynamical Systems, US Patent US 9,727,034 Bl. Issued Aug 8, 2017. https://calhoun.nps.edu/bitstream/handle/10945/55812/USPN%209727034.pdf?sequence=1&isAllowed=y
  4. ↑ Ross, I. Michael; Karpenko, Mark; Proulx, Ronald J. (2015). "Riemann-Stieltjes Optimal Control Problems for Uncertain Dynamic Systems". Journal of Guidance, Control, and Dynamics. AIAA. 38 (7): 1251–1263. Bibcode:2015JGCD...38.1251R. doi:10.2514/1.G000505. S2CID 121424228.
  5. ↑ Karpenko, Mark; Proulx, Ronald J. (2016). "Experimental Implementation of Riemann–Stieltjes Optimal Control for Agile Imaging Satellites". Journal of Guidance, Control, and Dynamics. 39 (1): 144–150. Bibcode:2016JGCD...39..144K. doi:10.2514/1.g001325. ISSN 0731-5090. S2CID 116887441.
  6. 1 2 Naoya Ozaki and Ryu Funase. "Tube Stochastic Differential Dynamic Programming for Robust Low-Thrust Trajectory Optimization Problems", 2018 AIAA Guidance, Navigation, and Control Conference, AIAA SciTech Forum, (AIAA 2018-0861) doi:10.2514/6.2018-0861
  7. ↑ "Robust Differential Dynamic Programming for Low-Thrust Trajectory Design: Approach with Robust Model Predictive Control Technique" (PDF).
  8. 1 2 Shaffer, R.; Karpenko, M.; Gong, Q. (July 2016). "Unscented guidance for waypoint navigation of a fixed-wing UAV". 2016 American Control Conference (ACC). pp. 473–478. doi:10.1109/acc.2016.7524959. ISBN 978-1-4673-8682-1. S2CID 11741951.
  9. ↑ Ross, I. Michael; Karpenko, Mark; Proulx, Ronald J. (July 2016). "Path constraints in tychastic and unscented optimal control: Theory, application and experimental results". 2016 American Control Conference (ACC). IEEE. pp. 2918–2923. doi:10.1109/acc.2016.7525362. ISBN 978-1-4673-8682-1. S2CID 1123147.
  10. 1 2 Aubin, Jean-Pierre; Saint-Pierre, Patrick (2008), A Tychastic Approach to Guaranteed Pricing and Management of Portfolios under Transaction Constraints, Progress in Probability, vol. 59, Basel: Birkhäuser Basel, pp. 411–433, doi:10.1007/978-3-7643-8458-6_22, ISBN 978-3-7643-8457-9, retrieved 2020-12-23
  11. ↑ Ross, I. M.; Proulx, R. J.; Karpenko, M. (July 2015). "Unscented guidance". 2015 American Control Conference (ACC). pp. 5605–5610. doi:10.1109/acc.2015.7172217. ISBN 978-1-4799-8684-2. S2CID 28136418.
  12. ↑ Ross, I. M.; Karpenko, M.; Proulx, R. J. (July 2016). "Path constraints in tychastic and unscented optimal control: Theory, application and experimental results". 2016 American Control Conference (ACC). pp. 2918–2923. doi:10.1109/acc.2016.7525362. ISBN 978-1-4673-8682-1. S2CID 1123147.
  13. ↑ Ng, Hok Kwan (2020-06-08), "Strategic Planning with Unscented Optimal Guidance for Urban Air Mobility", AIAA AVIATION 2020 FORUM, AIAA AVIATION Forum, American Institute of Aeronautics and Astronautics, doi:10.2514/6.2020-2904, ISBN 978-1-62410-598-2, S2CID 225658104, retrieved 2020-12-23


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