Unscented optimal control

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] [4]} It is a specific application of tychastic optimal control theory, ^{[1] [5] [6] [7]} which is a generalization of Riemmann-Stieltjes optimal control theory, ^{[8] [9]} a concept introduced by Ross and his coworkers.

Mathematical description

Suppose that the initial state ${\displaystyle x^{0}}$ of a dynamical system,

${\displaystyle {\dot {x}}=f(x,u,t)}$

is an uncertain quantity. Let ${\displaystyle \mathrm {X} ^{i}}$ be the sigma points. Then sigma-copies of the dynamical system are given by,

${\displaystyle {\dot {\mathrm {X} }}^{i}=f(\mathrm {X} ^{i},u,t)}$

Applying standard deterministic optimal control principles to this ensemble generates an unscented optimal control. ^{[10] [11] [12]} Unscented optimal control is a special case of tychastic optimal control theory. ^{[1] [5] [13]} According to Aubin ^[13] 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, ^{[12] [14]} spacecraft attitude control, ^[6] air-traffic control ^[15] and low-thrust trajectory optimization ^{[2] [10]}

References

1. 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. Ross, I. Michael; Proulx, Ronald; Karpenko, Mark (August 4–7, 2014). Unscented Optimal Control for Orbital and Proximity Operations in an Uncertain Environment: A New Zermelo Problem . AIAA/AAS Astrodynamics Specialist Conference. San Diego, CA: American Institute of Aeronautics and Astronautics. doi:10.2514/6.2014-4423 . Retrieved August 23, 2024.
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. Manchester, Zachary; Kuindersma, Scott (December 2016). "Derivative-free trajectory optimization with unscented dynamic programming". 2016 IEEE 55th Conference on Decision and Control (CDC). IEEE. pp. 3642–3647. doi:10.1109/cdc.2016.7798817. ISBN   978-1-5090-1837-6.
5. 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.
6. 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.
7. Ross, I. M.; Proulx, R. J.; Karpenko, M. (2024-05-04). "Unscented Trajectory Optimization". Journal of Guidance Control Dynamics. 48 (1): 20. arXiv: 2405.02753 . Bibcode:2025JGCD...48...20R. doi:10.2514/1.G007925.
8. Ross, I. Michael; Karpenko, Mark; Proulx, Ronald J. (2015). "Riemann-Stieltjes Optimal Control Problems for Uncertain Dynamic Systems" . Journal of Guidance, Control, and Dynamics. 38 (7). AIAA: 1251–1263. Bibcode:2015JGCD...38.1251R. doi:10.2514/1.G000505. hdl: 10945/48189 . S2CID   121424228.
9. 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. hdl: 10945/50355 . ISSN   0731-5090. S2CID   116887441.
10. Ozaki, Naoya; Funase, Ryu (January 8–12, 2018). Tube Stochastic Differential Dynamic Programming for Robust Low-Thrust Trajectory Optimization Problems . 2018 AIAA Guidance, Navigation, and Control Conference. Kissimmee, Florida. doi:10.2514/6.2018-0861.
11. "Robust Differential Dynamic Programming for Low-Thrust Trajectory Design: Approach with Robust Model Predictive Control Technique" (PDF).
12. 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.
13. Aubin, Jean-Pierre; Saint-Pierre, Patrick (2008). "A Tychastic Approach to Guaranteed Pricing and Management of Portfolios under Transaction Constraints". Seminar on Stochastic Analysis, Random Fields and Applications V. 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.
14. 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.
15. Ng, Hok Kwan (2020-06-08). "Strategic Planning with Unscented Optimal Guidance for Urban Air Mobility". AIAA Aviation 2020 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.