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 $x^{0}$ of a dynamical system,

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

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

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

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

^ ^a ^b ^c ^d 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.
^ ^a ^b 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.
^ 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
^ Manchester, Zachary; Kuindersma, Scott (December 2016). "Derivative-free trajectory optimization with unscented dynamic programming". 2016 IEEE 55th Conference on Decision and Control (CDC). IEEE. doi:10.1109/cdc.2016.7798817.
^ ^a ^b 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.
^ ^a ^b 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.
^ Ross, I. M.; Proulx, R. J.; Karpenko, M. (2024-05-04), Unscented Trajectory Optimization, doi:10.48550/arXiv.2405.02753, retrieved 2024-08-22
^ 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.
^ 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.
^ ^a ^b 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.
^ "Robust Differential Dynamic Programming for Low-Thrust Trajectory Design: Approach with Robust Model Predictive Control Technique" (PDF).
^ ^a ^b 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.
^ ^a ^b 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
^ 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.
^ 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

This mathematics-related article is a stub. You can help Wikipedia by expanding it.

[:3-1] 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.

[:0-2] 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. doi:10.1109/cdc.2016.7798817.

[:5-5] 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.

[:6-6] 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, doi:10.48550/arXiv.2405.02753, retrieved 2024-08-22

[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.

[:1-10] 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).

[:2-12] 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.

[:4-13] 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

[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

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]