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] 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 $x^{0}$ 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.^[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

^ ^a ^b ^c Ross, Isaac (2015). A primer on Pontryagin's principle in optimal control. San Francisco: Collegiate Publishers. pp. 75–82. ISBN 0-9843571-1-4.
^ ^a ^b 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
^ 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
^ 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.
^ Karpenko, Mark; Proulx, Ronald J. "Experimental Implementation of Riemann–Stieltjes Optimal Control for Agile Imaging Satellites". Journal of Guidance, Control, and Dynamics. 39 (1): 144–150. doi:10.2514/1.g001325. ISSN 0731-5090.
^ ^a ^b 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) https://doi.org/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): 473–478. doi:10.1109/acc.2016.7524959. ISBN 978-1-4673-8682-1.
^ 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. doi:10.1109/acc.2016.7525362. ISBN 978-1-4673-8682-1.
^ ^a ^b Aubin, Jean-Pierre; Saint-Pierre, Patrick, "A Tychastic Approach to Guaranteed Pricing and Management of Portfolios under Transaction Constraints", Progress in Probability, Basel: Birkhäuser Basel, pp. 411–433, 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): 5605–5610. doi:10.1109/acc.2015.7172217. ISBN 978-1-4799-8684-2.
^ 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): 2918–2923. doi:10.1109/acc.2016.7525362. ISBN 978-1-4673-8682-1.
^ 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, retrieved 2020-12-23

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[:3-1] Ross, Isaac (2015). A primer on Pontryagin's principle in optimal control. San Francisco: Collegiate Publishers. pp. 75–82. ISBN 0-9843571-1-4.

[:0-2] 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.

[5] Karpenko, Mark; Proulx, Ronald J. "Experimental Implementation of Riemann–Stieltjes Optimal Control for Agile Imaging Satellites". Journal of Guidance, Control, and Dynamics. 39 (1): 144–150. doi:10.2514/1.g001325. ISSN 0731-5090.

[:1-6] 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) https://doi.org/10.2514/6.2018-0861

[7] "Robust Differential Dynamic Programming for Low-Thrust Trajectory Design: Approach with Robust Model Predictive Control Technique" (PDF).

[:2-8] Shaffer, R.; Karpenko, M.; Gong, Q. (July 2016). "Unscented guidance for waypoint navigation of a fixed-wing UAV". 2016 American Control Conference (ACC): 473–478. doi:10.1109/acc.2016.7524959. ISBN 978-1-4673-8682-1.

[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. doi:10.1109/acc.2016.7525362. ISBN 978-1-4673-8682-1.

[:4-10] Aubin, Jean-Pierre; Saint-Pierre, Patrick, "A Tychastic Approach to Guaranteed Pricing and Management of Portfolios under Transaction Constraints", Progress in Probability, Basel: Birkhäuser Basel, pp. 411–433, 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): 5605–5610. doi:10.1109/acc.2015.7172217. ISBN 978-1-4799-8684-2.

[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): 2918–2923. doi:10.1109/acc.2016.7525362. ISBN 978-1-4673-8682-1.

[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, retrieved 2020-12-23

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