Unscented optimal control explained

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,[5] which is a generalization of Riemmann-Stieltjes optimal control theory,[6] [7] a concept introduced by Ross and his coworkers.

Mathematical description

Suppose that the initial state

x0

of a dynamical system,
•
x

=f(x,u,t)

is an uncertain quantity. Let

\Chii

be the sigma points. Then sigma-copies of the dynamical system are given by,
•
\Chi

i=f(\Chii,u,t)

Applying standard deterministic optimal control principles to this ensemble generates an unscented optimal control.[8] [9] [10] Unscented optimal control is a special case of tychastic optimal control theory.[11] [12] According to Aubin and Ross, 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,[13] spacecraft attitude control,[14] air-traffic control[15] and low-thrust trajectory optimization

Notes and References

  1. Book: Ross, Isaac. A primer on Pontryagin's principle in optimal control. Collegiate Publishers. 2015. 978-0-9843571-1-6. San Francisco. 75–82.
  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 4–7, 2014 . San Diego, CA . August 23, 2024 . AIAA/AAS Astrodynamics Specialist Conference . American Institute of Aeronautics and Astronautics . 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. Book: Manchester . Zachary . Kuindersma . Scott . Derivative-free trajectory optimization with unscented dynamic programming . December 2016 . 2016 IEEE 55th Conference on Decision and Control (CDC) . http://dx.doi.org/10.1109/cdc.2016.7798817 . 3642–3647 . IEEE . 10.1109/cdc.2016.7798817. 978-1-5090-1837-6 .
  5. Ross . I. M. . Unscented Trajectory Optimization . 2024-05-04 . 2405.02753 . Proulx . R. J. . Karpenko . M.. math.OC .
  6. 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. 1251–1263. AIAA. 10.2514/1.G000505. 2015JGCD...38.1251R . 121424228 . 10945/48189. free.
  7. Karpenko. Mark. Proulx. Ronald J.. Experimental Implementation of Riemann–Stieltjes Optimal Control for Agile Imaging Satellites. Journal of Guidance, Control, and Dynamics. 2016. 39. 1. 144–150. 10.2514/1.g001325. 2016JGCD...39..144K . 116887441 . 0731-5090. 10945/50355. free.
  8. Naoya . Ozaki . Ryu . Funase . Tube Stochastic Differential Dynamic Programming for Robust Low-Thrust Trajectory Optimization Problems . 2018 AIAA Guidance, Navigation, and Control Conference . January 8–12, 2018. Kissimmee, Florida . 10.2514/6.2018-0861.
  9. Web site: Robust Differential Dynamic Programming for Low-Thrust Trajectory Design: Approach with Robust Model Predictive Control Technique.
  10. Book: Shaffer. R.. Karpenko. M.. Gong. Q.. 2016 American Control Conference (ACC) . Unscented guidance for waypoint navigation of a fixed-wing UAV . July 2016. https://ieeexplore.ieee.org/document/7524959. 473–478. 10.1109/acc.2016.7524959. 978-1-4673-8682-1. 11741951 .
  11. Book: Ross. I. Michael. Karpenko. Mark. Proulx. Ronald J.. 2016 American Control Conference (ACC) . Path constraints in tychastic and unscented optimal control: Theory, application and experimental results . July 2016. http://dx.doi.org/10.1109/acc.2016.7525362. 2918–2923. IEEE. 10.1109/acc.2016.7525362. 978-1-4673-8682-1. 1123147 .
  12. Book: Aubin. Jean-Pierre. A Tychastic Approach to Guaranteed Pricing and Management of Portfolios under Transaction Constraints. http://dx.doi.org/10.1007/978-3-7643-8458-6_22. 411–433. Basel. Birkhäuser Basel. 978-3-7643-8457-9. 2020-12-23. Saint-Pierre. Patrick. Seminar on Stochastic Analysis, Random Fields and Applications V . Progress in Probability. 2008. 59. 10.1007/978-3-7643-8458-6_22.
  13. Book: Ross. I. M.. Proulx. R. J.. Karpenko. M.. 2015 American Control Conference (ACC) . Unscented guidance . July 2015. https://ieeexplore.ieee.org/document/7172217. 5605–5610. 10.1109/acc.2015.7172217. 978-1-4799-8684-2. 28136418 .
  14. Book: Ross. I. M.. Karpenko. M.. Proulx. R. J.. 2016 American Control Conference (ACC) . Path constraints in tychastic and unscented optimal control: Theory, application and experimental results . July 2016. https://ieeexplore.ieee.org/document/7525362. 2918–2923. 10.1109/acc.2016.7525362. 978-1-4673-8682-1. 1123147 .
  15. Book: Ng, Hok Kwan. Strategic Planning with Unscented Optimal Guidance for Urban Air Mobility. 2020-06-08. https://arc.aiaa.org/doi/10.2514/6.2020-2904. AIAA Aviation 2020 Forum. American Institute of Aeronautics and Astronautics. 10.2514/6.2020-2904. 978-1-62410-598-2. 225658104 . 2020-12-23.

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