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]} 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,• x = f(x, u, t)

is an uncertain quantity. Let

\Chi ^{i} be the sigma points. Then sigma-copies of the dynamical system are given by,• \Chi ^{i} = f(\Chi ^{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.^[9] 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,^[10] spacecraft attitude control,^[11] air-traffic control 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 2014, American Institute of Aeronautics and Astronautics (AIAA)

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. 38. 7. 1251–1263. AIAA. 10.2514/1.G000505. 2015JGCD...38.1251R . 121424228 . 10945/48189. free.
5. 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.
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)
7. Web site: Robust Differential Dynamic Programming for Low-Thrust Trajectory Design: Approach with Robust Model Predictive Control Technique.
8. 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 .
9. 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 .
10. 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 .
11. 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 .