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.{{Cite book|last=Ross|first=Isaac|title=A primer on Pontryagin's principle in optimal control|publisher=Collegiate Publishers|year=2015|isbn=978-0-9843571-1-6|location=San Francisco|pages=75–82}}{{Cite conference |title=Unscented Optimal Control for Orbital and Proximity Operations in an Uncertain Environment: A New Zermelo Problem |first1=I. Michael |last1=Ross |first2=Ronald |last2=Proulx |first3=Mark |last3=Karpenko |date=August 4–7, 2014 |location=San Diego, CA |access-date=August 23, 2024 |url=https://arc.aiaa.org/doi/abs/10.2514/6.2014-4423 |conference=AIAA/AAS Astrodynamics Specialist Conference |publisher=American Institute of Aeronautics and Astronautics |doi=10.2514/6.2014-4423|url-access=subscription }}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{{Cite book |last1=Manchester |first1=Zachary |last2=Kuindersma |first2=Scott |chapter=Derivative-free trajectory optimization with unscented dynamic programming |date=December 2016 |title=2016 IEEE 55th Conference on Decision and Control (CDC) |chapter-url=http://dx.doi.org/10.1109/cdc.2016.7798817 |pages=3642–3647 |publisher=IEEE |doi=10.1109/cdc.2016.7798817|isbn=978-1-5090-1837-6 }} It is a specific application of tychastic optimal control theory,{{cite arXiv |last1=Ross |first1=I. M. |title=Unscented Trajectory Optimization |date=2024-05-04 |eprint=2405.02753 |last2=Proulx |first2=R. J. |last3=Karpenko |first3=M.|class=math.OC }} which is a generalization of Riemmann-Stieltjes optimal control theory,{{Cite journal|last1=Ross|first1=I. Michael|last2=Karpenko|first2=Mark|last3=Proulx|first3=Ronald J.|date=2015|title=Riemann-Stieltjes Optimal Control Problems for Uncertain Dynamic Systems|url=https://doi.org/10.2514/1.G000505|journal=Journal of Guidance, Control, and Dynamics|volume=38|issue=7|pages=1251–1263|publisher=AIAA|doi=10.2514/1.G000505|bibcode=2015JGCD...38.1251R |s2cid=121424228 |hdl=10945/48189|hdl-access=free}}{{Cite journal|last1=Karpenko|first1=Mark|last2=Proulx|first2=Ronald J.|title=Experimental Implementation of Riemann–Stieltjes Optimal Control for Agile Imaging Satellites|url=http://dx.doi.org/10.2514/1.g001325|journal=Journal of Guidance, Control, and Dynamics|year=2016|volume=39|issue=1|pages=144–150|doi=10.2514/1.g001325|bibcode=2016JGCD...39..144K |s2cid=116887441 |issn=0731-5090|hdl=10945/50355|hdl-access=free}} 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 $\Chi^i$ be the sigma points. Then sigma-copies of the dynamical system are given by,

$\dot\Chi^i = f(\Chi^i, u, t)$

Applying standard deterministic optimal control principles to this ensemble generates an unscented optimal control.{{cite conference |first1=Naoya |last1=Ozaki |first2=Ryu |last2=Funase |title=Tube Stochastic Differential Dynamic Programming for Robust Low-Thrust Trajectory Optimization Problems |conference=2018 AIAA Guidance, Navigation, and Control Conference |url=https://arc.aiaa.org/doi/abs/10.2514/6.2018-0861 |date=January 8–12, 2018

|location=Kissimmee, Florida |doi=10.2514/6.2018-0861|url-access=subscription }}{{Cite web|url=http://issfd2017.org/paper/ISTS-2017-d-127__ISSFD-2017-127.pdf|title=Robust Differential Dynamic Programming for Low-Thrust Trajectory Design: Approach with Robust Model Predictive Control Technique}}{{Cite book|last1=Shaffer|first1=R.|last2=Karpenko|first2=M.|last3=Gong|first3=Q.|title=2016 American Control Conference (ACC) |chapter=Unscented guidance for waypoint navigation of a fixed-wing UAV |date=July 2016|chapter-url=https://ieeexplore.ieee.org/document/7524959|pages=473–478|doi=10.1109/acc.2016.7524959|isbn=978-1-4673-8682-1|s2cid=11741951 }} Unscented optimal control is a special case of tychastic optimal control theory.{{Cite book|last1=Ross|first1=I. Michael|last2=Karpenko|first2=Mark|last3=Proulx|first3=Ronald J.|title=2016 American Control Conference (ACC) |chapter=Path constraints in tychastic and unscented optimal control: Theory, application and experimental results |date=July 2016|chapter-url=http://dx.doi.org/10.1109/acc.2016.7525362|pages=2918–2923|publisher=IEEE|doi=10.1109/acc.2016.7525362|isbn=978-1-4673-8682-1|s2cid=1123147 }}{{cite book |last1=Aubin|first1=Jean-Pierre|chapter=A Tychastic Approach to Guaranteed Pricing and Management of Portfolios under Transaction Constraints|chapter-url=http://dx.doi.org/10.1007/978-3-7643-8458-6_22|pages=411–433|place=Basel|publisher=Birkhäuser Basel|isbn=978-3-7643-8457-9|access-date=2020-12-23|last2=Saint-Pierre|first2=Patrick|title=Seminar on Stochastic Analysis, Random Fields and Applications V |series=Progress in Probability|year=2008|volume=59|doi=10.1007/978-3-7643-8458-6_22}} 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,{{Cite book|last1=Ross|first1=I. M.|last2=Proulx|first2=R. J.|last3=Karpenko|first3=M.|title=2015 American Control Conference (ACC) |chapter=Unscented guidance |date=July 2015|chapter-url=https://ieeexplore.ieee.org/document/7172217|pages=5605–5610|doi=10.1109/acc.2015.7172217|isbn=978-1-4799-8684-2|s2cid=28136418 }} spacecraft attitude control,{{Cite book|last1=Ross|first1=I. M.|last2=Karpenko|first2=M.|last3=Proulx|first3=R. J.|title=2016 American Control Conference (ACC) |chapter=Path constraints in tychastic and unscented optimal control: Theory, application and experimental results |date=July 2016|chapter-url=https://ieeexplore.ieee.org/document/7525362|pages=2918–2923|doi=10.1109/acc.2016.7525362|isbn=978-1-4673-8682-1|s2cid=1123147 }} air-traffic control{{cite book|last=Ng|first=Hok Kwan|chapter=Strategic Planning with Unscented Optimal Guidance for Urban Air Mobility|date=2020-06-08|chapter-url=https://arc.aiaa.org/doi/10.2514/6.2020-2904|title=AIAA Aviation 2020 Forum|publisher=American Institute of Aeronautics and Astronautics|doi=10.2514/6.2020-2904|isbn=978-1-62410-598-2|s2cid=225658104 |access-date=2020-12-23}} and low-thrust trajectory optimization

References

Category:Optimal control