In this paper, we use a Circle Restricted Three-Body
Problem (CRTBP) to simulate the motion of a satellite.Then we
reformulate this problem with the controller into the description using Koopman
eigenfunction. Although the original dynamical system is nonlinear, the Koopman
eigenfunction behaves linearly. Choosing Jacobi integralas the
Koopman eigenfunction and using the zero velocity curve as the reference for
control, we are allowed to combine well-developed Linear Quadratic Regulator
(LQR) controller to design a nonlinear controller. Using thisapproach,
we design the low thrust orbit transfer strategy for the satellite flying from
the earth to the moon or from the earth to the sun.
Brunton, S.L., Brunton, B.W., Proctor, J.L. and Kutz, J.N. (2016) Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control. PLoS ONE, 11, e0150171.
https://doi.org/10.1371/journal.pone.0150171
[3]
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https://doi.org/10.1073/pnas.1517384113
[4]
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https://doi.org/10.2514/2.4862
[5]
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[6]
Kaiser, E., Kutz, J.N. and Brunton, S.L. (2017) Data-Driven Discovery of Koopman Eigenfunctions for Control. arXiv preprint arXiv:1707.01146
[7]
Koopman, B.O. (1931) Hamiltonian Systems and Transformation in Hilbert Space. Proceedings of the National Academy of Sciences of USA, 17, 315-318.
https://doi.org/10.1073/pnas.17.5.315
[8]
Topputo, F. and Zhang, C. (2014) Survey of Direct Transcription for Low-Thrust Space Trajectory Optimization with Applications. Abstract and Applied Analysis, 2014, Article ID 851720.
