In this paper, we use
Port-Hamiltonian framework to stabilize the Lagrange points in the Sun-Earth three-dimensional Circular Restricted Three-Body
Problem (CRTBP). Through rewriting the CRTBP into Port-Hamiltonian framework,
we are allowed to design the feedback controller through energy-shaping
and dissipation injection. The closed-loop Hamiltonian is a candidate of the Lyapunov function to establish
nonlinear stability of the designed equilibrium, which enlarges the application
region of feedback controller compared with that based on linearized dynamics.
Results show that the
Port-Hamiltonian approach
allows us to successfully stabilize the Lagrange points, where the Linear
Quadratic Regulator (LQR) may fail. The feedback system based on Port-Hamiltonian approach is also robust against white
noise in the inputs.
References
[1]
Musielak, Z.E. and Quarles, B. (2014) The Three-Body Problem. Reports on Progress in Physics, 77, Article ID: 065901. https://doi.org/10.1088/0034-4885/77/6/065901
[2]
Pilbratt, G.L., Riedinger, J.R., Passvogel, T., Crone, G., Doyle, D., Gageur, U., Heras, A.M., Jewell, C., Metcalfe, L., Ott, S., et al. (2010) Herschel Space Observatory. An ESA Facility for Far-Infrared and Submillimetre Astronomy. Astronomy & Astrophysics, 518, Article No. L1. https://doi.org/10.1051/0004-6361/201014759
[3]
Gardner, J.P., Mather, J.C., Clampin, M., Doyon, R., Greenhouse, M.A., Hammel, H.B., Hutchings, J.B., Jakobsen, P., Lilly, S.J., Long, K.S., et al. (2006) The James Webb Space Telescope. Space Science Reviews, 123, 485-606. https://doi.org/10.1007/s11214-006-8315-7
[4]
Farquhar, R. (1998) The Flight of ISEE-3/ICE-Origins, Mission History, and a Legacy. AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Boston, 10-12 August 1998, 4464. https://doi.org/10.2514/6.1998-4464
[5]
Stone, E.C., Frandsen, A.M., Mewaldt, R.A., Christian, E.R., Margolies, D., Ormes, J.F. and Snow, F. (1998) The Advanced Composition Explorer. Space Science Reviews, 86, 1-22. https://doi.org/10.1023/A:1005082526237
[6]
Sweetser, T.H., Broschart, S.B., Angelopoulos, V., Whiffen, G.J., Folta, D.C., Chung, M.K., Hatch, S.J. and Woodard, M.A. (2012) ARTEMIS Mission Design. In: Russell, C. and Angelopoulos, V., Eds., The ARTEMIS Mission, Springer, New York, 61-91. https://doi.org/10.1007/978-1-4614-9554-3_4
[7]
Shirobokov, M., Trofimov, S. and Ovchinnikov, M. (2017) Survey of Station-Keeping Techniques for Libration Point Orbits. Journal of Guidance, Control, and Dynamics, 40, 1085-1105. https://doi.org/10.2514/1.G001850
[8]
Meyer, K.R. and Schmidt, D.S. (1986) The Stability of the Lagrange Triangular Point and a Theorem of Arnold. Journal of Differential Equations, 62, 222-236. https://doi.org/10.1016/0022-0396(86)90098-7
[9]
Gómez, G., Jorba, A., Masdemont, J. and Simó, C. (1993) Study of the Transfer from the Earth to a Halo Orbit around the Equilibrium pointL1. Celestial Mechanics and Dynamical Astronomy, 56, 541-562. https://doi.org/10.1007/BF00696185
[10]
Richardson, D.L. (1980) Halo Orbit Formulation for the ISEE-3 Mission. Journal of Guidance, Control, and Dynamics, 3, 543-548. https://doi.org/10.2514/3.56033
[11]
Cielaszyk, D. and Wie, B. (1996) New Approach to Halo Orbit Determination and Control. Journal of Guidance, Control, and Dynamics, 19, 266-273. https://doi.org/10.2514/3.21614
[12]
Ardaens, J.S. and D’Amico, S. (2008) Control of Formation Flying Spacecraft at a Lagrange Point. No. 00-08.
[13]
Xu, M. and Xu, S.J. (2008) Trajectory and Correction Maneuver during the Transfer from Earth to Halo Orbit. Chinese Journal of Aeronautics, 21, 200-206. https://doi.org/10.1016/S1000-9361(08)60026-6
[14]
Yamato, H. and Spencer, D. (2004) Transit-Orbit Search for Planar Restricted Three-Body Problems with Perturbations. Journal of Guidance, Control, and Dynamics, 27, 1035-1045. https://doi.org/10.2514/1.4524
[15]
Lyapunov, A.M. (1992) The General Problem of the Stability of Motion. International Journal of Control, 55, 531-534. https://doi.org/10.1080/00207179208934253
[16]
Jacobi, C.G.J. (1836) Sur le mouvement d’un point et sur un cas particulier du probleme destrois corps. Comptes Rendus Chimie, 3, 59-61.
[17]
Van Der Schaft, A.J. and Schumacher, J.M. (2000) An Introduction to Hybrid Dynamical Systems. Springer, London, 251. https://doi.org/10.1007/BFb0109998
[18]
Van Der Schaft, A.J., Jeltsema, D. et al. (2014) Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends in Systems and Control, 1, 173-378. https://doi.org/10.1561/2600000002
[19]
Ortega, R., Van Der Schaft, A.J., Mareels, I. and Maschke, B. (2001) Putting Energy Back in Control. IEEE Control Systems Magazine, 21, 18-33. https://doi.org/10.1109/37.915398
[20]
Ortega, R., Van Der Schaft, A.J., Maschke, B. and Escobar, G. (2002) Interconnection and Damping Assignment Passivity-Based Control of Port-Controlled Hamiltonian Systems. Automatica, 38, 585-596. https://doi.org/10.1016/S0005-1098(01)00278-3
[21]
Ortega, R., Van Der Schaft, A.J., Castanos, F. and Astolfi, A. (2008) Control by Interconnection and Standard Passivity-Based Control of Port-Hamiltonian Systems. IEEE Transactions on Automatic Control, 53, 2527-2542. https://doi.org/10.1109/TAC.2008.2006930
[22]
Liu, C. (2019) Teaching Control Theory in Physics: The Port-Hamiltonian Framework. College Physics, 38, 1-7.
[23]
Liu, C. and Dong, L. (2019) Physics-Based Control Education: Energy, Dissipation, and Structure Assignments. European Journal of Physics, 40, Article ID: 035006. https://doi.org/10.1088/1361-6404/ab03e8
[24]
Liu, C. and Dong, L. (2019) Stabilization of Lagrange Points in Circular Restricted Three-Body Problem: A Port-Hamiltonian Approach. Physics Letters A, 383, 1907-1914. https://doi.org/10.1016/j.physleta.2019.03.033
[25]
LaSalle, J. (1960) Some Extensions of Liapunov’s Second Method. IRE Transactions on Circuit Theory, 7, 520-527. https://doi.org/10.1109/TCT.1960.1086720
[26]
Kwakernaak, H. and Sivan, R. (1972) Linear Optimal Control Systems. Wiley-Inter- science, New York.
[27]
Zhou, K., Doyle, J.C., Glover, K., et al. (1996) Robust and Optimal Control. Prentice Hall, Upper Saddle River.
