Optimal damping of deviations of angular speeds of an eightsymetric space aircraft

1Stenin, АA, 1Pasko, VP, 1Drozdovych, IG, 2Soldatova, MO
1National Technical University of Ukraine "Kyiv Polytechnic Institute named after Igor Sikorsky", Kyiv, Ukraine
2Institute of Telecommunications and Global Information Space of the National Academy of Sciences of Ukraine, Kyiv, Ukraine
Space Sci. & Technol. 2021, 27 ;(4):21-31
https://doi.org/10.15407/knit2021.04.021
Publication Language: Ukrainian
Abstract: 
This paper considers the problem of optimal fuel consumption damping of sudden deviations of angular velocities of an axisymmetric spacecraft with a constant speed of rotation around the main axis of symmetry. This assumption has some practical significance and may be due to the creation of artificial gravity on the spacecraft.
         The idea of artificial gravity due to the rotation of an axisymmetric cylindrical spacecraft is based on the principle of equivalence of the force of gravity and the force of inertia. The urgency of the fuel consumption optimization problem is due to the presence of its limited stock onboard the spacecraft.
The optimization problem is solved based on the maximum principle and the phase plane method. The authors of the article determine the structure of optimal fuel consumption processes with three levels of control, and the number of their switches depends on the initial conditions.
          Synthesized on the phase plane, the optimal switching curves divide the phase plane into eight curvilinear quadrants, which uniquely determine the values of the optimal control effects by the current values of the deviations of the angular velocities of the spacecraft. The problem of the possible presence of a delay in the control loop is proposed to be solved based on the Bess compensation method. To do this, the corresponding optimal curves of switching and disabling the controls are built as geometric locations of points remoted for the time of delay from the found curves of switching and the beginning of coordinates accordingly. It allows us to avoid the emergence of steady self-oscillations in a control contour and to provide a condition of keeping the spacecraft in a given final state after the completion of the stabilization process. Depending on the technical equipment of the spacecraft, two variants of the optimal damping algorithm are offered, namely: an autonomous device in the onboard control system of the spacecraft in the absence of a sufficiently powerful onboard computer, or the optimal damping algorithm, implemented entirely in the onboard computer of the spacecraft in case of its sufficient power.
Keywords: angular stabilization of spacecraft, axisymmetric spacecraft, Bess method, fuel consumption optimization, maximum principle, optimal switching curves, phase plane, the predictive models
References: 
1. Anuchin O. N., Komarova I. E., Porfiriev L. F. (2004). Onboard navigation and orientation systems of artificial earth satellites. Saint Petersburg: Central research Institute «Electropribor».
2. Brovkin A. G., Burdygov B. G.,Gordiyko S. V., et al. (2010). Onboard control systems for space vehicles: Textbook. Moscow: MAI-PRINT Publishing house.
3. .Zelenskiy K. H., Ignatenko V. M., Stenin O. A. (2017). Structural power of optimal vitrate control processes in dynamic systems. Adaptive automatic control systems, No 2 (31), 12—16.
4. Ignatenko N. M., Kobelev N. S., Gromkov A. S. (2015). Trends in the development of correcting rocket engines of spacecraft. Fundamental and applied research in the field of high space technologies of Russia and foreign countries. Eds S. N. Frolov et al. Kursk, 34—46.
5. Kravchuk S. V., Shatsky M. A., Koval A. Yu. (2010). Principles of construction of the spacecraft motion control system. Kosmicheskaya tekhnika i tekhnologiya, No 38, 1—5.
6. Kulik A. S., Luchenko O. A., Gavrilenko O. I. (2005). Solution of the problem of precesion orientation of a space aircraft. Radio electronics, computer science, control. Zaporozhye, 69—78.
7. Lovchakov V. I., Solovyov V. E., Yu. Yu. (2013). Dorokhov Modified method of phase space in solving speed problems. Izvestiya of Tula state University, series Technical Sci., 2, 217—224.
8. Pikina G. A., Kocharovsky D. N. (2006). Investigation of a system with a predictive algorithm of maximum speed. Teploenergetika, No 10, 49—52.
9. Pontryagin L. S., Boltyansky V. G., Gamkrelidze R. V., Mishchenko E. F. (1989). Mathematical theory of optimal processes: 4th edition. Moscow: Nauka.
10. Stenin A. A., Burlakov V. M., Strakhova N. V. (1996). Optimal fuel consumption control of the angular position of the spacecraft. Probl. management and Inform., No 5, 109—118.
11. Athans M., Peter L. Falb Optimal Control: An Introduction to the Theory and Its Applications. Courier Corporation, 2006.
12. Bass R. W. (1956). Improved on-off Missile Stabilization. Jet Propulsion, 26, 415—417.
13. Beard R. W., Hadaegh F. Y. Fuel Optimization for Unconstrained Rotation of Spacecraft Formations. J. Astronautical Sci., 47(3).
14. Fagerholt K., Laporte G., Norstad I. (2010). Reducing fuel emissions by optimizing speed on shipping routes. J. Operational Res. Soc., 61, No 3, 523—529.
15. Gulko F. B., Kogan B. Y., Lerner A. Y., Mikhailov N. N., Novoseltseva Z. A. (1964). Predictive control methods using high speed analog computers and its applications. Automatica and Telemekhanica, 25, 896—908.
16. Kim J.-G., Kim H.-J., Lee P. T.-W. (2014). Optimizing ship speed to minimize fuel consumption. Transportation Letters, 6 (3), 109—117.
17. Kostyuk V. I., Stenin A. A., Ignatenko V. N. (1977). Optimal fuel control of systems with delay. System Sci., 3 (2), 159—169.
18. Mikhalyov A. I., Stenin A. A., Ignatenko V. N., Soldatova M. A., Stenin A. S. (2018). Synthesis of optimal consumption fuels one class of linear nonstationary systems (the method of predicted control). System technologies. Regional interuniversity collection, No 6 (119), 64—72.
19. Polyak B., Shably L. (2019). Minimum fuel-consumption stabilization of a spacecraft at the Lagrangian points. Automat. and Remote Contr., 80(12), 2217—2228.