Comparative analysis of autonomous control methods for small spacecraft

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription or Fee Access

Abstract

A specialized methodology was developed, computer modeling was conducted, and a comparative analysis was performed for autonomous control methods for small spacecraft, differing in how the nonlinearity of the initial stabilization problem is taken into account:

— control constructed using a system of linear differential equations (SLDE);

— control constructed using SLDE applied to a more general system of differential equations in which nonlinear terms are taken into account in trajectory calculations;

— control using the SDRE method for a nonlinear system of differential equations. To evaluate the effectiveness of the three control methods under consideration under identical initial conditions, stabilization problems using the quadratic quality criterion (LQR) were considered.

A computational experiment showed that, in most cases, the SDRE method yields the lowest quality functional of the three control methods considered. However, the final quality functional values for control constructed using SLDE with nonlinear terms remain consistently higher than those for SDRE. This is due to the fact that the nonlinear term significantly contributes to the increase in the quality functional in the initial equations. Unlike linear optimal control, the SDRE method used does not provide the necessary condition for a minimum performance functional, but it does not require the linearity of the initial differential equation system, which is a significant advantage. Despite the lack of a mathematical proof of the minimal performance functional for all cases, numerical experiments nevertheless demonstrate the advantages of the SDRE method for calculating control. However, the lack of a proven minimum performance functional allows this method to be considered a suboptimal control method. The described approach, using a mathematical model of suboptimal control for small spacecraft, allows it to be used to solve energy-intensive problems that largely determine the operational life of such spacecraft

Full Text

Restricted Access

About the authors

M. A. Bubnova

National Research University Higher School of Economics

Author for correspondence.
Email: mbubnova@hse.ru
ORCID iD: 0009-0008-3710-6606

PhD Student, Assistant

Russian Federation, Moscow

V. M. Chetverikov

National Research University Higher School of Economics

Email: vchetverikov@hse.ru
ORCID iD: 0000-0003-3592-4441

Dr. of Phys.-Math. Sci., Professor

Russian Federation, Moscow

E. D. Pozhidaev

National Research University Higher School of Economics

Email: epozhidaev@hse.ru
ORCID iD: 0000-0002-3234-5236

Dr. of Tech. Sci., Professor

Russian Federation, Moscow

References

  1. Petrukovich A. A., Nikiforov O. V. Small satellites for space research, Raketno-kosmicheskoe priborostroenie i informatsionnye sistemy, 2016, no. 4, pp. 22—31, doi: 10.17238/issn2409-0239.2016.4.22.
  2. Chernyshov A. A. et al. Study of the inhomogeneous ionospheric structure using simultaneous measurements by CubeSat nanosatellites, Izvestiya vuzov. Priborostroenie, 2016, vol. 59, no. 6, pp. 443—449 (in Russian).
  3. Sining L., Panagiotis T., Raad R. et al. А Survey on CubeSat Missions and Their Antenna Designs, Electronics, 2022, no. 11, pp. 1—46, doi: 10.3390/electronics11132021.
  4. Klimenko A. G., Kudinov A. S., Olkhovskaya V. V., Suetin B. P., Yurchenko I. I. Modern trends in propulsion system development for CubeSat-class spacecraft, Kosmonavtika i raketostroenie, 2019, no. 1 (106), pp. 92—100 (in Russian).
  5. Kolganov I. V. Review of electric rocket engines for propulsion systems of small spacecraft, Omskiy nauchnyy vestnik. Seriya Aviatsionno-raketnoe i energeticheskoe mashinostroenie, 2025, vol. 9, no. 2, pp. 94—103, doi: 10.25206/2588-0373-2025-9-2-94-103 (in Russian).
  6. Kartsan I. N. Ground control complex for small spacecraft, Vestnik Sibirskogo gosudarstvennogo aerokosmicheskogo universiteta imeni akademika M. F. Reshetneva, 2009, pp. 89—92 (in Russian).
  7. Khanov V. Kh., Shakhmatov A. V., Chekmarev S. A., Vergazov M. Yu., Lukin F. A. Concept of onboard control complex development for small spacecraft, Vestnik Sibirskogo gosudarstvennogo aerokosmicheskogo universiteta imeni akademika M. F. Reshetneva, 2012, pp. 144—149 (in Russian).
  8. Konovalov V. N., Korlyakova M. O. Approach to developing control systems for small spacecraft on a neural-network basis, Inzhenernyy zhurnal: nauka i innovatsii, 2014, iss. 5, available at: http://engjournal.ru/catalog/it/nav/1271.html (date of access: 26.01.2026) (in Russian).
  9. Malyshkin I. A., Strelnikov S. V. Organization of control processes for multisatellite orbital constellations of small spacecraft, Raketno-kosmicheskoe priborostroenie i informatsionnye sistemy, 2025, vol. 12, no. 3, pp. 20—30 (in Russian).
  10. Bordovitsyna T. V., Avdyushev V. A. Theory of Artificial Earth Satellites Motion: Analytical and Numerical Methods, Tomsk, Tomsk State University Publishing House, 2007, 172 p. (in Russian).
  11. Ivanov D. S., Kushniruk M. S. Study of an algorithm for controlling the spatial motion of a satellite group using aerodynamic force, Preprints of Keldysh Institute of Applied Mathematics, 2017, no. 53, 32 p., doi: 10.20948/prepr-2017-53, available at: http://library.keldysh.ru/preprint.asp?id=2017-53 (date of access: 26.01.2026) (in Russian).
  12. Belokonov I. V., Timbay I. A., Orazbaeva U. M. Resonant motion of a CubeSat nanosatellite in low circular orbits, Izvestiya vuzov. Priborostroenie, 2018, vol. 61, no. 5, pp. 458—464 (in Russian).
  13. Bogdanov K. A., Timakov S. N., Zykov A. V., Subbotin А. V. Relay autonomous control system for a satellite constellation based on a low Earth orbit, Kosmicheskaya tekhnika i tekhnologii, 2020, no. 1 (28), pp. 98—110 (in Russian).
  14. Monakhova U. V., Shestakov S. A., Mashtakov Ya. V., Ivanov D. S. Decentralized motion control of a swarm of small spacecraft for maintaining communication connectivity, Kosmicheskie issledovaniya, 2024, vol. 62, no. 1, pp. 105—120 (in Russian).
  15. Andrievskiy B. R., Kuznetsov N. V., Popov А. M. Aerodynamic control algorithms for relative motion of two satellites in a near-circular orbit: linearized model equations with aerodynamic drag taken into account, Differentsialnye uravneniya i protsessy upravleniya, 2020, no. 4, available at: http://diffjournal.spbu.ru/ (date of access: 26.01.2026) (in Russian).
  16. Zhaldybina O. D., Mordanov M. R. Analysis of atmospheric influence on the motion of spacecraft in low Earth orbits, LI Samarskaya oblastnaya studencheskaya nauchnaya konferentsiya, 2025, vol. 1, pp. 413—415, available at: https://journals.eco-vector.com/osnk-sr2025/article/view/679827 (date of access: 26.01.2026) (in Russian).
  17. Iskenderov I. A., Bagirzade S. S. On improving control accuracy for a small spacecraft, Aviakosmicheskoe priborostroenie, 2019, no. 9, pp. 3—8, doi: 10.25791/aviakosmos.09.2019.862 (in Russian).
  18. Heydari A., Balakrishnan S. N. Approximate Closed-Form Solutions to Finite-Horizon Optimal Control of Nonlinear Systems, Proceedings of the 2012 American Control Conference (ACC), Montreal, QC, Canada, 27—29 June 2012.
  19. Afanasyev V. N., Kupriyanov A. O., Neusypin K. A. Navigation complex correction algorithm based on a differential game approach, Aviakosmicheskoe priborostroenie, 2025, no. 9, pp. 37—45 (in Russian).
  20. Basak K., Giri D. K. LQR based Optimal Control Design of Satellite Formation Flight in Earth-centered Circular Orbit, AIAA SciTech Forum, AIAA 2022-0763, doi: 10.2514/6.2022-0763.
  21. Afanasyev V. N. Control of Nonlinear Uncertain Dynamical Objects, Moscow, LENAND, 2015 (in Russian).
  22. Cimen T. State-Dependent Riccati Equation (SDRE) Control: А Survey, Proceedings of the 17th IFAC World Congress (IFAC’08), Seoul, Korea, 6—11 July 2008, pp. 3761—3775, doi: 10.3182/20080706-5-KR-1001.00635.
  23. Massari M., Zamaro M. Application of SDRE technique to orbital and attitude control of spacecraft formation flying, Acta Astronautica, 2013, doi: 10.1016/j.actaastro.2013.02.001.
  24. Heydari A., Balakrishnan S. N. Closed-Form Solution to Finite-Horizon Suboptimal Control of Nonlinear Systems, International Journal of Robust and Nonlinear Control, 2015, vol. 25, pp. 2687—2704, doi: 10.1002/rnc.3222.
  25. Ivanov D., Amaro G., Mashtakov Y., Ovchinnikov M., Guerman А. Formation Flying Lyapunov-Based Control Using Lorentz Forces, Aerospace, 2023, vol. 10, art. 39, doi: 10.3390/aerospace10010039.
  26. Jin Z., Bai L., Wang Z., Zhang P. Self-Triggered Distributed Formation Control of Fixed-Wing Unmanned Aerial Vehicles Subject to Velocity and Overload Constraints, IEEE Transactions on Automation Science and Engineering, 2024, vol. 21, pp. 4082—4093, doi: 10.1109/TASE.2023.3292176.
  27. Bakhtiari M., Panahyazdan A., Abbasali E. Finite-Time Control for Satellite Formation Reconfiguration and Maintenance in LEO: А Nonlinear Lyapunov-Based SDDRE Approach, Aerospace, 2025, vol. 12, art. 201, doi: 10.3390/aerospace12030201.
  28. Afanasyev V. N. Control of Nonlinear Uncertain Dynamic Objects, Moscow, LENAND, 2015.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download
2. Fig. 1. The relationship between coordinates in inertial and non-inertial reference frames [20]

Download (44KB)
Indexing metadata
3. Fig. 2. Diagram of the leading satellite's position (point O), moving in a circular orbit around the Earth with the center at the Earth's center — the point with coordinates {x, y} = {–1, 0}

Download (43KB)
Indexing metadata
4. Fig. 3. Spatial trajectories (a) and velocity change trajectories (b) during the movement of the follower satellite from the initial point {x0, y0, z0, v0, w0, V0} = {–0.45; 0.95; 0.1; 0.3; 0.3; 0} to the origin; changes in Lyapunov functions (c) and the performance index (d)

Download (85KB)
Indexing metadata
5. Fig. 4. Spatial trajectories (a) and velocity change trajectories (b) during the movement of the follower satellite from the initial point {x0, y0, z0, v0, w0, V0} = {0.55, 0.78, 0.1, 0.5, –0.5, 0} to the origin; changes in the Lyapunov functions (c) and the performance index (d)

Download (84KB)
Indexing metadata
6. Fig. 5. Spatial trajectories (a) and velocity change trajectories (b) during the movement of the follower satellite from the initial point {x0, y0, z0, v0, w0, V0} = {–0.55; 0.95; 0.1; 0.7; –0.7; 0} to the origin; change of the Lyapunov function (c) and the performance functional (d)

Download (81KB)
Indexing metadata

Copyright (c) 2026 Informacionnye Tehnologii


СМИ зарегистрировано Федеральной службой по надзору в сфере связи, информационных технологий и массовых коммуникаций (Роскомнадзор).
Регистрационный номер и дата принятия решения о регистрации СМИ: серия ПИ № 77 - 15565 от 02 июня 2003 г.