Abstract
The properties of the solution to the well-known problem of the optimal orientation of a satellite in the plane of a circular orbit and its steering to a gravitationally stable angular position, which was posed and considered earlier by Vladimir V. Beletskij, have been refined. The constraint on the control torque module is considered as the main parameter of the problem. It turned out that, even in the case when the admissible control torque exceeds the gravitational moment in magnitude, in the phase plane the existence of additional switching curves corresponding to relay control with two switchings is possible. As a result, a simple numerical algorithm is proposed, and with its help the threshold absolute control value is found at which the indicated switching curves are infinitesimal, and their coordinates in the phase plane are found.
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References
V. V. Beletskij, On Optimal Guiding of an Artificial Earth Satellite into a Gravitationally Stable Attitude, Cosmic Res. 9 (3), 337 (1971).
B. Friedland and P. Sarachik, Indifference Regions in Optimum Attitude Control, IEEE Trans. Automatic Control 9 (2), 180 (1964).
Chernousko F.L., Optimal Control of Two-Dimensional Motions of a Body by a Movable Mass, Proc. 9th Vienna International Conference on Mathematical Modelling. 21–23 February (2018); IFAC-PapersOnLine 51 (2), 232 (2018).
A. N. Sirotin. A Family of Trigonometric Extremals in the Problem of Reorienting a Spherically Symmetrical Body with Minimum Energy Consumption. J. Appl. Mat. Mech. 77 (2), 205 (2013).
M. V. Levskii, Analytic Controlling Reorientation of a Spacecraft Using a Combined Criterion of Optimality. J. Comp. Syst. Sci. Int. 57 (2), 283 (2018).
V. S. Aslanov and S. P. Bezglasnyi, Gravitational Stabilization of a Satellite Using a Movable Mass. J. Appl. Math. Mech. 76 (4), 405 (2012).
S. P. Bezglasnyi and A. A. Mukhametzyanova, Gravitational Stabilization and Reorientation of a Satellite- Dumbbell in the Circular Orbit under the Swing Principles, Avtomatizatsiya Protsessov Upravleniya, No. 1 (43), 91 (2016).
A. P. Markeev, Theoretical Mechanics (NITs “Regular and Chaotic Dynamics”, Moscow–Izhevsk, 2007).
S. A. Reshmin, Bifurcation in a Time-Optimal Problem for a Second-Order Non-Linear System. J. Appl. Math. Mech. 73 (4), 403 (2009).
S. A. Reshmin, Dispersal Curve Properties in the Time Minimization Problem for a Second-Order Nonlinear System. J. Comp. Syst. Sci. Int. 51 (3), 366 (2012).
S. A. Reshmin and F. L. Chernousko, Properties of the Time-Optimal Feedback Control for a Pendulum-Like System. J. Optimiz. Theory Appl. 163 (1), 320 (2014).
S. A. Reshmin, Estimate of the Control Threshold Value in the Problem on a Time-Optimal Satellite Attitude Transition Maneuver, Mech. Solids 52 (1), 9 (2017).
E. B. Lee and L. Markus, Foundations of Optimal Control Theory (Wiley, New York, 1967; Nauka, Moscow, 1972).
J. L. Garcia Almuzara and I. Flügge-Lotz, Minimum Time Control of a Nonlinear System. J. Differen. Equats 4 (1), 12 (1968).
S. A. Reshmin, Finding the Principal Bifurcation Value of the Maximum Control Torque in the Problem of Optimal Control Synthesis for a Pendulum. J. Comp. Syst. Sci. Int. 47 (2), 163 (2008).
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Original Russian Text © S.A. Reshmin, 2018, published in Doklady Akademii Nauk, 2018, Vol. 480, No. 6, pp. 671–675.
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Reshmin, S.A. The Threshold Absolute Value of a Relay Control Bringing a Satellite to a Gravitationally Stable Position in Optimal Time. Dokl. Phys. 63, 257–261 (2018). https://doi.org/10.1134/S1028335818060101
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DOI: https://doi.org/10.1134/S1028335818060101