The problem of the quasi-optimal stabilization of small oscillations of mechanical systems with many degrees of freedom with slowly varying parameters and small perturbations is considered. For the synthesis of approximately optimal controllers, the Bellman dynamic programming principle is used in combination with the averaging method. The averaging method is used to obtain an asymptotic solution of the Hamilton-Jacobi-Bellman equations. The proposed synthesis algorithm makes it possible to significantly simplify the problem of designing approximately optimal controllers, and to obtain an analytical solution for the particular cases under consideration. As an example, the problem of stabilizing the motion of a small spacecraft (SC) relative to the direction of the tether during the deployment of a space tether system is considered.
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