The method of synthesis of continuous locally optimal controls in reaction wheel attitude stabilization systems

The article is devoted to modeling a system of automatic control of rotational motion that employs reaction wheels to generate control torque. A method is formulated for discretizing the dynamics equations of the control object, which take into account the derivatives of control actions, for both piecewise-constant and piecewise-linear input actions. A method is proposed for calculating stabilizing controls that are optimal at each step and that accounts for the continuity constraint on piecewise-linear controls. The global continuity of the flywheels' angular velocities is ensured by taking into account, in the coefficients of their linear functions, the control increments at previous steps (assuming that only the flywheels' angular velocities affect the system dynamics). The locally optimal controls are calculated using the Lagrange multiplier method. As a result of the synthesis, it is established that the discrete system under consideration is nonstationary, as are the matrices of the total control quality functional. The formulated method for calculating the total quality functional accounts for the continuity constraint on piecewise-linear controls. A locally optimal control operator is calculated that delivers the minimum to the discrete quality functional at each step of the system simulation. The simulation results show that the control operator makes it possible to solve the problem of stabilizing the deflection angles of the control object in the vicinity of the equilibrium position within a specified finite time. Differences in the nature of the transient processes are demonstrated depending on how the matrices of the total process quality functional are defined. The results of the work are of theoretical significance for solving synthesis problems of discrete systems with globally continuous piecewise-linear controls.

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