Boundary control of flexible satellite vibration in planar motion

In this paper, the planar maneuver of a flexible satellite with regard to its flexible appendages vibration has been studied. The flexible satellite translates and rotates in a plane; in addition, the flexible appendages can also vibrate in that plane. The system governing equations, which are coupled partial and ordinary differential equations, are obtained based on Hamilton's principle. Then the original system converts to three equivalent subsystems, two of which contains one partial differential equation and one ordinary differential equation along with four boundary conditions, by using change of variables. Employing control forces and one control torque which are applied to the central hub, control objectives, tracking the desired angle and suppressing the satellite and its flexible appendages vibrations, are fulfilled. Furthermore, to eliminate spillover instability phenomenon and to exclude in-domain measurement and actuator usage problem, these control torque and forces are designed based on boundary control method. Lyapunov's direct method is employed to prove the asymptotic stability in absence of any damping effect in modeling the vibrations of flexible appendages. Eventually, in order to demonstrate the effectiveness of the designed boundary control, numerical simulations are presented.