In this paper, the robust superconductor flywheel energy storage system(SFESS) controller using $H_{infty}$ control theory was designed to damp low frequency oscillation of power system. The main advantage of the $H_{infty}$ controller is that uncertainties of power system can be included at the stage of controller design. Both disturbance attenuation and robust stability for the power system were treated simultaneously by using mixed sensitivity $H_{infty}$ problem. The robust stability and the performance for uncertainties of power system were represented by frequency weighted transfer function. To verify control performance of proposed SFESS controller using $H_{infty}$ control, the closed loop eigenvalue and the damping ratio in dominant oscillation mode of power system were analyzed and nonlinear simulation for one-machine infinite bus system was performed under disturbance for various operating conditions. The results showed that the proposed $H_{infty}$ SFESS controller was more robust than conventional power system stabilizer (PSS).