The linear motion of the Corner Cube Mechanism developed for the infrared sounder of the third generations of Meteosat weather satellites requires a high level of accuracy. The system is subject to external micro-vibration perturbations from surrounding instruments, which cannot be rejected with the current PID controllers with notch filters. A data-driven H-infinity robust controller design method is proposed to improve the control performance. The method uses only frequency-domain data and satisfies the constraints on the weighted infinity-norm of sensitivity functions using the convex optimization algorithms. The frequency response of the system is identified from the finite element model of the system. The designed controller is validated in simulation. The performance improvement with respect the PID controller with notch filters is illustrated via experimental results.