Hydropneumatic Passive Heave Compensator (PHC) for offshore drilling may exhibit undesirable effects due to nonlinearities of the friction seal, causing the so called Compensator-Induced Vibration (CIV) phenomenon, which induces longitudinal oscillations of frequencies higher than the heave motion. In this paper, a control system is proposed to attenuate the heave motion and to mitigate the CIV and compared to common practice, such as a linear feedforward controller. The proposed control is composed of a nonlinear feedforward to mitigate the nonlinear seal friction of PHC and a feedback CRONE (French acronym for fractional order robust control) controller to address the CIV. The CRONE control is designed based on a reduced order model of the drill sting and the linearized model of the PHC. This control drastically reduces the CIV and the transmission of heave motion, highlighting the limitation of the linear feedforward. Furthermore, it is shown that the proposed control strategy can satisfactory attenuate the heave motion (by 95%) and the CIV phenomenon (by 88%), despite noisy measurements by the inclusion of a Kalman and a Butterworth filters.