This paper presents issues related to the damping of torsional vibrations in a system with elastic coupling. A novel PI-based control structure with additional feedback from torsional torque and its derivative is proposed. For the estimation of the required variables, the integral observer is proposed. Analytical expressions are presented to enable the selection of parameters of the control system. The relationship between the considered system and popular structures with a PI controller and one additional feedback from torsional torque and the derivative of torsional torque is pointed out. The proposed control structure is tested under simulation and experimental studies.