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This paper deals with the application of the adaptive control structure for torsional vibration suppression in the drive system with an elastic coupling. The proportional-integral speed controller and gain factors of two additional feedback loops, from the shaft torque and load side speed, are tuned on-line according to the changeable load side inertia. This parameter, as well as other mechanical variables of the drive system (load side speed, torsional and load torques), are estimated with the use of the developed nonlinear extended Kalman filter (NEKF). The initial values of the Kalman filter covariance matrices are set using the genetic algorithm. Then, to ensure the smallest state and parameter estimation errors, the on-line adaptation law for the chosen element of the state covariance matrix of the NEKF is proposed. The described control strategy is tested in an open and a closed-loop control structure. The simulation results are confirmed by laboratory experiments.