This paper presents issues related to the adaptive control of the drive system with an elastic clutch connecting the main motor and the load machine. Firstly, the problems and the main algorithms often implemented for the mentioned object are analyzed. Then, the control concept based on the RNN (recurrent neural network) for the drive system with the flexible coupling is thoroughly described. For this purpose, an adaptive model inspired by the Elman model is selected, which is related to internal feedback in the neural network. The indicated feature improves the processing of dynamic signals. During the design process, for the selection of constant coefficients of the controller, the PSO (particle swarm optimizer) is applied. Moreover, in order to obtain better dynamic properties and improve work in real conditions, one model based on the ADALINE (adaptive linear neuron) is introduced into the structure. Details of the algorithm used for the weights’ adaptation are presented (including stability analysis) to perform the shaft torque signal filtering. The effectiveness of the proposed approach is examined through simulation and experimental studies.