Two different approaches to state estimation and state control of electrical drives with elastic transmission are presented. Since in many practical applications load speed and/or coupling torque (torsional angle) measurement is difficult or even impossible to achieve, it is reasonable to apply suitable state estimators (observers). Therefore, design procedures for two types of estimators and state controllers are presented in this paper: design of the state controller and the full-state estimator that includes speed estimation at motor and load side, as well as torsion angle and load torque estimation, and the design of the PI motor speed controller with additional coupling torque feedback (further referred to as PIm controller) obtained by the reduced coupling torque estimator. The control system is optimized according to the damping optimum, taking into account the influence of the estimator dynamics. The behavior of the designed control system is examined by simulation and experimentally on a laboratory model of the electrical drive which is accomplished by two synchronous motors with permanent magnets (SMPM) with variable inertia and driven by current controlled frequency converters. Motors are coupled by exchangeable axes with different stiffness coefficients.