In any drive system, there are always couplings between the motor and the load. Since the hardness of these couplings is finite, they have elastic properties, causing unwanted vibration and negatively affecting system quality. When the couplings are springs with nonlinear characteristics, control is particularly difficult because it is very difficult or impossible to define the parameters of the controlled object. To solve these difficulties, this article proposes an adaptive controller of the major functions for controlling a drive system with nonlinear elastic couplings of unidentified parameters. For the proposed control system, we measure the response speed of the object, use a Luenberger observer to estimate the state variables of the system, and use an adaptive controller to control the system. The experimental results demonstrate that the control object can be controlled without knowing the parameters: the control quality of the system is very good, close to that of a system with a hard coupling, there is no vibration or overshoot, and the transition time is small.