Control of mechanical systems with backlash is a topic well studied by many control practitioners. This interest has been motivated by the fact that backlash in mechanical systems can cause severe performance degradation and lead to instability of the control system. Furthermore, high impact-forces in backlash-systems can lead to a lower durability of the components and to strokes and peaks in the output. In this paper a mechanical benchmark system is presented to provide facilities for testing the identification and control of systems with backlash. For controller design a hybrid model of the system was derived and used in a model predictive control (MPC) scheme. Observer-based state-estimation was used to recover unmeasured states, particularly the backlash angle. Explicit solutions of a tracking controller were computed to control the mechanical benchmark system in real-time. Simulation as well as experimental results are presented to show the applicability of this hybrid control approach.