Torsional stick-slip vibrations decrease the performance, reliability and fail-safety of drilling systems used for the exploration and harvesting of oil, gas, min- erals and geo-thermal energy. Current industrial controllers regularly fail to eliminate stick-slip vibrations, especially when multiple torsional flexibility modes in the drill- string dynamics play a role in the onset of stick-slip vibrations. This chapter presents the experimental validation of novel robust output-feedback controllers designed to eliminate stick-slip vibrations in the presence of multiple dominant torsional flexibility modes. For this purpose, a representative experimental test setup is designed, using a model of a real-life drilling rig as a basis. The model of the dynamics of the experimental setup can be cast in Lure-type form with set-valued nonlinearities representing an (uncertain) model for the complex bit-rock interaction and the interaction between the drill-string and the borehole. The proposed controller design strategy is based on skewed-m-DK-iteration and aims at optimizing the robustness with respect to uncertainty in the non-smooth bit-rock interaction. Moreover, a closed-loop stability analysis for the non-smooth drill-string model is provided. Experimental results confirm that stick-slip vibrations are indeed eliminated using the designed controller in realistic drilling scenarios in which state-of-practice controllers have failed to achieve the same.