Stick-slip vibrations decrease the performance, reliability, and fail safety of drilling systems. The aim of this paper is to design a robust output-feedback control approach to eliminate torsional stick-slip vibrations in drilling systems. Current industrial controllers regularly fail to eliminate stick-slip vibrations, especially when multiple torsional flexibility modes play a role in the onset of stick-slip vibrations. As a basis for controller synthesis, a multimodal model of the torsional dynamics for a real drill-string system is employed. The proposed controller design strategy is based on skewed-\mu DK iteration and aims at optimizing the robustness with respect to uncertainty in the nonlinear bit-rock interaction. Moreover, a closed-loop stability analysis for the nonlinear drill-string model is provided. This controller design strategy offers several benefits compared with existing controllers. First, only surface measurements are employed, therewith avoiding the need for down-hole measurements. Second, multimodal drill-string dynamics are effectively dealt with in ways inaccessible to state-of-practice controllers. Third, robustness with respect to uncertainties in the bit-rock interaction is explicitly provided and closed-loop performance specifications are included in the controller design. Case study results confirm that stick-slip vibrations are indeed eliminated in realistic drilling scenarios using the designed controller in which state-of-practice controllers fail to achieve this.