The extensive use of frequency-domain tools for analyzing and controlling linear systems have become indispensable for the control systems engineer. However, due to the increased performance demands on today's industrial systems, the effects of certain nonlinearities can no longer be neglected in control applications, and the use of these tools becomes problematic. In the current literature, however, frequency-domain methods exist where the underlying linear dynamics of a nonlinear system can be captured in an identification experiment; in this manner, the nonlinear system is replaced by a linear model with a noise source where a best linear approximation of the nonlinear system is obtained with an associated frequency-dependent uncertainty. This allows the use of robust control algorithms to ensure performance for the underlying linear system. In this paper, a data-driven $\mathcal{H}_\infty$ robust control strategy is presented which implements a convex optimization algorithm to ensure the performance and closed-loop stability of a linear system that is subject to nonlinear distortions. A case study is presented to illustrate how the proposed method can be used to design controllers for this class of systems.