Repetitive control (RC) enables high control performance for systems subject to periodic disturbances. Many disturbances in mechatronic applications involve the sum of multiple periodic signals, which is not necessarily periodic over a small time interval. The contribution of this article is a design procedure for multiple repetitive controllers, each addressing a single periodic component that achieves high performance with fast convergence in the presence of multiple periodic disturbances. The developed approach involves sequential design of the controllers, and systematically addresses modeling errors and coupling between controllers. An experimental case study on an industrial roll-to-roll printer confirms the benefits of the proposed approach, including superior performance compared to preexisting RC designs.