Learning from past data enables substantial performance improvement for systems that perform repeating tasks. Achieving high accuracy and fast convergence in the presence of unknown disturbances typically imposes requirements on the available system knowledge. The aim of this paper is to develop a data-driven approach that achieves high tracking performance through learning for Linear Time-Invariant (LTI) systems whose dynamics are unknown and that are subject to unknown disturbances. This is achieved by developing an Iterative Inversion-based Control (IIC) framework that employs a nonlinear input updating strategy to ensure fast and robust convergence. The developed method is applied to an experimental desktop printer and is compared to a pre-existing approach, which shows that the performance is significantly improved by imposing smoothness properties on the iteration dynamics.