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In this paper, we consider the iterative learning control (ILC) framework to design a reference signal that directly cancels periodic disturbances in a feedback measurement. Cancellation of periodic disturbances is useful in reducing undesirable repeatable tracking errors in applications such as the two-stage servo track writing process for disk drives. A general problem description is given for a linear discrete-time periodic system and convergence results for the learning system are derived. A learning filter is designed with the use of a finite-impulse response model approximation for the inverse of the closed-loop sensitivity such that convergence is achieved in learning a reference signal that provides cancellation with periodic perturbations affecting the system measurement. The ILC algorithm is applied to a disk drive system where experimental results demonstrate the effectiveness of the method in reducing periodic measurement disturbances.