Iterative learning and repetitive control aim to eliminate the effect of unwanted disturbances over repeated trials or cycles. The disturbance-free system model, if known, can be used in a model-based iterative learning or repetitive control system to eliminate the unwanted disturbances. In the case of periodic disturbances, although the unknown disturbance frequencies may be the same from trial to trial, the disturbance amplitudes, phases, and biases do not necessarily repeat. Furthermore, the system may not return to the same initial state at the end of each trial before starting the next trial. In spite of these constraints, this paper shows how to identify the system disturbance-free dynamics from disturbance-corrupted input-output data collected over multiple trials without having to measure the disturbances directly. The system disturbance-free model can then be used to identify the disturbances as well, for use in learning or repetitive control. This paper represents the first extension of the interaction matrix approach to the multiple-trial environment of iterative learning control.