On the equivalence of causal LTI iterative learning control and feedback control

The goal of iterative learning control (ILC) is to improve the accuracy of a system that repeatedly follows a reference trajectory. This paper proves that for each causal linear time-invariant ILC, there is an equivalent feedback that achieves the ultimate ILC error with no iterations. Remarkably, this equivalent feedback depends only on the ILC operators and hence requires no plant knowledge. This equivalence is obtained whether or not the ILC includes current-cycle feedback. If the ILC system is internally stable and converges to zero error, there exists an internally stabilizing feedback that approaches zero error at high gain. Since conventional feedback requires no iterations, there is no reason to use causal ILC.