The equivalence of LTI iterative learning control and feedback control

Iterative learning control (ILC) improves the accuracy of a system that repeatedly tracks a reference trajectory. If certain convergence conditions are satisfied, the tracking error tends toward some ultimate error signal as the number of iterations grows. We show that for any causal LTI ILC, there exists a feedback control that, achieves an error arbitrarily close to the ultimate ILC error (in a single iteration). This feedback control requires no knowledge of the plant and can be constructed from the ILC parameters alone. This result is obtained even if the ILC algorithm itself includes feedback. Two cases are considered, corresponding to whether or not a certain transfer function matrix is singular. The nonsingular case is investigated for MIMO systems and it is proved that feedback control achieves the ultimate ILC error exactly. For the singular case, it is proved that feedback control can make the tracking error arbitrarily small in the SISO case. The results of this study suggest that ILC provides no benefit over conventional feedback control, even when no plant knowledge is available.