l1-optimal robust iterative learning controller design

In this paper we consider the robust iterative learning control (ILC) design problem for SISO discrete-time linear plants subject to unknown, bounded disturbances. Using the supervector formulation of ILC, we apply a Youla parameterization to pose a MIMO l₁-optimal control problem. The problem is analyzed for three situations: (1) the case of arbitrary ILC controllers that use current iteration tracking error (CITE), but without explicit integrating action in iteration, (2) the case of arbitrary ILC controllers with CITE and with explicit integrating action in iteration, and (3) the case of ILC controllers without CITE but that force an integral action in iteration. Analysis of these cases shows that the best ILC controller for this problem when using a non-CITE ILC algorithm is a standard Arimoto-style update law, with the learning gain chosen to be the system inverse. Further, such an algorithm will always be worse than a CITE-based algorithm. It is also found that a trade-off exists between asymptotic tracking of reference trajectories and rejection of unknown-bounded disturbances and that ILC does not help alleviate this trade-off. Finally, the analysis reinforces results in the literature noting that for SISO discrete-time linear systems, first-order ILC algorithms can always do as well as higher-order ILC algorithms.