Rohrs' Example Revisited: On the Robustness of Adaptive Iterative Learning Control

Adaptive feedback based methods in iterative learning control (ILC) have garnered much interest from researchers for some time now. Much as in adaptive feedback control, most of these methods use Lyapunov functions and positive real transfer functions to prove convergence and boundedness of system signals updated through iterative estimations. While Rohrs et al. have motivated further research on the design of robust adaptive feedback controllers by demonstrating in the early 1980's that the algorithms of the time were not robust in the presence of unmodeled dynamics, the topic of robustness has not been studied much in the adaptive iterative learning control (AILC) literature. Inspired by Rohrs' counterexample, we use a model reference AILC scheme to show the lack of robustness to unmodeled dynamics in AILC. We rigorously define the concept of stability in ILC via space concepts, and demonstrate the existence of unstable learning operators. We put forth linear systems arguments to explain how conditions leading to instability can occur, and support heuristic arguments with simulation examples. Our findings indicate that the shortcomings of AILC in terms of robustness are no different than those of adaptive feedback, with the robustness issue more severe in certain cases, and further research is necessary to design robust AILC schemes.