The phenomenon of apparent convergence followed by divergence in learning and repetitive control

It is a common experimental experience with learning and repetitive control, that the error decreases very substantially in the fast few repetitions or periods, and then starts to grow. In some cases this takes the form of a long term instability, for example in experiments cited here instability became evident only after 2,500 repetitions. Here we develop both time domain and frequency domain approaches to both learning and repetitive control, including use of root locus and Nyquist concepts. And these different viewpoints are used to explain in different ways this phenomenon of apparent initial convergence followed by divergence. It is seen that with typical distributions of tracking errors in the frequency spectrum relative to the bandwidth of the feedback controller, this phenomenon is easily generated. Three classes of approaches aze summarized for addressing this problem, either by stopping the mechanism of divergence, or by causing the convergence to continue to zero tracking error.