  Before AI and neural nets, the excitement was about iterative learning
control (ILC): the idea to train robots to perform repetitive tasks or train a
system to reject quasi-periodic disturbances. The excitement waned after the
discovery of "learning transients" in systems which satisfy the ILC asymptotic
convergence (AC) stability criteria. The transients may be of long duration,
persisting long after eigenvalues imply they should have decayed, and span
orders of magnitude. They occur both for causal and noncausal learning. The
field recovered with the introduction of tests for "monotonic convergence of
the vector norm"; but no deep and truly satisfying explanation was offered.
Here we explore solutions of the ILC equations that couple the iteration index
to the within-trial sample index. This sheds light on the causal learning - for
which the AC test gives a repeated eigenvalue. Moreover, since 2016, this
author has demonstrated that a new class of solutions, which are soliton-like,
satisfy the recurrence equations of ILC and offer additional insight to
long-term behaviour. A soliton is a wave-like object that emerges in a
dis-persive medium that travels with little or no change of shape at an
identifiable speed. This paper is the first public presentation of the soliton
solutions, which may occur for both causal (i.e. look back) and noncausal (i.e.
look ahead) learning func-tions that have diagonal band structure for their
matrix representation.
