  Iterative Learning Control (ILC) is a technique for adaptive feed-forward
control of electro-mechanical plant that either performs programmed periodic
behavior or rejects quasi-periodic disturbances. For example, ILC can suppress
particle-beam RF-loading transients in RF cavities for acceleration. This
paper, for the first time, explains the structural causes of ``bad learning
transients'' for causal and noncausal learning in terms of their eigen-system
properties. This paper underscores the fundamental importance of the linear
weighted-sums of the column elements of the iteration matrix in determining
convergence, and the relation to the convergence of sum of squares. This paper
explains how to apply the z-transform convergence criteria to causal and
noncausal learning. These criteria have an enormous advantage over the matrix
formulation because the algorithm scales as N^2 (or smaller) versus N^3, where
N is the length of the column vector containing the time series. Finally, the
paper reminds readers that there are also wave-like (soliton) solutions of the
ILC equations that may occur even when all convergence criteria are satisfied.
