  Discrete-time domain Iterative Learning Control (ILC) schemes inspired by
Repetitive control algorithms are proposed and analyzed. The well known
relation between a discrete-time plant (filter) and its Markov Toeplitz matrix
representation has been exploited in previous ILC literature. However, this
connection breaks down when the filters have noncausal components. In this
paper we provide a formal representation and analysis that recover the
connections between noncausal filters and Toeplitz matrices. This tool is then
applied to translate the anti-causal zero-phase-error prototype repetitive
control scheme to the design of a stable and fast converging ILC algorithm. The
learning gain matrices are chosen such that the resulting state transition
matrix has a \textit{Symmetric Banded Toeplitz} (SBT) structure. It is shown
that the well known sufficient condition for repetitive control closed loop
stability based on a filter's frequency domain $H_\infty$ norm is also
sufficient for ILC convergence and that the condition becomes necessary as the
data length approaches infinity. Thus the ILC learning matrix design can be
translated to repetitive control loop shaping filter design in the frequency
domain.
