Design and Implementation of Repetitive Control based Noncausal   Zero-Phase Iterative Learning Control
  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. 
K. Krishnamoorthy and Tsu-Chin Tsao
math.OC