Repetitive control using low order models

In repetitive control based on the internal model principle of Francis and Wonham (1976) much interest has been focused around the model comprising a time delay with a memoryless positive feedback around it. A definite merit of this model is that, when used in the closed loop, it will generate an arbitrary periodic signal. In discrete-time the model order increases proportionally to the rise in sampling rate, giving a high order model. The input (disturbance or reference) signal may however be concentrated to some frequencies rather than the whole band up to the Nyquist frequency. Then the model is too large in the sense that it models frequencies that are not present in the input. The inactive part of the model can produce oscillations in the closed loop. Oscillations can be reduced by introducing a lowpass filter (or Q-filter) in the feedback path of the model, but thereby the asymptotic zero error property is lost. A lower order model is shown to overcome this problem and feasibility illustrated by a simulation example.