Use of unstable repetitive control for improved tracking accuracy

The simplest form of repetitive control is based on application of integral control concepts applied in repetitions, in order to produce zero tracking error. This repetitive control law is attractive because it is easy to apply and is essentially system independent, so that no substantial design effort is involved in its use. However, it is shown that this repetitive control law often results in a significant decrease in tracking error during the early repetitions of the learning process, but after a certain repetition the error starts to diverge. The purpose of this paper is to develop ways to make use of this improvement in tracking accuracy, by turning off the update of the repetitive control after the tracking error has become a minimum, and before the instability becomes manifest. This requires the development of methods to keep the frozen repetitive control signal synchronized with the errors that it is intended to attenuate. Methods are developed to keep the phase adjusted, both when the period of the repetitive command or disturbance is not an integer number of sample times, and when imprecise knowledge of the period results in a drift.