Fractional repetitive control of nanopositioning stages for high-speed scanning using low-pass FIR variable fractional delay filter

The repetitive control (RC), capable of tracking periodic trajectories and rejecting periodic disturbances, is a promising technique to control the nanopositioning stages for high-speed raster scanning. In digital implementation of the RC scheme, the number of delay points has to be an integer, which implies that the sampling frequency should be an integer multiple of the desired tracking frequency. Clearly, this is a severe limitation on the range of the trajectory frequencies where the RC scheme can effectively be applied. To overcome this limitation, this article proposes a low-pass FIR variable fractional delay filter as an alternative to the conventional interpolating method employed in conventional fractional RC scheme. This filter combines the interpolating and the low-pass filtering that are employed in the fractional RC and its coefficients are analytically computed as a function of fractional delay; thereby, making it suitable for trajectories of all frequencies. The weighted-least-square method is employed to design the low-pass FIR variable fractional delay filter, where the weights are tuned to minimize the approximation errors within the bandwidth-of-interested. Experimental results are presented to demonstrate the advantages of the proposed method over the conventional RC scheme as well as the interpolating-based fractional RC scheme. These results show that the proposed RC scheme with low-pass FIR variable fractional delay filter improves the tracking performance of the nanopositioner significantly, especially for the trajectories with high-frequency.