Small gain stability theory for matched basis function repetitive control

Abstract

Many spacecraft suffer from jitter produced by periodic vibration sources such as momentum wheels, reaction wheels, or control moment gyros. Vibration isolation mounts are needed for fine pointing equipment. Active control methods directly addressing frequencies of interest have the potential to completely cancel the influence of these disturbances. Typical repetitive control methods initially address all frequencies of a given period. Matched basis function repetitive control individually addresses each frequency, finding error components at these frequencies using the projection algorithm, and can converge to zero error, using only frequency response knowledge at addressed frequencies. This results in linear control laws but with periodic coefficients. Frequency domain raising produces a time invariant pole/zero model of the control law. A small gain stability theory is developed, that exhibits very strong stability robustness properties to model error. For convergence to zero tracking error it needs only knowledge of the phase response at addressed frequencies, and it must be known within an accuracy of ±90°. Controllers are then designed by pole-zero placement, bypassing the complexity of original periodic coefficient equations. Compared to the usual repetitive control approaches, the approach here eliminates the need for a robustifying zero phase low pass filter, eliminates the need for interpolation in data, and handles multiple unrelated frequencies easily and naturally.