Design of repetitive controllers in the frequency domain for multi-input multi-output systems

Momentum wheels, CMG's, reaction wheels, and cryo pumps can have slight imbalances that create vibrations in a spacecraft structure. Fine pointing equipment must be isolated from such vibrations. Repetitive control (RC) is an active control method that theoretically can completely eliminate the effects of periodic disturbances. Very effective design methods have previously been developed to design repetitive controllers for single-input single-output (SISO) systems. These methods are generalized here to handle the multiple-input, multiple-output (MIMO) problem needed for such applications. A necessary and sufficient condition for asymptotic stability in MIMO RC is derived, and four sufficient conditions are created. One of these is the MIMO version of the approximate monotonic decay condition in SISO RC, and one is a necessary and sufficient condition for stability for all possible disturbance periods. Methods are developed to use the effective SISO design methods multiple times to produce a MIMO RC system. And an appropriate optimization criterion for direct design for MIMO is presented based on minimizing a Frobenius norm. This design process is very tractable, requiring only the solution of a linear algebraic equation.