Theory and evaluation of a stability condition for second order repetitive control

Repetitive Control (RC) is a control method for tracking a periodic command or eliminating a periodic disturbance to a control system, aiming for zero tracking error in each case. Applications in spacecraft address jitter from internal moving parts such as reaction wheels, control moment gyros, a momentum wheel, or a cryo pump. RC requires knowledge of the reference or disturbance period. RC performance suffers if the period fluctuates or is not precisely known. Second order RC (SORC) offers less sensitivity to accurate period knowledge, but stability analysis is difficult because of the high order governing difference equations. Sufficient stability conditions can be helpful in the design process. This paper presents a derivation of a sufficient condition, developed based on a partial fraction expansion, that makes SORC only require data from one period back. This can significantly reduce the memory requirements. This approach was originally developed in a companion paper treating magnetically suspended rotors, and used an unusual block diagram configuration. This approach is developed here for the standard block diagram structure for RC. It is then compared to the true stability boundary and to different sufficient conditions developed by the third author and coworkers. Which sufficient condition is closest to the true stability boundary is established for different situations. The designer may find it beneficial to examine multiple conditions.