Stability of Higher Order Repetitive Control

Abstract

Repetitive control (RC) and iterative learning control (ILC) apply to repeating situations, and iteratively adjust the command to a feedback control system aiming to converge on zero tracking error. Thus, these forms of control solve an inverse problem, but instead of doing so numerically with a mathematical model, it is done iteratively with the real world hardware. ILC restarts the system before each run, while repetitive control applies to executing a periodic command or eliminating the effects of a periodic disturbance. The ILC literature has many contributions developing what are called higher order ILC laws, where the control action in the current repetition is a function of the errors observed in multiple previous runs. It is the purpose of this paper to develop the higher order repetitive control, and in particular to develop the relevant theory of stability. It is proved that the simple frequency response based sufficient condition for convergence in first order RC, is also a sufficient condition for convergence in higher order RC, and it is independent of the order chosen. Furthermore, this condition is very close to being a necessary condition for stability. The result is that the typical design process in the frequency domain for first order RC can now be directly extended to the design of higher order RC as well.