Robustness Analysis of Repetitive Control Systems

Abstract

The main problem in control theory is controlling the output of a system to achieve the asymptotic tracking of desired signals and/or asymptotic rejection of disturbances. Among the existing approaches to asymptotic tracking and rejection, tracking via an internal model, which can handle the exogenous signal (The term “exogenous signal” is used to refer to both the desired signal and the disturbance when there is no need to distinguish them.) from a fixed family of functions of time, is one of the important approaches [1]. The basic concept of tracking via the internal model originated from the internal model principle (IMP) [2, 3]. The IMP states that if any exogenous signal can be regarded as the output of an autonomous system, then the inclusion of this signal model, i.e., the internal model, in a stable closed-loop system can ensure asymptotic tracking and asymptotic rejection of the signal. Given that the exogenous signals under consideration are often nonvanishing, the characteristic roots of these autonomous systems that generate these exogenous signals are neutrally stable. To produce asymptotic tracking and asymptotic rejection, if a given signal has a certain number of harmonics, then a corresponding number of neutrally stable internal models (one for each harmonic) should be incorporated into the closed-loop based on the IMP. Repetitive control (RC, or repetitive controller, also abbreviated as RC) is a specialized tracking method via an internal model for the asymptotic tracking and rejection of general T-periodic signals [4].