Asian Journal of Control, Vol. 20, No. 3, pp. 1047–1057, May 2018
Published online 7 December 2016 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.1437
IMPROVING ROBUSTNESS FILTER BANDWIDTH IN REPETITIVE
CONTROL BY CONSIDERING MODEL MISMATCH
Arnfinn A. Eielsen, Yik R. Teo, and Andrew J. Fleming
Repetitive control (RC) is used to track and reject periodic signals by including a model of a periodic signal in
the feedback path. The performance of RC can be improved by including an inverse plant response lter, but due to
modeling uncertainty at high frequencies, a low-pass robustness lter is also required to limit the bandwidth of the
signal model and ensure stability. The design of robustness lters is presently ad-hoc, which may result in excessively
conservative performance. This article proposes a new automatic method for designing the robustness lter based on
convex optimization and an uncertainty model. Experimental results on a nanopositioning system demonstrate that the
proposed method outperforms the traditional brick-wall lter approach.
Key Words: Repetitive control, learning control, uncertainty, optimization.
Repetitive control (RC) is a method suited to ref-
erence tracking and rejection of periodic signals .
The method is based on the internal model principle
 where an exogenous signal (a reference or distur-
bance) can be nulled in the error signal if a signal model
is contained in the feedback path. RC was developed
to reject the periodic disturbances that arise in power
supply control [3,4], but has since been used for machin-
ing , precision positioning [6–8], optical drives [9–11],
electro-hydraulics , power-converters , and scan-
ning probe microscopy [14–16].
Fig. 1 illustrates the signal model used in RC for
periodic signals of period L. This implementation is com-
putationally efcient and numerically stable as the model
consists of only positive feedback around a time-delay.
The corresponding transfer function is an innite num-
ber of marginally stable poles with innite gain at the
harmonics of the periodic reference.
The most common implementation of discrete-time
RC was rst proposed in , where the plant dynamics
are inverted using the zero-phase tracking error con-
trol (ZPETC) lter in order to improve the RC per-
formance. In principle, an inverse plant response lter
(IPRF) should provide a signal model bandwidth up
to the Nyquist-frequency. However, the ZPETC lter
Manuscript received April 13, 2016; revised August 4, 2016; accepted October
All authors are with the Precision Mechatronics Labs at the School of Elec-
trical Engineering and Computer Science, The University of Newcastle, 2308
Callaghan, New South Wales, Australia.
Yik Teo is the corresponding author (e-mail: firstname.lastname@example.org).
Fig. 1. A time-delay with positive feedback with the
appropriate initial function can model any periodic
relies on an accurately identied innite impulse response
(IIR) model, which is not always possible. Furthermore,
non-minimum phase zeros cannot be inverted; hence the
magnitude response of the ZPETC inverse can be inac-
As an alternative to an IIR lter, a nite impulse
response (FIR) lter can be used for the IPRF. Com-
pared to a ZPETC inverse, an FIR lter does not require
an explicit model structure and can alleviate the prob-
lems due to non-minimum phase zeros. However, FIR
lters can be more computationally demanding than IIR
lters. The IPRF as an FIR lter [18,19] can be found
using frequency domain optimization . The fore-
most difculties with this approach are the ad-hoc design
procedure and non-optimal performance. For example,
an error weighting function must be chosen to syn-
thesize a lter which ensures closed-loop stability. An
alternative method presented in [8,21] is a more direct
method for IPRF synthesis which uses the inverse dis-
crete Fourier transform (IDFT) of the inverse empiri-
cal transfer-function estimate (ETFE). This method is
© 2016 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd