Digital repetitive control under varying frequency conditions
Authors:   , Ramos Fuentes Germán Andrés 
Published: October 2023

The tracking/rejection of periodic signals constitutes a wide field of research in the control theory and applications area and
Repetitive Control has proven to be an efficient way to face this topic; however, in some applications the period of the signal to
be tracked/rejected changes in time or is uncertain, which causes and important performance degradation in the standard
repetitive controller. This thesis presents some contributions to the open topic of repetitive control working under varying
frequency conditions. These contributions can be organized as follows:
One approach that overcomes the problem of working under time varying frequency conditions is the adaptation of the
controller sampling period, nevertheless, the system framework changes from Linear Time Invariant to Linear Time-Varying
and the closed-loop stability can be compromised. This work presents two different methodologies aimed at analysing the
system stability under these conditions. The first one uses a Linear Matrix Inequality (LMI) gridding approach which provides
necessary conditions to accomplish a sufficient condition for the closed-loop Bounded Input Bounded Output stability of the
system. The second one applies robust control techniques in order to analyse the stability and yields sufficient stability
conditions. Both methodologies yield a frequency variation interval for which the system stability can be assured. Although
several approaches exist for the stability analysis of general time-varying sampling period controllers few of them allow an
integrated controller design which assures closed-loop stability under such conditions. In this thesis two design
methodologies are presented, which assure stability of the repetitive control system working under varying sampling period
for a given frequency variation interval: a mu-synthesis technique and a pre-compensation strategy.
On a second branch, High Order Repetitive Control (HORC) is mainly used to improve the repetitive control performance
robustness under disturbance/reference signals with varying or uncertain frequency. Unlike standard repetitive control, the
HORC involves a weighted sum of several signal periods. With a proper selection of the associated weights, this high order
function offers a characteristic frequency response in which the high gain peaks located at harmonic frequencies are
extended to a wider region around the harmonics. Furthermore, the use of an odd-harmonic internal model will make the
system more appropriate for applications where signals have only odd-harmonic components, as in power electronics
systems. Thus an Odd-harmonic High Order Repetitive Controller suitable for applications involving odd-harmonic type
signals with varying/uncertain frequency is presented. The open loop stability of internal models used in HORC and the one
presented here is analysed. Additionally, as a consequence of this analysis, an Anti-Windup (AW) scheme for repetitive
control is proposed. This AW proposal is based on the idea of having a small steady state tracking error and fast recovery
once the system goes out of saturation.
The experimental validation of these proposals has been performed in two different applications: the Roto-magnet plant and
the active power filter application. The Roto-magnet plant is an experimental didactic plant used as a tool for analysing and
understanding the nature of the periodic disturbances, as well as to study the different control techniques used to tackle this
problem. This plant has been adopted as experimental test bench for rotational machines. On the other hand, shunt active
power filters have been widely used as a way to overcome power quality problems caused by nonlinear and reactive loads.
These power electronics devices are designed with the goal of obtaining a power factor close to 1 and achieving current
harmonics and reactive power compensation.