IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 1, FEBRUARY2007 567 Analog Circuits to Implement Repetitive Controllers With Feedforward for Harmonic Compensation Gerardo Escobar, Member, IEEE, P. R. Martínez, and J. Leyva-Ramos, Member, IEEE Abstract—A feedforward modification for both positive- and negative-feedback schemes of repetitive control is described. It was shown that repetitive controllers can be a useful tool for tracking of periodic reference signals and compensation of periodic distur- bances, in other words, for harmonic compensation. It was shown that the feedforward modification considerably improves the fre- quency response and performance, providing higher gains with enhanced selectivity. Simple analog circuits are presented to im- plement both positive- and negative-feedback repetitive schemes. A description of the circuits and their corresponding experimental frequency responses are also given. Index Terms—Analog circuits, harmonic compensation, peri- odic disturbances, repetitive control. I. I NTRODUCTION T HE compensation of harmonic disturbances is a subject that has attracted the attention of many researchers in the last decades. In this sense, repetitive control arises as a practical solution to such issue and is based on the well-known internal model principle [1]. The development of repetitive control was motivated by a design of a magnet power supply for a proton synchrotron [2]. The first works were presented in [3] and [4], which were later formally gathered in a seminal paper [5] with a stability study of linear infinite-dimensional repetitive con- trollers. Other interesting theoretical developments of repetitive control can be found in [6] and the numerous references therein. The discrete-time formulation was presented in [7]. Repetitive control is a potential solution to many precision systems such as industrial robots, disc drives, numerical control machines, servo scanners, and roughly speaking, in almost every system that rotates or repeats the same task on a periodic basis. The harmonic-compensation issue has received special attention in power electronics and power systems applications where the disturbances to cancel and/or the reference signals to track are composed of specific higher harmonics of the fundamental fre- quency of the power supply. Applications of repetitive control on power electronic systems such as rectifiers, inverters, and active filters can be found in [8] and [9] and the references therein. Manuscript received November 23, 2004; revised April 25, 2005. Abstract published on the Internet September 15, 2006. This work was supported by the National Council of Science and Technology of Mexico (CONACYT) under Grant SEP-2003-C02-42643. G. Escobar is with the Instituto Potosino de Investigación Científica y Tec- nológica, San Luis Potosí 78216, Mexico (e-mail: gescobar@ipicyt.edu.mx). P. R. Martínez and J. Leyva-Ramos are with the Division of Applied Math- ematics, Instituto Potosino de Investigación Científica y Tecnológica, San Luis Potosí 78216, México (e-mail: panfilo@ipicyt.edu.mx; jleyva@ipicyt.edu.mx). Digital Object Identifier 10.1109/TIE.2006.885515 The internal model principle states that a controlled output can track a class of reference commands without a steady error if the generator (or the model) of the reference is included in the stable closed-loop system. Therefore, it can be used to provide exact asymptotic output tracking of periodic inputs or to reject periodic disturbances. It is well known that the generator of a sinusoidal signal that contains only one harmonic component is a harmonic oscillator, in other words, a resonant filter. Thus, following this idea, if a periodic signal has an infinite Fourier series (of harmonic components), then an infinite number of harmonic oscillators are required to track or reject such a periodic signal. Fortunately, in the repetitive-control approach, a simple delay line in a proper feedback array can be used to produce an infinite number of poles, thereby simulating a bank of an infinite number of harmonic oscillators, which leads to a system dynamics of infinite dimension. To construct such a delay line, which is also referred to as “transport delay,” several approaches have been proposed; among them are given as follows. 1) Digital approach. Its implementation is based on an ana- log/digital converter, a shift register, and a digital/analog converter. Unfortunately, this approach has several disad- vantages. a) It is not cost effective. b) There are some imprecisions due to inaccuracy con- versions. c) It may be difficult to tune. d) The signal-to-noise ratio can be low. 2) Analog approach. A combination of inductors and capac- itors can be used to build an approximated delay line. The main disadvantage in this approach is the difficulty to get the exact delay. In this paper, we study two feedback schemes for the delay line: positive- and negative-feedback compensators. We show that while the positive-feedback array is able to compensate for all harmonics, the negative-feedback scheme can only compen- sate for the odd harmonics. We propose to add a feedforward modification for both schemes. The motivation for this modifi- cation is to introduce zeros that lie in between the poles, thus improving the selective nature of the whole controller, which will, in principle, allow higher gains and better performance. The implementation of such compensators using an analog IC to implement the delay line is also presented in this paper. This IC, which is referred to as low-noise bucket brigade delay (BBD), is an analog device that is very simple to tune for the exact delay and has a high signal-to-noise ratio; therefore, precision is not lost during the delay. This IC is thoroughly used 0278-0046/$25.00 © 2007 IEEE