IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 53, NO. 4, AUGUST 2006 1383 Letters to the Editor ANegativeFeedbackRepetitiveControlSchemefor Harmonic Compensation G. Escobar, P. R. Martínez, J. Leyva-Ramos, and P. Mattavelli Abstract—In this letter, a different feedback structure of the repetitive control that apparently is more appropriate for applications in power electronicsisproposed.Moreover,asimpleanalog-circuitimplementation isproposedwhichissuitableforhigh-frequencypowerelectronicsapplica- tions, where digital control is unpractical due to cost and performance of availableDSPsandmicrocontrollers. Index Terms—Analogcircuits,harmoniccompensation,periodicdistur- bances,repetitivecontrol. I. I NTRODUCTION Tracking or rejection of periodic signals is an issue commonly found in power electronics applications, i.e., switching power supplies, ac/dc converters, motor speed fluctuation, synchronous rectifiers, un- interruptible power supplies (UPS) and active filters. In these cases, the disturbances and/or references are composed of specific higher harmonics of the fundamental frequency of the power source. Repet- itive control arises as a practical solution to such issues and is based on the internal model principle [1]. It is aimed to provide an exact asymptotic output tracking of periodic inputs or rejection of periodic disturbances. The internal model principle states that the controlled output can track a class of reference commands without a steady-state error if the generator, or the model, for the reference is included in the stable closed-loop system. It is well known that the generator of a sinusoidal signal, i.e., containing only one harmonic component, is a harmonic oscillator, i.e., a resonant filter. Therefore, according to the internal model principle, if a periodic disturbance has an infinite Fourier series (of harmonic components), then an infinite number of resonant filters are required to reject it. 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 and thereby simulating a bank of an infinite number of resonant filters, leading to a system dynamics of infinite dimension. First works on repetitive control were presented in [2] and [3]. Interesting theoretical developments of repetitive control can be found in [4] and [5], and the numerous references within where the discrete- time formulation has also been treated. See [6]–[9], and the references therein, for applications of repetitive control on power electronic systems such as rectifiers, inverters and active filters. Indeed, as shown in these papers, it is already well accepted in the power electronics community that repetitive techniques offer some advantages over Manuscript received August 17, 2004; revised October 5, 2005. Abstract published on the Internet May 18, 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 Cientifica y Tecnológica (IPICyT), San Luis Potosí 78216, Mexico (e-mail: gescobar@ ipicyt.edu.mx). P. R. Martínez and J. Leyva-Ramos are with the Division of Applied Mathematics, Instituto Potosino de Investigación Científica y Tecnológica (IPICyT), San Luis Potosí 78216, Mexíco (e-mail: panfilo@ipicyt.edu.mx; jleyva@ipicyt.edu.mx). P. Mattavelli is with the Dipartimento di Ingegneria Elettrica Gestionale e Meccanica (DIEGM), Udine University, 33100 Udine, Italy (e-mail: mattavelli@ uniud.it). Digital Object Identifier 10.1109/TIE.2006.878293 conventional solutions especially in active filters and inverters, which is demonstrated by the industrial applications in recent equipment. In most of these works, the authors use a positive feedback scheme to implement the repetitive controller. Some of them place the delay line in the direct path and others in the feedback path. It is important to notice that a positive feedback structure has the disadvantage of compensating for every single harmonic either odd or even, including the dc component. Moreover, depending on the position of the delay line in the structure, it may even modify the phase shift, which explains the need of some extra filters to alleviate this problem. Although the positive feedback-based scheme may apparently solve the harmonics compensation problem, it may lead to more distortion in certain cases. Consider, for instance, a system where even harmonics do not exist originally, like in many power electronic systems, in this case, the positive feedback repetitive controller would try to amplify, and indeed reinject, any small noise that has components on the even frequencies. This evidently has the danger of producing responses polluted with such harmonics, which were not present before. This letter shows that a negative feedback scheme, in contrast to the positive scheme, compensates only for the odd harmonics, thus, reducing the possibility of reinjecting unnecessary distortion into the system. Moreover, it has been found that placing a delay line in the feedback trajectory shows better phase characteristics. It is expected that this scheme will be specially useful, and will generate cleaner responses, than traditional positive feedback-based repetitive schemes in applications of power electronic systems containing mainly odd harmonics. For instance, the proposed scheme could replace the traditional repetitive schemes, and moreover, it may replace the bank of resonators reported in [10]. An analog implementation of the proposed negative feedback scheme is also presented for low-power high-frequency power elec- tronics applications. The proposed scheme is completely different from a more complex fully digital implementation, which would require costly a microcontroller and/or a DSP with analog-to-digital converters, with the inherent quantization errors, and, above all, large memory requirements. Thus, the proposed circuit seems to be an inter- esting solution for low-power high-frequency power-factor-correction circuits, inverters, active filters, etc., where cost constrains would not allow the use of the microcontrollers or the DSPs. Our experimental setup uses an analog integrated circuit (IC) of special purpose referred as low-noise bucket brigade delay (BBD) for the implementation of the analog delay. This circuit is an analog delay line which 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 circuit is thoroughly used in the music industry to create reverberation and echo effects. The circuitry presented here can reproduce the same frequency response as an infinite set of resonant filters tuned at higher odd harmonic frequencies of the fundamental. II. BLOCK DIAGRAM REPRESENTATION Consider the single-input–single-output (SISO) continuous-time system described by y(t)= u(t) − y(t − L) (1) where L is a positive real representing the time delay, y(t) is the output and u(t) is the input of the system. Application of the Laplace transform to (1) results in the block diagram shown in Fig. 1(b), where for simplicity zero initial conditions are considered. 0278-0046/$20.00 © 2006 IEEE