On the passivity properties of a new family of repetitive (hyperbolic) controllers

This paper studies the passivity properties of three recently reported repetitive schemes [1], [2]. They are referred as negative feedback, positive feedback and 6¿ ± 1 repetitive compensators. The first two controllers are composed of a feedback array of a single delay line, while the third controller comprises the feedback array of two delay lines. As most repetitive schemes, these three schemes are intended for the compensation or tracking of period signals, which are composed of harmonic components of a fundamental frequency. In particular, the negative feedback scheme is aimed for the compensation of odd-harmonic components, the positive feedback scheme for the compensation of all harmonics, and the 6¿ ±1 scheme for the compensation of 6¿ ±1 (l = 0, 1, 2, ...,¿) harmonics. It is shown here that all three schemes have also equivalent expressions in terms of hyperbolic functions. The main contribution of the present work is to show that these three schemes are discrete-time positive real and thus passive. Moreover, it is shown that, after a a modification, motivated by practical issues, these schemes become strictly passive.