ROBUST REJECTION OF PERIODIC AND ALMOST PERIODIC DISTURBANCES Vishwesh V. Kulkarni, ∗ Lucy Y. Pao, ∗ Hua Zhong ∗ ∗ Department of Electrical and Computer Engineering, University of Colorado, Boulder, CO. USA. Email: vishwesh@colorado.edu, pao@colorado.edu, zhongh@colorado.edu Abstract: We present algorithms to exponentially reject periodic and almost peri- odic disturbances, the motivating application being a rejection of reel eccentricity induced disturbances in tape-drive systems. The prevalent periodic disturbance rejection algorithms rely on a constant gain approximation of the system at a particular frequency. These are inadequate for this application because a tape-drive system has parametric uncertainties and because the disturbance is time-varying. We present a robust extension of an existing technique derived by Bodson et al. and further use quadratic and parameter-dependent Lyapunov functions to synthesize gain-scheduled feedback compensators. Copyright c  2005 IFAC Keywords: disturbance rejection, periodic disturbances, tape drive systems, nonlinear control systems, gain scheduling, linear matrix inequality 1. INTRODUCTION This paper is motivated by the problem of re- jecting reel eccentricity induced disturbances in tape drive storage systems. Reel eccentricities in- duce ripples in the tape tension and the dom- inant frequency of this disturbance is the same as the transport resonance frequency. The ripple magnitude is inversely proportional to the square root of the tape speed (Lu, 2002, Ch. 2) and is especially large at low speeds. Further, the dis- turbance frequency is a function of the tape speed and the pack radii. Thus, the disturbance is not periodic but almost periodic in the sense that the disturbance frequency varies with time, typically from 20 Hz to 80 Hz, as the tape runs end to end, albeit the rate of variation is small. 1 This work has been supported in part by the Colorado Center for Information Storage. Literature on the rejection of periodic distur- bances, dubbed repetitive control, abounds (Ghosh and Paden, 2000) and largely relies on the internal model principle (Francis and Wonham, 1976). The principle states that a plant output can track a class of reference signals, with the steady-state tracking error approaching zero asymptotically, if a model of the reference signal generator is inter- nally added to the stable closed-loop system; if the input signal has a finite Fourier series, only finitely many internal models need be added. A lin- ear infinite dimensional single-input single-output (SISO) repetitive controller was first derived in (Hara et al., 1988) and a corresponding discrete- time formulation was derived in (Tomizuka et al., 1988) with certain robustness considerations added in (Tsao and Tomizuka, 1994). The in- ternal model principle has been used to re- ject periodic disturbances in (Brown and Zhang, 2004; Narendra and Annaswamy, 1989; Feng and