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In this paper, a systematic method for designing a fixed-order controller for the track-following control of optical disc drives is proposed. The design specifications are given in the frequency domain and are expressed as frequency-domain inequalities. On the basis of the generalized Kalman-Yakubovich-Popov lemma, each frequency-domain inequality is converted into linear matrix inequalities of the controller parameters. The controller parameters can be computed efficiently by solving a convex optimization problem to maximize the loop gain around the rotational frequency while satisfying other frequency-domain specifications on the performance and stability. Experimental results show that a reasonable performance can be obtained by using a second-order controller, and the performance can be further improved by using a third-order controller.