This paper considers single-input/single-output systems whose transfer functions take the form of a strictly proper rational function times a delay. A closed-form expression is presented for the controller which is optimally robust with respect to perturbations measured in the gap metric. The formula allows the H∞ loop-shaping procedure of Glover—McFarlane to be carried out explicitly for this class of systems without the need to first find a rational approximation of the plant. The form of the controller involves a certain algebra of “pseudo-derivation" operators. These operators, and their matrix generalizations, play a central role in the derivation of the controller. A discussion of the main properties of these operators will be given. An example will be presented of a controller design to achieve disturbance attenuation and robust set-point following for a plant with two lightly damped poles and a nontrivial time delay. The performance is compared, and shown to be superior, to that of a Smith predictor.
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Manuscript received December 1, 1993; revised May 25, 1994 and July 23, 1994. Recommended by Associate Editor, B. Lehman. This work was supported in part by the NSF, AFOSR, SERC, and the Nuffield Foundation. H. Dym is with the Department of Theoretical Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel. T. T. Georgiou is with the Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455 USA. M. C. Smith is with the Department of Engineering, University of Cambridge, Cambridge, CB2 lPZ, U.K. IEEE Log Number 9408279.