Explicit formulas for optimally robust controllers for delay systems

H. Dym, T.T. Georgiou, M.C. Smith


This paper considers 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-infinity 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 non-trivial time delay. The performance is compared, and shown to be superior, to that of a Smith predictor.