Explicit formulas for optimally robust controllers for delay systems

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 is given. An example is 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