#
Explicit formulas for optimally robust controllers
for delay systems

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

## Abstract

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.