Explicit Formulas for Optimally Robust Controllers for Delay Systems

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 1 loop-shaping procedure of Glover-McFarlane to be carried out explicitly for this class of systems without the need to rst nd 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. Notation R and C denote the real and complex elds. H 2 , H 1 denote the standard Hardy spaces in the right half of the complex plane (RHP). L p denotes the standard Lebesgue space and k k p its norm. The inner product in L 2 is denoted by hh; i. For x 2 L 2 (?1; 1) we denote the Laplace transform of x by ^ x. If G(s) is a matrix function of s, G (s) := G(?s) T is the conjugate of G(s). If G(s) is real rational, G (s) = G(?s) T. H n 2 , L n 2 etc. denote corresponding spaces of vector-valued functions. (H n 2) ? denotes the orthogonal complement of H n 2 in L n 2 (?j1; j1). We denote by + (resp. ?) the orthogonal projection from