A finite-dimensional robust controller for systems in the CD-algebra

Robust multivariable controllers for stable infinite-dimensional systems in the Callier-Desoer (CD) algebra are discussed. In particular, the following robust regulation problem is solved. Given reference and disturbance signals, which are linear combinations of signals of the form t^j sin(ω_kt+φ_k), j⩾0, k=0, ···, n, find a low-order finite-dimensional controller so that the outputs asymptotically track the reference signals, asymptotically reject the disturbance signals, and the closed-loop system is stable and robust with respect to a class of perturbations in the plant. The proposed controller consists of a positive scalar gain ε and certain polynomial matrices K_k(s) for k=0, ···, n. The main result of the paper shows that the matrices K_k(s) must satisfy certain stability conditions involving the values of the plant transfer function only at the reference and disturbance signal frequencies ω_k for k=0, ···, n. The controller has unstable poles on the imaginary axis. The behavior of these poles as a function of the scalar parameter ε in the form of a Puiseux series is given