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/sup j/ sin(/spl omega//sub k/t+/spl phi//sub k/), j/spl ges/0, k=0, /spl middot//spl middot//spl middot/, 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 /spl epsi/ and certain polynomial matrices K/sub k/(s) for k=0, /spl middot//spl middot//spl middot/, n. The main result of the paper shows that the matrices K/sub k/(s) must satisfy certain stability conditions involving the values of the plant transfer function only at the reference and disturbance signal frequencies /spl omega//sub k/ for k=0, /spl middot//spl middot//spl middot/, n. The controller has unstable poles on the imaginary axis. The behavior of these poles as a function of the scalar parameter /spl epsi/ in the form of a Puiseux series is given.