Output regulation for infinite-dimensional linear systems periodic reference signals from Sobolev spaces

We study asymptotic tracking and rejection (i.e. regulation) of continuous periodic signals in the context of exponentially stabilizable linear infinite-dimensional systems. Our reference signals are in Sobolev type spaces H(/spl omega//sub n/, f/sub n/) and they (as well as the disturbance signals) are generated by an infinite-dimensional exogenous system. We show that there exists a feedforward controller which achieves output regulation if and only if the so called regulator equations are satisfied and a decomposability condition holds. We show that if the stabilized plant does not have transmission zeros at the frequencies i/spl omega//sub n/ of the reference signals, then all reference signals in H(/spl omega//sub n/, f/sub n/) can be asymptotically tracked in the presence of disturbances if and only if (H/sub K/(i/spl omega//sub n/)/sup -1/[1-H/sub d/(i/spl omega//sub n/)/spl phi//sub n/]fn/sup -1/)/sub n/spl epsiv/I/ /spl epsiv//spl lscr/ /sup 2/. Here H/sub K/(i/spl omega//sub n/) and H/sub d/(i/spl omega//sub n/) are certain transfer functions evaluated at points i/spl omega//sub n/ and the sequence (f/sub n/) consists of weights for the Fourier coefficients of the reference signals. Three examples are given to illustrate the theory.