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(ω_n, f_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 iω_n of the reference signals, then all reference signals in H(ω_n, f_n) can be asymptotically tracked in the presence of disturbances if and only if (H_K(iω_n)^-1[1-H_d(iω_n)φ_n]fn^-1)_nεI εℓ ². Here H_K(iω_n) and H_d(iω_n) are certain transfer functions evaluated at points iω_n and the sequence (f_n) consists of weights for the Fourier coefficients of the reference signals. Three examples are given to illustrate the theory.