An approximation theory leading to a design of a finite-dimensional compensator for control systems generated by analytic semigroups is presented. The novelty of this paper with respect to other results available in the literature is threefold : (i) it treats fully unbounded control/observation operators; (ii) it does not require compactness property of the underlined generator (an assumption that is often violated in practice); and (iii) the design of a finite-dimensional compensator is based on finite element approximation of the original model rather than on modal (eigenfunctions) approximations which, in turn, require the a priori knowledge of the eigenvalues for the system. Applications of the theory to heat equations and plate equations are provided.