Partially observed control systems described by analytic semigroup are considered. Finite-dimensional feedback control based on FEM approximations and accounting for incomplete observations is constructed. It is shown that this feedback control provides uniform stability (in time)) of the originally unstable system. The main novel feature of the problem is that both—control and observation operators—are modeled by fully unbounded operators as they frequently arise in modeling of “smart” sensors and actuators. This contributes to technical difficulties at the level of perturbation theory for analytic semigroups. It is shown that a careful and rather special approximation in the area of support of the unbounded control/observation operators allows to obtain the “right” stability estimates. Theoretical results are illustrated with several examples of control problems governed by heat and plate equations.