In this paper we consider two separate approaches to the development of finite dimensional control systems for approximating distributed parameter models. One method uses the “standard finite element” approximations to construct the basic system matrices. The resulting system can then be balanced by any of several balancing algorithms. The second method is based on truncating infinite dimensional balanced realizations of the input-output map. Both approaches are applied to a control problem governed by the heat equation. We present a comparison of the resulting finite dimensional models. / Ph. D.