This paper considers the problem of designing near-optimal finite-dimensional compensators for unstable infinite-dimensional plants. Standard weighted H∞ sensitivity measures are used to define the notion of optimality. The method of solution is based on finite-dimensional techniques applied to finite-dimensional approximants of the original plant. The difficulties which arise from such an approach can be attributed to two factors. First, there is the lack of continuity of the performance measures with respect to perturbations in the plant, even with the graph topology. Second, there are many infinite-dimensional plants which cannot be approximated uniformly in the graph topology. It is shown in this paper, for the sensitivity minimization problem, that it is sufficient to obtain approximants of the plant on compact sets, provided that the inner part of the plant's `numerator coprime factor' is approximated in some sense. Constructive algorithms are presented. New results on the convergence of actual closed loop transfer functions are also given.