Many systems of engineering importance are governed by partial differential equations (PDEs) in one or more spatial dimensions, and are therefore infinite dimensional. Controlling such spatially distributed plants is non-trivial, given that the bulk of established control theory and practice assumes plant models of finite and low state dimension. In order to obtain such a model it is necessary to approximate the plant dynamics, trading off a reduction in state dimension for an increase in plant/model mismatch. This paper describes a new technique for selecting a low order model that is a suitable approximation in a closed-loop sense to the spatially distributed plant we seek to control. Unlike model reduction, the new procedure starts from a coarse spatial discretization of the plant dynamics and increases in fidelity until a suitable control model is obtained, thus avoiding the numerical difficulties inherent in large-scale model reduction. We argue, through use of H?? loop-shaping and the ??-gap metric, that it is primarily the closed-loop design specifications and the method of spatial discretization that determine a suitable level of approximation. The main theoretical contribution of this work is a proof that, for plant models of successively finer spatial discretization, the order of convergence in the ??-gap metric is bounded by the order of convergence of their differences in the H?? norm. We also show how to easily compute reasonably tight upper bounds on the ??-gap between a finite dimensional model and an infinite dimensional plant. The ideas presented in the first part of this paper are demonstrated on a disturbance rejection problem for a 1D heat equation.