When is the discretization of a PDE good enough for control?

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.