Efficient frequency response computation for low-order modelling of spatially distributed systems

Motivated by the challenges of designing feedback controllers for spatially distributed systems, we present an efficient approach to obtaining the frequency response of such systems, from which low-order models can be identified. This is achieved by combining the frequency responses of constituent lower-order subsystems in a way that exploits the interconnectivity arising from spatial discretisation. This approach extends to the singular subsystems that arise upon spatial discretisation of systems governed by PDAEs, with fluid flows being a prime example. The main result of this paper is a proof that the computational complexity of forming the overall frequency response is minimised if the subsystems are merged in a particular fashion. Doing so reduces the complexity by several orders of magnitude; a result demonstrated upon the numerical example of a spatially discretised wave-diffusion equation. By avoiding the construction, storage, or manipulation of large-scale system matrices, this modelling approach is well conditioned and computationally tractable for spatially distributed systems consisting of enormous numbers of subsystems, therefore bypassing many of the problems with conventional model reduction techniques.