Efficient frequency response computation for loworder modelling of spatially distributed systems

Motivated by the challenges of designing feedback controllers for spatially distributed systems, this paper presents a computationally efficient approach to obtaining the point-wise frequency response of such systems, from which low-order models can be easily identified. This is achieved by sequentially combining the individual frequency responses of the constituent lower-order subsystems in a way that exploits the interconnectivity arising from spatial discretisation. Importantly, this approach extends to the singular subsystems that naturally arise upon spatial discretisation of systems governed by partial differential-algebraic equations, with fluid flows being a prime example. The main result of this paper is a proof that the computational complexity associated with forming the overall frequency response is minimised if the smallest subsystems are first merged into larger subsystems, before combining the frequency responses of the latter. This reduces the complexity by several orders of magnitude; a result that is demonstrated upon the numerical example of a spatially discretised two-dimensional wave-diffusion equation. By avoiding the necessity to construct, store, or manipulate large-scale system matrices, the modelling approach presented in this paper is well conditioned and computationally tractable for spatially distributed systems consisting of enormous numbers of subsystems. It therefore bypasses many of the problems associated with conventional model reduction techniques.