Fourier Series for Accurate, Stable, Reduced-Order Models in Large-Scale Linear Applications

A new method, Fourier model reduction, for obtaining stable, accurate, low-order models of very large linear systems is presented. The technique draws on traditional control and dynamical system concepts and utilizes them in a way which is computationally very efficient. Discrete-time Fourier coefficients of the large system are calculated and used to construct a reduced-order model that preserves stability properties of the original system. Many coefficients can be calculated, which results in a very accurate representation of the system dynamics, but only a single factorization of the large system is required. The resulting system can be further reduced using explicit formulae for balanced truncation. The method is applied to two computational fluid dynamic systems, which model unsteady motion of a two-dimensional subsonic airfoil and unsteady flow in a supersonic diffuser. In both cases, the new method is found to work extremely well. Results are compared to models developed using the proper orthogonal decomposition and Arnoldi method. In comparison with these widely used techniques, the new method is computationally more efficient, preserves the stability of the original system, uses both input and output information, and, for smooth transfer functions, is valid over a wide range of frequencies.