Structure‐preserving interpolation of quadratic‐bilinear systems via regular multivariate transfer functions

Abstract For the accurate modeling of real‐world phenomena, high‐dimensional nonlinear dynamical systems are indispensable. Quadratic‐bilinear nonlinearities are of particular importance since many systems with general nonlinearities can be remodeled into this form. Such nonlinear systems typically encode physical properties via their internal differential structure. This leads to a high demand for structure‐preserving model order reduction techniques for quadratic‐bilinear systems that allow for the accurate representation of their dynamics by a small number of differential equations while retaining the internal system structure. In this paper, we provide formulae for structured regular subsystem transfer functions of quadratic‐bilinear systems with internal differential structures and provide a corresponding interpolation theory for the efficient construction of the resulting structured reduced‐order models. The presented theoretical results are illustrated in numerical experiments including a time‐delayed nonlinear reaction–diffusion equation and a molecular simulation.