  In this paper we suggest a moment matching method for quadratic-bilinear
dynamical systems. Most system-theoretic reduction methods for nonlinear
systems rely on multivariate frequency representations. Our approach instead
uses univariate frequency representations tailored towards user-pre-defined
families of inputs. Then moment matching corresponds to a one-dimensional
interpolation problem, not to multi-dimensional interpolation as for the
multivariate approaches, i.e., it also involves fewer interpolation frequencies
to be chosen. Comparing to former contributions towards nonlinear model
reduction with univariate frequency representations, our approach shows
profound differences: Our derivation is more rigorous and general and reveals
additional tensor-structured approximation conditions, which should be
incorporated. Moreover, the proposed implementation exploits the inherent
low-rank tensor structure, which enhances its efficiency. In addition, our
approach allows for the incorporation of more general input relations in the
state equations - not only affine-linear ones as in existing system-theoretic
methods - in an elegant way. As a byproduct of the latter, also a novel
modification for the multivariate methods falls off, which is able to handle
more general input-relations.
