Data‐driven model order reduction of quadratic‐bilinear systems

Abstract

Summary We introduce a data‐driven model order reduction approach that represents an extension of the Loewner framework for linear and bilinear systems to the case of quadratic‐bilinear (QB) systems. For certain types of nonlinear systems, one can always find an equivalent QB model without performing any approximation whatsoever. An advantage of the Loewner framework is that information about the redundancy of the given data is explicitly available, by means of the singular values of the Loewner matrices. This feature is also valid for the proposed generalization. As for the linear and bilinear cases, these matrices can be directly computed by solving Sylvester equations. We begin by defining generalized higher‐order transfer functions for QB systems. These multivariate rational functions play an important role in the model order reduction process. We construct reduced‐order systems for which the associated transfer functions match those corresponding to the original system at selected tuples of interpolation points. Another benefit of the proposed approach is that it is data‐driven oriented, in the sense that one would only need computed or measured samples to construct a reduced‐order QB system. We illustrate the practical applicability of the proposed method by means of several numerical experiments.

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