Speaker
Description
In recent years, structured reduced-order modeling has become an essential component in meaningful applications across engineering and the physical sciences whenever mathematical models are unavailable, but input-output data are abundant. For linear time-invariant systems, the Loewner framework provides a non-intrusive methodology for the construction of minimal interpolants from rational transfer function data. Its key ingredients are the Loewner and shifted Loewner matrices, which are constructed solely from transfer function evaluations. Significantly, the rank of the Loewner matrix reveals the underlying order (McMillan degree) of the system that generated the data, enabling a trade-off between the accuracy-of-fit and the complexity of the data-based reduced-order model.
In this work, we generalize the Loewner framework to the class of linear quadratic-output systems
$$E\dot{x}(t)=Ax(t)+Bu(t),~~y(t)= M(x(t)\otimes x(t)).$$
Dynamical systems with quadratic-output functions arise naturally in applications where one is interested in observing or simulating response quantities computed as the product of time- or frequency-components of the state, such as in vibro-acoustic problems or energy-based modeling. We introduce appropriately defined Loewner and shifted Loewner matrices that are built from samples of the multivariate rational transfer function of the underlying quadratic-output model. These matrices retain the hallmark features of the linear time-invariant Loewner approach; namely, their rank exposes the order of the system used to generate the data, and the reduced-order model computed using the proposed Loewner and shifted Loewner matrices is guaranteed to satisfy certain multivariate rational interpolation conditions. We also discuss practical considerations, such as obtaining quadratic-output transfer function data from frequency-response measurements of an equivalent, multiple-output linear time-invariant system.