Model reduction of linear and nonlinear systems in the Loewner framework: A summary

The study of methods that involve rational interpolation is of interest. We provide an overview of the Loewner framework which is interpolatory but uses measured or computed data (e.g. measurements of the frequency response of a to-be approximated system) instead of the system matrices, and constructs reduced models based on a rank revealing factorization of appropriately constructed matrices. We concentrate on generalizing the already existing framework for linear to bilinear and quadratic bilinear systems. The primary reason behind using quadratic bilinear systems is that they represent the bridge between linear and nonlinear systems. For certain types of nonlinear systems, we can always find an equivalent quadratic bilinear model without performing any approximation.