Ion Victor Gosea

• The Loewner framework is an interpolatory model order reduction technique that uses measured or computed data, e.g., measurements of the frequency response of a to-be approximated dynamical system instead of the system matrices, and constructs reduced models based on a rank revealing factorization of appropriately constructed matrices. In this thesis, we propose extensions of the classical Loewner framework for reduction of linear systems to some specific applications such as reducing classes of mildly nonlinear systems. The later includes bilinear, quadratic-bilinear and linear switched systems. The motivation behind this endeavor is that some of these aforementioned classes of systems can be viewed as a bridge between linear and nonlinear systems. For example, one can always write an approximation of a nonlinear system by means of a bilinear system. Moreover, for certain types of nonlinear systems, we can always find an equivalent quadratic-bilinear model without performing any approximation. Linear switched systems have been extensively studied in the literature since they offer a valuable addition to the class of linear systems, although reduction of such systems is arguably new. They can also be viewed as an intermediate step towards hybrid systems. For all the classes of systems that were previously mentioned, the overall strategy for extending the Loewner framework is conceptually similar. After collecting samples of input/output frequency domain mappings, e.g., either by means of measuring or by direct computation, one makes use of a specific arrangement of the data in matrix format. Hence, following some theoretical considerations, one can build reduced order models directly from the given data. The reduced systems have similar response to the large-scale original systems. More exactly, the input/output mappings for both systems have similar characteristics in the frequency range where the samples were considered.