The Loewner framework for system identification and reduction

One of the main approaches to model reduction of both linear and nonlinear dynamical systems is by means of interpolation. This approach seeks reduced models whose transfer function matches that of the original system at selected interpolation points. Data-driven methods constitute an important special case. We start with an account of the Loewner framework in the linear case [52]. It constructs models from given data in a straightforward manner. An important attribute is that it provides a trade-off between accuracy of fit and complexity of the model. We compare this approach with other approximation methods and apply it to different test-cases. One of the case studies to which we apply the aforementioned methods is defined by the inverse of the Bessel function. We then turn our attention to the approximation of an Euler-Bernoulli beam model with Rayleigh damping. Further case studies include the approximation of two real valued functions with specific difficulties (discontinuity, sharp peaks). One computational tool is the SVD; its complexity is cubic in the number of data points. For large data sets the CUR factorization is a viable alternative. Note that its complexity is cubic as well but with respect to the dimension of the reduced order model (ROM). Another option is to use stochastic procedures such as randomized singular value decomposition (r-SVD) [41].