Generalizations of data-driven balancing: What to sample for different balancing-based reduced models

The quadrature-based balanced truncation (QuadBT) framework of arXiv:2104.01006 is a non-intrusive reformulation of balanced truncation (BT), a classical projection-based model-order reduction technique for linear systems. QuadBT is non-intrusive in the sense that it builds approximate balanced truncation reduced-order models entirely from system response data, e.g., transfer function measurements, without the need to reference an explicit state-space realization of the underlying full-order model. In this work, we generalize the QuadBT framework to other types of balanced truncation model reduction. Namely, we show what transfer function data are required to compute data-driven reduced models by balanced stochastic truncation, positive-real balanced truncation, and bounded-real balanced truncation. In each case, these data are evaluations of particular spectral factors associated with the system of interest. These results lay the theoretical foundation for data-driven reformulations of the aforementioned BT variants. Although it is not yet clear how to compute or obtain these spectral factor data in a practical real-world setting, examples using synthetic (numerically evaluated) transfer function data are included to validate the data-based reduced models.