The role of the state in model reduction with subspace and POD-based data-driven methods

The paper investigates the selection of state sequences in data-driven projection-based model reduction methods that compute parsimonious models by forming regression problems featuring low-order fictitious states. Specifically, subspace identification and dynamic mode decomposition techniques are considered. It is shown that, while sharing a seemingly equivalent structure, they differ profoundly in the way these states are selected. A theoretical characterization of the differences is given, including a parametrization of a new class of state transformations implicitly used in both approaches and a balanced transformation obtained directly from data. Numerical examples are proposed to show the impact of these differences on the accuracy of the extracted low-order representations.