Data-driven balanced truncation for second-order systems via the approximate Gramians

This paper studies the data-driven balanced truncation (BT) method for second-order systems based on the measurements in the frequency domain. The basic idea is to approximate Gramians via the numerical quadrature rules, and establish the relationship between the main quantities in the procedure of BT and the sample data, which paves the way for the execution of BT in a nonintrusive manner. We construct the structure-preserving reduced models approximately based on the sample data of second-order systems with proportional damping, and provide a detailed algorithm in real-valued arithmetic to establish the data-driven counterpart of BT. In order to address the issue of large amount of sample data, we exploit the fact that the main quantities satisfy a couple of Sylvester matrix equations. The low-rank approximation to the solution of Sylvester equations is employed to avoid the explicit calculation of the main quantities, leading to an acceleration of the process of the data-driven BT. The performance of our approach is illustrated in detail via two numerical examples.