Speaker
Description
In this talk, we present a data-driven approach to identify second-order systems of the form
\begin{equation}
\begin{array}{rcl}
\mathbf{M}\ddot{\mathbf{x}}(t) + \mathbf{D}\dot{\mathbf{x}}(t)+ \mathbf{K}\mathbf{x}(t) &=& \mathbf{B}\mathbf{u}(t),\qquad \mathbf{x}(0) = 0,~\dot{\mathbf{x}}(0) = 0, \
\mathbf{y}(t) &=& \mathbf{C}\mathbf{x}(t).
\end{array}
\end{equation}
These systems typically appear in order to perform various engineering studies, e.g., vibration analysis. The frequency response of the system can be given by the following rational structured function:
$$\mathbf{H}(s) = \mathbf{C}\left(s^2\mathbf{M}+ s\mathbf{D}+\mathbf{K}\right)^{-1}\mathbf{B}, $$
which is also known as the transfer function. The frequency response of a system can be measured in an experimental setup. As a consequence, given frequency samples, the identification of the underlying system relies on a rational structured approximation. To that aim, we propose two identification approaches inspired by the interpolation of the system transfer function. In the first proposed approach, we assume the internal Rayleigh damping hypothesis, i.e., the damping matrix is given as a linear combination of the mass and stiffness matrices, i.e., $\mathbf{D} = \alpha\mathbf{M} +\beta\mathbf{K}$. As a consequence, the identification problem is solved analytically using the frequency data. In the second approach, the Rayleigh damping hypothesis is no longer assumed, and the problem is solved finding low-rank matrices that best fit the given data ensuring the second order structure. Finally, the efficiency of the proposed methods is demonstrated by means of various numerical benchmarks.
