Speaker
Description
Learning dynamical systems from data has emerged as a pivotal area of research, bridging the realms of mathematics, engineering, and data science. Of particular importance is the construction of models with meaningful internal structure that allows interpretability and explainability of the results. In the case of mechanical and electro-mechanical processes, dynamical systems are typically described via ordinary differential equations with second-order time derivatives
$$
M \ddot{x}(t) + D \dot{x}(t) + K x(t) = B u(t), \quad
y(t) = C_{\mathrm{p}} x(t) + C_{\mathrm{v}} \dot{x}(t),
$$
with the system matrices $M, D, K \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$, $C_{\mathrm{p}}, C_{\mathrm{v}} \in \mathbb{R}^{p \times n}$, the inputs $u(t) \in \mathbb{R}^{m}$, the internal states $x(t) \in \mathbb{R}^{n}$, and the outputs $y(t) \in \mathbb{R}^{p}$. A popular approach in model order reduction to construct accurate low-dimensional surrogate models that retain the second-order internal structure is the second-order position-velocity balanced truncation method. Thereby, the system is balanced with respect to structured system Gramians and states corresponding to small singular values of these Gramians are truncated.
In this work, we present a new data-driven formulation of the second-order
balanced truncation method. Utilizing suitable structured Loewner matrices, we can directly compute low-dimensional balanced second-order systems with generalized proportional damping, $D(s) = f(s) M + g(s) K$ where $f(s), g(s) \in \mathbb{C}$, from given frequency domain measurements. While based on classical structure-preserving model reduction, our method can be used to enforce mechanical model structure independent of the data source. Several numerical examples demonstrate the effectiveness of the proposed method.