Structured barycentric forms for interpolation-based data-driven reduced modeling of second-order systems

TL;DR

The paper tackles data-driven, interpolatory reduced-order modeling of second-order dynamical systems from frequency-domain data by introducing structured barycentric forms that preserve the second-order structure. It develops three Loewner-like algorithms based on constrained stiffness or damping matrices (and zero-damping variants) to explicitly construct second-order systems whose transfer functions interpolate given measurements. The approach yields barycentric transfer functions with explicit parameters (weights and support points) that can be tuned to enforce stability or energy-preserving properties, and it is extended to real-valued realizations. Numerical experiments demonstrate competitive accuracy against unstructured methods and showcase the ability to preserve physical structure, with potential extensions to least-squares fitting and more flexible parameter choices.

Abstract

An essential tool in data-driven modeling of dynamical systems from frequency response measurements is the barycentric form of the underlying rational transfer function. In this work, we propose structured barycentric forms for modeling dynamical systems with second-order time derivatives using their frequency domain input-output data. By imposing a set of interpolation conditions, the systems' transfer functions are rewritten in different barycentric forms using different parametrizations. Loewner-like algorithms are developed for the explicit computation of second-order systems from data based on the developed barycentric forms. Numerical experiments show the performance of these new structured data driven modeling methods compared to other interpolation-based data-driven modeling techniques from the literature.

Figures (5)

• Figure 1: Butterfly gyroscope example: All methods recover reduced-order models with similar accuracy using the same amount of given interpolation data. The model learned by soLoewRayleigh performs slightly better for frequencies between $500$ and $5\,000$ rad/s.
• Figure 2: Artificial fishtail example: All methods provide a similar approximation accuracy. The model inferred by soBaryLoewD performs insignificantly worse for frequencies between $1$ and $100$ rad/s, while the model from soLoewRayleigh keeps the accuracy level also for higher frequencies.
• Figure 3: Flexible aircraft example: All methods construct reduced-order models that recover the given data set with sufficient accuracy. For higher frequencies, more interpolation data is used due to the presence of many local maxima and minima.
• Figure 4: Bone model example: For lower frequencies, the second-order methods produce models with at least one order of magnitude smaller relative errors than the classical Loewner framework. The curves of soBaryLoewKD0 and soLoewRayleigh are identical up to numerical round-off errors.
• Figure 5: Hysteretic plate example: For up to $50$ rad/s, the second-order methods soBaryLoewKD0 and soLoewRayleigh produce relative errors that are at least four orders of magnitude smaller than the classical BaryLoew. The curves of soBaryLoewKD0 and soLoewRayleigh are identical up to numerical round-off.

Theorems & Definitions (18)

• Proposition 1
• Lemma 1
• proof
• Corollary 1
• proof
• Lemma 2
• proof
• Remark 1
• Lemma 3
• proof