GraFIT: A toolbox for fast and accurate frequency response identification in Gravitational Wave Detectors

TL;DR

GraFIT tackles the need for accurate, data-efficient FRF identification in Gravitational Wave detectors by implementing the Local Rational Method within a local modelling framework. It jointly estimates the system FRF $G(\Omega_k)$ and the smooth transient $T(\Omega_k)$ in a frequency window, improving leakage suppression and noise handling for both open- and closed-loop data. Validation on Virgo experiments shows that GraFIT can outperform traditional Spectral Analysis with 2–3× lower variance and up to 10× data savings, enabling faster, more reliable controller design and diagnostics. The toolbox thus provides a practical, scalable solution for FRF identification in GW detectors, with quantified uncertainty and applicability to multi-input multi-output systems.

Abstract

Frequency response function (FRF) measurements are widely used in Gravitational Wave (GW) detectors, e.g., for the design of controllers, calibrating signals and diagnostic problems with system dynamics. The aim of this paper is to present GraFIT: a toolbox that enables fast, inexpensive, and accurate identification of FRF measurements for GW detectors compared to the commonly used approaches, including common spectral analysis techniques. The toolbox consists of a single function to estimate the frequency response function for both open-loop and closed-loop systems and for arbitrary input and output dimensions. The toolbox is validated on two experimental case studies of the Virgo detector, illustrating more than a factor 3 reduction in standard deviation of the estimate for the same measurement times, and comparable standard deviations with up to 10 times less data for the new method with respect to the currently implemented Spectral Analysis method.

Figures (6)

• Figure 1: Standard open-loop identification problem where the goal is to identify $G$ using the input signal $r(n)$ and noisy output signal $y(n)$ which is perturbed a disturbance $v(n)$.
• Figure 2: Standard closed-loop identification problem, where the goal is to identify $G$ using the input signal $r(n)$ and noisy output signals $y(n)$ and $u(n)$.
• Figure 3: FRF of $\widehat{G}$ for the $x$, $z$ and $t_y$ degrees of freedom of the top stage of the MultiSAS using SA estimation () and its standard deviation () and LRM estimation () and its standard deviation (). The LRM method obtains almost consistently a factor 2 to 3 lower standard deviation with even lower standard deviations at the resonance peaks and also has a factor $P=6$ higher frequency resolution.
• Figure 4: FRF of $\widehat{G}$ for the three longitudinal DoFs using SA estimation () and its standard deviation () for 120s of data, LRM estimation () and its standard deviation () for 120s of data and LRM estimation () and its standard deviation () for the first 12s of the same dataset. The standard deviation for SA is not available. Just 12s of data using the LRM method is sufficient to get a good quality FRF.
• Figure 5: FRF of $\widehat{G}_{ru}$ for the three longitudinal DoFs using SA estimation () and its standard deviation () for 120s of data, LRM estimation () and its standard deviation () for 120s of data and LRM estimation () and its standard deviation () for the first 12s of the same dataset. The LRM method is able to obtain an order lower variance with just a tenth of the data compared to SA, while having almost identical estimates, indicating marginal bias.