Multi-scale optimal control for Einstein Telescope active seismic isolation

TL;DR

The paper tackles the challenge of achieving low-frequency sensitivity in a third-generation gravitational-wave detector by addressing seismic and control-noise limitations and tilt-to-translation coupling. It proposes a multi-scale optimal-control framework that jointly optimizes feedback and blending filters within a generalized plant, using an acausal optimum $\xi(f)$ and sensor-noise weighting to target sensor-limited performance, while accommodating cross-coupled loops. The authors derive loop analyses, define a sensor-noise–based weighting $W_p(s)$ with a $f^{-\alpha}$ factor, and solve a mixed $H_{\infty}$/$H_2$ optimization via a subgradient-descent method, demonstrating on ET-relevant configurations (OmniSens and BRS-T360). The approach enables rapid re-optimization for different sensor configurations, projects sensor-noise into performance metrics, and offers a scalable design framework for ET active seismic isolation.

Abstract

We present a multi-scale optimal control framework for active seismic isolation in the Einstein Telescope, a third-generation gravitational-wave observatory. Our approach jointly optimizes feedback and blending filters in a cross-coupled opto-mechanical system using a unified cost function based on the "acausal optimum," which quantifies sensor signal-to-noise ratios across frequencies. This method enables efficient re-optimization under varying sensor configurations and environmental conditions. We apply the framework to two candidate sensing systems using their modeled sensitivity: OmniSens-a six-degree-of-freedom inertial isolation system-and BRS-T360, which combines Beam Rotation Sensor (BRS) as an inertial tilt sensor with T360 as a horizontal seismometer. We demonstrate superior low-frequency isolation with OmniSens, reducing platform motion by up to two orders of magnitude near the microseism. The framework allows for ready optimization and projection of sensor noise to metrics relevant to the performance of the instrument, aiding the design of the Einstein Telescope.

Figures (15)

• Figure 1: Active platform with two inertial sensor configurations. The left figure shows the OmniSens configuration using a single reference mass for rotation and translation. The right figure shows the BRS-T360 configuration using separate rotation and translation sensors. Both configurations use interferometric (HoQI) sensors to track the relative displacement of the platform.
• Figure 2: Control schemes for rotation (left, $\theta$) and horizontal translation (right, $x$). White noise inputs $w$ are shaped by noise coloring filters $\eta$ to model real noise sources. The transfer function $T_{\theta \to x}$ captures coupling from rotation to translation. Weighting functions $W$ normalize the closed-loop residual motions $\theta$ and $x$ in order to produce performance outputs $z_1$ and $z_2$. $G_{\theta}$ and $G_x$ are the plant transfer functions, while $K_{\theta}$ and $K_x$ are the feedback controllers for rotation and translation, respectively. $H_{\theta}$ and $H_x$ are the blending filters, with complementary filters $\bar{H}_{\theta}$ and $\bar{H}_x$. $\eta_{\mathrm{dist}}$, $\eta_{\mathrm{disp}}$, and $\eta_{\mathrm{sens}}$ represent the disturbance, displacement-sensor, and inertial-sensor noise coloring filters, respectively; the subscript $\theta$ or $x$ indicates the rotational or horizontal loop.
• Figure 3: Procedure for Setting Up the Optimization in the Given Control Problem. The process begins by modeling the noise and transforming it into a format suitable for numerical computation. Using these noise models, the acausal optimum is constructed and integrated into the generalized plant, which represents the multi-scale cross-coupled system. This procedure consistently generates performance predictions based on the input noise spectra.
• Figure 4: OmniSens acausal optimum $\xi$ decomposition for both $\theta$ and x loop. The acausal optimum represents the global minimum across all noise curves, as defined in Eq. \ref{['eq:acausal-optimum']}. The curves $\eta_{dist}$, $\eta_{disp}$, and $\eta_{sens}$ captures ground disturbance, displacement sensor noise, and inertial sensing noise, respectively.
• Figure 5: BRS-T360 acausal optimum $\xi$ decomposition for both $\theta$ and x loop. The curves $\eta_{dist}$, $\eta_{disp}$, and $\eta_{sens}$ captures ground disturbance, displacement sensor noise, and inertial sensing noise, respectively.