Robust Bilinear-Noise-Optimal Control for Gravitational-Wave Detectors: A Mixed LQG/$H_\infty$ Approach

TL;DR

This work tackles bilinear noise in gravitational-wave detectors by formulating benchmark costs and cast­ing the control design as a robust, mixed-sensitivity problem. It bridges classical LQG optimization with modern $\mathcal{H}_\infty$ techniques, using state-space Riccati equations to compute controllers that minimize a weighted RMS of bilinear noise while enforcing stability margins. A Pareto-front methodology is developed via a weight parameter $\zeta$, balancing the BNS-range impact against broadband motion, yielding a spectrum of near-optimal controllers. The approach delivers robust, implementable designs for alignment sensing controls and offers a pathway to reduce control-noise across LIGO subsystems, with potential applicability to next-generation detectors through automated, noise-adaptive controller design. The combination of $H_2$- and $H_\infty$-based optimization provides both performance gains and guaranteed margins, enabling more aggressive noise suppression without sacrificing stability.

Abstract

At its lowest frequencies, LIGO is limited by noise in its many degrees of freedom of suspended optics, which, in turn, introduce noise in the interferometer through their feedback control systems. Nonlinear interactions are a dominant source of low-frequency noise, mixing noise from multiple degrees of freedom. The lowest-order form is bilinear noise, in which the noise from two feedback-controlled subsystems multiplies to mask gravitational waves. Bilinear couplings require control trade-offs that simultaneously balance high- and low-frequency noise. Currently, there is no known lower limit to bilinear control noise. Here, we develop benchmark cost functions for bilinear noise and associated figures of merit. Linear-quadratic-Gaussian control then establishes aggressive feedback that saturates the lower bounds on the cost functions. We then develop a mixed LQG and $H_\infty$ approach to directly compute stable, robust, and optimal feedback, using the LIGO's alignment control system as an example. Direct computations are fast while ensuring a global optimum. By calculating optimal robust control, it is possible to construct the lower bound on controls noise along the Pareto front of practical controllers for LIGO. This method can be used to drastically improve controls noise in existing observatories as well as to set subsystem control noise requirements for next-generation detectors with parameterized design.

Figures (13)

• Figure 1: The layout of the modeled system with a controller attached. Here, the input white noise, $u_{n1}$, is being shaped to appear as environmental noise seen by the plant. The system has two FOM, which are simply transfer functions that weigh the output noise: a broadband FOM outputs the unshaped noise, i.e., every frequency has the same weight, and a BNS inspiral FOM, which represents the band of detection for BNS signals. In this model, the two plant blocks are identical and serve as a way to represent $S_{\mathrm{a}}$ for actuation-point controls noise minimization. These FOM are discussed further in Section \ref{['subsec:foms']}. The input associated with $u_{\infty}$, which has a dotted line, is used for the $\mathcal{H}_\infty$ minimization discussed in Section \ref{['subsec: h_inf simple explination']}.
• Figure 2: The noise spectrum of the DHARD yaw DOF at the LIGO Hanford Observatory decomposed into fits of measurement and environmental contributions. The spectrum are fit to filters $M$ and $E$, see relations of Eqs. \ref{['eq:EM_relations S_Env']}-\ref{['eq:EM_relations S_un1']}. The measured ASD of $y_\text{meas}$ is the measured spectrum of the cavity axis (see Fig. \ref{['fig:systemlayout']}) and is equal to $|EP+M|$. This spectrum assumes no control, $K=0$ (open-loop). After fitting a ZPK to $|EP|$, the contribution from $E$ was isolated by dividing by the model for the plant ($P$). The filters are using the Python package wield.iirrational to refine fits with gradient descent after making initial guesses using the AAA algorithm iirrationalNakatsukasaSJSC18AAAAlgorithm.
• Figure 3: The magnitude of two FOM filters used in the augmented system, as shown in Fig. \ref{['fig:systemlayout']}. Both of these FOM shape either $N_{\mathrm{p}}$ or $N_{\mathrm{a}}$ to create metrics for the controller to minimize. The broadband FOM results in an unshaped signal that must be below a specified threshold to keep the interferometer locked. The BNS FOM shapes $S_{\mathrm{a}}$ allowing for the direct optimization of LIGO's BNS detection range.
• Figure 4: The RMS noise at the two FOM outputs given the white noise disturbances for the closed-loop system, including the system shown in Fig. \ref{['fig:systemlayout']} and optimal LQG controllers calculated for specified $\zeta$ values. Each point represents the performance of a single controller that is the LQG optimal solution for the plant with a different FOM weighting. The RMS value is shown on the axis of the corresponding FOM. The BNS FOM's RMS is calibrated to represent BNS range lost by the noise from a given controller, while the flat FOM represents the RMS noise in radians for ASC DHARD yaw. The number next to each point represents the $\zeta$ value used to acquire the desired weighting. The color of each point corresponds to the phase margin of the controller. The triangle represents the current LIGO hand-tuned controller's performance and phase margin. All of the LQG optimal controllers whose performance beats the hand-tuned controller in both FOM have poor phase margins, and they get worse as the BNS range FOM is emphasized using a higher $\zeta$ factor. The margins can be improved by the LQG optimal solutions with an $\mathcal{H}_\infty$ performance bound discussed in Section \ref{['sec: LQG with arb margins']} and presented in Fig. \ref{['fig:RMSplot']}.
• Figure 5: The open-loop gain, $G$, of the unscaled plant and controller system shown for different values of $\zeta$. As $\zeta$ gets lower, the optimizer gives less weight to the $\tilde{R}_\mathrm{BNS}$. A hand-designed LIGO DHARD yaw controller is included for comparison. Each of these LQG controllers was optimized using an LQG solver with no enforced stability margins. The hand-tuned controller meets the stability requirements necessary for implementation in the interferometer.