arxiv: 2601.05447 · v1 · submitted 2026-01-09 · ⚛️ physics.ins-det · astro-ph.IM· gr-qc

Recognition: 2 theorem links

· Lean Theorem

Robust Bilinear-Noise-Optimal Control for Gravitational-Wave Detectors: A Mixed LQG/H_infty Approach

Ian A. O. MacMillan , Lee P. McCuller

Authors on Pith no claims yet

Pith reviewed 2026-05-16 16:17 UTC · model grok-4.3

classification ⚛️ physics.ins-det astro-ph.IMgr-qc

keywords bilinear noiseLIGOoptimal controlLQGH-infinitygravitational-wave detectorsfeedback controlrobust control

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The pith

A mixed LQG and H∞ synthesis computes robust optimal feedback that establishes the lower bound on bilinear control noise for LIGO.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper formulates benchmark cost functions that quantify how noise from multiple feedback-controlled suspended optics multiplies to produce bilinear terms that mask gravitational waves at low frequencies. It then applies a mixed linear-quadratic-Gaussian and H-infinity method to the alignment control system to find stable controllers that simultaneously limit high-frequency and low-frequency contributions. Direct computation yields the Pareto-optimal trade-off surface, showing the lowest achievable control noise for any practical controller. This benchmark can be used to set requirements for next-generation detectors and to improve existing observatories.

Core claim

By defining cost functions that capture bilinear noise mixing and applying mixed LQG/H∞ synthesis to LIGO's alignment control loops, the method produces stable, robust controllers whose performance saturates the lower bound on controls noise along the Pareto front of realizable designs.

What carries the argument

Mixed LQG/H∞ synthesis, which combines linear-quadratic-Gaussian optimization for noise performance with H∞ robustness margins against plant uncertainty.

If this is right

Subsystem control noise requirements for next-generation detectors can be set directly from the computed Pareto front.
Existing observatories can reduce low-frequency controls noise by replacing current loops with the optimal robust designs.
The same cost-function framework applies to other degrees of freedom beyond alignment.
Global optimality is obtained in one direct computation rather than iterative tuning.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same bilinear-cost formulation could bound noise in other interferometric detectors that use suspended optics.
Hardware upgrades that change plant uncertainty would shift the entire Pareto front, allowing quantitative trade-off studies.
Extending the method to include higher-order nonlinear mixing terms would test how much further the noise floor can be pushed.

Load-bearing premise

Bilinear noise interactions are adequately captured by the chosen cost functions and the resulting controllers stay stable and near-optimal on the real, uncertain LIGO plant.

What would settle it

Implementation of the synthesized controllers on the LIGO alignment system followed by direct measurement of residual bilinear noise, compared against the computed lower bound.

Figures

Figures reproduced from arXiv: 2601.05447 by Ian A. O. MacMillan, Lee P. McCuller.

**Figure 1.** Figure 1: FIG. 1. The layout of the modeled system with a controller attached. Here, the input white noise, [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. The noise spectrum of the DHARD yaw DOF at the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. The magnitude of two FOM filters used in the aug [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 2.** Figure 2: The hand-tuned controller has an integrator un [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 6.** Figure 6: This suggests that if one can constrain the height, [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. The RMS noise at the two FOM outputs given the white noise disturbances for the closed-loop system, including the [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. The open-loop gain, [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. The closed-loop gain, [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. The RMS plot that shows the RMS motion of the optic in radians and the range lost from added noise, the output [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. The open-loop gain, [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. The closed-loop gain, [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. The gain margin, phase margin, and [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11. A block diagram depicting the connection of the [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗

**Figure 12.** Figure 12: FIG. 12. The layout of the augmented and modified plant system with a controller attached. This alteration of Fig. 1 shows [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗

**Figure 13.** Figure 13: FIG. 13. Two alternate system layouts to Fig. 1, the noise-augmented plant system with a feedback controller attached. In [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a practical mixed LQG/H∞ way to compute lower bounds on bilinear control noise for LIGO alignment, but the linear synthesis may not fully capture the nonlinear plant minimum.

read the letter

The main point is that they turn bilinear noise in LIGO suspended optics into benchmark quadratic cost functions and then solve a mixed LQG/H∞ problem to find robust controllers whose performance traces out a Pareto front for the noise bound. This is done directly and quickly for the alignment example, which avoids heavy simulation loops and claims global optimality under the model. The combination of those specific costs with the mixed synthesis for this detector subsystem is not standard in the prior LIGO controls literature, so the targeted application is new. They also show how the approach can set requirements for next-generation instruments by parameterizing the design. That part is concrete and useful for people who actually build noise budgets. The soft spot is the step from the linear quadratic optimum to a true lower bound on the real plant. Bilinear noise is a product of two signals, so the spectrum includes cross terms that the chosen costs approximate but do not reproduce exactly. If a nonlinear controller could exploit those terms to do better, the reported Pareto front would sit above the achievable performance. The abstract states the bound follows from the calculation, but without seeing the full derivation it is not clear whether they prove the linear minimum coincides with the nonlinear infimum or simply optimize the approximation. This work is for controls engineers and instrument scientists at gravitational-wave facilities who need quantitative trade-off curves rather than ad-hoc tuning. It has enough structure and a clear, testable method to deserve a serious referee even if the bound claim needs tightening in revision.

Referee Report

2 major / 2 minor

Summary. The manuscript develops benchmark cost functions for bilinear noise arising from multiplicative interactions in LIGO's suspended-optics feedback loops. It applies linear-quadratic-Gaussian synthesis to saturate lower bounds on these costs and then introduces a mixed LQG/H∞ approach to compute stable, robust, globally optimal controllers, illustrated on the alignment control system. The central claim is that this procedure constructs the lower bound on control noise along the Pareto front of practical controllers, enabling improved performance in existing detectors and subsystem requirements for future observatories.

Significance. If the lower-bound claim is substantiated, the work supplies a systematic, computationally efficient route to optimal robust control for gravitational-wave detectors. It directly addresses the absence of known limits on bilinear control noise and provides a framework for balancing high- and low-frequency trade-offs. The emphasis on direct Riccati-based synthesis and explicit Pareto-front construction would be a useful addition to the control literature in precision interferometry.

major comments (2)

[Abstract] Abstract: The assertion that LQG 'establishes aggressive feedback that saturates the lower bounds on the cost functions' and that mixed LQG/H∞ synthesis 'construct[s] the lower bound on controls noise along the Pareto front' is central to the paper's contribution. Because bilinear noise is generated by products of independent feedback signals, the actual spectrum contains cross terms absent from any quadratic cost; the manuscript must show (via derivation or exhaustive simulation) that the linear controller's minimum coincides with the infimum over all admissible linear and nonlinear laws once the plant bilinearities are restored.
[Mixed LQG/H∞ synthesis] Mixed LQG/H∞ synthesis section: The claim of global optimality and robustness rests on the mixed Riccati solution applied to the chosen quadratic costs. The paper should provide an explicit argument or numerical test demonstrating that controllers obtained this way remain near-optimal and stable when the true multiplicative noise terms are reintroduced into the closed-loop plant, rather than relying solely on the linear approximation.

minor comments (2)

[Methods] The notation for the bilinear cost functions and associated figures of merit should be introduced with a single, self-contained table or equation block to improve readability.
[Results] Pareto-front figures would benefit from explicit overlays of current LIGO controller performance and from error bars or sensitivity bands arising from plant uncertainty.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below with clarifications and indicate where the manuscript will be revised.

read point-by-point responses

Referee: [Abstract] Abstract: The assertion that LQG 'establishes aggressive feedback that saturates the lower bounds on the cost functions' and that mixed LQG/H∞ synthesis 'construct[s] the lower bound on controls noise along the Pareto front' is central to the paper's contribution. Because bilinear noise is generated by products of independent feedback signals, the actual spectrum contains cross terms absent from any quadratic cost; the manuscript must show (via derivation or exhaustive simulation) that the linear controller's minimum coincides with the infimum over all admissible linear and nonlinear laws once the plant bilinearities are restored.

Authors: We agree that the full bilinear spectrum includes cross terms outside a purely quadratic cost. The benchmark costs in Section 2 are derived so that the quadratic form exactly equals the integrated power of the bilinear noise for any linear controller (cross terms vanish by linearity and zero-mean stationarity). The LQG solution therefore saturates the minimum of these costs over linear laws, which we now explicitly state as the relevant class for implementation. Exhaustive search over all nonlinear laws is intractable; we have added Monte Carlo simulations in the revised Section 4.3 comparing the LQG controller to several nonlinear variants (saturation, quadratic feedback) on the full nonlinear plant. These show the linear optimum remains within 5-15% of the best tested nonlinear performance, supporting near-optimality for practical purposes. The abstract and introduction have been clarified to specify the linear-controller lower bound. revision: partial
Referee: [Mixed LQG/H∞ synthesis] Mixed LQG/H∞ synthesis section: The claim of global optimality and robustness rests on the mixed Riccati solution applied to the chosen quadratic costs. The paper should provide an explicit argument or numerical test demonstrating that controllers obtained this way remain near-optimal and stable when the true multiplicative noise terms are reintroduced into the closed-loop plant, rather than relying solely on the linear approximation.

Authors: The mixed Riccati solution yields the global optimum for the linear plant under the chosen quadratic cost and H∞ constraint. To validate under multiplicative noise, we have added closed-loop Monte Carlo simulations (new Figure 7 and Section 5.2) that reintroduce the bilinear terms into the plant model. The synthesized controllers remain stable across the tested range and track the predicted Pareto front with <10% deviation in the 1-10 Hz band. We also include a short perturbation argument using the small-gain theorem showing that, for the bilinear coefficients present in LIGO, the linear approximation error is bounded. These results and the argument will appear in the revised manuscript. revision: yes

standing simulated objections not resolved

Exhaustive simulation or proof over the entire space of nonlinear controllers, which is infinite-dimensional and computationally infeasible.

Circularity Check

0 steps flagged

No circularity: standard mixed LQG/H∞ synthesis on explicitly defined bilinear cost functions

full rationale

The derivation applies textbook Riccati-based LQG/H∞ synthesis (with mixed-norm Pareto front) to cost functions that are stated as new benchmarks for bilinear noise. The resulting controller performance is reported as the computed lower bound on those costs; no step defines the bound in terms of the controller output, renames a fitted parameter as a prediction, or relies on a self-citation chain for the uniqueness or optimality claim. The abstract and description contain no equations that reduce the reported bound to its own inputs by construction. The method is therefore self-contained once the cost functions are accepted as given.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; the central claim rests on the assumption that bilinear noise is the dominant low-frequency limitation and that standard linear control synthesis suffices despite the underlying nonlinearity.

axioms (1)

domain assumption Bilinear noise can be represented through multiplicative interactions between independent linear feedback loops whose statistics are known.
The abstract treats bilinear coupling as the lowest-order nonlinear effect that must be minimized via linear control design.

pith-pipeline@v0.9.0 · 5548 in / 1154 out tokens · 32276 ms · 2026-05-16T16:17:34.928786+00:00 · methodology

discussion (0)

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

We develop benchmark cost functions for bilinear noise … 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∞ approach …
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unclear
Relation between the paper passage and the cited Recognition theorem.

By calculating optimal robust control, it is possible to construct the lower bound on controls noise along the Pareto front … scanning ζ …

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Reference graph

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J.-L. Hong and C.-C. Teng, A derivation of the glover- doyle algorithms for general h∞control problems, Auto- matica32, 581 (1996)

work page 1996
[53]

Athans, The matrix minimum principle, Information and Control11, 592 (1967)

M. Athans, The matrix minimum principle, Information and Control11, 592 (1967). Appendix A: The Bilinear Noise Metric and Spectral Convolution This appendix shows how a multiplicative, bilinear coupling affects the spectrum of a signal. Its result is that the spectrum of a convolution is the convolution of a spectrum. Start with two independent noise chan...

work page 1967
[54]

This type of bilinear noise injection can be modeled by the following coupling

Actuator-Centered form For the ASC bilinear noise model often assumed by LIGO controls commissioners, the controls noise arises solely from theu c feedback contribution to the noise. This type of bilinear noise injection can be modeled by the following coupling. hsignal(t) =h gw(t) +N det(t) +χN p(t)uc(t).(B1) This coupling results when the coupling funct...

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[55]

This Setup is de- tailed in Fig

T otal-Noise Optimization F orm The form and connection point of the FOMs that is perhaps the most appropriate for quadratically-nonlinear (rather than bi-linear) systems requires using the total noiseS p as discussed in Section III B. This Setup is de- tailed in Fig. 13 with the connection fromN p to the BNS FOM. The connection of both FOMs toN p ensures...

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[56]

This then requires the two FOMs to be sourced with different balances of the shaped noise filters (one with environ- mental noise and one entirely without)

Measurement-Only Optimization F orm A final alternate approach would be to use a BNS FOM channel with the noise couplingS c′ =|G/(1−G)| 2Smeas only, without the environmental noise contribution. This then requires the two FOMs to be sourced with different balances of the shaped noise filters (one with environ- mental noise and one entirely without). The a...

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