Reducing controls noise in gravitational wave detectors with interferometric local damping of suspended optics

Control noise is a limiting factor in the low-frequency performance of the LIGO gravitational wave detectors. In this paper we model the effects of using new sensors called HoQIs to control the suspension resonances. We show if we were to use HoQIs, instead of the standard shadow sensors, we can suppress resonance peaks up to tenfold more while simultaneously reducing the noise injected by the damping system. Through a cascade of effects this will reduce the resonant cross-coupling, allow for improved stability for feed-forward control, and result in improved sensitivity of the detector in the 10-20 Hz band. This analysis shows that local sensors such as HoQIs should be used in current and future detectors to improve low-frequency performance.


INTRODUCTION
Gravitational waves were predicted by Einstein's Theory of General Relativity, and were first observed in 2015 1 .Since then multiple events 2,3 have been detected by the Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) 4 and Advanced Virgo 5 interferometers.These interferometers precisely measure the Differential ARM length (DARM) changes of the long (3-4 km) arm cavities.Passing gravitational waves induce differential strain in the perpendicular arms, allowing interferometric detection.
The first detection of a binary neutron star inspiral 6,7 demonstrated the importance of gravitational wave detectors for multi-messenger astronomy.Gravitational wave detectors provide sky localization from triangulation with multiple detectors.Inspiral detection can provide early alerts for electromagnetic and particle observatories.
Improvements to sensitivity at 10−20 Hz enable earlier detections, greater signal-to-noise ratios, and further astrophysical reach into space 8 .All events observed so far have been inspirals, where the frequency increases until the objects collide and merge.However, the signals have much longer duration at lower frequencies, with the timeuntil-merger proportional to f −8/3 .Improvements to low frequency sensitivity are therefore especially important for earlier detections and increasing the time in the measurement band, which in turn improves sky localisation 9 .
One of the largest noise sources in earth-bound gravitational-wave detectors is ground vibration, which is ten orders of magnitude larger than the signal 1 and moves the optics of the interferometer.The LIGO observatory has seismic isolation systems consisting of cascaded passive 10 and active isolation 11,12 to suppress this movement noise, and facilitate gravitational wave detection.Passive isolation is achieved through the use of pendula, and mass-spring-systems.Active isolation employs a blend of relative displacement and inertial sensors for feedback control of the passive isolation systems.
Despite the success of these isolation systems in reducing direct vibration coupling, Advanced LIGO's lowfrequency sensitivity is limited at 10 − 20 Hz by 'global control noise' from the interferometer's Alignment Sensing and Control (ASC) and the auxiliary Length Sensing and Control systems 13 .
Global controls keep the optics of the interferometer correctly placed and oriented relative to each other.Local controls, on the other hand, minimize the transfer of ground motion to an individual optic.
In this paper we analyse how improved local sensors and controls can improve performance in a manner that improves the performance and predictability for global controls.Suspension chains with better local damping produce a quieter, simpler, and more stable plant, thereby reducing noise that is associated with non-linear, bi-linear, and non-stationary couplings that cannot currently be suppressed in post-processing 3,14 .
There is a large and growing community of instrumentation development for 3G observatories (for a subset of proposed sensors and measurement methods see references [15][16][17][18][19][20][21] and inertial sensors [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] ).In this paper we present a novel analysis of the projected quantitative effect of these instruments on the performance of a multi-stage suspension.This is the most detailed analysis of this kind.It includes the most important cross-couplings and all known input noise sources based on measurements from LIGO, and produces output metrics that are relevant for global interferometer controls.While there are currently no models that correctly predict the detector sensitivity based on local suspension performance, we qualitatively elaborate on the connection between local sensors and improved detector sensitivity at low frequencies.In particular, we show how local resonances below 3 Hz result in noise in the 10 − 20 Hz region.Our results support the statement that improved damping is one of the elements required for breaking the 'low-frequency wall' 9 The MIMO suspension model used in this work is a modification of previous models that includes the design parameters for the new Big BeamSplitter Suspension (BBSS), which is the first Advanced LIGO suspension to receive a major upgrade 42 .Previous suspensions show excellent agreement between the MIMO model and measured transfer functions, with the exception of cross-couplings, which are not included in the stiffness matrix 43 .We include the state-space suspension model, the control filters, and scripts for generating plots as supplemental material.
The evaluation of improved damping for the BBSS is a case-study applicable to other triple suspensions, and with some modifications, to the quadruple suspensions used for the test mass optics.Installing better sensors at the BBSS would be an effective technology demonstration.
One of the LIGO A+ upgrades, planned for after the upcoming fourth observing run, is the installation of a new, larger beamsplitter 42 .This necessitates the new Big BeamSplitter Suspension (BBSS) that contains slots for optional Homodyne Quadrature Interferometers (HoQIs) 44 to be used as relative displacement sensors.Compact interferometric sensors 17 , like HoQIs provide significant performance improvements 44 to the baseline Birmingham Optical Sensor and Electro-Magnetic actuator (BOSEM) 45,46 .This makes the BBSS a perfect test case for modeling how better sensors could affect the performance of the suspended optics.
Figure 1 shows an overview of the BBSS, a cascade of 3 masses.The suspension system isolates the beamsplitter optic (M3) from the residual ground motion of the Internal Seismic Isolation (ISI) platform.Every step of this chain reduces the motion transmitted to the lower mass, with an f −2 power law above its pendulum frequency.The chain's resonances must be damped to reduce the Root Mean Squared (RMS) motion of M3.The damping system works by measuring and actuating between the rigid cage and the suspended stages, depicted in In this paper we present a detailed investigation of the damping and noise performance if HoQIs are mounted between the beamsplitter intermediate mass (M2) and the cage, while using the BOSEMs for actuation.It is the first study showing that the improved sensitivity allows HoQIs to be used for local control at a stage closer to the optic, resulting in more control authority and improved damping without disturbing the sensitivity in the critical measurement band above 10 Hz.The combination of M2 HoQI sensing and M2 BOSEM actuation is referred to as 'M2 HoQI damping'.We will provide a detailed noise-budget breakdown of the contributions to optic motion for the BBSS in both Length and Pitch for both the BOSEM and HoQI damping scenarios.Four targets were identified as crucial for control design: a stable controller that reduces the suspension resonance peaks, meeting 10 Hz performance requirements and reducing RMS motion.

SIMULATING THE HOQI DAMPING PERFORMANCE
The purpose of the damping system is to lower the quality factor of the resonances of the suspension chain at its eigenfrequencies.This reduces the total motion at M3, provides a simpler plant for implementing global interferometer controls, and reduces the RMS motion of the suspended masses and corresponding non-linear effects.Active damping is preferred over passive damping to have more freedom in shaping the frequency response of the dissipation and to allow for fine-tuning after installation.In addition to damping performance, this system should not introduce noise into the sensing region of the detector (10 Hz and above).To design the damping system we need models of the damping performance and an understanding of the noise contributions.
The following method is used to show the damping performance of M2 HoQI damping in comparison to M1 BOSEM damping.A Matlab model for simulating the damping performance of the BOSEMs at M1 exists 47,48 .This model provides an open-loop (undamped) statespace for the suspension dynamics, and includes feedback paths and damping filters for sensing and actuating at M1.These produce the closed-loop BOSEM-damped state-space.
We expanded this implementation to include feedback paths from HoQIs sensors to BOSEMs actuators, all at M2, and designed appropriate new stable damping filters.This study is limited to the longitudinal (L) and pitch (P) degrees of freedom, which are strongly cross-coupled in the underlying mechanical equations of motion.Similar damping performance is expected for yaw (Y).
Three primary noise sources are identified: actuator noise, sensor noise, and inertial noise.
For actuator noise we re-used the validated actuator noise model of the LIGO beamsplitter suspension used in the current observing run 49 , and updated the geometry and DAC to those planned for the BBSS.For global control all actuators are used independent of the local controls, so in both the M2 HoQI and M1 BOSEM damped scenarios, actuator noise originates from both the M1 and M2 actuators.M3 actuator noise was not included since important design parameters are still missing.Actuator noise is primarily caused by the DAC voltage noise, which generates force noise at the suspended masses.
Sensor noise depends on the resolution of the individual sensors and their geometrical placement.Depending on the control filters and the loop gain, the sensor noise will be injected via actuators into the suspension.Both HoQIs and BOSEMs are relative position sensors.To measure translation and rotation, multiple sensors are used at different locations of M1 and M2.Common sensor output corresponds to translational movement, and the noise is lower than that of an individual sensor due to the multiplicity of sensors.Differential sensor outputs correspond to rotation and the noise is higher than for an individual sensor due to the short lever arm.Scripts were used to obtain the sensing noise in each degrees of freedom (dof) based on the geometric placement of BOSEMs for the M1 BOSEM damping model 50 and modified for use with the M2 HoQI damping model.
Inertial noise consists of two parts.The first part is 'ISI noise', the transmission of motion from the ISI (rigidly connected to SUSpension Point (SUSP)) to M3.The second part is 'cage noise'.At high frequencies, the masses M1-M3 move much less than the cage, but the sensors can only measure the relative motions.To model this we projected the ISI movement to the M1 and M2 sensor locations.It is non-trivial to accurately determine the inertial rotation of the ISI at low frequencies, and we used recent predictions of expected tilt, synthesised from several sensors 51 .
Since the BBSS hangs down from the ISI, the motion of the cage due to rotation of the ISI increases at each stage.However, damping performance increases substantially when measuring and actuating closer to the optic.With a total noise budget and the complete damped suspension model, the effect of M1 BOSEM versus M2 HoQI damping was compared quantitatively.The full matlab model of our simulations is shared online 53 .

HOQI VS BOSEM DAMPING PERFORMANCE
We close the control loops on the BBSS model with sensing and actuation at M2. Figure 3 shows the transfer function of ISI longitudinal motion to M3 longitudinal displacement with different damping configurations shown; the baseline sensing and damping at M1, our proposed sensing and damping at M2, and both damping methods simultaneously.It can be seen that the resonances around 0.4 Hz, 1.1 Hz and 1.8 Hz are suppressed by up to a factor of eight with M2 HoQI damping compared with (only) M1 BOSEM damping.The results for M2 HoQI damping, and M2 HoQI + M1 BOSEM damping are very similar in performance.The performance improvements in pitch (P) can be found in figure 4. The inpulse responses from the ISI to the mirror M3 in figure 7 show the improved settling time with the HoQI-damped suspension compared to BOSEM-damped.We have only considered configurations that are feasible for the BBSS.
Figures 5 and 6 present the closed-loop noise budgets, showing different noise sources contributing to optic longitudinal (L) and pitch (P) movement with either M1 BOSEM or M2 HoQI damping.When excluding the actuator noise it can be seen that above 1 Hz, the HoQI-damped system always has a lower total noise than the BOSEM-damped system.Above 1 Hz the BOSEMdamped system is limited by BOSEM sensor noise, while the HoQI damped system is always limited by ISI and cage noise.Finally, the accumulated RMS movement in the pitch (P) direction at 1 Hz is a factor 60 lower for the M2 HoQI-damped system when compared to the M1 BOSEM-damped system.When taking into account actuator noise the noise budget of both methods is compa- rable.
The results show that using M2 HoQI-damping obtains better damping performance without introducing more noise.Figure 3 shows that when using M2 HoQIdamping, additional M1 BOSEM-damping does not bring substantial improvements.M1 BOSEM-damping has two limitations: sensor noise and dynamic coupling.For sensor noise, if the gain is increased, motion at the optic will increase rather than decrease due to the injection of BOSEM sensor noise.For dynamic coupling, only some fraction of the total kinetic energy couples to  the top mass, creating an impedance-matching limit for damping.Increasing damping gain beyond this limit increases optic motion.Combining this information with the lower noise injection of the HoQI system, as shown in figures 5 and 6 provides strong motivation for deactivating BOSEM-damping in degrees of freedom where HoQI-damping can be used: longitudinal (L), pitch (P), yaw (Y).BOSEM sensor noise and actuator noise are driving the noise budget above 1 Hz in the M1 BOSEM-damped system.Actuator noise, cage noise and inertial motion of the ISI and the cage dominates the M2 HoQI-damped budget, especially in longitudinal (L).If the sensors were instead mounted on a suspended reaction chain, present in the quadruple suspensions used for suspending the test masses of main arm-cavity optics 4 , the cage noise will be strongly attenuated.This will result in even better performance for the HoQI damped system.Cage noise can be reduced with 'sensor correction', a feed-forward technique designed to subtract the inertial motion of the ISI from the HoQI sensor output.This is possible using the inertial sensors on the ISI, which measure motion in the relevant band with significant signal-to-noise ratio 51 .This analysis has also exposed that actuation noise is a dominant noise source around 1 Hz and above 10 Hz.To realise the full improvements possible with interferometric local sensors, actuation noise should be suppressed by more than two orders of magnitude at frequencies near 1.5 Hz.
It is difficult to further reduce the noise of LIGO's DACs and significant gains can only be made by reducing the actuation strength.In order to reduce the required actuation strength required for alignment, it is possible to offload static offsets in pitch (P) and yaw (Y) to stepper motors, as implemented successfully at Virgo 52 .Additionally, slow drift in the longitudinal (L) direction can be corrected further upstream from suspension chain, at either the ISI or Hydraulic External Pre-Isolation (HEPI).If the actuators only have to compensate for drifts on short timeframes, less movement and therefore less actuation force is needed resulting in less noise from the DAC propagating to the suspensions.

IMPACT OF SUSPENSION DAMPING ON GRAVITATIONAL-WAVE DETECTION
Current simulation models don't correctly predict the LIGO interferometers' total noise budget for the 10 − 20 Hz region.Steps are being taken to model non-linear couplings into DARM 54 and to use machine learning for non-linear noise prediction and suppression 14,55 .
Further efforts are being made to iterate on control filter design by closing an over-arching loop, with detailed technical noise projections 56 .Figure 8 illustrates some of the interconnections between distinct control systems.An integrated loop over the different control systems would allow for quickly evaluating the DARM performance gains seen by improving one element in the chain.
As these models are currently under development, we aren't able to quantify the improvements of better damping (for single or multiple optics) on DARM.Instead we outline the mechanisms that link suspension dynamics and optic motion with DARM sensitivity.Reducing the RMS motion of LIGO's suspended optics allows for a reduction of interferometric sensing and control bandwidths, and therefore reduced control forces and lower optical sensor noise injection.Lower resonance peaks mean lower cross-coupling into different degrees of freedom.A better damped BBSS (evidenced by 'smoother' transfer functions), enables more robust and simpler global control options, and less of a need for frequency dependent features, which affect phase.Better damping (figure 7) and 'smoother' phase features should improve the quality and stability of the feed-forward, used to de-couple degrees of freedom.
The effects of having multiple better damped and therefore easier to control core optics are outlined con-  The model adaptations we have made can be readily transferred to other triple suspensions in future upgrades.Initial studies showing the benefits for quadruple suspensions have been performed 60 and we expect that similar results can be obtained for all core suspended optics.The increased sensitivity of HoQIs deeply suppresses the contribution of sensor noise and the inertial movement of the cage (cage noise) is expected to dominate the motion of the optic for BBSS triple suspension.As such, lower total noise is anticipated if HoQIs are placed on the suspended 'reaction chain' of the quadruple suspensions that support the test masses.Even without a precise quantification, which is necessarily dependent on the (evolving) status of the interferometer, we have linked previously successful strategies that provide causal evidence that reduced motion via better damping should result in reduced noise in DARM at frequencies between 10 and 30 Hz.

CONCLUSIONS
The Big BeamSplitter Suspension design has slots for installing HoQIs, providing the potential to test improved sensors (and as such better damping) for this suspen-sion, and act as a technology demonstrator for future upgrades.We have simulated the damping performance if HoQIs are fitted and used the resulting performance as a case study for improved sensors in all suspensions.This kind of analysis is crucial for determining the improvements that can be realised in the detector, including existing limits imposed by other systems.
We have shown that by using HoQI damping at M2, instead of BOSEM damping at M1, we can reduce the resonance peaks of the plant by a factor of up to eight while simultaneously reducing the motion of the suspended optic.Reduced RMS motion means lower global control bandwidth, reduced control forces, and less non-linear, bi-linear and non-stationary couplings.A simpler plant allows for more flexible and robust global control options.This better plant stability permits improvements to the feed-forward control.These aspects will improve the sensitivity of the LIGO interferometers in the control-noiselimited 10 − 20 Hz region.

FIG. 1 .
FIG. 1. 3D CAD rendering of the BBSS triple.(a) the suspended masses; beamsplitter top mass (M1), beamsplitter intermediate mass (M2), beamsplitter optic (M3) that are mounted to the cage through the SUSP.(b) the suspended masses together with the cage which is fixed to the ISI platform.(c) detailed view showing the proposed sensor and actuator arrangement at M2, showing BOSEMs (used only for actuation) in the corners and HoQIs (only capable of sensing) along the vertical and horizontal axes.

Figure 2 .FIG. 2 .
FIG. 2. Simplified side-view schematic of the triple BBSS showing the sensing and actuation points for damping the remainder of ground motion not suppressed by the ISI.(a) Overview of the sensors used for damping, at M1, multiple BOSEMs are mounted on the cage, measuring the relative displacement between the cage and M1 in all six degrees of freedom.Similarly, at M2 four HoQIs are mounted on the cage, measuring the relative displacement between the cage and M2, but only in longitudinal (L), pitch (P), and yaw (Y).(b) Overview of the actuators used for damping, at both M1 and M2 there are multiple BOSEMs to actuate on the suspension.By default, the BOSEMs at M1 are used for local sensing and control and provide static offsets for global control, and the BOSEMs at M2 for actuation from global interferometer signals.We investigate using the BOSEMs at M2 for actuation based on both local and global signals.

FIG. 3 .FIG. 4 .
FIG.3.In loop damped transfer function from ISI translation to optic translation in the 'L' direction (parallel to the optical axis) for different input frequencies.A comparison is made between three local sensing and control configurations: only at M1 or M2, and both together.Outside of the displayed range the responses were identical.
FIG.5.Noise budget M3 translation in the 'L' direction, displaying the Cage, ISI and Sensor noise when using BOSEM damping at M1 (black and brown), or HoQI damping at M2 (light and dark blue).And for both systems the actuation noise (green).Below 0.6 Hz the noise budget is limited by ISI movement, above 6 Hz it is limited by actuator noise

FIG. 7 .
FIG. 7. Impulse response from ISI 'L' (SUSP drive location) resulting in (a) M3 'L' movement and (b) M3 'P' movement.Figures show purely the damped plant response, sensor and cage noise have not been taken into account.The M2 HoQI damped system exhibits a much shorter transient response.

FIG. 8 .
FIG.8.Simplified flow chart of the different stages of control and types of noise injections.The ISI actively lowers the ground movement propagating to the suspensions.Measurement noises are seen as actual motion such that counteracting this perceived motion will cause actual motion.Sensor noise and ground noise result in ISI noise.ISI noise, cage noise and HoQI and BOSEM noise aggregate to create optic movement.Optic movement and global sensor noise (such as shot noise) cause movement between optics, which couples to DARM.In the 10 − 20 Hz region these effects limit the detector performance.

FIG. 9 .
FIG.9.A simplified flow chart sketching how better suspension damping can improve DARM sensitivity.Better suspension damping reduces the RMS motion of the optic, and (especially) reduces eigenmode resonance peaks, making the plant easier to control.When global sensor noise in auxiliary degrees of freedom is limiting the performance, the reduced RMS motion allows for auxiliary control bandwidths to be lowered with the same closed-loop motion.Lower control bandwidth reduces the injection of optical sensor noise.Lower resonance peaks make a more robust plant, allowing for more accurate feed-forward and more aggressive roll-off.Finally, energy in resonance peaks is strongly cross-coupled between eigenmodes that are close in frequency.Damping therefore reduces the strength of the mechanical cross-coupling between resonances.Combined, these improvements result in improved L-to-angle and angle-to-L coupling, which will result in better gravitational wave sensitivity.