^{1,a)}and B. N. Shapiro

^{2,b)}

### Abstract

The mirrors of laser interferometric gravitational wave detectors hang from multi-stage suspensions. These support the optics against gravity while isolating them from external vibration. Thermal noise must be kept small so mechanical loss must be minimized and the resulting structure has high-Q resonances rigid-body modes, typically in the frequency range between about 0.3 Hz and 20 Hz. Operation of the interferometer requires these resonances to be damped. Active damping provides the design flexibility required to achieve rapid settling with low noise. In practice there is a compromise between sensor performance, and hence cost and complexity, and sophistication of the control algorithm. We introduce a novel approach which combines the new technique of modal damping with methods developed from those applied in GEO 600. This approach is predicted to meet the goals for damping and for noise performance set by the Advanced LIGO project.

The authors would like to thank members of the LSC-Virgo collaboration for their interest in this work. We would like to thank the NSF in the USA (Award Nos. PHY-05 02641 and PHY-07 57896). LIGO was constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation and operates under cooperative agreement PHY-0107417. In the UK, we are grateful for the financial support provided by Science and Technology Facilities Council (STFC) and the University of Glasgow. This paper has LIGO Document No. LIGO-P1200009. Public internal LIGO documents are found at https://dcc.ligo.org/cgi-bin/DocDB/DocumentDatabase/.

I. INTRODUCTION—SUSPENSIONS FOR INTERFEROMETRIC GRAVITATIONAL WAVE DETECTORS

II. LOCAL CONTROL OF THE ALIGO QUADRUPLE SUSPENSIONS

III. SENSOR AND ACTUATOR PLACEMENT, AND IMPLICATIONS FOR CONTROLLER TOPOLOGY

IV. THE ALIGO SUSPENSIONMODEL

V. MODAL DAMPING

A. Control design

B. State estimation

VI. MULTI-MODE DAMPING OF MULTI-STAGE SUSPENSIONS

A. Robustness of the multi-mode technique

VII. CONCLUSION: APPLICATION TO ALIGO

### Key Topics

- Suspensions
- 45.0
- Gravitational wave detectors
- 12.0
- Mirrors
- 12.0
- Interferometers
- 5.0
- Low pass filters
- 5.0

##### B60G

##### F16F

##### F16F15/00

##### G01B9/02

##### G02B5/08

##### G05D19/00

## Figures

Modified rendering of a CAD model of the main elements of an aLIGO quadruple suspension. There are two chains of 4 stages, numbered as shown. One supports the mirror (lowest mass in the front chain), the other provides a quiet platform at each level for actuation. The top 3 stages are supported on springs to improve vertical isolation. Stages 1 and 2 (and stage 3 of the reaction chain) incorporate adjustable/moveable mass to trim and balance the suspension. Stages 3 and 4 of the main chain are formed from fused silica and weigh 40 kg each. The test mass is polished and coated to form a mirror which hangs on 4 fused silica fibers of 0.2 mm radius and 600 mm length, to provide low thermal noise.

Modified rendering of a CAD model of the main elements of an aLIGO quadruple suspension. There are two chains of 4 stages, numbered as shown. One supports the mirror (lowest mass in the front chain), the other provides a quiet platform at each level for actuation. The top 3 stages are supported on springs to improve vertical isolation. Stages 1 and 2 (and stage 3 of the reaction chain) incorporate adjustable/moveable mass to trim and balance the suspension. Stages 3 and 4 of the main chain are formed from fused silica and weigh 40 kg each. The test mass is polished and coated to form a mirror which hangs on 4 fused silica fibers of 0.2 mm radius and 600 mm length, to provide low thermal noise.

A block diagram of a modal damping scheme for the 4 *x* modes. An estimator converts the incomplete sensor information into modal signals. The modal signals are then sent to damping filters, one for each DOF. The resulting modal damping forces are brought back into the Euler coordinate system through the transpose of the inverse of the eigenvector matrix Φ. Only stage 1 forces are applied to maximize sensor noise filtering to stage 4. Note that this figure applies to a four DOF system.

A block diagram of a modal damping scheme for the 4 *x* modes. An estimator converts the incomplete sensor information into modal signals. The modal signals are then sent to damping filters, one for each DOF. The resulting modal damping forces are brought back into the Euler coordinate system through the transpose of the inverse of the eigenvector matrix Φ. Only stage 1 forces are applied to maximize sensor noise filtering to stage 4. Note that this figure applies to a four DOF system.

The loop gain transfer function of an example 1 Hz modal oscillator with its damping filter. The plant contributes the large resonant peak and the damping filter contributes the remaining poles and zeros. The 10 Hz notch reduces the sensor noise amplification at the start of the gravitational wave detection band, where it is typically the worst. The large phase margin near the resonance permits tuning of the gain *k* to achieve a significant range of closed loop Qs. All the damping loops have the same basic shape but are shifted in frequency and gain (the notch remains at the same frequency).

The loop gain transfer function of an example 1 Hz modal oscillator with its damping filter. The plant contributes the large resonant peak and the damping filter contributes the remaining poles and zeros. The 10 Hz notch reduces the sensor noise amplification at the start of the gravitational wave detection band, where it is typically the worst. The large phase margin near the resonance permits tuning of the gain *k* to achieve a significant range of closed loop Qs. All the damping loops have the same basic shape but are shifted in frequency and gain (the notch remains at the same frequency).

The components of the cost function Eq. (12) for the *x* DOF as a function of **R** calculated by the optimization routine. At each value of **R** the closed loop system performance is simulated using the estimator design based on the LQR solution with that particular **R** value.

The components of the cost function Eq. (12) for the *x* DOF as a function of **R** calculated by the optimization routine. At each value of **R** the closed loop system performance is simulated using the estimator design based on the LQR solution with that particular **R** value.

An amplitude spectrum showing a simulation of the mirror displacement along the *x* DOF under the influence of the optimized modal damping loop with **R** = 0.06. The black dashed line is the sensor noise and the green line is its contribution to the mirror displacement. The solid black line is the ground disturbance and the blue line is its contribution to the mirror displacement. The red line is the uncorrelated stochastic sum of both contributions.

An amplitude spectrum showing a simulation of the mirror displacement along the *x* DOF under the influence of the optimized modal damping loop with **R** = 0.06. The black dashed line is the sensor noise and the green line is its contribution to the mirror displacement. The solid black line is the ground disturbance and the blue line is its contribution to the mirror displacement. The red line is the uncorrelated stochastic sum of both contributions.

Complex frequency (*s*)-plane plot showing the poles × of the closed loop modal damping system. The reference system is represented by the bold (black) symbols, while the 100 trials of perturbed systems are represented by the finer (red) symbols. Each trial represents a system modified from the ideal using the random parameters described in the text. In this test 16% of the cases are unstable.

Complex frequency (*s*)-plane plot showing the poles × of the closed loop modal damping system. The reference system is represented by the bold (black) symbols, while the 100 trials of perturbed systems are represented by the finer (red) symbols. Each trial represents a system modified from the ideal using the random parameters described in the text. In this test 16% of the cases are unstable.

Settling times for the four *x* modes of a quadruple suspension with pure velocity damping of variable strength at stage 1. The lowest mode (1) dominates yielding a shortest settling time of 18 s (to 2%), with damping strength 55 kg/s.

Settling times for the four *x* modes of a quadruple suspension with pure velocity damping of variable strength at stage 1. The lowest mode (1) dominates yielding a shortest settling time of 18 s (to 2%), with damping strength 55 kg/s.

Open loop Bode plot comparing damping laws for yaw. The damping law, low-pass filter, and mechanical plant are combined. The solid (blue) curves represent the differentiator law, with a suitable low-pass filter. The dashed (green) line shows the truncated differentiator with the pole at 3.5 Hz. The dotted (red) curve represents the interrupted differentiator. Finally the dashed-dotted (cyan) curve shows an example with resonant zeros and poles. The gains are adjusted to match at 10 Hz. Other filter parameters are given in the text.

Open loop Bode plot comparing damping laws for yaw. The damping law, low-pass filter, and mechanical plant are combined. The solid (blue) curves represent the differentiator law, with a suitable low-pass filter. The dashed (green) line shows the truncated differentiator with the pole at 3.5 Hz. The dotted (red) curve represents the interrupted differentiator. Finally the dashed-dotted (cyan) curve shows an example with resonant zeros and poles. The gains are adjusted to match at 10 Hz. Other filter parameters are given in the text.

The magnitude of the transmissibility from the sensor input to motion of the suspended mirror, in rad/m. The 4 curves correspond to the same 4 control laws as in the previous figure. The “upper-limit” line shows the maximum value allowed for the transmissibility above 10 Hz.

The magnitude of the transmissibility from the sensor input to motion of the suspended mirror, in rad/m. The 4 curves correspond to the same 4 control laws as in the previous figure. The “upper-limit” line shows the maximum value allowed for the transmissibility above 10 Hz.

## Tables

Noise amplitude spectral density limits for the aLIGO test masses. Upper limits are set a factor of 10 below the intended instrumental noise floor, allowing for cross-coupling to the sensitive direction. Each limit falls as 1/*f* ^{2} from 10 Hz to 30 Hz. The interferometer is insensitive to roll, though roll noise can couple into, e.g., *x* in the mechanical system.

Noise amplitude spectral density limits for the aLIGO test masses. Upper limits are set a factor of 10 below the intended instrumental noise floor, allowing for cross-coupling to the sensitive direction. Each limit falls as 1/*f* ^{2} from 10 Hz to 30 Hz. The interferometer is insensitive to roll, though roll noise can couple into, e.g., *x* in the mechanical system.

Settling time to 2% resulting from a unit impulse applied to the sensor input of the closed loop system. This is equivalent to the effect of an impulsive motion of the top mass, as observed at the test mass. The interrupted differentiator yields ≈15% quicker settling than the mean of the other methods.

Settling time to 2% resulting from a unit impulse applied to the sensor input of the closed loop system. This is equivalent to the effect of an impulsive motion of the top mass, as observed at the test mass. The interrupted differentiator yields ≈15% quicker settling than the mean of the other methods.

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