G1200694 - ACC 2012 - 27 June

Modal Damping of a Quad Pendulum for Advanced Gravitational Wave Detectors

Brett ShapiroNergis Mavalvala

Kamal Youcef-ToumiACC 2012 – Montreal

June 27

Massachusetts Institute of Technology

2

AbstractMotivation: Observe gravitational waves from astrophysical

sources (supernovae, pulsars, black hole mergers, etc) using the LIGO observatories.

Problem: Multi-DOF isolation systems enhance ground motion at high Q resonances. Damped using active feedback. This control introduces additional noise. Optimal control required achieve adequate trade-off.

Solution: Modal damping to simplify and decouple optimization of each mode’s damping. Also permits real-time tuning.

G1200694 - ACC 2012 - 27 June

3

Outline

1. LIGO and gravitational waves2. Seismic (vibration) isolation3. Competing damping control goals4. Method of modal damping5. Optimization of modal damping6. Results

G1200694 - ACC 2012 - 27 June

PulsarSupernova Merging Black Holes

• Supernovae Asymmetry required

• Coalescing Binaries Black Holes or Neutron Stars

Mergers

• Pulsars Asymmetry required

• Stochastic Background (Big bang, etc.)

Gravitational Waves

Wave of strain amplitude h

4G1200694

- ACC 2012 - 27 June

The Laser InterferometerGravitational-wave Observatory (LIGO)

Livingston, LA

Hanford, WA

• 3, 4 km interferometers at 2 sites in the US• Michelson interferometers with Fabry-Pérot arms• Optical path enclosed in vacuum• Sensitive to strains around 10-22 -> 10-19mrms• LIGO Budget ≈ $60 Million per year from NSF. • Operated by MIT and Caltech.

Funded by

FI

Test Masses:fused silica,

34 cm diam x 20 cm thick,40 kg

SRM

ITM

ETM

InputMode

Cleaner

OutputMode

Cleaner

PRM

BS

4 km

T = 3%

Laser Φ m

PDGW readout

FI

ITM ETM125 W

5.7 kW

815 kW

CP

Not to scale

5

•Advanced LIGO configuration•Under construction until ≈2015•Upgrade to Initial LIGO

If we put LIGO in Cambridge, MA

LIGO spans 16 km2. Cambridge, MA covers 16.65 km2 (wikipedia http://en.wikipedia.org/wiki/Cambridge,_Massachusetts). 6

Kenmore Square

MIT

Harvard

MGH

Porter Square

7

LIGO Scientific CollaborationAustralian Consortiumfor InterferometricGravitational AstronomyThe Univ. of AdelaideAndrews UniversityThe Australian National Univ.The University of BirminghamCalifornia Inst. of TechnologyCardiff UniversityCarleton CollegeCharles Sturt Univ.Columbia UniversityCSU FullertonEmbry Riddle Aeronautical Univ.Eötvös Loránd UniversityUniversity of FloridaGerman/British Collaboration forthe Detection of Gravitational WavesUniversity of GlasgowGoddard Space Flight CenterLeibniz Universität HannoverHobart & William Smith CollegesInst. of Applied Physics of the Russian Academy of SciencesPolish Academy of SciencesIndia Inter-University Centrefor Astronomy and AstrophysicsLouisiana State UniversityLouisiana Tech UniversityLoyola University New OrleansUniversity of MarylandMax Planck Institute for Gravitational Physics

University of MichiganUniversity of MinnesotaThe University of MississippiMassachusetts Inst. of TechnologyMonash UniversityMontana State UniversityMoscow State UniversityNational Astronomical Observatory of JapanNorthwestern UniversityUniversity of OregonPennsylvania State UniversityRochester Inst. of TechnologyRutherford Appleton LabUniversity of RochesterSan Jose State UniversityUniv. of Sannio at Benevento, and Univ. of SalernoUniversity of SheffieldUniversity of SouthamptonSoutheastern Louisiana Univ.Southern Univ. and A&M CollegeStanford UniversityUniversity of StrathclydeSyracuse UniversityUniv. of Texas at AustinUniv. of Texas at BrownsvilleTrinity UniversityTsinghua UniversityUniversitat de les Illes BalearsUniv. of Massachusetts AmherstUniversity of Western AustraliaUniv. of Wisconsin-MilwaukeeWashington State UniversityUniversity of Washington

ACC 2012 - 27 June 8

100

101

10210

-2110

-2010-1910-18

10-1710-1610-15

10-1410-1310-12

10-11

10-10

10-9

10-810-710-6

Frequency (Hz)

Dis

plac

emen

t Sen

sitiv

ity (m

/ H

z)

Advanced LIGO Goal

Projected Sensitivity for Advanced LIGO

100

101

10210

-2110

-2010-1910-18

10-1710-1610-15

10-1410-1310-12

10-11

10-10

10-9

10-810-710-6

Frequency (Hz)

Dis

plac

emen

t Sen

sitiv

ity (m

/ H

z)

Quantum NoiseSeismic NoiseGravity Gradient NoiseSuspension Thermal NoiseCoating Brownian NoiseAdvanced LIGO Goal

100

101

10210

-2110

-2010-1910-18

10-1710-1610-15

10-1410-1310-12

10-11

10-10

10-9

10-810-710-6

Frequency (Hz)

Dis

plac

emen

t Sen

sitiv

ity (m

/ H

z)

Quantum NoiseSeismic NoiseGravity Gradient NoiseSuspension Thermal NoiseCoating Brownian NoiseAdvanced LIGO Goal

ACC 2012 - 27 June 9

Suspensions and Seismic Isolation

active isolation platform (2 stages

of isolation)

hydraulic external pre-isolator (HEPI) (one

stage of isolation)

quadruple pendulum (four stages of isolation) with

monolithic silica final stage

Advanced LIGO test mass isolation

Ref: LIGO-G1000469-v1; Kissel PhD thesis

quadruple pendulum (four stages of isolation)

active isolation platform (2 stages

of isolation)

Installing prototype quad pendulum with glass optic on metal wires, Jan 2009 at MIT.10

Quadruple PendulumMain (test)

ChainReaction

Chain

Control• Damping –stage 1• Cavity length - all stages

Sensors/Actuators• BOSEMs at stage 1 & 2

• AOSEMs at stage 3

• Opt. levs. and interf. sigs. at stage 2• Electrostatic drive (ESD) at stage 4

Purpose• Test mass (stage 4) isolation. the test mass consists of a 40 kg high reflective mirror

Stage 1

Stage 2

Stage 3

Stage 4

11G1200694 - ACC 2012 - 27 June

IR laser

12

10-1 100 101 10210-15

10-13

10-11

10-9

10-7

10-5

10-3

10-1

101

103

105 Seismic Isolation along the Interferometer axis of Various Pendulum Styles

Isol

atio

n (m

/m)

Frequency (Hz)

Single

Multi-stage Isolation Performance

10-1 100 101 10210-15

10-13

10-11

10-9

10-7

10-5

10-3

10-1

101

103

105 Seismic Isolation along the Interferometer axis of Various Pendulum Styles

Isol

atio

n (m

/m)

Frequency (Hz)

Single Double

10-1 100 101 10210-15

10-13

10-11

10-9

10-7

10-5

10-3

10-1

101

103

105 Seismic Isolation along the Interferometer axis of Various Pendulum Styles

Isol

atio

n (m

/m)

Frequency (Hz)

Single Double Triple

10-1 100 101 10210-15

10-13

10-11

10-9

10-7

10-5

10-3

10-1

101

103

105 Seismic Isolation along the Interferometer axis of Various Pendulum Styles

Isol

atio

n (m

/m)

Frequency (Hz)

Single Double Triple Quadruple

Each stage provides 1/f2 isolation

1/f2

1/f4

1/f6

1/f8

G1200694 - ACC 2012 - 27 June

10-2 10-1 100 101 10210-16

10-14

10-12

10-10

10-8

10-6

10-4

Frequency (Hz)

Dis

plac

emen

t (m H

z)

Simulated Seismic Disturbance and Sensor Noise Amplitude Spectrum

Seismic Disturbance

10-2 10-1 100 101 10210-16

10-14

10-12

10-10

10-8

10-6

10-4

Frequency (Hz)

Dis

plac

emen

t (m H

z)

Simulated Seismic Disturbance and Sensor Noise Amplitude Spectrum

Seismic Disturbance Stage 1 Undamped Displacement

10-2 10-1 100 101 10210-16

10-14

10-12

10-10

10-8

10-6

10-4

Frequency (Hz)

Dis

plac

emen

t (m H

z)

Simulated Seismic Disturbance and Sensor Noise Amplitude Spectrum

Stage 1 Sensor Noise Seismic Disturbance Stage 1 Undamped Displacement

Two Competing Goals

13

10-2 10-1 100 101 10210-16

10-14

10-12

10-10

10-8

10-6

10-4

Frequency (Hz)

Dis

plac

emen

t (m H

z)

Simulated Seismic Disturbance and Sensor Noise Amplitude Spectrum

Stage 1 Sensor Noise Seismic Disturbance Stage 1 Undamped Displacement Stage 1 Damped Displacement

Damp modes

Limit sensor noise amplification beyond 10 Hz

G1200694 - ACC 2012 - 27 June

Along laser beam axis

Modal Damping with State Estimation

xf

mf

T1

CartesianCoordinates

(coupled dynamics)

ModalCoordinates(decoupled dynamics)

Estimator

q̂

Pendulum

•

• = pendulumeigenvector matrix• = Cartesian coordinates• = sensor sig.• = estimated modal coord.• = Cartesian damping forces• = modal damping forces• mode damping filter• mode damping gain

qx

mf

f

q̂

x

thii

thiGi

11G

22G

33G

44G

y

y

14 G1200694 - ACC 2012 - 27 June

-100-50

050

100150

Bode Diagram

Mag

nitu

de (d

B)

1 Hz modal plant: P

10-2

10-1

100

101

102

-360-315-270-225-180-135-90-45

04590

Pha

se (d

eg)

Frequency (Hz)

Modal Feedback Design

22ns

sP

15

2n10 ,,,

-100-50

050

100150

Bode Diagram

Mag

nitu

de (d

B)

1 Hz modal plant: PFeedback filter: G

10-2

10-1

100

101

102

-360-315-270-225-180-135-90-45

04590

Pha

se (d

eg)

Frequency (Hz)

-100-50

050

100150

Bode DiagramGM = 9.1865 (2.403 Hz), PM = 74.5731 (1.0464 Hz)

Mag

nitu

de (d

B)

1 Hz modal plant: PFeedback filter: GLoop gain: GP

10-2

10-1

100

101

102

-360-315-270-225-180-135-90-45

04590

Pha

se (d

eg)

Frequency (Hz)

2222

22

)20(20)5.2()10sin(5)20(2

ssss

ssGnn

16

0 5 10 15 20-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (seconds)

Sta

ge 4

dis

plac

emen

t (no

rmal

ized

)

Stage 41/e Decay Line

Modes 1 2 3 4

Freq (Hz) 0.443 0.996 2.001 3.416

0.040 0.018 0.009 0.001

Damped Response to Impulse from Gnd.

Assuming perfect state estimation

10 second damping

requirement

Impulse Response Simulation

Estimator Design

y

qq

CLf

Bqq

Aqq

ˆˆ

0ˆˆ

ˆˆ

mmmm

matrixfeedback Estimator matrixSensor

model Pendulum ,

m

m

LC

BA mm

m

J

dtJ

m

mTm

TT

LL

Rzzqq

Qqq

)min(arg

~~

~~0

ionamplificat noisesensor ~~

error estimation ˆ~

qq

Lz

qqq

Tmm

RQ ,

Weight cost of estimation error

Weight cost of using noisy sensor

?

17G1200694 - ACC 2012 - 27 June

Linear Quadratic Regulator (LQR) design

Choosing Q and R: Not Unique

18

n

n

n

n

Tn

Tn

T

qq

q

mm

qqq

~~...

~

0

...00

~~...~1

1

2

21

11

mR

RR

0...

0

2

1

R

R is still to be determined

Q

amplitude response impulse velocity modal

mode of mass modal 1

i

i

m

im

m

J

dtJ

m

mTm

TT

LL

Rzzqq

Qqq

)min(arg

~~

~~0

G1200694 - ACC 2012 - 27 June

19

Solving the R matrix for MIMO Modal Damping

)min(arg

maxmax)( 22,

R

ii

iisR

JR

NTRJ

seconds 10 DOFfor timesettling 4 Stage

,iT is

Hz 10at DOFfor t requiremen noise 4 StageHz 10at DOFfor noisesensor 4 Stage

iiN i

DOFs i are: x, y, z, yaw, pitch, roll

•

•

xz

Try a bunch of R matrices and see what works best

Measure ‘best’ with an auxiliary cost

function.

20

Modal Estimation Cost

10-10 10-9 10-8 10-7 10-6 10-5 10-410-4

10-3

10-2

10-1

100

101

102

103

R

Cos

t Val

ues

x Estimator Cost vs. R

Settling Cost, max(Ts2)

Sensor Noise Cost, max(N2)Total Cost, J

R

8109 R

G1200694 - ACC 2012 - 27 June

(x is the laser axis)

21

Optimal Noise Amplification

10-1 100 10110-22

10-20

10-18

10-16

10-14

10-12

10-10

10-8

10-6Contributions to the Test Mass Displacement along the x DOF with R = 9 10-8

Am

plitu

de m

/ H

z

Frequency (Hz)

Ground disturbanceOSEM sensor noiseStage 4 response to ground disturbanceStage 4 response to sensor noiseTotal Stage 4 response

Noise requirement

simulation

G1200694 - ACC 2012 - 27 June

22

Conclusions

• Modal damping provides an intuitive way to optimize a highly coupled, many DOF system, with strict noise performance.

• Real-time or adaptive tuning possible by adjusting gains on each mode.

• Future work to involve implementation on a true Advanced LIGO interferometer.

G1200694 - ACC 2012 - 27 June

Backups

LIGO spans 16 km2. Cambridge, MA covers 16.65 km2 (wikipedia http://en.wikipedia.org/wiki/Cambridge,_Massachusetts). 23

Kenmore Square

MIT

Harvard

MGH

Porter Square

FI

Test Masses :fused silica ,

34 cm diam x 20 cm thick ,40 kg

SRM

ITM

ETM

OutputMode

Cleaner

PRM

BS

4 km

T = 3%

Laser Φ

InputMode

Cleaner

m

PDGW readout

FI

ITM ETM125 W

5.7 kW

815 kW

CP

24

Five Pendulum Designs

Ref: G1100434

Backups: Optical Sensor ElectroMagnet (OSEM)

Birmingham OSEM (BOSEM) Advanced LIGO OSEM (AOSEM)- modified iLIGO OSEM

BOSEM SchematicACC 2012 - 27 June 25

Magnet Types (M0900034)• BOSEM – 10 X 10 mm, NdFeB , SmCo

10 X 5 mm, NdFeB, SmCo• AOSEM – 2 X 3 mm, SmCo

2 X 6 mm, SmCo 2 X 0.5 mm, SmCo

ACC 2012 - 27 June 26

Backups: Quadruple Suspension ESDMain (test)

ChainReaction

Chain The electrostatic drive (ESD) acts directly on the test ITM and ETM test masses.• ± 400 V (ΔV 800 V)

≈ 100 μN• Each quadrant has an independent control channel• Common bias channel over all quadrants

ULUR

LLLR

ACC 2012 - 27 June

Backups: Quadruple SuspensionMIT

monolithic quad in BSC

June 2010

27/33