LIGO- G060XXX-00-R E2E meeting, September 2006 1 How to design feedback filters? E2E meeting September 27, 2006 Osamu Miyakawa, Caltech

E2E meeting, September 2006 1LIGO- G060XXX-00-R

How to design

feedback filters?

E2E meetingSeptember 27, 2006

Osamu Miyakawa, Caltech


Single Fabry-Perot cavity

•P: cavity pole

•S: flat

•F: ??

•A: suspension TF

S P

Sensor

A Actuator

F

Feedback filter

N

ITM

Pickoff

ETM

Laser EOM

Suspension Suspension

Photo detector

Oscillator Mixer

Coil-magnetActuator

Seismic noise

Seismic noise

Feedback filter

Plant

Actuator

Sensor Feedback filter

Plant

Noise

16Hz:LIGO1.6kHz:40m

f -1

f -2

~1Hz

L

Sig

nal[

a.u.

]

Error signal

Lock point

Locking area


Condition for stable control

S P

Sensor

A Actuator

F

Feedback filter

NPlant

NoiseoutV

1. Phase delay of OLTF at UGF (unity gain frequency) must be less than 180 degree.

2. Enough gain at DC to suppress outloop noise into linear range.

function transfer loopOpen :SPAFG


Transfer function fromNoise N to signal Vout

SPNVSPAF

VFANPSV

OUT

OUTOUT

1

DOUT

CD

BC

AB

OUTA

VSV

VPV

VNV

VAV

VFV

function transfer loopOpen :SPAFG

S P

Sensor

A Actuator

F

Feedback filter

NPlant

NoiseoutV

AV

BV

CVDV

G

SP

SPAF

SP

N

V

11OUT


Transfer function fromoutloop noise to inloop noise

GSPN

V

N

V

1

11outC

S P

Sensor

A Actuator

F

Feedback filter

NPlant

NoiseoutV

AV

BV

CVDV

Outloop noise N is suppressed by OLG G to inloop noise N/(1+G), then it is multiplied by TF in loop.

G

G

G

PSFA

N

V

11B

G

P

N

V

1D


Basic concept :relationship between gain and phase

-180

180

-90

0

90

Pha

se [

deg]

Gai

n

1

10

100

1000

Frequency log f [Hz]1 10 100 1000

f -1

• 1pole: 90deg delay

-180

180

-90

0

90

Pha

se [

deg]

Gai

n

1

10

100

1000


f -2

• 2pole: 180deg delay

-180

180

-90

0

90

Pha

se [

deg]

Gai

n

1

10

100

1000


-180

180

-90

0

90

Pha

se [

deg]

Gai

n1

10

100

1000


f +1 f +2

• 1zero: 90deg advance • 2zero: 180deg advance


Conditoin 2: Phase delay of OLG at UGF must be less than 180 degree. (UGF: frequency at gain = 1)

1kHz

f -1

f -2

1Hz

P : cavity pole x S : flat x F : ?? x A : suspension = system TF

1kHz

f -1

f -2

1Hz

P : cavity pole x S : flat x F : x A : suspension = OLTF

f -1f +1

1kHz10Hz

zero@10Hzpole@1kHz,1kHz

-180-900

90

Pha

se [

deg]

Gai

n

10.01

10010000

Frequency log f [Hz]1 10 100 1k

f -2

10k0.1

0.0010.0001

1000000

-270

f -4

f -1

Phaserestore

Open loop TF

-180-900

90

Pha

se [

deg]

Gai

n

10.01

10010000

Frequency log f [Hz]1 10 100 1k 10k0.1

0.0010.0001

1000000

-270

Not stable!

f -2

f -3


Example of Feedback filterused in 40m arm cavity

F :

f -1f +1

4kHz10Hz


Phaserestore

85degreePhase

advance


Conditoin 2: Enough gain at DC to suppress outloop noise within linear range.

1kHz

f -1

f -2

1Hz

P : cavity pole x S : flat x F : x A : suspension = OLTF

f -1f +1

1kHz10Hz


PhaseRestore

+Boost

f -2

Wave length: ~ 10-6mFinesse: ~ 1000FWHM (full width half maximum):

~10-9m

Seismic noise N: ~10-6m @1HzQ of suspension: ~10

Required gain at 1Hz > 104

Wave length ~ 10-6m

FWHM ~ 10-9m length wave

FWHMFinesse

HzmN 1@101010: 56

HzmG

N1@10

)101(

10

19

4

5

Boost filter

-180-900

90

Pha

se [

deg]

Gai

n

10.01

10010000

Frequency log f [Hz]1 10 100 1k

f -4

10k0.1

0.0010.0001

1000000

-270

f -4

f -1


High gain: limited by phase delay due to cavity pole, circuit, DAC/ADC time delay, etc.

Low gain: limited by phase delay of boost, or by too low DC gain.

How to adjust feedback gain?

-180

-90

0

90

Pha

se [

deg]

Gai

n

1

0.01

100

10000

Frequency log f [Hz]

f -4

0.1

0.001

0.0001

1000000

-270

f -4

f -1

Once you get stable operation, it is very important to measure open loop TF to see how stable!

1 10 100 1k 10k


How to measure Open Loop TF?

1. Use closed transfer function:

• Measure

• Calculate G

2. Measure OLTF directly

• Measure

S P

Sensor

A Actuator

F

Feedback filter

NPlant

NoiseoutV

IN1VEXCVIN2V

GC

1

1

GV

V

1

1

EXC

IN2

11

C

G

GFAPSV

V

IN2

IN1


Example of measured Open loop TF

Phase margin 45degree

Generally, phase margin should be more than 30degree at least.

UGF

f -1 slope


Why low frequency measurement is dirty?

• Excitation signal is suppressed a lot with very high gain G>>1 at low frequency.

S P

Sensor

A Actuator

F

Feedback filter

NPlant

NoiseoutV

IN1VEXCVIN2V

Noise1

1EXCIN2

V

GV


What is Coherence?

•Coherence: coh(f)How much related between input and output

• 0 < coh(f) < 1

•Coherence is sometimes convenient to estimate whether the measurement is reliable.

•Generally, If coherence is smaller than 0.8 the measurement is not good.

yxfWfW

yxfW

fWfW

fWh

yyxx

xy

yyxx

xy

and of spectrumpower :)(),(

and of spectrum cross :)(

)()(

)()(coh


What does Closed Loop TF mean?

1. C1 :used to estimate gain oscillation

• Measure

• Gain oscillation is caused by small phase margin at UGF.

2. C2 :• Measure

S P

Sensor

A Actuator

F

Feedback filter

NPlant

NoiseoutV

IN1VEXCVIN2V

GC

1

11

GV

V

1

1

EXC

IN2

There are two definitions for closed loop TF

G

GC

12

Gai

n 1


G

G

V

V

1EXC

IN1G

ain 1


UGF

UGF


Example of measured Closed loop TF

Gain oscillation 6dB

Generally, gain oscillation should be less than 10dB at most.


Example of measured Closed loop TF


How to measure plant TF?

• Measure:

S P

Sensor

A Actuator

F

Feedback filter

NPlant

Noise

IN1VEXCVIN2V

SPAV

V

OUT

IN1

Once system become stable, you can measure given plant TF.

OUTV

-180-900

90

Pha

se [

deg]

Gai

n

10.01

10010000


0.0010.0001

1000000

-270

f -3

f -2


Example: actuator x optical plant x sensor

f -2 slope


Example: suspension Local damping•P: flat

•S: flat

•F: ??

•A: suspension TF

•OLTF S P

Sensor

A Actuator

F

Feedback filter

NPlantNoise

ETM

Suspension

Coil-magnetActuator

Seismic noise

Feedback filter

ActuatorSensor

Feedback filter

Plant

f +1

~10Hz

f -2

~1Hz

Q:~1000

-180-900

90

Pha

se [

deg]

Gai

n

10.01

10100

Frequency log f [Hz]

1 10 1000.1

0.0010.0001

1000

-270

f -2

f +1 f -1

Phase margin90degree

UGF

zero@0Hzpole@10Hz


Example in E2EQuad-suspension: local damping

Main Ref.

M0

M1

M2

M3

6(6)

0(3)

0(3)

0(0)

Damped DOFs(OSEM DOFs)

100

101

10-2

100

Open loop TF

Ga

in

100

101

10-2

100

Suspenstoin TF

Ga

in

rawdamped

100

101

10-1

100

101

102

Feedback filter TF

frequency [Hz]

Ga

in

M0: pitch

Zero @ 0HzPole @ 10,10Hz

S P Sensor

A Actuator F

Feedback filter

N

PlantNoise

IN1VEXCVOUTV

SPAV

V

OUT

IN1:raw

G

SPA

V

V

1

:damped

EXC

IN1


Advanced example: Double loop•P: cavity pole•S: flat•F1: cavity pole

restore

•F2: AC couple, 1pole, cavity pole restore

•A1: suspension TF

•A2: flat

•OLTF

ITM

Pickoff

ETM

Laser EOM

Suspension Suspension

Photo detector

Oscillator Mixer

Coil-magnetActuator

Seismic noise

Seismic noise

Feedback filter

Actuator 1

Sensor Feedback filter 1

Plant

f -2

~1Hz

PZT

Feedback filter 2

Actuator 2

S P

Sensor

A1 Actuator 1 F1

Feedback filter 1

Plant

A2

F2 Feedback filter 2

Actuator 2

-180-900

90P

hase

[de

g]G

ain

10.01

100

10000


1000000

-270

f -1

f -2

•Phase margin at UGF > 0 deg ( >30deg)•Relative phase margin at cross over frequency > 0deg ( >30deg)


Measured open loop gain of MC

-150

-100

-50

0

50

100

150

Pha

se[d

eg]

10-1

100

101

102

103

104

105

Frequency[Hz]

200

150

100

50

0

Mag

[dB

]

Calculated Mirror loop gainCalculated VCO loop gainCalculated total loop gainMeasured Mirror loop gainMeasured VCO loop gainMeasured total loop gain

C=10dB, B=16dB, L=30.4dBUnity gain freq.=67.2kHzPhase margin=28.4degCross over freq.=26.6Hz

Open loop transfer function of Mode Cleaner

Phase margin=28.4deg

Unity gain frequency

Cross over frequency

Relative phase = 165deg


+-

Em

El

++

Measured cross over frequency of MC

-150

-100

-50

0

50

100

150

Pha

se[d

eg]

12 3 4 5 6 7 8 9

102 3 4 5 6 7 8 9

1002 3 4 5 6 7 8 9

1000

Frequency[Hz]

-100

-80

-60

-40

-20

0

20

Mag

[dB

] 26.6Hz

Common gain =10dBBoost gain =16dBLength gain =30.4dB

Measurement of Cross over frequency of MC 12/23/2002

Cross over frequency = 26.6Hz Phase margin = 15degree

l

m

l

m

G

G

G1

G

V1

V2

Gl

Gm

VCO loop

Mirror loop

V1 V2


APPENDIX


Example: MC loop

bmcamcdtransmitte

bamcmc

pafmmmcb

pafldpsla

)1(

)(

FCFCF

FFDCv

vEEHFF

vEEHFF

d

d

d

mcml

mcpsldemod 1

CGG

FFF

Equation of servo topology

mcml

mclpsl

ml

mmcdtransmitte 1

1

1F

GG

CGF

GG

GCF

Fdemod

?

Fb

Fa

PSL

TO IFO


Approximation of Frequency noiseof MC Transmitted light

A B C

A

B

C

mcml

mclpsl

ml

mmcdtransmitte 1

1

1F

GG

CGF

GG

GCF

mc

l

mpsl

m

lmc

l

m

lpsl

m

ldtransmitte

mcml

1

1

1

11

11

1

1

1 ,1

F

GG

F

GG

F

GG

GF

GG

F

CGG

mcpsll

mcmc

l

mclpsl

l

mcdtransmitte

lm

11

1

1

1

FFG

CF

G

CGF

G

CF

GG

mcpslmcmcmcpslmcdtransmitte

mclm

)1(

11 1

FFCFCFCF

CGG


Which is better?

Calibration

Residual Frequency noise» In-loop noise

» Out-loop noise

Example:Error signal or Feedback signal?

Fresml

mcpslres GG1

FFF

)GG1(F

FFF

mlres

mcpslmes

Suppression factor by MC gain,estimated by another measurement

? Vor V dpsl

mcmcpafl

pslres CDE E

VF

Ne

x EpafEl

+ +

Ne

1+Gl+Gm<<Ne

• To avoid the electronic noise of feedback filter

Ne

1+Gl+Gm

-Nex(Gl+Gm)1+Gl+Gm

~ -Ne~


Example:Calibration with complicated optical response

C S

P pendulum

Cavityresponse

Sensing

F

Feedback filter

DARM_IN1

DARM_OUT

N G

N

1

CSFPG

G

CSN

1

G

PGN

G

CSFN

1

/

1

G

GN

1

DARM_IN2

EXC

Use DARM_IN1 : error signal

•Measure DARM_IN2/EXC=

•Estimate S•Measure (or estimate) C

Use DARM_OUT: feedback signal

•Measure DARM_IN1/EXC=

•Estimate P

G1

1

G

G

1


PL-

FL-

S

P: plant,IFOF: feedback filterS: suspensionC: coupling constant from l- to L- : shotnoise limited sensitivity of l-G: open loop gain

Pl-

Fl-

S

C

lPl sn

sn lL- loop

l- loop

Ll

l

GG

lCG

1

1

1sn

l

l

G

lP

1sn

L

L

Ll

l

G

G

P

C

G

lG

11sn

Calibrated noise

Ll

lLL

L

L

Ll

l

P

Cl

G

GGG

G

G

P

C

G

lGsn

sn

1/)1(

11

l

l

G

lG

1sn

L

L

l

l

G

GC

G

lG

111sn

1

2

3

45

6

Coupling between 2 loops

Documents

LIGO- G060XXX-00-R E2E meeting, September 2006 1 How to design feedback filters? E2E meeting September 27, 2006 Osamu Miyakawa, Caltech