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transcript
E2E meeting, September 2006 1LIGO- G060XXX-00-R
How to design
feedback filters?
E2E meetingSeptember 27, 2006
Osamu Miyakawa, Caltech
E2E meeting, September 2006 2LIGO- G060XXX-00-R
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
E2E meeting, September 2006 3LIGO- G060XXX-00-R
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
E2E meeting, September 2006 4LIGO- G060XXX-00-R
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
E2E meeting, September 2006 5LIGO- G060XXX-00-R
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
E2E meeting, September 2006 6LIGO- G060XXX-00-R
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
Frequency log f [Hz]1 10 100 1000
f -2
• 2pole: 180deg delay
-180
180
-90
0
90
Pha
se [
deg]
Gai
n
1
10
100
1000
Frequency log f [Hz]1 10 100 1000
-180
180
-90
0
90
Pha
se [
deg]
Gai
n1
10
100
1000
Frequency log f [Hz]1 10 100 1000
f +1 f +2
• 1zero: 90deg advance • 2zero: 180deg advance
E2E meeting, September 2006 7LIGO- G060XXX-00-R
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
E2E meeting, September 2006 8LIGO- G060XXX-00-R
Example of Feedback filterused in 40m arm cavity
F :
f -1f +1
4kHz10Hz
zero@10Hzpole@4kHz,4kHz
Phaserestore
85degreePhase
advance
E2E meeting, September 2006 9LIGO- G060XXX-00-R
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
zero@10Hzpole@1kHz,1kHz
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
E2E meeting, September 2006 10LIGO- G060XXX-00-R
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
E2E meeting, September 2006 11LIGO- G060XXX-00-R
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
E2E meeting, September 2006 12LIGO- G060XXX-00-R
Example of measured Open loop TF
Phase margin 45degree
Generally, phase margin should be more than 30degree at least.
UGF
f -1 slope
E2E meeting, September 2006 13LIGO- G060XXX-00-R
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
E2E meeting, September 2006 14LIGO- G060XXX-00-R
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
E2E meeting, September 2006 15LIGO- G060XXX-00-R
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
Frequency log f [Hz]1 10 100 1k 10k0.1
G
G
V
V
1EXC
IN1G
ain 1
Frequency log f [Hz]1 10 100 1k 10k0.1
UGF
UGF
E2E meeting, September 2006 16LIGO- G060XXX-00-R
Example of measured Closed loop TF
Gain oscillation 6dB
Generally, gain oscillation should be less than 10dB at most.
E2E meeting, September 2006 18LIGO- G060XXX-00-R
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
Frequency log f [Hz]1 10 100 1k 10k0.1
0.0010.0001
1000000
-270
f -3
f -2
E2E meeting, September 2006 19LIGO- G060XXX-00-R
Example: actuator x optical plant x sensor
f -2 slope
E2E meeting, September 2006 20LIGO- G060XXX-00-R
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
E2E meeting, September 2006 21LIGO- G060XXX-00-R
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
E2E meeting, September 2006 22LIGO- G060XXX-00-R
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
Frequency log f [Hz]1 10 100 1k 10k0.1
1000000
-270
f -1
f -2
•Phase margin at UGF > 0 deg ( >30deg)•Relative phase margin at cross over frequency > 0deg ( >30deg)
E2E meeting, September 2006 23LIGO- G060XXX-00-R
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
E2E meeting, September 2006 24LIGO- G060XXX-00-R
+-
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
E2E meeting, September 2006 26LIGO- G060XXX-00-R
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
E2E meeting, September 2006 27LIGO- G060XXX-00-R
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
E2E meeting, September 2006 28LIGO- G060XXX-00-R
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~
E2E meeting, September 2006 29LIGO- G060XXX-00-R
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
E2E meeting, September 2006 30LIGO- G060XXX-00-R
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