# How to design feedback filters

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```					                      How to design
feedback filters?

E2E meeting
September 27, 2006
Osamu Miyakawa, Caltech

LIGO- G060XXX-00-R       E2E meeting, September 2006   1
Single Fabry-Perot cavity
Seismic                                                                              Seismic                Error signal
noise                                                                                noise
Suspension                        Suspension                                        Locking area
Plant

Signal[a.u.]
Pick                                          Coil-magnet
Laser      EOM                                                          Actuator                                    Lock point
off

ITM                                ETM               Actuator
Photo
detector          Feedback                                                               L
filter
16Hz:LIGO
Oscillator   Mixer                                                                                           1.6kHz:40m
Sensor               Feedback filter
•P: cavity pole                                        f -1

N                                  P                S
•S: flat
Noise
Plant        Sensor
A                                                    •F: ??                                    ~1Hz

Actuator
F                            •A: suspension TF
f -2
Feedback filter

LIGO- G060XXX-00-R                              E2E meeting, September 2006                                                  2
Condition for stable control

N                           P            S               Vout
Noise
Plant       Sensor

A                                           G = SPAF : Open loop transfer function
Actuator
F
Feedback filter

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.
LIGO- G060XXX-00-R                      E2E meeting, September 2006              3
Transfer function from
Noise N to signal Vout
VC                VD                                         VA = F × VOUT
N                            P               S               Vout               VB = A × VA
Noise

VB
Plant          Sensor                               VC = N − VB
A
Actuator
VD = P × VC
F
VOUT = S × VD
Feedback filter

VA
VOUT = S × P × ( N − A × F × VOUT )
(1 + SPAF )VOUT = SPN
VOUT        SP       SP
∴      =            =
N     (1 + SPAF ) (1 + G )
G = SPAF : Open loop transfer function
LIGO- G060XXX-00-R                         E2E meeting, September 2006                         4
Transfer function from
outloop noise to inloop noise
VC                VD
VC Vout 1   1
=       =
N                            P               S               Vout        N   N SP (1 + G )
Noise
Plant
VB                                       Sensor
Outloop noise N is suppressed by
A
Actuator                                             OLG G to inloop noise N/(1+G),
F                              then it is multiplied by TF in loop.
Feedback filter

VA                                        VD   P
=
N (1 + G )

VB PSFA       G
=       =
N (1 + G ) (1 + G )

LIGO- G060XXX-00-R                         E2E meeting, September 2006                      5
Basic concept :
relationship between gain and phase
• 1pole: 90deg delay                                        • 2pole: 180deg delay
1000                                                               1000
f -1                                                                   f -2
100                                                                100
Gain

Gain
10                                                                 10
1                                                                  1
180                                                                180
Phase [deg]

Phase [deg]
90                                                                 90
0                                                                  0
-90                                                                -90
-180                                                               -180
1     10          100         1000                                 1   10          100           1000
Frequency log f [Hz]                                             Frequency log f [Hz]

1000                                                               1000
100                                                                100
Gain

Gain
f +1                                                                          f +2
10                                                                 10
1                                                                  1
180                                                                180
Phase [deg]

Phase [deg]

90                                                                 90
0                                                                  0
-90                                                                -90
-180                                                               -180
1     10          100         1000                                 1   10          100           1000
Frequency log f [Hz]                                             Frequency log f [Hz]
LIGO- G060XXX-00-R                                    E2E meeting, September 2006                                               6
Open loop TF
Conditoin 2: Phase delay of OLG at UGF must be less than 180 degree.
(UGF: frequency at gain = 1)
P : cavity pole           x S : flat   x F : ??                  x A : suspension                  =   system TF
1000000
1kHz                                                       1Hz             10000                       f -2
100

Gain
1
0.01                                        f -3
f -1                                                                 0.001
f -2     0.0001
90

Phase [deg]
0
-90              Not stable!
-180
-2700.1         1    10       100       1k   10k
Frequency log f [Hz]
Phase
P : cavity pole           x S : flat   x F : restore x A : suspension =                                  OLTF
1000000                                f -2
1kHz                                       1kHz            1Hz               10000                       f -1
10Hz                                              100

Gain
f +1          f -1                            1
0.01                                      f -4
f -1                                                                   0.001
f -2       0.0001
zero@10Hz                                          90

Phase [deg]
pole@1kHz,1kHz                                      0
-90
-180
-2700.1       1    10       100       1k   10k
Frequency log f [Hz]
LIGO- G060XXX-00-R                E2E meeting, September 2006                                                         7
Example of Feedback filter
used in 40m arm cavity

Phase
F : restore
4kHz
10Hz
f +1          f -1

zero@10Hz
pole@4kHz,4kHz

85degree
Phase

LIGO- G060XXX-00-R     E2E meeting, September 2006                           8
Boost filter
Conditoin 2: Enough gain at DC to suppress outloop noise within linear range.
Wave length:     ~ 10-6m                                                                         Finesse =
FWHM
FWHM ~ 10-9m
Finesse:         ~ 1000                                                                                       wave length
FWHM (full width half maximum):
~10-9m
Wave length ~ 10-6m
Seismic noise N: ~10-6m @1Hz
N : 10 −6 × 10 = 10 −5 m @ 1Hz
Q of suspension: ~10
N        10 −5
=           ≈ 10 −9 m @ 1Hz
Required gain at 1Hz > 104                                   (1 + G ) (1 + 10 )4

Phase
Restore
P : cavity pole           x S : flat   x F:      +
Boost
x A : suspension                  =         OLTF
1000000                  f -4
f -2      1kHz             1Hz
1kHz                                                                   10000                         f -1
10Hz                                      100

Gain
f +1      f -1                             1
0.01                                        f -4
f -1                                                             0.001
f -2    0.0001
zero@10Hz                                     90

Phase [deg]
pole@1kHz,1kHz                                 0
-90
-180
-2700.1           1    10    100        1k   10k
Frequency log f [Hz]
LIGO- G060XXX-00-R               E2E meeting, September 2006                                                      9
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.

1000000
f -4
10000
100                                f -1
Once you get stable
Gain

1
0.01                                                        f -4      operation, it is very
0.001
0.0001
important to measure open
90                                                                 loop TF to see how stable!
Phase [deg]

0
-90
-180
-270
0.1         1          10          100      1k      10k

Frequency log f [Hz]

LIGO- G060XXX-00-R                                E2E meeting, September 2006               10
How to measure Open Loop TF?

N                            P             S              Vout
Noise
Plant        Sensor                                                            1
1. Use closed transfer function: C ≡
A                                                                                           (1 + G )
Actuator
F                                               VIN2    1
• Measure            =
Feedback filter                                           VEXC (1 + G )
VIN2 VEXC VIN1                                     1
• Calculate G     G=     −1
C
2. Measure OLTF directly

• Measure       VIN1
= FAPS = G
VIN2

LIGO- G060XXX-00-R                      E2E meeting, September 2006                         11
Example of measured Open loop TF

f -1 slope

UGF

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

Phase margin 45degree

LIGO- G060XXX-00-R          E2E meeting, September 2006                         12
Why low frequency measurement is dirty?

N                          P              S           Vout
Noise
Plant         Sensor

A
Actuator
F
Feedback filter

VIN2 VEXC VIN1

• Excitation signal is suppressed
a lot with very high gain G>>1 at
low frequency.
1
VIN2 =          VEXC << Noise
(1 + G )
LIGO- G060XXX-00-R    E2E meeting, September 2006                                 13
What is Coherence?

• Coherence: coh(f)
How much related between input
and output
Wxy ( f )
coh (h) =
Wxx ( f ) ⋅W yy ( f )
Wxy ( f ) : cross spectrum of x and y
Wxx ( f ),W yy ( f ) : power spectrum of x and y

• 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.

LIGO- G060XXX-00-R   E2E meeting, September 2006                                       14
What does Closed Loop TF mean?
N                             P            S              Vout
Noise
Plant       Sensor
There are two definitions for
A                                                               closed loop TF
Actuator
F                                         1                                     G
C1 ≡                                  C2 ≡
Feedback filter
(1 + G )                              (1 + G )
VIN2 VEXC VIN1                                                UGF

1. C1 :used to estimate gain oscillation                                     1

Gain
VIN2    1
• Measure                 =
VEXC (1 + G )
0.1   1      10   100     1k     10k
• Gain oscillation is caused by small phase                                        Frequency log f [Hz]
margin at UGF.
UGF

2. C2 :                     VIN1    G
Gain   1

• Measure                =
VEXC (1 + G )
0.1   1      10   100     1k     10k
Frequency log f [Hz]
LIGO- G060XXX-00-R                      E2E meeting, September 2006                                             15
Example of measured Closed loop TF

Gain oscillation 6dB

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

LIGO- G060XXX-00-R          E2E meeting, September 2006                               16
Example of measured Closed loop TF

LIGO- G060XXX-00-R          E2E meeting, September 2006   17
How to measure plant TF?
N                            P             S
Noise
Plant        Sensor

A                                                    Once system become stable, you
Actuator
F                                 can measure given plant TF.
Feedback filter

VOUT                VIN2 VEXC VIN1

• Measure:                                                         1000000
10000                   f -2
100

Gain
1
VIN1                                                    0.01                               f -3
= A× P × S                                        0.001
VOUT                                                 0.0001
90

Phase [deg]
0
-90
-180
-2700.1       1   10     100   1k   10k
Frequency log f [Hz]

LIGO- G060XXX-00-R                      E2E meeting, September 2006                                              18
Example: actuator x optical plant x sensor

f -2 slope

LIGO- G060XXX-00-R   E2E meeting, September 2006   19
Example: suspension Local damping
Seismic
•P: flat
Plant
noise
Suspension                  •S: flat
Coil-magnet                                              ~10Hz
Actuator
•F: ??                f +1
zero@0Hz
pole@10Hz
Sensor                                      ETM               Actuator
•A: suspension TF
Feedback                                                                           Q:~1000
filter

Feedback filter                                                                            f -2
~1Hz
•OLTF
1000
N                               P                S
100
10 +1
UGF
f -1
1 f

Gain
Noise
Plant         Sensor                                           0.01
0.001
f -2
A                                                                          0.0001
Actuator                                                                    90        Phase margin

Phase [deg]
0          90degree
F                                                        -90
-180
Feedback filter                                                -2700.1    1        10           100
Frequency log f [Hz]
LIGO- G060XXX-00-R                            E2E meeting, September 2006                                           20
Example in E2E
Main Ref.         Damped DOFs
(OSEM DOFs)
M0: pitch               Open loop TF

M0                               6(6)
0
10
n
i
M1                               0(3)                           a
G

-2
10
0                       1
M2                               0(3)                                                   10                      10

Suspenstoin TF

raw
0
10                                     damped

M3                               0(0)                           n
i
a
G
-2
10

0                       1
10                      10

N                         P         S                                          10
2
Feedback filter TF
Noise                                                     VIN1
Plant     Sensor            raw :    = A× P × S                 Zero @ 0Hz
VOUT                  10
1

A
damped :              n
i
Pole @ 10,10Hz
a        0
G
Actuator                F                           VIN1 A × P × S
10

Feedback
=
VOUT V     filter
VEXC   1+ G            10
-1

VIN1
0                       1
EXC                                                                            10                      10
LIGO- G060XXX-00-R                         E2E meeting, September 2006                    frequency [Hz]    21
Seismic                                     Seismic
noise                             • P: cavity pole
noise
Suspension                       Suspension     • S: flat
Plant
Laser      EOM
Pick                                          Coil-magnet
Actuator     • F1: cavity pole restore
off

PZT                         ITM
• F2: AC couple, 1pole,
ETM Actuator 1
Actuator 2              Photo                                                                    cavity pole restore           ~1Hz
detector             Feedback
filter
• A1: suspension TF
f -2

Oscillator   Mixer
Sensor               Feedback filter 1                       • A2: flat
•Phase margin at UGF > 0 deg ( >30deg)
• OLTF                  •Relative phase margin at cross over
Feedback filter 2                                                              frequency > 0deg ( >30deg)
1000000                      f -2
10000
P                S

Gain
100                            f -1
Plant         Sensor
1
A2         A1                     Feedback filter 1                                       0.01
90
Actuator 2     Actuator 1
F1                                Phase [deg]      0
-90
F2                                                                -180
Feedback filter 2                                          -2700.1   1   10   100   1k    10k
LIGO- G060XXX-00-R                         E2E meeting, September 2006                                  Frequency log f [Hz]   22
Measured open loop gain of MC
Open loop transfer function of Mode Cleaner
200
Calculated Mirror loop gain
Calculated VCO loop gain
150                                                Calculated total loop gain
Measured Mirror loop gain
Measured VCO loop gain
Mag[dB]

Measured total loop gain
100
Cross over frequency
C=10dB, B=16dB, L=30.4dB
50         Unity gain freq.=67.2kHz
Phase margin=28.4deg                               Unity gain frequency
Cross over freq.=26.6Hz
0

150
100
Phase[deg]

50
0
Relative phase
-50                         = 165deg
-100
-150
Phase margin=28.4deg
-1          0           1          2         3            4               5
10           10          10          10        10        10              10
Frequency[Hz]
LIGO- G060XXX-00-R                       E2E meeting, September 2006                                    23
Measured cross over frequency of MC
-                      MC
+
Measurement of Cross over frequency of MC 12/23/2002
20                                                                        n
C om m on gai =10dB

0
n
B oost gai =16dB
h    n
Lengt gai =30.  4dB                           Gl
VCO loop
-20                                                                                                                         El
6H
26. z
Gm
Mag[dB]

-40                                                                                                      Mirror loop
+                     Em
-60                                                                                              +
-80
V1                   V2
-100
V2   Gm      G
150                                                                                                  =       ≈− m
100
V1 1 − G l   Gl
50
Phase[deg]

0

-50                                                                                         Cross over frequency = 26.6Hz
-100                                                                                         Phase margin = 15degree
-150
2   3   4   5 6 7 89    2    3   4   5 6 7 89        2    3   4   5 6 7 89
1                         10                       100                         1000
LIGO- G060XXX-00-R         Frequency[Hz]    E2E meeting, September 2006                                           24
APPENDIX

LIGO- G060XXX-00-R   E2E meeting, September 2006   25
Example: MC loop
Equation of servo topology
TO IFO
Fa = Fpsl − H d El Epaf vd
Fb = Fmc − H m Em Epaf vd
PSL
vd = Cmc Dmc ( Fa − Fb )
Ftransmitted = Cmc Fa + (1 − Cmc ) Fb

Fa                    Fdemod

Fb                                                         Cmc − Gm          1 + Gl − Cmc
Ftransmitted =               Fpsl +              Fmc
1 + Gl − Gm        1 + Gl − Gm

Fpsl − Fmc
Fdemod =                       Cmc
1 + Gl + Gm

?
LIGO- G060XXX-00-R                 E2E meeting, September 2006                                             26
Approximation of Frequency noise
of MC Transmitted light

Cmc − Gm          1 + Gl − Cmc
Ftransmitted =                  Fpsl +              Fmc
1 + Gl − Gm        1 + Gl − Gm

A             B          C
1 << Gl , Gm Cmc ≈ 1
A    Ftransmitted ≈
1
Fpsl +
Gl
Fmc ≈
1
Fpsl +
1
F
Gl             1 − Gm          Gl           Gm mc
1+               1+              1−           1−
1 − Gm             Gl            Gm           Gl

B    Gm << 1 << Gl
Cmc          1 + Gl − Cmc       C
Ftransmitted ≈          Fpsl +              Fmc ≈ mc Fpsl + Fmc
1 + Gl           1 + Gl         1 + Gl

C      Gm << Gl << 1          1 − Cmc ≈ 1
Ftransmitted ≈ Cmc Fpsl + (1 − Cmc ) Fmc ≈ Cmc Fpsl + Fmc

LIGO- G060XXX-00-R          E2E meeting, September 2006                                                       27
Example:
Error signal or Feedback signal?

Residual Frequency noise
»    In-loop noise
-Nex(Gl+Gm)
~ -N                          Fpsl − Fmc
Fres         1+Gl+Gm ~ e                    Fres =
1+ Gl + Gm
»    Out-loop noise

Fmes = Fpsl − Fmc
Ne
= Fres × (1 + G l + G m )
++
Suppression factor by MC gain,
estimated by another measurement
Ne
Ne                                                      <<N
xE E                                          1+Gl+Gm e
1+Gl+Gm paf l
Which is better?
Vpsl or Vd ?
• To avoid the electronic
Calibration
noise of feedback filter                                        Vpsl
Fres =
E l E paf D mc C mc
LIGO- G060XXX-00-R              E2E meeting, September 2006                                       28
Example:
Calibration with complicated optical response

N                                                           CS
N
1+ G                                                        1+ G
N                                     C                 S                DARM_IN1

−G                            Cavity          Sensing
N                              response                              EXC
1+ G
G = CSFP                               DARM_IN2
P       pendulum
Use DARM_IN1 : error signal
Feedback filter                                                     1
•Measure DARM_IN2/EXC=
1+ G
F
•Estimate S
•Measure (or estimate) C
DARM_OUT
Use DARM_OUT: feedback signal
CSF     −G/ P
N      =N
1+ G    1+ G                                    •Measure DARM_IN1/EXC=
G
1+ G
•Estimate P

LIGO- G060XXX-00-R                   E2E meeting, September 2006                              29
Gl −l− sn C − GL −
×
Coupling between 2 loops                                                                1 + Gl − 1 1 + GL −

5                                    4 CG l                  1
Gl −l− sn C    − GL −
l − − sn
×
×                         1 + Gl −       1 + GL −
1 + Gl − PL − 1 + GL −
P: plant,IFO
F: feedback filter
S: suspension                                                                                    PL-
C: coupling constant from l- to L-
l− sn: shotnoise limited sensitivity of l-                                                                     C
G: open loop gain                                                               S           L- loop

FL-
3 − Gl −l− sn                         1
1 + Gl −                          l− sn × Pl −
Pl-
2
Calibrated noise                                                                                                                Pl −l− sn
6                                    S          l- loop                             1 + Gl −
Gl −l− sn C    − GL −                           Gl −          C
×         × (1 + GL − ) / GL − =          l− sn
1 + Gl − PL − 1 + GL −                        1 + Gl −       PL −
Fl-
LIGO- G060XXX-00-R                            E2E meeting, September 2006                                                     30

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