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

2. LIGO- G060XXX-00-R E2E meeting, September 2006 2 Single Fabry-Perot cavity P: cavity pole S: flat F: ?? A: suspension TF S P Sensor A Actuator F Feedback filter ITM Pick off ETM Laser EOM Suspension Photo detector Oscillator Mixer Coil-magnet Actuator Seismic noise Feedback filter Plant Actuator Sensor Feedback filter Plant Noise 16Hz:LIGO 1.6kHz:40m f -1 f -2 ~1Hz L Signal[a.u.] Error signal Lock point Locking area

3. LIGO- G060XXX-00-R E2E meeting, September 2006 3 Condition for stable control S P Sensor A Actuator F Feedback filter Plant Noise 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.

4. LIGO- G060XXX-00-R E2E meeting, September 2006 4 Transfer function from Noise N to signal V out S P Sensor A Actuator F Feedback filter Plant Noise

5. LIGO- G060XXX-00-R E2E meeting, September 2006 5 Transfer function from outloop noise to inloop noise S P Sensor A Actuator F Feedback filter Plant Noise Outloop noise N is suppressed by OLG G to inloop noise N/(1+G), then it is multiplied by TF in loop.

6. LIGO- G060XXX-00-R E2E meeting, September 2006 6 Basic concept : relationship between gain and phase -180 180 -90 0 90 Phase [deg] Gain 1 10 100 1000 Frequency log f [Hz] 110 1001000 f -1 1pole: 90deg delay -180 180 -90 0 90 Phase [deg] Gain 1 10 100 1000 Frequency log f [Hz] 110 1001000 f -2 2pole: 180deg delay -180 180 -90 0 90 Phase [deg] Gain 1 10 100 1000 Frequency log f [Hz] 110 1001000 -180 180 -90 0 90 Phase [deg] Gain 1 10 100 1000 Frequency log f [Hz] 110 1001000 f +1 f +2 1zero: 90deg advance2zero: 180deg advance

7. LIGO- G060XXX-00-R E2E meeting, September 2006 7 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 -1 f +1 1kHz 10Hz zero@10Hz pole@1kHz,1kHz -180 -90 0 90 Phase [deg] Gain 1 0.01 100 10000 Frequency log f [Hz] 1101001k f -2 10k0.1 0.001 0.0001 1000000 -270 f -4 f -1 Phase restore Open loop TF -180 -90 0 90 Phase [deg] Gain 1 0.01 100 10000 Frequency log f [Hz] 1101001k10k0.1 0.001 0.0001 1000000 -270 Not stable! f -2 f -3

8. LIGO- G060XXX-00-R E2E meeting, September 2006 8 Example of Feedback filter used in 40m arm cavity F : f -1 f +1 4kHz 10Hz zero@10Hz pole@4kHz,4kHz Phase restore 85degree Phase advance

9. LIGO- G060XXX-00-R E2E meeting, September 2006 9 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 -1 f +1 1kHz 10Hz zero@10Hz pole@1kHz,1kHz Phase Restore + Boost f -2 Wave length: ~ 10 -6 m Finesse: ~ 1000 FWHM (full width half maximum): ~10 -9 m Seismic noise N: ~10 -6 m @1Hz Q of suspension: ~10 Required gain at 1Hz > 10 4 Wave length ~ 10 -6 m FWHM ~ 10 -9 m Boost filter -180 -90 0 90 Phase [deg] Gain 1 0.01 100 10000 Frequency log f [Hz] 1101001k f -4 10k0.1 0.001 0.0001 1000000 -270 f -4 f -1

10. LIGO- G060XXX-00-R E2E meeting, September 2006 10 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 Phase [deg] Gain 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! 1101001k10k

11. LIGO- G060XXX-00-R E2E meeting, September 2006 11 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 Plant Noise

12. LIGO- G060XXX-00-R E2E meeting, September 2006 12 Example of measured Open loop TF Phase margin 45degree Generally, phase margin should be more than 30degree at least. UGF f -1 slope

13. LIGO- G060XXX-00-R E2E meeting, September 2006 13 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 Plant Noise

14. LIGO- G060XXX-00-R E2E meeting, September 2006 14 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.

15. LIGO- G060XXX-00-R E2E meeting, September 2006 15 What does Closed Loop TF mean? 1.C 1 :used to estimate gain oscillation Measure Gain oscillation is caused by small phase margin at UGF. 2.C 2 : Measure S P Sensor A Actuator F Feedback filter Plant Noise There are two definitions for closed loop TF Gain 1 Frequency log f [Hz] 1101001k10k0.1 Gain 1 Frequency log f [Hz] 1101001k10k0.1 UGF

16. LIGO- G060XXX-00-R E2E meeting, September 2006 16 Example of measured Closed loop TF Gain oscillation 6dB Generally, gain oscillation should be less than 10dB at most.

17. LIGO- G060XXX-00-R E2E meeting, September 2006 17 Example of measured Closed loop TF

18. LIGO- G060XXX-00-R E2E meeting, September 2006 18 How to measure plant TF? Measure: S P Sensor A Actuator F Feedback filter Plant Noise Once system become stable, you can measure given plant TF. -180 -90 0 90 Phase [deg] Gain 1 0.01 100 10000 Frequency log f [Hz] 1101001k10k0.1 0.001 0.0001 1000000 -270 f -3 f -2

19. LIGO- G060XXX-00-R E2E meeting, September 2006 19 Example: actuator x optical plant x sensor f -2 slope

20. LIGO- G060XXX-00-R E2E meeting, September 2006 20 Example: suspension Local damping P: flat S: flat F: ?? A: suspension TF OLTF S P Sensor A Actuator F Feedback filter Plant Noise ETM Suspension Coil-magnet Actuator Seismic noise Feedback filter Actuator Sensor Feedback filter Plant f +1 ~10Hz f -2 ~1Hz Q:~1000 -180 -90 0 90 Phase [deg] Gain 1 0.01 10 100 Frequency log f [Hz] 110 100 0.1 0.001 0.0001 1000 -270 f -2 f +1 f -1 Phase margin 90degree UGF zero@0Hz pole@10Hz

21. LIGO- G060XXX-00-R E2E meeting, September 2006 21 Example in E2E Quad-suspension: local damping Main Ref. M0 M1 M2 M3 6(6) 0(3) 0(0) Damped DOFs (OSEM DOFs) M0: pitch Zero @ 0Hz Pole @ 10,10Hz S P Sensor A Actuator F Feedback filter Plant Noise

22. LIGO- G060XXX-00-R E2E meeting, September 2006 22 Advanced example: Double loop P: cavity pole S: flat F 1 : cavity pole restore F 2 : AC couple, 1pole, cavity pole restore A 1 : suspension TF A 2 : flat OLTF ITM Pick off ETM Laser EOM Suspension Photo detector Oscillator Mixer Coil-magnet Actuator Seismic noise Feedback filter Actuator 1 Sensor Feedback filter 1 Plant f -2 ~1Hz PZT Feedback filter 2 Actuator 2 S P Sensor A 1 Actuator 1 F 1 Feedback filter 1 Plant A 2 F 2 Feedback filter 2 Actuator 2 -180 -90 0 90 Phase [deg] Gain 1 0.01 100 10000 Frequency log f [Hz] 1101001k10k0.1 1000000 -270 f -1 f -2 Phase margin at UGF > 0 deg ( >30deg) Relative phase margin at cross over frequency > 0deg ( >30deg)

23. LIGO- G060XXX-00-R E2E meeting, September 2006 23 Measured open loop gain of MC Phase margin=28.4deg Unity gain frequency Cross over frequency Relative phase = 165deg

24. LIGO- G060XXX-00-R E2E meeting, September 2006 24 Measured cross over frequency of MC  Cross over frequency = 26.6Hz  Phase margin = 15degree GlGl GmGm VCO loop Mirror loop V1 V2

25. LIGO- G060XXX-00-R E2E meeting, September 2006 25 APPENDIX

26. LIGO- G060XXX-00-R E2E meeting, September 2006 26 Example: MC loop Equation of servo topology F demod ? FbFb FaFa PSL TO IFO

27. LIGO- G060XXX-00-R E2E meeting, September 2006 27 Approximation of Frequency noise of MC Transmitted light A B C A B C

28. LIGO- G060XXX-00-R E2E meeting, September 2006 28  Which is better?  Calibration  Residual Frequency noise »In-loop noise »Out-loop noise Example: Error signal or Feedback signal? F res Suppression factor by MC gain, estimated by another measurement NeNe x E paf E l + NeNe 1+G l +G m <<N e To avoid the electronic noise of feedback filter NeNe 1+G l +G m -N e x(G l +G m ) 1+G l +G m ~ -N e ~

29. LIGO- G060XXX-00-R E2E meeting, September 2006 29 Example: Calibration with complicated optical response C S P pendulum Cavity response Sensing F Feedback filter DARM_IN1 DARM_OUT 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

30. LIGO- G060XXX-00-R E2E meeting, September 2006 30 P L- F L- S P: plant,IFO F: feedback filter S: suspension C: coupling constant from l- to L- : shotnoise limited sensitivity of l- G: open loop gain P l- F l- S C L- loop l- loop Calibrated noise 1 2 3 4 5 6 Coupling between 2 loops