1. LIGO-G030187-00-DPage 1 Modal Analysis and Feedback Control of HAM MEPI Oct 16, 2002 Lei Zuo Osamah Rifai Samir Nayfeh

2. LIGO-G030187-00-DPage 2 Overview Review of loop transmissions vs. modal data Preliminary look at control –Multiple lightly damped modes close to crossover make control difficult Approaches –Modal control may decrease the relative magnitudes of the flexible modes. –Easiest approach to robust performance: structural damping. –Optimal MIMO Control can improve the performance, but requires a good model

3. LIGO-G030187-00-DPage 3 Typical TF from Horizontal Actuator to Geophones 40dB/dec 20dB/dec 18.4Hz 14Hz 16.4Hz 8.6Hz blue 1 red 2 mage 3 green 4 180°-15° Magnitude (dB) Phase

4. LIGO-G030187-00-DPage 4 Typical TF from Vertical Actuator to Geophones 40dB/dec 20dB/dec 27.7Hz 29.2Hz 18.5Hz 19.1Hz 14Hz 9.6Hz blue 1 red 2 mage 3 green 4 180°-85° Magnitude (dB) Phase

5. LIGO-G030187-00-DPage 5 Observation of this two typical TFs The horizontal-horizontal coupling is very strong, so is vertical-vertical coupling. The vertical -horizontal coupling is very strong at around 3Hz, 19Hz and 28Hz The actuator-sensor pairs seems to be collocated well For control consideration, the critical modes are the two modes around 19Hz, and two modes around 28Hz. (The modes around 14Hz are also critical)

6. LIGO-G030187-00-DPage 6 Modal Analysis

7. LIGO-G030187-00-DPage 7 Typical Measurement (x) 1.68Hz 2.95Hz 4.4Hz 5.0Hz 6.8Hz 8.4/9.3Hz 9.9/10.9Hz 13.4/13.7Hz 15.7Hz 18.3Hz 18.9Hz 20.38/20.41Hz 27.5/28.7Hz (2.5Hz) 1.3Hz

8. LIGO-G030187-00-DPage 8 Typical Measurement (y) 1.68Hz 2.5Hz 4.4Hz (5.0Hz) 6.8Hz 8.4/9.3Hz 9.9/10.9Hz 13.4/13.7Hz 15.7Hz 18.3Hz 18.9Hz 20.38/20.41Hz 27.5/28.7Hz (2.95Hz) (1.3Hz)

9. LIGO-G030187-00-DPage 9 Typical Measurement (z) (1.68Hz) 2.95Hz 4.4Hz 5.0Hz 6.8Hz 9.3Hz 9.9/10.9Hz 13.4/13.7Hz 15.7Hz 18.3/18.9Hz 20.38/20.41Hz 27.5/28.7Hz 2.5Hz 1.3Hz 8.4Hz

10. LIGO-G030187-00-DPage 10 Critical Modes 15.75Hz, in-plane bending 13.37Hz, in-plane bending + twist13.71Hz, in-plane bending (local) In-plane Beam Bending 18.32Hz, in-plane bending

11. LIGO-G030187-00-DPage 11 Critical Modes 18.85Hz, out-of-plane twist Twist

12. LIGO-G030187-00-DPage 12 Critical Modes 27.45Hz, tank 28.74Hz, tank Tank Related

13. LIGO-G030187-00-DPage 13 Mode Summary Modes at (2.5),3.0, 4.4, 5.0, 6.8, 8.4, and 9.3 Hz are rigid-body modes (compare with Dennis’s modeling?) Modes at 13.4, 13.7, 15.7 Hz, and 18.3 Hz are associated with in-plane bending Mode at 18.9 Hz is related to twist Modes at 27.5 and 28.7 Hz are associated with tank motion Two modes around 20.4 are out-of-plane (vertical) bending Modes at 1.3 and 1.7 Hz seems to be tank motion+frame rigid body (measurements is not so reliable at so low freq) Modes at 9.9, 10.9 are rigid-body plus some bending

14. LIGO-G030187-00-DPage 14 Feedback Control of HAM |G(jw)H(jw)| >>1 at low frequency, T(jw)  1/H(jw) |G(jw)H(jw)| <<1 at high frequency, T(jw)  G(jw) G(s), HAM dynamics H(s), sensing and control - + FdV F 1+G(s)H(s) T(s)= G(s)V Fd =

15. LIGO-G030187-00-DPage 15 -2 -2 +1+1 -90° - 180° |G(s)| A Control Loop (Concept Design) |H(s)| Phase Margin 180-15-90 + =75° - |T(s)| Closed Loop 15 reduction 1-3Hz |G(s)H(s)| Loop Gain (The low-frequency characteristics of geophones is not good  Position feedback will be used for <0.5Hz) +1+1 +2+2 +1+1

16. LIGO-G030187-00-DPage 16 F d  V and X 0  X    1 -2/dec -1/dec

17. LIGO-G030187-00-DPage 17 Further Consideration The previous concept design is for vertical channels based on the assumption of low plant uncertainties. See the previous control loop again: * Multiple crossing, not robust (unstable!) * PM is 15°, not 75° (add lead? no) See horizontal channels * Multiple crossing, not robust * PM is only 5°

18. LIGO-G030187-00-DPage 18 -2 -2 +1+1 -90° - 180° |G(s)| Vertical Loop (Concept Design) |H(s)| Phase Margin 180-15-90 + =75° - 180-75-90 + =15° - |T(s)| Closed Loop 15 reduction 1-3Hz |G(s)H(s)| Loop Gain Multiple Crossings After Structure Modification (Position feedback will be used for <0.5Hz)

19. LIGO-G030187-00-DPage 19 -2 -2 +1+1 -90° - 180° |G(s)| Horizontal Loop (Concept Design) |H(s)| Phase Margin 180-85-90 + =5° - |T(s)| Closed Loop 15 reduction 1-3Hz |G(s)H(s)| Loop Gain Multiple Crossings After Structure Modification (Position feedback will be used for <0.5Hz)

20. LIGO-G030187-00-DPage 20 Suggestion for Feedback Control * Modal Decomposition SISO Control. Questions (SISO): 1) how good is the decomposition? 2) how large are the peaks of flexible modes? Other rigid-body modes from non-perfect decomposition Flexible modes Dominant rigid-body mode  |·||·| 1

21. LIGO-G030187-00-DPage 21 Suggestions on Structure Modification (to make control easier) Add constrained layer (viscoelastic) damping to the beam structure (to damp the bending modes) Make the joints stiffer; add some damping to the joints; or add another beam Reduce the coupling between tank and the frame.

22. LIGO-G030187-00-DPage 22 Conclusions Multiple lightly damped modes close to crossover make control difficult –Instability –Amplification of disturbances Modal control may decrease their relative magnitudes. Easiest approach to robust performance: structural damping. Optimal MIMO Control can improve the performance, but requires a good model

23. LIGO-G030187-00-DPage 23 Other Control Schedules Discussed Osamah: low frequency crossing with high order controller Dave: notch filter Rich, David and Denis: high frequency crossing

24. LIGO-G030187-00-DPage 24 Appendix

25. LIGO-G030187-00-DPage 25 Typical TF from Horizontal Actuator to Positions Magnitude (dB) Phase

26. LIGO-G030187-00-DPage 26 Typical TF from Vertical Actuator to Positions Magnitude (dB) Phase

27. LIGO-G030187-00-DPage 27 Out-of-Plane Bending Modes 20.38Hz 20.41Hz Out-of-Plane Beam Bending Related

28. LIGO-G030187-00-DPage 28 Mode Shapes 1.3Hz 1.7Hz (low-freq, measurement is not reliable)

29. LIGO-G030187-00-DPage 29 Mode Shapes 2.50Hz 2.97Hz

30. LIGO-G030187-00-DPage 30 Mode Shapes 4.41Hz 4.99Hz

31. LIGO-G030187-00-DPage 31 Mode Shapes 6.86Hz 8.33Hz

32. LIGO-G030187-00-DPage 32 Mode Shapes 9.25Hz

33. LIGO-G030187-00-DPage 33 Mode Shapes 10.9Hz 9.82Hz