LIGO-G030187-00-D Page 1

Modal Analysis and Feedback Control of HAM MEPI

Oct 16, 2002

Lei ZuoOsamah Rifai Samir Nayfeh

LIGO-G030187-00-D Page 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

LIGO-G030187-00-D Page 3

100

101

10-4

10-2

100

Mag

nitu

de

Horizontal Actuator 1 to All Horizontal (solid) and Vertical (dashed) Velocities

100

101

-300

-200

-100

0

100

200

300

Frequency (Hz)

Pha

se

Typical TF from Horizontal Actuator to Geophones 40dB/dec

20dB/dec18.4Hz

14Hz16.4Hz

8.6Hz

blue 1red 2mage 3green 4

180°-15°

Mag

nitu

de (

dB)

Pha

se

LIGO-G030187-00-D Page 4

100

101

10-4

10-2

100

Mag

nitu

de

Vertical Actuator 3 to All Vertical (solid) and Horizontal (dashed) Velocities

100

101

-300

-200

-100

0

100

200

300

Frequency (Hz)

Pha

se

Typical TF from Vertical Actuator to Geophones 40dB/dec

20dB/dec

27.7Hz29.2Hz

18.5Hz19.1Hz14Hz

9.6Hz

blue 1red 2mage 3green 4

180°-85°

Mag

nitu

de (

dB)

P

hase

LIGO-G030187-00-D Page 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)

LIGO-G030187-00-D Page 6

Modal Analysis

LIGO-G030187-00-D Page 7

Typical Measurement (x)

1.68Hz

2.95Hz

4.4Hz

5.0Hz6.8Hz

8.4/9.3Hz9.9/10.9Hz

13.4/13.7Hz

15.7Hz

18.3Hz

18.9Hz

20.38/20.41Hz

27.5/28.7Hz

(2.5Hz)

1.3Hz

LIGO-G030187-00-D Page 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

15.7Hz18.3Hz

18.9Hz20.38/20.41Hz

27.5/28.7Hz

(2.95Hz)

(1.3Hz)

LIGO-G030187-00-D Page 9

Typical Measurement (z)

(1.68Hz)

2.95Hz

4.4Hz

5.0Hz6.8Hz

9.3Hz

9.9/10.9Hz

13.4/13.7Hz

15.7Hz

18.3/18.9Hz

20.38/20.41Hz

27.5/28.7Hz

2.5Hz1.3Hz

8.4Hz

LIGO-G030187-00-D Page 10

Critical Modes

15.75Hz, in-plane bending

13.37Hz, in-plane bending + twist 13.71Hz, in-plane bending (local)

In-plane Beam Bending

18.32Hz, in-plane bending

LIGO-G030187-00-D Page 11

Critical Modes

18.85Hz, out-of-plane twist

Twist

LIGO-G030187-00-D Page 12

Critical Modes

27.45Hz, tank

28.74Hz, tank

Tank Related

LIGO-G030187-00-D Page 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

LIGO-G030187-00-D Page 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

-+Fd V

F1+G(s)H(s)T(s)=

G(s)V

Fd=

LIGO-G030187-00-D Page 15

100

101

-200

-150

-100

-50

0

50

100

150

200

250

300

100

101

10-3

10-2

10-1

100

101

-1

-2

-1-2

+1

-90°

-180°

|G(s)|

A Control Loop (Concept Design)|H(s)|

Phase Margin180-15-90+=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

+2

+1

LIGO-G030187-00-D Page 16

FdV and X0X

)( od XXk

sX

F

V

ks

FVFV

X

X

d

d

o

||||dF

VT

||oX

X

1

-2/dec

-1/dec

LIGO-G030187-00-D Page 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°

LIGO-G030187-00-D Page 18

100

101

-200

-150

-100

-50

0

50

100

150

200

250

300

100

101

10-3

10-2

10-1

100

101

-1

-2

-1-2

+1

-90°

-180°

|G(s)|

Vertical Loop (Concept Design)|H(s)|

Phase Margin180-15-90+=75°-

180-75-90+=15°-

|T(s)|Closed Loop

15 reduction 1-3Hz

|G(s)H(s)|Loop Gain

Multiple Crossings

After StructureModification

(Position feedback will be used for <0.5Hz)

LIGO-G030187-00-D Page 19

100

101

-200

-100

0

100

200

300

100

101

10-3

10-2

10-1

100

101

100

101

-150

-100

-50

0

50

100

150

-1

-2

-1-2

+1

-90°

-180°

|G(s)|

Horizontal Loop (Concept Design)|H(s)|

Phase Margin180-85-90+=5°-

|T(s)|Closed Loop

15 reduction 1-3Hz

|G(s)H(s)|Loop Gain

Multiple Crossings

After StructureModification

(Position feedback will be used for <0.5Hz)

LIGO-G030187-00-D Page 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

|·|

dF

V

1oX

X

LIGO-G030187-00-D Page 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.

LIGO-G030187-00-D Page 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

LIGO-G030187-00-D Page 23

Other Control Schedules Discussed

• Osamah: low frequency crossing with high order controller

• Dave: notch filter

• Rich, David and Denis: high frequency crossing

LIGO-G030187-00-D Page 24

Appendix

LIGO-G030187-00-D Page 25

Typical TF from Horizontal Actuator to Positions

100

101

10-4

10-3

10-2

10-1

Ma

gn

itud

eHorizontal Actuator 1 to All Horizontal (solid) and Vertical (dashed) Positions

100

101

-2000

-1500

-1000

-500

0

500

Frequency (Hz)

Ph

ase

Mag

nitu

de (

dB)

Pha

se

LIGO-G030187-00-D Page 26

100

101

10-4

10-3

10-2

10-1

Ma

gn

itud

eVertical Actuator 3 to All Vertical (solid) and Horizontal (dashed) Positions

100

101

-2000

-1500

-1000

-500

0

500

Frequency (Hz)

Ph

ase

Typical TF from Vertical Actuator to Positions M

agni

tude

(dB

) P

hase

LIGO-G030187-00-D Page 27

Out-of-Plane Bending Modes

20.38Hz

20.41Hz

Out-of-Plane Beam Bending Related

LIGO-G030187-00-D Page 28

Mode Shapes

1.3Hz

1.7Hz

(low-freq, measurement is not reliable)

LIGO-G030187-00-D Page 29

Mode Shapes

2.50Hz

2.97Hz

LIGO-G030187-00-D Page 30

Mode Shapes

4.41Hz

4.99Hz

LIGO-G030187-00-D Page 31

Mode Shapes

6.86Hz

8.33Hz

LIGO-G030187-00-D Page 32

Mode Shapes

9.25Hz

LIGO-G030187-00-D Page 33

Mode Shapes

10.9Hz

9.82Hz