decentralized control applied to multi-dof tuned-mass damper design decentralized h2 control...

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Decentralized Control Applied Decentralized Control Applied to Multi-DOF Tuned-Mass to Multi-DOF Tuned-Mass Damper Design Damper Design • Decentralized H2 Control Decentralized H Control Decentralized Pole Shifting Decentralized H2 with Regional Pole Placement Lei Zuo Lei Zuo and and Samir Nayfeh Samir Nayfeh

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Page 1: Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized

Decentralized Control Decentralized Control Applied to Multi-DOF Applied to Multi-DOF Tuned-Mass Damper Tuned-Mass Damper

DesignDesign

• Decentralized H2 Control

• Decentralized H Control

• Decentralized Pole Shifting

• Decentralized H2 with Regional Pole Placement

Lei ZuoLei Zuo and and Samir NayfehSamir Nayfeh

Page 2: Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized

Control View of SDOF TMD

Spring: feedback relative displacement with gain k2

Damper: feedback relative velocity with gain c2

k2 c2 u

2

1

2

1

:State

x

x

x

x

x

u = k2(x2 - x1)+c2( )12 xx

SDOF TMD MDOF TMD: ---- To make use of other degree of freedoms ---- Better vibration suppression ---- To damp multiple modes with one mass damper

Page 3: Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized

Formulation for MDOF TMD Systems

The mass-spring-damper systems can be cast as a Decentralized Static Output Feedback problem

yFyck

ck

u

uDwDxCy

uDwDxCz

uBwBAxx

d

......22

11

22212

12111

21Cost Output

Measurement

Disturbance

00

Page 4: Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized

# Performance Disturbance Approach

1 Decentralized H2/LQ r.m.s. response

(impulse energy)

white noise gradient-based

2 Decentralized H peak in frequency domain

worst-case sinusoid

LMI iteration/ gradient-based

3 Pole shifting modal damping unknown -subgradient

4 Decentralized H2 + pole constraint

r.m.s. response

+transient char.

partially-known white noise

Methods of multipliers

decentralized control for different disturbances and performance requirements

Page 5: Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized

k1 k2c1 c22d

0 0.5 1 1.5 22

2.02

2.04

2.06

2.08

2.1

2.12Minimal ||H||2 of x0xs versus /d

M

inim

al |

|H|| 2

Radius of gyration / location (/d)

mass ratio md /ms=5%

2DOF TMD for Single Mode Vibration

3

/d=1: two separate SDOF TMDs /d=: traditional SDOF TMD

/d=1/ : 2DOF TMDs (uniform) /d=0.780: 2DOF TMDs (optimal)

k1 c2k1c2

Page 6: Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

1

2

3

4

5

6

7

Normalized frequency ( / s)

Ma

gn

itu

de

x

s( j

) / x 0(

j)

2DOF TMD: Decentralized H /d=

Normalized Frequency

Mag

nitu

de x

s /x

0

/d=1

/d=1/sqrt(3)

/d=0.751

k1 k2c1 c22d

mass ratio md /ms=5%

2DOF TMD can be better than the traditional SDOF TMD and two separate TMDs

Page 7: Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized

2DOF TMD: Negative Damping

Frequency (rad/sec)

Phas

e (

deg

)M

agn

itu

de

(dB

)

30

20

10

0

10

20

30

100

270

180

90

0

peak 6.071

mass ratio md /ms=5%, /d=0.2

Much better performance if one of the damper can be negative. A new reaction mass actuator

Page 8: Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized

Application: Beam Splitter of Lithography Machine

flexures

beam splitter (mockup)

table

(Acknowledgement: Thanks to Justin Verdirame for making this mockup)

Page 9: Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized

6DOF TMD for 6DOF Beam Splitter

excitation

accelerometer

spring-dashpotconnectionsmass damper

Page 10: Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized

50 100 150 200 250 300 350 400 450-60

-40

-20

0

20

40Frequency Response

Ma

gn

itu

de

(d

B) Original System

With One 6MDOF TMD

50 100 150 200 250 300 350 400 450

-270

-180

-90

0

Frequency (Hz)

Ph

ase

(d

eg

)

Measured T.F. of 6DOF TMD

SIX modes are damped well just by using ONE secondary mass

Phase

(deg)

Magnit

ude(d

B)

Page 11: Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized

Decentralized Pole Shifting

2DOF TMD for a free-free beam, 72.7" long

Objective: To maximize the minimal damping of some modesMethod: nonsmooth, Minimax (subgradient + eigenvalue sensitivity)

cup

plunger blade adjustable screw

Page 12: Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized

Experiment: 2DOF TMD for a free-free beam

Page 13: Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized

Vibration Vibration Isolation/SuspensionIsolation/Suspension

• Passive Vehicle Suspension: Decentralized H2 optimization

• 6DOF Active Isolation: Modal Control (collaborated with MIT/Catech LIGO)

• Dynamic Sliding Control for Active Isolation (with Prof Slotine)

Page 14: Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized

Passive Vehicle Suspension

Page 15: Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized
Page 16: Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized

Sliding Control for Frequency-Domain Performance

• Conventional Sliding Surface

• Frequency-Shaped Sliding Surface

ir

rrrdiiiiiiiii xxtfuxxxxx )()()()(2 002

0

iiii xxx )( 0

sx

x ii

0

0

iiiii xxxsL ))(( 0

0102

01

00

0

01

)(0)(

bsbas

bsb

x

xxxx

as

bsb

i

iiiii

We can design Li(s) to meet the frequency performance requirement

Control force Disturbance force

Coupling due to non-proportional damping In mode space:

Page 17: Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized

10

-210

-110

010

110

-4

10-3

10-2

10-1

100

101

Frequency (Hz)

Mag

nitu

de (d

B)

Physical Interpretation of the Frequency-Shaped Sliding Surface

002

0

0 bsas

b

x

x

i

i

Mag

nitu

de (

dB)

a0=2(0.12)0.7b0=(0.12)2

For another case

002

00

0 bsas

bsa

x

x

i

i

sky

Take b1=0, on the sliding surface

Skyhook !Frequency (Hz)

Page 18: Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized

Case Study: 2DOF IsolationM1=500 kg, I1=250 kg m2,

l1=1.0m, l2=1.4 m,

1=5.42 Hz, 1=1.01%

2=9.56 Hz, 2=1.41%

l1 l2

Mag

nitu

de (

dB)

Frequency (rad/sec)

Ma

gn

itu

de

(d

B)

101

100

101

102

103

120

100

80

60

40

20

0

20

40

target

x1/x0

x2/x0

Page 19: Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized

Simulation Results

0 2 4 6 8 10 12 14 16 18 20-2

0

2

4

6x 10

-4

x 1 outpu

t (m)

0 2 4 6 8 10 12 14 16 18 20-2

0

2

4

6x 10

-4

x 2 outpu

t (m)

0 2 4 6 8 10 12 14 16 18 20-2

0

2

4

6x 10

-4

time (sec)

ideal

outpu

t (m)

0 2 4 6 8 10 12 14 16 18 20-0.015

-0.01

-0.005

0

0.005

0.01

0.015

time (sec)

x 1 outpu

t (m)

Ground x0=0.01sin(1.232t) meter

X1 (

m)

X2 (

m)

Idea

l Out

put (

m)

X1 (

m)

6.610-5 m

Ideal output of “skyhook” system

red--without controlblue--with control

( 1.23Hz: one natural freq of the 2nd stage )