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
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
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
# 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
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
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
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
Application: Beam Splitter of Lithography Machine
flexures
beam splitter (mockup)
table
(Acknowledgement: Thanks to Justin Verdirame for making this mockup)
6DOF TMD for 6DOF Beam Splitter
excitation
accelerometer
spring-dashpotconnectionsmass damper
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)
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
Experiment: 2DOF TMD for a free-free beam
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)
Passive Vehicle Suspension
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:
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)
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
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 )