Transcript Lab Tutor

Decentralized Control
Applied to Multi-DOF
Tuned-Mass Damper Design
Lei Zuo and Samir Nayfeh
• Decentralized H2 Control
• Decentralized H Control
• Decentralized Pole Shifting
• Decentralized H2 with Regional Pole Placement
Control View of SDOF TMD
k2
c2
u
Spring: feedback relative
displacement with gain k2
Damper: feedback relative
velocity with gain c2
u = k2(x2 - x1)+c2(x2  x1 )
 x1 
x 
St at e: x   2 
 x1 
 
 x 2 
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
Disturbance
x  Ax  B1w  B2u
Cost Output
Measurement
z  C1 x  D11 w  D12 u
y  C2 x  D21 w  D22 u
 k1

u  

c1
0
k2
c2

 y  F y
d

... ...
0
The mass-spring-damper systems can be cast as a
Decentralized Static Output Feedback problem
decentralized control for different disturbances and performance requirements
#
Performance
Disturbance
Approach
1 Decentralized H2/LQ
r.m.s. response
(impulse energy)
white noise
gradient-based
2 Decentralized H
peak in frequency worst-case
domain
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 Methods of
white noise
multipliers
2DOF TMD for Single Mode Vibration
2.12
c1 2d
k2
mass ratio md /ms=5%
c2
2.08
2.06


2.04

2.02
k1
c2
k1
c2

2.1
Minimal ||H||2
k1
Minimal ||H||2 of x0xs versus /d
2
0
0.5
1
1.5
Radius of gyration / location (/d)
 /d=: traditional SDOF TMD
 /d=1: two separate SDOF TMDs
 /d=1/ 3: 2DOF TMDs (uniform)
 /d=0.780: 2DOF TMDs (optimal)
2
2DOF TMD: Decentralized H
7
 /d= 
mass ratio md /ms=5%
 /d=1
 /d=1/sqrt(3)
Magnitude xs /x0
Magnitude xs( j) / x0( j)
6
5
 /d=0.751
4
3
k1
c1
2
k2
2d
c2
1
0
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
Normalized
Frequency
Normalized
frequency
( /  s )
2DOF TMD can be better than the traditional SDOF TMD and two separate TMDs
2DOF TMD: Negative Damping
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
beam splitter
(mockup)
flexures
table
(Acknowledgement: Thanks to Justin Verdirame for making this mockup)
6DOF TMD for 6DOF Beam Splitter
mass damper
excitation
accelerometer
spring-dashpot
connections
Measured T.F. of 6DOF TMD
Frequency Response
0
-20
-40
-60
50
100
150
200
100
150
200
250
300
350
400
450
250
300
Frequency (Hz)
350
400
450
0
Phase (deg)
Phase (deg)
Original System
With One 6MDOF TMD
20
Magnitude (dB)
Magnitude(dB)
40
-90
-180
-270
50
SIX modes are damped well just by using ONE secondary mass
Decentralized Pole Shifting
2DOF TMD for a free-free beam, 72.7" long
Objective: To maximize the minimal damping of some modes
Method: nonsmooth, Minimax (subgradient + eigenvalue sensitivity)
plunger
cup
blade
adjustable
screw
Experiment: 2DOF TMD for a free-free beam
Vibration
Isolation/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
In mode space:
Coupling due to non-proportional damping
xi  2ii ( xi  x0i )  i2 ( xi  x0i )  ui  f d (t )    r ( xr  x0r )
Control force
r i
Disturbance force
• Conventional Sliding Surface  i   ( xi  x0i )  xi
xi

i  0  
x0 s  
• Frequency-Shaped Sliding Surface  i  Li (s)(xi  x0i )  xi
b1s  b0
xi
b1s  b0

i 
( xi  x0i )  xi  0 
 2
s  a0
x0i s  (a0  b1 ) s  b0
We can design Li(s) to meet the frequency performance requirement
Physical Interpretation
of the Frequency-Shaped Sliding Surface
1
10
a0=2(0.12)0.7
b0=(0.12)2
Take b1=0, on the sliding surface
xi
b0
 2
x0i s  a0 s  b0
xi
a0 s  b0
 2
x0i s  a0 s  b0
Skyhook !
(dB)
Magnitude
Magnitude (dB)
For another case
0
10
-1
10
sky
-2
10
-3
10
-4
10 -2
10
-1
10
0
10
Frequency
Frequency (Hz) (Hz)
1
10
Case Study: 2DOF Isolation
M1=500 kg, I1=250 kg m2,
l1=1.0m, l2=1.4 m,
Magnitude (dB)
l1
l2
1=5.42 Hz, 1=1.01%
2=9.56 Hz, 2=1.41%
x1/x0
x2/x0
target
Simulation Results
Ground x0=0.01sin(1.232t) meter
-4
6
x 10
output (m)
( 1.23Hz: one natural freq of the 2nd stage )
X1 (m)
4
1
2
x
0.015
6.610-5 m
red--without control
blue--with control
0
0.01
-2
output (m)
6
2
4
6
8
10
12
14
16
18
20
0.005
X1 (m)
output (m)
X2 (m)
4
2
0
0
x
1
2
x
0
-4
x 10
-2
0
2
4
6
8
10
12
14
16
18
20
-4
ideal output (m)
Ideal Output (m)
6
x 10
4
-0.01
Ideal output of “skyhook” system
2
0
-2
-0.005
-0.015
0
2
4
6
8
10
time (sec)
12
14
16
18
20
0
2
4
6
8
10
time (sec)
12
14
16
18
20