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** 김동현김동현 : KAIST : KAIST 토목공학과토목공학과 , , 박사후연구원박사후연구원 오주원오주원 : : 한남대학교 토목환경공학과한남대학교 토목환경공학과 , , 교수 교수 이규원이규원 : : 전북대학교 토목환경공학과전북대학교 토목환경공학과 , , 교수교수 이인원이인원 : KAIST : KAIST 토목공학과토목공학과 , , 교수교수
CMAC 신경망을 이용한 구조물의 진동제어
토목학회 토목학회 2000 2000 학술발표회학술발표회
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2 2Structural Dynamics & Vibration Control Lab., KAIST, Korea
1 INTRODUCTION
2 CMAC* FOR VIBRATION CONTROL
3 NUMERICAL EXAMPLES
4 CONCLUSIONS
CONTENTS
*Cerebellar Model Articulation Controller
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3 3Structural Dynamics & Vibration Control Lab., KAIST, Korea
1 INTRODUCTION1 INTRODUCTION
- mathematical model is not required in
designing controller
• Features of neural network control• Features of neural network control Background
• Application areas• Application areas
- control of structures with uncertainty or nonlinearity
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4 4Structural Dynamics & Vibration Control Lab., KAIST, Korea
structure
external load
neural networkneural
network
sensor
• Structural control using neural network• Structural control using neural network
response
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5 5Structural Dynamics & Vibration Control Lab., KAIST, Korea
• Multilayer Neural Network (MLNN)• Multilayer Neural Network (MLNN)
control forcecontrol force
state ofstructure
(displacement)(velocity)
state ofstructure
(displacement)(velocity)
WijWij
Wij : weightsWij : weights
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6 6Structural Dynamics & Vibration Control Lab., KAIST, Korea
1) H. M. Chen et al. (1995). ASCE J. Comp. in Civil Eng.
2) J. Ghaboussi et al. (1995). ASCE J. Eng. Mech.
3) K. Nikzad et al. (1996). ASCE J. Eng. Mech.
4) K. Bani-Hani et al. (1998). ASCE J. Eng. Mech.
5) J. T. Kim et al. (2000). ASCE J. Eng. Mech.
Previous studies
- All methods are based on multilayer neural network, whose learning speed is too slow
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7 7Structural Dynamics & Vibration Control Lab., KAIST, Korea
Objective and Scope
- To reduce learning time, we apply CMAC*
neural network for structural control
*Cerebellar Model Articulation Controller
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8 8Structural Dynamics & Vibration Control Lab., KAIST, Korea
CMAC
2 CMAC FOR VIBRATION CONTROL2 CMAC FOR VIBRATION CONTROL
- proposed by J. S. Albus(1975)
- a neural network with fast learning speed
- mainly used for manipulator control
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9 9Structural Dynamics & Vibration Control Lab., KAIST, Korea
input space output
space
x
memory space
W1
W2
W3
Wn-1
Wn
u
Procedure of CMAC
weights
displacementvelocity
control signal
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10 10Structural Dynamics & Vibration Control Lab., KAIST, Korea
• Output calculation (1)• Output calculation (1)
output W12+W22+W32+W42
x
W11 W12 W13 W14
W21 W22 W23 W24 W31 W32 W33 W34 W41 W42 W43 W44
x1
layer 1
layer 2
layer 3
layer 4
input
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11 11Structural Dynamics & Vibration Control Lab., KAIST, Korea
• Output calculation (2)• Output calculation (2)
output W13+W23+W32+W42
x
W11 W12 W13 W14
W21 W22 W23 W24 W31 W32 W33 W34 W41 W42 W43 W44
x1 x2
layer 1
layer 2
layer 3
layer 4
input
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12 12Structural Dynamics & Vibration Control Lab., KAIST, Korea
CMAC MLNN
memory size Large Small
learning speed Fast Slow
computing mode Local Global
• CMAC vs. MLNN• CMAC vs. MLNN
items
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13 13Structural Dynamics & Vibration Control Lab., KAIST, Korea
Vibration Control using CMAC
structure
external load
CMACCMAC
learning rule
sensor
response
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14 14Structural Dynamics & Vibration Control Lab., KAIST, Korea
• Control criterion: cost function• Control criterion: cost function
1N
0kk
Tk1k1k
f
2
1J RuuQzzT (1)
: state vector: control vector : relative weighting matrix: time step : final time step
: state vector: control vector : relative weighting matrix: time step : final time step
RQuz
,
fN
k
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15 15Structural Dynamics & Vibration Control Lab., KAIST, Korea
kTk1k
T1kk 2
1J RuuQzz
: learning rate: learning rateη
ki,ki,1ki, WWW
(2)
(3)
(5)
Ru
u
zQz T
kk
1kT1kki, ηW
• Learning rule• Learning rule
i
kki, W
JηW
(4)
proposedmethodproposedmethod
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16 16Structural Dynamics & Vibration Control Lab., KAIST, Korea
3. NUMERICAL EXAMPLES3. NUMERICAL EXAMPLES
Model structure
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17 17Structural Dynamics & Vibration Control Lab., KAIST, Korea
: Mass matrix: Damping matrix: Restoring force : Location vector
: displacement vector: ground acceleration: control force
(6) gxLu 1M)xF(x,xCxM
LFCM
uxgx
• Equation of motion• Equation of motion
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18 18Structural Dynamics & Vibration Control Lab., KAIST, Korea
dykxkxf 00 )1()(
)(1 1 pp
yxyyxxd
y
p
k
,,,
0
: linear stiffness
: contribution of k0
: constants
• Nonlinear restoring force (Bouc-Wen, 1981)• Nonlinear restoring force (Bouc-Wen, 1981)
(7)
(8)
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19 19Structural Dynamics & Vibration Control Lab., KAIST, Korea
• Effect of parameters
-0 .10 -0.05 0.00 0.05 0.10D isplacem ent (m )
-4 .0
-2.0
0.0
2.0
4.0R
esto
ring
forc
e (N
)
-0 .10 -0.05 0.00 0.05 0.10D isplacem ent (m )
-4 .0
-2.0
0.0
2.0
4.0
Res
torin
g fo
rce
(N)
-0 .10 -0.05 0.00 0.05 0.10D isplacem ent (m )
-4 .0
-2.0
0.0
2.0
4.0
Res
torin
g fo
rce
(N)
-0 .10 -0.05 0.00 0.05 0.10D isplacem ent (m )
-4 .0
-2.0
0.0
2.0
4.0
Res
torin
g fo
rce
(N)
3.0
6.0 600 k
390 k
395.055.0104.0
0
kp
d
6.05.055.0104.0
p
d
395.055.0104.0
0
kp
d
6.05.055.0104.0
p
d
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20 20Structural Dynamics & Vibration Control Lab., KAIST, Korea
mass
pump
• Active Mass Driver (AMD)• Active Mass Driver (AMD)
piston
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21 21Structural Dynamics & Vibration Control Lab., KAIST, Korea
mass : 200 kg (story)stiffness : 2.25105 N/m (inter-story)damping ratios : 0.6, 0.7, 0.3% (modal)
mass : 18 kg (3% of building total mass)stiffness : 3.71103 N/mdamping ratio : 8.65%
Structure
AMD
• Parameters• Parameters
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22 22Structural Dynamics & Vibration Control Lab., KAIST, Korea
CMAC structure
input: 2 (disp., vel. of 3rd floor)
output: 1 (control signal)
no. of divisions: 3 per variable
no. of layers: 200
no. of weights: 1800
input: 2 (disp., vel. of 3rd floor)
output: 1 (control signal)
no. of divisions: 3 per variable
no. of layers: 200
no. of weights: 1800
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23 23Structural Dynamics & Vibration Control Lab., KAIST, Korea
integration time: 0.25 ms
sampling time: 5.0 ms
delay time: 0.5 ms
Simulation
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24 24Structural Dynamics & Vibration Control Lab., KAIST, Korea
Case studiesearthquake simulation
El Centro trainEl Centro controlNorthridge controlKern County controlEl Centro trainEl Centro control Northridge controlKern County control
model
linear
nonlinear
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25 25Structural Dynamics & Vibration Control Lab., KAIST, Korea
• Linear cases (=1.0)• Linear cases (=1.0)
※1 Epoch = 0.005 s × 2000 steps ※1 Epoch = 0.005 s × 2000 steps
CMAC
MLNN
0 100 200 300 400 500Epoch
0.0
0.1
0.2
0.3
Cos
t fun
ctio
n • training under El Centro earthquake • training under El Centro earthquake
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26 26Structural Dynamics & Vibration Control Lab., KAIST, Korea
• Training results • Training results
Jmin epochJmin epoch
MLNN
CMAC
MLNN
CMAC
1.94 10-2 65 (1.09) (0.15)
1.94 10-2 65 (1.09) (0.15)
1.77 10-2 412 (1.00) (1.00)
1.77 10-2 412 (1.00) (1.00)
neuralnetworkneuralnetwork
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w/o controlw/ control
• El Centro earthquake (3rd floor)• El Centro earthquake (3rd floor)
0 5 10 15 20-0.10-0.050.000.050.10
0 5 10 15 20-1.00-0.500.000.501.00
Dis
plac
emen
t (m
)
Time (sec)
Vel
ocity
(m/s
ec)
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28 28Structural Dynamics & Vibration Control Lab., KAIST, Korea
w/o controlw/ control
0 5 10 15 20-20.0-10.0
0.010.020.0
• El Centro earthquake (3rd floor) - continued• El Centro earthquake (3rd floor) - continuedA
ccel
erat
ion
(m
/sec
2 )
Time (sec)
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29 29Structural Dynamics & Vibration Control Lab., KAIST, Korea
Dis
plac
emen
t (m
)
w/o controlw/ control
0 5 10 15 20-0.10-0.050.000.050.10
Time (sec)
0 5 10 15 20-1.00-0.500.000.501.00
Vel
ocity
(m/s
ec)
• Northridge earthquake (3rd floor)• Northridge earthquake (3rd floor)
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30 30Structural Dynamics & Vibration Control Lab., KAIST, Korea
0 5 10 15 20-20.0-10.0
0.010.020.0
Acc
eler
atio
n (
m/s
ec2 )
w/o controlw/ control
Time (sec)
• Northridge earthquake (3rd floor) - continued• Northridge earthquake (3rd floor) - continued
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31 31Structural Dynamics & Vibration Control Lab., KAIST, Korea
Time (sec)
0 5 10 15 20-0.10-0.050.000.050.10
Dis
plac
emen
t (m
)
0 5 10 15 20-1.00-0.500.000.501.00
w/o controlw/ control
Vel
ocity
(m/s
ec)
• Kern County earthquake (3rd floor)• Kern County earthquake (3rd floor)
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32 32Structural Dynamics & Vibration Control Lab., KAIST, Korea
0 5 10 15 20-20.0-10.0
0.010.020.0
Acc
eler
atio
n (
m/s
ec2 )
w/o controlw/ control
Time (sec)
• Kern County earthquake (3rd floor) - continued• Kern County earthquake (3rd floor) - continued
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33 33Structural Dynamics & Vibration Control Lab., KAIST, Korea
0 100 200 300 400 500Epoch
0.0
0.1
0.2
0.3
Cos
t fun
ctio
n • Learning under El Centro earthquake • Learning under El Centro earthquake
CMAC
MLNN
• Nonlinear cases (=0.5)• Nonlinear cases (=0.5)
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34 34Structural Dynamics & Vibration Control Lab., KAIST, Korea
Jmin epochJmin epoch
MLNN
CMAC
MLNN
CMAC
2.02 10-2 34 (1.06) (0.08)
2.02 10-2 34 (1.06) (0.08)
1.91 10-2 427 (1.00) (1.00)
1.91 10-2 427 (1.00) (1.00)
• Training results • Training results
neuralnetworkneuralnetwork
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35 35Structural Dynamics & Vibration Control Lab., KAIST, Korea
• El Centro earthquake (1st floor)• El Centro earthquake (1st floor)
-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0D isp lacem ent (cm )
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
Res
torin
g fo
rce
(kN
)
-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0D isp lacem ent (cm )
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
Res
torin
g fo
rce
(kN
)
w/o control w/ control
5.05,5.01,01.0
p
d
5.05,5.01,01.0
p
d
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36 36Structural Dynamics & Vibration Control Lab., KAIST, Korea
w/o control
-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0D isp lacem ent (cm )
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
Res
torin
g fo
rce
(kN
)
-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0D isp lacem ent (cm )
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
Res
torin
g fo
rce
(kN
)
w/ control
• Northridge earthquake (1st floor)• Northridge earthquake (1st floor)
5.05,5.01,01.0
p
d
5.05,5.01,01.0
p
d
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37 37Structural Dynamics & Vibration Control Lab., KAIST, Korea
-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0D isp lacem ent (cm )
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
Res
torin
g fo
rce
(kN
)
-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0D isp lacem ent (cm )
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
Res
torin
g fo
rce
(kN
)
• Kern County earthquake (1st floor)• Kern County earthquake (1st floor)
w/o control w/ control
5.05,5.01,01.0
p
d
5.05,5.01,01.0
p
d
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38 38Structural Dynamics & Vibration Control Lab., KAIST, Korea
0 5 10 15 20-0.04
-0.02
0.00
0.02
0.04
0 5 10 15 20-0.04
-0.02
0.00
0.02
0.04
0 5 10 15 20-0.04
-0.02
0.00
0.02
0.04
• Comparison of control results (linear, 3rd floor) • Comparison of control results (linear, 3rd floor)
El Centro El Centro
Northridge Northridge
Kern County Kern County
Dis
plac
emen
t (m
)
MLNNCMAC
Time (sec)
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39 39Structural Dynamics & Vibration Control Lab., KAIST, Korea
• Comparison of control results (nonlinear, 3rd floor) • Comparison of control results (nonlinear, 3rd floor)
El Centro El Centro
Northridge Northridge
Kern County Kern County
Dis
plac
emen
t (m
)
MLNNCMAC
Time (sec)
0 5 10 15 20-0.04
-0.02
0.00
0.02
0.04
0 5 10 15 20-0.04
-0.02
0.00
0.02
0.04
0 5 10 15 20-0.04
-0.02
0.00
0.02
0.04
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40 40Structural Dynamics & Vibration Control Lab., KAIST, Korea
• Maximum responses of 3rd floor (cm)• Maximum responses of 3rd floor (cm)
linear
nonlinear
5.01 2.06 1.65 (3.04) (1.24) (1.00)
6.15 2.14 1.38 (4.46) (1.55) (1.00)
3.42 0.97 0.72 (4.75) (1.35) (1.00)
3.48 2.54 2.34 (1.49) (1.09) (1.00)
3.94 2.20 1.63 (2.42) (1.35) (1.00)
2.68 0.97 0.80 (3.35) (1.21) (1.00)
Earthquake w/o controlw/ control
CMAC MLNN
El Centro
Northridge
Kern County
El Centro
Northridge
Kern County
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41 41Structural Dynamics & Vibration Control Lab., KAIST, Korea
4. CONCLUSIONS4. CONCLUSIONS
• Learning speed of CMAC is much faster
than that of MLNN.
• Response controlled by CMAC is slightly
larger than that by MLNN.
• Learning speed of CMAC is much faster
than that of MLNN.
• Response controlled by CMAC is slightly
larger than that by MLNN.
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42 42Structural Dynamics & Vibration Control Lab., KAIST, Korea
Future workFuture work
• Further reduction of response controlled
by CMAC with fast learning speed.
• Further reduction of response controlled
by CMAC with fast learning speed.
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43 43Structural Dynamics & Vibration Control Lab., KAIST, Korea
utqgg
tqgg
)(1
)(2121
21, gg
u
q
: oil flow rate: control signal: time constant: valve gains
• Pump dynamics• Pump dynamics
(9)
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44 44Structural Dynamics & Vibration Control Lab., KAIST, Korea
qfa
Vf
a
cxa
rr
lrr
2
: displacement of ram
: area of ram
: compression coefficient
: volume of cylinder
: leakage coefficientl
r
r
c
V
a
x
• Piston dynamics• Piston dynamics
(10)
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45 45Structural Dynamics & Vibration Control Lab., KAIST, Korea
BuAzz
B
A
u
z : state vector
: control force vector
: system matrix
: control matrix
: state vector
: control force vector
: system matrix
: control matrix)(
)(
)1(
)1(
mn
nn
m
n
(s-1)
• Sensitivity Evaluation• Sensitivity Evaluation
• State equation• State equation
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46 46Structural Dynamics & Vibration Control Lab., KAIST, Korea
kkk HuGzz 1
sTeAG
(s-2)
(s-3)
(s-4)
sT : sampling time: sampling time
BAH A 1 Ie sT
Hu
z
k
k 1 (s-5)
• Discretized equation using ZOH• Discretized equation using ZOH
• Sensitivity matrix• Sensitivity matrix
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47 47Structural Dynamics & Vibration Control Lab., KAIST, Korea
kkk HuGzz 1
][0z k
mjijif
ijifkj ~1
)(0
)(1,
u
ik hz 1
initial condition:initial condition:
loading condition:loading condition:
measurement: measurement:
(s-6)
(s-7)
(s-8)
(s-9)
• Computation of H• Computation of H
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48 48Structural Dynamics & Vibration Control Lab., KAIST, Korea
Method Time Method Time
Emulator minutes ~ hours Emulator minutes ~ hours
Proposed m sampling time Proposed m sampling time
Evaluation timeEvaluation time
mi hhhhH 21 (s-10)
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49 49Structural Dynamics & Vibration Control Lab., KAIST, Korea
1
1
2
1
1n
i
n
j
eji
ji
ee WW
JJJ
1
0,
fN
k
ekji
eji WW
(c-1)
(c-2)
(c-3)
1
0
fN
kkJJ
1
0
fN
k ji
k
ji W
J
W
J
ji
kekji
W
JW
,
(c-4)
(c-5)
• Convergence of learning rule• Convergence of learning rule
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50 50Structural Dynamics & Vibration Control Lab., KAIST, Korea
(c-6)
(c-7)
(c-8)
1
1
2
1
21
1
1n
i
n
j
N
k ji
keef
W
JJJ
eee JJJ 1
)0(1
1
2
1
21
1
n
i
n
j
N
k ji
kef
W
JJ
minlim JJ ee
(c-9)
Inserting (3), (4) into (2)