1 artificial neural networks for structural vibration control ju-tae kim: graduate student, kaist,...
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1
Artificial Neural Networks for
Structural Vibration Control
Ju-Tae Kim: Graduate Student, KAIST, Korea
Ju-Won Oh: Professor, Hannam University, Korea
In-Won Lee: Professor, KAIST, Korea
Aug. 23, 1999.
2
CONTENTS
1. Introduction
2. Neural Networks for Control
3. Numerical Examples
4. Conclusions
3
1. Introduction
required
impossible/hard
impossible/hard
Response based ANN control
Model based conventional control
Mathematicalmodel
Parametric uncertainty
Parametric variation
not required
simple/easy
simple/easy
Conventional Control vs. ANN Control
4
Previous Works on ANN Control in CE
H. M. Chen et al. (1995), J. Ghaboussi et al. (1995) - pioneering research in civil engineering
K. Nikzad (1996) - delay compensation
K. Bani-Hani et al. (1998) - nonlinear structural control
Condition : desired response is to be pre-determined.
5
• Training rule of controller neural network
• SDOF linear/nonlinear structural control
Scope
6
• Emulator neural network
- trained to imitate responses of unknown structures.
- used for training of controller neural network.
• Controller neural network - trained to make control force.
- used for controller.
2. Neural Networks for Control
Two Neural Networks
7
Controller(ANN)
Minimize error(E)
Emulator(ANN)
Structure
Load
Z-1
+
_
D (desired response)
E=D-X
Previous Studies
Weights of controller neural network(W) are updated
to minimize error function(E).
U X
8
Controller(ANN)
Minimize cost(J)
Emulator(ANN)
Structure
Load
Z-1
Proposed Method
Weights of controller neural network(W) are updated
to minimize cost function(J) instead of error function(E).
UX
9
n
nTnnTn
n
n
tRUUQXX
JJ
2
1
ˆˆ
(1)
RQ,
nn UX , : response, control force vector
: weighting matrices
• Cost function
where
10
• Controller neural network
)( jj netfh
L
iijij IWnet
1
)( kk netgu
L
ijkjk hWnet
1
hidden layer
Output layer
(2)
(3)
(4)
(5)
Ii uk
Wji Wkj
i=1~Lj=1~M
k=1~N
11
kj
nTn
n
nTn
kj
n
n
n
kj
n
n
n
n
n
kj
nn
kj
W
URU
U
XQXt
W
U
U
J
W
U
U
X
X
J
W
JW
ˆˆ
ˆ
• Learning rule: weights of output-hidden layer
11 nkj
nkj
nkj
nkj WWWW (6)
(7)
12
jknkkn
k
nTn
kj
nkn
kknk
nTnn
kj
hnetguru
XQXt
W
uur
u
XQXtW
)(
(8)
nj
nk
nkj hW
)( knkkn
k
nTnn
k netguru
XQXt
(9)
(10)where
13
(11)
(12)
• Learning rule: weights of hidden-input layer
11 nji
nji
nji
nji WWWW
ji
nTn
n
nTn
ji
n
n
n
ji
n
n
n
n
n
ji
nnji
W
URU
U
XQXt
W
U
U
J
W
U
U
X
X
J
W
JW
ˆˆ
ˆ
14
ni
nj
nji IW
)()(
)()(
)()( 11
jn
NjN
jn
kjk
jnj
Tnn
nTnn
j
netfWnetg
netfWnetg
netfWnetg
RUU
XQXt
(13)
(14)
where
15
k
c
m
uxmkxxcxm g
u
x
x
g
3. Numerical Examples
Control of Linear Structure
• Equation of motion
: mass
: damping
: stiffness
: displacement
: ground acceleration
: control force
(15)
16
TxxX
gxFBuAXX
1
0,
/1
0,
10F
mB
mcmkA
gxumx
x
mcmkx
x
1
0
/1
010
• State-space form
Let , then
(16)
(17)
17
017.00
01Q
41076.5 R
kgm 1
mNc sec25.1
mNk 39
• Parameters
• Controller neural network
)1( nx
)1( nx
)1( nxg
)(nu
18
(a) El Centro earthquake(1940) (b) California earthquake(1952)
(c) Northridge earthquake(1994)
• Ground accelerations( )
TRAINED UNTRAINED
UNTRAINED
gx
0 10 20 30 40Time (sec)
-4 .0
-2 .0
0.0
2.0
4.0
Acc
eler
atio
n (
m/s
ec 2 )
0 10 20 30 40Time (sec)
-2 .0
-1 .0
0.0
1.0
2.0
Acc
eler
atio
n (
m/s
ec 2 )
0 10 20 30 40Time (sec)
-4 .0
-2 .0
0.0
2.0
4.0
Acc
eler
atio
n (
m/s
ec 2 )
19
0 10 20 30 40 500.0
0.5
1.0
1.5
2.0
epoch
<
Cos
t fun
ctio
n(J)
• Minimization of cost function
20
(a) El Centro earthquake(trained)
(b) California earthquake(untrained)
• Control results
0 5 10 15 20T im e (sec)
-10.0
-5.0
0.0
5.0
10.0
Dis
plac
emen
t (cm
) uncontro lledcontro lled
0 5 10 15 20T im e (sec)
-80
-40
0
40
80
Dis
plac
emen
t (cm
) uncontro lledcontro lled
0 5 10 15 20T im e (sec)
-6 .0
-3.0
0.0
3.0
6.0
Dis
plac
emen
t (cm
) uncontro lledcontro lled
0 5 10 15 20T im e (sec)
-40
-20
0
20
40
Vel
ocity
(cm
/sec
) uncontro lledcontro lled
21
(c) Northridge earthquake(untrained)
0 5 10 15 20T im e (sec)
-10.0
-5.0
0.0
5.0
10.0
Dis
plac
emen
t (cm
) uncontro lledcontro lled
0 5 10 15 20T im e (sec)
-80
-40
0
40
80
Vel
ocity
(cm
/sec
) uncontro lledcontro lled
22
uxmxxfxcxm g ),(
kdwkxxxf )1(),(
)(1 1 pp
wxwwxxad
w
5.0
5.0
6.0
5.0
04.0
0.1
p
d
a
Control of Nonlinear Structure
(18)
(19)
(20)
• Equation of motion
• Parameters
23
-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 60k
39k
24
(a) El Centro earthquake(trained)
(b) California earthquake(untrained)
• Control results-1
0 5 10 15 20T im e (sec)
-10.0
-5.0
0.0
5.0
10.0
Dis
plac
emen
t (cm
) uncontro lledcontro lled
0 5 10 15 20T im e (sec)
-60
-30
0
30
60
Vel
ocity
(cm
/sec
) uncontro lledcontro lled
0 5 10 15 20T im e (sec)
-6 .0
-3.0
0.0
3.0
6.0
Dis
plac
emen
t (cm
) uncontro lledcontro lled
0 5 10 15 20T im e (sec)
-40
-20
0
20
40
Vel
ocity
(cm
/sec
) uncontro lledcontro lled
25
(c) Northridge earthquake(untrained)
0 5 10 15 20T im e (sec)
-10.0
-5.0
0.0
5.0
10.0
Dis
plac
emen
t (cm
) uncontro lledcontro lled
0 5 10 15 20T im e (sec)
-60
-30
0
30
60
Vel
ocity
(cm
/sec
) uncontro lledcontro lled
26
-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)
-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
)
-0.14 -0.07 0.00 0.07 0.14D isplacem ent (m )
-4 .0
-2.0
0.0
2.0
4.0
Res
torin
g fo
rce
(N)
-0.14 -0.07 0.00 0.07 0.14D isplacem ent (m )
-4 .0
-2.0
0.0
2.0
4.0
Res
torin
g fo
rce
(N)
(a) El Centro earthquake (b) California earthquake (c) Northridge earthquake
• Control results-2 c
ontr
olle
d
u
ncon
trol
led
27
4. Conclusions
• Training rule of neural network for optimal
control is proposed.
• Not only linear but nonlinear structure is
controlled successfully.