* 김동현 : kaist 토목공학과 , 박사후연구원 오주원 : 한남대학교...

** 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

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


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


structure

external load

neural networkneural

network

sensor

• Structural control using neural network• Structural control using neural network

response


• Multilayer Neural Network (MLNN)• Multilayer Neural Network (MLNN)

control forcecontrol force

state ofstructure

(displacement)(velocity)

state ofstructure

(displacement)(velocity)

WijWij

Wij : weightsWij : weights


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


Objective and Scope

- To reduce learning time, we apply CMAC*

neural network for structural control

*Cerebellar Model Articulation Controller


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


input space output

space

x

memory space

W1

W2

W3

Wn-1

Wn

u

Procedure of CMAC

weights

displacementvelocity

control signal


• 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


• 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


CMAC MLNN

memory size Large Small

learning speed Fast Slow

computing mode Local Global

• CMAC vs. MLNN• CMAC vs. MLNN

items


Vibration Control using CMAC

structure

external load

CMACCMAC

learning rule

sensor

response


• 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


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


3. NUMERICAL EXAMPLES3. NUMERICAL EXAMPLES

Model structure


: 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


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)


• 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


mass

pump

• Active Mass Driver (AMD)• Active Mass Driver (AMD)

piston


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


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


integration time: 0.25 ms

sampling time: 5.0 ms

delay time: 0.5 ms

Simulation


Case studiesearthquake simulation

El Centro trainEl Centro controlNorthridge controlKern County controlEl Centro trainEl Centro control Northridge controlKern County control

model

linear

nonlinear


• 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


• 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


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)



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)


Dis

plac

emen

t (m

)


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)


0 5 10 15 20-20.0-10.0

0.010.020.0

Acc

eler

atio

n (

m/s

ec2 )


Time (sec)

• Northridge earthquake (3rd floor) - continued• Northridge earthquake (3rd floor) - continued


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


Vel

ocity

(m/s

ec)

• Kern County earthquake (3rd floor)• Kern County earthquake (3rd floor)


0 5 10 15 20-20.0-10.0

0.010.020.0

Acc

eler

atio

n (

m/s

ec2 )


Time (sec)

• Kern County earthquake (3rd floor) - continued• Kern County earthquake (3rd floor) - continued


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)


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


• 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


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


-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


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)


• 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


• 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


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.


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.


utqgg

tqgg

)(1

)(2121

21, gg

u

q

: oil flow rate: control signal: time constant: valve gains

• Pump dynamics• Pump dynamics

(9)


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)


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


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


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


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)


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


(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)

* 김동현 : kaist 토목공학과 , 박사후연구원 오주원 : 한남대학교...

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