sang-won cho* : ph.d. candidate, kaist hyung-jo jung : research assistant professor, kaist

Sang-Won Cho* : Ph.D. Candidate, KAISTSang-Won Cho* : Ph.D. Candidate, KAIST Hyung-Jo Hyung-Jo Jung : Research Assistant Professor, KAISTJung : Research Assistant Professor, KAIST Ju-Won Oh : Professor, Hannam UniversityJu-Won Oh : Professor, Hannam University In-Won Lee : Professor, KAIST In-Won Lee : Professor, KAIST

Implementation of Modal Control for Seismically Implementation of Modal Control for Seismically

Excited Structures using MR Damper Excited Structures using MR Damper

KSCE Conference, KSCE Conference, Busan, KoreaBusan, KoreaNov. 8-9, 2002Nov. 8-9, 2002

2 2 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea

CONTENTSCONTENTS

IntroductionIntroduction

Implementation of Modal ControlImplementation of Modal Control

Numerical ExamplesNumerical Examples

ConclusionsConclusions

Further StudyFurther Study


BackgroundsBackgrounds

Introduction Introduction

• Semi-active control device has Semi-active control device has

reliability of passivereliability of passive and and adaptability of activeadaptability of active system. system.

• MR dampers are quite promising semi-active device forMR dampers are quite promising semi-active device for

small power requirement, reliability, and inexpensive to small power requirement, reliability, and inexpensive to manufacture. manufacture.

• It is It is not possible not possible toto directly control directly control the MR damper. the MR damper.

Control Force ofControl Force of MR Damper MR Damper

F Input Input

voltagevoltageStructuralStructuralResponseResponse= = ,,


Previous StudiesPrevious Studies• Karnopp et al. (1974) Karnopp et al. (1974)

““Skyhook” damper control algorithmSkyhook” damper control algorithm

• Feng and Shinozukah (1990) Feng and Shinozukah (1990)

Bang-Bang controller for a hybrid controller on bridBang-Bang controller for a hybrid controller on bridge ge

• Brogan (1991), Leitmann (1994) Brogan (1991), Leitmann (1994)

Lyapunov stability theory for ER dampersLyapunov stability theory for ER dampers

• McClamroch and Gavin (1995) McClamroch and Gavin (1995)

Decentralized Bang-Bang controllerDecentralized Bang-Bang controller

--

--

--

--


• Inaudi (1997) Inaudi (1997)

Modulated homogeneous friction algorithm for a Modulated homogeneous friction algorithm for a variable friction device variable friction device

• Dyke, Spencer, Sain and Carlson (1996) Dyke, Spencer, Sain and Carlson (1996)

Clipped optimal controller for semi-active devicesClipped optimal controller for semi-active devices

• Jansen and Dyke (2000) Jansen and Dyke (2000) - Formulate previous algorithms for use with MR dampers- Formulate previous algorithms for use with MR dampers

- Compare the performance of each algorithm- Compare the performance of each algorithm

--

--

Difficulties in designing phase of controllerDifficulties in designing phase of controller--

Efficient control design method is requiredEfficient control design method is required--


Objective and ScopeObjective and ScopeImplementation of modal controlImplementation of modal control for seismically for seismically

excited structure using MR dampers and excited structure using MR dampers and

comparison of performancecomparison of performance with previous algorithms with previous algorithms


Modal Control

Modal Control Scheme

• Equations of motion for MDOF systemEquations of motion for MDOF system

• Using modal transformationUsing modal transformation

• Modal equationsModal equations

gxMtftKxtxCtxM )()()()( gxMtftKxtxCtxM )()()()(

n

iiittx

1

)()(

n

iiittx

1

)()(

(1)

(2)

(3)g

Ti

Tiiiiii xfttt )()(2)( 2

gT

iT

iiiiii xfttt )()(2)( 2

),,1( ni ),,1( ni


• DisplacementDisplacement

wherewhere

• State space equationState space equation

wherewhere

• Control forceControl force

• Modal control is desirable for civil engineering structureModal control is desirable for civil engineering structure

gCCCCC xEtFBtwAtw )()()( gCCCCC xEtFBtwAtw )()()(

T

CCT

CC

CC

CC EB

IA

0,

0,

02

T

CCT

CC

CC

CC EB

IA

0,

0,

02

)()()( txtxtx RC )()()( txtxtx RC

.:)( dsipcontrolledtxC .:)( dsipcontrolledtxC )(,1

nlxm

iiiC

)(,1

nlxm

iiiC

.:)( dsipresidualtxR .:)( dsipresidualtxR

(4)

(5)

)()( twKtF CC )()( twKtF CC

- Involve hundred or thousand DOFs- Involve hundred or thousand DOFs- Vibration is dominated by the first few modes- Vibration is dominated by the first few modes

(6)


• Design of is based on optimal control theoryDesign of is based on optimal control theory

• Clipped-optimal algorithm is adopted for MR damperClipped-optimal algorithm is adopted for MR damper

• General cost functionGeneral cost function

• Cost function for modal controlCost function for modal control

Design of Optimal Controller

ft

TT dttRututQxtxJ0

)]()()()([2

1 ft

TT dttRututQxtxJ0

)]()()()([2

1

ft

CT

CCTCc dttFRtFtwQtwJ

0

)]()()()([2

1 ft

CT

CCTCc dttFRtFtwQtwJ

0

)]()()()([2

1

CKCK

(7)

(8)


• Comparing design efficiency of weighting matrixComparing design efficiency of weighting matrix

matrixnnQ : matrixnnQ :

matrixllQC : matrixllQC : (9)

- Weighting matrix is reduced- Weighting matrix is reduced

- Control force is focus on reducing responses of- Control force is focus on reducing responses of

the selected modes the selected modes


)]()(ˆ)([ tfDtwCtyL CCC

• In reality, sensors measure In reality, sensors measure notnot

• Modal state estimator (Kalman filter) forModal state estimator (Kalman filter) for

• and is changeable depending on the feedbackand is changeable depending on the feedback

Modal State Estimation from Various State Feedback

(10)gCCCCC xEtFBtwAtw )()(ˆ)(ˆ gCCCCC xEtFBtwAtw )()(ˆ)(ˆ

)]()(ˆ)([ tfDtwCtyL CCC )]()(ˆ)([ tfDtwCtyL CCC

)(ˆ twc )(ˆ twc

CCCC

- Modal state estimator forModal state estimator for is required is required)(ˆ twC )(ˆ twC

)(twc )(twc)(tx )(tx

CDCD

CCCC CDCD


-- Displacement feedbackDisplacement feedback

- Velocity feedback - Velocity feedback

- Acceleration feedback - Acceleration feedback

- Performance of each feedback is compared - Performance of each feedback is compared

• Various feedback cases for better performance Various feedback cases for better performance

(13)

(11)

(12)

]0[ CCC ]0[ CCC

]0[ CCC ]0[ CCC

C

CC CMKMC

0

0][ 11

C

CC CMKMC

0

0][ 11

0CD 0CD

0CD 0CD

1MDC 1MDC


• Rewrite the state space equationsRewrite the state space equations

• Observation spillover problem byObservation spillover problem by

• Control spillover problem by Control spillover problem by

g

C

R

C

C

R

C

CCR

CRRCR

CCCCC

C

R

C

x

tE

tE

tE

te

tw

tw

LCALC

KBAKB

KBKBA

te

tw

tw

)(

)(

)(

)(

)(

)(

0

0

)(

)(

)(

g

C

R

C

C

R

C

CCR

CRRCR

CCCCC

C

R

C

x

tE

tE

tE

te

tw

tw

LCALC

KBAKB

KBKBA

te

tw

tw

)(

)(

)(

)(

)(

)(

0

0

)(

)(

)(

RLC

RLCRLC

CR KB CR KB

CR KBCR KB

- Produce instability in the residual modes- Produce instability in the residual modes- Terminated by the - Terminated by the low-pass filterlow-pass filter

- Cannot destabilize the closed-loop system- Cannot destabilize the closed-loop system


Numerical ExamplesNumerical Examples Six-Story Building (Jansen and Dyke 2000)

ControlComputer

gx gx

LVDT

LVDTv1v1

v2v2MR Damper


System Data• Mass of each floor Mass of each floor : 0.277 N/(cm/sec2): 0.277 N/(cm/sec2)

• Stiffness Stiffness : 297 N/cm: 297 N/cm

• Damping ratioDamping ratio : each mode of 0.5%: each mode of 0.5%

• MR damperMR damper

- TypeType : Shear mode: Shear mode- Capacity- Capacity : Max. 29N: Max. 29N


Frequency Response Analysis• Under the scaled El Centro earthquakeUnder the scaled El Centro earthquake

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2.0

2.5

3.0

PS

D

frequency, Hz

102

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

x 105

PS

D

frequency, Hz0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

16

x 106

PS

D

frequency, Hz

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

PS

D

frequency, Hz0 1 2 3 4 5 6 7 8 9 10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

PS

D

frequency, Hz

104

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

x 106

PS

D

frequency, Hz

PSD of Displacement PSD of Displacement PSD of Velocity PSD of Velocity PSD of PSD of AccelerationAcceleration

11stst F

loor

Flo

or66thth

Flo

or F

loor


• In frequency analysis, the first mode is dominant.In frequency analysis, the first mode is dominant.

• Reduced weighting matrix (2Reduced weighting matrix (22) is chosen in cost function.2) is chosen in cost function.

-The responses can be reduced by modal control using The responses can be reduced by modal control using the lowest one mode. the lowest one mode.

mv

mdC q

qQ

0

0

mv

mdC q

qQ

0

0(14)

- : for modal displacement: for modal displacement- - : for modal velocity: for modal velocity

mdqmdq

mvqmvq


- Normalized maximum displacementNormalized maximum displacement

- Normalized maximum interstory drift- Normalized maximum interstory drift

- Normalized maximum peak acceleration - Normalized maximum peak acceleration

• Spencer et al 1997Spencer et al 1997

Evaluation Criteria

max,1

|)(|max

x

txJ i

it

max,1

|)(|max

x

txJ i

it

max,2

|/)(|max

n

ii

it d

htdJ

max,2

|/)(|max

n

ii

it d

htdJ

max,

|)(|max

a

ai

it x

txJ

3

max,

|)(|max

a

ai

it x

txJ

3


Weighting Matrix Design• Variations of evaluation criteriaVariations of evaluation criteria

• All 12 weighting matrixes are designed All 12 weighting matrixes are designed

- JJ11

- - JJ22

- - JJ33

- - JJ4 4 = = JJ11 + + JJ22 + + JJ33

with weighting parameters with weighting parameters

,mdq ,mdq mvqmvq

for each feedback casefor each feedback case

Acceleration feedbackAcceleration feedback

Displacement feedbackDisplacement feedback

Velocity feedbackVelocity feedback


• Weighting matrix design for the Weighting matrix design for the acceleration feedbackacceleration feedback

1500,400. mvmd qqatMin 1500,400. mvmd qqatMin 500,1. mvmd qqatMin 500,1. mvmd qqatMin

100,2200. mvmd qqatMin 100,2200. mvmd qqatMin

qqmdmd qqmvmv

JJ11

qqmdmd qqmvmv

JJ22

qqmdmd qqmvmv

JJ33


qqmdmd qqmvmv

JJT T =J=J11+J+J22+J+J33

AAJ1J1 AAJ2J2

AAJ3J3 AAJTJT


800700 mvmd qqatMin ,. 800700 mvmd qqatMin ,. 4001 mvmd qqatMin ,. 4001 mvmd qqatMin ,.

1001300 mvmd qqatMin ,. 1001300 mvmd qqatMin ,.

qqmdmd qqmvmv

JJ11

qqmdmd qqmvmv

JJ22

qqmdmd qqmvmv

JJ33

500600 mvmd qqatMin ,. 500600 mvmd qqatMin ,.

qqmdmd qqmvmv


• Weighting matrix design for the Weighting matrix design for the displacement feedbackdisplacement feedback

DDJ1J1 DDJ2J2

DDJ3J3 DDJTJT


4900,100. mvmd qqatMin 4900,100. mvmd qqatMin 4900,100. mvmd qqatMin 4900,100. mvmd qqatMin


qqmdmd qqmvmv

JJ11

qqmdmd qqmvmv

JJ22

qqmdmd qqmvmv

JJ33


qqmdmd qqmvmv


• Weighting matrix design for the Weighting matrix design for the velocity feedbackvelocity feedback

VVJ1J1 VVJ2J2

VVJ3J3 VVJTJT


Result• Controlled max. responsesControlled max. responses

– Under the scaled Under the scaled El Centro earthquakeEl Centro earthquake,,

– For all For all 12 designed weighting matrixes12 designed weighting matrixes,,

– Compared with Compared with previous 6 algorithmsprevious 6 algorithms

(Jansen and Dyke 2000)(Jansen and Dyke 2000)


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

J20

0.2

0.4

0.6

0.8

1

1.2

1.4

J3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

J1

• Normalized controlled max. responses of the Normalized controlled max. responses of the acceleration acceleration feedbackfeedback

Jansen and Dyke 2000Jansen and Dyke 2000 ProposedProposed


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

J1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

J20

0.2

0.4

0.6

0.8

1

1.2

1.4

J3

• Normalized controlled max. responses of the Normalized controlled max. responses of the displacement displacement

feedbackfeedback


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

J1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

J20

0.2

0.4

0.6

0.8

1

1.2

1.4

J3

• Normalized Controlled Max. Responses of the Normalized Controlled Max. Responses of the velocity feedbackvelocity feedback


ConclusionsConclusions• Modal control scheme is implemented to seismically Modal control scheme is implemented to seismically excited structures using MR dampersexcited structures using MR dampers

• Kalman filter for state estimation and low-pass filterKalman filter for state estimation and low-pass filter for spillover problem is included in modal control schemefor spillover problem is included in modal control scheme

• Weighting matrix in design phase is reducedWeighting matrix in design phase is reduced

• Modal controller achieve reductions resulting in the Modal controller achieve reductions resulting in the lowest value of all cases considered herelowest value of all cases considered here

• Controller AController AJTJT, V, VJT JT fail to achieve any lowest value, howeverfail to achieve any lowest value, however

have competitive performance in all evaluation criteria have competitive performance in all evaluation criteria

• Modal control scheme is implemented to seismically Modal control scheme is implemented to seismically excited structures using MR dampersexcited structures using MR dampers

• Kalman filter for state estimation and low-pass filterKalman filter for state estimation and low-pass filter for spillover problem is included in modal control schemefor spillover problem is included in modal control scheme

• Weighting matrix in design phase is reducedWeighting matrix in design phase is reduced

• Modal controller achieve reductions resulting in the Modal controller achieve reductions resulting in the lowest value of all cases considered herelowest value of all cases considered here

• Controller AController AJTJT, V, VJT JT fail to achieve any lowest value, howeverfail to achieve any lowest value, however

have competitive performance in all evaluation criteria have competitive performance in all evaluation criteria

- Controller AController AJ1J1 : 39% (in J1): 39% (in J1)

- Controller A- Controller AJ2J2 : 30% (in J2): 30% (in J2)

- Controller V- Controller VJ3J3 : 30% (in J3): 30% (in J3)


Future WorkFuture WorkFuture WorkFuture Work

• Examine the influence of the number of controlledExamine the influence of the number of controlled modemode

• Further improvement of design efficiency andFurther improvement of design efficiency and performance of modal control scheme performance of modal control scheme

• Examine the influence of the number of controlledExamine the influence of the number of controlled modemode

• Further improvement of design efficiency andFurther improvement of design efficiency and performance of modal control scheme performance of modal control scheme


Thank you for your attention.Thank you for your attention.Thank you for your attention.Thank you for your attention.

sang-won cho* : ph.d. candidate, kaist hyung-jo jung : research assistant professor, kaist

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