Download - 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
3 3 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
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= = ,,
4 4 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
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
--
--
--
--
5 5 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
• 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--
6 6 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
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
7 7 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
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
8 8 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
• 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)
9 9 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
• 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)
10 10 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
• 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
11 11 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
)]()(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
12 12 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
-- 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
13 13 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
• 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
14 14 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
Numerical ExamplesNumerical Examples Six-Story Building (Jansen and Dyke 2000)
ControlComputer
gx gx
LVDT
LVDTv1v1
v2v2MR Damper
15 15 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
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
16 16 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
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
17 17 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
• 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
0
0
mv
mdC q
0
0(14)
- : for modal displacement: for modal displacement- - : for modal velocity: for modal velocity
mdqmdq
mvqmvq
18 18 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
- 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
19 19 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
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
20 20 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
• 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
600,500. mvmd qqatMin 600,500. mvmd qqatMin
qqmdmd qqmvmv
JJT T =J=J11+J+J22+J+J33
AAJ1J1 AAJ2J2
AAJ3J3 AAJTJT
21 21 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
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
JJT T =J=J11+J+J22+J+J33
• Weighting matrix design for the Weighting matrix design for the displacement feedbackdisplacement feedback
DDJ1J1 DDJ2J2
DDJ3J3 DDJTJT
22 22 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
4900,100. mvmd qqatMin 4900,100. mvmd qqatMin 4900,100. mvmd qqatMin 4900,100. mvmd qqatMin
4900,200. mvmd qqatMin 4900,200. mvmd qqatMin
qqmdmd qqmvmv
JJ11
qqmdmd qqmvmv
JJ22
qqmdmd qqmvmv
JJ33
4400,3300. mvmd qqatMin 4400,3300. mvmd qqatMin
qqmdmd qqmvmv
JJT T =J=J11+J+J22+J+J33
• Weighting matrix design for the Weighting matrix design for the velocity feedbackvelocity feedback
VVJ1J1 VVJ2J2
VVJ3J3 VVJTJT
23 23 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
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)
24 24 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
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
25 25 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
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
26 26 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
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
27 27 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
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)
28 28 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
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
29 29 Structural Dynamics & Vibration Control Lab., KAIST, KoreaStructural Dynamics & Vibration Control Lab., KAIST, Korea
Thank you for your attention.Thank you for your attention.Thank you for your attention.Thank you for your attention.