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Smart Structures Technology Lab., UIUC 3 Introduction Hybrid control system (HCS) A combination of passive and active control devices Passive devices: insure the control system robustness Active devices: improve the control performances The overall system robustness may be negatively impacted by active device or active controller may cause instability due to small margins.
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Robust Analysis of a Hybrid System Controlled by a -Synthesis Method
Kyu-Sik Park, Post Doctoral Researcher, UIUC, USAHyung-Jo Jung, Assistant Professor, Sejong Univ., KoreaWoo-Hyun Yoon, Professor, Kyungwon Univ., KoreaIn-Won Lee, Professor, KAIST, Korea
The Third International Workshop on Advanced Smart Materials and Smart Structures Technology
Smart Structures Technology Lab., UIUC 2
IntroductionRobust hybrid control systemNumerical examplesConclusions
Contents
Smart Structures Technology Lab., UIUC 3
Introduction Hybrid control system (HCS)
A combination of passive and active control devices
• Passive devices: insure the control system robustness • Active devices: improve the control performances
The overall system robustness may be negatively impacted by active device or active controller may cause instability due to small margins.
Smart Structures Technology Lab., UIUC 4
Objective
Apply a hybrid control system for vibration control of a seismically excited cable-stayed bridge
Apply a -synthesis method to improve the controller robustness
Smart Structures Technology Lab., UIUC 5
Robust Hybrid Control System (RHCS) Control devices
Passive control devices • Lead rubber bearings (LRBs) • Design procedure: Ali and Abdel-Ghaffar (1995) • Bouc-Wen model
LRBF
rxyD
yF
ekpk
LRBF
rxyD
yF
ekpk
1
( , ) (1 )
1LRB r r e r e y
n ni r r r
y
F x x k x k D y
y A x x y y x yD
Smart Structures Technology Lab., UIUC 6
Active control devices • Hydraulic actuators (HAs) • An actuator capacity has a capacity of 1000 kN. • The actuator dynamics are neglected.
Smart Structures Technology Lab., UIUC 7
Control algorithm: -synthesis method
where : structured singular value: transfer function of closed-loop system : perturbation
Cost function
(1)
Advantages
• Combine uncertainty in the design procedure • Guarantee the stability and performance (robust performance)
supd dy wJ j
N
d dy wNΔ
Smart Structures Technology Lab., UIUC 8
Frequency dependent filters
• Kanai-Tajimi filter
(2)
10-2
100
102
104
106
10-10
10-8
10-6
10-4
10-2
100
102
Frequency (rad/sec)
Mag
initu
de
El Centro Mexico CityGebze K-T filter
20
2 2
2
2g g g
gg g g
S sW
s s
Smart Structures Technology Lab., UIUC 9
• High-pass and low-pass filters
(3), (4)
10-2
100
102
104
106
10-1
100
Frequency (rad/sec)
Mag
nitu
de
WzWu
10.2 1
601
1240
,u
sW
s
11
601
130
z
sW
s
10.2 1
601
1240
,u
sW
s
11
601
130
z
sW
s
Smart Structures Technology Lab., UIUC 10
• Additive uncertainty filter
(5)
10-1
100
101
102
103
104
100
101
102
103
Frequecy (rad/sec)
SV, m
agni
tude
Evaluation ModelDesign Model
10-1
100
101
102
103
104
10-4
10-3
10-2
10-1
100
101
102
Frequency (rad/sec)SV
-diff
, mag
nitu
de
SV-diff-xg2yWxg2y
• Multiplicative uncertainty filter
(6)
2 21 2 2 2
2 21 1 1
2
2gx
c s sW
s s
y
0.01W u u
Smart Structures Technology Lab., UIUC 11
Block diagram of robust hybrid control system
Bridge Model
Sensor-synthesis methodHA
LRB
MU
Xey
mysy
,LRB ,LRB,r rx x
HAuHAf
LRBf
fgx
Smart Structures Technology Lab., UIUC 12
Analysis model
Bridge model • Bill Emerson Memorial Bridge · Benchmark control problem · Located in Cape Girardeau, MO, USA · 16 shock transmission devices (STDs) are employed
between the tower-deck connections.
Numerical Examples
Smart Structures Technology Lab., UIUC 13
Configuration of control devices (LRBs+HAs)
(3+2)
(3+2)
(3+4)
(3+4)
(3+4)
(3+4)
(3+2)
(3+2)
Bent 1 Pier 2 Pier 3 Pier 4
142.7 m 350.6 m 142.7 m
Smart Structures Technology Lab., UIUC 14
0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0T im e (se c )
-3
-2
-1
0
1
2
3
4
Acc
eler
atio
n (m
/s2 )
E l Centro
PGA: 0.348gPGA: 0.348g
0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0T im e (se c )
-2
-1
0
1
2
Acc
eler
atio
n (m
/s2 )
M exico C ity
PGA: 0.143gPGA: 0.143g
0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0T im e (se c )
-2
-1
0
1
2
3
Acc
eler
atio
n (m
/s2 )
G ebze
PGA: 0.265gPGA: 0.265g
0 1 2 3 4 5 6 7 8 9 1 0F re q u en c y (H z )
0
1
2
3
4
5
6
7
8
Pow
er S
pect
ral D
ensi
ty
0 1 2 3 4 5 6 7 8 9 1 0F re q u e n c y (H z )
0
1
2
3
4
5
6
Pow
er S
pect
ral D
ensi
ty
0 1 2 3 4 5 6 7 8 9 1 0F re q u e n c y (H z )
0
1
2
3
4
5
6
7
8
9
Pow
er S
pect
ral D
ensi
ty
Historical earthquake excitations
Smart Structures Technology Lab., UIUC 15
Analysis results Control performances
Displacement under El Centro earthquake
(a) STDs (b) RHCS
Smart Structures Technology Lab., UIUC 16
Deviation of cable tension under El Centro earthquake
(a) STDs (b) RHCS
Smart Structures Technology Lab., UIUC 17
Shear force of tower under El Centro earthquake
(a) STDs (b) RHCS
El Centro Mexico City Gebze0
5
10
15
20
25
30
Earthquake
Max
imum
Dec
k D
ispl
acem
ent (
cm) STDs
LRBsHAsLRBs+HAs-LQGLRBs+HAs-
El Centro Mexico City Gebze0
1000
2000
3000
4000
5000
6000
Earthquake
Max
imum
Dec
k Sh
ear (
kN)
El Centro Mexico City Gebze0
2
4
6
8
10
12x 10
5
Earthquake
Max
imum
Bas
e M
omen
t (kN
-m)
0 20 40 60 80 100 1200
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Cable Number
Cab
le T
ensi
on (k
N)
Acceptable RegionSTDsLRBs+HAs-
• Important responses of bridge
El Centro
Deck Dis. Deck Shear
Base Mom. Deviation of Cable Tension
Smart Structures Technology Lab., UIUC 19
Controller robustness
• The dynamic characteristic of as-built bridge is not identical to the numerical model.
• There are large differences at high frequencies between evaluation and design models.
• There is a time delay of actuator introduced by the controller dynamics and A/D input and D/A output conversions.
Robust analysis should be performed to verify the applicability of the control system.
Smart Structures Technology Lab., UIUC 20
where : nominal stiffness matrix: perturbed stiffness matrix: perturbation amount (5, 10, 15, 20%)
• Stiffness matrix perturbation
• Mass matrix perturbation
· Additional snow loads (97.7 kg/m2, UBC) are added to the deck.
where : time delay: time delay amount: sampling time (0.02 sec)
• Time delay of actuator
(7)
(8)
pert (1 ) K K
pertKK
sT
sT
Smart Structures Technology Lab., UIUC 21
1 2 3 4 5 6 7 8 9 10 110
10
20
30
40
50
60
70
80
90
Evaluation Criteria
Varia
tion
(%)
5%10%15%20%
1 2 3 4 5 6 7 8 9 10 110
20
40
60
80
100
120
Evaluation Criteria
Varia
tion
(%)
5% w/ Snow Load10% w/ Snow Load15% w/ Snow Load20% w/ Snow Load
Limit of Controller Robustness
J1/J7 : Peak/Normed base shear; J2/J8 : Peak/Normed shear at deck levelJ3/J9 : Peak/Normed overturning moment; J4/J10 : Peak/Normed moment at deck level
J5/J11 : Peak/Normed cable tension deviation; J6: Peak Deck dis. at abutment`
• Stiffness perturbation
w/o snow load w/ snow load · Cable tension deviation (J5, J11) is more sensitive.
· Deck dis. (J6) is relatively insensitive.
· Normed cable tension deviation (J11) is varied more than 100%
when there are -20% stiffness perturbation with snow load under Gebze earthquake.
Smart Structures Technology Lab., UIUC 22
0 20 40 60 80 100 1200
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Cable Number
Cab
le T
ensi
on (k
N)
Acceptable ResignLRBs+HAs-
0 10 20 30 40 50 60 70 80 90 1002300
2400
2500
2600
2700
2800
2900
Time (sec)
Cab
le T
ensi
on (k
N)
Nominal System-20% Stiffness Perturbation w/ Snow Loads
Deviation of cable tensions Deviation of tension of cable no. 10
· However, the cable tension is well within the bounds and dies out after seismic event.
Smart Structures Technology Lab., UIUC 23
1 2 3 4 5 6 7 8 9 10 110
5
10
15
20
25
30
35
Evaluation Criteria
Varia
tion
(%)
0.004 sec0.008 sec0.012 sec0.016 sec0.020 sec
1 2 3 4 5 6 7 8 9 10 110
5
10
15
20
25
30
35
Evaluation Criteria
Varia
tion
(%)
0.004 sec w/ snow0.008 sec w/ snow0.012 sec w/ snow0.016 sec w/ snow0.020 sec w/ snow
• Time delay
w/o snow load w/ snow loadJ1/J7 : Peak/Normed base shear; J2/J8 : Peak/Normed shear at deck level
J3/J9 : Peak/Normed overturning moment; J4/J10 : Peak/Normed moment at deck levelJ5/J11 : Peak/Normed cable tension deviation; J6: Peak Deck dis. at abutment`
· Deck shear and dis. (J2,J6,J8) is more sensitive.
· Variation is much less than compare to stiffness perturbation.
Smart Structures Technology Lab., UIUC 24
• Stiffness perturbation and time delay
w/o snow load w/ snow load
· The effect of stiffness perturbation is larger than time delay. · As the stiffness perturbation increases, the effect of time delay
decreases.
Smart Structures Technology Lab., UIUC 25
0 20 40 60 80 100 120-2000
0
2000
4000
6000
8000
10000
Cable Number
Cab
le T
ensi
on (k
N)
Acceptable RegionSTDsLRBs+HAs-
Limit of unseating of cable
0 20 40 60 80 100 1200
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Cable NumberC
able
Ten
sion
(kN
)
• Additional earthquakes
· Chi-Chi (1999) – PGA = 0.42g > design PGA· Hachnohe (1968) – PGA = 0.23g < design PGA
Chi-Chi Hachinohe
Smart Structures Technology Lab., UIUC 26
Hybrid system controlled by a -synthesis method
shows similar performance with conventional one has excellent robustness without loss of control performances
could be proposed as an improved control strategy for a seismically excited cable-stayed bridges containing many uncertainties
Conclusions
Smart Structures Technology Lab., UIUC 27
Thank you for your attention!
Acknowledgements
This research was supported by the Korea Research Foundation (Grant no. KRF-2005-214-D00169) and author also acknowledges NSF for partial travel support.