Robust Analysis of a Hybrid System Controlled by a -Synthesis Method Kyu-Sik Park, Post Doctoral Researcher, UIUC, USA Hyung-Jo Jung, Assistant Professor,

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


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.


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


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


Active control devices • Hydraulic actuators (HAs) • An actuator capacity has a capacity of 1000 kN. • The actuator dynamics are neglected.


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


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


• 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


• 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


Block diagram of robust hybrid control system

Bridge Model

Sensor-synthesis methodHA

LRB

MU

Xey

mysy

,LRB ,LRB,r rx x

HAuHAf

LRBf

fgx


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


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


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


Analysis results Control performances

Displacement under El Centro earthquake

(a) STDs (b) RHCS


Deviation of cable tension under El Centro earthquake

(a) STDs (b) RHCS


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-


1000

2000

3000

4000

5000

6000

Earthquake

Max

imum

Dec

k Sh

ear (

kN)


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


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.


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


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.


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.


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.


• 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.


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


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


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.

Robust Analysis of a Hybrid System Controlled by a -Synthesis Method Kyu-Sik Park, Post Doctoral Researcher, UIUC, USA Hyung-Jo Jung, Assistant Professor,

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