State-Switched Control Algorithm for Multi Degree of Freedom Smart Base Isolation System Employing MR Elastomer

International Conference on Earthquake Engineering and Seismology (ICEES 2011), NUST, Islamabad, Pakistan

April 25-26, 2011

State-Switched Control Algorithm for Multi Degree of Freedom Smart Base Isolation System Employing MR Elastomer

Muhammad Usman1*, Sajjad Ahmad 2, Hyung-Jo Jung 3

1* School of Civil and Environmental Engineering, National University of Sciences & Technology, NUST, H-12, Islamabad, Pakistan

( [email protected]) 2 School of Civil and Environmental Engineering, National University of Sciences & Technology, NUST, H-12,

Islamabad, Pakistan

3 Department of Civil and Environmental Engineering, KAIST, Daejeon 305-701, Korea Abstract The vibration control of building structures under seismic loading is a very critical issue in the safety of the structure. Base isolation is quite an effective and practical solution to this issue. The proposed magneto-rheological elastomer(MRE)-based base isolation system eliminates the limitation of passive system by being adaptable to various loading conditions. The proposed MR elastomer based system traditionally uses the Linear Quadratic Regulator (LQR) algorithm which has its own limitations. State-Switched control algorithm was proposed by the author in 2008[10] for the single degree of freedom system. In this work the proposed algorithm has been upgraded for multi degree of freedom systems. This paper describes the application of State-Switched control algorithm to multi DOF system and the comparison of the response of a five degree of freedom system using proposed control algorithm, with that of a system using Linear Quadratic Regulator (LQR) and the uncontrolled case. To assess the behavior of State-Switched control algorithm simulations were performed by using historical data of El Centro, Hachinohi and Northridge earthquakes. Simulations result indicate state-switched control algorithm to be much better in terms of control performance and practicability as compared to the commonly used LQR algorithm. Keywords: Magneto-rheological Elastomers (MRE), Base isolation, Linear Quadratic Regulator (LQR), State-Switched Control

1. Introduction

Base isolation of building structures has been the popular structural control technique to mitigate the undesirable response of the structure under seismic loading. Initially the passive base isolation using some laminated rubber bearings, lead rubber bearings and friction bearing have been in practice to seismically isolate the building structure from the vibrating ground surface. The conventional base isolation systems reduce the response of the structure by lengthening the natural period of the structure. Also it reduces the inter-storey drift by increasing the base drift. But this excessive base drift sometimes causes some problems in the building such as damaging the utility ducts and sometimes it may be problematic to restore the structure to its original position. Active base isolation system reduces some of these problems by limiting the base drift, but it may add to destabilizing forces if the control force accidentally becomes in phase with the ground excitation. Also it requires large power input which may also be a disadvantage in real applications. Semi active base isolation system employing magneto rheological elastomers (MRE) overcome the disadvantages of both the passive and active systems. On one hand it reduces the peak base drift being actively controlled and


April 25-26, 2011

on the other hand it adds to the stability of the structure owing to its passive type mechanism. Additionally it needs very less external power and hence it may continue working even on low power.

Several numerical investigations and studies have been carried out to check out the feasibility of the semi active base isolation system employing MRE’s. Initially the typical structural models are studied numerically with and without the proposed control system. First of all ([6]) single degree of freedom structure was studied with conventional control algorithms. Then this study was further extended ([9]) to multi degree of freedom system again using conventional (LQR) control algorithm. Then in our previous work([10]) we proposed the state-switched control algorithm and studied that for the single degree of freedom structure but at that time the applicability seemed to be a bit limited as in case of single degree of freedom the structural stiffness is much larger than the isolator stiffness and any switching criteria can be used as the base and the structural responses are almost the same. In this work the state-switched control is proposed for the multi degree of freedom structure with semi active base isolation system employing MRE’s.

2. MRE based semi active base isolation

Magneto rheological elastomers are the basically solid state analogues of MR fluids. MRE is basically conventional elastomer, such as rubber etc., with micron sized iron particles embedded. When magnetic field is applied across the MR elastomer the embedded iron particles align themselves along the direction of the magnetic field as shown in Fig.1. This alignment of the particles causes the stiffness of the elastomer to increase and hence the response of the whole structure to be modified.

Fig.1 Schematic representation of Magnetorheological elastomers

When base isolation bearing is used for the building structure it naturally lengthens the natural period of the building and consequently reduces the response of the structure. In case of passive type bearings the base drift is normally more than desirable range but the when semi active base isolation bearings are used significantly reduced base drift is observed. This reduction obviously adds much to the efficiency of the structure. Here there is another important issue worth mentioning that the semi active system quite obviously implies the need of some sensors to measure the response of the structure in real time. Hence there is the need to minimize the number of necessary responses to be measured and sensors to make the system more compact and economical.

3. System model of the structure used for Numerical simulations

For the purpose of numerically investigating the proposed system a typical five storey building structure was considered with the schematics as depicted in the Fig.2.


April 25-26, 2011

Fig.2 Schematic representation of the building structures used

Here M, K and C represent the respective mass, stiffness and damping co-efficient for various levels. The building was considered with the passive base isolation as well as with MRE based semi active base isolation system. The structural parameters of the structure are given in the table 1.

Table 1: Structural parameters of the structural modal considered

Location Mass (kg) Stiffness (kN/m) Damping (kNs/m) Base 6800 232 3.74

1st Floor 5897 33732 67 2nd Floor 5897 29093 58 3rd Floor 5897 28621 57 4th Floor 5897 24954 50 5th Floor 5897 19059 38

4. Equations of Motion

Assuming the structural motion is sufficiently small such that nonlinear effects may be neglected, and denoting the base and structure displacements relative to the ground by [ ]Tsb xxx = , the equations of motion of the base-isolated system may be expressed as

gxMtKxCxM &&&&& Γ−=++ )( (1)

or

gxMfxKxCxM &&&&& Γ−Λ=++ 0 (2)

where x ( [ ]Tsb xxx = ) is displacement vector of the structure and base isolator; M, C and K are the mass, damping, and stiffness matrices, respectively; f is the supplemental force exerted by the MRE’s, Λ ( [ ]T01=Λ ) vector represents the position of the supplemental damper force; Γ vector are unity vector; gx&& is the ground acceleration. The system matrices can be expressed as

⎥⎦

⎤⎢⎣

⎡=

s

b

mm

M0

0

, ⎥⎦

⎤⎢⎣

⎡−

−+=

ss

ssb

ccccc

C,

⎥⎦

⎤⎢⎣

⎡−

−+=

ss

ssb

kkkkk

K0

(3)

M

5M

4M

3M

2M

M 1

M R E b ase -iso la tio n sy stem B ase-iso la tio n sy stem

1M

M 2

M 3

M 4

M 5

M b

K 5 5C

C 44K

C 33K

C 22K

C 11K

C bbK (t) K b bC

K 1 1C

K 2 2C

K 3 3C

K 4 4C

C 55K

b


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Defining the state vectors as [ ]Txxz &= and the output vector to be regulated as [ ]Txxxy &&&= , the state-space form of the equations of motion can be given by

gxEBfAzz &&& ++= (4)

gyyy xFfDzCy &&++=

(5)

where ⎥⎦

⎤⎢⎣

⎡−−

= −− CMKMI

A 11

0

, ⎥⎦

⎤⎢⎣

⎡Λ

= −1

0M

B,

⎥⎦

⎤⎢⎣

⎡Γ−

=0

E

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−=

−− CMKMI

IC y

11

00

, ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Λ=

−1

00

MDy

and

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Γ−= 0

0

yF

The state-state equations can be modeled using the MATLB/SIMULINK environment for numerical analysis of the system. The simulation model along with detailed system and simulation parameters are presented in the following section.

5. Semi-active control of MRE stiffness

The stiffness of the MRE should be varied in real time to reduce the response of the structure. For this purpose the various control algorithms can be selected. The most commonly used one is linear quadratic regulator (LQR) algorithm. In this work another type of algorithm is proposed for changing the stiffness in real time, which is state-switched control.

5.1. LQR control Algorithm

Linear quadratic regulator (LQR) control algorithm is basically optimal control algorithm. Basic concept behind this control algorithm is to minimize the following cost function,

dttRututQztzJt

TT∫ +=0

)]()()()([

with respect to the control input u(t) and subject to the constraining equation

0)0()()()()( zztHftButAztz =++=&

In regulator type algorithm problems the system is assumed to be in equilibrium and the purpose of the control algorithm is to maintain the equilibrium despite being subjected to the disturbances and also minimize the response of the structure due to the disturbances of any nature. The main disadvantage of the LQR algorithm is that to control the structure using this algorithm the full state response of the structure has to be measured. For a multi storey building structure full state means displacement, velocity and acceleration at each floor which is almost impossible in the real structure.

5.2. State-switched control Algorithm

Basically state-switched control considers two discrete values of the stiffness for the isolation device i.e. MRE and switches among them based on the structural response. Basically with two stiffness values we have two natural frequencies for the structure to switch between. Here we have selected two discrete stiffness values as 0.5K0 and 1.5K0, which are within the range mentioned above and therefore now the range of control force will be


April 25-26, 2011

)(5.0)()(5.0 00 txKtftxK ≤≤− (6)

Now the criterion based on which the stiffness values can be switched has to be selected. Kenneth et al. (2000) proposed the switching criterion based on the product of base velocity and relative velocity. In our case the relative velocity between structure and the base may not be an efficient factor to contribute in switching criterion. Therefore following criterion was adopted for our system to switch the stiffness values. First of all the control command u(t) is calculated by the simple product

{ }nnxxtu &&=)( (7)

Then it is saturated between 0 and 1 to get the modified output v(t) with well defined maximum and minimum values so that the switching value can be decided more conveniently around 0.5.

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

=<>

=5.0)(05.0)(5.0)(

))(( 2

1

tviftvifKtvifK

tvK s

s

(8)

Here Ks1 and Ks2 correspond to the extreme values of the control force range shown in the Equation (6) and K0 is the middle value of the range i.e. base line stiffness of the MRE without any magnetic field applied. The main advantage with the state-switched control is that instead of full state only the displacement and acceleration is required and that too at the top most level only. In this way it becomes most viable and practical control algorithm.

6. Numerical Simulation

The numerical simulations were performed to verify the effectiveness of the proposed state-switched control system using the numerical models mentioned above. The responses for the passive, semi-active with LQR and semi-active with SS control are compared. The figures show the base drift, top floor acceleration and the storey drift at first floor for all the three cases. And then the peak responses are compared in the following table. Here follow three sets of results for various earthquake input excitations.

EL CENTRO

0 5 10 15 20 25 30-200

-100

0

100

200

time(sec)

Bas

e dr

ift(m

m)

PassiveMRE-SSMRE-LQR

0 5 10 15 20 25 30-1.5

-1

-0.5

0

0.5

1

time(sec)

Stru

ctur

e ac

cele

ratio

n(m

/sec

2 )



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Fixed Base MRE (SS) MRE (LQR) Base Drift 158.45 90.87 116.97

Top Floor Acc. 0.988 0.398 0.784 1st Floor storey drift 0.889 0.255 0.776

Top Displ. 161.58 91.78 119.67

HACHINOHE



Top Displ. 272.45 143.67 129.21

0 5 10 15 20 25 30-1

-0.5

0

0.5

1

time(sec)

Stru

ctur

e dr

ift a

t 1st

floo

r(mm

)


0 5 10 15 20 25 30-400

-200

0

200

400

time(sec)

Bas

e dr

ift(m

m)


0 5 10 15 20 25 30-2

-1

0

1

2

time(sec)

Stru

ctur

e ac

cele

ratio

n(m

/sec

2 )


0 5 10 15 20 25 30-2

-1

0

1

2

time(sec)

Stru

ctur

e dr

ift a

t 1st

floo

r(mm

)



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NORTHRIDGE



Top Displ. 495.79 334.12 305.8967

Above simulation results for various input excitation levels quite obviously imply that some of the responses have improved significantly such as the peak acceleration and the inter-storey drift. Whereas the base drift in general is a bit more than the LQR case but still less than the passive base isolation system. In consequence of base drift the top floor displacement also exceeds the LQR case and again still less than the passive case. Here we can observe clearly that the state-switched control has proven to be much better than the LQR in terms of response as well as the fact that we only need to measure displacement and acceleration at the top floor. Also state-switched control is also much simpler than the conventional control algorithms and we only need to switch between two stiffness values, which enhances the practicability of the control system in whole.

7. Conclusions

This study concludes that the state-switched control system is far more practical than the conventional LQR algorithm and also numerical investigations show significant reduction in the response of the structure compared to the convention control system. As we know that the main disadvantage of the active control system lies in the chance of being accidentally unstable and requirement of large external power. The proposed control system eliminates both of these limitations

0 5 10 15 20 25 30-500

0

500

time(sec)

Bas

e dr

ift(m

m)


0 5 10 15 20 25 30-4

-2

0

2

4

time(sec)

Stru

ctur

e ac

cele

ratio

n(m

/sec

2 )


0 5 10 15 20 25 30-4

-2

0

2

4

time(sec)

Stru

ctur

e dr

ift a

t 1st

floo

r(mm

)



April 25-26, 2011

without being impractical as in the case of LQR where the measurement of full state response of the structure is almost impossible. Now the proposed system must be analyzed experimentally before it can be implemented in some real scale building structure. Besides the application to the multi storey structure it can find many other potential applications like isolation of building structure from some highly vibrating machine equipment installed within the structure.

8. Acknowledgements

This research was supported by the Smart Infra-Structure Technology Center (funded by the Korea Science and Engineering Foundation), the Construction Core Technology Program (funded by the Ministry of Land, Transport and Maritime Affairs of Korea), and Higher Education Commission, Government of Pakistan. Their financial supports are gratefully acknowledged.

9. References

[1] Deng, H., Gong, X. and Wang, L. (2006), “Development of an adaptive tuned vibration absorber with magneto rheological elastomer,” Smart. Mater. Struct. 15, 111-116.

[2] Gandhi, F. and Anusonti-Inthra, P. (2003), “Adaptive control of semiactive variable stiffness devices for narrow-band disturbance rejection,” Journal of Intelligent Material Systems and Structures, 14, 191-201.

[3] Ginder, J.M., Schlotter, W.F. and Nichols, M.E. (2001), “Magnetorheological elastomers in tunable vibration absorbers” Smart Structures and Materials 2001: Damping and Isolation, 4331, 103-110

[4] Kelly, J.M., Leitmann, G., and Soldatos, A.G. (1987), ‘‘Robust control of base-isolated structures under earthquake excitation,’’ J. Optim. Theory Appl., 53, 159–180.

[5] Kenneth, A.C., Sergio, D.R., Nader, S. and Gregg, L (2000), “State-switched absorber for semi-active structural control” Journal of Intelligent Material Systems and Structures, 11, 300-310.

[6] Koo, J.H., Sung, S.H., Lee, H.J., Jung, H.J., “Seismic Protection of Structures using Smart Base Isolation Systems Based on MR Elastomers” The 4th International Conference on Advances in Structural Engineering and Mechanics (ASEM08), May 26 – 28, 2008, Jeju, Korea

[7] Naeim, F., and Kelly, J. M. (1999), Design of seismic isolated structures: From theory to practice, Wiley, Chichester, England.

[8] Ramallo, J.C., Johnson, E.A. and Spencer, B.F., Jr. (2002), ‘‘Smart’’ base isolation systems,” Journal of Engineering Mechanics, ASCE, 128(10), 1088-1099.

[9] Usman, M., Sung, S.H., Jang, D.D., Jung, H.J. and Koo, J.H. (2008), “Numerical investigation of smart base isolation system employing MR Elastomer.” Proceedings of ’08 ERMR Dresden, Germany

[10] Usman M, Jung HJ, Park JW, Koo JH, “State-switched Controller for Smart Base Isolation System Employing MR Elastomer” Proceedings of 21st KKCNN 27th -29th Oct 2008 Singapore

[11] Yoshioka, H., Ramallo, J.C. and Spencer, B.F., Jr. (2002), “’Smart’ base isolation strategies employing magnetorheological damper.” Journal of Structural Engineering, ASCE, 128(5), 540-551.