research article optimal control of magnetorheological

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 251935 7 pageshttpdxdoiorg1011552013251935

Research ArticleOptimal Control of Magnetorheological Fluid Dampers forSeismic Isolation of Structures

Ameen H El-Sinawi1 Mohammad H AlHamaydeh2 and Ali A Jhemi1

1 Department of Mechanical Engineering American University of Sharjah Sharjah 26666 United Arab Emirates2 Department of Civil Engineering American University of Sharjah Sharjah 26666 United Arab Emirates

Correspondence should be addressed to Ameen H El-Sinawi aelsinawiausedu

Received 27 February 2013 Accepted 21 April 2013

Academic Editor Chengjin Zhang

Copyright copy 2013 Ameen H El-Sinawi et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents themodeling and control of amagnetorheological (MR) damper installed in Chevron configuration at the baseof a 20-story benchmark buildingThe building structural model is created using the commercial software package ETABSTheMRdamper model is derived from Bouc-Wen hysteresis model which provides the critical nonlinear dynamics that best represents theMR damper under a wide range of operating conditions System identification is used to derive a low-order nonlinear model thatbest mimics the nonlinear dynamics of the actual MR damper Dynamic behavior of this low-order model is tested and validatedover a range of inputs The damper model has proven its validity to a high degree of accuracy against the nonlinear model AKalman filter is designed to best estimate the state of the structure-damper system for feedback implementation purposes Usingthe estimated states an LQG-based compensator is designed to control the MR damper under earthquake loads To demonstratethe effectiveness of this control strategy four historical earthquakes are applied to the structure Controlled and uncontrolled flooraccelerations and displacements at key locations are compared Results of the optimally controlled model demonstrate superiorperformance in comparison to the uncontrolled model

1 Introduction

Protection of large structures against external disturbancessuch as earthquakes and wind has been a major concern toresearchers for decades Seismic isolation with and withoutsupplemental damping for energy dissipation has proven tobe very effective in protecting civil structures during seismicevents Most classical isolators are of the passive type suchas natural rubber bearings (NRBs) [1ndash3] These isolatorsare capable of providing adequate damping during low-to-moderate earthquakes With high-velocity pulses and highdisplacement demands many Near-Field (NF) situationsrequire impractical isolator bearing dimensions and designsIn such occasions utilizing high-damping rubber (HDR)bearings lead-rubber (LR) bearings or friction pendulumsystem (FPS) alone is not the best engineering solutionTypical FF response of base-isolated structures ismanageablecompared to the high demands of an NF event [4ndash8] The

combined isolation system of HDR or LR bearings withviscous dampers seems to work well in the NF regions wherethe ground shaking characteristics are capable of producingpulses with velocity of 05ndash15msec and durations of 1ndash3 secUnfortunately this combined system does not perform desir-ably in moderate or strong Far-Field (FF) events due to thesecondary forces produced by the dampers and their complexcoupling effects [9] Consequently supplemental dampingis needed to reduce the horizontal displacement demandsotherwise structural integrity could be jeopardized [10] Formoderate-to-severe earthquakes semiactive dampers suchas Magnetorheological (MR) dampers have proven to besignificantly more effective Jangid [11] investigated the opti-mum use of the FPS isolators for NF earthquake motion inmultistory buildings He evaluated the response of the systemto six records of NF earthquakes and derived the optimumfriction coefficient of the FPSThis was performed so that thetop floor acceleration and the total horizontal sliding distance

2 Mathematical Problems in Engineering

are minimized It was concluded that the optimum frictioncoefficient for FPS for NF earthquake motions is in the rangeof 005 to 015

When considering the challenge of limiting the totalmaximum displacement (119863TM) to practical limits especiallyin NF sites somtimes the designer relies on fluid viscousdampers (FVDs) The state of the practice involves carryingout preliminary calculations and analyses using typical HDRbearings These preliminary calculations could readily showwhether or not supplemental damping is required Oncesupplemental damping is deemed necessary many designerswould prefer utilizing the linear behavior of NRB isola-tors combined with the supplemental damping provided byFVDs the use of such a system often results in additionaluniformity in the induced superstructure story forces Thissystem has also been used in many projects in the USA [12ndash19] Recently AlHamaydeh et al [20] developed simplifieddesign equations for seismic isolation systems with dampersSeveral researchers [21ndash23] proposed a cost effective real-time hybrid simulation to evaluate different control strategiesfor advanced MR dampers Four semiactive control strate-gies based on the clipped-optimal controller were evaluatedexperimentally Force-tracking type controllers were foundto achieve excellent control performance while maintainingrelatively lowMRdamper forces Recently Zhao andZhu [24]introduced a stochastic optimal semiactive control law forcable-stayed bridges by solving the dynamical programmingequation produced by utilizing the Bingham model for anMR damper Since supplemental damping devices generallyprovide higher damping levels which are inversely propor-tional to their stiffness Hoslashgsberg [25] demonstrated thatMRdampers can be used to minimize dampers stiffness and evenhave equivalent negative stiffness Using linear equivalentmodels obtained by harmonic averaging improvement inresponse reduction is shown when compared to the corre-sponding case with optimal passive viscous dampers Mostrecently Assaleh et al [26] utilized group method of datahandling (GMDH) to model the MR damper behavior

This work presents a technique for seismic isolation of a20-story building adaptive control of anMRdamper installedin Chevron configuration between the base and first floorof a 20-story building The building structural model isderived from a benchmark structure model using ETABSTheMRdampermodel considered is derived fromWien-Bochysteresis model This model provides the critical nonlineardynamics that best represents the MR damper under a widerange of operating conditions System identification is usedto derive a low-order nonlinear model that best mimicsthe nonlinear dynamics of the actual MR damper Dynamicbehavior of this low order model is tested and validated overa wide range of inputs The damper model has proven itsvalidity to a high degree of accuracy against the nonlinearmodel A Kalman filter is designed to best estimate thestate of the structure-damper system for feedback imple-mentation purposes Using the estimated states an LQG-based compensator is designed to control the MR damperunder earthquake loads To demonstrate the effectiveness ofthis control strategy a wide range of historical earthquakesare applied to the structure and the MR damper is set

active Accelerations and drifts at all building floors arecomputed over the duration of each earthquake Results ofthe controlledmodel are compared to the uncontrolledmodeland the superior performance of the optimally controlledmodel is demonstrated

2 Structural Model

The proposed structure model was derived from a 20 storybenchmark building well studied in literature A full descrip-tion of the structure details is provided by others Spencer etal [27] and Ohtori et al [28] The building parameters werefed into ETABS and natural frequencies and mode shapeswere computed Figure 1 shows a sample mode of vibrationof the 20-story building under consideration

The structure is discretized into finite element modelforming an 119899-dimensional discrete spring-mass-damper sys-temwhose dynamics is described by the second-ordermatrixdifferential equation

119872 + 119862 + 119870119909 = 119906 (119905) (1)

where 119872 119862 and 119870 are the (20 times 20) square and symmetricmass stiffness and damping coefficient matrices respec-tively The variables 119909(119905) and 119906(119905) are the displacement andforce vectors respectively For systems with proportionaldamping the matrices 119872 119870 and 119862 can be diagonalized byemploying a proper normalized orthogonal transformationThis transformation yields

120578119894 (119905) + 2120589

119894120596119894120578119894 (119905) + 120596

119894120578119894 (119905) = 119881

119894119906 (119905) 119894 = 1 20 (2)

where 120578119894 120596119894 and 120589

119894represent the transformed coordinates

natural frequency and damping ratio of the structurersquos 119894thvibration mode respectively When the input is a pointforce (ie actuators) 119881

119894is the vector of the 119894th mode shape

evaluated at the force input locationFor flexible structures having point force(s) as the input(s)

and point displacement(s) as the measured output the state-space model of the flexible structures can be transformed asfollows

= [0 119868

minusΩ2

minus2120589Ω] 119911 + [

0

119881] 119906

119883119905= [119882 0] 119911 + 119863119906

(3)

where 119911(119905) = 120578(119905)

120578(119905) state vector 119873

119898 number of

modes 119873119906 number of inputs 119873

119910 number of outputs

120578(119905) = 1205781(119905) 1205782(119905) 120578

119873119898

(119905)119879 modal displacement

120578(119905) = 1205781(119905) 1205782(119905) 120578

119873119898

(119905)119879 modal velocity 119906(119905) =

1199061(119905) 1199062(119905) 119906

119873119906

(119905)119879 input vector 119903

119894 spatial coordi-

nates 119883119905(119905) = 119909(119903

1 119905) 119909(119903

2 119905) 119909(119903

119873119910

119905)119879 output vec-

tor Ω = diag1205961 1205962 120596

119873119898

natural frequency 120589 =

diag1205771 1205772 120577

119873119898

modal damping 120595119894119895 mode shape 119894 at

location 119895

Mathematical Problems in Engineering 3

1 1 1 1 1 1A B C D E F

119909

119911

Base

B minus 1

Ground

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

10th

11th

12th

13th

14th

15th

16th

17th

18th

19th

20th

lowast lowast

Figure 1 Representation of building modes of vibration (5thhorizontal translational mode)

input matrix

119881 =[[

[

12059511

sdot sdot sdot 1205951119873119906

d

1205951198731198981 sdot sdot sdot 120595

119873119898119873119906

]]

]

(4)

output matrix

119882 =[[

[

12059511

sdot sdot sdot 1205951198731198981

d

1205951119873119910

sdot sdot sdot 120595119873119898119873119910

]]

]

(5)

The state-spacemodel of (3) can be expressed in the followingcompact form

= 119860119904 (120579) 119911 + 119861

119904 (120579) 119906 (6)

119883119905= 119862119904 (120579) 119911 + 119863

119904 (120579) 119906 (7)

where 119883119905is a vector of nodal displacement(s) at sensor(s)

location(s) and 119860119904 119861119904 119862119904 and 119863

119904matrices are functions

of the system (natural frequency damping ratio and modeshapes) (ie if we assume 120579 = 119891(120596

119894 120589119894 and 120595

119894)119894=1119899

)Information needed to construct matrices 119860

119904 119861119904 119862119904 and119863

119904

of (6) and (7) (iemode shapes andnatural frequencies)wereall obtained using ETABS

3 MR Damper Model

To make the simulation realistic the MR damper has tobe properly modeled The damper model must be accurateenough to capture the dynamic characteristics of the realdamper yet simple enough carry the computation in real timeon a low-power microprocessor To bridge the gap betweenthese two competing requirements system identification (SI)was used to derive an 8th-order nonlinear auto regression(Narx) model Spencer et al [27] have presented a model oftheMR damper based on the Bouc-WenHysteresis model Inthis model the force displacement force velocity and forceas a function of time were computed The damper modelequations are presented here for convenience

119865 = 1198621

119910 + 1198961(119909 minus 119909

119900) (8)

119910 =1

119888119900+ 1198881

[120572119911 + 119888119900 + 119896119900(119909 minus 119910)] (9)

= minus1205741003816100381610038161003816 minus 119910

1003816100381610038161003816 119911|119911|119899minus1

minus 120573 ( minus 119910) |119911|119899+ 119860 ( minus 119910) (10)

120572 = 120572119886+120572119887119906 (11)

1198881= 1198881119886+1198621119887119906 (12)

119888119900= 119888119900119886 +119862

119900119887119906 (13)

= minus120578 (119906 minus V) (14)

In this work numerical solutions of (8)ndash(14) have beenperformed and time history of all states was validated bychecking against previously published solutions by Spenceret al [27]

System identification (SI) relating input to output hasbeen performed to derive a simple nonlinearmodel DifferentSI techniques using the Matlab System Identification tool-box were employed yielding excellent matchingThe derivedmodels have been thoroughly tested and proved to matchthe nonlinear model behavior to a high degree of accuracyover a wide range of inputs Among the different methodsemployed the Narx method provided the best matchingResponses of various models obtained by the different SItechniques are shown in Figure 2

The polynomial model obtained from Narx is convertedto state-space format and compared to the nonlinear model


0

500

1000

Forc

e (N

)

01 02 03 04 05 06 07 08 09Time (s)

minus1500

minus1000

minus500

Force (sim)

ze measuredNarx3 fit 4097Narx8 fit 9305

Fit 9306Fit 8131

Figure 2 MR damper response versus different identified models

01 02 03 04 05 06 07 08

0

500

1000

1500

2000

Time (s)

MR

dam

per f

orce

(N)

Nonlinear versus linear response

Narx modelNonlinear model

minus1500

minus1000

minus500

Figure 3 Comparison between the state-spacemodel andnonlinearmodel

The comparison is shown in Figure 3 The state-space modelof the damper is expressed as

119909119891= 119860119891119909119891+ 119861119891119906119891

119884119891= 119862119891119909119891+119863119891119906119891

(15)

where 119909119891

is the state vector of the MR damper state-space model 119906

119891is the input and 119884

119891is the output (ie

damper force) The quadruple (119860119891 119861119891 119862119891 119863119891) represents

the dynamic input output and direct input matrices of theMR damper model The following figure shows the resultsobtained from the Narx model and the nonlinear modelExcellent matching of dynamic behavior is demonstrated inFigure 3

The state-space model obtained from Narx polynomial isutilized in the control scheme of the multidegree of freedomstructure found in the preceding The controller design ispresented next

4 Controller Design

Assuming that the control effort will be utilized to isolatethe passive structure from ground excitation as shown inFigure 4 A schematic of the control process implemented onthe structure is shown in Figure 5

The (KAFB) dynamic model is formed from the com-bined dynamics of the MR damper transmitted force repre-sented by (15) and the dynamics of the structure representedby (6) and (7)

The equivalent continuous state-spacemodel of the trans-mitted forcewhere acceleration is the input to the forcemodelis

119891= 119860119891119909119891+ 119861119891119884119884

119865 = 119862119891119909119891+ 119863119891119884119884

(16)

where (119884119884(119905)) is the base acceleration and 119909119891is a vector of the

transmitted force states 119860119891 119861119891 119862119891 and 119863

119891are dynamics

input output and direct input matrices of the transmittedforce block of Figure 5 respectively The term 119865 denotes theforce transmitted to the structure through its elastic base andis a function of time

Using the above analysis we can express the continuousstate-space model of the structure described by (7) and (8) as

119891= 119860119904119911 + 119861119904119865

119883119905= 119862119904119911 + 119863

119904119865

(17)

where that the term 119906(119905) of (6) and (7) has been replacedby 119865 in (17) to indicate that the input to the structure is theforce transmitted to it through its elastic base and includesthe seismic excitation and the damper control force

A state-space model of the beam-base system (the trans-mitted force represented by (16) and the structure representedby (17)) can now be constructed by augmenting the two partstogether such that

119860119886= [

119860119891

0

119861119904119862119891

119860119904

]

119861119886= [

119861119891

119861119904119863119891

]

119862119886= lfloor119863119904119862119891 119862

119904rfloor

119863119886= lfloor119863119891119863119904rfloor

(18)

where 119860119886 119861119886 119862119886 and 119863

119886represent the state-space matrices

of the augmented base-beam system in which the first twostates belong to the transmitted force part and the remainingstates belong to the structure mounted on the base


11990911199092

Figure 4 Passive structure

Structure

Nonlinear MRdamper model

Gain

MR damper force

Structure model

Controller

1199091 1199092

119870119886

119870119888

sum

sum

Figure 5 Active structure

Matrices 119860119886 119861119886 and 119862

119886are used for designing the

(KAFB) matrix of gains (119870119886) such that

119870119886= 119878∘119862119879

119886119877minus1 (19)

The column vector 119870119886in this case is a (2 + 2119899) times 1 column

vector and the first two rows are the Kalman gains of thestates of the transmitted force and the remaining 2119899 gainsare those of the states of the structure 119878

∘is the steady-state

solution of the following filter algebraic Riccati equation

119878 = 119860119886119878 + 119878119860

119879minus 119878119862119879

119886119877minus1119862119886119878 + 119861119886119876119861119879

119886 (20)

Matrices 119877 and 119876 are positive definite and positive semidef-inite matrices respectively [29 30] Proper choice of 119877 and119876 is important because both matrices are heavily involved

0 5 10 15 20 25 30

0

5El Centro

0 5 10 15 20 25 30

0010203

Time (s)

Time (s)

minus5

Uncontrolled accelerationControlled acceleration

Acce

lera

tion

(ms2)

minus02

minus01

Disp

lace

men

t (m

)Uncontrolled displacementControlled displacement

Figure 6 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)El Centro earthquake input

in the construction of the cost function In this work thevalue of 119876119877 ranges from 1 to 100 This ratio is limitedby the maximum force that can be provided by the MRdamper and keep the control force within its saturationlimit If the 119877 is very high sensitivity to measurementnoise will be accentuated and the controller performancemight be degraded After many iterations and driven bythe objective of minimizing accelerations and displacementswhile maintaining damper force with allowable limits thefinal 119877 and 119876matrices were selected

For a specific value of 119877 and 119876 Kalman matrix of gains(119870119886) of (19) is

119870119886= [

[119870119891]

[119870119904]

] =

[[[[[[[

[

[1198701

1198702

]

[[

[

1198703

1198702119899+2

]]

]

]]]]]]]

]

(21)

Equation (21) shows that 119870119886is partitioned into two parts

namely 119870119891which corrects the estimates of 119909

119891in (16) and

119870119904which corrects the estimates beam states (ie119883

119905of (17))

In general the structure of the Kalman estimator takeson a particularly simple structure that closely resembles theoriginal dynamic system [29 30] The complete vibrationisolation scheme proposed by this study is shown in Figure 5119870119888in Figure 5 is the linear quadratic regulator (LQR) gain

obtained with a similar procedure used to obtain119870119886

It is well known that the Kalman estimator is subject toall deterministic inputs that the plant is subject to including


Time (s)0 5 10 15 20 25 30

0123

Hachinohe

0 5 10 15 20 25 30

00102

Time (s)


Acce

lera

tion

(ms2)

minus03

minus02

minus01

Disp

lace

men

t (m

)

Uncontrolled displacementControlled displacement

minus1

minus2

Figure 7 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Hachinohe earthquake

0 5 10 15 20 25 30

0

10

20 Kobe

0 5 10 15 20 25 30

0

05

1

Time (s)

Time (s)

minus20

minus10


Acce

lera

tion

(ms2)

minus1

minus05

Disp

lace

men

t (m

)


Figure 8 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Kobe earthquake

the estimated damper control force shown in Figure 5 Thisis why the realization of the structure inside the controllerin Figure 5 is subject to the estimated transmitted damperforce twice These two forces have the same magnitudeand like the two forces acting on the structure (plant) theyare opposite in sign nullifying the net force seen by the

Time (s)0 5 10 15 20 25 30

05

10Northridge

0 5 10 15 20 25 30

0

05

Time (s)


Acce

lera

tion

(ms2)

minus1

minus05

Disp

lace

men

t (m


minus15

minus10

minus5

Figure 9 Simulated acceleration and displacement response of topfloor of the structure with control (red) and without control (blue)Northridge earthquake

realization of the structure inside the controller Thereforethe Kalman estimate of the acceleration of any point onthe beam is identically zero which eliminates the need forrealizing (including) the structure inside the controller whichsubsequently yields a second-order control scheme regardlessof the order of the plant model This lowers the complexityof the controller and therefore significantly reduces thecomputational time Simulated results of the 20th floor arepresented in Figures 6 7 8 and 9 Each figure has two plotsone for acceleration and the other for displacement Each plotshows the uncontrolled and the controlled response

5 Conclusions

To demonstrate the advanced performance of a semiactiveMR damper used in the protection of a 20-story structurefrom earthquake damage a control strategy based on alinear quadratic Gaussian regulator is proposed First a linearmodel of the structure is derived using ETABS SecondSystem identification is used to derive a linear low-ordermodel of the MR damper from the Bouc-Wen nonlinearmodel Third a Kalman filter is designed to best estimatethe states of the system for feedback implementation pur-poses Finally an LQG controller is designed to minimizedynamic loads and structural damage Extensive simulationis performed to test and validate the effectiveness of theMR damper control strategy Four historical earthquakes areapplied to the structure and theMRdamper is set active Aftersuccessful simulation accelerations and drifts at all structurefloors are computed Proposed optimal control ofMRdamper


effectiveness is demonstrated by significantly reducing theaccelerations and displacements at all floors of the structure

Acknowledgment

The authors acknowledge the support of the AmericanUniversity of Sharjah

References

[1] R I Skinner W H Robinson and G H McVerryAn Introduc-tion to Seismic Isolation John Wiley amp Sons Chichester UK1993

[2] J M Kelly Earthquake-Resistant Design With Rubber SpringerLondon UK 2nd edition 1997

[3] F Naeim and J M Kelly Design of Seismic Isolated StructuresJohn Wiley amp Sons New York NY USA 1999

[4] R S Jangid and J M Kelly ldquoBase isolation for near-faultmotionsrdquoEarthquake Engineering and Structural Dynamics vol30 no 5 pp 691ndash707 2001

[5] A Rodriguez-Marek Near fault seismic site response [PhDthesis] Civil Engineering University of California BerkeleyCalif USA 2000

[6] G A MacRae D V Morrow and C W Roeder ldquoNear-faultground motion effects on simple structuresrdquo ASCE Journal ofStructural Engineering vol 127 no 9 pp 996ndash1004 2001

[7] A K Chopra and C Chintanapakdee ldquoComparing responseof SDF systems to near-fault and far-fault earthquake motionsin the context of spectral regionsrdquo Earthquake Engineering andStructural Dynamics vol 30 no 12 pp 1769ndash1789 2001

[8] N Makris and C J Black ldquoDimensional analysis of bilinearoscillators under pulse-type excitationsrdquo ASCE Journal of Engi-neering Mechanics vol 130 no 9 pp 1019ndash1031 2004

[9] C P Providakis ldquoEffect of LRB isolators and supplementalviscous dampers on seismic isolated buildings under near-faultexcitationsrdquo Engineering Structures vol 30 no 5 pp 1187ndash11982008

[10] C P Providakis ldquoEffect of supplemental damping on LRB andFPS seismic isolators under near-fault ground motionsrdquo SoilDynamics and Earthquake Engineering vol 29 no 1 pp 80ndash902009

[11] R S Jangid ldquoOptimum friction pendulum system for near-faultmotionsrdquoEngineering Structures vol 27 no 3 pp 349ndash3592005

[12] S M Hussain J W Asher and R D Ewing ldquoSeismic baseisolation design for the San Bernardino CountyMedical Centerreplacement projectrdquo in Proceedings of the Symposium onStructural Engineering in Natural Hazards Mitigation vol 1 pp760ndash765 Irvine Calif USA April 1993

[13] S Hussain ldquoPerformance of base isolated buildings in thenorthridge earthquakerdquo in Seismic Base Isolation State of thePractice Seminar Structural Engineers Association of SouthernCalifornia 1994

[14] S Hussain and E Retamal ldquoA hybrid seismic isolation systemmdashisolators with supplemental viscous dampersrdquo in Proceedings ofthe 1st World Conference on Structural Control vol 3 pp FA2-53ndashFA2-62 International Association for Structural ControlLos Angeles Calif USA August 1994

[15] M Al Satari and J Abdalla ldquoOptimization of a base-isolationsystem consisting of natural rubber bearings and fluid viscous

dampersrdquo in Proceedings of the 11thWorld Conference on SeismicIsolation Energy Dissipation and Active Vibration Control ofStructures Guangzhou China 2009

[16] S Hussain and M Al Satari ldquoInnovative design of a seismicisolation supplemental viscous damping systems of an essentialservices facility in a near-fault regionrdquo in Proceedings of the 14thWorld Conference on Earthquake Engineering Beijing China2008

[17] S Hussain and M Al Satari ldquoDesign of a seismic isolationsystem with supplemental viscous damping for a near-faultessential services facilityrdquo in Proceedings of the 76th StructuralEngineers Association of California Annual Convention SquawCreek Calif USA 2007

[18] S Hussain andM Al Satari ldquoViscous-damped seismic isolationsystem for a near-fault essential services facilityrdquo in Proceedingsof the 10th World Conference on Seismic Isolation EnergyDissipation and Active Vibration Control of Structures IstanbulTurkey 2007

[19] M AlHamaydeh and S Hussain ldquoInnovative design of aseismically-isolated building with supplemental dampingrdquo inProceedings of 14th European Conference on Earthquake Engi-neering (ECEE rsquo10) Ohrid Republic of Macedonia 2010

[20] M AlHamaydeh S Barakat and F Abed ldquoMultiple regressionmodeling of natural rubber seismic-isolation systems withsupplemental viscous damping for near-field ground motionrdquoJournal of Civil Engineering and Management In press

[21] J C Ramallo E A Johnson and B F Spencer Jr ldquolsquoSmartrsquo baseisolation systemsrdquo ASCE Journal of Engineering Mechanics vol128 no 10 pp 1088ndash1099 2002

[22] A Friedman J Zhang Y Cha et al ldquoAccommodating MRdamper dynamics for control of large scale structures systemsrdquoin Proceddings of the 5thWorld Conference on Structural Controland Monitoring 2010

[23] H J Jung B F Spencer Jr and I W Lee ldquoControl ofseismically excited cable-stayed bridge employing magnetorhe-ological fluid dampersrdquo Journal of Structural Engineering vol129 no 7 pp 873ndash883 2003

[24] M Zhao andWQ Zhu ldquoStochastic optimal semi-active controlof stay cables by using magneto-rheological damperrdquo Journal ofVibration and Control vol 17 no 13 pp 1921ndash1929 2011

[25] J Hoslashgsberg ldquoThe role of negative stiffness in semi-activecontrol of magneto-rheological dampersrdquo Structural Controland Health Monitoring vol 18 no 3 pp 289ndash304 2011

[26] K Assaleh T Shanableh and Y Kheil ldquoGroup method of datahandling for modeling magnetorheological dampersrdquo Intelli-gent Control and Automation vol 4 no 1 pp 70ndash79 2013

[27] B F Spencer S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoASCE Journal of Engineering Mechanics vol 123 no 3 pp 230ndash238 1997

[28] Y Ohtori R E Christenson B F Spencer Jr and S J DykeldquoBenchmark control problems for seismically excited nonlinearbuildingsrdquo ASCE Journal of Engineering Mechanics vol 130 no4 pp 366ndash385 2004

[29] AH El-Sinawi ldquoActive vibration isolation of a flexible structuremounted on a vibrating elastic baserdquo Journal of Sound andVibration vol 271 no 1-2 pp 323ndash337 2004

[30] A El-Sinawi and A R Kashani ldquoActive isolation using aKalman estimator-based controllerrdquo Journal of Vibration andControl vol 7 no 8 pp 1163ndash1173 2001

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Stochastic AnalysisInternational Journal of


are minimized It was concluded that the optimum frictioncoefficient for FPS for NF earthquake motions is in the rangeof 005 to 015

When considering the challenge of limiting the totalmaximum displacement (119863TM) to practical limits especiallyin NF sites somtimes the designer relies on fluid viscousdampers (FVDs) The state of the practice involves carryingout preliminary calculations and analyses using typical HDRbearings These preliminary calculations could readily showwhether or not supplemental damping is required Oncesupplemental damping is deemed necessary many designerswould prefer utilizing the linear behavior of NRB isola-tors combined with the supplemental damping provided byFVDs the use of such a system often results in additionaluniformity in the induced superstructure story forces Thissystem has also been used in many projects in the USA [12ndash19] Recently AlHamaydeh et al [20] developed simplifieddesign equations for seismic isolation systems with dampersSeveral researchers [21ndash23] proposed a cost effective real-time hybrid simulation to evaluate different control strategiesfor advanced MR dampers Four semiactive control strate-gies based on the clipped-optimal controller were evaluatedexperimentally Force-tracking type controllers were foundto achieve excellent control performance while maintainingrelatively lowMRdamper forces Recently Zhao andZhu [24]introduced a stochastic optimal semiactive control law forcable-stayed bridges by solving the dynamical programmingequation produced by utilizing the Bingham model for anMR damper Since supplemental damping devices generallyprovide higher damping levels which are inversely propor-tional to their stiffness Hoslashgsberg [25] demonstrated thatMRdampers can be used to minimize dampers stiffness and evenhave equivalent negative stiffness Using linear equivalentmodels obtained by harmonic averaging improvement inresponse reduction is shown when compared to the corre-sponding case with optimal passive viscous dampers Mostrecently Assaleh et al [26] utilized group method of datahandling (GMDH) to model the MR damper behavior

This work presents a technique for seismic isolation of a20-story building adaptive control of anMRdamper installedin Chevron configuration between the base and first floorof a 20-story building The building structural model isderived from a benchmark structure model using ETABSTheMRdampermodel considered is derived fromWien-Bochysteresis model This model provides the critical nonlineardynamics that best represents the MR damper under a widerange of operating conditions System identification is usedto derive a low-order nonlinear model that best mimicsthe nonlinear dynamics of the actual MR damper Dynamicbehavior of this low order model is tested and validated overa wide range of inputs The damper model has proven itsvalidity to a high degree of accuracy against the nonlinearmodel A Kalman filter is designed to best estimate thestate of the structure-damper system for feedback imple-mentation purposes Using the estimated states an LQG-based compensator is designed to control the MR damperunder earthquake loads To demonstrate the effectiveness ofthis control strategy a wide range of historical earthquakesare applied to the structure and the MR damper is set

active Accelerations and drifts at all building floors arecomputed over the duration of each earthquake Results ofthe controlledmodel are compared to the uncontrolledmodeland the superior performance of the optimally controlledmodel is demonstrated

2 Structural Model

The proposed structure model was derived from a 20 storybenchmark building well studied in literature A full descrip-tion of the structure details is provided by others Spencer etal [27] and Ohtori et al [28] The building parameters werefed into ETABS and natural frequencies and mode shapeswere computed Figure 1 shows a sample mode of vibrationof the 20-story building under consideration

The structure is discretized into finite element modelforming an 119899-dimensional discrete spring-mass-damper sys-temwhose dynamics is described by the second-ordermatrixdifferential equation

119872 + 119862 + 119870119909 = 119906 (119905) (1)

where 119872 119862 and 119870 are the (20 times 20) square and symmetricmass stiffness and damping coefficient matrices respec-tively The variables 119909(119905) and 119906(119905) are the displacement andforce vectors respectively For systems with proportionaldamping the matrices 119872 119870 and 119862 can be diagonalized byemploying a proper normalized orthogonal transformationThis transformation yields

120578119894 (119905) + 2120589

119894120596119894120578119894 (119905) + 120596

119894120578119894 (119905) = 119881

119894119906 (119905) 119894 = 1 20 (2)

where 120578119894 120596119894 and 120589

119894represent the transformed coordinates

natural frequency and damping ratio of the structurersquos 119894thvibration mode respectively When the input is a pointforce (ie actuators) 119881

119894is the vector of the 119894th mode shape

evaluated at the force input locationFor flexible structures having point force(s) as the input(s)

and point displacement(s) as the measured output the state-space model of the flexible structures can be transformed asfollows

= [0 119868

minusΩ2

minus2120589Ω] 119911 + [

0

119881] 119906

119883119905= [119882 0] 119911 + 119863119906

(3)

where 119911(119905) = 120578(119905)

120578(119905) state vector 119873

119898 number of

modes 119873119906 number of inputs 119873

119910 number of outputs

120578(119905) = 1205781(119905) 1205782(119905) 120578

119873119898

(119905)119879 modal displacement

120578(119905) = 1205781(119905) 1205782(119905) 120578

119873119898

(119905)119879 modal velocity 119906(119905) =

1199061(119905) 1199062(119905) 119906

119873119906

(119905)119879 input vector 119903

119894 spatial coordi-

nates 119883119905(119905) = 119909(119903

1 119905) 119909(119903

2 119905) 119909(119903

119873119910

119905)119879 output vec-

tor Ω = diag1205961 1205962 120596

119873119898

natural frequency 120589 =

diag1205771 1205772 120577

119873119898

modal damping 120595119894119895 mode shape 119894 at

location 119895


1 1 1 1 1 1A B C D E F

119909

119911

Base

B minus 1

Ground

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

10th

11th

12th

13th

14th

15th

16th

17th

18th

19th

20th

lowast lowast


input matrix

119881 =[[

[

12059511

sdot sdot sdot 1205951119873119906

d

1205951198731198981 sdot sdot sdot 120595

119873119898119873119906

]]

]

(4)

output matrix

119882 =[[

[

12059511

sdot sdot sdot 1205951198731198981

d

1205951119873119910

sdot sdot sdot 120595119873119898119873119910

]]

]

(5)


= 119860119904 (120579) 119911 + 119861

119904 (120579) 119906 (6)

119883119905= 119862119904 (120579) 119911 + 119863

119904 (120579) 119906 (7)





119894 120589119894 and 120595

119894)119894=1119899


119904 119861119904 119862119904 and119863

119904


3 MR Damper Model


119865 = 1198621

119910 + 1198961(119909 minus 119909

119900) (8)

119910 =1

119888119900+ 1198881

[120572119911 + 119888119900 + 119896119900(119909 minus 119910)] (9)

= minus1205741003816100381610038161003816 minus 119910

1003816100381610038161003816 119911|119911|119899minus1


120572 = 120572119886+120572119887119906 (11)

1198881= 1198881119886+1198621119887119906 (12)

119888119900= 119888119900119886 +119862

119900119887119906 (13)

= minus120578 (119906 minus V) (14)





0

500

1000

Forc

e (N

)

01 02 03 04 05 06 07 08 09Time (s)

minus1500

minus1000

minus500

Force (sim)


Fit 9306Fit 8131


01 02 03 04 05 06 07 08

0

500

1000

1500

2000

Time (s)

MR

dam

per f

orce

(N)



minus1500

minus1000

minus500



119909119891= 119860119891119909119891+ 119861119891119906119891

119884119891= 119862119891119909119891+119863119891119906119891

(15)

where 119909119891







4 Controller Design




119891= 119860119891119909119891+ 119861119891119884119884

119865 = 119862119891119909119891+ 119863119891119884119884

(16)



119891are dynamics



119891= 119860119904119911 + 119861119904119865

119883119905= 119862119904119911 + 119863

119904119865

(17)



119860119886= [

119860119891

0

119861119904119862119891

119860119904

]

119861119886= [

119861119891

119861119904119863119891

]

119862119886= lfloor119863119904119862119891 119862

119904rfloor

119863119886= lfloor119863119891119863119904rfloor

(18)

where 119860119886 119861119886 119862119886 and 119863




11990911199092


Structure


Gain

MR damper force

Structure model

Controller

1199091 1199092

119870119886

119870119888

sum

sum


Matrices 119860119886 119861119886 and 119862



119870119886= 119878∘119862119879

119886119877minus1 (19)





119878 = 119860119886119878 + 119878119860

119879minus 119878119862119879

119886119877minus1119862119886119878 + 119861119886119876119861119879

119886 (20)


0 5 10 15 20 25 30

0

5El Centro

0 5 10 15 20 25 30

0010203

Time (s)

Time (s)

minus5


Acce

lera

tion

(ms2)

minus02

minus01

Disp

lace

men

t (m





119870119886= [

[119870119891]

[119870119904]

] =

[[[[[[[

[

[1198701

1198702

]

[[

[

1198703

1198702119899+2

]]

]

]]]]]]]

]

(21)



119891in (16) and


119905of (17))





Time (s)0 5 10 15 20 25 30

0123

Hachinohe

0 5 10 15 20 25 30

00102

Time (s)


Acce

lera

tion

(ms2)

minus03

minus02

minus01

Disp

lace

men

t (m

)


minus1

minus2


0 5 10 15 20 25 30

0

10

20 Kobe

0 5 10 15 20 25 30

0

05

1

Time (s)

Time (s)

minus20

minus10


Acce

lera

tion

(ms2)

minus1

minus05

Disp

lace

men

t (m

)




Time (s)0 5 10 15 20 25 30

05

10Northridge

0 5 10 15 20 25 30

0

05

Time (s)


Acce

lera

tion

(ms2)

minus1

minus05

Disp

lace

men

t (m


minus15

minus10

minus5



5 Conclusions




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 1 1 1 1 1A B C D E F

119909

119911

Base

B minus 1

Ground

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

10th

11th

12th

13th

14th

15th

16th

17th

18th

19th

20th

lowast lowast


input matrix

119881 =[[

[

12059511

sdot sdot sdot 1205951119873119906

d

1205951198731198981 sdot sdot sdot 120595

119873119898119873119906

]]

]

(4)

output matrix

119882 =[[

[

12059511

sdot sdot sdot 1205951198731198981

d

1205951119873119910

sdot sdot sdot 120595119873119898119873119910

]]

]

(5)


= 119860119904 (120579) 119911 + 119861

119904 (120579) 119906 (6)

119883119905= 119862119904 (120579) 119911 + 119863

119904 (120579) 119906 (7)





119894 120589119894 and 120595

119894)119894=1119899


119904 119861119904 119862119904 and119863

119904


3 MR Damper Model


119865 = 1198621

119910 + 1198961(119909 minus 119909

119900) (8)

119910 =1

119888119900+ 1198881

[120572119911 + 119888119900 + 119896119900(119909 minus 119910)] (9)

= minus1205741003816100381610038161003816 minus 119910

1003816100381610038161003816 119911|119911|119899minus1


120572 = 120572119886+120572119887119906 (11)

1198881= 1198881119886+1198621119887119906 (12)

119888119900= 119888119900119886 +119862

119900119887119906 (13)

= minus120578 (119906 minus V) (14)





0

500

1000

Forc

e (N

)

01 02 03 04 05 06 07 08 09Time (s)

minus1500

minus1000

minus500

Force (sim)


Fit 9306Fit 8131


01 02 03 04 05 06 07 08

0

500

1000

1500

2000

Time (s)

MR

dam

per f

orce

(N)



minus1500

minus1000

minus500



119909119891= 119860119891119909119891+ 119861119891119906119891

119884119891= 119862119891119909119891+119863119891119906119891

(15)

where 119909119891







4 Controller Design




119891= 119860119891119909119891+ 119861119891119884119884

119865 = 119862119891119909119891+ 119863119891119884119884

(16)



119891are dynamics



119891= 119860119904119911 + 119861119904119865

119883119905= 119862119904119911 + 119863

119904119865

(17)



119860119886= [

119860119891

0

119861119904119862119891

119860119904

]

119861119886= [

119861119891

119861119904119863119891

]

119862119886= lfloor119863119904119862119891 119862

119904rfloor

119863119886= lfloor119863119891119863119904rfloor

(18)

where 119860119886 119861119886 119862119886 and 119863




11990911199092


Structure


Gain

MR damper force

Structure model

Controller

1199091 1199092

119870119886

119870119888

sum

sum


Matrices 119860119886 119861119886 and 119862



119870119886= 119878∘119862119879

119886119877minus1 (19)





119878 = 119860119886119878 + 119878119860

119879minus 119878119862119879

119886119877minus1119862119886119878 + 119861119886119876119861119879

119886 (20)


0 5 10 15 20 25 30

0

5El Centro

0 5 10 15 20 25 30

0010203

Time (s)

Time (s)

minus5


Acce

lera

tion

(ms2)

minus02

minus01

Disp

lace

men

t (m





119870119886= [

[119870119891]

[119870119904]

] =

[[[[[[[

[

[1198701

1198702

]

[[

[

1198703

1198702119899+2

]]

]

]]]]]]]

]

(21)



119891in (16) and


119905of (17))





Time (s)0 5 10 15 20 25 30

0123

Hachinohe

0 5 10 15 20 25 30

00102

Time (s)


Acce

lera

tion

(ms2)

minus03

minus02

minus01

Disp

lace

men

t (m

)


minus1

minus2


0 5 10 15 20 25 30

0

10

20 Kobe

0 5 10 15 20 25 30

0

05

1

Time (s)

Time (s)

minus20

minus10


Acce

lera

tion

(ms2)

minus1

minus05

Disp

lace

men

t (m

)




Time (s)0 5 10 15 20 25 30

05

10Northridge

0 5 10 15 20 25 30

0

05

Time (s)


Acce

lera

tion

(ms2)

minus1

minus05

Disp

lace

men

t (m


minus15

minus10

minus5



5 Conclusions




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

500

1000

Forc

e (N

)

01 02 03 04 05 06 07 08 09Time (s)

minus1500

minus1000

minus500

Force (sim)


Fit 9306Fit 8131


01 02 03 04 05 06 07 08

0

500

1000

1500

2000

Time (s)

MR

dam

per f

orce

(N)



minus1500

minus1000

minus500



119909119891= 119860119891119909119891+ 119861119891119906119891

119884119891= 119862119891119909119891+119863119891119906119891

(15)

where 119909119891







4 Controller Design




119891= 119860119891119909119891+ 119861119891119884119884

119865 = 119862119891119909119891+ 119863119891119884119884

(16)



119891are dynamics



119891= 119860119904119911 + 119861119904119865

119883119905= 119862119904119911 + 119863

119904119865

(17)



119860119886= [

119860119891

0

119861119904119862119891

119860119904

]

119861119886= [

119861119891

119861119904119863119891

]

119862119886= lfloor119863119904119862119891 119862

119904rfloor

119863119886= lfloor119863119891119863119904rfloor

(18)

where 119860119886 119861119886 119862119886 and 119863




11990911199092


Structure


Gain

MR damper force

Structure model

Controller

1199091 1199092

119870119886

119870119888

sum

sum


Matrices 119860119886 119861119886 and 119862



119870119886= 119878∘119862119879

119886119877minus1 (19)





119878 = 119860119886119878 + 119878119860

119879minus 119878119862119879

119886119877minus1119862119886119878 + 119861119886119876119861119879

119886 (20)


0 5 10 15 20 25 30

0

5El Centro

0 5 10 15 20 25 30

0010203

Time (s)

Time (s)

minus5


Acce

lera

tion

(ms2)

minus02

minus01

Disp

lace

men

t (m





119870119886= [

[119870119891]

[119870119904]

] =

[[[[[[[

[

[1198701

1198702

]

[[

[

1198703

1198702119899+2

]]

]

]]]]]]]

]

(21)



119891in (16) and


119905of (17))





Time (s)0 5 10 15 20 25 30

0123

Hachinohe

0 5 10 15 20 25 30

00102

Time (s)


Acce

lera

tion

(ms2)

minus03

minus02

minus01

Disp

lace

men

t (m

)


minus1

minus2


0 5 10 15 20 25 30

0

10

20 Kobe

0 5 10 15 20 25 30

0

05

1

Time (s)

Time (s)

minus20

minus10


Acce

lera

tion

(ms2)

minus1

minus05

Disp

lace

men

t (m

)




Time (s)0 5 10 15 20 25 30

05

10Northridge

0 5 10 15 20 25 30

0

05

Time (s)


Acce

lera

tion

(ms2)

minus1

minus05

Disp

lace

men

t (m


minus15

minus10

minus5



5 Conclusions




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










11990911199092


Structure


Gain

MR damper force

Structure model

Controller

1199091 1199092

119870119886

119870119888

sum

sum


Matrices 119860119886 119861119886 and 119862



119870119886= 119878∘119862119879

119886119877minus1 (19)





119878 = 119860119886119878 + 119878119860

119879minus 119878119862119879

119886119877minus1119862119886119878 + 119861119886119876119861119879

119886 (20)


0 5 10 15 20 25 30

0

5El Centro

0 5 10 15 20 25 30

0010203

Time (s)

Time (s)

minus5


Acce

lera

tion

(ms2)

minus02

minus01

Disp

lace

men

t (m





119870119886= [

[119870119891]

[119870119904]

] =

[[[[[[[

[

[1198701

1198702

]

[[

[

1198703

1198702119899+2

]]

]

]]]]]]]

]

(21)



119891in (16) and


119905of (17))





Time (s)0 5 10 15 20 25 30

0123

Hachinohe

0 5 10 15 20 25 30

00102

Time (s)


Acce

lera

tion

(ms2)

minus03

minus02

minus01

Disp

lace

men

t (m

)


minus1

minus2


0 5 10 15 20 25 30

0

10

20 Kobe

0 5 10 15 20 25 30

0

05

1

Time (s)

Time (s)

minus20

minus10


Acce

lera

tion

(ms2)

minus1

minus05

Disp

lace

men

t (m

)




Time (s)0 5 10 15 20 25 30

05

10Northridge

0 5 10 15 20 25 30

0

05

Time (s)


Acce

lera

tion

(ms2)

minus1

minus05

Disp

lace

men

t (m


minus15

minus10

minus5



5 Conclusions




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Time (s)0 5 10 15 20 25 30

0123

Hachinohe

0 5 10 15 20 25 30

00102

Time (s)


Acce

lera

tion

(ms2)

minus03

minus02

minus01

Disp

lace

men

t (m

)


minus1

minus2


0 5 10 15 20 25 30

0

10

20 Kobe

0 5 10 15 20 25 30

0

05

1

Time (s)

Time (s)

minus20

minus10


Acce

lera

tion

(ms2)

minus1

minus05

Disp

lace

men

t (m

)




Time (s)0 5 10 15 20 25 30

05

10Northridge

0 5 10 15 20 25 30

0

05

Time (s)


Acce

lera

tion

(ms2)

minus1

minus05

Disp

lace

men

t (m


minus15

minus10

minus5



5 Conclusions




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article optimal control of magnetorheological

Documents