Research ArticleFree Vibration Response of a Frame Structural ModelControlled by a Nonlinear Active Mass Driver System

Ilaria Venanzi and Filippo Ubertini

Department of Civil and Environmental Engineering University of Perugia Via G Duranti 93 06025 Perugia Italy

Correspondence should be addressed to Ilaria Venanzi ilariavenanziunipgit

Received 21 November 2013 Revised 24 February 2014 Accepted 24 February 2014 Published 27 March 2014

Academic Editor John Mander

Copyright copy 2014 I Venanzi and F Ubertini 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

Active control devices such as active mass dampers are mainly employed for the reduction of wind-induced vibrations in high-rise buildings with the final aim of satisfying vibration serviceability limit state requirements and of meeting appropriate comfortcriteria When such active devices normally operating under wind loads associated with short return periods are subjected toseismic events they can experience large amplitude vibrations and exceed stroke limitsThis may lead to a reduced performance ofthe control system that can even worsen the performance of the whole structure In this paper a nonlinear control strategy basedon a modified direct velocity feedback algorithm is proposed for handling stroke limits of an active mass driver (AMD) systemIn particular a suitable nonlinear braking term proportional to the relative AMD velocity is included in the control law in orderto slowdown the device in the proximity of the stroke limits Experimental and numerical free vibration tests are carried out on ascaled-down five-story frame structure equipped with an AMD to demonstrate the effectiveness of the proposed control strategy

1 Introduction

Active control systems are in principle very effective for themitigation of the structural response especially for high-rise buildings and flexible structures that may experiencesignificant wind-induced vibrations [1 2] However their usein practical applications is still limited by the physical boundsof the devices In the case of strong earthquakes the limits ofthe actuators may be exceeded forcing the system to operatein a nonlinear mode for which it was not designed thusworsening the performance of the controlled structure Thephysical bounds of the actuators include both the controlforce limits and the stroke limits

The problem of force saturation has been deeply studiedin the literature Some approaches deal with preventing sat-uration of the control signal by designing the control systemto always operate below its limits in the framework of linearcontrol [3] Another category of control methods accountsfor system limitations directly in the control algorithmChase et˜al [4] modified the H

infincontrol method through

the addition of nonlinear state-dependent terms in orderto model the actuators saturation and the uncertainties in

the parameters of the system Indrawan et al [5] developedthe bound-force control method which excludes the control-effort penalty from the performance index defined in thecase of LQR control defines it at the end of each timeinterval and seeks the optimal control force for each timeinterval Another control strategy based on solving in realtime the classic linear quadratic regulator (LQR) problemwith adaptive weights and system matrices is representedby the ldquostate-dependent Riccati equationrdquo (SDRE) [6 7]Materazzi and Ubertini [8] developed an application of theSDRE for fully constrained systems with both physical limitsand actuators saturation A state feedback control law and anobserver-based controller were proposed by Kim and Jabbari[9] The bang-bang control which minimizes a performanceindex subjected to the control force constraint has beenwidely investigated by several authorsThemain shortcomingof the bang-bang control is the undesirable control chatteringnear the origin of the state space due to high frequencyswitching of the control force To overcome this problemMongkol et al [10] proposed the linear saturation (LS) controlalgorithm that consists of a low-gain linear controller whenthe system is close to the zero state and a bang-bang controller

Hindawi Publishing CorporationAdvances in Civil EngineeringVolume 2014 Article ID 745814 11 pageshttpdxdoiorg1011552014745814

2 Advances in Civil Engineering

1

qn

qa

q2

q3

q1

mn

ma

J

m2

m3

m1

kn

k2

k3

k1

uT

(a)

2 3

4

56

l

120599

(b)

Figure 1 Multistory frame structure with AMD made of electric torsional servomotor (element 1) and ball screw (a) detailed view of ballscrew components (b) additional carried mass (2) return tube (3) screw (4) bearing balls (5) and ball nut (6)

otherwise Wu and Soong [11] introduced the suboptimalbang-bang control strategy described by a function of thestate where the control force is determined by minimizingthe time derivative of a quadratic Lyapunov function underthe control force constraint This method was found to beeffective under a certain range of control force but it can beunstable outside of this range To overcome this instabilityLim [12] proposed an adaptive bang-bang control algorithm

The problem of exceeding the stroke limits is perhaps themost important constraint for application of AMD systemsto actual structures but it has not been widely discussedin the literature Nagashima and Shinozaki [13] developed avariable gain feedback (VGF) control for buildings equippedwith AMD systems The variable feedback gain is a functionof a quantity representing the tradeoff between the reductionof the building response and the amplitude of themass strokeand this quantity is on-line controlled to keep the strokewithin its limits Yamamoto and Sone [14] utilized a linearlyvariable gain proportional to an index representing the activ-ity of an AMD that is a function of its stroke displacementthe first modal frequency of the controlled building and thestroke limit of the AMD Within the framework of the state-dependent Riccati equation Friedland [7] proposed to applyto the system a physical constraint modeled by a nonlinearspring which provides a state-dependent restoring force thatis accounted for in the controller design

Authors have recently started a research program oninnovative solutions for structural control On one sideanalytical studies on the definition of original control algo-rithms for constrained systems and on the optimizationof noncollocated control systems [15 16] were developedOn the other side laboratorial experimental work has beencarried out to implement a control system made of an activemass driver system [17] The prototype AMD is made ofa torsional electric servomotor that moves a small massthrough a ball screw

In this paper a nonlinear control strategy is proposed thatis capable of preventing crossing of the stroke limits of anAMD system To this aim a skyhook control algorithm is

modified by adding to the control force a nonlinear term thatincreases in the vicinity of the stroke bounds To reduce theimpulsive effect of this additional braking term the functionsmoothly varies within the fixed boundaries To demonstratethe effectiveness of the proposed approach experimental testswere carried out on a reduced-scale five-story frame structureequipped with the AMD and subjected to free vibrationtests The effect of a variation in the parameters defining thenonlinear control force is discussed and the optimal choiceof such parameters is identified Numerical analyses are alsocarried out to correctly interpret the experimental resultsTheeffectiveness of the proposed control algorithm is comparedto that of the classical skyhook algorithm highlighting thevalidity of the new approach

2 Modeling of the Systemrsquos Dynamics

An 119899-story planar frame structure equipped with an activemass driver system placed on top is considered as shown inFigure 1 Horizontal displacements of the stories relative tothe ground are denoted as 119902

1 1199022 119902

119899 while story masses

and stiffness are denoted as1198981 1198982 119898

119899and 1198961 1198962 119896

119899

respectivelyThe AMD system is composed of a ball screw that

converts the rotational motion of an electric torsional servo-motor into the translational motion of a mass 119898

119886 The total

moving mass of the AMD 119898119886 is the sum of the mass of the

ball nut and the additional carried mass (elements 2 and 6in Figure 1) The displacement of the movable mass relativeto the ground is denoted by 119902

119886 while the rotation of the ball

screw is denoted by 120599 As the motor rotates one revolutionthe movable mass is advanced one pitch 119897 of the screwThusthe following kinematic relation holds for the AMD

119897

2120587120599 = 119902119886minus 119902119899= 119902119886minus F119879q (1)

where q = [1199021 1199022 119902

119899]119879 and F = [0 0 1]119879 are the 119899-

dimensional collocation vector of the AMD

Advances in Civil Engineering 3

The servomotor is commanded by assigning an angularvelocity to the servodrive of the motor which amplifies sucha signal and provides proportional electric current to themotor An encoder reports the actual status (actual value ofthe angular velocity measured by the encoder) of the motorto the servodrive which corrects input current by calculatingthe deviation between commanded and actual status Becausethe device actuation is imperfect the demand value of theangular velocity ( 120599

119863) and its actual value ( 120599

119860) are in general

different This internal dynamics of the actuator can bemodeled by accounting for the transfer function betweenvelocity demand value and velocity actual value Howeverin the specific problem under consideration in which themovablemass and the inertia of the screw are small comparedto the torque capacity of the servomotor and the commandsignal is not excessively fast imperfect actuation of the deviceis neglected and demand and actual values of the angularvelocity are considered coincident

The free response of the system is governed by the fol-lowing equation of motion readily obtained using Lagrangersquosequations [17]

(M + 119898119886FF119879) q + Cq + Kq = minus119898119886119897

2120587F 120599

(119869 +1198981198861198972

41205872) 120599 = 119906

119879( 120599) minus

119898119886119897

2120587Fq

(2)

where M C and K are 119899 times 119899 mass damping and systemmatrices of the frame structure J is the torsional massmoment of inertia of the ball screw and 119906

119879( 120599) is the torque

provided by the servomotor for the commanded value of theangular velocity

For active damping of the dynamic response of thesubstructure it is particularly useful to regulate the inertialforce 119906 acting on the movable massThis last is simply givenby

119906 = 119898119886119902119886 (3)

Substituting (1) into (3) and integrating in time under theassumption of zero initial conditions yield the followingequation

120599 =2120587

119898119886119897int119905

0

119906 (120591) d120591 minus 2120587119897119902119899 (4)

which provides the angular velocity to be commanded to themotor for providing the desired control force 119906

3 Nonlinear Control Strategy forHandling Stroke Limits

The movement 119909 of real inertial actuators (in the presentcase 119909 = 119902

119886minus 119902119899) is limited by a maximum value 119909max

corresponding to the physical stroke extension Mathemat-ically such a limitation is represented by a nonholonomicconstraint that can be modeled by introducing a propernonlinear restoring force acting on the movable mass Such

a force can for instance have the following expression asproposed in [7]

120601 (119909) = (119909

119909max)

2119873+1

(5)

Equation (5) corresponds tomodeling the impact of the massagainst the stroke limit as an elastic (nondissipative) impactwhere 119873 is an integer parameter that increases with theincreasing rigidity of the physical constraint

The motion of the inertial mass is in general governed bya feedback control algorithm that accomplishes the task ofreducing the vibrations of the substructureWhen the relativedisplacement 119909 between the mass and the substructurereaches the value 119909max an impact occurs with a consequentexchange of the impulsive force 120601 between the mass andthe substructure This clearly reduces the performance ofthe control system and can produce damages to the controldevice as well as to the structural system

We propose to introduce the stroke limit 119909max as a param-eter in the control law by coupling a constant gain feedbackcontroller deputed to dampen structural vibrations with anonlinear controller deputed to handle stroke limitation Wedenote by 119906

0the control signal regulated by the constant gain

controller which is expressed as

1199060= minus119891 (119909) (6)

where 119909 = [119902 119902]119879 is the state vector of the system and

119891 is an appropriate function This last is usually a linearfunction that can be designed in such a way to guarantee theasymptotic stability of the system and to satisfy some optimalperformance criteria In order to handle the physical limita-tions imposed on the stroke extension being impossible to acton 119909max it is necessary to gradually stop the actuator when 119909approaches 119909maxThe easiest way to do it is to introduce in (2)a dissipative force that becomes effective close to the physicallimit and is nil elsewhere Such a force can be obtained witha derivative term with a variable gain given by a constant119866NL multiplied by a nonlinear 119909-dependent coefficient 0 le120576NL(119909) le 1

119906 = 1199060minus 119866NL120576NL (119909) (7)

Equation (7) is in the following referred to as ldquoNLC algo-rithmrdquo where NLC stands for nonlinear control

The simplest function 120576NL(119909) to introduce in (7) is a stepfunction that assumes a nil value when |119909| is less than 119896sdot119909maxwith 0 le 119896 le 1 and a unit value elsewhereUnfortunately thisfunction which can be expressed as 120576NL(119909) = 119867(|119909|minus119896sdot119909max)(119867 denoting the Heaviside step function) is discontinuousfor |119909| = 119896 sdot 119909max which determines the application of anabrupt brake force requiring a very large electrical power

In practice the step function is not applicable and does notsolve the problem of the abrupt stop of the actuator In orderto solve this problem it is necessary to consider a function120576NL(119909) which smoothly varies between 0 and 1 in an intervalcomprised between 119896

0119909max and 1198961119909max with 0 le 1198960 le 1198961 le 1

A similar function is here chosen as follows

4 Advances in Civil Engineering

120576NL (119909) =

1 |119909| ge 1198961119909max

1 minus exp( minus1

1 minus ((|119909| minus 1198960119909max) (1198961119909max minus 1198960119909max))

2+ 1) 119896

0119909max lt |119909| lt 1198961119909max

0 |119909| le 1198960119909max

(8)

It is worth noting that 120576NL(119909) given by (8) is an infinitelydifferentiable function In particular it does not exhibit anydiscontinuity and its slope is always finite This ensures thegradual application of the brake force and the reduction ofthe associated power A graphical representation of func-tion 120576NL(119909) is represented in Figure 2(a) It can be shownthat 120576NL(119909) tends in norm in the 1198712 functional space tothe Heaviside function when 119896

0and 119896

1tend to the same

value 119896 This circumstance is graphically represented inFigure 2(b)

It should be noticed that the linearization of (7) in theorigin of the state space here assumed as a fixed equilibriumpoint of the system coincides with the linearization of (6)Therefore the asymptotic stability of the closed-loop systemwith control algorithm given by (6) also ensures the localasymptotic stability of the NLC algorithm with 120576NL(119909) givenby (8)

A classic skyhook control algorithm is chosen in thiswork for specializing (6) This strategy results in applyingan inertial force to the movable mass that is proportionalto the velocity of the top floor relative to the groundand offers the main advantages of being relatively sim-ple to implement and resulting in a significant dampingeffectiveness

By regulating 1199060through a skyhook control algorithm (7)

is specialized as follows

119906 = 119866119871119902119899minus 119866NL120576NL (119909) (9)

where 119866119871is the constant gain of the skyhook strategy

The demand value of the angular velocity correspondingto (9) is obtained by substituting such equation into (4)and by time integration under the assumption of zero initialconditions The following control algorithm is thus obtainedwhere integration by parts of the nonlinear term in (9) hasbeen carried out

120599 =2120587

119897(119866119871119902119899

119898119886

minus119866NL120576NL (119909) 119909

119898119886

+119866NL int

119905

0

1205761015840NL119909 d120591119898119886

minus 119902119899)

(10)

In (10) 1205761015840NL is the derivative of 120576NL with respect to 119909 Thisquantity is shown in Figure 2(c) and is given by

1205761015840

NL (119909)

=

0 |119909| ge 1198961119909max

2 (|119909| minus 1198960119909max) sign (119909) exp(((minus1) (1 minus ((|119909| minus 1198960119909max) (1198961119909max minus 1198960119909max))

2

)minus1

) + 1)

(1 minus ((|119909| minus 1198960119909max) (1198961119909max minus 1198960119909max))

2

)2

(1198961119909max minus 1198960119909max)

2

1198960119909max lt |119909| lt 1198961119909max

0 |119909| le 1198960119909max

(11)

where sign(119909) is the signum function It should be noticedthat 1205761015840NL(119909) locally tends to Diracrsquos delta function for 119896

0and

1198961approaching the same value 119896In this work the integral term appearing in (10) is

disregarded under the assumption that it is small comparedto the term containing 120576NL(119909) This is true for 119896

0which

is sufficiently smaller than 1198961 that is for a small value of

1205761015840NL(119909) also because the product between 119909 and 119899is small

compared to 119909 Moreover 119902119899and 119902

119899in (10) are obtained by

online double and single integrations of acceleration signalsrespectively

4 Description of the Experimental Set-Up

The proposed control strategy was tested in the laboratoriesof the Department of Civil and Environmental Engineeringof University of Perugia A physical model of the structure-AMD system described in Section 2 was designed and con-structedThe experimental set-up shown in Figure 3 ismadeof the following components (i) frame structure (ii) activemass driver (iii) monitoring sensors and (iv) controller

The test structure is a five-story single-bay frame shownin Figure 3 having a total height of about 19mThe structureis made of steel S 235 whose nominal value of yield strength

Advances in Civil Engineering 5

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 10

05

1

120576 NL

k0 = 05xmax k1 = 09xmax

xxmax

(a)

0

05

1

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

120576 NL

k0 = 050xmax k1 = 090xmaxk0 = 060xmax k1 = 080xmaxk0 = 065xmax k1 = 075xmaxk0 = 069xmax k1 = 071xmax

xxmax

(b)

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus150

minus750

75150

120576998400 NL

xxmax

(c)

Figure 2 Nonlinear function in (8) for a particular choice ofparameters 119896

0and 119896

1(a) nonlinear function in (8) for 119896

0and

1198961approaching the same value 119896 (b) derivative of the nonlinear

function in (8) expressed by (11) for 1198960and 119896

1approaching the same

value 119896 (c)

is 235Nmm2 according to the Eurocode 3 [18] Each flooris made of 700 times 300 times 20mm steel plates of about 328 kgweightThe structural configuration consists of the use of 5 times30 times 340mm columns for the lowest two stories and 4 times 30times 340mm in the remaining ones The structure is fixed to asupport table

The AMD installed on the top floor is composed by a600mm long ball screw that converts the rotationalmotion ofaKollmorgenAKM33HAC servomotor into the translationalmotion of a 4 kgmass block (approximately 25 of the wholestructural mass) The maximum speed of the servomotor is8000 roots per minute and the peak torque is 1022Nm Themaximum stroke of the small mass is equal to plusmn300mmwhile the ball screw has a diameter of 25mm and a pitch of25mm The whole AMD system except for its control unit(drive of the servomotor) is mounted on an aluminum platedesigned for the purpose that also serves as top floor of thestructure

The position of the mass along the ball screw is measuredbymeans of a JX-P420 linear position transducer Two opticalsensors connected to a logical circuit disable the drive of the

motor when the mass exceeds the stroke limits in order toprevent damages of the actuator

The controller is a National Instruments PXIe-8133 Corei7-820QM 173GHz which mounts on board a PXIe-6361X Series Multifunction DAQ card with 8 channels in differ-ential input The software installed on the controller is theLabVIEW (LV) real-time The system fully stands alone andrequires the network connection with a host PC only forthe initial configuration of the LV code The communicationbetween the controller and the drive of the motor is based onthe EtherCAT protocol

Six PCB 393C accelerometers are mounted on each floorfor the purpose of monitoring structural accelerations Theaccelerometers are connected through short cables to theDAQ card by means of the SCB-68 noise rejecting connectorblock

5 Experimental and Numerical Tests

Free vibration experimental tests were carried out in orderto investigate the effectiveness of the control strategy andto compare experimental results with numerical predictionsTests were carried out by varying the parameters of the con-trol law to examine their effect on the control effectivenessThe experimental and numerical results were compared tothose obtained with the classical skyhook algorithm high-lighting the validity of the proposed strategy for handlingstroke limitations

51 Free Vibration Tests To excite the frame structure theAMD was used as a harmonic shaker located at the top floorof the structure and then after 10 forcing cycles theAMDwasswitched to control The frequency of excitation was 18Hzclose to the first natural frequency of the frame system

Two different control strategies were compared (1) thelinear ldquoskyhookrdquo control law (LC) (2) the proposed nonlinearcontrol law (NLC) The corresponding control forces areexpressed as

119906 = 119866119871119902119899(LC)

119906 = 119866119871119902119899minus 119866NL120576NL (119909) (NLC)

(12)

The corresponding demand values of the angular velocity areexpressed as

120599 =2120587

119897(119866119871119902119899

119898119886

minus 119902119899) (LC) (13)

120599 =2120587

119897(119866119871119902119899

119898119886

minus119866NL120576NL (119909) 119909

119898119886

+119866NL int

119905

0

1205761015840

NL119909 d120591119898119886

minus 119902119899) (NLC)

(14)

In Table 1 are summarized the parameters of the controlalgorithms The gains 119866

119871and 119866NL (12) were set equal to

150 and the stroke limit was assumed to be the 90 of the

6 Advances in Civil Engineering

Servomotor

(a)

(b)

(c)Potentiometer

Movable mass

Ball screw

Servomotor

Potentiometer

Movable mass

Ball screw

Figure 3 Experimental set-up overview of the system (a) detailed views of the AMD (b-c)

Table 1 Performance indices of the control effectiveness computedadopting the experimental results

Parameter Value119866119871

150119866NL 150119909max 68 cm1198961

081198960

02ndash07

maximum stroke of the movable mass obtained with the LCstrategy The coefficient 119896

1was selected as 08 and 119896

0was

varied between 02 and 07Figure 4 shows the top floor displacements of the uncon-

trolled and controlled system with LC and NLC strate-gies During the loading phase the structural displacementsincrease while when the control is switched on the structuralresponse is rapidly decreased by both of the adopted controlapproaches

Figure 5 shows the results in terms of AMD displace-ments and top floor displacements obtained with the LC andthe NLC algorithms for 119896

0= 02 and 119896

1= 08 It is also shown

for comparison with the nonlinear velocity demand that isactivated when the AMD stroke exceeds the value 119896

0119909max

Neglecting the integral term in (14) the nonlinear velocitydemand is the difference between the total velocity demandsexpressed by (14) and (13) and is written as

120599NL =2120587

119897

119866NL120576NL (119909) 119909

119898119886

(15)

It is possible to observe that adopting the NLC strategythe AMD stroke never exceeds 119909max Although the controlalgorithm is not recentering the nonlinear algorithm con-tributes to limiting the residual AMD displacement thusavoiding the need for manual or automatic recentering afterthe seismic event The small residual displacement regardsthe AMD only while the substructure elastically recoversits initial configuration after the vibration is dissipated The

0 2 4 6 8 10 12

0

001

002

Time (s)

UNCLC

NLCControl on

minus001

minus002

Top

floor

disp

lace

men

ts (m

)

Figure 4 Top floor displacements for the uncontrolled (UNC) andcontrolled systems with LC and NLC (119896

0= 02) strategies

difference between the structural response obtained withLC and NLC algorithms appears after the first peak of thecontrolled response when the AMD stroke has exceeded thevalue 119896

0119909max and the nonlinear restoring force is activated

As expected after the second peak the control performanceobtained with the NLC algorithm is slightly worse than thatobtained with the LC law However the damping providedto the structure is sufficient to rapidly decrease the structuralresponse in both cases (Figure 4)

Figure 6 shows the velocity demand obtained with boththe classic skyhook (LC) and the proposed nonlinear control(NLC) strategies (13)-(14) The difference of the two lines inFigure 6 represents the nonlinear velocity demand plottedat the bottom of Figure 5 A peak of the AMD velocity incorrespondence of the first braking of the movable mass isvisible After the first peak the velocity demand required bythe two algorithms is almost comparable

Advances in Civil Engineering 7

0 2 4 6 8 10 12minus005

0005

01

disp

lace

men

ts (m

)

LCNLC

Control on

Time (s)

xmax

AM

D

k1 lowastxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus002minus001

0

002001

Top

floor

LCNLCControl on

0 2 4 6 8 10 12Time (s)

disp

lace

men

ts (m

)

(b)

minus200

2000

600

1000800

400

Non

linea

r vel

ocity

NLCControl on

0 2 4 6 8 10 12Time (s)

dem

and

(rad

s)

(c)Figure 5 AMD displacements (a) top floor displacements (b)nonlinear velocity demand expressed by (15) (c) for the controlledsystem

In practical applications control force saturation canlikely occur as the nonlinear control system requires asignificant power demand to produce the first braking ofthe AMD To study the effect of force saturation additionaltests were carried out In particular by properly setting thecontroller the actual velocity supplied by the servomotor waslimited to 150 rads value well below the physical limit of thesystem Figure 7 shows that the proposed control algorithmis effective in reducing the structural response and avoidingthe stroke limits crossing even in presence of such forcesaturation

To quantify the effectiveness of the control strategy thefollowing performance indices are defined

1198691=

max119905gt1199050

1003816100381610038161003816119902119886 minus 11990251003816100381610038161003816NLC

max119905gt1199050

1003816100381610038161003816119902119886 minus 11990251003816100381610038161003816LC

1198692=

max119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

max119905gt1199050 119894

1003816100381610038161003816119902LC119894

1003816100381610038161003816

0 2 4 6 8 10 12

150

Time (s)

Velo

city

dem

and

valu

e (ra

ds)

LCNLCControl on

minus50

minus250

minus450

minus650

Figure 6 Velocity demand obtained with LC and NLC (1198960= 02)

strategies

Table 2 Performance indices of the control effectiveness computedadopting the experimental results

1198960

1198691

1198692

1198693

1198694

1198695

02 0749 0987 1102 0995 464203 0767 1001 1077 1004 480304 0843 1012 1010 1208 511705 0857 1008 1139 1268 533206 0862 1005 0947 1214 554107 0891 1059 1109 1226 5603

1198693=

rms119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

rms119905gt1199050 119894

1003816100381610038161003816119902LC119894

1003816100381610038161003816

1198694=

max119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

max119905gt1199050 119894

1003816100381610038161003816 119902LC119894

1003816100381610038161003816

1198695=

max119905gt1199050

10038161003816100381610038161003816120599NLC119863

10038161003816100381610038161003816

max119905gt1199050

10038161003816100381610038161003816120599LC119863

10038161003816100381610038161003816

(16)

where 1198691is the ratio of the maximum relative displacements

between the AMD (119902119886) and the top floor (119902

5) obtained with

the NLC and the LC strategies 1198692is the ratio of themaximum

floor displacements (119902119894) obtained with the NLC and the

LC strategies 1198693is the ratio of the root mean square of

the floor displacements obtained with the NLC and the LCstrategies 119869

4is the ratio of the maximum floor accelerations

( 119902119894) obtained with the NLC and the LC strategies 119869

5is the

ratio of the maximum velocity demand value ( 120599119863) obtained

with the NLC and the LC strategiesIn Table 2 are shown the performance indices corre-

sponding to experimental responses obtained for different

8 Advances in Civil Engineering

minus0050

00501

0 2 4 6 8 10 12

AM

D

LCNLC

Control on

Time (s)

xmax

disp

lace

men

ts (m

)

k1 lowastxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus002minus001

0

002001

LCNLCControl on

Top

floor

disp

lace

men

ts (m

)

0 2 4 6 8 10 12Time (s)

(b)

LCNLCControl on

minus300minus200minus100

100300300

0

0 2 4 6 8 10 12Time (s)

Velo

city

dem

and

valu

e (ra

ds)

(c)

Figure 7 AMD displacements (a) top floor displacements (b)nonlinear velocity demand (c) for the system controlled with LCand NLC algorithms (119896

0= 02) in case of force saturation

values of parameter 1198960The value 119896

0= 08was not considered

because as observed in Section 3 when 1198960= 1198961 120576NL(119909)

essentially coincides with Heavisidersquos step function whichcannot be reproduced in practice

The results in Table 2 show that if 1198960increases (i) the

braking efficacy decreases (as 1198691increases) (ii) the control

performance is slightly worsened (as 1198692increases) (iii) the

ratio of the root mean square of the top displacementsobtained with the NLC and the LC (119869

3) is generally higher

than one (iv) the structural acceleration induced by thebraking term increases (as 119869

4increases) and (v) due to the

additional nonlinear term the velocity demand increaseswith respect to the LC case (as 119869

5increases)

Based on the obtained results it can be concluded thatthe NLC is effective in reducing the structural responseand avoiding the stroke limit crossing at the expense of analmost negligible increase in structural displacements and aslight increase in structural accelerations For free response

Table 3 Identified modal characteristics of the system

Mode Natural frequency (Hz) Modal damping ratio ()1 176 032 492 043 734 114 962 125 1215 13

Table 4 Performance indices of the control effectiveness computedadopting the numerical results

1198960

1198691

1198692

1198693

1198694

1198696

02 0585 1000 1000 1000 194303 0619 1000 1000 1000 241204 0651 1000 1000 1000 239205 0681 1000 1000 1000 273706 0711 1000 1000 1000 326807 0740 1000 1000 1000 4097

vibrations the increase in power demand with respect to theLC case is important only for the first braking peak and thepossible force saturation does not significantly deteriorate thecontrol performance

52 Numerical Simulations and Discussion Numerical anal-yses were performed to better interpret the experimentalresults and further discuss the effectiveness of the controlstrategy

Preliminarily a numerical model was built and tuned tothe free vibration response of the uncontrolled system Amodal identification allowedobtaining an accurate numericalmodel of the system considered as a shear-type structurehaving 5 degrees of freedom The signals provided by the 5accelerometers located on the floors recorded with a sam-pling frequency of 1 kHz and a total duration of 1 hour wereused for modal identificationThe structural response signalswere acquired with excitation provided by microtremorsin the laboratory The Frequency domain decomposition(FDD) was used to perform the modal identification afterdecimation of themeasured time history signals to 50Hz andusing a frequency resolution to compute the power spectraldensity matrix of the measurements equal to 00122Hz Theresults of modal identification are summarized in Table 3After identifying the dynamic characteristics of the physicalsystem a model tuning was performed to obtain the massesand stiffness coefficients of the frame to be adopted in thenumerical model as detailed in [17] In Table 3 are alsoreported themodal damping ratios obtained by analyzing theenvelopes of the free decay responses of the five structuralmodes Single mode responses are obtained by harmonicallyexciting the structure with the AMD working as a shakerproviding excitation in resonance with a specific modeConsidering that the modal damping ratios vary slightlywith the amplitude of vibration their values are estimated inintermediate ranges of the response

Advances in Civil Engineering 9

0 2 4 6minus0125

minus01

minus0075

minus005

minus0025

0

0025

005

0075

01

0125

Time (s)

AM

D d

ispla

cem

ents

(m)

LC num LC expxmax minusxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus0125

minus01

minus0075

minus005

minus0025

0

0025

005

0075

01

0125

AM

D d

ispla

cem

ents

(m)

NLC num NLC expxmax minusxmax

0 2 4 6Time (s)

k0 lowastxmaxminusk0 lowastxmax

(b)

Figure 8 Comparison between numerical and experimental AMD displacements for LC (a) and NLC (b)

The experimental tests were reproduced numericallyapplying at the top of the structure a sinusoidal inertialforce corresponding to the acceleration of the AMD Figure 8shows the comparison between numerical and experimentalAMD displacements for both LC (GL = 150) and NLCstrategies (119866NL = 150 119896

0= 02 119896

1= 08) The compa-

rison is limited to the controlled part of the response Theexperimental and numerical results are close to each otherespecially for the first and second peaks of the responseNevertheless there are still some deviations in the rate ofdecay and the phase of the response due to the imperfectactuation of the AMD

Similar observations can be drawn for Figure 9 repre-senting the comparison between the numerical and experi-mental top floor displacements Although there is a relativelygood correspondence between the experimental and numer-ical results obtained at the beginning of the control phasethe experimental response decay appears different due to theinternal dynamics of the actuator

Figure 10 reports the comparison between the numer-ically predicted control force obtained with LC and NLCstrategies The NLC algorithm leads to an important increasein the control force at the beginning of the actuation

In Table 4 are summarized the performance indicesintroduced in Section 51 computed by adopting the numeri-cal resultsThe index 119869

5is computed by using the control force

instead of the velocity demand value as follows

1198696=max119905gt1199050

10038161003816100381610038161003816119906NLC

10038161003816100381610038161003816

max119905gt1199050

1003816100381610038161003816119906LC1003816100381610038161003816 (17)

where 119906 is given by (12) It is clear from the results that beingequal to the gain 119866NL the braking performance is reducedwith the increase of the coefficient 119896

0 The indices 119869

2 1198693 and

1198694are equal to 1 for all the analyzed cases meaning that at

the first peak the LC and NLC algorithms provide the samestructural response The index 119869

5suggests that the difference

between the control force yielded by NLC and LC strategiesincreases as 119896

0increasesThe discrepancy between the results

reported in Tables 2 and 4 can be ascribed to the imperfectexperimental actuation and the small differences between thephysical and numerical models

6 Conclusions

In this paper a nonlinear control strategy based on amodifieddirect velocity feedback algorithm is proposed for handlingstroke limits of an AMD system In particular a nonlinearbraking term proportional to the relative AMD velocity isincluded in the control algorithm in order to slowdown thedevice in the proximity of the stroke limits

Experimental and numerical tests were carried out ona scaled-down five-story building equipped with an AMDunder free vibrations

The proposed nonlinear control law proved to be effectivein reducing the structural response and avoiding the strokelimit crossing at the expense of a small increase of structuraldisplacements and a slight increase of structural accelera-tionsThe growth in power demand with respect to the linearcase is important only for the first braking peak and thepossible force saturation does not interfere significantly with

10 Advances in Civil Engineering

minus002

minus001

0

001

002

LC numLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(a)

minus002

minus001

0

001

002

NLC numNLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(b)

Figure 9 Comparison between numerical and experimental top displacements for LC (a) and NLC (b)

0 2 4 6

05

10152025

Time (s)

Con

trol f

orce

(N)

LCNLC

minus45

minus40

minus30

minus35

minus25

minus20

minus15

minus10

minus5

Figure 10 Control force obtained with the numerical analyses

the control performance The parametric analysis showedthat an increase of the ratio 119896

11198960between the parameters

defining the nonlinear restoring force results in an improvedperformance in terms of braking efficacy On the contrarythe same braking efficacy can be obtained with a lower 119896

11198960

ratio and a higher gain 119866NL at the expense of an increase inthe structural response especially in terms of accelerationsThe optimal choice of the ratio 119896

11198960should be obtained as a

compromise between the need of limiting the power demandand that of limiting the structural response

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial support ofthe ldquoCassa di Risparmio di Perugiardquo Foundation that fundedthis study through the project ldquoDevelopment of active controlsystems for the response mitigation under seismic excitationrdquo(Project no 20100110490) Suggestions and comments byDr Michele Moretti and Dr Matteo Becchetti University ofPerugia are also acknowledged with gratitude

References

[1] I Venanzi and A L Materazzi ldquoAcrosswind aeroelastic respo-nse of square tall buildings a semi-analytical approach based ofwind tunnel tests on rigid modelsrdquo Wind amp Structures vol 15no 6 pp 495ndash508 2012

[2] I Venanzi andA LMaterazzi ldquoMulti-objective optimization ofwind-excited structuresrdquo Engineering Structures vol 29 no 6pp 983ndash990 2007

[3] A Forrai S Hashimoto H Funato and K Kamiyama ldquoRobustactive vibration suppression control with constraint on thecontrol signal application to flexible structuresrdquo Earthquake

Advances in Civil Engineering 11

Engineering amp Structural Dynamics vol 32 no 11 pp 1655ndash1676 2003

[4] J G Chase S E Breneman andH A Smith ldquoRobustHinfinstatic

output feedback control with actuator saturationrdquo Journal ofEngineering Mechanics vol 125 no 2 pp 225ndash233 1999

[5] B Indrawan T KoboriM Sakamoto N Koshika and S OhruildquoExperimental verification of bounded-force control methodrdquoEarthquake Engineering amp Structural Dynamics vol 25 no 2pp 179ndash193 1996

[6] J R Cloutier ldquoState-dependent Riccati equation techniques anoverviewrdquo in Proceedings of the American Control Conferencepp 932ndash936 Albuquerque NM USA 1997

[7] B Friedland ldquoOn controlling systems with state-variable con-straintsrdquo in Proceedings of the American Control Conference pp2123ndash2127 Philadelphia Pa USA 1998

[8] A L Materazzi and F Ubertini ldquoRobust structural control withsystem constraintsrdquo Structural Control and Health Monitoringvol 19 no 3 pp 472ndash490 2012

[9] J-HKim andF Jabbari ldquoActuator saturation and control designfor buildings under seismic excitationrdquo Journal of EngineeringMechanics vol 128 no 4 pp 403ndash412 2002

[10] J Mongkol B K Bhartia and Y Fujino ldquoOn Linear-Saturation(LS) control of buildingsrdquo Earthquake Engineering amp StructuralDynamics vol 25 no 12 pp 1353ndash1371 1996

[11] Z Wu and T T Soong ldquoModified bang-bang control lawfor structural control implementationrdquo Journal of EngineeringMechanics vol 122 no 8 pp 771ndash777 1996

[12] C-W Lim ldquoActive vibration control of the linear structure withan active mass damper applying robust saturation controllerrdquoMechatronics vol 18 no 8 pp 391ndash399 2008

[13] I Nagashima and Y Shinozaki ldquoVariable gain feedback con-trol technique of active mass damper and its application tohybrid structural controlrdquo Earthquake Engineering amp StructuralDynamics vol 26 no 8 pp 815ndash838 1997

[14] M Yamamoto and T Sone ldquoBehavior of active mass damper(AMD) installed in high-rise building during 2011 earthquakeoff Pacific coast of Tohoku and verification of regeneratingsystem of AMD based on monitoringrdquo Structural Control andHealth Monitoring vol 21 no 4 pp 634ndash647 2013

[15] I Venanzi F Ubertini and A L Materazzi ldquoOptimal design ofan array of active tuned mass dampers for wind-exposed high-rise buildingsrdquo Structural Control and Health Monitoring vol20 pp 903ndash917 2013

[16] I Venanzi andA LMaterazzi ldquoRobust optimization of a hybridcontrol system for wind-exposed tall buildings with uncertainmass distributionrdquo Smart Structures and Systems vol 12 no 6pp 641ndash659 2013

[17] F Ubertini I Venanzi G Comanducci and A L MaterazzildquoOn the control performance of an active mass driver systemrdquoin Proceedings of the AIMETA Congress Turin Italy 2013

[18] EN 1993-1-1 Eurocode 3 Design of steel structuresmdashpart 1-1General rules and rules for buildings 2005

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

2 Advances in Civil Engineering

1

qn

qa

q2

q3

q1

mn

ma

J

m2

m3

m1

kn

k2

k3

k1

uT

(a)

2 3

4

56

l

120599

(b)

Figure 1 Multistory frame structure with AMD made of electric torsional servomotor (element 1) and ball screw (a) detailed view of ballscrew components (b) additional carried mass (2) return tube (3) screw (4) bearing balls (5) and ball nut (6)

otherwise Wu and Soong [11] introduced the suboptimalbang-bang control strategy described by a function of thestate where the control force is determined by minimizingthe time derivative of a quadratic Lyapunov function underthe control force constraint This method was found to beeffective under a certain range of control force but it can beunstable outside of this range To overcome this instabilityLim [12] proposed an adaptive bang-bang control algorithm

The problem of exceeding the stroke limits is perhaps themost important constraint for application of AMD systemsto actual structures but it has not been widely discussedin the literature Nagashima and Shinozaki [13] developed avariable gain feedback (VGF) control for buildings equippedwith AMD systems The variable feedback gain is a functionof a quantity representing the tradeoff between the reductionof the building response and the amplitude of themass strokeand this quantity is on-line controlled to keep the strokewithin its limits Yamamoto and Sone [14] utilized a linearlyvariable gain proportional to an index representing the activ-ity of an AMD that is a function of its stroke displacementthe first modal frequency of the controlled building and thestroke limit of the AMD Within the framework of the state-dependent Riccati equation Friedland [7] proposed to applyto the system a physical constraint modeled by a nonlinearspring which provides a state-dependent restoring force thatis accounted for in the controller design

Authors have recently started a research program oninnovative solutions for structural control On one sideanalytical studies on the definition of original control algo-rithms for constrained systems and on the optimizationof noncollocated control systems [15 16] were developedOn the other side laboratorial experimental work has beencarried out to implement a control system made of an activemass driver system [17] The prototype AMD is made ofa torsional electric servomotor that moves a small massthrough a ball screw

In this paper a nonlinear control strategy is proposed thatis capable of preventing crossing of the stroke limits of anAMD system To this aim a skyhook control algorithm is

modified by adding to the control force a nonlinear term thatincreases in the vicinity of the stroke bounds To reduce theimpulsive effect of this additional braking term the functionsmoothly varies within the fixed boundaries To demonstratethe effectiveness of the proposed approach experimental testswere carried out on a reduced-scale five-story frame structureequipped with the AMD and subjected to free vibrationtests The effect of a variation in the parameters defining thenonlinear control force is discussed and the optimal choiceof such parameters is identified Numerical analyses are alsocarried out to correctly interpret the experimental resultsTheeffectiveness of the proposed control algorithm is comparedto that of the classical skyhook algorithm highlighting thevalidity of the new approach

2 Modeling of the Systemrsquos Dynamics

An 119899-story planar frame structure equipped with an activemass driver system placed on top is considered as shown inFigure 1 Horizontal displacements of the stories relative tothe ground are denoted as 119902

1 1199022 119902

119899 while story masses

and stiffness are denoted as1198981 1198982 119898

119899and 1198961 1198962 119896

119899

respectivelyThe AMD system is composed of a ball screw that

converts the rotational motion of an electric torsional servo-motor into the translational motion of a mass 119898

119886 The total

moving mass of the AMD 119898119886 is the sum of the mass of the

ball nut and the additional carried mass (elements 2 and 6in Figure 1) The displacement of the movable mass relativeto the ground is denoted by 119902

119886 while the rotation of the ball

screw is denoted by 120599 As the motor rotates one revolutionthe movable mass is advanced one pitch 119897 of the screwThusthe following kinematic relation holds for the AMD

119897

2120587120599 = 119902119886minus 119902119899= 119902119886minus F119879q (1)

where q = [1199021 1199022 119902

119899]119879 and F = [0 0 1]119879 are the 119899-

dimensional collocation vector of the AMD

Advances in Civil Engineering 3

The servomotor is commanded by assigning an angularvelocity to the servodrive of the motor which amplifies sucha signal and provides proportional electric current to themotor An encoder reports the actual status (actual value ofthe angular velocity measured by the encoder) of the motorto the servodrive which corrects input current by calculatingthe deviation between commanded and actual status Becausethe device actuation is imperfect the demand value of theangular velocity ( 120599

119863) and its actual value ( 120599

119860) are in general

different This internal dynamics of the actuator can bemodeled by accounting for the transfer function betweenvelocity demand value and velocity actual value Howeverin the specific problem under consideration in which themovablemass and the inertia of the screw are small comparedto the torque capacity of the servomotor and the commandsignal is not excessively fast imperfect actuation of the deviceis neglected and demand and actual values of the angularvelocity are considered coincident

The free response of the system is governed by the fol-lowing equation of motion readily obtained using Lagrangersquosequations [17]

(M + 119898119886FF119879) q + Cq + Kq = minus119898119886119897

2120587F 120599

(119869 +1198981198861198972

41205872) 120599 = 119906

119879( 120599) minus

119898119886119897

2120587Fq

(2)

where M C and K are 119899 times 119899 mass damping and systemmatrices of the frame structure J is the torsional massmoment of inertia of the ball screw and 119906

119879( 120599) is the torque

provided by the servomotor for the commanded value of theangular velocity

For active damping of the dynamic response of thesubstructure it is particularly useful to regulate the inertialforce 119906 acting on the movable massThis last is simply givenby

119906 = 119898119886119902119886 (3)

Substituting (1) into (3) and integrating in time under theassumption of zero initial conditions yield the followingequation

120599 =2120587

119898119886119897int119905

0

119906 (120591) d120591 minus 2120587119897119902119899 (4)

which provides the angular velocity to be commanded to themotor for providing the desired control force 119906

3 Nonlinear Control Strategy forHandling Stroke Limits

The movement 119909 of real inertial actuators (in the presentcase 119909 = 119902

119886minus 119902119899) is limited by a maximum value 119909max

corresponding to the physical stroke extension Mathemat-ically such a limitation is represented by a nonholonomicconstraint that can be modeled by introducing a propernonlinear restoring force acting on the movable mass Such

a force can for instance have the following expression asproposed in [7]

120601 (119909) = (119909

119909max)

2119873+1

(5)

Equation (5) corresponds tomodeling the impact of the massagainst the stroke limit as an elastic (nondissipative) impactwhere 119873 is an integer parameter that increases with theincreasing rigidity of the physical constraint

The motion of the inertial mass is in general governed bya feedback control algorithm that accomplishes the task ofreducing the vibrations of the substructureWhen the relativedisplacement 119909 between the mass and the substructurereaches the value 119909max an impact occurs with a consequentexchange of the impulsive force 120601 between the mass andthe substructure This clearly reduces the performance ofthe control system and can produce damages to the controldevice as well as to the structural system

We propose to introduce the stroke limit 119909max as a param-eter in the control law by coupling a constant gain feedbackcontroller deputed to dampen structural vibrations with anonlinear controller deputed to handle stroke limitation Wedenote by 119906

0the control signal regulated by the constant gain

controller which is expressed as

1199060= minus119891 (119909) (6)

where 119909 = [119902 119902]119879 is the state vector of the system and

119891 is an appropriate function This last is usually a linearfunction that can be designed in such a way to guarantee theasymptotic stability of the system and to satisfy some optimalperformance criteria In order to handle the physical limita-tions imposed on the stroke extension being impossible to acton 119909max it is necessary to gradually stop the actuator when 119909approaches 119909maxThe easiest way to do it is to introduce in (2)a dissipative force that becomes effective close to the physicallimit and is nil elsewhere Such a force can be obtained witha derivative term with a variable gain given by a constant119866NL multiplied by a nonlinear 119909-dependent coefficient 0 le120576NL(119909) le 1

119906 = 1199060minus 119866NL120576NL (119909) (7)

Equation (7) is in the following referred to as ldquoNLC algo-rithmrdquo where NLC stands for nonlinear control

The simplest function 120576NL(119909) to introduce in (7) is a stepfunction that assumes a nil value when |119909| is less than 119896sdot119909maxwith 0 le 119896 le 1 and a unit value elsewhereUnfortunately thisfunction which can be expressed as 120576NL(119909) = 119867(|119909|minus119896sdot119909max)(119867 denoting the Heaviside step function) is discontinuousfor |119909| = 119896 sdot 119909max which determines the application of anabrupt brake force requiring a very large electrical power

In practice the step function is not applicable and does notsolve the problem of the abrupt stop of the actuator In orderto solve this problem it is necessary to consider a function120576NL(119909) which smoothly varies between 0 and 1 in an intervalcomprised between 119896

0119909max and 1198961119909max with 0 le 1198960 le 1198961 le 1

A similar function is here chosen as follows

4 Advances in Civil Engineering

120576NL (119909) =

1 |119909| ge 1198961119909max

1 minus exp( minus1

1 minus ((|119909| minus 1198960119909max) (1198961119909max minus 1198960119909max))

2+ 1) 119896

0119909max lt |119909| lt 1198961119909max

0 |119909| le 1198960119909max

(8)

It is worth noting that 120576NL(119909) given by (8) is an infinitelydifferentiable function In particular it does not exhibit anydiscontinuity and its slope is always finite This ensures thegradual application of the brake force and the reduction ofthe associated power A graphical representation of func-tion 120576NL(119909) is represented in Figure 2(a) It can be shownthat 120576NL(119909) tends in norm in the 1198712 functional space tothe Heaviside function when 119896

0and 119896

1tend to the same

value 119896 This circumstance is graphically represented inFigure 2(b)

It should be noticed that the linearization of (7) in theorigin of the state space here assumed as a fixed equilibriumpoint of the system coincides with the linearization of (6)Therefore the asymptotic stability of the closed-loop systemwith control algorithm given by (6) also ensures the localasymptotic stability of the NLC algorithm with 120576NL(119909) givenby (8)

A classic skyhook control algorithm is chosen in thiswork for specializing (6) This strategy results in applyingan inertial force to the movable mass that is proportionalto the velocity of the top floor relative to the groundand offers the main advantages of being relatively sim-ple to implement and resulting in a significant dampingeffectiveness

By regulating 1199060through a skyhook control algorithm (7)

is specialized as follows

119906 = 119866119871119902119899minus 119866NL120576NL (119909) (9)

where 119866119871is the constant gain of the skyhook strategy

The demand value of the angular velocity correspondingto (9) is obtained by substituting such equation into (4)and by time integration under the assumption of zero initialconditions The following control algorithm is thus obtainedwhere integration by parts of the nonlinear term in (9) hasbeen carried out

120599 =2120587

119897(119866119871119902119899

119898119886

minus119866NL120576NL (119909) 119909

119898119886

+119866NL int

119905

0

1205761015840NL119909 d120591119898119886

minus 119902119899)

(10)

In (10) 1205761015840NL is the derivative of 120576NL with respect to 119909 Thisquantity is shown in Figure 2(c) and is given by

1205761015840

NL (119909)

=

0 |119909| ge 1198961119909max

2 (|119909| minus 1198960119909max) sign (119909) exp(((minus1) (1 minus ((|119909| minus 1198960119909max) (1198961119909max minus 1198960119909max))

2

)minus1

) + 1)

(1 minus ((|119909| minus 1198960119909max) (1198961119909max minus 1198960119909max))

2

)2

(1198961119909max minus 1198960119909max)

2

1198960119909max lt |119909| lt 1198961119909max

0 |119909| le 1198960119909max

(11)

where sign(119909) is the signum function It should be noticedthat 1205761015840NL(119909) locally tends to Diracrsquos delta function for 119896

0and

1198961approaching the same value 119896In this work the integral term appearing in (10) is

disregarded under the assumption that it is small comparedto the term containing 120576NL(119909) This is true for 119896

0which

is sufficiently smaller than 1198961 that is for a small value of

1205761015840NL(119909) also because the product between 119909 and 119899is small

compared to 119909 Moreover 119902119899and 119902

119899in (10) are obtained by

online double and single integrations of acceleration signalsrespectively

4 Description of the Experimental Set-Up

The proposed control strategy was tested in the laboratoriesof the Department of Civil and Environmental Engineeringof University of Perugia A physical model of the structure-AMD system described in Section 2 was designed and con-structedThe experimental set-up shown in Figure 3 ismadeof the following components (i) frame structure (ii) activemass driver (iii) monitoring sensors and (iv) controller

The test structure is a five-story single-bay frame shownin Figure 3 having a total height of about 19mThe structureis made of steel S 235 whose nominal value of yield strength

Advances in Civil Engineering 5

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 10

05

1

120576 NL

k0 = 05xmax k1 = 09xmax

xxmax

(a)

0

05

1

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

120576 NL

k0 = 050xmax k1 = 090xmaxk0 = 060xmax k1 = 080xmaxk0 = 065xmax k1 = 075xmaxk0 = 069xmax k1 = 071xmax

xxmax

(b)

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus150

minus750

75150

120576998400 NL

xxmax

(c)

Figure 2 Nonlinear function in (8) for a particular choice ofparameters 119896

0and 119896

1(a) nonlinear function in (8) for 119896

0and

1198961approaching the same value 119896 (b) derivative of the nonlinear

function in (8) expressed by (11) for 1198960and 119896

1approaching the same

value 119896 (c)

is 235Nmm2 according to the Eurocode 3 [18] Each flooris made of 700 times 300 times 20mm steel plates of about 328 kgweightThe structural configuration consists of the use of 5 times30 times 340mm columns for the lowest two stories and 4 times 30times 340mm in the remaining ones The structure is fixed to asupport table

The AMD installed on the top floor is composed by a600mm long ball screw that converts the rotationalmotion ofaKollmorgenAKM33HAC servomotor into the translationalmotion of a 4 kgmass block (approximately 25 of the wholestructural mass) The maximum speed of the servomotor is8000 roots per minute and the peak torque is 1022Nm Themaximum stroke of the small mass is equal to plusmn300mmwhile the ball screw has a diameter of 25mm and a pitch of25mm The whole AMD system except for its control unit(drive of the servomotor) is mounted on an aluminum platedesigned for the purpose that also serves as top floor of thestructure

The position of the mass along the ball screw is measuredbymeans of a JX-P420 linear position transducer Two opticalsensors connected to a logical circuit disable the drive of the

motor when the mass exceeds the stroke limits in order toprevent damages of the actuator

The controller is a National Instruments PXIe-8133 Corei7-820QM 173GHz which mounts on board a PXIe-6361X Series Multifunction DAQ card with 8 channels in differ-ential input The software installed on the controller is theLabVIEW (LV) real-time The system fully stands alone andrequires the network connection with a host PC only forthe initial configuration of the LV code The communicationbetween the controller and the drive of the motor is based onthe EtherCAT protocol

Six PCB 393C accelerometers are mounted on each floorfor the purpose of monitoring structural accelerations Theaccelerometers are connected through short cables to theDAQ card by means of the SCB-68 noise rejecting connectorblock

5 Experimental and Numerical Tests

Free vibration experimental tests were carried out in orderto investigate the effectiveness of the control strategy andto compare experimental results with numerical predictionsTests were carried out by varying the parameters of the con-trol law to examine their effect on the control effectivenessThe experimental and numerical results were compared tothose obtained with the classical skyhook algorithm high-lighting the validity of the proposed strategy for handlingstroke limitations

51 Free Vibration Tests To excite the frame structure theAMD was used as a harmonic shaker located at the top floorof the structure and then after 10 forcing cycles theAMDwasswitched to control The frequency of excitation was 18Hzclose to the first natural frequency of the frame system

Two different control strategies were compared (1) thelinear ldquoskyhookrdquo control law (LC) (2) the proposed nonlinearcontrol law (NLC) The corresponding control forces areexpressed as

119906 = 119866119871119902119899(LC)

119906 = 119866119871119902119899minus 119866NL120576NL (119909) (NLC)

(12)

The corresponding demand values of the angular velocity areexpressed as

120599 =2120587

119897(119866119871119902119899

119898119886

minus 119902119899) (LC) (13)

120599 =2120587

119897(119866119871119902119899

119898119886

minus119866NL120576NL (119909) 119909

119898119886

+119866NL int

119905

0

1205761015840

NL119909 d120591119898119886

minus 119902119899) (NLC)

(14)

In Table 1 are summarized the parameters of the controlalgorithms The gains 119866

119871and 119866NL (12) were set equal to

150 and the stroke limit was assumed to be the 90 of the

6 Advances in Civil Engineering

Servomotor

(a)

(b)

(c)Potentiometer

Movable mass

Ball screw

Servomotor

Potentiometer

Movable mass

Ball screw

Figure 3 Experimental set-up overview of the system (a) detailed views of the AMD (b-c)

Table 1 Performance indices of the control effectiveness computedadopting the experimental results

Parameter Value119866119871

150119866NL 150119909max 68 cm1198961

081198960

02ndash07

maximum stroke of the movable mass obtained with the LCstrategy The coefficient 119896

1was selected as 08 and 119896

0was

varied between 02 and 07Figure 4 shows the top floor displacements of the uncon-

trolled and controlled system with LC and NLC strate-gies During the loading phase the structural displacementsincrease while when the control is switched on the structuralresponse is rapidly decreased by both of the adopted controlapproaches

Figure 5 shows the results in terms of AMD displace-ments and top floor displacements obtained with the LC andthe NLC algorithms for 119896

0= 02 and 119896

1= 08 It is also shown

for comparison with the nonlinear velocity demand that isactivated when the AMD stroke exceeds the value 119896

0119909max

Neglecting the integral term in (14) the nonlinear velocitydemand is the difference between the total velocity demandsexpressed by (14) and (13) and is written as

120599NL =2120587

119897

119866NL120576NL (119909) 119909

119898119886

(15)

It is possible to observe that adopting the NLC strategythe AMD stroke never exceeds 119909max Although the controlalgorithm is not recentering the nonlinear algorithm con-tributes to limiting the residual AMD displacement thusavoiding the need for manual or automatic recentering afterthe seismic event The small residual displacement regardsthe AMD only while the substructure elastically recoversits initial configuration after the vibration is dissipated The

0 2 4 6 8 10 12

0

001

002

Time (s)

UNCLC

NLCControl on

minus001

minus002

Top

floor

disp

lace

men

ts (m

)

Figure 4 Top floor displacements for the uncontrolled (UNC) andcontrolled systems with LC and NLC (119896

0= 02) strategies

difference between the structural response obtained withLC and NLC algorithms appears after the first peak of thecontrolled response when the AMD stroke has exceeded thevalue 119896

0119909max and the nonlinear restoring force is activated

As expected after the second peak the control performanceobtained with the NLC algorithm is slightly worse than thatobtained with the LC law However the damping providedto the structure is sufficient to rapidly decrease the structuralresponse in both cases (Figure 4)

Figure 6 shows the velocity demand obtained with boththe classic skyhook (LC) and the proposed nonlinear control(NLC) strategies (13)-(14) The difference of the two lines inFigure 6 represents the nonlinear velocity demand plottedat the bottom of Figure 5 A peak of the AMD velocity incorrespondence of the first braking of the movable mass isvisible After the first peak the velocity demand required bythe two algorithms is almost comparable

Advances in Civil Engineering 7

0 2 4 6 8 10 12minus005

0005

01

disp

lace

men

ts (m

)

LCNLC

Control on

Time (s)

xmax

AM

D

k1 lowastxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus002minus001

0

002001

Top

floor

LCNLCControl on

0 2 4 6 8 10 12Time (s)

disp

lace

men

ts (m

)

(b)

minus200

2000

600

1000800

400

Non

linea

r vel

ocity

NLCControl on

0 2 4 6 8 10 12Time (s)

dem

and

(rad

s)

(c)Figure 5 AMD displacements (a) top floor displacements (b)nonlinear velocity demand expressed by (15) (c) for the controlledsystem

In practical applications control force saturation canlikely occur as the nonlinear control system requires asignificant power demand to produce the first braking ofthe AMD To study the effect of force saturation additionaltests were carried out In particular by properly setting thecontroller the actual velocity supplied by the servomotor waslimited to 150 rads value well below the physical limit of thesystem Figure 7 shows that the proposed control algorithmis effective in reducing the structural response and avoidingthe stroke limits crossing even in presence of such forcesaturation

To quantify the effectiveness of the control strategy thefollowing performance indices are defined

1198691=

max119905gt1199050

1003816100381610038161003816119902119886 minus 11990251003816100381610038161003816NLC

max119905gt1199050

1003816100381610038161003816119902119886 minus 11990251003816100381610038161003816LC

1198692=

max119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

max119905gt1199050 119894

1003816100381610038161003816119902LC119894

1003816100381610038161003816

0 2 4 6 8 10 12

150

Time (s)

Velo

city

dem

and

valu

e (ra

ds)

LCNLCControl on

minus50

minus250

minus450

minus650

Figure 6 Velocity demand obtained with LC and NLC (1198960= 02)

strategies

Table 2 Performance indices of the control effectiveness computedadopting the experimental results

1198960

1198691

1198692

1198693

1198694

1198695

02 0749 0987 1102 0995 464203 0767 1001 1077 1004 480304 0843 1012 1010 1208 511705 0857 1008 1139 1268 533206 0862 1005 0947 1214 554107 0891 1059 1109 1226 5603

1198693=

rms119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

rms119905gt1199050 119894

1003816100381610038161003816119902LC119894

1003816100381610038161003816

1198694=

max119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

max119905gt1199050 119894

1003816100381610038161003816 119902LC119894

1003816100381610038161003816

1198695=

max119905gt1199050

10038161003816100381610038161003816120599NLC119863

10038161003816100381610038161003816

max119905gt1199050

10038161003816100381610038161003816120599LC119863

10038161003816100381610038161003816

(16)

where 1198691is the ratio of the maximum relative displacements

between the AMD (119902119886) and the top floor (119902

5) obtained with

the NLC and the LC strategies 1198692is the ratio of themaximum

floor displacements (119902119894) obtained with the NLC and the

LC strategies 1198693is the ratio of the root mean square of

the floor displacements obtained with the NLC and the LCstrategies 119869

4is the ratio of the maximum floor accelerations

( 119902119894) obtained with the NLC and the LC strategies 119869

5is the

ratio of the maximum velocity demand value ( 120599119863) obtained

with the NLC and the LC strategiesIn Table 2 are shown the performance indices corre-

sponding to experimental responses obtained for different

8 Advances in Civil Engineering

minus0050

00501

0 2 4 6 8 10 12

AM

D

LCNLC

Control on

Time (s)

xmax

disp

lace

men

ts (m

)

k1 lowastxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus002minus001

0

002001

LCNLCControl on

Top

floor

disp

lace

men

ts (m

)

0 2 4 6 8 10 12Time (s)

(b)

LCNLCControl on

minus300minus200minus100

100300300

0

0 2 4 6 8 10 12Time (s)

Velo

city

dem

and

valu

e (ra

ds)

(c)

Figure 7 AMD displacements (a) top floor displacements (b)nonlinear velocity demand (c) for the system controlled with LCand NLC algorithms (119896

0= 02) in case of force saturation

values of parameter 1198960The value 119896

0= 08was not considered

because as observed in Section 3 when 1198960= 1198961 120576NL(119909)

essentially coincides with Heavisidersquos step function whichcannot be reproduced in practice

The results in Table 2 show that if 1198960increases (i) the

braking efficacy decreases (as 1198691increases) (ii) the control

performance is slightly worsened (as 1198692increases) (iii) the

ratio of the root mean square of the top displacementsobtained with the NLC and the LC (119869

3) is generally higher

than one (iv) the structural acceleration induced by thebraking term increases (as 119869

4increases) and (v) due to the

additional nonlinear term the velocity demand increaseswith respect to the LC case (as 119869

5increases)

Based on the obtained results it can be concluded thatthe NLC is effective in reducing the structural responseand avoiding the stroke limit crossing at the expense of analmost negligible increase in structural displacements and aslight increase in structural accelerations For free response

Table 3 Identified modal characteristics of the system

Mode Natural frequency (Hz) Modal damping ratio ()1 176 032 492 043 734 114 962 125 1215 13

Table 4 Performance indices of the control effectiveness computedadopting the numerical results

1198960

1198691

1198692

1198693

1198694

1198696

02 0585 1000 1000 1000 194303 0619 1000 1000 1000 241204 0651 1000 1000 1000 239205 0681 1000 1000 1000 273706 0711 1000 1000 1000 326807 0740 1000 1000 1000 4097

vibrations the increase in power demand with respect to theLC case is important only for the first braking peak and thepossible force saturation does not significantly deteriorate thecontrol performance

52 Numerical Simulations and Discussion Numerical anal-yses were performed to better interpret the experimentalresults and further discuss the effectiveness of the controlstrategy

Preliminarily a numerical model was built and tuned tothe free vibration response of the uncontrolled system Amodal identification allowedobtaining an accurate numericalmodel of the system considered as a shear-type structurehaving 5 degrees of freedom The signals provided by the 5accelerometers located on the floors recorded with a sam-pling frequency of 1 kHz and a total duration of 1 hour wereused for modal identificationThe structural response signalswere acquired with excitation provided by microtremorsin the laboratory The Frequency domain decomposition(FDD) was used to perform the modal identification afterdecimation of themeasured time history signals to 50Hz andusing a frequency resolution to compute the power spectraldensity matrix of the measurements equal to 00122Hz Theresults of modal identification are summarized in Table 3After identifying the dynamic characteristics of the physicalsystem a model tuning was performed to obtain the massesand stiffness coefficients of the frame to be adopted in thenumerical model as detailed in [17] In Table 3 are alsoreported themodal damping ratios obtained by analyzing theenvelopes of the free decay responses of the five structuralmodes Single mode responses are obtained by harmonicallyexciting the structure with the AMD working as a shakerproviding excitation in resonance with a specific modeConsidering that the modal damping ratios vary slightlywith the amplitude of vibration their values are estimated inintermediate ranges of the response

Advances in Civil Engineering 9

0 2 4 6minus0125

minus01

minus0075

minus005

minus0025

0

0025

005

0075

01

0125

Time (s)

AM

D d

ispla

cem

ents

(m)

LC num LC expxmax minusxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus0125

minus01

minus0075

minus005

minus0025

0

0025

005

0075

01

0125

AM

D d

ispla

cem

ents

(m)

NLC num NLC expxmax minusxmax

0 2 4 6Time (s)

k0 lowastxmaxminusk0 lowastxmax

(b)

Figure 8 Comparison between numerical and experimental AMD displacements for LC (a) and NLC (b)

The experimental tests were reproduced numericallyapplying at the top of the structure a sinusoidal inertialforce corresponding to the acceleration of the AMD Figure 8shows the comparison between numerical and experimentalAMD displacements for both LC (GL = 150) and NLCstrategies (119866NL = 150 119896

0= 02 119896

1= 08) The compa-

rison is limited to the controlled part of the response Theexperimental and numerical results are close to each otherespecially for the first and second peaks of the responseNevertheless there are still some deviations in the rate ofdecay and the phase of the response due to the imperfectactuation of the AMD

Similar observations can be drawn for Figure 9 repre-senting the comparison between the numerical and experi-mental top floor displacements Although there is a relativelygood correspondence between the experimental and numer-ical results obtained at the beginning of the control phasethe experimental response decay appears different due to theinternal dynamics of the actuator

Figure 10 reports the comparison between the numer-ically predicted control force obtained with LC and NLCstrategies The NLC algorithm leads to an important increasein the control force at the beginning of the actuation

In Table 4 are summarized the performance indicesintroduced in Section 51 computed by adopting the numeri-cal resultsThe index 119869

5is computed by using the control force

instead of the velocity demand value as follows

1198696=max119905gt1199050

10038161003816100381610038161003816119906NLC

10038161003816100381610038161003816

max119905gt1199050

1003816100381610038161003816119906LC1003816100381610038161003816 (17)

where 119906 is given by (12) It is clear from the results that beingequal to the gain 119866NL the braking performance is reducedwith the increase of the coefficient 119896

0 The indices 119869

2 1198693 and

1198694are equal to 1 for all the analyzed cases meaning that at

the first peak the LC and NLC algorithms provide the samestructural response The index 119869

5suggests that the difference

between the control force yielded by NLC and LC strategiesincreases as 119896

0increasesThe discrepancy between the results

reported in Tables 2 and 4 can be ascribed to the imperfectexperimental actuation and the small differences between thephysical and numerical models

6 Conclusions

In this paper a nonlinear control strategy based on amodifieddirect velocity feedback algorithm is proposed for handlingstroke limits of an AMD system In particular a nonlinearbraking term proportional to the relative AMD velocity isincluded in the control algorithm in order to slowdown thedevice in the proximity of the stroke limits

Experimental and numerical tests were carried out ona scaled-down five-story building equipped with an AMDunder free vibrations

The proposed nonlinear control law proved to be effectivein reducing the structural response and avoiding the strokelimit crossing at the expense of a small increase of structuraldisplacements and a slight increase of structural accelera-tionsThe growth in power demand with respect to the linearcase is important only for the first braking peak and thepossible force saturation does not interfere significantly with

10 Advances in Civil Engineering

minus002

minus001

0

001

002

LC numLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(a)

minus002

minus001

0

001

002

NLC numNLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(b)

Figure 9 Comparison between numerical and experimental top displacements for LC (a) and NLC (b)

0 2 4 6

05

10152025

Time (s)

Con

trol f

orce

(N)

LCNLC

minus45

minus40

minus30

minus35

minus25

minus20

minus15

minus10

minus5

Figure 10 Control force obtained with the numerical analyses

the control performance The parametric analysis showedthat an increase of the ratio 119896

11198960between the parameters

defining the nonlinear restoring force results in an improvedperformance in terms of braking efficacy On the contrarythe same braking efficacy can be obtained with a lower 119896

11198960

ratio and a higher gain 119866NL at the expense of an increase inthe structural response especially in terms of accelerationsThe optimal choice of the ratio 119896

11198960should be obtained as a

compromise between the need of limiting the power demandand that of limiting the structural response

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial support ofthe ldquoCassa di Risparmio di Perugiardquo Foundation that fundedthis study through the project ldquoDevelopment of active controlsystems for the response mitigation under seismic excitationrdquo(Project no 20100110490) Suggestions and comments byDr Michele Moretti and Dr Matteo Becchetti University ofPerugia are also acknowledged with gratitude

References

[1] I Venanzi and A L Materazzi ldquoAcrosswind aeroelastic respo-nse of square tall buildings a semi-analytical approach based ofwind tunnel tests on rigid modelsrdquo Wind amp Structures vol 15no 6 pp 495ndash508 2012

[2] I Venanzi andA LMaterazzi ldquoMulti-objective optimization ofwind-excited structuresrdquo Engineering Structures vol 29 no 6pp 983ndash990 2007

[3] A Forrai S Hashimoto H Funato and K Kamiyama ldquoRobustactive vibration suppression control with constraint on thecontrol signal application to flexible structuresrdquo Earthquake

Advances in Civil Engineering 11

Engineering amp Structural Dynamics vol 32 no 11 pp 1655ndash1676 2003

[4] J G Chase S E Breneman andH A Smith ldquoRobustHinfinstatic

output feedback control with actuator saturationrdquo Journal ofEngineering Mechanics vol 125 no 2 pp 225ndash233 1999

[5] B Indrawan T KoboriM Sakamoto N Koshika and S OhruildquoExperimental verification of bounded-force control methodrdquoEarthquake Engineering amp Structural Dynamics vol 25 no 2pp 179ndash193 1996

[6] J R Cloutier ldquoState-dependent Riccati equation techniques anoverviewrdquo in Proceedings of the American Control Conferencepp 932ndash936 Albuquerque NM USA 1997

[7] B Friedland ldquoOn controlling systems with state-variable con-straintsrdquo in Proceedings of the American Control Conference pp2123ndash2127 Philadelphia Pa USA 1998

[8] A L Materazzi and F Ubertini ldquoRobust structural control withsystem constraintsrdquo Structural Control and Health Monitoringvol 19 no 3 pp 472ndash490 2012

[9] J-HKim andF Jabbari ldquoActuator saturation and control designfor buildings under seismic excitationrdquo Journal of EngineeringMechanics vol 128 no 4 pp 403ndash412 2002

[10] J Mongkol B K Bhartia and Y Fujino ldquoOn Linear-Saturation(LS) control of buildingsrdquo Earthquake Engineering amp StructuralDynamics vol 25 no 12 pp 1353ndash1371 1996

[11] Z Wu and T T Soong ldquoModified bang-bang control lawfor structural control implementationrdquo Journal of EngineeringMechanics vol 122 no 8 pp 771ndash777 1996

[12] C-W Lim ldquoActive vibration control of the linear structure withan active mass damper applying robust saturation controllerrdquoMechatronics vol 18 no 8 pp 391ndash399 2008

[13] I Nagashima and Y Shinozaki ldquoVariable gain feedback con-trol technique of active mass damper and its application tohybrid structural controlrdquo Earthquake Engineering amp StructuralDynamics vol 26 no 8 pp 815ndash838 1997

[14] M Yamamoto and T Sone ldquoBehavior of active mass damper(AMD) installed in high-rise building during 2011 earthquakeoff Pacific coast of Tohoku and verification of regeneratingsystem of AMD based on monitoringrdquo Structural Control andHealth Monitoring vol 21 no 4 pp 634ndash647 2013

[15] I Venanzi F Ubertini and A L Materazzi ldquoOptimal design ofan array of active tuned mass dampers for wind-exposed high-rise buildingsrdquo Structural Control and Health Monitoring vol20 pp 903ndash917 2013

[16] I Venanzi andA LMaterazzi ldquoRobust optimization of a hybridcontrol system for wind-exposed tall buildings with uncertainmass distributionrdquo Smart Structures and Systems vol 12 no 6pp 641ndash659 2013

[17] F Ubertini I Venanzi G Comanducci and A L MaterazzildquoOn the control performance of an active mass driver systemrdquoin Proceedings of the AIMETA Congress Turin Italy 2013

[18] EN 1993-1-1 Eurocode 3 Design of steel structuresmdashpart 1-1General rules and rules for buildings 2005

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DistributedSensor Networks

International Journal of

Advances in Civil Engineering 3

The servomotor is commanded by assigning an angularvelocity to the servodrive of the motor which amplifies sucha signal and provides proportional electric current to themotor An encoder reports the actual status (actual value ofthe angular velocity measured by the encoder) of the motorto the servodrive which corrects input current by calculatingthe deviation between commanded and actual status Becausethe device actuation is imperfect the demand value of theangular velocity ( 120599

119863) and its actual value ( 120599

119860) are in general

different This internal dynamics of the actuator can bemodeled by accounting for the transfer function betweenvelocity demand value and velocity actual value Howeverin the specific problem under consideration in which themovablemass and the inertia of the screw are small comparedto the torque capacity of the servomotor and the commandsignal is not excessively fast imperfect actuation of the deviceis neglected and demand and actual values of the angularvelocity are considered coincident

The free response of the system is governed by the fol-lowing equation of motion readily obtained using Lagrangersquosequations [17]

(M + 119898119886FF119879) q + Cq + Kq = minus119898119886119897

2120587F 120599

(119869 +1198981198861198972

41205872) 120599 = 119906

119879( 120599) minus

119898119886119897

2120587Fq

(2)

where M C and K are 119899 times 119899 mass damping and systemmatrices of the frame structure J is the torsional massmoment of inertia of the ball screw and 119906

119879( 120599) is the torque

provided by the servomotor for the commanded value of theangular velocity

For active damping of the dynamic response of thesubstructure it is particularly useful to regulate the inertialforce 119906 acting on the movable massThis last is simply givenby

119906 = 119898119886119902119886 (3)

Substituting (1) into (3) and integrating in time under theassumption of zero initial conditions yield the followingequation

120599 =2120587

119898119886119897int119905

0

119906 (120591) d120591 minus 2120587119897119902119899 (4)

which provides the angular velocity to be commanded to themotor for providing the desired control force 119906

3 Nonlinear Control Strategy forHandling Stroke Limits

The movement 119909 of real inertial actuators (in the presentcase 119909 = 119902

119886minus 119902119899) is limited by a maximum value 119909max

corresponding to the physical stroke extension Mathemat-ically such a limitation is represented by a nonholonomicconstraint that can be modeled by introducing a propernonlinear restoring force acting on the movable mass Such

a force can for instance have the following expression asproposed in [7]

120601 (119909) = (119909

119909max)

2119873+1

(5)

Equation (5) corresponds tomodeling the impact of the massagainst the stroke limit as an elastic (nondissipative) impactwhere 119873 is an integer parameter that increases with theincreasing rigidity of the physical constraint

The motion of the inertial mass is in general governed bya feedback control algorithm that accomplishes the task ofreducing the vibrations of the substructureWhen the relativedisplacement 119909 between the mass and the substructurereaches the value 119909max an impact occurs with a consequentexchange of the impulsive force 120601 between the mass andthe substructure This clearly reduces the performance ofthe control system and can produce damages to the controldevice as well as to the structural system

We propose to introduce the stroke limit 119909max as a param-eter in the control law by coupling a constant gain feedbackcontroller deputed to dampen structural vibrations with anonlinear controller deputed to handle stroke limitation Wedenote by 119906

0the control signal regulated by the constant gain

controller which is expressed as

1199060= minus119891 (119909) (6)

where 119909 = [119902 119902]119879 is the state vector of the system and

119891 is an appropriate function This last is usually a linearfunction that can be designed in such a way to guarantee theasymptotic stability of the system and to satisfy some optimalperformance criteria In order to handle the physical limita-tions imposed on the stroke extension being impossible to acton 119909max it is necessary to gradually stop the actuator when 119909approaches 119909maxThe easiest way to do it is to introduce in (2)a dissipative force that becomes effective close to the physicallimit and is nil elsewhere Such a force can be obtained witha derivative term with a variable gain given by a constant119866NL multiplied by a nonlinear 119909-dependent coefficient 0 le120576NL(119909) le 1

119906 = 1199060minus 119866NL120576NL (119909) (7)

Equation (7) is in the following referred to as ldquoNLC algo-rithmrdquo where NLC stands for nonlinear control

The simplest function 120576NL(119909) to introduce in (7) is a stepfunction that assumes a nil value when |119909| is less than 119896sdot119909maxwith 0 le 119896 le 1 and a unit value elsewhereUnfortunately thisfunction which can be expressed as 120576NL(119909) = 119867(|119909|minus119896sdot119909max)(119867 denoting the Heaviside step function) is discontinuousfor |119909| = 119896 sdot 119909max which determines the application of anabrupt brake force requiring a very large electrical power

In practice the step function is not applicable and does notsolve the problem of the abrupt stop of the actuator In orderto solve this problem it is necessary to consider a function120576NL(119909) which smoothly varies between 0 and 1 in an intervalcomprised between 119896

0119909max and 1198961119909max with 0 le 1198960 le 1198961 le 1

A similar function is here chosen as follows

4 Advances in Civil Engineering

120576NL (119909) =

1 |119909| ge 1198961119909max

1 minus exp( minus1

1 minus ((|119909| minus 1198960119909max) (1198961119909max minus 1198960119909max))

2+ 1) 119896

0119909max lt |119909| lt 1198961119909max

0 |119909| le 1198960119909max

(8)

It is worth noting that 120576NL(119909) given by (8) is an infinitelydifferentiable function In particular it does not exhibit anydiscontinuity and its slope is always finite This ensures thegradual application of the brake force and the reduction ofthe associated power A graphical representation of func-tion 120576NL(119909) is represented in Figure 2(a) It can be shownthat 120576NL(119909) tends in norm in the 1198712 functional space tothe Heaviside function when 119896

0and 119896

1tend to the same

value 119896 This circumstance is graphically represented inFigure 2(b)

It should be noticed that the linearization of (7) in theorigin of the state space here assumed as a fixed equilibriumpoint of the system coincides with the linearization of (6)Therefore the asymptotic stability of the closed-loop systemwith control algorithm given by (6) also ensures the localasymptotic stability of the NLC algorithm with 120576NL(119909) givenby (8)

A classic skyhook control algorithm is chosen in thiswork for specializing (6) This strategy results in applyingan inertial force to the movable mass that is proportionalto the velocity of the top floor relative to the groundand offers the main advantages of being relatively sim-ple to implement and resulting in a significant dampingeffectiveness

By regulating 1199060through a skyhook control algorithm (7)

is specialized as follows

119906 = 119866119871119902119899minus 119866NL120576NL (119909) (9)

where 119866119871is the constant gain of the skyhook strategy

The demand value of the angular velocity correspondingto (9) is obtained by substituting such equation into (4)and by time integration under the assumption of zero initialconditions The following control algorithm is thus obtainedwhere integration by parts of the nonlinear term in (9) hasbeen carried out

120599 =2120587

119897(119866119871119902119899

119898119886

minus119866NL120576NL (119909) 119909

119898119886

+119866NL int

119905

0

1205761015840NL119909 d120591119898119886

minus 119902119899)

(10)

In (10) 1205761015840NL is the derivative of 120576NL with respect to 119909 Thisquantity is shown in Figure 2(c) and is given by

1205761015840

NL (119909)

=

0 |119909| ge 1198961119909max

2 (|119909| minus 1198960119909max) sign (119909) exp(((minus1) (1 minus ((|119909| minus 1198960119909max) (1198961119909max minus 1198960119909max))

2

)minus1

) + 1)

(1 minus ((|119909| minus 1198960119909max) (1198961119909max minus 1198960119909max))

2

)2

(1198961119909max minus 1198960119909max)

2

1198960119909max lt |119909| lt 1198961119909max

0 |119909| le 1198960119909max

(11)

where sign(119909) is the signum function It should be noticedthat 1205761015840NL(119909) locally tends to Diracrsquos delta function for 119896

0and

1198961approaching the same value 119896In this work the integral term appearing in (10) is

disregarded under the assumption that it is small comparedto the term containing 120576NL(119909) This is true for 119896

0which

is sufficiently smaller than 1198961 that is for a small value of

1205761015840NL(119909) also because the product between 119909 and 119899is small

compared to 119909 Moreover 119902119899and 119902

119899in (10) are obtained by

online double and single integrations of acceleration signalsrespectively

4 Description of the Experimental Set-Up

The proposed control strategy was tested in the laboratoriesof the Department of Civil and Environmental Engineeringof University of Perugia A physical model of the structure-AMD system described in Section 2 was designed and con-structedThe experimental set-up shown in Figure 3 ismadeof the following components (i) frame structure (ii) activemass driver (iii) monitoring sensors and (iv) controller

The test structure is a five-story single-bay frame shownin Figure 3 having a total height of about 19mThe structureis made of steel S 235 whose nominal value of yield strength

Advances in Civil Engineering 5

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 10

05

1

120576 NL

k0 = 05xmax k1 = 09xmax

xxmax

(a)

0

05

1

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

120576 NL

k0 = 050xmax k1 = 090xmaxk0 = 060xmax k1 = 080xmaxk0 = 065xmax k1 = 075xmaxk0 = 069xmax k1 = 071xmax

xxmax

(b)

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus150

minus750

75150

120576998400 NL

xxmax

(c)

Figure 2 Nonlinear function in (8) for a particular choice ofparameters 119896

0and 119896

1(a) nonlinear function in (8) for 119896

0and

1198961approaching the same value 119896 (b) derivative of the nonlinear

function in (8) expressed by (11) for 1198960and 119896

1approaching the same

value 119896 (c)

is 235Nmm2 according to the Eurocode 3 [18] Each flooris made of 700 times 300 times 20mm steel plates of about 328 kgweightThe structural configuration consists of the use of 5 times30 times 340mm columns for the lowest two stories and 4 times 30times 340mm in the remaining ones The structure is fixed to asupport table

The AMD installed on the top floor is composed by a600mm long ball screw that converts the rotationalmotion ofaKollmorgenAKM33HAC servomotor into the translationalmotion of a 4 kgmass block (approximately 25 of the wholestructural mass) The maximum speed of the servomotor is8000 roots per minute and the peak torque is 1022Nm Themaximum stroke of the small mass is equal to plusmn300mmwhile the ball screw has a diameter of 25mm and a pitch of25mm The whole AMD system except for its control unit(drive of the servomotor) is mounted on an aluminum platedesigned for the purpose that also serves as top floor of thestructure

The position of the mass along the ball screw is measuredbymeans of a JX-P420 linear position transducer Two opticalsensors connected to a logical circuit disable the drive of the

motor when the mass exceeds the stroke limits in order toprevent damages of the actuator

The controller is a National Instruments PXIe-8133 Corei7-820QM 173GHz which mounts on board a PXIe-6361X Series Multifunction DAQ card with 8 channels in differ-ential input The software installed on the controller is theLabVIEW (LV) real-time The system fully stands alone andrequires the network connection with a host PC only forthe initial configuration of the LV code The communicationbetween the controller and the drive of the motor is based onthe EtherCAT protocol

Six PCB 393C accelerometers are mounted on each floorfor the purpose of monitoring structural accelerations Theaccelerometers are connected through short cables to theDAQ card by means of the SCB-68 noise rejecting connectorblock

5 Experimental and Numerical Tests

Free vibration experimental tests were carried out in orderto investigate the effectiveness of the control strategy andto compare experimental results with numerical predictionsTests were carried out by varying the parameters of the con-trol law to examine their effect on the control effectivenessThe experimental and numerical results were compared tothose obtained with the classical skyhook algorithm high-lighting the validity of the proposed strategy for handlingstroke limitations

51 Free Vibration Tests To excite the frame structure theAMD was used as a harmonic shaker located at the top floorof the structure and then after 10 forcing cycles theAMDwasswitched to control The frequency of excitation was 18Hzclose to the first natural frequency of the frame system

Two different control strategies were compared (1) thelinear ldquoskyhookrdquo control law (LC) (2) the proposed nonlinearcontrol law (NLC) The corresponding control forces areexpressed as

119906 = 119866119871119902119899(LC)

119906 = 119866119871119902119899minus 119866NL120576NL (119909) (NLC)

(12)

The corresponding demand values of the angular velocity areexpressed as

120599 =2120587

119897(119866119871119902119899

119898119886

minus 119902119899) (LC) (13)

120599 =2120587

119897(119866119871119902119899

119898119886

minus119866NL120576NL (119909) 119909

119898119886

+119866NL int

119905

0

1205761015840

NL119909 d120591119898119886

minus 119902119899) (NLC)

(14)

In Table 1 are summarized the parameters of the controlalgorithms The gains 119866

119871and 119866NL (12) were set equal to

150 and the stroke limit was assumed to be the 90 of the

6 Advances in Civil Engineering

Servomotor

(a)

(b)

(c)Potentiometer

Movable mass

Ball screw

Servomotor

Potentiometer

Movable mass

Ball screw

Figure 3 Experimental set-up overview of the system (a) detailed views of the AMD (b-c)

Table 1 Performance indices of the control effectiveness computedadopting the experimental results

Parameter Value119866119871

150119866NL 150119909max 68 cm1198961

081198960

02ndash07

maximum stroke of the movable mass obtained with the LCstrategy The coefficient 119896

1was selected as 08 and 119896

0was

varied between 02 and 07Figure 4 shows the top floor displacements of the uncon-

trolled and controlled system with LC and NLC strate-gies During the loading phase the structural displacementsincrease while when the control is switched on the structuralresponse is rapidly decreased by both of the adopted controlapproaches

Figure 5 shows the results in terms of AMD displace-ments and top floor displacements obtained with the LC andthe NLC algorithms for 119896

0= 02 and 119896

1= 08 It is also shown

for comparison with the nonlinear velocity demand that isactivated when the AMD stroke exceeds the value 119896

0119909max

Neglecting the integral term in (14) the nonlinear velocitydemand is the difference between the total velocity demandsexpressed by (14) and (13) and is written as

120599NL =2120587

119897

119866NL120576NL (119909) 119909

119898119886

(15)

It is possible to observe that adopting the NLC strategythe AMD stroke never exceeds 119909max Although the controlalgorithm is not recentering the nonlinear algorithm con-tributes to limiting the residual AMD displacement thusavoiding the need for manual or automatic recentering afterthe seismic event The small residual displacement regardsthe AMD only while the substructure elastically recoversits initial configuration after the vibration is dissipated The

0 2 4 6 8 10 12

0

001

002

Time (s)

UNCLC

NLCControl on

minus001

minus002

Top

floor

disp

lace

men

ts (m

)

Figure 4 Top floor displacements for the uncontrolled (UNC) andcontrolled systems with LC and NLC (119896

0= 02) strategies

difference between the structural response obtained withLC and NLC algorithms appears after the first peak of thecontrolled response when the AMD stroke has exceeded thevalue 119896

0119909max and the nonlinear restoring force is activated

As expected after the second peak the control performanceobtained with the NLC algorithm is slightly worse than thatobtained with the LC law However the damping providedto the structure is sufficient to rapidly decrease the structuralresponse in both cases (Figure 4)

Figure 6 shows the velocity demand obtained with boththe classic skyhook (LC) and the proposed nonlinear control(NLC) strategies (13)-(14) The difference of the two lines inFigure 6 represents the nonlinear velocity demand plottedat the bottom of Figure 5 A peak of the AMD velocity incorrespondence of the first braking of the movable mass isvisible After the first peak the velocity demand required bythe two algorithms is almost comparable

Advances in Civil Engineering 7

0 2 4 6 8 10 12minus005

0005

01

disp

lace

men

ts (m

)

LCNLC

Control on

Time (s)

xmax

AM

D

k1 lowastxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus002minus001

0

002001

Top

floor

LCNLCControl on

0 2 4 6 8 10 12Time (s)

disp

lace

men

ts (m

)

(b)

minus200

2000

600

1000800

400

Non

linea

r vel

ocity

NLCControl on

0 2 4 6 8 10 12Time (s)

dem

and

(rad

s)

(c)Figure 5 AMD displacements (a) top floor displacements (b)nonlinear velocity demand expressed by (15) (c) for the controlledsystem

In practical applications control force saturation canlikely occur as the nonlinear control system requires asignificant power demand to produce the first braking ofthe AMD To study the effect of force saturation additionaltests were carried out In particular by properly setting thecontroller the actual velocity supplied by the servomotor waslimited to 150 rads value well below the physical limit of thesystem Figure 7 shows that the proposed control algorithmis effective in reducing the structural response and avoidingthe stroke limits crossing even in presence of such forcesaturation

To quantify the effectiveness of the control strategy thefollowing performance indices are defined

1198691=

max119905gt1199050

1003816100381610038161003816119902119886 minus 11990251003816100381610038161003816NLC

max119905gt1199050

1003816100381610038161003816119902119886 minus 11990251003816100381610038161003816LC

1198692=

max119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

max119905gt1199050 119894

1003816100381610038161003816119902LC119894

1003816100381610038161003816

0 2 4 6 8 10 12

150

Time (s)

Velo

city

dem

and

valu

e (ra

ds)

LCNLCControl on

minus50

minus250

minus450

minus650

Figure 6 Velocity demand obtained with LC and NLC (1198960= 02)

strategies

Table 2 Performance indices of the control effectiveness computedadopting the experimental results

1198960

1198691

1198692

1198693

1198694

1198695

02 0749 0987 1102 0995 464203 0767 1001 1077 1004 480304 0843 1012 1010 1208 511705 0857 1008 1139 1268 533206 0862 1005 0947 1214 554107 0891 1059 1109 1226 5603

1198693=

rms119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

rms119905gt1199050 119894

1003816100381610038161003816119902LC119894

1003816100381610038161003816

1198694=

max119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

max119905gt1199050 119894

1003816100381610038161003816 119902LC119894

1003816100381610038161003816

1198695=

max119905gt1199050

10038161003816100381610038161003816120599NLC119863

10038161003816100381610038161003816

max119905gt1199050

10038161003816100381610038161003816120599LC119863

10038161003816100381610038161003816

(16)

where 1198691is the ratio of the maximum relative displacements

between the AMD (119902119886) and the top floor (119902

5) obtained with

the NLC and the LC strategies 1198692is the ratio of themaximum

floor displacements (119902119894) obtained with the NLC and the

LC strategies 1198693is the ratio of the root mean square of

the floor displacements obtained with the NLC and the LCstrategies 119869

4is the ratio of the maximum floor accelerations

( 119902119894) obtained with the NLC and the LC strategies 119869

5is the

ratio of the maximum velocity demand value ( 120599119863) obtained

with the NLC and the LC strategiesIn Table 2 are shown the performance indices corre-

sponding to experimental responses obtained for different

8 Advances in Civil Engineering

minus0050

00501

0 2 4 6 8 10 12

AM

D

LCNLC

Control on

Time (s)

xmax

disp

lace

men

ts (m

)

k1 lowastxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus002minus001

0

002001

LCNLCControl on

Top

floor

disp

lace

men

ts (m

)

0 2 4 6 8 10 12Time (s)

(b)

LCNLCControl on

minus300minus200minus100

100300300

0

0 2 4 6 8 10 12Time (s)

Velo

city

dem

and

valu

e (ra

ds)

(c)

Figure 7 AMD displacements (a) top floor displacements (b)nonlinear velocity demand (c) for the system controlled with LCand NLC algorithms (119896

0= 02) in case of force saturation

values of parameter 1198960The value 119896

0= 08was not considered

because as observed in Section 3 when 1198960= 1198961 120576NL(119909)

essentially coincides with Heavisidersquos step function whichcannot be reproduced in practice

The results in Table 2 show that if 1198960increases (i) the

braking efficacy decreases (as 1198691increases) (ii) the control

performance is slightly worsened (as 1198692increases) (iii) the

ratio of the root mean square of the top displacementsobtained with the NLC and the LC (119869

3) is generally higher

than one (iv) the structural acceleration induced by thebraking term increases (as 119869

4increases) and (v) due to the

additional nonlinear term the velocity demand increaseswith respect to the LC case (as 119869

5increases)

Based on the obtained results it can be concluded thatthe NLC is effective in reducing the structural responseand avoiding the stroke limit crossing at the expense of analmost negligible increase in structural displacements and aslight increase in structural accelerations For free response

Table 3 Identified modal characteristics of the system

Mode Natural frequency (Hz) Modal damping ratio ()1 176 032 492 043 734 114 962 125 1215 13

Table 4 Performance indices of the control effectiveness computedadopting the numerical results

1198960

1198691

1198692

1198693

1198694

1198696

02 0585 1000 1000 1000 194303 0619 1000 1000 1000 241204 0651 1000 1000 1000 239205 0681 1000 1000 1000 273706 0711 1000 1000 1000 326807 0740 1000 1000 1000 4097

vibrations the increase in power demand with respect to theLC case is important only for the first braking peak and thepossible force saturation does not significantly deteriorate thecontrol performance

52 Numerical Simulations and Discussion Numerical anal-yses were performed to better interpret the experimentalresults and further discuss the effectiveness of the controlstrategy

Preliminarily a numerical model was built and tuned tothe free vibration response of the uncontrolled system Amodal identification allowedobtaining an accurate numericalmodel of the system considered as a shear-type structurehaving 5 degrees of freedom The signals provided by the 5accelerometers located on the floors recorded with a sam-pling frequency of 1 kHz and a total duration of 1 hour wereused for modal identificationThe structural response signalswere acquired with excitation provided by microtremorsin the laboratory The Frequency domain decomposition(FDD) was used to perform the modal identification afterdecimation of themeasured time history signals to 50Hz andusing a frequency resolution to compute the power spectraldensity matrix of the measurements equal to 00122Hz Theresults of modal identification are summarized in Table 3After identifying the dynamic characteristics of the physicalsystem a model tuning was performed to obtain the massesand stiffness coefficients of the frame to be adopted in thenumerical model as detailed in [17] In Table 3 are alsoreported themodal damping ratios obtained by analyzing theenvelopes of the free decay responses of the five structuralmodes Single mode responses are obtained by harmonicallyexciting the structure with the AMD working as a shakerproviding excitation in resonance with a specific modeConsidering that the modal damping ratios vary slightlywith the amplitude of vibration their values are estimated inintermediate ranges of the response

Advances in Civil Engineering 9

0 2 4 6minus0125

minus01

minus0075

minus005

minus0025

0

0025

005

0075

01

0125

Time (s)

AM

D d

ispla

cem

ents

(m)

LC num LC expxmax minusxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus0125

minus01

minus0075

minus005

minus0025

0

0025

005

0075

01

0125

AM

D d

ispla

cem

ents

(m)

NLC num NLC expxmax minusxmax

0 2 4 6Time (s)

k0 lowastxmaxminusk0 lowastxmax

(b)

Figure 8 Comparison between numerical and experimental AMD displacements for LC (a) and NLC (b)

The experimental tests were reproduced numericallyapplying at the top of the structure a sinusoidal inertialforce corresponding to the acceleration of the AMD Figure 8shows the comparison between numerical and experimentalAMD displacements for both LC (GL = 150) and NLCstrategies (119866NL = 150 119896

0= 02 119896

1= 08) The compa-

rison is limited to the controlled part of the response Theexperimental and numerical results are close to each otherespecially for the first and second peaks of the responseNevertheless there are still some deviations in the rate ofdecay and the phase of the response due to the imperfectactuation of the AMD

Similar observations can be drawn for Figure 9 repre-senting the comparison between the numerical and experi-mental top floor displacements Although there is a relativelygood correspondence between the experimental and numer-ical results obtained at the beginning of the control phasethe experimental response decay appears different due to theinternal dynamics of the actuator

Figure 10 reports the comparison between the numer-ically predicted control force obtained with LC and NLCstrategies The NLC algorithm leads to an important increasein the control force at the beginning of the actuation

In Table 4 are summarized the performance indicesintroduced in Section 51 computed by adopting the numeri-cal resultsThe index 119869

5is computed by using the control force

instead of the velocity demand value as follows

1198696=max119905gt1199050

10038161003816100381610038161003816119906NLC

10038161003816100381610038161003816

max119905gt1199050

1003816100381610038161003816119906LC1003816100381610038161003816 (17)

where 119906 is given by (12) It is clear from the results that beingequal to the gain 119866NL the braking performance is reducedwith the increase of the coefficient 119896

0 The indices 119869

2 1198693 and

1198694are equal to 1 for all the analyzed cases meaning that at

the first peak the LC and NLC algorithms provide the samestructural response The index 119869

5suggests that the difference

between the control force yielded by NLC and LC strategiesincreases as 119896

0increasesThe discrepancy between the results

reported in Tables 2 and 4 can be ascribed to the imperfectexperimental actuation and the small differences between thephysical and numerical models

6 Conclusions

In this paper a nonlinear control strategy based on amodifieddirect velocity feedback algorithm is proposed for handlingstroke limits of an AMD system In particular a nonlinearbraking term proportional to the relative AMD velocity isincluded in the control algorithm in order to slowdown thedevice in the proximity of the stroke limits

Experimental and numerical tests were carried out ona scaled-down five-story building equipped with an AMDunder free vibrations

The proposed nonlinear control law proved to be effectivein reducing the structural response and avoiding the strokelimit crossing at the expense of a small increase of structuraldisplacements and a slight increase of structural accelera-tionsThe growth in power demand with respect to the linearcase is important only for the first braking peak and thepossible force saturation does not interfere significantly with

10 Advances in Civil Engineering

minus002

minus001

0

001

002

LC numLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(a)

minus002

minus001

0

001

002

NLC numNLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(b)

Figure 9 Comparison between numerical and experimental top displacements for LC (a) and NLC (b)

0 2 4 6

05

10152025

Time (s)

Con

trol f

orce

(N)

LCNLC

minus45

minus40

minus30

minus35

minus25

minus20

minus15

minus10

minus5

Figure 10 Control force obtained with the numerical analyses

the control performance The parametric analysis showedthat an increase of the ratio 119896

11198960between the parameters

defining the nonlinear restoring force results in an improvedperformance in terms of braking efficacy On the contrarythe same braking efficacy can be obtained with a lower 119896

11198960

ratio and a higher gain 119866NL at the expense of an increase inthe structural response especially in terms of accelerationsThe optimal choice of the ratio 119896

11198960should be obtained as a

compromise between the need of limiting the power demandand that of limiting the structural response

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial support ofthe ldquoCassa di Risparmio di Perugiardquo Foundation that fundedthis study through the project ldquoDevelopment of active controlsystems for the response mitigation under seismic excitationrdquo(Project no 20100110490) Suggestions and comments byDr Michele Moretti and Dr Matteo Becchetti University ofPerugia are also acknowledged with gratitude

References

[1] I Venanzi and A L Materazzi ldquoAcrosswind aeroelastic respo-nse of square tall buildings a semi-analytical approach based ofwind tunnel tests on rigid modelsrdquo Wind amp Structures vol 15no 6 pp 495ndash508 2012

[2] I Venanzi andA LMaterazzi ldquoMulti-objective optimization ofwind-excited structuresrdquo Engineering Structures vol 29 no 6pp 983ndash990 2007

[3] A Forrai S Hashimoto H Funato and K Kamiyama ldquoRobustactive vibration suppression control with constraint on thecontrol signal application to flexible structuresrdquo Earthquake

Advances in Civil Engineering 11

Engineering amp Structural Dynamics vol 32 no 11 pp 1655ndash1676 2003

[4] J G Chase S E Breneman andH A Smith ldquoRobustHinfinstatic

output feedback control with actuator saturationrdquo Journal ofEngineering Mechanics vol 125 no 2 pp 225ndash233 1999

[5] B Indrawan T KoboriM Sakamoto N Koshika and S OhruildquoExperimental verification of bounded-force control methodrdquoEarthquake Engineering amp Structural Dynamics vol 25 no 2pp 179ndash193 1996

[6] J R Cloutier ldquoState-dependent Riccati equation techniques anoverviewrdquo in Proceedings of the American Control Conferencepp 932ndash936 Albuquerque NM USA 1997

[7] B Friedland ldquoOn controlling systems with state-variable con-straintsrdquo in Proceedings of the American Control Conference pp2123ndash2127 Philadelphia Pa USA 1998

[8] A L Materazzi and F Ubertini ldquoRobust structural control withsystem constraintsrdquo Structural Control and Health Monitoringvol 19 no 3 pp 472ndash490 2012

[9] J-HKim andF Jabbari ldquoActuator saturation and control designfor buildings under seismic excitationrdquo Journal of EngineeringMechanics vol 128 no 4 pp 403ndash412 2002

[10] J Mongkol B K Bhartia and Y Fujino ldquoOn Linear-Saturation(LS) control of buildingsrdquo Earthquake Engineering amp StructuralDynamics vol 25 no 12 pp 1353ndash1371 1996

[11] Z Wu and T T Soong ldquoModified bang-bang control lawfor structural control implementationrdquo Journal of EngineeringMechanics vol 122 no 8 pp 771ndash777 1996

[12] C-W Lim ldquoActive vibration control of the linear structure withan active mass damper applying robust saturation controllerrdquoMechatronics vol 18 no 8 pp 391ndash399 2008

[13] I Nagashima and Y Shinozaki ldquoVariable gain feedback con-trol technique of active mass damper and its application tohybrid structural controlrdquo Earthquake Engineering amp StructuralDynamics vol 26 no 8 pp 815ndash838 1997

[14] M Yamamoto and T Sone ldquoBehavior of active mass damper(AMD) installed in high-rise building during 2011 earthquakeoff Pacific coast of Tohoku and verification of regeneratingsystem of AMD based on monitoringrdquo Structural Control andHealth Monitoring vol 21 no 4 pp 634ndash647 2013

[15] I Venanzi F Ubertini and A L Materazzi ldquoOptimal design ofan array of active tuned mass dampers for wind-exposed high-rise buildingsrdquo Structural Control and Health Monitoring vol20 pp 903ndash917 2013

[16] I Venanzi andA LMaterazzi ldquoRobust optimization of a hybridcontrol system for wind-exposed tall buildings with uncertainmass distributionrdquo Smart Structures and Systems vol 12 no 6pp 641ndash659 2013

[17] F Ubertini I Venanzi G Comanducci and A L MaterazzildquoOn the control performance of an active mass driver systemrdquoin Proceedings of the AIMETA Congress Turin Italy 2013

[18] EN 1993-1-1 Eurocode 3 Design of steel structuresmdashpart 1-1General rules and rules for buildings 2005

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International Journal of

4 Advances in Civil Engineering

120576NL (119909) =

1 |119909| ge 1198961119909max

1 minus exp( minus1

1 minus ((|119909| minus 1198960119909max) (1198961119909max minus 1198960119909max))

2+ 1) 119896

0119909max lt |119909| lt 1198961119909max

0 |119909| le 1198960119909max

(8)

It is worth noting that 120576NL(119909) given by (8) is an infinitelydifferentiable function In particular it does not exhibit anydiscontinuity and its slope is always finite This ensures thegradual application of the brake force and the reduction ofthe associated power A graphical representation of func-tion 120576NL(119909) is represented in Figure 2(a) It can be shownthat 120576NL(119909) tends in norm in the 1198712 functional space tothe Heaviside function when 119896

0and 119896

1tend to the same

value 119896 This circumstance is graphically represented inFigure 2(b)

It should be noticed that the linearization of (7) in theorigin of the state space here assumed as a fixed equilibriumpoint of the system coincides with the linearization of (6)Therefore the asymptotic stability of the closed-loop systemwith control algorithm given by (6) also ensures the localasymptotic stability of the NLC algorithm with 120576NL(119909) givenby (8)

A classic skyhook control algorithm is chosen in thiswork for specializing (6) This strategy results in applyingan inertial force to the movable mass that is proportionalto the velocity of the top floor relative to the groundand offers the main advantages of being relatively sim-ple to implement and resulting in a significant dampingeffectiveness

By regulating 1199060through a skyhook control algorithm (7)

is specialized as follows

119906 = 119866119871119902119899minus 119866NL120576NL (119909) (9)

where 119866119871is the constant gain of the skyhook strategy

The demand value of the angular velocity correspondingto (9) is obtained by substituting such equation into (4)and by time integration under the assumption of zero initialconditions The following control algorithm is thus obtainedwhere integration by parts of the nonlinear term in (9) hasbeen carried out

120599 =2120587

119897(119866119871119902119899

119898119886

minus119866NL120576NL (119909) 119909

119898119886

+119866NL int

119905

0

1205761015840NL119909 d120591119898119886

minus 119902119899)

(10)

In (10) 1205761015840NL is the derivative of 120576NL with respect to 119909 Thisquantity is shown in Figure 2(c) and is given by

1205761015840

NL (119909)

=

0 |119909| ge 1198961119909max

2 (|119909| minus 1198960119909max) sign (119909) exp(((minus1) (1 minus ((|119909| minus 1198960119909max) (1198961119909max minus 1198960119909max))

2

)minus1

) + 1)

(1 minus ((|119909| minus 1198960119909max) (1198961119909max minus 1198960119909max))

2

)2

(1198961119909max minus 1198960119909max)

2

1198960119909max lt |119909| lt 1198961119909max

0 |119909| le 1198960119909max

(11)

where sign(119909) is the signum function It should be noticedthat 1205761015840NL(119909) locally tends to Diracrsquos delta function for 119896

0and

1198961approaching the same value 119896In this work the integral term appearing in (10) is

disregarded under the assumption that it is small comparedto the term containing 120576NL(119909) This is true for 119896

0which

is sufficiently smaller than 1198961 that is for a small value of

1205761015840NL(119909) also because the product between 119909 and 119899is small

compared to 119909 Moreover 119902119899and 119902

119899in (10) are obtained by

online double and single integrations of acceleration signalsrespectively

4 Description of the Experimental Set-Up

The proposed control strategy was tested in the laboratoriesof the Department of Civil and Environmental Engineeringof University of Perugia A physical model of the structure-AMD system described in Section 2 was designed and con-structedThe experimental set-up shown in Figure 3 ismadeof the following components (i) frame structure (ii) activemass driver (iii) monitoring sensors and (iv) controller

The test structure is a five-story single-bay frame shownin Figure 3 having a total height of about 19mThe structureis made of steel S 235 whose nominal value of yield strength

Advances in Civil Engineering 5

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 10

05

1

120576 NL

k0 = 05xmax k1 = 09xmax

xxmax

(a)

0

05

1

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

120576 NL

k0 = 050xmax k1 = 090xmaxk0 = 060xmax k1 = 080xmaxk0 = 065xmax k1 = 075xmaxk0 = 069xmax k1 = 071xmax

xxmax

(b)

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus150

minus750

75150

120576998400 NL

xxmax

(c)

Figure 2 Nonlinear function in (8) for a particular choice ofparameters 119896

0and 119896

1(a) nonlinear function in (8) for 119896

0and

1198961approaching the same value 119896 (b) derivative of the nonlinear

function in (8) expressed by (11) for 1198960and 119896

1approaching the same

value 119896 (c)

is 235Nmm2 according to the Eurocode 3 [18] Each flooris made of 700 times 300 times 20mm steel plates of about 328 kgweightThe structural configuration consists of the use of 5 times30 times 340mm columns for the lowest two stories and 4 times 30times 340mm in the remaining ones The structure is fixed to asupport table

The AMD installed on the top floor is composed by a600mm long ball screw that converts the rotationalmotion ofaKollmorgenAKM33HAC servomotor into the translationalmotion of a 4 kgmass block (approximately 25 of the wholestructural mass) The maximum speed of the servomotor is8000 roots per minute and the peak torque is 1022Nm Themaximum stroke of the small mass is equal to plusmn300mmwhile the ball screw has a diameter of 25mm and a pitch of25mm The whole AMD system except for its control unit(drive of the servomotor) is mounted on an aluminum platedesigned for the purpose that also serves as top floor of thestructure

The position of the mass along the ball screw is measuredbymeans of a JX-P420 linear position transducer Two opticalsensors connected to a logical circuit disable the drive of the

motor when the mass exceeds the stroke limits in order toprevent damages of the actuator

The controller is a National Instruments PXIe-8133 Corei7-820QM 173GHz which mounts on board a PXIe-6361X Series Multifunction DAQ card with 8 channels in differ-ential input The software installed on the controller is theLabVIEW (LV) real-time The system fully stands alone andrequires the network connection with a host PC only forthe initial configuration of the LV code The communicationbetween the controller and the drive of the motor is based onthe EtherCAT protocol

Six PCB 393C accelerometers are mounted on each floorfor the purpose of monitoring structural accelerations Theaccelerometers are connected through short cables to theDAQ card by means of the SCB-68 noise rejecting connectorblock

5 Experimental and Numerical Tests

Free vibration experimental tests were carried out in orderto investigate the effectiveness of the control strategy andto compare experimental results with numerical predictionsTests were carried out by varying the parameters of the con-trol law to examine their effect on the control effectivenessThe experimental and numerical results were compared tothose obtained with the classical skyhook algorithm high-lighting the validity of the proposed strategy for handlingstroke limitations

51 Free Vibration Tests To excite the frame structure theAMD was used as a harmonic shaker located at the top floorof the structure and then after 10 forcing cycles theAMDwasswitched to control The frequency of excitation was 18Hzclose to the first natural frequency of the frame system

Two different control strategies were compared (1) thelinear ldquoskyhookrdquo control law (LC) (2) the proposed nonlinearcontrol law (NLC) The corresponding control forces areexpressed as

119906 = 119866119871119902119899(LC)

119906 = 119866119871119902119899minus 119866NL120576NL (119909) (NLC)

(12)

The corresponding demand values of the angular velocity areexpressed as

120599 =2120587

119897(119866119871119902119899

119898119886

minus 119902119899) (LC) (13)

120599 =2120587

119897(119866119871119902119899

119898119886

minus119866NL120576NL (119909) 119909

119898119886

+119866NL int

119905

0

1205761015840

NL119909 d120591119898119886

minus 119902119899) (NLC)

(14)

In Table 1 are summarized the parameters of the controlalgorithms The gains 119866

119871and 119866NL (12) were set equal to

150 and the stroke limit was assumed to be the 90 of the

6 Advances in Civil Engineering

Servomotor

(a)

(b)

(c)Potentiometer

Movable mass

Ball screw

Servomotor

Potentiometer

Movable mass

Ball screw

Figure 3 Experimental set-up overview of the system (a) detailed views of the AMD (b-c)

Table 1 Performance indices of the control effectiveness computedadopting the experimental results

Parameter Value119866119871

150119866NL 150119909max 68 cm1198961

081198960

02ndash07

maximum stroke of the movable mass obtained with the LCstrategy The coefficient 119896

1was selected as 08 and 119896

0was

varied between 02 and 07Figure 4 shows the top floor displacements of the uncon-

trolled and controlled system with LC and NLC strate-gies During the loading phase the structural displacementsincrease while when the control is switched on the structuralresponse is rapidly decreased by both of the adopted controlapproaches

Figure 5 shows the results in terms of AMD displace-ments and top floor displacements obtained with the LC andthe NLC algorithms for 119896

0= 02 and 119896

1= 08 It is also shown

for comparison with the nonlinear velocity demand that isactivated when the AMD stroke exceeds the value 119896

0119909max

Neglecting the integral term in (14) the nonlinear velocitydemand is the difference between the total velocity demandsexpressed by (14) and (13) and is written as

120599NL =2120587

119897

119866NL120576NL (119909) 119909

119898119886

(15)

It is possible to observe that adopting the NLC strategythe AMD stroke never exceeds 119909max Although the controlalgorithm is not recentering the nonlinear algorithm con-tributes to limiting the residual AMD displacement thusavoiding the need for manual or automatic recentering afterthe seismic event The small residual displacement regardsthe AMD only while the substructure elastically recoversits initial configuration after the vibration is dissipated The

0 2 4 6 8 10 12

0

001

002

Time (s)

UNCLC

NLCControl on

minus001

minus002

Top

floor

disp

lace

men

ts (m

)

Figure 4 Top floor displacements for the uncontrolled (UNC) andcontrolled systems with LC and NLC (119896

0= 02) strategies

difference between the structural response obtained withLC and NLC algorithms appears after the first peak of thecontrolled response when the AMD stroke has exceeded thevalue 119896

0119909max and the nonlinear restoring force is activated

As expected after the second peak the control performanceobtained with the NLC algorithm is slightly worse than thatobtained with the LC law However the damping providedto the structure is sufficient to rapidly decrease the structuralresponse in both cases (Figure 4)

Figure 6 shows the velocity demand obtained with boththe classic skyhook (LC) and the proposed nonlinear control(NLC) strategies (13)-(14) The difference of the two lines inFigure 6 represents the nonlinear velocity demand plottedat the bottom of Figure 5 A peak of the AMD velocity incorrespondence of the first braking of the movable mass isvisible After the first peak the velocity demand required bythe two algorithms is almost comparable

Advances in Civil Engineering 7

0 2 4 6 8 10 12minus005

0005

01

disp

lace

men

ts (m

)

LCNLC

Control on

Time (s)

xmax

AM

D

k1 lowastxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus002minus001

0

002001

Top

floor

LCNLCControl on

0 2 4 6 8 10 12Time (s)

disp

lace

men

ts (m

)

(b)

minus200

2000

600

1000800

400

Non

linea

r vel

ocity

NLCControl on

0 2 4 6 8 10 12Time (s)

dem

and

(rad

s)

(c)Figure 5 AMD displacements (a) top floor displacements (b)nonlinear velocity demand expressed by (15) (c) for the controlledsystem

In practical applications control force saturation canlikely occur as the nonlinear control system requires asignificant power demand to produce the first braking ofthe AMD To study the effect of force saturation additionaltests were carried out In particular by properly setting thecontroller the actual velocity supplied by the servomotor waslimited to 150 rads value well below the physical limit of thesystem Figure 7 shows that the proposed control algorithmis effective in reducing the structural response and avoidingthe stroke limits crossing even in presence of such forcesaturation

To quantify the effectiveness of the control strategy thefollowing performance indices are defined

1198691=

max119905gt1199050

1003816100381610038161003816119902119886 minus 11990251003816100381610038161003816NLC

max119905gt1199050

1003816100381610038161003816119902119886 minus 11990251003816100381610038161003816LC

1198692=

max119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

max119905gt1199050 119894

1003816100381610038161003816119902LC119894

1003816100381610038161003816

0 2 4 6 8 10 12

150

Time (s)

Velo

city

dem

and

valu

e (ra

ds)

LCNLCControl on

minus50

minus250

minus450

minus650

Figure 6 Velocity demand obtained with LC and NLC (1198960= 02)

strategies

Table 2 Performance indices of the control effectiveness computedadopting the experimental results

1198960

1198691

1198692

1198693

1198694

1198695

02 0749 0987 1102 0995 464203 0767 1001 1077 1004 480304 0843 1012 1010 1208 511705 0857 1008 1139 1268 533206 0862 1005 0947 1214 554107 0891 1059 1109 1226 5603

1198693=

rms119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

rms119905gt1199050 119894

1003816100381610038161003816119902LC119894

1003816100381610038161003816

1198694=

max119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

max119905gt1199050 119894

1003816100381610038161003816 119902LC119894

1003816100381610038161003816

1198695=

max119905gt1199050

10038161003816100381610038161003816120599NLC119863

10038161003816100381610038161003816

max119905gt1199050

10038161003816100381610038161003816120599LC119863

10038161003816100381610038161003816

(16)

where 1198691is the ratio of the maximum relative displacements

between the AMD (119902119886) and the top floor (119902

5) obtained with

the NLC and the LC strategies 1198692is the ratio of themaximum

floor displacements (119902119894) obtained with the NLC and the

LC strategies 1198693is the ratio of the root mean square of

the floor displacements obtained with the NLC and the LCstrategies 119869

4is the ratio of the maximum floor accelerations

( 119902119894) obtained with the NLC and the LC strategies 119869

5is the

ratio of the maximum velocity demand value ( 120599119863) obtained

with the NLC and the LC strategiesIn Table 2 are shown the performance indices corre-

sponding to experimental responses obtained for different

8 Advances in Civil Engineering

minus0050

00501

0 2 4 6 8 10 12

AM

D

LCNLC

Control on

Time (s)

xmax

disp

lace

men

ts (m

)

k1 lowastxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus002minus001

0

002001

LCNLCControl on

Top

floor

disp

lace

men

ts (m

)

0 2 4 6 8 10 12Time (s)

(b)

LCNLCControl on

minus300minus200minus100

100300300

0

0 2 4 6 8 10 12Time (s)

Velo

city

dem

and

valu

e (ra

ds)

(c)

Figure 7 AMD displacements (a) top floor displacements (b)nonlinear velocity demand (c) for the system controlled with LCand NLC algorithms (119896

0= 02) in case of force saturation

values of parameter 1198960The value 119896

0= 08was not considered

because as observed in Section 3 when 1198960= 1198961 120576NL(119909)

essentially coincides with Heavisidersquos step function whichcannot be reproduced in practice

The results in Table 2 show that if 1198960increases (i) the

braking efficacy decreases (as 1198691increases) (ii) the control

performance is slightly worsened (as 1198692increases) (iii) the

ratio of the root mean square of the top displacementsobtained with the NLC and the LC (119869

3) is generally higher

than one (iv) the structural acceleration induced by thebraking term increases (as 119869

4increases) and (v) due to the

additional nonlinear term the velocity demand increaseswith respect to the LC case (as 119869

5increases)

Based on the obtained results it can be concluded thatthe NLC is effective in reducing the structural responseand avoiding the stroke limit crossing at the expense of analmost negligible increase in structural displacements and aslight increase in structural accelerations For free response

Table 3 Identified modal characteristics of the system

Mode Natural frequency (Hz) Modal damping ratio ()1 176 032 492 043 734 114 962 125 1215 13

Table 4 Performance indices of the control effectiveness computedadopting the numerical results

1198960

1198691

1198692

1198693

1198694

1198696

02 0585 1000 1000 1000 194303 0619 1000 1000 1000 241204 0651 1000 1000 1000 239205 0681 1000 1000 1000 273706 0711 1000 1000 1000 326807 0740 1000 1000 1000 4097

vibrations the increase in power demand with respect to theLC case is important only for the first braking peak and thepossible force saturation does not significantly deteriorate thecontrol performance

52 Numerical Simulations and Discussion Numerical anal-yses were performed to better interpret the experimentalresults and further discuss the effectiveness of the controlstrategy

Preliminarily a numerical model was built and tuned tothe free vibration response of the uncontrolled system Amodal identification allowedobtaining an accurate numericalmodel of the system considered as a shear-type structurehaving 5 degrees of freedom The signals provided by the 5accelerometers located on the floors recorded with a sam-pling frequency of 1 kHz and a total duration of 1 hour wereused for modal identificationThe structural response signalswere acquired with excitation provided by microtremorsin the laboratory The Frequency domain decomposition(FDD) was used to perform the modal identification afterdecimation of themeasured time history signals to 50Hz andusing a frequency resolution to compute the power spectraldensity matrix of the measurements equal to 00122Hz Theresults of modal identification are summarized in Table 3After identifying the dynamic characteristics of the physicalsystem a model tuning was performed to obtain the massesand stiffness coefficients of the frame to be adopted in thenumerical model as detailed in [17] In Table 3 are alsoreported themodal damping ratios obtained by analyzing theenvelopes of the free decay responses of the five structuralmodes Single mode responses are obtained by harmonicallyexciting the structure with the AMD working as a shakerproviding excitation in resonance with a specific modeConsidering that the modal damping ratios vary slightlywith the amplitude of vibration their values are estimated inintermediate ranges of the response

Advances in Civil Engineering 9

0 2 4 6minus0125

minus01

minus0075

minus005

minus0025

0

0025

005

0075

01

0125

Time (s)

AM

D d

ispla

cem

ents

(m)

LC num LC expxmax minusxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus0125

minus01

minus0075

minus005

minus0025

0

0025

005

0075

01

0125

AM

D d

ispla

cem

ents

(m)

NLC num NLC expxmax minusxmax

0 2 4 6Time (s)

k0 lowastxmaxminusk0 lowastxmax

(b)

Figure 8 Comparison between numerical and experimental AMD displacements for LC (a) and NLC (b)

The experimental tests were reproduced numericallyapplying at the top of the structure a sinusoidal inertialforce corresponding to the acceleration of the AMD Figure 8shows the comparison between numerical and experimentalAMD displacements for both LC (GL = 150) and NLCstrategies (119866NL = 150 119896

0= 02 119896

1= 08) The compa-

rison is limited to the controlled part of the response Theexperimental and numerical results are close to each otherespecially for the first and second peaks of the responseNevertheless there are still some deviations in the rate ofdecay and the phase of the response due to the imperfectactuation of the AMD

Similar observations can be drawn for Figure 9 repre-senting the comparison between the numerical and experi-mental top floor displacements Although there is a relativelygood correspondence between the experimental and numer-ical results obtained at the beginning of the control phasethe experimental response decay appears different due to theinternal dynamics of the actuator

Figure 10 reports the comparison between the numer-ically predicted control force obtained with LC and NLCstrategies The NLC algorithm leads to an important increasein the control force at the beginning of the actuation

In Table 4 are summarized the performance indicesintroduced in Section 51 computed by adopting the numeri-cal resultsThe index 119869

5is computed by using the control force

instead of the velocity demand value as follows

1198696=max119905gt1199050

10038161003816100381610038161003816119906NLC

10038161003816100381610038161003816

max119905gt1199050

1003816100381610038161003816119906LC1003816100381610038161003816 (17)

where 119906 is given by (12) It is clear from the results that beingequal to the gain 119866NL the braking performance is reducedwith the increase of the coefficient 119896

0 The indices 119869

2 1198693 and

1198694are equal to 1 for all the analyzed cases meaning that at

the first peak the LC and NLC algorithms provide the samestructural response The index 119869

5suggests that the difference

between the control force yielded by NLC and LC strategiesincreases as 119896

0increasesThe discrepancy between the results

reported in Tables 2 and 4 can be ascribed to the imperfectexperimental actuation and the small differences between thephysical and numerical models

6 Conclusions

In this paper a nonlinear control strategy based on amodifieddirect velocity feedback algorithm is proposed for handlingstroke limits of an AMD system In particular a nonlinearbraking term proportional to the relative AMD velocity isincluded in the control algorithm in order to slowdown thedevice in the proximity of the stroke limits

Experimental and numerical tests were carried out ona scaled-down five-story building equipped with an AMDunder free vibrations

The proposed nonlinear control law proved to be effectivein reducing the structural response and avoiding the strokelimit crossing at the expense of a small increase of structuraldisplacements and a slight increase of structural accelera-tionsThe growth in power demand with respect to the linearcase is important only for the first braking peak and thepossible force saturation does not interfere significantly with

10 Advances in Civil Engineering

minus002

minus001

0

001

002

LC numLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(a)

minus002

minus001

0

001

002

NLC numNLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(b)

Figure 9 Comparison between numerical and experimental top displacements for LC (a) and NLC (b)

0 2 4 6

05

10152025

Time (s)

Con

trol f

orce

(N)

LCNLC

minus45

minus40

minus30

minus35

minus25

minus20

minus15

minus10

minus5

Figure 10 Control force obtained with the numerical analyses

the control performance The parametric analysis showedthat an increase of the ratio 119896

11198960between the parameters

defining the nonlinear restoring force results in an improvedperformance in terms of braking efficacy On the contrarythe same braking efficacy can be obtained with a lower 119896

11198960

ratio and a higher gain 119866NL at the expense of an increase inthe structural response especially in terms of accelerationsThe optimal choice of the ratio 119896

11198960should be obtained as a

compromise between the need of limiting the power demandand that of limiting the structural response

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial support ofthe ldquoCassa di Risparmio di Perugiardquo Foundation that fundedthis study through the project ldquoDevelopment of active controlsystems for the response mitigation under seismic excitationrdquo(Project no 20100110490) Suggestions and comments byDr Michele Moretti and Dr Matteo Becchetti University ofPerugia are also acknowledged with gratitude

References

[1] I Venanzi and A L Materazzi ldquoAcrosswind aeroelastic respo-nse of square tall buildings a semi-analytical approach based ofwind tunnel tests on rigid modelsrdquo Wind amp Structures vol 15no 6 pp 495ndash508 2012

[2] I Venanzi andA LMaterazzi ldquoMulti-objective optimization ofwind-excited structuresrdquo Engineering Structures vol 29 no 6pp 983ndash990 2007

[3] A Forrai S Hashimoto H Funato and K Kamiyama ldquoRobustactive vibration suppression control with constraint on thecontrol signal application to flexible structuresrdquo Earthquake

Advances in Civil Engineering 11

Engineering amp Structural Dynamics vol 32 no 11 pp 1655ndash1676 2003

[4] J G Chase S E Breneman andH A Smith ldquoRobustHinfinstatic

output feedback control with actuator saturationrdquo Journal ofEngineering Mechanics vol 125 no 2 pp 225ndash233 1999

[5] B Indrawan T KoboriM Sakamoto N Koshika and S OhruildquoExperimental verification of bounded-force control methodrdquoEarthquake Engineering amp Structural Dynamics vol 25 no 2pp 179ndash193 1996

[6] J R Cloutier ldquoState-dependent Riccati equation techniques anoverviewrdquo in Proceedings of the American Control Conferencepp 932ndash936 Albuquerque NM USA 1997

[7] B Friedland ldquoOn controlling systems with state-variable con-straintsrdquo in Proceedings of the American Control Conference pp2123ndash2127 Philadelphia Pa USA 1998

[8] A L Materazzi and F Ubertini ldquoRobust structural control withsystem constraintsrdquo Structural Control and Health Monitoringvol 19 no 3 pp 472ndash490 2012

[9] J-HKim andF Jabbari ldquoActuator saturation and control designfor buildings under seismic excitationrdquo Journal of EngineeringMechanics vol 128 no 4 pp 403ndash412 2002

[10] J Mongkol B K Bhartia and Y Fujino ldquoOn Linear-Saturation(LS) control of buildingsrdquo Earthquake Engineering amp StructuralDynamics vol 25 no 12 pp 1353ndash1371 1996

[11] Z Wu and T T Soong ldquoModified bang-bang control lawfor structural control implementationrdquo Journal of EngineeringMechanics vol 122 no 8 pp 771ndash777 1996

[12] C-W Lim ldquoActive vibration control of the linear structure withan active mass damper applying robust saturation controllerrdquoMechatronics vol 18 no 8 pp 391ndash399 2008

[13] I Nagashima and Y Shinozaki ldquoVariable gain feedback con-trol technique of active mass damper and its application tohybrid structural controlrdquo Earthquake Engineering amp StructuralDynamics vol 26 no 8 pp 815ndash838 1997

[14] M Yamamoto and T Sone ldquoBehavior of active mass damper(AMD) installed in high-rise building during 2011 earthquakeoff Pacific coast of Tohoku and verification of regeneratingsystem of AMD based on monitoringrdquo Structural Control andHealth Monitoring vol 21 no 4 pp 634ndash647 2013

[15] I Venanzi F Ubertini and A L Materazzi ldquoOptimal design ofan array of active tuned mass dampers for wind-exposed high-rise buildingsrdquo Structural Control and Health Monitoring vol20 pp 903ndash917 2013

[16] I Venanzi andA LMaterazzi ldquoRobust optimization of a hybridcontrol system for wind-exposed tall buildings with uncertainmass distributionrdquo Smart Structures and Systems vol 12 no 6pp 641ndash659 2013

[17] F Ubertini I Venanzi G Comanducci and A L MaterazzildquoOn the control performance of an active mass driver systemrdquoin Proceedings of the AIMETA Congress Turin Italy 2013

[18] EN 1993-1-1 Eurocode 3 Design of steel structuresmdashpart 1-1General rules and rules for buildings 2005

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DistributedSensor Networks

International Journal of

Advances in Civil Engineering 5

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 10

05

1

120576 NL

k0 = 05xmax k1 = 09xmax

xxmax

(a)

0

05

1

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

120576 NL

k0 = 050xmax k1 = 090xmaxk0 = 060xmax k1 = 080xmaxk0 = 065xmax k1 = 075xmaxk0 = 069xmax k1 = 071xmax

xxmax

(b)

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus150

minus750

75150

120576998400 NL

xxmax

(c)

Figure 2 Nonlinear function in (8) for a particular choice ofparameters 119896

0and 119896

1(a) nonlinear function in (8) for 119896

0and

1198961approaching the same value 119896 (b) derivative of the nonlinear

function in (8) expressed by (11) for 1198960and 119896

1approaching the same

value 119896 (c)

is 235Nmm2 according to the Eurocode 3 [18] Each flooris made of 700 times 300 times 20mm steel plates of about 328 kgweightThe structural configuration consists of the use of 5 times30 times 340mm columns for the lowest two stories and 4 times 30times 340mm in the remaining ones The structure is fixed to asupport table

The AMD installed on the top floor is composed by a600mm long ball screw that converts the rotationalmotion ofaKollmorgenAKM33HAC servomotor into the translationalmotion of a 4 kgmass block (approximately 25 of the wholestructural mass) The maximum speed of the servomotor is8000 roots per minute and the peak torque is 1022Nm Themaximum stroke of the small mass is equal to plusmn300mmwhile the ball screw has a diameter of 25mm and a pitch of25mm The whole AMD system except for its control unit(drive of the servomotor) is mounted on an aluminum platedesigned for the purpose that also serves as top floor of thestructure

The position of the mass along the ball screw is measuredbymeans of a JX-P420 linear position transducer Two opticalsensors connected to a logical circuit disable the drive of the

motor when the mass exceeds the stroke limits in order toprevent damages of the actuator

The controller is a National Instruments PXIe-8133 Corei7-820QM 173GHz which mounts on board a PXIe-6361X Series Multifunction DAQ card with 8 channels in differ-ential input The software installed on the controller is theLabVIEW (LV) real-time The system fully stands alone andrequires the network connection with a host PC only forthe initial configuration of the LV code The communicationbetween the controller and the drive of the motor is based onthe EtherCAT protocol

Six PCB 393C accelerometers are mounted on each floorfor the purpose of monitoring structural accelerations Theaccelerometers are connected through short cables to theDAQ card by means of the SCB-68 noise rejecting connectorblock

5 Experimental and Numerical Tests

Free vibration experimental tests were carried out in orderto investigate the effectiveness of the control strategy andto compare experimental results with numerical predictionsTests were carried out by varying the parameters of the con-trol law to examine their effect on the control effectivenessThe experimental and numerical results were compared tothose obtained with the classical skyhook algorithm high-lighting the validity of the proposed strategy for handlingstroke limitations

51 Free Vibration Tests To excite the frame structure theAMD was used as a harmonic shaker located at the top floorof the structure and then after 10 forcing cycles theAMDwasswitched to control The frequency of excitation was 18Hzclose to the first natural frequency of the frame system

Two different control strategies were compared (1) thelinear ldquoskyhookrdquo control law (LC) (2) the proposed nonlinearcontrol law (NLC) The corresponding control forces areexpressed as

119906 = 119866119871119902119899(LC)

119906 = 119866119871119902119899minus 119866NL120576NL (119909) (NLC)

(12)

The corresponding demand values of the angular velocity areexpressed as

120599 =2120587

119897(119866119871119902119899

119898119886

minus 119902119899) (LC) (13)

120599 =2120587

119897(119866119871119902119899

119898119886

minus119866NL120576NL (119909) 119909

119898119886

+119866NL int

119905

0

1205761015840

NL119909 d120591119898119886

minus 119902119899) (NLC)

(14)

In Table 1 are summarized the parameters of the controlalgorithms The gains 119866

119871and 119866NL (12) were set equal to

150 and the stroke limit was assumed to be the 90 of the

6 Advances in Civil Engineering

Servomotor

(a)

(b)

(c)Potentiometer

Movable mass

Ball screw

Servomotor

Potentiometer

Movable mass

Ball screw

Figure 3 Experimental set-up overview of the system (a) detailed views of the AMD (b-c)

Table 1 Performance indices of the control effectiveness computedadopting the experimental results

Parameter Value119866119871

150119866NL 150119909max 68 cm1198961

081198960

02ndash07

maximum stroke of the movable mass obtained with the LCstrategy The coefficient 119896

1was selected as 08 and 119896

0was

varied between 02 and 07Figure 4 shows the top floor displacements of the uncon-

trolled and controlled system with LC and NLC strate-gies During the loading phase the structural displacementsincrease while when the control is switched on the structuralresponse is rapidly decreased by both of the adopted controlapproaches

Figure 5 shows the results in terms of AMD displace-ments and top floor displacements obtained with the LC andthe NLC algorithms for 119896

0= 02 and 119896

1= 08 It is also shown

for comparison with the nonlinear velocity demand that isactivated when the AMD stroke exceeds the value 119896

0119909max

Neglecting the integral term in (14) the nonlinear velocitydemand is the difference between the total velocity demandsexpressed by (14) and (13) and is written as

120599NL =2120587

119897

119866NL120576NL (119909) 119909

119898119886

(15)

It is possible to observe that adopting the NLC strategythe AMD stroke never exceeds 119909max Although the controlalgorithm is not recentering the nonlinear algorithm con-tributes to limiting the residual AMD displacement thusavoiding the need for manual or automatic recentering afterthe seismic event The small residual displacement regardsthe AMD only while the substructure elastically recoversits initial configuration after the vibration is dissipated The

0 2 4 6 8 10 12

0

001

002

Time (s)

UNCLC

NLCControl on

minus001

minus002

Top

floor

disp

lace

men

ts (m

)

Figure 4 Top floor displacements for the uncontrolled (UNC) andcontrolled systems with LC and NLC (119896

0= 02) strategies

difference between the structural response obtained withLC and NLC algorithms appears after the first peak of thecontrolled response when the AMD stroke has exceeded thevalue 119896

0119909max and the nonlinear restoring force is activated

As expected after the second peak the control performanceobtained with the NLC algorithm is slightly worse than thatobtained with the LC law However the damping providedto the structure is sufficient to rapidly decrease the structuralresponse in both cases (Figure 4)

Figure 6 shows the velocity demand obtained with boththe classic skyhook (LC) and the proposed nonlinear control(NLC) strategies (13)-(14) The difference of the two lines inFigure 6 represents the nonlinear velocity demand plottedat the bottom of Figure 5 A peak of the AMD velocity incorrespondence of the first braking of the movable mass isvisible After the first peak the velocity demand required bythe two algorithms is almost comparable

Advances in Civil Engineering 7

0 2 4 6 8 10 12minus005

0005

01

disp

lace

men

ts (m

)

LCNLC

Control on

Time (s)

xmax

AM

D

k1 lowastxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus002minus001

0

002001

Top

floor

LCNLCControl on

0 2 4 6 8 10 12Time (s)

disp

lace

men

ts (m

)

(b)

minus200

2000

600

1000800

400

Non

linea

r vel

ocity

NLCControl on

0 2 4 6 8 10 12Time (s)

dem

and

(rad

s)

(c)Figure 5 AMD displacements (a) top floor displacements (b)nonlinear velocity demand expressed by (15) (c) for the controlledsystem

In practical applications control force saturation canlikely occur as the nonlinear control system requires asignificant power demand to produce the first braking ofthe AMD To study the effect of force saturation additionaltests were carried out In particular by properly setting thecontroller the actual velocity supplied by the servomotor waslimited to 150 rads value well below the physical limit of thesystem Figure 7 shows that the proposed control algorithmis effective in reducing the structural response and avoidingthe stroke limits crossing even in presence of such forcesaturation

To quantify the effectiveness of the control strategy thefollowing performance indices are defined

1198691=

max119905gt1199050

1003816100381610038161003816119902119886 minus 11990251003816100381610038161003816NLC

max119905gt1199050

1003816100381610038161003816119902119886 minus 11990251003816100381610038161003816LC

1198692=

max119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

max119905gt1199050 119894

1003816100381610038161003816119902LC119894

1003816100381610038161003816

0 2 4 6 8 10 12

150

Time (s)

Velo

city

dem

and

valu

e (ra

ds)

LCNLCControl on

minus50

minus250

minus450

minus650

Figure 6 Velocity demand obtained with LC and NLC (1198960= 02)

strategies

Table 2 Performance indices of the control effectiveness computedadopting the experimental results

1198960

1198691

1198692

1198693

1198694

1198695

02 0749 0987 1102 0995 464203 0767 1001 1077 1004 480304 0843 1012 1010 1208 511705 0857 1008 1139 1268 533206 0862 1005 0947 1214 554107 0891 1059 1109 1226 5603

1198693=

rms119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

rms119905gt1199050 119894

1003816100381610038161003816119902LC119894

1003816100381610038161003816

1198694=

max119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

max119905gt1199050 119894

1003816100381610038161003816 119902LC119894

1003816100381610038161003816

1198695=

max119905gt1199050

10038161003816100381610038161003816120599NLC119863

10038161003816100381610038161003816

max119905gt1199050

10038161003816100381610038161003816120599LC119863

10038161003816100381610038161003816

(16)

where 1198691is the ratio of the maximum relative displacements

between the AMD (119902119886) and the top floor (119902

5) obtained with

the NLC and the LC strategies 1198692is the ratio of themaximum

floor displacements (119902119894) obtained with the NLC and the

LC strategies 1198693is the ratio of the root mean square of

the floor displacements obtained with the NLC and the LCstrategies 119869

4is the ratio of the maximum floor accelerations

( 119902119894) obtained with the NLC and the LC strategies 119869

5is the

ratio of the maximum velocity demand value ( 120599119863) obtained

with the NLC and the LC strategiesIn Table 2 are shown the performance indices corre-

sponding to experimental responses obtained for different

8 Advances in Civil Engineering

minus0050

00501

0 2 4 6 8 10 12

AM

D

LCNLC

Control on

Time (s)

xmax

disp

lace

men

ts (m

)

k1 lowastxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus002minus001

0

002001

LCNLCControl on

Top

floor

disp

lace

men

ts (m

)

0 2 4 6 8 10 12Time (s)

(b)

LCNLCControl on

minus300minus200minus100

100300300

0

0 2 4 6 8 10 12Time (s)

Velo

city

dem

and

valu

e (ra

ds)

(c)

Figure 7 AMD displacements (a) top floor displacements (b)nonlinear velocity demand (c) for the system controlled with LCand NLC algorithms (119896

0= 02) in case of force saturation

values of parameter 1198960The value 119896

0= 08was not considered

because as observed in Section 3 when 1198960= 1198961 120576NL(119909)

essentially coincides with Heavisidersquos step function whichcannot be reproduced in practice

The results in Table 2 show that if 1198960increases (i) the

braking efficacy decreases (as 1198691increases) (ii) the control

performance is slightly worsened (as 1198692increases) (iii) the

ratio of the root mean square of the top displacementsobtained with the NLC and the LC (119869

3) is generally higher

than one (iv) the structural acceleration induced by thebraking term increases (as 119869

4increases) and (v) due to the

additional nonlinear term the velocity demand increaseswith respect to the LC case (as 119869

5increases)

Based on the obtained results it can be concluded thatthe NLC is effective in reducing the structural responseand avoiding the stroke limit crossing at the expense of analmost negligible increase in structural displacements and aslight increase in structural accelerations For free response

Table 3 Identified modal characteristics of the system

Mode Natural frequency (Hz) Modal damping ratio ()1 176 032 492 043 734 114 962 125 1215 13

Table 4 Performance indices of the control effectiveness computedadopting the numerical results

1198960

1198691

1198692

1198693

1198694

1198696

02 0585 1000 1000 1000 194303 0619 1000 1000 1000 241204 0651 1000 1000 1000 239205 0681 1000 1000 1000 273706 0711 1000 1000 1000 326807 0740 1000 1000 1000 4097

vibrations the increase in power demand with respect to theLC case is important only for the first braking peak and thepossible force saturation does not significantly deteriorate thecontrol performance

52 Numerical Simulations and Discussion Numerical anal-yses were performed to better interpret the experimentalresults and further discuss the effectiveness of the controlstrategy

Preliminarily a numerical model was built and tuned tothe free vibration response of the uncontrolled system Amodal identification allowedobtaining an accurate numericalmodel of the system considered as a shear-type structurehaving 5 degrees of freedom The signals provided by the 5accelerometers located on the floors recorded with a sam-pling frequency of 1 kHz and a total duration of 1 hour wereused for modal identificationThe structural response signalswere acquired with excitation provided by microtremorsin the laboratory The Frequency domain decomposition(FDD) was used to perform the modal identification afterdecimation of themeasured time history signals to 50Hz andusing a frequency resolution to compute the power spectraldensity matrix of the measurements equal to 00122Hz Theresults of modal identification are summarized in Table 3After identifying the dynamic characteristics of the physicalsystem a model tuning was performed to obtain the massesand stiffness coefficients of the frame to be adopted in thenumerical model as detailed in [17] In Table 3 are alsoreported themodal damping ratios obtained by analyzing theenvelopes of the free decay responses of the five structuralmodes Single mode responses are obtained by harmonicallyexciting the structure with the AMD working as a shakerproviding excitation in resonance with a specific modeConsidering that the modal damping ratios vary slightlywith the amplitude of vibration their values are estimated inintermediate ranges of the response

Advances in Civil Engineering 9

0 2 4 6minus0125

minus01

minus0075

minus005

minus0025

0

0025

005

0075

01

0125

Time (s)

AM

D d

ispla

cem

ents

(m)

LC num LC expxmax minusxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus0125

minus01

minus0075

minus005

minus0025

0

0025

005

0075

01

0125

AM

D d

ispla

cem

ents

(m)

NLC num NLC expxmax minusxmax

0 2 4 6Time (s)

k0 lowastxmaxminusk0 lowastxmax

(b)

Figure 8 Comparison between numerical and experimental AMD displacements for LC (a) and NLC (b)

The experimental tests were reproduced numericallyapplying at the top of the structure a sinusoidal inertialforce corresponding to the acceleration of the AMD Figure 8shows the comparison between numerical and experimentalAMD displacements for both LC (GL = 150) and NLCstrategies (119866NL = 150 119896

0= 02 119896

1= 08) The compa-

rison is limited to the controlled part of the response Theexperimental and numerical results are close to each otherespecially for the first and second peaks of the responseNevertheless there are still some deviations in the rate ofdecay and the phase of the response due to the imperfectactuation of the AMD

Similar observations can be drawn for Figure 9 repre-senting the comparison between the numerical and experi-mental top floor displacements Although there is a relativelygood correspondence between the experimental and numer-ical results obtained at the beginning of the control phasethe experimental response decay appears different due to theinternal dynamics of the actuator

Figure 10 reports the comparison between the numer-ically predicted control force obtained with LC and NLCstrategies The NLC algorithm leads to an important increasein the control force at the beginning of the actuation

In Table 4 are summarized the performance indicesintroduced in Section 51 computed by adopting the numeri-cal resultsThe index 119869

5is computed by using the control force

instead of the velocity demand value as follows

1198696=max119905gt1199050

10038161003816100381610038161003816119906NLC

10038161003816100381610038161003816

max119905gt1199050

1003816100381610038161003816119906LC1003816100381610038161003816 (17)

where 119906 is given by (12) It is clear from the results that beingequal to the gain 119866NL the braking performance is reducedwith the increase of the coefficient 119896

0 The indices 119869

2 1198693 and

1198694are equal to 1 for all the analyzed cases meaning that at

the first peak the LC and NLC algorithms provide the samestructural response The index 119869

5suggests that the difference

between the control force yielded by NLC and LC strategiesincreases as 119896

0increasesThe discrepancy between the results

reported in Tables 2 and 4 can be ascribed to the imperfectexperimental actuation and the small differences between thephysical and numerical models

6 Conclusions

In this paper a nonlinear control strategy based on amodifieddirect velocity feedback algorithm is proposed for handlingstroke limits of an AMD system In particular a nonlinearbraking term proportional to the relative AMD velocity isincluded in the control algorithm in order to slowdown thedevice in the proximity of the stroke limits

Experimental and numerical tests were carried out ona scaled-down five-story building equipped with an AMDunder free vibrations

The proposed nonlinear control law proved to be effectivein reducing the structural response and avoiding the strokelimit crossing at the expense of a small increase of structuraldisplacements and a slight increase of structural accelera-tionsThe growth in power demand with respect to the linearcase is important only for the first braking peak and thepossible force saturation does not interfere significantly with

10 Advances in Civil Engineering

minus002

minus001

0

001

002

LC numLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(a)

minus002

minus001

0

001

002

NLC numNLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(b)

Figure 9 Comparison between numerical and experimental top displacements for LC (a) and NLC (b)

0 2 4 6

05

10152025

Time (s)

Con

trol f

orce

(N)

LCNLC

minus45

minus40

minus30

minus35

minus25

minus20

minus15

minus10

minus5

Figure 10 Control force obtained with the numerical analyses

the control performance The parametric analysis showedthat an increase of the ratio 119896

11198960between the parameters

defining the nonlinear restoring force results in an improvedperformance in terms of braking efficacy On the contrarythe same braking efficacy can be obtained with a lower 119896

11198960

ratio and a higher gain 119866NL at the expense of an increase inthe structural response especially in terms of accelerationsThe optimal choice of the ratio 119896

11198960should be obtained as a

compromise between the need of limiting the power demandand that of limiting the structural response

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial support ofthe ldquoCassa di Risparmio di Perugiardquo Foundation that fundedthis study through the project ldquoDevelopment of active controlsystems for the response mitigation under seismic excitationrdquo(Project no 20100110490) Suggestions and comments byDr Michele Moretti and Dr Matteo Becchetti University ofPerugia are also acknowledged with gratitude

References

[1] I Venanzi and A L Materazzi ldquoAcrosswind aeroelastic respo-nse of square tall buildings a semi-analytical approach based ofwind tunnel tests on rigid modelsrdquo Wind amp Structures vol 15no 6 pp 495ndash508 2012

[2] I Venanzi andA LMaterazzi ldquoMulti-objective optimization ofwind-excited structuresrdquo Engineering Structures vol 29 no 6pp 983ndash990 2007

[3] A Forrai S Hashimoto H Funato and K Kamiyama ldquoRobustactive vibration suppression control with constraint on thecontrol signal application to flexible structuresrdquo Earthquake

Advances in Civil Engineering 11

Engineering amp Structural Dynamics vol 32 no 11 pp 1655ndash1676 2003

[4] J G Chase S E Breneman andH A Smith ldquoRobustHinfinstatic

output feedback control with actuator saturationrdquo Journal ofEngineering Mechanics vol 125 no 2 pp 225ndash233 1999

[5] B Indrawan T KoboriM Sakamoto N Koshika and S OhruildquoExperimental verification of bounded-force control methodrdquoEarthquake Engineering amp Structural Dynamics vol 25 no 2pp 179ndash193 1996

[6] J R Cloutier ldquoState-dependent Riccati equation techniques anoverviewrdquo in Proceedings of the American Control Conferencepp 932ndash936 Albuquerque NM USA 1997

[7] B Friedland ldquoOn controlling systems with state-variable con-straintsrdquo in Proceedings of the American Control Conference pp2123ndash2127 Philadelphia Pa USA 1998

[8] A L Materazzi and F Ubertini ldquoRobust structural control withsystem constraintsrdquo Structural Control and Health Monitoringvol 19 no 3 pp 472ndash490 2012

[9] J-HKim andF Jabbari ldquoActuator saturation and control designfor buildings under seismic excitationrdquo Journal of EngineeringMechanics vol 128 no 4 pp 403ndash412 2002

[10] J Mongkol B K Bhartia and Y Fujino ldquoOn Linear-Saturation(LS) control of buildingsrdquo Earthquake Engineering amp StructuralDynamics vol 25 no 12 pp 1353ndash1371 1996

[11] Z Wu and T T Soong ldquoModified bang-bang control lawfor structural control implementationrdquo Journal of EngineeringMechanics vol 122 no 8 pp 771ndash777 1996

[12] C-W Lim ldquoActive vibration control of the linear structure withan active mass damper applying robust saturation controllerrdquoMechatronics vol 18 no 8 pp 391ndash399 2008

[13] I Nagashima and Y Shinozaki ldquoVariable gain feedback con-trol technique of active mass damper and its application tohybrid structural controlrdquo Earthquake Engineering amp StructuralDynamics vol 26 no 8 pp 815ndash838 1997

[14] M Yamamoto and T Sone ldquoBehavior of active mass damper(AMD) installed in high-rise building during 2011 earthquakeoff Pacific coast of Tohoku and verification of regeneratingsystem of AMD based on monitoringrdquo Structural Control andHealth Monitoring vol 21 no 4 pp 634ndash647 2013

[15] I Venanzi F Ubertini and A L Materazzi ldquoOptimal design ofan array of active tuned mass dampers for wind-exposed high-rise buildingsrdquo Structural Control and Health Monitoring vol20 pp 903ndash917 2013

[16] I Venanzi andA LMaterazzi ldquoRobust optimization of a hybridcontrol system for wind-exposed tall buildings with uncertainmass distributionrdquo Smart Structures and Systems vol 12 no 6pp 641ndash659 2013

[17] F Ubertini I Venanzi G Comanducci and A L MaterazzildquoOn the control performance of an active mass driver systemrdquoin Proceedings of the AIMETA Congress Turin Italy 2013

[18] EN 1993-1-1 Eurocode 3 Design of steel structuresmdashpart 1-1General rules and rules for buildings 2005

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

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DistributedSensor Networks

International Journal of

6 Advances in Civil Engineering

Servomotor

(a)

(b)

(c)Potentiometer

Movable mass

Ball screw

Servomotor

Potentiometer

Movable mass

Ball screw

Figure 3 Experimental set-up overview of the system (a) detailed views of the AMD (b-c)

Table 1 Performance indices of the control effectiveness computedadopting the experimental results

Parameter Value119866119871

150119866NL 150119909max 68 cm1198961

081198960

02ndash07

maximum stroke of the movable mass obtained with the LCstrategy The coefficient 119896

1was selected as 08 and 119896

0was

varied between 02 and 07Figure 4 shows the top floor displacements of the uncon-

trolled and controlled system with LC and NLC strate-gies During the loading phase the structural displacementsincrease while when the control is switched on the structuralresponse is rapidly decreased by both of the adopted controlapproaches

Figure 5 shows the results in terms of AMD displace-ments and top floor displacements obtained with the LC andthe NLC algorithms for 119896

0= 02 and 119896

1= 08 It is also shown

for comparison with the nonlinear velocity demand that isactivated when the AMD stroke exceeds the value 119896

0119909max

Neglecting the integral term in (14) the nonlinear velocitydemand is the difference between the total velocity demandsexpressed by (14) and (13) and is written as

120599NL =2120587

119897

119866NL120576NL (119909) 119909

119898119886

(15)

It is possible to observe that adopting the NLC strategythe AMD stroke never exceeds 119909max Although the controlalgorithm is not recentering the nonlinear algorithm con-tributes to limiting the residual AMD displacement thusavoiding the need for manual or automatic recentering afterthe seismic event The small residual displacement regardsthe AMD only while the substructure elastically recoversits initial configuration after the vibration is dissipated The

0 2 4 6 8 10 12

0

001

002

Time (s)

UNCLC

NLCControl on

minus001

minus002

Top

floor

disp

lace

men

ts (m

)

Figure 4 Top floor displacements for the uncontrolled (UNC) andcontrolled systems with LC and NLC (119896

0= 02) strategies

difference between the structural response obtained withLC and NLC algorithms appears after the first peak of thecontrolled response when the AMD stroke has exceeded thevalue 119896

0119909max and the nonlinear restoring force is activated

As expected after the second peak the control performanceobtained with the NLC algorithm is slightly worse than thatobtained with the LC law However the damping providedto the structure is sufficient to rapidly decrease the structuralresponse in both cases (Figure 4)

Figure 6 shows the velocity demand obtained with boththe classic skyhook (LC) and the proposed nonlinear control(NLC) strategies (13)-(14) The difference of the two lines inFigure 6 represents the nonlinear velocity demand plottedat the bottom of Figure 5 A peak of the AMD velocity incorrespondence of the first braking of the movable mass isvisible After the first peak the velocity demand required bythe two algorithms is almost comparable

Advances in Civil Engineering 7

0 2 4 6 8 10 12minus005

0005

01

disp

lace

men

ts (m

)

LCNLC

Control on

Time (s)

xmax

AM

D

k1 lowastxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus002minus001

0

002001

Top

floor

LCNLCControl on

0 2 4 6 8 10 12Time (s)

disp

lace

men

ts (m

)

(b)

minus200

2000

600

1000800

400

Non

linea

r vel

ocity

NLCControl on

0 2 4 6 8 10 12Time (s)

dem

and

(rad

s)

(c)Figure 5 AMD displacements (a) top floor displacements (b)nonlinear velocity demand expressed by (15) (c) for the controlledsystem

In practical applications control force saturation canlikely occur as the nonlinear control system requires asignificant power demand to produce the first braking ofthe AMD To study the effect of force saturation additionaltests were carried out In particular by properly setting thecontroller the actual velocity supplied by the servomotor waslimited to 150 rads value well below the physical limit of thesystem Figure 7 shows that the proposed control algorithmis effective in reducing the structural response and avoidingthe stroke limits crossing even in presence of such forcesaturation

To quantify the effectiveness of the control strategy thefollowing performance indices are defined

1198691=

max119905gt1199050

1003816100381610038161003816119902119886 minus 11990251003816100381610038161003816NLC

max119905gt1199050

1003816100381610038161003816119902119886 minus 11990251003816100381610038161003816LC

1198692=

max119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

max119905gt1199050 119894

1003816100381610038161003816119902LC119894

1003816100381610038161003816

0 2 4 6 8 10 12

150

Time (s)

Velo

city

dem

and

valu

e (ra

ds)

LCNLCControl on

minus50

minus250

minus450

minus650

Figure 6 Velocity demand obtained with LC and NLC (1198960= 02)

strategies

Table 2 Performance indices of the control effectiveness computedadopting the experimental results

1198960

1198691

1198692

1198693

1198694

1198695

02 0749 0987 1102 0995 464203 0767 1001 1077 1004 480304 0843 1012 1010 1208 511705 0857 1008 1139 1268 533206 0862 1005 0947 1214 554107 0891 1059 1109 1226 5603

1198693=

rms119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

rms119905gt1199050 119894

1003816100381610038161003816119902LC119894

1003816100381610038161003816

1198694=

max119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

max119905gt1199050 119894

1003816100381610038161003816 119902LC119894

1003816100381610038161003816

1198695=

max119905gt1199050

10038161003816100381610038161003816120599NLC119863

10038161003816100381610038161003816

max119905gt1199050

10038161003816100381610038161003816120599LC119863

10038161003816100381610038161003816

(16)

where 1198691is the ratio of the maximum relative displacements

between the AMD (119902119886) and the top floor (119902

5) obtained with

the NLC and the LC strategies 1198692is the ratio of themaximum

floor displacements (119902119894) obtained with the NLC and the

LC strategies 1198693is the ratio of the root mean square of

the floor displacements obtained with the NLC and the LCstrategies 119869

4is the ratio of the maximum floor accelerations

( 119902119894) obtained with the NLC and the LC strategies 119869

5is the

ratio of the maximum velocity demand value ( 120599119863) obtained

with the NLC and the LC strategiesIn Table 2 are shown the performance indices corre-

sponding to experimental responses obtained for different

8 Advances in Civil Engineering

minus0050

00501

0 2 4 6 8 10 12

AM

D

LCNLC

Control on

Time (s)

xmax

disp

lace

men

ts (m

)

k1 lowastxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus002minus001

0

002001

LCNLCControl on

Top

floor

disp

lace

men

ts (m

)

0 2 4 6 8 10 12Time (s)

(b)

LCNLCControl on

minus300minus200minus100

100300300

0

0 2 4 6 8 10 12Time (s)

Velo

city

dem

and

valu

e (ra

ds)

(c)

Figure 7 AMD displacements (a) top floor displacements (b)nonlinear velocity demand (c) for the system controlled with LCand NLC algorithms (119896

0= 02) in case of force saturation

values of parameter 1198960The value 119896

0= 08was not considered

because as observed in Section 3 when 1198960= 1198961 120576NL(119909)

essentially coincides with Heavisidersquos step function whichcannot be reproduced in practice

The results in Table 2 show that if 1198960increases (i) the

braking efficacy decreases (as 1198691increases) (ii) the control

performance is slightly worsened (as 1198692increases) (iii) the

ratio of the root mean square of the top displacementsobtained with the NLC and the LC (119869

3) is generally higher

than one (iv) the structural acceleration induced by thebraking term increases (as 119869

4increases) and (v) due to the

additional nonlinear term the velocity demand increaseswith respect to the LC case (as 119869

5increases)

Based on the obtained results it can be concluded thatthe NLC is effective in reducing the structural responseand avoiding the stroke limit crossing at the expense of analmost negligible increase in structural displacements and aslight increase in structural accelerations For free response

Table 3 Identified modal characteristics of the system

Mode Natural frequency (Hz) Modal damping ratio ()1 176 032 492 043 734 114 962 125 1215 13

Table 4 Performance indices of the control effectiveness computedadopting the numerical results

1198960

1198691

1198692

1198693

1198694

1198696

02 0585 1000 1000 1000 194303 0619 1000 1000 1000 241204 0651 1000 1000 1000 239205 0681 1000 1000 1000 273706 0711 1000 1000 1000 326807 0740 1000 1000 1000 4097

vibrations the increase in power demand with respect to theLC case is important only for the first braking peak and thepossible force saturation does not significantly deteriorate thecontrol performance

52 Numerical Simulations and Discussion Numerical anal-yses were performed to better interpret the experimentalresults and further discuss the effectiveness of the controlstrategy

Preliminarily a numerical model was built and tuned tothe free vibration response of the uncontrolled system Amodal identification allowedobtaining an accurate numericalmodel of the system considered as a shear-type structurehaving 5 degrees of freedom The signals provided by the 5accelerometers located on the floors recorded with a sam-pling frequency of 1 kHz and a total duration of 1 hour wereused for modal identificationThe structural response signalswere acquired with excitation provided by microtremorsin the laboratory The Frequency domain decomposition(FDD) was used to perform the modal identification afterdecimation of themeasured time history signals to 50Hz andusing a frequency resolution to compute the power spectraldensity matrix of the measurements equal to 00122Hz Theresults of modal identification are summarized in Table 3After identifying the dynamic characteristics of the physicalsystem a model tuning was performed to obtain the massesand stiffness coefficients of the frame to be adopted in thenumerical model as detailed in [17] In Table 3 are alsoreported themodal damping ratios obtained by analyzing theenvelopes of the free decay responses of the five structuralmodes Single mode responses are obtained by harmonicallyexciting the structure with the AMD working as a shakerproviding excitation in resonance with a specific modeConsidering that the modal damping ratios vary slightlywith the amplitude of vibration their values are estimated inintermediate ranges of the response

Advances in Civil Engineering 9

0 2 4 6minus0125

minus01

minus0075

minus005

minus0025

0

0025

005

0075

01

0125

Time (s)

AM

D d

ispla

cem

ents

(m)

LC num LC expxmax minusxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus0125

minus01

minus0075

minus005

minus0025

0

0025

005

0075

01

0125

AM

D d

ispla

cem

ents

(m)

NLC num NLC expxmax minusxmax

0 2 4 6Time (s)

k0 lowastxmaxminusk0 lowastxmax

(b)

Figure 8 Comparison between numerical and experimental AMD displacements for LC (a) and NLC (b)

The experimental tests were reproduced numericallyapplying at the top of the structure a sinusoidal inertialforce corresponding to the acceleration of the AMD Figure 8shows the comparison between numerical and experimentalAMD displacements for both LC (GL = 150) and NLCstrategies (119866NL = 150 119896

0= 02 119896

1= 08) The compa-

rison is limited to the controlled part of the response Theexperimental and numerical results are close to each otherespecially for the first and second peaks of the responseNevertheless there are still some deviations in the rate ofdecay and the phase of the response due to the imperfectactuation of the AMD

Similar observations can be drawn for Figure 9 repre-senting the comparison between the numerical and experi-mental top floor displacements Although there is a relativelygood correspondence between the experimental and numer-ical results obtained at the beginning of the control phasethe experimental response decay appears different due to theinternal dynamics of the actuator

Figure 10 reports the comparison between the numer-ically predicted control force obtained with LC and NLCstrategies The NLC algorithm leads to an important increasein the control force at the beginning of the actuation

In Table 4 are summarized the performance indicesintroduced in Section 51 computed by adopting the numeri-cal resultsThe index 119869

5is computed by using the control force

instead of the velocity demand value as follows

1198696=max119905gt1199050

10038161003816100381610038161003816119906NLC

10038161003816100381610038161003816

max119905gt1199050

1003816100381610038161003816119906LC1003816100381610038161003816 (17)

where 119906 is given by (12) It is clear from the results that beingequal to the gain 119866NL the braking performance is reducedwith the increase of the coefficient 119896

0 The indices 119869

2 1198693 and

1198694are equal to 1 for all the analyzed cases meaning that at

the first peak the LC and NLC algorithms provide the samestructural response The index 119869

5suggests that the difference

between the control force yielded by NLC and LC strategiesincreases as 119896

0increasesThe discrepancy between the results

reported in Tables 2 and 4 can be ascribed to the imperfectexperimental actuation and the small differences between thephysical and numerical models

6 Conclusions

In this paper a nonlinear control strategy based on amodifieddirect velocity feedback algorithm is proposed for handlingstroke limits of an AMD system In particular a nonlinearbraking term proportional to the relative AMD velocity isincluded in the control algorithm in order to slowdown thedevice in the proximity of the stroke limits

Experimental and numerical tests were carried out ona scaled-down five-story building equipped with an AMDunder free vibrations

The proposed nonlinear control law proved to be effectivein reducing the structural response and avoiding the strokelimit crossing at the expense of a small increase of structuraldisplacements and a slight increase of structural accelera-tionsThe growth in power demand with respect to the linearcase is important only for the first braking peak and thepossible force saturation does not interfere significantly with

10 Advances in Civil Engineering

minus002

minus001

0

001

002

LC numLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(a)

minus002

minus001

0

001

002

NLC numNLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(b)

Figure 9 Comparison between numerical and experimental top displacements for LC (a) and NLC (b)

0 2 4 6

05

10152025

Time (s)

Con

trol f

orce

(N)

LCNLC

minus45

minus40

minus30

minus35

minus25

minus20

minus15

minus10

minus5

Figure 10 Control force obtained with the numerical analyses

the control performance The parametric analysis showedthat an increase of the ratio 119896

11198960between the parameters

defining the nonlinear restoring force results in an improvedperformance in terms of braking efficacy On the contrarythe same braking efficacy can be obtained with a lower 119896

11198960

ratio and a higher gain 119866NL at the expense of an increase inthe structural response especially in terms of accelerationsThe optimal choice of the ratio 119896

11198960should be obtained as a

compromise between the need of limiting the power demandand that of limiting the structural response

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial support ofthe ldquoCassa di Risparmio di Perugiardquo Foundation that fundedthis study through the project ldquoDevelopment of active controlsystems for the response mitigation under seismic excitationrdquo(Project no 20100110490) Suggestions and comments byDr Michele Moretti and Dr Matteo Becchetti University ofPerugia are also acknowledged with gratitude

References

[1] I Venanzi and A L Materazzi ldquoAcrosswind aeroelastic respo-nse of square tall buildings a semi-analytical approach based ofwind tunnel tests on rigid modelsrdquo Wind amp Structures vol 15no 6 pp 495ndash508 2012

[2] I Venanzi andA LMaterazzi ldquoMulti-objective optimization ofwind-excited structuresrdquo Engineering Structures vol 29 no 6pp 983ndash990 2007

[3] A Forrai S Hashimoto H Funato and K Kamiyama ldquoRobustactive vibration suppression control with constraint on thecontrol signal application to flexible structuresrdquo Earthquake

Advances in Civil Engineering 11

Engineering amp Structural Dynamics vol 32 no 11 pp 1655ndash1676 2003

[4] J G Chase S E Breneman andH A Smith ldquoRobustHinfinstatic

output feedback control with actuator saturationrdquo Journal ofEngineering Mechanics vol 125 no 2 pp 225ndash233 1999

[5] B Indrawan T KoboriM Sakamoto N Koshika and S OhruildquoExperimental verification of bounded-force control methodrdquoEarthquake Engineering amp Structural Dynamics vol 25 no 2pp 179ndash193 1996

[6] J R Cloutier ldquoState-dependent Riccati equation techniques anoverviewrdquo in Proceedings of the American Control Conferencepp 932ndash936 Albuquerque NM USA 1997

[7] B Friedland ldquoOn controlling systems with state-variable con-straintsrdquo in Proceedings of the American Control Conference pp2123ndash2127 Philadelphia Pa USA 1998

[8] A L Materazzi and F Ubertini ldquoRobust structural control withsystem constraintsrdquo Structural Control and Health Monitoringvol 19 no 3 pp 472ndash490 2012

[9] J-HKim andF Jabbari ldquoActuator saturation and control designfor buildings under seismic excitationrdquo Journal of EngineeringMechanics vol 128 no 4 pp 403ndash412 2002

[10] J Mongkol B K Bhartia and Y Fujino ldquoOn Linear-Saturation(LS) control of buildingsrdquo Earthquake Engineering amp StructuralDynamics vol 25 no 12 pp 1353ndash1371 1996

[11] Z Wu and T T Soong ldquoModified bang-bang control lawfor structural control implementationrdquo Journal of EngineeringMechanics vol 122 no 8 pp 771ndash777 1996

[12] C-W Lim ldquoActive vibration control of the linear structure withan active mass damper applying robust saturation controllerrdquoMechatronics vol 18 no 8 pp 391ndash399 2008

[13] I Nagashima and Y Shinozaki ldquoVariable gain feedback con-trol technique of active mass damper and its application tohybrid structural controlrdquo Earthquake Engineering amp StructuralDynamics vol 26 no 8 pp 815ndash838 1997

[14] M Yamamoto and T Sone ldquoBehavior of active mass damper(AMD) installed in high-rise building during 2011 earthquakeoff Pacific coast of Tohoku and verification of regeneratingsystem of AMD based on monitoringrdquo Structural Control andHealth Monitoring vol 21 no 4 pp 634ndash647 2013

[15] I Venanzi F Ubertini and A L Materazzi ldquoOptimal design ofan array of active tuned mass dampers for wind-exposed high-rise buildingsrdquo Structural Control and Health Monitoring vol20 pp 903ndash917 2013

[16] I Venanzi andA LMaterazzi ldquoRobust optimization of a hybridcontrol system for wind-exposed tall buildings with uncertainmass distributionrdquo Smart Structures and Systems vol 12 no 6pp 641ndash659 2013

[17] F Ubertini I Venanzi G Comanducci and A L MaterazzildquoOn the control performance of an active mass driver systemrdquoin Proceedings of the AIMETA Congress Turin Italy 2013

[18] EN 1993-1-1 Eurocode 3 Design of steel structuresmdashpart 1-1General rules and rules for buildings 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Civil Engineering 7

0 2 4 6 8 10 12minus005

0005

01

disp

lace

men

ts (m

)

LCNLC

Control on

Time (s)

xmax

AM

D

k1 lowastxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus002minus001

0

002001

Top

floor

LCNLCControl on

0 2 4 6 8 10 12Time (s)

disp

lace

men

ts (m

)

(b)

minus200

2000

600

1000800

400

Non

linea

r vel

ocity

NLCControl on

0 2 4 6 8 10 12Time (s)

dem

and

(rad

s)

(c)Figure 5 AMD displacements (a) top floor displacements (b)nonlinear velocity demand expressed by (15) (c) for the controlledsystem

In practical applications control force saturation canlikely occur as the nonlinear control system requires asignificant power demand to produce the first braking ofthe AMD To study the effect of force saturation additionaltests were carried out In particular by properly setting thecontroller the actual velocity supplied by the servomotor waslimited to 150 rads value well below the physical limit of thesystem Figure 7 shows that the proposed control algorithmis effective in reducing the structural response and avoidingthe stroke limits crossing even in presence of such forcesaturation

To quantify the effectiveness of the control strategy thefollowing performance indices are defined

1198691=

max119905gt1199050

1003816100381610038161003816119902119886 minus 11990251003816100381610038161003816NLC

max119905gt1199050

1003816100381610038161003816119902119886 minus 11990251003816100381610038161003816LC

1198692=

max119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

max119905gt1199050 119894

1003816100381610038161003816119902LC119894

1003816100381610038161003816

0 2 4 6 8 10 12

150

Time (s)

Velo

city

dem

and

valu

e (ra

ds)

LCNLCControl on

minus50

minus250

minus450

minus650

Figure 6 Velocity demand obtained with LC and NLC (1198960= 02)

strategies

Table 2 Performance indices of the control effectiveness computedadopting the experimental results

1198960

1198691

1198692

1198693

1198694

1198695

02 0749 0987 1102 0995 464203 0767 1001 1077 1004 480304 0843 1012 1010 1208 511705 0857 1008 1139 1268 533206 0862 1005 0947 1214 554107 0891 1059 1109 1226 5603

1198693=

rms119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

rms119905gt1199050 119894

1003816100381610038161003816119902LC119894

1003816100381610038161003816

1198694=

max119905gt1199050 119894

10038161003816100381610038161003816119902NLC119894

10038161003816100381610038161003816

max119905gt1199050 119894

1003816100381610038161003816 119902LC119894

1003816100381610038161003816

1198695=

max119905gt1199050

10038161003816100381610038161003816120599NLC119863

10038161003816100381610038161003816

max119905gt1199050

10038161003816100381610038161003816120599LC119863

10038161003816100381610038161003816

(16)

where 1198691is the ratio of the maximum relative displacements

between the AMD (119902119886) and the top floor (119902

5) obtained with

the NLC and the LC strategies 1198692is the ratio of themaximum

floor displacements (119902119894) obtained with the NLC and the

LC strategies 1198693is the ratio of the root mean square of

the floor displacements obtained with the NLC and the LCstrategies 119869

4is the ratio of the maximum floor accelerations

( 119902119894) obtained with the NLC and the LC strategies 119869

5is the

ratio of the maximum velocity demand value ( 120599119863) obtained

with the NLC and the LC strategiesIn Table 2 are shown the performance indices corre-

sponding to experimental responses obtained for different

8 Advances in Civil Engineering

minus0050

00501

0 2 4 6 8 10 12

AM

D

LCNLC

Control on

Time (s)

xmax

disp

lace

men

ts (m

)

k1 lowastxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus002minus001

0

002001

LCNLCControl on

Top

floor

disp

lace

men

ts (m

)

0 2 4 6 8 10 12Time (s)

(b)

LCNLCControl on

minus300minus200minus100

100300300

0

0 2 4 6 8 10 12Time (s)

Velo

city

dem

and

valu

e (ra

ds)

(c)

Figure 7 AMD displacements (a) top floor displacements (b)nonlinear velocity demand (c) for the system controlled with LCand NLC algorithms (119896

0= 02) in case of force saturation

values of parameter 1198960The value 119896

0= 08was not considered

because as observed in Section 3 when 1198960= 1198961 120576NL(119909)

essentially coincides with Heavisidersquos step function whichcannot be reproduced in practice

The results in Table 2 show that if 1198960increases (i) the

braking efficacy decreases (as 1198691increases) (ii) the control

performance is slightly worsened (as 1198692increases) (iii) the

ratio of the root mean square of the top displacementsobtained with the NLC and the LC (119869

3) is generally higher

than one (iv) the structural acceleration induced by thebraking term increases (as 119869

4increases) and (v) due to the

additional nonlinear term the velocity demand increaseswith respect to the LC case (as 119869

5increases)

Based on the obtained results it can be concluded thatthe NLC is effective in reducing the structural responseand avoiding the stroke limit crossing at the expense of analmost negligible increase in structural displacements and aslight increase in structural accelerations For free response

Table 3 Identified modal characteristics of the system

Mode Natural frequency (Hz) Modal damping ratio ()1 176 032 492 043 734 114 962 125 1215 13

Table 4 Performance indices of the control effectiveness computedadopting the numerical results

1198960

1198691

1198692

1198693

1198694

1198696

02 0585 1000 1000 1000 194303 0619 1000 1000 1000 241204 0651 1000 1000 1000 239205 0681 1000 1000 1000 273706 0711 1000 1000 1000 326807 0740 1000 1000 1000 4097

vibrations the increase in power demand with respect to theLC case is important only for the first braking peak and thepossible force saturation does not significantly deteriorate thecontrol performance

52 Numerical Simulations and Discussion Numerical anal-yses were performed to better interpret the experimentalresults and further discuss the effectiveness of the controlstrategy

Preliminarily a numerical model was built and tuned tothe free vibration response of the uncontrolled system Amodal identification allowedobtaining an accurate numericalmodel of the system considered as a shear-type structurehaving 5 degrees of freedom The signals provided by the 5accelerometers located on the floors recorded with a sam-pling frequency of 1 kHz and a total duration of 1 hour wereused for modal identificationThe structural response signalswere acquired with excitation provided by microtremorsin the laboratory The Frequency domain decomposition(FDD) was used to perform the modal identification afterdecimation of themeasured time history signals to 50Hz andusing a frequency resolution to compute the power spectraldensity matrix of the measurements equal to 00122Hz Theresults of modal identification are summarized in Table 3After identifying the dynamic characteristics of the physicalsystem a model tuning was performed to obtain the massesand stiffness coefficients of the frame to be adopted in thenumerical model as detailed in [17] In Table 3 are alsoreported themodal damping ratios obtained by analyzing theenvelopes of the free decay responses of the five structuralmodes Single mode responses are obtained by harmonicallyexciting the structure with the AMD working as a shakerproviding excitation in resonance with a specific modeConsidering that the modal damping ratios vary slightlywith the amplitude of vibration their values are estimated inintermediate ranges of the response

Advances in Civil Engineering 9

0 2 4 6minus0125

minus01

minus0075

minus005

minus0025

0

0025

005

0075

01

0125

Time (s)

AM

D d

ispla

cem

ents

(m)

LC num LC expxmax minusxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus0125

minus01

minus0075

minus005

minus0025

0

0025

005

0075

01

0125

AM

D d

ispla

cem

ents

(m)

NLC num NLC expxmax minusxmax

0 2 4 6Time (s)

k0 lowastxmaxminusk0 lowastxmax

(b)

Figure 8 Comparison between numerical and experimental AMD displacements for LC (a) and NLC (b)

The experimental tests were reproduced numericallyapplying at the top of the structure a sinusoidal inertialforce corresponding to the acceleration of the AMD Figure 8shows the comparison between numerical and experimentalAMD displacements for both LC (GL = 150) and NLCstrategies (119866NL = 150 119896

0= 02 119896

1= 08) The compa-

rison is limited to the controlled part of the response Theexperimental and numerical results are close to each otherespecially for the first and second peaks of the responseNevertheless there are still some deviations in the rate ofdecay and the phase of the response due to the imperfectactuation of the AMD

Similar observations can be drawn for Figure 9 repre-senting the comparison between the numerical and experi-mental top floor displacements Although there is a relativelygood correspondence between the experimental and numer-ical results obtained at the beginning of the control phasethe experimental response decay appears different due to theinternal dynamics of the actuator

Figure 10 reports the comparison between the numer-ically predicted control force obtained with LC and NLCstrategies The NLC algorithm leads to an important increasein the control force at the beginning of the actuation

In Table 4 are summarized the performance indicesintroduced in Section 51 computed by adopting the numeri-cal resultsThe index 119869

5is computed by using the control force

instead of the velocity demand value as follows

1198696=max119905gt1199050

10038161003816100381610038161003816119906NLC

10038161003816100381610038161003816

max119905gt1199050

1003816100381610038161003816119906LC1003816100381610038161003816 (17)

where 119906 is given by (12) It is clear from the results that beingequal to the gain 119866NL the braking performance is reducedwith the increase of the coefficient 119896

0 The indices 119869

2 1198693 and

1198694are equal to 1 for all the analyzed cases meaning that at

the first peak the LC and NLC algorithms provide the samestructural response The index 119869

5suggests that the difference

between the control force yielded by NLC and LC strategiesincreases as 119896

0increasesThe discrepancy between the results

reported in Tables 2 and 4 can be ascribed to the imperfectexperimental actuation and the small differences between thephysical and numerical models

6 Conclusions

In this paper a nonlinear control strategy based on amodifieddirect velocity feedback algorithm is proposed for handlingstroke limits of an AMD system In particular a nonlinearbraking term proportional to the relative AMD velocity isincluded in the control algorithm in order to slowdown thedevice in the proximity of the stroke limits

Experimental and numerical tests were carried out ona scaled-down five-story building equipped with an AMDunder free vibrations

The proposed nonlinear control law proved to be effectivein reducing the structural response and avoiding the strokelimit crossing at the expense of a small increase of structuraldisplacements and a slight increase of structural accelera-tionsThe growth in power demand with respect to the linearcase is important only for the first braking peak and thepossible force saturation does not interfere significantly with

10 Advances in Civil Engineering

minus002

minus001

0

001

002

LC numLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(a)

minus002

minus001

0

001

002

NLC numNLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(b)

Figure 9 Comparison between numerical and experimental top displacements for LC (a) and NLC (b)

0 2 4 6

05

10152025

Time (s)

Con

trol f

orce

(N)

LCNLC

minus45

minus40

minus30

minus35

minus25

minus20

minus15

minus10

minus5

Figure 10 Control force obtained with the numerical analyses

the control performance The parametric analysis showedthat an increase of the ratio 119896

11198960between the parameters

defining the nonlinear restoring force results in an improvedperformance in terms of braking efficacy On the contrarythe same braking efficacy can be obtained with a lower 119896

11198960

ratio and a higher gain 119866NL at the expense of an increase inthe structural response especially in terms of accelerationsThe optimal choice of the ratio 119896

11198960should be obtained as a

compromise between the need of limiting the power demandand that of limiting the structural response

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial support ofthe ldquoCassa di Risparmio di Perugiardquo Foundation that fundedthis study through the project ldquoDevelopment of active controlsystems for the response mitigation under seismic excitationrdquo(Project no 20100110490) Suggestions and comments byDr Michele Moretti and Dr Matteo Becchetti University ofPerugia are also acknowledged with gratitude

References

[1] I Venanzi and A L Materazzi ldquoAcrosswind aeroelastic respo-nse of square tall buildings a semi-analytical approach based ofwind tunnel tests on rigid modelsrdquo Wind amp Structures vol 15no 6 pp 495ndash508 2012

[2] I Venanzi andA LMaterazzi ldquoMulti-objective optimization ofwind-excited structuresrdquo Engineering Structures vol 29 no 6pp 983ndash990 2007

[3] A Forrai S Hashimoto H Funato and K Kamiyama ldquoRobustactive vibration suppression control with constraint on thecontrol signal application to flexible structuresrdquo Earthquake

Advances in Civil Engineering 11

Engineering amp Structural Dynamics vol 32 no 11 pp 1655ndash1676 2003

[4] J G Chase S E Breneman andH A Smith ldquoRobustHinfinstatic

output feedback control with actuator saturationrdquo Journal ofEngineering Mechanics vol 125 no 2 pp 225ndash233 1999

[5] B Indrawan T KoboriM Sakamoto N Koshika and S OhruildquoExperimental verification of bounded-force control methodrdquoEarthquake Engineering amp Structural Dynamics vol 25 no 2pp 179ndash193 1996

[6] J R Cloutier ldquoState-dependent Riccati equation techniques anoverviewrdquo in Proceedings of the American Control Conferencepp 932ndash936 Albuquerque NM USA 1997

[7] B Friedland ldquoOn controlling systems with state-variable con-straintsrdquo in Proceedings of the American Control Conference pp2123ndash2127 Philadelphia Pa USA 1998

[8] A L Materazzi and F Ubertini ldquoRobust structural control withsystem constraintsrdquo Structural Control and Health Monitoringvol 19 no 3 pp 472ndash490 2012

[9] J-HKim andF Jabbari ldquoActuator saturation and control designfor buildings under seismic excitationrdquo Journal of EngineeringMechanics vol 128 no 4 pp 403ndash412 2002

[10] J Mongkol B K Bhartia and Y Fujino ldquoOn Linear-Saturation(LS) control of buildingsrdquo Earthquake Engineering amp StructuralDynamics vol 25 no 12 pp 1353ndash1371 1996

[11] Z Wu and T T Soong ldquoModified bang-bang control lawfor structural control implementationrdquo Journal of EngineeringMechanics vol 122 no 8 pp 771ndash777 1996

[12] C-W Lim ldquoActive vibration control of the linear structure withan active mass damper applying robust saturation controllerrdquoMechatronics vol 18 no 8 pp 391ndash399 2008

[13] I Nagashima and Y Shinozaki ldquoVariable gain feedback con-trol technique of active mass damper and its application tohybrid structural controlrdquo Earthquake Engineering amp StructuralDynamics vol 26 no 8 pp 815ndash838 1997

[14] M Yamamoto and T Sone ldquoBehavior of active mass damper(AMD) installed in high-rise building during 2011 earthquakeoff Pacific coast of Tohoku and verification of regeneratingsystem of AMD based on monitoringrdquo Structural Control andHealth Monitoring vol 21 no 4 pp 634ndash647 2013

[15] I Venanzi F Ubertini and A L Materazzi ldquoOptimal design ofan array of active tuned mass dampers for wind-exposed high-rise buildingsrdquo Structural Control and Health Monitoring vol20 pp 903ndash917 2013

[16] I Venanzi andA LMaterazzi ldquoRobust optimization of a hybridcontrol system for wind-exposed tall buildings with uncertainmass distributionrdquo Smart Structures and Systems vol 12 no 6pp 641ndash659 2013

[17] F Ubertini I Venanzi G Comanducci and A L MaterazzildquoOn the control performance of an active mass driver systemrdquoin Proceedings of the AIMETA Congress Turin Italy 2013

[18] EN 1993-1-1 Eurocode 3 Design of steel structuresmdashpart 1-1General rules and rules for buildings 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Advances in Civil Engineering

minus0050

00501

0 2 4 6 8 10 12

AM

D

LCNLC

Control on

Time (s)

xmax

disp

lace

men

ts (m

)

k1 lowastxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus002minus001

0

002001

LCNLCControl on

Top

floor

disp

lace

men

ts (m

)

0 2 4 6 8 10 12Time (s)

(b)

LCNLCControl on

minus300minus200minus100

100300300

0

0 2 4 6 8 10 12Time (s)

Velo

city

dem

and

valu

e (ra

ds)

(c)

Figure 7 AMD displacements (a) top floor displacements (b)nonlinear velocity demand (c) for the system controlled with LCand NLC algorithms (119896

0= 02) in case of force saturation

values of parameter 1198960The value 119896

0= 08was not considered

because as observed in Section 3 when 1198960= 1198961 120576NL(119909)

essentially coincides with Heavisidersquos step function whichcannot be reproduced in practice

The results in Table 2 show that if 1198960increases (i) the

braking efficacy decreases (as 1198691increases) (ii) the control

performance is slightly worsened (as 1198692increases) (iii) the

ratio of the root mean square of the top displacementsobtained with the NLC and the LC (119869

3) is generally higher

than one (iv) the structural acceleration induced by thebraking term increases (as 119869

4increases) and (v) due to the

additional nonlinear term the velocity demand increaseswith respect to the LC case (as 119869

5increases)

Based on the obtained results it can be concluded thatthe NLC is effective in reducing the structural responseand avoiding the stroke limit crossing at the expense of analmost negligible increase in structural displacements and aslight increase in structural accelerations For free response

Table 3 Identified modal characteristics of the system

Mode Natural frequency (Hz) Modal damping ratio ()1 176 032 492 043 734 114 962 125 1215 13

Table 4 Performance indices of the control effectiveness computedadopting the numerical results

1198960

1198691

1198692

1198693

1198694

1198696

02 0585 1000 1000 1000 194303 0619 1000 1000 1000 241204 0651 1000 1000 1000 239205 0681 1000 1000 1000 273706 0711 1000 1000 1000 326807 0740 1000 1000 1000 4097

vibrations the increase in power demand with respect to theLC case is important only for the first braking peak and thepossible force saturation does not significantly deteriorate thecontrol performance

52 Numerical Simulations and Discussion Numerical anal-yses were performed to better interpret the experimentalresults and further discuss the effectiveness of the controlstrategy

Preliminarily a numerical model was built and tuned tothe free vibration response of the uncontrolled system Amodal identification allowedobtaining an accurate numericalmodel of the system considered as a shear-type structurehaving 5 degrees of freedom The signals provided by the 5accelerometers located on the floors recorded with a sam-pling frequency of 1 kHz and a total duration of 1 hour wereused for modal identificationThe structural response signalswere acquired with excitation provided by microtremorsin the laboratory The Frequency domain decomposition(FDD) was used to perform the modal identification afterdecimation of themeasured time history signals to 50Hz andusing a frequency resolution to compute the power spectraldensity matrix of the measurements equal to 00122Hz Theresults of modal identification are summarized in Table 3After identifying the dynamic characteristics of the physicalsystem a model tuning was performed to obtain the massesand stiffness coefficients of the frame to be adopted in thenumerical model as detailed in [17] In Table 3 are alsoreported themodal damping ratios obtained by analyzing theenvelopes of the free decay responses of the five structuralmodes Single mode responses are obtained by harmonicallyexciting the structure with the AMD working as a shakerproviding excitation in resonance with a specific modeConsidering that the modal damping ratios vary slightlywith the amplitude of vibration their values are estimated inintermediate ranges of the response

Advances in Civil Engineering 9

0 2 4 6minus0125

minus01

minus0075

minus005

minus0025

0

0025

005

0075

01

0125

Time (s)

AM

D d

ispla

cem

ents

(m)

LC num LC expxmax minusxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus0125

minus01

minus0075

minus005

minus0025

0

0025

005

0075

01

0125

AM

D d

ispla

cem

ents

(m)

NLC num NLC expxmax minusxmax

0 2 4 6Time (s)

k0 lowastxmaxminusk0 lowastxmax

(b)

Figure 8 Comparison between numerical and experimental AMD displacements for LC (a) and NLC (b)

The experimental tests were reproduced numericallyapplying at the top of the structure a sinusoidal inertialforce corresponding to the acceleration of the AMD Figure 8shows the comparison between numerical and experimentalAMD displacements for both LC (GL = 150) and NLCstrategies (119866NL = 150 119896

0= 02 119896

1= 08) The compa-

rison is limited to the controlled part of the response Theexperimental and numerical results are close to each otherespecially for the first and second peaks of the responseNevertheless there are still some deviations in the rate ofdecay and the phase of the response due to the imperfectactuation of the AMD

Similar observations can be drawn for Figure 9 repre-senting the comparison between the numerical and experi-mental top floor displacements Although there is a relativelygood correspondence between the experimental and numer-ical results obtained at the beginning of the control phasethe experimental response decay appears different due to theinternal dynamics of the actuator

Figure 10 reports the comparison between the numer-ically predicted control force obtained with LC and NLCstrategies The NLC algorithm leads to an important increasein the control force at the beginning of the actuation

In Table 4 are summarized the performance indicesintroduced in Section 51 computed by adopting the numeri-cal resultsThe index 119869

5is computed by using the control force

instead of the velocity demand value as follows

1198696=max119905gt1199050

10038161003816100381610038161003816119906NLC

10038161003816100381610038161003816

max119905gt1199050

1003816100381610038161003816119906LC1003816100381610038161003816 (17)

where 119906 is given by (12) It is clear from the results that beingequal to the gain 119866NL the braking performance is reducedwith the increase of the coefficient 119896

0 The indices 119869

2 1198693 and

1198694are equal to 1 for all the analyzed cases meaning that at

the first peak the LC and NLC algorithms provide the samestructural response The index 119869

5suggests that the difference

between the control force yielded by NLC and LC strategiesincreases as 119896

0increasesThe discrepancy between the results

reported in Tables 2 and 4 can be ascribed to the imperfectexperimental actuation and the small differences between thephysical and numerical models

6 Conclusions

In this paper a nonlinear control strategy based on amodifieddirect velocity feedback algorithm is proposed for handlingstroke limits of an AMD system In particular a nonlinearbraking term proportional to the relative AMD velocity isincluded in the control algorithm in order to slowdown thedevice in the proximity of the stroke limits

Experimental and numerical tests were carried out ona scaled-down five-story building equipped with an AMDunder free vibrations

The proposed nonlinear control law proved to be effectivein reducing the structural response and avoiding the strokelimit crossing at the expense of a small increase of structuraldisplacements and a slight increase of structural accelera-tionsThe growth in power demand with respect to the linearcase is important only for the first braking peak and thepossible force saturation does not interfere significantly with

10 Advances in Civil Engineering

minus002

minus001

0

001

002

LC numLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(a)

minus002

minus001

0

001

002

NLC numNLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(b)

Figure 9 Comparison between numerical and experimental top displacements for LC (a) and NLC (b)

0 2 4 6

05

10152025

Time (s)

Con

trol f

orce

(N)

LCNLC

minus45

minus40

minus30

minus35

minus25

minus20

minus15

minus10

minus5

Figure 10 Control force obtained with the numerical analyses

the control performance The parametric analysis showedthat an increase of the ratio 119896

11198960between the parameters

defining the nonlinear restoring force results in an improvedperformance in terms of braking efficacy On the contrarythe same braking efficacy can be obtained with a lower 119896

11198960

ratio and a higher gain 119866NL at the expense of an increase inthe structural response especially in terms of accelerationsThe optimal choice of the ratio 119896

11198960should be obtained as a

compromise between the need of limiting the power demandand that of limiting the structural response

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial support ofthe ldquoCassa di Risparmio di Perugiardquo Foundation that fundedthis study through the project ldquoDevelopment of active controlsystems for the response mitigation under seismic excitationrdquo(Project no 20100110490) Suggestions and comments byDr Michele Moretti and Dr Matteo Becchetti University ofPerugia are also acknowledged with gratitude

References

[1] I Venanzi and A L Materazzi ldquoAcrosswind aeroelastic respo-nse of square tall buildings a semi-analytical approach based ofwind tunnel tests on rigid modelsrdquo Wind amp Structures vol 15no 6 pp 495ndash508 2012

[2] I Venanzi andA LMaterazzi ldquoMulti-objective optimization ofwind-excited structuresrdquo Engineering Structures vol 29 no 6pp 983ndash990 2007

[3] A Forrai S Hashimoto H Funato and K Kamiyama ldquoRobustactive vibration suppression control with constraint on thecontrol signal application to flexible structuresrdquo Earthquake

Advances in Civil Engineering 11

Engineering amp Structural Dynamics vol 32 no 11 pp 1655ndash1676 2003

[4] J G Chase S E Breneman andH A Smith ldquoRobustHinfinstatic

output feedback control with actuator saturationrdquo Journal ofEngineering Mechanics vol 125 no 2 pp 225ndash233 1999

[5] B Indrawan T KoboriM Sakamoto N Koshika and S OhruildquoExperimental verification of bounded-force control methodrdquoEarthquake Engineering amp Structural Dynamics vol 25 no 2pp 179ndash193 1996

[6] J R Cloutier ldquoState-dependent Riccati equation techniques anoverviewrdquo in Proceedings of the American Control Conferencepp 932ndash936 Albuquerque NM USA 1997

[7] B Friedland ldquoOn controlling systems with state-variable con-straintsrdquo in Proceedings of the American Control Conference pp2123ndash2127 Philadelphia Pa USA 1998

[8] A L Materazzi and F Ubertini ldquoRobust structural control withsystem constraintsrdquo Structural Control and Health Monitoringvol 19 no 3 pp 472ndash490 2012

[9] J-HKim andF Jabbari ldquoActuator saturation and control designfor buildings under seismic excitationrdquo Journal of EngineeringMechanics vol 128 no 4 pp 403ndash412 2002

[10] J Mongkol B K Bhartia and Y Fujino ldquoOn Linear-Saturation(LS) control of buildingsrdquo Earthquake Engineering amp StructuralDynamics vol 25 no 12 pp 1353ndash1371 1996

[11] Z Wu and T T Soong ldquoModified bang-bang control lawfor structural control implementationrdquo Journal of EngineeringMechanics vol 122 no 8 pp 771ndash777 1996

[12] C-W Lim ldquoActive vibration control of the linear structure withan active mass damper applying robust saturation controllerrdquoMechatronics vol 18 no 8 pp 391ndash399 2008

[13] I Nagashima and Y Shinozaki ldquoVariable gain feedback con-trol technique of active mass damper and its application tohybrid structural controlrdquo Earthquake Engineering amp StructuralDynamics vol 26 no 8 pp 815ndash838 1997

[14] M Yamamoto and T Sone ldquoBehavior of active mass damper(AMD) installed in high-rise building during 2011 earthquakeoff Pacific coast of Tohoku and verification of regeneratingsystem of AMD based on monitoringrdquo Structural Control andHealth Monitoring vol 21 no 4 pp 634ndash647 2013

[15] I Venanzi F Ubertini and A L Materazzi ldquoOptimal design ofan array of active tuned mass dampers for wind-exposed high-rise buildingsrdquo Structural Control and Health Monitoring vol20 pp 903ndash917 2013

[16] I Venanzi andA LMaterazzi ldquoRobust optimization of a hybridcontrol system for wind-exposed tall buildings with uncertainmass distributionrdquo Smart Structures and Systems vol 12 no 6pp 641ndash659 2013

[17] F Ubertini I Venanzi G Comanducci and A L MaterazzildquoOn the control performance of an active mass driver systemrdquoin Proceedings of the AIMETA Congress Turin Italy 2013

[18] EN 1993-1-1 Eurocode 3 Design of steel structuresmdashpart 1-1General rules and rules for buildings 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Civil Engineering 9

0 2 4 6minus0125

minus01

minus0075

minus005

minus0025

0

0025

005

0075

01

0125

Time (s)

AM

D d

ispla

cem

ents

(m)

LC num LC expxmax minusxmax

k0 lowastxmaxminusk0 lowastxmax

(a)

minus0125

minus01

minus0075

minus005

minus0025

0

0025

005

0075

01

0125

AM

D d

ispla

cem

ents

(m)

NLC num NLC expxmax minusxmax

0 2 4 6Time (s)

k0 lowastxmaxminusk0 lowastxmax

(b)

Figure 8 Comparison between numerical and experimental AMD displacements for LC (a) and NLC (b)

The experimental tests were reproduced numericallyapplying at the top of the structure a sinusoidal inertialforce corresponding to the acceleration of the AMD Figure 8shows the comparison between numerical and experimentalAMD displacements for both LC (GL = 150) and NLCstrategies (119866NL = 150 119896

0= 02 119896

1= 08) The compa-

rison is limited to the controlled part of the response Theexperimental and numerical results are close to each otherespecially for the first and second peaks of the responseNevertheless there are still some deviations in the rate ofdecay and the phase of the response due to the imperfectactuation of the AMD

Similar observations can be drawn for Figure 9 repre-senting the comparison between the numerical and experi-mental top floor displacements Although there is a relativelygood correspondence between the experimental and numer-ical results obtained at the beginning of the control phasethe experimental response decay appears different due to theinternal dynamics of the actuator

Figure 10 reports the comparison between the numer-ically predicted control force obtained with LC and NLCstrategies The NLC algorithm leads to an important increasein the control force at the beginning of the actuation

In Table 4 are summarized the performance indicesintroduced in Section 51 computed by adopting the numeri-cal resultsThe index 119869

5is computed by using the control force

instead of the velocity demand value as follows

1198696=max119905gt1199050

10038161003816100381610038161003816119906NLC

10038161003816100381610038161003816

max119905gt1199050

1003816100381610038161003816119906LC1003816100381610038161003816 (17)

where 119906 is given by (12) It is clear from the results that beingequal to the gain 119866NL the braking performance is reducedwith the increase of the coefficient 119896

0 The indices 119869

2 1198693 and

1198694are equal to 1 for all the analyzed cases meaning that at

the first peak the LC and NLC algorithms provide the samestructural response The index 119869

5suggests that the difference

between the control force yielded by NLC and LC strategiesincreases as 119896

0increasesThe discrepancy between the results

reported in Tables 2 and 4 can be ascribed to the imperfectexperimental actuation and the small differences between thephysical and numerical models

6 Conclusions

In this paper a nonlinear control strategy based on amodifieddirect velocity feedback algorithm is proposed for handlingstroke limits of an AMD system In particular a nonlinearbraking term proportional to the relative AMD velocity isincluded in the control algorithm in order to slowdown thedevice in the proximity of the stroke limits

Experimental and numerical tests were carried out ona scaled-down five-story building equipped with an AMDunder free vibrations

The proposed nonlinear control law proved to be effectivein reducing the structural response and avoiding the strokelimit crossing at the expense of a small increase of structuraldisplacements and a slight increase of structural accelera-tionsThe growth in power demand with respect to the linearcase is important only for the first braking peak and thepossible force saturation does not interfere significantly with

10 Advances in Civil Engineering

minus002

minus001

0

001

002

LC numLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(a)

minus002

minus001

0

001

002

NLC numNLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(b)

Figure 9 Comparison between numerical and experimental top displacements for LC (a) and NLC (b)

0 2 4 6

05

10152025

Time (s)

Con

trol f

orce

(N)

LCNLC

minus45

minus40

minus30

minus35

minus25

minus20

minus15

minus10

minus5

Figure 10 Control force obtained with the numerical analyses

the control performance The parametric analysis showedthat an increase of the ratio 119896

11198960between the parameters

defining the nonlinear restoring force results in an improvedperformance in terms of braking efficacy On the contrarythe same braking efficacy can be obtained with a lower 119896

11198960

ratio and a higher gain 119866NL at the expense of an increase inthe structural response especially in terms of accelerationsThe optimal choice of the ratio 119896

11198960should be obtained as a

compromise between the need of limiting the power demandand that of limiting the structural response

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial support ofthe ldquoCassa di Risparmio di Perugiardquo Foundation that fundedthis study through the project ldquoDevelopment of active controlsystems for the response mitigation under seismic excitationrdquo(Project no 20100110490) Suggestions and comments byDr Michele Moretti and Dr Matteo Becchetti University ofPerugia are also acknowledged with gratitude

References

[1] I Venanzi and A L Materazzi ldquoAcrosswind aeroelastic respo-nse of square tall buildings a semi-analytical approach based ofwind tunnel tests on rigid modelsrdquo Wind amp Structures vol 15no 6 pp 495ndash508 2012

[2] I Venanzi andA LMaterazzi ldquoMulti-objective optimization ofwind-excited structuresrdquo Engineering Structures vol 29 no 6pp 983ndash990 2007

[3] A Forrai S Hashimoto H Funato and K Kamiyama ldquoRobustactive vibration suppression control with constraint on thecontrol signal application to flexible structuresrdquo Earthquake

Advances in Civil Engineering 11

Engineering amp Structural Dynamics vol 32 no 11 pp 1655ndash1676 2003

[4] J G Chase S E Breneman andH A Smith ldquoRobustHinfinstatic

output feedback control with actuator saturationrdquo Journal ofEngineering Mechanics vol 125 no 2 pp 225ndash233 1999

[5] B Indrawan T KoboriM Sakamoto N Koshika and S OhruildquoExperimental verification of bounded-force control methodrdquoEarthquake Engineering amp Structural Dynamics vol 25 no 2pp 179ndash193 1996

[6] J R Cloutier ldquoState-dependent Riccati equation techniques anoverviewrdquo in Proceedings of the American Control Conferencepp 932ndash936 Albuquerque NM USA 1997

[7] B Friedland ldquoOn controlling systems with state-variable con-straintsrdquo in Proceedings of the American Control Conference pp2123ndash2127 Philadelphia Pa USA 1998

[8] A L Materazzi and F Ubertini ldquoRobust structural control withsystem constraintsrdquo Structural Control and Health Monitoringvol 19 no 3 pp 472ndash490 2012

[9] J-HKim andF Jabbari ldquoActuator saturation and control designfor buildings under seismic excitationrdquo Journal of EngineeringMechanics vol 128 no 4 pp 403ndash412 2002

[10] J Mongkol B K Bhartia and Y Fujino ldquoOn Linear-Saturation(LS) control of buildingsrdquo Earthquake Engineering amp StructuralDynamics vol 25 no 12 pp 1353ndash1371 1996

[11] Z Wu and T T Soong ldquoModified bang-bang control lawfor structural control implementationrdquo Journal of EngineeringMechanics vol 122 no 8 pp 771ndash777 1996

[12] C-W Lim ldquoActive vibration control of the linear structure withan active mass damper applying robust saturation controllerrdquoMechatronics vol 18 no 8 pp 391ndash399 2008

[13] I Nagashima and Y Shinozaki ldquoVariable gain feedback con-trol technique of active mass damper and its application tohybrid structural controlrdquo Earthquake Engineering amp StructuralDynamics vol 26 no 8 pp 815ndash838 1997

[14] M Yamamoto and T Sone ldquoBehavior of active mass damper(AMD) installed in high-rise building during 2011 earthquakeoff Pacific coast of Tohoku and verification of regeneratingsystem of AMD based on monitoringrdquo Structural Control andHealth Monitoring vol 21 no 4 pp 634ndash647 2013

[15] I Venanzi F Ubertini and A L Materazzi ldquoOptimal design ofan array of active tuned mass dampers for wind-exposed high-rise buildingsrdquo Structural Control and Health Monitoring vol20 pp 903ndash917 2013

[16] I Venanzi andA LMaterazzi ldquoRobust optimization of a hybridcontrol system for wind-exposed tall buildings with uncertainmass distributionrdquo Smart Structures and Systems vol 12 no 6pp 641ndash659 2013

[17] F Ubertini I Venanzi G Comanducci and A L MaterazzildquoOn the control performance of an active mass driver systemrdquoin Proceedings of the AIMETA Congress Turin Italy 2013

[18] EN 1993-1-1 Eurocode 3 Design of steel structuresmdashpart 1-1General rules and rules for buildings 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Advances in Civil Engineering

minus002

minus001

0

001

002

LC numLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(a)

minus002

minus001

0

001

002

NLC numNLC exp

0 2 4 6Time (s)

Top

floor

disp

lace

men

ts (m

)

(b)

Figure 9 Comparison between numerical and experimental top displacements for LC (a) and NLC (b)

0 2 4 6

05

10152025

Time (s)

Con

trol f

orce

(N)

LCNLC

minus45

minus40

minus30

minus35

minus25

minus20

minus15

minus10

minus5

Figure 10 Control force obtained with the numerical analyses

the control performance The parametric analysis showedthat an increase of the ratio 119896

11198960between the parameters

defining the nonlinear restoring force results in an improvedperformance in terms of braking efficacy On the contrarythe same braking efficacy can be obtained with a lower 119896

11198960

ratio and a higher gain 119866NL at the expense of an increase inthe structural response especially in terms of accelerationsThe optimal choice of the ratio 119896

11198960should be obtained as a

compromise between the need of limiting the power demandand that of limiting the structural response

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial support ofthe ldquoCassa di Risparmio di Perugiardquo Foundation that fundedthis study through the project ldquoDevelopment of active controlsystems for the response mitigation under seismic excitationrdquo(Project no 20100110490) Suggestions and comments byDr Michele Moretti and Dr Matteo Becchetti University ofPerugia are also acknowledged with gratitude

References

[1] I Venanzi and A L Materazzi ldquoAcrosswind aeroelastic respo-nse of square tall buildings a semi-analytical approach based ofwind tunnel tests on rigid modelsrdquo Wind amp Structures vol 15no 6 pp 495ndash508 2012

[2] I Venanzi andA LMaterazzi ldquoMulti-objective optimization ofwind-excited structuresrdquo Engineering Structures vol 29 no 6pp 983ndash990 2007

[3] A Forrai S Hashimoto H Funato and K Kamiyama ldquoRobustactive vibration suppression control with constraint on thecontrol signal application to flexible structuresrdquo Earthquake

Advances in Civil Engineering 11

Engineering amp Structural Dynamics vol 32 no 11 pp 1655ndash1676 2003

[4] J G Chase S E Breneman andH A Smith ldquoRobustHinfinstatic

output feedback control with actuator saturationrdquo Journal ofEngineering Mechanics vol 125 no 2 pp 225ndash233 1999

[5] B Indrawan T KoboriM Sakamoto N Koshika and S OhruildquoExperimental verification of bounded-force control methodrdquoEarthquake Engineering amp Structural Dynamics vol 25 no 2pp 179ndash193 1996

[6] J R Cloutier ldquoState-dependent Riccati equation techniques anoverviewrdquo in Proceedings of the American Control Conferencepp 932ndash936 Albuquerque NM USA 1997

[7] B Friedland ldquoOn controlling systems with state-variable con-straintsrdquo in Proceedings of the American Control Conference pp2123ndash2127 Philadelphia Pa USA 1998

[8] A L Materazzi and F Ubertini ldquoRobust structural control withsystem constraintsrdquo Structural Control and Health Monitoringvol 19 no 3 pp 472ndash490 2012

[9] J-HKim andF Jabbari ldquoActuator saturation and control designfor buildings under seismic excitationrdquo Journal of EngineeringMechanics vol 128 no 4 pp 403ndash412 2002

[10] J Mongkol B K Bhartia and Y Fujino ldquoOn Linear-Saturation(LS) control of buildingsrdquo Earthquake Engineering amp StructuralDynamics vol 25 no 12 pp 1353ndash1371 1996

[11] Z Wu and T T Soong ldquoModified bang-bang control lawfor structural control implementationrdquo Journal of EngineeringMechanics vol 122 no 8 pp 771ndash777 1996

[12] C-W Lim ldquoActive vibration control of the linear structure withan active mass damper applying robust saturation controllerrdquoMechatronics vol 18 no 8 pp 391ndash399 2008

[13] I Nagashima and Y Shinozaki ldquoVariable gain feedback con-trol technique of active mass damper and its application tohybrid structural controlrdquo Earthquake Engineering amp StructuralDynamics vol 26 no 8 pp 815ndash838 1997

[14] M Yamamoto and T Sone ldquoBehavior of active mass damper(AMD) installed in high-rise building during 2011 earthquakeoff Pacific coast of Tohoku and verification of regeneratingsystem of AMD based on monitoringrdquo Structural Control andHealth Monitoring vol 21 no 4 pp 634ndash647 2013

[15] I Venanzi F Ubertini and A L Materazzi ldquoOptimal design ofan array of active tuned mass dampers for wind-exposed high-rise buildingsrdquo Structural Control and Health Monitoring vol20 pp 903ndash917 2013

[16] I Venanzi andA LMaterazzi ldquoRobust optimization of a hybridcontrol system for wind-exposed tall buildings with uncertainmass distributionrdquo Smart Structures and Systems vol 12 no 6pp 641ndash659 2013

[17] F Ubertini I Venanzi G Comanducci and A L MaterazzildquoOn the control performance of an active mass driver systemrdquoin Proceedings of the AIMETA Congress Turin Italy 2013

[18] EN 1993-1-1 Eurocode 3 Design of steel structuresmdashpart 1-1General rules and rules for buildings 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Civil Engineering 11

Engineering amp Structural Dynamics vol 32 no 11 pp 1655ndash1676 2003

[4] J G Chase S E Breneman andH A Smith ldquoRobustHinfinstatic

output feedback control with actuator saturationrdquo Journal ofEngineering Mechanics vol 125 no 2 pp 225ndash233 1999

[5] B Indrawan T KoboriM Sakamoto N Koshika and S OhruildquoExperimental verification of bounded-force control methodrdquoEarthquake Engineering amp Structural Dynamics vol 25 no 2pp 179ndash193 1996

[6] J R Cloutier ldquoState-dependent Riccati equation techniques anoverviewrdquo in Proceedings of the American Control Conferencepp 932ndash936 Albuquerque NM USA 1997

[7] B Friedland ldquoOn controlling systems with state-variable con-straintsrdquo in Proceedings of the American Control Conference pp2123ndash2127 Philadelphia Pa USA 1998

[8] A L Materazzi and F Ubertini ldquoRobust structural control withsystem constraintsrdquo Structural Control and Health Monitoringvol 19 no 3 pp 472ndash490 2012

[9] J-HKim andF Jabbari ldquoActuator saturation and control designfor buildings under seismic excitationrdquo Journal of EngineeringMechanics vol 128 no 4 pp 403ndash412 2002

[10] J Mongkol B K Bhartia and Y Fujino ldquoOn Linear-Saturation(LS) control of buildingsrdquo Earthquake Engineering amp StructuralDynamics vol 25 no 12 pp 1353ndash1371 1996

[11] Z Wu and T T Soong ldquoModified bang-bang control lawfor structural control implementationrdquo Journal of EngineeringMechanics vol 122 no 8 pp 771ndash777 1996

[12] C-W Lim ldquoActive vibration control of the linear structure withan active mass damper applying robust saturation controllerrdquoMechatronics vol 18 no 8 pp 391ndash399 2008

[13] I Nagashima and Y Shinozaki ldquoVariable gain feedback con-trol technique of active mass damper and its application tohybrid structural controlrdquo Earthquake Engineering amp StructuralDynamics vol 26 no 8 pp 815ndash838 1997

[14] M Yamamoto and T Sone ldquoBehavior of active mass damper(AMD) installed in high-rise building during 2011 earthquakeoff Pacific coast of Tohoku and verification of regeneratingsystem of AMD based on monitoringrdquo Structural Control andHealth Monitoring vol 21 no 4 pp 634ndash647 2013

[15] I Venanzi F Ubertini and A L Materazzi ldquoOptimal design ofan array of active tuned mass dampers for wind-exposed high-rise buildingsrdquo Structural Control and Health Monitoring vol20 pp 903ndash917 2013

[16] I Venanzi andA LMaterazzi ldquoRobust optimization of a hybridcontrol system for wind-exposed tall buildings with uncertainmass distributionrdquo Smart Structures and Systems vol 12 no 6pp 641ndash659 2013

[17] F Ubertini I Venanzi G Comanducci and A L MaterazzildquoOn the control performance of an active mass driver systemrdquoin Proceedings of the AIMETA Congress Turin Italy 2013

[18] EN 1993-1-1 Eurocode 3 Design of steel structuresmdashpart 1-1General rules and rules for buildings 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of