Research ArticleTest Verification of Two-Stage Adaptive Delay CompensationMethod for Real-Time Hybrid Simulation

Zhen Wang123 Xueqi Yan3 Xizhan Ning 4 and Bin Wu23

1Key Laboratory of Earthquake Engineering and Engineering Vibration Institute of Engineering MechanicsChina Earthquake Administration Harbin 150080 China2School of Civil Engineering and Architecture Wuhan University of Technology Wuhan 430070 China3School of Civil Engineering Harbin Institute of Technology Harbin 150090 China4College of Civil Engineering Huaqiao University Xiamen 361021 China

Correspondence should be addressed to Xizhan Ning xzninghqueducn

Received 21 February 2020 Revised 12 June 2020 Accepted 13 June 2020 Published 14 September 2020

Academic Editor Giuseppe Petrone

Copyright copy 2020 Zhen Wang et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Real-time hybrid simulation (RTHS) is a versatile testing technique for performance evaluation of structures subjected to dynamicexcitations Research revealed that compensation for the delay induced by the dynamics of the loading system and other factors isa critical issue for obtaining reliable test results Lately a two-stage adaptive delay compensation (TADC) method was conceivedand performed on the benchmark problem of RTHS For this method the main part of the system delay is coarsely compensatedby the classic polynomial extrapolation (PE) method the second stage represents a fine remedy for the remaining delay withadaptive compensation based on a discrete model of the loading system As an extension of this study this paper aims to furtherverify and reveal the performance of this method through real tests on a viscous damper specimen In particular loading tests witha swept signal and RTHS with sinusoidal and seismic excitations were carried out Investigations show that the TADC method isendowed with smaller parameter variation ranges simple yet effective initialization or a soft-start process less dependence oninitial parameter estimation accuracy and best compensation performance

1 Introduction

Real-time hybrid simulation (RTHS) [1ndash4] is a versatiletesting technique developed in the past three decades forperformance evaluation of structures subjected to earth-quake wind or other dynamic excitations is techniquedivides the emulated structure into the numerical sub-structure (NS) and experimental substructure (ES) and atransfer system that is loading systems of actuators andorshaking tables is employed to ensure the deformationcompatibility and force equilibrium at the interface betweenthe two substructures [4ndash7] Research reveals that loadingphase errorsndashtime delayndashcause unreliable test results andeven render the test instability [8]

In order to accurately reproduce the boundary condi-tions numerous efforts have been paid and great progresshas been made To be specific a polynomial extrapolation

(PE) method based on a constant delay assumption wasproposed and improved [8ndash11] various adaptive strategiesfor compensating variable delay have been conceived basedon online delay estimation [10 12 13] synchronizationerror [14ndash16] adaptive inverse control [17ndash19] updateddiscrete models of the testing system [20 21] and othertechniques [22ndash24] Additionally sophisticated controlstrategies such as robustness control [25ndash27] and nonlinearcontrol [28 29] have also drawn considerable attention inrecent years for addressing displacement tracking and delaycompensation problems

For the purpose of realizing high-performance com-pensation a two-stage adaptive delay compensation (TADC)method was recently developed and performed on theBenchmark problem of RTHS [30] In the first stage of thismethod the main part of the system delay is coarsely com-pensated by means of the classic polynomial extrapolation

HindawiShock and VibrationVolume 2020 Article ID 7848421 14 pageshttpsdoiorg10115520207848421

(PE) method the second stage represents a fine remedy forthe remaining delay with adaptive compensation based on adiscrete model of the loading system Virtual RTHS showedthe superiority of the TADC method As an extension of thisstudy this paper aims to further verify and reveal the per-formance of this method through real loading tests and RTHSon a viscous damper specimen

2 An Overview of Two-Stage Adaptive DelayCompensation (TADC) Method

In order to realize high performance for delay compen-sation in RTHS a novel compensation strategy called atwo-stage adaptive delay compensation (TADC) methodwas conceived and performed [30] with the benchmarkproblem of RTHS [31] is method consists of two stagesas shown in Figure 1 In the first stage a polynomial ex-trapolation (PE) compensation method is used to coarselycompensate the system delay whereas in the second stagean adaptive delay compensation based on a discrete loadingsystem model [21] is employed to finely compensate theremaining time delay

e conventional PE method is to establish a polynomialmodel based on structural displacements of the substructureinterface at current and previous steps and then to predictthe displacement response after a time delay is predicteddisplacement is sent to the actuator controller at the currentmoment Given the assumption that the loading systemdelay is constant the predicted displacement is achievedafter the time delay which means this displacement isimposed on the specimen at the right instant that is thesystem delay is compensated As an example the TADCmethod adopts the second-order PE for the first stage whichcan be expressed as

yci 1 +32η +

12η21113874 1113875yaci minus 2η + η21113872 1113873yaciminus1 +

12η +

12η21113874 1113875yaciminus2

(1)

in which yc and yac are the output displacements of the firstand second stage in the TADC method (refer to Figure 1)respectively i represents the time step and η is a constantvalue calculated by

η τΔt

(2)

where Δt is the time interval between two adjacent dis-placement points and τ is the estimated system delay It isworth mentioning that the size of Δt is often chosen at thesame level as the system delay [30]

e PE method exhibits satisfactory compensation per-formance provided the constant delay assumption is metHowever the loading system delay often varies owing to thenonlinearities of the loading system and the specimen be-havior and other reasons Additionally not only time delaybut also the amplitude errors of the loading system have to betackled to obtain accurate and reliable test results ereforean adaptive method based on discrete loading system modelsis adopted to compensate for the residual time delay andamplitude errors A three-parameter difference equation

model is employed to simulate the loading system com-pensated by the PE method namely

yaci θa1ymi + θa2ymiminus1 + θa3ymiminus2 (3)

where ym is the measured displacement of the loading systemand θa are the model parameters is model is utilized toidentify the system parameters in conjunction with thedisplacement data Subsequently the command is generatedusing the identified model namely

yaci 1113954θa1yai + 1113954θa2yaiminus1 + 1113954θa3yaiminus2 (4)

where ya is the desired displacement of the loading systemand 1113954θa are the estimated model parameters e recursiveleast square method with a forgetting factor is employed toonline estimate the parameters is algorithm featuressmall amount of calculation and fast speed which can ef-fectively overcome the data-saturation issue e algorithmexpression is

1113954θi 1113954θiminus1 +Piminus1ψi

ρ + ψTi Piminus1ψi

yaci minus ψTi

1113954θiminus11113960 1113961 (5)

Pi 1ρ

I minusPiminus1ψiψT

i

ρ + ψTi Piminus1ψi

1113890 1113891Piminus1 (6)

ψi ymi ymiminus 1 ymiminus 21113858 1113859T (7)

in which ρ is the forgetting factor satisfying 09le ρle 1 elarger the forgetting factor ρ is the greater effect the previousdata have on the current estimated parameters When ρ 1the algorithm degenerates to the recursive least-squaresmethod P is a covariance matrix while I denotes an identitymatrix

e parameters 1113954θi and Pi in (5) need to be initializationbefore RTHS Assuming that the PE method in the first stagecan achieve the ideal situation that is the time delay is fullycompensated the initial value of the parameters in thesecond stage can be taken as [1 0 0] which resultsinyaciequal to yai according to (4) As for the initial co-variance matrix one identity matrix times a value between100 and 1000 is usually employed Additionally a soft startof this identification process can also be designed by ini-tializing the parameters through loading prescribed desireddisplacements to the testing system prior to final RTHS

3 Test Verification of the TADC Method

31 Testing System is study carried out RTHS with adamper specimen using the MTS-dSpace Testing system atthe Structure and Seismic Experiment Center of HarbinInstitute of Technology As schematically shown in Figure 2this testing system consists of a dSpace 1103 board and anMTS loading system e actuator of this loading system ischaracterized in a capacity of plusmn100 kN and a piston stroke of254mm e dSpace system is responsible for the evaluationof the structural response and implementation of delaycompensationmethods whereas theMTS loading system is incharge of imposing the command displacement to thespecimen and measuring the actual displacement and

2 Shock and Vibration

damping force Note that the communication between thedSpace and MTS is achieved by digital-analog conversionand resampling e viscous damper is characterized by amaximum length of 830mm and a stroke of 256mm Tofurther reveal damper properties the relationship betweenthe damping force and its displacement under sinusoidalexcitation is depicted in Figure 3 Clearly It can be seen inFigure 3 the damper exhibits a stiffness indicating acombination of a viscous damper and spring

32 Structural Model e emulated structure is a three-story frame installed with a viscous damper e mass andinterstory stiffness of each story are assumed as 2times104 kgand 4times107Nm respectively ese parameters result in thestructural natural frequencies of 317Hz 888Hz and128Hz e Rayleigh damping model is adopted with thefirst two modal damping ratios of 2 and hence thestructural damping matrix is expressed as

C

47302 minus23486 0

minus23486 47302 minus23486

0 minus23486 23816

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ times 104N middot sm (8)

e viscous damper is installed in the first story andtested physically as ESe remaining part of the structure isnumerically simulated on the dSpace board with a timeinterval of 11024 s

33 Offline Estimation of the System Delay Prior to verifi-cation tests of the TADC method a preliminary test wascarried out to estimate the time delay of the testing systemusing a sinusoidal signal with a frequency of 2Hz and anamplitude of 4mm e displacement time histories aredepicted in Figure 4 From Figure 4(b) one can see that the

system delay is smaller than 15ms In order to accuratelyevaluate the system delay the index J1 expressed as

J1 argk

max 1113944i

ya(i)ym(i minus k)⎡⎣ ⎤⎦ (9)

was calculated and the system delay was found to be 137msin this case

34 Loading Test with a Swept Signal In this subsection aswept signal with a start frequency of 01Hz a stop frequencyof 10Hz and an amplitude of 1mm was imposed on thespecimen using the loading system e command and themeasured actuator displacement are plotted in Figure 5 It canbe seen from this figure that with the increase of the loadingfrequency the measured displacement appears to decay eaverage delay evaluated by means of the index J1 using data ofthe last one second is 156ms indicating an increase of 19mscompared with that in the sinusoidal command testis delayvariation is attributed to the flow nonlinearity of the hydraulicservo systems and uncertainties of the loading system and itimplies the necessity of adaptive delay compensation

Subsequently this signal was imposed on the specimenas a desired displacement with the system delay compen-sated by three schemes namely the traditional PE methodsingle-stage adaptive delay compensation (SADC) methodand the TADC method For the PE method the measureddelay of 137ms was set for compensation e initial valueof the parameter of the SADC method is calculated by therecursive least square method with a forgetting factor whichis [3187 0428 minus2660] e initial value of the parameter is[1 0 0] for the TADC method e forgetting factors forboth adaptive compensation methods are all taken asρ 09996 and the covariance is 1000 times an identitymatrix with a size of 3times 3

ya

yac

ym

yc

θa1 θa2 θa3

yaci = θaj yai+1ndashj

3j=1yaci = θaj ymi+1ndashj

Polynomialextrapolation

Stage 1Servo

loadingsystem

andspecimen

Two-stage adaptivedelay compensationAdaptive delay compensation

Stage 2

Figure 1 Schematic of the two-stage adaptive delay compensation method

Command

Actual force anddisplacementdSpace

MTS loading system Damping force Damper

Displacement

Figure 2 Schematic of the testing system for RTHS

Shock and Vibration 3

Obtained displacements with different compensationmethods are illustrated in Figure 6 It can be seen that the PEmethod results in large amplitude errors and slight phaseerrors between the desired and measured displacementswhereas the two adaptive methods exhibit good compen-sation performance Actually the PE method often amplifiesthe amplitude of high-frequency signals and hence it is notsuitable for compensation of relatively high-frequency sig-nals From Figure 6(a) one can summarize that the PEmethod is only suitable for signal compensation with afrequency of less than 6Hz (corresponding to 30 s) Addi-tionally the two adaptive methods slightly amplify thecommand amplitudes to interact with the response decayowing to the testing system and finally provides satisfactorycompensation accuracy as shown in Figures 6(d) and 6(f) Acareful comparison shows that the measured displacementprovided by the TADC method matches the desired onebetter than that provided by the SADC method

Figure 7 illustrates a comparison of measured displace-ments with different delay compensation methodsFigure 7(a) shows the large amplitude of the PEmethod at theearlier phase of the test while Figures 7(b) and 7(d) depict twoclose-up views of displacement peaks to clearly reflectcompensation performance Figure 7(c) plots zero-displace-ment points at the later test phase to demonstrate phaseerrors that is residual delays e method which providesmeasured displacements in better agreement with the desiredone exhibits better compensation performance It can be seenthat the PE method shows a significant displacement am-plitude and phase errors at the later phase of the test andcomparative amplitude errors at the earlier test phaseFigures 7(b)ndash7(d) show outstanding agreement between thedesired displacement and the measured displacement ob-tained with the TADC method In summary in terms oftracking accuracy the TADC method is superior to the othertwo methods and the PE method performs the worst

Figure 8 demonstrates the time histories of the estimatedparameters In the tests parameter updating started at about05 s As can be seen in the figure the variation ranges of theparameters of the SADCmethod are large reaching about 6

whereas the TADC method has a maximum parametervariation range of 3 at is to say the parameter variationfor the TADC method is smaller indicating that the diffi-culty in identifying them is decreased and hence moresatisfactory compensation performance is expected Smallerparameter variation indicates that the parameters are closerto constants and that it is easier to identify these nearlyconstant parameters Note that estimated parameter varia-tion does not necessarily mean property change of theloading system e three-parameter model for the loadingsystem might insufficiently simulate all dynamics of theloading system that is insignificant dynamics unmodeledOwing to the unmodeled dynamics even though the systemparameters are constant the estimated parameter can vary tofit different groups of the displacement data Notably theparameters of the TADC method were initialized by [1 0 0]which means that the TADC method depends less on itsinitial parameters and that it is considerably easy to set theseparameters

In order to quantitatively compare compensation per-formance two indexes for evaluating the tracking perfor-mance are employed defined as (Silva et al [31])

J2

1113936Ni1 ym(i) minus ya(i)1113858 1113859

2

1113936Ni1 ya(i)( 1113857

2 times 100

11139741113972

(10)

J3 max ym(i) minus ya(i)

11138681113868111386811138681113868111386811138681113868

max ya(i)1113868111386811138681113868

1113868111386811138681113868times 100 (11)

where ym and ya represent the measured and desired dis-placements respectively N is the total number of datapoints Clearly J2 is the normalized root mean square (RMS)of the tracking error of a compensator representing thedifference between ym and ya J3 is the peak tracking errornamely the normalized maximum synchronization errorbetween the measured and desired displacements eevaluation indexes of the three compensation methods arecollected in Table 1 For both J2 and J3 the TADCmethod issuperior to the other two methods and the PE method

15

10

5

0

ndash5

ndash10

ndash15ndash10 ndash5 0

Displacement (mm)

Forc

e (kN

)

5 10

Figure 3 Relationship between damping force and displacement

4 Shock and Vibration

performs the worst Note that the indexes are improved bythe TADC method compared with the SADC method by2222 ((261minus 203)261) and 3604 ((591minus 378)591) respectively ese results reveal the efficacy of theTADC method

35 RTHS with Sinusoidal Excitation In this subsection aseries of RTHS with different compensation methods wereperformed on the three-story frame structure excited by asinusoidal signal with a frequency of 3Hz and an amplitude of5times104Ne offline estimated delay that is 137ms was firstcompensated by the PE method e SADC method wasemployed as well to compensate for the system time delaye forgetting factor ρ was set as 09996 and the initial

covariance matrix P was defined as the identity matrixmultiplied by 1000 In order to softly start the algorithm asinusoidal signal with varying amplitudes and a frequency of3Hz was imposed as a prescribed desired displacement issoft start process led to the initial parameter values of [31648minus16895 minus054155] e TADCmethod was used to carry outRTHS In the first stage a PE method was used to roughlycompensate the system time delay of 137ms In the secondstage adaptive control was employed to finely compensate forthe residual time delay e initial covariance values andforgetting factors were the same as those for the SADCmethod e initial values of the parameters provided by thesoft start process were [13317minus057548 023305] e ap-proximation of these values to [1 0 0] validated the possibilityof initializing the parameters with [1 0 0]

Time (s)

ndash4

ndash2

0

2

4D

ispla

cem

ent (

mm

)

2 4 6 8 10

MeasuredCommand

(a)

549 55 551 552Time (s)

ndash04

ndash02

0

02

04

Disp

lace

men

t (m

m)

MeasuredCommand

(b)

Figure 4 Time histories of command and measured displacements for offline delay estimation (a) Global view (b) Close-up view

Disp

lace

men

t (m

m)

Time (s)

ndash1

ndash05

0

05

1

0 10 20 30 40 50

DesiredMeasured

(a)

Disp

lace

men

t (m

m)

4952 4953 4954 4955 4956Time (s)

01

02

03

04

05

DesiredMeasured

(b)

Figure 5 Time histories of commanded and measured swept displacements (a) Global view (b) Close-up view

Shock and Vibration 5

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

CommandDesiredMeasured

(a)

Time (s)

ndash15

ndash1

ndash05

0

05

1

15

4825 483 4835 484

Disp

lace

men

t (m

m)

CommandDesiredMeasured

(b)

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

CommandDesiredMeasured

(c)

Time (s)

ndash15

ndash1

ndash05

0

05

1

15

4825 483 4835 484

Disp

lace

men

t (m

m)

CommandDesiredMeasured

(d)

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

CommandDesiredMeasured

(e)

Time (s)4825 483 4835 484

ndash15

ndash1

ndash05

0

05

1

15

Disp

lace

men

t (m

m)

CommandDesiredMeasured

(f)

Figure 6 Displacement time histories with swept loading target (a) Displacements obtained with the PE method (b) Enlarged view of (a)(c) Displacements obtained with the SADC method (d) Enlarged view of (c) (e) Displacements obtained with the TADC method (f )Enlarged view of (e)

6 Shock and Vibration

e obtained displacement time histories are shown inFigure 9 Although global views are very similar to each otherenlarged views show different tracking performance FromFigure 9(b) it can be seen that the measured displacement(dash-dot line) and the desired displacement (solid line) ob-tainedwith the PEmethod are not in good agreement especiallyat the peaks is can be attributed to the prediction amplitudeerror of the PE method and the response amplitude error of theloading system By comparing Figures 9(d) and 9(f) with 9(b)one can conclude that both adaptivemethods are superior to thePE method owing to smaller synchronization errors is isbecause the adaptive strategies can compensate not only thephase error but also the amplitude error and can accommodateproperties variation and uncertainties

From the time histories of the estimated parametersshown in Figure 10 the parameters of the TADC methodhave much smaller absolute values compared with the cor-responding parameters of the SADC method is is becausethe SADCmethod is to compensate for the whole delay of theloading system whereas the second stage of the TADCmethod is to deal with the residual delay of the loading systemcompensated by the first stage that is the PE method eseresults indicate that the coarse compensation based on the PE

method effectively reduces the parameter variation and fa-cilitates the parameter identification In fact this is the reasonwhy the TADC method performs better Actually stableestimated parameters often mean more satisfactory com-pensation performance As shown in Figure 11 the TADCmethod provides results with smaller errors than the SADCmethod is also implies that the TADC method shows lessdependence on the initial parameter values namely morerobust than the SADC method

In order to more intuitively evaluate the performance ofthe compensation methods J2 and J3 in (10) and (11) arecalculated and presented in Table 2 Obviously RTHS withthe three methods under the excitation of a 3Hz sinusoidalsignal show good compensation effects Comparativelyspeaking the TADCmethod exhibits the best compensationaccuracy As the excitation is very regular compensation forthe delay is less complicated even so the TADC method isendowed with good robustness and good accuracy

36 Real-Time Hybrid Simulation with Seismic ExcitationIn this subsection RTHS with seismic excitation was con-ducted to examine the performance of different

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

DesiredPE

SADCTSDC

(a)

Time (s)688 698 769 692 696694 702

Disp

lace

men

t (m

m)

092

094

096

098

1

102

DesiredPE

SADCTSDC

(b)

Time (s)48716 48719 487248717 48718 48721 48722

Disp

lace

men

t (m

m)

ndash01

005

ndash005

0

DesiredPE

SADCTSDC

(c)

Time (s)4883 4886 48874884 4885 4888

Disp

lace

men

t (m

m)

05

11

08

07

06

09

1

DesiredPE

SADCTSDC

(d)

Figure 7 Comparison of measured displacements with different compensation methods (a) Global view (b) Close-up view of the earlierphase (c) Close-up view of zero-displacement points of the later phase (d) Close-up view of one peak of the later phase

Shock and Vibration 7

compensation methods In particular the El Centro (1940NS) earthquake record was adopted to excite the structurewith a peak ground acceleration of 7837 Gal e threeaforementioned compensation methods were carried outherein with the same parameters and settings as those in theprevious subsection e model parameters of the twoadaptive methods were initialized with the soft start schemeyielding [46278 minus44702 078133] and [10771 0081833minus017691] respectively Obviously the latter one is very closeto the common initial parameters namely [1 0 0] and thisvalidates the rationality of this initialization RTHS of amultiple DOF structure was implemented because they weremore challenging than previous tests owing to multiple-frequency-content structural responses and randomness ofthe seismic excitation

e displacement time histories obtained with the threedelay compensation methods are shown in Figure 12 It canbe seen from Figure 12(b) that the error of the PE method isrelatively large especially up to 067mm at 247 s When thevelocity approaches zero at the displacement peaks themethod predicts displacement responses based on the trendsof several past steps thereby causing errors in the dis-placement command Compared with the PE method theSADC method induces smaller peak errors as shown inFigure 12(d) is is attributed to its online updated discretemodel of the loading system which can effectively capturethe variation of the system characteristics and adjust actu-ator commands accordingly In Figure 12(f ) the desireddisplacement and measured displacement match very well

with the TADC method even at displacement peaks isresult shows that this RTHS of multiple degree-of-freedomstructures subjected to an earthquake can be remarkablycompensated by the TADC method

Figure 13 shows the parameter evolutions of adaptivecompensation methods rough comparison it can befound that the parameter variation ranges of the SADCmethod are much wider with a maximum value of about 9Conversely owing to the contribution of its first-stagecompensation that is the course compensation based on thePE method the parameters of the TADC method vary invery small ranges is is because the delay compensated bythe second stage of the TADC method that is the adaptivecompensation method is indeed the residual time delay ofthe first-stage compensation As shown in this figure theparameters with the TADC method are very close to con-stant ones and the identification of these values is easy andaccurate Consequently there is no doubt that the TADCmethod possesses favorable performance

Subspace plots of the measured and desired displace-ments of the actuator are illustrated in Figure 14 It can beseen from the figure that the PE method has the worstcompensation effect which is attributed to the varying timedelay and influence of multiple frequency contents of thedesired displacement e SADC method can realize betterperformance owing to its continuously updated systemmodel which can effectively capture the varying charac-teristics of the loading system and can compensate both theamplitude and phase errors e TADC method performsthe best because of its unique features such as coarse and finecompensation

Evaluation indexes are calculated and collected in Ta-ble 3 As can be seen from this table the TADC methodprovides results with the smallest J2 and J3 and hence issuperior to the other twomethodsis is consistent with theconclusion presented in Figures 12(b) and 12(c) Generallyindex values in this scenario are larger than those in

6

4

2

0θ

ndash2

ndash4

ndash60 10 20 30 40 50

Time (s)

θ1θ2θ3

(a)

θ

6

4

2

0

ndash2

ndash4

ndash60 10 20 30 40 50

Time (s)

θ1θ2θ3

(b)

Figure 8 Time histories of estimated parameters with swept loading target (a) e SADC method (b) e TADC method

Table 1 Evaluation indexes of 10Hz swept signal loading

Compensation method J2 () J3 ()

e PE method 793 2185e SADC method 261 591e TADC method 203 378

8 Shock and Vibration

Time (s)

ndash8

ndash4

0

4

8D

ispla

cem

ent (

mm

)

0 5 10 15 20

DesiredMeasuredCommand

(a)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

174 1745 175Time (s)

DesiredMeasuredCommand

(b)

0 5 10 15 20Time (s)

ndash8

ndash4

0

4

8

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(c)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

1706 1708 171 1712 1714 1716Time (s)

DesiredMeasuredCommand

(d)

0 5 10 15 20Time (s)

ndash8

ndash4

0

4

8

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(e)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

1706 1708 171 1712 1714 1716Time (s)

DesiredMeasuredCommand

(f )

Figure 9 Displacement time histories obtained in RTHS with sinusoidal excitation (a) Displacements obtained with the PE method (b)Enlarged view of (a) (c) Displacements obtained with the SADC method (d) Enlarged view of (c) (e) Displacements obtained with theTADC method (f ) Enlarged view of (e)

Shock and Vibration 9

Subsection 35 and smaller than those in Subsection 34 atis to say the RTHS with the sinusoidal excitation is the easiestone because of its regular input compensation for the sweptloading test is the most challenging one for its large frequencywidth of the desired displacement and compensation forRTHS with seismic excitation has a medium difficulty level

owing to its random earthquake input Among the three teststhe TADC method is consistently endowed with the bestindexes indicating the superiority of this method One mayargue that the improvement of this strategy is limited Ac-tually the SADC method performs relatively well and anyimprovement is considerably difficult Moreover in this

Time (s)

ndash6

ndash4

ndash2

0

2

4

6

θ

0 5 10 15 20

θ1θ2θ3

(a)

0 5 10 15 20Time (s)

ndash6

ndash4

ndash2

0

2

4

6

θ

θ1θ2θ3

(b)

Figure 10 Time histories of estimated parameters with sinusoidal excitation (a) e SADC method (b) e TADC method

3

2

1

0

ndash1

ndash2

Disp

lace

men

t (m

m)

Time (s)0 01 02 03 04 05

DesiredMeasuredCommand

(a)

0 01 02 03 04 05Time (s)

3

2

1

0

ndash1

ndash2

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(b)

Figure 11 Time histories of displacements at the beginning of tests (a) e SADC method (b) e TADC method

Table 2 RTHS evaluation index with 3Hz sine signal excitation

Method of compensation J2 () J3 ()

e PE method 184 261e SADC method 141 249e TADC method 065 121

10 Shock and Vibration

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(a)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(b)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(c)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(d)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(e)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(f )

Figure 12 Displacement time histories obtained in RTHS with seismic excitation (a) Displacements obtained with the PE method (b)Enlarged view of (a) (c) Displacements obtained with the SADCmethod (e) Displacements obtained with the TADCmethod (f ) Enlargedview of (e)

Shock and Vibration 11

scenario J2 and J3 are improved by 378 [(465minus 289)465]and 339 [(617minus 408)617] compared with the SADCmethod respectively indicating substantial improvement

4 Conclusions

is study carried out a series of verification tests of a two-stage adaptive delay compensation (TADC)method for real-time hybrid simulation in conjunction with the comparisonwith the polynomial extrapolation (PE) method and tradi-tional single-stage adaptive delay compensation (SADC)

method ese include loading tests with a prescribed sweptsignal as the desired displacement RTHS with a sinusoidalexcitation and RTHS with a seismic excitation From thisinvestigation the conclusions can be drawn as follows

(1) e estimated parameters of the TADC method varyin smaller ranges than those of the SADC methodowing to the first-stage compensation method whichreduces the difficulty in parameter estimation andhence results in better compensation performance

(2) e model parameters of the TADC method can beinitialized either as [1 0 0] or through a soft-startprocess e first-stage compensation of the TADCmethod reduces the dependence of the performanceon the parameter estimation accuracy especially atthe beginning of a test where the parameters varyapparently e compensation accuracy benefitsfrom this feature

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(a)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(b)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(c)

Figure 14 Desired-measured displacement plots with seismic excitation (a) e PE method (b) e SADC method (c) e TADCmethod

Table 3 Evaluation indexes of RTHS with seismic excitation

Compensation method J2 () J3 ()

e PE method 517 846e SADC method 465 617e TADC method 289 408

10

5

0

ndash5

ndash10

ndash150 10 20

Time (s)30

θ

θ1θ2θ3

(a)

10

5

0

ndash5

ndash10

ndash15

θ

0 10 20Time (s)

30

θ1θ2θ3

(b)

Figure 13 Estimated parameters in RTHS with seismic excitation (a) e SADC method (b) e ADC method

12 Shock and Vibration

(3) e TADC method exhibits the best tracking ac-curacy to the desired displacements among the threecompensation methods owing to its features

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e research and publication of this article were funded by theScientific Research Fund of Institute of EngineeringMechanicsChina Earthquake Administration (Grants nos 2018D10 and2020D14) the National Key Research and DevelopmentProgram of China (Grant no 2016YFC0701106) and theNational Natural Science Foundation of China (Grants nos51778190 and 51908231)

References

[1] M Nakashima H Kato and E Takaoka ldquoDevelopment ofreal-time pseudo dynamic testingrdquo Earthquake Engineering ampStructural Dynamics vol 21 no 1 pp 79ndash92 1992

[2] M S Williams and A Blakeborough ldquoLaboratory testing ofstructures under dynamic loads an introductory reviewrdquoPhilosophical Transactions of the Royal Society of LondonSeries A Mathematical Physical and Engineering Sciencesvol 359 no 1786 pp 1651ndash1669 2001

[3] B Wu H Bao J Ou and S Tian ldquoStability and accuracyanalysis of the central difference method for real-time sub-structure testingrdquo Earthquake Engineering amp Structural Dy-namics vol 34 no 7 pp 705ndash718 2005

[4] O S Bursi and D Wagg Modern Testing Techniques forStructural Systems Dynamics and Control Vol 502 SpringerScience amp Business Media Berlin Germany 2009

[5] O S Bursi Z Wang C Jia and B Wu ldquoMonolithic andpartitioned time integration methods for real-time hetero-geneous simulationsrdquo Computational Mechanics vol 52no 1 pp 99ndash119 2013

[6] X Cai C Yang and Y Yuan ldquoHybrid simulation of seismicresponses of a typical station with a reinforced concretecolumnrdquoApplied Sciences vol 10 no 4 Article ID 1331 2020

[7] Z Chen H Wang H Wang et al ldquoApplication of the hybridsimulation method for the full-scale precast reinforced con-crete shear wall structurerdquo Applied Sciences vol 8 no 2Article ID 252 2018

[8] T Horiuchi M Inoue T Konno and Y Namita ldquoReal-timehybrid experimental system with actuator delay compensa-tion and its application to a piping system with energy ab-sorberrdquo Earthquake Engineering amp Structural Dynamicsvol 28 no 10 pp 1121ndash1141 1999

[9] P A Bonnet C N Lim M S Williams et al ldquoReal-timehybrid experiments with Newmark integration MCSmdouter-loop control and multi-tasking strategiesrdquo EarthquakeEngineering amp Structural Dynamics vol 36 no 1 pp 119ndash1412007

[10] A P Darby M S Williams and A Blakeborough ldquoStabilityand delay compensation for real-time substructure testingrdquoJournal of Engineering Mechanics vol 128 no 12 pp 1276ndash1284 2002

[11] M Nakashima and N Masaoka ldquoReal-time on-line test forMDOF systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 28 no 4 pp 393ndash420 1999

[12] M Ahmadizadeh G Mosqueda and A M ReinhornldquoCompensation of actuator delay and dynamics for real-timehybrid structural simulationrdquo Earthquake Engineering ampStructural Dynamics vol 37 no 1 pp 21ndash42 2008

[13] Z Wang B Wu O S Bursi G Xu and Y Ding ldquoAn effectiveonline delay estimation method based on a simplified physicalsystem model for real-time hybrid simulationrdquo SmartStructures and Systems vol 14 no 6 pp 1247ndash1267 2014

[14] S Strano andM Terzo ldquoActuator dynamics compensation forreal-time hybrid simulation an adaptive approach by meansof a nonlinear estimatorrdquo Nonlinear Dynamics vol 85 no 4pp 2353ndash2368 2016

[15] M I Wallace J Sieber S A Neild D J Wagg andB Krauskopf ldquoStability analysis of real-time dynamic sub-structuring using delay differential equation modelsrdquoEarthquake Engineering amp Structural Dynamics vol 34no 15 pp 1817ndash1832 2005

[16] H Zhou D J Wagg and M Li ldquoEquivalent force controlcombined with adaptive polynomial-based forward predic-tion for real-time hybrid simulationrdquo Structural Control andHealth Monitoring vol 24 no 11 p e2018 2017

[17] Y Chae K Kazemibidokhti and J M Ricles ldquoAdaptive timeseries compensator for delay compensation of servo-hydraulicactuator systems for real-time hybrid simulationrdquo EarthquakeEngineering amp Structural Dynamics vol 42 no 11pp 1697ndash1715 2013

[18] C Chen J M Ricles and T Guo ldquoImproved adaptive inversecompensation technique for real-time hybrid simulationrdquoJournal of Engineering Mechanics vol 138 no 12 pp 1432ndash1446 2012

[19] V Nguyen and U Dorka ldquoPhase lag compensation in real-time substructure testing based on online system identifica-tionrdquo in Proceedings of the 14th World Conference onEarthquake Engineering Beijing China October 2008

[20] X Ning Z Wang C Wang et al ldquoAdaptive feedforward andfeedback compensation method for real-time hybrid simu-lation based on a discrete physical testing system modelrdquoJournal of Earthquake and Engineering 2020

[21] Z Wang G Xu Q Li et al ldquoAn adaptive delay compensationmethod based on a discrete systemmodel for real-time hybridsimulationrdquo Smart Structures and Systems vol 25 no 5pp 569ndash580 2020

[22] J E Carrion and B F Spencer Model-based Strategies forReal-Time Hybrid Testing 1940ndash9826 Newmark StructuralEngineering Laboratory University of Illinois at UrbanaChampaign IL USA 2007

[23] R-Y Jung P Benson Shing E Stauffer and B oenldquoPerformance of a real-time pseudodynamic test systemconsidering nonlinear structural responserdquo Earthquake En-gineering amp Structural Dynamics vol 36 no 12 pp 1785ndash1809 2007

[24] B Wu Z Wang and O S Bursi ldquoActuator dynamicscompensation based on upper bound delay for real-timehybrid simulationrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 12 pp 1749ndash1765 2013

[25] X Gao N Castaneda and S J Dyke ldquoReal time hybridsimulation from dynamic system motion control to

Shock and Vibration 13

experimental errorrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 6 pp 815ndash832 2013

[26] X Ning Z Wang H Zhou B Wu Y Ding and B XuldquoRobust actuator dynamics compensation method for real-time hybrid simulationrdquo Mechanical Systems and SignalProcessing vol 131 pp 49ndash70 2019

[27] G Ou A I Ozdagli S J Dyke and BWu ldquoRobust integratedactuator control experimental verification and real-timehybrid-simulation implementationrdquo Earthquake Engineeringamp Structural Dynamics vol 44 no 3 pp 441ndash460 2015

[28] N Nakata ldquoEffective force testing using a robust loop shapingcontrollerrdquo Earthquake Engineering amp Structural Dynamicsvol 42 no 2 pp 261ndash275 2013

[29] B Wu and H Zhou ldquoSliding mode for equivalent forcecontrol in real-time substructure testingrdquo Structural Controland Health Monitoring vol 21 no 10 pp 1284ndash1303 2014

[30] Z Wang X Ning G Xu et al ldquoHigh performance com-pensation using an adaptive strategy for real-time hybridsimulationrdquo Mechanical Systems and Signal Processingvol 133 Article ID 106262 2019

[31] C E Silva D Gomez A Maghareh et al ldquoBenchmark controlproblem for real-time hybrid simulationrdquoMechanical Systemsand Signal Processing vol 135 Article ID 106381 2020

14 Shock and Vibration

damping force Note that the communication between thedSpace and MTS is achieved by digital-analog conversionand resampling e viscous damper is characterized by amaximum length of 830mm and a stroke of 256mm Tofurther reveal damper properties the relationship betweenthe damping force and its displacement under sinusoidalexcitation is depicted in Figure 3 Clearly It can be seen inFigure 3 the damper exhibits a stiffness indicating acombination of a viscous damper and spring

32 Structural Model e emulated structure is a three-story frame installed with a viscous damper e mass andinterstory stiffness of each story are assumed as 2times104 kgand 4times107Nm respectively ese parameters result in thestructural natural frequencies of 317Hz 888Hz and128Hz e Rayleigh damping model is adopted with thefirst two modal damping ratios of 2 and hence thestructural damping matrix is expressed as

C

47302 minus23486 0

minus23486 47302 minus23486

0 minus23486 23816

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ times 104N middot sm (8)

e viscous damper is installed in the first story andtested physically as ESe remaining part of the structure isnumerically simulated on the dSpace board with a timeinterval of 11024 s

33 Offline Estimation of the System Delay Prior to verifi-cation tests of the TADC method a preliminary test wascarried out to estimate the time delay of the testing systemusing a sinusoidal signal with a frequency of 2Hz and anamplitude of 4mm e displacement time histories aredepicted in Figure 4 From Figure 4(b) one can see that the

system delay is smaller than 15ms In order to accuratelyevaluate the system delay the index J1 expressed as

J1 argk

max 1113944i

ya(i)ym(i minus k)⎡⎣ ⎤⎦ (9)

was calculated and the system delay was found to be 137msin this case

34 Loading Test with a Swept Signal In this subsection aswept signal with a start frequency of 01Hz a stop frequencyof 10Hz and an amplitude of 1mm was imposed on thespecimen using the loading system e command and themeasured actuator displacement are plotted in Figure 5 It canbe seen from this figure that with the increase of the loadingfrequency the measured displacement appears to decay eaverage delay evaluated by means of the index J1 using data ofthe last one second is 156ms indicating an increase of 19mscompared with that in the sinusoidal command testis delayvariation is attributed to the flow nonlinearity of the hydraulicservo systems and uncertainties of the loading system and itimplies the necessity of adaptive delay compensation

Subsequently this signal was imposed on the specimenas a desired displacement with the system delay compen-sated by three schemes namely the traditional PE methodsingle-stage adaptive delay compensation (SADC) methodand the TADC method For the PE method the measureddelay of 137ms was set for compensation e initial valueof the parameter of the SADC method is calculated by therecursive least square method with a forgetting factor whichis [3187 0428 minus2660] e initial value of the parameter is[1 0 0] for the TADC method e forgetting factors forboth adaptive compensation methods are all taken asρ 09996 and the covariance is 1000 times an identitymatrix with a size of 3times 3

ya

yac

ym

yc

θa1 θa2 θa3

yaci = θaj yai+1ndashj

3j=1yaci = θaj ymi+1ndashj

Polynomialextrapolation

Stage 1Servo

loadingsystem

andspecimen

Two-stage adaptivedelay compensationAdaptive delay compensation

Stage 2

Figure 1 Schematic of the two-stage adaptive delay compensation method

Command

Actual force anddisplacementdSpace

MTS loading system Damping force Damper

Displacement

Figure 2 Schematic of the testing system for RTHS

Shock and Vibration 3

Obtained displacements with different compensationmethods are illustrated in Figure 6 It can be seen that the PEmethod results in large amplitude errors and slight phaseerrors between the desired and measured displacementswhereas the two adaptive methods exhibit good compen-sation performance Actually the PE method often amplifiesthe amplitude of high-frequency signals and hence it is notsuitable for compensation of relatively high-frequency sig-nals From Figure 6(a) one can summarize that the PEmethod is only suitable for signal compensation with afrequency of less than 6Hz (corresponding to 30 s) Addi-tionally the two adaptive methods slightly amplify thecommand amplitudes to interact with the response decayowing to the testing system and finally provides satisfactorycompensation accuracy as shown in Figures 6(d) and 6(f) Acareful comparison shows that the measured displacementprovided by the TADC method matches the desired onebetter than that provided by the SADC method

Figure 7 illustrates a comparison of measured displace-ments with different delay compensation methodsFigure 7(a) shows the large amplitude of the PEmethod at theearlier phase of the test while Figures 7(b) and 7(d) depict twoclose-up views of displacement peaks to clearly reflectcompensation performance Figure 7(c) plots zero-displace-ment points at the later test phase to demonstrate phaseerrors that is residual delays e method which providesmeasured displacements in better agreement with the desiredone exhibits better compensation performance It can be seenthat the PE method shows a significant displacement am-plitude and phase errors at the later phase of the test andcomparative amplitude errors at the earlier test phaseFigures 7(b)ndash7(d) show outstanding agreement between thedesired displacement and the measured displacement ob-tained with the TADC method In summary in terms oftracking accuracy the TADC method is superior to the othertwo methods and the PE method performs the worst

Figure 8 demonstrates the time histories of the estimatedparameters In the tests parameter updating started at about05 s As can be seen in the figure the variation ranges of theparameters of the SADCmethod are large reaching about 6

whereas the TADC method has a maximum parametervariation range of 3 at is to say the parameter variationfor the TADC method is smaller indicating that the diffi-culty in identifying them is decreased and hence moresatisfactory compensation performance is expected Smallerparameter variation indicates that the parameters are closerto constants and that it is easier to identify these nearlyconstant parameters Note that estimated parameter varia-tion does not necessarily mean property change of theloading system e three-parameter model for the loadingsystem might insufficiently simulate all dynamics of theloading system that is insignificant dynamics unmodeledOwing to the unmodeled dynamics even though the systemparameters are constant the estimated parameter can vary tofit different groups of the displacement data Notably theparameters of the TADC method were initialized by [1 0 0]which means that the TADC method depends less on itsinitial parameters and that it is considerably easy to set theseparameters

In order to quantitatively compare compensation per-formance two indexes for evaluating the tracking perfor-mance are employed defined as (Silva et al [31])

J2

1113936Ni1 ym(i) minus ya(i)1113858 1113859

2

1113936Ni1 ya(i)( 1113857

2 times 100

11139741113972

(10)

J3 max ym(i) minus ya(i)

11138681113868111386811138681113868111386811138681113868

max ya(i)1113868111386811138681113868

1113868111386811138681113868times 100 (11)

where ym and ya represent the measured and desired dis-placements respectively N is the total number of datapoints Clearly J2 is the normalized root mean square (RMS)of the tracking error of a compensator representing thedifference between ym and ya J3 is the peak tracking errornamely the normalized maximum synchronization errorbetween the measured and desired displacements eevaluation indexes of the three compensation methods arecollected in Table 1 For both J2 and J3 the TADCmethod issuperior to the other two methods and the PE method

15

10

5

0

ndash5

ndash10

ndash15ndash10 ndash5 0

Displacement (mm)

Forc

e (kN

)

5 10

Figure 3 Relationship between damping force and displacement

4 Shock and Vibration

performs the worst Note that the indexes are improved bythe TADC method compared with the SADC method by2222 ((261minus 203)261) and 3604 ((591minus 378)591) respectively ese results reveal the efficacy of theTADC method

35 RTHS with Sinusoidal Excitation In this subsection aseries of RTHS with different compensation methods wereperformed on the three-story frame structure excited by asinusoidal signal with a frequency of 3Hz and an amplitude of5times104Ne offline estimated delay that is 137ms was firstcompensated by the PE method e SADC method wasemployed as well to compensate for the system time delaye forgetting factor ρ was set as 09996 and the initial

covariance matrix P was defined as the identity matrixmultiplied by 1000 In order to softly start the algorithm asinusoidal signal with varying amplitudes and a frequency of3Hz was imposed as a prescribed desired displacement issoft start process led to the initial parameter values of [31648minus16895 minus054155] e TADCmethod was used to carry outRTHS In the first stage a PE method was used to roughlycompensate the system time delay of 137ms In the secondstage adaptive control was employed to finely compensate forthe residual time delay e initial covariance values andforgetting factors were the same as those for the SADCmethod e initial values of the parameters provided by thesoft start process were [13317minus057548 023305] e ap-proximation of these values to [1 0 0] validated the possibilityof initializing the parameters with [1 0 0]

Time (s)

ndash4

ndash2

0

2

4D

ispla

cem

ent (

mm

)

2 4 6 8 10

MeasuredCommand

(a)

549 55 551 552Time (s)

ndash04

ndash02

0

02

04

Disp

lace

men

t (m

m)

MeasuredCommand

(b)

Figure 4 Time histories of command and measured displacements for offline delay estimation (a) Global view (b) Close-up view

Disp

lace

men

t (m

m)

Time (s)

ndash1

ndash05

0

05

1

0 10 20 30 40 50

DesiredMeasured

(a)

Disp

lace

men

t (m

m)

4952 4953 4954 4955 4956Time (s)

01

02

03

04

05

DesiredMeasured

(b)

Figure 5 Time histories of commanded and measured swept displacements (a) Global view (b) Close-up view

Shock and Vibration 5

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

CommandDesiredMeasured

(a)

Time (s)

ndash15

ndash1

ndash05

0

05

1

15

4825 483 4835 484

Disp

lace

men

t (m

m)

CommandDesiredMeasured

(b)

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

CommandDesiredMeasured

(c)

Time (s)

ndash15

ndash1

ndash05

0

05

1

15

4825 483 4835 484

Disp

lace

men

t (m

m)

CommandDesiredMeasured

(d)

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

CommandDesiredMeasured

(e)

Time (s)4825 483 4835 484

ndash15

ndash1

ndash05

0

05

1

15

Disp

lace

men

t (m

m)

CommandDesiredMeasured

(f)

Figure 6 Displacement time histories with swept loading target (a) Displacements obtained with the PE method (b) Enlarged view of (a)(c) Displacements obtained with the SADC method (d) Enlarged view of (c) (e) Displacements obtained with the TADC method (f )Enlarged view of (e)

6 Shock and Vibration

e obtained displacement time histories are shown inFigure 9 Although global views are very similar to each otherenlarged views show different tracking performance FromFigure 9(b) it can be seen that the measured displacement(dash-dot line) and the desired displacement (solid line) ob-tainedwith the PEmethod are not in good agreement especiallyat the peaks is can be attributed to the prediction amplitudeerror of the PE method and the response amplitude error of theloading system By comparing Figures 9(d) and 9(f) with 9(b)one can conclude that both adaptivemethods are superior to thePE method owing to smaller synchronization errors is isbecause the adaptive strategies can compensate not only thephase error but also the amplitude error and can accommodateproperties variation and uncertainties

From the time histories of the estimated parametersshown in Figure 10 the parameters of the TADC methodhave much smaller absolute values compared with the cor-responding parameters of the SADC method is is becausethe SADCmethod is to compensate for the whole delay of theloading system whereas the second stage of the TADCmethod is to deal with the residual delay of the loading systemcompensated by the first stage that is the PE method eseresults indicate that the coarse compensation based on the PE

method effectively reduces the parameter variation and fa-cilitates the parameter identification In fact this is the reasonwhy the TADC method performs better Actually stableestimated parameters often mean more satisfactory com-pensation performance As shown in Figure 11 the TADCmethod provides results with smaller errors than the SADCmethod is also implies that the TADC method shows lessdependence on the initial parameter values namely morerobust than the SADC method

In order to more intuitively evaluate the performance ofthe compensation methods J2 and J3 in (10) and (11) arecalculated and presented in Table 2 Obviously RTHS withthe three methods under the excitation of a 3Hz sinusoidalsignal show good compensation effects Comparativelyspeaking the TADCmethod exhibits the best compensationaccuracy As the excitation is very regular compensation forthe delay is less complicated even so the TADC method isendowed with good robustness and good accuracy

36 Real-Time Hybrid Simulation with Seismic ExcitationIn this subsection RTHS with seismic excitation was con-ducted to examine the performance of different

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

DesiredPE

SADCTSDC

(a)

Time (s)688 698 769 692 696694 702

Disp

lace

men

t (m

m)

092

094

096

098

1

102

DesiredPE

SADCTSDC

(b)

Time (s)48716 48719 487248717 48718 48721 48722

Disp

lace

men

t (m

m)

ndash01

005

ndash005

0

DesiredPE

SADCTSDC

(c)

Time (s)4883 4886 48874884 4885 4888

Disp

lace

men

t (m

m)

05

11

08

07

06

09

1

DesiredPE

SADCTSDC

(d)

Figure 7 Comparison of measured displacements with different compensation methods (a) Global view (b) Close-up view of the earlierphase (c) Close-up view of zero-displacement points of the later phase (d) Close-up view of one peak of the later phase

Shock and Vibration 7

compensation methods In particular the El Centro (1940NS) earthquake record was adopted to excite the structurewith a peak ground acceleration of 7837 Gal e threeaforementioned compensation methods were carried outherein with the same parameters and settings as those in theprevious subsection e model parameters of the twoadaptive methods were initialized with the soft start schemeyielding [46278 minus44702 078133] and [10771 0081833minus017691] respectively Obviously the latter one is very closeto the common initial parameters namely [1 0 0] and thisvalidates the rationality of this initialization RTHS of amultiple DOF structure was implemented because they weremore challenging than previous tests owing to multiple-frequency-content structural responses and randomness ofthe seismic excitation

e displacement time histories obtained with the threedelay compensation methods are shown in Figure 12 It canbe seen from Figure 12(b) that the error of the PE method isrelatively large especially up to 067mm at 247 s When thevelocity approaches zero at the displacement peaks themethod predicts displacement responses based on the trendsof several past steps thereby causing errors in the dis-placement command Compared with the PE method theSADC method induces smaller peak errors as shown inFigure 12(d) is is attributed to its online updated discretemodel of the loading system which can effectively capturethe variation of the system characteristics and adjust actu-ator commands accordingly In Figure 12(f ) the desireddisplacement and measured displacement match very well

with the TADC method even at displacement peaks isresult shows that this RTHS of multiple degree-of-freedomstructures subjected to an earthquake can be remarkablycompensated by the TADC method

Figure 13 shows the parameter evolutions of adaptivecompensation methods rough comparison it can befound that the parameter variation ranges of the SADCmethod are much wider with a maximum value of about 9Conversely owing to the contribution of its first-stagecompensation that is the course compensation based on thePE method the parameters of the TADC method vary invery small ranges is is because the delay compensated bythe second stage of the TADC method that is the adaptivecompensation method is indeed the residual time delay ofthe first-stage compensation As shown in this figure theparameters with the TADC method are very close to con-stant ones and the identification of these values is easy andaccurate Consequently there is no doubt that the TADCmethod possesses favorable performance

Subspace plots of the measured and desired displace-ments of the actuator are illustrated in Figure 14 It can beseen from the figure that the PE method has the worstcompensation effect which is attributed to the varying timedelay and influence of multiple frequency contents of thedesired displacement e SADC method can realize betterperformance owing to its continuously updated systemmodel which can effectively capture the varying charac-teristics of the loading system and can compensate both theamplitude and phase errors e TADC method performsthe best because of its unique features such as coarse and finecompensation

Evaluation indexes are calculated and collected in Ta-ble 3 As can be seen from this table the TADC methodprovides results with the smallest J2 and J3 and hence issuperior to the other twomethodsis is consistent with theconclusion presented in Figures 12(b) and 12(c) Generallyindex values in this scenario are larger than those in

6

4

2

0θ

ndash2

ndash4

ndash60 10 20 30 40 50

Time (s)

θ1θ2θ3

(a)

θ

6

4

2

0

ndash2

ndash4

ndash60 10 20 30 40 50

Time (s)

θ1θ2θ3

(b)

Figure 8 Time histories of estimated parameters with swept loading target (a) e SADC method (b) e TADC method

Table 1 Evaluation indexes of 10Hz swept signal loading

Compensation method J2 () J3 ()

e PE method 793 2185e SADC method 261 591e TADC method 203 378

8 Shock and Vibration

Time (s)

ndash8

ndash4

0

4

8D

ispla

cem

ent (

mm

)

0 5 10 15 20

DesiredMeasuredCommand

(a)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

174 1745 175Time (s)

DesiredMeasuredCommand

(b)

0 5 10 15 20Time (s)

ndash8

ndash4

0

4

8

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(c)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

1706 1708 171 1712 1714 1716Time (s)

DesiredMeasuredCommand

(d)

0 5 10 15 20Time (s)

ndash8

ndash4

0

4

8

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(e)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

1706 1708 171 1712 1714 1716Time (s)

DesiredMeasuredCommand

(f )

Figure 9 Displacement time histories obtained in RTHS with sinusoidal excitation (a) Displacements obtained with the PE method (b)Enlarged view of (a) (c) Displacements obtained with the SADC method (d) Enlarged view of (c) (e) Displacements obtained with theTADC method (f ) Enlarged view of (e)

Shock and Vibration 9

Subsection 35 and smaller than those in Subsection 34 atis to say the RTHS with the sinusoidal excitation is the easiestone because of its regular input compensation for the sweptloading test is the most challenging one for its large frequencywidth of the desired displacement and compensation forRTHS with seismic excitation has a medium difficulty level

owing to its random earthquake input Among the three teststhe TADC method is consistently endowed with the bestindexes indicating the superiority of this method One mayargue that the improvement of this strategy is limited Ac-tually the SADC method performs relatively well and anyimprovement is considerably difficult Moreover in this

Time (s)

ndash6

ndash4

ndash2

0

2

4

6

θ

0 5 10 15 20

θ1θ2θ3

(a)

0 5 10 15 20Time (s)

ndash6

ndash4

ndash2

0

2

4

6

θ

θ1θ2θ3

(b)

Figure 10 Time histories of estimated parameters with sinusoidal excitation (a) e SADC method (b) e TADC method

3

2

1

0

ndash1

ndash2

Disp

lace

men

t (m

m)

Time (s)0 01 02 03 04 05

DesiredMeasuredCommand

(a)

0 01 02 03 04 05Time (s)

3

2

1

0

ndash1

ndash2

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(b)

Figure 11 Time histories of displacements at the beginning of tests (a) e SADC method (b) e TADC method

Table 2 RTHS evaluation index with 3Hz sine signal excitation

Method of compensation J2 () J3 ()

e PE method 184 261e SADC method 141 249e TADC method 065 121

10 Shock and Vibration

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(a)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(b)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(c)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(d)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(e)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(f )

Figure 12 Displacement time histories obtained in RTHS with seismic excitation (a) Displacements obtained with the PE method (b)Enlarged view of (a) (c) Displacements obtained with the SADCmethod (e) Displacements obtained with the TADCmethod (f ) Enlargedview of (e)

Shock and Vibration 11

scenario J2 and J3 are improved by 378 [(465minus 289)465]and 339 [(617minus 408)617] compared with the SADCmethod respectively indicating substantial improvement

4 Conclusions

is study carried out a series of verification tests of a two-stage adaptive delay compensation (TADC)method for real-time hybrid simulation in conjunction with the comparisonwith the polynomial extrapolation (PE) method and tradi-tional single-stage adaptive delay compensation (SADC)

method ese include loading tests with a prescribed sweptsignal as the desired displacement RTHS with a sinusoidalexcitation and RTHS with a seismic excitation From thisinvestigation the conclusions can be drawn as follows

(1) e estimated parameters of the TADC method varyin smaller ranges than those of the SADC methodowing to the first-stage compensation method whichreduces the difficulty in parameter estimation andhence results in better compensation performance

(2) e model parameters of the TADC method can beinitialized either as [1 0 0] or through a soft-startprocess e first-stage compensation of the TADCmethod reduces the dependence of the performanceon the parameter estimation accuracy especially atthe beginning of a test where the parameters varyapparently e compensation accuracy benefitsfrom this feature

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(a)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(b)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(c)

Figure 14 Desired-measured displacement plots with seismic excitation (a) e PE method (b) e SADC method (c) e TADCmethod

Table 3 Evaluation indexes of RTHS with seismic excitation

Compensation method J2 () J3 ()

e PE method 517 846e SADC method 465 617e TADC method 289 408

10

5

0

ndash5

ndash10

ndash150 10 20

Time (s)30

θ

θ1θ2θ3

(a)

10

5

0

ndash5

ndash10

ndash15

θ

0 10 20Time (s)

30

θ1θ2θ3

(b)

Figure 13 Estimated parameters in RTHS with seismic excitation (a) e SADC method (b) e ADC method

12 Shock and Vibration

(3) e TADC method exhibits the best tracking ac-curacy to the desired displacements among the threecompensation methods owing to its features

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e research and publication of this article were funded by theScientific Research Fund of Institute of EngineeringMechanicsChina Earthquake Administration (Grants nos 2018D10 and2020D14) the National Key Research and DevelopmentProgram of China (Grant no 2016YFC0701106) and theNational Natural Science Foundation of China (Grants nos51778190 and 51908231)

References

[1] M Nakashima H Kato and E Takaoka ldquoDevelopment ofreal-time pseudo dynamic testingrdquo Earthquake Engineering ampStructural Dynamics vol 21 no 1 pp 79ndash92 1992

[2] M S Williams and A Blakeborough ldquoLaboratory testing ofstructures under dynamic loads an introductory reviewrdquoPhilosophical Transactions of the Royal Society of LondonSeries A Mathematical Physical and Engineering Sciencesvol 359 no 1786 pp 1651ndash1669 2001

[3] B Wu H Bao J Ou and S Tian ldquoStability and accuracyanalysis of the central difference method for real-time sub-structure testingrdquo Earthquake Engineering amp Structural Dy-namics vol 34 no 7 pp 705ndash718 2005

[4] O S Bursi and D Wagg Modern Testing Techniques forStructural Systems Dynamics and Control Vol 502 SpringerScience amp Business Media Berlin Germany 2009

[5] O S Bursi Z Wang C Jia and B Wu ldquoMonolithic andpartitioned time integration methods for real-time hetero-geneous simulationsrdquo Computational Mechanics vol 52no 1 pp 99ndash119 2013

[6] X Cai C Yang and Y Yuan ldquoHybrid simulation of seismicresponses of a typical station with a reinforced concretecolumnrdquoApplied Sciences vol 10 no 4 Article ID 1331 2020

[7] Z Chen H Wang H Wang et al ldquoApplication of the hybridsimulation method for the full-scale precast reinforced con-crete shear wall structurerdquo Applied Sciences vol 8 no 2Article ID 252 2018

[8] T Horiuchi M Inoue T Konno and Y Namita ldquoReal-timehybrid experimental system with actuator delay compensa-tion and its application to a piping system with energy ab-sorberrdquo Earthquake Engineering amp Structural Dynamicsvol 28 no 10 pp 1121ndash1141 1999

[9] P A Bonnet C N Lim M S Williams et al ldquoReal-timehybrid experiments with Newmark integration MCSmdouter-loop control and multi-tasking strategiesrdquo EarthquakeEngineering amp Structural Dynamics vol 36 no 1 pp 119ndash1412007

[10] A P Darby M S Williams and A Blakeborough ldquoStabilityand delay compensation for real-time substructure testingrdquoJournal of Engineering Mechanics vol 128 no 12 pp 1276ndash1284 2002

[11] M Nakashima and N Masaoka ldquoReal-time on-line test forMDOF systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 28 no 4 pp 393ndash420 1999

[12] M Ahmadizadeh G Mosqueda and A M ReinhornldquoCompensation of actuator delay and dynamics for real-timehybrid structural simulationrdquo Earthquake Engineering ampStructural Dynamics vol 37 no 1 pp 21ndash42 2008

[13] Z Wang B Wu O S Bursi G Xu and Y Ding ldquoAn effectiveonline delay estimation method based on a simplified physicalsystem model for real-time hybrid simulationrdquo SmartStructures and Systems vol 14 no 6 pp 1247ndash1267 2014

[14] S Strano andM Terzo ldquoActuator dynamics compensation forreal-time hybrid simulation an adaptive approach by meansof a nonlinear estimatorrdquo Nonlinear Dynamics vol 85 no 4pp 2353ndash2368 2016

[15] M I Wallace J Sieber S A Neild D J Wagg andB Krauskopf ldquoStability analysis of real-time dynamic sub-structuring using delay differential equation modelsrdquoEarthquake Engineering amp Structural Dynamics vol 34no 15 pp 1817ndash1832 2005

[16] H Zhou D J Wagg and M Li ldquoEquivalent force controlcombined with adaptive polynomial-based forward predic-tion for real-time hybrid simulationrdquo Structural Control andHealth Monitoring vol 24 no 11 p e2018 2017

[17] Y Chae K Kazemibidokhti and J M Ricles ldquoAdaptive timeseries compensator for delay compensation of servo-hydraulicactuator systems for real-time hybrid simulationrdquo EarthquakeEngineering amp Structural Dynamics vol 42 no 11pp 1697ndash1715 2013

[18] C Chen J M Ricles and T Guo ldquoImproved adaptive inversecompensation technique for real-time hybrid simulationrdquoJournal of Engineering Mechanics vol 138 no 12 pp 1432ndash1446 2012

[19] V Nguyen and U Dorka ldquoPhase lag compensation in real-time substructure testing based on online system identifica-tionrdquo in Proceedings of the 14th World Conference onEarthquake Engineering Beijing China October 2008

[20] X Ning Z Wang C Wang et al ldquoAdaptive feedforward andfeedback compensation method for real-time hybrid simu-lation based on a discrete physical testing system modelrdquoJournal of Earthquake and Engineering 2020

[21] Z Wang G Xu Q Li et al ldquoAn adaptive delay compensationmethod based on a discrete systemmodel for real-time hybridsimulationrdquo Smart Structures and Systems vol 25 no 5pp 569ndash580 2020

[22] J E Carrion and B F Spencer Model-based Strategies forReal-Time Hybrid Testing 1940ndash9826 Newmark StructuralEngineering Laboratory University of Illinois at UrbanaChampaign IL USA 2007

[23] R-Y Jung P Benson Shing E Stauffer and B oenldquoPerformance of a real-time pseudodynamic test systemconsidering nonlinear structural responserdquo Earthquake En-gineering amp Structural Dynamics vol 36 no 12 pp 1785ndash1809 2007

[24] B Wu Z Wang and O S Bursi ldquoActuator dynamicscompensation based on upper bound delay for real-timehybrid simulationrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 12 pp 1749ndash1765 2013

[25] X Gao N Castaneda and S J Dyke ldquoReal time hybridsimulation from dynamic system motion control to

Shock and Vibration 13

experimental errorrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 6 pp 815ndash832 2013

[26] X Ning Z Wang H Zhou B Wu Y Ding and B XuldquoRobust actuator dynamics compensation method for real-time hybrid simulationrdquo Mechanical Systems and SignalProcessing vol 131 pp 49ndash70 2019

[27] G Ou A I Ozdagli S J Dyke and BWu ldquoRobust integratedactuator control experimental verification and real-timehybrid-simulation implementationrdquo Earthquake Engineeringamp Structural Dynamics vol 44 no 3 pp 441ndash460 2015

[28] N Nakata ldquoEffective force testing using a robust loop shapingcontrollerrdquo Earthquake Engineering amp Structural Dynamicsvol 42 no 2 pp 261ndash275 2013

[29] B Wu and H Zhou ldquoSliding mode for equivalent forcecontrol in real-time substructure testingrdquo Structural Controland Health Monitoring vol 21 no 10 pp 1284ndash1303 2014

[30] Z Wang X Ning G Xu et al ldquoHigh performance com-pensation using an adaptive strategy for real-time hybridsimulationrdquo Mechanical Systems and Signal Processingvol 133 Article ID 106262 2019

[31] C E Silva D Gomez A Maghareh et al ldquoBenchmark controlproblem for real-time hybrid simulationrdquoMechanical Systemsand Signal Processing vol 135 Article ID 106381 2020

14 Shock and Vibration

Obtained displacements with different compensationmethods are illustrated in Figure 6 It can be seen that the PEmethod results in large amplitude errors and slight phaseerrors between the desired and measured displacementswhereas the two adaptive methods exhibit good compen-sation performance Actually the PE method often amplifiesthe amplitude of high-frequency signals and hence it is notsuitable for compensation of relatively high-frequency sig-nals From Figure 6(a) one can summarize that the PEmethod is only suitable for signal compensation with afrequency of less than 6Hz (corresponding to 30 s) Addi-tionally the two adaptive methods slightly amplify thecommand amplitudes to interact with the response decayowing to the testing system and finally provides satisfactorycompensation accuracy as shown in Figures 6(d) and 6(f) Acareful comparison shows that the measured displacementprovided by the TADC method matches the desired onebetter than that provided by the SADC method

Figure 7 illustrates a comparison of measured displace-ments with different delay compensation methodsFigure 7(a) shows the large amplitude of the PEmethod at theearlier phase of the test while Figures 7(b) and 7(d) depict twoclose-up views of displacement peaks to clearly reflectcompensation performance Figure 7(c) plots zero-displace-ment points at the later test phase to demonstrate phaseerrors that is residual delays e method which providesmeasured displacements in better agreement with the desiredone exhibits better compensation performance It can be seenthat the PE method shows a significant displacement am-plitude and phase errors at the later phase of the test andcomparative amplitude errors at the earlier test phaseFigures 7(b)ndash7(d) show outstanding agreement between thedesired displacement and the measured displacement ob-tained with the TADC method In summary in terms oftracking accuracy the TADC method is superior to the othertwo methods and the PE method performs the worst

Figure 8 demonstrates the time histories of the estimatedparameters In the tests parameter updating started at about05 s As can be seen in the figure the variation ranges of theparameters of the SADCmethod are large reaching about 6

whereas the TADC method has a maximum parametervariation range of 3 at is to say the parameter variationfor the TADC method is smaller indicating that the diffi-culty in identifying them is decreased and hence moresatisfactory compensation performance is expected Smallerparameter variation indicates that the parameters are closerto constants and that it is easier to identify these nearlyconstant parameters Note that estimated parameter varia-tion does not necessarily mean property change of theloading system e three-parameter model for the loadingsystem might insufficiently simulate all dynamics of theloading system that is insignificant dynamics unmodeledOwing to the unmodeled dynamics even though the systemparameters are constant the estimated parameter can vary tofit different groups of the displacement data Notably theparameters of the TADC method were initialized by [1 0 0]which means that the TADC method depends less on itsinitial parameters and that it is considerably easy to set theseparameters

In order to quantitatively compare compensation per-formance two indexes for evaluating the tracking perfor-mance are employed defined as (Silva et al [31])

J2

1113936Ni1 ym(i) minus ya(i)1113858 1113859

2

1113936Ni1 ya(i)( 1113857

2 times 100

11139741113972

(10)

J3 max ym(i) minus ya(i)

11138681113868111386811138681113868111386811138681113868

max ya(i)1113868111386811138681113868

1113868111386811138681113868times 100 (11)

where ym and ya represent the measured and desired dis-placements respectively N is the total number of datapoints Clearly J2 is the normalized root mean square (RMS)of the tracking error of a compensator representing thedifference between ym and ya J3 is the peak tracking errornamely the normalized maximum synchronization errorbetween the measured and desired displacements eevaluation indexes of the three compensation methods arecollected in Table 1 For both J2 and J3 the TADCmethod issuperior to the other two methods and the PE method

15

10

5

0

ndash5

ndash10

ndash15ndash10 ndash5 0

Displacement (mm)

Forc

e (kN

)

5 10

Figure 3 Relationship between damping force and displacement

4 Shock and Vibration

performs the worst Note that the indexes are improved bythe TADC method compared with the SADC method by2222 ((261minus 203)261) and 3604 ((591minus 378)591) respectively ese results reveal the efficacy of theTADC method

35 RTHS with Sinusoidal Excitation In this subsection aseries of RTHS with different compensation methods wereperformed on the three-story frame structure excited by asinusoidal signal with a frequency of 3Hz and an amplitude of5times104Ne offline estimated delay that is 137ms was firstcompensated by the PE method e SADC method wasemployed as well to compensate for the system time delaye forgetting factor ρ was set as 09996 and the initial

covariance matrix P was defined as the identity matrixmultiplied by 1000 In order to softly start the algorithm asinusoidal signal with varying amplitudes and a frequency of3Hz was imposed as a prescribed desired displacement issoft start process led to the initial parameter values of [31648minus16895 minus054155] e TADCmethod was used to carry outRTHS In the first stage a PE method was used to roughlycompensate the system time delay of 137ms In the secondstage adaptive control was employed to finely compensate forthe residual time delay e initial covariance values andforgetting factors were the same as those for the SADCmethod e initial values of the parameters provided by thesoft start process were [13317minus057548 023305] e ap-proximation of these values to [1 0 0] validated the possibilityof initializing the parameters with [1 0 0]

Time (s)

ndash4

ndash2

0

2

4D

ispla

cem

ent (

mm

)

2 4 6 8 10

MeasuredCommand

(a)

549 55 551 552Time (s)

ndash04

ndash02

0

02

04

Disp

lace

men

t (m

m)

MeasuredCommand

(b)

Figure 4 Time histories of command and measured displacements for offline delay estimation (a) Global view (b) Close-up view

Disp

lace

men

t (m

m)

Time (s)

ndash1

ndash05

0

05

1

0 10 20 30 40 50

DesiredMeasured

(a)

Disp

lace

men

t (m

m)

4952 4953 4954 4955 4956Time (s)

01

02

03

04

05

DesiredMeasured

(b)

Figure 5 Time histories of commanded and measured swept displacements (a) Global view (b) Close-up view

Shock and Vibration 5

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

CommandDesiredMeasured

(a)

Time (s)

ndash15

ndash1

ndash05

0

05

1

15

4825 483 4835 484

Disp

lace

men

t (m

m)

CommandDesiredMeasured

(b)

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

CommandDesiredMeasured

(c)

Time (s)

ndash15

ndash1

ndash05

0

05

1

15

4825 483 4835 484

Disp

lace

men

t (m

m)

CommandDesiredMeasured

(d)

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

CommandDesiredMeasured

(e)

Time (s)4825 483 4835 484

ndash15

ndash1

ndash05

0

05

1

15

Disp

lace

men

t (m

m)

CommandDesiredMeasured

(f)

Figure 6 Displacement time histories with swept loading target (a) Displacements obtained with the PE method (b) Enlarged view of (a)(c) Displacements obtained with the SADC method (d) Enlarged view of (c) (e) Displacements obtained with the TADC method (f )Enlarged view of (e)

6 Shock and Vibration

e obtained displacement time histories are shown inFigure 9 Although global views are very similar to each otherenlarged views show different tracking performance FromFigure 9(b) it can be seen that the measured displacement(dash-dot line) and the desired displacement (solid line) ob-tainedwith the PEmethod are not in good agreement especiallyat the peaks is can be attributed to the prediction amplitudeerror of the PE method and the response amplitude error of theloading system By comparing Figures 9(d) and 9(f) with 9(b)one can conclude that both adaptivemethods are superior to thePE method owing to smaller synchronization errors is isbecause the adaptive strategies can compensate not only thephase error but also the amplitude error and can accommodateproperties variation and uncertainties

From the time histories of the estimated parametersshown in Figure 10 the parameters of the TADC methodhave much smaller absolute values compared with the cor-responding parameters of the SADC method is is becausethe SADCmethod is to compensate for the whole delay of theloading system whereas the second stage of the TADCmethod is to deal with the residual delay of the loading systemcompensated by the first stage that is the PE method eseresults indicate that the coarse compensation based on the PE

method effectively reduces the parameter variation and fa-cilitates the parameter identification In fact this is the reasonwhy the TADC method performs better Actually stableestimated parameters often mean more satisfactory com-pensation performance As shown in Figure 11 the TADCmethod provides results with smaller errors than the SADCmethod is also implies that the TADC method shows lessdependence on the initial parameter values namely morerobust than the SADC method

In order to more intuitively evaluate the performance ofthe compensation methods J2 and J3 in (10) and (11) arecalculated and presented in Table 2 Obviously RTHS withthe three methods under the excitation of a 3Hz sinusoidalsignal show good compensation effects Comparativelyspeaking the TADCmethod exhibits the best compensationaccuracy As the excitation is very regular compensation forthe delay is less complicated even so the TADC method isendowed with good robustness and good accuracy

36 Real-Time Hybrid Simulation with Seismic ExcitationIn this subsection RTHS with seismic excitation was con-ducted to examine the performance of different

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

DesiredPE

SADCTSDC

(a)

Time (s)688 698 769 692 696694 702

Disp

lace

men

t (m

m)

092

094

096

098

1

102

DesiredPE

SADCTSDC

(b)

Time (s)48716 48719 487248717 48718 48721 48722

Disp

lace

men

t (m

m)

ndash01

005

ndash005

0

DesiredPE

SADCTSDC

(c)

Time (s)4883 4886 48874884 4885 4888

Disp

lace

men

t (m

m)

05

11

08

07

06

09

1

DesiredPE

SADCTSDC

(d)

Figure 7 Comparison of measured displacements with different compensation methods (a) Global view (b) Close-up view of the earlierphase (c) Close-up view of zero-displacement points of the later phase (d) Close-up view of one peak of the later phase

Shock and Vibration 7

compensation methods In particular the El Centro (1940NS) earthquake record was adopted to excite the structurewith a peak ground acceleration of 7837 Gal e threeaforementioned compensation methods were carried outherein with the same parameters and settings as those in theprevious subsection e model parameters of the twoadaptive methods were initialized with the soft start schemeyielding [46278 minus44702 078133] and [10771 0081833minus017691] respectively Obviously the latter one is very closeto the common initial parameters namely [1 0 0] and thisvalidates the rationality of this initialization RTHS of amultiple DOF structure was implemented because they weremore challenging than previous tests owing to multiple-frequency-content structural responses and randomness ofthe seismic excitation

e displacement time histories obtained with the threedelay compensation methods are shown in Figure 12 It canbe seen from Figure 12(b) that the error of the PE method isrelatively large especially up to 067mm at 247 s When thevelocity approaches zero at the displacement peaks themethod predicts displacement responses based on the trendsof several past steps thereby causing errors in the dis-placement command Compared with the PE method theSADC method induces smaller peak errors as shown inFigure 12(d) is is attributed to its online updated discretemodel of the loading system which can effectively capturethe variation of the system characteristics and adjust actu-ator commands accordingly In Figure 12(f ) the desireddisplacement and measured displacement match very well

with the TADC method even at displacement peaks isresult shows that this RTHS of multiple degree-of-freedomstructures subjected to an earthquake can be remarkablycompensated by the TADC method

Figure 13 shows the parameter evolutions of adaptivecompensation methods rough comparison it can befound that the parameter variation ranges of the SADCmethod are much wider with a maximum value of about 9Conversely owing to the contribution of its first-stagecompensation that is the course compensation based on thePE method the parameters of the TADC method vary invery small ranges is is because the delay compensated bythe second stage of the TADC method that is the adaptivecompensation method is indeed the residual time delay ofthe first-stage compensation As shown in this figure theparameters with the TADC method are very close to con-stant ones and the identification of these values is easy andaccurate Consequently there is no doubt that the TADCmethod possesses favorable performance

Subspace plots of the measured and desired displace-ments of the actuator are illustrated in Figure 14 It can beseen from the figure that the PE method has the worstcompensation effect which is attributed to the varying timedelay and influence of multiple frequency contents of thedesired displacement e SADC method can realize betterperformance owing to its continuously updated systemmodel which can effectively capture the varying charac-teristics of the loading system and can compensate both theamplitude and phase errors e TADC method performsthe best because of its unique features such as coarse and finecompensation

Evaluation indexes are calculated and collected in Ta-ble 3 As can be seen from this table the TADC methodprovides results with the smallest J2 and J3 and hence issuperior to the other twomethodsis is consistent with theconclusion presented in Figures 12(b) and 12(c) Generallyindex values in this scenario are larger than those in

6

4

2

0θ

ndash2

ndash4

ndash60 10 20 30 40 50

Time (s)

θ1θ2θ3

(a)

θ

6

4

2

0

ndash2

ndash4

ndash60 10 20 30 40 50

Time (s)

θ1θ2θ3

(b)

Figure 8 Time histories of estimated parameters with swept loading target (a) e SADC method (b) e TADC method

Table 1 Evaluation indexes of 10Hz swept signal loading

Compensation method J2 () J3 ()

e PE method 793 2185e SADC method 261 591e TADC method 203 378

8 Shock and Vibration

Time (s)

ndash8

ndash4

0

4

8D

ispla

cem

ent (

mm

)

0 5 10 15 20

DesiredMeasuredCommand

(a)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

174 1745 175Time (s)

DesiredMeasuredCommand

(b)

0 5 10 15 20Time (s)

ndash8

ndash4

0

4

8

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(c)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

1706 1708 171 1712 1714 1716Time (s)

DesiredMeasuredCommand

(d)

0 5 10 15 20Time (s)

ndash8

ndash4

0

4

8

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(e)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

1706 1708 171 1712 1714 1716Time (s)

DesiredMeasuredCommand

(f )

Figure 9 Displacement time histories obtained in RTHS with sinusoidal excitation (a) Displacements obtained with the PE method (b)Enlarged view of (a) (c) Displacements obtained with the SADC method (d) Enlarged view of (c) (e) Displacements obtained with theTADC method (f ) Enlarged view of (e)

Shock and Vibration 9

Subsection 35 and smaller than those in Subsection 34 atis to say the RTHS with the sinusoidal excitation is the easiestone because of its regular input compensation for the sweptloading test is the most challenging one for its large frequencywidth of the desired displacement and compensation forRTHS with seismic excitation has a medium difficulty level

owing to its random earthquake input Among the three teststhe TADC method is consistently endowed with the bestindexes indicating the superiority of this method One mayargue that the improvement of this strategy is limited Ac-tually the SADC method performs relatively well and anyimprovement is considerably difficult Moreover in this

Time (s)

ndash6

ndash4

ndash2

0

2

4

6

θ

0 5 10 15 20

θ1θ2θ3

(a)

0 5 10 15 20Time (s)

ndash6

ndash4

ndash2

0

2

4

6

θ

θ1θ2θ3

(b)

Figure 10 Time histories of estimated parameters with sinusoidal excitation (a) e SADC method (b) e TADC method

3

2

1

0

ndash1

ndash2

Disp

lace

men

t (m

m)

Time (s)0 01 02 03 04 05

DesiredMeasuredCommand

(a)

0 01 02 03 04 05Time (s)

3

2

1

0

ndash1

ndash2

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(b)

Figure 11 Time histories of displacements at the beginning of tests (a) e SADC method (b) e TADC method

Table 2 RTHS evaluation index with 3Hz sine signal excitation

Method of compensation J2 () J3 ()

e PE method 184 261e SADC method 141 249e TADC method 065 121

10 Shock and Vibration

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(a)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(b)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(c)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(d)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(e)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(f )

Figure 12 Displacement time histories obtained in RTHS with seismic excitation (a) Displacements obtained with the PE method (b)Enlarged view of (a) (c) Displacements obtained with the SADCmethod (e) Displacements obtained with the TADCmethod (f ) Enlargedview of (e)

Shock and Vibration 11

scenario J2 and J3 are improved by 378 [(465minus 289)465]and 339 [(617minus 408)617] compared with the SADCmethod respectively indicating substantial improvement

4 Conclusions

is study carried out a series of verification tests of a two-stage adaptive delay compensation (TADC)method for real-time hybrid simulation in conjunction with the comparisonwith the polynomial extrapolation (PE) method and tradi-tional single-stage adaptive delay compensation (SADC)

method ese include loading tests with a prescribed sweptsignal as the desired displacement RTHS with a sinusoidalexcitation and RTHS with a seismic excitation From thisinvestigation the conclusions can be drawn as follows

(1) e estimated parameters of the TADC method varyin smaller ranges than those of the SADC methodowing to the first-stage compensation method whichreduces the difficulty in parameter estimation andhence results in better compensation performance

(2) e model parameters of the TADC method can beinitialized either as [1 0 0] or through a soft-startprocess e first-stage compensation of the TADCmethod reduces the dependence of the performanceon the parameter estimation accuracy especially atthe beginning of a test where the parameters varyapparently e compensation accuracy benefitsfrom this feature

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(a)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(b)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(c)

Figure 14 Desired-measured displacement plots with seismic excitation (a) e PE method (b) e SADC method (c) e TADCmethod

Table 3 Evaluation indexes of RTHS with seismic excitation

Compensation method J2 () J3 ()

e PE method 517 846e SADC method 465 617e TADC method 289 408

10

5

0

ndash5

ndash10

ndash150 10 20

Time (s)30

θ

θ1θ2θ3

(a)

10

5

0

ndash5

ndash10

ndash15

θ

0 10 20Time (s)

30

θ1θ2θ3

(b)

Figure 13 Estimated parameters in RTHS with seismic excitation (a) e SADC method (b) e ADC method

12 Shock and Vibration

(3) e TADC method exhibits the best tracking ac-curacy to the desired displacements among the threecompensation methods owing to its features

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e research and publication of this article were funded by theScientific Research Fund of Institute of EngineeringMechanicsChina Earthquake Administration (Grants nos 2018D10 and2020D14) the National Key Research and DevelopmentProgram of China (Grant no 2016YFC0701106) and theNational Natural Science Foundation of China (Grants nos51778190 and 51908231)

References

[1] M Nakashima H Kato and E Takaoka ldquoDevelopment ofreal-time pseudo dynamic testingrdquo Earthquake Engineering ampStructural Dynamics vol 21 no 1 pp 79ndash92 1992

[2] M S Williams and A Blakeborough ldquoLaboratory testing ofstructures under dynamic loads an introductory reviewrdquoPhilosophical Transactions of the Royal Society of LondonSeries A Mathematical Physical and Engineering Sciencesvol 359 no 1786 pp 1651ndash1669 2001

[3] B Wu H Bao J Ou and S Tian ldquoStability and accuracyanalysis of the central difference method for real-time sub-structure testingrdquo Earthquake Engineering amp Structural Dy-namics vol 34 no 7 pp 705ndash718 2005

[4] O S Bursi and D Wagg Modern Testing Techniques forStructural Systems Dynamics and Control Vol 502 SpringerScience amp Business Media Berlin Germany 2009

[5] O S Bursi Z Wang C Jia and B Wu ldquoMonolithic andpartitioned time integration methods for real-time hetero-geneous simulationsrdquo Computational Mechanics vol 52no 1 pp 99ndash119 2013

[6] X Cai C Yang and Y Yuan ldquoHybrid simulation of seismicresponses of a typical station with a reinforced concretecolumnrdquoApplied Sciences vol 10 no 4 Article ID 1331 2020

[7] Z Chen H Wang H Wang et al ldquoApplication of the hybridsimulation method for the full-scale precast reinforced con-crete shear wall structurerdquo Applied Sciences vol 8 no 2Article ID 252 2018

[8] T Horiuchi M Inoue T Konno and Y Namita ldquoReal-timehybrid experimental system with actuator delay compensa-tion and its application to a piping system with energy ab-sorberrdquo Earthquake Engineering amp Structural Dynamicsvol 28 no 10 pp 1121ndash1141 1999

[9] P A Bonnet C N Lim M S Williams et al ldquoReal-timehybrid experiments with Newmark integration MCSmdouter-loop control and multi-tasking strategiesrdquo EarthquakeEngineering amp Structural Dynamics vol 36 no 1 pp 119ndash1412007

[10] A P Darby M S Williams and A Blakeborough ldquoStabilityand delay compensation for real-time substructure testingrdquoJournal of Engineering Mechanics vol 128 no 12 pp 1276ndash1284 2002

[11] M Nakashima and N Masaoka ldquoReal-time on-line test forMDOF systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 28 no 4 pp 393ndash420 1999

[12] M Ahmadizadeh G Mosqueda and A M ReinhornldquoCompensation of actuator delay and dynamics for real-timehybrid structural simulationrdquo Earthquake Engineering ampStructural Dynamics vol 37 no 1 pp 21ndash42 2008

[13] Z Wang B Wu O S Bursi G Xu and Y Ding ldquoAn effectiveonline delay estimation method based on a simplified physicalsystem model for real-time hybrid simulationrdquo SmartStructures and Systems vol 14 no 6 pp 1247ndash1267 2014

[14] S Strano andM Terzo ldquoActuator dynamics compensation forreal-time hybrid simulation an adaptive approach by meansof a nonlinear estimatorrdquo Nonlinear Dynamics vol 85 no 4pp 2353ndash2368 2016

[15] M I Wallace J Sieber S A Neild D J Wagg andB Krauskopf ldquoStability analysis of real-time dynamic sub-structuring using delay differential equation modelsrdquoEarthquake Engineering amp Structural Dynamics vol 34no 15 pp 1817ndash1832 2005

[16] H Zhou D J Wagg and M Li ldquoEquivalent force controlcombined with adaptive polynomial-based forward predic-tion for real-time hybrid simulationrdquo Structural Control andHealth Monitoring vol 24 no 11 p e2018 2017

[17] Y Chae K Kazemibidokhti and J M Ricles ldquoAdaptive timeseries compensator for delay compensation of servo-hydraulicactuator systems for real-time hybrid simulationrdquo EarthquakeEngineering amp Structural Dynamics vol 42 no 11pp 1697ndash1715 2013

[18] C Chen J M Ricles and T Guo ldquoImproved adaptive inversecompensation technique for real-time hybrid simulationrdquoJournal of Engineering Mechanics vol 138 no 12 pp 1432ndash1446 2012

[19] V Nguyen and U Dorka ldquoPhase lag compensation in real-time substructure testing based on online system identifica-tionrdquo in Proceedings of the 14th World Conference onEarthquake Engineering Beijing China October 2008

[20] X Ning Z Wang C Wang et al ldquoAdaptive feedforward andfeedback compensation method for real-time hybrid simu-lation based on a discrete physical testing system modelrdquoJournal of Earthquake and Engineering 2020

[21] Z Wang G Xu Q Li et al ldquoAn adaptive delay compensationmethod based on a discrete systemmodel for real-time hybridsimulationrdquo Smart Structures and Systems vol 25 no 5pp 569ndash580 2020

[22] J E Carrion and B F Spencer Model-based Strategies forReal-Time Hybrid Testing 1940ndash9826 Newmark StructuralEngineering Laboratory University of Illinois at UrbanaChampaign IL USA 2007

[23] R-Y Jung P Benson Shing E Stauffer and B oenldquoPerformance of a real-time pseudodynamic test systemconsidering nonlinear structural responserdquo Earthquake En-gineering amp Structural Dynamics vol 36 no 12 pp 1785ndash1809 2007

[24] B Wu Z Wang and O S Bursi ldquoActuator dynamicscompensation based on upper bound delay for real-timehybrid simulationrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 12 pp 1749ndash1765 2013

[25] X Gao N Castaneda and S J Dyke ldquoReal time hybridsimulation from dynamic system motion control to

Shock and Vibration 13

experimental errorrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 6 pp 815ndash832 2013

[26] X Ning Z Wang H Zhou B Wu Y Ding and B XuldquoRobust actuator dynamics compensation method for real-time hybrid simulationrdquo Mechanical Systems and SignalProcessing vol 131 pp 49ndash70 2019

[27] G Ou A I Ozdagli S J Dyke and BWu ldquoRobust integratedactuator control experimental verification and real-timehybrid-simulation implementationrdquo Earthquake Engineeringamp Structural Dynamics vol 44 no 3 pp 441ndash460 2015

[28] N Nakata ldquoEffective force testing using a robust loop shapingcontrollerrdquo Earthquake Engineering amp Structural Dynamicsvol 42 no 2 pp 261ndash275 2013

[29] B Wu and H Zhou ldquoSliding mode for equivalent forcecontrol in real-time substructure testingrdquo Structural Controland Health Monitoring vol 21 no 10 pp 1284ndash1303 2014

[30] Z Wang X Ning G Xu et al ldquoHigh performance com-pensation using an adaptive strategy for real-time hybridsimulationrdquo Mechanical Systems and Signal Processingvol 133 Article ID 106262 2019

[31] C E Silva D Gomez A Maghareh et al ldquoBenchmark controlproblem for real-time hybrid simulationrdquoMechanical Systemsand Signal Processing vol 135 Article ID 106381 2020

14 Shock and Vibration

performs the worst Note that the indexes are improved bythe TADC method compared with the SADC method by2222 ((261minus 203)261) and 3604 ((591minus 378)591) respectively ese results reveal the efficacy of theTADC method

35 RTHS with Sinusoidal Excitation In this subsection aseries of RTHS with different compensation methods wereperformed on the three-story frame structure excited by asinusoidal signal with a frequency of 3Hz and an amplitude of5times104Ne offline estimated delay that is 137ms was firstcompensated by the PE method e SADC method wasemployed as well to compensate for the system time delaye forgetting factor ρ was set as 09996 and the initial

covariance matrix P was defined as the identity matrixmultiplied by 1000 In order to softly start the algorithm asinusoidal signal with varying amplitudes and a frequency of3Hz was imposed as a prescribed desired displacement issoft start process led to the initial parameter values of [31648minus16895 minus054155] e TADCmethod was used to carry outRTHS In the first stage a PE method was used to roughlycompensate the system time delay of 137ms In the secondstage adaptive control was employed to finely compensate forthe residual time delay e initial covariance values andforgetting factors were the same as those for the SADCmethod e initial values of the parameters provided by thesoft start process were [13317minus057548 023305] e ap-proximation of these values to [1 0 0] validated the possibilityof initializing the parameters with [1 0 0]

Time (s)

ndash4

ndash2

0

2

4D

ispla

cem

ent (

mm

)

2 4 6 8 10

MeasuredCommand

(a)

549 55 551 552Time (s)

ndash04

ndash02

0

02

04

Disp

lace

men

t (m

m)

MeasuredCommand

(b)

Figure 4 Time histories of command and measured displacements for offline delay estimation (a) Global view (b) Close-up view

Disp

lace

men

t (m

m)

Time (s)

ndash1

ndash05

0

05

1

0 10 20 30 40 50

DesiredMeasured

(a)

Disp

lace

men

t (m

m)

4952 4953 4954 4955 4956Time (s)

01

02

03

04

05

DesiredMeasured

(b)

Figure 5 Time histories of commanded and measured swept displacements (a) Global view (b) Close-up view

Shock and Vibration 5

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

CommandDesiredMeasured

(a)

Time (s)

ndash15

ndash1

ndash05

0

05

1

15

4825 483 4835 484

Disp

lace

men

t (m

m)

CommandDesiredMeasured

(b)

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

CommandDesiredMeasured

(c)

Time (s)

ndash15

ndash1

ndash05

0

05

1

15

4825 483 4835 484

Disp

lace

men

t (m

m)

CommandDesiredMeasured

(d)

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

CommandDesiredMeasured

(e)

Time (s)4825 483 4835 484

ndash15

ndash1

ndash05

0

05

1

15

Disp

lace

men

t (m

m)

CommandDesiredMeasured

(f)

Figure 6 Displacement time histories with swept loading target (a) Displacements obtained with the PE method (b) Enlarged view of (a)(c) Displacements obtained with the SADC method (d) Enlarged view of (c) (e) Displacements obtained with the TADC method (f )Enlarged view of (e)

6 Shock and Vibration

e obtained displacement time histories are shown inFigure 9 Although global views are very similar to each otherenlarged views show different tracking performance FromFigure 9(b) it can be seen that the measured displacement(dash-dot line) and the desired displacement (solid line) ob-tainedwith the PEmethod are not in good agreement especiallyat the peaks is can be attributed to the prediction amplitudeerror of the PE method and the response amplitude error of theloading system By comparing Figures 9(d) and 9(f) with 9(b)one can conclude that both adaptivemethods are superior to thePE method owing to smaller synchronization errors is isbecause the adaptive strategies can compensate not only thephase error but also the amplitude error and can accommodateproperties variation and uncertainties

From the time histories of the estimated parametersshown in Figure 10 the parameters of the TADC methodhave much smaller absolute values compared with the cor-responding parameters of the SADC method is is becausethe SADCmethod is to compensate for the whole delay of theloading system whereas the second stage of the TADCmethod is to deal with the residual delay of the loading systemcompensated by the first stage that is the PE method eseresults indicate that the coarse compensation based on the PE

method effectively reduces the parameter variation and fa-cilitates the parameter identification In fact this is the reasonwhy the TADC method performs better Actually stableestimated parameters often mean more satisfactory com-pensation performance As shown in Figure 11 the TADCmethod provides results with smaller errors than the SADCmethod is also implies that the TADC method shows lessdependence on the initial parameter values namely morerobust than the SADC method

In order to more intuitively evaluate the performance ofthe compensation methods J2 and J3 in (10) and (11) arecalculated and presented in Table 2 Obviously RTHS withthe three methods under the excitation of a 3Hz sinusoidalsignal show good compensation effects Comparativelyspeaking the TADCmethod exhibits the best compensationaccuracy As the excitation is very regular compensation forthe delay is less complicated even so the TADC method isendowed with good robustness and good accuracy

36 Real-Time Hybrid Simulation with Seismic ExcitationIn this subsection RTHS with seismic excitation was con-ducted to examine the performance of different

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

DesiredPE

SADCTSDC

(a)

Time (s)688 698 769 692 696694 702

Disp

lace

men

t (m

m)

092

094

096

098

1

102

DesiredPE

SADCTSDC

(b)

Time (s)48716 48719 487248717 48718 48721 48722

Disp

lace

men

t (m

m)

ndash01

005

ndash005

0

DesiredPE

SADCTSDC

(c)

Time (s)4883 4886 48874884 4885 4888

Disp

lace

men

t (m

m)

05

11

08

07

06

09

1

DesiredPE

SADCTSDC

(d)

Figure 7 Comparison of measured displacements with different compensation methods (a) Global view (b) Close-up view of the earlierphase (c) Close-up view of zero-displacement points of the later phase (d) Close-up view of one peak of the later phase

Shock and Vibration 7

compensation methods In particular the El Centro (1940NS) earthquake record was adopted to excite the structurewith a peak ground acceleration of 7837 Gal e threeaforementioned compensation methods were carried outherein with the same parameters and settings as those in theprevious subsection e model parameters of the twoadaptive methods were initialized with the soft start schemeyielding [46278 minus44702 078133] and [10771 0081833minus017691] respectively Obviously the latter one is very closeto the common initial parameters namely [1 0 0] and thisvalidates the rationality of this initialization RTHS of amultiple DOF structure was implemented because they weremore challenging than previous tests owing to multiple-frequency-content structural responses and randomness ofthe seismic excitation

e displacement time histories obtained with the threedelay compensation methods are shown in Figure 12 It canbe seen from Figure 12(b) that the error of the PE method isrelatively large especially up to 067mm at 247 s When thevelocity approaches zero at the displacement peaks themethod predicts displacement responses based on the trendsof several past steps thereby causing errors in the dis-placement command Compared with the PE method theSADC method induces smaller peak errors as shown inFigure 12(d) is is attributed to its online updated discretemodel of the loading system which can effectively capturethe variation of the system characteristics and adjust actu-ator commands accordingly In Figure 12(f ) the desireddisplacement and measured displacement match very well

with the TADC method even at displacement peaks isresult shows that this RTHS of multiple degree-of-freedomstructures subjected to an earthquake can be remarkablycompensated by the TADC method

Figure 13 shows the parameter evolutions of adaptivecompensation methods rough comparison it can befound that the parameter variation ranges of the SADCmethod are much wider with a maximum value of about 9Conversely owing to the contribution of its first-stagecompensation that is the course compensation based on thePE method the parameters of the TADC method vary invery small ranges is is because the delay compensated bythe second stage of the TADC method that is the adaptivecompensation method is indeed the residual time delay ofthe first-stage compensation As shown in this figure theparameters with the TADC method are very close to con-stant ones and the identification of these values is easy andaccurate Consequently there is no doubt that the TADCmethod possesses favorable performance

Subspace plots of the measured and desired displace-ments of the actuator are illustrated in Figure 14 It can beseen from the figure that the PE method has the worstcompensation effect which is attributed to the varying timedelay and influence of multiple frequency contents of thedesired displacement e SADC method can realize betterperformance owing to its continuously updated systemmodel which can effectively capture the varying charac-teristics of the loading system and can compensate both theamplitude and phase errors e TADC method performsthe best because of its unique features such as coarse and finecompensation

Evaluation indexes are calculated and collected in Ta-ble 3 As can be seen from this table the TADC methodprovides results with the smallest J2 and J3 and hence issuperior to the other twomethodsis is consistent with theconclusion presented in Figures 12(b) and 12(c) Generallyindex values in this scenario are larger than those in

6

4

2

0θ

ndash2

ndash4

ndash60 10 20 30 40 50

Time (s)

θ1θ2θ3

(a)

θ

6

4

2

0

ndash2

ndash4

ndash60 10 20 30 40 50

Time (s)

θ1θ2θ3

(b)

Figure 8 Time histories of estimated parameters with swept loading target (a) e SADC method (b) e TADC method

Table 1 Evaluation indexes of 10Hz swept signal loading

Compensation method J2 () J3 ()

e PE method 793 2185e SADC method 261 591e TADC method 203 378

8 Shock and Vibration

Time (s)

ndash8

ndash4

0

4

8D

ispla

cem

ent (

mm

)

0 5 10 15 20

DesiredMeasuredCommand

(a)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

174 1745 175Time (s)

DesiredMeasuredCommand

(b)

0 5 10 15 20Time (s)

ndash8

ndash4

0

4

8

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(c)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

1706 1708 171 1712 1714 1716Time (s)

DesiredMeasuredCommand

(d)

0 5 10 15 20Time (s)

ndash8

ndash4

0

4

8

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(e)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

1706 1708 171 1712 1714 1716Time (s)

DesiredMeasuredCommand

(f )

Figure 9 Displacement time histories obtained in RTHS with sinusoidal excitation (a) Displacements obtained with the PE method (b)Enlarged view of (a) (c) Displacements obtained with the SADC method (d) Enlarged view of (c) (e) Displacements obtained with theTADC method (f ) Enlarged view of (e)

Shock and Vibration 9

Subsection 35 and smaller than those in Subsection 34 atis to say the RTHS with the sinusoidal excitation is the easiestone because of its regular input compensation for the sweptloading test is the most challenging one for its large frequencywidth of the desired displacement and compensation forRTHS with seismic excitation has a medium difficulty level

owing to its random earthquake input Among the three teststhe TADC method is consistently endowed with the bestindexes indicating the superiority of this method One mayargue that the improvement of this strategy is limited Ac-tually the SADC method performs relatively well and anyimprovement is considerably difficult Moreover in this

Time (s)

ndash6

ndash4

ndash2

0

2

4

6

θ

0 5 10 15 20

θ1θ2θ3

(a)

0 5 10 15 20Time (s)

ndash6

ndash4

ndash2

0

2

4

6

θ

θ1θ2θ3

(b)

Figure 10 Time histories of estimated parameters with sinusoidal excitation (a) e SADC method (b) e TADC method

3

2

1

0

ndash1

ndash2

Disp

lace

men

t (m

m)

Time (s)0 01 02 03 04 05

DesiredMeasuredCommand

(a)

0 01 02 03 04 05Time (s)

3

2

1

0

ndash1

ndash2

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(b)

Figure 11 Time histories of displacements at the beginning of tests (a) e SADC method (b) e TADC method

Table 2 RTHS evaluation index with 3Hz sine signal excitation

Method of compensation J2 () J3 ()

e PE method 184 261e SADC method 141 249e TADC method 065 121

10 Shock and Vibration

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(a)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(b)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(c)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(d)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(e)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(f )

Figure 12 Displacement time histories obtained in RTHS with seismic excitation (a) Displacements obtained with the PE method (b)Enlarged view of (a) (c) Displacements obtained with the SADCmethod (e) Displacements obtained with the TADCmethod (f ) Enlargedview of (e)

Shock and Vibration 11

scenario J2 and J3 are improved by 378 [(465minus 289)465]and 339 [(617minus 408)617] compared with the SADCmethod respectively indicating substantial improvement

4 Conclusions

is study carried out a series of verification tests of a two-stage adaptive delay compensation (TADC)method for real-time hybrid simulation in conjunction with the comparisonwith the polynomial extrapolation (PE) method and tradi-tional single-stage adaptive delay compensation (SADC)

method ese include loading tests with a prescribed sweptsignal as the desired displacement RTHS with a sinusoidalexcitation and RTHS with a seismic excitation From thisinvestigation the conclusions can be drawn as follows

(1) e estimated parameters of the TADC method varyin smaller ranges than those of the SADC methodowing to the first-stage compensation method whichreduces the difficulty in parameter estimation andhence results in better compensation performance

(2) e model parameters of the TADC method can beinitialized either as [1 0 0] or through a soft-startprocess e first-stage compensation of the TADCmethod reduces the dependence of the performanceon the parameter estimation accuracy especially atthe beginning of a test where the parameters varyapparently e compensation accuracy benefitsfrom this feature

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(a)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(b)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(c)

Figure 14 Desired-measured displacement plots with seismic excitation (a) e PE method (b) e SADC method (c) e TADCmethod

Table 3 Evaluation indexes of RTHS with seismic excitation

Compensation method J2 () J3 ()

e PE method 517 846e SADC method 465 617e TADC method 289 408

10

5

0

ndash5

ndash10

ndash150 10 20

Time (s)30

θ

θ1θ2θ3

(a)

10

5

0

ndash5

ndash10

ndash15

θ

0 10 20Time (s)

30

θ1θ2θ3

(b)

Figure 13 Estimated parameters in RTHS with seismic excitation (a) e SADC method (b) e ADC method

12 Shock and Vibration

(3) e TADC method exhibits the best tracking ac-curacy to the desired displacements among the threecompensation methods owing to its features

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e research and publication of this article were funded by theScientific Research Fund of Institute of EngineeringMechanicsChina Earthquake Administration (Grants nos 2018D10 and2020D14) the National Key Research and DevelopmentProgram of China (Grant no 2016YFC0701106) and theNational Natural Science Foundation of China (Grants nos51778190 and 51908231)

References

[1] M Nakashima H Kato and E Takaoka ldquoDevelopment ofreal-time pseudo dynamic testingrdquo Earthquake Engineering ampStructural Dynamics vol 21 no 1 pp 79ndash92 1992

[2] M S Williams and A Blakeborough ldquoLaboratory testing ofstructures under dynamic loads an introductory reviewrdquoPhilosophical Transactions of the Royal Society of LondonSeries A Mathematical Physical and Engineering Sciencesvol 359 no 1786 pp 1651ndash1669 2001

[3] B Wu H Bao J Ou and S Tian ldquoStability and accuracyanalysis of the central difference method for real-time sub-structure testingrdquo Earthquake Engineering amp Structural Dy-namics vol 34 no 7 pp 705ndash718 2005

[4] O S Bursi and D Wagg Modern Testing Techniques forStructural Systems Dynamics and Control Vol 502 SpringerScience amp Business Media Berlin Germany 2009

[5] O S Bursi Z Wang C Jia and B Wu ldquoMonolithic andpartitioned time integration methods for real-time hetero-geneous simulationsrdquo Computational Mechanics vol 52no 1 pp 99ndash119 2013

[6] X Cai C Yang and Y Yuan ldquoHybrid simulation of seismicresponses of a typical station with a reinforced concretecolumnrdquoApplied Sciences vol 10 no 4 Article ID 1331 2020

[7] Z Chen H Wang H Wang et al ldquoApplication of the hybridsimulation method for the full-scale precast reinforced con-crete shear wall structurerdquo Applied Sciences vol 8 no 2Article ID 252 2018

[8] T Horiuchi M Inoue T Konno and Y Namita ldquoReal-timehybrid experimental system with actuator delay compensa-tion and its application to a piping system with energy ab-sorberrdquo Earthquake Engineering amp Structural Dynamicsvol 28 no 10 pp 1121ndash1141 1999

[9] P A Bonnet C N Lim M S Williams et al ldquoReal-timehybrid experiments with Newmark integration MCSmdouter-loop control and multi-tasking strategiesrdquo EarthquakeEngineering amp Structural Dynamics vol 36 no 1 pp 119ndash1412007

[10] A P Darby M S Williams and A Blakeborough ldquoStabilityand delay compensation for real-time substructure testingrdquoJournal of Engineering Mechanics vol 128 no 12 pp 1276ndash1284 2002

[11] M Nakashima and N Masaoka ldquoReal-time on-line test forMDOF systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 28 no 4 pp 393ndash420 1999

[12] M Ahmadizadeh G Mosqueda and A M ReinhornldquoCompensation of actuator delay and dynamics for real-timehybrid structural simulationrdquo Earthquake Engineering ampStructural Dynamics vol 37 no 1 pp 21ndash42 2008

[13] Z Wang B Wu O S Bursi G Xu and Y Ding ldquoAn effectiveonline delay estimation method based on a simplified physicalsystem model for real-time hybrid simulationrdquo SmartStructures and Systems vol 14 no 6 pp 1247ndash1267 2014

[14] S Strano andM Terzo ldquoActuator dynamics compensation forreal-time hybrid simulation an adaptive approach by meansof a nonlinear estimatorrdquo Nonlinear Dynamics vol 85 no 4pp 2353ndash2368 2016

[15] M I Wallace J Sieber S A Neild D J Wagg andB Krauskopf ldquoStability analysis of real-time dynamic sub-structuring using delay differential equation modelsrdquoEarthquake Engineering amp Structural Dynamics vol 34no 15 pp 1817ndash1832 2005

[16] H Zhou D J Wagg and M Li ldquoEquivalent force controlcombined with adaptive polynomial-based forward predic-tion for real-time hybrid simulationrdquo Structural Control andHealth Monitoring vol 24 no 11 p e2018 2017

[17] Y Chae K Kazemibidokhti and J M Ricles ldquoAdaptive timeseries compensator for delay compensation of servo-hydraulicactuator systems for real-time hybrid simulationrdquo EarthquakeEngineering amp Structural Dynamics vol 42 no 11pp 1697ndash1715 2013

[18] C Chen J M Ricles and T Guo ldquoImproved adaptive inversecompensation technique for real-time hybrid simulationrdquoJournal of Engineering Mechanics vol 138 no 12 pp 1432ndash1446 2012

[19] V Nguyen and U Dorka ldquoPhase lag compensation in real-time substructure testing based on online system identifica-tionrdquo in Proceedings of the 14th World Conference onEarthquake Engineering Beijing China October 2008

[20] X Ning Z Wang C Wang et al ldquoAdaptive feedforward andfeedback compensation method for real-time hybrid simu-lation based on a discrete physical testing system modelrdquoJournal of Earthquake and Engineering 2020

[21] Z Wang G Xu Q Li et al ldquoAn adaptive delay compensationmethod based on a discrete systemmodel for real-time hybridsimulationrdquo Smart Structures and Systems vol 25 no 5pp 569ndash580 2020

[22] J E Carrion and B F Spencer Model-based Strategies forReal-Time Hybrid Testing 1940ndash9826 Newmark StructuralEngineering Laboratory University of Illinois at UrbanaChampaign IL USA 2007

[23] R-Y Jung P Benson Shing E Stauffer and B oenldquoPerformance of a real-time pseudodynamic test systemconsidering nonlinear structural responserdquo Earthquake En-gineering amp Structural Dynamics vol 36 no 12 pp 1785ndash1809 2007

[24] B Wu Z Wang and O S Bursi ldquoActuator dynamicscompensation based on upper bound delay for real-timehybrid simulationrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 12 pp 1749ndash1765 2013

[25] X Gao N Castaneda and S J Dyke ldquoReal time hybridsimulation from dynamic system motion control to

Shock and Vibration 13

experimental errorrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 6 pp 815ndash832 2013

[26] X Ning Z Wang H Zhou B Wu Y Ding and B XuldquoRobust actuator dynamics compensation method for real-time hybrid simulationrdquo Mechanical Systems and SignalProcessing vol 131 pp 49ndash70 2019

[27] G Ou A I Ozdagli S J Dyke and BWu ldquoRobust integratedactuator control experimental verification and real-timehybrid-simulation implementationrdquo Earthquake Engineeringamp Structural Dynamics vol 44 no 3 pp 441ndash460 2015

[28] N Nakata ldquoEffective force testing using a robust loop shapingcontrollerrdquo Earthquake Engineering amp Structural Dynamicsvol 42 no 2 pp 261ndash275 2013

[29] B Wu and H Zhou ldquoSliding mode for equivalent forcecontrol in real-time substructure testingrdquo Structural Controland Health Monitoring vol 21 no 10 pp 1284ndash1303 2014

[30] Z Wang X Ning G Xu et al ldquoHigh performance com-pensation using an adaptive strategy for real-time hybridsimulationrdquo Mechanical Systems and Signal Processingvol 133 Article ID 106262 2019

[31] C E Silva D Gomez A Maghareh et al ldquoBenchmark controlproblem for real-time hybrid simulationrdquoMechanical Systemsand Signal Processing vol 135 Article ID 106381 2020

14 Shock and Vibration

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

CommandDesiredMeasured

(a)

Time (s)

ndash15

ndash1

ndash05

0

05

1

15

4825 483 4835 484

Disp

lace

men

t (m

m)

CommandDesiredMeasured

(b)

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

CommandDesiredMeasured

(c)

Time (s)

ndash15

ndash1

ndash05

0

05

1

15

4825 483 4835 484

Disp

lace

men

t (m

m)

CommandDesiredMeasured

(d)

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

CommandDesiredMeasured

(e)

Time (s)4825 483 4835 484

ndash15

ndash1

ndash05

0

05

1

15

Disp

lace

men

t (m

m)

CommandDesiredMeasured

(f)

Figure 6 Displacement time histories with swept loading target (a) Displacements obtained with the PE method (b) Enlarged view of (a)(c) Displacements obtained with the SADC method (d) Enlarged view of (c) (e) Displacements obtained with the TADC method (f )Enlarged view of (e)

6 Shock and Vibration

e obtained displacement time histories are shown inFigure 9 Although global views are very similar to each otherenlarged views show different tracking performance FromFigure 9(b) it can be seen that the measured displacement(dash-dot line) and the desired displacement (solid line) ob-tainedwith the PEmethod are not in good agreement especiallyat the peaks is can be attributed to the prediction amplitudeerror of the PE method and the response amplitude error of theloading system By comparing Figures 9(d) and 9(f) with 9(b)one can conclude that both adaptivemethods are superior to thePE method owing to smaller synchronization errors is isbecause the adaptive strategies can compensate not only thephase error but also the amplitude error and can accommodateproperties variation and uncertainties

From the time histories of the estimated parametersshown in Figure 10 the parameters of the TADC methodhave much smaller absolute values compared with the cor-responding parameters of the SADC method is is becausethe SADCmethod is to compensate for the whole delay of theloading system whereas the second stage of the TADCmethod is to deal with the residual delay of the loading systemcompensated by the first stage that is the PE method eseresults indicate that the coarse compensation based on the PE

method effectively reduces the parameter variation and fa-cilitates the parameter identification In fact this is the reasonwhy the TADC method performs better Actually stableestimated parameters often mean more satisfactory com-pensation performance As shown in Figure 11 the TADCmethod provides results with smaller errors than the SADCmethod is also implies that the TADC method shows lessdependence on the initial parameter values namely morerobust than the SADC method

In order to more intuitively evaluate the performance ofthe compensation methods J2 and J3 in (10) and (11) arecalculated and presented in Table 2 Obviously RTHS withthe three methods under the excitation of a 3Hz sinusoidalsignal show good compensation effects Comparativelyspeaking the TADCmethod exhibits the best compensationaccuracy As the excitation is very regular compensation forthe delay is less complicated even so the TADC method isendowed with good robustness and good accuracy

36 Real-Time Hybrid Simulation with Seismic ExcitationIn this subsection RTHS with seismic excitation was con-ducted to examine the performance of different

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

DesiredPE

SADCTSDC

(a)

Time (s)688 698 769 692 696694 702

Disp

lace

men

t (m

m)

092

094

096

098

1

102

DesiredPE

SADCTSDC

(b)

Time (s)48716 48719 487248717 48718 48721 48722

Disp

lace

men

t (m

m)

ndash01

005

ndash005

0

DesiredPE

SADCTSDC

(c)

Time (s)4883 4886 48874884 4885 4888

Disp

lace

men

t (m

m)

05

11

08

07

06

09

1

DesiredPE

SADCTSDC

(d)

Figure 7 Comparison of measured displacements with different compensation methods (a) Global view (b) Close-up view of the earlierphase (c) Close-up view of zero-displacement points of the later phase (d) Close-up view of one peak of the later phase

Shock and Vibration 7

compensation methods In particular the El Centro (1940NS) earthquake record was adopted to excite the structurewith a peak ground acceleration of 7837 Gal e threeaforementioned compensation methods were carried outherein with the same parameters and settings as those in theprevious subsection e model parameters of the twoadaptive methods were initialized with the soft start schemeyielding [46278 minus44702 078133] and [10771 0081833minus017691] respectively Obviously the latter one is very closeto the common initial parameters namely [1 0 0] and thisvalidates the rationality of this initialization RTHS of amultiple DOF structure was implemented because they weremore challenging than previous tests owing to multiple-frequency-content structural responses and randomness ofthe seismic excitation

e displacement time histories obtained with the threedelay compensation methods are shown in Figure 12 It canbe seen from Figure 12(b) that the error of the PE method isrelatively large especially up to 067mm at 247 s When thevelocity approaches zero at the displacement peaks themethod predicts displacement responses based on the trendsof several past steps thereby causing errors in the dis-placement command Compared with the PE method theSADC method induces smaller peak errors as shown inFigure 12(d) is is attributed to its online updated discretemodel of the loading system which can effectively capturethe variation of the system characteristics and adjust actu-ator commands accordingly In Figure 12(f ) the desireddisplacement and measured displacement match very well

with the TADC method even at displacement peaks isresult shows that this RTHS of multiple degree-of-freedomstructures subjected to an earthquake can be remarkablycompensated by the TADC method

Figure 13 shows the parameter evolutions of adaptivecompensation methods rough comparison it can befound that the parameter variation ranges of the SADCmethod are much wider with a maximum value of about 9Conversely owing to the contribution of its first-stagecompensation that is the course compensation based on thePE method the parameters of the TADC method vary invery small ranges is is because the delay compensated bythe second stage of the TADC method that is the adaptivecompensation method is indeed the residual time delay ofthe first-stage compensation As shown in this figure theparameters with the TADC method are very close to con-stant ones and the identification of these values is easy andaccurate Consequently there is no doubt that the TADCmethod possesses favorable performance

Subspace plots of the measured and desired displace-ments of the actuator are illustrated in Figure 14 It can beseen from the figure that the PE method has the worstcompensation effect which is attributed to the varying timedelay and influence of multiple frequency contents of thedesired displacement e SADC method can realize betterperformance owing to its continuously updated systemmodel which can effectively capture the varying charac-teristics of the loading system and can compensate both theamplitude and phase errors e TADC method performsthe best because of its unique features such as coarse and finecompensation

Evaluation indexes are calculated and collected in Ta-ble 3 As can be seen from this table the TADC methodprovides results with the smallest J2 and J3 and hence issuperior to the other twomethodsis is consistent with theconclusion presented in Figures 12(b) and 12(c) Generallyindex values in this scenario are larger than those in

6

4

2

0θ

ndash2

ndash4

ndash60 10 20 30 40 50

Time (s)

θ1θ2θ3

(a)

θ

6

4

2

0

ndash2

ndash4

ndash60 10 20 30 40 50

Time (s)

θ1θ2θ3

(b)

Figure 8 Time histories of estimated parameters with swept loading target (a) e SADC method (b) e TADC method

Table 1 Evaluation indexes of 10Hz swept signal loading

Compensation method J2 () J3 ()

e PE method 793 2185e SADC method 261 591e TADC method 203 378

8 Shock and Vibration

Time (s)

ndash8

ndash4

0

4

8D

ispla

cem

ent (

mm

)

0 5 10 15 20

DesiredMeasuredCommand

(a)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

174 1745 175Time (s)

DesiredMeasuredCommand

(b)

0 5 10 15 20Time (s)

ndash8

ndash4

0

4

8

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(c)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

1706 1708 171 1712 1714 1716Time (s)

DesiredMeasuredCommand

(d)

0 5 10 15 20Time (s)

ndash8

ndash4

0

4

8

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(e)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

1706 1708 171 1712 1714 1716Time (s)

DesiredMeasuredCommand

(f )

Figure 9 Displacement time histories obtained in RTHS with sinusoidal excitation (a) Displacements obtained with the PE method (b)Enlarged view of (a) (c) Displacements obtained with the SADC method (d) Enlarged view of (c) (e) Displacements obtained with theTADC method (f ) Enlarged view of (e)

Shock and Vibration 9

Subsection 35 and smaller than those in Subsection 34 atis to say the RTHS with the sinusoidal excitation is the easiestone because of its regular input compensation for the sweptloading test is the most challenging one for its large frequencywidth of the desired displacement and compensation forRTHS with seismic excitation has a medium difficulty level

owing to its random earthquake input Among the three teststhe TADC method is consistently endowed with the bestindexes indicating the superiority of this method One mayargue that the improvement of this strategy is limited Ac-tually the SADC method performs relatively well and anyimprovement is considerably difficult Moreover in this

Time (s)

ndash6

ndash4

ndash2

0

2

4

6

θ

0 5 10 15 20

θ1θ2θ3

(a)

0 5 10 15 20Time (s)

ndash6

ndash4

ndash2

0

2

4

6

θ

θ1θ2θ3

(b)

Figure 10 Time histories of estimated parameters with sinusoidal excitation (a) e SADC method (b) e TADC method

3

2

1

0

ndash1

ndash2

Disp

lace

men

t (m

m)

Time (s)0 01 02 03 04 05

DesiredMeasuredCommand

(a)

0 01 02 03 04 05Time (s)

3

2

1

0

ndash1

ndash2

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(b)

Figure 11 Time histories of displacements at the beginning of tests (a) e SADC method (b) e TADC method

Table 2 RTHS evaluation index with 3Hz sine signal excitation

Method of compensation J2 () J3 ()

e PE method 184 261e SADC method 141 249e TADC method 065 121

10 Shock and Vibration

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(a)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(b)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(c)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(d)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(e)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(f )

Figure 12 Displacement time histories obtained in RTHS with seismic excitation (a) Displacements obtained with the PE method (b)Enlarged view of (a) (c) Displacements obtained with the SADCmethod (e) Displacements obtained with the TADCmethod (f ) Enlargedview of (e)

Shock and Vibration 11

scenario J2 and J3 are improved by 378 [(465minus 289)465]and 339 [(617minus 408)617] compared with the SADCmethod respectively indicating substantial improvement

4 Conclusions

is study carried out a series of verification tests of a two-stage adaptive delay compensation (TADC)method for real-time hybrid simulation in conjunction with the comparisonwith the polynomial extrapolation (PE) method and tradi-tional single-stage adaptive delay compensation (SADC)

method ese include loading tests with a prescribed sweptsignal as the desired displacement RTHS with a sinusoidalexcitation and RTHS with a seismic excitation From thisinvestigation the conclusions can be drawn as follows

(1) e estimated parameters of the TADC method varyin smaller ranges than those of the SADC methodowing to the first-stage compensation method whichreduces the difficulty in parameter estimation andhence results in better compensation performance

(2) e model parameters of the TADC method can beinitialized either as [1 0 0] or through a soft-startprocess e first-stage compensation of the TADCmethod reduces the dependence of the performanceon the parameter estimation accuracy especially atthe beginning of a test where the parameters varyapparently e compensation accuracy benefitsfrom this feature

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(a)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(b)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(c)

Figure 14 Desired-measured displacement plots with seismic excitation (a) e PE method (b) e SADC method (c) e TADCmethod

Table 3 Evaluation indexes of RTHS with seismic excitation

Compensation method J2 () J3 ()

e PE method 517 846e SADC method 465 617e TADC method 289 408

10

5

0

ndash5

ndash10

ndash150 10 20

Time (s)30

θ

θ1θ2θ3

(a)

10

5

0

ndash5

ndash10

ndash15

θ

0 10 20Time (s)

30

θ1θ2θ3

(b)

Figure 13 Estimated parameters in RTHS with seismic excitation (a) e SADC method (b) e ADC method

12 Shock and Vibration

(3) e TADC method exhibits the best tracking ac-curacy to the desired displacements among the threecompensation methods owing to its features

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e research and publication of this article were funded by theScientific Research Fund of Institute of EngineeringMechanicsChina Earthquake Administration (Grants nos 2018D10 and2020D14) the National Key Research and DevelopmentProgram of China (Grant no 2016YFC0701106) and theNational Natural Science Foundation of China (Grants nos51778190 and 51908231)

References

[1] M Nakashima H Kato and E Takaoka ldquoDevelopment ofreal-time pseudo dynamic testingrdquo Earthquake Engineering ampStructural Dynamics vol 21 no 1 pp 79ndash92 1992

[2] M S Williams and A Blakeborough ldquoLaboratory testing ofstructures under dynamic loads an introductory reviewrdquoPhilosophical Transactions of the Royal Society of LondonSeries A Mathematical Physical and Engineering Sciencesvol 359 no 1786 pp 1651ndash1669 2001

[3] B Wu H Bao J Ou and S Tian ldquoStability and accuracyanalysis of the central difference method for real-time sub-structure testingrdquo Earthquake Engineering amp Structural Dy-namics vol 34 no 7 pp 705ndash718 2005

[4] O S Bursi and D Wagg Modern Testing Techniques forStructural Systems Dynamics and Control Vol 502 SpringerScience amp Business Media Berlin Germany 2009

[5] O S Bursi Z Wang C Jia and B Wu ldquoMonolithic andpartitioned time integration methods for real-time hetero-geneous simulationsrdquo Computational Mechanics vol 52no 1 pp 99ndash119 2013

[6] X Cai C Yang and Y Yuan ldquoHybrid simulation of seismicresponses of a typical station with a reinforced concretecolumnrdquoApplied Sciences vol 10 no 4 Article ID 1331 2020

[7] Z Chen H Wang H Wang et al ldquoApplication of the hybridsimulation method for the full-scale precast reinforced con-crete shear wall structurerdquo Applied Sciences vol 8 no 2Article ID 252 2018

[8] T Horiuchi M Inoue T Konno and Y Namita ldquoReal-timehybrid experimental system with actuator delay compensa-tion and its application to a piping system with energy ab-sorberrdquo Earthquake Engineering amp Structural Dynamicsvol 28 no 10 pp 1121ndash1141 1999

[9] P A Bonnet C N Lim M S Williams et al ldquoReal-timehybrid experiments with Newmark integration MCSmdouter-loop control and multi-tasking strategiesrdquo EarthquakeEngineering amp Structural Dynamics vol 36 no 1 pp 119ndash1412007

[10] A P Darby M S Williams and A Blakeborough ldquoStabilityand delay compensation for real-time substructure testingrdquoJournal of Engineering Mechanics vol 128 no 12 pp 1276ndash1284 2002

[11] M Nakashima and N Masaoka ldquoReal-time on-line test forMDOF systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 28 no 4 pp 393ndash420 1999

[12] M Ahmadizadeh G Mosqueda and A M ReinhornldquoCompensation of actuator delay and dynamics for real-timehybrid structural simulationrdquo Earthquake Engineering ampStructural Dynamics vol 37 no 1 pp 21ndash42 2008

[13] Z Wang B Wu O S Bursi G Xu and Y Ding ldquoAn effectiveonline delay estimation method based on a simplified physicalsystem model for real-time hybrid simulationrdquo SmartStructures and Systems vol 14 no 6 pp 1247ndash1267 2014

[14] S Strano andM Terzo ldquoActuator dynamics compensation forreal-time hybrid simulation an adaptive approach by meansof a nonlinear estimatorrdquo Nonlinear Dynamics vol 85 no 4pp 2353ndash2368 2016

[15] M I Wallace J Sieber S A Neild D J Wagg andB Krauskopf ldquoStability analysis of real-time dynamic sub-structuring using delay differential equation modelsrdquoEarthquake Engineering amp Structural Dynamics vol 34no 15 pp 1817ndash1832 2005

[16] H Zhou D J Wagg and M Li ldquoEquivalent force controlcombined with adaptive polynomial-based forward predic-tion for real-time hybrid simulationrdquo Structural Control andHealth Monitoring vol 24 no 11 p e2018 2017

[17] Y Chae K Kazemibidokhti and J M Ricles ldquoAdaptive timeseries compensator for delay compensation of servo-hydraulicactuator systems for real-time hybrid simulationrdquo EarthquakeEngineering amp Structural Dynamics vol 42 no 11pp 1697ndash1715 2013

[18] C Chen J M Ricles and T Guo ldquoImproved adaptive inversecompensation technique for real-time hybrid simulationrdquoJournal of Engineering Mechanics vol 138 no 12 pp 1432ndash1446 2012

[19] V Nguyen and U Dorka ldquoPhase lag compensation in real-time substructure testing based on online system identifica-tionrdquo in Proceedings of the 14th World Conference onEarthquake Engineering Beijing China October 2008

[20] X Ning Z Wang C Wang et al ldquoAdaptive feedforward andfeedback compensation method for real-time hybrid simu-lation based on a discrete physical testing system modelrdquoJournal of Earthquake and Engineering 2020

[21] Z Wang G Xu Q Li et al ldquoAn adaptive delay compensationmethod based on a discrete systemmodel for real-time hybridsimulationrdquo Smart Structures and Systems vol 25 no 5pp 569ndash580 2020

[22] J E Carrion and B F Spencer Model-based Strategies forReal-Time Hybrid Testing 1940ndash9826 Newmark StructuralEngineering Laboratory University of Illinois at UrbanaChampaign IL USA 2007

[23] R-Y Jung P Benson Shing E Stauffer and B oenldquoPerformance of a real-time pseudodynamic test systemconsidering nonlinear structural responserdquo Earthquake En-gineering amp Structural Dynamics vol 36 no 12 pp 1785ndash1809 2007

[24] B Wu Z Wang and O S Bursi ldquoActuator dynamicscompensation based on upper bound delay for real-timehybrid simulationrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 12 pp 1749ndash1765 2013

[25] X Gao N Castaneda and S J Dyke ldquoReal time hybridsimulation from dynamic system motion control to

Shock and Vibration 13

experimental errorrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 6 pp 815ndash832 2013

[26] X Ning Z Wang H Zhou B Wu Y Ding and B XuldquoRobust actuator dynamics compensation method for real-time hybrid simulationrdquo Mechanical Systems and SignalProcessing vol 131 pp 49ndash70 2019

[27] G Ou A I Ozdagli S J Dyke and BWu ldquoRobust integratedactuator control experimental verification and real-timehybrid-simulation implementationrdquo Earthquake Engineeringamp Structural Dynamics vol 44 no 3 pp 441ndash460 2015

[28] N Nakata ldquoEffective force testing using a robust loop shapingcontrollerrdquo Earthquake Engineering amp Structural Dynamicsvol 42 no 2 pp 261ndash275 2013

[29] B Wu and H Zhou ldquoSliding mode for equivalent forcecontrol in real-time substructure testingrdquo Structural Controland Health Monitoring vol 21 no 10 pp 1284ndash1303 2014

[30] Z Wang X Ning G Xu et al ldquoHigh performance com-pensation using an adaptive strategy for real-time hybridsimulationrdquo Mechanical Systems and Signal Processingvol 133 Article ID 106262 2019

[31] C E Silva D Gomez A Maghareh et al ldquoBenchmark controlproblem for real-time hybrid simulationrdquoMechanical Systemsand Signal Processing vol 135 Article ID 106381 2020

14 Shock and Vibration

e obtained displacement time histories are shown inFigure 9 Although global views are very similar to each otherenlarged views show different tracking performance FromFigure 9(b) it can be seen that the measured displacement(dash-dot line) and the desired displacement (solid line) ob-tainedwith the PEmethod are not in good agreement especiallyat the peaks is can be attributed to the prediction amplitudeerror of the PE method and the response amplitude error of theloading system By comparing Figures 9(d) and 9(f) with 9(b)one can conclude that both adaptivemethods are superior to thePE method owing to smaller synchronization errors is isbecause the adaptive strategies can compensate not only thephase error but also the amplitude error and can accommodateproperties variation and uncertainties

From the time histories of the estimated parametersshown in Figure 10 the parameters of the TADC methodhave much smaller absolute values compared with the cor-responding parameters of the SADC method is is becausethe SADCmethod is to compensate for the whole delay of theloading system whereas the second stage of the TADCmethod is to deal with the residual delay of the loading systemcompensated by the first stage that is the PE method eseresults indicate that the coarse compensation based on the PE

method effectively reduces the parameter variation and fa-cilitates the parameter identification In fact this is the reasonwhy the TADC method performs better Actually stableestimated parameters often mean more satisfactory com-pensation performance As shown in Figure 11 the TADCmethod provides results with smaller errors than the SADCmethod is also implies that the TADC method shows lessdependence on the initial parameter values namely morerobust than the SADC method

In order to more intuitively evaluate the performance ofthe compensation methods J2 and J3 in (10) and (11) arecalculated and presented in Table 2 Obviously RTHS withthe three methods under the excitation of a 3Hz sinusoidalsignal show good compensation effects Comparativelyspeaking the TADCmethod exhibits the best compensationaccuracy As the excitation is very regular compensation forthe delay is less complicated even so the TADC method isendowed with good robustness and good accuracy

36 Real-Time Hybrid Simulation with Seismic ExcitationIn this subsection RTHS with seismic excitation was con-ducted to examine the performance of different

Disp

lace

men

t (m

m)

Time (s)0 20 3010 40 50

ndash15

ndash1

ndash05

0

05

1

15

DesiredPE

SADCTSDC

(a)

Time (s)688 698 769 692 696694 702

Disp

lace

men

t (m

m)

092

094

096

098

1

102

DesiredPE

SADCTSDC

(b)

Time (s)48716 48719 487248717 48718 48721 48722

Disp

lace

men

t (m

m)

ndash01

005

ndash005

0

DesiredPE

SADCTSDC

(c)

Time (s)4883 4886 48874884 4885 4888

Disp

lace

men

t (m

m)

05

11

08

07

06

09

1

DesiredPE

SADCTSDC

(d)

Figure 7 Comparison of measured displacements with different compensation methods (a) Global view (b) Close-up view of the earlierphase (c) Close-up view of zero-displacement points of the later phase (d) Close-up view of one peak of the later phase

Shock and Vibration 7

compensation methods In particular the El Centro (1940NS) earthquake record was adopted to excite the structurewith a peak ground acceleration of 7837 Gal e threeaforementioned compensation methods were carried outherein with the same parameters and settings as those in theprevious subsection e model parameters of the twoadaptive methods were initialized with the soft start schemeyielding [46278 minus44702 078133] and [10771 0081833minus017691] respectively Obviously the latter one is very closeto the common initial parameters namely [1 0 0] and thisvalidates the rationality of this initialization RTHS of amultiple DOF structure was implemented because they weremore challenging than previous tests owing to multiple-frequency-content structural responses and randomness ofthe seismic excitation

e displacement time histories obtained with the threedelay compensation methods are shown in Figure 12 It canbe seen from Figure 12(b) that the error of the PE method isrelatively large especially up to 067mm at 247 s When thevelocity approaches zero at the displacement peaks themethod predicts displacement responses based on the trendsof several past steps thereby causing errors in the dis-placement command Compared with the PE method theSADC method induces smaller peak errors as shown inFigure 12(d) is is attributed to its online updated discretemodel of the loading system which can effectively capturethe variation of the system characteristics and adjust actu-ator commands accordingly In Figure 12(f ) the desireddisplacement and measured displacement match very well

with the TADC method even at displacement peaks isresult shows that this RTHS of multiple degree-of-freedomstructures subjected to an earthquake can be remarkablycompensated by the TADC method

Figure 13 shows the parameter evolutions of adaptivecompensation methods rough comparison it can befound that the parameter variation ranges of the SADCmethod are much wider with a maximum value of about 9Conversely owing to the contribution of its first-stagecompensation that is the course compensation based on thePE method the parameters of the TADC method vary invery small ranges is is because the delay compensated bythe second stage of the TADC method that is the adaptivecompensation method is indeed the residual time delay ofthe first-stage compensation As shown in this figure theparameters with the TADC method are very close to con-stant ones and the identification of these values is easy andaccurate Consequently there is no doubt that the TADCmethod possesses favorable performance

Subspace plots of the measured and desired displace-ments of the actuator are illustrated in Figure 14 It can beseen from the figure that the PE method has the worstcompensation effect which is attributed to the varying timedelay and influence of multiple frequency contents of thedesired displacement e SADC method can realize betterperformance owing to its continuously updated systemmodel which can effectively capture the varying charac-teristics of the loading system and can compensate both theamplitude and phase errors e TADC method performsthe best because of its unique features such as coarse and finecompensation

Evaluation indexes are calculated and collected in Ta-ble 3 As can be seen from this table the TADC methodprovides results with the smallest J2 and J3 and hence issuperior to the other twomethodsis is consistent with theconclusion presented in Figures 12(b) and 12(c) Generallyindex values in this scenario are larger than those in

6

4

2

0θ

ndash2

ndash4

ndash60 10 20 30 40 50

Time (s)

θ1θ2θ3

(a)

θ

6

4

2

0

ndash2

ndash4

ndash60 10 20 30 40 50

Time (s)

θ1θ2θ3

(b)

Figure 8 Time histories of estimated parameters with swept loading target (a) e SADC method (b) e TADC method

Table 1 Evaluation indexes of 10Hz swept signal loading

Compensation method J2 () J3 ()

e PE method 793 2185e SADC method 261 591e TADC method 203 378

8 Shock and Vibration

Time (s)

ndash8

ndash4

0

4

8D

ispla

cem

ent (

mm

)

0 5 10 15 20

DesiredMeasuredCommand

(a)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

174 1745 175Time (s)

DesiredMeasuredCommand

(b)

0 5 10 15 20Time (s)

ndash8

ndash4

0

4

8

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(c)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

1706 1708 171 1712 1714 1716Time (s)

DesiredMeasuredCommand

(d)

0 5 10 15 20Time (s)

ndash8

ndash4

0

4

8

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(e)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

1706 1708 171 1712 1714 1716Time (s)

DesiredMeasuredCommand

(f )

Figure 9 Displacement time histories obtained in RTHS with sinusoidal excitation (a) Displacements obtained with the PE method (b)Enlarged view of (a) (c) Displacements obtained with the SADC method (d) Enlarged view of (c) (e) Displacements obtained with theTADC method (f ) Enlarged view of (e)

Shock and Vibration 9

Subsection 35 and smaller than those in Subsection 34 atis to say the RTHS with the sinusoidal excitation is the easiestone because of its regular input compensation for the sweptloading test is the most challenging one for its large frequencywidth of the desired displacement and compensation forRTHS with seismic excitation has a medium difficulty level

owing to its random earthquake input Among the three teststhe TADC method is consistently endowed with the bestindexes indicating the superiority of this method One mayargue that the improvement of this strategy is limited Ac-tually the SADC method performs relatively well and anyimprovement is considerably difficult Moreover in this

Time (s)

ndash6

ndash4

ndash2

0

2

4

6

θ

0 5 10 15 20

θ1θ2θ3

(a)

0 5 10 15 20Time (s)

ndash6

ndash4

ndash2

0

2

4

6

θ

θ1θ2θ3

(b)

Figure 10 Time histories of estimated parameters with sinusoidal excitation (a) e SADC method (b) e TADC method

3

2

1

0

ndash1

ndash2

Disp

lace

men

t (m

m)

Time (s)0 01 02 03 04 05

DesiredMeasuredCommand

(a)

0 01 02 03 04 05Time (s)

3

2

1

0

ndash1

ndash2

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(b)

Figure 11 Time histories of displacements at the beginning of tests (a) e SADC method (b) e TADC method

Table 2 RTHS evaluation index with 3Hz sine signal excitation

Method of compensation J2 () J3 ()

e PE method 184 261e SADC method 141 249e TADC method 065 121

10 Shock and Vibration

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(a)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(b)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(c)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(d)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(e)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(f )

Figure 12 Displacement time histories obtained in RTHS with seismic excitation (a) Displacements obtained with the PE method (b)Enlarged view of (a) (c) Displacements obtained with the SADCmethod (e) Displacements obtained with the TADCmethod (f ) Enlargedview of (e)

Shock and Vibration 11

scenario J2 and J3 are improved by 378 [(465minus 289)465]and 339 [(617minus 408)617] compared with the SADCmethod respectively indicating substantial improvement

4 Conclusions

is study carried out a series of verification tests of a two-stage adaptive delay compensation (TADC)method for real-time hybrid simulation in conjunction with the comparisonwith the polynomial extrapolation (PE) method and tradi-tional single-stage adaptive delay compensation (SADC)

method ese include loading tests with a prescribed sweptsignal as the desired displacement RTHS with a sinusoidalexcitation and RTHS with a seismic excitation From thisinvestigation the conclusions can be drawn as follows

(1) e estimated parameters of the TADC method varyin smaller ranges than those of the SADC methodowing to the first-stage compensation method whichreduces the difficulty in parameter estimation andhence results in better compensation performance

(2) e model parameters of the TADC method can beinitialized either as [1 0 0] or through a soft-startprocess e first-stage compensation of the TADCmethod reduces the dependence of the performanceon the parameter estimation accuracy especially atthe beginning of a test where the parameters varyapparently e compensation accuracy benefitsfrom this feature

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(a)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(b)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(c)

Figure 14 Desired-measured displacement plots with seismic excitation (a) e PE method (b) e SADC method (c) e TADCmethod

Table 3 Evaluation indexes of RTHS with seismic excitation

Compensation method J2 () J3 ()

e PE method 517 846e SADC method 465 617e TADC method 289 408

10

5

0

ndash5

ndash10

ndash150 10 20

Time (s)30

θ

θ1θ2θ3

(a)

10

5

0

ndash5

ndash10

ndash15

θ

0 10 20Time (s)

30

θ1θ2θ3

(b)

Figure 13 Estimated parameters in RTHS with seismic excitation (a) e SADC method (b) e ADC method

12 Shock and Vibration

(3) e TADC method exhibits the best tracking ac-curacy to the desired displacements among the threecompensation methods owing to its features

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e research and publication of this article were funded by theScientific Research Fund of Institute of EngineeringMechanicsChina Earthquake Administration (Grants nos 2018D10 and2020D14) the National Key Research and DevelopmentProgram of China (Grant no 2016YFC0701106) and theNational Natural Science Foundation of China (Grants nos51778190 and 51908231)

References

[1] M Nakashima H Kato and E Takaoka ldquoDevelopment ofreal-time pseudo dynamic testingrdquo Earthquake Engineering ampStructural Dynamics vol 21 no 1 pp 79ndash92 1992

[2] M S Williams and A Blakeborough ldquoLaboratory testing ofstructures under dynamic loads an introductory reviewrdquoPhilosophical Transactions of the Royal Society of LondonSeries A Mathematical Physical and Engineering Sciencesvol 359 no 1786 pp 1651ndash1669 2001

[3] B Wu H Bao J Ou and S Tian ldquoStability and accuracyanalysis of the central difference method for real-time sub-structure testingrdquo Earthquake Engineering amp Structural Dy-namics vol 34 no 7 pp 705ndash718 2005

[4] O S Bursi and D Wagg Modern Testing Techniques forStructural Systems Dynamics and Control Vol 502 SpringerScience amp Business Media Berlin Germany 2009

[5] O S Bursi Z Wang C Jia and B Wu ldquoMonolithic andpartitioned time integration methods for real-time hetero-geneous simulationsrdquo Computational Mechanics vol 52no 1 pp 99ndash119 2013

[6] X Cai C Yang and Y Yuan ldquoHybrid simulation of seismicresponses of a typical station with a reinforced concretecolumnrdquoApplied Sciences vol 10 no 4 Article ID 1331 2020

[7] Z Chen H Wang H Wang et al ldquoApplication of the hybridsimulation method for the full-scale precast reinforced con-crete shear wall structurerdquo Applied Sciences vol 8 no 2Article ID 252 2018

[8] T Horiuchi M Inoue T Konno and Y Namita ldquoReal-timehybrid experimental system with actuator delay compensa-tion and its application to a piping system with energy ab-sorberrdquo Earthquake Engineering amp Structural Dynamicsvol 28 no 10 pp 1121ndash1141 1999

[9] P A Bonnet C N Lim M S Williams et al ldquoReal-timehybrid experiments with Newmark integration MCSmdouter-loop control and multi-tasking strategiesrdquo EarthquakeEngineering amp Structural Dynamics vol 36 no 1 pp 119ndash1412007

[10] A P Darby M S Williams and A Blakeborough ldquoStabilityand delay compensation for real-time substructure testingrdquoJournal of Engineering Mechanics vol 128 no 12 pp 1276ndash1284 2002

[11] M Nakashima and N Masaoka ldquoReal-time on-line test forMDOF systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 28 no 4 pp 393ndash420 1999

[12] M Ahmadizadeh G Mosqueda and A M ReinhornldquoCompensation of actuator delay and dynamics for real-timehybrid structural simulationrdquo Earthquake Engineering ampStructural Dynamics vol 37 no 1 pp 21ndash42 2008

[13] Z Wang B Wu O S Bursi G Xu and Y Ding ldquoAn effectiveonline delay estimation method based on a simplified physicalsystem model for real-time hybrid simulationrdquo SmartStructures and Systems vol 14 no 6 pp 1247ndash1267 2014

[14] S Strano andM Terzo ldquoActuator dynamics compensation forreal-time hybrid simulation an adaptive approach by meansof a nonlinear estimatorrdquo Nonlinear Dynamics vol 85 no 4pp 2353ndash2368 2016

[15] M I Wallace J Sieber S A Neild D J Wagg andB Krauskopf ldquoStability analysis of real-time dynamic sub-structuring using delay differential equation modelsrdquoEarthquake Engineering amp Structural Dynamics vol 34no 15 pp 1817ndash1832 2005

[16] H Zhou D J Wagg and M Li ldquoEquivalent force controlcombined with adaptive polynomial-based forward predic-tion for real-time hybrid simulationrdquo Structural Control andHealth Monitoring vol 24 no 11 p e2018 2017

[17] Y Chae K Kazemibidokhti and J M Ricles ldquoAdaptive timeseries compensator for delay compensation of servo-hydraulicactuator systems for real-time hybrid simulationrdquo EarthquakeEngineering amp Structural Dynamics vol 42 no 11pp 1697ndash1715 2013

[18] C Chen J M Ricles and T Guo ldquoImproved adaptive inversecompensation technique for real-time hybrid simulationrdquoJournal of Engineering Mechanics vol 138 no 12 pp 1432ndash1446 2012

[19] V Nguyen and U Dorka ldquoPhase lag compensation in real-time substructure testing based on online system identifica-tionrdquo in Proceedings of the 14th World Conference onEarthquake Engineering Beijing China October 2008

[20] X Ning Z Wang C Wang et al ldquoAdaptive feedforward andfeedback compensation method for real-time hybrid simu-lation based on a discrete physical testing system modelrdquoJournal of Earthquake and Engineering 2020

[21] Z Wang G Xu Q Li et al ldquoAn adaptive delay compensationmethod based on a discrete systemmodel for real-time hybridsimulationrdquo Smart Structures and Systems vol 25 no 5pp 569ndash580 2020

[22] J E Carrion and B F Spencer Model-based Strategies forReal-Time Hybrid Testing 1940ndash9826 Newmark StructuralEngineering Laboratory University of Illinois at UrbanaChampaign IL USA 2007

[23] R-Y Jung P Benson Shing E Stauffer and B oenldquoPerformance of a real-time pseudodynamic test systemconsidering nonlinear structural responserdquo Earthquake En-gineering amp Structural Dynamics vol 36 no 12 pp 1785ndash1809 2007

[24] B Wu Z Wang and O S Bursi ldquoActuator dynamicscompensation based on upper bound delay for real-timehybrid simulationrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 12 pp 1749ndash1765 2013

[25] X Gao N Castaneda and S J Dyke ldquoReal time hybridsimulation from dynamic system motion control to

Shock and Vibration 13

experimental errorrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 6 pp 815ndash832 2013

[26] X Ning Z Wang H Zhou B Wu Y Ding and B XuldquoRobust actuator dynamics compensation method for real-time hybrid simulationrdquo Mechanical Systems and SignalProcessing vol 131 pp 49ndash70 2019

[27] G Ou A I Ozdagli S J Dyke and BWu ldquoRobust integratedactuator control experimental verification and real-timehybrid-simulation implementationrdquo Earthquake Engineeringamp Structural Dynamics vol 44 no 3 pp 441ndash460 2015

[28] N Nakata ldquoEffective force testing using a robust loop shapingcontrollerrdquo Earthquake Engineering amp Structural Dynamicsvol 42 no 2 pp 261ndash275 2013

[29] B Wu and H Zhou ldquoSliding mode for equivalent forcecontrol in real-time substructure testingrdquo Structural Controland Health Monitoring vol 21 no 10 pp 1284ndash1303 2014

[30] Z Wang X Ning G Xu et al ldquoHigh performance com-pensation using an adaptive strategy for real-time hybridsimulationrdquo Mechanical Systems and Signal Processingvol 133 Article ID 106262 2019

[31] C E Silva D Gomez A Maghareh et al ldquoBenchmark controlproblem for real-time hybrid simulationrdquoMechanical Systemsand Signal Processing vol 135 Article ID 106381 2020

14 Shock and Vibration

compensation methods In particular the El Centro (1940NS) earthquake record was adopted to excite the structurewith a peak ground acceleration of 7837 Gal e threeaforementioned compensation methods were carried outherein with the same parameters and settings as those in theprevious subsection e model parameters of the twoadaptive methods were initialized with the soft start schemeyielding [46278 minus44702 078133] and [10771 0081833minus017691] respectively Obviously the latter one is very closeto the common initial parameters namely [1 0 0] and thisvalidates the rationality of this initialization RTHS of amultiple DOF structure was implemented because they weremore challenging than previous tests owing to multiple-frequency-content structural responses and randomness ofthe seismic excitation

e displacement time histories obtained with the threedelay compensation methods are shown in Figure 12 It canbe seen from Figure 12(b) that the error of the PE method isrelatively large especially up to 067mm at 247 s When thevelocity approaches zero at the displacement peaks themethod predicts displacement responses based on the trendsof several past steps thereby causing errors in the dis-placement command Compared with the PE method theSADC method induces smaller peak errors as shown inFigure 12(d) is is attributed to its online updated discretemodel of the loading system which can effectively capturethe variation of the system characteristics and adjust actu-ator commands accordingly In Figure 12(f ) the desireddisplacement and measured displacement match very well

with the TADC method even at displacement peaks isresult shows that this RTHS of multiple degree-of-freedomstructures subjected to an earthquake can be remarkablycompensated by the TADC method

Figure 13 shows the parameter evolutions of adaptivecompensation methods rough comparison it can befound that the parameter variation ranges of the SADCmethod are much wider with a maximum value of about 9Conversely owing to the contribution of its first-stagecompensation that is the course compensation based on thePE method the parameters of the TADC method vary invery small ranges is is because the delay compensated bythe second stage of the TADC method that is the adaptivecompensation method is indeed the residual time delay ofthe first-stage compensation As shown in this figure theparameters with the TADC method are very close to con-stant ones and the identification of these values is easy andaccurate Consequently there is no doubt that the TADCmethod possesses favorable performance

Subspace plots of the measured and desired displace-ments of the actuator are illustrated in Figure 14 It can beseen from the figure that the PE method has the worstcompensation effect which is attributed to the varying timedelay and influence of multiple frequency contents of thedesired displacement e SADC method can realize betterperformance owing to its continuously updated systemmodel which can effectively capture the varying charac-teristics of the loading system and can compensate both theamplitude and phase errors e TADC method performsthe best because of its unique features such as coarse and finecompensation

Evaluation indexes are calculated and collected in Ta-ble 3 As can be seen from this table the TADC methodprovides results with the smallest J2 and J3 and hence issuperior to the other twomethodsis is consistent with theconclusion presented in Figures 12(b) and 12(c) Generallyindex values in this scenario are larger than those in

6

4

2

0θ

ndash2

ndash4

ndash60 10 20 30 40 50

Time (s)

θ1θ2θ3

(a)

θ

6

4

2

0

ndash2

ndash4

ndash60 10 20 30 40 50

Time (s)

θ1θ2θ3

(b)

Figure 8 Time histories of estimated parameters with swept loading target (a) e SADC method (b) e TADC method

Table 1 Evaluation indexes of 10Hz swept signal loading

Compensation method J2 () J3 ()

e PE method 793 2185e SADC method 261 591e TADC method 203 378

8 Shock and Vibration

Time (s)

ndash8

ndash4

0

4

8D

ispla

cem

ent (

mm

)

0 5 10 15 20

DesiredMeasuredCommand

(a)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

174 1745 175Time (s)

DesiredMeasuredCommand

(b)

0 5 10 15 20Time (s)

ndash8

ndash4

0

4

8

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(c)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

1706 1708 171 1712 1714 1716Time (s)

DesiredMeasuredCommand

(d)

0 5 10 15 20Time (s)

ndash8

ndash4

0

4

8

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(e)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

1706 1708 171 1712 1714 1716Time (s)

DesiredMeasuredCommand

(f )

Figure 9 Displacement time histories obtained in RTHS with sinusoidal excitation (a) Displacements obtained with the PE method (b)Enlarged view of (a) (c) Displacements obtained with the SADC method (d) Enlarged view of (c) (e) Displacements obtained with theTADC method (f ) Enlarged view of (e)

Shock and Vibration 9

Subsection 35 and smaller than those in Subsection 34 atis to say the RTHS with the sinusoidal excitation is the easiestone because of its regular input compensation for the sweptloading test is the most challenging one for its large frequencywidth of the desired displacement and compensation forRTHS with seismic excitation has a medium difficulty level

owing to its random earthquake input Among the three teststhe TADC method is consistently endowed with the bestindexes indicating the superiority of this method One mayargue that the improvement of this strategy is limited Ac-tually the SADC method performs relatively well and anyimprovement is considerably difficult Moreover in this

Time (s)

ndash6

ndash4

ndash2

0

2

4

6

θ

0 5 10 15 20

θ1θ2θ3

(a)

0 5 10 15 20Time (s)

ndash6

ndash4

ndash2

0

2

4

6

θ

θ1θ2θ3

(b)

Figure 10 Time histories of estimated parameters with sinusoidal excitation (a) e SADC method (b) e TADC method

3

2

1

0

ndash1

ndash2

Disp

lace

men

t (m

m)

Time (s)0 01 02 03 04 05

DesiredMeasuredCommand

(a)

0 01 02 03 04 05Time (s)

3

2

1

0

ndash1

ndash2

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(b)

Figure 11 Time histories of displacements at the beginning of tests (a) e SADC method (b) e TADC method

Table 2 RTHS evaluation index with 3Hz sine signal excitation

Method of compensation J2 () J3 ()

e PE method 184 261e SADC method 141 249e TADC method 065 121

10 Shock and Vibration

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(a)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(b)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(c)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(d)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(e)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(f )

Figure 12 Displacement time histories obtained in RTHS with seismic excitation (a) Displacements obtained with the PE method (b)Enlarged view of (a) (c) Displacements obtained with the SADCmethod (e) Displacements obtained with the TADCmethod (f ) Enlargedview of (e)

Shock and Vibration 11

scenario J2 and J3 are improved by 378 [(465minus 289)465]and 339 [(617minus 408)617] compared with the SADCmethod respectively indicating substantial improvement

4 Conclusions

is study carried out a series of verification tests of a two-stage adaptive delay compensation (TADC)method for real-time hybrid simulation in conjunction with the comparisonwith the polynomial extrapolation (PE) method and tradi-tional single-stage adaptive delay compensation (SADC)

method ese include loading tests with a prescribed sweptsignal as the desired displacement RTHS with a sinusoidalexcitation and RTHS with a seismic excitation From thisinvestigation the conclusions can be drawn as follows

(1) e estimated parameters of the TADC method varyin smaller ranges than those of the SADC methodowing to the first-stage compensation method whichreduces the difficulty in parameter estimation andhence results in better compensation performance

(2) e model parameters of the TADC method can beinitialized either as [1 0 0] or through a soft-startprocess e first-stage compensation of the TADCmethod reduces the dependence of the performanceon the parameter estimation accuracy especially atthe beginning of a test where the parameters varyapparently e compensation accuracy benefitsfrom this feature

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(a)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(b)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(c)

Figure 14 Desired-measured displacement plots with seismic excitation (a) e PE method (b) e SADC method (c) e TADCmethod

Table 3 Evaluation indexes of RTHS with seismic excitation

Compensation method J2 () J3 ()

e PE method 517 846e SADC method 465 617e TADC method 289 408

10

5

0

ndash5

ndash10

ndash150 10 20

Time (s)30

θ

θ1θ2θ3

(a)

10

5

0

ndash5

ndash10

ndash15

θ

0 10 20Time (s)

30

θ1θ2θ3

(b)

Figure 13 Estimated parameters in RTHS with seismic excitation (a) e SADC method (b) e ADC method

12 Shock and Vibration

(3) e TADC method exhibits the best tracking ac-curacy to the desired displacements among the threecompensation methods owing to its features

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e research and publication of this article were funded by theScientific Research Fund of Institute of EngineeringMechanicsChina Earthquake Administration (Grants nos 2018D10 and2020D14) the National Key Research and DevelopmentProgram of China (Grant no 2016YFC0701106) and theNational Natural Science Foundation of China (Grants nos51778190 and 51908231)

References

[1] M Nakashima H Kato and E Takaoka ldquoDevelopment ofreal-time pseudo dynamic testingrdquo Earthquake Engineering ampStructural Dynamics vol 21 no 1 pp 79ndash92 1992

[2] M S Williams and A Blakeborough ldquoLaboratory testing ofstructures under dynamic loads an introductory reviewrdquoPhilosophical Transactions of the Royal Society of LondonSeries A Mathematical Physical and Engineering Sciencesvol 359 no 1786 pp 1651ndash1669 2001

[3] B Wu H Bao J Ou and S Tian ldquoStability and accuracyanalysis of the central difference method for real-time sub-structure testingrdquo Earthquake Engineering amp Structural Dy-namics vol 34 no 7 pp 705ndash718 2005

[4] O S Bursi and D Wagg Modern Testing Techniques forStructural Systems Dynamics and Control Vol 502 SpringerScience amp Business Media Berlin Germany 2009

[5] O S Bursi Z Wang C Jia and B Wu ldquoMonolithic andpartitioned time integration methods for real-time hetero-geneous simulationsrdquo Computational Mechanics vol 52no 1 pp 99ndash119 2013

[6] X Cai C Yang and Y Yuan ldquoHybrid simulation of seismicresponses of a typical station with a reinforced concretecolumnrdquoApplied Sciences vol 10 no 4 Article ID 1331 2020

[7] Z Chen H Wang H Wang et al ldquoApplication of the hybridsimulation method for the full-scale precast reinforced con-crete shear wall structurerdquo Applied Sciences vol 8 no 2Article ID 252 2018

[8] T Horiuchi M Inoue T Konno and Y Namita ldquoReal-timehybrid experimental system with actuator delay compensa-tion and its application to a piping system with energy ab-sorberrdquo Earthquake Engineering amp Structural Dynamicsvol 28 no 10 pp 1121ndash1141 1999

[9] P A Bonnet C N Lim M S Williams et al ldquoReal-timehybrid experiments with Newmark integration MCSmdouter-loop control and multi-tasking strategiesrdquo EarthquakeEngineering amp Structural Dynamics vol 36 no 1 pp 119ndash1412007

[10] A P Darby M S Williams and A Blakeborough ldquoStabilityand delay compensation for real-time substructure testingrdquoJournal of Engineering Mechanics vol 128 no 12 pp 1276ndash1284 2002

[11] M Nakashima and N Masaoka ldquoReal-time on-line test forMDOF systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 28 no 4 pp 393ndash420 1999

[12] M Ahmadizadeh G Mosqueda and A M ReinhornldquoCompensation of actuator delay and dynamics for real-timehybrid structural simulationrdquo Earthquake Engineering ampStructural Dynamics vol 37 no 1 pp 21ndash42 2008

[13] Z Wang B Wu O S Bursi G Xu and Y Ding ldquoAn effectiveonline delay estimation method based on a simplified physicalsystem model for real-time hybrid simulationrdquo SmartStructures and Systems vol 14 no 6 pp 1247ndash1267 2014

[14] S Strano andM Terzo ldquoActuator dynamics compensation forreal-time hybrid simulation an adaptive approach by meansof a nonlinear estimatorrdquo Nonlinear Dynamics vol 85 no 4pp 2353ndash2368 2016

[15] M I Wallace J Sieber S A Neild D J Wagg andB Krauskopf ldquoStability analysis of real-time dynamic sub-structuring using delay differential equation modelsrdquoEarthquake Engineering amp Structural Dynamics vol 34no 15 pp 1817ndash1832 2005

[16] H Zhou D J Wagg and M Li ldquoEquivalent force controlcombined with adaptive polynomial-based forward predic-tion for real-time hybrid simulationrdquo Structural Control andHealth Monitoring vol 24 no 11 p e2018 2017

[17] Y Chae K Kazemibidokhti and J M Ricles ldquoAdaptive timeseries compensator for delay compensation of servo-hydraulicactuator systems for real-time hybrid simulationrdquo EarthquakeEngineering amp Structural Dynamics vol 42 no 11pp 1697ndash1715 2013

[18] C Chen J M Ricles and T Guo ldquoImproved adaptive inversecompensation technique for real-time hybrid simulationrdquoJournal of Engineering Mechanics vol 138 no 12 pp 1432ndash1446 2012

[19] V Nguyen and U Dorka ldquoPhase lag compensation in real-time substructure testing based on online system identifica-tionrdquo in Proceedings of the 14th World Conference onEarthquake Engineering Beijing China October 2008

[20] X Ning Z Wang C Wang et al ldquoAdaptive feedforward andfeedback compensation method for real-time hybrid simu-lation based on a discrete physical testing system modelrdquoJournal of Earthquake and Engineering 2020

[21] Z Wang G Xu Q Li et al ldquoAn adaptive delay compensationmethod based on a discrete systemmodel for real-time hybridsimulationrdquo Smart Structures and Systems vol 25 no 5pp 569ndash580 2020

[22] J E Carrion and B F Spencer Model-based Strategies forReal-Time Hybrid Testing 1940ndash9826 Newmark StructuralEngineering Laboratory University of Illinois at UrbanaChampaign IL USA 2007

[23] R-Y Jung P Benson Shing E Stauffer and B oenldquoPerformance of a real-time pseudodynamic test systemconsidering nonlinear structural responserdquo Earthquake En-gineering amp Structural Dynamics vol 36 no 12 pp 1785ndash1809 2007

[24] B Wu Z Wang and O S Bursi ldquoActuator dynamicscompensation based on upper bound delay for real-timehybrid simulationrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 12 pp 1749ndash1765 2013

[25] X Gao N Castaneda and S J Dyke ldquoReal time hybridsimulation from dynamic system motion control to

Shock and Vibration 13

experimental errorrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 6 pp 815ndash832 2013

[26] X Ning Z Wang H Zhou B Wu Y Ding and B XuldquoRobust actuator dynamics compensation method for real-time hybrid simulationrdquo Mechanical Systems and SignalProcessing vol 131 pp 49ndash70 2019

[27] G Ou A I Ozdagli S J Dyke and BWu ldquoRobust integratedactuator control experimental verification and real-timehybrid-simulation implementationrdquo Earthquake Engineeringamp Structural Dynamics vol 44 no 3 pp 441ndash460 2015

[28] N Nakata ldquoEffective force testing using a robust loop shapingcontrollerrdquo Earthquake Engineering amp Structural Dynamicsvol 42 no 2 pp 261ndash275 2013

[29] B Wu and H Zhou ldquoSliding mode for equivalent forcecontrol in real-time substructure testingrdquo Structural Controland Health Monitoring vol 21 no 10 pp 1284ndash1303 2014

[30] Z Wang X Ning G Xu et al ldquoHigh performance com-pensation using an adaptive strategy for real-time hybridsimulationrdquo Mechanical Systems and Signal Processingvol 133 Article ID 106262 2019

[31] C E Silva D Gomez A Maghareh et al ldquoBenchmark controlproblem for real-time hybrid simulationrdquoMechanical Systemsand Signal Processing vol 135 Article ID 106381 2020

14 Shock and Vibration

Time (s)

ndash8

ndash4

0

4

8D

ispla

cem

ent (

mm

)

0 5 10 15 20

DesiredMeasuredCommand

(a)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

174 1745 175Time (s)

DesiredMeasuredCommand

(b)

0 5 10 15 20Time (s)

ndash8

ndash4

0

4

8

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(c)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

1706 1708 171 1712 1714 1716Time (s)

DesiredMeasuredCommand

(d)

0 5 10 15 20Time (s)

ndash8

ndash4

0

4

8

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(e)

35

45

4

55

5

6

Disp

lace

men

t (m

m)

1706 1708 171 1712 1714 1716Time (s)

DesiredMeasuredCommand

(f )

Figure 9 Displacement time histories obtained in RTHS with sinusoidal excitation (a) Displacements obtained with the PE method (b)Enlarged view of (a) (c) Displacements obtained with the SADC method (d) Enlarged view of (c) (e) Displacements obtained with theTADC method (f ) Enlarged view of (e)

Shock and Vibration 9

Subsection 35 and smaller than those in Subsection 34 atis to say the RTHS with the sinusoidal excitation is the easiestone because of its regular input compensation for the sweptloading test is the most challenging one for its large frequencywidth of the desired displacement and compensation forRTHS with seismic excitation has a medium difficulty level

owing to its random earthquake input Among the three teststhe TADC method is consistently endowed with the bestindexes indicating the superiority of this method One mayargue that the improvement of this strategy is limited Ac-tually the SADC method performs relatively well and anyimprovement is considerably difficult Moreover in this

Time (s)

ndash6

ndash4

ndash2

0

2

4

6

θ

0 5 10 15 20

θ1θ2θ3

(a)

0 5 10 15 20Time (s)

ndash6

ndash4

ndash2

0

2

4

6

θ

θ1θ2θ3

(b)

Figure 10 Time histories of estimated parameters with sinusoidal excitation (a) e SADC method (b) e TADC method

3

2

1

0

ndash1

ndash2

Disp

lace

men

t (m

m)

Time (s)0 01 02 03 04 05

DesiredMeasuredCommand

(a)

0 01 02 03 04 05Time (s)

3

2

1

0

ndash1

ndash2

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(b)

Figure 11 Time histories of displacements at the beginning of tests (a) e SADC method (b) e TADC method

Table 2 RTHS evaluation index with 3Hz sine signal excitation

Method of compensation J2 () J3 ()

e PE method 184 261e SADC method 141 249e TADC method 065 121

10 Shock and Vibration

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(a)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(b)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(c)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(d)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(e)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(f )

Figure 12 Displacement time histories obtained in RTHS with seismic excitation (a) Displacements obtained with the PE method (b)Enlarged view of (a) (c) Displacements obtained with the SADCmethod (e) Displacements obtained with the TADCmethod (f ) Enlargedview of (e)

Shock and Vibration 11

scenario J2 and J3 are improved by 378 [(465minus 289)465]and 339 [(617minus 408)617] compared with the SADCmethod respectively indicating substantial improvement

4 Conclusions

is study carried out a series of verification tests of a two-stage adaptive delay compensation (TADC)method for real-time hybrid simulation in conjunction with the comparisonwith the polynomial extrapolation (PE) method and tradi-tional single-stage adaptive delay compensation (SADC)

method ese include loading tests with a prescribed sweptsignal as the desired displacement RTHS with a sinusoidalexcitation and RTHS with a seismic excitation From thisinvestigation the conclusions can be drawn as follows

(1) e estimated parameters of the TADC method varyin smaller ranges than those of the SADC methodowing to the first-stage compensation method whichreduces the difficulty in parameter estimation andhence results in better compensation performance

(2) e model parameters of the TADC method can beinitialized either as [1 0 0] or through a soft-startprocess e first-stage compensation of the TADCmethod reduces the dependence of the performanceon the parameter estimation accuracy especially atthe beginning of a test where the parameters varyapparently e compensation accuracy benefitsfrom this feature

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(a)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(b)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(c)

Figure 14 Desired-measured displacement plots with seismic excitation (a) e PE method (b) e SADC method (c) e TADCmethod

Table 3 Evaluation indexes of RTHS with seismic excitation

Compensation method J2 () J3 ()

e PE method 517 846e SADC method 465 617e TADC method 289 408

10

5

0

ndash5

ndash10

ndash150 10 20

Time (s)30

θ

θ1θ2θ3

(a)

10

5

0

ndash5

ndash10

ndash15

θ

0 10 20Time (s)

30

θ1θ2θ3

(b)

Figure 13 Estimated parameters in RTHS with seismic excitation (a) e SADC method (b) e ADC method

12 Shock and Vibration

(3) e TADC method exhibits the best tracking ac-curacy to the desired displacements among the threecompensation methods owing to its features

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e research and publication of this article were funded by theScientific Research Fund of Institute of EngineeringMechanicsChina Earthquake Administration (Grants nos 2018D10 and2020D14) the National Key Research and DevelopmentProgram of China (Grant no 2016YFC0701106) and theNational Natural Science Foundation of China (Grants nos51778190 and 51908231)

References

[1] M Nakashima H Kato and E Takaoka ldquoDevelopment ofreal-time pseudo dynamic testingrdquo Earthquake Engineering ampStructural Dynamics vol 21 no 1 pp 79ndash92 1992

[2] M S Williams and A Blakeborough ldquoLaboratory testing ofstructures under dynamic loads an introductory reviewrdquoPhilosophical Transactions of the Royal Society of LondonSeries A Mathematical Physical and Engineering Sciencesvol 359 no 1786 pp 1651ndash1669 2001

[3] B Wu H Bao J Ou and S Tian ldquoStability and accuracyanalysis of the central difference method for real-time sub-structure testingrdquo Earthquake Engineering amp Structural Dy-namics vol 34 no 7 pp 705ndash718 2005

[4] O S Bursi and D Wagg Modern Testing Techniques forStructural Systems Dynamics and Control Vol 502 SpringerScience amp Business Media Berlin Germany 2009

[5] O S Bursi Z Wang C Jia and B Wu ldquoMonolithic andpartitioned time integration methods for real-time hetero-geneous simulationsrdquo Computational Mechanics vol 52no 1 pp 99ndash119 2013

[6] X Cai C Yang and Y Yuan ldquoHybrid simulation of seismicresponses of a typical station with a reinforced concretecolumnrdquoApplied Sciences vol 10 no 4 Article ID 1331 2020

[7] Z Chen H Wang H Wang et al ldquoApplication of the hybridsimulation method for the full-scale precast reinforced con-crete shear wall structurerdquo Applied Sciences vol 8 no 2Article ID 252 2018

[8] T Horiuchi M Inoue T Konno and Y Namita ldquoReal-timehybrid experimental system with actuator delay compensa-tion and its application to a piping system with energy ab-sorberrdquo Earthquake Engineering amp Structural Dynamicsvol 28 no 10 pp 1121ndash1141 1999

[9] P A Bonnet C N Lim M S Williams et al ldquoReal-timehybrid experiments with Newmark integration MCSmdouter-loop control and multi-tasking strategiesrdquo EarthquakeEngineering amp Structural Dynamics vol 36 no 1 pp 119ndash1412007

[10] A P Darby M S Williams and A Blakeborough ldquoStabilityand delay compensation for real-time substructure testingrdquoJournal of Engineering Mechanics vol 128 no 12 pp 1276ndash1284 2002

[11] M Nakashima and N Masaoka ldquoReal-time on-line test forMDOF systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 28 no 4 pp 393ndash420 1999

[12] M Ahmadizadeh G Mosqueda and A M ReinhornldquoCompensation of actuator delay and dynamics for real-timehybrid structural simulationrdquo Earthquake Engineering ampStructural Dynamics vol 37 no 1 pp 21ndash42 2008

[13] Z Wang B Wu O S Bursi G Xu and Y Ding ldquoAn effectiveonline delay estimation method based on a simplified physicalsystem model for real-time hybrid simulationrdquo SmartStructures and Systems vol 14 no 6 pp 1247ndash1267 2014

[14] S Strano andM Terzo ldquoActuator dynamics compensation forreal-time hybrid simulation an adaptive approach by meansof a nonlinear estimatorrdquo Nonlinear Dynamics vol 85 no 4pp 2353ndash2368 2016

[15] M I Wallace J Sieber S A Neild D J Wagg andB Krauskopf ldquoStability analysis of real-time dynamic sub-structuring using delay differential equation modelsrdquoEarthquake Engineering amp Structural Dynamics vol 34no 15 pp 1817ndash1832 2005

[16] H Zhou D J Wagg and M Li ldquoEquivalent force controlcombined with adaptive polynomial-based forward predic-tion for real-time hybrid simulationrdquo Structural Control andHealth Monitoring vol 24 no 11 p e2018 2017

[17] Y Chae K Kazemibidokhti and J M Ricles ldquoAdaptive timeseries compensator for delay compensation of servo-hydraulicactuator systems for real-time hybrid simulationrdquo EarthquakeEngineering amp Structural Dynamics vol 42 no 11pp 1697ndash1715 2013

[18] C Chen J M Ricles and T Guo ldquoImproved adaptive inversecompensation technique for real-time hybrid simulationrdquoJournal of Engineering Mechanics vol 138 no 12 pp 1432ndash1446 2012

[19] V Nguyen and U Dorka ldquoPhase lag compensation in real-time substructure testing based on online system identifica-tionrdquo in Proceedings of the 14th World Conference onEarthquake Engineering Beijing China October 2008

[20] X Ning Z Wang C Wang et al ldquoAdaptive feedforward andfeedback compensation method for real-time hybrid simu-lation based on a discrete physical testing system modelrdquoJournal of Earthquake and Engineering 2020

[21] Z Wang G Xu Q Li et al ldquoAn adaptive delay compensationmethod based on a discrete systemmodel for real-time hybridsimulationrdquo Smart Structures and Systems vol 25 no 5pp 569ndash580 2020

[22] J E Carrion and B F Spencer Model-based Strategies forReal-Time Hybrid Testing 1940ndash9826 Newmark StructuralEngineering Laboratory University of Illinois at UrbanaChampaign IL USA 2007

[23] R-Y Jung P Benson Shing E Stauffer and B oenldquoPerformance of a real-time pseudodynamic test systemconsidering nonlinear structural responserdquo Earthquake En-gineering amp Structural Dynamics vol 36 no 12 pp 1785ndash1809 2007

[24] B Wu Z Wang and O S Bursi ldquoActuator dynamicscompensation based on upper bound delay for real-timehybrid simulationrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 12 pp 1749ndash1765 2013

[25] X Gao N Castaneda and S J Dyke ldquoReal time hybridsimulation from dynamic system motion control to

Shock and Vibration 13

experimental errorrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 6 pp 815ndash832 2013

[26] X Ning Z Wang H Zhou B Wu Y Ding and B XuldquoRobust actuator dynamics compensation method for real-time hybrid simulationrdquo Mechanical Systems and SignalProcessing vol 131 pp 49ndash70 2019

[27] G Ou A I Ozdagli S J Dyke and BWu ldquoRobust integratedactuator control experimental verification and real-timehybrid-simulation implementationrdquo Earthquake Engineeringamp Structural Dynamics vol 44 no 3 pp 441ndash460 2015

[28] N Nakata ldquoEffective force testing using a robust loop shapingcontrollerrdquo Earthquake Engineering amp Structural Dynamicsvol 42 no 2 pp 261ndash275 2013

[29] B Wu and H Zhou ldquoSliding mode for equivalent forcecontrol in real-time substructure testingrdquo Structural Controland Health Monitoring vol 21 no 10 pp 1284ndash1303 2014

[30] Z Wang X Ning G Xu et al ldquoHigh performance com-pensation using an adaptive strategy for real-time hybridsimulationrdquo Mechanical Systems and Signal Processingvol 133 Article ID 106262 2019

[31] C E Silva D Gomez A Maghareh et al ldquoBenchmark controlproblem for real-time hybrid simulationrdquoMechanical Systemsand Signal Processing vol 135 Article ID 106381 2020

14 Shock and Vibration

Subsection 35 and smaller than those in Subsection 34 atis to say the RTHS with the sinusoidal excitation is the easiestone because of its regular input compensation for the sweptloading test is the most challenging one for its large frequencywidth of the desired displacement and compensation forRTHS with seismic excitation has a medium difficulty level

owing to its random earthquake input Among the three teststhe TADC method is consistently endowed with the bestindexes indicating the superiority of this method One mayargue that the improvement of this strategy is limited Ac-tually the SADC method performs relatively well and anyimprovement is considerably difficult Moreover in this

Time (s)

ndash6

ndash4

ndash2

0

2

4

6

θ

0 5 10 15 20

θ1θ2θ3

(a)

0 5 10 15 20Time (s)

ndash6

ndash4

ndash2

0

2

4

6

θ

θ1θ2θ3

(b)

Figure 10 Time histories of estimated parameters with sinusoidal excitation (a) e SADC method (b) e TADC method

3

2

1

0

ndash1

ndash2

Disp

lace

men

t (m

m)

Time (s)0 01 02 03 04 05

DesiredMeasuredCommand

(a)

0 01 02 03 04 05Time (s)

3

2

1

0

ndash1

ndash2

Disp

lace

men

t (m

m)

DesiredMeasuredCommand

(b)

Figure 11 Time histories of displacements at the beginning of tests (a) e SADC method (b) e TADC method

Table 2 RTHS evaluation index with 3Hz sine signal excitation

Method of compensation J2 () J3 ()

e PE method 184 261e SADC method 141 249e TADC method 065 121

10 Shock and Vibration

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(a)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(b)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(c)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(d)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(e)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(f )

Figure 12 Displacement time histories obtained in RTHS with seismic excitation (a) Displacements obtained with the PE method (b)Enlarged view of (a) (c) Displacements obtained with the SADCmethod (e) Displacements obtained with the TADCmethod (f ) Enlargedview of (e)

Shock and Vibration 11

scenario J2 and J3 are improved by 378 [(465minus 289)465]and 339 [(617minus 408)617] compared with the SADCmethod respectively indicating substantial improvement

4 Conclusions

is study carried out a series of verification tests of a two-stage adaptive delay compensation (TADC)method for real-time hybrid simulation in conjunction with the comparisonwith the polynomial extrapolation (PE) method and tradi-tional single-stage adaptive delay compensation (SADC)

method ese include loading tests with a prescribed sweptsignal as the desired displacement RTHS with a sinusoidalexcitation and RTHS with a seismic excitation From thisinvestigation the conclusions can be drawn as follows

(1) e estimated parameters of the TADC method varyin smaller ranges than those of the SADC methodowing to the first-stage compensation method whichreduces the difficulty in parameter estimation andhence results in better compensation performance

(2) e model parameters of the TADC method can beinitialized either as [1 0 0] or through a soft-startprocess e first-stage compensation of the TADCmethod reduces the dependence of the performanceon the parameter estimation accuracy especially atthe beginning of a test where the parameters varyapparently e compensation accuracy benefitsfrom this feature

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(a)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(b)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(c)

Figure 14 Desired-measured displacement plots with seismic excitation (a) e PE method (b) e SADC method (c) e TADCmethod

Table 3 Evaluation indexes of RTHS with seismic excitation

Compensation method J2 () J3 ()

e PE method 517 846e SADC method 465 617e TADC method 289 408

10

5

0

ndash5

ndash10

ndash150 10 20

Time (s)30

θ

θ1θ2θ3

(a)

10

5

0

ndash5

ndash10

ndash15

θ

0 10 20Time (s)

30

θ1θ2θ3

(b)

Figure 13 Estimated parameters in RTHS with seismic excitation (a) e SADC method (b) e ADC method

12 Shock and Vibration

(3) e TADC method exhibits the best tracking ac-curacy to the desired displacements among the threecompensation methods owing to its features

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e research and publication of this article were funded by theScientific Research Fund of Institute of EngineeringMechanicsChina Earthquake Administration (Grants nos 2018D10 and2020D14) the National Key Research and DevelopmentProgram of China (Grant no 2016YFC0701106) and theNational Natural Science Foundation of China (Grants nos51778190 and 51908231)

References

[1] M Nakashima H Kato and E Takaoka ldquoDevelopment ofreal-time pseudo dynamic testingrdquo Earthquake Engineering ampStructural Dynamics vol 21 no 1 pp 79ndash92 1992

[2] M S Williams and A Blakeborough ldquoLaboratory testing ofstructures under dynamic loads an introductory reviewrdquoPhilosophical Transactions of the Royal Society of LondonSeries A Mathematical Physical and Engineering Sciencesvol 359 no 1786 pp 1651ndash1669 2001

[3] B Wu H Bao J Ou and S Tian ldquoStability and accuracyanalysis of the central difference method for real-time sub-structure testingrdquo Earthquake Engineering amp Structural Dy-namics vol 34 no 7 pp 705ndash718 2005

[4] O S Bursi and D Wagg Modern Testing Techniques forStructural Systems Dynamics and Control Vol 502 SpringerScience amp Business Media Berlin Germany 2009

[5] O S Bursi Z Wang C Jia and B Wu ldquoMonolithic andpartitioned time integration methods for real-time hetero-geneous simulationsrdquo Computational Mechanics vol 52no 1 pp 99ndash119 2013

[6] X Cai C Yang and Y Yuan ldquoHybrid simulation of seismicresponses of a typical station with a reinforced concretecolumnrdquoApplied Sciences vol 10 no 4 Article ID 1331 2020

[7] Z Chen H Wang H Wang et al ldquoApplication of the hybridsimulation method for the full-scale precast reinforced con-crete shear wall structurerdquo Applied Sciences vol 8 no 2Article ID 252 2018

[8] T Horiuchi M Inoue T Konno and Y Namita ldquoReal-timehybrid experimental system with actuator delay compensa-tion and its application to a piping system with energy ab-sorberrdquo Earthquake Engineering amp Structural Dynamicsvol 28 no 10 pp 1121ndash1141 1999

[9] P A Bonnet C N Lim M S Williams et al ldquoReal-timehybrid experiments with Newmark integration MCSmdouter-loop control and multi-tasking strategiesrdquo EarthquakeEngineering amp Structural Dynamics vol 36 no 1 pp 119ndash1412007

[10] A P Darby M S Williams and A Blakeborough ldquoStabilityand delay compensation for real-time substructure testingrdquoJournal of Engineering Mechanics vol 128 no 12 pp 1276ndash1284 2002

[11] M Nakashima and N Masaoka ldquoReal-time on-line test forMDOF systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 28 no 4 pp 393ndash420 1999

[12] M Ahmadizadeh G Mosqueda and A M ReinhornldquoCompensation of actuator delay and dynamics for real-timehybrid structural simulationrdquo Earthquake Engineering ampStructural Dynamics vol 37 no 1 pp 21ndash42 2008

[13] Z Wang B Wu O S Bursi G Xu and Y Ding ldquoAn effectiveonline delay estimation method based on a simplified physicalsystem model for real-time hybrid simulationrdquo SmartStructures and Systems vol 14 no 6 pp 1247ndash1267 2014

[14] S Strano andM Terzo ldquoActuator dynamics compensation forreal-time hybrid simulation an adaptive approach by meansof a nonlinear estimatorrdquo Nonlinear Dynamics vol 85 no 4pp 2353ndash2368 2016

[15] M I Wallace J Sieber S A Neild D J Wagg andB Krauskopf ldquoStability analysis of real-time dynamic sub-structuring using delay differential equation modelsrdquoEarthquake Engineering amp Structural Dynamics vol 34no 15 pp 1817ndash1832 2005

[16] H Zhou D J Wagg and M Li ldquoEquivalent force controlcombined with adaptive polynomial-based forward predic-tion for real-time hybrid simulationrdquo Structural Control andHealth Monitoring vol 24 no 11 p e2018 2017

[17] Y Chae K Kazemibidokhti and J M Ricles ldquoAdaptive timeseries compensator for delay compensation of servo-hydraulicactuator systems for real-time hybrid simulationrdquo EarthquakeEngineering amp Structural Dynamics vol 42 no 11pp 1697ndash1715 2013

[18] C Chen J M Ricles and T Guo ldquoImproved adaptive inversecompensation technique for real-time hybrid simulationrdquoJournal of Engineering Mechanics vol 138 no 12 pp 1432ndash1446 2012

[19] V Nguyen and U Dorka ldquoPhase lag compensation in real-time substructure testing based on online system identifica-tionrdquo in Proceedings of the 14th World Conference onEarthquake Engineering Beijing China October 2008

[20] X Ning Z Wang C Wang et al ldquoAdaptive feedforward andfeedback compensation method for real-time hybrid simu-lation based on a discrete physical testing system modelrdquoJournal of Earthquake and Engineering 2020

[21] Z Wang G Xu Q Li et al ldquoAn adaptive delay compensationmethod based on a discrete systemmodel for real-time hybridsimulationrdquo Smart Structures and Systems vol 25 no 5pp 569ndash580 2020

[22] J E Carrion and B F Spencer Model-based Strategies forReal-Time Hybrid Testing 1940ndash9826 Newmark StructuralEngineering Laboratory University of Illinois at UrbanaChampaign IL USA 2007

[23] R-Y Jung P Benson Shing E Stauffer and B oenldquoPerformance of a real-time pseudodynamic test systemconsidering nonlinear structural responserdquo Earthquake En-gineering amp Structural Dynamics vol 36 no 12 pp 1785ndash1809 2007

[24] B Wu Z Wang and O S Bursi ldquoActuator dynamicscompensation based on upper bound delay for real-timehybrid simulationrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 12 pp 1749ndash1765 2013

[25] X Gao N Castaneda and S J Dyke ldquoReal time hybridsimulation from dynamic system motion control to

Shock and Vibration 13

experimental errorrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 6 pp 815ndash832 2013

[26] X Ning Z Wang H Zhou B Wu Y Ding and B XuldquoRobust actuator dynamics compensation method for real-time hybrid simulationrdquo Mechanical Systems and SignalProcessing vol 131 pp 49ndash70 2019

[27] G Ou A I Ozdagli S J Dyke and BWu ldquoRobust integratedactuator control experimental verification and real-timehybrid-simulation implementationrdquo Earthquake Engineeringamp Structural Dynamics vol 44 no 3 pp 441ndash460 2015

[28] N Nakata ldquoEffective force testing using a robust loop shapingcontrollerrdquo Earthquake Engineering amp Structural Dynamicsvol 42 no 2 pp 261ndash275 2013

[29] B Wu and H Zhou ldquoSliding mode for equivalent forcecontrol in real-time substructure testingrdquo Structural Controland Health Monitoring vol 21 no 10 pp 1284ndash1303 2014

[30] Z Wang X Ning G Xu et al ldquoHigh performance com-pensation using an adaptive strategy for real-time hybridsimulationrdquo Mechanical Systems and Signal Processingvol 133 Article ID 106262 2019

[31] C E Silva D Gomez A Maghareh et al ldquoBenchmark controlproblem for real-time hybrid simulationrdquoMechanical Systemsand Signal Processing vol 135 Article ID 106381 2020

14 Shock and Vibration

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(a)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(b)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(c)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(d)

10

5

0

ndash5

Disp

lace

men

t (m

m)

0 10 20Time (s)

30

DesiredMeasuredCommand

(e)

10

5

0

ndash5

ndash10

Disp

lace

men

t (m

m)

23 24 25 26 27Time (s)

28

DesiredMeasuredCommand

(f )

Figure 12 Displacement time histories obtained in RTHS with seismic excitation (a) Displacements obtained with the PE method (b)Enlarged view of (a) (c) Displacements obtained with the SADCmethod (e) Displacements obtained with the TADCmethod (f ) Enlargedview of (e)

Shock and Vibration 11

scenario J2 and J3 are improved by 378 [(465minus 289)465]and 339 [(617minus 408)617] compared with the SADCmethod respectively indicating substantial improvement

4 Conclusions

is study carried out a series of verification tests of a two-stage adaptive delay compensation (TADC)method for real-time hybrid simulation in conjunction with the comparisonwith the polynomial extrapolation (PE) method and tradi-tional single-stage adaptive delay compensation (SADC)

method ese include loading tests with a prescribed sweptsignal as the desired displacement RTHS with a sinusoidalexcitation and RTHS with a seismic excitation From thisinvestigation the conclusions can be drawn as follows

(1) e estimated parameters of the TADC method varyin smaller ranges than those of the SADC methodowing to the first-stage compensation method whichreduces the difficulty in parameter estimation andhence results in better compensation performance

(2) e model parameters of the TADC method can beinitialized either as [1 0 0] or through a soft-startprocess e first-stage compensation of the TADCmethod reduces the dependence of the performanceon the parameter estimation accuracy especially atthe beginning of a test where the parameters varyapparently e compensation accuracy benefitsfrom this feature

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(a)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(b)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(c)

Figure 14 Desired-measured displacement plots with seismic excitation (a) e PE method (b) e SADC method (c) e TADCmethod

Table 3 Evaluation indexes of RTHS with seismic excitation

Compensation method J2 () J3 ()

e PE method 517 846e SADC method 465 617e TADC method 289 408

10

5

0

ndash5

ndash10

ndash150 10 20

Time (s)30

θ

θ1θ2θ3

(a)

10

5

0

ndash5

ndash10

ndash15

θ

0 10 20Time (s)

30

θ1θ2θ3

(b)

Figure 13 Estimated parameters in RTHS with seismic excitation (a) e SADC method (b) e ADC method

12 Shock and Vibration

(3) e TADC method exhibits the best tracking ac-curacy to the desired displacements among the threecompensation methods owing to its features

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e research and publication of this article were funded by theScientific Research Fund of Institute of EngineeringMechanicsChina Earthquake Administration (Grants nos 2018D10 and2020D14) the National Key Research and DevelopmentProgram of China (Grant no 2016YFC0701106) and theNational Natural Science Foundation of China (Grants nos51778190 and 51908231)

References

[1] M Nakashima H Kato and E Takaoka ldquoDevelopment ofreal-time pseudo dynamic testingrdquo Earthquake Engineering ampStructural Dynamics vol 21 no 1 pp 79ndash92 1992

[2] M S Williams and A Blakeborough ldquoLaboratory testing ofstructures under dynamic loads an introductory reviewrdquoPhilosophical Transactions of the Royal Society of LondonSeries A Mathematical Physical and Engineering Sciencesvol 359 no 1786 pp 1651ndash1669 2001

[3] B Wu H Bao J Ou and S Tian ldquoStability and accuracyanalysis of the central difference method for real-time sub-structure testingrdquo Earthquake Engineering amp Structural Dy-namics vol 34 no 7 pp 705ndash718 2005

[4] O S Bursi and D Wagg Modern Testing Techniques forStructural Systems Dynamics and Control Vol 502 SpringerScience amp Business Media Berlin Germany 2009

[5] O S Bursi Z Wang C Jia and B Wu ldquoMonolithic andpartitioned time integration methods for real-time hetero-geneous simulationsrdquo Computational Mechanics vol 52no 1 pp 99ndash119 2013

[6] X Cai C Yang and Y Yuan ldquoHybrid simulation of seismicresponses of a typical station with a reinforced concretecolumnrdquoApplied Sciences vol 10 no 4 Article ID 1331 2020

[7] Z Chen H Wang H Wang et al ldquoApplication of the hybridsimulation method for the full-scale precast reinforced con-crete shear wall structurerdquo Applied Sciences vol 8 no 2Article ID 252 2018

[8] T Horiuchi M Inoue T Konno and Y Namita ldquoReal-timehybrid experimental system with actuator delay compensa-tion and its application to a piping system with energy ab-sorberrdquo Earthquake Engineering amp Structural Dynamicsvol 28 no 10 pp 1121ndash1141 1999

[9] P A Bonnet C N Lim M S Williams et al ldquoReal-timehybrid experiments with Newmark integration MCSmdouter-loop control and multi-tasking strategiesrdquo EarthquakeEngineering amp Structural Dynamics vol 36 no 1 pp 119ndash1412007

[10] A P Darby M S Williams and A Blakeborough ldquoStabilityand delay compensation for real-time substructure testingrdquoJournal of Engineering Mechanics vol 128 no 12 pp 1276ndash1284 2002

[11] M Nakashima and N Masaoka ldquoReal-time on-line test forMDOF systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 28 no 4 pp 393ndash420 1999

[12] M Ahmadizadeh G Mosqueda and A M ReinhornldquoCompensation of actuator delay and dynamics for real-timehybrid structural simulationrdquo Earthquake Engineering ampStructural Dynamics vol 37 no 1 pp 21ndash42 2008

[13] Z Wang B Wu O S Bursi G Xu and Y Ding ldquoAn effectiveonline delay estimation method based on a simplified physicalsystem model for real-time hybrid simulationrdquo SmartStructures and Systems vol 14 no 6 pp 1247ndash1267 2014

[14] S Strano andM Terzo ldquoActuator dynamics compensation forreal-time hybrid simulation an adaptive approach by meansof a nonlinear estimatorrdquo Nonlinear Dynamics vol 85 no 4pp 2353ndash2368 2016

[15] M I Wallace J Sieber S A Neild D J Wagg andB Krauskopf ldquoStability analysis of real-time dynamic sub-structuring using delay differential equation modelsrdquoEarthquake Engineering amp Structural Dynamics vol 34no 15 pp 1817ndash1832 2005

[16] H Zhou D J Wagg and M Li ldquoEquivalent force controlcombined with adaptive polynomial-based forward predic-tion for real-time hybrid simulationrdquo Structural Control andHealth Monitoring vol 24 no 11 p e2018 2017

[17] Y Chae K Kazemibidokhti and J M Ricles ldquoAdaptive timeseries compensator for delay compensation of servo-hydraulicactuator systems for real-time hybrid simulationrdquo EarthquakeEngineering amp Structural Dynamics vol 42 no 11pp 1697ndash1715 2013

[18] C Chen J M Ricles and T Guo ldquoImproved adaptive inversecompensation technique for real-time hybrid simulationrdquoJournal of Engineering Mechanics vol 138 no 12 pp 1432ndash1446 2012

[19] V Nguyen and U Dorka ldquoPhase lag compensation in real-time substructure testing based on online system identifica-tionrdquo in Proceedings of the 14th World Conference onEarthquake Engineering Beijing China October 2008

[20] X Ning Z Wang C Wang et al ldquoAdaptive feedforward andfeedback compensation method for real-time hybrid simu-lation based on a discrete physical testing system modelrdquoJournal of Earthquake and Engineering 2020

[21] Z Wang G Xu Q Li et al ldquoAn adaptive delay compensationmethod based on a discrete systemmodel for real-time hybridsimulationrdquo Smart Structures and Systems vol 25 no 5pp 569ndash580 2020

[22] J E Carrion and B F Spencer Model-based Strategies forReal-Time Hybrid Testing 1940ndash9826 Newmark StructuralEngineering Laboratory University of Illinois at UrbanaChampaign IL USA 2007

[23] R-Y Jung P Benson Shing E Stauffer and B oenldquoPerformance of a real-time pseudodynamic test systemconsidering nonlinear structural responserdquo Earthquake En-gineering amp Structural Dynamics vol 36 no 12 pp 1785ndash1809 2007

[24] B Wu Z Wang and O S Bursi ldquoActuator dynamicscompensation based on upper bound delay for real-timehybrid simulationrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 12 pp 1749ndash1765 2013

[25] X Gao N Castaneda and S J Dyke ldquoReal time hybridsimulation from dynamic system motion control to

Shock and Vibration 13

experimental errorrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 6 pp 815ndash832 2013

[26] X Ning Z Wang H Zhou B Wu Y Ding and B XuldquoRobust actuator dynamics compensation method for real-time hybrid simulationrdquo Mechanical Systems and SignalProcessing vol 131 pp 49ndash70 2019

[27] G Ou A I Ozdagli S J Dyke and BWu ldquoRobust integratedactuator control experimental verification and real-timehybrid-simulation implementationrdquo Earthquake Engineeringamp Structural Dynamics vol 44 no 3 pp 441ndash460 2015

[28] N Nakata ldquoEffective force testing using a robust loop shapingcontrollerrdquo Earthquake Engineering amp Structural Dynamicsvol 42 no 2 pp 261ndash275 2013

[29] B Wu and H Zhou ldquoSliding mode for equivalent forcecontrol in real-time substructure testingrdquo Structural Controland Health Monitoring vol 21 no 10 pp 1284ndash1303 2014

[30] Z Wang X Ning G Xu et al ldquoHigh performance com-pensation using an adaptive strategy for real-time hybridsimulationrdquo Mechanical Systems and Signal Processingvol 133 Article ID 106262 2019

[31] C E Silva D Gomez A Maghareh et al ldquoBenchmark controlproblem for real-time hybrid simulationrdquoMechanical Systemsand Signal Processing vol 135 Article ID 106381 2020

14 Shock and Vibration

scenario J2 and J3 are improved by 378 [(465minus 289)465]and 339 [(617minus 408)617] compared with the SADCmethod respectively indicating substantial improvement

4 Conclusions

is study carried out a series of verification tests of a two-stage adaptive delay compensation (TADC)method for real-time hybrid simulation in conjunction with the comparisonwith the polynomial extrapolation (PE) method and tradi-tional single-stage adaptive delay compensation (SADC)

method ese include loading tests with a prescribed sweptsignal as the desired displacement RTHS with a sinusoidalexcitation and RTHS with a seismic excitation From thisinvestigation the conclusions can be drawn as follows

(1) e estimated parameters of the TADC method varyin smaller ranges than those of the SADC methodowing to the first-stage compensation method whichreduces the difficulty in parameter estimation andhence results in better compensation performance

(2) e model parameters of the TADC method can beinitialized either as [1 0 0] or through a soft-startprocess e first-stage compensation of the TADCmethod reduces the dependence of the performanceon the parameter estimation accuracy especially atthe beginning of a test where the parameters varyapparently e compensation accuracy benefitsfrom this feature

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(a)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(b)

10

5

0

ndash5

1050Desired (mm)

ndash5

Mea

sure

d (m

m)

(c)

Figure 14 Desired-measured displacement plots with seismic excitation (a) e PE method (b) e SADC method (c) e TADCmethod

Table 3 Evaluation indexes of RTHS with seismic excitation

Compensation method J2 () J3 ()

e PE method 517 846e SADC method 465 617e TADC method 289 408

10

5

0

ndash5

ndash10

ndash150 10 20

Time (s)30

θ

θ1θ2θ3

(a)

10

5

0

ndash5

ndash10

ndash15

θ

0 10 20Time (s)

30

θ1θ2θ3

(b)

Figure 13 Estimated parameters in RTHS with seismic excitation (a) e SADC method (b) e ADC method

12 Shock and Vibration

(3) e TADC method exhibits the best tracking ac-curacy to the desired displacements among the threecompensation methods owing to its features

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e research and publication of this article were funded by theScientific Research Fund of Institute of EngineeringMechanicsChina Earthquake Administration (Grants nos 2018D10 and2020D14) the National Key Research and DevelopmentProgram of China (Grant no 2016YFC0701106) and theNational Natural Science Foundation of China (Grants nos51778190 and 51908231)

References

[1] M Nakashima H Kato and E Takaoka ldquoDevelopment ofreal-time pseudo dynamic testingrdquo Earthquake Engineering ampStructural Dynamics vol 21 no 1 pp 79ndash92 1992

[2] M S Williams and A Blakeborough ldquoLaboratory testing ofstructures under dynamic loads an introductory reviewrdquoPhilosophical Transactions of the Royal Society of LondonSeries A Mathematical Physical and Engineering Sciencesvol 359 no 1786 pp 1651ndash1669 2001

[3] B Wu H Bao J Ou and S Tian ldquoStability and accuracyanalysis of the central difference method for real-time sub-structure testingrdquo Earthquake Engineering amp Structural Dy-namics vol 34 no 7 pp 705ndash718 2005

[4] O S Bursi and D Wagg Modern Testing Techniques forStructural Systems Dynamics and Control Vol 502 SpringerScience amp Business Media Berlin Germany 2009

[5] O S Bursi Z Wang C Jia and B Wu ldquoMonolithic andpartitioned time integration methods for real-time hetero-geneous simulationsrdquo Computational Mechanics vol 52no 1 pp 99ndash119 2013

[6] X Cai C Yang and Y Yuan ldquoHybrid simulation of seismicresponses of a typical station with a reinforced concretecolumnrdquoApplied Sciences vol 10 no 4 Article ID 1331 2020

[7] Z Chen H Wang H Wang et al ldquoApplication of the hybridsimulation method for the full-scale precast reinforced con-crete shear wall structurerdquo Applied Sciences vol 8 no 2Article ID 252 2018

[8] T Horiuchi M Inoue T Konno and Y Namita ldquoReal-timehybrid experimental system with actuator delay compensa-tion and its application to a piping system with energy ab-sorberrdquo Earthquake Engineering amp Structural Dynamicsvol 28 no 10 pp 1121ndash1141 1999

[9] P A Bonnet C N Lim M S Williams et al ldquoReal-timehybrid experiments with Newmark integration MCSmdouter-loop control and multi-tasking strategiesrdquo EarthquakeEngineering amp Structural Dynamics vol 36 no 1 pp 119ndash1412007

[10] A P Darby M S Williams and A Blakeborough ldquoStabilityand delay compensation for real-time substructure testingrdquoJournal of Engineering Mechanics vol 128 no 12 pp 1276ndash1284 2002

[11] M Nakashima and N Masaoka ldquoReal-time on-line test forMDOF systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 28 no 4 pp 393ndash420 1999

[12] M Ahmadizadeh G Mosqueda and A M ReinhornldquoCompensation of actuator delay and dynamics for real-timehybrid structural simulationrdquo Earthquake Engineering ampStructural Dynamics vol 37 no 1 pp 21ndash42 2008

[13] Z Wang B Wu O S Bursi G Xu and Y Ding ldquoAn effectiveonline delay estimation method based on a simplified physicalsystem model for real-time hybrid simulationrdquo SmartStructures and Systems vol 14 no 6 pp 1247ndash1267 2014

[14] S Strano andM Terzo ldquoActuator dynamics compensation forreal-time hybrid simulation an adaptive approach by meansof a nonlinear estimatorrdquo Nonlinear Dynamics vol 85 no 4pp 2353ndash2368 2016

[15] M I Wallace J Sieber S A Neild D J Wagg andB Krauskopf ldquoStability analysis of real-time dynamic sub-structuring using delay differential equation modelsrdquoEarthquake Engineering amp Structural Dynamics vol 34no 15 pp 1817ndash1832 2005

[16] H Zhou D J Wagg and M Li ldquoEquivalent force controlcombined with adaptive polynomial-based forward predic-tion for real-time hybrid simulationrdquo Structural Control andHealth Monitoring vol 24 no 11 p e2018 2017

[17] Y Chae K Kazemibidokhti and J M Ricles ldquoAdaptive timeseries compensator for delay compensation of servo-hydraulicactuator systems for real-time hybrid simulationrdquo EarthquakeEngineering amp Structural Dynamics vol 42 no 11pp 1697ndash1715 2013

[18] C Chen J M Ricles and T Guo ldquoImproved adaptive inversecompensation technique for real-time hybrid simulationrdquoJournal of Engineering Mechanics vol 138 no 12 pp 1432ndash1446 2012

[19] V Nguyen and U Dorka ldquoPhase lag compensation in real-time substructure testing based on online system identifica-tionrdquo in Proceedings of the 14th World Conference onEarthquake Engineering Beijing China October 2008

[20] X Ning Z Wang C Wang et al ldquoAdaptive feedforward andfeedback compensation method for real-time hybrid simu-lation based on a discrete physical testing system modelrdquoJournal of Earthquake and Engineering 2020

[21] Z Wang G Xu Q Li et al ldquoAn adaptive delay compensationmethod based on a discrete systemmodel for real-time hybridsimulationrdquo Smart Structures and Systems vol 25 no 5pp 569ndash580 2020

[22] J E Carrion and B F Spencer Model-based Strategies forReal-Time Hybrid Testing 1940ndash9826 Newmark StructuralEngineering Laboratory University of Illinois at UrbanaChampaign IL USA 2007

[23] R-Y Jung P Benson Shing E Stauffer and B oenldquoPerformance of a real-time pseudodynamic test systemconsidering nonlinear structural responserdquo Earthquake En-gineering amp Structural Dynamics vol 36 no 12 pp 1785ndash1809 2007

[24] B Wu Z Wang and O S Bursi ldquoActuator dynamicscompensation based on upper bound delay for real-timehybrid simulationrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 12 pp 1749ndash1765 2013

[25] X Gao N Castaneda and S J Dyke ldquoReal time hybridsimulation from dynamic system motion control to

Shock and Vibration 13

experimental errorrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 6 pp 815ndash832 2013

[26] X Ning Z Wang H Zhou B Wu Y Ding and B XuldquoRobust actuator dynamics compensation method for real-time hybrid simulationrdquo Mechanical Systems and SignalProcessing vol 131 pp 49ndash70 2019

[27] G Ou A I Ozdagli S J Dyke and BWu ldquoRobust integratedactuator control experimental verification and real-timehybrid-simulation implementationrdquo Earthquake Engineeringamp Structural Dynamics vol 44 no 3 pp 441ndash460 2015

[28] N Nakata ldquoEffective force testing using a robust loop shapingcontrollerrdquo Earthquake Engineering amp Structural Dynamicsvol 42 no 2 pp 261ndash275 2013

[29] B Wu and H Zhou ldquoSliding mode for equivalent forcecontrol in real-time substructure testingrdquo Structural Controland Health Monitoring vol 21 no 10 pp 1284ndash1303 2014

[30] Z Wang X Ning G Xu et al ldquoHigh performance com-pensation using an adaptive strategy for real-time hybridsimulationrdquo Mechanical Systems and Signal Processingvol 133 Article ID 106262 2019

[31] C E Silva D Gomez A Maghareh et al ldquoBenchmark controlproblem for real-time hybrid simulationrdquoMechanical Systemsand Signal Processing vol 135 Article ID 106381 2020

14 Shock and Vibration

(3) e TADC method exhibits the best tracking ac-curacy to the desired displacements among the threecompensation methods owing to its features

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

e research and publication of this article were funded by theScientific Research Fund of Institute of EngineeringMechanicsChina Earthquake Administration (Grants nos 2018D10 and2020D14) the National Key Research and DevelopmentProgram of China (Grant no 2016YFC0701106) and theNational Natural Science Foundation of China (Grants nos51778190 and 51908231)

References

[1] M Nakashima H Kato and E Takaoka ldquoDevelopment ofreal-time pseudo dynamic testingrdquo Earthquake Engineering ampStructural Dynamics vol 21 no 1 pp 79ndash92 1992

[2] M S Williams and A Blakeborough ldquoLaboratory testing ofstructures under dynamic loads an introductory reviewrdquoPhilosophical Transactions of the Royal Society of LondonSeries A Mathematical Physical and Engineering Sciencesvol 359 no 1786 pp 1651ndash1669 2001

[3] B Wu H Bao J Ou and S Tian ldquoStability and accuracyanalysis of the central difference method for real-time sub-structure testingrdquo Earthquake Engineering amp Structural Dy-namics vol 34 no 7 pp 705ndash718 2005

[4] O S Bursi and D Wagg Modern Testing Techniques forStructural Systems Dynamics and Control Vol 502 SpringerScience amp Business Media Berlin Germany 2009

[5] O S Bursi Z Wang C Jia and B Wu ldquoMonolithic andpartitioned time integration methods for real-time hetero-geneous simulationsrdquo Computational Mechanics vol 52no 1 pp 99ndash119 2013

[6] X Cai C Yang and Y Yuan ldquoHybrid simulation of seismicresponses of a typical station with a reinforced concretecolumnrdquoApplied Sciences vol 10 no 4 Article ID 1331 2020

[7] Z Chen H Wang H Wang et al ldquoApplication of the hybridsimulation method for the full-scale precast reinforced con-crete shear wall structurerdquo Applied Sciences vol 8 no 2Article ID 252 2018

[8] T Horiuchi M Inoue T Konno and Y Namita ldquoReal-timehybrid experimental system with actuator delay compensa-tion and its application to a piping system with energy ab-sorberrdquo Earthquake Engineering amp Structural Dynamicsvol 28 no 10 pp 1121ndash1141 1999

[9] P A Bonnet C N Lim M S Williams et al ldquoReal-timehybrid experiments with Newmark integration MCSmdouter-loop control and multi-tasking strategiesrdquo EarthquakeEngineering amp Structural Dynamics vol 36 no 1 pp 119ndash1412007

[10] A P Darby M S Williams and A Blakeborough ldquoStabilityand delay compensation for real-time substructure testingrdquoJournal of Engineering Mechanics vol 128 no 12 pp 1276ndash1284 2002

[11] M Nakashima and N Masaoka ldquoReal-time on-line test forMDOF systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 28 no 4 pp 393ndash420 1999

[12] M Ahmadizadeh G Mosqueda and A M ReinhornldquoCompensation of actuator delay and dynamics for real-timehybrid structural simulationrdquo Earthquake Engineering ampStructural Dynamics vol 37 no 1 pp 21ndash42 2008

[13] Z Wang B Wu O S Bursi G Xu and Y Ding ldquoAn effectiveonline delay estimation method based on a simplified physicalsystem model for real-time hybrid simulationrdquo SmartStructures and Systems vol 14 no 6 pp 1247ndash1267 2014

[14] S Strano andM Terzo ldquoActuator dynamics compensation forreal-time hybrid simulation an adaptive approach by meansof a nonlinear estimatorrdquo Nonlinear Dynamics vol 85 no 4pp 2353ndash2368 2016

[15] M I Wallace J Sieber S A Neild D J Wagg andB Krauskopf ldquoStability analysis of real-time dynamic sub-structuring using delay differential equation modelsrdquoEarthquake Engineering amp Structural Dynamics vol 34no 15 pp 1817ndash1832 2005

[16] H Zhou D J Wagg and M Li ldquoEquivalent force controlcombined with adaptive polynomial-based forward predic-tion for real-time hybrid simulationrdquo Structural Control andHealth Monitoring vol 24 no 11 p e2018 2017

[17] Y Chae K Kazemibidokhti and J M Ricles ldquoAdaptive timeseries compensator for delay compensation of servo-hydraulicactuator systems for real-time hybrid simulationrdquo EarthquakeEngineering amp Structural Dynamics vol 42 no 11pp 1697ndash1715 2013

[18] C Chen J M Ricles and T Guo ldquoImproved adaptive inversecompensation technique for real-time hybrid simulationrdquoJournal of Engineering Mechanics vol 138 no 12 pp 1432ndash1446 2012

[19] V Nguyen and U Dorka ldquoPhase lag compensation in real-time substructure testing based on online system identifica-tionrdquo in Proceedings of the 14th World Conference onEarthquake Engineering Beijing China October 2008

[20] X Ning Z Wang C Wang et al ldquoAdaptive feedforward andfeedback compensation method for real-time hybrid simu-lation based on a discrete physical testing system modelrdquoJournal of Earthquake and Engineering 2020

[21] Z Wang G Xu Q Li et al ldquoAn adaptive delay compensationmethod based on a discrete systemmodel for real-time hybridsimulationrdquo Smart Structures and Systems vol 25 no 5pp 569ndash580 2020

[22] J E Carrion and B F Spencer Model-based Strategies forReal-Time Hybrid Testing 1940ndash9826 Newmark StructuralEngineering Laboratory University of Illinois at UrbanaChampaign IL USA 2007

[23] R-Y Jung P Benson Shing E Stauffer and B oenldquoPerformance of a real-time pseudodynamic test systemconsidering nonlinear structural responserdquo Earthquake En-gineering amp Structural Dynamics vol 36 no 12 pp 1785ndash1809 2007

[24] B Wu Z Wang and O S Bursi ldquoActuator dynamicscompensation based on upper bound delay for real-timehybrid simulationrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 12 pp 1749ndash1765 2013

[25] X Gao N Castaneda and S J Dyke ldquoReal time hybridsimulation from dynamic system motion control to

Shock and Vibration 13

experimental errorrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 6 pp 815ndash832 2013

[26] X Ning Z Wang H Zhou B Wu Y Ding and B XuldquoRobust actuator dynamics compensation method for real-time hybrid simulationrdquo Mechanical Systems and SignalProcessing vol 131 pp 49ndash70 2019

[27] G Ou A I Ozdagli S J Dyke and BWu ldquoRobust integratedactuator control experimental verification and real-timehybrid-simulation implementationrdquo Earthquake Engineeringamp Structural Dynamics vol 44 no 3 pp 441ndash460 2015

[28] N Nakata ldquoEffective force testing using a robust loop shapingcontrollerrdquo Earthquake Engineering amp Structural Dynamicsvol 42 no 2 pp 261ndash275 2013

[29] B Wu and H Zhou ldquoSliding mode for equivalent forcecontrol in real-time substructure testingrdquo Structural Controland Health Monitoring vol 21 no 10 pp 1284ndash1303 2014

[30] Z Wang X Ning G Xu et al ldquoHigh performance com-pensation using an adaptive strategy for real-time hybridsimulationrdquo Mechanical Systems and Signal Processingvol 133 Article ID 106262 2019

[31] C E Silva D Gomez A Maghareh et al ldquoBenchmark controlproblem for real-time hybrid simulationrdquoMechanical Systemsand Signal Processing vol 135 Article ID 106381 2020

14 Shock and Vibration

experimental errorrdquo Earthquake Engineering amp StructuralDynamics vol 42 no 6 pp 815ndash832 2013

[26] X Ning Z Wang H Zhou B Wu Y Ding and B XuldquoRobust actuator dynamics compensation method for real-time hybrid simulationrdquo Mechanical Systems and SignalProcessing vol 131 pp 49ndash70 2019

[27] G Ou A I Ozdagli S J Dyke and BWu ldquoRobust integratedactuator control experimental verification and real-timehybrid-simulation implementationrdquo Earthquake Engineeringamp Structural Dynamics vol 44 no 3 pp 441ndash460 2015

[28] N Nakata ldquoEffective force testing using a robust loop shapingcontrollerrdquo Earthquake Engineering amp Structural Dynamicsvol 42 no 2 pp 261ndash275 2013

[29] B Wu and H Zhou ldquoSliding mode for equivalent forcecontrol in real-time substructure testingrdquo Structural Controland Health Monitoring vol 21 no 10 pp 1284ndash1303 2014

[30] Z Wang X Ning G Xu et al ldquoHigh performance com-pensation using an adaptive strategy for real-time hybridsimulationrdquo Mechanical Systems and Signal Processingvol 133 Article ID 106262 2019

[31] C E Silva D Gomez A Maghareh et al ldquoBenchmark controlproblem for real-time hybrid simulationrdquoMechanical Systemsand Signal Processing vol 135 Article ID 106381 2020

14 Shock and Vibration