Large-scale real-time hybrid simulation involving multiple experimental substructures and adaptive...

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2012; 41:549–569Published online 20 July 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/eqe.1144

Large‐scale real‐time hybrid simulation involving multipleexperimental substructures and adaptive actuator

delay compensation

Cheng Chen1,*,†,‡ and James M. Ricles2,§

1School of Engineering, San Francisco State University, San Francisco, CA 94132, U.S.A.2ATLSS Engineering Research Center, Bethlehem, PA 18015, U.S.A.

SUMMARY

Real‐time hybrid simulation provides a viable method to experimentally evaluate the performance ofstructural systems subjected to earthquakes. The structural system is divided into substructures, where part ofthe system is modeled by experimental substructures, whereas the remaining part is modeled analytically.The displacements in a real‐time hybrid simulation are imposed by servo‐hydraulic actuators to theexperimental substructures. Actuator delay compensation has been shown by numerous researchers to vitallyachieve reliable real‐time hybrid simulation results. Several studies have been performed on servo‐hydraulicactuator delay compensation involving single experimental substructure with single actuator. Research onreal‐time hybrid simulation involving multiple experimental substructures, however, is limited. The effect ofactuator delay during a real‐time hybrid simulation with multiple experimental substructures presentschallenges. The restoring forces from experimental substructures may be coupled to two or more degrees offreedom (DOF) of the structural system, and the delay in each actuator must be adequately compensated. Thispaper first presents a stability analysis of actuator delay for real‐time hybrid simulation of a multiple‐DOFlinear elastic structure to illustrate the effect of coupled DOFs on the stability of the simulation. An adaptivecompensation method then proposed for the stable and accurate control of multiple actuators for a real‐timehybrid simulation. Real‐time hybrid simulation of a two‐story four‐bay steel moment‐resisting frame withlarge‐scale magneto‐rheological dampers in passive‐on mode subjected to the design basis earthquake isused to experimentally demonstrate the effectiveness of the compensation method in minimizing actuatordelay in multiple experimental substructures. Copyright © 2011 John Wiley & Sons, Ltd.

Received 11 October 2010; Revised 30 March 2011; Accepted 18 April 2011

KEY WORDS: real‐time hybrid simulation; actuator delay; adaptive compensation; multiple experimentalsubstructures; servo‐hydraulic control; damper

1. INTRODUCTION

Real‐time hybrid simulation has been considered as a viable and economical method for experimentallyinvestigating the dynamic response of structural systems to earthquakes [1]. The method divides astructural system into experimental substructure(s) and analytical substructure(s), enabling the completestructural system to be considered in the simulation. During a real‐time hybrid simulation, thedisplacement response of the structural system is calculated by using an integration algorithm to solvethe dynamic equations of motion. The calculated displacement response is transformed to commanddisplacements for actuators and is imposed onto the experimental substructure(s). The integration

*Correspondence to: ChengChen, School of Engineering, San Francisco StateUniversity, San Francisco, CA94132,U.S.A.†E‐mail: [email protected]‡Assistant professor.§Bruce G. Johnston Professor.

Copyright © 2011 John Wiley & Sons, Ltd.

550 C. CHEN AND J. M. RICLES

algorithm requires the use of restoring forces that are developed in the substructures. In the experimentalsubstructure(s) the restoring forces are measured by load cells attached to the actuators. To realisticallysimulate the response of an entire structure, it is necessary to maintain the compatibility and equilibriumat the interface between the experimental and analytical substructures. This poses challenges foractuator control during a real‐time hybrid simulation. Inherent hydraulic dynamics exist in servo‐hydraulic actuators, which induce an inevitable time delay in the actuator response to the command(referred to hereafter as actuator delay). Actuator delay leads to a de‐synchronization between thedisplacement response from the integration algorithm and the measured restoring forces.

The effect of actuator delay on real‐time hybrid simulation has been studied by a number ofresearchers [2, 3]. Wallace et al. [4] and Mercan and Ricles [5] performed stability analysis of real‐timehybrid simulation of a single‐degree‐of‐freedom (SDOF) linear elastic system using a delay differentialequation to model the effects of actuator delay in the experimental substructure. Chen and Ricles [6]introduced discrete control theory to include explicit integration algorithms in the stability analysis andinvestigated the effect of actuator delay on the entire real‐time hybrid simulation system. These studiesshow that actuator delay introduces negative damping into the system, which, if not compensatedproperly, can destabilize a real‐time hybrid simulation. Various compensation methods have beenproposed to minimize the effect of actuator delay for real‐time testing. Horiuchi et al. [3] and Horiuchiand Konno [7] proposed two compensation schemes based on polynomial extrapolation and a linearacceleration assumption, respectively. Both methods intend to calculate predicted displacements so thatthe actuator can achieve the correct command displacements under actuator delay, where the formerassumes a second‐order polynomial relation for the command displacements and the latter utilizes theNewmark family of integration algorithms [8]. Other compensation methods originating from controlengineering practice have also been investigated, where the servo‐hydraulic system is treated as a time(phase) delay system and delay compensation methods such as phase lead compensator [9] orderivative feedforward [10, 11] are introduced. The determination of compensation parameters forthese two methods is usually based on an analysis in the frequency domain. Chen et al. [12] proposed asimplified discrete transfer function model for the servo‐hydraulic actuator and applied the inverse ofthe model to derive an actuator delay compensator for real‐time hybrid simulation.

Chen and Ricles [13] proposed an equivalent discrete transfer function approach to analyze actuatordelay compensation methods. Through the use of the discrete z‐transform from control theory,compensation methods for actuator delay were shown by Chen and Ricles to be an extrapolation in thetime domain or as an equivalent transfer function in the frequency domain. The performance of acompensation method can be analyzed through a frequency response analysis of the equivalent transferfunction for the compensation method. The aforementioned delay compensation methods have beenexperimentally demonstrated to be able to reduce the effect of actuator dynamics when the actuatordelay is accurately identified and does not vary much during a real‐time hybrid simulation. However,this may not always be the case because experimental substructures and various ground motions mightinduce different structural response, where the actuator delay may not be accurately identified andcould also vary during a simulation. Darby et al. [14] performed a series of real‐time hybridsimulations with different experimental substructures and found that actuator delay is dependent on theexperimental substructure stiffness. The compensation methods developed for constant delay thereforemight result in an overcompensation or undercompensation due to error in the estimated amount ofactuator delay used to establish the values for the compensation parameter(s) and any variability inactuator delay that may occur during the simulation due to a change in the stiffness of the experimentalsubstructure. These effects can lead to inaccurate simulation results. To improve the performance ofthe inverse compensation method for real‐time hybrid simulation, Chen and Ricles [15] proposed adual compensation technique, where the actuator control error is utilized as an auxiliary signal tominimize the effect of overcompensation or undercompensation for actuator delay.

Compensation methods based on adaptive control theory have also been proposed for real‐timehybrid simulations with variable actuator delay. Darby et al. [14] proposed an online procedure toestimate and compensate actuator delay during a real‐time hybrid simulation using a proportionalfeedback system. Bonnet et al. [16] applied adaptive minimal control synthesis to real‐time hybridsimulation, where this control method is a model reference adaptive controller. Carrion and Spencer [17]used a combined feedforward–feedback controller in conjunction with inverse modeling and bumpless

Copyright © 2011 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2012; 41:549–569DOI: 10.1002/eqe

REAL‐TIME HYBRID SIMULATION WITH MULTIPLE EXPERIMENTAL SUBSTRUCTURES 551

transfer in a real‐time hybrid simulation to compensate for variable actuator delay that occurswhen the experimental substructures include semi‐active controlled dampers. To minimize the effectof inaccurately estimated or variable actuator delay during a real‐time hybrid simulation, Chen andRicles [18] developed an adaptive inverse compensation (AIC) method based on the inversecompensation method. The real‐time hybrid simulation of an SDOF system with an elastomeric damperexperimentally demonstrated that the AIC method can effectively minimize the effect of variable orinaccurately estimated actuator delay for simulations involving a single actuator.

The developments on actuator delay compensation as well as advances in integration algorithmshave made real‐time hybrid simulation an appealing experimental technique and led to significantprogress in its application to structural engineering research. This is particularly true in thedevelopment and experimental validation of performance‐based design procedures for structures withrate‐dependent seismic devices [19, 20]. More than one rate‐dependent device is often used in astructure, requiring multiple experimental substructures and therefore multiple servo‐hydraulicactuators in a real‐time hybrid simulation. The experimental substructures often will have DOFs thatare linked to common DOFs of the structural system; therefore, the restoring forces at some of theDOFs of the structural system are coupled to two or more of the experimental substructures. In thesesimulations, the actuators are connected to different experimental substructures (i.e., rate‐dependentseismic devices) and are not physically coupled. Real‐time hybrid simulations involving physicallycoupled actuators have also been investigated by other researchers [10, 11]. A compensation methodthat can minimize the effect of actuator delay for multiple experimental substructures coupled throughthe structural system’s DOF is therefore necessary. This paper first discusses a stability analysis ofreal‐time hybrid simulation of a linear elastic structure to illustrate the effect of coupled DOFs. TheAIC method is then reviewed and applied to real‐time hybrid simulations involving multipleexperimental substructures to validate the AIC method for simulations involving multipleexperimental substructures with servo‐hydraulic actuators.

2. EFFECT OF ACTUATOR DELAY ON REAL‐TIME HYBRID SIMULATION OF MDOFSYSTEMS: ANALYTICAL CASE STUDY

The effect of actuator delay on the real‐time hybrid simulation of an MDOF system is first analyticallyassessed. The assessment is conducted by extending the discrete transfer function approach proposedby Chen and Ricles [6] to an MDOF linear elastic structure. For the purpose of illustrating the effectsof actuator delay in an MDOF system, the linear elastic two‐DOF system shown in Figure 1(a) isselected, where the mass M and stiffness matrices K are defined as

M ¼ 1 00 1

� �m; K ¼ k1e þ k1a þ k2e þ k2a −k2e−k2a

−k2e−k2a k2e þ k2a

� �(1)

mass m

mass m

DOF 2

DOF 1

story stiffness k

story stiffness k

spring in the first story

spring in the second story

mass m

mass m

substructure stiffness k2e

substructure stiffness k1e

(c)

substructurestiffness k2a

substructurestiffness k1a

x2, +1i

x1, +1i

d =x -x+1i

c(2)2, +1i 1, +1i

d =x+1i

c(1)1, +1i

x2, +1i

x1, +1i

(b)(a)

Figure 1. Linear elastic two‐DOF structure for theoretical investigation of stability limits associated withactuator delay in real‐time hybrid simulations with multiple experimental substructures: (a) prototype

structure, (b) experimental substructures, and (c) analytical substructure.



In Equation (1), k1e and k1a are the experimental and analytical substructure stiffness for the firststory, respectively; k2e and k2a are the experimental and analytical substructure stiffness for the secondstory, respectively. For real‐time hybrid simulation, the structural elements associated with k1e and k2eare isolated and appear in the experimental substructures (see Figure 1(b)), whereas the rest of thestructure is modeled as the analytical substructure (see Figure 1(c)). The braces are assumed to beaxially rigid.

For the analytical case study, the story mass m is selected to be 5E + 5 kg, and the story stiffnessk1e+ k1a and k2e + k2a are selected to be both equal to 5E + 7 kN/m. The resulting natural periods of thestructure are 1.02 and 0.39 s for the first and the second modes, respectively. Furthermore, k1e = k1a andk2e= k2a are used in the case study, which is similar to that of the structure in the hybrid simulationsperformed in the laboratory and discussed later. The structure is assumed to have an inherent dampingof 2% Rayleigh damping for both the first and the second modes. The equations of motion for thesystem can be formulated as

M⋅ ::x tð Þ þ C⋅ :x tð Þ þK⋅ x tð Þ ¼ F tð Þ (2a)

where C is the inherent viscous damping matrix of the structure; x(t), :x tð Þ, and ::x tð Þ are thedisplacement, velocity, and acceleration response vectors, respectively; and F(t) is the predefinedexternal excitation force vector. A numerical integration algorithm is utilized in a real‐time hybridsimulation to solve the temporally discretized form of Equation (2a), which can be expressed as

M⋅::xiþ1 þ C⋅:xiþ1 þK⋅ xiþ1 ¼ Fiþ1 (2b)

where xi+ 1,:xiþ1,

::xiþ1, Fi+ 1 are the displacement, velocity, acceleration, and excitation force vectors attime step i+ 1, respectively. Chen and Ricles [6] showed that the properties of an explicit integrationalgorithm will affect the stability of a real‐time hybrid simulation of SDOF systems with actuatordelay. In the present study, the unconditionally stable explicit CR integration algorithm [21] is selectedfor the analysis because the CR algorithm is also used in the experimental study presented later. Thevariation of the displacement and velocity over the time step for the CR integration algorithm aredefined as

:xiþ1 ¼ :xi þ Δt⋅β1⋅::xi (3a)

xiþ1 ¼ xi þ Δt⋅ :xi þ Δt2⋅β2⋅::xi (3b)

To attain unconditional stability for the CR integration algorithm for a linear elastic structure, theintegration parameter matrices β1 and β2 are defined as

β1 ¼ β2 ¼ 4⋅ 4⋅Mþ 2⋅Δt⋅Cþ Δt2⋅K� �−1⋅M (3c)

where Δt is the integration time step; β1 and β2 are the matrices of integration parameters; and xi,:xi,

and ::xi are the displacement, velocity, and acceleration at time step i, respectively.The command displacement dc jð Þ

iþ1 for the jth actuator at time step i + 1 is determined from thedisplacement response vector xi+ 1 that is calculated by the integration algorithm by relatingthe actuator DOF to the structure’s DOFs. For the stability analysis, the command displacement for theactuator of the experimental substructure for the first‐story damper, dc 1ð Þ

iþ1 , has a command displacementthat is equal to the displacements of DOF 1 inxi+ 1, whereas the command displacement for theactuator of the experimental substructure for the second‐story damper, dc 2ð Þ

iþ1 , is the difference betweenthe displacements in xi+ 1 associated with DOF 2 and DOF 1 (see Figure 1(b)).

The vector of restoring forces ri + 1 associated with DOFs 1 and 2 for the structure at time step i + 1are analytically computed based on the stiffness matrix K of the system and the vector of measuredactuator displacements dmiþ1.The measured actuator displacements are simulated by inducing aspecified amount of delay with respect to the actuator command displacements dc jð Þ

iþ1 for each actuator.



By determining the restoring forces analytically, the equation of motion for real‐time hybridsimulation can be rewritten as

M⋅::xiþ1 þ C⋅:xiþ1 þKa⋅xiþ1 þKe⋅dmiþ1 ¼ Fiþ1 (4)

where Ka and Ke the stiffness matrices relating to the analytical and experimental substructures,respectively; the vector dmiþ1 can be expressed as [dm 1ð Þ

iþ1 ; dm 1ð Þiþ1 þ dm 2ð Þ

iþ1 ]T, where dm 1ð Þiþ1 and dm 2ð Þ

iþ1 are themeasured displacements for the two actuators, respectively.

Actuator delay between dciþ1 and dmiþ1 is modeled using the approach by Chen and Ricles [6], where

a first‐order discrete transfer function is used to model servo‐hydraulic actuator response with actuatordelay. As will be explained later, in the laboratory, the command displacements for an actuator areinterpolated in between time steps using a ramp function in order to smoothly impose the commanddisplacement onto the test specimen. The substep size is typically equal to the sampling time of theservo‐hydraulic controller. For a single actuator j,

dm jð Þiþ1;q ¼ dm jð Þ

iþ1;q−1 þ1αj⋅ dc jð Þ

iþ1;q−dm jð Þiþ1;q−1

� �(5a)

In Equation (5a) dm jð Þiþ1;q and dm jð Þ

iþ1;q−1are the measured displacements at substeps q and q− 1 of the(i+ 1)th time step for the jth actuator, respectively; dc jð Þ

iþ1;q is the actuator command displacement forsubstep q of time step (i+ 1); and αj is the delay constant for the jth actuator defined as the ratio td / δt,where td is the actuator response time to achieve the target command displacement for the substep andδt the servo‐hydraulic controller sampling time (see Figure 2). For modern servo‐hydraulic controllers,1/1024 s is the sampling time (which is associated with a controller sampling speed of 1024Hz) and isthe value used for δt in this paper. From Figure 2, it is apparent that the jth actuator delay is equal to(αj− 1)δt. Applying the discrete z‐transform [22, 23] to Equation (5a) leads to the discrete transferfunction G jð Þ

d zð Þ relating the measured response to the command displacement for actuator j:

G jð Þd zð Þ ¼ Dm jð Þ zð Þ

Dc jð Þ zð Þ ¼ z

αj⋅z− αj−1� (5b)

InEquation (5b) z is the complexvariable in thediscrete z‐domain, andDm( j)(z) andDc( j)(z) are the discretez‐transforms of dm jð Þ

iþ1;q and dc jð Þiþ1;q, respectively. j is an index number for the actuator, which ranges from1 to

2 for the selected two‐DOF system in the present study.

di q+1,c ( )j

di q+1, -1m ( )j

di q+1,m ( )j

o Time

Displacement

t

t i q+1, -1 t i q+,

Command fromramp generator

Actuator response

td = tj

Figure 2. Actuator response with a time delay.



Applying the discrete z‐transformation to Equation (4) and substituting the measured displacementwith delay from Equation (5b) and the z‐transform of the acceleration ::xiþ1 into the result leads to theclosed‐loop discrete transfer function matrix Gcl(z). The z‐transform for the acceleration ::xiþ1 isobtained by rearranging and solving for ::xi from Equation (3a) and then using the real translationtheorem [22]. For the analytical case study, no substeps are necessary, permitting the use of the abovez‐transforms along with the application of the real translation theorem to obtain Gcl(z). Gcl(z) relatesthe z‐transform of the state vector response for displacements and velocities, X(z), to the z‐transform ofthe input force F(z) for the two‐DOF system, where

X zð Þ ¼X1 zð ÞX2 zð ÞX1 zð ÞX2 zð Þ

2664

3775 ¼

G11 zð ÞG21 zð ÞG31 zð ÞG41 zð Þ

G12 zð ÞG22 zð ÞG32 zð ÞG42 zð Þ

2664

3775⋅ F1 zð Þ

F2 zð Þ� �

¼ Gcl zð Þ⋅F zð Þ (6a)

Elements in Gcl(z), Gij(z) where i= 1 to 4; j =1 to 2, can be written in a general form as

Gij zð Þ ¼ np⋅zp þ⋯þ n1⋅zþ n0dq⋅zq þ⋯þ d1⋅zþ d0

(6b)

The corresponding characteristic equation is the common denominator among the elements Gij(z),and can be written in the following form:

a6⋅z6 þ a5⋅z5 þ a4⋅z4 þ a3⋅z3 þ a2⋅z2 þ a1⋅zþ a0 ¼ 0 (6c)

In Equation (6c), ai (i= 0 to 6) are coefficients expressed in terms of m, k, Δt, α1, and α2 as well ascomponents of the viscous damping matrix C (the characteristic equation is expanded where the termsin the matrices of integration parameters, β1 and β2, are expressed in Equation (6c) in terms of m, k1a,k2a k1e, k2e, c, and Δt from Equation (3c)). The coefficients ai (i= 0 to 6) are derived and tabulated inTable I for the case of zero damping for the two‐DOF system. The characteristic equation in Equation(6c) can be observed to have six roots for the complex variable z, four of which represent the apparentfrequencies of the prototype structure. The remaining two are spurious roots associated with actuatordelay.

The stability of the simulation with actuator delay can be investigated through the solutions for z ofEquation (6c), where the simulation remains stable if all the solutions of z have a magnitude less thanor equal to one. Otherwise, the simulation becomes unstable. For each value of α1 in Equation (6c), acorresponding value for α2 can be determined that results in the largest magnitude for z being equal to1.0, whereby the values for α1 and α2 are associated with the stability limit. Figure 3 shows thestability limit surface for α1 and α2 for the two‐DOF linear elastic system when multiple actuatordelays exist in the simulation. The time step for the integration algorithm is 0.01 s. For the selectedstructural properties (which were given previously), the stability limit surface is seen to have a linearrelationship between α1 and α2. The value for α1 decreases associated with the stability limit surfacewith an increase in α2 and vice versa. Figure 3 also indicates that when delay exists in both actuators,the stability limits for α1 and α2 are smaller than the limits when delay exists in only one actuator,implying that actuator delay in multiple experimental substructures can have a more detrimental effecton the stability of a real‐time hybrid simulation than delay in only a single actuator.

3. ADAPTIVE INVERSE COMPENSATION FOR ACTUATOR DELAY NEGATION

To minimize the destabilizing effect of actuator delay in multiple experimental substructures in a real‐time hybrid simulation, the AIC method developed by Chen and Ricles [18] is experimentallyevaluated in this study for multiple actuators. The AIC method is based on inverting Equation (5b) and


Table I. Coefficients of characteristic equation for undamped case.

Coefficient Expression

a6 Δt4k2ak1eα1α2 þ 16m2α1α2 þ Δt4k1ak2eα1α2 þ Δt4k1ek2eα1α2 þ 4Δt2k1amα1α2þ8⋅Δt2mα1k2eα2 þ Δt4k1ak2aα1α2 þ 4Δt2k1emα1α2 þ 8Δt2mα1k2aα2

a5

−6Δt4k1ek2eα1α2−6Δt4k2ak1eα1α2−24Δt2k1amα1α2−96m2α1α2−48Δt2mα1k2eα2−48Δt2mα1k2aα2−6Δt4k1ak2aα1α2 þ 24mΔt2k2eα1 þ 8mΔt2k2aα1 þ 4Δt2k1emα1

þ5Δt4k1ek2eα1 þ Δt4k1ek2aα1 þ 4Δt2k1amα1 þ 5Δt4k1ak2eα1 þ Δt4k1ak2aα1 þ 24mΔt2k2eα2 þ 8mΔt2k2aα2þ20Δt2k1emα2 þ 5Δt4k1ek2eα2 þ 5Δt4k1ek2aα2 þ 4Δt2k1amα2 þ Δt4k1ak2eα2Δt4k1ak2aα2 þ 16m2α1 þ 16m2α2

a4

16m2 þ 15Δt4k1ek2eα1α2 þ 15Δt4k2ak1eα1α2 þ 60Δt2k1amα1α2 þ 240m2α1α2 þ 120Δt2mα1k2eα2þ120Δt2mα1k2aα2 þ 60Δt2k1emα1α2 þ 15Δt4k1ak2eα1α2 þ 15Δt4k1ak2aα1α2 þ 40mΔt2k2e þ 8mΔt2k2a

þ20Δt2k1emþ 25Δt4k1ek2e þ 5Δt4k1ek2a þ 4Δt2k1amþ 5Δt4k1ak2e þ Δt4k1ak2a−88mΔt2k2eα1−40mΔt2k2aα1−20Δt2k1emα1−17Δt4k1ek2eα1−5Δt4k1ek2aα1−20Δt2k1amα1−17Δt4k1ak2eα1−5Δt4k1ak2aα1

−88mΔt2k2eα2−40mΔt2k2aα2−68Δt2k1emα2−17Δt4k1ek2eα2−17Δt4k1ek2aα2−20Δt2k1amα2−5Δt4k1ak2eα2−5Δt4k1ak2aα2−80m2α1−80m2α2

a3

−64m2−20Δt4k1ek2eα1α2−20Δt4k2ak1eα1α2−80Δt2k1amα1α2−320m2α1α2−160Δt2mα1k2eα2−160Δt2mα1k2aα2−80Δt2k1emα1α2−20Δt4k1ak2eα1α2−20Δt4k1ak2aα1α2−96mΔt2k2e−32mΔt2k2a−48Δt2k1em−20Δt4k1ek2e−12Δt4k1ek2a−16Δt2k1am−12Δt4k1ak2e−4Δt4k1ak2a þ 128mΔt2k2eα1

þ80mΔt2k2aα1 þ 40Δt2k1emα1 þ 22Δt4k1ek2eα1 þ 10Δt4k1ek2aα1 þ 40Δt2k1amα1 þ 22Δt4k1ak2eα1þ10Δt4k1ak2aα1 þ 128mΔt2k2eα2 þ 80mΔt2k2aα2 þ 88Δt2k1emα2 þ 22Δt4k1ek2eα2 þ 22Δt4k1ek2aα2

þ40Δt2k1amα2 þ 10Δt4k1ak2eα2 þ 10Δt4k1ak2aα2 þ 160m2α1 þ 160m2α2

a2

96m2 þ 15Δt4k1ek2eα1α2 þ 15Δt4k2ak1eα1α2 þ 60Δt2k1amα1α2 þ 240m2α1α2 þ 120Δt2mα1k2eα2þ120Δt2mα1k2aα2 þ 60Δt2k1emα1α2 þ 15Δt4k1ak2eα1α2 þ 15Δt4k1ak2aα1α2 þ 80mΔt2k2e

þ48mΔt2k2a þ 40Δt2k1emþ 14Δt4k1ek2e þ 10Δt4k1ek2a þ 24Δt2k1amþ 10Δt4k1ak2eþ6Δt4k1ak2a−96mΔt2k2eα1−80mΔt2k2aα1−40Δt2k1emα1−14Δt4k1ek2eα1−10Δt4k1ek2aα1−40Δt2k1amα1−14Δt4k1ak2eα1−10Δt4k1ak2aα1−96mΔt2k2eα2−80mΔt2k2aα2−56Δt2k1emα2

−14Δt4k1ek2eα2−14Δt4k1ek2aα2−40Δt2k1amα2−10Δt4k1ak2eα2−10Δt4k1ak2aα2−160m2α1−160m2α2

a1

−64m2−6Δt4k1ek2eα1α2−6Δt4k2ak1eα1α2−24Δt2k1amα1α2−96m2α1α2−48Δt2mα1k2eα2−48Δt2mα1k2aα2−24Δt2k1emα1α2−6Δt4k1ak2eα1α2−6Δt4k1ak2aα1α2−32mΔt2k2e

−32mΔt2k2a−16Δt2k1em−4Δt4k1ek2e−4Δt4k1ek2a−16Δt2k1am−4Δt4k1ak2e−4Δt4k1ak2a þ 40mΔt2k2eα1 þ 40mΔt2k2aα1 þ 20Δt2k1emα1 þ 5Δt4k1ek2eα1 þ 5Δt4k1ek2aα1

þ20Δt2k1amα1 þ 5Δt4k1ak2eα1 þ 5Δt4k1ak2aα1 þ 40mΔt2k2eα2 þ 40mΔt2k2aα2 þ 20Δt2k1emα2þ5Δt4k1ek2eα2 þ 5Δt4k1ek2aα2 þ 20Δt2k1amα2 þ 5Δt4k1ak2eα2 þ 5Δt4k1ak2aα2 þ 80m2α1 þ 80m2α2

a0

16m2 þ Δt4k1ek2eα1α2 þ Δt4k2ak1eα1α2 þ 4Δt2k1amα1α2 þ 16m2α1α2 þ 8Δt2mα1k2eα2 þ 8Δt2mα1k2aα2þ4Δt2k1emα1α2 þ Δt4k1ak2eα1α2 þ Δt4k1ak2aα1α2 þ 8mΔt2k2e þ 8mΔt2k2a þ 4Δt2k1em

þΔt4k1ek2e þ Δt4k1ek2a þ 4Δt2k1amþ Δt4k1ak2e þ Δt4k1ak2a−8mΔt2k2eα1−8mΔt2k2aα1−4Δt2k1emα1−Δt4k1ek2eα1−Δt4k1ek2aα1−4Δt2k1amα1−Δt4k1ak2eα1−Δt4k1ak2aα1−8mΔt2k2eα2−8mΔt2k2aα2

−4Δt2k1emα2−Δt4k1ek2eα2−Δt4k1ek2aα2−4Δt2k1amα2−Δt4k1ak2eα2−Δt4k1ak2aα2−16m2α1−16m2α2


expressing the actuator delay as a variable, whereby for actuator j, the discrete transfer function G jð Þc zð Þ,

relating the z‐transform of the compensated actuator command displacement Dp jð Þ zð Þ to the measuredactuator command displacement Dm jð Þ zð Þ, is expressed as follows:

G jð Þc zð Þ ¼ Dp jð Þ zð Þ

Dm jð Þ zð Þ ¼αes;j þ Δαj�

⋅z− αes;j þ Δαj−1� z

(7a)

In Equation (7a), αes,j is the initial estimate for the actuator delay constant αj, and Δαj is anevolutionary variable for actuator j. Δαj has an initial value of zero, whose value over time t isdetermined using the following adaptive control law:

Δαj tð Þ ¼ kp;j⋅TI jð ÞðtÞ þ ki;j⋅ ∫t

0TI jð Þ τð Þdτ (7b)


Figure 3. Effect of multiple actuator delay on real‐time hybrid simulation of two‐DOF structure.


In Equation (7b), kp, j and ki, j are proportional and integrative gains of the adaptive control law foractuator j, respectively, and TI(j) is the tracking indicator for actuator j based on the enclosed area ofthe hysteresis in the synchronized subspace plot, as shown in Figure 4.

As noted previously, the command displacements are interpolated and sent in substeps to theactuator servo‐hydraulic controller using a ramp function. The size of the substep is associated withdividing the time step Δt by the number of substeps. The compensation for actuator delay is applied tothe actuator command displacement command of each substep sent to the servo‐controller. The

command displacement dc jð Þiþ1;q for actuator j at substep q is as follows:

dc jð Þiþ1;q ¼

q

n⋅ dc jð Þ

iþ1−dc jð Þi

� �þ dc jð Þ

i (8)

In Equation (8), q is the index for the interpolation substep of the ramp generator within one singletime step, which ranges from 1 to n, where n is the number of substeps and equal to the integer ratio ofΔt / δt. dc jð Þ

iþ1 and dc jð Þi in Equation (8) are the command displacements for the jth actuator of the (i + 1)th

and ith time steps, respectively. The calculation of TI(j) for each actuator at the qth substep of the(i + 1)th time step can be formulated as [24]

d i+ ,q+1 1c

d

dm

d i+ ,q1c

dm

di+ ,q1m j( )

dAi+ ,q1

dTAi+1,q

( )j

( )j

i+ ,q+1 1

( )j

( )j

( )j

c

Figure 4. Hysteresis developed in synchronized subspace plot from actuator delay.



TI jð Þiþ1;q ¼ 0:5 A jð Þ

iþ1;q−TAjð Þ

iþ1;q

� �(9a)

A jð Þiþ1;q ¼ A jð Þ

iþ1;q−1 þ dA jð Þiþ1;q ¼ A jð Þ

iþ1;q−1 þ 0:5 dc jð Þiþ1;q þ dc jð Þ

iþ1;q−1

� �dm jð Þiþ1;q−d

m jð Þiþ1;q−1

� �(9b)

TA jð Þiþ1;q ¼ TA jð Þ

iþ1;q−1 þ dTA jð Þiþ1;q ¼ TA jð Þ

iþ1;q−1 þ 0:5 dm jð Þiþ1;q þ dm jð Þ

iþ1;q−1

� �dc jð Þiþ1;q−d

c jð Þiþ1;q−1

� �(9c)

where dm jð Þiþ1;q, d

m jð Þiþ1;q−1, and dc jð Þ

iþ1;q were defined previously and dc jð Þiþ1;q−1 is the command displacement of

actuator j at the (q− 1)th substep of time step (i + 1).When αes,j is set equal to 1.0 and kp, jand ki, j are both set equal to zero, no actuator delay

compensation occurs in the associated actuator during the real‐time hybrid simulation. Chen andRicles [18] showed that for real‐time hybrid simulations involving a single servo‐hydraulic actuatorand experimental substructure that the selection of kp = 0.4 and ki = 0.1kp was found to enable goodadaptation with a small control error. When the AIC method is applied for real‐time hybrid simulationsinvolving multiple experimental substructures, the integrative gain ki, j is also selected to be one‐tenthof the proportional gain kp,j for an actuator.

4. PROTOTYPE STRUCTURE FOR REAL‐TIME HYBRID SIMULATION CASE STUDIES

The prototype building shown in Figure 5 was used for the study reported in this paper. The building isa two‐story, six‐bay by six‐bay structure with four identical perimeter steel moment‐resisting frames(MRFs). The elevation of a typical perimeter MRF shown in Figure 5(b), where the MRF is four bays(the remaining two bays along each side of the building have beam‐to‐column gravity connections anddo not contribute to the lateral load resistance of the building). As indicated in Figure 5(b), 220 kNcapacity magneto‐rheological (MR) fluid dampers are located in the interior bays of the MRFs. Thebuilding is assumed to be located on a stiff soil site near Los Angeles. The prototype building wasdesigned using the procedure of Lee et al. [25], where the expected maximum story drift under thedesign basis earthquake [26] is limited to 1.7% by using 6 and 4 MR fluid dampers in passive mode inthe first and second stories of the MRF, respectively. Earthquake ground motions were considered inonly one direction. Taking advantage of symmetry, only one of the MRFs and the associated gravityframes and seismic mass within the tributary area of the MRF are considered in the hybrid simulations.For the real‐time hybrid simulation, the DOF associated with the equations of motion include in‐planehorizontal and vertical translations at each node of the structure, as well as rotations about the normalto the plane of the structure. At each floor level, the horizontal translational DOFs of the MRF andgravity frames were constrained to be equal to simulate a rigid floor diaphragm. The properties of the

(a)

Mom

ent-

resi

stin

g fr

ame

Mom

ent-

resi

stin

g fr

ame

6@9.

15=

54.9

m

Moment-resisting frame

Moment-resisting frame

dampers

dampers

dampers

dampers 4.57

m

3.96

m

(b)

Prototype4 @ 9.15m = 36.6m

Figure 5. Prototype building (a) plan view; (b) perimeter moment‐resisting frame with dampers and braces.



MRF without dampers are tabulated in Table II, where the column and beam section sizes,fundamental period of vibration T1, and story stiffness are given. The MRF has an inherent damping of2% for the first and second modes based on Rayleigh proportional damping before the MR dampersare added into the system.

The real‐time hybrid simulations were performed at the NEES Real‐Time Multi‐Directional Facilitylocated at Lehigh University. The unconditionally stable explicit CR integration algorithm presentedearlier is used in the real‐time hybrid simulations. As noted earlier, the MR dampers are assumed to bein passive mode and have a constant current of 2.5A. Two MR fluid dampers were available for thestudy, and each was placed in the laboratory to create the experimental substructures. Figure 6 showsthe experimental setup. Each experimental substructure had one MR fluid damper, one servo‐hydraulicactuator with supports and roller bearings, reaction frames, and a tie‐down beam securing the MR fluiddamper to the strong floor. The two actuators each have a 500‐mm stroke but a different maximumforce capacity of 1700 and 2300 kN, respectively. Two servo‐valves, each with a flow capacity of2500 L/min, are mounted on each actuator to enable them to achieve a maximum velocity of 760 and560mm/s, respectively. The MR fluid dampers have a stroke of 584mm, and an Advanced MotionControl PWM servo‐amplifier is utilized to control the electrical current input for the dampers. The2300‐ and 1700‐kN actuators in the experimental setup are connected to the MR dampers associatedwith the first and second stories of the MRF, respectively. The actuator DOF for the experimentalsubstructure for the first‐story damper is equal to the difference between the horizontal displacement ofthe first floor and that at the top of the first‐story diagonal bracing. For the experimental substructureassociated with the second‐story damper, the actuator DOF is equal to the difference between thehorizontal displacement of the second floor and that at the top of the second‐story diagonal bracing.

Because the MR fluid dampers at the same story level of an MRF are designed to be placed inparallel in the prototype building, the dampers at same story level are assumed to be subjected to thesame velocity and displacement during the real‐time hybrid simulation. Hence, each of the MR fluiddamper test setups in the laboratory represents all of the dampers in one story of the MRF. Themeasured restoring force from each MR fluid damper is multiplied by the number of dampers in astory to obtain the total restoring force of all the dampers at the story level of the prototype MRF. Inthe equations of motion for the structure, both the first‐story and second‐story damper forces contributeto the first‐floor restoring force associated with the horizontal translation of the structure for the hybridsimulation, whereas only the second‐story damper force contributes to the second‐floor restoring force

Table II. Properties of system without dampers for real‐time hybrid simulations.

Column Beams T1 (s) Story stiffness (kN/m)

1st and 2nd story 1st story 2nd story 1.48 1st story 2nd storyW14 × 120 W24× 55 W18× 40 36 007 23 894

1700-kN Actuator

MR dampers

2300-kN Actuator

Roller bearing

Tie-down beam

Reaction frames

Figure 6. Experimental setup for real‐time hybrid simulation. MR, magneto‐rheological.



associated with the horizontal translation of the structure. Thus, a delay in the actuator associated withthe first‐story MR damper experimental substructure affects the first‐floor restoring force, whereas anactuator delay in the experimental substructure for the second‐story MR damper affects both the first‐floor and second‐floor restoring forces for the real‐time hybrid simulation of the prototype buildingstructure. The analytical substructure for the hybrid simulations comprises the rest of the structure andmodeled using a nonlinear finite element program with a total of 122 DOFs and 71 elements [27]. Thegravity frames were modeled using a lean‐on column to capture both the elastic lateral stiffness and theP‐delta effects from the gravity loads acting on the gravity frames within the tributary area of the MRF.Inelastic behavior is modeled by means of a bilinear hysteretic lumped plasticity beam–columnelement with a 3% post‐yield hardening. In order to overcome the shortcomings of the lumpedplasticity model predicting accurate plastic hinge rotations, each member in the MRF (beams andcolumns) was modeled with three beam–column elements in series; that is, two elements were used tomodel the two plastic hinge regions at each end of the member with a length equal to 5% of themember length and one element with a length equal to the remaining 90% of the member length.

5. REAL‐TIME HYBRID SIMULATION CASE STUDIES

The N196E component of the 1994 Northridge earthquake recorded at Canoga Park is selected as theground motion for the real‐time hybrid simulations and scaled to the design basis earthquake (DBE) byemploying the scaling procedure of Somerville [28]. A total of 12 real‐time hybrid simulations wereconducted to evaluate the performance of the AIC method applied to the actuators of multipleexperimental substructure. Table III presents the actuator delay compensation parameter values for thereal‐time hybrid simulations presented in this study, where the subscripts 1 and 2 refer to the actuatorsof two experimental substructures shown in Figure 6. Values of Δt = 10/1024 s and n = 10 were used inthe real‐time hybrid simulations.

The maximum magnitude of tracking error (MTE), root mean square (RMS) of the tracking error,maximum magnitude of the tracking indicator (MTI), and the maximum magnitude of the energy error(MEE) are used to evaluate the actuator tracking of the two actuators. MTE, RMS, MTI, and MEE aredefined for each actuator as

MTE jð Þ ¼ max ðABSðdc jð Þ−dm jð ÞÞÞ (10a)

RMS jð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑N

i¼1dc jð Þi −dm jð Þ

i

h i2= ∑

N

i¼1

sdc jð Þi

h i2(10b)

Table III. Summary of compensation parameter values for real‐time hybrid simulations.

Simulation

Estimated delay Adaptive gain

αes,1 αes,2 kp,1 kp,2

1 1 1 0.0 0.02 1 1 0.4 0.43 30 30 0.0 0.04 30 30 0.4 0.45 60 60 0.0 0.06 60 60 0.4 0.47 1 45 0.0 0.08 1 60 0.0 0.09 1 60 0.4 0.410 30 1 0.0 0.011 60 1 0.0 0.012 60 1 0.4 0.4



MTI jð Þ ¼ sgn ⋅ð Þ max ABS TIj� �

(10c)

MEE jð Þ ¼ sgn ⋅ð Þ max 0:5ABS ∑N

i¼1rm jð Þi þ rm jð Þ

iþ1

� �⋅ dm jð Þ

i −dm jð Þiþ1

� �− dc jð Þ

i −dc jð Þiþ1

� �h i� �� (10d)

where N is the number of time steps in the simulation, rm jð Þi is the measured restoring force of actuator j

(i.e., either 1 or 2), and dm jð Þi are the command and measured displacements for actuator j, respectively.

The subscript i refers to the time step. The MTE, RMS, and MTI focus on actuator control, whereas theMEE evaluates the energy error induced by actuator delay. As the name implies, MEE is the maximumamplitude of the energy error during the simulation and is based on the energy error developed byMosqueda et al. [29]. For MTI and MEE, the sign is retained in Equation (10c) and (10d). The real‐time hybrid simulation results are presented and discussed below. Results for the aforementioned errorindicators are summarized in Table IV for each simulation. To prevent possible damage to thehydraulic equipment and the dampers if the simulation becomes unstable, actuator commanddisplacement and velocity amplitude limits of 200mm and 300mm/s, respectively, are used during asimulation. When either of these two limits is exceeded as the response begins to become unbounded,the hydraulic system is shut down, and the simulation is considered unstable.

5.1. Real‐time hybrid simulation with αes,1 = 1, αes,2 = 1, kp = 0, and ki = 0 (Simulation 1)

For Simulation 1, no actuator compensation was used, as discussed earlier. Unstable results areobserved to occur in the simulation, as shown in the time history of actuator displacement given inFigure 7. The magnitude of the command displacements from the integration algorithm began to growin amplitude, with a sudden increase occurring at about 1.25 s, which caused the displacement andvelocity limits to be exceeded and the servo‐hydraulic system to shut down. A time delay can beobserved in the response of both actuators, where the measured displacement is lagging behind thecommand displacement. This demonstrates that actuator delay in the multiple experimentalsubstructures can destabilize a real‐time hybrid simulation if not compensated properly even whenphysical damping is introduced into the system by the dampers.

5.2. Real‐time hybrid simulation with αes,1 = 1, αes,2 = 1, kp = 0.4, and ki = 0.04 (Simulation 2)

Simulation 2 involved adaptive compensation for both actuators, with values of αes,1 = 1 and αes,2 = 1,as well as kp = 0.4 and ki = 0.04 for both actuators. The response of the MRF has a maximum lateraldisplacement magnitude of 49.6 and 94.6mm at the first and second floors, respectively. The resulting

Table IV. Summary of actuator control error for real‐time hybrid simulation results.

Simulation

MTE (mm) RMS (%) MTI (mm2) MEE(kNm)

1st story 2nd story 1st story 2nd story 1st story 2nd story 1st story 2nd story

1 – – – – – – – –2 2.0 3.2 2.5 5.2 46.8 73.5 −5.5 −6.23 0.6 2.3 1.3 3.7 9.1 160.2 −2.4 −3.44 0.8 1.6 1.4 2.0 11.8 48.1 −2.6 −2.35 5.9 3.9 11.2 7.7 −790.7 −269.9 −2.9 −1.46 1.6 1.8 2.8 3.3 −53.0 −31.9 −3.2 −3.87 – – – – – – – –8 6.3 4.1 12.7 8.9 1304.2 −454.2 −22.7 −2.99 1.1 1.9 2.4 3.0 52.7 −31.2 −4.0 −2.110 – – – – – – – –11 6.8 9.3 12.2 18.2 −1088.8 1014.5 −5.0 −27.712 1.5 1.9 2.7 3.9 −55.5 90.0 −2.8 −4.6

Note: Simulations 1, 7, and 10 became unstable during testing.


Figure 7. Actuator displacement history for Simulation 1: αes1 = 1, αes2 = 1, kp = 0, and ki = 0.


peak story drifts are about 1.1% and 1.2% for the first and second story, respectively, which is smallerthan the design drift of 1.7%. The residual story drifts are near zero at the end of the simulation, whichmeans that the structure develops little, if any, inelastic response. The actuator displacement timehistories are shown in Figures 8(a) and (b), with the damper force‐deformation responses shown inFigures 8(c) and (d). Unlike the Simulation 1 results presented in Figure 7, the AIC stabilized the real‐time hybrid simulation when values of αes,1 = 1 and αes,2 = 1 were used, as shown in Figures 8(a) and(b). However, small amplitude high‐frequency oscillations can be observed in Figures 8(a) and (b) atthe beginning of the simulation. This is due to the fact that the adaptive compensation method tried tonegate the destabilizing effect induced by using estimates of 1.0 for αes,1 and αes,2. The adaptation ofthe actuator delay constant through the evolutionary variable for both actuators is accomplished inabout 2 s. Good agreement occurs between the command and the measured displacements in Figures 8(a) and (b) for the rest of the simulation. At about 19 s, a spike in the actuator displacement occurs,where the dampers are observed to develop maximum deformations of 50.2 and 45.8mm for the firstand second stories, respectively, see Figures 8(c) and (d). This leads to a sudden in actuator delaybecause it imposes higher demand on the actuators. However, the actuator delay is compensated for asthe simulation continues.

Figure 8. Experimental results for Simulation 2: αes1 = 1, αes2 = 1, kp = 0.4, and ki = 0.04.



The actuator tracking error shown in Figures 9(a) and (e) has a value for theMTE of 2.0 and 3.2mmfor the actuators of the experimental substructures for the first‐story and second‐story dampers,respectively (see Table IV). The maximum actuator control error in Figures 9(a) and (e) can also beobserved to occur at the beginning of the simulation and therefore can be attributed to the poor initialestimates for actuator delay. The RMS for tracking error for the two actuators is equal to 2.5% and5.2%, respectively, (see Table IV). Similarly, because of the poor estimates for actuator delay, thetracking indicator time history given in Figures 9(b) and (f) shows a rapid increase in the trackingindicator at the beginning of the simulation. However, the AIC adjusts the compensation parameters,which subsequently led to smaller changes and an almost constant set of values for the trackingindicator (indicating subsequent good actuator control). The MTI is 46.8 and 73.5mm2 for the twoactuators (see Table IV) and occurs at about 19 s when a large increase in damper deformation and themaximum displacements develops (see Figures 8(a) and (b)). After 19 s, the actuator motions shown inFigures 8(a) and (b) become smaller, and the values for the TI associated with the two actuators alsobecome smaller. The change in the evolutionary parameter Δα for the two actuators appears to subside(see Figures 9(d) and (h)), which implies that the actuator delay that is being compensated developsless variability after 19 s. The energy errors in Figures 9(c) and (g) are shown to have negative valuesfor both actuators with MEE values of −5.5 and −6.2 kNm, for the two dampers, respectively (seeTable IV). The negative energy errors shown in Figures 9(c) and (g) are consistent with the positivetracking indicators, indicating that artificial energy is added into the system by the lag in the responseof the servo‐hydraulic actuators during the simulation. The AIC method is observed to make fast andnoticeable adjustments to the compensation parameters throughout the simulation.

5.3. Real‐time hybrid simulation with αes,1 = 30 and αes,2 = 30 (Simulations 3 and 4)

Simulations 3 and 4 involved inverse and adaptive compensation, respectively, with values ofαes,1 = 30 and αes,2 = 30. The results for the simulations are presented in Figure 10. With bothproportional and integrative gains set equal to zero in Simulation 3, the adaptive compensation methodreduces to the inverse compensation method. For Simulation 3 the MTE values are 0.6 and 2.3mm forthe actuators of the experimental substructures for the first‐story and second‐story dampers (seeTable IV). The MTE values for both actuators developed at the maximum damper deformation, wherein the actuator of the experimental substructure for the second‐story damper, there is a spike in thetracking error at 19 s (see Figures 10(a) and (e)) when the maximum actuator displacement occurred.Figures 10(b) and (f) show the tracking indicator time history for the two actuators. Differentamplitudes of the tracking indicator are evident (where in Table IV, the MTI is 9.1 and 160.2mm2),

Figure 9. Actuator control results for Simulation 2: αes1 = 1, αes2 = 1, kp = 0.4, and ki = 0.04.


Figure 10. Actuator control results for Simulations 3 and 4: αes1 = 30 and αes2 = 30. AIC, adaptive inversecompensation.


indicating that the two actuators have a different amount of delay. Different compensation parameterstherefore should be used for the two actuators. The energy errors for both dampers are shown inFigures 10(c) and (g). The actuator in the experimental substructure for the second‐story damper has alarger MEE magnitude associated with the value of −3.4 kNm compared with the actuator of theexperimental substructure for the first‐story damper, where the MEE is −2.4 kNm (see Table IV).When AIC is used in Simulation 4, where kp = 0.4 and ki = 0.04 for both actuators, the tracking errorand tracking indicator time histories shown in Figures 10(e) and (f), respectively, and the MTE andMTI values in Table IV for the actuator in the experimental substructure for the second‐story damperare observed to be significantly reduced (to 1.6mm and 48.1 mm2, respectively) compared with theresults for Simulation 3. For the actuator in the experimental substructure for the first‐story damper,similar results for Simulations 3 and 4 are achieved for these quantities, see Figures 10(a) and (b) andTable IV. These results imply that αes,1 = 30 is a good estimate for the actuator of the experimentalsubstructure for the first‐story damper. The energy errors in the time histories shown in Figures 10(c)and (g) are observed to be negative, indicating that artificial energy is added into the structural systemand consistent with the fact that an undercompensation is observed for both actuators from the positivevalues for the tracking indicator. The actuator of the experimental substructure for the damper in thefirst story is shown in Figure 10(d) to have an almost zero value for the evolutionary variable Δα, witha small sudden increase occurring around 19 s, indicating again that 30 is a good estimate for αes,1. Onthe contrary, the evolutionary variable Δα for the actuator of the experimental substructure for thedamper in the second story is shown in Figure 10(h) to increase throughout the entire simulation, witha large sudden increase occurring around 19 s. These results imply that a value of 30 for αes,2 is anunderestimate for the actuator delay constant.

5.4. Real‐time hybrid simulation with αes,1 = 60, αes,2 = 60 (Simulations 5 and 6)

Simulations 5 and 6 involved inverse and adaptive compensation, respectively, with a value ofαes,1 = 60 and αes,2 = 60. The results for the simulations are presented in Figure 11. The real‐timehybrid simulation without AIC (Simulation 5, where kp = 0.0 and ki = 0.0 for both actuators) wasobserved to render stable results in the actuator tracking error time history results shown in Figures 11(a) and (e). The MTE values for Simulation 5 in Table IV are 5.9 and 3.9mm for the actuators of theexperimental substructures for the first‐story and second‐story dampers, respectively, whichcorrespond to 11.7% and 8.5% of the maximum damper deformation. These MTE values are largerthan those of Simulation 3 where αes,1 = 30, αes,2 = 30, kp = 0, and ki = 0, indicating that αes,1 = 30 andαes,2 = 30 represent better estimates for actuator delay. In Figures 11(a) and (e), it can be observed that




the maximum tracking error in Simulation 5 occurs when the experimental substructures (i.e., the MRfluid dampers) develop their maximum deformation at the time of 19 s. The spike in damperdeformation imposes higher demand on the actuators, leading to a sudden variation of actuator delayand a subsequent increase in actuator tracking error. The RMS values for actuator tracking error areequal to 11.2% and 7.7%, which are considered to be large and indicate that in spite of a stablesimulation, the experimental results are accurate. The negative values for the tracking indicator inFigures 11(b) and (f) for Simulation 5 show that an overcompensation for actuator delay is inducedwhen the initial estimate for αes,1 and αes,2 are both equal to 60. Different amplitudes for the trackingindicator imply different levels of overcompensation occur for the two actuators. The energy errors,however, in Figures 11(c) and (g) are observed to have negative values, which mean that artificialenergy is added to the structural system. This can be attributed to the fact that the tracking indicator isonly affected by the phase error (actuator lead or lag) induced by the actuator dynamics, whereas theenergy error is affected by both phase error and amplitude error in the actuator displacement due toactuator dynamics.

The results from Simulation 6, where the AIC is applied by using kp = 0.4 and ki = 0.04 for bothactuators, is observed in Figures 11(a) and (e) to have a smaller actuator tracking error than the resultsfrom Simulation 5. The two actuators in Simulation 6 are shown in Table IV to have MTE values of1.6mm and 1.8mm, which represent a 73% and 54% reduction when compared with the simulationwithout AIC (i.e., Simulation 5). The RMS values for the actuator tracking error in Simulation 6 arereduced to 2.8% and 3.3%. The reduction of the MTE and corresponding RMS imply that improvedactuator control is achieved in Simulation 6 with the use of AIC. When compared with the results fromSimulations 2 and 4, which had the same values for kp and ki as in Simulation 6 but different values ofαes,1 and αes,2, a small difference in magnitude for the respective MTE and maximum MTI values forthe simulations can be observed. This implies that the AIC can achieve good performance whendifferent estimates for actuator delay are used. The time history for the tracking indicators inFigures 11(b) and (f) also show much smaller values in Simulation 6 when compared with the inversecompensation results of Simulation 5. This also indicates that better actuator control is achieved inSimulation 6. The energy error due to actuator control errors in Simulation 6 is observed in Figures 11(c) and (g) to be similar to that of Simulation 5, where the negative energy error indicates that energy(i.e., negative damping) is added to the system during Simulation 6. Figures 11(d) and (h) show thetime history of the evolutionary variables, where the AIC adjusted the compensation parameters tominimize the effect of actuator delay. Spikes can be observed when the dampers developed theirmaximum deformation (near the time of 19 s).



5.5. Real‐time hybrid simulation with αes,1 = 1 and αes,2 = 45 (Simulation 7)

Similar to Simulation 1 with αes,1 = 1 and αes,2 = 1, Simulation 7 with αes,1 = 1, αes,2 = 45, kp = 0.0 andki = 0.0, was found to be unstable when the inverse compensation was used without any adaptation inthe actuator delay parameter for the compensator. The displacement command from the integrationalgorithm kept increasing (similar to that for Simulation 1 shown in Figure 6) and the simulation wasstopped around 3 s. The instability is attributed to the undercompensation in the actuator associatedwith the experimental substructure with the first‐story damper. The undercompensation in the actuatorintroduced artificial energy into the structural system that destabilized the real‐time hybrid simulation,although the actuator associated with the experimental substructure with the second‐story damper isproperly compensated with a good delay estimate for the actuator.


Simulations 8 and 9 involved inverse and adaptive compensation, respectively, with values of αes,1 = 1and αes,2 = 60. The results for the simulations are presented in Figure 12. Unlike Simulations 1 and 7,which have αes,1 = 1, kp = 0, and ki = 0, Simulation 8 with αes,1 = 1, αes,2 = 60, kp = 0, and ki = 0 isobserved to be stable. The tracking indicator time histories shown in Figures 12(b) and (f) illustratethat the actuators of the experimental substructures for the first‐story and second‐story dampers haveundercompensation and overcompensation, respectively. The use of a value that represents anoverestimate for actuator delay (i.e., αes,2 = 60) for the experimental substructure for the second‐storydamper stabilizes the simulation by developing overcompensation that introduces damping into thesystem, offsetting the energy introduced into the system by the undercompensation in the actuator ofthe experimental substructure for the first‐story damper. However, Simulation 8 is observed inTable IV to have MTE values of 6.3 and 4.1mm, which occur at 19 s when the tracking error is amaximum (see Figures 12(a) and (e)). These MTE values correspond to 12.5% and 9.0% of themaximum damper deformation, and the RMS of the tracking errors are observed to be 12.7% and8.9%. The use of the AIC method in Simulation 9, where kp = 0.4 and ki = 0.04, significantly reducesthe MTE and RMS values, where the MTE values are 1.1mm and 1.9mm and the RMS values for thetracking error are 2.4% and 3.0% (see Figures 12(a) and (e) and Table IV). The energy errors inFigures 12(c) and (g) again show negative values for both actuators for Simulation 9; however, theMEE values are reduced by 76.2% and 27.6% when compared with the MEE values of Simulation8 that did not use AIC. The time histories for the evolutionary variable are shown in Figures12(d) and(h) where the AIC adjusted the compensation parameters to minimize the effect of actuator delay forthe two actuators. It is apparent in Figure 12(d) that the evolutionary variable for the actuator of the

Figure 12. Actuator control results for Simulation 8 and 9: αes1 = 1 and αes2 = 60. AIC, adaptive inversecompensation.



experimental substructure for the first‐story damper adjusted the actuator delay parameter to avoidundercompensation by the use of αes,1 = 1, whereas for the actuator of the experimental substructurefor the second‐story damper Figure 12(h) shows that adjustments occur to avoid overcompensation bythe use of αes,2 = 60.

5.7. Real‐time hybrid simulation with αes,1 = 30 and αes,2 = 1 (Simulation 10)

Simulation 10 was conducted with no adaptive compensation (i.e., kp = 0, ki = 0) for both actuators andαes,1 = 30 and αes,2 = 1. A value of αes,1 = 30 is used for the actuator of the experimental substructurefor the first‐story damper because it is considered a good estimate from Simulation 3. An unstablesimulation occurred, where the test was stopped around 3.2 s due to a large growth in amplitude of theactuator displacement commands from the integration algorithm. The unstable simulation results againimply that improper compensation in one actuator (i.e., the actuator of the experimental substructurefor the second‐story damper) can destabilize the entire simulation by adding energy to the systemthrough the actuator delay in the associated experimental substructure.


Simulations 11 and 12 involved inverse and adaptive compensation, respectively, with values ofαes,1 = 60 and αes,2 = 1. The results of the simulations are shown in Figure 13. The MTE values for thesimulation with kp = 0 and ki = 0 (Simulation 11) are observed to be 6.8 and 9.3mm in Figures 13(a)and (e), as well as in Table IV, for the two actuators. These results correspond to 13.5% and 20.3% ofthe maximum damper deformation. The maximum tracking errors again occur when the dampersdevelop their maximum deformation at the time of 19 s. The RMS values of the actuator tracking errorsare 12.2% and 18.2%, respectively. The history of the TI values shown in Figures 13(b) and (f)indicates that the actuator of the experimental substructure for the first‐story damper hasovercompensation, whereas the actuator of the experimental substructure for the second‐storydamper has undercompensation. The energy errors due to actuator tracking are again negative for bothactuators with MEE values of −5.0 and −27.7 kNm, respectively, as given in Table IV. With the samevalues for the initial delay estimates, Simulation 12 with AIC has the MTE values reduced to 1.5 and1.9mm for the two actuators, respectively. Compared with Simulation 11 without adaptivecompensation, the MTE values for Simulation 12 are reduced by about 77.9% and 80.0%. Theamplitudes of the TI and energy error values are also significantly reduced, as shown in Figures 13(b)and (f), and 13(c) and (g), respectively. Because of the inaccurate estimates for actuator delay at thebeginning of Simulation 12, the evolutionary variables are observed in Figures 13(d) and (h) to




decrease for the actuator of the experimental substructure for the first‐story damper and increase for theactuator of the experimental substructure for the second‐story damper.

Simulation 11 was stable, whereas Simulation 10 was not because in the former simulation, anovercompensation occurred in the actuator control of the experimental substructure for the first‐floordamper, adding damping into the system that offset the effects of the undercompensation that occurredin the actuator of the experimental substructure for the second‐floor damper, where negative dampingis introduced. A similar phenomenon was observed in Simulations 7 and 8.

6. ASSESSMENT OF REAL‐TIME HYBRID SIMULATIONS

The simulations show that a large actuator tracking error and possible unstable simulation can occurwhen inaccurate estimates of actuator delays are used without adaptive compensation for real‐timehybrid simulations involving multiple experimental substructures (e.g., Simulations 1, 3, 5, 7, 8, 10,and 11). The varying slope in the time history plot of the tracking indicators for the experimentalresults implies varying amount of actuator delay. The AIC method is demonstrated to enable achievestable and good actuator control under these circumstances (e.g., Simulations 2, 4, 6, 9, and 12). Fromthe simulations presented earlier, it can be observed that αes,1 = 30 is a good estimate for the actuatordelay in the experimental substructure for the first‐story damper and αes,2 = 45 is a good estimate forthe actuator delay in the experimental substructure for the second‐story damper. Simulations 7 through12 demonstrate the effect of delay in the system restoring force among the different experimentalsubstructures during a real‐time hybrid simulation that is dependent on the compensation for theactuators attached to these experimental substructures. The coupling between different experimentalsubstructures can lead to an unstable simulation because of undercompensation in one actuator. Underthis circumstance, the restoring force coupling between DOFs could help stabilize the simulation ifsimultaneously overcompensation develops in the actuator(s) of another experimental substructure(s).However, large actuator control errors occur, and possibly, inaccurate simulation results exist.

Overall, the real‐time hybrid simulation with the AIC in Table IV are observed to have betteractuator control than the corresponding simulations without the AIC, with smaller magnitude of theMTE, MTI, and MEE. It can also be observed in Table IV that better estimates for the delay constantsαes,1 and αes,2 (e.g., other than 1.0) can help further improve the performance of the AIC method butare not necessary to achieve an accurate and reliable real‐time hybrid simulation result. It is alsoobserved that for some simulations presented in this study, the energy error gives inconsistent resultswith respect to the tracking indicator. As noted previously, this is attributed to the fact that the energyerror is affected by both phase error and amplitude error in the actuator displacement, whereas thetracking indicator is affected only by phase error.

7. SUMMARY AND CONCLUSIONS

The effect of actuator delay associated with multiple experimental substructures and its compensationare discussed in this paper for real‐time hybrid simulations. Analysis using a discrete transfer functionapproach for a linear elastic two‐DOF system, where the experimental substructures in the analysis hadisolated actuators that were not physically coupled, shows that actuator delay in either actuator cancause instability and that the stability limit of the system decreases when delay exists in both actuators.Hence, delay compensation for the real‐time hybrid simulation of an MDOF system presentschallenges for actuator control in that all actuators must be properly compensated. A recentlydeveloped AIC method is proposed for real‐time actuator control in multiple experimentalsubstructures. A structure consisting of two‐story four‐bay steel MRFs and large‐scale passive MRdampers subjected to a design basis earthquake is utilized to experimentally evaluate the effectivenessof the adaptive compensation method for real‐time hybrid simulation involving multiple experimentalsubstructures. Similar to the analytical study, in the real‐time hybrid simulations performed in thelaboratory, the dampers in the structure are placed in two separate experimental substructures, with theremaining components of the structure modeled analytically. Different estimates for delay constants



and adaptive gains are used to provide a systematic evaluation of actuator control error in terms oftracking error, RMS of the tracking error, tracking indicator, and energy error. The adaptivecompensation method is demonstrated to enable stable and good control of the actuators of the twoexperimental substructures to be achieved when actuator delay is not accurately estimated before thesimulation and when the actuator delay varies throughout the simulation. The experimental resultsconfirm that the actuator delay compensation is challenging for real‐time hybrid simulation involvingmultiple experimental substructures for the delay must be accurately compensated in the actuators ofall of the experimental substructures. Further research is required to evaluate the AIC method for real‐time hybrid simulation where more than one actuator exist in an experimental substructure that resultsin the physical coupling of the actuators’ DOF. This topic is to be investigated by the authors in afuture research study.

ACKNOWLEDGEMENTS

This paper is based upon work supported by grants from the Pennsylvania Department of Community andEconomic Development through the Pennsylvania Infrastructure Technology Alliance, and by the NationalScience Foundation under grant no. CMS‐0402490 within the George E. Brown, Jr., Network forEarthquake Engineering Simulation Consortium Operation. The MR fluid dampers were provided byDr. Richard Christenson from the University of Connecticut. The authors appreciate his support.

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