Stability analysis of SDOF real-time hybrid testing systems with explicit integration algorithms and...

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2008; 37:597–613Published online 10 December 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.775

Stability analysis of SDOF real-time hybrid testing systemswith explicit integration algorithms and actuator delay

Cheng Chen‡ and James M. Ricles∗,†

Department of Civil and Environmental Engineering, Lehigh University, Bethlehem PA 18015, U.S.A.

SUMMARY

Real-time hybrid testing is a method that combines experimental substructure(s) representing component(s)of a structure with a numerical model of the remaining part of the structure. These substructures arecombined with the integration algorithm for the test and the servo-hydraulic actuator to form the real-timehybrid testing system. The inherent dynamics of the servo-hydraulic actuator used in real-time hybridtesting will give rise to a time delay, which may result in a degradation of accuracy of the test, and possiblyrender the system to become unstable. To acquire a better understanding of the stability of a real-timehybrid test with actuator delay, a stability analysis procedure for single-degree-of-freedom structures ispresented that includes both the actuator delay and an explicit integration algorithm. The actuator delayis modeled by a discrete transfer function and combined with a discrete transfer function representingthe integration algorithm to form a closed-loop transfer function for the real-time hybrid testing system.The stability of the system is investigated by examining the poles of the closed-loop transfer function.The effect of actuator delay on the stability of a real-time hybrid test is shown to be dependent on thestructural parameters as well as the form of the integration algorithm. The stability analysis results canhave a significant difference compared with the solution from the delay differential equation, therebyillustrating the need to include the integration algorithm in the stability analysis of a real-time hybridtesting system. Copyright q 2007 John Wiley & Sons, Ltd.

Received 26 March 2007; Revised 22 October 2007; Accepted 23 October 2007

KEY WORDS: hybrid testing; real time; integration algorithm; stability; actuator delay; discrete transferfunction; closed loop

∗Correspondence to: James M. Ricles, ATLSS Research Center, Department of Civil and Environmental Engineering,Lehigh University, Bethlehem PA 18015, U.S.A.

†E-mail: [email protected]‡Graduate Research Assistant.

Contract/grant sponsor: Pennsylvania Department of Community and Economic DevelopmentContract/grant sponsor: National Science Foundation (NSF); contract/grant number: CMS-0402490

Copyright q 2007 John Wiley & Sons, Ltd.

598 C. CHEN AND J. M. RICLES

INTRODUCTION

Real-time hybrid testing is a viable testing technique for investigating the dynamic responseof structural systems. It divides a structural system into experimental and numerical substruc-tures and thus enables more readily the full-scale testing of component(s) of a structure underdynamic loading. In real-time hybrid testing, the coupling between the experimental and numericalsubstructures is achieved by maintaining the compatibility and equilibrium at the interface of thesesubstructures. The measured restoring force from the experimental substructure and the calculatedrestoring force for the numerical substructure are fed back to an integration algorithm and then usedto calculate the command displacements for the next time step. The experimental and analyticalsubstructures, the integration algorithm for the test, and the servo-hydraulic actuator combine toform the real-time hybrid testing system.

Numerous integration algorithms have been developed for structural dynamics. These include theNewmark family of integration algorithms [1] and the HHT �-method [2]. For explicit integrationalgorithms, the displacements are calculated based on the structural response from the previoustime steps. Unlike implicit integration algorithms, explicit integration algorithms do not requireiteration within the time step and therefore require less computational effort. Explicit integrationalgorithms such as the Newmark explicit method have been used by numerous researchers forreal-time testing [3–8].

The schematic representation of the procedure for a real-time hybrid test using an explicitintegration algorithm is shown in Figure 1. For a given time step i , the command displacementsxi are determined using the known displacements xi−1, velocities xi−1, accelerations xi−1. Thedisplacements xe associated with the degrees of freedom (DOF) of the experimental substructure aretypically imposed using servo-hydraulic actuators. Owing to the inherent effects of servo-hydraulicdynamics, a time delay exists between the time that the command displacement xe is sent to theactuator and the actual time the actuator achieves this command displacement, resulting in latencyin the measured restoring force re. This latency will increase if there is any communication delaywhen sending the restoring force back to the experiment coordinator. An additional delay, referredto as a computational delay, is associated with the time necessary for the state determination tocalculate the restoring forces ra of the numerical substructure. With the restoring forces re and raknown, the velocities xi and accelerations xi are determined using a selected integration algorithm,adding possible additional delay, and the test then proceeds to the next time step. The total delayduring the real-time hybrid test can, therefore, be a combination of actuator delay, communicationdelay, and computational delay.

For real-time testing using the central difference method, Nakashima et al. [3] proposed the useof extrapolation to ensure a continuous actuator movement between when the time to achieve thecommand displacement for the current time step has expired and when the command displacementfor the next time step is determined. The clock speed (i.e. sampling rate) of data acquisition (DAQ)systems utilized in a real-time test is usually several times faster than that of the servo-controller(e.g. 4096Hz for typical DAQs and 1024Hz for typical servo-controllers) and can therefore measurethe restoring force re within a clock tick or two of the faster clock speed of the DAQ (i.e. 1

4096to 2

4096 s). The use of shared common RAM network, which has a communication rate of about180 ns, but requires at least one clock tick of a controller ( 1

1024 s) to update the data structure totransfer the data, also reduces communication delay.

The amount of computation delay depends on the number of DOF and the complexity ofthe nonlinearity in the numerical substructure as well as the state determination algorithm.

Copyright q 2007 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2008; 37:597–613DOI: 10.1002/eqe

SDOF REAL-TIME HYBRID TESTING SYSTEMS 599

Figure 1. Schematic representation of procedure for real-time hybrid testing using anexplicit integration algorithm.

A computation delay will result in a time delay in the restoring force ra of the numerical substruc-ture. In this paper, it is assumed that the numerical substructure is not too complex and thatessentially no computational delay occurs in the state determination and in the computation ofthe velocities xi and accelerations xi . This paper focuses on the delay due to actuator dynamicsin a real-time hybrid testing, where the restoring force re of the experimental substructure has atime delay. However, the procedure developed to evaluate the effect of delay in this paper can beextended to account for other forms of delay.

The effects of actuator delay have been studied by many researchers. Horiuchi et al. [9] inves-tigated the energy introduced into real-time testing caused by an actuator delay and proposedthe polynomial extrapolation for the delay compensation. Wallace et al. [10] used a delay differ-ential equation model to perform stability analysis of a single-degree-of-freedom (SDOF) real-time substructure test and showed that the delay in restoring force introduces negative damping.Wu et al. [11] investigated the effect of actuator delay in real-time substructure testing for theoperator-splitting method (OSM). Based on the amplification matrix analysis method, the stabilitylimit of the OSM is found to decrease in the presence of actuator delay.



Most of the prior studies on the effect of actuator delay used differential equation models andassumed a continuous form of time. Integration algorithms, however, are programmed in a discretemanner, and most modern servo-controllers utilized in real-time testing are digital controllers.Integration algorithms have unique properties associated with accuracy and stability. The effect ofthese properties in prior research of the stability analysis of real-time hybrid testing systems hasreceived little attention.

This paper presents a procedure to assess the stability of real-time hybrid testing systems that useexplicit integration algorithms and that have an actuator delay. The focus is on SDOF test structures.The integration algorithms and the actuator delay due to the servo-hydraulic dynamics are modeledusing discrete transfer functions that are combined to form a closed-loop representation of the real-time hybrid testing system. The effect of the delay on the stability of the real-time hybrid testingsystem is then evaluated using concepts from the discrete control theory. Two explicit integrationmethods are selected for illustrating the procedure and the effect of the integration algorithm onthe stability of the system when an actuator delay is present. These integration algorithms are theexplicit form of the Newmark integration method [1] and the explicit CR algorithm [12].

BLOCK DIAGRAM REPRESENTATION OF REAL-TIME HYBRID TESTING

The SDOF mass–dashpot–spring system subjected to the excitation input F(t) shown in Figure 2(a)is considered in this paper. The SDOF system consists of a mass (m), dashpot (c), and two springs,where the latter are associated with a numerical and experimental substructure, respectively. Thedisplacement, velocity, and acceleration of the SDOF system are designated as x(t), x(t), andx(t), respectively.

The differential equation of motion for the SDOF system is

m · x(t)+c · x(t)+ra(t)+re(t)=F(t) (1)

(a)

(b) (c)

Figure 2. Schematic representation of an SDOF system.



Figure 3. Block diagram representation of real-time hybrid testing of an SDOF system.

where ra(t) and re(t) are the restoring forces of the springs, respectively. When the springs inFigure 2(a) are linear elastic, the restoring forces can be calculated as ra(t)=ka ·x(t) and re(t)=ke ·x(t), where ka and ke are the linear elastic stiffness of the numerical and experimental substructure,respectively. ke is equal to �·k and therefore ka equal to (1−�) ·k, where k is the total linear elasticstiffness of the SDOF system with k=ka+ke, and � is referred to as the stiffness proportionalityfactor.

To conduct real-time hybrid testing of the SDOF system shown in Figure 2(a), the spring relatedto restoring force re(t) is isolated and taken as the experimental substructure (Figure 2(b)). Theremainder of the structure (i.e. the damping c, mass m, and spring related to the restoring forcera(t)) is accounted for in the numerical substructure (Figure 2(c)). The equation of motion for thenumerical substructure is

m · x(t)+c · x(t)+ra(t)=F(t)−re(t)= P(t) (2)

The temporally discretized form of Equation (2) is shown below, where integration algorithmsare used to solve for the response of the system:

m · xi +c · xi +rai =Fi −rei = Pi (3)

where xi , xi , and xi are the displacement, velocity, and acceleration of the SDOF system at the i thtime step; rai and rei are the restoring forces of the springs at the i th time step; Fi is the value of theexternal excitation at the i th time step; and Pi is the excitation force for the numerical substructurein the real-time hybrid testing system. When the substructures are linear elastic, Equation (3) canbe expressed as

m · xi +c · xi +ka ·xi =Fi −ke ·xi = Pi (4)

The process of hybrid testing based on Equations (3) and (4) can be represented by the blockdiagram shown in Figure 3, where G(z) is the discrete transfer function of the integration algorithmutilized in the hybrid test that relates the displacement xi to the excitation force Pi , as discussedbelow. It is assumed in this paper that the calculation of command displacement xi by the inte-gration algorithm is based on the calculated displacements of the previous time steps instead ofthe measured displacements from the experimental substructure. Real-time testing conducted byMercan [13] and Chen et al. [14] has shown this to improve the accuracy of the test results ifaccurate actuator control is used.

The Newmark family of integration algorithms [1] is formulated in a general form as

xi = xi−1+[(1−�) ·�t]· xi−1+(�·�t) · xi (5a)

xi = xi−1+(�t) · xi−1+[(0.5−�) ·(�t)2]· xi−1+[�·(�t)2]· xi (5b)



Table I. Coefficients of G(z) for the Newmark family of integration algorithms.

Numerator Denominator

n2 2�·�t2 d2 2�·�t2 ·ka+2�·c ·�t+2mn1 �t2(2�−4�+1) d1 �t2(2�−4�+1) ·ka+c ·�t (2−4�)−4mn0 �t2(2�−2�+1) d0 �t2(2�−2�+1) ·ka+c ·�t (2�−2)+2m

The integration parameters � and � in Equations (5a) and (5b) define the variation of velocity anddisplacement over the time step �t and determine the stability and accuracy characteristics of theNewmark family of integration algorithms.

Ramirez [15], Mugan and Hulbert [16, 17], and Chen [18] showed that direct integration algo-rithms for linear elastic structural dynamic problems can be represented by a discrete transferfunction that relates an external excitation F to the integrated displacement response x . The prop-erties of these integration algorithms can also be investigated in the frequency domain using thecorresponding discrete transfer functions. Chen and Ricles [12] utilized a discrete transfer functionapproach to develop an unconditionally stable explicit direct integration algorithm for linear elasticstructural dynamic problems. Chen and Ricles [19] represented direct integration algorithms in aclosed-loop for, and used the root locus method from control theory to investigate the stability ofselected direct integration algorithms when applied to nonlinear structural dynamic problems.

The discrete transfer function G(z) for the Newmark family of integration algorithms used tosolve the equation of motion for a linear elastic structure (i.e. Equation (4)) has the followingformat:

G(z)= X (z)

P(z)= n2 ·z2+n1 ·z+n0

d2 ·z2+d1 ·z+d0(6)

where X (z) and P(z) are the discrete z-transforms of the displacement xi and excitation forcePi , respectively; ni and di (i=0–2) are coefficients of the numerator and denominator and aretabulated in Table I. The transfer function G(z) for the Newmark family of integration algorithmsis obtained by substituting Equations (5a) and (5b) into Equation (4) and applying the discretez-transform [20]. The solution of z that renders the denominator in Equation (6) to be zero isdefined as a pole of the discrete transfer function [21, 22]. When the poles of the discrete transferfunction for a discrete system are located within or on the unit circle of the discrete z-domain, thesystem is stable; otherwise it is unstable [22].

As noted above, this paper focuses on real-time hybrid testing using explicit integration algo-rithms. For the Newmark family of integration algorithms, the parameter � needs to be zero torender an explicit integration algorithm, which in combination with �= 1

2 results in the Newmarkexplicit method.

The explicit CR integration algorithm was developed and implemented for real-time testing byChen and Ricles [12] and Chen et al. [14]. It is an unconditionally stable integration algorithm forlinear and nonlinear softening structures [19]. The variations of displacement and velocity for theCR integration algorithm over the time step are

xi = xi−1+�1 ·�t · xi−1 (7a)

xi = xi−1+�t · xi−1+�2 ·�t2 · xi−1 (7b)



Table II. Coefficients of G(z) for the CR algorithm.


n2 0 d2 �t2(ka+ke)+2c ·�t+4mn1 4�t2 d1 2�t2(ka−ke)−8mn0 0 d0 �t2(ka+ke)−2c ·�t+4m

where �1 and �2 are integration parameters defined as

�1=�2= 4m

4m+4c ·�t+(ka+ke) ·�t2 (8)

Similarly, the CR integration algorithm can also be represented by a discrete transfer functionG(z), where the coefficients are tabulated in Table II.

DISCRETE TRANSFER FUNCTION ACTUATOR DELAY MODEL

In real-time hybrid testing, the command displacement xi is issued to the servo-hydraulic actuatoras shown in Figure 4. To ensure a smooth actuator response and reduce possible overshoot, aramp generator is typically used to interpolate the command displacement over the time step �t .In this paper, a linear ramp generator is used that results in a linear interpolation of the commanddisplacement increment, as shown in Figure 4. Linear ramp generators have successfully been usedin real-time testing by a number of researchers [13, 14].

Because of actuator servo-hydraulic dynamics, the actuator has a delay in achieving the inter-polated command displacement from the ramp generator. For a small time step �t , the actuatorresponse can be idealized as a linear response, as shown in Figure 4. The duration to achieve thecommand displacement xi and have the restoring force communicated back to the experimentalcoordinator is td and designated as ��t , where � is a value greater than or equal to 1.0. The delayis, therefore, equal to (�−1)�t and could vary in different time steps during a real-time hybridtest. The stability analysis in this paper is conducted in the discrete domain for a single time step;when the actuator delay causes an unstable response for the time step, the real-time hybrid testingsystem is considered unstable. The corresponding actuator delay is considered the critical value.

The amount of actuator delay that occurs during a real-time hybrid test depends on numerousitems, such as the mechanical characteristics of the equipment (e.g. bandwidth of the servo-valves),the displacement and velocity amplitude, and interaction between the servo-hydraulic actuator andthe experimental substructure. In real-time tests on a full-scale elastomeric damper, Chen et al.[14] reported a delay of 0.017 s. The actuator used by Carrion et al. [7] in real-time testing of anSDOF system was reported to have a delay of 0.012 s.

When an actuator delay of � is introduced into the real-time hybrid testing system, the feedbackrestoring force of the experimental substructure in Equation (2) becomes re(t−�), whereby

m · x(t)+c · x(t)+ra(t)=F(t)−re(t−�) (9)



Figure 4. Conceptual actuator response.

Figure 5. Block diagram representation of real-time hybrid testing of an SDOF system with actuator delay.

Equation (9) is similar to the delay differential equation that was analyzed by Wallace et al.[10] to identify the critical delay for a linear elastic SDOF system in a real-time hybrid test. Thetemporally discretized form of the delay differential equation in Equation (9) can be expressed as

m · xi +c · xi +rai =Fi −rdei (10)

where rdei is designated as the delayed restoring force of the experimental substructure that is fedback to the experimental coordinator. Therefore, the delay � in Equation (10) can be thought ofas including both actuator delay and any delay in communication when transmitting the restoringforce for solving Equation (10). The block diagram for a real-time hybrid test shown in Figure 3can be revised as shown in Figure 5 to incorporate the actuator delay into the closed-loop systemrepresentation of the real-time hybrid testing system.

By assuming that the actuator achieves the displacement xdi−1 at time ti−1, and with the linearactuator response shown in Figure 4, the delayed displacement response of xdi at the end of thei th time step can be expressed as

xdi = xdi−1+ 1

�(xi −xdi−1) (11)

where xi is the command displacement issued to the experimental substructure from the integrationalgorithm for the i th time step. The displacement xdi−1 is the delayed displacement achieved inthe prior time step i−1. In the case where the actuator command displacement is not interpolatedwithin the time step using a linear ramp generator, Equation (11) needs to be modified accordingly.



Applying the discrete z-transform to Equation (11), the discrete transfer function Gd(z) relatingthe delayed restoring force rdei to the command displacement xi for an linear elastic experimentalsubstructure is equal to

Gd(z)= rde (z)

X (z)= ke ·z

�·z−(�−1)(12)

where rde (z) and X (z) are the discrete z-transforms of rdei and xi , respectively.

STABILITY FORMULATION OF REAL-TIME HYBRID TESTINGWITH ACTUATOR DELAY

To evaluate the effect of actuator delay on the stability of a real-time hybrid testing system, thepoles of the discrete transfer function Gcl(z) for the closed-loop system shown in Figure 5 arestudied, where

Gcl(z)= X (z)

F(z)= G(z)

1+G(z) ·Gd(z)(13)

When � is equal to 1.0, there is no delay in the experimental restoring force, and Gd(z) reducesto a proportional gain of ke. The closed-loop transfer function in Equation (13) for this case isreduced to

Gcl(z)= G(z)

1+ke ·G(z)(14)

By substituting the coefficients of G(z) from Table I for the Newmark explicit method or fromTable II for the CR integration algorithm, it can be shown that Gcl(z) in Equation (14) is the sameas the discrete transfer function of the integration algorithm used to solve Equation (1) for a linearelastic structure. Thus, for the case of no actuator delay, the stability limit of the real-time hybridtesting system is the same as that of the corresponding integration algorithm.

For the case of �>1, substituting the coefficients for G(z) from either Tables I or II and Gd(z)from Equation (12) into Equation (13), the closed-loop transfer function Gcl(z) can be expressedin the following general form:

Gcl(z)= n3 ·z3+n2 ·z2+n1 ·z+n0d3 ·z3+d2 ·z2+d1 ·z+d0

(15)

The coefficients of the numerator and denominator of Gcl(z) for real-time hybrid testing using theNewmark explicit method and the CR integration algorithm are derived and tabulated in Tables IIIand IV, respectively.

Stability of the discrete transfer function of Gcl(z), representing the real-time hybrid testingsystem, requires that all of its poles be located inside or on the unit circle of the complex z-domain.The stability limit for the real-time hybrid testing system is expressed in this paper in terms of�n�t , where �n is the natural frequency of the SDOF system and for the linear elastic SDOF isequal to

√(ke+ka)/m.



Table III. Coefficients of Gcl(z) for the Newmark explicit method with actuator delay.


n3 0 d3 �(�t ·c+2m)

n2 2�·�t2 d2 �t2[2ka ·�+2ke]+c ·�t (1−�)+2m(1−3�)

n1 2�t2(1−�) d1 2�t2 ·ka ·�−c ·�t ·�+2m(3�−2)n0 0 d0 c ·�t (�−1)+2m(1−�)

Table IV. Coefficients of Gcl(z) for the CR algorithm with actuator delay.


n3 0 d3 �(ka+ke) ·�t2+2�·c ·�t+4�·mn2 4�·�t2 d2 �t2(�+1) ·ka+�t2(5−3�) ·ke+2c ·�t (1−�)+4m(1−3�)

n1 4(1−�) ·�t2 d1 (2−�) ·ka ·�t2+(3�−2) ·ke ·�t2−2�·c ·�t+4m(3�−2)n0 0 d0 (1−�) ·ka ·�t2+(1−�) ·ke ·�t2+2c ·�t (�−1)+4m(1−�)

STABILITY OF REAL-TIME HYBRID TESTING OF DAMPED SYSTEMSWITH ACTUATOR DELAY

Stability analyses were performed for the real-time hybrid testing of a lightly damped linear elasticSDOF system with actuator delay. As noted previously, the Newmark explicit method and theCR integration algorithm are considered. Values for the critical viscous damping ratio � of 2 and5% were used, with the stiffness proportionality factor � ranging from 0.25 to 1.0. The solutionfor the critical delay, also referred to herein as the stability limit for the real-time hybrid testingsystem, was obtained by solving numerically for the poles of Gcl(z) in Equation (15) that lie onthe unit circle in the complex z-domain. The results are compared below with the solution to thedelay equation developed by Wallace et al. [10], which assumed a constant delay and neglectedthe integration algorithm in the stability analysis. For the case of no viscous damping, it canbe shown using the discrete transfer function approach with c=0 that the closed-loop transferfunction Gcl(z) for both algorithms is stable only when no delay occurs (i.e. when �=1.0).

Using the first-order approximation of e−s� =1−s ·�, Wallace et al. [10] showed that the criticaldelay for Equation (9) is �cr=c/ke=2�/(�·�n), from which the stability limit of �n�t can beexpressed as

�n ·�t� 2�

(�−1)�(16)

The stability limit from Equation (16) is referred as the approximate solution in this paper. � inEquation (16) is the inherent viscous damping of the system.

Wallace et al. [10] also derived the exact solution of the critical delay for the delay differentialequation (9). Following the procedure and using the notation by Wallace et al. [10], the exact



solution can be expressed as

� = 1

�arccos

(�2−(1−�)

�

)+ 2n�

�(17a)

� =√

�2−[�2−(1−�)]22�

(17b)

where �=�/�n , �=�n ·�, and n is an integer number. The smallest value of � gives the criticaltime delay of �. The stability limit from Equation (17a) is referred as the exact solution in thispaper. Equation (17b) is the damping value corresponding to the stability limit for the selectedset of values for � and �. For the lightly viscous damped SDOF system considered in this paper,Wallace et al. [10] indicate that the approximate solution for the critical delay is nearly the sameas the exact solution.

Figures 6 and 7 show the numerical results of the stability limit of the real-time hybrid testingsystem in terms of �n�t for an inherent viscous damping ratio � of 2 and 5%, respectively. Theresults for the discrete transfer function approach are when the Newmark explicit method is usedfor the numerical integration algorithm in the real-time testing system. The results based on thediscrete transfer function approach are identified in the legend by the letter D. The approximateand exact solutions by Wallace et al. [10] are identified by the letters A and E , respectively.Figures 6 and 7 show that the approximate and exact solutions to the differential delay equationare almost identical to each other. It can also be observed that the stability limit of a real-timehybrid testing system is larger for an SDOF system when a greater amount of viscous dampingis present, indicating that viscous damping in the SDOF system helps to stabilize the real-timehybrid testing system when an actuator delay occurs.

The stiffness proportionality factor � is shown in Figures 6 and 7 to affect the stability limit ofthe real-time hybrid testing system. With increasing values of �, the stability limit of the real-timehybrid testing system with an actuator delay is shown to decrease for a fixed set of values for �and �. With the stability limit expressed in terms of �n�t , it is apparent from Figures 6 and 7 thatan SDOF system with a higher natural frequency �n is more susceptible to an instability, as thevalue of �n�t increases the amount of delay to cause instability decreases.

As the delay approaches zero (i.e. �=1.0), the stability limit of the real-time hybrid testingsystem based on the discrete transfer function solution approaches the value of 2.0 when theNewmark explicit integration algorithm is used. This is expected, for the stability of the systemwith no actuator delay reduces to the stability of the integration algorithm. Unlike the stabilitylimit from the discrete transfer function approach, the stability limit of the real-time hybrid testingsystem from the delay differential equation by Wallace et al. [10] goes to infinity as the delayapproaches zero. Consequently, the stability limit based on the discrete transfer function is smallerthan that derived in the solution by Wallace et al. when the delay is small. For a moderate value ofdelay, the solutions to the delay differential equation show good agreement with the solution basedon the discrete transfer function approach for the case of �=0.02, while for the case of �=0.05,the solutions to the delay differential equation have a smaller stability limit when compared withthe solution based on the discrete transfer function approach. For a large value of delay, all ofthe methods give almost the same solution for the stability limit for the real-time hybrid testingsystem.



1 1.5 2 2.50

0.5

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1.5

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bilit

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n∆t

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Sta

bilit

y lim

it ω

n∆t

ζ=0.02 Dζ=0.02 Aζ=0.02 E

(a) (b)

(c) (d)

Figure 6. Stability limits of real-time hybrid testing of linear elastic SDOF system usingthe Newmark explicit method, �=0.02.

The stability limits in Figures 6 and 7 also show that as the delay increases from �=1.0,the stability limit for the real-time hybrid testing system based on the discrete transfer functionapproach has a slight increase before a decrease occurs, particularly for the case of �=0.05 andsmaller values of �. This initial increase of the stability is due to the interaction between thedynamics of the integration algorithm and the actuator dynamics of the real-time hybrid testingsystem. The interaction effects on the equivalent damping of the system, �eq, are shown in Figure 8,where the results related to the explicit Newmark method with an inherent viscous damping of�=0.05 are plotted. In Figure 8(a), �eq is plotted as a function of �n�t for the cases of �=1.0(no delay), �=2.0, and 2.5, with �=0.25. In Figure 8(b) �eq is plotted as a function of �n�t forthe cases of �=1.0,1.05, and 1.2, with �=1.0. The results from the discrete transfer functionapproach (identified by D) and the delay differential equation first-order approximate solution(identified by A) are both included in Figure 8. The equivalent damping shown in Figure 8 for thediscrete transfer function approach, �deq, is based on discrete control theory and the position of thedominant pole:

�deq= − ln(2+2)

2tan−1(/)(18)



1 1.5 2 2.50

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bilit

y lim

it ω

n∆t

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(a) (b)

(c) (d)

Figure 7. Stability limits of real-time hybrid testing of linear elastic SDOF system usingthe Newmark explicit method, �=0.05.

0 0.5 1 1.5 2 2.50

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vale

nt d

ampi

ng

0 0.5 1 1.5 2 2.50

0.01

0.02

0.03

0.04

0.05

0.06

equi

vale

nt d

ampi

ng

D, α=2.5

A, α=2.5

D, α=2.0

no delay, α=1.0 no delay, α=1.0

A, α=2.0

A, α=1.2

A, α=1.05

D, α=1.2

D, α=1.05

(a) (b)

Figure 8. Equivalent damping of real-time hybrid testing system with actuator delay.

where and are the real and imaginary coordinates of the pole in the discrete z-domain. For thefirst-order approximation, the equivalent damping, �aeq, is determined from

�aeq= c−ke ·�2 ·m ·�n

=�−�·(�−1) ·�n ·�t/2 (19)



1 1.5 2 2.50

0.5

1

1.5

2

2.5

α

Sta

bilit

y lim

it ω

n∆t

1 1.5 2 2.50

0.5

1

1.5

2

2.5

α

Sta

bilit

y lim

it ω

n∆t

1 1.5 2 2.50

0.5

1

1.5

2

2.5

α

Sta

bilit

y lim

it ω

n∆t

1 1.5 2 2.50

0.5

1

1.5

2

2.5

α

Sta

bilit

y lim

it ω

n∆t

ζ=0.02 Dζ=0.02 Aζ=0.02 E

ζ=0.02 Dζ=0.02 Aζ=0.02 E

ζ=0.02 Dζ=0.02 Aζ=0.02 E

ζ=0.02 Dζ=0.02 Aζ=0.02 E

(a) (b)

(c) (d)

Figure 9. Stability limits of real-time hybrid testing of linear elastic SDOF system usingCR integration algorithm, �=0.02.

For no delay, the equivalent damping of the system is shown in Figure 8 to reduce from theinherent viscous damping value of 0.05 for the SDOF to a value of zero as �n�t increases, whereat �n�t=2 the equivalent damping is �eq=0. The value of �n�t=2 is the stability limit for thesystem with no actuator dynamics (i.e. the stability limit for the system is that of the integrationalgorithm). For the delay differential equation solution, which excludes the effects of the integrationalgorithm, the effect of the delay is shown to decrease the equivalent damping of the system,where the values of �n�t corresponding to where �eq=0 represent the stability limit. The discretetransfer function approach, which considers the interaction between the integration algorithm andactuator dynamics, is shown to have �eq decrease as �n�t increases, where the rate at which itdecreases and the value of �n�t when �eq=0 is shown in Figure 8 to depend on the amount ofdelay (i.e. value of �) as well as the value of �. The results in Figure 8(a) for �=2 show theequivalent damping to become zero when �n�t=2.15, and hence the stability limit is greater thanthat when the system has no delay. Likewise, in Figure 8(b), for �=1.05 the equivalent dampingbecomes zero when �n�t=2.10.

Figures 9 and 10 show the comparison of the stability limit of the real-time hybrid testingsystem when the explicit CR integration algorithm is used. The same values for � and the viscousdamping ratios considered in the stability analysis involving the Newmark explicit method areused in Figures 9 and 10. Although the original CR integration algorithm is unconditionally stable,



1 1.5 2 2.50

0.5

1

1.5

2

2.5

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Sta

bilit

y lim

it ω

n∆t

1 1.5 2 2.50

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2

2.5

α

Sta

bilit

y lim

it ω

n∆t

1 1.5 2 2.50

0.5

1

1.5

2

2.5

α

Sta

bilit

y lim

it ω

n∆t

1 1.5 2 2.50

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1

1.5

2

2.5

α

Sta

bilit

y lim

it ω

n∆t

ζ=0.05 D

ζ=0.05 A

ζ=0.05 E

ζ=0.05 D

ζ=0.05 A

ζ=0.05 E

ζ=0.05 D

ζ=0.05 A

ζ=0.05 E

ζ=0.05 D

ζ=0.05 A

ζ=0.05 E

(a) (b)

(c) (d)

Figure 10. Stability limits of real-time hybrid testing of linear elastic SDOF system usingCR integration algorithm, �=0.05.

it is observed in Figures 9 and 10 that the real-time hybrid testing system becomes conditionallystable when there is an actuator delay.

Similar to the results presented for the Newmark explicit method, the stability limit of a real-time hybrid testing system using the CR integration algorithm that is based on the discrete transferfunction approach decreases with an increase in � and �. The results in Figures 9 and 10 alsoindicate that the real-time hybrid testing system is more susceptible to instability when the SDOFhas a higher natural frequency, for the stability limit since actuator delay again is smaller for largervalues of �n�t . As the actuator delay becomes small, the stability limit of the real-time hybridtesting system from the discrete transfer function method is shown to approach that of the originalCR integration algorithm, namely infinity.

For the range of � investigated in this paper, the stability limit of the real-time hybrid testingsystem based on the discrete transfer function approach is shown in Figures 9 and 10 to always begreater than that based on the delay differential equation solution, implying that the delay differentialequation solution gives a conservative result for the stability limit when the CR integration algorithmis used. Note that the delay differential equation solutions are the same in Figures 6, 7, 9, and10, since they are independent of the integration algorithm used in the real-time hybrid testingsystem.

It can then be concluded from the comparison of the results presented in Figures 6, 7, 9, and10 that the integration algorithm has an effect on the stability limit of the real-time hybrid testing



system when an actuator delay occurs. The stability limit for the real-time hybrid testing systemis shown to be dependent on the integration algorithm used in the real-time test, particularly whenthe amount of inherent viscous damping in the SDOF system is large and the stiffness proportionfactor � is small.

SUMMARY AND CONCLUSIONS

A procedure to investigate the effect of actuator delay on a real-time hybrid testing system thatincludes the integration algorithm that is in explicit form was presented in this paper. The stabilityanalysis is conducted in the discrete time domain in order to take into account the properties ofintegration algorithms and the actuator delay. Actuator delay due to servo-hydraulics is idealizedand modeled by a discrete transfer function, which is combined with the discrete transfer functionfor explicit integration algorithms to form a closed-loop discrete transfer function for the real-timehybrid testing system. The stability of the real-time hybrid testing system under actuator delayis then investigated through the closed-loop discrete transfer function using concepts based ondiscrete control theory.

It is found that the real-time hybrid testing using the Newmark explicit method and the originallyunconditionally stable CR algorithm are both conditionally stable when an actuator delay existsthat causes a delay in the restoring force of the experimental substructure. The extent of the effectof actuator delay on the stability of the real-time hybrid testing system is dependent on the formof the explicit integration algorithm used in the real-time test. For the Newmark explicit method,the solutions from the delay differential equation give unconservative results for the stability limitfor a real-time hybrid testing system when a small amount of actuator delay is present and aconservative result for the stability limit for a moderate amount of actuator delay. However, forthe CR integration algorithm, the solution from the delay differential equation always gives aconservative stability limit for the real-time hybrid testing system.

The stability analyses show that a delay in the restoring force decreases the equivalent damping ofa real-time hybrid testing system and, therefore, has a detrimental effect on the stability of the test.The equivalent damping of the system is sensitive to the amount of actuator delay, the integrationalgorithm, and properties of the structure (inherent viscous damping, stiffness proportionalityfactor, and natural frequency). The stability analysis of the effect of actuator delay for a real-timehybrid test, therefore, should include the numerical integration algorithm in addition to the systemdynamics. The approach presented in this paper based on a discrete transfer function representationof a real-time hybrid testing system is useful for this purpose.

ACKNOWLEDGEMENTS

This paper is based upon work supported by a grant from the Pennsylvania Department of Community andEconomic Development through the Pennsylvania Infrastructure Technical Alliance and by the NationalScience Foundation (NSF) under grant no. CMS-0402490 within the George E. Brown, Jr. Network forEarthquake Engineering Simulation Consortium Operations. The support is gratefully appreciated. Anyopinions, findings, and conclusions or recommendations expressed in this paper are those of the authorsand do not necessarily reflect the views of the sponsors.



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