ANALYSIS OF DELAY COMPENSATION METHODS
FOR HYBRID SIMULATIONS
NUKOON MANGKALAKIRI*
University of California, Irvine
Program: Network for Earthquake Engineering Simulation (NEES)
Host Institution: University at Buffalo (SUNY)
REU Advisor: Dr. Gilberto Mosqueda*
PhD Student: Mehdi Ahmadizadeh
August 2006
ABSTRACT
The hybrid simulation test method is a versatile technique for evaluating and analyzing
the seismic performance of structures. Its ability to seamlessly integrate both a physical
and numerical simulation of a structure into a single model allows for a cost-effective test
method that can be chosen as an alternative to other tests, such as shake-table tests and
quasi-static tests.
A difficulty in the application of the hybrid simulation test method is that the
delay and dynamic response of the servo-hydraulic actuators can introduce errors into the
results. These actuator delays tend to add more energy to the structural system, and as
these errors propagate, the system may become unstable. The compensation of these
delays is critical to the stability and the accuracy of hybrid simulation. Various
procedures have been introduced to help compensate for these delays in order to improve
the effectiveness of hybrid simulations. In this study, four different delay compensation
methods were studied: the polynomial extrapolation method for both force and
displacement, kinematic equations used in explicit Newmark’s Beta Method, and the
Force Correction method. These tests were evaluated analytically, numerically, and
experimentally. The results of these tests showed that all of these methods greatly
improved the stability and accuracy of the simulation through modifications of the
command and measurement signals. The polynomial extrapolation procedure for
displacement and Newmark’s Beta method proved to be the most effective in
compensating for the actuator delays that occur during hybrid simulations.
Introduction
In the midst of catastrophic earthquakes affecting the Middle East and Southeast Asia,
and recognition of these risks closer to home, more and more Americans are shifting
focus to reducing the existing vulnerabilities in our nation’s infrastructure. Earthquakes
are known to cause devastating damage, and as engineers, we seek to lessen this damage.
More research is needed to improve our current knowledge of building behavior during
such severe earthquake shaking. The most effective method to understand the behavior
of structures and components is through experimental evaluation.
Three main experimental methods exist to evaluate the performance of structures
during seismic excitation. This includes shake table tests, quasi-static tests, and hybrid
simulations. Shake table tests allow for models to be subjected to the most realistic
seismic loadings. The disadvantage of using shake tables is that they are very expensive
to build and payload is limited to the size of the table and its payload capacity. Expenses
increase greatly when building larger shake tables with multi-degree of freedom
capabilities, so the reduction in scale required for models remains a major drawback for
using shake tables for seismic testing. An alternative is quasi-static testing which is
much simpler to perform at large scales, however, predefined displacement history is
required and is often difficult to relate to the seismic demands of the structure (Mahin and
Shing 1984; Mosqueda et al. 2005).
Hybrid simulation is another alternative that offers a cost-effective solution for
large-scale laboratory testing of structures subjected to seismic excitation. The principles
of hybrid simulation techniques are based on the pseudo-dynamic testing method
developed over the past 30 years (Magonette 2001; Mosqueda et al. 2005; Nakashima
1984; Nakashima and Takai 1985). Hybrid simulation combines numerical simulations
with that of physical simulations to evaluate the seismic performance of a given structure
when subjected to dynamic excitation.
The hybrid simulation test method consists of the physical and numerical parts
actively interacting with each other during the test. The first part consists of an
experimental test specimen that undergoes dynamic excitation from actuators as if under
the force of an earthquake. The second part is a computational subsystem which models
the remainder of the test structure. Hybrid simulations are efficient when modeling
structures that display complex non-linear behavior, especially when this behavior is
concentrated around certain areas of the structure. For instance, during strong
earthquakes, a base isolated building has non-linear isolation bearings at its base while
the rest of the structure remains linear. For such a case, a hybrid simulation would be
able to physically test the isolation bearings while simultaneously model the response of
the linear structure numerically on the computer. Components whose behavior can be
easily predicted are often modeled on the computer, while the subassemblages that are
difficult to model numerically are done so experimentally. This aspect of hybrid
simulations show that this method of testing is much more economical than shake table
tests since a full-size physical model of the entire structure is not needed. The test
specimen consists only of substructures since the rest of the structure can be modeled
numerically.
Figure 1. Example of Hybrid Testing
Fig. 1 shows a schematic diagram of a hybrid simulation test of a multi-story
structure. In such a structure, the columns at the ground floor may be tested
experimentally since they may potentially suffer the most severe damage during an
earthquake and not enough is known about their behavior and reactions of structural
elements under large deformations during seismic excitation (Sivaselvan 2005).
Numerical Methods for Hybrid Simulation
Hybrid simulations are used to determine the seismic response of a structure by
combining both experimental and numerical elements. The governing equation of motion
for the complete structure can be expressed as
Ma(t)  Cv(t)  R(t)  f (t)
(1)
where M is the mass, C is the damping coefficient, R is the restoring force, f is the applied
load, and a(t), v(t), and d(t) are the time-dependent acceleration, velocity, and

displacement, respectively.
One difficulty regarding Equation (1) is the difficulty in
determining the restoring force R. The problem can be simplified by doing a linear
elastic analysis and letting R(d) = Kd, where K is the stiffness factor. This allows the
problem to be reduced to a linear second order equation. However, this limits the scope
of the problem to linear elastic analysis. Structures are designed to respond in the
nonlinear range during strong earthquake shaking and this should be considered in the
analysis.
The main task of the numerical model is to integrate the equation of motion using
the restoring force data R(d) measured from the experimental setup. The solution to
getting this restoring force is based on a time-stepping integration procedure and enforces
the discretized equation of motion
mai  cvi  ri  f i
(2)
at time intervals ti  it for i=1 to N. The time-dependant variables are noted by the
subscript i and t is the integration time step and N is the number of integration steps.
Therefore, the displacementresponse time at time step ti1 is calculated based on the
response from time step ti (Mosqueda et al. 2005; Stojadinovic 2005).


For hybrid simulations, the most widely 
used method for solving the equation of
motion are explicit methods. One of the more popular explicit methods is the Central
Difference Method. The Central Difference Method is most widely used for simple to
moderate applications of hybrid tests (Magonette 2001; Nakashima 1999). For all
explicit methods, the limiting factor is their stability criteria, where the Central
Difference Method has the stability criterion of
t 
Tn

(3)
that limits the maximum allowable integration step size t that is based on the lowest
natural period of the structure, Tn.


The Central Difference Method is based on the difference approximations of
velocity and acceleration (Chopra 2000). Taking constant time steps, ti  t , the central
difference expressions for velocity and acceleration at time i are
vi 
di1  di1
d  2di  di1
, ai  i1
2t
t 2
(4)
By combining these two expressions with the descretized equation of motion (2), the
target displacement at the end of step i+1 can be seen as


2m

 m
1
c 
di1 
di   2 
(5)
di1  ri  f i 

2
m
c t


t
2t


t 2 2t
Now that the displacement at time ti+1 can be solved, the numerical side of the hybrid
simulation is functional and only needs to follow a simple stepwise procedure to advance

the hybrid simulation
for each time step until the simulation time has elapsed.
Figure 2. Flowchart of hybrid simulation procedures
The procedure shown in Fig. 2 describes the steps taken for each integration timestep until the number of iterations desired has been achieved. For each time-step, the
process starts with computing the desired displacement and then sending it to the actuator
as a command (step 1). In step 2, the actuator imposes the required target displacement
and holds it while measurements are taken and calculations are computed so that the next
target displacement can be found (steps 2-4). Step 5 is where the counter is incremented
by 1 so that the iteration process can repeat until the number of integration steps needed
is reached. Once the simulation time has elapsed, the computer exits the loop and the
hybrid simulation is complete (Mosqueda et al. 2005; Stojadinovic 2005).
Delays Associated with Hybrid Simulations
In a hybrid simulation where the integration steps rely on the controlled displacement,
there is delay between the actuator response and the displacement command. When the
rate of testing in increased, the delay becomes critical since the errors may propagate.
Delays also affect the structure by acting as a negative damping force, which in effect
adds more energy to the system, causing the structure to become unstable (Mosqueda et
al. 2005).
One of the most widely used method to compensate for this actuator delay was
introduced by Horiuchi et al. in 1999 (Ahmadizadeh et al. 2006). Horiuchi devised a
procedure using a polynomial extrapolation of the command displacement to help offset
the delay and to lessen the errors that may have been produced. Following Horiuchi’s
method, the last four data points are fitted with a third order polynomial and is used to
predict the command displacement ahead of the actual simulation time.
Figure 3. Polynomial extrapolation of command displacement.
As shown in Fig. 3, four data points for the command displacement are fitted with
a third order polynomial. The current displacement is xo and the desired displacement at
time t is x1. Knowing the actuator delay to be jt, the polynomial extrapolation indicates
that the next predicted displacement will be x  at time t + jt. Therefore, in order to
compensate for the actuator delay, the command displacement x  at time t + jt is sent to
the actuator instead of the desire displacement of x1. Horiuchi was able to verify the
reliability and the accuracy of the hybrid simulation with delay compensation by

comparing the results of his hybrid simulation
test with those obtained from shaking table

tests (Horiuchi et al. 1999).
Another method one may use to find the command displacement is to use
Newmark’s Beta method. Knowing the delay jt, the desired and command displacement
can be computed using simple kinematic equations that assume constant acceleration.
1
Ýo
x1  x o  t  xÝo  t 2  xÝ
2

1
Ýo
x  x o  (t  jt)  xÝo  (t  jt)2  xÝ
2
(6)
(7)
Plugging in the known quantities into Equation (6), the desire displacement can be found.
Similar to Equation (6), taking into account the delay jt and plugging in the other known

quantities, the command
displacement can be computed.
A newer method currently being studied for delay compensation involves the
correction of the force measurements based on the displacement measurements.
Figure 4. Estimation of force corresponding to the desired displacement.
Figure 4 illustrates this method where two polynomials are fitted to the last few force and
displacement measurements, creating a curve separate from the actual measured values.
When the desired displacement is known, the corresponding time is found by using the
fitted displacement polynomial. The force correction can then be found by using the
previously found time and replacing it into the fitted force polynomial.
Delay Compensation
The effects of actuator delays have been studied for many years in the areas of both
hybrid simulations and active structural control. The delay in a hybrid simulation in
effect can act as negative damping, which causes more energy to be added to the system.
In some cases, this negative damping can eventually cause the system to become
unstable. Compensation of delay is therefore very critical to the accuracy of both the
experimental and numerical simulations of hybrid experiments.
In this paper, established methods of actuator delay compensations have been
tested. The accuracy and reliability of these tests were studied analytically, numerically,
and experimentally. Based on these results, the effectiveness of each procedure was
evaluated.
Procedures
The behavior of the structure is modeled numerically during the hybrid testing by the
equation of motion (1) in discrete parameter form. The equation of motion (1) is solved
using numerical integration algorithms. These algorithms can also incorporate errors into
the model such as actuator delays and random noise from the load cells to better study the
actual experiment.
Proper care should be taken to effectively tune the actuator controllers and to
calibrate the measurement instrumentation. Hybrid simulations are very sensitive to
these errors since experimental measurements are used to compute future commands in
the time-stepping algorithms.
Errors caused by actuator delays and faulty
instrumentation may cause for errors to propagate as the simulation progresses, possibly
causing for the structure to become unstable. Methods have already been discussed to
help compensate for actuator delays in order to minimize the effects of experimental
errors. These methods will be explored further into detail later on in this report.
Simulink, a computer software often used for simulations and model-based
designs, was used to help model the system (Mathworks 2003). Shown below is the
Simulink model used to replicate the equation of motion.
Figure 5. Simulink model of equation of motion.
As shown in Figure 5, the input to the system is the displacement. The displacement is
multiplied by the stiffness factor while the first and second derivative of the displacement
is taken to yield the velocity and acceleration, respectively. The velocity is multiplied by
the damping factor while the acceleration is multiplied by the mass. The sum of these
three terms are then taken to yield the force, which is the output of this system.
The main Simulink model that was used for this hybrid simulation is shown
below in Figure 6. Following the flow of the diagram, the simulation begins with the
excitations caused by ground acceleration. Switches can be seen where different
simulations can be turned on or off, based on which delay compensation method is to be
tested. Subsystems, or blocks, are positioned to perform certain tasks, such as the delay
compensations methods that are being studied in this report. Two different polynomial
extrapolations are being used in this test, one for displacement and one for force.
Newmark’s Beta method is also used to compensate for the actuator delay as well as the
force correction method. The subsystem labeled “Experiment + Noise” simulates the test
while also incorporating random noise caused by the load cells that are often found in
these type of tests. The Simulink routine that houses the equation of motion is located in
the block labeled “Exact Specimen.” This block simulates the behavior of the structure
under exact conditions. Delays, random noise, and other discrepancies are not taken into
account when this block is used in the simulation.
Figure 6. Simulink model to numerically reproduce system behavior.
Shown below in Figure 7 is the Simulink model for delay compensation,
employing Horiuchi’s method of using a polynomial extrapolation of the displacement
curve. The unit delays are used to call previous displacement measurements. These four
points on the graph are then put through the “polyfitval” program where it fits a
polynomial to the previous four points and predicts the command displacement for the
next time step plus the delay time in order to compensate for the actuator delay. This
model is also used for the force measurements, where a polynomial is fit to the previous
four points and predicts a force command.
Figure 7. Simulink model for polynomial extrapolation of displacement curve.
Shown below in Figure 8 is a schematic drawing of the test setup used in these
experiments. Two actuators were installed to apply the displacements necessary while
two load cells, capable of measuring 5 kips of force and displacement transducers capable
of measuring up to 5 inches of displacement, were installed to help take measurements.
Coupons were used to connect the columns to each other and to the frame to allow the
coupons to fail before the actual columns yielded. This allowed for a more cost-efficient
test and allowed for multiple tests to be performed.
Figure 8. Schematic drawing of experimental test setup.
Results
Delays present during a hybrid simulation tend to add more energy to the system.
This can often lead to the structure becoming unstable. This unstable characteristic of the
structure can be seen in Figure 9.
Figure 9. Simulation without delay compensation
One of the first compensation methods studied was the Polynomial Extrapolation
method for displacement developed by Horiuchi. This method calls for a polynomial to
be fitted to the previous four displacement measurements and the corrected command
displacement is found by using the fitted polynomial. The results from this method can
be seen in Figure 10 where it is being compared to the exact simulation. This method can
be seen to efficiently compensate for the delays that are present during the simulation.
Figure 10. Simulation with compensation using
Polynomial Extrapolation for displacement
The second delay compensation method tested was the Polynomial Extrapolation
for force, which was also developed by Horiuchi. This method is comparable to the
extrapolation method for displacement since a polynomial is fitted to four previous force
measurements. The corrected force command is then found by using the fitted
polynomial. The results can be seen in Figure 11 where it is compared to the exact
simulation. Compared to the polynomial extrapolation for displacement, it can be seen
that this method is not as efficient and is not very accurate when trying to compensate for
the delay.
Figure 11. Simulation with compensation using Polynomial Extrapolation for force
The third method tested was Newmark’s Beta method. This method utilizes
Equation 7 which uses a basic kinematic equation assuming constant acceleration and
evaluates the equation at time t + jt to find the command displacement. This is a very
simple method that is very easy to use. The results shown in Figure 12 also show that
this method is efficient and very accurate in compensating for the delay. The graphs for
the experimental test and the exact test are very similar and indeed do show that this
method is accurate despite the noise that was introduced during the experimentation.
Figure 12. Simulation with compensation using Newmark’s Beta method.
The last method tested was the Force Correction method. This method calls for
two polynomials to be fitted to the displacement curve and the force curve. The time for
the command displacement is found by using the fitted curve and the corrected force is
then found by evaluating the fitted curve at the corresponding time. The results for this
method can be seen in Figure 13. It can be seen that when compared to the exact test,
this method did not perform as expected. The graph is seen to have many jumps and
outliers. These jumps in the graphs are due to the fact that often times, the quadratic
equation yields a complex number. Trying to avoid complex and imaginary numbers, the
uncorrected force is used, which results in the jumps seen in the graph. Looking at the
graph and the results, it can clearly be seen that this method is not the preferred method
for delay compensations due to its jumps and discrepancies.
Figure 13. Simulation with compensation using Force Correction method.
Conclusion
Hybrid simulations are considered to be very accurate and cost-effective method for
testing structures subjected to dynamic excitations. Delays present during the hybrid
simulation, however, tend to add more energy to the system, which can cause it to
become unstable. These delays can stem from measurement delays, calculations, or
actuator delays. More focus is often given to actuator delays.
This paper researched four different delay compensation methods analytically,
numerically, and experimentally. These methods included the Polynomial Extrapolation
method for both displacement and force, Newmark’s Beta method, and the Force
Correction method. It has been observed based on the tests that the delay compensation
methods that performed best were the Polynomial Extrapolation for displacement and
Newmark’s Beta method. Not only were these tests most accurate and effective, but they
were also very easy to employ and use during the simulation. Other methods that can be
used to better compensate for delays during the hybrid simulation can be the subject of
future studies.
Acknowledgements
Support of this work was made possible through a NSF grant within the George E. Brown
Jr. Network for Earthquake Engineering Simulation (NEES) program. I would like to
acknowledge the assistance of Dr. Gilberto Mosqueda, a professor at the State University
of New York at Buffalo, as well as Mehdi Ahmadizadeh, a PhD candidate also from the
University at Buffalo, for their assistance and guidance throughout the research program.
Much appreciation is also given to Dr. Reinhorn, Tom Albrechcinski, Carmella Gosden,
Sofia Tangalos, and Melanie Brown of NEESinc for all their help and support. Lastly, I
would like to thank NEESinc and NSF for giving me the opportunity to participate in this
NEESreu program.
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