System identification of non-linear hysteretic systems

5 th GRACM International Congress on Computational Mechanics
GRACM 05
Limassol, 29 June – 1 July, 2005
© GRACM
SYSTEM IDENTIFICATION OF NON-LINEAR HYSTERETIC SYSTEMS WITH
APPLICATION TO FRICTION PENDULUM ISOLATION SYSTEMS
Panos C. Dimizas and Vlasis K. Koumousis
Institute of Structural Analysis and Aseismic Research
National Technical University of Athens
NTUA, Zografou Campus GR-15773, Athens, Greece
e-mail: vkoum@central.ntua.gr, web page: http://users.ntua.gr/vkoum
Keywords: System Identification, Hysteresis, Friction Pendulum Systems, Seismic Isolation.
Abstract. In this work, the parameter identification of non-linear dynamical systems is presented. More
specifically the non-linear hysteretic behavior of seismic isolators modeled by the versatile Bouc-Wen model
such as FPS is determined based on given data. The primal objective of this paper is to develop a parametric
identification method for the modeling of FPS seismic isolator from periodic vibration experimental data. The
estimation of the model parameters based on measured data from periodic experiments is implemented using a
time domain method. The Bouc-Wen differential model is adopted to take into account the hysteretic frictional
damping of the FPS bearings. Then, the parameter identification problem is solved using nonlinear optimization
methods such as the Levenberg-Marquardt algorithm. The identification results, of the proposed method, are
verified by numerical simulations and the accuracy of the identified parameters is verified by comparing the
experimentally measured and the identified time histories.
1 INTRODUCTION
In recent years, the Friction Pendulum System (FPS) has become a widely accepted device for seismic
isolation of structures. The concept is to isolate the structure from ground shaking during strong earthquake.
Seismic isolation systems like the FPS are designed to lengthen the structural period far from the dominant
frequency of the ground motion and to dissipate vibration energy during an earthquake. The FPS consists of a
spherical stainless steel surface and a slider, covered by a Teflon-based composite material. During severe
ground motion, the slider moves on the spherical surface lifting the structure and dissipating energy by friction
between the spherical surface and the slider.
Friction Pendulum Systems exhibit nonlinear inelastic behavior under severe dynamic loading such as
earthquakes. In the case of cyclic loading the nonlinear restoring force of such isolators exhibits hysteretic loops
and is memory-dependent. It depends not only on the instantaneous deformation, but also on the history of
deformation. This memory nature makes modeling and analysis more difficult than other non-linear systems. A
widely used model that describes hysteretic behavior that belongs to the class of endochronic models is that of
Bouc-Wen [1] . This model consists of a system of nonlinear differential equations where the memory-depended
nature of hysteresis is taken into account with the use of an extra variable. Different values of the parameters of
this model reveal a wide range of different mechanical behavior. This accounts for softening/hardening behavior,
stiffness degradation, strength deterioration, pinching etc. An appropriate choice of parameters determined by the
identification algorithm on experimental data makes it possible to describe sufficiently the non-linear dynamic
behavior of a hysteretic system.
In the past two decades, several researchers have devoted their efforts to identify the parameters of the Bouc-
Wen model using experimental data. From a mathematical standpoint, determination of the hysteretic loop
parameters by identification is a problem of non-linear multivariate optimization. In this paper, this problem is
solved using a popular gradient optimization algorithm called the Levenberg-Marquardt method. In section 2, the
versatile Bouc-Wen model is presented to take into account the inelastic behavior of the FPS bearings. The time
domain identification algorithm is presented in section 3. The effectiveness and accuracy of the proposed
algorithm is addressed in section 4 with the aid of a numerical study. Finally, this paper concludes with the
presentation of identification results using the proposed algorithm and a summary of the findings in section 5.

2 HYSTERETIC MODEL
Panos C. Dimizas and Vlasis K. Koumousis
2.1 Bouc-Wen hysteretic model
In the context of a forced single degree of freedom hysteretic oscillator the equation of motion using the Bouc-
Wen model is as follows:
mx ̇̇ + cẋ + akx + (1 − a)
kz = f
(1)
where m, c, k,
f and a are the mass, damping coefficient, stiffness, external excitation and plastic to elastic
stiffness ratio respectively. The hysteretic auxiliary variable z , according to the Bouc-Wen model is given by:
The ultimate value of z is given by:
( γ β )
n
ż = Aẋ − | z | sign( xz ̇ ) + ẋ
(2)
z
MAX
⎡ A ⎤
= ⎢
β + γ ⎥
⎣ ⎦
1
n
(3)
In the case of system identification, apart from the mass m that can be measured, all the other parameters of the
model are to be identified resulting in the following unknown parameter vector:
[ β γ ] T
p = a k c A n
(4)
2.2 Identification issues and model limitations
The Bouc-Wen model parameters A, β , γ are the one that control the hysteretic loop shape, whereas
parameter n affects the smoothness of the hysteretic curves. A large variety of complex non-linear hysteretic
dynamic behavior can be represented by the Bouc-Wen model through an appropriate choice of these parameters.
Wong et al [2, 3] conducted an extensive study of the parameters effect on the response of the Bouc-Wen model.
On the other hand, Erlicher and Point [4] proved the constraints that must hold for the Bouc-Wen model
parameters so that it is thermodynamically admissible. Therefore, during Bouc-Wen model identification, one
must also take into account the restrictions and constraints regarding the Bouc-Wen model parameters.
Another issue that has to do with the identification of the Bouc-Wen model is the non-smoothness of the
hysteretic restoring force. The main advantage of the Bouc-Wen model is the ability to capture non-smooth
dynamic behavior as the sliding of an FPS seismic isolator using an analytic mathematical formulation. However,
the implementation of identification algorithms on rapidly changing dynamical systems, such as the FPS
isolators, renders the common identification methodologies unable to track them. As claimed by Ching and
Glaser [5] , the reason is that all these identification methodologies possess the inherent assumption that the degree
of change of the dynamical system studied is uniform in time. So, systems with frictional slip like the FPS are
difficult to be identified. Additionally, the problem identification of the Bouc-Wen model becomes harder
because of the fact that in certain cases, different combinations of some parameters may produce almost identical
hysteretic loops [6] .
2.3 Modelling of FPS seismic isolators
As explained in the last section, identification of the seven unknown parameters of equation possesses great
difficulties. However, in the case of FPS bearings, the identification problem can be simplified. The natural
period of a FPS bearing is only related to the radius of the concave surface R :
T
R
= 2π
(5)
g
where g is the acceleration of gravity.
With the aid of equation (5), the post yield (sliding) stiffness of the FPS can be calculated by:

Panos C. Dimizas and Vlasis K. Koumousis
g
ak = m (6)
R
It is common among researchers [7] to assume that in the case of FPS bearings, energy is dissipated by means
of frictional sliding of the isolator, so one can assume that there is very little or not at all viscous damping:
c ≈ 0
(7)
Finally, the value of parameter a usually, in the case of FPS isolators, is taken equal to:
Using equation (7), the equation of motion of the FPS becomes:
a ≈ 0.1
(8)
mx ̇̇ + akx + (1 − a)
kz = f
(9)
where the hysteretic auxiliary variable z is given by equation (2).
The post yield stiffness ak can be calculated by equation (6) resulting to the following unknown parameter
vector:
[ β γ ] T
p = A n
(10)
One can see that the unknown parameters were reduced from seven to four and that in the case of a periodic
vibration experiment, the hysteretic auxiliary variable can be easily measured.
3 TIME DOMAIN PARAMETER ESTIMATION
In the case of a periodic vibration test of a FPS bearing, both the cyclic excitation f and response x are
measured. One can then evaluate the post yield stiffness ak and the evolution of the hysteretic auxiliary variable
z using equations (6) and (9) respectively. The evolution of ż can then be calculated by numerical differentiation
of the variable z. Equation (2) can be rewritten as:
( γ β )
n
D( t) = ż − Aẋ + | z | sign( xz ̇ ) + ẋ
(11)
If the actual force-displacement relation conforms completely to the Bouc-Wen model and p are the true model
parameters, then equation (11) should be equal to zero regardless of time t. In fact, equation (11) never
approaches zero due to model errors and measurement noise. Hence, the identification problem becomes a nonlinear
optimization one, with equation (11) representing the error residuals. The objective function whose
minimum one seeks, in terms of non-linear least squares, becomes:
1
F p = ∑ D p
(12)
t
( )
2
2
( )
t
1
The resulting non-linear least-squares optimization problem can be solved using the Levenberg-Marquardt
algorithm [8] . The parameters that minimize the objective function (12) are found by the iteration formula of the
algorithm:
t+ 1 t T
1
T
⎡ µ ⎤
−
p = p − ⎣J J + I ⎦ J D
(13)
where J is the Jacobian matrix and µ the Levenberg – Marquardt parameter.
The Jacobian matrix J that is required for the implementation of the algorithm can be derived analytically by
the equations:

Panos C. Dimizas and Vlasis K. Koumousis
∂D( t)
= −ẋ
∂A
∂D( t)
= ẋ
| z |
∂β
∂D( t)
n
= ẋ
| z | sign( xz ̇ )
∂γ
n
∂D( t)
n
= ẋ
| z | ln( z ) sign( xz)
+
∂n
( γ ̇ β )
In the time domain implementation of the algorithm, the residuals are calculated at each iteration step with the
aid of equation (11) using the time series from the periodic vibration experiment. After that, the Jacobian matrix
is calculated analytically using equation (14). Finally, the new parameter values are estimated using the iteration
formula shown in equation (13). In the case where there is a range of different periodic vibration tests, the
algorithm is implemented in a similar way, but using as an objective function the sum of residuals of each
experiment.
The Levenberg-Marquardt algorithm is a popular gradient method for unconstrained optimization. To
improve convergence and stability, one can impose constraints on the algorithm by introducing new parameters
enforced by logistic transformations as Zhang et al. [9] suggest.
4 APPLICATION TO FRICTION PENDULUM SYSTEMS
The identification method presented in the previous sections was examined through identification of FPS
bearings. In order to evaluate the accuracy of the identification algorithm, a set of realistic parameters were used
to numerically generate experimental data. The experimental data were obtained by solving numerically the
system of non-linear differential equations (9) and (2) using a 4 th -5 th order Embedded Runge Kutta method under
sinusoidal excitation f. The Bouc-Wen model parameters used throughout the numerical simulation are
summarized in table 1.
m a k c Α β γ n
1 0.1 10 0 1 0.1 0.9 2
Table 1: Bouc-Wen model parameter values used throughout the numerical simulation
This set of parameters yields maximum value for the hysteretic auxiliary parameter z
MAX
= 1 and it was
chosen because it is suggested by other researchers [10] for capturing the dynamic behavior of FPS seismic
isolators in a sufficient way. According to this set of parameters, the natural period of such a system is
T = 6.3sec .
To verify the time domain identification algorithm, a numerical experiment was conducted using sinusoidal
excitation ( f = 15sin 0.1t
). The amplitude of the excitation was chosen so that strong nonlinearities and sliding
of the isolator was manifested. As claimed by other researchers [11] , the identification method gives best results
with a few cycles of non-linear response. The hysteretic loops and the evolution of the hysteretic auxiliary
variable z are shown in figure 1 and 2 respectively.
The identification method was conducted on the steady state part of the experimental data using a two-period
signal. To examine whether the algorithm was sensitive to initial parameter guesses, four sets of initial parameter
values were tried. The initial values used are shown in table 2. The results of identification algorithm are
summarized on table 3 and it becomes evident that the method behaved very well since it managed to converge to
the global minimum of the objective function.
Initial Parameter Values Sets
Model Parameters 1 st 2 nd 3 rd 4 th
A 0.1 1 5 10
β 0.1 1 5 10
γ 0.1 1 5 10
n 0.1 1 5 10
Table 2: Different initial parameter values sets.
(14)

Panos C. Dimizas and Vlasis K. Koumousis
Identified parameter using different initial sets
Model Parameters Real 1 st 2 nd 3 rd 4 th
A 1 1.002 1.002 1.002 1.002
β 0.1 0.1076 0.1076 0.1076 0.1076
γ 0.9 0.8944 0.8944 0.8944 0.8944
n 2 1.9962 1.9962 1.9962 1.9962
z 1 1 1 1 1
MAX
Table 3: Identification results.
In order to examine the effect of noise on the identification accuracy the numerical experimental data were
corrupted with noise:
x( t) = x( t)
+ εr
(15)
where ε represents the signal to noise ratio level and r is a random variable with zero mean and unit
variance. The time series of numerically obtained displacement and velocity were corrupted with discrete noise
of various levels and the identification method was implemented again using various initial conditions. The
results of the identification method using the second set of initial conditions are summarized in table 4.
Identified parameter using different noise levels
Model Parameters Real e=0.001 e= 0.005 e= 0.01 e=0.02
A 1 1.002 1.0095 1.036 1.1542
β 0.1 0.1078 0.1455 0.2591 0.5572
γ 0.9 0.8941 0.8626 0.7721 0.5839
n 2 1.9888 1.8091 1.3876 0.6507
z 1 1.0001 1.0008 1.0033 1.0176
MAX
Table 4: Identification results using noise corrupted data.
It becomes clear from table 4 that the identification accuracy is reduced with the increase of the level of noise.
In the case of substantial noise, identified parameter values may be completely different from the real ones. The
most sensitive parameter seems to be n. The values of the other parameters seem to deviate from the real ones,
but in a way such that the ultimate value of the hysteretic variable z is very close to the real one.
Zhang et al. [9] propose the use of digital filters in series into the identification algorithm for expanding the
capabilities of the optimization algorithm. Following their suggestion, the noisy experimental data were filtered
using two digital filters. The first was a median filter and the second was a least-squares lowpass filter also
known as the Savitzky-Golay FIR lowpass filter. The algorithm was implemented for various noise levels and the
results are summarized in table 5.
Identified parameter using different noise levels
Model Parameters Real e=0.01 e= 0.02 e= 0.05 e=0.1
A 1 1.004 1.0037 1.0092 0.9623
β 0.1 0.1064 0.1049 0.1482 0.0793
γ 0.9 0.8946 0.8952 0.8576 0.876
n 2 1.9889 1.9644 1.9212 1.8872
z 1 1.0015 1.0019 1.0018 1.0038
MAX
Table 5: Identification results using digitally filtered data.
A comparison between table 4 and table 5 shows that noise pre-filtering of the experimental data expands
significantly the accuracy and effectiveness of the proposed identification algorithm. A comparison between the
hysteretic loops obtained from noisy and noise pre-filtered data is shown in figures 3 and 4.

5 CONCLUSIONS
Panos C. Dimizas and Vlasis K. Koumousis
An identification method is proposed for the estimation of parameters of the Bouc-Wen model based on
experimental data. The identification problem is solved using nonlinear optimization methodologies developed in
the time domain. Parameter estimation was accomplished using the gradient Levenberg-Marquardt method. It
was shown that the identification method was able to capture the inelastic dynamic behavior in the case of
reliable experimental data. From the parametric studies conducted, the method was found to be insensitive to the
initial guesses of the parameter values.
The effect of noise on the accuracy of the method was studied extensively. It was found that the method is
sensitive to noise-corrupted data. As far as the Bouc-Wen model parameters are concerned, the presence of noise
seems to play an important role in the final parameter values, especially for the exponential parameter n. It was
also shown that a change in this parameter requires a significant change in the other parameter values in order to
fit the same hysteretic loops. However, it was shown that by digital filtering the noisy experimental data, one can
expand the capabilities and the accuracy of the identification method significantly.
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Performance analysis” J. Engineering Mechanics, 120, pp. 2299-2325.
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statistical method based on Wavelets” Earthquake Engineering & Structural Dynamics, 32, pp. 2377-2406.
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[7] Nagarajaiah, S. and Constantinou, M. C., (1989), “Nonlinear dynamic analysis of three dimensional base
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[9] Zhang, H., Foliente, G. C., Yang, Y. and Ma, F. (2002), “Parameter identification of inelastic structures under
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[10] Constantinou, M. C., Mokha, A. and Reinhorn, A. M., (1990), “Teflon bearings in base isolation II:
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J. Engrg. Mech., ASCE, 114, pp.833-846.

Panos C. Dimizas and Vlasis K. Koumousis
Figure 1. Hysteresis loops obtained by numerical simulation
Figure 2. Hysteresis loops obtained by numerical simulation

Panos C. Dimizas and Vlasis K. Koumousis
Figure 3. Comparison between hysteresis loops obtained from noisy data and noise pre-filtered data.
Figure 4. Comparison between hysteresis loops obtained from noisy data and noise pre-filtered data.