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DISPLACEMENT ESTIMATION OF NONLINEAR
SDOF SYSTEM UNDER SEISMIC EXCITATION
USING KALMAN FILTER FOR
STATE-PARAMETER ESTIMATION
Yaohua YANG1, Tomonori NAGAYAMA2, Di SU3
1Ph.D. candidate, Dept. of Civil Eng., The University of Tokyo
(7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan)
E-mail: [email protected] 2Member of JSCE, Associate professor, Dept. of Civil Eng., The University of Tokyo
(7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan) E-mail: [email protected] (Corresponding Author)
3Member of JSCE, Associate professor, Dept. of Civil Eng., The University of Tokyo
(7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan) E-mail: [email protected]
An extended Kalman filter (EKF) based displacement estimation method for nonlinear SDOF systems
under seismic excitation is proposed. In this method, time intervals where the system experiences signifi-
cant nonlinearity or not are firstly distinguished. For a time period when the system is in an elastic phase,
available observations for EKF are acceleration, displacement obtained via double integration of acceler-
ation, and residual displacement. During a time period with significant nonlinearity, acceleration and vir-
tual displacement measurement are employed as observations, and the displacement is estimated along with
time-variant stiffness using an augmented state vector in EKF. The results are further smoothed by ex-
tended Kalman smoother (EKS). The proposed method is studied on an SDOF system with a bi-linear
hysteresis model in detail and further verified considering various hysteresis models and earthquake exci-
tations. The estimated displacements are shown to be accurate.
Key Words: displacement estimation, extended Kalman filter, Robbins-Monro algorithm, seismic exci-
tation
1. INTRODUCTION
Post-earthquake structural assessment is important
to efficiently mitigate the impact of earthquakes. The
quick assessment of intact structures can minimize
unnecessary service stops while the severity as-
sessment of damaged structures helps evacuations
from unsafe structures and planning of repair and
retrofit. Though seismic resistance can be assessed
from design information, actual damages on struc-
tures depend on the ground motions and the current
condition of structures; the problem of aging infra-
structure1) and non-uniformity of ground motions
make the assessment difficult. Analysis based on
signals observed during earthquakes is considered to
achieve this assessment reflecting the health condi-
tion of structures under specific ground motions.
Displacement response is an important indicator
reflecting structural health conditions after an
earthquake excitation. For example, inter-story drift
ratio2), defined as inter-story drift normalized by
story height, is used as a judgment criterion on
buildings; hysteresis loop of an SDOF system, which
can be presented as displacement versus inertial
force, contains valuable energy dissipation infor-
mation of the system3). Besides, in the context of
system identification, displacement measurements
are preferred because consistent stiffness identifica-
tion results may not be easily obtained by observing
accelerations only as reported by Chatzis et al.4).
Generally, displacement measurement is beneficial
in the field of structural health monitoring.
Measuring structural displacement is usually a
challenging task in practice. Since displacement is a
relative physical quantity, a stationary platform is
needed for contact displacement sensors, e.g. LVDT.
1
The sensors might work well in indoor environments,
but the difficulty in finding a reliable platform can
become a significant obstacle for on-site applica-
tions. Non-contact technologies, such as global po-
sitioning systems, are alternatives. However, the
drawbacks of most of them include but not limited to
high equipment costs, low sampling frequency, and
low resolution. A comprehensive review of various
inter-story displacement measurement techniques is
found in Skolnik et al.5). Compared to a displacement
transducer, an accelerometer is preferable in practice
considering its advantages such as high sampling
frequency, robustness, and ease of deployment. By
double integrating an acceleration signal and apply-
ing baseline correction and appropriate high pass
filter, reasonable displacement can be obtained6).
Besides, novel displacement estimation methods
using acceleration measurement have been devel-
oped7) and applied to the wireless displacement
measurement system8). The acceleration integra-
tion-based methods perform well on linear systems5).
However, due to the high pass filter, they cannot be
applied to systems undergoing nonlinear defor-
mations since low-frequency components including
residual displacements are inevitably removed.
In this paper, an EKF based displacement estima-
tion method is proposed for nonlinear SDOF systems
under seismic excitation. Acceleration and residual
displacement are utilized. From a forward simulation
point of view, residual displacement is sensitive to
nonlinear system parameters, e.g. post-yielding
stiffness ratio α in a bi-linear model studied by Ka-
washima et al.9). It indicates that the estimation of
residual displacement may not be implemented eas-
ily in practice. Therefore, the residual displacement
is assumed as known in this study, which could be
obtained from inclinometers10) in practice.
In the proposed method, firstly, time-intervals of
responses with and without significant system non-
linearity are distinguished based on a time-variant
parameter identification method proposed by Kuleli
et al.11). For the time periods without nonlinearity,
e.g. the initial and ending parts of the time history,
observations in EKF include acceleration and dis-
placement which is obtained by combining dis-
placement from the integration of acceleration and
residual displacement. During the time interval with
significant nonlinearity, acceleration and virtual
displacement are employed as observations. In order
to consider the nonlinearity, displacement is esti-
mated along with the time-variant stiffness parame-
ter. The estimation process is based on an incre-
mental Newmark-β state-space model in EKF. The
result from EKF is further smoothed by EKS. The
EKF, which is one of the data assimilation techniques
utilized in data science applications, is implemented
in an adaptive way customizing the Robbins-Monro
(RM) stochastic approximation method. The method
is originally an algorithm to find the root of a func-
tion that may not be computed directly but estimated
from noisy observations. This method used in the
field of machine learning to estimate parameters12)13)
is investigated in the displacement estimation.
In the following, the displacement estimation
method is presented in section 2. In section 3, the
proposed method is firstly studied on a SDOF system
with a bi-linear hysteresis model in detail and further
investigated with three different hysteresis models,
i.e. bi-linear model, tri-linear model, and Bouc-Wen
model excited by four earthquakes of two levels of
amplitudes. Finally, the conclusion is given in sec-
tion 4.
2. METHODOLOGY
(1) Equation of motion and state-space model
The equation of motion of a nonlinear SDOF
system under seismic excitation in relative coordi-
nate is written as
, gmx cx f x x mu (1)
in which x , x and x are vectors of displacement,
velocity, and acceleration respectively; m and c are
mass and damping coefficient; f represents nonlinear
restoring force as a function of x and x , and deter-
mines the hysteresis rule of loading and unloading;
gu is input ground acceleration. The incremental
formulation is written as
T gm x c x k x m u (2)
in which Δ stands for incremental value and kT is
tangential stiffness for each time instance.
Usually, equation (2) is solved by a numerical in-
tegration algorithm in a step by step manner. Using
the incremental Newmark-β method, the equation is
transformed into a discrete state-space model as
2
1
2
1
11
10 0 0
10 1 2
0 1 1 2
1
1
d g
tx k
tx k
x kt
t
x k
x k t k m u
x k t
I + L +
(3)
2
in which β and δ are integration parameters of the
algorithm which are set as 1/6 and 1/2 in this study; I
is the unit matrix; k stands for a specific time instant,
L matrix is defined as below
1 0 12 2
d
m c t mk c t
t
L (4)
kd is equivalent dynamic stiffness calculated as
2
1d Tk m c k
t t
(5)
This state-space model is different from the
commonly used one where only displacement and
velocity variables are included in the state vector. In
the commonly used state-space model, dynamic
equilibrium is exerted implicitly on every point ac-
cording to its integration algorithm. Because of that,
the state-space model can only estimate displace-
ments of a linear elastic system. However, dynamic
equilibrium is not enforced in this state-space model
of equation (3), thus it can be applied to displacement
estimation of a nonlinear system.
In this study, if necessary, parameters can also be
identified by augmenting them in the state vector.
Specifically, the augmented state vector, system
equation, and observation equation in EKF are writ-
ten as
T
T T T T
a t t t
X x x x (6)
1 ,a a gk k u k X f X w (7)
1 1 1ak k k y h X v (8)
f function is basically the extension of equation (3)
with parameters augmented; h can be a simple se-
lection matrix containing only 0 or 1; w(k) and ν(k)
are process and measurement noise vectors consist-
ing of uncorrelated zero-mean noise.
Suppose ˆ |a k kX and |k kP are estimated
value and error covariance matrix of state vector at k
instant based on measurements from 1 to k instants.
EKF firstly calculates a prior estimation and covar-
iance matrix of state vector at k+1 instant based on
system equation (7)
ˆ ˆ1| | ,a a gk k k k u X f X (9)
1 | | Tk k k k k k k P F P F Q (10)
Combining the information from observation
equation (8), the estimation and covariance matrix of
state vector at k+1 instant can be obtained
ˆ ˆ1| 1 1 |
ˆ1 1| ,
a a
a g
k k k k
k k k u
X X
G y h X (11)
1 | 1 1 |
1 |T
y
k k k k
k k k k
P P
G P G (12)
In the formulation above, G(k) is the so-called
Kalman filter gain matrix and Py(k+1|k) is the prior
estimation of observation covariance matrix
1
1| 1 1 |T
yk k k k k k
G P H P (13)
1 |
1 1 | 1 1
y
T
k k
k k k k k
P
H P H R (14)
F(k) and H(k+1) are Jacobian matrices of function f
and h respectively
ˆ |
,a g
aa a k k
k
X X
f X uF
X (15)
ˆ 1|
,1
a g
aa a k k
k
X X
h X uH
X (16)
Q(k) and R(k+1) are the process and measurement
noise covariance matrices respectively
T
k E k k
Q w w (17)
1 1 1T
k E k k
R v v (18)
The aforementioned procedures are conducted
recursively, regarding estimation of previous instant
as the initial value for current instant.
(2) Robbins-Monro algorithm for process noise
adaption
The R matrix in equation (18) can be decided be-
forehand because it is usually related to sensor noise.
As for the Q matrix which actually reflects the error
of system equation (7) in EKF, a trial and error
manner is usually used to tune the value which is
time-consuming and subjective. Estimation of Q
matrix using the RM algorithm can be formulated as
below
+1 1
1 1
Q
T
Q
k k
diag k k
Q Q
d d (19)
ˆ ˆ1 1| 1 1|a ak k k k k d X X (20)
in which d(k+1) is the so-called state innovation
3
vector. The Q(k+1) matrix is constrained to have a
diagonal form assuming the process noise is uncor-
related. The αQ is a small positive value chosen be-
tween 0 and 1, serving as a forgetting factor to av-
erage estimations of Q over a period of time.
Based on the RM algorithm, time-variant param-
eters can be identified, as reported in Kuleli et al.11).
In the following, the combination of the EKF and
RM algorithm is referred to as the EKF-RM method.
(3) Observation scheme
Accelerations are, in practice, the most easily ac-
cessible system responses. Based on an acceleration
measurement, the displacement without a
low-frequency drift component can be obtained by
double integration and high pass filtering. Therefore,
the true displacement can be regarded as the combi-
nation of displacement from the integration of ac-
celeration and the low-frequency drift component.
Fig. 1 shows the displacement decomposition.
Because a structure under earthquake excitation
behaves as a linear system when the excitation level
is small at the beginning and end of the excitation,
displacement from integration can reproduce the true
displacement in these linear sections except for the
constant component, i.e., residual displacement
component. In the middle part, e.g. 3~10 s in Fig. 1,
because of the employment of high pass filter, dis-
placement from integration differs from the true one
significantly.
Fig. 1 Displacement decomposition
(a)
(b)
Fig. 2 Available observations: (a) displacement; (b) acceleration
The residual displacement in this study is assumed
as known value; in practice, it can be estimated by
inclinometer10) indirectly. Therefore, the available
observation information for displacement estimation
includes displacement values at the beginning and
end and the acceleration signal. The displacements in
the linear ranges are obtained by combining dis-
placement from integration and initial and residual
displacement. In order to estimate the displacement
of the nonlinear part, a virtual displacement obser-
vation shown in Fig. 2(a), i.e. a straight line con-
necting the two sides, is assumed. The virtual dis-
placement possesses large uncertainties due to the
assumption. This method was firstly proposed by
Chatzi et al.14) to suppress divergence for displace-
ment estimation using acceleration only. Fig. 2
shows the available observation information in the
following steps of EKF and EKS.
One remaining problem is how to define the time
interval where the virtual displacement should be
applied. This time interval is determined by identi-
fying the time-variant parameters k and c of the sys-
tem based on the EKF-RM method using displace-
ment from integration. Note that the commonly used
state-space model rather than the incremental New-
mark-β state-space model explained in Eq. (3) is
employed for the parameter identification. While the
parameters may not be very accurate especially
during the strong nonlinearity part, the time interval
with significant stiffness reduction is defined as the
virtual displacement observation part. It will be
demonstrated with a numerical example in the fol-
lowing section.
(4) Estimation scheme during the virtual dis-
placement observation part
The time interval with virtual displacement actu-
ally corresponds to the period when the system ex-
periences strong nonlinearity, i.e. the stiffness pa-
rameter is changing. In order to consider the non-
linearity, the proposal is to identify the time-variant
stiffness value along with system response as shown
in equation (6)~(8). The EKF-RM method is used
with the incremental Newmark-β state-space model
explained in Eq. (3). The stiffness identified is tan-
gential stiffness. The damping coefficient is obtained
in the previous step, i.e. when time interval with
virtual displacement is defined. Specifically, the
damping coefficient at the end of the identification
history is used. Alternatively, damping coefficient
values identified using small earthquake signals and
ambient vibration results can also be utilized. It is
assumed that when a system experiences strong
hysteresis behaviors, the viscous damping does not
play a significant role and the estimation accuracy of
the damping parameter is not important.
(5) Extended Kalman smoother
4
As mentioned in the last section, ˆ |a k kX is the
estimated value of the state vector based on meas-
urement information up to k instant, i.e. y1~k. In order
to utilize measurements of all time instants, espe-
cially the signals in the latter parts including residual
displacement, EKS based on Rauch-Tung-Striebel
smoothing15) is applied to the outcome of EKF. With
the estimated values and error-covariance of state
vector at the ending point, i.e. ˆa N | NX and
N | NP , the output of the smoother at k instant
(k=N-1, N-2,…, 1) is written as
1
1 1| 1 | 1T
k k k k k k
J = P F P (21)
1
ˆ ˆ1| 1 | 1
ˆ ˆ| | 1
a a
k a a
k N k k
k N k k
X X
J X X (22)
1 1
1 | 1 | 1
| | 1 T
k k
k N k k
k N k k
P P
J P P J (23)
in which N stands for the total number of time sam-
ples; J is the gain matrix of the Kalman smoother; all
other symbols are the same as those in section 2.(1).
Smoother outputs are more accurate than those
from the filter as shown in the following. Thus after
applying the EKF-RM method, the estimated result is
further smoothed by EKS. Overall, Fig. 3 shows the
flow chart of the displacement estimation method.
Fig. 3 Flowchart
3. NUMERICAL VERIFICATION
(1) Application to a bi-linear SDOF model The illustrative model considered here is a SDOF
system with a bi-linear hysteresis model. The pro-
totype of the model is simplified from a full-scale
bridge pier experiment, called C-1-1 experiment in
E-defense shaking database16), as shown in Fig. 4.
The basic parameters of the SDOF model include
mass m=252.5 ton, elastic stiffness k=1.625×107
N/m, damping coefficient c=4.05×104 Ns/m,
post-yielding stiffness ratio α=0.05, and yielding
displacement dy=0.05m.
The forward simulation is conducted using in-
cremental Newmark-β method with β=1/6 and the
time interval equals 0.005 s. Northridge earthquake
acceleration is treated as excitation. The simulated
displacement, acceleration, hysteresis loop as well as
the earthquake acceleration is shown in Fig. 5. The
simulated acceleration signal is then added with
white noise process whose RMS equals 5% of the
signal’s RMS (1.2 m/s2) to consider practical meas-
urement noise. Based on the noisy acceleration, as
mentioned earlier, displacement is obtained by dou-
ble integrating the acceleration and using a high pass
filter with a 0.3 Hz cutoff frequency. Fig. 5 (e) shows
the displacement from the integration of the acceler-
ation and low-frequency displacement drift, i.e. dif-
ference between the integrated and true displace-
ment. In this case, the residual displacement occurs
from about 15 s with 2.3 cm. Subsequently,
time-variant stiffness value can be identified using
the displacement from integration and the accelera-
tion. R matrix and initial Q matrix is set as
R=diag([10-5, 5×10-3]) and Q= diag([10-6, 10-4, 0, 0]);
initial parameter value and error covariance matrix
are set 1.0θreal and diag([0.1θreal]2) where θreal stands
for the true elastic parameter vector. Besides, the αQ
coefficient of the RM algorithm is set as 1/15. The
identified parameter time histories are shown in Fig.
6. As mentioned earlier, by using the displacement
from integration and the acceleration, the
time-variant parameters are identified based on the
commonly used state-space model, i.e. equivalent
stiffness and damping coefficient of the nonlinear
system are identified rather than tangential ones.
5
Except for the fluctuation during the initial period,
significant stiffness reduction occurs between 4~15 s
which matches the time interval of significant
low-frequency displacement drift well as shown in
Fig. 7(a). For the latter part of the identification, e.g.
after 18 s, the identified parameters coincide with
those true elastic parameter values well because the
system behavior is elastic.
Based on the identification results, the time in-
terval with virtual displacement is defined from 3.09
s to 15.7 s. The latter part of the displacement from
integration is summed with the residual displace-
ment, i.e. 2.3 cm. The displacement observation is
shown in Fig. 7(b).
(a)
(b)
Fig. 4 (a) C1-1 experiment structure; (b) simplified model
(d)
(a)
(b)
(c) (e)
Fig. 5 Bi-linear model simulation response: (a) earthquake acceleration; (b) displacement; (c) acceleration; (d) hysteresis loop; (e)
displacement from integration of acceleration and low frequency displacement drift
(a) (b)
Fig. 6 Time-variant parameter estimation history: (a) stiffness; (b) damping coefficient
(a) (b)
Fig. 7 (a) Displacement from integration of acceleration; (b) displacement observation
In terms of the measurement noise of virtual dis-
placement to be set in the R matrix, as studied in
Chatzi et al.14) and Naets et al.17), a wide range of
values is applicable as long as they are above some
lower threshold. When the measurement noise value
is too small, the filter will force the estimated dis-
6
placement to adhere to the virtual one. In this study,
the measurement noise value is selected based on the
maximum value of displacement from integration.
Specifically, 0.10 m is employed here because the
maximum value of the displacement from integration
is about 0.09 m. For the linear behavior part, the
displacement measurement noise value used in
time-variant parameter identification can be applied.
However, since the displacement from integration
cannot match the real displacement perfectly, e.g.
during the initial period of time, a relatively large
value could be employed for the displacement
component in the R matrix in order to estimate more
accurate displacement. In this case, it is selected as
0.01 m.
As discussed above, the displacement and accel-
eration observations are those of Fig. 7(b) and Fig.
5(c). The R matrix is set as diag([10-2, 5×10-3]) and
diag([10-4, 5×10-3]) for the virtual displacement and
the linear behavior displacement part respectively.
Initial Q matrix is set as diag([10-6, 10-4, 10-2, 0])
corresponding to displacement, velocity, accelera-
tion, and stiffness value. Note that the damping co-
efficient is regarded as a known value here because
the time-variant parameter identification step has
determined the value, i.e. Fig. 6(b). Other computa-
tion parameters remain unchanged.
Fig. 8(a) shows the estimated displacement from
EKF and EKS as well as true displacement signal,
while Fig. 8(b) shows the estimation errors of EKF
and EKS results. The EKS result is closer to the real
displacement of which the maximum error is about
1.5 cm while the error of EKF is about 3.7 cm. The
maximum displacement of 0.1056 m takes place at
12.28 s. The corresponding error is 0.98 cm and 2.89
cm with respect to EKS and EKF results.
The displacement from integration is presented in
Fig. 9(a). The error at the maximum displacement is
3.4 cm. In this case, it shows that by incorporating
the information of the latter residual displacement
part based on EKS, the estimation accuracy of the
virtual displacement part can be improved.
In this example, the virtual displacement is ini-
tially assumed as a straight line as shown in Fig. 7(b).
If the low-frequency drift component is extracted
from the former EKS result, it can be regarded as the
virtual displacement, and the updated displacement
measurement is shown in Fig. 9(b). Based on the
updated virtual displacement, the estimation is re-
peated and the displacement results are shown in Fig.
10. The estimation result based on the updated virtual
displacement is more accurate than the previous case.
Specifically, the maximum error is about 0.5 cm and
2.4 cm for EKS and EKF respectively. In terms of the
maximum displacement, the error is 0.25 cm and
1.56 cm. The improvement is due to the fact that the
updated virtual displacement is closer to the true
low-frequency drift part of displacement. For exam-
ple, from 8~15 s, the true low-frequency displace-
ment drift and the update virtual displacement both
present upward curves while it is originally assumed
as a straight line. This is also the reason that dis-
placement is underestimated between 8~15 s as
shown in Fig. 8(b). Intuitively, it seems that by
consecutive iterations, the estimated displacement
will be even closer to the true one. This is not the
case, because some parts are closer but other parts are
farther from the true displacement after several iter-
ations. Besides that, it is difficult to find an index
indicating when the iteration should stop based on
the currently available observation information.
Therefore, this 2 iteration computation scheme is
employed empirically.
(a)
(b)
Fig. 8 Using straight line virtual displacement: (a) estimated
displacement; (b) estimation error
(a)
(b)
Fig. 9 (a) Comparison of displacement from integration and true
displacement; (b) updated displacement measurement
(a)
7
(b)
Fig. 10 Using updated displacement measurement: (a) estimated
displacement; (b) estimation error
(a)
(b)
Fig. 11 (a) Time-variant stiffness estimation history; (b) real
tangential stiffness
As mentioned earlier, along with the displacement,
time-variant stiffness, specifically tangential stiff-
ness, is also identified in this method because of the
incremental Newmark-β state-space model. Howev-
er, this stiffness is not expected to be identified ac-
curately because only acceleration measurement has
a small observation noise level and is mainly func-
tioning for parameter identification during the non-
linear behavior period. Fig. 11(a) shows the identi-
fied time-variant stiffness while Fig. 11(b) presents
the true time history of the tangential stiffness. A
large discrepancy exists between the two. In fact, if
accurate tangential stiffness is expected to be ob-
tained, displacement measurement is suggested to be
included4). Therefore, by identifying the time-variant
stiffness here, the nonlinearity of the system is in-
tended to be considered as much as possible.
(2) Application to various hysteresis model and
earthquake excitations
In this section, the proposed methods are further
numerically investigated using various hysteresis
models and earthquake excitations. Specifically,
three hysteresis models including a bi-linear model, a
tri-linear model, and a Bouc-Wen model are consid-
ered with four earthquake excitations, i.e. Takatori,
Northridge, Imperial Valley, and San Fernando
earthquake. Additionally, two levels of peak ground
acceleration are employed here, i.e. 400 gal and 800
gal. The basic parameter values of these hysteresis
models and earthquake accelerations are summarized
in Table 1 and Fig. 12 respectively. The hysteresis
loops of the three models under 400 gal Northridge
earthquake excitations are shown in Fig. 13.
The forward simulations are conducted with 0.005
s time interval. 5% of RMS white noises are added to
the simulated accelerations. Besides displacement
estimated from the proposed method, displacements
from the integration of the noisy accelerations are
also compared in this section. Only the 2nd iteration
results are shown here.
Two errors based on EKS result, i.e. error at
maximum displacement and maximum error, nor-
malized by maximum displacement are employed
here to represent the estimation accuracies quantita-
tively. In the following part, they are referred to the
error1 and error2 respectively for simplicity. The bar
figures, i.e. Fig. 14 to Fig. 16, show the results of all
cases.
Table 1 summary of basic parameters of three hysteresis models
Parameter Bi-linear Tri-linear Bouc-Wen
Mass m (ton) 252.5
Elastic stiffness k (N/m) 1.625×107
Elastic damping factor c (Ns/m) 4.05×104
The 1st post-yielding displacement dy1 (m) 0.05 0.03 0.05
The 2nd post-yielding displacement dy2 (m) N.A. 0.05 N.A.
The 1st post-yielding stiffness ratio α1 0.05 0.20 0.05
The 2nd post-yielding stiffness ratio α2 N.A. 0.067 N.A.
Loop parameter n N.A. N.A. 5
Loop parameter μ N.A. N.A. 1
Stiffness degradation parameter δυ N.A. N.A. 20
Strength degradation parameter δη N.A. N.A. 20
8
(a) (b)
(c) (d)
Fig. 12 Earthquake excitations of 400 gal: (a) Takatori; (b) Northridge; (c) Imperial Valley; (d) San Fernando
(a) (b) (c)
Fig. 13 Hysteresis loops under 400 gal Northridge earthquake excitations:
(a) bi-linear model; (b) tri-linear model; (c) Bouc-Wen model
(a) (b)
(c) (d)
Fig. 14 Estimation errors of bi-linear model:
(a) error1 in 400 gal case; (b) error2 in 400 gal case; (c) error1 in 800 gal case; (d) error2 in 800 gal case
(a) (b)
(c) (d)
Fig. 15 Estimation errors of tri-linear model:
(a) error1 in 400 gal case; (b) error2 in 400 gal case; (c) error1 in 800 gal case; (d) error2 in 800 gal case
Generally, the proposed method presents good
performances and outperforms the double integration
method significantly. Usually, error1 is smaller than
error2, i.e. most of the error1 values are below 10%
while error2 values are generally lower than 20%.
Therefore, the largest error of an estimated signal
does not occur at the maximum displacement point,
which is important from a structural point of view.
9
(a) (b)
(c) (d)
Fig. 16 Estimation errors of Bouc-Wen model:
(a) error1 in 400 gal case; (b) error2 in 400 gal case; (c) error1 in 800 gal case; (d) error2 in 800 gal case
4. CONCLUSION
In this paper, an EKF based displacement estima-
tion method is proposed for nonlinear SDOF systems
under seismic excitation. With acceleration meas-
urement, displacement from the integration of ac-
celeration and residual displacement regarded as
known, displacement signal during significant sys-
tem nonlinearity is estimated along with time-variant
stiffness parameter in EKF, and further smoothed
using EKS. The effectiveness of the proposed
method is verified through a numerical bi-linear
SDOF model in detail and further investigated con-
sidering various hysteresis models and earthquake
excitations. Validation of the proposed method using
experimental data is expected to be conducted in
future studies.
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(Received June 30, 2020)
(Accepted July 31, 2020)
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