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5.4. Simulation Results
is the disturbance location matrix,
To demonstrate the effectiveness of the proposed
HRC MIMO ARX-TS fuzzy models, a three-story
building structure employing an MR damper is
investigated. Two input signals, which are a dis-
turbance signal and a control signal, are applied
to the benchmark three-story building structure to
generate output data. Figure 3 shows the first input
signal, which is an artificial earthquake used as
a disturbance input signal such that the spectrum
of the random signal includes the frequency char-
acteristics of the earthquake ground acceleration.
The second input is the MR damper force signal
as shown in Figure 4. On the other hand, the 3
rd
floor acceleration and the 1
st
floor displacement
responses are selected as output signals.
The model order of the MIMO ARX-TS fuzzy
model is chosen to be
n
=
m
= 2. Note that the
number and the type of the MFs are determined
via trial-and-errors. Although the architecture of
the MIMO ARX-TS fuzzy model can be optimized
via an optimization procedure (e.g., genetic algo-
rithm), it is beyond the scope of the present
chapter. However, the authors intend to optimize
the architecture of the MIMO ARX-TS fuzzy
model in near future. Note that the performance
of the identified model can be improved by in-
creasing either the order of the MIMO ARX-TS
fuzzy model or the number of the MFs, resulting
in the larger dimension of the fuzzy rule base.
Figure 5 and Figure 6 compare the displacement
and acceleration responses of the original simula-
tion model with those of the identified HRC MIMO
ARX-TS fuzzy model, respectively. As can be
seen from the figures, overall good agreements
between the original values and the identified
HRC MIMO ARX-TS fuzzy models are found in
the time histories of both displacement and ac-
celeration responses.
Table 2 shows the error of the identified MIMO
ARX-TS fuzzy model: represents the mean
−
1
1
0
F
=
−
(37)
0
1
1
−
0
0
1
is the location matrix that a Chevron brace is
located within the building structure,
n
is the
noise vector,
z
1
and
z
4
are the displacement and
the velocity at the 1
st
floor level of the three-story
building structure, respectively,
I
is the identity
matrix, and
0
is the zero matrix. Note that in the
earthquake engineering applications, the earth-
quake disturbance excites all the floor levels
within the building structure as the inertia forces,
i.e. is a 3 × 1 vector with the same component of
the 1940 El-Centro earthquake. Properties of the
three story building structure are adopted from a
scaled model (Dyke et al. 1996) of a prototype
building structure that was developed by Chung et
al. (1989). The mass of each floor
m
1
=
m
2
=
m
3
=
98.3 kg; the stiffness of each story
k
1
= 516,000
N/m,
k
2
= 684,000 N/m, and
k
3
= 684,000 N/m;
and the damping coefficients of each floor
c
1
=
125 Ns/m,
c
2
= 50 Ns/m, and
c
3
= 50Ns/m. In ad-
dition, a SD-1000 MR damper whose parameters
are given in Table 1 is installed on the 1
st
floor
level using a Chevron brace, which leads to a
nonlinear dynamic model, i.e., a building-MR
damper system. Based on the physical model, a
set of input-output data is generated for training
the proposed HRC MIMO ARX-TS fuzzy system
identification procedures. Note that it is challeng-
ing to identify
M
,
C
, and
K
matrices through a
linear time-invariant (LTI) model framework
because the building structures employing MR
dampers are nonlinear time-varying systems.
Therefore, it is recommended to develop a nonlin-
ear time-varying model framework for modeling
the building-MR damper system.
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