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Oyadiji 2008). However, few investigations are
found in the field of the nonlinear fuzzy SI for
civil building structures subjected to earthquake
loadings (Adeli & Jiang 2006; Jiang & Adeli
2005). Adeli and Jiang developed a fuzzy wavelet
neural network (FWNN) model for nonlinear SI
of high-rise building structures. In their work,
the multi-input-single-output (MISO) FWNN
was trained by a hybrid Levenberg-Marquardt
least-squares algorithm. However, their approach
does not adopt a fuzzy model as an input-output
mapping function, but uses a fuzzy C-mean
clustering technique only as one of data mining
methods for use in a neural network model. Fur-
thermore, no investigation has been conducted
on a nonlinear Takagi-Sugeno (TS) fuzzy SI
for use with building structures equipped with a
highly nonlinear hysteretic control device such as
magnetorheological (MR) damper. Without the
TS fuzzy model for the nonlinear building-MR
damper systems, it is very difficult to design a
parallel distributed compensation (PDC)-based
TS fuzzy control system for damage mitigation
of seismically excited civil structures equipped
with MR damper systems (Kim et al. 2009) and it
would not be easy to develop a damage detection
algorithm for the seismically excited structure-MR
damper system. The reason is that the PDC-based
TS fuzzy control design framework requires
that a TS fuzzy controller, i.e., a nonlinear fuzzy
controller, has the same premise parameters as
a TS fuzzy model, i.e., dynamic model of the
building-MR damper system (Kim et al. 2009;
Kim et al. 2010). Therefore, this chapter proposes
a fuzzy modeling framework based on the TS
fuzzy model for identifying nonlinear behavior
of the building-MR damper systems subjected to
earthquake disturbances: hierarchical clustering-
based (HRC) multi-input, multi-output (MIMO)
autoregressive exogenous (ARX) Takagi-Sugeno
(TS) fuzzy model: HRC MIMO ARX-TS fuzzy
model. The advantages of the proposed HRC
MIMO ARX-TS fuzzy model can be summarized
as follows: 1) The proposed modeling framework
can be directly applied to the structure-MR damper
system without the decoupling process because it is
a nonlinear SI method; 2) it is more appropriate to
identify incomplete and incoherent measurements
of large civil structures than typical parametric
SI approaches; and 3) it provides a systematic
design framework for the PDC-based nonlinear
TS fuzzy controller.
In the rest of this chapter, first the HRC MIMO
ARX-TS fuzzy identification framework is de-
scribed, following by simulation results involving
the time histories of the excitation input signals
and the associated system output responses.
2. FUZZY LOGIC MODEL
In general, it is still challenging to develop a math-
ematical model for a dynamic model equipped
with the MR damper system because the nonlinear
dynamic system has multiple operating regions. In
this section, a fuzzy modeling framework will be
presented to model nonlinear behavior of structure-
MR damper systems: first, the fundamentals on
fuzzy logic models are summarized; and then
HRCARXTS fuzzy model is introduced; and
finally, simulation results are discussed.
2.1. Membership Functions
and Fuzzy Sets
Membership functions (MFs) and fuzzy sets are
the cornerstone of a fuzzy logic-based system that
is appropriate for modeling complex nonlinear
systems with uncertain parameters. There exist
always a variety of uncertainties in engineering
problems, e.g., “the structural damage is very
large” and “the performance of an MR damper
is sensitive to high temperature.” However, ques-
tions would arise: “How much damage would be
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