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c
= ×
2
ξ
×
(
m k
×
)
(21)
i
R IF x is A AND x is A THEN y is Bi
:
d
tmd
d
d
1
1
i
2
2
i
(22)
Then
′
m
0
,
β
and
ξ
tmd
are considered as the
design variables in the multi-objective optimiza-
tion procedure.
The equation of motion of the building with
a semi active tuned mass damper (STMD) control
device is the same as the structure with TMD, but
in this case the damping ratio of STMD is a time
varying, and can be expressed as
ξ
stmd
( )
which
can be regulated by a fuzzy logic system, briefly
described in the following.
where
x
1
and
x
2
are input linguistic variables,
y
is
the output linguistic variable; and
A
1i
,
A
2i
, and
B
are the values for each input linguistic variables
and output linguistic variable, respectively. In this
study,
x
1
and
x
2
are the displacement and velocity
of the top floor, respectively; and
y
is the damp-
ing ratio of the STMD system. The design of a
fuzzy system involves decisions about a num-
ber of important design parameters that should
be determined before the actual system starts.
These parameters are the fuzzy sets in the rules,
the rules themselves, scaling factor in input and
output, inference methods, and defuzzification
procedures (Pourzeynali et al., 2007). Because
of a crisp number for real application, defuzzifier
maps the system output from the fuzzy domain into
the crisp domain. The center of area (COA) and
the mean of maximum (MOM) are the two most
commonly used methods in generating the crisp
system output (Shin & Xu, 2009). In this study,
the center of area method is selected to produce
the crisp system output in discrete universe of
discourse (Shin & Xu, 2009):
Fuzzy Logic Controller
In this study, damping ratio of the STMD is regu-
lated by a fuzzy logic controller. For the first time,
Fuzzy set theory was proposed by Lotfi Zadeh
in 1965 (Zadeh, 1965). Fuzzy set theory allows
objects to have a degree of membership within a
set, while traditional mathematics requires objects
to have either 0 or 100 per cent membership within
a set. As a result, fuzzy controller, which is based
on the fuzzy set theory, is a reliable method to deal
with the imprecision and uncertainty that is often
present in real-world applications. Nowadays,
fuzzy systems are used in a wide range of science
and technology such as control, signal process-
ing and etc. Important information of practical
systems originates from two sources: the first
one is experiences of human beings that define
their knowledge about the systems with natural
language; and the another source is measurement
and mathematical models derived from physical
rules. Fuzzy systems are knowledge-based or rule-
based systems. The main part of a fuzzy system
is a knowledge database which is composed of
IF-THEN rules based on classical control theories.
A fuzzy system consists of four parts, the fuzzifier,
the fuzzy rule base, the inference engine, and the
defuzzifier. The fuzzy rule base in this study is
based on a Mamdani linguistic fuzzy model which
can be written as:
q
( )
∑
x
.
µ
x
i
A
*
x
=
i
=
1
(23)
q
( )
∑
µ
x
A
i
i
=
1
where
q
is the number of the discreet elements in
the universe of discourse;
x
i
is the value of discrete
element; and
µ
A
( )
offers the corresponding
membership function value at the point
x
i
. To
achieve a fuzzy system with minimum design
variables, the ideas proposed by Park et al. (1995)
is used. In this method, the membership functions
are considered as triangular membership functions
and since often dynamic systems such as vibra-
i
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