Geology Reference
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where
R
j
is the
j
th
fuzzy rule;
z
i
FZ
are premise
variables that can be either input or output values;
pi,j are fuzzy sets centered at the jth operating
point;
y
thought as very large quantity?” or “Which degree
of temperature is high?” In reality, it is impossible
to model the uncertain variables in a conventional
way; while, MFs can be used for modeling such
variables as an element of a fuzzy set. Fuzzy sets
are constructed from MFs. For example, a fuzzy
set of the structural damage can be constructed as
three MFs, e.g., small, medium, and large. This
fuzzy set is used for constructing the premise part
of IF-THEN rules, i.e., IF STATEMENT.
=
(
)
1
z
i
, ...,
can be any type of func-
tion in terms of the premise vector
z
FZ
f z
FZ
FZ
=
, ,
. The TS fuzzy model-based
reasoning is to simply compute weighted mean
values
z
1
z
i
FZ
FZ
N
r
∑
w y
j
j
2.2. Fuzzy Rules
j
=
1
y
final
=
,
(2)
N
r
∑
w
A fuzzy rule base has a set of fuzzy IF-THEN rules;
e.g., “if a building structure has large damage, a
controller is operated such that an alarm is rung
twice”, “if the structural damage is medium, the
controller is operated such that the alarm is rung
once”, and “if there is no damage in the building
structure, the controller is not operated.” The set
of IF-THEN rules is blended into an integrated
system through fuzzy reasoning methods.
j
j
=
1
where
w
j
is the fuzzy interpolation parameters;
N
r
is the number of the fuzzy rules. To appropriate
model nonlinear behavior of dynamic systems,
ARX input models are applied to the consequence
part of the TS fuzzy model.
2.3. Fuzzy Reasoning
3. AUTOREGRESSIVE-
EXOGENOUS INPUT TAKAGI-
SUGENO FUZZY MODEL
Fuzzy reasoning is a mechanism to perform the
fuzzy inference system that derives conclusions
from a family of IF-THEN rules, i.e., fuzzy
reasoning is a methodology to organize a set of
the IF-THEN rules. Takagi and Sugeno (1985)
developed a systematic methodology for a fuzzy
reasoning using linear functions in the consequent
part. Because the TS-fuzzy model uses linear func-
tions in the consequent part, the defuzzification
procedure is not required. A typical fuzzy rule for
the TS fuzzy model has the form
A nonlinear dynamic system can be described
by the following multivariable nonlinear model
z
=
( ,
f
t
z u
, ),
(3)
where
t
is the time variable;
z
is a state vector;
u
is an input vector; and
f
represents a multivariable
nonlinear dynamic system. The nonlinear dynamic
model can be described by the multiple multi-input
multi output (MIMO) ARX input-based TS fuzzy
model (MIMO ARX-TS) fuzzy model
1
2
i
R
:
If
z
is
p
and
z
is
p
and
and
z
is
p
j
FZ
1
,
j
FZ
2
,
j
FZ
i j
,
(
)
1
z
i
FZ
Then
y
=
f z
, .
..,
,
FZ
(1)
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