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In-Depth Information
considered an implementation of the input-output fuzzy models of dynamic systems
proposed by Yager [17].
Here, we introduce the main concepts and elements of our paradigm for system
modeling allowing experts to build comprehensible fuzzy linguistic models in an
easier way. In our framework, a FFSM is a tuple
{
Q
,
U
,
f
,
Y
,
g
}
,where:
•
Q
is the state of the system.
•
U
is the input vector of the system.
•
f
is the transition function which calculates the state of the system.
•
Y
is the output vector of the system.
•
g
is the output function which calculates the output vector.
Each of these components is described in the following subsections. Further-
more, the interested reader can refer to [1-3, 14] for additional information and
applications.
Fuzzy States (
Q
)
The state of the system (
Q
) is defined as a linguistic variable [20] that takes its
values in the set of linguistic labels
, with
n
being the number of
fuzzy states. Every fuzzy state represents the pattern of a repetitive situation and it
is represented numerically by a state activation vector
S
{
q
1
,
q
2
,...,
q
n
}
[
t
]=(
s
1
[
t
]
,
s
2
[
t
]
,...,
s
n
[
t
])
,
n
∑
i
=
1
where
s
i
[
t
]
∈
[
0
,
1
]
and
s
i
[
t
]=
1.
S
0
is defined as the initial value of the state
activation vector, i.e.,
S
0
=
S
[
t
=
0
]
.
Input Vector (
U
)
(
,
, ...,
u
n
u
)
U
is the input vector
, with
n
u
being the number of input variables.
U
is a set of linguistic variables obtained after fuzzification of numerical data. Typi-
cally,
u
i
can be directly obtained from sensor data or by applying some calculations
to the raw measures, e.g., the derivative or integral of the signal, or the combination
of several signals. The domain of numerical values that
u
i
can take is represented
by a set of linguistic labels,
A
u
i
=
{
u
1
u
2
A
n
i
A
u
i
,
A
u
i
,...,
u
i
}
, with
n
i
being the number of
linguistic labels of the linguistic variable
u
i
.
Transition Function (
f
)
The transition function (
f
) calculates, at each time instant, the next value of the
state activation vector:
S
. It is implemented by means of a
fuzzy rule-based system (FRBS). Once the expert has identified the relevant states
in the model, she/he must define the allowed transitions among states. There are
rules
R
ii
to remain in a state
q
i
,andrules
R
ij
to change from state
q
i
to state
q
j
.Ifa
transition is forbidden in the FFSM, it will have no fuzzy rules associated.
The generic expression of a rule to remain in a state
q
i
(
R
ii
) is formulated as
follows: IF (
S
[
t
+
1
]=
f
(
U
[
t
]
,
S
[
t
])
(
u
1
is
A
u
1
)
(
u
n
u
is
A
u
n
u
)
[
t
]
is
q
i
)
∨
∨...∨
∨
(
d
i
is
T
st ay
i
)THEN
S
[
t
+
1
]
is
q
i
,where:
