Digital Signal Processing Reference
In-Depth Information
low
medium
1
1
1
α
1
β
β
y
1
1
x
2
y
1
=f
1
(x
1
,x
2
)
x
1
medium
high
1
1
1
β
2
α
α
2
2
y
x
1
x
2
y
2
=f
2
(x
1
,x
2
)
1
β
y
+
α
y
y
=
1
1
2
2
β
+
α
1
2
y
y
Fig. 11.15
Takagi-Sugeno FIS structure
y
k
¼
f
k
ð
X
Þ¼
X
n
a
ki
x
i
þ
a
k0
:
ð
11
:
30
Þ
i
¼
1
The weighting coefficients in the final output equation
y
¼
P
k
¼
1
d
k
y
k
P
k
¼
1
d
k
:
ð
11
:
31
Þ
are minimum values of the membership grades for given rule read out from the
MFs for all input values, i.e. d
1
¼
min
ð
a
1
;
b
1
Þ¼
b
1
and d
2
¼
min
ð
a
2
;
b
2
Þ¼
a
2
.
Experiences show that Sugeno reasoning schemes are computationally effective
and work well with optimization and adaptive techniques, which makes them very
attractive in control problems, particularly for dynamic and nonlinear systems.
However, their tuning (setting the factors in rule output equations (
11.30
)) is not
straightforward. The coefficients are difficult to be defined intuitively, they are
usually set on basis of the results of thorough simulation studies, which can be
seen as a serious disadvantage of this scheme.
On the contrast, the Mamdani scheme [
9
,
22
] (illustration in Fig.
11.16
) allows
to describe the expertise in more intuitive, more human-like manner. However,
Mamdani-type fuzzy inference entails a substantial computational burden. Here, as
results of rule firing the fuzzy output values are obtained that are represented by
areas being parts of output MFs lying under certain membership levels (here, with
application of min-type implication—min
ð
a
1
;
b
1
Þ¼
b
1
and min
ð
a
2
;
b
2
Þ¼
a
2
).
The schemes' final fuzzy output is reached by aggregation of particular rules'
outputs by taking maximum values over all partial output MFs, which gives an
area colored in red in Fig.
11.16
. The crisp output of the scheme is further obtained
by applying one of the available signal defuzzification methods.
Search WWH ::
Custom Search