Digital Signal Processing Reference
In-Depth Information
Therefore definition of fuzzy implication is as follows:
• with use of min for conjunction
l
p
!
q
ð
x
;
y
Þ¼
1
l
p
\
q
ð
x
;
y
Þ¼
1
min
½
l
p
ð
x
Þ;
1
l
q
ð
y
Þ:
ð
11
:
26
Þ
• with use of product for conjunction
l
p
!
q
ð
x
;
y
Þ¼
1
l
p
\
q
ð
x
;
y
Þ¼
1
l
p
ð
x
Þ½
1
l
q
ð
y
Þ;
ð
11
:
27
Þ
where l
p
ð
x
Þ;
l
q
ð
y
Þ
are membership values for the antecedent and consequent,
respectively.
In engineering applications, where the cause-effect relationships are usually
studied, the implications (
11.26
) and (
11.27
) are often applied in simplified forms
[
8
,
9
]:
• Mamdani definition
l
p
!
q
ð
x
;
y
Þ¼
min
½
l
p
ð
x
Þ;
l
q
ð
y
Þ:
ð
11
:
28
Þ
• Larsen definition
l
p
!
q
ð
x
;
y
Þ¼
l
p
ð
x
Þ
l
q
ð
y
Þ:
ð
11
:
29
Þ
that can be accepted without further proof for all causal engineering systems.
The effects of using (
11.28
) and (
11.29
) may be quite different, which is
illustrated in Fig.
11.12
. The Mamdani definition results in so called MF clipping
(Fig.
11.12
a), whereas with Larsen definition one may observe MF scaling
(Fig.
11.12
b). The application of either definition is dependent on the practical
problem to be solved.
11.3 Fuzzy Reasoning
A typical Fuzzy Inference System is composed of a number of fuzzy rules
(implications) as described above. The rules are usually stored in the so called
tables of truth (where all the rules with their inputs and outputs are listed) or in the
so called FAMs (Fuzzy Associative Memory) being a kind of relational matrix, as
illustrated in Fig.
11.13
. In the example shown the acceleration of a car is defined
with a set of rules expressed linguistically as functions of distance to trip desti-
nation and car velocity. From particular cells of the FAM one can read out the
required car acceleration; e.g. for very small distance (VS) and very high car
velocity (VH) one can see that the acceleration should be very negative (VN),
which means the car should significantly decelerate. The consequence of particular
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