Information Technology Reference
In-Depth Information
Definition 3.
A system S is a fuzzy system if (input) u
(
t
)
, output y
(
t
)
, or state s
(
t
)
of
S or any combination of them ranges over fuzzy sets. [104, p. 33].
Here, we have the same systems equations that hold for usual systems but with
different meanings for
u
,
y
,and
s
as specified in their definition (
t
=
0
,
1
,
2
,...
):
s
t
+
1
=
f
(
s
t
,
u
t
)
and
y
t
=
g
(
s
t
,
u
t
)
(3.9)
Zadeh explained that “these concepts relate to situations in which the source of
imprecision is not a random variable or a stochastic process but rather a class or
classes which do not possess sharply defined boundaries.” [104, p. 29]
In 1968, Zadeh presented “fuzzy algorithms”, a concept that “may be viewed as a
generalization, through the process of fuzzification, of the conventional (nonfuzzy)
conception of an algorithm.” [106, p. 94] Inspired by this idea, he wrote in the
article “Fuzzy Algorithms” that all people function according to fuzzy algorithms
in their daily life - they use recipes for cooking, consult the instruction manual to
fix a TV, follow prescriptions to treat illnesses or heed the appropriate guidance to
park a car. Even though activities like this are not normally called algorithms: “For
our point of view, however, they may be regarded as very crude forms of fuzzy
algorithms.” [106, p. 95]
In 1973, in his “Outline of a New Approach to the Analysis of Complex Systems
and Decision Processes” [110], he combined this concept of fuzzy algorithms with
a new approach that was supposed to bring about a completely new form of system
analysis based on his Fuzzy Set Theory: “The approach described in this paper
represents a substantial departure from the conventional quantitative techniques of
system analysis.” [110, p. 28] This new way of going about system analysis differed
from the conventional approach in the following new concepts:
•
Linguistic variables: i.e. variables whose values are words or terms from natural
or artificial languages. For instance,
not very large
,
very large
or
fat
,
not fat
or
fast
,
very slow
are terms of the linguistic variables size, fatness and speed. Zadeh
represented linguistic variables as fuzzy sets whose membership functions map
the linguistic terms onto a numerical scale of values (see Fig. 3.5.
•
Fuzzy IF-THEN Rules: i.e. composite statements of the form IF
A
THEN
B
,
where
A
and
B
are fuzzy expressions, “terms with a fuzzy meaning, e. g., 'IF
John is nice to you THEN you should be kind to him,' are used routinely in
everyday discourse. However, the meaning of such statements when used in
communication between humans is poorly defined.” [110, p. 29]
Zadeh often compared the strategies of problem solving by computers on the one
hand and by humans on the other hand. In a conference paper in 1970 he called it
a paradox that the human brain is always solving problems by manipulating “fuzzy
concepts” and “multidimensional fuzzy sensory inputs” whereas “the computing
power of the most powerful, the most sophisticated digital computer in existence”
is not able to do this. Therefore, he stated that “in many instances, the solution to
a problem need not be exact”, so that a considerable “measure of fuzziness in its
