Digital Signal Processing Reference
In-Depth Information
• high precision is not always needed and such fuzzy statements are good enough
to represent the main meaning of the statement,
• accurate measurement of certain variables is difficult, costly and may take a lot
of time,
• precise measurements bring not much when the model of the plant to be con-
trolled/protected is very complex,
• the decision is sometimes to be made quickly, even if the measurement of
criteria signals is not very precise.
Therefore, especially when the final result/decision is of classification type (like
an assessment of the state of protected plant by the protective relays), precise
statements are not better than the ones that are not that accurate (fuzzy). Moreover,
a very precise model may in some cases cause difficulties when some information
is missing or becomes corrupted or contaminated by noise. The well known
Zadeh's principle of incompatibility [
21
]: ''As the complexity of a system
increases, our ability to make precise and yet significant statements about its
behavior diminishes until a threshold is reached beyond which precision and
significance (or relevance) become almost mutually exclusive characteristics'',
leads to the conclusion that the usage of fuzzy sets/FL may become a mechanism
for abstraction of unnecessary or too complex details.
Before the application of fuzzy sets and logic for protection and control are
outlined, let us consider/recall the basic definitions and rules of the related theory.
Coming out from linguistic variables that correspond to the imprecise linguistic
statements, let us define the fuzzy set as follows. A fuzzy set A in the universe of
discourse X is expressed as a set of ordered pairs [
5
,
22
]
A
¼fð
x
;
l
A
ð
x
ÞÞj
x
2
X
g;
X :
!½
0
;
1
ð
11
:
1
Þ
where l
A
ð
x
Þ
is a membership function (MF), l
A
ð
x
Þ2½
0
;
1
.
A fuzzy set is totally characterized by its MF. It brings an information about the
degree (between 0 and 1, inclusive) to which an element belongs to the set A.
In many applications fuzzy sets provide an interface between a numerical scale
and a symbolic scale (composed of linguistic terms). The elements of X may be of
various natures, while the X itself may be discrete (with limited or countable
number of elements) or continuous. For example, the MF for the fuzzy set defined
linguistically as ''set of people being around 50 years old'' may be expressed by
1
1
þ
x
50
l
A
ð
x
Þ¼
10
4
:
ð
11
:
2
Þ
which is illustrated graphically in Fig.
11.1
.
The notion of fuzziness may be characterized as the object similarity to
imprecisely defined properties. Looking at Fig.
11.1
one can conclude that a
person belongs to the set ''around 50 years old'' with the degree of membership
higher than 0.5 only when his/her age is in the range (40, 60). According to (
11.2
)
a person being 65 years old belongs to A only with the degree of ca. 0.16, which is
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