Elements of Fuzzy Logic in Protective Relays - Digital Signal Processing in Power System Protection and Control - page 221

Fig. 11.1 Membership
function of the set of people
''around 50 years old''

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0 10 20 30 40 50 60 70 80 90 100

X

rather low. In other words, we would say that he/she ''rather does not belong to the
fuzzy set A'' .
One may enumerate a lot of examples of engineering terms that are quite often
used as fuzzy instead of providing precise/measured values. They are, e.g.,

• bandwidth: narrowband, broadband,
• correlation: low, medium, high, perfect,
• errors: large, medium, small, a lot of, not so great, very large, very small, almost
zero,
• frequency: low, high, ultra-high,
• resolution: low, high,
• sampling: low-rate, medium-rate, high-rate,
• stability: stable (lightly damped, highly damped, over-damped, critically

damped), unstable.

With regard to fuzzy sets the following set-theoretical operations can be
defined [ 5 ]:

• subset (containment of A in B)

A B , l A ð x Þ l B ð x Þ; x 2 X : ð 11 : 3 Þ

• complement (not A, defining how much do elements not belong to the set)

A ¼ X A , l A ð x Þ¼ 1 l A ð x Þ: ð 11 : 4 Þ

• union (sum of sets, determining how much of the element is in either set)

- logical sum

C ¼ A [ B , l C ð x Þ¼ max l A ð x Þ; l B ð x Þ ½ ¼ l A ð x Þ_ l B ð x Þ: ð 11 : 5 Þ