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ANALOG INPUTS

FUZZIFICATION
INPUT FUZZY
VARIABLES
RULE EVALUATION
OUTPUT FUZZY
VARIABLES
DEFUZZIFICATION
FIGURE 32.19 The block diagram of the fuzzy con-
troller. ANALOG OUTPUTS

57 O F 80 O F
COLD COOL NORMAL WARM HOT HOT 0 HOT 0
FUZZIFICATION 0.3 FUZZIFICATION 0.2
WARM
WARM
57 O F NORMAL 0 80 O F NORMAL 0.7
T COOL 0.5 COOL 0
O F COLD 0 COLD 0
(a) 20 30 40 50 60 70 80 90 100 110
FIGURE 32.20 Fuzzification process: (a) typical membership functions for the fuzzification and the defuzzification
processes, (b) example of converting a temperature into fuzzy variables.

Fuzzification
The purpose of fuzzification is to convert an analog variable input into a set of fuzzy variables. For higher
accuracy, more fuzzy variables will be chosen. To illustrate the fuzzification process, consider that the input
variable is the temperature and is coded into five fuzzy variables: cold, cool, normal, warm, and hot. Each
fuzzy variable should obtain a value between zero and one, which describes a degree of association of the
analog input (temperature) within the given fuzzy variable. Sometimes, instead of the term degree of associ-
ation, the term degree of membership is used. The process of fuzzification is illustrated in Fig. 32.20. Using
Fig. 32.20 we can find the degree of association of each fuzzy variable with the given temperature. For example,
for a temperature of 57°F, the following set of fuzzy variables is obtained: [0, 0.5, 0.2, 0, 0], and for T = 80°F,
it is [0, 0, 0.25, 0.7, 0]. Usually only one or two fuzzy variables have a value other than zero. In the example,
trapezoidal functions are used for calculation of the degree of association. Various different functions such
as triangular or Gaussian can also be used, as long as the computed value is in the range from zero to one.
Each membership function is described by only three or four parameters, which have to be stored in memory.
For proper design of the fuzzification stage, certain practical rules should be used:
• Each point of the input analog variable should belong to at least one and no more than two
membership functions.
• For overlapping functions, the sum of two membership functions must not be larger than one.
This also means that overlaps must not cross the points of maximum values (ones).
• For higher accuracy, more membership functions should be used. However, very dense functions
lead to frequent system reaction and sometimes to system instability.

Rule Evaluation
In contrary to boolean logic where variables can have only binary states, in fuzzy logic all variables may
have any values between zero and one. The fuzzy logic consists of the same basic: ∧—AND, ∨—OR, and
NOT operators:
A ∧ B ∧ C ⇒ min{A, B, C}—smallest value of A or B or C
A ∨ B ∨ C ⇒ max{A, B, C}—largest value of A or B or C
A ⇒ 1 1–A—one minus value of A

©2002 CRC Press LLC