Fuzzy Logic Approach - Computational Intelligence in Time Series Forecasting - page 146

partially with degree of membership equal to 0.1, but it does not belong to the
fuzzy set FAST at all. It is important to note that in any of the above rules the
summation of degree of membership of output variable in the consequent fuzzy
sets need not always be 1.0. From the above three rules it is easy to understand that
relational fuzzy rule can be regarded as a generalization of the Mamdani-type
fuzzy rules.

4.4 Inferencing the Fuzzy Logic System

Inferencing refers to the process of generating the output fuzzy set when the fuzzy
rules and the input set are given. Usually, inferencing of Mamdani-type linguistic
fuzzy rules and relational (Pedrycz) fuzzy rules produces an output fuzzy set that is
not directly compatible with a real-world signal (such as a control signal for an
actuator within the range 4 to 20 mA) as it is fuzzy in nature. If a crisp (numerical)
output value is required, which is directly compatible with a real-world signal, then
the output fuzzy set must be defuzzified. Defuzzification is a transformation
process that translates the output fuzzy set into a single numerical value
representative of that fuzzy set. For this purpose, preferably the centre of gravity
(COG) method is used.
Given a fuzzy set F represented in the point-wise form as

^ `

F P x x , P x x , " , P x x ,

F 1 1 F 2 2 F p p

the COG method helps in computing the x coordinate of the centre of gravity of the
fuzzy set F as follows:

p

P

¦ x x

i i

F

x c

i 1

p

¦ P

x

i

F

i 1

In contrast, fuzzification translates a crisp value into a corresponding fuzzy
value (degree of membership). If the computed degree of membership of the crisp
input in a fuzzy set F is exactly 1 or close to 1 or greater than some threshold value
the input (crisp) is considered to be equivalent to that fuzzy set F .

4.4.1 Inferencing a Mamdani-type Fuzzy Model

Inferencing the Mamdani type of fuzzy model basically consists of four steps. For
a single-input single-output model, however, if an input fuzzy set is given instead
of a crisp input value, then the procedure is slightly altered, as shown below. Given
the rule base, for instance with M fuzzy rules, as