An Introduction to Fuzzy Control - Fuzzy Logic: An Introductory Course for Engineering Students - page 176

8.2.1 Inference from Imprecise Rules

If a dynamic systems is considered, with input variables ( x 1 ,..., x n ) and output

variable y , whose behavior is known by m imprecise rules R i :

R 1 : If x 1 is P 11 and x 2 is P 12 ··· and x n is P 1 n , then y is Q 1

R 2 : If x 1 is P 21 and x 2 is P 22 ··· and x n is P 2 n , then y is Q 2

.

R m : If x 1 is P m 1 and x 2 is P m 2 ··· and x n is P mn , then y is Q m

The question is the following: If we observe that the input variables are in the
“states” “ x 1 is P 1 ”, “ x 2 is P 2 ”, ,“ x n is P n ”, respectively, what can be inferred

for variable y ? That is, supposing that it lies in the “state” “ y is Q ∗ ”, what is Q ∗ ?
Without loss of generality let us consider only the case n

...

= m = 2:

R 1 :If x 1 is P 11 and x 2 is P 12 ,then y is Q 1

R 2 :If x 1 is P 21 and x 2 is P 22 ,then y is Q 2

x 1 is P 1 and x 2 is P 2

y is Q ,whatis Q ?

where x 1 ∈ X 1 , x 2 ∈

∈

Y , and let us represent it in fuzzy logic by means of
the functions R 1 and R 2 . This leads to:

X 2 , y

for convenient continuous t-norms T 1 , T 2 and convenient implication functions J 1 ,
J 2 . Convenient, in the sense of adequate to the use of the conditional phrases R 1 and
R 2 relative to the problem under consideration within the given system S .
To find a solution to this problem, the problem of linguistic control, that is a part
of what can be called intelligent systems control, we need to pass throughout several
steps.

First Step: Functions J
Implication functions J are obtained through the interpre-

tation and representation of the rule's use, that is, from the actual meaning of these
conditional phrases.Thus, different types of protoform exist

:[ 0 , 1 ]×[ 0 , 1 ]ₒ[ 0 , 1 ]

a ₒ b ⃔ J ( a , b )

and are used as introduced previously in the topic. For example: