Fuzzy Logic Rule Base Assignment Help

Fuzzy Rules And Their Operations - Fuzzy Logic Rule Base

Fuzzy Logic Rule Base

Consider questions rose above. Firstly, consider, for illustration, the following fuzzy

IF-THEN rule containing an OR operation:

IF a1 is A1 AND a2 is A2 OR a3 is A3 AND a4 is A4 THEN b is B." By convention, it is understood in logic as

"(IF a1 is A1 AND a2 is A2) OR (a3 is A3 AND a4 is A4) THEN b is B."

With this understanding and convention, this is clear that this statement is Equivalent to the combination of the given two fuzzy IF-THEN rules as:

"IF a1 is A1 AND a2 is A2 THEN b is B." "IF a3 is A3 AND a4 is A4 THEN b is B."

Thus, the fuzzy logic OR operation is not necessary to employ: this may shorten a statement of a fuzzy IF-THEN rule; however it increases the format complexity of the rules.

Secondly, consider the fuzzy logic NOT operation. For a negative statement like "IF a is not A," one can always interpret it by a positive one "IF a¯ is A" or "IF a is A" where A means "not A" in logic theory and also "complement of A" in set theory. Moreover, the statement "a¯ is A" or "a is A" can be evaluated by

 μA (a¯) = μA (a) = 1 - μA (a)

Illustration 1

Specified a fuzzy logic implication statement

"IF a1 is A1 AND a2 is not A2 OR a3 is not A3 THEN b is B,"

How one can rewrite this as a set of Equivalent general fuzzy IF-THEN rules in the unified form?

Solution

One may firstly drop the fuzzy logic OR operation by rewriting the specified statement as

"IF a1 is A1 AND a2 is not A2 THEN b is B,"

"IF a3 is not A3 THEN b is B,"

One may then drop the fuzzy logic NOT operation by rewriting them like

"IF a1 is A1 AND a¯2 is not A2 THEN b is B,"

"IF a¯3  is A3 THEN b is B,"

Finally, these two general fuzzy IF-THEN rules can be evaluated as follows

μA1(a1) ∧ μA2 (a¯2 ) ⇒ μB(b)

μA3  (a¯3) ⇒ μB(b)

Hence, one only needs two general fuzzy IF-THEN rules, (1) and (2), and three membership ship values  μA1(a1), μA2(a2), μA3(a3) to infer the conclusion "b is B," namely, b ∈ B, along with the truth values μB(b) calculated from the three specified membership values.

Every other fuzzy logic operations can be simply expressed and defined only by the AND and OR operations. They can be evaluated through min and the max operations as given below:

μA(a1 ) ∧ μA (a2) : min {μ1(a1), μA(a2)};

Among

 μA(a¯) ∧ μA   (a2) : max {μ1 (a1 ), μA

μA (a ) = μA  (a) = 1 - μA(a) (a2)};

 μA (a ⇒ b) = μA  (a) ⇒ μA (b) = min {1, 1 + μA(a) - μA(b)}

μA (a ⇔ b) = μA  (a) ⇔ μA (b) = 1 -| μ A(a) - μ A(b) |

Consequently, each finite combinations of these fuzzy logic operations can be expressed only by the AND and OR operations, hence in any finite fuzzy logic inference statement

IF ... THEN ...,

While the condition part "IF ..." can be expressed only by the AND and OR operations.

Consequently, a finite fuzzy logic implication can usually be described by a set of general or common IF-THEN rules containing merely the fuzzy logic AND operation, in the given generic form as:

(a) "IF a11 is A11 AND . . . AND a1n is A1n THEN b1 is B1."

(b) "IF a21 is A21 AND . . . AND a2n is A2n THEN b2 is B2."

                                    ..........

 (c) "IF am1 is Am1 AND . . . AND amn is Amn THEN b2 is B2."

This family of general or common fuzzy IF_THEN rules is mostly called a fuzzy logic rule base.

This is remarked that the number of components in all rule needs not to be the similar. If n = 2 however a rule has one component in the situation part, say as,

"IF a11 is A11 THEN b1 is B1,"

One cam formally rewrites this as

""IF a11 is A11 AND a12 is I12 THEN b1 is B1." Here I12 is a fuzzy subset along with μ12 (a) = 1 for all a ∈ I12

Now, one actually inserts an "always true" or redundant condition into the "IF ... AND ..." part to fill in the gap of the statement. In a common discussion of the subject the format of a fuzzy logic rule base can be remained simple.

This is clear that such general forms of a fuzzy logic rule base includes the non-fuzzy logic rule base involve the non fuzzy case and the unconditional case with only "b is B" as special cases. Furthermore, this common fuzzy logic rule base with merely the fuzzy logic AND operation in the situation part too covers too unusual fuzzy logic implication statements, as like the one shown in the later illustration.


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