:rule r1

IF (speed is Slow AND distance is Medium) THEN (power is

Medium);

:rule r2

IF (speed is Medium AND distance is Short) THEN (power is

Medium);

and that rule r1 ¬res with antecedent con¬dence 0.8, and that rule r2 ¬res with ante-

cedent con¬dence 0.6. With monotonic reasoning, rule r1 would succeed in setting

the grade of membership of Medium to 0.8, even if rule r2 ¬res after rule r1. If we

allow each rule to set truth values regardless of their existing value, the last rule to

¬re would succeed.

There are, however, occasions when we wish to modify grades of membership in

different ways, employing non-monotonic reasoning; these will be discussed in

Section 7.7.

7.4 FUZZIFICATION AND DEFUZZIFICATION

We review brie¬‚y fuzzi¬cation and defuzzi¬cation, discussed more fully in

Chapter 3, from the point of view of the choices to be made and their speci¬cation

in an expert system language.

Overall, we have these steps to carry out; fuzzi¬cation and evaluation of antece-

dent con¬dence; modi¬cation of consequent membership functions; aggregation of

membership functions for each consequent linguistic variable; and defuzzi¬cation of

the aggregated membership functions.

7.4.1 Fuzzi¬cation and Evaluation of Antecedent Con¬dence

Fortunately, almost everyone agrees on how fuzzi¬cation should be carried out; in a

working expert system. Also, combination of truth values of two or more antecedent

clauses is almost always carried out using the Zadehian min “ max AND and OR

operations. It is probably satisfactory to take the conventional methods as defaults,

and not necessary to permit specifying any other choices here.

7.4.2 Modi¬cation of Consequent Membership Functions

The modi¬cation of membership functions can carried out by

B0j ¼ P AND Bj (7:7)

but there is not general agreement as to which AND operator should be used.

Common choices are the Zadehian min operator (which yields the Mamdani

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120 INFERENCE IN A FUZZY EXPERT SYSTEM II

method) and the product operator. The min operator is widely accepted, and is prob-

ably a good default, but it lends ¬‚exibility to a system if other operators can be

speci¬ed.

7.4.3 Aggregation of Consequent Membership Functions for Each

Consequent Linguistic Variable

Aggregation of all the modi¬ed membership functions belonging to a single linguis-

tic variable is usually carried out using the Zadehian max OR operator, and this is

probably acceptable as a default at present. However, an argument can be made

for employing a different OR operator. Let us take an example and follow it through.

Suppose we have a linguistic variable control in the consequent of a rule, with

three members N, Z, and P with these membership functions (Fig. 7.1):

We ¬re rules that give grades of membership 0 for N, 0.8 for Z and 0.6 for P, and

modify the membership functions using the Zadeh AND operator, giving these

modi¬ed membership functions (Fig. 7.2):

We now aggregate these modi¬ed functions using the Zadehian OR operator

(Fig. 7.3):

The aggregated membership functions in Figure 7.3 represent a new membership

function. We may ask”Membership in what? The answer is that Figure 7.3 rep-

resents the truth value that a number between 21 and þ1 is valid for this particular

instance of the linguistic variable control. But the resulting membership function is

Membership functions for linguistic variable control.

Figure 7.1

Modi¬ed membership functions for linguistic variable control.

Figure 7.2

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121

7.4 FUZZIFICATION AND DEFUZZIFICATION

Figure 7.3 Aggregated membership functions using Zadehian OR operator for linguistic

variable control.

not convex, and the notch in Figure 7.3 at an argument 0.5 seems quite counterin-

tuitive. Why should 0.5 have a lower grade of membership in control than arguments

of 0 and 1?

In Chapter 3 we showed that when ORing fuzzy numbers are logically incompa-

tible, such as 2 and NOT 2, there is a mathematical (and practical) argument for

using the bounded sum OR operator. In aggregating membership functions, we

deal with fuzzy numbers that are semantically incompatible.

Unfortunately, we do not have a mathematical argument to justify using the

bounded sum operator in this, but from a practical viewpoint it seems to work

out, at least in this case. In Figure 7.4, we aggregate the membership functions of

Figure 7.2 using the bounded sum operator with intuitively pleasing results; the

resulting membership function is convex and the annoying notch in Figure 7.3

has been removed.

Figures 7.3 and 7.4 indicate that it might be desirable to furnish a choice of other

OR operators than the Zadehian for aggregation of membership functions, provided

that the memberships of the unweighted aggregated membership functions satisfy

appropriate conditions, such as their sum being one at any point. More research is

needed on this point.

7.4.4 Determination of Defuzzi¬ed Value for Consequent Attribute

We denote the argument of the membership functions, the number whose defuzzi-

¬ed value is to be obtained by z; its grade of membership by m(z); and the defuzzi-

¬ed value by zdef .

Figure 7.4 Aggregated membership functions for linguistic variable control. using the

bounded sum OR operator.

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122 INFERENCE IN A FUZZY EXPERT SYSTEM II

Here again different methods are used, described in Klir and Yuan (1996) (books

on.fuzzy control). Here, we will describe only the most commonly used defuzzi¬ca-

tion method, the centroid method.

Probably the most commonly used defuzzi¬cation method is the centroid

method, equivalent to calculating the center of area of the aggregated membership

function, as shown in Figure 7.3 or 7.4. Analytically, we divide the integral over

the range of the membership function of the product of the argument and its

grade of membership, and divide this by the integral of the grade of membership;

this is very useful when the membership functions are given as parameterized

continuous functions, such as piecewise linear, piecewise quadratic or normalized

Gaussian:

Ð

Z zm(z)dz