The most common choices for the AND operator in (3.39) are the Zadehian

min(A, B), often known as the Mamdani method because of its early successful

use in process control by Mamdani (1976) and the product operator tv(A) . tv(B).

In Figure 3.10, we show the membership functions of Figure 3.9 modi¬ed to

re¬‚ect the memberships of their respective linguistic values.

Next, the individual membership functions in Figure 3.10 must be aggre-

gated into a single membership function for the entire linguistic variable. Aggrega-

tion operators resemble t-conorms, but with fewer restrictions (Klir and Yuan 1995,

p. 88 ff ); the Zadehian max OR operator is frequently used. Figure 3.11 shows the

aggregated membership functions of Figure 3.10.

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52 FUZZY LOGIC, FUZZY SETS, AND FUZZY NUMBERS: I

Figure 3.10 Membership functions of linguistic values in linguistic variable temperature,

modi¬ed by the grades of membership of the linguistic values A AND B ¼ min(A, B).

In the last step, we ¬nd a single number compatible with the membership function

for Temp in Figure 3.11. This number will be the output from this ¬nal step in the

defuzzi¬cation process.

There are several methods for calculating a single defuzzi¬ed number. We will

present three: the average maximum method, the weighted average maxima

method, and the method most commonly used, a centroid method. In the following,

let x represent the numbers from the real line, let m(x) be the corresponding grade of

membership in the aggregated membership function, let xmin be the minimum x

value at the maximum and xmax be the maximum x value at the maximum, and

let X be the defuzzi¬ed value of x.

The simplest is the average maximum method. In Figure 3.11, the maxi-

mum grade of membership stretches from x ¼ 43 to x ¼ 55. The average of

Figure 3.11 Aggregated membership functions of linguistic values in linguistic variable

temperature, using Zadehian max(A, B) OR operator.

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53

3.6 FUZZIFICATION AND DEFUZZIFICATION

these is 49; this is the defuzzi¬ed value by the average maximum method. The

formula is

X(average maximum) ¼ (x max1 þ x max2 )=2 (3:40)

Next is the weighted average maxima method. In Figure 3.11, we have two

maxima: one stretches from x ¼ zero to x ¼ 15 with grade of membership 0.42,

and the second stretches from x ¼ 43 to x ¼ 55 with grade of membership 0.88.

We take the average of these two maxima, weight each by its grade of membership,

and add the products, and divide this sum by the sum of the grades of membership.

The defuzzi¬ed value by this method is

(0 þ 15) 43 þ 55

Á 0:42 þ 0:88 °0:42 þ 0:88Þ ¼ 35:6

X¼

2 2

Denote the start and end of each local maximum by xmini and xmaxi. If we have n

local maxima, the general formula is

X (x maxi Ám(x maxi ))

n

P

X(weighted average maxima) ¼ (3:41)

m(x maxi )

i¼1

The ¬nal method, preferred by most fuzzy control engineers, is the centroid

method. It is

Ðb

a xm(x)dx

X (centroid) ¼ Ð b (3:42)

a m(x)dx

In these integrals, we have assumed that the support of the aggregated membership

function is the interval [a, b]. In our case, the defuzzi¬ed value by the centroid

method is 41.8.

The defuzzi¬cation process may be made much simpler by assigning singleton

membership functions to the output linguistic variable. Remember that a singleton

is a fuzzy number with grade of membership one at only one value of its argument,

and grade of membership zero everywhere else. In this case, the centroid method

reduces to a simple weighted average.

While not usually of interest to control engineers, the idea of reversible defuzzi-

¬cation is sometimes attractive to expert system modelers in other areas. By revers-

ible defuzzi¬cation, we mean that if an input number is fuzzi¬ed and immediately

defuzzi¬ed, the defuzzi¬ed value equals the input value. Reversible defuzzi¬cation

is assured if membership functions are triangular, adjacent membership functions

intersect at a membership of 0.5, as each membership function begins to decline

from 1 the next begins to ascend from 0, and memberships always add to 1 over

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54 FUZZY LOGIC, FUZZY SETS, AND FUZZY NUMBERS: I

the admissible range of input numbers. However, fuzzy control engineers routinely

shape response surfaces (plots of defuzzi¬ed value against the input variables)

very carefully to achieve speci¬c quantitative results, and may require the ¬‚exibility

offered by the numerous variations on the defuzzi¬cation theme.

Exercise Defuzz.par. Program Defuzz.par illustrates both fuzzi¬cation and defuz-

zi¬cation. A discrete fuzzy set with members Short, Medium, and Tall is declared,

and membership functions for these terms are de¬ned. The user enters a height.

Defuzz then fuzzi¬es the height into discrete fuzzy set size and prints out the

grades of membership. This newly de¬ned fuzzy set is now defuzzi¬ed into a

scalar number, which is then printed out. The membership functions are chosen

so that fuzzi¬cation and defuzzi¬cation are reversible.

Of course, non-numeric discrete fuzzy sets cannot be defuzzi¬ed since they do

not describe numbers.

3.7 QUESTIONS

Evaluate the following:

3.1

a. Proposition P ¼ A AND B; proposition Q ¼ A OR B. For all crisp truth

values for A and B, ¬rst construct a truth table for P, for Q, for P AND

Q, and for P OR Q.

b. Proposition R ¼ A IMPLIES B; proposition S ¼ A OR B. For all crisp

truth values for A and B, ¬rst construct a truth table for P, for Q, for P

AND Q, and for P OR Q.

Let W be a complex logical proposition made up of elementary (atomic) prop-

3.2

ositions P, Q, R . . . connected by and, or, implies and using not. For example,

W ¼ not (P implies (not Q))

W is a tautology if tv(W) ¼ 1 for all values of tv(P), tv(Q), . . .; the truth table

for a tautology W will contain only 1 in its last column. W is a contradiction

if tv(W) ¼ 0 for all values of tv(P), . . .; the truth table for a contradiction W