values.

7.1 MODIFICATION OF EXISTING DATA BY RULE

CONSEQUENT INSTRUCTIONS

Conventionally, a simple rule of the type

IF (A0 ¼ A) then (B0 ¼ B) (7:1)

where A, A0 , B, and B0 are fuzzy numbers, and B0 is evaluated setting up a fuzzy

relation [Ai AND Bj] and evaluating B0 by composing A0 with this relation,

usually using max “min composition:

B0 ¼ A0 ½Ai AND Bj (7:2)

where the AND operator is a selected t-norm. [In the Mamdani method, the

Zadehian t-norm min(Ai , Bj) is used.]

There are two problems with this approach. First, the method is dif¬cult to apply

to complex antecedents. Second, the method involves two nested loops, ¬rst over

rows, then over columns of the fuzzy relation. Both these problems are easily

solved, since (7.2) is precisely equivalent to formulation (7.4), which does not

require two nested loops. [See Klir and Yuan (1995), p. 316.]

Let p be the antecedent truth value. Then in rule (7.1),

p ¼ max(min(A0 (x), A(x))) over x (7:3)

Then,

B0 ¼ pB ¼ {min( p, B0j } (7:4)

We can then evaluate our rule (7.1) quite simply, using (7.4):

IF (A0 ¼ A) THEN (B0 ¼ pB) (7:5)

in which P is the antecedent truth value, B is the speci¬ed prior truth value of B, and

B0 is the new value of B resulting from ¬ring the rule. If B is a fuzzy set, then B0 must

be de¬ned on the same universe as B.

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117

7.2 MODIFICATION OF NUMERIC DISCRETE FUZZY SETS

7.2 MODIFICATION OF NUMERIC DISCRETE FUZZY SETS:

LINGUISTIC VARIABLES AND LINGUISTIC TERMS

Recall that a linguistic variable is a special kind of discrete fuzzy set, such as Speed,

that describes a numeric quantity. The members of this discrete fuzzy set are linguis-

tic terms such as Fast, Medium, and Slow. To each linguistic term a membership

function is attached that maps numbers from the real line onto grades of membership

of the linguistic terms.

In this section, we consider rules with linguistic terms in both antecedent and con-

sequent. This is almost always the case in fuzzy control, and sometimes in more

general fuzzy reasoning problems as well.

Consider a special case of rule (7.1) above:

if (speed is Slow AND distance is Far) then (power is High) (7:6)

where speed, distance, and power are numeric attributes.

This type of rule is found in most fuzzy control programs. There are two different

ways of de¬ning the syntax of this rule. In typical fuzzy control rule syntax, Slow,

Far, and High are membership functions in (unspeci¬ed) linguistic variables; speed,

distance, and power are ¬‚oating-point numeric variables. Such rules are almost

invariably coupled with many other similar rules.

Firing this rule involves a complex procedure. First, the numeric attributes speed

and distance are fuzzi¬ed to get the truth value of the antecedent clauses (speed is

Slow) and (power is High); these are combined to get the antecedent con¬dence.

The membership function for High is then modi¬ed using downward monotonic

reasoning. Next, the membership functions relating to power, including High and,

for example, Medium, Low, and Zero, are combined, that is, aggregated, using an

OR operator. Finally, the combined membership functions are defuzzi¬ed using,

for example, the centroid method to yield a numeric value for power. This yields

an extremely compact rule structure.

There are, however, disadvantages to lumping this entire procedure into one rule,

stemming in part from the fact that while linguistic terms are employed, the linguis-

tic variable of which they are members is unspeci¬ed. This means that the term

Medium, for example, should not be used except in association with speed, since

only one membership function can be associated with Medium. Since names of

the linguistic variables associated with speed, distance, and power are not speci¬ed,

they are inaccessible to the programmer if he wishes to use the grades of member-

ship of their members (such as High) in later rules; multistep reasoning becomes

more dif¬cult. Fuzzi¬cation and defuzzi¬cation are automatic and inescapable; it

is impossible to use a discrete fuzzy set of classi¬cation in the consequent. For

these reasons, we prefer to avoid this rule of syntax in general-purpose fuzzy reason-

ing, and to break out fuzzi¬cation and defuzzi¬cation (if any) as separate steps in

two other rules. For example,

declare Data

fspeed flt

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118 INFERENCE IN A FUZZY EXPERT SYSTEM II

speed fzset (Slow Medium Fast)

fdistance flt

distance fzset (Short Medium Far)

fpower flt

power fzset (Zero Low Medium High);

rule block 1

IF (in Data fspeed = <SP> AND fdistance = <D>)

THEN fuzzify 1 <SP> into speed,

fuzzify 1 <D> into distance;

After this rule has ¬red, the discrete fuzzy sets (linguistic variables) speed and

distance will have grades of membership assigned to their respective members.

Although the term Medium is used three times, each one belongs to a different

linguistic variable, and will have its own membership function.

We are now ready to ¬re our next set of rules, of the type

rule block 2

IF (speed is Slow AND distance is Far) THEN (power is High);

rule block 2

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

Zero);

When the rules in block 2 have ¬red (in parallel), the members of linguistic variable

power will have been assigned their grades of membership, and we are ready to

defuzzify if we wish;

rule block 3

IF(Data)

THEN defuzzify 1 power (centroid) into 1 fpower;

The grades of membership of the members of linguistic variables speed, dis-

tance, and power are accessible to the programmer after the block 2 rules have

¬red.

We now ask what type of reasoning is appropriate in this problem.

7.3 SELECTION OF REASONING TYPE AND

GRADE-OF-MEMBERSHIP INITIALIZATION

A critical element in deciding reasoning type is to decide what we want to do if two

or more rules have the same consequent discrete fuzzy set member or linguistic

term, ¬ring with different antecedent con¬dences.

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119

7.4 FUZZIFICATION AND DEFUZZIFICATION