Basic Concepts of Fuzzy Logic

Apparatus of fuzzy logic is built on:

Fuzzy sets: describe the value of variables

Linguistic variables: qualitatively and quantitatively described by fuzzy sets

Possibility distributions: constraints on the value of a linguistic variable

Fuzzy if-then rules: a knowledge*Fuzzy Logic: Intelligence, Control, and Information - J. Yen and R. Langari, Prentice Hall 1999

Linguistic variables

A fuzzy set can be used to describe the value of a variable.

- Temperature is high.

- The load is heavy.

Linguistic variables

The variable Temperature (x) is characterized both by a symbolic variable (“High”) and a numeric variable expressing its membership in the fuzzy set “High”.

Linguistic variables

A linguistic variable is “a variable whose values are words or sentences in a natural or artificial language”. Each linguistic variable may be assigned one or more linguistic values, which are in turn connected to a numeric value through the mechanism of membership functions.

Linguistic variables

Membership functions for the lingustic variable "Width"

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9

Measured width, cm

Fu

zzy

set

mem

ber

-sh

ip v

alu

e

Narrow Normal Wide

An example of a fuzzy linguistic variable and membership functions

Possibility Distribution

Assigning a fuzzy set to a linguistic variable constrains the value of the variable.

Possible vs. Impossible values of the variable are a matter of degree.

Possibility Distribution

Example: a suspect is a male between 20 and 30 years old.

A crisp set defines the age of this suspect as [20,30].

A 19-years old male would not be a suspect, as this age is an impossible value for this set.

Possibility Distribution

A fuzzy set defines the age of this suspect as (age) that may have a smooth boundary.

0

0.2

0.4

0.6

0.8

1

1.2

15 17 19 21 23 25 27 29 31 33 35

Possibility Distribution

A possibility that the suspect is 19 years old is 0.75

0

0.2

0.4

0.6

0.8

1

1.2

Possibility Distribution

In general, when we assign a fuzzy set A to a variable X, the assignment results in a possibility distribution of X, which is defined by A’s membership function.

X(x)=A(x)

0

0.2

0.4

0.6

0.8

1

1.2

15 17 19 21 23 25 27 29 31 33 35

If-Then rules

“If temperature is hot then AC_setting is high”

Provide fuzzy inference. Can be viewed as:

- Interpolation scheme- Multi-expert panel- Generalization of logic inference

If-Then rules

“If temperature is hot then AC_setting is high”

Provide fuzzy inference. Can be viewed as:

- Multi-expert panel- Interpolation scheme- Generalization of logic inference

If-Then rules

Multi-expert panel: A kingdom with 3 mathematicians.

1. Can sqrt numbers between 0 and 10002. Can sqrt numbers between 1001 and

20003. Can sqrt numbers between 2001 and

5000

The task: What is the sqrt of 1156.

If-Then rules

M1: 31.6 How sure? – 0M2: 34 How sure? – 1M3: 44.73 How sure? =0 The answer – 34.

If-Then rules

Interpolation:

If-Then rules

Inference:

Rule: if a person’s income is more than 100Kthen the person is rich

Fact: Jack’s income is 101K

Consequence: Jack is rich

If-Then rules

Structure of fuzzy rules:

IF <antecedent> THEN <consequent>

Example:

IF a person’s income is high THEN the person is rich

If-Then rules

An antecedent may combine multiple conditions using logic connectives

(AND, OR, NOT):

IF a person’s income is high ANDthe income figure is true

THEN the person is rich

If-Then rules

Consequent

1. Crisp: IF … THEN y=nonfuzzy_value2. Fuzzy: IF …THEN y is A_fuzzy_set3. Functional: IF x1 is A1 AND x2 is A2 …

AND xn is An THEN

n

i ii xaay10