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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.
Fuzzy set
membership value
Linguistic variables
1.5
Membership functions for the lingustic variable "Width"
1
0.5
0
0
1
2
3
Narrow
4
5
Normal
6
7
8
9
Measured w idth, cm
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.
1.2
1
0.8
0.6
0.4
0.2
0
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
1.2
1
0.8
0.6
0.4
0.2
35
33
31
29
27
25
23
21
19
17
15
0
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)
1.2
1
0.8
0.6
0.4
0.2
0
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 1000
2. Can sqrt numbers between 1001 and 2000
3. Can sqrt numbers between 2001 and 5000
The task: What is the sqrt of 1156.
If-Then rules
M1: 31.6
M2: 34
M3: 44.73
The answer – 34.
How sure? – 0
How sure? – 1
How sure? =0
If-Then rules
Interpolation:
If-Then rules
Inference:
Rule: if a person’s income is more than 100K
then 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 AND
the income figure is true
THEN the person is rich
If-Then rules
Consequent
1.
2.
3.
Crisp: IF … THEN y=nonfuzzy_value
Fuzzy: IF …THEN y is A_fuzzy_set
Functional: IF x1 is A1 AND x2 is A2 …
AND xn is An THEN
y  a0  i 1 ai  xi
n