C291-78
Tópicos Avanzados:
Inteligencia Computacional I
Dra. Ma. del Pilar Gómez Gil
Primavera 2014 [email protected]
1
Ver: 08-Mar-2016
Introducción a la Lógica Difusa
mailto:[email protected]
Este material ha sido tomado de varias fuentes,
y forma parte del “Tutorial en Lógica Difusa”
impartido por el Dr. Juan Manuel Ramírez C.
Coordinación de Electrónica, 2012
2
FUZZY LOGIC
In 1965 Lotfi Zadeh, published his famous paper “Fuzzy sets”.
Zadeh extended the work on possibility theory into a formal system of mathematical logic, and introduced a new concept for applying natural language terms.
This new logic for representing and manipulating fuzzy terms was called fuzzy logic.
3
Introduction
Many decision-making and problem-solving tasks are too complex to be understood quantitatively, however, people succeed by using knowledge that is imprecise rather than precise.
Fuzzy set theory resembles human reasoning in its use of approximate information and uncertainty to generate decisions.
4
Example: driving decisions
5
Introduction (2)
Since knowledge can be expressed in a more natural way by using fuzzy sets, many engineering and decision problems can be greatly simplified.
It was specifically designed to mathematically represent uncertainty and vagueness and provide formalized tools for dealing with the imprecision intrinsic to many problems.
Fuzzy theory may be seen as an extension to classical set theory, where a membership function is added to a set. This function is defined in
]1,0[
6
Membership function
(a) Boolean Logic. (b) Multi-valued Logic.
0 1 10 0.2 0.4 0.6 0.8 100 1 10
Fuzzy logic is a set of mathematical principles for knowledge representation based on degrees of membership.
Unlike two-valued Boolean logic, fuzzy logic is multi-valued. It deals with degrees of membership and degrees of truth.
Fuzzy logic uses the continuum of logical values between 0 (completely false) and 1(completely true). Instead of just black and white it accepts that things can be partly true and partly false at the same time.
7
Membership function (2)
A membership function
represents the “degree” in that variable t is
included in the concept represented by
the tag A.
)(tA
8
Example:
fuzzy variable “temperatura”
Linguistic value/variable:
temperatura
Fuzzy subsets:
{frío, fresco, normal, tibio, caliente}
Membership functions:
)}(),(),(),(),({ ttttt calientetibionormalfrescofrío
9
Example:
fuzzy variable “temperatura” (2)
)(tA
t
The t-axis represents the universe of discourse
Each colour line is a membership function
10
The membership function of a
“Crisp set”
11
The membership function of a
“Fuzzy set”
12
Crisp sets and characteristic
function
Ax
Axxf A
if0,
if 1,)(
Let X be the universe of discourse and its elements be
denoted as x. In the classical set theory, crisp set A of X
is defined as function fA(x) called the characteristic
function of A:
fA(x) : X {0, 1}, where
This set maps universe X to a set of two elements. For
any element x of universe X, characteristic function fA(x) is
equal to 1 if x is an element of set A, and is equal to 0 if x
is not an element of A.
13
Fuzzy sets and membership functions
In the fuzzy theory, fuzzy set A of universe X is defined by function µA(x) called the membership function of set A
µA(x) : X [0, 1], where µA(x) = 1 if x is totally in A;
µA(x) = 0 if x is not in A;
0 < µA(x) < 1 if x is partly in A.
This set allows a continuum of possible choices. For any element x of universe X, membership function µA(x) equals the degree to which x is an element of set A. This degree, a value between 0 and 1, represents the degree of membership, also called membership value, of element x in set A.
14
Fuzzy Set Representation
First, we determine the membership functions. In our
“height” example, we can obtain fuzzy sets of tall, short and
average men.
The universe of discourse – the person’s heights – is
represented by three sets: short, average and tall men. As you
will see, a man who is 184 cm tall is a member of the average
men set with a degree of membership of 0.1, and at the same
time, he is also a member of the tall men set with a degree of
0.4. (see graph on next page)
15
Fuzzy Set Representation
150 210170 180 190 200160
Height, cmDegree ofMembership
Tall Men
150 210180 190 200
1.0
0.0
0.2
0.4
0.6
0.8
160
Degree ofMembership
Short Average ShortTall
170
1.0
0.0
0.2
0.4
0.6
0.8
Fuzzy Sets
Crisp Sets
Short Average
Tall
Tall
16
Fuzzy Set Representation
Typical functions that can be used to represent a fuzzy set
are sigmoid, gaussian and pi. However, these functions
increase the time of computation. Therefore, in practice,
most applications use linear fit functions.
17
Funciones de membresía (1)
Función de membresía triangular
18
Función de membresía trapezoidal
Funciones de membresía (2)
19
Función de membresía gaussiana
Funciones de membresía (3)
20
Funciones de membresía sigmoide
Funciones de membresía (4)
21
Linguistic Variables and Hedges At the root of fuzzy set theory lies the idea of linguistic variables.
A linguistic variable is a fuzzy variable. For example, the statement “John is tall” implies that the linguistic variable John takes the linguistic value tall.
In fuzzy expert systems, linguistic variables are used in fuzzy rules. For example:
IF wind is strong
THEN sailing is good
IF project_duration is long
THEN completion_risk is high
IF speed is slow
THEN stopping_distance is short
22
Linguistic Variables and Hedges
The range of possible values of a linguistic variable represents the universe of discourse of that variable. For example, the universe of discourse of the linguistic variable speed might have the range between 0 and 220 km/h and may include such fuzzy subsets as very slow, slow, medium, fast, and very fast.
A linguistic variable carries with it the concept of fuzzy set qualifiers, called hedges.
Hedges are terms that modify the shape of fuzzy sets. They include adverbs such as very, somewhat, quite, more or less and slightly.
23
Linguistic Variables and Hedges
Short
Very Tall
Short Tall
Degree ofMembership
150 210180 190 200
1.0
0.0
0.2
0.4
0.6
0.8
160 170
Height, cm
Average
TallVery Short Very Tall
24
Linguistic Variables and Hedges
25
Linguistic Variables and Hedges
26
Membership Functions
For the sake of convenience, usually a fuzzy set is denoted as:
A = A(xi)/xi + …………. + A(xn)/xn where A(xi)/xi (a singleton) is a pair “grade of
membership” element, that belongs to a finite universe of discourse:
A = {x1, x2, .., xn}
27
Logic operators (1/2)
28
(Matlab, 2015)
Operations of Crisp Sets
Intersection Union
Complement
Not A
A
Containment
AA
B
BA BAA B
29
Operations of Fuzzy Sets
Complement
0 x
1
( x )
0 x
1
Containment
0 x
1
0 x
1
A B
Not A
A
Intersection
0 x
1
0 x
A B
Union
0
1
A B
A B
0 x
1
0 x
1
B
A
B
A
( x )
( x ) ( x )
30
Blue line is the
resulting
membership function
Logic operators (2/2)
31
(Matlab, 2015)
Complement Crisp Sets: Who does not belong to the set?
Fuzzy Sets: How much do elements not belong to the set?
If A is the fuzzy set, its complement A can be found as follows:
A(x) = 1 A(x)
The complement of a set is an opposite of this set. For example, if we have the set of tall men, its complement is the set of NOT tall men. When we remove the tall men set from the universe of discourse, we obtain the complement.
32
Containment
Crisp Sets: Which sets belong to which other sets?
Fuzzy Sets: How much sets belong to other sets?
Similar to a Chinese box, a set can contain other sets.
The smaller set is called the subset. For example, the
set of tall men contains all tall men; very tall men is a
subset of tall men. However, the tall men set is just a
subset of the set of men. In crisp sets, all elements of a
subset entirely belong to a larger set. In fuzzy sets,
however, each element can belong less to the subset
than to the larger set. Elements of the fuzzy subset have
smaller memberships in it than in the larger set.
33
Intersection Crisp Sets: Which element belongs to both sets?
Fuzzy Sets: How much of the element is in both sets?
In classical set theory, an intersection between two sets contains the elements shared by these sets. For example, the intersection of the set of tall men and the set of fat men is the area where these sets overlap. In fuzzy sets, an element may partly belong to both sets with different memberships.
A fuzzy intersection is the lower membership in both sets of each element. The fuzzy intersection of two fuzzy sets A and B on universe of discourse X:
AB(x) = min [A(x), B(x)] = A(x) B(x),
where xX
34
Union
Crisp Sets: Which element belongs to either set?
Fuzzy Sets: How much of the element is in either set?
The union of two crisp sets consists of every element that falls into
either set. For example, the union of tall men and fat men contains
all men who are tall OR fat.
In fuzzy sets, the union is the reverse of the intersection. That is,
the union is the largest membership value of the element in
either set. The fuzzy operation for forming the union of two fuzzy
sets A and B on universe X can be given as:
AB(x) = max [A(x), B(x)] = A(x) B(x),
where xX
35
Crisp and Fuzzy –union and
intersection
36
A B 𝑨 ∪ 𝑩 𝑨 ∩ 𝑩
¬𝑨 𝑨 ∪ ¬𝑨 = 𝑼 𝑨 ∩ ¬𝑨 = ∅
m𝑎𝑥(𝜇𝐴, 𝜇𝐵) min(𝜇𝐴, 𝜇𝐵) m𝑎𝑥(𝜇𝐴, 𝜇¬𝐴) min(𝜇𝐴, 𝜇¬𝐴)
0 0 0 = m𝑎𝑥(0,0)
0 = m𝑖𝑛(0,0) 1 1 = m𝑎𝑥(0,1) 0 = m𝑖𝑛(0,1)
0 1 1= m𝑎𝑥(0,1)
0 = m𝑖𝑛(0,1)
1 0 1= m𝑎𝑥(1,0)
0 = m𝑖𝑛(1,0) 0 1 = m𝑎𝑥(1,0) 0 = m𝑖𝑛(1,0)
1 1 1= m𝑎𝑥(1,1)
1 = m𝑖𝑛(1,1)
Crisp and Fuzzy – containment
(subset)
37
A B 𝑨 ∪ 𝑩 𝑨 ∩ 𝑩
m𝑎𝑥(𝜇𝐴, 𝜇𝐵) min(𝜇𝐴, 𝜇𝐵)
0 0 0 = m𝑎𝑥(0,0) 0 = m𝑖𝑛(0,0)
0 1 1= m𝑎𝑥(0,1) 0 = m𝑖𝑛(0,1)
1 0 1= m𝑎𝑥(1,0) 0 = m𝑖𝑛(1,0)
1 1 1= m𝑎𝑥(1,1) 1 = m𝑖𝑛(1,1)
If 𝐴 ⊆ 𝐵𝑡ℎ𝑒𝑛𝐴 ∪ 𝐵 = 𝐵 𝐴 ∩ 𝐵 = 𝐴,
Example:
A = “very tall”
B = “Tall”
You can’t be
“very tall” and
“not tall” at the
same time!
Properties of Fuzzy Sets
Equality of two fuzzy sets
Inclusion of one set into another fuzzy set
Cardinality of a fuzzy set
An empty fuzzy set
-cuts (alpha-cuts)
38
Equality
Fuzzy set A is considered equal to a fuzzy
set B, IF AND ONLY IF (iff):
A(x) = B(x), xX
39
Inclusion
Inclusion of one fuzzy set into another fuzzy set. Fuzzy
set A X is included in (is a subset of) another fuzzy
set, B X:
A(x) B(x), xX
Consider X = {1, 2, 3} and sets A and B
A = 0.3/1 + 0.5/2 + 1/3;
B = 0.5/1 + 0.55/2 + 1/3
then A is a subset of B, or A B
40
Cardinality Cardinality of a non-fuzzy set, Z, is the number of elements in Z.
BUT the cardinality of a fuzzy set A, the so-called SIGMA COUNT, is expressed as a SUM of the values of the membership function of A, A(x):
cardA = A(x1) + A(x2) + … A(xn) = ΣA(xi), for i=1..n
Consider X = {1, 2, 3} and sets A and B
A = 0.3/1 + 0.5/2 + 1/3;
B = 0.5/1 + 0.55/2 + 1/3
cardA = 1.8
cardB = 2.05
41
Empty Fuzzy Set
A fuzzy set A is empty, IF AND ONLY IF:
A(x) = 0, xX
Consider X = {1, 2, 3} and set A
A = 0/1 + 0/2 + 0/3
then A is empty
42
Alpha-cut
An -cut or -level set of a fuzzy set A X is an
ORDINARY SET A X, such that:
A={A(x), xX}.
Consider X = {1, 2, 3} and set A
A = 0.3/1 + 0.5/2 + 1/3
then A0.5 = {2, 3},
A0.1 = {1, 2, 3},
A1 = {3}
43
Fuzzy Set Normality
A fuzzy subset of X is called normal if there exists at least
one element xX such that A(x) = 1.
A fuzzy subset that is not normal is called subnormal.
All crisp subsets except for the null set are normal. In fuzzy
set theory, the concept of nullness essentially generalises to
subnormality.
The height of a fuzzy subset A is the large membership
grade of an element in A
height(A) = maxx(A(x))
44
Fuzzy Sets Core and Support
Assume A is a fuzzy subset of X:
the support of A is the crisp subset of X consisting of
all elements with membership grade:
supp(A) = {x A(x) 0 and xX}
the core of A is the crisp subset of X consisting of all
elements with membership grade:
core(A) = {x A(x) = 1 and xX}
45
Fuzzy Set Math Operations
aA = {aA(x), xX}
Let a =0.5, and
A = {0.5/a, 0.3/b, 0.2/c, 1/d}
then
Aa = {0.25/a, 0.15/b, 0.1/c, 0.5/d}
Aa = {A(x)a, xX}
Let a =2, and
A = {0.5/a, 0.3/b, 0.2/c, 1/d}
then
Aa = {0.25/a, 0.09/b, 0.04/c, 1/d}
…
46
Fuzzy Sets Examples
Consider two fuzzy subsets of the set X,
X = {a, b, c, d, e }
referred to as A and B
A = {1/a, 0.3/b, 0.2/c 0.8/d, 0/e}
and
B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e}
47
Fuzzy Sets Examples
Support:
supp(A) = {a, b, c, d }
supp(B) = {a, b, c, d, e }
Core:
core(A) = {a}
core(B) = {o}
Cardinality:
card(A) = 1+0.3+0.2+0.8+0 = 2.3
card(B) = 0.6+0.9+0.1+0.3+0.2 = 2.1
48
Fuzzy Sets Examples
Complement:
A = {1/a, 0.3/b, 0.2/c 0.8/d, 0/e}
A = {0/a, 0.7/b, 0.8/c 0.2/d, 1/e}
Union:
A B = {1/a, 0.9/b, 0.2/c, 0.8/d, 0.2/e}
Intersection:
A B = {0.6/a, 0.3/b, 0.1/c, 0.3/d, 0/e}
49
Fuzzy Sets Examples
aA: for a=0.5 aA = {0.5/a, 0.15/b, 0.1/c, 0.4/d, 0/e}
Aa: for a=2 Aa = {1/a, 0.09/b, 0.04/c, 0.64/d, 0/e}
a-cut:
A0.2 = {a, b, c, d}
A0.3 = {a, b, d}
A0.8 = {a, d}
A1 = {a}
50
Exercises
For A = {0.2/a, 0.4/b, 1/c, 0.8/d, 0/e} B = {0/a, 0.9/b, 0.3/c, 0.2/d, 0.1/e}
Draw the Fuzzy Graph of A and B Then, calculate the following: - Support, Core, Cardinality, and Complement for A and
B independently - Union and Intersection of A and B - the new set C, if C = A2 - the new set D, if D = 0.5B - the new set E, for an alpha cut at A0.5
51
Solutions A = {0.2/a, 0.4/b, 1/c, 0.8/d, 0/e} B = {0/a, 0.9/b, 0.3/c, 0.2/d, 0.1/e} Support
Supp(A) = {a, b, c, d} Supp(B) = {b, c, d, e}
Core
Core(A) = {c} Core(B) = {}
Cardinality
Card(A) = 0.2 + 0.4 + 1 + 0.8 + 0 = 2.4 Card(B) = 0 + 0.9 + 0.3 + 0.2 + 0.1 = 1.5
Complement
Comp(A) = {0.8/a, 0.6/b, 0/c, 0.2/d, 1/e} Comp(B) = {1/a, 0.1/b, 0.7/c, 0.8/d, 0.9/e}
52
Solutions A = {0.2/a, 0.4/b, 1/c, 0.8/d, 0/e} B = {0/a, 0.9/b, 0.3/c, 0.2/d, 0.1/e} Union
AB = {0.2/a, 0.9/b, 1/c, 0.8/d, 0.1/e} Intersection
AB = {0/a, 0.4/b, 0.3/c, 0.2/d, 0/e} C=A2
C = {0.04/a, 0.16/b, 1/c, 0.64/d, 0/e}
D = 0.5B
D = {0/a, 0.45/b, 0.15/c, 0.1/d, 0.05/e}
E = A0.5 E = {c, d}
53
Inference Fuzzy Rules
In 1973, Dr. Lotfi Zadeh published his second most influential paper. This
paper outlined a new approach to analysis of complex systems, in which
Zadeh suggested capturing human knowledge in fuzzy rules.
A fuzzy rule can be defined as a conditional statement in the form:
IF x is A
THEN y is B
where x and y are linguistic variables; and A and B are linguistic values
determined by fuzzy sets on the universe of discourses X and Y,
respectively.
54
Classical Vs Fuzzy Rules A classical IF-THEN rule uses binary logic, for example,
Rule: 1 Rule: 2
IF speed is > 90 km/h IF speed is < 40 km/h
THEN THEN
stopping_distance> 40 m stopping_distance < 15 m
The variable speed can have any numerical value between 0 and 150 km/h, and the variable stopping_distance can take any numerical value between 0 and 60 m. In other words, classical rules are expressed in the black-and-white language of Boolean logic.
55
Classical Vs Fuzzy Rules
A fuzzy IF-THEN rule uses fuzzy logic :
Rule: 1 Rule: 2
IF speed is fast IF speed is slow
THEN stopping_distance is long THEN stopping_distance is short
In fuzzy rules, the linguistic variable speed also has the range (the universe of discourse) between 0 and 150 km/h, but this range includes fuzzy sets, such as slow, medium and fast. The universe of discourse of the linguistic variable ‘stopping_distance’ can be between 0 and 60 m and may include such fuzzy sets as short, medium and long.
56
Classical Vs Fuzzy Rules
Fuzzy rules relate fuzzy sets.
In a fuzzy system, all rules fire to some extent, or in other words they fire partially. If the antecedent is true to some degree of membership, then the consequent is also true to the corresponding degree.
57
Firing Fuzzy Rules
Tall men Heavy men
180
Degree ofMembership
1.0
0.0
0.2
0.4
0.6
0.8
Height, cm
190 200 70 80 100160
Weight, kg
120
Degree ofMembership
1.0
0.0
0.2
0.4
0.6
0.8
These fuzzy sets provide the basis for a weight
estimation model. The model is based on a
relationship between a man’s height and his weight:
IF height is tall THEN weight is heavy
58
Firing Fuzzy Rules
Tall menHeavy men
180
Degree ofMembership
1.0
0.0
0.2
0.4
0.6
0.8
Height, cm
190 200 70 80 100160
Weight, kg
120
Degree ofMembership
1.0
0.0
0.2
0.4
0.6
0.8
The value of the output or a truth membership grade of the
rule consequent can be estimated directly from a
corresponding truth membership grade in the antecedent.
This form of fuzzy inference uses a method called
monotonic selection.
59
Firing Fuzzy Rules
A fuzzy rule can have multiple antecedents, for example:
IF service is excellent
OR food is delicious
THEN tip is generous
The consequent of a fuzzy rule can also include multiple parts, for instance:
IF temperature is hot
THEN hot_water is reduced;
cold_water is increased
60
1a. Specify the problem
Air-conditioning involves the delivery of air, which can be
warmed or cooled and have its humidity raised or lowered.
An air-conditioner is an apparatus for controlling, especially
lowering, the temperature and humidity of an enclosed
space. An air-conditioner typically has a fan which
blows/cools/circulates fresh air and has a cooler. The cooler
is controlled by a thermostat. Generally, the amount of air
being compressed is proportional to the ambient
temperature.
1b. Define linguistic variables
• Ambient Temperature
• Air-conditioner Fan Speed
Example: Air Conditioner
61
2. Determine Fuzzy Sets
Example: Air Conditioner
Fuzzy sets can have a variety of shapes. However, a triangle or a trapezoid can often provide an adequate representation of the expert knowledge, and at the same time, significantly simplifies the process of computation.
62
2. Determine Fuzzy Sets: Temperature Temp
(0C).
COL
D
COOL PLEASAN
T
WARM HO
T
0 Y* N N N N
5 Y Y N N N
10 N Y N N N
12.5 N Y* N N N
15 N Y N N N
17.5 N N Y* N N
20 N N N Y N
22.5 N N N Y* N
25 N N N Y N
27.5 N N N N Y
30 N N N N Y*
Temp
(0C).
COLD COOL PLEASANT WARM HOT
0
64
Temperature Fuzzy Sets
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25 30
Temperature Degrees C
Tru
th V
alu
e Cold
Cool
Pleasent
Warm
Hot
Temperature Fuzzy Sets
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25 30
Temperature Degrees C
Tru
th V
alu
e Cold
Cool
Pleasent
Warm
Hot
2. Determine Fuzzy Sets: Temperature (cont.)
Example: Air Conditioner
2. Determine Fuzzy Sets: Fan Speed Rev/sec
(RPM)
MINIMAL SLOW MEDIUM FAST BLAST
0 Y* N N N N
10 Y N N N N
20 Y Y N N N
30 N Y* N N N
40 N Y N N N
50 N N Y* N N
60 N N N Y N
70 N N N Y* N
80 N N N Y Y
90 N N N N Y
100 N N N N Y*
Example: Air Conditioner
65
2. Determine Fuzzy Sets: Fan Speed (cont.)
Example: Air Conditioner
Speed Fuzzy Sets
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60 70 80 90 100
Speed
Tru
th V
alu
e MINIMAL
SLOW
MEDIUM
FAST
BLAST
66
3. Bring out and construct fuzzy rules
Example: Air Conditioner
RULE 1: IF temp is cold THEN speed is minimal
RULE 2: IF temp is cool THEN speed is slow
RULE 3: IF temp is pleasant THEN speed is medium
RULE 4: IF temp is warm THEN speed is fast
RULE 5: IF temp is hot THEN speed is blast
To accomplish this task, we might ask the expert to describe how the problem can be solved using the fuzzy linguistic variables defined previously.
Required knowledge also can be collected from other sources such as books, computer databases, flow diagrams and observed human behaviour.
67
4. Encode the fuzzy sets, fuzzy rules
and procedures to perform fuzzy
inference into the expert system
To accomplish this task, we may choose an option:
to build our system using a programming language such as C/C++ or Pascal, or
to apply a fuzzy logic development tool such as MATLAB Fuzzy Logic Toolbox or Fuzzy Knowledge Builder.
etc
Example: Air Conditioner
68
5. Evaluate and tune the system
Example: Air Conditioner
The last, and the most laborious, task is to evaluate and tune the system. We want to see whether our fuzzy system meets the requirements specified at the beginning.
Evaluation of the system output is performed for test situations on the several representative values of input variables. Fuzzy Logic development tools often can generate surface to help us evaluate and analyze the system’s performance.
Tuning of the system consists of reviewing, adding and/or changing the membership functions and rules in order to increase the performance of the system.
69
5a. Evaluate the system
Consider a temperature of 16oC, use the system
to compute the optimal fan speed.
Example: Air Conditioner
RECALL: Operation of a fuzzy expert system:
Fuzzification: determination of the degree of membership of crisp inputs in appropriate fuzzy sets.
Inference: evaluation of fuzzy rules to produce an output for each rule.
Aggregation: combination of the outputs of all rules.
Defuzzification: computation of crisp output
70
• Fuzzification
Affected fuzzy sets: COOL and PLEASANT
COOL(T) = – T / 5 + 3.5
= – 16 / 5 + 3.5
= 0.3
PLSNT(T) = T /2.5 - 6
= 16 /2.5 - 6
= 0.4
Temp=16 COLD COOL PLEASANT WARM HOT
0 0.3 0.4 0 0
Example: Air Conditioner
71
• Inference
Example: Air Conditioner
RULE 1: IF temp is cold THEN speed is minimal
RULE 2: IF temp is cool THEN speed is slow
RULE 3: IF temp is pleasant THEN speed is medium
RULE 4: IF temp is warm THEN speed is fast
RULE 5: IF temp is hot THEN speed is blast
RULE 2: IF temp is cool (0.3) THEN speed is slow (0.3)
RULE 3: IF temp is pleasant (0.4) THEN speed is medium (0.4) 72
• Aggregation
speed is slow (0.3) speed is medium (0.4) +
Example: Air Conditioner
73
• Defuzzification
COG = 0.125(12.5) + 0.25(15) + 0.3(17.5+20+…+40+42.5) + 0.4(45+47.5+…+52.5+55) + 0.25(57.5)
0.125 + 0.25 + 0.3(11) + 0.4(5) + 0.25
= 45.54rpm
Example: Air Conditioner
74
Centroid of area zCOA
where A(z) is the aggregated output MF.
,dz)z(
zdz)z(
z
Z
A
Z
A
COA
Input – Output Plot
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.2
0.3
0.4
0.5
0.6
number_of_serversmean_delay
nu
mb
er_
of_
spa
res
0 5 10 15 20 25 300
10
20
30
40
50
60
70
80
90
100
temp
fan-s
peed
Example: Air Conditioner
one input – one output
gives nonlinear transfer
characteristic More general example:
two inputs – one output gives
3D transfer surface 76