Fuzzy logic

Presented by:
Bhanu Fix Poudyal
066BEL305
Department Of Electrical Engineering
Pulchowk Campus

2/20/2012 Institute Of Engineering, Pulchowk Campus

Agenda

 General Definition
 Applications
 Formal Definitions
 Operations
 Rules
 Fuzzy Air Conditioner
 Controller Structure


Fuzzy Logic

Definition

 Experts rely on common sense when they solve problems.

 How can we represent expert knowledge that uses vague and
ambiguous terms in a computer?

 Fuzzy logic is not logic that is fuzzy, but logic that is used to describe
fuzziness. Fuzzy logic is the theory of fuzzy sets, sets that calibrate
vagueness.

 Fuzzy logic is based on the idea that all things admit of degrees.
Temperature, height, speed, distance, beauty – all come on a sliding scale.
 The motor is running really hot.
 Tom is a very tall guy.


Fuzzy Logic

Definition

 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.
 It was specifically designed to mathematically represent uncertainty and
vagueness and provide formalized tools for dealing with the imprecision
intrinsic to many engineering and decision problems in a more natural
way.
 Boolean logic uses sharp distinctions. It forces us to draw lines between
members of a class and non-members. For instance, we may say, Tom is tall
because his height is 181 cm. If we drew a line at 180 cm, we would find
that David, who is 179 cm, is small.
 Is David really a small man or we have just drawn an arbitrary line in the
sand?


Fuzzy Logic

Bit of History

 Fuzzy, or multi-valued logic, was introduced in the 1930s by Jan
Lukasiewicz, a Polish philosopher. While classical logic operates with
only two values 1 (true) and 0 (false), Lukasiewicz introduced logic that
extended the range of truth values to all real numbers in the interval
between 0 and 1.

 For example, the possibility that a man 181 cm tall is really tall might be
set to a value of 0.86. It is likely that the man is tall. This work led to
an inexact reasoning technique often called possibility theory.

 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.


Fuzzy Logic

Why Fuzzy Logic?
 Why fuzzy?
As Zadeh said, the term is concrete, immediate and descriptive; we all
know what it means. However, many people in the West were repelled by
the word fuzzy, because it is usually used in a negative sense.
 Why logic?
Fuzziness rests on fuzzy set theory, and fuzzy logic is just a small part of
that theory.
 The term fuzzy logic is used in two senses:
 Narrow sense: Fuzzy logic is a branch of fuzzy set theory, which deals
(as logical systems do) with the representation and inference from
knowledge. Fuzzy logic, unlike other logical systems, deals with
imprecise or uncertain knowledge. In this narrow, and perhaps correct
sense, fuzzy logic is just one of the branches of fuzzy set theory.
 Broad Sense: fuzzy logic synonymously with fuzzy set theory


Applications

 ABS Brakes
 Expert Systems
 Control Units
 Bullet train between Tokyo and Osaka
 Video Cameras
 Automatic Transmissions
 Washing Machines


Formal Definitions
 Definition 1: Let X be some set of objects, with elements noted as x.
 X = {x}.
 Definition 2: A fuzzy set A in X is characterized by a membership function
mA(x) which maps each point in X onto the real interval [0.0, 1.0]. As
mA(x) approaches 1.0, the "grade of membership" of x in A increases.
 Definition 3: A is EMPTY iff for all x, mA(x) = 0.0.
 Definition 4: A = B iff for all x: mA(x) = mB(x) [or, mA = mB].
 Definition 5: mA' = 1 - mA.
 Definition 6: A is CONTAINED in B iff mA mB.
 Definition 7: C = A UNION B, where: mC(x) = MAX(mA(x), mB(x)).
 Definition 8: C = A INTERSECTION B where: mC(x) = MIN(mA(x),
mB(x)).


Fuzzy Logic Operators
 Fuzzy Logic:
 NOT (A) = 1 - A
 A AND B = min( A, B)
 A OR B = max( A, B)


Operations

A B

A B A B A


Fuzzy Logic NOT


Fuzzy Logic AND


Fuzzy Logic OR


Fuzzy Controllers
 Used to control a physical system


Controller Structure
 Fuzzification
 Scales and maps input variables to fuzzy sets
 Inference Mechanism
 Approximate reasoning
 Deduces the control action
 Defuzzification
 Convert fuzzy output values to control signals


Structure of a Fuzzy Controller


Fuzzification
 Conversion of real input to fuzzy set values
 e.g. Medium ( x ) = {
 0 if x >= 1.90 or x < 1.70,
 (1.90 - x)/0.1 if x >= 1.80 and x < 1.90,
 (x- 1.70)/0.1 if x >= 1.70 and x < 1.80 }


Inference Engine
 Fuzzy rules
 based on fuzzy premises and fuzzy consequences

 e.g.
 If height is Short and weight is Light then feet are Small
 Short( height) AND Light(weight) => Small(feet)


Fuzzification & Inference Example
 If height is 1.7m and weight is 55kg
 what is the value of Size(feet)


Defuzzification
 Rule base has many rules
 so some of the output fuzzy sets will have membership
value > 0

 Defuzzify to get a real value from the fuzzy outputs
 One approach is to use a centre of gravity method


Defuzzification Example
 Imagine we have output fuzzy set values
 Small membership value = 0.5
 Medium membership value = 0.25
 Large membership value = 0.0
 What is the deffuzzified value


Fuzzy Control Example


Rule Base

Air Temperature Fan Speed

 Set cold {50, 0, 0} • Set stop {0, 0, 0}
 Set cool {65, 55, 45} • Set slow {50, 30, 10}
 Set just right {70, 65, 60} • Set medium {60, 50, 40}
 Set warm {85, 75, 65} • Set fast {90, 70, 50}
 Set hot { , 90, 80} • Set blast { , 100, 80}


default: Membership function is
The truth of any
statement is a
a curve of the degree
of truth of a given
Rules
matter of degree
input value

Air Conditioning Controller Example:

 IF Cold then Stop
 If Cool then Slow
 If OK then Medium
 If Warm then Fast
 IF Hot then Blast


Fuzzy Air Conditioner
0

100
s t If Hot
90 Bla then
Blast

80 Fa
st If Warm
then
70 Fast

60
Med If Just Right
ium then
50 Medium

40 IF Cool
Sl
ow then
30 Slow

if Cold
20
then Stop

10
St
o p

0

1
m

t
ol

Ho
ar
Co

W
Co

Rig t
ht
Jus
ld

0

45 50 55 60 65 70 75 80 85 90

Mapping Inputs to Outputs
1
0

100
s t
90 Bla t

80 Fa
st
70

60
Med
iu m
50

40
Sl
ow
30

20

10
St
op

0

1

m

t
ol

Ho
ar
Co

W
Co

Rig t
ht
Jus
ld

0

45 50 55 60 65 70 75 80 85 90


Fuzzy logic

More Related Content

What's hot (20)

Viewers also liked (20)

Similar to Fuzzy logic (20)

Recently uploaded (20)

Fuzzy logic