Institute Of Engineering, Pulchowk Campus

Fuzzy Logic: An IntroductionPresented by:

Bhanu Fix Poudyal066BEL305

Department Of Electrical EngineeringPulchowk Campus

2/20/2012

Institute Of Engineering, Pulchowk Campus

Agenda

General DefinitionApplicationsFormal DefinitionsOperationsRulesFuzzy Air ConditionerController Structure

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Institute Of Engineering, Pulchowk Campus

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

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Institute Of Engineering, Pulchowk Campus

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?

Definition

Fuzzy Logic

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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

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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

Fuzzy Logic

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Applications

ABS BrakesExpert SystemsControl UnitsBullet train between Tokyo and

OsakaVideo CamerasAutomatic TransmissionsWashing Machines

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Institute Of Engineering, Pulchowk Campus

Formal DefinitionsDefinition 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)).

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Fuzzy Logic OperatorsFuzzy Logic:

NOT (A) = 1 - AA AND B = min( A, B)A OR B = max( A, B)

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Operations

A B

A B A B A

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Fuzzy Logic NOT

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Fuzzy Logic AND

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Fuzzy Logic OR

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Fuzzy ControllersUsed to control a physical system

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Controller Structure

FuzzificationScales and maps input variables to fuzzy

setsInference Mechanism

Approximate reasoningDeduces the control action

DefuzzificationConvert fuzzy output values to control

signals

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Structure of a Fuzzy Controller

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FuzzificationConversion of real input to fuzzy set valuese.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 }

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Inference EngineFuzzy rules

based on fuzzy premises and fuzzy consequences

e.g.If height is Short and weight is Light then feet

are SmallShort( height) AND Light(weight) =>

Small(feet)

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Fuzzification & Inference ExampleIf height is 1.7m and weight is 55kg

what is the value of Size(feet)

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DefuzzificationRule 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

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Defuzzification ExampleImagine we have output fuzzy set values

Small membership value = 0.5Medium membership value = 0.25Large membership value = 0.0

What is the deffuzzified value

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Fuzzy Control Example

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Rule Base

Air Temperature

Set cold {50, 0, 0}Set cool {65, 55, 45}Set just right {70, 65,

60}Set warm {85, 75, 65}Set hot {, 90, 80}

Fan Speed

• Set stop {0, 0, 0}• Set slow {50, 30, 10}• Set medium {60, 50, 40}• Set fast {90, 70, 50}• Set blast {, 100, 80}

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Rules

Air Conditioning Controller Example:

IF Cold then StopIf Cool then SlowIf OK then MediumIf Warm then FastIF Hot then Blast

default:

The truth of any statement is a matter of degree

default:

The truth of any statement is a matter of degree

Membership function is a curve of the degree of truth of a given input value

Membership function is a curve of the degree of truth of a given input value

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Fuzzy Air Conditioner

Stop

Slow

Medium

Fast

Blast

0

10

20

30

40

50

60

70

80

90

100

0

1

45 50 55 60 65 70 75 80

0C

old

C

ool

85 90

Just

Rig

ht

W

arm

Hot

if Coldthen Stop

IF CoolthenSlow

If Just Rightthen

Medium

If WarmthenFast

If HotthenBlast

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Institute Of Engineering, Pulchowk Campus

Mapping Inputs to Outputs

1

0

10

20

30

40

50

60

70

80

90

100

0

1

45 50 55 60 65 70 75 80

0

85 90

t

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