Fuzzy Logic and Neural Network

ByMrs. Shimi S.L

Assistant Professor,EENITTTR, Chandigarh

Fuzzy Logic using MATLAB

The term "fuzzy logic" was introduced withthe 1965 proposal of fuzzy settheory by Lotfi A. Zadeh.

Fuzzy logic is a formof many-valued logic; itdeals with reasoning that isapproximate rather thanfixed and exact.

Mrs. Shimi S.LAssistant Professor,EE

NITTTR, Chandigarh

Fuzzy Controllers

The Outputs of the Fuzzy Logic System Are the Command Variables of the Plant:

Fuzzification Inference Defuzzification

IF temp=low

AND P=high

THEN A=med

IF ...

Variables

Measured Variables

Plant

Command

Conventional (Boolean) Set Theory:

Fuzzy Set Theory

“Strong Fever”

40.1°C

42°C

41.4°C

39.3°C

38.7°C

37.2°C

38°C

Fuzzy Set Theory:

40.1°C

42°C

41.4°C

39.3°C

38.7°C

37.2°C

38°C

“More-or-Less” Rather Than “Either-Or” !

“Strong Fever”

Fuzzy Set vs Crisp Set

• X is a set of all real numbers from 1 to 10• Universe of Discourse• A is a set of real numbers between 5 and 8• Crisp or Classical Set• Membership Value 1 or 0

Fuzzy Set vs Crisp Set

• B is a set of young people

• Membership values between 0 and 1 – Fuzzy Set

Age 65 27 17 32 22 25

B 0 0.3 1 0 0.8 0.5

Fuzzy Set

• Another example of Fuzzy Set

• What season is it right now?

• Using the astronomical definitions for season, we get sharp boundaries.

• What we experience as seasons varies more or less continuously

Traditional Representation of Logic

Slow Fast

Speed = 0 Speed = 1

bool speed;

get the speed

if ( speed == 0) {

// speed is slow

}

else {

// speed is fast

}


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Fuzzy Logic Representation

Every problem must be represent in terms of fuzzy sets.

What are fuzzy sets?

Slowest

Fastest

Slow

Fast

[ 0.0 – 0.25 ]

[ 0.25 – 0.50 ]

[ 0.50 – 0.75 ]

[ 0.75 – 1.00 ]

Fuzzy Logic Representation

Slowest Fastestfloat speed;

get the speed

if ((speed >= 0.0)&&(speed < 0.25)) {

// speed is slowest

}

else if ((speed >= 0.25)&&(speed < 0.5))

{

// speed is slow

}

else if ((speed >= 0.5)&&(speed < 0.75))

{

// speed is fast

}

else // speed >= 0.75 && speed < 1.0

{

// speed is fastest

}

Slow Fast

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Fuzzy Linguistic Variables

• Fuzzy Linguistic Variables are used to represent qualities spanning a particular spectrum

• Temp: {Freezing, Cool, Warm, Hot}

• Membership Function

• Question: What is the temperature?

• Answer: It is warm.

• Question: How warm is it?


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Membership Functions• Temp: {Freezing, Cool, Warm, Hot}• Degree of Truth or "Membership“

• Each of these linguistic terms is associated with a fuzzy set defined by a corresponding membership function.

50 70 90 1103010

Temp. (F°)

Freezing Cool Warm Hot

0

1


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Membership Functions

• Membership function (MF) is a curve that defineshow each point in the input space is mapped to amembership value (or degree of membership)between 0 and 1 and is often given the designationof µ.

• µA(x) is called the membership function (or MF) of xin A.

• Thus membership functions are subjective measuresfor linguistic terms.

• There are many types of membership functions.

Types of Membership Functions

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• How cool is 36 F° ?

50 70 90 1103010

Temp. (F°)


0

1


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• How cool is 36 F° ?• It is 30% Cool and 70% Freezing

50 70 90 1103010

Temp. (F°)


0

1

0.7

0.3


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Fuzzy Logic

• How do we use fuzzy membership functions in predicate logic?

• Fuzzy logic Connectives:

– Fuzzy Conjunction,

– Fuzzy Disjunction,

• Operate on degrees of membership in fuzzy sets


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Fuzzy Disjunction (Union)

• AB max(A, B)

• AB = C "Quality C is the disjunction of Quality A and B"

0

1

0.375

A

0

1

0.75

B

(AB = C) (C = 0.75)


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Fuzzy Conjunction (Intersection)

• AB min(A, B)

• AB = C "Quality C is the conjunction of Quality A and B"

0

1

0.375

A

0

1

0.75

B

(AB = C) (C = 0.375)


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Fuzzy Set Operations• There are three basic operation on fuzzy sets: negation, intersection,

and union

• Negation

membership_value(not x)= 1- membership_value(x) where x is the fuzzy set being negated

• Intersection

membership_value(x and y) = minimum{membership_value(x), membership_value(y)}

where x and y are the fuzzy sets involved in the intersection

• Union

membership_value(x or y) = maximum{membership_value(x), membership_value(y)}

where x and y are the fuzzy sets involved in the union

• minimum operator for intersection and the

maximum operator for union

Fuzzy Set Operations

• Let A be a fuzzy interval between 5 and 8 • B be a fuzzy number about 4.


Union of A and B

A OR B

Intersection of A and B

A AND B


• Negation of A • De Morgan’s Laws

BABA

BABA


100

0

80

0

40

0

20

40

10

70

1

1

..A

100

0

80

20

40

50

20

01

10

01

1

30

.....B

BABAvi

BABAv

Biv

Aiii

BAii

BAi

)(

)(

)(

)(

)(

)(

Fuzzy Relations• A crisp relation between two sets X, Y is a

binary relation.

• Binary relations are represented by relation

matrices and also by sagittal diagrams.

• R={(1,a) (2,c) (3,b) (4,c)}

• Sagittal Diagram

• Relation Matrix

1004

0103

1002

0011

cba

Fuzzy Relations• Relation between two or more fuzzy sets is

obtained by the Cartesian product.

yxyxyx BAAxBR ,min,,

321 x

1+

x

50+

x

20=A

..

21 y

90+

y

30=B

..

9030x

5030x

2020x

yy

3

2

1

21

..

..

..

Fuzzy Relations

• Let us describe the relationship between the

colour of a fruit, x and the grade of maturity, y.

• x= {green, yellow, red}

y={verdant, half-mature, mature}

• Considering the relation between the linguistic

terms red and mature, and representing them

by the membership functions, a fruit can be

characterized by the property of red and

mature.

Fuzzy Relations• Characteristics of a red and mature fruit

4.08.0

5.07.0=

2

1

21

x

x

yy

R

Fuzzy Compositions

T = R o S - max-min composition

T = R S - max-product composition

Chain-strength analogy for max-min composition

5.07.01.0

2.06.09.0=

2

1

321

y

y

zzz

S

( ) ( ) ( )( ){ }zyyxzx SRT ,,,minmax=,

( ) ( ) ( )( ){ }zyyxzx SRT ,•,max=,

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Composition of fuzzy relations

33

• Example

(1, ) max[min(0.1,0.9),min(0.2,0.2),min(0.0,0.8),min(1.0,0.4)]

max[0.1,0.2,0.0,0.4] 0.4

S R

g


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• Example

(1, ) max[min(0.1,0.0),min(0.2,1.0),min(0.0,0.0),min(1.0,0.2)]

max[0.0,0.2,0.0,0.2] 0.2

S R

g


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Fuzzy Relations• Three variables of interest in power transistors are the

amount of current that can be switched, the voltage that can

be switched, and the cost. The following membership

functions for power transistors were developed from

hypothetical components catalog:

• Average current

• Average voltage

• Power is defined by the algebraic operation P = VI

(a) Let us find the Cartesian Product P = VxI.

{ }21

60+

11

80+

1

1+

90

70+

80

40=

.

.

.

.

.

.

.

.I

{ }90

70+

75

90+

60

1+

45

80+

30

20=

....V

Fuzzy Relations• The Cartesian Product expresses the relationship between

Vi and Ij , where Vi and Ij are individual elements in the

fuzzy set V and I.

• Now let us define a fuzzy set for the cost C in rupees, of a

transistor

(b)Using a fuzzy Cartesian Product, find T = IxC.

(c) Using max-min composition find E = PoT

(d) Using max-product composition find E = PoT

{ }70

50+

60

1+

50

40=

.

.

..

.C

Fuzzy Control

Using a procedure originated by Ebrahim Mamdani

in the late 70s, three steps are taken to create a

fuzzy controlled machine:

Fuzzification (Using membership functions to

graphically describe a situation)

Rule Evaluation (Application of fuzzy rules)

Defuzzification (Obtaining the crisp results)

Fuzzy Control Fuzzification is the process of making a crisp quantity

fuzzy.

Membership functions characterize the fuzziness in a fuzzy set.

Six procedures to build membership functions

Intuition

Inference

Rank Ordering

Neural Networks

Genetic Algorithm

Inductive Reasoning

Fuzzy Control Defuzzification is the conversion of a fuzzy quantity to a

precise quantity.

Output of a fuzzy process can be the logical union of two or more fuzzy membership functions defined on the universe of discourse. .

Methods of defuzzification

Max-membership principle

Centroid method

Weighted average method

Mean max membership

Center of sums

Center of largest area

First (or last) of maxima

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Fuzzy Control

• Fuzzy Control combines the use of fuzzy linguistic variables with fuzzy logic

• Example: Speed Control

• How fast am I going to drive today?

• It depends on the weather.

• Disjunction of Conjunctions


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Inputs: Temperature, Cloud Cover

• Temp: {Freezing, Cool, Warm, Hot}

• Cover: {Sunny, Partly, Overcast}

50 70 90 1103010

Temp. (F°)


0

1

40 60 80 100200

Cloud Cover (%)

OvercastPartly CloudySunny

0

1


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Output: Speed

• Speed: {Slow, Fast}

50 75 100250

Speed (mph)

Slow Fast

0

1


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Rules

• If it's Sunny and Warm, drive FastSunny(Cover)Warm(Temp) Fast(Speed)

• If it's Cloudy and Cool, drive SlowCloudy(Cover)Cool(Temp) Slow(Speed)

• Driving Speed is the combination of output of these rules...


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Example Speed Calculation

• How fast will I go if it is

– 65 F°

– 25 % Cloud Cover ?


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Fuzzification:Calculate Input Membership Levels

• 65 F° Cool = 0.4, Warm= 0.7

• 25% Cover Sunny = 0.8, Cloudy = 0.2

50 70 90 1103010

Temp. (F°)


0

1

40 60 80 100200

Cloud Cover (%)

OvercastPartly CloudySunny

0

1


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...Calculating...

• If it's Sunny and Warm, drive FastSunny(Cover)Warm(Temp)Fast(Speed)

0.8 0.7 = 0.7 Fast = 0.7

• If it's Cloudy and Cool, drive SlowCloudy(Cover)Cool(Temp)Slow(Speed)

0.2 0.4 = 0.2 Slow = 0.2

AB = min(A, B)


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Defuzzification: Constructing the Output

• Speed is 20% Slow and 70% Fast

• Find centroids: Location where membership is 100%

50 75 100250

Speed (mph)

Slow Fast

0

1


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Defuzzification: Constructing the Output

• Speed is 20% Slow and 70% Fast

• Speed = weighted mean =

= (2*25+7*75)/(9)

= 63.8 mph

50 75 100250

Speed (mph)

Slow Fast

0

1


NITTTR, Chandigarh

Artificial Neural Network

● Artificial neural network (ANN) is a machinelearning approach that models human brain andconsists of a number of artificial neurons.

● An Artificial Neural Network is specified by:− neuron model: the information processing unit

of the NN,− an architecture: a set of neurons and links

connecting neurons. Each link has a weight,− a learning algorithm: used for training the NN

by modifying the weights in order to model aparticular learning task correctly on thetraining examples.

● The aim is to obtain a NN that is trained andgeneralizes well.

● It should behaves correctly on new instances ofthe learning task.

The Biological Neural Network

Characteristics of Human Brain• Ability to learn from experience

• Ability to generalize the knowledge it possess

• Ability to perform abstraction

• To make errors.

• A neuron fires when the sum of its collective inputs reaches a threshold

• There are about 10^11 neurons per person

• Each neuron may be connected with up to10^5 other neurons

Consists of three sectionscell bodydendritesaxon

• Nerve impulses which pass down the axon, jump from node to node, thus saving energy.

• There are about 10^16 synapses. Usually no physical or electrical connection made at the synapse.

Human neurons Artificial neurons

Neurons Neurons

Axon, Synapse Wkj (weight)

Synaptic terminalsto next neuron

output terminals

Synaptic terminalstaking input

input terminals (Xj)

human response time=1 ms silicon chip response time=1ns

Input

values

weights

Summing

function

Bias

b

Activation

functionInduced

Field

vOutput

y

x1

x2

xm

w2

wm

w1

)(

m

1jj xwu

j

Perceptron: Neuron Model(Special form of single layer feed forward)

Neuron

● The neuron is the basic information processing unit of a NN. It consists of:1 A set of links, describing the neuron inputs, with weights W1, W2,

…, Wm

2 An adder function (linear combiner) for computing the weighted sum of the inputs:

(real numbers)

3 Activation function for limiting the amplitude of the neuron output. Here ‘b’ denotes bias.

m

1jj xwu

j

)(u y b

Bias of a Neuron

● The bias b has the effect of applying a transformation tothe weighted sum u

v = u + b

● The bias is an external parameter of the neuron. It can bemodeled by adding an extra input.

● v is called induced field of the neuron

bw

xwv j

m

j

j

0

0

Activation Function

● The choice of activation function determines the neuron model.

Examples:

● step function:

● ramp function:

● sigmoid function with z,x,y parameters

● Gaussian function:

2

2

1exp

2

1)(

vv

)exp(1

1)(

yxvzv

otherwise ))/())(((

if

if

)(

cdabcva

dvb

cva

v

cvb

cvav

if

if )(

Training

Training is accomplished by sequentially applying input vectors while

adjusting network weights according to a predetermined procedures.

Supervised Training

requires the pairing of each input vector with a target vector representing

the desired output.

Unsupervised Training

requires no target vector for the output and no comparisons to

predetermined ideal responses. The training algorithm modifies network

weights to produce output vectors that are consistent. Also called self-

organizing networks.

Gradient descent or Steepest Descent

ɳ is the learning rate

global minimum

X1

1 true true

false true

0 1 X2

Boolean function OR – Linearly separable

These two classes (true and false) cannot be separated using a

line. Hence XOR is non linearly separable.

Input Output X1 X2 X1 XOR X2

0 0 0

0 1 1

1 0 1

1 1 0

X1

1 true false

false true

0 1 X2

Multi layer feed-forward NN (FFNN)

● FFNN is a more general network architecture, where there arehidden layers between input and output layers.

● Hidden nodes do not directly receive inputs nor send outputs tothe external environment.

● FFNNs overcome the limitation of single-layer NN.

● They can handle non-linearly separable learning tasks.

Input

layer

Output

layer

Hidden Layer

3-4-2 Network

FFNN for XOR

● The ANN for XOR has two hidden nodes that realizes this non-linearseparation and uses the sign (step) activation function.

● Arrows from input nodes to two hidden nodes indicate the directions ofthe weight vectors (1,-1) and (-1,1).

● The output node is used to combine the outputs of the two hiddennodes.

Input nodes Hidden layer Output layer Output

H1 –0.5

X1 1

–1 1

Y

–1 H2

X2 1 1

Inputs Output of Hidden Nodes Output

Node

X1 XOR X2 X1 X2 H1 H2 0 0 0 0 –0.5 0 0

0 1 –1 0 1 0.5 1 1

1 0 1 –1 0 0.5 1 1

1 1 0 0 –0.5 0 0

Since we are representing two states by 0 (false) and 1 (true), we

will map negative outputs (–1, –0.5) of hidden and output layers

to 0 and positive output (0.5) to 1.

Input nodes Hidden layer Output layer Output

H1 –0.5

X1 1

–1 1

Y

–1 H2

X2 1 1

Hardware Implementation• Dspace

• Quad-Core AMD

Opteron processor

Opal RT

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Thank you.

Questions, Comments, …?

[email protected]

mailto:[email protected]

• Human can identify a person through thoughts.which means humans neurons are getting trained itself. Therefore through Artificial Neural Network we can train artificial neurons using computer programming . using neural network we are trying to build a network between neurons to transfer the electrical signals.which are consists of neural commands .

• usually Computer response time is 10^6 times faster than humans response time because of the silicon Integrated chips.

• silicon chip response time :- 1 nanosecond• human response time :- 1 millisecond•

• but human can perform faster than chips because human has massively parallel neural structure. If we consider human neuron structure it has synaptic terminals, cell body(neurons), basal dendrite and axon. Each components has some function to transfer signal to neurons.

• Bias neurons are added to neural networks to help them learn patterns. A bias neuron is nothing more than a neuron that has a constant output of one. Because the bias neurons have a constant output of one they are not connected to the previous layer. The value of one, which is called the bias activation, can be set to values other than one. However, one is the most common bias activation.

Engineering

Fuzzy Logic and Neural Network