Dr.-Ing. Erwin Sitompul President University Lecture 8 Introduction to Neural Networks and Fuzzy...

Dr.-Ing. Erwin SitompulPresident University

Lecture 8

Introduction to Neural Networksand Fuzzy Logic

President University Erwin Sitompul NNFL 8/1

http://zitompul.wordpress.com


( )T T( )Tl T ( )Tm T

( C)T

1

“Temperature is low” AND “Temperature is middle”

( )T T( )Tl T ( )Tm T

( C)T

1

“Temperature is low” OR “Temperature is middle”

Min

Max

Membership FunctionFuzzy Logic

Homework 5


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( C)T

1


( )T T( )Tl T ( )Tm T

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1


Algebraic product

Algebraic sum

Homework 5Membership FunctionFuzzy Logic


( )T T( )Tl T ( )Tm T

( C)T

1


( )T T( )Tl T ( )Tm T

( C)T

1


Bounded product

Bounded sum

Homework 5Membership FunctionFuzzy Logic


Further Fuzzy Set Operations

2

1/ 2

3

( ) ( )

( ) ( )

( ) ( )

very A A

morl A A

extremely A A

x x

x x

x x

1/3

( ) 1 ( )

( ) ( )not A A

slightly A A

x x

x x

Fuzzy ControlFuzzy Logic

Dilation

Concentration


Fuzzy Control LoopFuzzy ControlFuzzy Logic


Prior to fuzzy control, the followings must be defined: Fuzzy membership functions Fuzzy logic operators Fuzzy rules, including fuzzy linguistic value and

linguistic variable

The processing steps in a fuzzy control include: Fuzzification Implication / Inference Core Accumulation Defuzzification


Fuzzy Inference



Fuzzy Rules Example of a fuzzy rule while “Driving a Car”:

“IF the distance to the car in front is small, AND the distance is decreasing slowly, THEN decelerate quite big”

The question that arises:Given a certain distance and a certain change of distance, what (crisp) value of acceleration should we select?


Definition of Fuzzy Membership Functions

v. small

Distance

small perfect big v. big moderate

Distance decrease

slow fast very fastv. slow

Acceleration

–small zero+small +big–big



Fuzzification

Observation/measurement


• Distance between small and perfect

• Distance decrease can be moderate or fast

• What acceleration should be applied?


v. small

Distance

small perfect big v. big moderate

Distance decrease


Acceleration


10 m 4m s


RULE 1:IF distance is small THEN decelerate small

0.55

Inference core: ClippingClip the fuzzy membership function of “–small” at the height given by the premises (0.55). Later, the clipped area will be considered in the final decision

Implication of RulesFuzzy ControlFuzzy Logic

v. small

Distance

small perfect big v. big


Acceleration

–small zero +small +big–big

10 m


RULE 2:IF distance decrease is moderate THEN keep the speed

Inference core: ClippingClip the fuzzy membership function of “zero” at the height given by the premises (0.7). Later, the clipped area will be considered in the final decision

Implication of RulesFuzzy ControlFuzzy Logic


Acceleration

–small zero+small +big–bigmoderate

Distance decrease


0.7

4m s


From each rule, a clipped area is obtained. But, in the end only one single output is wanted. How do we make a final decision?


Acceleration

Accumulation

Rule 1

Rule 2


In the accumulation (aggregation) step, all clipped areas are merged into one merged area (taking the union).

Rules with high premises will contribute large clipped area to the merged area. These rules will “pull” that merged area towards their own central value.



Acceleration

In this last step, the returned value is the wanted acceleration.

Out of many possible ways, the center of gravity is the commonly used method in defuzzification.

Crisp value

2. . 2.3m si e Center of gravity


Defuzzification


0.55

acceleration

1. Clipping approach:

0.55

acceleration

2. Scaling approach:

Impl , ( ) min , ( )A B A By y

A

( )B y

Fuzzification value Membership function

Impl , ( ) ( )A B A By y

A

( )B y

Min-Operator

Algebraic Product


Inference Core There are two approaches that can be used for

inference core:


Rectangle Triangle


Review on Center of Gravity


Isosceles Trapezoid Trapezoid


Review on Center of Gravity


Summary of Fuzzy ControlFuzzy ControlFuzzy Logic

1. Fuzzify inputs, determine the degree of membership for all terms in the premise.

2. Apply fuzzy logic operators, if there are multiple terms in the premise (min-max, algebraic, bounded).

3. Apply inference core (clipping, scaling, etc.)4. Accumulate all outputs (union operation i.e. max,

sum, etc.)5. Defuzzify (center of gravity of the merged outputs,

max-method, modified center of gravity, height method, etc)


Limitations of Fuzzy ControlFuzzy ControlFuzzy Logic

Definition and fine-tuning of membership functions need experience (covered range, number of MFs, shape).

Defuzzification may produce undesired results (needs redefinition of membership functions).


Homework 6

v. small

Distance to next car [m]

small perfect big v. big

0 5 10 15 20 25

1

Speed change [m/s2]

constant growingdeclining

–small zero +small +big–big

Acceleration adj. [m/s2] –2 –1 0 1 2

1

–10 –5 0 5 10

1


A fuzzy controller is to be used in driving a car. The fuzzy membership functions for the two inputs and one output are defined as below.


Homework 6 (Cont.)Fuzzy ControlFuzzy Logic

A fuzzy controller is to be used in driving a car. The fuzzy rules are given as follows.Rule 1: IF distance is small AND speed is declining,

THEN maintain acceleration.Rule 2: IF distance is small AND speed is constant,

THEN acceleration adjustment negative small.Rule 3: IF distance is perfect AND speed is declining,

THEN acceleration adjustment positive small.Rule 4: IF distance is perfect AND speed is constant,

THEN maintain acceleration.


Homework 6 (Cont.)Fuzzy ControlFuzzy Logic

Using Min-Max as fuzzy operators, clipping as inference core, union operator as accumulator, and center of gravity method as defuzzifier, find the output of the controller if the measurements confirms that distance to next car is 13 m and the speed is increasing by 2.5 m/s2.


Homework 6 NewFuzzy ControlFuzzy Logic

NEW

A driver of an open-air car determine how fast he drives based on the air temperature and the sky conditions. The corresponding fuzzy membership functions can be seen here.

50 70 90 1103010Temperature (°F)

Freezing Cool Warm Hot

0

1

0 40 60 80 100200Cloud Cover (%)

OvercastPartly CloudySunny

0

1

50 75 100250Speed (km/h)

Slow Fast

0

1


Homework 6 New(Cont.)Fuzzy ControlFuzzy Logic

NEW

After years of experience, he summarizes his personal driving rules as follows:Rule 1: IF it is sunny AND warm, THEN drive fast.Rule 2: IF it is partly cloudy AND hot, THEN drive slow.Rule 3: IF it is partly cloudy, THEN drive fast.

You are now assigned to design a fuzzy control with the following requirements: Fuzzy logic operators: algebraic sum / product Inference core: scaling Accumulator: union operator Defuzzification: center of gravity method

The speed limit is 120 km/h. How fast will the driver go if in one day the temperature is 65 °F and the cloud cover is 25 %?

Dr.-Ing. Erwin Sitompul President University Lecture 8 Introduction to Neural Networks and Fuzzy...

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