Fuzzy Logic

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Fundamentals of Fuzzy Logic

• Introduction• Fuzzy Set and example• Fuzzy Terminology• Fuzzy Logic Control and case study of

Room Cooler• Fuzzy Regions, Fuzzy Profiles and Fuzzy

Rules• Fuzzification• Defuzzifier

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Introduction

• Fuzzy systems, Neural networks and Genetic Algorithms are a part of soft computing technologies

• Assume that the problems to be solved belong to a multidimensional input-output space or search space; for example a two input and one output space where the inputs and output are related with nonlinear function

• The objective is to find the best input that produces the required output

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Introduction contd..

• Fuzzy systems and Neural networks model such complex nonlinearity by combining multiple simple functions

• Neural networks use sigmoid or other simple functions and synaptic weights

• Fuzzy systems use several rules and membership functions

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Introduction to fuzzy logic

• Uncertainty is inherent in accessing information from large amount of data; for example words like near and slow in sentences like” My house is near to the office, “He drivels slowly”

• If we set slow as speeds <=20 and fast otherwise, then is 20.1 is fast?

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Introduction to fuzzy logic contd..

• Fuzzy logic deals with techniques to capture the essence of comprehension and embed it on the system

• Thus using fuzzy logic a gradual transition from slow to high speed is allowed

• Due to the comprehension, fuzzy logic provides higher intelligence quotient to machines

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Crisp sets and Fuzzy sets

Crisp sets: In a crisp set, members belong to the group identified by the set or not

slow = {s such that 0 <= s <= 40}

fast = {s such that 40 < s <70}

40.1 belongs to set fast, hence 40.1 is not slow

Drawback of crisp sets: Suppose a physical system has to apply brakes if the speed of the vehicle is fast and release the brake if the speed is slow. If the speed is in the interval [39, 41], such a system would continuously keep jerking which is not desired

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Fuzzy Sets

• To reduce the complexity of comprehension, vagueness is introduced in crisp sets

• Fuzzy set contains elements; each element signifies the degree or grade of membership to a fuzzy aspect

• Membership values denote the sense of belonging of a member of a crisp set to a fuzzy set

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Example of a fuzzy set

• Consider a crisp set A with elements representing ages of a set of people in years

• A = { 2, 4, 10, 15, 20, 30, 35, 40, 45, 60, 70}

• Classify the age in terms of six fuzzy variables or names given to fuzzy sets as: infant, child, adolescent, adult, young and old

• Membership is different from probabilities• Memberships do not necessarily add up to one

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Ages and their memberships

Table 1.

Age Infant Child Adolescent

Young Adult Old

2 1 0 0 1 0 0

4 0.1 0.5 0 1 0 0

10 0 1 0.3 1 0 0

15 0 0.8 1 1 0 0

21 0 0 0.1 1 0.8 0.1

30 0 0 0 0.6 1 0.3

0 0 0 0 0.5 1 0.35

40 0 0 0 0. 4 1 0.4

45 0 0 0 0.2 1 0.6

60 0 0 0 0 1 0.8

70 0 0 0 0 1 1

Explanation of Example

• How to categorize a person with age 30?• A person with age 40 is old?• The table 1. shows the fuzzy sets namely ages,

infant, child, adolescent, adult, young and old• The values in the table indicate the memberships to

the fuzzy sets• For example, consider the fuzzy set child.• A child with age 4 belongs to the fuzzy set child with

0.5 membership value and a child with age 10 is 100% member

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Explanation of Example contd..

• As per the table 1. a person with age 30 is 60% young and 100% adult

• A person with age 40 is 40% young and 100% adult

• A person with age 60 is 100% adult and 80% old

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Features of Fuzzy Sets

1. A complex nonlinear input-output relation is represented as a combination of simple input-output relations

2. The simple input-output relation is described in each rule

3. The system output from one rule area to the next rule area gradually changes

4. Fuzzy logic systems are augmented with techniques that facilitate learning and adaptation to the environment; thus logic and fuzziness are separate in fuzzy systems

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Features of Fuzzy Sets contd..

• In Conventional two value logic based systems logic and fuzziness are not different

• fuzzy logic systems modify rules when logic is to be changed and change membership functions when fuzziness is to be changed

Some Fuzzy Terminology

• Universe of Discourse (U): The range of all possible values that comprise the input to the fuzzy system

• Fuzzy set: A set that has members with membership (real) values in the interval [0,1]

• Membership function: It is the basis of a fuzzy set. The membership function of the fuzzy set A is given by µA: U [0,1]

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Fuzzy Terminology contd..

• Support of a fuzzy set (Sf): The support S of a fuzzy set f in a universal crisp set U is that set which contains all elements of the set U that have a non-zero membership value in f

the support of the fuzzy set adult S adult is given by

S adult = {21,30,35,40,45,60,70}

Depiction of a fuzzy set: A fuzzy set in a universal crisp set U is written as

f =µ1/s1 + µ2/s2+…+ µn/sn wher µi is the membership, si

is the corresponding term in the support set ; + and / are only user for representation purpose; fuzzy set OLD is depicted as

Old =0.1/21+0.3/30+0.35/35+0.4/40+0.6/45+0.8/60+1/70 15

Fuzzy Set Operations

• Union: The membership function of the union of two fuzzy sets A and B is defined as the maximum of the two individual membership functions. It is equivalent to the Boolean OR operation µ AUB = max( µA, µ B)

• Intersection: The membership function of the Intersection of two fuzzy sets A and B is defined as the minimum of the two individual membership functions. It is equivalent to the Boolean AND operation µ A^B = min(µA, µ B)

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Fuzzy Set Operations contd..

Complement: The membership function of the complement of a fuzzy set A is defined as the negation of the specified membership function It is equivalent to the Boolean NOT operation µ Ac = (1- µA)

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Fuzzy Inference Processing

• There are three models for Fuzzy processing based on the expressions of consequent parts in fuzzy rules

Suppose xi are inputs and y is the consequents in fuzzy rules

1.Mamdani Model: y = A

where A is a fuzzy number to reflect fuzziness

• Though it can be used in all types of systems, the model is more suitable for knowledge processing systems than control systems 18

Fuzzy Inference Processing contd..

2. TSK (Takagi-Sugano-Kang) model:

y = a0 + Ʃ ai xi where ai are constants

The output is the weighted linear combination of input variables (it can be expanded to nonlinear combination of input variables)

Used in fuzzy control applications

3. Simplified fuzzy model: y = c

where c is a constant

Thus consequents are expressed by constant values

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Applications of Fuzzy Logic

• Fuzzy logic has been used in many applications including

- Domestic appliances like washing machines and cameras

-Sophisticated applications such as turbine control, data classifiers etc.

- Intelligent systems that use fuzzy logic employ techniques for learning and adaptation to the environment

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Case Study: Controlling the speed of a motor in a room cooler

• Through this case study we can understand fuzzy logic, defining fuzzy rules and fuzzy inference and control mechanisms

• Mamdani style of inference processing is used• Problem: A room cooler has a fan encased in a box

with wool or hay. The wool is continuously moistened by water that flows through a pump connected to a motor. The rate of flow of water is to be determined; it is a function of room temperature and the speed of motor

• The speed of the motor is based on two parameters: temperature and humidity; humidity is increased to reduce temperature 21

Case Study: Operation of a room cooler contd..

• Two input variables –room temperature and cooler fan speed control the output variable – flow rate of the water. The fuzzy regions using fuzzy terms for input-output are defined as follows

Variable name Fuzzy terms

Temperature Cold, Cool, Moderate, Warm and Hot

Fan speed Slack, Low, Medium, Brisk, fast

((rotations per minute)

Flow rate of water Strong Negative (SN), Negative (N), Low-Negative (LN), Medium (M), Low-Positive (LP), Positive (P), and High-Positive (HP)

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• Fuzzy profiles are defined for each of the three parameters by assigning memberships to their respective values

• The profiles have to be carefully designed after studying the nature and desired behavior of the system

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1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Temperature

Degree of membership

Cold Cool Moderate Warm

Fig.1. Fuzzy relationships for the inputs Temperature

1.2

1

0.8

0.6

0.4

0.2

0

Hot


Figure 2. Fuzzy relationships for the inputs Fan Motor speed

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1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Motor speed RPM


Slack Low Medium Brisk Fast

1.2

1

0.8

0.6

0.4

0.2

0

Slack Low Medium Brisk Fast


Figure 3. Fuzzy relationships for the outputs Water Flow Rate

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0 0.2 0.4 0.6 0.8 1 1.2 14 1.6

Flow rate (ml/Sec)


SN N LN M LP P HP

1.2

1

0.8

0.6

0.4

0.2

0

Fuzzy Rules for fuzzy room cooler• The fuzzy rules form the triggers of the fuzzy engine

• After a study of the system, the rules could be written as follows

R1: If temperature is HOT and fan motor speed is SLACK then the flow-rate is HIGH-POSITIVE

R2: If temperature is HOT and fan motor speed is LOW then the flow-rate is HIGH-POSITIVE

R3: If temperature is HOT and fan motor speed is MEDIUM then the flow-rate is POSITIVE

R4: If temperature is HOT and fan motor speed is BRISK then the flow-rate is HIGH-POSITIVE

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Fuzzy Rules for fuzzy room cooler contd..

• R5: If temperature is WARM and fan motor speed is MEDIUM then the flow-rate is LOW-POSITIVE

• R6: If temperature is WARM and fan motor speed is BRISK then the flow-rate is POSITIVE

• R7: If temperature is COOL and fan motor speed is LOW then the flow-rate is NEGATIVE

• R8: If temperature is MODERATE and fan motor speed is LOW then the flow-rate is MEDIUM

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Fuzzification

• The fuzzifier that performs the mapping of the membership values of the input parameters temperature and fan speed to the respective fuzzy regions is known as fuzzification. This is the most important step in fuzzy systems

• Suppose that at some time t, the temperature is 42 degrees and fan speed is 31 rpm. The corresponding membership values and the fuzzy regions are shown in Table 2

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Example of fuzzification

• From Figure 1., the temperature 42 degrees correspond to two membership values 0.142 and 0.2 that belong to WARM and HOT fuzzy regions respectively

• Similarly From Figure 2., the fan speed 31 rpm corresponds to two membership values 0.25 and 0.286 that belong to MEDIUM and BRISK fuzzy regions respectively Table 2

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Parameters Fuzzy Regions Memberships

Temperature Warm, hot 0.142, 0.2

Fan Speed medium, brisk 0.25, 0.286

Example of fuzzification contd..

• From Table 2, there are four combinations possible• If temperature is WARM and fan speed is MEDIUM• If temperature is WARM and fan speed is BRISK• If temperature is HOT and fan speed is MEDIUM• If temperature is HOT and fan speed is BRISK• Comparing the above combinations with the left side

of fuzzy rules R5, R6, R3, and R4 respectively, the flow-rate should be LOW-POSITIVE, POSITIVE, POSITIVE and HIGH-POSITIVE

• The conflict should be resolved and the fuzzy region is to be given as a value for the parameter water flow-rate

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Defuzzification• The fuzzy outputs LOW-POSITIVE, POSITIVE, and HIGH-

POSITIVE are to be converted to a single crisp value that is provided to the fuzzy cooler system; this process is called defuzzification

• Several methods are used for defuzzification

• The most common methods are

1. The centre of gravity method and

2. The Composite Maxima method

The centroid, of a two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line or the average of all points of X. (Moment is a quantitative measure of the shape of a set of points.)

In both these methods the composite region formed by the portions A, B, C, and D (corresponding to rules R3, R4, R5 and R6 respectively) on the output profile is to be computed31

Defuzzification contd..

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• ttttt 1 4 7… 37 40 43 46 48

1 4 … 13…. 31 34 37 40 43 46 48

1.2

1

.8

.6

. 4

.2

0

1.2

1

.8

.6

.4

.2

0

Hot

Medium

Temperature 42 D Centigrade Motor speed (RPM) 31

0.25

1.2

1

0.8

0.6

0.4

0.2

0

Rule R3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

P

0.2

Flow rate (ml/Sec)

Min(0.2,0.25) = 0.2

C

Figure 4.1

Figure 4.2

Figure 4.3


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• ttttt 1 4 7… 37 40 43 46 48

1 4 … 28 ..31.. 37 40 43 46 48

1.2

1

.8

.6

. 4

.2

0

1.2

1

.8

.6

.4

.2

0

Hot

Brisk


0.286

1.2

1

0.8

0.6

0.4

0.2

0

Rule R4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

HP

0.2

Flow rate (ml/Sec)

Min(0.2,0.286) = 0.2D

Figure 5.1

Figure 5.2

Figure 5.3


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• ttttt 1 4 7..28.. 40 43 46 48

1 4 … 13…. 31 34 37 40 43 46 48

1.2

1

.8

.6

. 4

.2

0

1.2

1

.8

.6

.4

.2

0

Warm

Medium


0.25

1.2

1

0.8

0.6

0.4

0.2

0

Rule R5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

LP

0.142

Flow rate (ml/Sec)

Min(0.142,0.25) = 0.25

A

Figure 6.1

Figure 6.2

Figure 6.3


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• ttttt 1 4 7..28.. 40 43 46 48

1 4 …13.. 28..31 34 37 40 43 46 48

1.2

1

.8

.6

. 4

.2

0

1.2

1

.8

.6

.4

.2

0

Warm

Brisk


0.286

1.2

1

0.8

0.6

0.4

0.2

0

Rule R6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

P

0.142

Flow rate (ml/Sec)

Min(0.142,0.286) =0.142B

Figure 7.1

Figure 7.2

Figure 7.3


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1.2

1

0.8

0.6

0.4

0.2

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

P

Flow rate (ml/Sec)

LPHP

Centroid

A B is within C as it is a subset of the region C

D

Figure 8When parameters are connected by AND the minimum of their memberships is taken

The area C is the region formed by the application of rule R3 as shown in Figure 4.3

The area D is the region formed by the application of rule R4 as shown in Figure 5.3

The area A is the region formed by the application of rule R5 as shown in Figure 6.3

The area B is the region formed by the application of rule R6 as shown in Figure 7.3

The composite region formed by the portions A, B, C and D on the output profile is shown in Figure 8.

The centre of gravity of this composite region is the crisp output or the desired flow rate value

Steps in Fuzzy logic based system

• Formulating fuzzy regions

• Fuzzy rules

• Embedding a Defuzzification procedure

In Defuzzification procedure, depending on the application, either the centre of gravity or the composite maxima is found to obtain the crisp output

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