mathematical comparison of defuzzification of fuzzy logic

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International Journal of Scientific Research in ___________________________ Research Paper. Mathematical and Statistical Sciences

Volume-8, Issue-2, pp.29-37, April (2021) E-ISSN: 2348-4519

Mathematical Comparison of Defuzzification of Fuzzy Logic Controller

for Intelligence Air Conditioning System

M.A. Islam1*

, M.S. Hossain2, I.S.M. Haque

3

1,2,3

Department of Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh

*Corresponding Author: [email protected], Tel.: +8801773387844

Available online at: www.isroset.org

Received: 27/Feb/2021, Accepted: 07/Apr/2021, Online: 30/Apr/2021

Abstract—The objective of this paper is to compare several defuzzification techniques for the application of an air cooler

control design in order to predict the error difference between different methods. The four input parameters are considered

to be user temperature (UT), temperature difference (Tdf), dew point (Td), and number of peoples (NP). However,

Compressor Speed (CS), Fan Speed (FS) and Fin Direction (FD) are used as outputs. The simulation of this paper is carried

out by using MATLAB.

Keywords—Fuzzy logic controller, defuzzification techniques, fuzzy rule base, MATLAB simulation.

1. INTRODUCTION

In 1965, the idea of Fuzzy Logic was proposed by L.A.

Zadeh in one of his research papers under the name "Fuzzy

Logic or Fuzzy Sets"[1, 2]. In 1974, Mamdani [3] applied

the fuzzy logic in a practical application to control an

automatic steam engine. Today, Fuzzy Logic makes a

tremendous change in the development of uncertainty in

artificial intelligence system. However, it is easy to

implement and its impact in modern science is very ground

breaking. Its main objective is to simulate human brain on

decision making mechanism in intelligence system while

working in a vague, precise or uncertain environment.

Although it doesn’t need total knowledge about the

characteristic of the model, it helps machine to think like

human and respond instantly. The contribution of FIS

(Fuzzy Inference System) in this manner is not doubtful.

Its working principle can be divided into four mechanisms

like Fuzzification, Fuzzy Rules Based System, Fuzzy

Inference Engine and Defuzzification [4]. With the

introduction of linguistic variable, it is easy to describe

fuzzy knowledge-based system where its transition is very

smooth between rules-based systems. Besides,

Fuzzification and Defuzzification system refer to the

process of making fuzzy and crisp output. In order to get

the target output of an air conditioner by manipulating

temperature and humidity to save the energy of compressor

and fan while using all the resources, fuzzy logic rules-

based system is used for fuzzification, defuzzification for

the acquiring the target output [5]. In 2012, S. K, Dash et

al. [6] proposed an automated intelligent air conditioning

system while utilizing all available resources including

climatic condition in the efficient manner in order to

provide the user comfortable cooling level and also

optimized energy consumption. In 2015, S. M. Sabhy et al.

[7] developed fuzzy logic Control system of an air

conditioner where he used several linguistic variables and

showed how membership function works in air

conditioning system. Another researcher A. Saepullah [1]

modified FIS to save the energy of air conditioner by using

Mamdani, Sugeno, and Tsukamoto method in 2015.

Further, R. Roshmi et al. [8], S. M. Sofi et al. [9] Rajkumar

et al. [10], R. Kiruthika et al. [11], A. Sandanasamy et al.

[12] and Mohammad Sammany et al. [13] are also working

on fuzzy.

Our aim is to develop fuzzy logic Control System, in order

to make it automated by using triangular and trapezoidal

fuzzy number. This paper provides the idea of improving

air conditioner’s air quality. Here, we demonstrated a little

bit about FIS and in the introductory section while the

simulation results and graphical representation which was

obtained by using MATLAB are discussed afterwards.

Also, this paper describes how automated air conditioning

system can be improved which is based on the imprecise

input sensors.

2. AIR ORGANIZATION SYSTEM

Due point temperature indicates both Temperature and

Relative Humidity (RH) in a certain place. In a certain due

point temperature, if RH increases than Temperature will

decrease. Again, if Temperature increases than RH will

decrease. By controlling due point, material’s decay can be

decreased. Moreover, it is used to measure humidity. The

table 1 shown below shows human’s reaction on a standard

dew point.

Table 1. A standard Dew Point Human Reaction Table

Dew point (Td) Human Reaction

Less than 8° Very Dry

8°- 14° Dry

14°-18° Refreshing

19°-20° Humid

20°-25° Comfortable

Above 25° Painful

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Int. J. Sci. Res. in Mathematical and Statistical Sciences Vol. 8, Issue.2, 2021


Air Condition Fuzzy Logic control system takes four

variables into consideration showing in the following

block:

(1) User temperature (14°C-30° continuous control).

(2) Temperature difference (Tdf).

(3) Dew point (Td).

(4) Number of people (NP).

User temperature subtracted from temperature difference

before sending data for fuzzification step. Fuzzy arithmetic

and criterion step is applied on these variables and final

result is defuzzified step to get following crisp results as

showing in the following figure 1.

(1) Compressor Speed (CS).

(2) Fan Speed (FS).

(3) Fin Direction (FD).

Figure 1. FLC for Air Conditioning System.

3. FUZZY LOGIC CONTROLLER (FLC)

If-Then rules-based format is solved by fuzzy logic

controller in four steps as follows:

Figure 2. Fuzzy Logic Controller (FLC) System

Step-1 Input Variables: Firstly, a set of MF of linguistic

variables are taken as inputs where input variables are

basically words or sentences. In order to take any fuzzy or

crisp output by fuzzification or defuzzification respectably,

fuzzy MF are used.

Step-2 Fuzzification: Fuzzification is a process which

provides fuzzy output. Here, crisp values are fuzzified for

fuzzy output.

Step-3 Fuzzy Inference System: Fuzzy rule-based is

applied here. For each antecedent, there is a consequent

which is the resulted output for every rule-base. Several

operators like or, and, else, not are used in rule-based

system to connect multiple linguistic variables so that

target output is obtained inside the inference system.

Step-4 Defuzzification: In the defuzzification process,

crisp output is obtained from fuzzy set. Hardware

applications are greatly dependent on defuzzification

system.

4. DEFUZZIFICATION TECHNIQUES

In general, the fuzzification process involves the union of

two or more fuzzy sets, noted as, input sets” and the sets

are defined as the universe of discourse. This process

generates outputs from the precise values of input which

are in fuzzy form. The outputs of the fuzzification process

depend on the rule base and inference engine.

This process of converting the fuzzy rule based output to

crisp value for the designed system is known as

defuzzification. The defuzzification method cannot be

chosen systematically following the applications. It

depends on the need of the application.

4.1 CENTROID METHOD The "centroid method" also known as "center of gravity" or

"center of area" method is a technique which is most

commonly employed and familiar for defuzzification. It

reduces the area to smaller regions and a combined

operation is performed to obtain the final output. It is

stated by the expression below [14]:

1

1

.n

i ii

n

ii

xx x

x

Here, n represents the number of elements in the sample,

i.e., ix are the elements and ix are their

corresponding membership functions.

4.2 BISECTOR METHOD The area is divided into two regions by the bisector method

which may or may not coincide with the centroid line of

the given area.

4.3 MEAN OF MAXIMA (MOM) METHOD Actually, the easiest way of defuzzification is to take the

nearby crisp value with highest membership function. The

arithmetic average of all mean values of the intervals that

contains fuzzy set including zero length intervals is called

mean of maxima method. General equation of this method

is given by [14]



| |

i

iMxxM

x

Where M height of the fuzzy set and

| |M Cardinality of the fuzzy set .M

5. FUZZY MEMBERSHIP FUNCTION

Membership function is used to represent a fuzzy set

graphically. Triangular and trapezoidal MF (Membership

Function) are used in FIS. In this paper, we will use

triangular and trapezoidal membership function to develop

Fuzzy Logic Controller (FLC). Each input and output

including their linguistic variable are discussed below.

5.1 FUZZY INPUT VARIABLES

5.1.1 User Temperature (UT)

From the several sensors like thermostat or electronic,

user’s temperature can be collected which has three

linguistic variables named low, medium and high as shown

in table 2.

Table 2: User Temperature (UT) Classification

Range Fuzzy Set

User Temperature (UT) 14-26 Low

22-28 Medium

26-34 High

5.1.2 Temperature Difference (Tdf)

The difference between user’s temperature and room’s

temperature is the temperature difference (Tdf). In this

paper, the range of Tdf is taken between -4 to +4. As soon

as the difference goes out of the range, the air conditioner

switched off since it can’t be worked as a heat pump. Four

linguistic variables are used here which are neg (negative),

zero, pos (positive), hpos (high positive) as shown in table

3.

Table 3: Temperature Difference (Tdf) Classification

Range Fuzzy Set

-4 – 0 neg

Temperature Difference (Tdf) -1.5 - +1.5 zero

0 – +2.5 pos

+2 - +4 hpos

5.1.3 Dew Point (Td)

Dew point temperature is the temperature of captive room

where air conditioner is placed. Here, two linguistic

variables are taken named Low and High as shown in the

table 4.

Table 4: Dew Point (Td) Classification

Range Fuzzy Set

Dew Point (Td) 12-20 Low

17-29 High

5.1.4 Number of People (NP)

The number of people who are staying inside the room are

the number of people. If there is no one inside the room

then the air conditioner will remain off. Three linguistic

variables are taken here which are Low, Medium and High

as shown in the table 5.

Table 5: Number of People (NP) Classification

Range Fuzzy Set

Number of People (NP) 1-4 Low

3-9 Medium

7-12 High

5.2 FUZZY OUTPUT VARIABLES

5.2.1 Compressor Speed (CS)

Actually, the compressor speed varies from 0 to 100%

depending on the inputs that are taken by the sensors from

the room. By controlling the compressor speed, room

temperature can be optimized. Three linguistic variables

are taken here which are Slow, Medium and Fast as shown

in the table 6.

Table 6: Compressor Speed (CS) Classification

Range Fuzzy Set

Compressor Speed (CS) 0-55 Slow

45-75 Medium

70-100 Fast

5.2.2 Fan Speed (FS)

Inside an air conditioner fan is running with a speed that

varies from 0 to 100% depending on the inputs. Here,

Slow, Medium and Fast are taken as linguistic variables as

shown in the table 7.

Table 7: Fan Speed (FS) Classification

Range Fuzzy Set

Fan Speed (FS) 0-50 Slow

45-80 Medium

70-100 Fast

5.1.1 Fin Direction (FD))

The fin direction indicates the direction of the cold air flow

that comes from the fin which is a bunch of blades

bounded to the air conditioner. Its direction is either

towards (Td) the user or away (Aw) from the user.

Table 8: Fin Direction (FD) Classification

Range Fuzzy Set

Fin Direction (FD) 0-50 Slow

45-80 Medium

70-100 Fast

Here in the Fig. 3 to 9 inputs and outputs membership

functions of fuzzy logic controller shown:

Fig. 3: Input of FLC “User Temperature (UT)”



Fig. 4: Input of FLC “Temperature Difference (Tdf)”

Fig. 5: Input of FLC “Dew Point (Td)”

Fig. 6: Input of FLC “Number of People (NP)”

Fig. 7: Output of FLC “Compressor Speed (CS)”

Fig. 8: Output of FLC “Fan Speed (FS)”

Fig. 9: Output of FLC “Fin Direction (FD)”

6. FUZZY RULE BASE

Fuzzy Rules based are applied in the FLC by selecting the

appropriate sequence in the IF-Then rules which are based

on natural language. It is designed to make any automated

decision. It is formed by keeping the relationship among

Input and output in mind. The input variables (UT, Tdf,

Td, and NP) make a total of 3*4*2*3=72 rules as shown in

the followings:

1. If (UT is Low) and (Tdf is neg) and (Td is Low)

and (NP is Low) then (CS is Slow) (FS is Slow)

(FD is towards).

2. If (UT is Low) and (Tdf is zero) and (Td is Low)


(FD is towards).

3. If (UT is Low) and (Tdf is pos) and (Td is Low)


(FD is towards).

4. If (UT is Low) and (Tdf is hpos) and (Td is Low)


(FD is towards).


and (NP is Medium) then (CS is Slow) (FS is

Slow) (FD is towards).






Medium) (FD is away).




and (NP is Medium) then (CS is Medium) (FS is



and (NP is High) then (CS is Slow) (FS is Slow)

(FD is away).



(FD is towards).


and (NP is High) then (CS is Medium) (FS is



and (NP is High) then (CS is Fast) (FS is Fast)

(FD is away).

13. If (UT is Low) and (Tdf is neg) and (Td is High)


(FD is towards).

14. If (UT is Low) and (Tdf is zero) and (Td is High)


(FD is towards).

15. If (UT is Low) and (Tdf is pos) and (Td is High)

and (NP is Low) then (CS is Medium) (FS is

Slow) (FD is away).

16. If (UT is Low) and (Tdf is hpos) and (Td is High)

and (NP is Low) then (CS is Slow) (FS is




Slow) (FD is away).












(FD is away).



(FD is towards).



(FD is away).



(FD is away).

25. If (UT is Medium) and (Tdf is neg) and (Td is

Low) and (NP is Low) then (CS is Slow) (FS is


26. If (UT is Medium) and (Tdf is zero) and (Td is



27. If (UT is Medium) and (Tdf is pos) and (Td is



28. If (UT is Medium) and (Tdf is hpos) and (Td is




Low) and (NP is Medium) then (CS is Slow) (FS

is Slow) (FD is towards).






is Medium) (FD is away).


Low) and (NP is Medium) then (CS is Medium)

(FS is Fast) (FD is away).


Low) and (NP is High) then (CS is Slow) (FS is



Low) and (NP is High) then (CS is Slow) (FS is



Low) and (NP is High) then (CS is Medium) (FS



Low) and (NP is High) then (CS is Fast) (FS is

Fast) (FD is away).


High) and (NP is Low) then (CS is Slow) (FS is



High) and (NP is Low) then (CS is Slow) (FS is



High) and (NP is Low) then (CS is Medium) (FS

is Slow) (FD is away).


High) and (NP is Low) then (CS is Medium) (FS



High) and (NP is Medium) then (CS is Slow) (FS



High) and (NP is Medium) then (CS is Slow) (FS



High) and (NP is Medium) then (CS is Medium)

(FS is Medium) (FD is away).


High) and (NP is Medium) then (CS is Medium)

(FS is Fast) (FD is away).


High) and (NP is High) then (CS is Slow) (FS is



High) and (NP is High) then (CS is Medium) (FS



High) and (NP is High) then (CS is Fast) (FS is

Fast) (FD is away).





Fast) (FD is away).

49. If (UT is High) and (Tdf is neg) and (Td is Low)


(FD is towards).

50. If (UT is High) and (Tdf is zero) and (Td is Low)


(FD is towards).

51. If (UT is High) and (Tdf is pos) and (Td is Low)


(FD is towards).

52. If (UT is High) and (Tdf is hpos) and (Td is Low)


(FD is towards).











and (NP is Medium) then (CS is Fast) (FS is




(FD is towards).



(FD is towards).






(FD is away).

61. If (UT is High) and (Tdf is neg) and (Td is High)


(FD is towards).

62. If (UT is High) and (Tdf is zero) and (Td is High)


(FD is towards).

63. If (UT is High) and (Tdf is pos) and (Td is High)

and (NP is Low) then (CS is Slow) (FS is


64. If (UT is High) and (Tdf is hpos) and (Td is High)

and (NP is Low) then (CS is Medium) (FS is

Fast) (FD is away).








and (NP is Medium) then (CS is Fast) (FS is


68. If (UT is High) and (Tdf is hpos) and (Td is

High) and (NP is Medium) then (CS is Fast) (FS

is Fast) (FD is away).



(FD is towards).



Fast) (FD is away).



(FD is away).

72. If (UT is High) and (Tdf is hpos) and (Td is


Fast) (FD is away).

7. FUZZY LOGIC CONTROLLER (FLC) IMP-

PLEMENTATION IN AIR CONDITIONING

USING MATLAB

7.1 FUZZY BASE CLASS

Mamdani method is used to create system control rules

obtained from experienced human operators [15]. In this

paper, Mamdani method is used to illustrate and centroid

method is used for defuzzification. Here, FIS editor FIS

Editor defines the Fuzzy Base Class, the various inputs,

i.e.User temperature (UT), Temperature Difference (Tdf),

Dew Point (Td), and Number of People (NP) and the

various output variables like Compressor Speed (CS), Fan

Speed (FS) and, Fin Direction (FD) [16] as shown.

Fig. 10: Fuzzy Base Class

7.2 FUZZY RULE BASE

By a user or deliberately, the rule of fuzzy can be planned

manually, for all situations the rule editor propagates rules

of the marked input variable and a user fills consequent

fuzzy items. The illustration of inputs and the fuzzy

outputs depicted on the basis of [16] as shown.



Fig. 12: Fuzzy Base Rules

7.3 SOURFACE PLOTS

By applying three defuzzification methods like- centroid

method, bisector method and mean of maxima method. We

get the surface plots which are shown as [17] based on the

above effectuation.

Fig: 13 Surface plots for FLC of Compressor Speed using the

Centroid method

Fig. 14: Surface plot for FLC of Fan Speed using the Centroid

method

Fig. 15: Surface plot for FLC of Fin Direction using the

Centroid method

Fig. 16: Surface plot for FLC of Compressor Speed using

the Bisector method

Fig. 17: Surface plot for FLC of Fan Speed using the

Bisector method

Fig. 18: Surface plot for FLC of Fin Direction using the Bisector

method

Fig. 19: Surface plot for FLC of Compressor Speed using the

MOM method

Fig. 20: Surface plot for FLC of Fan Speed using the MOM

method

Fig. 21: Surface plot for FLC of Fin Direction using the MOM

method



8.RESULT AND DISCUSSION

Based on the user's temperature, temperature difference,

dew point and number of people it can be said that this

method will have different compressor speed, fan speed

and fin direction. Here, a large number of values of user of

temperature, temperature difference, dew point and number

of people can be used to evaluate the difference between

these methods. Clearly it can be said that, these three

techniques shown in the above table which is used for

comparison, provides compressor speed, fan speed and fin

direction as output. Hence, by using “if” and “then” rules

during the defuzzification of the air conditioning system,

it’s output can be modified and also verified the same

result. Any modification in the rule base model will require

modify the if” and “then” rules to get a correct output.The

compressor speed, fan speed and fin direction outputs for

varying input parameters as shown in the tables 9, 10 and

11 respectively.

Table 9: Compressor Speed output for varying input parameters

UT

Tdf

Td

NP

Compressor Speed (CS)

Centroid Bisector MOM

24 0 20 6 28.1 28 28.5

22 2 24 7 39.1 33 29.5

24 1 25 5 39.3 40 60

Table 10: Fan Speed output for varying input parameters

UT

Tdf

Td

NP

Fan Speed (FS)


24 0 20 6 28.1 28 28.5

22 2 24 7 69.5 67 64.5

24 1 25 5 44.9 49 64.5

Table 11: Fin Direction output for varying input parameters

UT

Tdf

Td

NP

Fin Direction (FD)


24 0 20 6 36.2 36 37.4

22 2 24 7 60 60.3 59.5

24 1 25 5 49.1 52.2 59.5

For compressor speed, we can see if Dew point increases

then compressor speed, increases drastically.

Hence, mathematically center of gravity method (COG) is

the best option for this sequence. Now, for Fan Speed, if

Dew point and Number of people increases then Fan Speed

also increase at the same time where bisector method is the

best option by our observation.

Finally, for Fin Direction, we can see if Temperature

difference and Number of people increases then Fin

Direction also increases sequentially. Here, we can use one

of these two method (bisector, center of gravity) for the

best output.

Fig. 22: Defuzzification output for FLC using the Centroid

method

Fig. 23: Defuzzification output for FLC using the Bisector

method

Fig. 24: Defuzzification output for FLC using the MOM method

9.CONCLUSION

From the above discussion, we can clearly see that, it is

better to use Center of gravity method and bisector method

for the defuzzification of Compressor Speed and Fan

Speed respectively. However, for the defuzzification of Fin

Direction, we can use bisector method or Center of gravity

method on our own interest.

ACKNOWLEDGMENT

Authors of this paper are thankful to other authors whose

names are included in the references section for their

suggestions which helps us to modify this paper.

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AUTHORS PROFILE

Md. Azharul Islam was born in

Rangpur, Bangladesh, in 1995. He

received his B.Sc. and M.Sc. degree in

Mathematics from University of

Rajshahi, Faculty of Science,

Bangladesh, in 2017 and 2018,

respectively. His research interest

includes Fuzzy Algebra, Fuzzy

Relation, Fuzzy Optimization, Fuzzy logic, Fuzzy

inference, system, Fuzzy control system, Machine learning

and networked control system, Artificial Intelligence (AI).

Md. Sahadat Hossain was born in

Rangpur, Bangladesh, in 1977. He

obtained his M.Sc. in 1999, M.Phil. in

2005, Ph. D. in 2012, in Fuzzy

Bitopology, from Rajshahi University,

Department of Mathematics, Faculty

of Science. Currently, he is employed

at the same university in the

department of Mathematics as a professor. He is the author

of 1 book and more than 40 other publications, including

more than 15 journal articles. His research interest includes

Fuzzy Algebra, Fuzzy Topology, Fuzzy logic, Fuzzy

control system, Fuzzy algorithm, and Fuzzy inference

system.

Ibna Sina Munzurul Haque was born

in 1995 in Meherpur, Khulna,

Bangladesh. He received his B.Sc.

degree in Mathematics from the

University of Rajshahi, Bangladesh in

2018. His research interests are Fuzzy

Logic, Fuzzy Algebra, Fuzzy Inference

System, Fuzzy Control System, Fuzzy

relation, Artificial Intelligence (Ai), Neural Network

(ANN) and, Machine learning (ML).

mathematical comparison of defuzzification of fuzzy logic

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