On Development of Fuzzy Controller: The Case of Gaussian and Triangular Membership Functions

doi:10.4236/jsip.2011.24036

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Journal of Signal and Information Processing, 2011, 2, 257-265

doi:10.4236/jsip.2011.24036 Published Online November 2011 (http://www.SciRP.org/journal/jsip)

On Development of Fuzzy Controller: The Case of

Gaussian and Triangular Membership Functions

Vincent O. S. Olunloyo, Abayomi M. Ajofoyinbo, Oye Ibidapo-Obe

Department of Systems Engineering, Faculty of Engineering Complex, University of Lagos, Lagos, Nigeria.

Email: [email protected]om, [email protected], [email protected]

Received September 30th, 2011; revised October 31st, 2011; accepted November 16th, 2011.

ABSTRACT

In recent years, the use of Fuzzy set theory has been popularised for handling overlap domains in control engineering

but this has mostly been within the context of triangular membership functions. In actual practice however, such do-

mains are hardly triangular and in fact for most engineering applications the membership functions are usually Gaus-

sian and sometimes cosine. In an earlier paper, we derived explicit Fourier series expressions for systematic and dy-

namic compu tation of g rade of me mbership in the overlap a nd non-o verlap region s of triangu lar Fu zzy sets. In ano ther

paper, we extended the methodology to cover cases of cosine, exponential and Gaussian Fuzzy sets by presenting ex-

plicit Fourier series representation for encoding fuzziness in the overlap and non-overlap domains of Fuzzy sets. This

current paper presents the development of a “Fuzzy Controller” device, which incorporates the formal mathematical

representation for computing grade of membership of Gaussian and triangular Fuzzy sets. It is shown that triangular

approximation of Gaussian membersh ip function in Fuzzy control can lead to wrong lingu istic classification which may

have adverse effects on operational and control decisions. The development of the Fuzzy controller demonstrates that

the proposed technique can indeed be incorporated in engineering systems for dynamic and systematic computation of

grade of membership in th e overlap and non-overlap regio ns of Fuzzy sets; and thus provides a basis for th e design of

embedded Fuzzy controller for mission critical applications.

Keywords: Fuzzy Controller, Triangular, Gaussian, Fou r ier Series Representation, Membership Functions

1. Introduction

The key elements in human thinking are not numbers,

but labels of Fuzzy sets, viz: classes of objects in which

the transition from membership to non-membership is

gradual rather than abrupt (Zadeh [1]). Fuzzy logic has

found applications for control and analysis purposes, as

for example recorded in the work of Bellman and Zadeh

[2], Berenji and Khedar [3]. Ruan and Fantoni [4] also

reported industrial applications of Fuzzy logic. Olunloyo

and Ajofoyinbo [5] applied hybrid Fuzzy-stochastic

methodology for maintenance optimization. Araujo,

Sandri and Macau [6], Marinke and Araujo [7], and

Moura, Rodrigues and Araujo [8] presented some other

industrial applications of Fuzzy systems/logic most of

which are related to thermal-vacuum processes, usually

encountered in particular, in the qualification of space

devices. Savkovic [9] studied Fuzzy logic theory and

applied it to the process control system. Ji and Wang [10]

developed an adaptive Neural Fuzzy Controller for active

vibration suppression in flexible structures. Researchers

generally treat the overlap region as intersection or union

of two or more Fuzzy sets and have invoked the Min and

Max operators, respectively, as needed. Olunloyo, Ajo-

foyinbo and Badiru [11] proposed an algorithm for the

treatment of overlap of adjoining Fuzzy sets based on

partitioned grids. In view of the importance of this Fuzzy

overlap region, especially where there is need to monitor

and ensure smooth transition between the adjoining

Fuzzy sets in relation to the design of mission critical

applications, Olunloyo and Ajofoyinbo [12] proposed an

alternative approach for computing membership function

based on the Fourier series representation of the envelope

of the Fuzzy patch. In the literature, for example, as in

the work of King and Mamdani [13], and Zimmermann

[14], most control applications use triangular and trape-

zoidal profiles for membership functions. However, such

triangular or trapezoidal assumptions, in most applica-

tions are generally poor approximations of the prevailing

Gaussian membership function that governs most engi-

neering processes. The Gaussian membership function

applies in engineering problem domain, especially for

On Development of Fuzzy Controller: The Case of Gaussian and Triangular Membership Functions

258

engineering measurements; as it gives actual representa-

tion at every point.

According to Ross [15], membership function essen-

tially embodies all fuzziness in a particular Fuzzy set,

and its description is the essence of a Fuzzy property or

operation. Watanabe [16] asserted that the statistical tec-

hniques for determining membership functions fall into

two broad categories viz: use of frequencies and direct

estimation. The two methods were analysed by Turksen

[17] when he reviewed the various methods and their

methodology for implementation. The determination of

membership function can also be categorized as either

being manual or automatic. The automatic generation of

membership function emphasises the use of modern soft

computing techniques (in particular Genetic Algorithm

and Neural Networks). Meredith, Karr and Krishnakumar

[18] applied Genetic Algorithm (GA) to the fine tuning

of membership functions in a Fuzzy logic controller for a

helicopter. Karr [19] applied GA to the design of Fuzzy

logic controller for the Cart Pole problem. Lee and Ta-

kagi [20] also tackled the Cart problem. In their case,

they took a holistic approach by using GA to design the

whole system (determination of the optimal number of

rules as well as the membership functions). Moreover,

Ross [21] reported on six methods for developing mem-

bership functions namely: intuition, inference, rank or-

dering, neural networks, genetic algorithm and inductive

reasoning. The manual and automatic techniques for de-

termining membership functions of Fuzzy sets are non-

systematic and suffer from certain deficiencies. On the

one hand, most of the existing automatic techniques are

heuristic in nature; which implies that different values

can be obtained for same input values presented at dif-

ferent times. On the other hand, the manual techniques

suffer from the deficiency that they rely on subjective

interpretation of words and the peculiarities of the en-

gaged human expert.

By analyzing the nature of the overlap patches defined

by the intersection and union of a typical grade of mem-

bership function for a linguistic variable, it is shown that

the resultant signal does fall into the class of functions

for which a Fourier series representation can be written.

The problem then is to construct such a series and com-

pute the corresponding coefficients. Furthermore, in or-

der to align the results with the properties of membership

functions, some element of normalization and standardi-

zation is introduced. To be more specific, starting with

triangular Fuzzy sets, Olunloyo, Ajofoyinbo and Badiru

[22] formulated explicit Fourier series representation for

computing the grade of membership in the overlap and

non-overlap regions. Ajofoyinbo [23] derived explicit

Fourier series expressions for encoding fuzziness in the

overlap and non-overlap domains of membership func-

tions of different Fuzzy sets. This methodology was ex-

tended by obtaining explicit Fourier series expressions

for computing the union and intersection of the Gaussian,

cosine and exponential Fuzzy sets. In [24], Olunloyo,

Ajofoyinbo and Ibidapo-Obe presented an implementa-

tion of embedded “Fuzzy Controller” via simulation. In

the current work, the development of Fuzzy controller

based on Fourier series representation for computing

grade of membership of Gaussian and triangular Fuzzy

sets are presented. This paper also investigates the per-

formance of Fuzzy controller based on Gaussian and

triangular membership functions, in classifying data val-

ues in the universe of discourse. The remainder of this

paper is organized as follows: Problem formulation is

presented in Chapter 2. This is followed by Systems de-

sign and implementation in Chapter 3. Discussion of

sample results is presented in Chapter 4. Chapter 5 con-

cludes the paper.

2. Problem Formulation

Fundamental conditions for Fourier series representation

are: 1) Function must be periodic, 2) Function must have

finite number of discontinuities, and 3) Function must be

bounded.

We note that the universe of discourse in a Fuzzy

plane consists of one or more data points. Each of the

data points in a given universe of discourse has some

form of data distribution around it in the form of some

distribution profile, whether Gaussian, exponential, tri-

angular or any other. Since all data points in the universe

of discourse would have same form of data distribution

around every data point, we could therefore derive an

explicit Fourier series expression for the envelope of the

Fuzzy patch since we can be assured of the repetition of

the assumed distribution pattern around each data point.

Moreover, in as much as the distribution around the data

points has same shape, then appropriate normalisation

can be introduced to transform the union and intersection

of such Fuzzy sets into functions that are amenable to

Fourier series representation. Although various func-

tional profiles of membership functions could be used,

the triangular and trapezoidal have in the past served as

approximations of the others in the first instance. In fact,

the trapezoidal form can, further, be approximated by the

triangular form since the end-points of the ‘tolerance’

interval in a trapezoidal distribution have the same grade

of membership and could therefore be assigned a point

value that represents the peak of the triangular profile. In

Sections 2.1 and 2.2 below, we present Fourier series

representation for computing union and intersection of

Gaussian and triangular Fuzzy sets respectively.

On Development of Fuzzy Controller: The Case of Gaussian and Triangular Membership Functions

259

2.1. Fourier Series Representation for Gaussian

Membership Function

  



cos sin

xakwx bkwx



 

 (1)

The membership function of union of Gaussian Fuzzy

sets is computed as follows: w

here

12π4π14π16π

π2π

2618 2618

1ede

28π

xx xx





 



  

 



 























(2)

Recall from Abramowitz and Stegun [25]:



221π

ed*e**

bac

atbt cab

tterfa tConst



 

 





 (3)

Thus,

2π2π

166

0.5*2π0.5 * π

28π0.5 0.5

4π4π

0.5*2π0.5 * 2π0.5*π

0.5 0.5

aerf erf

erf erf





 



 



 























 

 

 



 































(4)

From Equation (4), we define 1

and 2

as:

2π2π

0.5 2π*0.5*π

0.5 0.5

Ierf erf





 



















(5)

and

4π4π

0.5 2π*0.5*2π0.5 * π

0.5 0.5

Ierf erf





 



 



 





























(6)

We compute coefficients and as follows:

band



12π4π

π2

2618

1416

2π2

2618

1ecos

2π

ecosd

kx x





 













 















dxx













(7)



12π4π

π2

2618

14π16π

2π2

2618

1esin

2π

esind

kx x





 









 



























(10)



 



111cos1 1

2π

kII

bkk









(11)

By recalling 1

and 2

from Equations (5) and (6)

respectively, we can re-write Equation (7) as follows:

 



 



1sin πsin 0

2π

sin2πsin π

Ikk



















(8)

Similarly, we obtain the expression for computing

membership function in the overlap region of Gaussian

Fuzzy sets (i.e., intersection of the Gaussian Fuzzy sets)

as follows:

 



cos sin

f xakwxbkwx



 

 (12)

a (9)

On Development of Fuzzy Controller: The Case of Gaussian and Triangular Membership Functions

260

where

4π

14π16π12π4π

π22

26182618

2ππ

11 1

ede

22π2π2

xx xx







 



  

 



 





















(13)

By invoking Equation (3), we can express Equation (13) as:

4π4π

2π

0.5*2π0.5 * 2π0.5 *3

0.5 0.5

22π2π2π

4π66

0.5* 2π0.5 *30.5 0.5

erf erf







 

 





 



 









 







 







 

















 

 



 





























(14)

From Equation (14), we define 1

and 2

as:

4π4π

2π

0.5*2π0.5 * 2π0.5 *3

0.5 0.5

Jerf erf



 

 

 



 



























(15)

and

2π2π

4π6

0.5*2π0.5 *30.50.5

Jerf erf





 

 



 























(16)

Coefficients and are then computed as follows:

 

4π

14π16π12π4π

π22

26182618

2ππ

11 1

ecosdecos

π2π2π

xx xx

akxx

 



 

 

  

 

 













dkxx

(17)

Upon substituting 1

and 2

from Equations (15) and (16) respectively, we can re-write Equation (17) as:

14π2π

sin sin

π33

kkk

aJ J





 







 

 



(18)

and

 

12π4π

coscos πcos πcos

π33

bJ kJk





 







 

 





k



(19)

2.2. Fourier Series Representation for

Triangular Membership Function

The Fourier series representation for computing the grade

of membership of the intersection of triangular Fuzzy

sets (i.e. triangular pulses) is given by:

  



cos sin

xakxb



 

kx

(20)

where

a (21)

For the case k odd,

  

1sin

fx kx











 (22)

and for the case k even

  

11cos

4π





 





x (23)

Similarly, the Fourier series representation for the un-

ion of triangular Fuzzy sets (i.e. polygonal waveform) is

given by:

On Development of Fuzzy Controller: The Case of Gaussian and Triangular Membership Functions261

  

cos sin

okk

Gxakxbkx





 









(24)

where

224



 (25)

for the case k odd,

 

1cos

24 π

Gx kx







 





 (26)

while, for the case k even,



 

184148cos

Gx kx









 















(27)

We note that 0



is the point of overlap of the two ad-

joining triangular Fuzzy sets. Thus, 0



represents the

maximum grade of membership of the intersection of the

triangular Fuzzy sets. In compliance with the require-

ments of the membership function of Fuzzy set, in respect

of the intersection, we normalize



x as follows:

 







(28)

where . i.e. maximum grade of

membership for the intersection of triangular Fuzzy sets.



0max INTERSECTION





3. Systems Design and Implementation

We briefly describe the development of a “Fuzzy con-

troller” to measure temperature and pressure, and pro-

duce some output that can represent input to other sub-s

ystems or systems. Itemised below are some of the de-

tails of systems design and the implementation of the

Fuzzy Controllers. The electrical circuit is presented in

Figure 1, while the corresponding photo-image of the

device is presented in Figure 2.

The circuit in Figure 1 consists of the following major

hardware components:

a) Microchip 40-Pin Enhanced Flash PIC16F877A

Microcontrollers

b) LM35D precision integrated-circuit temperature

sensor

c) MPX4115A piezoresistive pressure sensor, and

d) LCM-S01602DSF/C Liquid Crystal Display (HD

44780-compliant LCD)

There are only four units of the PIC16F877A Micro-

controllers deployed in the circuit. Each Microcontroller

is configured with XT 4MHz Crystal. Moreover, the cir-

cuit incorporates the LCM-S01602DSF/C Liquid Crystal

Display (LCD) output unit capable of displaying 2 × 16

characters.

The four (4) Microcontrollers are grouped into two

functional sections namely:

a) Section 1: Temperature

This section consists of two (2) Microcontrollers. Mi-

crocontroller #1 executes the Program code for tempera-

ture input from the LM35D Sensor; it also conditions and

convert signals to digital form, and computes grade of

membership of the Gaussian Fuzzy sets. Similarly, Mi-

crocontroller #2 executes the corresponding program

code for temperature input from the LM35D Sensor,

digitises the signals, and computes membership grades of

the triangular Fuzzy sets.

E5V

TPM

E5V

TPM

E5V

TPM

PSN PSN

IC1

PSN PSN

IC3 IC4IC2

PSN

IC1

IC2

IC3

IC4

LED

RA0/AN0

RA1/AN1

RA2/AN2/VREF-

RA4/T0CKI

RA5/AN4/SS

RE0/AN5/RD

RE1/AN6/WR

RE2/AN7/CS

OSC1/CLKIN

OSC2/CLKOUT

RC1/T1OSI/CCP2 16

RC2/CCP1 17

RC3/SCK/SCL 18

RD0/PSP0 19

RD1/PSP1 20

RB7/PGD 40

RB6/PGC 39

RB5 38

RB4 37

RB3/PGM 36

RB2 35

RB1 34

RB0/INT 33

RD7/PSP7 30

RD6/PSP6 29

RD5/PSP5 28

RD4/PSP4 27

RD3/PSP3 22

RD2/PSP2 21

RC7/RX/DT 26

RC6/TX/CK 25

RC5/SDO 24

RC4/SDI/SDA 23

RA3/AN3/VREF+

RC0/T1OSO/T1CKI 15

MCLR/Vpp/THV

MCU #1 (GAUSSIAN MF)

PIC1 6 F8 77A

20k

15.4

3 4 5 621

SENSOR

MPX 5 100D

14 D6

13 D5

12 D4

11 D3

10 D2

9D1

8D0

6RW

5RS

VSS

VDD

VEE

LCD

LCMS01602DSF/C

RA0/ AN0

RA1/ AN1

RA2/AN2/VREF-

RA4/ T0CKI

RA5/ AN4/SS

RE0/ AN5/RD

RE1/ AN6/WR

RE2/ AN7/CS

OSC1 /C L K IN

OSC2 /C L K OUT

RC1/T1OSI/CCP2 16

RC2/CCP117

RC3/SCK/SCL18

RD0/PSP019

RD1/PSP120

RB7/PGD40

RB6/PGC39

RB5 38

RB4 37

RB3/PGM 36

RB2 35

RB1 34

RB0/INT 33

RD7/PSP730

RD6/PSP629

RD5/PSP528

RD4/PSP427

RD3/PSP322

RD2/PSP221

RC7/RX/DT26

RC6/TX/CK25

RC5/SDO 24

RC4/SDI/SDA 23

RA3/AN3/VREF+

RC0/T1OSO/T1CKI 15

MCLR/Vpp/THV

MCU #4 (TRI ANGULAR MF)

PIC16F877A

RA0/AN0

RA1/AN1

RA2/AN2/VREF-

RA4/T0CKI

RA5/AN4/SS

RE0/AN5/RD

RE1/AN6/WR

RE2/AN7/CS

OSC1/CLKIN

OSC2/CLKOUT

RC1/T1OSI/CCP2 16

RC2/CCP1 17

RC3/SCK/SCL18

RD0/PSP019

RD1/PSP120

RB7/PGD 40

RB6/PGC 39

RB5 38

RB4 37

RB3/PGM 36

RB2 35

RB1 34

RB0/INT 33

RD7/PSP730

RD6/PSP629

RD5/PSP528

RD4/PSP427

RD3/PSP322

RD2/PSP221

RC7/RX/DT 26

RC6/TX/CK 25

RC5/SDO 24

RC4/SDI/SDA 23

RA3/AN3/VREF+

RC0/T1OSO/T1CKI 15

MCLR/Vpp/THV

MCU #3 (GAUSSIAN MF)

PIC16F877A

RA0/AN0

RA1/AN1

RA2/AN2/VREF-

RA4/T0CKI

RA5/AN4/SS

RE0/AN5/RD

RE1/AN6/WR

RE2/AN7/CS

OSC1/CLKIN

OSC2/CLKOUT

RC1/T1OSI/CCP2 16

RC2/CCP117

RC3/SCK/SCL18

RD0/PSP0 19

RD1/PSP1 20

RB7/PGD 40

RB6/PGC 39

RB5 38

RB4 37

RB3/PGM 36

RB2 35

RB1 34

RB0/INT 33

RD7/PSP7 30

RD6/PSP6 29

RD5/PSP5 28

RD4/PSP4 27

RD3/PSP3 22

RD2/PSP2 21

RC7/RX/DT 26

RC6/TX/CK 25

RC5/SDO 24

RC4/SDI/SDA 23

RA3/AN3/VREF+

RC0/T1OSO/T1CKI 15

MCLR/Vpp/THV

MCU #2 (TRIANGULAR MF)

PIC1 6 F8 77A

GND

3+VS 1

VOUT

U6 (SENSOR)

LM35

RV2

2.2k

LED-RED

220

Volts

+88.8

Volts

+88.8

IC SEL

TEMP/ PR E S

U1:A

4081

U1:B

4081

U1:C

4081

U1:D

4081

3 2

U2:A

4049

5 4

U2:B

4049

7 6

U2:C

4049

33p

Figure 1. Electrical circuit of the fuzzy controller.

On Development of Fuzzy Controller: The Case of Gaussian and Triangular Membership Functions

262

Figure 2. Photo image of the fuzzy controller device.

b) Section 2: Pressure

This section consists of two (2) additional Microcon-

trollers to handle the pressure readings from the MPX

4115A Sensor. Microcontroller #3 computes grade of

membership of the Gaussian Fuzzy sets, while the Mi-

crocontroller #4 computes the grade of membership of

the triangular Fuzzy sets.

Switching between Sections 1 and 2 is achieved with

the Switch labelled TEMP/PRES; while switching be-

tween the two Microcontrollers in each Section is

achieved with the Switch labelled IC SEL.

3.1. Use of Fourier Series Representations in the

Fuzzy Controllers

We note that the Program code for the implementation of

the Fuzzy controller is written in HITECH ANSI C Lan-

guage and programmed onto the Microcontrollers using

the Microchip PICSTART Plus Programmer.

3.2. Working Principle of the Fuzzy Controller

The Fuzzy controller starts by obtaining the real tem-

perature/pressure value and executes the Program code

for temperature/pressure input. The Fuzzy Controller

conditions and converts the input signal to digital form.

The conversion result is subsequently passed to the

Function in the Program code that does further process-

ing of the result, computes the grade of membership (i.e.

Gaussian or triangular) based on the Fourier representa-

tion and relates this value to appropriate linguistic value.

The final output (i.e. Very Low, Low, Low Normal,

Normal, High Normal, High, Very High) and the corre-

sponding input value from the sensor, which is converted

to characters, are then displayed on the LCD.

Sample Fuzzy rules for the Fuzzy Controller for the

case of union of Gaussian Fuzzy sets are presented be-

low:

IF ((a0 <= t) AND (t < a1) AND MF <= 0.3)) THEN

Output = “Very Low”

IF ((a1 <= t) AND (t < a2) AND MF > 0.3)) THEN

Output = “Low”;

IF ((a2 <= t) AND (t < a3) AND MF >0.3)) THEN Out-

put = “Low Normal”

IF ((a3 <= t) AND (t < m) AND MF <= 0.3)) THEN

Output = “Normal”

IF ((m <= t) AND (t < a4) AND MF <= 0.3)) THEN

Output = “Normal”

IF ((a4 <= t) AND (t < b2) AND MF > 0.3)) THEN

Output = “High Normal”

IF ((b2 <= t) AND (t < b3) AND MF > 0.3)) THEN

Output = “High”

IF ((b3 <= t) AND (t <= b4) AND MF <= 0.3)) THEN

Output = “Very High”

where:

MF Computed grade of membership

t Temperature value

a0…a4 Data points in the first Fuzzy set

b0…b4 Data Points in the second Fuzzy set

m Data value at the point of overlap of

the two adjoining Fuzzy sets

For the purpose of linguistic analysis or classification

in the Fuzzy plane, we chose 0.3 as the baseline grade of

membership.

3.3. System Flowchart

We present the system flowchart for the operations of the

Fuzzy Controller in Figure 3.

4. Discussion of Sample Results Obtained

from Device for Temperature

Measurements

We present in Table 1, sample results obtained from the

Fuzzy Controller device for temperature measurements.

The linguistic classifications are based on Gaussian and

triangular membership functions for same range of tem-

perature measurements. We have used a baseline mem-

bership grade of 0.3. The observed differences in the

band for linguistic classifications indicate effect of ap-

proximation errors. Whereas, for example, 44˚C - 46˚C is

classified as Normal on the basis of Gaussian mem-

bership function, 44˚C - 46˚C is not classified as belong-

ing to any linguistic class on the basis of triangular

membership function. Similar disparities in linguistic

classifications are noted in the other data ranges.

For mission-critical applications, such wrong classify-

cations may have adverse effects on operational and con-

trol decisions. For example, a decision rule that would

have related to Normal linguistic class, would by virtue

of wrong classifications, be related to others.

On Development of Fuzzy Controller: The Case of Gaussian and Triangular Membership Functions263

Initialise Microcontrollers and Liquid Crystal

Display (LCD) unit.

Read input value from LM35D sensor or MPX

4115A Sensor.

Input value (signal) conditioning and

normalization of the data value.

Use normalized data value to compute Fourier

coefficients

, a

, b

)

Compute grade of membership

(union/intersection) of the Gaussian/triangular

Fuzzy sets. That is, Fourier series, f(x).

Use the computed grade of membership, (f(x)),

and defined linguistic classifications to produce

output as appropriate.

Display Output on the LCD.

New sensor reading?

Stop

Start

Choose Fuzzy set (Gaussian or triangular)

Figure 3. System flowchart.

5. Summary and Conclusions

Fuzzy logic is very relevant in machine, process or sys-

tems control, and particularly as a means of making ma-

chines more capable and responsive by resolving inter-

mediate categories in between states hitherto classified

on bivalent logic. In recent years, the use of Fuzzy set

theory has been popularised for handling overlap do-

mains in control engineering but this has mostly been in

the context of triangular membership functions. In actual

practice however, such domains are hardly triangular and

Table 1. Results—Linguistic classification.

(˚C)

Gaussian

Membership

Function f(x)

—Union of

Fuzzy sets

Linguisitic

classification

based on Gaus-

sian Member-

ship Function

Linguisitic clas-

sification base-

don Triangular

Membership

Function

Triangular

Membership

Function f(x)

—Union of

Fuzzy sets

20 0.1111250360.00533707

21 0.222743982

Very Low

(MF <=0.3) Very Low

(MF <=0.3) 0.082072208

22 0.3326050360.163304225

23 0.438975173

No linguistic

classification 0.245415925

24 0.540176440.326837778

25 0.6346124180.40862068

26 0.7207934120.490380506

27 0.797359940.571886187

28 0.8631041910.653809756

29 0.9169890730.73530292

30 0.9581645660.81705902

31 0.9859811410.898952519

321

Low

(MF > 0.3)Low

(MF > 0.3)

0.979741815

331 1

34 0.9859811410.959996217

35 0.9581645660.918978777

36 0.9169890730.878081149

37 0.8631041910.837379693

38 0.797359940.796377493

39 0.7207934120.755647716

40 0.6346124180.714763085

41 0.540176440.673863999

42 0.4389751730.633164403

43 0.332605036

Low-Normal

(MF > 0.3)Low-Normal

(MF > 0.3)

0.592101584

44 0.2227439820.551480941

45 0.1111250360.513252964

46 0.222743982

Normal

(MF <=0.3)

No linguistic

classification 0.551760132

47 0.3326050360.592363279

48 0.4389751730.633427333

49 0.540176440.675010158

50 0.6346124180.715023007

51 0.7207934120.755916334

52 0.797359940.796640786

53 0.8631041910.837642549

54 0.9169890730.878351756

55 0.9581645660.919233939

56 0.9859811410.960273135

571

High-Normal

(MF>0.3) High-Normal

(MF > 0.3)

1.000259219

581 0.979220042

59 0.9859811410.898408762

60 0.9581645660.816541304

61 0.9169890730.734765832

62 0.8631041910.653282651

63 0.797359940.571359212

64 0.7207934120.489844469

65 0.6346124180.408099566

66 0.54017644

High

(MF > 0.3)

0.326300678

67 0.4389751730.244889464

68 0.332605036

High

(MF > 0.3)

No linguistic

classification 0.162781266

69 0.2227439820.081513341

70 0.111125036

Very High

(MF <=0.3) Very High

(MF<=0.3) 0.005348055

On Development of Fuzzy Controller: The Case of Gaussian and Triangular Membership Functions

264

in fact for most engineering applications are usually Gau-

ssian and sometimes cosine. In this paper, we presented

Fourier series representation for the systematic computa-

tion of membership functions for Gaussian and triangular

Fuzzy sets. We also presented the development of a

“Fuzzy Controller” to measure temperature and pressure

and produce output that can represent input to additional

sub-systems or systems. By way of comparative analysis,

it is shown that triangular approximation of Gaussian

membership function in Fuzzy control can lead to wrong

linguistic classification(s) which may have adverse ef-

fects on operational and control decisions. The develop-

ment of the Fuzzy controller device clearly demonstrates

that the proposed technique can indeed be incorporated

in engineering systems for the dynamic and systematic

computation of grade of membership in the overlap and

non-overlap regions of Fuzzy sets; and thus provides a

basis for the design of embedded Fuzzy Controller for

mission critical applications.

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