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)
1
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.
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
   
0
1
cos sin
2
n
kk
k
a
f
xakwx bkwx

 
 (1)
The membership function of union of Gaussian Fuzzy
sets is computed as follows: w
here
22
22
12π4π14π16π
π2π 22
2618 2618
0
3
0
1ede
28π
xx xx
ad
x
x

  
   
 

 













(2)
Recall from Abramowitz and Stegun [25]:


2
221π
ed*e**
2
bac
atbt cab
f
tterfa tConst
aa

 
 


 (3)
Thus,
0
3
2π2π
166
0.5*2π0.5 * π
28π0.5 0.5
4π4π
66
0.5*2π0.5 * 2π0.5*π
0.5 0.5
aerf erf
erf erf


 

  
 
 









 
 
  
 



 















(4)
From Equation (4), we define 1
I
and 2
I
as:
1
2π2π
66
0.5 2π*0.5*π
0.5 0.5
Ierf erf


 

 











 




(5)
and
2
4π4π
6
0.5 2π*0.5*2π0.5 * π
0.5 0.5
Ierf erf


 

  
 





 


6







(6)
We compute coefficients and as follows:
k
ak band


2
2
2
2
12π4π
π2
2618
3
0
1416
2π2
2618
π
1ecos
2π
ecosd
xx
k
xx
ak
kx x



 







  









dxx







(7)


2
2
2
2
12π4π
π2
2618
3
0
14π16π
2π2
2618
π
1esin
2π
esind
xx
k
xx
bk
kx x



 







 









d
xx







(10)


 

2
12
3
111cos1 1
2π
kk
kII
bkk




k
(11)
By recalling 1
I
and 2
I
from Equations (5) and (6)
respectively, we can re-write Equation (7) as follows:
  
 

1
3
2
1sin πsin 0
2π
sin2πsin π
kI
ak
k
Ikk
k









(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:
  
0
1
cos sin
2
n
kk
k
a
f xakwxbkwx

 
 (12)
0
k
a (9)

On Development of Fuzzy Controller: The Case of Gaussian and Triangular Membership Functions
260
where
22
4π
14π16π12π4π
π22
3
26182618
0
2ππ
3
11 1
ede
22π2π2
xx xx
ad
x
x



 

  
 

 












(13)
By invoking Equation (3), we can express Equation (13) as:
0
4π4π
2π 66
0.5*2π0.5 * 2π0.5 *3
0.5 0.5
1
22π2π2π
4π66
0.5* 2π0.5 *30.5 0.5
erf erf
a
erf erf



 
 



  
 



 
 




 

 

  







   

 






















(14)
From Equation (14), we define 1
J
and 2
J
as:
1
4π4π
2π 6
0.5*2π0.5 * 2π0.5 *3
0.5 0.5
Jerf erf

 
 
  
 
 







6







(15)
and
2
2π2π
4π6
0.5*2π0.5 *30.50.5 Jerf erf


 
  
 
 



 


6






(16)
Coefficients and are then computed as follows:
k
ak b
 
22
22
4π
14π16π12π4π
π22
3
26182618
2ππ
3
11 1
ecosdecos
π2π2π
xx xx
k
akxx
 
  
 
  
 
 
 
 







dkxx
(17)
Upon substituting 1
J
and 2
J
from Equations (15) and (16) respectively, we can re-write Equation (17) as:
21
14π2π
sin sin
π33
kkk
aJ J
k


 



 
 

(18)
and
 
12
12π4π
coscos πcos πcos
π33
kk
bJ kJk
k


 



 
 


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:
  

0
1
cos sin
2
n
kk
k
a
f
xakxb

 
kx
(20)
where
01
24
a (21)
For the case k odd,
  
1
2
22
1
41
1sin
4
k
N
k
fx kx
k 




 (22)
and for the case k even
  
2
22
1
14
11cos
4π
Nk
k
f
xk
k


 


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
  
1
cos sin
2
okk
k
a
Gxakxbkx


  



(24)
where
0
1
224
a0

 (25)
for the case k odd,
 
00
22
1
4 1cos
24 π
N
k
Gx kx
k



 


 (26)
while, for the case k even,


 
02
00
1
184148cos
24
Nk
k
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

f
x as follows:
 
0
1
fx
fx




(28)
where . i.e. maximum grade of
membership for the intersection of triangular Fuzzy sets.


0max INTERSECTION
fx  
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.
D7 D6 D5 D4 D3 D2 D1 D0 E RW RS
RS
RW
E5V
D0
D1
D2
D3
D4
D5
D6
D7
TPM
RS
RW
E5V
D0
D1
D2
D3
D4
D5
D6
D7
TPM
RS
RW
E5V
D0
D1
D2
D3
D4
D5
D6
D7
TPM
D7
D6
D5
D4
D3
D2
D1
D0
RS
RW
E
TPM
5V
TPM
5V
PSN PSN
IC1
PSN PSN
IC3 IC4IC2
PSN IC1 IC2 IC3 IC4
LED
RA0/AN0
2
RA1/AN1
3
RA2/AN2/VREF-
4
RA4/T0CKI
6
RA5/AN4/SS
7
RE0/AN5/RD
8
RE1/AN6/WR
9
RE2/AN7/CS
10
OSC1/CLKIN
13
OSC2/CLKOUT
14
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+
5
RC0/T1OSO/T1CKI 15
MCLR/Vpp/THV
1
MCU #1 (GAUSSIAN MF)
PIC1 6 F8 77A
R9
20k
15.4
3 4 5 621
SENSOR
MPX 5 100D
D7
14 D6 13 D5 12 D4 11 D3 10 D2
9D1 8D0 7
E
6RW 5RS 4
VSS
1
VDD
2
VEE
3
LCD
LCMS01602DSF/C
RA0/ AN0
2
RA1/ AN1
3
RA2/AN2/VREF-
4
RA4/ T0CKI
6
RA5/ AN4/SS
7
RE0/ AN5/RD
8
RE1/ AN6/WR
9
RE2/ AN7/CS
10
OSC1 /C L K IN
13
OSC2 /C L K OUT
14
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+
5
RC0/T1OSO/T1CKI 15
MCLR/Vpp/THV
1
MCU #4 (TRI ANGULAR MF)
PIC16F877A
RA0/AN0
2
RA1/AN1
3
RA2/AN2/VREF-
4
RA4/T0CKI
6
RA5/AN4/SS
7
RE0/AN5/RD
8
RE1/AN6/WR
9
RE2/AN7/CS
10
OSC1/CLKIN
13
OSC2/CLKOUT
14
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+
5
RC0/T1OSO/T1CKI 15
MCLR/Vpp/THV
1
MCU #3 (GAUSSIAN MF)
PIC16F877A
RA0/AN0
2
RA1/AN1
3
RA2/AN2/VREF-
4
RA4/T0CKI
6
RA5/AN4/SS
7
RE0/AN5/RD
8
RE1/AN6/WR
9
RE2/AN7/CS
10
OSC1/CLKIN
13
OSC2/CLKOUT
14
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+
5
RC0/T1OSO/T1CKI 15
MCLR/Vpp/THV
1
MCU #2 (TRIANGULAR MF)
PIC1 6 F8 77A
GND
3+VS 1
VOUT
2
U6 (SENSOR)
LM35
RV2
2.2k
D2
LED-RED
R3
220
Volts
+88.8
Volts
+88.8
IC SEL TEMP/ PR E S
1 2
3
U1:A
4081
5 6
4
U1:B
4081
8 9
10
U1:C
4081
12 13
11
U1:D
4081
3 2
U2:A
4049
5 4
U2:B
4049
7 6
U2:C
4049
C1
33p
C2
33p
C3
33p
C4
33p
C5
33p
C6
33p
C7
33p
C8
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
0
, a
n
, b
n
)
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)
Y
N
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.
T
(˚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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