ON-LINE MEASUREMENTS USING NEURAL NETWORKS AND SOFT
COMPUTING FOR THE CONTROL OF GLASS PRODUCTION FURNACE
Sukanya Sirinonrang
B.S., King Mongkut’s University of Technology Thonburi, Thailand, 2002
THESIS
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF SCIENCE
in
MECHANICAL ENGINEERING
at
CALIFORNIA STATE UNIVERSITY, SACRAMENTO
FALL
2010
ii
ON-LINE MEASUREMENTS USING NEURAL NETWORKS AND SOFT
COMPUTING FOR THE CONTROL OF GLASS PRODUCTION FURNACE
A Thesis
by
Sukanya Sirinonrang
Approved by:
__________________________________, Committee Chair
Tien-I Liu, Ph.D.
__________________________________, Second Reader
Akihiko Kumagai, Ph.D.
____________________________
Date
iii
Student: Sukanya Sirinonrang
I certify that this student has met the requirements for format contained in the University
format manual, and that this thesis is suitable for shelving in the Library and credit is to
be awarded for the thesis.
________________________, Graduate Coordinator
Kenneth Sprott, Ph.D.
Department of Mechanical Engineering
iv
_____________________
Date
Abstract
of
ON-LINE MEASUREMENTS USING NEURAL NETWORKS AND SOFT
COMPUTING FOR THE CONTROL OF GLASS PRODUCTION FURNACE
by
Sukanya Sirinonrang
Liquefied petroleum (LP) gas is used as a backup energy system for glass
production furnace. LP gas is mixed with air at a desired ratio in order to get a proper
gravity. The previous research was conducted to improve a performance of monitoring
and diagnosis of glass production furnace.
The objective of this research is to apply adaptive-network based fuzzy inference
system (ANFIS) and counterpropagation neural networks (CPN) for on-line monitoring
and measurements of LP gas for glass production. Three inputs, air inlet absolute
pressure (PSIA), air/mixed differential pressure (PSID), and propane/mixed differential
pressure (PSID), were selected for on-line measurements.
Three to five ANFIS membership functions were used for each input of ANFIS
for on-line measurements of the specific gravity of the LP gas. The ANFIS using
Generalized Bell membership functions yielded the best performance with an average
v
absolute error of 0.23%, a maximum error of 1.78%, and a minimum of 0%, while an
average error of 1.7%, a maximum of 4.58%, and a minimum of 0%, were obtained using
3x12x1 CPN.
____________________________, Committee Chair
Tien-I Liu, Ph.D.
____________________________
Date
vi
ACKNOWLEDGMENTS
I would like to thank Professor Tien-I Liu for his guidance. I would also like to
thank Professor Akihiko Kumagai for his help in ANFIS. In addition, I would like to
thank Carl S. Lyons for data acquisition and neural networks.
I would specially like to thank my family for their emotional and financial support
during my graduate studies in Mechanical engineering at CSUS.
vii
TABLE OF CONTENTS
Page
Acknowledgements ........................................................................................................... vii
List of Tables ...................................................................................................................... x
List of Figures ................................................................................................................... xii
Chapter
1. INTRODUCTION .......................................................................................................... 1
2. EXPERIMENTATION ................................................................................................... 3
3. FEATURE SELECTION ................................................................................................ 5
4. NEURAL NETWORKS AND SOFT COMPUTING .................................................... 7
4.1 Neural Networks .................................................................................................. 7
4.1.1 Counterpropagation Neural Networks ..................................................... 7
4.2 Soft Computing .................................................................................................. 11
4.2.1 Adaptive Network-Based Fuzzy Inference System ............................... 11
5. ON-LINE MEASUREMENTS FOR GLASS PRODUCTION FURNACE ................ 16
5.1 Counterpropagation Neural Networks ............................................................... 16
5.1.1 Training Process of Counterpropagation Neural Networks ................... 16
5.1.2 On-Line Tests Using Counterpropagation Neural Networks ................. 18
5.2 Adaptive Network-Based Fuzzy Inference System (ANFIS) ............................ 24
5.2.1 Training Process of ANFIS .................................................................... 24
viii
6. CONCLUSIONS........................................................................................................... 27
Appendix A: ANFIS Training and On-Line Test Data ..................................................... 30
Appendix B: Triangular Membership Functions (Trimf) ................................................. 35
Appendix C: Trapezoidal Membership Functions (Trapmf) ............................................ 38
Appendix D: Generalized Bell Membership Functions (Gbellmf) ................................... 41
Appendix E: Gaussian Membership Functions (Gaussmf) ............................................... 46
Appendix F: Two-sided Gaussian Membership Functions (Gauss2mf) ........................... 51
Appendix G: Pi-Shaped Membership Functions (Pimf) ................................................... 56
Appendix H: Difference of Sigmoid Membership Functions (Dsigmf) ........................... 61
Appendix I: Product of Sigmoid Membership Functions (Psigmf) .................................. 66
Appendix J: Counterpropagation Neural Network Weights ............................................. 71
Appendix K: Properties of Gases ...................................................................................... 74
References ......................................................................................................................... 75
ix
List of Tables
Page
1. Table 1: CPN Architecture versus Error………..………………………….…….17
2. Table 2: CPN Interpolation Number versus Error…………………………….…18
3. Table 3: CPN Outputs for 3x12x1 Architecture……………………………....…21
4. Table 4: Statistic of CPN Outputs………………………...…………………..….23
5. Table 5: Absolute Average Error Percentage of ANFIS Architectures……….....25
6. Table 6: On-Line Test Data..………………………………….............................31
7. Table 7: Training Data………………………………………………….….....….33
8. Table 8: Outputs of 3 x 5 Trimf……………………………………...….…….....36
9. Table 9: Outputs of 3 x 5 Trapmf…….…………………………………...…......39
10. Table 10: Outputs of 3 x 4 Gbellmf………….…………...………….………......42
11. Table 11: Outputs of 3 x 5 Gbellmf………...……………………………..…..…44
12. Table 12: Outputs of 3 x 4 Gaussmf………………………..…………...…....….47
13. Table 13: Outputs of 3 x 5 Gaussmf……………..……………………....…...….49
14. Table 14: Outputs of 3 x 4 Gauss2mf………………………………..…….…….52
15. Table 15: Outputs of 3 x 5 Gauss2mf………………………………..…….…….54
16. Table 16: Outputs of 3 x 4 Pimf…………………..………………...…….....…...57
17. Table 17: Outputs of 3 x 5 Pimf……………………………………...……….….59
18. Table 18: Outputs of 3 x 4 Dsigmf……………………...……………..…….…..62
Page
x
19. Table 19: Outputs of 3 x 5 Dsigmf…………………………………..…...….…..64
20. Table 20: Outputs of 3 x 4 Psigmf…………………………………...…....……..67
21. Table 21: Outputs of 3 x 5 Psigmf………………………………………...……..69
22. Table 22: 3x12x1 CPN Weights Hidden Kohonen Layer Weight Vector……….72
23. Table 23: Grossberg Outputs Layer Weight Vector………………………..…....73
24. Table 24: Properties of Gasses……………………………………….……....…..74
xi
List of Figures
Page
1. Figure 1: Propane Air Mixer Schematic…………………………………….4
2. Figure 2: A 3x8x1 CPN Architecture.……………………………………....8
3. Figure 3: A 3x3 ANFIS Architecture…………………….………..……..…13
4. Figure 4: Mean Squared Error versus Number of Active Kohonen Nodes
for a 3x12x1 CPN……………………………………………….…………..20
5. Figure 5: Mean Squared Error versus CPN Architecture……….…….….....21
6. Figure 6: Error of Actual versus Estimated Specific Gravity of 3x12x1
CPN…………………………………………………………………….…...21
7. Comparison of Absolute Error Percentage versus ANFIS Architectures..…27
xii
1
Chapter 1
INTRODUCTION
Liquefied Petroleum (LP) gas is used as one kind of fuel supplies in the glass
production furnaces. Air is blended with propane vapor in a desired ratio by mixing
process (T.I. Liu et al. 1993) to accomplish a proper heating value to supply to glass
furnace. This research aims to analyze LP gas as a back-up fuel supply for glass
production furnace.
Mixer control system is required to regulate the heating value of the propane and
air mixture. The mixture control system is accomplished by sampling the mixed gas
from process. The mixed gas is then measured its specific gravity which is proportional
to heating value of the gas. The specific gravity is measured by a gravitometer that sends
an output signal to the control system for gravity adjustment. The control system
supervises the position of the series of flow and pressure control valves. The control
valves, in turn, regulate the relative amounts of air and propane flowing into a common
mixed gas manifold. The mixed gas is then available for the glass production furnaces.
Information from propane mixed differential pressure, air mixed differential
pressure and air inlet pressure into the final control valve affect the specific gravity of the
mixed gas. The specific gravity of the mixed gas stream depends upon the ratio of
propane and air flow in the control system. These parameters are extracted for neural
ANFIS analysis.
The data sets for this study were collected using impulse wheel gravitometer (T.I.
Liu, et al. 1993). The previous research on-line monitoring and diagnosis of glass
2
production was conducted using backpropagation (BPN) and counterpropagation (CPN)
neural networks. In this research an ANFIS and CPN are employed in analyzing the
control system for glass production.
3
Chapter 2
EXPERIMENTATION
This research is demonstrated on analysis of propane/air mixing system that
designed to supply fuel gas to a glass production furnace. The system illustrated in
Figure 1, consisting of a modulating proportional mixer regulating mixed gas specific
gravity and pressure. The mixer operation requires a continuous supply of propane gas
and dry compressed air for operation.
The mixer operation requires propane gas and clean, dry compressed air supply
for glass furnace. Compressed air flows through a control system, from a split check
valve, then air is introduced to a sensing and pneumatic control system. From the check
valve, the air stream flows through a modulating control valve, in which a linear control
valve. This valve adjusts air flow in proportion to its actuator stem position. By
modulating the valve position, the relative proportion of air mixing with propane is
regulated. Next, the air is introduced to the mixed gas pressure control valve before it
enters the mixing manifold vanes. This pressure control valve is coupled to its propane
control valve counterpart.
4
Figure 1: Propane Air Mixer Schematic (T.I. Liu et al 1993)
Mechanically inter-linked control valve configuration is used for controlling the
mixed gas pressure level. Finally, when air flows out of the final control valve and is
mixed with propane stream via a series of turbulence vanes. Using this flow sequence,
the air pressure level and air proportion are regulated for satisfactory gas blending.
5
Chapter 3
FEATURE SELECTION
The propane/mix differential pressure, air/mix differential pressure, and air inlet
pressure into the final control valve affect the specific gravity of the mixed gas. The
three parameters that determine process changes have been identified. The specific
gravity of the mixed gas stream depends upon the ratio of propane/air flow. Since the
gravity depending upon the flow of the two different compressible fluids, the theory of
control valve flow is introduced (T.I. Liu et al 1993). The important flow parameters for
analytical of compressible control valve flow are listed in the following:
Q
=
volumetric flow rate (standard cubic feet per hour)
Fp
=
piping geometry factor
Cv
=
valve flow coefficient for incompressible fluids
P1
=
upstream absolute pressure (psia)
P2
=
downstream absolute pressure (psia)
_P
=
(P1 – P2), pressure drop
X
=
_P/P1, pressure drop ratio
Xt
=
terminal pressure drop ratio
Y
=
[1 – (X/3Xt)] expansion factor
G
=
specific gravity of gas, ρgas/ρair
Tt
=
inlet absolute temperature ( °R)
Z
=
gas compressibility factor
6
For compressible flow through a control valve, equation 1, defines volumetric
flow for pressure ratios below the critical pressure ratio, X t . Regarding the propane/air
blending process, the flow parameters that remain constant are f t , G , T1, and z , while
variable parameters include Cv , P1, Y , and X.
X
Q  1360Fp Cv P1Y
GT1z
  
for
  
X  Xt

(1)
Flow coefficient, Cv , varies depending upon valve type, size, and area of flow

 (T.I. Liu et al. 1993). Though, the
opening
Cv parameter varies with flow, since it

changes accordingly
with valve position, it is not used as a neural network input. The
 for compressible flow. Since expansion factor, Y ,
factor Y allows correction of Cv value
cannot be directly measured. Thus, substituting pressure conditions as neural networks


input, the expansionfactor is included. Pressure drop ratio, X, is measured by sensing
differential pressure drop, _P, and inlet pressure, P1, in order to define process flows. The
process flows results the mixed gas gravity.
Since ANFIS has capability to 
adapt to conditions of incomplete information; thus,
the omission of the flow coefficient, Cv , as network input is justified. Therefore,
differential pressures across the control valves and inlet pressure into the control valves
 that that are extracted from the process and are used as
become the three parameters
ANFIS inputs.
7
Chapter 4
NEURAL NETWORKS AND SOFT COMPUTING
4.1 Neural Networks
4.1.1 Counterpropagation Neural Networks
Counterpropagation neural networks (CPN) is invented by Hecht Nielsen (1987)
are suitable for statistical analysis, function approximation and look-up tables. The CPN
architecture, illustrated in Figure 2, consists of three layers of neurons called; input,
Kohonen, and Grossberg layers. The neurons of input and Kohonen layers, as well as
Kohonen and Grossberg layers are fully connected.
Figure 2: A 3x8x1 CPN Architecture
8
The CPN parameters are as follow:
N
=
Kohonen neural node number
n
=
input neural node number
m
=
output neural node number
Xi
=
input vector of Kohonen layer
Yi
=
Grossberg layer neural node output i
Wi
=
weight vector of the ith Kohonen neural node
Uj
=
weight vector of the jth Grossberg neural node
bi
=
bias for the ith Kohonen node
pi
=
win frequency for the ith Kohonen neural node
CPN Training is a two-step process. In the first step, the training of the Kohonen
layer is performed. Euclidean distances between the input vector and the weight vector
of each Kohonen node are calculated. After the distances are determined, the node with
the least distance is recognized as the winning node. After many training iterations, each
Kohonen node moves to a region, which is closer to the input in terms of Euclidean
distances:
n
di  W
i
X 
(W
j 1
where,
 d
=Euclidean distance
ii
X i ) 2
(2)
9
Wi
= Weight vector of the ith Kohonen node
X
= Node values of the input layer
The winner is selected according to:
zi 1 if d
i  di
for all j
zi  0 otherwise
 After thewinner is selected, the weight vectors are modified according to:

W inew  W iold   (x  W iold )  (x  W iold )(1  zi )
 where

 and  are parameters to be assigned.
 the minimum distance, do, each Kohonen's node output, zi , is calculated as
Using
follows:
ei 
do
di
ei  0
 
ei 1

ei  0

f i  eir

if
do  0
if
di  do  0

if
if
di  do  0
i is not among the winning nodes

(typically r = 1)

Afterthe winner is selected, the weight vectors are modified according to:
(3)
10
zi 
fi
(4)
N
f
i
j 1
 of the training of a counterpropagation network is training of the
The second stage
t h
Grossberg layer. The output of the k Grossberg node is calculated by:

N
Yk U k z  U ki zi
(5)
j 1

where z is a vector
containing all the zi values.


During training, equations
(4) and (5) determine the weight update. The
differences between the training value, Yk and the Grossberg weight, Uk , iare determined
for each connection.
U kinew  U kiold   (Ykiold 
U ki )


U kinew  U kiold
if
zi = 1
 if
zi  0
 
(6)
(7)
h
 rate and Y stands for the kt element
The factor a, is the Grossberg layer's learning
of the
k
training vector.


11
4.2 Soft Computing
4.2.1 Adaptive Network-Based Fuzzy Inference System
Adaptive network-based fuzzy inference system (ANFIS) has been found to
possess an excellent ability to learn from the available information. Adaptive-networkbased fuzzy inference system is a mathematical representation of fuzzy ‘if–then’ rules,
which maps the relationship between the input and the output variables utilizing the
excellent learning ability of Artificial Neural Networks (ANN) (N. Vora et al. 1997).
The ANFIS combines the two approaches, neural networks and fuzzy systems. If
both these two intelligent approaches are combined, good reasoning will be achieved in
quality and quantity. In other words, both fuzzy reasoning and network calculation will
be available simultaneously. ANFIS can construct fuzzy rules with membership functions
to generate an input-output pair (T.I. Liu et al. 2005).The ANFIS architecture in Figure 3,
is 3 inputs and 3 membership functions (3x3).
12
Figure 3: A 3x3 ANFIS Architecture
After the input layer, the first layer is the layer of nodes representing all features
selected input membership functions. Example of the membership functions that are
used for on-line monitoring and diagnosis for glass production furnace is represented as
follow:
The Gaussian membership function (gaussmf) can be described as follow:
(x c )2
f (x;a,c)  e
2 2
(8)
The symmetric Gaussian function depends on two parameters  and c. Where  controls
the slopes
 and c adjusts the center of the corresponding membership function.
13
The Difference of Sigmoid membership function (dsigmf) can be described as
follow:
f (x;a,c) 
1
1 e
a(x c )
(9)
 a is the slope of the sigmoid function and c is the node offset value. The
Where
difference of sigmoid membership function depends on four parameters, a1, a2, c1, and c2,
and is the difference between two of these two sigmoid functions:
f1(x;a1,c1)  f 2 (x;a2,c2 )
(10)
The Trapezoidal membership function (trapmf) can be described as follow:
The trapezoidal curve is a function of a vector, x, and depends on four parameters,
a, b, c, and d, as given by:
 x  a  d  x  
f (x;a,b,c,d)  max min 
,1,
, 0 
 b  a  d  c  
(11)
 Where a and d locate the “feet” of the trapezoid and the parameters b and c locate
the shoulders.
The second layer after the input layer contains the nodes of all rules in the fuzzy
inference system. In Figure 3, a 3x3 ANFIS architecture, there are twenty-seven rules (33)
are acquired in the training. The format of 27 rules is expressed by:
IF in1mfp AND in2mfq AND in3mfr THEN outmfs
(p

1

3
,q

1

3
,r
1

3
;s
1

27
)

(12)
14
The antecedent of rules is generated by all of the combinations of in1mfp, in2mfq,
and in2mfr with the given constraints of p
13
,q13
,r1.3The fuzzy operator
used in the inference system is “AND”. In the consequent (THEN part), a Sugeno type
membership function was used. Each node multiplies the incoming signals, and the

product Pk is called the firing strength of the rule k.
1
2
3
P
k M
pM
q M
r
(13)

where, M1p , M q2 , and M r3 are membership functions of input 1, 2, and 3, respectively.

The output of the node containing the rule k,Pk is the ratio of the firing strength


of k th rule to the sum of all rules’ firing strength. It is called the normalized firing

strength:

Pk 
pk
(14)
27
p
k
k 1
The third layer represents the output membership functions. The output

membership
function is determined by the types of interference system selected, in which
either Mamdani or Sugeno. In the Mamdani inference system, the output membership
function is a fuzzy set. In the Sugeno inference system, the output membership function
is linear or constant. In this research a Sugeno inference system is used. The output layer
is expressed by:
Qk  pk Mk

(15)
15
where Pk is the normalized firing strength and Mk is the membership function of the
output kt hrule.


The node in the forth layer is theoutput
layer. The results of all the 27 rules are
 applied into a single fuzzy set. The output value is acquired by adding the output of all
the nodes from third layer:
27
O  Qk
(16)
k 1
 value has been achieved, the defuzzification method is applied to transfer
After the output
a fuzzy set to a single number.
The training process of the inference system is carried out based on the difference
between the desired output value and the actual output. The learning algorithm used a
hybrid method, which is the combination of the backpropagation and the least square
method. As the iteration increases, the parameters of the membership functions are
adjusted so that the fuzzy inference system represents a nonlinear function to correlate
the input and output relationship of the training data (K.Y. Chen et al. 2000).
16
Chapter 5
ON-LINE MEASUREMENTS FOR GLASS PRODUCTION FURNACE
5.1 Counterpropagation Neural Networks
5.1.1 Training Process of Counterpropagation Neural Networks
The 63 data sets shown in Table 7 were presented to counterpropagation neural
networks for the purpose of learning. Data were normalized from 0.1 to 0.9 before being
presented to neural networks.
A 3x12x1 counterpropagation neural network was used. The Kohonen layer
learning rate, A = 0.02, and Grossberg layer learning rate, a = 0.3, are defined.
Probability factor, b, is given a value of 0.0083 and the bias threshold barrier, T, is given
a value of 0.0056, and the bias distance, di , is allotted a value of -10. All three
parameters are kept constant during network training.

The training of counterpropagation
networks, being two-stage in nature, requires
that the hidden Kohonen layer be trained first. The Kohonen layer learning rate is
gradually reduced from α = 0.02 to α = 0.001 overtime until it ceases to adjust its weights
any further. At this time the learning of the Grossberg layer may begin.
The Grossberg learning rate starting value of a = 0.3 was reduced to a = 0.05 after
2,000 iterations and finally after 4,000 more training iterations to a=0.01. At the same
time probability error factor, c, is reduced from a starting value of 2.6 gradually to a
value of 0.2. This is a result of the increased in equal probable weight adjustments
occurring on the Kohonen layer. Table 1 is shown outputs error versus CPN
17
architectures. The 3x12x1 CPN architecture estimated a smallest mean squared error of
normalized output.
Table 1: CPN Architecture versus Error
α = 0.02, 0.005, 0.001
a = 0.30, 0.05, 0.01 (No Interpolation)
3x3x1
Mean Squared Error of
Normalized Output
0.0301
3x5x1
0.0164
3x10x1
0.0170
3x12x1
0.0124
3x13x1
0.0130
3x15x1
0.0148
3x16x1
0.0136
3x18x1
0.0160
3x20x1
0.0138
3x24x1
0.0176
Architecture
18
5.1.2 On-Line Tests Using Counterpropagation Neural Networks
Another 63 data sets shown in Table 2 were used for the on-line tests. Ten
different CPN architectures with different number of Kohonen nodes were used. A
3x12x1 CPN yields the best results for the on-line measurement of specific gravity of the
LP gas. The comparison of the mean squared error of the normalized output for various
CPN architectures is shown in Figure 4, Figure 5, and Table 3. Therefore, the 3x12x1
CPN was selected. The performance of the 3x12x1 CPN with interpolation was then
compared with the same network architecture without using the interpolation. The
interpolation feature uses a proportional relationship to calculate Grossberg output for
more than one of the Kohonen nodes that was measured to be closest to the input set.
Interestingly, the best performance was achieved by the 3x12x1
counterpropagation neural network with an interpolation of three Kohonen nodes in
closest proximity to the input set. This counterpropagation neural network resulted in an
average gravity estimation error of 1.7% with an error standard deviation of 0.0326. The
result was obtained after 6,000 training iterations with the absolute maximum error is
4.58%, and the minimum error is 0%. The comparison is shown in Table 2 and Figure 4.
Table 2: CPN Interpolation Number versus Error (T.I., Liu et al 1993)
Number of Kohonen Node Interpolations
On Grossberg Node 3x12x1
Architecture
Mean Squared Error of
Normalized Output
1
0.0124
2
0.0116
3
0.0114
4
0.0117
Mean Squared Error of
Normallized Outputs
19
0.0126
0.0124
0.0122
0.012
0.0118
0.0116
0.0114
0.0112
0.011
0.0108
1
2
3
4
Nomber of Active Nodes on Kohonen Layer
Figure 4: Mean Squared Error versus Number of Active Kohonen Nodes for a 3x12x1 CPN
(A=0.02,a=0.3)
Interestingly, the 3x12x1 CPN yields the best performance. A graphical
comparison of error versus node number of CPN is presented in Figure 5. This CPN
estimated an average gravity estimation error of 1.7% with an error standard deviation of
0.0326. The results were obtained after 6,000 training iterations with a maximum error
of 4.58% as shown in Table 3 and Figure 6.
Mean Squared Error of the Nomalized
Outputs
20
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
3
5
10
12
13
15
16
18
20
24
Number of Kohonen Nodes
Figure 5: Mean Squared Error versus CPN Architecture
% Error of Estimated Specific Gravity
5
4
3
2
1
0
-1
-2
-3
-4
-5
1.2
1.25
1.3
1.35
1.4
Specific Gravity
Figure 6: Error of Actual versus Estimated Specific Gravity of a 3x12x1 CPN
1.45
21
Table 3: CPN Outputs for 3x12x1 Architecture
(A = 0.02 a = 0.30 b = 0.0083 c = 2.60 T = 0.0056 INTERP. = -3) (Table continues)
Measured
Estimated
Abs. %
Specific
Specific
Error
Error
Gravity
Gravity
1.22
1.25
0.03
2.46
1.23
1.25
0.02
1.63
1.23
1.25
0.02
1.63
1.23
1.27
0.04
3.25
1.24
1.25
0.01
0.81
1.24
1.26
0.02
1.61
1.24
1.26
0.02
1.61
1.25
1.26
0.01
0.80
1.25
1.28
0.03
2.40
1.25
1.26
0.01
0.80
1.25
1.29
0.04
3.20
1.25
1.25
0.00
0.00
1.25
1.28
0.03
2.40
1.25
1.26
0.01
0.80
1.25
1.29
0.04
3.20
1.26
1.27
0.01
0.79
1.26
1.28
0.02
1.59
1.26
1.29
0.03
2.38
1.27
1.29
0.02
1.57
1.27
1.27
0.00
0.00
1.27
1.26
-0.01
0.79
1.27
1.28
0.01
0.79
1.27
1.28
0.01
0.79
1.27
1.28
0.01
0.79
1.27
1.26
-0.01
0.79
1.27
1.28
0.01
0.79
1.28
1.29
0.01
0.78
1.28
1.29
0.01
0.78
1.28
1.28
0.00
0.00
1.28
1.28
0.00
0.00
1.28
1.28
0.00
0.00
1.28
1.29
0.01
0.78
1.28
1.28
0.00
0.00
1.28
1.33
0.05
3.91
1.28
1.28
0.00
0.00
1.28
1.30
0.02
1.56
1.28
1.30
0.02
1.56
1.28
1.28
0.00
0.00
1.29
1.29
0.00
0.00
1.29
1.30
0.01
0.78
1.29
1.28
-0.01
0.78
1.29
1.28
-0.01
0.78
Table 3: CPN Outputs for 3x12x1 Architecture (Continued)
Data
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
22
(A = 0.02 a = 0.30 b = 0.0083 c = 2.60 T = 0.0056
Data
Number
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
Measured
Specific
Gravity
1.29
1.29
1.30
1.30
1.30
1.30
1.31
1.31
1.32
1.32
1.33
1.33
1.37
1.38
1.38
1.38
1.38
1.39
1.40
1.41
1.42
Estimated
Specific
Gravity
1.30
1.28
1.27
1.30
1.30
1.30
1.25
1.30
1.28
1.36
1.30
1.35
1.38
1.36
1.36
1.36
1.34
1.35
1.36
1.37
1.39
Error
0.01
-0.01
-0.03
0.00
0.00
0.00
-0.06
-0.01
-0.04
0.04
-0.03
0.02
0.01
-0.02
-0.02
-0.02
-0.04
-0.04
-0.04
-0.04
-0.03
Maximum
Minimum
Average
INTERP. = -3)
Abs. %
Error
0.78
0.78
2.31
0.00
0.00
0.00
4.58
0.76
3.03
3.03
2.26
1.50
0.73
1.45
1.45
1.45
2.90
2.88
2.86
2.84
2.11
4.58
0.00
1.41
23
Table 4: Statistic of CPN Outputs
Mean
Squared
Error
Error
Standard
Deviation
Maximum
Error
Maximum
percent
Error
Mean
Absolute
Error
Absolute
Percent
Error
0.00105
0.03260
-0.06000
-4.58000
0.02220
1.41
24
5.2 Adaptive Network-Based Fuzzy Inference System (ANFIS)
5.2.1 Training Process of ANFIS
In ANFIS training process, the number of membership functions has been
changed to determine different ANFIS architectures. Three features were used as the
inputs of ANFIS. These three features are the absolute air inlet pressure, the propane/mix
differential pressure, and the propane/air differential pressure. The number of
membership functions used was from three to five. The membership functions used were
Triangular, Trapezoidal, Generalized bell, Gaussian, Two-Sided Gaussian, Pi-Shaped,
Difference of Sigmoid, and Product of Sigmoid membership functions. Therefore, the
training process was carried out for each of the 24 ANFIS architectures (3 inputs x 8
ANFIS) for on-line measurements of specific gravity of the LP gas. The combinations
were denoted by j x l, where j is the number of inputs and l is the number of membership
functions of each input. For example, the case of 3 x 5 means the case of 3 inputs and 5
membership functions. The data of three inputs for the training process are also shown in
Table 6. The number of rules in the ANFIS training process can be generated
automatically. For example, there are 27 (33) rules for 3 x 3 architecture, and 243 (35)
rules for the 3 x 5 architecture. The Gaussian, Trapezoid, or Difference of Sigmoid
membership function was assigned to each input so as to compare their performance for
on-line measurements. The hybrid-learning algorithm (a combination of the
backpropagation and least square methods) was used. The epoch in the training process
was set to be 65. The error tolerance was 0.
25
5.2.2 On-Line Tests Using ANFIS
Another set of 63 data was introduced to a 3 x 5 ANFIS architecture for on-line
measurements of glass furnace. The epoch in the testing process was set to be 65. The
error tolerance was set to be 0.
After the 3 x 5 ANFIS architecture with the generalized bell membership function
has been trained, it is then can be used for on-line measurements of glass furnace. The
results of the 3 x 5 ANFIS architecture with a generalized bell membership functions was
very successful. The absolute average error for on-line measurements of glass furnace
was 0.23% while the minimum error of 0%, and a maximum error of 1.78% were
achieved. Table 5 shown the absolute average percentage error outputs using different
ANFIS architectures.
Table 5: Outputs Absolute Average Error Percentage of ANFIS Architectures
Absolute Error Percentage of ANFIS Architectures
Membership
Functions
3 x 3 ANFIS
3 x 4 ANFIS
3 x 5 ANFIS
Max.
Min.
Avg.
Max.
Min.
Avg.
Max.
Min.
Avg.
Triangular
3.09
0.01
1.04
2.3
0
0.64
1.92
0
0.37
Trapezoid
6.3
0
1.41
4.56
0
1.04
4.08
0
0.8
Generalized
Bell
3.57
0
0.92
2.54
0
0.37
1.78
0
0.23
Gaussian
3.71
0
0.93
2.82
0
0.5
2.27
0
0.79
2-sided
Gaussian
4.38
0
0.93
3.29
0
0.62
4.93
0
0.96
Pi-Shaped
6.37
0
1.45
5.21
0
1.05
4.85
0
1.04
Difference of
Sigmoid
3.43
0
1.21
3.32
0
0.6
5.01
0
0.96
Product of
Sigmoid
3.43
0
1.21
3.32
0
0.6
5.01
0
0.96
26
Based on architecture set up at 65 epochs, tolerance error of 0. Interestingly, the
percentages error of Triangular, Trapezoidal, and Generalized bell membership functions
were decreased when membership function increased whereas the rest of membership
functions that were used result of increasing of percentage error when membership
Absolute Average Specific Gravity Error (%)
functions changed from 4 to 5 membership functions as shown in Figure 7.
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Trimf
Trapmf
Gbellmf
Gaussmf Gauss2mf
Pimf
Dsigmf
Psigmf
Membership Functions
3x3 Membership Functions
3x4 Membership Function
3x5 Membership Functions
Figure 7: Comparison of Absolute Error Percentage versus ANFIS Architectures
27
Chapter 6
CONCLUSIONS
The compressible gas flow system that is used in the gas blending process of the
glass production furnace is nonlinear in nature. Variables such orifices and compressible
flow coefficients make the analysis of the process extremely difficult. Neural networks
and neuro-fuzzy system can perform on-line measurements without any prior knowledge
of the process. Based on this research, the following conclusions can be drawn:
1.
Three features were selected from glass production control system for on-line
monitoring and diagnosis. These three features are air inlet pressure, propane
differential pressure, and air differential pressure. Interestingly all these three
features as the input of neural networks and neuro-fuzzy systems generates the
best on-line specific gravity estimation.
2. A 3x12x1 CPN can measure the specific gravity of the LP gas on-line an absolute
average of 1.7%, a minimum error of 0%, and a maximum of 4.58%.
3. A 3 x 5 ANFIS architecture with Triangular membership function can measure
the specific gravity of the LP gas on-line an absolute average of 0.37%, a
minimum error of 0%, and a maximum of 1.92%.
4. A 3 x 5 ANFIS architecture with Trapezoidal membership function can measure
the specific gravity of the LP gas on-line an absolute average of 0.80%, a
minimum error of 0%, and a maximum of 4.08%.
5. A 3 x 5 ANFIS architecture with Generalized bell membership function can
measure the specific gravity of the LP gas on-line an absolute average of 0.23%, a
28
minimum error of 0%, and a maximum of 1.78%.
6. A 3 x 5 ANFIS architecture with Gaussian membership function can measure the
specific gravity of the LP gas on-line an absolute average of 0.79%, a minimum
error of 0%, and a maximum of 2.27%.
7. A 3 x 5 ANFIS architecture with Two-Sided Gaussian membership function can
measure the specific gravity of the LP gas on-line an absolute average of 0.96%, a
minimum error of 0%, and a maximum of 4.93%.
8. A 3 x 5 ANFIS architecture with Difference of Sigmoid membership function can
measure the specific gravity of the LP gas on-line an absolute average of 0.96%, a
minimum error of 0%, and a maximum of 5.01%.
9. A 3 x 5 ANFIS architecture with Product of Sigmoid membership function can
measure the specific gravity of the LP gas on-line an absolute average of 0.96%, a
minimum error of 0%, and a maximum of 5.01%.
10. 3 x 5 ANFIS architecture with Generalized bell membership function yields the
best results. Therefore, it is selected for the on-line measurements of the specific
gravity of the LP gas for the glass production furnace. This can achieve much
higher energy efficiency, which is very important for glass production.
11. The instrument lag time for this intelligent system is negligible. This high-speed
response results in an exceedingly rapid process control, which is extremely
essential for glass production.
29
APPENDICES
30
APPENDIX A
ANFIS Training and On-Line Test Data
31
Table 6: On-Line Test Data (Table continues)
Data
Number
Air Inlet
Pressure
(PSIA)
Air/Mix
Pressure
(PSID)
Propane/Mix
Pressure
(PSID)
Mix Specific
Gravity
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
30.80
37.90
38.80
37.80
31.30
39.50
33.80
29.40
30.20
33.60
40.70
31.30
30.40
35.90
32.80
42.30
28.20
35.70
42.10
31.40
29.30
31.10
34.20
33.20
34.20
30.90
31.80
37.70
31.30
31.40
31.20
31.20
42.80
31.70
32.30
31.30
31.70
31.60
0.10
0.20
0.60
1.10
0.10
1.80
0.10
0.00
0.50
0.90
3.00
1.40
0.20
2.20
0.10
4.60
0.50
0.00
5.40
1.20
0.60
0.40
1.50
0.00
1.50
0.20
1.10
3.00
1.10
1.20
1.00
1.00
5.10
1.00
2.60
1.10
2.00
1.90
1.30
0.20
0.10
0.00
1.30
0.10
1.90
1.30
1.80
2.00
0.20
2.20
2.10
1.20
1.90
0.20
2.10
1.50
1.20
2.00
2.80
1.50
1.20
1.20
1.30
1.50
1.80
0.90
1.90
1.80
2.00
1.80
0.20
1.30
2.00
2.00
1.80
1.80
1.22
1.23
1.23
1.23
1.24
1.24
1.24
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.26
1.26
1.26
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
Table 6: On-Line Test Data (Continued)
32
Data
Number
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
Air Inlet
Pressure
(PSIA)
30.90
37.10
33.00
31.40
32.20
32.70
30.30
31.70
32.40
32.00
32.50
30.70
33.30
32.20
33.60
32.50
34.50
33.50
34.70
34.60
33.70
34.30
34.30
34.40
33.70
Air/Mix
Pressure
(PSID)
1.20
2.40
2.30
1.20
1.00
2.00
1.10
1.00
1.90
1.80
1.80
0.00
2.10
1.50
3.40
1.80
3.30
3.80
4.00
3.90
2.50
2.60
3.10
3.20
4.00
Propane/Mix
Pressure
(PSID)
2.00
1.20
1.20
2.10
1.30
1.80
2.10
1.40
1.60
1.90
2.00
2.00
1.50
1.60
1.80
2.00
1.50
2.00
1.40
1.40
2.00
1.40
1.50
1.80
2.00
Mix Specific
Gravity
1.29
1.29
1.29
1.29
1.29
1.29
1.30
1.30
1.30
1.30
1.31
1.31
1.32
1.32
1.33
1.33
1.37
1.38
1.38
1.38
1.38
1.39
1.40
1.41
1.42
33
Table 7: Training Data (Table continues)
Data
Number
Air Inlet
Pressure
(PSIA)
Air/Mix
Pressure
(PSID)
Propane/Mix
Pressure
(PSID)
Mix Specific
Gravity
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
29.90
30.20
29.90
37.60
34.90
30.40
32.10
38.70
34.00
30.20
32.90
33.00
31.80
30.90
30.70
39.60
35.00
33.50
33.30
29.10
31.30
31.90
30.00
30.90
30.50
42.70
31.40
43.70
31.20
44.40
31.70
31.70
32.80
32.00
30.80
31.70
30.90
31.60
31.90
30.90
31.70
32.70
0.20
0.00
0.20
0.90
2.20
0.20
0.40
2.00
2.30
0.50
0.20
0.80
0.10
0.20
0.50
2.40
0.30
0.80
0.60
0.40
1.10
1.20
0.30
0.20
0.80
5.50
1.20
6.00
1.00
6.70
2.00
2.00
2.10
1.80
1.10
1.00
0.70
0.90
1.20
1.20
1.00
2.00
2.00
2.00
2.00
0.00
1.60
1.40
1.60
0.10
2.50
2.30
2.10
1.80
1.90
1.50
1.90
0.10
1.30
0.00
1.90
2.60
2.00
2.10
2.10
2.00
1.90
1.00
2.00
0.20
1.90
0.00
2.20
2.00
3.20
2.00
2.00
1.50
1.20
1.80
1.50
2.20
1.00
2.50
1.19
1.20
1.22
1.22
1.22
1.22
1.22
1.23
1.23
1.23
1.23
1.24
1.25
1.25
1.25
1.25
1.25
1.26
1.26
1.26
1.27
1.27
1.27
1.27
1.27
1.27
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.29
1.29
1.29
1.29
1.29
1.29
1.30
34
Table 7: Training Data (Continued)
Data
Number
Air Inlet
Pressure
(PSIA)
Air/Mix
Pressure
(PSID)
Propane/Mix
Pressure
(PSID)
Mix Specific
Gravity
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
30.90
32.70
32.20
30.70
32.80
30.70
33.70
33.70
33.20
32.20
32.50
31.70
32.00
33.20
33.90
34.20
35.20
33.90
33.50
33.60
34.20
1.20
2.00
2.00
1.00
2.10
1.00
1.00
3.00
2.50
2.00
2.00
1.50
1.30
3.00
3.70
2.50
4.50
3.70
3.80
3.90
3.50
2.10
1.80
1.00
1.80
1.20
2.00
1.20
2.20
2.00
2.40
2.20
1.60
2.00
1.60
2.00
1.60
1.20
2.00
1.90
2.00
2.00
1.30
1.30
1.30
1.31
1.32
1.32
1.32
1.33
1.33
1.33
1.34
1.35
1.37
1.37
1.38
1.38
1.38
1.39
1.41
1.41
1.42
35
APPENDIX B
Triangular Membership Functions (Trimf)
Description: The triangular curve is a function of a vector, x, and depends on three scalar
parameters, a, b, and c, as given by:
 x  a c  x  
f (x;a,b,c)  max min 
,
, 0 
 b  a c  b  
the parameters a and c are located at the “feet” of the triangle, and the parameter b
Where
locates at the peak of the triangle.
36
Table 8: Outputs of 3 x 5 Trimf (Table continue)
Specific Gravity
Data
Number
Measured
Estimated
Error
% Error
Abs.
Error
Abs %
Error
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
1.22
1.23
1.23
1.23
1.24
1.24
1.24
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.26
1.26
1.26
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.23
1.23
1.23
1.23
1.23
1.24
1.23
1.25
1.26
1.26
1.25
1.25
1.25
1.25
1.26
1.26
1.26
1.26
1.27
1.28
1.27
1.27
1.27
1.27
1.27
1.27
1.30
1.28
1.28
1.28
1.28
1.27
1.28
1.28
1.29
0.01
0.00
0.00
0.00
-0.01
0.00
-0.01
0.00
0.01
0.01
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.02
0.00
0.00
0.00
0.00
-0.01
0.00
0.00
0.01
1.07
0.00
0.00
0.00
-0.75
0.00
-0.49
-0.25
0.57
0.43
0.00
0.06
0.20
-0.02
0.60
0.00
0.26
0.02
0.00
1.10
-0.06
-0.21
0.11
-0.02
-0.21
-0.13
1.92
0.00
-0.07
0.16
0.05
-0.63
0.00
-0.23
0.53
0.01
0.00
0.00
0.00
0.01
0.00
0.01
0.00
0.01
0.01
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.02
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.01
1.07
0.00
0.00
0.00
0.75
0.00
0.49
0.25
0.57
0.43
0.00
0.06
0.20
0.02
0.60
0.00
0.26
0.02
0.00
1.10
0.06
0.21
0.11
0.02
0.21
0.13
1.92
0.00
0.07
0.16
0.05
0.63
0.00
0.23
0.53
37
Table 8: Outputs of 3 x 5 Trimf (Continued)
Specific Gravity
Data
Number
Measured
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
1.28
1.28
1.28
1.29
1.29
1.29
1.29
1.29
1.29
1.30
1.30
1.30
1.30
1.31
1.31
1.32
1.32
1.33
1.33
1.37
1.38
1.38
1.38
1.38
1.39
1.40
1.41
1.42
Estimated
Error
% Error
Abs.
Error
Abs %
Error
1.28
1.28
1.28
1.27
1.29
1.29
1.29
1.30
1.31
1.30
1.30
1.29
1.29
1.31
1.30
1.32
1.31
1.33
1.31
1.37
1.39
1.38
1.38
1.37
1.39
1.39
1.42
1.41
Max.
Min.
Avg.
0.00
0.00
0.00
-0.02
0.00
0.00
0.00
0.01
0.02
0.00
0.00
-0.01
-0.01
0.00
-0.01
0.00
-0.01
0.00
-0.02
0.00
0.01
0.00
0.00
-0.01
0.00
-0.01
0.01
-0.01
0.02
-0.02
0.00
0.15
-0.22
0.15
-1.48
0.01
-0.02
-0.21
0.40
1.40
0.26
-0.37
-0.49
-0.69
0.11
-0.85
0.20
-0.50
0.15
-1.39
0.12
0.43
0.06
-0.07
-0.52
0.09
-0.61
0.54
-0.51
1.92
-1.48
0.01
0.00
0.00
0.00
0.02
0.00
0.00
0.00
0.01
0.02
0.00
0.00
0.01
0.01
0.00
0.01
0.00
0.01
0.00
0.02
0.00
0.01
0.00
0.00
0.01
0.00
0.01
0.01
0.01
0.02
0.00
0.00
0.15
0.22
0.15
1.48
0.01
0.02
0.21
0.40
1.40
0.26
0.37
0.49
0.69
0.11
0.85
0.20
0.50
0.15
1.39
0.12
0.43
0.06
0.07
0.52
0.09
0.61
0.54
0.51
1.92
0.00
0.37
38
APPENDIX C
Trapezoidal Membership Functions (Trapmf)
Description: The Trapezoid curve is a function of a vector, x, and depends on four
parameters a, b, c, and d, as given by:
 x  a  d  x  
f (x;a,b,c,d)  max min 
,1,
, 0 
 b  a  d  c  
 Where a and d locate the “feet” of the trapezoid and the parameters b and c locate
the shoulders.
39
Table 9: Outputs of 3 x 5 Trapmf (Table continues)
Specific Gravity
Data
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
Measured
Estimated
Error
% Error
Abs. Error
Abs % Error
1.22
1.23
1.23
1.23
1.24
1.24
1.24
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.26
1.26
1.26
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.25
1.23
1.23
1.23
1.25
1.24
1.24
1.25
1.25
1.25
1.25
1.28
1.27
1.30
1.27
1.26
1.26
1.26
1.27
1.28
1.27
1.26
1.26
1.25
1.26
1.25
1.29
1.28
1.29
1.29
1.28
1.29
1.28
1.30
1.28
0.03
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.03
0.02
0.05
0.02
0.00
-0.01
0.00
0.00
0.01
0.00
-0.01
-0.01
-0.02
-0.01
-0.02
0.01
0.00
0.01
0.01
0.00
0.01
0.00
0.02
0.00
2.43
0.00
0.00
0.00
0.78
0.00
-0.01
0.00
0.00
0.06
0.00
2.75
1.25
4.08
1.82
0.00
-0.40
0.27
0.00
1.13
0.00
-1.01
-1.12
-1.39
-1.12
-1.60
0.83
0.00
0.58
0.83
0.34
0.83
0.00
1.25
0.22
0.03
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.03
0.02
0.05
0.02
0.00
0.01
0.00
0.00
0.01
0.00
0.01
0.01
0.02
0.01
0.02
0.01
0.00
0.01
0.01
0.00
0.01
0.00
0.02
0.00
2.43
0.00
0.00
0.00
0.78
0.00
0.01
0.00
0.00
0.06
0.00
2.75
1.25
4.08
1.82
0.00
0.40
0.27
0.00
1.13
0.00
1.01
1.12
1.39
1.12
1.60
0.83
0.00
0.58
0.83
0.34
0.83
0.00
1.25
0.22
40
Table 9: Outputs of 3 x 5 Trapmf (Continued)
Specific Gravity
Data
Number
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
Measured
1.28
1.28
1.28
1.29
1.29
1.29
1.29
1.29
1.29
1.30
1.30
1.30
1.30
1.31
1.31
1.32
1.32
1.33
1.33
1.37
1.38
1.38
1.38
1.38
1.39
1.40
1.41
1.42
Estimated
Error
% Error
Abs. Error
Abs %
Error
1.28
1.29
1.29
1.28
1.29
1.30
1.28
1.30
1.29
1.31
1.30
1.30
1.29
1.28
1.28
1.29
1.30
1.33
1.28
1.38
1.38
1.38
1.38
1.37
1.38
1.38
1.42
1.42
Max.
Min.
Avg.
0.00
0.01
0.01
-0.01
0.00
0.01
-0.01
0.01
0.00
0.01
0.00
0.00
-0.01
-0.03
-0.03
-0.03
-0.02
0.00
-0.05
0.01
0.00
0.00
0.00
-0.01
-0.01
-0.02
0.01
0.00
0.05
-0.05
0.00
0.34
0.82
0.83
-0.43
0.00
0.61
-0.43
0.47
0.04
0.39
-0.31
-0.31
-0.97
-1.95
-2.22
-2.07
-1.82
0.00
-3.43
0.71
0.01
-0.09
-0.09
-0.60
-0.71
-1.42
0.65
-0.02
4.08
-3.43
0.02
0.00
0.01
0.01
0.01
0.00
0.01
0.01
0.01
0.00
0.01
0.00
0.00
0.01
0.03
0.03
0.03
0.02
0.00
0.05
0.01
0.00
0.00
0.00
0.01
0.01
0.02
0.01
0.00
0.05
0.00
0.01
0.34
0.82
0.83
0.43
0.00
0.61
0.43
0.47
0.04
0.39
0.31
0.31
0.97
1.95
2.22
2.07
1.82
0.00
3.43
0.71
0.01
0.09
0.09
0.60
0.71
1.42
0.65
0.02
4.08
0.00
0.80
41
APPENDIX D
Generalized Bell Membership Functions (Gbellmf)
Description: The generalized bell function depends on three parameters a, b, and c as
given by:
f (x;a,b,c) 
1
x c
1
a
2b
Where a determines
the width, and c adjusts the center of the corresponding membership
functions; the parameter b controls the slope at the crossover points.
42
Table 10: Outputs of 3 x 4 Gbellmf (Table continues)
Specific Gravity
Data
Number
Measured
Estimated
Error
% Error
Abs. Error
Abs % Error
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
1.22
1.23
1.23
1.23
1.24
1.24
1.24
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.26
1.26
1.26
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.24
1.23
1.23
1.23
1.23
1.24
1.23
1.24
1.26
1.25
1.25
1.25
1.27
1.25
1.27
1.26
1.26
1.26
1.27
1.29
1.27
1.27
1.27
1.27
1.27
1.27
1.28
1.28
1.28
1.28
1.28
1.27
1.28
1.29
1.28
1.29
1.28
0.02
0.00
0.00
0.00
-0.01
0.00
-0.01
-0.01
0.01
0.00
0.00
0.00
0.02
0.00
0.02
0.00
0.00
0.00
0.00
0.02
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
-0.01
0.00
0.01
0.00
0.01
0.00
1.30
0.00
0.00
0.00
-0.45
0.00
-0.73
-0.49
0.74
0.10
-0.01
0.14
1.51
0.03
1.34
0.02
0.05
0.05
0.00
1.39
-0.24
0.13
-0.18
-0.07
0.37
0.14
-0.03
0.02
-0.28
0.37
0.24
-1.02
-0.02
0.59
-0.01
0.41
-0.22
0.02
0.00
0.00
0.00
0.01
0.00
0.01
0.01
0.01
0.00
0.00
0.00
0.02
0.00
0.02
0.00
0.00
0.00
0.00
0.02
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.01
0.00
0.01
0.00
1.30
0.00
0.00
0.00
0.45
0.00
0.73
0.49
0.74
0.10
0.01
0.14
1.51
0.03
1.34
0.02
0.05
0.05
0.00
1.39
0.24
0.13
0.18
0.07
0.37
0.14
0.03
0.02
0.28
0.37
0.24
1.02
0.02
0.59
0.01
0.41
0.22
43
Table 10: Outputs of 3 x 4 Gbellmf (Continued)
Specific Gravity
Data
Number
Measured
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
1.28
1.29
1.29
1.29
1.29
1.29
1.29
1.30
1.30
1.30
1.30
1.31
1.31
1.32
1.32
1.33
1.33
1.37
1.38
1.38
1.38
1.38
1.39
1.40
1.41
1.42
Estimated
Error
% Error
Abs. Error
Abs % Error
1.28
1.29
1.29
1.29
1.28
1.28
1.29
1.30
1.30
1.31
1.30
1.32
1.28
1.32
1.31
1.33
1.32
1.37
1.38
1.38
1.38
1.38
1.39
1.40
1.41
1.42
Max.
Min.
Avg.
0.00
0.00
0.00
0.00
-0.01
-0.01
0.00
0.00
0.00
0.01
0.00
0.01
-0.03
0.00
-0.01
0.00
-0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.23
-0.33
-0.07
0.06
-0.80
-1.02
0.19
-0.01
-0.09
0.45
-0.30
0.67
-2.54
-0.15
-0.43
0.02
-0.84
0.18
0.02
-0.14
0.17
0.01
-0.09
-0.10
-0.06
-0.03
0.00
0.00
0.00
0.00
0.01
0.01
0.00
0.00
0.00
0.01
0.00
0.01
0.03
0.00
0.01
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.23
0.33
0.07
0.06
0.80
1.02
0.19
0.01
0.09
0.45
0.30
0.67
2.54
0.15
0.43
0.02
0.84
0.18
0.02
0.14
0.17
0.01
0.09
0.10
0.06
0.03
0.02
-0.03
0.00
1.51
-2.54
-0.01
0.03
0.00
0.00
2.54
0.00
0.37
44
Table 11: Outputs of 3 x 5 Gbellmf (Table continues)
Specific Gravity
Data
Number
Measured
Estimated
Error
% Error
Abs. Error
Abs %
Error
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
1.22
1.23
1.23
1.23
1.24
1.24
1.24
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.26
1.26
1.26
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.23
1.23
1.23
1.23
1.23
1.24
1.24
1.25
1.25
1.25
1.25
1.26
1.25
1.25
1.25
1.26
1.26
1.26
1.27
1.29
1.27
1.27
1.27
1.27
1.27
1.27
1.28
1.28
1.28
1.28
1.29
1.28
1.28
1.28
1.28
1.29
0.01
0.00
0.00
0.00
-0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.02
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.01
0.89
0.00
0.00
0.00
-0.80
0.00
-0.03
-0.10
-0.06
0.23
0.00
0.45
0.30
-0.01
-0.13
0.00
-0.03
0.01
0.00
1.78
0.00
-0.06
-0.02
0.01
0.04
0.13
-0.09
0.00
-0.02
0.15
0.50
0.02
0.00
-0.02
0.34
0.73
0.01
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.02
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.01
0.89
0.00
0.00
0.00
0.80
0.00
0.03
0.10
0.06
0.23
0.00
0.45
0.30
0.01
0.13
0.00
0.03
0.01
0.00
1.78
0.00
0.06
0.02
0.01
0.04
0.13
0.09
0.00
0.02
0.15
0.50
0.02
0.00
0.02
0.34
0.73
45
Table 11: Outputs of 3 x 5 Gbellmf (Continued)
Specific Gravity
Data
Number
Measured
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
1.28
1.28
1.29
1.29
1.29
1.29
1.29
1.29
1.30
1.30
1.30
1.30
1.31
1.31
1.32
1.32
1.33
1.33
1.37
1.38
1.38
1.38
1.38
1.39
1.40
1.41
1.42
Estimated
Error
% Error
Abs. Error
Abs % Error
1.28
1.28
1.28
1.29
1.29
1.28
1.29
1.29
1.30
1.30
1.30
1.29
1.31
1.31
1.32
1.32
1.33
1.31
1.37
1.38
1.38
1.38
1.38
1.39
1.40
1.41
1.42
Max.
Min.
Avg.
0.00
0.00
-0.01
0.00
0.00
-0.01
0.00
0.00
0.00
0.00
0.00
-0.01
0.00
0.00
0.00
0.00
0.00
-0.02
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
-0.10
0.05
-0.92
0.00
0.01
-0.95
0.05
0.20
-0.08
-0.03
0.35
-0.76
0.15
-0.31
-0.24
-0.17
0.00
-1.35
0.04
-0.02
0.01
-0.02
0.01
0.04
-0.03
-0.01
0.01
0.00
0.00
0.01
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.02
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.10
0.05
0.92
0.00
0.01
0.95
0.05
0.20
0.08
0.03
0.35
0.76
0.15
0.31
0.24
0.17
0.00
1.35
0.04
0.02
0.01
0.02
0.01
0.04
0.03
0.01
0.01
0.02
-0.02
0.00
1.78
-1.35
0.01
0.02
0.00
0.00
1.78
0.00
0.23
46
APPENDIX E
Gaussian Membership Functions (Gaussmf)
Description: The symmetric Gaussian function depends on two parameters  and c as
given by:
(x c )2
f (x;a,c)  e
2 2

Where  controls the slopes and c adjusts the center of the corresponding membership
function.
47
Table 12: Outputs of 3 x 4 Gaussmf (Table continues)
Specific Gravity
Data
Number
Measured
Estimated
Error
% Error
Abs.
Error
Abs %
Error
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
1.22
1.23
1.23
1.23
1.24
1.24
1.24
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.26
1.26
1.26
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.24
1.23
1.23
1.23
1.23
1.24
1.23
1.24
1.27
1.26
1.25
1.27
1.27
1.25
1.26
1.26
1.26
1.26
1.27
1.28
1.27
1.26
1.27
1.27
1.28
1.26
1.29
1.28
1.28
1.29
1.28
1.28
1.28
1.28
1.28
1.28
0.02
0.00
0.00
0.00
-0.01
0.00
-0.01
-0.01
0.02
0.01
0.00
0.02
0.02
0.00
0.01
0.00
0.00
0.00
0.00
0.01
0.00
-0.01
0.00
0.00
0.01
-0.01
0.01
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
1.42
0.00
0.00
0.00
-0.43
0.00
-0.72
-0.43
1.31
0.50
-0.01
1.74
1.46
0.08
1.00
0.01
0.22
0.05
0.00
1.09
-0.06
-0.46
-0.33
-0.18
0.65
-0.46
1.09
0.02
0.27
0.53
-0.03
-0.06
-0.01
0.11
-0.10
0.13
0.02
0.00
0.00
0.00
0.01
0.00
0.01
0.01
0.02
0.01
0.00
0.02
0.02
0.00
0.01
0.00
0.00
0.00
0.00
0.01
0.00
0.01
0.00
0.00
0.01
0.01
0.01
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
1.42
0.00
0.00
0.00
0.43
0.00
0.72
0.43
1.31
0.50
0.01
1.74
1.46
0.08
1.00
0.01
0.22
0.05
0.00
1.09
0.06
0.46
0.33
0.18
0.65
0.46
1.09
0.02
0.27
0.53
0.03
0.06
0.01
0.11
0.10
0.13
48
Table 12: Outputs of 3 x 4 Gaussmf (Continued)
Specific Gravity
Data
Number
Measured
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
1.28
1.28
1.29
1.29
1.29
1.29
1.29
1.29
1.30
1.30
1.30
1.30
1.31
1.31
1.32
1.32
1.33
1.33
1.37
1.38
1.38
1.38
1.38
1.39
1.40
1.41
1.42
Estimated
Error
% Error
Abs.
Error
Abs %
Error
1.28
1.28
1.28
1.29
1.29
1.28
1.29
1.30
1.29
1.30
1.31
1.30
1.30
1.27
1.31
1.31
1.33
1.30
1.38
1.38
1.38
1.38
1.39
1.39
1.40
1.40
1.42
Max.
Min.
Avg.
0.00
0.00
-0.01
0.00
0.00
-0.01
0.00
0.01
-0.01
0.00
0.01
0.00
-0.01
-0.04
-0.01
-0.01
0.00
-0.03
0.01
0.00
0.00
0.00
0.01
0.00
0.00
-0.01
0.00
0.02
-0.04
0.00
0.08
0.10
-1.02
-0.09
0.05
-0.98
0.02
0.92
-0.98
-0.24
0.86
-0.06
-0.63
-2.82
-0.70
-0.65
0.01
-2.12
0.91
0.07
-0.07
-0.03
0.51
-0.34
-0.26
-0.48
-0.04
1.74
-2.82
-0.01
0.00
0.00
0.01
0.00
0.00
0.01
0.00
0.01
0.01
0.00
0.01
0.00
0.01
0.04
0.01
0.01
0.00
0.03
0.01
0.00
0.00
0.00
0.01
0.00
0.00
0.01
0.00
0.04
0.00
0.01
0.08
0.10
1.02
0.09
0.05
0.98
0.02
0.92
0.98
0.24
0.86
0.06
0.63
2.82
0.70
0.65
0.01
2.12
0.91
0.07
0.07
0.03
0.51
0.34
0.26
0.48
0.04
2.82
0.00
0.50
49
Table 13: Outputs of 3 x 5 Gaussmf (Table continues)
Specific Gravity
Data
Number
Measured
Estimated
Error
% Error
Abs.
Error
Abs %
Error
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
1.22
1.23
1.23
1.23
1.24
1.24
1.24
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.26
1.26
1.26
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.21
1.21
1.21
1.22
1.22
1.22
1.21
1.23
1.23
1.23
1.23
1.24
1.26
1.25
1.23
1.25
1.25
1.26
1.26
1.26
1.27
1.28
1.26
1.27
1.27
1.27
1.28
1.28
1.29
1.28
1.28
1.28
1.28
1.29
1.31
1.29
-0.01
-0.02
-0.02
-0.01
-0.02
-0.02
-0.03
-0.02
-0.02
-0.02
-0.02
-0.01
0.01
0.00
-0.02
-0.01
-0.01
0.00
-0.01
-0.01
0.00
0.01
-0.01
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.01
0.03
0.01
-1.21
-1.58
-2.02
-0.81
-1.50
-1.74
-2.05
-1.60
-1.54
-1.28
-1.74
-1.14
0.42
0.34
-1.24
-0.79
-0.79
0.00
-0.50
-0.83
0.24
1.02
-0.57
-0.30
0.13
0.00
-0.26
0.00
1.08
0.00
-0.16
-0.30
0.08
0.69
2.08
1.09
0.01
0.02
0.02
0.01
0.02
0.02
0.03
0.02
0.02
0.02
0.02
0.01
0.01
0.00
0.02
0.01
0.01
0.00
0.01
0.01
0.00
0.01
0.01
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.01
0.03
0.01
1.21
1.58
2.02
0.81
1.50
1.74
2.05
1.60
1.54
1.28
1.74
1.14
0.42
0.34
1.24
0.79
0.79
0.00
0.50
0.83
0.24
1.02
0.57
0.30
0.13
0.00
0.26
0.00
1.08
0.00
0.16
0.30
0.08
0.69
2.08
1.09
50
Table 13: Outputs of 3 x 5 Gaussmf (Continued)
Specific Gravity
Data
Number
Measured
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
1.28
1.28
1.29
1.29
1.29
1.29
1.29
1.29
1.30
1.30
1.30
1.30
1.31
1.31
1.32
1.32
1.33
1.33
1.37
1.38
1.38
1.38
1.38
1.39
1.40
1.41
1.42
Estimated
Error
% Error
Abs.
Error
Abs %
Error
1.29
1.30
1.29
1.29
1.29
1.29
1.29
1.30
1.30
1.30
1.32
1.30
1.32
1.33
1.33
1.33
1.35
1.35
1.34
1.37
1.38
1.38
1.38
1.38
1.41
1.41
1.42
Max.
Min.
Avg.
0.01
0.02
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.02
0.00
0.01
0.02
0.01
0.01
0.02
0.01
-0.03
-0.01
0.00
0.00
0.00
-0.01
0.01
0.00
0.00
0.03
-0.03
0.00
0.78
1.50
-0.08
-0.13
-0.02
0.31
0.36
1.12
0.00
0.36
1.56
0.26
0.75
1.43
0.67
0.98
1.23
1.13
-2.27
-0.70
0.17
-0.17
0.00
-0.55
0.69
0.18
0.25
2.08
-2.27
-0.11
0.01
0.02
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.02
0.00
0.01
0.02
0.01
0.01
0.02
0.01
0.03
0.01
0.00
0.00
0.00
0.01
0.01
0.00
0.00
0.03
0.00
0.01
0.78
1.50
0.08
0.13
0.02
0.31
0.36
1.12
0.00
0.36
1.56
0.26
0.75
1.43
0.67
0.98
1.23
1.13
2.27
0.70
0.17
0.17
0.00
0.55
0.69
0.18
0.25
2.27
0.00
0.79
51
APPENDIX F
Two-sided Gaussian Membership Functions (Gauss2mf)
Description: The Gaussian function depending upon two parameters σ and c as given by:
(x c )2
f (x;,c)  e
2 2
 Gaussian membership function is a combination of two of these.
The two-sided
The first function, specified by σ1 and c1, determines the shape of the leftmost curve. The
second function specified by σ2 and c2 determines the shape of the right-most curve.
Whenever c1<c2, the two-sided Gaussian membership function reaches a maximum value
of one. Otherwise, the maximum value is less than one.
52
Table 14: Outputs of 3 x 4 Gauss2mf (Table continues)
Specific Gravity
Data
Number
Measured
Estimated
Error
% Error
Abs.
Error
Abs %
Error
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
1.22
1.23
1.23
1.23
1.24
1.24
1.24
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.26
1.26
1.26
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.24
1.23
1.23
1.23
1.25
1.24
1.24
1.24
1.27
1.26
1.25
1.25
1.26
1.27
1.27
1.26
1.27
1.26
1.27
1.29
1.27
1.26
1.27
1.27
1.28
1.26
1.29
1.28
1.28
1.29
1.27
1.27
1.28
1.30
1.28
1.28
-0.02
0.00
0.00
0.00
-0.01
0.00
0.00
0.01
-0.02
-0.01
0.00
0.00
-0.01
-0.02
-0.02
0.00
-0.01
0.00
0.00
-0.02
0.00
0.01
0.00
0.00
-0.01
0.01
-0.01
0.00
0.00
-0.01
0.01
0.01
0.00
-0.02
0.00
0.00
-1.98
-0.02
0.02
0.00
-0.56
0.00
0.16
0.94
-1.33
-0.47
0.00
-0.25
-1.09
-1.50
-1.50
0.00
-0.49
0.00
0.00
-1.61
0.02
0.45
0.35
0.24
-0.57
0.49
-0.57
-0.30
-0.25
-0.98
0.57
0.48
0.00
-1.21
-0.02
-0.15
0.02
0.00
0.00
0.00
0.01
0.00
0.00
0.01
0.02
0.01
0.00
0.00
0.01
0.02
0.02
0.00
0.01
0.00
0.00
0.02
0.00
0.01
0.00
0.00
0.01
0.01
0.01
0.00
0.00
0.01
0.01
0.01
0.00
0.02
0.00
0.00
1.98
0.02
0.02
0.00
0.56
0.00
0.16
0.94
1.33
0.47
0.00
0.25
1.09
1.50
1.50
0.00
0.49
0.00
0.00
1.61
0.02
0.45
0.35
0.24
0.57
0.49
0.57
0.30
0.25
0.98
0.57
0.48
0.00
1.21
0.02
0.15
53
Table 14: Outputs of 3 x 4 Gauss2mf (Continued)
Specific Gravity
Data
Number
Measured
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
1.28
1.28
1.29
1.29
1.29
1.29
1.29
1.29
1.30
1.30
1.30
1.30
1.31
1.31
1.32
1.32
1.33
1.33
1.37
1.38
1.38
1.38
1.38
1.39
1.40
1.41
1.42
Estimated
Error
% Error
Abs.
Error
Abs %
Error
1.28
1.28
1.28
1.27
1.29
1.28
1.26
1.31
1.30
1.29
1.31
1.30
1.30
1.27
1.32
1.31
1.33
1.30
1.38
1.38
1.38
1.37
1.38
1.39
1.40
1.40
1.42
Max.
Min.
Avg.
0.00
0.00
0.01
0.02
0.00
0.01
0.03
-0.02
0.00
0.01
-0.01
0.00
0.01
0.04
0.00
0.01
0.00
0.03
-0.01
0.00
0.00
0.01
0.00
0.00
0.00
0.01
0.00
0.04
-0.02
0.00
0.24
-0.31
0.51
1.44
0.29
0.83
1.98
-1.54
0.28
0.78
-0.49
-0.16
0.46
3.29
0.08
0.98
-0.21
1.95
-0.80
0.14
-0.18
0.40
-0.01
-0.10
-0.04
0.79
-0.02
3.29
-1.98
0.01
0.00
0.00
0.01
0.02
0.00
0.01
0.03
0.02
0.00
0.01
0.01
0.00
0.01
0.04
0.00
0.01
0.00
0.03
0.01
0.00
0.00
0.01
0.00
0.00
0.00
0.01
0.00
0.04
0.00
0.01
0.24
0.31
0.51
1.44
0.29
0.83
1.98
1.54
0.28
0.78
0.49
0.16
0.46
3.29
0.08
0.98
0.21
1.95
0.80
0.14
0.18
0.40
0.01
0.10
0.04
0.79
0.02
3.29
0.00
0.62
54
Table 15: Outputs of 3 x 5 Gauss2mf (Table continues)
Specific Gravity
Data
Number
Measured
Estimated
Error
% Error
Abs.
Error
Abs %
Error
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
1.22
1.23
1.23
1.23
1.24
1.24
1.24
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.26
1.26
1.26
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.22
1.22
1.22
1.22
1.22
1.22
1.25
1.23
1.23
1.25
1.23
1.27
1.24
1.23
1.24
1.25
1.25
1.26
1.25
1.26
1.30
1.30
1.24
1.23
1.27
1.27
1.30
1.28
1.29
1.28
1.28
1.28
1.28
1.31
1.31
0.00
0.01
0.01
0.01
0.02
0.02
-0.01
0.02
0.02
0.00
0.02
-0.02
0.01
0.02
0.01
0.01
0.01
0.00
0.02
0.01
-0.03
-0.03
0.03
0.04
0.00
0.00
-0.02
0.00
-0.01
0.00
0.00
0.00
0.00
-0.03
-0.03
-0.07
0.76
0.75
0.81
1.60
1.29
-0.50
1.60
1.60
-0.17
1.96
-1.51
0.66
1.66
1.10
0.79
0.79
0.00
1.20
0.81
-2.17
-2.09
2.07
3.17
0.06
0.00
-1.76
0.00
-0.84
0.00
-0.25
0.13
0.02
-2.29
-2.03
0.00
0.01
0.01
0.01
0.02
0.02
0.01
0.02
0.02
0.00
0.02
0.02
0.01
0.02
0.01
0.01
0.01
0.00
0.02
0.01
0.03
0.03
0.03
0.04
0.00
0.00
0.02
0.00
0.01
0.00
0.00
0.00
0.00
0.03
0.03
0.07
0.76
0.75
0.81
1.60
1.29
0.50
1.60
1.60
0.17
1.96
1.51
0.66
1.66
1.10
0.79
0.79
0.00
1.20
0.81
2.17
2.09
2.07
3.17
0.06
0.00
1.76
0.00
0.84
0.00
0.25
0.13
0.02
2.29
2.03
55
Table 15: Outputs of 3 x 5 Gauss2mf (Continued)
Specific Gravity
Data
Number
Measured
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
1.28
1.28
1.28
1.29
1.29
1.29
1.29
1.29
1.29
1.30
1.30
1.30
1.30
1.31
1.31
1.32
1.32
1.33
1.33
1.37
1.38
1.38
1.38
1.38
1.39
1.40
1.41
1.42
Estimated
Error
% Error
Abs.
Error
Abs %
Error
1.29
1.29
1.28
1.30
1.29
1.29
1.31
1.29
1.31
1.30
1.30
1.32
1.30
1.32
1.33
1.34
1.32
1.33
1.31
1.30
1.36
1.39
1.38
1.38
1.39
1.41
1.41
1.42
Max.
Min.
Avg.
-0.01
-0.01
0.00
-0.01
0.00
0.00
-0.02
0.00
-0.02
0.00
0.00
-0.02
0.00
-0.01
-0.02
-0.02
0.00
0.00
0.02
0.07
0.02
-0.01
0.00
0.00
0.00
-0.01
0.00
0.00
0.07
-0.03
0.00
-0.52
-0.77
0.13
-0.64
0.13
-0.01
-1.23
-0.31
-1.85
0.00
0.20
-1.54
0.28
-0.75
-1.56
-1.64
0.20
0.10
1.47
4.93
1.49
-0.76
0.12
0.00
-0.04
-0.40
-0.01
0.20
4.93
-2.29
0.14
0.01
0.01
0.00
0.01
0.00
0.00
0.02
0.00
0.02
0.00
0.00
0.02
0.00
0.01
0.02
0.02
0.00
0.00
0.02
0.07
0.02
0.01
0.00
0.00
0.00
0.01
0.00
0.00
0.07
0.00
0.01
0.52
0.77
0.13
0.64
0.13
0.01
1.23
0.31
1.85
0.00
0.20
1.54
0.28
0.75
1.56
1.64
0.20
0.10
1.47
4.93
1.49
0.76
0.12
0.00
0.04
0.40
0.01
0.20
4.93
0.00
0.96
56
APPENDIX G
Pi-Shaped Membership Functions (Pimf)
Description: The spline-based curved is so named because of its  shape. This
membership function is evaluated at the points determined by the vector, x. The
parameters a and b locate the “feet” of the curve while b and c locate at the “shoulders.”
57
Table 16: Outputs of 3 x 4 Pimf (Table continues)
Specific Gravity
Data
Number
Measured
Estimated
Error
% Error
Abs. Error
Abs %
Error
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
1.22
1.23
1.23
1.23
1.24
1.24
1.24
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.26
1.26
1.26
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.25
1.23
1.23
1.23
1.26
1.24
1.25
1.24
1.27
1.27
1.25
1.25
1.26
1.27
1.25
1.26
1.27
1.26
1.27
1.29
1.27
1.26
1.27
1.26
1.28
1.26
1.28
1.28
1.29
1.29
1.28
1.28
1.28
1.27
1.35
1.29
1.30
0.03
0.00
0.00
0.00
0.02
0.00
0.01
-0.01
0.02
0.02
0.00
0.00
0.01
0.02
0.00
0.00
0.01
0.00
0.00
0.02
0.00
-0.01
0.00
-0.01
0.01
-0.01
0.00
0.00
0.01
0.01
0.00
0.00
0.00
-0.01
0.07
0.01
0.02
2.50
-0.01
-0.01
0.04
1.41
-0.02
1.19
-1.03
1.22
1.52
0.00
0.00
1.10
1.40
0.38
0.00
0.54
0.00
0.00
1.63
0.00
-1.04
0.13
-0.75
0.71
-0.93
0.32
0.36
0.40
0.84
-0.11
-0.11
0.00
-0.88
5.21
0.40
1.61
0.03
0.00
0.00
0.00
0.02
0.00
0.01
0.01
0.02
0.02
0.00
0.00
0.01
0.02
0.00
0.00
0.01
0.00
0.00
0.02
0.00
0.01
0.00
0.01
0.01
0.01
0.00
0.00
0.01
0.01
0.00
0.00
0.00
0.01
0.07
0.01
0.02
2.50
0.01
0.01
0.04
1.41
0.02
1.19
1.03
1.22
1.52
0.00
0.00
1.10
1.40
0.38
0.00
0.54
0.00
0.00
1.63
0.00
1.04
0.13
0.75
0.71
0.93
0.32
0.36
0.40
0.84
0.11
0.11
0.00
0.88
5.21
0.40
1.61
58
Table 16: Outputs of 3 x 4 Pimf (Continued)
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
1.28
1.29
1.29
1.29
1.29
1.29
1.29
1.30
1.30
1.30
1.30
1.31
1.31
1.32
1.32
1.33
1.33
1.37
1.38
1.38
1.38
1.38
1.39
1.40
1.41
1.42
1.30
1.29
1.27
1.30
1.29
1.27
1.30
1.30
1.27
1.30
1.30
1.30
1.26
1.30
1.30
1.38
1.30
1.37
1.38
1.37
1.40
1.34
1.37
1.39
1.38
1.38
Max.
Min.
Avg.
0.02
0.00
-0.02
0.01
0.00
-0.02
0.01
0.00
-0.03
0.00
0.00
-0.01
-0.05
-0.02
-0.02
0.05
-0.03
0.00
0.00
-0.01
0.02
-0.04
-0.02
-0.01
-0.03
-0.04
0.07
-0.05
0.00
1.62
0.35
-1.49
0.55
0.05
-1.64
0.82
-0.36
-1.98
0.05
0.05
-0.72
-3.77
-1.55
-1.48
3.96
-2.21
0.30
0.20
-0.73
1.40
-3.25
-1.37
-0.76
-2.09
-2.63
5.21
-3.77
0.04
0.02
0.00
0.02
0.01
0.00
0.02
0.01
0.00
0.03
0.00
0.00
0.01
0.05
0.02
0.02
0.05
0.03
0.00
0.00
0.01
0.02
0.04
0.02
0.01
0.03
0.04
0.07
0.00
0.01
1.62
0.35
1.49
0.55
0.05
1.64
0.82
0.36
1.98
0.05
0.05
0.72
3.77
1.55
1.48
3.96
2.21
0.30
0.20
0.73
1.40
3.25
1.37
0.76
2.09
2.63
5.21
0.00
1.05
59
Table 17: Outputs of 3 x 5 Pimf (Table continues)
Specific Gravity
Data
Number
Measured
Estimated
Error
% Error
Abs.
Error
Abs %
Error
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
1.22
1.23
1.23
1.23
1.24
1.24
1.24
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.26
1.26
1.26
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.22
1.22
1.22
1.22
1.22
1.22
1.25
1.23
1.23
1.25
1.23
1.27
1.25
1.23
1.23
1.25
1.25
1.26
1.25
1.25
1.30
1.30
1.25
1.23
1.27
1.27
1.30
1.28
1.29
1.28
1.31
1.31
1.28
1.31
1.31
1.29
0.00
-0.01
-0.01
-0.01
-0.02
-0.02
0.01
-0.02
-0.02
0.00
-0.02
0.02
0.00
-0.02
-0.02
-0.01
-0.01
0.00
-0.02
-0.02
0.03
0.03
-0.02
-0.04
0.00
0.00
0.02
0.00
0.01
0.00
0.03
0.03
0.00
0.03
0.03
0.01
0.16
-0.64
-0.65
-0.81
-1.59
-1.54
0.44
-1.60
-1.58
0.25
-1.79
1.53
-0.37
-1.53
-1.80
-0.79
-0.79
0.00
-1.47
-1.38
2.06
2.02
-1.38
-3.02
-0.21
0.00
1.70
0.00
0.73
0.00
2.07
2.16
0.00
2.13
2.02
0.45
0.00
0.01
0.01
0.01
0.02
0.02
0.01
0.02
0.02
0.00
0.02
0.02
0.00
0.02
0.02
0.01
0.01
0.00
0.02
0.02
0.03
0.03
0.02
0.04
0.00
0.00
0.02
0.00
0.01
0.00
0.03
0.03
0.00
0.03
0.03
0.01
0.16
0.64
0.65
0.81
1.59
1.54
0.44
1.60
1.58
0.25
1.79
1.53
0.37
1.53
1.80
0.79
0.79
0.00
1.47
1.38
2.06
2.02
1.38
3.02
0.21
0.00
1.70
0.00
0.73
0.00
2.07
2.16
0.00
2.13
2.02
0.45
60
Table 17: Outputs of 3 x 5 Pimf (Continued)
Specific Gravity
Data
Number
Measured
Estimated
Error
% Error
Abs.
Error
Abs %
Error
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
1.28
1.28
1.29
1.29
1.29
1.29
1.29
1.29
1.30
1.30
1.30
1.30
1.31
1.31
1.32
1.32
1.33
1.33
1.37
1.38
1.38
1.38
1.38
1.39
1.40
1.41
1.42
1.29
1.28
1.30
1.29
1.29
1.31
1.29
1.31
1.31
1.30
1.31
1.30
1.32
1.33
1.34
1.31
1.31
1.31
1.30
1.39
1.39
1.38
1.38
1.39
1.39
1.41
1.42
0.01
0.00
0.01
0.00
0.00
0.02
0.00
0.02
0.01
0.00
0.01
0.00
0.01
0.02
0.02
-0.01
-0.02
-0.02
-0.07
0.01
0.01
0.00
0.00
0.00
-0.01
0.00
0.00
0.03
-0.07
0.00
0.78
-0.16
0.68
0.29
0.09
1.29
0.29
1.35
0.62
-0.27
0.60
-0.36
0.64
1.50
1.33
-1.02
-1.76
-1.67
-4.85
0.63
0.36
-0.26
0.00
-0.36
-0.81
-0.01
0.11
2.16
-4.85
-0.14
0.01
0.00
0.01
0.00
0.00
0.02
0.00
0.02
0.01
0.00
0.01
0.00
0.01
0.02
0.02
0.01
0.02
0.02
0.07
0.01
0.01
0.00
0.00
0.00
0.01
0.00
0.00
0.07
0.00
0.01
0.78
0.16
0.68
0.29
0.09
1.29
0.29
1.35
0.62
0.27
0.60
0.36
0.64
1.50
1.33
1.02
1.76
1.67
4.85
0.63
0.36
0.26
0.00
0.36
0.81
0.01
0.11
4.85
0.00
1.04
Max.
Min.
Avg.
61
APPENDIX H
Difference of Sigmoid Membership Functions (Dsigmf)
Description: The sigmoidal membership function that used for this function depends
upon the two parameters a and c and is given by:
f (x;a,c) 
1
1 e
a(x c )
 slope of sigmoid membership function depends on four parameters, a1, c1,
Where a is the
a2, and c2, and is the difference between two of these sigmoidal functions:
f1(x; a1, c1) – f2 (x; a2, c2)
62
Table 18: Outputs of 3x4 Dsigmf (Table continues)
Specific Gravity
Data
Number
Measured
Estimated
Error
% Error
Abs.
Error
Abs %
Error
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
1.22
1.23
1.23
1.23
1.24
1.24
1.24
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.26
1.26
1.26
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.24
1.23
1.23
1.23
1.25
1.24
1.24
1.24
1.27
1.26
1.25
1.25
1.26
1.26
1.27
1.26
1.26
1.26
1.27
1.29
1.27
1.26
1.26
1.27
1.28
1.26
1.29
1.28
1.28
1.29
1.27
1.28
1.28
1.29
1.28
1.28
1.28
0.02
0.00
0.00
0.00
0.01
0.00
0.00
-0.01
0.02
0.01
0.00
0.00
0.01
0.01
0.02
0.00
0.00
0.00
0.00
0.02
0.00
-0.01
-0.01
0.00
0.01
-0.01
0.01
0.00
0.00
0.01
-0.01
0.00
0.00
0.01
0.00
0.00
0.00
2.00
0.03
-0.02
-0.02
0.51
0.01
-0.37
-0.82
1.28
0.89
0.00
0.29
1.13
0.66
1.33
0.01
0.39
-0.01
0.00
1.47
-0.02
-0.54
-0.65
-0.13
0.96
-0.61
0.43
0.28
0.21
0.94
-0.44
-0.32
0.00
1.13
-0.02
0.09
-0.15
0.02
0.00
0.00
0.00
0.01
0.00
0.00
0.01
0.02
0.01
0.00
0.00
0.01
0.01
0.02
0.00
0.00
0.00
0.00
0.02
0.00
0.01
0.01
0.00
0.01
0.01
0.01
0.00
0.00
0.01
0.01
0.00
0.00
0.01
0.00
0.00
0.00
2.00
0.03
0.02
0.02
0.51
0.01
0.37
0.82
1.28
0.89
0.00
0.29
1.13
0.66
1.33
0.01
0.39
0.01
0.00
1.47
0.02
0.54
0.65
0.13
0.96
0.61
0.43
0.28
0.21
0.94
0.44
0.32
0.00
1.13
0.02
0.09
0.15
63
Table 18: Outputs of 3x4 Dsigmf (Contined)
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
1.28
1.29
1.29
1.29
1.29
1.29
1.29
1.30
1.30
1.30
1.30
1.31
1.31
1.32
1.32
1.33
1.33
1.37
1.38
1.38
1.38
1.38
1.39
1.40
1.41
1.42
1.29
1.29
1.28
1.29
1.28
1.26
1.31
1.30
1.29
1.31
1.30
1.30
1.27
1.32
1.31
1.33
1.30
1.38
1.38
1.38
1.38
1.38
1.39
1.41
1.40
1.42
Max.
Min.
Avg.
0.01
0.00
-0.01
0.00
-0.01
-0.03
0.02
0.00
-0.01
0.01
0.00
-0.01
-0.04
0.00
-0.01
0.00
-0.03
0.01
0.00
0.00
0.00
0.00
0.00
0.01
-0.01
0.00
0.02
-0.04
0.00
0.52
-0.30
-0.79
0.13
-0.81
-2.09
1.61
-0.34
-0.78
0.46
0.15
-0.55
-3.32
-0.02
-1.05
-0.06
-2.05
0.48
0.02
0.15
-0.33
-0.02
-0.32
0.37
-0.51
-0.01
2.00
-3.32
-0.01
0.01
0.00
0.01
0.00
0.01
0.03
0.02
0.00
0.01
0.01
0.00
0.01
0.04
0.00
0.01
0.00
0.03
0.01
0.00
0.00
0.00
0.00
0.00
0.01
0.01
0.00
0.04
0.00
0.01
0.52
0.30
0.79
0.13
0.81
2.09
1.61
0.34
0.78
0.46
0.15
0.55
3.32
0.02
1.05
0.06
2.05
0.48
0.02
0.15
0.33
0.02
0.32
0.37
0.51
0.01
3.32
0.00
0.60
64
Table 19: Outputs of 3x5 Dsigmf (Table continues)
Specific Gravity
Data
Number
Measured
Estimated
Error
%
Error
Abs.
Error
Abs %
Error
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
1.22
1.23
1.23
1.23
1.24
1.24
1.24
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.26
1.26
1.26
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.21
1.22
1.21
1.22
1.22
1.22
1.25
1.23
1.23
1.23
1.23
1.27
1.25
1.23
1.24
1.25
1.25
1.26
1.25
1.26
1.30
1.29
1.27
1.23
1.27
1.27
1.30
1.28
1.29
1.28
1.28
1.28
1.28
1.31
1.31
-0.01
-0.01
-0.02
-0.01
-0.02
-0.02
0.01
-0.02
-0.02
-0.02
-0.02
0.02
0.00
-0.02
-0.01
-0.01
-0.01
0.00
-0.02
-0.01
0.02
0.02
0.00
-0.04
0.00
0.00
0.02
0.00
0.01
0.00
0.00
0.00
0.00
0.03
0.03
-0.45
-1.11
-1.26
-0.81
-1.61
-1.44
0.90
-1.60
-1.61
-1.61
-1.74
1.47
-0.27
-1.51
-1.17
-0.79
-0.79
0.00
-1.47
-0.79
1.97
1.28
-0.13
-2.93
0.31
0.00
1.67
0.00
0.74
0.00
0.13
-0.22
0.01
2.44
2.02
0.01
0.01
0.02
0.01
0.02
0.02
0.01
0.02
0.02
0.02
0.02
0.02
0.00
0.02
0.01
0.01
0.01
0.00
0.02
0.01
0.02
0.02
0.00
0.04
0.00
0.00
0.02
0.00
0.01
0.00
0.00
0.00
0.00
0.03
0.03
0.45
1.11
1.26
0.81
1.61
1.44
0.90
1.60
1.61
1.61
1.74
1.47
0.27
1.51
1.17
0.79
0.79
0.00
1.47
0.79
1.97
1.28
0.13
2.93
0.31
0.00
1.67
0.00
0.74
0.00
0.13
0.22
0.01
2.44
2.02
65
Table 19: Outputs of 3x5 Dsigmf (Continued)
Specific Gravity
Data
Number
Measured
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
1.28
1.28
1.28
1.29
1.29
1.29
1.29
1.29
1.29
1.30
1.30
1.30
1.30
1.31
1.31
1.32
1.32
1.33
1.33
1.37
1.38
1.38
1.38
1.38
1.39
1.40
1.41
1.42
Estimated
Error
%
Error
Abs.
Error
Abs %
Error
1.28
1.29
1.28
1.30
1.29
1.29
1.30
1.30
1.32
1.30
1.30
1.32
1.30
1.32
1.33
1.34
1.33
1.33
1.31
1.30
1.36
1.39
1.38
1.38
1.39
1.41
1.41
1.42
Max.
Min.
Avg.
0.00
0.01
0.00
0.01
0.00
0.00
0.01
0.01
0.03
0.00
0.00
0.02
0.00
0.01
0.02
0.02
0.01
0.00
-0.02
-0.07
-0.02
0.01
0.00
0.00
0.00
0.01
0.00
0.00
0.03
-0.07
0.00
0.34
0.79
-0.11
0.42
0.02
0.03
0.93
0.42
1.99
-0.04
-0.12
1.61
-0.23
0.74
1.53
1.81
0.55
0.30
-1.17
-5.01
-1.63
0.53
-0.11
0.00
-0.19
0.67
-0.06
-0.05
2.44
-5.01
-0.14
0.00
0.01
0.00
0.01
0.00
0.00
0.01
0.01
0.03
0.00
0.00
0.02
0.00
0.01
0.02
0.02
0.01
0.00
0.02
0.07
0.02
0.01
0.00
0.00
0.00
0.01
0.00
0.00
0.07
0.00
0.01
0.34
0.79
0.11
0.42
0.02
0.03
0.93
0.42
1.99
0.04
0.12
1.61
0.23
0.74
1.53
1.81
0.55
0.30
1.17
5.01
1.63
0.53
0.11
0.00
0.19
0.67
0.06
0.05
5.01
0.00
0.96
66
APPENDIX I
Product of Sigmoid Membership Functions (Psigmf)
Description: The sigmoid membership functions that are used depend upon two
parameters, a and c and is given by:
f (x;a,c) 
1
1 e a(x c )
Where a is 
the slope of the sigmoid function and c is the node offset value.
The product of sigmoid membership function is simply the product of two such curves
plotted for the values of the vector x:
f1(x;a1,c1)  f 2 (x;a2,c2 )

67
Table 20: Outputs of 3x4 Psigmf (Table continues)
Specific Gravity
Data
Number
Measured
Estimated
Error
% Error
Abs.
Error
Abs % Error
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
1.22
1.23
1.23
1.23
1.24
1.24
1.24
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.26
1.26
1.26
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.24
1.23
1.23
1.23
1.25
1.24
1.24
1.24
1.27
1.26
1.25
1.25
1.26
1.26
1.27
1.26
1.26
1.26
1.27
1.29
1.27
1.26
1.26
1.27
1.28
1.26
1.29
1.28
1.28
1.29
1.27
1.28
1.28
1.29
1.28
1.28
1.28
0.02
0.00
0.00
0.00
0.01
0.00
0.00
-0.01
0.02
0.01
0.00
0.00
0.01
0.01
0.02
0.00
0.00
0.00
0.00
0.02
0.00
-0.01
-0.01
0.00
0.01
-0.01
0.01
0.00
0.00
0.01
-0.01
0.00
0.00
0.01
0.00
0.00
0.00
2.00
0.03
-0.02
-0.02
0.51
0.01
-0.37
-0.82
1.28
0.89
0.00
0.29
1.13
0.66
1.33
0.01
0.39
-0.01
0.00
1.47
-0.02
-0.54
-0.65
-0.13
0.96
-0.61
0.43
0.28
0.21
0.94
-0.44
-0.32
0.00
1.13
-0.02
0.09
-0.15
0.02
0.00
0.00
0.00
0.01
0.00
0.00
0.01
0.02
0.01
0.00
0.00
0.01
0.01
0.02
0.00
0.00
0.00
0.00
0.02
0.00
0.01
0.01
0.00
0.01
0.01
0.01
0.00
0.00
0.01
0.01
0.00
0.00
0.01
0.00
0.00
0.00
2.00
0.03
0.02
0.02
0.51
0.01
0.37
0.82
1.28
0.89
0.00
0.29
1.13
0.66
1.33
0.01
0.39
0.01
0.00
1.47
0.02
0.54
0.65
0.13
0.96
0.61
0.43
0.28
0.21
0.94
0.44
0.32
0.00
1.13
0.02
0.09
0.15
68
Table 20: Outputs of 3x4 Psigmf (Continued)
Specific Gravity
Data
Number
Measured
Estimated
Error
% Error
Abs.
Error
Abs % Error
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
1.28
1.29
1.29
1.29
1.29
1.29
1.29
1.30
1.30
1.30
1.30
1.31
1.31
1.32
1.32
1.33
1.33
1.37
1.38
1.38
1.38
1.38
1.39
1.40
1.41
1.42
1.29
1.29
1.28
1.29
1.28
1.26
1.31
1.30
1.29
1.31
1.30
1.30
1.27
1.32
1.31
1.33
1.30
1.38
1.38
1.38
1.38
1.38
1.39
1.41
1.40
1.42
Max.
Min.
Avg.
0.01
0.00
-0.01
0.00
-0.01
-0.03
0.02
0.00
-0.01
0.01
0.00
-0.01
-0.04
0.00
-0.01
0.00
-0.03
0.01
0.00
0.00
0.00
0.00
0.00
0.01
-0.01
0.00
0.02
-0.04
0.00
0.52
-0.30
-0.79
0.13
-0.81
-2.09
1.61
-0.34
-0.78
0.46
0.15
-0.55
-3.32
-0.02
-1.05
-0.06
-2.05
0.48
0.02
0.15
-0.33
-0.02
-0.32
0.37
-0.51
-0.01
2.00
-3.32
-0.01
0.01
0.00
0.01
0.00
0.01
0.03
0.02
0.00
0.01
0.01
0.00
0.01
0.04
0.00
0.01
0.00
0.03
0.01
0.00
0.00
0.00
0.00
0.00
0.01
0.01
0.00
0.04
0.00
0.01
0.52
0.30
0.79
0.13
0.81
2.09
1.61
0.34
0.78
0.46
0.15
0.55
3.32
0.02
1.05
0.06
2.05
0.48
0.02
0.15
0.33
0.02
0.32
0.37
0.51
0.01
3.32
0.00
0.60
69
Table 21: Outputs of 3 x 5 Psigmf (Table continues)
Specific Gravity
Data
Number
Measured
Estimated
Error
% Error
Abs.
Error
Abs %
Error
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
1.22
1.23
1.23
1.23
1.24
1.24
1.24
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.26
1.26
1.26
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.27
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.28
1.21
1.22
1.21
1.22
1.22
1.22
1.25
1.23
1.23
1.23
1.23
1.27
1.25
1.23
1.24
1.25
1.25
1.26
1.25
1.26
1.30
1.29
1.27
1.23
1.27
1.27
1.30
1.28
1.29
1.28
1.28
1.28
1.28
1.31
1.31
-0.01
-0.01
-0.02
-0.01
-0.02
-0.02
0.01
-0.02
-0.02
-0.02
-0.02
0.02
0.00
-0.02
-0.01
-0.01
-0.01
0.00
-0.02
-0.01
0.02
0.02
0.00
-0.04
0.00
0.00
0.02
0.00
0.01
0.00
0.00
0.00
0.00
0.03
0.03
-0.45
-1.11
-1.26
-0.81
-1.61
-1.44
0.90
-1.60
-1.61
-1.61
-1.74
1.47
-0.27
-1.51
-1.17
-0.79
-0.79
0.00
-1.47
-0.79
1.97
1.28
-0.13
-2.93
0.31
0.00
1.67
0.00
0.74
0.00
0.13
-0.22
0.01
2.44
2.02
0.01
0.01
0.02
0.01
0.02
0.02
0.01
0.02
0.02
0.02
0.02
0.02
0.00
0.02
0.01
0.01
0.01
0.00
0.02
0.01
0.02
0.02
0.00
0.04
0.00
0.00
0.02
0.00
0.01
0.00
0.00
0.00
0.00
0.03
0.03
0.45
1.11
1.26
0.81
1.61
1.44
0.90
1.60
1.61
1.61
1.74
1.47
0.27
1.51
1.17
0.79
0.79
0.00
1.47
0.79
1.97
1.28
0.13
2.93
0.31
0.00
1.67
0.00
0.74
0.00
0.13
0.22
0.01
2.44
2.02
70
Table 21: Outputs of 3 x 5 Psigmf (Continued)
Specific Gravity
Data
Number
Measured
Estimated
Error
% Error
Abs.
Error
Abs %
Error
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
1.28
1.28
1.28
1.29
1.29
1.29
1.29
1.29
1.29
1.30
1.30
1.30
1.30
1.31
1.31
1.32
1.32
1.33
1.33
1.37
1.38
1.38
1.38
1.38
1.39
1.40
1.41
1.42
1.28
1.29
1.28
1.30
1.29
1.29
1.30
1.30
1.32
1.30
1.30
1.32
1.30
1.32
1.33
1.34
1.33
1.33
1.31
1.30
1.36
1.39
1.38
1.38
1.39
1.41
1.41
1.42
Max.
Min.
Avg.
0.00
0.01
0.00
0.01
0.00
0.00
0.01
0.01
0.03
0.00
0.00
0.02
0.00
0.01
0.02
0.02
0.01
0.00
-0.02
-0.07
-0.02
0.01
0.00
0.00
0.00
0.01
0.00
0.00
0.03
-0.07
0.00
0.34
0.79
-0.11
0.42
0.02
0.03
0.93
0.42
1.99
-0.04
-0.12
1.61
-0.23
0.74
1.53
1.81
0.55
0.30
-1.17
-5.01
-1.63
0.53
-0.11
0.00
-0.19
0.67
-0.06
-0.05
2.44
-5.01
-0.14
0.00
0.01
0.00
0.01
0.00
0.00
0.01
0.01
0.03
0.00
0.00
0.02
0.00
0.01
0.02
0.02
0.01
0.00
0.02
0.07
0.02
0.01
0.00
0.00
0.00
0.01
0.00
0.00
0.07
0.00
0.01
0.34
0.79
0.11
0.42
0.02
0.03
0.93
0.42
1.99
0.04
0.12
1.61
0.23
0.74
1.53
1.81
0.55
0.30
1.17
5.01
1.63
0.53
0.11
0.00
0.19
0.67
0.06
0.05
5.01
0.00
0.96
71
APPENDIX J
Counterpropagation Neural Network Weights
72
Table 22: 3x12x1 CPN Weights Hidden Kohonen Layer Weight Vector
Node
Layer
Weight
Vector
Layer
Weight
Vector
Layer
Weight
Vector
1
1
0.277
2
0.125
3
0.563
2
1
0.287
2
0.188
3
0.519
3
1
0.247
2
0.181
3
0.615
4
1
0.556
2
0.274
3
0.110
5
1
0.415
2
0.522
3
0.556
6
1
0.207
2
0.157
3
0.263
7
1
0.361
2
0.333
3
0.488
8
1
0.725
2
0.747
3
0.281
9
1
0.365
2
0.359
3
0.669
10
1
0.326
2
0.223
3
0.566
11
1
0.372
2
0.230
3
0.416
12
1
0.267
2
0.136
3
0.395
73
Table 23: Grossberg Outputs Layer Weight Vector
Node
Weight
1
0.248
2
0.376
3
0.340
4
0.268
5
0.779
6
0.000
7
0.485
8
0.391
9
0.462
10
0.426
11
0.434
12
0.286
74
APPENDIX K
Properties of Gases
Table 24: Properties of Gases
Properties
N-Butane
Propane
Natural Gas
Vapor pressure at
100 F-psig
36.9
175.8
…
Boiling point at
14.7 psia –F
31.1
-43.7
-258.7
Heat of
vaporization
BTU/GAL at B.P.
797
774
712
Higher heating
value –BTU/CF at
60 F
3368
2558
1012
Specific gravity at
60 F - /air
2.01
1.52
0.6
75
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