Creating Fuzzy Models from Limited Data
SAŠO BLAŽIČ
Faculty of Electrical Engineering,
University of Ljubljana,
Tržaška 25, Ljubljana,
SLOVENIA
Abstract: - The design of experiments is a methodological approach in which measurement experiments are
carefully planned to obtain highly informative data. This paper addresses the challenge of constructing
mathematical models for complex nonlinear processes when the available measurement data have low
information content. This problem often arises when data are collected without the guidance of an experimental
modeling expert. We examine two practical examples to illustrate this issue: a textile wastewater decolorization
process and atmospheric corrosion of structural metal materials. In both cases, the measured data were
insufficient to construct highly accurate models. It is, therefore, necessary to make a trade-off between model
complexity and accuracy by adapting modeling techniques to work effectively with the limited data available.
The main aim of the paper is, therefore, to focus on simple but effective techniques that allow as much
information as possible to be extracted from low-quality measurements and to maximize the usefulness of the
model for its intended purpose.
Key-Words: - design of experiments, low information content, decolorization process, atmospheric corrosion,
fuzzy model, radial basis network.
Received: May 24, 2023. Revised: May 21, 2024. Accepted: June 22, 2024. Published: July 24, 2024.
1 Introduction
This paper addresses a crucial issue in the field of
modeling from measurement data. High-quality,
informative data are essential for the construction of
accurate mathematical models. Achieving this level
of data quality requires careful experimental design.
The challenge is even greater when working with
nonlinear models, such as fuzzy models or artificial
neural networks. These types of models involve a
large number of parameters that need to be
estimated or trained, making the need for highly
informative data even more critical. Without
sufficient and well-structured data, the reliability
and accuracy of the resulting models can be
significantly compromised.
Design of Experiments (DOE) is a systematic
approach to the challenge of obtaining high-quality
data for modeling. The process begins by
identifying the key influencing (input) variables and
the corresponding consequence (output) variables.
Once these variables are identified, the next step is
to define the range of their variation. DOE then
involves strategically spreading the input variables
across the entire experimental space to ensure that
the model covers all potential scenarios it will
encounter in real-world applications.
By applying DOE, researchers can minimize the
number of experiments required while still capturing
the essential properties of the system. This efficient
approach not only reduces cost and time but also
increases the reliability of the model over a wide
range of input variable variations. Effective DOE
ensures that the experimental design is
comprehensive and robust, resulting in models that
are both accurate and generalizable.
Design of experiments is a well-established
methodology, dating back to the seminal work of
[1]. Since then, the benefits of DOE have been well
documented in the literature. For example, [2]
highlights that DOE provides a powerful tool for
maximizing information extraction from
experimental data while minimizing resource
expenditure. Similarly, [3] emphasizes that carefully
designed experiments can lead to significant
improvements in the accuracy and efficiency of
model building. These principles are particularly
important when dealing with complex, nonlinear
models such as fuzzy models and artificial neural
networks, where the large number of parameters
requires carefully structured data to ensure
successful training and estimation.
Design of experiments can be applied in various
fields, including computer-aided circuit design [4],
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production system design [5], evaluation of
experiments [6], fuzzy modeling [7], [8], control
design [9], tableting process optimization [10],
servo system control design [11], to name a few.
However, DOE becomes significantly more
challenging when developing dynamic models of
processes, as the frequency content of the excitation
must be carefully considered to ensure accurate
results.
Unfortunately, it is often necessary to construct
a mathematical model of a process using data
collected without the supervision of an experimental
modeling expert. This typically results in data that is
less informative, making the modeling process
significantly more challenging. In such cases, it is
crucial to reduce expectations about the level of
precision and accuracy that the model can achieve.
In practice, it is important for modeling to focus on
the dominant dynamics in such cases. This can be
achieved by selecting simple but robust modeling
algorithms and parameterizing them appropriately.
An algorithm with a relatively small number of
tunable and design parameters can often cope well
with high levels of uncertainty and noise in the data.
When expectations for the model are adjusted and
suitable methods are chosen, it is possible to
develop a model that captures the properties of the
system despite the inherent limitations of the data.
This paper presents two practical problems.
Both cases are similar in that they involve complex
nonlinear chemical processes, the measurement
databases are small and include significant
uncertainty, and new data could no longer be
collected due to practical reasons. The objective is
to build a nonlinear model that extracts the available
information despite the measurement database being
much smaller than typically expected for problems
of this complexity.
This paper is organized as follows: Sections 2
and 3 deal with the decolorization process and the
atmospheric corrosion process, while Sections 4 and
5 provide the discussion and conclusions.
2 Decolorization Process
This section examines the process of decolorization
of textile wastewater from industrial dyeing. There
are many techniques for decolorization, [12]. In our
case, we measure the absorbance of the wastewater
(A), which indicates its “dirtiness” – zero means that
the water is completely transparent, while higher
values indicate increased opacity. The
decolorization process involves adding hydrogen
peroxide (H2O2) and exposing the wastewater to
ultraviolet light (UV) for a certain duration.
We developed a model of this process that was
later used for control design. While a fuzzy model
approach to model and control this process was used
in [13], our work focuses on model development
based on a smaller database.
Before proceeding with control design, it is
essential to have a robust model of the process to be
controlled. This is particularly important when using
model-based control methods. The main challenge
in this scenario is the quality of the data from which
the model is derived. It is well known that various
model forms (e.g. fuzzy models, artificial neural
networks, spline models, piecewise linear models,
etc.) can describe nonlinear processes with arbitrary
small modeling errors. However, none of these
models can reliably extrapolate information to parts
of space where little or no measured data is
available.
2.1 Model Structure
When deciding on the model structure, the first
consideration is the choice of model inputs and
outputs. A primary concern is the determination of
the model output, which presents at least two
possible choices in this particular example:
The decolorization factor D. It is given by the
formula
if
i
AA
DA
Where Ai represents the initial absorbance
(before the decolorization process), and Af
represents the final absorbance (after the
decolorization process). Therefore, this means
that the process was completely unsuccessful,
while
1D
indicating perfect decolorization.
The final absorbance Af.
As the model is intended for control design
purposes, the focus of the decision-making process
should be on the control aspect of the problem. The
objective of the control system is to achieve
effective decolorization of the effluent. It is
therefore essential to establish a specific reference
point. In our view, the definition of a target final
absorbance serves as an appropriate control
objective.
If this approach were not adopted, and the
decolorization factor was chosen as the system
output (and consequently the control reference), all
wastewater treatments would be treated equally
regardless of their initial “dirtiness”. This would
lead to almost identical control actions in different
scenarios, undermining the rationale for developing
a complex nonlinear model.
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In conclusion, the optimal solution for the
system output is the final absorbance, Af.
The choice of model inputs may seem
straightforward, but there are still some open
questions. Potential candidates for system inputs
include: the concentration of the peroxide H2O2
(Cp), the power of the UV lamp (PUV), and the time
of exposure to the UV lamp (T).
Since the final absorbance is likely to be non-
linearly dependent on the initial absorbance, the
initial absorbance must be included as one of the
system inputs. However, adding a fourth input
instead of three increases the amount of data
required to build the model. Therefore, to streamline
the process, we avoid a 4-dimensional input space.
Consequently, the model will take the following
form:
( , , )
p UV
D f C P T
Note that the decolorization factor is used as the
output, as we do not include the dependence on the
initial absorbance in our model. The final output Af
is then derived from equation (1):
(1 ) (1 ( , , ))
f i p UV i
A D A f C P T A
It is important to note the inclusion of the
exposure time T in the model described by equation
(2). Because the measurements are taken at
somewhat arbitrary intervals, the model is structured
to explicitly include time as a model input.
2.2 RBN Model of the Decolorization
Process
The Radial Basis Network (RBN) is used to
approximate the mapping described by (2). The
model for the process is based on measured data of
the dye Irgalan Gelb 3R KWL: PUV [W], Cp [mg/l],
T [min], and the absorbance A (measured only at a
few different times T). The data collected include
measurements from three replicate sets of 15
experiments each. The sample size is relatively
small due to the manual collection and analysis of
wastewater samples, which also contributes to
higher measurement errors. Obvious outliers were
manually removed from the data set. With three
inputs to the network and one output, the results are
plotted as three-dimensional planes: the x and y axes
correspond to two inputs, while the third input is
fixed, and the decolorization factor is plotted on the
z axis.
Fig. 1: RBN approximation for PUV = 1200 W
Fig. 2: RBN approximation for PUV = 1400 W
Fig. 3: RBN approximation for PUV = 1600 W
Figure 1, Figure 2 and Figure 3 show the
decolorization factor plotted against peroxide
concentration and time, with the UV lamp power
held constant at 1200, 1400, and 1600 W,
respectively. The measured data points are
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represented by circles, while the network outputs at
these inputs are represented by crosses. Figure 4,
Figure 5 and Figure 6 show similar plots for fixed
values of peroxide concentration. Despite the
obvious limitations of the data set (missing
important data points under certain operating
conditions and high variability in repeated
experiments), some qualitative characteristics of the
process are visible.
Fig. 4: RBN approximation for Cp = 0.7 mg/l
Fig. 5: RBN approximation for Cp = 4.5 mg/l
Fig. 6: RBN approximation for Cp = 8.3 mg/l
2.3 Control Design
In this case, the control problem is to determine the
actions that lead to purified water. Increasing these
actions – Cp, PUV, and T – results in better control
results, but on the other hand it also contributes to
higher control costs.
We propose optimal control as the solution to
the control problem. A crucial aspect of applying
optimal control is defining an appropriate cost
function that comprehensively captures all relevant
aspects of the problem. In our approach we use the
following cost function.
() pUV
d f p e
max max max
CPT
J k g A k k
C P T
Here, g(Af) represents the function defining the
cost associated with unsatisfactory final absorbance:
0
() f satis
ff satis f satis
AA
gA A A A A
If the final absorbance falls below a certain
threshold Asatis (the acceptable level), the first term
of the cost function, which penalizes unsatisfactory
decolorization, will be zero. Conversely, if the final
absorbance exceeds this threshold, the first term will
have a positive penalty. The second and third terms
in (4) represent the costs associated with the
consumption of H2O2 and energy, respectively,
where Cmax, Pmax, and Tmax are the maximum values
for Cp, PUV, and T, respectively.
Choosing the appropriate weights kd, kp, and ke
for the decolorization cost, peroxide cost, and
energy cost, respectively, is a delicate task.
However, this specific problem is beyond the scope
of this paper.
Substituting (3) into (4) gives the control cost
function:
( , , , ) ((1 ( , , )) ) pUV
max max max
CPT
p UV i d p UV i p e
C P T
J C P T A k g f C P T A k k
The cost function is based on four variables,
with the mapping function f approximated by the
RBN model. To achieve optimal control, we
minimize the cost function (6) in the space of the
three control variables (Cp, PUV, and T). As the
minimization is performed in a high-dimensional
space (Cp, PUV, T, and Ai), three optimal control
commands depend only on the initial absorbance,
which is known before the decolorization process.
By minimizing the cost function (6) off-line, the
control functions can be derived:
1
2
3
()
()
()
UV i
i
pi
P h A
T h A
C h A
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The control functions described can be readily
implemented on dedicated hardware, for example, in
tabular format. It is crucial to understand that the
shape of these control functions is significantly
influenced by the chosen design parameters kd, kp,
and ke. Experts in textile engineering have validated
the correct behavior of the control system.
3 Atmospheric Corrosion Process
In this section, we model the process of atmospheric
corrosion using a Takagi-Sugeno fuzzy model. The
resulting model can be used to predict atmospheric
corrosion if the near future under simulated climate
changes, such as increased atmospheric
acidification, lower SO2 concentrations in Europe,
higher O3 and NOx concentrations, and global
warming.
Atmospheric corrosion of structural metals is a
complex and nonlinear process influenced by
several meteorochemical and material factors. The
full extent of the effects of these factors on material
degradation is not yet fully understood. Fortunately,
long-term climate programs in Europe (e.g.
ECE/EMEP and ICP) and Asia/Africa (e.g.
RAPIDC) provide daily measurements of
meteorochemical variables such as temperature,
relative humidity, precipitation, and major pollutant
concentrations. In addition, annual corrosion mass
loss data are recorded for metals such as carbon
steel, copper, zinc, and aluminum. Corrosion
measurements for various metals are also available
from many sites around the world.
Previous research has addressed the modeling of
these phenomena, [14], [15]. However, the results
from existing large databases are not permitted for
publication. This paper uses a very small, publicly
available database, but still provides significant
insights. The analysis of this database also
highlights the major challenges typically
encountered in the study of atmospheric corrosion.
The main problem in this case is that the model
is based on a small database that includes 32
measurements of corrosion taken in different parts
of the world. Each set of measurements contains 7
variables, one of which is the corrosion mass. The
aim is to build a model that predicts the corrosion
mass based on the other 6 measurements, although
this is difficult given the sparsity of the data in a
high dimensional input space.
3.1 Linear Static Model
As mentioned above, this paper is based on a
relatively small atmospheric corrosion database
consisting of only 32 measurements. Each
measurement consists of exposition time t (in years),
temperature T (in 0C), relative humidity HR (in %),
SO2 concentration CSO2 (in µg.m-3), precipitation p
(in mm), pH pH, and corrosion mass C (in g.m-2).
We aim to develop a model of this system with
corrosion mass C as the output and the other six
variables as inputs. Given the limited number of
data points in this six-dimensional space and the
potentially poor quality of the measurements, we
started the modeling process by identifying a static
linear system:
1
2
6
ˆ, 1 32
i
T
i i i Ri SiO i H i
C t T H C p p i
ψθ
where the hat symbol represents the estimated
output of the linear system, and the tilde symbol
denotes a normalized variable, which is obtained by
subtracting the mean and dividing by the standard
deviation of that variable. Normalizing the variables
ensures that all variables have a mean of zero and a
variance of one, thereby equalizing their
contributions to the final model.
The solution in this case can be obtained by a
simple method of least squares:
0 8162 0 0564 0 0913 0 4277 0 0123 0 1084
T. . . . . . θ
It is important to note that the data set is very
small, making it impractical to split it into separate
identification and validation subsets. In addition, the
data spans a large region of the input space, making
extrapolation to other regions difficult. The key
point is that while the identification of a linear
model can be done with a smaller dataset, the
identification of a Takagi-Sugeno model requires
significantly more data.
Due to the lack of a proper validation set, we
will conduct a form of verification. The model error
is defined as follows:
32,2,1,
ˆ
~ iCe T
iii θψ
The mean-square error (MSE) is then defined as:
32 2
1
1
32 i
i
Me
For the linear model (9), the mean-square error is
0 1096M.
3.2 Takagi-Sugeno Static Model
The first step in building the model is structure
identification, which involves selecting the
appropriate system structure. Due to the limited
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number of measurements, we cannot afford a very
complex structure. Therefore, we decided to test
each of the six variables as potential antecedents of
the fuzzy model. Thus, in the k-th iteration
( 1,2,3,4,5,6)k
, not only did xk serve as an
antecedent variable, but we also replaced it with
three fuzzy variables f1xk, f2xk and f3xk (f1+ f2+ f3=1
by design) in the regressor vector. Three Gaussian
membership functions
2
()
ki
xc
ie
were used for
calculating fi. Since all the signals are normalized
and roughly span the interval [-3, 3], the
membership functions had centers at
1 2 3
2, 0, 2c c c
, with the σ parameter set to 1 in
all cases. Three fuzzy variables are therefore given
by
2 2 2
1 2 3
( ) ( ) ( )
1 2 3
3
12
1 2 3
1 2 3 1 2 3 1 2 3
k k k
x c x c x c
e e e
f f f
Finally, the regressor vector takes the following
form:
1 1 1 2 3 1 6
T
k k k k k k
x x x f x f x f x x
ψ
Similar to the linear model approach, the signals
used for identification were normalized. The vector
ˆk
θ
consisting of eight estimated parameters (3
“fuzzy” and 5 “linear”) was obtained by least
squares. Of particular interest is the mean square
error calculated over the identification data set,
following a method analogous to that of the linear
model. The results are detailed in Table 1 (second
row), which shows the results for different values of
k, representing different antecedent variables.
In addition, an experiment was conducted using
7 membership functions centred at -3, -2, -1, 0, 1, 2,
3, all with σ values of 1. The mean square errors
from this experiment are presented in Table 1 (third
row).
Table 1. Mean-Square Errors of the T-S Model (3
and 7 MFs) and Antecedent Variable xk
k
1
2
3
4
5
6
M3k
0.0790
0.0928
0.0975
0.1095
0.0907
0.1092
M7k
0.0642
0.0669
0.0708
0.0945
0.0789
0.0959
Comparing the second and third rows of Table
1, the most significant reduction in variance occurs
when x1 is used as the antecedent variable (in both
rows). In the second row x5, x2 and x3 are the
following candidates for the antecedent. In the third
row, x1 is followed by x2, x3 and x5. Both sequences
are similar but not identical due to the specific
placement of the membership functions.
Based on the results of this simple analysis, the
recommended antecedent variables would be x1
when using a single antecedent variable, and x1 and
x2 when using two. The final Takagi-Sugeno model
based on the original data uses x1 and x2 as
antecedent variables (each fuzzified with 3
membership functions, as in the case of a single
antecedent variable), resulting in 9 fuzzy variables
fi. The next step was to select one of the six
variables (xk) to be replaced by nine regressors (f1xk
to f9xk). It turned out that the best results were
obtained by selecting x2. In the end, the regressor
vector is made up of the other original variables (x1
and x3 to x6) and the nine variables mentioned above
(f1x2 to f9x2). In this configuration, the MSE is
0.0582, with a total of 3x3 + 5 = 14 estimated
parameters.
Figure 7 compares the measured outputs with
the outputs of three models: the linear model (6
estimated parameters), the T-S model with x1 as the
antecedent variable with 7 membership functions
(12 estimated parameters), and the T-S model with
x1 and x2 as the antecedent variables (each fuzzified
with 3 membership functions, for a total of 14
estimated parameters). Ideally, it would be
preferable to plot in the original high-dimensional
space, but as the input space is 6-dimensional, this is
impractical. Therefore, Figure 7 shows the sample
index i on the x-axis. We can see that the Takagi-
Sugeno model reduces the error significantly as
expected. Any validation of the model is impossible
in this case due to the scarcity of data, which
prevents the data set from being split into a training
set and a validation set.
Fig. 7: The comparison between the measured
output, the output of the linear model, and the
outputs of two T-S models. The data are plotted
against the sample index i. Each data set,
represented by an index i, also contains 6 input
variables, which are not plotted here
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4 Discussion
Both cases of modeling nonlinear complex
processes share some common features. In both
cases, there was a relatively small number of
measurements available and these measurements
were of poor quality due to inherent problems in the
measurement process.
Section 2 discusses the decolorization process.
It was shown that the first crucial step is to define
the structure that leads to more favorable model
properties. Specifically, in this case, the model
becomes “less” nonlinear when a particular input is
chosen over another option. In addition, as with any
modeling approach, the purpose of modeling is
essential for validating the model obtained. In our
case, the model is used for subsequent optimal
control, which selects the control inputs to purify
the water while minimizing the use of energy and
chemicals. A Radial Basis Network model is
therefore used to “filter out” the measurements and
produce smooth control laws.
The aim of modeling the atmospheric corrosion
process, described in Section 3, is to predict
atmospheric corrosion in the near future based on
measurements of some influential atmospheric
parameters. In this case, a Takagi-Sugeno model
was used. The main challenge is the selection of the
antecedent variables. Working with an extremely
small database in a high-dimensional space poses
some problems, but a simple method of selecting
one or two antecedent variables is proposed, where
the structure of the regressor vector is adapted
accordingly.
5 Conclusion
This paper addresses a well-known challenge:
constructing a model for a nonlinear process using
data with limited information content. While it does
not provide a definitive solution, it does stimulate
discussion of possible approaches. In such
scenarios, the balance between model complexity
and accuracy is crucial. Ideally, new experiments
could improve data quality, but in the cases
presented here, additional measurements were not
feasible. This highlights the difficulty of modeling
complex systems with sparse data and underlines
the need for innovative strategies in such
constrained environments.
As it is impossible to provide general guidelines
for modeling an arbitrary process based on
measured data, it is essential to consider the purpose
of the model and adapt the techniques to the amount
and quality of data available. In the examples
discussed, the most sensible decision would have
been to collect more data and focus on trustworthy
data. However, as this was not possible, the methods
had to be simplified and the resulting model had to
be robust to the uncertainty of individual
measurements by properly tuning the design
parameters of radial basis networks and Takagi-
Sugeno models. As a result, the output of the model
changes slowly as the inputs vary; in other words,
the output of the function has small derivatives with
respect to individual inputs. Although other
approaches could have been used, this basic
philosophy should always be followed.
Acknowledgement:
This work has been supported by the Slovenian
Research Agency (ARIS) under Research Program
P2-0219.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This work has been supported by the Slovenian
Research Agency (ARIS) under Research Program
P2-0219.
Conflict of Interest
The authors have no conflicts of interest to declare.
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WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2024.19.22
Sašo Blažič
E-ISSN: 2224-2856
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Volume 19, 2024
