Año 23
/ Nº 35 / 2021 /
Modelado
con lógica difusa del circuito RC en serie con hardware y software de bajo
costo
Fuzzy logic modelling for the RC series circuit with
low cost hardware and software
Edgar
Serrano Pérez
Universidad
Tecnológica de México  UNITEC MÉXICO  Campus Atizapán
*Email:
edgar_serrano@my.unitec.edu.mx
Recibido:
06/11/2020; Aprobado 19/04/2021
Resumen
En este trabajo se presenta el
desarrollo del modelado de la respuesta a un escalón de 5 Volts de un circuito
RC conectado en serie. Se presenta el método de identificación del sistema
utilizando una tarjeta basada en Arduino Nano como
sistema de adquisición de datos. A partir de los datos experimentales
obtenidos, se definieron las funciones de membresía de la entrada y salida del
sistema difuso, cuya implementación se realizó a través de la caja de
herramientas de lógica difusa en Scilab. Un conjunto
de 11 reglas permitió relacionar las entradas con las salidas en un sistema de
inferencia de tipo Mamdani. El trabajo engloba la
combinación de elementos de hardware de bajo costo con programas de uso libre
para la integración didáctica que permite introducir conceptos fundamentales de
la lógica difusa en un enfoque de aplicación física práctica. Los resultados
del modelado difuso se comparan tanto con los resultados obtenidos
experimentales, como los resultados numéricos que se obtienen utilizando la
formulación analítica clásica para el sistema de circuito RC conectado en
serie.
Palabras
Claves: Lógica
difusa; Arduino; Modelado; Low
cost; Hardware.
Abstract
This work presents the development
of modeling for the response to a 5 Volt step of a RC
series circuit. The system identification method is presented
using an Arduino Nano based board as the data acquisition system. From the
experimental data obtained, the input and output membership functions are defined. The implementation was
carried out through the Fuzzy Logic Toolbox in Scilab.
A set of 11 rules relate inputs to outputs in a Mamdani
type inference system. The work includes the combination of lowcost hardware
elements with open source software for the didactic integration that allows
introducing fundamental concepts of fuzzy logic in a practical physical
application approach. The results of the fuzzy logic model are
compared with the experimental, numerical and the classical analytical
formulation for the RC series circuit.
Keywords: Fuzzy Logic; Arduino; Modelling; Low cost; Hardware.
Today, we are immersed in an
increasingly digitized society, numerous applications and technological devices
have emerged from applications of artificial intelligence. The development of
electronics and household appliances that use fuzzy logic as a decisionmaking
method has increased steadily. Numerous technological advances have been
benefited from the development of systems based on fuzzy logic [1]–[3] The
learning of various branches of artificial intelligence such as fuzzy logic and
neural networks, are generally taught in optional specialization courses in
careers related to mechatronics. Theorical sessions
and laboratory experiments provide a handson approach to developing manual
skills and abilities. Through laboratory experiments it is possible to verify
theoretical results, make comparisons and contrast them, furthermore where is
appropriate, make some adjustments to the models based on the experimental
results.
The exploration of the RC circuit has
been used to address fundamental concepts of physics related to electric
current, resistance and capacitance with various approaches [4]. In this work it has been proposed to model the response of a
resistorcapacitor series circuit, when a unit step input of 5 volts of direct
current is connected. The use of fuzzy logic is encouraged for this purpose,
implemented through lowcost hardware and open software elements such as
Arduino [5] and Scilab for system identification. In
the current moments of contingency there is a growing
need to transfer educational and academic environments to the home, where
engineering students take virtual sessions. It is of great importance the
development of laboratory practices in the engineering area [6], they can be carried out using available and lowcost
hardware elements, as well as the use of free software that facilitates the
processing and interpretation of data.
This engineering experiment could be adapted for
students to introduce and gain an understanding of the fuzzy logic modeling
process, through a reallife experiment using open hardware instrumentation and
open source software. This situation could reduce the inconveniences observed
during the transfer of facetoface engineering education into educational
centers, to home confined spaces of virtual distance education.
The modeling of response to a 5 volt
step input of a circuit formed by a resistor (100 kΩ)
and a capacitor (100 µF) using fuzzy logic start with the experimental
identification of the system. The objective is to know the behavior and
evolution of the voltage at capacitor terminal as a variable output in time. An
Arduino Nano based board has been used for the
experiment. The microcontroller board has low cost and it is possible to
acquire it with some ease in establishments dedicated to the commercialization
of electronic components. Through analog pin A0 it is
possible to measure the analog voltage of the capacitor over time. The
connection diagram for the physical implementation can be
observed in Figure 1.
At the beginning of the test, it must
be ensured that the capacitor is completely discharged. To ensure that
it starts from 0 volts. The capacitor terminals can be
connected to the same resistance for a period of 2 minutes without the
power source, in order to ensure the initial conditions of the experiment. The
data transfer between the Arduino Nano based board and a personal computer is carried out between the Serial monitor enabled by the
program, as shown in Figure 2.
Fig. 2.
Arduino language program for data acquisition.
The implementation of the program in Arduino language, performs a data sampling for the analog signal
using a time delay 1000 ms, that is, an analog
voltage reading is made about one second. Considering that the circuit consists
of a 100 kΩ resistor and a
100 µF capacitor, it is observed that the constant of the system is reached
around 10 seconds, that is, 63% of the expected final value. The circuit
presented is simple, the values of the resistance and the capacitor can be modified and adapted for other educational purposes, where
it is necessary to increase or decrease the establishment time of the transient
response of the circuit.
The use of the board based on Arduino nano has allowed to obtain
experimental results that allow to create a model based on fuzzy logic. The
main purpose is estimating the response of the circuit over time, using
linguistic variables in an approximate decisionmaking system. The analog
voltage between the capacitor terminals is read by the
analogRead function through the A0 physical pin of
the board with microcontroller. The analog reading and digital conversion,
allows to obtain voltages values between 0 and 5
volts. It is noted that the microcontroller board have an analogtodigital
converter with 10 bit resolution. The experimental
results obtained from the reading of analog voltages over time, corresponding
to the evolution of the voltage across the capacitor terminals over time, are shown in Table 1. The data were obtained directly from
the interface of the serial monitor with the that
found in the Arduino program.
Table
1. Experimental data acquisition using the Arduino Nano based board for
the RC series circuit.
Time (seconds) 
Voltage (volts) 
0 
0 
5 
1.68 
10 
2.9 
15 
3.65 
20 
4.14 
25 
4.44 
30 
4.63 
35 
4.76 
40 
4.84 
45 
4.89 
50 
4.91 
The data obtained experimentally represent the
evolution in time of the capacitor charge, from when the capacitor is totally discharged, until reaching values close to the
total charge, that is, 5 volts.
IV. Fuzzy
Logic modelling through Scilab
The 11 data points obtained from the identification of
the system, were used to create a Mamdanitype
fuzzy logic system in Scilab [7], through the fuzzy
logic toolbox. The system consists of a single "Time" input
and a single "Voltage" output. The input variable consists of
11 triangular membership functions distributed in the interval that represents
the range from 0.1 to 55 seconds as shown in Figure 3. This range is known as the universe of the discourse of the variable.
Fig. 3.
Membership distribution for the Input variable “Time”.
The highest point of each of the triangles represents
the highest degree of membership of each function, that is, the core and
corresponds to each of the points obtained in the identification of the
experimental system through the use of the Arduino
Nano board, used as a low cost data acquisition system [8]. The 3 parameters of each proposed triangular functions are: the
lower limit, core and upper limit, they can be consulted in Table 2.
Table
2. Triangular membership
parameters for the input variable “Time”.
Input Membership function name 
Triangular Membership parameters 

Time1 
0.1 
0 
5 
Time2 
0 
5 
10 
Time3 
5 
10 
15 
Time4 
10 
15 
20 
Time5 
15 
20 
25 
Time6 
20 
25 
30 
Time7 
25 
30 
35 
Time8 
30 
35 
40 
Time9 
35 
40 
45 
Time10 
40 
45 
50 
Time11 
45 
50 
55 
For the case of the output variable, 11 triangular
membership functions are proposed, which were
distributed throughout the universe of the variable's discourse, from 0.1 to
5.01 volts, as can be seen in Figure 4. For the case of selecting the upper
limit of each membership function, it was proposed
that it must coincide with the core value of the following membership function.
The Figure 2 and Figure 4 were
created using Open Office Calc for graphical
presentation of the data. It is observed that each
core of the outgoing membership functions correspond to the point values
obtained experimentally from the identification of the system. The distribution
of the lower and upper limits of each of the membership functions, for the
output variable was made in such a way that the lower
limit of the membership function coincides with the core of the membership
function. The individual parameters of each of the membership functions, which
correspond to the lower, core and upper limits, can be seen
in Table 3.
Table
3. Triangular membership parameters for the output variable “Voltage”.
Output Membership function name 
Triangular membership parameters 

Volt1 
0.1 
0 
1.68 
Volt2 
0.1 
1.68 
2.9 
Volt3 
1.68 
2.9 
3.65 
Volt4 
2.9 
3.65 
4.14 
Volt5 
3.65 
4.14 
4.44 
Volt6 
4.14 
4.44 
4.63 
Volt7 
4.44 
4.63 
4.76 
Volt8 
4.63 
4.76 
4.84 
Volt9 
4.76 
4.84 
4.89 
Volt10 
4.84 
4.89 
4.91 
Volt11 
4.89 
4.91 
5.01 
It is noted that there is a bigger number of
membership functions located in the right half of the universe of discourse of
the output variable, this situation corresponds to that experimentally, the data obtained for the last time interval are also closer
between them, at the end of the time interval. This situation refers to the
exponential behavior that first order systems present when a unit step input is
applied. Once the input and output variables of the fuzzy system have been defined, a rule base is required for relating each
of the inputs with each of the outputs. In this sense, the rule base allows
taking a specific action according to a specific input. For this work, 11 rules
have been proposed that assign each of the inputs to
each of the outputs as can be seen in the Table 4.
Table
4. Fuzzy Logic rule set
for the Mamdani Inference system.
Rule 

1 
If (Time is time1) Then (Voltage is volt1) 
2 
If (Time is time2) Then (Voltage is volt2) 
3 
If (Time is time3) Then (Voltage is volt3) 
4 
If (Time is time4) Then (Voltage is volt4) 
5 
If (Time is time5) Then (Voltage is volt5) 
6 
If (Time is time6) Then (Voltage is volt6) 
7 
If (Time is time7) Then (Voltage is volt7) 
8 
If (Time is time8) Then (Voltage is volt8) 
9 
If (Time is time9) Then (Voltage is volt9) 
10 
If (Time is time10) Then (Voltage is volt10) 
11 
If (Time is time11) Then (Voltage is volt11) 
With the definition of the membership functions of
both the input variable and the output variable, the set of rules that relate
each of the inputs to each of the outputs, we proceed to construct the fuzzy
system using the Mamdani type Scilab
Toolbox. The implication method selected was the Minimum and the aggregation
method was the maximum. The defuzzification method
was the centroid, which allows obtaining a numerical value as the output of the
system.
V. Results
and Discusion
When implementing the fuzzy system in Scilab and using the surface view function of the fuzzy
logic toolbox, the output response of the system is computed, that is, the
relationships that exist between the input and the output of the system. The fuzzy logic modeling, generated from the experimental
identification of the system with the microcontroller board, shows that the
results obtained with the fuzzy system implemented in Scilab
are approximate; the analytical solution is close to the experimental data
obtained by mean of the microcontroller board, and also close to the values
obtained with the fuzzy logic approach proposed, as shown in Figure 5.
Fig. 5.
Comparison of the data obtained using the fuzzy model, experimental data and
analytic data.
It is worth highlighting the approximation obtained to
model physical systems implementing a fuzzy logic system. Different parameters can be modified to obtain a closer tuning in the modeling,
such as membership functions definitions for both, the input and output. The
implication and aggregation methods can be used, as
well as different defuzzification methods. These
characteristics make fuzzy logic a very versatile modeling tool that can be
adapted for the analysis of more complex systems with multiple inputs and / or
multiple outputs.
The implementation of a Mamdani
type fuzzy system has been described, to model the
response in time to a unit step of 5 volts in a system composed of a resistor
and a capacitor connected in series. Low cost and easily acquired hardware
elements are used, as well as free software for its
implementation through the Scilab tool. This
combination favors the implementation of this type of systems for the analysis
of the behavior of physical systems and their evolution in time before
different physical conditions. It is intended to
promote the understanding of fundamental concepts of fuzzy logic with a practical
approach that combines the manipulation of hardware elements as well as free
software. The work presented could be adapted as an educational tool that
allows engineering students and practitioners to develop skills and abilities
for the construction of data acquisition systems. This process is fundamental
for the identification of systems and the consequent development of control
systems in engineering. An emphasis is placed on the
use of accessible and low cost devices and software. This situation can be
adapted to circumstances where distance education is
transferred to individual virtual environments such as the current
situation experienced by virus pandemic.
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