Simulation, bifurcation, and stability analysis of a SEPIC converter controlled with ZAD

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 10, No. 1, February 2020, pp. 728~737
ISSN: 2088-8708, DOI: 10.11591/ijece.v10i1.pp728-737  728
Journal homepage: http://ijece.iaescore.com/index.php/IJECE
Simulation, bifurcation, and stability analysis of a SEPIC
converter controlled with ZAD
Aquiles J. Morelo1
, Simeón Casanova Trujillo2
, Fredy E. Hoyos3
1,2
Grupo de investigación: Cálculo Científico y Modelamiento Matemático,
Universidad Nacional de Colombia - Sede Manizales, Colombia
3
Faculty of Science, School of Physics, Universidad Nacional de Colombia-Sede Medellín,
Carrera 65 Nro. 59A-110, Medellín 050034, Colombia
Article Info ABSTRACT
Article history:
Received Jun 11, 2019
Revised Sep 30, 2019
Accepted Oct 7, 2019
This article presents some results of SEPIC converter dynamics when
controlled by a center pulse width modulator controller (CPWM). The duty
cycle is calculated using the ZAD (Zero Average Dynamics) technique.
Results obtained using this technique show a great variety of non-linear
phenomena such as bifurcations and chaos, as parameters associated with
the switching surface. These phenomena have been studied in the present paper
in numerical form. Simulations were done in MATLAB.
Keywords:
Bifurcation
Non-linear phenomena
SEPIC converter
ZAD technique
Copyright © 2020 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Fredy E. Hoyos,
Facultad de Ciencias - Escuela de Física,
Universidad Nacional de Colombia - Sede Medellín,
Carrera 65 No. 59A-110, Medellín, Colombia.
Email: fehoyosve@unal.edu.co
1. INTRODUCTION
Research on dynamic systems has been applied to different fields such as biology, power converters,
impact oscillators, mechanical systems, etc. where a large number of phenomena [1] of a non-linear
nature [2, 3] are presented. Dynamic systems defined in pieces are very important topics of study in theoretical
and experimental matters, being investigated in depth in recent years. An example of systems defined in pieces,
are DC–DC voltage converters, which allow the control of output voltage from a given voltage source; that is,
they act as bridges for energy transfers between sources and loads, both of direct current [4]. This leads
naturally to the question of how to transfer energy from the source to the load with 𝑉𝑖𝑛 amplitude, which needs
a 𝑥1𝑟𝑒𝑓 voltage, with the minimum loss of power. Multiple applications are presented by these converters
including power sources in computers, distributed power systems, power systems in electric vehicles,
etc [5, 6]. Therefore, this study has been a source of research in the fields of dynamic systems. Power converters
introduce a series of non-linearities in the switching process, which is why they have been studied as variable
structure systems. In [7] controllers were designed in sliding mode to work with this type of converter.
Later, Carpita [8] designed a controller based on a sliding surface given by a linear proportion of the error and
the derivative of the error. These two results allow working with a robust, stable, and efficient controller.
However, by generating a discontinuous action of this controller, a "chattering" phenomenon arises in
the system [9], which implies an increase in the ripple and distortion at the exit.

Int J Elec & Comp Eng ISSN: 2088-8708 
Simulation, bifurcation, and stability analysis of a SEPIC converter controlled with ZAD (Aquiles J. Morelo)
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With the purpose of eliminating the "chattering" phenomenon, several techniques have been proposed
to find a control scheme that guarantees a fixed switching frequency. For example, in [7], it is proposed to
synthesize a controller that guarantees a zero average of voltage error through a technique known as Zero
Average Current Error (ZACE). Fossas and his colleagues proposed a new control technique for power
converters in which an auxiliary output is set and a digital control action is defined that guarantees zero average
in the auxiliary output in each iteration, maintaining a fixed frequency of commutation, robustness,
and stability. This technique is known as ZAD (Zero Average Dynamics) and consists of the definition of
a switching surface on which the system is evolved on average. In [10] , it has been implemented making use
of the switching surface 𝑠(x(𝑡)) = (𝑥1(𝑡) − 𝑥1𝑟𝑒𝑓) + 𝑘 𝑠(𝑥̇1(𝑡) − 𝑥̇1𝑟𝑒𝑓), where good results are shown in
terms of robustness and low output error. In [11, 12] it is also applied to analyze the dynamics present in
the boost converter to study present non-linear phenomena, driven by a center aligned pulse width modulation
converter (CPWM).
In the present article, the ZAD technique has been implemented to control a SEPIC converter, which
has been used to control boost and buck converters in previous Works [11, 13, 14]. A linear combination of
the error in voltage and current has also been taken as the switching surface 𝑠(x(𝑡)) = 𝑘1(𝑥1(𝑡) − 𝑥1𝑟𝑒𝑓) +
𝑘2(𝑥2(𝑡) − 𝑥2𝑟𝑒𝑓) + 𝑘3(𝑥3(𝑡) − 𝑥3𝑟𝑒𝑓) + 𝑘4(𝑥4(𝑡) − 𝑥4𝑟𝑒𝑓) and from this, the calculation of duty cycle has
been done with which the system is evolved in a period of time T [15]. Finally, bifurcations that arise in
the evolution of 1-periodic orbits have been characterized and the presence of chaos has been determined from
certain values of constants associated with the commutation surface, which were taken as bifurcation
parameters [16, 17].
2. RESEARCH METHOD
An SEPIC converter [18] is a DC to DC converter belonging to the family of fourth-order
converters [19]. This device can supply more or less voltage than the input voltage. The basic scheme of
a SEPIC converter is shown in Figure 1, where 𝑉𝑖𝑛 is the input voltage, 𝑖1 is the current in 𝐿1 inductor, 𝑆 is
the switch, 𝐷 is the diode, 𝑣1 is the voltage in 𝐶1 capacitor, 𝑅 is load resistance, 𝑖2 𝑖s the current in 𝐿2
inductor, 𝑣2 is the voltage in 𝐶2 capacitor. The basic principle of the SEPIC converter consists of two different
states, depending on the state of the switch 𝑆.
Figure 1. Basic scheme of an SEPIC converter
When switch 𝑆 is closed, the status is 𝑂𝑁 and the input source 𝑉𝑖𝑛 connects to 𝐿1 coil at the same time
as the diode 𝐷 is polarized inversely. As a consequence, the intensity that circulates through 𝐿1 inductance
grows linearly, storing energy. In this situation, the 𝐶1 capacitor feeds the 𝐿2 inductor and the tension of 𝐶2 is
delivered to the load. When switch 𝑆 is open, the status 𝑂𝐹𝐹 andthe energy previously stored in 𝐿1coil together
with the input is transferred to the 𝐶1 input capacitor and the energy stored in the 𝐿2 inductor is transferred to
𝐶2 and to the load.
Two modes of operation are distinguished in the SEPIC converter, depending on the currents by
the inductors canceled during the operation period 𝑇: Continuous Driving Mode (𝑀𝐶𝐶) and Discontinuous
Driving Mode (𝑀𝐶𝐷). In this article, we will study the dynamics of the SEPIC converter in 𝑀𝐶𝐶.
The dynamics of the SEPIC converter are governed by the solution of this system of differential equations:

 ISSN: 2088-8708
Int J Elec & Comp Eng, Vol. 10, No. 1, February 2020 : 728 - 737
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𝐿1
𝑑𝑖1
𝑑𝜏
= −(1 − 𝑢)(𝑣1 + 𝑣2) + 𝑉𝑖𝑛
𝐶1
𝑑𝑣1
𝑑𝜏
= (1 − 𝑢)𝑖1 − 𝑢𝑖2
𝐿2
𝑑𝑖2
𝑑𝜏
= 𝑢𝑣1 − (1 − 𝑢)𝑣2
𝐶2
𝑑𝑣2
𝑑𝜏
= (1 − 𝑢)(𝑖1 + 𝑖2) −
𝑣2
𝑅
.
(1)
In this system, 𝑖1, 𝑣1, 𝑖2, and 𝑣2 are status variables 𝐿1, 𝐿2, 𝐶1, 𝐶2, and ;, 𝑅 are the parameters and
𝑢 ∈ {0,1} is the control variable. In system (1), if 𝑢 = 1, then SEPIC is 𝑂𝑁 (topology 1) and when 𝑢 = 0,
SEPIC is 𝑂𝐹𝐹 (topology 2). Making the change of variables:
𝑥1 =
𝑖1
𝑉 𝑖𝑛
√
𝐿1
𝐶1
𝑥2 =
𝑣1
𝑉 𝑖𝑛
𝑥3 =
𝑖2
𝑉 𝑖𝑛
√
𝐿1
𝐶1
𝑥4 =
𝑣2
𝑉 𝑖𝑛
𝑡 =
𝜏
√𝐿1 𝐶1
,
(2)
if 𝛼 =
𝐿2
𝐿1
, 𝛽 =
𝐶2
𝐶1
, and 𝛾 = 𝑅√
𝐶1
𝐿1
are defined, then a dimensional system is obtained for the dynamics of
the SEPIC converter:
𝑥̇1 = −(1 − 𝑢)(𝑥2 + 𝑥4) + 1
𝑥̇2 = (1 − 𝑢)𝑥1 − 𝑢𝑥3
𝛼𝑥̇3 = 𝑢𝑥2 − (1 − 𝑢)𝑥4
𝛽𝑥̇4 = (1 − 𝑢)(𝑥1 + 𝑥3) −
𝑥4
𝛾
,
(3)
where the new parameters of the system are 𝛼, 𝛽, and 𝛾. Each topology of the system can be expressed in
compact form by
𝐱̇ = 𝐀 𝑖 𝐱(𝑡) + 𝐛, (1)
where 𝑖 ∈ {1,2} and
𝐀1 =
[
0 0 0 0
0 0 −1 0
0
1
𝛼
0 0
0 0 0 −
1
𝛽𝛾]
, 𝐀2 =
[
0 −1 0 −1
1 0 0 0
0 0 0
−1
𝛼
1
𝛽
0
1
𝛽
−
1
𝛽𝛾]
, 𝐛 = [
1
0
0
0
].
The solution of each topology can be expressed as
𝐱𝑖 = 𝜙𝑖(𝑡 − 𝑡0)𝐱(𝑡0) + 𝝍𝑖(𝑡 − 𝑡0), (2)
where 𝜙𝑖(𝑡 − 𝑡0) = 𝑒 𝐀 𝑖(𝑡−𝑡0)
and 𝝍𝑖(𝑡 − 𝑡0) = ∫
𝑡
𝑡0
𝑒 𝐀 𝑖(𝑡−𝜏)
𝐛𝑑𝜏
2.1. Pulse width modulation
The control scheme [20] that will be used in this study corresponds to center-aligned modulation by
pulse width in such a way that a time interval [nT, (n + 1)T] is divided into three subintervals, where the first
and the last have the same length, as shown in Figure 2. Commutations are made according to the scheme
{1,0,1} and, therefore, the system will operate as follows:
𝒙̇ =
{
𝐀1 𝐱 + 𝐛 si nT ≤ t ≤ nT +
d
2
𝐀2 𝐱 + 𝐛 si nT +
d
2
≤ t ≤ (n + 1)T −
d
2
𝐀1 𝐱 + 𝐛 si (n + 1)T −
d
2
≤ t ≤ (n + 1)T,
(6)
where 𝑇 is the period, 𝑑 is the duty cycle and the time during which the system is operated in status 𝑂𝑁.

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Figure 2. Symmetrical center pulse
After choosing the control scheme, we must decide how to calculate the time that the system must
remain in conduction; that is, we must choose a criterion that allows us to calculate (period to period)
the duty cycle 𝑑. In this paper, we will calculate it using ZAD (Zero Average Dynamics) control technique [2]
and based on the fact that being a variable structure system, the principles of control in sliding modes can be
applied in such a way that the error dynamic is zero on average in each iteration [21, 22].
3. CONTROL STRATEGY
3.1. ZAD control strategy
This technique consists of defining a switching surface 𝑠(𝐱(𝑡)) = 0 in which the system will evolve
on average [12]. In this paper, the switching surface given by the equation 𝑠(𝐱(𝑡)) = (𝑥1(𝑡) − 𝑥1𝑟𝑒𝑓) +
𝑘 𝑠(𝑥̇1(𝑡) − 𝑥̇1𝑟𝑒𝑓), using this switching surface and the technical control ZAD, we obtain the next expression
for the duty cycle:
𝑑 =
2𝑠(𝐱(𝑛𝑇))+𝑇𝑠̇2(𝐱(𝑛𝑇))
𝑠̇2(𝐱(𝑛𝑇))−𝑠̇1(𝐱(𝑛𝑇))
, (7)
where 𝑑 is a real number between 0 and T, if 𝑑 < 0 or 𝑑 > 0. Expression (7) is redefined, saying that
the system is saturated. For this situation, the following selection is made in each period:
𝑑 = (
0 if d ≤ 0
𝑇 if d ≥ T
. (8)
3.2. Poincaré map
The Poincaré map of the SEPIC converter controlled with CPWM {1,0,1} and ZAD technique is given
by the following:
1. If 𝑑 𝑛 ∈ (0, 𝑇) (duty cycle does not saturate)
𝑃(𝐱 𝑛, 𝑑 𝑛) = 𝜙1 (
𝑑 𝑛
2
) 𝜙2(𝑇 − 𝑑 𝑛)𝜙1 (
𝑑 𝑛
2
) 𝐱(𝑛𝑇)
+𝜙1 (
𝑑 𝑛
2
) 𝜙2(𝑇 − 𝑑 𝑛)𝜓1 (
𝑑 𝑛
2
)
+𝜙1 (
𝑑 𝑛
2
) 𝜓2(𝑇 − 𝑑 𝑛) + 𝜓1 (
𝑑 𝑛
2
) , and
(9)
2. If 𝑑 𝑛 = 0 (duty cycle saturates)
𝑃(𝐱 𝑛, 𝑑 𝑛) = 𝜙2(𝑇)𝐱(𝑛𝑇) + 𝜓2(𝑇) (10)
3. If 𝑑 𝑛 = 𝑇 (duty cycle saturates)
𝑃(𝐱 𝑛, 𝑑 𝑛) = 𝜙1(𝑇)𝐱(𝑛𝑇) + 𝜓1(𝑇) (11)
4. BIFURCATIONS
In this section, a study of the qualitative change of the SEPIC converter is made by varying one of the
parameters associated with the switching surface. To characterize the type of bifurcation we find, we will use
the bifurcation diagram, which is obtained from the Poincaré map given by the relations (9), (10), and (11),
and proper values of the Jacobian matrix evaluated at equilibrium points of the system.

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4.1. Flip bifurcation
The simulation described below is presented when we consider the SEPIC converter as a reducer;
in addition, we take as reference values the vector (0.0544 1 0.1237 0.44) 𝑇
. Figure 3 shows a configuration
of the state variables for which there is a 1T-periodic orbit that goes from stable to unstable, which indicates
a bifurcation point. The point of interest is when 𝑘3 ∈ [40, 60]. The diagrams of these figures were obtained
by varying 𝑘3 in the specified range with 𝑘1 = 25, 𝑘2 = −15, 𝑘4 = −10, 𝑇 = 0.18, 𝛼 = 0.2683, 𝛽 = 0.7021,
and 𝛾 = 3.5583. Reviewing the eigenvalues of the Jacobian matrix associated with the Poincaré application in
Table 1 and observing Figure 3, we find the resulting bifurcation is of the flip type [23] because one of these
values goes from being stable to unstable, crossing through −1 for a value of the parameter 𝑘3= 51.96.
This type of bifurcation is characterized by a period doubling; that is, the system goes from having
a 1T-periodic orbit to having 2T-periodic orbits. Table 1 presents the values of the Poincaré map. An analysis
of these allows confirmation that this is a flip-type bifurcation [24, 25] because the proper value that goes from
stable to unstable does it crossing by −1 in the interval 𝑘3 ∈ (51.40, 52.30) as shown in Figure 4. This type of
bifurcation occurs because the orbit 1𝑇-periodic becomes unstable and an orbit 2 𝑇-periodic is born, i.e.,
a period doubling occurs [12].
Figure 3. Bifurcation diagram varying 𝑘3
Table 1. Eigenvalues of the jacobian matrix near the stability limit when varying 𝑘3
𝑘3 𝜆1 𝜆2,3 𝜆4
51.40 -0.99970 0.98106± 0.15499𝑖 0.95968
51.58 -0.99979 0.98107 ± 0.15495𝑖 0.95959
51.76 -0.99989 0.98107 ± 0.15492𝑖 0.95950
51.94 -0.99998 0.98107± 0.15489𝑖 0.95942
52.12 -1.00008 0.98108± 0.15485𝑖 0.95933
52.30 -1.00017 0.981084± 0.15482𝑖 0.95925

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Figure 4. Graphic variation of eigenvalues by varying 𝑘3
4.2. Neimar–saker bifurcation
When 𝑘1 is varied in the interval [−10, −2] and we take as fixed values 𝑘2 = 1, 𝑘3 = −6, 𝑘4 = 2.5,
𝑇 = 0.18, 𝛼 = 0.2683, 𝛽 = 0.7021, 𝛾 = 3.5583, and, as a condition, initial (1.1241 1 0.5621 2) 𝑇
, we also
find a bifurcation in the dynamics of the SEPIC converter as seen in Figure 5. In the bifurcation diagrams of
Figure 5 a change in the dynamics of the system is observed, which is characteristic of a Neimar-Saker type
bifurcation. To characterize this bifurcation, we analyze the evolution of the eigenvalues of the Jacobian matrix
of the Poincaré map near the point of bifurcation 𝑘1 = −2.42. We observe in Table 2 and Figure 6 that
the complex and conjugated eigenvalues of the Poincaré application near the bifurcation point, its module
approaches 1, which characterizes the Neimar–Saker-type bifurcation.
Figure 5. Bifurcation diagram varying 𝑘1

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Table 2. Eigenvalues of the poincaré map near the stability limit when varying 𝑘1
𝑘1 𝜆1 𝜆2,3 𝜆4
-3.0 -0.98014 0.9918± 0.1470i 0.95786
-2.75 -0.97929 0.9911± 0.1447i 0.96092
-2.5 -0.97843 0.9902 ± 0.1423i 0.96426
-2.25 -0.97757 0.9891± 0.139i 0.96792
-2.0 -0.97669 0.9879± 0.1372i 0.97194
Figure 6. Graphic variation of eigenvalues by varying 𝑘1
5. EXISTENCE AND CHAOS CONTROL
To guarantee the presence of chaos in the system, we use the exponents of Lyapunov, which determine
the proximity or divergence of two orbits that were initially close. The i-th Lyapunov exponent is given by
the expression:
𝐿𝑖 = lim
𝑘→∞
(
1
𝑘
∑ 𝑘
𝑛=0 𝑙𝑜𝑔|𝜆𝑖(𝐽𝑃(𝐱 𝑛))|), (12)
where 𝐱 𝑛 is the i-th value of Poincaré map, 𝐽𝑃 its Jacobian, and 𝜆𝑖 is the i-th eigenvalue of 𝐽𝑃.
The presence of a positive Lyapunov exponent in a system whose trajectories evolve within a finite zone of
the state space guarantees chaotic behavior [26]. On the other hand, the sum of all Lyapunov exponents in
a chaotic attractor must be negative [27].
To determine the presence of chaos [16, 28] and its respective control, we will use the values with
which the flip bifurcation was obtained. Figure 7 shows the existence of positive Lyapunov exponents for
values of 𝑘3 greater than 51.96 . Additionally, Figure 3 shows that for values of 𝑘3 ∈ (51.40,52.30),
the system evolves in a bounded region of the state space. Therefore, we can see that the system presents
chaotic behavior for values of 𝑘3 greater than 51.96.
Figure 7. Lyapunov exponents by varying 𝑘3

735
5.1. Chaos control with FPIC technique
Because our system is non-autonomous, it is excited with an external signal 𝑢. Any method of chaos
control must stabilize the unstable orbits, and for this it must necessarily assure that proper values of
the Jacobian matrix of the Poincaré map are within the unit circle (stability border). In this sense, several control
strategies have been designed. To control the chaos presented by the SEPIC converter with ZAD, we use
the FPIC (Fixed Point Induced Control) technique [29], which was designed by [14] and has been used
numerically in [13, 30].
This is based on the continuity of proper values theorem and helps stabilize period one or more orbits
in unstable and/or chaotic systems and does not require the measurement of state variables. It forces the system
to evolve to the fixed point; therefore, it is necessary to have prior knowledge of the control signal equilibrium
point. The equilibrium point obtained for the system is given by the transposed vector:
𝐱∗
= (
𝑥4𝑟𝑒𝑓
2
𝛾
, 1,
𝑥4𝑟𝑒𝑓
𝛾
, 𝑥4𝑟𝑒𝑓)
𝑡
,
that corresponds to a steady-state duty cycle
𝑑∗
= 𝑇
𝑥4𝑟𝑒𝑓
1+𝑥4𝑟𝑒𝑓
(13)
The duty cycle 𝑑 𝑍𝐹 is to apply ZAD and FPIC control techniques is given by:
𝑑 𝑍𝐹 =
𝑑+𝑁𝑑∗
𝑁+1
, (14)
where 𝑑 is determined by (7) and 𝑑∗
by (13).
Next we apply the FPIC technique to the dynamics of the SEPIC converter. Figures 8 (a) shows that
by choosing 𝑁 = 0.003, the area in which the system exhibits chaotic behavior decreases. Figure 8 (b) shows
that for the FPIC constant 𝑁 = 0.005 the area in which the system exhibits chaotic behavior continues to
decrease. Finally in Figure 8 (c) when the FPIC constant 𝑁 = 0.006 the area in which the system exhibits
chaotic behavior has almost completely disappeared.
(a) (b)
(c)
Figure 8. Bifurcation diagram varying 𝑘3 with FPIC control (a) 𝑁 = 0.003, (b) 𝑁 = 0.005, (c) 𝑁 = 0.006

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6. CONCLUSION
By analyzing the dynamics of the SEPIC converter numerically using the ZAD technique with
CPWM, we can observe the presence of non-linear phenomena such as quasi-periodicity and chaos.
The presence of bifurcations was detected by varying the parameters 𝑘1 and 𝑘3 associated to the switching
surface, being the flip and Neimar–Saker bifurcations, respectively. Chaotic behavior can be controlled by
introducing the FPIC technique. However, it is important to see how the FPIC technique influences other
behaviors of the system, such as regulation.
ACKNOWLEDGEMENTS
This paper was supported by the Universidad Nacional de Colombia, Sede Medellín under
the projects HERMES- 36911 and HERMES- 34671. The authors thank the School of Physics for their valuable
support to conduct this research.
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BIOGRAPHIES OF AUTHORS
Aquiles José Morelo Sanchez: received his B.S. degree in Matematics in 1999, M.S in Sciences-
Applied Mathematics in 2015, rector of the Educational Institucion Canutalito, Ovejas-Sucre,
Colombia, E-mail: ajmorelos@unal.edu.co. His research interests include didactics in the teaching
of mathematics and dynamic systems.
Simeon Casanova Trujillo: received his B.S. degree in Mathematics and Physics in 1993, M.S in
Mathematics in 1997 and Ph.D in Automation in 2011. Dr. Casanova is currently an Associate
Professor of the Exact and Natural Sciences Faculty, at National University of Colombia,
at Manizales, Colombia. His research interests include nonlinear dynamics analysis, control of
nonsmooth systems, and differentiability in topological groups.
Fredy Edimer Hoyos: received his BS and MS degree from the National University of Colombia,
at Manizales, Colombia, in Electrical Engineering and Industrial Automation, in 2006 and 2009,
respectively, and an Industrial Automation Ph.D. in 2012. Dr. Hoyos is currently an Associate
Professor of the Science Faculty, School of Physics, at National University of Colombia,
at Medellin, Colombia. His research interests include nonlinear control, system modeling,
nonlinear dynamics analysis, control of nonsmooth systems, and power electronics, with
applications extending to a broad area of technological processes. Dr. Hoyos is an Associate
Researcher in Colciencias and member of the Applied Technologies Research Group (GITA) at
the Universidad Nacional de Colombia. https://orcid.org/0000-0001-8766-5192

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