Scientific Paper / Artículo Científico |
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https://doi.org/10.17163/ings.n29.2023.06 |
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pISSN: 1390-650X / eISSN: 1390-860X |
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GAIN-SCHEDULING CONTROL APPLIED TO A DC-DC CONVERTER
FOR DIMMING LIGHT INTENSITY IN A LED LAMP |
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CONTROL POR GANANCIAS PROGRAMADAS APLICADO A UN CONVERTIDOR DC-DC PARA LA REGULACIÓN DE LA INTENSIDAD LUMINOSA DE UNA LÁMPARA LED |
Received: 12-11-2022, Received after review:
01-12-2022, Accepted: 09-12-2022, Published: 01-01-2023 |
Abstract |
Resumen |
Nowadays there are several applications of
LED lighting, but some of them require to precisely regulate
lighting in a wide range of operation, mainly in tracking tasks. In these
cases, it is necessary to consider the nonlinearity of the LED for the design
of control schemes; moreover, an efficient power source is
needed to supply the voltage and current variations required by the
lamp. This paper presents the design of a DC-DC converter capable of
implementing these variations, as well as the design, simulation and comparison
of classical PI, fuzzy PI and gainscheduling
control schemes for these applications. In order to validate the described
control schemes and to comply with the recommendations of the World Health
Organization, the control of the illuminance produced by an eye protection
lamp is taken as a case study, where the controller
varies the duty cycle of the converter to adjust the voltage of the lamp, and
consequently regulate the luminous intensity. Comparing the control schemes,
the gain-scheduling control has a better performance for the case study
described above, presenting a steady state error of 0% and lower overshoot. |
En la actualidad existen diversas aplicaciones de la iluminación LED, sin embargo, algunas de estas requieren regular la iluminación con precisión y en un amplio rango de operación, principalmente en tareas de seguimiento. Es en estos casos en donde es necesario considerar la no linealidad del LED para el diseño de esquemas de control, además, se necesita de una fuente de alimentación eficiente ante los cambios de tensión y corriente requeridos por la lámpara. En este trabajo se presenta el diseño de un convertidor DC-DC capaz de realizar y soportar estas variaciones, así como el diseño, simulación y comparación de esquemas de control PI clásico, PI difuso y ganancias programadas para estas aplicaciones. Con el fin de validar los esquemas de control descritos, y con el propósito de cumplir las recomendaciones de la Organización Mundial de la Salud, se toma el control de la iluminancia producida por una lámpara de protección ocular como caso de estudio, en donde el controlador varía el ciclo de trabajo del convertidor, ajustando de esta manera la tensión de la lámpara, y en consecuencia regulando la intensidad luminosa. Comparando los esquemas de control, el desempeño del sistema con el control por ganancias programadas tiene mejores características para el caso de estudio descrito con anterioridad, presentándose un error en estado estable de 0 % y menor sobretiro en comparación con los otros esquemas de control desarrollados. |
Keywords: control, converter, dimming,
Gain- scheduling, LED, nonlinear |
Palabras clave: control, convertidor, ganancias programadas, LED, no lineal, regulación |
1,* Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, Cuernavaca, México. Corresponding
author ✉: m21ce085@cenidet.tecnm.mx. Suggested citation: Olivar- Castellanos, G. S.; Vela- Valdés, L. G. and Aguayo- Alquicira, J. “Gain-scheduling control applied to a DC-DC converter for dimming light intensity in a LED lamp”. Ingenius, Revista de Ciencia y Tecnología. N.◦ 29, (january-june). pp. 66-78. 2023. doi: https://doi.org/10.17163/ings.n29.2023.06. |
1. Introduction Over time, lightning has
experienced various technological advances, such as the incorporation of the
LED (light emitting diode) in different applications. This is mainly because
LEDs have low consumption, high reliability and a larger useful life compared
to other technologies, which make them a sustainable, practical and
functional alternative to save energy [1]. In this technology,
some applications are so simple that an On-Off control of the LED lamp is
enough, but many others often require to regulate
light intensity between 0% and 100% with a fine resolution [2]. There are different
architectures for LED controllers, whose objective is to address the
requirements and limitations posed by different lightning applications, such
as indoor/outdoor lightning, greenhouse lamps, traffic signals, etc. LED
controllers are often developed and adapted from basic topologies of AC-DC
and/or DC-DC converters, to integrate functionalities such as precise power
control or regulation of light intensity [3]. Regardless of their
complexity, all power conversion systems have at least one DC-DC converter
stage. The topologies of DC-DC converters most commonly used in the
development of LED drivers include buck, boost, flyback,
sepic and/or half-bridge converters [3]. However, there are
applications in which it is essential to incorporate control schemes with the
aim of improving the precision and response of the system. For this purpose,
classical control schemes have been commonly used in [4–7] for regulation
applications, i.e., where there is a single operating point of the LED, since
they are enough to guarantee the desired dynamics around such point. On the other hand,
there are more complex applications that require stability
along the entire operating range of the LED, giving rise to tracking control.
Since the LED is a nonlinear device, it is required a nonlinear control
strategy to have it work with precision in its entire operating range. To address this
issue, some authors have incorporated schemes based on neural networks [8],
fuzzy logic [9–11], or even a combination of control schemes such as the
fuzzy PID [12]. Nevertheless, most of these works focus on guaranteeing a low
steady-state error, and for these cases it is enough
to implement a unique nonlinear control, such as the work presented in [2].
However, there are applications where it is required to watch the maximum
overshoot, besides guaranteeing a low error percentage. For this reason, another
alternative has been presented in the literature to address the LED
nonlinearities, such as the gain-scheduling technique together with a Flyback converter proposed in [13] for light intensity
control in a LED arrangement; however, there is no specific application, and
the controller is tuned empirically. |
The gain-scheduling
technique uses linear tools that approximate the nonlinear dynamics of the
system, enabling to adjust a controller for every different operating point
in advance, and subsequently update the parameters from such designs and
according to the operating point of the process [14, 15]. The objective of
this work is to design a supply source that regulates the light intensity of
a LED lamp in a wide operating range, with low steady-state error and taking
care not to damage the lamp with large overshoots. The eye protection lamps
are used a case study. The rest of the
paper is organized as follows. Section 2 presents
the methodology for developing the system, which comprises from lamp characterization
to tuning the control schemes. Section 3 analyzes the results obtained, and
finally section 4 addresses conclusions and future works. 1.1.
Case study:
Eye protection lamp According to the World Health
Organization (WHO), at least 2200 million people have vision impairment or
blindness, and more than 1000 million could have prevented it [16]. Some
studies demonstrate that for 2050, 50% of the world population will suffer
myopia, and human vision will face an increasingly severe test [17]. One of the main
causes of visual impairment is myopia, which may be
prevented having the appropriate lighting levels. For this reason, the
WHO recommends to maintain an illuminance level of 500 lx in the study and
working areas, preferably with white light sources, in order to prevent this
condition [18]. This is not a simple
task, since external light sources, such as natural lighting, or even other
sources of artificial lightning, disturb lighting in the desired area,
causing that the recommended levels are not reached. Based on this, it is
necessary a system capable of regulating the light intensity provided by an
eye protection lamp, such that it complements the external lighting,
maintaining the illuminance level at the 500 lx recommended by the WHO. Taking this into
account, a DC-DC converter is designed with control schemes such as classical
PI, fuzzy PI and gain-scheduling, with the purpose of selecting the control
scheme suitable for this case study, and thus demonstrate that a linear
control scheme is not appropriate for tracking tasks in LED lighting. 2. Materials and methods A LED is a device whose light intensity depends on the current that circulates through it [13]. Nevertheless, this current depends on the voltage across its terminals. Therefore, for varying the light intensity of a LED it is practical to vary the voltage. For this purpose, it is |
necessary a supply source capable of
performing and withstanding these variations. Because
of this, it is proposed to implement a DC-DC
converter that fulfills these requirements. In addition, a control scheme should be incorporated to guarantee the levels recommended
by the WHO. This is represented by the block diagram shown
in Figure 1. The
required illuminance level is assured if it is
guaranteed that the lamp voltage is the desired one. Therefore, in order to
simplify the system, the different controllers are tuned
considering the voltage value as reference. The
entire process is described in subsequent sections,
using Simulink/Matlab to simulate the system under
the action of the controllers. Figure 1. Block diagram of the system proposed 2.1. Lamp
characterization In order to have the eye
protection lamp supplementing natural light, it is necessary to design a
supply source with a controller that enables to regulate the illuminance
depending on the varying conditions of natural light. The
first step is to characterize the lamp. The schematic diagram shown in Figure
2 is used for this purpose. It is
known that the lamp requires an input voltage of 12 VDC; hence, such
voltage is applied and then progressively reduced, measuring the changes in
voltage, current and illuminance. Figure 2. Circuit for characterizing the lamp Figure
3 shows the V-I (voltage-current) curve and Figure 4 shows the V-E (voltage-illuminance)
curve, both obtained plotting the measured data. |
Figure 3.
V-I curve of the eye protection lamp
Figure 4.
V-E curve of the eye protection lamp 2.2.
Design of the
DC-DC converter The Buck converter is used in applications where the load requires voltages
lower than the input voltage of the converter. Its implementation is simple
due to the few components; in addition, its control is simple because the
relationship between the output and input voltages is proportional to the
duty cycle of the control signal [13]. For this reason, the
Buck converter is chosen as the supply source for
the lamp. The specifications shown in Table 1, which are
proposed from the characterization of the lamp, are defined for the
design of the converter. Figure 5 shows the topology of this converter. Table 1.
Design specifications of the converter
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Figure 5.
Topology of the Buck converter From the analysis
and the equations showed by Gamboa [19], it is made the calculation and selection of the components.
Their values are shown in Table 2. Table 2. Calculation of the
components and variables of the converter
2.3. Transfers functions To obtain the transfer functions
(TF) it should be considered that the converter load
is variable, because the LED works at different operating points according to
the requirements. Therefore, it is not possible to obtain a unique linear TF
that represents the entire dynamics of the system. Based on the above and to facilitate the design of the controllers,
the V-E curve is divided into five ranges approximately linear, as shown in
Figure 6. After the curve is divided, it is possible
to obtain five TFs that approximate the dynamics in each of those ranges.
Figure 6.
V-E curve divided into five ranges Afterwards,
since the operating points are defined, it is possible to obtain the TF from
the control-output |
characteristic
function of a Buck converter, shown in Equation (1).
Since
the controller performs in the entire operating range of the lamp, the value
of R in Equation (1) is variable. Therefore, an average R is
calculated in each of the ranges defined, thus obtaining a different
TF for each operating range. The parameters for these TFs are
shown in Table 3, whereas the transfer functions are shown in
Equations (2) - (6).
Table 3. Parameters of the transfer functions
It is important to remark that having a different transfer function for each operating range implies that different dynamics are achieved in each of them. This is due to the fact that there is a different damping ratio (ζ) in each range, despite the natural frequency is the same (ωn). This is detailed in Table 4. |
Table 4.
Damping ratio and natural frecuency in the operating
ranges
2.4. Tuning of a PI controller Since the PI control
scheme is one of the most widely used in applications with DC-DC converters,
such control scheme is tuned to validate and compare
its performance in the regulation of the illuminance of an eye protection
lamp. The
controller is tuned for the intermediate operating
range, corresponding to the TF of Equation (4). The
pole assignment methodology is used for the tuning,
considering that the characteristic equation of a second order TF may be
simplified as shown in Equation (7).
In
addition, it is known that the TF of a PI controller
is of the form shown in Equation (8). Then,
closing the loop of the plant with the controller results in the TF shown in
Equation (9). Therefore,
a pole assignment should be implemented based on a characteristic equation of
the form shown in Equation (10), corresponding to a third order system with
two complex conjugate poles and a real pole, where β represents a factor of
proportionality that relates the distance of the real pole with respect to
the complex conjugate poles. Then,
setting (10) equal to the denominator of the closed-loop TF of Equation (9)
and solving for the unknowns, results in three equations useful to calculate
the gains of the PI controller. is calculated using
Equation (11), whereas Equations (12) and (13) are used to calculate the
proportional gain (Kc) and the integral gain (Ki),
respectively. |
In
this manner, the gains of the PI controller may be calculated
from the desired damping ratio (ζ) and natural frequency (ωn). It is important to mention that
it is possible to calculate ζ and ωn
from Equations (14) and (15), defining a maximum overshoot (Mp) and a settling time (tss). In this case, it is proposed an Mp not larger than 5 % and to search for a tss of 1 ms.
Therefore,
it is obtained that β = 1.608, Kc = 0.247 and Ki =
168.7537. This is implemented as shown in Figure 7.
Figure 7.
Block diagram of the implementation of the PI control 2.5.
Tuning of a
fuzzy PI controller It is common that
fuzzy logic control has a level of uncertainty, since it is
better expressed as a control through words that interpret common
sense, instead of numbers, or sentences instead of equations [20]. However,
the process variables are not measured in common
sense, but in numbers. To
overcome this issue, it is convenient to incorporate the gains of a PI
controller to numerically correct the weaknesses of
the fuzzy interpretation. The block |
Figure 8. Block diagram of the
implementation of the fuzzy PI control The design of the
fuzzy controller requires to clearly know the input and
output variables. In this case, the input variables are the error and
the integral of the error, and the output is the duty cycle, since this is
the variable required by the plant. Since the error is
obtained through the comparison of the output of the converter, which
corresponds to a maximum voltage of 12 V, it is defined a universe of
discourse between {–12, 12} for both inputs. It is important to mention that
the universe defined contains negative numbers, because it is possible to
obtain a positive or a negative difference between the output and the
reference. The linguistic
variables defined for the error signal are: · NE: Negative Error · ZE: Zero Error ·
PE: Positive Error On the other hand, the linguistic
variables defined for the integral of the error are: ·
NIE: Negative
Integral of the Error ·
ZIE: Zero Integral of
the Error ·
PIE: Positive
Integral of the Error |
Once the different linguistic
variables for the inputs have been established, as
well as the associated universes of discourse, the corresponding fuzzy sets
are defined, as shown in Figure 9 and Figure Figura
10, respectively.
Figure 9.
Fuzzy sets for the error
Figure 10.
Fuzzy sets for the integral of the error On the other hand,
the output of the fuzzy block is the duty cycle, whose universe of discourse
is defined between {0, 1} because this is its interval of operation. The
linguistic variables defined for the duty cycle are: ·
S: Small ·
I: Ideal ·
L: Large The fuzzy sets of
the duty cycle are shown in Figure 11.
Figure 11.
Fuzzy sets for the duty cycle Once the different fuzzy sets have been defined, the control rules are assigned. For each output it is assigned an appropriate linguistic variable, based on the possible combinations of the inputs. The set of rules is shown in Table 5. |
Table 5. Fuzzy
rules for the output variable of the controller In order to verify
the performance of the fuzzy PI controller, the fuzzy sets and rules defined are uploaded in the fuzzyLogicDesigner
tool. 2.6.
Gain-
scheduling control Gain-scheduling control consists
of designing a controller for every different operating point in advance, and
further selecting the controller to be implemented
from such designs according to the operating point of the process [14, 15]. One of the most
widely used methods for selecting the gains is fuzzy logic, which gives rise
to a control strategy known as fuzzy gain scheduling [13]. In this strategy
fuzzy logic is responsible for varying the gains in real-time, with the added
benefit of enabling a smooth transition between the controllers determining
intermediate values of the gains, and thus reducing drastic changes that may
affect the controller and the plant. Based on the above,
the block diagram of Figure 12 shows the system to be
implemented. The gain-scheduling control designed is
constituted by a fuzzy logic block, which is responsible of choosing
the gains of the PI controller depending on the operating point.
Figure 12.
Block diagram of a gain-scheduling control Since the operating
range of the LED lamp has been divided into five intervals, there are 5
different TFs for a which a different PI controller
should be designed. Considering that
each operating range has different dynamics, distinct design specifications are proposed for each of those ranges. An important point
to be taken into account is that the overshoot
should not be very large in the operating range from 11.3 V to 12 V, because
the LED lamp could be damaged. Based on this, Table 6 shows the design
specification for each controller. |
Table 6.
Design specifications for the PI controllers
The pole assignment
methodology described in section 2.4 is used for
tuning the controllers, which gives the gains shown in Table 7. Table 7.
Gains of the PI controllers
On the other hand, a
fuzzy logic control is designed for selecting the
gains of the PI controller. For this case, it is
convenient to have the reference signal as input variable, since it defines
the working operating range. Similarly, the Kc and Ki
gains are defined as output variables, since these
variables vary depending on the operating range. Taking into account
the operating range, the universe of discourse of the input variable is defined as {9, 12}. On the other hand, the linguistic
variables for the operating ranges are defined as: · RI: {9.2 – 9.7} · R2: {9.7 – 10.2} · R3: {10.2 – 10.7} · R4: {10.7 – 11.3} · R5: {11.3 – 12} Once the linguistic
variables of the input have been established, the
fuzzy sets shown in Figure 13 are defined.
Figure 13.
Fuzzy sets of the operating ranges Regarding the fuzzy
sets of the output, the universe of discourse for Kc is
{0.004, 0.022} and for Ki is {10, 110}. Meanwhile, it is
observed for linguistic variables that there are two ranges where the gains
are very similar; consequently, these two were averaged and |
defined as a single
one, and thus the linguistic variables for Kc are defined as: · Very small- MP: 0.0059 · Small – S: 0.0079 · Medium – M: 0.0203 · Large – L: 0.02075 On the other hand, the linguistic
variables for Ki are defined as: · Small –S: 16.725 · Medium – M: 22.6430 · Large – L: 30.7392 · Very Large – VL: 106.0908 Then, the fuzzy sets for the Kc
gain are defined as shown in Figure 14, whereas the
fuzzy sets for the Ki gain is shown in Figure
15.
Figure 14. Fuzzy sets for the proportional gain Kc
Figure 15. Fuzzy sets for the integral gain Ki For
the set of rules, a linguistic variable is assigned
based on the input for each output, as shown in Table 8. Table 8. Set of rules for selecting the gains
|
It
should be recalled that it is desired to control the illuminance level in a
study or working area, to maintain the 500 lx recommended by WHO. Voltage levels were used to facilitate the design of these
controllers, and thus it is required to convert from voltage to illuminance.
This may be carried out based on lamp characterization, modifying the block diagram of
the system as shown in Figure 16.
Figure 16.
Block diagram of the system including the lamp 3.
Results and discussion For validating the control schemes
designed, they are simulated with three different transfer
functions: the intermediate transfer function, given by equation (4), and the
transfer functions at the ends of the operating range of the lamp, given by
equations (2) and (6). Figure 17 shows the
response obtained when simulating the intermediate transfer function with the
PI controller. It is observed that there is a steady-state error of 0%, a
maximum overshoot of 4.6% and a settling time of approximately 1.8 ms. Figure 17. Response of the transfer
function corresponding to the intermediate operating range with a PI control
scheme Nevertheless, some
undesired dynamics appear when the same control scheme is
simulated with the same gains in the operating ranges at the ends; in particular,
the maximum overshoot exceeds the 12 V corresponding to the supply voltage of
the lamp. The responses of these TFs under the action of the control scheme are shown in Figures 18 and 19 respectively. |
Figure 18. Response of the transfer
function corresponding to the lower operating range with a PI control scheme Although in these
three cases a steady-state error of 0% is reached,
an overshoot above 12 V may damage the lamp, and thus a unique PI control is
not very suitable for the application. Figure 19. Response of the transfer
function corresponding to the upper operating range with a PI control On the other hand,
an improvement in the response within the tuning range is
obtained when simulating the fuzzy PI control scheme, since the
maximum overshoot is significantly reduced, as shown in Figure 20. In
addition, a steady-state error of 0% is guaranteed
and a settling time of 1.2 ms is achieved. |
Figure 20. Response of the transfer function
corresponding to the intermediate operating range with a fuzzy PI control
scheme Nevertheless,
undesired responses are obtained when the same controller is applied in the
operating ranges at the ends, since even though there is no overshoot, the steady-state error is greater than 2%, as
shown in Figures 21 and 22. Figure 21. Response of the transfer
function corresponding to the lower operating range with a fuzzy PI control
scheme |
Figure 22. Response of the transfer
function corresponding to the upper operating range with a fuzzy PI control
scheme On the other hand,
the desired response is obtained for each of the ranges
when using the gain-scheduling control scheme. Figure 23 shows the response
for the intermediate operating range, whereas the responses for the lower and
upper operating ranges are shown in Figures 24 and
25, respectively.
Figure 23. Response of the transfer
function corresponding to the intermediate operating range with a
gain-scheduling control schem
Figure 24. Response of the transfer
function corresponding to the lower operating range with a gain-scheduling control
scheme |
Figure 25. Response of the transfer
function corresponding to the upper operating range with a gain-scheduling
control scheme It is seen that all responses have a steady-state error of 0%,
and maximum overshoots smaller than 2%. However, the settling time increases. A comparison of the
responses obtained with the three control schemes is shown
in Table 9. Table 9. Comparison of the
responses obtained with the PI, fuzzy PI and gain-scheduling control
schemes
It is important to
mention that the maximum overshoot and the settling time for which the
gain-scheduling control was tuned are different from the design specifications.
This is mainly due to the effects of the real pole and the zero that appears
when the loop is closed. However, these
effects do not harm the desired dynamics but benefit it, achieving very small
overshoots. The effect of the real pole and the zero is
mostly noted in the settling time; however, this time is in the order
of milliseconds, which for the case of lighting and of the converter is not
relevant. Based on the above,
the gain-scheduling control shows a better response in the entire operating
range of the lamp, and thus the simulations incorporate the model of the
lamp. The external illuminance pattern is taken from the levels of solar
irradiance during a spring day, in the city of Cuernavaca, Morelos, Mexico,
collected from the database of the Instituto
Nacional |
de Ecología y Cambio Climático [21]. This pattern is shown
in Figure 26. The
eye protection lamp should supplement the lighting in the case of changes in
the external illuminance, in order to maintain the 500 lx recommended by the
WHO. The response of the system in the presence of variations in the external
illuminance is shown in Figure 27. It
is seen that despite the changes in the external lighting, the level of total
illuminance is maintained at the 500 lx recommended by the WHO. However, it is observed a higher ripple when the external lighting is
minimum. This ripple is shown in Figure 28, and it appears because at this point the converter should supply a higher voltage to the
lamp, and consequently there is a higher voltage ripple due to the effects of
the converter output capacitor. However, this ripple is around 1 %, which is
within the margins that are acceptable from the point of view of control
theory.
Figure 26. Pattern of external illuminance
Figure 27. Total illuminance of the system |
Figure 28. Effects of the voltage ripple on
the illuminance On
the other hand, Figure 29 shows the voltage supplied by the designed
converter. It is seen that the voltage remains
within the operating ranges, which guarantees not damaging the lamp. In
addition, it is shown that the lamp does not work in
a single operating point, but it demands different voltage levels to the
converter according to the changes in the level of external illuminance.
Figure 29.
Voltage supplied to the lamp by the DC-DC converter 4. Conclusions At present, there
are different applications of LED lamps. Some of them require precision when varying
the light intensity, such as eye protection lamps. For
that reason, this work presented the simulation of a DC-DC converter with
three control schemes, which vary the illuminance of an eye protection lamp
to maintain the 500 lx recommended by the WHO to prevent eye diseases, and
considering that external light sources disturb the lighting provided by the
lamp. classical PI
control scheme is capable of guaranteeing a steady-state error of 0 % in the
entire operating range of the lamp, but it does not guarantee appropriate
levels of maximum overshoot. |
On
the other hand, the designed fuzzy PI control scheme is capable of
guaranteeing that the voltage levels supported by the lamp are not exceeded;
however, if the operating point changes the steady-state
error increases. Conversely,
the designed gain-scheduling control scheme is a combination of the PI and
fuzzy logic controllers, where the latter is used as
gain selector to enable the adjustment of the controller gain depending on
the operating point of the system. In this manner, it is
guaranteed to have a steady-state error of 0 % and to reduce as much
as possible the maximum overshoot. In
the simulation of the gain-scheduling control scheme including the model of
the lamp, it was possible to maintain the illuminance level at 500 lx, and
thus the lamp supplements external lighting to fulfill the recommendation by
the WHO. In addition, it is guaranteed that the DC-DC
converter does not supply voltage levels that exceed the operating range of
the lamp, and thus it is verified that the light intensity of the LED lamp is
regulated precisely and guaranteed not to damage it, as opposed to other
works that focus on the control at only one operating point using a linear
controller. As
a future work, it is worth mentioning that it is possible to improve the
ripple present in the total illuminance, by increasing the capacitance value
of the converter output capacitor; however, this modifies the TFs, and thus
the dynamics obtained. On
the other hand, gain-scheduling control may be incorporated
in different applications, and it is thus recommended its validation with
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