Design of an adaptive I-PD controller for a five-level resonant inverter with selective harmonic elimination technique
Introduction. Resonant inverters are indispensable in demanding applications such as induction heating, wireless energy transfer, and high-frequency power conversion systems. Problem. The main topologies for realizing resonant inverters are the half-bridge and full-bridge configurations, but the mul...
Збережено в:
| Дата: | 2026 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
National Technical University "Kharkiv Polytechnic Institute" and Аnatolii Pidhornyi Institute of Power Machines and Systems of NAS of Ukraine
2026
|
| Теми: | |
| Онлайн доступ: | https://eie.khpi.edu.ua/article/view/342998 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Electrical Engineering & Electromechanics |
| Завантажити файл: |  |
Репозитарії
Electrical Engineering & Electromechanics| _version_ | 1869562799209840640 |
|---|---|
| author | Garmat, A. Toubal Maamar, A. E. Abdelouahed, T. Bensafi, N. |
| author_facet | Garmat, A. Toubal Maamar, A. E. Abdelouahed, T. Bensafi, N. |
| author_institution_txt_mv | [
{
"author": "A. Garmat",
"institution": "Ziane Achour University of Djelfa"
},
{
"author": "A. E. Toubal Maamar",
"institution": "University of M’hamed Bougara of Boumerdes"
},
{
"author": "T. Abdelouahed",
"institution": "University of M’hamed Bougara of Boumerdes"
},
{
"author": "N. Bensafi",
"institution": "University of M’hamed Bougara of Boumerdes"
}
] |
| author_sort | Garmat, A. |
| baseUrl_str | http://eie.khpi.edu.ua/oai |
| collection | OJS |
| datestamp_date | 2026-07-01T21:42:56Z |
| description | Introduction. Resonant inverters are indispensable in demanding applications such as induction heating, wireless energy transfer, and high-frequency power conversion systems. Problem. The main topologies for realizing resonant inverters are the half-bridge and full-bridge configurations, but the multilevel topology is not well-known for resonant inverters because their modeling and control design are challenging steps. The goal of this study is to investigate a five-level resonant inverter combined with the selective harmonic elimination (SHE) technique to eliminate the third harmonic and minimize the total harmonic distortion (THD). Methodology. The structure of the proposed inverter and the SHE modulation technique are presented to illustrate harmonic reduction in the applied voltage. To address the inherent nonlinearities of the system, the extended describing function (EDF) method is employed to derive a generalized small-signal state-space model from any defined input to any desired output. This model enables accurate prediction of system behavior around the operating point. Based on this model, an adaptive I-PD controller incorporating a model reference adaptive control (MRAC) mechanism, designed according to the Massachusetts Institute of Technology (MIT) rule, is developed. The adaptive mechanism continuously tunes the proportional, derivative, and integral gains to maintain the desired performance despite load and parameter changes. Results. Numerical simulations validate the accuracy of the developed model and demonstrate that the adaptive I-PD control significantly ensures the system’s robustness. The results indicate that the THD of voltage and current are 25.46 %, and 9.43 %, respectively. The third harmonic is well eliminated. The model prediction error, when compared to full MATLAB/Simulink nonlinear simulations, did not exceed 4.1 %, thereby validating the effectiveness and precision of the modeling approach. The presented MRAC-based adaptive I-PD controller demonstrates high performance in tracking reference signal and responds to abrupt changes of load parameter (30 % change of the resistance value), highlighting its effectiveness for current control in five-level resonant inverter system. Scientific novelty. The proposed framework combines SHE-based harmonic mitigation, EDF-based modeling, and MRAC-based adaptive I-PD control for multilevel resonant inverters. This integration provides a generalized and flexible approach for handling system nonlinearities and improving dynamic performance. Practical value. The results confirm the feasibility of implementing adaptive I-PD control for five-level resonant inverters. The proposed scheme ensures high efficiency, stable power regulation, and reliable operation, paving the way for industrial applications requiring precise temperature control and robust performance under varying load conditions. References 28, table 1, figures 17. |
| doi_str_mv | 10.20998/2074-272X.2026.4.07 |
| first_indexed | 2026-07-02T01:00:24Z |
| format | Article |
| fulltext |
Electrical Engineering & Electromechanics, 2026, no. 4 47
© A. Garmat, A.E. Toubal Maamar, T. Abdelouahed, N. Bensafi
UDC 621.314 https://doi.org/10.20998/2074-272X.2026.4.07
A. Garmat, A.E. Toubal Maamar, T. Abdelouahed, N. Bensafi
Design of an adaptive I-PD controller for a five-level resonant inverter
with selective harmonic elimination technique
Introduction. Resonant inverters are indispensable in demanding applications such as induction heating, wireless energy transfer, and high-
frequency power conversion systems. Problem. The main topologies for realizing resonant inverters are the half-bridge and full-bridge
configurations, but the multilevel topology is not well-known for resonant inverters because their modeling and control design are
challenging steps. The goal of this study is to investigate a five-level resonant inverter combined with the selective harmonic elimination
(SHE) technique to eliminate the third harmonic and minimize the total harmonic distortion (THD). Methodology. The structure of the
proposed inverter and the SHE modulation technique are presented to illustrate harmonic reduction in the applied voltage. To address the
inherent nonlinearities of the system, the extended describing function (EDF) method is employed to derive a generalized small-signal state-
space model from any defined input to any desired output. This model enables accurate prediction of system behavior around the operating
point. Based on this model, an adaptive I-PD controller incorporating a model reference adaptive control (MRAC) mechanism, designed
according to the Massachusetts Institute of Technology (MIT) rule, is developed. The adaptive mechanism continuously tunes the
proportional, derivative, and integral gains to maintain the desired performance despite load and parameter changes. Results. Numerical
simulations validate the accuracy of the developed model and demonstrate that the adaptive I-PD control significantly ensures the system’s
robustness. The results indicate that the THD of voltage and current are 25.46 %, and 9.43 %, respectively. The third harmonic is well
eliminated. The model prediction error, when compared to full MATLAB/Simulink nonlinear simulations, did not exceed 4.1 %, thereby
validating the effectiveness and precision of the modeling approach. The presented MRAC-based adaptive I-PD controller demonstrates
high performance in tracking reference signal and responds to abrupt changes of load parameter (30 % change of the resistance value),
highlighting its effectiveness for current control in five-level resonant inverter system. Scientific novelty. The proposed framework combines
SHE-based harmonic mitigation, EDF-based modeling, and MRAC-based adaptive I-PD control for multilevel resonant inverters. This
integration provides a generalized and flexible approach for handling system nonlinearities and improving dynamic performance. Practical
value. The results confirm the feasibility of implementing adaptive I-PD control for five-level resonant inverters. The proposed scheme
ensures high efficiency, stable power regulation, and reliable operation, paving the way for industrial applications requiring precise
temperature control and robust performance under varying load conditions. References 28, table 1, figures 17.
Key words: resonant multilevel inverter, adaptive control, selective harmonic elimination, induction heating.
Вступ. Резонансні інвертори є невід’ємною складовою сучасних високотехнологічних застосувань, зокрема систем індукційного
нагрівання, бездротового передавання енергії та високочастотних систем перетворення електроенергії. Проблема. Основними
топологіями реалізації резонансних інверторів є напівмостові та мостові схеми. Водночас багаторівнева топологія для
резонансних інверторів залишається недостатньо дослідженою через складність математичного моделювання та синтезу
систем керування. Мета. Дослідження п’ятирівневого резонансного інвертора у поєднанні з методом селективного усунення
гармонік (SHE) для пригнічення третьої гармоніки та мінімізації повного гармонічного спотворення (THD). Методика.
Представлено структуру запропонованого інвертора та метод модуляції SHE для демонстрації зменшення гармонічних
складових прикладеної напруги. Для врахування притаманних системі нелінійностей застосовано метод розширеної описувальної
функції (EDF), за допомогою якого отримано узагальнену малосигнальну модель у просторі станів для довільно визначених входів
і виходів. Розроблена модель забезпечує точне прогнозування поведінки системи поблизу робочої точки. На її основі синтезовано
адаптивний I-PD-регулятор із механізмом адаптивного керування за еталонною моделлю (MRAC), побудованим відповідно до
правила Массачусетського технологічного інституту (MIT). Адаптивний механізм забезпечує безперервне налаштування
пропорційного, диференціального та інтегрального коефіцієнтів для підтримання необхідних показників роботи за зміни
навантаження та параметрів системи. Результати. Чисельне моделювання підтвердило точність розробленої моделі та
показало, що адаптивне I-PD-керування суттєво підвищує робастність системи. Отримані результати свідчать, що THD
напруги та струму становлять відповідно 25,46 % і 9,43 %. Третя гармоніка ефективно усувається. Похибка прогнозування
моделі порівняно з повною нелінійною моделлю MATLAB/Simulink не перевищувала 4,1 %, що підтверджує ефективність і
точність запропонованого підходу до моделювання. Представлений адаптивний I-PD-регулятор на основі MRAC демонструє
високу якість відстеження опорного сигналу та ефективно реагує на різкі зміни параметрів навантаження (зміна опору на 30
%), що підтверджує його придатність для керування струмом у системі п’ятирівневого резонансного інвертора. Наукова
новизна. Запропонований підхід поєднує метод селективного усунення гармонік SHE, моделювання на основі EDF та адаптивне I-
PD-керування на основі MRAC для багаторівневих резонансних інверторів. Така інтеграція забезпечує узагальнений і гнучкий
підхід до врахування нелінійностей системи та покращення її динамічних характеристик. Практична значимість. Отримані
результати підтверджують можливість практичної реалізації адаптивного I-PD-керування для п’ятирівневих резонансних
інверторів. Запропонована схема забезпечує високу ефективність, стабільне регулювання потужності та надійну роботу, що
відкриває перспективи її застосування у промислових системах, які потребують точного температурного регулювання та
робастної роботи за змінних умов навантаження. Бібл. 28, табл. 1, рис. 17.
Ключові слова: резонансний багаторівневий інвертор, адаптивне керування, селективне усунення гармонік,
індукційне нагрівання.
Introduction. High-frequency power electronic
converter, typically a resonant inverter, is a widely adopted
circuit in industrial manufacturing, such as
telecommunication systems, the induction heating, wireless
energy transfer, and high-frequency power conversion
systems [1, 2]. However, the most high-frequency
applications are highly nonlinear. The electrical parameters
of these systems, such as their equivalent resistance and
inductance, vary significantly with temperature and
magnetic field intensity [3]. These variations shift the
resonant frequency and the load impedance, leading to
fluctuations in output power and potential detuning from
the optimal zero voltage switching (ZVS) operation, which
is crucial for efficiency [4]. Consequently, a fixed-gain
controller, such as a standard PID controller, is often
inadequate to maintain consistent performance across the
entire operating range. Its performance degrades under
parameter variations, leading to poor disturbance rejection
and potential instability [5].
48 Electrical Engineering & Electromechanics, 2026, no. 4
Therefore, the problem addressed in this paper is the
design of a robust control strategy for a multilevel
resonant inverter that can ensure stable output power and
dynamic performance despite the nonlinear and time-
varying nature of the load. Solving this problem is highly
relevant for advancing industrial manufacturing processes
that demand precise control and high reliability.
Multilevel converters have garnered significant attention
in recent years for medium and high power applications
[6]. Compared to traditional inverters, they offer superior
performances, including reduced output voltage harmonic
distortion, lower electromagnetic interference, and the
ability to operate at higher voltages with devices of lower
voltage rating [7]. For high-frequency applications,
multilevel topologies can provide a stepped voltage
waveform that more closely approximates a sinusoid,
reducing the stress on the resonant tank components.
Review of recent publications demonstrate that an
extensive research have been conducted on both
multilevel inverter topologies and resonant inverters. In
terms of topologies, reduced-switch-count multilevel
inverters are an active area of research to minimize cost
and complexity [8]. The topology proposed in [9] presents
a 7-level inverter with 6 switches, which is less than a
conventional cascaded H-bridge. This work aligns with
the trend of topological simplification. The authors [10]
discussed the resent advances in resonant converters. The
study establishes the current level of knowledge,
technology, and advancement in converters, which helps
researchers identify knowledge gaps, and confirm the
novelty of their own research. In [11] the authors
proposed an extended topology for series resonant
inverter with high quality control for induction heating
applications. Furthermore, authors [12] presented the
analysis of 5-level LLC resonant converter with design
method for battery chargers. The discussed results
demonstrated the performance of the resonant converter
topologies. For inverters modulation, various techniques
are employed, such as the carrier-based pulse width
modulation (PWM) technique, space vector PWM
technique, and selective harmonic elimination (SHE)
technique are employed [13–15]. SHE is the well-
established, offering the advantage of direct elimination
of specific order harmonics. Recent works, such as [16],
employ optimization algorithms like particle swarm
optimization (PSO) to solve the nonlinear SHE equations
efficiently. Regarding modeling, the nonlinear nature of
resonant converters necessitates advanced modeling
techniques for controller design. The extended describing
function (EDF) method is a powerful tool for deriving
large-signal and small-signal models of resonant
converters, accurately predicting behavior around the
operating point [17]. The small-signal modeling approach
has demonstrated its effectiveness across a wide range of
systems, including LLC converters [18] and chopper [19].
In the domain of control, adaptive control techniques have
been proposed to handle system uncertainties. The model
reference adaptive control (MRAC) scheme, particularly
using the Massachusetts Institute of Technology (MIT)
rule for adaptation, is a direct method to make a system’s
output track that of a reference model [20]. This approach
has been explored for DC-DC converters [21] and motor
drives [22]. Integral-Proportional-Derivative (I-PD)
controller, a variant of the PID structure where the
integral action is on the reference and the proportional-
derivative actions are on the feedback, is known to reduce
overshoot and improve set-point response compared to
the parallel form [23–25].
Despite these advancements, several challenges
remain unsolved or are not fully addressed in the current
literature, including: Lack of comprehensive studies that
integrate a reduced-switch-count multilevel inverter, SHE
modulation for harmonic mitigation, an accurate EDF-
based dynamic model, and an advanced adaptive
controller for the specific challenges of resonant inverters.
Also, the application of MRAC-based adaptive I-PD
control, which combines the benefits of improved set-
point response I-PD with online parameter tuning MRAC,
to multilevel resonant inverters for induction heating as
example is not thoroughly explored. Thus, the unsolved
problem is the development and validation of a unified
control strategy that synergistically combines these
advanced techniques to achieve robust, stable, and high-
performance operation of a multilevel resonant inverter
under realistic, varying operating conditions.
The goal of this study is to investigate a five-level
resonant inverter combined with SHE technique to
eliminate the third harmonic and minimize the total
harmonic distortion (THD). In addition, this work
presents the development of a small-signal model to
handle the intrinsic nonlinear behavior of the system. This
model is then used for the design of an adaptive I-PD
controller through a simple and accurate approach.
Adaptive control is employed to ensure the system’s
robustness in the presence of parameter variations.
To achieve the goal, an emerging inverter topology
employing only 6 power switches is used, which is fewer
than the usual number of switches in a cascaded H-bridge
multilevel inverter. To generate a high-quality output
voltage waveform with minimized lower-order
harmonics, SHE technique is suggested. To address the
modeling issue, the EDF method is employed to create a
small signal model of the load with the proposed five-
level multilevel resonant inverter. This model makes it
possible to get a good idea of how the system will behave
around the operating point when the load changes. The
second objective of this paper is to synthesize an adaptive
I-PD controller based on MRAC structure, with
adaptation mechanisms governed by the MIT rule, to
autonomously adjust controller gains and maintain desired
performance. Finally purpose is to validate the accuracy
of the developed model and the efficacy of the proposed
control strategy through numerical simulations,
demonstrating improved disturbance rejection and robust
tracking performance under load variations.
The proposed five-level inverter. Figure 1 shows the
proposed single phase 5-levels multilevel resonant inverter
topology used in this study. The proposed topology
includes a DC power supply (Vi), 4 main switches T1 – T4
and 2 auxiliary switches T5 and T6. IGBT switches have
external freewheeling diodes. The oscillating circuit has a
series inductor Ls and a parallel resonant capacitor Cp. The
workpiece is put inside the N turns cooper coil with a
certain amount of air space. A capacitor Cb is put in series
with the primary of the transformer.
Electrical Engineering & Electromechanics, 2026, no. 4 49
Fig. 1. Proposed five-level multilevel resonant inverter
To make sure that the ZVS works during the heating
cycle, the converter is run above the system’s natural
frequency. The proposed inverter’s main task is to create
5 different levels of output voltage based on the state of
switches is on or off. Table 1 illustrates the output voltage
levels.
Table 1
The switches on-off condition
uab T1 T2 T3 T4 T5 T6
–Vi 0 1 1 0 1 0
–0.5Vi 0 1 1 0 0 1
0 0 1 0 1 0 0
0 0 1 0 1 0 0
0.5Vi 1 0 0 1 0 1
Vi 1 0 0 1 1 0
The typical waveforms for the five-level inverter
output voltage uab(t) with PWM operation modes are
illustrated in Fig. 2, where T is the period, f is frequency,
θ1, θ2 are the switching angles.
Fig. 2. Typical output voltage waveform
SHE strategy. The output voltage in Fig. 2 is a
stepped waveform, and θ1 and θ2 are the angles at which
the waveform switches. These angles need to be set so
that certain harmonics (3, 5, 7, etc.) are removed. Using
Fourier series analysis, we can write the output voltage as:
1 21 )sin()cos()cos(
2
)(
n s
i
ab tnnn
n
V
tu
, (1)
where n is the natural number; s is the angular switching
frequency; t is the time. The switching angles θ1 and θ2
must meet the following condition:
20 21 .
We use SHE to get rid of some harmonic components.
In this instance, the analysis of the output voltage waveform
through Fourier theory yields a collection of non-linear
transcendental equations. If these equations have a solution,
it gives the switching angles needed for a certain
fundamental component and a chosen harmonic profile. In
5-level multilevel inverter, to eliminate the 3rd harmonic
order the angles θ1 and θ2 are chosen to satisfy:
,0)3cos()3cos(
;
1
)cos()cos(
21
21
m
k (2)
where m is the modulation index (m = πh1/2kVi); k is the
number of switching angles (in our study k = 2); h1 is the
fundamental harmonic.
This equation is not linear and has trigonometric
terms. To solve these sets of equations, iterative methods
like Newton-Raphson or Gauss Jordan are well used. But,
the initial values must be chosen near to the optimal
solution. In this study PSO method is applied to find the
desired switching angles: θ1 and θ2. The PSO method is
well detailed in several previous researches, including
[16]. Figure 3 shows the switching angles as a function of
the modulation index m.
m
Fig. 3. Switching angles θ1 and θ2 vs modulation index m
The five-level resonant inverter under SHE control is
simulated, where the simulation parameters are
summarized in Appendix A. The applied output voltage
waveform uab(t) and the corresponding harmonic spectrum
that revealed elimination of 3rd harmonic are shown in
Fig. 4, 5, respectively. The output voltage has 5 levels:
Vi, 0.5Vi, 0, –0.5Vi, –Vi, and the wave shape is near
sinusoidal. The output voltage has a THD of 25.46 %.
From the harmonic spectrum the 3rd harmonic is absent
from the waveform as predicted. Figure 6 shows the
resulting current is(t) from applying the voltage of Fig. 4
to the LLC load. The corresponding harmonic spectrum is
shown in Fig. 7. From the harmonic spectrum, the THD
of the current waveform equals 9.43 %.
t, ms
uab, V
Fig. 4. Output voltage waveform of the proposed inverter
Fig. 5. Harmonic spectrum of the applied voltage
50 Electrical Engineering & Electromechanics, 2026, no. 4
t, ms
is, A
Fig. 6. Output current waveform of the proposed inverter
Fig. 7. Harmonic spectrum of the output current is(t)
Generalized small signal model. To create the
control system for the five-level multilevel resonant
inverter shown in Fig. 1, it’s necessary to make a linear
mathematical model of the real system. It is possible to
model the induction heating load (the copper coil and the
workpiece) by putting together its equivalent resistance R0
and equivalent inductance L0 [26]. The equivalent circuit
of the proposed inverter is given in Fig. 8.
Fig. 8. The equivalent circuit of the proposed inverter
The primary side of the transformer is denoted by
the following equivalents R, L and C:
pC
n
n
CL
n
n
LR
n
n
R
2
1
2;0
2
2
1;0
2
2
1
,
where (n1/n2) refers to the turns ratio of the transformer.
The resonant frequency ω0, the quality factor Q and the
inductance ratio Le of the system are given by:
L
L
L
C
L
R
Q
LCL
LL s
e
s
s
,
1
,0 .
Theoretically, the current gain of the inverter can be
given as:
22
0)(
QLL
QL
I
I
H
ee
e
s
i
. (3)
Figure 9, which was obtained from (3), shows the
values of |Hi| for different inductance ratio Le and Q
factors.
From Fig. 9, when ωs ≈ ω0 and Q >> 1 (Q ≈ 30), the
current is amplified over the heating coil with gain which
can be simplified by: |Hi| ≈ Le.
The circuit’s differential nonlinear equations can be
expressed using the symbols in Fig. 8, as follows:
Le
Hi
Fig. 9. Current gain as function of Le and Q
.d)()(
1
)(
;
d
)(d
)()(
);(
d
)(d
)(
ttiti
c
tv
t
ti
LtRitv
tv
t
ti
Ltu
sc
c
c
s
sab
(4)
The state vector is chosen as:
tcs titvtitx )()()()( .
The input u(t) and the output y(t) variables are given
as: u(t) = uab(t), y(t) = vc(t).
In order to provide the generalized small signal
model, we need to calculate the coefficients of Fourier
series for the circuit state variables. In this case, the state
variables are complex values which can be defined as:
.
;
;
331
221
111
qd
qdc
qds
jxxi
jxxv
jxxi
(5)
The 1st harmonic of the input voltage is given by:
211
coscos
2
i
ab
V
u . (6)
The initial harmonic approximation of is(t), vc(t) and
i(t) is valid under the conditions: Q >> 1 and ωs ≈ ω0.
We can break down 12 into 6 equations by replacing
(5) and (6) into (4) and using the harmonic balance
method. The large signal model of the system is
constructed as follows:
.
1
;
1
;
11
;
11
;
1
;)cos()cos(
21
2333
2333
3122
3122
211
21211
qdsqq
dqsdd
qqdsq
ddqsd
q
s
dsq
s
i
d
s
qsd
x
L
xx
L
R
x
x
L
xx
L
R
x
x
C
x
C
xx
x
C
x
C
xx
x
L
xx
L
V
x
L
xx
(7)
The new state variables become:
tqdqdqd xxxxxxtx 332211)( .
The input controls are defined as:
tis Vtu 21)( .
The output variable is expressed as:
2
3
2
3)( qd xxty .
Electrical Engineering & Electromechanics, 2026, no. 4 51
Digital simulations were conducted in
MATLAB/Simulink to validate the accuracy of the
developed large-signal model. The responses of the
proposed mathematical model and the simulated circuit
were compared under identical operating conditions. The
comparative results are presented in Fig. 10, 11.
t, ms
vc, V
Fig. 10. Comparison of voltage between the large signal model
and the simulated circuit
t, ms
i, A
Fig. 11. Comparison of current between the large signal model
and the simulated circuit
These results show the good concordance existing
between the developed mathematical model and the
simulated circuit. Keep in mind that the large signal
model is a nonlinear equation because it has the product
(ωs xid) and (ωs xiq), where ωs, xid and xiq are all
independent variables. It is hard to use this model to make
the control system. Once the system’s large signal model
has been found, the nonlinear equations in (7) can be
linearized around the operating point. This lets us make
the small signal transfer function from any input Vi, θ1, θ2,
ωs to any desired output is, vc, i, P.
To find the system’s equilibrium operation point, we
can solve the equation above by setting the derivatives of
the system equations (7) equal to zero. Using Taylor
series, we get the small signal model of the system by
perturbing and linearizing the large signal model around
the operating point (U0, X0, Y0):
.)(~,)(~)(
;)(~,)(~)(
;)(~,)(~)(
00
00
00
YtytyYty
UtutuUtu
XtxtxXtx
(8)
The small signal model’s state equations are written
in matrix form as follows:
),(~)(~
);(~)(~)(~
txCty
tuBtxAtx
s
ss
(9)
where:
)(
~
)(~
;~~~~)(~
;~~~~~~)(~
21
332211
tity
vtu
xxxxxxtx
t
si
t
qdqdqd
The matrix’s As, Bs and Cs are given in Appendix B.
Adaptive I-PD control. Adaptive control has been
successfully utilized in control practice for a diverse range
of engineering challenges, including DC-DC converters.
This method is used when the system is hard to model
because it is complicated or not linear. The suggested
resonant inverters induction heating systems are in this
group because they have a structure that changes over
time and parts that are not linear.
Putting 1
~ , 2
~ , s
~ equal to zeros, the small signal
transfer function from the input voltage )(~ svi to the
output inductor current )(
~
si is derived as follow:
sss
i
BASIC
sv
si
sG 1
)(~
)(
~
)( . (10)
In order to confirm the accuracy of the resulting
small-signal model, the dynamic response )(
~
ti of G(s) is
compared with the response of the simulated circuit for
the same random input signal where the magnitude is
varied between –5 % and +5 % of Vi. Simulation results
are given in Fig. 12.
t, ms
i, A
Fig. 12. Comparison between the small signal model
and the simulated circuit
From the results shown in Fig. 12, there is a small
deviation between the response output of the resulting
mathematic model and the response of the simulated
circuit. This confirms that around the operation point, the
resulting model G(s) agrees well with the dynamics of the
simulated resonant inverter.
The transfer function (10) is a 6 order system, which
has 6 poles:
.100152.00315.0
;107071.00127.0
;103990.10315.0
5
6,5
5
4,3
5
2,1
jp
jp
jp
To determine graphically the phase margin and the
gain margin of the open loop system (8), the magnitude vs
phase plot is sketched on the Nichols chart given in Fig. 13.
From the Nichols charts, the minimum gain margin
Gm, the phase margin Pm, and the associated frequencies
ωgm and ωpm are given by:
Gm = 0.106 dB;
Pm = [94.6°, 64.5°, –109.8°, –135.5°, 100.9°];
ωgm = 7.01104 rad/s;
ωpm = [1.2104, 5.6104, 8.4104, 1.3105, 1.4105].
Figure 14 shows the schematic diagram of the
proposed adaptive I-PD control structure. The system has
2 loops. The inner loop is consisting of the system G(s)
and the I-PD controller. In order to force the system to
52 Electrical Engineering & Electromechanics, 2026, no. 4
track the reference the outer loop (adaptive mechanism) is
used the adjust the controller parameters Kp, Ki, and Kd. In
this instance, the required performance specifications are
articulated through a reference model GRM(s).
Fig. 13. Nichols chart of the open loop system G(s)
Fig. 14. MRAC I-PD controller block diagram
The output signal )(~ svi is calculated as:
)(
~
1
)()(~ si
sT
sK
Ks
s
K
sv
f
d
p
i
i
, (11)
where Tf is the filter time constant.
Based on the popular MIT rule, the adaptive
mechanism needs the following signals ε(s), e(s) and
)(
~
si to adjust the I-PD controller [27, 28]. In our case,
the controller parameter β is adjusted to minimizing the
cost function J given by:
)(
2
1
)( 2 eJ , (12)
where:
)(
~
)(
~
)( sisise m ; dip KKK ,, .
The MIT rule affirms that the rate of change of β is
directly related to the negative gradient of J [28]. The
controller parameter β is calculated by:
)()(
d
d e
e
J
t
, (13)
where )(e is the sensitivity derivative of error with
regard to β; γ is the adaptation gain. In practice, the choice
of adaptation gain is very important, and its value depends
on the levels of the signals.
The closed-loop transfer function of the system with
the suggested I-PD structure in control theory is given by:
)(
)(
)(
~
)(
~
sD
sN
si
si
r
, (14)
where:
)()()( sGsGKsN fi ;
)()()()()( 2 sGsKssGKsGKssGsD dfpfif ;
1)( sTsG ff .
Using (13), the controller parameters Kp, Ki and Kd
can be determinate by:
.
~
~
d
d
;
~
~
d
d
;
~
~
d
d
d
d
d
i
i
i
p
p
p
K
i
i
e
e
J
t
K
K
i
i
e
e
J
t
K
K
i
i
e
e
J
t
K
(15)
From (15), the adaptation I-PD law can be described
with respect to the original controller parameters as:
).(
~
)(
)(
d
d
;)(
~
)(
~
)(
)()1(
d
d
);(
~
)(
)()1(
d
d
2
si
sD
sGs
e
t
K
sisi
sD
sGsT
e
t
K
si
sD
sGsTs
e
t
K
d
d
r
f
i
i
f
p
p
(16)
It is important to choose the reference model so that
the closed loop response has an overshoot of less than
5 % and a settling time < 5 ms.
To test how well the proposed adaptive I-PD
controller works, we compare the response results of the
real system and the reference model. Figures 15, 16
display the results of the comparison and the associated
tracking error e(t), respectively.
t, ms
i, A
Fig. 15. Output current responses i(t) under sinus input reference
t, ms
e
Fig. 16. Tracking error variation with adaptive I-PD controller
These figures show that the suggested control method
can make the system output current follow the reference
model. This is justified by the correct selection of the
adaptation gains γp, γi and γd, which ensure the transient
performance of the output current regarding rise time,
overshoot, and settling time. The controller demonstrates a
fast and well-damped response, with an estimated rise time
of 1.4 ms, a low overshoot of approximately 5 %, and a
settling time of about 5 ms. These results confirm the
Electrical Engineering & Electromechanics, 2026, no. 4 53
ability of the controller to achieve rapid tracking while
maintaining system stability and robustness.
We tested the proposed adaptive I-PD control under
load variation, where the load changes from 0.107 Ω to
0.0749 Ω at t = 40 ms, to make sure it was sufficiently
robust. Figure 17 shows that the tracking works well and
the disturbances are very well rejected.
t, ms
i, A
Fig. 17. Output current responses i(t) with disturbance rejection
To reject the disturbance, the system has to follow
the output of the reference model by changing the input
the input )(~ svi . In this case, the control parameters come
together to form a constant value, and the tracking error
e(t) is getting closer to zero.
Conclusions. This paper describes the modeling and
control of a five-level resonant inverter using SHE
technique and an adaptive I-PD strategy based on MRAC.
A five-level resonant inverter topology with a
reduced number of switches (6 switches instead of 8 in
the conventional H-bridge topology) was successfully
tested. This resulted in a 25 % reduction in power
semiconductor devices compared to the conventional
cascaded H-bridge structure, leading to lower hardware
complexity and reduced cost.
SHE technique successfully eliminated the 3rd
harmonic component from the output voltage waveform.
THD of the inverter output voltage equals 25.46 %, while
the load current THD equals 9.43 %.
An accurate generalized small-signal dynamic model
of the inverter was derived using the EDF method. The
model prediction error, when compared to full
MATLAB/Simulink nonlinear simulations, did not
exceed 4.1 %, thereby validating the effectiveness and
precision of the proposed modeling approach.
The proposed MRAC-based adaptive I-PD controller
demonstrates excellent performance in tracking reference
signal. The controller ensures a fast and well-damped
response, with an estimated rise time of 1.4 ms, a low
overshoot of approximately 5 %, and a settling time of about
5 ms. The controller demonstrates high responds to abrupt
changes of load parameter (30 % change of the resistance
value). These results confirm the ability of the controller to
achieve rapid tracking while maintaining system stability and
robustness, highlighting its effectiveness for current control
in five-level resonant inverter system.
The future work will focus on building a hardware
prototype of the proposed circuit to validate the simulation
results and assess practical implementation challenges. The
next step is developing more mathematically rigorous
adaptation laws, potentially grounded in Lyapunov stability
theory to reinforce the theoretical stability guarantees of the
proposed control strategy, and extending the control
strategy to simultaneously optimize multiple performance
indices, such as efficiency and tracking error. On the other
hand, to explore new research topics, it is paving the way to
study other related systems such as the extension of the
system with other converter topologies and the integration
of renewable energy such as photovoltaic solar, wind, or
hybrid systems like photovoltaic-wind, which have
garnered significant attention due to their sustainability.
Appendix A. The equivalent parameters used in
numerical simulation are: resistor R = 0.107 Ω;
inductance L = 14.13 μH; capacitance C = 17.7 μF;
inductance Ls = 70 μH; DC power source Vi = 540 V;
switching angles θ1 = 25°, θ2 = 34°; factor Q = 8.35;
ωs = 7.07104 rad/s.
Appendix B. The matrices As, Bs, Cs are:
L
R
L
L
R
L
CC
CC
L
L
A
s
s
s
s
s
s
s
s
s
0
0
0
0
0
0
1
000
0
1
00
1
00
1
0
0
1
00
1
00
1
00
000
1
0
;
t
svs BBBBB ][
21 ;
2
03
2
03
03
2
03
2
03
030000
qd
q
qd
d
s
xx
x
xx
x
C ,
where:
t
s
v L
B
00000coscos
4
2010
;
t
s
i
L
V
B
00000cos
4
101
;
t
s
i
L
V
B
00000cos
4
202
;
tdqdqdqs XXXXXXB 030302020101 .
Conflict of interest. The authors declare that they
have no conflicts of interest.
REFERENCES
1. Jaafari B., Namadmalan A. Design and parameter estimation of
series resonant induction heating systems using self-oscillating tuning
loop. Journal of Applied Research in Electrical Engineering, 2022, vol.
1, no. 1, pp. 42-49. doi: https://doi.org/10.22055/jaree.2021.36904.1025.
2. Irwanto M., Kita L.K.W. An application of multi-magnetic circular
planar spiral relay to improve the performance of wireless power transfer
system. Electrical Engineering & Electromechanics, 2024, no. 6, pp. 19-
26. doi: https://doi.org/10.20998/2074-272X.2024.6.03.
3. Namadmalan A. Universal Tuning System for Series-Resonant
Induction Heating Applications. IEEE Transactions on Industrial
Electronics, 2017, vol. 64, no. 4, pp. 2801-2808. doi:
https://doi.org/10.1109/TIE.2016.2638399.
4. Yachiangkam S., Sangswang A., Naetiladdanon S., Koompai C.,
Chudjuarjeen S. Steady-State Analysis of ZVS and NON-ZVS Full-
Bridge Inverters with Asymmetrical Control for Induction Heating
54 Electrical Engineering & Electromechanics, 2026, no. 4
Applications. Journal of Power Electronics, 2015, vol. 15, no. 2, pp. 544-
554. doi: https://doi.org/10.6113/JPE.2015.15.2.544.
5. Borase R.P., Maghade D.K., Sondkar S.Y., Pawar S.N. A review of
PID control, tuning methods and applications. International Journal of
Dynamics and Control, 2021, vol. 9, no. 2, pp. 818-827. doi:
https://doi.org/10.1007/s40435-020-00665-4.
6. Ebrahimi F., Wndarko N.A., Gunawan A.I. Wild horse optimization
algorithm implementation in 7-level packed U-cell multilevel inverter to
mitigate total harmonic distortion. Electrical Engineering &
Electromechanics, 2024, no. 5, pp. 34-40. doi:
https://doi.org/10.20998/2074-272X.2024.5.05.
7. Priyanka G., Surya Kumari J., Lenine D., Srinivasa Varma P., Sneha
Madhuri S., Chandu V. MATLAB-Simulink environment based power
quality improvement in photovoltaic system using multilevel inverter.
Electrical Engineering & Electromechanics, 2023, no. 2, pp. 43-48. doi:
https://doi.org/10.20998/2074-272X.2023.2.07.
8. Parimalasundar E., Muthukaruppasamy S., Dharmaprakash R.,
Suresh K. Performance investigations of five-level reduced switches
count Η-bridge multilevel inverter. Electrical Engineering &
Electromechanics, 2023, no. 6, pp. 58-62. doi:
https://doi.org/10.20998/2074-272X.2023.6.10.
9. Ebrahimi F., Windarko N.A., Gunawan A.I. Minimizing Total
Harmonic Distortion of 7-Level Packed U-Cell Multilevel Inverter Using
Whale Optimization Algorithm. Indonesian Journal of Computer
Science, 2023, vol. 12, no. 6, pp. 3217-3231. doi:
https://doi.org/10.33022/ijcs.v12i6.3507.
10. Alatai S., Salem M., Ishak D., Das H.S., Alhuyi Nazari M.,
Bughneda A., Kamarol M. A Review on State-of-the-Art Power
Converters: Bidirectional, Resonant, Multilevel Converters and Their
Derivatives. Applied Sciences, 2021, vol. 11, no. 21, art. no. 10172. doi:
https://doi.org/10.3390/app112110172.
11. Herasymenko P., Pavlovskyi V., Yurchenko O. A series-resonant
inverter with extended topology and pulse-density-modulation control for
induction heating applications. International Journal of Power
Electronics and Drive Systems (IJPEDS), 2022, vol. 13, no. 1, pp. 348-
367. doi: https://doi.org/10.11591/ijpeds.v13.i1.pp348-367.
12. Alatai S., Salem M., Alhamrouni I., Ishak D., Bughneda A., Kamarol
M. Design Methodology and Analysis of Five-Level LLC Resonant
Converter for Battery Chargers. Sustainability, 2022, vol. 14, no. 14, art.
no. 8255. doi: https://doi.org/10.3390/su14148255.
13. Elamri O., Toubal Maamar A.E., Oukassi A., El Kharki A.,
Hammoudi A., Mekhilef S. SVPWM-Based Control of a Three-Phase
Five-Level NPC Inverter for Grid-Connected Solar Power System. 2025
5th International Conference on Innovative Research in Applied Science,
Engineering and Technology (IRASET), 2025, pp. 1-6. doi:
https://doi.org/10.1109/IRASET64571.2025.11008300.
14. Mohamed O., Jalil A., Bouazza E.M., Yassine L., Mohamed B.
Study and Realization of a Single-Phase Solar Inverter with Harmonics
Rejection. Advances in Science, Technology and Innovation, 2024, pp.
59-65. doi: https://doi.org/10.1007/978-3-031-51796-9_7.
15. Akaaboune J., El Mourabit B., Oulaaross M., Benchagra M. A New
SPWM Approach for High-Performance Single-Phase Half-Bridge
Inverters with Pure Sine Wave. Advances in Science, Technology and
Innovation, 2024, pp. 49-57. doi: https://doi.org/10.1007/978-3-031-
51796-9_6.
16. Toubal Maamar A.E. Analysis and Experimental Validation of
Selective Harmonic Elimination in Single-Phase Five-Level Inverter
using Particle Swarm Optimization Algorithm. Electronics, 2022, vol. 26,
no. 2, pp. 65-72. doi: https://doi.org/10.53314/ELS2226065M.
17. Yue X., Wang X., Blaabjerg F. Review of Small-Signal Modeling
Methods Including Frequency-Coupling Dynamics of Power Converters.
IEEE Transactions on Power Electronics, 2019, vol. 34, no. 4, pp. 3313-
3328. doi: https://doi.org/10.1109/TPEL.2018.2848980.
18. Abbasi A., Ghayebloo A. Dynamic Modeling of LLC Resonant
Converter Using Extracted Two‐Harmonic Approximation and
Comparative Analysis With Other Approaches. International Journal of
Circuit Theory and Applications, 2025, vol. 53, no. 9, pp. 5457-5466. doi:
https://doi.org/10.1002/cta.4395.
19. Ayachit A., Kazimierczuk M.K. Averaged Small-Signal Model of
PWM DC-DC Converters in CCM Including Switching Power Loss.
IEEE Transactions on Circuits and Systems II: Express Briefs, 2019, vol.
66, no. 2, pp. 262-266. doi: https://doi.org/10.1109/TCSII.2018.2848623.
20. Gupta D., Kumar A., Kumar G. Investigations on MIT and
Lyapunov rule-based modified MRAC for noninteracting and interacting
two-tank coupled systems. FME Transactions, 2024, vol. 52, no. 2, pp.
305-318. doi: https://doi.org/10.5937/fme2402305G.
21. Kahani R., Jamil M., Iqbal M.T. Direct Model Reference Adaptive
Control of a Boost Converter for Voltage Regulation in Microgrids.
Energies, 2022, vol. 15, no. 14, art. no. 5080. doi:
https://doi.org/10.3390/en15145080.
22. Nesri M., Benkadi H., Sifelislam G., Idir A., Benkhoris M.F. Robust
Adaptive Speed Control for the DC Motor Based on a Modified MRAC.
Communications - Scientific Letters of the University of Zilina, 2024, vol.
26, no. 4, pp. 38-46. doi: https://doi.org/10.26552/com.C.2024.055.
23. Shanthini C., Devi V.S.K., Rajendran S., Jena D. Comparative
analysis of PID, I-PD and fractional order PI-PD for a DC-DC converter.
2022 IEEE North Karnataka Subsection Flagship International
Conference (NKCon), 2022, pp. 1-5. doi:
https://doi.org/10.1109/NKCon56289.2022.10126840.
24. Shiryayeva O., Suleimenov B., Kulakova Y. Optimal Design of I-PD
and PI-D Industrial Controllers Based on Artificial Intelligence
Algorithm. Algorithms, 2024, vol. 17, no. 7, art. no. 288. doi:
https://doi.org/10.3390/a17070288.
25. Raja G.L. Robust I-PD Controller Design with Case Studies on
Boiler Steam Drum and Bioreactor. 2023 15th International Conference
on Computer and Automation Engineering (ICCAE), 2023, pp. 486-491.
doi: https://doi.org/10.1109/ICCAE56788.2023.10111205.
26. Ahmed N.A. High-Frequency Soft-Switching AC Conversion
Circuit With Dual-Mode PWM/PDM Control Strategy for High-Power
IH Applications. IEEE Transactions on Industrial Electronics, 2011, vol.
58, no. 4, pp. 1440-1448. doi: https://doi.org/10.1109/TIE.2010.2050752.
27. Sachit S., Vinod B.R. MRAS Based Speed Control of DC Motor
with Conventional PI Control – A Comparative Study. International
Journal of Control, Automation and Systems, 2022, vol. 20, no. 1, pp. 1-
12. doi: https://doi.org/10.1007/s12555-020-0470-1.
28. Swarnkar P., Jain S., Nema R.K. Effect of Adaptation Gain in Model
Reference Adaptive Controlled Second Order System. Engineering,
Technology & Applied Science Research, 2011, vol. 1, no. 3, pp. 70-75.
doi: https://doi.org/10.48084/etasr.11.
Received 02.03.2026
Accepted 12.05.2026
Published 02.07.2026
A. Garmat1, PhD, Associate Professor,
A.E. Toubal Maamar2, PhD, Associate Professor,
T. Abdelouahed 2, PhD, Associate Professor,
N. Bensafi 2, PhD, Associate Professor,
1 Ziane Achour University of Djelfa, Algeria,
e-mail: a.garmat@univ-djelfa.dz (Corresponding Author)
2 University of M’hamed Bougara of Boumerdes, Algeria.
How to cite this article:
Garmat A., Toubal Maamar A.E., Abdelouahed T., Bensafi N. Design of an adaptive I-PD controller for a five-level resonant inverter
with selective harmonic elimination technique. Electrical Engineering & Electromechanics, 2026, no. 4, pp. 47-54. doi:
https://doi.org/10.20998/2074-272X.2026.4.07
|
| id | eiekhpieduua-article-342998 |
| institution | Electrical Engineering & Electromechanics |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-02T01:00:24Z |
| publishDate | 2026 |
| publisher | National Technical University "Kharkiv Polytechnic Institute" and Аnatolii Pidhornyi Institute of Power Machines and Systems of NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | eiekhpieduua/c8/edc891588c278b7dc171f9c8217b0dc8.pdf |
| spelling | eiekhpieduua-article-3429982026-07-01T21:42:56Z Design of an adaptive I-PD controller for a five-level resonant inverter with selective harmonic elimination technique Design of an adaptive I-PD controller for a five-level resonant inverter with selective harmonic elimination technique Garmat, A. Toubal Maamar, A. E. Abdelouahed, T. Bensafi, N. резонансний багаторівневий інвертор адаптивне керування селективне усунення гармонік індукційне нагрівання resonant multilevel inverter adaptive control selective harmonic elimination induction heating Introduction. Resonant inverters are indispensable in demanding applications such as induction heating, wireless energy transfer, and high-frequency power conversion systems. Problem. The main topologies for realizing resonant inverters are the half-bridge and full-bridge configurations, but the multilevel topology is not well-known for resonant inverters because their modeling and control design are challenging steps. The goal of this study is to investigate a five-level resonant inverter combined with the selective harmonic elimination (SHE) technique to eliminate the third harmonic and minimize the total harmonic distortion (THD). Methodology. The structure of the proposed inverter and the SHE modulation technique are presented to illustrate harmonic reduction in the applied voltage. To address the inherent nonlinearities of the system, the extended describing function (EDF) method is employed to derive a generalized small-signal state-space model from any defined input to any desired output. This model enables accurate prediction of system behavior around the operating point. Based on this model, an adaptive I-PD controller incorporating a model reference adaptive control (MRAC) mechanism, designed according to the Massachusetts Institute of Technology (MIT) rule, is developed. The adaptive mechanism continuously tunes the proportional, derivative, and integral gains to maintain the desired performance despite load and parameter changes. Results. Numerical simulations validate the accuracy of the developed model and demonstrate that the adaptive I-PD control significantly ensures the system’s robustness. The results indicate that the THD of voltage and current are 25.46 %, and 9.43 %, respectively. The third harmonic is well eliminated. The model prediction error, when compared to full MATLAB/Simulink nonlinear simulations, did not exceed 4.1 %, thereby validating the effectiveness and precision of the modeling approach. The presented MRAC-based adaptive I-PD controller demonstrates high performance in tracking reference signal and responds to abrupt changes of load parameter (30 % change of the resistance value), highlighting its effectiveness for current control in five-level resonant inverter system. Scientific novelty. The proposed framework combines SHE-based harmonic mitigation, EDF-based modeling, and MRAC-based adaptive I-PD control for multilevel resonant inverters. This integration provides a generalized and flexible approach for handling system nonlinearities and improving dynamic performance. Practical value. The results confirm the feasibility of implementing adaptive I-PD control for five-level resonant inverters. The proposed scheme ensures high efficiency, stable power regulation, and reliable operation, paving the way for industrial applications requiring precise temperature control and robust performance under varying load conditions. References 28, table 1, figures 17. Вступ. Резонансні інвертори є невід’ємною складовою сучасних високотехнологічних застосувань, зокрема систем індукційного нагрівання, бездротового передавання енергії та високочастотних систем перетворення електроенергії. Проблема. Основними топологіями реалізації резонансних інверторів є напівмостові та мостові схеми. Водночас багаторівнева топологія для резонансних інверторів залишається недостатньо дослідженою через складність математичного моделювання та синтезу систем керування. Мета. Дослідження п’ятирівневого резонансного інвертора у поєднанні з методом селективного усунення гармонік (SHE) для пригнічення третьої гармоніки та мінімізації повного гармонічного спотворення (THD). Методика. Представлено структуру запропонованого інвертора та метод модуляції SHE для демонстрації зменшення гармонічних складових прикладеної напруги. Для врахування притаманних системі нелінійностей застосовано метод розширеної описувальної функції (EDF), за допомогою якого отримано узагальнену малосигнальну модель у просторі станів для довільно визначених входів і виходів. Розроблена модель забезпечує точне прогнозування поведінки системи поблизу робочої точки. На її основі синтезовано адаптивний I-PD-регулятор із механізмом адаптивного керування за еталонною моделлю (MRAC), побудованим відповідно до правила Массачусетського технологічного інституту (MIT). Адаптивний механізм забезпечує безперервне налаштування пропорційного, диференціального та інтегрального коефіцієнтів для підтримання необхідних показників роботи за зміни навантаження та параметрів системи. Результати. Чисельне моделювання підтвердило точність розробленої моделі та показало, що адаптивне I-PD-керування суттєво підвищує робастність системи. Отримані результати свідчать, що THD напруги та струму становлять відповідно 25,46 % і 9,43 %. Третя гармоніка ефективно усувається. Похибка прогнозування моделі порівняно з повною нелінійною моделлю MATLAB/Simulink не перевищувала 4,1 %, що підтверджує ефективність і точність запропонованого підходу до моделювання. Представлений адаптивний I-PD-регулятор на основі MRAC демонструє високу якість відстеження опорного сигналу та ефективно реагує на різкі зміни параметрів навантаження (зміна опору на 30 %), що підтверджує його придатність для керування струмом у системі п’ятирівневого резонансного інвертора. Наукова новизна. Запропонований підхід поєднує метод селективного усунення гармонік SHE, моделювання на основі EDF та адаптивне I-PD-керування на основі MRAC для багаторівневих резонансних інверторів. Така інтеграція забезпечує узагальнений і гнучкий підхід до врахування нелінійностей системи та покращення її динамічних характеристик. Практична значимість. Отримані результати підтверджують можливість практичної реалізації адаптивного I-PD-керування для п’ятирівневих резонансних інверторів. Запропонована схема забезпечує високу ефективність, стабільне регулювання потужності та надійну роботу, що відкриває перспективи її застосування у промислових системах, які потребують точного температурного регулювання та робастної роботи за змінних умов навантаження. Бібл. 28, табл. 1, рис. 17. National Technical University "Kharkiv Polytechnic Institute" and Аnatolii Pidhornyi Institute of Power Machines and Systems of NAS of Ukraine 2026-07-02 Article Article application/pdf https://eie.khpi.edu.ua/article/view/342998 10.20998/2074-272X.2026.4.07 Electrical Engineering & Electromechanics; No. 4 (2026); 47-54 Электротехника и Электромеханика; № 4 (2026); 47-54 Електротехніка і Електромеханіка; № 4 (2026); 47-54 2309-3404 2074-272X en https://eie.khpi.edu.ua/article/view/342998/351644 Copyright (c) 2025 A. Garmat, A. E. Toubal Maamar, T. Abdelouahed, N. Bensafi http://creativecommons.org/licenses/by-nc/4.0 |
| spellingShingle | resonant multilevel inverter adaptive control selective harmonic elimination induction heating Garmat, A. Toubal Maamar, A. E. Abdelouahed, T. Bensafi, N. Design of an adaptive I-PD controller for a five-level resonant inverter with selective harmonic elimination technique |
| title | Design of an adaptive I-PD controller for a five-level resonant inverter with selective harmonic elimination technique |
| title_alt | Design of an adaptive I-PD controller for a five-level resonant inverter with selective harmonic elimination technique |
| title_full | Design of an adaptive I-PD controller for a five-level resonant inverter with selective harmonic elimination technique |
| title_fullStr | Design of an adaptive I-PD controller for a five-level resonant inverter with selective harmonic elimination technique |
| title_full_unstemmed | Design of an adaptive I-PD controller for a five-level resonant inverter with selective harmonic elimination technique |
| title_short | Design of an adaptive I-PD controller for a five-level resonant inverter with selective harmonic elimination technique |
| title_sort | design of an adaptive i-pd controller for a five-level resonant inverter with selective harmonic elimination technique |
| topic | resonant multilevel inverter adaptive control selective harmonic elimination induction heating |
| topic_facet | резонансний багаторівневий інвертор адаптивне керування селективне усунення гармонік індукційне нагрівання resonant multilevel inverter adaptive control selective harmonic elimination induction heating |
| url | https://eie.khpi.edu.ua/article/view/342998 |
| work_keys_str_mv | AT garmata designofanadaptiveipdcontrollerforafivelevelresonantinverterwithselectiveharmoniceliminationtechnique AT toubalmaamarae designofanadaptiveipdcontrollerforafivelevelresonantinverterwithselectiveharmoniceliminationtechnique AT abdelouahedt designofanadaptiveipdcontrollerforafivelevelresonantinverterwithselectiveharmoniceliminationtechnique AT bensafin designofanadaptiveipdcontrollerforafivelevelresonantinverterwithselectiveharmoniceliminationtechnique |