Real Time Discrete Optimized Adaptive Control for Ionic Polymer
Metal Composites
KYRIAKOS TSIAKMAKIS, VASILEIOS DELIMARAS, ARGYRIOS T. HATZOPOULOS,
MARIA S. PAPADOPOULOU
Department of Information and Electronic Engineering,
International Hellenic University,
Sindos, 57400,
GREECE
Abstract: - This paper describes a proposed method for optimizing the parameters of a Model Reference
Adaptive Control (MRAC) system. The MRAC system uses a reference model to control a plant with unknown
dynamics and continuously updates its parameters to improve control accuracy. The system requires an
adjustment of parameter , which participates in the feedback of the system but cannot be adjusted in real time
through trial and error. The proposed method uses optimization techniques to adjust the parameters in real
time, specifically at the start of the control process, when the maximum deviation of the plant from the
reference model is observed. The optimization technique varies the parameters and seeks the best solution to
quickly reduce the error. Once the optimal solution is found, the optimization is turned off, allowing the MRAC
to continue efficiently reducing the error. In the case of sudden changes in the error due to endogenous or
exogenous factors, optimization is activated again to redefine the parameters. The magnitude of the change
depends on the rate of error changes. The response of the IPMC was measured and compared against a
reference signal using three different control techniques MRAC, Model Reference Adaptive Control (MRAC),
MRAC-Taguchi, and MRAC-Taguchi-DCT, and the results show that the last penalizes frequencies beyond the
fundamental frequency through the cost function, resulting in negligible harmonic distortion.
Key-Words: - IPMC, Taguchi Optimization, Model Reference Adaptive Control, DCT.
Received: September 27, 2022. Revised: January 4, 2023. Accepted: February 7, 2023. Published: March 7, 2023.
1 Introduction
An Ionic Polymer Metal Composite (IPMC), is a
type of smart material that is made by sandwiching
an ion-exchange membrane between two metal
electrodes. The ion-exchange membrane is
composed of a hydrated polymer, typically Nafion,
which is capable of transporting ions. When an
electric potential is applied across the metal
electrodes, it creates an electric field within the ion-
exchange membrane, causing the ions to move and
create an actuation force. On the other hand, IPMC
sensors use IPMC to detect changes in mechanical
stress, pressure, or other physical parameters [1-2].
IPMCs have several interesting properties, such as
fast response time, high sensitivity, low driving
voltage, and excellent durability. These properties
have made IPMCs attractive for a wide range of
applications, including artificial muscles, MEMS,
actuators, grippers, sensors, and energy harvesting,
owing to their fast response time, high sensitivity,
low driving voltage, and excellent durability [3-5].
IPMCs can be modeled and controlled using several
different techniques, including electrical and
mechanical models. These models can be used to
design and optimize IPMC systems for various
applications, as well as to predict the behavior of
IPMCs under different operating conditions [6-8].
All these important manufacturing factors influence
the dynamic response of the material when
functioning as an actuator, making it challenging to
find an analytical model of the system.
Developing a control strategy is a viable option
for ensuring a certain dynamic reaction from the
IPMC. Various authors have worked on the design
of control techniques for IPMC materials with
various goals [10].
PID, MRAC, and LQR control are among the
main control algorithms used for controlling ionic
polymer composite (IPMC) actuators. However,
there are some differences in their effectiveness and
complexity. PID control is the most common and
simplest control algorithm used in the IPMC
actuators. It is a linear control algorithm that uses a
proportional term, integral term, and derivative term
to control the position of the IPMC actuator. The
PID controller is simple to implement and can be
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Kyriakos Tsiakmakis, Vasileios Delimaras,
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easily adjusted to satisfy the requirements of the
system. However, its linearity makes it less effective
in controlling nonlinear systems, such as IPMC
actuators [10]. MRAC control is a nonlinear control
algorithm that uses a model reference adaptive
control approach. It adjusts its control strategy in
real-time to accommodate changes in the system
and improve its performance. This makes MRAC
more effective than PID control in controlling
nonlinear systems [11]. However, MRAC requires
more computational resources and is more complex
to implement than PID control. LQR control is a
linear quadratic control algorithm that optimizes the
performance of a control system based on a
mathematical model of the system [12]. LQR
control uses a weighting matrix to balance the trade-
off between the control effort and control
performance. LQR control is more effective than
PID control [13] and MRAC control in controlling
IPMC actuators because it takes into account the
nonlinearity of the system. However, the LQR
control is also more complex and computationally
intensive than the other two algorithms.
The choice of control algorithm for IPMC
actuators depends on the requirements of the system
and available computational resources. PID control
is the simplest and easiest to implement, whereas
LQR control is the most effective but also the most
complex, and it is very difficult to implement them
in microcontrollers. The MRAC control is a good
compromise between the two, providing good
performance and ease of implementation.
The purpose of this control is to ensure the same
reaction, even if the dynamic behavior of the beam
changes over time. The proposed control technique
is based on a reference model-adaptive control
MRAC scheme. This type of strategy adjusts its
control rule to account for the fact that the actuator
parameters change over time. The control technique
involves compelling the entire system to conform to
a user-defined model reference system behavior,
even though the IPMC actuator response varies over
time.
Several authors have attempted to reduce the
effect of an actuator's dynamic drifts on the final
closed-loop response by employing robustness
strategies [14-15]. The MRAC control used in our
study was virtually insensitive to model dynamic
changes.
If γ remains constant, then for various frequencies,
the error takes longer to reach zero or can even
cause the system to become unstable.
For different frequencies, we need to adjust γ-
MRAC to avoid instability and increase the speed of
adaptation. In addition, a discrete form of the
control system is necessary for applying
optimization and analysis algorithms.
To improve the real-time system of adjustment at
various frequencies, optimization using the Taguchi
algorithm [16] and discrete cosine transform (DCT)
analysis were performed [17].
Discrete cosine transform (DCT) is a type of
mathematical transform that is widely used in digital
signal processing, image compression, and audio
compression. It converts a signal from its original
time or spatial domain into a transformed domain,
where the frequency components of the signal are
represented by a series of coefficients.
DCT is a powerful tool for determining the
frequencies present in a set of discrete data points.
DCT provides a clear representation of the
frequency content of the data by transforming the
data into a set of cosine functions with different
frequencies.
In this study, an improved MRAC system for
the fast and stable control of an IPMC strip was
implemented for use in embedded systems. One
potential use of our control with an IPMC strip is to
control the motion of a robotic fish or robotic arm
that uses artificial muscles made of IPMC materials.
Using the proposed controller, the system could
adapt its control parameters faster based on the
changing dynamics of the IPMC actuators, allowing
for more precise and robust control.
2 Subject & Methods
2.1 IPMC Model and Setup
The transfer function model identification of an
ionic polymer–metal composite (IPMC) involves
measuring the voltage response of the IPMC to a
known input signal, such as a step function, sweep
wave, or mixed signal, and then using mathematical
techniques to fit a transfer function model to the
measured response. The transfer function model
represents the relationship between the input signal
and output response and can be used to predict the
performance of the IPMC under different
conditions.
The transfer function obtained from the model
identification is of 4th order, but to be used in the
MRAC system, this means needed to be easily
implementable in discrete-time systems such as
microcontrollers, the 2nd order reduced version was
preferred, which is shown below in the z-domain.
(1)
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Measuring the displacement of an ionic polymer
metal composite (IPMC) material using a laser is a
common technique used in the research and
development of these materials.
In general, the displacement of an IPMC can be
measured using a laser-displacement sensor. This
type of sensor uses a laser to measure the distance
between the sensor and target surface. The
displacement can be calculated by reflecting the
laser off the surface of the IPMC and measuring the
time it takes for the light to return to the sensor.
Displacement can be measured in real time as the
IPMC deforms and changes its shape.
It is important to note that the accuracy of
displacement measurement can be influenced by
several factors, including the resolution of the laser
sensor, reflectivity of the surface of the IPMC, and
stability of the laser beam. To ensure accurate
measurements, it may be necessary to use high-
resolution laser sensors and calibrate the sensors
before use.
The experimental setup is shown in Fig.1.
Fig. 1: The experimental setup.
2.1 MRAC Analysis
Model Reference Adaptive Control (MRAC) is a
technique that uses a reference model to control a
plant with unknown dynamics. The reference model
represents the desired behavior of the plant, and the
control algorithm adjusts the control input to ensure
that the plant follows the reference model as closely
as possible. In the MRAC, the controller
continuously updates its parameters to improve the
accuracy of the control process. This results in a
more accurate control signal, even in the presence of
changes in plant dynamics over time.
In a previous work [11], it was mentioned that
, which participates in four points in the feedback
of the system, can be adjusted by the user. However,
this cannot occur in a real system, where the trial-
and-error method is difficult to apply. In addition,
the strain of the material and environmental
conditions of the system operation are all factors
that lead to a change in the response of the IPMC.
Thus, the IPMC should not receive a one-time
constant value for the entire duration of system
operation. Furthermore, the parameter is set for a
specific operating frequency and needs to be set to a
different value when the frequency is changed. This
creates the need for an adjustment of in real time
and for any changes that occur in the system. The
error between the output of the plant and reference
model is shown in Fig.2, when MRAC attempts to
fit the response of the plant to the reference model
for different frequencies and with the same value of
.
Fig. 2: Error between the output of the plant and the
reference model for and various
frequencies.
A value of , it does not work properly for
all frequencies. Even at a low frequency of
100 mHz, where the error approaches zero, the time
required for the error to be considered negligible
exceeds 30 s.
The error between the plant and reference model for
, is shown in Fig.3.
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Fig. 3: Error between the output of the plant and the
reference model for and various
frequencies.
For frequencies of 100 mHz and 2 Hz, we observe
that with this value of γ, the error decreases faster in
contrast to frequencies of 5 Hz and 10 Hz, but still
takes a long time to approach zero. In addition,
when is high, MRAC works more aggressively on
the system and with fast responses, which can also
lead to vibrations and mechanical stress. Fast
responses of the system are also seen in the
100 mHz error changes, where the error changes not
only at the input frequency but also at higher
frequencies. In Fig.4, the DFT spectrum of the error
is shown for an operating frequency of 100 mHz
input when γ has values -10 and -150, respectively.
Fig. 4: DFT spectrum of the error for 100 mHz
input for and .
The system response produces harmonics owing to
high values. This occurs when the MRAC
attempts to adjust the plant input appropriately such
that the plant's output matches that of the reference
model. Fig.5 presents the output of the plant
compared to the reference model for the above
operating frequency and values.
Fig. 5: The output of the plant has been compared
to the reference model for 100 mHz input,
and respectively.
It should be emphasized that parameter cannot
have a fixed value and should be adjusted with the
rest of the endogenous parameters of the system, but
also with various exogenous factors that affect the
system. Optimization of some parameters of a real-
time controller is achieved with many iterations of
the input before the controller starts working in real
time for the system it is intended for.
This occurs due to changes in the controller
parameters cause the system output to diverge from
the target. As a result, as the controller tries to return
the error to zero, it becomes increasingly difficult to
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do so as the parameters of the controller change.
Owing to the aforementioned factors, the
optimization of these parameters is not applied in
real time.
On the one hand, this can cause strain on the plant;
on the other hand, for every change in the
parameters, the system must be shut down to re-
adjust its parameters.
2.1 Improved MRAC using Taguchi
Optimization
In the proposed method, optimization techniques are
used and will act in real time during the system
startup. The principle of operation is based on the
fact that the MRAC requires some time to adjust the
output of the plant with the reference model, and at
the start of the control, the maximum deviation
(error) of the plant from the reference model is
observed. During this initial period, the proposed
method was exploited to optimize the system
parameters, specifically . Because optimization
techniques and not the trial-and-error method will
be used, it is easier in the four different feedbacks in
which participates to optimize four different
(, , , and ) instead of a common one to
provide the maximum flexibility in the system to
find a better solution, which will lead to a desired
output.
At the initial time interval, where the maximum
deviation occurs, that is, the maximum error,
optimization techniques can vary the parameters and
examine how quickly the error decreases or not. In
addition, within this interval, the best solution for
the parameter values that lead to a faster reduction
in the error is sought. Once this solution is found,
the optimization is turned off, and with the optimal
parameter values, the MRAC is free to continue
reducing the error further and more efficiently. The
continuous operation of the optimization algorithm
throughout the MRAC causes continuous deviations
in the error-zeroing process and creates continuous
oscillations in the system.
In the proposed technique, the optimization
method starts dynamically and more aggressively to
change the parameters within a short period of
time. Over time, the effect of the optimization
technique on the parameters decreased, leaving
the normal operation of the MRAC free.
Any endogenous or exogenous factors can cause
a sudden and significant change in error when the
MRAC is operating in normal mode (where the
error tends to be zero). Subsequently, the
optimization is activated again to redefine the
parameters. The optimization technique adjusts the
values of the parameters. The magnitude of the
change depends on the rate of error changes , from
a predetermined number of samples , according to
the following relationship (2):
(2)
where is the new value of the parameters and
is an initial scaling factor of the parameters,
where .
In addition, during the process, the optimization
technique reduces its effect on the variation of
parameters . This is achieved by continuously
increasing the coefficient ; thus, the new value of
is given by (3).
(3)
where is a scaling factor of the factor and
. As the coefficient approaches unity,
the change in parameters will be smaller, until
finally, no change occurs.
Taguchi optimization is an iterative
optimization method based on orthogonal arrays
developed by Taguchi [18]. Each row of the array is
a combination of values of the parameters to be
optimized. In each column of the array, the values
of each parameter, which were defined in levels,
were placed. Each level is a percentage change in
the value of the parameter, from its initial value
when the array was initialized.
For example, in Table 1, a Taguchi orthogonal
array of four factors (in our case, the parameters)
with three levels for each factor, named L9, is
depicted.
Table 1. The Taguchi L9 orthogonal array of four
factors.
Experiment
Numbers
1
1
1
1
1
2
1
2
2
2
3
1
3
3
3
4
2
1
2
3
5
2
2
3
1
6
2
3
1
2
7
3
1
3
2
8
3
2
1
3
9
3
3
2
1
Initially, a value (even random) is defined for each
factor that constitutes level 2, and the number 2 of
each column in the array is replaced by these initial
values.
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Level 1 is a negative percentage change of central
level 2, on the other hand, level 3 is a positive
percentage change of central level 2. This is exactly
the meaning of the coefficient, the percentage
increase or decrease of each factor to create the
three levels.
This array constitutes the design of experiments
(DOE), that is, each row of the array is a
combination of factor values to be tested and
evaluated by a fitness function (or cost function,
objective function, etc.). The array contains a subset
of all possible combinations to run in a certain
region to locate the value on which each factor must
converge to achieve a better score in the fitness
function.
The algorithm determines the effect of each
variable on output. It identifies the value (level) of
each factor that offers the best solution through the
relationship expressing the signal-to-noise ratio or
SN:
(4)
where is the experiment number (array row), is
the mean value of the Cost Function, and
is the
variance. The average value
is given by (5):
(5)
where is the number of trials, and is the value
of the Cost Function for the given experiment and
trial . Finally, is the total number of trials in the
experiment .
The calculation of variance
is given by (6):
(6)
The Cost Function score is evaluated for each array
combination. Either the score value should be
increased (positive or negative) as much as possible,
either to tend to zero. Therefore, the "minimum is
better" and "maximum is better" cases are
distinguished, depending on the problem. Thus,
relation (4) is formulated as follows for the
"minimum is better" case:
(7)
and for the case “maximum is better”:
(8)
The algorithm then calculates the mean value of SN
for each factor and level according to equation (9):
(9)
where is the number of the factor (column of the
array), is the number of the level and is the total
number of experiments (total number of rows of the
array).
Table 2 contains the mean value for each
factor and level.
Table 2. The mean value for each factor and
level.
Level
1
2
3
Rank
For each factor, the range of variation of , is
calculated, with the relation (10):
(10)
and is filled in on a new line of Table 2.
The greater the range in a parameter, the
greater the effect of the variable on the system and,
consequently, on the optimization process.
This occurs because the same percentage change in
a parameter (signal) causes a greater effect on the
output of the system and consequently on the Cost
Function.
In the "Rank" line of Table 2, the position of the
most important parameter is filled. All are
sorted in descending order, the one that will be first,
and therefore the largest, will get the number 1, and
the rest are filled in the same way.
The levels of each parameter for the
optimization process are chosen, where is the
maximum or the minimum, depending on the
“maximum is better” or “minimum is better” case.
This is because the values of these levels have the
maximum effect on the Cost Function.
The values of the levels of each factor that offer
a better score in the Cost Function are placed as
central (level 2) and the process is repeated again
from the beginning, until the goals set are achieved.
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2.2 Improved MRAC using Taguchi-DCT
Optimization
The Taguchi optimization chooses the appropriate
, but it does not consider the spectral content of
the response, resulting in an optimization stack into
a local minimum of the cost function without further
improvement. The choice of using the Taguchi
method may lead to a response with high-frequency
oscillations in the system.
The next step of the proposed methodology is to
apply the discrete cosine transform (DCT) technique
to the Taguchi optimization process, consider the
spectral content of the response to the cost function,
and determine the optimal values of to avoid
oscillations in the output of the system.
3 Problem Solution
3.1 Simulation Results
When the optimization is performed on the MRAC
while the process lasts, the error does not converge
to zero in a short time.
The MRAC attempts to minimize the error, but
the Taguchi optimization continuously changes the
parameters , , , and of the system. This
results in the error continuously deviating from zero,
occurring just in time when the Taguchi
optimization changes the parameters .
In Fig.6, the optimization procedure is
demonstrated, which exploits the initial time space
where the error takes its maximum value and is
described using markers. The error is depicted by a
blue line, and the upper boundary of the error
located at its peaks is indicated by a dashed red line.
The green circles describe the points where the
Taguchi optimization is activated and operates, that
is, the combinations of each row of the orthogonal
array. With red circles, the moment when all the
experimental runs of the orthogonal array are
completed, and the selection of the best parameters
that minimize the Cost Function are made.
The calculation of the DCT and imposition of
the penalty were performed at the points marked
with black asterisks. The calculation of the DCT and
finding the average of all peaks with a frequency
greater than the first peak participates in the
calculation of the Cost Function. It should be noted
that the first peak is the fundamental frequency of
the system's operation. This average for the highest
frequencies imposes an increase in the Cost
Function as a penalty that punishes the appearance
of high frequencies in the system so that the
parameters can be appropriately adjusted so that
rapid changes are not displayed in the system and
the system's response becomes smoother.
Fig. 6: The optimization procedure using Taguchi
and DCT.
Fig. 7: A zoomed region of the optimization
procedure using Taguchi and DCT.
In Fig.7, which is a small area in Fig.6. is presented,
where the operating points of the system are clearly
visible.
Taguchi-DCT optimization is performed in real
time at the system startup to adjust the
parameters. When the optimization objectives are
achieved, the optimization algorithm is turned off,
and the system is left free to operate only with
MRAC to adapt the plant to the reference model.
In Fig.8 the comparative results of the proposed
MRAC-Taguchi-DCT and simple MRAC methods
are presented, when the constant parameter is
and the frequency is 100 mHz.
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Fig. 8: Comparative results of the proposed
MRAC-Taguchi-DCT and simple MRAC methods,
and 100 mHz.
With the proposed optimization method, the error
converges to zero faster and the parameters of the
MRAC system converge and stabilize faster. In
Fig.9, the comparison results with MRAC-Taguchi-
DCT and simple MRAC for a frequency of 2 Hz are
presented. In Fig.10 comparison results for 10 Hz
are shown.
Fig. 9: Comparative results of the proposed
MRAC-Taguchi-DCT and simple MRAC methods,
and 2 Hz.
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Fig. 10: Comparative results of proposed MRAC-
Taguchi-DCT and simple MRAC methods,
and 10 Hz.
In Fig.11, the results of the MRAC-Taguchi
function without DCT optimization for a frequency
of 10 Hz are presented. The significant presence of
harmonic frequencies in the system is shown in
Fig.12 in contrast with Fig.13 where DCT
optimization is enabled and the harmonics have a
smaller appearance in the spectrum. Furthermore, as
shown in Fig.12, the amplitude of the fundamental
frequency of the error is suppressed. In both graphs,
the discrete Fourier transform (DFT) spectrum is
presented with the FFT and DCT for comparison.
Fig. 11: The MRAC-Taguchi operation without
DCT optimization at a frequency of 10 Hz.
Fig. 12: Presence of other harmonic frequencies in
the system without DCT optimization.
Fig. 13: The spectrum with DCT optimization.
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The values of for all the operating
frequencies presented in the graphs above are
listed in Table 3.
Table 3. Values of for all operating frequencies
of the tested methods.
Method
f (Hz)
MRAC
0.1/2/10
10
10
10
10
MRAC-
Taguchi
-DCT
0.1
10.31
85.93
9.45
8.59
MRAC-
Taguchi
-DCT
2
180.45
226.85
137.49
85.93
MRAC-
Taguchi
10
53163
147.68
116.16
660.99
MRAC-
Taguchi
-DCT
10
498.39
523.31
88.51
119.44
In particular, for the case of 10 Hz, a large
difference was observed in the values of the
parameters between MRAC-Taguchi and MRAC-
Taguchi-DCT.
3.2 Experimental Results
The results of the experimental setup are presented
in Fig.A in Appendix. The reference signal that the
IPMC should follow with its movement is shown at
the top of the graph, followed by the response of the
IPMC using only MRAC (with a fixed ), the
response with MRAC-Taguchi, and finally the
response with MRAC-Taguchi-DCT. The target is
approached rapidly by the MRAC-Taguchi
response, but the choices of made through the
optimization do not consider the frequency content
of the response, and the optimization can lock into a
local minimum without further improving . In
contrast, it is observed that with the MRAC-
Taguchi-DCT technique, frequencies beyond the
fundamental frequency are penalized through the
Cost Function, and the result appears with negligible
harmonic distortion, mainly due to low-amplitude
frequencies adjacent to the fundamental frequency.
Moreover, it appears that in the time interval from 0
to approximately 0.2 s, harmonic frequencies are
present, and it is this time where the necessary
number of samples is collected, in order for the
DCT to output more accurately the coefficients that
will penalize the presence of harmonic frequencies
and will direct the optimization to better choices of
, which can be seen by a rapid decrease in
harmonic distortion after 0.2 s.
4 Conclusion
The MRAC technique is a control method that uses
a reference model to control a plant with unknown
dynamics. The controller continuously updates its
parameters to improve control accuracy. However,
parameter γ cannot have a fixed value and should be
adjusted in real time with the rest of the system
parameters and various exogenous factors. The
proposed method uses real-time optimization
techniques to optimize the parameters during system
startup and whenever the error suddenly changes.
The MRAC-Taguchi-DCT optimization method
has been proposed to improve the performance of
MRAC (Model Reference Adaptive Control)
system. This optimization technique continuously
changes parameters of the MRAC system to
minimize the error. The calculation of the DCT and
imposition of the penalty were performed to ensure
a smoother response of the system. The comparative
results of the proposed MRAC-Taguchi-DCT
method have been presented and show that the error
converges to zero faster, and the parameters of the
MRAC system converge and stabilize faster when
compared to MRAC with one and constant
parameter . Overall, the MRAC-Taguchi-DCT
optimization method improved the performance of
the MRAC system and achieved the fastest
convergence and stabilization of the parameters.
This leads to a more accurate control signal and
reduces strain on the plant.
One possible future development of this work is
the use of machine learning techniques, such as
deep learning and reinforcement learning, to
enhance the adaptive capabilities of the control
system. These techniques have shown promising
results in other applications, and there is potential
for their application to MRAC control. These
techniques can be implemented, compared, or
integrated into the existing system, leading to more
robust and adaptable control systems that can handle
a wider range of uncertainties and disturbances.
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WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.3
Kyriakos Tsiakmakis, Vasileios Delimaras,
Argyrios T. Hatzopoulos, Maria S. Papadopoulou
E-ISSN: 2224-2856
35
Volume 18, 2023
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Kyriakos Tsiakmakis, has implemented the
algorithms and control system
Vasileios Delimaras, carried out the simulations and
the optimization algorithms.
Argyrios Xatzopoulos, has organized the
simulations and experiments
Maria Papadopoulou, has organized the simulations
and experiments
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.3
Kyriakos Tsiakmakis, Vasileios Delimaras,
Argyrios T. Hatzopoulos, Maria S. Papadopoulou
E-ISSN: 2224-2856
36
Volume 18, 2023
Appendix
Fig. A: The reference signal (top and in all following graphs) and the responses of MRAC, MRAC-Taguchi,
and MRAC-Taguchi-DCT, respectively.
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.3
Kyriakos Tsiakmakis, Vasileios Delimaras,
Argyrios T. Hatzopoulos, Maria S. Papadopoulou
E-ISSN: 2224-2856
37
Volume 18, 2023
