=Paper=
{{Paper
|id=Vol-3842/paper1
|storemode=property
|title=The controller synthesis automation using a dynamic mathematical model and genetic algorithms
|pdfUrl=https://ceur-ws.org/Vol-3842/paper1.pdf
|volume=Vol-3842
|authors=Victoria Vysotska,Vasyl Lytvyn,Serhii Vladov,Oleksandr Muzychuk,Oleksii Kryshan
|dblpUrl=https://dblp.org/rec/conf/bait/VysotskaLVMK24
}}
==The controller synthesis automation using a dynamic mathematical model and genetic algorithms==
https://ceur-ws.org/Vol-3842/paper1.pdf
The controller synthesis automation using a dynamic
mathematical model and genetic algorithms
Victoria Vysotska1,†, Vasyl Lytvyn1,†, Serhii Vladov2,∗,†, Oleksandr Muzychuk3,† and
Oleksii Kryshan4,†
1
Lviv Polytechnic National University, Stepan Bandera Street 12 79013 Lviv, Ukraine
2
Kremenchuk Flight College of Kharkiv National University of Internal Affairs, Peremohy Street 17/6 39605
Kremenchuk, Ukraine
3
Kharkiv National University of Internal Affairs, L. Landau Avenue 27 61080 Kharkiv, Ukraine
4
Interregional Academy of Personnel Management, Frometivska Street 2 03039 Kyiv, Ukraine
Abstract
This research develops a dynamic mathematical model for controller synthesis using genetic
algorithms, surpassing traditional methods by integrating adaptive optimization with evolutionary
principles. Unlike conventional fixed algorithms and manual tuning, this model dynamically
explores a broader parameter space and autonomously adjusts parameters, enhancing performance
in complex systems. The research involved a computer experiment applying this model to a PID
controller implemented with a dynamic neural network. The results demonstrated significant
improvements, including adjustments to PID coefficients that reduced transient process time and
overshoot, increased modeling accuracy from 99.523 to 99.783 %, and minimized losses from 2.5 to
0.5 %. These findings suggest that incorporating the developed model into the automatic control
system for helicopter turboshaft engines free turbine rotor speed could significantly enhance system
performance and reliability. These findings suggest that incorporating the developed model into the
automatic control system for helicopter turboshaft engines free turbine rotor speed could
significantly enhance system performance and reliability. Future work will focus on validating
these results under diverse operational conditions and exploring additional optimization techniques.
Keywords
PID controller, genetic algorithms, mathematical model, optimal control, neural network, helicopter
turboshaft engine, free turbine rotor speed 1
1. Introduction
Automation of controller synthesis is a critical task in control complex dynamic systems,
where traditional design methods may be insufficiently effective [1]. In recent years, there has
been growing interest in the application of optimization methods based on evolutionary
algorithms, among which genetic algorithms hold a prominent position [2, 3]. These
algorithms, inspired by the natural selection and genetic evolution processes, enable effective
1
BAIT’2024: The 1st International Workshop on “Bioinformatics and applied information technologies”, October 02-04,
2024, Zboriv, Ukraine
∗
Corresponding author.
†
These authors contributed equally.
victoria.a.vysotska@lpnu.ua (V. Vysotska); vasyl.v.lytvyn@lpnu.ua (V. Lytvyn); serhii.vladov@univd.edu.ua (S.
Vladov); o.muzychuk23@gmail.com (O. Muzychuk); krishan@ki-maup.com.ua (O. Kryshan)
0000-0001-6417-3689 (V. Vysotska); 0000-0002-9676-0180 (V. Lytvyn); 0000-0001-8009-5254 (S. Vladov); 0000-
0001-8367-2504 (O. Muzychuk); 0000-0002-2967-0126 (O. Kryshan)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
solutions to optimization tasks in multidimensional spaces, where traditional methods may
encounter difficulties [4, 5]. In the controller synthesis context, genetic algorithms provide
flexibility and adaptability, allowing for the multiple criteria consideration and modern
control systems constraints characteristic.
This research relevance is driven by the modern dynamic systems increasing complexity
that require highly efficient and adaptive control methods. Traditional approaches to
controller synthesis are often limited due to the systems and multi-criteria requirements
complexity, which can lead to suboptimal solutions or excessive design costs. In this context,
genetic algorithms, with their ability to effectively search for global optima in
multidimensional spaces and adapt to changing conditions [6–8], offer new opportunities for
automating controller synthesis. The genetic algorithms application significantly enhances the
controller synthesis accuracy and reliability, which is particularly important for control
critical systems such as aviation and energy complexes, where failures can have severe
consequences.
2. Related Works
Existing research in the controller synthesis automation field using genetic algorithms has
shown significant progress in solving optimization tasks for complex dynamic systems. Many
researches emphasize the genetic algorithms flexibility and adaptability, allowing the various
quality criteria consideration and constraints inherent in real-world control systems [9].
Research demonstrates that genetic algorithms are successfully applied in various industries,
including aerospace [10–12], energy [13, 14], robotics [15, 16], and others [17–19]. Their
ability to effectively solve multi-objective optimization tasks [20] makes them an attractive
tool for automating controller synthesis, especially under conditions of high uncertainty and
the numerous local optima presence.
One important area of research is the hybrid methods development that combine genetic
algorithms with other optimization techniques, such as neural networks [21–23] and particle
swarm [24, 25] methods. These approaches improve convergence and the finding optimal
solutions speed, as well as enhance the algorithms against local minima resilience. Notably,
successful results have been obtained when applying hybrid methods for synthesizing
controllers in complex nonlinear systems, where traditional methods prove to be less effective
[26–28]. However, such hybrid approaches require more precise tuning and complicate the
process of model development.
Despite the advances in applying genetic algorithms for controller synthesis, there remain
several unresolved issues related to these processes mathematical modeling. Specifically, the
genetic algorithms scalability when working with large and high-dimensional systems
remains insufficiently studied. Additionally, there are limitations in assessing algorithm
performance in the synthesis early stages, which can lead to increased computation time and
reduced automation efficiency. Moreover, the literature highlights a lack of development in
methods for adapting genetic algorithms to changing environmental conditions and parameter
instability, which limits their application in real-time systems.
Therefore, further development of mathematical models for automating controller
synthesis using genetic algorithms is necessary, focusing on improving the scalability and
adaptability of algorithms, as well as reducing computational costs. Research in this direction
may include developing new approaches to the genetic algorithm parameters adaptive tuning,
enhancing hybridization methods with other optimization techniques, and exploring ways to
integrate genetic algorithms into real-time control systems. These research gaps present
opportunities for developing more effective and versatile tools for automating controller
synthesis, capable of handling control tasks in increasingly complex and uncertain
environments.
3. Proposed technique
Genetic algorithms have proven their effectiveness in solving optimization tasks, especially in
cases where classical methods encounter difficulties due to the multidimensionality and
nonlinearity of tasks [29, 30]. This research presents an innovative dynamic mathematical
model for the controllers’ synthesis automating based on the genetic algorithms use. The
model includes several stages: the target function formation, the controller parameters
determination, the genetic algorithm development for the controller optimization and
dynamic adaptation during the system operation [31].
Let y(t) be the system output variable, yr(t) be the output variable desired value (reference).
The output variable from the reference deviation is defined as:
e(t) = yr(t) − y(t). (1)
The optimization task is to minimize the objective function J, which is defined as the
integral of the squared error over a certain time interval T:
T
J =∫ e 2 ( t ) dt . (2)
0
Let us consider an example of a PID controller for the helicopter turboshaft engines (TE)
free turbine rotor speed nFT, whose transfer function has the form [32, 33]:
Ki
C ( s ) =K p + +Kd∙ s , (3)
s
where Kp, Ki, and Kd are the proportional, integral, and differential components coefficients,
respectively. The genetic algorithm task is to find such values of Kp, Ki, and Kd that minimize
the objective function J.
The genetic algorithm application first stage is the initialization of the population. Let N be
the population size, then the initial population P(0) can be defined as a set of random values
Kp, Ki, and Kd:
P ( 0 ) = {( K p 1 , K i 1 , K d 1 ) , … , ( K pN , K ¿ , K dN ) } . (4)
Next, the fitness function is evaluated. Each population member is evaluated by the
objective function J value as:
T
J i =∫ e i2 ( t ) dt ,i=1 … N , (5)
0
Next, it performs a crossover as follows: two parent solutions Pi and Pj produce an
offspring Pk using a parameters combination:
Pk = α ∙ Pi + (1 − α) ∙ Pj, α ∈ [0, 1], (6)
Next, mutation is carried out by randomly changing one or more parameters of the
offspring:
Pk = Pk + δ, δ ∼ N(0, σ2). (7)
For the selection process, where M best solutions that minimize the objective function J
survive, an algorithm is applied in which there is a solutions population {P1, P2, …, PN}, where
N is the solutions total number. For each solution, the objective function J(Pi) value is
calculated according to (5).
At the next stage, the solution is sorted by the objective function J(Pi) value in ascending
order:
P(1), P(2), …, P(N), (8)
where Pi is the solution for which the J(P(i)) value is less than for all previous solutions:
J(P(1)) ≤ J(P(2)) ≤ … ≤ J(P(N)). (9)
After sorting, M best solutions are selected for which J(P(i)) is minimal:
J(P(1)) ≤ J(P(2)) ≤ … ≤ J(P(M)), (10)
where M ≤ N, and the selected solutions {P(1), P(2), …, P(M)} become the basis for forming a new
population in the algorithm next step. A new population is formed from descendants and
parents:
P(t + 1) = {P(1), P(2), …, P(M), new descendants}. (11)
This process continues until a stopping criterion is reached, such as when the minimum
value is reached or when the iterations maximum number is reached.
In real systems, the system parameters may change over time, which requires the regulator
to adapt in real time. For this aim, an algorithm is proposed that consists of modeling the
change in system parameters, controller adaptive adjustment, and feedback. Let the system
parameters a(t), b(t), and c(t) change over time:
ẋ ( t ) = A ( t ) ∙ x ( t ) + B ( t ) ∙ u ( t ) , (12)
y(t) = C(t) ∙ x(t),
where A(t) = a(t), B(t) = b(t), C(t) = c(t) are the system matrices that change over time.
The controller adaptive tuning assumes that the controller parameters can change over
time in response to changes in the control object dynamics. Let us consider a mathematical
model for the PID controller adaptive tuning that updates the controller coefficients Kp(t), Ki(t),
and Kd(t) at each time step. The helicopter TE free turbine rotor speed nFT PID controller is
presented in the form [32, 33]:
t
de ( t )
u ( t ) =K p ( t ) ∙ e ( t ) + K i ( t ) ∙∫ e ( τ ) dτ + K d ( t ) ∙ , (13)
0 dt
where e(t) = yr(t) – y(t) is the control error calculated according to (1), yr(t) is the specified
output value (reference).
To estimate the system parameters A(t), B(t), and C(t) in real time, the recursive least
squares (RLS) method [34] can be used:
θ^ ( t ) = θ^ ( t −1 ) + L ( t ) ∙ [ y ( t ) − ^y ( t ) ] , (14)
where θ^ ( t ) is the system parameters estimate at time t, L(t) is the adaptation matrix,
^ ( t ) ∙ x^ ( t ) is the system output estimate, x^ ( t ) is the system state estimate.
^y ( t ) =C
The controller parameters are updated using gradient descent:
∂ J (t )
K p ( t +1 ) =K p ( t ) −α ∙ , (18)
∂Kp
∂ J (t )
K i ( t +1 ) =K i ( t ) −α ∙ , (19)
∂ Ki
∂ J (t )
K d ( t +1 ) =K d ( t ) −α ∙ , (20)
∂ Kd
where J(t) is the objective function (for example, the squared error integral), α is the training
step.
The objective function gradients for each of the parameters can be calculated as follows:
t
∂ J (t ) ∂u(τ )
=−2∙∫ e ( τ ) ∙ dτ , (21)
∂Kp 0 ∂Kp
t
∂ J (t ) ∂u(τ )
=−2∙∫ e ( τ ) ∙ dτ , (22)
∂ Ki 0 ∂ Ki
t
∂ J (t ) ∂u(τ )
=−2∙∫ e ( τ ) ∙ dτ . (23)
∂ Kd 0 ∂ Kd
Because:
∂ u (t )
=e ( t ) , (24)
∂Kp
t
∂ u (t )
=∫ e ( τ ) dτ , (25)
∂ Ki 0
∂ u (t ) ∂ e (t )
= . (26)
∂ Kd ∂t
Substituting expressions (24)–(26) into the formulas for gradients (21)–(23), we obtain:
t t
∂ J (t )
=−2∙∫ e ( τ ) ∙ e ( τ ) dτ =−2∙∫ e 2 ( τ ) dτ , (27)
∂Kp 0 0
( )
t τ
∂ J (t )
=−2 ∙∫ e ( τ ) ∙ ∫ e ( ξ ) dξ dτ , (28)
∂ Ki 0 0
t
∂ J (t ) ∂e (τ )
=−2∙∫ e ( τ ) ∙ dτ . (29)
∂ Kd 0 ∂t
Thus, at each time step, the parameters Kp(t), Ki(t), and Kd(t) are updated taking into
account the control error current values and its derivatives, which allows the controller to
adapt to changes in the system dynamics.
Feedback is a control system key element that allows the system behavior to be adjusted
based on the system current state deviation from a given value. To control a nonlinear system,
the input-output linearization method is used, which consists of selecting the control action so
that the closed system becomes linear. The control signal is selected as:
u ( t )=
1
g ( x (t ))
∙
(
−d ( r ) ∙ h ( x ( t ) )
dt ( r ) )
+v ( t ) , (30)
where r is the system relative degree, and v(t) is the linear system new control action.
Substituting u(t) into the original equation, we obtain the linear system:
d(r ) ∙ y ( t )
v ( t )= . (31)
dt ( r )
The control task is reduced to choosing v(t) so that the output signal y(t) desired trajectory
is achieved.
The developed dynamic mathematical model for the controllers synthesis automating
using genetic algorithms represents a significant advancement over traditional approaches by
integrating adaptive optimization mechanisms with evolutionary principles. Unlike
conventional methods that rely on fixed algorithmic structures and heuristic tuning, this
model employs genetic algorithms to dynamically explore and optimize a broader parameter
space, facilitating the more effective and robust controller configurations discovery. The
model's innovation lies in its ability to autonomously adjust controller parameters through a
continuous evolutionary process, thereby enhancing performance in complex and variable
system environments. This adaptive capability contrasts with traditional methods that often
require manual intervention and lack the flexibility to respond to real-time changes in system
dynamics, thus offering a more versatile and efficient solution for modern control
applications.
4. Results
Based on [32, 33], the work solves the task of synthesizing a PID controller (Figure 1) for the
helicopters TE free turbine rotor speed nFT controlling task.
Figure 1: Updated block diagram illustrating the system for the helicopter turboshaft engines
free turbine rotor speed controlling using a PID controller (author’s research).
The input parameters are the atmospheric parameters (h is the flight altitude, TN is the
temperature, PN is the pressure, ρ is the air density) values. The parameters recorded on board
of the helicopter (nFT is the free turbine rotor speed) reduced to absolute values according to
the gas-dynamic similarity theory (table 1). We assume in this research that the atmospheric
parameters are constant (h is the flight altitude, TN is the temperature, PN is the pressure, ρ is
the air density) [35–37].
Table 1
Training dataset fragment
Number nFT Number nFT
1 0.983 132 0.992
… … … …
28 0.979 … …
The training
dataset … … 256 0.974
homogeneity evaluation, as described in [35–37], employed the Fisher-Pearson [38] and
Fisher-Snedecor [39] tests. Based on these criteria, the dataset was found to be homogeneous,
with the computed Fisher-Pearson and Fisher-Snedecor statistics falling below their respective
2 2
critical limits, specifically χ =5.365< χ critical =6.6 and F=2.177< F critical =2.58. To assess
the dataset's representativeness, cluster analysis using the k-means method [40–42] was
conducted. The dataset was divided into training and testing subsets in a 2:1 ratio,
corresponding to 67 % (172 samples) and 33 % (84 samples), respectively. The cluster analysis
(Table 1) identified seven distinct classes (I...VII), confirming their presence and demonstrating
consistency between the training and test datasets (Figure 2). These results allowed for the
optimal sample datasets determination: the training dataset consists of 256 elements (100 %),
the validation dataset comprises 172 elements (67 % of the training dataset), and the test
dataset contains 84 elements (33% of the training dataset).
Figure 2: The cluster analysis results, where “left figure” is the training dataset, “right figure”
is the test (author’s research) (author’s research).
As detailed in [32, 33], an improved standard configuration for the helicopter TE free
turbine rotor speed controlling has been introduced, featuring a PID controller. This
enhancement is realized by integrating a dynamic neural network with direct data
transmission, where the first layer consists of neurons with a radial-basis activation function,
and the second layer includes adalines are the neurons with a linear activation function. Since
maintaining the main rotor rotational speed is a critical objective during helicopter flight, this
modification is especially relevant. In [32, 33], a neural network was employed to adjust the
PID controller Kp, Kd, and Ki coefficients, which ultimately enabled the main rotor speed
dynamic regulation.
In [33] it is implemented a discrete PID controller using a dynamic neural network with
direct data flow, where only the activation function in the second layer is linear, while the
activation functions in the first layer are nonlinear, specifically radial-basis functions (see
Figure 3, where Δ represents a one-step delay).
Figure 3: The nonlinear PID controller neural network-based representation (author’s
research).
To fine-tune the PID controller, a dynamic neural network with direct data flow was
utilized. This network comprises neurons with a radial basis activation function in the first
layer and adalines is the neurons with a linear activation function is the in the second layer.
The network's structural parameters include a learning rate set to 1.5, 20 neurons in the
hidden layer, a delay line length of 5 for input signals, and 1000 training epochs. The initial
coefficients were set at Kp = 0.5, Ki = 5, Kd = 0.01. For initial training, recorded data from the
control object’s operation were used, specifically the first 50 data points, with a window size
of 10 and a training accuracy of 0.00005. If during operation the control error exceeded 2 over
ten time cycles, the neural network underwent further training, with a training accuracy of
0.0001 and a limit of 10 training iterations. As per [32, 33], the first step involved optimizing
the PID controller coefficients (Figure 1). The transient response after optimization is depicted
by curve 1 in Figure 4, where the linear control law resulted in significant overshoot and a
large static error. The transient response is shown as “blue curve” in Figure 4, which indicates
that while the response became aperiodic, the static error remained, and the rise time
increased slightly. In the third step, the neuron's activation function parameters,
corresponding to the integral component of the PID controller, were optimized (Figure 3).
This optimization resulted in the transient response shown as “red curve” in Figure 5, where
the response became oscillatory again. It is evident that with changes in the task, the transient
response remained consistent, maintaining a low overshoot due to the PID coefficients static
nature. The overshoot was minimal, the slew rate significantly improved, and the static error
was negligible.
Figure 4: The PID controller computer simulation results (author’s research).
According to Figure 4 initial coefficient values are: Kp = 0.5, Ki = 5, Kd = 0.01, after the add-
on Kp = 0.45, Ki = 4.985, Kd = 0.007, overshoot value does not exceed 0.3 %.
To evaluate the neural network's performance in the next training phase, both accuracy
(Figure 5) and loss (Figure 6) are measured. The Accuracy metric represents the correct
predictions percentage, while the Loss metric shows the predictions average squared error,
indicating how much they deviate from the true values. To evaluate the precise calculations
proportion for the free turbine rotor speed nFT, the Accuracy metric is used (Figure 5), and it is
computed at training epoch t using the following expression [35–37]:
N
1
∙ ∑ I ( n^ TCi =nTC ) .
t
Accuracy t = (32)
N i=1
Figure 5: The accuracy metric diagram (author’s research).
Figure 6: The loss metric diagram (author’s research).
As shown in Figures 5 and 6, these metrics indicate that the neural network model delivers
high prediction accuracy (99.77 %) and performs efficiently, with the mean squared error
staying under 2.5 %. Furthermore, the significant reduction in the loss function from 2.5 to 0.5
% reflects an enhancement in the model's quality over the training process course.
Similarly to the method described in [32, 33], the approach quality is assessed using
classification metrics derived from the confusion matrix presented in Table 2. In this matrix,
TP denotes true positives (instances correctly classified as defects by the model), FP refers to
false positives (non-defects mistakenly classified as defects), TN represents true negatives
(cases accurately identified as non-defective data), and FN indicates false negatives (defects
incorrectly identified as non-defective) [43, 44].
Table 2
The error matrix [43, 44]
Category Positive Negative
Predicted positive TP FP
Predicted negative FN TN
To assess the developed method quality in this research, the following metrics were chosen
TP+TN
[45–47]: Accuracy= is the objects percentage calculates for which the
TP+TN + FP+ FN
TP
classifier accurately made decisions, Accuracy= is the relevant parameters percent
TP+ FP
TP
among all researched, Recall= is the crucial parameter in defect detection is
TP+ FN
precision, as the detected defects ratio signifies to the defective instances overall number,
Precision ∙ Recall
F 1=2∙ is the F-measure, which is the “harmonic” average between
Precision+ Recall
Precision and Recall. Table 3 presents the model's training outcomes average results, including
the mean and variance for the accuracy metrics.
Table 3
The testing indicators average values
Value
Metric PID-controller developed on PID-controller developed on
the neural network basis [33] the neural network basis with
genetic algorithms
Accuracy 0.99523 0.99783
Precision 0.96238 0.98472
Recall 1.0 1.0
F-measure 0.98165 0.99362
Average time, seconds 1201.99 1095.38
Average Accuracy 0.99319 0.99525
Dispersion Accuracy 0.00000886 0.00000322
Table 4 provides an accuracy comparative analysis provided by each of the evaluated
controllers, highlighting the Type I and Type II errors [48–50] probabilities in identifying the
optimal parameter nFT.
Table 4
The 1st and 2nd types errors determining results
Error probability in determining the
parameter nFT optimal value
Controller type
Type 1st error Type 2nd error
Linear PD-controller 1.95 1.42
PD-controller with reduced Kd 1.74 1.21
Quadratic controller 1.46 1.03
PD-controller with a variable amplification 1.32 0.95
factor
Fuzzy logical P-controller 1.08 0.77
Fuzzy logical P-controller with a corrective
0.97 0.64
differential link
PID-controller developed on the neural
0.58 0.22
network basis [32]
Modified PID-controller developed on the
0.36 0.14
neural network basis [33]
PID-controller developed on the neural
0.22 0.10
network basis with genetic algorithms
As shown in Table 4, incorporating a dynamic neural network with direct data flow into
the PID controller with genetic algorithms, where the first layer consists of neurons with a
radial basis activation function and the second layer includes adalines with a linear activation
function is led to a decrease in Type I and Type II errors by 30 to 40 % compared to the PID-
controller described in [33].
5. Discussions
The research is aimed at creating a dynamic mathematical model (1)–(31) for the controllers
(see Figure 1) synthesis using genetic algorithms. The new dynamic mathematical model for
controller synthesis, utilizing genetic algorithms, surpasses traditional methods by integrating
adaptive optimization with evolutionary principles. Unlike fixed algorithms and manual
tuning, this model dynamically explores a broader parameter space, autonomously adjusting
parameters for improved performance in complex systems. This approach offers greater
flexibility and efficiency, responding to real-time changes more effectively than conventional
methods.
In this research, a computer experiment was conducted to determine the helicopter TE free
turbine rotor speed transient process by introducing the developed mathematical model (1)–
(31) into a PID controller implemented utilizing a dynamic neural network with direct data
flow, where only the second layer has a linear activation function, while the first layer
features nonlinear, radial-basis activation functions (see Figure 3).
According to the obtained results (see Figure 4), the developed mathematical model (1)–
(31) application made it possible to adjust the PID controller coefficients: the Kp coefficient
from 0.5 to 0.45, the Ki coefficient from 5.0 to 4.985, the Kd coefficient from 0.01 to 0.007, which
made it possible to minimize the transient process time and overshoot from 0.5 to 0.3%
compared to the PID controller developed in [32, 33].
The developed mathematical model (1)–(31) application made it possible to increase the
accuracy of the helicopter TE free turbine rotor speed transient process modeling from 99.523
to 99.783 % (see Figure 5 and Table 3), and also to minimize loss from 2.5 to 0.5% (see Figure 6),
and also reduce the first and second types errors by 30...40 % compared to the PID controller
developed in [32, 33].
Thus, it seems appropriate to implement the developed mathematical model (1)–(31) into
the helicopter TE free turbine rotor speed automatic control system, including a neural
network, which task is to determine the PID controller coefficients (Figure 7).
Figure 7: Modified PID controller design with an auto-tuning block based on a neural
network, where TD represents the delay operator (author’s research).
The limitations of the obtained results may be as follows. First, the proposed mathematical
model (1)–(31) and optimization methods based on genetic algorithms might not account for
all possible external and internal influences, limiting their applicability to real-world
operational conditions. Second, the improvement in modeling accuracy and reduction in
losses achieved by adjusting PID controller coefficients may be due to the specific parameters
and neural network architecture used, which does not necessarily guarantee similar results in
other systems or under different operating conditions. Third, the experimental data used in
the study may not fully capture complex dynamic processes, potentially affecting the
proposed method accuracy and reliability in various real-world scenarios. Specifically, the
controller's performance accuracy is contingent on the sensor data precision, and any
significant deviation in sensor readings can negatively affect the dynamic response.
Additionally, the genetic algorithm computational complexity, combined with the neural
network, can lead to increased processing time, particularly in real-time applications, which
may necessitate further optimization for onboard systems.
Future research prospects include several key directions. First, additional experiments are
needed to verify the proposed mathematical model and optimization methods universality
under various operational conditions and different system types. Second, integrating the
model with more advanced adaptive control algorithms should be considered to enhance its
ability to handle dynamic changes and unforeseen disturbances. Third, exploring new neural
network architectures that may improve modeling accuracy and efficiency is promising.
Additionally, investigating the model's application in real operational environments will be
valuable for assessing its practical utility and reliability.
6. Conclusions
The research demonstrates the newly developed dynamic mathematical model (1)–(31)
effectiveness for synthesizing controllers using genetic algorithms. The model offers
significant advantages over traditional methods by integrating adaptive optimization with
evolutionary principles, allowing for the broader parameter space dynamic exploration and
parameters autonomous adjustment. This approach enhances flexibility, efficiency, and
responsiveness to real-time changes.
The computer experiment results show that applying the model to a PID controller
significantly improved performance. Adjustments to the PID coefficients led to a reduction in
transient process time and overshoot, with modeling accuracy increasing from 99.523 to
99.783 % and losses minimized from 2.5 to 0.5 %. These improvements indicate the
incorporating potential benefits the developed model into the automatic control system for
helicopter TE free turbine rotor speed.
However, limitations include potential gaps in accounting for all external and internal
influences, the neural network parameters specificity affecting generalizability, and the
experimental data potential inadequacy in capturing complex dynamics. Future research
should focus on validating the model's universality under diverse conditions, integrating it
with advanced adaptive control algorithms, exploring new neural network architectures, and
assessing its practical applicability in real operational environments.
Acknowledgements
The research was supported by the Ministry of Internal Affairs of Ukraine “Theoretical and
applied aspects of the development of the aviation sphere” under Project No. 0123U104884.
The research was carried out with the grant support of the National Research Fund of Ukraine
“Methods and means of active and passive recognition of mines based on deep neural
networks”, project registration number 273/0024 from 1/08/2024 (2023.04/0024). Also, we
would like to thank the reviewers for their precise and concise recommendations that
improved the presentation of the results obtained.
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