=Paper=
{{Paper
|id=Vol-3861/paper1
|storemode=property
|title=The helicopter turboshaft engines parametric debugging using neural network technology
|pdfUrl=https://ceur-ws.org/Vol-3861/paper1.pdf
|volume=Vol-3861
|authors=Victoria Vysotska,Serhii Vladov,Yevhen Volkanin,Andrii Siora,Maryna Bulakh,Oleksandr Muzychuk,Oleksandr Koren
|dblpUrl=https://dblp.org/rec/conf/ciaw/VysotskaVVSBMK24
}}
==The helicopter turboshaft engines parametric debugging using neural network technology==
https://ceur-ws.org/Vol-3861/paper1.pdf
Victoria Vysotska1,†, Serhii Vladov2,∗,†, Yevhen Volkanin2,†, Andrii Siora2,†, Maryna Bulakh3,†,
Oleksandr Muzychuk4,† and Oleksandr Koren5,†
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
Rzeszow University of Technology, Kwiatkowskiego Street 4 37-450 Stalowa Wola, Poland
4
Kharkiv National University of Internal Affairs, L. Landau Avenue 27 61080 Kharkiv, Ukraine
5
Interregional Academy of Personnel Management, Frometivska Street 2 03039 Kyiv, Ukraine
Abstract
This research presents a mathematical model designed to optimize the helicopter turboshaft engines
parametric tuning by accurately predicting engine performance characteristics through the integration of
key operational parameters such as rotor speeds, fuel consumption rates, and temperature profiles. A neural
network model is developed to capture the complex nonlinear relations between input parameters and
engine performance outputs, employing a supervised training algorithm and an adaptive training rate to
enhance convergence efficiency. The model demonstrates impressive performance metrics, achieving a
prediction accuracy of 99.25 % and a mean squared error below 2.5 %. While the results are promising, the
research identifies limitations related to the reliance on historical performance data and the potential for
overfitting. Future studies are recommended to explore the various factors influence on engine
performance, develop more adaptive neural network architectures, and conduct extensive field testing to
ensure model robustness and effectiveness in real-world conditions. Ultimately, the integration of advanced
predictive models into helicopter control systems will significantly enhance flight safety and operational
efficiency.
Keywords
helicopter turboshaft engine, mathematical model, optimization, neural network, training algorithm,
operation data and deviations 1
1. Introduction
Parametric debugging for helicopter turboshaft engines (TE) is a critical aspect in optimizing
performance and ensuring reliability during flight operations [1]. Helicopter TE operate under
varying conditions, which demand precise calibration and adjustment across multiple parameters to
maintain efficiency and safety [2, 3]. The process involves fine-tuning various engine parameters,
such as rotational speeds, temperatures, and fuel consumption rates, to align the engine's
performance with expected operational standards [4]. This approach helps identify potential faults
early, enhances the engine's operational longevity, and contributes to more effective engine control
systems.
The helicopter TE parametric debugging importance stems from the increasing complexity of
modern aviation engines and the demand for higher reliability in dynamic flight conditions [5]. As
CIAW-2024: Computational Intelligence Application Workshop, October 10-12, 2024, Lviv, Ukraine
∗
Corresponding author.
†
These authors contributed equally.
victoria.a.vysotska@lpnu.ua (V. Vysotska); serhii.vladov@univd.edu.ua (S. Vladov); volkanin@ukr.net (Y. Volkanin);
siora.andrey@gmail.com (A. Siora); m.bulakh@prz.edu.pl (M. Bulakh); o.muzychuk23@gmail.com (O. Muzychuk);
koren@ki-maup.com.ua (O. Koren)
0000-0001-6417-3689 (V. Vysotska); 0000-0001-8009-5254 (S. Vladov); 0000-0003-3507-1987 (Y. Volkanin); 0000-0002-
2934-7281 (A. Siora); 0000-0003-4264-2303 (M. Bulakh); 0000-0001-8367-2504 (O. Muzychuk); 0000-0002-7406-7400 (O.
Koren)
© 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
helicopters are often used in critical operations, ensuring optimal engine performance is paramount
for both safety and operational efficiency. Advanced methods for parametric debugging allow for
more accurate diagnostics, early detection of deviations in engine behavior, and timely corrective
measures. This enhances overall flight safety, reduces maintenance costs, and supports the
development of more robust systems for engine monitoring and control [6].
2. Related works
Research on helicopter TE has significantly advanced over the years, focusing on areas such as
engine modeling, fault diagnosis, and control systems. Traditional parametric debugging approaches
primarily rely on physical models based on thermodynamic principles, which describe the engine's
behavior under various operating conditions [7, 8]. These models are often calibrated using real-
world data and are effective for steady-state conditions. However, in transient modes, such as during
takeoff or acceleration, physical models encounter limitations due to the complexity and nonlinearity
of engine dynamics. Studies [9–11] emphasize the need for enhanced methods to capture these
transient behaviors more accurately, which remain underrepresented in classical models.
Recent developments in helicopter TE fault diagnosis have shifted towards model-based and data-
driven techniques [12]. Model-based approaches, such as Kalman filters [13, 14] or observer-based
fault detection [15, 16], have been widely used for identifying deviations in engine performance.
These methods, although effective in controlled environments, often struggle when faced with
uncertainties in real-time operations or when sensors provide incomplete or noisy data. Data-driven
approaches [17, 18] have gained traction, using historical performance data to detect anomalies
through statistical or machine learning techniques. However, these methods are usually limited by
the available data quality and quantity and often fail to account for rare or unexpected faults.
In the dynamic flight conditions context, where engine parameters change rapidly, traditional
methods fall short in their predictive accuracy. Studies [19, 20] that these approaches do not fully
exploit the complex relations between multiple engine variables, especially in nonlinear and
nonstationary environments. Additionally, the increasing advanced control systems use, such as
adaptive control and fault-tolerant systems [21, 22], requires faster and more reliable diagnostic
techniques. This gap in real-time diagnostic capability highlights the need for more sophisticated
methods capable of data large amounts processing and adapting to changing conditions during flight.
The neural networks application in the helicopter TE diagnostics and control is an emerging field
that promises to the traditional methods limitations address many. Neural networks, particularly
recurrent neural networks (RNN) [23] and long short-term memory (LSTM) networks [24], have the
capacity to model complex temporal dependencies in engine behavior. Researches [25, 26] have
demonstrated the potential of neural networks to improve fault detection and parameter estimation
by training patterns directly from operational data. These models offer a more flexible and scalable
solution, particularly in capturing transient and nonlinear behaviors that are difficult to model using
conventional techniques.
Despite the promising results in the literature, there are still gaps in the application of neural
networks for helicopter TE. Most existing studies focus on steady-state conditions or specific fault
scenarios, while few address the full range of operating conditions, especially during transient
modes. Additionally, the integration of neural networks with traditional diagnostic systems has been
limited, with most implementations remaining experimental. These gaps highlight the need for
further research into hybrid models that combine the strengths of physical and data-driven
approaches.
The neural network approach offers significant advantages for the helicopter TE real-time
diagnostics and control, particularly in handling the nonlinearity, noise, and uncertainties inherent
in engine performance data. By continuously training from operational data, neural networks can
improve predictive accuracy and provide earlier warnings for potential faults. Moreover, their ability
to generalize from past data enables them to detect rare or complex fault patterns that might be
missed by conventional methods. This justifies the need for a more comprehensive application of
neural networks in the field, as they hold the potential to revolutionize engine diagnostics and
enhance flight safety.
3. Materials and methods
This research proposes the helicopter TE parametric debugging mathematical model, which takes
into account both static and dynamic engine characteristics. The model is based on the time series
use of engine parameters recorded on board the helicopter: the gas-generator rotor r.p.m. nTC(t), the
free turbine rotor speed nFT(t), gas temperature in front of the compressor turbine 𝑇 ∗ (𝑡), the fuel
consumption Qf(t) and other parameters [27, 28]. The model takes into account both standard engine
operating parameters and deviations caused by external and internal influences. Additionally, the
model allows for real-time analysis and adjustment to enhance engine performance and reliability.
The main engine dynamics are described by the nonlinear differential equations system in the
form: this system models the complex interactions between various engine components and
operational parameters.
Thus,
𝑑𝑛 (𝑡)
= 𝑓 𝑛 (𝑡), 𝑛 (𝑡), 𝑇 ∗ (𝑡), 𝑄 (𝑡), 𝑃 (𝑡) ,
𝑑𝑡
𝑑𝑛 (𝑡)
= 𝑓 𝑛 (𝑡), 𝑛 (𝑡), 𝑇 ∗ (𝑡), 𝑄 (𝑡), 𝑃 (𝑡) ,
𝑑𝑡
(1)
𝑑𝑇 ∗ (𝑡) ∗
= 𝑓 𝑛 (𝑡), 𝑛 (𝑡), 𝑇 (𝑡), 𝑄 (𝑡), 𝑃 (𝑡) ,
𝑑𝑡
𝑑𝑄 (𝑡)
= 𝑓 𝑛 (𝑡), 𝑛 (𝑡), 𝑇 ∗ (𝑡), 𝑄 (𝑡), 𝑃 (𝑡) .
𝑑𝑡
Functions f1, f2, f3, f4 describe interactions between engine parameters depending on its state, while
Pext(t) represents external factors such as atmospheric pressure and turbulence.
To minimize deviations actual rotor speeds nTC(t), nFT(t), gas temperature 𝑇 ∗ (𝑡), and fuel
consumption Qf(t) from their nominal values nTC_nom, nFT_nom, 𝑇 ∗_ , Qf_nom the following deviations
are introduced [29]:
∆𝑛 (𝑡) = 𝑛 (𝑡) − 𝑛 _ ,
∆𝑛 (𝑡) = 𝑛 (𝑡) − 𝑛 _ ,
∗ (𝑡) ∗ (𝑡) (2)
∆𝑇 =𝑇 − 𝑇 ∗_ ,
∆𝑄 (𝑡) = 𝑄 (𝑡) − 𝑄 _ .
The objective function for minimizing deviations takes the form:
𝐽= 𝑤 ∙ ∆𝑛 (𝑡) + 𝑤 ∙ ∆𝑛 (𝑡) + 𝑤 ∙ ∆𝑇 ∗ (𝑡) + 𝑤 ∙ ∆𝑄 (𝑡) 𝑑𝑡, (3)
where w1, w2, w3, w4 weighting coefficients defining the significance each parameter, and T is the
final moment time. The parametric adjustment task reduces to minimizing the functional J, aiming
to decrease deviations across key engine parameters.
To adaptively correct engine parameters in real-time, a neural network corrector [30] is
introduced. Let the network NN receive deviations ΔnTC(t), ΔnFT(t), Δ𝑇 ∗ (𝑡), ΔQf(t) as inputs,
generating a corrective signal u(t), which adjusts the engine control system:
𝑢(𝑡) = 𝑁𝑁 ∆𝑛 (𝑡), ∆𝑛 (𝑡), ∆𝑇 ∗ (𝑡), ∆𝑄 (𝑡) . (4)
The neural network is trained on historical engine operation data and deviations, allowing for
more precise parameter adjustments.
The corrective signal u(t), generated by the neural network, is used to adapt control inputs for
the engine. This can be described by the equation:
𝑈(𝑡 + 1) = 𝑈(𝑡) + 𝛼 ∙ 𝑢(𝑡), (5)
where U(t) is the control input vector for the system (such as fuel consumption rate, turbine blade
angle), and α is the training coefficient that defines the speed adjustment.
To account for random disturbances (such as changes in external conditions or airflow
instability), a stochastic component is introduced [31, 32]. External influences can be described as:
𝑃 (𝑡) = 𝑃 _ (𝑡) + 𝜉(𝑡), (6)
where ξ(t) is white noise with zero mean and variance σ2. This accounts for random deviations
and allows real-time adjustments to control parameters.
The overall control system for helicopter TE dynamics, considering the neural network corrector
and stochastic perturbations, can be written as:
𝑑𝑿(𝑡)
= 𝑭 𝑿(𝑡), 𝑼(𝑡) + 𝑁𝑁 ∆𝑿(𝑡) , (7)
𝑑𝑡
where 𝐗(𝑡) = 𝑛 (𝑡), 𝑛 (𝑡), 𝑇 ∗ (𝑡), 𝑄 (𝑡) is the engine state vector, and U(t) is the control
input vector. The neural network corrector NN adjusts the deviations ∆𝐗(𝑡) =
∆𝑛 (𝑡), ∆𝑛 (𝑡), ∆𝑇 ∗ (𝑡), ∆𝑄 (𝑡) , ensuring precise engine control under uncertainty.
In this research, the developed mathematical model is implemented in a neural network basis
[33–37]. The proposed neural network (Figure 1) model consists of several layers, each serving a
specific function to achieve optimal performance in helicopter TE parametric adjustment. The
architecture includes an input layer, multiple hidden layers, and an output layer. Each layer consists
of neurons that process information, passing it to the next layer through activation functions.
Hidden
layers
Input Output
layer layer
nTC nopt
TC
parameters
TG* TG*opt
Engine
nFT nopt
FT
opt
Qf Qf
...
Figure 1: The proposed neural network architecture (author’s research).
The architecture consists of an input layer that receives deviations in parameters and external
influences, followed by multiple fully connected hidden layers equipped with activation functions to
effectively capture complex relations among the input data. Finally, the output layer generates
corrective signals aimed at adjusting engine parameters, ensuring optimal performance and
responsiveness to identified deviations.
The input layer is composed of four neurons, each corresponding to a specific input signal
representing deviations from nominal values: ΔnTC(t) for the deviation in gas generator rotor speed,
ΔnFT(t) for the deviation in free turbine rotor speed, Δ𝑇 ∗ (𝑡) for the deviation in gas temperature, and
ΔQf(t) for the deviation in fuel consumption. This layer processes these deviation signals, serving as
the initial stage for capturing critical information necessary for subsequent computations and
adjustments within the system.
The neural network architecture includes three hidden layers, each designed to progressively
refine and abstract the input features. The first hidden layer comprises 64 neurons with a
SmoothReLU (Rectified Linear Unit) activation function, developed by this authors group in [27],
capturing non-linear relationships and providing an initial level of abstraction from the input signals.
The second hidden layer, consisting of 32 neurons and utilizing SmoothReLU activation, further
refines these representations, enhancing the model's ability to generalize across varying input
scenarios. The third hidden layer, with 16 neurons and SmoothReLU activation, reduces
dimensionality while preserving essential information in preparation for output generation. The
output layer features a single neuron with a linear activation function, producing a corrective signal
u(t) that is applied to the control inputs for engine adjustments, ensuring optimal performance and
response to deviations.
The neural network training follows a supervised training approach, using historical data of
engine operations and their corresponding corrective actions. For training, historical engine
performance data ΔnTC(t), ΔnFT(t), Δ𝑇 ∗ (𝑡), ΔQf(t) is collected and target outputs u(t) (corrective
signals) are taken. Data preprocessing involves normalizing the input data to scale it between 0 and
1, followed by splitting the dataset into training, validation, and test sets to facilitate effective model
training and evaluation. The output signal u(t) is calculated using the current weights and biases:
𝑢 = 𝑊 ∙ 𝑆𝑚𝑜𝑜𝑡ℎ𝑅𝑒𝐿𝑈(𝑊 ∙ 𝑆𝑚𝑜𝑜𝑡ℎ𝑅𝑒𝐿𝑈(𝑊 ∙ 𝑿 + 𝑏 ) + 𝑏 ) + 𝑏 , (8)
where Wi are weight matrices, bi are bias vectors, and X is the input vector.
The loss function is represented as the mean square error [38, 39] and is defined as:
1
𝐿= ∙ 𝑢 (𝑖) − 𝑢 (𝑖) , (9)
𝑛
where utrue is the actual corrective signal and upred is the predicted signal.
The gradients are calculated using backpropagation as:
𝜕𝐿 𝜕𝐿 𝜕𝑢
= ∙ . (10)
𝜕𝑊 𝜕𝑢 𝜕𝑊
Parameter updating with adaptive training rate using the Adam optimizer [40] is performed as:
𝜕𝐿 𝜕𝐿
𝑚 = (𝛽 ∙ 𝑚 ) + (1 − 𝛽 ) ∙ , 𝑣 = (𝛽 ∙ 𝑣 ) + (1 − 𝛽 ) ∙ ,
𝜕𝑊 𝜕𝑊
(11)
𝑚 𝑣 𝜂
𝑚 = ,𝑣 = ,𝑊 = 𝑊 − ∙𝑚 ,
1−𝛽 1−𝛽 𝑣 +𝜖
where mt is the first moment estimate, vt is the second moment estimate, 𝑚 and 𝑣 are the bias-
corrected first and second moment estimates, ηt is the initial learning rate adjusted based on the
parameter update.
It is noted that at the neural network (see Figure 1) training initial stage, adaptive parameters for
adaptive training algorithms are initialized. For Adam, for example, the parameters are initialized as:
m = 0 (first moment vector), v = 0 (second moment vector), β1 = 0.9, β2 = 0.999, ϵ = 10–8 (to prevent
division by zero) [38–40].
The training process involves repeating the parameters initialize, forward pass and loss
calculation for a specified number of epochs until the loss converges, ensuring that the model
effectively learns from the data. Following training, model performance is evaluated on validation
and test datasets, with hyperparameters adjusted as necessary to mitigate the risk of overfitting.
Upon successful training and validation, the model can be deployed within the onboard control
system, facilitating real-time corrective adjustments to engine parameters, thereby enhancing overall
performance and operational efficiency.
The proposed innovative model for the parametric adjustment of helicopter gas turbine engines
combines a robust mathematical framework with an advanced neural network architecture, enabling
real-time optimization and adaptive control of engine parameters. The mathematical model
effectively captures the dynamic relationships between critical operational variables, such as rotor
speeds, gas temperature, and fuel flow, using precise equations that characterize engine behavior
under varying conditions. Complementing this, the neural network leverages a multi-layered
architecture with adaptive learning rates, allowing for efficient learning from historical data and
improving the model's capability to generalize and respond to unforeseen operational scenarios. This
hybrid approach not only enhances the accuracy of performance predictions and corrective actions
but also contributes to improved engine reliability and efficiency, positioning the model as a cutting-
edge solution in aviation technology.
4. Results
The subject of this study is the TV3-117 TE [41, 42], which powers the Mi-8MTV helicopter and its
various modifications. This engine is widely used in both civil and military aviation. The recorded
parameters onboard the helicopter include: nTC(t), representing the gas-generator rotor speed
(measured by the D-2M sensor); nFT(t), indicating the free turbine rotor speed (measured by the D-
1M sensor); and 𝑇 ∗(t), representing the gas temperature before the compressor turbine (measured by
a set of 14 T-101 thermocouples) (Table 1) [43–45]. Additionally, atmospheric conditions such as
flight altitude (h), temperature (TN), pressure (PN), and air density (ρ) are considered as input variables.
For this study purposes, these atmospheric parameters are assumed to remain constant. Furthermore,
the engine's dynamic behavior under various flight conditions is thoroughly analyzed to optimize its
performance and reliability. The collected data allows for accurate modeling of the engine's
operational behavior, serving as a foundation for further improvements in control and diagnostics
systems.
Table 1
The training dataset fragment
Number The gas-generator The gas temperature in The engine The fuel
rotor r.p.m. nTC front of the compressor inlet consumption
turbine 𝑇 ∗, pressure 𝑃∗ GT
1 0.973 0.961 0.983 0.973
… … … … …
42 0.983 0.966 0.988 0.977
… … … … …
139 0.988 0.950 0.992 0.970
… … … … …
256 0.985 0.952 0.984 0.971
During the training dataset pre-processing phase, homogeneity is assessed, followed by the
division into control and test subsets, along with an evaluation of their representativeness through
cluster analysis. To evaluate the homogeneity of the training dataset, the Fisher-Pearson criterion
[46] is employed, utilizing observed frequencies and comparing them against critical values of χ²,
where the degrees of freedom r – k – 1 = 13 and the significance level α = 0.01. This methodology
enables the statistical significance determination, which is accepted only when the likelihood of
obtaining these or more extreme outcomes under the null hypothesis is less than 1 %. The computed
value of χ² = 5.721 remains below the critical threshold of 6.6, thereby the samples consistency
validating and the normal distribution hypothesis supporting.
To further affirm homogeneity, the Fisher-Snedecor criterion [47] is applied, which calculates the
ratio of the larger variance to the smaller variance, with degrees of freedom r – k – 1 = 13 and
significance level α α = 0.01. The calculated value of F = 2.224 does not surpass the critical value of
2.58, reinforcing the samples' consistency and the normal distribution hypothesis. The training and
test subsets representativeness is examined using cluster analysis, which aims to partition the input
data set X (refer to Table 1) into k distinct clusters, where k is a clusters pre-defined number. Each
cluster comprises objects deemed more similar to one another than to those from other clusters. The
k-means clustering method is utilized, focusing on minimizing the total squared distances between
the objects in a cluster and their centroids. Each object xi in the set X is allocated to the nearest
centroid according to the equation 𝐶 = arg min 𝑥 − 𝜇 , where μj represents the initial centroids
and 𝑥 − 𝜇 signifies the Euclidean distance between object xi and centroid μj. Subsequently,
centroids are recalibrated as the objects average within each cluster using 𝜇 = ∙ ∑ ∈ 𝑥 , where
𝐶 denotes the objects quantity in the j-th cluster. The calculations for Ci and μj are reiterated until
variations in cluster distribution become minimal. The algorithm concludes when none of the
centroids undergo significant alterations or upon reaching the predetermined iteration count [48,
49].
The results from the cluster analysis conducted on the training sample data (see Table 1) revealed
eight distinct classes (I…VIII). Following a random selection process, training and test samples were
formed in a 2:1 ratio (67 % training and 33 % testing). The cluster analysis performed on both subsets
unveiled the presence of eight groups, indicating a similarity in composition between the training
and test samples. The inter-group distances are nearly identical in both subsets, confirming the
comparability of their compositions (refer to Figure 2). Thus, an optimal sample size was established,
consisting of 256 elements for training (100 %), 172 elements for control (67 % of the training sample),
and 84 elements for testing (33 % of the training sample).
Figure 2: The cluster analysis results, where “left figure” denotes the training dataset, “right figure”
denotes the test (author’s research).
To assess the neural network's efficacy in the subsequent training stage, both accuracy (Figure 3)
and loss (Figure 4) are quantified. The accuracy metric reflects the percentage of correct predictions,
whereas the loss metric represents the average squared error of the predictions, illustrating the
extent to which they differ from the actual values. To determine the precise calculations ratio for
ΔnTC(t), ΔnFT(t), Δ𝑇 ∗ (𝑡), ΔQf(t), the accuracy metric is employed (Figure 3) and is calculated at training
epoch t using the following expression [50, 51]:
1
𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = ∙ 𝐼 𝑛 =𝑛 . (12)
𝑁
As illustrated in Figures 3 and 4, these metrics demonstrate that the neural network model
achieves a remarkable prediction accuracy of 99.25 % and operates effectively, with the mean squared
error remaining below 2.5 %. Moreover, the substantial decrease in the loss function from 2.5 to 0.5
% signifies an improvement in the model's performance throughout the training process.
Figure 3: The accuracy metric diagram (author’s research).
Figure 4: The loss metric diagram (author’s research).
The performance evaluation of the developed neural network (Figure 1) is carried out using
essential quality metrics, including accuracy, precision, recall, F1 score, and AUC-ROC. These
metrics provide a comprehensive assessment of the model's ability to make accurate predictions,
reduce errors, accurately identify relevant instances, and maintain a balance between precision and
recall. The F1 score provides insight into the harmonic mean of precision and recall, while AUC-
ROC evaluates the model's ability to differentiate between classes at various thresholds, ensuring
resilience in diverse operational contexts. These metrics are computed using the following
expressions [52–54]:
𝑇𝑃
𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 = , (13)
𝑇𝑃 + 𝐹𝑃
𝑇𝑃
𝑅𝑒𝑐𝑎𝑙𝑙 = ,
𝑇𝑃 + 𝐹𝑁
2 ∙ 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 ∙ 𝑅𝑒𝑐𝑎𝑙𝑙
𝐹1 − 𝑠𝑐𝑜𝑟𝑒 = ,
𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 + 𝑅𝑒𝑐𝑎𝑙𝑙
𝐴𝑈𝐶 − 𝑅𝑂𝐶 = 𝑇𝑃𝑅 ∙ 𝐹𝑃𝑅 (𝑡) 𝑑𝑡.
In this context, several key terms are utilized to assess the model's performance: True Positives
(TP) refer to instances correctly identified as positive, representing relevant cases accurately detected
by the model. False Positives (FP) indicate instances incorrectly classified as positive, highlighting
irrelevant cases mistakenly identified as relevant. Conversely, True Negatives (TN) signify instances
accurately classified as negative, while False Negatives (FN) represent relevant instances the model
failed to identify. The True Positive Rate (TPR) measures the proportion of actual positives correctly
identified, calculated as 𝑇𝑃𝑅 = . Meanwhile, the False Positive Rate (FPR) assesses the
proportion of actual negatives incorrectly identified as positives, computed as 𝐹𝑃𝑅 = .
Collectively, these metrics offer valuable insights into the classification performance of the model,
aiding in the evaluation of its effectiveness in detecting relevant instances while minimizing
erroneous classifications.
The evaluation metrics reveal significant insights into the model's performance: a precision score
of 0.989 indicates that 98.9% of the instances classified as positive are indeed relevant, reflecting a
high level of accuracy in the model's positive predictions. A recall score of 1.0 signifies perfect
sensitivity, meaning the model successfully identifies all relevant instances without missing any,
showcasing its comprehensive detection capability. Finally, the F1-score of 0.994, which is the
harmonic mean of precision and recall, highlights the model's balanced performance, indicating that
it maintains both high precision and recall rates effectively. Collectively, these scores suggest that
the model operates with exceptional reliability and accuracy in identifying relevant instances within
the dataset.
The evaluation metrics provide important insights into the model's classification performance:
the TPR = 0.833 indicates that the model correctly identifies 83.3 % of actual positive instances,
demonstrating a strong sensitivity in detecting relevant cases. The FPR = 0.0136 signifies that only
1.36 % of actual negative instances are incorrectly classified as positive, reflecting a low level of
erroneous positive predictions and enhancing the model's reliability. The FNR = 0.0095 shows that
the model fails to identify only 0.95 % of actual positives, which is a minimal proportion, indicating
high effectiveness in recognizing relevant instances. Lastly, an AUC-ROC score of 0.844 suggests
that the model has a good capability to distinguish between positive and negative classes across
various thresholds, with a value closer to 1 indicating better performance. Collectively, these metrics
reveal that the model is effective in achieving a balance between sensitivity and specificity while
maintaining robust discrimination power in classifying instances.
The results obtained in this study enabled the optimal helicopter TE parameter values prediction
(Table 2), ensuring acceptable performance for safe flight operations. Through the analysis of
recorded onboard data, including gas-generator rotor speed, free turbine rotor speed, and gas
temperature, combined with constant atmospheric parameters, the developed models provide a
reliable framework for predicting engine behavior under various flight conditions. These predicted
parameters are crucial for maintaining engine stability, minimizing risk, and enhancing the
helicopter operations overall safety and reliability in both civil and military aviation.
Table 2
The optimal helicopter TE parameter values predicted values
Set The gas-generator The gas temperature in The engine The fuel
number rotor r.p.m. nTC front of the compressor inlet consumption
turbine 𝑇 ∗, pressure 𝑃∗ GT
1 0.985 0.972 0.984 0.972
2 0.986 0.973 0.987 0.974
3 0.984 0.971 0.983 0.973
4 0.985 0.972 0.986 0.971
5 0.983 0.978 0.983 0.971
6 0.987 0.977 0.987 0.977
7 0.990 0.979 0.991 0.972
8 0.985 0.973 0.983 0.975
9 0.986 0.973 0.987 0.973
10 0.985 0.978 0.984 0.974
5. Discussions
In this research, a mathematical model (1)–(7) has been developed to optimize the helicopter TE
parametric tuning. This model focuses on accurately predicting engine performance characteristics
by integrating various operational parameters, including rotor speeds, fuel consumption rates, and
temperature profiles. By employing a systematic approach to data analysis and parameter estimation,
the model enhances the helicopter TE understanding behavior under different operating conditions.
A neural network model (see Figure 1) has been developed to implement the mathematical
framework for the helicopter TE parametric tuning optimizing. This model is designed to capture
complex nonlinear relations between various input parameters and engine performance outputs,
leveraging an architecture that typically includes multiple layers, such as input, hidden, and output
layers. The training process (8)–(11) involves a supervised learning algorithm, where the model is
exposed to a dataset comprising historical engine performance data and corresponding operational
conditions. Utilizing backpropagation, the model adjusts its weights and biases through iterative
optimization, minimizing the loss function that quantifies the difference between predicted and
actual outputs. An adaptive training rate is incorporated to enhance convergence efficiency, allowing
for dynamic adjustments based on the model’s performance during training. By employing this
approach, the neural network not only trains to predict engine behavior accurately but also improves
its capability to generalize across various operational scenarios, thus facilitating effective parametric
tuning in real-time applications.
A homogeneous and representative training dataset (see Table 1 and Figure 2) has been
formulated, consisting of input parameters crucial for the helicopter TE optimal tuning. This dataset
includes key operational variables such as rotor speeds, fuel consumption rates, temperature
readings, and other relevant metrics that influence engine performance.
A computational experiment established that the evaluation metrics reveal the neural network
model's prediction accuracy of 99.25 % (see Figure 3) and effective operation, with a mean squared
error below 2.5 %. A notable reduction in the loss function from 2.5 to 0.5 % (see Figure 4) signifies
significant performance enhancement during training. Key metrics indicate that a precision score of
0.989 reflects high accuracy in positive predictions, while a recall score of 1.0 confirms the model's
ability to identify all relevant instances, showcasing comprehensive detection capability. The F1-
score of 0.994 highlights the model's balanced performance in maintaining both precision and recall.
Further analysis shows a True Positive Rate (TPR) of 0.833, indicating strong sensitivity, and a
False Positive Rate (FPR) of 0.0136, which enhances reliability by showing that only 1.36 % of actual
negatives are misclassified. A False Negative Rate (FNR) of 0.0095 signifies high effectiveness in
recognizing relevant instances. Lastly, an AUC-ROC score of 0.844 illustrates robust discrimination
between positive and negative classes across thresholds. Collectively, these metrics confirm the
model's effectiveness in balancing sensitivity and specificity while maintaining strong classification
power.
The neural network model quality assessing obtained results made it possible to obtain the
helicopter TE optimal thermogas-dynamic parameters set (see Table 2), at which the flight will be as
safe as possible. The results obtained in this research, while promising, are subject to several
limitations that warrant consideration. The developed mathematical model (1)–(7) for optimizing
helicopter TE parametric tuning primarily relies on historical performance data, which may not
encompass all potential operating conditions, leading to reduced generalizability in real-world
scenarios. Furthermore, the neural network model's architecture, despite its capability to capture
complex nonlinear relationships, may be sensitive to overfitting, particularly if the training dataset
does not adequately represent the operational variables full spectrum, such as variations in
environmental conditions or anomalies during engine operation.
Additionally, the use of an adaptive training rate, while beneficial for convergence, may introduce
instability if not carefully managed, potentially affecting the model's reliability. The metrics
indicating high prediction accuracy (99.25 %) and low mean squared error (< 2.5 %) suggest effective
performance; however, these figures must be interpreted with caution, as they do not account for
potential biases in the training dataset or limitations in the model's assumptions regarding engine
behavior. Lastly, while the quality assessment of the neural network enabled the optimal thermogas-
dynamic parameters identification for safe flight (see Table 2), the practical implementation of these
parameters in diverse operational environments necessitates further validation through extensive
field testing to ensure their robustness and effectiveness under varying conditions.
The prospects for further research in helicopter TE parameter optimization involve a deeper
exploration into the influence that various factors have on engine performance under dynamic
operating conditions. Future studies may focus on developing more complex and adaptive neural
network architectures capable of efficiently processing and analyzing data in real time, which would
improve prediction quality and enhance model resilience to external disturbances. Additionally,
comparing different machine learning algorithms and their combinations would be beneficial in
identifying the most effective optimization approaches. It is essential to consider the impact resulting
from changing climatic and operational conditions on engine behavior, which will require expanding
the database and incorporating additional variables. Finally, integrating developed models into
helicopter control systems and testing them in real flight conditions will be crucial for verifying the
reliability and effectiveness of proposed solutions, thereby contributing to enhanced aviation engine
safety and efficiency.
6. Conclusions
The research developed a mathematical model to optimize the helicopter turboshaft engines
parametric tuning, demonstrating a high level of accuracy in predicting engine performance
characteristics. By integrating key operational parameters such as rotor speeds, fuel consumption
rates, and temperature profiles, this model significantly enhances the helicopter TE understanding
behavior across varying operating conditions. The neural network inclusion further strengthens this
framework by effectively capturing complex nonlinear relationships between input variables and
engine performance outputs, allowing for more precise parametric tuning in real-time applications.
The model's training process, leveraging a representative dataset of historical engine performance
data, has resulted in impressive performance metrics. With a prediction accuracy of 99.25 % and a
low mean squared error below 2.5 %, the neural network demonstrates its capability to generalize
well across different operational scenarios. Moreover, metrics such as precision, recall, and F1-score
indicate a robust ability to identify relevant instances while maintaining a balance between
sensitivity and specificity. These findings highlight the potential for applying this model to enhance
safety and efficiency in helicopter operations.
Despite the promising results, the research identifies several limitations that must be addressed.
The reliance on historical performance data may limit the model's generalizability to all potential
operating conditions, and the neural network’s architecture could be prone to overfitting if the
training dataset is not sufficiently diverse. Additionally, the adaptive training rate, while beneficial,
requires careful management to avoid introducing instability. Therefore, the high prediction
accuracy should be interpreted with caution, considering potential biases and limitations inherent in
the training data.
Future research avenues include exploring the influence of various factors on engine performance
and developing more complex neural network architectures for real-time data analysis. A
comparative study of different machine learning algorithms may also yield insights into the most
effective optimization approaches. Expanding the database to encompass a wider range of climatic
and operational conditions is essential for improving model robustness. Ultimately, integrating these
developed models into helicopter control systems and conducting extensive field tests will be critical
for verifying their reliability and enhancing aviation engine safety and efficiency.
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.
References
[1] J. Song, Y. Wang, C. Ji, H. Zhang, Real-time optimization control of variable rotor speed based
on Helicopter/ turboshaft engine on-board composite system, Energy 301 (2024) 131701. doi:
10.1016/j.energy.2024.131701.
[2] H. Wang, W. Jiang, X. Deng, J. Geng, A new method for fault detection of aero-engine based on
isolation forest, Measurement 185 (2021) 110064. doi: 10.1016/j.measurement.2021.110064.
[3] S. M. Hosseinimaab, A. M. Tousi, A new approach to off-design performance analysis of gas
turbine engines and its application, Energy Conversion and Management 243 (2021) 114411. doi:
10.1016/j.enconman.2021.114411.
[4] X. Zheng, H. Zeng, B. Wang, M. Wen, H. Yang, and Z. Sun, Numerical simulation method of
surge experiments on gas turbine engines, Chinese Journal of Aeronautics 36:3 (2023) 107–120.
doi: 10.1016/j.cja.2022.08.007.
[5] B. Jiang, K. Zhang, Y. Lu, Q. Miao, Fault Diagnosis and Fault-Tolerant Control of Helicopters,
Reference Module in Materials Science and Materials Engineering. Elsevier, 2024. doi:
10.1016/b978-0-443-14081-5.00006-4.
[6] C. Zhang, V. Gümmer, Multi-objective optimization and system evaluation of recuperated
helicopter turboshaft engines, Energy 191 (2020) 116477. doi: 10.1016/j.energy.2019.116477.
[7] O. Balli, Exergetic, sustainability and environmental assessments of a turboshaft engine used on
helicopter, Energy 276 (2023) 127593. doi: 10.1016/j.energy.2023.127593.
[8] Y. Ye, Z. Wang, X.Zhang, Cascade ensemble-RBF-based optimization algorithm for aero-engine
transient control schedule design optimization, Aerospace Science and Technology 115 (2021)
106779. doi: 10.1016/j.ast.2021.106779.
[9] Z. Li, T. Nikolaidis, D. Nalianda, Recursive Least Squares for Online Dynamic Identification on
Gas Turbine Engines, Journal of Guidance, Control, and Dynamics 39(11) (2016) 1–8. doi:
10.2514/1.G000408.
[10] H. Aygun, O. Turan, Application of genetic algorithm in exergy and sustainability: A case of
aero-gas turbine engine at cruise phase, Energy 238 (part A) (2022) 121644. doi:
10.1016/j.energy.2021.121644.
[11] E. Tsoutsanis, I. Qureshi, M. Hesham, Performance diagnostics of gas turbines operating under
transient conditions based on dynamic engine model and artificial neural networks, Engineering
Applications of Artificial Intelligence 236 (part B) (2023) 106936. doi:
10.1016/j.engappai.2023.106936.
[12] S. Pang, Q. Li, B. Ni, Improved nonlinear MPC for aircraft gas turbine engine based on semi-
alternative optimization strategy, Aerospace Science and Technology 118 (2021) 106983. doi:
10.1016/j.ast.2021.106983.
[13] S. Vladov, Y. Shmelov, R. Yakovliev, M. Petchenko, Helicopters Turboshaft Engines Parameters
Identification Using Neural Network Technologies Based on the Kalman Filter, Communications
in Computer and Information Science 1980 (2023) 82–97. doi: 10.1007/978-3-031-48325-7_7.
[14] F. Lu, H. Ju, J. Huang, An improved extended Kalman filter with inequality constraints for gas
turbine engine health monitoring, Aerospace Science and Technology 58 (2016) 36–47. doi:
10.1016/j.ast.2016.08.008.
[15] S. Kim, J.H. Im, M. Kim, J. Kim, Y.I. Kim, Diagnostics using a physics-based engine model in aero
gas turbine engine verification tests, Aerospace Science and Technology 133 (2023) 108102. doi:
10.1016/j.ast.2022.108102.
[16] N. Ahmadian, A. Khosravi, P. Sarhadi, Adaptive control of a jet turboshaft engine driving a
variable pitch propeller using multiple models, Mechanical Systems and Signal Processing 92
(2017) 1–12. doi: 10.1016/j.ymssp.2017.01.023
[17] S. Zhang, A. Ma, T. Zhang, N. Ge, X. Huang, A Performance Simulation Methodology for a
Whole Turboshaft Engine Based on Throughflow Modelling, Energies 17(2) (2024) 494. doi:
10.3390/en17020494
[18] E.N. Osegi, Z.O.O. Jagun, C.C. Chujor, V.I.E. Anireh, B.A. Wokoma, O. Ojuka, An evolutionary
programming technique for evaluating the effect of ambient conditions on the power output of
open cycle gas turbine plants - A case study of Afam GT13E2 gas turbine, Applied Energy 349
(2023) 121661. doi: 10.1016/j.apenergy.2023.121661
[19] X. Han, J. Huang, X. Zhou, Z. Zou, F. Lu, W. Zhou, A novel, reduced-order optimization method
for nonlinear model correction of turboshaft engines, Journal of Mechanical Science and
Technology 38 (2024) 2103–2122. doi: 10.1007/s12206-024-0340-5
[20] Y. Sha, J. Zhao, X. Luan, X. Liu, Fault feature signal extraction method for rolling bearings in
gas turbine engines based on threshold parameter decision screening, Measurement 231 (2024)
114567. doi: 10.1016/j.measurement.2024.114567
[21] S.D. Leoshchenko, H.V. Pukhalska, S.A. Subbotin, A.O. Oliinyk, Ye.O. Gofman, Neural network
diagnostics of aircraft parts based on the results of operational processes, Radio Electronics,
Computer Science, Control 2 (2022) 69–79. doi: 10.15588/1607-3274-2022-2-7
[22] J. Zhao, J. Zhang, Y. Li, Y. Wu, A Control Design of Rotor Speed Regulation for an Aero-engine
based on Smooth Switching Strategy, IFAC-PapersOnLine 54:10 (2021) 324–329. doi:
10.1016/j.ifacol.2021.10.183
[23] Y. Shen, K. Khorasani, Hybrid Multi-Mode Machine Learning-Based Fault Diagnosis Strategies
with Application to Aircraft Gas Turbine Engines, Neural Networks (130) 2020 126–142. doi:
10.1016/j.neunet.2020.07.001
[24] A. Boujamza, S. Lissane Elhaq, Attention-Based LSTM for Remaining Useful Life Estimation of
Aircraft Engines, IFAC-PapersOnLine 55 (2022) 450–455. doi: 10.1016/j.ifacol.2022.07.353.
[25] B. Li, Y.-P. Zhao, Y.-B. Chen, Unilateral Alignment Transfer Neural Network for Fault Diagnosis
of Aircraft Engine, Aerospace Science and Technology 118 (2021) 107031. doi:
10.1016/j.ast.2021.107031.
[26] J. Liu, J. Bai, S. Liu, Real-Time Monitoring of Aircraft Engines Using a Feedforward Deep Neural
Network. Lecture Notes in Mechanical Engineering: Proceedings of the 6th China Aeronautical
Science and Technology Conference 3 (2023) 390–400. doi: 10.1007/978-981-99-8861-7_40.
[27] S. Vladov, L. Scislo, V. Sokurenko, O. Muzychuk, V. Vysotska, S. Osadchy, A. Sachenko, Neural
Network Signal Integration from Thermogas-Dynamic Parameter Sensors for Helicopters
Turboshaft Engines at Flight Operation Conditions, Sensors 24:13 (2024) 4246. doi:
10.3390/s24134246.
[28] W. Shabbir, L. Aijun, C. Yuwei, Neural Network-Based Sensor Fault Estimation and Active Fault-
Tolerant Control for Uncertain Nonlinear Systems, Journal of the Franklin Institute 360 (2023)
2678–2701. doi: 10.1016/j.jfranklin.2022.12.044.
[29] A. Didier, M. N. Zeilinger, Generalised Regret Optimal Controller Synthesis for Constrained
Systems, IFAC-PapersOnLine 56:2 (2023) 2576–2582. doi: 10.1016/j.ifacol.2023.10.1341.
[30] Y. Wang, A. Li, S. Yang, H. Tian, A model reference adaptive control scheme of a high-order
nonlinear helicopter subject to input and state constraints, Journal of the Franklin Institute
359:13 (2022) 6709–6734. doi: 10.1016/j.jfranklin.2022.07.011.
[31] G. Ghazi, B. Gerardin, M. Gelhaye, R. M. Botez, New Adaptive Algorithm Development for
Monitoring Aircraft Performance and Improving Flight Management System Predictions,
Journal of Aerospace Information Systems 17:2 (2020) 97–112. doi: 10.2514/1.i010748.
[32] S. Vladov, Y. Shmelov, R. Yakovliev, Method for Forecasting of Helicopters Aircraft Engines
Technical State in Flight Modes Using Neural Networks, CEUR Workshop Proceedings 3171
(2022) 974–985. URL: https://ceur-ws.org/Vol-3171/paper70.pdf
[33] H. Feng, B. Liu, M. Xu, M. Li, Z. Song, Model-Based Deduction Learning Control: A Novel
Method for Optimizing Gas Turbine Engine Afterburner Transient, Energy 292 (2024) 130512.
doi: 10.1016/j.energy.2024.130512.
[34] B. Rusyn, B. Rusyn, O. Lutsyk, R. Kosarevych, O. Kapshii, O. Karpin, T. Maksymyuk, J. Gazda,
Rethinking Deep CNN Training: A Novel Approach for Quality-Aware Dataset Optimization,
IEEE Access 12 (2024) 137427–137438. doi: 10.1109/access.2024.3414651.
[35] E. M. Cherrat, R. Alaoui, H. Bouzahir, Score fusion of finger vein and face for human recognition
based on convolutional neural network model, International Journal of Computing 19:1 (2020)
11–19. doi: 10.47839/ijc.19.1.1688.
[36] T. E. Romanova, P. I. Stetsyuk, A. M. Chugay, S. B. Shekhovtsov, Parallel Computing
Technologies for Solving Optimization Problems of Geometric Design, Cybernetics and Systems
Analysis 55:6 (2019) 894–904. doi: 10.1007/s10559-019-00199-4.
[37] A. R. Marakhimov, K. K. Khudaybergenov, Approach to the synthesis of neural network
structure during classification, International Journal of Computing 19:1 (2020) 20–26.
doi: 10.47839/ijc.19.1.1689.
[38] K. Andriushchenko, V. Rudyk, O. Riabchenko, M. Kachynska, N. Marynenko, L. Shergina, V.
Kovtun, M. Tepliuk, A. Zhemba, O. Kuchai, Processes of managing information infrastructure
of a digital enterprise in the framework of the «Industry 4.0» concept, Eastern-European Journal
of Enterprise Technologies 1 (3–97) (2019) 60–72. doi: 10.15587/1729-4061.2019.157765.
[39] R. Zhou, Y. Guo, Y. Wang, Z. Sun, X. Liu, Safe control of logical control networks with random
impulses, Neural Networks 165 (2023) 884–895. doi: 10.1016/j.neunet.2023.06.035.
[40] F. Zare, P. Mahmoudi-Nasr, R. Yousefpour, A real-time network based anomaly detection in
industrial control systems, International Journal of Critical Infrastructure Protection 45 (2024)
100676. doi: 10.1016/j.ijcip.2024.100676.
[41] L. Gebrehiwet, Y. Nigussei, T. Teklehaynanot, A Review-Differentiating TV2 and TV3 Series
Turbo Shaft Engines, International Journal of Research Publication and Reviews 3(8) (2022)
1822–1839. doi: 10.55248/gengpi.2022.3.8.55.
[42] R. M. Catana, G. Dediu, Analytical Calculation Model of the TV3-117 Turboshaft Working
Regimes Based on Experimental Data, Applied Sciences 13:19 (2023) 10720.
doi: 10.3390/app131910720.
[43] S. Vladov, Y. Shmelov, R. Yakovliev, M. Petchenko, S. Drozdova, Neural Network Method for
Helicopters Turboshaft Engines Working Process Parameters Identification at Flight Modes. In
Proceedings of the 2022 IEEE 4th International Conference on Modern Electrical and Energy
System (MEES), Kremenchuk, Ukraine, October 20–22, 2022, pp. 604–609. doi:
10.1109/MEES58014.2022.10005670.
[44] S. Vladov, R. Yakovliev, O. Hubachov, J. Rud, Neuro-Fuzzy System for Detection Fuel
Consumption of Helicopters Turboshaft Engines, CEUR Workshop Proceedings 3628 (2023) 55–
72. URL: https://ceur-ws.org/Vol-3628/paper5.pdf
[45] S. Vladov, L. Scislo, V. Sokurenko, O. Muzychuk, V. Vysotska, A. Sachenko, A. Yurko, Helicopter
Turboshaft Engines’ Gas Generator Rotor R.P.M. Neuro-Fuzzy On-Board Controller
Development, Energies 17:16 (2024) 4033. doi: 10.3390/en17164033.
[46] H.-Y. Kim, Statistical notes for clinical researchers: Chi-squared test and Fisher’s exact test,
Restorative Dentistry & Endodontics 42:2 (2017) 152. doi: 10.5395/rde.2017.42.2.152.
[47] C. M. Stefanovic, A. G. Armada, X. Costa-Perez, Second Order Statistics of -Fisher-Snedecor
Distribution and Their Application to Burst Error Rate Analysis of Multi-Hop Communications,
IEEE Open Journal of the Communications Society 3 (2022) 2407–2424. doi:
10.1109/ojcoms.2022.3224835.
[48] Z. Hu, E. Kashyap, O.K. Tyshchenko, GEOCLUS: A Fuzzy-Based Learning Algorithm for
Clustering Expression Datasets. Lecture Notes on Data Engineering and Communications
Technologies 134 (2022) 337–349. doi: 10.1007/978-3-031-04812-8_29.
[49] S. Babichev, J. Krejci, J. Bicanek, V. Lytvynenko, Gene expression sequences clustering based on
the internal and external clustering quality criteria. In Proceedings of the 2017 12th International
Scientific and Technical Conference on Computer Sciences and Information Technologies
(CSIT), Lviv, Ukraine, 05–08 September 2017. doi: 10.1109/STC-CSIT.2017.8098744.
[50] D. Vlasenko, O. Inkarbaieva, M. Peretiatko, D. Kovalchuk, O. Sereda, Helicopter Radio System
for Low Altitudes and Flight Speed Measuring with Pulsed Ultra-Wideband Stochastic Sounding
Signals and Artificial Intelligence Elements, Radioelectronic and Computer Systems 3 (2023) 48–
59. doi: 10.32620/reks.2023.3.05.
[51] V. Kovtun, T. Altameem, M. Al-Maitah, W. Kempa, Entropy-Metric Estimation of the Small Data
Models with Stochastic Parameters, Heliyon 10 (2024) e24708. doi:
10.1016/j.heliyon.2024.e24708.
[52] M. Pasieka, N. Grzesik, K. Kuźma, Simulation modeling of fuzzy logic controller for aircraft
engines. International Journal of Computing 16(1) (2017) 27–33. doi: 10.47839/ijc.16.1.868.
[53] B. Rusyn, O. Lutsyk, R. Kosarevych, Y. Obukh, Application Peculiarities of Deep Learning
Methods in the Problem of Big Datasets Classification, Lecture Notes in Electrical Engineering
831 (2021) 493–506. doi: 10.1007/978-3-030-92435-5_28.
[54] S. Vladov, Y. Shmelov, R. Yakovliev, Helicopters Aircraft Engines Self‐Organizing Neural
Network Automatic Control System, CEUR Workshop Proceedings 3137 (2022) 28–47. doi:
10.32782/cmis/3137-3.