=Paper=
{{Paper
|id=Vol-3896/paper12
|storemode=property
|title=The optimal controller parametric synthesis using variational calculus for a dynamic system general mathematical model
|pdfUrl=https://ceur-ws.org/Vol-3896/paper12.pdf
|volume=Vol-3896
|authors=Victoria Vysotska,Vasyl Lytvyn,Serhii Vladov,Viktor Vasylenko,Oleksii Kryshan
|dblpUrl=https://dblp.org/rec/conf/ittap/VysotskaLVVK24
}}
==The optimal controller parametric synthesis using variational calculus for a dynamic system general mathematical model==
https://ceur-ws.org/Vol-3896/paper12.pdf
The optimal controller parametric synthesis
using variational calculus for a dynamic
system general mathematical model
Victoria Vysotska1,†, Vasyl Lytvyn1,†, Serhii Vladov2,∗,†, Viktor Vasylenko3,† 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
A novel method for parametric synthesis of an optimal controller, grounded in variational calculus,
has been developed, assuming system mathematical model is expressed by a differential equation in
operator form, with both sides articulated as algebraic polynomials in variable p=d/dt. The
mathematical model of the controller is represented in a similar manner. An Euler-Poisson equation
system for the extremal variational problem was derived, leading to an analytical expression for
determining the settings for the specified controller structure by equating polynomial coefficients
from the optimization task with those from the original system and controller equations. A computer
experiment involving neural network implementation of this parametric synthesis method
demonstrated high effectiveness, with accuracy reaching 99.3% and losses not exceeding 0.5%. These
results confirm the superior performance of the neural network model in prediction accuracy and
error minimization compared to traditional methods. ROC analysis of the neural network controller,
neuro-fuzzy controller, and “classical” PID controller showed that both neural network and neuro-
fuzzy controllers achieved high accuracy with minimal false positives, underscoring their reliability
in precision-critical applications. Conversely, cubic spline interpolation and the "classical" PID
controller displayed lower accuracy and higher false positives, emphasizing the advanced control
methods' advantages for complex systems.
Keywords ⋆1
optimal controller, parametric synthesis, dynamic system, control object, neural network, helicopter
turboshaft engine, gas-generator rotor r.p.m.
⋆
ITTAP’2024: 4th International Workshop on Information Technologies: Theoretical and Applied Problems, October 23-
25, 2024, Ternopil, Ukraine, Opole, Poland
1∗
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); klk.vonrgp@gmail.com (V. Vasylenko); 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-
0002-9313-861X (V. Vasylenko); 0000-0002-2967-0126 (O. Kryshan)
© 2023 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
1. Introduction
The parametric synthesis of optimal controllers for dynamic systems represents a critical
challenge in control theory, particularly when aiming to achieve desired system performance
under varying operational conditions [1–3]. This research leverages the principles of variational
calculus to develop a general mathematical model for dynamic systems, enabling the precise
formulation and solution of optimization problems associated with controller design [4, 5]. By
systematically deriving the optimal control parameters, this approach ensures enhanced system
stability, performance, and adaptability, making it a robust framework for the controllers’
synthesis in complex dynamic environments [6].
The need for highly effective methods to synthesize optimal control parameters for dynamic
systems is driven by increasing complexity and demands in modern technical systems. With
growing variability in external influences and uncertainty in dynamic processes, ensuring
stability and performance in controlled systems is crucial. Variational calculus enables
enhanced adaptability, precision, and stability across a wide range of operating conditions,
making it particularly relevant for fields such as aviation [7], energy [8], and robotics [9].
2. Related works
Research in the field of optimal controller parametric synthesis using variational calculus for
dynamic systems has evolved over several decades, with foundational researches establishing
the core principles. Initial research focused on applying variational techniques to derive optimal
control laws for both linear and nonlinear dynamic systems. These early efforts demonstrated
the potential for variational calculus to achieve precise control parameterization, leading to
significant improvements in system stability and performance [10, 11]. Researchers successfully
applied these methods to relatively simple systems, paving the way for more complex
applications [12, 13].
As the field progressed, researchers began to address the challenges associated with real-
world dynamic systems, including nonlinearity, time-varying parameters, and external
disturbances [14, 15]. Researches explored the variational calculus extension to more
sophisticated dynamic models, incorporating constraints such as actuator limitations, time
delays, and uncertainties [16, 17]. This phase of research highlighted the versatility and
robustness of variational calculus in handling complex dynamic behaviors, making it applicable
to a wide range of engineering systems [18, 19]. Advances in computational methods also played
a critical role during this period, enabling more efficient implementation of variational
techniques for optimal controller synthesis [20].
In recent years, the variational calculus with modern optimization algorithms integration
has led to further enhancements in the synthesis process. Researchers have combined these
methods with gradient-based optimization, evolutionary algorithms, and machine learning
techniques to address multi-dimensional and multi-agent systems [21–23]. These innovations
have expanded the applicability of variational calculus to more complex scenarios, such as
aerospace systems [24], robotics [25], and large-scale industrial processes [26]. Current research
continues to explore new avenues for optimizing controller parameters, focusing on improving
adaptability, computational efficiency, and real-time implementation in dynamic environments.
The necessity for parametric synthesis of controllers using variational calculus arises from
the need to optimize control laws in complex dynamic systems, ensuring enhanced stability,
adaptability, and performance across diverse operational scenarios.
3. Proposed technique
Based on [26, 27], it is assumed that the mathematical model describing the control object is
represented by a differential equation in operator form with the right-hand side defined by the
d
nominal value of the variable p= :
dt
( p +d ∙ p +…+d ∙ p+d ) ∙ y=⏟
⏟
n
n−1
n−1
( b ∙ p +b ∙ p +…+b ∙ p+b )
1 0 x
x
x−1
x−1
1 0
(1)
d ( p) b ( p)
in this case n > x; d(p)y = b(p)u. The controller mathematical model is also presented in general
form as:
( g ∙ p +…+ g ∙ p+ g ) ∙ u=⏟
⏟ x−1
x−1
( r +r ∙ p+…+r ∙ p )
1 0 1 2 n
n−1
(2)
g ( p) r ( p)
in this case g(p)u = r(p)y.
As a quality functional, the integral-quadratic criterion is adopted in a generalized form,
taking into account the constraint on energy costs for control [28] in the form:
( )
∞ γ
J =∫ ∑ q ii ∙ y 2 ∙ ( i−1 ) +u2 dt . (3)
0 i=1
It is required to the controller structure (2) and the settings rj determine such that the
functional (3) reaches its minimum value. The controller structure is defined by rj values that are
non-zero, zero, or close to zero. To solve the optimization task [26, 28, 29], i.e., to determine the
transient response to a disturbance given by non-zero initial conditions y(0) = y10, y’(0) = y20, ...
y(n)(0) = yn0, while considering boundary conditions that ensure asymptotic stability:
y ( t → ∞ )= y ' ( t → ∞ )=…= y ( n ) ( t → ∞ )=0. (4)
Based on [26, 30, 31], the Lagrange function [32] is introduced. Since the constraints in this
case are represented by a differential equation: d(p)y = b(p)u, the Lagrange multiplier λ should be
replaced by a time-dependent Lagrange variable λ(t) in the Lagrange function. Thus, the
Lagrange function is expressed as:
γ
L=∑ q ii ∙ y 2 ∙ ( i−1 ) +u2 + λ ( t ) ∙ ( d ( p ) −b ( p ) ) .
y u
(5)
i=1
Let us compose the Euler-Poisson equation for the functional L(y, u) of two functions y(t) and
u(t) and obtain a system of two equations:
{
∂L d ∂L d ∂L
+… (−1 ) ∙ β ∙ ( β ) =d (− p ) λ+2 ∙ q ∙ ( p2 ) ∙ y=0
β
− ∙
∂ y dt ∂ y ' dt ∂ y
(6)
∂L d ∂L α d ∂L
− ∙ ' +… (−1 ) ∙ α ∙ ( α ) =2 ∙ u−b (− p ) λ=0
∂ u dt ∂ u d t ∂u
γ
where q ( p )=∑ q ii ∙ y 2 ∙ ( i−1 ) ∙ (−1 )
2 ( i−1 )
.
i=1
By eliminating the variables u and λ from these equations, the equation for the variational
task extremals has been obtained.
( d ( p ) ∙ d (− p ) +b ( p ) ∙ b (− p ) ∙ q ( p2 ) ) y=0. (7)
Characteristic polynomial
∆ ( p )=d ( p ) ∙ d (− p ) +b ( p ) ∙ b (− p ) ∙ q ( p )
2
(8)
is an equation of degree 2∙β, where β=max {n , ( γ + x−1 ) }.
For simplicity, let x + γ – 1 ≤ n. This is a polynomial of even degrees P, which can be
expressed in factored form:
∆ ( p )=δ ( p ) ∙ δ (− p ) , (9)
where δ ( p ) is the polynomial of degree β containing the roots of a polynomial Δp with a
negative real part.
On the other hand, the closed-loop system characteristic polynomial is given by:
D ( p )=d ( p ) ∙ q ( p ) +b ( p ) ∙ r ( p ) . (10)
We form the identities δ ( p ) = D(p) and compare the same degrees coefficients of P in these
identities, resulting in equations for determining the desired controller settings.
To summarize, the formula for the optimal control u(t) is obtained by solving the resulting
polynomial equations and substituting the values back into the control law:
r ( p)
u ( t )= ∙ y (t ) . (11)
g ( p)
where the coefficients rj are determined from:
∆ ( p )=D ( p ) . (12)
Thus, solving these equations will provide the optimal controller settings and structure. This
mathematical model presents a novel approach to optimizing dynamic systems control through
several key aspects:
1. The model utilizes polynomials in the frequency domain to describe system and
controller dynamics. This enables efficient optimization by considering both nominal
system parameters and controller parameters.
2. Employing time-dependent Lagrange multipliers λ(t) to account for control constraints
provides flexibility in solving optimization problems and allows integration of stability
conditions and energy expenditure constraints.
3. The introduction of an integral-quadratic criterion (quality criterion) allows for
consideration of both dynamic system error and control costs, ensuring a more
comprehensive and accurate analysis of control quality.
4. Introducing the characteristic polynomial Δ(p) and its factorization simplifies the
optimal controller settings determination, streamlining the solution and interpretation
process.
Optimal control u(t) can be obtained by solving the characteristic polynomial equation and
substituting the derived coefficients into the control expression. Solving these equations and
substituting values yields optimal settings for rj, minimizing functional J and satisfying all
stability conditions. This ensures optimal system control considering given constraints and
requirements.
Thus, the proposed model and method for optimal control represent an innovative approach
to controlling dynamic systems, integrating modern polynomial analysis and Lagrange
multipliers to achieve optimal outcomes.
4. Experiment
The work presents a computer experiment involving the proposed technique application in the
helicopter turboshaft engines (TE) gas-generator rotor r.p.m. nTC controlling task. The proposed
neural network (Figure 1) is a tool for implementing the developed technique and is
implemented using the TensorFlow and Keras libraries [33, 34].
Figure 1: The proposed neural network (author’s research).
The input layer takes four input parameters: the current gas-generator rotor r.p.m. nTC, the
¿
gas temperature in front of the compressor turbine T G , the engine inlet pressure P ¿¿, and the fuel
consumption GT [35, 36], that is:
x=[ nTC ,T G¿ , P ¿¿ ,G T ] . (13)
Hidden layers with ReLU activation function allow the model to capture nonlinear
dependencies between input and output data.
The first hidden layer is defined as:
h1=ReLU ( W 1 ∙ x +b1 ) , (14)
where W1 is the first hidden layer weight matrix, b1 is the first hidden layer bias, and ReLU is the
ReLU activation function. A modified Smooth ReLU function described in [37] can be used,
which preserves the ReLU advantages, such as the gradient absence for positive values, while
adding smoothness for negative values.
The second hidden layer is defined as:
h2=ReLU ( W 2 ∙ h1 +b2 ) , (15)
where W2 is the second hidden layer weight matrix, b2 is the second hidden layer bias.
The output layer returns the desired gas-generator rotor r.p.m. nTC predicted value, i.e.:
y=W 3 ∙ h2 +b3 , (16)
where W3 is the output hidden layer weight matrix, b2 is the output hidden layer bias.
A loss function is used to minimize the mean squared error (MSE), allowing the model to
train from the difference between the predicted and actual gas-generator rotor r.p.m. nTC values.
5. Results
The focus of this research is the TV3-117 TE, an integral component of the Mi-8MTV
helicopter's propulsion system and its variants, extensively utilized in civil and military aviation
[35–38]. Flight tests provided the gas-generator rotor r.p.m. nTC, the gas temperature in front of
¿
the compressor turbine T G, the engine inlet pressure P ¿¿, and the fuel consumption GT, which
formed a training dataset with 256 values for each parameter (Table 1).
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 G , 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
The training dataset homogeneity assessment, detailed in [35–38], utilized Fisher-Pearson
[39] and Fisher-Snedecor [40] criteria. According to these measures, the dataset is
homogeneous, as the calculated Fisher-Pearson and Fisher-Snedecor values are below their
2 2
respective critical thresholds, specifically χ =5.717< χ critical =6.6 and
F=2.221< F critical =2.58. To evaluate the representativeness of the dataset, cluster analysis
was performed using k-means [41, 42]. The dataset was split into training and test subsets in a
2:1 ratio (67 and 33 %, equating to 172 and 84 samples, respectively). The cluster analysis (Table
1) revealed eight distinct classes (I...VIII), confirming these groups presence and demonstrating
consistency between the training and test datasets (Figure 2). These findings facilitated the
optimal sample datasets determination: the training dataset comprises 256 elements (100 %), the
validation dataset includes 172 elements (67 % of the training set), and the test dataset includes
84 elements (33 % of the training set).
a b
Figure 2: The cluster analysis results: a is the training dataset, b is the test (author’s research).
During the neural network training initial phase, the epochs number impacts the MSE,
calculated as [43]:
N
1
∙ ∑ ( n^ TCi
2
MSE t = t
−nTC ) , (17)
N i=1
where N represents the count observations (i.e., the elements number in the training dataset),
t
nTCi denotes the actual value, and n^ TCi indicates the predicted value at training epoch t.
This is achieved by examining the MSEt values at each epoch, as computed using (17), and
evaluating the relations between MSEt and the epochs number (Figure 3). Notably, the MSE
metric in this research is influenced by the neural network’s loss function, with the model error
magnitude reflecting the difference between predicted and actual values.
To identify the neural network training epochs optimal number, a training curve is analyzed
to pinpoint the epoch where the validation dataset error is minimized or reaches stability,
indicating that further training yields no significant improvement. For each epoch t ∈ [1, t], the
MSE on the validation dataset MSEval(t) is calculated. The epochs optimal number t∗ is determined
as:
{| }
t ¿ =min t ∀ t ' ,|MSE val ( t ' )−MSE val ( t ' )|¿ ϵ , (18)
where t∗ denotes the initial epoch from which the change in error on the validation dataset
remains below the threshold value ϵ for all subsequent epochs.
Figure 3: Diagram illustrating the epochs number impact on the mean squared error (author’s
research).
The findings suggest that 160 training epochs are adequate to reach the minimum value of
MSEmin = 0.969.
To assess neural network performance during the subsequent stage of training, accuracy
(Figure 4) and loss (Figure 5) are calculated over 160 training epochs. The Accuracy metric
quantifies the proportion of correctly predicted values, whereas the Loss metric indicates the
mean squared error of predictions, reflecting the extent of deviation from actual values.
To estimate the accurate calculations proportion for the gas-generator rotor r.p.m. nTC, the
Accuracy metric is employed (Figure 4), which is calculated at training epoch t using the
following expression:
N
1
Accuracy t = ∙ ∑ I ( n^ TCi =nTC ) ,
t
(19)
N i=1
where I represents an indicator function that equals 1 if the predicted value matches the true
value, and 0 otherwise.
As illustrated in Figures 4 and 5, these metrics suggest that the neural network model
achieves high accuracy (99.3 %) in predictions and demonstrates efficiency, as the mean squared
error remains below 2.5 %. Additionally, a substantial decrease in the loss function from 2.5 to
0.5 % indicates an improvement in model quality throughout the training process.
Figure 4: The accuracy metric diagram (author’s research).
Figure 5: The loss metric diagram (author’s research).
Simulation results for the automatic control system of the gas-generator rotor r.p.m. nTC
using various PID controllers are shown in Figure 6. The neural network regulator is
represented by a blue transient response, the neuro-fuzzy controller developed in [43] is
depicted in black, and the “classical” PID controller [44] is shown in red.
The transient responses of the neural network and neuro-fuzzy controllers exhibit identical
speed (transient process time of 1.5 seconds), which is notably shorter compared to the transient
process time of 2.5 seconds observed with the “classical” controller.
In practice, the helicopter TE model is described by a high-order differential equation (up to
the 30th order) [45]. This poses challenges in synthesizing controllers. Simplified models,
typically described by the 2nd to 4th order equations, are used in the control systems
development. However, not all controllers can ensure proper system performance under real
flight conditions. Therefore, the helicopter TE various operating modes are considered in the
control systems research and simulation.
Figure 6: The diagrams of transient processes for the gas-generator rotor r.p.m. nTC during
engine operation in the helicopter turboshaft engines cruise mode (author’s research).
The helicopter TE model presented in [35–38] is relevant for the cruising (primary) mode.
The research also examines engine models in two additional modes: emergency (climbing) and
idle (ground) conditions.
The gas-generator rotor r.p.m. nTC transient characteristics with various PID controllers
during the helicopter TE emergency operation are shown in Figure 7.
Figure 7: The diagrams illustrating the transient processes during helicopter turboshaft engines
operation in climb mode (author’s research).
Figure 8 shows the simulation results for the system in idle mode. In maximum operating
conditions, both neural network and neuro-fuzzy PID controllers achieve a transient response
time of 2 seconds. In contrast, the system with a classical controller has a transient response
time of 6 seconds, significantly longer than with the neural network and neuro-fuzzy PID
controllers. In idle mode, the system with the “classical” controller exhibits overshooting and
oscillations, which are unacceptable. The neural network and neuro-fuzzy PID controllers
achieve a transient response time of 1 second.
Figure 8: The diagrams illustrating the transient processes during helicopter turboshaft engines
operation in ground mode (author’s research).
6. Discussion
A method for parametric synthesis of an optimal controller based on variational calculus has
been developed, assuming that the mathematical model of the system is represented by a
differential equation in operator form, with both sides expressed as algebraic polynomials in the
d
variable p= . The controller's mathematical model is similarly represented. Based on this, an
dt
Euler-Poisson equation system for the extremal variational task was derived. By equating the
polynomials coefficients in the optimization task with those obtained from the system and the
controller original equations, an analytical expression was obtained to determine the settings
for the specified controller structure.
A computer experiment was conducted involving the neural network implementation of a
parametric synthesis method for optimal control based on variational calculus (see Figure 1).
The obtained results (accuracy up to 99.3 % as shown in Figure 3 and losses not exceeding 0.5 %
as depicted in Figure 4) demonstrate the effectiveness and high precision of the developed neural
network-based approach. These findings indicate that the neural network model achieves
superior performance in terms of prediction accuracy and error minimization compared to
traditional methods, validating its suitability for optimizing control systems in complex
applications.
For ROC analysis across the three approaches (the proposed neural network controller, the
neuro-fuzzy controller [43], and the “classical” PID controller [44]), true positive and false
positive rates were computed for each class and method, followed by the generation of
corresponding ROC curves. This process required establishing a binary classification for each
class (distinguishing this class from all others).
A confusion matrix was constructed for the four classification categories (True Positives,
True Negatives, False Positives, False Negatives). Each cell within the confusion matrix
represents the frequency with which the actual class (rows) was predicted as the corresponding
class (columns) for each method. For each class, True Positive Rate (TPR) and False Positive Rate
(FPR) are calculated as [46–48]:
TP FP
TPR= , FPR= . (20)
FP+TN FP+TN
The area under the ROC curve (AUC) can be approximated using the trapezoidal rule. Given
that (TPRi and FPRi) represent the ROC curve points coordinates, the AUC is determined as [49–
51]:
P−1
TPR i +TPR i+1
AUC= ∑ ∙ ( FPR i+1−FPR i ) , (21)
i=1 2
where P denotes the ROC curve points number [52]. The ROC analysis results are presented in
Table 2.
Table 5
ROC analysis results (author's research)
Actual \ The proposed The neuro-fuzzy The “classical” PID
Predicted neural network controller [43] controller [44]
controller
True Positives 94 94 0
True Negatives 4 3 11
False Positives 269 270 280
False Negatives 23 22 110
TPR 0.832 0.832 0
FPR 0.012 0.012 0.426
AUC 0.856 0.857 0.214
As a result, the neural network controller proposed in this work, along with the neuro-fuzzy
controller [43], provides high accuracy with a minimal level false positive. The use cubic spline
interpolation, on the contrary, leads to low accuracy and the highest number false positives.
This highlights the effectiveness and reliability the neural network and neuro-fuzzy
controllers in scenarios where precision is critical. Their ability to minimize false positives
without sacrificing accuracy makes them superior choices for applications requiring robust and
accurate control mechanisms. Conversely, the “classical” PID controller’s [44] higher rate false
positives underscore its limitations in achieving the desired accuracy. The significant contrast in
AUC values further emphasizes the advantages these advanced control methods bring to the
table, making them preferable options in complex and dynamic systems.
These findings suggest that the neural network and neuro-fuzzy controllers are not only
more reliable but also more adaptable to varying conditions, ensuring stable performance across
different operational modes. In contrast, the “classical” PID controller’s [44] limitations could
lead to suboptimal decisions in critical scenarios, potentially compromising system safety and
efficiency.
The difference in the areas under the AUC curve is clearly illustrated in Figure 9.
a b
с
Figure 9: The AUC-ROC curve: a is the proposed neural network controller; b is the neuro-
fuzzy controller [43]; c is the “classical” PID controller [44] (author's research).
Future research in this domain should focus on several key areas to further enhance the
developed method's applicability and robustness. One avenue is the extension of the parametric
synthesis approach to multi-variable systems, where the interaction between multiple control
variables can significantly impact overall system performance. Additionally, integrating
advanced machine learning techniques, such as reinforcement learning or hybrid models
combining neural networks with fuzzy logic, could further improve the controller's adaptability
and precision in dynamic environments. Another promising direction involves real-time
implementation and validation in practical applications, particularly in systems with high
nonlinearity and uncertainty. This would not only confirm the theoretical results but also
provide insights into potential modifications required for real-world deployment. Lastly,
exploring the method's scalability and computational efficiency for larger, more complex
systems could open new possibilities for its use in cutting-edge technologies, such as
autonomous vehicles or advanced robotics.
7. Conclusions
This research scientific novelty lies in the development of a method for parametric synthesis of
an optimal controller using variational calculus, with the system and controller models
represented by differential equations in operator form as algebraic polynomials in the variable
d
p= . This approach led to the derivation of an Euler-Poisson equation system for extremal
dt
variational tasks, providing an analytical expression for determining optimal controller settings.
Furthermore, the implementation of this method through neural networks demonstrated
remarkable accuracy, achieving prediction precision up to 99.3 % and minimal losses not
exceeding 0.5 %. These results underscore the superiority of the proposed neural network-based
approach in optimizing control systems, particularly in complex and dynamic applications,
marking a significant advancement over traditional control methods.
The ROC analysis conducted for the three approaches is the proposed neural network
controller, the neuro-fuzzy controller, and the “classical” PID controller is involved computing
true positive and false positive rates for each class, followed by generating corresponding ROC
curves based on binary classification for each class. The results demonstrated that both the
neural network and neuro-fuzzy controllers achieved high accuracy with minimal false
positives, highlighting their effectiveness and reliability in precision-critical scenarios. In
contrast, cubic spline interpolation resulted in lower accuracy and the false positives highest
number, while the "classical" PID controller exhibited limitations due to its higher rate of false
positives. The marked difference in AUC values underscores the superiority of the neural
network and neuro-fuzzy controllers, making them preferable for applications requiring robust
and accurate control mechanisms in complex and dynamic systems.
Acknowledgements
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). 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. 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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