<!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Archiving and Interchange DTD v1.0 20120330//EN" "JATS-archivearticle1.dtd">
<article xmlns:xlink="http://www.w3.org/1999/xlink">
  <front>
    <journal-meta>
      <journal-title-group>
        <journal-title>Z. Jiang, S. Yang, X. Wang, Y. Long, An Onboard Adaptive Model for Aero-Engine Performance
Fast Estimation, Aerospace</journal-title>
      </journal-title-group>
    </journal-meta>
    <article-meta>
      <article-id pub-id-type="doi">10.1016/j.pnucene.2023.104836</article-id>
      <title-group>
        <article-title>Hybrid model for predicting and correcting readings of the helicopter turboshaft engines gas temperature sensor using neural networks and CPS infrastructure⋆</article-title>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author">
          <string-name>Serhii Vladov</string-name>
          <email>serhii.vladov@univd.edu.ua</email>
          <xref ref-type="aff" rid="aff1">1</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Anatoliy Sachenko</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
          <xref ref-type="aff" rid="aff4">4</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Victoria Vysotska</string-name>
          <email>victoria.a.vysotska@lpnu.ua</email>
          <xref ref-type="aff" rid="aff2">2</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Nataliia Vladova</string-name>
          <xref ref-type="aff" rid="aff3">3</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Viktor Ostroverkhov</string-name>
          <email>v.ostroverkhov@wunu.edu.ua</email>
          <xref ref-type="aff" rid="aff4">4</xref>
        </contrib>
        <aff id="aff0">
          <label>0</label>
          <institution>Casimir Pulaski Radom University</institution>
          ,
          <addr-line>Malczewskiego Street 29 26-600 Radom</addr-line>
          ,
          <country country="PL">Poland</country>
        </aff>
        <aff id="aff1">
          <label>1</label>
          <institution>Kharkiv National University of Internal Affairs</institution>
          ,
          <addr-line>L. Landau Avenue 27 61080 Kharkiv</addr-line>
          ,
          <country country="UA">Ukraine</country>
        </aff>
        <aff id="aff2">
          <label>2</label>
          <institution>Lviv Polytechnic National University</institution>
          ,
          <addr-line>Stepan Bandera Street 12 79013 Lviv</addr-line>
          ,
          <country country="UA">Ukraine</country>
        </aff>
        <aff id="aff3">
          <label>3</label>
          <institution>Ukrainian State Flight Academy</institution>
          ,
          <addr-line>Chobanu Stepana Street 1 25005 Kropyvnytskyi</addr-line>
          ,
          <country country="UA">Ukraine</country>
        </aff>
        <aff id="aff4">
          <label>4</label>
          <institution>West Ukrainian National University</institution>
          ,
          <addr-line>Lvivska Street 11 46009 Ternopil</addr-line>
          ,
          <country country="UA">Ukraine</country>
        </aff>
      </contrib-group>
      <pub-date>
        <year>1913</year>
      </pub-date>
      <volume>9</volume>
      <issue>12</issue>
      <fpage>0000</fpage>
      <lpage>0001</lpage>
      <abstract>
        <p>The research develops a hybrid model for the helicopter turboshaft engine gas temperature sensor readings predicting and correcting, which combines machine learning algorithms and a cyber-physical infrastructure (CPS) to improve the monitoring accuracy and adaptive response to changes in the engine operating mode. into account both the measurement noise stochastic components and the therocouple slow thermodynamic sh -processing, which smooths out high-frequency by the network through weighted fusion with a confidence coefficient k, which ensures the accumulated drift is almost completely eliminated while maintaining performance. The CPS infrastructure is used, which includes a secure communication channel shown that the telemetry delivery time grows linearly fro n amount of S at S experiments have demonstrated the proposed architecture's high efficiency: the adjusted temperature cur with traditional LSTM, RBF network, and three-layer perceptron confirmed the proposed solution's superiority in accuracy (0.991), recall (0.985), and F1 measure (0.986) terms. The obtained results show that the hybrid approach provides reliable engine health diagnostics and can form the basis for developing a ICyberPhyS 5: 2nd International Workshop on Intelligent &amp; CyberPhysical Systems, July 04, 2025, Khmelnytskyi, Ukraine 1 Corresponding author. These authors contributed equally.</p>
      </abstract>
      <kwd-group>
        <kwd>helicopter turboshaft engine</kwd>
        <kwd>LSTM with drift gate</kwd>
        <kwd>CPS infrastructure</kwd>
        <kwd>thermocouple drift1</kwd>
      </kwd-group>
    </article-meta>
  </front>
  <body>
    <sec id="sec-1">
      <title>1. Introduction</title>
      <p>Modern helicopter turboshaft engines (TE) place high demands on the operating process, monitoring
key parameters for accuracy and efficiency [1]. One of the most critical indicators is the gas
temperature in front of the compressor turbine, which directly affects the engine efficiency, its
components' service life, and flight safety [2]. Traditional thermocouple sensors [3, 4], despite their
production technologies' development, are subject to errors, aging, and an aggressive environmental
influence, which entails measurement distortion and a decrease in the engine's actual state
assessment reliability.
readings is becoming especially relevant. On the one hand, neural networks [5] are capable of taking
al flight and test data, forming more
accurate estimates of probable temperature values. On the other hand, the CPS infrastructure
(cyberphysical systems) use [6] ensures continuous data exchange between physical sensors and
computing modules in real time, which allows not only to correct current readings but also to
adaptively respond to changes in the engine operating mode.</p>
      <p>The neural network technologies and CPS infrastructure integration are becoming critically
important for increasing helicopter TE reliability: by continuously combining analytical models and
real data, it is possible to promptly identify deviations in the
emergencies, and adaptively adjust engine operating parameters. The introduction of such hybrid
solutions not only helps reduce the unscheduled repairs and extend the components' service life but
also increases the aircraft's overall cost-effectiveness.</p>
      <p>Thus, the hybrid model development for predicting and correcting temperature readings directly
meets modern requirements for helicopter TE safety, efficiency, and intellectualization control.</p>
    </sec>
    <sec id="sec-2">
      <title>2. Related works</title>
      <p>In studies [2, 4, 7], the traditional approach to modeling and assessing the gas temperature in
helicopter GTEs is based on the heat exchange and gas flow dynamics physical laws. Such models
(gas-dynamic, thermophysical) provide an approximation in steady-state modes at the 90...92 % level
but demonstrate significant errors (up to 5 %) under transient conditions and due to the
thermocouples' own imperfection (drift due to corrosion, thermal shock, etc.). At the same time,
classical error compensation algorithms [8] usually rely on linear calibration and do not take into
account complex nonlinear interactions, which leads to insufficient accuracy (up to 80%) and a slow
response to changes in the engine operating mode.</p>
      <p>With the machine learning methods, studies have emerged on the use of neural networks to
correct temperature sensor data. In particular, multilayer perceptrons [9] and recurrent neural
networks [10] trained on historical data from takeoff, idle, and cruise modes are considered. Such
complex nonlinearities, but they often have difficulty generalizing beyond the training dataset and
require large labeled data sets, which acquisition in aviation conditions is associated with high costs.</p>
      <p>In parallel, the cyber-physical systems (CPS) direction for aviation equipment is actively
developing: distributed architectures are being created that are capable of combining telemetry
streams from various sensors, onboard computing modules, and headquarters storages in real time
[11, 12]. The CPS infrastructure ensures reliable and scalable data transmission, flexible deployment
of analysis algorithms (edge computing on board and cloud analytics), and a high degree. However,
existing CPS solutions [11,</p>
      <p>Research combining machine learning and CPS is still mostly conceptual or demonstrated in
laboratory setups: helicopter TE digital twins are created with embedded neural network blocks to
predict failures and modes of destabilization [13]. However, most of them focus on large-sized
stationary installations (thermal power plants [14], gas pumping stations [15]) and are poorly
adapted to the helicopter TE-specific features of limited computing resources of onboard computers,
high vibration levels and thermal effects, and strict requirements for data transmission and
processing delays.</p>
      <p>Among the unanswered questions, the following are particularly relevant: how to automatically
conditions change; how to optimally distribute computing resources between onboard and ground
systems (edge-cloud), taking into account delays and transmitted data amounts; and how to
guarantee the reliability and the CPS infrastructure cybersecurity, ensuring the telemetry protection
and model updates even in combat or emergency situations.</p>
      <p>Thus, the hybrid model combining the development of a neural network (to account for complex
nonlinearities and predict error-free values) and a CPS infrastructure (to collect, transmit, and
flexibly process data in real time) allows us to close these gaps. Such a system will make it possible
to automatically compensate for thermocouple drift, continuously adapt to new modes, and perform
the helicopter TE gas temperature monitoring accuracy and reliability.</p>
    </sec>
    <sec id="sec-3">
      <title>3. Proposed model</title>
      <sec id="sec-3-1">
        <title>3.1. The basic model development</title>
        <p>This study proposes a hybrid model structural diagram for the helicopter TE gas temperature sensor
readings predicting and correcting using neural networks and a CPS infrastructure (Figure 1),
predicting
adaptation, which ensures resistance to sensor degradation and adaptability to operating conditions.</p>
        <p>Gas
temperature
sensor</p>
        <sec id="sec-3-1-1">
          <title>Data collection and preprocessing module (Edge)</title>
        </sec>
        <sec id="sec-3-1-2">
          <title>Sensor drift detection module (Edge)</title>
        </sec>
        <sec id="sec-3-1-3">
          <title>Temperature prediction neural network (Edge)</title>
        </sec>
        <sec id="sec-3-1-4">
          <title>Located on board a helicopter</title>
        </sec>
        <sec id="sec-3-1-5">
          <title>Data correction and merging module</title>
        </sec>
        <sec id="sec-3-1-6">
          <title>Exit to engine control system</title>
        </sec>
        <sec id="sec-3-1-7">
          <title>Retraining and model update module (Cloud)</title>
        </sec>
        <sec id="sec-3-1-8">
          <title>Transferring the updated model back to the Edge device</title>
        </sec>
        <sec id="sec-3-1-9">
          <title>Cloud analytics and data warehousing (Cloud)</title>
        </sec>
        <sec id="sec-3-1-10">
          <title>Secure telemetry channel (Edge</title>
        </sec>
        <sec id="sec-3-1-11">
          <title>Cloud)</title>
          <p>preprocessing module (Edge) filters, normalizes, and aggregates the data. It removes anomalies and
determines the signal's validity. It prepares inputs for intelligent algorithms. The sensor drift
detection module (Edge) compares the current signals with the reference profiles, identifies
deviations caused by thermocouple aging or environmental influences, and activates correction or
transmits an alarm. The temperature prediction neural network (Edge) uses historical and current
combines
the measured value, the neural network predicting, and the drift estimate. It produces a final
corrected temperature value suitable for use by the engine control system. The output to the engine
management system transmits the corrected temperature value to the engine regulation and
monitoring system.
telemetry between on-board and cloud modules, cloud analytics with data storage for long-term
analysis, trend detection and fault diagnostics, and a module for automatic retraining of the neural
network on new data with adaptation to new operating modes and subsequent delivery of the
updated model back to the on-board computers.</p>
          <p>Based on [16], in this study the noise is represented as Gaussian:
ymeas (t ) = ytrue (t ) + d (t ) + n(t ).</p>
          <p>n(t )</p>
          <p>N (0, n2 ).</p>
          <p>The drift d(t) is described, for example, by a stochastic diffusion process [17] in the form:
dd (t )
dt
=  d (t ) +  + (t ),  (t )</p>
          <p>N (0,2 ).</p>
          <p>Preprocessing (filtering) includes low-pass approximation using an RC filter [18], represented by
the expression:
 
dz (t )
dt</p>
          <p>+ z (t ) = ymeas (t ), z (0) = ymeas (0),
where z(t) is the smoothed signal.</p>
          <p>In the drift detection module, a comparison with the reference profile yref(t) is performed as:
The study developed a hybrid model mathematical formulation, reflecting all the diagram key
blocks (Figure 1). It is assumed that ytrue(t) is the true gas temperature at time t, ymeas(t) is the
d(t) is the sensor drift, n(t) is the measurement noise, d (t ) and ynn (t )
are the drift and true temperature estimates issued by the corresponding modules.</p>
          <p>The sensor reading model is described by the expression:
ed (t ) = z (t ) − yref (t ).</p>
          <p> ed (t ),if ed (t )   ,
d (t ) = </p>
          <p>d (t − t ),otherwise.</p>
          <p>x(t ) = z (t ), z (t ), P(t ),... ,
In this case, the threshold logic is defined as:</p>
          <p>The neural network predicts the gas temperature according to the feature vector:
where P(t) are other engine parameters.</p>
          <p>Based on [19, 20], in this study, the prediction neural network is implemented by a recurrent
model (LSTM) as:</p>
          <p>The neural network is trained by minimizing the loss function:
In the correction and merge module, the final score is determined as:
h(t ) = LSTM(x(t ),h(t − t ), W),</p>
          <p>T 2
L ( W, V,b) =  ( ynn (t ) − ytrue (t )) dt +   W 2 .</p>
          <p>
            0
neural
network
ycorr (t ) = ynn (t ) + d (t ) + k  ( z (t ) − ynn (t )),
drift
local adaptive correction
where k ∈ [0, 1] is the confidence coefficient for the raw data.
(
            <xref ref-type="bibr" rid="ref1">1</xref>
            )
(
            <xref ref-type="bibr" rid="ref2">2</xref>
            )
(
            <xref ref-type="bibr" rid="ref3">3</xref>
            )
(
            <xref ref-type="bibr" rid="ref4">4</xref>
            )
(
            <xref ref-type="bibr" rid="ref5">5</xref>
            )
(
            <xref ref-type="bibr" rid="ref6">6</xref>
            )
(
            <xref ref-type="bibr" rid="ref7">7</xref>
            )
(
            <xref ref-type="bibr" rid="ref8">8</xref>
            )
(
            <xref ref-type="bibr" rid="ref9">9</xref>
            )
(10)
3.2. The CPS infrastructure and retraining model development
The data flow from the board to the cloud is encrypted using the SSL/TLS protocol, and the channel
throughput and latency are described by the equation:
          </p>
          <p>SSL/TLS
x(t ), z (t ), d (t ), ynn (t ) → CloudStorage,
and the channel throughput and delay are described by the equation:</p>
          <p>Latency (t ) = L0 +</p>
          <p>S (t )</p>
          <p>B</p>
          <p>+  (t ),</p>
          <p>D − i−N:i , i
Di = i
 i−N:i</p>
          <p>M,
where L0 is the basic protocol delay, S(t) is the amount of transmitted packets during the interval t,
B is the channel throughput, and (t) is the delay's random component.</p>
          <p>Received on-board data D = x(t ), z (t ), d (t ), ynn (t ) are aggregated into a buffer of size N and
normalized over a sliding window:
where i N:i and i N:i are the mean and standard deviation of the last N points.</p>
          <p>Tretrain hours in the cloud, a retraining optimization problem of the form
is solved:</p>
          <p>Wm,Vin,b  M1  iM=1  ynn  ti ; W*, U*,V*,b*,by  − ycorr (ti ) 2 +   W 2 +  R (W) ,
(11)
(12)
(13)
(14)
(15)
(16)
(17)
new data, and
is the
L2where R(W) is a regularizer to prevent overfitting in changing modes,
neural network parameters, 0 &lt;
regularization coefficient.</p>
          <p>The weight update is performed using stochastic gradient descent as:</p>
          <p>Wk+1 = Wk − k  W L, kli→m k = 0,  k = .</p>
          <p>k</p>
          <p>To quickly respond to new conditions, a forgetting factor is introduced into the loss function,
represented as:</p>
          <p>M 2
L =  M −i  ( ynn (ti ) − ycorr (ti )) ,
i=1</p>
          <p>L = Lprev − L*   th ,
where 0 &lt; &lt; 1 specifies the new data priority degree.</p>
          <p>After convergence, {W*, V*, b} are sent back on board via the telemetry channel. The transmission
condition is represented as:
where th is the minimum gain in model quality that requires updating.</p>
          <p>Thus, the CPS infrastructure use not only ensures reliable bidirectional data exchange but also
implements the neural network adaptive retraining full cycle, taking into account the new
measurements and communication channel limitations priority.
3.3. The neural network predicting model development
To develop a neural network predicting model (Figure 2), it is assumed that the input vector consists
of the filtered temperature value z(t) and its derivative z (t ) , the latest drift estimate d (t − t ) , and
the remaining engine operating parameters vector P(t). Thus,</p>
          <p> z (t ) 
xt =  Pz((tt))   m.</p>
          <p> 
 d (t − t ) 
ct–
ht–
xt</p>
          <p>Whf
Wxf
Whi
Wxi
Whc
Wxc
Whg
Wxg
Who
Wxo</p>
        </sec>
        <sec id="sec-3-1-12">
          <title>Forget gate</title>
          <p>(sig)</p>
        </sec>
        <sec id="sec-3-1-13">
          <title>Input gate</title>
          <p>(sig)</p>
        </sec>
        <sec id="sec-3-1-14">
          <title>Candidate cell state (tanh)</title>
        </sec>
        <sec id="sec-3-1-15">
          <title>Drift module</title>
          <p>(sig)
Output gate
(sig)
ft
it
ct
gt
ot
tanh
ct
ht
it = (Wi  xt + Ui  ht−1 + bi ),
ft = (Wf  xt + U f  ht−1 + bf ),
ot = (Wo  xt + Uo  ht−1 + bo ),
ct = tanh (Wc  xt + Uc  ht−1 + bc ),
gt = (Wg  xt + Ug  ht−1 + bg ),</p>
          <p>W*  nm , U*  nn , b*  n .</p>
          <p>To account for and compensate for gradual shifts in the thermocouple characteristics, a built-in
Based on the current discrepancy between the filtered signal and the previous drift estimate, it
generates a correction value for updating the memory state and is described by the expression:
(18)
(19)
(20)
(21)
(22)
(23)
where ⊙ is the element-wise multiplication, and δt adds to the cell memory information about the
current discrepancy between the filtered signal and the drift estimate.</p>
          <p>Thus, the output vector is defined as:
ct = ft
ct−1 + it</p>
          <p>ct + t ,
ht = ot</p>
          <p>tanh (ct ),
ynn (t ) = V  h(t ) + by ,
k+1 = k −  L(k ).
where V  1n , by </p>
          <p>.</p>
          <p>The weights are updated using stochastic gradient descent:</p>
          <p>The resulting architecture is an extended LSTM cell that takes as input the preprocessed z(t), its
derivative, engine parameters, and a previous drift estimate, within which the standard input, forget,
which the combined memory state produces an output hidden vector that is transformed by a linear
layer into a predicted gas temperature value.</p>
        </sec>
      </sec>
    </sec>
    <sec id="sec-4">
      <title>4. Case study</title>
      <sec id="sec-4-1">
        <title>4.1. The experimental setup description</title>
        <p>In this study, the developed hybrid model structural diagram for predicting and correcting gas
temperature sensor readings (see Figure 1) is implemented in the Matlab R2014b software
environment (Figure 3).
gas temperature sensor) reproduces the
ytrue(t), sensor drift d(t), and
noise component n(t) sum. It includes a source ytrue is an organic sinusoidal (or tabular) signal
reflecting the real temperature dynamics; a drift module implemented through an integrator
receiving low-amplitude white noise at the input (generated by the Random Number block), which
simulates a slowly increasing drift; and a measuring noise generator n(t) in the Gaussian random
ymeas(t) =
ytrue(t) + d(t) + n(t), which is then fed for filtering and subsequent processing.</p>
        <p>The Filtering/Preprocessing block takes the raw reading ymeas(t) as input and applies a continuous
1
RC filter with the transfer function , where is selected based on the smoothing required
  s + 1
degree, thereby filtering out high-frequency noise and outputting z(t
in parallel to the filter output is the Derivative block, which calculates z (t ) for further use in the
predictive model. If necessary, a normalization or scaling block can be added on the RC filter top, but
the basic circuit is limited to a series connection of the Transfer Fcn (Numerator = [1], Denominator
= [ , 1]) and Derivative blocks, which ensures smoothing of the input signal and its derivative
formation for the following processing stages.</p>
        <p>The Drift Detection block compares the smoothed value z(t) with a preset reference profile yn ·
en(t), calculating the difference draw(t) = z(t) yn · en(t) and its absolute value e(d(t)) = |draw(t)|; then,
via the Compare To Constant block, it is checked whether e(d(t)) exceeds a specified threshold , and
if so, the logical flag Drift_Flag = 1 is set, signaling the drift presence, and the drift estimate dest(t) =
draw(t) is passed on for correction algorithms.</p>
        <p>The neural network temperature prediction (NN Prediction) block takes as input a feature vector
including the current smoothed value z(t), its derivative z (t ) , drift estimate d (t ) , and additional
engine parameters P(t), and then loads a pre-trained LSTM network (netLSTM) via a custom
MATLAB Function and performs a one-step prediction
ytrue (t ) = predict (netLSTM, z (t ), z (t ), P (t ),...) ). Inside the MATLAB Function, the first call loads
the trainedNet.mat file containing the LSTM weights and architecture using a persistent variable,
calculated, which is output as ypred, providing adaptive prediction taking into account current and
historical data.</p>
        <p>The Correction &amp; Fusion block combines the raw readings ymeas, the predicting ypred and the drift
estimate d , computing the corrected predicting ycporerrd = ypred − d , and then mixes it with the raw
data via weighted addition: the signal ymeas is multiplied by the confidence coefficient k (Gain k), then
the difference ycporerrd k k), and both results are summed (Sum), forming
the final value ycorr = k  ymeas + (1− k )  ( ypred − d ) , which is output for further use by the control
system.
ycorr signal generated
in the previous step as input and transmits it to the external controller via the output port (Outport),
providing integration with the engine control system; in addition, the Scope block is connected in
parallel for the ycorr dynamics visual monitoring during modeling, and if necessary, the Drift_Flag
logical signal from the drift detection module can be output to a separate Outport (for example,
"Drift_Flag_Out") to indicate the sensor alarming state.
4.2. The input data analysis and preprocessing
The computational experiment used data on the gas temperature in front of the compressor turbine
ymeas(t) of the TV3-117 engine, recorded by the standard Mi-8MTV helicopter sensor (14 dual
thermocouples T-102 [5, 19]) in the nominal operating mode. The tests were carried out at the 2500
meters altitude, measuring every 0.25 seconds for 320 seconds. According to the data in Figure 4, the
maximum gas temperature reached 1140 K.</p>
        <p>The data on the gas temperature, obtained during flight tests of the Mi-8MTV helicopter using
on-board monitoring, were cleared of interference and anomalies and then transformed into
timeordered series [21]. To unify the scales, z-normalization [22] was used:
z ( ymeas )i =
ym(ie)as −
1  N ym(ie)as</p>
        <p>N i=1
1  N ym(ie)as − 1
N i=1 </p>
        <p>N
  ym(ie)as 
N i=1</p>
        <p>To assess the training dataset (see Table 1) representativeness, the k-means cluster analysis
method was used [24]. The training dataset was randomly divided into training and test datasets in
a ratio of 2:1 (67% are 858 elements and 33% are 422 elements). The training dataset clustering
identified eight groups (classes I VIII), which confirms the two datasets' structural similarity
(Figure 5). Based on this, the gas temperature dataset amounts were determined: out of a total
training dataset of 1280 elements, 858 (67%) constitute the control dataset, and 422 (33%) constitute
the test dataset.
4.3. The computational experiment results
For reproducible LSTM training, the Adam optimizer was used with learning rate=10 3 (can be
reduced to 10 4 if necessary), batch size=32 64, and seeds were fixed (numpy.random.seed(42),
torch.manual_seed(42)).</p>
        <p>The deterministic mode was enabled (torch.backends.cudnn.deterministic=True,
torch.backends.cudnn.benchmark=False), and a fixed number of epochs and a training rate reduction
scheme were set (ReduceLROnPlateau with a 0.1 factor and patience=5).</p>
        <p>As the computational experiment part, diagrams
corrected gas temperature readings (Figure 6), the sensor drift estimate and its predicted estimate
(Figure 7), the data transmission delay and the telemetry amount dependence (Figure 8).</p>
        <p>In Figure 6, the gas temperature ranges from 1080 K to 1150 K, with the blue thin curve showing
the raw data, which fluctuates with about 30 K amplitude and contains significant noise and drift,
while the red curve shows the corrected readings: they have a smoother shape, the drift trend is
removed, and the noise is smoothed out, allowing us to observe the true temperature dynamics
without sensor artifacts.</p>
        <p>According to Figure 7, the thermocouple drift in degrees Celsius ranges from about 0 to 2 °C. The
true drift is represented by the red solid line and consists of a linear trend increasing at about a 0.02
°C/s rate, superimposed on a sine wave with a 0.5 °C amplitude and about a 62-second period, plus
some random noise with a 0.05 °C variance. The blue dotted line is phased out with the true drift by
about 1 second and has a slightly smaller sine wave amplitude (about 0.45 °C) due to smoothing, and
the noise in its values is limited to about 0.03 °C variance. Thus, the dotted line accurately follows
the general growth and the drift oscillation, but with a small delay and reduced sawtooth peaks.</p>
        <p>According to Figure 8, the transmitted telemetry amount covers the range from 10 to 2000 kbps,
and the measured transmission delay varies from approximately 50 ms to 1400 ms.
ms (base L0 = 50 ms plus minor random noise), then with an increase to 500 kbps, the delay increases
linearly to approximately 350 ms, and by 2000 kbps, it is already approaching 1400 ms.</p>
        <p>The curve shows small fluctuations of ±10 ms due to random components of (t), but the overall
S
(kilobits), the transmission time increases almost proportionally, showing the channel bandwidth
influence.</p>
        <p>Despite random fluctuations in , the overall curve shows a linear increase in delay as the data
amount increases: first, at S &lt; 100 kbps, a 60 70 ms plateau is visible, then a predictable rise to 400
ms for S 2000 kbps maximum load.</p>
        <p>This clearly illustrates the channel capacity limitation and the basic L0 delay impact.
4.4. The results obtained effectiveness evaluation
To evaluate the predicted gas temperature values obtained, the various neural network architectures
(traditional LSTM network, three-layer perceptron, and neural network on radial basis functions)
were compared using accuracy, precision, recall, and F1-score metrics (Table 3), where:
Accuracy =</p>
        <p>TP + TN
TP + TN + FP + FN
, Precision =</p>
        <p>TP
TP + FP</p>
        <p>TP
TP + FN
, F1-score = 2 </p>
        <p>Precision  Recall
Precision + Recall</p>
        <p>(25)</p>
        <p>The comparative analysis results of the four models demonstrate the modified LSTM network's
advantage over the others: its accuracy = 0.991 exceeds the traditional LSTM (0.979), the radial basis
function network (0.951), and the three-layer perceptron (0.922) indicator; the classification accuracy
(precision) of the proposed LSTM is 0.987, while that of the traditional LSTM is 0.973, RBF is 0.944,
and that of the perceptron is 0.909; the recall indicator (0.985) for the proposed model also exceeds
the traditional LSTM (0.962), RBF (0.934), and the perceptron (0.883).</p>
        <p>Finally, the F1-score of the proposed LSTM (0.986) is the highest compared to 0.967 of the
traditional LSTM, 0.939 of RBF, and 0.896 of the three-layer perceptron, indicating a consistently
higher balance between precision and recall in the proposed architecture.</p>
        <p>The neural network model's stability to external interference was analyzed. For this aim, additive
interference in the white noise with zero mathematical expectation i = 0.025 form was introduced,
which corresponds to 2.5 % [25]. Table 4 shows the calculating results, the standard deviation, and
the absolute error without noise and in the white noise when applying the above neural network
architectures to the problem of predicting gas temperatures.</p>
        <p>From Table 4 it can be seen that the modified LSTM network demonstrates the best performance
among the considered architectures: without noise its standard deviation is 0.008, and with noise it
is only 0.135, while the absolute error without noise is 0.9%, and with noise it is 1.623%; the traditional
LSTM is inferior with deviations of 0.021 (without noise) and 0.216 (with noise) and errors of 2.1%
and 3.779%, respectively; the network on radial basis functions shows higher deviations of
0.063/0.335 and errors of 4.9%/8.117%.</p>
        <p>In turn, the three-layer perceptron has the worst results, with deviations of 0.119 (without noise)
and 0.519 (with noise) and absolute errors of 7.8% and 14.035%, which emphasizes the stability and
noise resistance of the proposed LSTM architecture.</p>
      </sec>
    </sec>
    <sec id="sec-5">
      <title>5. Discussions</title>
      <p>
        A hybrid model for the helicopter TE gas temperature sensor readings predicting and correcting
block diagram has been developed using a neural network and the CPS infrastructure (see Figure 1).
predicting
raw data ymeas(t) are collected, which contain the true
temperature value ytrue(t), drift d(t), and noise n(t), according to model (
        <xref ref-type="bibr" rid="ref1">1</xref>
        ).
      </p>
      <p>
        The noise is approximated by a Gaussian process (
        <xref ref-type="bibr" rid="ref2">2</xref>
        ), and the drift is described by a stochastic
model (
        <xref ref-type="bibr" rid="ref3">3</xref>
        ). Preprocessing includes smoothing using an RC filter (
        <xref ref-type="bibr" rid="ref4">4</xref>
        ), and drift detection is based on
comparing the signal with the reference profile yref(t) (
        <xref ref-type="bibr" rid="ref5">5</xref>
        ) and applying threshold logic (
        <xref ref-type="bibr" rid="ref6">6</xref>
        ).
derivative, drift, and engine operating parameters P(t), as shown in (
        <xref ref-type="bibr" rid="ref7">7</xref>
        ) (
        <xref ref-type="bibr" rid="ref8">8</xref>
        ). The readings correction
is implemented by merging the prediction, measurement, and drift estimate according to (10) with a
confidence weighting coefficient k ∈ [0,1].
      </p>
      <p>
        The CPS infrastructure use ensures secure two-way exchange between the edge and cloud
modules, where data storage, analytics, and the neural network automatic retraining on new data
with regular resending of the updated model on board are implemented (see equations (11) (17)).
(16), while the weights are updated using the stochastic gradient descent method (15), and the
resending occurs when a specified thr
adaptive, drift- and noise-resistant temperature predicting system capable of self-correction and
realtime learning.
readings ymeas(t), including the true value ytrue(t), drift d(t), and noise n(t) (
        <xref ref-type="bibr" rid="ref1">1</xref>
        ) comparison, with the
corrected values ycorr obtained taking into account the neural network predicting and drift estimate.
The original signal (thin blue curve) fluctuates within 1080 1150 K with noise spikes and increasing
drift, while the corrected curve (thick red line) effectively eliminates drift and smooths out noise,
confirming the filtering correctness (
        <xref ref-type="bibr" rid="ref4">4</xref>
        ) and data fusion algorithm (10). This is achieved through the
RC filter and weighted correction algorithm (confidence coefficient k
temperature estimates for the control system. Figure 7 shows the drift estimate d (t ) based on the
comparison with the reference profile (
        <xref ref-type="bibr" rid="ref5">5</xref>
        ) and the threshold logic (
        <xref ref-type="bibr" rid="ref6">6</xref>
        ), which accurately reproduces
econds),
with a small delay (~1 second) and smoothed amplitude. This confirms the algorithm's high
sensitivity and its suitability for drift compensation in real time.
      </p>
      <p>Figure 8 shows that the telemetry transmission delay in the CPS infrastructure depends on the
from
100 to 2000 Kbit, the delay increases from ~100 to ~1400 ms. This indicates the need to optimize the
computations between the edge and cloud modules (14) (17).</p>
      <p>
        Table 3 demonstrates that the modified LSTM with a drift gate (20), (21) outperforms traditional
LSTM, RBF networks, and perceptrons in accuracy, recall, and F1-score due to adaptive weight
updating (16), (23) and drift accounting (
        <xref ref-type="bibr" rid="ref8">8</xref>
        ).
from 0.9% to 1.6%), which is significantly better than that of analogues. This is explained by the new
data priority ( in (16)) and built-in distortion compensation.
      </p>
      <p>As a result, the integration of the extended LSTM cell (19)
the CPS infrastructure (11) (17) ensures high accuracy and reliability in predicting and correcting
the helicopter TE gas temperature.</p>
      <p>Along with the significant results of the study, the following limitations should be noted,
presented in Table 5. Table 6 presents prospects for further research.</p>
      <p>Description
The model was trained and tested on data obtained from
TV3117 engines on the Mi-8MTV helicopter in the nominal
operating mode and at a 2500 meters fixed altitude. Under other
2
3
4
types and
operating</p>
      <p>modes
Dependence on
reference profile
and training
data quality
Limitations on
computational
resources and
latencies in the</p>
      <p>CPS
infrastructure
Simplified a</p>
      <p>priori
assumptions
about the noise
and drift nature
conditions (different engines, changed flight modes, extreme
temperatures, high loads), the noise, drift, and gas temperature
dynamics parameters may differ significantly, which will
require additional training or adaptation of the neural network
architecture and preprocessing algorithms.</p>
      <p>The drift detection module requires a predefined reference
profile yref(t) to function correctly. If the reference profile does
not accurately reflect the actual engine dynamics (e.g., due to
component wear or incorrect thermocouple calibration), the
algorithm might misinterpret natural temperature variations as
drift or, conversely, miss real shifts in readings. Training an
LSTM network requires a high-quality labeled data; insufficient
or biased representation in the training dataset limits the
prediction accuracy outside the training domain.</p>
      <p>On board the helicopter, computing resources (CPU, memory)
are limited. A complex LSTM model with a "drift gate" and
additional calculations for drift estimation may require more
resources than are available in real time, which leads to an
increase in the prediction and correction delay. Considering that
the model shows that the data transmission delay via the CPS
channel can reach more than 1 second with a load of about 2000
Kbps, in extreme scenarios (e.g., in emergency modes), the
updated model may simply not have time to be delivered on
board in time, and the readings correction may be performed
with a delay, which reduces its practical effectiveness.</p>
      <p>In the study, the measurement noise n(t) is represented as a
Gaussian process, and the thermocouple drift d(t) is modeled by
a stochastic diffusion process with a specific sinusoidal
superposition. In real engine conditions, more complex and
nonstationary sources of errors are possible: nonlinear effects of
corrosion, thermal shocks, microcracks in the thermocouple,
and electromagnetic interference. Accordingly, the algorithm
may underestimate or misinterpret anomalies that go beyond
the adopted a priori models, which reduces the estimates'
accuracy in real field tests.</p>
      <p>Action
Adaptation and validation of the proposed hybrid drift
detection model for other gas turbine engines and for
different flight regimes.</p>
      <p>Study of the altitude, temperature, and load variations'
influence on predicting accuracy and methods development
for the model parameters' automatic reconfiguration.</p>
      <p>Develop algorithms for automatic generation and adaptive
updating of the reference profile (yref) based on current
operational data to reduce the impact of outdated or
inaccurate calibration curves.</p>
      <p>Expanding the training set through field testing and
simulating scenarios with different sensor wear states;
implementing data augmentation methods to model
unexpected temperature fluctuations.</p>
      <p>Research into more compact neural network architectures
(e.g., lightweight LSTM, TCN, or Transformer-Lite) and
weight quantization to reduce memory and CPU costs.</p>
      <p>Development of a prototype taking into account the on-board
-board</p>
      <p>Optimizing
computational
complexity and
implementing it
on the onboard
computing</p>
      <p>system
Modeling and
accounting for
more complex
nonlinear sources
of noise and drift</p>
      <p>integrating it
with the general
monitoring
system
data exchange via the CPS channel.</p>
      <p>The stochastic models' construction takes into account
nonstationary corrosion processes, transient thermal effects
(thermal shocks), electromagnetic interference, and
destruction of thermocouple insulation.</p>
      <p>To investigate methods for anomaly adaptive detection that
are not specific to predefined sinusoidal patterns using hybrid
approaches (e.g., combining LSTM with variational
autoencoders or GANs).</p>
      <p>Integrating Develop online learning methods that enable the model to
feedback and self- adjust its parameters based on continuously incoming data
learning on real measurements and maintenance results.
mechanisms in The adaptive algorithm implementation automatically
real time reconfigures itself after maintenance or when systematic
deviations are detected, reducing the need for manual
reconfiguration.</p>
      <p>Developing the A single platform creation combining engine thermodynamic
concept of a models, drift detection algorithms, and other diagnostic
modules (e.g., vibration analysis, wear assessment of parts).</p>
      <p>Ways to investigate transmitting aggregated data to
groundbased maintenance centers and using cloud computing
resources for deeper analytics and unit life prediction.</p>
    </sec>
    <sec id="sec-6">
      <title>6. Conclusions</title>
      <p>The helicopter turboshaft engines' gas temperature sensor readings hybrid model for predicting and
correcting structural diagrams has been developed and implemented. It combines an extended LSTM
el takes into account stochastic
descriptions of noise and thermocouple drift, applies an RC filter for preliminary processing, and a
builtaccount the accumulated bias.</p>
      <p>The CPS infrastructure use ensures encrypted
twoguarantees the model adaptability to changes in engine operating modes.</p>
      <p>As the computational experiments result, the corrected readings demonstrate effective drift
removal and noise spike smoothing compared to the raw data, which confirms the preprocessing
algorithms' (RC filter) correctness and correction via weighted fusion. The drift estimation accurately
reproduces the linear trend and the true drift sinusoidal oscillations with about 1 second delay and
reduced amplitude (~0.45 °C), which proves the proposed detection logic's high sensitivity and
adequacy.</p>
      <p>The neural network architectures comparative analysis showed that the modified LSTM with a
-layer perceptron in all
metrics: the accuracy (0.991), recall (0.985), and F1-measure (0.986) of the proposed model are
LSTM retains low RMSE (0.135) and absolute error (1.623%), which confirms its resistance to low
external interference.</p>
      <p>The CPS channel characteristics study showed that the data transmission delay increases linearly
emphasizes the need to balance computations between onboard modules and cloud analytics. The
th is exceeded, but the obtained
latencies indicate potential limitations under extreme loads, requiring further optimization of
transmission protocols and the neural network architecture simplification.</p>
    </sec>
    <sec id="sec-7">
      <title>Acknowledgements</title>
      <p>0123U104884.</p>
      <p>The research was carried out with the grants support of the National Research Fund of Ukraine:
project registration number 273/0024 from 1/08/2024 (2023.04/0024), and "Information system
development for automatic detection of misinformation sources and inauthentic behaviour of chat
users", project registration number 187/0012 from 1/08/2024 (2023.04/0012).</p>
    </sec>
    <sec id="sec-8">
      <title>Declaration on Generative AI</title>
      <p>The author(s) have not employed any Generative AI tools.</p>
    </sec>
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