The paper presents a model for the helicopter turbosha engine gas temperature sensor thermocouple dynamics modeling based on a heat transfer physical model combination, a recurrent LSTM neural network with an attention mechanism, and an adaptive Kalman lter. The thermocouple discrete model is obtained that takes into account the heat capacity and thermal resistance, using the expansion of a nonlinear dependence in the operating point vicinity through a Taylor series. A modi ed LSTM architecture is developed that accepts engine parameters as input, transformed through an attention mechanism to extract relevant features. The neural network output is corrected by an adaptive Kalman lter, which is able to adjust the noise variance at each step and also suppresses out-of-band signal components using a bandpass lter. A computational experiment showed that the reconstructed temperature in front of the compressor turbine curve accurately reproduces the reference signal: the values do not deviate by more than 2...3 K, and the absolute error is kept within ± 1 K in the vast majority of points. To evaluate the training characteristics, two integral indicators were used: the e ciency coe cient Ke and the quality coe cient Kquality, which, in comparison with classical LSTM, GRU, and RNN, showed the adaptability and convergence best balance (Ke up to 0.991, Kquality up to 0.989). Additional analysis on the accuracy, precision, recall, and F1-score metrics con rmed the model's superiority (up to 0.993, 0.988, 0.986, and 0.987, respectively).
In modern helicopter turbosha engines (TE), accurate gas temperature measurement is a key factor in ensuring optimal operation, reducing component wear, and increasing overall ight safety [1–3]. Thermocouples have traditionally been used as the main sensing element [4, 5] for such measurements, but with rapid load changes and pressure uctuations, classical signal processing algorithms [6, 7] exhibit signi cant delays and errors. This is especially critical in helicopter ight conditions [8, 9], where the speed and accuracy of the engine control systems e ciency and overheating protection data are directly a ected.
The research relevance is due to increased requirements for aviation equipment e ciency and environmental friendliness, as well as the need for helicopter resource capabilities to expand in severe and changeable weather conditions [10]. The intelligent adaptive thermocouple model's use will reduce the risk of thermal damage to turbine blades, optimize fuel consumption, and reduce accidents associated with the temperature conditions incorrect diagnostics. The intelligent adaptive thermocouple model integration into on-board monitoring systems creates the prerequisites for the predictive maintenance development and the operating helicopter TE process further automation [11, 12].
In this research, an intelligent adaptive thermocouple model is built on modern machine learning methods [13, 14] and digital ltering [15, 16], which allows the temperature estimation parameters automatic adjustment in real time. Due to self-adjustment to the gas ow current dynamic characteristics, including pressure pulsations, variations in fuel composition, and temperature gradients in the channel [17], the developed model can signi cantly reduce constant and dynamic measurement errors. As a result, the response speed during transient engine operation modes increases, and the thermal state control system improves stability.
Modern research in the helicopter TE gas temperature parameters measuring eld emphasizes the data coming from thermocouples accuracy and critical importance [4, 5, 10]. Under the pressure pulsations, vibrations, and high temperatures in uence, traditional signal processing methods, such as uncontrolled noise ltering [15, 16, 18] or static calibration tables [6, 7], demonstrate signi cant errors and delays in response time. The elimination of these errors is relevant in transient engine operating modes, when timely adjustments are important to prevent overheating and component wear.
Machine learning methods, in particular neural networks [19–23] and Kalman lters [24–27], are increasingly used to improve the measurements accuracy and e ciency. Thus, in [19], a multilayer neural network with a closed dynamic compensation loop was proposed, which, a er 280 training epochs, achieved a 99.5% accuracy when integrating thermocouple signals, signi cantly surpassing classical median-recursive [28] and recursive lters [29] in the rst and second kinds of reducing errors. Similarly, the radial-basis neural networks used in combination with a multidimensional Kalman lter made it possible to ensure the accuracy of the identifying helicopter TE parameters at a 99.75% level [24].
Along with neural networks, hybrid approaches to digital ltering are also being developed [30– 32]. Adaptive lters based on classical median and averaging algorithms, in combination with digital ltering algorithms with a variable sliding window function, demonstrate better noise suppression characteristics without signi cant dynamic response loss [33]. However, their e ciency decreases with strong nonlinearities and sharp transient processes under conditions that stimulate the search for new ltering structures and self-tuning algorithms in real time.
Despite the successes achieved, there are still unanswered questions that require further development of intelligent adaptive models. Balancing the onboard computers computing resources and the model complexity remains a problem, since many methods require GPU acceleration, which is not available in aircra controllers. At the same time, the algorithm's stability in extreme weather conditions and sensor degradation in the eld has not been tested extensively enough. Also, preventing issues of over tting with limited training datasets and ensuring the model's ability to train online on ying platforms are open.
Thus, there is an urgent need to develop an intelligent adaptive thermocouple model that would combine deep learning methods (e.g., recurrent neural networks with attention mechanisms) and digital lters (adaptive Kalman lter with frequency domain ltering), taking into account the avionics real limitations. The developed model should provide dynamic compensation and selftuning for changing engine operating modes, minimize response delays, and maintain high measurement accuracy even with insu cient training data and limited processing resources.
Based on [4, 5], the thermocouple is modeled as a single-circuit heat-capacitive element with thermal capacitance C and thermal resistance R relative to the gas as: where TС(t) is the thermocouple junction temperature, TG is the helicopter TE gas in front of the turbine compressor true value.
For the explicit Euler scheme further application (
+ dt R
=0 , d T C (t ) 1
dt = τ ∙(T C (t )−T G (t )) , f (N +1) ( ξ ) ( N +1) !
∙ ∆ T N +1 , ξ ∈ (T 0 , T C ) . where τ = C · R.
Discretization (
T C [ k +1]=T C [ k ]+∆ t ∙(−τ1 ∙ (T C [ k ]−T G [ k ])).
To account for small temperature deviations around the operating point T0, when high accuracy does not require taking into account all higher orders of nonlinearity, the TC dependence local approximation using a polynomial is used. Assuming that f(TC) = exp(−a · TC), for an arbitrary smooth function f(TC) around the operating point T0, its Taylor series consists of the following components:
f (T C )=∑n=∞0 f (nn)(T! 0) ∙ (T C−T 0)n=f (T 0)+ f ' (T 0) ∙ ∆ T + f ' '2(T! 0) ∙ ( ∆ T )2+ f ' ' 3'(T! 0) ∙ ( ∆ T )3+… , (
In (
Denoting the increment as ΔT = TC − T0 and the series coe cients as a1 = f′(T0), a2= a3= f ' ' 3'(T! 0) ,… the linear term contribution to the total change f(TC) − f(T0) is equal to a1 · ΔT, and this approximation “quality” is estimated by the relative contribution as:
|a1 ∙ ∆ T| |a1 ∙ ∆ T| |f ' (T 0) ∙ ∆ T| n=1 For small ΔT the second approximation gives η1= ∞ ≈ ∑|an ∙ ∆ T n| |a1 ∙ ∆ T|+|a2 ∙ ∆ T 2|=|f ' (T 0) ∙ ∆ T|+|f ' '2(T! 0) ∙ ( ∆ T )2| . , which clearly shows how the linear term relative “strength” decreases with increasing ∣ΔT∣ and the second derivative.
Similarly, taking a1 = f′(T0), a2= f ' '2(T! 0) , a3= f ' ' 3'(T! 0) ,…, the quadratic term contribution to the total change f(TC) − f(T0) is equal to a2 · ΔT2, and its relative “strength” is estimated as the rst two terms sum fraction: |a2 ∙ ∆ T 2|
|a2 ∙ ∆ T 2| η2= ∞ ≈ ∑|an ∙ ∆ T n| |a1 ∙ ∆ T|+|a2 ∙ ∆ T 2|=|f ' (T 0) ∙ ∆ T|+|f ' ' (T 0)|∙|∆ T|2 n=1 2 = |f ' ' (T 0)|∙|∆ T|2
2
From (
. |∆ T|≫ 2 ∙|f ' (T 0)| |f ' ' (T 0)| f ' ' (T 0) 2 !
,
(
For n ≥ 3, the contribution of each n-th term an ∙ ∆ T n= f (n) (T 0) ∙ (T C−T 0)n to the total change Δf
n !
can be estimated through this term relative “strength” as:
|f (n) (T 0)|∙|∆ T|n
|an ∙ ∆ T n| n !
ηn= ∞ ≈ , (
For the remainder term rigorous estimate a er truncating the series to order N, using the
Lagrange form (
max f (N +1) (u ) |RN|≤ u∈ [T0,TC] ( N +1) ! ∙|∆ T|n+1 .
From (
Thus, when expanding the thermocouple response function in a Taylor series, the zeroth order n = 0 speci es a constant “biased” component, the rst order n = 1 ensures proportional (linear) following of the output signal to the temperature deviation, the second order n = 2 introduces a correction for the response curvature and thus takes into account average nonlinear e ects, and the higher-order terms n ≥ 3 describe more subtle, high-order nonlinearities, signi cant mainly for extreme variations in ΔT. In practice, for the helicopter TE gas temperature range, it is usually su cient to truncate the series at n = 2 or n = 3, since the remainder term RN becomes negligibly small.
It is assumed that at each k-th step the vector acts at the neural network input:
xk=[T C [ k ] , uk ] ,
(
In this research, the developed recurrent neural network with attention mechanisms and a Kalman lter basis is the LSTM network (Figure 1), whose use is justi ed by the high accuracy (99% and higher) in solving applied problems of monitoring helicopter TE [19, 25, 35].
It is assumed that hk∈ Rn is a hidden state, ck is the LSTM network cell state. Then: ik=σ (W i ∙ xk +U i ∙ hk−1+bi) , f k=σ (W f ∙ xk +U f ∙ hk−1+bf ) , ok=σ (W o ∙ xk +U o ∙ hk−1+bo) , ~ck=tanh (W c ∙ xk +U c ∙ hk−1+bc) , ck=f k⊙ ck−1+ik⊙ ~ck , hk=ok⊙ tanh (ck ) .
To focus on the most “important” past states {h1, …, hk−1}, an attention mechanism is introduced,
presented in the context:
(
j=1 ∑ exp (ek , j) m=1 Then the “enriched” hidden state is de ned as: Taking into account the above, the gas temperature prediction is determined as: ~hk=tanh (W h [ hk ; ckatt ]).
T^G [ k ]=ωTo ∙ ~hk +bo .
The adaptive Kalman lter combines the neural network output and the thermocouple dynamic model in the linear state system form sk=[T C [ k ] , T˙ C [ k ] ]T and measurements zk=T^G [ k ], which allows the temperature estimate recursive prediction and correction. Then the prediction is carried out according to the expression:
sk|k−1=F ∙ sk−1|k−1 , Pk|k−1= F ∙ Pk−1|k−1 ∙ FT +Qk , 1 where F=(0 1− ∆ t ∆ t ), and the correction is according to the expression: τ K k=Pk|k−1 ∙ H T ∙( H ∙ Pk|k−1 ∙ H T + Rk)−1 , sk|k =sk−1|k−1+ K k ∙ ( zk− H ∙ sk|k−1) ,
Pk|k =( I − K k ∙ H ) ∙ Pk|k−1 , H =[ 1 0 ] , where Rk is the measurement noise estimate, adaptively adjusted, according to the residuals recurrent estimate:
Rk= λ ∙ Rk−1+(1− λ ) ∙ ( zk− H ∙ sk|k−1)2 , 1 ∙ j ∙ ω ω0 where 0 < λ < 1.
Taking into account the helicopter avionics limitations (passing low-frequency thermal oscillations, damping high-frequency noise), to isolate in the thermocouple signal the frequency range corresponding to the gas temperature change actual dynamics (from ω1 to ω2), and to suppress noise components outside this interval, a bandpass lter is used, whose transfer function is described by the expression:
H ( j ∙ ω)=
∙ exp (− j ∙ ω ∙ τ d ) , with a regularizer for smooth changes in Kalman coe cients. Backpropagation takes a step along the gradient [36, 37]: θ ← θ−η ∙ ∂ L , ∂ L =∑ 2 ∙ (~TG [ k ]−T G [ k ])∙ ∂ θ ∂ θ k .
The developed model is implemented in the Matlab Simulink R2014b so ware environment (Figure
2), where the input signal TС[k] and the additional parameters uk vector are fed to the subsystem
“Thermocouple dynamic model,” implemented by the MATLAB Function block with the equation
T C [ k +1]=T C [ k ]+ ∆ t ∙(−τ1 ∙ (T C [ k ]−T G [ k ])). Normalized data in the sequence form are fed to the
LSTM cell implemented by the MATLAB Function block with the corresponding equations (
In the computational experiment for the helicopter TE's thermal dynamics modeling in the nominal operating mode, the TV3-117 engine [8, 9, 11, 19, 24, 35, 36] gas temperature in front of the compressor turbine TG(t) real measurement number was used. The measurements were carried out on board the serial Mi-8MTV helicopter using a standard sensor, which is a set of 14 dual chromealumel thermocouples of the T-102 type [8, 9, 11, 19, 24, 35, 36]. The tests were carried out at a 2500-meter altitude above sea level under standard atmospheric conditions (air temperature ≈ 268 K, pressure ≈ 74 kPa). The signals were recorded with a Δt = 0.25 seconds periodicity (sampling frequency 4 Hz) for 320 seconds, which yielded 1280 readings in the nal sample. The thermocouple signals were preprocessed by the onboard controller. A two-stage algorithm was used to lter out noise: smoothing with a sliding window of average length 11 readings (Savitzky– Golay, third-order polynomial [38]) and removing outliers according to the “±3σ” principle, followed by the gaps linear interpolation. When correcting for systematic errors, the thermocouples calibration characteristic (error ≤ 1.5 K) and a correction for ow velocity (up to 20 m/s) were taken into account. A er cleaning and checking for homogeneity (Shapiro–Wilk and Durbin–Watson [39] tests), the time series was reduced to a single scale using classical znormalization: z (T G)i=
2 ,
T (Gi)meas− N1 ∙ ∑i=N1 T (Gi)meas 1 ∙ ∑N (T (Gi)meas− N1 ∙ ∑i=N1 T (Gi)meas) N i=1 (24) where N = 4 · 320 = 1280.
According to the data in Figure 3, the gas temperature in front of the turbine maximum absolute temperature reached 1140 K in the exposure approximately 130...160 seconds interval; a er that, a smooth decline to 1090 K is observed by the record end. The resulting normalized series z(t) was then used in adaptive algorithms for estimating the heat transfer model and for constructing correlation function parameters with the engine output parameters.
To form the training dataset, the gas temperature in front of the compressor turbine normalized values were used. This dataset fragment is presented in Table 1. The tests conducted con rmed that the dataset complies with the Fisher-Pearson [40, 41] and Fisher-Snedecor [42, 43] homogeneity criteria, and these tests detailed results are recorded in Table 2.
It is noted that the signi cance level α = 0.01 was adopted in the research, since such a strict threshold allows us to signi cantly reduce the error rst type probability (the null hypothesis false rejection) and thereby increase the obtained conclusion’s reliability regarding the training dataset homogeneity. When analyzing the helicopter TE's gas temperature, the statistically signi cant e ect of incorrect recognition could lead to erroneous engineering solutions and emergency operating modes, so the stricter criterion choice is justi ed by the need for strict safety control and the results reliability. The α = 0.01 use provides su cient tests of statistical power with a 1280 readings data amount, which makes the conclusions about the gas temperature time series behavior justi ed.
To assess the training dataset's (see Table 1) representativeness, the k-means clustering algorithm [44, 45] was used. The original data were randomly divided in a 2:1 ratio, yielding 67 % (858 objects) for training and 33 % (422 objects) for validation. When analyzing the training part using the k-means method, the cluster number was xed in advance at eight, which made it possible to identify eight stable groups (classes I…VIII) and, thus, con rm the training and test subdatasets similar internal structure (Figure 4). It is noted that the clustering quality can be assessed, for example, through the squared distances intracluster sum:
k W k=∑ ∑ ‖x j−μi‖2 , i=1 xj∈ Ci (25) Gas temperature normalized value where Ci is the i-th cluster, μi is its center. The clusters optimal number was chosen from the “elbow” [46] on the diagram of Wk versus k.
Thus, from the total training dataset of 1280 normalized gas temperature values, 858 (67 %) were de ned as the main training subdataset, and 422 (33 %) as the test subdataset.
The computational experiment's main result is the gas temperature in front of the compressor turbine signal resulting diagram (Figure 5), obtained using the developed model (see Figures 2 and 3).As can be seen from Figure 5, the resulting model reproduces the gas temperature dynamics reference curve's key characteristics: a smooth rise from the initial 1105 K to peak values of about 1140 K in the 120...160 seconds range, followed by a uniform decrease to about 1090 K by 310 seconds, while the uctuation amplitude and the oscillation frequency composition practically coincide with the original experimental data. Deviations for each reading do not exceed 2...3 K, and not only the general trajectory preservation but also small peaks and troughs indicate that the model adequately takes into account the main heat exchange and dynamic processes in the thermogasdynamic ow, which indicates high accuracy and low modeling error (Figure 6).
Figure 6 shows that the approximated temperature from the “reference” gas temperature absolute deviations at most moments do not exceed ±1 K and are concentrated in a narrow range near zero, which indicates a small shear system. Single bursts up to +3…+3.5 K are observed around 120 seconds and at the experiment's end, and the maximum negative emissions up to –3…–4.5 K areapproximately at 180 and 250 seconds and are probably associated with transient phenomena and abrupt changes in heat exchange. Time intervals from 50 to 100 seconds and from 200 to 240 seconds are characterized by higher-frequency but small-scale oscillations (±1…2 K), which indicates the ow dynamic uctuations adequately account. The average error value is close to zero, and rare large emissions are easily compensated for by the model's local re nement.
Figure 7 shows the linear term (n = 1) contribution diagram, which is the value a1 · (TC(t) − T0) time diagram, illustrating the proportional component and its dynamics over the entire ΔT range.
Figure 7 shows the parameter a1 · (TС(t) − T0) evolution, which characterizes the thermocouple output signal relative contribution dependence on the temperature increment ΔT Taylor expansion approximation linear term, experimental recording over 320 seconds. In the initial phase (0…100 seconds), a steady increase in the parameter a1 · (TС(t) − T0) is observed from negative values to about +25 K peak, which corresponds to an increase in ΔT in the warm-up mode and the linear response predominance at small deviations from the operating point. In the 100…150 seconds region, the linear term drops sharply to negative values contribution (about –8…–12K), re ecting the transition to stronger nonlinear e ects and the quadratic in uence and higher orders when approaching the maximum gas temperature. In about 150 seconds, there is an instantaneous jump in the parameter a1 · (TС(t) − T0), probably associated with the engine operating mode restructuring or switching the ltering scheme in the Simulink model, a er which, in the 150…320 seconds interval, the contribution of the linear term steadily decreases to –30…–35 K, which indicates the nonlinear corrections dominance and the error accumulation at large temperature deviations from the base point. During the entire experiment, a1 · (TС(t) − T0) (± 2…3 K) small uctuations re ect ow pulsations and residual noise, e ectively smoothed by the Kalman lter.
Figure 8 shows the parameter a1 · (TС(t) − T0) evolution, which characterizes the thermocouple output signal relative contribution dependence on the temperature increment ΔT Taylor expansion approximation linear term, experimental recording over 320 seconds.
Figure 8 shows that the approximation a2 · (TC(t) − T0)2 changes quadratic term contribution in a wide range from 0 to 1000 K2 depending on the operating mode: in the initial heating phase (0…50 seconds) with small temperature increments, the quadratic contribution is almost absent, then in the 50…90 seconds range it quickly increases to ~600 K2, re ecting the increase in nonlinearity during the transition to moderate ΔT. Upon reaching stabilization (90…140 seconds), it again decreases to zero, to a 550 K2 peak at about 160 seconds during the second short-term accelerated acceleration (probably associated with a change in the compressor load). During the nal intensive run (200…300 seconds), the quadratic contribution steadily increases, reaching about 950 K2 maximum before the experiment end, which indicates that with large temperature di erences, it is the ΔT second power that becomes the approximation error dominant source. Small-scale uctuations (±20 K2) correspond to residual ow pulsations and sensor noise.
Figure 9 shows the residual term R2(t) diagram, which is the residual term R2(t) estimate showing where and when the higher terms become noticeable.
In Figure 9, the second-order Taylor series R2(t) residual term, proportional to the gas temperature increment ΔT3 cube, demonstrates a clear dependence on the heat ux dynamics: in intervals relative to the steady-state regime (approximately 0…40 seconds, 100…150 seconds, 180… 240 seconds), the R2 value remains close to zero, which indicates the rst two expansion terms dominance and the high-order nonlinear e ects negligibility. During transient processes (peaks in the 50…80 seconds region, a sharp jump around 160 seconds, and the most pronounced rise in the 260…300 seconds range), the R2 value increases to 2…3 · 104 K3, indicating the cubic term's signi cant contribution and the need to take into account higher orders with rapid temperature variability. The obtained results are fully consistent with the conclusions about the adaptive thermocouple model self-tuning, where in transient modes the model should adjust the lter and neural network parameters to compensate for the signal nonlinear and dynamic distortions.
Figure 10 shows the adaptive Kalman coe cient evolution diagram, which is the parameter Kk time diagram for the aim of analyzing the lter adjustment to changes in noise and model during helicopter ight.
In Figure 10, the adaptive Kalman coe cients evolution, it is evident that in the initial phase (0…≈150 seconds) and with relatively smooth temperature changes, both gain components, the temperature K tk (solid curve) and the derivative K kd (dash-dotted curve), uctuate in an approximately 0.1…0.15 narrow range, which provides a compromise between the trust in the model and the sensor signal. With a sharp transition at about 150…160 seconds, both coe cients tend to zero, which corresponds to a sharp increase in the residuals and a decrease in the noisy measurements in uence on the adaptive estimate Rk according to (19).
Figure 11 shows the di erence diagram in gas temperature approximations for N and N + 1. Figure 11 illustrates how much adding each subsequent term in the series improves (or does not) the model accuracy.
The presented diagram of the di erence in the rst (N = 1) and second (N = 2) order
approximation error squares (Figure 11) shows that the quadratic term inclusion in the Taylor
expansion provides the greatest gain precisely at the sharp transient process moments, when ΔT
reaches comparatively large values: the di erence peaks up to 900…1000 K2 fall on the rapid
increase and decrease intervals in temperature (approximately 50…80 seconds, 140…155 seconds,
and 260…305 seconds). At the same time, in the smooth change phases (0…30 seconds, 110…130
seconds, 180…200 seconds), the second-order advantage tends to zero (< 50 K2). This is completely
consistent with the quadratic term η2 relative contribution estimate from (
To evaluate the developed model with an LSTM network, an attention mechanism, and an adaptive Kalman lter (Figure 1) e ciency, used to the helicopter TE gas temperature sensor thermocouple model, two key indicators were selected: the e ciency coe cient and the quality coe cient [47, 48]. The e ciency coe cient (Ke ) shows how quickly and adequately the network adapts to errors in the optimization process and is calculated as the change ratio in the loss function at the current iteration to the change in the network parameters in the same iteration. The quality coe cient (Kquality) evaluates the approximation accuracy and the model convergence stability, calculated as the decrease ratio in the loss function at the current step to the previous iterations total losses. These indicators provide the LSTM network e ciency comprehensive assessment, facilitating its training characteristics objective analysis and the gas temperature sensor signals approximation accuracy. The e ciency and quality coe cients are calculated as [47, 48]: |E (θk )− E (θk−1)| , K quality= E (θk−1)− E (θk ) , ‖θk−θk−1‖ E (θ0)−E (θk−1) (26) where E(θ0) is the loss function initial value, E(θk) is the loss function value at the current iteration, E(θk–1) is the loss function value at the previous iteration, and ‖θk–1 – θk‖ is the LSTM network parameters change rate at the current iteration.
Table 3 presents a comparative analysis of the helicopter TE gas temperature in front of the compressor turbine sensor signals approximating e ciency using the developed model with an LSTM network, an attention mechanism, and an adaptive Kalman lter, as well as the recurrent neural networks and other traditional architectures adapted to similar problems [8, 9, 11, 19, 24, 35, 36, 47]: a traditional LSTM network [49, 50], a traditional GRU network [51], and a traditional RNN network [52].
Comparative analysis (Table 3) showed that the developed model with LSTM network, attention mechanism, and adaptive Kalman lter outperforms traditional recurrent network architectures in both metrics both in the noise absence (Ke = 0.991; Kquality = 0.989) and with the white noise addition (σ = 0.25) (Ke = 0.982; Kquality = 0.980). In turn, the traditional LSTM network demonstrated slightly lower values (without noise: Ke = 0.982; Kquality = 0.977; with noise: Ke = 0.965; Kquality = 0.961), followed by the GRU network (Ke = 0.965, Kquality = 0.960 and Ke = 0.944, Kquality = 0.938) and a simple RNN network (Ke = 0.943, Kquality = 0.938 and Ke = 0.929, Kquality = 0.921). Thus, the proposed architecture provides faster adaptation to errors and more stable convergence in the helicopter TE gas temperature signals approximating.
At the developed model with an LSTM network, attention mechanism, and adaptive Kalman lter (Figure 1), e ectiveness is evaluated in the next stage. The traditional metrics of accuracy, precision, recall, and F1-score values are compared with the traditional LSTM network [49, 50], the traditional GRU network [51], and the traditional RNN network [52], which are determined according to the expressions:
A ccuracy=
TP+TN TP +TN + FP+ FN , P reci sion= , F 1=2 ∙ P reci sion ∙ R ecall ,
P reci sion+ R ecall (27) where T P=|i : yi=1 ∩ ^yi=1| corresponds to the number of cases when the algorithm correctly identi ed the anomaly in the thermocouple signal (i.e., the real temperature deviation was detected); T N =|i : yi=0 ∩ ^yi=0| corresponds to the number of moments when the model correctly recognized the signal as normal (there is no anomaly and the model did not detect it); F P=|i : yi=0 ∩ ^yi=1| re ects the false alarms number, when in the real anomaly absence the model erroneously signaled a failure; F N =|i : yi=1 ∩ ^yi=0| shows the missed errors number, when the real temperature deviation remained undetected by the model.
The results of the comparative analysis (Table 4) show that the proposed model based on the LSTM network with an attention mechanism and an adaptive Kalman lter provides the highest performance among the considered architectures: accuracy = 0.993, precision = 0.988, recall = 0.986, and F1-score = 0.987. The classical LSTM network is characterized by lower metric values (0.983; 0.972; 0.971; 0.972, respectively), followed by the GRU network (0.980; 0.969; 0.965; 0.967) and the simple RNN network (0.954; 0.950; 0.951; 0.951). These results demonstrate the developed model's excellent ability to accurately classify normal and abnormal thermocouple signals, minimizing both false alarms and missed real deviations.
The research describes the thermocouple representation as a single-loop heat-capacity element
with heat capacity C and thermal resistance R, which is formalized by di erential equation (
The proposed approach is implemented by a recurrent neural network with attention
mechanisms and a Kalman lter by modifying the LSTM architecture (Figure 1), which takes the
vector xk (
A computational experiment performed using the discrete thermocouple model (
The e ciency and quality coe cients used for the LSTM network with an attention mechanism and a Kalman lter adaptability and convergence comprehensive assessment are substantiated, and then speci c results of the comparative analysis are presented. Table 3 shows the Ke and Kquality values, where the developed model signi cantly outperforms the classical LSTM, GRU, and RNN architectures both without noise and with the white noise σ = 0.25 addition. Table 4 presents traditional classi cation metrics (accuracy, precision, recall, and F1 score), con rming the developed model's high accuracy (accuracy = 0.993; precision = 0.988; recall = 0.986; F1 score = 0.987).
Table 5 presents the developed model's main limitations and corresponding directions for further research.
Computational The LSTM network with attention The lightweight architectures complexity and adaptive Kalman lter high (ghost networks [53], weight load makes implementation on quantization [54]) development onboard computers di cult. and hardware-accelerated
solutions.
dataset limitation
The representative data lack for The transfer learning [55] and extreme conditions (transients, synthetic data generation [56] extreme temperatures). methods application. The continuous online learning organization.
Sensitivity to Accuracy decreases due to the The sensor failure detection [57] sensor degradation thermocouples aging or partial and signal reconstruction failure. mechanisms integration (hybrid neuro- lter schemes). 4 5
Assumption of Truncating the approximation The adaptive choice approximation small deviations series to a low order (N = 2…3) order research and non-regularized in Taylor series may not take into account strong expansions implementation [58, nonlinearities. 59] for extreme ΔT.
Instability The model's resistance to sudden Conducting eld tests [60] in under extreme pressure changes, vibration and various climatic zones and weather humidity changes has not been adapting lters to the external conditions su ciently tested. disturbance dynamics.
The developed dynamic thermocouple model, based on the heat balance equation discretization and supplemented by an LSTM network with an attention mechanism and an adaptive Kalman lter, demonstrated the ability to take into account in detail the main heat exchange processes in a thermogasdynamic ow. The model e ectively integrates a Taylor series expansion nonlinear approximation with recurrent learning, providing the temperature deviations adequate estimate even at small ∆T and cleaning the signal from out-of-band noise before feeding it to the Kalman lter.
In the computational experiment, the reconstructed gas temperature in front of the compressor turbine signal almost completely coincided with the reference curve: an increase from 1105 to ~1140 K in 120...160 seconds and a decrease to ~1090 K by 310 seconds, while the deviations for each reading did not exceed 2...3 K and the absolute modeling error was kept within ±1 K.
To objectively evaluate the proposed architecture's adaptivity and accuracy, two integral indicators were introduced: the e ciency coe cient (Ke ) and the quality coe cient (Kquality), and their comparative analysis with traditional RNN architectures was performed. It is shown that the developed LSTM network with an attention mechanism and an adaptive Kalman lter outperforms classical LSTM, GRU, and RNN in both metrics both in a clean signal (Ke = 0.991; Kquality = 0.989) and when adding white noise σ = 0.25 (Ke = 0.982; Kquality = 0.980).
The results comparison based on the Accuracy, Precision, Recall and F1-score metrics con rmed the developed model leading position: the 0.993; 0.988; 0.986; 0.987 values, respectively, are signi cantly ahead of traditional LSTM (0.983; 0.972; 0.971; 0.972), GRU (0.980; 0.969; 0.965; 0.967) and RNN (0.954; 0.950; 0.951; 0.951).
The research was carried out with the grants 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), 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).
The research was supported by the Ministry of Internal A airs of Ukraine “Theoretical and applied aspects of the development of the aviation sphere” under Project No. 0123U104884.
The authors have not employed any Generative AI tools. [18] S. Cao, H. Zuo, X. Zhao, C. Xia, Real-Time Gas Path Fault Diagnosis for Aeroengines Based on Enhanced State-Space Modeling and State Tracking, Aerospace 12:7 (2025) 588. doi: 10.3390/aerospace12070588. [19] S. Vladov, L. Scislo, V. Sokurenko, O. Muzychuk, V. Vysotska, A. Sachenko, A. Yurko, Helicopter Turbosha Engines’ Gas Generator Rotor R.P.M. Neuro-Fuzzy On-Board Controller Development, Energies, 17:16 (2024), 4033. doi: 10.3390/en17164033. [20] L. Shen, Y. Wang, B. Du, H. Yang, H. Fan, Remaining Useful Life Prediction of Aero-Engine Based on Improved GWO and 1DCNN, Machines 13:7 (2025) 583. doi: 10.3390/machines13070583. [21] G. Li, L. Zhang, G. Xia, S. Li, Structural Reliability Analysis of Aero-Engine Turbine Components Based on Particle Swarm Optimization Back Propagation Neural Network, Applied Sciences 15:6 (2025) 3160. doi: 10.3390/app15063160. [22] S. Du, W. Han, Z. Kang, F. Luo, Y. Liao, and Z. Li, A Peak-Finding Siamese Convolutional Neural Network (PF-SCNN) for Aero-Engine Hot Jet FT-IR Spectrum Classi cation, Aerospace 11:9 (2024) 703. doi: 10.3390/aerospace11090703. [23] W. Gao, M. Pan, W. Zhou, F. Lu, J.-Q. Huang, Aero-Engine Modeling and Control Method with Model-Based Deep Reinforcement Learning, Aerospace 10:3 (2023) 209. doi: 10.3390/aerospace10030209. [24] S. Vladov, Y. Shmelov, R. Yakovliev, M. Petchenko, Helicopters Turbosha Engines Parameters Identi cation Using Neural Network Technologies Based on the Kalman Filter, Communications in Computer and Information Science (CCIS) book series 1980 (2023) 82–97. doi: 10.1007/978-3-031-48325-7_7 [25] X. Zhang, H. Liang, J. Feng, H. Tan, Kalman Filter Based High Precision Temperature Data
Processing Method, Frontiers in Energy Research 10 (2022). doi: 10.3389/fenrg.2022.832346. [26] Y. Wang, R.-Q. Sun, L.-F. Gou, Two-Stage Hyperelliptic Kalman Filter-Based Hybrid Fault Observer for Aeroengine Actuator under Multi-Source Uncertainty, Aerospace 11:9 (2024) 736. doi: 10.3390/aerospace11090736. [27] H. Guo, Y. Li, C. Liu, Y. Ni, K. Tang, A Deformation Force Monitoring Method for Aero-Engine Casing Machining Based on Deep Autoregressive Network and Kalman Filter, Applied Sciences 12:14 (2022) 7014. doi: 10.3390/app12147014. [28] C. Raikar, R. Ganguli, Denoising Signals Used in Gas Turbine Diagnostics with Ant Colony Optimized Weighted Recursive Median Filters, INAE Letters 2:3 (2017) 133–143. doi: 10.1007/s41403-017-0023-y. [29] F. Lu, Y. Wang, J. Huang, Y. Huang, Gas Turbine Transient Performance Tracking Using Data Fusion Based on an Adaptive Particle Filter, Energies 8:12 (2015) 13911–13927. doi: 10.3390/en81212403. [30] P. Cosenza, A.-L. Fauchille, D. Prêt, S. Hedan, A. Giraud, Statistical representative elementary area of shale inferred by micromechanics, International Journal of Engineering Science 142 (2019) 53–73. doi: 10.1016/j.ijengsci.2019.05.012. [31] M. Komar, V. Golovko, A. Sachenko, S. Bezobrazov, Development of neural network immune detectors for computer attacks recognition and classi cation. In Proceedings of the 2013 IEEE 7th International Conference on Intelligent Data Acquisition and Advanced Computing Systems (IDAACS), Berlin, Germany, 2013, pp. 665–668. doi: 10.1109/IDAACS.2013.6663008. [32] A. R. Marakhimov, K. K. Khudaybergenov, Approach to the synthesis of neural network structure during classi cation, International Journal of Computing 19:1 (2020) 20–26. doi: 10.47839/ijc.19.1.1689. [33] Z. Wang, Y. Wang, X. Wang, K. Yang, Y. Zhao, A Novel Digital Twin Framework for Aeroengine Performance Diagnosis, Aerospace 10:9 (2023) 789. doi: 10.3390/aerospace10090789. [34] T. Castiglione, D. Perrone, J. Song, L. Strafella, A. Ficarella, S. Bova, Linear model of a turbosha aero-engine including components degradation for control-oriented applications, Energies 16:6 (2023) 2634. doi: 10.3390/en16062634. [35] Y. Shmelov, S. Vladov, Y. Klimova, M. Kirukhina, Expert system for identi cation of the technical state of the aircra engine TV3-117 in ight modes. In Proceedings of the System Analysis & Intelligent Computing : IEEE First International Conference on System Analysis & Intelligent Computing (SAIC), 08–12 October 2018, pp. 77–82. doi: 10.1109/SAIC.2018.8516864 [36] S. Vladov, A. Banasik, A. Sachenko, W. M. Kempa, V. Sokurenko, O. Muzychuk, P. Pikiewicz, A. Molga, V. Vysotska, Intelligent Method of Identifying the Nonlinear Dynamic Model for Helicopter Turbosha Engines, Sensors 24:19 (2024) 6488. doi: 10.3390/s24196488 [37] V. Turchenko, E. Chalmers, A. Luczak, A deep convolutional auto-encoder with pooling – unpooling layers in ca e, International Journal of Computing 18:1 (2019) 8–31. doi: 10.47839/ijc.18.1.1270. [38] Y. Ge, H.-H. Ma, L.-Q. Wang, On the smoothing explosion pressure curves using SavitzkyGolay method, Journal of Loss Prevention in the Process Industries 80 (2022) 104929. doi: 10.1016/j.jlp.2022.104929. [39] V. Turchenko, E. Chalmers, A. Luczak, A deep convolutional auto-encoder with pooling – Unpooling layers in ca e, International Journal of Computing 18:1 (2019) 8–31. doi: 10.47839/ijc.18.1.1270. [40] 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. [41] A. Benaceur, B. Ver ürth, Statistical variational data assimilation, Computer Methods in
Applied Mechanics and Engineering 432 (2024) 117402. doi: 10.1016/j.cma.2024.117402. [42] 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. [43] S. Leoshchenko, A. Oliinyk, S. Subbotin, M. Ilyashenko, T. Kolpakova, Neuroevolution methods for organizing the search for anomalies in time series, CEUR Workshop Proceedings 3392 (2023) 164–176. URL: https://ceur-ws.org/Vol-3392/paper14.pdf [44] 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 Scienti c and Technical Conference on Computer Sciences and Information Technologies (CSIT), Lviv, Ukraine, 05–08 September 2017. doi: 10.1109/STCCSIT.2017.8098744. [45] Y. V. Bodyanskiy, O. K. Tyshchenko, A Hybrid Cascade Neuro–Fuzzy Network with Pools of Extended Neo–Fuzzy Neurons and Its Deep Learning, International Journal of Applied Mathematics and Computer Science 29:3 (2019) 477–488. doi: 10.2478/amcs-2019-0035. [46] I. Perova, Y. Bodyanskiy, Adaptive human machine interaction approach for feature selectionextraction task in medical data mining, International Journal of Computing 17:2 (2018) 113– 119. doi: 10.47839/ijc.17.2.997. [47] S. Vladov, Y. Shmelov, R. Yakovliev, Y. Stushchankyi, Y. Havryliuk, Neural Network Method for Controlling the Helicopters Turbosha Engines Free Turbine Speed at Flight Modes, CEUR Workshop Proceedings 3426 (2023) 89–108. URL: https://ceur-ws.org/Vol-3426/paper8.pdf [48] C. Romesis, N. Aretakis, and K. Mathioudakis, Model-Assisted Probabilistic Neural Networks for E ective Turbofan Fault Diagnosis, Aerospace 11:11 (2024) 913. doi: 10.3390/aerospace11110913. [49] S. Xiang, Y. Qin, J. Luo, H. Pu, and B. Tang, Multicellular LSTM-based deep learning model for aero-engine remaining useful life prediction, Reliability Engineering & System Safety 216 (2021) 107927. doi: 10.1016/j.ress.2021.107927. [50] L. Peng, N. Wei, Applications of LSTM Model for Aeroengine Forecasting. In Proceedings of the 2020 7th International Conference on Dependable Systems and Their Applications (DSA), Xi'an, China, 28–29 November 2020. doi: 10.1109/DSA51864.2020.00030. [51] J. Zou, P. Lin, Multichannel Attention-Based TCN-GRU Network for Remaining Useful Life
Prediction of Aero-Engines, Energies 18:8 (2025) 1899. doi: 10.3390/en18081899.