The article develops an intelligent adaptive model for correcting the helicopter turboshaft engine gas temperature in front of the compressor turbine sensor readings, based on a hybrid variational-Bayesian MLP network with two hidden layers and Smooth ReLU activation. The model combines a physicalstatistical component, a first-order dynamic error, the calibration drift approximation by a sensor time polynomial, a nonlinear characteristic expanded in a Taylor series up to the third order, and machine learning methods: the recursive least squares (RLS) method for initialising parameters and Bayes by Backprop for estimating the weight distribution uncertainty. The variational approximation μ and Σ is initialised based on the RLS results, which ensures fast convergence and regularisation of training based on physical assumptions. The experimental validation was performed in MATLAB Simulink 2014b using real flight data of the TV3-117 engine on the Mi-8MTV helicopter (clock step 0.25 seconds, total time 320 seconds, maximum temperature 1140 K). The results showed that the third-order nonlinear characteristic approximation provides an error of no more than 0.1 K within 1080-1150 K, and the RLS estimates for the drift parameters and dynamics τ and α reach 50% convergence in the first 20 steps. With variational training, ELBO increased from -440 to -45 in 100 epochs and stabilised by the 50th iteration. The developed model demonstrated high indicators: Accuracy = 0.992; recall = 0.997; precision = 0.988; F1-score = 0.987; average training time is about 182 seconds with an accuracy variance of about 1.08 · 10⁻⁶. Comparative analysis with 1D-CNN, LSTM/GRU, and extended Kalman filter confirmed the developed model's improvement in terms of resistance to noise and sensor drift in real time up to 5%.
The development of the modern helicopter turboshaft engine (TE) requires precise control of its
operating parameters [
0000-0001-8009-5254 (S. Vladov); 0000-0001-6417-3689 (V. Vysotska); 0000-0002-9676-0180 (V. Lytvyn);
0000-00034837-4688 (Y. Sahun); 0009-0009-7957-7497 (N. Vladova); 0000-0002-2196-252X (M. Luchkevych)
consumption, and provokes the components' premature wear. The introduction of the intelligent
adaptive correction model [
The research relevance is due to the growing requirements for helicopter flight reliability and
safety [
Existing methods for compensating errors in helicopter TE temperature sensors [
In recent years, adaptive filters have become widespread, in particular the Kalman filter [20] and its extended modifications [21–24], used to estimate the engine's actual thermal state based on data from several sensors. These methods demonstrate high accuracy in the presence of a correct mathematical model of the process dynamics. Still, they are sensitive to incorrect initialisation of covariance matrices and require significant computational resources when processing multidimensional vector states in real time.
Machine learning (ML) methods have begun to be implemented for the analysis of large datasets obtained during the helicopter TE ground and flight tests [25–28]. In the last decade, researchers, for example, in [29–32], have proposed neural network correlators [29, 30] that are capable of training sensors with nonlinear characteristics without explicitly specifying a physical model, as well as models based on gradient boosting [31, 32] for predicting correction values. However, such solutions are often "black box" [33] and do not provide a transparent understanding of the error causes. They can also be retrained in specific modes, losing accuracy when moving to new conditions.
Bayesian calibration methods [
Hybrid solutions at the Bayesian and ML models junction propose using neural networks [38, 39] to set the correction functional form and to train the parameters using the variational Bayes method. Such approaches demonstrate better results in the incomplete data conditions and allow one to estimate the correction signal uncertainty. However, the studies are still limited to laboratory data, and the algorithms' scalability issues for integration into onboard computing systems remain open.
In the helicopter TE sensors failure diagnostics and residual life prediction field, methods for detecting anomalies in the measurement stream are being actively developed [40–44]. Algorithms based on autoencoders [40] and LSTM networks [41] allow detecting sharp and hidden "chatter" in readings associated with mechanical damage or electrical failures. However, most existing studies, including [42–44], focus on classifying failures that have already occurred, rather than on continuous data correction in real time to prevent error accumulation.
The following key issues remain unresolved, requiring the intelligent adaptive correction model development:
Continuous online calibration should be provided, taking into account ageing, contamination, and vibration effects without taking the sensor service out.
Adaptation to rapidly changing thermal flight conditions with minimal computational delays and limited resources of onboard computers.
Reasonable unification of helicopter TE dynamics physical models with flexible ML structures and Bayesian uncertainty processing.
Development of methods for assessing the confidence in corrected readings and automatic transition to emergency mode when parameters go beyond safe limits.
The solution to these problems will make it possible to create a reliable intelligent system for correcting sensory data, which will significantly increase the helicopter's TE thermal state monitoring accuracy.
Based on [
T *meas t TG*true t dyn t drift t t .
G
The dynamic error model creates, it is assumed that the dynamic component depends on the
temperature change rate and load u(t) [
Substituting (2) into (1) yields:
T *meas t TG*true t
G
u t drift t t . dTG*meas dt
N drift t bk t k ,
k0
It is assumed that the drift Δdrift(t) changes slowly and can be approximated by a time polynomial of the form: which allows adaptive estimation of the coefficients bk.
Often the temperature sensor has a nonlinear characteristic f(•) [
G dTG*meas dt
N u t bk t k t , k0 Let us expand f into a Taylor series around the operating point T0: f T f T0 f T0 T T0 f T0 T T0 2 ... f M T0 T T0 M RM 1, 2! M ! where RM+1 is the remainder term.
Introducing the notation T TG*true T0 , we obtain
M f TG*true ai T i , ai k0 f i T0 .
i!
M T *meas ai T i
G k0 dTG*meas dt
N u t bk t k t , k0 For TG*true restoration, a correction function E is determined:
E TG*meas ,u,t M ai TG*meas T0 i N bk t k dTG*meas u t ,
k0 k0 dt Then the assessment is defined as:
For the adaptive parameter estimation purposes, it is assumed that a parameter vector of the form is formed: (7) (8) (9) (10) (11) (12) (13) (14) (15) where λ ∈ (0, 1] is the "forgetting" factor, P is the covariance matrix.
The developed adaptive model combines a dynamic part (time constant), calibration drift (time polynomial), and a nonlinear characteristic (Taylor series). The recursive least squares method provides automatic updating of parameters during operation, which allows the helicopter TE gas temperature to be reduced in front of the compressor turbine measurement error.
T *true TG*meas E TG*meas ,u,t ,
G θ a0 , a1,..., a M ,b 0 ,...,b N , , T ,
T *Gtrue TG*meas T t θ, K t
P t 1 t T t P t 1 t
P t 1 I K t T t P t 1, and the input regressor vector has the form:
t 1,TG*meas T0 ,...,TG*meas T0 M ,t0 ,...,t N ,TG*meas ,u t T ,
Let the reference temperature TG*ref be measured (from the reference device), then the error is defined as:
t TG*ref t T *Gtrue t , For adaptive parameter estimation, the recursive least squares (RLS) method is used: The sensor characteristic function expansion in a Taylor series around the operating point T0 allows us to approximate an arbitrary smooth nonlinear dependence f(T) by a polynomial with a controlled order, which simplifies the model structure (linearisation by parameters) and applies efficient adaptive algorithms (e.g., RLS); the polynomial approximation of the calibration drift over time bk tk reflects long-term changes in the sensor sensitivity; the RM+1 series residual term provides the approximation uncertainty analytical estimate, which allows us to balance between accuracy and computational complexity; and error control within the operating temperature region will enable us to select the series optimal order to minimise measurement errors.
In order to automatically correct the helicopter TE temperature sensor readings under changing operating conditions, a neural network architecture (Figure 1) is proposed, implementing a hybrid adaptive model for fixing the helicopter TE temperature sensor readings based on the developed model components: the basic measurement model (1), the dynamic component (2), the calibration drift (4), and the sensor nonlinear characteristic approximation by the Taylor series (6).
Hidden layers
The neural network (Figure 1) input features are: 1 t y t , 2 t dt example, by a moving average), 3 t u t , 3k t tk , k = 0, 1, …, K, 3k t tk , , (approximation, for dy t K 4m t y t T0 m , m = 0, 1, …, M.
Then the general feature vector (11) takes the form: t 1,2 ,3 ,4 ,...,K 4 ,K 5 ,...,K M 5 T . (16)
The neural network architecture contains an input layer of dimension D = 3 + (K + 1) + (M + 1), two hidden layers of size H with the Smooth ReLU activation function [45, 46], and an output layer without activation, giving the adjusted value: 1 t
... K 4m t
Input layer Output layer * T G t where θ is the neural network parameter.
A variational Bayesian approach is used to take into account the neural network parameter uncertainties. For this aim, an a priori distribution of the form is introduced: and a variational approximation is selected:
Then, training the neural network is reduced to minimising the negative ELBO (evidence lower
bound) [
L , KL q ; , p Eq ; , log p D , where is the dataset D ti ,TG*ref ti i1 , and the likelihood is given by:
N * T G t y t t; , t; NN t ; ,
p N 0, 02 I , q ; , N , .
The gradients with respect to μ and Σ are calculated using Bayes by Backprop and updated using stochastic gradient descent [48]: where where (17) (18) (19) (20) (21) (22) (23) (24) (25)
N p D N TG*ref ti y ti NN ti ; , 2 .
i1 L 1 0 Eq log p D ,
L 1 1 1 Eq T 1 , 2 L , L , t TG*ref t y t T t . where the parameters θ are estimated by the recursive least squares (RLS) method according to (15), and the prediction error is defined as
In the developed hybrid model, the RLS method is used as an effective way to estimate the neural network parameters initially. Specifically, the θ(0) and the covariance matrix P(0) values obtained from RLS according to (15) are used to initialise the variational approximation parameters μ = θ(0), Σ = P(0). The proposed approach accelerates the training and provides convergence of the neural network a priori regularisation based on physical assumptions and the linear structure of error.
In this research, a computational experiment was carried out based on the developed adaptive model implementation for correcting the helicopter TE gas temperature in front of the compressor turbine sensor readings in the MATLAB Simulink 2014b environment (Figure 2).
The "Signals Preprocessing" subsystem receives raw readings from the temperature sensor Traw(t) and normalises and smooths them before feeding them to the model further stages: first, znormalization is performed through the Z-normalization and Sum blocks T Traw , the signal is then passed through a moving average filter to remove high frequency noise and produce a smoothed value T ; both signals received are normalised T and smoothed T are output and then distributed to the dynamic error and polynomial drift subsystems, providing a single scale of input data and preliminary filtering of noise components.
The "Dynamic Error" subsystem receives a normalised value T as input and evaluates its rate of dT change through the "Derivative block" u
, after which the coefficient α is applied using "Gain" dt and the signal is sent to the "Discrete-Time Integrator", which implements the dynamics d dyn dyn u t , the final value Δdyn(t) is formed by the "Sum" block, which sums the dt integrator output and the weighted signal, and is then transferred to the neural network regressor vector, taking into account the sensor response inertial properties and time delay.
The "Polynomial Drift" subsystem tracks the slow drift of sensor readings using a polynomial calibration model: the "Clock" block generates the current time t, after which the basis vectors tk for k = 0…M are calculated via the "Math Function" and "Gain" chain, then the "MATLAB Function" recursively updates the least squares method coefficients bk, and the Sum block generates the final
M drift value drift t bk t k , which is fed into the neural network regressor vector to compensate k0 for long-term calibration changes. is multiplied by the corresponding Taylor series coefficient ai by the "Gain" block, after which the "Sum" block sums all the terms up to a given order N, forming an approximate nonlinear value fTaylor T , which is then fed into the neural network input vector to take into account the sensor physical characteristics.
The "Variational Bayesian MLP" subsystem generates an input regressor vector via the "Bus Creator" block, combining T , T , Δdyn, Δdrift and fTaylor. The μ and Σ parameters initial estimates obtained from RLS are loaded into the "MATLAB Function", after which the signal passes two Fully Connected layers with SmoothReLU (Narma L2-controller) activation, and the "Custom MATLAB Function" block performs a variational Bayesian training step (Bayes by Backprop), updating μ and Σ using ELBO gradients; the neural network output yNN is subtracted from the original Traw via "Sum", generating a corrected value Tcorr, which is simultaneously transferred to "Scope" for logging and evaluating quality metrics.
To conduct the computational experiment, the TV3-117 engine gas temperature in front of the
compressor turbine data were used, recorded in the Mi-8MTV helicopter flight mode by a standard
sensor consisting of 14 dual thermocouples T-102 in the nominal engine operating mode [
The gas temperature in front of the compressor turbine's TG* initial measurements, obtained during the Mi-8MTV helicopter flight tests using the onboard monitoring system, were preliminarily cleared of noise interference and abnormal emissions. Then these data were transformed into a time series, and the parameter sequences were ordered by time [50]. To bring time series of different scales to a comparable form, the z-normalisation procedure was applied: z TG* i
TG*i N1 iN1 TG*i N1 iN1 TG*i N1 iN1 TG*i where N = 4 · 320 = 1280.
The gas temperature in front of the compressor turbine TG* and normalised values formed a training sample, which is given in Table 1. It is noted that this sample meets the homogeneity requirements according to the Fisher–Pearson [51, 52] and Fisher–Snedecor [53, 54] criteria (the homogeneity test results are in Table 2).
To assess the training set representativeness (see Table 1), the k-means cluster analysis method was used [55–57]. The training and test sets were obtained by randomly dividing the data. Their ratio was 2:1, i.e., 67% (858 elements) and 33% (422 elements), respectively. The training dataset clustering revealed eight groups (classes I–VIII), which indicates division into eight clusters (see Table 1). It confirms the training and test datasets' similar structure (Figure 4). Based on the obtained results, the optimal dataset sizes for gas temperatures in front of the compressor turbine were determined. The total training dataset is 1280 elements (100%). The control dataset is 858 elements (67% of the training dataset), and the test dataset is 422 elements (33% of the training dataset). a b
As the computational experiment part, diagrams of uncertainty assessment (Figure 5), error control within the working area (Figure 6), drift influence analysis (Figure 7), ELBO (evidence lower bound) convergence curve for variational learning (Figure 8), and the RLS assessment parameters evolution (drift polynomial coefficients and τ, α) (Figure 9) were obtained.
Figure 5 illustrates the sensor readings' dependence on the actual gas temperature: the blue line is the true nonlinear characteristic f(T), the red line is the 1st order Taylor diagram, the orange line is the 2nd order, and the green line is the 3rd order. At the 1080 K lower limit, the 1st order underestimates the readings by approximately 0.7… 0.8 K, the 2nd by ≈ 0.25 K, the 3rd by less than 0.1 K, and up to 1150 K by ≈ 0.6 K, ≈ 0.2 K, and practically 0 K, respectively. The enlarged inset (1110…1120 K) shows that within ±5 K, the 1st order is ≈ 0.05 K, the 3rd is ≤ 0.01 K, which clearly demonstrates how the approximation uncertainty decreases with increasing Taylor series order.
Figure 6 shows the Taylor series expansion absolute error in the temperature range 1080… 1150 K: the red curve (1st order) forms a symmetric parabola with a maximum of about 2.5 K at 1080 K and about 2.1 K at 1150 K; the blue (2nd order) does not exceed ≈ 0.18 K at the range ends and decreases to ≈ 0.03 K near and remains within ≤ 0.02 K throughout the entire range, and stays within ≤ 0.02 K throughout the range; the dotted black line indicates the operating temperature T0 = 1115 K, where all orders errors are reduced to zero.
Figure 7 illustrates the change in the sensor reading error through the calibration drift over time, where the blue curve is the theoretical model, growing almost linearly with ≈ 0.02 K/s slope to ≈ 6.3 K peak around 270 seconds and then smoothly decreasing to ≈ 6.0 K, ± red 0.05…0.1 K, demonstrating the measurements real scattering around the theoretical trend.
Figure 8 shows a rapid increase in the proof lower bound in the first 20 epochs: from about 440 at epoch 1 to –200 at epoch 20, corresponding to an average per-epoch improvement of about +12 ELBO. Between epochs 20 and 50, the growth rate slows to about +2 ELBO per epoch, reaching –100 by epoch 50. In the remaining 50 iterations, ELBO continues to increase slowly, reaching about –45 by epoch 100, indicating that training has stabilised and the variance distribution is approaching the optimal solution. The variation around the trend (standard deviation of noise fluctuations ~5 ELBO) indicates moderate stochasticity in the gradient estimates. decreases from ≈2 to 0.1, and the forgetting factor α increases from ≈ –1.5 and stabilises around 0.9, with a noticeable exponential convergence rate: in the first 20 steps, each curve reaches more than 50 % of the distance to its asymptote. Barely perceptible noise fluctuations (σ ≈ 0.05) reflect the RLS operation under stochastic disturbances.
True Positive (TP) is the cases when the sensor actually gave an erroneous reading (out of range or significant systematic error) and the neural network correctly classified it as faulty: True Negative (TN) is the cases when the sensor reading was normal and the neural network correctly identified it as usual: 3. False Positive (FP) is the cases where the sensor gave a correct reading, but the network mistakenly classified it as requiring correction: 4. False Negative (FN) is the case when the sensor actually gave an erroneous reading, and the neural network did not detect the deviation (classified it as usual):
The comparative metrics analysis in Table 4 demonstrates that the proposed hybrid variationalBayesian MLP model consistently outperforms all alternative architectures: its accuracy (0.992) and recall (0.997) are the highest, indicating the lowest number of misses in detecting false positives, while precision (0.988) and F1-score (0.987) also remain comparable to the best competitors, indicating an excellent balance between false positives and misses. Traditional MLP and deeper MLPs show only minor losses in precision and recall (up to –0.004 in recall and –0.005 in precision), while 1D-CNN, LSTM/GRU, and TCN are slightly worse (–0.007…–0.010 in accuracy), highlighting the Bayesian regularisation contribution, RLS initialisation, and Smooth ReLU to the robustness to sensor noise and drift. VAE and EKF, being conceptually different approaches, show even lower values (accuracy ≈ 0.984 and 0.980), confirming the integration of variational Bayesian methods directly into the MLP architecture for solving the problem of online temperature correction.
A physical-statistical model of measurement errors has been developed, given by expression (1), in which the dynamic error is described by the first order of the linear differential equation (2), and the calibration drift is approximated by the polynomial (4).
A hybrid variational-Bayesian MLP network (see Figure 1) with two hidden layers of dimension H, Smooth ReLU activation, and an input given by expression (16) is developed. The output without * activation gives the adjusted value T G (17). To take into account the uncertainty, a Bayesian prior (19) and a variational approximation (20) are introduced, trained through minimisation of the negative ELBO (formula (21)) and implemented by the Bayes by Backprop method with the parameters μ, Σ initialisation from the recursive least squares (RLS) according to (15), (25).
Experimental validation was performed in the MATLAB Simulink environment (see Figure 2) using real data on the gas temperature in front of the compressor turbine during the Mi-8MTV flight (320 seconds at a 0.25-second step, see Figure 2). To assess homogeneity and representativeness, a training sample (1280 elements) was constructed, divided 67/33% into training (858 elements) and test (422 elements) subsamples; k-means cluster analysis showed eight similar groups (see Figure 4).
Based on the training results, the developed neural network demonstrated the following indicators (see Table 3): accuracy = 0.992, precision = 0.988, recall = 0.997, F1-score = 0.987, average training time = 182 seconds, and accuracy variance ≈ 1.08 · 10⁻⁶.
During the computational experiment, graphs of uncertainty estimation (see Figure 5), error control (see Figure 6) and drift influence (see Figure 7) were obtained. The ELBO convergence curve (see Figure 8) and the evolution of RLS parameter estimates (see Figure 9) showed a rapid growth stage in the first 20 epochs and stabilisation by the 100th epoch, extremely close to the optimum.
Despite the significant results obtained, the following are key limitations and unresolved issues identified in the research: 1. Experimental validation was performed on data from one type of gas temperature sensor in the MATLAB Simulink laboratory environment and on a limited range of temperatures and dynamic modes. The model's stability under vibration, pollution, and extreme climatic influences typical for real helicopter flights was not tested. 2. The model's current version adapts to calibration drift polynomially in time but does not take into account the sensitive element and its ageing contamination variability under the vibration influence and aggressive environments without decommissioning the sensor. 3. For implementation on onboard computers, it is necessary to minimise the inference time and overhead costs of variational-Bayesian training. The average training time (~182–217 seconds) indicators and the RLS initialisation needs are not yet adapted to the aviation controllers' strict resource and time constraints. 4. The model includes an uncertainty assessment through ELBO. Still, no criterion has been developed for automatic transition to emergency mode when the adjusted readings go beyond safe limits, and there are no mechanisms for the quantitative confidence calibration in current estimates. 1. Validate the model in a broader range of sensors (different manufacturers and designs) and real flight/production conditions, including vibrations, hostile environments (dust, moisture) and extreme temperatures. 2. Integrate data from multiple geographically distributed sensors (multi-sensor fusion) to improve the reliability and accuracy of the assessment. 1. Develop a continuous online learning mechanism taking into account sensor ageing and contamination, while exploiting the computational resource limitations of onboard controllers (online RLS + low-power Bayesian updates). 2. Investigate adaptive thresholds for switching to emergency mode based on ELBO uncertainty estimates and dynamic control of confidence in predictions. 1. Formulate lightweight versions of the model (model pruning, knowledge distillation) or alternative "global-local" architectures (e.g., combined PINN + shallow MLP) for implementation on limited platforms (microcontrollers, FPGA).
approximation (e.g., Smooth ReLU → piecewise-linear) on accuracy and performance.
Interpretability 1. Implement explainable AI methods (SHAP, LIME) for and adjustments, local analysis, and "physical" pattern identification explainability of in network parameters (τ, α, drift coefficients).
decisions [65, 2. Investigate the variational approximation and its relations, 66] visualising latent variables with fundamental changes in sensor
properties over time.
Extension to 1. Adapt and transfer the approach to correcting readings of other types of industrial sensors of other types (pressure, flow, vibration) and sensor systems to multiphysical systems (combination of temperature and and applications mechanical signals).
[67, 68] 2. Investigate the hybrid variational-Bayesian MLPs use in predictive diagnostics and failure prevention problems based on the "anomalous" behaviour of time series.
Integration with 1. Synchronise the model with the gas turbine digital twin using digital twins telemetry streams and cloud computing for centralised and industrial monitoring and condition prediction.
IoT [69, 70] 2. Assess the distributed training capabilities (federated training) to update the model across multiple onboard nodes without transmitting raw data.
Further research in this area will not only improve the developed model's practical applicability and reliability but also expand its functionality within the framework of Industry 4.0 and IoT modern tasks.
An intelligent adaptive model for the helicopter TE gas temperature in front of the compressor turbine sensor, correcting readings, has been developed. It is a hybrid variational-Bayesian MLP network with two hidden layers and Smooth ReLU activation. The model combines a physicalstatistical component (first-order dynamic error, polynomial calibration drift, and nonlinear characteristic approximation by a Taylor series) and machine learning methods: the recursive least squares method (RLS) and Bayes by Backprop for estimating parameter uncertainty. The variational approximation μ and Σ initialisation is performed based on the RLS results, which ensures fast convergence and training regularisation based on physical assumptions.
Experimental validation was performed in the MATLAB Simulink 2014b environment using data on gas temperature in front of the compressor turbine during Mi-8MTV flight (320 seconds, 0.25 seconds step, maximum temperature 1140 K). The nonlinear characteristic approximation by the 3rd order Taylor series gave an error ≤ 0.1 K in the 1080…1150 K range, while the 2nd order is 0.25 K and the 1st 0.8 K at interval boundaries. RLS estimates of drift coefficients and τ–α parameters converge by 50% in the first 20 steps, and ELBO with variational training increased from –440 to – 45 in 100 epochs, reaching stabilisation around the 50th iteration.
According to the training results, the developed model demonstrated the following indicators: Accuracy = 0.992, recall = 0.997, precision = 0.988, and F1-score = 0.987 with an average training time of ≈ 182 seconds and an accuracy variance of ≈ 1.08 · 10⁻⁶. Comparative analysis (1D-CNN, LSTM/GRU, classic EKF) showed the proposed approach's superiority due to Bayesian regularisation and RLS initialisation, providing resistance to noise and sensor drift in online mode.
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.
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