An innovative method for the helicopter turbosha engines gas temperature measuring has been developed. It is based on a dynamic mathematical model with error compensation. The original nonlinear temperature dependence described by the function is linearly approximated by the Taylor expansion method, which allows taking into account the system's dynamic characteristics through a rst-order di erential equation. To eliminate the model's static and dynamic errors, a correction system with an adaptive PI controller has been implemented, whose parameters are selected using a neural network implementing online training. Simulation experiments conducted in the Matlab Simulink environment on real ight test data for the TV3-117 engine demonstrated high accuracy of parameter identi cation (up to 0.9975) and signi cant improvement in the transient process's dynamics in the model errors present in the ± 3% range. The experimental results con rm the proposed method's correctness and high e ciency, ensuring the system's stable operation in real time and demonstrating the prospects for further development of intelligent control methods.
In the rapid development context of helicopter aviation and increasing requirements for the helicopter turbosha engines (TE) safety and e ciency, the gas temperature's accurate measurement is becoming critical to optimize the unit’s operation [1–3]. Traditional measurement methods [4–6] are o en subject to errors, which can negatively a ect the engines’ performance characteristics and service life. In this regard, the intelligent gas temperature measurement method with model error compensation developing relevance is due to the need to integrate modern arti cial intelligence technologies and self-correction algorithms that can ensure the system’s high accuracy and adaptability in real time, which will ultimately improve the helicopter TE reliability and safety.
In modern research on the helicopter TE temperature conditions measuring, for example [7, 8], classical methods based on the pyrometers, thermocouples, and optical sensors used are widely used. These methods are o en combined with mathematical models describing thermal processes in combustion chambers and gas ows. However, despite signi cant progress in sensor technology [9, 10] and modeling [11–13], the measurements accuracy remains limited due to the dynamic changes in uence on engine operating conditions and the inevitable errors inherent in the models.
In recent years, there has been a trend towards integrating arti cial intelligence and machine learning methods to improve the accuracy and adaptability of measurement systems. Research in this area is focused on the neural networks [14–17] and self-calibration algorithms [18, 19] used that allow real-time data correction and compensation for systematic errors. Such approaches demonstrate promising results, but they require signi cant computing resources and large training datasets, which complicates their practical application in limited response time conditions in aviation systems.
In addition to the above approaches, a number of researchers focus on the multi-sensor analysis and data fusion methods integration to improve the temperature measurements accuracy. In [20– 22], big data processing algorithms are used that allow combining information from sensors’ di erent types to minimize the noise and random errors impact. An integrated approach combining physical models, statistical analysis, and arti cial intelligence algorithms opens up new possibilities for adaptive monitoring. However, their adaptation for real-time operation in helicopter TE remains an unsolved problem requiring further research.
The studies [23, 24] showed that ber optic sensors provide high accuracy and protection against electromagnetic interference, and infrared thermography allows for contactless measurements. At the same time, [25, 26] states that CFD modeling and experimental data form accurate predictive models, and Kalman lters e ectively correct dynamic indicators, reducing the noise impact. However, the these method’s high cost and computational complexity make them di cult to use in real time.
Despite the successes achieved, issues related to dynamic compensation of model errors remain unresolved. Existing approaches [12, 16, 20] o en cannot adequately respond to sudden changes in operating conditions, which leads to the errors accumulating in temperature measurements. In addition, the models’ insu cient adaptability [7, 9, 21] and the sensor’s universal self-correction algorithms [23, 25] lack of limits limit their application possibilities in helicopter TE real operating conditions.
Thus, there is a need to develop a new intelligent method for measuring gas temperature, compensating for model errors in real time. This requires the exible adaptive systems integrating traditional measurement methods [3, 4] with innovative arti cial intelligence algorithms creation [10, 14, 15, 27], which will signi cantly improve the helicopter TE operating parameters monitoring reliability and accuracy.
A goal of the paper is to provide measuring the helicopter TE gas temperature with compensating the model errors in real time. To reach this goal we formulated the following objectives: (i) to develop the intelligent method of measuring gas temperature; (ii) to create the neural network controller for implementation of the developed method.
It is known [1, 4, 28, 29] that in general the gas temperature in front of the compressor turbine model is represented as a function:
T G=f (nTC , nFT , PN , T N ) , τ ∙ d T G +T G=f (nTC , nFT , PN , T N ) , dt τ nTC ∙ ddntTC +nTC=nTmCeas , τ nFT ∙ ddntFT +nFT =nmFTeas , where nTC is the gas generator rotor speed (recorded on board the helicopter by the standard D-2M sensor); nFT is the free turbine rotor speed (recorded on board the helicopter by the standard D-1M sensor); PN= P0N ∙ σrest ∙(1+ M 2 ∙ γ −1 ) is the air pressure PN inhibited value at ight altitude h = h(t), 2 PN is the pressure at the ambient pressure sensor output (recorded on board the helicopter by the standard DP-11 sensor); T N =T 0N ∙(1+ M 2 ∙ γ −1 ) is the air temperature T 0N inhibited value at ight 2 altitude h = h(t), P0N is the pressure at the ambient temperature sensor output (recorded on board the helicopter by the standard TT-11 sensor); σrest is the total pressure recovery coe cient in the helicopter TE inlet air section, M = M(t) is the Mach number at ight altitude h = h(t), γ is the adiabatic index [29, 30].
This model does not take into account the static error in uence in calculating the gas
temperature on the on-board automated control system (ACS) time constant setting accuracy [31,
32], which leads to a deterioration in the transient processes quality in its operation. To analyze
small deviations around the operating point (nTC0, nFT0, PN0, TN0) [33–35], function (
T G ≈ T G 0+ ∂∂nfTC ∙ (nTC−nTC 0)+ ∂∂nfFT ∙ (nFT −nFT 0)+ ∂ f ∂ PN ∂ f ∂ T N ∙ ( PN− PN 0)+ ∙ (T N−T N 0) .
(
The obtained approximation (
Since the quantities nTC, nFT, PN and TN are also subject to dynamic changes, we introduce their
dynamics’ equations:
(
To eliminate the static error in uence in measuring gas temperature, compensation is introduced. Let TG,ref be the temperature measured by a thermocouple (standard); then the model error is de ned as:
When switching to a new dynamic mode, the adjustment is carried out according to the rule: where K is the correction coe cient (at steady state, o en K = 1). In this case, a condition is introduced on the model’s change rate to ensure the transient processes stability: with a threshold ε0, which speci es the maximum permissible change in temperature per unit time.
In addition, the error compensation dynamics are speci ed through the integral action as: τ P ∙ d PN + PN=PmNeas ,
N dt
d T N +T N=T mNeas , τ T N ∙ dt ∆ T G=T G,ref −T G .
T corr=T G+ K ∙ ∆ T G ,
|d T G|≤ ε0
dt
t
I (t )=∫ ∆ T G ( ξ ) dξ ,
t0
(
T corr=T G+ K P ∙ ∆ T G+ K I ∙ I (t ) , where KP and KI are proportional and integral coe cients, respectively.
To improve accuracy, adaptive change of correction coe cient K is provided ac-cording to the
following law:
dK =α ∙ ∆ T G− β ∙ K ,
dt
(
In this case, the nal temperature value, taking into account error compensation, is determined
as:
(
T G , final=T G+ K ∙ ∆ T G ,
t T G ,final=T G+ K P ∙ ∆ T G+ K I ∙∫ ∆ T G (ξ ) dξ .
t0
The proposed controller’ general structure (Figure 1) is based on the corrected error, which is represented as: ∆ T G (t )=T G ,ref (t )−T G (t ) ,
t uPI (t )= K P ∙ ∆ T G+ K I ∙∫ ∆ T G ( τ ) dτ .
0
where TG,ref(t) is the reference temperature measured by the thermocouple, and TG(t) is the model
output (see (
The proposed neural network PI controller extends this scheme by adaptively determining the
coe cients KP and KI based on a feature vector re ecting the entire system’s dynamics. To take into
account the model’ speci c features, the input signals vector x(t) includes: the error’ current value
d ∆ T G (t )
∆TG(t), the error’ derivative ∆ T˙ G (t )= dt , the gas generator and free turbine nTC(t) and nFT(t)
operating parameters, the aerodynamic parameters: the Mach number M(t), the pressure P0N (t ) and
the temperature T 0N (t ) (or their “slowed down” versions PN and TN. In this case, the model’s change
rate is additionally taken into account, for example, the value |ddTt G|, limited according to (
x (t )=(∆ T G (t ) ∆ T˙ G (t ) nTC (t ) nFT (t ) M (t ) PN (t ) T N (t ) |ddTt G|),
It is further assumed that a neural network with parameters w de nes the input vector x(t) nonlinear mapping into a coe cients [KP(t), KI(t)] pair:
K P (t )=φP ( x (t ) ; wP) , K I (t )=φI ( x (t ) ; wI ) .
In the simplest case, one can use a single-layer neural network with nonlinear activation, for example, hyperbolic tangent (tanh). Then for each hidden neuron hj(t) (j = 1, …, Nh) we obtain: and the outputs are calculated as:
n h j (t )=tanh(∑ ωij ∙ xi (t )+b j),
i=1
Nh Nh K P (t )=∑ v P, j ∙ h j (t )+bP , K I (t )=∑ v I , j ∙ h j (t )+bI .
j=1 j=1 (19) (20) (21)
For the controller parameters’ adaptive adjustment, the neural network is trained online using error backpropagation [39, 40]. The objective function can be de ned as the mean square error between the reference and output temperatures:
The rules for updating the weights wP and wI are given by gradient descent:
1 J (t )= 2 ∙ (T G ,ref (t )−T G ,final (t ))2 .
∂ J (t ) ∂ wP ∆ wP=−η ∙ , ∆ wP=−η ∙ ∂ J (t ) ∂ wP , (23) (24) where η is the training rate. This approach allows the neural network to “train” from model errors and adjust the controller coe cients in accordance with changes in the system dynamics.
Thus, the neural network adapts the PI controller coe cients depending on the system’s current state. The original model features (sensor dynamics, aerodynamic parameters in uence, total pressure recovery, limitation on the temperature change rate) are implemented due to:
The input vector x(t), which includes not only the control error, but also the parameters nTC, nFT, M, PN, TN and |ddTt G|.
The coe cients KP(t) and KI(t) adaptive selections, which change depending on the model’s current state, which allows for the transient processes’ correction under changing operating mode conditions.
Training on real data. When integrating with measuring signals from thermocouples and sensors, the neural network constantly adjusts its weights, which improves the ACS time constant adjustment quality.
Thus, the neural network PI controller’s nal control action is represented as: t T G , final (t )=T G (t )+φP ( x (t ) ; wP) ∙ ∆ T G (t )+φI ( x (t ) ; wI )∙∫ ∆ T G ( τ ) dτ . 0 (25)
The developed neural network PI controller allows for the gas temperature’s adaptive control, taking into account the model’s both static and dynamic features, as well as limitations on the transient processes speed. Thus, the developed neural network PI controller is integrated into the original model, enhancing its features due to the coe cients’ adaptive selection, which allows for achieving higher accuracy and stability of helicopter TE changing dynamics control under conditions.
To test the gas temperature measurement developed model with the model error compensation, implemented using a PI controller with neural network tuning (see Figure 1), a computational experiment was conducted, which research object was the TV3-117 engine [10, 14, 15, 27, 37, 41], which is the Mi-8MTV helicopter power plant part. According to the authors' team to the Ministry of Internal A airs of Ukraine o cial request, within the research project “Theoretical and Applied Aspects of the Development of the Aviation Sphere” framework, registered in Ukraine under state number 0123U104884, the Mi-8MTV helicopter ight test data (nTC, nFT, TG, PN and TN) were Number 1 ... ... ... 373 562 797 ... 965 ... 1280 nTC with the parameters’ di erent scales, z-normalization xi= obtained, which were carried out at a ight altitude of 2500 meters above sea level for 320 seconds with a sampling interval of 0.25 seconds. The data (nTC, nFT, TG, PN and TN at M = 0.7) obtained during the Mi-8MTV helicopter ight tests using the onboard monitoring system were preprocessed with the noise interference and anomalous values removal, a er which they were transformed into time series is the parameters sequences ordered by time [42]. To ensure the time series comparability 1 N xi− N ∙ ∑i=1 xi √ N i=1 1 N ∙ ∑ ( xi− 1 N N ∙ ∑i=1 xi) 2 applied, which brings their values to a single range, setting the middle at zero and the standard deviation equal to one. This made it possible to form a parameters nTC, nFT, TG, PN and TN training dataset, which fragment is presented in Table 1.
As the research’ part, the training dataset’s homogeneity was assessed using the Fisher-Pearson [44] and Fisher-Snedecor statistical tests [45], with a signi cance level of α = 0.01. This analysis revealed that the Fisher-Pearson test values were 13.225 for nTC, 13.208 for nFT, 13.215 for TG, 13.295 for PN, 13.297 for TN, which is below the threshold value of 13.3. Similarly, the Fisher-Snedecor test values were 1.124 for nTC, 1.123 for nFT, 1.115 for TG, 1.133 for PN, 1.134 for TN, which also does not exceed the critical value of 1.139. These results con rm both the training dataset homogeneity and the parameters’ variances consistency under research. In addition, a cluster analysis was performed using the k-means method [46, 47], which made it possible to identify 8 separate groups, thereby con rming the both the training and test datasets represent the population. Note that these datasets were formed by randomly splitting in a ratio of 67 to 33 %, which ensures objectivity and TG balance in the data distribution. Based on the obtained results, the dataset-amounts optimizing process for sensor signals was carried out: the training dataset includes 1280 elements, the control dataset includes 858 elements, and the test dataset includes 422 elements. Such optimization creates a solid foundation for further research and timely prevention of engine failures during ight.
The research involved a computational experiment, whose main objective was to obtain the transient process results of the developed PI controller with neural network tuning with the gas temperature model error of ± 3 % and with the gas temperature model error correction of ± 3 %. These deviations (± 3 %) of the gas temperature value were selected based on the modern helicopter TE performance characteristics analysis, where this value is taken as the limit value during the unit’s normal operation. The permissible limit of ± 3% is due to the fact that it is with such deviations that the aerodynamic and thermodynamic processes stability is maintained, which ensures optimal engine operation and minimizes the excessive wear or emergency situations risk [48–50].
To conduct a computational experiment in the Matlab Simulink so ware environment, an integrated computational model was developed, including a helicopter TE dynamic model, a PI controller algorithm with neural network tuning, and a module for compensating for the gas temperature model error (Figure 2).
Figure 3 shows the gas temperature model transient processes with a gas temperature model error of ± 3% (Figure 3 a, b) and with a gas temperature model error correction of ± 3% (Figure 3 c, d), from which it is evident that the gas temperature model error has a negative e ect on the transient processes’ quality (curves 1 and 2 di er in Figure 3 a, b).
This di erence is due to the fact that the gas temperature model (
Curve 1 shows the transient response at the controller output implemented according to the classical self-adjusting gas temperature controller diagram. Curve 2 illustrates the transient response at the developed PI controller with neural network adaptation output (see Figure 1). Curve 3 represents the signal received from the thermocouple output. Thus, it can be concluded that the additive error a ects only the measuring device's dynamic operating mode. One of the ways to eliminate this inaccuracy is to correct the gas temperature model error. It is evident from Figure 3, c, d that the gas temperature model error is compensated, and when applying a disturbance (the transient process noted at the 8th second), the model-corrected value is used. It is obvious that the transient process's quality is not determined by the gas temperature model error.
Figure 4 shows the gas temperature model transient processes with an admitted error of ± 3% when the temperature changes according to a sinusoidal law. It is clear that initially the 2% error had a negative impact on the transient processes’ quality (curves 1 and 2 di er). Then, at the 3.5 second mark, the model error is eliminated, and then the model signal without the error is applied (curves 1 and 2 coincide).
Figure 5 shows the signal at the developed PI controller output (see Figure 1) with single jumps in gas temperature and the stabilization mode zone allocation in relation to the process shown in Figure 3, c.
According to Figure 5, when single jumps in the input signal (changes in gas temperature) occur, a quick response is observed at the controller output. A er each jump, the signal gradually moves into the stabilization zone, where a new stable level is established. This zone re ects the correction mechanism’s operation, which, thanks to the coe cients (KP and KI) adaptive adjustment through the neural network, minimizes the model error consequences and allows the system to return to a stable state. Comparison with the process without compensation (as can be seen, for example, in Figure 3, from which the initial data are taken) shows that the neural network adaptive PI controller implementation can signi cantly reduce the model error impact. The signal at the controller output becomes smoother, which indicates the gas temperature measurement dynamics and increased accuracy adjustment. Rapid adaptation to abrupt changes and the stable mode’s subsequent establishment are important for maintaining optimal operating conditions for helicopter TE.
The model error value (ε0) choice is determined using the free turbine rotor speed sensor. Depending on the free turbine rotor speed nFT, the gas temperature model error in the current operating mode is calculated from the model error dependence on the frequency diagram (Figure 6). The curve in Figure 6 is the model errors real values approximation based on [51–53].
An example of the helicopter TE gas temperature model error determining the free turbine rotor speed's certain value is presented based on the following data: 11300 rpm is the rst cruising mode (model error value +1.48 %); 11500 rpm is the nominal mode (model error value +1.79 %). To calculate the error value between the rst cruising and nominal operating modes, for example, at a free turbine rotor speed of 11400 rpm, the following analytical expression is used:
According to the obtained results, at 11300 rpm (the rst cruising mode), the model error is +1.48 %, and at 11500 rpm (nominal mode), it is +1.79 %. Linear interpolation for 11400 rpm yielded an error value of 1.635 %, which indicates a uniform change in the error between the modes. These results demonstrate the accurate model correction possibility depending on the free turbine rotor speed.
In this research, the helicopter TE gas temperature measuring method has been developed, which is
based on a dynamic model, where the initial temperature dependence is speci ed by function (
The obtained results demonstrate that the adaptive method for compensating for model errors using a neural network implementation to adjust the PI controller coe cients allows for a signi cant increase in the helicopter TE gas temperature measuring accuracy (parameter identi cation up to 0.9975), which is con rmed by simulation experiments: in the model error of ± 3 % presence, the out-put signal dynamics is signi cantly improved–a rapid recovery a er temperature jumps and the signal transition to the stability zone are observed (Figures 3–5), which speci es the method’s e ectiveness in minimizing the model errors impact and the automated control system ensures reliable operation in real conditions.
The obtained results demonstrate the developed method's high accuracy and adaptability; however, a limited number of limitations require additional analysis. The developed measurement model is based on a nonlinear dependence linear approximation (Taylor expansion), which can lead to a decrease in the assessment correctness with abrupt changes in operating conditions or in the strong disturbances event. The experimental validation was carried out under conditions close to the normal operating mode (model error ± 3 %), which limits the extrapolating possibility of the results to more extreme or unpredictable scenarios. The computational costs associated with the neural network online training for the PI controller coe cients adaptive adjustment require optimization for real implementation in systems with limited resources.
Future research prospects include expanding the model to include more complex nonlinear e ects and dynamic modes and adapting it to di erent engine types and operating conditions [54]. Field testing to verify the system’s performance in real-world conditions is recommended, as is research into the more advanced machine learning algorithms use [55] to improve adaptability and reduce computational load. An important area for future work is the multisensor analysis integration [56], which will improve error compensation algorithms and ensure the ACS more stable operation.
The helicopter turbosha engine's gas temperature measuring innovative method has been
developed, which is based on a dynamic model using a nonlinear dependencies linear
approximation and compensation for static and dynamic model errors using an adaptive PI
controller with a neural network. The applied approach, implemented through (
Simulation experiments conducted in the Matlab Simulink environment demonstrated that the proposed method signi cantly improves the transient processes dynamics: a er sharp temperature jumps, a quick response and subsequent establishment of a stable mode are observed. Thus, the developed method allows for the model errors to in uence e ective compensation and adaptation to changing operating conditions, which is an important factor for increasing the helicopter turbosha engine's reliability and safety.
The research was carried out with the grant support of the National Research Fund of Ukraine, "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 Ministry of Internal A airs of Ukraine supported the research entitled "Theoretical and applied aspects of the development of the aviation sphere" under Project No. 0123U104884.
The authors have not employed any Generative AI tools.
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