The Helicopter Turboshaft Engine’s Reconfigured Dynamic Model for Functional Safety Estimation

Abstract

This research substantiates the necessity for developing and implementing structural reconfiguration methods for automatic control systems in the event of a parametric sensor failure to enhance the helicopter turboshaft engine’s overall reliability and safety. The research aim is the substantiation of the helicopter turboshaft engine’s mathematically reconfigured automatic control system in the event of the failure of a standard sensor, which will ensure the helicopter turboshaft engine’s stable operation under failure conditions, minimizing the impact on engine control and performance. A theorem was developed and proven concerning the reconfiguration of the helicopter turboshaft engine’s automatic control system structure, defining the system’s new mathematical form using nonlinear thermogas-dynamic parameters. A method was proposed to determine the values of these parameters that keep the reconfigured control system stable. This method uses numerical optimization to find the best thermogas-dynamic parameters to ensure system stability. Experimental results showed that for slow changes, using parameters from the previous step works best, while for fast changes, restarting is more effective due to significant differences in the system states. The accuracy of the proposed mathematical model for the reconfigured control system was confirmed through mean square error analysis (within 0.4% and 0.77% under white noise), regression analysis (with a determination coefficient of 0.986), and cross-validation (with a metric deviation from the maximum mean square error of 3.88%).

1. Introduction

1.1. Relevance of the Research

In modern helicopter operating conditions, the helicopter turboshaft engine’s (TE) reliability and safety are of paramount importance [1,2]. One critical aspect is the automatic control system’s (ACS) ability to adapt to changing conditions [3], including the occurrence of parametric sensor failure [4]. A parametric sensor failure, characterized by its output data deviating from the nominal values, can significantly affect the ACS’s operation accuracy and adequacy [5]. In such situations, the ACS’s structural reconfiguration becomes a key element to ensure the helicopter TE’s operation stability and reliability.

The ACS’s structural reconfiguration involves the redistribution of functional tasks between the system components and the corrective mechanism’s introduction that provides compensation for erroneous data [6]. This may include the redundancy algorithms used, corrective filters, or adaptive models that can automatically identify and account for changes in system behavior [7]. This approach allows helicopter TE functionality to be maintained during sensor failures and ensures optimal engine operating modes, minimizing risks and preventing potential emergencies [8].

Thus, the development relevance and the implementation of ACS structural reconfiguration methods in parametric sensor failure conditions is due to the need to increase the helicopter TE operation’s overall reliability and safety. Modern technologies [9] and artificial intelligence methods, such as neural networks [10,11] and machine learning algorithms [12], provide a wide range of tools for creating adaptive ACS [13] capable of coping effectively with emerging challenges and ensuring a high degree of helicopter TE automation and autonomy.

1.2. State-of-the-Art

According to statistical data, up to 40% of aviation system failures are related to sensor malfunctions, highlighting the importance of developing effective methods for their detection and compensation. The current state technologies for fault detection involve the use of neural networks and machine learning algorithms that can process large amounts of data and identify complex dependencies between system parameters. Research shows that applying such methods can increase diagnostic accuracy by 25…30% compared to traditional methods like Kalman filtering. Reconfiguration of the ACS in the case of failure requires adaptive control algorithms capable of automatically adjusting to changing conditions and restoring system functionality in real time. Thus, the integration of advanced mathematical models and artificial intelligence methods into ACS detection and reconfiguration processes enhances flight reliability and safety, which is particularly relevant given the increasing complexity of aviation systems.

The ACS reconfiguration is a complex task that requires the development and application of complex mathematical models and algorithms [14]. The reconfiguration’s main aim is to ensure the system’s stability and reliability under various failure conditions, which is especially important for mission-critical applications such as helicopter TE control [15,16]. In this regard, the task on the mathematical side includes several key aspects: failure diagnostics, identification of system parameters, control algorithms adaptation and performance restoration optimization [17,18].

The primary task is the development of an effective method for diagnosing failure, which includes the mathematical models used to identify anomalies in the system’s operation. Such methods can be based on the statistical analysis theory [19], data filtering [20] and machine learning methods [21]. For example, Kalman filtering [22,23] and its extensions [24] are widely used to assess the system state and identify deviations that indicate possible failures. More modern approaches include the use of neural networks [25,26] and other artificial intelligence algorithms [27] that can process data in large amounts and identify complex dependencies between system parameters.

After diagnosing a failure, there is a need to adapt control algorithms to compensate for the identified faults [28]. Here, adaptive control methods play an important role; these allow the system to automatically adapt to changing conditions. One key approach is the recursive algorithms used for identifying system parameters [29], which may include least squares, gradient methods, and other optimization techniques. These algorithms allow you to quickly assess the system’s current parameters and adjust control actions in real-time.

Mathematical models used for ACS adaptation and reconfiguration must take into account the system’s dynamic nature and possible external disturbances. In this context, optimal control methods are important, such as linear and nonlinear [30,31] predictive control models that take into account limited resources and system response time. Control actions’ optimization, taking into account the constraints and system performance requirements, is the number one key task for reconfiguration.

Systems with a high degree of autonomy also require algorithm development that can predict possible failures and plan actions to prevent them. In this direction, prediction methods are actively being developed based on time-series analysis and probabilistic predicting models [32,33]. Such approaches enable not only a quick response to emerging failures but also measures to be taken to prevent them in advance, which significantly increases the system’s overall reliability.

It is important to note that the mathematical side of the ACS reconfiguration task includes not only the complex algorithms, model development and applications but also their integration into real systems, taking into account the limitations and requirements for reliability and performance. Modern technologies and methods, such as artificial intelligence and machine learning, open up new opportunities for creating an adaptive and intelligent ACS [34,35], which is effectively capable of coping with emerging challenges and providing a high degree of automation and autonomy.

Thus, the research relevance in the helicopter TE ACS reconfiguration field, especially in the event of one of the sensors’ parametric failures [5], is due to the need to increase aircraft operational reliability and safety [36]. The mathematical methods and models used can significantly improve ACS stability and flexibility, ensuring continuous and stable operation even in the event of failure, which is important for ensuring flight safety and increasing the helicopter’s operational efficiency in various conditions [37].

This issue is particularly critical today due to the increasing complexity and autonomy of aviation systems, as well as the growing intensity of their operation under various conditions, which increases the likelihood of failures and malfunctions, especially at the sensor level. In modern helicopter usage, where reliability and flight safety are top priorities, sensor malfunctions can lead to significant consequences, including loss of control over the system and the occurrence of emergency situations. Modern approaches to ACS reconfiguration, based on mathematical models and artificial intelligence algorithms, play a key role in ensuring a system resilient to failure, allowing for timely diagnosis and compensation malfunctions. This is especially important in the context of heightened system complexity and autonomy, where human intervention may be limited, and the demands for reliability and stability in ACS operation continue to rise

1.3. Main Attributes of the Research

The object of the research is the helicopter TE ACS.

The subject of the research includes mathematical models for the helicopter TE’s reconfigured ACS in the event of a standard sensor failure.

The research aim is the mathematical substantiation of the helicopter TE’s reconfigured ACS in the event of standard sensor failure, which will ensure the helicopter TE’s stable operation under failure conditions, minimizing the impact on engine control and performance.

To achieve this aim, the following scientific and practical tasks were solved:

• Formulation and proof of the theorem “On the helicopter TE ACS structural reconfiguration”.
• Development of the helicopter TE reconfigured-ACS mathematical model in the event of an example of one standard sensor failure.
• Development of the helicopter TE reconfigured-ACS optimal parameters for the finding method in the event of one standard sensor failure.
• Development of the neural network architecture and training algorithm to implement the developed methods and models regarding the helicopter TE reconfigured-ACS in the event of one standard sensor failure.
• Conducting a computational experiment consisting of the helicopter TE reconfigured-ACS optimal parameter calculation in the event of the failure of the standard sensor.
• Determination of the helicopter TE reconfigured-ACS noise resistance in the event of standard sensor failure.
• Assessing the accuracy of the results obtained by the helicopter TE reconfigured-ACS optimal parameters calculation in the event of one of the standard sensors failing.

In summary, the main contribution of the research is the helicopter TE reconfigured-ACS mathematical model development in the event of standard sensor failure. The developed mathematical model will allow the following:

• The helicopter TE ensures reliable operation by the standard sensors in the event of failure of one, minimizing the associated failure risks.
• Increase the helicopter TE ACS’s stability and adaptability, providing the ability to automatically reconfigure the control system.
• Improve the helicopter TE’s accuracy and stability under failure conditions using additional correction algorithms.

Reduce dependence on individual sensors by integrating data from other sensors and using intelligent information processing methods.

The research’s practical implication is directly linked to addressing critical challenges in the helicopter industry, particularly the need for enhancing the helicopter TE’s reliability and safety under sensor failure conditions. By developing advanced ACS-reconfiguration capabilities in real-time to compensate for sensor anomalies, this research contributes to ensuring continuous and stable engine operation, thereby reducing the risk of potential in-flight emergencies and improving overall operational efficiency.

2. Materials and Methods

The recording indicators from a helicopter TE’s sensor process (Figure 1a) include data collection and display from key and additional sensors that provide the engine operation monitoring. The main sensors are the gas-generator rotor, r.p.m. (n_TC) and the free turbine rotor speed (n_FT) sensors, as well as the gas temperature sensor in front of the compressor turbine ($T_G^{*}$) [38,39]. The n_TC sensor measures the first rotor speed, which is important for assessing its performance and correct operation. Similarly, the free turbine rotor speed sensor monitors the second rotor speed, which provides information about the turbine’s condition and energy conversion efficiency. The $T_G^{*}$ sensor records the temperature, which is critical for controlling the gas temperature and preventing overheating.

Additionally, for complete engine operation monitoring, atmospheric pressure (P_N) and ambient temperature (T_N) sensors are installed. The atmospheric pressure sensor measures the current atmospheric pressure, which is important for correct calibration and engine performance calculation, since pressure affects air characteristics and combustion efficiency. The ambient temperature sensor monitors the external temperature, which also affects engine operation and is necessary for adjusting operating parameters. The gas-generator installation angle guide blades (α) are set to optimize air flow for efficient compression and engine operation. The electronic engine governor (h) manages all of these factors, providing automatic adjustments to optimize engine performance and reliability.

The helicopter TE ACS plays a key role in the sensor recording and optimizing the operation process, providing data integration from various sensors and engine operating parameters’ dynamic regulation. The helicopter TE ACS automatically processes readings from n_TC, n_FT, $T_G^{*}$, P_N and T_N sensors, as well as data on the gas-generator installation angle guide blades, to adjust engine operating modes in real time. The helicopter TE ACS analyzes this data, compares it with specified parameters and optimization algorithms to ensure stable operation and engine efficiency, minimizing deviation and preventing potential malfunctions or overheating. An electronic regulator integrated into the helicopter TE ACS automatically adjusts operating parameters, taking into account changing conditions and external factors, which helps to improve helicopter TE reliability and performance.

If one of the helicopter TE sensors suddenly fails, for example, an n_TC sensor, the recording indicators and controlling engine operation process undergoes significant changes (Figure 1b). To systematize the sensor’s failure effects on the system, a failure matrix is provided (

Table 1. The proposed failure matrix.

Failure Type\Impact Aspect	Measurement Accuracy	System Reliability	Response Time	Operation Under Extreme Conditions	Economic Impact	Safety
Instantaneous failure	High	Reduced	Immediate disruption	Not relevant	High costs	High risk
Intermittent failure	Medium	Unstable	Delays	May be hindered	Potential additional costs	Medium risk
Partial failure	Accuracy degradation	Partial	Possible delays	Depends on the situation	Additional costs for correction	Medium risk
Multiple failure	Very high error	Critical	Significant delays	Critical	Significant economic losses	Critical risk
Drift failure	Gradual degradation	Gradual degradation	Delays	May be problematic	Gradual additional costs	Medium risk
Systematic failure	Constant degradation	Constant degradation	Stable slowdown	May be hindered	High economic losses	Medium risk
Random failure	Unpredictable impact	Unpredictable impact	Uncertain	May be problematic	Unpredictable costs	Medium risk
Material degradation failure	Gradual degradation	Gradual degradation	Delays	May be problematic	Gradual additional costs	Medium risk
Thermal failure	Accuracy degradation	Temporary or permanent	Slowdown	Critical	High replacement costs	High risk
Vibration failure	Unpredictable impact	Unpredictable	Possible delays	Critical	Potential additional costs	High risk
Moisture failure	Gradual degradation	Gradual degradation	Slowdown	Critical	High repair costs	Medium risk
Electromagnetic interference failure	Accuracy degradation	Possible issues	Possible delays	May be hindered	Additional shielding costs	Medium risk
Software failure	Unpredictable impact	Unpredictable	Slowdown	May be problematic	High software correction costs	Medium risk
Leakage current failure	Accuracy degradation	Temporary or permanent	Slowdown	Not relevant	High replacement costs	High risk

).

The matrix rows represent failure modes and the columns represent impact potential aspects on the system. The matrix cells indicate the impact degree or nature of each failure type on specific aspects of the system.

The helicopter TE ACS must quickly recognize the data absence from one of the key sensors and activate restoring algorithms [5], such as data interpolation or use of backup sensor, if available. In the absence of backup sensors, the helicopter TE switches to emergency control mode, based on previous data and state assessment algorithms, which may include predicting behavior based on the engine model. Actual engine operating parameters may be adjusted based on incomplete data, which requires an increased accuracy level in the calculations and monitoring to prevent potential overheating or other critical conditions. Sudden sensor failure requires rapid system adaptation and can lead to a decrease in control accuracy, which emphasizes the importance of having an emergency control and diagnostic system during helicopter TE operation.

Under these conditions, the need for the helicopter ‘s TE ACS structural reconfiguration arises in the event of critical sensor failure, since it requires function reassessment and redistribution and control algorithms to ensure reliable engine operation under the changed conditions and to minimize the risk of unforeseen situations.

In the development of a helicopter TE’s reconfigurable ACS, it is assumed that if a sensor fails, the signals it transmits to the control object are considered to be zero. This approach allows the modeling of extreme scenarios, ensuring control system reliability during failures. Consequently, adjustments to control algorithms are required to maintain stability and flight safety when reliable data from the failed sensor are unavailable.

In this regard, two calibrated sensors can be used: the air temperature at the engine inlet (T_N) and the gas temperature in front of the compressor turbine ($T_G^{*}$). Three sensors are also required for atmospheric pressure (P_N), free turbine rotor speed (n_FT) and the gas-generator installation angle guide blades (α), as well as the electronic engine governor (h) presence [38,39]. Based on the above, an on-board algorithm F is used to calculate and limit the gas-generator rotor r.p.m. by influencing the engine fuel supply (fuel mass flow).

Features of the helicopter TE ACS structural reconfiguration task solving after failure detection and localization [5] made it possible to formulate the theorem “On the helicopter TE ACS structural reconfiguration”:

Theorem 1. 

When a failure is detected and localized, the gas-generator rotor r.p.m. sensor (n_TC) is structurally reconfigured to ensure the helicopter systems’ uninterrupted operation and maintain stability.

Proof of Theorem 1. 

Table 2. The proof of the theorem “On the helicopter TE ACS structural reconfiguration” main stages.

Stage	Description
Structural reconfiguration definition	It is assumed that the helicopter TE ACS has an initial structure that includes temperature sensors and other serviceable sensors. When the gas-generator rotor r.p.m. sensor failure is detected, the system switches to an alternate structure that uses other available sensors and controls.
Stability analysis	The control and observability theory are applied to ensure that the new structure of the helicopter TE ACS stability system is ensured. The temperature, atmospheric pressure, free turbine speed and the gas-generator guide device position sensors should be used to obtain important system parameters.
Fuel supply control	The electronic engine governor (h) and on-board algorithm F, which take into account the new input from the replaced sensors, must be adjusted to effectively control the fuel delivery to avoid over-revving the gas-generator rotor.
Conducting experiments	The helicopter TE and their ACS used mathematical and neural network models to carry out simulations and experiments to confirm the new structure’s effectiveness in detecting failure.

shows the main stages of proving the theorem “On the helicopter TE ACS structural reconfiguration”. □

To confirm the theorem formulation correctness, as well as to determine the structural reconfiguration, it should be assumed that the system state is described by the state vector x, while u is the input command, y is the output value, and d is the failure parameters vector. Then the system’s mathematical model is described by a differential equation system in the following form:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial x}{\partial t}}=f\left(x,u,d\right),\\ y=h\left(x,u,d\right).\end{array}\right.$$

(1)

When taking into account the gas-generator rotor r.p.m. sensor failure influence, the differential equations system (1) will take the form:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial x}{\partial t}}=f\left(x,u,d,∆n_{TC}\right),\\ y=h\left(x,u,d,∆n_{TC}\right),\end{array}\right.$$

(2)

where Δn_TC is the gas-generator rotor r.p.m. deviation.

Consequently, when a gas-generator rotor r.p.m. sensor failure is detected and localized, the helicopter TE ACS switches to an alternative structure described by new differential equations:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial x_{new}}{\partial t}}=f_{new}\left(x_{new},u_{new},d_{new}\right),\\ y_{new}=h_{new}\left(x_{new},u_{new},d_{new}\right),\end{array}\right.$$

(3)

where x_new, y_new, and d_new are new state vectors, input commands and failure parameters, respectively.

At the next stage, specific mathematical expressions are introduced for the new system structure, taking into account the gas-generator rotor r.p.m. sensor failure effect. These expressions can be derived based on the system’s physical laws analysis. For example, it is possible to form a new differential equation for the gas-generator rotor r.p.m. in the identified failure event:

$${\displaystyle \frac{\partial n_{TCnew}}{\partial t}}=g\left(x_{new},u_{new},d_{new},∆n_{TC}\right).$$

(4)

Similarly, new equations for the system and other important parameters can be derived.

To obtain mathematical expressions for the system’s new structure, taking into account the gas-generator rotor r.p.m. sensor failure, the next step is to simplify the system model. It is assumed that the failure affects only the differential equation describing the gas-generator rotor r.p.m. Let us introduce a new value δn_TC as the deviation from the normal n_TC value:

$$\delta n_{TC}=n_{TC}-n_{TCnorm},$$

(5)

where n_TCnorm is the gas-generator rotor r.p.m. normal value.

Accordingly, the model for the δn_TC value has the form:

$${\displaystyle \frac{\partial \delta n_{TC}}{\partial t}}=g\left(x,u,d,∆n_{TC}\right).$$

(6)

Taking into account (6), the differential equation system (3) will take the form:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial x_{new}}{\partial t}}=f_{new}\left(x_{new},u_{new},d_{new},\delta n_{TC}\right),\\ y_{new}=h_{new}\left(x_{new},u_{new},d_{new},\delta n_{TC}\right),\end{array}\right.$$

(7)

The differential equation system (7) is the helicopter TE ACS’s new structure, taking into account the gas-generator rotor r.p.m. sensor failure effect. It can be used for dynamic modelling and analysis of system stability after a failure is detected.

At the next stage, the new system stability is analyzed and it is confirmed that it ensures the helicopter TE ACS uninterrupted operation when the gas-generator rotor r.p.m. sensor failure is detected. This analysis may include linear or nonlinear stability, important control constraints and system input verification.

The new system stability analysis after gas-generator rotor r.p.m. sensor-failure detection can be carried out using linear approximation or nonlinear analysis, depending on the equation’s nature. For linear analysis, linear approximations around the equilibrium state are used. Let us denote the system equilibrium state as ${\overline{x}}_{new}$, ${\overline{u}}_{new}$, ${\overline{d}}_{new}$, $\overline{\delta }{n}_{TC}$. Then linear deviations can be defined as:

$${\tilde{x}}_{new}=x_{new}-{\overline{x}}_{new}, {\tilde{u}}_{new}=u_{new}-{\overline{u}}_{new}, {\tilde{d}}_{new}=d_{new}-{\overline{d}}_{new}, \tilde{\delta }{n}_{TC}=\delta n_{TC}-\overline{\delta }{n}_{TC}.$$

(8)

Linear equations can be defined using the differential equation system (7) linear approximations and output signal functions. For example:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial {\tilde{x}}_{new}}{\partial t}}=A·{\tilde{x}}_{new}+B·{\tilde{u}}_{new}+E·{\tilde{d}}_{new}+F·\tilde{\delta }{n}_{TC},\\ {\tilde{y}}_{new}=C·{\tilde{x}}_{new}+D·{\tilde{u}}_{new}+G·{\tilde{d}}_{new}+H·\tilde{\delta }{n}_{TC},\end{array}\right.$$

(9)

where matrices A, B, C, D, E, F, G, and H are the state linear functions, input, failure parameters and deviation from the gas-generator rotor r.p.m. normal value.

For stability analysis, one can use linear stability theory and stability criteria such as a matrix A eigenvalue. If a matrix A of all eigenvalues has negative real parts, the system will be linearly stable. If the analysis is performed for a nonlinear model, methods such as Lyapunov analysis should be used to determine the system stability along the nonlinear differential equation trajectories. This analysis will help determine whether the new system structure ensures the helicopter TE ACS stable operation after gas-generator rotor r.p.m. sensor failure detection.

To analyze the system stability using the Lyapunov analysis method, it is necessary to determine the Lyapunov function and derive stability conditions. In this case, we will use the new system linear approximation after gas-generator rotor r.p.m. sensor detecting. It is assumed that ${\overline{x}}_{new}$, ${\overline{u}}_{new}$, ${\overline{d}}_{new}$, $\overline{\delta }{n}_{TC}$ have equilibrium values. Consider the quadratic Lyapunov function for the system:

$$V\left({\tilde{x}}_{new},{\tilde{u}}_{new},{\tilde{d}}_{new},\tilde{\delta }{n}_{TC}\right)={\tilde{x}}_{new}^T·P·{\tilde{x}}_{new}+{\tilde{u}}_{new}^T·Q·{\tilde{u}}_{new}+{\tilde{d}}_{new}^T·R·{\tilde{d}}_{new}+\tilde{\delta }{n}_{TC}^2,$$

(10)

where P, Q, and R are the symmetric positive definite matrices.

Let us calculate the Lyapunov function derivative concerning time:

$${\displaystyle \frac{\partial V}{\partial t}}={\displaystyle \frac{\partial V}{\partial {\tilde{x}}_{new}}}·{\displaystyle \frac{\partial {\tilde{x}}_{new}}{\partial t}}+{\displaystyle \frac{\partial V}{\partial {\tilde{u}}_{new}}}·{\displaystyle \frac{\partial {\tilde{u}}_{new}}{\partial t}}+{\displaystyle \frac{\partial V}{\partial {\tilde{d}}_{new}}}·{\displaystyle \frac{\partial {\tilde{d}}_{new}}{\partial t}}+2·\tilde{\delta }{n}_{TC}·{\displaystyle \frac{\partial \tilde{\delta }{n}_{TC}}{\partial t}}.$$

(11)

Given the system’s linear equations, the derivative of some parts can be expressed:

$$\begin{gathered} {\displaystyle \frac{\partial V}{\partial t}}={\tilde{x}}_{new}^T·\left(A^T·P+P·A\right)·{\tilde{x}}_{new}+{\tilde{u}}_{new}^T·\left(B^T·P+Q·B\right)·{\tilde{u}}_{new}+{\tilde{d}}_{new}^T·\left(E^T·P+R·E\right)·{\tilde{d}}_{new} \\ +2·\tilde{\delta }{n}_{TC}·{\displaystyle \frac{\partial \tilde{\delta }{n}_{TC}}{\partial t}}. \end{gathered}$$

(12)

To ensure the new system stability after failure identification, the Lyapunov function derivative (12) must be negative definite, with the equilibrium points exception, i.e.,

$$\begin{gathered} {\displaystyle \frac{\partial V}{\partial t}}={\tilde{x}}_{new}^T·\left(A^T·P+P·A\right)·{\tilde{x}}_{new}+{\tilde{u}}_{new}^T·\left(B^T·P+Q·B\right)·{\tilde{u}}_{new}+{\tilde{d}}_{new}^T·\left(E^T·P+R·E\right)·{\tilde{d}}_{new} \\ +2·\tilde{\delta }{n}_{TC}·{\displaystyle \frac{\partial \tilde{\delta }{n}_{TC}}{\partial t}}<0. \end{gathered}$$

(13)

To analyze the system stability using the Lyapunov method, consider the Lyapunov function, which includes quadratic expressions for the state deviations from the equilibrium state. Let us mark the deviation vector as

$$\mathbf{z}={\left({\tilde{x}}_{new},{\tilde{u}}_{new},{\tilde{d}}_{new},\tilde{\delta }{n}_{TC}\right)}^T.$$

(14)

Taking P as a symmetric positive definite matrix, the Lyapunov function is introduced:

$$V\left(\mathbf{z}\right)={\mathbf{z}}^T·P·\mathbf{z},$$

(15)

after which the Lyapunov function derivative along the system trajectory is calculated:

$${\displaystyle \frac{\partial V}{\partial t}}={\displaystyle \frac{\partial V}{\partial \mathbf{z}}}·{\displaystyle \frac{\partial \mathbf{z}}{\partial t}}={\mathbf{z}}^T·P·\mathbf{z}.$$

(16)

Using the system linear equations, we express the derivative:

$${\displaystyle \frac{\partial V}{\partial t}}={\mathbf{z}}^T·\left(A^T·P+P·A\right)·\mathbf{z}+2·\tilde{\delta }{n}_{TC}·{\displaystyle \frac{\partial \tilde{\delta }{n}_{TC}}{\partial t}}.$$

(17)

For stability, all terms except the last one must be negative or non-negative:

$${\displaystyle \frac{\partial V}{\partial t}}\le 0\Rightarrow {\mathbf{z}}^T·\left(A^T·P+P·A\right)·\mathbf{z}\le -2·\tilde{\delta }{n}_{TC}·{\displaystyle \frac{\partial \tilde{\delta }{n}_{TC}}{\partial t}}.$$

(18)

To ensure this, it is necessary to choose the matrix P so that it is positive definite and so that the main diagonal (A^T ∙ P + P ∙ A) has negative elements. This condition can be used to determine the matrix P parameters and establish the conditions for system stability concerning the gas-generator rotor r.p.m. sensor failure. However, this analysis is linear and is based on the real system linear approximation. Taking into account the stability conditions and taking into account Lyapunov analysis, we can assume that the system is stable. With stability determined by Lyapunov analysis, using the quadratic Lyapunov function, we obtain the following:

$${\mathbf{z}}^T·\left(A^T·P+P·A\right)·\mathbf{z}\le -2·\tilde{\delta }{n}_{TC}·{\displaystyle \frac{\partial \tilde{\delta }{n}_{TC}}{\partial t}}.$$

(19)

Inequality (19) indicates that the deviation from the equilibrium state (z) increases or remains stable, which the system stable functioning guarantees. For an even more detailed and expanded proof of the theorem, taking into account nonlinearity and other aspects that may affect stability, additional mathematical apparatus is used (

Table 3. Additional mathematical apparatus for extended proof of the theorem “On the helicopter TE ACS structural reconfiguration”.

Name	Description
Lyapunov extended analysis	1. The higher-order Lyapunov functions to use or take into account the system’s nonlinear elements. 2. Taking into account the system’s internal connection dynamics and their mutual influence on stability.
Sensitivity analysis	Studying the system sensitivity to parametric variables and the optimal parameter selection to maximize stability.
The Lyapunov–Krasovsky method	The methods used take into account nonlinearity in the system and ensure stability under the specified criteria conditions.
The disorder impact analysis	Studying the possible disruption impact and developing control strategies to ensure resilience in unforeseen circumstances.
Constraint analysis	Taking into account control constraints and system initial values during stability analysis.
Constructive analysis method	The constructive analysis methods are used for the constructive control procedures development and system reconfiguration.

).

The additional mathematical apparatus (Table 2) use allows us to take into account more details and improve and expand the proof of the theorem, taking into account the helicopter TE ACS’s specific properties and the gas-generator rotor r.p.m. sensor failure.

Let us consider the system’s probable nonlinear elements. For example, it is possible to take into account a nonlinear limiting function for the gas-generator rotor r.p.m. or nonlinear relations between different system states. Let us assume that the speed limit function has the following form:

$$f\left(\delta n_{TC}\right)=k·{\left(\delta n_{TC}\right)}^n,$$

(20)

where k is a constant.

Let us modify the system Equation (4) by adding this nonlinear element:

$${\displaystyle \frac{\partial n_{TC}}{\partial t}}=g\left(x,u,d,∆n_{TC}\right)+k·{\left(\delta n_{TC}\right)}^n.$$

(21)

Let us introduce the Lyapunov function for the nonlinear case:

$$V\left(\mathbf{z}\right)={\mathbf{z}}^T·P·\mathbf{z}+k·{\left(\delta n_{TC}\right)}^n.$$

(22)

Let us calculate the Lyapunov function derivative based on the nonlinear element:

$${\displaystyle \frac{\partial V}{\partial t}}={\mathbf{z}}^T·\left(A^T·P+P·A\right)·\mathbf{z}+n·k·{\left(\delta n_{TC}\right)}^{\left(n-1\right)}·\left(g\left(x,u,d,∆n_{TC}\right)+k·{\left(\delta n_{TC}\right)}^n\right).$$

(23)

The system stability conditions taking into account nonlinear elements will take the following form:

$${\mathbf{z}}^T·\left(A^T·P+P·A\right)·\mathbf{z}+n·k·{\left(\delta n_{TC}\right)}^{\left(n-1\right)}·\left(g\left(x,u,d,∆n_{TC}\right)+k·{\left(\delta n_{TC}\right)}^n\right)\le 0.$$

(24)

This is an extended stability condition that takes into account the cubic nonlinear element in the system. Taking into account the nonlinear element k ∙ (δn_TC)ⁿ, expression (24) characterizes this element’s influence on the Lyapunov function derivative. The stability condition requires that the expression ${\displaystyle \frac{\partial V}{\partial t}}$ have a negative term. To do this, the nonlinear element and other members’ interaction are taken into account:

$$n·k·{\left(\delta n_{TC}\right)}^{\left(n-1\right)}·\left(g\left(x,u,d,∆n_{TC}\right)+k·{\left(\delta n_{TC}\right)}^n\right)\le -{\mathbf{z}}^T·\left(A^T·P+P·A\right)·\mathbf{z}.$$

(25)

Expression (25) is an extended stability condition for the system taking into account nonlinear elements.

Let us consider additional conditions for the system stability study. Let us add a condition related to the function g(x, u, d, ∆n_TC) gradient to get a more detailed analysis. Let us assume that this function gradient is limited by some constant M:

$$‖\nabla g\left(x,u,d,∆n_{TC}\right)‖\le M.$$

(26)

Condition (26) indicates that the function g gradient does not increase indefinitely and allows the nonlinear effects to have additional control in the system. Taking this condition into account, the stability condition (24) is expanded as follows:

$${\mathbf{z}}^T·\left(A^T·P+P·A\right)·\mathbf{z}+n·k·{\left(\delta n_{TC}\right)}^{\left(n-1\right)}·\left(g\left(x,u,d,∆n_{TC}\right)+k·{\left(\delta n_{TC}\right)}^n\right)+\lambda ·‖\nabla g\left(x,u,d,∆n_{TC}\right)‖\le 0,$$

(27)

where λ is a parameter that affects the gradient condition weight.

An additional condition for stability may be taking into account the system’s own limitations or its derivatives’ limitations. For example, you can introduce the state vector z or its derivatives boundedness condition. Suppose that there is a constant B such that ‖z‖ ≤ B. This condition indicates the system state has limited amplitude. Thus, the additional condition looks like the following:

$$‖\mathbf{z}‖+\beta \le 0,$$

(28)

where β is a parameter that takes into account the system-limited state condition weight. This condition indicates that the state amplitude sum and the additional term weighted by the parameter β must remain negative for the system to be stable.

In the additional condition view that the system state is bounded, the stability condition (27) is expanded as follows:

$$\begin{gathered} {\mathbf{z}}^T·\left(A^T·P+P·A\right)·\mathbf{z}+n·k·{\left(\delta n_{TC}\right)}^{\left(n-1\right)}·\left(g\left(x,u,d,∆n_{TC}\right)+k·{\left(\delta n_{TC}\right)}^n\right)+\lambda ·‖\nabla g\left(x,u,d,∆n_{TC}\right)‖ \\ +\left(‖\mathbf{z}‖+\beta \right)\le 0, \end{gathered}$$

(29)

where λ > 0 and β are parameters that influence the gradient condition and state boundedness weight, respectively.

Let us assume that there is a constant α > 0 so that for all z ≠ 0, the following holds:

$${\mathbf{z}}^T·\left(A^T·P+P·A\right)·\mathbf{z}+\alpha ·{‖\mathbf{z}‖}^2\le 0.$$

(30)

This condition ensures that the system has constant sensitivity and is asymptotically stable. Integrating this into the stability condition, we obtain the following:

$$\begin{gathered} {\mathbf{z}}^T·\left(A^T·P+P·A\right)·\mathbf{z}+n·k·{\left(\delta n_{TC}\right)}^{\left(n-1\right)}·\left(g\left(x,u,d,∆n_{TC}\right)+k·{\left(\delta n_{TC}\right)}^n\right)+\lambda ·‖\nabla g\left(x,u,d,∆n_{TC}\right)‖ \\ +\left(‖\mathbf{z}‖+\beta \right)+\alpha ·{‖\mathbf{z}‖}^2\le 0, \end{gathered}$$

(31)

where λ > 0, α > 0 and β are parameters that take into account the gradient condition, state boundedness and robust sensitivity weight, respectively.

For structural reconfiguration to ensure uninterrupted operation, it is assumed that the helicopter TE ACS has the following generalized structure:

$$n_{TC}\to h\to F\to G_T,$$

(32)

where n_TC is the gas-generator rotor r.p.m. sensor; h is the electronic engine governor; F is the on-board algorithm for calculating and limiting gas-generator rotor r.p.m.; G_T is the mass fuel consumption.

According to the condition, the n_TC sensor has failed. It is assumed that the following dependencies are also known:

$$\left\{\begin{array}{c}F\left(n_{TC},T_N,T_G^{*},P_N,n_{FT},\alpha \right)=n_{TC},\\ h\left(n_{TC},T_N,T_G^{*},P_N,n_{FT},\alpha \right)=h\left(n_{TC}\right),\\ G_T\left(n_{TC},T_N,T_G^{*},P_N,n_{FT},\alpha \right)=G_T\left(n_{TC}\right).\end{array}\right.$$

(33)

Then the helicopter TE ACS new structure after reconfiguration will look like this:

$$\left(T_N,T_G^{*},P_N,n_{FT},\alpha \right)\to h\to F\to G_T,$$

(34)

It is assumed that the new dependences F and h after reconfiguration will be presented in the following form:

$$\left\{\begin{array}{c}F\left(T_N,T_G^{*},P_N,n_{FT},\alpha \right)=n_{TC},\\ h\left(T_N,T_G^{*},P_N,n_{FT},\alpha \right)=h\left(n_{TC}\right).\end{array}\right.$$

(35)

Then the helicopter TE ACS new system will be stable if the new system F and h are stable.

Let the new system F and h be stable. Then, for n_TC any initial value, the new system will tend to the n_TC value at which the system F and h will be stable. Let this n_TC value be $n_{TC}^{*}$. Then, for G_T any initial value, the new system will tend to m value at which the F and h system will be stable. Let this G_T value be equal to $G_T^{*}$. Consequently, the helicopter TE ACS new system will be stable.

Example 1. 

Let the new system F and h look like this:

$$\left\{\begin{array}{c}F\left(T_N,T_G^{*},P_N,n_{FT},\alpha \right)=n_{TC}^{*}=n_{TC}^{*}-k·\left(T_N-T_G^{*}\right)+k·\left(P_N-P_N^{*}\right),\\ h\left(T_N,T_G^{*},P_N,n_{FT},\alpha \right)=h\left(n_{TC}^{*}\right)=h\left(n_{TC}^{*}\right)-k·\left(\alpha -a^{*}\right).\end{array}\right.$$

(36)

where k is the reconfiguration coefficient;  $n_{TC}^{*}$ is the n_TC value at which the system F and h will be stable; $P_N^{*}$ is the P_N value at which the system F and h will be stable; α^* is the α value at which the system F and h will be stable.

The solution to this equation’s system will be the parameters T_N, $T_G^{*}$, P_N, n_FT and α values set, at which the $n_{TC}^{*}$ values and the function h will be stable. Equations F and h define the relations between these parameters, including an adjustment with a reconfiguration factor k, which allows the system to adapt to changes in input variables. The system’s stable state is achieved when $n_{TC}^{*}$ and the function h($n_{TC}^{*}$) are equal to their stable values, corrected for the T_N, P_N and α deviations from their stable values $T_N^{*}$, $P_N^{*}$ and α^*.

To solve the equation system (36), a numerical optimization method is proposed. To achieve this, the objective function that needs to be minimized is determined. In this case, the objective function will express the system parameters’ total deviation from their stable values:

$$L\left(T_N,T_G^{*},P_N,n_{FT},\alpha \right)=\left|n_{TC}^{*}-\left(n_{TC}^{*}-k·\left(T_N-T_G^{*}\right)\right)+k·\left(P_N-P_N^{*}\right)\right|+\left|h\left(n_{TC}^{*}\right)-\left(h\left(n_{TC}^{*}\right)-k·\left(\alpha -a^{*}\right)\right)\right|,$$

(37)

where $n_{TC}^{*}$, $P_N^{*}$ and α^* are the specified stable values.

The objective function constraints are written as:

$$\left\{\begin{array}{c}{T}_{N\mathrm{m}\mathrm{i}\mathrm{n}}\le T_N\le T_{N\mathrm{m}\mathrm{a}\mathrm{x}}\\ T_{G\mathrm{m}\mathrm{i}\mathrm{n}}^{*}\le T_G^{*}\le T_{G\mathrm{m}\mathrm{a}\mathrm{x}}^{*}\\ P_{N\mathrm{m}\mathrm{i}\mathrm{n}}\le P_N\le P_{N\mathrm{m}\mathrm{a}\mathrm{x}}\\ n_{FT\mathrm{m}\mathrm{i}\mathrm{n}}\le n_{FT}\le n_{FT\mathrm{m}\mathrm{a}\mathrm{x}}\\ {\alpha }_{\mathrm{m}\mathrm{i}\mathrm{n}}\le \alpha \le {\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}\end{array}\right.$$

(38)

To minimize the objective function (37), taking into account restrictions (38), the Lagrange multiplier method was used. Then the Lagrange function for (37) is as follows:

$$L\left(T_N,T_G^{*},P_N,n_{FT},\alpha \right)=L\left(T_N,T_G^{*},P_N,n_{FT},\alpha \right)+\sum _i{\lambda }_i·\left(g_i\left(T_N,T_G^{*},P_N,n_{FT},\alpha \right)-0\right),$$

(39)

where g_i are the functions representing constraints:

$$\left\{\begin{array}{c}{g_1\left(T_N\right)=T}_{N\mathrm{m}\mathrm{i}\mathrm{n}}\le T_N\le T_{N\mathrm{m}\mathrm{a}\mathrm{x}}\\ g_2\left(T_G^{*}\right)=T_{G\mathrm{m}\mathrm{i}\mathrm{n}}^{*}\le T_G^{*}\le T_{G\mathrm{m}\mathrm{a}\mathrm{x}}^{*}\\ g_3\left(P_N\right)=P_{N\mathrm{m}\mathrm{i}\mathrm{n}}\le P_N\le P_{N\mathrm{m}\mathrm{a}\mathrm{x}}\\ g_4\left(n_{FT}\right)=n_{FT\mathrm{m}\mathrm{i}\mathrm{n}}\le n_{FT}\le n_{FT\mathrm{m}\mathrm{a}\mathrm{x}}\\ g_5\left(\alpha \right)={\alpha }_{\mathrm{m}\mathrm{i}\mathrm{n}}\le \alpha \le {\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}\end{array}\right.$$

(40)

In this case, the objective function L gradient is defined as follows:

$$\nabla L=\left({\displaystyle \frac{\partial L}{\partial T_N}},{\displaystyle \frac{\partial L}{\partial T_G^{*}}},{\displaystyle \frac{\partial L}{\partial P_N}},{\displaystyle \frac{\partial L}{\partial n_{FT}}},{\displaystyle \frac{\partial L}{\partial \alpha }}\right).$$

(41)

To find optimal values (39), taking into account restrictions (40), the Kuhn–Tucker equations’ system is solved:

$${\displaystyle \frac{\partial L}{\partial T_N}}=0,{\displaystyle \frac{\partial L}{\partial T_G^{*}}}=0,{\displaystyle \frac{\partial L}{\partial P_N}}=0,{\displaystyle \frac{\partial L}{\partial n_{FT}}}=0,{\displaystyle \frac{\partial L}{\partial \alpha }}=0.$$

(42)

Similarly, to find the function h($n_{TC}^{*}$), a differential equation in the following form is used:

$${\displaystyle \frac{\partial h\left(n_{TC}^{*}\right)}{\partial n_{TC}^{*}}}=f\left(n_{TC}^{*}\right),$$

(43)

where h($n_{TC}^{*}$) is the given function.

The solution to the Kuhn–Tucker Equations (42) and (43) system is carried out using numerical methods, such as Newton’s method for nonlinear systems.

Newton’s method for the nonlinear equations system F(X) = 0 is reduced to an iterative process:

$$X_{k+1}=X_k-J_F{\left(X_k\right)}^{-1}·F\left(X_k\right),$$

(44)

where X is the variable T_N, $T_G^{*}$, P_N, n_FT α, λ_i vector, J_F is the equations F(X) system Jacobian.

For (41), and (42), the Lagrange gradients and the Jacobian of the system of equations have the following form:

$$\begin{gathered} {\displaystyle \frac{\partial L}{\partial T_N}}={\displaystyle \frac{\partial L}{\partial T_N}}+\sum _i{\lambda }_i·{\displaystyle \frac{\partial g_i}{\partial T_N}}, {\displaystyle \frac{\partial L}{\partial T_G^{*}}}={\displaystyle \frac{\partial L}{\partial T_G^{*}}}+\sum _i{\lambda }_i·{\displaystyle \frac{\partial g_i}{\partial T_G^{*}}},{\displaystyle \frac{\partial L}{\partial P_N}}={\displaystyle \frac{\partial L}{\partial P_N}}+\sum _i{\lambda }_i·{\displaystyle \frac{\partial g_i}{\partial P_N}}, \\ {\displaystyle \frac{\partial L}{\partial n_{FT}}}={\displaystyle \frac{\partial L}{\partial n_{FT}}}+\sum _i{\lambda }_i·{\displaystyle \frac{\partial g_i}{\partial n_{FT}}},{\displaystyle \frac{\partial L}{\partial \alpha }}={\displaystyle \frac{\partial L}{\partial \alpha }}+\sum _i{\lambda }_i·{\displaystyle \frac{\partial g_i}{\partial \alpha }}, {\displaystyle \frac{\partial L}{\partial {\lambda }_i}}={\displaystyle \frac{\partial L}{\partial {\lambda }_i}}+\sum _i{\lambda }_i·{\displaystyle \frac{\partial g_i}{\partial {\lambda }_i}}, \end{gathered}$$

(45)

$$J_F=\left(\begin{array}{cccccc}{\displaystyle \frac{\partial F_1}{\partial T_N}}& {\displaystyle \frac{\partial F_1}{\partial T_G^{*}}}& {\displaystyle \frac{\partial F_1}{\partial P_N}}& {\displaystyle \frac{\partial F_1}{\partial n_{FT}}}& {\displaystyle \frac{\partial F_1}{\partial \alpha }}& \dots \\ {\displaystyle \frac{\partial F_2}{\partial T_N}}& {\displaystyle \frac{\partial F_2}{\partial T_G^{*}}}& {\displaystyle \frac{\partial F_2}{\partial P_N}}& {\displaystyle \frac{\partial F_2}{\partial n_{FT}}}& {\displaystyle \frac{\partial F_2}{\partial \alpha }}& \dots \\ \dots & \dots & \dots & \dots & \dots & \dots \\ {\displaystyle \frac{\partial F_i}{\partial T_N}}& {\displaystyle \frac{\partial F_i}{\partial T_G^{*}}}& {\displaystyle \frac{\partial F_i}{\partial P_N}}& {\displaystyle \frac{\partial F_i}{\partial n_{FT}}}& {\displaystyle \frac{\partial F_i}{\partial \alpha }}& \dots \end{array}\right),$$

(46)

where F_i are the function’s vector elements that make up the Kuhn–Tucker equations system.

The iterative process is carried out by initializing X₀ as the initial approximation for T_N, $T_G^{*}$, P_N, n_FT α, λ_i, J_F and applying iterations in the form (44). The iterations continue until ‖X_k+1 − X_k‖ becomes less than a given threshold.

To solve optimization problems using Kuhn–Tucker conditions, an addition to the NSGA-II method [40] is proposed, which integrates processing and compliance with these conditions. The Kuhn–Tucker term’s inclusion in NSGA-II suggests the following addition:

• The optimization problem with Kuhn–Tucker conditions is formulated as a multi-objective optimization task, where it is necessary to take into account both objective functions and restrictions specified by inequalities and equalities. The Kuhn–Tucker terms, which are a set of equations and inequalities, must be included in the fitness function.
• To integrate the Kuhn–Tucker conditions into the NSGA-II method, the fitness function f(x) is extended by adding penalty functions such that the Kuhn–Tucker conditions penalize violations. Let g(x) ≤ 0 and h(x) = 0 denote inequalities and equalities, respectively. Then the fitness function can be written as follows:

$$f_{adapted}\left(x\right)=f\left(x\right)+{\lambda }_1·\sum _{i=1}^m\max 0,g_i\left(x\right)+{\lambda }_2·\sum _{j=1}^p\left|h_j\left(x\right)\right|,$$

(47)

where λ₁ and λ₂ are penalty coefficients for inequalities and equalities violation, respectively.

3.

The selection, crossover and mutation process in NSGA-II is also modified to accommodate Kuhn–Tucker conditions. During crossover and mutation operations, it is necessary to maintain the solutions admissibility that meet the Kuhn–Tucker conditions by adjusting new solutions so that they minimize the condition violations. This can be achieved using additional local search procedures or constraints that adjust the decision values according to the Kuhn–Tucker terms.

4.

In the evolution process, in addition to the traditional dominant sorting and selection in NSGA-II, an additional checking step, the Kuhn–Tucker conditions compliance, is carried out. This may involve selecting the best solutions based on their suitability to Kuhn–Tucker conditions and additionally using optimization techniques such as the Lagrange method to solve local optimization problems within the feasible space.

3. Results and Discussion

In this work, a computer experiment was carried out, which consisted of finding the optimal parameters T_N, $T_G^{*}$, P_N, n_FT (these parameters are recorded on board the helicopter by standard sensors) and α, at which the helicopter TE’s reconfigured ACS (36) will be stable. For this purpose, we used the helicopter TE (the TV3-117 engine, using the example) thermogas-dynamic parameters’ data array (256 values), obtained during the Mi-8MTV helicopter flight tests (the TV3-117 engine is the type of power plant part for this helicopter) [41,42] (Figure 2). The sensors used to record these parameters on board the helicopter and their description are given in

Table 4. The sensors used in the work [43,44].

Parameter	The Sensor Used	Sensor Figure	The Received Data Processing, Filtering and Analysis Description
TN	PT100 sensor		The ambient temperature on the helicopter is recorded by a PT100 sensor, providing an analog signal. This signal undergoes initial filtering to remove electrical noise and interference, followed by analog-to-digital conversion and calibration to ensure accurate measurements. The data are then analyzed for consistency and accuracy, ensuring reliable use in mathematical models related to the helicopter’s dynamics and operation.
PN	Honeywell RSC-24 sensor		The ambient atmospheric pressure on the helicopter is recorded by the RSC-24 sensor, providing an analog signal. This signal is first filtered to remove electromagnetic interference and noise, then undergoes analog-to-digital conversion and calibration to ensure accuracy. The data are then analyzed for systematic error correction and consistency, ensuring reliable use in mathematical models for evaluating helicopter dynamics and performance.
nFT	D-1M sensor		The free turbine rotor speed, recorded by the D-1M sensor, is presented as an analog signal. This signal undergoes initial filtering to remove low-frequency noise and drift, followed by analog-to-digital conversion. The data are then further processed, including interpolation and smoothing, to ensure their accuracy and reliability for use in the engine’s dynamic model.
$T_G^{*}$	14 dual thermocouples T-101		Data on the gas temperature before the compressor turbine, recorded by T-101 dual thermocouples, are fed into the data acquisition system in analog form. These signals undergo initial filtering to remove high-frequency noise, followed by analog-to-digital conversion and calibration. The data are then analyzed to detect deviations from normal behavior, allowing for the engine’s operational model validation and adjustment.

[43,44].

The training sample is formed from the helicopter TE thermogas-dynamic parameters values according to the presented oscillogram (Figure 2). The main criterion for the time series identifying “reference” sections when forming the training sample is the engine control lever position. In the future, from the thermogas-dynamic parameters general set presented on the digital oscillogram obtained in the flight mode of a twin-engine helicopter (for example, the Mi-8MTV), only those data that relate to the first engine are considered. The preliminary processing training sample data results (the homogeneity [45,46] and representativeness [47,48] assessment) are given in [26,38,39,41,42].

In the helicopter TE ACS optimization process, the reconfiguration coefficient k defines the system’s ability to adapt to changing conditions. Values n_FT, P_N, T_N and α^∗ serve as target parameters where system functions F and h remain stable, ensuring steady operation across different modes. These parameters are crucial for maintaining desired characteristics and minimizing deviations from a stable state during control.

Figure 3 shows the target functions (37) value distribution surfaces at T_N = const, P_N = const (a), T_N = const, α = const (b), P_N = const, α = const (c).

The need for the goal function values to visualize the distribution surfaces for various fixed parameters (T_N = const, P_N = const; T_N = const, α = const; P_N = const, α = const) is due to the need for visual analysis of goal function behavior depending on changes in these parameters in flight conditions. The visualization reveals how target functions respond to changes under fixed conditions and demonstrates that $T_G^{*}$ and n_FT values, as critical parameters, cannot remain constant or fluctuate under different flight conditions.

It is noted that the $T_G^{*}$ and n_FT values cannot be constant and have no deviations under any flight conditions. The values T_N = const and P_N = const can be constant under constant flight conditions (for example, horizontal flight at a constant altitude in a certain area). The results obtained indicate that the goal function (37) does not exceed 0.802 at T_N = const, P_N = const. However, flight conditions are not always constant.

Of particular interest is the helicopter TE ACS (36) behavior research at T_N ≠ const, P_N ≠ const. Then the helicopter’s reconfigured-ACS (36) multicriteria optimization task consists of the parameter values T_N, $T_G^{*}$, P_N, n_FT and α optimal Pareto set [49] finding, at which the objective function (37) under constraints (40) will be minimal. Based on previous studies in the field of helicopter TE working process parameters optimization [49], the use of the NSGA-II algorithm is proposed to find the optimal Pareto set.

The study of the helicopter TE ACS’s behavior under varying parameters T_N and P_N (T_N ≠ const, P_N ≠ const) revealed critical insights into optimizing the system’s performance. Specifically, it demonstrated that the goal function exhibits significant sensitivity to changes in these parameters, necessitating a multicriteria optimization approach. The application of the NSGA-II algorithm was proposed to identify the optimal Pareto set for parameters T_N, $T_G^{*}$, P_N, n_FT, and α, ensuring minimal objective function values under specified constraints. The results underscore the complexity of maintaining stable control across fluctuating flight conditions, highlighting the importance of adaptive optimization techniques in enhancing system reliability.

The proposed modified NSGA-II algorithm is implemented in the proposed neural network by using specialized blocks to evaluate and sort solutions based on their dominance and diversity. The network performs partial derivative calculations and Kuhn–Tucker conditions to enforce constraints and uses ranking and edge distance mechanisms to maintain solution diversity. These components enable the neural network to efficiently find and select the optimal Pareto set in multicriteria optimization.

Table 5. An example of the found optimal parameters T_N, $T_G^{*}$, P_N, n_FT and α.

Number	TN	${\mathit{T}}_{\mathit{G}}^{\mathit{*}}$	PN	nFT	α
Pareto set 1	1.000	0.938	1.000	0.983	1.000
Pareto set 2	0.997	0.935	0.997	0.987	0.998
Pareto set 3	0.998	0.935	0.998	0.987	0.999
Pareto set 4	1.000	0.937	1.000	0.990	1.000
Pareto set 5	0.997	0.932	0.997	0.976	0.998
Pareto set 6	0.999	0.936	0.999	0.988	0.999
Pareto set 7	0.999	0.936	0.999	0.988	0.999
Pareto set 8	0.997	0.934	0.997	0.979	0.998
Pareto set 9	0.998	0.937	0.998	0.992	0.998
Pareto set 10	1.000	0.939	1.000	0.997	1.000

shows an example of the found optimal parameters T_N, $T_G^{*}$, P_N, n_FT and α, at which the objective function (37) under restrictions (40) is minimal.

As can be seen from Table 5, values T_N ∈ [0.997…1] ≈ const, P_N ∈ [0.997…1] ≈ const, $T_G^{*}$ ∈ [0.932,0.939], n_FT ∈ [0.979…0.997], α ∈ [0.979…0.997], which emphasize the possibility of finding the helicopter TE’s reconfigured-ACS optimal parameters in horizontal flight conditions. The results obtained emphasize that in horizontal flight conditions, it is possible to achieve the helicopter TE’s reconfigured ACS with the optimal parameters. The key parameter stability confirms the proposed model’s effectiveness and its potential for practical application.

The parameters listed in Table 4 are indicative of an optimized configuration for the helicopter TE’s reconfigured ACS under horizontal flight conditions. While these parameters do represent an effective configuration that meets the criteria for minimizing the objective function and adhering to the constraints, they are not necessarily the only optimal solution. The use of the Pareto optimization in the selection process ensured that these parameters provide a balance between competing objectives such as system stability and performance. Therefore, these parameters reflect one of potentially many viable solutions that satisfy the optimization criteria. The observed stability of these parameters highlights the proposed model’s effectiveness and its practical applicability, but it is important to acknowledge that other parameter combinations could also be optimized within the constraints and the system requirements.

The helicopter TE ACS’s parameters optimization process for horizontal flight considered trade-offs between competing objectives, such as system stability and effectiveness. By using Pareto optimization, a balanced set of parameters was found, allowing for the objective function L(T_N, $T_G^{*}$, P_N, n_FT, α) minimization while adhering to the given constraints (40). During the formation of the Pareto set, each parameter combination represented a solution optimizing both system stability and effectiveness. The parameters listed in Table 4 fall within narrow ranges, indicating their values’ stability in horizontal flight and their equivalence, as they are recorded on board the helicopter and used for the helicopter TE thermodynamic calculation according to the universal model [50].

The chosen parameters do indeed represent an optimal configuration within the given constraints and aims, as they were obtained using the Pareto optimization approach, which enables finding the best solutions in the competing objectives context. However, it is important to note that they are not necessarily the only optimal solution but rather one among many possible solutions that satisfy the criteria for minimizing the objective function L(T_N, $T_G^{*}$, P_N, n_FT, α). These parameters represent a compromise solution, achieved by considering various factors such as system stability and effectiveness. Thus, while they represent an effective solution for the current conditions and constraints, other parameter combinations may also meet the requirements and the task constraints, offering alternative optimization options.

The adequacy of the proposed helicopter reconfigured-ACS mathematical model is confirmed in the results using a mean square error (MSE) metric for results, as shown in Figure 4.

Figure 4 shows diagrams of changes in the error in finding the optimal parameters T_N, $T_G^{*}$, P_N, n_FT and α, at which the objective function (37) under restrictions (40) is minimal. From Figure 4, it follows that the calculation error is in the range from –0.4 to +0.4% (blue diagrams in Figure 4). Under white noise conditions, specified as a random error with zero mathematical expectation and standard deviation σ_i = 0.025, the calculation error increases by 0.38% and ranges from –0.78 to +0.78% (red diagrams on Figure 4).

In analyzing these results, it becomes clear that the increase in calculation error under noise conditions may have a cumulative effect on the overall system performance, particularly during prolonged operations or in environments where external disturbances are persistent. This cumulative effect could lead to gradual degradation in system accuracy, potentially affecting the critical function reliability within the helicopter’s control system. Therefore, addressing these issues is crucial to ensure that the ACS can maintain consistent performance over time, even in less-than-ideal operational scenarios.

In discussing these results, it is evident that while the model demonstrates robust performance under controlled conditions, its accuracy diminishes in the presence of stochastic noise. This raises important considerations for the helicopter TE ACS’s practical implementation in dynamic flight environments, where unpredictable factors may introduce variability into the system. The observed increase in error under noisy conditions suggests a need for enhanced robustness in the optimization algorithms to handle real-world complexities. Additionally, incorporating real-time error correction mechanisms and adaptive adjustment strategies could further improve the model’s resilience and ensure more reliable performance in varied operational scenarios.

Regression analysis is carried out, which shows the proposed mathematical model’s prediction accuracy depending on various input parameters (T_N, $T_G^{*}$, P_N, n_FT and α), and also helps to evaluate how well the model explains the data variability. This is critical for verifying that the model correctly reflects the dynamics and characteristics of the system, and can highlight possible inconsistencies or areas for improvement, thereby improving the reliability and accuracy of the control system in real-world operating conditions. According to Table 4, the regression model has the following form:

$$f_{adapted}\left(x\right)=0.936·n_{TC}+0.972·n_{FT}+0.964·T_G^{*}+0.995·P_N+0.999·\alpha .$$

(48)

The determination coefficient R² = 0.986 means that the variability in the model output variables of 98.6% is explained by the selected independent variables and model parameters. This indicates a very high level of accuracy in describing and predicting the system behavior in the context of the adequacy of the helicopter TE’s reconfigured-ACS mathematical model. Such a high determination coefficient shows that the model effectively captures the system’s main patterns and dynamics, confirming its adequacy and reliability for solving control and optimization tasks. However, it is important to consider other aspects, such as possible model retraining and checking its stability with changing input data.

Also, to assess the computational experiment’s effectiveness, a k-fold cross-validation procedure was conducted. For this, k = 5 equal partitions (folds) of 51 elements each were randomly selected from the training sample. The following metric values were obtained: Metric(k₁) = 0.0036, Metric(k₂) = 0.0038, Metric(k₃) = 0.0037, Metric(k₄) = 0.0039, Metric(k₅) = 0.0038. Consequently, the average metric was calculated as 0.00376, which is 3.88% higher than MSE_min = 0.00391. This indicates that the metric (MSE) average value after cross-validation is close to the MSE_min obtained from the first fold (in this case, fold k₁). The 4.65% difference reflects how much the average metric deviates from the best individual result achieved during the cross-validation iterations. In this case, ${\mathrm{M}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}}_i=\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{e}\left(M^{\left(i\right)},D_{val}^{\left(i\right)}\right),$ $\mathrm{A}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e} \mathrm{M}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}={\displaystyle \frac{1}{k}}·\sum _{i=1}^k{\mathrm{M}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}}_i$, where M⁽ⁱ⁾ is a model trained on $D_{val}^{\left(i\right)}=D_i$, D_i is the training set part (fold).

To assess the accuracy of the solutions obtained from the multicriteria optimization tasks, the IGD (inverted generational distance) measure is used, which shows the difference degree between the parameters T_N, $T_G^{*}$, P_N, n_FT and α original and found values [51,52]. For non-stationary multicriteria optimization tasks, the averaged MIGD measure is used [53] as follows:

$$MIGD={\displaystyle \frac{1}{T}}·\sum _{t=1}^{T}IGD\left(X_t^{*},X_t\right)={\displaystyle \frac{1}{T}}·\sum _{t=1}^T\sum _{i=1}^{n_{X_t}}{\displaystyle \frac{d_t^i}{n_{X_t}}},$$

(49)

where $n_{X_t}=\left|X_t\right|$, $d_t^i$ is the Euclidean distance between the i-th element X_t and the element closest to it $X_t^{*}$, T is the task iteration number, X_t makes parameters T_N, $T_G^{*}$, P_N, n_FT and α sense.

The hypervolume measure is also used, which shows the front distance degree from some worst point (reference point), which is set manually [51,54]. For nonstationary multicriteria optimization tasks, the averaged MHV measure is also used:

$$MHV={\displaystyle \frac{1}{T}}·\sum _{t=1}^T{HV}_t\left(X_t^{*}\right),$$

(50)

where HV_t is the hypervolume operator. In [51,55], it is proposed to set the worst point as (L₁ + 0.5, L₂ + 0.5, …, L_m + 0.5), where L_j is the j-th objective function maximum value for the parameters X_t initial values at the moment time t; m is the objective function number.

When conducting numerical experiments, the following test task parameter values were used: the intensity of the change was 10; the task number of iterations was T = 50; the task variable number X_t = 5. Numerical experiments were carried out using two different change rates: 10 (fast changes) and 50 (slow changes) to test whether no restart gives a better result for slower changes in the task. Each test task was solved 40 times, and the results were averaged.

Table 6. The MIGD metric resulting values depending on the X_t parameters’ initial values with different initialization for slow changes.

Number	No Restart	Restart 50%	With Restart
Pareto set 1	0.737	0.738	0.859
Pareto set 2	0.719	0.736	0.838
Pareto set 3	0.865	0.868	0.944
Pareto set 4	0.833	0.853	0.927
Pareto set 5	0.818	0.821	0.899
Pareto set 6	0.701	0.703	0.777
Pareto set 7	0.685	0.687	0.764
Pareto set 8	0.709	0.717	0.791
Pareto set 9	0.735	0.737	0.840
Pareto set 10	0.799	0.805	0.863

and

Table 7. The MIGD metric resulting values depending on the X_t parameters’ initial values with different initialization for fast changes.

Number	No Restart	Restart 50%	With Restart
Pareto set 1	0.884	0.886	1.031
Pareto set 2	0.863	0.883	1.006
Pareto set 3	1.038	1.042	1.133
Pareto set 4	1.000	1.024	1.112
Pareto set 5	0.982	0.986	1.079
Pareto set 6	0.841	0.844	0.932
Pareto set 7	0.822	0.824	0.917
Pareto set 8	0.851	0.860	0.949
Pareto set 9	0.882	0.884	1.008
Pareto set 10	0.959	0.966	1.036

present the MIGD metric average values for each test task at slow and fast change rates using the proposed modified NSGA-II algorithm. Results are shown for three scenarios: using a restart (the X_t parameters random initialization), without a restart (initializing X_t parameters with values from the previous time point), and with a 50% restart (the P_N and T_N parameters random initialization and the remaining parameters a random selection from their previous values).

Table 8. The MHV metric resulting values depending on the X_t parameters’ initial values with different initialization for slow changes.

Number	No Restart	Restart 50%	With Restart
Pareto set 1	1.564	1.565	1.819
Pareto set 2	1.535	1.547	1.803
Pareto set 3	2.072	2.073	2.788
Pareto set 4	1.998	2.004	2.735
Pareto set 5	1.765	1.766	2.336
Pareto set 6	1.423	1.425	1.693
Pareto set 7	1.373	1.375	1.629
Pareto set 8	1.406	1.412	1.685
Pareto set 9	1.589	1.590	1.746
Pareto set 10	1.735	1.743	2.252

and

Table 9. The MHV metric resulting values depending on the X_t parameters’ initial values with different initialization for fast changes.

Number	No Restart	Restart 50%	With Restart
Pareto set 1	2.190	2.191	2.547
Pareto set 2	2.149	2.166	2.524
Pareto set 3	2.901	2.902	3.903
Pareto set 4	2.797	2.806	3.829
Pareto set 5	2.471	2.472	3.270
Pareto set 6	1.992	1.995	2.370
Pareto set 7	1.922	1.928	2.281
Pareto set 8	1.968	1.977	2.359
Pareto set 9	2.225	2.226	2.444
Pareto set 10	2.429	2.440	3.378

shows the MHV metric average values. Results were analyzed using the Mann–Whitney statistical test. The best statistically significant metric values, according to the algorithm, are highlighted in green. If multiple bars are highlighted in the same color, the differences between them are not statistically significant.

From the results presented in Table 6, Table 7, Table 8 and Table 9, it can be observed that in the case of slow changes when solving most test tasks, using the parameters T_N, $T_G^{*}$, P_N, n_FT and α values, obtained in the previous step, is the most effective. However, in the case of rapid change, you can see an increase in the optimal Pareto set number, at which the restart (either full or 50%) use is most effective. This is because, with fast changes, the task state at the previous step differs from the state at the current step to a greater extent in comparison with slow changes.

When analyzing rapid changes in Pareto-optimal sets No. 1 and No. 9, it was found that in the restart absence, the error when changes occur increased significantly compared to the restart use. In Pareto-optimal set No. 9, with slow changes in the NSGA-II algorithm initial iterations, a low error value was observed without restart, but the error increased at the NSGA-II algorithm’s subsequent steps. These results indicate the need for different approaches depending on the change degree within a single task.

Metrics such as MIGD and MHV, as shown in Table 6, Table 7, Table 8 and Table 9, can vary over time or under changing operating conditions due to dynamic external disturbances. For instance, in the modified NSGA-II algorithm, the MIGD metric average values might fluctuate significantly when the change rates are altered from slow to fast, reflecting how the algorithm’s performance adapts to different rates of problem evolution. This variation can be attributed to the dynamic nature of external disturbances that may cause the metric to deviate from a stable value. Despite these fluctuations, the metric is often reported as a single average value without accounting for these temporal changes. For example, during a test, if the algorithm’s MIGD metric shows improved performance with a 50% restart strategy compared to a complete restart, this suggests that the metric’s sensitivity to initialization conditions can influence its effectiveness under different operational scenarios. Such analysis underscores the importance of considering dynamic and time-varying factors when evaluating metric performance.

Future research should focus on analyzing how temporal dynamics and external disturbances impact the MIGD and MHV metric values over time. This may involve employing time-series analysis techniques and adaptive algorithms to better understand and manage these variations, resulting in more accurate assessments of metric performance and enhanced stability of the algorithm under changing conditions.

Thus, this article mathematically substantiates the helicopter TE ACS reconfiguration in the event of a single standard sensor failure (using, the example, the n_TC sensor). As an example, the helicopter TE reconfigured ACS (36) is considered, the parameters of which were determined by solving a multicriteria optimization task. The direction of further research lies in obtaining the helicopter TE reconfigured ACS (36) transfer function numbers, to analyze the frequency and transient characteristics to confirm its stability in particular tasks.

In the helicopter TE functional safety context, the proposed reconfigured ACS mathematical model plays a key role in ensuring the reliability and stability of the system. It can integrate and function across several critical aspects in the following ways:

• The proposed reconfigured ACS mathematical model serves as a backup system that can take over functions in the event of a single sensor failure. This includes not only switching to redundant components (calculating the optimal Pareto set), but also the ability for self-diagnosis and restoration. The model is automatically detecting likely failures and problems in the system (when one value of the engine parameters is zero), as well as taking action to eliminate them and restore normal engine operation (calculate the necessary engine parameters in case of sensor failure).
• The stability metric (external disturbances metric, EDM) for impacts, such as sensor failure, for the proposed helicopter TE reconfigured ACS mathematical model is presented as the ability of the model to maintain adequate and reliable engine operating characteristics when the recorded engine parameters change. EDM are defined as:

$$EDM={\displaystyle \frac{1}{N}}·\sum _{k=1}^K\left|{\displaystyle \frac{P_k^{dist}-P_k^{nom}}{P_k^{nom}}}\right|·100\%,$$

(51)

where $P_k^{dist}$ is the parameter value under disturbance, $P_k^{nom}$ is the parameter nominal value, K is the parameter number considered in the analysis [56,57].

Figure 5 shows a diagram of the EDM resilience metric determined when one, two, three, four or five sensors fail. As can be seen from Figure 5, if one, two or three sensors fail, the helicopter TE’s reconfigured ACS is stable, since the value of the EDM metric changes slightly, within 4.75%. If four sensors fail, the EDM metric value changes by 7.2%, and if all five sensors fail, by 9.3%.

This indicates that the system maintains fairly good stability and functionality even when the sensors fail in significant numbers. However, the observed increase in the metric for four and five sensor failures indicates a gradual decrease in system resilience, which may require additional measures to improve its ability to recover and is redundant under conditions of more significant failures. If all five sensors fail, the system uses data interpolation and extrapolation algorithms based on previous measurements and models to estimate and reconstruct the missing information, which have limited accuracy and may not fully compensate for the data loss.

3.

An important aspect is also the model’s compatibility with other helicopter systems, including control and navigation systems. The proposed reconfigured ACS mathematical model can effectively interact with these systems to ensure synchronous operation and coordination by continuously recording engine parameters even if one of the sensors fails. This interoperability allows redundant functions to be integrated into the overall helicopter control context, ensuring that if a single component fails, the system can seamlessly switch to redundant functions and continue reliable operation without disrupting interaction with control and navigation systems.

The presented outcomes demonstrate a significant advancement in the helicopter TE’s ACS structural reconfiguration under sensor failure conditions. However, the generalizability of these results may be constrained by the specific engine models and operational conditions studied. Future research should focus on validating the proposed mathematical model across various helicopter engine types and failure scenarios, as well as exploring potential limitations, such as the model’s robustness in real-time applications and under different environmental conditions. Additionally, further work is required to refine the optimization methods for more complex fault scenarios and to ensure the model’s adaptability to evolving operational requirements. In this context, further research prospects include improving system integration and coordination algorithms to increase its ability to adapt and restore from failures, as well as developing more accurate and reliable methods for monitoring and control under conditions of external disturbances and system failures. Further research will also focus on the works [58,59,60].

4. Conclusions

The article seeks to substantiate the mathematical reconfiguration of a helicopter turboshaft engine’s automatic control system in the event of the failure of one standard sensor. The results obtained in this article ensure the engine’s stable operation under fault conditions, minimizing the impact on control and performance. To achieve the results obtained in the article:

• It is substantiated that the methods for the development and implementation of the automatic control system’s structural reconfiguration in the event of sensor parametric failures are relevant due to the need to increase the overall reliability and safety of the helicopter turboshaft engine’s operation.
• The theorem “On the helicopter turboshaft engines automatic control system structural reconfiguration” was formulated and proven, which made it possible to establish the helicopter turboshaft engine’s reconfigured automatic control system’s mathematical form, presented in the form of the thermogas-dynamic parameters nonlinear dependencies system.
• A method is proposed for the helicopter turboshaft engine’s thermogas-dynamic parameter values to determine at which points the reconfigured automatic control system is stable. The proposed method application, which combines numerical optimization methods, made it possible to obtain many optimal thermogas-dynamic parameters that ensure the stability of the reconfigured automatic control system.
• An assessment of the solution of the multicriteria optimization task in finding optimal thermogas-dynamic parameter accuracy was made using the MIGD and MHV metrics under slow and fast change rate conditions. It was experimentally determined that when solving a multicriteria optimization task with slow changes, the best results are achieved using the thermogas-dynamic parameters obtained at the previous step; while with fast changes, methods using restart (full or partial) are optimal, due to significant differences between the task states at the previous and current steps.
• The adequacy of the proposed mathematical model for a helicopter turboshaft engine’s reconfigured automatic control system was confirmed using the mean square error (the solution to the helicopter turboshaft engines parameters multicriteria optimization task did not exceed 0.4%, and in white noise conditions was 0.77%), regression analysis (the determination coefficient was 0.986) and cross-validation (the resulting metric differed from the maximum mean square error by 3.88%).