## 3 Case Study

Table 2: The developed algorithm for the proposed neural network training. of the Mi-8 MTV helicopter power plant [39; 40], was chosen as the research object for the computational experiment. The main diagnostic parameter recorded by the standard onboard sensor D-2M [32; 41] is the gas-generator rotor speed, \(n_{\mathrm{TC}}\). The \(n_{\mathrm{TC}}\) values were obtained exclusively from the Mi-8 MTV helicopter flight test data. Data were collected for 256 s under actual flight conditions at the nominal engine operating mode with a sampling interval of 1 s. The base frequency value, \(n_{\mathrm{TC}\mathit{ruj}}\), was set equal to 15,900 rpm [31; 40]. The data were provided to the authors' team upon an official request to the Ministry of Internal Affairs of Ukraine within the framework of the project \"Theoretical and Applied Aspects of Aviation Sphere Development\", registered under number 0123U104884. The changes in the \(n_{\mathrm{TC}}\) frequency recorded under the operating conditions of the TV3-117 engine demonstrate the time series complexity (Figure 4), and the constructed diagrams emphasize the importance of the current parameter values and their accumulation in the model's memory [42; 43]. In accordance with the recommendations provided in [32; 33; 34; 35; 36; 37; 38], the 256 \(n_{\mathrm{TC}}\) values presented in Figure 4 were selected for analysis. The time sampling and adaptive quantization with a dynamic range applied to the reconstructed \(n_{\mathrm{TC}}\) signal [33] resulted in its discretized form (points in Figure 4). The generated diagrams (Figure 4) play an essential role in the formation of the training dataset. According to the Nyquist theorem, the sampling frequency should be at least twice the maximum frequency to reconstruct the signal accurately. With 256 measurements over 256 s, the sampling frequency is 1 measurement per second, which is optimal for tracking slowly changing parameters and provides sufficient detail without an excessive amount of data [32; 33; 34; 35; 36; 37; 38]. Thus, the choice of 256 measurements is a compromise between the sampling frequency, data amount, and computing resources, which allows for achieving a balance between data processing accuracy and efficiency. To form a training dataset consisting of the system error, \(e(t)\), and the reference action, \(r(t)\), values, \(z\)-normalization of the \(n_{\mathrm{TC}}\) values was carried out [44], with the help of which each \(n_{\mathrm{TC}}\) value was transformed in such a way that it was expressed in standard deviation units from the mean. This allows us to compile the \(n_{TC}\) values into a form with a mean  
Figure 4: The gas-generator rotor speed parameter recorded onboard the helicopter during the 256 s research interval. value of 0 and a standard deviation of 1, which is essential for improving the developed neural network training algorithm's (Table 2) convergence and for correct comparison of the data with different scales. The \(n_{TC}\) values' \(z\)-normalization was carried out according to the following expression:  
\[z(n_{TC})_{i}=\frac{n_{TC}^{(i)}-\frac{1}{N}\cdot\sum_{i=1}^{N}n_{TC}^{(i)}}{ \sqrt{\frac{1}{N}\cdot\sum_{i=1}^{N}\left(n_{TC}^{(i)}-\frac{1}{N}\cdot\sum_{i= 1}^{N}n_{TC}^{(i)}\right)^{2}}}, \tag{23}\]  
where \(N=256\). To determine the system error, \(\epsilon(t)\), for \(n_{TC}\) values normalized using \(z\)-normalization, the mean standard error was applied, calculated as follows:  
\[\epsilon^{(i)}=\frac{\sigma_{z}}{\sqrt{N}}, \tag{24}\]  
where \(\sigma_{z}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(z(n_{TC})_{i}-\overline{z(n_ {TC})}\right)^{2}}\) is the standard deviation of the \(z\)-normalized \(n_{TC}\) values, and \(\overline{z(n_{TC})}\) is the mean of the z-normalized \(n_{TC}\) values, typically \(\overline{z(n_{TC})}=0\) if the data are normalized correctly. The reference action is the \(n_{TC}\) value in the helicopter TE's current operating mode [42].