## 3 Vibration-Based Monitoring Technique: Development Steps  
A signal processing workflow was developed to improve the monitoring technique.  
To capture the full system vibrational behavior, the magnitude of the 3D space vector was computed by combining the signal components along the three Cartesian axes, and several time-features (statistical indexes) were calculated on the previously mentioned signal. The calculated statistical indexes were mean, median, standard deviation, variance, asymmetry, kurtosis, RMS, crest factor, quadratic oscillation index, harmonic mean, range, amplitude between consecutive points, average between consecutive points, shape factor, impulse factor, clarity factor, vibration index, Pearson correlation index, mode, absolute deviation from mean, absolute deviation from median, and synchrony index. These indexes can describe and synthesize a HEPS' vibrational behavior.  
The HEPS' vibrational behavior was similar in the three operating modes considering a case in which the torque delivered or absorbed by the EM was close to zero. The features were filtered using a lowpass Finite Impulse Response (FIR) filter to reduce the noise [58]. The indexes were divided into three datasets (training, validation, and testing dataset) using the Holdout method [59]. The mean and the standard deviation were calculated only on the training set for data standardization. The final datasets used as input for the network were reduced with the PCA technique [60; 61]. The accuracy of the ML prediction was improved by employing the FIR filter and the PCA. To prevent the unbalance of the model towards a single class, the appearance probability of each class was set equal for all before training the model. The workflow process is summarized in Figure 3.  
Figure 3: Training, validation, and testing dataset partition workflow.  
A Feedforward Neural Network (FNN) [62, 63] was developed to predict the HEPS' operating mode. To minimize the model loss function, the best hyperparameter combination was selected using Bayesian Optimization performed by the Expected Improvement (EI) function [64, 65] as the method to model the objective function through a Gaussian Process (GP). Finally, the model accuracy was evaluated.  
The workflow process for the FNN training, optimization, and testing phases is shown in Figure 4.  
The vibrational signals sampled under operating conditions outside the limits of Figure 2 were processed similarly to the validation and test datasets. This dataset was only used for tests and not to train the FNN: the ML model was only trained and optimized on the dataset obtained from the acquisition in the \(\pm\)5% limits of the propeller torque curve.  
After assigning a 0 value for incorrect model prediction and a 1 value for correct classification, a logical vector was obtained. For every second, an RMS value [66] was calculated to monitor the equilibrium point position in the angular speed-torque plane: if the RMS tended to 1, then the equilibrium point was inside the \(\pm\)5% limits of the propeller torque curve, while if the RMS decreased and tended to 0, then the equilibrium point was outside the limits. This assumption was validated in experimental tests.