## 2 HEPS Architecture and Testing Protocol  
The proposed HEPS architecture is a parallel type (Figure 1) for aerospace applications consisting of an ICE type CMD 22 and an EM (Electric Machine) type EMRAX 268 Low Voltage (EMMAX, Kamnik, Slovenia), paired with a DC/AC Inverter type GVI-H650 (Parker, Mayfield Heights, Ohio, USA) and a Rectifier type ITECH 500-90 (ITECH, Sassuolo, Italy). Finally, the propeller torque was simulated using a brake type SCHENCK 260 (SCHENCK, Darmstadt, Germany) managed by an APICom MP2030 (APICom, Cento, Italy) bench control system.  
Figure 1: Hybrid electric architecture for aircraft application.  
The useful drive torque transferred to the propeller \(P_{prop}\) is given by the sum of ICE torque \(P_{ICE}\) and the EM torque \(P_{EM}\):  
\[P_{prop}=P_{ICE}+P_{EM} \tag{1}\]  
where the torque \(P_{EM}\) is positive if the Electric Machine acts as a motor and thus adds useful axle torque or negative if it subtracts a portion of useful axle torque to generate the electrical energy needed to recharge the batteries. During take-off or climb, the EM operates as a motor, while during descent or landing, the EM functions as a generator. During flight phases such as descent and landing, when low torque is required from the propeller as the aircraft is in the glide phase, the excess torque available from the ICE heat engine could be used to generate electric power and recharge the battery via the generator mode of the EM machine [54; 55]. The hybrid configuration in Figure 1 is a parallel type, and the mechanical coupling between the ICE and EM engines is achieved through one-way bearing interposition. This device allows the two engines to be coupled in the propeller positive rotation direction, thereby summing the ICE and EM torques in the hybrid operating mode. More details on the entire experimental setup are provided in [56].  
Tests were conducted in the following configurations:  
* ICE: only the ICE contributes to balancing the propeller resistance torque (\(P_{EM}=0\) and \(P_{ICE}=P_{prop}\));
* HYBM: the EM operates as a motor, contributing to balancing the propeller resistance torque (\(P_{EM}>0\) and \(P_{ICE}=P_{prop}-P_{EM}\));
* HYBG: the EM operates as a generator to recharge the batteries, and the ICE must balance both the propeller and the EM torque (\(P_{EM}<0\) e \(P_{ICE}=P_{prop}+P_{EM}\))  
The tests were performed by alternating the three different HEPS operating modes, ensuring that the equilibrium point was located inside the \(\pm 5\%\) limits of propeller torque. Further tests were carried out: the equilibrium point was located outside the limits. These tests were conducted to validate the proposed monitoring technique. Because the HEPS was designed for aircraft that do not fly at considerable altitudes, it is possible to assume that the resistance torque curve is always inside the \(\pm 5\%\) limits of the propeller torque curve. For this reason, any deviation of the equilibrium point from the limits is attributed to a change in air properties [57]. During the tests, it was possible to change the equilibrium point because the propeller was simulated by the brake. Figure 2 shows the propeller torque curve and its \(\pm 5\%\) limits.  
Figure 2: Propeller torque curve.  
The propeller angular speed was set between 1000 rpm and 2500 rpm, while the total torque was set between 50 and 300 Nm, and the EM torque ranged from \(-60\) to \(+60\) Nm. Vibrational signals were sampled using a triaxial accelerometer, Bruel and Kjaer 4321 - type [56], at a frequency of 1 kHz, while the angular speed at the brake and the torques (of the ICE, EM, and brake) were recorded at a frequency of 10 Hz. The choice of this sampling frequency was due to the future goal of testing the proposed algorithm in real time.