## 4 Stability Analysis of the Powertrain
### Influence of the Gas Turbine Control Dynamic on the Stability of the Powertrain  
After having considered the stability analysis with respect to the generator, the issue is to analyze the influence of the gas turbine speed control dynamic on the system stability.  
The gas turbine speed control dynamic, originally with a bandwidth set to 1 Hz, varies here between the following values (indicated in Hz): {0.9; 1; 1.25; 1.5; 1.75; 2}.  
As in the previous analysis, the following figures show the results of the different means of study depending on the generator speed range but for a bandwidth of 0.9 Hz: Figure 10 presents the new root locus; time domain simulation is shown on Figure 11 and experimental performances on Figure 12.  
Figure 8: Time simulation with respect to variable stages of the generator speed with a bandwidth of 1 Hz for the generator speed control.  
For the other control dynamics, the results are summarized in Table 2. In that table, the speed range (\(\Delta\)) with unstable behavior is displayed for the left column (related to the root locus analysis).  
These different results firstly confirm the correct concordance between the different means of study of the AC electric propulsion architecture, except for the case of a bandwidth equal to 1.75 Hz, for which slight differences appear between the analysis tools. Indeed, for such a control dynamic, the system is \"quasi stable\" with slight oscillations appearing in experiments and being also predicted from the root locus. Contrarily these oscillations are not visible in the time simulation for this control dynamic. This can be explained by the fact that the unstable poles are very close to the ordinate axis, which characterizes the limit of stability.  
Figure 10: Small-signal model-based root locus with respect to the generator speed (blue crosses) with a bandwidth of 0.9 Hz for the gas turbine speed control.  
Figure 9: Experimental test with respect to variable stages of the generator speed with a bandwidth of 1 Hz for the generator speed control.  
Figure 11: Time simulation with respect to variable stages of the generator speed with a bandwidth of 0.9 Hz for the gas turbine speed control.  
Figure 12: Experimental test with respect to variable stages of the generator speed with a bandwidth of 0.9 Hz for the gas turbine speed control.  
These results also reveal a huge trend related to the tuning of the gas turbine control dynamic: \"the higher the speed control dynamic the more stable the powertrain\".  
However, this conclusion should be considered carefully, being only valid for low values of the control bandwidth. Indeed, this trend is reversed at high frequency of the control bandwidth, as it can be seen in Figure 13.  
For a low gas turbine dynamic value, increasing its value tends to bring the mechanical poles back into the stable domain of the pole location and thus stabilize the system, as seen with the previous results. However, this is no longer the case at the highest bandwidth value, where an increase in the bandwidth value brings the mechanical poles back into the unstable domain of the pole location. Thus, a notion of optimal bandwidth value emerges, allowing to consider a setting offering the greatest possible stability to the AC electric propulsion architecture.