## 3 Modelling, Analysis Tools and Experimental Support for Stability Analysis
### Nonlinear Analytic State Model
#### 3.1.1 State Model Derivation
Thus, the speed control of the gas turbine--generator association can be added to complete the model. In our study, as shown on Figure 4, an integral proportional (IP) controller structure was retained:  
This control structure sets an additional state equation which links the gas turbine torque \(T_{TAG}\) and the reference generator speed \(\omega_{S_{ref}}\), which becomes the new system input:  
\[ \begin{array}{c}\frac{dT_{TAG}}{dt}=\frac{k_{IP}}{I_{IP}}*\frac{\omega_{S_{ ref}}}{F_{g}}-\frac{k_{IP}}{I_{IP}}*\frac{\omega_{g}}{F_{g}}-\frac{k_{IP}}{P_{g}}* \frac{\omega_{g}}{P_{g}}-\frac{k_{IP}}{P_{g}}*\frac{\omega_{g}}{dt}\\ =\frac{k_{IP}}{I_{IP}}*\frac{\omega_{S_{ref}}}{p_{g}}-\frac{k_{IP}}{I_{IP}}* \frac{\omega_{g}}{P_{g}}-\frac{k_{IP}}{P_{g}}*\frac{\omega_{g}}{P_{g}}*\frac{ \omega_{g}}{P_{g}}\\ \end{array}  \tag{10}\]  
where: \(\omega_{S_{ref}}=p_{g}*\Omega_{S_{ref}}\)  
Finally, a sixth order analytic state model can be derived, where the generator reference speed and the propeller load torque become the new inputs:  
\[ \begin{array}{c}\frac{d}{dt}\\ \omega_{m}\\ \omega_{g}\\ \omega_{g}\\ T_{TAG}\end{array} \left( \begin{array}{cccc}\begin{array}{cccc}l_{d}\\ I_{q}\\ \delta\\ \omega_{m}\\ \omega_{g}\end{array}\right)=\left(\begin{array}{cccc}0&0\\ 0&0\\ 0&0\\ 0&0&-1&1\\ \frac{3p_{m}^{2}}{2I_{\omega_{m}}}\phi_{\omega_{m}}\sin(\delta)\frac{3p_{m}^{2 }}{2I_{\omega_{m}}}\phi_{\omega_{m}}\cos(\delta)0&\frac{-f_{m}}{I_{\omega_{m}} }&0\\ 0&-\frac{3p_{g}^{2}}{2I_{\omega_{g}}}\phi_{\omega_{g}}&0&\frac{-f_{g}}{I_{ \omega_{g}}}&\frac{p_{g}}{I_{\omega_{g}}}\\ 0&0&-1&1\\ \frac{3p_{g}^{2}}{2I_{\omega_{g}}}\phi_{\omega_{g}}\sin(\delta)\frac{3p_{m}^{2 }}{2I_{\omega_{g}}}\phi_{\omega_{g}}\cos(\delta)0&\frac{-f_{g}}{I_{\omega_{g}} }&0\\ 0&-\frac{3p_{g}^{2}}{2I_{\omega_{g}}}\phi_{\omega_{g}}&0&\frac{-f_{g}}{I_{ \omega_{g}}}&\frac{p_{g}}{I_{\omega_{g}}}\\ 0&\frac{k_{IP}}{I_{\omega_{g}}}\left(\frac{3p_{g}^{2}}{2I_{\omega_{g}}}\phi_{ \omega_{g}}\right)0&0&-\frac{k_{IP}}{P_{g}}\left(\frac{1}{I_{IP}}-\frac{f}{I_ {\omega_{g}}}\right)\frac{-k_{IP}}{I_{\omega_{g}}}\\ \end{array}\right)\left(\begin{array}{c}I_{d}\\ I_{q}\\ \delta\\ \omega_{m}\\ \omega_{g}\end{array}\right)+\left(\begin{array}{c}0&0\\ 0&0\\ 0&0\\ 0&0\\ \frac{k_{IP}}{I_{\omega_{g}}}&0\end{array}\right)\left(\begin{array}{c}0&0\\ 0&0\\ 0&0\\ 0&0\\ \frac{k_{IP}}{I_{\omega_{g}}}&0\end{array}\right)\end{array}  \tag{11}\]  
Figure 4: IP speed control of the gas turbine—generator association. In that model, the state matrix involves all nonlinearities of the powertrain (in red) with sine and cosine terms and also with bilinearities (products of two state variables). Let us note that the same derivation process may be applied for a multi-motor association with \(n\) propellers, leading to an order extension of the state model:  
\[\frac{d}{dt}\left( \begin{array}{c}I_{d_{1}}\\ I_{q_{1}}\\ I_{d_{2}}\\ I_{q_{2}}\\ \vdots\\ \delta_{1}\\ \delta_{2}\\ \vdots\\ \omega_{1}\\ \omega_{2}\\ \vdots\\ \omega_{g}\\ T_{TAG}\end{array} \right)=\textbf{A}\cdot\left( \begin{array}{c}I_{d_{1}}\\ I_{q_{1}}\\ I_{d_{2}}\\ I_{q_{2}}\\ \vdots\\ \delta_{1}\\ \delta_{2}\\ \vdots\\ \omega_{1}\\ \omega_{2}\\ \vdots\\ \omega_{g}\\ T_{PRO_{n}}\end{array} \right)+\textbf{B}\cdot\left( \begin{array}{c}\omega_{ gnsf}\\ T_{prop_{1}}\\ T_{prop_{2}}\\ \vdots\\ T_{prop_{n}}\end{array} \right) \tag{12}\]