## 3 Adaptive Inversion Control
### Stability Analysis  
The stability of the whole system depends on the stability of each subsystem. The Lyapunov function of the whole control system is defined as:  
\[V=\frac{1}{2}e_{0}^{2}+V2+V3=\frac{1}{2}e_{0}^{2}+\frac{1}{2}e_{\omega}^{2}+ \frac{1}{2}\gamma_{1}^{-1}\widetilde{\tau}_{L}^{2}+\frac{1}{2}e_{iq}^{2}+ \frac{1}{2}e_{id}^{2} \tag{53}\]  
Derivating the Lyapunov function gives:  
\[\dot{V}=-k_{\theta}^{2}-k_{1}e_{\omega}^{2}-k_{4}e_{id}^{2}-\left(\sqrt{\frac {k_{2}}{2}}\widetilde{\tau}_{L}+\frac{n}{\sqrt{2k_{2}}}e_{iq}\right)^{2}- \left(\sqrt{\frac{k_{2}}{2}}\widetilde{\tau}_{L}+\frac{m}{\sqrt{2k_{2}}}e_{iq }\right)^{2}-\left(k_{3}-\frac{m^{2}+n^{2}}{2k_{2}}\right)e_{iqwind load torque. When calculating the gravity torque, the weight of the heavy object was set to 10 tons. The parameters used in calculating the wind load torque are shown in Table 1 (assuming that the application site is Urumqi and that the local temperature is 20 \({}^{\circ}\)C).  
The mathematical model for deriving the load torque is simulated in Simulink and the results are shown in Figure 4. Figure 4a is the part of the gravitational torque. It can be seen from the figure that the torque first decreases with time during the erection process, and then the reverse increases. Figure 4b is the wind load torque part, it can be seen from the figure that the torque increases first and then decreases with time during the erection process. Take the sum of the two parts of the torque as the load torque of the system.  
Given the displacement command of 1.7 m and the simulation time of 15 s, the load torque is the sum of the calculated gravity torque and the wind load torque. The simulation comparison results of the electromechanical actuation system under the traditional PID and adaptive inversion controller are shown in Figure 5. When the displacement is 1.7 m, the system can reach the system displacement command in 9.8 s under the adaptive inversion controller with a relative error of 0.012%, while it is stable in 12.4 s under the traditional PID control, with a relative error of 0.71%.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Parameters Value** \\ \hline aerodynamic drag coefficient \(Cx\) & 0.6 \\ correction factor \(R_{H}\) & 1.52 \\ gust dynamic coefficient \(\beta\) & 1.5 \\ roughness index \(\alpha\) & 0.12 \\ local wind speed \(\gamma\) (m/s) & 2.5 \\ air density \(\rho\) (kg/m\({}^{3}\)) & 1.35 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Calculation parameters of wind load torque.  
Figure 4: Variation of the load torque and wind load torque during the erection process relative to point O\({}_{2}\). (**a**) Gravitational torque; (**b**) wind load torque.  
The load torque is composed of the two parts of torque in Figure 4. The load of the system according to the load torque is shown in Figure 6. The figure shows that the torque provided by the motor first decreases to zero and then increases in the opposite direction during the system operation.  
Figure 7 is a graph showing the speed comparison of the motor when the electromechanical actuation system is running. It can be seen that the adaptive inversion method can reach the target command faster with higher precision, indicating that it has better dynamic performance and verifies the feasibility and reliability of the design.  
Figure 5: Comparison of displacement simulation between traditional PID and adaptive inversion controller.  
Figure 6: Comparison of electromagnetic torque curve between traditional PID and adaptive inversion controller.