## 3 Adaptive Inversion Control  
### Design of Adaptive Inversion Controller  
This paper adopts \(i_{d}=0\), and the control realizes the decoupling control and position tracking control of PMSM. Figure 2 is the system control block diagram. The PMSM control system includes adaptive backstepping position controller, load torque adaptive law, coordinate transformation module, SVPWM module, inverter, and PMSM. The reference position is \(\theta_{r}\), and the given d-axis current is \(i_{dr}=0\); the main purpose of the controller is to obtain the appropriate control voltage to realize the position tracking control. According to the inversion control principle, the position error and its first derivative are defined as:  
\[e_{\theta}=\theta-\theta_{r} \tag{22}\]  
\[\dot{e}_{\theta}=\dot{\theta}-\dot{\theta}_{r}=\omega-\dot{\theta}_{r} \tag{23}\]  
Introduce speed virtual control quantity:  
\[\omega_{d}^{*}=\dot{\theta}_{r}-ke_{\theta},k>0 \tag{24}\]  
Then, \(\dot{e}_{\theta}=ke_{\theta}\) can be obtained by combining Formulas (22) and (23). Define the speed tracking error and its first derivative as:  
\[e_{\omega}=\omega-\omega_{d}^{*} \tag{25}\]  
\[\dot{e}_{\omega}=\dot{\omega}-\dot{\omega}_{d}^{*} \tag{26}\]  
Because the load torque is unknown, we define the Lyapunov function as:  
\[V_{1}=\frac{1}{2}e_{\omega}^{2}+\frac{1}{2}\gamma_{1}^{-1}\tilde{\tau}_{L}^{2} \tag{27}\]  
where \(\gamma_{1}\) is a strictly positive definite adaptive gain, \(\tilde{\tau}_{L}=\tau_{L}-\hat{\tau}_{L}\) is the load error, and \(\hat{\tau}_{L}\) represents the estimated value of the load torque, \(\tau_{L}\). Although the load torque is an unknown variable, if the sampling interval changes fast enough compared with the load disturbance time, the disturbance model can be considered as \(\tilde{\tau}_{L}=0\). So, the derivative of the load torque error is \(\tilde{\tilde{\tau}}_{L}=-\hat{\tau}_{L}\). According to the above analysis, the first derivative of Equation (27) is:  
\[\dot{V}_{1}=e_{\omega}\dot{e}_{\omega}+\gamma_{1}^{-1}\tilde{\tau}_{L}\dot{ \tilde{\tau}}_{L}=e_{\omega}\Big{(}\dot{\omega}-\dot{\omega}_{d}^{*}\Big{)}- \gamma_{1}^{-1}\tilde{\tau}_{L}\dot{\tilde{\tau}}_{L} \tag{28}\]  
According to the inversion control principle, the virtual control current \(i_{qr}\) and q-axis current error \(e_{iq}\) are introduced:  
\[\left\{ \begin{array}{l}i_{qr}=\frac{J}{n_{p}\beta}\Big{(}\alpha+\dot{\omega }_{d}^{*}\Big{)}\\ e_{iq}=i_{q}-i_{qr}\end{array} \right. \tag{29}\]  
Figure 2: Block diagram of adaptive inversion control system for PMSM. where \(\alpha\) is the stabilization function, which can make \(e_{iq}\) asymptotically stable. Combining Equations (1), (24), (28) and (29), we can obtain:  
\[\dot{V}_{1}=\frac{1}{J}n_{p}\phi e_{\omega}e_{iq}+e_{\omega}\underbrace{\left( \alpha-\frac{1}{J}\tau_{L}\right)}_{-k_{1}\epsilon_{\omega}}+\widetilde{\tau} _{L}\underbrace{\left(-\frac{\epsilon_{\omega}}{J}-\gamma_{1}^{-1}\dot{\tau} _{L}\right)}_{-k_{2}\widetilde{\tau}_{L}} \tag{30}\]  
where the feedback gain is \(k_{1}>0,k_{2}>0\). The stabilization function \(\alpha\) and the load torque adaptive law \(\dot{\hat{\tau}}_{L}\) are selected as:  
\[\alpha=-k_{1}\epsilon_{\omega}+\frac{1}{J}\hat{\tau}_{L} \tag{31}\]  
\[\dot{\hat{\tau}}_{L}=\gamma_{1}k_{2}\widetilde{\tau}_{L}-\frac{1}{J}\gamma_{1 }\epsilon_{\omega} \tag{32}\]  
Equation (28) can now be transformed into:  
\[\dot{V}_{1}=\frac{1}{J}n_{p}\phi e_{\omega}e_{iq}-k_{1}\epsilon_{\omega}^{2}- k_{2}\widetilde{\tau}_{L}^{2} \tag{33}\]  
Before proceeding to the next design step, we need to clarify the derivation process of the load disturbance estimation \(\hat{\tau}_{L}\).