## 5 The Fuzzy Sliding-Mode Method for Fractional-Order PMSM Vector Control  
The actual model of fractional-order PMSM is given according to [33], and the vector control state-space representation of PMSM can be derived under parameter uncertainties and mismatches in (40). \[\left\{ \begin{array}{l}{{}^{C}D^{a}\omega_{e}=\frac{3}{2}\frac{n^{2}}{f}i_{ q}[(L_{d}-L_{q})i_{d}+\psi_{f}]-\frac{n_{p}}{f}T_{L}-\frac{B}{f}\omega_{e}+g_{e}} \\ \end{array} \right. \tag{40}\]  
where  
\[\left\{ \begin{array}{l}g_{d}=-\frac{\Delta L_{d}}{L_{d}(L_{d}+\Delta L_{d})} (u_{d}-Ri_{d}-\Delta Ri_{d}+L_{q}\omega_{e}i_{q}+\Delta L_{q}\omega_{e}i_{q}+ \omega_{e}\Delta\psi_{f})\\ g_{q}=-\frac{\Delta L_{d}}{L_{q}(L_{d}+\Delta L_{q})}(u_{q}-Ri_{q}-\Delta Ri_ {q}-L_{d}\omega_{e}i_{d}-\Delta L_{q}\omega_{e}i_{d}-\omega_{e}\Delta\psi_{f} -\omega_{e}\psi_{f})\\ g_{e}=\frac{3}{2}\frac{n^{2}}{f}i_{q}[(\Delta L_{d}-\Delta L_{q})i_{d}+\Delta \psi_{f}]\\ \end{array} \right. \tag{41}\]  
Figure 9: (**a**) Tracking error \(e1\) of the first state function. (**b**) Tracking error \(e2\) of the second state function. (**c**) Tracking error \(e3\) of the third state function. (**d**) State tracking of each state. The control model of fractional-order PMSM includes the \(d\)-\(q\) axis current dynamic equation and the speed dynamic equation. In the current state equation, the disturbances \(d_{d}\) and \(d_{q}\) may come from inverter nonlinearity (such as dead time effect), current sensor noise, resistance or inductance parameter perturbation, or harmonic interference, and they belong to the mechanical property disturbances. As for \(d_{e}\), the disturbance that belongs to the mechanical properties may come from the load mutations such as mechanical vibration, friction disturbance, moment of inertia uncertainty, and so on. A well-conceived design of the outer speed ring can mitigate the impact of disturbances on the system, minimize speed fluctuations, and enable the system to operate in a stable state. \(i_{d}=0\) is a vector control method that is widely used. In this control method, \(i_{d}=0\) is maintained and the motor torque is controlled by controlling \(i_{q}\). Given condition \(i_{d}=0\), the system states (40) are shown as follows:  
\[\left\{ \begin{array}{l}{}^{C}D^{\alpha}i_{q}=\frac{1}{L_{q}}u_{q}-\frac{R}{L _{q}}i_{q}-\frac{1}{L_{q}}\omega_{e}\psi_{f}+g^{\prime}_{q}+d_{q}(t)\\ {}^{C}D^{\alpha}\omega_{e}=\frac{3}{2}\frac{n_{p}^{2}}{T}i_{q}\psi_{f}-\frac{ n_{p}}{T}T_{L}-\frac{R}{J}\omega_{e}+g^{\prime}_{e}+d_{e}(t)\end{array} \right. \tag{42}\]  
and the internal interferences can be rewritten as  
\[\left\{ \begin{array}{l}g^{\prime}_{q}=-\frac{\Delta L_{q}}{L_{q}(L_{q}+ \Delta L_{q})}(u_{q}-Ri_{q}-\Delta Ri_{q}-\omega_{e}\Delta\psi_{f}-\omega_{e} \psi_{f})\\ g^{\prime}_{e}=\frac{3}{2}\frac{n_{p}^{2}}{T}i_{q}\Delta\psi_{f}\end{array} \right. \tag{43}\]  
There is an expected value of the velocity \(\omega_{e}^{*}\), and we take the space error as the system state:  
\[x_{1}=\omega_{e}^{*}-\omega_{e} \tag{44}\] \[{}^{C}D_{t}^{\alpha}x_{1}=-{}^{C}D_{t}^{\alpha}\omega_{e} \tag{45}\]  
In order to make the actual electrical angular velocity of the motor track the reference value \(\omega_{e}^{*}\), it is necessary to change the electromagnetic torque output of the motor by adjusting \(i_{q}\), thus affecting the speed and electrical angular velocity of the motor. In this case, \(i_{q}\) is used as a control variable. By adjusting the value of \(i_{q}\) reasonably, the running state of the motor can be changed, and, finally, the error will be gradually reduced until a satisfactory control accuracy is achieved. The state function can be described as follows:  
\[\left\{ \begin{array}{l}{}^{C}D^{\alpha}x_{1}=-\frac{3}{2}\frac{n_{p}^{2}}{T} \psi_{f}i_{q}+\frac{n_{p}}{T}T_{L}+\frac{R}{J}\omega_{e}-g^{\prime}_{e}-d_{e} (t)\\ y=i_{q}\end{array} \right. \tag{46}\]  
Design a simple sliding-mode surface:  
\[s_{e}=ax_{1} \tag{47}\]  
where \(a>0\).