## 5 The Fuzzy Sliding-Mode Method for Fractional-Order PMSM Vector Control
\begin{split}{}^{C}D_{t}^{\alpha}V_{1}&\leq as_{e}(- \tilde{\theta_{e}}^{T}\xi(x)-\delta+\tilde{d_{e}}-\frac{\lambda}{a}signs_{e})- \frac{1}{\alpha_{1}}\tilde{\theta_{e}}^{T}{}^{C}D_{t}^{\alpha}\theta_{e}+ \frac{1}{\alpha_{2}}\tilde{d_{e}}^{C}D_{t}^{\alpha}d_{e}^{*}\\ &=\tilde{\theta_{e}}^{T}(-as_{e}\xi(x)-\frac{1}{\alpha_{1}}{}^{C}D _{t}^{\alpha}\theta_{e})+\tilde{d_{e}}(as_{e}+\frac{1}{\alpha_{2}}{}^{C}D_{t}^ {\alpha}d_{e}^{*})-as_{e}\delta\\ &=-\lambda s_{e}sign(s_{e})-as_{e}\delta\\ &=-\lambda|s_{e}|-as_{e}\delta\end{split}  \tag{58}\]  
where \(\delta\) is the minimum approximation error between \(g_{e}^{\prime}\) and \(\tilde{g_{e}}^{*}\). Design \(\lambda>a\delta\), then \({}^{C}D_{t}^{\alpha}V_{1}\) can be guaranteed to be negative. According to [PERSON]'s second method, \(V_{1}(x)\) is positive definite and \({}^{C}D_{t}^{\alpha}V_{1}(x)<0\), and then the system is asymptotically stable. In the process of simulation, we keep parameters of fractional-order PMSM in [33] and \(a=2\), \(\alpha_{1}=\alpha_{2}=2\), \(\lambda=8\), \(\delta=3\). The selection rule of fuzzy rules is to keep the error 0 at the zero of the rule, and then the width of the membership function is reduced in order to improve the accuracy. As for the uncertain nonlinear function \(g_{e}\), we hope that this uncertainty term does not exist, so we keep the expected value 0 at the zero of the rule, too. \[{}^{C}D_{t}^{0.95}\theta_{e}=-4s_{e}\xi(x) \tag{59}\]  
\[{}^{C}D_{t}^{0.95}d_{e}^{*}=-4s_{e} \tag{60}\]  
From the above analysis, we obtain the expected value \(i_{q}^{*}\), and next consider the system as a second-order system with states \(i_{q}\) and \(i_{d}\). \[\left\{ \begin{array}{l}{}^{C}D^{\alpha}i_{d}=\frac{1}{L_{d}}u_{d}-\frac{R}{L _{d}}i_{d}+\omega_{e}\frac{L_{d}}{L_{d}}i_{q}+g_{d}\\ {}^{C}D^{\alpha}i_{q}=\frac{1}{L_{q}}u_{q}-\frac{R}{L_{q}}i_{q}+\omega_{e} \frac{L_{d}}{L_{d}}i_{d}-\frac{1}{L_{q}}\omega_{e}\psi_{f}+g_{q}\end{array} \right. \tag{61}\]  
We can obtain through function (32) that the sliding-mode surfaces are  
\[\left\{ \begin{array}{l}s_{q}(t)=e_{q}(t)+10l_{t}^{0.95}e_{q}(t)-(e_{q}(0)+1 0l_{t}^{0.95}e_{q}(0))e^{-4t}\\ s_{d}(t)=e_{d}(t)+10l_{t}^{0.95}e_{d}(t)-(e_{d}(0)+10l_{t}^{0.95}e_{d}(0))e^{-4 t}\end{array} \right. \tag{62}\]  
Design the fuzzy adaptive laws as follows:  
\[\left\{ \begin{array}{l}{}^{C}D_{t}^{0.95}\theta_{\delta q}=s_{q}\xi(x)\\ {}^{C}D_{t}^{0.95}\theta_{\delta d}=s_{d}\xi(x)\end{array} \right. \tag{63}\]  
\[\left\{ \begin{array}{l}{}^{C}D_{t}^{0.95}\theta_{u_{q}}=200s_{q}\phi(s_{q}) \\ {}^{C}D_{t}^{0.95}\theta_{u_{q}}=300s_{q}\phi(s_{d})\end{array} \right. \tag{64}\]  
\[\left\{ \begin{array}{l}{}^{C}D_{t}^{0.95}\delta_{q}=4s_{q}\\ {}^{C}D_{t}^{0.95}\delta_{d}=3s_{d}\end{array} \right. \tag{65}\]  
In Table 2, it is easy to conclude that the errors of AFSMC are the smallest and the adjustment time is the largest, which is basically consistent with the previous analysis results. From Figure 10, it is clear that all three methods can track the desired rotational speed very well, even with a step here. As we use SMC with sign switching function in the design of the speed ring, the advantage of the SMC with fuzzy switching function is not