## 5 The Fuzzy Sliding-Mode Method for Fractional-Order PMSM Vector Control
Design \(i_{q}^{*}\), that is, the control rate \(u\) is  
\[i_{q}^{*}=\frac{2J}{3n_{p}^{2}\psi_{f}}(\frac{n_{p}}{J}T_{L}+\frac{R}{J}\omega _{e}-g^{\prime}_{e}-d_{e}(t)+\frac{\lambda}{a}sgn(s_{e})) \tag{48}\]  
where \(d_{e}(t)\) is uncertain and \(g^{\prime}_{e}\) is the unknown internal nonlinear disturbance. In order to ensure the smooth progress of the control, adaptive control and FLSs must be used. Assume that \(d_{e}(t)\), \(d_{q}(t)\), and \(d_{d}(t)\) are bounded and \(\|d_{e}(t)\|\leq d_{e}\), \(\|d_{q}(t)\|\leq d_{q}\), and \(\|d_{d}(t)\|\leq d_{d}\). \(d_{e}^{*}\), \(d_{q}^{*}\), and \(d_{d}^{*}\) are the expected values of uncertain external disturbances. \(\widetilde{d_{e}}\), \(\widetilde{d_{q}}\), and \(\widetilde{d_{d}}\) are the estimation errors of uncertain external disturbances. \[\left\{ \begin{array}{l}\widetilde{d_{e}}=d_{e}^{*}-d_{e}\\ \widetilde{d_{q}}=d_{q}^{*}-d_{q}\\ \widetilde{d_{d}}=d_{d}^{*}-d_{d}\end{array} \right. \tag{49}\]  
Assume that \(\hat{g_{e}^{\prime}}\) is the fuzzy approximation form of nonlinear function \(g_{e}^{\prime}\). \[\hat{g_{e}^{\prime}}=\theta_{e}^{T}\hat{g}(x)=g_{e}(x|\theta_{e}) \tag{50}\]  
where \(x=\left[ \begin{array}{ccc}\omega_{e}&i_{q}&i_{d}\end{array} \right]^{T}\). It is also necessary to take adaptive control due to unknown function \(g_{e}^{\prime}\). We design \(\hat{g_{e}^{\prime*}}\) as the optimal function of \(\hat{g_{e}^{\prime}}\). \[\hat{g_{e}^{\prime*}}=g_{e}(x|\theta_{e}^{*}), \tag{51}\]  
\[\delta={g_{e}}^{\prime}-\hat{g_{e}^{\prime*}}={g_{e}}^{\prime}-g_{e}(x|\theta _{e}^{*}); \tag{52}\]  
thus, \(\widetilde{\theta_{e}}=\theta_{e}^{*}-\theta_{e}\) is the error between the fuzzy approximation and optimal fuzzy approximation. According to the above two estimates, the function (48) can be improved as  
\[i_{q}^{*}=\frac{2J}{3n_{p}^{2}\psi_{f}}(\frac{n_{p}}{J}T_{L}+\frac{B}{J}\omega _{e}-g_{e}(x|\theta_{e})-d_{e}^{*}+\frac{\lambda}{a}sign(s_{e})) \tag{53}\]  
And the adaptive control laws are designed as  
\[{}^{C}D_{t}^{\alpha}d_{e}^{*}=-a\alpha_{2}s_{e} \tag{54}\] \[{}^{C}D_{t}^{\alpha}\theta_{e}=-a\alpha_{1}s_{e}g(x) \tag{55}\]  
It is clear that the system (46) is stable under the control rate. The Lyapunov function is selected as \(V_{1}=\frac{1}{2}s_{e}^{2}+\frac{1}{2\alpha_{1}}\widetilde{\theta}_{e}^{2}+ \frac{1}{\alpha_{2}}\widetilde{d}_{e}^{2}\). The fractional differential form is  
\[ \begin{split}{}^{C}D_{t}^{\alpha}V_{1}&={s_{e}}^{C}D _{t}^{\alpha}s_{e}+\frac{1}{\alpha_{1}}\tilde{\theta_{e}}^{T}{}^{C}D_{t}^{ \alpha}\widetilde{\theta_{e}}+\frac{1}{\alpha_{2}}\widetilde{d}_{e}^{C}D_{t}^ {\alpha}\widetilde{d_{e}}\\ &={s_{e}}^{C}D_{t}^{\alpha}s_{e}-\frac{1}{\alpha_{1}}\tilde{ \theta_{e}}^{T}{}^{C}D_{t}^{\alpha}\theta_{e}+\frac{1}{\alpha_{2}}\widetilde{ d}_{e}^{C}D_{t}^{\alpha}d_{e}^{*}\\ &={s_{e}}^{C}D_{t}^{\alpha}x_{1}-\frac{1}{\alpha_{1}}\tilde{ \theta_{e}}^{T}{}^{C}D_{t}^{\alpha}\theta_{e}+\frac{1}{\alpha_{2}}\widetilde{ d}_{e}^{C}D_{t}^{\alpha}d_{e}^{*}\end{split}  \tag{56}\]  
Because of the control rate (53), we can obtain \({}^{C}D_{t}^{\alpha}x_{1}\):  
\[ \begin{split}{}^{C}D_{t}^{\alpha}x_{1}&\leq-\frac{3 }{2}\frac{n_{p}^{2}}{J}\psi_{f}i_{q}^{*}+\frac{n_{p}}{J}T_{L}+\frac{B}{J}\omega _{e}-g_{e}^{\prime}-d_{e}\\ &=g_{e}(x|\theta_{e})-(g_{e}(x|\theta_{e}^{*})+\delta)+(d_{e}^{*}- d_{e})-\frac{\lambda}{a}sign(s_{e})\\ &=\theta_{e}^{T}\xi(x)-\theta_{e}^{*T}\tilde{\xi}(x)-\delta+ \widetilde{d_{e}}-\frac{\lambda}{a}sign(s_{e})\\ &=-\widetilde{\theta_{e}}^{T}\xi(x)-\delta+\widetilde{d_{e}}- \frac{\lambda}{a}sign(s_{e})\end{split}  \tag{57}\]Thus, \({}^{C}D_{t}^{\alpha}V_{1}\) is  
\[