## 4 Simulation and Results  
In this section, an example is presented to show the effectiveness of the proposed method using the algorithmic solution of Caputo fractional-order differential equations. PMSM plays an important part in the electric system, and it is of urgency for mechanical and electrical engineers to research and manufacture the PMSM servosystems of high performance.  
Because of the stator voltages and the mechanical angular velocity, the three-dimensional fractional-order state function of the PMSM is given as follows:  
\[\left\{\begin{array}{l}\mathbb{C}D_{t}^{\alpha}i_{d}=-\frac{R_{s}}{L_{d}}i_ {d}+\frac{L_{d}}{L_{d}}\omega_{e}i_{q}+\frac{1}{L_{d}}u_{d}\\ \mathbb{C}D_{t}^{\alpha}i_{q}=-\frac{R_{s}}{L_{d}}i_{q}-\frac{L_{d}}{L_{d}} \omega_{e}i_{d}+\frac{V_{e}}{L_{d}}\omega_{e}+\frac{1}{L_{d}}u_{q}\\ \mathbb{C}D_{t}^{\alpha}\omega_{e}=\frac{n_{p}}{f}\Big{[}\frac{3}{2}n_{p}\big{(} \psi/i\eta_{i}+\big{(}L_{d}-L_{q}\big{)}i\dot{a}_{i}\big{)}\Big{]}-\frac{n_{p}} {f}T-\frac{B}{f}\omega_{e}\end{array}\right. \tag{31}\]  
where \(i_{d}\), \(i_{q}\), and \(\omega_{e}\) are the state variables of the system, representing the stator current and rotor angular velocity of the \(d\) and \(q\) axes, respectively; \(u_{d}\), \(u_{q}\), \(L_{d}\), and \(L_{q}\) represent the \(d\)-axis voltage, \(q\)-axis voltage, and stator inductance, respectively; \(\psi_{f}\) and \(T_{l}\) are permanent magnetic flux and external mechanical torque, respectively; \(n_{p}\), \(R_{s}\), \(J\), and \(B\) are the number of magnetic pole pairs, stator resistance, rotational inertia, and viscous damping coefficient, respectively.  
Considering [14], the parameters of this system can be designed as \(B=7.2\), \(\psi_{f}=2.667\), \(J=4\), \(u_{d}=-12.7\), \(n_{p}=4\), \(u_{q}=2.34\), \(R_{s}=2.857\), \(L_{d}=6.45\), \(L_{q}=7.125\), and \(T=0.525\). Let the initial values of the variables be \(0.1\), and the reference signals be \(x_{1d}=x_{2d}=x_{3d}=\sin(t)\). At this time, the MIMO system is constructed by the state equation structure of PMSM, and the model does not represent the actual model of PMSM.  
Incorporate the above parameters into (31), where the fractional-order operator \(\alpha\) is chosen as \(0.95\), and the external disturbances are selected as \(d_{1}=\sin(t)\), \(d_{2}=2\), and \(d_{3}=\cos(t)\) after ten seconds of the control process. Especially in the first ten seconds, the disturbances are zero. In this simulation model, there are three state variables, and the fuzzy of the nonlinear functions has three inputs and a single output. Let \(r_{1}=r_{2}=r_{3}=1\), \(l_{1}=500\), \(l_{2}=600\), \(l_{3}=800\), \(r_{1}=2\), \(r_{2}=6\), and \(r_{3}=4\). Taking the first nonlinear function \(g_{1}(x)\) as an example, the FLSs of \(g_{1}(x)\) have five rules for each state variable, which can be described as \(A_{1}^{1}=\exp(-((x+1.2-0.6(l-1))/0.3)^{2}),l=1\cdots 5\) and \(125\) fuzzy basis functions. The five rules of FLSs of \(s_{1}\) are \(A_{1}^{1}=\exp(-((x+0.5-0.25(l-1))/0.125)^{2}),l=1\cdots 5\).  
The sliding-mode surfaces are as follows:  
\[s_{i}(t)=e_{i}(t)+10l_{i}^{0.95}e_{i}(t)-(e_{i}(0)+10l_{i}^{0.95}e_{i}(0))e^{-4 t},i=1,2,3 \tag{32}\]Design the fuzzy adaptive laws as follows:  
\[{}^{C}D_{t}^{0.95}\theta_{ig}=s_{i}\xi(x),i=1,2,3 \tag{33}\]  
\[{}^{C}D_{t}^{0.95}\theta_{1u}=500s_{1}\phi(s_{1}) \tag{34}\]  
\[{}^{C}D_{t}^{0.95}\theta_{2u}=600s_{2}\phi(s_{2}) \tag{35}\]  
\[{}^{C}D_{t}^{0.95}\theta_{3u}=800s_{3}\phi(s_{3}) \tag{36}\]  
\[{}^{C}D_{t}^{0.95}\delta_{1}=2s_{1} \tag{37}\]  
\[{}^{C}D_{t}^{0.95}\delta_{2}=6s_{2} \tag{38}\]  
\[{}^{C}D_{t}^{0.95}\delta_{3}=2s_{3} \tag{39}\]