# A Singular Perturbation Approach-Based Non-Cascade Sliding Mode Control for Surface-Mounted PMSMs
## 4 Simulation Results
\begin{array}{c|cc}1.2165& 0&0\\ \hline 0&0.1469&-0.0083\\ 0&-0.0083&0.148\end{array} \right] \tag{51}\]  
which indicates that \(\lambda(\mathbf{P})=\lambda(P_{\text{s}})\cup\lambda(\mathbf{P}_{\text{f}})=\{1.2165,0.1558,0.1391\}\), which are all positive, thus guaranteeing the Theorem 1. Figure 3: The recursive procedures of solving the transformation matrices \(\mathbf{L}\) and \(\mathbf{H}\). Finally, the corresponding coefficients for constructing the sliding mode surface \(\mathcal{S}_{\mathrm{c}}\) (37) and the different SMC laws are individually calculated, which are as follows. \[\mathcal{S}_{1}=\left[ \begin{array}{cc}-0.4069\\ 24.562\end{array} \right],\qquad\mathcal{S}_{2}=\left[ \begin{array}{cc}0.3236& -0.0183\\ -0.0183&2.5455\end{array} \right]\] \[\left(\varepsilon\mathcal{S}_{1}\mathcal{B}_{1}+\mathcal{S}_{2} \mathcal{B}_{2}\right)^{-1}=\left[ \begin{array}{cc}1.4037&0.0101\\ 0.0101&0.1784\end{array} \right],\qquad\varepsilon\mathcal{S}_{1}A_{11}+ \mathcal{S}_{2}\mathcal{A}_{21}=\left[ \begin{array}{cc}0.0286\\ -3.5508\end{array} \right] \tag{52}\] \[\varepsilon\mathcal{S}_{1}\mathcal{A}_{12}+\mathcal{S}_{2} \mathcal{A}_{22}=\left[ \begin{array}{cc}-0.3236&-1.2331\\ 0.0183&0.0183\end{array} \right],\qquad\varepsilon\mathcal{S}_{1}\mathcal{D}_ {1}+\mathcal{S}_{2}\mathcal{D}_{2}=\left[ \begin{array}{cc}-1.4534&-0.0403\\ 87.7341&5.6067\end{array} \right]\]  
On the other hand, in order to implement the whole block control scheme for a PMSM servo system shown as Figure 2, the DC-link capacitor voltage is set as \(U_{\mathrm{dc}}=U_{\mathrm{n}}\times\sqrt{2}\mathcal{V}\), while the carrier frequency is determined as \(f_{\mathrm{PWM}}=10\) kHz. Meanwhile, the associated design parameters of the different control laws are listed in Table 2, where their output magnitude values are limited as \(\pm 0.9\times U_{\mathrm{n}}\). For the purpose of realizing the TD-based eventual SMC law (46), we carefully explore the tracking performance by employing different velocity factor, as shown in Figure 4.  
It should be emphasized that the response time of the favourable transition dynamic \(\hat{\omega}_{\mathrm{m}}^{*}\) can be adjusted by determining an appropriate \(r\), while the corresponding high quality differential signal \(\hat{\omega}_{\mathrm{m}}^{*}\) is subsequently obtained. On the other hand, for the presented nonlinear function \(\mathrm{f}\mathrm{d}(\cdot)\), its characteristic curves with respect to \(\mathrm{sgn}(\cdot)\) are comprehensively compared with different \(\alpha_{\mathrm{n}}\), which are shown as Figure 5. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{\Gamma}\) & \(\gamma+\sigma\) & \(T_{\mathbf{o}}\) & \(\mathbf{h}\) & \(\mathbf{r}\) & \(\mathbf{\alpha_{\mathrm{n}}}\) \\ \hline
100 & 10 & \(1\times 10^{-6}\) & \(10\times T_{\mathrm{o}}\) & \(2\times 10^{4}\) & 3.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Associated design parameters of the different control laws. Figure 4: The tracking performance and differential signal of the presented TD. It can be seen form Figure 5 that the function \(\mathrm{fal}(\cdot)\) is smoother than \(\mathrm{sgn}(\cdot)\) along with the input variable \(e_{\mathrm{o}}\), thus improving the chattering phenomenon and the speed regulation performances, simultaneously. According to the above-mentioned discussions, the important parameters are selected as \(r=2\times 10^{4}\) and \(\alpha_{\mathrm{n}}=3.5\), respectively, which are also listed in the above-presented Table 2.  
In order to validate the effectiveness and advantages of the presented approaches, the reference velocity value \(\omega_{\mathrm{m}}^{*}\) is set as a step signal, which changes from 50 rad/s (initial value) to 80 rad/s at 0.2 s. In addition, a sudden load torque \(1.5\,\mathrm{N}\cdot\mathrm{m}\) is employed at 0.3 s, which is aimed at illustrating the speed regulation system robustness against the external disturbances.