# A Singular Perturbation Approach-Based Non-Cascade Sliding Mode Control for Surface-Mounted PMSMs
## 4 Simulation Results  
A surface-mounted PMSM is considered in this section to illustrate the effectiveness and advantages of the presented approach, whose specification parameters has been exhibited in Table 1. The subsequent research results are conducted on the well-known Matlab/Simulink platform, which involves the discrete-time model. According to the corresponding parameters, one can calculate the individual time constants as \(T_{\rm c}=9.89\times 10^{-3}\) and \(T_{\rm s}=0.7309\), which indicates the relationship (7) is strictly guaranteed. Then, the singular perturbation parameter is determined as \(\varepsilon=T_{\rm c}\). In addition, the matrices presented in Assumption 1 are as follows. \[A_{0}=-394.3564,\qquad\mathbf{B}_{0}=\left[ \begin{array}{cc}0&684.6483\end{array} \right] \tag{47}\]  
According to Lemma 1, one can artificially assign the individual eigenvalues of the corresponding slow and fast subsystem matrices as \(\lambda(A_{0}+\mathbf{B}_{0}\mathbf{K}_{0})=-4.1068\) and \(\lambda(\mathbf{A}_{22}+\mathbf{B}_{2}\mathbf{K}_{2})=\{-34.0396,-34.0396\}\), respectively. The above-mentioned eigenvalues can be realized by employing the following state feedback gain matrices:  
\[\mathbf{K}_{0}=\left[ \begin{array}{c}0.57\\ 0.57\end{array} \right],\qquad\mathbf{K}_{2}=-15\times\left[ \begin{array}{cc}1&0 \\ 0&1\end{array} \right],\qquad\mathbf{K}_{1}=\left[ \begin{array}{c}19.4026\\ 0.4378\end{array} \right] \tag{48}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Value** & **Unit** \\ \hline \(R_{\rm s}\) & 0.454 & \(\Omega\) \\ \(L_{\rm s}\) & 4.492 & mH \\ \(\Phi_{\rm f}\) & 0.1435 & Wb \\ Rated voltage (\(U_{\rm n}\)) & 220 & V \\ \(J\) & \(2.77\times 10^{-3}\) & kg\(\cdot\) m\({}^{2}\) \\ \(F\) & \(3.79\times 10^{-3}\) & – \\ \(p_{\rm n}\) & 4 & – \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specification parameters of a surface-mounted PMSM. Figure 2: The whole block control structure for a PMSM servo system. In order to illustrate the effectiveness of the fixed-point recursive algorithm presented in Remark 2, we introduce the norm-variables \(\|\Delta\mathbf{L}\|\) and \(\|\Delta\mathbf{H}\|\) (both of their initial values are set as 0.1) to denote the recursive errors between the current and last iteration values. In the procedure of program realization, the iteration operations will be immediately finished if the above-mentioned norm-variables are greater than \(10^{-5}\). As a result, the corresponding iteration procedures are exhibited as Figure 3.  
It can be concluded form Figure 3 that the transformation matrix calculation results will gradually converge their actual values when the iteration numbers are \(i=2\) and \(j=3\), respectively. Eventually, the following transformation matrices can be obtained:  
\[\mathbf{L}=\left[ \begin{array}{cc}-1.257\\ 0.0088\end{array} \right],\qquad\varepsilon\mathbf{R}(\mathbf{L})=10^{-6}\times\left[ \begin{array}{cc}0.2023\\ -0.0014\end{array} \right] \tag{49}\] \[\mathbf{H}=\left[ \begin{array}{cc}0&-9.1496\end{array} \right], \qquad\mathbf{R}(\mathbf{H})=10^{-8}\times\left[ \begin{array}{cc}0&0.1058\end{array} \right]\]  
Thus, the system matrix and control input matrix presented in (23) can be obtained according to the above calculation results, which are as follows. \[\mathbf{\bar{A}}=-\left[ \begin{array}{cc}4.1101&0&0\\ \hline 0&34.0396&3.8659\\ 0&0&34.0125\end{array} \right],\qquad\mathbf{B}=\left[ \begin{array}{cc}0&20.1534 \\ \hline 2.2026&0\\ 0&2.026\end{array} \right] \tag{50}\]  
Furthermore, it can found that \(\lambda(\mathbf{\bar{A}})=\{-4.1101,-34.0396,-34.0125\}\), which are very approximate to the above-mentioned subsystem eigenvalues \(\lambda(A_{0}+\mathbf{B}_{0}\mathbf{K}_{0})\) and \(\lambda(\mathbf{A}_{22}+\mathbf{B}_{2}\mathbf{K}_{2})\). By choosing the matrix as \(\mathbf{Q}=10\mathbf{I}\), we can obtain the solution of the Lyapunov Equation (24), which is as follows. \[\mathbf{P}=\left[ \begin{array}{cc}P_{\text{s}}&\mathbf{0}\\ \mathbf{0}&\mathbf{P}_{\text{f}}\end{array} \right]=\left[