# A Singular Perturbation Approach-Based Non-Cascade Sliding Mode Control for Surface-Mounted PMSMs
## 2 System Modeling and Preliminaries
\begin{array}{l}\int\frac{\mathrm{d}}{\mathrm{d}t}e_{\mathrm{w}}=K_{ \mathrm{T}}i_{\mathrm{q}}-Fe_{\mathrm{w}}+f_{\mathrm{m}}\\ L_{\mathrm{s}}\frac{\mathrm{d}}{\mathrm{d}t}i_{\mathrm{d}}=u_{\mathrm{do}}-R_{ \mathrm{s}}i_{\mathrm{d}}\\ L_{\mathrm{s}}\frac{\mathrm{d}}{\mathrm{d}t}i_{\mathrm{q}}=u_{\mathrm{q}\mathrm{ o}}-R_{\mathrm{s}}i_{\mathrm{q}}-p_{\mathrm{n}}e_{\mathrm{w}}\psi_{\mathrm{f}}+f_{ \mathrm{q}}\end{array} \right. \tag{6}\]  
where \(f_{\mathrm{m}}=-\left(J\mathrm{d}\omega_{\mathrm{m}}^{*}/\mathrm{d}t+F\omega_{ \mathrm{m}}^{*}+T_{\mathrm{m}}\right)\) and \(f_{\mathrm{q}}=-p_{\mathrm{n}}\omega_{\mathrm{m}}^{*}\psi_{\mathrm{f}}\) represent the mechanical and \(q\) axis disturbances, respectively. To our best knowledge, the electrical transients (namely, currents \(i_{\mathrm{d}}\) and \(i_{\mathrm{q}}\)) are rather faster comparing with the mechanical response (i.e., angular velocity \(\omega_{\mathrm{m}}\)) [10], which can also be characterized by the following inequality:  
\[T_{\mathrm{c}}\ll T_{\mathrm{s}} \tag{7}\]  
where \(T_{\mathrm{c}}=L_{\mathrm{s}}/R_{\mathrm{s}}\) and \(T_{\mathrm{s}}=J/F\) denote the electrical and mechanical time constants, respectively. According to the above relationship, it can be concluded that the surface-mounted PMSM is an typical dual-time-scale system. As a result, by selecting the singular perturbation parameter \(\varepsilon=T_{\mathrm{c}}\), the Equation (6) can be modeled as the following SPS:  
\[E(\varepsilon)\frac{\mathrm{d}}{\mathrm{d}t}\boldsymbol{\psi}=\boldsymbol{A} \boldsymbol{\psi}+\boldsymbol{B}\boldsymbol{u}+\boldsymbol{D}\boldsymbol{f} \tag{8}\]  
where the system matrices and parameters are presented as follows. \[ \begin{array}{l}E(\varepsilon)=\left[\begin{array}{c|cc}1&0&0\\ \hline 0&\varepsilon&0\\ 0&0&\varepsilon\end{array}\right],\qquad\boldsymbol{\psi}=\left[\begin{array}[ ]{c}x\\ z\end{array}\right]=\left[\begin{array}{c}e_{\mathrm{w}}\\ \hline i_{\mathrm{d}}\\ i_{\mathrm{q}}\end{array}\right]\\ \boldsymbol{A}=\left[\begin{array}{cc}A_{11}&A_{12}\\ A_{21}&A_{22}\end{array}\right]=\left[\begin{array}{cc}-F/J&0&K_{ \mathrm{T}}/J\\ \hline 0&-1&0\\ -p_{\mathrm{n}}\psi_{\mathrm{f}}/R_{\mathrm{s}}&0&-1\end{array}\right]\\ \boldsymbol{B}=\left[\begin{array}{c}\boldsymbol{B}_{1}\\ \boldsymbol{B}_{2}\end{array}\right]=\left[\begin{array}{cc}0&0\\ \hline 1/R_{\mathrm{s}}&0\\ 0&1/R_{\mathrm{s}}\end{array}\right],\qquad\boldsymbol{u}=\left[\begin{array} []{c}u_{\mathrm{do}}\\ u_{\mathrm{q}\mathrm{o}}\end{array}\right]\\ \boldsymbol{D}=\left[\begin{array}{c}\boldsymbol{D}_{1}\\ \boldsymbol{D}_{2}\end{array}\right]=\left[\begin{array}{cc}1/J&0\\ \hline 0&0\\ 0&1/R_{\mathrm{s}}\end{array}\right],\qquad\boldsymbol{f}=\left[\begin{array}[ ]{c}f_{\mathrm{m}}\\ f_{\mathrm{q}}\end{array}\right]\end{array}  \tag{9}\]where \(\mathbf{\psi}\) is the system state vector, the separation degree between the slow mode \(x=e_{\rm w}\) and fast-time state variables \(\mathbf{z}=\left[ \begin{array}{cc}i_{\rm d}&i_{\rm q}\\ \end{array} \right]^{\rm T}\) is indicated by the small parameter \(\varepsilon\); \(\mathbf{f}\) denotes the disturbances, which satisfying \(\|\mathbf{f}\|\leq M\) (\(M>0\) is the upper bound), and \(\|\cdot\|\) represents the Euclidean norm. It can be concluded from time constant difference (7) that the surface-mounted PMSM speed regulation system is characterized by the obvious dual-time-scale, which can be formulated as a typical SPS (8). This study presents an alternative way to build the mathematical model, and thus the non-cascade control scheme can be subsequently performed.