# A Singular Perturbation Approach-Based Non-Cascade Sliding Mode Control for Surface-Mounted PMSMs
## 2 System Modeling and Preliminaries  
The classical mathematical model of a surface-mounted PMSM can be established in terms of the two-phase synchronous rotating orthogonal \(d-q\) reference coordinate system [14], which is comprised by electrical dynamics  
\[\left\{ \begin{array}{l}L_{\mathrm{s}}\frac{\mathrm{d}}{\mathrm{d}t}i_{ \mathrm{d}}=u_{\mathrm{d}}-R_{\mathrm{s}}i_{\mathrm{d}}+\omega_{\mathrm{e}}L_{ \mathrm{s}}i_{\mathrm{q}}\\ L_{\mathrm{s}}\frac{\mathrm{d}}{\mathrm{d}t}i_{\mathrm{q}}=u_{\mathrm{q}}-R_{ \mathrm{s}}i_{\mathrm{q}}-\omega_{\mathrm{e}}L_{\mathrm{s}}i_{\mathrm{d}}- \omega_{\mathrm{e}}\psi_{\mathrm{f}}\end{array} \right. \tag{1}\]  
and mechanical dynamic equation  
\[J\frac{\mathrm{d}}{\mathrm{d}t}\omega_{\mathrm{m}}=K_{\mathrm{T}}i_{\mathrm{q} }-F\omega_{\mathrm{m}}-T_{\mathrm{m}} \tag{2}\]  
where \(L_{\mathrm{s}}\) is the stator inductance; \(i_{\mathrm{d}}\) and \(i_{\mathrm{q}}\) denote \(d\) and \(q\) axes stator currents, respectively; \(u_{\mathrm{d}}\) and \(u_{\mathrm{q}}\) represent \(d\) and \(q\) axes stator voltages, respectively; \(R_{\mathrm{s}}\) is stator resistance; \(\psi_{\mathrm{f}}\) is the flux linkage of permanent magnets; \(J\) is the moment of the rotational inertia; \(K_{\mathrm{T}}=3p_{\mathrm{n}}\psi_{\mathrm{f}}/2\) is the electromagnetic torque coefficient, and \(p_{\mathrm{n}}\) is the number of pole pairs; \(F\) is the viscous friction coefficient; \(T_{\mathrm{m}}\) represents the load torque disturbance; \(\omega_{\mathrm{e}}\) and \(\omega_{\mathrm{m}}\) are electrical and mechanical angular velocities, respectively, which satisfying \(\omega_{\mathrm{e}}=p_{\mathrm{n}}\omega_{\mathrm{m}}\). It is worth mentioning that the nonlinear back electromotive forces (EMFs) produced by products of electrical angular velocity \(\omega_{\mathrm{e}}\) with \(d\) and \(q\) axes stator currents \(i_{\mathrm{d}}\) and \(i_{\mathrm{q}}\), increasing the complexity of controller design and analysis. In order to eliminate the adverse effect of the above-mentioned back EMFs, we can actively compensate their influences through the feedforward channel, and thus the following simplified linear electrical dynamics can be easily obtained:  
\[\left\{ \begin{array}{l}L_{\mathrm{s}}\frac{\mathrm{d}}{\mathrm{d}t}i_{\mathrm{d}} =u_{\mathrm{do}}-R_{\mathrm{s}}i_{\mathrm{d}}\\ L_{\mathrm{s}}\frac{\mathrm{d}}{\mathrm{d}t}i_{\mathrm{q}}=u_{\mathrm{q}\mathrm{ q}}-R_{\mathrm{s}}i_{\mathrm{q}}-p_{\mathrm{n}}\omega_{\mathrm{m}}\psi_{\mathrm{f}}\end{array} \right. \tag{3}\]  
where \(u_{\mathrm{do}}\) and \(u_{\mathrm{q}\mathrm{o}}\) are the nominal control inputs in the absence of \(\omega_{\mathrm{e}}L_{\mathrm{s}}i_{\mathrm{q}}\) and \(\omega_{\mathrm{e}}L_{\mathrm{s}}i_{\mathrm{d}}\), respectively, which also satisfying the following relationship:  
\[\left[ \begin{array}{c}u_{\mathrm{d}}\\ u_{\mathrm{q}}\end{array} \right]=\left[ \begin{array}{c}u_{\mathrm{do}}\\ u_{\mathrm{q}\mathrm{o}}\end{array} \right]+\omega_{\mathrm{e}}L_{\mathrm{s}} \left[ \begin{array}{c}-i_{\mathrm{q}}\\ i_{\mathrm{d}}\end{array} \right] \tag{4}\]  
The objective of this study is to design the novel SMC laws \(u_{\mathrm{do}}\) and \(u_{\mathrm{q}\mathrm{o}}\) for a surface-mounted PMSM, such that \(\omega_{\mathrm{m}}\) can be regulated to its desired signal \(\omega_{\mathrm{m}}^{*}\) in the presences of the external disturbances. To this end, we can introduce the following velocity tracking error:  
\[e_{\mathrm{w}}=\omega_{\mathrm{m}}-\omega_{\mathrm{m}}^{*} \tag{5}\]  
Substituting the velocity tracking error \(e_{\mathrm{w}}\) (5) into the mechanical dynamic Equation (2) and electrical dynamics (3), yields  
\[\left\{