# A Singular Perturbation Approach-Based Non-Cascade Sliding Mode Control for Surface-Mounted PMSMs
## 3 Singular Perturbation Approach-Based SMC
### TD-Based Eventual SMC for a PMSM Servo System  
In this subsection, the TD-based eventual SMC strategy for a surface-mounted PMSM will be synthesized in details. It is worth mentioning that the presented disturbances \(\mathbf{f}\) in (9) can be rewritten as  
\[ \begin{split}-\left[\begin{array}{c}\mathrm{J}\mathrm{d}\omega_ {\rm m}^{*}\big{/}\mathrm{d}t+F\omega_{\rm m}^{*}+T_{\rm m}\\ p_{\rm n}\omega_{\rm m}^{*}\psi_{\rm f}\end{array}\right]&=-\left[\begin{array} []{c}\mathrm{J}\mathrm{d}\omega_{\rm m}^{*}\big{/}\mathrm{d}t+F\omega_{\rm m }^{*}\\ p_{\rm n}\omega_{\rm m}^{*}\psi_{\rm f}\end{array}\right]+\left[\begin{array} []{c}-T_{\rm m}\\ 0\end{array}\right]\\ \mathbf{f}&=\mathbf{f}_{\rm o}+\mathbf{f}_{\rm T}\end{split}  \tag{42}\]  
where \(\mathbf{f}_{\rm o}\) can be defined as the nominal disturbances in the absence of external load torque \(T_{\rm m}\). For the above-mentioned \(\mathbf{f}_{\rm o}\), we can incorporate it into the SMC law (36), if the favorable differential signal is provided. In addition, it is an effective way to improve the tracking performance by arranging a smooth transition dynamic for reference velocity [26]. To these ends, an optimal nonlinear function-based TD is employed in this study, which is as follows:  
\[\left[\begin{array}{c}\hat{x}_{1}(k+1)-\hat{x}_  
where \(\hat{x}_{1}\) and \(\hat{x}_{2}\) are the tracking estimation values for \(x_{1}\) and its differential signal \(x_{2}\), respectively; \(T_{0}\) is the discrete step; \(e(k)=\hat{x}_{1}(k)-x_{1}(k)\) is the estimation error; \(r\) is the velocity factor; \(h\) is the filtering factor; The nonlinear function \(u_{\rm r}=\text{fhan}(\cdot)\) is presented as  
\[ \begin{array}{c}d=rh,\ \ \ d_{\rm o}=hd\\ y=e(k)+h\hat{x}_{2}(k)\\ a_{\rm o}=\sqrt{d^{2}+8r|y|}\\ a=\left\{\begin{array}{c}\hat{x}_{2}(k)+\frac{a_{\rm o}-d}{2}\text{sgn}(y), \ \ \ |y|>d_{\rm o}\\ \hat{x}_{2}(k)+\frac{y}{h},\ \ \ |y|\leq d_{\rm o}\\ \text{fhan}(e(k),\hat{x}_{2}(k),r,h)=-\left\{\begin{array}{c}r\text{sgn}(a),\ \ \ |a|>d\\ r\frac{d}{d},\ \ \ |a|\leq d\end{array}\right.\end{array}\right.\end{array}  \tag{44}\]  
On the other hand, it is well known that the discontinuous function \(\text{sgn}(\cdot)\) is the essential reason to cause the chattering phenomenon, which is a challenging problem in SMC community. In order to address the existing issue, a nonlinear function \(\text{Fal}(\cdot)\) is adopted as  
\[\text{Fal}(e_{\rm o},\alpha_{\rm n})=\left\{ \begin{array}{c}\ \ \ |e_{\rm o}|^{ \alpha_{\rm n}}\text{sgn}(e_{\rm o}),\ \ \ |e_{\rm o}|>0.1\\ 0.1^{\alpha_{\rm n}-1}e_{\rm o},\ \ \ |e_{\rm o}|\leq 0.1\end{array} \right. \tag{45}\]  
where \(e_{\rm o}\) is the input variable, and \(\alpha_{\rm n}>1\) is the design parameter to be determined later. For the above introduced function \(\text{Fal}(\cdot)\), if we constrain the corresponding output within the bound value \(\pm 1\), resulting in \(\text{fal}(\cdot)=\text{Sat}(\text{Fal})\), where \(\text{Sat}(\cdot)\) is the saturation function. By taking the above-mentioned improvements into account, the TD-based eventual SMC can be synthesized as follows. \[ \begin{array}{c}\mathbf{u}=-(\epsilon\mathcal{S}_{1}\mathbf{B}_{1}+\mathcal{S}_{2} \mathbf{B}_{2})^{-1}[(\epsilon\mathcal{S}_{1}\mathbf{A}_{11}+\mathcal{S}_{2}\mathbf{A}_{ 21})x+(\epsilon\mathcal{S}_{1}\mathbf{A}_{12}+\mathcal{S}_{2}\mathbf{A}_{22})\mathbf{z}\\ +(\epsilon\mathcal{S}_{1}\mathbf{D}_{1}+\mathcal{S}_{2}\mathbf{D}_{2})\mathbf{f}_{\rm o}+ \Gamma\mathcal{S}_{\rm c}+(\gamma+\sigma)\text{fal}(\mathcal{S}_{\rm c})] \end{array}  \tag{46}\]  
where \(\gamma=\|\epsilon\mathcal{S}_{1}\mathbf{D}_{1}+\mathcal{S}_{2}\mathbf{D}_{2}\|\cdot\| \mathbf{f}_{\rm T}\|\). As a result, the corresponding schematic block diagrams of the different non-cascade SMC strategy for a PMSM regulation system are individually shown as Figure 1, which are distinguished with each other by using the abbreviations of \"SMC\" and \"TD-SMC\", respectively. **Remark 5**.