# A Singular Perturbation Approach-Based Non-Cascade Sliding Mode Control for Surface-Mounted PMSMs
## 3 Singular Perturbation Approach-Based SMC
### Sliding Mode Controller Design and Reachability Analysis  
In this subsection, we will mainly focus on the design of the sliding mode controller, while the reachability condition is guaranteed. According to the commonly employed exponential reaching law [11], one can design the following sliding mode controller:  
\[\mathbf{u}=-(\varepsilon\mathbf{S}_{1}\mathbf{B}_{1}+\mathbf{S}_{2}\mathbf{B}_{2})^{-1}[( \varepsilon\mathbf{S}_{1}A_{11}+\mathbf{S}_{2}\mathbf{A}_{21})x+(\varepsilon\mathbf{S}_{1}\mathbf{ A}_{12}+\mathbf{S}_{2}\mathbf{A}_{22})\mathbf{z}+\Gamma\mathbf{S}_{\rm c}+(\gamma+\sigma)\text{sgn}(\mathbf{S}_{ \rm c})] \tag{36}\]  
where \(\mathbf{S}_{1}=\mathbf{B}_{\rm s}^{\rm T}P_{\rm s}(1-\varepsilon\mathbf{H}\mathbf{L})+\mathbf{B}_{ \rm f}^{\rm T}\mathbf{P}_{\rm f}\mathbf{L}\) and \(\mathbf{S}_{2}=-\varepsilon\mathbf{B}_{\rm s}^{\rm T}P_{\rm s}\mathbf{H}+\mathbf{B}_{\rm f}^{ \rm T}\mathbf{P}_{\rm f}\), respectively; \(\Gamma>0\) and \(\sigma>0\) are the exponential and switching gains, respectively; \(\gamma=\|\varepsilon\mathbf{S}_{1}\mathbf{D}_{1}+\mathbf{S}_{2}\mathbf{D}_{2}\|M\); \(\mathbf{S}_{\rm c}=\mathbf{B}^{\rm T}\mathbf{S}\); \(\text{sgn}(\cdot)\) denotes the signum function. For the the proposed SMC (36), we have the following theorem. **Theorem 3**: _Under the designed SMC law (36), the system state variables will be globally driven onto the above-mentioned composite sliding mode surface \(\mathbf{S}_{\rm c}\) in a finite time. Namely, the reachability condition can be guaranteed._  
First of all, it can be easily obtained the following formulation from (31):  
\[ \begin{array}{c}\bar{\mathbf{B}}^{\rm T}\mathbf{P}N(\varepsilon)=\left[\begin{array} []{cc}\mathbf{B}_{\rm s}^{\rm T}P_{\rm s}(1-\varepsilon\mathbf{H}\mathbf{L})+\mathbf{B}_{ \rm f}^{\rm T}\mathbf{P}_{\rm f}\mathbf{L}&-\varepsilon\mathbf{B}_{\rm s}^{\rm T}P_{\rm s }\mathbf{H}+\mathbf{B}_{\rm f}^{\rm T}\mathbf{P}_{\rm f}\end{array}\right]\\ \Downarrow\\ \mathbf{S}_{\rm c}=\left[\begin{array}{cc}\mathbf{S}_{1}&\mathbf{S}_{2}\end{array}\right] \mathbf{\psi}=\mathbf{S}_{1}x+\mathbf{S}_{2}\mathbf{z}\end{array}  \tag{37}\]  
On the other hand, it can be concluded from (8) that  
\[\varepsilon\frac{\rm d}{\rm d t}\left[ \begin{array}{c}x\\ \mathbf{z}\end{array} \right]=\left[ \begin{array}{cc}\varepsilon A_{11}&\varepsilon A _{12}\\ \mathbf{A}_{21}&\mathbf{A}_{22}\end{array} \right]\left[ \begin{array}{c}x\\ \mathbf{z}\end{array} \right]+\left[ \begin{array}{c}\varepsilon\mathbf{B}_{1}\\ \mathbf{B}_{2}\end{array} \right]\mathbf{u}+\left[ \begin{array}{c}\varepsilon\mathbf{D}_{1} \\ \mathbf{D}_{2}\end{array} \right]\mathbf{f} \tag{38}\]And then, taking the time-derivative of the \(S_{\rm c}\) in terms of \(\psi\), yields  
\[ \begin{split}\varepsilon\dot{S}_{\rm c}&=S_{1}\varepsilon \dot{x}+S_{2}\varepsilon\dot{z}\\ &=(\varepsilon S_{1}A_{11}+S_{2}A_{21})x+(\varepsilon S_{1}A_{12} +S_{2}A_{22})z+(\varepsilon S_{1}B_{1}+S_{2}B_{2})u+(\varepsilon S_{1}D_{1}+ S_{2}D_{2})f\end{split}  \tag{39}\]  
Introducing the following Lyapunov function:  
\[V_{\rm C}=\frac{\varepsilon}{2}S_{\rm c}^{\rm T}S_{\rm c} \tag{40}\]  
Calculating the time-derivative of the \(V_{\rm C}\), leads to  
\[ \begin{split}\dot{V}_{\rm C}&=\mathbf{S}_{\rm c}^{\rm T }(\varepsilon\dot{S}_{\rm c})\\ &=-\Gamma S_{\rm c}^{\rm T}S_{\rm c}-\sigma S_{\rm c}^{\rm T}{\rm sgn }(S_{\rm c})+S_{\rm c}^{\rm T}(\varepsilon S_{1}D_{1}+S_{2}D_{2})f-\gamma S_{ \rm c}^{\rm T}{\rm sgn}(S_{\rm c})\\ &\leq-\Gamma\|\mathbf{S}_{\rm c}\|^{2}-\sigma\|\mathbf{S}_{\rm c}\|-\| \mathbf{S}_{\rm c}\|(\gamma-\|\varepsilon S_{1}D_{1}+S_{2}D_{2}\|M)\end{split}  \tag{41}\]  
where \(\|\mathbf{S}_{\rm c}\|^{2}=S_{\rm c}^{\rm T}S_{\rm c}\). The inequality (41) imply that \(\dot{V}_{\rm C}\leq 0\), while \(V_{\rm C}\) is a positive scalar function. According to the Lyapunov stability theory, it can be concluded that the composite sliding mode surface (37) can be attained in a finite time. This completes the proof. **Remark 4**.