# A Singular Perturbation Approach-Based Non-Cascade Sliding Mode Control for Surface-Mounted PMSMs
## 3 Singular Perturbation Approach-Based SMC
### Composite Sliding Mode Surface Design and Stability Analysis
In order to guarantee the desired system performance, we construct a novel composite sliding mode surface associating with the system dynamics (22), which results the following theorem. Based on the Lyapunov approach, a novel composite sliding mode surface is constructed for original SPS (8), which is as follows  
\[\mathbf{S}=\mathbf{P}\mathbf{N}(\varepsilon)\mathbf{\psi} \tag{31}\]  
Meanwhile, the system (22) is asymptotically stable on the above designed sliding mode surface during the sliding mode. Proof. : First of all, we can construct the following sliding mode surface:  
\[\mathbf{S}=\mathbf{P}\mathbf{\varphi} \tag{32}\]  
When the sliding mode is arrived, the equivalent control method will be commonly adopted to analyze the system stability. As a result, the following formulation will be obtained:  
\[\mathbf{S}=\mathbf{\dot{S}}=\mathbf{0} \tag{33}\]  
Introducing the following Lyapunov function:  
\[V_{\rm S}=\mathbf{\varphi}^{\rm T}\mathbf{P}E(\varepsilon)\mathbf{\varphi} \tag{34}\]  
Calculating the time-derivative of the \(V_{\rm S}\), leads to \[\dot{V}_{\rm S} =[E(\varepsilon)\dot{\mathbf{\phi}}]^{\rm T}P\mathbf{\varphi}+\mathbf{\varphi}^{ \rm T}P[E(\varepsilon)\dot{\mathbf{\phi}}]\] \[=\left[ \begin{array}{c}\mathbf{\varphi}\\ \mathbf{v}\\ \mathbf{f}\end{array} \right]^{\rm T}\left[ \begin{array}{cc}\bar{A}^{\rm T}\\ \bar{B}^{\rm T}\\ \bar{D}^{\rm T}\end{array} \right]\left[ \begin{array}{cc}\mathbf{P}&\mathbf{0}&\mathbf{0 }\end{array} \right]\left[ \begin{array}{c}\mathbf{\varphi}\\ \mathbf{v}\\ \mathbf{f}\end{array} \right]+\left[ \begin{array}{c}\mathbf{\varphi}\\ \mathbf{v}\\ \mathbf{f}\end{array} \right]^{\rm T}\left[ \begin{array}{cc}\mathbf{P}\\ \mathbf{0}\\ \mathbf{0}\end{array} \right]\left[ \begin{array}{cc}\bar{A}&\bar{B}&\bar{D}\\ \mathbf{f}\end{array} \right]\left[ \begin{array}{c}\mathbf{\varphi}\\ \mathbf{v}\\ \mathbf{f}\end{array} \right]\] \[=\left[ \begin{array}{c}\mathbf{\varphi}\\ \mathbf{v}\\ \mathbf{f}\end{array} \right]^{\rm T}\left(\left[ \begin{array}{cc}\bar{A}^{\rm T} \\ \mathbf{B}^{\rm T}\\ \mathbf{D}^{\rm T}\end{array} \right]\left[ \begin{array}{cc}\mathbf{P}&\mathbf{0}&\mathbf{0 }\end{array} \right]+\left[ \begin{array}{c}\mathbf{P}\\ \mathbf{0}\\ \mathbf{0}\\ \mathbf{0}\end{array} \right]\left[ \begin{array}{cc}\bar{A}&\bar{B}&\bar{D}\\ \mathbf{f}\end{array} \right]\right)\left[ \begin{array}{c}\mathbf{\varphi}\\ \mathbf{v}\\ \mathbf{f}\end{array} \right] \tag{35}\] \[=\left[ \begin{array}{cc}\mathbf{\varphi}^{\rm T}&\mathbf{v}^{\rm T}& \mathbf{f}^{\rm T}\end{array} \right]\left[ \begin{array}{ccc}\bar{A}^{\rm T}P+P \bar{A}&P\bar{B}&P\bar{D}\\ \bar{B}^{\rm T}P&\mathbf{0}&\mathbf{0}\end{array} \right]\left[ \begin{array}{c}\mathbf{ \varphi}\\ \mathbf{v}\\ \mathbf{f}\end{array} \right]\] \[=\mathbf{\varphi}^{\rm T}\Big{(}\bar{A}^{\rm T}P+P\bar{A}\Big{)}\mathbf{ \varphi}+\mathbf{v}^{\rm T}\bar{B}^{\rm T}P\mathbf{\varphi}+\mathbf{f}^{\rm T}\bar{D}^{\rm T }P\mathbf{\varphi}+\mathbf{\varphi}^{\rm T}P\mathbf{B}\mathbf{v}+\mathbf{\varphi}^{\rm T}P\mathbf{D}f\] \[=-\mathbf{\varphi}^{\rm T}Q\mathbf{\varphi}+2\Big{(}\mathbf{v}^{\rm T}\bar{B}^ {\rm T}+\mathbf{f}^{\rm T}\bar{D}^{\rm T}\Big{)}S\]  
According to the Lyapunov Equation (30) and the formulation (33), it can be concluded that the system (22) is asymptotically stable, when the system state trajectory is driven and strictly restricted onto the sliding mode surface (32). On the other hand, based on the Chang transformation matrix \(N(\varepsilon)\) presented in (20), we can rewrite the sliding mode surface (32) into the original coordinate, which is shown as (31). This completes the proof.