# 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
\[\mathbf{u}_{\rm o}=\mathbf{u}_{\rm s}+\mathbf{u}_{\rm f}=\mathbf{K}_{0}x+\mathbf{K}_{2}\mathbf{z}_{\rm f }=\mathbf{K}_{0}x+\mathbf{K}_{2}(\mathbf{z}-\mathbf{z}_{\rm s}) \tag{18}\]  
Substituting quasi-steady solution (12) into (18) leads to the following nominal control law for the full-order SPS (8) in the original coordinate:  
\[\mathbf{u}_{\rm o} =\mathbf{K}_{0}x+\mathbf{K}_{2}\mathbf{A}_{22}^{-1}(A_{21}x+\mathbf{B}_{2}\mathbf{u}_ {\rm s})+\mathbf{K}_{2}\mathbf{z} \tag{19}\] \[=\left(\mathbf{K}_{0}+\mathbf{K}_{2}\mathbf{A}_{22}^{-1}\mathbf{A}_{21}\right)x+ \mathbf{K}_{2}\mathbf{A}_{22}^{-1}\mathbf{B}_{2}\mathbf{K}_{0}x+\mathbf{K}_{2}\mathbf{z}\] \[=\left[ \begin{array}{cc}\mathbf{K}_{0}+\mathbf{K}_{2}\mathbf{A}_{22}^{-1} \mathbf{B}_{2}\mathbf{K}_{0}+\mathbf{K}_{2}\mathbf{A}_{22}^{-1}\mathbf{A}_{21}&\mathbf{K}_{2}\end{array}  \right]\left[ \begin{array}{c}x\\ \mathbf{z}\end{array} \right]\] \[=\left[ \begin{array}{cc}\mathbf{K}_{1}&\mathbf{K}_{2}\end{array}  \right]\left[ \begin{array}{c}x\\ \mathbf{z}\end{array} \right]\]  
This completes the proof. **Lemma 2** (Singular Perturbation Approach). : _Incorporating an extra control input \(\mathbf{v}\) into the nominal control law (11) results in the composite control law \(\mathbf{u}=\mathbf{u}_{\rm o}+\mathbf{v}\). Substituting the actual \(\mathbf{u}\) into the original full-order system (8), and introducing the following Chang transformation:_  
\[\mathbf{\varphi}=\left[ \begin{array}{c}\xi\\ \mathbf{\eta}\end{array} \right]=\mathbf{N}(\varepsilon)\mathbf{\psi}=\left[ \begin{array} []{cc}1-\varepsilon\mathbf{H}\mathbf{L}&-\varepsilon\mathbf{H}\\ \mathbf{L}&\mathbf{I}\end{array} \right]\left[ \begin{array}{c}x\\ \mathbf{z}\end{array} \right] \tag{20}\]  
_where \(\mathbf{\varphi}\) is the completely decoupled system state vector comprising by the new slow mode \(\xi\) and fast dynamics \(\mathbf{\eta}\); \(\mathbf{\psi}=\mathbf{N}^{-1}(\varepsilon)\mathbf{\varphi}\), and \(\mathbf{N}^{-1}(\varepsilon)=\left[ \begin{array}{cc}1&\varepsilon\mathbf{H}\\ -\mathbf{L}&\mathbf{I}-\varepsilon\mathbf{L}\mathbf{H}\end{array} \right]\); \(\mathbf{I}\) is the identity matrix with appropriate dimensions; \(\mathbf{L}\in R^{2\times 1}\) and \(\mathbf{H}\in R^{1\times 2}\) are the corresponding solutions of the following algebraic equations:_  
\[\varepsilon\mathbf{R}(\mathbf{L})=\mathbf{T}_{21}-\mathbf{T}_{22}\mathbf{L}+\varepsilon\mathbf{L}\mathbf{ A}_{\rm s}=\mathbf{0},\qquad\mathbf{R}(\mathbf{H})=\varepsilon\mathbf{A}_{\rm s}\mathbf{H}-\mathbf{H}\mathbf{A}_{ \rm f}+\mathbf{T}_{12}=\mathbf{0} \tag{21}\]  
_where \(\mathbf{T}_{11}=A_{11}+\mathbf{B}_{1}\mathbf{K}_{1}\), \(\mathbf{T}_{12}=\mathbf{A}_{12}+\mathbf{B}_{1}\mathbf{K}_{2}\), \(\mathbf{T}_{21}=\mathbf{A}_{21}+\mathbf{B}_{2}\mathbf{K}_{1}\) and \(\mathbf{T}_{22}=\mathbf{A}_{22}+\mathbf{B}_{2}\mathbf{K}_{2}\); \(\mathbf{A}_{\rm s}=\mathbf{T}_{11}-\mathbf{T}_{12}\mathbf{L}\) and \(\mathbf{A}_{\rm f}=\mathbf{T}_{22}+\varepsilon\mathbf{L}\mathbf{T}_{12}\); \(\mathbf{0}\) is the zero matrix with appropriate dimensions.__As a result, the exactly complete decoupling subsystems with the principal diagonal form can be eventually derived as_  
\[E(\varepsilon)\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{\varphi}=\bar{\mathbf{A}}\mathbf{\varphi }+\bar{\mathbf{B}}\mathbf{v}+\bar{\mathbf{D}}\mathbf{f} \tag{22}\]  
_where the corresponding matrices are listed as follows_  
\[\bar{\mathbf{A}}=\left[ \begin{array}{cc}A_{\mathrm{s}}&\mathbf{0}\\ \mathbf{0}&\mathbf{A}_{\mathrm{f}}\end{array} \right],\qquad\bar{\mathbf{B}}=\left[ \begin{array}{c}\mathbf{B}_{\mathrm{s}}\\ \mathbf{B}_{\mathrm{f}}\end{array} \right]=\left[ \begin{array}{c}(1-\varepsilon \mathbf{HL})\mathbf{B}_{\mathrm{1}}-\mathbf{HB}_{\mathrm{2}}\\ \varepsilon\mathbf{LB}_{\mathrm{1}}+\mathbf{B}_{\mathrm{2}}\end{array} \right] \tag{23}\] \[\bar{\mathbf{D}}=\left[ \begin{array}{c}\mathbf{D}_{\mathrm{s}}\\ \mathbf{D}_{\mathrm{f}}\end{array} \right]=\left[