# 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 the subsection, we will mainly concern on the design and analysis of a novel composite sliding mode surface, which is integrated by decoupled subsystem dynamics. For the purpose of satisfying individual desired performances, the Lyapunov equations are involved for singular perturbation decomposition-based exact subsystems. For convenience, we can obtain the following assumption and lemmas motivated by [20]:  
The matrix pairs (\(A_{0}\), \(\mathbf{B}_{0}\)) and (\(\mathbf{A}_{22}\), \(\mathbf{B}_{2}\)) are controllable, where  
\[A_{0}=A_{11}-\mathbf{A}_{12}A_{22}^{-1}\mathbf{A}_{21},\qquad\mathbf{B}_{0}=\mathbf{B}_{1}-\bm {A}_{12}\mathbf{A}_{22}^{-1}\mathbf{B}_{2} \tag{10}\]  
According to eigenvalue placement technique, it is reasonable to assume that there exist state feedback gain matrices \(\mathbf{K}_{0}\) and \(\mathbf{K}_{2}\), thus stabilizing the slow and fast subsystems, respectively. That is to say, their system matrices \(A_{0}+\mathbf{B}_{0}\mathbf{K}_{0}\) and \(\mathbf{A}_{22}+\mathbf{B}_{2}\mathbf{K}_{2}\) will be arbitrarily assigned to become Hurwitz stable. As a result, the following nominal control law can be synthesized:  
\[\mathbf{u}_{\rm o}=\left[ \begin{array}{cc}\mathbf{K}_{1}&\mathbf{K}_{2}\\ \end{array} \right]\left[ \begin{array}{c}x\\ \mathbf{z}\\ \end{array} \right] \tag{11}\]  
where \(\mathbf{K}_{1}=\mathbf{K}_{0}+\mathbf{K}_{2}\mathbf{A}_{22}^{-1}\mathbf{B}_{2}\mathbf{K}_{0}+\mathbf{K}_{ 2}\mathbf{A}_{22}^{-1}\mathbf{A}_{21}\). For the SPS (8), if we do not take the disturbances \(\mathbf{f}\) into account, the corresponding nominal system can be obtained. According to the quasi-steady theory, one can firstly derive the following quasi-steady solution by setting \(\varepsilon=0\):  
\[\mathbf{z}_{\rm s}=-\mathbf{A}_{22}^{-1}(\mathbf{A}_{21}x+\mathbf{B}_{2}\mathbf{u}_{\rm s}) \tag{12}\]  
where the subscript \"s\" represents the slow variable components of the corresponding physical quantities, and \(x=x_{\rm s}\). Based on (12), the original full-order SPS (8) can be approximately equivalent by the following slow-time subsystem:  
\[\frac{\rm d}{\rm d\,t}x=A_{0}x+\mathbf{B}_{0}\mathbf{u}_{\rm s} \tag{13}\]  
and the fast-time subsystem, which is as follows. \[\varepsilon\frac{\rm d}{\rm d\,t}\mathbf{z}_{t}=\mathbf{A}_{22}\mathbf{z}_{t}+\mathbf{B}_{2} \mathbf{u}_{t} \tag{14}\]where the subscript \"\(t\)\" denotes the fast variable components of the above-mentioned corresponding physical quantities, which satisfying \(\mathbf{z}=\mathbf{z}_{\rm s}+\mathbf{z}_{\rm f}\) and \(\mathbf{u}_{\rm o}=\mathbf{u}_{\rm s}+\mathbf{u}_{\rm f}\), respectively. According to Assumption 1, one can design the following slow-time control effort:  
\[\mathbf{u}_{\rm s}=\mathbf{K}_{0}x \tag{15}\]  
which is employed to stabilize the slow-time subsystem (13), resulting in  
\[\lambda(A_{0}+\mathbf{B}_{0}\mathbf{K}_{0})<0 \tag{16}\]  
where \(\lambda(\cdot)\) represent the eigenvalue. Obeying the same procedure as for the slow-time scale subsystem, we can also find the following fast-time control law:  
\[\mathbf{u}_{\rm f}=\mathbf{K}_{2}\mathbf{z}_{\rm f} \tag{17}\]  
which will stabilize the fast-time subsystem (14), namely the closed-loop system matrix \(\mathbf{A}_{22}+\mathbf{B}_{2}\mathbf{K}_{2}\) is asymptotically stable. Finally, we can synthesize the nominal controller by combining the individual slow-time and fast-time linear feedback control law (15) and (17), which is as follows.