# 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
\begin{array}{c}(1-\varepsilon \mathbf{HL})\mathbf{D}_{\mathrm{1}}-\mathbf{HD}_{\mathrm{2}}\\ \varepsilon\mathbf{LD}_{\mathrm{1}}+\mathbf{D}_{\mathrm{2}}\end{array} \right]\]  
**Remark 2**: _According to the algebraic Equation (21), we can easily obtain that \(\mathbf{T}_{21}-\mathbf{T}_{22}\mathbf{L}+\varepsilon\mathbf{LA}_{\mathrm{s}}=\mathbf{T}_{21}-\bm {T}_{22}\mathbf{L}+O(\varepsilon)\) and \(\varepsilon\mathbf{A}_{\mathrm{s}}\mathbf{H}-\mathbf{H}(\mathbf{T}_{22}+\varepsilon\mathbf{LT}_{ \mathrm{12}})+\mathbf{T}_{\mathrm{12}}=\mathbf{T}_{\mathrm{12}}-\mathbf{HT}_{22}+O(\varepsilon)\), resulting in the initial matrix values \(\mathbf{L}^{(0)}=\mathbf{T}_{22}^{-1}\mathbf{T}_{21}\) and \(\mathbf{H}^{(0)}=\mathbf{T}_{\mathrm{12}}\mathbf{T}_{22}^{-1}\), respectively, where \(O(\varepsilon)\) represent the \(\varepsilon\)-dependent infinitesimal value. In addition, by employing the fixed-point recursive algorithm, one can calculate the solutions of the above formulations, which are as follows: \(\mathbf{L}^{(i+1)}=\mathbf{T}_{22}^{-1}\Big{[}\mathbf{T}_{21}+\varepsilon\mathbf{L}^{(i)} \mathbf{T}_{\mathrm{11}}-\varepsilon\mathbf{L}^{(i)}\mathbf{T}_{\mathrm{12}}\mathbf{L}^{(i)} \Big{]}\) and \(\mathbf{H}^{(j+1)}=\left[\varepsilon(\mathbf{T}_{\mathrm{11}}-\mathbf{T}_{\mathrm{12}}\bm {L})\mathbf{H}^{(j)}+\mathbf{T}_{\mathrm{12}}\right]\big{(}\mathbf{T}_{22}+\varepsilon\mathbf{ LT}_{\mathrm{12}}\big{)}^{-1}\), where \(i\geq 0\) and \(j\geq 0\) are the iteration numbers (non-negative integers) for \(\mathbf{L}\) and \(\mathbf{H}\), respectively._  
For the presented exact decoupled SPSs (22), we have the following theorem. **Theorem 1**: _Under Lemmas 1 and 2, the diagonal system matrix \(\bar{\mathbf{A}}\) in (22) is asymptotically stable. To this end, there exists a positive definite symmetric matrix \(\mathbf{P}\) satisfying the following Lyapunov equation for the any given positive definite symmetric matrix \(\mathbf{Q}\):_  
\[\bar{\mathbf{A}}^{\mathrm{T}}\mathbf{P}+\mathbf{P}\bar{\mathbf{A}}=-\mathbf{Q} \tag{24}\]  
Proof. : First of all, we can easily obtain the following closed-loop system matrix from the reduced-order subsystem dynamic \(\xi\):  
\[A_{\mathrm{s}}  
It can be concluded form Lemma 1 that the matrix \(A_{22}+B_{2}K_{2}\) is asymptotically stable, and thus there exists a small \(\varepsilon^{*}>0\) such that the eigenvalues of \(A_{\rm f}\) have negative real parts for all \(\varepsilon\in(0,\varepsilon^{*}]\). Therefore, for the any given positive definite symmetric matrix \(\mathbf{Q}_{\rm f}\), there exists a positive definite symmetric matrix \(\mathbf{P}_{\rm f}\) satisfying the following Lyapunov equation:  
\[\mathbf{A}_{\rm f}^{\rm T}\mathbf{P}_{\rm f}+\mathbf{P}_{\rm f}\mathbf{A}_{\rm f}=-\mathbf{Q}_{\rm f} \tag{29}\]  
As a result, according to the well-known separation principle, the closed-loop system matrix \(\mathbf{\bar{A}}\) is asymptotically stable. At the same time, according to individual Lyapunov Equations (27) and (29), it can be summarized that there exists the following Lyapunov equation for system (22):  
\[\mathbf{\bar{A}}^{\rm T}\mathbf{P}+\mathbf{P}\mathbf{\bar{A}}=-\mathbf{Q} \tag{30}\]  
where the diagonal matrix \(\mathbf{P}=\text{diag}\{P_{\rm s},\mathbf{P}_{\rm f}\}\) and \(\mathbf{Q}=\text{diag}\{Q_{\rm s},\mathbf{Q}_{\rm f}\}\), respectively. This completes the proof. The above-mentioned theorem indicates that the eigenvalues of the diagonal system matrix \(\mathbf{\bar{A}}\) can be approximately equivalent to the eigenvalue combinations of the corresponding slow-subsystem matrix \(A_{\rm s}\) and fast-subsystem matrix \(\mathbf{A}_{\rm f}\). According to Assumption 1, both of their matrix pairs can be stabilized based on the eigenvalue placement technique. As a result, the corresponding subsystem dynamics are asymptotically stable, simultaneously. According to Theorem 1, we can conclude that the singular perturbation decomposition method-based exact decoupled system state vector \(\mathbf{\varphi}\) are asymptotically stable with satisfying the Lyapunov Equation (24).