## 4 Design of Two-Time-Scale SMO for Five-Phase IM
### Stability Analysis  
Via utilizing the two-time-scale technique [52], checking the stability of the system under study and which is described by the matrix gains \(G_{z12},\ G_{z2},\ G_{z3}\), and \(G_{z4}\) of the measured state variables can be effectively accomplished. Subsequently, \(G_{x1}\) and \(G_{x2}\) of the slow subsystem or unmeasured states are evaluated so that the reduced-order model when \(S\equiv\dot{S}=0\) is internally stable. Using the value of \(\tau=(t-t_{0})/\zeta\) and after setting the value of \(\zeta=0\) in (29), the observation errors can be represented in the form of fast reduced order system as follows,  
\[\left\{ \begin{array}{c}\frac{de_{z12}}{d\tau}=Ae_{x}-\Delta\omega B\hat{x}+ \Delta A_{r}(\hat{x}-L_{m}z_{12})+G_{z12}I_{smo}\\ \dot{e}_{z34}=G_{z34}T_{smo}\\ \frac{de_{z12}}{d\tau}=0\end{array} \right. \tag{32}\]  
**Remark 1**. : _To obtain \(S=S^{\prime}=e_{z12}=e_{z34}=0\), a precise selection of the matrix gains \(G_{z1},\ G_{z2},\ G_{z3}\), and \(G_{z4}\) must be performed._  
Assuming that the flux errors \(e_{x1}\) and \(e_{x2}\) are boundaried in this time-scale and \(x\) changes slowly, then by considering the first part of (32), the observer gains matrices can be defined by  
\[G_{z12}=\left( \begin{array}{cc}-\gamma_{1}&0\\ 0&-\gamma_{2}\end{array} \right) \tag{33}\]  
\[G_{z13}=\left( \begin{array}{cc}-\delta_{1}&0\\ 0&-\delta_{2}\end{array} \right) \tag{34}\]  
The convergence condition of the sliding surface \(S=0\), which is verified by the inequality in (36), can be then expressed by the following formulations. \[S^{T}\frac{dS}{d\tau}<0 \tag{35}\]  
\[\left\{ \begin{array}{c}\gamma_{1}>\left|A_{r}e_{x1}+\omega e_{x2}+\Delta \omega\hat{x}_{2}+\Delta A_{r}(\hat{x}_{1}-L_{m}z_{1})\right|\\ \gamma_{2}>\left|A_{r}e_{x2}-\omega e_{x1}-\Delta\omega\hat{x}_{1}+\Delta A_{ r}(\hat{x}_{2}-L_{m}z_{2})\right|\\ \delta_{1}>max\left|e_{z3}\right|\\ \delta_{2}>max\left|e_{z4}\right|\end{array} \right. \tag{36}\]  
**Proof. ** Using the definite positive Lyapunov function expressed in (37) and its derivate in the time scale given by (38),  
\[V=\frac{S^{T}S+S^{\prime T}S^{\prime}}{2} \tag{37}\]\[\dot{V}=S^{T}\frac{dS}{d\tau}+S^{\prime T}\frac{dS^{\prime}}{d\tau} \tag{38}\]  
\(\Box\)  
Then, via replacing (30) into (38), \(\dot{V}\) becomes  
\[\dot{V}=S^{T}(A\epsilon_{x}-\Delta\omega B\hat{x}+\Delta A_{r}(\hat{x}-L_{m}z_{ 12})+G_{z12}I_{smo})+S^{\prime T}G_{z34}T_{smo} \tag{39}\]  
Using the values of \(G_{z12}\) and \(G_{z34}\), which are given, respectively, in Equations (33) and (34), the relationship (39) becomes  
\[ \begin{array}{rl}\dot{V}=&-S_{1}(\gamma_{1}sign(S_{1})-A_{r}\epsilon_{x1}- \omega e_{x2}-\Delta\omega\hat{x}_{2}-\Delta A_{r}(\hat{x}_{1}-L_{m}z_{1}))\\ &-S_{2}(\gamma_{2}sign(S_{2})-A_{r}\epsilon_{x2}+\omega e_{x1}+\Delta\omega x_ {1}-\Delta A_{r}(\hat{x}_{2}-L_{m}z_{2}))\\ &-S_{1}^{\prime}\big{(}\Omega_{1}sign\big{(}S_{1}^{\prime}\big{)}\big{)}-S_{2 }^{\prime}\big{(}\Omega_{2}sign\big{(}S_{2}^{\prime}\big{)}\big{)}\end{array}  \tag{40}\]  
Taking into account the constraints of all parameters and states of five-phase IM, the derivative of Lyapunov function can be written by  
\[\dot{V}=S^{T}\frac{dS}{d\tau}+S^{\prime T}\frac{dS^{\prime}}{d\tau}<0 \tag{41}\]  
**Remark 2**. : _Once the stator currents access the sliding surface \(S=S^{\prime}=\epsilon_{z}=0\), the system errors can be evaluated by (29)._  
Using the system equation of (29) and setting \(\zeta=0\) for slow error dynamics, one can write  
\[\left\{ \begin{array}{c}0=A\epsilon_{x}-\Delta\omega B\hat{x}+\Delta A_{r}( \hat{x}-L_{m}z_{12})+G_{z12}I_{smo}\\ \dot{\epsilon}_{z34}=G_{z34}T_{smo}\\ \dot{\epsilon}_{x12}=-A\epsilon_{x}+\Delta\omega B\hat{x}-\Delta A_{r}(\hat{x }-L_{m}z_{12})+G_{x12}I_{smo}\end{array} \right. \tag{42}\]  
Choosing matrices \(H=G_{z12}I_{smo}\), \(D=G_{z34}T_{smo}\), and \(M=G_{x12}G_{z12}{}^{-1}\) represented in the real time scale, the relationship (42) becomes  
\[\left\{