## 4 Design of Two-Time-Scale SMO for Five-Phase IM
### Stability Analysis
\begin{array}{c}0=A\epsilon_{x}-\Delta\omega B\hat{x}+\Delta A_{r}( \hat{x}-L_{m}z_{12})+H\\ \dot{\epsilon}_{z34}=D\\ \dot{\epsilon}_{x12}=-A\epsilon_{x}+\Delta\omega B\hat{x}-\Delta A_{r}(\hat{x }-L_{m}z_{12})+MH\end{array} \right. \tag{43}\]  
Now, by considering the speed as a varying parameter, the Lyapunov function can be then chosen as  
\[V_{l}=\frac{1}{2}(\epsilon_{x}^{T}\epsilon_{x}+\frac{1}{g_{1}}(\Delta\omega)^ {2}+\frac{1}{g_{2}}(\Delta A_{r})^{2}) \tag{44}\]  
where \(g_{1}\) and \(g_{2}\) are two positive constants. According to the singular perturbation hypothesis, the speed \(\omega\) is assumed to be with a fixed value and it varies with a very slow rate respecting to the change of the rotor flux and stator current. Thus, the derivative of the Lyapunov function \(V_{l}\) is obtained from (44) by  
\[\dot{V}_{l}=\dot{\epsilon}_{x}^{T}\epsilon_{x}+\frac{1}{g_{1}}\Delta\omega \frac{d\omega}{dt}+\frac{1}{g_{2}}\Delta A_{r}\frac{d\hat{A}_{r}}{dt} \tag{45}\]  
It is clear that the Lyapunov function \(V_{l}\) expressed by (44) is positive definite. Then, by using the first part of (42), the error \(\epsilon_{x}\) is expressed by  
\[\epsilon_{x}=-A^{-1}H+\Delta\omega A^{-1}B\hat{x}-\Delta A_{r}A^{-1}(\hat{x}- L_{m}z_{12}) \tag{46}\]Using (43), the error dynamics \(\dot{e}_{x}\) can be derived by  
\[\dot{e}_{x}=(I+M)H=RH \tag{47}\]  
with \(R=(I+M)\) By substituting from (46) into (45), the following is obtained,  
\[\dot{V}_{I}=-H^{T}R^{T}A^{-1}H+Q_{1}+Q_{2} \tag{48}\]  
where  
\[\left\{ \begin{array}{c}Q_{1}=\Delta\omega(\frac{1}{S_{1}}\frac{d\omega}{dt}+ H^{T}R^{T}A^{-1}B\dot{x})\\ Q_{2}=\Delta A_{r}(\frac{1}{S_{2}}\frac{d\omega_{r}}{dt}-H^{T}R^{T}A^{-1}(\dot{ x}-L_{m}z_{12}))\end{array} \right. \tag{49}\]  
To satisfy the stability condition of Lyapunov approach, the values \(Q_{1}\) and \(Q_{2}\) in (48) must be expressed by  
\[\left\{ \begin{array}{c}Q_{1}=\Delta\omega\Big{(}\frac{1}{S_{1}}\frac{d\omega }{dt}+H^{T}R^{T}A^{-1}B\dot{x}\Big{)}=0.0\\ Q_{2}=\Delta A_{r}\Big{(}\frac{1}{S_{2}}\frac{d\lambda_{r}}{dt}-H^{T}R^{T}A^{ -1}(\dot{x}-L_{m}z_{12})\Big{)}=0.0\\ -H^{T}R^{T}A^{-1}H<0\end{array} \right. \tag{50}\]  
The condition of the third part of (50) is satisfied by choosing  
\[R^{T}A^{-1}=g_{0}I \tag{51}\]  
with \(g_{0}\) is positive constant. A simple adaptive law for the rotor time constant \(\hat{A}_{r}\) and motor speed \(\hat{\omega}\) is given by substituting (51) into the first and second parts of (50), and this yields o  
\[\left\{ \begin{array}{c}\frac{d\hat{A}_{r}}{dt}=g_{0}g_{2}(G_{z12}Is_{smo})^ {T}B\dot{x}\\ \frac{d\hat{\omega}}{dt}=-g_{0}g_{1}(G_{z12}Is_{smo})^{T}(\dot{x}-L_{m}z_{12}) \end{array} \right. \tag{52}\]  
After further simplification of (52), the rotor time constant and speed observer's expressions can be finally defined as the following,  
\[\left\{ \begin{array}{c}\frac{d\hat{A}_{r}}{dt}=g_{0}g_{2}(\gamma_{1}\hat{x} _{2}\text{sign}\ (S_{1})-\gamma_{2}\hat{x}_{1}\text{sign}(S_{2}))\\ \frac{d\hat{\omega}}{dt}=g_{0}g_{1}(\gamma_{1}(\hat{x}_{1}-L_{m}z_{1})\text{ sign}\ (S_{1})\\ +\gamma_{2}(\hat{x}_{2}-L_{m}z_{2})\text{sign}\ (S_{2}))\end{array} \right. \tag{53}\]  
The estimated signal \(\hat{\omega}\) is considered as a switching function which includes high and low frequency components. The speed can be extracted from the low frequency component [53]. Thus, a low-pass filter can be used to estimate the value of \(\hat{\omega}\) as follows,  
\[\hat{\omega}_{lpf}=\frac{1}{1+\kappa.s}\hat{\omega} \tag{54}\]  
where \(\kappa\) is the filter's time constant.