## 4 Design of Two-Time-Scale SMO for Five-Phase IM
### Sliding Mode Observer (SMO)  
Using the general approach of singular perturbation of five-phase IM model without incorporating the mechanical part, the mathematical expression of the SMO can be expressed by  
\[\left\{ \begin{array}{c}\zeta\dot{z_{2}}=-L_{m}\hat{A}_{r}z_{1}+\hat{A}_{r} \hat{x}_{1}+\hat{\alpha}\hat{x}_{2}+\frac{L_{r}}{L_{m}}(v_{\text{s}\alpha}-R_{ \text{s}}z_{1})+z_{i\alpha}\\ \dot{\zeta\dot{z_{2}}}=-L_{m}A_{r}z_{2}+\hat{A}_{r}\hat{x}_{2}-\hat{\alpha}\hat {x}_{1}+\frac{L_{r}}{L_{m}}(v_{\text{s}\beta}-R_{\text{s}}z_{2})+z_{i\beta}\\ \dot{\xi_{1}}=L_{m}\hat{A}_{r}z_{1}-\hat{A}_{r}\hat{x}_{1}-\hat{\alpha}\hat{x} _{2}+z_{\text{s}\alpha}\\ \dot{\xi_{2}}=L_{m}\hat{A}_{r}z_{2}+\hat{\alpha}\hat{x}_{1}-\hat{A}_{r}x_{2}+z_{ x\beta}\\ \dot{z_{3}}=-A_{\text{s}}z_{3}+\frac{1}{L_{\text{is}}}v_{\text{s}x}+z_{x}\\ \dot{z_{4}}=-A_{s}z_{4}+\frac{1}{L_{\text{is}}}v_{\text{s}y}+z_{y}\end{array} \right. \tag{24}\]where the accent \"\(\cdot\)\" denotes to the estimated variables, and \(\hat{A}_{r}\), \(\hat{\alpha}_{1}\), \(\hat{x}_{2}\), \(\hat{\omega}\), \(\hat{z}_{1}\), and \(\hat{z}_{2}\) are the estimated values of \(A_{r},x_{1},x_{2}\)\(\omega\), \(z_{1}\), and \(z_{2}\), whereas \(z_{i\alpha},z_{i\beta}\), \(z_{i\alpha}\), \(z_{x\beta}\), \(z_{x}\), and \(z_{y}\) are the correction terms which are used for achieving the desired performance, these terms can be expressed as follows. \[\left\{ \begin{array}{l}Z_{i\alpha}=G_{z1}I_{smo}\\ Z_{i\beta}=G_{z2}I_{smo}\\ Z_{x\alpha}=G_{x1}I_{smo}\\ Z_{x\beta}=G_{x2}I_{smo}\\ Z_{x}=G_{z3}T_{smo}\\ Z_{y}=G_{z4}T_{smo}\end{array} \right. \tag{25}\]  
The switching vectors \(I_{smo}\) and \(T_{smo}\) are chosen as  
\[\left\{I_{smo}=\left( \begin{array}{c}sign(S_{1})\\ sign(S_{2})\end{array} \right)\right.=\left( \begin{array}{c}\hat{z}_{1}-z_{1 }\\ \hat{z}_{2}-z_{2}\end{array} \right)=\left( \begin{array}{c}e_{z1}\\ e_{z2}\end{array} \right) \tag{26}\]  
\[\left\{T_{smo}=\left( \begin{array}{c}sign\left(S_{1}^{\prime}\right)\\ sign\left(S_{2}^{\prime}\right)\end{array} \right)=\left( \begin{array}{c} \hat{z}_{3}-z_{3}\\ \hat{z}_{4}-z_{4}\end{array} \right)=\left( \begin{array}{c}e_{z3}\\ e_{z4}\end{array} \right) \tag{27}\]  
Now, for estimating the speed and flux, the following hypotheses are considered. **Hypothesis 1**. : _Stator voltages and currents are measurable._  
**Hypothesis 2**. : _Rotor resistance is the only unknown parameter and the rest of motor parameters are known and constant._  
**Hypothesis 3**. : _The rotor resistance \(R_{r}\) is assumed to be unknown quantity and its nominal value is \(R_{m}\)._  
**Hypothesis 4**. : _Motor speed is considered as a time varying unknown bounded variable._  
**Hypothesis 5**. : _The variation of rotor resistance \(R_{r}\) is assumed to be very slow, so that \(\hat{R}_{r}\approx 0\)._  
By setting \(\varepsilon_{zj}=\hat{z}_{j}-z_{j}\) as the estimation current errors for \(j\in[1\ 2\ 3\ 4]\), \(\varepsilon_{xi}=\hat{x}_{j}-x_{j}\) as the estimation flux errors for \(i\in[1\ 2]\), \(\Delta\omega=\hat{\omega}-\omega\) as the estimation speed error, and \(\Delta A_{r}=\hat{A}_{r}-A_{r}\) as the rotor time constant estimation error, the estimation error dynamics can be evaluated by subtracting (23) from (24), which results in  
\[\left\{