## 3 Modelling of the Proposed Controller and Observer
### Design of the SMO  
The design parameters of the SMO are presented in Table 1 (see in Appendix A.3). The sliding mode observer states for the rotor displacements are modelled from individual suspension currents and voltages and are given in Equation (15):  
\[\overset{\wedge}{X}_{2}=\frac{k}{m}x_{1}+\mathrm{K}_{x}I_{x}+\mathrm{K}_{y}I_{y }+\mathrm{K}_{\mathrm{x}2}\mathrm{sign}\left(\overset{\wedge}{I}_{x}-\mathrm{I} _{x}\right);\overset{\wedge}{Y}_{2}=\frac{k}{m}y_{1}+\mathrm{K}_{y}I_{y}+ \mathrm{K}_{x}I_{x}+\mathrm{K}_{y2}\mathrm{sign}\left(\overset{\wedge}{I}_{y}- \mathrm{I}_{y}\right) \tag{15}\]  
Similarly, the motor equivalent observer state vectors' current, torque, flux linkages, speed and positions are modelled from the actual motor main winding phase voltages and currents, which are given collectively in Equations (16) to (20):  
\[\overset{\wedge}{i}=\mathrm{N}(\overset{\wedge}{\theta})\overset{\wedge}{ \psi} \tag{16}\]  
\[\overset{\wedge}{T}_{e}=\overset{\wedge}{T}_{e}(\overset{\wedge}{i},\theta) \tag{17}\]  
\[\frac{d\overset{\wedge}{\psi}}{\mathrm{dt}}=-\mathrm{r}\mathrm{N}(\theta) \hat{\psi}+\mathrm{V}+\mathrm{K}_{\psi}\mathrm{sign}(\overset{\wedge}{i}- \mathrm{i}) \tag{18}\]  
\[\frac{d\overset{\wedge}{w}}{\mathrm{dt}}=\frac{\overset{\wedge}{\mathrm{Te} }}{J}-\frac{B}{J}\overset{\wedge}{w}+\mathrm{K}_{w}\mathrm{sign}(\overset{ \wedge}{i}-\mathrm{i}) \tag{19}\]  
\[\frac{d\overset{\wedge}{\theta}}{\mathrm{dt}}=\overset{\wedge}{w}+\mathrm{K }_{\theta}\mathrm{sign}(\overset{\wedge}{i}-\mathrm{i}) \tag{20}\]  
From the above SMO equations, the notation \(\wedge\) indicates the measured value. The \(\mathrm{K}_{\Psi}=200\), \(\mathrm{K}_{\mathrm{W}}=30\) and \(\mathrm{K}_{\theta}=25\) are the design parameters for the SMO, which are called the switching gain constants used as gain multipliers to the current error signal [21; 22; 23]. The equivalent structure of the sliding mode observer is shown in Figure 5. The sufficient condition of the sliding mode surface  
Figure 4: Block diagram of the dynamic sliding mode control.  
\((S^{\tau}\overset{\bullet}{S}\ \leq\ 0)\) is considered to get the perfect system sliding mode property, even though under the presence of any faults. The switching dynamics are defined as shown in Equations (21) and (22):  
\[\text{S}=\overset{\wedge}{i}-\text{i}=\left[\begin{array}{cccc}N_{0}&0&M_{0} \end{array}\right]\left[\begin{array}{cccc}\overset{\wedge}{\psi}-\psi& \overset{\wedge}{w}-\text{w}&\overset{\wedge}{\theta}-\theta\end{array}\right]^{T} \tag{21}\]  
\[\text{S}=\text{N}_{0}e_{\psi}+\text{M}_{0}e_{\theta},\ \overset{\bullet}{S}=\text{N}_{0}e_{\psi}^{ \bullet}+\text{M}_{0}e_{\theta}^{\bullet} \tag{22}\]  
The detailed block diagram of the sensorless control BSRM is shown in Figure 6. The reference displacements are started at (0, 0), and the reference speed is fixed at 9000 rpm.  
Figure 5: Block diagram of the sliding mode observer.  
Figure 6: Overall sensorless control block diagram of the BSRM.  
The resultant error signals are sent to the controller and generate the controlled command force signals of F\({}_{\text{x}}\) * and F\({}_{\text{y}}\) *. These command force signals are given to the current hysteresis control logic block, which excites the four-phase asymmetric converter. According to the generated current hysteresis control signals, the four-phase converter directs the controlled currents to the suspension windings. Finally, the controlled suspension magnetic force is produced, and it suspends the rotor to the centre position. The speed is calculated from the rotor position using the speed estimator block, which is further compared with the speed command W *(reference speed). The proposed controller in the speed loop produces the current command signal Im* according to the resultant speed error input signal. The current hysteresis control block generates the sign-enabled signals, which are applied to the two-phase asymmetric converter. This two-phase converter excites the stator main windings according to the control signals.