## 3 Theoretical Background of the Predictive Torque Control  
### Predictive Torque Control Based on PWM  
Many control strategies have been proposed to enhance the torque control performance of the SRM [31; 33]. The core idea of the MPTC is to carry out the control process in steps, which is very suitable for the discrete working mode of the SRM. The PWM technology allows precise adjustment of the phase output torque at each sampling time. Nevertheless, the operating states of the converter are discretized into a finite set making the conventional MPTC unsuitable for PWM. Therefore, it is necessary to propose an improved MPTC method. The structure diagram of the improved MPTC method is shown in Figure 2.  
In Figure 2, the total reference torque signal \(T_{ref}\) is output by the speed controller and used as an input to the torque controller. Then, the duty cycle of each phase is calculated using the phase reference torque and output to the PWM unit. Finally, the converter controls the SRM to output the corresponding torque. In this paper, the speed controller is a proportional-integral (PI) controller, and the torque controller is designed by the control algorithm proposed in this section. In the torque controller, the phase reference torque is  
Figure 2: Block diagram of SRM drive system based on the improved MPTC. output by the torque prediction sharing unit. Therefore, to make the phase torque track its reference signal, (5) is usually used to calculate the duty cycle. \[\alpha=\frac{V_{p}}{V_{dc}} \tag{5}\]  
Therefore, it is possible to set the phase voltage to any value between \(-V_{dc}\) and \(V_{dc}\) by adjusting the duty cycle. The phase electric balance equation of the SRM is  
\[V_{p}=i_{p}R+\frac{d\psi_{p}}{dt} \tag{6}\]  
Since the phase resistance is very small, the phase voltage in Equation (6) can be considered to be mainly relevant to the change rate of the phase flux linkage. Furthermore, Ref. [20] shows that the phase torque can be controlled by adjusting the phase flux linkage. Using the standard forward-Euler approximation, the discrete expression of (6) is obtained as  
\[V_{p}(k)=i_{p}(k)R+\frac{\Delta\psi_{p}(k+1)}{\Delta t} \tag{7}\]  
where \(\Delta t\) is the discrete control period, \(i_{p}(k)\) is the sampling current at the starting of the control period \(k\), and \(\Delta\psi_{p}(k+1)\) is the expected change of flux linkage at the end of this control period (or the starting of the next control period \(k+1\)). Therefore, it is necessary to extrapolate the rotor position at the end of the period. Assuming that the phase reference torque \(T_{p,ref}\) and speed \(\omega_{r}\) are constant in each control period, we have  
\[\theta_{p}(k+1)=\theta_{p}(k)+\omega_{r}\Delta t \tag{8}\]  
The phase reference flux linkage \(\psi_{p,ref}(k+1)\) and the phase flux linkage \(\psi_{p}(k)\) are obtained from the LUT \(i_{p}(T_{p},\theta_{p})\) and LUT \(\psi(i_{p},\theta_{p})\). The expected change of the flux linkage is described as  
\[\Delta\psi_{p}(k+1)=\psi_{p,ref}(k+1)-\psi_{p}(k) \tag{9}\]  
Further, by combining (9) and (7), the duty cycle can be rewritten as  
\[\alpha=\frac{V_{p}}{V_{dc}}=\frac{i_{p}(k)R+\frac{\psi_{p,ref}(k+1)-\psi_{p}(k )}{\Delta t}}{V_{dc}} \tag{10}\]  
Therefore, with the reference phase torque given, the phase torque can track its reference signal through the duty cycle calculated from (10). Based on the excitation sequence of the three-phase 12/8-pole SRM, its excitation period (360\({}^{\circ}\)/8) can be divided into six parts, as shown in Figure 3. In the single-phase excitation interval, the total reference torque is served as the reference signal for the excitation phase. During commutation, the two consecutive phases overlap, and the relation (11) should be satisfied. \[T_{ref}=T_{in,ref}(k+1)+T_{ou,ref}(k+1) \tag{11}\]  
where \(T_{ref}\) is the total reference torque, and \(T_{in,ref}\) and \(T_{ou,ref}\) are the reference torques of the incoming phase (excitation phase) and outgoing phase (demagnetization phase), respectively.