# An Improved Hybrid Control Scheme of a Switched Reluctance Motor for Torque Ripple Reduction
## 3 Model Predictive Torque Control  
Figure 4 shows the control diagram of the MPTC scheme, whereby the hysteresis control is replaced by MPC to further reduce the torque ripple. First, the current and position at the next sampling moment (\(k+1\)) are predicted by current and position at the current moment (\(k\)) according to the discrete model. Then, the phase torque at the sampling moment (\(k+1\)) can be achieved by the torque modeling method. Lastly, the cost function is established based on a comparison between the predicted and the reference values, thereby selecting the optimal voltage vector and corresponding switching state.  
According to the phase voltage Equation (1), which is executed for each phase at any time,  
\[\frac{\mathrm{d}i_{p}}{\mathrm{d}t}=\frac{1}{\partial\phi_{p}/\partial\theta}( U_{p}-R_{p}i_{p}-\frac{\partial\Psi_{p}}{\partial\theta}\omega) \tag{8}\]  
Discretizing (8), the phase current and position at the sampling moment (\(k+1\)) can be predicted by [28]  
\[\left\{\begin{array}{l}\theta_{k+1}=\theta_{k}+\omega_{k}T_{s}\\ i_{p,k+1}=i_{p,k}+\frac{T_{s}}{\partial\Psi_{p}/\partial\theta_{k}^{\prime}}( U_{p,k}-R_{p}i_{p,k}-\frac{\partial\Psi_{p,k}}{\partial\theta_{k}}\omega_{k}) \end{array}\right. \tag{9}\]  
where \(T_{s}\) is the sampling time; \(\psi_{p,k}\), \(\theta_{k}\), \(\omega_{k}\), and \(i_{p,k}\) are the phase flux linkage, rotor position, angular speed, and phase current, respectively, at the current moment (\(k\)).  
Next, the phase torque at the sampling moment (\(k+1\)) can be calculated as  
\[T_{p,k+1}=[\frac{L_{\mathrm{b}}-L_{\mathrm{c}}}{2}i_{p,k+1}{}^{2}+Aip_{k+1}- \frac{A}{B}(1-e^{-Bi_{p,k+1}})]f(\theta_{k+1}) \tag{10}\]  
The phase voltages differ depending on the switching state, resulting varying generated torques. A typical asymmetric half-bridge converter is utilized in this study, so three switching states of power converter can be analyzed, as shown in Figure 5.  
Figure 4: The control block of the MPTC scheme.  
Figure 3: The cosine and proposed TSF.  
As shown, states \"1\", \"0\", and \"\(-1\)\" indicate that both the switches are on, only one of the switches is off, and both switches are off, respectively. Accordingly, the phase voltage \(U_{p}\) is defined as  
\[U_{p}=\left\{\begin{array}{ll}U_{\rm b}-2U_{\rm T},&1\\ -U_{\rm T}-U_{\rm D},&0\\ -U_{\rm b}-2U_{\rm D},&-1\end{array}\right. \tag{11}\]  
where \(U_{\rm b}\), \(U_{\rm T}\), and \(U_{\rm D}\) are the bus voltage, voltage drop in the power switch, and voltage drop in the diode, respectively.  
In order to better track the desired torque, it is necessary to define the selection criteria of the optimal control quantity and evaluate a series of predicted values using the cost function. Considering torque ripple and copper loss, the cost function can be expressed as [41]  
\[J=\sum_{p=1}^{N_{p}}\left(q_{1}(T_{p,k+1}-{T_{p}}^{*})^{2}+q_{2}i_{p,k+1}{}^{2}\right) \tag{12}\]  
where \(q_{1}\) and \(q_{2}\) are the weight coefficients of torque and current, respectively.  
As for MPTC, the desired torque corresponding to different positions of each phase is determined by the torque distribution function; the current value of the determined position of each phase is also determined. Therefore, consideration of the current in the objective function is not necessary. The cost function can be simplified, and only the torque term is retained. There is almost no difference between the instantaneous torque at time \(k+1\) and \(k\) because the sampling time is sufficiently short, so the cost function (\(J\)) is expressed as  
\[J=\sum_{p=1}^{N_{p}}\left(T_{p,k+1}-{T_{p}}^{*}\right)^{2} \tag{13}\]  
The optimal switching state can be obtained by the minimum value of the cost function for the next sampling period.