## 3 Optimization of Controllable Rectification Based on Advanced Control Strategies
### MPSMC Control Strategy
#### 3.3.1 Design of Model Predictive Controller  
The principle of model predictive control is shown in Figure 11, and its basic process is:  
1. Establish a predictive model. Establish a predictive model based on the current system model and predict the future state of the system based on its current state. 2. Perform rolling optimization. Build a suitable cost function to determine the best control quantity for upcoming systems. 3. System feedback correction. Model predictive control can provide feedback correction for errors based on the current state of system. The model predictive control-based rectification strategy's main process is shown in Figure 12.  
Figure 10: System double closed loop control block diagram. In the control strategy of three-phase controlled rectification, coupling occurs between the d-axis current and q-axis current. To better design the model predictive control loop, the mathematical equations of the current loop need to be rewritten as follows:  
\[ \begin{cases}\frac{d{{i_{d}}}}{{{H}}}=-\frac{R}{L}{{i_{d}}}+\frac{{{U_{d}}}}{{{L }}}\\ \frac{d{{i_{q}}}}{{{d{t}}}}=-\frac{R}{L}{{i_{q}}}+\frac{{{U_{q}}}}{{{L}}}\\ \end{cases}  \tag{8}\]  
In the formula, \(R\) is the resistance of the coil in the linear motor, \(L\) is the inductance in the linear motor, assuming a sampling period of \(T_{s}\),  
Among them, \(U_{d}\) and \(U_{q}\) are the voltage drops on the \(dq\) axis of the linear motor coil, which can be expressed as:  
\[ \begin{cases}U_{d}=e_{d}-V_{d}+\omega Li_{q}\\ U_{q}=e_{q}-V_{q}-\omega Li_{d}\\ \end{cases}  \tag{9}\]  
In the formula, \(V_{d}\) and \(V_{q}\) represent the components along the \(dq\) axis on the AC side of the linear motor, which can be expressed as:  
\[ \begin{cases}V_{d}=U_{dc}\cdot S_{d}\\ V_{q}=U_{dc}\cdot S_{q}\\ \end{cases}  \tag{10}\]  
Figure 11: Model Predictive Control Structure. Figure 12: Model predictive control flow chart. Based on the above analysis, the above equation is rewritten into discrete form. \[ \begin{cases}i_{d}(k+1)=\Big{(}1-R\frac{T_{s}}{L}\Big{)}i_{d}(k)+\frac{T_{s}}{L}U_ {d}(k)\\ i_{q}(k+1)=\Big{(}1-R\frac{T_{s}}{L}\Big{)}i_{q}(k)+\frac{T_{s}}{L}U_{q}(k)\end{cases}  \tag{11}\]  
Given a prediction interval length of \(L_{P}\), taking the \(I_{d}\) component as an example, the current expression within the prediction interval is:  
\[ \begin{bmatrix}\hat{i}_{d}(k+1)\\ \hat{i}_{d}(k+2)\\ \vdots\\ \hat{i}_{d}(k+L_{p})\end{bmatrix} = \begin{bmatrix}1-\frac{RT_{s}}{L}\\ \Big{(}1-\frac{RT_{s}}{L}\Big{)}^{2}\\ \vdots\\ \Big{(}1-\frac{RT_{s}}{L}\Big{)}^{i-1}\frac{T_{s}}{L}\end{bmatrix} i_{d}(k)+ \begin{bmatrix}\frac{T_{s}}{L}&0&\cdots&0\\ \Big{(}1-\frac{RT_{s}}{L}\Big{)}\frac{T_{s}}{L}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ \Big{(}1-\frac{RT_{s}}{L}\Big{)}^{i-1}\frac{T_{s}}{L}\end{bmatrix}\begin{bmatrix} U_{d}(k)\\ U_{d}(k+1)\\ \vdots\\ U_{d}(k+L_{p}-2)\end{bmatrix}  \tag{12}\]  
To ensure stable and adjustable output voltage while maintaining good system control performance, model predictive control must incorporate output constraints. Consequently, its cost function is formulated as follows:  
\[J=\sum\big{[}y(k+i)-\omega(k+i)\big{]}^{2}Q(i)+\sum\big{[}u(k+j-1)\big{]}^{2} \lambda(j) \tag{13}\]  
In the formula, \(Q_{i}\) and \(\lambda_{j}\) represent error weighting coefficients and control weighting coefficients respectively; \(Y_{i}\) and \(\omega_{i}\) represent the actual input and expected input of the controller respectively; \(L_{u}\) represents the length of the control interval, \(J\) is the cost function, and is the optimization objective function of the MPSMC algorithm. The relevant switch vector is found and applied to the rectifier bridge by using the cost function's minimization as the goal to obtain a precise control effect.