# Decoupled MPC Power Balancing Strategy for Coupled Inductor Flying Capacitor DC-DC Converter
## 3 Decoupled MPC Power Balancing Strategy for Coupled Inductor FCDC
### Multi-Objective MPC Strategy Design
\tag{12}\]  
However, in order to achieve the minimum value of (13), it is necessary to calculate the predicted current values with different duty ratios according to (10). Even if the two cost functions are independent of each other, there is still a significant computational load. Actually, (13) is a non-negative function. Assuming \(u_{\text{i}}=100\) V, \(u_{\text{o}}=60\) V, \(i_{\text{a}}^{*}=i_{\text{ak}}=5\) A and \(u_{\text{fa}}=50\) V, the curves of different duty ratios based on (13) can be obtained and are shown in Figure 4.  
Figure 4: Curves of \(J_{\text{a}}^{\text{k}}\) with different duty ratios. It can be seen from Figure 4 that when the derivative value of \({J_{\text{a}}}^{\text{k}}\) is zero, the minimum value is obtained. So, taking derivatives of \({J_{\text{a}}}^{\text{k}}\) for \({d_{\text{a}1}}^{\text{k}}\), \({d_{\text{a}2}}^{\text{k}}\) and taking derivatives of \({J_{\text{b}}}^{\text{k}}\) for \({d_{\text{b}1}}^{\text{k}}\) and \({d_{\text{b}2}}^{\text{k}}\), respectively, we can have:  
\[\left\{ \begin{aligned} \frac{\partial{j_{\text{a}}^{k}}}{ \partial{d_{\text{a}}^{k}}}\bigg{|}_{{d_{\text{a}2}^{k}}^{\text{k}}=1}& =2(i_{\text{a}}^{*}-\dot{i}_{\text{a}}^{k+1})(\frac{\partial{j_{ \text{a}}^{k+1}}}{\partial{d_{\text{a}1}^{k}}}-\frac{\partial{j_{\text{a}}^{k }}}{\partial{d_{\text{a}1}^{k}}})\\ \frac{\partial{j_{\text{a}}^{k}}}{\partial{d_{\text{a}2}^{k}}} \bigg{|}_{{d_{\text{a}1}^{k}}^{\text{k}}=1}&=2(i_{\text{a}}^{*} -\dot{i}_{\text{a}}^{k+1})(\frac{\partial{j_{\text{a}}^{k+1}}}{\partial{d_{ \text{a}2}^{k}}}-\frac{\partial{i_{\text{a}}^{k}}}{\partial{d_{\text{a}2}^{k}} })\\ \end{aligned} \right. \tag{13}\]  
\[\left\{ \begin{aligned} \frac{\partial{j_{\text{b}}^{k}}}{ \partial{d_{\text{b}1}^{k}}}\bigg{|}_{{d_{\text{b}2}^{k}}^{\text{k}}=1}& =2(i_{\text{b}}^{*}-\dot{i}_{\text{b}}^{k+1})(\frac{\partial{j_{ \text{b}}^{k+1}}}{\partial{d_{\text{b}1}^{k}}}-\frac{\partial{i_{\text{b}}^{k }}}{\partial{d_{\text{b}1}^{k}}})\\ \frac{\partial{j_{\text{b}}^{k}}}{\partial{d_{\text{b}2}^{k}}} \bigg{|}_{{d_{\text{b}1}^{k}}^{\text{k}}=1}&=2(i_{\text{b}}^{*} -\dot{i}_{\text{b}}^{k+1})(\frac{\partial{j_{\text{b}}^{k+1}}}{\partial{d_{ \text{b}2}^{k}}}-\frac{\partial{i_{\text{b}}^{k}}}{\partial{d_{\text{b}2}^{k}} })\\ \end{aligned} \right. \tag{14}\]  
When the derivative value of \({J_{\text{a}}}^{\text{k}}\) and \({J_{\text{b}}}^{\text{k}}\) to \({d_{\text{a}1}}^{\text{k}}\), \({d_{\text{a}2}}^{\text{k}}\) and \({d_{\text{b}1}}^{\text{k}}\), \({d_{\text{b}2}}^{\text{k}}\) is zero, the minimum value of the cost function shown in (13) can be obtained, and thus the duty ratio to control \({i_{\text{a}}}^{\text{k}}\) and \({i_{\text{b}}}^{\text{k}}\) tracking the current reference can be derived. Assuming the equations shown in (14) are 0, we can have:  
\[\left\{ \begin{aligned} (\frac{\partial{d_{\text{a}}^{k+1}}}{ \partial{d_{\text{a}1}^{k}}}-\frac{\partial{i_{\text{a}}^{k}}}{\partial{d_{ \text{a}1}^{k}}})&\
eq 0\\ (\frac{\partial{j_{\text{a}}^{k+1}}}{\partial{d_{\text{a}2}^{k}}}- \frac{\partial{i_{\text{a}}^{k}}}{\partial{d_{\text{a}2}^{k}}})& \
eq 0\\ \end{aligned} \right. \tag{15}\]  
Due to the predicted value being approximately equal to the steady-state reference, this yields:  
\[i_{x}^{*}\approx i_{x}^{k}-\frac{i_{x}^{k}r_{x}T_{s}}{L_{x}\pm M}+\frac{(d_{x2 }-d_{x1})T_{s}}{L_{x}\pm M}u_{tx}^{k}+\frac{T_{s}}{L_{x}\pm M}\times\left(d_{x1 }u_{i}^{k}-u_{\text{o}}^{k}\right) \tag{16}\]  
According to (17), it can be seen that the predicted value is approximately equal to the reference value, so:  
\[\left\{ \begin{aligned} {i_{\text{a}}^{*}}-{i_{\text{a}}^{k+1}}& \approx 0\\ {i_{\text{b}}^{*}}-{i_{\text{b}}^{k+1}}&\approx 0\\ \end{aligned} \right. \tag{17}\]  
Thus, \({d_{\text{a}1}}^{\text{k}}\) and \({d_{\text{a}2}}^{\text{k}}\) can be derived from (15) and (18). And the expression of \({d_{\text{a}1}}^{\text{k}}\) and \({d_{\text{a}2}}^{\text{k}}\) under Boost mode is shown in Appendix A. \[\left\{