# Decoupled MPC Power Balancing Strategy for Coupled Inductor Flying Capacitor DC-DC Converter
## 3 Decoupled MPC Power Balancing Strategy for Coupled Inductor FCDC  
### Decoupled Mathematical Model of Coupled Inductor FCDC  
The mathematical model of the coupled inductor FCDC should be firstly derived to propose the decoupled MPC power balancing strategy. Taking \(0<D<0.5\) as an example, as shown in Table 2, the converter consists of eight operating states. Taking \(i_{\rm a}\), \(i_{\rm b}\), \(u_{\rm fa}\), and \(u_{\rm fb}\) as the state variables, \(T_{\rm s}\) is the switching period. Thus, the operation time of mode i, h, g, f, m, l, k, and j in each switching period can be derived as (0.75-\(d_{\rm b1}\))\(T_{\rm s}\), (\(d_{\rm b1}\)-0.5)\(T_{\rm s}\), (\(0.75\)-\(d_{\rm a1}\))\(T_{\rm s}\), (\(d_{\rm a1}\)-0.5)\(T_{\rm s}\), (\(0.75\)-\(d_{\rm b2}\))\(T_{\rm s}\), (\(d_{\rm b2}\)-0.5)\(T_{\rm s}\), (\(0.75\)-\(d_{\rm a2}\))\(T_{\rm s}\), (\(d_{\rm a2}\)-0.5)\(T_{\rm s}\). According to the operating states shown in Figure 2, writing the circuit state equations and combine them with the corresponding operating time, the average state equation of the coupled inductor FCDC with \(0<D<0.5\) can be derived as:  
\[\frac{d}{dt} \begin{bmatrix}i_{\rm a}\\ i_{\rm b}\\ u_{\rm fa}\\ u_{\rm fb}\end{bmatrix} = \begin{bmatrix}-\frac{r_{\rm a}}{L_{\rm a}}&0&\frac{d _{\rm a2}-d_{\rm a1}}{L_{\rm a}}&0\\ 0&-\frac{r_{\rm b}}{L_{\rm b}}&0&\frac{d_{\rm b2}-d_{\rm b1}}{L_{\rm b}}\\ \frac{d_{\rm a1}-d_{\rm a2}}{C_{\rm fa}}&0&0&0\\ 0&\frac{d_{\rm b1}-d_{\rm b2}}{C_{\rm b}}&0&0\\ \end{bmatrix} \times \begin{bmatrix}i_{\rm a}\\ i_{\rm b}\\ u_{\rm fa}\\ u_{\rm fb}\end{bmatrix} + \begin{bmatrix}\frac{d_{\rm a1}}{L_{\rm a}}&-\frac{1 }{L_{\rm a}}\\ \frac{d_{\rm b1}}{L_{\rm b}}&-\frac{1}{L_{\rm b}}\\ 0&0\\ 0&0\\ \end{bmatrix}\begin{bmatrix}u_{\rm i}\\ u_{\rm i}\\ u_{\rm fb}\end{bmatrix} - \begin{bmatrix}\frac{M}{L_{\rm a}}\frac{d_{\rm i}}{ dt}\\ \frac{M}{L_{\rm b}}\frac{d_{\rm i}}{dt}\\ 0\\ 0\end{bmatrix}  \tag{2}\]  
where \(u_{\rm fa}\) and \(u_{\rm fb}\) are the voltages of flying capacitors \(C_{\rm fa}\) and \(C_{\rm fb}\), respectively. Similarly, the average state equation of the coupled inductor FCDC operating in other duty ratio intervals can be obtained by the above method, and the result is the same as (2). From (2), the mutual inductance M leads to a coupling relationship between the two phases, affecting the power control performance and complicating the control strategy design. Therefore, it is necessary to decouple the converter model. Writing (1) with Laplace transform, it yields:  
\[\mathbf{U}(\rm s)=\mathbf{Z}(\rm s)\mathbf{I}(\rm s) \tag{3}\]  
where Z(s) is the impedance matrix and it is expressed as:  
\[\mathbf{Z}(\rm s) = \begin{bmatrix}r_{\rm a}+L_{\rm a}s&M\rm s\\ M\rm s&r_{\rm b}+L_{\rm b}s\end{bmatrix}  \tag{4}\] \begin{table}
\begin{tabular}{c c c} \hline
**Item** & **Value** & **Unit** \\ \hline Maximum input voltage & 100 & V \\ Maximum operating power & 300 & W \\ Switching frequency & 10 & kHz \\ Self-inductance of phase a and b & 1 & mH \\ Mutual inductance & \(-\)0.33 & mH \\ Flying capacitor & 220 & μF \\ Output voltage under Buck mode & 50 & V \\ Load & 12 & \(\Omega\) \\ \hline \end{tabular}
\end{table} 
Table 2: Experimental parameters. For the interleaved converter with independent inductor, the inverse matrix of the impedance matrix is a diagonal matrix. Thus, in order to decouple the mathematical model of the coupled inductor FCDC, the inverse matrix of \(\mathbf{Z}\)(s) shown in (4) should be transformed into a second-order diagonal matrix. Performing a similar diagonalization transformation on (4) yields:  
\[\mathbf{Y}(s)=\mathbf{Z}^{-1}(s)=\mathbf{P}^{-1}\mathbf{H}(s)\mathbf{P} \tag{5}\]  
where:  
\[\mathbf{P}= \begin{bmatrix}1&-1\\ 1&1\end{bmatrix} ,\ \mathbf{P}^{-1}= \begin{bmatrix}1/2&1/2\\ -1/2&1/2\end{bmatrix} \] \[\mathbf{H}(s)= \begin{bmatrix}r_{\text{a}}/\big{[}r_{\text{a}}+s(L_{ \text{a}}+M)\big{]}&0\\ 0&r_{\text{b}}/\big{[}r_{\text{b}}+s(L_{\text{b}}-M)\big{]}\end{bmatrix}