# Decoupled MPC Power Balancing Strategy for Coupled Inductor Flying Capacitor DC-DC Converter
## 3 Decoupled MPC Power Balancing Strategy for Coupled Inductor FCDC
### Stability Analysis  
The direct Lyapunov method is employed in this paper to verify the stability of the proposed MPC strategy. This method indicates that when the function \(L(\textbf{x})\) defined according to the system state variables **x** satisfies that \(L(\textbf{x})\) is positive, \(\dot{L}(\text{x})\) is negative and \(\lim_{\|\textbf{x}\|-\infty}L(\text{x})=\infty\), and thus the system is stable. So, according to the cost function, the discrete Lyapunov function can be defined as follows:  
\[L(\textbf{x})^{k}=\frac{1}{2}[\textbf{x}_{\text{L-err}}^{k+1}]^{\top}[\textbf{ x}_{\text{L-err}}^{k+1}] \tag{25}\]  
where **x** is \([i_{\text{a}}\ i_{\text{b}}]^{\top}\).  
Define the error variables between predicted and reference values as:  
\[\left\{\begin{array}{l}\dot{i}_{\text{a--err}}^{k+1}=i_{\text{a}}^{k+1}-i_{ \text{a}}^{*}\\ \dot{i}_{\text{b--err}}^{k+1}=i_{\text{b}}^{k+1}-i_{\text{b}}^{*}\end{array}\right. \tag{26}\]  
According to (10), (26), and (27), the variation rate of \(L(\textbf{x})\) as a function of the inductor current can be expressed as:  
\[\begin{array}{l}\Delta L(i_{x})^{k}\\ =L(i_{x})^{k+1}-L(i_{x})^{k}\\ =\frac{1}{2}\left[\begin{array}{c}i_{x}^{k}-\frac{\dot{i}_{x}T_{x}T_{x}}{L_{ x}\pm M}+\frac{(d_{x1}-d_{x2})T_{x}}{L_{x}\pm M}u_{\text{fr}}^{k}\\ +\frac{T_{x}}{L_{x}\pm M}\times\left[u_{\text{i}}^{k}+(d_{x2}-1)u_{\text{o}}^{k }\right]-i_{x}^{*}\end{array}\right]\\ \times\left[\begin{array}{c}i_{x}^{k}-\frac{i_{x}T_{x}T_{x}}{L_{ x}\pm M}+\frac{(d_{x1}-d_{x2})T_{x}}{L_{x}\pm M}u_{\text{fr}}^{k}\\ +\frac{T_{x}}{L_{x}\pm M}\times\left[u_{\text{i}}^{k}+(d_{x2}-1)u_{\text{o}}^{k }\right]-i_{x}^{*}\end{array}\right]\\ -\frac{1}{2}\left[\dot{i}_{x\text{,err}}^{k}\right]\left[\dot{i}_{x\text{,err}}^ {k}\right]\end{array} \tag{27}\]  
where x represents phase a and b, respectively. When x represents phase a, the symbol \(\pm\) should be written as \(+\), and when x represents phase b, the symbol \(\pm\) should be written as \(-\).  
When \(\Delta{d_{\text{f1}}}^{k}\) and \(\Delta{d_{\text{f2}}}\) do not reach the limit, the predicted value is approximately equal to the steady-state reference due to the tracking characteristics of MPC, as shown in (17). Substituting (17) into (28), we can have:  
\[\Delta L(i_{x})^{k}=-\frac{1}{2}\left[\dot{i}_{x\text{,err}}^{k}\right]\left[ \dot{i}_{x\text{,err}}^{k}\right] \tag{28}\]  
However, when \(\Delta{d_{\text{f1}}}^{k}\) or \(\Delta{d_{\text{f2}}}^{k}\) reaches the limit, the predicted current value will fail to track the reference. To address this issue, the adjustment of \(\Delta{i_{\text{f1 im}}}\) should align with the  
Figure 5: Control block diagram of the proposed MPC power balancing strategy.  
changes in \(\Delta L(\mathbf{x})\) to guarantee the stability and convergence of the variable's control. The maximum current deviation can be optimized as follows:  
\[\Delta i_{L} \tag{29}\]  
where \(s\) represents the increment step of current deviation.  
The relaxation of \(\Delta i_{L,\mathrm{lim}}\) leads to a corresponding relaxation of \(\Delta D_{\mathrm{lim}}\), thereby ensuring the sustained validity of Equation (29) within the system under these conditions.  
As shown in (26) and (29), \(L(\mathbf{x})>0\) and \(\dot{L}(\mathbf{x})<0\). Thus, based on the direct Lyapunov method, it can be known that the MPC strategy based on the cost function shown in (13) is stable.