## 3 Continuous-Time Average State-Space Model
### Isolation Stage (DC-DC)
To this end, given the value of parameters defined in Equations (34)-(37) are placed into Equation (33), then the state-space model can be expressed as in Equation (39) while the corresponding values are given in A, B, C, and D matrix form. Further, the transfer function of the DAB converter can be expressed using a conversion formula, which is given in Equation (40), and the values are assigned and given in Equation (41), where \(\mathrm{G}_{\mathrm{open-loop}}\) is the open-loop transfer function of the DAB. \[\triangle\mathfrak{D}=\mathfrak{D}-\mathrm{d} \tag{34}\]  
\[\triangle\mathrm{U}_{\mathrm{o}}^{0}=\mathrm{U}_{\mathrm{o}}^{0}-\mathrm{U}_{ \mathrm{o}}^{0} \tag{35}\]  
\[\triangle_{L}^{\mathrm{Ireal}}=i_{L}^{\mathrm{Ireal}}-I_{L}^{\mathrm{Ireal}} \tag{36}\]  
\[\triangle_{L}^{\mathrm{Iimag}}=i_{L}^{\mathrm{Iimag}}-I_{L}^{\mathrm{Iimag}} \tag{37}\]  
\[ \begin{split}\mathrm{in}(\pi\mathfrak{D})\mathrm{U}_{\mathrm{o}}^{0}= \sin(\pi\mathfrak{D})\triangle\mathrm{U}_{\mathrm{o}}^{0}+\mathrm{U}_{\mathrm{o} }^{0}\mathrm{sin}(\pi\mathfrak{D})\mathrm{cos}(\pi\Delta\mathfrak{D})+\mathrm{U}_{ \mathrm{o}}^{0}\mathrm{cos}(\pi\mathrm{D})\mathrm{sin}(\pi\Delta\mathfrak{D})\\ =\sin(\pi\mathfrak{D})\triangle\mathrm{U}_{\mathrm{o}}^{0}+\mathrm{U}_{\mathrm{o} }^{0}\mathrm{sin}(\pi\mathfrak{D})+\mathrm{U}_{\mathrm{o}}^{0}\mathrm{cos}(\pi \mathfrak{D})+\mathrm{U}_{\mathrm{o}}^{0}\mathrm{cos}(\pi\mathfrak{D})\end{split}  \tag{38}\]  
\[ \begin{split}\frac{\mathrm{d}}{\mathrm{dt}}\begin{bmatrix}\triangle \mathrm{U}_{\mathrm{o}}\\ \triangle_{L}^{\mathrm{Ireal}}\\ \triangle_{L}^{\mathrm{Iimag}}\end{bmatrix}=\begin{bmatrix}-\frac{1}{\mathrm{RC} _{0}}&-\frac{4\sin(\mathfrak{D}\pi)}{\pi\mathrm{C}_{0}}&-\frac{4\cos( \mathfrak{D}\pi)}{\pi\mathrm{C}_{0}}\\ \frac{\sin(\mathfrak{D}\pi)}{\pi\mathrm{L}_{\mathrm{f}}}&-\frac{\mathrm{R}_{ \mathrm{f}}}{\mathrm{L}_{\mathrm{f}}}&\omega_{\mathrm{s}}\\ \frac{2\cos(\mathfrak{D}\pi)}{\pi\mathrm{L}_{\mathrm{f}}}&-\omega_{\mathrm{s}}&-\frac{ \mathrm{R}_{\mathrm{f}}}{\mathrm{L}_{\mathrm{f}}}\end{bmatrix}\times\begin{bmatrix} \triangle\mathrm{U}_{\mathrm{o}}\\ \triangle_{L}^{\mathrm{Ireal}}\\ \triangle_{L}^{\mathrm{Iimag}}\end{bmatrix}\\ +\begin{bmatrix}\frac{4}{\mathrm{C}_{0}}\left(\mathrm{I}_{ \mathrm{I}_{\mathrm{O}}\mathrm{mag}}\sin(\pi\mathfrak{d})-\mathrm{I}_{\mathrm{O }}\mathrm{real}\mathrm{cos}(\pi\mathfrak{d})\right)\\ \frac{2\times\mathrm{U}_{\mathrm{o}}^{0}\mathrm{\geq_{\mathrm{o}}}}{\mathrm{L}_{ \mathrm{f}}}\cos(\pi\mathfrak{d})\\ -\frac{2\times\mathrm{U}_{\mathrm{o}}^{0}\mathrm{\geq_{\mathrm{o}}}}{\mathrm{L}_{ \mathrm{f}}}\sin(\pi\mathfrak{d})\end{bmatrix}\times\triangle\mathfrak{D}\\ -\frac{2\times\mathrm{U}_{\mathrm{o}}^{0}\mathrm{\geq_{\mathrm{o}}}}{\mathrm{L}_{ \mathrm{f}}}\sin(\pi\mathfrak{d})\end{bmatrix}\times\triangle\mathfrak{D}\\ \end{split}  \tag{39}\]\[\mathrm{A}= \begin{bmatrix}-178&-363.7827&2.227\times 10^{-14}\\ 1.8346\times 10^{5}&-4.611\times 10^{4}&1.256\times 10^{5}\\ 1.1234\times 10^{-11}&-1.256\times 10^{5}&-4.611\times 10^{4}\end{bmatrix} _{3 \times 3}\]  
\[\mathrm{B}= \begin{bmatrix}4.5\times 10^{3}\\ 7.0585\times 10^{-9}\\ -1.1527\times 10^{8}\end{bmatrix} _{3\times 1}\]  
\[\mathrm{C}= \begin{bmatrix}10&0\end{bmatrix} _{1\times 3}\mathrm{D}= \begin{bmatrix} 0\end{bmatrix} _{1\times 1}\]  
\[\mathrm{G}_{\mathrm{open-loop}}= \begin{bmatrix}\mathrm{C}(\mathrm{sI}- \mathrm{A})^{-1}\mathrm{B}+\mathrm{D}\end{bmatrix}  \tag{40}\]  
\[\mathrm{G}_{\mathrm{open-loop}}=\frac{4.5\times 10^{4}\mathrm{s}^{2}+4.15 \times 10^{9}\mathrm{s}+6.072\times 10^{15}}{\mathrm{s}^{3}+9239\mathrm{s} ^{2}+1.798\times 10^{10}\mathrm{s}+6.264\times 10^{12}} \tag{41}\]