## 3 Continuous-Time Average State-Space Model
### Isolation Stage (DC-DC)
In this design, the converters are operating in 50% of the duty cycle. Thus, the first-order terms of \(\rm T_{\rm 1}\) (t) and \(T_{\rm 2}\) (t) are similar to zero order and are presented in Equations (26)-(29). Moreover, in Equations (17) and (18), the independent state variables are the inductor current and capacitor voltage, which form the state-space model of the DAB converter with the state vector x(t) and the control vector u(t). Further, Equations (30) and (31) represent the state-space model with the input and output signals of the DAB converter. Therefore, the model parameters are input voltage source (\(\rm U_{\rm dc}\)), inductor current (\(i_{L}\)), and duty cycle (\(\mathbb{D}\)). When \(T_{\rm 1}^{\rm 1 imag}\) and \(T_{\rm 1}^{\rm 1 real}\) are introduced into Equations (20) and (25), the state-space model becomes Equation (31). Furthermore, the zero initial states are assumed in Equation (32), which are, \(\rm i_{L}^{0}\), \(\rm U_{\rm o}^{\rm 1 real}\), \(\rm U_{\rm o}^{\rm 1 imag}\) are similar to zero order, and Equation (32) becomes Equation (33). Figure 4:6  
Figure 4:7  
Figure 4:8  
Figure 4:9  
Figure 3:0  
Figure 3:1\[\frac{\mathrm{d}}{\mathrm{dt}} \begin{bmatrix}\triangle\mathrm{U}_{ \mathrm{o}}\\ \triangle_{L}^{\mathrm{Ireal}}\\ \triangle_{L}^{\mathrm{Ireal}}\end{bmatrix} =\] \[ \begin{bmatrix}-\frac{1}{\mathrm{RC}_{0}}&-\frac{4\sin\pi(\mathrm{D} \pi)}{\pi\mathrm{C}_{0}}&-\frac{4\cos\pi(\mathrm{D}\pi)}{\pi\mathrm{C}_{0}}&0 &0&0\\ \frac{2\sin\pi(\mathrm{D}\pi)}{\pi\mathrm{L}_{\mathrm{f}}}&-\frac{\mathrm{R} _{\mathrm{f}}}{\mathrm{L}_{\mathrm{f}}}&\omega_{\mathrm{s}}&0&0&0\\ \frac{2\cos\pi(\mathrm{D}\pi)}{\pi\mathrm{L}_{\mathrm{f}}}&-\omega_{\mathrm{s} }&-\frac{\mathrm{R}_{\mathrm{f}}}{\mathrm{L}_{\mathrm{f}}}&0&0&0\\ 0&0&0&-\frac{\mathrm{R}_{\mathrm{f}}}{\mathrm{L}_{\mathrm{f}}}&\frac{4\sin \pi(\mathrm{D}\pi)}{\pi\mathrm{L}_{\mathrm{f}}}&\frac{4\cos\pi(\mathrm{D}\pi)} {\pi\mathrm{L}_{\mathrm{f}}}\\ 0&0&0&-\frac{2\sin\pi(\mathrm{D}\pi)}{\pi\mathrm{C}_{0}}&-\frac{1}{\mathrm{R} _{\mathrm{C}}}&\omega_{\mathrm{s}}\\ 0&0&0&-\frac{2\cos\pi(\mathrm{D}\pi)}{\pi\mathrm{C}_{0}}&\omega_{\mathrm{s}}& -\frac{1}{\mathrm{R}_{\mathrm{C}}}\end{bmatrix} \times \tag{32}\]  
\[ \begin{bmatrix}\triangle\mathrm{U}_{\mathrm{o}}\\ \triangle_{L}^{\mathrm{Ireal}}\\ \triangle_{L}^{\mathrm{Iimag}}\end{bmatrix} + \begin{bmatrix}0&\frac{1}{\mathrm{C} _{0}}\\ 0&\frac{1}{\mathrm{C}_{0}}\\ -\frac{2}{\pi\mathrm{L}_{\mathrm{f}}}&0\end{bmatrix} \times \begin{bmatrix} \mathrm{U}_{\mathrm{d}c}\\ \mathrm{i}_{\mathrm{L}}\end{bmatrix} \]  
\[\frac{\mathrm{d}}{\mathrm{dt}} \begin{bmatrix}\triangle\mathrm{U}_{\mathrm{o}}\\ \triangle_{L}^{\mathrm{Ireal}}\\ \triangle_{L}^{\mathrm{Iimag}}\end{bmatrix} = \begin{bmatrix}-\frac{1}{\mathrm{RC} _{0}}&-\frac{4\sin(\mathrm{D}\pi)}{\pi\mathrm{C}_{0}}&-\frac{4\cos(\mathrm{D} \pi)}{\pi\mathrm{C}_{0}}\\ \frac{2\sin(\mathrm{D}\pi)}{\pi\mathrm{L}_{\mathrm{f}}}&-\frac{\mathrm{R}_{ \mathrm{f}}}{\mathrm{L}_{\mathrm{f}}}&\omega_{\mathrm{s}}\\ \frac{2\cos(\mathrm{D}\pi)}{\pi\mathrm{L}_{\mathrm{f}}}&-\omega_{\mathrm{s} }&-\frac{\mathrm{R}_{\mathrm{f}}}{\mathrm{L}_{\mathrm{f}}}\end{bmatrix}  \times \begin{bmatrix}\triangle\mathrm{U}_{\mathrm{o}}\\ \triangle_{L}^{\mathrm{Ireal}}\\ \triangle_{L}^{\mathrm{Iimag}}\end{bmatrix} \times \begin{bmatrix}0&-\frac{1}{ \mathrm{C}_{0}}\\ 0&0\\ -\frac{2}{\pi\mathrm{L}_{\mathrm{f}}}&0\end{bmatrix} \times \begin{bmatrix} \mathrm{U}_{\mathrm{d}c}\\ \mathrm{i}_{\mathrm{L}}\end{bmatrix}  \tag{33}\]  
A small signal control for the output transfer function is needed to measure the system stability of the DAB converter with the controller structure. Thus, a small deviation in \(\mathfrak{D}\) causes a distinction in the values of \(\mathrm{U}_{\mathrm{o}}^{0}\) from its steady-state value. Hence, the corresponding small signal model is defined in Equations (34)-(37). In addition, the nonlinearity of the system is introduced when the control and state variables are multiplied. The approximation solution of nonlinear terms is presented in Equation (38).