## 4 Envelope Model Order Reduction  
The system under consideration (cf. Figure 3) includes a single AC-side capacitor and a single AC-side inductor, indicating second-order AC-side dynamics. Considering the DC-side capacitor, third-order overall system dynamics are expected. Nonetheless, a fifth-order envelope model was derived in the preceding section. This stemmed from the nature of the sine/cosine components' separation process adopter for the envelope dynamics capturing process, leading to system order augmentation. Therefore, original (third) model order restoration would be beneficial. The fourth-order set of equations describing the AC-side model given by (23)-(26) may be represented by the following general form:  
\[\frac{d\mathbf{x}(t)}{dt}=\mathbf{A}(t)\mathbf{x}(t)+\mathbf{B}(t)u(t), \tag{38}\]  
Figure 4: Full-order envelope model of the system. where  
\[\mathbf{x}(t)=\left( \begin{matrix}\vec{r}\,(t)&\vec{r}\,(t)&v_{\mathbb{C}}^{s}(t)&v_ {\mathbb{C}}^{s}(t)\end{matrix} \right)^{\mathsf{T}}, \tag{39}\]  
\[\mathbf{A}(t)=\left( \begin{matrix}-\frac{\frac{R(t)+\frac{dL(t)}{dt}}{L(t)}}{ \omega_{s}(t)}&-\omega_{s}(t)&-\frac{1}{L(t)}&0\\ \omega_{s}(t)&-\frac{R(t)+\frac{dL(t)}{dt}}{L(t)}&0&-\frac{1}{L(t)}\\ \frac{1}{C(t)}&0&-\frac{\frac{\mathcal{C}(t)}{dt}}{\mathbb{C}(t)}&-\omega_{s}( t)\\ 0&\frac{1}{C(t)}&\omega_{s}(t)&-\frac{\frac{\mathcal{C}(t)}{dt}}{\mathbb{C}(t)} \end{matrix} \right), \tag{40}\]  
\[\mathbf{B}(t)=\left( \begin{matrix}0&\frac{4}{\pi L(t)}&0&0\end{matrix} \right) ^{\mathsf{T}},\quad u(t)=\overline{v}_{in}(t). \tag{41}\]  
The order of the system (38) is dictated by the size of matrix \(\mathbf{A}(t)\). In order to reduce it, the elimination of some state equations is needed. Note that the set given in (23)-(26) may also be expressed as  
\[\left( \begin{matrix}\frac{d\mathbf{x}_{1}(t)}{dt}\\ \frac{d\mathbf{x}_{2}(t)}{dt}\end{matrix} \right)=\left( \begin{matrix}\mathbf{ A_{11}}(t)&\mathbf{A_{12}}(t)\\ \mathbf{A_{21}}(t)&\mathbf{A_{22}}(t)\end{matrix} \right)\left( \begin{matrix} \mathbf{x}_{1}(t)\\ \mathbf{x}_{2}(t)\end{matrix} \right)+\left( \begin{matrix}\mathbf{B_{1}}(t)\\ \mathbf{B_{2}}\end{matrix} \right)u(t), \tag{42}\]  
where  
\[\mathbf{x}_{1}(t)=\left( \begin{matrix}\vec{r}\,(t)&\vec{r}\,(t)\end{matrix}  \right)^{\mathsf{T}},\quad\mathbf{x}_{2}(t)=\left(v_{\mathbb{C}}^{s}(t)&v_{ \mathbb{C}}^{s}(t)\right)^{\mathsf{T}}, \tag{43}\]  
\[\mathbf{A_{11}}(t)=\left( \begin{matrix}-\frac{R(t)+\frac{dL(t)}{dt}}{L(t)}&- \omega_{s}(t)\\ \omega_{s}(t)&-\frac{R(t)+\frac{dL(t)}{dt}}{L(t)}\end{matrix} \right),\quad \mathbf{A_{12}}(t)=\left( \begin{matrix}-\frac{1}{L(t)}&0\\ 0&-\frac{1}{L(t)}\end{matrix} \right), \tag{44}\]  
\[\mathbf{A_{21}}(t)=\left( \begin{matrix}\frac{1}{C(t)}&0\\ 0&\frac{1}{C(t)}\end{matrix} \right),\quad\mathbf{A_{22}}(t)=\left( \begin{matrix} -\frac{\mathcal{C}(t)}{C(t)}&-\omega_{s}(t)\\ \omega_{s}(t)&-\frac{\mathcal{C}(t)}{C(t)}\end{matrix} \right), \tag{45}\]  
\[\mathbf{B_{1}}(t)=\left( \begin{matrix}0\\ \frac{4}{\pi L(t)}\end{matrix} \right),\,\,\,\mathbf{B_{2}}=\left( \begin{matrix} 0\\ 0\end{matrix} \right). \tag{46}\]  
The main interest is in the DC-side-to-AC-side power transfer by means of the AC-side current, i.e., the AC-side current-related state variables \(\mathbf{x}_{1}(t)\) are the main interest, while the AC-side capacitor voltage-related state variables \(\mathbf{x}_{2}(t)\) are of secondary importance.