## 4 Envelope Model Order Reduction
Hence, in case a linear function \(\mathbf{F}(t)\) satisfying  
\[\frac{d\mathbf{x}_{2}(t)}{dt}=\mathbf{F}(t)\frac{d\mathbf{x}_{1}(t)}{dt} \tag{47}\]  
exists, substituting it into (42) would yield  
\[\mathbf{x}_{2}(t)=\mathbf{A_{22}^{-1}}(t)\left(\mathbf{F}(t)\frac{d\mathbf{x _{1}}(t)}{dt}-\mathbf{A_{21}}(t)\mathbf{x}_{1}(t)-\mathbf{B_{2}}u(t)\right). \tag{48}\]  
Combining (48) with (42) would eliminate the AC-side capacitor voltage-related state variables \(\mathbf{x}_{2}(t)\) from (42), reducing it to be a function of the state variable vector of interest \(\mathbf{x}_{1}(t)\) only:  
\[\frac{d\mathbf{x}_{1}(t)}{dt}=\,\,(\mathbf{I}-\mathbf{A_{12}}(t)\mathbf{A_{22 }}(t)\mathbf{F}(t))^{-1}\Big{[}\Big{(}\mathbf{A_{11}}(t)-\mathbf{A_{12}}(t) \mathbf{A_{22}^{-1}}(t)\mathbf{A_{21}}(t)\Big{)}\mathbf{x}_{1}(t)+\Big{(} \mathbf{B_{1}}(t)-\mathbf{A_{12}}(t)\mathbf{A_{22}^{-1}}(t)\mathbf{B_{2}} \Big{)}u(t)\Big{]}. \tag{49}\]  
In order to obtain the function \(\mathbf{F}(t)\), a methodology assuming the slow-varying amplitude's derivative and phase (SVADP) proposed in [21] is adopted hereafter. In the system under consideration, the SVADP is adopted by assumed that the capacitor voltage phase and derivative of the capacitor voltage magnitude are slow-varying functions of time. The derivative of (9) is given by  
\[\frac{dv_{C}(t)}{dt}=\frac{dV_{M}(t)}{dt}\sin(\theta_{s}(t)+\varphi_{v}(t))+V_{M} (t)\bigg{(}\omega_{s}(t)+\frac{d\varphi_{v}(t)}{dt}\bigg{)}\cos(\theta_{s}(t)+ \varphi_{v}(t)), \tag{50}\]  
reducing to  
\[\frac{dv_{C}(t)}{dt}\approx V_{M}(t)\omega_{s}(t)\cos(\theta_{s}(t)+\varphi_{ v}(t)), \tag{51}\]  
under the SVADP assumption. Reformulating (51) according to (13) yields  
\[\frac{dv_{C}(t)}{dt}\approx v_{C}^{s}(t)\omega_{s}(t)\cos(\theta_{s}(t))-v_{C }^{c}(t)\omega_{s}(t)\sin(\theta_{s}(t)). \tag{52}\]  
Further combining this with (5) while adopting the basic assumptions given in Section 2 yields  
\[i(t)\approx C(t)\frac{dv_{C}(t)}{dt}=C(t)\omega_{s}(t)v_{C}^{s}(t)\cos(\theta _{s}(t))-C(t)\omega_{s}(t)v_{C}^{c}(t)\sin(\theta_{s}(t)). \tag{53}\]  
Reformulating (53) according to (14) yields  
\[v_{C}^{s}(t)\cos(\theta_{s}(t))-v_{C}^{c}(t)\sin(\theta_{s}(t))=\frac{1}{C(t )\omega_{s}(t)}[i^{c}(t)\cos(\theta_{s}(t))+i^{s}(t)\sin(\theta_{s}(t))]. \tag{54}\]  
Separating into sine and cosine terms and equating the coefficients on both sides of each equation gives  
\[v_{C}^{s}(t)=\tfrac{1}{C(t)\omega_{s}(t)}i^{c}(t),\hskip 14.226378 ptv_{C}^{c}(t) =-\tfrac{1}{C(t)\omega_{s}(t)}i^{s}(t). \tag{55}\]  
Applying the derivative to (55) results in  
\[\frac{dv_{C}^{s}(t)}{dt}=-\frac{\Big{(}\frac{dC(t)}{dt}\omega_{s}(t)+\frac{d \omega_{s}(t)}{dt}C(t)\Big{)}}{\big{(}C(t)\omega_{s}(t)\big{)}^{2}}i^{c}(t)+ \frac{1}{C(t)\omega_{s}(t)}\frac{di^{c}(t)}{dt} \tag{56}\]  
and  
\[\mathbf{x}_{2}(t)=\mathbf{A_{22}^{-1}}(t)\left(\mathbf{F}(t)\frac{d\mathbf{x _{1}}(t)}{dt}-\mathbf{A_{21}}(t)\mathbf{x}_{1}(t)-\mathbf{B_{2}}u(t)\right). \tag{48}\]0  
respectively. Applying the SVADP assumption, (56) and (57) reduce to  
\[\mathbf{x}_{2}(t)=\mathbf{A_{22}^{-1}}(t)\left(\mathbf{F}(t)\frac{d\mathbf{x _{1}}(t)}{dt}-\mathbf{A_{21}}(t)\mathbf{x}_{1}(t)-\mathbf{B_{2}}u(t)\right). \tag{48}\]1  
\[\mathbf{x}_{2}(t)=\mathbf{A_{22}^{-1}}(t)\left(\mathbf{F}(t)\frac{d\mathbf{x _{1}}(t)}{dt}-\mathbf{A_{21}}(t)\mathbf{x}_{1}(t)-\mathbf{B_{2}}u(t)\right). \tag{48}\]2  
Comparing (58) with (47), the corresponding function \(\mathbf{F}(t)\) is obtained as  
\[\mathbf{F}(t)=\left( \begin{array}{cc}0&-\frac{1}{C(t)\omega_{s}(t)}\\ \frac{1}{C(t)\omega_{s}(t)}&0\end{array} \right). \tag{60}\]  
Combining (43)-(46) with (49) and (60) yields a set of two differential equations related to the AC-side current state variables:  
\[