## 3 Main Results  
This section designs the desired reduced-order model to approximate the original high-order one. Firstly, we realize the parameterization of unknown matrices in (2) by providing an alternative characterization for the system (3). We then formulate the design conditions for a reduced-order model in terms of BMIs, guaranteeing the stability and \(H_{\infty}\) specification of the error system. Finally, an iterative procedure based on the inner-approximation strategy is proposed to achieve the desired reduced-order model. Denote the augmented matrices:  
\[\bar{A} = \begin{bmatrix}A&\mathbf{0}\\ \mathbf{0}&\mathbf{0}\end{bmatrix} ,\bar{B}= \begin{bmatrix}B\\ \mathbf{0}\end{bmatrix} ,\bar{C}= \begin{bmatrix}C&\mathbf{0}\end{bmatrix} , \bar{D}=D,\] \[\bar{F} = \begin{bmatrix}\mathbf{0}&\mathbf{0}\\ \mathbf{I}&\mathbf{0}\end{bmatrix} ,\bar{H}= \begin{bmatrix}\mathbf{0}&-\mathbf{ I}\end{bmatrix} ,\bar{M}= \begin{bmatrix}\mathbf{0}&\mathbf{I}\\ \mathbf{0}&\mathbf{0}\end{bmatrix} ,\bar{N}= \begin{bmatrix}\mathbf{0}\\ \mathbf{I}\end{bmatrix} ,\]and the error system (3) can be described by  
\[A_{e}=\bar{A}+\bar{F}G_{r}\bar{M},B_{e}=\bar{B}+\bar{F}G_{r}\bar{N},C_{e}=\bar{C}+ \bar{H}G_{r}\bar{M},D_{e}=\bar{D}+\bar{H}G_{r}\bar{N}\]  
with \(G_{r}= \begin{bmatrix}A_{r}&B_{r}\\ C_{r}&D_{r}\end{bmatrix} \). To render the problem of positivity-preserving model reduction tractable, the matrices \(A_{r},B_{r},C_{r},D_{r}\) to be designed are reorganized as a single matrix \(G_{r}\). Notice that \(G_{r}\) is embedded between two known matrices, and by Lemma (4), the design condition for \(G_{r}\) is recast into a BMI involving the coupled term between \(G\) and \(G_{r}\). Furthermore, \((A_{r},B_{r},C_{r},D_{r})\in\mathbb{S}_{r}\) implies \(G_{r}\succeq 0\), and the additional positivity constraint on \(G_{r}\) sharpens the difficulty in searching for feasible solutions for the non-convex optimization problem. For a given scalar \(\gamma>0\) and known matrices \(G^{\kappa},G^{\kappa}_{r},U^{\kappa},V^{\kappa}\), the problem of positivity-preserving \(H_{\infty}\) model reduction is solvable if there exist matrices \(P,G,G_{r},U,V\) such that \(P=P^{T}>0,G_{r}\succeq 0\) and  
\[\Gamma=\left[ \begin{array}{cc}\Phi&\Psi\\ \Psi^{T}&\Sigma\end{array} \right]<\mathbf{0}, \tag{9}\]  
where  
\[\Phi= \begin{bmatrix}P-G-G^{T}&\mathbf{0}&G^{T}(\bar{A}+\bar{F}G^{\kappa}_{r} \bar{M})&G^{T}(\bar{B}+\bar{F}G^{\kappa}_{r}\bar{N})\\ \star&-\mathbf{I}&\bar{C}+\bar{H}G_{r}\bar{M}&\bar{D}+\bar{H}G_{r}\bar{N}\\ \star&\star&-P&\mathbf{0}\\ \star&\star&\star&-\gamma^{2}\mathbf{I}\end{bmatrix} \]  
\[\Psi= \begin{bmatrix}\beta(G-G^{\kappa})^{T}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}\\ \bar{M}^{T}(G_{r}-G^{\kappa}_{r})\bar{F}^{T}(U^{\kappa})^{T}&\bar{M}^{T}(G_{ r}-G^{\kappa}_{r})\bar{F}^{T}V^{\kappa}&\mathbf{0}\\ \bar{N}^{T}(G_{r}-G^{\kappa}_{r})\bar{F}^{T}(U^{\kappa})^{T}&\bar{N}^{T}(G_{ r}-G^{\kappa}_{r})\bar{F}^{T}V^{\kappa}&\mathbf{0}\end{bmatrix} ,\]  
\[\Xi= \begin{bmatrix}-\beta(U+U^{T})&\mathbf{0}&U-U^{\kappa}\\ \star&-2V^{\kappa}+V&\mathbf{0}\\ \star&\star&-V\end{bmatrix} .\]  
Proof of Theorem 1.: From the above analysis, (7) is presented in the form of BMI, which can be further reformulated as  
\[\mathbb{L}+\mathrm{He}\{\mathbb{X}N\mathbb{Y}\}<\mathbf{0}, \tag{10}\]  
where  
\[\mathbb{L}= \begin{bmatrix}P-G-G^{T}&\mathbf{0}&G^{T}\bar{A}&G^{T}\bar{B}\\ \star&-\mathbf{I}&\bar{C}+\bar{H}G_{r}\bar{M}&\bar{D}+\bar{H}G_{r}\bar{N}\\ \star&\star&-P&\mathbf{0}\\ \star&\star&\star&-\gamma^{2}\mathbf{I}\end{bmatrix} ,\]  
\[\mathbb{X}= \begin{bmatrix}G&\mathbf{0}&\mathbf{0}&\mathbf{0}\end{bmatrix} ^{T },\mathbb{Y}= \begin{bmatrix}\mathbf{0}&\mathbf{0}&\bar{F}G_{r}\bar{M}&\bar{F}G _{r}\bar{N}\end{bmatrix}