# Frequency-Limited Model Reduction for Linear Positive Systems: A Successive Optimization Method
## 3 Main Results  
This section presents a method to design an appropriate reduced-order model. Firstly, an equivalent characterization for the system (3) is developed to parameterize the unknown matrices \(A_{r},B_{r},C_{r},D_{r}\). Then, the design conditions in the form of BMIs are established, ensuring that the reduced-order model maintains positivity and the error system has a given finite-frequency performance level. On this basis, we propose an iterative procedure to obtain a reduced-order model. To simplify the problem, we denote \(G_{r}= \begin{bmatrix}A_{r}&B_{r}\\ C_{r}&D_{r}\end{bmatrix} \) and  
\[\bar{A} = \begin{bmatrix}A&\mathbf{0}\\ \mathbf{0}&\mathbf{0}\end{bmatrix} ,\bar{B}= \begin{bmatrix}B\\ \mathbf{0}\end{bmatrix} ,\bar{C}= \begin{bmatrix}C&\mathbf{0}\end{bmatrix} , \bar{D}=D,\] \[\bar{F} = \begin{bmatrix}\mathbf{0}&\mathbf{0}\\ \mathbf{I}&\mathbf{0}\end{bmatrix} ,\bar{H}= \begin{bmatrix}\mathbf{0}&-\mathbf{ I}\end{bmatrix} ,\bar{M}= \begin{bmatrix}\mathbf{0}&\mathbf{I}\\ \mathbf{0}&\mathbf{0}\end{bmatrix} ,\bar{N}= \begin{bmatrix}\mathbf{0}\\ \mathbf{I}\end{bmatrix} ,\]  
and, thus, (3) can be reformulated as \(A_{e}=\bar{A}+F\bar{G}_{r}\bar{M},B_{e}=\bar{B}+\bar{F}G_{r}\bar{N},C_{e}=\bar{ C}+\bar{H}G_{r}\bar{M},D_{e}=\bar{D}+\bar{H}G_{r}\bar{N}\). According to Remark 1, the condition (7) cannot be directly solved for which the constrained \(G_{r}\in\mathbb{P}\) is coupled with \(P\) and \(Q\). Therefore, we will provide feasible conditions in the following theorem. **Theorem 1**. : _Given a specified finite frequency range \(\Omega\) and matrices \(G_{r}^{x},P_{s}^{x},X^{x},Y^{x},Z^{x},U_{s}^{x},V_{s}^{x},\)\(U^{x},V^{x},\) a desired reduced-order model \(G_{r}\) can be obtained if there exist matrices \(P_{s},X,Y,Z,P,Q,U_{s},\)\(V_{s},U,V\) such that \(P_{s}=P_{s}^{T}>\mathbf{0},Q=Q^{T}>\mathbf{0}\) and_  
\[G_{r}= \begin{bmatrix}A_{r}&B_{r}\\ C_{r}&D_{r}\end{bmatrix} \in\mathbb{P}, \tag{8}\]  
\[\Gamma_{s}= \begin{bmatrix}\Gamma_{s1}&\Gamma_{s2}\\ \Gamma_{s2}^{T}&\Gamma_{s3}\end{bmatrix} <\mathbf{0} \tag{9}\]  
\[\Gamma= \begin{bmatrix}\Gamma_{1}&\Gamma_{2}\\ \Gamma_{2}^{T}&\Gamma_{3}\end{bmatrix} <\mathbf{0} \tag{10}\]  
_where_  
\[\Gamma_{s1} =\mathrm{He}\{P_{s}A_{r}^{x}+P_{s}^{x}(A_{r}-A_{r}^{x})\},\] \[\Gamma_{s2} = \begin{bmatrix}\beta(P_{s}-P_{s}^{x})+(A_{r}-A_{r}^{x})^{T}(U_{ s}^{x})^{T}&(A_{r}-A_{r}^{x})^{T}V_{s}^{x}&\mathbf{0}\end{bmatrix} ,\] \[\Gamma_{s3} = \begin{bmatrix}-\beta(U_{s}+U_{s}^{T})&\mathbf{0}&U_{s}-U_{s}^{x}\\ \star&-2V_{s}^{x}+V_{s}&\mathbf{0}\\ \star&\star&-V_{s}\end{bmatrix} ,\] \[\Gamma_{1} = \begin{bmatrix}\Xi&\mathbf{0}&\mathbf{0}\\ \hdashline-1&\mathbf{0}&\mathbf{C}^{T}\\ \hdashline-1&-\gamma\bar{\mathbf{1}}&\bar{D}_{r}^{x}\\ \hdashline-1&D_{e}&-\mathbf{1}\end{bmatrix} +\mathrm{He} \begin{Bmatrix}\begin{bmatrix} -X&X(\bar{A}+\bar{F}G_{r}^{x}\bar{M})&X(\bar{B}+\bar{F}G_{r}^{x}\bar{N})&\mathbf{0 }\\ -Y&Y(\bar{A}+\bar{F}G_{r}^{x}\bar{M})&Y(\bar{B}+\bar{F}G_{r}^{x}\bar{N})&\mathbf{0} \\ -Z&Z(\bar{A}+\bar{F}G_{r}^{x}\bar{M})&Z(\bar{B}+\bar{F}G_{r}^{x}\bar{N})&\mathbf{0} \\ \mathbf{0}&\mathbf{0}&\mathbf{0}\end{bmatrix}\end{Bmatrix} \] \[+\mathrm{He}\begin{Bmatrix} \begin{bmatrix}(X^{x})\\ (Y^{x})\\ (Z^{x})\\ \mathbf{0}\end{bmatrix}\begin{bmatrix}\mathbf{0}&\bar{F}(G_{r}-G_{r}^{x})\bar{M}&\bar {F}(G_{r}-G_{r}^{x})\bar{N}&\mathbf{0}\end{bmatrix} ,\] \[\Gamma_{2} = \begin{bmatrix}\beta(X-X^{x})\\ \beta(Y-Y^{x})+\bar{M}^{T}(G_{r}-G_{r}^{x})\bar{F}^{T}(U^{x})^{T}&\bar{M}^{T} (G_{r}-G_{r}^{x})\bar{F}^{T}(V^{x})^{T}&\mathbf{0}\\ \beta(Z-Z^{x})+\bar{N}^{T}(G_{r}-G_{r}^{x})\bar{F}^{T}(U^{x})^{T}&\bar{N}^{T} (G_{r}-G_{r}^{x})\bar{F}^{T}(V^{x})^{T}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}\end{bmatrix} ,\] \[\Gamma_{3} = \begin{bmatrix}-\beta(U+U^{T})&\mathbf{0}&U-U^{x}\\ \star&-2V^{x}+V&\mathbf{0}\\ \star&\star&-V\end{bmatrix} \]  
**Proof of Theorem 1.