# Frequency-Limited Model Reduction for Linear Positive Systems: A Successive Optimization Method
## 3 Main Results
** By Lyapunov stability theory, the asymptotic stability of the reduced-order system (2) can be ensured if, and only if, there exists a Lyapunov matrix \(P_{s}=P_{s}^{T}\) satisfying  
\[A_{r}^{T}P_{s}+P_{s}A_{r}<\mathbf{0} \tag{11}\]  
and (7) can be further expressed as  
\[ \begin{bmatrix}A_{e}&B_{e}\\ \mathbf{1}&\mathbf{0}\\ \mathbf{0}&\mathbf{1}\end{bmatrix} ^{T}\underbrace{\left(\mathrm{diag}\{\Xi,\mathbf{0}\} +\mathrm{diag}\{\mathbf{0}, \begin{bmatrix}C_{e}&D_{e}\\ \mathbf{0}&\mathbf{1}\end{bmatrix} ^{T}\Pi \begin{bmatrix}C_{e}&D_{e}\\ \mathbf{0}&\mathbf{1}\end{bmatrix} \}\right)}_{\mathrm{Y}} \begin{bmatrix}A_{e}&B_{e}\\ \mathbf{1}&\mathbf{0}\\ \mathbf{0}&\mathbf{1}\end{bmatrix} <\mathbf{0} \tag{12}\]  
Let \(\mathcal{H}= \begin{bmatrix}-\mathbf{1}&A_{e}&B_{e}\end{bmatrix} ^{T}\), and its null space denoted as \(\mathcal{H}^{\perp}= \begin{bmatrix}A_{e}&B_{e}\\ \mathbf{1}&\mathbf{0}\\ \mathbf{0}&\mathbf{1}\end{bmatrix} ^{T}\). Thus, (14) is equivalent to  
\[\mathcal{H}^{\perp}\Gamma(\mathcal{H}^{\perp})^{T}<\mathbf{0} \tag{13}\]By [PERSON]'s lemma, we have  
\[\mathcal{H}^{\perp}\Gamma(\mathcal{H}^{\perp})^{T}<\mathbf{0}\Leftrightarrow \Gamma+\mathcal{H}\mathcal{X}+\mathcal{X}^{T}\mathcal{H}^{T}<\mathbf{0} \tag{14}\]  
where \(\mathcal{X}= \begin{bmatrix}X^{T}&Y^{T}&Z^{T}\end{bmatrix} \). It follows that (11) and (14) can be presented in a unified form of  
\[\mathbb{L}_{J}+\mathrm{He}\{\mathbb{A}_{J}N_{j}\mathbb{B}_{J}\}<\mathbf{0}_{J} =1,2. \tag{15}\]  
where  
\[\mathbb{L}_{1}=\mathbf{0},\mathbb{A}_{1}=P_{s},\mathbb{B}_{1}=A_{r},N_{1}= \mathbf{I},\]  
\[\mathbb{L}_{2}= \begin{bmatrix}-Q&P&\mathbf{0}&\mathbf{0}\\ \star&\alpha_{1}^{2}Q&\mathbf{0}&C_{e}^{T}\\ \star&\star&-\gamma^{2}\mathbf{I}&D_{e}^{T}\\ \star&\star&\star&-\mathbf{I}\end{bmatrix} +\mathrm{He} \begin{cases}-X&X \bar{A}&X\bar{B}&\mathbf{0}\\ -Y&Y\bar{A}&Y\bar{B}&\mathbf{0}\\ -Z&Z\bar{A}&Z\bar{B}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}\end{cases} \end{cases},\]  
\[\mathbb{A}_{2}= \begin{bmatrix}X&Y&Z&\mathbf{0}\end{bmatrix} ^{T},\mathbb{B}_{2 }= \begin{bmatrix}\mathbf{0}&\bar{F}G_{r}\bar{M}&\bar{F}G_{r}\bar{N}&\mathbf{0} \end{bmatrix} ,N_{2}=\mathbf{I}.\]  
It is observed that \(\mathbb{A}_{t}\) is coupled with \(\mathbb{B}_{t}\). For given feasible matrices \(P_{s}^{x},G_{r}^{x},X^{x},Y^{x},Z^{x}\), we can reformulate (15) as  
\[\mathbb{L}+\mathrm{He}\{\mathbb{A}N\mathbb{B}^{x}+A^{x}N(\mathbb{B}-\mathbb{B} ^{x})\}+\mathrm{He}\{(\mathbb{A}-A^{x})N(\mathbb{B}-\mathbb{B}^{x})\}<\mathbf{ 0}. \tag{16}\]  
We notice that \(\mathbb{L}+\mathrm{He}\{\mathbb{A}N\mathbb{B}^{x}+\mathbb{A}^{x}N\ (\mathbb{B}-\mathbb{B}^{x})\}\) is linear, whereas \(\mathrm{He}\{(\mathbb{A}-A^{x})N(\mathbb{B}-\mathbb{B}^{x})\}\) is bilinear. According to [23], the bilinear term can be further decomposed into  
\[ \begin{split}&\mathrm{He}\{(\mathbb{A}-A^{x})N(\mathbb{B}- \mathbb{B}^{x})\}\\ &=\begin{bmatrix}(\mathbb{A}-A^{x})^{T}\\ (\mathbb{B}-\mathbb{B}^{x})\end{bmatrix}^{T}\begin{bmatrix}\mathbf{0}&N\\ N^{T}&\mathbf{0}\end{bmatrix}\begin{bmatrix}(\mathbb{A}-A^{x})^{T}\\ (\mathbb{B}-\mathbb{B}^{x})\end{bmatrix}\\ &=\begin{bmatrix}(\mathbb{A}-A^{x})^{T}\\ (\mathbb{B}-\mathbb{B}^{x})\end{bmatrix}^{T}\begin{bmatrix}\beta N&\beta N\\ U^{T}&-U\end{bmatrix}\begin{bmatrix}\frac{1}{\beta}(U+U^{T})^{-1}&\mathbf{0} \\ \mathbf{0}&-\frac{1}{\beta}(U+U^{T})^{-1}\end{bmatrix}\begin{bmatrix}\beta N& \beta N\\ U^{T}&-U\end{bmatrix}^{T}\begin{bmatrix}(\mathbb{A}-A^{x})^{T}\\ (\mathbb{B}-\mathbb{B}^{x})\end{bmatrix}\\ &\leq\frac{1}{\beta}\Big{(}\beta(\mathbb{A}-A^{x})N+(\mathbb{B}- \mathbb{B}^{x})^{T}U^{T}\Big{)}\left(U+U^{T}\right)^{-1}\Big{(}\beta(\mathbb{A }-A^{x})N+(\mathbb{B}-\mathbb{B}^{x})^{T}U^{T}\Big{)}^{T}\end{split}  \tag{17}\]  
where \(U\) serves as an auxiliary variable and \(U+U^{T}>\mathbf{0}\), without loss of generality. Defining \(\Delta A^{x}\triangleq\mathbb{A}-A^{x},\Delta\mathbb{B}^{x}\triangleq\mathbb{B }-\mathbb{B}^{x}\), and applying the above approximation strategy, results in  
\[