# Frequency-Limited Model Reduction for Linear Positive Systems: A Successive Optimization Method
## 3 Main Results
\begin{bmatrix}\mathbb{L}+\mathrm{He}\{\mathbb{A}N\mathbb{B}^{x}+\mathbb{A}^{x }N\Delta\mathbb{B}^{x}\}&\beta\Delta A^{x}N+(U\Delta\mathbb{B}^{x})^{T}\\ \star&-\beta(U+U^{T})\end{bmatrix} <\mathbf{0}. \tag{18}\]  
Similarly, we decompose \(U\) as \(U=U^{x}+\Delta U^{x}\), and thus  
\[ \begin{bmatrix}\mathbb{L}+\mathrm{He}\{\mathbb{A}N\mathbb{B}^{x}+\mathbb{A}^{x }N\Delta\mathbb{B}^{x}\}&\beta\Delta A^{x}N+(U^{x}\Delta\mathbb{B}^{x})^{T}\\ \star&-\beta(U+U^{T})\end{bmatrix} +\mathrm{He}\bigg{\{} \begin{bmatrix}(\Delta \mathbb{B}^{x})^{T}\\ \mathbf{0}\end{bmatrix}\begin{bmatrix}\mathbf{0}\ \ (\Delta U^{x})^{T}\end{bmatrix} \right\}<\mathbf{0}. \tag{19}\]  
According to \(\mathrm{He}\{\mathbb{U}V\}\leq\mathbb{U}\mathbb{M}\mathbb{U}^{T}+\mathbb{V}^{T} \mathbb{M}^{-1}\mathbb{V}\), one obtains that  
\[ \begin{split}&\begin{bmatrix}\mathbb{L}+\mathrm{He}\{\mathbb{A}N \mathbb{B}^{x}+\mathbb{A}^{x}N\Delta\mathbb{B}^{x}\}&\beta\Delta A^{x}N+(U^{x} \Delta\mathbb{B}^{x})^{T}\\ \star&-\beta(U+U^{T})\end{bmatrix}\\ &+\begin{bmatrix}(\Delta\mathbb{B}^{x})^{T}\\ \mathbf{0}\end{bmatrix}V\big{[}(\Delta\mathbb{B}^{x})\ \mathbf{0}\big{]}+\begin{bmatrix} \mathbf{0}\\ (\Delta U^{x})\end{bmatrix}V^{-1}\big{[}\mathbf{0}\ \ (\Delta U^{x})^{T}\big{]}<\mathbf{0}.\end{split}  \tag{20}\]  
which is equivalent to  
\[ \begin{bmatrix}\mathbb{L}+\mathrm{He}\{\mathbb{A}N\mathbb{B}^{x}+\mathbb{A}^{x }N\Delta\mathbb{B}^{x}\}&\beta\Delta A^{x}N+(U^{x}\Delta\mathbb{B}^{x})^{T}&( \Delta\mathbb{B}^{x})^{T}&\mathbf{0}\\ \star&-\beta(U+U^{T})&\mathbf{0}&(\Delta U^{x})\\ \star&\star&-V^{-1}&\mathbf{0}\\ \star&\star&\star&-V\end{bmatrix} <\mathbf{0} \tag{21}\]We observe that \(-V^{-1}\) is convex, implying \(-V^{-1}\leq-2(V^{\kappa})^{-1}+(V^{\kappa})^{-1}V(V^{\kappa})^{-1}\) for a given \(V^{\kappa}\). Replacing \(-V^{-1}\) with \(-2V^{\kappa}+(V^{\kappa})^{-1}V(V^{\kappa})^{-1}\) and performing the congruent transformation \(\operatorname{diag}\{\mathbf{I},\mathbf{I},V^{\kappa},\mathbf{I}\}\) on (21), we can obtain (9) and (10). In Theorem 1, we first establish sufficient and necessary conditions (11) and (14) for the existence of a desired model \(\Sigma_{r}\) in the form of BMIs. Given feasible solutions \(P_{s}^{\kappa},G_{r}^{\kappa},\)\(X^{\kappa},\)\(Y^{\kappa},\)\(Z^{\kappa}\) satisfying \(\mathbb{L}_{s}+\operatorname{He}\{\mathbb{A}_{s}^{\kappa}N_{\mathbb{B}} \mathbb{R}_{s}^{\kappa}\}<\mathbf{0},\mathbb{L}+\operatorname{He}\{\mathbb{A} ^{\kappa}N\mathbb{B}^{\kappa}\}<\mathbf{0}\), the BMI conditions can be rewritten as the sum of linear and residual terms. By virtue of the convex-approximation strategy, we propose an SCO algorithm that iteratively solves a series of convex optimization sub-problems,  
\[ \begin{cases}\min\gamma\\ \text{s.t. }P_{s}=P_{s}^{T}>\mathbf{0},Q=Q^{T}>\mathbf{0},(8)\text{--}(10). \end{cases}  \tag{22}\]  
each of which is an inner convex approximation of the original non-convex constraint. At each iteration, the solution to the previous step is used as a starting point for the next iteration until the convergence is reached. To be concise, the above procedures can be summarized in the following algorithm. **Remark 2**. : _As pointed out in Algorithm 1, it is crucial to have an admissible \(G_{r}^{0}\) for optimizing the subsequent solutions. However, one can see that obtaining such an initial condition is equally as intricate as solving the original problem. It can be shown that the objective function decreases with each iteration of the SCO algorithm, and according to the convergence property, we can obtain an initial solution by simply choosing a stable model \(G_{r}^{0}\in\mathbb{P}\). This significantly asses the challenge of implementing the proposed algorithm._  
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0:\(\epsilon\): tolerable bound; \(1 Max\): maximal iteration limit
1: Given a feasible solution \(G_{r}^{0}\), solve the convex optimization problem \[