# Frequency-Limited Model Reduction for Linear Positive Systems: A Successive Optimization Method
## 2 Problem Statement and Fundamental Results  
Consider a stable system (\(\Sigma\)) with a relatively high order:  
\[\Sigma: \begin{cases}\lambda[x(t)]=Ax(t)+Bu(t),\\ y(t)=Cx(t)+Du(t),\end{cases}  \tag{1}\]where \(x\in\mathbb{R}^{n_{x}},u\in\mathbb{R}^{n_{u}},y\in\mathbb{R}^{n_{y}}\) are the state, input and output vectors, respectively, and \([A,B;C,D]\) are the given constant matrices. The model \(\Sigma\) provides a unified description for linear systems, that is, \(\lambda[x(t)]\triangleq\dot{x}(t)\) for the continuous-time (CT) case and \(\lambda[x(t)]\triangleq x(t+1)\) for the discrete-time (DT) case. Suppose that \(\Sigma\) is a positive system whose definition is provided as follows:  
**Lemma 1** ([26]). : _The system \(\Sigma\) is positive if all the state and output trajectories are non-negative, that is, \(\forall t\geq 0,x(t)\in\mathbb{R}^{n_{x}}_{+}\) and \(y(t)\in\mathbb{R}^{n_{u}}_{+}\), provided that \(x(0)\in\mathbb{R}^{n_{x}}_{+}\) and \(u(t)\in\mathbb{R}^{n_{u}}_{+}\)._  
**Lemma 2** ([19]). : _For the CT case, the system \(\Sigma\) is positive if, and only if, \(A\) is Metzler and \(B\succeq 0\), \(C\succeq 0,D\succeq 0\), whereas, for the DT case, the system \(\Sigma\) is positive if, and only if, \(A\succeq 0,B\succeq 0\), \(C\succeq 0,D\succeq 0\)._  
We intend to investigate an available model \(\Sigma_{r}\) with a smaller order  
\[\Sigma_{r}: \begin{cases}\lambda[x_{r}(t)]=A_{r}x_{r}(t)+B_{r}u(t),\\ y_{r}(t)=C_{r}x_{r}(t)+D_{r}u(t),\end{cases}  \tag{2}\]  
for approximating \(\Sigma\) with a sufficiently small error, where \(x_{r}\in\mathbb{R}^{n_{r}}(0<n_{r}<n_{x}),y_{r}\in\mathbb{R}^{n_{y}}\) are reduced states and approximated outputs, respectively, and \([A_{r},B_{r};C_{r},D_{r}]\) are unknown matrices. By defining \(x_{e}(t)= \begin{bmatrix}x^{T}(t)&x_{r}^{T}(t)\end{bmatrix} ^{T}\) and \(e(t)\triangleq y(t)-y_{r}(t)\), the approximation error system can be expressed as  
\[\Sigma_{e}: \begin{cases}\lambda[x_{e}(t)]=A_{e}x_{e}(t)+B_{e}u(t),\\ e(k)=C_{e}x_{e}(t)+D_{e}u(t),\end{cases}  \tag{3}\]  
where  
\[A_{e}= \begin{bmatrix}A&\mathbf{0}\\ \mathbf{0}&A_{r}\end{bmatrix} ,B_{e}= \begin{bmatrix}B\\ B_{r}\end{bmatrix} ,C_{e}= \begin{bmatrix}C&-C_{r}\end{bmatrix} ,D_{e}=D-D_{r}.\]  
The transfer function of (3) is presented as:  
\[\bar{G}_{ue}(\lambda)=C_{e}(\lambda\mathbf{I}-A_{e})^{-1}B_{e}+D_{e} \tag{4}\]  
where \(\lambda\) denote the Laplace operator for the CT case and the \(z\) operator for the DT case, respectively. For brevity, we give the following set to denote a positive system. **Definition 1**. : _The reduced-order system \(\Sigma_{r}\) is positive if, and only if, \( \begin{bmatrix}A_{r}&B_{r}\\ C_{r}&D_{r}\end{bmatrix} \in\mathbb{P}\), where_  
\[\mathbb{P}\triangleq \begin{cases}\begin{bmatrix}A_{r}&B_{r}\\ C_{r}&D_{r}\end{bmatrix}:A_{r}\text{ is Metzler },B_{r}\succeq 0,C_{r}\succeq 0,D_{r} \succeq 0\\ \begin{bmatrix}A_{r}&B_{r}\\ C_{r}&D_{r}\end{bmatrix}:A_{r}\succeq 0,B_{r}\succeq 0,C_{r}\succeq 0,D_{r} \succeq 0\end{cases}  \tag{5}\]  
There are two concerns when developing a reduced-order model \(\Sigma_{r}\): First, it is expected that \( \begin{bmatrix}A_{r}&B_{r}\\ C_{r}&D_{r}\end{bmatrix} \in\mathbb{P}\) to preserve the positivity of \(\Sigma\). Second, in practical applications, it suffices to evaluate the approximation performance within a finite-frequency range. To provide an exact characterization for the property, we employ a frequency-limited \(H_{\infty}\) specification  
\[\|G_{ue}(\lambda)\|_{\infty}^{\Omega}\triangleq \begin{cases}\sup_{\omega\in \Omega}\sigma_{max}[G_{ue}(j\omega)]<\gamma,\text{(CT case)}\\ \sup_{\omega\in\Omega}\sigma_{max}[G_{ue}(e^{j\omega})]<\gamma,\text{(DT case)} \end{cases}  \tag{6}\]throughout the paper, where \(\Omega\) gives the frequency interval of interest, \(\sigma_{max}\) denotes the maximum singular value, and \(\gamma>0\) is an index to be minimized.