## 2 Preliminaries and Problem Formulation  
Consider a class of stable systems, whose dynamics can be described as follows:  
\[ \begin{cases}x(k+1)=Ax(k)+Bu(k),\\ y(k)=Cx(k)+Du(k),\end{cases}  \tag{1}\]where \(x\in\mathbb{R}^{n_{x}},u\in\mathbb{R}^{n_{u}},y(k)\in\mathbb{R}^{n_{y}}\) are the vectors of state, controlled input and output, respectively, and \(A,B,C,D\) are known constant matrices. In this paper, we assume that the system (1) is positive, and the definition of positive systems is given as follows. **Lemma 1** ([25]). : _Given the initial condition \(x(0)=x_{0}\in\mathbb{R}^{n_{x}}_{+}\) and the input \(u(k)\in\mathbb{R}^{n_{u}}_{+}\), the system (1) is said to be positive if \(\forall k>0,x(k)\in\mathbb{R}^{n_{x}}_{+},y(k)\in\mathbb{R}^{n_{y}}_{+}\)._  
In the following, we present a well-established condition for checking whether the system (1) is positive. **Lemma 2** ([26]). : _The system (1) is positive if and only if \(A\succeq 0,B\succeq 0,C\succeq 0,D\succeq 0\)._  
In this paper, we aim at constructing a lower-order model to approximate the system (1), and the dynamics of the reduced-order model can be formulated in the following form  
\[ \begin{cases}x_{r}(k+1)=A_{r}x_{r}(k)+B_{r}u(k),\\ y_{r}(k)=C_{r}x_{r}(k)+D_{r}u(k),\end{cases}  \tag{2}\]  
where \(x_{r}(k)\in\mathbb{R}^{n_{x}}(0<n_{r}<n_{x}),y_{r}\in\mathbb{R}^{n_{y}}\) are the states and approximated outputs of the reduced-order system (2), and \(A_{r},B_{r},C_{r},D_{r}\) are constant matrices to be designed. Denote the augmented vector \(x_{e}(k)= \begin{bmatrix}x^{T}(k)&x_{r}^{T}(k)\end{bmatrix} ^{T}\) and the approximation error \(e(k)=y(k)-y_{r}(k)\). Substituting (2) into (1) results in the following error system  
\[ \begin{cases}x_{e}(k+1)=A_{e}x_{e}(k)+B_{e}u(k),\\ e(k)=C_{e}x_{e}(k)+D_{e}u(k),\end{cases}  \tag{3}\]  
where  
\[A_{e}= \begin{bmatrix}A&\mathbf{0}\\ \mathbf{0}&A_{r}\end{bmatrix} ,B_{e}= \begin{bmatrix}B\\ B_{r}\end{bmatrix} ,C_{e}= \begin{bmatrix}C&-C_{r}\end{bmatrix} ,D_{e}=D-D_{r}.\]  
For the error system (3), the transfer function from \(u(k)\) to \(e(k)\) is given by  
\[\mathcal{G}_{ue}(z)=C_{e}(z\mathbf{I}-A_{e})^{-1}B_{e}+D_{e}. \tag{4}\]  
The primary objective of the conventional \(H_{\infty}\) model reduction problem is to search for a reduced-order model (2) such that  
\[\|G_{ue}(z)\|_{\infty}<\gamma, \tag{5}\]  
where \(\gamma>0\) is a specified scalar. Nevertheless, a positive model is more likely to capture the dynamics of (1) with a sufficiently small error. That is, apart from the \(H_{\infty}\) performance (5), it is expected that the reduced-order model is preserved as positive.To ensure positivity, it follows from Lemma 2 that \(A_{r}\succeq 0,B_{r}\succeq 0,C_{r}\succeq 0,D_{r}\succeq 0\). For convenience, we denote \(\mathbb{S}_{r}\triangleq\{(A_{r},B_{r},C_{r},D_{r}):A_{r}\succeq 0,B_{r} \succeq 0,C_{r}\succeq 0,D_{r}\succeq 0\}\). The problem formulation for the positivity-preserving model reduction can be stated as follows:  
**Positivity-preserving \(H_{\infty}\) model reduction:** Given a stable positive system (1), construct a positive model with a lower order such that  
1. \((A_{r},B_{r},C_{r},D_{r})\in\mathbb{S}_{r}\). 2. The resulted error system (3) is asymptotically stable with an \(H_{\infty}\) performance \(\|\mathcal{G}_{ue}(z)\|_{\infty}<\gamma\). To characterize the \(H_{\infty}\) performance in (5), we present the following bounded real lemma. **Lemma 3** ([27]). : _Given the state-space realization (3) and a scalar \(\gamma>0\), the following statements are equivalent:_1. _The error system in (_3_) is asymptotically stable with an_ \(H_{\infty}\) _performance_ \(\|G_{ue}\|_{\infty}<\gamma\)_._
2. _There exists matrix_ \(P=P^{T}>\mathbf{0}\) _satisfying_ \[ \begin{bmatrix}-P&\mathbf{0}&PA_{e}&PB_{e}\\ \star&-\mathbf{I}&C_{e}&D_{e}\\ \star&\star&-P&\mathbf{0}\\ \star&\star&\star&-\gamma^{2}\mathbf{I}\end{bmatrix} <\mathbf{0}.\] (6)  
To facilitate the construction of the reduced-order model, an efficient condition equivalent to (6) is provided.