## 4 Simulation  
Compartmental networks consist of a finite number of homogeneous, well-mixed subsystems, called compartments, which exchange with each other and the environment [28]. In this example, we compare the proposed model-reduction method with [26]. Consider the compartmental model of six states with the following system matrices:  
\[A =\begin{bmatrix}0.7&0.12&0.2&0&0&0\\ 0.06&0.6&0.04&0&0&0\\ 0.04&0.1&0.4&0.2&0&0\\ 0&0&0.1&0.4&0.15&0.1\\ 0&0&0&0.08&0.6&0.06\\ 0.05&0&0&0.12&0.1&0.68\end{bmatrix},B=\begin{bmatrix}0.5&0\\ 0&0.7\\ 0&0.0\\ 0.2&0\\ 0&0\end{bmatrix},\] \[C =\begin{bmatrix}0.3&0.2&0.5&1&0.6&0.9\end{bmatrix},D=\begin{bmatrix} 0&0.3\end{bmatrix}.\]  
and we observe that the system is stable. The aim of this example is to construct reduced-order positive models with different orders to approximate the original system.  
Case 1: \(n_{r}=4\)  
According to Remark 3, the initial condition \(A_{r}^{0}=\mathbf{0},B_{r}^{0}=\mathbf{0},C_{r}^{0}=\mathbf{0},D_{r}^{0}= \mathbf{0}\) is adopted, and by Algorithm 1, one can iteratively obtain the reduced-order model as:  
\[G_{r}^{(4)}=\left[\begin{array}{cccc}0.8201&0.0054&0.0032&0.0022 \text{'}.0811&0.0385\\ 0.3838&0.3638&0.0610&0.0327\text{'}.0709&0.0439\\ 0.2039&0.1697&0.4350&0.0262\text{'}.0662&0.0471\\ 0.0451&0.1008&0.3554&0.2962\text{'}.0502&0.0605\\ 0.0234&0.0355&0.0510&5.2130\text{'}.0004&0.3000\\ \end{array}\right].\]  
Moreover, by adopting the model reduction method in [26], we obain  
\[G_{r}^{(4)^{\prime}}=\left[\begin{array}{cccc}0.5602&0.2264&0.0558&0.1771 \text{'}.09126&0.0231\\ 0.1634&0.2966&0.1163&0.2773\text{'}.0689&0.0386\\ 0.0326&0.1058&0.4542&0.1454\text{'}.0230&0.7114\\ 0.1125&0.2105&0.1469&0.2337\text{'}.0545&0.0474\\ 0.2828&0.3093&0.3349&0.3084\text{'}.0110&0.3529\\ \end{array}\right].\]  
Similarly, the corresponding second-order model for Algorithm 1 and [26] can be readily obtained as follows.  
Case 2: \(n_{r}=2\) with  
\[G_{r}^{(2)}=\left[\begin{array}{cccc}0.4774&0.3713\text{'}&0.0566&0.0684\\ 0.0000&0.8342\text{'}&0.0799&0.0397\\ 4.6078&0.0555&0.0015&0.3005\\ \end{array}\right],\]  
\[G_{r}^{(2)^{\prime}}=\left[\begin{array}{cccc}0.3106&0.1945&0.1020&0.4109\\ 0.5039&0.6506\text{'}&0.7127&0.1763\\ \text{'}.04228&0.3956&0.0035&0.3694\\ \end{array}\right].\]  
Moreover, the \(H_{\infty}\) performance indices \(\gamma\) with respect to the above cases are listed in Table 1. It can be readily observed from the attained results that the provided algorithm can achieve smaller \(H_{\infty}\) performance indices than [26]. To evaluate the approximation effects of different model reduction methods, Figures 1 and 2 provide the frequency response gains of the original and reduced-order systems, and Figures 3 and 4 depict the frequency response gains of the associated error systems with different orders. We can easily infer that compared with \(G_{r}^{(4)^{\prime}}\) and \(G_{r}^{(2)^{\prime}}\), the frequency gains of proposed forth-order \(G_{r}^{(4)}\) model and second-order one \(G_{r}^{(2)}\) are much closer to the original system, and the associated error system with \(G_{r}^{(4)}\) and \(G_{r}^{(2)}\) have smaller gains than that for corresponding \(G_{r}^{(4)^{\prime}}\) and \(G_{r}^{(2)^{\prime}}\), which illustrates the benifits of the proposed algorithm.  
\begin{table}
\begin{tabular}{c c c} \hline
**Different Methods** & **Algorithm 1** & [26] \\ \hline \(n_{r}=4\) & 0.004 & 0.15 \\ \(n_{r}=2\) & 0.004 & 0.18 \\ \hline \end{tabular}
\end{table}
Table 1: The \(H_{\infty}\) performance indices achieved by different methods.  
Figure 1: Frequency response gains of the original and reduced-order systems with \(n_{r}=4\).  
Figure 3: Frequency response gains of the associated error systems with \(n_{r}=4\).  
Figure 2: Frequency response gains of the original and reduced-order systems with \(n_{r}=2\).  
Figure 8: The approximation errors between the original model and \(G_{r}^{(2)},G_{r}^{(2)^{\prime}}\).  
Figure 6: The output trajectories of the original model and \(G_{r}^{(2)},G_{r}^{(2)^{\prime}}\).