## 5 Simulation Results  
This section shows the effectiveness of the equivalent ARE approach described in Section 4 and Theorem 2. The simulation is realized in MATLAB/Simulink. Consider a class of Multiple Autonomous Unmanned homogeneous linear systems referring to [5] which is shown as follows  
\[\dot{x}_{i}=Ax_{i}+B_{i}u_{i}\]  
where \(A=\begin{bmatrix}0&1\\ -1&0\end{bmatrix}\), \(B_{1}=\begin{bmatrix}2\\ 1\end{bmatrix}\), \(B_{2}=\begin{bmatrix}2\\ 3\end{bmatrix}\), \(B_{3}=\begin{bmatrix}2\\ 2\end{bmatrix}\) and \(B_{3}=\begin{bmatrix}1\\ 1\end{bmatrix}\), with the Leader dynamics \(\dot{x}_{0}=Ax_{0}\). The communication digraph structure is shown in Figure 1. The edge weights and the pinning gains are taken equal to 1.  
The selected the weight matrices in (11) are \(R_{11}=9\), \(R_{14}=1\), \(R_{22}=9\), \(R_{21}=1\), \(R_{33}=9\), \(R_{32}=1\), \(R_{44}=9\), \(R_{43}=1\) and \(Q_{1}=Q_{2}=Q_{3}=Q_{4}=\begin{bmatrix}8&0\\ 0&8\end{bmatrix}\). The bounded \(L_{2}\)-gain coefficient in (11) for each agent are preset as \(\gamma_{1}=1.75\), \(\gamma_{2}=3.75\), \(\gamma_{3}=4.5\) and \(\gamma_{4}=6.25\). The cooperative control protocol of each agent is implemented, as in Section 4.2, where the solution of ARE (33) is  
\[P_{1}=\begin{bmatrix}4.8954&-5.6084\\ -5.6084&9.4934\end{bmatrix},P_{2}=\begin{bmatrix}41.9228&-7.3129\\ -7.3129&2.6742\end{bmatrix},P_{3}=\begin{bmatrix}14.4257&-9.363\\ -9.363&10.4565\end{bmatrix}P_{4}=\begin{bmatrix}17.4696&-0.5066\\ -0.5066&13.6\end{bmatrix}.\]  
**Remark 4**.: _In the process of a simulation design, the \(\gamma_{i}\) should be gradually reduced to search for a feasible and high coupling attenuation level under the premise that (33) has a positive definite solution \(P_{i}\). Using the ARE solver in MATLAB, it is very convenient to solve (33) and obtain \(P_{i}\). Then, the design of the coupling attenuation controller can be completed according to (30)._  
To elevate the Bounded \(L_{2}\)-gain problem for the coupling attenuation, the following variable \(C_{\gamma}\) is introduced based on (29)  
\[C_{\gamma}=\int_{0}^{T}\ \left(\delta_{i}^{T}Q_{i}\delta_{i}+u_{i}^{T}R_{ii}u_{ i}\right)\ dt-\gamma_{i}^{2}\int_{0}^{T}\sum_{j\in N_{i}}u_{j}^{T}R_{ji}u_{j}\ dt-V_{i}^{*}(\delta_{i}(0)) \tag{39}\]  
That is, \(C_{\gamma}\leq 0\), which means that the Bounded \(L_{2}\)-gain condition (10) is satisfied.  
The local neighborhood error vector of each agent is shown in Figure 2. Figure 3 is the 3-D phase plane plot of the system's evolution for agents 1, 2, 3, 4 and leader 0. The \(C_{\gamma}\) of the node 1 agent is shown in Figure 4. As can be seen from Figures 2 and 3, the neighborhood error vector can converge asymptotically and uniformly to the origin and all agents in the communication digraph are eventually synchronized, which is also consistent with Remark 2. Figure 4 shows that \(C_{\gamma}\) is always negative, which is equivalent to that the node 1 agent satisfies the bounded \(L_{2}\)-gain condition with \(\gamma_{1}=1.75\) for the coupling attenuation. The effectiveness of the proposed method is thus verified.  
Figure 1: The communication digraph structure.  
Figure 3: The 3-D phase plane plot of the system’s evolution for agents 1, 2, 3, 4 and leader 0.  
Figure 2: The local neighborhood error vector of each agent: (**a**) The node 1 agent; (**b**) The node 2 agent; (**c**) The node 3 agent; (**d**) The node 4 agent.