# Impulsive Multiple-Bipartite Consensus Control for Networked Second-Order Multi-Agent Systems
## 4 Simulations  
In this section, two simulation examples will be assessed to illustrate the aforementioned theoretical results. For convenience, we assume \(N=1\) in this section. Consider a network consisting of nine agents, and the information communication graph topology is shown in Figure 1. Obviously, node set \(\mathcal{V}=\{1,2,\cdots,9\}\) has an acyclic partition as \(\mathcal{V}_{1}=\{1,2,3\}\), \(\mathcal{V}_{2}=\{4,5,6\}\) and \(\mathcal{V}_{3}=\{7,8,9\}\). The corresponding graphs \(\mathcal{G}_{1}\), \(\mathcal{G}_{2}\), and \(\mathcal{G}_{3}\) are all structurally balanced. Following the definition of \(\phi_{i}\), we have \(\phi_{1}=\phi_{4}=\phi_{6}=\phi_{7}=\phi_{8}=1\) and \(\phi_{2}=\phi_{3}=\phi_{5}=\phi_{9}=-1\). Then, \(\Phi=\mathrm{diag}\{1,-1,-1,1,-1,1,1,-1\}\). Furthermore, the eigenvalues of matrix \(\Phi\mathcal{L}\Phi\) can be obtained by direct calculation as \(0\), \(0\), \(0\), \(1\)\(1\), \(1\), \(1.5+0.866\mathbf{i}\) and \(1.5-0.866\mathbf{i}\), where \(\mathbf{i}\) is an imaginary unit. The corresponding vectors \(\xi_{i}\), \(i=1,2,\cdots,3\), defined in Equation (8) have the form \(\xi_{1}=[1,0,0,0,0,0,0,0,0]^{\mathrm{T}}\), \(\xi_{2}=[\frac{2}{3},-\frac{1}{3},-\frac{1}{3},\frac{1}{3},\frac{1}{3},0,0,0]^{ \mathrm{T}}\) and \(\xi_{3}=[0,0,0,0,0,0,0,1,0]^{\mathrm{T}}\). The initial values \(r(0)\) and \(v(0)\) are set as \(r(0)=[r_{1}(0),r_{2}(0),\cdots,r_{9}(0)]^{\mathrm{T}}=[2,-1,-3,-1,2,-1.5,-2,3,-5] ^{\mathrm{T}}\) and \(v(0)=[v_{1}(0),v_{2}(0),\cdots,v_{9}(0)]^{\mathrm{T}}=[2,1,-2,3,-7,-5,10,6,-3] ^{\mathrm{T}}\). The control gains \(\alpha\) and \(\beta\) and control interval \(h\) are set as \(\alpha=1\), \(\beta=0.1\) and \(h=0.02\), respectively. Then, all the conditions of Theorem 1 are satisfied. Therefore, according to Theorem 1, through simple calculations, we can obtain the critical value of \(h=\min_{1\leq i\leq n-k}\frac{4\beta^{2}\mathbf{Re}(\lambda_{i})|\lambda_{i}|^ {2}-2\beta^{3}|\lambda_{i}|^{4}}{4\mathbf{Im}^{2}(\lambda_{i})+\alpha\beta^{2 }|\lambda_{i}|^{4}}=0.1108\). Accordingly, if \(h<0.1108\), multiple-bipartite consensus should be reached; and the final multiple-bipartite consensus states are \(\lim_{t\to\infty}r_{1}(t)=2+2t\), \(\lim_{t\to\infty}r_{2}(t)=-2t\), \(\lim_{t\to\infty}r_{3}(t)=-2-2t\), \(\lim_{t\to\infty}r_{4}(t)=-\frac{3}{2}+\frac{3}{8}t\), \(\lim_{t\to\infty}r_{5}(t)=\frac{3}{2}-\frac{8}{8}t  
Figure 3: The time evolution trajectories of \(v_{i}(t)\), \(i=1,2,\cdots,9\), with \(h=0.02\) in Example 1. Figure 2: The time evolution trajectories of \(r_{i}(t)\), \(i=1,2,\cdots,9\), with \(h=0.02\) in Example 1. **Example 2**. : _The interesting result of some subgroups being cooperative networks will be assessed through the simulation of this example. Consider the networks consisting of nine agents and the graph topology presented in Figure 6. It is easy to see that the node set \(\mathcal{V}=\{1,2,\cdots,9\}\) has an AP as \(\mathcal{V}_{1}=\{1,2,3\}\), \(\mathcal{V}_{2}=\{4,5,6\}\) and \(\mathcal{V}_{3}=\{7,8,9\}\). Since it is different from Example 1, subgraph  
Figure 4: The time evolution trajectories of \(r_{i}(t)\), \(i=1,2,\cdots,9\), with \(h=0.2\) in Example 1. Figure 5: The time evolution trajectories of \(v_{i}(t)\), \(i=1,2,\cdots,9\), with \(h=0.2\) in Example 1. \(\mathcal{G}_{1}\) corresponding to the first subgroup \(\mathcal{V}_{1}=\{1,2,3\}\) is cooperative. In addition, the second and third subgraphs \(\mathcal{G}_{2}\) and \(\mathcal{G}_{3}\) are structurally balanced. The control gains \(\alpha\) and \(\beta\) and control interval \(h\) are set as \(\alpha=1\), \(\beta=0.1\) and \(h=0.02\), respectively. In this case, the final convergence state should be that the first group reaches a complete consensus, and the second and third groups reach bipartite consensus. Figures 7 and 8 display the trajectories of the nine agents. As can be seen from Figures 7 and 8, the first group reaches a complete consensus, and the second and third groups reach bipartite consensus, which coincides with our deduction.