# Impulsive Multiple-Bipartite Consensus Control for Networked Second-Order Multi-Agent Systems
## 2 Preliminaries
### Problem Formulation  
Assume that the \(i\)th \((i=1,2,\cdots,n)\) agent dynamics which are described by the following second-order differential equation  
\[ \begin{cases}\dot{r}_{i}(t)=v_{i}(t),\\ \dot{v}_{i}(t)=u_{i}(t),\end{cases}  \tag{1}\]  
where \(r_{i}(t)\in\mathbb{R}^{N}\), \(v_{i}(t)\in\mathbb{R}^{N}\), and \(u_{i}(t)\in\mathbb{R}^{N}\) are the position, velocity and control input of the \(i\)th agent, respectively. Let the graph \(\mathcal{G}\) have a partition \(\Xi=\{\mathcal{P}_{1},\mathcal{P}_{2},\cdots,\mathcal{P}_{k}\}\), that is, \(\cup_{w=1}^{k}\mathcal{P}_{w}=\mathcal{V}\), \(\mathcal{P}_{w}\cap\mathcal{P}_{z}=\emptyset\); each of \(\mathcal{P}_{w}\) is not an empty set; and \(w,z=1,2,\cdots,k\). Let \(\tilde{i}\) be the index of the subset associated with the aforementioned partition to which the \(i\)th agent belongs, i.e., \(i\in\mathcal{P}_{\tilde{i}}\). Moreover, we assume \(\mathcal{P}_{\tilde{i}}=\{h_{\tilde{i}-1}+1,h_{\tilde{i}-1}+2,\cdots,h_{ \tilde{i}}\}\), \(h_{0}=0\), \(h_{k}=n\) and \(h_{\tilde{i}}-h_{\tilde{i}-1}=n_{\tilde{i}}\in\mathbb{Z}^{+}\), \(\tilde{i}=1,2,\cdots,k\). That is, \"\(h_{i}\)\" denotes the node number sum of the first \(\tilde{i}\) subgroups, and \"\(n_{i}\)\" denotes the node number of the \(i\)th subgroup. This paper aimed to construct a multiple symmetric control task for second-order MASs (1); and accordingly, the mathematical model of this control task should be an organic unity of cluster consensus and bipartite consensus. By considering this background, we formulated the definition of a multiple bipartite consensus for second-order MASs (1). **Definition 1**. : _Under the control input \(u_{i}(t)\) and the partition \(\Xi\), the second-order MASs (1) are referred to reach multiple-bipartite consensus if the following conditions hold: (1) Each \(\mathcal{P}_{\tilde{i}}\) can be divided into two subgroups as \(\mathcal{P}_{\tilde{i}}^{(1)}\) and \(\mathcal{P}_{\tilde{i}}^{(2)}\), satisfying that (i) \(\mathcal{P}_{\tilde{i}}^{(1)}\) and \(\mathcal{P}_{\tilde{i}}^{(2)}\) are all nonempty sets; (ii) \(\mathcal{P}_{\tilde{i}}^{(1)}\cup\mathcal{P}_{\tilde{i}}^{(2)}=\mathcal{P}_{ \tilde{i}}\); (iii) \(\mathcal{P}_{\tilde{i}}^  
bipartite consensus of second-order MASs. Therefore, the structurally balanced topological condition should be introduced as in many existing references, such as [14,31]. If all \(\mathcal{G}_{i}\) are structurally balanced, then each \(\mathcal{P}_{i}\) can be divided into two subgroups as \(\mathcal{P}_{i}^{(1)}\) and \(\mathcal{P}_{\tilde{i}}^{(2)}\), satisfying that \(\mathcal{P}_{\tilde{i}}^{(1)}\) and \(\mathcal{P}_{\tilde{i}}^{(2)}\) are all nonempty sets, \(\mathcal{P}_{\tilde{i}}^{(1)}\cup\mathcal{P}_{\tilde{i}}^{(2)}=\mathcal{P}_{ \tilde{i}}\) and \(\mathcal{P}_{\tilde{i}}^{(1)}\cap\mathcal{P}_{\tilde{i}}^{(2)}=\emptyset\)._  
If all \(\mathcal{G}_{i}\), \(\tilde{i}\in\{1,2,\cdots,k\}\), are structurally balanced, then for \(i\in\{1,2,\cdots,n\}\), \(\exists\phi_{i}\in\{1,-1\}\), such that \(\Phi_{i}\mathcal{L}_{i}\overline{\phi}_{\tilde{i}}\) is a traditional standard Laplacian matrix, i.e., the off-diagonal elements of \(\mathcal{L}_{i\overline{i}}\) are all non-positive, where \(\Phi_{\tilde{i}}=\mathrm{diag}\{\phi_{\tilde{i}_{i-1}+1},\cdots,\phi_{\tilde{ i}_{i}}\}\) and \(\mathcal{A}_{i\overline{i}}\) is the weighted matrix associated with the agent set \(\mathcal{V}_{\tilde{i}}\). Now, the Laplacian matrix \(\mathcal{L}\) associated with graph \(\mathcal{G}\) can be defined as \(\mathcal{L}=[l_{ij}]_{n\times n}\), \(i,j\in\{1,2,\cdots,n\}\), where \(l_{ij}=-a_{ij}\), \(i\
eq j\) and \(l_{ii}=\sum_{j\
eq P_{i}}\phi_{j}a_{ij}+\sum_{j\in P_{j}}|a_{ij}|\), \(i,j=1,2,\cdots,n\). In addition to the structurally balanced topological structure, an acyclic partition [37] is an essential condition ensuring the multiple-bipartite consensus of second-order MASs (1). If partition is an acyclic partition, then the Laplacian matrix \(\mathcal{L}\) has the form of [37]  
\[\mathcal{L}=\left(