# Event-Triggered MFAILC Bipartite Formation Control for Multi-Agent Systems Under DoS Attacks
## 2 Preliminaries and Problem Formulation  
### Algebraic Graph Theory  
Graph theory is an effective tool for describing the communication relationships among agents in MASs. Taking Figure 1 as an example, it illustrates the communication topology among the agents. The communication topology of this system can be represented as a directed graph \(\overline{\mathcal{G}}=(\overline{\mathcal{V}},\overline{\mathcal{E}},A)\). \(\bar{\mathcal{V}}=\mathcal{V}\cup\{0\}\) presents the node set of the directed graph \(\bar{\mathcal{G}}\), where \(\mathcal{V}=\{1,2,\ldots,N\}\), N represents the number of agents, and \(\mathcal{V}\) consists of two subsets \(\mathcal{V}_{1}\) and \(\mathcal{V}_{2}\), satisfying \(\mathcal{V}=\mathcal{V}_{1}+\mathcal{V}_{2}\) and \(\mathcal{V}_{1}\cap\mathcal{V}_{2}\
eq\emptyset\). \(\mathcal{E}\subseteq\bar{\mathcal{V}}\times\bar{\mathcal{V}}\) denotes the edge set of the directed graph \(\bar{\mathcal{G}}\). As shown in Figure 1, the topology represents a bipartite Formation Multi-Agent System (BFMAS). The direction of information flow between nodes is indicated by the arrows. Nodes within the same subset are considered to have a cooperative relationship, and the information flow arrows for such nodes are colored black. Conversely, nodes in different subsets are regarded as having a competitive relationship, and the arrows indicating information flow between these nodes are colored red. In the adjacency matrix A, the diagonal elements \(a_{pp}=0\), while the off-diagonal elements \(a_{pj}\) are determined by the relationship between the nodes and their respective subsets. If node \(p\) can receive signals from node \(j\) and both nodes belong to the same subset, then \(a_{pp}=1\). If the two nodes belong to different subsets, then \(a_{pp}=-1\). If there is no information flow between the two nodes, \(a_{ij}=0\). The Laplacian matrix of the directed graph \(\bar{\mathcal{G}}\) can be expressed as \(L_{A}=B-A\), \(L_{|A|}=B-\left|A\right|\). Here, \(B=diag(b_{1},b_{2},\ldots,b_{N})\), \(b_{p}=\sum_{j=1}^{N}|a_{pj}|\). Therefore, matrix \(\delta\)= diag\((\delta_{1},\delta_{2},\ldots,\delta_{N})\) is defined to describe the subset attributes of node\(p\in\mathcal{V}\). For example, when node \(p\in\mathcal{V}\), \(\delta_{p}=1\); if node \(p\in V_{2}\), \(\delta_{p}=-1\).  
For leaderless BFMASs, each agent adjusts its state solely based on the information from its neighboring agents, without a clear goal. This may lead to local optimal solutions or chaotic movements among the agents. Introducing a virtual leader as a reference point can provide a common goal or direction for all agents, enabling better coordination. In Figure 1, node 0 represents the virtual leader. The connection matrix \(D=diag(d_{1},d_{2},\ldots,d_{N})\) describes the information flow between the virtual leader and the agents. If node \(p\) can receive signals from the virtual leader, \(d_{p}=1\); otherwise, \(d_{p}=0\).  
Furthermore, for BFMASs with time-varying and switching communication topologies, the time-varying communication topology can be described by a directed graph \(\overline{\mathcal{G}}(\mathsf{k})=(\overline{\mathcal{V}}(k),\overline{ \mathcal{E}}(k),A(k))\). Here, \(\overline{\mathcal{G}}(k)=\overline{\mathcal{G}}^{t}=\{\overline{\mathcal{G}} ^{1},\overline{\mathcal{G}}^{2},\ldots\}\) represents all possible communication topologies of the BFMAS. The Laplacian matrix of the directed graph \(\mathcal{G}(k)\) is denoted as \(L_{{}_{A}}(k)=B(k)-A(k)\), where \(A(k)=\left[\begin{array}{c}a_{p,j}^{t}(k)\\ \end{array}\right]\in\mathbb{R}^{N\times N}\), \(B(k)=diag(b_{1}^{t}(k),b_{2}^{t}(k),\ldots,b_{N}^{t}(k))\) and \(D(k)=diag(d_{1}(k),d_{2}(k),\ldots,d_{N}(k))\) is the connection matrix between the virtual leader and the agents.