# Flocking of Multi-Agent System with Nonlinear Dynamics via Distributed Event-Triggered Control
## 2 Preliminaries and Problems Formulation
### Controller Design and Stability Analysis
Taking the time derivative of \(Q(t)\) on time interval \([T_{0},T_{1})\) gives  
\[-c\left(\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)(\hat{v}_{i}(t)- \hat{v}_{j}(t))+\epsilon_{i}^{x}(t)\right).\]1 \[-c\left(\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)(\hat{v}_{i}(t)- \hat{v}_{j}(t))+\epsilon_{i}^{x}(t)\right).\]2 \[-c\left(\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)(\hat{v}_{i}(t)- \hat{v}_{j}(t))+\epsilon_{i}^{x}(t)\right).\]3 \[-c\left(\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)(\hat{v}_{i}(t)- \hat{v}_{j}(t))+\epsilon_{i}^{x}(t)\right).\]4 \[-c\left(\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)(\hat{v}_{i}(t)- \hat{v}_{j}(t))+\epsilon_{i}^{x}(t)\right).\]5 \[-c\left(\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)(\hat{v}_{i}(t)- \hat{v}_{j}(t))+\epsilon_{i}^{x}(t)\right).\]6 \[-c\left(\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)(\hat{v}_{i}(t)- \hat{v}_{j}(t))+\epsilon_{i}^{x}(t)\right).\]7 \[-c\left(\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)(\hat{v}_{i}(t)- \hat{v}_{j}(t))+\epsilon_{i}^{x}(t)\right).\]8  
Please note that the inequality \(|xy|\leq a\pi^{2}/2+y^{2}/2a\) holds for \(\forall a>0\). According to the properties of Laplacian matrix, one has \(\sum_{i=1}^{N}\bar{\sigma}_{i}^{T}\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)(\bar {\sigma}_{i}(t)-\bar{\sigma}_{j}(t))\geq\sum_{i=1}^{N}\lambda_{2}(L(0))\|\bar{ \sigma}_{i}(t)\|^{2}\), then  
\[-c\left(\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)(\hat{v}_{i}(t)- \hat{v}_{j}(t))+\epsilon_{i}^{x}(t)\right).\]9 \[t_{k+1}^{i}=\inf\left\{t>t_{k}^{i}:g_{1}\left(\epsilon_{i}^{x}(t),\epsilon_{i} ^{y}(t),\tilde{v}_{i}(t)\right)\geq 0\right\}, \tag{9}\]0 \[t_{k+1}^{i}=\inf\left\{t>t_{k}^{i}:g_{1}\left(\epsilon_{i}^{x}(t),\epsilon_{i} ^{y}(t),\tilde{v}_{i}(t)\right)\geq 0\right\}, \tag{9}\]1  
Since \(a_{1},a_{2}\) and \(c\) satisfy \(c>(2\rho+(a_{1}+ca_{2})/2)\)\(/\lambda_{2}(L)\), enforcing the triggering condition (10), one has  
\[t_{k+1}^{i}=\inf\left\{t>t_{k}^{i}:g_{1}\left(\epsilon_{i}^{x}(t),\epsilon_{i} ^{y}(t),\tilde{v}_{i}(t)\right)\geq 0\right\}, \tag{9}\]2  
which implies that \(Q(t)<Q(T_{1}),\forall t\in[T_{0},T_{1})\). 1. If \(T_{1}=\hat{t}_{1}\), \(Q\) is continuous and \(T_{1}^{+}=T_{1}^{-}<Q^{*}\);
2. If \(T_{1}=\hat{t}_{1}\), which implies the first topology switching occurs at instant \(T_{1}\). Since \(Q(t)<Q(T_{0}),\forall t\in[T_{0},T_{1})\), from the definition of the potential function, one has \(Q_{0}<\psi(r)\), which implies that no edge will tend to \(r\) for \(t\in[T_{0},T_{1})\). Therefore, new edges must be added into the network atinstant \(T_{1}\). For a system consists of \(N\) agents, there are at most \(N(N-1)/2\) edges. At the initial instant \(t_{0}\), the system contains \(m_{0}\) edges, then \(Q(\hat{t}_{1}^{+})\leq Q_{0}+(N(N-1)/2-m_{0})\psi(\|r-\tau\|)<Q^{*}\), no edges are lost. In addition, since \(\psi(\|r_{0}\|)\geq Q^{*}\), we can also derive that no collision occurs during \(t\in[t_{0},\hat{t}_{1})\). Similarly, taking the time derivative of \(Q(t)\) on \(\forall t\in[T_{\mu^{\prime}},T_{\mu^{\prime}+1})\), one can have \(\dot{Q}(t)\leq 0\), which implies \(Q(t)\) is decreasing monotonously as far as the topology of the network keeps fixed; thus, no edge-distance will tend to \(r\) for \(t\in[T_{\mu^{\prime}},T_{\mu^{\prime}+1})\), which means no edges will be lost at time \(\hat{t}_{k^{\prime}}\). Then by Lemma 1, \(\lambda_{2}(L(0))\leq\lambda_{2}(L(t))\) for \(\forall t\geq 0\). Thus, the triggering function (10) is available in the whole process. As the network is initially connected, \(\mathcal{G}(t)\) will be connected for \(\forall t\geq 0\). Furthermore, from the definition of potential function, \(\psi(r_{0})\geq Q^{*}\), we can deduce no edge-distance will tend to \(r_{0}\) for \(\forall t\geq 0\) similar to the above analysis, thus collisions are avoided during the whole process. This completes the proof of part i). Next, we prove part ii). Assume that there are \(m_{k}\) new edges that can be added into \(\mathcal{G}(t)\) at \(\hat{t}_{k^{\prime}}\).