# Flocking of Multi-Agent System with Nonlinear Dynamics via Distributed Event-Triggered Control
## 2 Preliminaries and Problems Formulation
### Controller Design and Stability Analysis
The time instant \(t_{k}^{i}\) for each agent \(i\) is updated by  
\[t_{k+1}^{i}=\inf\left\{t>t_{k}^{i}:g_{1}\left(\epsilon_{i}^{x}(t),\epsilon_{i} ^{y}(t),\tilde{v}_{i}(t)\right)\geq 0\right\}, \tag{9}\]  
where  
\[g_{1}\left(\epsilon_{i}^{x}(t),\epsilon_{i}^{y}(t),\tilde{v}_{i} (t)\right)= \frac{1}{2a_{1}}\|\epsilon_{i}^{x}(t)\|^{2}+\frac{c}{2a_{2}}\| \epsilon_{i}^{y}(t)\|^{2}-\eta\|\tilde{v}_{i}(t)\|^{2} \tag{10}\] \[\eta= \theta\left(c\lambda_{2}\left(L(0)\right)-\left(2\rho+\frac{a_{ 1}+ca_{2}}{2}\right)\right),\]  
with \(0<\theta<1\), constants \(a_{1},a_{2}\) and \(c\) satisfy \(c>\left(2\rho+\left(a_{1}+ca_{2}\right)/2\right)/\lambda_{2}(L(0))\) and \(\rho\) is defined in (2). The condition \(g_{1}\left(\epsilon_{i}^{x}(t),\epsilon_{i}^{y}(t),\tilde{v}_{i}(t)\right)\geq 0\) is called the triggering condition. Without loss of generality, we assume \(t_{0}^{i}=0,\forall i\). **Remark 1**. : _It must be pointed out that the triggering condition (10) of each agent is associated with the average position and velocity of all agents. However, since_  
\[\sum_{j\in\mathcal{N}_{i}(t)}\phi\left(\big{\|}\hat{x}_{i}(t)- \hat{x}_{j}(t)\big{\|}\right)\mathbf{n}_{ij}(t)\] \[= \sum_{j\in\mathcal{N}_{i}(t)}\phi\left(\big{\|}\left(\hat{x}_{i} (t)-\bar{x}(t)\right)-\left(\hat{x}_{j}(t)-\bar{x}(t)\right)\big{\|}\right) \mathbf{n}_{ij}(t)\] \[= \sum_{j\in\mathcal{N}_{i}(t)}\phi\left(\big{\|}x_{i}(t)-x_{j}(t) \big{\|}\right)\mathbf{n}_{ij}(t)\]  
_and similarly_  
\[\sum_{j\in\mathcal{N}_{i}(t)}\left(\tilde{v}_{i}(t)-\tilde{v}_{j}(t)\right)= \sum_{j\in\mathcal{N}_{i}(t)}\left(v_{i}(t)-v_{j}(t)\right),\]  
_one can see that the average position and velocity are not needed for the detection of triggering condition (10). Therefore, the proposed ETC strategy is in a distributed manner._  
Define \(\hat{x}_{ij}=\hat{x}_{i}-\hat{x}_{j}\), the constant \(Q_{0}\) in (5) is given as  
\[Q_{0}=\frac{1}{2}\sum_{i=1}^{N}\left\{\sum_{j\in\mathcal{N}_{i}(0)}\psi\big{(} \big{\|}\hat{x}_{ij}(0)\big{\|}\big{)}+\tilde{v}_{i}^{T}(0)\tilde{v}_{i}(0) \right\}, \tag{11}\]  
then the following results are obtained. **Theorem 1**. : _Consider the MAS of N agents with dynamics expressed by (1). The controller for each agent is given by (6), which is updated by (9). Suppose that the network is initially connected and \(Q_{0}\) is bounded. Then, the following statements hold._  
_(i) \(\mathcal{G}(t)\) is connected and no collisions occur for \(\forall t\geq 0\);_  
_(ii)  Flocking motion is achieved asymptotically._  
**Proof. ** Define an energy-like Lyapunov function as  
\[-c\left(\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)(\hat{v}_{i}(t)- \hat{v}_{j}(t))+\epsilon_{i}^{x}(t)\right).\]0  
Denote the increasing topology switching time sequence as \(\hat{t}_{k^{\prime}},k^{\prime}\in\mathcal{Z}^{+}\), then \(\{\hat{t}_{k^{\prime}}\}\) are the set of discontinuous points of \(Q\). The increasing controller update time sequence of all the agents is denoted by \(\bar{t}_{k^{\prime\prime}}^{i}=\cup_{i=1}^{N}t_{k^{\prime}}^{i},k^{\prime \prime}\in\mathcal{Z}\). Let \(T_{k}=\hat{t}_{k^{\prime}}\cup\hat{t}_{k^{\prime\prime}},k\in\mathcal{Z}\), where \(T_{0}<T_{1}<\cdots\). Without loss of generality, we assume \(\hat{t}_{0}=0\), \(T_{0}=0\).