# Signal Source Localization of Multiple Robots Using an Event-Triggered Communication Scheme
## 2 Preliminaries
### Convergence Analysis
\[\sum_{i=0}^{n}\phi_{i}(t)^{2}\geq k_{1}\|\epsilon(t)\|_{2}^{2} =k_{1}\sum_{i=1}^{n}(\|\alpha y_{i}(t_{s}^{i})\|^{2}+\|\beta z_{i}(t_ {s}^{i})\|^{2})) \tag{24}\]  
Finally, by combining (22) with (24), the inequality (25) holds. \[\|M\|^{2}\|\alpha(e_{x}(t))+\beta(e_{v}(t))\|^{2}+\|D(\alpha e_{x0}(t)+\beta e _{v0}(t))\|^{2}\leq\frac{2\gamma^{2}}{k_{1}}\|\phi(t)\|^{2} \tag{25}\]  
**Lemma 2**. : _For a multi-robot system (19) with an event-triggered communication scheme (16), the following inequality is established._  
\[\|\alpha e^{x}(t)+\beta e^{y}(t)\|\leq\sqrt{\frac{8\gamma^{2}}{k_{1}-8\gamma^{2}} }\|\alpha y(t)+\beta z(t)\|\]  
_where \(k_{1}>8\gamma^{2}\) and is constant._  
Proof. : From the definitions of \(e^{x},e^{y}\) and \(M\), the following inequalities are derived. \[ \begin{split}\|\alpha e^{x}(t)+\beta e^{y}(t)\|& \leq\|M(\alpha e_{x}(t)+\beta e_{y}(t))\|+\|B((\alpha e_{x0}(t)+ \beta e_{y0}(t))\|\\ &\leq\|M\|\|(\alpha e_{x}(t)+\beta e_{y}(t))\|+\|B((\alpha e_{x0} (t)+\beta e_{y0}(t))\|\end{split}  \tag{26}\]  
Further, according to the definition of \(\|\phi(t)\|\) in (19), we can establish a new inequality. \[ \begin{split}\|\phi(t)\|^{2}&=\sum_{i=1}^{n}( \alpha y_{i}(t)+\beta z_{i}(t)+\alpha e_{i}^{x}(t)+\beta e_{i}^{y}(t))^{2}\\ &\leq\sum_{i=1}^{n}(2(\alpha y_{i}(t)+\beta z_{i}(t))^{2}+2( \alpha e_{i}^{x}(t)+\beta e_{i}^{y}(t))^{2})\\ &\leq 2\|\alpha y(t)+\beta z(t)\|^{2}+2\|\alpha e^{x}(t)+\beta e^{y}( t))\|^{2}\\ &\leq 2\|\alpha y(t)+\beta z(t)\|^{2}+4(\|M\|^{2}\|\alpha e_{x}(t)+ \beta e_{y}(t))\|^{2}\\ &+\|B(\alpha e_{x0}(t)+\beta e_{y0}(t))\|^{2})\end{split}  \tag{27}\]  
By Lemma 1 and the inequality (27), we obtain the inequality (28). \[ \begin{split}\|M\|^{2}\|\alpha(e_{x}(t))&+\beta(e_{ y}(t))\|^{2}+\|B(\alpha e_{x0}(t)+\beta e_{y0}(t))\|^{2}\leq\frac{2\gamma^{2}}{k_{1}} \|\phi(t)\|^{2}\\ &\leq\frac{4\gamma^{2}}{k_{1}}\|\alpha y(t)+\beta z(t)\|^{2}+ \frac{8\gamma^{2}}{k_{1}}(\|M\|^{2}\|ae_{x}(t)+\beta e_{y}(t))\|^{2}\\ &+\|B(\alpha e_{x0}(t)+\beta e_{y0}(t))\|^{2})\end{split}  \tag{28}\]  
Simplify the inequality (28) as:  
\[ \begin{split}\|M\|\|ae_{x}(t)+\beta e_{y}(t))\|&+\|B (\alpha e_{x0}(t)+\beta e_{y0}(t))\|\\ &\leq\sqrt{\frac{8\gamma^{2}}{k_{1}-8\gamma^{2}}}\|\alpha y(t)+ \beta z(t)\|\end{split}  \tag{29}\]  
Since \(k_{1}>8\gamma^{2}\), in terms of (26) and (29), the following inequality holds. \[\|\alpha e^{x}(t)+\beta e^{y}(t)\|\leq\sqrt{\frac{8\gamma^{2}}{k_{1}-8\gamma^ {2}}}\|\alpha y(t)+\beta z(t)\|\]  
Finally, we can give the following theorem for the multi-robot system (2) with the proposed communication scheme and controller. In addition, Zeno-behaviors denote that there is an infinite number of discrete transitions in a finite period of time in the multi-robot system. The following theorem can guarantee that the multi-robot system (2) with the proposed communication scheme and controller does not show Zeno-behaviors before consensus is achieved. **Theorem 1**. : _Consider the event-triggered communication scheme (16) and the cooperative controller (18) for a multi-robot system (2). Suppose that the undirected communication topology \(G_{n}(W,E,A)\) is connected with at least one \(a_{i0}\) not being zero. Let \(k=\frac{1}{2}(\sum_{j=1}^{n}a_{ij}-\sum_{j=1}^{n}a_{ji})+a_{i0}\). The variable \(u_{min}\) denotes the minimum eigenvalue of \(M+M^{T}\). The positive constant \(k_{1}\) can be found in Lemma 1. If the inequalities \(\beta>\sqrt{\alpha/u_{min}}\), \(\gamma<\sqrt{k_{1}/8}\), \(\delta<\frac{\beta^{2}u_{min}-\alpha}{2\beta^{2}}\) where \(\delta=\|M\|\sqrt{\frac{8\gamma^{2}}{k_{1}-8\gamma^{2}}}\) hold, the cooperative controller (18) with the event-triggered communication scheme (16) can guarantee that \(x_{i}(t)\to x_{0}(t)\) and \(v_{i}(t)\to v_{0}(t),\forall i\in 1,\cdots,n\). In addition, the multi-robot system does not show Zeno-behaviors before consensus is achieved._  
Proof. : We have three steps to prove the theorem. First, it is proven that the following function \(V(t)\) in (30) is a Lyapunov function.