# Signal Source Localization of Multiple Robots Using an Event-Triggered Communication Scheme
## 2 Preliminaries
### Convergence Analysis  
In order to illustrate the position and velocity consensus for the multi-robot system (2) under the controller (18) with the event-triggered communication scheme (16), we first transform the model (2) in the following. Let \(\bar{x}_{i}(t)=x_{i}(t)-x_{0}(t)\) and \(\bar{v}_{i}(t)=v_{i}(t)-v_{0}(t)\). Then, the system (2) with the controller (18) can be rewritten as:  
\[\left\{ \begin{array}{l}\dot{x}_{i}(t)=\bar{v}_{i}(t)\\ \dot{v}_{i}(t)=\sum\limits_{j=0}^{n}a_{ij}\alpha(\bar{x}_{j}(t)-\bar{x}_{i}(t) )\\ +\sum\limits_{j=0}^{n}a_{ij}\alpha(e_{xj}(t)-e_{xi}(t))\\ +\sum\limits_{j=0}^{n}a_{ij}\beta(\bar{v}_{j}(t)-\bar{v}_{i}(t))\\ +\sum\limits_{j=0}^{n}a_{ij}\beta(e_{vj}(t)-e_{vi}(t))\end{array} \right.\]  
Furthermore, set:  
\[ \begin{array}{l}y_{i}(t)=\sum\limits_{j=0}^{n}a_{ij}((\bar{x}_{j}(t)-\bar{x }_{i}(t))\\ z_{i}(t)=\sum\limits_{j=0}^{n}a_{ij}((\bar{v}_{j}(t)-\bar{v}_{i}(t))\\ e_{i}^{x}(t)=\sum\limits_{j=0}^{n}a_{ij}(e_{xj}(t)-e_{xi}(t))\\ e_{i}^{v}(t)=\sum\limits_{j=0}^{n}a_{ij}(e_{vj}(t)-e_{vi}(t))\end{array} \]  
Hence, the dynamics of a multi-robot system can be deduced as:  
\[\left\{ \begin{array}{l}\dot{y}(t)=z(t)\\ \dot{z}(t)=-M\phi(t)\\ \phi(t)=\alpha y(t)+\beta z(t)+\alpha e^{x}(t)+\beta e^{v}(t)\end{array} \right. \tag{19}\]where \(y(t)=[y_{1}(t),y_{2}(t),\cdots y_{n}(t)]^{T}\) and \(z(t)\), \(e^{x}(t)\), \(e^{v}(t)\) are similar. The following lemmas are given in order to illustrate the convergence proof. **Lemma 1**. : _For a multi-robot system (19) with an event-triggered communication scheme (16), the following inequality is established._  
\[\|M\|^{2}\|\alpha(e_{x}(t))+\beta(e_{v}(t))\|^{2}+\|B(\alpha e_{x0}(t)+\beta e _{v0}(t))\|^{2}\leq\frac{2\gamma^{2}}{k_{1}}\|\phi(t)\|^{2} \tag{20}\]  
_where \(B=diag\{a_{10},\cdot sa_{n0}\}\) and \(k_{1}\) is a positive constant._  
Proof. : The event-triggered communication scheme (16) is listed as:  
\[\|M\|\|\alpha e_{xi}(t)+\beta e_{vi}(t)\|+a_{i0}\|\alpha e_{x0}(t)+\beta e_{v0} (t)\|\leq\gamma(\|\alpha y_{i}(t_{s}^{i})\|+\|\beta z_{i}(t_{s}^{i})\|)\quad for \quad t\in[t_{s}^{i},t_{s+1}^{i}) \tag{21}\]  
According to the inequalities \(a_{1}^{2}+b_{1}^{2}\leq(a_{1}+b_{1})^{2},a_{1}>0,b_{1}>0\) and \(2a_{1}^{2}+2b_{1}^{2}\geq(a_{1}+b_{1})^{2}\), the inequality (21) can be further changed as:  
\[\|M\|^{2}\|\alpha(e_{xi}(t))+\beta(e_{vi}(t))\|^{2}+a_{i0}\|\alpha e_{x0}(t)+ \beta e_{v0}(t))\|^{2}\leq 2\gamma^{2}(\|\alpha y_{i}(t_{s}^{i})\|^{2}+\| \beta z_{i}(t_{s}^{i})\|^{2}) \tag{22}\]  
Notice the definition of \(\phi_{i}(t)\). The variable \(\phi_{i}(t)^{2}\) is rewritten using a matrix-vector form. \[\phi_{i}(t)^{2}=[\alpha y_{i}(t_{s}^{i})\quad\beta z_{i}(t_{s}^{i})]Q[\alpha y _{i}(t_{s}^{i})\quad\beta z_{i}(t_{s}^{i})]^{T}\]  
where \(Q= \begin{bmatrix}1&1\\ 1&1\end{bmatrix} \) which is a semi-positive definite matrix. We consider the sum of \(\phi_{i}(t)^{2}\), \(i=1,\ldots,n\). \[ \begin{split}\sum_{i=0}^{n}\phi_{i}(t)^{2}&=\sum_{i=0}^{n}[ \alpha y_{i}(t_{s}^{i})\quad\beta z_{i}(t_{s}^{i})]Q[\alpha y_{i}(t_{s}^{i}) \quad\beta z_{i}(t_{s}^{i})]^{T}\\ &=\epsilon(t)^{T}I_{n}\otimes Q\epsilon(t)\end{split}  \tag{23}\]  
where \(\epsilon(t)=[[\alpha y_{1}(t_{s}^{1})\quad\beta z_{1}(t_{s}^{1})],\cdots,[ \alpha y_{n}(t_{s}^{n})\quad\beta z_{n}(t_{s}^{n})]]\). For the set \(U=\{\sigma\in\mathbb{R}^{2n}:\sigma^{T}\sigma=1\}\), which is bounded and closed, one can know \(\frac{\epsilon(t)}{\|\epsilon(t)\|_{2}}\in U\), and there exists a positive constant \(k_{1}>0\) for \((\frac{\epsilon(t)}{\|\epsilon(t)\|_{2}})^{T}I_{n}\otimes Q\frac{\epsilon(t)} {\|\epsilon(t)\|_{2}}\). \[k_{1}=\min_{\frac{\epsilon(t)}{\|\epsilon(t)\|_{2}}\in U}\left(\frac{\epsilon( t)}{\|\epsilon(t)\|_{2}}\right)^{T}I_{n}\otimes Q\frac{\epsilon(t)}{\| \epsilon(t)\|_{2}}\]  
Then, in terms of Equation (23) and the minimum value \(k_{1}\), the following inequality is established.