# Signal Source Localization of Multiple Robots Using an Event-Triggered Communication Scheme
## 2 Preliminaries
### Convergence Analysis
Second, it is proven that the system (2) with the event-triggered communication scheme (16) and the cooperative controller (18) is asymptotically stable. Finally, it is proven that the multi-robot system does not show Zeno-behaviors before consensus is reached. We construct a Lyapunov functional as:  
\[V(t)=0.5\xi(t)^{T} \begin{pmatrix}\alpha\beta(M+M^{T})&\alpha I\\ \alpha I&\beta I\end{pmatrix} \xi(t) \tag{30}\]  
where \(\xi(t)=[y(t)^{T},z(t)^{T}]\) and \(I\) is a unit matrix of \(n\) order. Let:  
\[\Omega= \begin{pmatrix}\alpha\beta(M+M^{T})&\alpha I\\ \alpha I&\beta I\end{pmatrix} \]  
where \(M+M^{T}\) is a real symmetric matrix, and we can diagonalize it as \(\beta^{-1}\Lambda\beta\), where \(\Lambda=diag\left\{u_{1},u_{2},\cdots,u_{n}\right\}\) is a diagonal matrix and \(u_{i}\) is the eigenvalue of \(M+M^{T}\). Thus, \(\Omega\) can be written as:  
\[\Omega= \begin{pmatrix}\beta&0\\ 0&\beta\end{pmatrix} ^{-1}\bar{\Omega} \begin{pmatrix}\beta&0\\ 0&\beta\end{pmatrix} \]  
where:  
\[\Omega= \begin{pmatrix}\alpha\beta\Lambda&\alpha I\\ \alpha I&\beta I\end{pmatrix} \]  
Then, we solve its eigenvalue by:  
\[det(\lambda I_{2n}-\Omega)=det\left( \begin{array}{cc}\lambda I-\alpha\beta \Lambda&-\alpha I\\ -\alpha I&\lambda I-\beta I\end{array} \right)\]  
The eigenvalues of \(\bar{\Omega}\) are:  
\[\lambda_{i\pm}=\frac{\beta+\alpha\beta u_{i}\pm\sqrt{(\beta+\alpha\beta u_{i} )^{2}-4(\alpha\beta^{2}u_{i}-\alpha^{2})}}{2}\]  
where \(\lambda_{i+}\) and \(\lambda_{i-}\) are the eigenvalues of \(\bar{\Omega}\), which are associated with \(u_{i}\). Thus, if the condition \(\beta>\sqrt{\alpha/u_{min}}\) is satisfied, the matrix \(\Omega\) is a positive definite matrix, that is the Lyapunov function \(V(t)\geqslant 0\). The derivative of \(\dot{V}(t)\) is as:\[\dot{V}(t) =y(t)^{T}\alpha\beta(M+M^{T})z(t)+z(t)^{T}\alpha Iz(t)\] \[+y(t)^{T}\alpha Iz(t)+z(t)^{T}\beta Iz(t)\] \[=-z(t)^{T}(\beta^{2}M-\alpha I)z(t)-y(t)^{T}\alpha^{2}My(t)\] \[-(\alpha y(t)^{T}+\beta z(t)^{T})M(\alpha e^{z}(t)+\beta e^{v}(t))\]  
We can get the following inequality as:  
\[\dot{V}(t) \leq-z(t)^{T}(\beta^{2}M-\alpha I)z(t)-y(t)^{T}\alpha^{2}My(t) \tag{31}\] \[+\|\alpha y(t)^{T}+\beta z(t)^{T}\|\|M\|\|\omega e^{z}(t)+\beta e^ {v}(t)\|\]  
From Lemma 1 and (30), we can give the following result. \[\dot{V}(t) \leq-z(t)^{T}(\beta^{2}M-\alpha I)z(t)-y(t)^{T}\alpha^{2}My(t)+ \delta\|\alpha y(t)+\beta z(t)\|^{2}\] \[\leq-z(t)^{T}(\beta^{2}M-\alpha I-2\delta^{2}I)z(t)-y(t)^{T}( \alpha^{2}M-2\delta\alpha^{2}I)y(t)\]  
where \(\delta=\|M\|\sqrt{\frac{8\gamma^{2}}{k_{1}-8\gamma^{2}}}\). Since \(\beta>\sqrt{\alpha/u_{min}}\), \(\gamma<\sqrt{k_{1}/8}\) and \(\delta<\frac{\beta^{2}u_{min}-\alpha}{2\beta^{2}}\), the inequality \(\dot{V}(t)\leq 0\) holds. It shows that the system \((y(t),z(t))\) will asymptotically converge to \((0_{n},0_{n})\). It is assumed that the velocity and acceleration of the robot are bounded by \(s_{v}\) and \(s_{a}\). The variable \(e_{xi}(t_{s}^{i})\) is zero, and \(x_{i}(t_{s}^{i})\) is constant for \(t\in[t_{s}^{i},t_{s+1}^{i})\). Then, the following inequality is established. \[|e_{xi}(t)|\leq|\int_{t_{s}^{i}}^{t}e_{xi}(t)d\tau|\leq\int_{t_{s}^{i}}^{t}|e_ {xi}(t)|d\tau=\int_{t_{s}^{i}}^{t}|\dot{x}_{i}(t)|d\tau\leq s_{v}(t-t_{s}^{i}),t\in[t_{s}^{i},t_{(s+1)}^{i})\]  
In the same way, the following inequality is established. \[\dot{V}(t) =y(t)^{T}\alpha\beta(M+M^{T})z(t)+z(t)^{T}\alpha Iz(t)\]0  
Moreover, we have \(\|M\|*|\alpha(e_{xi}(t))+\beta(e_{vi}(t))|+a_{i0}|\alpha(x_{0}(t_{s}^{i})-x_{0 }(t))+\beta(v_{0}(t_{s}^{i})-v_{0}(t))|\leq((\|M\|+1)\alpha s_{v}+(\|M\|+1) \beta s_{a})(t-t_{s}^{i})\). According to the event-triggered communication scheme, we obtain \(\|M\|*|\alpha(e_{xi}(t))+\beta(e_{vi}(t))|+a_{i0}|\alpha(x_{0}(t_{s}^{i})-x_{0 }(t))+\beta(v_{0}(t_{s}^{i})-v_{0}(t))|-\gamma(|\alpha y_{i}(t_{s}^{i})|+| \beta z_{i}(t_{s}^{i})|)>0\) at \(t=t_{s+1}^{i}\). Hence, we derive \((t_{s+1}^{i}-t_{s}^{i})>\frac{\gamma(|\alpha y_{i}(t_{s}^{i})|+|\beta z_{i}(t_ {s}^{i})|)}{(\|M\|+1)\alpha s_{v}+(\|M\|+1)\beta s_{a}}>0\).