## 3 Main Results
### Distributed Formation Protocol  
From the proposed dynamic event triggering mechanism, we then present a control law of each agent based on its neighborhood position and velocity with the formation description. The distributed formation control input between two consecutive triggering times of the \(i\)th agent is given by the following:  
\[u_{i}(t)= \gamma_{p}\sum_{j\in\mathcal{N}_{i}}\ell_{ij}\big{(}\eta_{p,i}(t _{k_{i}}^{i})-\eta_{p,j}(t_{k_{j}}^{i})\big{)}+\gamma_{v}\sum_{j\in\mathcal{ N}_{i}}\ell_{ij}\big{(}\eta_{v,i}(t_{k_{i}}^{i})-\eta_{v,j}(t_{k_{j}}^{i})\big{)}\] \[\qquad-\gamma_{p}\delta_{i}\big{(}\eta_{p,i}(t_{k_{i}}^{i})-r(t_{ k_{i}}^{i})\big{)}-\gamma_{v}\delta_{i}\big{(}\eta_{v,i}(t_{k_{i}}^{i})-v_{0} \big{)}+f_{a,i}(t_{k_{i}}^{i}), \tag{12}\]  
for all \(t\in\big{[}t_{k_{i}}^{i},t_{k_{i}+1}^{i}\big{)}\), where \(f_{a,i}(t)=\dot{f}_{v,i}(t)\), \(\gamma_{p},\gamma_{v}\in\mathbb{R}_{+}\), and \(t_{k_{j}}^{i}\) is the last triggering time of the \(j\)th agent. Additionally, when \(t>t_{k_{i}}^{i}\), \(t_{k_{i}+1}^{i}\) can be determined from the dynamic event triggering mechanism (6). Hence, each agent takes into account the last updated value of each of its neighbors in its control law. Further, the control law of the \(i\)th agent is updated both at its own event times \(t_{0}^{i},t_{1}^{i},\ldots,t_{k_{i}}^{i},\ldots\) and at the event times of its neighbors \(t_{0}^{i},t_{1}^{j},\ldots,t_{k_{j}}^{i},\ldots,j\in\mathcal{N}_{i}\). It should be noted that the definition of \(t_{k_{j}}^{j}\) implies that \(\eta_{p,j}(t_{k_{j}}^{j})=\eta_{p,j}(t)+\widetilde{\eta}_{p,j}(t)\) and \(\eta_{v,j}(t_{k_{j}}^{j})=\eta_{v,j}(t)+\widetilde{\eta}_{v,j}(t)\). Then, the closed loop error dynamics of the \(i\)th agent is given as follows:  
\[\dot{e}_{p,i}(t)= e_{v,j}(t),\] \[\dot{e}_{v,i}(t)= \gamma_{p}\sum_{j\in\mathcal{N}_{i}}\ell_{ij}\big{(}e_{p,i}(t)-e_{ p,j}(t)\big{)}\!+\!\gamma_{p}\sum_{j\in\mathcal{N}_{i}}\ell_{ij}\big{(}\widetilde{ \eta}_{p,i}(t)-\widetilde{\eta}_{p,j}(t)\big{)}\!+\!\gamma_{v}\sum_{j\in \mathcal{N}_{i}}\ell_{ij}\big{(}e_{v,i}(t)-e_{v,j}(t)\big{)}\] \[+\!\gamma_{v}\sum_{j\in\mathcal{N}_{i}}\ell_{ij}\big{(} \widetilde{\eta}_{v,j}(t)-\widetilde{\eta}_{v,j}(t)\big{)}\!-\!\gamma_{p}\delta _{i}\big{(}e_{p,i}(t)+\widetilde{\eta}_{p,j}(t)-\widetilde{\tau}_{i}(t)\big{)} \!-\!\gamma_{v}\delta_{i}\big{(}e_{v,i}(t)+\widetilde{\eta}_{v,i}(t)\big{)}\] \[+f_{a,i}(t_{k_{j}}^{i})-f_{a,i}(t),\;t\in\big{[}t_{k_{j}}^{i},\;t _{k_{i}+1}^{i}\big{)}, \tag{13}\]  
where \(e_{p,i}(t)=\eta_{p,i}(t)-r(t),e_{v,i}(t)=\eta_{v,i}(t)-v_{0}\), and \(\widetilde{r}_{i}(t)=r(t_{k_{i}}^{i})-r(t)\). From (13), the closed loop error dynamics of \(N\) agents follows:  
\[\dot{\mathbf{e}}_{p}(t)=\mathbf{e}_{v}(t),\] \[\dot{\mathbf{e}}_{v}(t)=-\gamma_{p}\mathcal{M}\mathbf{e}_{p}(t)-\gamma_{ v}\mathcal{M}\mathbf{e}_{v}(t)+\widetilde{f}_{a}(t)-\gamma_{p}\mathcal{M}\widetilde{ \mathbf{\eta}}_{p}(t)-\gamma_{v}\mathcal{M}\widetilde{\mathbf{\eta}}_{v}(t)+\gamma_{p} (\Delta\otimes I_{n})\widetilde{\mathbf{r}}(t), \tag{14}\]  
where:  
\[\mathbf{e}_{p}(t)=\big{[}e_{1,1}^{T}(t),e_{p,2}^{T}(t),\ldots,e_{p,N} ^{T}(t)\big{]}^{T},\;\mathbf{e}_{v}(t)=\big{[}e_{v,1}^{T}(t),e_{v,2}^{T}(t),\ldots,e_{v,N}^{T}(t)\big{]}^{T},\] \[\widetilde{f}_{a}(t)=\big{[}f_{a,1}^{T}(t_{k_{1}}^{1})-f_{a,1}^{T }(t),f_{a,2}^{T}(t_{k_{2}}^{2})-f_{a,2}^{T}(t),\ldots,f_{a,N}^{T}(t_{N}^{1})- f_{a,N}^{T}(t)\big{]}^{T},\] \[\qquad-\gamma_{p}\delta_{i}\big{(}\eta_{p,i}(t_{k_{i}}^{i})-r(t_{ k_{i}}^{i})\big{)}-\gamma_{v}\delta_{i}\big{(}\eta_{v,i}(t_{k_{i}}^{i})-v_{0} \big{)}+f_{a,i}(t_{k_{i}}^{i}), \tag{12}\]0 \[\qquad-\gamma_{p}\delta_{i}\big{(}\eta_{p,i}(t_{k_{i}}^{i})-r(t_{ k_{i}}^{i})\big{)}-\gamma_{v}\delta_{i}\big{(}\eta_{v,i}(t_{k_{i}}^{i})-v_{0} \big{)}+f_{a,i}(t_{k_{i}}^{i}), \tag{12}\]1  
The following theorem presents the control synthesis conditions formulated in terms of linear matrix inequalities. **Theorem 1**. : _Consider the multi-agent system (2).