## 3 Pull Based Event-Triggered Strategy  
A sampled-date based way-point control law was designed in [26], shown as:  
\[u_{i}(t)=\frac{d_{i^{-}}}{d_{i}+d_{i^{-}}}y_{i}(t)-\frac{d_{i}}{d_{i}+d_{i^{-}}} y_{i^{-}}(t), \tag{7}\]  
where \(i=0,1,2,...,N\). It has been approved that the continuous updating control protocol (7) can drive all agents to their equilibrium point \(x^{*}\), but typically costs unnecessary transmission energy and communication bandwidth. To address this problem, we propose a pull-based event-triggered strategy. Particularly, the controllers of agents only update at discrete event instants. Meanwhile, continuous communication between neighboring agents maintain. Let an increasing sequence \((t_{0}^{i},t_{1}^{i},...,t_{k}^{i},...)\) denotes the event instants of agent \(i\), such that \(y_{i}(t_{k}^{i})\) is the state of of agent \(i\) at the \(k\)-th event instants. It is noteworthy that since all agents are asynchronously triggered, each agent has its sequence of events. Based on the event triggered control, a distributed circle formation control law for agent \(i\) is designed as:  
\[u_{i}(t)=\frac{d_{i^{-}}}{d_{i^{-}}+d_{i}}y_{i}(t_{k}^{i})-\frac{d_{i}}{d_{i^{ -}}+d_{i}}y_{i^{-}}(t_{k}^{i^{-}}),t\in[t_{k}^{i},t_{k+1}^{i}), \tag{8}\]  
Control law (8) suggests that the controller of agent \(i\) updates at its own event sequence \((t_{0}^{i},t_{1}^{i},...,t_{k}^{i},...)\) and transmits this renewed state \(y_{i}(t_{k}^{i})\) to all its out-neighbours at the same time. For simplicity, assume \(\hat{y}_{i}(t)=y_{i}(t_{k}^{i})\), \(\hat{\delta}_{i}(t)=\hat{y}_{i}(t)/d_{i}\), consequently, control law (8) can be rewritten as:  
\[u_{i}(t)=\frac{d_{i}d_{i^{-}}}{d_{i^{-}}+d_{i}}(\hat{\delta}_{i}(t)-\hat{ \delta}_{i^{-}}(t)),t\in[t_{k}^{i},t_{k+1}^{i}). \tag{9}\]  
Substituting (5) and (9) into (4), a closed-loop form of agent \(i\) can be rewritten by \(\delta_{i}\) as:  
\[\dot{\delta}_{i}(t)=\sum_{j\in N_{i}^{ni}}\frac{d_{j}}{d_{i}+d_{j}}\left(\hat {\delta}_{j}(t)-\hat{\delta}_{i}(t)\right),t\geq 0. \tag{10}\]  
Define \(e_{i}(t)=\hat{\delta}_{i}(t)-\delta_{i}(t)\), then a compact form of the system is derived as:  
\[\dot{\delta}(t)=-L_{d}^{\rm T}(\delta(t)+e(t)),t\in[t_{k}^{i},t_{k+1}^{i}) \tag{11}\]  
where \(\delta(t)=[\delta_{1}(t),\delta_{2}(t),...,\delta_{N}(t)]\in\mathbb{R}^{N}\), and \(e(t)=[e_{1}(t),e_{2}(t),...,e_{N}(t)]\in\mathbb{R}^{N}\), and  
\[L_{d}=\left[ \begin{array}{cccc}\frac{d_{2}}{d_{2}+d_{1}}+\frac{d_{N}}{d_{N} +d_{1}}&-\frac{d_{1}}{d_{2}+d_{1}}&\ldots&0&-\frac{d_{1}}{d_{N}+d_{1}}\\ -\frac{d_{2}}{d_{2}+d_{1}}&\frac{d_{3}}{d_{3}+d_{2}}+\frac{d_{1}}{d_{2}+d_{1} }&\ldots&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&\ldots&\frac{d_{N}}{d_{N}+d_{N-1}}+\frac{d_{N-2}}{d_{N-1}+d_{N-2}}&-\frac{ d_{N-1}}{d_{N}+d_{N-1}}\\ -\frac{d_{N}}{d_{N}+d_{1}}&0&\ldots&-\frac{d_{N}}{d_{N}+d_{N-1}}&\frac{d_{1}}{d_{ N}+d_{1}}+\frac{d_{N-1}}{d_{N}+d_{N-1}}\end{array} \right]. \tag{12}\]  
Define:  
\[\Lambda=\frac{1}{2}(L_{d}\Theta+\Theta L_{d}^{T}), \tag{13}\]where according to Lemma 2, \(\Lambda\) is semi-positive definite. The eigenvalues of \(\Lambda\) can be arranged in an ascending order as \(0=\gamma_{1}\leq\gamma_{2}\leq...\leq\gamma_{N}\). Let  
\[\Psi=\Theta-\xi\xi^{T}. \tag{14}\]  
From (14), we can obtain that \(\Psi\) has a zero eigenvalue and its eigenvalues increasingly ordered as \(0=\
u_{1}\leq\
u_{2}\leq...\leq\
u_{N}\). We define the eigenvalues of \(L_{d}L_{d}^{T}\) by \(0=\omega_{1}\leq\omega_{2}\leq...\leq\omega_{N}=\rho(L_{d}L_{d}^{T})\).