## 2 Preliminary Knowledge and a Description of the Issue
### Problem Description  
In this study, we address the case of several leaders within an \(N\)-agent linear multi-agent system, where \(M(M<N)\) agents are followers and the leaders constitute the remaining \(N-M\) agents. We specify the set of leaders as \(E=\{M+1,\ldots,N\}\) and the set of followers as \(F=\{1,\ldots,M\}\). The following is a representation of the dynamics of the follower \(i\):  
\[\dot{x}_{i}(t)=Ax_{i}(t)+Bu_{i}(t),\quad i\in F \tag{3}\]where the status and input controls for agent \(i\) are represented by \(x_{i}(t)\) and \(u_{i}(t)\), respectively, in \(\in\mathbb{R}^{n_{x}}\) and \(\in\mathbb{R}^{n_{u}}\). Matrices \(A\in\mathbb{R}^{n_{x}\times n_{x}}\) and \(B\in\mathbb{R}^{n_{x}\times n_{x}}\) are system matrices. **Assumption 2**. : \((A,B)\) _is a stable matrix._  
For each leader, assuming their control input is \(u_{i}(t)=0\), the dynamics equation for the leaders can be stated as follows:  
\[\dot{x}_{i}(t)=Ax_{i}(t),\quad i\in E \tag{4}\]  
In the given equation, \(x_{i}(t)\in\mathbb{R}^{n_{x}}\) is an indication of the leaders' condition. Since the leaders do not have neighbors, the \(\mathcal{G}\) Laplacian matrix of the graph can be broken down into:  
\[L= \begin{bmatrix}L_{1}&L_{2}\\ 0_{(N-M)\times M}&0_{(N-M)\times(N-M)}\end{bmatrix}  \tag{5}\]  
where \(L_{1}\) is expressed in \(\mathbb{R}^{M\times M}\) and \(L_{2}\) is expressed in \(\mathbb{R}^{M\times(N-M)}\). Furthermore, matrix \(L_{1}\) represents the information flow between the \(M\) followers, and the information flow from the \(N-M\) leaders to the \(M\) followers is represented by matrix \(L_{2}\). **Assumption 3**. : _To ensure how connected graph \(\mathcal{G}\) is, each follower must have a guided path to at least one leader, who in turn must have at least one follower. In other words, at least one leader agent is globally reachable._  
**Lemma 2** ([33]). : _According to Assumption 1, each member of \(-L_{1}^{-1}L_{2}\) is non-negative, and positive real components can be found in each \(L_{1}\)'s eigenvalue. In addition, the matrix \(-L_{1}^{-1}L_{2}\) has row sums of \(1\)._  
**Definition 2** ([32]). : _Let the actual vector space \(\mathbb{R}^{n}\) be represented by \(X=\{x_{1},x_{2},\ldots,x_{n}\}\). The convexity of the hull of \(X\) is represented by \(C_{O}(X)\), and defined as:_  
\[C_{O}(X)=\left\{\sum_{i=1}^{n}\lambda_{i}x_{i}\,\Big{|}\,x_{i}\in X,\lambda_{i }>0,\sum_{i=1}^{n}\lambda_{i}=1\right\} \tag{6}\]  
**Definition 3**. : _Multi-agent systems (3) and (4) achieve bilateral leader-follower consensus if agent \(i\)\((i=1,\ldots,M)\) satisfies the following conditions:_  
\[\left\{ \begin{aligned} \lim_{t\to t_{f}}\left\|\sum_{j=1}^{N}a_{ij}(x _{i}(t)-x_{j}(t))\right\|&\leq c,\\ \left\|\sum_{j=1}^{N}a_{ij}(x_{i}(t)-x_{j}(t))\right\|& \leq c,\quad t>t_{f}\\ \lim_{t\to\infty}\left\|\sum_{j=1}^{N}a_{ij}(x_{i}(t)-x_{j}(t)) \right\|&=0.\end{aligned} \right. \tag{7}\]  
_where \(c\) is a known constant._  
The control protocol is designed using a completely distributed event-triggered technique to reduce the amount of system resources used. According to the definition:  
\[\bar{x}_{i}(t)=\sum_{j=1}^{N}a_{ij}(x_{i}(t)-x_{j}(t)),\quad i\in F \tag{8}\]  
\[e_{i}(t)=\bar{x}_{i}(t_{k}^{i})-\bar{x}_{i}(t) \tag{9}\]where the event-triggered condition determines the \(k\)th sample instant for follower \(i\). The following describes the event-triggering conditions for follower \(i\)  
\[f_{i}=\|Ke_{i}(t)\|^{2}-\frac{\theta}{\varphi(t)+1}\|K\bar{x}_{i}(t)\|^{2} \tag{10}\]  
\(\theta\) is a constructive constant, and a time-varying function is \(\varphi(t)\). Equation (10) determines the sample period,  
\[t_{k+1}^{i}=\inf_{t>t_{k}^{i}}\{t\,|\,f_{i}(e_{i}(t),\bar{x}_{i}(t),t)=0\} \tag{11}\]  
where \(i=1,\ldots,M\), \(k\in\mathbb{N}\), and \(t_{0}^{i}=0\). In order to achieve containment control, fixed time, and reduce communication burden, the controller is designed by using a time reference generator and event trigger.