## 3 Main Results  
This section suggests an event-triggered containment control procedure based on relative information perception for the realistic consensus with a predetermined leader. The follower (ii) control protocol is created as  
\[u_{i}(t)=-(\varphi(t)+1)K\bar{x}_{i}(t_{k}^{i}) \tag{12}\]  
where \(K\) is an adequate dimensional constant matrix. So,  
\[u(t)=-(\varphi(t)+1)(I_{M}\otimes K)(e(t)+\bar{x}(t)) \tag{13}\]  
where \(u(t)=\text{col}\{u_{1}(t),\ldots,u_{M}(t)\},e(t)=\text{col}\{e_{1}(t),\ldots,e_ {M}(t)\},\text{and}\ \bar{x}(t)=\text{col}\{\bar{x}_{1}(t),\ldots,\)\(\bar{x}_{M}(t)\}\). The control protocol can address the system containment control issue, given the following conditions. Theorem 1: _If there is a control protocol \(u_{i}\), such that all follower states asymptotically converge to the convex hull created by the leader states, then system (3) solves the containment control issue for any beginning conditions \(x_{i}(0)\), \(i\in F\), i.e.,_  
\[\lim_{t\to\infty}(x_{f}(t)-\omega(t))=0 \tag{14}\]  
_where \(\omega(t)=(-L_{1}^{-1}L_{2}\otimes e^{At}) \begin{bmatrix}x_{M+1}(0)\\ \vdots\\ x_{N}(0)\end{bmatrix} \)._  
By substituting Equation (4) into Equation (3), we obtain:  
\[ \begin{cases}\dot{x}_{i}(t)=Ax_{i}(t)+Bu_{i}(t),&i\in F\\ \dot{x}_{i}(t)=Ax_{i}(t),&i\in E\end{cases}  \tag{15}\]  
where \(x_{f}= \begin{bmatrix}x_{1}^{T}&\ldots&x_{M}^{T}\end{bmatrix} ^{T}\), \(x_{l}= \begin{bmatrix}x_{M+1}^{T}&\ldots&x_{N}^{T}\end{bmatrix} ^{T}\). By substituting Equation (13) into Equation (15), it can be stated succinctly as follows:  
\[ \begin{cases}\dot{x}_{f}=(I_{M}\otimes A)x_{f}-(\varphi(t)+1)(I_{M}\otimes BK )(e(t)+\bar{x}(t)),\\ \dot{x}_{l}=(I_{M}\otimes A)x_{l},\end{cases}  \tag{16}\]  
where \(l\in E,f\in F\). In accordance with the meaning of \(\bar{x}_{i}(t)\), we have  
\[\bar{x}=(L_{1}\otimes I_{M})\dot{x}_{f}+(L_{2}\otimes I_{M})\dot{x}_{l} \tag{17}\]  
where \(\bar{x}= \begin{bmatrix}\bar{x}_{1}^{T},\ldots,\bar{x}_{M}^{T}\end{bmatrix} ^{T}\). Taking the derivative of Equation (16), we have  
\[ \begin{split}\dot{\bar{x}}&=(L_{1}\otimes I_{M})\dot{x}_{f}+(L_ {2}\otimes I_{M})\dot{x}_{l}\\ &=(I_{M}\otimes A)\bar{x}-(\varphi(t)+1)(L_{1}\otimes BK)(e+\bar{x})\end{split}  \tag{18}\]  
where \(e= \begin{bmatrix}\bar{e}_{1}^{T},\ldots,\bar{e}_{M}^{T}\end{bmatrix} ^{T}\)  
Theorem 2: _To achieve an actual fixed-time leader following consensus under Assumptions 1 and 2, the requirements that follow have to be satisfied:_  
1. _Let us find a positive definite matrix_ \(P\in\mathbb{R}^{n_{x}\times n_{x}}\) _that satisfies:_  
\[A^{T}P+PA-\lambda_{min}(L1)(1-\theta)PBB^{T}P\leq-Q \tag{19}\]\(K=B^{T}P\). 3. \(\varphi(t)=\frac{\lambda_{min}(P)\xi(t)}{\lambda_{min}(L1)(1-\xi(t)+\delta)q},\quad q =\frac{\lambda_{min}(Q)}{\lambda_{min}(L1)(1-\theta)}\). _where the constant \(0<\theta<1\) and \(Q>0\) is a matrix with a positive definite, respectively._  
**Proof of Theorems 1 and 2. ** The candidate function for the Lyapunov described below is  
\[V(t)=\frac{1}{2}\bar{x}^{T}(t)P\bar{x}(t) \tag{20}\]  
The \(V(t)\) derivative in Equation (18), which occurs when \(0\leq t\leq t_{f}\), is given as  
\[\dot{V}(t) =\bar{x}^{T}(t)(I_{M}\otimes P)\hat{\mathfrak{x}}(t) \tag{21}\] \[=\frac{1}{2}\bar{x}^{T}(t)(I_{M}\otimes(A^{T}P+PA))\bar{x}(t)\] \[\quad-(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P)e(t)\] \[\quad-(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P)\bar{x} (t)\]  
In addition,  
\[ \begin{split}-(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T}P )e(t)\\ \leq\frac{1}{2}(\varphi(t)+1)\bar{x}^{T}(t)(L_{1}\otimes PBB^{T} P)\bar{x}(t)\\ +\frac{1}{2}(\varphi(t)+1)e^{T}(t)(L_{1}\otimes PBB^{T}P)e(t) \end{split}