## 4 Simulations  
This section presents an example to validate the effectiveness of our proposed control protocol. Consider the MASs with one leader agent and seven following agents in this example, wherein the dynamics of the leader and the followers are represented by (3) and (4), respectively. Figure 3 shows the information flow among all agents, with which Assumption 1 is satisfied. From Figure 3, the Laplace matrix \(L\), the adjacency matrix \(\mathcal{A}\), and matrix \(B\) are given as  
\[L= \begin{bmatrix}0&0&0&0&0&0&0\\ -1&1&0&0&0&0&0\\ 0&-1&1&0&0&0&0\\ 0&0&-1&1&0&0&0\\ 0&0&0&-1&1&0&0\\ -1&0&0&0&-1&2&0\\ 0&0&0&0&0&-1&1\end{bmatrix} \]  
\[\mathcal{A}= \begin{bmatrix}0&0&0&0&0&0&0\\ 1&0&0&0&0&0&0\\ 0&1&0&0&0&0&0\\ 0&0&1&0&0&0&0\\ 0&0&0&1&0&0&0\\ 1&0&0&0&1&0&0\\ 0&0&0&0&0&1&0\end{bmatrix} \]  
\(B=diag\{1,0,0,0,0,0,1\}\). Let \(p_{i}=[x_{i},y_{i}]^{T}\), \(v_{i}=[v_{x_{i}},v_{y_{i}}]^{T},i\in\{1,\cdots,7\}\), be the position and the velocity of each follower \(i\), respectively. The initial values of the followers are set as \(p_{1}(0)=[0.2,-1]^{T}\), \(p_{2}(0)=[-1,1]^{T}\), \(p_{3}(0)=[1,0]^{T}\), \(p_{4}(0)=[1,1]^{T}\), \(p_{5}(0)=[-1,-0.5]^{T}\), \(p_{6}(0)=[1,-1]^{T}\), \(p_{7}(0)=[-1,1]^{T}\), \(v_{1}(0)=[1,1]^{T}\), \(v_{2}(0)=[-0.5,0.5]^{T}\), \(v_{3}(0)=[1,-1]^{T}\), \(v_{4}(0)=[1.5,1]^{T}\), \(v_{5}(0)=[0,0]^{T}\), \(v_{6}(0)=[-1,-1]^{T}\), \(v_{7}(0)=[0.5,1]^{T}\). The vector of disturbance for followers on x-dimension and y-dimension are both set as \(d(t)=0.1\times[cos(0.6t),sin(0.6t),sin(0.6t),sin(0.6t),sin(0.6t),sin(0.6t),sin(0. 6t)]^{T}\). Figure 3: Topology in the simulation. Figure 2: Consensus control block diagram for each follower \(i\). Let \(p_{0}=[x_{0},y_{0}]^{T}\), \(v_{0}=[\sin(t),\sin(t)]^{T}\) be the position and the velocity of the leader. The values of the leader's initial position and initial velocity are set as \(p_{0}(0)=[0,0]^{T}\) and \(v_{0}(0)=[0,0]^{T}\), respectively. The designed parameters in (12) are selected as \(t_{f}=6s\), \(\varsigma_{i}(0)=3.2\), \(\varsigma_{i}(t_{f})=0.0095\) (\(i=1,\cdots,7\)). Finally, by using Theorem 1, the control parameters are selected as \(k_{1}=64\), \(k_{2}=150\). In order to prove that the controller proposed in this paper achieves the appointed-time leader-follower consensus for the MASs with the prescribed performance, the simulation results will be compared with the existing controller without PPC [6], whose controller is given as follows:  
\[u_{i}(t)=-k_{1}(\sum\limits_{j=1}^{N}a_{ij}(x_{i}-x_{j})-b_{i}(x_{i}-x_{0}))-k_ {2}(\sum\limits_{j=1}^{N}a_{ij}(v_{i}-v_{j})-b_{i}(v_{i}-v_{0})) \tag{30}\]  
where \(k_{1}=64\) and \(k_{2}=150\), and [PERSON]'s controller in [35], whose controller is given as follows:  
\[u_{i}(t)=(l_{ii}+b_{i})^{-1}\Bigg{(}\sum\limits_{j\in\mathbb{N}_{i}}-l_{ij}u_ {j}+f_{i}\Bigg{)} \tag{31}\]  
where \(f_{i}=-(\frac{k_{1}}{l_{f}}-\frac{\bar{p}_{fi}}{p_{fi}})(\frac{1}{2}(e_{iv}+i \varepsilon_{ix})+\frac{\partial e_{i}(\varpi_{i})}{\partial\varpi_{i}}\frac{ 1}{p_{fi}(t)}\varepsilon_{i})-i\varepsilon_{iv}-\frac{1}{2p_{fi}}(e_{iv}+i \varepsilon_{ix})+b_{i}\bar{x}_{0}\), \(\varpi_{i}=\frac{e_{ix}}{p_{fi}},\varepsilon_{i}=\ln(\frac{1+\varpi_{i}}{1- \varpi_{i}})\) and \(p_{fi}\) is the performance function designed in [35]. The designed parameters in (31) are selected as \(k_{1}=1\), \(t_{f}=6s\), \(i_{i}=100\), \(p_{fi}(0)=3.2\), \(p_{fi}(\infty)=0.0095\), \(\alpha=0.95\) and the \(\alpha\) was defined in [35]. Simulations are carried out based on the MATLAB R2020b platform. The simulation results are illustrated in Figures 4-10. In Figure 4, we can see the comparison with respect to the transient performance of MASs using controller (14) and controller (30). It is worth noting that controller (30) can only deal with the MASs without disturbance, and controller (14) deals with the MASs whose disturbance is \(d\) set above.