## 4 Simulations
We can see that the controller (30) cannot guarantee the prescribed transient performance, i.e., overshoot and convergence speed, and it is also difficult to determine when the system can reach the tracking accuracy. Looking back at Figure 4, using controller (14), seven followers' cumulative position difference evolves within the performance function boundary, and the error converges to an arbitrarily small expected region within the appointed time, proving that the proposed controller can ensure that each follower satisfies the prescribed performance constraints. Define the average control effort as \(\frac{1}{N}\int\limits_{t=0}^{t_{f}}\sum\limits_{i=1}^{N}|u_{i}(x)|\,dx\). It should be noted that Table 2 and Figure 5 are obtained with the disturbance equaling \(2d\) rather than \(d\). Table 2 shows the maximum amplitude of the control input and control effort of followers using both controller (14) and controller (31). We can see that the proposed controller (14) reduces the maximum control input and control effort, while the appointed-time and tracking accuracy are the same as (31). Figure 5 shows the seven followers' cumulative position difference trajectories using both controller (14) and (31). Although using controller (31) can achieve faster convergence rate, the control input also increases accordingly. Furthermore, the controller (31) of follower \(i\) needs the control input \(u_{j}\) of its neighbor \(j\), which needs more information, and thus it is more difficult to implement in practice. Figure 6 shows the seven followers' cumulative velocity difference converges to a bounded region. The trajectories of the agents in the x-y dimension are pictured in Figure 7, and it can be seen from the figure that all followers can follow the leader's movements within 6 s. As demonstrated by Figure 8, the velocity of each follower is bounded, and before the appointed time, the follower's velocity converges to the leader's velocity, which is also consistent with the convergence of the followers' position, as shown in Figure 7. Figure 9 further shows that the position tracking error converges to an arbitrarily small expected region within the appointed time, which verifies that the system achieves the practically leader-following appointed-time consensus. Figure 4: The evolution of seven followers’ cumulative position difference on x-dimension and y-dimension with performance bounds (dashed lines) with \(t_{f}=6\) s. (**a**) Case of the consensus protocol with PPC and \(d(t)\
eq 0\). (**b**) Case of the consensus protocol without PPC and \(d(t)\equiv 0\). Figure 5: The evolution of seven followers’ cumulative position difference on x-dimension with \(t_{f}=6\) s. (**a**) Case of the proposed protocol (14). (**b**) Case of the protocol (31). \begin{table}
\begin{tabular}{c c c} \hline
**Controller** & **Maximum Control Input** & **Control Effort** \\ \hline controller (14) with \(k_{1}=64\), \(k_{2}=150\) & 318.4 & 720.8 \\ controller (31) & 345.4 & 770.1 \\ \hline \end{tabular}
\end{table} 
Table 2: Maximum control input and control effort for different controllers in x-dimension. Figure 6: The evolution of seven followers’ cumulative velocity difference on x-dimension and y-dimension. Figure 8: The evolution of the velocity of seven followers and one leader: (**a**) x-dimension and (**b**) y-dimension. Figure 7: Trajectory of the agents in the x–y dimension. Figure 9: The evolution of seven followers’ position tracking error with \(t_{f}=6\,\)s.  
In order to analyze the sensitivities of the control parameters for the system performance, we choose different control parameters for the proposed control scheme. In both three cases, except for \(k_{1}\) and \(k_{2}\), all other parameters remain the same. The maximum amplitude of the control input and the control effort under different parameters on x-dimension are shown in Table 3. Figure 10a,b show the cumulative position difference trajectories with controller parameters set to \(k_{1}=24\), \(k_{2}=70\) and \(k_{1}=120\), \(k_{2}=300\), respectively.