## 5 Numerical Simulations  
In this section, numerical simulations are given to demonstrate the effectiveness and superiority of the proposed cooperative-guidance law. Consider the situation in which five vehicles simultaneously intercept a maneuvering target. The initial conditions of the multiple vehicles and target are presented in Table 1. Due to the physical constraints of the vehicles, the maximum accelerations in all directions are limited to 25 g, where g denotes the gravitational acceleration, and \(g=9.8\,\mathrm{m/s^{2}}\). The attack angle of the leader is set as \(10^{\circ}\). To realize the encirclement interception, the followers are arranged on both  
The control parameters are designed as follows:  
\[ \begin{array}{l}h_{1}=2,h_{2}=1.5,\alpha=0.8,\beta=1.2,\epsilon=0.005\\ k_{r1}=2,\gamma=0.7\\ k_{1}=1.3,k_{2}=6,h=1,\lambda=1.3,\eta=0.8\\ k_{3}=0.015,k_{4}=0.03,n=6\end{array} \]  
The simulation results with our guidance law are exhibited in Figures 4-9. From Figure 4, it can be observed that the multiple vehicles can intercept the maneuvering target along different trajectories. Figure 5 shows the time-to-go of the interceptors. To further demonstrate the convergence process, the tracking error between the followers and leader is provided in picture-in-picture. The \(t_{go}\) of each follower converges to the value of the leader rapidly, which means that the cooperative attack has been completed. Figure 6 shows that the vehicles attack the target with the different desired angles, and that the predetermined encirclement tactics can be implemented. Figure 7 gives the control command in the LOS direction during the engagement. The control command in the LOS direction is used to adjust the \(t_{go}\) of the multiple vehicles. In Figure 8, owing to the initial attack-angle errors of the vehicles, the control commands in the normal direction of the LOS are relatively large at the beginning of the guidance. As is shown in Figure 9, the flight speeds of the five vehicles are constantly adjusted within a relatively small range. In the initial phase of the flight, a large overload is applied in the normal direction of the LOS to control the attack angle, which leads to an increase in the speeds of the multiple vehicles. Meanwhile, the overload in the direction of the LOS is applied to track the flight time of the leader, as well as to compensate for the normal overload. With the rapid convergence of the attack angle and flight time, the control overloads tend to be gentle. The variation in the speeds of the multiple vehicles tends to be stable. Figure 4: Trajectories of the vehicles and target. Figure 5: Time-to-go of multiple vehicles. Figure 6: Attack angles of the LOS. Figure 7: Control commands in LOS direction. The estimation performance of our fixed-time distributed disturbance observer is shown in Figures 10 and 11. It can be seen that the disturbance error of the DDOB can converge to the neighborhood of the origin in time. Figure 8: Control commands in normal direction of LOS. Figure 10: Actual and estimated values of disturbance. Figure 9: Velocities of multiple vehicles. To validate the superiority of the proposed cooperative-guidance law, a series of contrast experiments were presented as follows. The comparison between our guidance law and the finite-time consensus-guidance (FTCG) law based on finite-time control [36] without the DDOB is shown in Figure 12. Owing to the maneuver of the target, the control commands of the followers in the LOS direction will change rapidly when the interceptors are closed to the target. The proposed DDOB can significant decrease the effect caused by the uncertain disturbance. In Figure 13, an estimation simulation with an ESO based on the disturbance observer in [22] is carried out for comparison. It can obviously be seen that large saturation shocks are avoided by using the DDOB. Figure 11: Tracking error of DDOB. Figure 12: Control-command comparison in LOS direction. Figure 5:0 Estimation-performance comparison between DDOB and ESO.