# Consensus Tracking Control of Multiple Unmanned Aerial Vehicles Subject to Distinct Unknown Delays
## 4 Simulation Results  
A numerical simulation was performed to validate the performance of the control scheme based on an unknown input observer for the formation flight of three UAVs. For this formation, consider the communication graph \(\mathcal{G}\) in Figure 2, where node 0 is associated with the leader agent and the other nodes are associated with the follower agents. Then, the Laplacian matrix of the graph \(\mathcal{G}\) is defined as follows:  
\[\mathcal{L}=\left[ \begin{array}{ccc}0&0&0\\ -1&2&-1\\ -1&-1&2\end{array} \right] \tag{59}\]  
The formation structure is shown in Figure 3, where the followers (gray) have a distance of \(d=1\,\mathrm{m}\) in the \(y\) coordinate from the leader (red). Therefore, because the leader is the origin of the formation, the vector \(h_{0}=\left[0\;\;0\;\;0\;\;0\;\;0\;\;0\right]^{T}\). For the follower agents, constant vectors that describe the formation structure are \(h^{1}=-h^{2}=\left[0\;\;0\;\;0\;\;0\;\;d\;\;\;0\right]^{T}\). Furthermore, the linear dynamic model of altitude and attitude is given by (5), and the control scheme was implemented according to (6)-(9). The parameters used in the simulation are presented in Table 1. The translation movement is described by Equation (15) with input delays as shown in Table 2. The observer-based consensus control for the coordinates \(x\) and \(y\) was designed according to Equations (40) and (44), the feedback gain matrices \(K_{1}=-K_{2}=\left[2.5775\;\;\;3.3450\;\;1.1890\right]\) were obtained through Theorem 4 with \(\alpha_{1}=6\) and \(\alpha_{2}=5\). In addition, the observer gain selected was as follows:  
\[\mathbf{L}=\left[ \begin{array}{cccccc}1&1&0&0&0&0\\ 0&2.7&1&0&0&0\\ 0&0&0.81&0&0&0\\ 0&0&0&1&1&0\\ 0&0&0&0&2.7&1\\ 0&0&0&0&0&0.81\end{array} \right] \tag{60}\]  
Figure 3: Formation structure of the UAVs. Figure 2: Communication graph \(\mathcal{G}\). Figure 4: Trajectories of all agents in formation. Figure 5: Time-varying delay trajectories. Figure 6: Tracking error of all agents in \(z\) and \(\psi\) coordinates. Figure 9 shows the behavior of follower Agent 1. We can observe that it is evident that the input time-varying delay impacted the follower more significantly than the leader. The trajectories of \(x\), \(y\), \(\phi\), and \(\theta\) remained unaffected, except when the desired position changed. Moreover, the \(x\) coordinate was more affected by time-varying delays than the \(y\) coordinate; the reason for this behavior was that, during the time intervals when the \(x\) coordinate changed position, the functions \(\tau_{0}(t)\), \(\tau_{1}(t)\) and \(\tau_{2}(t)\) had periods of greater amplitude, as shown in Figure 5, thus resulting in a more significant delay in the control input. Figure 10 illustrates the results of follower Agent 2. Notably, the behavior closely mirrored that of Agent 1, except for the initial part of the simulation, due to the initial conditions. This is characteristic of such control schemes, wherein any signal delay in the leader agent's state inevitably ripples through the entire agent system. In Figure 11, the estimation errors between \(\xi^{1}\) and \(\xi^{1}\) of all agents are depicted. It is evident that the error approached zero, indicating the convergence of \(\xi_{1}\) to \(\xi_{1}\). Finally, Figure 12 shows the estimation errors between \(\xi^{2}\) and \(\xi^{2}\) of all agents, we can see that the proposed algorithm caused the error to converge to zero; thus, the effectiveness of the designed control schemes based on state estimation was verified by isolating these errors in all agents, despite the presence of distinct unknown delays. Figure 8: Trajectories of Agent 0 (Leader) in \(x\), \(y\), \(\phi\) and \(\theta\) coordinates. Figure 7: Tracking error of all agents in \(x\), \(y\), \(\phi\), and \(\theta\) coordinates. Figure 11: Error estimation between \(\hat{\xi}^{1}\) and \(\hat{\xi}^{1}\) of all agents. Figure 10: Trajectories of Agent 2 in \(x\), \(y\), \(\phi\) and \(\theta\) coordinates.