# Consensus Tracking Control of Multiple Unmanned Aerial Vehicles Subject to Distinct Unknown Delays
## 3 Control Strategy
### Translation Control
#### Follower agent control
In addition, the following assumption must be satisfied:  
\[\mathbf{M}\tilde{A}+\mathbf{N}\mathbf{E}\tilde{C}-\mathbf{J} \tilde{C}+\mathbf{M}\tilde{K} =\mathbf{N}\] \[\mathbf{M}\tilde{A}-\mathbf{L}\tilde{C} =\mathbf{N}-\mathbf{M}\tilde{K}\] \[\mathbf{M}\tilde{A}-\mathbf{L}\tilde{C} =\mathbf{\bar{N}} \tag{50}\]  
where \(\mathbf{L}=(\mathbf{J}-\mathbf{N}\mathbf{E})\) and the matrix \(\mathbf{\tilde{N}}=\mathbf{N}-\mathbf{M}\tilde{K}\). It is worth recalling that the matrix \(\tilde{K}\) depends on the gain matrix of the consensus control, which will be further detailed below. Therefore, to guarantee the convergence of the observation error \(\tilde{e}\), one should satisfy that \(\mathbf{\tilde{N}}=\mathbf{M}\tilde{A}-\mathbf{L}\tilde{C}\) is a Hurwitz matrix, to ensure convergence of the observation error. Therefore, \(\mathbf{\tilde{N}}\) should be placed anywhere in the complex \(s\)-plane by a suitable selection of matrix \(\mathbf{L}\). Based on the above, the dynamics of the observation error is given by \(\dot{\delta}(t)=\mathbf{\tilde{N}}\tilde{e}(t)\), where the Hurwitz matrix \(\mathbf{\tilde{N}}\) achieves exponential convergence of the observation error. This means that the followers' observers will be not subject to the unknown time-varying delays \(\tau_{i}(t)\). Now, it is possible to design an observer-based control strategy to stabilize the follower agents' system, where the following observer-based control is considered:  
\[u_{i}(t)= K\sum_{j=0}^{N}a_{ij}(\hat{\xi}_{i}(t)-\hat{\xi}_{j}(t)-h_{i}+h_{j}) \tag{51}\] \[\left(I-\mathbf{E}\tilde{C}\right)\tilde{B} =0 \tag{47}\]0 \[\left(I-\mathbf{E}\tilde{C}\right)\tilde{B} =0 \tag{47}\]1 \[\left(I-\mathbf{E}\tilde{C}\right)\tilde{B} =0 \tag{47}\]2  
Since all estimation errors \(\tilde{e}_{i}(t)\) and \(\tilde{e}_{j}(t)\) tend to zero, the control law can be simplified as  
\[\left(I-\mathbf{E}\tilde{C}\right)\tilde{B} =0 \tag{47}\]3  
For the follower agents, let  
\[\left(I-\mathbf{E}\tilde{C}\right)\tilde{B} =0 \tag{47}\]4 \[\left(I-\mathbf{E}\tilde{C}\right)\tilde{B} =0 \tag{47}\]5  
According to Equations (29), (41) and (51), for \(i=1,\dots,N\), we have  
\[\left(I-\mathbf{E}\tilde{C}\right)\tilde{B} =0 \tag{47}\]6 \[\left(I-\mathbf{E}\tilde{C}\right)\tilde{B} =0 \tag{47}\]7 \[\left(I-\mathbf{E}\tilde{C}\right)\tilde{B} =0 \tag{47}\]8Considering the new variables \(\bar{u}_{i}(t)=\bar{u}_{i}(t)-\bar{u}_{0}(t)\) and \(\bar{A}=A-A_{0}\), where \(A_{0}=[A+BK_{0}]\) is a stable matrix according to the solution of the LMI in Theorem 2. Then, \(\hat{\xi}_{i}(t)\) is now defined as follows:  
\[\left(I-\mathbf{E}\tilde{C}\right)\tilde{B} =0 \tag{47}\]9  
It is important to note that the achievable formation structures must satisfy the constraints \(Al_{i}=0\). Therefore, the closed-loop dynamic of the followers is  
\[\dot{\tilde{e}}(t)= \xi(t)-\hat{\xi}(t)\]0  
where \(\bar{\xi}(t)=\left[\bar{\xi}_{1}(t),\ldots,\bar{\xi}_{N}(t)\right]^{T}\) and \(\bar{u}(t)=\left[\bar{u}_{1}(t),\ldots,\bar{u}_{N}(t)\right]^{T}\). Now, as happened with the leader agent, it is evident that the dynamics of the followers are influenced by the unknown signal \(\bar{u}(t)\). Therefore, the stability of the closed-loop system must also be guaranteed. Theorem 4: _For a direct communications graph \(\mathcal{G}\), the agents (41) reach the formation \(\bar{H}\) under the protocol (52) if there exists a \(P_{c}>0\) such that the following inequation holds for \(\alpha_{1}>\alpha_{2}>0\):_  
\[ \begin{bmatrix}I_{N}\otimes P_{c}\bar{A}^{T}+\mathcal{L}_{1}\otimes W^{T}B^{T} +I_{N}\otimes\bar{A}P_{c}+\mathcal{L}_{1}\otimes BW+I_{N}\otimes\alpha_{1}P_{ c}&I_{N}\otimes B\\ I_{N}\otimes B^{T}&I_{N}\otimes-\alpha_{2}I_{N}\end{bmatrix} <0, \tag{55}\]  
_where the control gain is \(K=WP_{c}^{-1}\), and it satisfies that \(Ah_{i}=0,\forall i=1,\ldots,N\). Thus, the observer-based control (40) of the follower agent system (41) is exponentially stable, even in the presence of unknown time-varying delays \(\tau_{i}\).