# Consensus Tracking Control of Multiple Unmanned Aerial Vehicles Subject to Distinct Unknown Delays
## 3 Control Strategy
### Translation Control
#### Follower agent control  
For the follower agents, the consensus control for formation is based on the estimation of the states \(\xi_{i}(t)\) of each agent, which are error states in the \(x_{i}\) and \(y_{i}\) coordinates. In the consensus protocols, the communication between agents in a MAS can be described by the graph \(\mathcal{G}\). Let \(\mathcal{G}=(\mathcal{V},\mathcal{E})\), where \(\mathcal{V}=\{v_{1},\ldots,v_{N}\}\) is a nonempty finite node set and \(\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}\) is an edge set. The edge \((v_{i},v_{j})\) in the edge set \(\mathcal{E}\) denotes that agent \(v_{j}\) can obtain information from agent \(v_{i}\), but not vice versa. The adjacency matrix \(\mathcal{A}= \begin{bmatrix}a_{ij}\end{bmatrix} \in\mathbb{R}^{N\times N}\) associated to the graph \(\mathcal{G}\) is defined such that \(a_{ii}=0\) and \(a_{ij}=1\Leftrightarrow(j,i)\in\mathcal{E}\) for \(i,j=0,1,\ldots,N\). The Laplacian matrix \(\mathcal{L}= \begin{bmatrix}\mathcal{L}_{ij}\end{bmatrix} \in\mathbb{R}^{N \times N}\) of the graph \(\mathcal{G}\) is defined as \(\mathcal{L}_{ii}=\sum_{j\in i}a_{ij}\) and \(\mathcal{L}_{ij}=-a_{ij}\), \(i\
eq j\). In leader-follower schemes, \(\mathcal{L}\) can be partitioned as \(\mathcal{L}= \begin{bmatrix}0&0_{1\times N}\\ \mathcal{L}_{2}&\mathcal{L}_{1}\end{bmatrix} \), where \(\mathcal{L}_{2}\in\mathbb{R}^{N\times 1}\) and \(\mathcal{L}_{1}\in\mathbb{R}^{N\times N}\). In addition, the constant formation structure of the agents in the graph \(\mathcal{G}\) in a reference coordinate frame is denoted by \(\tilde{H}=(h_{0},h_{1},\ldots,h_{N})\in\mathbb{R}^{n\times N}\), where \(h_{i}\in\mathbb{R}^{n}\) is the formation variable corresponding to agent \(i\). Assuming that each agent has access to its own state and the states of its neighbors, the agents achieve consensus in the sense of \(\lim_{t\rightarrow\infty}\|(\xi_{i}(t)-h_{i})-(\xi_{j}(t)-h_{j})\|=0,\forall i,j=0,\ldots,N\). Thus, the following law of observer-based consensus control for the coordinates \(x\) and \(y\) is considered:  
\[\theta_{di}(t)= K_{1}\sum_{j=0}^{N}a_{ij}(\xi_{ik}^{1}(t)-\xi_{jk}^{1}(t)-h_{i}^{1 }+h_{j}^{1}),\quad i=1,\ldots,N. \tag{38}\] \[\phi_{di}(t)= K_{2}\sum_{j=0}^{N}a_{ij}(\xi_{ik}^{2}(t)-\xi_{jk}^{2}(t)-h_{i}^{ 2}+h_{j}^{2}),\quad i=1,\ldots,N. \tag{39}\]  
The terms \(K_{1}\in\mathbb{R}^{1\times 3}\) and \(K_{2}\in\mathbb{R}^{1\times 3}\) are the feedback gain matrices, which will be defined later; \(a_{ij}\) is the \((i,j)\)-th element of the adjacency matrix associated with \(\mathcal{G}\), and \(\xi_{ik}^{1}(t)\), \(\xi_{ik}^{2}(t)\), \(\xi_{jk}^{2}(t)\), and \(\xi_{jk}^{2}(t)\) are the state estimations of \(\xi_{ik}^{1}(t)\), \(\xi_{ik}^{2}(t)\), \(\xi_{jk}^{1}(t)\), and \(\xi_{jk}^{2}(t)\). Moreover, \(h_{i}^{1}\), \(h_{i}^{2}\), \(h_{j}^{1}\) and \(h_{j}^{2}\) are constant vectors that describe the formation structure in coordinates \(x\) and \(y\). Considering the vectors \(h_{i}= \begin{bmatrix}h_{i}^{1}&h_{i}^{2}\end{bmatrix} ^{T}\), \(\hat{\xi}_{i}(t)= \begin{bmatrix}\xi_{ik}^{1}(t)&\xi_{ik}^{2}(t)\end{bmatrix} ^ {T}\) and \(\hat{\xi}_{j}(t)= \begin{bmatrix}\hat{\xi}_{jk}^{1}(t)&\hat{\xi}_{jk}^{2}(t) \end{bmatrix} ^{T}\), the consensus control law based on state estimation can be formulated as a single control input for each agent:  
\[u_{i}(t)=K\sum_{j=0}^{N}a_{ij}(\hat{\xi}_{i}(t)-\hat{\xi}_{j}(t)-h_{i}+h_{j}), \quad i=1,\ldots,N. \tag{40}\]  
where  
\[K=\left[ \begin{array}{cc}K_{1}&0\\ 0&K_{2}\end{array} \right]\]  
Then, the follower agent dynamics are given:  
\[\left\{ \begin{array}{l}\hat{\xi}_{i}(t)=A\xi_{i}(t)+Bu_{i}(t)+B\bar{u}_{i}(t) \\ \bar{y}_{i}(t)=C\xi_{i}(t)\end{array} \right. \tag{41}\]  
where \(\bar{u}_{i}(t)\) is the unknown input of the follower agent, which is affected by time-varying delay \(\tau_{i}(t)\). Substitute the observer-based control law (40) into the follower agent system (41) of agent \(i\). By making use of the properties of the Kronecker product, one can rewrite this in the following form:\[\left\{