# Consensus Tracking Control of Multiple Unmanned Aerial Vehicles Subject to Distinct Unknown Delays
## 3 Control Strategy
### Translation Control
#### Follower agent control
\begin{array}{l}\dot{\xi}(t)=(I_{N}\otimes A)\xi(t)+(\mathcal{L}_{1} \otimes BK)\hat{\xi}(t)+(I_{N}\otimes B)\tilde{u}(t)\\ \tilde{y}(t)=(I_{N}\otimes C)\hat{\xi}(t)\end{array} \right. \tag{42}\]  
where \(\xi(t)=\left[\xi_{1}^{T}(t),\ldots,\xi_{N}^{T}(t)\right]^{T}\), \(\tilde{u}(t)=\left[\tilde{u}_{1}^{T}(t),\ldots,\tilde{u}_{N}^{T}(t)\right]^{T}\), \(\tilde{y}(t)=\left[\tilde{y}_{1}^{T}(t),\ldots,\tilde{y}_{N}^{T}(t)\right]^{T}\) and \(\hat{\xi}(t)=\left[\xi_{1}^{T}(t),\ldots,\xi_{N}^{T}(t)\right]^{T}\) denotes the follower agent's state estimation. This yields the following follower agent system:  
\[\left\{ \begin{array}{l}\dot{\xi}(t)=\tilde{A}\xi(t)+\tilde{K}\hat{\xi}(t)+ \tilde{B}\tilde{u}(t)\\ \tilde{y}(t)=\tilde{C}\hat{\xi}(t)\end{array} \right. \tag{43}\]  
with \(\tilde{A}=(I_{N}\otimes A)\), \(\tilde{K}=(\mathcal{L}_{1}\otimes BK)\), \(\tilde{B}=(I_{N}\otimes B)\), \(\tilde{C}=(I_{N}\otimes C)\). Now, the same estimation approach is used in the follower agent system; thus, unknown input observers are applied to isolate the external disturbances with the unknown boundary \(\tilde{u}_{i}(t)\), where the unknown delayed input of each agent is present \(u_{i}(t-\tau_{i}(t))\). For isolation of each delayed control input, an observer like a nonlinear unknown input observer on agent \(i\) is constructed as follows:  
\[\left\{ \begin{array}{l}\dot{w}_{i}(t)=\mathbb{N}w_{i}(t)+\mathbb{J}g_{i}(t) \\ \hat{\xi}_{i}(t)=w_{i}(t)+\mathbb{E}g_{i}(t)\end{array} \right. \tag{44}\]  
where \(w_{i}(t)\in\mathbb{R}^{N}\) and \(\hat{\xi}_{i}(t)\in\mathbb{R}^{N}\) are the states of the observer, and the estimations of the system states \(\xi_{i}(t)\) for each agent \(i\). The constant matrices \(\mathbb{N},\mathbb{J}\) and \(\mathbb{E}\) are chosen to make \(\hat{\xi}_{i}(t)\) converge to \(\hat{\xi}_{i}(t)\) in a delay-free case. Now, based on the properties of the Kronecker product, one can formulate the following unknown input observer on the agent:  
\[\left\{ \begin{array}{l}\dot{w}(t)=\mathbf{N}w(t)+\mathbf{J}g(t)\\ \hat{\xi}(t)=w(t)+\mathbf{E}\tilde{y}(t)\end{array} \right. \tag{45}\]  
where \(w(t)=\left[w_{1}^{T}(t),\ldots,w_{N}^{T}(t)\right]^{T}\), \(\mathbf{N}=(I_{N}\otimes\mathbb{N})\), \(\mathbf{J}=(I_{N}\otimes\mathbb{J})\), \(\mathbf{E}=(I_{N}\otimes\mathbb{E})\). Now, the following theorem is provided to ensure convergence of the proposed observer:  
Theorem 3.2: _The unknown inputs \(\tilde{u}_{i}(t)\) of the follower agent system (41) can be decoupled from the state estimation under the unknown input observer (45) if the following conditions are satisfied:_  
\[\left(\tilde{A}-\mathbf{E}\tilde{C}\tilde{A}+\mathbf{N}\mathbf{E} \tilde{C}-\mathbf{J}\tilde{C}\right) =\mathbf{N}-\left(I-\mathbf{E}\tilde{C}\right)\tilde{K} \tag{46}\] \[\left(I-\mathbf{E}\tilde{C}\right)\tilde{B} =0 \tag{47}\]  
_where \(\left[\mathbf{N}-\left(I-\mathbf{E}\tilde{C}\right)\tilde{K}\right]\) is a Hurwitz matrix to achieve an exponential convergence of the observation error \(\tilde{e}(t)\). This means the follower observers are not subject to the unknown time-varying delay \(\tau_{i}(t)\)._  
Proof. : Now, the observation error is \(\tilde{e}(t)=\xi(t)-\hat{\xi}(t)\), and using Equations (43) and (45), one obtains the following dynamics:  
\[\dot{\tilde{e}}(t)= \xi(t)-\hat{\xi}(t)\] \[= \big{(}\mathbf{N}-\left(I-\mathbf{E}\tilde{C}\right)\tilde{K} \big{)}\tilde{e}(t)+\left(\tilde{A}-\mathbf{E}\tilde{C}\tilde{A}+\mathbf{N} \mathbf{E}\tilde{C}-\mathbf{J}\tilde{C}-\mathbf{N}+\left(I-\mathbf{E}\tilde{C }\right)\tilde{K}\right)\xi(t)\] \[+\left(I-\mathbf{E}\tilde{C}\right)\tilde{B}\tilde{u}(t) \tag{48}\]  
Then,  
\[\dot{\tilde{e}}(t)= \big{(}\mathbf{N}-\mathbf{M}\tilde{K}\big{)}\tilde{e}(t)+\big{(} \mathbf{M}\tilde{A}+\mathbf{N}\mathbf{E}\tilde{C}-\mathbf{J}\tilde{C}-\mathbf{N }+\mathbf{M}\tilde{K}\big{)}\tilde{\xi}(t)+\mathbf{M}\tilde{B}\tilde{u}(t) \tag{49}\]where \(\mathbf{M}=\left(I-\mathbf{E}\tilde{C}\right)\). Since \(\mathbf{M}\tilde{B}=0\), using Lemma 1, one can easily obtain the matrix \(\mathbf{E}\).