# Consensus Tracking Control of Multiple Unmanned Aerial Vehicles Subject to Distinct Unknown Delays
## 3 Control Strategy
### Translation Control
#### Leader agent control
Based on the above, if these conditions are fulfilled, the following dynamics of the error of observation can be obtained:  
\[((I-EC)B-H) =0\]3  
where \(N\) should be a Hurwitz matrix for achieving exponential convergence of the observation error. This means the observer is not subject to an unknown delayed input \(\tau_{0}(t)\). This ends the proof. Now, it is possible to design an observer-based control strategy to stabilize the leader agent system (18), where the following observer-based control for the leader agent is considered:  
\[((I-EC)B-H) =0\]4 \[((I-EC)B-H) =0\]5 \[((I-EC)B-H) =0\]6 \[((I-EC)B-H) =0\]7  
Then, the closed-loop dynamics of the leader are  
\[((I-EC)B-H) =0\]8  
Now, it is evident that there is an influence of the unknown signal \(\bar{u}_{0}(t)\) on the closed-loop system (29). Therefore, the stability of the closed-loop system must be guaranteed. **Theorem 2**. : _Consider the disturbance observer-based control (28) of the leader agent system (18), which is exponentially stable if there exists a \(P_{c}>0\) such that the following inequation holds for \(\mu_{1}>\mu_{2}>0\):_  
\[\left[P_{c}A^{T}+WB^{T}+AP_{c}+BW+\mu_{1}P_{c}\qquad \begin{array}{c}B\\ B^{T}\end{array} \right]<0, \tag{30}\]  
_where the control gain is \(K_{o}=WP_{c}^{-1}\). Thus, the system's stability is guaranteed, even in the presence of the unknown time-varying delay \(\tau_{0}(t)\)._  
Proof. : Considering the quadratic function \(V(\xi_{0}(t))=\xi_{0}^{T}(t)P\xi_{0}(t)\), one obtains the following dynamics:\[\dot{V}(\xi_{0}(t)) =\xi_{0}^{T}(t)P\xi_{0}(t)+\xi_{0}^{T}(t)P\xi_{0}^{t}(t)\] \[=((A+BK_{0})\xi_{0}(t)+B\bar{u}_{0}(t))^{T}P\xi_{0}(t)+\xi_{0}(t)^{ T}P((A+BK_{0})\xi_{0}(t)+B\bar{u}_{0}(t))\] \[=\xi_{0}(t)^{T}((A+BK_{0})^{T}P+P(A+BK_{0}))\xi_{0}(t)+\bar{u}_{0} (t)^{T}B^{T}P\xi_{0}(t)+\xi_{0}(t)^{T}PB\bar{u}_{0}(t))\] \[= \begin{bmatrix}\xi_{0}(t)\\ \bar{u}_{0}(t)\end{bmatrix} ^{T} \begin{bmatrix}((A+BK_{0})^{T}P+P(A+BK_{0}))& PB\\ B^{T}P&0\end{bmatrix}\begin{bmatrix}\xi_{0}(t)\\ \bar{u}_{0}(t)\end{bmatrix}  \tag{31}\]  
The following proposition provides a criterion that guarantees the exponential stability of the disturbance system (18). **Proposition 1**. : _Taking into account the system (29). If there exist matrix \(P>0\) and positive scalars \(\mu_{1}>\mu_{2}\) such that the following inequality holds:_  
\[\dot{V}(\xi_{0}(t))+\mu_{1}V(\xi_{0}(t))-\mu_{2}\bar{u}_{0}(t)^{T}P\bar{u}_{0} (t)\leq 0 \tag{32}\]  
_where \(V(\xi_{0}(t))\) is set \(V(\xi_{0}(t))=\xi_{0}^{T}(t)P\xi_{0}(t)\) and \(\dot{V}(\xi_{0}(t))\) is its derivative, then the closed-loop system (29) is exponentially stable._  
From Proposition 1, one can rewrite (31) as follows:  
\[\dot{V}(\xi_{0}(t))+\mu_{1}V(\xi_{0}(t))-\mu_{2}\bar{u}_{0}(t)^{T}P\bar{u}_{0} (t)=\] \[ \begin{bmatrix}\xi_{0}(t)\\ \bar{u}_{0}(t)\end{bmatrix} ^{T} \begin{bmatrix}((A+BK_{0})^{T}P+P(A+BK_{0}))+ \mu_{1}P&PB\\ B^{T}P&-\mu_{2}P\end{bmatrix}\begin{bmatrix}\xi_{0}(t)\\ \bar{u}_{0}(t)\end{bmatrix} \]  
Therefore, the following inequality must hold  
\[ \begin{bmatrix}(A+BK_{0})^{T}P+P(A+BK_{0})+\mu_{1}P&PB\\ B^{T}P&-\mu_{2}P\end{bmatrix} \leq 0 \tag{33}\]  
Now, the above equation is a bilinear matrix inequality; thus, a non-singular transformation, such as \(T= \begin{bmatrix}P^{-1}&0\\ 0&I_{n}\end{bmatrix} >0\), must be introduced:  
\[T \begin{bmatrix}(A+BK_{0})^{T}P+P(A+BK_{0})+\mu_{1}P&PB\\ B^{T}P&-\mu_{2}P\end{bmatrix} T= \tag{34}\] \[ \begin{bmatrix}P^{-1}(A+BK_{0})^{T}+(A+BK_{0})P^{-1}+\mu_{1}P^{-1}& B\\ B^{T}&-\mu_{2}I_{n}\end{bmatrix} \leq 0 \tag{35}\]  
Defining \(P^{-1}=P_{c}\), this yields  
\[ \begin{bmatrix}P_{c}(A+BK_{0})^{T}+(A+BK_{0})P_{c}+\mu_{1}P_{c}&BP_{c}\\ P_{c}B^{T}&-\mu_{2}I_{n}\end{bmatrix} <0. \tag{36}\]  
Thus, the following LMI is obtained,  
\[ \begin{bmatrix}P_{c}A^{T}+WB^{T}+AP_{c}+BW+\mu_{1}P_{c}&B\\ B^{T}&-\mu_{2}I_{n}\end{bmatrix} <0, \tag{37}\]  
where \(K_{o}=WP_{c}^{-1}\). Using Proposition 1, the closed-loop dynamics of the leader (29) is exponentially stable under (37).