# Consensus Tracking Control of Multiple Unmanned Aerial Vehicles Subject to Distinct Unknown Delays
## 3 Control Strategy
### Translation Control
#### Leader agent control  
Thus, for the leader agent, the control inputs are as follows:  
\[\theta_{d_{0}}(t)=K_{0}^{1}\xi_{0k}^{1}(t)\qquad\phi_{d_{0}}(t)=-K_{0}^{2}\xi_ {0k}^{2}(t) \tag{16}\]  
where \(\xi_{0k}^{1}\) and \(\xi_{0k}^{2}\) are the estimations of \(\xi_{0k}^{1}\) and \(\xi_{0k}^{2}\). The terms \(K_{0}^{1}\in\mathbb{R}^{3}\) and \(K_{0}^{2}\in\mathbb{R}^{3}\) are the gain feedback matrices. Then, the signal \(u_{0}\) can be expressed as  
\[u_{0}=K_{0}\xi_{0}(t) \tag{17}\]where  
\[K_{0}=\left[ \begin{array}{cc}K_{0}^{1}&0\\ 0&-K_{0}^{2}\end{array} \right]\quad\hat{\xi}_{0}(t)=\left[ \begin{array}{cc}\hat {\xi}_{0k}^{1}&\hat{\xi}_{0k}^{2}\end{array} \right]^{T}\]  
Therefore, the dynamics of the leader agent are given by  
\[ \begin{split}\dot{\xi}_{0}(t)=&A\xi_{0}(t)+Bu_{0}(t)+B\bar{u}_{0} (t)\\ \bar{g}_{0}(t)=&C\xi_{0}(t)\end{split}  \tag{18}\]  
where \(\bar{u}_{0}(t)\) is the unknown input of the leader agent, which is affected by the time-varying delay \(\tau_{0}(t)\). Now, the following unknown input observer is proposed for the translation movement of the leader agent under an unknown time-varying delay:  
\[\left\{ \begin{array}{l}\dot{w}_{0}(t)=Nw_{0}(t)+Hu_{0}(t)+J\bar{g}_{0}(t) \\ \dot{\xi}_{0}(t)=w_{0}(t)+E\bar{g}_{0}(t)\end{array} \right. \tag{19}\]  
where \(w_{0}(t)\in\mathbb{R}^{Nn}\) and \(\hat{\xi}_{0}(t)\in\mathbb{R}^{Nn}\) are the states of the observer, and the estimations of the system states \(\xi_{0}(t)\). The constant matrices \(N,H,J\) and \(E\) are chosen to make \(\hat{\xi}_{0}(t)\) converge to \(\hat{\xi}_{0}(t)\) in the delay-free case. Now, the following lemma is used in the design of the unknown input observer. Lemma 1: _Given any matrices \(C\) and \(B\), equation \(ECB=B\) has a solution if and only if \(\operatorname{rank}(CB)=\operatorname{rank}(B)\)._  
Then, the following theorem provides sufficient conditions that guarantee the convergence of the observation error \(\bar{e}_{0}(t)\). Theorem 1: _The unknown input \(\bar{u}_{0}(t)\) of the leader agent system (18) can be decoupled form the state estimation under the unknown input observer (19) if the following conditions are satisfied:_  
\[(A-ECA+NEC-JC-N) =0 \tag{20}\] \[((I-EC)B-H) =0\] (21) \[(I-EC)B =0 \tag{22}\]  
_where \(N\) is a Hurwitz matrix used to achieve exponential convergence of the observation error \(\bar{e}_{0}(t)\). This means the observer is not subject to an unknown time-varying delay \(\tau_{0}(t)\)._  
Proof: Now, the observation error is \(\bar{e}_{0}(t)=\zeta_{0}(t)-\hat{\xi}_{0}(t)\); then, the dynamics of the estimation error is  
\[\dot{\xi}_{0}(t)= \hat{\xi}_{0}(t)-\hat{\xi}_{0}(t) \tag{23}\] \[= (A-ECA)\xi_{0}(t)-Nw_{0}(t)-JC\xi_{0}(t)\] \[+((I-EC)B-H)u_{0}(t)+(I-EC)B\bar{u}_{0}(t)\] \[= N\bar{e}_{0}(t)+(A-ECA+NEC-JC-N)\xi_{0}(t)\] \[+((I-EC)B-H)u_{0}(t)+(I-EC)B\bar{u}_{0}(t)\] \[= N\bar{e}_{i}(t)+(A-ECA+NEC-JC-N)\xi_{0}(t)\] \[+((I-EC)B-H)u_{0}(t)+(I-EC)B\bar{u}_{0}(t) \tag{24}\]  
Now, in order to obtain \(E\) and \(H\) matrices, from Lemma 1, one can deduce that the general solution of \(E\) is as follows:  
\[((I-EC)B-H) =0\]0where \(X\) is an arbitrary matrix. Thus, \(X=0\) is selected for ease of calculation, and one obtains \(E=B(CB)^{\dagger}\). From the matrix \(E\), \(H\) is obtained as follows: \(H=(I-EC)B\). Now, this denotes that \(M=(I-EC)\); thus, Equation (20) can be rewritten as  
\[((I-EC)B-H) =0\]1  
Then,  
\[((I-EC)B-H) =0\]2  
where \(L=(J-NE)\). Now, \(N=MA-LC\) is a Hurwitz matrix used to ensure convergence of the observation error. Thus, the matrix \((MA-LC)\) should be placed anywhere in the complex s-plane by a suitable selection of matrix \(L\), where \((C,YA)\) must be observable. Subsequently, it is possible to find \(J\), computed as \(J=(L+NE)\). Then, the unknown input observer provides an easy-to-implement algorithm that eliminates the influence of the unknown delayed input in the state estimation process.