# Consensus Tracking Control of Multiple Unmanned Aerial Vehicles Subject to Distinct Unknown Delays
## 3 Control Strategy
### Translation Subsystem  
From the linear quadrotor multi-agent system (5), the translation movement of the quadcopter can be obtained:  
\[ \begin{split}\dot{x}_{i}(t)=& g\big{[}\theta_{i}(t)c_{ \psi_{ii}}+\phi_{i}(t)s_{\psi_{ii}}\big{]}\\ \dot{y}_{i}(t)=& g\big{[}\theta_{i}(t)s_{\psi_{ii}}- \phi_{i}(t)c_{\psi_{ii}}\big{]}\end{split}  \tag{10}\]  
Now, since all orientation errors of the quadrotor multi-agent system tend to be zero, it can be assumed that \(\theta_{i}\approx\theta_{d_{i}}\), \(\phi_{i}\approx\phi_{d_{i}}\). Moreover, if \(\psi_{d_{i}}\) is considered a constant equal to zero, the dynamics corresponding to the \(x_{i}\) and \(y_{i}\) coordinates are described in the following form:  
\[ \begin{split}\dot{x}_{i}(t)=& gc_{\psi_{d}}\theta_ {di}(t)\\ \dot{y}_{i}(t)=&-gc_{\psi_{d}}\phi_{di}(t)\end{split}  \tag{11}\]  
Then, one can define the following state vector \(\dot{\varsigma}_{i}(t)=\big{[}\dot{\varsigma}_{ik}^{1}(t)\quad\dot{\varsigma }_{ik}^{2}(t)\big{]}^{T}\in\mathbb{R}^{6}\) for each agent, where  
\[ \begin{split}\dot{\varsigma}_{ik}^{1}(t)=& \left[\int e_{xi}(t)dt\quad e_{xi}(t)\quad\dot{\varsigma}_{xi}(t)\right]^{T} \in\mathbb{R}^{3},\quad k=1,2,3\\ \dot{\varsigma}_{ik}^{2}(t)=&\left[\int e_{yi}(t) dt\quad e_{yi}(t)\quad\dot{\varsigma}_{yi}(t)\right]^{T}\in\mathbb{R}^{3}, \quad k=1,2,3\end{split}  \tag{12}\]  
Thus, the \(x\) and \(y\) coordinate errors are defined as follows:  
\[ \begin{split}\dot{\varsigma}_{xi}^{1}(t)=& x_{d_{i}}(t)-x_{i}(t)\quad\quad e_{yi}(t)=y_{d_{i}}(t)-y_{i}(t)\\ \dot{\varsigma}_{xi}(t)=&\dot{x}_{d_{i}}(t)-\dot{x }_{i}(t)\quad\quad\dot{\varsigma}_{yi}(t)=\dot{y}_{d_{i}}(t)-\dot{y}_{i}(t) \end{split} \]  
Considering \(x_{d_{i}}\) and \(y_{d_{i}}\) as constants, the elements of the derivative \(\dot{\varsigma}_{i}(t)=\big{[}\dot{\varsigma}_{ik}^{1}(t)\quad\dot{\varsigma }_{ik}^{2}(t)\big{]}^{T}\in\mathbb{R}^{6}\) are represented as follows:  
\[ \begin{split}\dot{\varsigma}_{xi}^{1}_{i1}=& \varsigma_{i2}^{1}\\ \dot{\varsigma}_{i2}^{1}=&\varsigma_{i3}^{1}\\ \dot{\varsigma}_{i3}^{1}=&-gc_{\psi_{d}}\theta_{di} \\ \dot{\varsigma}_{i1}^{2}=&\varsigma_{i2}^{2}\\ \dot{\varsigma}_{i2}^{2}=&\varsigma_{i3}^{2}\\ \dot{\varsigma}_{i3}^{2}=& gc_{\psi_{d}}\phi_{di} \end{split} \]  
Therefore, the translation movement \(x\)-\(y\) of the quadcopter multi-agent dynamics can be expressed in the following state-space form:  
\[ \begin{split}\dot{\xi}_{i}(t)=& A\dot{\varsigma}_{i} (t)+Bu_{i}(t)\\ \dot{y}_{i}(t)=& C\dot{\varsigma}_{i}(t)\end{split}  \tag{13}\]  
where matrix \(C=I_{6\times 6}\) and  
\[A=\left[ \begin{array}{cccc}0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\\ 0&0&0&0&0\end{array} \right]\quad B=\left[ \begin{array}{cccc}0&0\\ 0&0\\ -gc_{\psi_{d}}&0\\ 0&0\\ 0&0\\ 0&g\dot{\varsigma}_{vi}\end{array} \right]\quad u_{i}(t)=\left[ \begin{array}{ c}\theta_{di}(t)\\ \phi_{di}(t)\end{array} \right]\]  
A practical problem in the position tracking and formation control of quadrotor multi-agent systems is related to the time difference between the translation sensor as a globalpositioning system (GPS) that detects a position measurement and the force input that applies torque to each motor (input signal) of the aerial vehicle. This time difference is due to the communication or synchronization time between each servo controller, which entails UAVs not receiving the proper real-time input, meaning the system has an inherent input delay. Therefore, it is necessary to address the effect of variable time-varying delay in the input application on the UAV tracking control design. Thus, an observer-based consensus control is proposed, to ensure that all UAV agents can reach consensus with the leader, despite the presence of distinct unknown time-varying delays. Considering the previous problem, the lineal multi-agent system of the position tracking error dynamics (13) has inherent input delays. Therefore, the following delayed lineal multi-agent system is formulated:  
\[