## 4 Stability Analysis
\right]\]  
\[G_{2}=[L,M,N]^{T}\]  
\[L=[l_{\delta_{d}},l_{\delta_{dr}},l_{\delta_{dr}},l_{\delta_{dr}},0,0,l_{ \delta_{r}}]^{T}\]  
\[M=[m_{\delta_{d}},m_{\delta_{dr}},m_{\delta_{dr}},m_{\delta_{dr}},m_{\delta_{ dr}},m_{\delta_{dr}},m_{\delta_{dr}}]^{T}\]  
\[N=[n_{\delta_{d}},n_{\delta_{dr}},n_{\delta_{dr}},0,0,n_{\delta_{r}}]^{T}\]  
\[f_{3}(X_{i})=q_{i}\cos\phi_{i}-r_{i}\sin\phi_{i}\]  
_where \(X_{i}=\left(X_{i,1}^{T},X_{i,2}^{T}\right)^{T}=\left(\phi_{i},\alpha_{i},\beta _{i},p_{i},q_{i},r_{i},\theta_{i}\right)^{T}\) are the roll angle, attack angle, sideslip angle, roll angular velocity, pitching angular velocity, yaw angular velocity and pitch angle of fighter \(i\), respectively, \(y_{i}=X_{i,1}=\left[\phi_{i},\alpha_{i},\beta_{i}\right]^{T}X_{i,2}=\left[p_{i },q_{i},r_{i}\right]^{T}X_{i,3}=\theta_{i}\). \(u_{i}=\left[\delta_{i,\delta_{ir}},\delta_{ial},\delta_{iar},\delta_{ilef}, \delta_{ilef},\delta_{ir}\right]^{T}\) are the left and right elevators, left and right ailerons, front and rear flaps, and rudder, respectively. Detailed explanations for the parameters and variables of this model can be found in [26]. Suppose that they are all flying at an altitude of 40,000 feet, at a speed of 0.6 Mach. The desired output for these fighters is \(y_{d}=\left[y_{d,1},y_{d,2},y_{d,3}\right]^{T}=\left[20,30,0\right]^{T}\). The signal \(y_{d}\) is only available for fighter 1._  
According to Theorem 1, we design the distributed flight controller as follows  
\[\xi_{i,1}=G_{1}^{-1}(X_{i,1})diag\left\{-\lambda_{i,1}\frac{e_{ i,\theta}}{1-e_{i,\theta}^{-}},-\lambda_{i,1}\frac{e_{i,\theta}}{1-e_{i,\theta}^{-}},-\lambda_{i,1}\frac{e_{i,\theta}}{1-e_{i,\theta}^{-}}\right\}\] \[u_{i}=G_{2}^{+}diag\left\{-\lambda_{i,2}\frac{e_{i,p}}{1-e_{i,p} ^{-}},-\lambda_{i,2}\frac{e_{i,\theta}}{1-e_{i,\theta}^{-}},-\lambda_{i,2} \frac{e_{i,\theta}}{1-e_{i,\theta}^{-}}\right\}\]  
with  
\[ \begin{array}{l}e_{i,\phi}=\phi_{i}-q_{d,1},e_{i,\alpha}=\alpha_{i}-q_{d,2},e_{i,\beta}=\beta_{i}-q_{d,3}\\ \left[e_{i,p},e_{i,q},e_{i,r}\right]^{T}=\left[p_{i},q_{i},r_{i}\right]^{T}- \xi_{i,1}\end{array} \]  
where \(q_{d,1}\), \(q_{d,2}\) and \(q_{d,3}\) are the signals produced by filter (6) with \(y_{d,i},i=1,2,3\) being the filter inputs, respectively. \(\lambda_{i,1}=1\) and \(\lambda_{i,2}=2\) for \(i=1,2,3,4\), and \(G_{2}^{+}\) represents the pseudo-inverse for \(G_{2}\). For the purposes of comparison, we use the control method of [17] under the same conditions. Following [17], the controller for the distributed flight controller is designed as follows  
\[ \begin{array}{l}\xi_{i,1}=G_{1}^{-1}(X_{i,1})diag\left\{-\lambda_{i,1}e_{i, \theta}-\lambda_{i,1}e_{i,\alpha}-\lambda_{i,1}e_{i,\beta}\right\}\\ u_{i}=G_{2}^{+}diag\left\{-\lambda_{i,2}e_{i,p}-\lambda_{i,2}e_{i,\theta}- \lambda_{i,2}e_{i,r}\right\}\end{array} \]  
where the variables and controller parameters are the same as in our proposed methods. The simulation results are then reported in Figures 6-10. In Figure 6, the dotted curves  
Figure 5: Communication topology for four fighters. denote the outputs of fighters under the control of the method in [17], while the solid curves denote the outputs of fighters under the control of method in this paper. It can be seen from Figure 6 that our control performance is better than [17] since the outputs of ours track the desired value more accurately. Figures 7-10 show the actions of actuators of four fighters under our method. Figure 11 show the controller performance of our method and that from [17]. In Figure 11, the blue curves denote the control efforts \(E_{1}\) of the fighters with our method, while the red curves denote the control efforts \(E_{2}\) of Fighters in the method from [17], where \(E_{1}\) and \(E_{2}\) are defined as  
\[ \begin{array}{l}E_{k}=\sqrt{\delta_{itel}^{2}+\delta_{ier}^{2}+\delta_{ial}^{ 2}+\delta_{iar}^{2}+\delta_{ilef}^{2}+\delta_{itef}^{2}+\delta_{ir}^{2}}\\ k=1,2\;and\;i=1,2,3,4\end{array}