## 3 Strategy for Timed Cooperative Guidance of Multiple Missiles
### Cooperative Strategy of Follower Missiles  
The remaining time error between the leader missile and follower missiles is defined as follows:  
\[\xi=\frac{r_{0}}{V_{m0}}-\frac{r_{i}}{V_{mi}} \tag{6}\]  
By deriving the remaining time error with respect to time, we can get:  
\[\dot{\xi}=\cos\theta_{i}\cos\psi_{i}-\cos\theta_{0}\cos\psi_{0} \tag{7}\]  
According to Equation (7), \(A_{yi}\) indirectly controls \(\xi\) by controlling \(\psi_{i}\). Therefore, the control system can be divided into two sub-control systems according to direct control \(\psi_{i}\) and indirect control \(\xi\), namely, a nonlinear slow sub-system and a nonlinear fast sub-system. Therefore, the dynamic design method of time-scale separation is used to design the control instructions \(A_{yi}\). The desired dynamics of the slow sub-system is expressed as:  
\[\dot{\xi}_{des}=-K_{R}\xi \tag{8}\]  
where \(K_{R}\) is the bandwidth of the slow sub-system.  
If \(\dot{\xi}=\dot{\xi}_{des}\) is obtained through the control, considering the command value of \(\psi_{i}\) is \(\psi_{i}{}^{c}\), then:  
\[\cos\psi_{i}{}^{c}=\frac{\cos\theta_{0}\cos\psi_{0}-K_{R}\xi}{\cos\theta_{i}} \tag{9}\]  
Because the value range of \(\psi_{i}\) is \((-\pi/2,\pi/2)\), \(\psi_{i}{}^{c}\) is within the value range. Combining the above equation and the value range, the value range of KR can be obtained:  
\[\frac{\cos\theta_{0}\cos\psi_{0}-\cos\theta_{i}}{\xi}\leq K_{R}\leq\frac{\cos \theta_{0}\cos\psi_{0}}{\xi} \tag{10}\]  
if  
\[K_{R}=\frac{\cos\theta_{0}\cos\psi_{0}-\cos\theta_{i}+c_{1}\frac{\xi}{\xi+c_{2 }}\cos\theta_{i}}{\xi} \tag{11}\]  
where \(c_{1}\), \(c_{2}\) are constants and satisfy the relationship \(0<c_{1}\), \(c_{2}<1\).  
Herein, \(\psi_{i}{}^{c}\) can be expressed as:  
\[\psi_{i}^{c}=f_{\psi}\times\arccos\left(\frac{\cos\theta_{0}\cos\psi_{0}-K_{R }\xi}{\cos\theta_{i}}\right) \tag{12}\]  
where \(\psi_{i}\in[-\pi/2,0)\), \(f_{\psi}=-1\); \(\psi_{i}\in[0,\pi/2]\), \(f_{\psi}=1\).  
Taking the derivative of Equation (12) with respect to time, we have:  
\[\dot{\psi}_{i}^{c}=f_{\psi}\times\left(\begin{array}{c}K_{R}\cos\psi_{i}- \frac{\cos\psi_{i}^{c}\sin\theta_{i}}{\cos\theta_{i}}-\\ \frac{K_{R}\cos\theta_{0}\cos\psi_{0}-\theta_{0}\sin\theta_{0}\cos\psi_{0}- \vartheta_{0}\cos\theta_{0}\sin\psi_{0}}{\cos\theta_{i}\sqrt{1-(\cos\psi_{i}^{ c})^{2}}}\end{array}\right) \tag{13}\]  
If \(\psi_{i}\in[-\pi/2,0)\), \(f_{\psi}=-1\); \(\psi_{i}\in[0,\pi/2]\), \(f_{\psi}=1\).  
In order to make \(\zeta\) satisfy the relation in Equation (8), \(\psi_{i}\) should quickly converge to near its command value \(\psi_{i}^{c}\). Therefore, the following fast dynamics sub-system is designed, and its specific expression is given as:  
\[\dot{\psi}_{i,des}-\dot{\psi}_{i}^{c}=-K_{\psi}(\psi_{i}-\psi_{i}^{c}) \tag{14}\]  
where \(K_{\psi}\) is the bandwidth of the fast sub-system. Under the control \(\dot{\psi}_{i}=\dot{\psi}_{i,des}\) united with Equation (4), the final designed maneuvering control command in the turning plane can be obtained as:  
\[A_{yi}=\frac{{V_{m}}^{2}}{r_{i}}\sin\psi_{i}(1-\cos\theta_{i}\sin\theta_{i} \cos\psi_{i}\tan\theta_{Li})-V_{mi}\cos\theta_{i}\dot{\psi}_{i}^{c}+K_{\psi}V_ {mi}\cos\theta_{i}(\psi_{i}-\psi_{i}^{c}) \tag{15}\]  
Equation (14) can be satisfied by maneuvering control commands on the lateral plane, so that \(\psi_{i}\rightarrow\psi_{i}^{c}\) is obtained; \(\psi_{i}^{c}\) can make the slow dynamics sub-system meet Equation (8). Under this condition, the time cooperative control of multiple missiles can be realized. Longitudinal control commands \(A_{z0}\) and \(A_{zi}\) can make \(\theta_{0}\to 0\) and \(\theta_{i}\to 0\), along with the control of the yaw channel, \(\xi\to 0\), so as to realize \(\psi_{i}\rightarrow\psi_{0}\) (or \(\psi_{i}\rightarrow-\psi_{0}\)). The control flow chart for cooperative guidance of multiple missiles can be seen form Figure 3.  
Cooperative Prediction Guidance Based on Distributed Model Prediction Control (DMPC) for Multiple Missiles