## 3 Strategy for Timed Cooperative Guidance of Multiple Missiles
### Algorithm for Cooperative Guidance Considering Impact Time and Angle Constraints  
The framework idea of the distributed MPC is as follows: At each control moment \(t_{c}\), the current missile control variable is initialized with the optimal control amount of the previous prediction period \(\left[t_{c-1},t_{c-1}+T_{p}\right]\). After that, it exchanges the information with the adjacent missiles and receives the estimated value of the control variable at the moment \(t_{c-1}\). Subsequently, it sends out the estimated value of its own control variable at the moment \(t_{c-1}\) to estimate the state of the adjacent missile in the prediction period \(\left[t_{c},t_{c}+T_{p}\right]\) and evaluates the cost function \(\mathcal{J}_{i}\) in the prediction period \(\left[t_{c},t_{c}+T_{p}\right]\) and uses the particle swarm optimization (PSO) algorithm to solve the optimal control variable sequence in the prediction period \(\left[t_{c},t_{c}+T_{p}\right]\). Based on the rolling optimization idea, the optimal control amount of the first update interval is taken to obtain the optimal control sequence, and the missile state at \(\left[t_{c},t_{c+1}\right]\) is updated to enter the next control moment \(t_{c+1}\). The solution process of the estimated value is shown in Figure 5.  
The basic principle of the PSO-based distributed cooperative model prediction framework for multiple missiles is shown in Figure 6. The cost function of the cooperative  
Figure 5: Solution process of the predicted state value.  
control of missiles includes two parts, namely, the operation function, F, and the final state function, \(\Phi\):  
\[J_{i}=\int_{t_{c}}^{t_{c}+T_{p}}\mathrm{F}_{i}\Big{(}\mathrm{z}_{i}^{p}(s;t_{c}), \hat{z}_{-i}(s;t_{c}),u_{i}^{p}(s;t_{c})\Big{)}ds+\Phi_{i}\Big{(}\mathrm{z}_{i}^ {p}\big{(}t_{c}+T_{p};t_{c}\big{)}\Big{)} \tag{26}\]  
The operation function is defined as:  
\[\mathrm{F}_{i}\Big{(}\mathrm{z}_{i}^{p}(s;t_{c}),\hat{z}_{-i}(s;t_{c}),u_{i}^{p }(s;t_{c})\Big{)}=\underset{j\in N_{i}}{\sum}\left\|t_{go,i}^{p}(s;t_{c})-\hat{t }_{go,j}(s;t_{c})\right\|^{2}+(1-\alpha)\left\|u_{i}^{p}(s;t_{c})\right\|^{2} \tag{27}\]  
The final state function is defined as:  
\[\Phi_{i}\Big{(}\mathrm{z}_{i}^{p}\big{(}t_{c}+T_{p};t_{c}\big{)}\Big{)}=\beta \left\|r_{i}\big{(}t_{c}+T_{p};t_{c}\big{)}\right\|^{2}+\gamma\left\|\theta \big{(}t_{c}+T_{p};t_{c}\big{)}-\theta_{\mathrm{exp}\,ct}+\psi\big{(}t_{c}+T_{ p};t_{c}\big{)}-\psi_{\mathrm{exp}\,ct}\right\| \tag{28}\]  
where \(\alpha,\beta,\gamma\) are the weight constants, \(\left\|\cdots\right\|\) represents the modulus of the space vector, \(N_{i}\) is the set of missiles other than missile \(i\),\(r_{i}\) is the distance between the missile and the target, \(t_{go,i}^{p}\) is the predicted time of arrival, and \(\hat{t}_{go,i}\) is the estimated time of arrival, which can be solved by the following equation:  
\[\hat{t}_{go,j}(s;t_{c})=\frac{\hat{r}(s;t_{c})}{v_{j}},j\in N_{i} \tag{29}\]  
where \(\hat{r}(s;t_{c})\) is the estimated distance between the missile and the target at the moment \(t_{c}\), \(\theta_{\mathrm{expect}}\) and \(\psi_{\mathrm{expect}}\) represent the desired pitch angle and trajectory deflection angle that are used to control the angle of the missile at the terminal landing point to achieve the impact angle constraints.  
Figure 6: PSO-DMPC model prediction framework.