## 3 Strategy for Timed Cooperative Guidance of Multiple Missiles
### Introduction to Dmpc  
The application of the MPC method in a multi-agent cluster system has three structures: centralized, decentralized, and distributed MPC. As shown in Figure 4, the centralized MPC considers the overall performance of the cluster as an optimization index. In each sampling period, a central model predictor performs the rolling optimization on the overall control performance index of the cluster system to obtain the control sequence set with the best overall performance of the cluster system. Each sub-system of the decentralized and distributed MPC has an independent model predictor, which reduces the dimensionality of the system. It also reduces the amount of calculation for the online optimization of each sub-controller and increases the reliability of the system. The difference between the distributed and decentralized MPC is that there is no information interaction between the sub-controllers of the decentralized MPC. As a result, it cannot control the problem of the multi-agent system with coupling between the sub-systems. Figure 3: Control flow chart for cooperative guidance of multiple missiles. From the above-mentioned missile motion model, the motion equation of each missile can be expressed as:  
\[\dot{z}_{l}(t)=f_{l}(z_{i}(t),u_{i}(t)),t\geq t_{0},z_{i}(0)=z_{0} \tag{16}\]  
Therefore, the motion equations of its adjacent missiles can be defined as follows:  
\[\dot{z}_{-l}(t)=f_{-i}(z_{-i}(t),u_{-i}(t)),t\geq t_{0} \tag{17}\]  
The prediction time domain is defined as \(T_{p}\in(0,\infty),s\in\left[t_{c},t_{c}+T_{p}\right]\). The control moment of the receding horizon control is defined as \(t_{c}=t_{0}+\delta_{c},c\in\left\{0,1,2,\cdots\right\}\), where \(\delta\) is the control interval and satisfies \(\delta\in\left(0,T_{p}\right]\), \(\dot{z}_{i}(s;t_{c})\), \(\dot{u}_{i}(s;t_{c})\) is the estimated state sequence and control sequence, respectively, while \(z_{i}^{p}(s;t_{c})\) and \(u_{i}^{p}(s;t_{c})\) are the predicted state sequence and control sequence, respectively. For any time, \(t_{c}\), the optimal control sequence of the \(i\)th missile can be expressed as:  
\[u_{i}^{s}(s;t_{c})=\text{argmin}_{u_{i}^{s}(s;t_{c})}\mathcal{J}_{i}(z_{i}(s; t_{c}),z_{-i}(s;t_{c}),u_{i}(s;t_{c})) \tag{18}\]  
The cost function can be expressed as:  
\[\mathcal{J}_{i}=\int_{t_{c}}^{t_{c}+T_{p}}\text{F}_{i}\Big{(}z_{i}^{p}(s;t_{c }),\dot{z}_{-i}(s;t_{c}),u_{i}^{p}(s;t_{c})\Big{)}ds+\Phi_{i}\Big{(}z_{i}^{p} \big{(}t_{c}+T_{p};t_{c}\big{)}\Big{)} \tag{19}\]  
where \(\mathcal{J}_{i}\) is the operation function and \(\Phi\) is the final state function. The following constraints are satisfied:  
\[\dot{z}_{i}^{p}(s;t_{c})=f_{i}\Big{(}z_{i}^{p}(s;t_{c}),u_{i}^{p}(s;t_{c}) \Big{)} \tag{20}\]  
\[\dot{\underline{z}}_{-i}(s;t_{c})=f_{-i}(\underline{z}_{-i}(s;t_{c}),\dot{u} _{-i}(s;t_{c})) \tag{21}\]  
\[z_{i}^{p}(s;t_{c}),\dot{z}_{-i}(s;t_{c})\in Z \tag{22}\]  
\[u_{i}^{p}(s;t_{c}),\dot{u}_{-i}(s;t_{c})\in U \tag{23}\]  
Figure 4: Classification of MPC: (**a**) Centralized MPC; (**b**) Decentralized MPC; (**c**) Distributed MPC. According to the definition of the motion equations of the adjacent missiles, \(s\in\left[t_{c},t_{c}+T_{p}\right]\) in the prediction time domain and the estimated control sequence \(\hat{u}_{-i}(s;t_{c})\) can be obtained by:  
\[\dot{z}_{l}(t)=f_{l}(z_{i}(t),u_{i}(t)),t\geq t_{0},z_{i}(0)=z_{0} \tag{16}\]0  
In the prediction time domain \(s\in\left[t_{c},t_{c}+T_{p}\right]\), the estimated control sequence \(\hat{u}_{j}(s;t_{c})\) consists of two parts. In the time domain \(s\in\left[t_{c},t_{c-1}+T_{p}\right]\), the first part is the optimal control sequence \(u_{j}^{*}(s;t_{c})\) of the previous prediction time domain \(s\in\left[t_{c-1},t_{c-1}+T_{p}\right]\), while in the time domain \(s\in\left[t_{c-1}+T_{p},t_{c}+T_{p}\right]\), the second part is the optimal control sequence derived from \(u_{j}^{*}(s;t_{c-1})\) at the time \(s=t_{c-1}+T_{p}\). The expression of \(\hat{u}_{j}(s;t_{c})\) is as follows:  
\[\hat{u}_{j}(s;t_{c})=