## 3 Problem Formulation  
In this paper, the combat scenario of multiple 2D vehicles intercepting a maneuver target is studied. The goal is to simultaneously intercept the target by deploying multiple interceptors. Consider a group of N intercepting vehicles: \(M_{i}\), \(i\in\{1,2,3,\ldots,N\}\) denote the interceptors and \(T\) denotes the target. The planar one-to-one engagement between the \(i^{th}\) interceptor and a target may be depicted as in Figure 1. Both the interceptors and the target are assumed to be point masses, and their velocity vectors are denoted by \(v_{i}\) and \(v_{T}\), respectively. Moreover, the corresponding speeds of them (\(v_{i}\) and \(v_{T}\), respectively) are constants. Their LOS angles are \(q_{i}\) and \(q_{T}\), respectively. The flight path angles are \(\theta_{i}\) and \(\theta_{T}\), respectively. The term \(r_{i}\) denotes the distance between the vehicle (\(M_{i}\)) and the target (\(T\)). The relative separation between the target and the \(i^{th}\) interceptor and the LOS angle are denoted by \(r_{i}\) and \(\theta_{i}\), respectively:  
In an interception scenario, according to the principles of kinematics, the relative-motion equation can be deduced as:  
\[\dot{r}_{i} =v_{T}\cos(q_{i}-\theta_{T})-v_{i}\cos(q_{i}-\theta_{i}) \tag{9}\] \[r_{i}\dot{q}_{i} =-v_{T}\sin(q_{i}-\theta_{T})+v_{i}\sin(q_{i}-\theta_{i})\] (10) \[\dot{\theta}_{i} =\frac{a_{i}}{v_{i}}\] (11) \[\dot{\theta}_{T} =\frac{a_{T}}{v_{T}} \tag{12}\]  
To guarantee a salvo attack of the target by multiple interceptors, it is desirable that the interceptors achieve agreement on the time-to-go. Furthermore, this paper provides an encirclement-guidance strategy by utilizing a leader-follower topology. Considering N interceptors containing one leader and \(N-1\) followers, the vehicles can exchange information with communication networks, expressed by the weighted adjacency matrix: \(\mathbf{A}=\left[a_{ij}\right]\in\mathbb{R}_{n\times n}\). The guidance geometry is illustrated in Figure 2. Let \(M_{1}\) denote the leader, and \(M_{i}\), \(i\in\{2,3,\ldots,N\}\) denote the followers. The directed communication link is described by the blue arrows:  
Figure 1: Planar one-to-one engagement. Figure 2: Encirclement salvo attack engagement. **Note 1. ** As is shown in Figure 2, the leader and followers play different roles in the communication topology. The leader can send guidance information to adjacent followers. Each follower can work as an information transfer station. The mathematical models for the leader and followers are different, and the guidance laws for the leader and followers should be designed accordingly. The derivatives of (9) and (10) can be obtained as:  
\[\tilde{r}_{i}=r_{i}\dot{q}_{i}^{2}+u_{ri}-w_{ri} \tag{13}\]  
\[\ddot{q}_{i}=-2\frac{\dot{r}_{i}}{r_{i}}\dot{q}_{i}-\frac{u_{qi}}{r_{i}}+\frac {w_{qi}}{r_{i}} \tag{14}\]  
The motion equations in the LOS direction and its normal are given by (13) and (14), where \(u_{ri}\) and \(w_{ri}\) denote the components of the acceleration of the interceptor and the target in the LOS direction, respectively. \(u_{qi}\) and \(w_{qi}\) denote the components of the acceleration of the interceptor and the target in the normal direction of the LOS, respectively. In order for multiple vehicles to simultaneously attack the target at preset angles, the following nonlinear state equation is now established. Define the state variables as \(x_{1i}=r_{i}\), \(x_{1i}=r_{i}\), \(x_{2i}=\dot{r}_{i}\), \(x_{3i}=q_{i}-q_{di}\), \(x_{4i}=\dot{q}_{i}-\dot{q}_{di}\), where \(q_{di}\) denotes the desired terminal LOS angle. When \(i=1\), the interceptor is the leader, and \(q_{di}\) is a predefined constant. There is no need to design control command in the LOS direction. The cooperative-guidance model can be described as:  
\[\dot{r}_{i} =v_{T}\cos(q_{i}-\theta_{T})-v_{i}\cos(q_{i}-\theta_{i}) \tag{9}\]0  
\[\dot{r}_{i} =v_{T}\cos(q_{i}-\theta_{T})-v_{i}\cos(q_{i}-\theta_{i}) \tag{9}\]1  
When \(i\
eq 1\), the interceptor is the follower.