# Research on Head Pursuit Interception Strategy for Hypersonic Target
## 3 Design of Head Pursuit Interception Guidance Strategy
### Design of Midcourse Guidance Law  
The purpose of midcourse guidance is to guide the interceptor to the distance that the terminal guidance seeker can intercept the target and to make the seeker complete the interception of angle and distance. An excellent midcourse guidance law should make the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\Delta X\)/km & **200** & **250** & **300** & **350** & **400** \\ \hline \(X_{TP\text{min}}\)/km & 31.28 & 32.77 & 33.67 & 34.26 & 34.25 \\ \(X_{TP\text{max}}\)/km & 29.5 & 30.28 & 30.56 & 30.34 & 29.62 \\ \(X_{TPopt}\)/km & 29.3 & 29.3 & 29.3 & 29.3 & 29.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of \(X_{TP}\). interceptor work in the best state of its overload performance throughout the midcourse guidance process and make the interceptor lock the target, that is, at the beginning of the terminal guidance phase, the geometric relationship between the interceptor and the target reaches the optimal state. The guidance method based on impact point prediction estimates the collision position between the interceptor and the target, changes the real target flying at high speed into a false target moving at low speed, fully reduces the demand overload of the interception trajectory, and improves the hit accuracy. It is always used to intercept high-speed targets by head-on interception. The method of head pursuit interception in this paper learns from the idea of the traditional guidance method of impact point prediction, takes this point as the fixed point, and designs the midcourse guidance strategy based on optimal control theory. (1) Longitudinal channel  
The longitudinal channel guidance strategy must guide the interceptor to the same flight altitude as the target and maintain this altitude until it reaches the predicted impact point. Based on the sliding mode control theory, a longitudinal channel midcourse guidance law is designed. The difference between the interceptor altitude and target altitude is assumed to be \(e\), and the altitude command signal is \(H_{c}\). \[e=H_{c}-Y_{M} \tag{19}\]  
The derivative is as follows. \[\dot{e}=\dot{H_{c}}-\dot{Y}_{M} \tag{20}\]  
The sliding mode surface is designed as follows. \[S_{H\_mid}=ce+\dot{e} \tag{21}\]  
where \(c\) is a sliding mode guidance parameter, which is a constant greater than 0. The reaching law is designed as follows. \[\dot{S}_{H\_mid}=-k_{3}\text{sgn}(S_{H\_mid})-\varepsilon S_{H\_mid} \tag{22}\]  
where \(k_{3}\) and \(\varepsilon\) are sliding mode guidance parameters, which are constants and typically range from 0 to 1. According to the Lyapunov stability criterion, the function is defined:  
\[V=\frac{1}{2}S_{H\_mid}^{2} \tag{23}\]  
Then,  
\[ \begin{split}\dot{V}&=S_{H\_mid}\dot{S}_{H\_mid}\\ &=S_{H\_mid}(-k_{3}\text{sgn}(S_{H\_mid})-\varepsilon S_{H\_mid })\\ &=-k_{3}|S_{H\_mid}|-\varepsilon S_{H\_mid}^{2}<0\end{split}  \tag{24}\]  
According to Equation (24), the designed sliding mode surface meets the Lyapunov stability theorem, and the feasibility of the designed guidance law is verified theoretically. The following equation can be obtained by substituting the derivative of the surface into Equation (22):  
\[\dot{c}\dot{e}+\left(\dot{H}_{c}-\dot{Y}_{M}\right)=-\text{sgn}(S_{H\_mid})-k_ {3}S_{H\_mid} \tag{25}\]  
Then,  
\[\ddot{Y}_{M}=\dot{c}\dot{e}+\dot{H}_{c}+\epsilon\text{sgn}(S_{H\_mid})+k_{3}S_ {H\_mid} \tag{26}\]From Equation (4),  
\[\dot{\varphi}_{M}=\frac{(N_{MY}-\cos\varphi_{M})g}{V_{M}} \tag{27}\]  
\[\dot{Y}_{M}=\dot{V}_{M}\sin\varphi_{M}+\dot{\varphi}_{M}V_{M}\cos\varphi_{M} \tag{28}\]  
The longitudinal channel midcourse guidance law can be obtained. \[N_{MY}=\frac{c\dot{e}+\bar{H}_{c}+\epsilon\text{sgn}(S_{H\_mid})+k_{3}S_{H\_mid }+g\text{cos}^{2}\varphi_{M}-\dot{V}_{M}\sin\varphi_{M}}{g\cos\varphi_{M}} \tag{29}\]  
(2) Lateral channel  
According to Equation (10), the following is obtained.