## 2 HGV Tracking with Modified Aerodynamic Model
### Modified Aerodynamic Model Tracking Algorithm  
It is known by us that the existing radar equipment cannot directly measure the attack angle and bank angle of HGV. Furthermore, the change law of the two control variables is unknown by defenders as an important secret of a strike weapon. Thus, we need to find an effective method for modeling the change law of the two control variables. Because HGV tracking is still in primary stage, there is not an existing method for modeling the change law. To develop the MAM algorithm, we analyze the characters of HGV in near space first. HGV maneuvers by aerodynamic force with a Mach number bigger than 5 in near space and HGV can bear damage to the vehicle structure when bank angle and attack angle change strongly. Furthermore, in the practical control of HGV, the control variables usually change continuously in a certain degree for control stability. Thus, we think the rates of change of control variables in a short time are not very large and unknown by defenders. Then, the two control variables can be modeled as constant Gaussian Markov process and we can use additive white noise to model the rates of change of attack angle and bank angle [27] as  
\[\left\{ \begin{array}{c}\frac{dx}{dt}=w_{t,x}\approx 0\\ \frac{dv}{dt}=w_{t,\
u}\approx 0\end{array} \right. \tag{15}\]  
where \(w_{t,x}\) and \(w_{t,\
u}\) are white noise with an certain effect on attack angle and bank angle that accounts for the unknown control input. Thus, the complete process equation can be established. The augmented state variables at discrete time \(k\) can be defined as  
\[X_{k}=\left[ \begin{array}{cccccccc}V_{k}&\theta_{k}&\sigma_{k}&r_{k}&\lambda _{k}&\phi_{k}&\alpha_{k}&\
u_{k}\\ \end{array} \right] \tag{16}\]The basic process equations can be written by making Equation (3) discretization as  
\[\left\{ \begin{array}{l}V_{k+1}=V_{k}+\left(-D-g\text{sin}\theta_{k}\right)T+Tw_{ V,k}\\ \theta_{k+1}=\theta_{k}+\left[\frac{L\text{cos}\
u_{k}}{V_{k}}+\left(\frac{V_{k} }{V_{k}}-\frac{g}{V_{k}}\right)\text{cos}\theta_{k}+2\omega_{e}\text{cos} \phi_{k}\text{sin}\sigma_{k}\right]T+Tw_{\Theta,k}\\ \sigma_{k+1}=\sigma_{k}+\left[\frac{L\text{sin}\
u_{k}}{V_{k}\text{cos}\theta_ {k}}+\frac{\
u_{k}\text{cos}\theta_{k}\text{sin}\sigma_{k}}{r_{k}}-2\omega_{e} \left(\text{cos}\phi_{k}\text{tan}\theta_{k}\text{cos}\sigma_{k}-\text{sin} \phi_{k}\right)\right]T+Tw_{\sigma,k}\\ \phi_{k+1}=\phi_{k}+\frac{V_{k}\text{cos}\theta_{k}\text{cos}\lambda}{r_{k}}T +Tw_{\phi,k}\\ \lambda_{k+1}=\lambda_{k}+\frac{V_{k}\text{cos}\theta_{k}\text{sin}\sigma_{k }}{r_{k}\text{cos}\theta_{k}}T+Tw_{\lambda,k}\\ r_{k+1}=r_{k}+V_{k}\text{sin}\theta_{k}T+Tw_{r,k}\end{array} \right. \tag{17}\]  
Furthermore, the augmented equations about the control variables can be written by make Equation (15) discretization as  
\[\left\{ \begin{array}{l}\alpha_{k+1}=\alpha_{k}+Tw_{k,\alpha}\\ \
u_{k+1}=\
u_{k}+Tw_{k,\
u}\end{array} \right. \tag{18}\]  
where \(T\) is the time interval. \(w_{V,k}\), \(w_{\Theta,k}\), \(w_{\varphi,k}\), \(w_{\vartheta,k}\), \(w_{\lambda,k}\), \(w_{r,k}\), \(w_{\alpha,k}\) and \(w_{\
u,k}\) are the components of \(w_{k}\). We consider that the changes of the attack angle and the bank angle at the discrete time \(k\) and \(\left(k{-}1\right)\) are caused by the time interval \(T\) multiplied by white noise \(w_{k-1,\alpha}\) and \(w_{k-1,\
u}\), respectively. The method to model the rates of change of the two control variables is similar to the method in CV model to model the rate of change of velocity [28]. In CV model the ideal equation is modified as  
\[\dot{x}\left(t\right)=w\left(t\right)\approx 0 \tag{19}\]  
where the white noise \(w\left(t\right)\) has a \"small\" effect on \(x\) that accounts for unpredictable modeling errors due to turbulence, etc. After getting the process and measurement equations, according to CKF, we can get the predicated state \(\widehat{X}_{k}\) at discrete time \(k\) using the state \(\widehat{X}_{k-1}\) at discrete time \(k{-}1\) and the new measurement \(Y_{k}\).