## 4 AIMM Algorithm
### SVD Algorithm  
The selection of the IMM model set needs to meet the highest matching degree between the model and the motion state at any time. The existing models include the CV model, CA model, constant turning (CT) model, Singer model, CS model, and Jerk model, which are applicable to different scenarios. Among them, the CV and CA models are applicable to the non-maneuvering or weak maneuvering movement state of the target, CT models are applicable to the turning movement of the target, the Singer and CS models can describe the normal movement, and the Jerk model, as the model with the highest order, can describe the acceleration of the target, which is mainly used to track strong maneuvering targets. Literature [25] proposed the single model to target tracking, but the hypersonic target has high maneuverability, which is difficult to tracking. Therefore, this paper selected three models with high maneuverability, namely, the Singer model, CS model, and Jerk model, to build the IMM model set. The state equation of the target motion model can be expressed as  
\[\mathbf{x}_{k}=\mathbf{F}_{1}\mathbf{x}_{k-1}+\mathbf{w}_{k-1} \tag{30}\]  
where \(\mathbf{x}_{k-1}\) is the motion state of the target at the previous time, \(\mathbf{x}_{k}\) is the motion state of the current target, and \(\mathbf{F}_{1}\) is the state transition matrix. Singer model: Singer model is time related, that is, the target acceleration can be set as a first-order time correlation function with the mean value of 0, and its corresponding function can be expressed as:  
\[R_{a}(\tau)=E[a(t)a(t+\tau)]=\sigma_{a}^{\ 2}e^{-a|\tau|}\quad(\alpha\geq 0) \tag{31}\]  
where \(a(t)\) is the target acceleration, \(\sigma_{a}^{\ 2}\) is the variance, \(\alpha\) is the maneuvering frequency. The probability density function can be used to solve \(\sigma_{a}^{\ 2}\):  
\[\sigma_{a}^{\ 2}=\int_{-a_{\max}}^{a_{\max}}P(a)a^{2}da=\frac{a_{\max}^{\ 2}}{3}(1+4P_{\max}-P_{0}) \tag{32}\]  
The state transition matrix and state noise covariance matrix in one dimension are as follows:  
\[\mathbf{F}_{1}= \begin{bmatrix}1&T&(\alpha T-1+e^{-\alpha T})/\alpha^{2}\\ 0&1&(1-e^{-\alpha T})/\alpha\\ 0&0&e^{-\alpha T}\end{bmatrix}  \tag{33}\]  
In the formula, \(\alpha\) refers to the maneuvering frequency, which is usually selected according to experience: generally, 1, 1/20, and 1/60 are taken according to the maneuverability of the target. The state noise covariance matrix can be found in [26]. CS model: The state transition matrix and state noise covariance matrix are the same as in the Singer model. Compared with the Singer model, the CS model has a control input matrix, and its discrete state equation can be expressed as:  
\[\mathbf{x}_{k}=\mathbf{F}_{1}\mathbf{x}_{k-1}+\mathbf{U}_{k}\mathbf{\overline{a}}+\mathbf{w}_{k-1} \tag{34}\]  
where \(\mathbf{\overline{a}}\) is the average value of acceleration, and the one-step prediction of the acceleration state can be taken as the average value of the current acceleration, i.e., \(\overline{a}_{x}=\tilde{x_{k}}\) ; according to the average value of the \"current\" acceleration, the variance of the current acceleration can be obtained:  
\[\sigma_{a}^{2}= \begin{cases}(4-\pi)/\pi\cdot(a_{\text{max}}-\overline{a})^{2},&\overline{a}>0\\ (4-\pi)/\pi\cdot(\overline{a}-a_{\text{min}})^{2},&\overline{a}<0\end{cases}  \tag{35}\]  
where \(a_{\text{max}}\) and \(a_{\text{min}}\) are the maximum and minimum accelerations set for each axis, which are generally set according to prior conditions. Jerk model: Based on the Singer model, zero-mean colored noise is introduced to express the acceleration. The exponential autocorrelation function of target acceleration \(j(t)\) is:  
\[r_{j}(\tau)=E[j(t)j(t+\tau)]=\sigma_{j}^{2}e^{-a|\tau|}\quad(\alpha\geq 0) \tag{36}\]  
where, \(\sigma_{j}^{2}\) is the variance of acceleration of maneuvering target, and \(\alpha_{\text{m}}\) is the maneuvering frequency.