## 4 AIMM Algorithm
### AIMM Algorithm  
In the IMM algorithm, model switching is a Markov process, so it is very dependent on the Markov matrix of the model transformation. In the standard IMM algorithm, the Markov matrix is given in advance. Due to the uncertainty of prior information such as the mobility of the target, it is generally determined based on experience. Therefore, for highly maneuvering targets such as hypersonic targets, the fixed Markov matrix cannot sufficiently reflect the real motion state of the target and the switching process between the models. Therefore, based on traditional IMM algorithm, we proposed to use the posterior information to adaptively modify the Markov matrix.  
The model likelihood function value \(\Lambda_{j}(k)\) contains the residual and residual covariance between the predicted state and the real state at time \(k\), so it can sufficiently reflect the matching degree between model \(j\) and the current motion state. The degree of model matching is proportional to the value of the likelihood function. Therefore, based on the model likelihood function, this paper proposes a transfer probability matrix adaptive algorithm based on the likelihood function. The calculation process is as follows:  
\[\varphi_{ij}(k)=\left(\frac{\Lambda_{j}(k)}{\Lambda_{i}(k)}\right)^{\gamma} \tag{26}\]  
\[P_{ij}(k)=P^{\prime}_{ij}(k)/\sum_{j=1}^{u}P^{\prime}_{ij}(k) \tag{27}\]  
\[P^{\prime}_{ij}(k)=\varphi_{ij}(k)P_{ij}(k-1) \tag{28}\]  
where \(\varphi_{ij}(k)\) is the ratio of the model likelihood function, which is used as the adjustment factor to adjust the model conversion rate, and \(\gamma\) is the correction rate, \(0\leq\gamma\leq 1\), which is used to control the speed of correction. It can be seen from Equation (24) that at time \(k\), when the likelihood function index \(\Lambda_{j}(k)\) is greater than \(\Lambda_{i}(k)\), model \(j\) has a higher matching degree with the real motion than model \(i\), and the probability from model \(j\) to model \(i\) increases after the correction. Therefore, the model with a larger likelihood function value accounts for a larger proportion in the next moment. On the contrary, the proportion of the model with a small likelihood function decreases in the next moment. Therefore, it can be concluded that this method enhances the proportion of matching models by using the measurement information to adaptively modify the transfer probability matrix.  
However, it can be found through substitution calculation that Formula (25) has defects. During iterative calculation, some elements of the main diagonal may become larger and larger, and the transition probability between models may be reduced. Therefore, when the model state needs to be switched, the efficiency will be reduced. Therefore, the thresholds \(M\) and \(N\) are set for the main diagonal element and the non-main diagonal element, respectively, and the following improvements are made.  
Figure 4: Computational steps of the interactive multi-model algorithm. Each filter is an independent model. The observation information is introduced, the target motion state is obtained by filtering, and finally the state fusion is carried out by weighting.  
If the main diagonal element is too large, i.e., \(P_{ii}(k)>M\), let \(P^{\prime}_{ii}(k)=M\), and redistribute the non-main diagonal elements of its row according to the original ratio:  
\[P^{\prime\prime}_{ij}(k)=P_{ij}(k)\left[1-\frac{M-P_{ii}(k)}{\sum\limits_{j=1}^{ m}P_{ii}(k)-P^{\prime\prime}_{ii}(k)}\right] \tag{29}\]  
Similarly, if the non-main diagonal element is too small, i.e., \(P_{ij}(k)<N(i\
eq j)\), let \(P^{\prime}_{ij}(k)=N(i\
eq j)\), and redistribute the other elements of its row according to the original proportion (refer to Formula (27)).  
The correction of the elements by Equation (27) still satisfies \(\sum\limits_{j=1}^{m}P_{ij}(k)=1\) and \(0\leq P_{ij}(k)\leq 1\), and can enhance the stability and adaptability of the model transformation.