## 2 Description of the IMM-UKF Filtering Problem
### Non-Gaussian Distribution and Sensor Fault Problems for IMM-UKF  
#### 2.3.1 Non-Gaussian Distribution  
For the external IMM-UKF links, \(\hat{\mathbf{X}}_{j,k}^{o}\) is a weighted sum of \(m\) Gaussian random variables and no longer obeys the pure Gaussian distribution because many peaks will exist in the probability density, which is similar to the probability density of the mixed Gaussian distribution [19]. The case when \(\hat{\mathbf{X}}_{i,k-1}\) is scalar and \(u_{i,k-1}\) is constant is explained in this section. It is assumed that \(\hat{\mathbf{X}}_{j,k-1}^{o}\) is the weighted sum of three Gaussian random variables with normal distribution, and \(S_{j,k-1}^{o}\) is the square root of \(\mathbf{P}_{j,k-1}^{o}\), where \([\hat{X}_{1},\hat{X}_{2},\hat{X}_{3}]=[-3,0,5]\), \([S_{1},S_{2},S_{3}]=[3,2.25,1.8]\), \([u_{1},u_{2},u_{3}]=[0.3,0.4,0.3]\). The probability density of the mixed Gaussian distribution is shown in Figure 2.  
In Figure 2, the thick line represents \(\hat{\mathbf{X}}_{j,k-1}^{o}\) and has multiple peaks; the mixture no longer follows Gaussian distribution.  
After the interaction, [PERSON] completes the approximation of the probability density through the Gaussian distribution [19]:  
\[\Pr\left(\hat{\mathbf{X}}_{j,k-1}^{o}\right)=\sum\limits_{i=1}^{m}u_{i,k-1}N\left\{ \hat{\mathbf{X}}:\hat{\mathbf{X}}_{i,k-1},\ S_{i,k-1}(S_{i,k-1})^{T}\right\} \tag{10}\]  
where \(S_{i,k-1}\) is the square root of \(\mathbf{P}_{i,k-1}\).  
On the basis of the above distribution, MSE estimates the covariance matrix of the interacting random variables, where the mean and covariance matrix are computed by [16]  
\[\begin{array}{ll}\overline{\hat{\mathbf{X}}}_{j,k-1}^{o}&=\int\hat{\mathbf{X}}_{j,k- 1}^{o}\Pr\left(\hat{\mathbf{X}}_{j,k-1}^{o}\right)d\hat{\mathbf{X}}_{j,k-1}^{o}=\sum \limits_{i=1}^{m}u_{i,k-1}\int\hat{\mathbf{X}}_{j,k-1}^{o}N\left\{\hat{\mathbf{X}}:\hat {\mathbf{X}}_{i,k-1},\ S_{i,k-1}(S_{i,k-1})^{\mathrm{T}}\right\}d\hat{\mathbf{X}}_{j,k -1}^{o}\\ &=\sum\limits_{i=1}^{m}u_{i,k-1}\hat{\mathbf{X}}_{i,k-1}\end{array} \tag{11}\]  
Figure 2: The probability density of the mixed Gaussian distribution.  
\[\mathbf{P}_{j,k-1}^{MSE}=\sum_{i=1}^{m}u_{i,k-1}\bigg{(}\mathbf{S}_{i,k-1}\mathbf{S}_{i,k-1}^{T }+(\hat{\mathbf{X}}_{i,k-1}-\overline{\mathbf{X}}_{j,k-1}^{o})(\hat{\mathbf{X}}_{i,k-1}- \overline{\mathbf{X}}_{j,k-1}^{o})^{T}\bigg{)}. \tag{12}\]  
From Equations (11) and (12), we know \(\mathbf{P}_{j,k-1}^{MSE}\) is larger than the actual covariance of \(\hat{\mathbf{X}}_{j,k-1}^{o}\) (\(\mathbf{P}_{j,k-1}^{actual}\)). An example of the above one-dimensional case is available, and the Monte Carlo method (MC) is used with 10,000 random sampling points to compute the mean and covariance transferred. The results are shown in Table 1.  
From Table 1, \(\mathbf{P}_{j,k-1}^{MSE}>\mathbf{P}_{j,k-1}^{actual}\) is verified. Because \(\mathbf{P}_{j,k-1}^{MSE}\) will participate in the filtering iteration, it is equivalent to an increase of the system noise in the process of filtering. Then, the gain matrix will be larger, and the filtering effect declines, especially when the actual system noise is small and the measurement noise is large.  
Therefore, for the external IMM-UKF link, the filtering problem to be solved is how to make a precise estimate of the mean and covariance matrix of the random variables after the interaction for IMM.