# Non-Gaussian Pseudolinear Kalman Filtering-Based Target Motion Analysis with State Constraints
## 2 PL-MMSE Kalman Filter Under Gaussian Mixture Noise  
In the bearing-only two-dimensional (2D) plane TMA, the target-sensor model is established, as shown in Figure 1.  
As shown in Figure 1, the moving target position and velocity are \(T_{k}=\left[t_{x,k}\ t_{y,k}\right]^{T}\) and \(V_{k}=\left[v_{x,k}\ v_{y,k}\right]^{T}\), respectively, where  
\[t_{x,k} =t_{x,k-1}+v_{x,k-1}T, \tag{1}\] \[t_{y,k} =t_{y,k-1}+v_{y,k-1}T. \tag{2}\]  
\(T\) is the sampling interval. The sensor locates at \(S_{k}=\left[s_{x,k}\ s_{y,k}\right]^{T}\). The real angle information received from the sensor is given by  
\[\varphi_{k}=\tan^{-1}\left(\frac{t_{y,k}-s_{y,k}}{t_{x,k}-s_{x,k}}\right). \tag{3}\]  
The bearing measurement is  
\[\hat{\varphi}_{k}=\varphi_{k}+e_{k}, \tag{4}\]  
Figure 1: Schematic diagram of 2D bearing-only TMA.  
which indicates that the sensor measurement is corrupted by the mixed Gaussian noise \(e_{k}\) with zero mean at time \(kT\)\((k=1,2,3,...,n)\). The Gaussian mixture noise  
\[e_{k}\sim\sum_{j=1}^{n}\lambda_{j}\mathcal{N}(0,\sigma_{j}^{2}) \tag{5}\]  
is composed of \(n\) independent distributed Gaussian noises with zero mean and variance \(\sigma_{j}^{2}\), respectively, with  
\[\lambda_{j}>0\;\;\text{and}\;\;\sum_{j=1}^{n}\lambda_{j}=1. \tag{6}\]  
The moving target state vector \(X_{k}\) is given by  
\[X_{k}=\Big{[}t_{x,k}\;t_{y,k}\;v_{x,k}\;v_{y,k}\Big{]}^{T}. \tag{7}\]  
According to (3) and (4), the pseudolinear measurement equation between the target and the sensor is  
\[\sin\phi_{k}t_{x,k}-\cos\phi_{k}t_{y,k}= \sin\phi_{k}s_{x,k}-\cos\phi_{k}s_{y,k}+\|T_{k}-S_{k}\|\sin e_{k}. \tag{8}\]  
The pseudolinear state-space model for bearings-only TMA is  
\[X_{k} = FX_{k-1}+\omega_{k-1}, \tag{9}\] \[Z_{k} = H_{k}X_{k}+\tau_{k}, \tag{10}\]  
where \(X_{k}\) and \(X_{k-1}\) are the motion states at time \(kT\) and \((k-1)T\), respectively. It is assumed that \(\omega_{k-1}\) is the Gaussian mixture noise composed of n independent distributed Gaussian noises with zero mean and variance \(\xi_{i}^{2}\), respectively. According to the motion model (8), the state transition matrix F and the process noise Jacobian D are  
\[F = \begin{bmatrix}1&0&T&0\\ 0&1&0&T\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}, \tag{11}\] \[D = \begin{bmatrix}0&0&T^{2}/2&0\\ 0&0&0&T^{2}/2\\ 0&0&T&0\\ 0&0&0&T\end{bmatrix}. \tag{12}\]  
Ref. [39]. In the PLKF algorithm, the estimated state of the target at time \(kT\) is  
\[\hat{X}_{k|k-1}=F\hat{X}_{k-1}. \tag{13}\]  
The updated covariance matrix is  
\[P_{k|k-1}=FP_{k-1|k-1}F^{T}+DQ_{k-1}D^{T} \tag{14}\]  
where the Gaussian mixture noise variance \(Q_{k-1}\) is  
\[Q_{k-1}=\sum_{i=1}^{n}\rho_{i}\xi_{i}^{2} \tag{15}\]  
with  
\[\rho_{j}>0\;\;\text{and}\;\;\sum_{i=1}^{n}\rho_{i}=1. \tag{16}\]The observation matrix is  
\[H_{k}=[\sin\hat{\varphi}_{i}\ -\cos\hat{\varphi}_{i}\ 0\ 0]^{T}. \tag{17}\]  
The Kalman gain is  
\[K_{k}=P_{k|k-1}H_{k}^{T}\Big{[}H_{k}P_{k|k-1}H_{k}^{T}+R_{k}\Big{]}^{-1} \tag{18}\]  
where the pseudolinear noise variance \(R_{k}\) is given by  
\[R_{k}=\|T_{k}-S_{k}\|\left(\sum_{i=1}^{n}\lambda_{i}e^{\frac{1-e^{-\frac{2\pi ^{2}}{2}}}{2}}\right). \tag{19}\]  
The updated pseudolinear measurement is  
\[\hat{Z}_{k|k-1}=H_{k}\hat{X}_{k|k-1}. \tag{20}\]  
Therefore, the target state and covariance update equation at time \(kT\) can be described by  
\[\hat{X}_{k|k} =\hat{X}_{k|k-1}+K_{k}(Z_{k}-\hat{Z}_{k|k-1}), \tag{21}\] \[P_{k|k} =P_{k|k-1}-K_{k}H_{k}P_{k|k-1}. \tag{22}\]  
Based on the PLKF, the PL-MMSE for Gaussian mixture noise can be rewritten as shown in Table 1 where the pseudolinear observation matrix is given by  
\[H_{k}^{1}=[\tan^{-1}\left(\hat{X}_{k|k-1}(2)-S_{k}(2)\right)\ \tan^{-1}\left(\hat{X}_{k|k-1}(1)-S_{k}(1)\right)\ 0\ 0]. \tag{23}\]