## 2 Methodologies
### Maximum Correntropy Extended Kalman Filter  
Next, we use the MCC to develop two unique EKFs since correntropy greatly improves the accuracy of the filters for situations with non-Gaussian noise. As we have the approximate measurement equation as (6), then  
\[\mathbf{y}_{k}\approx h(\widetilde{\mathbf{x}}_{k})+\mathbf{H}_{k}(\mathbf{x }_{k}-\widetilde{\mathbf{x}}_{k})+\boldsymbol{\varphi}_{k} \tag{13}\]  
By combining Equations (4) and (5) with (7) and (8), we obtain  
\[ \begin{bmatrix}\widetilde{\mathbf{x}}_{k}\\ \mathbf{y}_{k}-h(\widetilde{\mathbf{x}}_{k})+\mathbf{H}_{k}\widetilde{\mathbf{ x}}_{k}\end{bmatrix} = \begin{bmatrix}\mathbf{I}\\ \mathbf{H}_{k}\end{bmatrix} \mathbf{x}_{k}+\boldsymbol{\varphi}_{k} \tag{14}\]  
Then the nonlinear form of covariance matrix \(\boldsymbol{\varphi}_{k}\) can be written as  
\[ \begin{bmatrix}\widetilde{\mathbf{x}}_{k}\\ \mathbf{y}_{k}\end{bmatrix} = \begin{bmatrix}\mathbf{x}_{k}\\ h(\mathbf{x}_{k})\end{bmatrix} +\boldsymbol{\varphi}_{k}, \tag{15}\]where \(\mathbf{\varphi}_{k}= \begin{bmatrix}-(\mathbf{x}_{k}-\widetilde{\mathbf{x}}_{k})\\ \mathbf{r}_{k}\end{bmatrix} \). The covariance matrix \(\mathbf{\varphi}_{k}\) is composed of \(\widetilde{\mathbf{P}}_{k}\) and \(\mathbf{R}_{k}\) matrices, and its covariance matrix \(\mathbf{B}_{k}\) can be calculated from the Cholesky decomposition method. \[ \begin{split}& E\big{[}\mathbf{\varphi}_{k}\mathbf{\varphi}_{k}^{T} \big{]}=\begin{bmatrix}\widetilde{\mathbf{P}}_{k}&0\\ 0&\mathbf{R}_{k}\end{bmatrix}=\begin{bmatrix}\mathbf{B}_{k,P}\mathbf{B}_{k,P}^{ T}&0\\ 0&\mathbf{B}_{k,R}\mathbf{B}_{k,R}^{T}\end{bmatrix}\\ &=\begin{bmatrix}\mathbf{B}_{k,P}&0\\ 0&\mathbf{B}_{k,R}\end{bmatrix}\begin{bmatrix}\begin{bmatrix}\mathbf{B}_{k,P}&0 \\ 0&\mathbf{B}_{k,R}\end{bmatrix}^{T}\end{bmatrix}\\ &=\mathbf{B}_{k}\mathbf{B}_{k}^{T}\end{split}  \tag{16}\]  
Multiplying \(\mathbf{B}_{k}^{-1}\) at both sides of the Equation (12) by, we will obtain the nonlinear regression model as  
\[\varepsilon_{k}=d(\mathbf{x}_{k})+\mathbf{e}_{k}, \tag{17}\]  
where we have  
\[\mathbf{z}_{k}=\mathbf{B}_{k}^{-1} \begin{bmatrix}\widetilde{\mathbf{x}}_{k} \\ \mathbf{y}_{k}\end{bmatrix} ,d(\mathbf{x}_{k})=\mathbf{B}_{k}^{-1} \begin{bmatrix} \mathbf{x}_{k}\\ \mathbf{h}(\mathbf{x}_{k})\end{bmatrix} ,\mathbf{e}_{k}=\mathbf{B}_{k}^{-1} \mathbf{\varphi}_{k} \tag{18}\]  
Next, the cost function based on MCC, \(J_{\text{MCC}}\), can be expressed as  
\[J_{\text{MCC}}(\mathbf{x}_{k})=\frac{1}{L}\sum_{i=1}^{L}G_{\sigma}(\mathbf{ \varepsilon}_{k,i}-d_{i}(\mathbf{x}_{k})) \tag{19}\]  
where the \(i^{\text{th}}\) elements of \(d(\mathbf{x}_{k})\), \(\mathbf{\varepsilon}_{k}\) are \(d_{i}(\mathbf{x}_{k})\) and \(\mathbf{\varepsilon}_{k,i}\), respectively. The dimension of \(D_{k}\), \(L=n+m\). Then the corresponding MCC-based optimal estimation state can be obtained as  
\[\hat{\mathbf{x}}_{k}=\underset{\mathbf{x}_{k}}{\text{argmax}}J_{\text{MCC}}( \mathbf{x}_{k})=\underset{\mathbf{x}_{k}}{\text{argmax}}\sum_{i=1}^{L}G_{ \sigma}(\mathbf{e}_{k,i}) \tag{20}\]  
where  
\[e_{k,i}=\mathbf{\varepsilon}_{k,i}-d_{i}(\mathbf{x}_{k}) \tag{21}\]  
Therefore, the optimal solution of \(\mathbf{x}_{k}\) can be written as  
\[\bigg{(}\frac{\partial d}{d\mathbf{x}_{k}}\bigg{)}^{T}\mathbf{C}_{k}(d( \mathbf{x}_{k})-\mathbf{\varepsilon}_{k})=0 \tag{22}\]  
By performing the linearization of \(d(\mathbf{x}_{k})\) around the estimation \(\hat{\mathbf{x}}_{k}^{T}\) derived from the fixed-point algorithm at each iteration step, we obtain  
\[\hat{\mathbf{x}}_{k}^{(t)}=(\mathbf{D}^{T}(\hat{\mathbf{x}}_{k}^{(t-1)}) \mathbf{C}_{k}^{(t-1)}\mathbf{D}(\hat{\mathbf{x}}_{k}^{(t-1)}))^{-1}\mathbf{D} ^{T}(\hat{\mathbf{x}}_{k}^{(t-1)})\mathbf{C}_{k}^{(t-1)}\times(\mathbf{D}^{T} (\hat{\mathbf{x}}_{k}^{(t-1)})\hat{\mathbf{x}}_{k}^{(t-1)}-d(\hat{\mathbf{x} }_{k}^{(t-1)})+\mathbf{z}_{k}) \tag{23}\]  
where \(\mathbf{D}(\hat{\mathbf{x}}_{k}^{(t-1)})=\frac{\partial d}{\partial\mathbf{x }}\big{|}_{\mathbf{x}=\hat{\mathbf{x}}_{k}^{(t-1)}}\). Further,  
\[\mathbf{D}(\hat{\mathbf{x}}_{k}^{(t-1)})=\mathbf{M}_{k}^{-1}