## 3 Robust Square-Root Continuous-Discrete Cubature Kalman Filter Algorithms
### Rsrcd-Mcckf Algorithm Based on Maximum Correntropy Criterion
Then, the optimal estimation of \(\mathbf{x}_{k}\) is  
\[\hat{\mathbf{x}}_{k}=\text{argmin}_{\mathbf{x}_{k}}J_{\text{MCC}}. \tag{25}\]  
The optimal solution of the above equation can be obtained by solving the derivative  
\[\frac{\partial J_{\text{MCC}}}{\partial\mathbf{x}_{k}}=\gamma\mathbf{P}_{k|k-1}^{-1} \Big{(}\mathbf{x}_{k}-\hat{\mathbf{x}}_{k|k-1}\Big{)}-\beta\frac{\text{L}_{k}\overline {\mathbf{H}}_{k}^{\text{T}}\overline{\mathbf{R}}_{k}^{-1}}{-2\sigma^{2}}\Big{(}\mathbf{z} _{k}-\overline{\mathbf{H}}_{k}\mathbf{x}_{k}-\hat{\mathbf{z}}_{k|k-1}+\overline{\mathbf{H}}_{ k}\hat{\mathbf{x}}_{k|k-1}\Big{)}=0 \tag{26}\]  
With  
\[\kappa(x,y)=G_{\sigma}(e)=\exp\biggl{(}-\frac{e^{2}}{2\sigma^{2}}\biggr{)}, \tag{18}\]0  
This correntropy criterion follows the idea of \(M\)-estimation, applying the Gaussian kernel into each element of the estimation error matrix. The adjustment factor obtained by the Gaussian kernel function (denoted as \(\text{L}_{k}\)) will be a scalar, which is easily separated and avoids the numerical calculation problem of zero matrix inversion. When \(\sigma\rightarrow\infty\), in order to guarantee the algorithm can converge to the traditional CD-CKF algorithm, we set \(\gamma=1\), \(\beta=-2\sigma^{2}\) in this paper. Then, the state estimate and the new filter gain can be obtained by the fixed-point iteration method:  
\[\kappa(x,y)=G_{\sigma}(e)=\exp\biggl{(}-\frac{e^{2}}{2\sigma^{2}}\biggr{)}, \tag{18}\]1  
\[\kappa(x,y)=G_{\sigma}(e)=\exp\biggl{(}-\frac{e^{2}}{2\sigma^{2}}\biggr{)}, \tag{18}\]2  
The update of the error covariance matrix can be calculated by the following equation:  
\[\kappa(x,y)=G_{\sigma}(e)=\exp\biggl{(}-\frac{e^{2}}{2\sigma^{2}}\biggr{)}, \tag{18}\]3  
Considering that in the fixed-point iteration, \(\overline{\mathbf{H}}_{k}\mathbf{x}_{k}\) can be replaced by \(\overline{\mathbf{H}}_{k}\hat{\mathbf{x}}_{k|k-1}\)[20], \(\text{L}_{k}\) can be rewritten as  
\[\kappa(x,y)=G_{\sigma}(e)=\exp\biggl{(}-\frac{e^{2}}{2\sigma^{2}}\biggr{)}, \tag{18}\]4  
where \(G_{\sigma}(\cdot)\) can be calculated by Equation (18). Therefore, the RSRCD-MCCKF Algorithm 4 can be summarized as follows:  
```
Step 1 Repeat Step 1 of the SRCD-CKF algorithm. Steps 2.1-3.7 are equivalent to Steps 2.1-3.7 of the SRCD-CKF algorithm. Step 3.8 Calculate \(\overline{\mathbf{H}}_{k}\), \(\overline{\mathbf{R}}_{k}\), \(\text{L}_{k}\) using Equations (21), (22), and (31). Step 3.9 Obtain the new Kalman filter gain \(\overline{\hat{\mathbf{K}}}_{k}\) using Equation (29). Step 3.10 Complete the estimation of the state value and the update of the error covariance matrix using Equations (28) and (30). ```  
**Algorithm 4** RSRCD-MCCKF. In the RSRCD-MCCKF, note that the Gaussian kernel bandwidth \(\sigma\) greatly affects the accuracy of the filtering algorithm. Studies have shown that [25] when \(\sigma\) is small, it is more robust to non-Gaussian measurement noise; however, when \(\sigma\) is too small, the filtering performance of the algorithm will be more severely degraded, and the filtering may even diverge. For the actual situation of continuous-discrete systems, how to select the proper value of \(\sigma\) according to different scenarios is also of great significance.