## 4 The CSVSF Based on Multi-Kernel Maximum Correntropy Criterion
 \tag{53}\]  
Given the challenge in ascertaining the system state, the \(x(k)\) is assumed equal to \(\hat{x}(k|k-1)\), thus:  
\[\tilde{\Phi}_{x}(k)=S_{k|k-1}\cdot D_{x}(k)\cdot S^{T}_{k|k-1}=\hat{P}(k|k-1) \tag{54}\]  
Meanwhile, the noise covariance is modified as:  
\[\tilde{\Phi}_{z}(k)=\tilde{R}(k)=\tilde{S}_{R}(k)D_{z}(k)\tilde{S}_{R}^{T}(k) \tag{55}\]  
**Remark 1**. : _It is clear that the option for the kernel bandwidth significantly influences the effectiveness of the MKMC. Choosing a smaller kernel bandwidth can enhance the algorithm's robustness against outliers. However, excessively small values can lead to filter divergence or deterioration. Conversely, selecting a significantly larger kernel bandwidth renders the MKMC ineffective, reverting the algorithm to its original filter state. Previous studies [29] have relied on trial-and-error methods for choosing the kernel bandwidth, which can be complex and time-consuming. To address this issue, we suggest automatically adjusting the kernel bandwidth using fuzzy rules to achieve an adaptive kernel bandwidth._  
Consequently, the algorithmic process of MKMC-CSVSF which combines the SR-CKF and the SVSF based on the MKMC can be outlined as follows:  
1. Determine the cubature point and propagate it with respect to the state function by (13) and (14). 2. Calculating the priori state estimate and the square-root covariance matrix through (15) and (16). 3. Calculating the measurement predictive estimation through the estimate function as (19), (20). 4. Considering the MKMC, calculate the modified noise covariance \(\tilde{R}(k)\) by (43)-(55). 5. Then, considering the MKMC, and the modified measurement covariance in (21), the state error covariance of the SR-CKF in (27) and the state error covariance of the SVSF in (40) are used to replace the original measurement covariance for the robust to the heavy-tailed outliers. \[S_{zz,k|k-1}=Tria([\gamma_{k|k-1}\quad\tilde{S}_{R,k-1}])\] (56) \[S_{k|k}=Tria([\chi_{k|k-1}-K_{k}\gamma_{k|k-1}\quad K_{k}\tilde{S}_{R,k}])\] (57) \[P_{SVSF,k|k}=(I-K_{SVSF,k}H_{k-1})P_{SVSF,k|k-1}(I-K_{SVSF,k}H_{k-1})^{T}+K_{ SVSF,k}\tilde{R}_{SVSF,k}{K^{T}}_{SVSF,k}\] (58) 6. Then, the predicted state estimation and state error covariance are obtained as (26), (27). 7. Calculating the predicted measurement in (31), measurement errors in (32) and the gain of the SVSF in (33). 8. Comparing the constant and time-varying layer, in the event that the fixed boundary layer's width surpasses the width of its time-varying counterpart, the SR-CKF based on the MKMC is used to refine the precision of the filter. The filter gain matrix and the modified state estimation are calculated as (25), (26). When the width of the time-varying smoothing boundary layer exceeds that of the constant one, the SVSF based on the MKMC is utilized to confirm a reliable estimate. The filter gain matrix and the modified state estimation are calculated as (39), (40). 9. Then, update the measurement estimate and estimation error as (41), (42). **Remark 2**. : _Consequently, it can be deduced that the MKMC-CSVSF, as proposed in this study, exhibits improved precision and robust performance in dealing with uncertain dynamics and heavy-tailed non-Gaussian noises. By setting the kernel bandwidths to a specific value as \(\sigma_{i}\rightarrow\infty\), the MKMC-CSVSF is converted to the conventional CSVSF, which effectively handles Gaussian noises and uncertain dynamics. Conversely, when different kernel bandwidths are employed as \(\sigma_{i}=\sigma\), the proposed filter reverts back to the CSVSF based on the MCC. Remarkably, this novel MKMC-CSVSF approach not only mitigates Gaussian noise but also attenuates non-Gaussian noise simultaneously. It outperforms the CKF and CSVSF under conditions where dynamic uncertainties coexist with heavy-tailed noises. Therefore, the algorithm proposed holds significant practical applicability and can be widely used in a practical condition._