## 4 The CSVSF Based on Multi-Kernel Maximum Correntropy Criterion  
It is widely recognized that the presented filtering algorithm may suffer from a decrease in performance when confronting non-Gaussian disturbances. The CSVSF algorithm is improved to better withstand external disturbances and heavy-tailed non-Gaussian noise by integrating the MKMC criterion, which considers higher-order moments of errors. Thus, the algorithm proposed achieves high filtering accuracy and robustness against parametric uncertainties and heavy-tailed non-Gaussian noise simultaneously [28]. According to the discrete-time system (6) and (7), which may operate in heavy-tailed non-Gaussian noise and external disturbances, we can obtain:  
\[ \begin{bmatrix}\hat{x}(k|k-1)\\ z(k)\end{bmatrix} = \begin{bmatrix}I\\ H(k)\end{bmatrix} x(k)+\varsigma(k) \tag{43}\]  
where \(I\) denotes the unit matrix of \(n\times n\), \(\delta(k)\) is defined as:  
\[\varsigma(k)= \begin{bmatrix}-x(k)-\hat{x}(k|k-1)\\ v(k)\end{bmatrix}  \tag{44}\]  
\[ \begin{split}\Phi(k)=E[\varsigma(k)\varsigma^{T}(k)]=\left[ \begin{array}{cc}P(k|k-1)&0\\ 0&R(k)\end{array}\right]\\ =\left[\begin{array}{cc}S_{k|k-1}\varsigma^{T}{}_{k|k-1}&0\\ 0&S_{R,k}\varsigma^{T}{}_{R,k}\end{array}\right]=A(k)A^{T}(k)\end{split}  \tag{45}\]  
\(A(k)\) is gained by the Cholesky decomposition of the \(E[\varsigma(k)\varsigma^{T}(k)]\). Then, multiplying both sides of (45) by \(A^{-1}(k)\), one can obtain:  
\[B(k)=C(k)x(k)+\varsigma(k) \tag{46}\]  
\[B(k)=A^{-1}(k) \begin{bmatrix}\hat{x}(k|k-1)\\ z(k)\end{bmatrix} ,\;C(k)=A^{-1}(k) \begin{bmatrix}I\\ H(k)\end{bmatrix} ,\;\varsigma(k)=A^{-1}(k)\varsigma(k) \tag{47}\]  
The MKMC-based cost function is gained considering the multiple kernel bandwidths:  
\[J_{(n+m)}(x(k))=\frac{1}{(n+m)}\sum_{i=1}^{n+m}G_{\varsigma_{i}}[b_{i}(k)-c_{ i}(k)x(k)] \tag{48}\]  
Figure 1: The state trajectory of the CSVSF. \(G_{\sigma_{i}}(\varepsilon_{i,k})\) denotes the Gaussian function with the kernel bandwidth \(\sigma_{i}\), \(b_{i}(k)\) means the \(i\)th item of \(B(k)\), \(c_{i}(k)\) represents the \(i\)th item of \(C(k)\), and \(\xi_{i}(k)=b_{i}(k)-c_{i}(k)x(k)\) means the error at time step \(k\). The optimal estimation of the state is obtained:  
\[\hat{x}(k)=\underset{x(k)}{\text{argmax}}J_{(n+m)}(x(k))=\frac{1}{(n+m)}\sum_{ i=1}^{n+m}G_{\sigma_{i}}(\xi_{i}(k)) \tag{49}\]  
Differentiating the (48), we can establish that:  
\[\frac{\partial J_{(n+m)}(x(k))}{\partial x(k)}=0 \tag{50}\]  
Substituting the (2) into (50), then the function with multiple kernel bandwidths is gained:  
\[\sum_{i=1}^{n+m}G_{\sigma_{i}}(\varepsilon_{i}(k))\cdot\xi_{i}(k)\frac{ \partial_{\xi_{i}}^{2}(k)}{\partial x(k)}=0 \tag{51}\]  
Defining \(G_{\sigma_{i}}(\xi_{i}(k))\cdot\xi_{i}(k)=\eta(\xi_{i}(k))\), then \(\eta(\xi_{i}(k))/\xi_{i}(k)=d_{i}(k)\), the Function (51) can be expressed as:  
\[(\frac{\partial\xi(k)}{\partial x(k)})^{T}D(k)[B(k)-C(k)x(k)]=0 \tag{52}\]  
where \(D(k)\) is defined as \(D(k)= \begin{bmatrix}D_{x}(k)&0\\ 0&D_{z}(k)\end{bmatrix} \) for convenience. \(D_{x}(k)=\text{diag}(G_{\sigma_{1}}(\xi_{1}(k)),\)\(\ldots G_{\sigma_{n}}(\xi_{n}(k)))\), \(D_{z}(k)=\text{diag}(\tilde{G}_{\sigma_{n+1}}(\xi_{n+1}(k)),\ldots G_{\sigma_{ n+n}}(\xi_{n+m}(k)))\). The \(\sigma_{1},\sigma_{2},\ldots\sigma_{n+m}\) mean multi-kernel bandwidths. \(\sigma_{1},\sigma_{2},\ldots\sigma_{n}\) denote the bandwidths of the system process. \(\sigma_{n+1},\sigma_{n+2},\ldots\sigma_{n+m}\) are the bandwidths of the measurement. Therefore, the modified covariance is gained as \(\tilde{\Phi}(k)=A(k)D(k)A^{T}(k)\); for the sake of analysis, the \(\tilde{\Phi}(k)\) is modified to diagonal form:  
\[\tilde{\Phi}(k)= \begin{bmatrix}\tilde{\Phi}_{x}(k)&0\\ 0&\tilde{\Phi}_{z}(k)\end{bmatrix}