## 3 Robust Square-Root Continuous-Discrete Cubature Kalman Filter Algorithms
### Rsrcd-Mcckf Algorithm Based on Maximum Correntropy Criterion  
Correntropy is a new method used to measure the similarity between two random variables. Given the random variables \(X\) and \(Y\), their joint probability density function is \(F_{X,Y}(x,y)\). The correntropy can be defined as follows [22]:  
\[V(X,Y)=E[\kappa(X,Y)]=\iint\kappa(x,y)\mathrm{d}F_{X,Y}(x,y), \tag{17}\]  
where \(E[\cdot]\) is the mathematical expectation, and \(\kappa(\cdot)\) is the Messer-type positive definite kernel function. In this paper, we select the Gaussian kernel as the kernel function of the correntropy, which is defined as follows:  
\[\kappa(x,y)=G_{\sigma}(e)=\exp\biggl{(}-\frac{e^{2}}{2\sigma^{2}}\biggr{)}, \tag{18}\]  
where \(e=x-y\), and \(\sigma>0\) is the kernel bandwidth of the kernel function. In practice, the joint probability density function is usually difficult to obtain, and the data sampling points are also limited. Therefore, the correntropy can be estimated by the approach of \(T\) data sampling points:  
\[\hat{V}(X,Y)=\frac{1}{T}\sum_{i=1}^{T}G_{\sigma}(e(i)). \tag{19}\]  
where \(e(i)=x(i)-y(i)\), \(\{x(i),y(i)\}_{i=1}^{T}\) represents the related sampling point sequence of the joint probability density function \(F_{X,Y}(x,y)\). Expanding the Gaussian kernel function using the Taylor series, we can obtain  
\[V(X,Y)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2^{n}\sigma^{2n}n!}E\Bigl{[}(X-Y)^{ 2n}\Bigr{]}. \tag{20}\]  
It can be seen from Equation (20) that the correntropy information contains all the even-order moments of the random variable error \(X-Y\). Therefore, as long as the appropriate kernel bandwidth is selected, higher-order information can be obtained. For the systems contaminated by noise, compared with the minimum mean square error criterion, the correntropy can be used to describe its statistical information. On this basis, this paper introduces MCC into the continuous-discrete system. Firstly, \(\overline{\mathbf{H}}_{k}\) is defined as the pseudo-measurement matrix, \(\overline{\mathbf{R}}_{k}\) is the estimated covariance matrix of the observation noise, and their calculation methods are as follows:  
\[\overline{\mathbf{H}}_{k}=P_{xz,k|k-1}^{T}P_{k|k-1}^{-1}, \tag{21}\]  
\[\overline{\mathbf{R}}_{k}=P_{zz,k|k-1}-\overline{\mathbf{H}}_{k}P_{k|k-1}\overline\mathbf{{H}}_{k}^{\Gamma}. \tag{22}\]The cost function based on the MCC filtering can be defined by  
\[\mathbf{J}_{\text{MCC}}=G_{\sigma}\Bigg{(}\left\|\mathbf{x}_{k}-\hat{\mathbf{x}}_{k|k-1} \right\|_{\mathbf{P}_{k|k-1}^{-1}}^{2}\Bigg{)}+G_{\sigma}\bigg{(}\left\|\mathbf{z}_{k}- \overline{\mathbf{H}}_{k}\mathbf{x}_{k}-\hat{\mathbf{z}}_{k|k-1}+\overline{\mathbf{H}}_{k}\hat {\mathbf{x}}_{k|k-1}\right\|_{\overline{\mathbf{R}}_{k}^{-1}}\bigg{)}. \tag{23}\]  
In this paper, the process noise is assumed to be Gaussian, and we modify the cost function as follows:  
\[J_{\text{MCC}}=\gamma\Big{\|}\mathbf{x}_{k}-\hat{\mathbf{x}}_{k|k-1}\Big{\|}_{\mathbf{P}_{ k|k-1}^{-1}}^{2}-\beta G_{\sigma}\bigg{(}\left\|\mathbf{z}_{k}-\overline{\mathbf{H}}_{k} \mathbf{x}_{k}-\hat{\mathbf{z}}_{k|k-1}+\overline{\mathbf{H}}_{k}\hat{\mathbf{x}}_{k|k-1} \right\|_{\overline{\mathbf{R}}_{k}^{-1}}\bigg{)}, \tag{24}\]  
where \(\gamma\) and \(\beta\) are adjusting weights, and \(\|\mathbf{x}\|_{\mathbf{A}}^{2}=\mathbf{x}^{T}\mathbf{A}\mathbf{x}\).