## 3 Estimation Algorithm
### Combined SR-CKF and SVSF  
To strengthen the system's reliability of the SR-CKF against system uncertainties in the practical applications. An algorithm that combines the SR-CKF with an SVSF is defined as CSVSF and the procedure is summarized as follows [23]. Initially, the SVSF is presented through the integration of the principles of a variable structure concept and sliding mode theory. \[\hat{x}_{k|k-1}=f(\hat{x}_{k-1|k-1},u_{k-1}) \tag{28}\]  
\[P_{k|k-1}=B_{k-1}P_{k-1|k-1}B^{T}{}_{k-1}+Q_{k-1} \tag{29}\]  
\[B_{k-1}=\left.\frac{\partial f}{\partial x}\right|_{x_{k-1|k-1}} \tag{30}\]  
The predicted measurement and measurement errors of the SVSF are gained as:  
\[\hat{z}_{k|k-1}=H_{k-1}\hat{x}_{k|k-1} \tag{31}\]  
\[\widehat{e}_{x,k|k-1}=z_{k}-\hat{z}_{k|k-1} \tag{32}\]Subsequently, the SVSF gain can be determined by taking into account both the initial measurement error \(\widehat{e}_{z,k-1|k-1}\) and the updated measurement error \(\varepsilon_{z,k|k-1}\). \[\overline{K}_{k}=H^{+}{}_{k}diag\big{[}\big{(}\big{|}\widehat{e}_{z,k|k-1} \big{|}+\lambda\big{|}\widehat{e}_{z,k-1|k-1}\big{|}\big{)}\circ sat(\varphi^{-1 }\widehat{e}_{z,k|k-1})\big{]}diag^{-1}(\widehat{e}_{z,k|k-1}) \tag{33}\]  
where \(H^{+}{}_{k}\) denotes the pseudoinverse of the measurement matrix, \(\lambda\) represents the convergence rate, \(\circ\) means the Schur multiplication, and \(\varphi\) denotes the boundary layer of the SVSF. To mitigate the emergence of excessive noise, the saturation function, which includes an optimal boundary layer, is employed instead of the sign function. The saturation function is defined as:  
\[sat(\varphi^{-1}\widehat{e}_{z,k|k-1})= \begin{cases}\begin{array}{ccc}1&,& \varphi_{i}{}^{-1}\widehat{e}_{z,k|k-1}\geq 1\\ \varphi_{i}{}^{-1}\widehat{e}_{z,k|k-1}&,&-1\leq\varphi_{i}{}^{-1}\widehat{e}_ {z,k|k-1}\leq 1\\ -1&,&\varphi_{i}{}^{-1}\widehat{e}_{z,k|k-1}\leq-1\end{array}\end{cases}  \tag{34}\]  
As the boundary layers are coupled to each other, to achieve a more precise state estimation, a coupled smooth boundary layer matrix is introduced, which encompasses interconnected boundary layers:  
\[\overline{\varphi}= \begin{bmatrix}\varphi_{11}&\varphi_{12}&\dots&\varphi_{1 m}\\ \varphi_{21}&\varphi_{22}&\dots&\varphi_{2m}\\ \vdots&\vdots&\ddots&\vdots\\ \varphi_{m1}&\varphi_{m2}&\dots&\varphi_{mm}\end{bmatrix}  \tag{35}\]  
The derivation of the predictive covariance to the boundary layer enables the attainment of an ideal boundary layer [22]:  
\[\frac{\partial(trace[P_{k-1|k-1}])}{\partial\varphi}=0 \tag{36}\]  
According to [20], the smoothing boundary needs to exceed the uncertain dynamics of the system to maintain the stability and eliminate the chattering. Nevertheless, employing a much broader width for the SVSF could result in slower convergence and performance degradation. Therefore, we can determine the optimal boundary layer of the CSVSF by employing the following calculation method:  
\[\varphi_{k}=(\overline{\psi}^{-1}H_{k-1}\overline{P}_{k|k-1}H^{T}{}_{k-1}F^{- 1}{}_{k})^{-1} \tag{37}\]  
\[\overline{\psi}=\big{(}\big{|}e_{k|k-1}\big{|}+\delta\big{|}e_{k-1|k-1}\big{|} \big{)} \tag{38}\]  
where \(\overline{\psi}\) is the diagonal matrix of \(\psi\). Revisions are applied to both the state estimate and its error covariance matrix, resulting in:  
\[\hat{x}_{k|k}=\hat{x}_{k|k-1}+\overline{K}_{k}e_{z,k|k-1} \tag{39}\]  
\[\overline{P}_{k|k}=(I-K_{k}H_{k-1})\overline{P}_{k|k-1}(I-K_{k}H_{k-1})^{T}+K_ {k}R_{k}K^{T}{}_{k} \tag{40}\]  
Ultimately, the measurement estimate and the measurement error are calculated:  
\[\hat{z}_{k|k}=H_{k}\hat{x}_{k|k} \tag{41}\]  
\[e_{z,k|k}=z_{k}-\hat{z}_{k|k} \tag{42}\]Considering the relationship between the SR-CKF and the SVSF through the implementation of a smooth boundary layer in the SVSF, it is observed that a wider constant smoothing boundary layer width compared to the time-varying one may lead to inaccuracies in the SVSF. Therefore, the SR-CKF is employed to optimize the accuracy of the filter.