## 2 Methodology
### Selection of Robust Estimator  
Under the standard data reconciliation process, the weighted least-squares (WLS) function is applied as the estimator. If the normalized correction for a known parameter is denoted as \(\xi\), WLS estimator should give the following:  
\[\rho_{\text{WLS}}(\xi)=\frac{1}{2}\xi^{2} \tag{5}\]  
Under the WLS estimator, one can see the surge of \(\rho_{\text{WLS}}\) when the correction scale \(\xi\) increases, which would cause excessive influence on the objective function. For robust data reconciliation purposes, the estimators should be applied to imper the scale of the objective function when specific elements (the measurements with gross errors) of \(\mathbf{v}\) become too large. Over the decades, a number of robust estimators have been proposed by different researchers. In this paper, the most representative robust estimators were compared for evaluation in S-CO\({}_{2}\) systems, which are listed below. Fair estimator: It was proposed by [PERSON] and [PERSON] [31] and is convex and has continuous first and second derivatives. It is shown in the following equation, where \(c_{\text{F}}\) is a constant parameter:  
\[\rho_{\text{fair}}(\xi)=2c_{\text{F}}^{2}\left[\frac{|\xi|}{c_{\text{F}}}- \ln\left(1+\frac{|\xi|}{c_{\text{F}}}\right)\right] \tag{6}\]  
Logistic estimator: The definition of the logistic estimator is given below, where \(c_{\text{L}}\) is a constant parameter:  
\[\rho_{\text{logistic}}(\xi)=2\ln\left[1+\exp\left(\frac{\xi}{c_{\text{L}}} \right)\right]-\frac{\xi}{c_{\text{L}}} \tag{7}\]  
Cauchy estimator: It can be derived from a Cauchy distribution, where \(c_{\text{C}}\) is a constant parameter:  
\[\rho_{\text{cauchy}}(\xi)=c_{\text{C}}^{2}\ln\left(1+\frac{\xi^{2}}{c_{ \text{C}}^{2}}\right) \tag{8}\]Welsch estimator: This is an estimator suitable for data with a large deviation, where \(c_{\rm W}\) is a constant parameter:  
\[\rho_{\rm welsch}(\xi)=\frac{c_{\rm W}^{2}}{2}\left[1-\exp\left(-\frac{\xi^{2}}{c _{\rm W}^{2}}\right)\right] \tag{9}\]  
To compare these different estimators, their coefficient should be decided \"impartially\", which means they should have the same asymptotical efficiency when \(\xi\) is under the standard normal distribution. In this work, the tuned parameters given by [PERSON] and [PERSON] [32], as well as [PERSON] et al. [25], were applied to decide the coefficient of each estimator, which made the efficiency of every estimator 95%. The tuned coefficients for each estimator are listed in Table 1.  
The relations between the forementioned estimators and the correction value are shown in Figure 1. It was clear that the WLS estimator increased significantly as the input magnitude grew, and it was not robust compared with the other estimators. As for fair and logistic estimators, the slope was approaching a constant when \(\xi\) increased. For Cauchy and Welsch estimators, they tended to show a constant value after receiving a large \(\xi\). This implied their robustness when facing gross error measurements. This implied a relatively large (and effective) correction for therored sensors under those estimators compared with the WLS estimator, as the rapid growth of \(\rho_{\rm WLS}\) can inhibit the meaningful correction of these sensors. On the other hand, when the errors were relatively small, Cauchy and Welsch estimators may be less competitive than the WLS estimator, as excessive corrections may be generated due to the slow growth of these estimators. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Robust Estimator** & **Coefficient** & **Value** \\ \hline Fair & \(c_{\rm F}\) & 1.3998 \\ Logistic & \(c_{\rm L}\) & 0.602 \\ Cauchy & \(c_{\rm C}\) & 2.3849 \\ Welsch & \(c_{\rm W}\) & 2.9846 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Tuned robust estimator coefficient to achieve 95% efficiency. Figure 1: Curves of the estimators applied for S-CO\({}_{2}\) systems. To summarize their characteristics, the influence function (IF) can be introduced to show the global robustness of different estimators.