## 4 Extensions of RUMVRHF Smoother
### Extension to Reduced Computational Complexity Receding Horizon Fixed-Lag Smoother
Second, to highlight the effects of the horizon length and delay size on the proposed RHF smoothers, we compare the RMSE errors of the smoothers with respect to various horizon lengths and lag sizes, as shown in Figure 6.

In contrast to all RMSE errors, it can be observed that the proposed RCCRF and RTSRHF smoothers and the previous optimal RHF smoother can be optimal choices for short lag sizes. In particular, in Figure 6c,e, it can be observed that RHF smoothers with long horizon lengths and short delay sizes perform better than RTSF Kalman smoother, even in the absence of modeling uncertainties. In addition, the RHF smoothers with large horizon lengths yield better estimation results than those with small horizon lengths in the case of an accurate system model. However, other proposed smoothers are also acceptable for use. From Figure 6b,d,f, it is observed that the proposed RCCRHF smoother with a short delay size provides better estimation performance than others in the case of a model mismatch because its averaged horizon length is shorter than other RHF smoothers. However, the RCCRHF smoother with a long delay size (longer than half of the horizon length) demonstrates poor estimation results owing to its increased horizon length. Thus, it can be stated, in the case of a model mismatch, a short lag size should be adopted for the proposed RCCRHF smoother. As observed in Figure 6a,c,e, the estimation performance of the previous optimal RHF smoother is better than the other proposed RHF smoothers. On the contrary, observing the results in Figure 6b,d,f, it can easily noticed that the proposed smoothers demonstrate better estimation performance than the previous optimal RHF smoother except for small lag size intervals. In particular, the proposed RTSRHF smoother guarantees its estimation performance for most lag sizes. Thus, we can infer that the proposed smoothers are robust and provide more exact estimate than the previous optimal RHF smoother. Furthermore, we can observe that there is best lag size, and it is close to half of the horizon length. Third, to demonstrate the computational efficiency of the proposed approaches, the time-averaged computation times of RHF smoothers are compared, as presented in Table 4 and Figure 7. For fair comparisons, the matrix gains of the batch RHF smoothers are calculated at every iteration, although the smoothers are designed for time-invariant systems. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Horizon Length and Lag Size** & \(\mathbf{N=20}\), \(\mathbf{h=5}\) & \(\mathbf{N=30}\), \(\mathbf{h=5}\) & \(\mathbf{N=40}\), \(\mathbf{h=5}\) \\ \hline RTSF Kalman smoother & 7.386 & 7.386 & 7.386 \\ Prev. optimal RHF smoother (Priori) & 7.848 & **7.321** & **7.341** \\ RUNVRHF smoother (Posteriori) & 8.031 & 7.355 & 7.357 \\ RUNVRHF smoother (Posteriori) & 7.968 & 7.345 & 7.352 \\ RTSRHF smoother & 7.892 & 7.323 & 7.345 \\ RCCRHF smoother & 7.956 & 7.393 & 7.378 \\ FMS smoother & 324.983 & 342.122 & 343.178 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: RMSE errors for the time interval \([50\;\;200]\)\((\times 10^{-3})\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Horizon Length and Lag Size** & \(\mathbf{N=20}\), \(\mathbf{h=5}\) & \(\mathbf{N=30}\), \(\mathbf{h=5}\) & \(\mathbf{N=40}\), \(\mathbf{h=5}\) \\ \hline RTSF Kalman smoother & 2.312 & 2.312 & 2.312 \\ Prev. optimal RHF smoother & 0.225 & 0.729 & 1.184 \\ RUNVRHF smoother (Priori) & 0.310 & 0.805 & 1.235 \\ RUNVRHF smoother (Posteriori) & 0.276 & 0.784 & 1.228 \\ RTSRHF smoother & 0.233 & 0.695 & 1.104 \\ RCCRHF smoother & **0.189** & **0.642** & **1.080** \\ FMS smoother & 1.051 & 15.677 & 1.817 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: RMSE errors for the time interval \([201\;\;350]\). As observed from these results, the proposed UMVRHF smoothers in a separated batch-form have better computational performance than the previous optimal RHF smoothers in batch form.