## 4 Extensions of RUMVRHF Smoother
### Extension to Reduced Computational Complexity Receding Horizon Fixed-Lag Smoother
Additionally, as an extension of the proposed recursive RHF smoothing algorithm, a Rauch-Tung-Striebel type RHF smoothing algorithm and reduced computational complexity RHF smoother were also proposed to improve the computational efficiency. The proposed RHF smoothers could provide optimality, unbiasedness, and bounded-input bounded-output stability. Moreover, the proposed RHF smoothers are independent of any a priori information of the initial state and assumption of a non-singular transition system matrix. Furthermore, the proposed RHF smoothers could significantly improve the computational efficiency, which is major disadvantage of RHF smoothers. Finally, it has been demonstrated via numerical experiments that the RHF smoothers proposed in this paper can provide better estimation results than fixed-lag Kalman smoothers in the case of a model mismatch. Furthermore, \begin{table}
\begin{tabular}{c c} \hline \hline
**Smoothers** & **Average Computation Time (\(msec\))** \\ \hline Prev. optimal RHF smoother & 3.0829 \\ RUMVRHF smoother (Posteriori) & 1.1896 \\ RTSRHF smoother & 1.0909 \\ RCCCHF smoother & **0.4852** \\ FMS smoother & 0.6780 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Average computation times of RHF smoothers. Figure 8: Estimation errors of smoothers. it was also shown that the proposed methods are more computational efficient than the existing RHF smoothers and applicable for practical application. However, it is possible to improve the estimation performance of the RH smoothing algorithms by choosing the appropriate fixed lag size and horizon length. Thus, we plan to investigate the horizon length adjustment method and how to find optimal fixed-lag size for further improvement of the proposed RH smoothers as future work. Moreover, in the future, it would be meaningful to extend and evaluate the proposed RH smoothing algorithms for more practical applications. Conceptualization, B.K. ; methodology, B.K. ; software, B.K. ; validation, B.K. and P.S.K. ; formal analysis, B.K. ; investigation, B.K. and P.S.K. ; resources, B.K. and P.S.K. ; data curation, B.K. and P.S.K. ; writing--original draft preparation, B.K. ; writing--review and editing, B.K. ; visualization, B.K. ; All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The authors declare no conflict of interest. The following abbreviations are used in this manuscript:

References

## References

* (1) [PERSON] _Optimal Estimation: With An Introduction to Stochastic Control Theory_; John Wiley and Sons: Hoboken, NJ, USA, 1986. * (2) [PERSON]; [PERSON] A joint introduction to Gaussian Processes and Relevance Vector Machines with connections to Kalman filtering and other kernel smoothers. _Inf. Fusion_**2021**, _74_, 17-38. [CrossRef]
* (3) [PERSON]; [PERSON] _Practical Smoothing: The Joys of P.-Splines_; Cambridge University Press: Cambridge, UK, 2021. * (4) [PERSON]; [PERSON] Adaptive Lag Smoother for State Estimation. _Sensors_**2022**, _22_, 5310. [CrossRef] [PubMed]
* (5) [PERSON]; [PERSON]; [PERSON]; [PERSON] Denoising ECG Signals Using Unbiased FIR Smoother and Harmonic State-Space Model. In Proceedings of the 28 th European Signal Processing Conference (EUSIPCO), Amsterdam, The Netherlands, 18-21 January 2021; pp. 1279-1283. * (6) [PERSON] LPPCNN: A Laplacian Pyramid-based Pulse Coupled Neural Network Method for Medical Image Fusion. _J. Appl. Sci. Eng._**2021**, _24_, 299-305. * (7) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Multisensor-Based Target-Tracking Algorithm with Out-of-Sequence-Measurements in Cluttered Environments. _Sensors_**2018**, _18_, 4043. [CrossRef] [PubMed]
* (8) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Research on Tracking Foreign Objects in Railway Tracks Based on Hidden Markov Kalman Filter. _J. Appl. Sci. Eng._**2022**, _25_, 893-903. * (9) [PERSON] Finite Memory Structure Filtering and Smoothing for Target Tracking in Wireless Network Environments. _Appl. Sci._**2019**, \(9\), 2872.