## 4 Extensions of RUMVRHF Smoother
### Extension to Reduced Computational Complexity Receding Horizon Fixed-Lag Smoother
In Figure 7, we can observe that the computational efficiency of the proposed UMVRHF smoothers in a separated batch form can be improved by selecting the lag size near the half of the horizon length. Moreover, the proposed RHF smoothers in a recursive-form have much smaller computation times than RHF smoothers in a batch-form. It is clearly noticeable that the proposed RCCRHF smoothers involving short lag sizes (shorter than half of horizon length) have much smaller averaged computation times than other smoothers. In addition, for a long lag size (larger than the half of the horizon length), the computation time is still smaller than that of the other proposed RHF smoothers, although their horizon sizes are longer than the others. Furthermore, the proposed RTSRHF smoothers have better computational performance than the proposed RUMVRHF smoothers. ### Direct Current Motor System

In this subsection, numerical experiment results are given for direct current (DC) motor system to verify the applicability of the proposed methods for more practical application. The discrete-time DC motor model with model uncertainty is given as follows [28; 29]:

\[x_{k+1} = \left[ \begin{array}{cc}0.8178+\delta_{k}&-0.0011\\ 0.0563&0.3678+\delta_{k}\end{array} \right]x_{k}+\left[ \begin{array}{cc}0.00 6&0\\ 0&0.0057\end{array} \right]w_{k}, \tag{103}\] \[y_{k} = \left[ \begin{array}{cc}1+0.1\delta_{k}&0\end{array} \right]x_{k}+ v_{k}, \tag{104}\]

where the DC motor is assumed to be operated without any payload and drive voltage is assumed to be encountered as an external unit step source. Covariances for system and measurement



For RH smoothers, the horizon length and fixed-lag size are taken as \(N=35\) and \(h=3\), respectively. As shown in the previous experiment, the estimation performance of RUMVRHF smoother for posteriori priori state and batch form are similar to that of the RUMVRHF smoother for posteriori. Thus, in this section, we compare the estimation results of the RUMVRHF smoother (posteriori), RCCRF smoother, RTSRHF smoother, FMS smoother, and RTSF Kalman smoother. Figure 6: RMSE errors with respect to the various horizon lengths and lag sizes. Figure 8 and Table 6 present estimation errors and time averaged values of RMSE errors of smoothers for \(200\leq k\leq 350\). In addition, the average computation times of RHF smoothers are presented in Table 7.

As shown in these results, the proposed RHF smoothers estimate the real state well and provide more exact estimates than RTSF Kalman smoother for the modeling uncertainty. Moreover, the proposed RHF smoothers have faster convergence speed than RTSF Kalman smoother as we expected. Furthermore, the proposed RCCRF smoother has better computational performance than other compared RHF smoothers. These observations show that the proposed RHF smoothers can work well and have robustness against temporarily modeling uncertainty for practical application. \begin{table}
\begin{tabular}{c c c} \hline
**Smoothers** & **RMSE Error (\(\mathbf{x_{1}}\))** & **RMSE Error (\(\mathbf{x_{2}}\))** \\ \hline RTSF Kalman smoother & 0.0540 & 6.3353 \\ Prev. optimal RHF smoother & 0.0044 & 3.1536 \\ RUMVRHF smoother (Posteriori) & 0.0051 & 3.2190 \\ RTSBHF smoother & **0.0034** & **3.0504** \\ RCCHF smoother & 0.0040 & 3.0770 \\ FMS smoother & 0.0282 & 4.8560 \\ \hline \end{tabular}
\end{table} 
Table 6: RMSE errors of smoothers for the time interval [200 - 350]. Figure 7: Time-averaged computation time with respect to the horizon lengths and lag sizes. ## 6 Conclusions

In this paper, novel optimal RHF smoothers have been proposed in batch and recursive forms for linear discrete time-varying state space models. The proposed RHF smoothers were obtained by optimally combining local optimal receding horizon filters and smoothers for the separated sub-horizons. The proposed recursive RHF smoothing algorithm was formulated as a finite horizon Kalman filtering and information smoothing.