## 4 Extensions of RUMVRHF Smoother
### Extension to Reduced Computational Complexity Receding Horizon Fixed-Lag Smoother
The concept of periodical switching scheme for RCCRHF smoother is depicted in Figure 4.  
The process for obtaining the local estimate and its error covariance of RUMVRHF smoother can be eliminated by sharing the mid estimate and its error covariance of RTSRHF smoother at the \(h\) lag time. Thus, the RCCRHF smoother could have less computation time than the proposed recursive RHF smoothers. Figure 3: The concept of RCCRHF smoothing. ```
1:Initialization:\(S_{k}^{+}\gets 0\) and \(\hat{w}_{k|k}^{+}\gets 0\)
2:for\(l\gets 0\) to \(N\)do \[S_{k-l} \leftarrow C_{k-l}^{T}R_{k-l}^{-1}C_{k-l}+S_{k-l}^{+}\] \[S_{k-l-1}^{+} \leftarrow A_{k-l-1}^{T}S_{k-l}A_{k-l-1}-A_{k-l-1}^{T}S_{k-l}G_{k-l}\] Figure 2:0 Figure 2:1 Figure 2:2 Figure 2:3
3:endfor
4:\(\hat{x}_{1,k-N|k}\gets S_{k-N}^{-1}\hat{w}_{k-N|k}\) and \(\bar{P}_{1,k-N}\gets S_{k-N}^{-1}\)
5:for\(i\gets 0\) to \(N-h-2\)do Figure 2:4 Figure 2:5 Figure 2:6 Figure 2:7 Figure 2:8 Figure 2:9
6:endfor
7: \[\hat{x}^{+}_{k-h|k} = U_{p1,k-h}\hat{x}^{+*}_{k-h|k-h}+U_{p2,k-h}\hat{x}^{+}_{2,k-h|k}, \tag{97}\]0 \[\hat{x}^{+}_{k-h|k} = U_{p1,k-h}\hat{x}^{+*}_{k-h|k-h}+U_{p2,k-h}\hat{x}^{+}_{2,k-h|k}, \tag{97}\]1 \[\hat{x}^{+}_{k-h|k} = U_{p1,k-h}\hat{x}^{+*}_{k-h|k-h}+U_{p2,k-h}\hat{x}^{+}_{2,k-h|k}, \tag{97}\]2 \[\hat{x}^{+}_{k-h|k} = U_{p1,k-h}\hat{x}^{+*}_{k-h|k-h}+U_{p2,k-h}\hat{x}^{+}_{2,k-h|k}, \tag{97}\]3
8:for\(i\gets 0\) to \(h\)do \[\hat{x}^{+}_{k-h|k} = U_{p1,k-h}\hat{x}^{+*}_{k-h|k-h}+U_{p2,k-h}\hat{x}^{+}_{2,k-h|k}, \tag{97}\]4 \[\hat{x}^{+}_{k-h|k} = U_{p1,k-h}\hat{x}^{+*}_{k-h|k-h}+U_{p2,k-h}\hat{x}^{+}_{2,k-h|k}, \tag{97}\]5 \[\hat{x}^{+}_{k-h|k} = U_{p1,k-h}\hat{x}^{+*}_{k-h|k-h}+U_{p2,k-h}\hat{x}^{+}_{2,k-h|k}, \tag{97}\]6 \[\hat{x}^{+}_{k-h|k} = U_{p1,k-h}\hat{x}^{+*}_{k-h|k-h}+U_{p2,k-h}\hat{x}^{+}_{2,k-h|k}, \tag{97}\]7
12:endfor
13:Return:\(\hat{x}_{k-h|k}^{+}\) and \(P_{k-h}^{+}\)
```

However, if the lag size \(h\) is greater than half of the horizon length \(N-h\), the RSTRHF smoother cannot be defined. In this situation, the overall horizon lengths of the RSTRHF and RUMVRHF smoothers are taken as \(N\) and \(N+h\), respectively. Although the horizon length of the RSTRHF smoother is increased by \(h\), the RCCHF smoother has less computation time of other proposed recursive RHF smoothers because \(2N\) iterations to obtain local estimate and its covariance of RUMVRHF smoother is omitted. ## 5 Numerical Experiments and Discussion

In this section, numerical experimental results present the performance and effectiveness of the proposed RHF smoothing algorithms. \[\hat{x}^{+}_{k-h|k} = U_{p1,k-h}\hat{x}^{+*}_{k-h|k-h}+U_{p2,k-h}\hat{x}^{+}_{2,k-h|k}, \tag{97}\]8 Concept of periodical switching scheme for RHCCRHF smoother. ### F-404 Gas Turbine Aircraft Engine System

We consider the following discrete time F-404 gas turbine aircraft engine model [21]. \[x_{k+1} = \left[ \begin{array}{ccc}0.9305+\delta_{k}&0&0.1107\\ 0.0077&0.9802+\delta_{k}&-0.0173\\ 0.0142&0&0.8953+\delta_{k}\end{array} \right]x_{k}+\left[\begin{array}{c}1\\ 1\\ 1



By comparing the results for the RTSRHF and RUMVRHF smoothers, it can be stated that the estimation performance is affected, albeit not severely, by the information state, information matrix, and their initial conditions. Moreover, the structure of the RTSRHF smoother is better than that of the RUMVRHF smoother. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Horizon Length and Lag Size** & \(N=\textbf{20}\), \(h=\textbf{5}\) & \(N=\textbf{30}\), \(h=\textbf{5}\) & \(N=\textbf{40}\), \(h=\textbf{5}\) \\ \hline RTSRAlman smoother & 1.386 & 1.386 & 1.386 \\ Prev. optimal RHF smoother & 0.135 & 0.437 & 0.709 \\ RUMVRHF smoother (Priori) & 0.186 & 0.482 & 0.740 \\ RUMVRHF smoother (Posteriori) & 0.166 & 0.469 & 0.735 \\ RTSRHF smoother & 0.140 & 0.417 & 0.661 \\ RCCHF smoother & **0.113** & **0.384** & **0.647** \\ FMS smoother & 0.674 & 0.977 & 1.125 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: RMSE errors for the time interval \([50\;\;500]\). Figure 5: Estimation errors with respect to various horizon lengths (\(l=5\)).