## 4 Extensions of RUMVRHF Smoother
### Extension to Reduced Computational Complexity Receding Horizon Fixed-Lag Smoother  
Although the proposed RUMVRHF and RTSRHF smoothers can provide more computational efficiency than RH smoothers in batch form, the computational complexity can be further improved by combining and sharing their estimates. In this section, we proposed the reduced computational complexity RHF (RCCRF) smoother that periodically switches between RTSRHF and RUMVRHF smoothers to reduce the computational complexity of the proposed recursive smoothing algorithms. The concept of RCCRF smoothing algorithm is shown in Figure 3. As can be seen in the figure, the basic concept of the RCCRF smoothing algorithm is a periodical switching RTSRHF and RUMVRHF smoothing algorithms and sharing their estimates. To illustrate the proposed RCCRF smoothing algorithm, let us suppose that the RTSRHF smoother is selected at time \(k-h\) and it estimates the lag state \(x_{k-2h}\) by using measurements in the horizon \([k-N\;\;k-h]\). The RCCRF smoothing algorithm is started from the RTSRHF smoother. The horizon initial state \(\hat{x}^{+}_{1,k-N|k-h}\) and its error covariance \(P^{+}_{1,k-N}\) are estimated through steps 1-4 of Algorithm 2. If we define the mid estimate and its error covariance as \(\hat{x}^{+*}_{k-h|k-h}=\hat{x}^{+}_{1,k-h|k-h}\) and \(\bar{P}^{+*}_{k-h}=P^{+}_{1,k-h}\), respectively, these are obtained through the steps 5-9 in Algorithm 2. Thereafter, the final estimate \(\hat{x}^{+}_{k-2h|k-h}\) and its error covariance \(P^{+}_{k-2h}\) are finally obtained through the steps 10-13 in Algorithm 2), with boundary conditions \(\hat{x}^{+*}_{k-h|k-h}\) and \(\bar{P}^{+*}_{k-h}\). It is noted that this process is exactly same as the RHRTS smoothing algorithm except that the overall horizon length is taken as \(N-h\), not \(N\). Moreover, it is also noted that the mid estimate \(\hat{x}^{+*}_{k-h|k-h}\) and its error covariance \(\bar{P}^{+*}_{k-h}\) are the same as the estimation results of minimum variance RH (MVRH) filter for the horizon \([k-N\;\;k-h]\). Figure 2: Concept of the RHRTSF smoother. Now, suppose that the smoothing mode is changed to RUMVRHF smoother at time \(k\) and it estimates the lag state \(x_{k-h}\) by using measurements on the horizon \([k-N\;\;k]\). In the RUMVRHF smoothing process, the local estimate \(\hat{x}^{+}_{1,k-h|k}\) and its error covariance \(P^{+}_{1,k-h}\) for the sub-horizon \([k-N\;\;k-h]\) should be obtained through the MVRH filter. This estimation process is equivalent to obtain the mid estimate and its error covariance of RTSRHF smoother. This means that the mid estimate and its error covariance of RTSRHF smoother at time \(k-h\) can be used as the local estimate and error covariance of RUMVRHF smoother at time \(k\) for the sub-horizon \([k-N\;\;k-h]\), i.e., \(\hat{x}^{+}_{1,k-h|k}=\hat{x}^{+*}_{k-h|k-h}\) and \(P^{+}_{1,k-h}=P^{+*}_{k-h}\) respectively. Thus, the overall smoothed estimate \(\hat{x}^{+}_{k-h|k}\) and its error covariance \(P^{+}_{k-h}\) of RUMVRHF smoother are obtained as follows:  
\[\hat{x}^{+}_{k-h|k} = U_{p1,k-h}\hat{x}^{+*}_{k-h|k-h}+U_{p2,k-h}\hat{x}^{+}_{2,k-h|k}, \tag{97}\] \[P^{+}_{k-h} = \bigg{\{}\Big{(}P^{+*}_{k-h}\Big{)}^{-1}+\Big{(}P^{+}_{2k-h} \Big{)}^{-1}\bigg{\}}^{-1},\] (98) \[U_{p1,k-h} = P^{+}_{2,k-h}\Big{(}P^{+*}_{k-h}+P^{+}_{2,k-h}\Big{)}^{-1},\] (99) \[U_{p2,k-h} = P^{+*}_{k-h}\Big{(}P^{+*}_{k-h}+P^{+}_{2,k-h}\Big{)}^{-1}, \tag{100}\]  
where \(\hat{x}^{+}_{2,k-h|k}\) and \(P^{+}_{2,k-h}\) are the local estimate and its error covariance matrix for the sub-horizon \([k-h\;\;k]\), respectively, and \(\hat{x}^{+*}_{k-h|k-h}\) and \(P^{*+}_{k-h}\) are obtained from the RTSRHF smoother on the overall horizon \([k-N-h\;\;k-h]\), not \([k-N\;\;k-h]\). Therefore, RCCRHF smoother can be designed by switching the roles of RTSRHF and RUMVRHF smoothers in every \(h\) time step periodically.