## 2 An Adaptive Fading-Memory Receding-Horizon Filter
### An Adaptive Fading-Memory Receding-Horizon Filter
A comparison of all RMSE errors reveals that RMSE errors of the proposed AFMRH filter with the AHLA method and SAFMRH filter are smaller than other fading-memory RH filters irrespective of whether model uncertainty exists. For \(k<200\), i.e., in the situation where the model uncertainty does not exist, the proposed SAFMRH filter has the smallest value of RMSE error. For \(200\leq k\leq 350\), i.e., when the model mismatch exists, the RMSE errors of the proposed AFMRH filter with the AHLA method and SAFMRH filter are much smaller than that of others. The reasons for this result are as follows: because the switching condition could be met when the model mismatch does not exist and the Kalman filter is used instead of the RH filter with a long horizon length, the SAFMRH filter could provide a more accurate estimate than other fading-memory RH filters. Moreover, because the horizon length in the AHLA method is considered as a design parameter, the AHLA method-based filters yield better estimation performances than the AFMRH filter with constant horizon length. Furthermore, when the AHLA method is incorporated in its algorithm, the proposed SAFMRH filter could also provide better estimates concerning whether the model uncertainty exists or not. Thus, the combination of the AHLA method and switch filtering scheme with AFMRH filtering improves the robustness and estimation accuracy of the proposed AFMRH filter. Additionally, in the SAFMRH filtering algorithm, the computational cost for backward recursion to find the initial horizon conditions is less than that of the AFMRH filter successively about three times. The computationally efficient Kalman filtering algorithm is used instead of the proposed AFMRH filter with a long horizon length when the conditions are met. Thus, the proposed SAFMRH filtering algorithm is expected to reduce the computational cost of the AFMRH filter. The time-averaged computation times of all the RH filters are compared in Table 2 to show the improvement in computational efficiency. As expected, the results show that the proposed AFMRH filter with AHLA method and SAFMRH filtering algorithm considerably reduces the computation time compared with the AFMRH filter. Moreover, the proposed SAFMRH filter consumes much less computation time than other fading-memory RH filters. Therefore, the proposed AFMRH filter with AHLA method and SAFMRH filter are helpful and appropriate for a system with a temporary model uncertainty. Finally, the improved performance of the proposed AFMRH smoother is verified by comparing the simulation results of the proposed AFMRH filter, recursive optimal RH smoother [14], and CFMRH smoother [11]. The horizon length and fixed lag size of all the RH smoothers described in this section are taken as \(N=20\) and \(h=7\), respectively. Further, the constant fading factor for the CFMRH smoother is taken as 0.65. The estimation errors of estimation algorithms are shown in Figure 7. \begin{table}
\begin{tabular}{c c} Standard Kalman filter & 0.037 \\ AFM Kalman filter & 0.056 \\ Recursive optimal RH filter & 0.506 \\ CFMRH filter & 1.856 \\ AFMRH filter & 0.841 \\ AFMRH filter + AHLA & 0.575 \\ SAFMRH filter & **0.359** \\ \end{tabular}
\end{table} 
Table 2: Average computation time (ms). As indicated, the proposed AFMRH smoother provides a more robust estimate than recursive optimal RH and CFMRH smoothers for the model mismatch. Moreover, a comparison of the estimation errors of the proposed AFMRH filter and smoother revealed that the proposed AFMRH smoother exhibited better estimation performance than the proposed AFMRH filter, as expected. Thus, the proposed AFMRH smoother is more efficient and appropriate than the AFMRH filter for a system with model uncertainty if some time delay is tolerable. Furthermore, the proposed AFMRH smoother provides significant improved estimation performance compared to the nonfading and constant-fading-memory RH smoother. ## 6 Conclusions

In this study, adaptive RH filters and a smoother were proposed for linear discrete time-varying systems.