## 2 An Adaptive Fading-Memory Receding-Horizon Filter
### An Adaptive Fading-Memory Receding-Horizon Filter
The proposed AFMRH filter was first obtained by combining receding-horizon and adaptive fading-memory methods. The adaptive fading factor was optimally obtained using the measurements and estimates in the recent finite horizon, and applied to the recursive RH filtering algorithm to adjust the state error covariance. Moreover, as an extension of the proposed AFMRH filter, an AFMRHF smoother based on the RTS smoothing algorithm was also proposed to improve the estimation accuracy of the proposed RH filter. Furthermore, an AFLA method and a switching AFMRH filtering method were proposed to improve the estimation accuracy and reduce the computational cost of the proposed AFMRH filter. Owing to the RH estimation structure, the proposed approaches guarantee BIBO stability and may have a fast convergence speed and robustness against temporary model uncertainties. Moreover, if the fading factor is taken as one, the proposed AFMRH filter and smoother are equivalent to the existing recursive optimal RH filter and smoother. Notably, the proposed AFMRH estimation algorithms can provide a more general solution than the previous optimal recursive RH filter. Furthermore, as the proposed AHLA method uses results in the estimation process, it may have less computational burden and provide a more suitable method for online and real-time applications compared with previous approaches. The proposed algorithms can be applied to state estimation problems in linear discrete-time systems with unknown noise statistics, model uncertainties, and external disturbances. The simulation results show that the adaptive RH estimators proposed in this study can exhibit better estimation performance compared with the Kalman-filter-based AFM filter and existing RH estimators. Figure 7: Estimation errors of smoothers. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data sharing is not applicable to this article. Conflicts of Interest: The author declares no conflict of interest. Abbreviations The following abbreviations are used in this manuscript: RH & Receding Horizon FM & Fading Memory CFM & Constant Fading Memory AFM & Adaptive Fading Memory AHLA & Adaptive Horizon Length Adjustment CFMRH & Constant Fading Memory Receding Horizon AFMRH & Adaptive Fading Memory Receding Horizon SAFMRH & Switching Adaptive Fading Memory Receding Horizon RTS & Rauch-Tung-Striebel RMSE & Root Mean Square Estimation

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