## 1 Introduction
Therefore, this study focuses on a comprehensive estimation algorithm by combining the advantages of the AFM estimation and the RH structure. Based on the concept of the AFM technique and the recursive RH filtering structure, an AFMRH filtering algorithm is proposed in this study. Additionally, an RH fixed-lag smoother can provide a more accurate estimate than an RH filter [7]. Because there is no report on an adaptive RH smoother and the filtering result can be expanded to the fixed-lag smoothing problem, an AFMRH smoother based on the Rauch-Tung-Striebel (RTS) algorithm [28; 29] is also proposed for a linear discrete time-varying state-space model. If all measurements have the same weight, the proposed AFMRH filter and smoother are reduced to optimal RH filter and smoother, respectively. Therefore, it is significant that the proposed filter and smoother can provide a more general solution than the previous optimal RH filter and smoother. Furthermore, the estimation performance of the proposed AFMRH estimators can be improved by choosing the appropriate horizon length, because the horizon length is a key design parameter of the RH filter. For example, if the model uncertainty or numerical error exists, the RH estimator with a short horizon length can provide better robustness compared to the RH filter with a long one. Moreover, it can enable fast-tracking ability and computational efficiency. Conversely, if the model accurately represents the real system, the estimate of the RH filter with a long horizon length is more accurate than that of the RH filter with a short one. Although the RH filter with a long horizon length is the proper choice for an accurate system model, the computational complexity considerably increases as the horizon length grows. Thus, the horizon length should be properly chosen to compromise between the estimation accuracy and the computational efficiency. Several algorithms for determining an appropriate value of the horizon length have been suggested for linear time-invariant systems [30; 31]. The proposed algorithms provide an effective way for linear time-invariant systems. However, they suffer from heavy computational burden. Moreover, if there are unpredictable temporary model uncertainties, the fixed and constant horizon lengths of the proposed algorithms are not satisfactory even for the linear time-invariant systems. Furthermore, these algorithms cannot guarantee the solution for time-varying systems and are difficult to apply in practice [32]. For temporarily uncertain systems, switching an RH filter that comprises two RH filters with different horizon lengths was proposed for time-invariant systems [33]. Because the temporary uncertainty is detected using the chi-square test statistic, and a suitable RH filter is selected and operated to estimate the state, the proposed switching RH filter performs well and provides good estimation results for both nominal and temporarily uncertain systems. However, the threshold value used to determine the temporary uncertainty should be precomputed, such that it is difficult to apply on time-varying systems. For time-varying systems, the switching RH filter-bank-based adaptive horizon-selection method was proposed in [34]. Because the horizon length is adjusted each time to adapt to the changes in system characteristics, the proposed algorithm exhibits better estimation performance than previous approaches with a fixed-constant horizon length. However, this method incurs a heavy computational cost because several different RH filters should always be operated simultaneously, and the calculation should be repeated until an appropriate value of the horizon length is achieved. Therefore, a novel adaptive-horizon-length adjustment (AHLA) algorithm was also proposed in this study to improve the computational efficiency and estimation performance of the proposed AFMRH estimators.