## 1 Introduction
To the best of the authors' knowledge, there are few results on the recursive RHF smoothing problem. In [19], an unbiased recursive RHF smoother based on receding horizon filtering was proposed for discrete time-invariant systems. Because the proposed unbiased RHF smoother is designed to ignore the statistical information of processes and measurement noise, it affords a more robust estimate than the FK smoother when the horizon length is selected appropriately. However, the proposed recursive unbiased RHF smoother requires a heavy computational load to find the optimal horizon length, in addition to obtaining the initial conditions and smoothed estimates from the filtering results via batch calculations. Moreover, because the inverse of the state transition matrix is required to establish the estimator gain matrix, it cannot be applied to systems with a singular state transition matrix, making the estimation problem unfeasible. A different type of recursive RHF smoother, finite-memory structure (FMS) smoother, was proposed in [21]. In the derivation of the RHF smoother, smoothed estimates are obtained from a one-step-ahead estimate, which affords a fast estimation ability. Although the structure of recursive estimation seems appropriate, the performance might be degraded by the system and measurement noises because the noise information is not considered in the derivation. Owing to its IIR iteration structure, the estimation stability cannot be guaranteed, resulting in the divergence of the estimation result due to accumulated errors such as estimation and numerical errors in the processing. Furthermore, a batch calculation is required during the estimate update and it also makes the matrix assumption of an invertible state transition matrix. Additionally, although the recursive RHF smoothing algorithm provides computational efficiency for time-varying systems, the previous recursive RHF smoothing algorithms were solely proposed for time-invariant systems. Therefore, in this study, we propose a UMVRHF smoother in batch and recursive forms involving separated sub-horizons for time-varying systems. The proposed UMVRHF smoother is designed by combining two independent receding-horizon local estimators for separated sub-horizons. The local estimates are obtained based on optimal receding horizon filtering and smoothing in terms of the unbiased minimum variance; they are then optimally combined using [PERSON]'s theorem to obtain an overall unbiased optimal estimate. We also propose a recursive algorithm for the proposed UMVRHF smoother based on the one-step predicted finite horizon Kalman filter. The proposed UMVRHF smoothers provide unbiased optimal estimates of the state on the finite estimation horizon. In addition, the proposed smoothers do not require any assumptions about the a priori information of the horizon initial conditions and inverse of the state transition matrix. Moreover, because the recursive form of the proposed UMVRHF smoother is derived using the Kalman filtering algorithm, it is easy to understand and extend. Thus, as extensions of the proposed recursive UMVRHF (RUMVRHF) smoother, we introduce a method for a posteriori state estimation. We also propose an additional recursive RHF smoother based on the Rauch-Tung-Striebel smoothing algorithm and computational complexity reduction strategy. The main contributions of this study are as follows:  
* Novel optimal unbiased RHF smoothers are proposed in batch and recursive forms for linear discrete time-varying state-space models. * The proposed RHF smoothers are obtained by optimally combining two local optimal receding horizon filters for the separated sub-horizons. * The Rauch-Tung-Striebel-type RHF smoother and reduced computational complexity RHF smoother are proposed to improve the computational efficiency. * The proposed RHF smoothers exhibit good properties, such as computational efficiency, optimality, unbiasedness, robustness against temporary modeling uncertainty, bounded-input and bounded-output stability, and fast convergence speed.