## 3 Unbiased Minimum Variance Receding-Horizon Fixed-Lag Smoother in Recursive Form  
In this section, we derive the recursive form of optimal estimators for each sub-horizon. For the optimal estimator on the sub-horizon \([k-N\ \ k-h-1]\), we introduce the one-step predicted Kalman filter and its batch form. The dynamic equation of the one-step predicted Kalman filter in recursive form is expressed as [1]  
\[\hat{x}_{1,k-N+i+1|k-1} = A_{k-N+i}\hat{x}_{1,k-N+i|k-1}-A_{k-N+i}\tilde{P}_{1,k-N+i}C_{k -N+i}^{T}\ \Big{(}C_{k-N+i}\tilde{P}_{1,k-N+i} \tag{57}\] \[\times C_{k-N+i}^{T}+R_{k-N+i}\Big{)}^{-1}\Big{(}y_{k-N+i}-C_{k-N+ i}\hat{x}_{1,k-N+i|k-1}\Big{)},\] \[\tilde{P}_{1,k-N+i+1} = A_{k-N+i}\tilde{P}_{1,k-N+i}A_{k-N+i}^{T}-A_{k-N+i}\tilde{P}_{1,k -N+i}C_{k-N+i}^{T}\Big{(}C_{k-N+i}\] (58) \[\times\tilde{P}_{1,k-N+i}C_{k-N+i}^{T}+R_{k-N+i}\Big{)}^{-1}C_{k -N+i}\tilde{P}_{1,k-N+i}A_{k-N+i}^{T}\] \[+G_{k-N+i}\tilde{Q}_{k-N+i}\tilde{G}_{k-N+i}^{T},\]for \(0\leq i\leq N-h-1\), where \(\hat{x}_{1,k-N+i|k-1}\) and \(\bar{P}_{1,k-N+i}\) denote the \(i\)-th estimate and its error covariance matrix on the sub-horizon \([k-N\;\;k-h-1]\), respectively. The batch form of the one-step predicted Kalman filter (57) and (58) for the finite subhorizon \([k-N\;\;k-h-1]\) can be represented as follows:  
**Lemma 2**. : _[_24_]_ _For the system model (1) and (2), the finite horizon one-step predicted Kalman filter in batch form can be represented on the horizon \([k-N\;\;k-h-1]\) as follows:_  
\[\hat{x}_{1,k-h|k-1}\] \[=\left[ \begin{array}{cc}\tilde{A}_{N,h+1}&\tilde{M}_{N,h+1}\\ \end{array} \right]\left[ \begin{array}{cc}\tilde{C}_{N,h+1}^{T}\tilde{R}_{N,h +1}^{-1}\tilde{C}_{N,h+1}+P_{k-N}^{-1}&\tilde{C}_{N,h+1}^{T}\tilde{R}_{N,h+1}^ {-1}\tilde{G}_{N,h+1}\\ \tilde{G}_{N,h+1}^{T}\tilde{R}_{N,h+1}^{-1}\tilde{C}_{N,h+1}&\tilde{G}_{N,h+1} ^{T}\tilde{R}_{N,h+1}^{-1}\tilde{G}_{N,h+1}+\tilde{Q}_{N,h+1}^{-1}\end{array}  \right]^{-1} \tag{59}\] \[\times\Bigg{(}\left[ \begin{array}{c}P_{k-N}^{-1}\\ 0\end{array} \right]x_{k-N}+\left[ \begin{array}{c}\tilde{C}_{N,h+1}^{T}\\ \tilde{G}_{N,h+1}^{T}\end{array} \right]\tilde{R}_{N,h+1}^{-1}Y_{k-N,k-h+1} \Bigg{)},\]  
_where_ \(x_{k-N}\) _and_ \(P_{k-N}\) _are the horizon initial state and its error covariance matrix, respectively, and the solution of the Riccati equation for the Kalman filter is given as follows:_  
\[P_{1,k-h}=\bar{P}_{1,k-h}\] \[=\left[ \begin{array}{cc}\tilde{A}_{N,h+1}&\tilde{M}_{N,h+1}\\ \end{array} \right]\left[ \begin{array}{cc}\tilde{C}_{N,h+1}^{T}\tilde{R}_{N,h +1}^{-1}\tilde{C}_{N,h+1}+P_{k-N}^{-1}&\tilde{C}_{N,h+1}^{T}\tilde{R}_{N,h+1 }^{-1}\tilde{G}_{N,h+1}\\ \tilde{G}_{N,h+1}^{T}\tilde{R}_{N,h+1}^{-1}\tilde{C}_{N,h+1}&\tilde{G}_{N,h+1} ^{T}\tilde{R}_{N,h+1}^{-1}\tilde{G}_{N,h+1}+\tilde{Q}_{N,h+1}^{-1}\end{array}  \right]^{-1} \tag{60}\] \[\times\left[ \begin{array}{cc}\tilde{A}_{N,h+1}&\tilde{M}_{N,h +1}\end{array} \right]^{T}.\]  
_With the following initial conditions,_  
\[x_{k-N} = \hat{x}_{1,k-N|k-1}=(\tilde{C}_{N,h+1}^{T}\tilde{\Pi}_{N,h+1}^{-1 }\tilde{C}_{N,h+1})^{-1}\tilde{C}_{N,h+1}^{T}\tilde{\Pi}_{N,h+1}^{-1}Y_{k-N,k- h-1}, \tag{61}\] \[P_{k-N} = \bar{P}_{1,k-N}=(\tilde{C}_{N,h+1}^{T}\tilde{\Pi}_{N,h+1}^{-1} \tilde{C}_{N,h+1})^{-1}, \tag{62}\]  
_the finite horizon one-step predicted Kalman filter (59) is equivalent to the optimal estimator (28) involving the optimal gain matrix (32)._  
Proof of Lemma 2.: We refer the reader to [24] for the proof of Lemma 2. The recursive form of the optimal estimator for the sub-horizon \([k-N\;\;k-h-1]\) can be obtained if the initial conditions (61) and (62) are calculated recursively. Moreover, the local estimate (55) and its error covariance matrix (56) for the sub-horizon \([k-h\;\;k-1]\) have the same equation structure because they are also the initial conditions of the sub-horizon. The following theorem proves that the initial estimate and its error covariance matrix on the finite horizon can be obtained recursively. **Theorem 2**.