## 3 Unbiased Minimum Variance Receding-Horizon Fixed-Lag Smoother in Recursive Form
: _On the horizon \([k-n\;\;k-m-1]\), the optimal estimate of the horizon initial state \(\hat{x}_{k-n|k-m-1}\) and its error covariance \(P_{k-n}\) in batch form are given as follows:_  
\[\hat{x}_{k-n|k-m-1} = (\tilde{C}_{n,m+1}^{T}\tilde{\Pi}_{n,m+1}^{-1}\tilde{C}_{n,m+1})^{ -1}\tilde{C}_{n,m+1}^{T}\tilde{\Pi}_{n,m+1}^{-1}Y_{k-n,k-m-1}, \tag{63}\] \[P_{k-n} = \bar{P}_{k-n}=(\tilde{C}_{n,m+1}^{T}\tilde{\Pi}_{n,m+1}^{-1} \tilde{C}_{n,m+1})^{-1}, \tag{64}\]_where the initial conditions are obtained from_  
\[\hat{x}_{k-n|k-m-1} = S_{k-n}^{-1}\hat{w}_{k-n|k-m-1}, \tag{65}\] \[\hat{P}_{k-n} = S_{k-n}^{-1}, \tag{66}\]  
_where the priori information matrix \(S_{k-n}\) and the priori information estimate \(\hat{w}_{k-n|k-m-1}\) are calculated from the following recursive equations:_  
\[S_{k-m-l} = C_{k-m-l}^{T}R_{k-m-l}^{-1}C_{k-m-l}+S_{k-m-l}^{+}, \tag{67}\] \[S_{k-m-l-1}^{+} = A_{k-m-l-1}^{T}S_{k-m-l}A_{k-m-l-1}-A_{k-m-l-1}^{T}S_{k-m-l}G_{k- ml}\Big{(}Q_{k-m-l}^{-1}\] (68) \[+G_{k-m-l}^{T}S_{k-m-l}G_{k-m-l}\Big{)}^{-1}G_{k-m-l}^{T}S_{k-m-l} A_{k-m-l-1},\]  
_and_  
\[\hat{w}_{k-m-l|k-m-1} = C_{k-m-l}^{T}R_{k-m-l}^{-1}y_{k-m-l}+\hat{w}_{k-m-l|k-m-1}^{+}, \tag{69}\] \[P_{k-N} = \bar{P}_{1,k-N}=(\tilde{C}_{N,h+1}^{T}\tilde{\Pi}_{N,h+1}^{-1} \tilde{C}_{N,h+1})^{-1}, \tag{62}\]0 (70) \[P_{k-N} = \bar{P}_{1,k-N}=(\tilde{C}_{N,h+1}^{T}\tilde{\Pi}_{N,h+1}^{-1} \tilde{C}_{N,h+1})^{-1}, \tag{62}\]1  
_respectively, for \(0\leq l\leq n-m-1\), with the initial conditions \(S_{k-m-1}^{+}=0\) and \(\hat{w}_{k-m-1|k-m-1}^{+}=0\), where \(S_{k-n}^{+}\) and \(\hat{w}_{k-n|k-m-1}^{+}\) are the posteriori information matrix and the posteriori information estimate, respectively._  
**Proof of Theorem 2. ** With the definition of \(\hat{C}_{i,j}\), \(\hat{G}_{i,j}\), \(\hat{\Pi}_{i,j}\), \(S_{k-m-l}\), and \(\hat{w}_{k-m-l|k-m-1}\) as  
\[\hat{C}_{i,j} = \left[ \begin{array}{c}C_{k-i+1}\tilde{A}_{i,i}\\ C_{k-i+1}\tilde{A}_{i,i-1}\\ \vdots\\ C_{k-j}\tilde{A}_{i,j+1}\end{array} \right]=\left[ \begin{array}{c}C_{k-i+1}\\ \tilde{C}_{i-1,j}\end{array} \right]A_{k-i}, \tag{71}\] \[\hat{G}_{i,j} = \left[ \begin{array}{c}C_{k-i+1}G_{k-i+1}\\ C_{k-i+2}\tilde{A}_{i-1,j-1}G_{k-i+1}\\ \vdots\\ C_{k-j}\tilde{A}_{i-1,j+1}G_{k-i+1}\end{array} \right]=\left[ \begin{array}{ ccc}C_{k-i+1}G_{k-i+1}&0\\ \tilde{C}_{i-1,j}G_{k-i+1}&\tilde{G}_{i-1,j}\end{array} \right],\] (72) \[\hat{\Pi}_{i,j} = \tilde{G}_{i,j}\tilde{Q}_{i-1,j}\tilde{G}_{i,j}^{T}+\tilde{R}_{i-1,j},\] (73) \[S_{k-m-l} = \tilde{C}_{m+l,m+1}^{T}\hat{\Pi}_{m+l,m+1}^{-1}\hat{C}_{m+l,m+1},\] (74) \[\hat{w}_{k-m-l|k-m-1} = \tilde{C}_{m+l,m+1}^{T}\hat{\Pi}_{m+l,m+1}^{-1}Y_{k-m-l+1,k-m-1}, \tag{75}\]  
for \(2\leq l\leq n-m\), respectively, \(\tilde{C}_{n,m+1}\) and \(\hat{\Pi}_{n,m+1}\) can be rewritten as follows:\[\tilde{C}_{n,m+1} = \left[ \begin{array}{c}\tilde{C}_{k-n}\\ \tilde{C}_{n,m+1}\end{array} \right], \tag{76}\] \[\tilde{\Pi}_{n,m+1} = \tilde{G}_{n,m+1}\tilde{Q}_{n,m+1}\tilde{G}_{n,m+1}^{T}+\tilde{R}_ {n,m+1}\] (77) \[= \left[ \begin{array}{c|c}0&0\\ \tilde{G}_{n,m+1}&0\end{array} \right]\tilde{Q}_{n,m+1}\left[ \begin{array}{c |c}0&0\\ \tilde{G}_{n,m+1}&0\end{array} \right]^{T}+\tilde{R}_{n,m+1}\] \[= \left[ \begin{array}{c|c}R_{k-n}&0\\ \hline 0&\tilde{\Pi}_{n,m+1}\end{array} \right],\]  
respectively. From (76) and (77), the estimated horizon initial error covariance \(P_{k-n}\) (64) and state \(\hat{x}_{k-n|k-m-1}\) (63) can be represented as follows:  
\[P_{k-n} = (C_{k-n}^{T}R_{k-n}^{-1}C_{k-n}+S_{k-n}^{+})^{-1}=S_{k-n}^{-1}, \tag{78}\] \[\hat{x}_{k-n|k-m-1} = P_{k-n}\left(C_{k-n}^{T}R_{k-n}^{-1}y_{k-n}+\tilde{C}_{n,m+1}^{T }\tilde{\Pi}_{n-m+1}^{-1}Y_{k-n+1,k-m-1}\right),\] (79) \[= (C_{k-n}^{T}R_{k-n}^{-1}C_{k-n}+S_{k-n}^{+})^{-1}\left(C_{k-n}^{ T}R_{k-n}^{-1}y_{k-n}+\tilde{w}_{k-n|k-m-1}^{+}\right),\] \[= S_{k-n}^{-1}\tilde{w}_{k-n|k-m-1},\]  
where \(S_{k-n}\) and \(\tilde{w}_{k-n|k-m-1}\) are denoted as  
\[S_{k-n} = C_{k-n}^{T}R_{k-n}^{-1}C_{k-n}+S_{k-n}^{+}, \tag{80}\] \[\tilde{w}_{k-n|k-m-1} = C_{k-n}^{T}R_{k-n}^{-1}y_{k-n}+\tilde{w}_{k-n|k-m-1}^{+}, \tag{81}\]  
respectively.