## 3 Unbiased Minimum Variance Receding-Horizon Fixed-Lag Smoother in Recursive Form
To obtain the recursive form of \(P_{k-n}\) and \(\hat{x}_{k-n|k-m-1}\), we must know the recursive calculations of \(S_{k-n}\), \(S_{k-n}^{+}\), \(\tilde{w}_{k-n|k-m-1}\), and \(\tilde{w}_{k-n|k-m-1}^{+}\). Before deriving the recursive form, we introduce the useful equality as follows:  
\[\Pi_{i+1,j}^{-1} = \left(\tilde{G}_{i+1,j}\tilde{Q}_{i,j}\tilde{G}_{i+1,j}^{T}+ \tilde{R}_{i,j}\right)^{-1} \tag{82}\] \[= \left(\left[ \begin{array}{c|c}\frac{C_{k-i}G_{k-i}}{\tilde{C}_ {i,j}\tilde{G}_{k-i}}\end{array} \right]\tilde{Q}_{i,j}\left[ \begin{array}{c |c}\frac{C_{k-i}G_{k-i}}{\tilde{C}_{i,j}\tilde{G}_{k-i}}\end{array} \right]^{T }+\tilde{R}_{i,j}\right)^{-1}\] \[= \left(\left[ \begin{array}{c|c}\frac{C_{k-i}}{\tilde{C}_{i,j} \tilde{G}_{k-i}}\end{array} \right]\tilde{Q}_{i,j}\left[ \begin{array}{c|c} \frac{C_{k-i}G_{k-i}}{\tilde{C}_{i,j}\tilde{G}_{k-i}}\end{array} \right]^{T }+\left[ \begin{array}{c|c}\frac{0}{\tilde{G}_{i,j}}\end{array} \right]\tilde {Q}_{i,j}\left[ \begin{array}{c|c}\frac{0}{\tilde{G}_{i,j}}\end{array} \right]^{T }+\tilde{R}_{i,j}\right)^{-1}\] \[= \left(\left[ \begin{array}{c|c}\frac{C_{k-i}}{\tilde{C}_{i,j} }\end{array} \right]G_{k-i}\tilde{Q}_{k-i}\tilde{G}_{k-i}^{T}\left[ \begin{array}[ ]{c|c}\frac{C_{ki}}{\tilde{C}_{i,j}}\end{array} \right]^{T}+\left[ \begin{array}[ ]{c|c}\frac{R_{k-i}}{0}&0\\ \hline\tilde{\Pi}_{i,j}\end{array} \right]\right)^{-1}\] \[= \left[ \begin{array}{c|c}\frac{R_{k-i}}{0}&0\\ \hline 0&\tilde{\Pi}_{i,j}\end{array} \right]^{-1}-\left[ \begin{array}{c|c} \frac{R_{k-i}}{0}&0\\ \hline\tilde{\Pi}_{i,j}\end{array} \right]^{-1}\left[ \begin{array}{c|c}\frac{C_ {k-i}}{\tilde{C}_{i,j}}\end{array} \right]G_{k-i}\left(Q_{k-i}^{-1}+G_{k-i}^{T}\] \[\times\left[ \begin{array}{c|c}\frac{C_{k-i}}{\tilde{C}_{i,j} }\end{array} \right]^{T}\left[ \begin{array}{c|c}\frac{R_{k-i}}{0}&0\\ \hline\tilde{\Pi}_{i,j}\end{array} \right]^{-1}\] \[\times\left[ \begin{array}{c|c}\frac{R_{k-i}}{0}&0\\ \hline\end{array} \right]^{-1}.\]Using (82), the recursive equations of \(S^{+}\) and \(\hat{w}^{+}_{\cdot\cdot}\) can be obtained as follows:  
\[S^{+}_{k-m-l-1} = \bar{C}^{T}_{m+l+1,m+1}\bar{\Gamma}^{-1}_{m+l+1,m+1}\bar{C}_{m+l+1,m +1}\] \[= A^{T}_{k-m-l-1}\bigg{[} \begin{array}{c}\bar{C}_{k-m-l}\\ \bar{C}_{m+l,m+1}\end{array} \bigg{]}^{T}\Big{(}\bar{G}_{m+l+1,m+1}\bar{Q}_{m+l, m+1}\bar{G}^{T}_{m+l+1,m+1}\] \[+\bar{R}_{m+l,m+1}\Big{)}^{-1}\bigg{[} \begin{array}{c}\bar{C}_{ k-m-l}\\ \bar{C}_{m+l,m+1}\end{array} \bigg{]}A_{k-m-l-1}\] \[= A^{T}_{k-m-l-1}\Big{(}C^{T}_{k-m-l}R^{-1}_{k-m-l}C_{k-m-l}+S_{k-m -l}{}^{+}\Big{)}A_{k-m-l-1}\] \[-A^{T}_{k-m-l-1}\Big{(}C^{T}_{k-m-l}R^{-1}_{k-m-l}C_{k-m-l}+S_{k-m -l}{}^{+}\Big{)}G_{k-m-l}\Big{\{}Q^{-1}_{k-m-l}\] \[+G^{T}_{k-m-l}\Big{(}C^{T}_{k-m-l}R^{-1}_{k-m-l}C_{k-m-l}+S_{k-m-l }{}^{+}\Big{)}G_{k-m-l}\Big{\}}^{-1}G^{T}_{k-m-l}\] \[\times\Big{(}C^{T}_{k-m-l}R^{-1}_{k-m-l}C_{k-m-l}+S^{+}_{k-m-l} \Big{)}A_{k-m-l-1},\] \[= A^{T}_{k-m-l-1}S_{k-m-l}A_{k-m-l-1}-A^{T}_{k-m-l-1}S_{k-m-l}G_{k -m-l}\Big{(}Q^{-1}_{k-m-l}\] \[+G^{T}_{k-m-l}S_{k-m-l}G_{k-m-l}\Big{)}^{-1}G^{T}_{k-m-l}S_{k-m-l }A_{k-m-l-1},\]  
and  
\[\hat{w}^{+}_{k-m-l-1|k-m-1} = \bar{C}^{T}_{m+l+1,m+1}\bar{\Gamma}^{-1}_{m+l+1,m+1}Y_{k-m-l,k-m-1}, \tag{84}\] \[= A^{T}_{k-m-l-1}\bigg{[} \begin{array}{c}\bar{C}_{k-m-l}\\ \bar{C}_{m+l,m+1}\end{array} \bigg{]}^{T}\Big{(}\bar{G}_{m+l+1,m+1}\bar{Q}_{m+l }\bar{G}^{T}_{m+l+1,m+1}\] \[+\bar{R}_{m+l,m+1}\Big{)}^{-1}\bigg{[} \begin{array}{c}y_{k-m-l} \\ Y_{k-m-l+1,k-m-1}\end{array}