## 2 Unbiased Minimum Variance Receding-Horizon Fixed-Lag Smoother in Batch Form
### Unbiased Minimum Variance Estimation on the Sub-Horizon \([k-N\ k-h-1]\)  
In this section, we derive an unbiased minimum variance estimator to estimate the state at time \(k-h\) using given finite measurements on the sub-horizon \([k-N\ k-h-1]\). Before we derive the optimal estimator, we define the following matrices:  
\[\tilde{A}_{i,j}=A_{k-j}\cdots A_{k-i+1}A_{k-i}\ (j\leq i) \tag{19}\]\[\tilde{C}_{i,j}\stackrel{{\triangle}}{{=}}\left[ \begin{array}{c}C_{k-i} \\ C_{k-i+1}\tilde{A}_{i,i}\\ C_{k-i+2}\tilde{A}_{i,j-1}\\ \vdots\\ C_{k-j-1}\tilde{A}_{i,j+2}\\ C_{k-j}\tilde{A}_{i,j+1}\end{array} \right], \tag{20}\]  
\[\tilde{M}_{i,j}=\left[ \begin{array}{cccc}\tilde{A}_{i,j+1}G_{k-i}&\tilde{A}_ {i-1,j+1}G_{k-i+1}&\cdots&\tilde{A}_{j+1,j+1}G_{k-j-1}&G_{k-j}\end{array}  \right], \tag{21}\]  
\[\tilde{G}_{i,j}\stackrel{{\triangle}}{{=}}\left[ \begin{array}{ ccccc}0&0&\cdots&0&0\\ C_{k-i+1}G_{k-i+1}&0&\cdots&0&0\\ C_{k-i+2}\tilde{A}_{i-1,i-1}G_{k-i+1}&C_{k-i+2}G_{k-i+2}&\cdots&0&0\\ C_{k-i+3}\tilde{A}_{i-1,j-2}G_{k-i+1}&C_{k-i+3}\tilde{A}_{i-2,j-2}G_{k-i+2}& \cdots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ C_{k-j-1}\tilde{A}_{i-1,j+2}G_{k-i+1}&C_{k-j-1}\tilde{A}_{i-2,j+2}G_{k-i+2}& \cdots&0&0\\ C_{k-j}\tilde{A}_{i-1,j+1}G_{k-i+1}&C_{k-j}\tilde{A}_{i-2,j+1}G_{k-i+2}&\cdots &C_{k-j}G_{k-j}&0\end{array} \right], \tag{22}\]  
\[\tilde{Q}_{i,j}=diag\left(\frac{\mathrm{i}}{\tilde{Q}_{k-i},Q_{k-i+1},\cdots,Q _{k-j}}\right), \tag{23}\]  
\[\tilde{R}_{i,j}=diag\left(\frac{\mathrm{i}}{\tilde{R}_{k-i},R_{k-i+1},\cdots, R_{k-j}}\right), \tag{24}\]  
\[\tilde{\Pi}_{i,j}=\tilde{G}_{i,j}\tilde{Q}_{i,j}\tilde{G}_{i,j}^{T}+\tilde{R}_ {i,j}, \tag{25}\]  
\[W_{k-i,k-j}=\left[ \begin{array}{cccc}w_{k-i}^{T}&w_{k-i+1}^{T}&\cdots&w_{k-j -1}^{T}&w_{k-j}^{T}\end{array} \right]^{T}, \tag{26}\]  
\[V_{k-i,k-j}=\left[ \begin{array}{cccc}v_{k-i}^{T}&v_{k-i+1}^{T}&\cdots&v_{k-j -1}^{T}&v_{k-j}^{T}\end{array} \right]^{T}. \tag{27}\]  
On the sub-horizon \([k-N\;\;k-h-1]\), an optimal estimator in terms of unbiased minimum variance with batch form for the state at \(k-h\) can be expressed as a linear function of the finite measurements, as follows:  
\[\hat{x}_{1,k-h|k-1}=H_{1,k-h}Y_{k-N,k-h-1}. \tag{28}\]  
where the optimal gain matrix \(H_{1,k-h}\) is determined by following lemma. Lemma 1: _On the horizon \([k-N\;\;k-h-1]\), for the system models (1) and (2) and the following unbiased minimum variance estimation problem:_  
\[\min_{H_{1,k-h}}\mathrm{E}\Big{[}e_{1,k-h}^{T}e_{1,k-h}\Big{]}, \tag{29}\]  
_subject to_  
\[\mathrm{E}\Big{[}\hat{x}_{1,k-h|k-1}\Big{]}=\mathrm{E}[x_{k-h}], \tag{30}\]  
_where the local estimation error \(e_{1,k-h}\) is defined as_  
\[e_{1,k-h}=\hat{x}_{1,k-h|k-1}-x_{k-h}. \tag{31}\]  
_The optimal gain matrix \(H_{1,k-h}\) of the optimal estimator (28) is given by [24]_\[H_{1,k-h} = \left[ \begin{array}{cc}\tilde{A}_{N,h+1}&\tilde{M}_{N,h+1}\end{array}  \right]\left[ \begin{array}{cc}\tilde{C}_{N,h+1}^{T}\tilde{R}_{N,h+1}^{-1} \tilde{C}_{N,h+1}&\tilde{C}_{N,h+1}^{T}\tilde{R}_{N,h+1}^{-1}\tilde{G}_{N,h+1}\\ \tilde{G}_{N,h+1}^{T}\tilde{R}_{N,h+1}^{-1}\tilde{C}_{N,h+1}&\tilde{G}_{N,h+1}^ {T}\tilde{R}_{N,h+1}^{-1}\tilde{G}_{N,h+1}+\tilde{Q}_{N,h+1}^{-1}\end{array}  \right]^{-1}\] \[\times\left[ \begin{array}{c}\tilde{C}_{N,h+1}^{T}\\ \tilde{G}_{N,h+1}^{T}\end{array} \right]\tilde{R}_{N,h+1}^{-1}.\]  
**Proof of Lemma 1. ** For a proof of Lemma 1, we refer the reader to [24].