## 2 Unbiased Minimum Variance Receding-Horizon Fixed-Lag Smoother in Batch Form
### Unbiased Minimum Variance Estimation on Sub-Horizon \([k-h\;\;k-1]\)  
In this section, the optimal estimator in terms of unbiased minimum variance is derived to estimate the state at time \(k-h\) on the sub-horizon \([k-h\;\;k1]\). An unbiased minimum variance estimator to estimate the state at time \(k-h\) for the finite measurements on the sub-horizon \([k-h\;\;k-1]\) can be expressed as follows:  
\[\hat{\tau}_{2,k-h|k-1}=H_{2,k-h}Y_{k-h,k-1}. \tag{39}\]  
The finite number of measurements \(Y_{k-h,k-1}\) can be expressed in terms of the horizon initial state \(x_{k-h}\) as follows:  
\[Y_{k-h,k-1} = \tilde{C}_{h,1}x_{k-h}+\tilde{G}_{h,1}W_{k-h,k-1}+V_{k-h,k-1}, \tag{40}\]and the following constraint is required to satisfy the unbiased condition, i.e., \(\mathrm{E}[x_{x-h}]=\mathrm{E}\Big{[}\tilde{x}_{2,k-h|k-1}\Big{]}\)[20]:  
\[H_{2,k-h}\tilde{C}_{h,1}=I. \tag{41}\]  
From (40) and (41), the local estimation error \(e_{2,k-h}\) can be expressed as  
\[ \begin{array}{rcl}e_{2,k-h}&=&\tilde{x}_{2,k-h|k-1}-x_{k-h}\\ &=&H_{2,k-h}\tilde{Y}_{k-h,k-1}-x_{k-h}\\ &=&\big{(}H_{2,k-h}\tilde{C}_{h,1}-I\big{)}x_{k-h}+H_{2,k-h}\tilde{C}_{h,1}W_{k -h,k-1}+H_{2,k-h}V_{k-h,k-1}\\ &=&H_{2,k-h}\tilde{G}_{h,1}W_{k-h,k-1}+H_{2,k-h}V_{k-h,k-1}.\end{array}  \tag{42}\]  
Thereafter, the objective is to determine the optimal gain matrix \(H_{2,k-h}\) subject to the unbiased constraint (41), which can be obtained from the following theorem:  
**Theorem 1**. : _On the horizon \([k-h\;\;k-1]\), for the following unbiased minimum variance estimation problem,_  
\[\min_{H_{2,k-h}}\mathrm{E}\Big{[}e_{2,k-h}^{T}e_{2,k-h}\Big{]}, \tag{43}\]  
_subject to_  
\[\mathrm{E}\Big{[}\tilde{x}_{2,k-h|k-1}\Big{]}=\mathrm{E}[x_{k-h}], \tag{44}\]  
_the optimal gain matrix \(H_{2,k-h}\) of the optimal estimator (39) is given by_  
\[H_{2,k-h}=\Big{(}\tilde{C}_{h,1}^{T}\tilde{\Pi}_{h,1}^{-1}\tilde{C}_{h,1} \Big{)}^{-1}\tilde{C}_{h,1}^{T}\tilde{\Pi}_{h,1}. \tag{45}\]  
**Proof of Theorem 1. ** For convenience, the gain matrix \(H_{2,k-h}\) in (39) is partitioned as follows:  
\[H_{2,k-h}^{T}=\big{[}\;\;h_{1,k-h}\quad h_{2,k-h}\quad\cdots\quad h_{h,k-h}\; \;\big{]}, \tag{46}\]  
where \(h_{i,k-h}^{T}\) is the \(i\)-th row vector of the gain matrix \(H_{2,k-h}\). By denting the \(i_{i}^{T}\) as the \(i\)-th row vectors of identity matrix \(I\), then the \(i\)-th unbiased constraint of (41) can be expressed as follows:  
\[\tilde{C}_{h,1}^{T}h_{i,k-h}=i_{i}, \tag{47}\]  
for \(1\leq i\leq h\). Moreover, by denoting \(e_{2(i),k-h}\) as the \(i\)-th row vectors of the estimation error \(e_{2,k-h}\), calculating \(e_{2(i),k-h}^{2}\) and taking expectation, we obtain  
\[\mathrm{E}\Big{[}e_{2(i),k-h}^{2}\Big{]}=(h_{i,k-h}^{T}\tilde{C}_{h,1})Q_{h}( h_{i,k-h}^{T}\tilde{C}_{h,1})^{T}+h_{i,k-h}^{T}\tilde{R}_{h,1}h_{i,k-h}. \tag{48}\]  
Note that the \(i\)-th component of the estimation error only depends on \(h_{i,k-h}\). Hence, the following performance criterion is established:  
\[\mathrm{E}\Big{[}e_{2(i),k-h}^{2}\Big{]} = h_{i,k-h}^{T}\Big{(}\tilde{C}_{h,1}\tilde{Q}_{h,1}\tilde{C}_{h,1 }^{T}+\tilde{R}_{h,1}\Big{)}h_{i,k-h}+\lambda_{i,k-h}^{T}\Big{(}\tilde{C}_{h,1 }^{T}h_{i,k-h}-i_{i}\Big{)}\] \[= h_{i,k-h}^{T}\tilde{\Pi}_{h,1}h_{i,k-h}+\lambda_{i,k-h}^{T} \Big{(}\tilde{C}_{h,1}^{T}h_{i,k-h}-i_{i}\Big{)},\]  
where \(\lambda_{i,k-h}\) is \(i\)-th row vectors of the Lagrange multiplier, which is associated with the \(i\)-th unbiased constraint.