## 2 An Adaptive Fading-Memory Receding-Horizon Filter
### An Adaptive Fading-Memory Receding-Horizon Filter for a Posteriori State Estimation

We first introduce the AFMRH filter for the a priori and a posteriori state estimation, which is the forward-pass filtering algorithm of the proposed smoother. On the horizon \([k-N\;\;k]\), a posteriori estimate \(\hat{x}^{+}_{k-N+i|k}\) and its error covariance matrix \(P^{+}_{k-N+i|k}\) can be obtained as follows:

\[= \mathsf{E}\Big{[}\big{(}y_{k-N+i}-C_{k-N+i}\hat{x}_{k-N+i|k-1} \big{)}\big{(}y_{k-N+i}-C_{k-N+i}\hat{x}_{k-N+i|k-1}\big{)}^{T}\Big{]}\]7 \[= \mathsf{E}\Big{[}\big{(}y_{k-N+i}-C_{k-N+i}\hat{x}_{k-N+i|k-1} \big{)}\big{(}y_{k-N+i}-C_{k-N+i}\hat{x}_{k-N+i|k-1}\big{)}^{T}\Big{]}\]8 (29) \[= \mathsf{E}\Big{[}\big{(}y_{k-N+i}-C_{k-N+i}\hat{x}_{k-N+i|k-1} \big{)}\big{(}y_{k-N+i}-C_{k-N+i}\hat{x}_{k-N+i|k-1}\big{)}^{T}\Big{]}\]9 (30) \[= C_{k-N+i}P_{k-N+i|k-1}C_{k-N+i}^{T}+R_{k-N+i}.\]0 (31) \[= C_{k-N+i}P_{k-N+i|k-1}C_{k-N+i}^{T}+R_{k-N+i}.\]1 (32) \[= C_{k-N+i}P_{k-N+i|k-1}C_{k-N+i}^{T}+R_{k-N+i}.\]2

where \(\lambda_{k-N+i}\) is obtained from (21), (22), and (27) with the estimated residual covariance \(\hat{\Delta}_{k-N+i|k}\) obtained by using the recent finite measurements on the horizon \([k-N\;\;k]\) as

\[= C_{k-N+i}P_{k-N+i|k-1}C_{k-N+i}^{T}+R_{k-N+i}.\]3

Similar to the design of the filter, the predicted residual covariance can be sequentially obtained for reducing the computational cost as

\[= C_{k-N+i}P_{k-N+i|k-1}C_{k-N+i}^{T}+R_{k-N+i}.\]4

where the initial value is obtained from the result of the previous horizon as \(\hat{\Delta}_{k-N|k}=\hat{\Delta}_{k-1|k-1}\) and

\[= C_{k-N+i}P_{k-N+i|k-1}C_{k-N+i}^{T}+R_{k-N+i}.\]5

for the first horizon \([1\;\;N]\). During the time interval \([k-h\;\;k]\), the a priori and a posteriori estimates and their corresponding covariance matrices are stored in memory for the backward RTS smoothing, where h is the fixed-lag size. The initial conditions of forward recursion are calculated from the following recursive equations [14]:

\[= C_{k-N+i}P_{k-N+i|k-1}C_{k-N+i}^{T}+R_{k-N+i}.\]6 \[= C_{k-N+i}P_{k-N+i|k-1}C_{k-N+i}^{T}+R_{k-N+i}.\]7where \(\hat{w}_{k-N-1|k}^{+}\) and \(S_{k-N-1|k}^{+}\) are the a posteriori estimate of the information state and the a posteriori information matrix, respectively, obtained from

\[= C_{k-N+i}P_{k-N+i|k-1}C_{k-N+i}^{T}+R_{k-N+i}.\]8 \[= C_{k-N+i}P_{k-N+i|k-1}C_{k-N+i}^{T}+R_{k-N+i}.\]9 (40) \[\mathsf{E}\big{[}\delta_{k-N+i+j|k-1}\delta_{k-N+i|k-1}^{T}\big{]} \tag{13}\]0 \[\mathsf{E}\big{[}\delta_{k-N+i+j|k-1}\delta_{k-N+i|k-1}^{T}\big{]} \tag{13}\]1 (41) \[\mathsf{E}\big{[}\delta_{k-N+i+j|k-1}\delta_{k-N+i|k-1}^{T}\big{]} \tag{13}\]2 (42) \[\mathsf{E}\big{[}\delta_{k-N+i+j|k-1}\delta_{k-N+i|k-1}^{T}\big{]} \tag{13}\]3

with the initial conditions \(S_{k|k}^{+}=0\) and \(\hat{w}_{k|k}^{+}=0\). ### Backward RTS Smoothing

The backward recursion of the RTS smoothing algorithm runs backward from current time \(k\) down to the fixed-lag time \(h\) to estimate the fixed-lag state \(\hat{x}_{k-h|k}^{b+}\), in which the superscript \(b\) denotes backward estimation. The backward state and its error covariance matrix conditions of backward smoothing are initialized as \(\hat{x}_{k|k}^{b+}=\hat{x}_{k|k}^{+}\) and \(P_{k|k}^{b+}=P_{k|k}^{+}\), respectively, which are obtained from the forward filtering pass. Then, the smoothed estimates and their corresponding covariance matrices are calculated recursively from \(k\) down to \(h\) as [14]

\[\mathsf{E}\big{[}\delta_{k-N+i+j|k-1}\delta_{k-N+i|k-1}^{T}\big{]} \tag{13}\]4 \[\mathsf{E}\big{[}\delta_{k-N+i+j|k-1}\delta_{k-N+i|k-1}^{T}\big{]} \tag{13}\]5 (44) \[\mathsf{E}\big{[}\delta_{k-N+i+j|k-1}\delta_{k-N+i|k-1}^{T}\big{]} \tag{13}\]6

for \(1\leq l\leq h\), where \(x_{k-l|k}^{+b}\) and \(P_{k-l|k}^{+b}\) are the backward state estimate and corresponding covariance matrix, respectively, and \(K_{k-l|k}^{b}\) is the gain matrix of the backward smoother.