## 2 An Adaptive Fading-Memory Receding-Horizon Filter
### An Adaptive Fading-Memory Receding-Horizon Filter  
Because the recursive optimal RH filter estimates the current state using recent finite-horizon measurements, it guarantees BIBO stability, fast tracking, and robustness against modeling uncertainties. It is still based on precise mathematical models and can only deal with dynamic linear systems. Although the recursive optimal RH filter has good properties, it is derived from a precise system model. Thus, when the system model and prior statistical information of noises are unknown or inaccurate, its estimation performance can degrade. As the mismatched system model and inaccurate noise covariances can be adjusted according to measurements, an AFMRH filter is proposed by combining recursive optimal RH filtering with an adaptive filtering algorithm based on the adaptive fading factor. The equations of the AFMRH filter can be described with those of the recursive optimal RH filter in Equations (3)-(9), except the error covariance update equation, as  
\[P_{k-N+i+1|k-1} = \lambda_{k-N+i|k-1}A_{k-N+i}\big{(}P_{1,k-N+i|k-1}-K_{k-N+i|k-1} C_{k-N+i}P_{k-N+i|k-1}\big{)} \tag{10}\] \[\times A_{k-N+i}^{T}+G_{k-N+i}Q_{k-N+i}G_{k-N+i}^{\mathsf{T}},\]  
where \(\lambda_{k-N+i|k-1}\geq 1\) is the adaptive fading factor. By adjusting the adaptive fading factor, most recent measurements are adaptively overweighted so that the estimation error due to incomplete information can be reduced. Now, the algorithm for selecting the optimal adaptive fading factor is presented. The residual, which is the difference between the measurement and predicted measurements, is represented as  
\[\delta_{k-N+i|k-1}=y_{k-N+i}-C_{k-N+i}\hat{x}_{k-N+i|k-1}. \tag{11}\]By substituting (11) into Equation (3) and taking the expectation on both sides, the covariance of the residual is formulated as  
\[\Delta_{k-N+i|k-1} = \mathsf{E}\big{[}\delta_{k-N+i|k-1}\delta_{k-N+i|k-1}^{T}\big{]} \tag{12}\] \[= \mathsf{E}\Big{[}\big{(}y_{k-N+i}-C_{k-N+i}\hat{x}_{k-N+i|k-1} \big{)}\big{(}y_{k-N+i}-C_{k-N+i}\hat{x}_{k-N+i|k-1}\big{)}^{T}\Big{]}\] \[= C_{k-N+i}P_{k-N+i|k-1}C_{k-N+i}^{T}+R_{k-N+i}.\]  
and the auto-covariance of the residual is represented as  
\[\mathsf{E}\big{[}\delta_{k-N+i+j|k-1}\delta_{k-N+i|k-1}^{T}\big{]} \tag{13}\] \[= \mathsf{E}\Big{[}\big{(}y_{k-N+i+j}-C_{k-N+i+j}\hat{x}_{k-N+i+j|k -1}\big{)}\big{(}y_{k-N+i}-C_{k-N+i}\hat{x}_{k-N+i|k-1}\big{)}^{T}\Big{]}\] \[= C_{k-N+i+j}A_{k-N+i+j-1}\big{(}I-K_{k-N+i+j-1}\big{)}A_{k-N+i+j- 2}\big{(}I-K_{k-N+i+j-2}\big{)}\] \[\times A_{k-N+i+j-3}\cdots\big{(}I-K_{k-N+1}\big{)}A_{k-N+i} \Lambda_{k-N+i},\]  
for \(1\leq j\leq N-i-1\), where \(\Lambda_{k-N+i}\) is definded as  
\[\times A_{k-N+i}^{T}+G_{k-N+i}Q_{k-N+i}G_{k-N+i}^{\mathsf{T}},\]0  
When the gain matrix \(K_{k-N+i}\) is optimal, i.e., Equation (5) holds,  
\[\times A_{k-N+i}^{T}+G_{k-N+i}Q_{k-N+i}G_{k-N+i}^{\mathsf{T}},\]1 \[\times A_{k-N+i}^{T}+G_{k-N+i}Q_{k-N+i}G_{k-N+i}^{\mathsf{T}},\]2 \[\times A_{k-N+i}^{T}+G_{k-N+i}Q_{k-N+i}G_{k-N+i}^{\mathsf{T}},\]3  
the auto-covariance of the residual is zero for \(0\leq i\leq N-1\) and \(1\leq j\leq N-i-1\). In practice, however, \(\Lambda_{k-N+i}\) in Equation (15) may not be zero owing to the incomplete information; thus the accuracy of the recursive optimal RH filter can be evaluated by using the following scalar function:  
\[\times A_{k-N+i}^{T}+G_{k-N+i}Q_{k-N+i}G_{k-N+i}^{\mathsf{T}},\]4  
where \(\Lambda_{n,m,k-N+i|k-1}\) represents the \((n,m)th\) elements of \(\Lambda_{k-N+1}\). The absolute minimum of \(J_{k-N+i|k-1}(\lambda_{k-N+i})\) means that the estimation result is mostly close to the optimal one. Therefore, the AFMRH filtering problem can be defined to find the adaptive fading factor selected to minimize the following cost function:  
\[\times A_{k-N+i}^{T}+G_{k-N+i}Q_{k-N+i}G_{k-N+i}^{\mathsf{T}},\]5  
However, the calculation to find the optimal solution of Equation (17) requires a heavy computational load and is not unsuitable for the online state estimation. Thus the following algorithm is introduced in [6].