## 2 An Adaptive Fading-Memory Receding-Horizon Filter
### An Adaptive Fading-Memory Receding-Horizon Filter
Note that \(\hat{x}_{k-1|k-2}\) is obtained from the AFMRH filter with the horizon length \(N_{k-2,k}\)

\[\mathsf{E}\big{[}\delta_{k-N+i+j|k-1}\delta_{k-N+i|k-1}^{T}\big{]} \tag{13}\]8 The concept of the AHLA method. At time \(k\), \(\hat{x}_{k|k-1}\) is obtained from the Kalman filtering recursion given by Equations (3) and (4) with the initial conditions \(\hat{x}_{k-N_{k-2}|k-2}\) and \(P_{k-N_{k-2}|k-2}\), which are already obtained at \(k-2\). At the same time, \(\hat{x}_{k-1|k-1}\) is stored in the estimation process. Moreover, the absolute values of residuals \(|\delta_{k-1|k-2}^{+}|\), \(|\delta_{k-1|k-2}^{c}|\), and \(|\delta_{k-1|k-1}^{-}|\) corresponding to \(\hat{x}_{k-1|k-2}\), \(\hat{x}_{k-1|k-2}\), and \(\hat{x}_{k-1|k-2}\), respectively, are obtained from Equation (11). Note that \(\hat{x}_{k-1|k-1}\) is obtained from the AFMRH filter with the horizon length \(N_{k-2,k}-1\). 4. Thereafter, the horizon length of the estimate that has the minimum of the absolute residual value is selected as the horizon length \(N_{k+1,k+3}\) for the next cycle \([k+1\ \ k+3]\). Note that the horizon length must be in the interval \([N_{min},N_{max}]\), where \(N_{min}\) is the order of the system and \(N_{max}\) is the maximum allowed horizon length considered as the design parameter. Second, the SAFMRH filtering algorithm is introduced by combining the proposed AHLA method and switching filter strategy. Although the horizon length is selected properly by the AHLA method, a large horizon length may cause an unacceptable computational load for the RH filter.Because the estimate of the RH filter converges to that of the Kalman filter as the horizon increases and Kalman filter has less computational complexity than the RH filter, the computation efficiency and estimation accuracy could be improved by using the Kalman filter instead of the RH filter when the model uncertainty does not exist. Thus, the switching rule of the Kalman filter and the proposed AFMRH filter with the AHLA method is as follows:

* In cases in which \(\lambda\leq 1\) and \(N\geq N_{max}\) indicate that the system model is reliable, the estimate and its error covariance are obtained directly from the Kalman filter given by Equations (3) and (4). * By contrast, for \(\lambda>1\) or \(N<N_{max}\), the estimate and its error covariance matrix are obtained from the proposed AFMRH filter with the AHLA method. If the filter is switched from the Kalman filter to the AFMRH filter, the initial value of the horizon length is always taken as the nominal one, which is the design parameter. ## 5 Simulation Results and Discussion

In this section, several simulation results are presented to verify the effectiveness and performance of the proposed state-estimation algorithms. To verify the adaptiveness and robustness of the proposed estimation algorithms, numerical simulations were performed for the discrete F-404 gas turbine aircraft engine model with a temporary modeling uncertainty [33] as

\[x_{k+1} = \left[ \begin{array}{ccc}0.9305+\delta_{k}&0&0.1107\\ 0.0077&0.9802+\delta_{k}&-0.0173\\ 0.0142&0&0.8953+\delta_{k}\end{array} \right]x_{k}+\left[ \begin{array}{c}1\\ 1\\ 1\end{array} \right]w_{k}, \tag{46}\] \[y = \left[ \begin{array}{ccc}1+0.1\delta_{k}&0&0\\ 0&1+0.1\delta_{k}&0\end{array} \right]x_{k}+v_{k}, \tag{47}\]

where the initial state is taken as \(x_{0}=\left[ \begin{array}{ccc}0&0&0\end{array} \right]^{T}\) and the actual process and measurement noise covariance matrices are taken as \(Q=0.2^{2}\) and \(R=0.01^{2}I_{2\times 2}\), respectively. To show the estimation performance of the proposed methods in the case of model mismatch, the model's uncertain parameter \(\delta_{k}\) is defined as

\[\delta_{k}=\left\{ \begin{array}{cc}0.1,&200\leq k\leq 250,\\ 0,&\textit{otherwise}.\end{array} \right.\]

All simulations presented in this section were performed using MATLAB R2020b on an Intel i5-9400 CPU, 2.90-GHz, 64-bit machine with 8-GB RAM.