## 2 An Adaptive Fading-Memory Receding-Horizon Filter
### An Adaptive Fading-Memory Receding-Horizon Filter
First, to verify the effectiveness of the proposed AFMRH filter, the proposed AFMRH filter was compared with a standard Kalman filter [3], recursive optimal RH filter [13], and AFM Kalman filter [22]. The initial state and its error covariance for both the standard Kalman filter and AFM Kalman filter were set as \(x_{0}= \begin{bmatrix}1&1&1\end{bmatrix} ^{T}\) and \(P_{0}=10^{3}I_{3\times 3}\), respectively, and the horizon length for both the standard recursive optimal RH filter and the proposed AFMRH filter was taken as \(N=20\). Moreover, the minimum and maximum allowed horizon lengths were taken as \(N_{min}=3\) and \(N_{max}=25\), respectively. The second state-estimation errors of four filters are shown in Figure 2.

Figure 2 shows that the proposed AFMRH filter successfully estimates the real state and has better estimation performance compared to others in the case of model mismatch. Moreover, for the model mismatch, both the adaptive filters have small estimation errors and fast convergence property compared with the nonadaptive filter. In particular, the proposed AFMRH filter progressively reduces the estimation error compared with that of the AFM Kalman filter when model uncertainty exists. Moreover, the estimation errors of the proposed AFMRH filter rapidly decrease even though the model uncertainty exists, whereas those of the AFM Kalman filter continuously increase. This result shows that the proposed AFMRH filter could possess early adaptability to changes in the system because its adaptive fading factors and the estimates are obtained by using the finite measurements in the future (based on the estimation time). In Figure 3, the changes in adaptive fading factors of the proposed AFMRH filter are also compared with those of the AFM Kalman filter. The results reveal that the adaptive fading factor could slowly converge to 1 because the accumulated estimation error of the AFM Kalman filter affects the determination of the adaptive fading factor due to its estimation structure. This means that the estimate of the AFM Kalman filter slowly converges to the optimal one. However, as the proposed filter is designed with an RH estimation structure, which estimates the state using only the recent finite measurements, the estimates of the RH filter are not affected by previous estimation errors. Thus, the proposed AFMRH filter is expected to possess better adaptiveness than the AFM Kalman filter. As expected, the adaptive fading factor of the proposed filter converges to 1 much faster than that of the AFM Kalman filter. This result shows that the AFM method enhances the estimation performance of the RH structured filter and facilitates better adaptiveness than the Kalman filter with the AFM method. Moreover, the filtering structure was also shown to affect the determination of the adaptive fading factor. Figure 2: Estimation errors of filters. Third, to verify the accuracy and computational efficiency of the proposed AFMRH filter with AHLA method and SAFMRH filter, their estimation results are compared with those of the proposed AFMRH filter and a constant fading RH (CFMRH) filter [8]. The constant fading factor for the CFMRH filter is taken as 1.02. The second state-estimation errors and time-averaged values of the root-mean-square estimation (RMSE) errors are compared in Figure 6 and Table 1, respectively. In Figure 6, it can be easily observed that the estimation errors of the proposed AFMRH filter with AHLA method and SAFMRH filter are remarkably smaller than those of CFMRH

Figure 5: Adjusted horizon length using the AHLA method. Figure 6: Estimation errors of filters. filter in the case of model mismatch. Moreover, it is also observed that the estimation errors of the proposed AFMRH filter with AHLA method and SAFMRH filter are smaller than the proposed AFMRH filter, as expected. In particular, the estimation errors of the proposed AFMRH filter with AHLA method and SAFMRH filter quickly converge to zero compared with others RH filters after the model uncertainty disappear.