# Sage-Husa Algorithm Based on Adaptive Double Forgetting Factors
## 1 Introduction  
Kalman filtering is a widely used recursive algorithm for state estimation in dynamic systems [1, 2]. However, its applicability is limited to linear systems. For nonlinear systems, alternative approaches such as Extended Kalman Filtering (EKF) [3, 4, 5], Unscented Kalman Filtering (UKF) [6, 7, 8], and Particle Filtering (PF) [9, 10, 11] are typically employed. Among these, EKF offers a favorable balance between accuracy and real-time performance, making it a promising approach for practical applications. In the application of EKF for state estimation, it is commonly assumed that the statistical characteristics of process and measurement noise are known and constant. However, in practical situations, the noise parameters often exhibit uncertainty due to the variability and unpredictability of the operational environment. This uncertainty can significantly degrade the filtering accuracy and may even result in filter divergence. To address this challenge, various adaptive filtering techniques have been proposed, including the Maximum Likelihood Method [12, 13], Strong Tracking Filter [14, 15], Artificial Intelligence-based approaches [16, 17], and the Sage-Husa algorithm [18] based on the Maximum A Posteriori criterion. Among these, the Sage-Husa algorithm, with its simplicity and computational efficiency, has emerged as a leading approach in adaptive filtering research and has been extensively studied and applied. Theoretically, the traditional Sage-Husa algorithm can simultaneously update the statistical properties of both process noise and measurement noise. However, in practical applications, estimating process noise and measurement noise concurrently presents significant challenges, often resulting in issues such as the process noise losing positive semi-definiteness and the measurement noise losing definiteness. Moreover, in systems with higher orders, filter divergence frequently occurs. To overcome the aforementioned problems, extensive research has been carried out to explore various enhancements to the Sage-Husa algorithm. [PERSON] et al. [19] used a simplified Sage-Husa estimator to achieve an online measurement noise covariance matrix, which was based on the adaptive current statistical model with velocity prediction. [PERSON] et al. [20] adopted the improved algorithm solely to estimate the measurement noise, in which the negative part of the predicted error covariance matrix was removed, as it could lead to filter divergence, thus ensuring the positive definiteness of the measurement noise. [PERSON] et al. [21] used the fuzzy logic control method to achieve the adaptive adjustment of the forgetting factor of the simplified Sage-Husa filter parameters. The Sage-Husa adaptive filter was updated on the basis of covariance matching technology applied in the navigation of a translational sprinkler irrigation machine [22]. [PERSON] et al. [23] introduced the filtering convergence criterion for the estimator into the Sage-Husa adaptive Kalman filter, which was applied to filter the FOG random drift errors in real time, effectively reducing FOG errors and improving its accuracy. [PERSON] et al. [24] proposed a modified strong tracking-based expended Sage-Husa adaptive robust Kalman filter algorithm that combined the strong tracking KF and the Sage-Husa robust adaptive KF to reduce the influence of transient disturbances caused by waves and ship oscillations. [PERSON] et al. [25] introduced a new type of weighted adjustment coefficient and adaptively adjusted the process noise and observation noise from the perspective of covariance matching, improving the accuracy of the FSI system. [PERSON] et al. [26] proposed a federated Sage-Husa adaptive filter, which used a federated filter as the framework of the multi-source integrated navigation, and the local filters chose the improved Sage-Husa adaptive filter. [PERSON] et al.