A stability-adaptive Kalman filtering algorithm for uncertain and time-varying noise using the measurement difference method

Abstract As a cornerstone of modern measurement and estimation systems, the Kalman filter exhibits degraded performance and can even diverge when confronted with unknown or time-varying process and measurement noise covariances. Although existing adaptive Kalman filters (AKFs) aim to overcome this, they often introduce numerical instability due to the inherent coupling between state and noise estimation. This paper proposes a stable AKF based on the measurement difference method (SAKF-MDM) to overcome these challenges in dynamic measurement environments. The method incorporates two main features: (i) a partitioned measurement construction that reduces the computational complexity of augmented formulations while enhancing numerical stability, and (ii) an exponentially weighted recursive update that enables reliable tracking of time-varying noise covariances under nonstationary conditions. By decoupling noise estimation from state estimation, SAKF-MDM improves robustness and measurement reliability. Theoretical analysis establishes the convergence and asymptotic stability of the proposed method. Simulation studies further demonstrate that it achieves accuracy close to an ideal Kalman filter with perfect noise knowledge, while surpassing existing adaptive filters in accuracy and computational efficiency. These results highlight SAKF-MDM as a viable solution for real-time measurement systems operating under uncertain noise conditions.