Adaptive Kalman Filters With Small-Magnitude and Inaccurate Process Noise Covariance Matrix—Part I: Theoretical Results

There has been a great deal of attention on the online estimation of noise covariance matrices in Kalman filtering during recent decades. However, the existing methods face challenges in the estimation of small-magnitude process noise covariance matrices (PNCMs), which often occur in engineering applications. The purpose of this article, along with its companion article (Part II), is to tackle the aforementioned challenges encountered in practical applications. In this article, two adaptive Kalman filters are proposed to improve the estimation accuracy of small-magnitude PNCMs. We propose a nonadjacent state transition model to accumulate multiple small-magnitude PNCMs into a large-magnitude equivalent PNCM. Then, the adaptive Kalman filters are derived within the framework of variational Bayesian, where the cases are divided into two categories, i.e., single-coefficient estimation and multiple-coefficient estimation, based on the prior information of the PNCM. For the first case, the adaptive Kalman filter can be analytically derived. For the other case, the adaptive Kalman filter is developed based on numerical optimization, where the variational Bayesian and expectation maximization methods are integrated to address the absence of the conjugate prior distribution of the PNCM coefficients. The relative means and mean square errors of the estimated PNCM coefficients in the proposed filters are derived, and an optimal method for selecting the length of the nonadjacent state transition model is also given. Simulation results demonstrate the superior performance of the proposed adaptive filters compared with the existing state-of-the-art methods when the PNCM is small-magnitude and inaccurate. These two filters will be further applied to inertial-based integrated navigation in Part II.