Kalman Filter with Recursive Process Noise Covariance Estimation

Abstract

Kalman filter has been found to be useful in vast areas. However, it is well known that successful use of standard Kalman filter is greatly restricted by the strict requirements on a priori information of the model structure and statistics information of the process and measurement noises. Generally speaking, the covariance matrix of process noise is harder to be determined than that of the measurement noise by routine experiments since the statistical property of process noise cannot be obtained directly by collecting a large number of sensor data due to the intrinsic coupling of process noise and system dynamics. Considering such background of wide applications, this chapter introduces one algorithm, named as recursive covariance estimation (RCE) algorithm, to estimate the unknown covariance matrix of noise from a sample of signals corrupted with the noise. Based on this idea, for a class of discrete-time linear time-invariant (LTI) systems where the covariance matrix of process noise is completely unknown, a new Kalman filtering algorithm named as Kalman filter with recursive covariance estimation (KF-RCE) is presented to resolve this challenging problem of state estimation without the statistical information of process noise, and the rigorous stability analysis is given to show that this algorithm is optimal in sense that the covariance matrix and state estimations are asymptotically consistent with the ideal Kalman filter when the exact covariance matrix of process noise is completely known a priori. Extensive simulation studies have also verified the theoretical results and the effectiveness of the proposed algorithm.