Noise Covariance Identification for Nonlinear Systems using Expectation Maximization and Moving Horizon Estimation

In order to estimate states from a noise-driven state space system, the state estimator requires a priori knowledge of both process and output noise covariances. Unfortunately, noise statistics are usually unknown and have to be determined from output measurements. Current expectation maximization (EM) based algorithms for estimating noise covariances for nonlinear systems assume the number of additive process and output noise signals are the same as the number of states and outputs, respectively. However, in some applications, the number of additive process noises could be less than the number of states. In this paper, a more general nonlinear system is considered by allowing the number of process and output noises to be smaller or equal to the number of states and outputs, respectively. In order to estimate noise covariances, a semi-definite programming solver is applied, since an analytical solution is no longer easy to obtain. The expectation step in current EM algorithms rely on state estimates from the extended Kalman filter (EKF) or smoother. However, the instability and divergence problems of the EKF could cause the EM algorithm to converge to a local optimum that is far away from true values. We use moving horizon estimation instead of the EKF/smoother so that the accuracy of the covariance estimation in nonlinear systems can be significantly improved