An EM algorithm for nonlinear state estimation with model uncertainties

In most solutions to state estimation problems like, for example target tracking, it is generally assumed that the state evolution and measurement models are known a priori. The model parameters include process and mea-surement matrices or functions as well as the corresponding noise statistics. However, there are situations where the model parameters are not known a priori or are known only partially (i.e., with some uncertainty). More-over, there are situations that the measurement is biased. In these scenarios, standard estimation algorithms like Kalman filter and extended Kalman Filter (EKF), which assume perfect knowledge of the model parame-ters, are not accurate anymore. The problem with uncertain model parameters is considered as a special case of maximum likelihood estimation with incomplete-data, for which a standard solution called the expectation-maximization (EM) algorithm exists. In this paper a new extension to the EM algorithm is proposed to solve the more general problem of joint state estimation and model parameter identification for nonlinear systems with possibly non-Gaussian noise. In the expectation (E) step, it is shown that the best variational distribution over the state variables is the conditional posterior distribution of states given all the available measurements and inputs. Therefore, a particular type of particle filter is used to estimate and update the posterior distribution. In the maximization (M) step the nonlinear measurement process parameters are approximated using a nonlinea