Computing Arrival Cost Parameters in Moving Horizon Estimation Using Sampling Based Filters
Moving horizon estimation (MHE) is a numerical optimization based approach to state estimation, where the joint probability density function (pdf) of a finite state trajectory is sought, which is conditioned on a moving horizon of measurements. The joint conditional pdf depends on the a priori state pdf at the start of the horizon, which is a prediction pdf based on historical data outside the horizon. When the joint pdf is maximized, the arrival cost is a penalty term based on the a priori pdf in the MHE objective function. Traditionally, the a priori pdf is assumed as a multivariate Gaussian pdf and the extended Kalman filter (EKF) and smoother are used to recursively update the mean and covariance. However, transformation of moments through nonlinearity is poorly approximated by linearization, which can result in poor initialization of MHE. Sampling based nonlinear filters completely avoid Taylor series approximations of nonlinearities and attempt to approximate the non-Gaussian state pdf using samples and associated weights or probability mass points. The performance gains of sampling based filters over EKF motivate their use to formulate the arrival cost in MHE. The a priori mean and covariance are more effectively propagated through nonlinearities and the resulting arrival cost term can help to keep the horizon small. It is also possible to find closed-form approximations to the non-Gaussian a priori pdf from the sampling based filters. Thus, more realistic nonparametric arrival cost terms can be included by avoiding the Gaussian assumption. In this paper the use of the deterministic sampling based unscented Kalman filter, the class of random sampling based particle filter and the aggregate Markov chain based cell filter are discussed for initializing MHE. Two simulation examples are included to demonstrate the benefits of these methods over the traditional EKF approach.