Maximum likelihood state and parameter estimation via derivatives ofthe V-Lambda filter

Abstract

Applying the method of maximum likelihood to the problem of parameter and state estimation in linear dynamical systems requires the implementation of a recursive algorithm that consists of a Kalman filter and its derivatives with respect to each of the unknown parameters (the sensitivity equations). Since the conventional Kalman filtering algorithm has been shown to be numerically unstable, a different approach is taken in this paper, which is based on using the K-Lambda square root filtering technique. Equations are developed for the recursive computation of the log-likelihood function gradient (score) and the Fisher information matrix (FIM) in terms of the K-Lambda filter variables, which are the eigenfactors (eigenvalues and eigenvectors) of the estimation error covariance matrix. Based on the singular value decomposition, the recently introduced KLambda filters have been shown to be numerically stable and accurate. Therefore, their usage renders the resulting maximum likelihood scheme numerically robust. Moreover, making the covariance eigenfactors available to the user at all estimation stages, which is an inherent and unique property of the K-Lambda class, adds invaluable insight into the heart of the estimation process.