Stabilized BFGS approximate Kalman filter
Alexander Bibov  LUT Mafy  Department of Mathematics and Physics, Lappeenranta University Of Technology, P.O. Box 20 FI53851, Finland (email) Abstract:
The Kalman filter (KF) and Extended Kalman filter (EKF) are wellknown tools for assimilating data and model predictions. The filters require storage and multiplication of $n\times n$ and $n\times m$ matrices and inversion of $m\times m$ matrices, where $n$ is the dimension of the state space and $m$ is dimension of the observation space. Therefore, implementation of KF or EKF becomes impractical when dimensions increase. The earlier works provide optimizationbased approximative lowmemory approaches that enable filtering in high dimensions.
However, these versions ignore numerical issues that deteriorate performance of the approximations: accumulating errors may cause the covariance approximations to lose nonnegative definiteness, and approximative inversion of large closetosingular covariances gets tedious. Here we introduce a formulation that avoids these problems. We employ LBFGS formula to get lowmemory representations of the large matrices that appear in EKF, but inject a stabilizing correction to ensure that the resulting approximative representations remain nonnegative definite. The correction applies to any symmetric covariance approximation, and can be seen as a generalization of the Joseph covariance update.
Keywords: Extended Kalman filter, approximate Kalman filter, lowmemory storage, BFGS update, observationdeficient inversion, chaotic dynamics.
Received: August 2014; Revised: May 2015; Available Online: October 2015. 
2015 Impact Factor.951
