On the Identification of Noise Covariances and Adaptive Kalman
Filtering: A New Look at a 50 Year-old Problem
Abstract
The Kalman filter requires knowledge of the noise statistics; however,
the noise covariances are generally unknown. Although this
problem has a long history, reliable algorithms for their estimation are
scant, and necessary and sufficient conditions for identifiability of
the covariances are in dispute. We address both of these issues in this
paper. We first present the necessary and sufficient condition for
unknown noise covariance estimation; these conditions are related to the
rank of a matrix involving the auto and cross-covariances of a weighted
sum of innovations, where the weights are the coefficients of the the
minimal polynomial of the closed-loop system transition matrix of a
stable, but not necessarily optimal, Kalman filter. We present an
optimization criterion and a novel six-step approach based on a
successive approximation, coupled with a gradient algorithm with
adaptive step sizes, to estimate the steady-state Kalman filter gain,
the unknown noise covariance matrices, as well as the state prediction
(and updated) error covariance matrix. Our approach enforces the
structural assumptions on unknown noise covariances and ensures symmetry
and positive definiteness of the estimated covariance matrices. We
provide several approaches to estimate the unknown measurement noise
covariance R via post-fit residuals, an approach not yet
exploited in the literature. The validation of the proposed method on
five different test cases from the literature demonstrates that the
proposed method significantly outperforms previous state-of-the-art
methods. It also offers a number of novel machine learning motivated
approaches, such as sequential (one sample at a time) and
mini-batch-based methods, to speed up the computations.