Kalman filtering with no a priori information about noise--White noise case: Identification of covariances

Abstract

Kalman iiltering in the presence of white process and measurement noises having unknown means and covariances is considered. Only stationary linear discrete stochastic systems are considered. It is shown that the identification of noise covariances can be done without the knowledge of noise means. This means that the problem of identifying the noise statistics can be decomposed into two separate subproblems, namely, 1) identiiication of noise covariances and 2) identification of noise means, and that these two subproblems can be solved in that order.A procedure for identifying noise covariances is developed in this paper. It is a nontrivial extension of Mehra's results to the case where the process and measurement noises have unlolown means, and are correlated with each other. This procedure, like Mehra's, can be used either in a nonrecursive mode, or in a batch-recursive mode.Solution of subproblem 2), and the standard Kalman filter algorithm are not discussed since they are well known in the literature.

Noise covariance matrices in state‐space models: A survey and comparison of estimation methods—Part I

“…The test assessing an optimality of the estimated CMs proposed by Mehra was modified using the whiteness test in the work of Alspach . In the work of Godbole, the ICM was extended to estimate the cross‐CM between the state and measurement noises. In the work of Wojcik, the ICM was extended to estimate parameters in the deterministic part of the model, and in the work of Noriega and Pasupathy, a method generalising the ICM for the LTV models was proposed.…”

“…The noise properties, namely, the noise CMs, have to be found on the basis of the measured data. Therefore, during the past 5 decades, various methods for the noise CM estimation have been proposed in the literature . The development of the noise CM estimation methods is thus closely tied with the advent of the SS models of dynamical systems being used in optimal and adaptive state estimation and control .…”

“…The feedback‐free noise CM estimation methods cover the correlation methods and the MLMs . The correlation methods are based on an analysis of the innovation sequence properties of a suboptimal linear filter in a way similar to the illustration in this subsection. The MLMs are based on optimisation techniques from the system identification designed for input‐output models …”

“…The methods differ in assumptions related to the considered model, underlying ideas and principles, properties of the estimates, and number and essence of the design parameters. Traditionally, CM estimation methods are divided into 4 groups: the correlation methods , the maximum‐likelihood methods (MLMs) , the covariance matching methods (CMMs) , and the Bayesian methods …”

Adaptive unscented Kalman filter for parameter and state estimation of nonlinear high-speed objects

“…Therefore, a wavelet transform is utilized to estimate the statistical information of the measurement noise in this subsection.We only need to estimate the noise variance, regardless of the mean [36].…”