Variational Kalman Filter Research Articles

Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants

This work studies the effects of sampling variability in Monte Carlo-based methods to estimate very high-dimensional systems. Recent focus in the geosciences has been on representing the atmospheric state using a probability density function, and, for extremely high-dimensional systems, various sample-based Kalman filter techniques have been developed to address the problem of real-time assimilation of system information and observations. As the employed sample sizes are typically several orders of magnitude smaller than the system dimension, such sampling techniques inevitably induce considerable variability into the state estimate, primarily through prior and posterior sample covariance matrices. In this article, we quantify this variability with mean squared error measures for two Monte Carlo-based Kalman filter variants: the ensemble Kalman filter and the ensemble square-root Kalman filter. Expressions of the error measures are derived under weak assumptions and show that sample sizes need to grow proportionally to the square of the system dimension for bounded error growth. To reduce necessary ensemble size requirements and to address rank-deficient sample covariances, covariance-shrinking (tapering) based on the Schur product of the prior sample covariance and a positive definite function is demonstrated to be a simple, computationally feasible, and very effective technique. Rules for obtaining optimal taper functions for both stationary as well as non-stationary covariances are given, and optimal taper lengths are given in terms of the ensemble size and practical range of the forecast covariance. Results are also presented for optimal covariance inflation. The theory is verified and illustrated with extensive simulations.

Fuzzy variant of a statistical test point Kalman filter

In this paper, we propose the conceptual use of fuzzy clustering techniques as iterative spatial methods to estimate a posteriori statistics in place of the weighted averaging scheme of the Unscented Kalman filter. Specifically, instead of a linearization methodology involving the statistical linear regression of the process and measurement functions through some deterministically chosen set of test points (sigma points) contained within the “uncertainty region” around the state estimate, we present a variant of the Unscented transformation involving fuzzy clustering techniques which will be applied to the test points yielding “degrees of membership” in which Gaussian shapes can be “fit” using a least squares scheme. Implementation into the Kalman methodology will be shown along with simple state and parameter estimation examples.

An Adaptive Kalman Filter for Dynamic Harmonic State Estimation and Harmonic Injection Tracking

Knowledge of the process noise covariance matrix Q is essential for the application of Kalman filtering. However, it is usually a difficult task to obtain an explicit expression of Q for large time varying systems. This paper looks at an adaptive Kalman filter method for dynamic harmonic state estimation and harmonic injection tracking. The method models the system as a linear frequency independent state model and does not require an exact knowledge of the noise covariance matrix Q. As an alternative, the proposed adaptive Kalman filter switches between the two basic Q models for steady-state and transient estimation. Its adaptive function allows for the resetting of the Kalman gain to avoid Kalman filter divergence problems under steady-state and allow fast tracking of system variations in transient conditions. Simulation results on the 220 kV network of the lower South Island of New Zealand are presented to validate this approach.

Kalman filters for non-linear systems: a comparison of performance

The Kalman filter is a well-known recursive state estimator for linear systems. In practice, the algorithm is often used for non-linear systems by linearizing the system's process and measurement models. Different ways of linearizing the models lead to different filters. In some applications, these ‘Kalman filter variants’ seem to perform well, while for other applications they are useless. When choosing a filter for a new application, the literature gives us little to rely on. This paper tries to bridge the gap between the theoretical derivation of a Kalman filter variant and its performance in practice when applied to a non-linear system, by providing an application-independent analysis of the performances of the common Kalman filter variants. Correlated uncertainties can be dealt with by augmenting the state vector, this is the original formulation of the KF (Kalman 1960). Expressed in this new state vector, the process and measurement models are of the form (1) and (2) with uncorrelated uncertainties. This paper separates performance evaluation of Kalman filters into (i) consistency, and (ii) information content of the estimates; and it separates the filter structure into (i) the process update step, and (ii) the measurement update step. This decomposition provides the insights supporting an objective and systematic evaluation of the appropriateness of a particular Kalman filter variant in a particular application.

Sensor Fusion for Dead-Reckoning Mobile Robot Navigation

In positioning and localization problems, two or more different sensors are often used to obtain the best estimation data for control system. Extended Kalman Filter (EKF) is widely used to fuse those data to obtain one optimal result. One consideration when using EKF is the signal used during navigation is a white noise signals. This consideration is hardly found in real application. This paper presents the sensor fusion for dead-reckoning mobile robot navigation. Odometry and sonar signals are fused using Extended Kalman Filter (EKF) and Adaptive Fuzzy Logic System (AFLS). The AFLS was used to adapt the gain and therefore prevent the Kalman filter divergence. The fused signal is more accurate than any of the original signals considered separately. The enhanced, more accurate signal is used to guide and navigate the robot.

Odometry and Sonar Data Fusion for Mobile Robot Navigation

To solve the problems in guidance, navigation, and control for an autonomous robot, its accurate positioning and localization are needed. Two or more different sensors are often used to obtain reliable data useful for control system. Extended Kalman Filter (EKF) is widely used to fuse those data to obtain one optimal result. The signals used during navigation cannot be always considered as white noise signals. On the other hand, colored signals will cause the EKF to diverge. This paper presents the data fusion system for mobile robot navigation. Odometry and sonar signals are fused using Extended Kalman Filter (EKF) and Adaptive Fuzzy Logic System (AFLS). The AFLS was used to adapt the gain and therefore prevent the Kalman filter divergence. The fused signal is more accurate than any of the original signals considered separately. The enhanced, more accurate signal is used to guide and navigate the robot.

Convergence analysis for inexact mechanization of Kalman filtering

A computational aspect of real-time estimation is considered, in which the estimation algorithm to be used has the standard optimal Kalman filtering structure, but the actual inverse matrix within the Kalman gain is replaced by an expedient approximation at each instant. In real-time applications, most Kalman filtering schemes are approximate to a degree as a consequence of numerical roundoff matrix inversion. The convergence properties and error estimates of such schemes are obtained to provide a theoretical basis for gauging the utility of using the above approximations of the Kalman gain matrix at each time instant. A new exponentially convergent scheme is also suggested for approximating the inverse matrix within the Kalman gain. Conditions are determined under which online approximate matrix inversion can be eliminated as the cause of Kalman filter divergence in real-time implementations. >

Realtime digital integration for continuous Kalman filtering in nonlinear systems

The utilization of different numerical integration formulas for on-line continuous Kalman filtering is investigated for nonlinear systems. Variational and extended versions of the Kalman filter are compared on the basis of accuracy and computational speed tradeoffs as input noise levels and system nonlinearity characteristics are varied. Digital simulations are performed on a typical nonlinear system for several combinations of filtering configurations and integration formulas with different step sizes. It is shown by ensemble-averaged Monte Carlo simulations that the second-order Adams-Bashforth formula (AB2) and the variational Kalman filter should be used for the mildly nonlinear system considered.

Instability of optimal stochastic control systems under parameter variations

The deterioration of a linear optimal stochastic control scheme, designed under the assumptions of the certainty-equivalence principle (the optimal filter and controller, determined independently, combine to give a totally optimal system), is investigated when the parameters of the actual system do not coincide with the design values. This linear suboptimal stochastic system is described by a covariance matrix composed of covariances of the estimates of the state variables, the errors in the estimates of the state variables, and the correlation between these errors and the estimates. In particular, this paper is concerned with the covariance matrix resulting from a single state dynamical system and a scalar linear measurement function of both the state variable and the control variable (e.g., accelerometer measurements). A modeling error in the control variable coefficient of the measurement function may induce instability in the stochastic system with either unstable or stable dynamics. Furthermore, the absolute magnitude of the error in the control variable coefficient directly influences system stability, not the relative error. Thus, relatively small errors compared to the design value of this coefficient may be quite important. 4 LTHOUGH there are many studies on divergence of op-£^- tirnal filters, little attention has been given to the effect of modeling inaccuracies on optimal linear stochastic control systems. Here we extend Fitzgerald's1 investigation of Kalman filter divergence to optimal linear stochastic control systems. These systems are designed under the certainty equivalence principle2 which states that if the expected value of a quadratic function of the state and the control variables is to be minimized subject to linear dynamics, the optimal system is composed of an optimal filter in cascade with an optimal controller. This separation is possible because the estimate in the state is uncorrelated with the error in this estimate. If the parameters in the assumed model of the dynamics or the measurement device deviate from the parameters of the actual system, the estimate and the error in the estimate become correlated. The behavior of the system because of the gains based on an inaccurate model is studied by considering the coupled matrix covariance equation composed of the Covariances of the error in the estimate, the estimate, and the estimate with its error. Some of the characteristics of this linear matrix equation are studied through a scalar linear dynamic equation. The errors in system parameters enter into the 2X2 covariance equation in a dimensionless form allowing the following general results to be obtained: 1) The stochastic control system may be unstable when the nonoptimal filter and deterministic control systems individually are stable. 2) Instability occurs only when the error in the parameter exceeds a finite threshold value. 3) If the measurement is a linear function of the control variable as well as the state (e.g., accelerometer measurements) and there are errors in the coefficient of the control, then instability of the total system may occur for both stable and unstable dynamical systems. 4) The filter or control gains are not functions of the coefficient of the control variable in the measurement function. Consequently, Presented as Paper 70-36 at the AIAA 8th Aerospace Sciences