Ondrej Straka

Stochastic Integration Filter

The technical note deals with state estimation of nonlinear stochastic dynamic systems. Traditional filters providing local estimates of the state, such as the extended Kalman filter, unscented Kalman filter, or the cubature Kalman filter, are based on computationally efficient but approximate integral evaluations. On the other hand, the Monte Carlo based Kalman filter takes an advantage of asymptotically exact integral evaluations but at the expense of substantial computational demands. The aim of the technical note is to propose a new local filter that utilises stochastic integration methods providing the asymptotically exact integral evaluation with computational complexity similar to the traditional filters. The technical note will demonstrate that the unscented and cubature Kalman filters are special cases of the proposed stochastic integration filter. The proposed filter is illustrated by a numerical example.

Sampling Densities of Particle Filter: A Survey and Comparison

The paper deals with the particle filter in discrete-time nonlinear non-Gaussian system state estimation. One of the key parameters affecting estimation quality of the particle filter is the sampling density (also called importance function or proposal density). In the literature, there are many sampling density proposals based on various ideas. The goal of the paper is to provide a survey of sampling densities, to classify them and to compare estimation quality of the particle filter with various sampling densities in an illustration example.

Random-point-based filters: analysis and comparison in target tracking

This paper compares state estimation techniques for nonlinear stochastic dynamic systems, which are important for target tracking. Recently, several methods for nonlinear state estimation have appeared utilizing various random-point-based approximations for global filters (e.g., particle filter and ensemble Kalman filter) and local filters (e.g., Monte-Carlo Kalman filter and stochastic integration filters). A special emphasis is placed on derivations, algorithms, and commonalities of these filters. All filters described are put into a common framework, and it is proved that within a single iteration, they provide asymptotically equivalent results. Additionally, some deterministic-point-based filters (e.g., unscented Kalman filter, cubature Kalman filter, and quadrature Kalman filter) are shown to be special cases of a random-point-based filter. The paper demonstrates and compares the filters in three examples, a random variable transformation, re-entry vehicle tracking, and bearings-only tracking. The results show that the stochastic integration filter provides better accuracy than the Monte-Carlo Kalman filter and the ensemble Kalman filter with lower computational costs.

Nonlinear estimation framework in target tracking

The goal of the article is to describe a software framework designed for nonlinear state estimation of discrete-time dynamic systems. The framework was designed with the aim to facilitate implementation, testing and use of various nonlinear state estimation methods. The main strength of the framework is its versatility due to the possibility of either structural or probabilistic model description. Besides the well-known basic nonlinear estimation methods such as the extended Kalman filter, the divided difference filters and the unscented Kalman filter, the framework implements the particle filter with advanced features. As the framework is designed on the object oriented basis, further extension by user-specified nonlinear estimation algorithms is extremely easy. The paper describes the individual components of the framework, their key features and use. The paper demonstrates easy and natural application of the framework in target tracking which is illustrated in two examples - tracking a ship with unknown control and tracking three targets based on raw data.

Scaling parameter in unscented transform: Analysis and specification

The paper deals with an analysis of the scaling factor of the unscented transform as a method to provide approximate means and covariance matrices of random variables in nonlinear systems. It is a basis of the unscented Kalman filter, which provides a state estimate of nonlinear stochastic dynamic systems. The scaling factor represents a design parameter significantly affecting quality of the approximation. The analysis provides an important insight into the parameter role and its impact on the quality. It also justifies recently published techniques for adaptive setting of the scaling parameter within the unscented Kalman filter. Usage of such an adaptive setting of the scaling parameter is illustrated in a bearings-only tracking example.

Multiple-model filtering with multiple constraints

The paper deals with state estimation of nonlinear stochastic systems, where the state is subject to nonlinear equality constraints reflecting some physical or technological limitations. Usually, this problem of constrained state estimation is solved within the Kalman filtering framework. The goal of the paper is to provide a generalization of the solution to a multiplemodel multiple-constraint problem, where the two-step method for constraint application is adopted. In addition, the model weight computation is analyzed and a weight correction for the constrained estimation is proposed. The proposed method is illustrated in a numerical example.

On Autocovariance Least-Squares Method for Noise Covariance Matrices Estimation

The technical note focuses on the estimation of the noise covariance matrices of the state space models. Stress is laid on the autocovariance least-squares method providing unbiased estimates of the noise covariance matrices of linear systems. In particular, two topics are discussed; first, selection of the predictor gain as a key parameter of the method, second, generalization of the method for linear systems with a time-varying measurement equation. The theoretical results are illustrated in numerical examples.

Covariance intersection in track-to-track fusion with memory

In several practical applications, an economical processing of multi-sensor data is realised by the fusion of estimates. A very cheap solution, which is also very conservative, is the Covariance Intersection (CI) fusion. Basic information configurations of the estimation system have been inspected by the authors in a preceding paper [1]. This paper provides an extension of the previous discussion to the fusion with memory. It shows that the adding of memory cannot improve the fusions without memory given by partial or full information feedback. Conversely, it shows that the fusion without memory and no information feedback can be in some cases improved by the adding of memory.

FRAMEWORK FOR IMPLEMENTING AND TESTING NONLINEAR FILTERS

Abstract The aim of this paper is to present a software framework facilitating implementation, testing and use of various nonlinear estimation methods. This framework is designed to offer an easy to use tool for state estimation of discrete time dynamic stochastic systems. Besides implementation of various local and global state estimation methods it contains procedures for system design and simulation. Its strength is in the fact that it provides means that help students get acquainted with nonlinear state estimation problem and to be able to test features of various estimation methods. Another considerable advantage of proposed framework is its high modularity and extensibility. The paper briefly describes nonlinear estimation problem and its general solution using the Bayesian approach leading to the Bayesian recursive relations. Then it presents key features of the software framework designed in MATLAB environment that supports straightforward implementation of estimation methods based on the Bayesian approach. The strengths of the framework are demonstrated on implementation of the Divided difference filter 1st order.

Self-assessment of local filters by non-Gaussianity measures

The paper deals with state estimation of stochastic nonlinear systems by means of local filters. A new technique is designed to provide a self-assessment of the filter with respect to its estimate quality. It uses a non-Gaussianity measure based on conditional third moment of the state to indicate a possible decrease of estimate quality. The technique is proposed for general local filters with detailed specification for three selected filters in the Kalman filtering framework. An application of the technique is illustrated in a numerical example.