Jindrich Dunik

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.

Sigma point Gaussian sum filter design using square root unscented filters

Abstract Local and global estimation approaches are discussed, above all the Unscented Kalman Filter and the Gaussian Sum Filter. The square root modification of the Unscented Kalman Filter is derived and it is used in the Gaussian Sum Filter framework. The new Sigma Point Gaussian Sum Filter is designed and some aspects of the filter are presented. Estimation quality and computational demands of the filter are illustrated in a numerical 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.

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.

Application of estimation techniques on queue lengths estimation in traffic network

The paper deals with the application of the state and parameter estimation techniques in the area of traffic control. The most important properties of the traffic system are described and the model of the traffic system, based on the traffic flow conservation principle, is presented. Various estimation and identification techniques are briefly introduced and applied for three types of roads and micro-regions, namely for city ring road, peripheral road, and city centre. Performance of estimation techniques is validated, using the derived models on real and synthetic data coming from Prague, with respect to accuracy and complexity.

Off-line Estimation of System Noise Covariance Matrices by a Special Choice of the Filter Gain

Estimation of noise covariance matrices for linear or nonlinear stochastic dynamic systems is treated. The stress is laid on the case when the initial state mean and covariance matrix are exactly known. The properties of the innovation sequence of the Kalman Filter and the local filters are discussed and the new offline method for estimation of the covariance matrices of the state and the measurement noise is designed. The proposed method is based on special choice of the filter gain and it takes an advantage of the well-known standard relations from the area of state estimation techniques and least square method. The theoretical results are verified in numerical examples.

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.