Jindřich Duník

Brief paper: Derivative-free estimation methods: New results and performance analysis

The derivative-free nonlinear estimation methods exploiting the Stirlings interpolation and the unscented transformation for discrete-time nonlinear stochastic systems are treated. The divided difference and unscented filters, smoothers, and predictors based on the methods are introduced in the unified framework. The new relations among the first order Stirlings interpolation, the second order Stirlings interpolation, and the unscented transformation are derived and their impact on the covariance matrices of the state estimates of the corresponding filters is analysed. The theoretical results are illustrated and used for the explanation of the unexpected behaviour of the sigma point Gaussian sum filters given as a mixture of the derivative-free filters.

Unscented Kalman Filter: Aspects and Adaptive Setting of Scaling Parameter

This technical note deals with the unscented Kalman filter for state estimation of nonlinear stochastic dynamic systems with a special focus on the scaling parameter of the filter. Its standard choice is analyzed and its impact on the estimation quality is discussed. On the basis of the analysis, a novel method for adaptive setting of the parameter in the unscented Kalman filter is proposed. The results are illustrated in a numerical example.

Truncation nonlinear filters for state estimation with nonlinear inequality constraints

The paper focuses on the state estimation problem of nonlinear non-Gaussian systems with state subject to a nonlinear inequality constraint. Taking into account the available additional information about the state given by the constraint increases the estimate quality compared to classical state estimation methods which cannot utilize the information. Considering the constraint in the form of an inequality involving a nonlinear function of the state makes the state estimation problem difficult and hence treated only marginally. In this paper, a generic local filter for the inequality constrained estimation problem is proposed. It covers the extended Kalman filter, unscented Kalman filter, and divided difference filter as special cases and enforces the constraint by truncating the conditional density of the state. The truncation is computationally cheap, yet it provides high estimate quality of the constrained estimate. The same idea is then utilized in a truncation Gaussian mixture filter which is also proposed in the paper to increase the estimate quality further by providing a global constrained estimate. Superior estimate quality and computational efficiency of the proposed filters are illustrated in two numerical examples.

Methods for Estimating State and Measurement Noise Covariance Matrices: Aspects and Comparison

Abstract The paper deals with estimation of noise covariance matrices in state and measurement equations of linear discrete-time stochastic dynamic systems. In the last decade several novel methods for noise covariance matrices estimation, which are based on state estimation techniques, have been proposed. Unfortunately, the novel methods have been compared mainly with classical methods proposed in the seventies only. The aim of the paper is to analyse identifiability of state noise parameters by means of the Bayesian approach and to summarise and compare the novel methods from both theoretical and numerical point of view.

Performance analysis of derivative-free filters

Nonlinear state estimation by the derivative-free Sigma Point Kalman Filters is treated. Particularly, impact of the derivative-free Kalman filters on estimation quality of the Sigma Point Gaussian Sum Filters is discussed. New relations between the Unscented Kalman Filter and the Divided Difference Filters are derived. The main stress is laid on the covariance matrixes which have crucial role for the behaviour explanation of the Sigma Point Gaussian Sum Filters. The theoretical results are illustrated in some numerical examples.

Adaptive choice of scaling parameter in derivative-free local filters

The paper deals with adaptive choice of the scaling parameter in derivative-free local filters. In the last decade several novel local derivative-free filtering methods have been proposed. These methods exploiting Stirlings interpolation and the unscented transformation are, however, conditioned by specification of a scaling parameter significantly influencing the quality of the state estimate. Surprisingly, almost no attention has been devoted to a suitable choice of the parameter. In fact, only a few basic recommendations have been provided, which are rather general and do not respect the particular system description. The choice of the parameter thus remains mainly on a user. The goal of the paper is to provide a technique for adaptive choice of the scaling parameter of the derivative-free local filters.

Unscented Kalman filter with advanced adaptation of scaling parameter

The paper deals with state estimation of the nonlinear stochastic systems by means of the unscented Kalman filter with a focus on specification of the ? -points. Their position is influenced by two design parameters-the scaling parameter determining the spread of the ? -points and a covariance matrix decomposition determining rotation of the ? -points. In this paper, a choice of the scaling parameter is analyzed. It is shown that considering other values than the standard choice may lead to increased quality of the estimate, especially if the scaling parameter is adapted. Several different criteria for the adaptation are proposed and techniques to reduce computational costs of the adaptation are developed. The proposed algorithm of the unscented Kalman filter with advanced adaptation of the scaling parameter is illustrated in a numerical example.

The Development of a Randomised Unscented Kalman Filter

Abstract The paper deals with state estimation of nonlinear stochastic dynamic systems. Traditional filters providing local estimates of the states, such as the extended Kalman filter, unscented Kalman filter or the cubature Kalman filter, are based on approximations which lead to biased estimates of the state and measurement statistics. The aim of the paper is to propose a new local filter that utilises a randomised unscented transformation which is a special case of stochastic integration rules providing an unbiased estimate of an integral. The new filter provides estimates of higher quality than the traditional filters and renders a randomised version of the unscented Kalman filter. The proposed filter is illustrated in a numerical example.

Multitarget tracking performance analysis using the non-credibility index in the Nonlinear Estimation Framework (NEF) toolbox

Target tracking, nonlinear control, and fault detection are typically evaluated with only a Root Mean Square (RMS). RMS is an absolute measurement of the system performance and does not provide a statistic as to the tracker, controller, or fault detection algorithmic performance. For this paper, we investigate the non-credibility index (NCI) and average normalized estimation error square (ANEES) for nonlinear estimation for the Kalman Filter (KF), the Central Difference Filter (DD1), the unscented Kalman filter (UKF), and the particle filter (PF). Fault detection and target track performance is dependent on target maneuvers, sensor errors, model parameters, and state estimation which need to be understood relative to the filter performance versus the absolute performance (i.e. root mean square) of the system. Utilizing the developments of the Nonlinear Estimation Framework (NEF) toolbox, we develop methods of nonlinear relative comparison performance between nonlinear filters in a unified scenario.

Estimation of State and Measurement Noise Covariance Matrices by Multi-Step Prediction

Abstract Estimation of noise covariance matrices for linear or nonlinear stochastic dynamic systems is treated. The novel off-line technique for estimation of the covariance matrices of the state and measurement noises is designed. The technique is based on the multi-step prediction error and on knowledge of the system initial condition 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 illustrated in numerical examples.