Unscented Filtering from Delayed Observations with Correlated Noises

A filtering algorithm based on the unscented transformation is proposed to estimate the state of a nonlinear system from noisy measurements which can be randomly delayed by one sampling time. The state and observation noises are perturbed by correlated nonadditive noises, and the delay is modeled by independent Bernoulli random variables.This work has been partially supported by the Ministerio de Ciencia e Innovación and the Junta de Andalucía through Projects MTM2008-05567 and P07-FQM-02701, respectively

Scaled unscented transform Gaussian sum filter: theory and application

In this work we consider the state estimation problem in nonlinear/non-Gaussian systems. We introduce a framework, called the scaled unscented transform Gaussian sum filter (SUT-GSF), which combines two ideas: the scaled unscented Kalman filter (SUKF) based on the concept of scaled unscented transform (SUT), and the Gaussian mixture model (GMM). The SUT is used to approximate the mean and covariance of a Gaussian random variable which is transformed by a nonlinear function, while the GMM is adopted to approximate the probability density function (pdf) of a random variable through a set of Gaussian distributions. With these two tools, a framework can be set up to assimilate nonlinear systems in a recursive way. Within this framework, one can treat a nonlinear stochastic system as a mixture model of a set of sub-systems, each of which takes the form of a nonlinear system driven by a known Gaussian random process. Then, for each sub-system, one applies the SUKF to estimate the mean and covariance of the underlying Gaussian random variable transformed by the nonlinear governing equations of the sub-system. Incorporating the estimations of the sub-systems into the GMM gives an explicit (approximate) form of the pdf, which can be regarded as a "complete" solution to the state estimation problem, as all of the statistical information of interest can be obtained from the explicit form of the pdf ... This work is on the construction of the Gaussian sum filter based on the scaled unscented transform

Latent state estimation in a class of nonlinear systems

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The problem of estimating latent or unobserved states of a dynamical system from observed data is studied in this thesis. Approximate filtering methods for discrete time series for a class of nonlinear systems are considered, which, in turn, require sampling from a partially specified discrete distribution. A new algorithm is proposed to sample from partially specified discrete distribution, where the specification is in terms of the first few moments of the distribution. This algorithm generates deterministic sigma points and corresponding probability weights, which match exactly a specified mean vector, a specified covariance matrix, the average of specified marginal skewness and the average of specified marginal kurtosis. Both the deterministic particles and the probability weights are given in closed form and no numerical optimization is required. This algorithm is then used in approximate Bayesian filtering for generation of particles and the associated probability weights which propagate higher order moment information about latent states. This method is extended to generate random sigma points (or particles) and corresponding probability weights that match the same moments. The algorithm is also shown to be useful in scenario generation for financial optimization. For a variety of important distributions, the proposed moment-matching algorithm for generating particles is shown to lead to approximation which is very close to maximum entropy approximation. In a separate, but related contribution to the field of nonlinear state estimation, a closed-form linear minimum variance filter is derived for the systems with stochastic parameter uncertainties. The expressions for eigenvalues of the perturbed filter are derived for comparison with eigenvalues of the unperturbed Kalman filter. Moment-matching approximation is proposed for the nonlinear systems with multiplicative stochastic noise

Theory, Design, and Implementation of Landmark Promotion Cooperative Simultaneous Localization and Mapping

Simultaneous Localization and Mapping (SLAM) is a challenging problem in practice, the use of multiple robots and inexpensive sensors poses even more demands on the designer. Cooperative SLAM poses specific challenges in the areas of computational efficiency, software/network performance, and robustness to errors. New methods in image processing, recursive filtering, and SLAM have been developed to implement practical algorithms for cooperative SLAM on a set of inexpensive robots. The Consolidated Unscented Mixed Recursive Filter (CUMRF) is designed to handle non-linear systems with non-Gaussian noise. This is accomplished using the Unscented Transform combined with Gaussian Mixture Models. The Robust Kalman Filter is an extension of the Kalman Filter algorithm that improves the ability to remove erroneous observations using Principal Component Analysis (PCA) and the X84 outlier rejection rule. Forgetful SLAM is a local SLAM technique that runs in nearly constant time relative to the number of visible landmarks and improves poor performing sensors through sensor fusion and outlier rejection. Forgetful SLAM correlates all measured observations, but stops the state from growing over time. Hierarchical Active Ripple SLAM (HAR-SLAM) is a new SLAM architecture that breaks the traditional state space of SLAM into a chain of smaller state spaces, allowing multiple robots, multiple sensors, and multiple updates to occur in linear time with linear storage with respect to the number of robots, landmarks, and robots poses. This dissertation presents explicit methods for closing-the-loop, joining multiple robots, and active updates. Landmark Promotion SLAM is a hierarchy of new SLAM methods, using the Robust Kalman Filter, Forgetful SLAM, and HAR-SLAM. Practical aspects of SLAM are a focus of this dissertation. LK-SURF is a new image processing technique that combines Lucas-Kanade feature tracking with Speeded-Up Robust Features to perform spatial and temporal tracking. Typical stereo correspondence techniques fail at providing descriptors for features, or fail at temporal tracking. Several calibration and modeling techniques are also covered, including calibrating stereo cameras, aligning stereo cameras to an inertial system, and making neural net system models. These methods are important to improve the quality of the data and images acquired for the SLAM process

Nonparametric Identification of nonlinear dynamic Systems

In der vorliegenden Arbeit wird eine nichtparametrische Identifikationsmethode für stark nichtlineare Systeme entwickelt, welche in der Lage ist, die Nichtlinearitäten basierend auf Schwingungsmessungen in Form von allgemeinen dreidimensionalen Rückstellkraft-Flächen zu rekonstruieren ohne Vorkenntnisse über deren funktionale Form. Die Vorgehensweise basiert auf nichtlinearen Kalman Filter Algorithmen, welche durch Ergänzung des Zustandsvektors in Parameterschätzer verwandelt werden können. In dieser Arbeit wird eine Methode beschrieben, die diese bekannte parametrische Lösung zu einem nichtparametrischen Verfahren weiterentwickelt. Dafür wird ein allgemeines Nichtlinearitätsmodell eingeführt, welches die Rückstellkräfte durch zeitvariable Koeffizienten der Zustandsvariablen beschreibt, die als zusätzliche Zustandsgrößen geschätzt werden. Aufgrund der probabilistischen Formulierung der Methode, können trotz signifikantem Messrauschen störfreie Rückstellkraft-Charakteristiken identifiziert werden. Durch den Kalman Filter Algorithmus ist die Beobachtbarkeit der Nichtlinearitäten bereits durch eine Messgröße pro Systemfreiheitsgrad gegeben. Außerdem ermöglicht diese Beschreibung die Durchführung einer vollständigen Identifikation, wobei die restlichen konstanten Parameter des Systems zusätzlich geschätzt werden. Die Leistungsfähigkeit des entwickelten Verfahrens wird anhand von virtuellen und realen Identifikationsbeispielen nichtlinearer mechanischen Systeme mit ein und drei Freiheitsgraden demonstriert

Recursive joint Cramér‐Rao lower bound for parametric systems with two‐adjacent‐states dependent measurements

Joint Cramér-Rao lower bound (JCRLB) is very useful for the performance evaluation of joint state and parameter estimation (JSPE) of non-linear systems, in which the current measurement only depends on the current state. However, in reality, the non-linear systems with two-adjacent-states dependent (TASD) measurements, that is, the current measurement is dependent on the current state as well as the most recent previous state, are also common. First, the recursive JCRLB for the general form of such non-linear systems with unknown deterministic parameters is developed. Its relationships with the posterior CRLB for systems with TASD measurements and the hybrid CRLB for regular parametric systems are also provided. Then, the recursive JCRLBs for two special forms of parametric systems with TASD measurements, in which the measurement noises are autocorrelated or cross-correlated with the process noises at one time step apart, are presented, respectively. Illustrative examples in radar target tracking show the effectiveness of the JCRLB for the performance evaluation of parametric TASD systems