Particle Filters for Set-membership State Estimation

Data assimilation is the process of estimating the state of dynamic systems (linear or nonlinear, Gaussian or non-Gaussian) as accurately as possible from noisy observational data. Although the Three Dimensional Variational (3D-VAR) methods, Four Dimensional Variational (4D-VAR) methods and Ensemble Kalman filter (EnKF) methods are widely used and effective for linear and Gaussian dynamics, new methods of data assimilation are required for the general situation, that is, nonlinear non-Gaussian dynamics. General Bayesian recursive estimation theory is reviewed in this thesis. The Bayesian estimation approach provides a rather general and powerful framework for handling nonlinear, non-Gaussian, as well as linear, Gaussian estimation problems. Despite a general solution to the nonlinear estimation problem, there is no closed-form solution in the general case. Therefore, approximate techniques have to be employed. In this thesis, the sequential Monte Carlo (SMC) methods, commonly referred to as the particle filter, is presented to tackle non-linear, non-Gaussian estimation problems. In this thesis, we use the SMC methods only for the nonlinear state estimation problem, however, it can also be used for the nonlinear parameter estimation problem. In order to demonstrate the new methods in the general nonlinear non-Gaussian case, we compare Sequential Monte Carlo (SMC) methods with the Ensemble Kalman Filter (EnKF) by performing data assimilation in nonlinear and non-Gaussian dynamic systems. The models used in this study are referred to as state-space models. The Lorenz 1963 and 1966 models serve as test beds for examining the properties of these assimilation methods when used in highly nonlinear dynamics. The application of Sequential Monte Carlo methods to different fixed parameters in dynamic models is considered. Four different scenarios in the Lorenz 1063 [sic] model and three different scenarios in the Lorenz 1996 model are designed in this study for both the SMC methods and EnKF method with different filter siz

Nonlinear dynamic state estimation in instrumented structures with conditionally linear Gaussian substructures

Location tracking is being increasingly used across many applications. While GPS is the most widely used location tracking technology, it is unavailable in many environments such as indoors and underground. Local positioning systems (LPS) that use time of arrival based ranging can provide high accuracy location tracking for many applications. Tracking location using range measurements is a non-linear state estimation problem and the measurement noise is often non-Gaussian in environments where LPS are typically used. Hence a particle filter is an appropriate state estimator for location tracking in LPS. Particle filters are computationally complex and have a serial bottleneck that prevents straightforward parallel implementation. In this paper we present a parallel architecture for the particle filter that can be efficiently implemented in a field programmable gate array or a fixed-point digital signal processor. We show that processing can be divided into up to twenty parallel particle filters to massively increase the processing speed. Mixing between the filters is essential and we present a new algorithm for this that minimises computational complexity and memory bandwidth. Finally, for efficient hardware implementation fixed point arithmetic should be used and we empirically determine the required precision.

This paper presents an innovative application of the particle filtering techniques to the vertical gyro problem, which is a highly nonlinear and non Gaussian recursive state estimation problem. The classical approach to recursive state estimation problem of nonlinear and/or non Gaussian systems is based on the extended Kalman filter (EKF), which is a heuristic method based on the linearization of the state dynamics and observations near a nominal path. Depending on the amount of nonlinearities and non Gaussian noise, the EKF often provides a solution far from acceptable. The vertical gyro is an instrument adopted to measure the tilt angles (pitch and roll) of an aircraft with respect to local vertical. It is composed by three orthogonal gyros that are used as process sensors and by three orthogonal accelerometers that are used as aiding sensors. The output of accelerometers is filtered using a proper low-pass filter to allow for the estimation of the local direction of the gravity vector that is assumed the same of the local vertical. In the application to the vertical gyro, Gordon's version of the particle filter was used. The relevant dynamical model is based on non-linear attitude rate equation using Euler angles rather than quaternions. The algorithm, developed in Matlab environment, was first calibrated and tested in simulations, obtaining very appealing performances. Subsequently, it was implemented using real data, acquired by means of an inertial unit produced by Crossbow <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">TM</sup> (model DMU-HDX <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">TM</sup> ), and its results were compared to those provided by the proprietary EKF embedded in the unit. First of all, stationary tests were performed. Sensor data were input to the particle filter that outputs the tilt angles (roll and pitch) with respect to the local geodetic reference. The results were compared to those provided by the internal EKF. Subsequently, the same operation was done using data that were acquired with moving platform. The accuracy of the filter was tested also for degraded operating conditions (lower data rates) and large initial errors. The main advantage of the methodology is that, while the internal filter works with 7 states (quaternions, biases and temperature), the particle filter ensures the convergence with 3 states, and provides acceptable performances with 1000 particles only, appearing promising for a real time application.

This paper proposes a Bayesian unscented Kalman filter with simplified Gaussian mixtures (BUKF-SGM) for dynamic state space estimation of nonlinear and non-Gaussian systems. In the BUKF-SGM, the state and noise densities are approximated as finite Gaussian mixtures, in which the mean and covariance for each component are recursively estimated using the UKF. To avoid the exponential growth of mixture components, a Gaussian mixture simplification algorithm is employed to reduce the number of mixture components, which leads to lower complexity in comparing with conventional resampling and clustering techniques. Experimental results show that the proposed BUKF-SGM can achieve better performance compared with the particle filter (PF)-based algorithms. This provides an attractive alternative for nonlinear state estimation problem.

The requirement for Terrain Referenced Navigation (TRN) also known as Terrain Aided Navigation (TAN) arises when Global Navigation Satellite System (GNSS) signals are unavailable, jammed or blocked. In recent years, research on the application of TRN to aerial and underwater vehicles has been increased rapidly with the developments in the accuracy of digital terrain elevation database (DTED). Since the land and sea floor profiles are inherently nonlinear, TRN becomes a nonlinear estimation problem. Because of the highly nonlinear and non-Gaussian problem, linear or linearized estimation techniques such as Kalman or Extended Kalman Filter (EKF) do not work properly for many terrain profiles. Hence, this paper focuses on the Sequential Monte Carlo (SMC) method namely the particle filter (PF) for dealing with nonlinearities and different types of probability distributions even multi modal. Two different PF implementations are studied, Sequential Importance Sampling with effective resampling (SIS-R) and Sampling Importance Resampling (SIR). The major difference between these implementations is that in SIS-R algorithm, resampling is only performed using effective sample size when actually required. Therefore, the SIS-R algorithm has less computational demands than the SIR algorithm for a given number of particles. Both algorithms are tested for an aircraft scenario over a commercial map. Simulations with different number of particles and inertial measurement units (IMUs) having various error measures are performed and investigated. Monte Carlo (MC) simulations show that both algorithms yield similar results in terms of horizontal position accuracy and convergence. Hereby with this paper, it is presented that besides requiring less computational load, the SIS-R implementation works accurately on TRN problem.

This paper develops a Rao-Blackwellised particle filtering algorithm for blind system identification. The state space model under consideration uses a time-varying autoregressive (AR) model for the sources, and a time-varying finite impulse response (FIR) model for the channel. The multi-sensor measurements result from the convolution of the sources with the channels in the presence of additive noise. A numerical approximation to the optimal Bayesian solution for the nonlinear sequential state estimation problem is implemented using sequential Monte Carlo (SMC) methods. The Rao-Blackwellisation technique is applied to improve the efficiency of the particle filter by marginalizing out the AR and FIR coefficients from the joint posterior distribution. Simulation results are given to verify the performance of the proposed method.

This paper presents efficient object tracking in video sequences using multiple features by embedding mean shift into particle filters. When clutter background and occlusions are present. Particle filtering is used because it is very robust and performs well for non-linear and non-Gaussian dynamic state estimation problems. The image features, such as shape, texture, color, contours, and random motion appearance can be used to track the moving object(s) in videos. We proposed a smart video surveillance system with real-time as well as offline stored video database for moving object detection, classification and tracking capabilities. The graphical user interface of the proposed system using MATLAB 2012b is implemented that operates on both color and gray scale video images from a stationary camera. In this method, we proposed real time moving object detection and tracking system using static webcam that can processes 800*600, 640*480, and 320*240 resolution video sequences for capturing live scene as well as stored database of segmented videos. Issues related with object detection and tracking is jointly employed using mean shift and particle filters. Tracker will estimate the dynamic shape and random appearance of objects. Tracking requires location and shape of object in every segmented frame. Rectangular bounding box is used to improve the estimate of its shape and appearance. Target object may experience partial occlusions, intersection with other objects with similar color distributions, abrupt motion speed changes, and cluttered background. Test result shows improvement in terms of robustness, tracking drifts, accuracy of tracked bounding box to occlusions.

This paper investigates the problem of nonlinear state estimation for control moment gyroscope (CMG) inverted pendulum. The dynamics of CMG inverted pendulum result into stiff nonlinearities and are included here. Both Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) algorithms are utilized for state estimation to investigate their effectiveness. The computer simulations are included for comparison of the two algorithms. There are inherent difficulties in state estimation for systems with stiff nonlinearities. In this research, the results show that EKF performs far better than UKF in most cases for this system. However, this is an attempt to explore the nonlinear estimation problem for this challenging system.

Motivated by state estimation problems in power distribution networks (PDN), the paper proposes a fusion based, reduced order, distributed implementation of the particle filter (FR/DPF) for large scale, nonlinear dynamical systems with localized sensor observations. Direct application of the centralized particle filter is computationally challenging due to the high dimensions of the state-space dynamics. Based on partitioning the overall system into N localized but mathematically coupled subsystems, the near-optimal FR/DPF provides computational savings of a factor of N over the centralized particle filter implementation. By introducing distributed state and observation fusion steps, the proposed FR/DPF does not require a fusion centre and maintains consistency between the local sub-systems. In our Monte Carlo simulations of a simplified PDN, the performance of the FR/DPF is consistently close to that of the centralized implementation.

The past several decades have witnessed the successful application of sequential Monte Carlo method (or particle filter) to a variety of fields. It has grown to be a popular method in solving different kinds of nonlinear Bayesian estimation problems. This paper introduces a two-stage particle filter for nonlinear filtering problem. In the proposed particle filter, each particle will be propagated and updated through two stages. At time step <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> , the first stage refers to using the unscented Kalman filtering equations to propagate the particles from time step <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t-1$ </tex-math></inline-formula> in order to obtain the preliminary estimations. Then, at the second stage, the particles will be updated again by the iterated extended Kalman filter to yield the final updated particles. In this way, the estimation accuracy of particle filter can be improved, which is validated through simulation experiments and real-world application experiments.

A particle filter deals with state estimation problem or nonlinear models with non-Gaussian noise. In the framework of a particle filter, a resampling scheme is used to decrease the degeneracy phenomenon, however it also introduces the problem or sample impoverishment, which can be reduced by using the Markov Chain Monte Carlo (MCMC) method, such as the Metropolis-Hastings (M-H) algorithm. However, there are many possible choices within the family of M-H algorithms, and the performance of particle filters with MCMC moves is closely related to the choice of the M-H algorithm. This paper discusses the implementation of a particle filter with various M-H algorithms. A numerical example is presented, and the simulation results are given for discussion.

The paper deals with the problem of simultaneous state and process fault estimation for uncertain dynamic systems. Contrarily to the approaches presented in the literature, the nonlinear estimation problem is reduced to the linear one by introducing a suitable system reparameterization and new estimator structure. Instead of estimating the fault directly, its product with state and the state itself are estimated. To tackle this problem, a robust design procedure is proposed that takes into account uncertainties acting onto the system being diagnosed. The approach is based on the quadratic boundedness approach allowing convergence analysis of uncertain systems with bounded uncertainties. Subsequently, a simple algebraic approach is proposed to derive the fault estimate. The final part of the paper shows a numerical example concerning state and pitch actuator component fault estimation of a wind turbine.

Parallel Kalman filter group integrated particle filter method for the train nonlinear operational status high-precision estimation under non-Gaussian environment

We present differentiable particle filters (DPFs): a differentiable implementation of the particle filter algorithm with learnable motion and measurement models. Since DPFs are end-to-end differentiable, we can efficiently train their models by optimizing end-to-end state estimation performance, rather than proxy objectives such as model accuracy. DPFs encode the structure of recursive state estimation with prediction and measurement update that operate on a probability distribution over states. This structure represents an algorithmic prior that improves learning performance in state estimation problems while enabling explainability of the learned model. Our experiments on simulated and real data show substantial benefits from end-to- end learning with algorithmic priors, e.g. reducing error rates by ~80%. Our experiments also show that, unlike long short-term memory networks, DPFs learn localization in a policy-agnostic way and thus greatly improve generalization. Source code is available at this https URL .