Showing papers on "Invariant extended Kalman filter published in 2005"

Critical Evaluation of Extended Kalman Filtering and Moving-Horizon Estimation

Abstract: The goal of state estimation is to reconstruct the state of a system from process measurements and a model. State estimators for most physical processes often must address many different challenges, including nonlinear dynamics, states subject to hard constraints (e.g. nonnegative concentrations), and local optima. In this article, we compare the performance of two such estimators: the extended Kalman filter (EKF) and moving horizon estimation (MHE). We illustrate conditions that lead to estimation failure in the EKF when there is no plant-model mismatch and demonstrate such failure via several simple examples. We then examine the role that constraints, the arrival cost, and the type of optimization (global versus local) play in determining how MHE performs on these examples. In each example, the two estimators are given exactly the same information, namely tuning parameters, model, and measurements; yet MHE consistently provides improved state estimation and greater robustness to both poor guesses of the initial state and tuning parameters in comparison to the EKF.

Estimating dynamic equilibrium economies: linear versus nonlinear likelihood

Abstract: This paper compares two methods for undertaking likelihood-based inference in dynamic equilibrium economies: a sequential Monte Carlo filter and the Kalman filter. The sequential Monte Carlo filter exploits the nonlinear structure of the economy and evaluates the likelihood function of the model by simulation methods. The Kalman filter estimates a linearization of the economy around the steady state. We report two main results. First, both for simulated and for real data, the sequential Monte Carlo filter delivers a substantially better fit of the model to the data as measured by the marginal likelihood. This is true even for a nearly linear case. Second, the differences in terms of point estimates, although relatively small in absolute values, have important effects on the moments of the model. We conclude that the nonlinear filter is a superior procedure for taking models to the data. Copyright © 2005 John Wiley & Sons, Ltd.

A Road-Matching Method for Precise Vehicle Localization Using Belief Theory and Kalman Filtering

Abstract: This paper describes a novel road-matching method designed to support the real-time navigational function of cars for advanced systems applications in the area of driving assistance. This method provides an accurate estimation of position for a vehicle relative to a digital road map using Belief Theory and Kalman filtering. Firstly, an Extended Kalman Filter combines the DGPS and ABS sensor measurements to produce an approximation of the vehicle's pose, which is then used to select the most likely segment from the database. The selection strategy merges several criteria based on distance, direction and velocity measurements using Belief Theory. A new observation is then built using the selected segment, and the approximate pose adjusted in a second Kalman filter estimation stage. The particular attention given to the modeling of the system showed that incrementing the state by the bias (also called absolute error) of the map significantly increases the performance of the method. Real experimental results show that this approach, if correctly initialized, is able to work over a substantial period without GPS.

An adaptive Kalman filter for dynamic harmonic state estimation and harmonic injection tracking

Abstract: Knowledge of the process noise covariance matrix Q is essential for the application of Kalman filtering. However, it is usually a difficult task to obtain an explicit expression of Q for large time varying systems. This paper looks at an adaptive Kalman filter method for dynamic harmonic state estimation and harmonic injection tracking. The method models the system as a linear frequency independent state model and does not require an exact knowledge of the noise covariance matrix Q. As an alternative, the proposed adaptive Kalman filter switches between the two basic Q models for steady-state and transient estimation. Its adaptive function allows for the resetting of the Kalman gain to avoid Kalman filter divergence problems under steady-state and allow fast tracking of system variations in transient conditions. Simulation results on the 220 kV network of the lower South Island of New Zealand are presented to validate this approach.

Aircraft Turbofan Engine Health Estimation Using Constrained Kalman Filtering

Abstract: Kalman ¯lters are often used to estimate the state variablesof a dynamic system. However, in the application of Kalman¯lters some known signal information is often either ignored ordealt with heuristically. For instance, state variable constraints(which may be based on physical considerations) are often ne-glected because they do not ¯t easily into the structure of theKalman ¯lter. This paper develops an analytic method of in-corporating state variable inequality constraints in the Kalman¯lter. The resultant ¯lter is a combination of a standard Kalman¯lter and a quadratic programming problem. The incorporationof state variable constraints increases the computational e®ort ofthe ¯lter but signi¯cantly improves its estimation accuracy. Theimprovement is proven theoretically and shown via simulationresults obtained from application to a turbofan engine model.This model contains 16 state variables, 12 measurements, and 8component health parameters. It is shown that the new algo-rithms provide improved performance in this example over un-constrained Kalman ¯ltering.INTRODUCTION

Square Root SAM

Abstract: Solving the SLAM problem is one way to enable a robot to explore, map, and navigate in a previously unknown environment. We investigate smoothing approaches as a viable alternative to extended Kalman filter-based solutions to the problem. In particular, we look at approaches that factorize either the associated information matrix or the measurement matrix into square root form. Such techniques have several significant advantages over the EKF: they are faster yet exact, they can be used in either batch or incremental mode, are better equipped to deal with non-linear process and measurement models, and yield the entire robot trajectory, at lower cost. In addition, in an indirect but dramatic way, column ordering heuristics automatically exploit the locality inherent in the geographic nature of the SLAM problem. In this paper we present the theory underlying these methods, an interpretation of factorization in terms of the graphical model associated with the SLAM problem, and simulation results that underscore the potential of these methods for use in practice.

Nonlinear Attitude Filtering Methods

Abstract: This paper provides a survey of modern nonlinear filtering methods for attitude estimation. Early applications relied mostly on the extended Kalman filter for attitude estimation. Since these applications, several new approaches have been developed that have proven to be superior to the extended Kalman filter. Several of these approaches maintain the basic structure of the extended Kalman filter, but employ various modifications in order to provide better convergence or improve other performance characteristics. Examples of such approaches include: filter QUEST, extended QUEST, the super-iterated extended Kalman filter, the interlaced extended Kalman filter, and the second-order Kalman filter. Filters that propagate and update a discrete set of sigma points rather than using linearized equations for the mean and covariance are also reviewed. A two-step approach is discussed with a first-step state that linearizes the measurement model and an iterative second step to recover the desired attitude states. These approaches are all based on the Gaussian assumption that the probability density function is adequately specified by its mean and covariance. Other approaches that do not require this assumption are reviewed, including particle filters and a Bayesian filter based on a non-Gaussian, finite-parameter probability density function on SO(3). Finally, the predictive filter, nonlinear observers and adaptive approaches are shown. The strengths and weaknesses of the various approaches are discussed.

On the equivalence between Kalman smoothing and weak‐constraint four‐dimensional variational data assimilation

Abstract: The fixed-interval Kalman smoother produces optimal estimates of the state of a system over a time interval, given observations over the interval, together with a prior estimate of the state and its error covariance at the beginning of the interval. At the end of the interval, the Kalman smoother estimate is identical to that produced by a Kalman filter, given the same observations and the same initial state and covariance matrix. For an imperfect model, the model error term in the covariance evolution equation acts to reduce the dependence of the estimate on observations and prior states that are well separated in time. In particular, if the assimilation interval is sufficiently long, the estimate at the end of the interval is effectively independent of the state and covariance matrix specified at the beginning of the interval. In this case, the Kalman smoother provides estimates at the end of the interval that are identical to those of a Kalman filter that has been running indefinitely. For a linear model, weak-constraint four-dimensional variational data assimilation (4D-Var) is equivalent to a fixed-interval Kalman smoother. It follows that, if the assimilation interval is made sufficiently long, the 4D-Var analysis at the end of the assimilation interval will be identical to that produced by a Kalman filter that has been running indefinitely. The equivalence between weak-constraint 4D-Var and a long-running Kalman filter is demonstrated for a simple analogue of the numerical weather-prediction (NWP) problem. For this nonlinear system, 4D-Var analysis with a 10-day assimilation window produces analyses of the same quality as those of an extended Kalman filter. It is demonstrated that the current ECMWF operational 4D-Var system retains a memory of earlier observations and prior states over a period of between four and ten days, suggesting that weak-constraint 4D-Var with an analysis interval in the range of four to ten days may provide a viable algorithm with which to implement an unapproximated Kalman filter. Whereas assimilation intervals of this length are unlikely to be computationally feasible for operational NWP in the near future, the ability to run an unapproximated Kalman filter should prove invaluable for assessing the performance of cheaper, but suboptimal, alternatives.

Backward-Smoothing Extended Kalman Filter

Abstract: The principle of the iterated extended Kalman filter has been generalized to create a new filter that has superior performance when the estimation problem contains severe nonlinearities. The new filter is useful when nonlinearities might significantly degrade the accuracy or convergence reliability of other filters. The new filter solves a nonlinear smoothing problem for the current and past sample intervals using iterative numerical techniques. This approach retains the nonlinearities of a fixed number of stages that precede the stage of interest, and it processes information from earlier stages in an approximate manner. The algorithm has been simulation tested on a difficult spacecraft attitude estimation problem that includes sensing of fewer than three axes and significant dynamic model uncertainty. The filter compensates for this uncertainty via simultaneous estimation of moment of inertia parameters. The new filter exhibits markedly better convergence reliability and accuracy than an extended Kalman filter and an unscented Kalman filter for estimation problems that start with large initial attitude or attitude rate errors.

Ensemble kalman filter experiments with a primitive-equation global model

Abstract: Title of dissertation: ENSEMBLE KALMAN FILTER EXPERIMENTS WITH A PRIMITIVE-EQUATION GLOBAL MODEL Takemasa Miyoshi, Doctor of Philosophy, 2005 Dissertation directed by: Professor Eugenia Kalnay Department of Meteorology The ultimate goal is to develop a path towards an operational ensemble Kalman filtering (EnKF) system. Several approaches to EnKF for atmospheric systems have been proposed but not systematically compared. The sensitivity of EnKF to the imperfections of forecast models is unclear. This research explores two questions: 1. What are the relative advantages and disadvantages of two promising EnKF methods? 2. How large are the effects of model errors on data assimilation, and can they be reduced by model bias correction? Chapter 2 contains a theoretical review, followed by the FORTRAN development and testing of two EnKF methods: a serial ensemble square root filter (serial EnSRF, Whitaker and Hamill 2002) and a local EnKF (LEKF, Ott et al. 2002; 2004). We reproduced the results obtained by Whitaker and Hamill (2002) and Ott et al. (2004) on the Lorenz (1996) model. If we localize the LEKF error covariance, LEKF outperforms serial EnSRF. We also introduce a method to objectively estimate the optimal covariance inflation. In Chapter 3 we apply the two EnKF methods and the three-dimensional variational method (3DVAR) to the SPEEDY primitive-equation global model (Molteni 2003), a fast but relatively realistic model. Perfect model experiments show that EnKF greatly outperforms 3DVAR. The 2-day forecast ”errors of the day” are very similar to the analysis errors, but they are not similar among different methods except in low ensemble dimensional regions. Overall, serial EnSRF outperforms LEKF, but their difference is substantially reduced if we localize the LEKF error covariance or increase the ensemble size. Since LEKF is much more efficient than serial EnSRF when using parallel computers and many observations, LEKF would be the only feasible choice in operations. In Chapter 4 we remove the perfect model assumption using the NCEP/NCAR reanalysis as the ”nature” run. The advantage of EnKF to 3DVAR is reduced. When we apply the model bias estimation proposed by Dee and da Silva (1998), we find that the full dimensional model bias estimation fails. However, if instead we assume that the bias is low dimensional, we obtain a substantial improvement in the EnKF analysis. ENSEMBLE KALMAN FILTER EXPERIMENTS WITH A PRIMITIVE-EQUATION GLOBAL MODEL

A kalman filter based background updating algorithm robust to sharp illumination changes

Abstract: A novel algorithm, based on Kalman filtering is presented for updating the background image within video sequences. Unlike existing implementations of the Kalman filter for this task, our algorithm is able to deal with both gradual and sudden global illumination changes. The basic idea is to measure global illumination change and to use it as an external control of the filter. This allows the system to better fit the assumptions about the process to be modeled. Moreover, we propose methods to estimate measurement noise variance and to deal with the problem of saturated pixels, to improve the accuracy and robustness of the algorithm. The algorithm has been successfully tested in a traffic surveillance task by comparing it to a background updating algorithm, based on Kalman filtering, taken from literature.

Traffic matrix tracking using Kalman filters

Abstract: In this work we develop a new approach to monitoring origin-destination flows in a large network. We start by building a state space model for OD flows that is rich enough to fully capture temporal and spatial correlations. We apply a Kalman filter to our linear dynamic system that can be used for both estimation and prediction of traffic matrices. We call our system a traffic matrix tracker due to its lightweight mechanism for temporal updates that enables tracking traffic matrix dynamics at small time scales. Our Kalman filter approach allows us to go beyond traffic matrix estimation in that our single system can also carry out traffic prediction and yield confidence bounds on the estimates, the predictions and the residual error processes. We show that these elements provide key functionalities needed by monitoring systems of the future for carrying out anomaly detection. Using real data collected from a Tier-1 ISP, we validate our model, illustrate that it can achieve low errors, and that our method is adaptive on both short and long timescales.

A comparison of error subspace Kalman filters

Abstract: Three advanced filter algorithms based on the Kalman filter are reviewed and presented in a unified notation. They are the well-known ensemble Kalman filter (EnKF), the singular evolutive extended Kalman (SEEK) filter, and the less common singular evolutive interpolated Kalman (SEIK) filter. For comparison, the mathematical formulations of the filters are reviewed in relation to the extended Kalman filter as error subspace Kalman filters. The algorithms are presented in their original form and possible variations are discussed. A comparison of the algorithms shows their theoretical capabilities for efficient data assimilation with large-scale non-linear systems. In particular, problems of the analysis equations are apparent in the original EnKF algorithm due to the Monte Carlo sampling of ensembles. Theoretically, the SEIK filter appears to be a numerically very efficient algorithm with high potential for use with non-linear models. The superiority of the SEIK filter is demonstrated on the basis of identical twin experiments using a shallow-water model with non-linear evolution. Identical initial conditions for all three filters allow for a consistent comparison of the data assimilation results. These show how choices of particular state ensembles and assimilation schemes lead to significant variations of the filter performance. This is related to different qualities of the predicted error subspaces, as is demonstrated in an examination of the predicted state covariance matrices.

Polynomial extended Kalman filter

Abstract: This work presents a polynomial version of the well-known extended Kalman filter (EKF) for the state estimation of nonlinear discrete-time stochastic systems. The proposed filter, denoted polynomial EKF (PEKF), consists in the application of the optimal polynomial filter of a chosen degree /spl mu/ to the Carleman approximation of a nonlinear system. When /spl mu/=1 the PEKF algorithm coincides with the standard EKF. For the filter implementation the moments of the state and output noises up to order 2/spl mu/ are required. Numerical simulations compare the performances of the PEKF with those of some other existing filters, showing significant improvements.

The iterated extended Kalman particle filter

Abstract: Particle filtering shows great promise in addressing a wide variety of non-linear and /or non-Gaussian problem. A crucial issue in particle filtering is the selection of the importance proposal distribution. In this paper, the iterated extended Kalman filter (IEKF) is used to generate the proposal distribution. The proposal distribution integrates the latest measurements into system state transition density, so it can match the posteriori density well. The simulation results show that the new particle filter superiors to the standard particle filter and the other filters such as the unscented particle filter (UPF), the extended Kalman particle filter (PF-EKF), the EKF.

Particle filters for tracking with out-of-sequence measurements

Abstract: An extension is presented to the particle filtering toolbox that enables nonlinear/non-Gaussian filtering to be performed in the presence of out-of-sequence measurements (OOSMs) with arbitrary lag, without the need to adopt linearising approximations in the filter and without the degradation of performance that would occur if the OOSMs were simply discarded. An estimate of the performance of the OOSM particle filter (OOSM-PF) is obtained for bearings-only tracking scenarios with a single target and a small number of sensors. These performance estimates are then compared with the posterior Cramer-Rao lower bound (CRLB) for the state estimate rms error and similar performance estimates obtained from the oosm extended Kalman filter (OOSM-EKF) algorithms recently introduced in the literature. For a mildly nonlinear bearings-only tracking problem the OOSM-PF and OOSM-EKF are shown to achieve broadly similar performance.

Particle filter for sensor fusion in a land vehicle navigation system

Abstract: Generally, the extended Kalman filter (EKF) is used for sensor fusion in a land vehicle navigation system. However, defects of the first-order linearization of the nonlinear model in the EKF can introduce large estimated errors, and may lead to sub-optimal performance. In order to yield higher accuracy of navigation, in this paper, a novel particle filter (PF) for sensor fusion is proposed and the sampling importance resampling particle filter (SIR-PF) is applied to address the nonlinear measurement model and it shows better performances when compared with the EKF. The basic theories and application of the general PF and the SIR-PF for a global position system/dead reckoning (GPS/DR) integrated navigation system are discussed.

Tracking time-varying parameters in software systems with extended Kalman filters

Abstract: Autonomic control of a service system can take advantage of a performance model only if a way can be found to track the changes in the system. A Kalman Filter provides a framework for integrating various kinds of measured data, and for tracking changes in any time-varying system. This work evaluates the effectiveness of such a filter in tracking changes in performance parameters of a software system that occur at different rates and amplitudes. The time-varying system is a Web application deployed in a data centre with layered queuing resources, in which parameter variations happen at random instants. The tracking filter is based on a layered queuing model of this system, with parameters representing CPU demands and the user load intensity. Experiments were performed to evaluate the effectiveness of the filter in tracking the changes, and the requirements for the filter settings for fast and slow variations in the parameters. The target application is autonomic control of a service centre.

Optimal linear mean square filter for continuous-time jump linear systems

Abstract: We consider a class of hybrid systems which is modeled by continuous-time linear systems with Markovian jumps in the parameters (LSMJP). Our aim is to derive the best linear mean square estimator for such systems. The approach adopted here produces a filter which bears those desirable properties of the Kalman filter: A recursive scheme suitable for computer implementation which allows some offline computation that alleviates the computational burden. Apart from the intrinsic theoretical interest of the problem in its own right and the application-oriented motivation of getting more easily implementable filters, another compelling reason why the study here is pertinent has to do with the fact that the optimal nonlinear filter for our estimation problem is not computable via a finite computation (the filter is infinite dimensional). Our filter has dimension Nn, with n denoting the dimension of the state vector and N the number of states of the Markov chain.

A SDRE-based asymptotic observer for nonlinear discrete-time systems

Abstract: A nonlinear asymptotic observer for a discrete-time nonlinear system is considered. The observer is based on a Kalman filter that uses the state dependent Riccati equation (SDRE) to obtain the filter gain. Unlike the extended Kalman filter, the SDRE-based Kalman filter does not involve the evaluation of a Jacobian at every time step. The convergence properties of the SDRE-based Kalman filter when used as an observer in a deterministic setting are analyzed. A few simulation examples are provided to demonstrate the performance and implementation of the SDRE-based observer in both deterministic and stochastic settings.

Unscented Kalman filter design for curvilinear motion models suitable for automotive safety applications

Abstract: Research in automotive safety leads to the conclusion that modern vehicle should utilize active and passive sensors for the recognition of the environment surrounding them. Thus, the development of tracking systems utilizing efficient state estimators is very important. In this case, problems such as moving platform carrying the sensor and maneuvering targets could introduce large errors in the state estimation and in some cases can lead to the divergence of the filter. In order to avoid sub-optimal performance, the unscented Kalman filter is chosen, while a new curvilinear model is applied which takes into account both the turn rate of the detected object and its tangential acceleration, leading to a more accurate modeling of its movement. The performance of the unscented filter using the proposed model in the case of automotive applications is proven to be superior compared to the performance of the extended and linear Kalman filter.

Consistency of three Kalman filter extensions in hybrid navigation

Abstract: A filter is consistent if predicted errors are at least as large as actual errors. In this paper, we evaluate the consistency of three filters and illustrate what could happen if filters are inconsistent. Our application is hybrid positioning which is based on signals from satellites and from mobile phone network base stations. Examples show that the consistency of a filter is very important. We evaluate three filters: EKF, EKF2 and PKF. Extended Kalman Filter (EKF) solves the filtering problem by linearizing functions. EKF is very commonly used in satellite-based positioning and it has also been applied in hybrid positioning. We show that nonlinearities are insignificant in satellite measurements but often significant in base station measurements. Because of this, we also apply Second Order Extended Kalman Filter (EKF2) in hybrid positioning. EKF2 is an elaboration of EKF that takes into consideration the nonlinearity of the measurement models. The third filter is called Position Kalman Filter (PKF), which filters a sequence of static positions and velocities. We also check what kind of measurement combinations satisfy CGALIES and FCC requirements for location.

A Comparative Study of Kalman Filtering for Sensorless Control of a Permanent-Magnet Synchronous Motor Drive

Abstract: This paper presents a comparative study of the novel unscented Kalman filter (UKF) and the extended Kalman filter (EKF) for estimation of the rotor speed and position of a permanent-magnet synchronous motor (PMSM) drive. The general structure of the EKF and the UKF are reviewed. The various system vectors, matrices, models and algorithm programs are presented. Simulation studies on the two Kalman filters are carried out using Matlab and Simulink to explore the usability of the UKF in a sensorless PMSM drive. In order to compare the estimation performances of the observers, both filters are designed for the same motor model and control system and run with the same covariances. The simulation results indicate that the UKF is capable of tracking the actual rotor speed and position provided that the elements of the covariance matrices are properly selected. Since covariance tuning of the Kalman filter is often a trial-and-error process, an unconventional, asymmetric way of setting the model covariance parameters is introduced. It is shown that tuning is easier and the method gives a significant improvement in performance and filter stability

Study on GPS attitude determination system aided INS using adaptive Kalman filter

Abstract: A marine INS/GPS (inertial navigation system/global positioning system) adaptive navigation system is presented in this paper. The GPS with two antennae providing vessel attitude is selected as the auxiliary system to fuse with INS. The Kalman filter is the most frequently used algorithm in the integrated navigation system, which is capable of estimating INS errors online based on the measured errors between INS and GPS. The conventional Kalman filter (CKF) assumes that the statistics of the noise of each sensor are given. As long as the noise distributions do not change, the Kalman filter will give the optimal estimation. However, the GPS receiver will be disturbed easily and thus temporally changing measurement noise will join into the outputs of GPS, which will lead to performance degradation of the Kalman filter. Many researchers introduce a fuzzy logic control method into innovation-based adaptive estimation Kalman filtering (IAE-AKF) algorithm, and accordingly propose various adaptive Kalman filters. However, how to design the fuzzy logic controller is a very complicated problem, which is still without a convincing solution. A novel IAE-AKF is proposed herein, which is based on the maximum likelihood criterion for the proper computation of the filter innovation covariance and hence of the filter gain. The approach is direct and simple without having to establish fuzzy inference rules. After having deduced the proposed IAE-AKF algorithm theoretically in detail, the approach is tested in the developed INS/GPS integrated marine navigation system. Real field test results show that the adaptive Kalman filter outperforms the CKF with higher accuracy and robustness. It is demonstrated that this proposed approach is a valid solution for the unknown changing measurement noise existing in the Kalman filter.

Using data-driven discrete-time models and the unscented Kalman filter to estimate unobserved variables of nonlinear systems.

Abstract: This paper addresses the problem of state estimation for nonlinear systems by means of the unscented Kalman filter UKF. Compared to the traditional extended Kalman filter, the UKF does not require the local linearization of the system equations used in the propagation stage. Important results using the UKF have been reported recently but in every case the system equations used by the filter were considered known. Not only that, such models are usually considered to be differential equations, which requires that numerical integration be performed during the propagation phase of the filter. In this paper the dynamical equations of the system are taken to be difference equations—thus avoiding numerical integration—and are built from data without prior knowledge. The identified models are subsequently implemented in the filter in order to accomplish state estimation. The paper discusses the impact of not knowing the exact equations and using data-driven models in the context of state and joint state-and-parameter estimation. The procedure is illustrated by means of examples that use simulated and measured data.

Radiation belt data assimilation with an extended Kalman filter

Abstract: [1] Kalman filtering provides an elegant framework for assimilating observational data into time-dependent theoretical models. This paper explores the application of this approach to specify and forecast the radiation belt particle distribution. The Kalman filter is first outlined in a general form. A data assimilation algorithm is then derived for a simple radiation belt forecast model driven by radial diffusion. The model assimilates particle flux measurements from spacecraft in the equatorial plane, using an external magnetic field model to calculate adiabatic invariants and phase space density. The algorithm is tested in a series of virtual experiments, with data from an idealized magnetic storm simulation. Compared to assimilation by direct insertion of data, the Kalman filter more accurately reconstructs the global particle distribution from sparse observational data. We examine the response of the filter to errors in the observations, magnetic field model, and forecast model and discuss the application of this approach to more realistic models and data sets.