Showing papers on "Invariant extended Kalman filter published in 2008"

Distributing the Kalman Filter for Large-Scale Systems

Abstract: This paper presents a distributed Kalman filter to estimate the state of a sparsely connected, large-scale, n -dimensional, dynamical system monitored by a network of N sensors. Local Kalman filters are implemented on nl-dimensional subsystems, nl Lt n, obtained by spatially decomposing the large-scale system. The distributed Kalman filter is optimal under an Lth order Gauss-Markov approximation to the centralized filter. We quantify the information loss due to this Lth-order approximation by the divergence, which decreases as L increases. The order of the approximation L leads to a bound on the dimension of the subsystems, hence, providing a criterion for subsystem selection. The (approximated) centralized Riccati and Lyapunov equations are computed iteratively with only local communication and low-order computation by a distributed iterate collapse inversion (DICI) algorithm. We fuse the observations that are common among the local Kalman filters using bipartite fusion graphs and consensus averaging algorithms. The proposed algorithm achieves full distribution of the Kalman filter. Nowhere in the network, storage, communication, or computation of n-dimensional vectors and matrices is required; only nl Lt n dimensional vectors and matrices are communicated or used in the local computations at the sensors. In other words, knowledge of the state is itself distributed.

Adaptive estimation of multiple fading factors in Kalman filter for navigation applications

Abstract: Kalman filter is the most frequently used algorithm in navigation applications. A conventional Kalman filter (CKF) assumes that the statistics of the system noise are given. As long as the noise characteristics are correctly known, the filter will produce optimal estimates for system states. However, the system noise characteristics are not always exactly known, leading to degradation in filter performance. Under some extreme conditions, incorrectly specified system noise characteristics may even cause instability and divergence. Many researchers have proposed to introduce a fading factor into the Kalman filtering to keep the filter stable. Accordingly various adaptive Kalman filters are developed to estimate the fading factor. However, the estimation of multiple fading factors is a very complicated, and yet still open problem. A new approach to adaptive estimation of multiple fading factors in the Kalman filter for navigation applications is presented in this paper. The proposed approach is based on the assumption that, under optimal estimation conditions, the residuals of the Kalman filter are Gaussian white noises with a zero mean. The fading factors are computed and then applied to the predicted covariance matrix, along with the statistical evaluation of the filter residuals using a Chi-square test. The approach is tested using both GPS standalone and integrated GPS/INS navigation systems. The results show that the proposed approach can significantly improve the filter performance and has the ability to restrain the filtering divergence even when system noise attributes are inaccurate.

Performance Improvement of Industrial Drives With Mechanical Elasticity Using Nonlinear Adaptive Kalman Filter

Abstract: This paper deals with the application of the adaptive control structure for torsional vibration suppression in the drive system with an elastic coupling The proportional-integral speed controller and gain factors of two additional feedback loops, from the shaft torque and load side speed, are tuned on-line according to the changeable load side inertia This parameter, as well as other mechanical variables of the drive system (load side speed, torsional and load torques), are estimated with the use of the developed nonlinear extended Kalman filter (NEKF) The initial values of the Kalman filter covariance matrices are set using the genetic algorithm Then, to ensure the smallest state and parameter estimation errors, the on-line adaptation law for the chosen element of the state covariance matrix of the NEKF is proposed The described control strategy is tested in an open and a closed-loop control structure The simulation results are confirmed by laboratory experiments

A New Approximate Solution of the Optimal Nonlinear Filter for Data Assimilation in Meteorology and Oceanography

Abstract: This paper introduces a new approximate solution of the optimal nonlinear filter suitable for nonlinear oceanic and atmospheric data assimilation problems. The method is based on a local linearization in a low-rank kernel representation of the state's probability density function. In the resulting low-rank kernel particle Kalman (LRKPK) filter, the standard (weight type) particle filter correction is complemented by a Kalman-type correction for each particle using the covariance matrix of the kernel mixture. The LRKPK filter's solution is then obtained as the weighted average of several low-rank square root Kalman filters operating in parallel. The Kalman-type correction reduces the risk of ensemble degeneracy, which enables the filter to efficiently operate with fewer particles than the particle filter. Combined with the low-rank approximation, it allows the implementation of the LRKPK filter with high-dimensional oceanic and atmospheric systems. The new filter is described and its relevance demonstrated through applications with the simple Lorenz model and a realistic configuration of the Princeton Ocean Model (POM) in the Mediterranean Sea.

Nonconcurrent multiple speakers tracking based on extended Kalman particle filter

Abstract: Acoustic reverberation introduces multipath components into an audio signal, and therefore changes the source signal statistical properties. This causes problems for source localisation and tracking since reverberation generates spurious peaks in the time delay functions, and makes the subsequent location estimator hard to track the motion trajectory. Previous time delay based tracking methods, such as the extended Kalman filter and the particle filter, are sensitive to reverberation and are unable to follow sharp changes in the source positions. In this paper, the extended Kalman filter and the particle filter are combined to solve this problem. One of the advantages of this approach is that the optimal importance function can be obtained after extended Kalman filtering. Thus, the position samples are distributed in a more accurate area than using a prior importance function. Experiment results show that the proposed algorithm outperforms the sequential importance resampling particle filter by reducing the estimation error and following the switch of speakers quickly under a moderate reverberant environment (reverberation time T60 < 0.3s).

Square-Root Quadrature Kalman Filtering

Abstract: The quadrature Kalman filter (QKF) is a recursive, nonlinear filtering algorithm developed in the Kalman filtering framework It computes the mean and covariance of all conditional densities using the Gauss-Hermite quadrature rule In this correspondence, we develop a square-root extension of the quadrature Kalman filter using matrix triangularizations The square-root quadrature Kalman filter (SQKF) propagates the mean and the square root of the covariance Although equivalent to the QKF algebraically, the SQKF exhibits excellent numerical characteristics, but at the expense of increased computational complexity We also present possible refinements of the generic SQKF

Convergence properties of a decentralized Kalman filter

Abstract: We consider the problem of decentralized Kalman filtering in a sensor network. Each sensor node implements a local Kalman filter based on its own measurements and the information exchanged with its neighbors. It combines the information received from other sensors through using a consensus filter as proposed in [14]. For a time-invariant process and measurement model, we show that this algorithm guarantees that the local estimates of the error covariance matrix converge to the centralized error covariance matrix and that the local estimates of the state converge in mean to the centralized Kalman filter estimates. However, due to the use of the consensus filter, the local estimates of the state do not converge to the least-squares estimate that would be obtained from a centralized Kalman filter.

Adaptive robust extended Kalman filter for nonlinear stochastic systems

Abstract: The authors analyse the error behaviour of the robust extended Kalman filter (REKF) for nonlinear stochastic systems. On the basis of some standard results about the boundedness of stochastic processes, it is specified that stability of the REKF cannot be guaranteed. In order to solve this problem, a novel method is proposed to design the REKF so that the sufficient conditions to ensure filter stability will be fulfilled. Furthermore, an adaptive scheme is adopted to automatically tune the error covariance matrix in response to the changing environment. Numerical example shows the superiority of the proposed adaptive REKF over the usual extended Kalman filter (EKF), the REKF and the adaptive EKF.

Optimal Full-Order and Reduced-Order Estimators for Discrete-Time Systems With Multiple Packet Dropouts

Abstract: This paper is concerned with the estimation problem for discrete-time stochastic linear systems with multiple packet dropouts. Based on a recently developed model for multiple-packet dropouts, the original system is transferred to a stochastic parameter system by augmentation of the state and measurement. The optimal full-order linear filter of the form of employing the received outputs at the current and last time instants is investigated. The solution to the optimal linear filter is given in terms of a Riccati difference equation governed by packet arrival rate. The optimal filter is reduced to the standard Kalman filter when there are no packet dropouts. The steady-state filter is also studied. A sufficient condition for the existence of the steady-state filter is given and the asymptotic stability of the optimal filter is analyzed. At last, a reduced-order filter is investigated.

Constrained state estimation using the Unscented Kalman Filter

Abstract: A simple procedure to include state inequality constraints in the unscented Kalman filter is proposed. With this procedure, the information of active state constraints influences the state covariance matrix, resulting in better estimates. In a numerical example, the approach outperforms the extended Kalman filter implemented with constraint handling via ldquoclippingrdquo.

Gain-Constrained Kalman Filtering for Linear and Nonlinear Systems

Abstract: This paper considers the state-estimation problem with a constraint on the data-injection gain. Special cases of this problem include the enforcing of a linear equality constraint in the state vector, the enforcing of unbiased estimation for systems with unknown inputs, and simplification of the estimator structure for large-scale systems. Both the one-step gain-constrained Kalman predictor and the two-step gain-constrained Kalman filter are presented. The latter is extended to the nonlinear case, yielding the gain-constrained unscented Kalman filter. Two illustrative examples are presented.

Computational aspects of continuous---discrete extended Kalman-filtering

Abstract: This paper elaborates how the time update of the continuous---discrete extended Kalman-filter (EKF) can be computed in the most efficient way. The specific structure of the EKF-moment differential equations leads to a hybrid integration algorithm, featuring a new Taylor---Heun-approximation of the nonlinear vector field and a modified Gauss---Legendre-scheme, generating positive semidefinite solutions for the state error covariance. Furthermore, the order of consistency and stability behavior of the outlined procedure is investigated. The results are incorporated into an algorithm with adaptive controlled step size, assuring a fixed numerical precision with minimal computational effort.

Electric power system static state estimation through Kalman filtering and load forecasting

Abstract: Static state estimation in electric power systems is normally accomplished without the use of time-history data or prediction. This paper presents preliminary work on the use of the discrete-time Kalman filter to incorporate time history and power demand prediction into state estimators. The problem of state estimation combined with the knowledge of the forecasted load is posed as a Kalman filtering problem using a novel discrete-time model. The model relates current and previous states using the electric power flow equations. An IEEE 14-bus test system example is used to illustrate the potential for enhanced performance of such Kalman filter-based state estimation.

The soft iron and hard iron calibration method using extended kalman filter for attitude and heading reference system

Abstract: The calibration method of the soft iron and hard iron distortion based on attitude and heading reference system (AHRS) can boil down to the estimation of 12 parameters of magnetic deviation, normally using 12-state Kalman filter (KF) algorithm. The performance of compensation is limited by the accuracy of local inclination angle of magnetic field and initial heading. A 14-state extended Kalman filter (EKF) algorithm is developed to calibrate magnetic deviation, local magnetic inclination angle error and initial heading error all together. The calibration procedure is to change the attitude of AHRS and rotate it two cycles. As the strapdown matrix can hold high precision after initial alignment of AHRS in short time for the gyropsilas short-term precision, the magnetic field vector can be projected onto the body frame of AHRS. The experiment results demonstrate that 14-state EKF outperforms 12-state KF, with measurement errors exist in the initial heading and local inclination angle. The heading accuracy (variance) after compensation is 0.4 degree for tilt angle ranging between 0 and 60 degree.

The Kalman Filter

Abstract: The Kalman filter and its variants are some of the most popular tools in statistical signal processing and estimation theory. In this chapter, we introduce the Kalman filter, providing a succinct, yet rigorous derivation thereof, which is based on the orthogonality principle. We also introduce several important variants of the Kalman filter, namely various Kalman smoothers, a Kalman predictor, a nonlinear extension (the extended Kalman filter), and adaptation to cases of temporally correlated measurement noise.

Target trajectory generator for predictive control of nonlinear systems using extended Kalman filter

Abstract: A model predictive controller (MPC) for predictive control of nonlinear processes utilizing an EKF (Extended Kalman Filter) and a nominal trajectory generator. The nominal trajectory generator includes another instance of EKF, a linear corrector, and a time-varying deviation model. A nominal control trajectory can be predicted and an optimal solution for the time-varying deviation model can be computed based on an approximation of a system inverse known as signal de-convolution. The EKF can be utilized to estimate a current process state by supplying a measured output and to predict a future nominal trajectory by supplying a reference output. A Kalman smoother can also be utilized for the signal de-convolution in order to obtain enhanced trajectory estimates.

Quaternion attitude estimation for miniature air vehicles using a multiplicative extended Kalman filter

Abstract: Utilizing the Euler-Rodrigues symmetric parameters (attitude quaternion) to describe vehicle orientation, we develop a multiplicative, nonlinear variation of the Kalman filter to fuse data from low-cost sensors. The sensor suite is comprised of gyroscopes, accelerometers, and a GPS receiver. Our filter states consist of the three components of an Euler attitude error vector. In parallel with the state time update, we utilize the gyroscope measurements for the time propagation of the attitude quaternion. The accelerometer and the GPS sensors are used independently for the measurement update portion of the filter. For both sensors, a vector arithmetic approach is used to determine the Euler attitude error vector. Following each measurement update, a multiplicative reset operation moves the attitude error information from the state into the attitude estimate. This reset operation utilizes quaternion algebra to implicitly maintain the unity-norm constraint. We demonstrate the effectiveness of our attitude estimation algorithm through flight simulations of aggressive maneuvers such as loops and small-radius circles.

A battery State of Charge estimation method with extended Kalman filter

Abstract: In this paper, a battery state of charge (SOC) estimation method based on the extended Kalman filter is proposed. In some known battery SOC estimation methods, it is assumed that the relationship between battery open circuit voltage and SOC is linear and static. However, this relationship is only piece wisely linear in practice and varies with the ambient temperature, as assumed in this work. The proposed model assumption matches better with the real battery behavior. A battery is modeled as a nonlinear system, with the SOC defined as a system state. The extended Kalman filter is applied to estimate SOC directly for a lithium battery pack. The effectiveness of the proposed method is verified on a power transmission line inspection robot. The experimental results verify the effectiveness of the proposed method.

Unscented Kalman filter with nonlinear dynamic process modeling for GPS navigation

Abstract: This paper preliminarily investigates the application of unscented Kalman filter (UKF) approach with nonlinear dynamic process modeling for Global positioning system (GPS) navigation processing. Many estimation problems, including the GPS navigation, are actually nonlinear. Although it has been common that additional fictitious process noise can be added to the system model, however, the more suitable cure for non convergence caused by unmodeled states is to correct the model. For the nonlinear estimation problem, alternatives for the classical model-based extended Kalman filter (EKF) can be employed. The UKF is a nonlinear distribution approximation method, which uses a finite number of sigma points to propagate the probability of state distribution through the nonlinear dynamics of system. The UKF exhibits superior performance when compared with EKF since the series approximations in the EKF algorithm can lead to poor representations of the nonlinear functions and probability distributions of interest. GPS navigation processing using the proposed approach will be conducted to validate the effectiveness of the proposed strategy. The performance of the UKF with nonlinear dynamic process model will be assessed and compared to those of conventional EKF.

Applicable filtering framework for online multiclass freeway network estimation

Abstract: Real-time traffic flow estimation is important for online traffic control and management. The traffic state estimator optimally matches traffic measurements from detectors with traffic flow predictions from a dynamic traffic model under a certain control strategy. The current and widely used estimator is based on the Extended Kalman Filter algorithm (EKF). Basically, EKF is developed from the recursive Bayesian estimation technique for Gaussian random distribution of the state. This approximation may result in large errors in the estimation and even lead to divergence of the filter in highly non-linear dynamic system such as heterogeneous traffic flow operations. The aims of this paper are therefore twofold. On the one hand, we present a generalized stochastic macroscopic traffic model for multiclass freeway networks. The model is developed in the form that can be applied by filtering methods. On the other hand, we implement an accurate probabilistic framework to the real-time multiclass freeway network estimation. The framework uses a variation of Kalman Filter, namely Unscented Kalman Filter, and a different filter that is based on a sequential Monte Carlo method, namely Unscented Particle Filter. We investigate the performance of the proposed framework with respect to accuracy and computational effort using real-life data collected in a freeway network in England. We expect that the developed tool is useful for traffic operators and planners in controlling large-scale multiclass freeway networks.

Using the Kalman filter for parameter estimation in biogeochemical models

Abstract: We investigate application of nonlinear variants of the Kalman filter (KF) to sequential parameter estimation in biogeochemical models, with particular focus on two components of the statistical model: Q, the covariance of the stochastic forcing which we use to represent model error, and R, the observation error covariance matrix. We explored sensitivity of parameter estimates from the extended and ensemble Kalman filters (EKF and EnKF) to the choice of Q, R, initial parameters and ensemble size using pseudo-data from a simple yet highly nonlinear test model with many characteristics similar to real terrestrial biogeochemistry models. We found for our application that the use of inflated observation uncertainties led to the best and most stable parameter estimates. Although this reduced the rate of convergence to a solution, it also reduced the sensitivity of the solution to model error or ensemble size in the EnKF. Neither the use of model error for the parameters nor inflation of the state error covariance was particularly successful. Copyright © 2008 John Wiley & Sons, Ltd.

Vehicle Tracking by non-Drifting Mean-shift using Projective Kalman Filter

Abstract: Robust vehicle tracking is essential in traffic monitoring because it is the groundwork to higher level tasks such as traffic control and event detection. This paper describes a new technique for tracking vehicles with mean-shift using a projective Kalman filter. The shortcomings of the mean-shift tracker, namely the selection of the bandwidth and the initialization of the tracker, are addressed with a fine estimation of the vehicle scale and kinematic model. Indeed, the projective Kalman filter integrates the non-linear projection of the vehicle trajectory in its observation function resulting in an accurate localization of the vehicle in the image. The proposed technique is compared to the standard Extended Kalman filter implementation on traffic video sequences. Results show that the performance of the standard technique decreases with the number of frames per second whilst the performance of the projective Kalman filter remains constant.