Extended Kalman Filter In Terms Research Articles

The Weighted Kalman Filter

This paper proposes a new version of the Kalman filter, referred to as weighted Kalman filter (WKF). In the WKF some recent results on the weighted linearization of nonlinear systems are exploited to incorporate modifications in the equations of the extended Kalman filter (EKF). More specifically, the computation of the Jacobian matrices at the current mean of the estimated state is replaced by the multiple integral over the state space of the Jacobian matrix functions multiplied by a weighting function. Similar modifications are introduced in the equations used to account for the available nonlinear model and compute the so-called a priori state and output estimates. The weighting function is chosen to be a multivariable Gaussian function where the generalized variance is selected as proportional to the current covariance matrix of the state estimate. An illustrative example is used to describe the step-by-step derivation of the WKF equations and compare its performance against the EKF in terms of convergence properties and estimation error performance.

A Time-Efficient Complementary Kalman Gain Filter Derived From Extended Kalman Filter and Used for Magnetic and Inertial Measurement Units

Magnetic and inertial measurement units (MIMUs) are mainly used for determining the attitude of moving bodies. The extended Kalman filter (EKF) and the complementary filters (CFs) are the most commonly applied algorithms for calculating the attitude of the body that an MIMU is attached. Despite its much higher accuracy than CFs, the time-consuming EKF may not be competent for the work of attitude calculation if a microcomputer with limited computation power is used. To any attitude calculating algorithm, its posterior attitude (PA) is computed as: PA = prior attitude + gain matrix (GM) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times $ </tex-math></inline-formula> innovation. The prior attitude and the innovation are derived from the measurements of the gyroscope, the accelerometer, and the magnetometer. The detailed analysis in this study found that the reason for the performance difference between EKF and CFs is their different structures of the GMs, which promotes the authors to construct a new GM. Based on some reasonable assumptions, the GM of EKF is simplified under the principle that the computation accuracy of PA is decreased as little as possible, but its computation burden is reduced as much as possible. After the replacement of the GM of the EKF by the innovatively designed GM, a new CF, named time-efficient complementary Kalman gain filter (TCF), is proposed. The simulation and experimental results show that TCF performs better than two CFs and is close to EKF in terms of attitude estimation accuracy but uses nearly the same computation time as the two CFs and about half of the computation time required by EKF.

Terrain estimation via vehicle vibration measurement and cubature Kalman filtering

The extent of vibrations experienced by a vehicle driving over natural terrain defines its ride quality. Generally, surface irregularities, ranging from single discontinuities to random variations of the elevation profile, act as a major source of excitation that induces vibrations in the vehicle body through the tire–soil interaction and suspension system. Therefore, the ride response of off-road vehicles is tightly connected with the ground properties. The objective of this research is to develop a model-based observer that estimates automatically terrain parameters using available onboard sensors. Two acceleration signals, one coming from the vehicle body and one from the wheel suspension, are fed into a dynamic vehicle model that takes into account tire/terrain interaction to estimate ground properties. To solve the resulting nonlinear simultaneous state and parameter estimation problem, the cubature Kalman filter is used, which is shown to outperform the standard extended Kalman filter in terms of accuracy and stability. An extensive set of simulation tests is presented to assess the performance of the proposed estimator under various surface roughness and deformability conditions. Results show the potential of the proposed observer to estimate automatically terrain properties during operations that could be implemented onboard of a general family of intelligent vehicles, ranging from off-road high-speed passenger cars to lightweight and low-speed planetary rovers.

State of charge estimation of vanadium redox battery based on improved extended Kalman filter

The accurate state of charge (SOC) estimation can protect the battery from overcharging and over-discharging, and it is useful to make an effective dispatching strategy. The extended Kalman filter (EKF) method is used to estimate SOC widely. But it does not consider the SOC constraints. Moreover, the convergence is influenced by the uncertain initial SOC, which may lead to false alarm, unwanted operation of protection, error dispatching and poor robustness of the system. This paper presents an improved extended Kalman filter (IEKF) method to estimate SOC for vanadium redox battery (VRB) by introducing a gain factor. It can be adjusted automatically according to the output error and SOC boundary. To implement IEKF estimator, a VRB state space model is established and its parameters are identified by recursive least square (RLS) method. Then a VRB of 5kW/30kWh experimental platform is built. Finally, the IEKF method is validated and compared with EKF against unknown initial value through the experiments. The results have shown that IEKF method is superior to EKF in terms of accuracy, convergence speed and robustness. And the estimated SOC remains bounded by using IEKF method. It is more suitable for SOC estimation than EKF algorithm in the industrial applications.

Distributed estimation for spatial rigid motion based on dual quaternions

SummaryThis paper proposes 2 distributed optimization algorithms for the estimation of spatial rigid motion using multiple image sensors in a connected network. The objective is to increase the estimation precision of translational and rotational motion based on dual quaternion models and the cooperation between connected sensors. The dual decomposition subgradient method and distributed Newton optimization method are applied to decompose the filtering task into a series of suboptimal problems and then solve them individually to achieve the global optimality. Our approach assumes that each sensor can communicate with its neighboring sensors to update the individual estimates. Discussion on converging speed of both methods are provided. Simulation examples are demonstrated to compare the 2 distributed algorithms with the traditional extended Kalman filter in terms of estimation accuracy and converging rate.

Robustness Evaluation of Extended and Unscented Kalman Filter for Battery State of Charge Estimation

In this paper, the robustness of model-based state observers including extended Kalman filter (EKF) and unscented Kalman filter (UKF) for state of charge (SOC) estimation of a lithium-ion battery against unknown initial SOC, current noise, and temperature effects is investigated. To more comprehensively evaluate the performance of EKF and UKF, two battery models including the first-order resistor-capacitor equivalent circuit and combined model are considered. A novel method is proposed to identify the parameters of the equivalent circuit model. The performance of SOC estimation is evaluated by employing measurement data from a commercial lithium-ion battery cell. The experiment results show that UKF generally outperforms EKF in terms of estimation accuracy and convergence rate for each battery model. However, the advantages of UKF over EKF with the combined model is not as significant as with the equivalent circuit model. Both EKF and UKF demonstrate strong robustness against current noise. The updates of model parameters corresponding to operational temperatures generally improve the estimation accuracy of EKF and UKF for both models.

Estimating model error covariance matrix parameters in extended Kalman filtering

Abstract. The extended Kalman filter (EKF) is a popular state estimation method for nonlinear dynamical models. The model error covariance matrix is often seen as a tuning parameter in EKF, which is often simply postulated by the user. In this paper, we study the filter likelihood technique for estimating the parameters of the model error covariance matrix. The approach is based on computing the likelihood of the covariance matrix parameters using the filtering output. We show that (a) the importance of the model error covariance matrix calibration depends on the quality of the observations, and that (b) the estimation approach yields a well-tuned EKF in terms of the accuracy of the state estimates and model predictions. For our numerical experiments, we use the two-layer quasi-geostrophic model that is often used as a benchmark model for numerical weather prediction.

Unscented Kalman filter performance for closed-loop nonlinear state estimation: a simulation case study

In this paper, unscented Kalman filter (UKF) is used for state estimation of nonlinear system. The main advantage of UKF is that it does not need any linearization for calculating the state transition matrix like extended Kalman filter (EKF). This study includes the combination of the nonlinear estimation and the optimal control strategy. Simulation results for a proportional integral derivative and linear quadratic regulator controlled inverted pendulum are presented and compared with the EKF under the measurement and process noise. The results show that the performance of the UKF is better than the EKF in terms of robustness, computation time and speed of convergence.

Online State Estimation of a Synchronous Generator Using Unscented Kalman Filter From Phasor Measurements Units

The most important reference quantities for monitoring and controlling transient stability in real time are the rotor angle and speed of the synchronous generators. If these quantities can be estimated with sufficient accuracy, they can be used in global and local control methods. In the classic state estimation methods, such as the extended Kalman filter (EKF) technique, the linear approximations of the system at a given moment in time may introduce errors in the states. In order to overcome the drawbacks of the EKF, the authors of this paper have applied the unscented Kalman filter (UKF) to estimating and predicting the states of a synchronous machine, including rotor angle and rotor speed, using phasor measurement unit (PMU) quantities. The UKF algorithm propagates the pdf of a random variable in a simple and effective way and is accurate up to the second order in estimating the mean and covariance. The overall impression is that the performance of the UKF is better than the EKF in terms of robustness, speed of convergence, and also different levels of noise. Simulation results including saturation effects and grid faults show the accuracy and efficiency of the UKF method in state estimation of the system, especially at higher noise ratios.

Observability-based consistent EKF estimators for multi-robot cooperative localization

In this paper, we investigate the consistency of extended Kalman filter (EKF)-based cooperative localization (CL) from the perspective of observability. We analytically show that the error-state system model employed in the standard EKF-based CL always has an observable subspace of higher dimension than that of the actual nonlinear CL system. This results in unjustified reduction of the EKF covariance estimates in directions of the state space where no information is available, and thus leads to inconsistency. To address this problem, we adopt an observability-based methodology for designing consistent estimators in which the linearization points are selected to ensure a linearized system model with observable subspace of correct dimension. In particular, we propose two novel observability-constrained (OC)-EKF estimators that are instances of this paradigm. In the first, termed OC-EKF 1.0, the filter Jacobians are calculated using the prior state estimates as the linearization points. In the second, termed OC-EKF 2.0, the linearization points are selected so as to minimize their expected errors (i.e., the difference between the linearization point and the true state) under the observability constraints. The proposed OC-EKFs have been tested in simulation and experimentally, and have been shown to significantly outperform the standard EKF in terms of both accuracy and consistency.