Evolutionary Computational Tools Aided Extended Kalman Filter for Ballistic Target Tracking

Real-time state of charge estimation for electric vehicle power batteries using optimized filter

Real-time water demand pattern estimation using an optimized extended Kalman filter

The electrification of the transport sector increases the demand for a battery management system that allows the Li-ion battery to work in a sophisticated way to reach its maximum. The main focus of the battery management system is the estimation of the battery’s state of charge (SOC) which is an indicator to determine the driving range of an electric vehicle. However, battery modeling has a direct influence on SOC estimation. In this article, the equivalent circuit model (ECM) and extended Kalman filter (EKF) algorithm are combined for SOC estimation of the Li-ion battery. Since the performance of Kalman filter algorithms strictly depends on noise covariance matrices, incorrect values reduce the accuracy and make divergence in state estimation. Therefore, it is motivated to analyze the impact of process noise and measurement noise covariance matrix values on SOC estimation using Kalman filter algorithms. Ten different combinations of noise covariance matrices are employed to analyze the effect of these matrices on SOC estimation. As a result, the tuning of the filter is critical for improvement in state estimation regardless of the type of the battery model and filtering technique. The matrix values are chosen to take into account the convergence rate and estimation error. To validate the effectiveness of the chosen noise covariance matrices, the simulation results of the ECM based EKF estimation method is superimposed with the discharge curves available in the data sheet. The simulation results show minimal estimation error and a good convergence rate for tuned EKF.

The paper proposes a new Kalman filtering (KF) algorithm called VBI-MCKF that combines the variational Bayesian inference (VBI)-based KF algorithm and the maximum correntropy KF (MCKF) for visual tracking problem. The VBI-based KF algorithm works to estimate the unknown measurement noise covariance matrix (MNCM) and process noise covariance matrix (PNCM), and the MCKF works to deal with non-Gaussian process noise and measurement noise. In order to cater for the visual tracking target scenario, where the MNCM and PNCM are unknown and non-Gaussian, we modify the VBI-based KF algorithm to work for suddenly occurring outliers and modify the MCKF work for small bandwidth. Simulations show that the proposed VBI-MCKF algorithm performs well when both covariance matrices of Gaussian process noise and non-Gaussian measurement noise are unknown or inaccurate.

This paper presents an extended Kalman filter for pose estimation using noise covariance matrices based on sensor output. Compact and lightweight nine-axis motion sensors are used for motion analysis in widely various fields such as medical welfare and sports. A nine-axis motion sensor includes a three-axis gyroscope, a three-axis accelerometer, and a three-axis magnetometer. Information obtained from the three sensors is useful for estimating joint angles using the Kalman filter. The extended Kalman filter is used widely for state estimation because it can estimate the status with a small computational load. However, determining the process and observation noise covariance matrices in the extended Kalman filter is complicated. The noise covariance matrices in the extended Kalman filter were found for this study based on the sensor output. Postural change appears in the gyroscope output because the rotational motion of the joints produces human movement. Therefore, the process noise covariance matrix was determined based on the gyroscope output. An observation noise covariance matrix was determined based on the accelerometer and magnetometer output because the two sensors’ outputs were used as observation values. During a laboratory experiment, the lower limb joint angles of three participants were measured using an optical 3D motion analysis system and nine-axis motion sensors while participants were walking. The lower limb joint angles estimated using the extended Kalman filter with noise covariance matrices based on sensor output were generally consistent with results obtained from the optical 3D motion analysis system. Furthermore, the lower limb joint angles were measured using nine-axis motion sensors while participants were running in place for about 100 s. The experiment results demonstrated the effectiveness of the proposed method for human pose estimation.

Electrical models of power distribution grids are used in applications such as state estimation and Optimal Power Flow (OPF), the reliability of which depends on the accuracy of the model. This work presents an approach for estimating distribution line parameters from Remote Terminal Unit (RTU) measurements which are subject to measurement device tolerances and random noise. Building upon an earlier work which introduced a measurement tolerance compensation model, we aim to improve a) the robustness towards noisy data and b) the estimate of the parallel susceptance. For this purpose, we employ an Extended Kalman Filter (EKF) whose measurement noise covariance matrix is modified in order to account for all noisy variables in the overdetermined system. Simulations confirm the advantages of the EKF over the previously used Least-Squares (LSQ) estimator. In the low random noise cases considered in this paper, the EKF yields a four-fold improvement over the LSQ for the parallel susceptance across all quantization ranges. For the highest levels of random and quantization noise, the EKF performs about 1.5 to 3 times better than the LSQ for all line parameters. Furthermore, the EKF shows more consistent behavior when applied to data obtained from a laboratory distribution grid, which exhibits uncertainties that are not accounted for in the models.

Traffic state estimation problems must deal with imperfect information. Imperfect information may occur as a result of misalignments between reality and the assumptions or simplifications made when developing the models as well as incomplete or corrupted data. One of the well-known methods of handling imperfect information is the Kalman filter algorithm and its variations, such as the extended Kalman filter and the unscented Kalman filter. While the standard Kalman filter and the extended Kalman filter have been widely applied to the state estimation of linear systems, the unscented Kalman filter has been reported the better choice in the multiclass traffic state estimation. In many applications of Kalman filters to traffic estimation problems, the model and measurement noise covariance matrices are normally estimated. When there is a mismatch between the true and the assumed noise distribution, however, the filter often suffers from performance degradation and even divergence in certain situations. To this end, this paper presents a more efficient and accurate algorithm embedded in the unscented Kalman filter to simultaneously estimate the traffic state and the model noise distribution statistics. The proposed method is facilitated through a simultaneous update of the model noise covariance matrix in the predicted covariance equations of the unscented Kalman filter algorithm. It is found through simulation that the proposed algorithm may improve the model performance over the standard algorithm.

An Attitude and Heading Reference System (AHRS) comprising accelerometers, gyroscopes and magnetometers can provide roll, pitch and heading information. AHRS is utilized in many applications such as navigation, augmented/virtual reality, and mobile mapping. The AHRS mechanization involves integration of angular rate measurement to provide high rate orientation but with unbounded drifts due to accumulation of random noise. To reduce drifts, mechanization output is combined with absolute measurement from magnetometer and accelerometer using Extended Kalman Filter(EKF). EKF accuracy is greatly affected by process covariance matrix (Q) and measurement noise covariance matrix(R). Conventional stochastic modeling approaches to determine Q and R parameters do not guarantee best performance. This paper proposes a systematic procedure for EKF parameters optimization using a hybrid statistical and genetic algorithms (GA) approach. The proposed approach has been verified on real data collected by an inertial measurement unit. Results showed that the Q and R can be optimized within few GA iterations outperforming conventional EKF parameter estimation methods.

Flowmeter is an instrument for measuring the flow rate of fluid, which is widely used in industrial, agricultural, environmental and other fields. However, the signal of flowmeter is often affected by various noises and interferences, which reduces the measurement accuracy. In order to improve the signal quality of flowmeter, this paper proposes a flowmeter signal processing method based on adaptive Kalman filter. This method uses the advantages of Kalman filter, combined with adaptive algorithm, dynamically adjusts the process noise covariance matrix and measurement noise covariance matrix, and realizes the effective filtering and estimation of flowmeter signal. This research takes the water flow signal processing based on Kalman filter as an example, and verifies the effectiveness and superiority of this method through simulation and experiment. The results show that this method can effectively suppress the noise and interference in the flowmeter signal, improve the measurement accuracy and stability of the flowmeter, and is suitable for different working conditions and different types of flowmeters.

Localization improvement in wireless sensor networks using a new statistical channel model

The proper selection of the model error covariance matrix and the measurement noise covariance matrix of Kalman filter is an optimization problem. Some scholars have studied this, but there is relatively little research on the selection of the two covariance matrices for Kalman filters with an unknown input. Recently, the authors proposed a modified directed bat algorithm (MDBA) which introduces the best historical location of individuals and the elimination strategy to effectively prevent falling into local optimal solution. So, two methods are proposed in this paper to optimize the model error covariance matrix and measurement noise covariance matrix of Kalman filter with unknown inputs (KF-UI) and extended Kalman filter with unknown inputs (EKF-UI) by MDBA, respectively. The objective functions are constructed using the measurement vectors and the corresponding estimated values, and MDBA is adopted to optimize the two covariance matrices of KF-UI and EKF-UI. To validate the effectiveness of proposed methods, two simple structure examples and a benchmark example are adopted. The influence of structural parameter uncertainties on KF-UI is also considered. The result shows that the MDBA-optimized KF-UI has a strong convergence and can take into account the effect of parameter uncertainties. Then, the effectiveness of the proposed MDBA-optimized EKF-UI method is validated by comparing it with EKF-UI with empirically selected covariance values through trial-and-error. The identification results showed that the proposed methods achieved better identification accuracy and enhanced convergence compared to KF-UI and EKF-UI with empirical covariance values.

Inertial Navigation Systems (INS) offer self-contained and highly autonomous navigation but suffer from unbounded error accumulation over time. Extended Kalman filter (EKF) is widely used to estimate and correct INS errors. However, EKF-INS encounters challenges in accuracy and adaptability caused by modeling assumption errors under complex motion modes, especially in Pedestrian Inertial Navigation System (PINS). To overcome these limitations, this paper proposes a Deep Adaptive Kalman Filter (DAKF) framework enhanced by a DualAttNet module. DualAttNet evaluates measurement confidence from raw inertial data in real time and converts it into a measurement noise covariance matrix. The system continuously performs measurement updates for error correction while adaptively adjusting correction weights to improve robustness. The Kalman filtering equations are embedded into the loss function, enabling end-to-end optimization that implicitly learns the mapping from deep features to noise covariance, thus mitigating the need for manually labeled ground truth. Experiments on a PINS dataset demonstrate that the proposed DAKF achieves real-time inference and reduces average position error to 0.48 m within a 100 m trajectory—50% lower than conventional methods.

Multiple fading factors-based strong tracking variational Bayesian adaptive Kalman filter

Object-Oriented Environment for Assessing Tracking Algorithms

Accurate and reliable vehicle state estimation results are very significant to the active safety, energy optimization, and the intelligent control of vehicles. In this paper, to improve the accuracy and adaptability of vehicle running state estimation, the vehicle running states fused estimation strategy is presented for in-wheel motor drive electric vehicle using the Kalman filters and tire force compensation method. The concept of electric drive wheel model (EDWM) is developed and deduced, and then, considering that the EDWM is a nonlinear model with an unknown input, the design concept of high-order sliding mode observer is used to construct the state space equation of longitudinal force. To improve the accuracy and the reliability of vehicle state estimation, an overall estimation strategy with information fusion and tire force compensation is designed, in which a weighted square-root cubature Kalman filter with an adaptive covariance matrix of measurement noise is developed for observer design. Finally, the simulations in CarSim-Simulink co-simulation model and experiments are carried out, and the effectiveness of the designed estimation strategy is validated.