Standard Kalman Filter Research Articles

Non-Gaussian Deterministic Assimilation of Radar-Derived Precipitation Accumulations

Abstract Data assimilation (DA) approaches currently used for operational numerical weather prediction (NWP) generally assume that errors in the background state are Gaussian. At the same time, approaches that make no assumptions regarding the background state probability distribution are gaining attention in research. Most such approaches, including the particle filter, are ensemble DA methods that produce an ensemble of analysis states consistent with the background and observation distributions. The present study instead proposes a non-Gaussian deterministic (NGD) DA method for producing a single deterministic analysis state. Consequently, the usual challenge of maintaining an ensemble with sufficient spread and diversity is avoided. The NGD approach uses background ensembles generated by a standard ensemble Kalman filter. A series of noncycled DA experiments is conducted to evaluate the NGD approach for assimilating precipitation derived from North American weather radars to initialize limited-area deterministic forecasts. The resulting forecasts are compared with those produced using either a local ensemble transform Kalman filter (LETKF) deterministic analysis or latent heat nudging (LHN). The experimental results indicate that, for forecast lead times beyond 1.5 h, the NGD approach improves precipitation forecasts relative to LHN. The NGD approach also leads to better temperature and zonal wind forecasts at lead times up to 12 h when compared to those obtained with either LHN or the LETKF. For precipitation, the NGD and LETKF approaches produce forecasts that are of comparable quality. Finally, simple strategies are demonstrated that combine the NGD approach for assimilating radar-derived precipitation accumulations with the ensemble–variational approach for assimilating all other observations.

Online State of Charge Estimation of Lithium-Ion Cells Using Particle Filter-Based Hybrid Filtering Approach

Filtering based state of charge (SOC) estimation with an equivalent circuit model is commonly extended to Lithium-ion (Li-ion) batteries for electric vehicle (EV) or similar energy storage applications. During the last several decades, different implementations of online parameter identification such as Kalman filters have been presented in literature. However, if the system is a moving EV during rapid acceleration or regenerative braking or when using heating or air conditioning, most of the existing works suffer from poor prediction of state and state estimation error covariance, leading to the problem of accuracy degeneracy of the algorithm. On this account, this paper presents a particle filter-based hybrid filtering method particularly for SOC estimation of Li-ion cells in EVs. A sampling importance resampling particle filter is used in combination with a standard Kalman filter and an unscented Kalman filter as a proposal distribution for the particle filter to be made much faster and more accurate. Test results show that the error on the state estimate is less than 0.8% despite additive current measurement noise with 0.05 A deviation.

LTI energy-maximising control for the Wave Star wave energy converter: identification, design, and implementation

Considering the Wave Star wave energy converter (WEC), which is a standard point absorber prototype, this study addresses the complete energy maximising control design procedure. The WEC model is obtained using system identification routines. Then, using the identified model, a LTI energy maximising control strategy, recently presented in the literature, is designed. Additionally, for the wave excitation force estimation, a standard Kalman filter with a harmonic oscillator is considered. Finally, for the assessment of the performance of the complete system, the controller and estimator are implemented in the numerical WEC simulation environment WEC-Sim. The system is tested under realistic conditions and satisfying performance of the LTI controller is shown. Thus, through a non-conventional approach, and considering a realistic software environment, a novel energy maximising controller is implemented, obtaining results which indicate the feasibility of the approach.

Rocket tracking impact point prediction using α-β, standard Kalman, extended, Kalman, and unscented Kalman filters: a comparative analysis

Accurate information about the impact point (IP) of a suborbital rocket on Earth’s surface during a launch is an important requirement for range safety operations. Four different estimators, i.e., the α-β filter, standard Kalman filter (SKF), extended Kalman filter (EKF), and unscented Kalman filter (UKF), are considered for the suborbital rocket tracking problem, whose data are used specifically for improving the accuracy of the IP prediction (IPP) of these vehicles. This paper presents a comparative analysis between the results of the estimators. Rocket flight data are discussed to demonstrate the advantages and disadvantages of the estimators and to determine the inherent limitations in predicting the aerodynamic effects found in certain flight situations. We discuss the appropriate mathematical model of a filter capable of running the real-time algorithm for the estimation of target position and velocity. This work uses actual data from a radar sensor to evaluate the tracking algorithms. We insert the filter result into the mathematical model developed to predict the rocket IP on Earth's surface. The main goal of this study is to evaluate the performance of four different estimators when specifically applied for the improvement of the IPP of suborbital rockets. It is demonstrated that the UKF outperforms all other tracking algorithms in terms of the accuracy and robustness of IP estimation.

Non-Contact Respiratory Rate Estimation in Real-Time With Modified Joint Unscented Kalman Filter.

It can be life-saving to monitor the respiratory rate (RR) even for healthy people in real-time. It is reported that the infected people with coronavirus disease 2019 (COVID-19), generally develop mild respiratory symptoms in the early stage. It will be more important to continuously monitor the RR of people in nursing homes and houses with a non-contact method. Conventional, contact-based, methods are not suitable for long-term health monitoring especially in-home care services. The potentials of wireless radio signals for health care applications, such as fall detection, etc., are examined in literature. In this paper, we focus on a device-free real-time RR monitoring system using wireless signals. In our recent study, we proposed a non-contact RR monitoring system with a batch processing (delayed) estimation method. In this paper, for real-time monitoring, we modify the standard joint unscented Kalman filter (JUKF) method for this new and time-critical problem. Due to the nonlinear structure of the RR estimation problem with respect to the measurements, a novel modification is proposed to transform measurement errors into parameter errors by using the hyperbolic tangent function. It is shown in the experiments conducted with the real measurements taken using healthy volunteers that the proposed modified joint unscented Kalman filter (ModJUKF) method achieves the highest accuracy according to the windowing-based methods in the time-varying RR scenario. It is also shown that the ModJUKF not only reduces the computational complexity approximately 8.54% but also improves the accuracy 36.7% with respect to the standard JUKF method.

Distributed Error Correction of EKF Algorithm in Multi-Sensor Fusion Localization Model

In order to solve the problem that the standard extended Kalman filter (EKF) algorithm has large errors in Unmanned Aerial Vehicle (UAV) multi-sensor fusion localization, this paper proposes a multi-sensor fusion localization method based on adaptive error correction EKF algorithm. Firstly, a multi-sensor navigation localization system is constructed by using gyroscopes, acceleration sensors, magnetic sensors and mileage sensors. Then the information detected by the sensor is compared and adjusted, to reduce the influence of error on the estimated value. The nonlinear observation equation is linearized by Taylor, and the normal distribution hypothesis is carried out in two steps of prediction and correction respectively. Finally, the parameters of system noise and measurement noise covariance in EKF are optimized by using the evolutionary iteration mechanism of genetic algorithm. The adaptive degree is obtained according to the absolute value of the difference between the estimated value and the real value of EKF. The individual evaluation results of EKF algorithm parameters are used as the measurement standard for iteration to obtain the optimal value of EKF algorithm parameters. Experimental simulation results show that the improved algorithm proposed has higher real-time localization accuracy and higher robustness than those of the standard EKF algorithm.

Markovian robust filtering and control applied to rehabilitation robotics

In this article, we deal with rehabilitation robot robust control in order to obtain best performance in the human-robot interaction. This class of system is subject to abrupt changes due to exogenous inputs and parameter variations caused, mainly, by the human dynamic behavior. In this context, Markovian jump linear systems offer suitable tools to model and control of this class of interactions. We propose force and impedance controllers based on robust recursive Markovian versions of the standard Kalman filter and the linear quadratic regulator. Different levels of the individual's actuation during the interaction with the robot define the Markov states. The measurement of electromyographic signals is used as jump parameter among Markov states. Also, a serious game is used to generate visual feedback, to promote active user participation, and to guide his movement routine. We compare our proposal with two other Markovian-based approaches and show the effectiveness our method through experiments with an ankle rehabilitation platform.

Kalman-based interacting multiple-model wind speed estimator for wind turbines

The use of state estimation technique offers a means of inferring the rotor-effective wind speed based upon solely standard measurements of the turbine. For the ease of design and computational concerns, such estimators are typically built based upon simplified turbine models that characterise the turbine with rigid blades. Large model mismatch, particularly in the power coefficient, could lead to degradation in estimation performance. Therefore, in order to effectively reduce the adverse impact of parameter uncertainties in the estimator model, this paper develops a wind sped estimator based on the concept of interacting multiple-model adaptive estimation. The proposed estimator is composed of a bank of extended Kalman filters and each filter model is developed based on different power coefficient mapping to match the operating turbine parameter. Subsequently, the algorithm combines the wind speed estimates provided by each filter based on their statistical properties. In addition, the proposed estimator not only can infer the rotor-effective wind speed, but also the uncertain system parameters, namely, the power coefficient. Simulation results demonstrate the proposed estimator achieved better improvement in estimating the rotor-effective wind speed and power coefficient compared to the standard Kalman filter approach.

Real Time Coding and Estimation of Linear Discrete Time System Over Networks

This study deals with the real time coding and estimation of linear discrete time scalar system over communication networks. With the mean-squared error (MSE) distortion criterion, the information rate distortion function describing the performance limit of the coding-estimation system is analyzed and discussed. To achieve near instantaneous encoding and decoding, an asymptotic design scheme has been presented as a realization of real time coding-estimation system. The outputs of standard Kalman filter relying on unquantized observations are encoded using the Lloyd-Max quantization rules and transmitted over the noiseless channel. The decoder side runs the corresponding decoding and reconstruction algorithms to produce the optimal real time state estimate of system. To synchronize the encoder and decoder, a double-predictor regime on the updating rules of the encoder-decoder pair is proposed, and it does not require any feedback information. The rate distortion function of the proposed scheme is derived and when comparing with the information theoretical lower bound, there is only a factor discrepancy related to the quantization rules. The rate distortion performance results of various design schemes are compared and demonstrated with numerical simulations.

Probabilistic Kalman filter for moving object tracking

Kalman filter has been successfully applied to tracking moving objects in real-time situations. However, the filter cannot take into account the existing prior knowledge to improve its predictions. In the moving object tracking, the trajectories of multiple targets in the same environment could be available, which can be viewed as the prior knowledge for the tracking procedure. This paper presents the probabilistic Kalman filter (PKF) that is able to take into account the stored trajectories to improve tracking estimation. The PKF has an extra stage after two steps of the Kalman filter to refine the estimated position of the targets. The refinement is obtained by applying the Viterbi algorithm to a probabilistic graph, that is constructed based on the observed trajectories. The graph is built in the offline situation and could be adapted in the online tracking. The proposed tracker has higher accuracy compared to the standard Kalman filter and could handle widespread problems such as occlusion. Another significant achievement of the proposed tracker is to track an object with anomalous behaviors by drawing an inference based on the constructed probabilistic graph. The PKF was applied to several manually-built videos and several other video-bases containing severe occlusions, which demonstrates a significant performance in comparison with other state-of-the-art trackers.

LQG control for sampled-data systems under stochastic sampling

This paper studies the linear-quadratic-Gaussian (LQG) problem for sampled-data systems with a stochastic sampling interval obeying a certain probability distribution. An optimal estimator of the system state is presented by the standard Kalman filter, and the Vandermonde matrix and Kronecker product operation are used to calculate the mathematical expectation caused by stochastic sampling in the process of designing the LQG controller. Moreover, it was proved that the controller can ensure the system is exponentially mean square stable. Finally, some simulation results are given to verify the effectiveness and practicability of the proposed controller design method.

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.

Derivation of Modified Input Estimation Filter Using Bayesian Framework

A new approach to derive the modified input estimation is proposed using Bayesian framework. Tracking manoeuvring targets is a tough problem and many algorithms are proposed to deal with it. Among these algorithms, modified input estimation (MIE) technique has been proved to be an effective method. MIE technique treats acceleration as additive input term in the state vector and estimates the original state and acceleration vectors simultaneously with a standard Kalman filter. Analogous to standard Kalman filter, however, the MIE filter equations which have a modification on Kalman gain is given without a cohesive derivation. Using a Bayesian framework, we will present a conceptually cohesive roadmap that starts at first principles and leads directly to derivation of MIE filter equations.

A Coarse-to-Fine Adaptive Kalman Filter for Weak GNSS Signals Carrier Tracking

The tracking performance of the standard Kalman filter (KF)-based carrier tracking depends heavily on the potentially insufficient priori noise covariance. The inadequate prior knowledge will degrade the tracking sensitivity when the receiver experiences weak signal conditions. To improve the tracking performance, a coarse-to-fine adaptive Kalman filter (AKF) carrier tracking approach is proposed. In this method, the process and measurement noise covariance are updated in real-time according to the measurement information. The AKF-based fine tracking is initialized by the coarse tracking results, which can improve the convergence speed of the AKF tracking. The proposed method has been conducted in the software receiver and compared with standard KF-based tracking and conventional tracking algorithms. The test results show that the proposed AKF-based tracking can improve tracking sensitivity under weak signal conditions. Moreover, the proposed method can reduce the convergence time compared with the AKF-based tracking without the coarse tracking stage.

A Novel Hybrid Unscented Particle Filter based on Firefly Algorithm for Tightly-Coupled Stereo Visual-Inertial Vehicle Positioning

This paper presents a hybrid unscented particle filter (UPF) based on the firefly algorithm for tightly-coupled stereo visual-inertial vehicle positioning systems (VIVPS). Compared with standard UPF, this novel approach can achieve similar estimation accuracy with much less computational complexity. To reduce the computational complexity, the time updating of the hybrid unscented Kalman filter is conducted via the formula of standard linear Kalman filter on the basis of the constructed linear/nonlinear mixed filter model. The particle updating of the particle filter is optimised by modified firefly algorithm to reduce the number of particles needed by means of moving particles towards high likelihood regions via the attraction and movement of fireflies, leading to a significant reduction of computational complexity. Experimental results show the average execution time of the proposed approach is 23·8% that of the standard UPF with similar accuracy, indicating the designed method for tightly-coupled stereo VIVPS can better satisfy the real-time requirement of the system.

Constrained Kalman Filter for Identification of Semiphysical Building Thermal Models

Model-based energy management of buildings through a model predictive control framework proved to be the promising solution for improving the energy efficiency of the building sector. The keystone for further improvements and real implementation is to have reliable and accurate mathematical models that are simple enough to be used in real-time control. In the work presented, a general identification framework for acquiring a control-oriented semiphysical thermal model of a building based on a modified constrained unscented Kalman filter algorithm is proposed. Experiments performed on a living-lab skyscraper building show that the adopted algorithm applied to short-term operation data outperforms the standard Kalman filter forms, both in convergence rate and numerical stability. The proposed method thus paves the way to the fast deployment of model-based energy management strategies in buildings.

A Novel Compensation Method for Magnetic Distortion Field With Noise Uncertainty

The measurement noise of the standard Kalman filter (SKF) and related filtering algorithms are usually required to obey Gaussian distribution. However, it is hard to be satisfied in practice and the reliability of the filtering results is reduced when the measurement noise is not normally distributed. The robust Kalman filter (RKF) has the ability to bound the abnormal observations by weighing the influence of the updated parameters in accordance with the magnitude of discrepancy between the updated parameters and the robust estimates obtained from measurements at each time step. However, there is a limitation that it wastes the useful information of the state model sometimes. In order to solve the problem of magnetic interference compensation under noise uncertainty, this paper addresses a new method based on the robust M estimation to make the uncertain noise suppressed. The double-model adaptive estimation (DMAE) approach is proposed, which consists of a SKF and a RKF. Two filters are selected adaptively by a conditional probability according to the innovation covariance at each time step. More specifically, the DMAE has ability to resist the measurement noise and make full use of effective information at the same time. Compared with the SKF and the RKF method, the DMAE achieves a better performance in terms of compensation accuracy and robustness for parameter estimation. After compensation, errors of components are reduced from 502, 613, 493 to 3.9, 2.1, and 23.9 nT, respectively, with the proposed method, the ideal accuracy of total field is achieved as well.

Identification of earthquake ground motion based on limited acceleration measurements of structure using Kalman filtering technique

Identification of earthquake ground motion from structural health monitoring (SHM) data provides a good means to reconstruct seismic loads that are essential for postearthquake safety assessments and disaster simulations of structures. Because the data measured by an SHM system are structural absolute response, they cannot be directly applied to the structural motion equation, which is established in relative coordinate system. As such, this paper originally derives the motion equation in absolute coordinate system and then expands the equation into modal space. In addition, the proposed method allows for identifying earthquake ground motion using incomplete modal information and limited measurements through the standard Kalman filter. Subsequently, a numerical two-dimensional frame is used to validate the feasibility of the proposed method, and the influences of modal parameters and measurement noise on the identification accuracy are also fully investigated. The results show that the proposed method is sensitive to frequency and measurement noise but insensitive to modal shape and damping ratio. It is also found that the identified ground motion subjected to certain measure noise can still be reliably employed for postseismic response calculations of medium- and long-period structures. Finally, a shaking table test performing on a five-floor frame further demonstrates the effectiveness and accuracy of the proposed identification algorithm for practical application.

Short-Term Prediction of Energy Consumption in Demand Response for Blocks of Buildings: DR-BoB Approach

Load forecasting plays a major role in determining the prices of the energy supplied to end customers. An accurate prediction is vital for the energy companies, especially when it comes to the baseline calculations that are used to predict the energy load. In this paper, an accurate short-term prediction using the Exponentially Weighted Extended Recursive Least Square (EWE-RLS) algorithm based upon a standard Kalman filter is implemented to predict the energy load for blocks of buildings in a large-scale for four different European pilot sites. A new software tool, namely Local Energy Manager (LEM), is developed to implement the RLS algorithm and predict the forecast for energy demand a day ahead with a regular meter frequency of a quarter of an hour. The EWE-RLS algorithm is used to develop the LEM in demand response for blocks of buildings (DR-BOB), this is part of a large-scale H2020 EU project with the aim to generate the energy baselines during and after running demand response (DR) events. This is achieved in order to evaluate and measure the energy reduction as compared with historical data to demonstrate the environmental and economic benefits of DR. The energy baselines are generated based on different market scenarios, different temperature, and energy meter files with three different levels of asset, building, and a whole pilot site level. The prediction results obtained from the Mean Absolute Percentage Error (MAPE) offer a 5.1% high degree of accuracy and stability at a UK pilot site level compared to the asset and whole building scenarios, where it shows a very acceptable prediction accuracy of 10.7% and 19.6% respectively.

Improved Extended Kalman Filter Based on Fuzzy Adaptation for SLAM in Underground Tunnels

This study aims to investigate autonomous location and environment mapping of moving objects under conditions of dust and weak illumination in underground tunnels. The standard extended Kalman filter (EKF) algorithm has a issue that system noise and the prior statistical characteristics of the observed noise cannot be accurately predicted. Thus, we propose an improved EKF algorithm to perform fuzzy adaptive simultaneous localization and mapping (SLAM). Laser matching is added to EKF prediction phase to predict the position, and the weighted average position is used as the final position of the predicted part. By observing the change of the mean value and covariance, the system noise and the weighted value of the observed noise are fuzzy adjusted. The improved filtering algorithm is applied to a SLAM simulation experiment, and the influence of four different landmark arrangements on position estimation is considered. The results show that the positioning and composition accuracy can be improved using the new algorithm, and the accuracy of positioning and composition is increased by more than 53.8% compared with standard EKF in y-direction. It is also found that a landmark layout along the center line of a roadway roof is superior to other arrangement methods.