Extended Kalman Filter Estimator Research Articles

Advancing state estimation for lithium-ion batteries with hysteresis through systematic extended Kalman filter tuning

Knowledge of remaining battery charge is fundamental to electric vehicle deployment. Accurate measurements of state-of-charge (SOC) cannot be obtained directly and estimation methods must be used instead. This requires both a good model of a battery and a well-designed state estimator. Here, hysteretic reduced-order battery models and adaptive extended Kalman filter estimators are shown to be highly effective, accurate predictors of SOC. A battery model parameterisation framework is proposed, which enhances standardised methods to capture hysteresis effects. The hysteretic model is parameterised for three independent NMC811 lithium-ion cells and is shown to reduce voltage RMS error by 50% across 18 h automotive drive-cycles. Parameterised models are used alongside an extended Kalman filter, which demonstrates the value of adaptive filter parameterisation schemes. When used alongside an extended Kalman filter, adaptive covariance matrices yield highly accurate SOC estimates, reducing SOC estimation error by 85%, compared to the industry standard battery model.

Q-Learning-Based Noise Covariance Matrices Adaptation in Kalman Filter for Inertial Navigation

Velocity estimation of a rigid body using measurements from low-cost inertial and magnetic sensors plays an important role in various applications. The extended Kalman filter (EKF) is widely used for this purpose. However, EKF's estimation performance relies on the knowledge of process and measurement noise covariance matrices, and this information is generally unavailable. In this work, we introduce a solution that combines two techniques: the generation of velocity pseudo-measurements using a Bidirectional Long Short-Term Memory (BiLSTM) network, and the Q-learning method for online adaptation of noise covariance matrices. The performance of the proposed solution is validated using real experimental datasets, demonstrating that Q-learning can select appropriate noise covariance matrices to enhance velocity estimation.

Unmanned aerial vehicle localization using angle of departure from a single base station and dead‐reckoning

AbstractLocalization in GNSS‐denied environments for unmanned aerial vehicles (UAVs) has recently gained significant interest from the research community. Most of the research is focused primarily on visual localization. This paper examines an algorithm which employs angle of departure and UAVs payload sensor data for UAV localization. First the algorithm uses multiple angle of departures from a single base station and a travel calculated by applying dead‐reckoning on the UAVs inertial measurement unit (IMU), to compute UAV location in 2D coordinates. The 2D location estimate is then fed into a modified extended Kalman filter, which employs the estimate, IMU and barometer data to compute the 3D coordinates for UAV. For the simulation, simulation‐in‐the‐loop accompanied by Arducopter and MAVLink is used to simulate different trajectories and collect the required data for the algorithm. The algorithm is validated by comparing the extended Kalman filter estimates with IMU dead‐reckoned positions. Three simulations were performed, consisting of linear, zigzag, and curved trajectories. A percentile error of 2.5 and 4 m is achieved for the x‐coordinate and y‐coordinate, respectively, on the zigzag and curved trajectories. Interestingly, the linear trajectory showed a larger localization error in its y‐coordinate.

Background-extracted extended Kalman filter-based phase shift estimation algorithm for phase shifting profilometry system

The accuracy of phase shift in the phase-measuring profilometry (PMP) directly affect the result of the three-dimensional reconstruction. A robust phase shifting estimation method based on the extended Kalman filter (EKF) is proposed to address the imprecise phase shift induced by the mechanical projector and dynamic object. Unlike the traditional fringe projection profilometry (FPP) in which the zero frequency is suppressed and the fundamental frequency is extracted, the zero-frequency component is extracted by using two-dimensional variational mode decomposition and bidimensional empirical mode decomposition (2D-VMD-BEMD) algorithm, and is used to establish the predicted observations of the measurement process in the EKF. The coefficient of the quadratic phase in a selected window is employed to estimate the actual phase step by using EKF and subsequently the accurate reconstructed phase map is obtained. This paper develops a novel 2D-VMD-BEMD EKF estimator to evaluate the actual phase shift and enhance the accuracy of the phase reconstruction, and simultaneously provides a reliable and effective method for the extraction of background under unfavorable conditions. Simulation and experimental results demonstrate that the proposed region-wise estimation method effectively removes the effect of the unexpected phase shift, and can be widely available for N-step phase shifting algorithm in phase shifting profilometry.

State of charge estimation for the vanadium redox flow battery based on the Sage–Husa adaptive extended Kalman filter

SummaryFocusing on improving the accuracy of vanadium redox flow battery (VRFB) state of charge (SOC) estimation, this paper will combine the extended Kalman filter (EKF) estimator with the Sage–Husa adaptive method, referred to as the SAEKF estimator. Firstly, a second‐order Equivalent Circuit Model (ECM) is established to provide the correct parameterization and degrees of freedom. Next, under a hybrid pulse power characterization (HPPC) text, the parameter estimator in MATLAB/Simulink is utilized to identify the ECM parameters by using nonlinear least squares method with Trust‐Region‐Reflective (TRR) iterative algorithm. Then, during the HPPC test, the estimators implemented on the ARM‐Cortex STM32F103 estimate the VRFB SOC under both the 0°C and 5°C conditions. Finally, the mean absolute error (MAE) of the IEKF estimator are 4.66% at the 0°C and 0.895% at the 5°C. In the meantime, the MAE of the SAEKF estimator are 4.15% at the 0°C and 0.703% at the 5°C. The evaluation factors of the 0°C and 5°C experiments in the SAEKF estimator are smaller than those of the IEKF and EKF estimators, indicating that the SAEKF estimator outperforms the IEKF estimator in VRFB SOC estimation.

Bayesian Noise Modelling for State Estimation of the Spread of COVID-19 in Saudi Arabia with Extended Kalman Filters.

The epistemic uncertainty in coronavirus disease (COVID-19) model-based predictions using complex noisy data greatly affects the accuracy of pandemic trend and state estimations. Quantifying the uncertainty of COVID-19 trends caused by different unobserved hidden variables is needed to evaluate the accuracy of the predictions for complex compartmental epidemiological models. A new approach for estimating the measurement noise covariance from real COVID-19 pandemic data has been presented based on the marginal likelihood (Bayesian evidence) for Bayesian model selection of the stochastic part of the Extended Kalman filter (EKF), with a sixth-order nonlinear epidemic model, known as the SEIQRD (Susceptible-Exposed-Infected-Quarantined-Recovered-Dead) compartmental model. This study presents a method for testing the noise covariance in cases of dependence or independence between the infected and death errors, to better understand their impact on the predictive accuracy and reliability of EKF statistical models. The proposed approach is able to reduce the error in the quantity of interest compared to the arbitrarily chosen values in the EKF estimation.

Intelligent Deep Learning Estimators of a Lithium-Ion Battery State of Charge Design and MATLAB Implementation—A Case Study

The main objective of this research paper was to develop two intelligent state estimators using shallow neural network (SNN) and NARX architectures from a large class of deep learning models. This research developed a new modelling design approach, namely, an improved hybrid adaptive neural fuzzy inference system (ANFIS) battery model, which is simple, accurate, practical, and well suited for real-time implementations in HEV/EV applications, with this being one of the main contributions of this research. On the basis of this model, we built four state of charge (SOC) estimators of high accuracy, assessed by a percentage error of less than 0.5% in a steady state compared to the 2% reported in the literature in the field. Moreover, these estimators excelled by their robustness to changes in the model parameters values and the initial “guess value” of SOC from 80–90% to 30–40%, performing in the harsh and aggressive realistic conditions of the real world, simulated by three famous driving cycle procedure tests, namely, two European standards, WLTP and NEDC, and an EPA American standard, FTP-75. Furthermore, a mean square error (MSE) of 7.97 × 10−11 for the SOC estimation of the NARX SNN SOC estimator and 5.43 × 10−6 for voltage prediction outperformed the traditional SOC estimators. Their effectiveness was proven by the performance comparison with a traditional extended Kalman filter (EKF) and adaptive nonlinear observer (ANOE) state estimators through extensive MATLAB simulations that reveal a slight superiority of the supervised learning algorithms by accuracy, online real-time implementation capability, in order to solve an extensive palette of HEV/EV applications.

Online time-varying navigation ratio identification and state estimation of cooperative attack

An online navigation ratio identification model based on the gated recurrent unit (GRU) and a state estimation extended Kalman filter (EKF) are proposed under the scenario in which multiple enemy missiles attack a stationary target using a time-cooperative guidance law. The navigation ratio identification is solved as a dynamic problem, and the time-varying navigation ratios of each missile, instead of the effective navigation constants and cooperative gains, are identified in this paper. In other words, the simplified assumption that the true value is within a known finite set, which is generally adopted in a conventional identification-estimation scheme such as multiple-model adaptive estimators (MMAEs) or interacting multiple-models (IMMs), is discarded. To increase the training speed and identification accuracy, the improved multiple-model mechanism (IMMM) is adopted, and a multiple-model layer, in which regimes representing different values are set, is connected behind a conventional neural network. Since the navigation ratios are identified online, the connections between missiles are decoupled, and only one filter is required for each missile. This could greatly reduce the computational burden of onboard computers. The effectiveness of the proposed online identification model and the performance of the state estimation filter are demonstrated through numerical simulations.

HUID: DBN-based fingerprint localization and tracking system with hybrid UWB and IMU

High-precision localization technology is attracting widespread attention in harsh indoor environments. In this paper, we present a fingerprint localization and tracking system to estimate the locations of the tag based on a deep belief network (DBN). In this system, we propose using coefficients as fingerprints to combine the ultra-wideband (UWB) and inertial measurement unit (IMU) estimation linearly, termed as a HUID system. In particular, the fingerprints are trained by a DBN and estimated by a radial basis function (RBF). However, UWB-based estimation via a trilateral method is severely affected by the non-line-of-sight (NLoS) problem, which limits the localization precision. To tackle this problem, we adopt the random forest classifier to identify line-of-sight (LoS) and NLoS conditions. Then, we adopt the random forest regressor to mitigate ranging errors based on the identification results for improving UWB localization precision. The experimental results show that the mean square error (MSE) of the localization error for the proposed HUID system reduces by 12.96%, 50.16%, and 64.92% compared with that of the existing extended Kalman filter (EKF), single UWB, and single IMU estimation methods, respectively.

Fuzzy Logic Control of Maximum Power Point Tracking Controller in an Autonomous Hybrid Power Generation System by Extended Kalman Filter for Battery State of Charge Estimation

Autonomous power generation systems are designed to operate independently from the public power grid. Batteries constitute the important element in stand-alone PV system. They are used to store electricity produced by solar energy at overnight or for emergency use during the non-constant load demand. This paper has three major parts.The first pertains the design of an intelligent method for maximum power point tracking based on fuzzy logic controller to improve the efficiency of a standalone solar energy system. The second part describes the battery state of charge (SOC). The proposed model, which better reflects the real SOC response of the lithium battery, is constructed by using the extended Kalman Filter (EKF) states estimator. This proposed method can be considered as a more accurate and reliable method to estimate the battery state of charge. The third part integrates a management system for the above two renewable energy sources. The performance of the proposed management system by using a fuzzy logic controller based maximum power point tracking FLC-MPPT and the EKF estimator have been simulated in Matlab/Simulink at different solar irradiation and temperature for a given no constant load energy request.

Adaptively Adjusted EKF-Based Magnet Tracking Method for Fast-Moving Object

Permanent magnet tracking (PMT) can be employed in robots and precision control due to its advantage of wireless, without optical occlusion, and simultaneous position and orientation tracking. However, existing PMT studies based on optimization algorithms are limited to stationary or low-speed motions, where the tracking accuracy rapidly decreases with the increasing magnet speed. This study presents an extended Kalman filter (EKF)-based PMT. The magnetic field distribution around the cylindrical permanent magnet is approximated with a magnetic dipole model, where a random walk model is utilized as the system motion model. The state update model of the EKF estimation is adaptively adjusted according to different magnet speeds, while the initial magnet pose is approximated by the statistical analysis of magnetometer array outputs. The proposed PMT method hence can track the object at a higher speed while ensuring an effective pose accuracy than the previous studies. The proposed method was tested in technical verification experiments and a practical application. In addition, the relationship between the magnet speed and the position accuracy is investigated. When the magnet speed reaches 40 mm/s, experimental results show that the average pose error and algorithm latency are (1.83 ± 0.25 mm, 1.296 ± 0.091°) and 0.485 ± 0.064 ms, respectively. In the robot parking experiment for charging, the pose errors are (2.85 ± 0.41mm, 1.302 ± 0.023°) when the robot speed reaches 40 mm/s. The proposed method can greatly improve tracking performance for fast-moving object is greatly improved by performing the proposed method, thereby expanding the application scenarios of PMT.

Robust extended state estimation for nonlinear complex networks with colored noise

In this article, a robust extended state Kalman filter estimation algorithm for nonlinear complex networks is designed, fully considering the state coupling and colored noise among system nodes. The innovative measurement information subtraction method at adjacent moments deals with colored noise, making it more in line to meet engineering requirements. The states information of system nodes is redundant, and the collected data are processed by the quantizer, which greatly reduces the impact of network bandwidth constraints. Based on the bounds of the dynamic change rate of the nonlinear function rather than the bounds of the procedure itself, the upper bound of the error covariance is obtained, and real-time gain optimization is realized. Finally, the rationality and accuracy of the estimator are verified by simulation. According to the control variable method, the effects of the quantization density and quantization level of the quantizer on the stability of system are briefly demonstrated.

A Bidirectional Active-Alignment Control System for LED Communication

LED-based free-space optical communication is becoming a promising alternative to the traditional acoustics mode of underwater communication because of its low cost, high data rate, and low power consumption. However, a fundamental challenge associated with the free space optical communication systems is to establish and maintain line-of-sight (LOS) between the two communicating agents. It becomes even more challenging when the agents are in motion, requiring an alignment system that enables each agent to point itself towards the LOS direction continuously. This paper presents a bi-directional active alignment control approach for the LED-based communication system of two robots, where the robots navigate in a three-dimensional (3D) space. We propose an extended Kalman filter (EKF) based alignment approach, where the estimates of azimuthal and elevation components of heading bias with the LOS are used to correct the alignment. We introduce and implement a new circular scanning technique on a two-degree-of-freedom (2DOF) rotational system mounted on each robot that enables consecutive independent measurements from a single photo-diode to satisfy the observability constraints of the EKF. Furthermore, we explore a synchronized alternating scheme to address the bidirectional nature of the problem. The scanning amplitude is further adjusted based on the EKF estimation covariance, to balance the trade-off between estimation accuracy and actuation effort. The presented results support the efficacy of the proposed method in the presence of slow relative motion between two robots, and demonstrate its superiority over an extremum-seeking (ES) based approach across a wide range of distances between the robots.

Time-Correlated Window-Carrier-Phase-Aided GNSS Positioning Using Factor Graph Optimization for Urban Positioning

This article proposes an improved global navigation satellite system (GNSS) positioning method that explores the time correlation between consecutive epochs of the code and carrier-phase measurements, which significantly increases the robustness against outlier measurements. Instead of relying on the time difference carrier phase which only considers two neighboring epochs using an extended Kalman filter estimator, this article proposed to employ the carrier-phase measurements inside a window, the so-called window carrier phase (WCP), to constrain the states inside a factor graph. A left null space matrix is employed to eliminate the shared unknown ambiguity variables and, therefore, correlate the associated states inside the WCP. Then, the pseudorange, Doppler, and the constructed WCP measurements are integrated simultaneously using factor graph optimization to estimate the state of the GNSS receiver. We evaluated the performance of the proposed method in two typical urban canyons in Hong Kong, achieving the mean positioning errors of 1.76 and 2.96 m, respectively, using the automobile-level GNSS receiver. Meanwhile, the effectiveness of the proposed method is further evaluated using a low-cost smartphone-level GNSS receiver, and similar improvement is also obtained when compared with several existing GNSS positioning methods.

Dynamic modeling and robust nonlinear control of a laboratory gas turbine engine

In this paper, a physics-based mathematical model of a mini SR-30 gas turbine engine (GTE) is presented using a state variable modeling method. This model is developed by utilizing approximated maps of the engine components and generic governing equations that describe the engine's behavior. The mathematical model captures both the steady-state and transient behavior of the engine. Using the developed model, a modern control system design approach is proposed to satisfy the stability and performance of the GTE. The objective of this control design is to adjust the required fuel flow rate to the combustion chamber such that the shaft speed of the engine tracks its command signal as closely as possible. The proposed robust nonlinear fuel flow controller is developed based on the nonlinear dynamic inversion (NDI) technique, augmented with a dual extended Kalman filter (DEKF) estimation design. DEKF-based state and parameter estimation are used here to ensure robustness against model inaccuracies due to not accurately known component maps, as well as process and measurement noises. To validate the developed mathematical model, the realistic data extracted from the SR-30 GTE are utilized. The simulation results of the proposed control design approach indicate that it is effective in achieving satisfactory performance of the engine. Also, the proposed NDI controller is compared with the well-established proportional-integral-derivative (PID) controller to illustrate its superiority.

Effect of Noise Covariance Matrices on State of Charge Estimation Using 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.

Evaluation of the Integrity Risk for Precise Point Positioning

Global Navigation Satellite System (GNSS) Precise Point Positioning (PPP) is an attractive positioning technology due to its high precision and flexibility. However, the vulnerability of PPP brings a safety risk to its application in the field of life safety, which must be evaluated quantitatively to provide integrity for PPP users. Generally, PPP solutions are processed recursively based on the extended Kalman filter (EKF) estimator, utilizing both the previous and current measurements. Therefore, the integrity risk should be qualified considering the effects of all the potential observation faults in history. However, this will cause the calculation load to explode over time, which is impractical for long-time missions. This study used the innovations in a time window to detect the faults in the measurements, quantifying the integrity risk by traversing the fault modes in the window to maintain a stable computation cost. A non-zero bias was conservatively introduced to encapsulate the effect of the faults before the window. Coping with the multiple simultaneous faults, the worst-case integrity risk was calculated to overbound the real risk in the multiple fault modes. In order to verify the proposed method, simulation and experimental tests were carried out in this study. The results showed that the fixed and hold mode adopted for ambiguity resolution is critical to an integrity risk evaluation, which can improve the observation redundancy and remove the influence of the biased predicted ambiguities on the integrity risk. Increasing the length of the window can weaken the impact of the conservative assumption on the integrity risk due to the smoothing effect of the EKF estimator. In addition, improving the accuracy of observations can also reduce the integrity risk, which indicates that establishing a refined PPP random model can improve the integrity performance.

Experimental tests of hybrid VTOL unmanned aerial vehicle designed for surveillance missions and operations in maritime conditions from ship‐based helipads

AbstractThis article presents research on a hybrid vertical take‐off and landing (VTOL) unmanned aerial vehicle (UAV) supported by a relative positioning system, enabling the deployment of autonomous missions from ship‐based helipads in maritime conditions. A crucial issue to be solved is ensuring precise positioning of the UAV relative to the landing pad in the take‐off and landing phases. To achieve this, an extended Kalman filter (EKF) is implemented on the UAV's onboard computer, which integrates the positioning data from the UAV's global navigation satellite system receiver and the positioning data of the landing pad broadcast by the landing pad navigation station (LNS). The EKF estimates both the absolute and relative position of the UAV, which are required for autonomous take‐off and landing on the moving landing pad. In unfavorable weather conditions, EKF also uses data from an optional local positioning system to keep the accuracy within the range 1–3 m. The research was concluded by experimental verification during a ferry cruise over the Baltic Sea. During the research, the VTOL UAV performed two fully autonomous flight missions at the range about 1 km from the moving ferry in international waters. Each of them ended with a successful landing back on the helipad with an accuracy matching its required level, which was already achieved in the previous research carried out in inland conditions. Data recorded during real flights confirm that the developed system, consisting of a hybrid VTOL UAV and a LNS, is ready to be utilized in autonomous missions at sea. Factors having a critical impact on the safety of use of the VTOL UAV in marine conditions were also identified, which were not observed during the research in inland conditions, and are related directly to turbulence around the landing pad.

An extended Kalman filter approach for real-time state estimation in multi-region MFD urban networks

The problem of traffic state estimation for large-scale urban networks modeled with MFD dynamics is studied here. Given a network partitioned in a number of regions, aggregated traffic dynamics describe the vehicle accumulation in each region, as well as transfer flows to and from neighboring regions. Considering that MFD accumulation-based models have been integrated in perimeter control approaches, this work tackles the real-time estimation problem when limited data is available. An estimation engine is developed according to the Extended Kalman Filter (EKF) theory; it seeks to estimate the real state of the multi-region dynamic system based on traffic sensors’ measurements. First, a stochastic model is presented for the dynamics of the process (plant). Then, the EKF estimation scheme is described based on a simpler aggregated model of dynamics and some real-time measurements. Estimation accuracy is investigated through detailed micro-simulation of downtown Barcelona by studying a realistic configuration of real-time measurement availability through loop detector data; however, the developed methodology is generic. The state vector we seek to estimate, as well as the available measurements configuration, can be altered according to the application. The proposed methodology is tested both in macro- and micro-simulation; resulting estimated traffic states (i.e., regional accumulations, demands, and distribution of outflows) are compared to actual ones obtained from the stochastic plant. The developed algorithm can be utilized by closed-loop online urban traffic management strategies to feed the estimated traffic state back to the controller.

Aero-engine health degradation estimation based on an underdetermined extended Kalman filter and convergence proof

In order to improve the reliability of aero-engine, reduce maintenance cost, and promote aircraft safety, lots of attention is paid to health monitoring of aero-engine. The aero-engine gas components involve flow and efficiency parameters, which are key health parameters to obtain the aero-engine’ performance degradation. A challenge has to be faced is that these health parameters needed to know are more than the available sensors, which cannot be estimated by the ordinary estimator like Kalman Filter (KF) and Extended Kalman Filter (EKF). In this paper, a system approach is raised to use model tuning parameter to solve the estimation problem mentioned before. To implement it, an underdetermined EKF estimator is constructed from previous achievement and applied to an aero-engine for health state estimation, to address the problem that there are fewer sensor data available with more unknown health parameters. And convergence proof of underdetermined EKF is also provided to make sure that the experimental result is deterministic rather than occasional, deducing that the convergence of this estimator can be verified with some mild constraints. It is found in this study that the covariance matrices Qk and Rk can meet the conditions of linear matrix inequality (LMI) by designing and setting specific ranges, leading to rapid convergence of the estimator. In addition, semi-physical experiments are shown to verify the feasibility of the proposed method.