Central Difference Kalman Filter Research Articles

State of Charge Estimation of Lithium‐Ion Batteries with Adaptive Square Root Central Difference Kalman Filter

AbstractLithium‐ion batteries have been a major energy source in electric vehicles because of their strong adaptability in operating conditions. Accurate estimation of state of charge (SOC) for lithium‐ion batteries can efficiently improve the efficiency of battery energy utilization. However, SOC estimation is complicated in operating conditions with unknown model parameters. This study proposes an adaptive square root central difference Kalman filter (ASRCDKF) algorithm based on the equivalent circuit model to achieve high‐precision estimation of SOC. First of all, to avoid an open‐circuit voltage test, a linear Kalman filter is constructed to realize real‐time estimation of unknown parameters in the measurement equation. Then, to improve the stability of the algorithm, a square root method is used to ensure a positive semi‐definite of the error covariance matrix that is based on the adaptive central difference Kalman filter algorithm. Model parameters are considered as the state to be estimated, and the joint estimation of the model parameters and SOC is realized by an ASRCDKF algorithm. After that, the linear Kalman filter is coupled with the ASRCDKF to realize the accurate estimation of SOC in the case of both the state equation and the measurement equation including unknown parameters. Last, the ASRCDKF algorithm is compared with the adaptive central difference Kalman filter algorithm and the adaptive cubature Kalman filter algorithm under two sets of operating conditions. The results show that the SOC estimation of the ASRCDKF algorithm is more significantly accurate than other algorithms under different operating conditions.

Electric Vehicle Battery State of Charge Estimation With an Ensemble Algorithm Using Central Difference Kalman Filter (CDKF) and Non-Linear Autoregressive With Exogenous Input (NARX)

Electric Vehicle Battery State of Charge Estimation With an Ensemble Algorithm Using Central Difference Kalman Filter (CDKF) and Non-Linear Autoregressive With Exogenous Input (NARX)

A novel generalized prognostic method of proton exchange membrane fuel cell using multi-point estimation under various operating conditions

The proton exchange membrane fuel cell has been introduced into the fields of transportation, power production, and mobile devices. Especially, priority is given to promoting the application in commercial vehicles. However, more severe fluctuations in power demands give new challenges to the lifespan of fuel cells. The accurate state estimation and reasonable degradation prediction can assist in improving the lifetime of fuel cell devices. To realize the on-road prediction, herein, a novel generalized prognostic method called the Multi-point Square-root central difference Kalman filter (MP-SRCDKF) is proposed. First of all, an extended mathematical model is introduced. Later, the sensitivity analysis and parameter recognition are conducted based on the Sobol’ method and the jellyfish searching algorithm. The performance of the SRCDKF method is compared under static and quasi-dynamic conditions, which is further extended into the road condition and verified under the dynamic cycle condition temporarily. In the end, the remaining useful life prediction under various operating conditions is discussed. Furthermore, the generalized method can provide an approach for prognostic decision-making and the updating of the dynamic polarization curve to expand the lifetime and optimize the control system.

Performance evaluation of a central difference Kalman filter applied to attitude determination

Performance evaluation of a central difference Kalman filter applied to attitude determination

Combined Estimation of Vehicle Dynamic State and Inertial Parameter for Electric Vehicles Based on Dual Central Difference Kalman Filter Method

Distributed drive electric vehicles (DDEVs) possess great advantages in the viewpoint of fuel consumption, environment protection and traffic mobility. Whereas the effects of inertial parameter variation in DDEV control system become much more pronounced due to the drastic reduction of vehicle weights and body size, and inertial parameter has seldom been tackled and systematically estimated. This paper presents a dual central difference Kalman filter (DCDKF) where two Kalman filters run in parallel to simultaneously estimate vehicle different dynamic states and inertial parameters, such as vehicle sideslip angle, vehicle mass, vehicle yaw moment of inertia, the distance from the front axle to centre of gravity. The proposed estimation method only integrates and utilizes real-time measurements of hub torque information and other in-vehicle sensors from standard DDEVs. The four-wheel nonlinear vehicle dynamics estimation model considering payload variations, Pacejka tire model, wheel and motor dynamics model is developed, the observability of the DCDKF observer is analysed and derived via Lie derivative and differential geometry theory. To address system nonlinearities in vehicle dynamics estimation, the DCDKF and dual extended Kalman filter (DEKF) are also investigated and compared. Simulation with various maneuvers are carried out to verify the effectiveness of the proposed method using Matlab/Simulink-Carsim®. The results show that the proposed DCDKF method can effectively estimate vehicle dynamic states and inertial parameters despite the existence of payload variations and variable driving conditions. This research provides a boot-strapping procedure which can performs optimal estimation to estimate simultaneously vehicle system state and inertial parameter with high accuracy and real-time ability.

Online parameters identification and state of charge estimation for lithium-ion batteries based on improved central difference particle filter

Accurate SOC is of great significance to alleviate the endurance anxiety of electric vehicle drivers. To improve the accuracy of SOC estimation, a method of combining FFRLS and GPR for parameter identification and an improved particle filter for lithium battery SOC estimation are proposed. The proposed collaborative parameter identification method uses the forgetting factor recursive least squares (FFRLS) to identify the parameters in the middle and high SOC region. In the low SOC region, the parameters identified by the FFRLS are used as the training data of Gaussian process regression (GPR) for model parameter prediction to improve the parameter identification accuracy in the low SOC region; in the SOC estimation stage, the central difference Kalman filter is used to generate the importance density function to overcome the particle degradation, and the step size h of the central difference is calculated suboptimal to further improve the prediction accuracy. Finally, the proposed method is compared with other methods under different working conditions and different temperatures, and the effectiveness and adaptability of the proposed method are verified.

Comparison of Kalman Filters for State Estimation Based on Computational Complexity of Li-Ion Cells

Over the last few decades, lithium-ion batteries have grown in importance for the use of many portable devices and vehicular applications. It has been seen that their life expectancy is much more effective if the required conditions are met. In one of the required conditions, accurately estimating the battery’s state of charge (SOC) is one of the important factors. The purpose of this research paper is to implement the probabilistic filter algorithms for SOC estimation; however, there are challenges associated with that. Generally, for the battery to be effective the Bayesian estimation algorithms are required, which are recursively updating the probability density function of the system states. To address the challenges associated with SOC estimation, the research paper goes further into the functions of the extended Kalman filter (EKF) and sigma point Kalman filter (SPKF). The function of both of these filters will be able to provide an accurate estimation. Further studies are required for these filters’ performance, robustness, and computational complexity. For example, some filters might be accurate, might not be robust, and/or not implementable on a simple microcontroller in a vehicle’s battery management system (BMS). A comparison is made between the EKF and SPKF by running simulations in MATLAB. It is found that the SPKF has an obvious advantage over the EKF in state estimation. Within the SPKF, the sub-filter, the central difference Kalman filter (CDKF), can be considered as an alternative to the EKF for state estimation in battery management systems for electric vehicles. However, there are implications to this which include the compromise of computational complexity in which a more sophisticated micro-controller is required.

A robust three-stage central difference Kalman filter for nonlinear discrete-time systems considering faults and unknown inputs

In this study, a novel robust three-stage central difference Kalman filter (ThSCDKF) is proposed for nonlinear discrete-time systems involving unknown inputs (uncertainties) and fault terms in the state and measurement equations. It is assumed that the evolution models of the fault and unknown inputs are not available and there is no prior information about the statistical properties of these factors. The proposed method is developed by decoupling an Augmented CDKF (ACDKF) via U-V transformation. By modifying the measurement stage, a robust ThSCDKF is proposed to achieve a robust state estimation when we do not have prior knowledge about the statistical model of the fault or uncertainties. The stability of the proposed filters is proved by the Lyapunov theory. The proposed methods have been applied to the on-board calibration of a star sensor. Finally, the effectiveness of the proposed filters is demonstrated by conducting different simulations.

A Robust Kalman Filter-Based Approach for SoC Estimation of Lithium-Ion Batteries in Smart Homes

Battery energy systems are playing significant roles in smart homes, e.g., absorbing the uncertainty of solar energy from root-top photovoltaic, supplying energy during a power outage, and responding to dynamic electricity prices. For the safe and economic operation of batteries, an optimal battery-management system (BMS) is required. One of the most important features of a BMS is state-of-charge (SoC) estimation. This article presents a robust central-difference Kalman filter (CDKF) method for the SoC estimation of on-site lithium-ion batteries in smart homes. The state-space equations of the battery are derived based on the equivalent circuit model. The battery model includes two RC subnetworks to represent the fast and slow transient responses of the terminal voltage. Moreover, the model includes the nonlinear relationship between the open-circuit voltage (OCV) and SoC. The proposed robust CDKF method can accurately estimate the SoC in the presence of the time-varying model uncertainties and measurement noises. Being able to cope with model uncertainties and measurement noises is essential, since they can lead to inaccurate SoC estimations. An experiment test bench is developed, and various experiments are conducted to extract the battery model parameters. The experimental results show that the proposed method can more accurately estimate SoC compared with other Kalman filter-based methods. The proposed method can be used in optimal BMSs to promote battery performance and decrease battery operational costs in smart homes.

An improved central difference Kalman filter for satellite attitude estimation with state mutation

AbstractThere are three problems in applying central difference Kalman filter (CDKF) to attitude estimation of low‐cost satellite: (1) the negative definiteness and asymmetry of the covariance matrix caused by limited computing power and storage space of the onboard microprocessor may result in calculation divergence, (2) the system model uncertainty including model mismatch and state mutation caused by environmental disturbances from inside and outside the satellite will lead to filter divergence, (3) the addition operation on quaternions violates the normalization constraint, while enforced unitization will introduce approximation errors. To solve the above problems, this article proposes the multiplicative quaternion square root strong tracking CDKF (MQSRSTCDKF), in which three improvements to basic CDKF are made: (1) square roots of covariance matrices are adopted for information propagation in place of covariance matrices themselves, which always guarantees the symmetry and positive definiteness of covariance matrices, (2) multiple fading factors are employed to adjust square root of the covariance matrix of predicted state variables, and then adjust the gain matrix to make the filter attach more weight to sensor measurements than to corrupt state predictions caused by model uncertainties, (3) multiplication operation on quaternions is conducted instead of the addition one, which makes the normalization constraint on quaternions naturally satisfied without enforced unitization. Numerical simulation based on raw telemetry data from the on‐orbit CubeSat NJUST‐1 shows that MQSRSTCDKF has higher attitude estimation accuracy, faster convergence speed, and stronger robustness to filter divergence and calculation divergence than basic CDKF.

Online SoC Estimation of Lithium-Ion Batteries Using a New Sigma Points Kalman Filter

The accurate state of charge (SoC) online estimation for lithium-ion batteries is a primary concern for predicting the remaining range in electric vehicles. The Sigma points Kalman Filter is an emerging SoC filtering technology. Firstly, the charge and discharge tests of the battery were carried out using the interval static method to obtain the accurate calibration of the SoC-OCV (open circuit voltage) relationship curve. Secondly, the recursive least squares method (RLS) was combined with the dynamic stress test (DST) to identify the parameters of the second-order equivalent circuit model (ECM) and establish a non-linear state-space model of the lithium-ion battery. Thirdly, based on proportional correction sampling and symmetric sampling Sigma points, an SoC estimation method combining unscented transformation and Stirling interpolation center difference was designed. Finally, a semi-physical simulation platform was built. The Federal Urban Driving Schedule and US06 Highway Driving Schedule operating conditions were used to verify the effectiveness of the proposed estimation method in the presence of initial SoC errors and compare with the EKF (extended Kalman filter), UKF (unscented Kalman filter) and CDKF (central difference Kalman filter) algorithms. The results showed that the new algorithm could ensure an SoC error within 2% under the two working conditions and quickly converge to the reference value when the initial SoC value was inaccurate, effectively improving the initial error correction ability.

An adaptive central difference Kalman filter approach for state of charge estimation by fractional order model of lithium-ion battery

The key issue of the model-based state of charge estimation approach is the accuracy of the battery model. In this paper, a fractional order model is built to simulate the electrochemistry dynamics of lithium-ion battery, whose model parameters are identified by adaptive genetic algorithm. Based on the computation simplification of central difference algorithm, an adaptive central difference Kalman filter by fractional order model is designed to estimate the state of charge. The designed approach is modelled by simulink and translated into C code, and then embedded in the battery management system for the validation by two dynamic cycles. Comparing experiments adopt two approaches, i.e. the central difference Kalman filter by fractional order model, the adaptive central difference Kalman filter by Thevenin model. Experimental results indicate that the designed approach has the better accuracy and robustness, and also show that fractional order model is more accurate than Thevenin model. With respect ot the ability to deal with noise, the robustness of the designed approach is verified by adding artificial noise. Experimental results show that the proposed approach has the best robustness to noise. Therefore, the proposed approach is a good candidate for the state of charge estimation in engineering practice.

Design of the modified fractional central difference Kalman filters under stochastic colored noises

For state estimation of discrete nonlinear fractional stochastic systems, this study presents two innovative modified fractional central difference Kalman filters. We consider a complicated scenario where the process noise or measurement noise in the system become colored noise. Firstly, the nonlinear function is linearized by utilizing the Stirling polynomial interpolation formula. Thus there is no need to calculate the Jacobi matrix for both algorithms, which means very few application limitations. Then, based on the augmented-state method, we develop an augmented state fractional central difference Kalman filter under the scenario of colored process noise. Afterwards, a state estimation algorithm for handling stochastic systems containing colored measurement noise is put forward by using the measurement expansion method. Finally, to perform the superiority of the developed algorithms, several simulations are carried out. As well, the algorithms derived in this paper are contrasted with the original fractional central difference Kalman filter and three other algorithms. Notably, a simulation with engineering significance for the state-of-charge estimation for lithium-ion batteries is also introduced, Aside from the commonly used numerical simulation. The results verify the superiority of the developed algorithms in sense of estimation accuracy and real-time performance.

Strong tracking central difference Kalman filter for CubeSat attitude estimation with status mutation

This paper addressed a strongly nonlinear problem caused by status mutation in CubeSat attitude estimation system. The multiple fading factors are employed to make different state channels have separate adjustment ability, which enhances the tracking performance for status mutation. The second-order difference transformation is adopted to improve the approximation accuracy of state posterior mean and covariance. Therefore, a multiple fading second-order central difference Kalman filter (MFSCDKF) is formed for CubeSat attitude estimation system in the presence of status mutation. Compared with the single fading factor first-order central difference Kalman filter, the proposed one can improve tracking performance and estimation accuracy simultaneously. Simulation results based on real telemetry data from the on-orbit CubeSat NJUST-1 verify the effectiveness of the proposed MFSCDKF.

Modified Strong Tracking System Identification Method Based on Square Root Center Difference Kalman Filter for Civil Structures

Civil engineering structures will exhibit hysteretic behavior due to damage caused by dynamic loads. Identifying the hysteretic behavior of structures is a practical and challenging problem that involves observing vibration data to determine strength and stiffness degradation. This paper proposes a modified square root central difference Kalman filter (MSRCD-KF) method to track this behavior. By combining the QR decomposition and strong tracking filtering technology, the proposed method makes the recursive calculation process unconditionally stable, while enabling the tracking of abrupt changes in structural parameters. A three degree-of-freedom (3-DOF) Duffing system is used in the simulation to verify the effectiveness of the proposed method. Numerical results show that the proposed method can converge to the true value quickly and accurately. Then, the proposed method is used to identify the structural parameters of a two-story concrete frame structure under different seismic loading sequences. In the first example, the structure is simplified as a 2-DOF linear system for which the equivalent stiffness and damping under different damage levels are identified. This information is then used to obtain the stiffness variation trend, damping ratio, and frequency. The second example uses the Bouc–Wen model to consider the stiffness and strength degradation of the structure. Finally, the experimental results demonstrate that the proposed method can accurately identify the structural parameters of nonlinear systems, and the identified hysteresis curves are in good agreement with the experimental ones.

Adaptive Central Difference Kalman Filter With Unknown Measurement Noise Covariance and Its Application to Airborne POS

Position and Orientation system (POS), a loosely integrated inertial navigation system (INS) and global positioning system (GPS), can provide high-accuracy motion information for the airborne remote sensing loads, which plays a crucial role in airborne remote sensing imaging. However, the airborne POS often suffers from the harsh environment, such as aircraft maneuver mode and other external disturbance, which will lead to measurement noise unknown and further affects the accuracy of motion parameters. In this paper, an adaptive central difference Kalman filter method based on expectation maximization algorithm is proposed, which can estimate measurement noise adaptively and further improve the performance of POS. A flight experiment is conducted and the results show that the proposed method achieves higher-accuracy motion information by compared with the traditional CDKF method and covariance matching.

Robust Three Stage Central Difference Kalman Filter for Helicopter Unmanned Aerial Vehicle Actuators Fault Estimation

This paper proposes state and fault estimations for uncertain time-varying nonlinear stochastic systems with unknown inputs. we suppose, the information about the fault and unknown inputs is not perfectly known. For this purpose, in this manuscript, we developed a robust three-stage central difference Kalman filter (RThSCDKF). We used RThSCDKF for model-based fault detection and identification (FDI) in nonlinear hover mode of helicopter unmanned aerial vehicle (HUAV) in the presence of external disturbance. In this system, actuator faults are affected by each other. The proposed method estimates and decouples actuator faults in the presence of external disturbances. This model can detect stuck and floating faults that are important to detect. At the end, this method is compared with the three-stage extended Kalman filter (ThSEKF). Simulation results show the effectiveness of the proposed robust method for detection and isolation of various actuator faults and also this shows more accuracy with respect to ThSEKF.

Robust Central Difference Kalman Filter With Mixture Correntropy: A Case Study for Integrated Navigation

In this paper, a robust central difference Kalman filter is proposed to address the process uncertainty and non-Gaussian measurement noise induced by the vehicle's severe maneuver and abnormal measurements in MEMS-SINS/GNSS integrated navigation system. Compared with the state-of-the-art noise distribution based robust filter, in the proposed filter, the process uncertainty and measurement uncertainty are simultaneously suppressed based on a new constructed cost function, which is independent of noise distribution and more insensitive to the non-Gaussian noise. To be specific, the statistical linearization approach is first presented to derive a linear-like regression model. Then, by resorting to the innovation orthogonal theory and Cholesky triangular decomposition, the fading factor of cost function is adaptively and robustly determined in the process of iteration, where the filtering performance and the stability of the algorithm under the condition of process uncertainty are extremely enhanced. Furthermore, the correntropy using the mixture of two Gaussian functions as the kernel function is incorporated into the cost function to prevent the non-Gaussian measurement noise. Our extensive simulation and car-mounted experiment demonstrate that the proposed filter can achieve higher estimation accuracy and better robustness as compared with the related state-of-the-art methods.

Online probabilistic model class selection and joint estimation of structures for post-disaster monitoring

Online selection of the appropriate model and identifying its parameters based on measured vibrational data are among the challenging issues in dynamic system identification. After a severe earthquake, quick monitoring and assessment of structural health status play a crucial role in effective critical risk management for the building owners and decision-makers. The Bayesian multiple modeling approach is a suitable tool for optimal model class selection, which is used in this article mainly for improving data fitting precision, decreasing dimensions of structural unknown vector through removing unnecessary parameters, detecting the occurrence and type of predominant phenomenon related to degradation and pinching, and finally increasing stability and convergence rate of the used identification algorithm. In this study, a reliable and effective time-domain method is proposed for online probabilistic model class selection and joint estimation of structures using the central difference Kalman filter and Bayesian multiple modeling approach. Also, in contrast to existing studies, the performance and efficiency of the proposed algorithm are investigated in nonlinear system identification with a large number of unknowns. The proposed algorithm is implemented on three moment-frame buildings with 8, 54, and 60 unknowns. In addition, a numerical example is provided to analyze the case in which the exact model is not among the plausible models. To verify the performance of the hybrid identification method, robust simulations with synthetic measurement noises and modeling errors were generated using the Monte Carlo random simulation method. The result shows the method can be used to simultaneously select the optimal model class and identify its unknown states and parameters in an online manner.

Online jointly estimation of hysteretic structures using the combination of central difference Kalman filter and Robbins–Monro technique

Rapid assessment of structural safety and performance right after the occurrence of significant earthquake shaking is crucial for building owners and decision-makers to make informed risk management decisions. Hence, it is vital to develop online and pseudo-online health monitoring methods to quantify the health of the building right after significant earthquake shaking. Many Bayesian inference–based methods have been developed in the past which allow the users to estimate the unknown states and parameters. However, one of the most challenging part of the Bayesian inference–based methods is the determination of the parameter noise covariance matrix. It is especially difficult when the number of unknown states and parameters is large. In this study, an effective online joint estimation method for nonlinear hysteretic structures, with consideration of degradation and pinching phenomena, is proposed. Simultaneous estimation of states and parameters is conducted using the combination of a central difference Kalman filter as an effective estimator and the Robbins–Monro stochastic approximation technique as the parameters noise covariance matrix regulator. The proposed algorithm is implemented on three shear buildings with 36, 54, and 90 unknown states and parameters. To verify the performance of the system identification method, robust simulations with synthetic measurement noises and modeling errors were generated using the Monte Carlo random simulation method. The result shows the proposed method can be used to estimate the unknown parameters and states of highly nonlinear systems efficiently and effectively.