Extended Kalman Smoother Research Articles

An improved long short‐term memory based on global optimization square root extended Kalman smoothing algorithm for collaborative state of charge and state of energy estimation of lithium‐ion batteries

SummaryState of charge and state of energy are essential performance indicators of the battery management system and the key to reflecting the remaining capacity of batteries. Aiming at the problems of low precision, long time, and strongly nonlinear system estimation of state of charge and state of energy of lithium‐ion batteries based on traditional algorithm under complex working conditions, this paper proposes a hybrid method consisting of the long short‐term memory neural network and square root extended Kalman smoothing. The long short‐term memory neural network can enhance the memory ability of the previous time data. The sliding window technology is introduced into the network to improve the correlation between the last time and the subsequent time estimation. Based on the traditional Kalman filtering algorithm, the square root and reverse smoothing algorithms are introduced to solve the risk of the negative covariance matrix and the problems of slow convergence and significant estimation deviation caused by a strongly nonlinear system. According to experiments, under the hybrid pulse power characterization working condition at 25°C, the maximum absolute errors of state of charge and state of energy are 1.779% and 1.487%, and the mean absolute errors are 0.352% and 0.894%, respectively. Under the Beijing bus dynamic stress test working condition at 25°C, the maximum absolute errors of state of charge and state of energy are 2.703% and 2.369%, and the mean absolute errors are 0.462% and 0.621%, respectively. The experimental results show that this algorithm can obtain reliable state of charge and state of energy under different complex working conditions with high accuracy, convergence, and robustness.

Optimal frequency control for wind power-integrated power system based on parameter identification

Frequency control based on wind power is required by the grid code for maintaining system inertia level as well as frequency stability. However, synchronous generator (SG)-emulated frequency control of wind power will result in a nadir and second dip of frequency trajectory after a disturbance, which may threaten system safety. Optimizing the frequency trajectory of system to improve the nadir and avoid second dip usually relies on grid-wide information. But the information is hardly perfectly acquired in advance for real power system since it may change anytime. To address these problems, an optimal frequency control method for wind power-integrated power system based on parameter identification is proposed. Frequency characteristics of wind power-integrated are firstly analyzed based on the common frequency theory. Then the control framework of proposed method to eliminate the nadir and second dip of frequency trajectory is introduced. To realize this framework, system equivalent parameters like inertia and damp coefficient are identified by the extended Kalman smoother algorithm after each disturbance. These parameters are furtherly utilized to shape the transfer function of wind power-integrated power system as a first-order system. Finally, the implementation steps of proposed method are presented. The effectiveness of proposed method is verified by case studies.

Multi-pass Extended Kalman Smoother with Partially-known Constraints for Estimation of Vapor Compression Cycles

State and parameter estimation methodologies have the potential to make a significant impact in the development of broad array of capabilities for widely-used vapor compression cycles, including advanced controls, performance monitoring, data-driven modeling, and deployment of digital twin technologies. However, the nonlinearity and numerical stiffness of large physics-based models of these systems pose challenges for the practical implementation of estimators that must also satisfy the physical state constraints. We present a three-pass fixed-interval smoothing method developed in the extended Kalman estimation formalism that incorporates linear inequality and partially-known nonlinear equality constraints defined in terms of unknown parameters of the system. The smoothing method is demonstrated to have high estimation accuracy during joint state and parameter estimation of the cycle model representing a realistic system that is implemented in Julia language leveraging automatic differentiation capabilities.

On application of Kalman filters in cohort models

On example of walleye pollock at East Kamchatka, one of possible approaches is considered to assess the state of stocks for marine commercial species with usage of the data on age structure in catches. Algorithms for suboptimal filtering and interpolation (extended Kalman smoother and unscented Kalman smoother) are presented for a cohort model of an exploited stock, in presence of uncertainty about true value of the vector of the system parameters.

Bayesian ODE solvers: the maximum a posteriori estimate

There is a growing interest in probabilistic numerical solutions to ordinary differential equations. In this paper, the maximum a posteriori estimate is studied under the class of nu times differentiable linear time-invariant Gauss–Markov priors, which can be computed with an iterated extended Kalman smoother. The maximum a posteriori estimate corresponds to an optimal interpolant in the reproducing kernel Hilbert space associated with the prior, which in the present case is equivalent to a Sobolev space of smoothness nu +1. Subject to mild conditions on the vector field, convergence rates of the maximum a posteriori estimate are then obtained via methods from nonlinear analysis and scattered data approximation. These results closely resemble classical convergence results in the sense that a nu times differentiable prior process obtains a global order of nu , which is demonstrated in numerical examples.

Black-Box Expectation–Maximization Algorithm for Estimating Latent States of High-Speed Vehicles

Tracking an object in a noisy environment is difficult, especially when unknown parameters affect the object’s behavior. In the case of a high-speed ballistic object, its trajectory is affected by changes in atmospheric conditions as well as various parameters of the object itself. To filter these latent factors of the dynamics model, this paper proposes a black-box expectation–maximization algorithm to estimate the latent parameters and enhance the accuracy of object tracking. The expectation step involves calculating the likelihood of observations through extended Kalman smoothing, which reflects the forward–backward probability combination. The maximization step involves optimizing the unknown parameter, which is the object mass, to maximize the likelihood through Bayesian optimization with Gaussian process regressions. Our simulation experiments show that our algorithm reduces the error of tracking a ballistic object’s position given noisy observations with unknown parameters.

Robot Localisation Using UHF-RFID Tags: A Kalman Smoother Approach †.

Autonomous vehicles enable the development of smart warehouses and smart factories with an increased visibility, flexibility and efficiency. Thus, effective and affordable localisation methods for indoor vehicles are attracting interest to implement real-time applications. This paper presents an Extended Kalman Smoother design to both localise a mobile agent and reconstruct its entire trajectory through a sensor-fusion employing the UHF-RFID passive technology. Extensive simulations are carried out by considering the smoother optimal-window length and the effect of missing measurements from reference tags. Monte Carlo simulations are conducted for different vehicle trajectories and for different linear and angular velocities to evaluate the method accuracy. Then, an experimental analysis with a unicycle wheeled robot is performed in real indoor scenario, showing a position and orientation root mean square errors of 15 cm, and 0.2 rad, respectively.

Kinematic State Estimation Using Multiple DGPS/MEMS-IMU Sensors

Inertial measurement unit based motion capture (MOCAP) technology has put an end to subjective gait analysis and has allowed researchers to obtain quantitative measurements of the kinematic and kinetic variables associated with gait. However, these systems are prone to drift in the inertial frame, especially under conditions of high acceleration, which naturally occur during rapid maneuveres. In an effort to circumvent this, we explore the possibility of integrating differential global positioning systems (DGPSs) on each limb of interest. This novel configuration is compared to a traditional inertial based system as well as a popular method of differential global positioning systems (DGPS) processing (RTKLIB) by means of an extended Kalman smoother. The configurations are experimentally validated on a mechanical test rig with the effect of nonline of sight of the global navigation satellite systems antennae explored. The results demonstrate that the combination of multiple DGPS receivers facilitate improved accuracy of the multibody system during rapid acceleration movements and can improve the accuracy of future Motion capture (MOCAP) systems.

Generalized pseudo Bayesian algorithms for tracking of multiple model underwater maneuvering target

The strength of Generalized Pseudo Bayesian (GPB) algorithms is exploited in the presented study to enhance the target tracking precision, effective model approximation and rapid convergence of multimodel maneuvering object tracking. The GPB methods are considered to be suitable for approximating systems whose dynamics follow discrete-time and fixed state Markov process. Underwater maneuvering target tracking problems are usually solved with nonlinear Bayesian algorithms, in which kinetics of object are associated with passive bearings using state-space modeling. Here accuracy and convergence of GPB methods based on Interacting Multiple Model Extended Kalman Filter (IMMEKF), Interacting Multiple Model Extended Kalman Smoother (IMMEKS), Interacting Multiple Model Unscented Kalman Filter (IMMUKF) and Interacting Multiple Model Unscented Kalman Smoother (IMMUKS) are efficiently analyzed for tracking of multimodel maneuvering target in complex ocean environment. Application of these algorithms is systematically presented for estimating the real-time state of a maneuvering object that follows a coordinated turn trajectory. Performance analysis of IMM Kalman filters and smoothers is done with variations in the standard deviation of white Gaussian measurement noise by following Bearings Only Tracking (BOT) phenomena. Least Mean Square Error (MSE) between approximated and the real position of maneuvering target in rectangular coordinates is calculated for analyzing the performance of filtering and smoothing techniques. Simulation results of the Monte Carlo runs validate the effectiveness of IMMEKS and IMMUKS over IMMEKF and IMMUKF for scenario of given framework.

Iterated Extended Kalman Smoother-Based Variable Splitting for $L_1$-Regularized State Estimation

In this paper, we propose a new framework for solving state estimation problems with an additional sparsity-promoting $L_1$-regularizer term. We first formulate such problems as minimization of the sum of linear or nonlinear quadratic error terms and an extra regularizer, and then present novel algorithms which solve the linear and nonlinear cases. The methods are based on a combination of the iterated extended Kalman smoother and variable splitting techniques such as alternating direction method of multipliers (ADMM). We present a general algorithmic framework for variable splitting methods, where the iterative steps involving minimization of the nonlinear quadratic terms can be computed efficiently by iterated smoothing. Due to the use of state estimation algorithms, the proposed framework has a low per-iteration time complexity, which makes it suitable for solving a large-scale or high-dimensional state estimation problem. We also provide convergence results for the proposed algorithms. The experiments show the promising performance and speed-ups provided by the methods.

NLIN Mitigation Using Turbo Equalization and an Extended Kalman Smoother

We develop a method for inter-channel nonlinear interference noise (NLIN) mitigation, based on Turbo equalization and an extended Kalman smoother. The method exploits both the unique statistical properties of NLIN and soft-decision information from the code structure, to obtain improved channel state estimation and symbol detection. The error-free information rate is improved by up to 0.3 b/symbols/polarization, for a 1000 km, 64-QAM, 32 Gbaud transmission over standard fiber.

An extended approach of a Kalman smoothing technique applied to a transient nonlinear two-dimensional inverse heat conduction problem

An extended Kalman smoother (EKS) introducing the concept of future measurement information is developed to handle nonlinear inverse heat conduction problems. In the present study, the formulation of the EKS algorithm is generalized in the case of a two-dimensional problem in order to reconstruct a non-homogeneous heating condition on the front surface of a cylindrical sample. This leads to apply the new method in a multiple sensor case for the simultaneous reconstruction of several parameters at each time. The analyzed inverse problem is nonlinear due to the variability of the thermophysical properties with temperature and to the presence of radiation boundary conditions. Inverse estimation is successfully performed where temporal and spatial variations of the front surface heat flux are recovered based on non intrusive transient temperature measurements made on the back surface. Numerical experiments show that the use of an optimal number of future data greatly improves the solution of the EKS compared to the extended Kalman filter (EKF) estimates, where a noticeable reduction of time lag and sensitivity to measurement errors is observed. Inversion results show that the optimal number of future data, chosen on the basis of a minimum measure of the heat flux bias, depends on the modeling error, the measurement time step, the distance between the sensors and the front heated surface and the measurement noise level. The proposed algorithm is robust in recovering different heat flux profiles and provides for all the examined patterns symmetric and stable solutions that superposes well with the exact functions. The EKS formulation based on an augmented state vector allows to efficiently recover the front surface temperature simultaneously with the heat flux.

Iterative Filtering and Smoothing in Nonlinear and Non-Gaussian Systems Using Conditional Moments

This letter presents the development of novel iterated filters and smoothers that only require specification of the conditional moments of the dynamic and measurement models. This leads to generalizations of the iterated extended Kalman filter, the iterated extended Kalman smoother, the iterated posterior linearization filter, and the iterated posterior linearization smoother. The connections to the previous algorithms are clarified and a convergence analysis is provided. Furthermore, the merits of the proposed algorithms are demonstrated in simulations of the stochastic Ricker map where they are shown to have a similar or superior performance to competing algorithms.

P and T wave detection and delineation of ECG signal using differential evolution (DE) optimization strategy.

Generally, P and T waves in an electrocardiogram (ECG) signal are lower in amplitude compared to amplitude of QRS complex and contaminated with noises from various sources. Due to these problems and lack of universal delineation rule, the automated detection and delineation of T and P waves (on, off, and peak position of T and P wave) in the ECG signal are challenging task. The effectiveness for detection of on, off, and peak position of T and P wave by using differential evolution (DE) algorithm with the denoising technique has been verified in this manuscript. The denoising operation of the ECG signal has been performed by extended Kalman smoother (EKS) framework. DE algorithm is used for selection of optimized width and phase of five waves of the ECG signal. These parameters are used in EKS for initialization of the process noise covariance matrix and also development of the state equation. The new algorithm (an intelligent process of searching and subtraction) for detection of on, off and peak location of P and T waves without using amplitude threshold is developed by using the optimized parameters computed by the DE algorithm and denoised ECG signal with the help of the EKS framework. The effectiveness of the proposed technique has been validated using real-time QT database. Our proposed method shows better sensitivity, predicitvity and accuracy compared to other well-known methods for detection of on, off, peak location of P and T wave.

Tracking the Cheetah Tail Using Animal-Borne Cameras, GPS, and an IMU

The cheetah (Acinonyx Jubatus) is the fastest terrestrial animal and is also highly maneuverable. An investigation into the whole-body motion dynamics of this specialized predator will illuminate various factors that influence and affect performance in legged animals as well as provide insight for the design of future bio-inspired robots. Presently, animal sensor collars can capture the gross animal behavior successfully but do not provide information about the animal's complete motion. Here, in an effort towards whole-body motion estimation, we demonstrate the use of animal mounted cameras, as well as the sensor suite of a smartphone that are attached to captive cheetahs during maneuvers. The various sensors are fused by an Extended Kalman Smoother (EKS) to provide high-bandwidth state estimates of the position, velocity, and acceleration of the cheetah spine and tail.

Extraction of fetal ECG signal by an improved method using extended Kalman smoother framework from single channel abdominal ECG signal.

This paper proposes a five-stage based methodology to extract the fetal electrocardiogram (FECG) from the single channel abdominal ECG using differential evolution (DE) algorithm, extended Kalman smoother (EKS) and adaptive neuro fuzzy inference system (ANFIS) framework. The heart rate of the fetus can easily be detected after estimation of the fetal ECG signal. The abdominal ECG signal contains fetal ECG signal, maternal ECG component, and noise. To estimate the fetal ECG signal from the abdominal ECG signal, removal of the noise and the maternal ECG component presented in it is necessary. The pre-processing stage is used to remove the noise from the abdominal ECG signal. The EKS framework is used to estimate the maternal ECG signal from the abdominal ECG signal. The optimized parameters of the maternal ECG components are required to develop the state and measurement equation of the EKS framework. These optimized maternal ECG parameters are selected by the differential evolution algorithm. The relationship between the maternal ECG signal and the available maternal ECG component in the abdominal ECG signal is nonlinear. To estimate the actual maternal ECG component present in the abdominal ECG signal and also to recognize this nonlinear relationship the ANFIS is used. Inputs to the ANFIS framework are the output of EKS and the pre-processed abdominal ECG signal. The fetal ECG signal is computed by subtracting the output of ANFIS from the pre-processed abdominal ECG signal. Non-invasive fetal ECG database and set A of 2013 physionet/computing in cardiology challenge database (PCDB) are used for validation of the proposed methodology. The proposed methodology shows a sensitivity of 94.21%, accuracy of 90.66%, and positive predictive value of 96.05% from the non-invasive fetal ECG database. The proposed methodology also shows a sensitivity of 91.47%, accuracy of 84.89%, and positive predictive value of 92.18% from the set A of PCDB.

An Extended Bayesian Framework for Atrial and Ventricular Activity Separation in Atrial Fibrillation.

An extended nonlinear Bayesian filtering framework is introduced for the analysis of atrial fibrillation (AF), in particular with single-channel electrocardiographical (ECG) recordings. It is suitable for simultaneously tracking the fundamental frequency of atrial fibrillatory waves (f-waves), and separating signals, linked to atrial and ventricular activity, during AF. In this framework, high-power ECG components, i.e., Q, R, S, and T waves, are modeled using sum of Gaussian functions. The atrial activity dynamical model is instead based on a trigonometrical function, with a fundamental frequency (the inverse of the dominant atrial cycle length), and its harmonics. The state variables of both dynamical models (QRS-T and f-waves) are hidden and, then estimated, sample by sample, using a Kalman smoother. Remarkably, the scheme is capable of separating ventricular and atrial activity signals, while contemporarily tracking the atrial fundamental frequency in time. The proposed method was evaluated using synthetic signals. In 290 ECGs in sinus rhythm from the PhysioNet PTB Diagnostic ECG Database, the P-waves were replaced with a synthetic f-wave. Broadband noise at different signal-to-noise ratio (SNR) (from 0 to 40 dB) was added to study the performance of the filter, under different SNR conditions. The results of the study demonstrated superior results in atrial and ventricular signal separation when compared with traditional average beat subtraction (ABS), one of the most widely used method for QRS-T cancellation (normalized mean square error = 0.045 for extended Kalman smoother (EKS) and 0.18 for ABS, SNR improvement was 21.1 dB for EKS and 12.2 dB for ABS in f-wave extraction). Various advantages of the proposed method have been addressed and demonstrated, including the problem of tracking the fundamental frequency of f-waves (root mean square error (RMSE) Hz for gradually changing frequency at SNR=15 dB) and of estimating robust QT/RR values during AF (RMSE at SNR=10 dB, ).

A BP–MF–EP Based Iterative Receiver for Joint Phase Noise Estimation, Equalization, and Decoding

In this work, with combined belief propagation (BP), mean field (MF) and expectation propagation (EP), an iterative receiver is designed for joint phase noise (PN) estimation, equalization and decoding in a coded communication system. The presence of the PN results in a nonlinear observation model. Conventionally, the nonlinear model is directly linearized by using the first-order Taylor approximation, e.g., in the state-of-the-art soft-input extended Kalman smoothing approach (soft-in EKS). In this work, MF is used to handle the factor due to the nonlinear model, and a second-order Taylor approximation is used to achieve Gaussian approximation to the MF messages, which is crucial to the low-complexity implementation of the receiver with BP and EP. It turns out that our approximation is more effective than the direct linearization in the soft-in EKS with similar complexity, leading to significant performance improvement as demonstrated by simulation results.

Extended Kalman smoother with differential evolution technique for denoising of ECG signal.

Electrocardiogram (ECG) signal gives a lot of information on the physiology of heart. In reality, noise from various sources interfere with the ECG signal. To get the correct information on physiology of the heart, noise cancellation of the ECG signal is required. In this paper, the effectiveness of extended Kalman smoother (EKS) with the differential evolution (DE) technique for noise cancellation of the ECG signal is investigated. DE is used as an automatic parameter selection method for the selection of ten optimized components of the ECG signal, and those are used to create the ECG signal according to the real ECG signal. These parameters are used by the EKS for the development of the state equation and also for initialization of the parameters of EKS. EKS framework is used for denoising the ECG signal from the single channel. The effectiveness of proposed noise cancellation technique has been evaluated by adding white, colored Gaussian noise and real muscle artifact noise at different SNR to some visually clean ECG signals from the MIT-BIH arrhythmia database. The proposed noise cancellation technique of ECG signal shows better signal to noise ratio (SNR) improvement, lesser mean square error (MSE) and percent of distortion (PRD) compared to other well-known methods.

ECG Denoising Using Marginalized Particle Extended Kalman Filter With an Automatic Particle Weighting Strategy.

In this paper, a model-based Bayesian filtering framework called the "marginalized particle-extended Kalman filter (MP-EKF) algorithm" is proposed for electrocardiogram (ECG) denoising. This algorithm does not have the extended Kalman filter (EKF) shortcoming in handling non-Gaussian nonstationary situations because of its nonlinear framework. In addition, it has less computational complexity compared with particle filter. This filter improves ECG denoising performance by implementing marginalized particle filter framework while reducing its computational complexity using EKF framework. An automatic particle weighting strategy is also proposed here that controls the reliance of our framework to the acquired measurements. We evaluated the proposed filter on several normal ECGs selected from MIT-BIH normal sinus rhythm database. To do so, artificial white Gaussian and colored noises as well as nonstationary real muscle artifact (MA) noise over a range of low SNRs from 10 to -5dB were added to these normal ECG segments. The benchmark methods were the EKF and extended Kalman smoother (EKS) algorithms which are the first model-based Bayesian algorithms introduced in the field of ECG denoising. From SNR viewpoint, the experiments showed that in the presence of Gaussian white noise, the proposed framework outperforms the EKF and EKS algorithms in lower input SNRs where the measurements and state model are not reliable. Owing to its nonlinear framework and particle weighting strategy, the proposed algorithm attained better results at all input SNRs in non-Gaussian nonstationary situations (such as presence of pink noise, brown noise, and real MA). In addition, the impact of the proposed filtering method on the distortion of diagnostic features of the ECG was investigated and compared with EKF/EKS methods using an ECG diagnostic distortion measure called the "Multi-Scale Entropy Based Weighted Distortion Measure" or MSEWPRD. The results revealed that our proposed algorithm had the lowest MSEPWRD for all noise types at low input SNRs. Therefore, the morphology and diagnostic information of ECG signals were much better conserved compared with EKF/EKS frameworks, especially in non-Gaussian nonstationary situations.