Standard Kalman Filter Research Articles

High‐dimensional ensemble Kalman filter with localization, inflation, and iterative updates

AbstractAccurate estimation of forecast‐error covariance matrices is an essential step in data assimilation, which becomes a challenging task for high‐dimensional data assimilation. The standard ensemble Kalman filter (EnKF) may diverge due to both the limited ensemble size and the model bias. In this article, we propose to replace the sample covariance in the EnKF with a statistically consistent high‐dimensional tapering covariance matrix estimator to counter the estimation problem under high dimensions. A high‐dimensional EnKF scheme combining covariance localization with the inflation method and an iterative update structure is developed. The proposed assimilation scheme is tested on the Lorenz‐96 model with spatially correlated observation systems. The results demonstrate that the proposed method could improve the assimilation performance under multiple settings.

A New Data Fusion Method for GNSS/INS Integration Based on Weighted Multiple Criteria

The standard Kalman filter and most of its enhancements are typically designed based on the criterion that minimizes the mean squared error, with little discussion of multiple criteria in the positioning and navigation fields. Therefore, a novel data fusion method that takes into account weighted multiple criteria is proposed in this paper, implementing a filtering algorithm based on integrated criteria with different weights determined by a weight adjustment factor. The proposed algorithm and conventional filtering algorithms were utilized for data fusion in GNSS/INS integration. Experiments were conducted using actual data collected from an urban environment. Comparative analysis revealed that, when utilizing the proposed algorithm, the precision of the position, velocity, and attitude of the tested land vehicle could be improved by approximately 24%, 48%, and 35%, respectively. Furthermore, a series of filtering algorithms with different weight adjustment factors was performed to test their influence on the filtering. The application of the proposed algorithm should be accompanied by an appropriate weight adjustment factor.

Improving the stochastic model for code pseudorange observations from Android smartphones

In recent years, there has been increasing attention to positioning, navigation, and timing applications with smartphones. Because of frequently disrupted carrier phase observations, code observations remain critical for smartphone-based positioning. Considering a realistic stochastic model is mandatory to obtain the utmost positioning performance, this study proposes a sound stochastic approach for code observations from Android smartphones. The proposed approach includes a modified version of the SIGMA-ɛ variance model with different coefficients for each GNSS constellation and a robust Kalman filter method. First the noise characteristics of observations from the Xiaomi Mi 8 smartphone are analyzed utilizing code-minus-phase combinations to estimate the coefficients for each GNSS constellation. This includes the determination of a variance model as well as a check of the probability distribution. Finally, the proposed approach is validated in the positioning domain using single-frequency code observation-based real-time standalone positioning. The results show that more than 95% of observations follow the normal distribution when the proposed approach is applied. Compared with the conventional stochastic approach, including a C/N0-dependent model and standard Kalman filter, it improves the positioning accuracy by 45.8% in a static experiment, while its improvement is equal to 26.6% in a kinematic experiment. For the static and kinematic experiments, in 50% of the epochs, the 3D positioning errors are smaller than 3.0 m and 3.4 m for the proposed stochastic approach. The results exhibit that the stochastic properties of code observations from smartphones can be successfully represented by the proposed approach.

Iterative Ensemble Smoothers for Data Assimilation in Coupled Nonlinear Multiscale Models

Abstract This paper identifies and explains particular differences and properties of adjoint-free iterative ensemble methods initially developed for parameter estimation in petroleum models. The aim is to demonstrate the methods’ potential for sequential data assimilation in coupled and multiscale unstable dynamical systems. For this study, we have introduced a new nonlinear and coupled multiscale model based on two Kuramoto–Sivashinsky equations operating on different scales where a coupling term relaxes the two model variables toward each other. This model provides a convenient testbed for studying data assimilation in highly nonlinear and coupled multiscale systems. We show that the model coupling leads to cross covariance between the two models’ variables, allowing for a combined update of both models. The measurements of one model’s variable will also influence the other and contribute to a more consistent estimate. Second, the new model allows us to examine the properties of iterative ensemble smoothers and assimilation updates over finite-length assimilation windows. We discuss the impact of varying the assimilation windows’ length relative to the model’s predictability time scale. Furthermore, we show that iterative ensemble smoothers significantly improve the solution’s accuracy compared to the standard ensemble Kalman filter update. Results and discussion provide an enhanced understanding of the ensemble methods’ potential implementation and use in operational weather- and climate-prediction systems.

A novel residual-based Bayesian expectation–maximization adaptive Kalman filter with inaccurate and time-varying noise covariances

In this study, we introduce a novel residual-based Bayesian expectation–maximization adaptive Kalman filter (RBEMAKF) for dynamic state estimation with inaccurate and time-varying noise covariance matrices. The proposed scheme presents a novel maximum a-posteriori (MAP) estimator for characterizing process and measurement noises, leveraging the residual information derived from the Kalman filter. Simultaneously, the MAP is addressed through the application of the expectation–maximization (EM) algorithm. Subsequently, the standard Kalman filter is executed based on the estimated posterior of process and measurement noises (PMNs) to correct the dynamic state in real-time. Additionally, RBEMAKF is extended to propose Laplacian-RBEMAKF (L-RBEMAKF) and Student’s t-RBEMAKF (ST-RBEMAKF) to accommodate outlier environments. These methods assume Laplacian and Student’s t distributions for the prior of PMNs in RBEMAKF, respectively. Extensive simulations and real-world results demonstrate the effectiveness of the proposed RBEMAKF and its extensions (L-RBEMAKF and ST-RBEMAKF) in dynamic state estimation. The availability of our codes can be found at https://github.com/Gaitxh/RBEMAKF.

An improved two-phase robust distributed Kalman filter

To enhance the efficacy of the distributed filter in mitigating heavy-tailed non-Gaussian noise and accommodating intricate environments, this study investigates the utilization of absolute and relative measurement information for the realization of multi-target cooperative positioning and proposes an improved robust distributed Kalman filter in this paper. The method is divided into two phases. Firstly, absolute measurement information undergoes processing via the standard Kalman filter, yielding the Kalman estimation value for the first phase, subsequently serving as the time update value for the second phase. Next, a robust Kalman filter based on Student’s t distribution is used to process relative measurement information for measurement updates, resulting in more accurate estimates. Simulation results demonstrate that the proposed approach effectively approximates the posterior probability distribution, leading to enhanced stability and accuracy in positioning within environments characterized by both Gaussian and heavy-tailed non-Gaussian noise.

Adaptive Kalman Filter for Real-Time Visual Object Tracking Based on Autocovariance Least Square Estimation

Real-time visual object tracking (VOT) may suffer from performance degradation and even divergence owing to inaccurate noise statistics typically engendered by non-stationary video sequences or alterations in the tracked object. This paper presents a novel adaptive Kalman filter (AKF) algorithm, termed AKF-ALS, based on the autocovariance least square estimation (ALS) methodology to improve the accuracy and robustness of VOT. The AKF-ALS algorithm involves object detection via an adaptive thresholding-based background subtraction technique and object tracking through real-time state estimation via the Kalman filter (KF) and noise covariance estimation using the ALS method. The proposed algorithm offers a robust and efficient solution to adapting the system model mismatches or invalid offline calibration, significantly improving the state estimation accuracy in VOT. The computation complexity of the AKF-ALS algorithm is derived and a numerical analysis is conducted to show its real-time efficiency. Experimental validations on tracking the centroid of a moving ball subjected to projectile motion, free-fall bouncing motion, and back-and-forth linear motion, reveal that the AKF-ALS algorithm outperforms a standard KF with fixed noise statistics.

Robust Cubature Kalman Filter for Moving-Target Tracking with Missing Measurements.

Handling the challenge of missing measurements in nonlinear systems is a difficult problem in various scientific and engineering fields. Missing measurements, which can arise from technical faults during observation, diffusion channel shrinking, or the loss of specific metrics, can bring many challenges when estimating the state of nonlinear systems. To tackle this issue, this paper proposes a technique that utilizes a robust cubature Kalman filter (RCKF) by integrating Huber's M-estimation theory with the standard conventional cubature Kalman filter (CKF). Although a CKF is often used for solving nonlinear filtering problems, its effectiveness might be limited due to a lack of knowledge regarding the nonlinear model of the state and noise-related statistical information. In contrast, the RCKF demonstrates an ability to mitigate performance degradation and discretization issues related to track curves by leveraging covariance matrix predictions for state estimation and output control amidst dynamic disruption errors-even when noise statistics deviate from prior assumptions. The performance of extended Kalman filters (EKFs), unscented Kalman filters (UKFs), CKFs, and RCKFs was compared and evaluated using two numerical examples involving the Univariate Non-stationary Growth Model (UNGM) and bearing-only tracking (BOT). The numerical experiments demonstrated that the RCKF outperformed the EKF, EnKF, and CKF in effectively handling anomaly errors. Specifically, in the UNGM example, the RCKF achieved a significantly lower ARMSE (4.83) and ANCI (3.27)-similar outcomes were observed in the BOT example.

Can We Intercalibrate Satellite Measurements by Means of Data Assimilation? An Attempt on LEO Satellites

AbstractLow Earth Orbit satellites offer extensive data of the radiation belt region, but utilizing these observations is challenging due to potential contamination and difficulty of intercalibration with spacecraft measurements at Highly Elliptic Orbit that can observe all equatorial pitch‐angles. This study introduces a new intercalibration method for satellite measurements of energetic electrons in the radiation belts using a Data assimilation (DA) approach. We demonstrate our technique by intercalibrating the electron flux measurements of the National Oceanic and Atmospheric Administration (NOAA) Polar‐orbiting Operational Environmental Satellites (POES) NOAA‐15,‐16,‐17,‐18,‐19, and MetOp‐02 against Van Allen Probes observations from October 2012 to September 2013. We use a reanalysis of the radiation belts obtained by assimilating Van Allen Probes and Geostationary Operational Environmental Satellites observations into 3‐D Versatile Electron Radiation Belt (VERB‐3D) code simulations via a standard Kalman filter. We compare the reanalysis to the POES data set and estimate the flux ratios at each time, location, and energy. From these ratios, we derive energy and L* dependent recalibration coefficients. To validate our results, we analyze on‐orbit conjunctions between POES and Van Allen Probes. The conjunction recalibration coefficients and the data‐assimilative estimated coefficients show strong agreement, indicating that the differences between POES and Van Allen Probes observations remain within a factor of two. Additionally, the use of DA allows for improved statistics, as the possible comparisons are increased 10‐fold. Data‐assimilative intercalibration of satellite observations is an efficient approach that enables intercalibration of large data sets using short periods of data.

Analytical Bounds for an Interval Kalman Filter

This paper is concerned with analytical developments of results firstly introduced by the authors in [1]. These developments are devoted to the optimization of upper bounds of the interval covariance matrices appearing in the Interval Kalman Filter [2]. The proposed study is mainly highlighted through two aspects. Firstly, the optimization is further performed by considering a class of upper bounds and minimizing the traces of these bounds in two stages (in terms of a gain matrix and then with respect to a scalar parameter). Secondly, the paper provides conditions under which the optimal trace value is controlled and hence the proposed Algorithm in [1], namely Optimal Upper Bound Interval Kalman Filter (OUBIKF), is ensured to perform with stability (i.e. without width explosion of the resulting interval estimators). Also under these conditions, the OUBIKF Algorithm, having a similar structure of the Standard Kalman Filter (SKF), is ensured to get a smaller trace upper bound of the covariance matrices in the correction step than the one in the prediction step. Numerical simulations based on an automotive model is performed to illustrate the developed results.

A Quantile-Conserving Ensemble Filter Framework. Part II: Regression of Observation Increments in a Probit and Probability Integral Transformed Space

Abstract Traditional ensemble Kalman filter data assimilation methods make implicit assumptions of Gaussianity and linearity that are strongly violated by many important Earth system applications. For instance, bounded quantities like the amount of a tracer and sea ice fractional coverage cannot be accurately represented by a Gaussian that is unbounded by definition. Nonlinear relations between observations and model state variables abound. Examples include the relation between a remotely sensed radiance and the column of atmospheric temperatures, or the relation between cloud amount and water vapor quantity. Part I of this paper described a very general data assimilation framework for computing observation increments for non-Gaussian prior distributions and likelihoods. These methods can respect bounds and other non-Gaussian aspects of observed variables. However, these benefits can be lost when observation increments are used to update state variables using the linear regression that is part of standard ensemble Kalman filter algorithms. Here, regression of observation increments is performed in a space where variables are transformed by the probit and probability integral transforms, a specific type of Gaussian anamorphosis. This method can enforce appropriate bounds for all quantities and deal much more effectively with nonlinear relations between observations and state variables. Important enhancements like localization and inflation can be performed in the transformed space. Results are provided for idealized bivariate distributions and for cycling assimilation in a low-order dynamical system. Implications for improved data assimilation across Earth system applications are discussed.

State of Charge Estimation of Lithium-Ion Batteries Based on Vector Forgetting Factor Recursive Least Square and Improved Adaptive Cubature Kalman Filter

Accurate online parameter identification and state of charge (SOC) estimation are both very crucial for ensuring the operating safety of lithium-ion batteries and usually the former is a base of the latter. To achieve accurate and stable SOC estimation results, this paper proposes a model-based method, which incorporates a vector forgetting factor least square (VFFLS) algorithm and an improved adaptive cubature Kalman filter (IACKF). Firstly, considering it is difficult for the traditional forgetting factor recursive least square (FFRLS) algorithm to balance the accuracy, convergence, and stability for multiple parameters with different time-varying periods, an improved VFFLS method is employed to determine the multiple parameters of the first-order RC battery model online. It supersedes the single forgetting factor in the FFRLS with multiple forgetting factors in a vector form for improving adaptive capability to multiple time-varying parameters. Secondly, aiming at the fact that the standard cubature Kalman filter (CKF) cannot operate properly when the error covariance matrix is non-positive definite, which is caused by disturbance, initial error, and the limit of the computer word length, the UR decomposition rather than the Cholesky decomposition is applied, thus improving the algorithm stability. In addition, an adaptive update strategy is added to the CKF to enhance accuracy and convergence speed. Finally, comparative experiments with different operating patterns, positive and non-positive definite error covariance matrices, and temperatures are carried out. Experimental results showed that the proposed method can estimate the SOC accurately and stably.

Spectral Graph Filtering for Noisy Signals Using the Kalman filter

Noise is unwanted electrical or electromagnetic radiation that degrades the quality of the signal and the data. It can be difficult to denoise a signal that has been acquired in a noisy environment, but doing so may be necessary in a number of signal processing applications. This paper extends the issue of signal denoising from signals with regular structures, which are affected by noise, to signals with irregular structures by applying the graph signal processing (GSP) technique and a very wellknown filter, the standard Kalman filter, after adjusting it. When the modified Kalman filter is compared to the standard Kalman filter, the modified one performs better in situations where there are uncertain observations and/or processing noise and shows the best results. Also, the modified Kalman filter showed a higher efficiency when we compared it with other filters for different types of noise, which are not only standard Gaussian noises but also uniform distribution noise across two intervals for uncertain observation noise.

State Space Estimation: from Kalman Filter Back to Least Squares

This note reviews a direct least squares estimation of a state space model and highlights its advantages over the standard Kalman filter in some applications. Although there is a close relationship between these two concepts, dual understanding of the estimation problem seems to be little appreciated by the mainstream econometric literature as well as applied researchers. Due to computational and theoretical advancements, the least squares estimation of a state space model has become a viable alternative in many fields, showing great potential in solving otherwise difficult problems. This note gathers and discusses some possible applications to illustrate the point and contribute to their wider use in practice.

A Static Reduced-Order Multiple-Model Adaptive Estimator for Noise Identification

A reduced-order multiple-model (MM) estimator for noise identification in dynamic stochastic systems is developed. The unknown noise statistics are assumed to be static. While a standard static MM estimator does not grow exponentially over time, its computational complexity grows exponentially with the number of modes. The proposed algorithm reduces the computational complexity of MM estimation from exponential to polynomial by constructing a significantly smaller set of mode models which are updated every time step. It is assumed that the constructed mode models do not change significantly between time steps, which in turn holds if the smoothness of the mode probabilities is guaranteed. It is shown that in the case where there is “enough” statistical distinction between the noise modes, the proposed reduced-order MM estimator converges to the standard MM estimator. The proposed reduced-order MM estimator is evaluated using Monte Carlo simulations, showing that it performs nearly similar to the standard MM estimator with a fraction of its complexity. The numerical example shows less than a 2% increase in the root mean-squared errors (RMSEs) of the reduced-order MM estimator from the standard one, while the reduction in the number of filters in the reduced-order MM estimator is 300%. To further validate the proposed filter, experimental results are presented of an unmanned aerial vehicle (UAV) navigating with terrestrial signals of opportunity. Opportunistic navigation serves a relevant application for MM-based estimation, as system parameters, namely the statistics of the clock error dynamics of opportunistic sources, are unknown and must be adaptively estimated. The experimental results show a UAV navigating for more than 5 minutes over a trajectory of more than 3 km, achieving a final position error of 6.21 m obtained using the standard MM estimator versus a final position error of 6.25 m obtained using the proposed reduced-order MM estimator. A standard extended Kalman filter (EKF) was implemented for comparative analysis, showing a final error of 40.03 m. In the experiments, the reduced-order MM estimator was implemented with 16 filters, while the standard MM was implemented with 256 filters.

On Quaternion Modeling and Kalman Filtering from Vector Observations

This paper presents novel results on quaternion state-space modeling and estimation from rate gyro measurements and vector observations. The spectral properties of the Lyapunov differential equation for the quaternion second-order moment are investigated and expressions for its solution are developed. The properties of a four-dimensional rank-degenerate measurement model of the quaternion are analyzed. Four different approaches are suggested in order to reduce the dimension of the measurement equation, hinging on information optimization or on the spectral properties of the measurement matrix. While the measurement dimension’s reduction saves computations, it does not impair the exact statistical treatment of state-dependent process and measurement noises. As a result, the proposed filters are optimal and statistically consistent without tuning. The performance of the maximum information rate quaternion filter is verified via extensive Monte Carlo simulations. The test cases of spinning and rapidly tumbling rotations of a small satellite on a low Earth orbit are simulated. The filters process measurements of the sun vector, of the Earth magnetic field, and of rate gyros. The novel filters converge where a standard extended Kalman filter fails to do so under similar conditions.

Optimal Linear Multilateration Combined With the Kalman Filter for Range-Only Tracking

This paper presents the optimal estimation for the extensively concerned linear multilateral positioning issues and further improves the accuracy of range-only tracking significantly. The traditional linear multilateration method has been used to achieve target positioning for range-only measurements, but the completeness of the theory and implementation effect are limited due to the imperfect model and the lack of proper statistical error analysis. Moreover, the performance of the representative nonlinear filters is not satisfactory because of the strong nonlinearity between the range observations and kinematic states of the moving target. This paper reveals the essence of the linear fusion of multiple distance sensors from a geometric perspective. The multi-sensor fusion model is reconstructed in the linear feature space by transforming the likelihood function into the probability function and separating the expectation of the compound random variable and residual. After that, a new linear multilateration method is developed based on the precise derivation of statistical characteristics corresponding to the linear fusion model, and also minimizing the squared Mahalanobis distance is introduced to replace minimizing the mean square error. The first-order moment, as a pseudo-measurement used to improve the positioning accuracy, and the second-order moment, as the covariance of the pseudo-measurement noise, are provided by this method, which has been proved to be best linear unbiased according to the Gauss–Markov theorem. Combined with the pseudo-measurement, a standard linear Kalman filter is capable of tracking the maneuvering target. Several simulations and experiments verify the superior performance of the proposed method.

State Parameter Estimation of Intelligent Vehicles Based on an Adaptive Unscented Kalman Filter

The premise of vehicle intelligent decision making is to obtain vehicle motion state parameters accurately and in real-time. Several state parameters cannot be measured directly by vehicle sensors, so estimation algorithms based on filtering are effective solutions. The most representative algorithm is the Kalman filter, especially the standard unscented Kalman filter (UKF) that has been widely used in vehicle state estimation because of its superiority in dealing with nonlinear filtering problems. However, although the UKF assumes that the noise statistics of the system are known, due to the complex and changeable operating conditions, sensor aging and other factors, these noises vary. In order to realize high-precision vehicle state estimation, a noise-adaptive UKF algorithm is proposed in this article. The maximum a posteriori (MAP) algorithm is used to dynamically update the noise of the vehicle system, and it is embedded into the update step of the UKF to form an adaptive unscented Kalman filter (AUKF). The system will dynamically update the noise when noise statistics are unknown and prevent filter divergence by adjusting the mean and covariance of the estimated noise to improve accuracy. On this basis, the proposed method is verified by the joint simulation of CarSim and Matlab/Simulink, confirming that the AUKF performs better than the standard UKF in estimation accuracy and stability under different degrees of noise disturbance, and the estimation accuracy for the yaw rate, side slip angle and longitudinal velocity is improved by 20.08%, 40.98% and 89.91%, respectively.

A Descriptor System Approach for the Nonlinear State Estimation of Li-Ion Battery Series/Parallel Arrangements

The nonlinear state estimation for a series/parallel arrangement of lithium-ion (Li-ion) battery cells is addressed in this article assuming limited information. First, a unified model for series/parallel arrangements in terms of a nonlinear descriptor model is proposed in order to account for Kirchhoff’s laws characterizing the interconnection. It relies on a simplified electrochemical model of the individual cells, the so-called equivalent-hydraulic model. Then, assuming that the system and output equation nonlinearities locally satisfy Lipschitz-like constraints, a linear matrix inequality (LMI)-based technique is proposed for designing a nonlinear state observer, which estimates the state of each individual cell (such as the state-of-charge and inner temperature) as well as the algebraic variables (currents or voltages for respectively parallel or series configurations). The proposed design conditions also provide an estimate of the region of guaranteed convergence while ensuring a peak-to-peak performance with respect to exogenous inputs and model uncertainties. The effectiveness of the approach is demonstrated on a detailed battery electrochemical simulator (based on the Doyle–Fuller–Newman model) for a series arrangement of two cells. Comparisons with a standard extended Kalman filter demonstrate the superior performance of the proposed approach.

Modified Kalman Filtering for Stochastic Nonlinear Systems Under Non-Gaussian–Lévy Noise and Cyber Attacks

This article studies the filtering problem for a class of stochastic nonlinear system under non-Gaussian–Lévy noise and cyber attacks, where the denial-of-service (DoS) attacks and the false data-injection (FDI) attacks are both considered. Since the covariance of the Lévy noise is unknown and infinite, the standard Kalman filter fails to estimate system states. By exploiting saturation functions, a modified Kalman filter is proposed, where the extremely large values of the measurement outputs caused by the Lévy noises can be clipped. In the presence of Lévy noise and cyber attacks, an upper bound for the error covariance is guaranteed and can be minimized via designing the filter parameter. Besides, a sufficient condition is provided to guarantee the boundedness of the upper bound, and the convergence analysis of the filtering error is presented. Finally, the simulation results are given to verify the algorithm.