Error Covariance Matrix Research Articles

An Integrated Navigation Method Based on an Adaptive Federal Kalman Filter for a Rice Transplanter

HighlightsAn integrated GPS/INS/VNS navigation system was developed to improve navigation accuracy.An adaptive federal Kalman filter with information distribution factors was used to fuse navigation information.Detection of seedling row lines was achieved based on subregional feature points clustering.A modified rice transplanter was developed as an experimental platform for automatic navigation.Abstract. In this article, an integrated global positioning system (GPS), inertial navigation system (INS), and visual navigation system (VNS) navigation method based on an adaptive federal Kalman filter (KF) is presented to improve positioning accuracy for a rice transplanter operating in a paddy field. The proposed method used GPS/VNS to aid the INS and reduce the influence of the accumulated error of the INS on navigation accuracy. An adaptive federal KF algorithm was designed to fuse navigation information from different sensors. The information distribution factor of each local filter was obtained adaptively on the basis of its own error covariance matrix. Computer simulation and transplanter tests were conducted to verify the proposed method. Results showed that the proposed method provided accurate and reliable navigation information outputs and achieved better navigation performance compared with single GPS navigation and an integrated method based on a conventional federal KF. Keywords: Federal Kalman filter, GPS/INS/VNS, Information distribution factor, Information fusion, Integrated navigation.

The Optimal Distributed Kalman Filtering Fusion With Linear Equality Constraint

In this paper, the optimal distributed Kalman filtering fusion with linear equality constraint (LEC) is proposed. When the Kalman filter subject to LEC is applied in state estimation of distributed linear dynamic system, all local error covariance matrices are singular, which is a challenge for most of the existing distributed fusion algorithms. However, the new fusion algorithm is rigorously derived with matrix analysis technique on the Moore-Penrose inverse. Updating state estimate and error covariance matrix, the new fusion algorithm only uses local estimates and their corresponding error covariances. We have proved theoretically that the distributed Kalman filtering with LEC (DKF-LEC) just using all local measurements is equivalent to the centralized Kalman filtering with LEC (CKF-LEC), which means it is optimal. Furthermore, we also prove the optimality of the proposed fusion formula for Kalman filtering with LEC and feedback. In addition, the simulation results show that the proposed fusion formula for Kalman filtering with LEC have the same performance of the centralized Kalman filtering with LEC whether considering feedback or not. It is consistent with the theoretical result that the DKF-LEC is equivalent to CKF-LEC as the estimate of CKF-LEC is disassembled into local estimates in distributed system. And the performance of the new fusion formula for Kalman filtering with LEC and feedback is obviously better than those without feedback.

State of Charge Estimation of Lithium-Ion Batteries Based on Cubature Kalman Filters with Different Matrix Decomposition Strategies

State of charge (SOC) is one key status to indicate the safety and reliability of lithium-ion batteries in electric vehicles (EVs). Cubature Kalman filter (CKF) is an intensively considered model-based approach for SOC estimation because of its merits in accuracy, convergence rate, and robustness compared with other Kalman filter methods. Nevertheless, CKF suffers from a non-positive definite error covariance matrix because of abnormal perturbations, inaccurate initial values and limited computer word length, causing the divergence of the CKF and the failure of SOC estimation. To address this issue, this paper introduces three typical matrix decomposition strategies, namely, singular value decomposition (SVD), UR decomposition and LU decomposition, to replace the Cholesky decomposition in the traditional CKF. The second-order RC equivalent circuit model is utilized to simulate the dynamics of a lithium-ion battery. The theoretical errors of the methods are formulated by F-norms. The results indicate that three matrix decomposition strategies can overcome the problem of a non-positive definite error covariance matrix and improve the convergence rate of the CKF. In particular, the UR decomposition exhibits the best comprehensive performance because it has a moderate convergence rate and computational cost, and it is robust against the initial error covariance matrix.

An adaptive multi-sensor resource allocation algorithm based on efficiency function

In this paper, combined with the Kalman filtering algorithm, the linear weighted sum of the normalized prediction information increment DI and the norm of target filter error covariance matrix ║ cP ║ is used to represent the pairing function of the target-sensor combination and the assignment and automatic updating of the pairing function between sensor and target are realized. Based on the analytic hierarchy process (AHP), an adaptive sensor resource allocation algorithm is proposed. The simulation results show that the algorithm can effectively integrate various evaluation factors and is compatible with different types of targets. The algorithm has simple structure and good versatility. The limited sensor resources can be dynamically adjusted according to the priority of targets, making the sensor allocation strategy more reasonable and effective.

Prediction based Event Triggered Estimation for Networked Systems

This work focuses on development of prediction-based event-triggered estimator for the networked system (NS) with packet delay and packet loss in sensor to estimator network channel. An event-based triggering mechanism limits the number of data transmitted through the network channel. The stochastic model of the sensor to estimator channel is developed using Bernoulli distributed random variables. The one-step predicted measurement is used by an estimator when the data packet is lost or when the event is not triggered. An augmented NS model with the one-step ahead predicted information is developed and the recursive estimator is designed using the orthogonal projection principle. The steady-state property of the proposed event-triggered based estimator is analyzed by showing the trace of prediction error covariance matrix that converges to a unique positive semi-definite value with the given initial conditions. The designed estimator is validated on a mass-spring-damper system model. It is inferred from the simulation results that the performance of the proposed estimator is satisfactory.

Алгоритм полиномиальной фильтрации в задачах обработки навигационной информации при квадратичных нелинейностях в уравнениях динамики и измерений. Часть I. Описание и сопоставление с алгоритмами калмановского типа

The paper considers the filtering problems solved in navigation data processing under quadratic nonlinearities both in system and measurement equations. A Kalman type recursive algorithm is proposed, where the predicted estimate and gain at each step are calculated based on the assumption on the Gaussian posterior proba-bility density function of the estimated vector at the previous step and minimization of estimation error covariance matrix using a linear procedure with respect to the current measurement. The similarities between this algorithm and other Kalman type algorithms such as extended and secondorder Kalman filters are discussed. The procedure for estimating the performance and comparing the algorithms is presented.

State estimation via prediction-based scheme for linear time-varying uncertain networks with communication transmission delays and stochastic coupling

In this paper, the design problem of recursive state estimation algorithm based on delay-prediction compensation is considered for a class of linear time-varying uncertain dynamical networks with network-induced communication transmission delays and stochastic coupling phenomenon. The random variables obeying Bernoulli distribution under uncertain occurrence probabilities are selected to depict the stochastic coupling phenomenon. Moreover, the modelling uncertainties and communication transmission delays are taken into account. Afterwards, a feasible prediction equation is constructed to compensate the communication transmission delays, and an updated time-varying state estimator is designed. Subsequently, the upper bound expression of the state estimation error covariance matrix is given by using the stochastic analysis technique and matrix theory, and the gain matrix of time-varying state estimator is obtained via minimizing the trace of such an upper bound. Furthermore, we present the mathematical proof which illustrates the monotonic relationship between the occurrence probabilities of stochastic coupling and the upper bound of error covariance. Finally, a simulation example is utilized to demonstrate the effectiveness of the proposed estimation strategy based on the delay prediction idea.

Planar array capacitance imaging based on adaptive Kalman filter

Planar array capacitance imaging system has the characteristics of uneven distribution of sensitive field, serious ill posed problem and measurement data vulnerable to external interference, and these characteristics will make the image artifacts particularly serious, affect the quality of the reconstructed image, and even determine the number of defects with difficulty. In order to solve the problem that the edge electric field and ill conditioned characteristics of planar array electrode seriously affect the quality of capacitance image reconstruction, an improved image reconstruction algorithm based on adaptive Kalman filter is proposed to reduce the noise of capacitance data and dielectric constant matrix. On the basis of constructing the state model of planar array capacitance imaging with noise, the maximum likelihood criterion is used to estimate and modify the noise variance matrix of dielectric constant matrix on-line, and the noise variance matrix of dielectric constant matrix is modified in real time. In order to restrain the filtering divergence and accelerate the convergence speed, different weighting coefficients are provided for the error covariance matrix with time going by. Through designing four kinds of samples from simple to complex structure, the defect detection experiment of composite structure is carried out. The experimental results show that compared with linear back projection (LBP), Tikhonov regularization (TR) algorithm and Kalman filtering algorithm, the image error of adaptive Kalman filtering algorithm can be reduced by about 20%, the image correlation coefficient is as high as 0.79 and the convergence speed can be improved by about 15%, the image artifacts of the four samples are greatly reduced. The experimental data show that the proposed adaptive Kalman filter image reconstruction algorithm can effectively reduce the noise of capacitance and permittivity matrix, enhance the stability of planar array capacitance imaging, and reduce the image error, so that the quality of the image can be significantly improved. This study provides a strong technical basis for improving the quantization accuracy of planar array capacitance imaging detection. In the future, we will further consider the image reconstruction under the condition of complex object field.

Move a Tropical Cyclone with 4D-Var and Vortex Dynamical Initialization in WRF Model

Previous studies showed that 4D-Var technique used for data assimilation could be modified for weather control. This study demonstrates the ability of 4D-Var to influence the future path of a tropical cyclone by calculating perturbations in WRF simulation. Given the background error covariance matrix, the initial field is improved by the vortex dynamic initialization technique. Our results show that 4D-Var can be applied to control the trajectory of simulated tropical cyclones by producing "optimal" perturbations. In the numerical simulation experiment of Typhoon Mitag in 2019, after this kind of weather control similar to data assimilation, the tropical cyclone moved obviously, and the damaging wind over the coastline weakened. The prediction results after the initial field modified by 4D-Var have a great change, and the position of the tropical cyclone moved about 0.5° southeastward after assimilation, which misses the southeast coast of China. Moreover, the damaging wind is also weakened. Since the 4D-Var is premised on the assumption that the model is perfect and does not consider the model error, then the research plan to consider model error and introduce new methods is discussed in the paper.

Parameter identification and state-of-charge prediction of decommissioned lithium batteries

Aiming at the problem that different temperatures and working modes affect the parameter identification and state of charge (SOC) estimation of decommissioned lithium batteries, a new method based on the second-order RC equivalent circuit model combined with the recursive least square method (RLS) is proposed to introduce the forgetting factor, and combined with the extended Kalman filter algorithm (EKF) to realize the method of online parameter identification of decommissioned lithium batteries and the optimal estimation of SOC. In order to solve the problem of obtaining the optimal solution of the error covariance matrix and the measurement noise covariance matrix in EKF, the particle swarm optimization algorithm (PSO) is used to optimize online to further improve the SOC prediction accuracy. The results show that the joint optimization algorithm can accurately identify the parameters and SOC values of retired lithium batteries, which is helpful to realize the echelon utilization of retired lithium batteries.

A Full-State Robust Extended Kalman Filter for Orientation Tracking During Long-Duration Dynamic Tasks Using Magnetic and Inertial Measurement Units.

Accurate and robust orientation estimation using magnetic and inertial measurement units (MIMUs) has been a challenge for many years in long-duration measurements of joint angles and pedestrian dead-reckoning systems and has limited several real-world applications of MIMUs. Thus, this research aimed at developing a full-state Robust Extended Kalman Filter (REKF) for accurate and robust orientation tracking with MIMUs, particularly during long-duration dynamic tasks. First, we structured a novel EKF by including the orientation quaternion, non-gravitational acceleration, gyroscope bias, and magnetic disturbance in the state vector. Next, the a posteriori error covariance matrix equation was modified to build a REKF. We compared the accuracy and robustness of our proposed REKF with four filters from the literature using optimal filter gains. We measured the thigh, shank, and foot orientation of nine participants while performing short- and long-duration tasks using MIMUs and a camera motion-capture system. REKF outperformed the filters from literature significantly (p < 0.05) in terms of accuracy and robustness for long-duration tasks. For example, for foot MIMU, the median RMSE of (roll, pitch, yaw) were (6.5, 5.5, 7.8) and (22.8, 23.9, 25) deg for REKF and the best filter from the literature, respectively. For short-duration trials, REKF achieved significantly (p < 0.05) better or similar performance compared to the literature. We concluded that including non-gravitational acceleration, gyroscope bias, and magnetic disturbance in the state vector, as well as using a robust filter structure, is required for accurate and robust orientation tracking, at least in long-duration tasks.

Position Report Enhancement Using Bayesian Estimator

In this article, a Bayesian estimator for a target position report is proposed. It is based on a maximum a priori algorithm, where the user's device knowledge about its location is used to deduce the prior probability density function. The algorithm does not require knowledge about the signal and noise levels, meaning, noninformative priors are used. It is well known that with a higher number of antenna elements in an array, narrower beams can be formed. In beamforming, task narrow beams are useful for serving many users simultaneously, whereas in direction-of-arrival (DOA) estimation, we are not interested in narrow beams as such; instead, estimation accuracy is important. So, reducing the number of antenna elements used for DOA estimation is beneficial from a system complexity point of view. In this article, signal source location report is used to enhance the estimated DOA, for the task, MAP estimator is developed. We will show that in the case of small array size and large array covariance matrix error values, the proposed estimator is the only one capable of improving the prior knowledge about the transmitter when comparing it with other popular algorithms, such as the maximum likelihood estimator and Root MUSIC. The algorithm could be used in a variety of different scenarios, but its advantages emerge in the case of complex signal propagation environments, such as urban canyons and large airports.

A Quadratic Interpolation-Based Variational Bayesian Algorithm for Measurement Information Lost in Underwater Navigation

The main challenges of sequential estimations of underwater navigation applications are the internal/external measurement noise and the missing measurement situations. A quadratic interpolation-based variational Bayesian filter (QIVBF) is proposed to solve the underwater navigation problem of measurement information missing or insufficiency. The quadratic interpolation is used to improve the observed vector for the precision and stability of sequential estimations when the environment is changed or the measurement information is lost. The state vector, the predicted error covariance matrix, and the measurement noise matrix are derived based on the variational Bayesian method. Simulation results demonstrate the superiority of the proposed QIVBF compared with the traditional algorithm under the condition of measurement information lost by autonomous underwater vehicles.

Robust Vision-Aided Inertial Navigation System for Protection Against Ego-Motion Uncertainty of Unmanned Ground Vehicle

The vision-aided inertial navigation system is of vital importance and great urgency to guarantee accurate and reliable navigation information for unmanned ground vehicles (UGV) in Global Navigation Satellite System-challenged/denied environments. Yet, it is almost inevitable to occur unpredictable stochastic disturbances caused by ego-motion uncertainty due to uneven pavements, unstructured and dynamic scenes, which may lead to the divergence of estimation algorithms and the unreliability of positioning. In this article, we aim at developing a robust strategy for vision-aided inertial navigation to resist uncertainties, and modifying unscented Kalman filter by formulating a novel adaptive factor according to the degree of abnormity (DoA). Applying the robust strategy and algorithms, a monocular camera and an inertial measurement unit (IMU) are tightly fused to construct a robust vision-aided inertial navigation system. During the fusion process of integrated IMU and camera, the new method can automatically estimate the DoA of UGV ego-motion uncertainty online, considering the relationship between the factual and estimated innovation covariances. Thus, we can adaptively adjust the estimation error covariance matrix and the Kalman gain for different scenarios on the basis of DoA levels. As the modification is made in the framework of Kalman filter, the real-time performance, robustness, and high-accuracy positioning can be achieved using the proposed adjustment strategy. Experiments with the KITTI dataset are carried out under different road conditions. The experimental results show that the robust vision-aided inertial navigation system possesses good robustness and self-adaptation for protection against ego-motion uncertainty, and can improve the positioning accuracy of UGV.

A Data-Driven Method for Hybrid Data Assimilation with Multilayer Perceptron

Abstract Accurate and timely weather prediction is of significance for autonomous vehicles, such as designing more appropriate sensors or other configurations and developing safer driving strategies. Generally, as the mainstream weather prediction method, numerical weather prediction (NWP) relies on high-quality spatio-temporal observations. However, the precise state of the real world is not measurable. Thus, how to obtain a proper initial condition estimation based on big geospatial-temporal data is a crucial procedure for NWP. Data assimilation (DA) has been a traditional solution to the problem, for the better performance of which various mathematical-physics models have been used. However, the computational effectiveness and efficiency are still largely compromised by the complicated and nonparallel integration process in existing DA methods. In this paper, we propose a novel data-driven method named HDA-MLP to address the DA problem. We first constructed a customized MLP by introducing the temporal peculiarities of the state variables to simulate and optimize pure 3DVar and EnKF. Then we blended the optimized analysis fields directly by implicitly updating the background error covariance matrix through another neural network model to alleviate the dependence on traditional DA methods. We conducted extensive experiments to investigate the effectiveness and efficiency of the proposal by utilizing two classical nonlinear dynamic models. Results reveal that our approach has better robustness and enhanced capability to capture the variation of state variables. Notably, the analysis quality and computational efficiency are significantly improved.

Macroeconomic effects of the ECB’S forward guidance

This paper evaluates the macroeconomic effects of the European Central Bank’s (ECB) forward guidance (FG) in the euro area and analyses its interaction with asset purchases. To that end, we employ a battery of structural vector autoregressions (VARs) with both constant and time-varying parameters and/or the error covariance matrix to explore the propagation of the FG shock over time and account for the changing nature of the ECB’s FG. The FG shock is identified via traditional sign and zero restrictions of Arias et al. (Econometrica 86(2): 685–720, 2018), narrative sign restrictions of Antolin-Diaz and Rubio-Ramirez (Am Econ Rev 108(10):2802–2829, 2018) and high-frequency information approach akin to Andrade and Ferroni (J Monet Econ, 2020) and Jarocinski and Karadi (Am Econ J Macroecon 12(2):1–43, 2020). We find that the ECB’s forward guidance on interest rates has been an effective policy tool as its announcement causing a 5 bps drop in interest rate expectations increases output by 0.09–0.12% and the price level by 0.035%. In addition, multiple evidence suggests that the introduction of the Asset Purchase Programme (APP) in 2015 considerably enhanced the credibility of FG. Regarding the transmission mechanism, we find that FG significantly lowered uncertainty in the euro area as well as borrowing costs for both households and firms.

Time Difference of Arrival Passive Localization Sensor Selection Method Based on Tabu Search.

This paper proposes a time difference of arrival (TDOA) passive positioning sensor selection method based on tabu search to balance the relationship between the positioning accuracy of the sensor network and system consumption. First, the passive time difference positioning model, taking into account the sensor position errors, is considered. Then, an approximate closed-form constrained total least-squares (CTLS) solution and a covariance matrix of the positioning error are provided. By introducing a Boolean selection vector, the sensor selection problem is transformed into an optimization problem that minimizes the trace of the positioning error covariance matrix. Thereafter, the tabu search method is employed to solve the transformed sensor selection problem. The simulation results show that the performance of the proposed sensor optimization method considerably approximates that of the exhaustive search method. Moreover, it can significantly reduce the running time and improve the timeliness of the algorithm.

A Variational Method for Analyzing Vortex Flows in Radar-Scanned Tornadic Mesocyclones. Part I: Formulations and Theoretical Considerations

AbstractA variational method is formulated with theoretical considerations for analyzing vortex flows in Doppler radar–scanned tornadic mesocyclones. The method has the following features. (i) The vortex center axis (estimated as a continuous function of time and height in the four-dimensional space) is used as the vertical coordinate, so the coordinate system used for the analysis is slantwise curvilinear and nonorthogonal in general. (ii) The vortex flow (VF), defined by the three-dimensional vector wind minus the horizontal moving velocity of vortex center axis, is expressed in terms of the covariant basis vectors (tangent to the coordinate curves), so its axisymmetric part can be properly defined in that slantwise-curvilinear coordinate system. (iii) To satisfy the mass continuity automatically, the axisymmetric part is expressed by the scalar fields of azimuthally averaged tangential velocity and cylindrical streamfunction and the remaining asymmetric part is expressed by the scalar fields of streamfunction and vertically integrated velocity potential. (iv) VF-dependent covariance functions are formulated for these scalar variables and then deconvoluted to construct the square root of background error covariance matrix analytically with the latter used to transform the control vector to precondition the cost function. (v) The deconvoluted covariance functions and their transformed control variables satisfy two required boundary conditions (i.e., zero vertical velocity at the lower rigid boundary and zero cross-axis velocity along the vortex center axis), so the analyzed VF satisfies not only the mass continuity but also the two boundary conditions automatically.

Feasible generalized least squares for panel data with cross-sectional and serial correlations

This paper considers generalized least squares (GLS) estimation for linear panel data models. By estimating the large error covariance matrix consistently, the proposed feasible GLS estimator is more efficient than the ordinary least squares in the presence of heteroskedasticity, serial and cross-sectional correlations. The covariance matrix used for the feasible GLS is estimated via the banding and thresholding method. We establish the limiting distribution of the proposed estimator. A Monte Carlo study is considered. The proposed method is applied to an empirical application.

A 1D Var Retrieval of Relative Humidity Using the ERA5 Dataset for the Assimilation of Raman Lidar Measurements

AbstractWe present a one-dimensional variational (1D Var) retrieval of fifth-generation European Centre for Medium-Range Weather Forecast reanalysis (ERA5) temperature and relative humidity profiles above Payerne, Switzerland, assimilating raw backscatter measurements from the MeteoSwiss Raman Lidar for Meteorological Observations (RALMO). Our reanalysis is called ERA5-reRH. We use an optimal estimation method to perform the 1D Var data retrieval. The forward model combines the Raman lidar equation with the Hyland and Wexler expression for water vapor saturation pressure. The error covariance matrix of ERA5 was derived from the differences between ERA5 and a set of 50 special radiosoundings that have not been assimilated into ERA5. We validate ERA5-reRH, ERA5, and RALMO temperature and relative humidity profiles against the same set of special radiosoundings and found the best agreement was with our reanalysis, with a bias of less than 2% relative humidity with respect to water (%RHw) and a spread of less than 8%RHw below 8 km in terms of relative humidity. Improvements for temperature in our reanalysis are only found in the boundary layer, as ERA5 assimilates a large number of upper-air temperature observations. Our retrieval also provides a full uncertainty budget of the reanalyzed temperature and relative humidity including both random and systematic uncertainties.