Error Covariance Research Articles

Intelligent navigation for the cruise phase of solar system boundary exploration based on Q-learning EKF

With the continuous advancement of deep space exploration missions, the solar system boundary exploration mission is established as one of the China's most important deep space scientific exploration missions. The mission of the solar system boundary exploration has many challenges such as ultra-remote detection distance, ultra-long operation time, and ultra-long communication delay. Therefore, the problem of high-precision autonomous navigation needs to be solved urgently. This paper designs an autonomous intelligent navigation method based on X-ray pulsars in the cruise phase, which estimate the motion state of the probe in real time. The proposed navigation method employs the Q-learning Extended Kalman filter (QLEKF) to improve navigation accuracy during long periods of self-determining running. The QLEKF selects automatically the error covariance matrix parameter of the process noise and the measurement noise by the reward mechanism of reinforcement learning. Compared to the traditional EKF and AEKF, the QLEKF improves the estimation accuracy of position and velocity. Finally, the simulation result demonstrates the effectiveness and the superiority of the intelligent navigation algorithm based on QLEKF, which can satisfy the high-precision navigation requirements in the cruise phase of the solar system boundary exploration.

Model based optimal magnetic feedback control of multiple resistive wall modes with discrete coil and sensor arrays

A model-based optimal control method for multiple resistive wall mode (RWM) feedback stabilization has been developed and tested in plasma experiments at the EXTRAP T2R reversed field pinch (RFP) device. The controller is designed to target issues that arise in connection with RWM magnetic feedback stabilization systems based on discrete control coil and sensor arrays in tokamak and reversed field pinch devices. Multi-mode control capabilities in these systems is limited due to coupling of modes induced by the control system. The coupling originates from the generation of side-band control field harmonics and from aliasing of multiple harmonics in the sensor measurements. These couplings naturally leads to a multiple-input, multiple-output (MIMO) control problem. A model based state space controller has been designed based on a relatively simple physics model of the RWM plasma response. The physics model, which is applicable for the high aspect ratio RFP, is the basis for Fourier decoupling of the MIMO control problem into a set of single-input, multiple-output (SIMO) systems using the discrete Fourier transform (DFT). The linear, time-invariant physics model allows for the design of a state space model with states representing physical quantities; in this case the Fourier harmonics of the radial field at the resistive wall. Since the states cannot be directly measured, a Kalman filter is used for estimation of the states from the aliased sensor array measurements. A linear–quadratic (LQ) optimal state controller has been implemented. Design parameters, such as the LQ control cost function state weights and the Kalman filter input error covariances have been used to optimize the control operation in various ways. The controller has enhanced multi-mode control capabilities compared to earlier designs. For example it allows the prioritizing of suppression of one of the multiple magnetic field Fourier harmonics produced by a given control current DFT component. The controller has been tested in plasma experiments at EXTRAP T2R device, utilizing a newly installed extended sensor array, and the enhanced capabilities for multiple RWM feedback stabilization has been demonstrated.

Data assimilation for the Model for Prediction Across Scales – Atmosphere with the Joint Effort for Data assimilation Integration (JEDI-MPAS 2.0.0-beta): ensemble of 3D ensemble-variational (En-3DEnVar) assimilations

Abstract. An ensemble of 3D ensemble-variational (En-3DEnVar) data assimilations is demonstrated with the Joint Effort for Data assimilation Integration (JEDI) with the Model for Prediction Across Scales – Atmosphere (MPAS-A) (i.e., JEDI-MPAS). Basic software building blocks are reused from previously presented deterministic 3DEnVar functionality and combined with a formal experimental workflow manager in MPAS-Workflow. En-3DEnVar is used to produce an 80-member ensemble of analyses, which are cycled with ensemble forecasts in a 1-month experiment. The ensemble forecasts approximate a purely flow-dependent background error covariance (BEC) at each analysis time. The En-3DEnVar BECs and prior ensemble-mean forecast errors are compared to those produced by a similar experiment that uses the Data Assimilation Research Testbed (DART) ensemble adjustment Kalman filter (EAKF). The experiment using En-3DEnVar produces a similar ensemble spread to and slightly smaller errors than the EAKF. The ensemble forecasts initialized from En-3DEnVar and EAKF analyses are used as BECs in deterministic cycling 3DEnVar experiments, which are compared to a control experiment that uses 20-member MPAS-A forecasts initialized from Global Ensemble Forecast System (GEFS) initial conditions. The experimental ensembles achieve mostly equivalent or better performance than the off-the-shelf ensemble system in this deterministic cycling setting, although there are many obvious differences in configuration between GEFS and the two MPAS ensemble systems. An additional experiment that uses hybrid 3DEnVar, which combines the En-3DEnVar ensemble BEC with a climatological BEC, increases tropospheric forecast quality compared to the corresponding pure 3DEnVar experiment. The JEDI-MPAS En-3DEnVar is technically working and useful for future research studies. Tuning of observation errors and spread is needed to improve performance, and several algorithmic advancements are needed to improve computational efficiency for larger-scale applications.

Toward a strongly coupled assimilation in the Earth System Prediction Capability system

AbstractWe assess a possibility to efficiently represent the strongly coupled increment in an ocean–atmosphere coupled data assimilation (DA) system by applying an iterative procedure involving uncoupled solvers and the weakly coupled analysis as a first guess approximation to the strongly coupled increment. Using the output of the ensemble‐based weakly coupled DA system, we explore convergence of the approximations to the strongly coupled DA solution by applying the uncoupled solver to a sequence of innovation vectors at various spacetime locations over the global ocean grid. The results demonstrate that, in general, fewer than two iterations are required to approximate the coupled increment in the majority of the locations tested with sufficient (3%) accuracy given the uncertainty of the background error covariance estimated from the limited number of the ensemble members. We assess the impact of data thinning and hybridization of the background error covariance model on the convergence of the iterative approximations to the strongly coupled increment. An empirical relationship between the spectral radius of the expansion matrix and convergence rate is obtained.

Encryption-Decryption-Based State Estimation With Multirate Measurements Against Eavesdroppers: A Recursive Minimum-Variance Approach

This paper is concerned with the secure state estimation problem for a networked system with multi-rate measurements. The measurement data are transmitted to the estimator over a communication network, where an eavesdropper could intercept the transmitted information and generate the state estimates based on the overheard signals. In order to protect the privacy of the system state from information leakage, a novel encryptor is proposed to encrypt the measurement signals into codewords before being transmitted, and a decryptor is then adopted at the user side to decrypt the received codewords into the desired measurement signals, based on which a recursive state estimator is developed whose parameter is calculated by solving several coupled difference equations. Here, the secret key for decryption is available to the user but is unknown to the eavesdropper. The encryptor and the estimator are co-designed such that 1) at the user side, the estimation error covariance has an upper bound that is subsequently minimized in the sense of the matrix trace, where the uniform boundedness of the error covariance is also examined; and 2) at the eavesdropper side, the corresponding state estimation error is divergent. Two simulation examples are provided to demonstrate the effectiveness of our developed secure state estimation scheme.

Distributed Recursive Filtering Over Sensor Networks Under Random Access Protocol: When State Saturation Meets Censored Measurement.

In this article, a new distributed filtering problem is studied for a class of state-saturated time-varying systems over sensor networks under measurement censoring, where the censored measurements are described by the Tobit measurement model. To curb the data collision and ease communication burden, a random access protocol (RAP) is implemented onto the sensor-to-filter channels to orchestrate the transmission sequence of multiple sensor nodes. The purpose of the addressed problem is to construct a state-saturated distributed filter such that upper bounds (on filtering error covariances) are guaranteed and filter parameters are determined to accommodate both measurement censoring and state saturation under the RAP. By means of matrix difference equations, the desired upper bounds are first acquired and later minimized through appropriately designing filter parameters. Particularly, the sparsity issue with respect to the network topology is tackled via the employing certain matrix simplification technique. A simulation example is finally presented to showcase the applicability of the proposed state-saturated distributed filtering algorithm.

Outlier-Resistant Recursive Security Filtering for Multirate Networked Systems Under Fading Measurements and Round-Robin Protocol

This paper investigates the recursive filtering problem with fading measurements and cyber attacks for multisensor multirate networked systems (MRNSs) under Round-Robin protocol (RRP). By exploiting the lifting technique, the sampling periods for both sensors and the state of the system are uniformed. It is assumed that the phenomenon of fading measurements which better describes practical engineering arises stochastically, and the attenuation coefficients of which are described by a set of random variables with known statistical properties. In order to fully utilize the limited communication resources, RRP is introduced in the sensor-to-filter channel. Considering the measurement outliers, a saturation function is adopted in the filter structure to suppress the anomalous innovations, so as to reduce the negative impact of the outliers. By means of matrix difference equation, an upper bound is first obtained on the filtering error covariance, and the filter gain is designed to minimize the obtained upper bound by partial derivation. Moreover, the exponential boundedness of the filtering error dynamics is analyzed in the mean square sense. Finally, a numerical simulation example is given to demonstrate the validity of the proposed recursive security filtering scheme.

On Information Fusion in Optimal Linear FDI Attacks Against Remote State Estimation

This paper studies the problem of false data injection (FDI) attacks against remote state estimation. The scenario that malicious attackers can intercept original data packets and also eavesdrop on some side information of system states with extra sensors is considered. To clarify a counter-intuitive issue in existing work, a different innovation-based linear attack policy fusing all available information is proposed. First, the evolution of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> estimation error covariance with FDI attacks is derived. Then, explicit solutions of optimal stealthy attack coefficients are obtained without solving optimization problems numerically. The condition under which there exist multiple optimal attacks is analyzed. Additionally, an easy-to-check criterion for comparing two information fusion methods in scalar systems is given. Simulation results show that, compared with existing work, the proposed attack strategy can completely deceive the anomaly detector and cause more severe performance degradation in remote state estimation.

Multi-Measurement Kalman-Filtering-Based Neural Network Estimator for SOC of Lithium Batteries

State of charge (SOC) refers to the remaining capacity of the battery, which cannot be measured directly. A multi-measurement Kalman filter which is composed of two sub Kalman filters is constructed to improve the estimation accuracy of SOC. The two sub filters share the same state function but have different measurements, namely the terminal voltage and the SOC estimation from neural network, respectively. Based on minimizing the trace of error covariance, an optimal weighted matrix is computed to fuse the estimates of the two sub filters. The training dataset of neural network is collected from mixed discharging cycles experiment and corresponding charging process. By comparing the results with model-based methods, such as H-infinity filter, unscented Kalman filter, data-driven methods, like neural networks and hybrid method, the multi-measurement Kalman filter is verified by both the root mean square error and mean absolute error that are less than 2% in different drive cycles.

Distributed Anti-Eavesdropping Fusion Estimation Under Energy Constraints

This paper studies the distributed fusion estimation problem with energy-constrained sensors in the presence of eavesdroppers, where smart sensors send their local estimates to a remote fusion center (FC). To enhance privacy level, a novel encryption strategy is proposed by establishing an optimization problem, where the optimization objective is constructed by maximizing a combination of the fusion terminal estimation error covariance and the cost of encryption process in finite time domain. Meanwhile, the established problem is decomposed into several independent sub-optimization problems under a relaxation condition, and a sufficient condition that depends on the system parameters is derived such that the sub-optimization problems have optimal solutions. In this case, an optimal encryption strategy is designed with an analytical solution. Finally, a simulation example is given to show the effectiveness of the proposed methods.

Multi-sensor information fusion localization of rare-earth suspended permanent magnet maglev trains based on adaptive Kalman algorithm.

Since the positioning accuracy of sensors degrades due to noise and environmental interference when a single sensor is used to localize a suspended rare-earth permanent magnetically levitated train, a multi-sensor information fusion method using multiple sensors and self-correcting weighting is proposed for permanent magnetic levitated train localization. A decay memory factor is introduced to reduce the weight of the influence of historical measurement data on the fusion estimation, thus enhancing the robustness of the fusion algorithm. The Kalman filtering results suffer from inaccuracy when process noise is present in the system. In this paper, we use a covariance adaptive scheme that replaces the prediction step of the Kalman filter with covariance. It uses the covariance adaptive scheme to search the posterior sequence online and reconstruct the prior error covariance. Since the process noise covariance is not used in the new adaptive scheme, the negative impact of the mismatch noise statistics is greatly reduced. Simulation and experimental results show that the use of multi-sensor information fusion and covariance adaptive Kalman algorithm has significant advantages in terms of adaptability, accuracy and simplicity.

Research on the development of AloT sensing module for new energy power information collection

Abstract In order to improve the ability of power communication technology to support large-scale and massive services of smart electricity consumption, this paper designs a sensing model to collect information on new energy electricity consumption. Using the arrival rate calculation method of intelligent IoT, the information on new energy electricity users is collected, and the environmental measurement data is collected with the help of intelligent sensors. Monitoring the concurrency factor of all sensors in the operation state, collecting queuing theory packet loss rate through utilization maximization objective function, and setting the average queuing length of communication nodes. On this basis, the voltage amplitude of the electrical energy sampling point is measured, and the Gaussian perceptual loss vector with non-zero mean and heteroskedasticity is performed to iteratively converge the constraint matrix and error covariance. The results show that the maximum relative errors for service delay and packet loss rate are 16.8% and 7.0%, respectively. As bandwidth increases, the queuing delay and packet loss rate decrease. The broadband power line carrier can reach up to 210 Mbps, and the micropower wireless rate can reach 38.5 kbps. The annual cycle characteristics fluctuate widely, but the cumulative error of electricity load prediction results is lower, and the prediction accuracy is high. It shows that the electricity consumption information collection sensing model provides an important reference for building a new communication network with low cost, high speed and reliability for smart electricity consumption.

Recursive filtering for an uncertain system with wireless switching channels and fading measurements: A microseismic event detection method

In this paper, the microseismic event detection problem is analyzed for an array of microseismic systems by developing a novel recursive filtering algorithm. The state-space model of the microseismic system with parameter uncertainty is constructed on the basis of the seismic wavelet and the ambient noise. In view of the actual situation of the wireless transmission, the switching channels and fading measurements are drawn into the microseismic system. The main purpose of the addressed problem is to devise a filter so as to precisely and automatically pick the first arrival time of the microseismic P wave and S wave. In addition, an upper bound of the filtering error covariance matrix is given and then minimized by properly designing the filter parameter. Furthermore, the mean-square exponential boundedness of the filtering error is verified. Finally, an illustrative example is provided to examine the effectiveness of the algorithm on microseismic event detection.

Observation impact metrics in NWP: A theoretical study. Part II: Systems with suboptimal observation errors

AbstractTwo methods are widely used to assess the impact of observations in global numerical weather prediction (NWP): data denial experiments (DDEs) and the forecast sensitivity‐based observation impact (FSOI) method. A DDE measures the impact on forecast accuracy of removing an observation type from the system, whereas FSOI estimates the amount by which an observation type reduces the short‐range forecast error within a system containing all observation types. This article describes the second part of a two‐part study. In the first part, the theory behind DDE and FSOI metrics was presented and then applied to a simple model with two state variables, all in the context of optimal data assimilation (DA), for which the error covariances used in the DA system match reality. The article showed why and under what conditions the DDE and FSOI metrics give different results, even for an optimal DA system. In this second part, we extend the theory to suboptimal systems, and specifically to systems that are suboptimal in their specification of observation errors, and then apply it to a very simple model, in this case with one state variable. As expected, DDE impacts are reduced when the system is suboptimal. By contrast, relative FSOI impacts (i.e. relative to those of other observation types) increase for an observation type for which the errors are underestimated. This gives the erroneous impression that the change in assumed errors has led to an improvement, whereas the opposite is the case. These results provide some insights into the interpretation of FSOI results from a suboptimal DA system.

A distributed diffusion Kalman filter with event‐triggered mechanism and guaranteed stability

AbstractIn this article, a distributed diffusion Kalman filtering algorithm with event‐triggered communication (DDKF‐E) is studied for discrete‐time nonlinear systems. According to the event‐triggered communication protocol, the data between sensors and estimators are transmitted only when the predefined conditions are satisfied. Considering the characteristic of event‐triggered method and truncated error by linearization, an upper bound of the estimation error covariance matrix is obtained by using the variance‐constrained method. The Kalman gain is designed to minimize the upper bound and then two Riccati equations are obtained. Furthermore, the stochastic stability theory is used to prove the stability of DDKF‐E, and it is derived that the estimation error of DDKF‐E is exponentially bounded in mean square. Finally, numerical simulations validate the effectiveness of the DDKF‐E algorithm.

ТОЧНІСТЬ КООРДИНАТ ВИЗНАЧЕННЯ МІСЦЯ СУДНА, РОЗРАХОВАНИХ МЕТОДОМ НАЙМЕНШИХ КВАДРАТІВ, ПРИ НАЯВНОСТІ НАДЛИШКОВИХ ЛІНІЙ ПОЛОЖЕННЯ

This article discusses the accuracy of the vessel’s position coordinates, calculated using the least squares method, in the presence of redundant line positions (LP) over time, assuming the LP error is normally distributed. It is stated that for a normal error distribution, the least squares method is the maximum likelihood method, thus the norm of the positional error covariance matrix is minimized, ensuring the highest accuracy of the vessel’s position coordinates. Field observation results are presented, where experimental data were obtained, demonstrating the possibility of altering navigational parameters. Following the adjustment of navigational parameters, a series of tests were conducted, exceeding 100 in total. The analysis of these results indicates that deviations in navigational parameterization, achieved over a limited time interval, conform to the normal law of variability distribution. Adjustments to the parameters over a larger interval are subject to changes in the laws of change, with the degree of power of some of the normal law proportional to the interval of the series value of the navigational parameter. It is demonstrated that the efficiency of observing the vessel’s coordinates, calculated by the least squares method during times of redundant LP, the errors of which can be calculated according to different laws, is less than one. To evaluate the efficiency of coordinate observation derived from overly large LP and gaps using the least squares method, a computer simulation was performed. During this simulation, efficiency values were compared with the highest theoretically achievable efficiency values, as outlined in the article, and the difference between them in the case of high-technological parameters was less than 22.0%. The simulations conducted showed good agreement between the efficiency assessments derived from analytical expressions and those obtained from the simulations, thereby confirming the validity of the analytical method for evaluating the efficiency of observed coordinates, calculated by the least squares method. Keywords: navigational safety, accuracy of coordinates assignment, change of law and subdivision, simulation modelling.

Formulation and use of 3D‐hybrid and 4D‐hybrid ensemble covariances in the Météo‐France global data assimilation system

AbstractThe global data assimilation (DA) system at Météo‐France is currently based on a 4D‐Var formulation relying on wavelet‐based 3D background‐error covariances. These covariances are specified at the beginning of the DA window and are evolved implicitly in the DA window through tangent linear and adjoint model integrations. Further research and development steps on data assimilation at Météo‐France are conducted in the framework of the Object‐Oriented Prediction System (OOPS), which is developed in collaboration with the European Centre for Medium‐Range Weather Forecasts (ECMWF). For instance, 3D background‐error covariances can be made hybrid through a linear combination between wavelet‐based covariances and ensemble‐based covariances that are filtered through spatial localisation. This allows covariances to be made more anisotropic in a flow‐dependent way, and implementation of this hybridation in the OOPS framework is shown to have general positive impacts on the forecast quality. This 3D‐hybrid approach can also be extended to a 4D‐hybrid approach in the OOPS framework: this relies on a linear combination between 4D ensemble covariances on the one hand and 4D linearly propagated covariances on the other hand, corresponding to initial covariances that are evolved more explicitly by tangent linear and adjoint versions of the model. This provides a unifying framework for implementations of DA schemes that correspond to 4DEnVar, 4D‐Var, and new 4D‐hybrid formulations. This is thus considered as a novel way to combine the respective attractive features of 4D‐Var and 4DEnVar approaches, leading in particular to a new 4D‐hybrid formulation of 4DEnVar. Its properties and implementation in the OOPS framework are presented, and first experimental results show that this new formulation of 4DEnVar is competitive with 4D‐Var, in relation with the improved hybridisation.

Robust estimation for function-on-scalar regression models

For the functional linear models in which the dependent variable is functional and the predictors are scalar, robust regularization for simultaneous variable selection and regression parameter estimation is an important yet challenging issue. In this paper, we propose two types of regularized robust estimation methods. The first estimator adopts the ideas of reproducing kernel Hilbert space, least absolute deviation and group Lasso techniques. Based on the first method, the second estimator applies the pre-whitening technique and estimates the error covariance function by using functional principal component analysis. Simulation studies are conducted to examine the performance of the proposed methods in small sample sizes. The method is also applied to the Canadian weather data set, which consists of the daily average temperature and precipitation observed by 35 meteorological stations across Canada from 1960 to 1994. Numerical simulations and real data analysis show a good performance of the proposed robust methods for function-on-scalar models.

Component-wise iterative ensemble Kalman inversion for static Bayesian models with unknown measurement error covariance

The ensemble Kalman filter (EnKF) is a Monte Carlo approximation of the Kalman filter for high dimensional linear Gaussian state space models. EnKF methods have also been developed for parameter inference of static Bayesian models with a Gaussian likelihood, in a way that is analogous to likelihood tempering sequential Monte Carlo (SMC). These methods are commonly referred to as ensemble Kalman inversion (EKI). Unlike SMC, the inference from EKI is asymptotically biased if the likelihood is non-linear and/or non-Gaussian and if the priors are non-Gaussian. However, it is significantly faster to run. Currently, a large limitation of EKI methods is that the covariance of the measurement error is assumed to be fully known. We develop a new method, which we call component-wise iterative EKI (CW-IEKI), that allows elements of the covariance matrix to be inferred alongside the model parameters at negligible extra cost. This novel method is compared to SMC on a linear Gaussian example as well as four examples with non-linear dynamics (i.e. non-linear function of the model parameters). The non-linear examples include a set of population models applied to synthetic data, a model of nitrogen mineralisation in soil that is based on the Agricultural Production Systems Simulator, a model predicting seagrass decline due to stress from water temperature and light, and a model predicting coral calcification rates. On our examples, we find that CW-IEKI has relatively similar predictive performance to SMC, albeit with greater uncertainty, and it has a significantly faster run time.

On the empirical validity of “Gendered reactions to terrorist attacks can cause slumps not bumps” (Holman et al. 2022)

AbstractM. R. Holman, J. L. Merolla and A. Zechmeister (2022) propose women (compared to men) political leaders experience significant drops in public approval ratings after a transnational terrorist attack. After documenting how survey‐based evaluations of then‐Prime Minister Theresa May suffered after the 2017 Manchester Arena attack, Holman et al. (2022) assemble a country–quarter level panel database to explore the generality of their hypothesis. They report evidence suggesting women (compared to men) leaders systematically experience decreased public approval rates after major transnational terrorist attacks (‐value of 0.020). We find that result disappears once any of the following adjustments is implemented: (i) excluding election quarter covariates (), (ii) correcting objective coding errors in the election quarter covariates (), (iii) excluding the May–Manchester observation () or (iv) clustering standard errors at the country level (). Exploring all combinations of the five control groups Holman et al. (2022) incorporate in their specification, none of them clears the 5% threshold of statistical significance once the corrected election quarter variables are employed. We conclude that the empirical evidence does not provide sufficient support for Holman et al.'s (2022) abstract claim that “conventional theory on rally events requires revision: women leaders cannot count on rallies following major terrorist attacks.”