Noise Covariance Research Articles

Lithium battery state of charge estimation based on improved variable forgetting factor recursive least squares method and adaptive Kalman filter joint algorithm

As one of the core functions of the battery management system, battery state of charge (SOC) estimation is crucial to battery life and safety. As the traditional recursive least squares method cannot adapt well to complex variable current conditions, this paper proposes a SOC estimation method that combines the improved variable forgetting factor recursive least squares (IVFFRLS) and the adaptive extended Kalman filter (AEKF). First, IVFFRLS removes the rounding operation of the original algorithm to improve the sensitivity to errors and uses a genetic algorithm to optimize the upper and lower bounds of the variable forgetting factor and the sensitivity coefficient to improve the accuracy of the VFFRLS algorithm. Meanwhile, AEKF introduces a noise covariance adaptive update link based on the covariance matching method to correct the system's process noise Q and measurement noise R in real time to improve the SOC estimation accuracy. Based on the second-order RC equivalent circuit model, the method proposed in this paper has good accuracy and robustness under four different working conditions. The experimental results show that compared with traditional methods, the proposed IVFFRLS-AEKF algorithm has higher SOC estimation accuracy and better robustness.

Robust Direction-of-Arrival Estimation in the Presence of Outliers and Noise Nonuniformity

In direction-of-arrival (DOA) estimation with sensor arrays, the background noise is usually modeled to be uncorrelated uniform white noise, such that the related algorithms can be greatly simplified by making use of the property of the noise covariance matrix being a diagonal matrix with identical diagonal entries. However, this model can be easily violated by the nonuniformity of sensor noise and the presence of outliers that may arise from unexpected impulsive noise. To tackle this problem, we first introduce an exploratory factor analysis (EFA) model for DOA estimation in nonuniform noise. Then, to deal with the outliers, a generalized extreme Studentized deviate (ESD) test is applied for outlier detection and trimming. Based on the trimmed data matrix, a modified EFA model, which belongs to weighted least-squares (WLS) fitting problems, is presented. Furthermore, a monotonic convergent iterative reweighted least-squares (IRLS) algorithm, called the iterative majorization approach, is introduced to solve the WLS problem. Simulation results show that the proposed algorithm offers improved robustness against nonuniform noise and observation outliers over traditional algorithms.

Orbit determination for spacecraft with continuous low-thrust using nonsingular Thrust-Fourier-Coefficients and filtering-through approach

Orbit determination using optical survey measurements for cataloging spacecraft encounters new obstacles from mega-constellations. These challenges arise from the sparse observations, and the frequent occurrences of unknown maneuvers which are difficult to model using traditional methods. By utilizing 14 Fourier coefficients, the Thrust-Fourier-Coefficients (TFC) model can effectively represent the averaged variations of orbital elements caused by arbitrary maneuvers. However, this model contains systematic offsets that will reduce the accuracy of orbit determination, and is unsuitable for spacecraft in nearly circular orbits. This paper first replaces the classical Keplerian elements of the previous TFC model with the nonsingular elements. Then detailed analytical expressions for model offsets have been derived based on Kozai’s orbit-averaging method. A process noise covariance matrix related to the model offsets has been added to the iterated extended Kalman filter to compensate for the unmodeled dynamics and enhance the accuracy. Simulations based on this modified method demonstrate that the accuracy of orbit determination for maneuvering satellites can meet the requirements of catalog maintenance even with sparse observations.

Online numerical simulation method based on adaptive unscented Kalman filter for shear wall structures

Hybrid test method is an effective approach to evaluate the seismic performance of engineering structures. Particularly, online numerical simulation method (ONSM) based on unscented Kalman filter (UKF) model updating significantly improves the accuracy of the hybrid test results. However, the sensitivity of UKF to the initial values of the state vector, state covariance, and observation noise covariance may lead to the distortion of hybrid test results. To address these issues, one novel ONSM based on adaptive UKF (ONSM-AUKF) is proposed in this paper. The AUKF algorithm incorporates an adaptive variance module to maintain equilibrium. This not only reduces the sensitivity of UKF to the initial parameter settings, prevents filter divergence, but also enhances the accuracy and stability of estimation. The proposed ONSM-AUKF accurately identifies the constitutive model parameters by utilizing measured data from specimens through the AUKF algorithm, which improves the accuracy and the stability of parameter estimation. The simulation results indicate that AUKF algorithm has the capability to decrease the sensitivity of initial parameter settings, thus enhancing the stability and robustness of parameter estimation in comparison to the UKF algorithm. The experimental results validate the feasibility and effectiveness of the proposed ONSM-AUKF, as well as the advantages of the ONSM-AUKF over the ONSM-UKF. The results indicate that the proposed ONSM-AUKF may have broad application prospect in evaluating the seismic performance of various of complex engineering structures.

Examining the influence of 2.5-D ultra-low velocity zone morphology on ScP waveforms and estimated elastic parameters

Summary Ultra-low velocity zones (ULVZs) have been identified as regions of extremely low velocity anomalies in the Earth's lowermost mantle using seismic observations from reflected, refracted, and diffracted arrivals along the mantle side of the core-mantle boundary (CMB). Estimation of ULVZ geometrical (i.e., shape and size) and elastic (i.e., velocity and density) parameters with uncertainties is crucial in understanding the role of ULVZs in the ongoing dynamic processes within the Earth's mantle; however, these parameters are still poorly known due to uncertainties and tradeoffs of the seismically resolved ULVZ geometries and elastic parameters. Computation of synthetic waveforms for 2-D and 3-D ULVZs shapes is currently computationally feasible, but past studies utilize higher dimensional waveform modeling of mostly only low-frequency diffracted waves. Most studies focusing on high-frequency core-reflected waveforms (e.g., ScP) still use 1-D modeling approaches to determine ULVZ properties. This approach might lead to wrong results if the imaged structures have inherently 3-D geometries. This study investigates high-frequency synthetic ScP waveforms for various 2.5-D ULVZ geometries showing that additional seismic arrivals are generated even when the ScP geometrical ray path does not directly strike the location of the ULVZ. The largest amplitude additional phases in the 2.5-D models are post-cursor arrivals that are generated at the edges of the finite-length ULVZs. These newly identified ScP post-cursors can arrive within the ScsP post-cursor time window traditionally analyzed in 1-D ULVZ studies. These post-cursors might then be misidentified or constructively/destructively interfere with the ScsP postcursor, leading to incorrect estimation of ULVZ parameters. In this study we investigate the bias introduced by the 2.5-D morphologies on the 1D estimated ULVZ elastic properties in a Bayesian waveform inversion scheme. We further expand the Bayesian method by including the data noise covariance matrix in the inversion and compare it to an autoregressive noise model that was utilized in previous studies. From the application to the observed ScP data, we find that the new approach converges faster, particularly for the inversion of data from multiple events, and the new algorithm retrieves ULVZ parameters with more realistic uncertainties. The inversion of 2.5-D synthetic ScP waveforms suggests that the retrieved ULVZ parameters can be misleading with unrealistically high confidence if we do not consider the data noise covariance matrix in the inversion. Our new approach can also retrieve the shape of a multi-dimensional Gaussian ULVZ if its length is 12o or longer in the great circle arc direction. However, 2.5-D synthetic waveforms show additional waveform complexity which can constructively interfere with the ScP wavefield. Hence, in many cases the estimation of ULVZ properties using 1-D forward modeling can provide incorrect ULVZ parameters. Hence previous ULVZ modeling efforts using 1-D parameter estimation methods may be incorrect.

A variational Bayesian based robust filter for unknown measurement bias and inaccurate noise statistics

In many practical fields, the unknown time-varying measurement biases (additive and multiplicative bias) and heavy-tailed measurement noise caused by some unpredictable anomalous behaviors may degrade the performance of conventional Kalman filter seriously. To solve the state estimation problem of systems with time-varying measurement biases and heavy-tailed measurement noise, this paper proposes a new variational Bayesian (VB) based robust filter. Firstly, the non-Gaussian measurement likelihood probability density function (ML-PDF) with multiplicative and additive measurement bias is built. Then, the conjugate prior distributions for unknown bias and noise scale parameters are selected, and the VB method is utilized to jointly infer the system state, unknown measurement biases and inaccurate measurement noise covariance matrix. Finally, a VB based robust filter is derived and its effectiveness is verified by the numerical simulations.

Adaptive IMM Smoothing Algorithms for Jumping Markov System With Mismatched Measurement Noise Covariance Matrix

Adaptive IMM Smoothing Algorithms for Jumping Markov System With Mismatched Measurement Noise Covariance Matrix

Noise covariance estimation in multi-task high-dimensional linear models

Noise covariance estimation in multi-task high-dimensional linear models

Subspace-based noise covariance estimation for Kalman filter in virtual sensing applications

The accuracy of the Kalman filter in state estimation depends on the knowledge of the process and measurement noise covariances. These are usually treated as tuning parameters and adjusted in a heuristic manner to fine-tune the state predictions. While several methods to identify the noise covariance from data exist, some require the use of optimization algorithms, or inversion of large matrices, which is numerically inefficient. In this work we explore a direct approach to estimate the covariance of possibly correlated process and measurement noises, which is based on subspace identification. It is shown that the subspace-based method outperforms the established autocovariance least-squares scheme and provides a good initial guess on the noise covariance in case the system is subjected to model errors. We validate the proposed scheme on a laboratory experiment, where it is shown that the predictions of the system outputs at sensor locations that are not used as observations in the identification procedure match well with the actual measurements.

A novel adaptive unscented kalman filter algorithm for SOC estimation to reduce the sensitivity of attenuation coefficient

A new exponential adaptive strong tracking unscented Kalman filtering algorithm (ASTRUKF) is proposed to address the issues of slow tracking speed, untimely response of model parameter changes, and inaccurate noise statistical characteristics when estimating the State of Charge (SOC) of batteries at different temperatures, in order to reduce the sensitivity of attenuation coefficients. This algorithm improves upon the traditional method of calculating attenuation factor by incorporating the correlation between current and past residuals. It dynamically updates the Kalman gain and contribution rate using an exponential model, which increases the impact of current data. This algorithm also ensures the stability of the noise covariance matrix by stabilizing the attenuation factor within a fixed range. The contribution rate is utilized to estimate the statistical characteristics of detection noise, however it may exhibit bias. The ASTRUKF method exhibits a maximum estimation error of 10−8 at 0 °C and 10−12 at 15 and 25 °C. This value is well below the levels of the UKF and EKF algorithms. The ASTRUKF algorithm outperforms the UKF and EKF algorithms in terms of prediction accuracy at different temperatures. The ASTRUKF method is well-suited for estimating SOC and has excellent adaptability to temperature variations.

Robust adaptive beamforming with partly calibrated arrays in the presence of gain-phase errors

The performance of the beamformer, especially the beamformer based on the interference plus noise covariance matrix (INCM) reconstruction which needs accurate array manifold knowledge, will degrade dramatically in the presence of gain-phase errors. To solve this problem, we propose a robust adaptive beamforming method based on partly calibrated arrays (PCA). The main idea of this method is to estimate the gain-phase errors more accurately using the higher signal-to-noise ratio (SNR) signal sources and the PCA. After that, we calibrate the interference signal steering vector (SV) and reconstruct the more accurate INCM based on the estimation results. Finally, the SV of the desired signal is estimated by the intersection of two special subspaces. On the basis of the above research, the optimal weight vector is obtained. Numerical simulation results demonstrate the superior performance of the proposed beamformer relative to other existing beamformers in the presence of gain-phase errors.

Weight fluctuations in deep linear neural networks and a derivation of the inverse-variance flatness relation

We investigate the stationary (late-time) training regime of single- and two-layer underparameterized linear neural networks within the continuum limit of stochastic gradient descent (SGD) for synthetic Gaussian data. In the case of a single-layer network in the weakly underparameterized regime, the spectrum of the noise covariance matrix deviates notably from the Hessian, which can be attributed to the broken detailed balance of SGD dynamics. The weight fluctuations, are in this case, generally anisotropic but effectively experience an isotropic loss. For an underparameterized two-layer network, we describe the stochastic dynamics of the weights in each layer and analyze the associated stationary covariances. We identify the interlayer coupling as a distinct source of anisotropy for the weight fluctuations. In contrast to the single-layer case, the weight fluctuations are effectively subject to an anisotropic loss, the flatness of which is inversely related to the fluctuation variance. We thereby provide an analytical derivation of the recently observed inverse-variance flatness relation in a model of a deep linear neural network. Published by the American Physical Society 2024

Fast Bayesian filtering for wastewater treatment plants with inaccurate process noise statistics

Accurate state estimation of wastewater treatment plants is critical for optimizing wastewater treatment processes and reducing operating costs and energy consumption. Due to their large size and numerous state variables, these wastewater treatment plants are considered as high-dimensional systems. The complexity of wastewater treatment plants results in varying and complex process noise statistics, posing challenges for state estimation. This paper proposes a novel state estimation method for wastewater treatment plants subject to inaccurate process noise statistics. The high-dimensional state vector is partitioned into multiple state blocks based on the system architecture, and lost correlations between blocks are compensated by considering time-series correlations. Real-time modification of the process noise covariance matrix is applied to adaptively adjust the inaccurate process noise statistics and compensate for errors from block division. It is verified through simulations that the proposed Bayesian algorithm can achieve satisfactory estimation results while the computational cost is moderate.

Research on stochastic error suppression method of FOG based on quasi-proportional information

Abstract In order to improve the precision of fiber optic gyro (FOG), it is necessary to suppress the stochastic error effectively. Therefore, the stochastic error suppression of FOG is studied. Firstly, the data of FOG at room temperature and variable temperature are analyzed. Then, a general form of FOG random error model is derived, and an adaptive filtering method based on quasi-proportional information is proposed to estimate the process noise covariance effectively. At last, the room temperature experiment, the variable temperature experiment and the dynamic experiment are carried out. Compared with the recently proposed adaptive filter algorithm, the variance reduction of the proposed algorithm at the average time of 33 s, 54 s and 55 s is more than 70%, and the suppression effect of the algorithm on the random walk coefficient is improved by more than 2%, showing strong robustness.

Robust Attitude Estimation for Low-Dynamic Vehicles Based on MEMS-IMU and External Acceleration Compensation.

Attitude determination based on a micro-electro-mechanical system inertial measurement unit (MEMS-IMU) has attracted extensive attention. The non-gravitational components of the MEMS-IMU have a significant effect on the accuracy of attitude estimation. To improve the attitude estimation of low-dynamic vehicles under uneven soil conditions or vibrations, a robust Kalman filter (RKF) was developed and tested in this paper, where the noise covariance was adaptively changed to compensate for the external acceleration of the vehicle. The state model for MEMS-IMU attitude estimation was initially constructed using a simplified direction cosine matrix. Subsequently, the variance of unmodeled external acceleration was estimated online based on filtering innovations of different window lengths, where the acceleration disturbance was addressed by tradeoffs in time-delay and prescribed computation cost. The effectiveness of the RKF was validated through experiments using a three-axis turntable, an automatic vehicle, and a tractor tillage test. The turntable experiment demonstrated that the angle result of the RKF was 0.051° in terms of root mean square error (RMSE), showing improvements of 65.5% and 29.2% over a conventional KF and MTi-300, respectively. The dynamic attitude estimation of the automatic vehicle showed that the RKF achieves smoother pitch angles than the KF when the vehicle passes over speed bumps at different speeds; the RMSE of pitch was reduced from 0.875° to 0.460° and presented a similar attitude trend to the MTi-300. The tractor tillage test indicated that the RMSE of plough pitch was improved from 0.493° with the KF to 0.259° with the RKF, an enhancement of approximately 47.5%, illustrating the superiority of the RKF in suppressing the external acceleration disturbances of IMU-based attitude estimation.

Irregular extended target tracking with unknown measurement noise covariance

To address the challenge of tracking irregular extended target while adapting to unknown measurement noise, this paper presents a novel tracking algorithm for irregular extended target. Considering the flexibility of the Gaussian process (GP) model for representing shape, in the proposed algorithm, we utilize the GP model to delineate the radial function of the tracked extended target’s contour. Then, we incorporate the GP model into the variational Bayesian technique to formulate the posterior distribution for both the unknown measurement noise and the shape state of the tracked target. Subsequently, the square-root cubature Kalman filter is employed to tackle the nonlinearity filtering in the problem of irregular extended target tracking (ETT), enabling simultaneous estimation of intricate shape details and the unknown measurement noise covariance. Through extensive experiments using both simulated and real-time lidar data, the proposed algorithm demonstrates superior performance compared to the existing ETT algorithms for extended target tracking in both accuracy and robustness.

Signals of detailed balance violation in nonequilibrium stationary states: subtle, manifest, and extraordinary

The evolution of physical systems are often modeled by simple Markovian processes. When settled into stationary states, the probability distributions of such systems are time independent, by definition. However, they do not necessarily fall within the framework of equilibrium statistical mechanics. Instead, they may be non-equilibrium steady states (NESS). One distinguishing feature of NESS is the presence of time reversal asymmetry (TRA) and persistent probability current loops. These loops lead naturally to the notion of probability angular momenta, which play a role on the same footing as the noise covariance in stochastic processes. Illustrating with simulations of simple models and physical data, we present ways to detect these signals of TRA, from the subtle to the prominent.

Eigenvalues of the noise covariance matrix in ocean waveguides.

The eigenvalue (EV) spectra of the theoretical noise covariance matrix (CM) and observed sample CM provide information about the environment, source, and noise generation. This paper investigates these spectra for vertical line arrays (VLAs) and horizontal line arrays (HLAs) in deep and shallow water numerically. Empirically, the spectra are related to the width of the conventional beamforming output in angle space. In deep water, the HLA noise CM tends to be isotropic regardless of the sound speed profile. Thus, the EV spectrum approaches a step function. In contrast, the VLA noise CM is non-isotropic, and the EVs of the CM jump in two steps. The EVs before the first jump are due to sea surface noise, while those between the first and second jump are due to bottom-reflected noise. In shallow water, the VLA noise CM is affected by the environment (sound speed profile and seabed density, sound speed, attenuation, and layers) and array depth, the EVs have a more complicated structure. For Noise09 VLA experimental data, the noise sample CM EVs match the waveguide noise model better than the three-dimensional isotropic noise model.

Robust UKF orbit determination method with time-varying forgetting factor for angle/range-based integrated navigation system

The angle/range-based integrated navigation system is a favorable navigation solution for deep space explorers. However, the statistical characteristics of the measurement noise are time-varying, leading to inaccuracies in the derived measurement covariance even causing filter divergence. To reduce the gap between theoretical and actual covariances, some adaptive methods use empirically determined and unchanged forgetting factors to scale innovations within the sliding window. However, the constant weighting sequence cannot accurately adapt to the time-varying measurement noise in dynamic processes. Therefore, this paper proposes an Adaptive Robust Unscented Kalman Filter with Time-varying forgetting factors (TFF-ARUKF) for the angle/range integrated navigation system. Firstly, based on a statistically linear regression model approximating the nonlinear measurement model, the M-estimator is adopted to suppress the interference of outliers. Secondly, the covariance matching method is combined with the Huber linear regression problem to adaptively adjust the measurement noise covariance used in the M-estimation. Thirdly, to capture the time-varying characteristics of the measurement noise in each estimation, a new time-varying forgetting factors selection strategy is designed to dynamically adjust the adaptive matrix used in the covariance matching method. Simulations and experimental analysis compared with EKF, AMUKF, ARUKF, and Student’s t-based methods have validated the effectiveness and robustness of the proposed algorithm.

Contact Force Estimation of Robot Manipulators With Imperfect Dynamic Model: On Gaussian Process Adaptive Disturbance Kalman Filter

This paper is concerned with the contact force estimation problem of robot manipulators based on imperfect dynamic models of the manipulator and the contact force. To handle the imperfect dynamic information of the manipulator, a hybrid model, consisting of the nominal model and the residual dynamics, is established for the manipulator, and the Gaussian process regression (GPR) technique is employed to learn the mean and covariance of the residual dynamics. On this basis, a virtual measurement equation is established for contact force estimation and a Gaussian process adaptive disturbance Kalman filter (GPADKF) is developed where the variational Bayes technique is employed to achieve online identification of the noise statistics in the force dynamics. The GPADKF is capable of decoupling the contact force from residual dynamics and system noises, thereby reducing the dependence on accurate dynamic models of the manipulator and the contact force. Simulation and experimental results demonstrate that the proposed scheme outperforms the state-of-art methods. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Contact force estimation for robot manipulators can be achieved by fusing the dynamic models of the manipulator and the contact force. When both models are imprecise, the traditional inverse dynamics-based and disturbance Kalman filter-based approaches can no longer provide accurate force estimates. To handle this challenge, a computationally efficient hybrid dynamic model is established for the manipulator, which consists of the nominal model and a residual dynamics compensation term learned from offline data via the GPR. On this basis, an adaptive disturbance Kalman filter is constructed by using the variational Bayes technique to deal with the inaccurate noise covariance matrix in the force dynamic model. Compared with the existing approaches, the force estimate obtained via the proposed scheme is more accurate and reliable, as refined noise covariance matrices (provided by both the GPR and the variational Bayes procedure) have been adopted in the Kalman gain calculation. The proposed GPADKF method is the extension of the composite disturbance filtering (CDF) framework. With the proposed scheme, the dependency on the perfect dynamic models in contact force estimation can be significantly reduced, and this makes our approach especially suitable for contact force estimation problems under unfamiliar and complicated environments.