Process Noise Covariance Matrices Research Articles

A Novel Adaptive Noise Covariance Matrix Estimation and Filtering Method: Application to Multiobject Tracking

This paper presents a novel online adaptive method for estimating the process and measurement noise covariance matrices in Kalman filters (KFs) to address the challenge of varying noise characteristics in practical applications. Specifically, the proposed method decomposes the noise covariance matrix into an element distribution matrix and a noise intensity and employs an improved Sage filter to estimate the element distribution matrix. Additionally, a calibration and correction method is introduced to accurately determine and adaptively correct the online bias of the noise intensity. The unbiasedness and convergence of the proposed method are mathematically proven under the condition that the system is detectable. Moreover, this method is applied to multiobject tracking (MOT) based on KFs and light detection and ranging (LiDAR), and it is evaluated on the KITTI dataset and the official KITTI server. The experimental results demonstrate that the proposed method achieves significantly improved MOT performance based on KFs and outperforms other LiDAR-based methods on the KITTI leaderboard. This method provides a new approach for enhancing the performance of KFs and assisting with MOT, and it has practical feasibility for real-world applications.

Attitude filtering with uncertain process and measurement noise covariance using SVD‐aided adaptive UKF

AbstractIt is presented in this article how to simultaneously alter the process and measurement noise covariance matrices for nontraditional attitude filtering technique. The unscented Kalman filter (UKF) and singular value decomposition (SVD) methods are integrated in the nontraditional attitude filtering algorithm to estimate a nanosatellite's attitude with an inherent robustness feature. The SVD approach determines the attitude of the nanosatellite and provides one estimate at a single frame utilizing measurements from the magnetometer and Sun sensor as the initial stage of the algorithm. These attitude terms are subsequently fed into the UKF with their error covariances, which makes the filter robust inherently. The attitude estimations of the satellite are compared between the filters presented. The Q (process noise covariance) adaption approach with multiple scale factors is specifically suggested for differences in between the process channels. Performance of the multiple scale factors‐based adaptive SVD‐aided UKF is examined in the event of process noise increase, which may result from changes in the environment or satellite dynamics.

Robust forecasting-aided state estimation of power system based on extended Kalman filter with adaptive kernel risk-sensitive loss

State estimation (SE) plays a pivotal role in the development of modern power system. Accurate forecasting-aided state estimation (FASE) can track the sudden changes of power system state and maintain the safe operation of modern power system. However, the performance of existing FASE methods is affected by anomalies in real power system, such as sudden state changes and bad data. To address this problem, a robust algorithm based on extended Kalman filter (EKF) with the kernel risk-sensitive loss (KRSL) (called KRSL-EKF) is proposed for FASE. The KRSL-EKF, taking KRSL as the cost function of the original EKF algorithm, can overcome the limitations of the EKF to perform higher estimation accuracy under non-Gaussian noise cases. In addition, an adaptive method is further introduced into the proposed KRSL-EKF algorithm to adjust the covariance matrices of process noise and measurement noise, and we denote it as AKRSL-EKF. The novel AKRSL-EKF algorithm can effectively adapt the noise to system state variations and achieves better estimation accuracy. The effectiveness of the proposed algorithms for FASE is verified on IEEE 14-bus, IEEE 30-bus, and IEEE 57-bus systems. The results show that the estimation accuracy of the proposed algorithms is 30% higher than other traditional algorithms, with high estimation accuracy.

A modified adaptive Kalman filtering method for maneuvering target tracking of unmanned surface vehicles

The filtering methods are crucial for an unmanned surface vehicle (USV) to realize target tracking. Due to the poor observation caused by the strong vibration during the navigation of the USV, the target tracking accuracy of the traditional filtering method has been significantly degraded. A modified strong tracking-based expended Sage-Husa adaptive robust Kalman filter (MST-ESHARKF) algorithm is proposed to overcome this problem in this paper. In the proposed algorithm, a modified fading factor for the strong tracking Kalman filter is introduced to eliminate disturbance-induced filter divergence. In addition, the adaptive factor of robust Kalman filtering is designed to balance the predicted and observed states dedicated to improving the robustness of the algorithm. Finally, the biased and unbiased estimators for measurement and process noise covariances are merged, and the measurement noise covariance matrix’s interval is constrained, resulting in a simultaneous evaluation of measurement and process noise covariance matrices with improved dependability of the proposed algorithm. The simulation and experiment results show that the proposed MST-ESHARKF outperforms the existing filters in target tracking.

An improved adaptive FastSLAM algorithm with time‐varying noise estimator

AbstractIn this study, an adaptive FastSLAM (AFastSLAM) algorithm, which is obtained by estimating the time‐varying noise statistics and improving FastSLAM algorithm, is proposed. This improvement was accomplished by using maximum likelihood estimation and expectation maximization criterion and a one‐step smoothing algorithm in importance sampling. In addition, innovation covariance estimation (ICE) method was used to prevent loss of positive definiteness of the process and measurement noise covariance matrices. The proposed method was compared with FastSLAM by calculating the root mean square error (RMSE) using different particle numbers at varying initial process and measurement noise values. Simulation studies have shown that AFastSLAM provides much more accurate, consistent, and successful estimates than FastSLAM for both robot and landmark positions.

Estimation of time-varying noise parameters for unscented Kalman filter

The unscented Kalman filter (UKF) is a promising method for system state and structural parameters estimation. However, its performance depends on the process noise and measurement noise covariance matrices, which are usually unknown in practice. Arbitrary selection of these covariance matrices may lead to unreliable or even diverging estimation results. To resolve this critical problem, we propose a Bayesian probabilistic algorithm for the estimation of the noise covariance matrices based on the response measurement. The proposed Noise-Parameters-Identified Unscented Kalman Filter (NPI-UKF) has the following salient features: (1) the divergence problem is resolved; (2) reliable estimation results including uncertainty quantification can be obtained; and (3) NPI-UKF is applicable to nonstationary situations. These salient features are illustrated through the numerical applications to a bridge structure and a laboratory experiment to a shear building model. The efficacy and robustness of NPI-UKF will be validated.

Time-Varying System Identification of Precast Segmental Columns Subjected to Seismic Excitations

Precast segmental columns, as one structures, have attracted increasing attention in bridge and building construction. To understand the performance and dynamic characteristics of precast segmental columns subjected to earthquake excitations, this paper develops a novel approach for time-varying system stiffness identification of precast segmental columns under seismic excitations, based on the adaptive unscented Kalman filtering (AUKF) method. In the proposed approach, a precast segmental column model is considered as an equivalent single-degree-of-freedom (SDOF) system with time-varying system parameters. The AUKF method is employed to identify the time-varying stiffness parameters by using measured acceleration responses and then calculate dynamic displacement responses of the SDOF system to verify the accuracy of the identified stiffness parameters. Since the system parameter identification process and measurement noise covariance matrices are updated at each time step, the estimated results can be reliably obtained to evaluate the conditions of segmental columns subjected to seismic excitations. To verify the feasibility and effectiveness of using the proposed approach for time-varying system stiffness identification of segmental columns, numerical studies on a planar precast segmental column subjected to different intensities of ground motion excitations are conducted. Shake table tests on a scaled precast segmental column and a scaled monolithic column subjected to earthquake excitations are conducted in the laboratory to further verify the performance of the proposed approach. Measured acceleration responses are used for time-varying system identification. Both numerical and experimental results demonstrate that the proposed approach can successfully identify the time-varying stiffness values of precast segmental columns subjected to earthquake excitations, and the displacement responses of these columns are predicted accurately as compared with the measured ones.

Tensor Kalman Filter and Its Applications

Kalman filter is one of the most important estimation algorithms, which estimates certain unknown variables given the measurements observed over time subject to a dynamic system, for many applications in science and engineering including environmental science, ecometrics, robotics, financial analysis, data mining, etc. It is often necessary to characterize multiple relationships among various kinds of signals/data in tensor form. The conventional Kalman filter paradigm is based on the low-dimensional state-space representation, which is restricted by the state-transition, observation-model, process-noise covariance, and observation-noise covariance matrices. However, we often need to express some or all of them in terms of tensors in practice. Very lately, the aforementioned Kalman filter in tensor form was tackled using tensor decomposition but the exact estimator has never been established so far. In this work, we propose a new generalized Kalman filter framework consisting of state, state-transition model, observation-model, process-noise covariance, and observation-noise covariance tensors of arbitrary orders by applying the ShermanMorrisonWoodbury identity and block tensor inverse, which we call "Tensor Kalman Filter" (TKF). Our proposed new approach can produce the exact Kalman filter estimator without any need of tensor decomposition (approximation). The pertinent computational- and memory-complexity studies are also provided in this paper.

Robust adaptive multi‐target tracking with unknown measurement and process noise covariance matrices

A robust adaptive probability hypothesis density (PHD) filter is proposed to address the degradation of PHD performance due to an unknown process noise and measurement noise covariance matrix. Therefore, the inverse Wishart distribution is introduced to model the prior distribution of process noise and measurement noise. Meanwhile, the multi-target posterior intensity is approximated as a mixture of the inverse Wishart distribution and Gaussian distribution. The closed solution of the robust PHD filter is derived by the variational Bayes approach. Simulation results show that the proposed algorithm outperforms the Gaussian-mixed PHD filter and the variational Bayesian PHD filter in terms of target number estimation accuracy and optimal sub-pattern assignment distance.

Lateral directional aircraft aerodynamic parameter estimation using adaptive stochastic nonlinear filter

AbstractThis paper aims to accurately estimate the lateral directional aerodynamic parameters in real time irrespective of the variations in the process and measurement covariance matrices. The proposed algorithm for parameter estimation is based on the integration of adaptive techniques into a stochastic nonlinear filter. The proposed adaptive estimation algorithm is applied to flight test data, and the lateral directional derivatives are estimated in real time. The estimates are compared with those obtained from the Filter Error Method (FEM), an offline parameter estimation method accounting for process noise. The estimation results are observed to be very comparable, and the supremacy of the adaptive filter is illustrated by varying the covariance matrices of both process and measurement noises. The parameters estimated by the adaptive filter are found to converge to their actual values, whereas the estimates of the regular filter are observed to diverge from the actual values when changing the noise covariance matrices. The proposed adaptive algorithm can estimate the lateral directional aerodynamic derivatives more accurately without prior knowledge of either process or measurement noise covariance matrices. Hence, it is of great value in online implementations.

A Tale of Three Priors

The histogram-probabilistic multihypothesis tracker (H-PMHT) is an efficient, track-before-detect algorithm that can operate directly on detection surfaces such as data from a focal-plane array (FPA). It provides good performance for low computational cost but is sensitive to sensor parameter changes, e.g. number of channels, gain, normalization, etc. This is because H-PMHT interprets the measured data as a histogram of point measurements; this sensor model is analogous to an FPA counting the number of photons that fall within each of its pixels. Typically this photon count is enormous, which can overwhelm the prior distribution describing target motion. The H-PMHT algorithm addresses this issue by applying a target's prior distribution to every measurement. For a linear, Gaussian target, this results in a data-dependent scaling term on both the measurement and the process noise covariance matrices. However, if the size or gain of an FPA is changed, the photon count changes significantly, and these scaling terms can cause unpredictable performance. One solution to this problem is to change the H-PMHT so that a target's prior distribution is only applied to photons from the same target. This article describes two different approaches to constructing target prior distributions that satisfy this constraint. The benefits of adopting either of these new target prior distributions are demonstrated using simulation.

Variational Bayesian Adaptive Unscented Kalman Filter for RSSI-based Indoor Localization

Most existing localization schemes necessitate a priori statistical characteristic of measurement noise, which may be unrealistic in practical applications. This paper investigates the variational Bayesian adaptive unscented Kalman filtering (VBAUKF) for received signal strength indication (RSSI) based indoor localization under inaccurate process and measurement noise covariance matrices. First, an inaccurate and slowly varying measurement noise covariance matrix can be estimated by choosing appropriate conjugate prior distribution for an indoor localization model with inaccurate process and measurement noise covariance matrices. By choosing inverse Wishart priors distribution, the state, predicted error and measurement noise covariance matrices are inferred on each time separately. Second, a parameter optimization algorithm is designed to minimize the localization error of VBAUKF until it less than the threshold set in advance. Finally, experimental validation is presented to demonstrate the accuracy and effectiveness of the proposed filtering method for indoor localization.

Robot Calibration Method Based on Extended Kalman Filter–Dual Quantum Behaved Particle Swarm Optimization and Adaptive Neuro-Fuzzy Inference System

This paper combines the extended Kalman filter (EKF), dual quantum-behaved particle swarm optimization (DQPSO), and adaptive neuro-fuzzy inference system (ANFIS) to propose a novel robot calibration method. Robot precision is influenced by kinematic and non-kinematic error sources. The EKF algorithm is robust for the nonlinear system with Gaussian noise to identify kinematic parameter errors for kinematic calibration. However, if inappropriate covariance matrices are selected, the EKF algorithm may converge to an incorrect solution. To increase the effectiveness of the EKF algorithm, we propose a DQPSO algorithm that consists of the QPSO-1 and QPSO-2 algorithms. The QPSO-1 algorithm adapts the covariance matrices of measurement noise and process noise, while the QPSO-2 algorithm optimizes the kinematic parameter errors estimated by the EKF algorithm. In addition, the used ANFIS predicts and compensates the non-kinematic error for non-kinematic calibration. Experiments have been performed on a five-bar parallel robot to confirm the effectiveness of the proposed method. The experimental results demonstrate that the proposed method significantly improves the positional accuracy, and is better than the previous methods.

Multi-Fading Factor and Updated Monitoring Strategy Adaptive Kalman Filter-Based Variational Bayesian.

Aiming at the problem that the performance of adaptive Kalman filter estimation will be affected when the statistical characteristics of the process and measurement of the noise matrices are inaccurate and time-varying in the linear Gaussian state-space model, an algorithm of multi-fading factor and an updated monitoring strategy adaptive Kalman filter-based variational Bayesian is proposed. Inverse Wishart distribution is selected as the measurement noise model and the system state vector and measurement noise covariance matrix are estimated with the variational Bayesian method. The process noise covariance matrix is estimated by the maximum a posteriori principle, and the updated monitoring strategy with adjustment factors is used to maintain the positive semi-definite of the updated matrix. The above optimal estimation results are introduced as time-varying parameters into the multiple fading factors to improve the estimation accuracy of the one-step state predicted covariance matrix. The application of the proposed algorithm in target tracking is simulated. The results show that compared with the current filters, the proposed filtering algorithm has better accuracy and convergence performance, and realizes the simultaneous estimation of inaccurate time-varying process and measurement noise covariance matrices.

Pose estimation by extended Kalman filter using noise covariance matrices based on sensor output

This paper presents an extended Kalman filter for pose estimation using noise covariance matrices based on sensor output. Compact and lightweight nine-axis motion sensors are used for motion analysis in widely various fields such as medical welfare and sports. A nine-axis motion sensor includes a three-axis gyroscope, a three-axis accelerometer, and a three-axis magnetometer. Information obtained from the three sensors is useful for estimating joint angles using the Kalman filter. The extended Kalman filter is used widely for state estimation because it can estimate the status with a small computational load. However, determining the process and observation noise covariance matrices in the extended Kalman filter is complicated. The noise covariance matrices in the extended Kalman filter were found for this study based on the sensor output. Postural change appears in the gyroscope output because the rotational motion of the joints produces human movement. Therefore, the process noise covariance matrix was determined based on the gyroscope output. An observation noise covariance matrix was determined based on the accelerometer and magnetometer output because the two sensors’ outputs were used as observation values. During a laboratory experiment, the lower limb joint angles of three participants were measured using an optical 3D motion analysis system and nine-axis motion sensors while participants were walking. The lower limb joint angles estimated using the extended Kalman filter with noise covariance matrices based on sensor output were generally consistent with results obtained from the optical 3D motion analysis system. Furthermore, the lower limb joint angles were measured using nine-axis motion sensors while participants were running in place for about 100 s. The experiment results demonstrated the effectiveness of the proposed method for human pose estimation.

Toward security monitoring of industrial Cyber-Physical systems via hierarchically distributed intrusion detection

Industrial Cyber-physical systems (ICPSs), integrating communication, computation and control of industrial processes are referred to as a core technology to approach the Industry 4.0. Ensuring the ICPS security is of paramount importance in smart manufacturing. Considering the characteristics of large-scale, geographically-dispersed and multi-dimensional heterogeneous, federated and life-critical natures of ICPSs, this paper investigates a hierarchically distributed intrusion detection scheme that seeks to achieve the all-round safety protection of ICPSs according to the system structure and attacking types of each ICPS layer. For physical system-relevant perceptual executive layer, potential and covert attacks are detected by the clustered sensory system state residual anomaly monitoring based on a process noise and measurement noise-adaptive Kalman filter (PNMN-AKF). PNMN-AKF can perform a joint recursive estimation of dynamic system states, time-varying process and measurement noise covariance matrices by the variational Bayes approximation framework. In cyberspace, potential cyber-attacks are detected by the anomaly monitoring of the statistical distribution of the network transmission characteristics of data transmission layer by introducing a forgetting factor-induced recursive Gaussian mixture model (FF-RGMM). In the application control layer, a regularized sparse deep belief network model is introduced to characterize the misuse behavior for detecting potential attacks. Extensive validation and comparative experiments have been conducted on a numerical simulation system and a comprehensive ICPS simulation platform by using OPNET and a commonly-used benchmark simplified Tennessee Eastman process (STEP) based on Matlab/Simulink. Experimental results demonstrate that the proposed hierarchically distributed intrusion detection method can efficiently recognize potential and covert cyber-attacks in each ICPSs link with low false alarm rate and missing detection rate, which lays a foundation for the overall security monitoring of ICPSs.

Improved GOES-R ABI image navigation and registration using maximum likelihood parameter estimation

Image navigation and registration (INR) for the GOES-R series advanced baseline imager (ABI) is performed in ground processing of the raw image data. A Kalman filter processes ABI star measurements to compute time-varying optical misalignment parameters that allow real-time alignment of image samples to the GOES-R Fixed Grid. For the system to meet INR requirements, the filter measurements noise covariance (R) and process noise covariance (Q) matrices must closely approximate behavior of the true system. Prelaunch estimates of the Q and R parameters did not match actual GOES-16 performance. Several attempts to manually determine improved Q&R parameters were not completely successful. Instead, the Kalman filter was calibrated (tuned) using maximum likelihood (ML) estimation of the parameters: computed utilizing partial derivatives of the Kalman equations, Gauss–Newton iteration, and nonlinear optimization logic. The updated parameters significantly improved performance and are currently operational for GOES-16 and 17. ML estimation of Kalman filter parameters has been used successfully for other systems but has not widely been documented. Greater use is encouraged. We also include a review of ABI INR theory and implementation.

Robust Particle Filtering With Time-Varying Model Uncertainty and Inaccurate Noise Covariance Matrix

This article proposes a robust particle filtering (PF) approach for a generic class of nonlinear systems with both additive time-varying uncertainty (ATVU) in the state transition equation and inaccurate process noise covariance matrices. To avoid sampling efficiency degradation of the PF approach caused by ATVU, we employ the disturbance observer-based PF (DOBPF) approach where the effect of ATVU is compensated in the particle generation stage. Different from the existing DOBPF method where disturbance estimation is achieved via the Kalman filter, the disturbance observer adopted in this article is in the form of variational Bayesian adaptive Kalman filter (VBAKF) which deals with the inaccurate process noise covariance matrices in both the dynamic models of the state and the ATVU. Compared with conventional PF approaches, the proposed method, named VBAKF-PF, exhibits enhanced robustness against both the ATVU in the state transition equation and the uncertainties of process noise covariance matrices. The simulation results demonstrate the superiority of VBAKF-PF over both the VBAKF and DOBPF methods.

A robust structural parameter estimation method using seismic response measurements

Although successful applications of data assimilation techniques such as Kalman filter to structural parameter identification problems exist in literature, they have been explored mostly via simulations because real-world application poses challenges. Current practice is to set the process and measurement noise covariance matrices by manual tuning and keeping them constant over the estimation period. However, when the matrices are not correct, Kalman filters are prone to suboptimal performance and/ or even divergence. In this paper, use of a stochastic approximation scheme along with the unscented Kalman filter is explored. The sensitivity of the initial filter parameter selection has been investigated through both simulations and full-scale laboratory experiments. Results show that by automatically updating the filter statistics, system states, including the structural parameters, are successfully estimated with a fast convergence rate, good accuracy, and trackability in a stable and objective manner. The method is further applied to in situ seismic response records of a box-girder bridge. The stiffness values of the rubber bearings are successfully identified.

Optimizing a Kalman filter with an evolutionary algorithm for nonlinear quadrotor attitude dynamics

Kalman filter is a linear minimum – mean square error estimator. However, process and measurement noise matrices of the Kalman filter must be known a priori to achieve this optimality condition. But in most cases, these parameters are guessed or tuned by trial and error approach both of which do not guarantee optimality and convergence. So in this work, an evolutionary algorithm is proposed to estimate the process and measurement noise covariance matrices of the Kalman filter to improve the performance of the sub – optimal filter. A fitness function is also proposed to achieve multi – dimensional optimization. Results are validated for both linear and nonlinear quadrotor attitude dynamics. Proposed method compared with optimization algorithms such as optimal Kalman filter, covariance – matrix adaptation evolution strategy and simulated annealing. Monte Carlo analysis showed that proposed algorithm is capable of tuning the state estimator better than the other algorithms. Thus, proposed algorithm reduces sensor errors and ensures efficient estimation of the quadrotor.