Process Noise Covariance Matrix Research Articles

A Bayesian model calibration framework for stochastic compartmental models with both time-varying and time-invariant parameters

We consider state and parameter estimation for compartmental models having both time-varying and time-invariant parameters. In this manuscript, we first detail a general Bayesian computational framework as a continuation of our previous work. Subsequently, this framework is specifically tailored to the susceptible-infectious-removed (SIR) model which describes a basic mechanism for the spread of infectious diseases through a system of coupled nonlinear differential equations. The SIR model consists of three states, namely, the susceptible, infectious, and removed compartments. The coupling among these states is controlled by two parameters, the infection rate and the recovery rate. The simplicity of the SIR model and similar compartmental models make them applicable to many classes of infectious diseases. However, the combined assumption of a deterministic model and time-invariance among the model parameters are two significant impediments which critically limit their use for long-term predictions. The tendency of certain model parameters to vary in time due to seasonal trends, non-pharmaceutical interventions, and other random effects necessitates a model that structurally permits the incorporation of such time-varying effects. Complementary to this, is the need for a robust mechanism for the estimation of the parameters of the resulting model from data. To this end, we consider an augmented state vector, which appends the time-varying parameters to the original system states whereby the time evolution of the time-varying parameters are driven by an artificial noise process in a standard manner. Distinguishing between time-varying and time-invariant parameters in this fashion limits the introduction of artificial dynamics into the system, and provides a robust, fully Bayesian approach for estimating the time-invariant system parameters as well as the elements of the process noise covariance matrix. This computational framework is implemented by leveraging the robustness of the Markov chain Monte Carlo algorithm permits the estimation of time-invariant parameters while nested nonlinear filters concurrently perform the joint estimation of the system states and time-varying parameters. We demonstrate performance of the framework by first considering a series of examples using synthetic data, followed by an exposition on public health data collected in the province of Ontario.

Application of optimized Kalman filtering in target tracking based on improved Gray Wolf algorithm

High precision is a very important index in target tracking. In order to improve the prediction accuracy of target tracking, an optimized Kalman filter approach based on improved Gray Wolf algorithm (IGWO-OKF) is proposed in this paper. Since the convergence speed of traditional Gray Wolf algorithm is slow, meanwhile, the number of gray wolves and the choice of the maximum number of iterations has a great influence on the algorithm, a nonlinear control parameter combination adjustment strategy is proposed. An improved Grey Wolf Optimization algorithm (IGWO) is formed by determining the best combination of adjustment parameters through the fastest iteration speed of the algorithm. The improved Grey Wolf Optimization algorithm (IGWO) is formed, and the process noise covariance matrix and observation noise covariance matrix in Kalman filter are optimized by IGWO. The proposed approach is applied into. The experiment results show that the proposed IGWO-OKF approach has low error, high accuracy and good prediction effect.

Adaptively tracking hypersonic gliding vehicles

The motion model plays a key role in tracking hypersonic gliding vehicles (HGVs). The process noise covariance matrix is used to scale the errors of the motion model, and its design has significant influence on the tracking accuracy. In this paper, a dynamics model with an adaptive process noise covariance matrix is proposed for tracking HGVs. The dynamics model is described by a nonlinear stochastic differential equation (SDE), and its corresponding process noise covariance matrix is calculated by adopting the modified Gauss-Legendre scheme. Furthermore, an adaptive adjustment method for process noise covariances of the augmented states is designed by analyzing the variation characteristics of the model errors. Specifically, the process noise covariances are adjusted by an adaptive factor: the square of the ratio of the estimated dynamic pressure to a prior dynamic pressure. Simulation results show that the proposed model can provide higher tracking accuracy than the dynamics model with fixed process noise covariances.

Autonomous navigation method of satellite constellation based on adaptive forgetting factors

To address the problem that model uncertainty and unknown time-varying system noise hinder the filtering accuracy of the autonomous navigation system of satellite constellation, an autonomous navigation method of satellite constellation based on the Unscented Kalman Filter with Adaptive Forgetting Factors (UKF-AFF) is proposed. The process noise covariance matrix is estimated online with the strategy that combines covariance matching and adaptive adjustment of forgetting factors. The adaptive adjustment coefficient based on squared Mahalanobis distance of state residual is employed to achieve online regulation of forgetting factors, equipping this method with more adaptability. The intersatellite direction vector obtained from photographic observations is introduced to determine the constellation satellite orbit together with the distance measurement to avoid rank deficiency issues. Considering that the number of available measurements varies online with intersatellite visibility in practical applications such as time-varying constellation configurations, the smooth covariance matrix of state correction determined by innovation and gain is adopted and constructed recursively. Stability analysis of the proposed method is also conducted. The effectiveness of the proposed method is verified by the Monte Carlo simulation and comparison experiments. The estimation accuracy of constellation position and velocity of UKF-AFF is improved by 30% and 44% respectively compared to those of the extended Kalman filter, and the method proposed is also better than other several adaptive filtering methods in the presence of significant model uncertainty.

Bayesian Optimization for Fine-Tuning EKF Parameters in UAV Attitude and Heading Reference System Estimation

In various applications, the extended Kalman filter (EKF) has been vital in estimating a vehicle’s translational and angular motion in 3-dimensional (3D) space. It is also essential for the fusion of data from multiple sensors. However, for the EKF to perform effectively, the optimal process noise covariance matrix (Q) and measurement noise covariance matrix (R) must be chosen correctly. The use of EKF has been challenging due to the need for an easy mechanism to select Q and R values. As a result, this research focused on developing an algorithm that can be easily applied to determine Q and R, allowing us to harness the full potential of EKF. Accordingly, an EKF innovation consistency statistics-driven Bayesian optimization algorithm was employed to achieve this goal. Q and R values were tuned until the expected result met the performance requirement for minimum error through improved measurement innovation consistency. The comprehensive results demonstrate that when the optimum Q and R, as tuned by the suggested technique, were used, the performance of the EKF significantly improved.

Soc Estimation of Li-ion Battery Based on Adaptive CKF Algorithm

A second-or der RC equivalent circuit model was established to improve the estimation accuracy of state of charge (SOC) of power Li-ion batteries, and the model parameters were identified by the recursive least square method with forgetting factor (FFRLS). On this basis, an adaptive cubature kalman filter (ACKF) algorithm was proposed to adaptively modify the process noise covariance matrix and the measurement noise covariance matrix to improve the SOC estimation accuracy. Finally, the SOC estimation algorithm was verified by MATLAB simulations. The results show that compared with UKF and CKF algorithms, the proposed algorithm has higher estimation accuracy and robustness, and can meet the application requirements.

Underwater Doppler-bearing maneuvering target motion analysis based on joint estimated adaptive unscented Kalman filter

Noncooperative maneuvering target motion analysis is one of the challenging tasks in the field of underwater target localization and tracking for passive sonar. Underwater noncooperative targets often perform various maneuvers, and the targets are commonly modeled as a combination of constant-velocity models and coordinate-turn models with unknown turning rates. Traditional algorithms for Doppler-bearing target motion analysis are incapable of processing noncooperative maneuvering targets because the algorithms rely on a priori information of the turning rate and the center frequency. To address these shortcomings, this paper proposes the joint estimated adaptive unscented Kalman filter (JE-AUKF) algorithm. The JE-AUKF places the center frequency and turning rate into the state vector and constructs a time-varying state model that self-adapts to a maneuvering target. The JE-AUKF also introduces a time-varying fading factor into the process noise covariance matrix to improve the tracking performance. Simulations and sea trials are conducted to compare the performance of the JE-AUKF with the iterative unscented Kalman filter, the interacting multiple model-unscented Kalman filter, the interacting multiple model-iterative unscented Kalman filter, and the interacting multiple model-joint estimated unscented Kalman filter. The result shows that the JE-AUKF achieves better tracking performance for noncooperative maneuvering targets.

Adaptive Neural-Network-Based Unscented Kalman Filter for Robust Pose Tracking of Noncooperative Spacecraft

This paper presents a neural-network-based unscented Kalman filter (UKF) to estimate and track the pose (i.e., position and orientation) of a known, noncooperative, tumbling target spacecraft in a close-proximity rendezvous scenario. The UKF estimates the target’s orbit and attitude relative to the servicer based on the pose information provided by a multitask convolutional neural network (CNN) from incoming monocular images of the target. To enable reliable tracking, the process noise covariance matrix of the UKF is tuned online using adaptive state noise compensation, which leverages a newly developed closed-form process noise model for relative attitude dynamics. This paper also introduces the Satellite Hardware-In-the-loop Rendezvous Trajectories (SHIRT) dataset to enable comprehensive analyses of the performance and robustness of the proposed pipeline. SHIRT comprises the labeled images of two representative rendezvous trajectories in a low Earth orbit created using both a graphics renderer and a robotic testbed. Specifically, the CNN is solely trained on synthetic data, whereas functionality and performance of the complete navigation pipeline are evaluated on real images from the robotic testbed. The proposed UKF is evaluated on SHIRT and is shown to have subdecimeter-level position and degree-level orientation errors at steady state.

A novel variational Bayesian adaptive Kalman filter with mismatched process noise covariance matrix

AbstractThis paper proposes a novel variational Bayesian (VB) adaptive Kalman filter with mismatched process noise covariance matrix (PNCM). Firstly, this paper explains the reason why the predicted error covariance matrix (PECM) is chosen for variational inference. Secondly, compared with the earlier VB adaptive Kalman filter (VB‐AKF‐Q), the proposed filter calculate the dynamic model of the PECM with its historical estimation information. Therefore, the proposed filter can overcome the influence of mismatched PNCM on the initial value setting of PECM in the VB‐AKF‐Q. Finally, we use the evidence lower bound for the proposed filter and give the convergence criterion on this basis. Some examples with a target tracking simulation are carried out to demonstrate the superiority of the proposed filter.

State-of-Charge and State-of-Health Joint Estimation of Lithium-Ion Battery Based on Iterative Unscented Kalman Particle Filtering Algorithm With Fused Rauch–Tung–Striebel Smoothing Structure

Abstract Traditional particle filtering has a large estimation error in the state of charge and Lithium-ion battery health of electric vehicle lithium batteries. For the above-mentioned problems, the lithium battery second-order resistance capacitance (RC) equivalent circuit model is established, and then, the model parameters are identified using the multi-innovation least square algorithm (MILS). Finally, an iterative unscented Kalman particle filtering algorithm with fused Rauch–Tung–Striebel Smoothing Structure (RTS-IUPF) applied to Li-ion battery state-of-charge (SOC) and state-of-health (SOH) joint estimation is proposed. The algorithm is based on the identification of battery parameters; the controller reads the sensor data and predicts the state results. RTS smoothing structure can do posterior estimation, and a significant probability density function is generated to select the optimal particle, and unscented Kalman algorithm regularized particles. The algorithm reduces the effect of the process noise covariance matrix and the measured noise covariance matrix on the filter accuracy and response time in traditional unselected Kalman filters. The algorithm proposed in the paper improves particle degradation and increases the estimation accuracy. Finally, the RTS-IUPF algorithm performs simulation analysis in Pulse current discharge condition and dynamic current condition (NEDC), respectively. The pulse current experimental results show that the mean absolute value error of UKF and particle filter (PF (number of particles N is 300)) is 1.26% and 1.24%, respectively, while the error of the RTS-IUPF is 0.748%. The root-mean-square error (RMSE) of the RTS-IUPF is reduced by 66.5% and 77.8% compared with UKF and PF. Furthermore, the error of joint estimation using this algorithm is smaller than that of single estimation. The RMSE of the RTS-IUPF joint is reduced by 27.4% compared with RTS-IUPF. The feasibility and effectiveness of the algorithm for the joint estimation of SOC and SOH of lithium batteries were verified.

Invariant Error-Based Integrated Solution for SINS/DVL in Earth Frame: Extension and Comparison

The error propagation of traditional strapdown inertial navigation system (SINS)/Doppler velocity log (DVL) is not autonomous because its error state model is trajectory-dependence. Recently, the invariant error defined on the Lie group has raised much attention due to its trajectory-independent and autonomous error propagation. In this article, the invariant error-based Kalman filter for the SINS/DVL integration solution is investigated with a main focus on its extension and comparison. The contributions of this study are threefold. First, the invariant error defined on the matrix Lie group (group of double direct isometrics) is extended to model the nongroup affine traditional SINS mechanism and the group affine transformed SINS mechanism; both of them are derived in Earth frame and augmented with the drift and bias of the inertial measurement units (IMUs). Then, the observation equations for different invariant error-based state models are derived for SINS/DVL application, a theoretical analysis is performed, and a comprehensive evaluation is conducted under different maneuvering conditions by lake field trial. Finally, the variational Bayesian approach is introduced into the invariant error-based Kalman filter to infer the inaccurate process noise covariance matrix (PNCM) and time-varying measurement noise covariance matrix (MNCM) from a practical perspective; experimental results demonstrate that it can improve the navigation accuracy significantly. This study is expected to facilitate the selection of appropriate invariant error to SINS/DVL applications.

Interacting Multiple Model Adaptive Unscented Kalman Filter for Accurate Localization of a High-Dynamic Motion Mobile Robot Using Wireless Sensor Network

Accurate and precise positioning of a mobile robot is a critical problem in the navigation system, and it has become difficult for localization indoors and in GPS-denied areas. Odometry, which is based on dead reckoning, cannot usually provide long-term accurate position estimates. As a result, integrating encoder-based odometry with Wireless Sensor Network (WSN) measurements is recommended as a good solution for minimizing divergences due to uncertainties. The Kalman filtering technique is used to improve state estimation and measurement fusion. Therefore, this paper describes the design and implementation of an Unscented Kalman Filter (UKF) with an adaptive tuning technique to adjust the process noise covariance matrix, resulting in better position estimation, particularly in high-dynamic motions of the mobile robot. Furthermore, for a mobile robot, this work proposes an Interacting Multiple Model (IMM) method with two dynamic motion models, i.e. Constant Velocity (CV) and Coordinated Turn (CT), to deliver high performance with minimal computational overhead. Several experiments on two types of mobile robots were conducted to evaluate and compare the effectiveness of the proposed localization algorithm. The proposed localization technique outperformed the other three methods and demonstrated its effectiveness in high dynamic motions due to the adaptive tuning system.

Estimation of Step Length With Wearable Thigh Sensor Using an Unscented Kalman Filter.

The determination of step length, an important gait parameter, has been a challenging task. Although unobtrusive sensors (inertial measurement units) have been developed recently, they cannot facilitate the automatic estimation of step length. In this article, we use a model-based technique to determine the step length using the Unscented Kalman Filter with angular velocity from a gyroscope inside the thigh pocket. We then propose a novel covariance estimation algorithm based on a screening technique that performs a search for the optimal Process Noise Covariance matrix. Upon implementing the Unscented Kalman Filter, the step length is found using the horizontal position of the foot relative to the hip using a patient-independent robust peak detection algorithm. This research article paves the way for algorithms that are computationally much faster than black box methods, with more scope for the development of better algorithms for covariance estimation using the one proposed in this article as a foundation.

Adaptive Fading-Memory Receding-Horizon Filters and Smoother for Linear Discrete Time-Varying Systems

In this paper, an adaptive fading-memory receding-horizon (AFMRH) filter is proposed by combining the receding-horizon structure and the adaptive fading-memory method. In the recent finite horizon, state error covariance is adapted with an adaptive fading factor; then the process noise covariance matrix adaption is realized by adjusting the properties of systems. An AFMRH fixed-lag smoother is also proposed by combining the proposed AFMRH filtering algorithm and a Rauch–Tung–Striebel smoothing algorithm to improve the estimation accuracy. Because the proposed AFMRH filter and smoother are reduced to the optimal receding-horizon (RH) filter and smoother when all measurements have the same weight, the proposed adaptive RH estimators could provide a more general solution than the optimal RH filter and smoother. To reduce the complexity and improve the estimation performance of the proposed RH estimators, an adaptive horizon adjustment method and a switching filtering algorithm based on an adaptive fading factor are also proposed. In particular, the proposed adaptive horizon adjustment method is designed to be computationally efficient, which makes it suitable for online and real-time applications. Through computer simulation, the performance and adaptiveness of the proposed approaches were verified by comparing them with existing fading-memory approaches.

Real-time state of charge estimation for electric vehicle power batteries using optimized filter

The main focus of the battery management system is the estimation of the battery's State of Charge (SOC), which is an indicator to determine the driving range of an electric vehicle. The Extended Kalman Filter (EKF) algorithm is the most promising for SOC estimation when the system is running. The EKF state estimation algorithm is sensitive to the process noise covariance matrix Q and measurement noise covariance matrix R. Inappropriate noise covariance matrices reduce the accuracy and cause divergence in state estimation. This paper uses the Sunflower Optimization algorithm (SFO) to find the optimal values of the noise covariance matrices before applying EKF for online SOC estimation. It is clearly stated that the iterative SFO does not affect the instantaneous response of EKF in the online estimate because the SFO is only performed once to determine the optimal values. The effectiveness of the proposed identification is examined through the constant discharge rate test and dynamic driving profile. The proposed algorithm's performance indices, such as maximum error, Mean Absolute Error, Mean Square Error, and Root Mean Square Error of SOC estimation, are low compared to the trial-error method, genetic algorithm, and adaptive extended Kalman filter. Besides accuracy, the proposed method quickly converges even when the initial SOC is inaccurate. The feasibility and benefits of the proposed SFO-EKF algorithm are demonstrated by qualitative and quantitative findings on five challenging datasets. To assess the performance of the SFO-EKF for SOC estimation in constant-current discharge and extreme drive cycle situations, three public datasets and two datasheets are employed. The simulation results show that the proposed method has high accuracy and a better convergence rate in estimating SOC under static and dynamic operating conditions.

Denoising algorithm of ground-airborne time-domain electromagnetic method based on Variational Bayesian-based adaptive Kalman filter (VBAKF)

The ground-airborne time-domain electromagnetic (GATEM) method, as a new type of electromagnetic method, offers detection efficiency and large investigation depth. It is an efficient electromagnetic detection technique for geological exploration and groundwater exploration. The field data are disturbed by motion noise and mixed noises, such as white noise and electromagnetic noises, which will reduce the data quality and accuracy of data interpretation. The characteristics of GATEM responses are analysed, and a denoising algorithm based on variational Bayesian-based adaptive Kalman filter (VBAKF) is proposed. Considering the attenuation characteristics of the GATEM response, the exponentially fitted data input into the VBAKF. Then the VBAKF with inverse Wishart priors is used to denoise GATEM field data, and the unknown process noise covariance matrix (PNCM) and measurement noise covariance matrix (MNCM) are estimated by variational Bayesian. The VBAKF algorithm is tested on the responses with different noises. It is verified on a layered geological model and compared with the adaptive Kalman filter (AKF) algorithm and the combined wavelet filter (CWF) algorithm. It is also tested on synthetic data of an anomalous model. Through synthetic data, the VBAKF algorithm can effectively suppress noise of GATEM data. Finally, the algorithm is applied to field data in Leshan, Jilin province, China. It effectively improves the quality of field data.

State Estimation of Time-Varying MRI with Radial Golden Angle Sampling

We propose a state estimation approach to time-varying magnetic resonance imaging utilizing a priori information. In state estimation, the time-dependent image reconstruction problem is modeled by separate state evolution and observation models. In our method, we compute the state estimates by using the Kalman filter and steady-state Kalman smoother utilizing a data-driven estimate for the process noise covariance matrix, constructed from conventional sliding window estimates. The proposed approach is evaluated using radially golden angle sampled simulated and experimental small animal data from a rat brain. In our method, the state estimates are updated after each new spoke of radial data becomes available, leading to faster frame rate compared with the conventional approaches. The results are compared with the estimates with the sliding window method. The results show that the state estimation approach with the data-driven process noise covariance can improve both spatial and temporal resolution.

Full adaptive Kalman filters for nonlinear fractional-order systems containing unknown parameters and fractional-orders

In this study, full adaptive Kalman filters are designed for continuous-time nonlinear fractional-order systems (FOSs) containing unknown parameters and fractional-orders. First, the estimated FOS is discretized using the Grünwald–Letnikov difference method to transform the fractional-order differential equation into a difference equation. Then, in terms of the nonlinear function contained in the investigated system, the Taylor expansion formula is adopted to linearize the discretized equation. Based on the method of augmented vector, an augmented state equation is established by state equations, unknown parameter equations, and fractional-order equation to achieve the state estimation. Besides, the sigmoid function is brought to ensure that the estimation of the fractional-order is performed in a suitable range. Because the covariance matrices of noises are difficult to be measured in physical systems, we also concern the problems on the state estimation, parameter estimation, and fractional-order estimation under the cases that the covariance matrix of process noise is unknown or the covariance matrix of measurement noise is unknown. Considering that the initial value can produce an error of state estimation and parameter identification for the FOS defined under the Caputo sense, the augmented vector method is used to achieve the initial value compensation. Finally, the effectiveness of the proposed algorithms is validated by four examples.

Consensus variational Bayesian moving horizon estimation for distributed sensor networks with unknown noise covariances

This paper deals with state estimation over distributed sensor networks with unknown process noise covariance matrix (PNCM) and measurement noise covariance matrices (MNCMs). The unknown PNCM and MNCMs are modeled with constrained inverse Wishart distributions and estimated together with the state trajectory within a moving horizon window. Each local node computes its local estimates by combining variational Bayes (VB) inference with Moving Horizon Estimation (MHE). Then, in order to diffuse the information through the network, at each time instant three consensus tasks are performed in parallel: the first one is a consensus on the parameters of the local approximated PNCM posterior; the second one is a consensus on the prior information on the state trajectory over the moving window; the third one is a consensus on the measurements over the moving window. The resulting consensus-based distributed VBMHE (D-VBMHE) algorithm generalizes existing consensus filters to the problem of jointly estimating the state trajectory over a window as well as to unknown PNCM and MNCMs. Simulation and experimental results on target tracking case-studies demonstrate the outperformance of the proposed D-VBMHE compared to state-of-the-art conventional and adaptive consensus filters.

Effect of Noise Covariance Matrices on State of Charge Estimation Using Extended Kalman Filter

The electrification of the transport sector increases the demand for a battery management system that allows the Li-ion battery to work in a sophisticated way to reach its maximum. The main focus of the battery management system is the estimation of the battery’s state of charge (SOC) which is an indicator to determine the driving range of an electric vehicle. However, battery modeling has a direct influence on SOC estimation. In this article, the equivalent circuit model (ECM) and extended Kalman filter (EKF) algorithm are combined for SOC estimation of the Li-ion battery. Since the performance of Kalman filter algorithms strictly depends on noise covariance matrices, incorrect values reduce the accuracy and make divergence in state estimation. Therefore, it is motivated to analyze the impact of process noise and measurement noise covariance matrix values on SOC estimation using Kalman filter algorithms. Ten different combinations of noise covariance matrices are employed to analyze the effect of these matrices on SOC estimation. As a result, the tuning of the filter is critical for improvement in state estimation regardless of the type of the battery model and filtering technique. The matrix values are chosen to take into account the convergence rate and estimation error. To validate the effectiveness of the chosen noise covariance matrices, the simulation results of the ECM based EKF estimation method is superimposed with the discharge curves available in the data sheet. The simulation results show minimal estimation error and a good convergence rate for tuned EKF.