Process Noise Covariance Matrices Research Articles

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.

A Novel Adaptive Kalman Filtering Approach to Human Motion Tracking With Magnetic-Inertial Sensors

This article presents an adaptive Kalman filtering approach to estimate the orientation of human body segments by using magnetic-inertial measurement units (MIMUs) with three-axis gyroscope, accelerometer, and magnetometer. In order to mitigate the negative impact of the external accelerations and ferromagnetic disturbances, the disturbances are modeled by first-order Gauss–Markov processes, and two parallel adaptive Kalman filters containing disturbance models are implemented to eliminate the disturbances. An adaptive factor is introduced to adjust the process noise covariance matrix to compensate for modeling errors induced by unmodeled gyroscope biases and the erroneous process noise covariance matrices. Furthermore, the outliers are checked and discarded by hypothesis tests on the innovation, and the measurement is rejected when the innovation exceeds a given confidence interval. Then, the estimated gravity acceleration and geomagnetic field are used to calculate the orientation quaternion using triaxial attitude determination algorithm. Finally, experiments under different motion scenarios are carried out to evaluate the effectiveness of the proposed method, and the performance of the proposed method is discussed in comparison with existing ones. A forward kinematics three-dimensional (3-D) human motion reconstruction method is proposed to drive the human skeleton model to reproduce human motion with a visual interface based on ROS platform.

Adaptive Kalman Filter for Detectable Linear Time-Invariant Systems

A novel covariance matching technique is proposed for estimating the states and unknown entries of the process and measurement noise covariance matrices for additive white Gaussian noise elements in a linear Kalman filter. Under this assumption of detectability (that is, unobservable modes remain stable), the stability and convergence properties of the covariance matching Kalman filter are established. It is shown that the measurement covariance matrix cannot be unambiguously estimated if the measurement model contains linearly dependent measurements. Monte Carlo simulations evaluate the numerical properties of the proposed algorithm.

Robust adaptive unscented Kalman filter for bearings-only tracking in three dimensional case

This paper proposes an improved version of Unscented Kalman Filter (UKF), namely Robust Adaptive UKF (RAUKF), with a special focus on Bearings-Only Target Tracking for three-dimensional case (3DBOT). The automatic tuning of the noise covariance matrices and the robust estimation of the target states form a critical point for the performance of the Kalman-type filtering algorithms, especially in the variable environmental conditions exposed in underwater. The key idea of the proposed filter is to combine robust aspects of UKF and adaption of the process and measurement noise covariance matrices with low computational complexity. The main contribution of this paper is to adjust these matrices by means of the steepest descent algorithm, and the H∞ technique is embedded to achieve superior performance in terms of accuracy and robustness against initial conditions and model uncertainties. Different experiments are performed to evaluate the performance of the proposed algorithm in the 3DBOT problem with a single moving observer. Simulations demonstrate that the proposed filter produce more accurate results with satisfactory computational burden in comparison with other methods.

A Novel Adaptive Kalman Filter With Inaccurate Process and Measurement Noise Covariance Matrices

In this paper, a novel variational Bayesian (VB)-based adaptive Kalman filter (VBAKF) for linear Gaussian state-space models with inaccurate process and measurement noise covariance matrices is proposed. By choosing inverse Wishart priors, the state together with the predicted error and measurement noise covariance matrices are inferred based on the VB approach. Simulation results for a target tracking example illustrate that the proposed VBAKF has better robustness to resist the uncertainties of process and measurement noise covariance matrices than existing state-of-the-art filters.

Design of measurement difference autocovariance method for estimation of process and measurement noise covariances

The paper deals with the estimation of the process and measurement noise covariance matrices of a system described by the linear time-varying state–space model. In particular, the stress is laid on the correlation methods and a novel method, the measurement difference autocovariance method, is designed. The proposed method is based on the statistical analysis of an augmented measurement prediction error leading to a system of linear matrix equations for the elements of the noise covariance matrices. Compared to other correlation methods, the proposed method provides unbiased estimates even for a finite number of measurements. The theoretical results are discussed and illustrated in a numerical example.

Uniqueness Conditions for ALS Problems

Abstract Knowledge of the process, measurement, and cross noise covariance matrices (denoted Q, R, and S, respectively) is necessary for tasks such as state estimation and performance monitoring. Several different types of algorithms have been developed to estimate these parameters from plant output data. Chief among them are the so-called correlation methods, such as autocovariance least squares (ALS). Despite the advances in covariance estimation algorithms, relatively little attention has been given to the topic of parameter identifiability. This paper discusses the limitations of when Q, R, and S are identifiable from output data. In particular, it is shown that for stable, linear time-invariant systems, the Kalman predictor gain, but not Q, R, and S, can be uniquely identified from the steady-state output autocovariance. Constrained ALS problems and the extension of the ALS problem to nonlinear and linear time-varying systems are also discussed.

Robust Adaptive Observer based on Multi wheeled Mobile Robot Cooperation Algorithm

FastSLAM2.0 is a framework for simultaneous localization of robot using a Rao-Blackwellized particle filter (RBPF). One of the problems of FastSLAM2.0 relates to the design of RBPF. The performance and quality of the estimation of RBPF depends heavily on the correct a priori knowledge of the process and measurement noise covariance matrices that are in most real-life applications unknown. On the other hand, an incorrect a priori knowledge may seriously degrade their performance. This paper presents an intelligent RBPF to solve this problem. In this method, two adaptive Neuro-Fuzzy inference systems (ANFIS) are used for tuning the process and measurement noise covariance matrices and for increasing acuuracy and consistency. In addition, we use particle swarm optimization (PSO) to optimize the performance of sampling. Experimental results demonstrate that the proposed algorithm is effective.

Six Object Tracking Algorithms: A Comparative Study

Objectives: To compare five different objects tracking algorithms performance wise with the proposed algorithm and to find out the best one among them for tracking of an object in occlusion and Background clutter condition i.e. when background contains target features. Methods/Analysis: In the proposed algorithm, the object’s position is obtained by Mean Shift tracking and then the prediction of object’s position is done through Kalman Filter. The Proposed method has Kalman Filter which consists of an adaptive system matrix and adaptive process error covariance and measurement error covariance matrices respectively. Adaptive system matrix of Kalman Filter is getting updated online depending on the quality of observation by Mean Shift algorithm and adaptive process and measurement noise covariance matrices are getting updated according to the variation in Bhattacharya Coefficient respectively. Findings: Proposed algorithm has a maximum value (0.2177 and 0.4821) of tracking efficiency metric i.e. Bhattacharya Coefficient in both video dataset among all the algorithms and it is taking less execution time (0.002761 sec and 0.005431 sec) than other Kalman filter based tracking algorithms. Applications/Improvements: Proposed algorithm is able to track the target in occlusion and background variation condition with more efficiency and less execution time than rest of the algorithms.

A Robust Finite-Horizon Kalman Filter for Uncertain Discrete Time-Varying Systems with State-Delay and Missing Measurements

In this paper, a robust kalman filter is designed for the uncertainty time-varying discrete systems with state delay in process and output matrices combined with the possibility of missing measurements. The uncertainties are expected in the process, output and white noise covariance matrices. A formula for a candidate upper bound on the actual state estimation error variances for all admissible parameter uncertainties and possible missing measurements is obtained. The filter parameters are optimized to give a minimal upper bound on the state estimation error covariance for all admissible uncertainties and missing measurements.

Tuning of GPS aided attitude estimation using evolutionary algorithms

Purpose – The purpose of this paper is to solve the problem of tuning of EKF parameters (process and measurement noise co-variance matrices) designed for attitude estimation using Global Positioning System (GPS) aided inertial sensors by employing a Human Opinion Dynamics (HOD)-based optimization technique and modifying the technique using maximum likelihood estimators and study its performance as compared to Particle Swarm Optimization (PSO) and manual tuning. Design/methodology/approach – A model for the determination of attitude of flight vehicles using inertial sensors and GPS measurement is designed and experiments are carried out to collect raw sensor and reference data. An HOD-based model is utilized to estimate the optimized process and measurement noise co-variance matrix. Added to it, few modifications are proposed in the HOD model by utilizing maximum likelihood estimator and finally the results obtained by the proposed schemes analysed. Findings – Analysis of the results shows that utilization of evolutionary algorithms for tuning is a significant improvement over manual tuning and both HOD and PSO-based methods are able to achieve the same level of accuracy. However, the HOD methods show better convergence and is easier to implement in terms of tuning parameters. Also, utilization of maximum likelihood estimator shows better search during initial iterations which increases the robustness of the algorithm. Originality/value – The paper is unique in its sense that it utilizes a HOD-based model to solve tuning problem of EKF for attitude estimation.

Extended state space recursive least squares

A new extended state space recursive least squares (ESSRLS) algorithm is proposed for state estimation of nonlinear systems. It is based on state space recursive least squares (SSRLS) approach and uses first order linearization of the system. It inherits the capability of obtaining state estimate without knowledge of process and measurement noise covariance matrices (Q and R respectively). The proposed approach is considered to provide new design option for scenarios where noise statistics and system dynamics vary. ESSRLS is initialized using delayed recursion method and a forgetting factor λ is employed to optimize the performance. The selection of λ can be problem specific as shown through experimental validations. However a value closer to and less than unity is generally recommended. Theoretical bases are validated by applying this algorithm to problems of tracking a non-conservative oscillator, a damped system with amplitude death and a signal modeled by mixture of Gaussian kernels. Simulation results show an MSE performance gain of 20 dB and 23 dB over extended Kalman filter (EKF) and unscented Kalman filter (UKF) while tracking van der Pol oscillator without knowledge about noise variances. The computational complexity of ESSRLS falls within that of EKF and UKF.

속도정합 및 매개변수 조정을 사용한 전달정렬의 성능 및 가관측성 분석

This paper considers the transfer alignment in the inertial navigation system which has lever-arm and the time delay in the velocity measurement. We suggest a method to improve the performance of the velocity matching. First, we analyze the estimation performance of the velocity matching through the tuning of the two covariance matrices of process noise and measurement noise. Next we provide some maneuvering conditions of the vehicles to improve the estimation performance using the observability analysis. The analysis results are verified using the computer simulations, which show that cruise movements do not provide the azimuth estimation of the vehicles, while east or north accelerating movement can provide.

NOx sensor ammonia cross-sensitivity estimation with adaptive unscented Kalman filter for Diesel-engine selective catalytic reduction systems

In order to meet the stringent Diesel-engine emission regulations, an aftertreatment system has become more and more popular in many countries. Specifically, to reduce the NOx emissions, a selective catalytic reduction (SCR) system has been successfully applied to many Diesel-engine applications. In the SCR system, NOx sensors are used to monitor the NOx concentration for the control strategies design and fault diagnosis. However, studies show that current NOx sensors are often cross-sensitive to ammonia. Thus, the reading of NOx sensor located at downstream of the SCR catalyst may not be accurate, due to the existing of ammonia slip in the system. Aiming to address this problem, some observers based on Kalman filtering theory are utilized to estimate the cross-sensitivity factor and the actual NOx concentration. Inspired by the fact that the adaptive unscented Kalman filter (AUKF) has the advantages of sample calculation and the capacity to deal with nonlinear system, the AUKF-based observer is investigated to estimate the factor in this paper. In addition, the process noise and measurement noise are usually unknown and time-varying in actual conditions. The AUKF would be effective to update the process and measurement noise covariance matrices online to adapt the practical SCR system. Simulation results through an experimentally-validated full vehicle simulator cX-Emission prove that the performance of designed observer based on AUKF is excellent.

Robust unscented Kalman filter with adaptation of process and measurement noise covariances

Unscented Kalman filter (UKF) has been extensively used for state estimation of nonlinear stochastic systems, which suffers from performance degradation and even divergence when the noise distribution used in the UKF and the truth in a real system are mismatched. For state estimation of nonlinear stochastic systems with non-Gaussian measurement noise, the Masreliez–Martin extended Kalman filter (EKF) gives better state estimates in relation to the standard EKF. However, the process noise and the measurement noise covariance matrices should be known, which is impractical in applications. This paper presents a robust Masreliez–Martin UKF which can provide reliable state estimates in the presence of both unknown process noise and measurement noise covariance matrices. Two numerical examples involving relative navigation of spacecrafts demonstrate that the proposed filter can provide improved state estimation performance over existing robust filtering approaches. Vision-aided robot arm tracking experiments are also provided to show the effectiveness of the proposed approach.

Propensity of Soot Deposition in a Rectangular Exhaust Gas Recirculation Cooler Using Kalman Filter

Abstract The detection of fouling in exhaust gas recirculation (EGR) coolers of diesel engines should be fast and accurate. This would facilitate deciding an effective strategy to combat fouling and to prolong the lifetime of EGR coolers. In the present study, the propensity of soot deposition in a rectangular EGR cooler is modeled using Kalman filters. Noises, coherent feature of many deposition processes which can be resulted from measurement sensors such as thermocouples or incidental deposit flake-off, are also considered in the model. The Kalman filter minimizes the estimation error covariance by considering the measurement and process noise covariance matrices while it can simultaneously handle the noisy data. The results are characterized with measurement process noise covariance. The relation between these two defines the smoothness and shape of the estimated trend of fouling resistance. Comparisons of the experimental data and the resultant model confirmed the usefulness of the applied method for various operating conditions of an EGR cooler prone to particulate deposition of soot particles. The paper proceeds with the impact of such models in monitoring fouling and taking an appropriate mitigation approach in diesel engines.

Multiple Model Adaptive Estimator for Nonlinear System with Unknown Disturbance

AbstractA multiple model adaptive estimator (MMAE) is presented for nonlinear systems with unknown disturbances. Multiple models are constructed with a set of process noise covariance matrices, such that the algorithm can adapt to different levels of unknown disturbances. The performance of the MMAE is analyzed for the considered system. It is proved that, under certain assumptions, the MMAE keeps the dynamics of its estimation error stable. A performance comparison among different estimators is carried out for space surveillance, where the position of a space target is estimated by using double line‐of‐sight measurements. Simulation studies illustrate that the presented algorithm outperforms the extended Kalman filter and the nonlinear robust filter.

Constant Modulus Blind Adaptive Beamforming Based on Unscented Kalman Filtering

An unscented Kalman filter-based constant modulus adaptation algorithm (UKF-CMA) is proposed for blind uniform linear beamforming. The proposed algorithm is obtained by first developing a model of the constant modulus (CM) criterion and then fitting that model into the Kalman filter-style state space model by using an auxiliary parameter. The proposed algorithm does not require a priori information about the process noise and measurement noise covariance matrices and hence it can be applied readily. Simulation results demonstrate that the proposed algorithm offers improved performance compared to the recursive least square-based CM (RLS-CMA) and least-mean square-based CM (LMS-CMA) algorithms for adaptive blind beamforming.

Hybrid state of charge estimation for lithium‐ion batteries: design and implementation

This study introduces a novel hybrid method for state of charge (SOC) estimation of lithium-ion battery types using extended H∞ filter and radial basis function (RBF) networks. The RBF network's parameters are adjusted off-line by acquired data from the battery in charging step. This kind of neural network approximates the non-linear function utilised in the state-space equations of the extended H∞ filter. The advantages of the proposed method are 3-fold: (i) it is not necessary to require the measurement and process noise covariance matrices as Kalman filter, (ii) the SOC is directly estimated and (3) it is a robust estimator in the sense of H∞ criteria. The state variables are composed of the SOC and the battery terminal voltage. The experimental results illustrate the feasibility of the proposed method in terms of robustness, accuracy and convergence speed.