Relative Study of Measurement Noise Covariance R and Process Noise Covariance Q of the Kalman Filter in Estimation (original) (raw)
Related papers
Process and Measurement Noise Estimation for Kalman Filtering
Conference Proceedings of the Society for Experimental Mechanics Series, 2011
The Kalman filter gain can be extracted from output signals but the covariance of the state error cannot be evaluated without knowledge of the covariance of the process and measurement noise Q and R. Among the methods that have been developed to estimate Q and R from measurements the two that have received most attention are based on linear relations between these matrices and: 1) the covariance function of the innovations from any stable filter or 2) the covariance function of the output measurements. This paper reviews the two approaches and offers some observations regarding how the initial estimate of the gain in the innovations approach may affect accuracy.
Noise variance estimation for Kalman filter
E3S Web of Conferences
In this paper, we propose an algorithm that evaluates noise variance with a numerical integration method. For noise variance estimation, we use Krogh method with a variable integration step. In line with common practice, we limit our study to fourth-order method. First, we perform simulation tests for randomly generated signals, related to the transition state and steady state. Next, we formulate three methodologies (research hypotheses) of noise variance estimation, and then compare their efficiency.
This paper presents an investigation towards developing a better understanding of the Kalman filtration process by adding an updating equation to the covariance of the random vector process noise in the main algorithm of Kalman. This updating equation results from the theoretical proof of the relationship between Unified Least Square Technique and the Kalman algorithm. Two numerical examples are used to illustrate the effect of the new step added to Kalman algorithm. In the first example, statistical analysis is applied on the original observations. No outliers were detected in the original observations. Three solutions were applied on the data. First, Kalman Filtration without updating the covariance of random vector process noise. Second, Kalman filtration with updating equation is added to the algorithm. Third, Recursive least square technique is used. In the second numerical example, original observations were collected from GPS observations to determine the deformation of two towers supporting a Tianjin Yong Highway cable -stayed bridge in China. Original observations were suffering from outliers. Using the same previous strategy to estimate the state vector and its variance. Finally we conclude that, when the original observations suffering from outliers, Updating the equation of the covariance of the random process noise must be added to Kalman algorithm to improve the performance of filtration process and to overcome the existence of outliers. Adding the new equation improves the variance of the estimated state vector to be identical with Recursive least Square Technique.
In this research we have demonstrated Kalman Filter (KF) that improves the quality of the measurement of sensor signal. Kalman Filter has long been used to eliminate the process error and measurement noise. Bearing in mind that almost all industrial automation and control systems are stock with process errors and measurement noises, we tried to implement Kalman Filtering algorithm to typical processes that measure the height of the water level of a tank and the angle of deviation of the wheel of a Mobile Robot (MR) from a predefined guided path. First a simulation study was conducted using the developed Kalman Filter algorithm. The algorithm was then translated and transferred to a real-word implementation domain which is an electronic computing module. While a level detector (pressure sensor) was used to sense the height of the real-time water levels under filling, dropping, both conditions, the LVDT transducer, developed in the laboratory was used to measure the angle of deviation of a MR's track in a lab room experimental setting. It was observed from the results that process error and measurement noise can be eliminated using KF. The paper systematically presents the results after reviewing the theoretic model of the KF and the application of families of KFs. We were able to reduce the errors and noise from about 15% to 5% using KF technique.
2010 20th International Conference on Pattern Recognition, 2010
This paper introduces a reformulation of the extended Kalman Filter using the Gauss-Helmert model for least squares estimation. By proving the equivalence of both estimators it is shown how the methods of statistical analysis in least squares estimation can be applied to the prediction and update process in Kalman Filtering. Especially the efficient computation of the reliability (or redundancy) matrix allows the implementation of self supervising systems. As an application an unparameterized method for estimating the variances of the filters process noise is presented.
Updating Noise Parameters of Kalman Filter using Bayesian Approach
In the present study, the Bayesian probabilistic approach is proposed to estimate the process noise and measurement noise parameters in the Kalman filter for the case when the input is a zero-mean Gaussian white noise process and limited output measurements are available. The methodology presented in this study is based on the idea that the process noise and measurement noise parameters directly affect the state estimate and the covariance matrix of the Kalman filter. Therefore, the noise parameters can be estimated by expressing the likelihood factor of the measurements as a function of the state estimate and the covariance matrix for a given set of process noise and measurement noise parameters. By evaluating the likelihood factor for all the noise parameters within the search domain, the optimal estimates of the noise parameters are chosen by the maximum likelihood criterion. Through the demonstration of an example, the optimal estimates of the noise parameters are close to the a...
A new process noise covariance matrix tuning algorithm for Kalman based state estimators
7th IFAC International Symposium on Advanced Control of Chemical Processes (2009), 2009
A suitable design of state estimators requires a representative model for capturing the plant behavior and knowledge about the noise statistics, which are generally not known in practical applications. While the measurement noise covariance can be directly derived from the measurement device reproducibility, the choice of the process noise covariance is much less straightforward. Further, processes such as continuous process with grade transitions and batch or semi-batch process are characterized by time-varying structural uncertainties which are, in many cases, partially and indirectly reflected in the uncertainty of the model parameters. It has been shown that the robust performance of state estimators significantly enhances with a time-varying and non-diagonal process noise covariance matrix, which explicitly takes parameter uncertainty into account. For this case, the parameter uncertainty is quantified through the parameter covariance matrix. This paper presents a direct and a sensitivity method for the parameter covariance matrix computation. In the direct method, the parameter covariance matrix is found during the parameter estimation step of the SELEST algorithm, while in the sensitivity method, the parameter covariance matrix is obtained through a time-varying sensitivity matrix. The results have shown the efficacy of these methods in improving the performance of an extended Kalman filter (EKF) for a semi-batch reactor process.
A Modified Kalman Filter for Non-gaussian Measurement Noise
A novel modification is proposed to the Kalman filter for the case of non-Gaussian measurement noise. We model the non-Gaussian data as outliers. Measurement data is robustly discriminated between Gaussian (valid data) and outliers by Robust Sequential Estimator (RSE). The measurement update is carried out for the valid data only. The modified algorithm proceeds as follows. Initially, the robust parameter and scale estimates of the measurement data are obtained for a sample of data using maximum likelihood estimates for a t-distribution error model through Iteratively Reweighted Least Squares (IRLS). The sample is dynamically updated with each new observation. Sequential classification of each new measurement is decided through a weighting scheme determined by RSE. State updates are carried out for the valid data only. Simulations provide satisfactory results and a significant improvement in mean square error with the proposed scheme.
This report provides a brief historical evolution of the concepts in the Kalman filtering theory since ancient times to the present. A brief description of the filter equations its aesthetics, beauty, truth, fascinating perspectives and competence are described. For a Kalman filter design to provide optimal estimates tuning of its statistics namely initial state and covariance, unknown parameters, and state and measurement noise covariances is important. The earlier tuning approaches are reviewed. The present approach is a reference recursive recipe based on multiple filter passes through the data without any optimization to reach a 'statistical equilibrium' solution. It utilizes the a priori, a posteriori, and smoothed states, their corresponding predicted measurements and the actual measurements help to balance the measurement equation and similarly the state equation to help form a generalized likelihood cost function. The filter covariance at the end of each pass is heuristically scaled up by the number of data points is further trimmed to statistically match the exact estimates and Cramer Rao Bounds (CRBs) available with no process noise provided the initial covariance for subsequent passes. During simulation studies with process noise the matching of the input and estimated noise sequence over time and in real data the generalized cost functions helped to obtain confidence in the results. Simulation studies of a constant signal, a ramp, a spring, mass, damper system with a weak non linear spring constant, longitudinal and lateral motion of an airplane was followed by similar but more involved real airplane data was carried out in MATLAB R. In all cases the present approach was shown to provide internally consistent and best possible estimates and their CRBs. i ACKNOWLEDGEMENTS It is a pleasure to thank many people with whom the authors interacted over a period of time in the topic of Kalman filtering and its Applications. Decades earlier this topic was started as a course in the Department of Aerospace Engineering and a Workshop was conducted along with Prof. S. M. Deshpande who started it and then moved over full time to Computational Fluid Dynamics. Subsequently MRA taught the course and spent many years carrying out research and development in this area during his tenure at the IISc, Bangalore. The problem of filter tuning has always been intriguing for MRA since most people in the area tweaked rather than tuned most of the time with the result that there is no one procedure that could be used routinely in applying the Kalman filter in its innumerable applications. The PhD thesis of RMOG has been the only effort for a proper tuning of the filter parameters but this was not too well known. In the recent past for a couple of years the interaction among the present authors helped to reach the present method which we believe is quite good for such routine applications. The report has been written in such a way to be useful as a teaching material. Our grateful thanks are due to
arXiv: Optimization and Control, 2015
Traditional statements of the celebrated Kalman filter algorithm focus on the estimation of state, but not the output. For any outputs, measured or auxiliary, it is usually assumed that the posterior state estimates and known inputs are enough to generate the minimum variance output estimate, given by yn|n = Cxn|n + Dun. Same equation is implemented in most popular control design toolboxes. It will be shown that when measurement and process noises are correlated, or when the process noise directly feeds into measurements, this equation is no longer optimal, and a correcting term is needed in above output estimation. This natural extension can allow designer to simplify noise modeling, reduce estimator order, improve robustness to unknown noise models as well as estimate unknown input, when expressed as an auxiliary output. This is directly applicable in motion control applications which exhibits such feed-through, such as estimating disturbance thrust affecting the accelerometer mea...