Ordinary Kalman Filter Research Articles

Research on Information Fusion Technology of MEMS Gyro Array

The measurement accuracy of MEMS gyroscope is relatively low. Starting from the software level, the multi-sensor information fusion technology of MEMS gyroscope array (MGA) is used to reduce the drift of MEMS gyroscope. Firstly, a signal acquisition and processing system for communicating with ADXRS810 gyroscope based on field-programmable gate array (FPGA) is designed. Secondly, the collected drift data of the gyroscope array is preprocessed. long-term drift trend terms are obtained by ensemble empirical mode decomposition (EEMD) and the linear function fitting, and the trend terms are filtered out to obtain a smooth and normal drift signal. Then, the data fusion model based on time series analysis is built for the gyroscope array and the error analysis is performed by Allan variance. Finally, based on the AR (1) model, the moving horizon optimization Kalman filter method is used to filter the drift data of the gyroscope array. The experiment shows that when the time domain length N=3, the bias instability of gyroscope 1, 2, 3 and 4 decreases from 9.716 o/h, 8.5682 o/h, 13.484 o/h and 26.414 o/h to 1.2922 o/h, 0.61147 o/h, 1.4184 o/h and 1.6964 o/h, respectively, and the average noise coefficient decreases by more than 85%. Compared with the ordinary Kalman filter method, the bias instability has been significantly improved.

Zonotopic Kalman filtering for stability augmentation and flight envelope estimation

This article introduces a novel methodology based on zonotopic Kalman filtering for stabilizing attitude dynamics and estimating flight envelopes of an unmanned helicopter via an observer-based (output) feedback control law and a reachability analysis, respectively. The helicopter dynamics is represented by a linear state-space model, based on which the feedback control law is designed. Since not all state variables are measurable, state information is acquired by applying a zonotopic Kalman filter to yield sets of state-variable estimates of the helicopter in terms of zonotopes. Not only are the resulting zonotopes used for the observer-based feedback control, but also for estimating the flight envelopes of the helicopter based on the reachability analysis. This approach is useful for enhancing pilot’s awareness about dynamic responses of the helicopter, which may undergo unsafe flight conditions due to actuator faults triggered by undesirable perturbations. The efficacy of the proposed methodology is demonstrated via an example, where we also expose the benefits of applying the zonotopic Kalman filter as compared with an ordinary stochastic Kalman filter.

Kalman Filtering With Delayed Measurements in Non-Gaussian Environments

Traditionally, Kalman filter (KF) is designed with the assumptions of non-delayed measurements and additive white Gaussian noises. However, practical problems often fail to satisfy these assumptions and the conventional Kalman filter suffers from poor estimation accuracy. This paper proposes a modified Kalman filter to address both the problems of delayed measurements and non-Gaussian noises. The proposed filter is updated using correntropy maximization criterion, which is suitable for non-Gaussian noise environments. It falls short of a closed-form solution due to analytically complex equations that appear during the filtering. We use fixed-point iterative method to find an approximate solution. The delayed measurement problem is addressed by implementing a likelihood-based approach to identify the delay. Based on the identified delay information, the measurement is used to update the desired state in the subsequent past instant. To perform real-time filtering, the estimated state is further updated up to the current time instant using the process dynamics. The performance analysis validates the improved accuracy of the proposed method compared to the ordinary Kalman filter and its existing extensions.

Fractionally delayed Kalman filter

The conventional Kalman filter is based on the assumption of non-delayed measurements. Several modifications appear to address this problem, but they are constrained by two crucial assumptions: 1&#x0029 the delay is an integer multiple of the sampling interval, and 2&#x0029 a stochastic model representing the relationship between delayed measurements and a sequence of possible non-delayed measurements is known. Practical problems often fail to satisfy these assumptions, leading to poor estimation accuracy and frequent track-failure. This paper introduces a new variant of the Kalman filter, which is free from the stochastic model requirement and addresses the problem of fractional delay. The proposed algorithm fixes the maximum delay &#x0028 problem specific &#x0029 , which can be tuned by the practitioners for varying delay possibilities. A sequence of hypothetically defined intermediate instants characterizes fractional delays while maximum likelihood based delay identification could preclude the stochastic model requirement. Fractional delay realization could help in improving estimation accuracy. Moreover, precluding the need of a stochastic model could enhance the practical applicability. A comparative analysis with ordinary Kalman filter shows the high estimation accuracy of the proposed method in the presence of delay.

Unscented Kalman Filter-Based Online State-of-Charge Estimation in LiFePO4 Battery-Powered Electric Vehicles

Lithium iron phosphate (LiFePO4) battery has gained popularity due to its desirable characteristics such as low pollution, long cycling life, large power density, high safety and relatively low production costs. The estimation precision of State of Charge (SoC) of lithium iron phosphate (LiFePO4) battery determines the life and working efficiency of energy storage system. In practical applications, the battery capacity deteriorates with temperature drift and charge/discharge cycles, which make the accurate estimation of SoC a challenging issue. In this paper, a novel online estimation technique for SoC has been developed. Based on a simplified RC model, the Open Circuit Voltage (OCV) of the battery is calculated through two adaptive filtering stage, which is then used to determine the SoC via a dynamic OCV–SoC curve. In the first stage a variable step-size Least Mean Square (VSS-LMS) algorithm is employed to adaptively estimate the model parameters in real-time; in the second stage, an Unscented Kalman Filter (UKF) is used to estimate the OCV from the parameters. UKF can provide a more accurate and stable performance compared to Extended Kalman Filter at an equal computational complexity by utilizing deterministic sample points to calculate mean and covariance terms which are propagated through the true nonlinearity, without approximation. Further, the influence of polarization potential on OCV parameter measurement is analyzed and the averaged battery terminal voltage and current are introduced to avoid the large error of OCV estimate causing by the inconsistency of batteries. The impact of polarization over potential variation in battery capacity is eliminated through a smooth filter so as to achieve a stable online estimation process. Plenty of experiments under data acquisition systems with different precisions were carried out in order to confirm the effectiveness of the proposed system. The errors in estimating the battery’s SoC were below 2% compared to more than 5% and 9% in the case of using an extended Kalman filter and ordinary Kalman filter. The proposed SOC estimation method with its simplified model was transferred to DSP system also to verify the accuracy, simplicity and feasibility of SoC estimation for real-time application.

New estimation methodologies for well logging problems via a combination of fuzzy Kalman filter and different smoothers

New smoothers are proposed in this article to estimate the spectrum of radioactivity of Gamma ray in well-log data problems. Natural gamma-ray tool (NGT) is one of the most important instruments in well logging. The natural gamma-ray tool detects changes in natural radioactivity emerging from the variations in concentrations of elements as uranium (U), thorium (Th), and potassium (K). A computer is utilized to convert the data obtained by NGT to show concentrations of the elements in various depths. The aim of this research is to estimate the concentrations of these elements. Regular techniques have some uncertainties in the estimation of the elements of reservoir evaluation. To solve this problem, a fuzzy Kalman filter is designed to overcome uncertainty in comparison with the ordinary Kalman filter. Although the estimation result is better than Kalman filter, there is still a high level of noise on the estimation. Therefore, two types of smoothers including Rauch-Tung-Striebel and wavelet smoothers are used to reduce the noise level. Results show the better performances of the suggested smoothers in comparison with fuzzy Kalman filters.

Fading Kalman filter-based real-time state of charge estimation in LiFePO4 battery-powered electric vehicles

A novel online estimation technique for estimating the state of charge (SoC) of a lithium iron phosphate (LiFePO4) battery has been developed. Based on a simplified model, the open circuit voltage (OCV) of the battery is estimated through two cascaded linear filtering stages. A recursive least squares filter is employed in the first stage to dynamically estimate the battery model parameters in real-time, and then, a fading Kalman filter (FKF) is used to estimate the OCV from these parameters. FKF can avoid the possibility of large estimation errors, which may occur with a conventional Kalman filter, due to its capability to compensate any modeling error through a fading factor. By optimizing the value of the fading factor in the set of recursion equations of FKF with genetic algorithms, the errors in estimating the battery’s SoC in urban dynamometer driving schedules-based experiments and real vehicle driving cycle experiments were below 3% compared to more than 9% in the case of using an ordinary Kalman filter. The proposed method with its simplified model provides the simplicity and feasibility required for real-time application with highly accurate SoC estimation.

LakeMetabolizer: an R package for estimating lake metabolism from free-water oxygen using diverse statistical models

Metabolism is a fundamental process in ecosystems that crosses multiple scales of organization from individual organisms to whole ecosystems. To improve sharing and reuse of published metabolism models, we developed LakeMetabolizer, an R package for estimating lake metabolism from in situ time series of dissolved oxygen, water temperature, and, optionally, additional environmental variables. LakeMetabolizer implements 5 different metabolism models with diverse statistical underpinnings: bookkeeping, ordinary least squares, maximum likelihood, Kalman filter, and Bayesian. Each of these 5 metabolism models can be combined with 1 of 7 models for computing the coefficient of gas exchange across the air–water interface (k). LakeMetabolizer also features a variety of supporting functions that compute conversions and implement calculations commonly applied to raw data prior to estimating metabolism (e.g., oxygen saturation and optical conversion models). These tools have been organized into an R package that contains example data, example use-cases, and function documentation. The release package version is available on the Comprehensive R Archive Network (CRAN), and the full open-source GPL-licensed code is freely available for examination and extension online. With this unified, open-source, and freely available package, we hope to improve access and facilitate the application of metabolism in studies and management of lentic ecosystems.

다중경로 환경에서 형상필터를 이용한 CDGPS 성능 향상 기법

The quality of float solution deeply influences the performance of CDGPS because the theories being used in the integer ambiguity resolution method are derived under the assumption of AWGN. But in real world, the properties of noises are far from AWGN, especially when multipath are concerned. It results in the bias in float solution which affects the success rate of integer ambiguity and the precision of position in CDGPS. This paper designs an augmented Kalman filter using shaping filter and Kalman filter for the performance improvement of CDGPS on multipath. The experimental results with real measurements show that the correct integer ambiguity is always found while the success rates of WLSQ and ordinary Kalman filter are 5% and 18%, respectively. Eventually, the position accuracy is also improved by using the proposed algorithm.

Analysis of the US Real Estate Market: Time-Varying Estimation and Forecast of the S&P Case-Shiller Composite 20 Cities

This study investigates the determinants which drive the evolution of the American real estate prices. Housing prices are modelling by a Standard&Poor index known under the name of Case-Shiller Index Composite 20 which aims to quantify the residential housing market in 20 US metropolitan regions across the United States. Using different regression methods (Ordinary Least Squares and Kalman filter), we examine the time-varying sensibility of the selected risk factors to the Case-Shiller 20 index. Then, an econometric model is proposed to anticipate its monthly time series evolution over the 1991-2009 periods thanks to time-varying betas obtain with OLS and Kalman filter. One of the main difficulty concerns the consideration and the detection of the shift between a positive and a negative price return period (and vis-versa) by the modelling. Hence, the criterion decision about the choice of the best forecasting model is the one which capture the most regime switching. Based on monthly US market data from 1991 to 2009 period, we seek to compute objective and robust forecasts of the American housing prices.

A new approach to real‐time training of dynamic neural networks

AbstractA fast, efficient and new way of real‐time training of dynamic neural networks is presented in this paper. The proposed method would be suitable for training neural networks for real‐time input–output modelling of uncertain dynamic systems. The training method proposed in this paper is unique in the sense that only the outer layer weights are updated; the hidden layer weights are chosen randomly at the beginning of the process, and left unchanged. The outer layer weights are estimated with an ordinary Kalman filter. This provides a learning rate which is an optimally time‐varying vector. Results of computer simulation experiments, including a comparison with conventional backpropagation, are presented. Copyright © 2003 John Wiley & Sons, Ltd.

Analysis of an Adaptive Parameter Estimation and Fault Detection Technique

This paper presents the probabilistic analysis of a combined parameter estimation and fault detection method. The method is applicable in situations where a plant model is not available, the plant is subject to sudden changes, and real-time parameter estimation is required. An ordinary Kalman filter, in conjunction with a three-stage decision rule for fault detection, is used to estimate an input-output model of the unknown plant.

Estimation, Prediction, and Interpolation for Nonstationary Series with the Kalman Filter

Abstract We show how our definition of the likelihood of an autoregressive integrated moving average (ARIMA) model with missing observations, alternative to that of Kohn and Ansley and based on the usual assumptions made in estimation of and forecasting with ARIMA models, permits a direct and standard state-space representation of the nonstationary (original) data, so that the ordinary Kalman filter and fixed point smoother can be efficiently used for estimation, forecasting, and interpolation. In this way, the problem of estimating missing values in nonstationary series is considerably simplified. The results are extended to regression models with ARIMA errors, and a computer program is available from the authors.

Efficient L-D factorization algorithms for PDA, IMM, and IMMPDA filters

Efficient algorithms exist for the square-root probabilistic data association filter (PDAF). The same approach is extended to develop square-root versions of the interacting multiple model (IMM) Kalman filter and the IMMPDAF algorithms. The computational efficiency of the method stems from the fact that the terms needed in the overall covariance updates of PDAF, IMM, and IMMPDAF can be obtained as part of the square-root covariance update of an ordinary Kalman filter. In addition, a new square-root covariance prediction algorithm that is substantially faster than the usual modified weighted Gram-Schmidt (MWG-S) algorithm, whenever the process noise covariance matrix is time invariant, is proposed. >

Robust Kalman filter as a chemometric method for analytical data processing

The Kalman filter has been made robust by altering the scheme of the information feedback in the recursive algorithm of the ordinary Kalman filter. A limiting transformation which operates on the innovation term has been defined to eliminate or reduce the influence of outliers on the performance of the Kalman filter. The behaviour of the proposed robust Kalman filter was studied by computer simulation and the robustness to outliers was demonstrated.

INITIALIZING THE KALMAN FILTER FOR NONSTATIONARY TIME SERIES MODELS

Abstract. The problem of initializing the Kalman filter for nonstationary time series models is considered. Ansley and Kohn have developed a ‘modified Kalman filter’ for use with nonstationary models to produce estimates from what they call a ‘transformation approach’. We show that the same results can be obtained with a suitable initialization of the ordinary Kalman filter. Assuming there are d starting values for the nonstationary series, we initialize the Kalman filter using data through time d with the transformation approach estimate of the state vector and its associated error covariance matrix at time d. We give details of the initialization for ARIMA models, ARIMA component models and dynamic linear models. We present an example to illustrate how the results may differ from results obtained under more naive initializations that have been suggested.

GDOP, Ridge Regression and the Kalman Filter

Multicollinearity and its effect on parameter estimators such as the Kalman filter is analysed using the navigation application as a special example. All position-fix navigation systems suffer loss of accuracy when their navigation landmarks are nearly collinear. Nearly collinear measurement geometry is termed the geometric dilution of position (GDOP). Its presence causes the errors of the position estimates to be highly inflated. In 1970 Hoerl and Kennard developed ridge regression to combat near collinearity when it arises in the predictor matrix of a linear regression model. Since GDOP is mathematically equivalent to a nearly collinear predictor matrix, Kelly suggested using ridge regression techniques in navigation signal processors to reduce the effects of GDOP. The original programme intended to use ridge regression not only to reduce variance inflation but also to reduce bias inflation. Reducing bias inflation is an extension of Hoerl's ridge concept by Kelly. Preliminary results show that ridge regression will reduce the effects of variance inflation caused by GDOP. However, recent results (Kelly) conclude it will not reduce bias inflation as it arises in the navigation problem, GDOP is not a mismatched estimator/model problem. Even with an estimator matched to the model, GDOP may inflate the MSE of the ordinary Kalman filter while the ridge recursive filter chooses a suitable biased estimator that will reduce the MSE. The main goal is obtaining a smaller MSE for the estimator, rather than minimizing the residual sum of squares. This is a different operation than tuning the Kalman filter's dynamic process noise covariance Q, in order to compensate for unmodelled errors. Although ridge regression has not yielded a satisfactory solution to the general GDOP problem, it has provided insight into exactly what causes multicollinearity in navigation signal processors such as the Kalman filter and under what conditions an estimator's performance can be improved.

Efficient generalized cross-validation for state space models

For a signal plus noise model with a state space representation, an efficient procedure is given for obtaining the trace of the influence matrix, where the influence matrix expresses the estimated signal vector as a linear combination of the observed data. This allows an O(n) evaluation of the generalized cross-validation criterion function. Our approach is very efficient, requiring the addition of only one equation to the ordinary Kalman filter, and extends to models with linear regressors in the observation equation. Important applications are to spline smoothing in regression and to variance components models for seasonal time series.

State Estimation of a Scalar Dynamic Precipitation Model From Time‐Aggregate Observations

Developed is an exact, recursive, state estimator for linear, scalar systems whose observables are time integrals, over an interval Δt, of linear functions of the state. In general, implementation of the estimator results in excessive computations. The computationally inexpensive, ordinary Kalman filter with inflated observation error variance is used to approximate the developed estimator in terms of prediction accuracy. Comparison of the exact estimator and of the Kalman filter, using precipitation data of various aggregation intervals Δt and a scalar dynamic precipitation model, shows that the Kalman filter develops prediction bias as Δt increases.

Extended recursive maximum-likelihood identification algorithm

An extension of the well known recursive maximum-likelihood or extended least-squares identification algorithm to improve its rate of convergence is presented. The extension is to include explicitly in the model giving rise to the algorithm the errors in the noise-generating sequence normally ignored. The extended algorithm is related to the original, much as an extended Kalman filter is related to an ordinary Kalman filter. The paper compares the algorithm with a similar, but differently motiviated, one, recently described by Ljung. The performance of the algorithm and its optimisation are discussed with reference to computational results from Monte Carlo tests.