Central Difference Kalman Filter Research Articles

Square-Root Sigma-Point Filtering Approach to State Estimation for Wind Turbine Generators in Interconnected Energy Systems

The internal states of generators obtained by dynamic states estimation (DSE) may provide additional information for the control performances. However, conventional sigma-point Kalman filter (SPKF)-based DSE may experience the loss of positive definiteness and symmetricalness of state noise covariances. This article proposes three numerically stable square-root SPKF (SR-SPKF) algorithms and proposes a novel derivative-free SR-SPKF-based DSE framework to estimate the dynamic states for doubly fed induction generator (DFIG) wind turbines in an interconnected power network. While this article investigates the dynamic behavior of the power grid at a system-level, the DSE of DFIG is achieved in a decentralized manner, which is made possible by the use of phasor measurement units (PMUs) to acquire and transmit voltage and current phasors at DFIG terminal. By utilizing the SR-SPKF-based DSE framework and PMUs data, a comparison study is conducted for square-root unscented Kalman filter, square-root Cubature Kalman filter, square-root central difference Kalman filter, and their conventional versions. The computational burden, estimation accuracy, and mathematical capability of each filtering algorithm are compared and analyzed through simulation studies.

Speed-sensorless control system of a bearingless induction motor based on iterative central difference Kalman filter

ABSTRACT In order to obtain speed self-detecting with low cost for a bearingless induction motor (BIM) a speed-sensorless control strategy based on the iterative central difference Kalman filter (ICDKF) is proposed. Firstly, on the basis of the BIM mathematical model, the nonlinear state equation is established and its order is reduced from fifth-order to fourth-order using the stator terminal voltage and current as input. Then, a sterling interpolation formulation is used in the filter to reduce the model error, and an iteration loop link is adopted to improve the filter accuracy. Finally, the online speed of the BIM is identified through the filter rotor speed estimation. Theoretical analysis, simulation and experimental results by UKF and CDKF method have been compared. The results show that the proposed speed-sensorless control system not only has good speed tracking performance and reduce the load disturbance but also improves the BIM suspension performance.

Design of a robust central difference kalman filter in the presence of uncertainties and unknown measurement errors

In this study, a novel RCDKF11Robust Central Difference Kalman Filter. is proposed for nonlinear plants. The system under consideration is a comprehensive case where the norm bounded uncertainties are supposed in both the state and output equations. Since the filtering methodology has been suggested based on the Stirling formula, it requires no derivative or Jacobian matrix computations. Furthermore, the mean value of uncertainties has been eliminated in the development of the filter structure. Different types of uncertainties are incorporated in the standard formulation of CDKF and it is demonstrated that the upper bound of covariance matrices can be derived according to them. In contrast to the similar works, the stability of the proposed filtering strategy is proved by the Lyapunov theory and the upper bound of state estimation error covariance is found for all admissible uncertainties. To demonstrate the performance of the introduced estimation algorithm, it will be evaluated on the attitude determination system of a three-axis satellite including a star camera and gyro sensor. The simulation results of the proposed filter are compared with the conventional CDKF,22Central Difference Kalman Filter. NRF,33Nonlinear Robust Filter. EKF44Extended Kalman Filter. and PF55Particle Filter. and the better efficiency of the developed method is established.

Posterior Inference on Parameters in a Nonlinear DSGE Model via Gaussian-Based Filters

This paper studies Gaussian-based filters within the pseudo-marginal Metropolis Hastings (PM-MH) algorithm for posterior inference on parameters in nonlinear DSGE models. We implement two Gaussian-based filters to evaluate the likelihood of a DSGE model solved to second and third order and embed them into the PM-MH: Central Difference Kalman filter (CDKF) and Gaussian mixture filter (GMF). The GMF is adaptively refined by splitting a mixture component into new mixture components based on Binomial Gaussian mixture. The overall results indicate that the estimation accuracy of the CDKF and the GMF is comparable to that of the particle filter (PF), except that the CDKF generates biased estimates in the extremely nonlinear case. The proposed GMF generates the most accurate estimates among them. We argue that the GMF with PM-MH can converge to the true and invariant distribution when the likelihood constructed by infinite Gaussian mixtures weakly converges to the true likelihood. In addition, we show that the Gaussian-based filters are more efficient than the PF in terms of effective computing time. Finally, we apply the method to real data.

Low-Power Centimeter-Level Localization for Indoor Mobile Robots Based on Ensemble Kalman Smoother Using Received Signal Strength

How to provide a low-cost but accurate localization solution for the indoor mobile robots are essential in many Internet of Things applications, such as smart home and asset tracking. To achieve this goal, this paper originally proposes a modified two-filter smoother based on ensemble Kalman filter (KF) (denoted as EnKS) for the localization of indoor mobile robots. The proposed EnKS algorithm consists of both a forward part of an ensemble KF (EnKF) with statistical linear regression and a backward part of a modified information KF with state error vector. The EnKS based on stochastic sampling with ensemble members can achieve better positioning accuracy than other Kalman smoothers. When compared to EnKF, the proposed EnKS combines a backward filter to compensate for the estimation error of EnKF and further improves the accuracy. Furthermore, the implementation of the proposed EnKS is conducted in the real world visible light positioning (VLP) system using pre-existing LED lights for low-cost robot localization. To make a performance comparison, this paper also uses baseline smoothers based on extended KF and central difference KF in the VLP system. Preliminary experimental results imply that the proposed EnKS is able to achieve the best positioning accuracy, as high as 11.18 cm on average, but with a comparable computational complexity, which enables to meet the demands of many robot applications.

A novel STSOSLAM algorithm based on strong tracking second order central difference Kalman filter

Simultaneous Localization and Mapping (SLAM) is an effective technique in the field of robot location and navigation. However, when the existing SLAM algorithm is applied in harsh terrain, such as the terrain found in coal mines, accuracy suffers, and on-line adaptive adjustment capability is poor. Furthermore, the system suffers from low robustness and is susceptible to random noise. In order to solve these problems, we propose an innovative Strong Tracking Second Order Central Difference SLAM (STSOSLAM) algorithm that combines a Strong Tracking Filter (STF), a Second-Order Central Differential Filter (SOCDF), and a Particle Filter (PF). The new algorithm utilizes the second order sterling interpolation formula to deal with the nonlinear system problem using the Cholesky decomposition technique, which propagates directly by using the covariance square root factor in the SLAM probabilistic estimation. This technique not only guarantees the positive definite property of the covariance matrix, but also reduces the truncation error of local linearization. In addition, STF is introduced into the algorithm. It updates every sigma point using an adaptive algorithm and obtains optimized filter gain through the STF online adjustment factor and suppresses uncertain noise and the influence of initial value selection. Through simulation and experiments, STSOSLAM algorithm shows much better performance in terms of estimation accuracy, robustness and reliability than FastSLAM2.0 and Central Difference FastSLAM (CDFastSLAM) algorithms, establishing the foundation of applying the STSOSLAM algorithm in the harsh terrain of coal mines.

Available power prediction limited by multiple constraints for LiFePO4 batteries based on central difference Kalman filter

The available power of vehicle-mounted batteries needs to be real-time predicted to accommodate prospective driving demands of overtaking, gradient climbing, constant-speed cruising, and regenerative braking. Generally, battery power capabilities are limited by multiple constraints, for example, terminal voltage, current, state-of-charge (SoC), and State-of-Energy (SoE). This paper constructs SoC and SoE estimators resorting to the square-root central difference Kalman filter (SR-CDKF) and an equivalent circuit model with online updated parameters. In addition, a battery thermal evolution model is formulated, whereby the ordinarily ignored temperature constraint is taken into account. Based on above achievements, a battery power prediction scheme conforming to multiple constraints is realized. Finally, experimental verifications are conducted on a LiFePO4 battery pack subject to consecutive Federal Urban Driving Schedule profiles. The SR-CDKF-based SoC and SoE estimators give accurate results under different conditions. In contrast to the conventional PNGV-HPPC method, the proposed method behaves with more reliability and robustness at different time horizons, temperatures, and aging levels. The assessment results justify the effectiveness of the battery modeling and algorithm utilization efforts.

Fractional central difference Kalman filter with unknown prior information

Abstract In this paper, a generalized fractional central difference Kalman filter for nonlinear discrete fractional dynamic systems is proposed. Based on the Stirling interpolation formula, the presented algorithm can be implemented as no derivatives are needed. Besides, in order to estimate the state with unknown prior information, a maximum a posteriori principle based adaptive fractional central difference Kalman filter is derived. The adaptive algorithm can estimate the noise statistics and system state simultaneously. The unbiasedness of the proposed algorithm is analyzed. Several numerical examples demonstrate the accuracy and effectiveness of the two Kalman filters.

A strong robust observer of distributed drive electric vehicle states based on strong tracking-iterative central difference Kalman filter algorithm

This article proposed a strong robust observer of distributed drive electric vehicle states, including yaw rate, sideslip angle, and longitudinal velocity. Based on strong tracking filter algorithm framework, the proposed observer realized a strong tracking-iterative central difference Kalman filter by introducing a time-varying fade factor into iterative central difference Kalman filter. The introducing of time-varying fade factor assigns approximate orthogonality to residual error, which improves robustness of the observer at mutation conditions. Calculation efficiency and accuracy are improved by applying central difference transformation to approach posterior mean and posterior co-variance. By correcting variance and co-variance with combination of states updating and Gauss–Newton iteration, the observer also achieves high estimation accuracy and convergence rate. Finally, the observer was simulated in vehicle dynamics simulator veDYNA at slalom test and double lane change test with high and low road friction coefficients, respectively. Simulation results have verified that the observer has higher estimation accuracy as well as robustness comparing to extended Kalman filter.

Hybrid consensus sigma point approximation nonlinear filter using statistical linearization

In this paper, a hybrid consensus sigma point approximation nonlinear filter is proposed for state estimation in collaborative sensor network, where hybrid consensus of both measurement and information is utilised. Statistical linearization of nonlinear functions is used in sigma point filters, that is, unscented Kalman filter (UKF), cubature Kalman filter (CKF), and central difference Kalman filter (CDKF). Stability of the proposed algorithm is also analysed with the help of linearization operation and some conservative assumptions. Two typical target tracking examples are used to demonstrate the effectiveness of the proposed algorithms. Simulation results show that the proposed algorithms are more stable than existing algorithms, and among our proposed algorithms, CKF- and CDKF-based algorithms are more accurate and stable than the UKF-based one.

EGP-CDKF for Performance Improvement of the SINS/GNSS Integrated System

Position and orientation system (POS), which typically integrates strapdown inertial navigation system (SINS) and global navigation satellite system (GNSS), serves as a key sensor in airborne remote sensing, mobile mapping, and vehicle localization. POS can provide reliable, high-frequency and high-precision motion parameters using nonlinear Kalman Filter models based on fusion methods, such as extended Kalman filter, unscented Kalman filter, central difference Kalman filter (CDKF), and square root CDKF (SR-CDKF). Although the nonlinear parametric models are of high efficiency, there are also limitation on their capabilities of prediction and estimation, as it is often impossible to model all aspects of POS. In this paper, Gaussian processes (GP)-based is presented to enhance the capabilities of prediction and estimation for parametric CDKF. On one hand, it can estimate the state vector of POS with the nonlinear parametric CDKF on condition that trained data is limited; on the other hand, GP can take both the noise and the uncertainty in the nonlinear parametric CDKF into consideration. Consequently, the incorporation of GP into CDKF can result in further performance improvement. The proposed approach is verified in the real experiment, and shows that large performance benefits are achieved through applying the enhanced GP-CDKF(EGP-CDKF) into the SINS/GNSS integrated system.

Investigation of optical current transformer signal processing method based on an improved Kalman algorithm

This paper explores the problem of signal processing in optical current transformers (OCTs). Based on the noise characteristics of OCTs, such as overlapping signals, noise frequency bands, low signal-to-noise ratios, and difficulties in acquiring statistical features of noise power, an improved standard Kalman filtering algorithm was proposed for direct current (DC) signal processing. The state-space model of the OCT DC measurement system is first established, and then mixed noise can be processed by adding mixed noise into measurement and state parameters. According to the minimum mean squared error criterion, state predictions and update equations of the improved Kalman algorithm could be deduced based on the established model. An improved central difference Kalman filter was proposed for alternating current (AC) signal processing, which improved the sampling strategy and noise processing of colored noise. Real-time estimation and correction of noise were achieved by designing AC and DC noise recursive filters. Experimental results show that the improved signal processing algorithms had a good filtering effect on the AC and DC signals with mixed noise of OCT. Furthermore, the proposed algorithm was able to achieve real-time correction of noise during the OCT filtering process.

Improved estimation and fault detection scheme for a class of non‐linear hybrid systems using time delayed adaptive CD state estimator

An improved fault detection scheme for a non-linear hybrid system with delayed measurement by using a modified non-linear adaptive state estimator is proposed. The proposed estimator performs acceptably even when the (i) covariance of the measurement noise is unknown and also (ii) when the measurements are delayed. The algorithm for the proposed estimator called time-delayed R-adaptive central difference Kalman filter (TD-RACDKF) uses a modified R-adaptive 2nd order Central Difference estimator, also called Central Difference Kalman Filter (CDKF) in some literature. Algorithm for the proposed TD-RACDKF has been presented and its performance evaluated as well as characterised with Monte Carlo simulation on two standard non-linear systems with delayed measurements. The characterisation includes comparison with existing estimators. Having demonstrated the improved performance of the proposed state estimator, its use for fault detection of non-linear hybrid systems was investigated. The performance of the fault detection scheme has been illustrated with the help of a benchmark non-linear hybrid system, namely ‘three tank system’ and by comparison with a previously available extended KF-based estimator.

A New Sigma-point Filter–Uniform Random Sampling Kalman Filter

This paper is motivated by the problem of more accurately estimating the hidden state in nonlinear dynamic system. We propose a uniform random sampling Kalman filter(URSKF) which can be regarded as a Sigma-point filter based on the statistical linearization method. Compared with other Sigma-point filters, the uniform random sampling method used to obtain the deterministic points can match the any-order moment of the prior distribution with the moderately increasing sampling points and automatically capture the more rich statistical properties of the system after nonlinear function mapping. Moreover, the URSKF is a derivative and square-rooting free operation which avoid computing Jacobian matrix and failing the filtering process for not always guaranteeing the covariance matrix to be positive definite, such as UKF. Besides, the computation complexity is linearly related to the system dimension avoiding the curse of high dimension in the Gauss-Hermite Quadratute Filter(GHQF). The performance of this filter is demonstrated by an aircraft tracking model. The simulation results show the URSKF performs higher accurately than the Unscented Kalman Filter(UKF), Central Difference Kalman Filter(CDKF) and Cubature Kalman Filter(CKF).

Hybrid consensus sigma point approximation nonlinear filter using statistical linearization

In this paper, a hybrid consensus sigma point approximation nonlinear filter is proposed for state estimation in collaborative sensor network, where hybrid consensus of both measurement and information is utilised. Statistical linearization of nonlinear functions is used in sigma point filters, that is, unscented Kalman filter (UKF), cubature Kalman filter (CKF), and central difference Kalman filter (CDKF). Stability of the proposed algorithm is also analysed with the help of linearization operation and some conservative assumptions. Two typical target tracking examples are used to demonstrate the effectiveness of the proposed algorithms. Simulation results show that the proposed algorithms are more stable than existing algorithms, and among our proposed algorithms, CKF- and CDKF-based algorithms are more accurate and stable than the UKF-based one.

Observation bootstrapping CDK-GMPHD filter based on consensus fusion strategy

Aiming at the adverse effect caused by the nonlinear, the randomness of observation noise and the uncertainty of external disturbances on the precision of state estimation and target number identification in multi-target tracking system, we propose a novel observation bootstrapping CDK-GMPHD filter based on consensus fusion strategy in this paper. Firstly, for the nonlinear of estimated system, we introduce central difference Kalman filter(CDKF) into the realization framework of Gaussian mixture probability hypothesis density (GMPHD) filter, and designs a new improved GMPHD filter based on CDKF. Secondly, considering the extraction and utilization of redundancy and complementary information form multiple observations can improve the stability and reliability of observations, we construct the bootstrapped observation set by the data assimilation technique. And on this basis, combined with the realization mechanism of membership function in fuzzy set theory, the consensus distance and the consensus matrix are further designed on the basis of confidence distance. Finally, according to the weighted fusion strategy, we realize the effective utilization of bootstrapped observations. The theoretical analysis and experimental results show the feasibility and efficiency of proposed algorithm.

A Strong Tracking Square Root Central Difference FastSLAM for Unmanned Intelligent Vehicle With Adaptive Partial Systematic Resampling

An improved fast simultaneous localization and mapping (FastSLAM) algorithm based on the strong tracking square root central difference Kalman filter (STSRCDKF) with adaptive partial systematic resampling is proposed in this paper to solve the large-scale simultaneous localization and mapping (SLAM) problem for unmanned intelligent vehicle. In the proposed algorithm, STSRCDKF is composed of a strong tracking filter and a square root central difference Kalman filter. STSRCDKF is used to design an adaptive adjusting proposal distribution of the particle filter and to estimate the Gaussian densities of the landmarks. Moreover, an adaptive partial systematic resampling operation is carried out to reduce the degree of particle degeneracy and maintain the diversity of particles. The performance of the proposed algorithm is compared with that of central difference FastSLAM and FastSLAM2.0; the simulation results based on the simulator and two benchmark data sets verify that the proposed algorithm has better adaptability and robustness to respond with time-varying measurement noise. In addition, it reduces computational cost and improves state estimation accuracy and consistency. Furthermore, the validity of the proposed algorithm is verified by the experimental result in campus test site of Beijing University of Technology.

MACV/Radio integrated navigation for Mars powered descent via robust desensitized central difference Kalman filter

An innovative integrated navigation scheme based on MCAV/Radio measurement information during Mars powered descent phase, and a robust desensitized central difference Kalman filter (DCDKF) for systems with uncertain parameters or biases are proposed to improve the navigation descent accuracy. Based on the altitude and velocity information of the Miniature Coherent Altimeter and Velocimeter (MCAV), the radio-range information is added into the integrated navigation system to correct the horizontal position error of the vehicle during the Mars powered descent phase. Based the central difference transform, the sensitivity propagation of the state estimate errors in the DCDKF is described, and a designed desensitized cost function is minimized to obtain the gain matrix of the DCDKF. The performances of the innovative navigation scheme and the proposed DCDKF are all demonstrated by two Monte Carlo simulations with the Inertial Measurement Unit biases during the Mars power descent phase.

Variants of extended Kalman filtering approaches for Bayesian tracking

SummaryWe provide a tutorial for a number of variants of the extended Kalman filter (EKF). In these methods, so called, sigma points are employed to tackle the nonlinearity of problems. The sigma points exactly represent the mean and the variance of the state distribution function in a dynamic state equation. The initially developed EKF variant, that is, unscented Kalman filter (UKF) (also called sigma point Kalman filter) shows enhanced performance compared with that of conventional EKF in the literature. Another variant, which is not well known, is central difference Kalman filter (CDKF) whose way to approximate the nonlinearity is based on the Sterling's polynomial interpolation formula instead of the Taylor series. Endeavor to reduce the computational load resulted in the development of square root versions of both UKF and CDKF, that is, square root unscented Kalman filter and square root central difference Kalman filter (SR‐CDKF). These SR‐versions are supposed to be numerically more stable than their original versions because the state covariance is guaranteed to be positive definite by avoiding the step of matrix decomposition. In this paper, we provide the step‐by‐step algorithms of above‐mentioned EKF variants with their pros and cons. We apply these filtering methods to a number of problems in various disciplines for performance assessment in terms of both mean squared error (MSE) and processing speed. Furthermore, we show how to optimize the filters in terms of MSE performance depending on diverse scenarios. According to simulation results, CDKF and SR‐CDKF show the best MSE performance in most scenarios; particularly, SR‐CDKF shows faster processing speed than that of CDKF. Therefore, we justify that SR‐CDKF is the most efficient and the best approach among the Kalman variants including the EKF for various nonlinear problems. The motivation of this paper targets at the contribution to the disseminative usage of the Kalman variants approaches, particularly, SR‐CDKF taking advantage of its estimating performance and high processing speed. Copyright © 2016 John Wiley & Sons, Ltd.

Estimation and Filtering of Nonlinear MS-DSGE Models

This article suggests and compares the properties of some nonlinear Markov-switching filters. Two of them are sigma point filters: the Markov switching central difference Kalman filter (MSCDKF) and MSCDKFA. Two of them are Gaussian assumed filters: Markov switching quadratic Kalman filter (MSQKF) and MSQKFA. A small scale financial MS-DSGE model is used for tests. MSQKF greatly outperforms other filters in terms of computational costs. It also is the first or the second best according to most tests of filtering quality (including the quality of quasi-maximum likelihood estimation with use of a filter, RMSE and LPS of unobserved variables).