Robust Kalman Filter Research Articles

Robust fractional order singular Kalman filter

AbstractIn this article, the state estimation problem of linear fractional order singular (FOS) systems subject to matrix uncertainties is investigated where a recursive robust algorithm is derived. Considering an uncertain discrete‐time linear FOS system with added process and measurement noises, we aim to design a robust Kalman‐type state estimation algorithm based on an optimal data fitting approach with a given sequence of observations. As a substitute for the stochastic formulation, this general filter is obtained by minimizing a completely deterministic regularized residual norm in its worst‐possible form at each step over admissible uncertainties. Analysis of the algorithm shows that not only does the proposed robust filter cover the traditional robust Kalman filters (KFs), but it also represents an extension of the nominal fractional singular KF (FSKF) when the system is not subject to uncertainties. Furthermore, besides giving a sufficient condition for the existence of the robust filter, we derive conditions for the asymptotic properties of the filter, where we demonstrate that the filter and the Riccati equation are stable and converge when an equivalent system is detectable and stabilizable. A numerical example is included to demonstrate the performance of the introduced filter.

A Robust Adaptive Extended Kalman Filter Based on an Improved Measurement Noise Covariance Matrix for the Monitoring and Isolation of Abnormal Disturbances in GNSS/INS Vehicle Navigation

Global Navigation Satellite Systems (GNSS) integrated with Inertial Navigation Systems (INS) have been widely applied in many Intelligent Transport Systems. However, due to the influence of various factors, such as complex urban environments, etc., accurately describing the measurement noise statistics of GNSS receivers and inertial sensors is difficult. An inaccurate definition of the measurement noise covariance matrix will lead to the rapid divergence of the position error of the integrated navigation system. To overcome this problem, this paper proposed a Robust Adaptive Extended Kalman Filter (RAKF) method based on an improved measurement noise covariance matrix. By analyzing and considering the position accuracy factors, measurement factor, and position standard deviation in GNSS measurement results, this paper constructed the optimal measurement noise covariance matrix. Based on the Huber model, this paper constructed a two-stage robust adaptive factor expression and obtained the robust adaptive factors with and without abnormal disturbances. And robust adaptive filtering was carried out. To assess the performance of this method, the author conducted experiments on land vehicles by using a self-developed POS system (GNSS/INS combined navigation system). The classic Extended Kalman Filter algorithm (EKF), Adaptive Kalman Filter (AKF) algorithm, Robust Kalman Filter (RKF) algorithm, and the proposed method were compared through data processing. Experimental results show that compared with the classical EKF, AKF, and RKF, the positioning accuracies of the proposed method were improved by 72.43%, 2.54%, and 47.82%, respectively, in the vehicle land experiment. In order to further evaluate the performance of this method, the vehicle data were subjected to different times and degrees of disturbance experiments. Experimental results show that compared with EKF, AKF, and RKF, the heading angle accuracy had obvious advantages, and its accuracy was improved by 34.65%, 31.53%, and 18.36%, respectively. Therefore, this method can effectively monitor and isolate disturbance and improve the robustness, reliability, accuracy, and stability of GNSS/INS integrated navigation systems in complex urban environments.

Robust weighted fusion Kalman filters under linearly correlated noise and mixed uncertainties of noise variances, multiplicative noises, and multiple networked inducements

AbstractThe paper solves the robust weighted fusion Kalman filtering problem for systems with linearly correlated noise and mixed uncertainties of noise variances, multiplicative noises, and multiple networked inducements including missing measurements, packets dropouts, and two‐step random measurement delays. It is assumed that system noise variances are uncertain but bounded above, and the other four uncertainties are compensated to fictitious white noise by the proposed model‐transformation method. For the transformed local multimodel system with correlated fictitious noise, the robust local time‐varying recursive Kalman filters are presented by decorrelation technique and minimax robust filtering principle. Then the six weighted fusion robust Kalman filters are presented in a unified form. The robustness of local and fused robust Kalman filters is proved by the extended Lyapunov equation approach, matrix factorization, and elementary transformation. Further, the local and fused steady‐state robust Kalman filters are designed. Finally, a simulation study applied to F404 aircraft engine system is provided to examine effectiveness and applicability of the proposed algorithm.

A Robust Parameter Identification Approach with Anti-Outlier Characteristics for Lithium-Ion Batteries

Accurate acquisition of model parameters of lithium-ion batteries (LIBs) is imperative for precious estimation state of charge. However, due to the interference of noise, the inevitable uncertainty may lead to outliers appearing in measurement data, which often leads to inaccurate parameter identification. To solve the problem, a robust Kalman filter (RKF) that uses the Mahalanobis Distance Criterion (MDC) is proposed for the parameter identification of LIBs. Firstly, the one-order equivalent circuit model of the LIBs is created. Subsequently, the gain of robust factor is implemented into the Kalman filter to decrease filter gain and swell the measurement noise covariance when facing the outlier measurements. Finally, the dynamic stress test condition of the electric vehicle is used to prove the effectiveness of the RKF-MDC approach. The simulation reveals that the RKF-MDC can reduce the influence of outlier values for the parameter identification of LIBs.

An Improved Multiple-Outlier Robust Filter Based on Maximum Correntropy Criterion for Integrated Navigation

In the integrated navigation of global navigation satellite system (GNSS) and Inertial navigation system (INS), the Kalman filter is frequently employed for state estimation. GNSS signals are subject to interference from the outside environment, and it is difficult for the classical Kalman filter to handle observations with outliers. Numerous robust filters have been proposed based on the maximum correntropy criterion (MCC), which is insensitive to non-Gaussian noise, to address the issue of state estimation in the presence of outliers. However, a fixed prior covariance will restrict the use of a maximum correntropy criterion Kalman filter (MCKF) in integrated navigation systems because GNSS signals can be affected by both time-varying noise and outliers. Existing filters consider the prior measurement noise covariance matrix (MNCM) as a deterministic matrix and can only amplify signals unidirectionally during iteration cycles. The weight of effective observations is reduced despite these filters having strong robustness. To overcome this shortcoming, this paper proposes a τ -based maximum correntropy criterion Kalman filter (τ -MCKF). A strategy based on the Chisquare test is designed to optimize the covariance of GNSS observations. By introducing τ to bi-directionally adjust the initial MNCM, τ -MCKF makes full use of effective observations. Then, the τ -corrected MNCM is used to weigh different elements of the innovation vector, and a robust estimation is calculated based on the multiple-outlier robust Kalman filter (MORKF) framework. The GNSS/INS integrated navigation system is taken as an example to verify the effectiveness of the filter through simulation and vehicular experiments.

Stochastic modeling with robust Kalman filter for real-time kinematic GPS single-frequency positioning

The centimeter-level positioning accuracy of real-time kinematic (RTK) depends on correctly resolving integer carrier-phase ambiguities. To improve the success rate of ambiguity resolution and obtain reliable positioning results, an enhanced Kalman filtering procedure has been developed. Based on a posteriori residuals of measurements and state predictions, the measurement noise variance–covariance matrix for double-differenced measurements is adaptively estimated, rather than approximated by an empirical function which uses satellite elevation angle as input. Since, in real-world situations, unexpected outliers and carrier-phase outages can degrade the filter performance, a stochastic model based on robust Kalman filtering is proposed, for which the double-differenced measurement noise variance–covariance matrix is computed empirically with a modified version of the IGG (Institute of Geodesy and Geophysics) III method in order to detect and identify outliers. The performance of the proposed method is assessed by two tests, one with simulated data and one with real data. In addition, the performance of F-ratio and W-ratio tests as proxies for the success of ambiguity fixing is investigated. Experimental results reveal that the proposed method can improve the reliability and robustness of relative kinematic positioning for simulation scenarios as well as in a real urban test.

Robust Filter Method for SINS/DVL/USBL Tight Integrated Navigation System

In order to solve the problem of SINS/DVL/USBL system interference in complex underwater environment. An asynchronous sequential robust filter method is proposed in this paper. The original output azimuth, slant range and altitude of the ultra-short baseline as the measurement information, a compact combination model is established. The problem of large amount of computation caused by the high measurement dimension of multi-sensor information fusion is solved by introducing asynchronous sequential filter technology. Meanwhile, a robust Kalman filter algorithm is proposed, which based on Mahalanobis distance. The outlier detection is designed based on statistical characteristics. Finally, the effectiveness of the method proposed in this paper is verified by simulation and River experiments. The experimental results show that the method proposed in this paper can effectively eliminate outliers and improve the navigation accuracy while reducing the amount of computation.

Fault Diagnosis and Its Applications to Fault Tolerant Control of a Turbojet Engine

This paper presents a comprehensive study of model-based fault diagnosis (FD) and a fault-tolerant control (FTC) scheme for sensor and actuator faults of turbojet engines. For actuator FD, an unbiased estimation scheme with a modified Kalman filter (KF) was developed. For sensor FD, two approaches, the generalized likelihood ratio with robust KF and the pseudo actuator model with modified KF, were investigated in a comparative study. For fault detection and isolation, test statistics are commonly employed to detect fault behavior. For FTC, integral-type sliding mode control using control reconfiguration and the reconstruction of the sensor signal was adopted with the FD schemes. The effectiveness of the employed methods was demonstrated in this study and discussed with numerical simulations.

A Hypothesis Test-Constrained Robust Kalman Filter for INS/GNSS Integration With Abnormal Measurement

This paper presents a hypothesis test-constrained robust Kalman filter for INS/GNSS (inertial navigation system/global navigation satellite system) integrated navigation in the presence of measurement outliers. This method estimates measurement noise covariance by combining hypothesis test with the maximum likelihood theory to handle measurement outliers. A chi square test and an improved sequential probability ratio test are established to characterize abrupt and slow-growing measurement outliers, respectively. Subsequently, these two hypothesis tests are used to constrain the maximum likelihood estimation of measurement noise covariance to accommodate measurement outliers. Based on the hypothesis test-constrained maximum likelihood estimation of measurement noise covariance, a robust Kalman filter is developed for INS/GNSS integrated navigation in the presence of measurement outliers. Simulation and experimental results demonstrate that the proposed method can effectively deal with measurement outliers. The resultant navigation accuracy is about 46% and 30% higher than that of the Kalman filter and maximum likelihood-based robust Kalman filter for INS/GNSS integrated navigation.

Robust Kalman Filter for Linear System With Convex Polytopic Uncertainties

This brief considers robust Kalman filtering problem for linear discrete-time systems with convex polytopic uncertain parameters. First, we characterize the uncertain parameter matrices as a combination of several vertex matrices. Then, based on the mean square stability and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {H}_{2}$ </tex-math></inline-formula> and variance-constrained performance criteria, we design a variance-constrained based robust Kalman filter. The parametric matrices of the filter can be directly obtained by the variance-constrained optimization and MATLAB Toolbox. A classical instance is provided to verify the effectiveness of the proposed filter.

Tracking the trajectory of an object in a noisy environment with unknown statistics: A novel robust Kalman filter residue-based approach

A novel robust Kalman filter (KF)-based controller design approach is proposed to accurately track a specified trajectory under unknown stochastic disturbance, known deterministic disturbance, and measurement noise. The system is nonlinear and is approximated by a piecewise-linear dynamic model. The Box–Jenkins model is an augmented model of the signal and the disturbance, is non-controllable and observable, while the signal model is controllable and observable. An emulator-based two-stage identification is employed to obtain an accurate system model needed to design the robust controller. The system and its KF are identified, and the signal and output errors are estimated. From the identified models, the signal, its KF, the disturbance model, and the whitening filter are all obtained using balanced model reduction techniques. It is shown that the signal model is a transfer matrix description relating the system output and KF residual, and the residual is the whitened output error. The disturbance model is identified by inverse filtering. A new combined feedforward–feedback controller is designed and implemented using an internal model of the reference driven by both the error between the reference and the signal estimate, and by the feedforwarded reference signal. The proposed scheme was successfully evaluated on a simulated autonomously guided drone.

Object Tracking in SWIR Imaging Based on Both Correlation and Robust Kalman Filters

Object Tracking in SWIR Imaging Based on Both Correlation and Robust Kalman Filters

Performance Analysis of Robust Cooperative Positioning Based on GPS/UWB Integration for Connected Autonomous Vehicles

The accurate position is a key requirement for autonomous vehicles. Although Global Navigation Satellite Systems (GNSS) are widely used in many applications, their performance is often disturbed, particularly in urban areas. Therefore, many studies consider multi-sensor integration and cooperative positioning (CP) approaches to provide additional degrees of freedom to address the shortcomings of GNSS. However, few studies adopted real-world datasets and internode ranging outliers within CP is left untouched, leading to unexpected challenges in practical applications. To address this, we propose a Robust Cooperative Positioning (RCP) scheme that augments the GPS with the Ultra-Wideband (UWB) system. A field experiment is conducted to generate a real-world dataset to evaluate the RCP scheme. Moreover, the analysis of the collected dataset enables us to optimise a simple but effective Robust Kalman Filter (RKF) to mitigate the influence of outlier measurements and improve the robustness of the proposed solution. Finally, a simulated dataset is derived from the real-world data to comprehensively study the performance of the proposed RCP method in urban canyon scenarios. Our results demonstrate that the proposed solution can crucially improve positioning performance when the number of visible GPS satellite is limited and is robust against various adverse effects in such environments.

Augmented Underwater Acoustic Navigation with Systematic Error Modeling Based on Seafloor Datum Network

Underwater acoustic navigation technology is an important approach to achieving high precision ocean navigation. One of the critical issues of the technology is to correct systematic errors, which are related to time delays and time-varying sound speed errors. In this study, we propose an augmented underwater acoustic navigation with systematic error model based on seafloor datum network. The proposed algorithm first selects data sets of piece-wise systematic error modeling by extracting the main periodic term of systematic errors based on the Fourier transform. Before that, the wavelet transform is used for denoising to better extract the main periodic term. Then the systematic error correction model is constructed by using the polynomial fitting method. After that, an augmented observation equation of underwater acoustic navigation with systematic error correction is constructed. Finally, an adaptive robust Kalman filter is developed for underwater acoustic navigation. The proposed algorithm is verified by an experiment in the South China Sea. The three-dimensional root mean square values of underwater acoustic navigation are 1.010 and 1.502 m in the operating range of 2.7 and 8.7 km. The results demonstrate that the proposed algorithm can efficiently reduce the influence of systematic error, thus improving underwater acoustic navigation accuracy.

Centered Error Entropy-Based Variational Bayesian Adaptive and Robust Kalman Filter

In this brief, a centered error entropy based variational Bayesian adaptive and robust Kalman filter (CEEVBKF) is proposed to suppress outlier noise and estimate the unknown noise covariance adaptively. The derived CEEVBKF contains three steps: one-step prediction, centered error entropy (CEE) based outlier suppression, and variational Bayesian (VB) inference. The CEE criterion is first used to suppress outlier noise and obtain rough state estimation value, then they are set as a priori value in VB inference step for accurate a posteriori state estimation. The joint estimation of CEE and VB improves the iterative efficiency and reduces the parameter sensitivity. The simulation results show the effectiveness of CEEVBKF.

A Sequential Student’s t-Based Robust Kalman Filter for Multi-GNSS PPP/INS Tightly Coupled Model in the Urban Environment

The proper stochastic model of a global navigation satellite system (GNSS) makes a significant difference on the precise point positioning (PPP)/inertial navigation system (INS) tightly coupled solutions. The empirical Gaussian stochastic model is easily biased by massive gross errors, deteriorating the positioning precisions, especially in the severe GNSS blockage urban environment. In this paper, the distributional characteristics of the gross-error-contaminated observation noise are analyzed by the epoch-differenced (ED) geometry-free (GF) model. The Student’s t distribution is used to express the heavy tails of the gross-error-contaminated observation noise. Consequently, a novel sequential Student’s t-based robust Kalman filter (SSTRKF) is put forward to adjust the GNSS stochastic model in the urban environment. The proposed method pre-weights all the observations with the a priori residual-derived robust factors. The chi-square test is adopted to distinguish the unreasonable variances. The proposed sequential Student’s t-based Kalman filter is conducted for the PPP/INS solutions instead of the standard Kalman filter (KF) when the null hypothesis of the chi-square test is rejected. The results of the road experiments with urban environment simulations reveal that the SSTRKF improves the horizontal and vertical positioning precisions by 57.5% and 62.0% on average compared with the robust Kalman filter (RKF).

A Novel Robust Kalman Filtering Framework Based on Normal-Skew Mixture Distribution

In this article, a novel normal-skew mixture (NSM) distribution is presented to model the normal and/or heavy-tailed and/or skew nonstationary distributed noises. The NSM distribution can be formulated as a hierarchically Gaussian presentation by leveraging a Bernoulli distributed random variable. Based on this, a novel robust Kalman filtering framework can be developed utilizing the variational Bayesian method, where the one-step prediction and measurement-likelihood densities are modeled as NSM distributions. For implementation, several exemplary robust Kalman filters (KFs) are derived based on some specific cases of NSM distribution. The relationships between some existing robust KFs and the presented framework are also revealed. The superiority of the proposed robust Kalman filtering framework is validated by a target tracking simulation example.

A modified adaptive Kalman filtering method for maneuvering target tracking of unmanned surface vehicles

The filtering methods are crucial for an unmanned surface vehicle (USV) to realize target tracking. Due to the poor observation caused by the strong vibration during the navigation of the USV, the target tracking accuracy of the traditional filtering method has been significantly degraded. A modified strong tracking-based expended Sage-Husa adaptive robust Kalman filter (MST-ESHARKF) algorithm is proposed to overcome this problem in this paper. In the proposed algorithm, a modified fading factor for the strong tracking Kalman filter is introduced to eliminate disturbance-induced filter divergence. In addition, the adaptive factor of robust Kalman filtering is designed to balance the predicted and observed states dedicated to improving the robustness of the algorithm. Finally, the biased and unbiased estimators for measurement and process noise covariances are merged, and the measurement noise covariance matrix’s interval is constrained, resulting in a simultaneous evaluation of measurement and process noise covariance matrices with improved dependability of the proposed algorithm. The simulation and experiment results show that the proposed MST-ESHARKF outperforms the existing filters in target tracking.

A Polar Robust Kalman Filter Algorithm for DVL-Aided SINSs Based on the Ellipsoidal Earth Model.

Autonomous underwater vehicles (AUVs) play an increasingly essential role in the field of polar ocean exploration, and the Doppler velocity log (DVL)-aided strapdown inertial navigation system (SINS) is widely used for it. Due to the rapid convergence of the meridians, traditional inertial navigation mechanisms fail in the polar region. To tackle this problem, a transverse inertial navigation mechanism based on the earth ellipsoidal model is designed in this paper. Influenced by the harsh environment of the polar regions, unknown and time-varying outlier noise appears in the output of DVL, which makes the performance of the standard Kalman filter degrade. To address this issue, a robust Kalman filter algorithm based on Mahalanobis distance is used to adaptively estimate measurement noise covariance; thus, the Kalman filter gain can be modified to weight the measurement. A trial ship experiment and semi-physical simulation experiment were carried out to verify the effectiveness of the proposed algorithm. The results demonstrate that the proposed algorithm can effectively resist the influence of DVL outliers and improve positioning accuracy.

A Novel Robust Inertial and Ultra-Short Baseline Integrated Navigation Strategy Under the Influence of Motion Effect

Ultra-short baseline (USBL) is a mature navigation method for underwater applications. Acoustic outliers and the slant distance calculation error are the main challenges for improving positioning accuracy. The latter, in particular, is often overlooked. In this paper, the slant distance calculation errors due to the motion effect and acoustic outliers will be mainly investigated. To address the problem, the influence of the motion effect on localization in dynamic environments is analyzed first. Aiming at the two challenges, a new measurement model based on the round trip time delay and bearing angles is derived, which can avoid the influence of the motion effect. Besides, to improve the robustness against outliers, a novel robust Kalman filter is designed based on the mixture distribution model. Simulation tests prove that the designed Kalman filter can improve the positioning accuracy in a dynamic environment and overcome the influence of motion effect on slant distance calculation. The proposed method also plays superior performance than the traditional ones under the influence of outliers. The algorithm has also been verified by the river test, and it has better robustness and higher positioning accuracy under dynamic conditions.