Nonlinear Kalman Filter Research Articles

A non-linear Kalman filter for track parameters estimation in high energy physics

The Kalman Filter is a widely used approach for the linear quadratic estimation of dynamical systems and is frequently employed within nuclear and particle physics experiments for the reconstruction of charged particle trajectories, known as tracks. Implementations of this formalism often make assumptions on the linearity of the underlying dynamic system and the Gaussian nature of the process noise and measurement model, which are violated in a number of track reconstruction applications. This paper introduces an implementation of a Non-Linear Kalman Filter (NLKF) within the ACTS track reconstruction toolkit. The NLKF addresses the issue of non-linearity by using a set of representative sample points during the projection of the track state to the measurement. In a typical use case, the NLKF outperforms the so-called Extended Kalman Filter in the accuracy and precision of the track parameter estimates obtained, with an increase in CPU time below a factor of two. It is therefore a promising approach for use in applications where precise estimation of track parameters is a key concern.

AE localization issues in heavy reinforced post-tensioned concrete beams

Damage localization is a relevant issue in structural health monitoring (SHM) of concrete structures. A greater level of insight into the structure can be so reached enabling more informed maintenance decisions and reducing operation and maintenance costs. The basic approach to acoustic emission localization, which is often referred to as the Time of Arrival (TOA) method, is based on detecting an AE source at a number of spatially-distributed sensors. However, in anisotropic media such as reinforced concrete, the basic assumptions for the TOA method become largely invalid. Different robust methods suitable to anisotropic media, incorporating an angular-dependent velocity term into the minimization process have been developed in the past. However, the wave-velocity profile is not always known. More recently a number of other authors have adopted various Bayesian approaches as part of AE localization strategies, including the use of a Markov chain Monte Carlo inference scheme as well as nonlinear Kalman filters. The use of a probabilistic approach in source location has become increasingly applied for AE source localization in complex large structures. In heavy reinforced post-tensioned concrete structures acoustic wave path can be significantly influenced by rebars as well as metallic post-tensioning ducts. Cracks opening can further influence acoustic propagation paths and greatly influence the reliability of AE source localization algorithms. In the present paper, different localization procedures have been developed and tested on post-tensioned concrete beams during loading and unloading cycles as well as on small homogeneous concrete blocks. In a highly emissive environment, significant difficulties have been reported also in event identification thus further reducing localization procedure reliability. EWGAE 35, Ljubljana, Slovenia, 13th – 16th Sep. www.ewg

Accurate Estimation of Effective Wind Speed for Wind Turbine Control Using Linear and Nonlinear Kalman Filters

Accurate Estimation of Effective Wind Speed for Wind Turbine Control Using Linear and Nonlinear Kalman Filters

M-estimation based Robust Approach for Hybrid Dynamic State Estimation in Power Systems

Background: The state estimation (SE) process in power systems estimates bus voltage magnitude and phase angles vital for operating the system securely and reliably. The power systems state estimation problem has been extensively solved through a weighted least squares (WLS) based static approach that fails to track the system dynamics. Furthermore, those approaches are not inherently robust against outliers, yielding a separate bad data processing (BDP) technique. Popular Dynamic state estimation (DSE) schemes which mainly employ nonlinear Kalman filters like Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF), also suffer from providing a reasonable estimation of states in the presence of bad data. Although several generalized maximum (GM) likelihood- based DSE approaches are robust against outliers, they are mainly based on nonlinear Kalman filters, which yield an iterative process. Therefore, this article focuses on developing a robust DSE approach that gives a good estimation of states against outliers in a single iteration. Objective: This article aims to propose a robust hybrid non-iterative DSE approach that gives robust SE results in the presence of bad data. Methods: The proposed novel robust hybrid DSE (NRHDSE) approach combines the robust Mestimation with the original novel hybrid DSE (NHDSE) approach. The proposed scheme implements a suitable linear relationship between integrated hybrid measurements and complex states. The proposed method uses a linear measurement model and thus employs an optimal linear Kalman filter to correct or estimate states. Results: The efficacy of the proposed approach has been demonstrated by applying it on IEEE 57, 118 bus test systems and one more extensive 246 Indian utility bus system, namely, Northern Regional Power Grid (NRPG), and after that comparing it with the original NHDSE, and DSE methods based on traditional EKF and M-estimation based robust version of EKF (REKF). The simulation result demonstrates the superiority of the proposed approach. Conclusions: Obtained results clearly show the superiority of the proposed approach.

Review of the Applications of Kalman Filtering in Quantum Systems

State variable and parameter estimations are important for signal sensing and feedback control in both traditional engineering systems and quantum systems. The Kalman filter, which is one of the most popular signal recovery techniques in classical systems for decades, has now been connected to the stochastic master equations of linear quantum mechanical systems. Various studies have invested effort on mapping the state evolution of a quantum system into a set of classical filtering equations. However, establishing proper evolution models with symmetry to classical filter equation for quantum systems is not easy. Here, we review works that have successfully built a Kalman filter model for quantum systems and provide an improved method for optimal estimations. We also discuss a practical scenario involving magnetic field estimations in quantum systems, where non-linear Kalman filters could be considered an estimation solution.

Implementation of a C Library of Kalman Filters for Application on Embedded Systems

Having knowledge about the states of a system is an important component in most control systems. However, an exact measurement of the states cannot always be provided because it is either not technically possible or only possible with a significant effort. Therefore, state estimation plays an important role in control applications. The well-known and widely used Kalman filter is often employed for this purpose. This paper describes the implementation of nonlinear Kalman filter algorithms, the extended and the unscented Kalman filter with square-rooting, in the programming language C, that are suitable for the use on embedded systems. The implementations deal with single or double precision data types depending on the application. The newly implemented filters are demonstrated in the context of semi-active vehicle damper control and the estimation of the tire–road friction coefficient as application examples, providing real-time capability. Their per-formances were evaluated in tests on an electronic control unit and a rapid-prototyping platform.

A Robust Nonlinear Filter Strategy Based on Maximum Correntropy Criterion for Multi-GNSS and Dual-Frequency RTK

The multi-constellation, multi-frequency Global Navigation Satellite System (GNSS) has the potential to empower precise real-time kinematics (RTK) with higher accuracy, availability, continuity, and integrity. However, to enhance the robustness of the nonlinear filter, both the measurement quality and efficiency of parameter estimation need consideration, especially for GNSS challenging or denied environments where outliers and non-Gaussian noise exist. This study proposes a nonlinear Kalman filter with adaptive kernel bandwidth (KBW) based on the maximum correntropy criterion (AMC-KF). The proposed method excavates data features of higher order moments to enhance the robustness against noise. With the wide-lane and ionosphere-free combination, a dual frequency (DF) data-aided ambiguity resolution (AR) method is also derived to improve the measurement quality. The filtering strategy based on the DF data-aided AR method and AMC-KF is applied for multi-GNSS and DF RTK. To evaluate the proposed method, the short baseline test, long baseline test, and triangle network closure test are conducted with DF data from GPS and Galileo. For the short baseline test, the proposed filter strategy could improve the positioning accuracy by more than 30% on E and N components, and 60% on U. The superiority of the proposed adaptive KBW is validated both in efficiency and accuracy. The triangle network closure test shows that the proposed DF data-aided AR method could achieve a success rate of more than 93%. For the long baseline test, the integration of the above methods gains more than 40% positioning accuracy improvement on ENU components. This study shows that the proposed nonlinear strategy could enhance both robustness and accuracy without the assistance of external sensors and is applicable for multi-GNSS and dual-frequency RTK.

Research on Gradient-Descent Extended Kalman Attitude Estimation Method for Low-Cost MARG.

Aiming at the problem of the weak dynamic performance of the gradient descent method in the attitude and heading reference system, the susceptibility to the interference of accelerometers and magnetometers, and the complex calculation of the nonlinear Kalman Filter method, an extended Kalman filter suitable for a low-cost magnetic, angular rate, and gravity (MARG) sensor system is proposed. The method proposed in this paper is a combination of a two-stage gradient descent algorithm and the extended Kalman filter (GDEKF). First, the accelerometer and magnetometer are used to correct the attitude angle according to the two-stage gradient descent algorithm. The obtained attitude quaternion is combined with the gyroscope measurement value as the observation vector of EKF and the calculated attitude of the gyroscope and the bias of the gyroscope are corrected. The elimination of the bias of the gyroscope can further improve the stability of the attitude observation results. Finally, the MARG sensor system was designed for mathematical model simulation and hardware-in-the-loop simulation to verify the performance of the filter. The results show that compared with the gradient descent method, it has better anti-interference performance and dynamic performance, and better measurement accuracy than the extended Kalman filter.

Implicit particle filtering via a bank of nonlinear Kalman filters

The implicit particle filter seeks to mitigate particle degeneracy by identifying particles in the target distribution’s high-probability regions. This study is motivated by the need to enhance computational tractability in implementing this approach. We investigate the connection of the particle update step in the implicit particle filter with that of the Kalman filter and then formulate a novel realization of the implicit particle filter based on a bank of nonlinear Kalman filters. This realization is more amenable and efficient computationally.

A cubature Kalman filter for online state-of-charge estimation of lithium-ion battery using a gas-liquid dynamic model

A novel state of charge (SOC) estimation method is proposed based on a gas-liquid dynamic model using a Cubature Kalman filter (CKF) with state constraints to improve the accuracy of estimations. The constraints derived from the principle of aerodynamic are introduced, which does not need the temperature correction coefficient and can achieve rapid SOC convergence due to its unique iterative form. Non-linear Kalman filter used to eliminate the violent jitter of the original algorithm when switching working conditions are also incorporated. Then, the proposed method is tested using a cylindrical-format ternary lithium-ion battery with a nominal capacity of 2.6 Ah and the estimated SOC were compared with the experimental results. As a result, comparative studies of two non-linear filters revealed that the CKF has the most outstanding performance based on concerning estimation accuracy, recovery time from an initial offset, and computational time. That is 36.4 % more accurate and the convergence time is 97.5 % shorter than the Extended Kalman filter (EKF) when the initial offset is 60 %, respectively.

Comparison of the performance of nonlinear Kalman filter based algorithms for state‐parameter identification of base isolated structures

Comparison of the performance of nonlinear Kalman filter based algorithms for state‐parameter identification of base isolated structures

Modeling and simulation of high energy density lithium-ion battery for multiple fault detection

Lithium-ion battery, a high energy density storage device has extensive applications in electrical and electronic gadgets, computers, hybrid electric vehicles, and electric vehicles. This paper presents multiple fault detection of lithium-ion battery using two non-linear Kalman filters. A discrete non-linear mathematical model of lithium ion battery has been developed and Unscented Kalman filter (UKF) is employed to estimate the model parameter. Occurrences of multiple faults such as over-charge, over-discharge and short circuit faults between inter cell power batteries, affects the parameter variation of system model. Parallel combinations of some UKF (bank of filters) compare the model parameter variation between the normal and faulty situation and generates residual signal indicating different fault. Simulation results of multiple numbers of statistical tests have been performed for residual based fault diagnosis and threshold calculation. The performance of UKF is then compared with Extended Kalman filter (EKF) with same battery model and fault scenario. The simulation result proves that UKF model responses better and quicker than that of EKF for fault diagnosis.

Highly Maneuverable Target Tracking Under Glint Noise via Uniform Robust Exact Filtering Differentiator With Intrapulse Median Filter

Highly maneuverable target tracking under glint noise and nonlinear uncertainties during course changes and terminal maneuvers has been solved suboptimally for many years via the interacting multiple model algorithm with the use of the Kalman filter (KF), unscented Kalman filter, and the extended Kalman filter (EKF). Also, nonlinear KFs such as the cubature Kalman filter have been proposed to improve nonlinear tracking without being able to filter out glint noise. To this end, the particle filter and some KF based on variational Bayesian approach have been proposed with very good results in filtering out glint noise. Nonetheless, it is difficult for state-of-the-art methods to achieve efficient filtering of glint noise and nonlinear tracking at the same time. On the other hand, robust exact differentiators, based on the super-twisting algorithm, have been used for many years in output-feedback control and state observation in order to obtain the derivatives of an input signal with theoretical finite-time exactness. However, their potential for target tracking applications has not been explored sufficiently. In this article, a uniform robust exact filtering differentiator with intrapulse median filtering is proposed to filter out glint noise at the sliding manifold, while offering nonlinear tracking robustness via high-degree super-twisting terms outside the sliding manifold. Numerical simulations comparing the proposed solution to other state-of-the-art methods were conducted, showing promising results.

Generalized Correntropy Sparse Gauss–Hermite Quadrature Filter for Epidemic Tracking on Complex Networks

The dynamic estimation of epidemic spreading on networks is essential for controlling morbidity. Nonlinear Kalman filters, which are capable of estimating the hidden state of a nonlinear system, can be utilized for dynamic state estimation of epidemic spreading. Traditional nonlinear Kalman filters perform optimization using the minimum mean square error (MMSE) criterion. Since observable measurements are generally corrupted by non-Gaussian noise, maximum correntropy criterion (MCC)-based nonlinear Kalman filters have been used for improving robustness against non-Gaussian noise. In comparison with the MCC, generalized correntropy is more robust and flexible in the presence of non-Gaussian noise. In this article, epidemic spreading is described by a compartmental model, i.e., susceptible-infected-recovered-susceptible model. Based on the compartmental model, the generalized correntropy-based sparse Gauss-Hermite quadrature filter (GCSGHQF) is proposed for epidemic tracking on homogeneous networks. The GCSGHQF approximates the Gaussian-weighted integrals by utilizing the sparse Gauss-Hermite quadrature rule. In addition, the generalized correntropy is applied to enhancing robustness in the presence of non-Gaussian noise. Simulation results show that the GCSGHQF exhibits a higher filtering accuracy and improves robustness compared to traditional nonlinear Kalman filters.

A Review of Non-Linear Kalman Filtering for Target Tracking

Target Tracking (TT) with Non-Linear (NL) Kalman Filtering (NLKF) has recently become a hot research hotspot, particularly in the field of Marine Engineering and air traffic control. This paper presents a comprehensive investigation of NLKF algorithms, with emphases on a proposed theoretical framework to significantly improve its implementation results with regards to accuracy and efficiency. Further, the proposed framework demonstrates potential superior performance in terms of robustness, convergence speed, effective computation and tracking accuracy, comparatively with prior state-of-the-art NLKF techniques. It is anticipated that this study will be beneficial to researchers studying Kalman Filtering (KF) algorithms and also serve as the bedrock for future research, especially for those pursuing their career in Electronics and Information Engineering. Some conclusions and possible research directions of NLKF are proposed in the end.

Flatness‐based disturbance observer for exoskeleton robots under time‐delayed contact forces

AbstractThe article proposes flatness‐based control and a Kalman filter‐based disturbance observer for solving the control problem of a robotic exoskeleton under time‐delayed exogenous disturbances. A two‐link lower‐limb robotic exoskeleton is used as a case study. It is proven that this robotic system is differentially flat. The robot is considered to be subject to unknown contact forces at its free‐end which in turn generate unknown disturbance torques at its joints. It is shown that the dynamic model of the robotic exoskeleton can be transformed into the input–output linearized form and equivalently into the linear canonical Brunovsky form. This linearized description of the exoskeleton's dynamics is both controllable and observable. It allows for designing a stabilizing feedback controller with the use of the pole‐placement (eigenvalues assignment) method. Moreover, it allows for solving the state estimation problem with the use of Kalman Filtering (the use of the Kalman filter on the flatness‐based linearized model of nonlinear dynamical systems is also known as derivative‐free nonlinear Kalman filtering). Furthermore, (i) by extending the state vector of the exoskeleton after considering as additional state variables the additive disturbance torques which affect its joints and (ii) by redesigning the Kalman filter as a disturbance observer, one can achieve the real‐time estimation of the perturbations that affect this robotic system. Finally, by including in the controller of the exoskeleton additional terms that compensate for the estimated disturbance torques, the perturbations' effects can be eliminated and the precise tracking of reference trajectories by the joints of this robot can be ensured.

Experimental 2D extended Kalman filter sensor fusion for low-cost GNSS/IMU/Odometers precise positioning system

The development of satellite techniques and the availability of mobile devices with built-in multi-GNSS (Global Navigation Satellite System) receivers allow the determination of position with increasing accuracy. At the same time, the requirements of users as to the accuracy of positioning are increasing, while the low production costs of the device must be maintained. This paper presents a loosely coupled integration of low-cost sensors (GNSS, IMU (Inertial Measurement Unit), and an odometer) with the use of a nonlinear Kalman filter and a dynamic weight matrix. The integration model was developed for horizontal (2D) components with the simultaneous determination of the azimuth of the test platform. The tests were carried out in the conditions of an open horizon, with partial obscuring of the horizon (passage under an open-work steel structure) and along walls. In this way, the working conditions for an autonomous lawn mower, which are now increasingly used by citizens, were simulated. The position accuracy obtained in these tests is better than 5 cm for horizontal components and better than 1 degrees for the azimuth.

Cubature Kalman Filter Under Minimum Error Entropy With Fiducial Points for INS/GPS Integration

Traditional cubature Kalman filter (CKF) is a preferable tool for the inertial navigation system (INS)/global positioning system (GPS) integration under Gaussian noises. The CKF, however, may provide a significantly biased estimate when the INS/GPS system suffers from complex non-Gaussian disturbances. To address this issue, a robust nonlinear Kalman filter referred to as cubature Kalman filter under minimum error entropy with fiducial points (MEEF-CKF) is proposed. The MEEF-CKF behaves a strong robustness against complex non-Gaussian noises by operating several major steps, i.e., regression model construction, robust state estimation and free parameters optimization. More concretely, a regression model is constructed with the consideration of residual error caused by linearizing a nonlinear function at the first step. The MEEF-CKF is then developed by solving an optimization problem based on minimum error entropy with fiducial points (MEEF) under the framework of the regression model. In the MEEF-CKF, a novel optimization approach is provided for the purpose of determining free parameters adaptively. In addition, the computational complexity and convergence analyses of the MEEF-CKF are conducted for demonstrating the calculational burden and convergence characteristic. The enhanced robustness of the MEEF-CKF is demonstrated by Monte Carlo simulations on the application of a target tracking with INS/GPS integration under complex non-Gaussian noises.

A single foot-mounted pedestrian navigation algorithm based on the maximum gait displacement constraint in three-dimensional space

Inertial navigation technology composed of inertial sensors is widely used in foot-mounted pedestrian positioning. However, inertial sensors are susceptible to noise, which affects the performance of the system. Zero-velocity update (ZUPT) is a traditional method utilized to suppress the cumulative error. Unfortunately, the walking distance calculated by a Kalman filter still has positional error. To improve the positional accuracy, in this work we propose a nonlinear Kalman filter with a spatial distance inequality constraint for a single foot. Since the stride distance between adjacent stance phases has an upper bound in plane and height, an inertial navigation system established by one inertial measurement unit is adopted to constrain the stride process. Eventually, the performance of the proposed method is verified by experiments. Compared with the single foot-mounted ZUPT method, the proposed method suppresses the plane error and the height error by 46.04% and 65.48%, respectively. For the dual-foot constraint method, the proposed constraint method can reduce the number of sensors while ensuring positioning accuracy. Moreover, the height error is reduced by 59.98% on average by optimizing the constraint algorithm. The experimental results show that the trajectory estimated by the proposed method is closer to the actual path.

A Review of Dynamic Phasor Estimation by Non-Linear Kalman Filters

Phasor estimation under dynamic conditions has been under study recently by relaxing the amplitude and phase of the static phasor. This paper will review some methods to estimate dynamic phasor by nonlinear Kalman filters. Nonlinear Kalman filters have found an application in the tracking of time-varying amplitude and phase. Five nonlinear Kalman filtering methods for dynamic phasor estimation are examined in this paper. These methods are: EKF1 stands for first-order Extended Kalman filter, EKF2 stands for second-order Extended Kalman, UKF stands for Unscented Kalman filter, GHKF stands for Gauss Hermite Kalman filter, and finally CKF stands for Cubature Kalman filter. This paper describes the theoretical processes of these methods and demonstrates their effectiveness in dynamic phasor estimation by some test signal simulations in MATLAB. The simulation section shows that nonlinear Kalman filters give more accuracy than linear Kalman filters when the phasor is relaxed by modulated amplitude and phase. Moreover, comparative assessments among the performance of five nonlinear Kalman filters are done for dynamic phasor estimation, and also their performances are compared with the other methods which have already been published. According to the simulation results, EKF1 gives the highest accuracy during steady-state ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2 \times 10^{-13}$ </tex-math></inline-formula> ) because the signal model is more similar to the estimation model of EK1 during the steady-state condition. However, the other non-linear Kalman filters show better performances in dynamic conditions. When the phasor is a time-varying amplitude and phase, filters give the same accuracy ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$TVE=0.5 \%$ </tex-math></inline-formula> ). A step-change in amplitude and phase creates different overshoot and response times, but EKF2 shows the least overshoot (3.2%) and the longest response time (7.6 ms). Computation burden and noise indices can discriminate the methods from other viewpoints. The computation burden of GHKF is drastically increased when the number of states gives rise. CKF shows an appropriate performance when the number of samples and the number of the state increased in the input signal. EKF1 is not a good solution for noise infiltration when SNR is less than 40 dB, but CKF gives the highest accuracy in high noise levels. Compared to the six already pblished methods, CK shows the best performance with a reasonable estimation error ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$TVE=0.1101 \%$ </tex-math></inline-formula> ) and simulation time (0.6146 ms).