Sigma-point Kalman Filters Research Articles

A New Battery SOC/SOH/eSOH Estimation Method Using a PBM and Interconnected SPKFs: Part I. SOC and Internal Variable Estimation

Battery management systems (BMSs) are required to estimate many non-measurable values that describe the actual operating condition of batteries; such as the state of charge (SOC) or the state of health (SOH). In order to improve this evaluation, many physical states and parameters can be estimated using physics-based models (PBMs). These estimates could be used to improve the control and prognosis of batteries. In a series of papers we propose a new method to estimate the internal physical states, the SOC, the SOH and the electrode-specific state of health (eSOH) parameters of a lithium-ion battery, using interconnected sigma-point Kalman filters (SPKFs) and a single-particle model with electrolyte dynamics (SPMe). This first paper focuses on state estimation for non-aged cells. To begin, we describe and validate our electrochemical model against a high-fidelity P2D model. The interconnected SPKF algorithm is then described and the observability of our system is analyzed, showing that the interconnected estimator approach improves an observability measure of the system. Finally, the results of the estimator are discussed, comparing the estimated variables with the truth values under initialization, measurement and modeling uncertainties. The results show that the algorithm can estimate the internal battery states with high accuracy.

An experimental comparative study of adaptive sigma-point Kalman filters: Case study of a rigid–flexible four-bar linkage mechanism and a servo-hydraulic actuator

This paper investigates the application, design, and implementation of different adaptive sigma-point Kalman filters for nonlinear dynamic systems. It is a well-known fact that the performance of nonlinear Kalman filters depends on a prior knowledge of system noise statistics. In real-time applications, this knowledge is difficult and in some cases impossible to predefine. Therefore, an adaptation mechanism based on the random weighting method is proposed in this work to estimate, simultaneously, noise covariance and the state of the system. On the other hand, for real time execution, the computational efficiency of such filters should also be considered to minimize the burden on computational resources. Based on these two factors, in this paper, four different variants of the adaptive sigma-point Kalman filter are studied in detail: (I) an Adaptive Unscented Kalman filter (AUKF), (II) an Adaptive Square-root Unscented Kalman filter (ASr-UKF) with an Unscented Transformation sigma-point selection strategy, (III) an Adaptive Scaled Spherical Simplex Unscented Kalman filter (AS3UKF), and (IV) an Adaptive Square-root Scaled Spherical Simplex Unscented Kalman filter (ASr-S3UKF) with scaled spherical simplex sigma-point selection strategy. The performance of these adaptive sigma-point Kalman filters is analyzed here in depth and discussed for application to a simulated rigid–flexible four-bar linkage mechanism and a practical servo-hydraulic actuator setup under different scenarios. The simulation and the experimental results indicated that, for the cases studied in this article, the filters are competitive and present advantages and disadvantages that should be dealt with according to the requirements of the problem.

State of charge estimation for lithium-ion batteries connected in series using two sigma point Kalman filters

This paper proposes a method to estimate state of charge (SoC) for Lithium-ion battery pack (LIB) with 𝑁 series-connected cells. The cell’s model is represented by a second-order equivalent circuit model taking into account the measurement disturbances and the current sensor bias. By using two sigma point Kalman filters (SPKF), the SoC of cells in the pack is calculated by the sum of the pack’s average SoC estimated by the first SPKF and SoC differences estimated by the second SPKF. The advantage of this method is the SoC estimation algorithm performed only two times instead of 𝑁 times in each sampling time interval, so the computational burden is reduced. The test of the proposed SoC estimation algorithm for 7 samsung ICR18650 Lithium-ion battery cells connected in series is implemented in the continuous charge and discharge scenario in one hour time. The estimated SoCs of the cells in the pack are quite accurate, the 3-sigma criterion of estimated SoC error distributions is 0.5%.

Adaptive Transformed Unscented Simplex Cubature Kalman Filter for Target Tracking

ABSTRACT This paper peruses the Doppler-bearing passive target tracking (DBT), a task that is subject to the nonlinear measurement equations. These measurement equations form the target tracking a nontrivial duty. In order to defeat these problems, many studies have reported various nonlinear estimation techniques, such as sigma-point Kalman filters. These types of filters have demonstrated superior performances for nonlinear system parameter estimation. Consequently, the main contribution of this paper is to improve the performance of the sigma-point Kalman filters. Thereafter, this paper introduces a transformed unscented simplex cubature Kalman filter, in which adaptation techniques are utilized to reduce the dependency of the proposed filter performance on the priori statistical information of the process and measurement noises. Moreover, the proposed filter is initialized by the range-parameterised Unscented Kalman filter. The Cramér–Rao lower bound (CRLB) determines the best accessible accuracy of estimators, which is utilized in this paper to accredit the performance of the proposed filter. The validity and feasibility of the proposed filter are confirmed by Monte Carlo simulations.

Root tracking using time-varying autoregressive moving average models and sigma-point Kalman filters

Root tracking is a powerful technique that provides insight into the mechanisms of various time-varying processes. The poles and the zeros of a signal-generating system determine the spectral characteristics of the signal under consideration. In this work, time-frequency analysis is achieved by tracking the roots of time-varying processes using autoregressive moving average (ARMA) models in cascade form. A cascade ARMA model is essentially a high-order infinite impulse response (IIR) filter decomposed into a series of first- and second-order sections. Each section is characterized by real or conjugate pole/zero pairs. This filter topology allows individual root tracking as well as immediate stability monitoring and correction. Also, it does not suffer from high round-off error sensitivity, as is the case with the filter coefficients of the direct-form ARMA structure. Instead of using conventional gradient-based recursive methods, we investigate the performance of derivative-free sigma-point Kalman filters for root trajectory tracking over time. Based on simulations, the sigma-point estimators provide more accurate estimates, especially in the case of tightly clustered poles and zeros. The proposed framework is applied to real data, and more specifically, it is used to examine the time-frequency characteristics of raw ultrasonic signals from medical ultrasound images.

Performance Enhancement of Robust Cubature Kalman Filter for GNSS/INS Based on Gaussian Process Quadrature

The sigma-point Kalman filters are generally considered to outperform extended Kalman filter in the application of GNSS/INS, where cubature Kalman filter (CKF) is widely approved because of its rigorous mathematic derivation. In order to improve the robustness of GNSS/INS under GNSS-challenged environment, a robust CKF (RCKF) is developed based on novel sigma-point update framework (NSUF) in our previous work, whereas the efficiency of NSUF is still plagued by the unknown process model uncertainty. In this paper, an enhanced RCKF is proposed based on Gaussian process quadrature (GPQ), where the uncertainty consisted in sigma points transform is processed by GPQ conditioning on the approximated posterior PDF. Experiment result on loosely coupled GNSS/INS demonstrates the superiority of proposed method, where the heading error and roll error are reduced by 60.5% and 37.5% respectively compared with RCKF, and it achieves better position result than GP-CKF under GNSS outage.

A new perspective on Gaussian sigma-point Kalman filters

ABSTRACTThere are a variety of Gaussian sigma-point Kalman filters (GSPKF) existing in the literature which are based on different quadrature rules. Their performances are always compared with each other on accuracy and robustness from the numerical-integration perspective, where the number of the sigma points and their corresponding weights are the main reasons resulting in the different accuracy. A new perspective on the GSPKF is proposed in this paper, which is the Mahalanobis distance ellipsoid (MDE). From the MDE perspective, GSPKFs differ from each other on accuracy and robustness mainly because they enclose different probability concentrations. This characteristic is evident when using the high-degree GSPKFs to filter the low-dimensional nonlinear systems. Two classical nonlinear system examples are used to demonstrate the proposed point of this paper. Moreover, some suggestions are given on how to select an appropriate GSPKF for a given nonlinear system.

Converted Measurement Sigma Point Kalman Filter for Bistatic Sonar and Radar Tracking

Tracking with bistatic sonar or radar measurements is challenging due to the fact that the measurements are nonlinear functions of the Cartesian state. The performance of existing approaches, including the extended Kalman filter and sigma point Kalman filters, such as the unscented Kalman filter and cubature Kalman filter, may not be acceptable in terms of mean square error or tracker consistency (i.e., the tracker's state estimation error covariance is not statistically consistent with the actual estimation errors). This paper generalizes the sigma point Kalman filter (SPKF) as a converted measurement SPKF. The resulting converted measurement sigma point Kalman filter is demonstrated to have improved performance over the conventional SPKF when employed in a bistatic tracking scenario.

A Novel and Computationally Efficient Joint Unscented Kalman Filtering Scheme for Parameter Estimation of a Class of Nonlinear Systems

Unscented Kalman filter (UKF) is one type of the sigma point Kalman filters and it is based on unscented transformation. UKF is used for parameter estimation of various dynamic systems and for such purpose either joint UKF (JUKF) or dual UKF (DUKF) schemes are considered. JUKF is based on estimating states and parameters together by using only one filter. For DUKF, states and parameters are decoupled and two separate filters are considered. In this paper, a modification to standard JUKF is proposed for parameter estimation which is based on decoupling parameter vector and updating parameter estimates by considering the error transformation between measurements and transformed sigma points during measurement update into the parameter errors. A linear transformation is proposed for such a purpose. Thus, the computational complexity of the standard JUKF is reduced significantly since parameters are decoupled from the state vector while the convergence of parameter estimate(s) is guaranteed. The new modified JUKF scheme is promising to be used for the parameter estimation of dynamic systems for which a linear transformation between measurement and parameter errors can be obtained. The effectiveness of this new scheme is proven by applying it to two nonlinear dynamic systems.

Observability Analysis for Horizontal Axis Wind Turbines using Empirical Gramian Matrices

This contribution deals with the state observability and parameter identifiability analysis for nonlinear wind turbine control (WTC) systems based on empirical observability Gramian (EOG) matrices. The concepts of observability and Gramian matrices are introduced to investigate the inverse condition number (ICN) and the singular value decomposition (SVD) of the EOG matrices and to evaluate different sensor configurations with respect to their degree of observability. The obtained results are then reviewed for practical plausibility over state estimates, provided by sigma-point Kalman filters (SPKF), in order to relate the observability measures directly to the expectable estimation error. The investigation reveals indeed a correlation between the employed measures for individual state observability and the estimates produced by the nonlinear filters. The joint assessment of ICN, SVD and SPKF results is found as a strong tool when aiming at a holistic and practically relevant observability discussion.

Erratum to: Joseph covariance formula adaptation to Square-Root Sigma-Point Kalman filters

Erratum to: Joseph covariance formula adaptation to Square-Root Sigma-Point Kalman filters

Joseph covariance formula adaptation to Square-Root Sigma-Point Kalman filters

The development of a reliable Sense And Avoid (SAA) system is one of the limiting aspects for the integration into civil airspace of unmanned aerial vehicles, for which the market demand is becoming viral in many fields. To overcome these limitations, it is required that the SAA system perform equally or even better than the human eye. This can be achieved integrating information from different sensors using data fusion algorithms, like Bayesian estimators or neural network techniques. SAA system degradation could arise from both single sensor shortcomings and bad numerical behaviours, injected by the specific fusion algorithm, such as real machine round-off errors or divergences introduced by approximating strongly nonlinear functions. An alternative formulation of the Square-Root unscented Kalman filter (SRUKF) based on the Joseph form of the state covariance update step, is used in order to avoid numerical instabilities induced by ill-conditioned matrix problems. The novelty of this technique lies in the exploitation of the Sigma-Point Kalman filters, which ensure a higher-order accuracy in the nonlinear inference problem solving, and in the application of the Joseph update equation, which improves numerical robustness. Moreover, this approach prevents the algorithm from failing, avoiding the downdating process of the Cholesky factor when the SRUKF is used and to take advantage by the higher numerical stability assured by a lower matrix condition number.

Sigma-Point Filters in Robotic Applications

Sigma-Point Kalman Filters (SPKFs) are popular estimation techniques for high nonlinear system applications. The benefits of using SPKFs include (but not limited to) the following: the easiness of linearizing the nonlinear matrices statistically without the need to use the Jacobian matrices, the ability to handle more uncertainties than the Extended Kalman Filter (EKF), the ability to handle different types of noise, having less computational time than the Particle Filter (PF) and most of the adaptive techniques which makes it suitable for online applications, and having acceptable performance compared to other nonlinear estimation techniques. Therefore, SPKFs are a strong candidate for nonlinear industrial applications, i.e. robotic arm. Controlling a robotic arm is hard and challenging due to the system nature, which includes sinusoidal functions, and the dependency on the sensors’ number, quality, accuracy and functionality. SPKFs provide with a mechanism that reduces the latter issue in terms of numbers of required sensors and their sensitivity. Moreover, they could handle the nonlinearity for a certain degree. This could be used to improve the controller quality while reducing the cost. In this paper, some SPKF algorithms are applied to 4-DOF robotic arm that consists of one prismatic joint and three revolute joints (PRRR). Those include the Unscented Kalman Filter (UKF), the Cubature Kalman Filter (CKF), and the Central Differences Kalman Filter (CDKF). This study gives a study of those filters and their responses, stability, robustness, computational time, complexity and convergences in order to obtain the suitable filter for an experimental setup.

Improvement Schemes for Indoor Mobile Location Estimation: A Survey

Location estimation is significant in mobile and ubiquitous computing systems. The complexity and smaller scale of the indoor environment impose a great impact on location estimation. The key of location estimation lies in the representation and fusion of uncertain information from multiple sources. The improvement of location estimation is a complicated and comprehensive issue. A lot of research has been done to address this issue. However, existing research typically focuses on certain aspects of the problem and specific methods. This paper reviews mainstream schemes on improving indoor location estimation from multiple levels and perspectives by combining existing works and our own working experiences. Initially, we analyze the error sources of common indoor localization techniques and provide a multilayered conceptual framework of improvement schemes for location estimation. This is followed by a discussion of probabilistic methods for location estimation, including Bayes filters, Kalman filters, extended Kalman filters, sigma-point Kalman filters, particle filters, and hidden Markov models. Then, we investigate the hybrid localization methods, including multimodal fingerprinting, triangulation fusing multiple measurements, combination of wireless positioning with pedestrian dead reckoning (PDR), and cooperative localization. Next, we focus on the location determination approaches that fuse spatial contexts, namely, map matching, landmark fusion, and spatial model-aided methods. Finally, we present the directions for future research.

Entropy-Based Approach for Uncertainty Propagation of Nonlinear Dynamical Systems

Uncertainty propagation of dynamical systems is a common need across many domains and disciplines. In nonlinear settings, the extended Kalman filter is the de facto standard propagation tool. Recently, a new class of propagation methods called sigma-point Kalman filters was introduced, which eliminated the need for explicit computation of tangent linear matrices. It has been shown in numerous cases that the actual uncertainty of a dynamical system cannot be accurately described by a Gaussian probability density function. This has motivated work in applying the Gaussian mixture model approach to better approximate the non-Gaussian probability density function. A limitation to existing approaches is that the number of Gaussian components of the Gaussian mixture model is fixed throughout the propagation of uncertainty. This limitation has made previous work ill-suited for nonstationary probability density functions either due to inaccurate representation of the probability density function or computational burden given a large number of Gaussian components that may not be needed. This work examines an improved method implementing a Gaussian mixture model that is adapted online via splitting of the Gaussian mixture model components triggered by an entropy-based detection of nonlinearity during the probability density function evolution. In doing so, the Gaussian mixture model approximation adaptively includes additional components as nonlinearity is encountered and can therefore be used to more accurately approximate the probability density function. This paper introduces this strategy, called adaptive entropy-based Gaussian-mixture information synthesis. The adaptive entropy-based Gaussian-mixture information synthesis method is demonstrated for its ability to accurately perform inference on two cases of uncertain orbital dynamical systems. The impact of this work for orbital dynamical systems is that the improved representation of the uncertainty of the space object can then be used more consistently for identification and tracking.

Adaptive “Cubature and Sigma Points” Kalman Filtering Applied to MEMS IMU/GNSS Data Fusion during Measurement Outlier

In this paper, adaptive sensor fusion INS/GNSS is proposed to solve specific problem of non linear time variant state space estimation with measurement outliers, different algorithms are used to solve this specific problem generally occurs in intentional and non-intentional interferences caused by other radio navigation sources, or by the GNSS receiver’s deterioration. Non linear approximation techniques such as Extended Kalman filter EKF, Sigma Point Kalman Filters such as UKF and CDKF are computed to estimate the navigation states for UAV flight control. Several comparisons are conduced and analyzed in order to compare the accuracy and the convergence of different approaches usually applied in navigation data fusion purposes. The last non linear filter algorithm developed is the Cubature Kalman Filter CKF which provides more accurate estimation with more stability in Tracking data fusion application. In this work, CKF is compared with SPKF and EKF in ideal conditions and during GNSS outliers supposed to occur during specific interval of time, innovation based adaptive approach is selected and used to modify the covariance calculation of the non linear filters performed in this paper. Interesting results are observed, discussed with real perspectives in navigation data fusion for real time applications. Three parallel modified algorithms are simulated and compared to non-adaptive forms according to Root Mean Square Error (RMSE) criteria.

A Non-Pilot Aided Iterative Carrier Frequency Offset Estimator Using Optimal Filtering for an Interleaved OFDMA Uplink System

In an uplink transmission of a coded orthogonal frequency division multiple access (C-OFDMA) system, channel estimation, time and frequency synchronization has to be addressed. For this purpose a control data, i.e. a known training sequence called “preamble” and pilot sub-carriers are used. As an alternative to the classic scheme and in order to maximize the data rate, we propose a non-pilot aided estimator based on an iterative architecture that does not require pilot sub-carriers. Our approach combines 1/ the so-called minimum mean square error successive detector to estimate the signal sent by each user 2/ a recursive method estimating the CFOs. Various algorithms such as the extended Kalman filter, the sigma-point Kalman filters and the extended H ∞ filter are tested and their performances are compared in terms of convergence speed and estimation accuracy. When considering an interleaved OFDMA uplink system over a Rayleigh fading channel, simulation results clearly show the efficiency of the proposed algorithm in terms of CFO estimation and bit error rate performances.

Sigma-point particle filter for parameter estimation in a multiplicative noise environment

[1] A pre-requisite for the “optimal estimate” by the ensemble-based Kalman filter (EnKF) is the Gaussian assumption for background and observation errors, which is often violated when the errors are multiplicative, even for a linear system. This study first explores the challenge of the multiplicative noise to the current EnKF schemes. Then, a Sigma Point Kalman Filter based Particle Filter (SPPF) is presented as an alternative to solve the issues associated with multiplicative noise. The classic Lorenz '63 model and a higher dimensional Lorenz '96 model are used as test beds for the data assimilation experiments. Performance of the SPPF algorithm is compared against a standard EnKF as well as an advanced square-root Sigma-Point Kalman Filters (SPKF). The results show that the SPPF outperforms the EnKF and the square-root SPKF in the presence of multiplicative noise. The super ensemble structure of the SPPF makes it computationally attractive compared to the standard Particle Filter (PF).

Square-root sigma-point Kalman filtering for spacecraft relative navigation

Abstract A variant of sigma-point Kalman filters family called square-root unscented Kalman filter is derived to estimate the relative attitude and position of two spacecrafts referred to as the leader and follower. The square-root forms of unscented Kalman filter have a consistently increased numerical stability because all resulting covariance matrices are guaranteed to stay semi-positive definite. The general six degrees of freedom relative equations of motion are developed based upon the tensors. All leader states are assumed known, whereas the relative states are estimated using available line-of-sight observations between the vehicles along with acceleration and angular velocity measurements of the follower. The quaternion is used to describe the spacecraft relative attitude kinematics, while a three-dimensional generalized Rodrigues parameter is used to maintain the quaternion normalization constraint in the filter formulation. The simulation results indicate that the proposed filter can provide lower relative attitude and position estimation errors with faster convergence rates than the standard extended Kalman filter.

Adaptive exact linearization control of batch polymerization reactors using a Sigma-Point Kalman Filter

The Chylla–Haase polymerization reactor is widely accepted as a benchmark process for the evaluation of control strategies for batch reactors. In this contribution a control concept based on Exact I/O-Linearization is proposed and compared to a conventional cascade control structure. In order to adapt the exact linearization control strategy to various polymerization products and batch conditions, an advanced probabilistic inference algorithm (Sigma-Point Kalman Filter) is applied and investigated. Sigma-Point Kalman Filters have the major improvement of simplified implementation compared to local linearization methods (i.e. Extended Kalman Filter) because no analytical Jacobians are required. Stochastic simulation studies are introduced and show the effectiveness, accuracy and benefit of the control concept. Within several scenarios a satisfying robustness against structural errors in the underlying model equations for the nonlinear control law and the inference algorithm is demonstrated. Furthermore it is pointed out, that with little effort in reassembling the plant design, control performance can be improved significantly.