Showing papers on "Invariant extended Kalman filter published in 2015"

Quaternion kinematics for the error-state Kalman filter

Abstract: A tutorial of quaternion algebra, especially suited for the error-state Kalman filter, with the aim of building Visual-Inertial SLAM and odometry systems.

Maximum Correntropy Kalman Filter

Abstract: Traditional Kalman filter (KF) is derived under the well-known minimum mean square error (MMSE) criterion, which is optimal under Gaussian assumption. However, when the signals are non-Gaussian, especially when the system is disturbed by some heavy-tailed impulsive noises, the performance of KF will deteriorate seriously. To improve the robustness of KF against impulsive noises, we propose in this work a new Kalman filter, called the maximum correntropy Kalman filter (MCKF), which adopts the robust maximum correntropy criterion (MCC) as the optimality criterion, instead of using the MMSE. Similar to the traditional KF, the state mean and covariance matrix propagation equations are used to give prior estimations of the state and covariance matrix in MCKF. A novel fixed-point algorithm is then used to update the posterior estimations. A sufficient condition that guarantees the convergence of the fixed-point algorithm is given. Illustration examples are presented to demonstrate the effectiveness and robustness of the new algorithm.

Kalman Filter and Its Application

Abstract: Kalman filter is a minimum-variance estimation for dynamic systems and has attracted much attention with the increasing demands of target tracking. Various algorithms of Kalman filter was proposed for deriving optimal state estimation in the last thirty years. This paper briefly surveys the recent developments about Kalman filter (KF), Extended Kalman filter (EKF) and Unscented Kalman filter (UKF). The basic theories of Kalman filter are introduced, and the merits and demerits of them are analyzed and compared. Finally relevant conclusions and development trends are given.

An Adaptive Unscented Kalman Filtering Approach for Online Estimation of Model Parameters and State-of-Charge of Lithium-Ion Batteries for Autonomous Mobile Robots

Abstract: In this brief, to get a more accurate and robust state of charge (SoC) estimation, the lithium-ion battery model parameters are identified using an adaptive unscented Kalman filtering method, and based on the updated model, the battery SoC is estimated consequently. An adaptive adjustment of the noise covariances in the estimation process is implemented using a technique of covariance matching in the unscented Kalman filter (UKF) context. The effectiveness of the proposed method is evaluated through experiments under different power duties in the laboratory environment. The obtained results are compared with that of the adaptive extended Kalman filter, extended Kalman filter, and unscented Kalman filter-based algorithms. The comparison shows that the proposed method provides better accuracy both in battery model parameters estimation and the battery SoC estimation.

Dynamic State Estimation of a Synchronous Machine Using PMU Data: A Comparative Study

Abstract: Accurate information about dynamic states is important for efficient control and operation of a power system. This paper compares the performance of four Bayesian-based filtering approaches in estimating dynamic states of a synchronous machine using phasor measurement unit data. The four methods are extended Kalman filter, unscented Kalman filter, ensemble Kalman filter, and particle filter. The statistical performance of each algorithm is compared using Monte Carlo methods and a two-area-four-machine test system. Under the statistical framework, robustness against measurement noise and process noise, sensitivity to sampling interval, and computation time are evaluated and compared for each approach. Based on the comparison, this paper makes some recommendations for the proper use of the methods.

Distributed Kalman Filtering With Dynamic Observations Consensus

Abstract: This paper studies distributed estimation of unstable dynamic random fields observed by a sparsely connected network of sensors. The field dynamics are globally detectable, but not necessarily locally detectable. We propose a consensus+innovations distributed estimator, termed Distributed Information Kalman Filter. We prove under what conditions this estimator is asymptotically unbiased with bounded mean-squared error, smaller than for other alternative distributed estimators. Monte Carlo simulations confirm our theoretical error asymptotic results.

Security in stochastic control systems: Fundamental limitations and performance bounds

Abstract: This work proposes a novel metric to characterize the resilience of stochastic cyber-physical systems to attacks and faults. We consider a single-input single-output plant regulated by a control law based on the estimate of a Kalman filter. We allow for the presence of an attacker able to hijack and replace the control signal. The objective of the attacker is to maximize the estimation error of the Kalman filter - which in turn quantifies the degradation of the control performance - by tampering with the control input, while remaining undetected. We introduce a notion of e-stealthiness to quantify the difficulty to detect an attack when an arbitrary detection algorithm is implemented by the controller. For a desired value of e-stealthiness, we quantify the largest estimation error that an attacker can induce, and we analytically characterize an optimal attack strategy. Because our bounds are independent of the detection mechanism implemented by the controller, our information-theoretic analysis characterizes fundamental security limitations of stochastic cyber-physical systems.

Convergence Analysis of Extended Kalman Filter for Sensorless Control of Induction Motor

Abstract: This paper deals with convergence analysis of the extended Kalman filters (EKFs) for sensorless motion control systems with induction motor (IM). An EKF is tuned according to a six-order discrete-time model of the IM, affected by system and measurement noises, obtained by applying a first-order Euler discretization to a six-order continuous-time model. Some properties of the discrete-time model have been explored. Among these properties, the observability property is relevant, which leads to conditions that can be directly linked with the working conditions of the machine. Starting from these properties, the convergence of the stochastic state estimation process, in mean square sense, has been shown. The convergence is also explored with reference to the difference between the samples of the state of the continuous-time model and that estimated by the EKF. The results theoretically achieved have been also validated by means of experimental tests carried out on an IM prototype.

Continuous-Discrete Extended Kalman Filter on Matrix Lie Groups Using Concentrated Gaussian Distributions

Abstract: In this paper we generalize the continuous-discrete extended Kalman filter (CD-EKF) to the case where the state and the observations evolve on connected unimodular matrix Lie groups. We propose a new assumed density filter called continuous-discrete extended Kalman filter on Lie groups (CD-LG-EKF). It is built upon a geometrically meaningful modeling of the concentrated Gaussian distribution on Lie groups. Such a distribution is parametrized by a mean and a covariance matrix defined on the Lie group and in its associated Lie algebra respectively. Our formalism yields tractable equations for both non-linear continuous time propagation and discrete update of the distribution parameters under the assumption that the posterior distribution of the state is a concentrated Gaussian. As a side effect, we contribute to the derivation of the first and second order differential of the matrix Lie group logarithm using left connection. We also show that the CD-LG-EKF reduces to the usual CD-EKF if the state and the observations evolve on Euclidean spaces. Our approach leads to a systematic methodology for the design of filters, which is illustrated by the application to a camera pose filtering problem with observations on Lie group. In this application, the CD-LG-EKF significantly outperforms two constrained non-linear filters (one based on a linearization technique and the other on the unscented transform) applied on the embedding space of the Lie group.

Intrinsic Filtering on Lie Groups With Applications to Attitude Estimation

Abstract: This paper proposes a probabilistic approach to the problem of intrinsic filtering of a system on a matrix Lie group with invariance properties. The problem of an invariant continuous-time model with discrete-time measurements is cast into a rigorous stochastic and geometric framework. Building upon the theory of continuous-time invariant observers, we introduce a class of simple filters and study their properties (without addressing the optimal filtering problem). We show that, akin to the Kalman filter for linear systems, the error equation is a Markov chain that does not depend on the state estimate. Thus, when the filter's gains are held fixed, the noisy error's distribution is proved to converge to a stationary distribution, under some convergence properties of the filter with noise turned off. We also introduce two novel tools of engineering interest: the discrete-time invariant extended Kalman filter, for which the trusted covariance matrix is shown to converge, and the invariant ensemble Kalman filter. The methods are applied to attitude estimation, allowing to derive novel theoretical results in this field, and illustrated through simulations on synthetic data.

State-of-Charge Estimation of Lithium-Ion Battery Using Square Root Spherical Unscented Kalman Filter (Sqrt-UKFST) in Nanosatellite

Abstract: State-of-charge (SOC) estimation is an important aspect for modern battery management system Dynamic and closed loop model-based methods such as extended Kalman filter (EKF) have been extensively used in SOC estimation However, the EKF suffers from drawbacks such as Jacobian matrix derivation and linearization accuracy In this paper, a new SOC estimation method based on square root unscented Kalman filter using spherical transform (Sqrt-UKFST) with unit hyper sphere is proposed The Sqrt-UKFST does not require the linearization for nonlinear model and uses fewer sigma points with spherical transform, which reduces the computational requirement of traditional unscented transform The square root characteristics improve the numerical properties of state covariance The proposed method has been experimentally validated The results are compared with existing SOC estimation methods such as Coulomb counting, portable fuel gauge, and EKF The proposed method has an absolute root mean square error (RMSE) of 142% and an absolute maximum error of 496% These errors are lower than the other three methods When compared with EKF, it represents 37% and 44% improvement in RMSE and maximum error respectively Furthermore, the Sqrt-UKFST is less sensitive to parameter variation than EKF and it requires 32% less computational requirement than the regular UKF

Dynamic State Estimation for Multi-Machine Power System by Unscented Kalman Filter with Enhanced Numerical Stability

Abstract: In this paper, in order to enhance the numerical stability of the unscented Kalman filter (UKF) used for power system dynamic state estimation, a new UKF with guaranteed positive semidifinite estimation error covariance (UKF-GPS) is proposed and compared with five existing approaches, including UKF-schol, UKF-$\kappa$, UKF-modified, UKF-$\Delta Q$, and the square-root unscented Kalman filter (SR-UKF). These methods and the extended Kalman filter (EKF) are tested by performing dynamic state estimation on WSCC 3-machine 9-bus system and NPCC 48-machine 140-bus system. For WSCC system, all methods obtain good estimates. However, for NPCC system, both EKF and the classic UKF fail. It is found that UKF-schol, UKF-$\kappa$, and UKF-$\Delta Q$ do not work well in some estimations while UKF-GPS works well in most cases. UKF-modified and SR-UKF can always work well, indicating their better scalability mainly due to the enhanced numerical stability.

Huber’s M-Estimation-Based Process Uncertainty Robust Filter for Integrated INS/GPS

Abstract: The integration of the inertial navigation system and the global positioning system (INS/GPS) is a widely used procedure for position and attitude determination applications. The Kalman type filter (KTF) is the primary mechanism to perform the integration. In the KTF, the process noise is always assumed to be Gaussian distribution, which may be violated by the vehicle’s severe maneuver, resulting in a much degraded performance. In this paper, the Huber’s M-estimation methodology is investigated to suppress the process uncertainty, founded on the cascaded form of the M-estimation-based Kalman filter. An iterated algorithm is designed to construct the weighted matrix to rescale the prior state estimate covariance. The proposed process uncertainty robust algorithm is embedded into the newly derived modified unscented quaternion estimator to perform the standard inertial navigation equations-based INS/GPS integration. The car-mounted experiments are carried out to validate the proposed method against the process uncertainty.

A Contraction Theory-Based Analysis of the Stability of the Deterministic Extended Kalman Filter

Abstract: The contraction properties of the extended Kalman filter, viewed as a deterministic observer for nonlinear systems, are analyzed. The approach relies on the study of an auxiliary “virtual” dynamical system. Some conditions under which exponential convergence of the state error can be guaranteed are derived. Moreover, contraction provides a simple formalism to study some robustness properties of the filter, especially with respect to measurement errors, as illustrated by a simplified inertial navigation example. This technical note sheds another light on the theoretical properties of this popular observer.

State of Charge Estimation of Lithium-Ion Batteries Using an Adaptive Cubature Kalman Filter

Abstract: Accurate state of charge (SOC) estimation is of great significance for a lithium-ion battery to ensure its safe operation and to prevent it from over-charging or over-discharging. However, it is difficult to get an accurate value of SOC since it is an inner sate of a battery cell, which cannot be directly measured. This paper presents an Adaptive Cubature Kalman filter (ACKF)-based SOC estimation algorithm for lithium-ion batteries in electric vehicles. Firstly, the lithium-ion battery is modeled using the second-order resistor-capacitor (RC) equivalent circuit and parameters of the battery model are determined by the forgetting factor least-squares method. Then, the Adaptive Cubature Kalman filter for battery SOC estimation is introduced and the estimated process is presented. Finally, two typical driving cycles, including the Dynamic Stress Test (DST) and New European Driving Cycle (NEDC) are applied to evaluate the performance of the proposed method by comparing with the traditional extended Kalman filter (EKF) and cubature Kalman filter (CKF) algorithms. Experimental results show that the ACKF algorithm has better performance in terms of SOC estimation accuracy, convergence to different initial SOC errors and robustness against voltage measurement noise as compared with the traditional EKF and CKF algorithms.

Distributed Kalman consensus filter with intermittent observations

Abstract: We consider the problem of distributed state estimation for linear time-varying systems with intermittent observations. An optimal Kalman consensus filter has been developed by minimizing the mean-squared estimation error for each node. To derive a scalable algorithm for the covariance matrices update, a suboptimal filter is proposed by omitting the edge covariance matrices among nodes. By using the Lyapunov-based approach, a sufficient condition is presented for ensuring the stochastic stability of the suboptimal filter. Two numerical examples are provided to verify the effectiveness of the proposed filter.

Sensor Placement for Optimal Kalman Filtering: Fundamental Limits, Submodularity, and Algorithms ✩

Abstract: In this paper, we focus on sensor placement in linear dynamic estimation, where the objective is to place a small number of sensors in a system of interdependent states so to design an estimator with a desired estimation performance. In particular, we consider a linear time-variant system that is corrupted with process and measurement noise, and study how the selection of its sensors affects the estimation error of the corresponding Kalman filter over a finite observation interval. Our contributions are threefold: First, we prove that the minimum mean square error of the Kalman filter decreases only linearly as the number of sensors increases. That is, adding extra sensors so to reduce this estimation error is ineffective, a fundamental design limit. Similarly, we prove that the number of sensors grows linearly with the system's size for fixed minimum mean square error and number of output measurements over an observation interval; this is another fundamental limit, especially for systems where the system's size is large. Second, we prove that the logdet of the error covariance of the Kalman filter, which captures the volume of the corresponding confidence ellipsoid, with respect to the system's initial condition and process noise is a supermodular and non-increasing set function in the choice of the sensor set. Therefore, it exhibits the diminishing returns property. Third, we provide efficient approximation algorithms that select a small number sensors so to optimize the Kalman filter with respect to this estimation error ---the worst-case performance guarantees of these algorithms are provided as well. Finally, we illustrate the efficiency of our algorithms using the problem of surface-based monitoring of CO2 sequestration sites studied in Weimer et al. (2008).

Random-point-based filters: analysis and comparison in target tracking

Abstract: This paper compares state estimation techniques for nonlinear stochastic dynamic systems, which are important for target tracking. Recently, several methods for nonlinear state estimation have appeared utilizing various random-point-based approximations for global filters (e.g., particle filter and ensemble Kalman filter) and local filters (e.g., Monte-Carlo Kalman filter and stochastic integration filters). A special emphasis is placed on derivations, algorithms, and commonalities of these filters. All filters described are put into a common framework, and it is proved that within a single iteration, they provide asymptotically equivalent results. Additionally, some deterministic-point-based filters (e.g., unscented Kalman filter, cubature Kalman filter, and quadrature Kalman filter) are shown to be special cases of a random-point-based filter. The paper demonstrates and compares the filters in three examples, a random variable transformation, re-entry vehicle tracking, and bearings-only tracking. The results show that the stochastic integration filter provides better accuracy than the Monte-Carlo Kalman filter and the ensemble Kalman filter with lower computational costs.

Model predictive-based shunt active power filter with a new reference current estimation strategy

Abstract: This study presents a new reference current estimation method using proposed robust extended complex Kalman filter (RECKF) together with model predictive current (MPC) control strategy in the development of a three-phase shunt active power filter (SAPF). A new exponential function embedded into the RECKF algorithm helps in the estimation of in phase fundamental component of voltage ( v h ) at the point of common coupling considering grid perturbations such as distorted voltage, measurement noise and phase angle jump and also for the estimation of fundamental amplitude of the load current ( i h ). The estimation of these two variables ( v h , i h ) is used to generate reference signals for MPC. The proposed RECKF-MPC needs less number of voltage sensors and resolves the difficulty of gain tuning of proportional-integral (PI) controller. The proposed RECKF-MPC approach is implemented using MATLAB/SIMULINK and also Opal-RT was used to obtain the real-time results. The results obtained using the proposed RECKF together with different variants of Kalman filters (Kalman filter (KF), extended KF (EKF) and extended complex KF (ECKF)) and PI controller are analysed both in the steady state as well as transient state conditions. From the above experimentation, it was observed that the proposed RECKF-MPC control strategy outperforms over PI controller and other variants of Kalman filtering approaches in terms of reference tracking error, power factor distortion and percentage total harmonic distortion in the SAPF system.

Extended Kalman Filter for Spacecraft Pose Estimation Using Dual Quaternions

Abstract: Based on the highly successful quaternion multiplicative extended Kalman filter for spacecraft attitude estimation using unit quaternions, this paper proposes a dual quaternion multiplicative extended Kalman filter for spacecraft pose (i.e., attitude and position) and linear and angular velocity estimation using unit dual quaternions. By using the concept of error unit dual quaternion, defined analogously to the concept of error unit quaternion in the quaternion multiplicative extended Kalman filter, this paper proposes, as far as the authors know, the first multiplicative extended Kalman filter for pose estimation. The state estimate of the dual quaternion multiplicative extended Kalman filter can directly be used by recently proposed pose controllers based on dual quaternions, without any additional conversions, thus providing an elegant solution to the output dynamic compensation problem of the full six degree-of-freedom motion of a rigid body. Three formulations of the dual quaternion multiplicative e...

Modified strong tracking unscented Kalman filter for nonlinear state estimation with process model uncertainty

Abstract: This paper presents a modified strong tracking unscented Kalman filter MSTUKF to address the performance degradation and divergence of the unscented Kalman filter because of process model uncertainty. The MSTUKF adopts the hypothesis testing method to identify process model uncertainty and further introduces a defined suboptimal fading factor into the prediction covariance to decrease the weight of the prior knowledge on filtering solution. The MSTUKF not only corrects the state estimation in the occurrence of process model uncertainty but also avoids the loss of precision for the state estimation in the absence of process model uncertainty. Further, it does not require the cumbersome evaluation of Jacobian matrix involved in the calculation of the suboptimal fading factor. Experimental results and comparison analysis demonstrate the effectiveness of the proposed MSTUKF. Copyright © 2015John Wiley & Sons, Ltd.

Quaternion-Based Kalman Filter for AHRS Using an Adaptive-Step Gradient Descent Algorithm:

Abstract: This paper presents a quaternion-based Kalman filter for real-time estimation of the orientation of a quadrotor. Quaternions are used to represent rotation relationship between navigation frame and body frame. Processing of a 3-axis accelerometer using Adaptive-Step Gradient Descent (ASGD) produces a computed quaternion input to the Kalman filter. The step-size in GD is set in direct proportion to the physical orientation rate. Kalman filter combines 3-axis gyroscope and computed quaternion to determine pitch and roll angles. This combination overcomes linearization error of the measurement equations and reduces the calculation cost. 3-axis magnetometer is separated from ASGD to independently calculate yaw angle for Attitude Heading Reference System (AHRS). This AHRS algorithm is able to remove the magnetic distortion impact. Experiments are carried out in the small-size flight controller and the real world flying test shows the proposed AHRS algorithm is adequate for the real-time estimation of the orien...

Design of adaptive H ∞ filter for implementing on state-of-charge estimation based on battery state-of-charge-varying modelling

Abstract: This study suggests a new method for modelling lithium-ion battery types and state-of-charge (SOC) estimation using adaptive H∞ filter (AHF). First, a universal linear model with some free parameters is considered for dynamical behaviour of the battery. The battery voltage and SOC are used as states of the model. Then for every period in the charge/discharge process the free parameters of the model are identified. Each period of process is associated with a specific SOC value, hence the parameters can be regarded as functions of SOC in the entire process. The functions are determined based on polynomial approximation and least squares method. The proposed SOC-varying model is incorporated in AHF for SOC estimation. Moreover, a new method for adjusting the tuning parameters of the filter is suggested. The proposed method is verified by experimental tests on a lithium-ion battery and is compared with adaptive extended Kalman filter and square-root unscented Kalman filter

A Kalman Filter-Based Framework for Enhanced Sensor Fusion

Abstract: Sensor fusion has found a lot of applications in today’s industrial and scientific world with Kalman filtering being one of the most practiced methods. Despite their simplicity and effectiveness, Kalman filters are usually prone to uncertainties in system parameters and particularly system noise covariance. This paper proposes a Kalman filtering framework for sensor fusion, which provides robustness to the uncertainties in the system parameters such as noise covariance and state initialization. Two methods are developed based on the proposed approach. The effectiveness of the proposed methods is verified through numerous simulations and experiments.