Gauss-Hermite Quadrature Filter Research Articles

Compressed Positive Quadrature Filter

A compressed positive quadrature filter (CPQF) is proposed in this technical note. Accuracy, efficiency, and numerical stability are the main concerns for quadrature-based nonlinear Gaussian filtering, but desirable performance in all these three aspects can hardly be simultaneously achieved in previously proposed nonlinear Gaussian filters, such as unscented Kalman filter, Gauss–Hermite quadrature filter (GHQF), and sparse grid quadrature filter. To improve the overall capacity of nonlinear Gaussian filtering techniques, the CPQF is proposed by numerically solving the constrained moment matching equations. By enforcing the positivity of the quadrature weights, the CPQF is numerically as stable as the GHQF. Through exploiting the underlying sparsity while solving the moment matching equations, the CPQF can also achieve the same level of accuracy as the GHQF using far fewer quadrature points. Simulations are conducted to evaluate the effectiveness of the CPQF, and the numerical results demonstrate its superior performance over conventional Gaussian filters.

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.

Maximum correntropy sparse Gauss–Hermite quadrature filter and its application in tracking ballistic missile

A novel robust filter referred as maximum correntropy sparse Gauss–Hermite quadrature filter (MCSGHQF) is proposed. The novel algorithm makes use of the sparse Gauss–Hermite quadrature (SGHQ) rule to numerically compute Gaussian-weighted integrals, which are propagated through non-linear state equation, and then a weighted means and covariance is taken. As the sensor measurements are always corrupted by non-Gaussian noise which is typically glint noise or mixed Gaussian-impulsive noise, the measurement update is redesigned based on the maximum correntropy criterion instead of minimum mean square error. Therefore, the MCSGHQF could exhibit robustness to the non-Gaussian measurement noise, especially impulsive noise. In addition, the study proposes the use of MCSGHQF for ballistic missile tracking during midcourse phase to deal with the non-Gaussian noise. The tracking performance is compared with that of SGHQ filter (SGHQF), Huber-based filter and extended Kalman filter by Monte-Carlo simulations. The simulation results demonstrate that the MCSGHQF is effective in glint noise case and exhibits superior to Huber-based filter in mixed Gaussian-impulsive noise case.

An Estimation Strategy of Lithium Battery SOC Based on Gauss-Hermite Filter

Battery state of charge (SOC) can directly reflect its sustainable power supply capacity and health. It is important basis to formulate balance and energy management strategy for battery management system (BMS). In this paper, a second-order RC battery model with factors affected battery characteristics was established based on the Gauss-Hermite quadrature filter (GHQF) theory. According to Kirchhoffs law a state space model was established and the parameters in the model were identified. The GHQF algorithm and model were verified, and comparison was made to an extended Kalman filter (EKF) algorithm on the Matlab and Simulink simulation platform.

Vision-Based Spacecraft Relative Navigation Using Sparse-Grid Quadrature Filter

In this paper, vision-based relative navigation of two spacecraft is addressed using the sparse-grid quadrature filter. The relative navigation provides the estimates of the relative orbit and relative attitude as well as the gyro biases. It is a challenging problem because of its high nonlinearity and dimensionality. The extended Kalman filter (EKF) and the unscented Kalman filter (UKF) have been used in the past to solve this problem. However, these filters are not accurate enough in the presence of large initial uncertainties or high nonlinearities. Moreover, although other filters, such as the Gauss-Hermite quadrature filter and the particle filter, can be more accurate than the EKF and UKF, they are hard to use in this high-dimensional estimation problem since a large number of quadrature points or particles are required and therefore the computation complexity is prohibitive. It is shown in this paper that the new sparse-grid quadrature filter can achieve much higher estimation accuracy than EKF, UKF, and the cubature Kalman filter without excessive computation load.

Adaptive sparse grid quadrature filter for spacecraft relative navigation

This paper explores a novel adaptive sparse grid quadrature filter. The sparse grid quadrature approach has been recently developed for nonlinear estimation problems to alleviate the curse-of-dimensionality issue of the Gauss–Hermite quadrature filter. Accuracy level of the sparse grid quadrature filter is an important tuning factor that affects desired performance. The proposed filter autonomously adjusts the accuracy level of the sparse grid quadrature rule in both prediction and update steps by increasing the level gradually until an adaptation criterion is satisfied. The adaptation criterion is derived based on a quadrature error estimator. The nestedness property of sparse grid quadrature rule enables efficient computation in adaptation by reusing quadrature points of the previous level sparse grid quadrature. An application to spacecraft relative navigation has been made to demonstrate the adaptive spare grid quadrature filter outperforming the extended Kalman filter and the unscented Kalman filter.

High-degree cubature Kalman filter

The cubature Kalman filter (CKF), which is based on the third degree spherical–radial cubature rule, is numerically more stable than the unscented Kalman filter (UKF) but less accurate than the Gauss–Hermite quadrature filter (GHQF). To improve the performance of the CKF, a new class of CKFs with arbitrary degrees of accuracy in computing the spherical and radial integrals is proposed. The third-degree CKF is a special case of the class. The high-degree CKFs of the class can achieve the accuracy and stability performances close to those of the GHQF but at lower computational cost. A numerical integration problem and a target tracking problem are utilized to demonstrate the necessity of using the high-degree cubature rules to improve the performance. The target tracking simulation shows that the fifth-degree CKF can achieve higher accuracy than the extended Kalman filter, the UKF, the third-degree CKF, and the particle filter, and is computationally much more efficient than the GHQF.

Sparse-grid quadrature nonlinear filtering

In this paper, a novel nonlinear filter named Sparse-grid Quadrature Filter (SGQF) is proposed. The filter utilizes weighted sparse-grid quadrature points to approximate the multi-dimensional integrals in the nonlinear Bayesian estimation algorithm. The locations and weights of the univariate quadrature points with a range of accuracy levels are determined by the moment matching method. Then the univariate quadrature point sets are extended to form a multi-dimensional grid using the sparse-grid theory. Compared with the conventional point-based methods, the estimation accuracy level of the SGQF can be flexibly controlled and the number of sparse-grid quadrature points for the SGQF is a polynomial of the dimension of the system, which alleviates the curse of dimensionality for high dimensional problems. The Unscented Kalman Filter (UKF) is proven to be a subset of the SGQF at the level-2 accuracy. The performance of this filter is demonstrated by an orbit estimation problem. The simulation results show that the SGQF achieves higher accuracy than the Extended Kalman Filter (EKF), the UKF, and the Cubature Kalman Filter (CKF). In addition, the SGQF is computationally much more efficient than the multi-dimensional Gauss–Hermite Quadrature Filter (GHQF) with the same performance.

Sparse Gauss-Hermite Quadrature Filter with Application to Spacecraft Attitude Estimation

A novel sparse Gauss―Hermite quadrature filter is proposed using a sparse-grid method for multidimensional numerical integration in the Bayesian estimation framework. The conventional Gauss―Hermite quadrature filter is computationally expensive for multidimensional problems, because the number of Gauss―Hermite quadrature points increases exponentially with the dimension. The number of sparse-grid points of the computationally efficient sparse Gauss―Hermite quadrature filter, however, increases only polynomially with the dimension. In addition, it is proven in this paper that the unscented Kalman filter using the suggested optimal parameter is a subset of the sparse Gauss―Hermite quadrature filter. The sparse Gauss-Hermite quadrature filter is therefore more flexible to use than the unscented Kalman filter in terms of the number of points and accuracy level, and it is more efficient than the conventional Gauss―Hermite quadrature filter. The application to the spacecraft attitude estimation problem demonstrates better performance of the sparse Gauss―Hermite quadrature filter in comparison with the extended Kalman filter, the cubature Kalman filter, and the unscented Kalman filter.